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+{"text":"\\section{Introduction}\n\\label{sec:Intro}\n\n\n    With the increased use of electric vehicles (EVs) over the past decade, a large network of EV charging stations has been installed on the electric grid.\n    Data collected from city-wide deployment of EV charging stations can be used for both academic and industrial purposes, e.g.,  through statistical analysis \\cite{nasrinEV2016} and modeling. Studies require reliable sessions data for understanding behaviors and exploring flexibility.  The scarcity of reliable data has been discussed previously \\cite{EVDSreview}, and its necessity has been pointed out for further research purposes. Even when data exists, it may be protected under confidentiality for private collectors, and not freely available for academic or public use. Lack of wide-spread availability and accessibility of realistic EV charging sessions data pose a significant hurdle, impeding further research in the field.\n    \n    We propose a Synthetic Data Generator (SDG), that can be used to generate a sample of EV charging sessions. This implies temporal statistical modeling of arrivals and modeling of departures and the associated electrical load (i.e.,  required energy). We define a parametric model that can be trained from a real-world dataset. This trained model can be subsequently used to generate realistic data samples of EV sessions. \n    In this paper, we contribute with:\n        \\begin{itemize}[topsep=0pt]\n            \\setlength\\itemsep{0em}\n            \\item An approach to model sessions data for EVs over a group of charging stations defined as the SDG (\\secref{sec:SDG});\n            \\item An outline of the model parameters, and discussion over benefits and drawbacks of different models;\n            \\item Answers to our main research questions, being:\n           \n        \n        \t\\begin{question}\\label{q:SDGdefination}~Which parametric models can be used to describe sessions of EVs. What are the input parameters and latent variables for these models? \\end{question}\n        \n        \t\\begin{question}\\label{q:generate}~How can we generate synthetic samples of EV sessions data from these parametric models? \\end{question}\n    \t   \n        \\end{itemize}\n\n\\section{Synthetic Data Generator}\n\\label{sec:SDG}\n\nEach EV session can be described using three parameters:\n\\begin{enumerate*}[(i)]\n        \\item \\textit{Arrival time},\n        \\item \\textit{Departure time}, and \n        \\item \\textit{Energy charged} (in kWh).\n\\end{enumerate*}\nWe can define three models for each of these as:\n\\begin{itemize}[topsep=0pt]\n    \\item \\textit{Arrivals} = $AM$(Horizon)\n    \\item \\textit{Connected times} = $MM_{c}$(Arrivals) \n    \\item \\textit{Energy} = $MM_{e}$(Arrivals)\n\\end{itemize}\nWhere $AM$ means arrival model, $MM_{c}$ means mixture model for connection times and $MM_{e}$ means mixture model for the energy charged. `Horizon' is the time horizon for which the data needs to be generated and is an input parameter. Departure time in an EV charging session can be calculated as the sum of its arrival time and connected time.\n\n\\subsection{Arrival Models}\n\\label{sec:arrivalmodel}\n    \n    Arrivals of EVs in a group of charging stations can be considered as events on a continuous timescale. Supported by our large-scale dataset, we assume that the inter-arrival time of EVs\n    follows an exponential distribution,\n    characterized by a parameter $\\lambda$ representing the arrival rate (EVs per time unit).\n    This means we can either model the times in between successive events, or rather the number of events in a given time interval:\n    \\begin{itemize}[topsep=0pt]\n        \\item \\textit{Exponential IAT distribution:} we generate the arrival of the next EV using $t_{i}  = t_{i-1}  + \\Delta t $. Where $t_{i} $ is the time of $i^{th}$ arrival. $\\Delta t $ is the time difference between the $i^\\textrm{th}$ and ${i-1}^\\textrm{th}$ arrivals. The probability density of $\\Delta t $ (inter-arrival time, IAT) is exponentially distributed, characterized by $\\lambda = f(\\textrm{month}, \\textrm{daytype}, \\textrm{time-of-day})$, where `daytype' is either weekday or weekend.\n        \\item \\textit{Poisson process:} we generate the number of arrivals in a given time slot (e.g.,  slots of 60 minutes). The number of arrivals $N_\\textit{arr}$ in a given time slot can be generated using sampling from a Poisson distribution\n        with mean $\\lambda \\cdot T$ (with $T$ the duration of a timeslot).\n    \\end{itemize}\n   \n   \n   \n   \n   \n   \n   \n   \n   \n        \n        \n   \n        \n   \n    \n    \n    \\noindent In \\figref{fig:expdensity} we plot density of inter-arrival times where the original data refers to real-world data collected by ELaadNL. A fitted exponential distribution validates this assumption that inter-arrival times are exponentially distributed: Kolmogorov\u2013Smirnov test results for all combinations of month, daytype and time-of-day slots have $p$-value $< 0.05$.\n    \n    \\begin{figure}[!t]\n        \\centering\n        \\includegraphics[width=\\columnwidth,height=6cm]{Resources\/exponential-dist-excel.png}\n       \n        \\caption{Inter-arrival time probability density for real world data collected by ELaadNL.\n        \\label{fig:expdensity}\n        \\setlength{\\belowcaptionskip}{-00pt}\n    \\end{figure} \n    \n\n    We can model $\\lambda$ (in function of month, daytype, time-of-day) using averaging (discontinuous) over each timeslot or rather fitting a continuous function of time (continuous). While fitting continuous curves, capturing the peaks during the day becomes very important. Also, fitted curves could result in negative values of $\\lambda$, thus we need to impose lower and upper bounds on it.\n    \n    \n    Using the IAT approach, the time of next EV arrival is determined relatively to the time of the previous EV. In case $\\lambda$ at a given time-of-day is very low, this implies the next EV arrival may be be generated very late (with a large $\\Delta t $), thus skipping several consecutive timeslots. This is problematic if those next timeslots exhibit a much higher $\\lambda$, and thus have a high probability of EV arrivals --- which would not be generated because of the very large  $\\Delta t $.\n   \n   \n    \n   \n    The second modeling approach, as a Poisson process, circumvents the above problem, since when certain timeslots have low $\\lambda$, we will likely generate 0 arrivals, but still proceed to the immediately next timeslot (with possibly high $\\lambda$ to generate possible arrivals there). A potential drawback still is that the variance and mean of the Poisson distribution are equal. In case this assumption does not hold, we adopt a negative binomial distribution instead.\n \n\n\n\n\n\n\n\n\n\n\n\\subsection{Mixture Models for Connected Times and Energy} \n\\label{sec:departuremodel}\n\nAside from arrival events, also EV departure events, as well as the EV charging load (or total energy charged) need to be modeled.\nArrivals are conditional on previous arrivals, obviously, and we model the probability distribution for connected times ($t_\\textit{depart} - t_\\textit{arr}$) as Gaussian Mixture Models (GMMs). Also for EV charging energy we use GMMs.\nAs in the arrival generation models, we define and fit different models for each month and time-of-day.\n\n\n\n\n\n\nIn summary, answering \\qref{q:SDGdefination} (Which models can be used in SDG?), we propose\n\\begin{enumerate*}[(i)]\n    \\item exponential IAT distribution or Poisson distribution for arrivals per timeslot ($\\textit{AM}$),\n    \\item GMMs for both the duration of EV sessions ($\\textit{MM}_c$) and \n    \\item their associated energy charged ($\\textit{MM}_e$).\n\\end{enumerate*}\n\n\n\n\\section{Generating Samples}\n\\label{sec:training}\n\nAfter fitting the SDG on a real-world data, we have $AM$, $MM_{c}$ and $MM_{e}$ as parametric models for arrival times, connection times and energy required. For synthetic generation of arrival times, e.g.,  $\\lambda$ values can be used to generate $\\Delta t $ (and hence a series of arrivals). For connection times and energy required, a sample from the PDFs can be used as a synthetically generated connection time and required energy. This answers \\qref{q:generate}, on how we can generate samples.\n\nWith our SDG (AM, $MM_{c}$ and $MM_{e}$) can be saved as a separate file. These models will be supplied with the code (that we plan to make publicly available) to generate new synthetic data samples. These models do not include the actual real-world EV session data that the SDG was trained on. Hence, these models can be shared without violating confidentiality constraints. Generated samples from these models can be subsequently used for flexibility analysis, load balancing and other research purposes.  \n\n\n\n\\section{Conclusion and Future Work}\n\\label{sec:future work}\n    Our paper summarized the modeling approach of EV charging sessions. \n    We adopted these models for training a synthetic data generator (SDG) with real-world data, that can thus generate synthetic samples of EV sessions data.\n   \n    We plan to release the source code including the \\emph{training} scripts, but also \\emph{generation} code (including the settings thereof reflecting the trained model characteristics based on a large-scale real-world dataset) to produce synthetic EV session data reflecting real-world behavior. We believe this fits a strong need of researchers in both academic and industrial settings.\n   \n    \n    As future work, we aim to propose modeling methods for the time-varying arrival distribution model parameters.\n    Further, we will tackle the following challenges in depth:\n    \\begin{enumerate*}[label={(\\arabic*)}]\n    \\item Studying the properties of the real-world data with the goal to define evaluations metrics for comparing real-world data with generated samples.\n    \\item Correctly modeling peaks of arrivals during the day; study effective methods that avoid negative values in continuous $\\lambda$ curves. \n\\end{enumerate*}\n    \n    \n\\begin{acks}\nThis research received funding from the Flemish Government under the \"Onderzoeksprogramma Artifici\u00eble Intelligentie (AI) Vlaanderen\" programme.\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\verb|acmsmall|}: The default journal template style.\n\\item {\\verb|acmlarge|}: Used by JOCCH and TAP.\n\\item {\\verb|acmtog|}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\verb|acmconf|}: The default proceedings template style.\n\\item{\\verb|sigchi|}: Used for SIGCHI conference articles.\n\\item{\\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\verb|sigplan|}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\verb|anonymous,review|}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\verb|\\acmSubmissionID| command to print the\n  submission's unique ID on each page of the work.\n\\item{\\verb|authorversion|}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\verb|screen|}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. Multiple authors may share one affiliation. Authors'\nnames should not be abbreviated; use full first names wherever\npossible. Include authors' e-mail addresses whenever possible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for all short- and\nfull-length articles, and optional for two-page abstracts.\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{The 1907 Franklin Model D roadster.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader. Figure captions go below the figure. Your figures should\n{\\bfseries also} include a description suitable for screen readers, to\nassist the visually-challenged to better understand your work.\n\nFigure captions are placed {\\itshape below} the figure.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\verb|acmsmall|}: The default journal template style.\n\\item {\\verb|acmlarge|}: Used by JOCCH and TAP.\n\\item {\\verb|acmtog|}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\verb|acmconf|}: The default proceedings template style.\n\\item{\\verb|sigchi|}: Used for SIGCHI conference articles.\n\\item{\\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\verb|sigplan|}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\verb|anonymous,review|}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\verb|\\acmSubmissionID| command to print the\n  submission's unique ID on each page of the work.\n\\item{\\verb|authorversion|}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\verb|screen|}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. Multiple authors may share one affiliation. Authors'\nnames should not be abbreviated; use full first names wherever\npossible. Include authors' e-mail addresses whenever possible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for all short- and\nfull-length articles, and optional for two-page abstracts.\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{The 1907 Franklin Model D roadster.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader. Figure captions go below the figure. Your figures should\n{\\bfseries also} include a description suitable for screen readers, to\nassist the visually-challenged to better understand your work.\n\nFigure captions are placed {\\itshape below} the figure.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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}%\n}\n\n\\newcommand{w}{w}\n\\newcommand{\\mathfrak{p}}{\\mathfrak{p}}\n\\newcommand{$g$-equivalent}{$g$-equivalent}\n\\newcommand{$g$-equivalence}{$g$-equivalence}\n\\newcommand{G^g}{G^g}\n\\newcommand{\\Psi}{\\Psi}\n\\newcommand{\\Upsilon}{\\Upsilon}\n\\newcommand{(\\sieve,\\siftable)}{(\\Psi,\\Upsilon)}\n\n\\newenvironment{epigraph}\n{\\hfill\\begin{minipage}{0.6\\linewidth}\\raggedleft\\footnotesize}{\\end{minipage}\\bigskip\\bigskip}\n\n\n\\newcommand{\\mathcal{M}}{\\mathcal{M}}\n\\newcommand{\\mathcal{L}}{\\mathcal{L}}\n\\newcommand{\\mathcal{S}}{\\mathcal{S}}\n\\newcommand{\\mathcal{C}}{\\mathcal{C}}\n\\newcommand{\\mathcal{P}}{\\mathcal{P}}\n\\newcommand{\\mathrm{P}}{\\mathrm{P}}\n\\newcommand{\\mathrm{L}}{\\mathrm{L}}\n\\newcommand{\\mathcal{F}}{\\mathcal{F}}\n\\newcommand{\\mathcal{Q}}{\\mathcal{Q}}\n\\newcommand{\\mathcal{K}}{\\mathcal{K}}\n\\newcommand{\\mathcal{R}}{\\mathcal{R}}\n\\newcommand{\\mathcal{J}}{\\mathcal{J}}\n\\newcommand{\\mathcal{I}}{\\mathcal{I}}\n\\newcommand{\\mathcal{G}}{\\mathcal{G}}\n\\newcommand{\\mathcal{B}}{\\mathcal{B}}\n\\newcommand{\\mathcal{E}}{\\mathcal{E}}\n\n\\newcommand{\\mathfrak{a}}{\\mathfrak{a}}\n\\newcommand{\\mathfrak{p}}{\\mathfrak{p}}\n\n\\newcommand{\\lambda_f}{\\lambda_f}\n\\newcommand{\\rho_f}{\\rho_f}\n\\newcommand{\\lambda_g}{\\lambda_g}\n\\newcommand{\\rho_g}{\\rho_g}\n\n\\newcommand{\\varphi}{\\varphi}\n\n\\renewcommand{\\geq}{\\geqslant}\n\\renewcommand{\\leq}{\\leqslant}\n\\renewcommand{\\Re}{\\mathfrak{Re}\\,}\n\\renewcommand{\\Im}{\\mathfrak{Im}\\,}\n\n\\newcommand{\\eqref}{\\eqref}\n\\newcommand{\\backslash}{\\backslash}\n\n\\newcommand{\\ov}[1]{\\overline{#1}}\n\n\\newcommand{\\norm}[1]{\\|{#1}\\|}\n\n\\newcommand{\\peter}[1]{\\langle{#1}\\rangle}\n\\newcommand\\sumsum{\\mathop{\\sum\\sum}\\limits}\n\\newcommand\\sumsumsum{\\mathop{\\sum\\sum\\sum}\\limits}\n\\newcommand\\sumsumnd{\\mathop{{\\sum\\sum}^{nd}}\\limits}\n\\newcommand\\delval{1\/8}\n\\newcommand\\delvaln{1\/16}\n\\newcommand\\finalexponent{1\/24}\n\\newcommand\\rpfree{1\/144}\n\n\\begin{document}\n\n\\title{Periodic twists of $\\GL_3$-automorphic forms}\n\n\\author{Emmanuel  Kowalski}\n\\address{ETHZ, Switzerland }\n\\email{kowalski@math.ethz.ch}\n\n\\author{Yongxiao Lin}\n\\address{EPFL\/MATH\/TAN, Station 8, CH-1015 Lausanne, Switzerland }\n\\email{yongxiao.lin@epfl.ch}\n\n\\author{Philippe Michel}\n\\address{EPFL\/MATH\/TAN, Station 8, CH-1015 Lausanne, Switzerland }\n\\email{philippe.michel@epfl.ch}\n\n\\author{Will Sawin}\n\\address{Columbia University, USA }\n\\email{sawin@math.columbia.edu}\n\n\\date{\\today,\\ \\thistime} \n\n\\subjclass[2010]{11F55,11M41,11L07, 11T23, 32N10}\n\n\\keywords{Automorphic forms on $\\GL_3$, Fourier coefficients, Hecke\n  eigenvalues, discrete Fourier transform, trace functions,\n  subconvexity}\n\n\\begin{abstract}\n  We prove that sums of length about $q^{3\/2}$ of Hecke eigenvalues of\n  automorphic forms on~$\\SL_3(\\mathbf{Z})$ do not correlate with $q$-periodic\n  functions with bounded Fourier transform. This generalizes the earlier\n  results of Munshi and Holowinsky--Nelson, corresponding to\n  multiplicative Dirichlet characters, and applies in particular to\n  trace functions of small conductor modulo primes.\n\\end{abstract}\n\n\\thanks{Y. L., Ph.\\ M.\\ and E.\\ K.\\ were partially supported by a\n  DFG-SNF lead agency program grant (grant number\n  200020L\\_175755). W. S. was partially supported by the Clay Mathematics\n  Institute. \\today\\ \\currenttime}\n\n\\maketitle \n\n\n\\section{Introduction}\n\nLet $\\varphi$ be a cusp form for $\\SL_3(\\mathbf{Z})$ which is an eigenfunction\nof all Hecke operators.\nFor any prime number~$q$ and any primitive Dirichlet character~$\\chi$\nmodulo~$q$, we can then define the twisted $L$-function\n$L(\\varphi\\otimes\\chi,s)$, which is an entire function satisfying a\nfunctional equation relating $s$ to $1-s$.\nIn a recent breakthrough, Munshi~\\cite{Munshi,Munshi1} solved the\nsubconvexity problem for these twisted $L$-functions\n$L(\\varphi\\otimes \\chi,s)$ in the conductor aspect:\n\n\\begin{theorem}[Munshi]\\label{th-munshi}\n  Let~$s$ be a complex number such that $\\Re s=1\/2$. For any\n  prime~$q$, any primitive Dirichlet character~$\\chi$ modulo~$q$, and\n  for any~$\\varepsilon>0$, we have\n\\begin{equation}\\label{eq:subconvex}\n  L(\\varphi\\otimes \\chi,s)\\ll q^{3\/4-1\/308+\\varepsilon},\n\\end{equation}\nwhere the implied constant depends on $\\varphi$, $s$ and~$\\varepsilon$.\n\\end{theorem}\n\nThis result was recently analyzed in depth by Holowinsky and\nNelson~\\cite{HN}, who discovered a remarkable simplification (and\nstrenghtening) of Munshi's ideas. They proved:\n\n\\begin{theorem}[Holowinsky--Nelson]\\label{th-hn}\n  With notation and assumptions as in Theorem~\\emph{\\ref{th-munshi}},\n  we have\n  \\begin{equation}\\label{eq:hn}\n    L(\\varphi\\otimes \\chi,s)\\ll q^{3\/4-1\/36+\\varepsilon}\n  \\end{equation}\n  where the implied constant depends on $\\varphi$, $s$ and~$\\varepsilon$.\n\\end{theorem}\n\n\\begin{remark}\n  We mention further variants, simplifications and improvements, by\n  Aggarwal, Holowinsky, Lin and Sun~\\cite{AHLS}, Holowinsky, Munshi\n  and Qi~\\cite{HMQ}, Lin~\\cite{Lin}, Sun and Zhao ~\\cite{SZ}.\n\\end{remark}\n\nLet $(\\lambda(m,n))$ denote the Hecke-eigenvalues of~$\\varphi$.  By\nthe approximate functional equation for the twisted $L$-functions, the\nbound \\eqref{eq:hn} is essentially equivalent to the bound\n\\begin{equation}\\label{eq:sumbound}\n  \\sum_{n\\geq 1}\\lambda(1,n)\\chi(n)\n  V\\Bigl(\\frac{n}{q^{3\/2}}\\Bigr)\\ll q^{3\/2-\\delta},\n\\end{equation}\nfor~$\\delta<1\/36$, where $V$ is any smooth compactly supported\nfunction and the implied constant depends on~$\\varphi$, $\\delta$\nand~$V$.\n\nFrom the perspective of such sums, motivated by the previous work of\nFouvry, Kowalski and Michel~\\cite{FKM1}, which relates to automorphic\nforms on~$\\GL_2$, it is natural to ask whether this\nbound~\\eqref{eq:sumbound} holds when $\\chi$ is replaced by a more\ngeneral trace function $K:{\\mathbf{F}_q}\\to \\mathbf{C}$. Our main result shows that this\nis the case, and in fact extends the result to a much wider range of\n$q$-periodic functions by obtaining estimates only in terms of the\nsize of the discrete Fourier transform modulo~$q$.\n\nPrecisely, for any function~$V$ with compact support on~$\\mathbf{R}$, we set\n\\begin{equation}\\label{defSKX}\n  S_{V}(K,X):=\\sum_{n\\geq 1}\\lambda(1,n)K(n)V\\Bigl(\\frac{n}{X}\\Bigr).\n\\end{equation}\n\nWe will assume that $V:\\mathbf{R}\\to \\mathbf{C}$ satisfies the following conditions\nfor some parameter~$Z\\geq 1$:\n\\begin{equation}\\label{eq:Vprop}\n  \\mathrm{supp}(V)\\subset ]1,2[,\\text{ and }V^{(i)}(x)\\ll Z^i\\text{\n    for all $i\\geq 0$},\n\\end{equation}\nwhere the implied constant depends only on~$i$.\n\nFor any integer~$q\\geq 1$ and any $q$-periodic function\n$K\\colon \\mathbf{Z}\\to \\mathbf{C}$, we denote by\n\\begin{equation}\\label{eq:fourierK}\n  \\widehat K(n)=\\frac{1}{q^{1\/2}}\\sum_{x\\in{\\mathbf{F}_q}}K(x)e\\Bigl(\\frac{nx}{q}\\Bigr),\n\\end{equation}\nfor~$n\\in\\mathbf{Z}$, its (unitarily normalized) discrete Fourier transform\nmodulo~$q$.  We write~$\\norm{\\widehat{K}}_{\\infty}$ for the maximum of\n$|\\widehat{K}(n)|$ for~$n\\in\\mathbf{Z}$. We then have the discrete Fourier\ninversion formula\n$$\nK(x)=\\frac{1}{q^{1\/2}}\\sum_{n\\in{\\mathbf{F}_q}} \\widehat{K}(n)\ne\\Bigl(-\\frac{nx}{q}\\Bigr)\n$$\nfor~$x\\in\\mathbf{Z}$.\n\nOur main result is a general bound for \\eqref{defSKX} which matches\nprecisely the bound of Holowinsky--Nelson \\cite{HN} in the case of a\nmultiplicative character:\n\n\\begin{theorem}\\label{thm:main}\n  Let $\\varphi$ be an $\\SL_3(\\mathbf{Z})$-invariant cuspidal Hecke-eigenform\n  with Hecke eigenvalues $(\\lambda(m,n))$.  Let~$q$ be a prime number,\n  and $K\\colon \\mathbf{Z}\\to\\mathbf{C}$ be a $q$-periodic function.\n \n \n  Let $V$ be a smooth, compactly supported function satisfying\n  \\eqref{eq:Vprop} for some $Z\\geq 1$.  Assume that\n$$\n    Z^{2\/3}q^{4\/3}\\leq X \\leq Z^{-2}q^{2}.\n$$\t\n For any~$\\varepsilon>0$, we have\n \\begin{equation}\\label{eq:sumboundK}\n   S_V(K,X) \\ll\n   \\norm{\\widehat{K}}_{\\infty}Z^{10\/9}q^{2\/9+\\varepsilon}X^{5\/6},\n \\end{equation}\n where the implied constant depends only on~$\\varepsilon$, on~$\\varphi$, and on\n the implicit constants in~\\emph{(\\ref{eq:Vprop})}.\n\\end{theorem}\n\n\\begin{remark}\n  (1)  Suppose that we vary~$q$ and apply this bound with functions~$K$\n  modulo~$q$ that have absolutely bounded Fourier transforms.  Take\n  $X=q^{3\/2}$. We then obtain the bound\n  $$\n  S_V(K,q^{3\/2}) \\ll Z^{10\/9}q^{3\/2-1\/36+\\varepsilon}\n  $$\n  for any~$\\varepsilon>0$.\n  \\par\n  (2) For the bound \\eqref{eq:sumboundK} to be non-trivial (i.e.,\n  assuming~$K$ to be absolutely bounded, better than $X$), it is\n  enough that\n  $$\n  X\\geq Z^{20\/3}q^{4\/3+\\delta}\n  $$\n  for some~$\\delta>0$.\n  \\par\n  (3) As in the paper~\\cite{short-sums} of Fouvry, Kowalski, Michel,\n  Raju, Rivat and Soundararajan, where the main estimate is also\n  phrased in Fourier-theoretic terms only,\\footnote{Although the size\n    of~$K$ enters in~\\cite{short-sums} as well as that of its Fourier\n    transform.} the motivating examples of functions~$K$ satisfying\n  uniform bounds on their Fourier transforms are the trace functions\n  of suitable $\\ell$-adic sheaves modulo~$q$. The simplest example\n  is~$K(n)=\\chi(n)$, which recovers the bound of Munshi (up to the\n  value of the exponent) and Holowinsky--Nelson, since the values of\n  the Fourier transform are normalized Gauss sums of modulo~$\\leq\n  1$. We recall some other examples below in\n  Section~\\ref{sec-examples}.\n\\end{remark}\n\nWe can  deduce from Theorem~\\ref{thm:main} a weak but non-trivial\nbound for the first moment of the twisted central $L$-values, with an\nadditional twist by a discrete Mellin transform. We first recall the\ndefinition\n$$\n\\Kl_3(n)=\\frac{1}{q}\\sum_{\\substack{x,y,z\\in{\\mathbf{F}^\\times_q}\\\\xyz=n}}\ne\\Bigl(\\frac{x+y+z}{q}\\Bigr)\n$$\nfor a hyper-Kloosterman sum with two variables modulo a prime~$q$.\n\n\\begin{corollary}\\label{cor-average}\n  Let $\\varphi$ be an $\\SL_3(\\mathbf{Z})$-invariant cuspidal Hecke-eigenform\n  with Hecke eigenvalues $(\\lambda(m,n))$.  Let~$q$ be a prime number\n  and let~$\\chi\\mapsto M(\\chi)$ be a function of Dirichlet characters\n  modulo~$q$.\n  \\par\n  Let~$K$ and~$L$ be the $q$-periodic functions defined\n  by~$K(0)=L(0)=0$ and\n  \\begin{align*}\n    K(n)&=\\frac{q^{1\/2}}{q-1}\\sum_{\\chi\\mods{q}}\\chi(n)M(\\chi)\\\\\n    L(n)&=\\frac{1}{q^{1\/2}} \\sum_{x\\in{\\mathbf{F}_q}}K(x)\\Kl_3(nx)\n  \\end{align*}\n  for~$n$ coprime to~$q$.  We then have\n  $$\n  \\frac{1}{q-1} \\sum_{\\chi\\mods{q}}M(\\chi)L(\\varphi\\otimes\\chi, 1\/2) \\ll\n  \\Bigl(\\norm{\\widehat{K}}_{\\infty}+\\norm{\\widehat{L}}_{\\infty}\\Bigr) q^{2\/9\n    + \\varepsilon},\n  $$\n  for any~$\\varepsilon>0$, where the  implied constant depends\n  on~$\\varphi$ and,~$\\varepsilon$.\n \n \n \n \n \n \n \n \n\\end{corollary}\n\n\n\nA further natural application concerns the symmetric square lift, $\\mathrm{sym}_2(\\psi)$, of a\n$\\GL_2$-cusp form of level~$1$. Precisely, let $\\psi$ be a cuspidal\nHecke-eigenform for~$\\SL_2(\\mathbf{Z})$ with Hecke eigenvalues\n$(\\lambda(n))_{n\\geq 1}$. This implies the following\n\\begin{corollary}\\label{cor-gl2}\n  Let $K$ and $V$ be as above and assume that\n  $Z^{2\/3}q^{4\/3}\\leq X \\leq Z^{-2}q^{2}$. Then, for any~$\\varepsilon>0$, we\n  have\n  $$\n  \\sum_{n\\geq 1}\\lambda(n^2)K(n)V\\Bigl(\\frac{n}{X}\\Bigr) \\ll\n  \\norm{\\widehat{K}}_{\\infty}Z^{10\/9}q^{2\/9+\\varepsilon}X^{5\/6}+\n  \\norm{K}_{\\infty}Z^{1\/3}q^{2\/3}X^{1\/2+\\varepsilon},\n  $$\n  where the implied constant depends only on~$\\varepsilon$, on~$\\psi$, and on\n  the implicit constants in~\\emph{(\\ref{eq:Vprop})}.\n\\end{corollary}\n\\begin{remark}\\label{blomerremark} As pointed out to us by V. Blomer, when $K=\\chi$ is a Dirichlet character a stronger bound should be available: for $\\chi$ quadratic,  one has (see \\cite{Blomer}) the stronger  subconvex bound for the central value\n\\begin{equation}\\label{blomerbound}\n\tL(\\mathrm{sym}_2(\\psi)\\otimes\\chi,s)\\ll_s q^{3\/4-1\/8+o(1)},\\ \\Re s=1\/2.\n\\end{equation}\nThis would amount to a bound of the shape\n$$\n  \\sum_{n\\geq 1}\\lambda(n^2)\\chi(n)V\\Bigl(\\frac{n}{q^{3\/2}}\\Bigr) \\ll_Z\n  q^{3\/2-1\/8+\\varepsilon}.\n  $$\nThe bound \\eqref{blomerbound} actually extends to any character $\\chi\\mods q$ by the same method, using the Petrow-Young variant of the Conrey-Iwaniec method \\cite{CI,PY}. However, since this approach uses positivity of central values, it is not entirely clear yet whether this could be extended to general trace functions. \n\\end{remark}\n\n\nFrom this corollary, one can easily derive an estimate for twists of\nthe arithmetic function~$\\lambda(n)^2=|\\lambda(n)|^2$, which is related\nto~$\\lambda(n^2)$ by the convolution identity\n\\begin{equation}\\label{convol}\n  \\lambda(n)^2=\\sum_{ab=n}\\lambda(a^2).\t\n\\end{equation}\n\nHowever, in terms of $L$-functions, a straightforward estimate\nconcerns sums of length close to~$q^2$, and not~$q^{3\/2}$ anymore (it\namounts, when~$K=\\chi$, to a subconvexity estimate\nfor~$L(f\\otimes f\\otimes \\chi,{\\textstyle{\\frac{1}{2}}})$, which results directly from the\nfactorization of this $L$-function of degree~$4$).\n\nOne can however recover a bound for sums of length about~$q^{3\/2}$\nwith more work, and here we require that $K$ be a trace function (more\nprecisely, a \\emph{non-exceptional} trace function, in the sense\nof~\\cite[p. 1686]{FKM2}).\n\n\n\\begin{corollary}\\label{cor2-gl2}\\label{RScor}\n  Let $V$ be as above.  Let $K$ be a the trace function of an\n  $\\ell$-adic sheaf $\\mathcal{F}$ modulo~$q$ which is a geometrically\n  irreducible middle-extension sheaf, pure of weight~$0$, on the\n  affine line over~$\\mathbf{F}_q$. Assume that the sheaf $\\mathcal{F}$ is not\n  geometrically isomorphic to the tensor product\n  $\\mathcal{L}_\\psi\\otimes\\mathcal{L}_\\chi$ of an Artin-Schreier sheaf and a Kummer\n  sheaf.\n  \\par\n  If $Z^{-4\/3}q^{4\/3+8\\gamma\/3} \\leq X \\leq Z^{-2}q^{2}$, then  we have\n  $$\n  \\sum_{n\\geq 1}\\lambda(n)^2K(n)V\\Bigl(\\frac{n}{X}\\Bigr) \\ll\n  X^{2\/3+\\varepsilon}q^{1\/3}+Z^{5\/6}X^{7\/8+\\varepsilon}q^{1\/6}+X^{1+\\varepsilon}q^{-\\gamma}\n  $$\n  for any~$\\varepsilon>0$, where the implied constant depends only\n  on~$\\psi~$, $\\varepsilon$ and on the conductor~$\\cond(\\mathcal{F})$ of~$\\mathcal{F}$.\n\\end{corollary}\n\n\\begin{remark}\n  (1) Suppose that~$Z$ is fixed. The estimate is then non-trivial as\n  long as $X\\gg q^{4\/3+\\delta}$; for $X=q^{3\/2}$, it saves a factor\n  $q^{1\/48}$ over the trivial bound.\n  \\par\n  (2) The assumption that~$\\mathcal{F}$ is not exceptional means intuitively\n  that $K$ is not proportional to the product of an additive and a\n  multiplicative character modulo~$q$. We then have in particular\n  $$\n  \\|K\\|_{\\infty}+\\|\\widehat{K}\\|_{\\infty}\\ll 1\n  $$\n  where the implied constant depends only on the conductor of~$\\mathcal{F}$.\n\\end{remark}\n\n\\begin{remark}\\label{remcor17}\n  (1) The reader may wonder why this paper is much shorter\n  than~\\cite{FKM1}, and (with the exception of Corollary \\ref{RScor})\n  requires much less input from algebraic geometry in the case of\n  trace functions. One reason is that we are considering (essentially)\n  sums of length~$q^{3\/2}$ whereas the coefficients functions~$K$ are\n  $q$-periodic. This means that periodicity properties of the\n  summand~$K(n)$ have a non-trivial effect, whereas they do not for\n  the sums of length about~$q$ which are considered in~\\cite{FKM1} in\n  the context of~$\\GL_2$.\n  \\par\n  Moreover, observe that an analogue of Theorem~\\ref{thm:main}, with\n  an estimate that depends (in terms of~$K$) only on the size of the\n  Fourier transform~$\\widehat{K}$, is \\emph{false} in the setting\n  of~\\cite{FKM1}, i.e., for sums\n  $$\n  \\sum_{n\\geq 1}\\lambda(n)K(n)V\\Bigl(\\frac{n}{X}\\Bigr)\n  $$\n  with~$X$ of size about~$q$, where $\\lambda(n)$ are the\n  Hecke-eigenvalues of a cusp forms~$\\psi$ for $\\SL_2(\\mathbf{Z})$ (as in\n  Corollary~\\ref{cor-gl2}). Indeed, if we take $X=q$ and define~$K$ to\n  be the $q$-periodic function that coincides with the (real-valued)\n  function $n\\mapsto \\lambda(n)$ for $1\\leq n\\leq q$, then~$K$ has\n  discrete Fourier transform of size~$\\ll \\log q$ by the well-known\n  Wilton estimate (see, e.g.,~\\cite[Th. 5.3]{iwaniec}, when~$\\psi$ is\n  holomorphic), and yet\n  $$\n  \\sum_{n\\leq q}K(n)\\lambda(n)=\\sum_{n\\leq q}|\\lambda(n)|^2\\asymp q\n  $$\n  by the Rankin--Selberg method.\n  \\par\n  On the other hand, the same bound of Wilton combined with discrete\n  Fourier inversion implies quickly that if~$K$ is any $q$-periodic\n  function, then\n  $$\n  \\sum_{n\\leq q^{3\/2}}\\lambda(n) K(n)\\ll\n  q^{1+1\/4+\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n  $$\n  for any~$\\varepsilon>0$. However, the natural length for applications\n  is~$q$ in the $\\GL_2$ case.\n  \\par\n  (2) The most obvious function $K$ for which Theorem~\\ref{thm:main}\n  gives trivial results is an additive character $K(n)=e(an\/q)$ for\n  some integer~$a\\in\\mathbf{Z}$, since the Fourier transform takes one value\n  of size~$q^{1\/2}$. However, a useful estimate also exists in this\n  case: Miller~\\cite{Miller} has proved that\n  $$\n  \\sum_{n\\geq 1}\\lambda(1,n)e(\\alpha\n  n)V\\Bigl(\\frac{n}{X}\\Bigr)\\ll_{\\varphi,Z} X^{3\/4+\\varepsilon}\n  $$\n  for~$X\\geq 2$, any~$\\alpha\\in\\mathbf{R}$ and any~$\\varepsilon>0$, where the\n  implied constant is independent of~$\\alpha$. This is the\n  generalization to~$\\GL_3$ of the bound of Wilton mentioned in the\n  first remark.\n  \\par\n  (3) Using either the functional equation for the $L$-functions\n  $L(\\varphi\\otimes\\chi,s)$, or the Voronoi summation formula, one can\n  show that the estimate of Miller implies a bound of the shape\n  $$\n  S_V(\\Kl_2(a\\cdot;q),X)\\ll_{\\varphi,Z} (qX)^{\\varepsilon}X^{1\/4}q^{3\/4}\n  $$\n  for any~$\\varepsilon>0$, where\n  $$\n  \\Kl_2(n;q)=\\frac{1}{q^{1\/2}}\\sum_{x\\in{\\mathbf{F}^\\times_q}}e_q(\\ov x+nx)\n  $$ \n  is a normalized Kloosterman sum. This bound is non-trivial as long\n  as $X\\geq q$. Since~$\\Kl_2$ is a trace function that is bounded\n  by~$2$ and has Fourier transform bounded by~$1$, this gives (in a\n  special case) a stronger bound than what follows from\n  Theorem~\\ref{thm:main}.\n  \\par\n  (4) Remark (2) suggests a direct approach by the discrete Fourier\n  inversion formula, which gives\n  $$\n  \\sum_{n\\leq X}\\lambda(1,n)K(n)=\\frac{1}{\\sqrt{q}} \\sum_{0\\leq h<q}\n  \\widehat{K}(h) \\sum_{n\\leq X}\\lambda(1,n)e\\Bigl(\\frac{nh}{q}\\Bigr).\n  $$\n  A non-trivial bound for~$X\\approx q^{3\/2}$ in terms\n  of~$\\norm{\\widehat{K}}_{\\infty}$ would then follow from a bound\n  $$\n  \\sum_{n\\leq X}\\lambda(1,n)e\\Bigl(\\frac{nh}{q}\\Bigr) \\ll X^{\\alpha}\n  $$\n  for additive twists of the Fourier coefficients\n  where~$\\alpha<2\/3$.\n  \\par\n  Unsurprisingly, in the case of~$\\GL_2$, although we have the best\n  possible estimate of Wilton (with the analogue of~$\\alpha$\n  being~$1\/2$), the resulting estimate for a sum of length~$q$ is\n  trivial.\n\\end{remark}\n\nThe plan of the paper is as follows: we will explain the idea and\nsketch the key steps of the proof in\nSection~\\ref{sec-principle}. Section~\\ref{sec-examples} recalls the\nmost important examples of trace functions, for which~$K$ has small\nFourier transform and hence for which Theorem~\\ref{thm:main} is\nnon-trivial. Section~\\ref{sec-reminders} presents a key\nFourier-theoretic estimate and some reminders concerning automorphic\nforms and the Voronoi summation formula for~$\\GL_3$. Then the last\nsections complete the proof of Theorem~\\ref{thm:main} following the\noutline presented previously, and explain how to deduce\nCorollaries~\\ref{cor-average}, \\ref{cor-gl2} and~\\ref{RScor} (the last\nof which requires further ingredients).\n\n\\subsection*{Acknowledgements}\n\nWe are very grateful to R. Holowinsky and P. Nelson for sharing with\nus and explaining their work \\cite{HN} which has directly inspired the\npresent paper. We are also very grateful to V. Blomer for Remark \\ref{blomerremark} and to the referees for their\ncareful reading of the manuscript, comments and suggestions and in particular for pointing out\na serious error in the first version of Corollary~\\ref{RScor}.\n\n\n\\subsection*{Notation}\n\nFor any~$z\\in\\mathbf{C}$, we define $e(z)=\\exp(2\\pi i z)$.  If~$q\\geq 1$,\nthen we denote by $e_q(x)$ the additive character modulo~$q$ defined\nby $e_q(x)=e(x\/q)$ for~$x\\in\\mathbf{Z}$. We often identify a $q$-periodic\nfunction defined on~$\\mathbf{Z}$ with a function on~$\\mathbf{Z}\/q\\mathbf{Z}$.\n\nFor any finite abelian group~$A$, we use the notation~$\\widehat{f}$ for\nthe unitary Fourier transform defined on the character\ngroup~$\\widehat{A}$ of~$A$ by\n$$\n\\widehat{f}(\\psi)=\\frac{1}{\\sqrt{|A|}}\n\\sum_{x\\in A}f(x)\\psi(x).\n$$\nWe have then the Plancherel formula~$\\norm{f}_2=\\norm{\\widehat{f}}_2$,\nwhere\n$$\n\\|f\\|_2=\\sum_{x\\in A}|f(x)|^2,\\quad\\quad \\|\\widehat{f}\\|_2=\\sum_{\\psi\\in\n  \\widehat{A}}|\\widehat{f}(\\psi)|^2.\n$$\n\n\nFor any integrable function on~$\\mathbf{R}$, we denote its Fourier transform\nby\n$$\n\\widehat{V}(y)=\\int_{\\mathbf{R}}V(x)e(-xy)dx.\n$$\nWe recall the Poisson summation formula when performed with a\n$q$-periodic function~$K$ in addition to a smooth function~$V$ with\nfast decay at infinity: for any~$X\\geq 1$, we have\n\\begin{equation}\\label{eq-poisson}\n  \\sum_{n\\in\\mathbf{Z}}K(n)V\\Bigl(\\frac{n}{X}\\Bigr)= \\frac{X}{q^{1\/2}} \\sum_{h\\in\\mathbf{Z}}\n  \\widehat{K}(h)\\widehat{V}\\Bigl(\\frac{hX}{q}\\Bigr).\n\\end{equation}\nThis follows directly from the usual Poisson formula and the\ndefinition of $\\widehat{K}$ after splitting the sum into congruence\nclasses modulo~$q$.\n\n\n\n\\section{Principle of the proof}\\label{sec-principle}\n\nWe use a direct generalisation of the method of Holowinsky and\nNelson~\\cite{HN} that led to Theorem~\\ref{th-hn}. Although it was\nmotivated by Munshi's approach, based on the use of the Petersson\nformula as a tool to express the delta symbol, there is no remaining\ntrace of this point of view; however, we refer to~\\cite[App. B]{HN}\nfor a detailed and insightful description of the origin of this\nstreamlined method, starting from Munshi's.\n\n\\subsection{Amplification}\\label{ssec-amplification}\n\nThe first step is to realize the $q$-periodic function~$K$ within a\none-parameter family of $q$-periodic functions. Precisely, let\n$\\widehat K$ be the Fourier transform of $K$ (see \\eqref{eq:fourierK})\nand define\n\\begin{equation}\\label{whatKzdef}\n\\widehat K(z,h):= \\begin{cases} \\widehat K(z)e_q(-h\\ov z)& q\\nmid z\\\\\n  \\widehat K(0)& q\\mid z\n\\end{cases}\n\\end{equation}\nfor $(z,h)\\in \\mathbf{Z}^2$. Then put\n\\begin{equation}\\label{eq-kn}\nK(n,h)=\\frac{1}{q^{1\/2}}\\sum_{z\\in{\\mathbf{F}^\\times_q}}\\widehat K(z,h)e_q(-nz).\n\\end{equation}\nfor $(n,h)\\in\\mathbf{Z}^2$.  By the discrete Fourier inversion formula, we\nhave\n\\begin{equation}\\label{eq-detector}\nK(n,0)=K(n)-\\frac{\\widehat K(0)}{q^{1\/2}}\n\\end{equation}\nMore generally, for any probability measure $\\varpi$ on~${\\mathbf{F}^\\times_q}$, the\naverage\n$$\nK_{\\varpi}(n,h)=\\sum_{l\\in{\\mathbf{F}^\\times_q}}\\varpi(l)K(n,\\ov lh)\n$$ \nsatisfies $K_{\\varpi}(n,0)=K(n)-\\frac{\\widehat K(0)}{q^{1\/2}}$.  It follows\nthat, for any parameter $H\\geq 1$, we can express the sum $S_V(K,X)$\nas the difference of double sums\n$$\nS_V(K,X)=\\sum_{l\\in\\mathcal{L}}\\varpi(l)\\sum_{|h|\\leq H}S_{V}(K(\\cdot,h\\ov\nl),X)- \\sum_{l\\in\\mathcal{L}}\\varpi(l)\\sum_{0<|h|\\leq H}S_{V}(K(\\cdot,h\\ov\nl),X),\n$$\nup to an error $\\ll X\/q^{1\/2}$.  We write this difference as\n$$\nS_V(K,X)=\\mathcal{F}-\\mathcal{O},\n$$\nsay. One then needs to select a suitable probability measure~$\\varpi$,\nand then the two terms are then handled by different methods. It\nshould be emphasized that no main term arises (which would have to be\ncanceled in the difference between the two terms).\n\n\\begin{remark}\n  The argument is reminiscent of the \\emph{amplification method}, the\n  function $K(n)=K(n,0)$ being ``amplified'' (up to a small error)\n  within the family $(K(n,h))_{|h|\\leq H}$.\n\\end{remark}\n\n\\subsection{Bounding $\\mathcal{F}$}\n\nAs in~\\cite{HN}, we consider a probability measure~$\\varpi$\ncorresponding to a product structure: we average over pairs $(p,l)$ of\nprimes such that $p\\sim P$ and $l\\sim L$, and take $\\varpi(x)$\nproportional to the number of representations $x=\\ov p l\\mods q$,\nwhere $p\\sim P$ and $l\\sim L$ are primes (their sizes being parameters\n$1\\leq P,L<q\/2$ to be chosen later).\n\nThe treatment of $\\mathcal{F}$ is essentially the same as\nin~\\cite{HN, Lin}. By applying the Poisson summation formula to the\n$h$-variable, with dual variable~$r$, we see the function\n$$\n(n,r,p,l)\\mapsto \\widehat K(-p\\ov{l r})\\lambda(1,n)e_q(np\\ov{l r}),\n$$\nappear. We then appeal to the classical ``reciprocity law'' for\nadditive exponentials, namely\n$$\ne_q(np\\ov {l r})\\approx e_{rl}(-np\\ov q),\n$$\ntrading the modulus $q$ for the modulus $rl$, which will be\nsignificantly smaller than $q$. We then apply the Voronoi summation\nformula for the cusp form $\\varphi$ on the $n$-variable (this is the\nonly real automorphic input), which transforms the additive phase\n$e_{rl}(-np\\ov q)$ into Kloosterman sums of modulus $rl$. We then\nobtain further cancellation by smoothing out the resulting variable\n(dual to $n$) by using Cauchy--Schwarz and detecting cancellations on\naverages of products of Kloosterman sums, where the product structure\nof the averaging set is essential.\n\nIn this part of the argument, the coefficient function~$K$ plays very\nlittle role, and we could just more or less quote the\ncorresponding statements in~\\cite{HN, Lin}, if the parameter~$Z$ was\nfixed. Since we wish to keep track of its behavior (for the purpose of\nflexibility for potential applications), we have to go through the\ncomputations anew. This is done in detail in Section~\\ref{sec:Fsum}.\n\n\\subsection{Bounding $\\mathcal{O}$}\n\nIn the sum~$\\mathcal{O}$, with the averaging performed in the same way as\nfor~$\\mathcal{F}$, the key point is that the $n$-variable in the sum\n$S_V(K(\\cdot, h\\bar{l},X)$ is very long compared to $q$. We apply\nCauchy's inequality to smooth it, keeping the other variables $h,p,l$\nin the inside, thus eliminating the automorphic coefficients\n$\\lambda(1,n)$ (for which we only require average bounds, which we\nborrow from the Rankin--Selberg theory, our second important\nautomorphic input). This leads quickly to the problem of estimating\nthe sum\n$$\n\\sumsum_{p_1,h_1,l_1,p_2,h_2,l_2}\\sum_{n\\sim X}K(n,h_1p_1\\ov\nl_1)\\ov{K(n,h_2p_2\\ov l_2)}.\n$$\nWe apply the Poisson formula in the $n$-variable; since $X$ is\ntypically much larger than $q$, only the zero frequency in the dual\nsum contributes. This yields a key sum of the shape\n$$\n\\sumsum_{p_1,h_1,l_1,p_2,h_2,l_2}\\sum_{u\\in{\\mathbf{F}^\\times_q}}|\\widehat\nK(u)|^2e_q((h_1p_1\\bar{l}_1-h_2p_2\\bar{l}_2)u^{-1})=\n\\sumsum_{p_1,h_1,l_1,p_2,h_2,l_2}\nK_2(h_1p_1\\bar{l}_1-h_2p_2\\bar{l}_2),\n$$\nsay.\n\\par\nWhen $K$ is a multiplicative character, as in the work of\nHolowinsky--Nelson, the proof is essentially finished then, since\n$\\widehat K(u)$ is a normalized Gauss sum, with a constant modulus,\nhence~$K_2$ is simply a Ramanujan sum, which we can evaluate\nexplicitly.\n\\par\nIn general, we obtain cancellation using a very general\nFourier-theoretic bound for general bilinear forms\n$$\n\\sum_{m\\in{\\mathbf{F}_q}}\\sum_{n\\in{\\mathbf{F}_q}}\\alpha_m\\beta_nK_2(m-n),\n$$\nwhich involves only $L^2$-norm bounds for the coefficients and\n$L^{\\infty}$-norm bounds for the Fourier transform of~$K_2$\n(see~Proposition~\\ref{willprop}). The latter, it turns out, is\nessentially~$|\\widehat{K}|^2$, and we can obtain a good estimate purely\nin terms of~$\\norm{\\widehat{K}}_{\\infty}$.  This part of the argument is\nperformed in Section~\\ref{sec:O}.\n\n\n\\section{Examples of trace functions}\\label{sec-examples}\n\nTheorem~\\ref{thm:main} certainly applies to ``random'' $q$-periodic\nfunctions $K\\colon \\mathbf{Z}\\to\\mathbf{C}$, for all reasonable meanings of the word\n``random'', but the basic motivating examples in number theory are\noften provided by \\emph{trace functions}.  Since there are by now a\nnumber of surveys and discussions of important examples (see,\ne.g.,~\\cite[\\S 10]{FKM1} or~\\cite[\\S 2.2]{short-sums} or~\\cite{pisa}),\nwe only recall some of them for concreteness.\n\n\\begin{itemize}\n\\item If $r\\geq 1$ is a fixed integer and $\\chi_1$, \\ldots, $\\chi_r$\n  are distinct non-trivial Dirichlet characters modulo~$q$, of\n  order~$d_i\\geq 2$, and if $f_1$, \\ldots, $f_r$, $g$ are polynomials\n  in $\\mathbf{Z}[X]$ such that either $\\deg(g\\mods q)\\geq 2$, or one of\n  the~$f_i\\mods q$ is not proportional to a $d_i$-th power\n  in~$\\bar{\\Ff}_q[X]$, then\n  $$\n  K(n)=\\chi_1(f_1(n))\\cdots \\chi_r(f_r(n))e\\Bigl(\\frac{g(n)}{q}\\Bigr)\n  $$\n  has Fourier transform of size bounded only in terms of~$r$ and the\n  degrees of the polynomials~$f_i$ and~$g$. (This is a consequence of\n  the Weil bounds for exponential sums in one variable).\n  \\par\n  Moreover (as is relevant only for Corollary~\\ref{RScor} in this\n  paper), $K$ is a trace function, and it is non-exceptional,\n  unless~$g$ is of degree~$1$, $r=1$ and~$f_1$ is of degree~$\\leq 1$.\n\\item  Let~$r\\geq 2$. Define~$\\Kl_r(0)=0$ and\n  $$\n  \\Kl_r(n)=\\frac{1}{q^{(r-1)-2}}\n  \\sum_{\\substack{x_1,\\ldots,x_r\\in {\\mathbf{F}_q}\\\\\n      x_1\\cdots x_r=n}}e\\Bigl(\\frac{x_1+\\cdots+x_r}{q}\\Bigr)\n  $$\n  for~$n\\in{\\mathbf{F}^\\times_q}$ (these are hyper-Kloosterman sums). Then\n  $\\norm{\\widehat{\\Kl}_r}_{\\infty}\\leq c_r$, where~$c_r$ depends only\n  on~$r$ (this depends on Deligne's general proof of the Riemann\n  Hypothesis over finite fields and on the construction and basic\n  properties of Kloosterman sheaves).\n  \\par\n  For all~$r\\geq 2$, the function~$\\Kl_r$ is a trace function of a\n  non-exceptional sheaf.\n\\end{itemize}\n\nWe also mention one important principle: if~$K$ is the trace function\nof a Fourier sheaf~$\\mathcal{F}$ (in the sense of~\\cite{ESDE}),\nthen~$\\widehat{K}$ is also such a function for a sheaf~$\\ft(\\mathcal{F})$;\nmoreover, if~$\\mathcal{F}$ has conductor~$c$ (in the sense of~\\cite{FKM1}),\nthen $\\ft(\\mathcal{F})$ has conductor $\\leq 10c^2$, and in\nparticular~$\\norm{\\widehat{K}}_{\\infty}\\leq 10c^2$. \n\nFinally, one example that is not usually discussed explicitly\n(formally, because it arises from a skyscraper sheaf) is\n$K(n)=q^{1\/2}\\delta_{n=a\\mods{q}}$, the $L^2$-normalized delta\nfunction at a point~$a\\in\\mathbf{Z}$. In this case, the Fourier transform is\nan additive character, hence is bounded by one, and dividing\nby~$q^{1\/2}$, we obtain the bound\n$$\n\\sum_{\\substack{n\\geq 1\\\\n\\equiv a\\mods{q}}}\n\\lambda(1,n)V\\Bigl(\\frac{n}{X}\\Bigr) \\ll\nZ^{10\/9}q^{-5\/18+\\varepsilon}X^{5\/6},\n$$\nunder the assumptions of Theorem~\\ref{thm:main}; in particular,\nif~$X=q^{3\/2}$ and~$V$ satisfies~(\\ref{eq:Vprop}) for~$Z=1$, we get\n$$\n\\sum_{\\substack{n\\geq 1\\\\n\\equiv a\\mods{q}}}\n\\lambda(1,n)V\\Bigl(\\frac{n}{q^{3\/2}}\\Bigr) \\ll q^{35\/36+\\varepsilon}\n$$\nfor any~$\\varepsilon>0$.  Note that, under the generalized\nRamanujan--Petersson conjecture $\\lambda(1,n)\\ll n^{\\varepsilon}$, we would\nobtain the stronger bound $q^{1\/2+\\varepsilon}$ (and knowing the\napproximation~$\\lambda(1,n)\\ll n^{\\theta}$ for some $\\theta<1\/3$ would\nbe enough to get a non-trivial bound).  We discuss this case in\nfurther details in Remark \\ref{lastremark}, in the context of\nCorollary \\ref{cor-average}.\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec-reminders}\n\n\n\\subsection{A Fourier-theoretic estimate}\n\nA key estimate in Section~\\ref{sec:O} will arise from the following\ngeneral bound (special cases of which have appeared before, e.g. in\nthe case of multiplicative characters for problems concerning sums\nover sumsets).\n \n\\begin{proposition}\\label{willprop}\n  Let $A$ be a a finite abelian group, with group operation denoted\n  additively.  Let $\\alpha$, $\\beta$ and $K$ be functions from~$A$\n  to~$\\mathbf{C}$. We have\n  $$\n  \\Bigl|\\sum_{m,n\\in A}\\alpha(m)\\beta(n)K(m-n)\\Bigr|\\leq\n  |A|^{1\/2}\\|\\widehat K\\|_\\infty \\|\\alpha\\|_2 \\|\\beta \\|_2.\n  $$\n\\end{proposition}\n\n\\begin{proof}\n  Using orthogonality of characters, we write\n  $$\n  \\sum_{m,n\\in A}\\alpha(m)\\beta(n) K(m-n)=\n  \\sum_{m,n,h\\in A}\\alpha(m)\\beta(n) K(h)\\frac{1}{|A|}\n  \\sum_{\\psi\\in\\widehat{A}}\\psi(h-(m-n)).\n  $$\n  Moving the sum over~$\\psi$ to the outside, this is equal to\n  $$\n  |A|^{1\/2}\\sum_{\\psi\\in\\widehat{A}}\n  \\widehat{\\alpha}(\\psi^{-1})\\widehat{\\beta}(\\psi)\\widehat{K}(\\psi),\n  $$\n  whose absolute value is\n  $$\n  \\leq |A|^{1\/2}\\norm{\\widehat{K}}_{\\infty}\n  \\sum_{\\psi\\in\\widehat{A}}|\\widehat{\\alpha}(\\psi^{-1})\\widehat{\\beta}(\\psi)|\n  \\leq |A|^{1\/2}\n  \\|\\widehat K\\|_\\infty \\|\\alpha\\|_2 \\|\\beta \\|_2,\n  $$\n  by the Cauchy--Schwarz inequality and the discrete Plancherel\n  formula.\n\\end{proof}\n\n\n\n\\subsection{Background on $\\GL_3$-cusp forms}\n\nWe refer to \\cite[Chap. 6]{Goldfeld} for notations. Let $\\varphi$ be a\ncusp form on~$\\GL_3$ with level~$1$ and with Langlands parameters\n$\\mu=(\\mu_1,\\mu_2,\\mu_3)\\in\\mathbf{C}^3$.  We denote by\n$(\\lambda(m,n))_{m,n\\not=0}$ its Fourier--Whittaker coefficients, and\nassume that\n$\\varphi$ is an eigenform of the Hecke operators $T_n$ and $T_n^*$,\nnormalized so that $\\lambda(1,1)=1$. The eigenvalue of~$T_n$ is\nthen~$\\lambda(1,n)$ for~$n\\geq 1$.\n\n\nLet $\\theta_3=5\/14$. The archimedean parameters and the Hecke\neigenvalues are bounded individually by\n$$\n|\\Re(\\mu_{i})|\\leq \\theta_3.\\quad\\quad |\\lambda(1,p)|\\leq 3p^{\\theta_3}\n$$\nfor any~$i$ and any prime number~$p$ (see ~\\cite{KimSar}).\n\nAverage estimates follow from the Rankin--Selberg method. We have\n\\begin{equation}\n  \\label{eq:RS}\n  \\sum_{1\\leq n\\leq X}|\\lambda(1,n)|^2\\ll X^{1+\\varepsilon},\n\\end{equation}\nand\n\\begin{equation}\n  \\label{eq-RS2}\n  \\sum_{1\\leq m^2n\\leq X}m|\\lambda(m,n)|^2\\ll X^{1+\\varepsilon},\n\\end{equation}\nfor $X\\geq 2$ and any $\\varepsilon>0$, where the implied constant depends\nonly on $\\varphi$ and $\\varepsilon$. (See~\\cite{Molteni} and ~\\cite[Lemma 2]{Munshi1}.)\n\nThe key analytic feature of $\\GL_3$-cusp forms that we use (as in\nprevious works) is the Voronoi summation formula for~$\\varphi$\n(originally due to Miller--Schmid, and Goldfeld--Li\nindependently). Since our use of the ``archimedean'' part of the\nformula is quite mild, we use the same compact formulation as\nin~\\cite[\\S 2.3]{HN}, where references are given.\n\nLet~$q\\geq 1$ be an integer (not necessarily prime). For $n\\in\\mathbf{Z}$, we\ndenote\n$$\n\\Kl_2(n;q)=\\frac{1}{\\sqrt{q}}\n\\sum_{x\\in (\\mathbf{Z}\/q\\mathbf{Z})^{\\times}}\ne\\Bigl(\\frac{nx+\\bar{x}}{q}\\Bigr)\n$$\nwhere~$\\bar{x}$ is the inverse of~$x$ modulo~$q$.\n\n\n\\begin{lemma}[Voronoi summation formula]\\label{Voronoi}\n  For $\\sigma\\in\\{-1,1\\}$, there exist functions $\\mathcal{G}^{\\sigma}$,\n  meromorphic on~$\\mathbf{C}$, holomorphic for~$\\Re(s)>\\theta_3$, with\n  polynomial growth in vertical strips~$\\Re(s)\\geq \\alpha$ for\n  any~$\\alpha>\\theta_3$, such that the following properties hold.\n  \\par\n  Let $a$ and~$q\\geq 1$ be coprime integers, let $X>0$, and let $V$ be\n  a smooth function on~$]0,+\\infty[$ with compact support.\n \n  We have\n  $$\n  \\sum_{n\\geq 1}\\lambda(1,n)e_q(an)V\\Bigl(\\frac{n}{X}\\Bigr) = q^{3\/2}\n  \\sum_{\\sigma\\in\\{-1,1\\}} \\sum_{n\\geq 1} \\sum_{m\\mid q}\n  \\frac{\\lambda(n,m)}{nm^{3\/2}} \\Kl_2\\Bigl(\\sigma n\\ov\n  a;\\frac{q}{m}\\Bigr) \\mathcal{V}_{\\sigma}\\Bigl(\\frac{m^2n}{q^3\/X}\\Bigr),\n  $$\n  where \n  $$\n  \\mathcal{V}_{\\sigma}(x)= \\frac{1}{2\\pi i}\\int_{(1)}x^{-s}\\mathcal{G}^{\\sigma} (s+1)\n  \\Bigl(\\int_{0}^{+\\infty}V(y)y^{-s}\\frac{dy}{y}\\Bigr)ds.\n  $$\n\\end{lemma}\n\nNote that the functions~$\\mathcal{G}^{\\sigma}$ depend (only) on the\narchimedean parameters of~$\\varphi$. We record some properties of the\nfunctions $\\mathcal{V}_{\\sigma}(x)$; for~$Z$ fixed they are already explained\nin~\\cite[\\S 2.3]{HN}.\n\n\\begin{lemma}\\label{bounds-for-V}\n  Let $\\sigma\\in\\{-1,1\\}$. For any $j\\geq 0$, any $A\\geq 1$ and\n  any~$\\varepsilon>0$, we have\n$$\nx^j\\mathcal{V}_{\\sigma}^{(j)}(x)\\ll \\min\\Bigl( Z^{j+1}x^{1-\\theta_3-\\varepsilon},\nZ^{j+5\/2+\\varepsilon}\\Bigl(\\frac{Z^3}{x}\\Bigr)^A\\Bigr)\n$$\nfor~$x>0$, where the implied constant depends on~$(j,A,\\varepsilon)$.\nMoreover, for $x\\geq 1$, we have\n$$\nx^j\\mathcal{V}_{\\sigma}^{(j)}(x)\\ll x^{2\/3}\\min(Z^j, x^{j\/3})\n$$\nwhere the implied constant depends on~$j$.\n\\end{lemma}\n\n\\begin{proof}\n  The first inequality in the first bound follows by shifting the\n  contour in $\\mathcal{V}_{\\pm}(x)$ to $\\Re s=\\theta_3-1+\\varepsilon$, while the\n  second one follows by shifting contour to the far right.  The second\n  bound follows from \\cite[Lemma 6]{Blomer}.\n\\end{proof}\n\nIn particular, we see from the lemma that the functions\n$\\mathcal{V}_{\\sigma}(x)$ decay very rapidly as soon as\n$x\\geq X^{\\delta}Z^{3}$ for some~$\\delta>0$.\n\n\\begin{remark}\n  The bound $x^j\\mathcal{V}_{\\sigma}^{(j)}(x)\\ll Z^{j+1}x^{1-\\theta_3-\\varepsilon}$\n  can be replaced by $x^j\\mathcal{V}_{\\sigma}^{(j)}(x)\\ll Z^{j}x^{1-\\varepsilon}$,\n  under the Ramanujan-Selberg conjecture, i.e., if $\\Re (\\mu_i)=0$ for\n  all~$i$.\n\\end{remark}\n\n\\begin{remark}\n  Let~$N\\geq 1$, and define a congruence\n  subgroup~$\\Gamma_N\\subset \\SL_3(\\mathbf{Z})$ by\n  $$\n  \\Gamma_N=\\Bigl\\{\\gamma\\in\\SL_3(\\mathbf{Z})\\,\\mid\\, \\gamma\\equiv\\begin{pmatrix}\n    *&*&*\\\\ *&*&*\\\\0&0&*\n  \\end{pmatrix}\\mods N\n  \\Bigr\\}.\n  $$\n  Zhou~\\cite{FZ} has established an explicit Voronoi summation\n  formula for $\\GL_3$-cuspforms that are invariant under~$\\Gamma_N$,\n \n \n \n \n \n \n \n \n  for additive twists by~$e_q(an)$ when either $(q,N)=1$ or $N\\mid\n  q$. It should then be possible to use this formula to generalise\n  Theorem~\\ref{thm:main} to such cuspforms by slight adaptations of\n  the argument below.\n \n \n \n \n \n \n\\end{remark}\n\n\n\\section{Amplification of the trace function}\n\nWe now begin the proof of Theorem~\\ref{thm:main}. Let $q$ be a prime\nnumber and $K$ a $q$-periodic function on $\\mathbf{Z}$. Let $\\widehat{K}$ be its\ndiscrete Fourier transform~(\\ref{eq:fourierK}), which is also a\n$q$-periodic function on~$\\mathbf{Z}$.  \n\nLet $P,L\\geq 1$ be two parameters to be chosen later, with $2P<q$ and\n$2L<q$. We define auxiliary sets\n\\begin{align*}\n  \\mathrm{P}&:=\\{p\\in[P,2P[\\,\\mid\\, p\\equiv 1\\mods{4}, \\text{ prime}\\}\\\\\n  \\mathrm{L}&:=\\{l\\in[L,2L[\\,\\mid\\, l\\equiv 3\\mods{4}, \\text{ prime}\\}.\n\\end{align*}\nNote that these sets are disjoint.  We denote\n\\begin{equation}\\label{eq-H}\nH=\\frac{q^2L}{XP}.\n\\end{equation}\nIn the sequel, we assume that $H\\geq 1$, that is\n\\begin{equation}\\label{Hcond}\n  XP\\leq q^2L.\n\\end{equation}\nLet $W$ be a smooth function on $\\mathbf{R}$ that satisfies \\eqref{eq:Vprop}\nwith $Z=1$ and furthermore $\\widehat W(0)=1$.\n\nWe now use the notation~$K(n,h)$ and~$\\widehat{K}(z,h)$ and the basic\namplicatifon idea discussed in Section~\\ref{ssec-amplification}\n(see~\\ref{whatKzdef} and~\\ref{eq-kn}).  We define\n\\begin{align*}\n  \\mathcal{F}\n  &=\n    \\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n    \\sum_{h\\in \\mathbf{Z}}S_{V}(K(\\cdot,hp\\ov l),X)\n    \\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)\\\\\n  &=\\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n    \\sum_{h\\in\\mathbf{Z}}\\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)\n    \\sum_{n\\geq 1}\\lambda(1,n)K(n,hp\\ov l)V\\Bigl(\\frac{n}{X}\\Bigr).\n\\end{align*}\n\nSeparating the contribution of $h=0$ and applying~(\\ref{eq-detector}),\nwe can write\n\\begin{equation}\\label{an-identity}\n  \\mathcal{F}=S_{V}(K,X)+\\mathcal{O}+\n  O\\Bigl(\\frac{q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}X}{q^{1\/2}}\\Bigr),\n\\end{equation}\nfor any~$\\varepsilon>0$, where\n\\begin{equation}\\label{eq-m2}\n\\mathcal{O}= \\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n\\sum_{h\\neq 0}\\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)\n\\sum_{n\\geq 1}\\lambda(1,n)K(n,hp\\ov l)V\\Bigl(\\frac{n}{X}\\Bigr).\n\\end{equation}\nIndeed, the contribution of $h=0$ is\n\\begin{align*}\n  \\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n  S_{V}(K(\\cdot,0),X)\\widehat{W}(0)\n  &=S_{V}(K,X)-\n    \\frac{\\widehat K(0)}{|\\mathrm{P}||\\mathrm{L}|q^{1\/2}}\n    \\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n    \\sum_{n\\geq 1}\\lambda(1,n)V\\Bigl(\\frac{n}{X}\\Bigr)\\\\\n  &=S_{V}(K,X)+O\\Bigl(\\frac{\\norm{\\widehat{K}}_{\\infty}X^{1+\\varepsilon}}{q^{1\/2}}\\Bigr),\n\\end{align*}\nfor any $\\varepsilon>0$, by \\eqref{eq:RS}.\n\n\\section{Evaluation of $\\mathcal{F}$}\\label{sec:Fsum}\n\nThe evaluation of~$\\mathcal{F}$ is close to the arguments of~\\cite{HN}\nand~\\cite[\\S 6]{Lin}. In fact, we could extract the desired bounds\nfrom these sources (especially~\\cite{Lin}) in the important special\ncase when the parameter~$Z$ is fixed as~$q$ varies. The reader who is\nfamiliar with one of these references may therefore wish to skip the\nproof of the next proposition in a first reading.\n\n\n\\begin{proposition}\\label{proposition-for-F}\nLet $\\eta>0$. Assume that \n\\begin{equation}\n  \\label{XZlower}\n  X\/Z\\geq q^{1+\\eta}.\n\\end{equation}\nand\n\\begin{equation}\\label{boundsPL}\nL\\leq P^4.\n\\end{equation}\nThen for any $\\varepsilon>0$, we have\n$$\n\\mathcal{F}\\ll q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n\\Bigl(\\frac{Z^{2}X^{3\/2}P}{qL^{1\/2}}+Z^{3\/2}X^{3\/4}(qPL)^{1\/4}\\Bigr),\n$$\nwhere the implied constant depends on $\\varphi$, $\\varepsilon$ and $\\eta$.\n\\end{proposition}\n\nThe remainder of this section is dedicated to the proof of this\nproposition. We fix $\\eta$ satisfying~(\\ref{XZlower}).\n \nWe apply the Poisson summation formula to the sum over $h$ in\n$\\mathcal{F}$, for each $(p,l)$. We obtain\n$$\n\\sum_{h\\in \\mathbf{Z}}K(n,hp\\ov\nl)\\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)=\\frac{H}{q^{1\/2}}\\sum_{(r,q)=1}\\widehat\nK(-p\\ov l\\ov r)e_q(np\\ov l\\ov r)W\\Bigl(\\frac{r}{R}\\Bigr),\n$$\nwhere\n\\begin{equation}\\label{eq-R}\nR=q\/H=\\frac{XP}{qL}.\n\\end{equation}\nHence it follows that\n$$\n\\mathcal{F}= \\frac{q^{3\/2}L}{XP|\\mathrm{P}||\\mathrm{L}|}\n\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}} \\sum_{(r,q)=1}\\widehat K(-p\\ov l\\ov r)\n\\sum_{n\\geq 1}e_q(n p\\ov l\\ov r) \\lambda(1,n)V\\Bigl(\\frac{n}{X}\\Bigr)\nW\\Bigl(\\frac{r}{R}\\Bigr).\n$$\n\nSince $l\\leq 2L<q$, we have $(q,rl)=1$ in the sums. By reciprocity, we\nhave\n$$\ne_q(n p\\ov l\\ov r)=e_{rl}(-np\\ov q)e_{qrl}(np)\n$$\nfor $n\\geq 1$.\n\n\\begin{remark}\n  Note that for $n\\asymp X$, we have\n  $$\n  \\frac{np}{qrl}\\asymp \\frac{XP}{qLXP\/(qL)}\\asymp 1,\n  $$\n  so that the additive character $e_{qrl}(np)$ doesn't oscillate.\n\\end{remark}\n\nWe define\n$$\nV_1(x)=e\\Bigl(\\frac{xXp}{qrl}\\Bigr)V(x).\n$$\nWe can then rephrase the above as\n$$\n\\sum_{n\\geq 1}\\lambda(1,n)e_{rl}(-np\\ov\nq)e_{qrl}(np)V\\Bigl(\\frac{n}{X}\\Bigr)= \\sum_{n\\geq\n  1}\\lambda(1,n)e_{rl}(-np\\ov q)V_1\\Bigl(\\frac{n}{X}\\Bigr),\n$$\nand\n$$\n\\mathcal{F}\n=\\frac{q^{3\/2}L}{XP|\\mathrm{P}||\\mathrm{L}|}\\sum_{p,l}\\sum_{r\\geq 1}\n\\widehat K(-p\\ov l\\ov r)W\\Bigl(\\frac{r}{R}\\Bigr)\n\\sum_{n\\geq 1}\\lambda(1,n)e_{rl}(-np\\ov q)V_1\\Bigl(\\frac{n}{X}\\Bigr).\n$$\n\nLet $\\mathcal{F}'$ be the contribution to the last expression of\nthose $(p,r,l)$ such that $(p,rl)=1$, and let $\\mathcal{F}''$ be the\nremaining contribution.\n\nIn the case $p\\mid r$, we can apply the Voronoi formula with modulus\n$rl\/p$; estimating the resulting expression directly, one obtains an\nestimate for the contribution $\\mathcal{F}''$ to $\\mathcal{F}$ that is bounded by\n$$\n\\mathcal{F}''\\ll \\frac{\\norm{\\widehat{K}}_{\\infty}Z^2X^{3\/2+\\varepsilon}}{qP}\n$$\nfor any $\\varepsilon>0$ (see~\\cite[\\S 6]{Lin} for a similar computation,\nwhere such contribution is denoted $\\mathcal{F}_1^{\\sharp}$).\n\nNow let $p$ be such that $(p,rl)=1$.  By the Voronoi summation\nformula (Lemma \\ref{Voronoi}), we have\n$$\n\\sum_{n\\geq 1}\\lambda(1,n)e_{rl}(-np\\ov q)\nV_1\\Bigl(\\frac{n}{X}\\Bigr)=(rl)^{3\/2} \\sum_{\\sigma\\in\\{-1,1\\}}\n\\sum_{n\\geq 1} \\sum_{m\\mid rl} \\frac{\\lambda(n,m)}{nm^{3\/2}}\n\\Kl_2(\\sigma\\ov\npqn;rl\/m)\\mathcal{V}_{1,\\sigma}\n\\Bigl(\\frac{m^2n}{r^3l^3\/X}\\Bigr).\n$$\nTherefore $\\mathcal{F}'=\\mathcal{F}'_{1}+\\mathcal{F}'_{-1}$, where\n\\begin{multline*}\n  \\mathcal{F}'_{\\sigma}\n  =\\frac{q^{3\/2}L}{XP|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n  \\sum_{r\\geq 1}\\widehat K(-p\\ov l\\ov\n  r)\n  W\\Bigl(\\frac{r}{R}\\Bigr)(rl)^{3\/2}\n  \\\\\n  \\sum_{n\\geq 1}\\sum_{m\\mid rl}\n  \\frac{\\lambda(n,m)}{nm^{3\/2}}\\Kl_2(\\sigma \\ov\n  pqn;rl\/m)\\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2n}{r^3l^3\/X}\\Bigr).\n\\end{multline*}\nWe re-arrange the sums to get\n\\begin{multline*}\n  \\mathcal{F}'_{\\sigma} =\\frac{(qRL)^{3\/2}L}{XP|\\mathrm{P}||\\mathrm{L}|}\n  \\sum_{r\\geq 1}\\Bigl(\\frac{r}{R}\\Bigr)^{3\/2}\n  W\\Bigl(\\frac{r}{R}\\Bigr) \\sum_{n,m}\n  \\frac{\\lambda(n,m)}{\\sqrt{nm}} \\\\\n  \\sum_{p\\in\\mathrm{P}}\\sum_{\\substack{l\\in\\mathrm{L}\\\\m\\mid rl}}\n  \\frac{(l\/L)^{3\/2}}{\\sqrt{n}m}\\widehat K(-p\\ov l\\ov r)\\Kl_2(\\sigma \\ov\n  pqn;rl\/m)\\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2n}{r^3l^3\/X}\\Bigr).\n\\end{multline*}\n\nLet~$\\delta>0$ be a small parameter. For fixed $r$ and $l$, using the\nbounds from Lemma \\ref{bounds-for-V} with a suitably large value\nof~$A$, the contribution to the sum over~$m$ and~$n$ of $(m,n)$\nsuch that\n\\begin{equation}\\label{eq-truncate}\n  m^2n\\geq q^{\\delta}\\frac{Z^3(rl)^3}{X}\\asymp \\frac{q^{\\delta}Z^3X^2P^3}{q^3}\n\\end{equation}\nis $\\ll \\norm{\\widehat{K}}_{\\infty}q^{-10}$ (say).\n\\par\nTo handle the remaining part of the sum, we apply the Cauchy--Schwarz\ninequality to the sum over $(m,n)$, and we obtain\n\\begin{equation}\\label{eq-mcfsigma}\n  \\mathcal{F}'_{\\sigma} \\ll \\frac{(qRL)^{3\/2}L}{XP|\\mathrm{P}||\\mathrm{L}|}\n  \\Bigl(\\sum_{r\\sim R} \\sum_{\\substack{n,m\\geq 1\\\\m^2n<\n      q^{\\delta}Z^3X^2P^3\/q^3}}\n  \\frac{|\\lambda(n,m)|^2}{nm}\\Bigr)^{1\/2}\\mathcal{N}_{\\sigma}^{1\/2}\n  +\\norm{\\widehat{K}}_{\\infty}q^{-1},\n\\end{equation}\nwhere\n\\begin{multline*}\n  \\mathcal{N}_{\\sigma}= \\sum_{r,m\\geq 1} W\\Bigl(\\frac{r}{R}\\Bigr)\n  \\frac{1}{m^2}\n  \\sumsum_{\\substack{p_1,p_2,l_1,l_2\\\\p_i\\in\\mathrm{P},l_i\\in\\mathrm{L}\\\\m\\mid\n      (rl_1,rl_2)}} \\Bigl(\\frac{l_1l_2}{L^2}\\Bigr)^{3\/2}\n  \\widehat K(-p_1\\ov l_1\\ov r)\\ov{\\widehat K(- p_2\\ov l_2\\ov r)}\\\\\n  \\times\\sum_{n\\geq 1}\\frac{1}{n}\\Kl_2(\\sigma \\ov\n  p_1qn;rl_1\/m)\\ov{\\Kl_2(\\sigma \\ov p_2qn;rl_2\/m)}\n  \\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2n}{r^3l_1^3\/X}\\Bigr)\n  \\ov{\\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2n}{r^3l_2^3\/X}\\Bigr)}.\n\\end{multline*}\nWe will prove the bound\n\\begin{equation}\\label{Nsigmabound}\n\\mathcal{N}_{\\sigma}\\ll\nq^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}^2\n\\Bigl(\nZ^4RPL\n+\\frac{Z^3R^{3\/2}q^3L^3}{X^2P}\\Bigr).\t\n\\end{equation}\nfor any~$\\varepsilon>0$.\nIf we select~$\\delta>0$ small enough in terms of~$\\varepsilon$, then by \nthe Rankin--Selberg bound~(\\ref{eq-RS2}), we deduce that\n\\begin{equation}\\label{eq-fpsigma}\n\\mathcal{F}'_{\\sigma} \\ll \\frac{q^{3\/2+\\varepsilon}Z^{\\varepsilon}L^{5\/2}R^2}\n{XP|\\mathrm{P}||\\mathrm{L}|}\\mathcal{N}_{\\sigma}^{1\/2}+\n\\norm{\\widehat{K}}_{\\infty}q^{-1}\n\\end{equation}\nfor any $\\varepsilon>0$.\nWe conclude, using~(\\ref{eq-mcfsigma}) and recalling\nthat~$R=XP\/(qL)$, that\n\\begin{align*}\n  \\mathcal{F}'_{\\sigma}\n  &\\ll \n    \\frac{q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n    R^2(qL)^{3\/2}}{XP^2}\\bigg(Z^4RPL+\\frac{Z^3R^{3\/2}q^3L^3}{X^2P}\n    \\bigg)^{1\/2}\n  \\\\\n  &\\ll\n    q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n    \\Bigl(\\frac{Z^{2}X^{3\/2}P}{qL^{1\/2}}+Z^{3\/2}X^{3\/4}(qPL)^{1\/4}\\Bigr).\n\\end{align*}\nfor any~$\\varepsilon>0$. Assuming \\eqref{Nsigmabound}, this concludes the proof of\nProposition~\\ref{proposition-for-F}.\n\n\\subsection{Proof of \\eqref{Nsigmabound}} We will now investigate the inner sum over~$n$\nin~$\\mathcal{N}_{\\sigma}$, and then perform the remaining summations\n(over $r$, $m$, $p_i$, $l_i$) essentially trivially. We let\n$$\nU=\\frac{q^{\\delta\/2}Z^{3\/2}XP^{3\/2}}{q^{3\/2}},\n$$\nso that the sum over~$m$ has been truncated to~$m\\leq U$.\n\nLet~$F$ be a smooth non-negative function on $\\mathbf{R}$ which is supported\non $[1\/2,3]$ and equal to $1$ on $[1,2]$. Let $Y\\geq 1$ be a parameter\nwith\n\\begin{equation}\\label{eq-boundy}\nY\\leq \\frac{q^{\\delta}Z^3X^2P^3}{m^2q^3},\n\\end{equation}\nand define\n$$\n\\mathcal{W}_{Y}(x)=\\frac{1}{x}\n\\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2xY}{r^3l_1^3\/X}\\Bigr)\n\\ov{\\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2xY}{r^3l_2^3\/X}\\Bigr)} F(x).\n$$\nWe study the sums\n$$\n\\mathcal{P}_Y= \\frac{1}{Y}\\sum_{n\\geq 1}\\Kl_2(\\ov\np_1qn;rl_1\/m)\\ov{\\Kl_2(\\ov p_2qn;rl_2\/m)}\n\\mathcal{W}_{Y}\\Bigl(\\frac{n}{Y}\\Bigr),\n$$\nand their combinations\n\\begin{equation}\\label{eq-py}\n  \\mathcal{N}_{Y,\\sigma}= \\sum_{r\\geq 1}\\sum_{1\\leq m\\leq U}\n  W\\Bigl(\\frac{r}{R}\\Bigr)\n  \\frac{1}{m^2}\n  \\sumsum_{\\substack{p_1,p_2,l_1,l_2\\\\p_i\\in\\mathrm{P},l_i\\in\\mathrm{L}\\\\m\\mid\n      (rl_1,rl_2)}} \\Bigl(\\frac{l_1l_2}{L^2}\\Bigr)^{3\/2}\n  \\widehat K(-p_1\\ov l_1\\ov r)\\ov{\\widehat K(- p_2\\ov l_2\\ov\n    r)}\\,\\mathcal{P}_Y.\n\\end{equation}\n\nWe will prove the following bound: for any $\\varepsilon>0$, if $\\delta$ is chosen small enough we have\n\\begin{equation}\\label{NYbound}\n\\mathcal{N}_{Y,\\sigma}\\ll q^{\\varepsilon}Z^4\\norm{\\widehat{K}}_{\\infty}^2RPL+q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}^2\\frac{Z^3R^{3\/2}q^3L^3}{X^2P}.\n\\end{equation}\nPerforming a dyadic partition of unity on the $n$ variable in $\\mathcal{N}_{\\sigma}$ we deduce \\eqref{Nsigmabound}.\n\\subsection{Bounding $\\mathcal{P}_Y$} \nWe apply the Poisson\nsummation formula~(\\ref{eq-poisson}) with modulus $r[l_1,l_2]\/m$, to\nget\n\\begin{equation}\\label{after-poisson}\n  \\mathcal{P}_Y\n  =\\frac{1}{r[l_1,l_2]\/m}\\sum_{n\\in\\mathbf{Z}}C(n,p_1,p_2, l_1,l_2,r, m)\n  \\widehat{\\mathcal{W}}_Y\\Bigl(\\frac{nY}{r[l_1,l_2]\/m}\\Bigr),\n\\end{equation}\nwhere\n\\begin{eqnarray*}\n  C(n,p_1,p_2,l_1,l_2, r,m)=\n  \\sum_{\\beta \\mods {r[l_1,l_2]\/m}}\n  \\Kl_2(\\ov p_1q\\beta;rl_1\/m)\n  \\ov{\\Kl_2(\\ov p_2q\\beta;rl_2\/m)} \\, e_{r[l_1,l_2]\/m}(\\beta n),\n\\end{eqnarray*}\nwith~$\\ov{p}_i$ denoting the inverse of~$p_i$ modulo~$rl_i\/m$. We write\n$$\n\\mathcal{P}_Y=\\mathcal{P}_0+\\mathcal{P}_1\n$$\nwhere\n$$\n\\mathcal{P}_{0}=\\frac{1}{r[l_1,l_2]\/m} C(0,p_1,p_2,l_1,l_2,r,\nm)\\widehat{\\mathcal{W}}_Y(0)\n$$\nis the contribution of the term~$n=0$ and~$\\mathcal{P}_1$ is the\nremainder in \\eqref{after-poisson}. We show below that for any $\\varepsilon>0$, if $\\delta$ is chosen small enough we have\n\\begin{equation}\\label{P0bound}\n\\mathcal{P}_{0}\\ll \\delta_\\stacksum{l_1=l_2}{p_1=p_2}(qr)^{\\varepsilon}Z^4+\\delta_\\stacksum{l_1=l_2}{p_1\\not=p_2}\\mathcal{P}_0\\ll\nq^{\\varepsilon}Z^4\\frac{m}{rl}\n\\Bigl(\\frac{rl}{m},p_2-p_1\\Bigr)\t\n\\end{equation}\nand that\n\\begin{equation}\\label{P1bound}\\mathcal{P}_{1}\\ll q^{2\\varepsilon}Z^3\\Bigl(\\frac{r[l_1,l_2]1}{m}\\Bigr)^{1\/2}\\frac{m^2q^3}{X^2P^3}.\\end{equation}\nUsing \\eqref{P0bound} in  the sum~(\\ref{eq-py}), we find that the contribution to~$\\mathcal{N}_{Y,\\sigma}$ of $\\mathcal{P}_0$ is bounded by \n\\begin{gather}\\nonumber\n  \\sum_{r\\geq 1} W\\Bigl(\\frac{r}{R}\\Bigr) \\sum_{1\\leq m\\leq U}\n  \\frac{1}{m^2}\n  \\sumsum_{\\substack{p_1,p_2,l\\\\p_i\\in\\mathrm{P},l\\in\\mathrm{L}\\\\m\\mid rl}}\n  \\Bigl(\\frac{l}{L}\\Bigr)^{3} \\widehat K(-p_1\\ov l\\ov r)\\ov{\\widehat K(-\n    p_2\\ov l\\ov r)}\\,\\mathcal{P}_0\n  \\\\ \\nonumber\n  \\ll q^{\\varepsilon}Z^4\\norm{\\widehat{K}}_{\\infty}^2 \\sum_{r\\asymp R}\n  \\sum_{1\\leq m\\leq U}\\frac{1}{m^2} \\sum_{\\substack{l\\in\\mathrm{L}\\\\m\\mid\n      rl}} \\Bigl(\\sum_{p\\in\\mathrm{P}}1+\n  \\frac{m}{rl}\\sum_{\\substack{p_1,p_2\\in\\mathrm{P}\\\\p_1\\not=p_2}}\\Bigl(\\frac{rl}{m},p_2-p_1\\Bigr)\n  \\Bigr)\\\\ \\nonumber\n  \\ll q^{\\varepsilon}Z^4\\norm{\\widehat{K}}_{\\infty}^2 \\Bigl(RPL+ \\sum_{r\\asymp\n    R} \\frac{1}{r}\\sum_{1\\leq m\\leq U}\\frac{1}{m}\n  \\sum_{\\substack{l\\in\\mathrm{L}\\\\m\\mid rl}}\\frac{1}{l} \\sum_{d\\mid rl\/m}\\varphi(d)\n  \\sum_{\\substack{p_1,p_2\\in\\mathrm{P}\\\\p_1\\equiv p_2\\mods{d}}}1\n  \\Bigr)\\\\\n  \\ll q^{\\varepsilon}Z^4\\norm{\\widehat{K}}_{\\infty}^2 (RPL+P^2)\\ll\n  q^{\\varepsilon}Z^4\\norm{\\widehat{K}}_{\\infty}^2RPL.\\label{NYP0}\n\\end{gather}\nHere $RPL=XP^2\/q\\gg P^2$, since $X$ satisfies \\eqref{XZlower}.\nUsing \\eqref{P1bound} we find that the contribution of~$\\mathcal{P}_1$\nto~$\\mathcal{N}_{\\sigma,Y}$ is bounded by\n\\begin{multline}\\label{NYP1}\n  \\ll q^{\\varepsilon}Z^3\\norm{\\widehat{K}}_{\\infty}^2\\sum_{r\\asymp R}\n  \\sum_{1\\leq m\\leq U}\n  \\frac{1}{m^2}\n  \\sum_{\\substack{p_1,p_2,l_1,l_2\\\\m\\mid (rl_1,rl_2)}}\n  \\Bigl(\\frac{r[l_1,l_2]}{m}\\Bigr)^{1\/2}\\frac{m^2q^3}{X^2P^3}\\\\\n  \\ll q^{\\varepsilon}Z^3\\norm{\\widehat{K}}_{\\infty}^2\n  R\\frac{q^3}{X^2P^3}(P^2L^2)(RL^2)^{1\/2}\n  \\ll q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}^2\\frac{Z^3R^{3\/2}q^3L^3}{X^2P}.\n\\end{multline}\n\n\n\nCombining \\eqref{NYP0} and \\eqref{NYP1} we obtain \\eqref{NYbound}. \n\n\\subsection{Proof of \\eqref{P0bound} and \\eqref{P1bound}}\n\nThe next two lemmas evaluate the archimedan and non-archimedan Fourier transforms which occur in \\eqref{after-poisson} :\n\n\\begin{lemma}\\label{lm-trunc}\n  With notation as above, in particular~\\emph{(\\ref{eq-boundy})},\n  let~$j\\geq 0$ be an integer and let~$\\varepsilon>0$.\n  \\par\n  \\emph{(1)} We have\n \n \n \n \n  \\begin{equation}\\label{eq-wyzero}\n    \\widehat{\\mathcal{W}}_Y(0)\\ll q^{\\delta}Z^4.\n  \\end{equation}\n  \\par \n  \\emph{(2)} We have\n  \\begin{equation}\\label{derivatives-of-F}\n    x^j\\mathcal{W}^{(j)}_{Y}(x)\\ll\n    \\begin{cases}\n      Z^{2+j}\\Bigl(\\frac{m^2Yq^3}{X^2P^3}\\Bigr)^{2-2\\theta_3-\\varepsilon}\n      & \\quad \\text{if } Y<\\frac{X^2P^3}{m^2q^3}\\\\\n      \\Bigl(\\frac{m^2Yq^3}{X^2P^3}\\Bigr)^{4\/3+j\/3} & \\quad \\text{if }\n      Y\\geq \\frac{X^2P^3}{m^2q^3},\n     \n    \\end{cases}\t\n  \\end{equation}\n  where the implied constants depends on~$(\\varphi,j,\\varepsilon)$.\n  \\par\n  \\emph{(3)} If $1\\leq |n|\\leq q^{\\delta}Z\\frac{r[l_1,l_2]}{mY}$\n \n \n  then we have\n  $$\n  \\widehat{\\mathcal{W}}_{Y}\\Bigl(\\frac{nY}{r[l_1,l_2]\/m}\\Bigr)\\ll\n  q^{\\delta}Z^{2}\\frac{m^2Yq^3}{X^2P^3}.\n  $$\n\\end{lemma}\n\n\\begin{proof}\n  Since~$F_Y$ has support in $[1\/2,3]$, part~(1) follows from the\n  bound~$\\mathcal{V}_{\\sigma}(x)\\ll x^{2\/3}$ for~$x\\geq 1$ and the\n  fact that\n  $$\n  \\frac{m^2xY}{r^3l_i^3\/X}\\asymp \\frac{m^2Y}{X^2P^3\/q^3}\\ll\n  q^{\\delta}Z^3.\n  $$\n  \\par\n  Part~(2) is obtained using the estimates\n  \\begin{gather*}\n    x^j\\mathcal{V}_{\\pm}^{(j)}(x)\\ll Z^{j+1}x^{1-\\theta_3-\\varepsilon}\\quad \\text{\n      if }\n    0<x<1,\\\\\n    x^j\\mathcal{V}_{\\pm}^{(j)}(x)\\ll x^{2\/3+j\/3}\\quad \\text{ if } x\\geq 1\n  \\end{gather*}\n  (see Lemma \\ref{bounds-for-V}), noting again that\n  $\\frac{(rl)^3}{X}\\sim \\frac{X^2P^3}{q^3}$.\n  \\par\n \n \n \n \n \n \n \n \n \n \n \n \n  From \\eqref{derivatives-of-F}, for any $n$ such that\n  $1\\leq |n|\\leq q^{\\delta}Z\\frac{r[l_1,l_2]}{mY}$, we get the estimates\n  $$\n  \\widehat{\\mathcal{W}}_Y\\Bigl(\\frac{nY}{r[l_1,l_2]\/m}\\Bigr)\\ll\n   \\begin{cases}\n     Z^{2}\\Bigl(\\frac{m^2Yq^3}{X^2P^3}\\Bigr)^{2-2\\theta_3-\\varepsilon},&\n     \\quad\n     \\mathrm{if}\\, Y<\\frac{X^2P^3}{m^2q^3}\\\\\n     \\Bigl(\\frac{m^2Yq^3}{X^2P^3}\\Bigr)^{4\/3},& \\quad \\mathrm{if}\\,\n     Y\\geq \\frac{X^2P^3}{m^2q^3}\n   \\end{cases}\n   $$\n   Since $m^2Yq^3\/(X^2P^3)\\ll q^{\\delta}Z^3$, the second bound is also\n   $$\n   \\ll q^{\\delta\/3}Z\\frac{m^2Yq^3}{X^2P^3}.\n   $$\n   Together with the first bound, this implies part (3) of the lemma.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lm-delta}\n  Let $n\\in\\mathbf{Z}$, $p_1$, $p_2$ primes, $l_1$, $l_2$ primes, $r\\geq 1$\n  and $m\\geq 1$ be integers with $m\\mid rl_i$.\n  \\par\n  \\emph{(1)} We have\n  $$\n  C(0,p_1,p_2,l_1,l_2,r,m)=0\n  $$\n  unless $l_1=l_2$.\n  \\par\n  \\emph{(2)} For $l$ prime with $m\\mid rl$, we have\n  $$\n  |C(0,p_1,p_2,l,l,r,m)|\\leq (rl\/m,p_2-p_1)\n. $$\n  \\par\n  \\emph{(3)} Let\n  $$\n  \\Delta=q\\frac{l_2^2p_2-l_1^2p_1}{(l_1,l_2)^2}.\n  $$\n \n \n  We have\n  $$\n  |C(n,p_1,p_2,l_1,l_2,r,m)|\\leq\n  2^{O(\\omega(r))}\\Bigl(\\frac{r[l_1,l_2]}{m}\\Bigr)^{1\/2}\n  \\frac{(\\Delta,n,m\/rl_1,m\/\/rl_2)}{(n,m\/rl_1,m\/rl_2)^{1\/2}}.\n  $$\n  \\par\n  \\emph{(4)} Suppose that $\\Delta=0$. If $(p_1,p_2)$ are\n  $\\equiv 1\\mods{4}$ and $(l_1,l_2)$ are $\\equiv 3\\mods{4}$, then\n  $p_1=p_2$ and $l_1=l_2$. For $p$ prime and $l$ prime with\n  $m\\mid rl$, we have\n$$\n|C(n,p,p,l,l,r,m)|\\leq 2^{O(\\omega(r))}\n\\Bigl(\\frac{rl}{m}\\Bigr)^{1\/2}\\Bigl(n,\\frac{rl}{m}\\Bigr)^{1\/2},\n$$\nIn particular, $C(0,p,p,l,l,r,m)\\ll r^{\\varepsilon}\\frac{rl}{m}$ for\nany~$\\varepsilon>0$.\n\\end{lemma}\n\n\\begin{proof}\n  Part (1) follows by direct computation (the sum vanishes unless\n  $[l_1,l_2]=l_1$ and $[l_1,l_2]=l_2$). If $n=0$ and $l_1=l_2$, then\n$$\n|C(0,p_1,p_2,l,l,r,m)|=\\bigg|\\sum_{\\substack{x\\bmod rl\/m\\\\(x,rl\/m)=1}}\ne\\Bigl(\\frac{( p_2-p_1)x}{rl\/m}\\Bigr)\\bigg| = \\bigg| \\sum_{d|(rl\/m, p_2-p_1)} d \\mu\\Bigl(\\frac{rl}{md} \\Bigr) \\bigg| \\leq (rl\/m,p_2-p_1)\n$$\nby a classical bound for Ramanujan's sum, which proves (2). Finally, part (3) is a special case of~\\cite[Lemma\nA.2 (A.3)]{HN} (applied with \n$(\\xi,s_1,s_2)=(n,rl_1\/m,rl_2\/m)$, and\n$(a_1,b_1,a_2,b_2)=(q,p_1,q,p_2)$ in the definition of~$\\Delta$).\nIf $\\Delta=0$, then necessarily $p_1=p_2$ and $l_1=l_2$, and we\nobtain~(4) immediately.\n\\end{proof}\n\n\\subsubsection{Estimation of $\\mathcal{P}_0$}\n\nNote that~$\\mathcal{P}_0=0$ unless $l_1=l_2$. If that is the case, we\ndenote~$l=l_1=l_2$. We then have two bounds for~$\\mathcal{P}_{0}$. If\nwe have also~$p_1=p_2$, then the quantity~$\\Delta$ of\nLemma~\\ref{lm-delta} (3) is zero.\nSince~$\\widehat{\\mathcal{W}}_Y(0)\\ll q^{\\varepsilon}Z^4$ for any~$\\varepsilon>0$\n(provided~$\\delta>0$ is chosen small enough) by Lemma~\\ref{lm-trunc}\n(1), we obtain\n$$\n\\mathcal{P}_0\\ll q^{\\varepsilon}Z^4\\frac{m}{rl}|C(0,p_1,p_1,l,l,r,m)|\\ll\n(qr)^{\\varepsilon}Z^4\n$$\nby the last part of Lemma~\\ref{lm-delta} (4).\n\\par\nOn the other hand, if~$p_1\\not=p_2$, we have~$\\Delta\\not=0$ hence\n$$\n\\mathcal{P}_0\\ll\nq^{\\varepsilon}Z^4\\frac{m}{rl}\n\\Bigl(\\frac{rl}{m},p_2-p_1\\Bigr)\n$$\nby Lemma~\\ref{lm-delta} (1) (which shows that the sum\n$C(0,p_1,p_2,l_1,l_2,r,m)$ is zero unless~$l_1=l_2$) and (2).\n\\par\n\n\\subsubsection{Estimation of $\\mathcal{P}_1$}\nUsing Lemma~\\ref{lm-trunc} (2) for a\nsuitable value of~$j$, we obtain first\n$$\n\\mathcal{P}_{1}= \\frac{1}{r[l_1,l_2]\/m}\\sum_{1\\leq |n|\\leq\n  q^{\\delta}Z\\frac{r[l_1,l_2]}{mY}} C(n,p_1,p_2,l_1,l_2,r,m)\n\\widehat{\\mathcal{W}}_Y\\Bigl(\\frac{nY}{r[l_1,l_2]\/m}\\Bigr) +O(q^{-1})\n$$\nfor any~$\\varepsilon>0$ if~$\\delta$ is chosen small enough.  Then, by\nLemma~\\ref{lm-delta} and Lemma~\\ref{lm-trunc} (3), we deduce that\n\\begin{align*}\n  \\mathcal{P}_1\n  &\\ll q^{\\varepsilon+2\\delta}\\frac{1}{(r[l_1,l_2]\/m)^{1\/2}}\n    \\sum_{1\\leq |n|\\leq q^{\\delta}Z\\frac{r[l_1,l_2]}{mY}}\n    \\frac{(\\Delta,n,rl_1\/m,rl_2\/m)}{(n,rl_1\/m,rl_2\/m)^{1\/2}}\n    \\frac{Z^{2}m^2Yq^3}{X^2P^3}\n  \\\\\n  &\\ll q^{2\\varepsilon}Z^3\\Bigl(\\frac{r[l_1,l_2]1}{m}\\Bigr)^{1\/2}\\frac{m^2q^3}{X^2P^3}\n\\end{align*}\nif $\\delta<\\varepsilon\/2$.\n\n\n\\section{Estimate of $\\mathcal{O}$}\\label{sec:O}\n\nIn this section, we bound the sum~$\\mathcal{O}$ defined\nin~(\\ref{eq-m2}). Our goal is:\n\n\\begin{proposition}\\label{corollary-for-O}\n  Let~$\\eta>0$ be a parameter such that \\eqref{XZlower} holds.\n  Let~$\\varepsilon>0$. If~$\\delta$ is a sufficiently small positive real\n  number and if $P,L,X$ satisfy\n  \\begin{equation}\\label{eqboundsPHLX}\n    XP\\leq q^2L,\\quad q^{1+\\delta}L^2<X\/8,\\quad q^{\\delta}PHL<q\/8,\n  \\end{equation}\n  then we have\n  \\begin{equation}\\label{final-bound-for-O}\n    \\mathcal{O}\\ll q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\\frac{qX^{1\/2}}{P},\n  \\end{equation}\n  where the implied constant depends on~$\\varphi$ and~$\\varepsilon$.\n\\end{proposition}\n\n\n\nWe start by decomposing $\\mathcal{O}$ into\n$$\n\\mathcal{O}=\\mathcal{O}_1+\\mathcal{O}_2\n$$\naccording to whether the prime $l$ divides $h$ or not, in other words\n\\begin{align*}\n  \\mathcal{O}_1\n  &=\\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n    \\sum_{h\\not=0}\\widehat{W}\\Bigl(\\frac{hl}{H}\\Bigr)\n    \\sum_{n\\geq 1}\\lambda(1,n)K(n,hp)V\\Bigl(\\frac{n}{X}\\Bigr)\\\\\n  \\mathcal{O}_2\n  &=\\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\n    \\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n    \\sum_\\stacksum{h\\neq 0}{(h,l)=1}\n    \\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)\n    \\sum_{n\\geq 1}\\lambda(1,n)K(n,hp\\ov l)V\\Bigl(\\frac{n}{X}\\Bigr).\n\\end{align*}\n\nBoth of these sums will be handled in a similar way in the next two\nsubsections, beginning with the most difficult one.\n\n\\subsection{Bounding $\\mathcal{O}_2$}\n\nIn the sum $\\mathcal{O}_2$, we first use the bound\n$$\n\\widehat{W}(x)\\ll (1+|x|)^{-A}\n$$\nfor any~$A\\geq 1$ and~$x\\in\\mathbf{R}$, and\n$$\n\\sum_{n\\geq 1}\\lambda(1,n)K(n,hp\\ov l)V\\Bigl(\\frac{n}{X}\\Bigr)\\ll\nX^{1+\\varepsilon}q^{1\/2}\\norm{\\widehat{K}}_{\\infty}\\ll\nq^{5\/2+2\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n$$\nfor any~$\\varepsilon>0$ (by~(\\ref{eq:RS}) and discrete Fourier inversion) to\ntruncate the sum over~$h$ to~$|h|\\leq q^{\\delta}H$, for\nsome~$\\delta>0$ that may be arbitrarily small.\n\n\nLet~$T\\geq 0$ be a smooth function with compact support such that\n$T(x)=\\norm{V}_{\\infty}$ for $x\\in [1\/2,3]$ and such that~$T$\nsatisfies \\eqref{eq:Vprop} with a fixed value of~$Z$. We then\nhave~$|V|\\leq T$.\n\nIn the sum~$\\mathcal{O}_2$, we split the $h$-sum into $O(\\log q)$ dyadic\nsums. We then apply the Cauchy--Schwarz inequality to smooth the\n$n$-variable, and we obtain\n$$\n\\mathcal{O}_2\\ll \\frac{\\log^3 q}{PL} \\Bigl(\\sum_{n\\sim\n  X}|\\lambda(1,n)|^2\\Bigr)^{1\/2} \\mathop{\\mathrm{Max}}\\limits_{1\\leq H'\\leq\n  q^{\\delta}H}\\mathcal{R}_{H'}^{1\/2}\\ll \\frac{X^{1\/2}\\log^3 q}{PL}\n\\mathop{\\mathrm{Max}}\\limits_{1\\leq H'\\leq q^{\\delta}H}\\mathcal{R}_{H'}^{1\/2},\n$$\nby~(\\ref{eq:RS}) again, where\n$$\n\\mathcal{R}_{H'}= \\sumsum_{p_1,h_1,l_1,p_2,h_2,l_2} \\sum_{n\\geq\n  1}K(n,h_1p_1\\ov l_1)\\ov{K(n,h_2p_2\\ov l_2)}\n\\widehat{W}\\Bigl(\\frac{h_1}{H}\\Bigr)\n\\ov{\\widehat{W}\\Bigl(\\frac{h_2}{H}\\Bigr)}\nT\\Big(\\frac{n}X\\Bigr),\n$$\nwith the variables in the sums constrained by the conditions\n$$\np_i\\in\\mathrm{P},\\quad l_i\\in\\mathrm{L},\\quad\nH'<h_i\\leq 2H',\\quad (l_i,h_i)=1.\n$$\n\nFor $x\\in{\\mathbf{F}_q}$, we define\n\\begin{equation}\\label{nudef}\n  \\nu(x)=\n  \\sum_{\\substack{\n      (p,h,l)\\in \\mathrm{P}\\times[H',2H'[  \\times\\mathrm{L},\\\\\n      (h,l)=1\\\\\n      ph\\ov l\\equiv x\\mods q}}\n  \\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)\n\\end{equation}\nso that we have\n\\begin{equation}\\label{eq:nsum}\n  \\mathcal{R}_{H'}=\\sumsum_{x_1,x_2\\in{\\mathbf{F}_q}}\n  \\nu(x_1)\\nu(x_2)\\sum_{n\\geq 1}K(n,x_1)\\ov{K(n,x_2)}\n  T\\Bigl(\\frac{n}X\\Bigr).\n\\end{equation}\n\nWe apply the Poisson summation formula~(\\ref{eq-poisson}) for the sum\nover~$n$. This results in the formula\n$$\n\\sum_{n\\geq 1}K(n,x_1)\\ov{K(n,x_2)}T\\Bigl(\\frac{n}X\\Bigr)\n=\\frac{X}{\\sqrt{q}}\n\\sum_{h\\in\\mathbf{Z}} \\Bigl(\\frac{1}{\\sqrt{q}}\n\\sum_{n\\mods{q}}K(n,x_1)\\ov{K(n,x_2)}e\\Bigl(\\frac{nh}{q}\\Bigr)\n\\Bigr)\n\\widehat{T}\\Bigl(\\frac{hX}{q}\\Bigr).\n$$\n\nObserve that for any~$h\\in\\mathbf{Z}$, we have\n$$\n\\frac{1}{\\sqrt{q}}\n\\sum_{n\\mods{q}}K(n,x_1)\\ov{K(n,x_2)}e\\Bigl(\\frac{nh}{q}\\Bigr)\n=\\frac{1}{\\sqrt{q}}\\sum_{u\\mods{q}} \\widehat K(u,x_1)\\ov{\\widehat\n  K(u+h,x_2)}\n$$\nwhere $ \\widehat K(u,x)$ is defined as in \\eqref{whatKzdef}. In\nparticular, this quantity is bounded by\n$q^{1\/2}\\norm{\\widehat{K}}_{\\infty}^2$.\n\nNow, for all $h\\not=0$ and all~$A\\geq 1$, we have\n$$\n\\widehat{T}\\Bigl(\\frac{hX}{q}\\Bigr)\\ll_A\n\\Bigl(\\frac{qZ}{hX}\\Bigr)^A\\leq \\Bigl(\\frac{qZ}{X}\\Bigr)^A\\leq\nq^{-A\\eta},\n$$\nby~(\\ref{XZlower}), where the implied constant depends on~$A$. Hence,\ntaking~$A$ large enough in terms of~$\\eta$, the contribution of\nall~$h\\not=0$ to the sum over~$n$ is\n$\\ll \\norm{\\widehat{K}}_{\\infty}^2q^{-5}$, and the total contribution\nto~$\\mathcal{R}_{H'}$ is (using very weak bounds on~$\\nu(x)$)\n$$\n\\ll \\norm{\\widehat{K}}_{\\infty}^2 q^{-3}(PHL)^2\\ll\n\\norm{\\widehat{K}}_{\\infty}^2q^{-1}\n$$\nby~(\\ref{eqboundsPHLX}).\n\n\nThe remaining contribution to~$\\mathcal{R}_{H'}$ from the\nfrequency~$h=0$ is\n$$\n\\frac{X}{\\sqrt{q}}\\sumsum_{x_1,x_2\\in{\\mathbf{F}_q}}\n\\nu(x_1)\\nu(x_2)\\frac{1}{\\sqrt{q}}\\sum_{u\\in{\\mathbf{F}_q}} \\widehat\nK(u,x_1)\\ov{\\widehat K(u,x_2)}\\widehat{T}(0).\n$$\n\n\\begin{lemma}\n  For any~$(x_1,x_2)\\in{\\mathbf{F}_q}\\times{\\mathbf{F}_q}$, we have\n  $$\n  \\frac{1}{\\sqrt{q}}\\sum_{u\\in{\\mathbf{F}_q}}\n  \\widehat K(u,x_1)\\ov{\\widehat K(u,x_2)}=\n  L(x_1-x_2)\n  $$\n  where\n  $$\n  L(x)=\\frac{1}{\\sqrt{q}}\\sum_{u\\in{\\mathbf{F}^\\times_q}}|\\widehat{K}(u)|^2e_q(-\\bar{u}x)\n  +\\frac{1}{\\sqrt{q}}|\\widehat{K}(0)|^2.\n  $$\n  Moreover, we have\n  $$\n  \\widehat{L}(h)=|\\widehat{K}(0)|^2\\delta_{h\\equiv 0\\mods{q}}+\n  |\\widehat{K}(\\bar{h})|^2\\delta_{h\\not\\equiv 0\\mods{q}},\n  $$\n  and in particular~$|\\widehat{L}(h)|\\leq \\norm{\\widehat{K}}_{\\infty}^2$ for\n  all~$h\\in{\\mathbf{F}_q}$.\n\\end{lemma}\n\n\\begin{proof}\n  The first formula is an immediate consequence of the\n  definition~(\\ref{whatKzdef}), and the second results from a\n  straightforward computation.\n\\end{proof}\n\n\\begin{lemma}\n  We have\n  $$\n  \\norm{\\nu}_2^2=\\sum_{x\\in{\\mathbf{F}_q}}\\nu(x)^2\\ll q^{\\varepsilon+\\delta}PHL\n  $$\n  for any~$\\varepsilon>0$.\n\\end{lemma}\n\n\\begin{proof}\nFrom the last condition in~(\\ref{eqboundsPHLX}), we have the\nimplications\n\\begin{equation}\\label{PHLbound}\n  h_2p_2\\ov l_2=h_1p_1\\ov l_1\\mods q\\Longleftrightarrow\n  l_1h_2p_2\\equiv l_2h_1p_1\\mods q\\Longleftrightarrow l_1h_2p_2=l_2h_1p_1.\n\\end{equation}\nTherefore, if $(p_1,h_1,l_2)$ are given, the number of possibilities\nfor $(p_2,h_2,l_1)$ is $\\ll q^{\\varepsilon}$ for any~$\\varepsilon>0$.\nThe bound\n$$\n\\sum_{x\\in{\\mathbf{F}^\\times_q}}\\nu(x)^2\\ll q^{\\varepsilon}PH'L\\leq q^{\\varepsilon+\\delta}PHL\n$$\nfollows immediately.\n\\end{proof}\n\nWe can now combine these two lemmas with Proposition~\\ref{willprop} to\ndeduce that\n\\begin{align*}\n  \\frac{X}{\\sqrt{q}}\\Bigl|\\sumsum_{x_1,x_2\\in{\\mathbf{F}_q}}\n  \\nu(x_1)\\nu(x_2)\\frac{1}{\\sqrt{q}}\\sum_{u\\in{\\mathbf{F}_q}}\n  \\widehat K(u,x_1)\\ov{\\widehat K(u,x_2)}\\widehat{T}(0)\\Bigr|\n  &\\leq X\\norm{\\widehat{L}}_{\\infty}\\norm{\\nu}_2^2|\\widehat{T}(0)|\\\\\n  &\\ll q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}^2XPHL\n\\end{align*}\nfor any~$\\varepsilon>0$, by taking~$\\delta$ small enough in terms of~$\\varepsilon$.\nHence we obtain\n\\begin{equation}\\label{diagonal1total}\n  \\mathcal{O}_2\\ll q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}X\\Bigl(\\frac{H}{PL}\\Bigr)^{1\/2}\\ll\n  q^{1+\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\\frac{X^{1\/2}}{P}.\n\\end{equation}\n\n\\subsection{Bounding $\\mathcal{O}_1$ and end of the proof of Proposition~\\ref{corollary-for-O}}\n\nThe treatment of $\\mathcal{O}_1$ is similar to that of~$\\mathcal{O}_2$, but simpler,\nso we will be brief. We have\n$$\n\\mathcal{O}_1=\\frac{1}{|\\mathrm{L}|}\\sum_{l\\in\\mathrm{L}}\\frac{1}{|\\mathrm{P}|}\n\\sum_{p\\in\\mathrm{P}}\\sum_{h\\not=0} \\widehat{W}\\Bigl(\\frac{h}{H\/l}\\Bigr)\n\\sum_{n\\geq 1}\\lambda(1,n)K(n,hp)V\\Bigl(\\frac{n}{X}\\Bigr).\n$$\nWe bound the sum over~$p$ for each individual~$l\\in\\mathrm{L}$, with\n$h\\ll H\/l\\asymp H\/L$, by repeating the arguments of the previous\nsection with $H$ replaced by $H\/l$ and $L$ replaced by $1$. We\nobtain\n\\begin{equation}\\label{O1bound}\n  \\mathcal{O}_1\\ll \\norm{\\widehat{K}}_{\\infty}\n  q^{\\varepsilon}X\\Bigl(\\frac{H}{PL}\\Bigr)^{1\/2}\n  \\ll q^{1+\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\\frac{X^{1\/2}}{P}\n\\end{equation}\nfor any~$\\varepsilon>0$, as in the previous case.\n\nFinally, since~$\\mathcal{O}=\\mathcal{O}_1+\\mathcal{O}_2$, this bound combined\nwith~(\\ref{diagonal1total}) implies Proposition~\\ref{corollary-for-O}.\n\n\n\\section{End of the proof}\n\nWe can now finish the proof of our main theorem.  We recall that $X,Z$\nare such that\n  \\begin{equation}\n    \\label{eqcondZXq}\nZ^{2\/3}q^{4\/3}\\leq X \\leq Z^{-2}q^{2}.\n\\end{equation}\nIn particular $Z\\leq q^{1\/4}$ and\n$$X\\geq Z^{2\/3}q^{4\/3}\\geq Zq^{1+1\/4}$$\ntherefore \\eqref{XZlower} holds for~$\\eta=1\/4$.\n\nAssuming that the conditions \\eqref{eqboundsPHLX} hold,\ncombining~\\eqref{an-identity}, Proposition~\\ref{proposition-for-F} and\nProposition~\\ref{corollary-for-O}, we deduce the estimate\n$$\nS_V(K,X) \\ll\nq^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n\\Biggl(\\frac{Z^{2}X^{3\/2}P}{qL^{1\/2}}+Z^{3\/2}X^{3\/4}(qPL)^{1\/4}\n+\\frac{qX^{1\/2}}{P}\\biggr)\n$$\nfor any~$\\varepsilon>0$.  When $L=Z^{2\/3}XP\/q^{5\/3}$, the first two terms are\nequal to $Z^{5\/3}XP^{1\/2}\/q^{1\/6}$. For $P=q^{7\/9}\/(X^{1\/3}Z^{10\/9})$,\nthey are also equal to the third term which is\n$Z^{10\/9}q^{2\/9}X^{5\/6}$.  Moreover, the conditions \\eqref{eqcondZXq}\nand~$Z\\leq q^{1\/4}$ imply then by simple computations that\n$$\n1\\leq P,\\ 1\\leq L,\\  L\\leq P^4,\\ XP\\leq q^2L\n$$\n(for instance, $X^3Z^{10}\\leq Z^{10}(q^2\/Z^2)^3=Z^4q^6\\leq q^7$\ngives~$P\\geq 1$), and then we get\n$$\nq^{1+\\delta}L^2<\\frac{X}{8}\n$$\nfor~$\\delta=1\/18$ provided~$q$ is large enough (since\n$qL^2=q^{-7\/9}Z^{-8\/9}X^{4\/3}\\leq X(X^{1\/3}q^{-7\/9})\\leq Xq^{-1\/9}$\nusing~$X\\leq q^2$). By~(\\ref{eq-H}), this also implies\nthat~$q^{\\delta}PHL<q\/8$. Hence this choice of the parameters\nsatisfies~(\\ref{eqboundsPHLX}). We finally conclude that\n$$\nS_V(K,X) \\ll \\norm{\\widehat{K}}_{\\infty}\nZ^{10\/9}q^{2\/9+\\varepsilon}X^{5\/6}\n$$\nfor any~$\\varepsilon>0$.\n\n\n\\section{Applications}\n\nIn this section, we prove \nCorollaries~\\ref{cor-average}, \\ref{cor-gl2}.\n\n\\subsection{ Proof of Corollary~\\ref{cor-average}} Applying the approximate functional\nequation for~$L(\\varphi\\otimes\\chi,s)$ in balanced form, we\nimmediately express the first moment\n$$\n\\frac{1}{q-1} \\sum_{\\chi\\mods{q}}M(\\chi)L(\\varphi\\otimes\\chi, 1\/2)\n$$\nin terms of the sums\n$$\n\\frac{1}{\\sqrt{q}}\n\\sum_{n\\geq 1} \\frac{\\lambda(1,n)}{\\sqrt{n}}\nK(n)V\\Bigl(\\frac{n}{q^{3\/2}}\\Bigr)\n$$\nand\n$$\n\\frac{1}{\\sqrt{q}}\\sum_{n\\geq 1}\n\\frac{\\overline{\\lambda(1,n)}}{\\sqrt{n}}\nL(n)V\\Bigl(\\frac{n}{q^{3\/2}}\\Bigr),\n$$\nfor suitable test functions satisfying~(\\ref{eq:Vprop}) for~$Z=1$,\nwhere\n$$\nK(n)=\\frac{q^{1\/2}}{q-1}\\sum_{\\chi\\mods{q}}M(\\chi){\\chi(n)},\\quad\nL(n)=\\frac{q^{1\/2}}{q-1}\\sum_{\\chi\\mods{q}}\\tau(\\chi)^3M(\\chi)\\ov{\\chi(n)},\n$$\nin terms of the normalized Gauss sum~$\\tau(\\chi)$. An elementary\ncomputation shows that this function~$L$ coincides with the function\nin the statement of Corollary~\\ref{cor-average}. Since moreover\nthe~$\\overline{\\lambda(1,n)}$ are the Hecke-eigenvalues of the dual\ncusp form~$\\widetilde{\\varphi}$, the corollary follows from\nTheorem~\\ref{thm:main} applied to~$K$ and~$L$.\n\n\\begin{remark}\\label{lastremark}\n  (1) If\n$$\nM(\\chi)=\\frac{1}{\\sqrt{q}}\\sum_{x\\in{\\mathbf{F}^\\times_q}}K(x)\\ov{\\chi(x)}\n$$\nis the discrete Mellin transform of the trace function~$K$ of a\nFourier sheaf~$\\mathcal{F}$ that is a middle-extension sheaf on~$\\mathbf{G}_m$ of\nweight~$0$, and if no sheaf of the\nform~$[x\\mapsto x^{-1}]^*\\dual(\\mathcal{K}\\ell_3)$ is among the geometrically\nirreducible components of~$\\mathcal{F}$, then \nboth~$\\norm{\\widehat{K}}_{\\infty}$ and~$\\norm{\\widehat{L}}_{\\infty}$ are\nbounded in terms of the conductor of~$\\mathcal{F}$ only and we obtain\n$$\n\\frac{1}{q-1} \\sum_{\\chi\\mods{q}}M(\\chi)L(\\varphi\\otimes\\chi,\n1\/2)\\ll q^{2\/9+\\varepsilon}\n$$\nfor any~$\\varepsilon>0$, where the implied constant depends only\non~$\\varepsilon$,~$\\varphi$ and the conductor of~$\\mathcal{F}$.\n\\par\n(2) Applying the approximate functional equation in a balanced form\nmay not always be the best move. For instance, consider the important\nspecial case where $M(\\chi)=1$. We are then evaluating the first\nmoment\n\\begin{equation}\\label{eqfirstmoment}\n\\frac{1}{q-1}\\sum_{\\chi\\mods q}L(\\varphi\\otimes\\chi,1\/2)\n\\end{equation}\nof the central values of the twisted $L$-functions. Then we are\nworking with the functions\n$$\nK(n)=q^{1\/2}\\delta_{n\\equiv 1\\mods q},\\quad L(n)=\\Kl_3( n;q),\n$$\nwhose Fourier transforms are bounded by absolute constants. Hence the\nabove leads to\n$$\n\\frac{1}{q-1} \\sum_{\\chi\\mods{q}}L(\\varphi\\otimes\\chi, 1\/2)\n\\ll q^{2\/9 + \\varepsilon}\n$$\nfor any~$\\varepsilon>0$, where the implied constant depends on~$\\varphi$\nand~$\\varepsilon$.\n\\par\nOn the other hand, the approximate functional equation in unbalanced\nform yields sums of the shape\n$$\n\\sum_{n\\equiv 1\\mods q} \\frac{\\lambda(1,n)}{\\sqrt{n}}\nV\\Bigl(\\frac{n}{Yq^{3\/2}}\\Bigr)\n\\quad\\text{ and }\\quad\n\\frac{1}{\\sqrt{q}}\\sum_{n\\geq 1}\n\\frac{\\overline{\\lambda(1,n)}}{\\sqrt{n}}\n\\Kl_3(n;q)V\\Bigl(\\frac{nY}{q^{3\/2}}\\Bigr),\n$$\nfor some parameter $Y>0$ at our disposal. Assuming the\nRamanujan--Petersson conjecture for~$\\varphi$\nand~$\\widetilde{\\varphi}$, and using Deligne's bound\n$|\\Kl_3(n;q)|\\leq 3$ for $(n,q)=1$, we obtain the much stronger bound\n$$\n\\frac{1}{q-1}\\sum_{\\chi\\mods\n  q}L(\\varphi\\otimes\\chi,1\/2)=1+(qY)^{\\varepsilon}\n\\bigl({Y^{1\/2}}\/{q^{1\/4}}+q^{1\/4}\/Y^{1\/2}\\bigr)\\ll\nq^{\\varepsilon}\n$$\nfor any~$\\varepsilon>0$, on choosing $Y=q^{1\/2}$.\n\\par\nNote that, again under the Ramanujan--Petersson conjecture\nfor~$\\varphi$ and its dual, we would obtain an \\emph{asymptotic\n  formula} for the first moment \\eqref{eqfirstmoment} \\emph{provided}\nwe could obtain an estimate for $S_V(\\Kl_3,X)$ with a power-saving in\nterms of~$q$, when $X$ is a bit smaller than $q$. Results of this type\nare however currently only known if $\\varphi$ is an Eisenstein series\n(starting from the work~\\cite{FI} of Friedlander and Iwaniec for the\nternary divisor function; see also the papers of Fouvry, Kowalski and\nMichel~\\cite{FKMd3}, of Kowalski, Michel and Sawin~\\cite{KMS} and of\nZacharias~\\cite{Zac}).\n\nThis illustrates the importance of the problem of obtaining\nnon-trivial bounds for short sums in Theorem \\ref{thm:main}. However,\nwe expect that much more refined properties of trace functions and\ntheir associated sheaves will be necessary for such a purpose (as\nindicated by Remark~\\ref{remcor17}).\n\\end{remark}\n\n\\subsection{Proof of Corollary~\\ref{cor-gl2}}\n\nThe symmetric square~$\\varphi$ of~$\\psi$ has Hecke eigenvalues\n\\begin{equation}\\label{lambda(n2)}\n\\lambda(1,n)=\\sum_{d^2\\mid n}\\lambda\\Bigl(\\frac{n^2}{d^2}\\Bigr),\n\\end{equation}\nand hence, by M\\\"obius inversion, we have\n$$\n\\lambda(n^2)=\\sum_{d^2\\mid n}\\mu(d)\\lambda\\Bigl(1,\\frac{n}{d^2}\\Bigr).\n$$\nWe deduce that\n$$\n\\sum_{n\\geq 1}\\lambda(n^2)K(n)V\\Bigl(\\frac{n}{X}\\Bigr)= \\sum_{d\\geq\n  1}\\mu(d) \\sum_{n\\geq\n  1}K(nd^2)\\lambda(1,n)V\\Bigl(\\frac{nd^2}{X}\\Bigr).\n$$\nFor\n$$\n1\\leq d\\leq \\frac{X^{1\/2}}{Z^{1\/3}q^{2\/3}},\n$$\nwe can apply Theorem~\\ref{thm:main} to the sum over~$n$ and the\n$q$-periodic function $L(n)=K(nd^2)$, with~$X$ replaced\nby~$X\/d^2$. Since~$q\\nmid d$, we have\n$\\widehat{L}(x)=\\widehat{K}(\\bar{d}^2x)$ for any~$x\\in \\mathbf{Z}$, so\nthat~$\\norm{\\widehat{L}}_{\\infty}=\\norm{\\widehat{K}}_{\\infty}$, and we get\n\\begin{align*}\n  \\sum_{d\\leq X^{1\/2}\/(Z^{1\/3}q^{2\/3})}\\mu(d) \\sum_{n\\geq\n  1}K(nd^2)\\lambda(1,n)V\\Bigl(\\frac{nd^2}{X}\\Bigr)\n  &\\ll\n    \\norm{\\widehat{K}}_{\\infty}Z^{10\/9}q^{2\/9+\\varepsilon} \\sum_{d\\geq\n    1}\\frac{X^{5\/6}}{d^{5\/3}}\n  \\\\\n  &\\ll\n    \\norm{\\widehat{K}}_{\\infty}Z^{10\/9}q^{2\/9+\\varepsilon}X^{5\/6}\n\\end{align*}\nfor any~$\\varepsilon>0$.\n\\par\nSince~$V$ has compact support in $[1\/2,3]$, the sum over~$n$ is empty\nif $d\\geq \\sqrt{3X}$. Since\n$$\n\\sum_{n\\geq 1}K(nd^2)\\lambda(1,n)V\\Bigl(\\frac{nd^2}{X}\\Bigr) \\ll\n\\norm{K}_{\\infty}\\Bigl(\\frac{X}{d^2}\\Bigr)^{1+\\varepsilon}\n$$\nfor any~$\\varepsilon>0$, by the Rankin--Selberg bound~(\\ref{eq:RS}), we can\nestimate the remaining part of the sum as follows:\n\\begin{align*}\n  \\sum_{X^{1\/2}\/(Z^{1\/3}q^{2\/3})<d\\leq \\sqrt{3X}}\\mu(d) \\sum_{n\\geq\n  1}K(nd^2)\\lambda(1,n)V\\Bigl(\\frac{nd^2}{X}\\Bigr)\n  &\\ll\n    \\norm{K}_{\\infty}X^{1+\\varepsilon}\n    \\sum_{X^{1\/2}\/(Z^{1\/3}q^{2\/3})<d\\leq\n    \\sqrt{3X}}\\frac{1}{d^{2+2\\varepsilon}}\n  \\\\\n  &\\ll\n    \\norm{K}_{\\infty}X^{1\/2+\\varepsilon}Z^{1\/3}q^{2\/3}\n\\end{align*}\nfor any~$\\varepsilon>0$.\n\n\n\\begin{remark}\n  The additional dependency on~$\\norm{K}_{\\infty}$ seems to be\n  unavoidable in Corollary~\\ref{cor-gl2}.\n\\end{remark}\n\n\\section{Proof of Corollary~\\ref{RScor}}\n\nThe proof of Corollary~\\ref{RScor} requires additional ingredients\nbesides Theorem~\\ref{thm:main}.  We will be somewhat brief in handling\nthese additional arguments (especially standard analytic arguments),\nsince similar computations have been performed in a number of other\npapers (e.g.~\\cite{FKM2}).\n\nFirst, in terms of the\nHecke-eigenvalues $\\lambda(m,n)$ of the symmetric square~$\\psi$\nof~$\\varphi$, we have the identity\n$$\n\\lambda(n)^2=\\sum_{d^2bc=n}\\mu(d)\\lambda(1,c)\n$$\n(see~(\\ref{lambda(n2)}) and~(\\ref{convol})).  Thus we have\n$$\n\\sum_{n\\geq 1}\\lambda(n)^2K(n)V\\Bigl(\\frac{n}{X}\\Bigr)=\n\\sum_{d,m,n\\geq\n  1}\\mu(d)\\lambda(1,n)K(d^2mn)V\\Bigl(\\frac{d^2mn}{X}\\Bigr).\n$$\n\\par\nWe bound the contribution of the integers~$n$ divisible by~$q$ using\nthe Kim--Sarnak bound~\\cite{KimSar} for~$\\lambda(1,n)$, which shows\nthat it is\n$$\n\\ll\n\\|K\\|_\\infty X^{1+\\varepsilon}q^{-1+7\/32},\n$$\nfor any~$\\varepsilon>0$, and hence is negligigle. We may therefore restrict\nthe sum on the right-hand side to integers such that $(dmn,q)=1$.\n\nFor a fixed value of $d\\leq D$, coprime to~$q$, we consider the sum\n\\begin{equation}\\label{eq-Td}\n  T_{d,x}=\\sum_{m,n\\geq\n    1}\\lambda(1,n)K(d^2mn)V\\Bigl(\\frac{d^2mn}{X}\\Bigr).\n\\end{equation}\nWe need to estimate the sum of~$T_{d,x}$ over~$d\\geq 1$.\n\nLet $D\\geq 1$ be some small parameter to be fixed later. The\ncontribution of the integers $d> D$ is bounded trivially and is\n$$\n\\sum_{d>D}T_{d,X}\\ll \\frac{\\|K\\|_\\infty X^{1+\\varepsilon}}{D}\n$$\nfor any~$\\varepsilon>0$.\n\nWe now fix $d\\leq D$, coprime to~$q$. We handle the sum~$T_{d,X}$ by a\nsmooth dyadic partition of unity on the $m$-variable. This reduces the\nproblem to estimates of $O(\\log X)$ sums of the form\n\\begin{equation}\\label{DMsum}\n  S_{d,M}=  \\sum_{\\substack{m,n\\geq 1\\\\(mn,q)=1}}\n  \\lambda(1,n)K(d^2mn)V\\Bigl(\\frac{d^2mn}{X}\\Bigr)\n  W\\Bigl(\\frac{m}M\\Bigr)\n\\end{equation}\nwhere $W$ is smooth and compactly supported in $[1\/2,5\/2]$. We\nset\n$$\nN=\\frac{X}{d^2M},\n$$\nso that $n\\sim N$ in the sum.\n\nThe estimate for~(\\ref{DMsum}) splits in three cases, depending on the\nsize of~$M$.\n\n\\subsection{When $M$ is small}\n\nWe assume that $\\frac{X}{d^2m}\\geq \\frac{X}{D^2M}\\geq\nZ^{2\/3}q^{4\/3}$, or in another words, that\n\\begin{equation}\\label{condition-on-M}\nD^2M\\leq \\frac{X}{Z^{2\/3}q^{4\/3}}.\n\\end{equation}\nWe can then apply Theorem~\\ref{thm:main}, and we derive\n\\begin{align}\\nonumber\n  S_{d,M}&\\ll \\|K\\|_\\infty q^{7\/32-1}\\frac{X^{1+\\varepsilon}}{d^2}\n           +\\|\\widehat K\\|_\\infty Z^{10\/9}q^{2\/9+\\varepsilon}\n           \\sum_{m\\sim M}\\Bigl(\\frac{X}{d^2m}\\Bigr)^{5\/6}\\\\\n         &\\ll \\|K\\|_\\infty X^{\\varepsilon}q^{7\/32-1}\\frac{X}{d^2}\n           +\\|\\widehat K\\|_\\infty X^{\\varepsilon}Z^{10\/9}\n           \\Bigl(\\frac{X}{d^2}\\Bigr)^{5\/6}q^{2\/9}{M^{1\/6}}\n\\label{bound1}\n\\end{align}\nfor any~$\\varepsilon>0$ (the first term corresponds to removing the\nconstraint $(n,q)=1$).\n\n\\subsection{When $M$ is in the Fourier range}\n\nIf $M\\geq q^{1\/2}$, then it is\nbeneficial to apply the Poisson summation formula to the\n$m$-variable. As in the previous case, the cost of removing the\ncondition $(m,q)=1$ is $\\ll \\|K\\|_\\infty q^{7\/32-1}X^{1+\\varepsilon}\/d^2$\nfor~$\\varepsilon>0$. The Poisson summation formula implies that\n$$\n\\sum_{m\\geq\n  1}K(d^2mn)V\\Bigl(\\frac{d^2mn}{X}\\Bigr)W\\Bigl(\\frac{m}M\\Bigr)\\ll\n\\|\\widehat K\\|_\\infty \\Bigl(\\frac{M}{q^{1\/2}}+q^{1\/2}\\Bigr)\n$$\nand therefore\n\\eqref{DMsum} is bounded by\n\\begin{equation}\\label{bound2}\n  S_{d,M}\\ll \\|K\\|_\\infty X^{\\varepsilon}q^{7\/32-1}\\frac{X}{d^2}\n  +\\|\\widehat K\\|_\\infty X^{\\varepsilon}\\frac{X}{d^2}\n  \\Bigl(\\frac{1}{q^{1\/2}}+\\frac{q^{1\/2}}{M}\\Bigr)\n\\end{equation}\nfor any~$\\varepsilon>0$.\n\n\\subsection{When $M$ is large but not in Fourier range}\n\nIf~$M\\leq q^{1\/2}$, thinking of the prototypical case when\n$X\\sim q^{3\/2}$ and $D$ is close to one, the $n$-sum is of length\nclose to $q$, so the natural move is to smooth the $n$-sum, and then\nuse the Poisson summation formula on the resulting sums.\n\nThus we apply the Cauchy--Schwarz inequality to \\eqref{DMsum}, leaving\nthe $n$ variable outside, namely\n\\begin{equation}\\label{eqCS}\n  |S_{d,M}|^2\n  \\ll\\sum_{n\\sim X\/d^2M}|\\lambda(1,n)|^2\\times\n  \\sumsum_\\stacksum{m_i\\sim\n    M}{(m_i,q)=1}\\sum_{n\\geq 1}K(d^2m_1n)\\ov{K(d^2m_2n)}\n  V\\Bigl(\\frac{d^2m_1n}X\\Bigr)\\ov{V}\\Bigl(\\frac{d^2m_2n}X\\Bigr).\n\\end{equation}\n\nHere, we have dropped the constraint $(n,q)=1$ on the right-hand side\nby positivity, and replaced the expressions $W\\Bigl(\\frac{m_i}M\\Bigr)$\nby the summation conditions $ m_i\\sim M$.\n\nBy the Poisson summation formula, we have\n\\begin{equation}\\label{applypoisson}\n  \\sum_{n\\geq 1}\n  K(d^2m_1n)\\ov{K(d^2m_2n)}\n  V\\Bigl(\\frac{d^2m_1n}X\\Bigr)\\ov{V}\\Bigl(\\frac{d^2m_2n}X\\Bigr)\n  =\\frac{N}{q^{1\/2}}\\sum_{h\\in\\mathbf{Z}}\n  \\widehat{K}_{(2)}(h)\\mathcal{W}\\Bigl(\\frac{h}{q\/(X\/d^2M)}\\Bigr),\n\\end{equation}\nwhere $\\mathcal{W}(y)$ is a smooth function depending on $d,m_1,m_2$,\nrapidly decaying as $y\\rightarrow\\infty$, and\n$$\n\\widehat{K}_{(2)}(h)=\\frac{1}{\\sqrt{q}}\\sum_{n\\in\\mathbf{F}_q}\nK(d^2m_1n)\\ov{K(d^2m_2n)}e\\Bigl(\\frac{nh}{q}\\Bigr).\n$$\n\\par\nTo go further, we use the assumption of Corollary~\\ref{RScor} that $K$\nis the trace function of a middle-extension $\\ell$-adic sheaf~$\\mathcal{F}$\nthat is not exceptional. Indeed, from \\cite[Theorem 6.3]{FKM2}, we can\ndeduce that there exists a set $B\\subset {\\mathbf{F}^\\times_q}$ such\nthat $|B|$ is bounded in terms of the conductor of~$\\mathcal{F}$ only, and\nsuch that whenever\n\\begin{equation}\\label{eqdiag}\n  m_1\/m_2\\mods q\\not\\in B,\n\\end{equation}\nthen we have\n$$\n\\|\\widehat{K}_{(2)}\\|_\\infty\\ll 1\n$$\nwhere the implied constant depends on the conductor of~$\\mathcal{F}$ only.\n\nReturning to \\eqref{eqCS}, we apply the bound \\eqref{applypoisson} to\nthe pairs pairs $(m_1,m_2)$ which satisfy \\eqref{eqdiag}, and apply\nthe trivial bound otherwise.\n\nWe see then that the contribution to the second factor of \\eqref{eqCS}\nof the ``diagonal'' pairs not satisfying \\eqref{eqdiag} is bounded by\n$$\n\\ll X^{\\varepsilon}M\\Bigl(\\frac{M}q+1\\Bigr)\\frac{X\/M}{d^2}\n$$\nfor any~$\\varepsilon>0$, while the contribution of the pairs $(m_1,m_2)$\nsatisfying \\eqref{eqdiag} is bounded by\n$$\n\\ll X^{\\varepsilon}M^2\\Bigl(\\frac{X\/M}{d^2q^{1\/2}}+q^{1\/2}\\Bigr),\n$$\nfor any~$\\varepsilon>0$, where in both cases the implied constant depends\nonly on~$\\varepsilon$ and on the conductor of~$\\mathcal{F}$.\n\n\nCollecting these bounds, we obtain from \\eqref{eqCS} the bound\n\\begin{equation}\\label{bound3}\n  S_{d,M} \\ll\n  \\frac{X^{1+\\varepsilon}}{d^2}\n  \\Bigl(\\frac{1}{M^{1\/2}}+\\frac{1}{q^{1\/4}}+q^{1\/4}M^{1\/2}\n  \\frac{d}{X^{1\/2}}\\Bigr),\n\\end{equation}\nfor any~$\\varepsilon>0$, where the implied constant depends only on~$\\varepsilon$\nand on the conductor of~$\\mathcal{F}$.\n\n\n\\subsection{End of the proof}\n\nNow we can combine the previous bounds.  Let~$\\eta>0$ and~$\\delta$\nwith~$0<\\delta<1\/4$ be parameters to be determined later.\n\n\\subsubsection*{-- If $M\\leq q^{2\\delta}$,} we then apply the bound\n\\eqref{bound1} (and the dyadic decomposition of~$T_{d,x}$ in a\ncombination of sums~$S_{d,M}$) to derive\n\\begin{equation}\\label{bound-c}\n  \\sum_{d\\leq D}T_{d,X}\\ll X^{1+\\varepsilon}q^{7\/32-1}\n  +Z^{10\/9}X^{5\/6+\\varepsilon}q^{2\/9+\\delta\/3},\n\\end{equation}\nunder the condition that\n\\begin{equation}\\label{condition-X}\nX\\geq Z^{2\/3}D^2q^{4\/3+2\\delta}\n\\end{equation}\n(see \\eqref{condition-on-M}).\n\n\\subsubsection*{-- If $M\\geq q^{1\/2+\\eta}$,} we apply the bound\n\\eqref{bound2} and sum over $d\\leq D$, to find that\n\\begin{equation}\\label{bound-a}\n  \\sum_{d\\leq D} T_{d,X}\n  \\ll X^{1+\\varepsilon}\\Bigl(\\frac{1}{q^{1\/2}}+\\frac{q^{1\/2}}{M}\\Bigr)\n  \\ll X^{1+\\varepsilon}q^{-\\eta}\n\\end{equation}\nin that case.\n\n\\subsubsection*{-- If $q^{2\\delta}\\leq M< q^{1\/2+\\eta}$,} we then\napply the bound \\eqref{bound3} and sum over $d\\leq D$, obtaining\n\\begin{equation}\\label{bound-b}\n  \\sum_{d\\leq D}T_{d,X}\\ll\n  X^{1+\\varepsilon}\\Bigl(\\frac{1}{q^{\\delta}}\n  +\\frac{1}{q^{1\/4}}+\\frac{q^{1\/2+\\eta\/2}}{X^{1\/2}}\\Bigr)\n  \\ll X^{1+\\varepsilon}\\Bigl(q^{-\\delta}+\\frac{q^{1\/2+\\eta\/2}}{X^{1\/2}}\\Bigr).\n\\end{equation}\n\nThis covers all of the ranges for $M$. We now choose $\\eta, \\delta>0$\nsuch that the bound in \\eqref{bound-a} is equal to the second term in\n\\eqref{bound-b}, and the first term in \\eqref{bound-b} is consistent\nwith the second term in \\eqref{bound-c}. That is, we choose\n$q^{\\eta}=(X\/q)^{1\/3}$ and\n$q^{\\delta}=\\frac{X^{1\/8}}{Z^{5\/6}q^{1\/6}}$. Therefore we have in all\ncases the estimate\n$$\n\\sum_{d\\leq D}T_{d,X} \\ll\nX^{2\/3+\\varepsilon}q^{1\/3}+Z^{5\/6}X^{7\/8+\\varepsilon}q^{1\/6}+X^{1+\\varepsilon}q^{7\/32-1},\n$$\nfor any~$\\varepsilon>0$, under the assumption that\n$$\nX\\gg D^{8\/3}q^{4\/3}Z^{-4\/3},\n$$\nand the implied constant depends only on~$\\varepsilon$ and the conductor\nof~$\\mathcal{F}$.\n\nFinally we combine this with the previously noted estimate\n$$\n\\sum_{d>D}T_{d,X}\\ll\n\\frac{\\|K\\|_\\infty X^{1+\\varepsilon}}{D}\n$$\n(recall that for a non-exceptional trace function, we\nhave~$\\|\\widehat{K}\\|_{\\infty}\\ll 1$ where the implied constant depends\nonly on the conductor of~$\\mathcal{F}$), to conclude that\n$$\n\\sum_{n\\geq 1}\\lambda(n)^2K(n)V\\Bigl(\\frac{n}{X}\\Bigr)\\ll\nX^{2\/3+\\varepsilon}q^{1\/3}+Z^{5\/6}X^{7\/8+\\varepsilon}q^{1\/6}+X^{1+\\varepsilon}\/D.\n$$\nWe take $D=q^{\\gamma}$ for some small~$\\gamma>0$, and then we have\n$$\n\\sum_{n\\geq 1}\\lambda(n)^2K(n)V\\Bigl(\\frac{n}{X}\\Bigr)\\ll\nX^{2\/3+\\varepsilon}q^{1\/3}+Z^{5\/6}X^{7\/8+\\varepsilon}q^{1\/6}+X^{1+\\varepsilon}q^{-\\gamma},\n$$\nprovided that\n$$\nX\\gg q^{4\/3+8\\gamma\/3}\/Z^{4\/3}, \n$$\nwhere the implied constant depends only on~$\\varepsilon$ and the conductor\nof~$\\mathcal{F}$.\n\nThis concludes the proof of Corollary \\ref{cor2-gl2}.\n\n\n\n\\begin{bibdiv}\n\n\\begin{biblist}\n\n\\bib{AHLS}{article}{\n  author={Aggarwal, K.},\n  author={Holowinsky, R.},\n  author={Lin, Y.},\n  author={Sun, Q.},\n  title={The Burgess bound via a trivial delta method},\n  note={\\url{arXiv:1803.00542v1}},\n  date={2018},\n}   \n\n\\bib{Blomer}{article}{\n   author={Blomer, V.},\n   title={Subconvexity for twisted $L$-functions on ${\\rm GL}(3)$},\n   journal={Amer. J. Math.},\n   volume={134},\n   date={2012},\n   number={5},\n   pages={1385--1421},\n\n}\n \\bib{CI}{article}{\n   author={Conrey, J. B.},\n   author={Iwaniec, H.},\n   title={The cubic moment of central values of automorphic $L$-functions},\n   journal={Ann. of Math. 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Journal},\n   date={2014},\n   volume={163},\n   pages={1683-1736},\n   number={9},\n }\n\n\n \\bib{pisa}{article}{\n   author={Fouvry, {\\'E}.},\n   author={Kowalski, E.},\n   author={Michel, Ph.},\n   title={Trace functions over finite fields and their applications},\n   book={\n     series={Colloquio de Giorgi},\n     publisher={Springer},\n   },\n   date={2014},\n }\n \n\n \\bib{short-sums}{article}{\n   author={Fouvry, {\\'E}.},\n   author={Kowalski, E.},\n   author={Michel, Ph.},\n   author={Raju, C.},\n   author={Rivat, J.},\n   author={Soundararajan, K.},\n   title={On short sums of trace functions},\n   journal={Ann. Inst. Fourier},\n   date={2017},\n   volume={67},\n   pages={423--449},\n }\n \n \\bib{FI}{article}{\n  author={Friedlander, J.B.},\n  author={Iwaniec, H.},\n  title={Incomplete Kloosterman sums and a divisor problem},\n  note={(with an appendix by\n  B. J. Birch and E. Bombieri)},\n  journal={Ann. of Math. 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+{"text":"\n\\section{Shape Priors for Object SLAM}\n\nIn this section, we explain in more detail the benefit of using learnt shape priors in object SLAM. We evaluate the related object-oriented SLAM methods under three important properties: \\textit{1. Can the system reconstruct previously unseen objects? 2. Is the reconstruction complete? 3. Does the reconstruction preserve detailed shape?} \n\nThe first line of works~\\cite{SLAM++, Tateno2016When2.5D} relies on a pre-scanned CAD model database to perform online detection and registration. The reconstructed object shapes are complete and detailed, but the system cannot handle new objects. The second category of works~\\cite{maskfusion, fusion++} leverages 2D segmentation masks, treat each segmented object as individual identities and perform reconstruction on-the-fly. This reconstruction-by-segmentation strategy can reconstruct arbitrary unseen object shapes in great detail, but the reconstructed shape is incomplete due to partial observation and missing depth. \nThe last line of research represents objects as simple geometric shapes, such as ellipsoids~\\cite{quadricslam} or cuboids~\\cite{cubeslam}. \nThis kind of geometric representation can be applied to arbitrary unseen shapes and preserve some important properties like scale and orientation, but we've lost the level of details in the shapes.\n\nWith learnt shape priors, all the three properties can be achieved at the same time. Its generative property means it can generalize to unseen shapes. Object shape decoded from latent code is guaranteed to be detailed and complete, even from a single partial view. The compact representation also benefits the optimization process. Table \\ref{tab:related_work} provides an overview of the related works under the different properties. Only NodeSLAM~\\cite{node-slam} and DSP-SLAM can achieve all the three important properties. Unlike NodeSLAM, DSP-SLAM can also deal with large scale environments and monocular RGB inputs.\n\n\\section{Full Derivation of Jacobians}\n\nAs mentioned in the main paper, one of our contribution is the fast Gauss-Newton optimizer, which is crucial to achieve real-time performance. This section provides full derivation of Jacobians of each individual residual term.\n\n\\subsection{Jacobian of Surface Term Residual \\label{sec:J_surf}}\n\nAs shown in Eq. 1 in the main paper, the surface term residual at pixel $\\matr{u} \\in \\mathbf{\\Omega}_s$ is simply the SDF value of the back-projected point under object coordinate:\n\n\\begin{equation}\n    e_s(\\matr{u}) = G(\\matr{T}_{oc} \\pi^{-1}(\\matr{u}, {\\mathcal{D}}), \\matr{z}) = G(^o\\matr{x}, \\matr{z})\n\\end{equation}\n\nIts Jacobian with respect to object pose and shape $[\\boldsymbol{\\xi}_{oc};\\matr{z}]$ is defined as:\n\n\\begin{equation} \\label{eq:J_surf}\n    \\matr{J}_s = \\frac{\\partial e_s(\\matr{u})}{\\partial [\\boldsymbol{\\xi}_{oc};\\matr{z}]}\n\\end{equation}\n\nwhere $\\boldsymbol{\\xi}_{oc} \\in \\mathbb{R}^7$ is the Cartesian representation (twist coordinate) of the corresponding element in Lie Algebra $\\mathfrak{sim}(3)$. The Jacobian with respect to the shape code $\\frac{\\partial e_s(\\matr{u})}{\\partial \\matr{z}}$ could be obtained directly via back-propagation. To derive the Jacobian with respect to object pose $\\boldsymbol{\\xi}_{oc}$, we first factorize it using chain rule:\n\n\\begin{table}[tbp]\n  \\center\n  \\setlength\\tabcolsep{4pt}\n    \\begin{tabular}{c||l|l|l||l|l}\n    Method       & \\specialcell[b]{Unseen\\\\Objects}  & \\specialcell[b]{Full\\\\Shape}  & \\specialcell[b]{Detailed\\\\Shape} & \\specialcell[b]{RGB\\\\Input} & \\specialcell[b]{Large\\\\Scene}  \\\\ \\hline \\hline\n    \\specialcell[b]{2.5D is not\\\\enough\\cite{Tateno2016When2.5D}} &  ~ & \\checkmark & \\checkmark & ~ & ~ \\\\ \\hline\n    \\specialcell[b]{SLAM++\\\\\\cite{SLAM++}} & ~ & \\checkmark & \\checkmark & ~ & ~\\\\ \\hline\\hline\n    \\specialcell[b]{Mask-\\\\Fusion \\cite{maskfusion}} & \\checkmark & ~ & \\checkmark & ~ & ~ \\\\ \\hline\n    \\specialcell[b]{Fusion++\\\\\\cite{fusion++}} & \\checkmark & ~ & \\checkmark & ~ & ~\\\\ \\hline\\hline\n     \\specialcell[b]{Quadric-\\\\SLAM \\cite{quadricslam}} & \\checkmark & \\checkmark & ~ & \\checkmark &\\checkmark \\\\ \\hline\n    \\specialcell[b]{Cube-\\\\SLAM \\cite{cubeslam}} & \\checkmark & \\checkmark & ~ & \\checkmark & \\checkmark\\\\ \\hline\\hline\n    \\specialcell[b]{Node-\\\\SLAM \\cite{node-slam}} & \\checkmark & \\checkmark & \\checkmark & ~ & ~ \\\\ \\hline\n    \\textbf{\\specialcell[b]{DSP-\\\\SLAM}} & \\checkmark & \\checkmark & \\checkmark & \\checkmark & \\checkmark\n    \\end{tabular}\n    \\caption{Comparison of the properties of DSP-SLAM with respect\n      to other object-oriented SLAM systems. \n      \\vspace{-1.0cm}}\n    \\label{tab:related_work}\n\\end{table}\n\n\\begin{equation} \\label{eq:J_surf_pose_1}\n    \\frac{\\partial e_s(\\matr{u})}{\\partial \\boldsymbol{\\xi}_{oc}} = \\frac{\\partial G(^o\\matr{x}, \\matr{z})}{\\partial ^o\\matr{x}} \\frac{\\partial ^o\\matr{x}}{\\partial \\boldsymbol{\\xi}_{oc}}\n\\end{equation}\n\nThen the first term $\\frac{\\partial G(^o\\matr{x}, \\matr{z})}{\\partial ^o\\matr{x}}$ can also be obtained via back-propagation. The second Jacobian term could be computed by applying a left perturbation to the pose $\\matr{T}_{oc}$:\n\n\\begin{align}\n    \\frac{\\partial ^o\\matr{x}}{\\partial \\boldsymbol{\\xi}_{oc}} & = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\exp{\\big(\\delta \\boldsymbol{\\xi}^\\wedge}\\big) \\matr{T}_{oc} {^c\\matr{x}} - \\matr{T}_{oc} {^c\\matr{x}}}{\\delta \\boldsymbol{\\xi}} \\\\\n    & = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\big(\\matr{I} + \\delta \\boldsymbol{\\xi}^\\wedge\\big) \\matr{T}_{oc} {^c\\matr{x}} - \\matr{T}_{oc} {^c\\matr{x}}}{\\delta \\boldsymbol{\\xi}} \\\\\n    & = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\delta \\boldsymbol{\\xi}^\\wedge \\matr{T}_{oc} {^c\\matr{x}}}{\\delta \\boldsymbol{\\xi}} = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\delta \\boldsymbol{\\xi}^\\wedge {^o\\matr{x}}}{\\delta \\boldsymbol{\\xi}}  \\label{eq:J_surf_pose_2}\n\\end{align}\n\nwhere $\\exp(\\cdot)$ is the exponential map from Lie Algebra $\\mathfrak{sim}(3)$ to the corresponding Lie Group $\\mathbf{Sim}(3)$, and $\\cdot ^\\wedge$ is the hat-operator that maps a twist coordinate in $\\mathbb{R}^7$ to $\\mathfrak{sim}(3)$:\n\n\\begin{equation} \\label{eq:hat}\n    \\boldsymbol{\\xi}^ \\wedge = \\begin{bmatrix} \\boldsymbol{\\nu} \\\\ \\boldsymbol{\\phi} \\\\ s \\end{bmatrix}^ \\wedge = \\begin{bmatrix} \\boldsymbol{\\phi}_{\\times} + s\\matr{I} & \\boldsymbol{\\nu} \\\\ \\matr{0} & 0\\end{bmatrix}\n\\end{equation}\n\n$\\boldsymbol{\\nu} \\in \\mathbb{R}^3$, $\\boldsymbol{\\phi} \\in \\mathbb{R}^3$ and $s \\in \\mathbb{R}$ represent the translation, rotation and scale part of the twist coordinate respectively. $(\\cdot)_\\times$ maps from $\\mathbb{R}^3$ to $\\mathfrak{so}(3)$ (skew-symmetric matrices). With the equation above, the Jacobian term in Eq. 6 can be computed as: \\vspace{-0.3cm}\n\n\\begin{align}\n    \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\delta \\boldsymbol{\\xi}^\\wedge {^o\\matr{x}}}{\\delta \\boldsymbol{\\xi}} & = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\delta \\boldsymbol{\\phi}_\\times {^o\\matr{x}} + \\delta s{^o\\matr{x}} + \\delta \\boldsymbol{\\nu}}{\\delta \\boldsymbol{\\xi}} \\\\\n    & = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{ \\delta \\boldsymbol{\\nu} - {^o\\matr{x}}_\\times \\delta \\boldsymbol{\\phi}+ \\delta s{^o\\matr{x}}}{\\delta \\boldsymbol{\\xi}} \\\\\n    & = \\begin{bmatrix} \\matr{I} & -^o\\matr{x}_{\\times} & ^o\\matr{x} \\end{bmatrix}  \\label{eq:J_surf_pose_3}\n\\end{align}\n\nThe full Jacobian of the surface consistency term residual with respect to the object pose can be obtained by combining Eq. \\ref{eq:J_surf_pose_1}, \\ref{eq:J_surf_pose_2}, \\ref{eq:J_surf_pose_3}. Note that here we derive the Jacobian with slight abuse of notations and neglected the notation of homogeneous representation. For a more detailed explanation of Lie Theory, please refer to ~\\cite{barfoot_2017} or \\cite{sola2018_micro_lie}. \n\n\\subsection{Jacobian of Rendering Term Residual}\n\nAs stated in the main paper, the rendering term residual of pixel $\\matr{u}$ is\n\n\\begin{equation}\n    e_r(\\matr{u}) = d_\\matr{u} - \\hat{d}_{\\matr{u}}\n\\end{equation}\n\nTo compute the Jacobian of the rendering terms $\\matr{J}_{r}$, we can expand it using chain rule:\n\n\\begin{align} \n    \\matr{J}_{r} & = \\frac{\\partial e_{r}}{\\partial \\left[\\boldsymbol{\\xi}_{oc}; \\matr{z} \\right]} \\\\\n    & = \\sum_{k=1}^M \\frac{\\partial e_{r}}{\\partial o_k} \\frac{\\partial o_k}{\\partial s_k} \\frac{\\partial{G({^o\\matr{x}_k}, \\matr{z})}}{\\partial \\left[ \\boldsymbol{\\xi}_{oc}; \\matr{z} \\right]} \\label{eq:J_render}\n\\end{align}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/rendering_values.png}\n\\caption{Demonstration of the depth rendering process. Only very few ray points contribute to the overall Jacobian term.\n\\vspace{-0.5cm}}\n\\label{fig:render_values}\n\\end{figure}\n\nwhere $\\{^o\\matr{x}_k\\}_{k=1}^M$ is the depth-ordered set of sampled points along the ray of back-projecting pixel $\\matr{u}$. The third term $\\frac{\\partial{G( ^o\\matr{x}_k, \\matr{z})}}{\\partial \\left[ \\boldsymbol{\\xi}_{oc}; \\matr{z} \\right]}$ has exactly the same form as the surface term Jacobian in Eq.~\\ref{eq:J_surf}, thus it can be computed following Sec.~\\ref{sec:J_surf} . The second term $\\frac{\\partial o_k}{\\partial s_k}$ is either a constant value or $0$, as the sdf-occupancy conversion is linear inside the cut-off threshold and constant elsewhere. \n\n\\begin{equation} \\label{eq:conversion_grad}\n    \\frac{\\partial o_k}{\\partial s_k} =   \n    \\begin{cases}\n        -\\frac{1}{2\\sigma} & |s_k| < \\sigma\\\\\n        0 & \\text{elsewhere}\n    \\end{cases}\n\\end{equation}\n\nTo compute the first term, we expand the residual term using Eq. 3 and Eq. 4 in the main paper following~\\cite{multi-view_supervision}: \\vspace{-0.3cm}\n\n\\begin{equation*}\n    \\begin{aligned}\n        e_r & = d_\\matr{u} - \\bigg( \\sum_{i=1}^M d_i \\ o_i \\prod_{j=1}^{i-1} (1-o_j) + d_{M+1} \\prod_{j=1}^M (1 - o_j) \\bigg) \\\\\n        & = \\sum_{i=1}^M \\psi(i) \\ o_i \\prod_{j=1}^{i-1} (1-o_j) + \\psi(M+1) \\prod_{j=1}^M (1 - o_j) \\\\\n       \n        & = \\sum_{i=1}^{M+1} \\psi(i) \\prod_{j=1}^{i-1} (1-o_j) - \\sum_{i=1}^M \\psi(i) \\prod_{j=1}^i (1-o_j) \\\\\n        & = \\psi(1) + \\sum_{i=1}^M (\\psi(i+1) - \\psi(i)) \\prod_{j=1}^i (1-o_j)\n    \\end{aligned}\n\\end{equation*}\n\nwhere $\\psi(i) = d_{\\matr{u}} - d_i$ is the depth residual for each of the sampled points along the ray. Differentiating with respect to the sample point, we reach: \\vspace{-0.2cm}\n\n\\begin{align} \n    \\frac{\\partial e_{r}}{\\partial o_k} \n    & = \\sum_{i=1}^M (\\psi(i+1) - \\psi(i)) \\frac{\\partial}{\\partial o_k}\\prod_{j=1}^i (1-o_j) \\\\\n    & = \\sum_{i=k}^M (\\psi(i) - \\psi(i+1)) \\prod_{j=1, j \\neq k}^i (1-o_j) \\\\\n    & = \\Delta d\\sum_{i=k}^M \\prod_{j=1, j \\neq k}^i (1-o_j) \\label{eq:dr_do}\n\\end{align}\n\nAs we are sampling uniformly along the ray, the coefficient $\\psi(i) - \\psi(i+1) = d_{i+1} - d_i$ reduces to $\\Delta d$. To understand this expression we can multiply $(1-o_k)$ on both sides, of Eq.~\\ref{eq:dr_do}, which gives us:\n\n\\begin{equation}\n        \\frac{\\partial e_{r}}{\\partial o_k}(1 - o_k)  = \\Delta d\\sum_{i=k}^M t_i\n\\end{equation}\n\nwhere $t_i = \\prod_{j=1}^i (1 - o_j)$ represents the accumulated transmittance at point $^o\\matr{x}_i$ along the ray. Before hitting the surface, $o_k$ is zero, so the term $(1-o_k)$ has no effect, and the Jacobian term becomes the sum of transmittance of all the points starting from point k. And after the ray entering the solid body, this Jacobian term will be zero. This means only points before entering the solid body contribute to the rendered depth value.\n\nNow from Eq.~\\ref{eq:conversion_grad} and Eq.~\\ref{eq:dr_do}, we already know that many Jacobian terms that make up the final Jacobian in Eq.~\\ref{eq:J_render} should be zero. Therefore, we could further speed up the optimization by not evaluating the third term $\\frac{\\partial{G( ^o\\matr{x}_k, \\matr{z})}}{\\partial \\left[ \\boldsymbol{\\xi}_{oc}; \\matr{z} \\right]}$ at those points. Figure~\\ref{fig:render_values} is a demonstration of the rendering process and the sparse nature of the resulting Jacobians.\n\n\\subsection{Jacobian of Prior Terms}\n\nBased on the shape regularisation energy defined in Eq. 6 in the main paper. The residual of this energy term is simply the shape code vector itself:\n\n\\begin{equation} \\label{eq:E_code}\n    e_c = \\matr{z}\n\\end{equation}\n\nBased on Eq.~\\ref{eq:E_code}, we have\n\n\\begin{equation}\n    \\frac{\\partial e_c}{\\partial \\boldsymbol{\\xi}_{oc}} = \\matr{0}, \\frac{\\partial e_c}{\\partial \\matr{z}} = \\matr{I}\n\\end{equation}\n\nOptionally, we can also apply structural priors such as constraining the object orientation to be aligned with the ground plane, as in~\\cite{wang2020directshape}. We can define the rotation prior residual as\n\n\\begin{align}\n    e_{rot} & = 1 - \\matr{R}_{co}(0:2,1) \\cdot \\matr{n}_g \\\\\n    & = 1 - \\begin{bmatrix} 0 & 1 & 0 \\end{bmatrix}\\matr{R}_{oc}\\matr{n}_g\n\\end{align}\n\n\\begin{figure}[tpb]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{figures\/redwood\/gt_together.png}\n\\includegraphics[width=0.48\\linewidth]{figures\/redwood\/no_gt_together.png}\n\\caption{Results with GT ($\\leftarrow$) vs. MaskRCNN ($\\rightarrow$) masks \\vspace{-0.6cm}}\n\\label{fig:gt_mask}\n\\end{figure}\n\n\nwhere $\\matr{R} \\in \\mathbf{SO}(3)$ denotes the rotation part of the transformation matrix, and $(0:2, 1)$ represents the operation of getting the second column of the matrix. $\\matr{n}_g$ is the normal direction to the ground plane under camera coordinate frame. The Jacobian with respect to pose can be easily obtained following Eq.~\\ref{eq:J_surf_pose_3}:\n\n\\begin{align}\n    \\frac{\\partial e_{rot}}{\\partial \\boldsymbol{\\xi}_{oc}} &= - \\begin{bmatrix} 0 & 1 & 0 \\end{bmatrix}\\frac{\\partial \\matr{R}_{oc}\\matr{n}_g}{\\partial \\boldsymbol{\\xi}_{oc}} \\\\\n    & = \\begin{bmatrix} 0 & 1 & 0 \\end{bmatrix} \\begin{bmatrix} \\matr{0} & (\\matr{R}_{oc}\\matr{n}_g)_{\\times} & \\matr{0} \\end{bmatrix} \\\\\n    & = \\begin{bmatrix} 0 & 0 & 0 & (\\matr{R}_{oc}\\matr{n}_g)_{\\times}(1,0:2) & 0 \\end{bmatrix}\n\\end{align}\n\nwhere $(0:2, 1)$ represents the operation of getting the second row of the matrix. As this residual has no effect on object shape, we have:\n\n\\begin{equation}\n    \\frac{\\partial e_{rot}}{\\partial \\matr{z}} = \\matr{0}\n\\end{equation}\n\n\\section{Experiment Details and Run-time Analysis}\n\nWe evaluate the run-time performance of our full SLAM system (stereo+LiDAR) on a Linux system with Intel Core i9 9900K CPU at 3.60GHz, and an nVidia GeForce RTX2080 GPU with 8GB of memory. The 2D detection boxes and masks are obtained using MaskRCNN~\\cite{maskrcnn},\\footnote{\\url{https:\/\/github.com\/facebookresearch\/detectron2}} throughout all the experiments. Initial object poses are inferred using the LiDAR-based 3D b-box detector SECOND~\\cite{second}.\\footnote{\\url{https:\/\/github.com\/traveller59\/second.pytorch}} Our object reconstruction pipeline is implemented in Python with PyTorch. The back-end object-SLAM framework is implemented in C++ as an extension of the original ORB-SLAM2 implementation. The Python part is embedded into C++ using pybind11. \n\nTable~\\ref{tab:timing} lists the breakdown of the run-time performance of different major components. Note that all those components are performed only at key-frames. \nFor KITTI sequences, we adopted two design choices to achieve real-time performance. Firstly, we noticed shape estimation did not improve much over time in KITTI, thus we perform full optimization only for new objects and only update poses for existing objects (as stated in the main paper). Secondly, we abort new object optimization whenever necessary to ensure new key-frame is inserted in time. These make our system able to operate at $10$Hz on KITTI. In sequences such as Freiburg and Redwood-OS, it is beneficial to update the shape more frequently, with fewer GN iterations to guarantee real-time performance. Object meshes can be extracted optionally using marching cube with an extra GPU, for visualization only.\n\n\\begin{table}[tbp]\n\\centering\n\\begin{tabular}{l|c}\n\\textbf{Component} & \\textbf{Time (ms)} \\\\ \\hline\nMask R-CNN Detector & 70 \/ frame \\\\\n3D LiDAR Detector & 60 \/ frame \\\\\nPose-shape Optimization & 20$\\times$10 \/ new object  \\\\\nPose-only Optimization & 4$\\times$5 \/ vis. object  \\\\ \\hline\n\\end{tabular}\n\\caption{Run-time analysis with system components \\vspace{-0.4cm}}\n\\label{tab:timing}\n\\end{table}\n\n\n\\section{Ablation with GT masks}\nWe have shown promising reconstruction results on cars on both KITTI and Freiburg dataset. Chairs have thin structures and a complex topology and are more challenging to reconstruct than cars. We noticed the thin legs of the last chair in Fig.~10 in the main paper were not properly reconstructed. This is because our reconstruction makes use of Mask-RCNN segmentation masks, which are noisy and affected by shadows. This can result in background points being associated to the chair, leading to incorrect results. We conducted ablation study using ground-truth masks as input. Fig.~\\ref{fig:gt_mask} illustrates an upper bound quality that could be achieved with clean GT masks.\n\n\\section{More Qualitative Results}\n\nWe show more qualitative reconstruction results in Fig.~\\ref{fig:recon_results}. For each scene the RGB image is shown on the top and the reconstruction results are shown underneath. We also show an example of reconstructed map in Fig.~\\ref{fig:map_result}. \n\n\n\\begin{figure*}[tpb]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/00-000117.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/00-001190.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-00-000117.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-00-001190.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/00-001848.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/00-003264.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-00-001848.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-00-003264.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/00-003516.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/07-000195.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-00-003516.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-07-000195.png}\n\\caption{More qualitative results on object reconstruction from a single view-point \\vspace{-1cm}}\n\\label{fig:recon_results}\n\\end{figure*}\n\n\\begin{figure*}[tpb]\n\\centering\n\\includegraphics[width=0.90\\textwidth]{figures\/supp\/kitti-07-map.png}\n\\caption{Reconstructed map of KITTI-07. Note the dense reconstruction of the objects and the shape variations. \\vspace{-1cm}}\n\\label{fig:map_result}\n\\end{figure*}\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\section{Introduction}\nSimultaneous Localization and Mapping (SLAM) is the process of estimating the trajectory of a moving camera while reconstructing its surrounding environment. From a purely geometric perspective, SLAM is often regarded as a well-understood or even solved problem. Many state-of-the-art dense SLAM algorithms can achieve accurate trajectory estimation and create high-quality geometric reconstructions that can be used in obstacle avoidance or path planning for mobile robots. However, when it comes to more complex tasks that require scene understanding, geometry-only scene representations fall short of providing key semantic information. \nTaking advantage of recent deep learning breakthroughs in semantic segmentation and object detection algorithms \\cite{maskrcnn, faster-rcnn, yolo} semantic SLAM systems augment geometric low-level map primitives by fusing semantic labels into the 3D reconstruction~\\cite{semantic-fusion, semantic-stereo-fusion,Chen2019SuMa++}. However, the resulting scene maps merely consist of a set of labelled 3D points where reasoning about the scene at the level of objects to infer meaningful information such as the number of objects of each category, their size, shape or relative pose remains a challenging task. Better use of the semantic information is required in the form of an object-centric map representation that allows detailed shape estimation and meaningful instantiation of scene objects. \n\n\nOur proposed approach forms part of a more recent family of \\emph{object-aware} SLAM methods that reconstruct object-centric maps grouping all the low-level geometric primitives (voxels, points ...) that make up the same object into a single instance. Front-end camera tracking and back-end optimization are both performed at the level of object instances. While the first object-level methods, such as SLAM++~\\cite{SLAM++}, mapped previously known object instances, more recent systems have taken advantage of instance level semantic segmentation masks \\cite{maskrcnn} to achieve object level reconstruction for unknown objects \\cite{fusion++} even in the presence of dynamic objects  \\cite{maskfusion, midfusion}.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.235\\textwidth]{figures\/08-2758.png}\n\\includegraphics[width=0.235\\textwidth]{figures\/recon-08-2758.png}\n\\caption{Qualitative shape and pose results on a stereo+LiDAR KITTI sequence. A very sparse set of LiDAR points was used to reconstruct each car. LiDAR points on the road are only shown for illustration.\\vspace{-0.5cm}}\n\\label{fig:single-view}\n\\end{figure}\n\nHowever, these early object level SLAM systems exhibit major drawbacks: They either require a pre-known database of object instances \\cite{SLAM++}; or reconstruct objects from scratch without exploiting shape priors \\cite{maskfusion, midfusion, fusion++}, which results in partial or incomplete object reconstructions. We improve this by exploiting the regularity of shapes within an object category in the form of learned shape priors, defined as a latent code $\\matr{z}$ and a generative model $G(\\matr{z})$ that decodes it into its full geometry. This brings us several advantages; object shapes decoded from latent codes are guaranteed to be detailed and complete, regardless of partial observations or changes in view-points, they provide a compact representation and they can be optimized using the gradients obtained through back-propagation. \n\nUsing ORB-SLAM2~\\cite{orbslam2} as a sparse camera tracking and mapping backbone, DSP-SLAM takes the reconstructed 3D point-cloud as input and fits a latent code to each detected object instance, using a combination of 3D surface consistency and rendered depth losses. Foreground objects, background features and camera poses are further refined via bundle adjustment using a joint factor graph. We show DSP-SLAM  operating in $3$ different modes: monocular, stereo, and stereo+LiDAR. The monocular and stereo systems use the respective ORB-SLAM2 modalities as the SLAM backbone and the reconstructed 3D point-clouds to reconstruct the detected objects. The stereo+LiDAR system uses stereo ORB-SLAM2 as the SLAM backbone but in addition it incorporates a sparse set of LiDAR measurements (as few as 50 per object) for object reconstruction and pose-only optimization. \n\n\\noindent\\textbf{Contributions:} While DSP-SLAM is not the first approach to leverage shape priors for 3D reconstruction~\\cite{frodo,node-slam} from image sequences, it innovates in various ways. \nFirstly, unlike~\\cite{frodo,node-slam}, our map does not only represent objects, but also reconstructs the background as sparse feature points, optimizing them together in a joint factor graph, marrying the best properties of feature-based~\\cite{orbslam2} (highly accurate camera tracking) and object-aware SLAM (high level semantic map). Secondly, although Node-SLAM \\cite{node-slam} also incorporates shape priors within a real-time SLAM system~\\cite{node-slam}, it uses dense depth images for shape optimization, while DSP-SLAM can operate with RGB-only monocular streams and requires as few as $50$ 3D points per object to obtain accurate shape estimates. Finally, although both FroDO~\\cite{frodo} and DSP-SLAM can operate in a monocular RGB setting, FroDO is a slow batch approach that requires all frames to be acquired in advance and associated with their camera poses, while  \nDSP-SLAM is an online, sequential method that can operate at $10$ frames per second. \n\nIn terms of object shape and pose estimation, we improve quantitative and qualitatively over auto-labelling~\\cite{Zakharov2020Autolabeling3D}, a state-of-the-art prior-based object reconstruction method. Experiments on the KITTI odometry \\cite{kitti_odom} dataset show that, with stereo+LiDAR input our joint bundle adjustment offers improvements in trajectory estimation over the feature-only stereo system ORB-SLAM2~\\cite{orbslam2}, used as our backbone. Moreover, DSP-SLAM offers comparable tracking performance to state-of-the-art stereo~\\cite{Wang2017StereoDSO}, LiDAR-only~\\cite{Chen2019SuMa++} and dynamic~\\cite{Bescos2018DynaSLAM} SLAM systems, while providing rich dense object reconstructions. DSP-SLAM also achieves promising qualitative reconstruction results with monocular input on Freiburg Cars \\cite{freiburg-cars} and Redwood-OS \\cite{choi2016redwood} dataset.\n\n\n\n\n\n\\section{\\label{system-overview}System Overview}\n\nDSP-SLAM is a sequential localisation and mapping method that reconstructs the complete detailed shape of detected objects while representing the background coarsely as a sparse set of feature points. Each object is represented as a compact and optimizable code vector $\\matr{z}$. We employ DeepSDF \\cite{deepsdf} as the shape embedding, that takes as input a shape code $\\matr{z} \\in \\mathbb{R}^{64}$ and a 3D query location $\\matr{x} \\in \\mathbb{R}^{3}$, and outputs the signed distance function (SDF) value $s = G(\\matr{x}, \\matr{z})$ at the given point. An overview of DSP-SLAM is shown in Fig.~\\ref{fig:system}. DSP-SLAM runs at almost real time ($10$ frames per second) and can operate on different modalities: monocular, stereo or stereo with LiDAR; depending on the available input data. \n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/000140.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/000168.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/000428.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/000655.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/140r.png} \\hspace{-0.6em}\n\\includegraphics[width=0.1115\\textwidth]{figures\/single_view_shape\/168r.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/428r.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/655r.png} \\hspace{-0.6em}\n\\caption{Shape reconstruction: qualitative results. \\vspace{-0.5cm}}\n\\label{fig:single-view-shape}\n\\end{figure}\n\n\\noindent\\textbf{Sparse SLAM backbone:} ORB-SLAM2~\\cite{orbslam2} is used as the tracking and mapping backbone, a feature-based SLAM framework that can operate on monocular or stereo sequences. While the tracking thread estimates camera pose at frame-rate from correspondences, the mapping thread builds a sparse map by reconstructing 3D landmarks.\n\n\\noindent\\textbf{Detections:}  We perform object detection at each key-frame, to jointly infer 2D bounding boxes and segmentation masks. In addition, an initial estimate for the object pose estimation is obtained via 3D bounding box detection~\\cite{smoke,second}.\n\n\\noindent\\textbf{Data association:} New detections will either be associated to existing map objects, or instantiated as a \\textit{new} object via object-level data association. \nEach detected object instance $I$ consists of a 2D bounding box ${\\mathcal{B}}$, a 2D mask ${\\mathcal{M}}$, the dpeth observation of sparse 3D point cloud ${\\mathcal{D}}$, and the initial object pose $\\matr{T}_{co, 0}$. \n\n\\noindent\\textbf{Prior-based object reconstruction:} Newly instantiated objects will be reconstructed following the object reconstruction pipeline described in Sec.~\\ref{object-recon}. DSP-SLAM takes the set of sparse 3D point observations ${\\mathcal{D}}$, which can come from reconstructed SLAM points (in monocular and stereo modes) or LiDAR input (in stereo+LiDAR mode) and optimises the shape code and object pose to minimise surface consistency and depth rendering losses. Objects already present in the map will only have their 6-dof pose updated via pose-only optimization. \n\n\n\\noindent\\textbf{Joint map optimisation:} A joint factor graph of point features (from SLAM), objects and camera poses is optimised via bundle adjustment to maintain a consistent map and incorporate loop closure. New objects are added as nodes to the joint factor graph and their relative pose estimates $\\matr{T}_{co}$ as camera-object edges. Object-level data association and joint bundle adjustment will be discussed in Sec.~\\ref{object-slam}.\n\n\n\n\n\n\n\\section{\\label{object-recon}Object Reconstruction with  Shape Priors}\n\nWe aim to estimate the full dense shape $\\matr{z}$ and 7-DoF  pose $\\matr{T}_{co}$, represented as a homogeneous transformation matrix $\\matr{T}_{co} = [s\\matr{R}_{co}, \\matr{t}_{co}; \\matr{0}, 1] \\in \\matr{Sim}(3)$, for an object with detections $I = \\{{\\mathcal{B}}, {\\mathcal{M}}, {\\mathcal{D}}, \\matr{T}_{co, 0}\\}$. We formulate this as a joint optimization problem, which iteratively refines the shape code and object pose from an initial estimate. We propose two energy terms $E_{surf}$ and $E_{rend}$ and formulate a Gauss-Newton solver with analytical Jacobians.\n\n\\subsection{Surface Consistency Term}\nThis term measures the alignment between observed 3D points and the reconstructed object surface. \n\\begin{equation} \\label{E_surf}\n    E_{surf} = \\frac{1}{\\left| \\mathbf{\\Omega}_s \\right|}\\sum_{\\matr{u} \\in \\mathbf{\\Omega}_s} G^2(\\matr{T}_{oc} \\pi^{-1}(\\matr{u}, {\\mathcal{D}}), \\matr{z})\n\\end{equation}\nwhere $\\mathbf{\\Omega}_s$ denotes the pixel coordinates of the set of sparse 3D points ${\\mathcal{D}}$, which can come from reconstructed SLAM points (in monocular and stereo modes) or LiDAR input (in stereo+LiDAR mode). Ideally, the back-projected point at pixel $\\matr{u}$ should perfectly align with the object surface resulting in zero SDF value, giving a zero error residual. In practice, we observed that the surface consistency term alone is not sufficient for correct shape and pose estimation in the case of partial observations. Fig.~\\ref{fig:shape-ablation} illustrates a case where only points on the back and the right side of a car are observed (shown in green). Using  the surface consistency alone term leads to incorrect shape estimation -- much larger than its actual size. To address this issue, we propose a rendering loss, that provides point-to-point depth supervision and enforces silhouette consistency to penalize shapes that grow outside of the segmentation mask. \n\n\\begin{figure}[tbp]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{figures\/shape_ablation_success.png}\n    \\caption{Illustration of the effectiveness of the rendering term in the presence of partial observations. \\textbf{Left:} Detected object and partial surface point observations (green). \\textbf{Middle:} Optimisation result with $E_{surf}$ only. The loss is minimised but the shape grows larger than its actual size. \\textbf{Right:} Optimisation result with the rendering term. Enforcing the silhouette constraint results in the correct scale. \\vspace{-0.6cm}}\n    \\label{fig:shape-ablation}\n\\end{figure}\n\n\\subsection{Differentiable SDF Renderer}\n\nFollowing \\cite{multi-view_supervision, node-slam}, we build our SDF renderer via differentiable ray-tracing. For each pixel $\\matr{u}$, we back-project a ray $^c\\matr{x} = \\matr{o} + d \\matr{K}^{-1} \\dot{\\matr{u}}$ parameterized by the depth value $d$ under camera coordinate space, with $\\matr{o}$ being the camera optical centre and $\\matr{K}$ being camera intrinsic matrix. We sample $M$ discrete depth values $\\{d_i\\}$ along each ray within the range $[d_{min}, d_{max}]$, with $d_i = d_{min} + (i-1) \\Delta d$, and $\\Delta d = (d_{max} - d_{min}) \/ (M-1)$. The bounds of the depth range are determined by the current estimation of object translation and scale, and are re-computed at each iteration.\n\n\\noindent\\textbf{Occupancy Probabilities} \nThe SDF value $s_i$ at each sampled point can be obtained by transforming sampled points to the object coordinate frame and passing through the DeepSDF decoder. The SDF value encodes the probability that a given point is occupied by the object or belongs to free space. We apply a piecewise linear function to the predicted SDF values to indicate the occupancy probability $o_i$, defined in Eq.~\\ref{sdf2occ}, where $\\sigma$ represents the cut-off threshold which controls the smoothness of the transition. We fix $\\sigma = 0.01$ throughout our experiments. \\vspace{-0.4cm}\n\n\\begin{equation} \\label{sdf2occ}\n    s_i = G(\\matr{T}_{oc} {^c\\matr{x}},\\, \\matr{z}) \\quad \\textrm{and} \\quad \n    o_i = \n    \\begin{cases}\n        1 & s_i < -\\sigma\\\\\n        -\\frac{s_i}{2 \\sigma} &  \\left|s_i\\right| \\leq \\sigma \\\\    \n        0 & s_i > \\sigma\n    \\end{cases}\n\\end{equation}\n\n\n\\noindent\\textbf{Event Probabilities} When tracing points along the ray, the ray either terminates or escapes without hitting other points. These $M+1$ event probabilities can be defined as: \n\\begin{eqnarray} \\label{term_prob}\n    & \\phi_i = o_i \\prod_{j=1}^{i-1} (1- o_j), i = 1, \\dots, M \\nonumber \\\\\n    & \\phi_{M + 1} = \\prod_{j=1}^{M} (1- o_j)\n\\end{eqnarray}\n\n\\noindent\\textbf{Rendered Depth and Rendering Term} With the probabilities defined above, the rendered depth value at each pixel $\\matr{u}$ can be computed as the expected depth value of the terminating point as in Eq.~\\ref{render_depth}. To make it consistent, we set $d_{M+1}$, the depth value associated with escape probability, to a constant value $1.1 d_{max}$, as in \\cite{node-slam}.\n\\begin{equation} \\label{render_depth}\n    \\hat{d}_{\\matr{u}} = \\sum_{i=1}^{M+1} \\phi_i d_i\n\\end{equation}\nSince the rendering is fully differentiable, it can be integrated in our optimization. Unlike~\\cite{node-slam, multi-view_supervision}, we perform  ray-tracing in continuous space and do not require to discretize the object model. The final rendering term is as follows:\n\\begin{equation}\n    E_{rend} = \\frac{1}{\\left| \\mathbf{\\Omega}_r \\right|} \\sum_{\\matr{u} \\in \\mathbf{\\Omega}_r} (d_\\matr{u} - \\hat{d}_{\\matr{u}})^2\n\\end{equation}\nwhere $\\mathbf{\\Omega}_r = \\mathbf{\\Omega}_s \\cup \\mathbf{\\Omega}_b$ is the union of surface pixels and pixels not on object surface but inside the 2D bounding box ${\\mathcal{B}}$. Surface pixels $\\mathbf{\\Omega}_s$ are the same set of pixels used in Eq.~\\ref{E_surf}, obtained by projecting the 3D reconstucted SLAM points onto the image masks as discussed in Sec.~\\ref{system-overview}. The pixels in $\\mathbf{\\Omega}_b$ are assigned the same depth value as $d_{M+1} = 1.1 d_{max}$ and provide important silhouette supervision for our optimization since they penalize renderings that lie outside the object boundary, forcing empty space. As the pixels in $\\mathbf{\\Omega}_b$ do not require a depth measurement, we perform uniform sampling inside the 2D bounding box and filter out those inside the segmentation mask.\n\n\\subsection{Optimization details}\n\nOur final energy is the weighted sum of the surface and rendering terms and a shape code regularization term:\n\\begin{equation}\n    E = \\lambda_s E_{surf} + \\lambda_r E_{rend} + \\lambda_c \\norm{\\matr{z}}^2\n\\end{equation}\nThe hyperparameter values used for optimization $\\lambda_s = 100$, $\\lambda_r = 2.5$ and $\\lambda_c = 0.25$ were tuned such that the Hessian matrices of the energy terms are of the same order of magnitude. Since all terms are quadratic, we adopt a  Gauss-Newton optimisation approach with analytical Jacobians (Please refer to supplemental material for detail), initialized from a zero shape code $\\matr{z} = \\matr{0}$. The initialisation for the object pose \n$\\matr{T}_{co, 0}$  is given by a LiDAR 3D detector~\\cite{second} when LiDAR is available. In the monocular\/stereo case, it is given by an image-based 3D detector~\\cite{smoke} or by performing PCA on the sparse object point cloud. \n\n\n\n\\section{\\label{object-slam}Object SLAM}\n\nAs an object-based SLAM system, DSP-SLAM builds a joint factor graph of camera poses, 3D feature points and object locations and poses. As Fig.~\\ref{system-overview} shows, the factor graph introduces object nodes and camera-object edges. \n\n\n\\subsection{\\label{data-association}Object Data Association}\n\nData association between new detections and reconstructed objects is an important step in object-level SLAM. We aim to associate each detection $I$ to its \\textit{nearest} object $o$ in the map, adopting different strategies depending on the different input modalities. When LiDAR input is available we compare the distance between 3D bounding box and reconstructed object. When only stereo or monocular images are used as input, we count the number of matched feature points between the detection and object. If multiple detections are associated with the same object, we  keep the \\textit{nearest} one and reject others. Detections not associated with any existing objects are initialised as new objects and their shape and pose  optimised following Sec.~\\ref{object-recon}. For stereo and monocular input modes, reconstruction only happens when enough surface points are observed. For detections associated with existing objects, only the pose is optimised by running pose-only optimization and a new camera-object edge added to the joint factor-graph.\n\n\\begin{table*}\n\\renewcommand{\\arraystretch}{1.0}\n\\centering\n\\begin{tabular}{c|c c c c|c c c c}\n\\hline\n\\multirow{2}{*}{Diff.} & \\multicolumn{4}{c|}{Auto-labelling\\cite{Zakharov2020Autolabeling3D}} & \\multicolumn{4}{c}{Ours} \\\\\n\\cline{2-9}\n & BEV@0.5 & 3D@0.5 & NS@0.5 & NS@1.0 & BEV@0.5 & 3D@0.5 & NS@0.5 & NS@1.0 \\\\\n\\hline\nE & 80.70 & \\textbf{63.96} & 86.52 & 94.31 & \\textbf{83.31} & 62.58 & \\textbf{88.01} & \\textbf{96.86} \\\\ \nM & 63.36 & 44.79 & 64.44 & 85.24 & \\textbf{75.28} & \\textbf{47.76} & \\textbf{76.15} & \\textbf{89.97} \\\\\n\\hline\n\\end{tabular}\n\\caption{Quantitative comparison of object cuboid prediction quality with Auto-labelling on KITTI3D on Easy and Moderate samples. Results of Auto-labelling are taken from their paper. Best results are shown as bold numbers.\n\\vspace{-0.1cm}}\n\\label{tab:object_detection}\n\\end{table*}\n\n\n\\begin{figure*}[tbp]\n\\centering\n\\includegraphics[width=1.00\\textwidth]{figures\/labelling_ours\/ours-labelling-cropped.png}\n\\caption{A qualitative comparison of shape reconstruction and pose estimation against Auto-labelling~\\cite{Zakharov2020Autolabeling3D}. \\textbf{Left:} input RGB image. \\textbf{Middle:} result with DSP-SLAM \\textbf{Right:} result with auto-labelling~\\cite{Zakharov2020Autolabeling3D}\\vspace{-0.5cm}}\n\\label{fig:comparison-auto-labelling}\n\\end{figure*}\n\n\\subsection{\\label{joint_ba}Joint Bundle Adjustment}\n\nOur joint map consists of a set of camera poses $C = \\{\\matr{T}_{wc_i}\\}_{i=1}^{M}$, object poses $O = \\{\\matr{T}_{wo_j}\\}_{j=1}^{N}$ and map points $P = \\{^w\\matr{p}_k\\}_{k=1}^{K}$. Our joint BA can be formulated as a non-linear least squares optimization problem:\n\\begin{eqnarray}\n    C^*, O^*, P^* &{} = {}& \\mathop{\\arg\\min}_{\\{C, O, P\\}} \\sum_{i, j} \\norm{\\matr{e}_{co}(\\matr{T}_{wc_i}, \\matr{T}_{wo_j})}_{\\Sigma_{i,j}} \\nonumber \\\\\n    && {+}\\:\\sum_{i, k} \\norm{\\matr{e}_{cp}(\\matr{T}_{wc_i}, ^w\\matr{p}_k)}_{\\Sigma_{i,k}}\n\\label{eq:ba}\n\\end{eqnarray}\nwhere $\\matr{e}_{co}$ and $\\matr{e}_{cp}$ represent the residuals for camera-object and camera-point measurements and $\\Sigma$ is the co-variance matrix of measurement residuals. Objects act as additional landmarks, which results in improvements in tracking performance as shown in our evaluation on KITTI. The optimization is solved with Levenberg-Marquardt in g2o \\cite{g2o}.\n\n\\noindent\\textbf{Camera-Object Measurements:} Object-camera poses $\\matr{T}_{co}$ are evaluated by minimising the surface alignment term in Eq. \\ref{E_surf} while keeping  the shape code and scale fixed. New pose observations serve as edges between camera pose $\\matr{T}_{wc}$ and object pose $\\matr{T}_{wo}$, and the residual is defined as:\n $   \\matr{e}_{co} = \\log (\\matr{T}^{-1}_{co} \\cdot \\matr{T}^{-1}_{wc} \\cdot \\matr{T}_{wo})$\nwhere $\\log$ is the logarithm mapping from $\\matr{SE}(3)$ to $\\mathfrak{se}(3)$. Poses in the factor graph are 6-DoF, as object scale is only optimised when first detected.\n\n\\noindent\\textbf{Camera-Point Measurements:} We use the standard formulation of re-projection error used in ORB-SLAM2~\\cite{orbslam2}:\n$    \\matr{e}_{cp} = \\pi(\\matr{T}_{wc}^{-1} {^w\\matr{p}}) - \\matr{\\Tilde{u}}$, \nwhere $\\matr{\\Tilde{u}}$ is the measured pixel coordinate of map point $\\matr{p}$. We follow a similar strategy as ORB-SLAM2 to tune $\\Sigma_{ij}$ such that the two energy terms contribute roughly the same to the overall optimization. \n\\section{Related work} \n\n\\noindent\\textbf{Object-aware SLAM:}\nSLAM++ \\cite{SLAM++} pioneered object-aware RGB-D SLAM, representing the scene at the level of objects using a joint pose-graph for camera and object poses. A database of pre-scanned objects was created in advance and object instances were detected and mapped using a pre-trained 3D detector, ICP losses and pose-graph optimization. In later work, Tateno \\emph{et al.}~\\cite{Tateno2016When2I} aligned object instances from a pre-trained database to volumetric maps while Stuckler \\emph{et al.}~\\cite{stuckler2012model} performed online exploration, learning object models on the fly and tracking them in real time. An important drawback of instance-based approaches is their inability to scale to a large number of objects and their need for object models to be known in advance. More recent object-aware RGB-D SLAM systems have dropped the requirement for known models and instead take advantage of state-of-the art 2D instance-level semantic segmentation masks~\\cite{maskrcnn} to obtain object-level scene graphs~\\cite{semantic-fusion} and per-object reconstructions via depth fusion, even in the case of dynamic scenes~\\cite{maskfusion,midfusion}.\n\nExtensions of object-aware SLAM to the case of monocular video input deal with the additional challenge of relying only on RGB-only information~\\cite{mono-object-slam, mono-object-sparse-slam, quadricslam, cubeslam, category-specific-models} which results in the use of simplistic shape representations. In QuadricSLAM \\cite{quadricslam} objects are represented as ellipsoids and fit to monocular observations while in CubeSLAM \\cite{cubeslam} cuboid proposals generated from single-view detections are optimized in a joint bundle adjustment optimization. \n\nWhile the above SLAM systems represent an important step forward towards equipping robots with the capability of building semantically meaningful object-oriented maps, they fall short of exploiting semantic priors for object reconstruction. In this paper we take this direction of using a category-specific learnt shape prior and embed this within an object-aware SLAM system. \n\n\\noindent\\textbf{3D Reconstruction with Shape Priors:}\nThe use of learnt compact shape embeddings as priors for 3D reconstruction has a long tradition in computer vision. From 3D morphable models for the reconstruction of faces or bodies~\\cite{blanz1999morphable,loper2015smpl}, to PCA models to represent category specific object shape priors~\\cite{directshape}.\nOther examples of the use of embedding spaces for single or multi-view shape reconstruction include GPLVMs \\cite{dame2013dense, prisacariu2011shared, prisacariu2012simultaneous} or neural representations~\\cite{hu2018dense-embedding, zhu2018object} such as a variational autoencoder~\\cite{node-slam},  AltlasNet~\\cite{AtlasNet,2019-lin} or DeepSDF~\\cite{deepsdf,frodo,2019-xu,Zakharov2020Autolabeling3D}. DeepSDF \\cite{deepsdf} provides a powerful implicit learnt shape model that encapsulates the variations in shape across an object category, in the form of an auto-decoder network that regresses the signed distance function (SDF) values of a given object and has been used as a shape prior for single-view~\\cite{2019-xu} and multi-view~\\cite{frodo} reconstruction. Similarly to~\\cite{Zakharov2020Autolabeling3D} DSP-SLAM adopts DeepSDF as the shape prior and takes sparse LiDAR and images as input, however~\\cite{Zakharov2020Autolabeling3D} takes single frames and is not a SLAM method. DOPS~\\cite{Najibi2020DOPSLT} is a single-pass 3D object detection architecture for LiDAR that estimates both 3D bounding boxes and shape.  \n\nOur approach is most closely related to those that build consistent multi-object maps over an entire sequence such as FroDO~\\cite{frodo} and Node-SLAM~\\cite{node-slam}. Unlike FroDO ~\\cite{frodo} ours is a sequential SLAM system and not a batch method. Unlike Node-SLAM \\cite{node-slam}, in our system low-level point features and high-level objects are jointly optimized to bring the best of both worlds: accurate tracking and rich semantic shape information. DeepSLAM++~\\cite{hu2019deep-slam++} leverages shape priors in a SLAM pipeline by selecting 3D shapes predicted by Pix3D~\\cite{Sun_2018_CVPR}, but forward shape generation is often unstable and lead to poor results on real data. \n\\section{Experimental Results}\n\n\\begin{table*}\n\\centering\n\\setlength{\\arrayrulewidth}{0.5mm}\n\\resizebox{1.00\\textwidth}{!}{\n\\begin{tabular}{c c c c c c c c c c c c c}\n\\hline\n\\multirow{3}{*}{Approach} & \\multicolumn{11}{c}{Sequence} & \\multirow{3}{*}{Average} \\\\\n& 00* & 01 & 02* & 03 & 04 & 05* & 06* & 07* & 08* & 09* & 10 & \\\\\n& rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & \\\\\n\\hline\nSuMa++ \\cite{Chen2019SuMa++} & \\textbf{0.64}\/\\textbf{0.22} & 1.60\/0.46 & 1.00\/0.37 & 0.67\/0.46 & \\textbf{0.37}\/0.26 & 0.40\/0.20 & 0.46\/0.21 & \\textbf{0.34}\/\\textbf{0.19} & 1.10\/0.35 & \\textbf{0.47}\/\\textbf{0.23} & \\textbf{0.66}\/0.28 & \\textbf{0.70}\/0.29 \\\\\nOurs St+LiDAR (250pts) & 0.75\/\\textbf{0.22} & \\textbf{1.49}\/0.20 & \\textbf{0.79}\/\\textbf{0.23} & \\textbf{0.60}\/\\textbf{0.18} & 0.47\/0.11 & \\textbf{0.32}\/\\textbf{0.15} & 0.39\/0.21 & 0.52\/0.28 & \\textbf{0.94}\/\\textbf{0.27} & 0.79\/0.28 & 0.69\/\\textbf{0.26} & \\textbf{0.70}\/\\textbf{0.22} \\\\\nOurs St+LiDAR (50pts) & 0.80\/0.24 & 1.50\/\\textbf{0.15} & 0.84\/0.26 & 0.61\/0.18 & 0.44\/\\textbf{0.10} & 0.32\/0.16 & \\textbf{0.35}\/\\textbf{0.15} & 0.57\/0.24 & 1.03\/0.30 & 0.78\/0.27 & 0.67\/0.30 & 0.72\/\\textbf{0.22} \\\\\n\\hline\nORB-SLAM2 \\cite{orbslam2} & 0.70\/0.25 & \\textbf{1.38}\/0.20 & 0.76\/0.23 & 0.71\/0.17 & 0.45\/0.18 & \\textbf{0.40}\/\\textbf{0.16} & 0.51\/0.15 & \\textbf{0.50}\/\\textbf{0.28} & 1.07\/0.31 & \\textbf{0.82}\/0.25 & 0.58\/0.28 & \\textbf{0.72}\/0.22 \\\\\nSt DSO \\cite{Wang2017StereoDSO} & 0.84\/0.26 & 1.43\/\\textbf{0.09} & 0.78\/\\textbf{0.21} & 0.92\/\\textbf{0.16} & 0.65\/0.15 & 0.68\/0.19 & 0.67\/0.20 & 0.83\/0.36 & \\textbf{0.98}\/\\textbf{0.25} & 0.98\/\\textbf{0.18} & \\textbf{0.49}\/\\textbf{0.18} & 0.84\/\\textbf{0.20} \\\\\nSt LSD-SLAM \\cite{engel2015stereo-lsd} & \\textbf{0.63}\/0.26 & 2.36\/0.36 & 0.79\/0.23 & 1.01\/0.28 & \\textbf{0.38}\/0.31 & 0.64\/0.18 & 0.71\/0.18 & 0.56\/0.29 & 1.11\/0.31 & 1.14\/0.25 & 0.72\/0.33 & 0.91\/0.27 \\\\\nDynaSLAM \\cite{Bescos2018DynaSLAM} & 0.74\/0.26 & 1.57\/0.22 & 0.80\/0.24 & 0.69\/0.18 & 0.45\/\\textbf{0.09} & \\textbf{0.40}\/\\textbf{0.16} & 0.50\/0.17 & 0.52\/0.29 & 1.05\/0.32 & 0.93\/0.29 & 0.67\/0.32 & 0.76\/0.23 \\\\\nOurs St only & 0.71\/\\textbf{0.24} & 1.45\/0.30 & \\textbf{0.75}\/0.23 & 0.73\/0.19 & 0.47\/0.11 & 0.57\/0.23 & 0.57\/0.22 & 0.51\/0.29 & 1.02\/0.32 & 0.87\/0.26 & 0.65\/0.31 & 0.75\/0.25 \\\\\nOurs St only (5Hz) & 0.71\/0.26 & 1.43\/0.23 & 0.78\/0.24 & \\textbf{0.67}\/0.18 & 0.46\/\\textbf{0.09} & \\textbf{0.40}\/\\textbf{0.16} & \\textbf{0.47}\/\\textbf{0.14} & 0.52\/0.29 & 0.99\/0.31 & 0.90\/0.28 & 0.63\/0.31 & \\textbf{0.72}\/0.22 \\\\\n\\hline\n\\end{tabular}}\n\\caption{Comparison of camera tracking accuracy - average $t_{rel}$ [\\%] and $r_{rel}$ [\\si{\\degree}\/100m] against state-of-the-art stereo and LiDAR SLAM systems. Sequences marked with * contain loops. Note that Stereo-DSO is a purely visual odometry system, so their result is without loop closing. We keep it in the table for completeness. \\vspace{-0.5cm}}\n\\label{tab:traj-lp}\n\\end{table*}\n\nWe perform a quantitative evaluation of our novel prior-based object reconstruction optimisation, using LiDAR input on the KITTI3D Dataset \\cite{kitti3d}, comparing with auto-labelling~\\cite{Zakharov2020Autolabeling3D}, the most related approach. In addition, we evaluate the camera trajectory errors of our full DSP-SLAM system on both stereo+LiDAR and stereo-only input on the  KITTI Odometry~\\cite{kitti_odom} benchmark, comparing with state-of-the-art approaches.  We also provide qualitative results of our full SLAM system on pure monocular input on Freiburg Cars \\cite{freiburg-cars} and Redwood-OS \\cite{choi2016redwood} Chairs dataset. \n\n\\subsection{\\label{det3d}3D Object Reconstruction}\n\nWe conduct a quantitative comparison of our object pose estimation on the KITTI3D benchmark,  against auto-labeling~\\cite{Zakharov2020Autolabeling3D}, a recent approach to prior-based object shape and pose reconstruction based on image and LiDAR inputs, and using the same shape prior embedding (DeepSDF~\\cite{deepsdf}) and similar level of supervision (object masks and sparse depth from the LiDAR measurements).\n\n\n\\noindent\\textbf{Experimental Setting:} For a fair comparison, we evaluate our approach using a single image and LiDAR input and take the 2D segmentation masks and initial pose estimates from the  auto-labelling code release~\\cite{Zakharov2020Autolabeling3D} as initialization for our own optimization approach. We evaluate the results of pose estimation on the trainval split of KITTI3D which consists of 7481 frames, using  the same metrics  proposed in~\\cite{Zakharov2020Autolabeling3D}: BEV AP @ 0.50, 3D AP @ 0.50, and the distance threshold metric (NS) from the nuscenes dataset~\\cite{caesar2020nuscenes}.\n\n\\noindent\\textbf{Results:} We report quantatitive results in Tab.~\\ref{tab:object_detection}. Our method achieves better performance under almost all metrics, especially on harder samples. We also visualize the comparison of reconstructed shapes and pose in Fig.~\\ref{fig:comparison-auto-labelling}. Auto-labelling~\\cite{Zakharov2020Autolabeling3D} does not capture shape accurately for several vehicles: The first two cars on the left side are sedans, but auto-labelling~\\cite{Zakharov2020Autolabeling3D} reconstructs them as \"beetle\"-shaped. In addition, some of the cars on the right side are reconstructed with incorrect poses which do not align with the image. In contrast, DSP-SLAM obtains accurate shape and pose.\n\n\n\n\\noindent\\textbf{Timing Analysis:} To achieve close to real-time performance, we employ a Gauss-Newton solver with faster convergence than first-order methods during our optimization, leading to significant speed-ups.\nTab.~\\ref{tab:timing} shows a run-time comparison between a first-order optimizer and our Gauss-Newton solver with analytical gradients. Our method is approximately one order of magnitude faster to complete a single iteration, and requires fewer iterations to converge.\n\\begin{table}[tbp]\n\\centering\n\\begin{tabular}{c|l|c|c}\nMethod & Energy Terms & msec. \/ iter & \\# of iter \\\\ \\hline\n1st order & $E_{surf} + E_{rend}$ & 183 & 50 \\\\\n1st order & $E_{surf}$ & 88 & 50 \\\\ \\hline\nOurs GN & $E_{surf} + E_{rend}$ & 20 & 10 \\\\\nOurs GN & $E_{surf}$ & 4 & 10 \\\\ \\hline\n\\end{tabular}\n\\caption{Speed comparison between first-order optimization and our Gauss-Newton method with analytical Jacobians\\vspace{-0.5cm}}\n\\label{tab:timing}\n\\end{table}\n\n\\noindent\\textbf{Ablation Study:} We conducted an ablation study for DSP-SLAM with stereo+LiDAR input to analyse the effect of the number of LiDAR points used for shape optimization  on the reconstruction error.  Fig.~\\ref{fig:recon-num-pts} shows that there is no significant difference when reducing the number of LiDAR points from 250 to 50. The reconstruction quality starts to degrade when the number of points is further reduced to 10. \n\n\\subsection{\\label{full_slam}KITTI Odometry Benchmark}\n\nWe evaluate the camera trajectory error for our full DSP-SLAM system on the KITTI odometry benchmark with both stereo+LiDAR and stereo-only input. We evaluate on the 11 training sequences and compare with state-of-the-art SLAM systems of different input modalities using relative translation error $t_{rel}$ (\\%) and relative rotation error $r_{rel}$ (degree per 100m). Quantitative results are shown in Table~\\ref{tab:traj-lp}.\n\n\\noindent\\textbf{Stereo+LiDAR input:} \nThe upper part of Tab.~\\ref{tab:traj-lp} shows trajectory errors of our system with stereo+LiDAR input. Results suggest our method achieves comparable results with SuMa++, a state-of-the-art LiDAR-based semantic SLAM system~\\cite{Chen2019SuMa++}. Note however, that our method only takes very few LiDAR points (several hundred per frame) while SuMa++ uses a full LiDAR point-cloud. It is interesting to see the comparison between our stereo+LiDAR system and stereo ORB-SLAM2, which is used as our backbone system. With our LiDAR-enhanced object reconstruction and joint BA, tracking accuracy improves on most sequences, especially 03, 05, 06, 08 where adequate number of static objects are observed throughout the sequence.\nHowever, our system performs slightly worse on some sequences which contain only moving objects (01, 04) or long trajectory segments where no static objects are observed (02, 10).\nThe table also shows the effect on the  camera trajectory error when using 250 vs 50 points for object reconstruction. The results suggest that the impact of reducing the number of points on camera tracking accuracy is minimal.\n\n\\noindent\\textbf{Stereo-only input:} The lower part of Tab.~\\ref{tab:traj-lp} contains the results of our stereo-only system. It can be seen that our stereo-only system performs slightly worse than stereo ORB-SLAM2, which means dense shape reconstruction and joint BA does not help improve tracking accuracy with stereo-only input. We argue that the reason is two-fold. Firstly, 3D measurements based on stereo images are noisier than LiDAR-based measurements, giving rise to lower accuracy in object pose estimates.\nSecondly, in the stereo-only case, the surface points are obtained from the SLAM system, where the same features are repeatedly measured and not from multiple (LiDAR) measurements.\nWe also noticed that, to guarantee timings, we were performing bundle-adjustment less frequently than ORB-SLAM2. We re-ran DSP-SLAM, at a slightly reduced frame-rate (5Hz), performing BA after every key-frame (as ORB-SLAM2) and the average performance increased, matching ORB-SLAM2 at $0.72\/0.22$. \nA comparison with state-of-the-art stereo SLAM systems is also included in Tab.~\\ref{tab:traj-lp}. \n\n\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=1.00\\textwidth]{figures\/reconstruction_number_pts_single_row.png}\n\\caption{Object reconstruction results when using different number of LiDAR points per object (N=250, 50, 10). There is no noticeable difference when the number of points is reduced from 250 to 50. The reconstruction quality starts to degrade when further reducing to 10. The degraded parts are marked with a red circle.\n\\vspace{-0.1cm}}\n\\label{fig:recon-num-pts}\n\\end{figure*}\n\n\\subsection{Freiburg Cars \\& Redwood-OS Dataset}\n\n\\begin{figure*}[tbph!]\n\\centering\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/Frame-001.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/Frame-002.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/Frame-003.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/Frame-010.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/render_001.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/render_002.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/render_003.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/render_010.png}\n\\caption{Qualitative results on Freiburg Cars dataset\n\\vspace{-0.1cm}}\n\\label{fig:freiburg}\n\\end{figure*}\n\n\\begin{figure*}[htbp!]\n\\centering\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/Frame-01053.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/Frame-02484.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/Frame-09374.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/Frame-09647.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/01053-front-side.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/02484-new-front-side.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/09374-front-side.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/09647-front-side.png}\n\\caption{Qualitative results on Redwood-OS Chairs dataset\n\\vspace{-0.4cm}\n}\n\\label{fig:redwood}\n\\end{figure*}\n\nFinally, we evaluate our SLAM system with monocular input on the Freiburg Cars dataset \\cite{freiburg-cars} and Redwood-OS Chairs dataset. Both datasets consist of object-centric sequences with the camera moving around the object. Demonstrations can be seen on Fig.~\\ref{fig:freiburg} and~\\ref{fig:redwood} and in the supplementary video.\n\n\\noindent\\textbf{Experimental Setting:}\n3D Bounding boxes are estimated using PCA on the reconstructed surface points. Note that this approach cannot differentiate between the front and back side of the car. To address this issue, we initialize with two flipped hypothesis and keep the orientation that yields a smaller loss.\n\n\\noindent\\textbf{Results:} Fig.~\\ref{fig:freiburg} provides qualitative reconstruction results on 4 Freiburg Cars sequences. Our system is capable of reconstructing dense, accurate and high-quality shapes for cars solely from monocular input at 10-20 fps. Fig.~\\ref{fig:redwood} illustrates results on chairs from the Redwood-OS \\cite{choi2016redwood} dataset. Reconstruction accuracy is slightly worse than on cars as chairs have more complex shape variations. Results are promising nonetheless -- our method still produces dense meshes that capture the overall object shape from monocular RGB-only sequences, in quasi-real time. \n\n\n\n\n\n\n\\section{Conclusions}\n\nWe have presented DSP-SLAM, a new object-aware real-time SLAM system that exploits deep  shape priors for object reconstruction, produces a joint map of sparse point features for the background and dense shapes for detected objects. We show almost real-time performance on challenging real-world datasets such as KITTI (stereo and stereo+LiDAR), and even on monocular setting Freiburg cars and Redwood-OS. Our quantitative comparisons with competing approaches on camera trajectory estimation and shape\/pose reconstruction show comparable or superior performance to state of the art methods.\n\n\\vspace{-0.1cm}\n\n\\section*{Acknowledgements}\n\n\\vspace{-0.1cm}\n\nResearch presented here has been supported by the UCL Centre for Doctoral Training in Foundational AI under UKRI grant number EP\/S021566\/1. We thank Wonbong Jang and Adam Sherwood for fruitful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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The thermal component probes gas (including ionized and neutral components) \nin thermal equilibrium, while the non-thermal emission is emitted by particles such as leptons (e.g. electrons and positrons) \nor hadrons (e.g., protons and ions of heavier elements) that are not in thermal equilibrium.\nThanks to multi-wavelength observations of various astrophysical objects such as supernova\nremnants \\citep[SNRs,][]{2012Helder}, active galactic nuclei \\citep{2010Abdo},\ngamma-ray bursts \\citep{2009Abdo}, galaxy clusters \\citep{Brunetti2014,2019vanWeeren}, and galaxies \\citep{Beck2015}, observational insights\ninto the emission processes and the radiating particle\ndistributions in collisionless astrophysical plasmas is now possible.\nObserved non-thermal emission in our Universe spans a vast range of energies: from radio to TeV gamma-ray energies, and occurs in \nvarious astrophysical objects spanning many length scales: from sub-AU to Mpc scales.\nThe observed non-thermal power-law emission indicates that these plasmas are far from thermal equilibrium \nand suggests an efficient particle acceleration process giving rise to a population of cosmic rays\n(CRs) composed of electrons, protons, and heavier ions \\citep[][]{1987Blandford,Marcowith2016}. In our Galaxy, the acceleration of \nparticles near SNR shocks is thought to be the main mechanism for producing CRs up to particle energies of $10^{16}$eV \\citep[e.g.,][]{Gaisser2016}.\n\n\n\nThe interaction of plasmas on both sides of the shock wave produces electromagnetic perturbations.\nProtons and heavier ions, with speeds larger than a few times the shock speed, interact resonantly with these electromagnetic perturbations and, as a result, are transported diffusively in the vicinity of shock waves.\nBecause these magnetic perturbations are converging at the shock front, this leads to proton and ion acceleration through a process known as diffusive shock acceleration \\citep[DSA,][]{Krymskii1977,Axford1977,Blandford1978,Bell1978a,Bell1978b}. \nThe level of magnetic fluctuations determines the acceleration efficiency of particles, and vice versa, \naccelerated particles excite plasma instabilities \\citep{Bell2004} which further amplify the magnetic field near shock fronts.\n\nObservational evidences for electron acceleration at shocks are numerous. These include puzzling giant radio shocks that are observed at the outskirts of merging galaxy clusters.\nThese diffuse radio-emitting structures are often coincident with weak merger-driven shocks \\citep{2019vanWeeren}, and interpreted as synchrotron emitting regions powered by relativistic CR electrons that underwent the DSA process \\citep{Ensslin1998,Pfrommer2008,Kang2012,Pinzke2013}.\nMoreover, X-ray observations show that the non-thermal emission component at the edges of SNRs such as SN 1006 is due to accelerated CR electrons emitting synchrotron radiation \\citep{Willingale+1996}.\n\nThe particular morphology of SN 1006 in radio and X-rays has raised the question about the impact of the orientation of the upstream magnetic field on the acceleration mechanism of CR electrons. Radio polarization observations suggest a large-scale magnetic field that is aligned along the north-east to south-west direction \\citep{Reynoso2013}. Hence, the magnetic field seems to be aligned with the polar caps visible in radio \\citep{Dyer2009}, X-rays \\citep{Cassam-Chenaie2008} and TeV gamma rays \\citep{Acero2010}, suggesting preferentially quasi-parallel acceleration \\citep{Voelk2003}. Azimuthal variations of X-ray cutoff energies \\citep{Rothenflug2004, Katsuda2010} and the radius ratio of the forward shock to the tangential discontinuity \\citep{Cassam-Chenaie2008} reinforce this picture. Three-dimensional magneto-hydrodynamics (MHD) simulations support the interpretation of preferred quasi-parallel shock acceleration of electrons \\citep{Bocchino2011,Schneiter2015,Winner+2020} and protons \\citep{Winner+2020,Pais2020a,Pais2020b}.\n\n\n\n\nHowever, we still lack a detailed understanding of how electrons are accelerated to high energies at these shocks: for electrons and protons traveling with the same non-relativistic speed, the kinetic energy of electrons is much smaller than that of protons owing to their substantially smaller mass in comparison to protons. Hence, unlike protons, these electrons can not directly scatter off of the magnetic perturbations near shock waves and DSA is not possible for electrons without some process(es) that can provide and sustain pre-acceleration of incoming upstream electrons.\nThis is known as the electron injection problem in shocks \\citep{Amano+2010,Guo+2015}.\nThe importance of understanding the acceleration mechanism of electrons is also reinforced by their much higher radiative yield in comparison to that of protons at the same energy because the Thompson cross section scales inversely with particle mass squared.\n\nIn general, electron pre-acceleration proceeds in two steps: 1.\\ electrons are heated in the downstream region to energies much higher than their initial kinetic energy, and 2.\\ an acceleration phase where electrons scatter off of electromagnetic waves with a wavelength shorter than the shock width, which further increases the electron energy and enables them to participate in the DSA process.\nThe nature of these waves, which accelerate electrons in step 2, depends on the relative angle, $\\theta_{B}$, between the direction of the large-scale background magnetic field and the direction of the propagation of the shock \\citep[see, e.g.,][]{Guo+2015}.\nMoreover, recent global MHD modeling of the multi-wavelength emission from the supernova remnant SN 1006 shows that efficiency of electron acceleration at quasi-parallel ($0^{\\circ}<\\theta_{B} < 45^{\\circ}$) shocks has to be at least an order of magnitude higher in comparison to quasi-perpendicular ($45^{\\circ}<\\theta_{B} < 90^{\\circ}$) shock configurations \\citep{Winner+2020}.\n\n\nIn quasi-perpendicular shocks, electron pre-acceleration is thought to occur via shock drift acceleration (SDA) where electrons are\nreflected and accelerated via a combination of the cross shock electric potential and the shock rest frame motional electric field \\citep{Wu1984,Krauss+Wu1989}\nand\/or shock surfing acceleration (SSA) where electron acceleration occurs when they  interact with electrostatic waves at the shock front \\citep{Shimada+2000,Xu2020,KumarReville2021}.\nTheoretically, the SDA mechanism is shown to be inefficient for accelerating electrons at planar shocks in the so-called scattering-free limit \\citep{Ball+Melrose_2001}. However, in the presence of strong pitch-angle scattering, this mechanism is modified to become stochastic SDA, which could result in efficient acceleration \\citep{Katou+2019}. \nOn the other hand, particle-in-cell (PIC) simulations have shown that the SSA mechanism mediated by waves that are generated due to the Buneman instability is efficient in pre-accelerating electrons \\citep[see, e.g.,][]{Bohdan2019}. While these simulations used unrealistically low ion-to-electron mass ratios and (too) high {\\alf} speeds, it remains to be shown how these results can carry over to more realistic astrophysical conditions.\n\n\nFor quasi-parallel shocks, it is usually assumed that hot electrons generate whistler waves that lead to pitch-angle scattering and acceleration of these electrons. Efficient wave production in this scenario requires high values of {\\alf}ic Mach number, $\\mathcal{M}_{\\rm A}$. However, when tested with PIC simulations, this scenario did not result in any significant electron acceleration \\citep{Riquelme+2011,Niemiec+2012}. \nRecently, \\citet{sharp2} discovered a new instability (called the intermediate-scale instability) that drives comoving ion-cyclotron waves unstable in the vicinity of the shock front. This presents {\\it a new mechanism for generating waves} that can scatter electrons and potentially enables an efficient DSA mechanism for electrons. Unlike the whistler-mediated case, this mechanism requires low values of $\\mathcal{M}_{\\rm A}$ which is a condition for the new instability to operate. \n\nIn this paper we test this mechanism using 1D3V PIC simulations (i.e., one spatial and three velocity-space dimensions) of parallel electron-ion shocks and show that it indeed leads to very efficient electron acceleration.\nThe paper is organized as follows.\nIn Section~\\ref{sec::setup}, we present the setup for our simulations, and compute the linear growth rates for the expected unstable wavemodes at the shock front region.\nThe growth of these wavemodes is responsible for the formation of the shock and for creating non-thermal populations of electrons and ions.\nIn Section~\\ref{sec:Bamp}, we present the evolution of density and magnetic field amplification  in our simulations.\nWe study the impact of the intermediate-scale instability driven wavemodes on the acceleration of electrons in the downstream region of the shock in Section~\\ref{sec:nth1}.\nIn Section~\\ref{sec:thermal}, we discuss the heating of the shocked plasmas in the downstream and the shock front region. We also compare this heating to analytical heating models at shocks.\nThe fraction of energy that is channeled into non-thermal  electrons and ions is quantified and its evolution is presented in Section~\\ref{sec:nothermal}. We discuss our findings and their implications in Section~\\ref{sec:dis}, and conclude the paper in Section~\\ref{sec:concl}.\nThroughout this paper, we assume the SI system of units.\n\n\n\n\\section{Non-relativistic shock simulations}\n\\label{sec::setup}\n\n\\begin{figure}\n\\includegraphics[width=8.8cm]{Plots\/Fig01.pdf}\n\\caption{The formation of a strong parallel shock in the contact discontinuity rest frame (simulation frame).\nLeft-hand side: the reflection of an incident electron-ion plasma drifting to the left side of the simulation with speed $\\varv_u$ leads to the formation of the shock.\nRight-hand side: after shock formation, the shock speed relative to the upstream plasma is $\\varv_{\\rm sh} \\sim 4 \\varv_u\/3$ (assuming a strong shock and adopting the Rankine-Hugoniot jump conditions), and the density jump at the shock front is on average $\\sim 4 n_0$, where $n_0$ is the number density of the far upstream plasma.\nThat is, at the shock front the {\\alf} speed of ions $\\varv_{\\rm A}$ is $0.5$ of that in the far upstream, which means that the {\\alf}ic Mach number, $\\mathcal{M}_{\\rm A}$, in this region is twice of that in the far upstream.\n\\label{fig:ShockSketch}}\n\\end{figure}\n\nHere we discuss the setup for our shock simulations and compute the scales and the growth rates for the expected unstable linear wavemodes at the shock front regions.\n\n\\subsection{Simulation setup}\n\nWe perform 1D3V particle-in-cell (PIC) simulations using the SHARP code \\citep{sharp,resolution-paper,sharp2}, where the shock is formed by reflecting the upstream electron-ion plasma (initially moving towards the left, i.e., negative $\\bs{\\hat{\\vec{x}}}$ direction) at $x=0$.\nThat is, the shock is formed by the interaction of two exact replica of the same plasma moving in opposite directions, and the simulations are performed in the rest-frame of the contact discontinuity. A sketch for the initial configuration and the resulting shock formation is shown in Figure~\\ref{fig:ShockSketch}.\n\nIn all simulations, the upstream plasma skin-depth is resolved by 10 computation cells, and the CFL stability condition on the computational time step, $\\Delta t$, is such that $c \\Delta t = 0.45   ~ \\Delta x$, where $c$ is the speed of light in vacuum.\nWe use 200 particles per species in each computational cell.\nThe boundary of the computational domain expands with time on the right-hand side to save computational cost while containing all energetic particles in the precursor of the shock.\nThe upstream  plasma is initially moving with velocity $\\varv_u = -0.1 c$, and both electrons (with mass $m_e$) and ions (with mass $m_i$) have the same physical temperature  $T_e = T_p = 4 \\times 10^{-8} m_i c^2\/k_{\\rm B}$, where $k_{\\rm B}$ is Boltzmann's constant.\nInitially, both species have the same uniform number density $n_i=n_e=n_0$. \n\n\nFor such shocks, the expected shock speed (in the rest-frame of upstream plasma) is \n$\\varv_{\\rm sh} = 4 |\\varv_u| \/3 \\sim 0.133 c$.\nTherefore, the sonic Mach number $ \\mathcal{M}_s = \\varv_{\\rm sh} \/ \\varv_s \\sim \n365$, where the sonic speed is $\\varv_s = \\sqrt{\\Gamma_{\\rm ad} k_{\\rm B} (T_e+T_p)\/m_i \n}$, and  $\\Gamma_{\\rm ad} = 5\/3$ is the adiabatic index. That is, all shock simulations presented here have high sonic Mach number.\nThe initial (large-scale) background magnetic field $\\vec{ B}_0 = B_0  \\bs{\\hat{\\vec{x}}}$ is such that the {\\alf}ic Mach number $\\mathcal{M}_{\\rm A} = \\varv_{\\rm sh}\/\\varv_{\\rm A}$, where $\\varv_{\\rm A} = B_0\/\\sqrt{ \\mu_0 n_i m_i}$ is the {\\alf} speed of ions.\n\n\nIt is important to note here that collisionless shocks found in the intracluster medium have $\\mathcal{M}_s \\sim 1$ to 3 \\citep{Ryu+2003, Pfrommer+2006,Vazza+2009,Schaal+2016}, and for such low values of the sonic Mach number, the intermediate-scale growth rates are much larger in comparison to that at the gyro-scale \\citep{sharp2}. This implies a stronger impact of the intermediate-scale unstable modes on the acceleration of electrons in the intracluster medium.\nHowever, we leave a demonstration of this point in simulations to future works.\n\n\n\\subsection{Unstable linear wavemodes at the shock front}\n\\label{sec::theory}\n\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{Plots\/Fig02.pdf}\n\\caption{Top: growth rates ($\\Gamma$) normalized to the non-relativistic ion gyro-frequency, $\\Omega_i$.\nBottom: the phase velocity ($\\varv_{\\rm ph}$) of the fastest growing parallel (unstable) electromagnetic wavemodes created by the penetration of upstream cold plasma drifting  into the denser plasma at the shock front (in the simulation frame).\nThe solutions are shown for $\\mathcal{M}_{\\rm A} = 21.3$, with a realistic ion-to-electron mass ratio, $m_r=m_i\/m_e$ (blue curves).\nIn the case of $\\mathcal{M}_{\\rm A} = 5.3$, solutions are shown for a realistic value of $m_r$ (red curves) and a reduced $m_r$ (black curves).\nOnly the case with $\\mathcal{M}_{\\rm A}=5.3$ and a realistic mass ratio is expected to excite the intermediate-scale instability~\\citep{sharp2} and thus the small scale ion-cyclotron wave modes that are comoving with the upstream plasma with velocity $\\varv_u = -0.1 c$ are amplified, i.e., come from solutions of $D^{+}=0$ (see Equation~\\ref{eq:disp}).\nSmall scale unstable wavemodes in the other (blue and black) cases are whistler waves, i.e., they derive from solutions of $D^{-}=0$ (see Equation~\\ref{eq:disp}).\n\\label{fig:GrowthRate}}\n\\end{figure}\n\n\nTo study the impact of the intermediate-scale instability on the heating and the acceleration of electrons at parallel non-relativistic shocks, we present simulations that differ in their ability to excite the intermediate-scale instability, which manifests itself by exciting ion-cyclotron waves that are comoving with the upstream plasma~\\citep{sharp2}.\nThe upstream drifting plasma drives wavemodes unstable at the shock front\\footnote{In the linear dispersion calculation we present here, the background number density is assumed to be uniform.\nNumber density non-uniformity could change the growth rates if the non-uniformity scale is close to the wavelengths  of the unstable modes.\nHowever studying this theoretically, even in the linear regime, is tedious \\citep[see, e.g.,][]{sim_inho_18, th_inho_20}.}. \nAssuming gyrotropic momentum distributions for various species, the \ndispersion relation, in the simulation frame, for parallel (w.r.t.\\ the background magnetic field $B_0$) propagating electromagnetic wavemodes with wavenumber $k$ and complex frequency \n$\\omega$ can be written as~\\citep{Schlickeiser+2002}\n\\begin{eqnarray}\nD^{\\pm}\n&=& 1- \\frac{k^2c^2}{\\omega ^2} \n\\nonumber \\\\\n&&+\n\\sum_{s=1}^{4}\n\\frac{ \\omega_s^2 }{ \\gamma_s  \\omega ^2 }\n\\left[\n\\frac{ \\omega -k \\varv_{\\parallel,s} }\n{  k \\varv_{\\parallel,s}  - \\omega  \\pm \\Omega_s}\n-\n\\frac{ \\varv_{\\perp,s}^2  \\left(k^2c^2-\\omega ^2\\right) \/c^2 }\n{2 \\left(k \\varv_{\\parallel,s}  -\\omega \\pm \\Omega_s \\right)^2}\n\\right] ~\\hspace{-0.16cm} .\n\\label{eq:disp}\n\\end{eqnarray} \nHere, $s=\\{1,2\\}$ denote the shocked electron and ion plasmas at the shock front, respectively, and $s=\\{3,4\\}$ denote the incoming (upstream) cold electron and ion plasmas, respectively. The drift speeds are $\\varv_{\\parallel,s} = \\{ - \\varv_{u}\/3, - \\varv_{u}\/3, \\varv_{u}, \\varv_{u} \\}$, \nand the relativistic gyro-frequencies $\\Omega_s = \\{- m_r \\Omega_0, \n\\Omega_0, -m_r \\Omega_0, \\Omega_0\\}\/\\gamma_s$, where \n$\\Omega_0 = e B_0\/m_i $ is the non-relativistic ion gyro-frequency, and \nthe Lorentz factor is $\\gamma_s = (1 - \\varv_{\\parallel,s}^2 - \\varv_{\\perp,s}^2)^{-1\/2}$.\n\nSolutions to $D^{+}=0$ ($D^{-}=0$) are solutions for the left (right) handed polarized electromagnetic wavemodes.  When solving for the maximum growth rates at some $k$, we solve the dispersion relation with both signs, and find the fastest growing mode regardless of its polarization.\nFor various species $s$, the perpendicular speeds are $\\varv_{\\perp,s} = \\varv_s\/\\sqrt{2} ~ \\forall s $,\nthe plasma frequencies are $\\omega_s = \\{ \\sqrt{3 m_r} ~\\omega_i, \n\\sqrt{3}~ \\omega_i, \\sqrt{m_r} ~\\omega_i, \\omega_i \\}$, where $\\omega^2_i= e^2 n_i\/(m_i \\epsilon_0)$ is the square of the ion plasma frequency in the far upstream of the shock, $e$ is the elementary charge, and $\\epsilon_0$ is the permittivity of free space. \n\nWe present a simulation that can excite the intermediate-scale instability at the shock front region. It has an upstream {\\alf}ic Mach number $\\mathcal{M}_{\\rm A} = 5.3$ and uses a realistic mass ratio $m_r = m_i\/m_e=1836.$\nThat is, the shock front {\\alf}ic Mach number \n\\begin{eqnarray}\n\\mathcal{M}^f_{\\rm A} \\sim 10.6 < \\sqrt{m_r}\/2 \\sim  21.4 .\n\\end{eqnarray}\nThis condition represents the necessary criteria for driving such comoving ion-cyclotron wave modes \\citep{sharp2}.\nThe solution of the dispersion relation (Equation~\\ref{eq:disp}) for \nthis case is shown in Figure~\\ref{fig:GrowthRate} (red curves).\nThe lower panel shows that the phase velocity of the unstable wave modes (at $kc\/\\omega_i \\sim 5$) are comoving with the upstream drifting plasmas ($\\varv_{\\rm ph} \\sim -0.1 c$) for the simulation with $\\mathcal{M}_{\\rm A} = 5.3$ and $m_r=1836$ (red curve).\nTo demonstrate the importance of this instability in electron-heating, we present two more simulations where the condition of this instability is violated.\n\n\\begin{deluxetable}{ cccccc }\n\\tablewidth{8.7cm}\n\\tabletypesize{\\footnotesize}\n\\tablecolumns{5} \n\\tablecaption{\nParameters of our electron-ion parallel shock simulations \\vspace{-0.2cm}\n\\label{table:Sims}\n}\n\\tablehead{ Name & $\\varv_u\/c$\\tablenotemark{a} &  $\\mathcal{M}_{\\rm A}$\\tablenotemark{b}\n&   $\\mathcal{M}_s$\\tablenotemark{c}\n&    $m_i\/m_e$ \n& Condition\\tablenotemark{d} \n}\n\\startdata\n\\rule{0pt}{12pt}  Ma5Mr1836  & -0.1 & 5.3 & 365 &  1836 & \\checkmark \n\\\\\n\\rule{0pt}{12pt}  Ma5Mr100\t & -0.1 & 5.3 & 365 &  100\t  & $\\times$ \n\\\\\n\\rule{0pt}{12pt}  Ma21Mr1836 & -0.1 & 21.3 & 365 & 1836 & $\\times$\n\\enddata\n\\tablenotetext{a}{Upstream plasma velocity in the contact-discontinuity rest frame.}\n\\tablenotetext{b}{Upstream {\\alf}ic Mach number.}\n\\tablenotetext{c}{Upstream sonic Mach number.}\n\\tablenotetext{d}{This shows whether the condition ($\\varv_{\\rm sh}\/\\varv_{\\rm A} < \\sqrt{m_r}\/2 $) for exciting the intermediate-scale instability at the shock front is satisfied.}\n\\end{deluxetable}\n\nThe first simulation has $\\mathcal{M}_{\\rm A} = 21.3$, with a realistic mass ratio, $m_r=1836$.\nIn this case, $\\mathcal{M}_{\\rm A}^f \\sim 40.6 > \\sqrt{m_r}\/2 \\sim  21.4$. \nIndeed, the solutions of the dispersion relation for this case (blue curves in Figure~\\ref{fig:GrowthRate}) show that sub ion-skin-depth unstable whistler wavemodes are  not comoving with the upstream plasma and thus can be easily quenched.\nFor this simulation, we increase $\\mathcal{M}_{\\rm A}$ by decreasing $B_0$, i.e., by lowering $\\varv_{\\rm A}$. \n\n\n\n\n\nThe second simulation has $\\mathcal{M}_{\\rm A} = 5.3$, but with a reduced mass ratio, $m_r=100$. In this case, $\\mathcal{M}_{\\rm A}^f \\sim 10.3 > \\sqrt{m_r}\/2 =  5 $.\nAgain, the solutions of the dispersion relation for this case (black curves in Figure~\\ref{fig:GrowthRate}) show that the sub ion-skin-depth unstable whistler wavemodes are not comoving with the upstream plasma and thus can also be easily quenched.\nFor this simulation, we increase $T_e=T_i$ such that $T_i\/m_i$ and thus the sonic speed $\\varv_s$ is unchanged. \nConsequently, the sonic Mach number $\\mathcal{M}_s$ is also unchanged.\n\n\n\n\nA summary of parameters of the three simulation that we present here is given in Table~\\ref{table:Sims}.\nIt is important to note that larger values of the number density at the shock front, i.e., $n > 4n_0$, increases $\\mathcal{M}_{\\rm A}^f$ in simulations. Hence, we have added a safety margin to our chosen values of $\\mathcal{M}_{\\rm A}^f$ that can in principle account for on additional factor of four enhancement in density over its MHD prediction without changing our conclusion on the condition of whether the intermediate-scale instability is excited in our simulations, as given in Table~\\ref{table:Sims}.\n\n\n\n\\section{Shock formation and magnetic field amplification; self-driven scattering centers}\n\\label{sec:Bamp}\n\n\\begin{figure*}\n\\includegraphics[width=18.5cm]{Plots\/Fig03.png}\n\\caption{Left: evolution of the number density normalized to the far upstream number density ($n_0$).\nRight: evolution of $B_y$ normalized to the far upstream parallel magnetic field ($B_0$). \nThese are the evolution in the simulation Ma5Mr1836 (top),  Ma5Mr100 (middle), and Ma20Mr1836 (bottom). \n\\label{fig:XT-evol}\n}\n\\end{figure*}\n\n\n\nThe interaction of the drifting and reflected plasma results in instabilities on scales both longer and shorter than the ion-skin depth.\nSuch unstable modes slow down the reflected plasma and thus create an over-density ($n> 2 n_0$) behind the shock transition region\\footnote{Here, $n$ is the number density of both ions and electrons together.}, where $n_0$ is the far upstream number density.\nAfter the formation of the shock, the interaction of the incoming upstream plasma with the denser plasma behind the shock also forms an unstable plasma configuration.\nThis leads to particle scattering and a constant heating and acceleration of the upstream plasma as it is swept over by the shock.\nThe left-hand side of Figure~\\ref{fig:XT-evol} shows the time evolution of the number density, $n$, normalized to $n_0$, in all the simulations.\nThe right-hand side shows the time evolution of $B_y$ normalized to the initial background (parallel) magnetic fields $B_0$ (the $B_z$ evolution is very similar to that of $B_y$, albeit shifted slightly in space).\nFigure~\\ref{fig:XT-evol} shows the formation of the shock via the excitation of unstable magnetic field wavemodes in various simulations.\nAfter the formation of the shock, wavemodes on small and large scales are unstable as seen in Figure~\\ref{fig:GrowthRate}. These modes are continuously driven at the shock front region as the shock swipes through the upstream plasma.\n\nIn Figure~\\ref{fig:XT-evol}, the white lines that separate the shock front and the upstream regions indicates the location of the shock in various simulations. While the exact position of the white line is determined visually, we note that its slope is approximately given by \n\\begin{eqnarray} \n\\frac{ \\Delta t \\Omega_i }{  \\Delta x \\omega_i\/c }  =\n\\frac{ \\Omega_i\/\\omega_i }{ \\varv_{\\rm sh}  \/c }   \n= \\frac{ 1}{\\mathcal{M}_{\\rm A} }.\n\\end{eqnarray}\nThat is, the slope for the Ma20Mr1836 simulation is smaller by a factor of $\\sim 4$ in comparison to the other simulations, and thus the range in  $x$-direction is larger by the same factor.\nWe define the shock front (transition) to be a region that is 200 $c\/\\omega_i$ wide behind the location of the shock, i.e., it is the region between the two white lines in various panels of Figure~\\ref{fig:XT-evol}.\n\nAs particles (especially ions) scatter back and forth across the shock front (transition) region, they are accelerated to higher energies, and thus some particles escape from this region toward the upstream plasma. \nThese excites unstable wavemodes, which  in turn scatter most of these counter-streaming energetic particles back toward the shock front region \\citep{Bell1978a}.\nThis process generates the so called shock precursor region ahead of the shock in the upstream plasma as seen in the right-hand panels of Figure~\\ref{fig:XT-evol}.\nIn the next sections, we closely look at the nature of these driven modes and their impact on particle acceleration and heating for both ions and electrons.\n\nThe overall density jump between the far upstream and far downstream region ($n\/n_0$) is notably larger than expected from the MHD jump condition that predicts $n\/n_0= 4$ (for a strong parallel shock). Instead, our simulations with $\\mathcal{M}_{\\rm A}=5.3$ show an overall density jump of $n\/n_0 \\sim 6$.\nThis has already been seen in a number of PIC simulation before as summarized by \\citet{Bret2020}.\nIn hybrid-PIC simulations, where the jump in the density is not constrained by using a very stiff electron equation of state, \n\\citet{Colby+2019} shows that for a simulation with $\\mathcal{M}_{\\rm A}=20$ the density jump grows to $n\/n_0  \\gtrsim 5.5$.\n\n\n\n\\section{Impact of intermediate-scale instability driven wavemodes on acceleration}\n\\label{sec:nth1}\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{Plots\/Fig04.pdf}\n\\caption{Top: downstream electron (solid) and ion (dashed) momentum spectra at $t ~\\Omega_i= 260$ from Ma5Mr1836 (red),  Ma5Mr100 (black),  Ma20Mr1836 (blue) simulations.\nThe downstream is defined to be at a distance that is larger than 200 $c\/\\omega_i$ from the shock front; see Figure~\\ref{fig:XT-evol}.\nBottom: downstream perpendicular magnetic energy in Fourier space at $t ~\\Omega_i= 260$ .\nThis shows that the co-moving (traveling towards the downstream) unstable waves that are driven at the shock front in the Ma5Mr1836 simulation generate a much higher level of small-scale magnetic fields. These increased magnetic fluctuations imply a much stronger scattering and thus, more efficient acceleration of electrons in comparison to the other simulations.\n\\label{fig:spectra300}}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18.5cm]{Plots\/Fig05.png}\n\\caption{Time evolution of the average perpendicular magnetic field power, $\\langle \\delta B^k_{\\perp} \\rangle$, in the downstream (left) and shock front (right) regions.\nThe dashed lines indicate the small-scale ($k c\/\\omega_i > 2$) time evolution while solid lines show the large-scale ($k c\/\\omega_i< 2$) time evolution. \nDifferent simulations are indicated with different colors; Ma5Mr1836 with red,  Ma5Mr100 with black, and   Ma20Mr1836 with blue.\n\\label{fig:KLS}\n}\n\\end{figure*}\n\nThe importance of the intermediate-scale instability on the particle acceleration process can be quantified by comparing simulations which differ in their ability to excite comoving ion-cyclotron modes via the intermediate-scale instability.\nAs seen in Figure~\\ref{fig:GrowthRate}, all simulations are predicted to have unstable wavemodes on scales smaller than the ion skin depth at the shock front (transition) region.\nHowever, only in the simulation Ma5Mr1836, these  modes are comoving with the incoming flow (red, see the bottom panel of Figure~\\ref{fig:GrowthRate}), and hence their mode power can be transferred to the downstream region where they can scatter electrons.\n\nThe top panel of Figure~\\ref{fig:spectra300} shows the electron and proton spectra as a function of dimensionless velocity $u_s\/c$. To obtain these spectra, we first transform to the frame in which the plasma is at rest before we compute the momentum spectra from which we estimate the plasma temperature as laid out in Appendix~\\ref{app:temp}. \nAs shown in the top panel of Figure~\\ref{fig:spectra300}, the simulation Ma5Mr1836 (where the intermediate-scale instability can grow) has a much higher efficiency in converting the incoming flow kinetic energy into non-thermal energy in electrons in the downstream region. In fact, it is more efficient (by about 2 orders of magnitude) in accelerating electrons in comparison to the simulation with a higher $\\mathcal{M}_{\\rm A}$ (blue), suggesting that there is a much more efficient process at work that enables electrons to be efficiently accelerated in our low-$\\mathcal{M}_{\\rm A}$ model (red).\n\nMoreover, the Ma5Mr100 simulation (black) has the same $\\mathcal{M}_{\\rm A}$ and $\\mathcal{M}_{\\rm s}$ as the simulation shown in red but uses a lower mass ratio: $m_r=100$.\nAs shown in the top panel of Figure~\\ref{fig:spectra300}, this results in a much smaller electron acceleration efficiency in this simulation.\nAdditionally, the heating of electrons in the downstream region of the simulation with a reduced mass ratio (black) is much smaller compared to the simulations with a realistic mass ratio (red and blue)\\footnote{In Appendix~\\ref{app:temp}, we show that the normalized temperature, $k_{\\rm B} T_s\/m_s c^2$, of the thermal part of the plasma population of species $s$ is proportional to the value of $u\/c$ where the $u^4 f(u)$ is maximized, see Equation~\\eqref{Eq:TR}.}.\nThat is, the top panel of Figure~\\ref{fig:spectra300} demonstrates that using a lower mass ratio results not only in an artificially suppressed electron acceleration efficiency but also leads to an erroneous electron heating in the downstream region.\n\nAs we will now argue, the differences in the acceleration efficiency of electrons as well as for particle heating at shocks is a direct consequence of the differences in the nature of the unstable small-scale modes.\nThe bottom panel of Figure~\\ref{fig:spectra300} shows that the small-scale power of the perpendicular magnetic field in the downstream region is much larger in the simulation model Ma5Mr1836 in comparison to the simulations Ma5Mr100 (black) and Ma21Mr1836 (blue).\nIn the latter two cases, the small-scale unstable modes are not comoving with the incoming flow, and thus can be easily quenched, which leads to a much smaller amount of small-scale perpendicular magnetic power in the downstream regions.\n\nTo demonstrate the growth of these small scale wavemodes, in Figure~\\ref{fig:KLS} we plot the time evolution of the average perpendicular magnetic field for small (dashed) and large (solid) scale wave modes in various simulations. To this end, we take the power of the perpendicular magnetic field as shown in the bottom panel of Figure~\\ref{fig:spectra300} and compute the average for larger scales, i.e., for $kc\/\\omega_i<2$ and for small scales, i.e., for $2<kc\/\\omega_i<20$.\nThe top panel of Figure~\\ref{fig:GrowthRate} predicts that in the linear regime, both small and large scale magnetic perturbations grow with similar rates in all simulations at the shock transition region. \nThe right-hand panel of Figure~\\ref{fig:KLS} demonstrates that this prediction is in an excellent agreement with the growth of magnetic perturbations in various simulations at the shock transition region. \n\n\nOn the other hand, the left-hand panel of Figure~\\ref{fig:KLS} demonstrates the crucial importance of the difference in the  nature of the grown small scale wavemodes.\nAs shown in the bottom panel of Figure~\\ref{fig:GrowthRate}, only for the Ma5Mr1836 simulation (red), the small-scale wavemodes are comoving with the incoming flow. This means that these modes are not easily quenched in the shock transition region, which is demonstrated by the ability of maintaining a high level of small-scale fluctuations as a result of continued growth of the instability, which explains the much larger efficiency of electron acceleration in this simulations, see Figure~\\ref{fig:spectra300}. \nMoreover, much of this small-scale electro-magnetic power is transferred to the downstream region as can be inferred from the left-hand panel of Figure~\\ref{fig:KLS}.\nIn the case of the simulations without the intermediate-scale instability (Ma5Mr100 and Ma21Mr1836), much of the small-scale power is quenched and thus a much smaller fraction of this is transferred to the downstream region of the shock.\n\n\n\n\nIn the next sections, we show that the use of a lower mass ratio also leads to erroneous heating and acceleration efficiency  in the shock front (transition) region, and that holds true throughout the evolution of the simulations (as seen in Figures~\\ref{fig:Temp} and \\ref{fig:Eff}).\n\n\n\\section{Heating of the shocked plasmas}\n\\label{sec:thermal}\n\n\\begin{figure*}\n\\includegraphics[width=18.5cm]{Plots\/Fig06.pdf}\n\\caption{Top: shown is the time evolution of downstream (left) and shock front (right) temperatures of ions (dashed) and electrons (solid) in various simulations. Dotted red (black) line in the top left-hand panel indicate the predicted MHD ion temperature for $m_r = 1836 ~ (100)$ as given in Equation~\\eqref{eq:kTiV}. \nBottom: evolution of the electron-to-ion temperature ratio of the plasma in the downstream (left) and shock front (right) regions in various simulations.\n\\label{fig:Temp}\n}\n\\end{figure*}\n\n\nAll simulations presented here only have a parallel background magnetic field and are characterized by the same sonic Mach number $\\mathcal{M}_s \\sim 365$.\nAfter shock formation, electrons and ions in the downstream and shock front regions are heated. Modeling the saturation temperatures for electron and ions  in such collisionless shocks is very important for understanding the observations of SNRs~\\citep{Vink+2015}, clusters of galaxies~\\citep{Russell+2012}, and the warm hot intergalactic medium which contains a significant fraction of the baryon in the universe~\\citep{Bykov+2008}.\n\n\\subsection{Models for saturation temperatures}\n\nThe MHD model for strong parallel shocks predicts that all the upstream drifting kinetic energy is converted into thermal energy in the post-shock rest frame.\nThe expected temperature can be computed using the Rankine-Hugoniot jump conditions for a strong, parallel MHD shock in the regime of high sonic Mach numbers, $\\mathcal{M}_s\\gg1$ \\citep[e.g.,][]{boyd}, so that  the temperature of the shocked regimes (downstream and shock front regimes) is given by\n\\begin{align}\nk_{\\rm B} T =  \\frac{3}{16} \\mu m_i \\varv_\\mathrm{sh}^2\n  = \\frac{1}{3} \\mu m_i \\varv_u^2 = \\frac{1}{600} m_r m_e c^2,\n  \\label{eq:kTi}\n\\end{align}\nwhere $\\varv_\\mathrm{sh}=4\\varv_u \/3$ is the upstream velocity in the shock frame, $\\varv_u=0.1c$ is our adopted value for the upstream velocity in the simulation frame, we adopted the adiabatic index $\\Gamma_\\mathrm{ad}=5\/3$ appropriate for our case, and $\\mu = 1\/2$ is the mean molecular weight of an electron-proton fluid.\nThe MHD assumption implies that the ion and electron temperatures equilibrate on negligible timescales so that $T_e=T_i$ in this model.\nThat is, we expect the temperature in the shocked regimes to depend on the adopted mass ratio as follows:\n\\begin{equation}\nk_{\\rm B}  T_{i} =  k_{\\rm B}  T_{e} =  \\left\\{ \n  \\begin{aligned}\n    &3.06~~~ m_e c^2\n    &&\\mbox{  for  } m_r = 1836,\\\\\n    &0.166~ m_e c^2\n    &&\\mbox{  for  } m_r = 100.\\\\\n  \\end{aligned}\n  \\right.\n    \\label{eq:kTiV}\n\\end{equation}\nThese values are shown with dotted red ($m_r=1836$) and black ($m_r=100$) lines in the top left-hand panel of Figure~\\ref{fig:Temp}.\n\nBecause these shocks are collisionless, the MHD assumption of strong collisional relaxation is not fulfilled and the electrons may not equilibrate with the ions.\nTherefore, another theoretical model for saturation temperatures can be obtained by assuming that thermalization of the initial flow kinetic energy occurs separately for ions and electron \\citep{Vink+2015}.\nThis implies that the electron-to-ion temperature $T_e\/T_i = m_e\/m_i$, and for our simulations this predicts $k_{\\rm B} T_e  = 0.0016 ~ m_e c^2$ (where we used Equation~\\ref{eq:kTi}). \n\n\n\\subsection{Saturation temperatures in simulations}\n\nIn Figure~\\ref{fig:Temp} we show the evolution of the temperature in the downstream (left) and shock front (right) regions in various simulations.\nThe top panel of Figure~\\ref{fig:Temp} shows that a significant fraction of the incoming flow kinetic energy is converted to thermal heating. However, unlike the prediction of MHD shocks, the electron and ion temperatures differ.\nIn all simulations, the ion temperatures in the downstream and shock front regions saturate slightly below the MHD prediction because a fraction of the incoming flow kinetic energy is channeled into magnetic energy (as seen in Section~\\ref{sec:Bamp}) and non-thermal energy of cosmic ray ions and electrons (as we show in Section~\\ref{sec:nothermal}).\n\nFor ions, we find $k_{\\rm B}T_{i}\\sim2.5~m_e c^2$ (for $m_r = 1836$, which is only 22.4\\% lower than the MHD prediction) and $k_{\\rm B}T_{i}\\sim0.1~m_e c^2$ (for $m_r = 100$, which is 66\\% lower than the MHD prediction).\nOn the other hand, for electrons, saturation temperatures have a larger mismatch with the MHD predicted temperatures.\n \nWhile the mismatch of the post-shock temperature in the case of a realistic mass ratio can be approximately accounted for by efficient ion acceleration and energy retained in electromagnetic fluctuations, the offset in the case of a mass ratio of $m_r = 100$ is puzzling and cannot be explained by energy in non-thermal components, which only add up to 5\\%. We conjecture that the missing plasma instabilities as a result of the artificially small mass ratio precludes exciting sufficient electromagnetic modes that provide the necessary scattering centers for dissipating all the kinetic energy of ions.\n\nFigure~\\ref{fig:Temp} also shows that assuming an independent evolution for ions and electrons is not correct since the initial flow kinetic energy (mainly from the ions) excites magnetic field wavemodes which both heat and accelerate electrons to an energy density that, in total,  is much larger than that of the initial flow kinetic energy for the electrons.\nThe top panel of Figure~\\ref{fig:Temp} shows that the saturation temperature for ions (electrons) is lower (larger) than predicted from such models, and the bottom panels show that the prediction for the electron-to-ions temperature ratio of such models is inconsistent with all simulations.\n\nIn agreement with the prediction from MHD shocks, simulations with a realistic ion-to-electron mass ratio have saturation temperatures of ions and electrons that are independent of the {\\alf}ic Mach number $\\mathcal{M}_{\\rm A}$. This is demonstrated by the excellent agreement between blue and red (dashed and solid) curves in Figure~\\ref{fig:Temp} albeit with $T_e < T_i$.\nThe saturation temperature of electrons is the same in the downstream and shock front regions, which is also approximately true for ions. This means that all heating occurs in the shock front region and {\\it no further} heating occurs in the downstream region for ions or electrons.\nThe lower panels of Figure~\\ref{fig:Temp} show that, at saturation, $T_e\/T_i = 0.4$ in the downstream regime for the two simulations that use a realistic mass ratio and $T_e\/T_i = 1.4$ for the simulation with $m_r=100$.\n\nIn comparison to simulations with a realistic mass ratio, the simulation that uses a low mass ratio of $m_r=100$ results in {\\it incorrect} saturation temperatures for ions (by factor of $\\sim 25$) and electrons (by factor of $\\sim 7.4$) at both, the downstream and shock front regions as shown in the top panel of Figure~\\ref{fig:Temp}. \nThis low saturation temperature for electrons leads to strong suppression in electron acceleration efficiency as we show in Section~\\ref{sec:nothermal}.\nMoreover, the use of a lower mass ratio results in a wrong electron-to-ion temperature ratio at saturation, i.e., $T_e>T_i$ as can be seen from the bottom panel of Figure~\\ref{fig:Temp}.\nSaturation temperatures where $T_e < T_i$ are observed for example in Balmer-dominated shocks~\\citep{van-Adelsberg+2008}.\nThat is, $T_e\/T_i>1$ found in the simulation with low ion-to-electron mass ratio disagrees with observed saturation temperature ratios.\nSurprisingly, for the simulation with $m_r=100$, the ion and electron temperatures continue to increase from the shock front region to the downstream region. This indicate that further heating occurs in the downstream region of this simulation in disagreement with the results from the simulations with a realistic mass ratio.\n\n\n\\section{Formation of non-thermal particles}\n\\label{sec:nothermal}\n\n\\begin{figure*}\n\\includegraphics[width=18.5cm]{Plots\/Fig07.pdf}\n\\caption{Top: the time evolution of the acceleration efficiency (as defined in Equation~\\ref{Eq:Eff-sh}) in the downstream (left) and shock front (right) region for ions (dashed) and electrons (solid) in various simulations.\nBottom: the time evolution of the fraction of non-thermal energy of electrons relative to that of ions ($K_{\\rm ei}$) in the downstream (left) and shock front (right) regions in various simulations.\n\\label{fig:Eff}\n}\n\\end{figure*}\n\n\n\nAs shown above, the upstream kinetic energy is channeled into magnetic energy (Section~\\ref{sec:Bamp}), non-thermal energy (Section~\\ref{sec:nth1}) and thermal energy (Section~\\ref{sec:thermal}) in the downstream and shock front regions of the system. \nHere we quantify the amount of energy carried by non-thermal ions and electrons in these regions in Figure~\\ref{fig:Eff}.\nIn Appendix~\\ref{app:temp}, we show that the fraction of particles with non-thermal energy can be quantified in various ways, and here we focus on how much energy density is carried by these particles when compared to the upstream kinetic energy density, i.e., using Equation~\\eqref{Eq:Eff-sh}.\nWe define non-thermal particles to be particles with $u>5u_m$~\\citep{Caprioli2014a,Xu2020}, where $u=\\gamma \\varv$ is the spatial-part of the 4-velocity, $\\gamma$ is the Lorenz factor, and $u_m$ is defined as the value of $u$ where $u^4 f(u)$ is maximized.\nAs an example, particles to the right of the blue vertical lines in Figure~\\ref{fig:Thermal} are considered to form the non-thermal part of the momentum distribution. \nThis quantification is done for different regions as a function of time and was used to compute the time evolution of the efficiency shown in the top panel of Figure~\\ref{fig:Eff}.\nThe top panel shows the time evolution of the efficiency $\\epsilon_{\\rm sh}$, i.e., the percentage of upstream kinetic energy density that is channeled into non-thermal electrons and ions in the downstream (right) and shock front regions (left).\nBy contrast, the bottom panel shows the time evolution of the ratio of non-thermal energy of electrons to that of ions, $K_{\\rm ei}$, in various simulations also in the downstream (left) and shock front (right) regions.\n\nThe non-thermal energy density of ions (dashed curves) is about 20-30\\% of the upstream kinetic energy density, and this result is roughly independent of the ion-to-electron mass ratio $m_r$ at the shock front region.\nThe simulation with a higher $\\mathcal{M}_{\\rm A}$ has a slightly higher efficiency in producing non-thermal ions at the shock front (transition) region.\nMoreover, in the downstream region, it contains a higher fraction of non-thermal ions indicating that this simulation is more efficient in scattering ions that are accelerated in the shock front (transition) region back to the downstream region.\n\nOn the other hand, the fraction that is channeled into non-thermal electrons is strongly dependent on both, $\\mathcal{M}_{\\rm A}$ and $m_r$.\nThe top panel of Figure~\\ref{fig:Eff} shows that, at the shock front, the simulation with $\\mathcal{M}_{\\rm A}=5.2$ and a realistic mass ratio efficiently accelerates electrons, leading to 0.05--0.1\\% of initial flow kinetic energy converted into non-thermal electrons, while in the simulation with a higher $\\mathcal{M}_{\\rm A}$ and a realistic mass ratio, a smaller fraction (0.01--0.05\\%) of the upstream kinetic energy is channeled into non-thermal electrons.\nThe simulation with a reduced mass ratio leads to about 2-3 orders of magnitude reduction in electron acceleration efficiency.\n\nClearly, the simulation where the intermediate-scale instability is allowed to grow at the shock front region (red) is much more efficient in confining the accelerated electrons by scattering them back to the downstream regions, leading to a much larger non-thermal electron energy in the downstream region in comparison to the simulation with a higher $\\mathcal{M}_{\\rm A}$ as seen in the left-hand panel of Figure~\\ref{fig:Eff}.\nMoreover, the simulation with a low mass ratio (black) leads to significant reduction in the energy channeled into non-thermal electrons in the downstream region.\n\nAs shown in the lower left-hand panel of Figure~\\ref{fig:Eff}, in the downstream region, the simulation that allows for growth of comoving ion-cyclotron waves (red) has a 2-3 orders of magnitude larger non-thermal electron-to-ion energy ratio, $K_{\\rm ei}$, in comparison to that in the simulations where the condition for the intermediate-scale instability is not satisfied.\n\n\n\\section{Discussion}\n\\label{sec:dis}\n\nIn this section, we discuss the impact of a finite thermal temperature on the growth of the intermediate-scale instability, followed by a comparison of our proposed electron pre-acceleration mechanism in comparison to the mechanism proposed by \\citet{2015phrvl.114h5003p}. Finally, we discuss how our results can be used to possibly infer a lower limit on the amplitude of the magnetic field at shocks of supernova remnants.\n\n \n\n\\subsection{Impact of finite plasma beta}\n\n\nThe dispersion relation used in Section~\\ref{sec::theory} to predict the unstable modes at the shock transition region assumes a cold background of electrons and ions.\nThus, it is of great importance to consider the impact of the finite plasma beta $\\beta_s$ on the growth of the intermediate-scale instability, where  $\\beta_s \\equiv 2 \\varv_{{\\rm th},s}^2\/\\varv_{{\\rm A},s}^2$, the square of the thermal speed is $\\varv_{{\\rm th},s}^2 \\equiv k_B T_s\/m_s$ and $s=e,i$ for electrons and ions respectively.\n\nThe dispersion relation studied in \\citet{sharp2} and presented in Section~\\ref{sec::theory} are computed assuming $\\beta_i =\\beta_e=0$ while the simulation presented by \\citet{sharp2} has $\\beta_e = \\beta_i = 2$. Interestingly, the growth rates of the analytical derivation in the cold-background limit and the simulations with the finite $\\beta$ values show an excellent agreement, thus indicating that there is no impact of  $\\beta_e$ or $\\beta_i$ on the maximum growth rate of the intermediate-scale instability.\n\n\\citet{sharp2} argue that the damping due to thermal ion-cyclotron damping (at $k \\varv_{\\rm th,i} = (1$--$2) \\Omega_i$) could at most impact one peak of the instability if parameters are fine-tuned and thus typically it cannot impact instability growth for typical astrophysical plasma conditions. Similarly, it is strainght-forward to show that this is also true for thermal electron-cyclotron damping because the separation between the two peaks of the intermediate-scale instability does not solely depend on the ion-to-electron mass ratio.\n\nFor the shock problem studied in the present paper, there are two very different regimes of the beta factor: a low-beta regime in the pre-shock region and a high-beta regime in the shock transition zone. In the pre-shock region, we have $\\beta_e = \\beta_i = 1.28\\times10^{-4}$ ($\\mathcal{M}_{\\rm A}=5.3$)  and $\\sim 2 \\times 10^{-3}$ $(\\mathcal{M}_{\\rm A} = 21.3$). The case of $\\beta_e \\gtrsim1$ would imply a much faster growth of the intermediate-scale instability in comparison to the growth rate of the gyro-scale instability because of the dependence of the growth rate on $\\varv_\\perp$ as given in equation (6) of \\citet{sharp2}. Thus, we expect that in this case the instability will have an even larger impact on the mechanism of electron acceleration but postpone a detailed study of this to future works.\n\nAt the shock transition zone of the simulations with $m_r=1836$, the electron are hot and the temperature is such that $k_{\\rm B} T_e \\sim m_e c^2$. Therefore, we obtain $\\beta_e = 1.74$ for the $\\mathcal{M}_{\\rm A} = 5.3$ simulation and $\\beta_e = 27.88$ for the $\\mathcal{M}_{\\rm A} = 21.3$ simulation, i.e., in both cases, the electron beta is larger than one. However, only in the simulation with $\\mathcal{M}_{\\rm A} = 5.3$, the relative drift is such that the instability condition is fulfilled and thus allows for the growth of the intermediate-scale instability. This allows resonant interactions between electrons and unstable modes and results in much larger acceleration efficiency as shown in the simulation.\n\n\n\\subsection{On the electron acceleration mechanism}\n\nThe proposed mechanism for pre-accelerating electrons works by exciting intermediate-scale waves at the shock transition region.\nThese sub-ion skin-depth (ion-cyclotron) waves are comoving with the incoming upstream plasma (see bottom panel of Figure~\\ref{fig:GrowthRate}) and hence are coherently transported to the downstream region (see Figure~\\ref{fig:KLS}).\nIn the downstream and shock transition regions, the hot electrons scatter off of these unstable waves  and the waves reflected at the contact discontinuity (see top-left panel of Figure~\\ref{fig:XT-evol}). This leads to the high acceleration efficiency shown in the red simulation (with $\\mathcal{M}_{\\rm A} = 5.3$) as seen in Figures~\\ref{fig:spectra300} and \\ref{fig:Eff}, possibly in a first-order Fermi type process.\n\nThe simulation with $\\mathcal{M}_A=21.3$ (depicted in blue) shows excitation of sub-ion skin-depth (Whistler) waves at the shock front region. However, these modes are not transported to the downstream region (see Figure~\\ref{fig:KLS}), which thus results in a much lower electron acceleration efficiency.\nThat is, this simulation shows that for a parallel shock geometry, electron pre-acceleration due to scattering with Whistler waves has a much lower efficiency in comparison to that in the red simulation (see Figure~\\ref{fig:Eff}).\n\n\nWe emphasize that our proposed mechanism for electron pre-acceleration does not depend on ion acceleration.\nThis is clearly manifested by the large fraction of pre-accelerated electrons before we observe any significant ion acceleration in the  simulation with $\\mathcal{M}_{\\rm A} = 5.3$, as shown in the top-panel (red curves) of Figure~\\ref{fig:spectra300}.\nOn the other hand, the mechanism proposed by~\\citet{2015phrvl.114h5003p} for electron pre-acceleration relies on the amplification of Bell modes in the shock pre-cursor driven by the propagation of accelerated ions and thus the combination of SDA and scattering with Bell modes resulted in electron pre-acceleration and injection in DSA cycles.\n\n\\subsection{Applications to supernova remnants}\n\\label{sec:SNR}\nWhile we have clearly demonstrated the importance of the intermediate-scale instability for thermalizing and accelerating electrons in our PIC simulations, we will now turn to discuss the potential relevance of this instability in observations of astrophysical shocks. Perhaps the cleanest astrophysical object is the supernova remnant SN~1006, which enables testing our ideas of the prevailing plasma instabilities that are responsible for electron scattering and acceleration at the quasi-parallel shock conditions we encounter at the polar cap regions of that remnant shock. \\citet{Winner+2020} perform three-dimensional MHD simulations of the Sedov Taylor explosion and evolve the electron distribution function while accounting for magnetic obliquity-dependent electron acceleration. To this end, their method finds and characterizes the MHD shocks, injects a pre-defined electron power-law spectrum into the post-shock region \\citep{Winner+2019}, and explores the effects of varying the magnetic amplification factor on the surface brightness maps as well as the multi-frequency spectrum of SN~1006.\n\nMatching the radial emission profiles and the gamma-ray spectrum  requires a volume-filling, turbulently amplified magnetic field of $B\\approx35~\\mu$G in the downstream region of the parallel shock and that the Bell-amplified magnetic field \\citep{Bell2004} is starting to get damped in the further post-shock region \\citep[see figure~2 of][]{Winner+2020}. The exact value of the Bell amplification factor $f_\\mathrm{B}$ barely changes the multi-frequency spectrum so that we obtain a post-shock Alfv\\'en velocity of $\\varv_\\mathrm{A}=B f_\\mathrm{B}\/\\sqrt{\\mu_0\\mu m_p n}\\approx200~f_\\mathrm{B}~\\mathrm{km~s}^{-1}$, where $\\mu=1.25$ is the mean molecular weight of the warm interstellar medium, $m_p$ is the proton rest mass, and $n=0.12~\\mathrm{cm}^{-3}$. While the Bell amplified field is maximized in the immediate post-shock regime \\citep{Caprioli2014b}, the turbulently amplified magnetic field keeps rising in the post-shock regime \\citep{Ji+2016}, so that it is appropriate to set $f_\\mathrm{B}=1$ while noting that the turbulently amplifying field on its route to saturation is replacing the Bell-amplified field as it is damping further downstream.\n\nAdopting the lab frame shock velocity of SN~1006 at the current epoch, $\\varv_s=3000~\\mathrm{km~s}^{-1}$, we obtain a post-shock Alfv\\'enic Mach number of\n\\begin{align}\n  \\mathcal{M}_\\mathrm{A}=\\frac{\\varv_s}{\\varv_\\mathrm{A}}=15.2<\\frac{\\sqrt{m_r}}{2},\n\\end{align}\nwhich obeys the condition for exciting the intermediate-scale instability and thus should enable efficient electron acceleration at the polar cap regions of the parallel shocks of SN~1006 \\citep[see figure~4 of][]{Winner+2020}.\n\nIn fact, efficient electron acceleration at parallel shocks enables us to put a lower limit on the post-shock magnetic field, \n\\begin{align}\n  \\label{eq:Bmin}\n  B&>2 \\varv_s \\sqrt{\\frac{\\mu_0\\rho}{m_r A}}=B_\\mathrm{min}\\\\\n  &= 22.7~\\left(\\frac{\\varv_s}{3000~\\mathrm{km~s}^{-1}}\\right)\n  \\left(\\frac{n}{A\\, 0.1~\\mathrm{cm}^{-3}}\\right)^{1\/2}\\,\\mu\\mathrm{G},\n\\end{align}\nwhere $m_r$ is the proton-to-electron mass ratio and $A$ is the atomic mass number of the element responsible for driving the intermediate~scale instability. Hence, if the plasma is composed of abundant ${}^{56}\\mathrm{Fe}$ ions, the minimum post-shock magnetic field is lowered to $3~\\mu$G for the same shock parameters.\nFor heavy ions to dominate the growth of the intermediate instability, we require them to be very abundant because the growth rate, $\\Gamma$, of the intermediate-scale instability depends on the number density of ions via $\\Gamma \\propto n_{\\rm Fe}^{1\/3}$.\n\n\n\\section{Conclusions}\n\n\\label{sec:concl}\n\n\nIn collisionless shocks, electrons are heated to energies much larger than the kinetic energy of upstream  electrons impinging on the shock. However, in non-relativistic shocks, their Larmor radii fall short of those of ions so that another acceleration mechanism is needed to boost their energies to the point where they can actively participate in the process of DSA.\nPreviously suggested mechanisms, which were based on driving whistler waves unstable by hot downstream or cold upstream electrons, require high values of the {\\alf}ic Mach number and were shown to not work in PIC simulations \\citep{Niemiec+2012}. \nIn this paper we consider a new mechanism for electron pre-acceleration in quasi-parallel, non-relativistic electron-ion shocks that is based on driving ion-cyclotron waves unstable by means of ions that are drifting with their upstream velocity through the shock transition zone. The corresponding intermediate-scale instability \\citep{sharp2} only works for low values of the {\\alf}ic Mach number at the shock front, $\\mathcal{M}_{\\rm A}^f < \\sqrt{m_i\/m_e} \\approx 21$, which is the condition for the instability to operate.\n\nWe present results from three 1D3V PIC simulations for parallel electron-ion shocks with sonic Mach number $\\mathcal{M}_{\\rm s} \\sim 365$:\nthe first (red) simulation uses a realistic ion-to-electron mass ratio of $m_r=1836$ and provides favorable conditions for exciting the intermediate scale instability.\nThe second (black) simulation employs identical physical initial conditions but has an artificially lower value for the ion-to-electron mass ratio, $m_r=100$, which violates the instability condition.\nIn the third (blue) simulation, the condition is also violated by using a higher value of $\\mathcal{M}_{\\rm A}$ and $m_r=1836$.\nHighlight results include:\n\\begin{itemize}\n\n\\item Only the simulation that grows the intermediate scale instability scatters hot electrons in the downstream off of driven ion-cyclotron waves. We demonstrate that this efficiently pre-accelerates electrons to energies that enable them to participate in DSA cycles and yields a non-thermal power-law distribution of electrons (see Figure~\\ref{fig:spectra300}).\n\n\\item This effective electron acceleration comes about because the excited ion-cyclotron waves at the shock front are comoving with the upstream plasma and hence, survive advection into the downstream. In consequence, the amplitude of perpendicular magnetic field fluctuations, which are resonantly scattering hot electrons, is substantially increased in comparison to the other simulations that preclude instability growth (see Figures~\\ref{fig:spectra300} and \\ref{fig:KLS}).\n\n\\item The simulation with the higher value of $\\mathcal{M}_{\\rm A}$ shows a reduction, by more than 2 orders of magnitude, in the efficiency of electron acceleration (see Figure~\\ref{fig:Eff}). However, the electrons in the downstream are heated to the same temperature as the red simulation (see Figure~\\ref{fig:Temp}).\n  \n\\item The simulation with the lower mass ratio ($m_r=100$) results in much lower heating of electrons in the downstream region and thus an even lower electron acceleration efficiency (see Figures~\\ref{fig:Eff} and \\ref{fig:Temp}). We conclude that accurate PIC simulations of collisionless electron-ion shocks require realistic mass ratios.\n  \n\\end{itemize}\n\nThese findings put the magnetic amplification processes at shocks into the limelight because of the strict instability criterion that favors low values of $\\mathcal{M}_{\\rm A}$ and as such, is able to relate efficient electron acceleration with a lower limit on the downstream magnetic field strength (see Equation~\\ref{eq:Bmin}). Most importantly, this paper provides an important cornerstone in our understanding of diffusive shock acceleration of electrons, and potentially enables us to construct an analytical subgrid model for including electron acceleration into modern MHD models.\n\n\n\n\n\\section*{Acknowledgements}\n\nWe would like to thank Anatoly Spitkovsky for discussions on various aspects of the simulations. We also acknowledge comments by Damiano Caprioli and the anonymous referee.\nM.S., R.L., T.T., and C.P. acknowledge support by the European Research Council under ERC-CoG grant CRAGSMAN-646955.\nThe authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre (www.lrz.de). This research was in part supported by the Munich Institute for Astro- and Particle Physics (MIAPP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany\u00b4s Excellence Strategy \u2013 EXC-2094 \u2013 390783311.\n\n\n\\begin{appendix}\n\n\n\n\n\\section{Distribution of relativistic particles}\n\\label{app:temp}\n\nHere we review the shape of the momentum distribution of charged particles in thermal equilibrium, and the impact of the existence of a non-thermal tail.\nWe discuss how it can be characterized in such cases, and different ways to quantify the fraction of density and energy in its non-thermal tail.\n\n\\subsection{Average thermal energy}\n\nIn thermal equilibrium, the rest-frame (R) isotropic momentum distribution of relativistic particles (with elementary mass $m_s$) is given by the Maxwell-J{\\\"u}ttner distribution~\\citep{Wright1975}\n\\begin{eqnarray}\n\\label{Eq:distribution}\nf_{\\rm th}(\\vec{u}) = \\frac{ n_R e^{-\\gamma\/\\tilde{T}_R}  }{4 \\pi \\tilde{T}_R K_2(1\/\\tilde{T}_R)} ,\n\\end{eqnarray}\nwhere $K_2$ is the modified Bessel function of the second kind, the Lorentz-factor is $\\gamma = \\sqrt{1+ \\vec{u} \\cdot \\vec{u}}$, and $\\tilde{T}_R$ is the thermodynamical equilibrium temperature in the rest frame\\footnote{\nIt is worth mentioning here that because the phase-space volume is invariant under Lorentz transformation, this distribution takes the same form in any other frame, e.g., in the laboratory frame \\citep{Wright1975}.} normalized by $m_s c^2\/k_{\\rm B}$, all velocities are normalized with the speed of light $c$, and $n_R$ is the rest-frame number density of the particles.\nTherefore, the average (thermal + rest-mass) energy per particle, normalized by $m_s c^2$, is\n\n\\begin{eqnarray}\n\\langle \\bar{E} \\rangle &=&  \\int d^3u \\gamma \\frac{f_{\\rm th}(\\vec{u})}{n_R} = 4 \\pi  \\int_0^{\\infty}  du  \\frac{u^2 \\gamma e^{-\\gamma\/\\tilde{T}_R}  }{4 \\pi \\tilde{T}_R K_2(1\/\\tilde{T}_R)}\n\\nonumber \\\\\n&=&\n\\frac{1}{ \\tilde{T}_R K_2(1\/\\tilde{T}_R)} \\int_0^{\\infty}  du u^2 \\gamma e^{-\\gamma\/\\tilde{T}_R} \n\\nonumber \\\\\n&=&\n\\frac{-1}{ \\tilde{T}_R K_2(1\/\\tilde{T}_R)} \\frac{d}{d(1\/\\tilde{T}_R)} \\int_0^{\\infty}  du u^2  e^{-\\gamma\/\\tilde{T}_R}\n\\nonumber \\\\\n&=&\n\\frac{-1}{ \\tilde{T}_R K_2(1\/\\tilde{T}_R)} \\frac{d ( \\tilde{T}_R K_2(1\/\\tilde{T}_R) )}{d(1\/\\tilde{T}_R)} \n=3 \\tilde{T}_R +\\frac{K_1 \\left(1\/\\tilde{T}_R \\right)}{K_2\\left(1\/\\tilde{T}_R \\right)}.\n~~~~~~~ \n\\end{eqnarray}\nThat is, the average thermal energy per particle is\n\\begin{eqnarray}\n\\label{Eq:Eth}\n\\langle E_{\\rm th} \\rangle = \\left( 3 \\tilde{T}_R +\\frac{K_1 \\left(1\/\\tilde{T}_R \\right)}{K_2\\left(1\/\\tilde{T}_R \\right)} - 1 \\right) m_s c^2.\n\\end{eqnarray}\n\n\n\\subsection{Equilibrium rest-frame temperature}\nFor a distribution of particle momenta, we can find $\\tilde{T}_R$ (assuming a thermal equilibrium of most low energy particles) as follows:\n\\begin{eqnarray}\n\\label{Eq:du4fu}\n\\frac{d }{du} \\left[  u^4 f_{\\rm th}  \\right]  =  \\frac{ n_R  u^3 \\left(4 \\sqrt{u^2+1} \\tilde{T}_R-u^2\\right)  }{4 \\pi \\tilde{T}_R^2 K_2(1\/\\tilde{T}_R) \\sqrt{u^2+1}}  e^{-\\frac{\\sqrt{u^2+1}}{\\tilde{T}_R}} .\n\\end{eqnarray}\nThat is, using the value of $u$ for which $u^4 f(u)$ is maximum ($u_m$), we can determines $\\tilde{T}_R$ using\n\\begin{eqnarray}\n\\label{Eq:TR}\n\\tilde{T}_R = \\frac{  u_m^2 }{ 4 \\sqrt{1+u_m^2}}.\n\\end{eqnarray} \n\n\n\\subsection{Momentum distribution with non-thermal particles}\n\n\n\nHere, we consider the full momentum distribution $f_{\\rm full}$ that contains non-thermal and high-energy particles. \nThe main assumption adopted here is that {\\it{all low energy particles are in thermal equilibrium}}.\nThat is, all non-thermal particle are such that $u \\gg u_m$, which means the value of $u$ at which the derivative in Equation~\\eqref{Eq:du4fu} is zero is always outside the range for which the non-thermal parts of the distribution is non-zero.\nTherefore, the value of $u_m$ and hence $\\tilde{T}_R$, computed from Equation~\\eqref{Eq:TR}, is still the same.\nAn example of such a case is shown in Figure~\\ref{fig:Thermal}.\n\nAssuming that the non-thermal part of the distribution follows a power law with a typical index of -$4$, the full distribution can be approximated as follows,\n\\begin{eqnarray}\n\\label{Eq:Ffull}\nf_{\\rm full}(u) = \\frac{n_R}{1+\\alpha}\n\\left[\n\\frac{e^{-\\gamma\/\\tilde{T}_R}}{4 \\pi k_2 (1\/\\tilde{T}_R)}\n+\n\\alpha\n\\frac{u_1 u_2 u^{-4}}{4 \\pi (u_1 - u_2) }\\theta_1 \\theta_2\n\\right],\n\\end{eqnarray}\nwhere $\\alpha$ is the fractional number density of non-thermal particles which is implicitly assumed to be small, i.e., $\\alpha \\ll 1$, $\\theta_1 = \\theta(u-u_1)$, $\\theta_2 = \\theta(u_2 -u)$, and $u_1$ and $u_2$ ($u_2>u_1$) is the range in which the distribution takes a power-law form.\nThe distribution in Equation~\\eqref{Eq:Ffull} is discontinuous at $u=u_1$. However, in reality the thermal and $u^{-4}$ parts of the distribution are connected with a suprathermal distribution (with a logarithmic slope steeper than $-4$) that starts at $u \\sim 4 u_m$ to $ 5 u_m$ \\citep{Caprioli2014a}.\nTherefore, the suprathermal distribution does not change the relation between $u_m$ and $\\tilde{T}_R$ (Equation~\\ref{Eq:TR}).\n\nThe inclusion of the non-thermal part leads to a lower fraction of particles in thermal equilibrium and hence lowers proportionately the value of $E_{\\rm th}$ at a given $\\tilde{T}_R$. Namely\n\\begin{eqnarray}\n\\label{Eq:Eth2}\n\\langle E_{\\rm th} \\rangle_{\\rm full}  \\approx \\frac{1}{1+\\alpha} \\left( 3 \\tilde{T}_R +\\frac{K_1 \\left(1\/\\tilde{T}_R \\right)}{K_2\\left(1\/\\tilde{T}_R \\right)} - 1 \\right) m_s c^2.\n\\end{eqnarray}\n\nThe value of $\\alpha$ can be determined by comparing the value of $f(u_m)$ and $f_{\\rm full}(u_m)$, where the value of \n$f(u_m)$ is computed from Equation~\\eqref{Eq:distribution} at $u_m$ where the expression $u^4 f_{\\rm full}(u)$ is maximized, and\n$f_{\\rm full}(u_m)$ is computed from the normalized histogram of the particles' normalized momenta ($u$) in the simulation.\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{Plots\/Fig08.pdf}\n\\caption{\nMomentum distribution for ions (red dashed) and electrons (red solid) at $t~ \\Omega_i=325$ for the Ma5Mr1836 simulation, where we first transform momenta of particles to the plasma rest frame before we compute the momentum spectra.\nSolid black lines show the analytical Maxwell-J{\\\"u}ttner distribution scaled appropriately, $u^4 f(u)$ (as given by Equation~\\ref{Eq:distribution}). The normalized temperature (given by Equation~\\ref{Eq:TR}) is determined from the location of the peaks ($u_m^i$ for ions and $u_m^e$ for electrons).\nParticle energies with $u>5u_m$ (to the right of blue lines) are considered to be non-thermal.\n\\label{fig:Thermal}\n}\n\\end{figure}\n\n\n\\subsection{Acceleration efficiency}\n\n\n\nWe note here that $\\alpha = n_{\\rm non-th}\/n_{\\rm th}$, where $n_{\\rm th}$ is the number density of particles in thermal equilibrium, and $n_{\\rm non-th}$ is the number density of non-thermal particles. Thus, \nthe rest-frame total number density is $n_R = n_{\\rm th} + n_{\\rm non-th}$.\nThat is, we can define the efficiency of acceleration as the fraction of non-thermal particles ($\\epsilon_n$) measured in per cent via\n\\begin{eqnarray}\n\\label{Eq:EffN}\n\\epsilon_n \\equiv \\frac{n_{\\rm non-th}}{n_R} \\times 100 = \\frac{\\alpha}{1+\\alpha} \\times 100  .\n\\end{eqnarray}\n\nWe can also define the acceleration efficiency by the fractional energy carried by non-thermal particles ($\\epsilon_E$) in per cent as\n\\begin{eqnarray}\n\\label{Eq:EffE}\n\\epsilon_E & \\equiv & \n\\frac{ E_{\\rm tot} - \\langle E_{\\rm th} \\rangle_{\\rm full} }{ E_{\\rm tot} } \\times 100  \n=\n\\frac{E_{\\rm non-th}}{E_{\\rm tot}} \\times 100 ,\n\\end{eqnarray}\nwhere $ E_{\\rm tot} = \\langle \\gamma -1 \\rangle m_s c^2$ is the average of $(\\gamma-1)$ of all thermal and non-thermal particles.\n\nAn equally important definition of the acceleration efficiency is how much of the upstream plasma kinetic energy is channeled into non-thermal energy. In this case, the efficiency of acceleration in per cent is defined as\n\\begin{eqnarray}\n\\label{Eq:Eff-sh}\n\\epsilon_{\\rm sh}\n& \\equiv &\n\\frac{ E_{\\rm non-th}   }{ \n0.5  ~ (m_e + m_i) \\varv_{\\rm sh}^2 } \\times 100 ,\n\\end{eqnarray}\nwhere $\\varv_{\\rm sh}$ is the upstream plasma drifting speed in the shock rest-frame.\nWe compute the non-thermal energy density, $E_{\\rm non-th}$, as the average (per particle) energy with $u>5u_m$ \\citep[][]{Xu2020}. The vertical blue lines in Figure~\\ref{fig:Thermal} show such cuts.\nThat is, particles with velocities $u>5u_m$ are assumed to form the non-thermal part of the distribution function.\n\n\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe classification problem for group actions on operator algebras has a long history dating back to the hallmark results of Connes \\cite{Connes73, Connes74, Connes75, Connes76, Connes77} about injective factors that employed ideas of a dynamical nature in a crucial way.\nThe subsequent pioneering work by Jones \\cite{Jones80}, Ocneanu \\cite{Ocneanu85} and others \\cite{SutherlandTakesaki89, KawahigashiSutherlandTakesaki92, KatayamaSutherlandTakesaki98, Masuda13} completely unraveled the structure of countable amenable group actions on injective factors and paved the way for related further developments in subfactor theory \\cite{Jones83, Popa94, Popa10, Masuda99}.\nWhen the acting group is nonamenable, then the structure of its actions on the hyperfinite II$_1$-factor is already less managable, as a theorem of Jones \\cite{Jones83oc} and other subsequent stronger \\emph{no-go theorems} such as \\cite{Popa06, BrothierVaes15} later demonstrated.\n\nConnes' seminal paper \\cite{Connes76} was the first of many influential works in operator algebras to make use of a kind of touchstone object (in his case the hyperfinite II$_1$-factor $\\mathcal{R}$) and to begin the classification approach by showing that every object to be classified absorbs this touchstone object.\nMore specifically, Connes' approach begins with a structural result asserting that every injective II$_1$-factor is \\emph{McDuff} --- i.e., tensorially absorbs $\\mathcal R$; see \\cite{McDuff70} --- which is then used further to show that each such factor is isomorphic to $\\mathcal R$.\nIn Ocneanu's pioneering work to classify outer actions of an arbitrary countable amenable group $G$ on $\\mathcal{R}$, he likewise proves at an early point in the theory that each such action (even without assuming outerness) absorbs the trivial $G$-action on $\\mathcal R$ tensorially, which he then exploits for his more sophisticated classification theorem.\nAlthough one would generally need injectivity of a factor to arrive at a satisfactory classification theory about (outer) $G$-actions on it, this early part of Ocneanu's theory works in remarkable generality.\nThe precise statement and a (comparably) self-contained proof using the methods of this article, which we included for the reader's convenience, is Theorem~\\ref{theorem:model-absorption}.\n\nIf one is concerned with C$^*$-algebraic considerations related to the \\emph{equivariant Jiang--Su stability problem} (see \\cite[Conjecture A]{Szabo21si}), the current methods always find a way to exploit Oceanu's aforementioned theorem in one form or another, usually involving to some degree Matui--Sato's property (SI) technique \\cite{MatuiSato12acta, MatuiSato12, MatuiSato14, Sato19}.\nLooking at the state-of-the-art at present \\cite{GardellaHirshbergVaccaro, Wouters21}, the key difficulties arise from pushing these methods to the case where a group action $G\\curvearrowright A$ on a C$^*$-algebra induces a complicated $G$-action on the traces of $A$.\nIn particular, it is generally insufficient for such considerations to only understand $G$-actions on $\\mathcal{R}$, but one rather needs to have control over $G$-actions on more general tracial von Neumann algebras.\nThis C$^*$-algebraically motivated line of investigation led us to ask the following question that is intrinsic to von Neumann algebras:\n\n\\begin{question}\nLet $G$ be a countable amenable group and $M$ a separably acting finite von Neumann algebra with $M\\cong M\\bar{\\otimes}\\mathcal R$.\nIs it true that every action $\\alpha: G\\curvearrowright M$ is cocycle conjugate to $\\alpha\\otimes\\operatorname{id}_{\\mathcal R}: G\\curvearrowright M\\bar{\\otimes}\\mathcal R$?\n\\end{question}\n\nAlthough Ocneanu's original results confirm this in full generality when $M$ is a factor,\\footnote{While Ocneanu's work \\cite{Ocneanu85} only contains an explicit proof for so-called centrally free actions $\\alpha$, his comment following \\cite[Theorem 1.2]{Ocneanu85} suggests an alternative approach to avoid this assumption. In several papers, the more general version without central freeness is also attributed to Ocneanu.} it turned out to be not so straightforward to resolve this question, despite common folklore wisdom in the subject suggesting that the factor case ought to imply the general case.\nSome classification results in the literature \\cite{JonesTakesaki84, SutherlandTakesaki85} imply that the above has a positive answer when $M$ is injective, but relying on this has two drawbacks.\nFirstly, the question we are trying to answer is by design much weaker than a hard classification result, so it would be desirable to have a proof not relying on such a powerful theorem, in particular when an assumption such as injectivity may not even be needed.\nSecondly, there is a subtle gap in the proof of \\cite[Lemma 4.2]{SutherlandTakesaki85}.\nWe are indebted to Stefaan Vaes for pointing this out to us in the context of the above question and for outlining a sketch of proof on how to correct this, which became a sort of blueprint for the main result of the fourth section.\n\nIn contemporary research by C$^*$-algebraists, the aforementioned results by Sutherland--Takesaki are still used to provide a partial answer to the above question, for example in \\cite{GardellaHirshbergVaccaro}. In light of the previous discussion, the present article aims to give a self-contained and --- dare we say also relatively elementary --- approach to answer this question instead.\nIn fact we can treat it in greater generality than posed above, without restrictions on the type of $M$ and in the setting of amenable actions of arbitrary discrete groups.\nThe following can be viewed as our main result; see Theorem~\\ref{thm:general-amenable-McDuff}.\n\n\\begin{theoremi} \\label{theorem-A}\nLet $G$ be a countable discrete group and $M$ a von Neumann algebra with separable predual such that $M \\cong M\\bar{\\otimes} \\mathcal{R}$. \nThen every amenable action $\\alpha\\colon G \\curvearrowright M$ is cocycle conjugate to $\\alpha \\otimes \\mathrm{id}_\\mathcal{R}\\colon G\\curvearrowright M\\bar{\\otimes} \\mathcal{R}$.\n\\end{theoremi}\n\nAlong the way, our methodology employs dynamical variants of McDuff-type properties analogous to the theory of strongly self-absorbing C$^*$-dynamics \\cite{Szabo18ssa}, which can and is treated in the more general setting of continuous actions of locally compact groups; see Definitions~\\ref{def:strong-absorption} and \\ref{def:ssa-action}.\n\n\\begin{defii}\nLet $G$ be a second-countable locally compact group.\nAn action $\\delta: G\\curvearrowright\\mathcal R$ is called \\emph{strongly self-absorbing}, if there exists an isomorphism $\\Phi: \\mathcal R\\to\\mathcal R\\bar{\\otimes}\\mathcal R$, a $(\\delta\\otimes\\delta)$-cocycle $\\mathbb{U}: G\\to\\mathcal{U}(\\mathcal R\\bar{\\otimes}\\mathcal R)$ and a sequence of unitaries $v_n\\in\\mathcal{U}(\\mathcal R\\bar{\\otimes}\\mathcal R)$ such that\n\\[\nv_n(x\\otimes 1_{\\mathcal R})v_n^* \\to \\Phi(x) \\quad\\text{and}\\quad v_n(\\delta\\otimes\\delta)_g(v_n)^* \\to \\mathbb{U}_g\n\\]\nin the strong operator topology for all $x\\in\\mathcal R$ and $g\\in G$, the latter uniformly over compact subsets in $G$.\n\\end{defii}\n\nFor such actions we prove the following dynamical generalization of McDuff's famous theorem \\cite{McDuff70}; see Theorem~\\ref{thm:general-McDuff}.\n\n\\begin{theoremi} \\label{theorem-C}\nLet $G$ be a second-countable locally compact group.\nLet $\\alpha: G \\curvearrowright M$ be an action on a von Neumann algebra with separable predual and let $\\delta: G \\curvearrowright \\mathcal{R}$ be a strongly self-absorbing action on the hyperfinite II$_1$-factor.\nThen $\\alpha$ is cocycle conjugate to $\\alpha\\otimes\\delta$ if and only if there exists a unital equivariant $*$-homomorphism $(\\mathcal R,\\delta) \\to (M_{\\omega,\\alpha},\\alpha_\\omega)$, where the latter denotes the induced $G$-action on the asymptotic centralizer algebra of $M$.\n\\end{theoremi}\n\nOur initial methodology inspired by the theory of C$^*$-dynamics is only well-suited to build all the aforementioned (and other related) theory in the setting of (actions on) semi-finite von Neumann algebras.\nAfter the first preliminary section, the second and third section are dedicated to proving Theorem~\\ref{theorem-C} in the special case of semi-finite von Neumann algebras.\nThe fourth section then builds on some of this theory, combined with the original ideas by Sutherland--Takesaki \\cite{SutherlandTakesaki85} related to disintegrating a $G$-action to an action of its transformation groupoid induced on the center. This culminates in our main technical result of that section --- a kind of measurable local-to-global principle for absorbing a given strongly self-absorbing action, Theorem~\\ref{thm:main_technical} --- which is then used to prove a stronger version of Theorem~\\ref{theorem-A} for actions on semi-finite von Neumann algebras. \n \nThe general main results are then obtained in the fifth section with the help of Tomita--Takesaki theory.\nIt is in this step that it becomes obvious why we want to treat Theorem~\\ref{theorem-C} beyond the case of discrete groups.\nNamely, if $\\alpha\\colon G\\curvearrowright M$ is an action as in Theorem~\\ref{theorem-A} on a von Neumann algebra that is not semi-finite, we may consider the extended action $\\tilde{\\alpha}\\colon G\\curvearrowright\\tilde{M}$ on the (semi-finite) continuous core.\nHowever, in order to conclude that $\\alpha$ absorbs $\\delta$ with the help of Tomita--Takesaki theory, it is not sufficient to argue that $\\tilde{\\alpha}$ absorbs $\\delta$, but one actually needs to verify this absorption for certain enlargements of these actions to continuous actions of $G\\times\\mathbb{R}$, which in any case requires Theorem~\\ref{theorem-C} for non-discrete groups.\nFortunately this can all be arranged to work and we end the last section with the proofs of our main results.\n\n\n\\section{Preliminaries}\n\nThroughout the paper, $\\omega$ denotes a fixed free ultrafilter on $\\mathbb{N}$ and $G$ denotes a second-countable, locally compact group.\nLet $M$ be a von Neumann algebra with predual $M_*$. For $x \\in M$ and $\\phi \\in M_*$ we define elements $x\\phi, \\phi x$ and $[x,\\phi] \\in M_*$ by $(x\\phi)(y) = \\phi(yx)$, $(\\phi x)(y) = \\phi(xy)$ for all $y \\in M$ and $[x,\\phi] = x\\phi - \\phi x$. Moreover, for $x \\in M$ and $\\phi \\in M_*$ we set $\\|x\\|_{\\phi} = \\phi(x^*x)^{1\/2}$ and $\\|x\\|_{\\phi}^\\# = \\phi(x^*x + xx^*)^{1\/2}$. When $\\phi$ is a faithful normal state, the norms $\\|\\cdot\\|_\\phi$ and $\\|\\cdot\\|_\\phi^\\#$ induce the strong and strong-$*$ topology on bounded sets, respectively.\nMore generally, when $\\phi$ is a normal weight on $M$, we define $\\|x\\|_{\\phi}:= \\phi(x^*x)$ for all $x$ contained in the left-ideal\n\\[\\{x  \\in M \\mid \\phi(x^*x) < \\infty\\}.\\]\nRecall that $M$ is called $\\sigma$-finite if it admits a faithful normal state.\n\\subsection{Ultrapowers of von Neumann algebras}\n\nWe start with a reminder on the Ocneanu ultrapower of a von Neumann algebra and related concepts.\nThis originates in \\cite[Section 2]{Connes74} and \\cite[Section 5]{Ocneanu85}, but the reader is also referred to \\cite{AndoHaagerup14} for a thorough exposition on ultrapower constructions.\n\n\\begin{definition}\nLet $M$ be a $\\sigma$-finite von Neumann algebra. We define the subset $\\mathcal{I}_\\omega(M) \\subset \\ell^\\infty(M)$ by\n\\begin{align*}\n\\mathcal{I}_\\omega(M) &= \\{(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M) \\mid x_n \\rightarrow 0 \\text{ $*$-strongly as } n \\rightarrow \\omega\\}\\\\\n&= \\{(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M) \\mid \\lim_{n \\rightarrow \\omega}\\|x_n\\|_{\\phi}^\\# =0 \\text{ for some faithful normal state } \\phi \\text{ on } M\\}.\n\\end{align*}\nDenote \\[\\mathcal{N}_\\omega(M)=\\{(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M) \\mid (x_n)_{n \\in \\mathbb{N}} \\mathcal{I}_\\omega(M) \\subset \\mathcal{I}_\\omega(M), \\text{ and } \\mathcal{I}_\\omega(M)(x_n)_{n \\in \\mathbb{N}} \\subset \\mathcal{I}_\\omega(M)\\},\\] \n\\[\\mathcal{C}_\\omega(M) =\\{(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M) \\mid \\lim_{n \\rightarrow \\omega}\\|[x_n,\\phi]\\| = 0 \\text{ for all } \\phi \\in M_*\\}.\\]\nThen \\[{\\mathcal{I}_\\omega(M) \\subset \\mathcal{C}_\\omega(M) \\subset \\mathcal{N}_\\omega(M)}.\\] \nThe \\emph{Ocneanu} ultrapower $M^\\omega$ of $M$ is defined as\n\\[M^\\omega := \\mathcal{N}_\\omega(M)\/\\mathcal{I}_\\omega(M),\\]\nand the \\emph{asymptotic centralizer} $M_\\omega$ of $M$ is defined as\n\\[M_\\omega := \\mathcal{C}_\\omega(M)\/\\mathcal{I}_\\omega(M).\\]\nThese are both von Neumann algebras. \nAny faithful normal state $\\phi$ on $M$ induces a faithful normal state $\\phi^\\omega$ on $M^\\omega$ via the formula\n\\[\\phi^\\omega((x_n)_{n \\in \\mathbb{N}}) = \\lim_{n \\rightarrow \\omega} \\phi(x_n).\\]\nThe restriction of $\\phi^\\omega$ to $M_\\omega$ is a tracial state.\n\\end{definition}\n\n\\begin{remark}\nSince the constant sequences are easily seen to be contained in $\\mathcal{N}_\\omega(M)$, one considers $M$ as a subalgebra of $M^\\omega$.\nIf $\\lim_{n \\rightarrow \\omega}\\|[x_n,\\phi]\\| = 0$ for all $\\phi \\in M_*$, then $\\lim_{n \\rightarrow \\omega}\\|[x_n,y]\\|_\\phi^\\# = 0$ for all $y \\in M$ by  \\cite[Proposition 2.8]{Connes74}.\n  In this way we get a natural inclusion $M_\\omega \\subset M^\\omega \\cap M'$.\n   That same proposition also shows that in order to check whether a sequence $(x_n)_n$ in $\\ell^\\infty(M)$ satisfies $\\lim_{n \\rightarrow \\omega}\\|[x_n,\\psi]\\| = 0$ for all $\\psi \\in M_*$, it suffices to check if this is true for just a single faithful normal state $\\phi$ and to check if $\\lim_{n \\rightarrow \\omega}\\|[x_n,y]\\|^\\#_\\phi =0$ for all $y \\in M$.\n    This shows that $M_\\omega = M^\\omega \\cap M'$ whenever $M$ admits a faithful normal tracial state. The same is then true for all semi-finite von Neumann algebras with separable predual (for example by \\cite[Lemma~2.8]{MasudaTomatsu16}).\n\\end{remark}\n\n\\begin{definition}\nA continuous action $\\alpha\\colon G \\curvearrowright M$ of a second-countable locally compact group on a von Neumann algebra is a homomorphism $G \\to \\mathrm{Aut}(M)$, $g \\mapsto \\alpha_g$ such that\n\\[\n\\lim_{g \\rightarrow 1_G}\\|\\varphi \\circ \\alpha_g - \\varphi\\|=0 \\text{ for all } \\varphi \\in M_*.\n\\]\nBy \\cite[Proposition~X.1.2]{Takesaki03}, this is equivalent to the map being continuous for the point-weak-$*$ (or equivalently, point-weak, point-strong,$\\hdots$) topology.\nIn most contexts we omit the word ``continuous'' as it will be implicitly understood that we consider some actions to be continuous.\nIn contrast, we will explicitly talk of an algebraic $G$-action when we are considering an action of $G$ viewed as a discrete group.\n\\end{definition}\n\nGiven an action $\\alpha\\colon G \\curvearrowright M$, the induced algebraic actions $\\alpha^\\omega\\colon G \\rightarrow \\mathrm{Aut}(M^\\omega)$ and $\\alpha_\\omega \\colon G \\rightarrow \\mathrm{Aut}(M_\\omega)$ are usually not continuous.\nThe remainder of this subsection is devoted (for lack of a good literature reference) to flesh out the construction of their `largest' von Neumann subalgebras where the action is sufficiently well-behaved for our needs, called the \\emph{$(\\alpha, \\omega)$-equicontinuous parts} (see Definition \\ref{def:equicontinuous_parts}).\nThese constructions are based on \\cite[Section 3]{MasudaTomatsu16}, where the special case $G=\\mathbb{R}$ is considered. \n\n\\begin{definition}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with an action $\\alpha\\colon G \\curvearrowright M$.\nFix a faithful normal state $\\phi$ on $M$. An element $(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M)$ is called \\emph{$(\\alpha, \\omega)$-equicontinuous} if for every $\\varepsilon>0$, there exists a set $W \\in \\omega$ and an open neighborhood $1_G\\in U \\subset G$ such that\n\\[\n\\sup_{n\\in W} \\sup_{g\\in U} \\|\\alpha_g(x_n) - x_n\\|_\\phi^\\# < \\varepsilon .\n\\]\nWe denote the set of $(\\alpha, \\omega)$-equicontinuous sequences by $\\mathcal{E}^\\omega_\\alpha$.\n\\end{definition}\n\n\\begin{remark}\nThe definition above does not depend on the faithful normal state chosen.\nWhenever $\\phi$ and $\\psi$ are two faithful normal states on $M$, one has for every $\\varepsilon>0$ some $\\delta>0$ such that for all $x \\in (M)_1$, $\\|x\\|_\\phi^\\# < \\delta$ implies $\\|x\\|_\\psi^\\# < \\varepsilon$. \n\\end{remark}\n\n\\begin{lemma}\\label{lemma:epsilon_delta_sequences}\nLet $M$ be a von Neumann algebra with faithful normal state $\\phi$. For all $(x_n)_{n \\in \\mathbb{N}} \\in \\mathcal{N}_\\omega(M)$ the following holds: \nFor any $\\varepsilon >0$ and compact set $\\Psi \\subset M_*^+$ there exists a $\\delta >0$ and $W \\in \\omega$ such that if $y \\in (M)_1$ and $\\|y\\|_\\phi^\\#< \\delta$, then $\\sup_{\\psi\\in \\Psi} \\|x_n y\\|_{\\psi}^\\#  <\\varepsilon$ and $\\sup_{\\psi\\in \\Psi} \\|y x_n\\|_{\\psi}^\\#  <\\varepsilon$ for all $n \\in W$. \n\\end{lemma}\n\\begin{proof} We prove this by contradiction. \nSuppose that there exists $\\varepsilon >0$ and a compact set $\\Psi \\subset M_*^+$ such that for any $k \\in \\mathbb{N}$ there exists a $y_k \\in (M)_1$ with $\\|y_k\\|_\\phi^\\# < 1\/k$ but the following set belongs to $\\omega$: \n\\[\nA_k := \\Big\\{ n \\in\\mathbb{N} \\ \\Big| \\ \\sup_{\\psi \\in \\Psi}\\|x_ny_k\\|_\\psi^\\#\\geq \\varepsilon \\text{ or } \\sup_{\\psi \\in \\Psi}\\|y_k x_n\\|_\\psi^\\# \\geq \\varepsilon \\Big\\}.\n\\]\nDefine $W_0 := \\mathbb{N}$ and $W_k := A_1 \\cap \\hdots \\cap A_k \\cap [k, \\infty)$ for $k \\geq 1$. \nThese all belong to $\\omega$. \nFor each $n \\in \\mathbb{N}$ define $k(n)$ as the unique number $k \\geq 0$ with $n \\in W_k \\setminus W_{k+1}$. Put $z_n := y_{k(n)}$ if $k(n) \\geq 1$, else put $z_n:=1_M$. \nNote that for all $n \\in W_m$ we get that $\\|z_n\\|_\\phi^\\# = \\|y_{k(n)}\\|_\\phi^\\# <\\frac{1}{k(n)} \\leq \\frac{1}{n}$. \nTherefore, it holds that $(z_n)_{n\\in \\mathbb{N}} \\in \\mathcal{I}_\\omega(M)$. \nSince $(x_n)_{n \\in \\mathbb{N}} \\in \\mathcal{N}_\\omega(M)$, it follows that also $(x_nz_n)_{n\\in \\mathbb{N}}$ and $(z_nx_n)_{n\\in \\mathbb{N}}$ belong to $\\mathcal{I}_\\omega(M)$. \nHence we get that for all $\\psi \\in \\Psi$ \n\\[\\lim_{n \\rightarrow \\omega} \\left(\\|x_nz_n\\|_\\psi^\\# + \\|z_nx_n\\|_\\psi^\\#\\right) = 0.\\]\nSince $\\Psi$ is compact, we also have \n\\[\\lim_{n \\rightarrow \\omega} \\sup_{\\psi \\in \\Psi}\\left(\\|x_nz_n\\|_\\psi^\\# + \\|z_nx_n\\|_\\psi^\\#\\right) = 0.\\]\nThis gives a contradiction, since our choice of $z_n$ implies that for all $n \\in W_1$ \n\\[\\sup_{\\psi \\in \\Psi}\\left(\\|x_nz_n\\|_\\psi^\\# + \\|z_nx_n\\|_\\psi^\\#\\right)\\geq \\varepsilon.\\]\n\\end{proof}\n\n\\begin{lemma}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with action $\\alpha:G \\curvearrowright M$.\nFor any two sequences $(x_n)_{n \\in \\mathbb{N}}, (y_n)_{n \\in \\mathbb{N}} \\in  \\mathcal{E}_\\alpha^\\omega \\cap \\mathcal{N}_\\omega(M)$ it follows that $(x_ny_n)_{n\\in \\mathbb{N}} \\in \\mathcal{E}_\\alpha^\\omega$. \n\\end{lemma}\n\\begin{proof}\nWithout loss of generality we may assume $\\sup_{n \\in \\mathbb{N}} \\|x_n\\| \\leq \\frac{1}{2}$ and $\\sup_{n \\in \\mathbb{N}} \\|y_n\\| \\leq \\frac{1}{2}$.\nFix a faithful normal state $\\phi$ on $M$. Let $K \\subset G$ be a compact neighbourhood of the neutral element.\nTake $\\varepsilon >0$ arbitrarily.\nBy Lemma \\ref{lemma:epsilon_delta_sequences} there exists $\\delta>0$ and $W_1 \\in \\omega$ such that for every $z \\in (M)_1$ with $\\|z\\|_\\phi^\\# < \\delta$ one has \n\\[\n\\sup_{g \\in K}\\|x_n z\\|_{\\phi \\circ \\alpha_g}^\\#< \\frac{\\varepsilon}{2} \\text{ and } \\|zy_n\\|_\\phi^\\# < \\frac{\\varepsilon}{2} \\text{ for all } n \\in W_1.\n\\] \nSince $(x_n)_{n\\in \\mathbb{N}}$ and $(y_n)_{n\\in \\mathbb{N}}$ both belong to $\\mathcal{E}_\\alpha^\\omega$, we can find an open $U \\subset K$ containing the neutral element, and a $W_2 \\in \\omega$ such that\n\\[\n\\sup_{n \\in W_2}\\sup_{ g \\in U}\\|\\alpha_g(x_n) - x_n\\|_\\phi^\\# < \\delta \\text{, and}\n\\]\n\\[\\sup_{n \\in W_2}\\sup_{ g \\in U}\n\\|\\alpha_{g^{-1}}(y_n) - y_n\\|_\\phi^\\# < \\delta.\n\\]\nThen for $g \\in U$ and $n \\in W_1 \\cap W_2$ we have\n\\begin{align*}\n\\|\\alpha_g(x_n) \\alpha_g(y_n) - x_ny_n\\|_\\phi^\\# &\\leq \\|\\alpha_g(x_n)(\\alpha_g(y_n) - y_n)\\|_\\phi^\\# + \\|(\\alpha_g(x_n) - x_n)y_n\\|_\\phi^\\#\\\\\n&= \\|x_n(\\alpha_{g^{-1}}(y_n) - y_n)\\|_{\\phi \\circ \\alpha_g}^\\# + \\|(\\alpha_g(x_n) - x_n)y_n\\|_\\phi^\\#\\\\\n& < \\frac{\\varepsilon}{2} + \\frac{\\varepsilon}{2} = \\varepsilon.\n\\end{align*}\nThis ends the proof.\n\\end{proof}\n\n\\begin{lemma}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with an action $\\alpha: G \\curvearrowright M$.\nThen:\n\\begin{enumerate}[label=\\textup{(\\arabic*)},leftmargin=*]\n\\item \\label{lem:equicont-algebra:1}\nSuppose $(x_n)_{n\\in \\mathbb{N}}, (y_n)_{n\\in \\mathbb{N}} \\in \\ell^\\infty(M)$ satisfy $(x_n-y_n)_{n\\in \\mathbb{N}} \\in \\mathcal{I}_\\omega(M)$.\nThen $(x_n)_{n\\in \\mathbb{N}} \\in \\mathcal{E}^\\omega_\\alpha$ if and only if $(y_n)_{n\\in \\mathbb{N}} \\in \\mathcal{E}^\\omega_\\alpha$.\n\\item \\label{lem:equicont-algebra:2}\n$\\mathcal{E}_\\alpha^\\omega \\cap \\mathcal{N}_\\omega(M)$ is an $\\alpha$-invariant C$^*$-subalgebra of $\\ell^\\infty(M)$. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} Fix a faithful normal state $\\phi$ on $M$.\nWe first prove \\ref{lem:equicont-algebra:1}.\nLet $\\varepsilon>0$.\nWe can choose $W \\in \\omega$ and an open neighborhood $1_G \\in U \\subset G$ such that \n\\[\n\\sup_{n\\in W} \\sup_{g\\in U} \\|\\alpha_g(x_n) - x_n\\|_\\phi^\\# < \\frac{\\varepsilon}{2}.\n\\] \nWithout loss of generality we may assume that $K=\\overline{U}$ is compact.\nConsider $s_n := \\sup_{g \\in K} \\|x_n - y_n\\|_{\\phi \\circ \\alpha_g}^\\#$. Since $K$ is compact and ${(x_n-y_n)_{n\\in \\mathbb{N}} \\in \\mathcal{I}_\\omega(M)}$, we have $\\lim_{n \\rightarrow \\omega} s_n = 0$.\nHence, after possibly replacing $W$ by a smaller set in the ultrafilter, we can assume that $s_n < \\varepsilon\/4$ for all $n \\in W$.\nWe may conclude for all $g \\in U$ and $n \\in W$ that\n\\begin{align*}\n\\|\\alpha_g(y_n) - y_n\\|_\\phi^\\# &\\leq \\|\\alpha_g(y_n) - \\alpha_g(x_n)\\|_\\phi^\\# + \\|\\alpha_g(x_n)- x_n\\|_\\phi^\\# + \\|x_n - y_n\\|_\\phi^\\#\\\\\n&\\leq 2s_n + \\frac{\\varepsilon}{2} < \\varepsilon.\n\\end{align*}\nSince $\\varepsilon>0$ was arbitrary, $(y_n)_{n\\in \\mathbb{N}}$ belongs to $\\mathcal{E}_\\alpha^\\omega$.\n\nLet us prove \\ref{lem:equicont-algebra:2}.\nClearly $\\mathcal{E}_\\alpha^\\omega$ is a $*$-closed, norm-closed linear subspace of $\\ell^\\infty(M)$. \nThe previous lemma shows that $\\mathcal{E}_\\alpha^\\omega \\cap \\mathcal{N}_\\omega(M)$ is closed under multiplication. \nTo see that $\\mathcal{E}_\\alpha^\\omega$ is $\\alpha$-invariant, take $(x_n)_{n \\in \\mathbb{N}} \\in \\mathcal{E}_\\alpha^\\omega$ and $h \\in G$. Take $\\varepsilon >0$. \nWe can find an open neighborhood $1_g \\in U \\subset G$ and $W \\in \\omega$ such that one has\n\\[\\sup_{n \\in W}\\sup_{g \\in U}\\|\\alpha_g(x_n) - x_n\\|_{\\phi \\circ \\alpha_h}^\\# < \\varepsilon.\\] \nThen for all $g \\in hUh^{-1}$ and $n \\in W$ we observe\n\\[\n\\|\\alpha_g(\\alpha_h(x_n)) - \\alpha_h(x_n)\\|_\\phi^\\# = \\|\\alpha_{h^{-1}gh}(x_n) - x_n\\|_{\\phi \\circ \\alpha_h}^\\# < \\varepsilon.\n\\]\nThis shows that $(\\alpha_h(x_n))_{n\\in \\mathbb{N}} \\in \\mathcal{E}_\\alpha^\\omega$.\n\\end{proof}\n\n\\begin{definition}\\label{def:equicontinuous_parts}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with an action $\\alpha\\colon G \\curvearrowright M$. \nWe define ${M_\\alpha^\\omega := (\\mathcal{E}_\\alpha^\\omega \\cap \\mathcal{N}_\\omega(M))\/\\mathcal{I}_\\omega}$ and ${M_{\\omega, \\alpha}:= M_\\alpha^\\omega \\cap M_\\omega}$. \nWe call them the \\emph{$(\\alpha,\\omega)$-equicontinuous parts} of $M^\\omega$ and $M_\\omega$, respectively. \n\\end{definition}\n\n\\begin{lemma}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with an action $\\alpha\\colon G \\curvearrowright M$.\nThen $M_\\alpha^\\omega$ and $M_{\\omega, \\alpha}$ are von Neumann algebras.  \n\\end{lemma}\n\\begin{proof}\nWe show that $M_\\alpha^\\omega$ is a von Neumann algebra by showing that its unit ball is closed with respect to the strong operator topology in $M^\\omega$. \nThen it automatically follows that $M_{\\omega, \\alpha} = M_\\alpha^\\omega \\cap M_\\omega$ is also a von Neumann algebra. \nTake a sequence $(X_k)_k \\in (M_\\alpha^\\omega)_1$ that strongly converges to $X \\in (M^\\omega)_1$. Fix a faithful normal state $\\phi$ on $M$ and a compact neighbourhood of the neutral element $K \\subset G$. \nThen the function $K \\rightarrow (M^\\omega)_*$ given by $g \\mapsto \\phi^\\omega \\circ \\alpha_g^\\omega$ is continuous (because $\\phi^\\omega \\circ \\alpha_g^\\omega = (\\phi \\circ \\alpha_g)^\\omega)$. \nHence, the set $\\{\\phi^\\omega \\circ \\alpha_g^\\omega\\}_{g \\in K}$ is compact and thus $\\lim_{n \\rightarrow \\infty} \\sup_{g \\in K} \\|X_n - X\\|^\\#_{\\phi^\\omega \\circ \\alpha_g^\\omega} = 0$.\nFix $\\varepsilon >0$. \nPick representing sequences $(x_k(n))_{n\\in \\mathbb{N}}$ and $(x(n))_{n\\in \\mathbb{N}}$ for the elements $X_k$ and $X$, respectively, such that $\\|x_k(n)\\| \\leq 1$, $\\|x(n)\\| \\leq 1$, for all $k,n \\in \\mathbb{N}$.\nThen we can find $k_0 \\in \\mathbb{N}$  and $W_1 \\in \\omega$ such that \n\\[\n\\sup_{n\\in W_1} \\sup_{g\\in K} \\|x_{k_0}(n) - x(n)\\|_{\\phi\\circ \\alpha_g}^\\# < \\frac{\\varepsilon}{3}.\n\\]\nSince $(x_{k_0}(n))_{n\\in \\mathbb{N}} \\in \\mathcal{E}_\\alpha^\\omega$, we can find an open neighborhood $1_G \\in U \\subset K$ and $W_2 \\in \\omega$ such that\n\\[\n\\sup_{n\\in W_2} \\sup_{g\\in U} \\|\\alpha_g(x_{k_0}(n)) - x_{k_0}(n)\\|_\\phi^\\# < \\frac{\\varepsilon}{3}.\n\\]\nThen for all $g \\in U$ and $n \\in W_1 \\cap W_2$ it holds that\n\\begin{align*}\n\\|\\alpha_g(x(n)) - x(n)\\|_\\phi^\\# & \\leq \\|x(n) - x_{k_0}(n)\\|_{\\phi \\circ \\alpha_g}^\\# + \\|\\alpha_g(x_{k_0}(n)) - x_{k_0}(n)\\|_\\phi^\\# + \\|x_{k_0}(n) - x(n)\\|_\\phi^\\#\\\\\n&< \\frac{\\varepsilon}{3}+\\frac{\\varepsilon}{3}+\\frac{\\varepsilon}{3} = \\varepsilon. \n\\end{align*} \nThis shows that $(x(n))_{n \\in \\mathbb{N}} \\in \\mathcal{E}_\\alpha^\\omega$, or in other words $X \\in M_\\alpha^\\omega$.\n\\end{proof}\n\n\\begin{lemma}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with an action $\\alpha\\colon G \\curvearrowright M$ of a second-countable locally compact group. \nThen $\\alpha^\\omega$ restricted to $M_\\alpha^\\omega$ and $\\alpha_\\omega$ restricted to $M_{\\omega,\\alpha}$ are continuous $G$-actions.\n\\end{lemma}\n\\begin{proof}\nFix a faithful normal state $\\phi$ on $M$. Since $\\phi^\\omega$ is faithful, $\\{a\\phi^\\omega \\mid a \\in M_\\alpha^\\omega\\}$ is dense in $(M_\\alpha^\\omega)_*$. For $a \\in M_\\alpha^\\omega$ and $g \\in G$ one has\n\\begin{align*}\n\\|(a\\phi^\\omega)  \\circ \\alpha^\\omega_g - a \\phi^\\omega \\|_{(M_\\alpha^\\omega)_*} & \\leq \\|\\alpha_{g^{-1}}^\\omega(a)(\\phi^\\omega \\circ \\alpha_g^\\omega - \\phi^\\omega)\\|_{(M_\\alpha^\\omega)_*} \\|+ \\|(\\alpha_{g^{-1}}^\\omega(a) - a)\\phi^\\omega\\|_{(M_\\alpha^\\omega)_*}\\\\\n& \\leq \\|a\\| \\, \\|\\phi \\circ \\alpha_g - \\phi\\|_{M_*} + \\|\\alpha_{g^{-1}}^\\omega(a) - a\\|_{\\phi^\\omega}.\n\\end{align*}\nWhen $g \\rightarrow 1_G$, this expression converges to zero because $\\alpha$ is a continuous $G$-action and $a \\in M_\\alpha^\\omega$. \nThis shows that $\\alpha^\\omega$ restricts to a genuine continuous $G$-action on $M_\\alpha^\\omega$, so the same is true for the restriction of $\\alpha_\\omega$ to $M_{\\omega, \\alpha}$. \n\\end{proof}\n\n\\begin{lemma} \\label{lemma:lifting_invariance_compact_sets}\nLet $M$ be a von Neumann algebra with a faithful normal state $\\phi$ and an action $\\alpha\\colon G \\curvearrowright M$. \nLet $z \\in M_\\alpha^\\omega$, $\\varepsilon >0$, $K \\subset G$ a compact set and suppose that $\\| \\alpha_g^\\omega(z)-z\\|_{\\phi^\\omega}^\\# \\leq \\varepsilon$ for all $g \\in K$.\nIf $(z_n)_{n\\in \\mathbb{N}}$ is any bounded sequence representing $z$, then \n\\[\n\\lim_{n \\rightarrow \\omega} \\max_{g \\in K} \\| \\alpha_g(z_n)-z_n\\|_\\phi^\\# \\leq \\varepsilon.\n\\]\n\\end{lemma}\n\\begin{proof}\nLet $\\delta>0$.\nThen for each $g \\in K$ there exists an open neighborhood $g \\in U \\subset G$ and $W_g \\in \\omega$ such that\n\\[\\sup_{n \\in W_g}\\sup_{h \\in U}\\|\\alpha_h(z_n) - \\alpha_g(z_n)\\|_\\phi^\\# < \\delta.\\]\nSince this obviously yields an open cover of $K$ and $K$ is compact, we can find finitely many elements $g_1, \\hdots, g_N \\in K$ and an open covering $K \\subset \\cup_{i=1}^N U_j$ with $g_j \\in U_j$ and some $W_1 \\in \\omega$ such that for $j=1, \\hdots, N$ we have\n\\[\\sup_{n \\in W_1}\\sup_{g \\in U_j}\\|\\alpha_{g}(z_n) - \\alpha_{g_j}(z_n)\\|_\\phi^\\# < \\delta.\\]\nSince $\\max_{g \\in K}\\| \\alpha_g^\\omega(z)-z\\|_{\\phi^\\omega}^\\# \\leq \\varepsilon$, there exists $W_2 \\in \\omega$ such that for all $n \\in W_2$ and $j=1, \\hdots, N$ \n\\[\\|\\alpha_{g_j}(z_n) - z_n\\|_\\phi^\\# \\leq \\varepsilon+\\delta.\\]\nHence, for an arbitrary $g \\in K$, there is some $j \\in \\{1, \\hdots, N\\}$ such that $g \\in U_j$ and\n\\[\\| \\alpha_g(z_n)-z_n\\|_\\phi^\\# \\leq \\|\\alpha_g(z_n) - \\alpha_{g_j}(z_n)\\|_\\phi^\\# + \\|\\alpha_{g_j}(z_n) - z_n\\|_\\phi^\\# \\leq 2\\delta+ \\varepsilon\\]\nfor all $n \\in W_1 \\cap W_2.$\nSince $\\delta$ was arbitrary, this proves the claim. \n\\end{proof}\n\n\\subsection{Cocycle morphisms}\n\n\\begin{definition}[cf.\\ {\\cite[Definition 1.10]{Szabo21cc}}]\nLet $\\alpha\\colon G \\curvearrowright M$ and $\\beta\\colon G \\curvearrowright N$ be two actions of a second-countable locally compact group on von Neumann algebras. \nA \\emph{cocycle morphism} from $(M,\\alpha)$ to $(N,\\beta)$ is a pair $(\\phi,\\mathbbm{u})$, where $\\phi\\colon M \\rightarrow N$ is a unital normal $*$-homomorphism and $\\mathbbm{u}:G\\rightarrow \\mathcal{U}(N)$ is a continuous map (in the strong operator topology) such that for all $g,h \\in G$ we have\n\\[\n\\mathrm{Ad}(\\mathbbm{u}_g) \\circ \\beta_g \\circ \\phi = \\phi \\circ \\alpha_g \\quad\\text{and}\\quad \\mathbbm{u}_g \\beta_g(\\mathbbm{u}_h) = \\mathbbm{u}_{gh}.\n\\]\nIn the special case where $\\mathbbm{u}$ is the trivial map, we identify $\\phi$ with $(\\phi,1)$ and call $\\phi$ equivariant.\n\\end{definition}\n\n\\begin{remark} \\label{rem:cocycle-category}\nAs the arguments in \\cite[Subsection 1.3]{Szabo21cc} show, the above endows the class of continuous $G$-actions on von Neumann algebras with a categorical structure, whereby the Hom-sets are given by cocycle morphisms.\nThe composition is given via\n\\[\n(\\psi,\\mathbbm{v}) \\circ (\\phi,\\mathbbm{u}) := (\\psi \\circ \\phi, \\psi(\\mathbbm{u}) \\mathbbm{v})\n\\]\nfor any pair of cocycle morphisms\n\\[\n(M, \\alpha) \\overset{(\\phi,\\mathbbm{u})}{\\longrightarrow} (N, \\beta) \\quad \\text{ and } \\quad (N, \\beta)  \\overset{(\\psi,\\mathbbm{v})}{\\longrightarrow} (L, \\gamma).\n\\]\nWe see furthermore that a cocycle morphism $(\\phi, \\mathbbm{u})\\colon(M, \\alpha) \\rightarrow (N, \\beta)$ is invertible if and only if $\\phi$ is a $*$-isomorphism of von Neumann algebras, in which case we have $(\\phi, \\mathbbm{u})^{-1} = (\\phi^{-1}, \\phi^{-1}(\\mathbbm{u})^*)$.\nIf this holds, we call $(\\phi, \\mathbbm{u})$ a \\emph{cocycle conjugacy}.\nWe call two actions $\\alpha$ and $\\beta$ \\emph{cocycle conjugate}, denoted as $\\alpha \\simeq_{\\mathrm{cc}} \\beta$, if there exists a cocycle conjugacy between them.\n\\end{remark}\n\n\\begin{example} \\label{ex:inner-cc}\nLet $\\alpha\\colon G \\curvearrowright M$ be an action.\nThen every unitary $v \\in \\mathcal{U}(M)$ gives rise to a cocycle conjugacy \n\\[\n\\big( \\mathrm{Ad}(v), (v \\alpha_g(v)^*)_{g \\in G} \\big) \\colon (M, \\alpha) \\rightarrow (M, \\alpha).\\]\nWe will also write this simply as $\\mathrm{Ad}(v)$ when it is clear from context that we are talking about cocycle morphisms. \nWhen $\\beta\\colon G \\curvearrowright N$ is another action and ${(\\phi, \\mathbbm{u})\\colon (M, \\alpha) \\rightarrow (N, \\beta)}$ is a cocycle conjugacy, then\n\\[ (\\phi,\\mathbbm{u}) \\circ \\mathrm{Ad}(v) = \\mathrm{Ad}(\\phi(v)) \\circ (\\phi, \\mathbbm{u}).\\] \n\\end{example}\n\n\n\\begin{definition}\nLet $\\alpha:G \\curvearrowright M$ and $\\beta\\colon G \\curvearrowright N$ be two actions on finite von Neumann algebras $M$ and $N$.\nLet $\\tau_N$ be a faithful normal tracial state on $N$.\nLet $(\\phi,\\mathbbm{u})$ and $(\\psi,\\mathbbm{v})$ be two cocycle morphisms from $(M,\\alpha)$ to $(N,\\beta)$. We say that $(\\phi,\\mathbbm{u})$ and $(\\psi,\\mathbbm{v})$ are \\emph{approximately unitarily equivalent} if there exists a net of unitaries $w_\\lambda \\in \\mathcal{U}(N)$ such that $\\|w_\\lambda\\phi(x)w_\\lambda^*-\\psi(x)\\|_{\\tau_N} \\to 0$ for all $x\\in M$ and $\\max_{g\\in K} \\| w_\\lambda \\mathbbm{u}_g \\beta_g(w_\\lambda)^* - \\mathbbm{v}_g\\|_{\\tau_N} \\rightarrow 0$ for every compact set $K\\subseteq G$.\nWe denote the relation of approximately unitary equivalence by $\\approx_{\\mathrm{u}}$.\n\\end{definition}\n\n\\section{One-sided intertwining}\nIn this section we prove a version of \\cite[Lemma~2.1]{Szabo18ssa} for group actions on semi-finite von Neumann algebras. First we prove the following intermediate lemma:\n\n\\begin{lemma}\\label{lemma:point_strong_dense_subset}\nLet $M, N$ be von Neumann algebras, and let $\\tau_N$ be  a faithful, normal, semi-finite trace on $N$. \nConsider a sequence of $*$-homomorphisms $(\\theta_n\\colon M \\rightarrow N)_{n \\in \\mathbb{N}}$ and a $*$-isomorphism $\\theta:M \\rightarrow N$ such that $\\tau_N \\circ \\theta = \\tau_N \\circ \\theta_n$ for all $n \\in \\mathbb{N}$. \nLet $X \\subset (M)_1$ be a dense subset in the strong operator topology that contains a sequence of projections $(p_n)_{n \\in \\mathbb{N}}$ converging strongly to $1_M$ with $\\tau_N (\\theta(p_n)) < \\infty$. \nIf $\\theta_n (x) \\rightarrow \\theta(x)$ strongly as $n \\rightarrow \\infty$ for every $x \\in X$, then $\\theta_n \\rightarrow \\theta$ in the point-strong topology as $n \\rightarrow \\infty$.\n\\end{lemma}\n\\begin{proof}\nTake $y \\in (M)_1$. Since the sequence $(\\theta(p_n))_{n \\in \\mathbb{N}}$ converges strongly to $1_N$, it suffices to show that for all $k \\in \\mathbb{N}$\n\\[ (\\theta(y)-\\theta_n(y)) \\theta(p_k) \\rightarrow 0 \\text{ strongly as } n \\rightarrow \\infty.\\]\nFix $k \\in \\mathbb{N}$ and $a \\in N$ such that $\\tau_N(a^*a) < \\infty$. Given $\\varepsilon>0$, there exists $x \\in X$ such that \n\\[\\|\\theta(x-y)\\theta(p_k)\\|_{\\tau_N} < \\frac{\\varepsilon}{4\\|a\\|}.\\]\n Then there exists $n_0 \\in \\mathbb{N}$ such that for all $n \\geq n_0$ \n \\[\\|(\\theta(x p_k) - \\theta_n ( x p_k))a\\|_{\\tau_N} < \\frac{\\varepsilon}{4} \\text{ and } \\|(\\theta(p_k)-\\theta_n(p_k))a\\|_{\\tau_N} < \\frac{\\varepsilon}{4}.\\] \nFor all $n \\geq n_0$ we then get that\n\\begin{align*}\n\\|(\\theta(y) - \\theta_n(y))\\theta(p_k)a\\|_{\\tau_N} &\\leq \\|\\theta(x-y)\\theta(p_k)a\\|_{\\tau_N} + \\|\\theta_n(x-y)\\theta_n(p_k)a\\|_{\\tau_N}\\\\\n&\\quad + \\|(\\theta(xp_k) - \\theta_n(xp_k))a\\|_{\\tau_N} + \\|\\theta_n(y)(\\theta(p_k) - \\theta_n(p_k))a\\|_{\\tau_N}\\\\\n&< 2\\|a\\| \\|\\theta(x-y)\\theta(p_k)\\|_{\\tau_N} +\\varepsilon\/4 + \\|(\\theta(p_k)-\\theta_n(p_k))a\\|_{\\tau_N} \\\\\n&< \\varepsilon.\n\\end{align*}\nAs $k$ and $a$ were arbitrary, this proves the claim.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:one-sided_intertwining} \nLet $M$ and $N$ be two von Neumann algebras with separable predual and faithful normal semi-finite traces $\\tau_{M}$ and $\\tau_{N}$, respectively. \nLet $\\alpha\\colon G \\curvearrowright M$ and $\\beta\\colon G \\curvearrowright N$ be two actions.\nLet $\\rho\\colon (M, \\alpha) \\rightarrow (N, \\beta)$ be a unital equivariant normal $*$-homomorphism with $\\tau_N \\circ \\rho = \\tau_M$.\nSuppose there exists a faithful normal state $\\phi$ on $N$ and a sequence of unitaries $(w_n)_{n \\in \\mathbb{N}}$ in $\\mathcal{U}(N)$ satisfying\n\\begin{enumerate}[leftmargin=*,label=$\\bullet$]\n\\item $\\mathrm{Ad}(w_n) \\circ \\rho \\to \\rho$ in the point-strong topology; \n\\item For all $y \\in (N)_1$ there exists a sequence $(x_n)_{n \\in \\mathbb{N}} \\subset (M)_1$ such that $y -  w_n\\rho(x_n)w_n^* \\rightarrow 0$ in the strong operator topology;\n\\item $\\max_{g \\in K} \\|\\beta_g(w_n^*) - w_n^*\\|_\\phi \\rightarrow 0$ for every compact subset $K \\subseteq G$.\n\\end{enumerate}\nThen $\\rho(\\mathcal{Z}(M))=\\mathcal{Z}(N)$ and there exists a cocycle conjugacy $(\\theta,\\mathbbm{v})$ between $\\alpha$ and $\\beta$ with $\\theta|_{\\mathcal{Z}(M)}=\\rho|_{\\mathcal{Z}(M)}$.\nIn case $\\tau_N$ is finite, the existence of such a sequence of unitaries for $\\phi = \\tau_N$ is equivalent to the condition that $\\rho$ is approximately unitarily equivalent to a cocycle conjugacy. \n\\end{lemma}\n\\begin{proof}\nWe note right away that the first two conditions above can always be tested on self-adjoint elements, hence one can equivalently state them with the strong-$*$ topology. \nDenote\n\\[\\mathfrak{m} := \\{x \\in M \\mid \\tau_M(x^*x) < \\infty\\} \\subset M.\\]\nWe let $L^2(M,\\tau_M)$ denote the GNS-Hilbert space of $M$ with respect to $\\tau_M$.\nSimilarily, we use the notation $L^2(N, \\tau_N)$.\n\nChoose a countable subset $X = \\{x_n\\}_{n \\in \\mathbb{N}}$ in $(M)_1$ such that $X \\cap \\mathfrak{m}$ is $\\|\\cdot\\|_{\\tau_M}$-dense in $(\\mathfrak{m})_1$. \nTake a strongly dense sequence $\\{y_n\\}_{n \\in \\mathbb{N}}$ in $(N)_1$.\nChoose an increasing sequence of compact subsets $K_n \\subseteq G$ such that the union is all of $G$. \n\nWe are going to create a map $\\theta: M\\to N$ via an inductive procedure. \nFor the first step, we choose $x_{1,1} \\in (M)_1$ and $z_1 \\in \\mathcal{U}(N)$ such that\n\\begin{itemize}\n\\item $ \\|z_1 \\rho(x_1) z_1^* - \\rho(x_1)\\|^\\#_\\phi \\leq 1\/2$;\n\\item $\\|y_1 - z_1\\rho(x_{1,1})z_1^*\\|_\\phi \\leq 1\/2$;\n\\item $\\max_{g \\in K_1} \\|\\beta_g(z_1^*) - z_1^*\\|_\\phi \\leq 1\/2$.\n\\end{itemize}\nNow assume that after the $n$-th step of the induction we have found $z_1, \\hdots, z_n \\in \\mathcal{U}(N)$ and ${\\{x_{l,j}\\}_{j \\leq l \\leq n} \\subset (M)_1}$ such that\n\\begin{enumerate}\n\\item $\\|z_n \\rho(x_j) z_n^* - \\rho(x_j)\\|^\\#_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_{n-1})} \\leq 2^{-n}$ for $j= 1, \\hdots, n$; \\label{eq:commutation_dense}\n\\item $\\|z_n \\rho(x_{l,j}) z_n^* - \\rho(x_{l,j})\\|^\\#_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_{n-1})} \\leq 2^{-n}$ for $l=1, \\hdots, n-1$ and $j = 1, \\hdots, l$; \\label{eq:commutation_close_elements}\n\\item $ \\|  z_{n-1}^* \\hdots z_1^*  y_j z_1 \\hdots z_{n-1} - z_n\\rho(x_{n,j})z_n^*\\|_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_{n-1})} \\leq 2^{-n}$ for $j=1, \\hdots, n$; \\label{eq:close_elements}\n\\item \n$\\max_{g \\in K_n} \\|\\beta_g(z_n^*) - z_n^*\\|_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_{n-1})} \\leq 2^{-n}$ and\n\n$\\max_{g \\in K_n} \\|\\beta_g(z_n^*) - z_n^*\\|_{\\phi \\circ \\mathrm{Ad}(\\beta_g(z_1 \\hdots z_{n-1}))} \\leq 2^{-n}$.\n \\label{eq:invariance}\n\n\\label{eq:approximate_fixedness}\n\\end{enumerate}\nThen by our assumptions we can find $z_{n+1} \\in \\mathcal{U}(N)$ and $\\{x_{n+1,j}\\}_{j \\leq n+1}\\subset (M)_1$ such that\n\\begin{itemize}\n\\item $\\|z_{n+1} \\rho(x_j)z_{n+1}^* - \\rho(x_j)\\|^\\#_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_n)} \\leq 2^{-(n+1)}$ for $j= 1, \\hdots, n+1$;\n\\item $\\|z_{n+1} \\rho(x_{l,j})z_{n+1}^* - \\rho(x_{l,j})\\|^\\#_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_n)} \\leq 2^{-(n+1)}$ for $l=1, \\hdots, n$ and $j = 1, \\hdots, l$;\n\\item $ \\|z_{n}^* \\hdots z_1^*y_jz_1 \\hdots z_{n} - z_{n+1}\\rho(x_{n+1,j})z_{n+1}^*\\|_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_n)} \\leq 2^{-(n+1)}$ for $j=1, \\hdots, n+1$;\n\\item $\\max_{g \\in K_{n+1}} \\|\\beta_g(z_{n+1}^*)- z_{n+1}^*\\|_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_n)} \\leq 2^{-(n+1)}$ and\n\n$\\max_{g \\in K_{n+1}} \\|\\beta_g(z_{n+1}^*)- z_{n+1}^*\\|_{\\phi \\circ \\mathrm{Ad}(\\beta_g(z_1 \\hdots z_n))} \\leq 2^{-(n+1)}$. \n\\end{itemize}\nWe carry on inductively and obtain a sequence of unitaries $(z_n)_{n \\in \\mathbb{N}}$ in $\\mathcal{U}(N)$ and a family $\\{ x_{n,j} \\}_{n\\in\\mathbb{N}, j\\leq n}\\subset (M)_1$.\nFor each $n \\in \\mathbb{N}$, we define $u_n=z_1 \\hdots z_n$ and the normal $*$-homomorphism ${\\theta_n\\colon M \\rightarrow N}$ by $\\theta_n = \\mathrm{Ad}(u_n) \\circ \\rho$.\n\nFor $n > m$ and $j=1, \\hdots, m+1$ we get\n\\begin{align*}\n\\|\\theta_n(x_j) - \\theta_m(x_j)\\|^\\#_{\\phi} &\\leq \n\\sum_{k=m}^{n-1} \\|\\theta_{k+1}(x_j) - \\theta_k(x_j)\\|^\\#_\\phi\\\\\n&= \\sum_{k=m}^{n-1} \\|z_{k+1}\\rho(x_j)z_{k+1}^* - \\rho(x_j)\\|^\\#_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_k)}\\\\\n\\overset{\\ref{eq:commutation_dense}}&{\\leq}  \\sum_{k=m}^{n-1} 2^{-k-1}.\n\\end{align*}\nWe see that for all $j\\in \\mathbb{N}$ the sequence $(\\theta_n(x_j))_{n \\in \\mathbb{N}}$ is norm-bounded and Cauchy with respect to $\\|\\cdot\\|_\\phi^\\#$.\n This means that it converges to some element in $N$ in the strong-$*$-operator topology.\n  A similar calculation using \\ref{eq:commutation_close_elements} shows that\nfor $n > m \\geq l \\geq j$  \n\\begin{equation}\\label{eq:convergence_close_elements}\\|\\theta_n(x_{l,j}) - \\theta_m(x_{l,j})\\|^\\#_{\\phi} < \\sum_{k=m}^{n-1} 2^{-k-1}, \\end{equation}\nso the sequence $(\\theta_n(x_{l,j}))_{n \\in \\mathbb{N}}$ also converges in the strong-$*$-operator topology for all $j \\leq l$.\n Since $\\theta_n$ is a $*$-homomorphism for all $n \\in \\mathbb{N}$, we conclude that, restricted to the C$^*$-algebra $A\\subset M$ generated by $\\{x_n\\}_{n \\in \\mathbb{N}} \\cup \\{x_{l,j}\\}_{j \\leq l}$, the sequence $(\\theta_n)_{n \\in \\mathbb{N}}$ converges point-$*$-strongly to a $*$-homomorphism $\\theta'\\colon A \\rightarrow N$.\nSince $A$ contains a $\\|\\cdot\\|_{\\tau_M}$-dense subset of $\\mathfrak{m}$, and clearly $\\tau_N\\circ\\theta'=\\tau_M|_A$, there is a unique isometry $T\\colon L^2(M, \\tau_M) \\rightarrow L^2(N, \\tau_N)$ induced from the formula $T[a]=[\\theta'(a)]$ for all $a\\in A\\cap\\mathfrak{m}$.\n Then the normal $*$-homomorphism\n\\[\\theta \\colon M \\rightarrow N \\colon x \\mapsto T x T^*\\]\n extends $\\theta'$ and $\\left(\\theta_n \\big \\lvert_\\mathfrak{m}\\right)_{n \\in \\mathbb{N}}$ converges point-strongly to $\\theta\\lvert_\\mathfrak{m}$.\n\nWe claim that $\\theta$ is an isomorphism.\nClearly $\\tau_N\\circ\\theta=\\tau_M$ and so $\\theta$ is injective.\nBy applying \\ref{eq:close_elements} we find for all $m \\geq j$ that\n\\begin{equation*}\n\\|\\theta_m(x_{m,j}) - y_j\\|_\\phi = \\|z_m\\rho(x_{m,j})z_m^* -  z_{m-1}^* \\hdots z_1^*  y_j z_1 \\hdots z_{m-1}\\|_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_{m-1})} < 2^{-m}.\n\\end{equation*}\nCombining this with \\eqref{eq:convergence_close_elements} for $l=m$ and $n \\rightarrow \\infty$ we find that\n\\[\\|\\theta(x_{m,j}) - y_j\\|_\\phi \\leq \\|\\theta'(x_{m,j}) - \\theta_m(x_{m,j})\\|_\\phi + \\|\\theta_m(x_{m,j}) - y_j\\|_\\phi \\leq 2^{-m} + 2^{-m} = 2^{-m+1}.\\]\nSince the $y_j$ are strongly dense in the unit ball of $N$ and $\\theta$ is normal, this implies surjectivity of $\\theta$.\nBy Lemma~\\ref{lemma:point_strong_dense_subset} it then follows that $\\theta_n\\to \\theta$ point-strongly as $n\\to\\infty$.\nSince $\\theta_n$ is a unitary perturbation of $\\rho$ for each $n$, this implies $\\rho|_{\\mathcal{Z}(M)}=\\theta_n|_{\\mathcal{Z}(M)}\\to\\theta|_{\\mathcal{Z}(M)}$ and in particular $\\rho(\\mathcal{Z}(M))=\\theta(\\mathcal{Z}(M))=\\mathcal{Z}(N)$.\n\nFor $n > m$ and $g \\in K_{m+1}$ we have\n\\begin{align*}\n&\\|z_1 \\hdots z_n \\beta_g(z_n^* \\hdots z_1^*) - z_1 \\hdots z_m \\beta_g(z_m^*\\hdots z_1^*)\\|_{\\phi}^\\#\\\\\n&\\leq \\sum_{k=m}^{n-1} \\|z_1 \\hdots z_k(z_{k+1}\\beta_g(z_{k+1}^*) - 1) \\beta_g(z_k^*\\hdots z_1^*)\\|_{\\phi}^\\#\\\\\n&=\\sum_{k=m}^{n-1} \\big( \\|\\beta_g(z_{k+1}^*) - z_{k+1}^*\\|^2_{\\phi \\circ \\mathrm{Ad}(\\beta_g(z_1 \\hdots z_k))} + \\|\\beta_g(z_{k+1}^*) - z_{k+1}^*\\|^2_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_k)} \\big)^{1\/2}\\\\\n\\overset{\\ref{eq:invariance}}&{\\leq} \\sqrt{2} \\sum_{k=m}^{n-1} 2^{-(k+ 1)}.\n\\end{align*}\nFrom this calculation we see that for every $g \\in G$ the sequences $(z_1 \\hdots z_n \\beta_g(z_n^*\\hdots z_1^*))_{n \\in \\mathbb{N}}$ are Cauchy with respect to $\\|\\cdot\\|^\\#_\\phi$, with uniformity on compact sets. \nIt follows that for every $g \\in G$, the strong-$*$ limit $\\mathbbm{v}_g = \\lim_{n \\rightarrow \\infty} u_n \\beta_g(u^*_n)$ exists in $\\mathcal{U}(N)$ and that this convergence is uniform (w.r.t.\\ $\\|\\cdot\\|^\\#_\\phi$) on compact sets. \nSince $\\beta$ is point-strong continuous, this implies the continuity of the assignment $g \\mapsto \\mathbbm{v}_g$. \n\nMoreover, for each $g \\in G$ and $x \\in M$ we have the equalities of limits with respect to the strong operator topology:\n\\begin{align*}\n(\\theta \\circ \\alpha_g) (x) &= \\lim_{n \\rightarrow \\infty} (\\mathrm{Ad}(u_n) \\circ \\rho \\circ \\alpha_g)(x)\\\\\n&= \\lim_{n \\rightarrow \\infty} (\\mathrm{Ad}(u_n) \\circ \\beta_g \\circ \\rho) (x)\\\\\n&= \\lim_{n \\rightarrow \\infty} u_n \\beta_g(u_n^*)\\beta_g(u_n \\rho(x) u_n^*) \\beta_g(u_n) u_n^*\\\\\n&= (\\mathrm{Ad}(\\mathbbm{v}_g) \\circ \\beta_g \\circ \\theta) (x).\n\\end{align*}\nIt follows that $(\\theta,\\mathbbm{v})$ is a cocycle conjugacy. \n\nFor the last part of the statement, assume that $\\tau_N$ is finite.\nThen our previous calculations show that in the above situation, $\\rho$ is approximately unitarily equivalent to $\\theta$.\nConversely, suppose $\\rho$ is approximately unitarily equivalent to a cocycle conjugacy $(\\theta,\\mathbbm{v})$.\nIn particular, there exists a sequence $(u_n)_{n \\in \\mathbb{N}} \\in \\mathcal{U}(N)$ such that $\\|u_n\\rho(x)u_n^* -  \\theta(x)\\|_{\\tau_N} \\rightarrow 0$ for all $x \\in M$ and $\\|u_n \\beta_g(u_n^*) - \\mathbbm{v}_g\\|_{\\tau_N} \\rightarrow 0$ uniformly over compact subsets of $G$.\nChoose a sequence $\\{y_n\\}_{n \\in \\mathbb{N}} \\subset (N)_1$ that is strongly dense in $(N)_1$. For all $k,n \\in \\mathbb{N}$ define $x_{n,k} = \\theta^{-1}(u_ny_ku_n^*)$. Then choose an increasing sequence $(m(n))_{n \\in \\mathbb{N}} \\subset \\mathbb{N}$ such that\n \\begin{equation*}\n \\lim_{n \\rightarrow \\infty} \\| \\theta(x_{n,k}) - u_{m(n)} \\rho (x_{n,k}) u_{m(n)}^*\\|_{\\tau_N} = 0 \\quad \\text{for } k \\in \\mathbb{N}.\\end{equation*}\nDefine $w_n := u_n^*u_{m(n)}$. One can check that these satisfy the assumptions in the lemma. \n\\end{proof}\n\n\\section{Strongly self-absorbing actions}\n\n\\begin{definition}[cf.\\ {\\cite[Definition 5.1]{Szabo21cc}}] \\label{def:strong-absorption}\nLet $\\alpha:G \\curvearrowright M$ and $\\delta\\colon G \\curvearrowright N$ be two actions of a second-countable locally compact group on finite von Neumann algebras $M$ and $N$ with separable predual.\nWe say that $\\alpha$ \\emph{strongly absorbs} $\\delta$ if the equivariant embedding\n\\[\\mathrm{id}_M \\otimes 1_N\\colon (M, \\alpha) \\rightarrow (M \\bar{\\otimes} N, \\alpha \\otimes \\delta)\\]\nis approximately unitarily equivalent to a cocycle conjugacy.\n\\end{definition}\n\n\\begin{definition} \\label{def:ssa-action}\nLet $\\delta\\colon G \\curvearrowright \\mathcal{R}$ be an action on the hyperfinite II$_1$-factor.\nWe say that $\\delta$ is \\emph{strongly self-absorbing}, if $\\delta$ strongly absorbs $\\delta$.\n\\end{definition}\n\n\\begin{definition}\nLet $\\alpha\\colon G \\curvearrowright \\mathcal{R}$ be an action on the hyperfinite II$_1$-factor.\nWe say $\\alpha$ has \\emph{approximately inner half-flip} if the two equivariant embeddings\n\\[\n\\mathrm{id}_\\mathcal{R} \\otimes 1_\\mathcal{R}, 1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R}\\colon (\\mathcal{R}, \\alpha) \\rightarrow (\\mathcal{R} \\bar{\\otimes} \\mathcal{R}, \\alpha \\otimes \\alpha)\n\\]\nare approximately unitarily equivalent (as cocycle morphisms).\n\\end{definition}\n\n\\begin{remark}\nIt is well-known that any type II$_1$ von Neumann algebra $N$ with approximately inner half-flip in the above sense (with $G=\\{1\\}$) must be isomorphic to $\\mathcal R$. Indeed, it is clear that $N$ must have trivial center. Then $N \\cong \\mathcal{R}$ follows from \\cite[Theorem 5.1]{Connes76} under the stronger condition that the flip automorphism on $N\\bar{\\otimes} N$ is approximately inner, but the weaker condition is seen to be enough via Connes' theorem and the obvious modification of the proof of \\cite[Proposition 2.8]{EffrosRosenberg78} that shows the semi-discreteness of $N$.\n\\end{remark}\n\n\\begin{example}\\label{example:trivial_action_R}\nFor any second-countable locally compact group $G$, the trivial action $\\mathrm{id}_\\mathcal{R}\\colon G \\curvearrowright \\mathcal{R}$ has approximately inner half-flip as a consequence of the flip automorphism on a tensor product of matrix algebras $M_n(\\mathbb{C}) \\otimes M_n(\\mathbb{C})$ being inner. \nIt is also seen to be a strongly self-absorbing action.\n\\end{example}\n\n\\begin{theorem} \\label{theorem:sufficient_criterium_strong_absorption}\nLet $\\alpha\\colon G \\curvearrowright M$ be an action on a semi-finite von Neumann algebra with separable predual.\nSuppose that $\\delta\\colon G \\curvearrowright \\mathcal{R}$ is an action with approximately inner half-flip such that there exists a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow (M_{\\omega,\\alpha}, \\alpha_\\omega)$.\nThen there exists a cocycle conjugacy $(\\theta,\\mathbbm{v}): (M,\\alpha) \\to (M\\bar{\\otimes}\\mathcal{R},\\alpha \\otimes \\delta)$ with $\\theta|_{\\mathcal{Z}(M)}=\\operatorname{id}_{\\mathcal{Z}(M)}\\otimes 1_{\\mathcal R}$.\nIf $M$ is finite, then $\\alpha$ strongly absorbs $\\delta$.\n\\end{theorem}\n\\begin{proof}\nFix a faithful normal state $\\phi$ on $M$.\nLet $\\pi\\colon (\\mathcal{R},\\delta) \\rightarrow (M_{\\omega,\\alpha},\\alpha_{\\omega})$ be a unital equivariant $*$-homomorphism.\nWe obtain an induced a map on the algebraic tensor product\n\\[\n\\mathcal{R} \\odot M \\rightarrow M^\\omega_\\alpha \\text{ via } x \\otimes m \\mapsto \\pi(x) m.\n\\]\nSince for each $m \\in M_+$ the map $x \\mapsto \\phi^\\omega(\\pi(x)m)$ defines a positive tracial functional on $\\mathcal{R}$,\n we see that it must be equal to some multiple of the unique tracial state $\\tau$ on $\\mathcal{R}$ and hence,\n  we get for each $x \\in \\mathcal{R}$ and $m \\in M$ that \n\\[\\phi^\\omega(\\pi(x) m) = \\tau(x) \\phi(m) = (\\tau \\otimes \\phi)(x \\otimes m).\\]\nSo we see that the map sends the faithful normal state $\\tau \\otimes \\phi$ to $\\phi^\\omega$ and hence,\n it extends to a unital normal $*$-homomorphism $\\mathcal{R} \\bar{\\otimes} M \\rightarrow M^\\omega_\\alpha$,\n which moreover is $(\\delta \\otimes \\alpha)$-to-$\\alpha^\\omega$ equivariant.   \nIn this way we get a unital equivariant normal $*$-homomorphism\n\\[(\\mathcal{R} \\bar{\\otimes} \\mathcal{R} \\bar{\\otimes}M, \\delta \\otimes \\delta \\otimes \\alpha) \\rightarrow (\\mathcal{R} \\bar{\\otimes} M^\\omega_\\alpha, \\delta \\otimes \\alpha^\\omega),\\]\ngiven by $x_1 \\otimes x_2 \\otimes m \\mapsto x_1 \\otimes (\\phi(x_2) m)$. \nComposing with the canonical inclusion map $\\iota\\colon \\mathcal{R} \\bar{\\otimes} M^\\omega_\\alpha \\mapsto (\\mathcal{R} \\bar{\\otimes} M)^\\omega_{\\delta\\otimes\\alpha}$ we get a unital and equivariant normal $*$-homomorphism\n\\[\\Phi\\colon (\\mathcal{R} \\bar{\\otimes} \\mathcal{R} \\bar{\\otimes} M, \\delta \\otimes \\delta \\otimes \\alpha) \\rightarrow ((\\mathcal{R} \\bar{\\otimes} M)^\\omega_{\\delta \\otimes \\alpha}, (\\delta \\otimes \\alpha)^\\omega)\\]\nsuch that\n\\[\n\\Phi(x \\otimes 1_\\mathcal{R} \\otimes m) = x \\otimes m \\text{ for all } x \\in \\mathcal{R}, m \\in M,\n\\]\nand\n\\[\n\\Phi(1_\\mathcal{R} \\otimes \\mathcal{R} \\otimes M) \\subset \\iota(1_\\mathcal{R} \\otimes M^\\omega_\\alpha).\n\\]\nSince $\\delta$ has approximately inner half-flip,\n we can choose a sequence of unitaries $(v_n)_{n \\in \\mathbb{N}}$ in $\\mathcal{R} \\bar{\\otimes} \\mathcal{R}$ such that \n $\\max_{g \\in K}\\|v_n - (\\delta \\otimes \\delta)_g(v_n)\\|_{\\tau \\otimes \\tau} \\rightarrow 0$ for all compact subsets $K \\subseteq G$ and \n ${\\|x \\otimes 1_\\mathcal{R} - v_n (1_\\mathcal{R} \\otimes x)v_n^*\\|_{\\tau \\otimes \\tau}  \\rightarrow 0}$ for all $x \\in \\mathcal{R}$. \nDefine $u_n := \\Phi(v_n \\otimes 1_M) \\subset (\\mathcal{R} \\bar{\\otimes} M)^\\omega_{\\delta \\otimes \\alpha}$. This sequence of unitaries satisfies\n\\begin{itemize} \n\\item $[u_n, 1_\\mathcal{R} \\otimes m] = \\Phi([v_n \\otimes 1_M, 1_{\\mathcal{R} \\bar{\\otimes} \\mathcal{R}} \\otimes m]) = 0$ for all $m \\in M$;\n\\item  $\\Phi(1_\\mathcal{R} \\otimes x \\otimes m) \\in \\iota(1_\\mathcal{R} \\otimes M^\\omega_\\alpha)$ and\n\\begin{align*}\n\\lim_{n \\rightarrow \\infty} u_n \\Phi(1_\\mathcal{R} \\otimes x \\otimes m) u_n^* &=\\lim_{n \\rightarrow \\infty} \\Phi((v_n \\otimes 1_M)(1_\\mathcal{R} \\otimes x \\otimes m)(v_n^* \\otimes 1_M))\\\\\n&=\\Phi(x \\otimes 1_\\mathcal{R} \\otimes m) \\\\&= x \\otimes m\n\\end{align*}\nwhere the limit is taken with respect to the strong operator topology;\n\\item $\\displaystyle \\max_{g \\in K} \\|u_n^* - (\\delta \\otimes \\alpha)^\\omega_g(u_n^*)\\|_{(\\tau \\otimes \\phi)^\\omega} = \\max_{g \\in K} \\|(v_n^* - (\\delta \\otimes \\delta)_g(v_n^*)) \\otimes 1_M\\|_{\\tau \\otimes \\tau \\otimes \\phi} \\to 0$ for all compact $K \\subseteq G$.\n\\end{itemize}\n\n Each $u_n$ can be lifted to a sequence of unitaries $(z_n^{(k)})_{k\\in \\mathbb{N}}$ in $\\mathcal{E}_{\\delta \\otimes \\alpha}^\\omega \\cap \\mathcal{N}_\\omega(\\mathcal{R} \\bar{\\otimes} M)$. Applying a diagonal sequence argument to the $(z_n^{(k)})_{k\\in \\mathbb{N}}$ and using Lemma~\\ref{lemma:lifting_invariance_compact_sets}, we can obtain a sequence of unitaries $(w_n)_{n \\in \\mathbb{N}}$ in $\\mathcal{R} \\bar{\\otimes} M$ such that\n\\begin{itemize}\n\\item $\\mathrm{Ad}(w_n)(1_\\mathcal{R} \\otimes m) - 1_\\mathcal{R} \\otimes m \\rightarrow 0$ strongly for all $m \\in M$.\n\\item $\\inf_{m \\in (M)_1}\\|x - w_n(1_\\mathcal{R} \\otimes m)w_n^*\\|_{\\tau \\otimes \\phi} \\rightarrow 0$ for $x \\in (\\mathcal{R} \\bar{\\otimes}M)_1$.\n\\item $\\max_{g \\in K} \\|w_n^* - (\\delta \\otimes \\alpha)_g(w_n^*)\\|_{\\tau \\otimes \\phi} \\rightarrow 0$ for every compact subset $K \\subseteq G$.\n\\end{itemize}\nWe conclude that the map $1_\\mathcal{R} \\otimes \\mathrm{id}_M\\colon (M, \\alpha) \\rightarrow (\\mathcal{R} \\bar{\\otimes} M, \\delta \\otimes \\alpha)$ satisfies all the necessary conditions to apply Lemma \\ref{lemma:one-sided_intertwining}. This completes the proof.\n\\end{proof}\n\n\\begin{theorem} \\label{theorem:equivalence_ssa}\nLet $\\delta:G \\curvearrowright \\mathcal{R}$ be an action on the hyperfinite II$_1$-factor.\nThen $\\delta$ is strongly self-absorbing if and only if it has approximately inner half-flip and there exists a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow (\\mathcal{R}_{\\omega,\\delta}, \\delta_\\omega)$. \n\\end{theorem}\n\\begin{proof}\nThe `if' direction follows immediately from the previous proposition.\nTo prove the other direction, we assume that $\\delta$ is strongly self-absorbing and reproduce an argument analogous to \\cite[Proposition 1.4]{TomsWinter07} and \\cite[Proposition 5.5]{Szabo21cc}.\nDenote the unique tracial state on $\\mathcal{R}$ by $\\tau$. Let $(\\phi,\\mathbbm{u})\\colon (\\mathcal{R}, \\delta) \\rightarrow (\\mathcal{R} \\bar{\\otimes} \\mathcal{R}, \\delta \\otimes \\delta)$ be a cocycle conjugacy and $u_n \\in \\mathcal{U}(\\mathcal{R} \\bar{\\otimes} \\mathcal{R})$ a sequence of unitaries such that\n\\begin{equation}\\label{eq:approx_coboundary} \\lim_{n \\rightarrow \\infty} \\max_{g \\in K} \\| u_n(\\delta \\otimes \\delta)_g(u_n^*) - \\mathbbm{u}_g\\|_{\\tau \\otimes \\tau} = 0 \\text{ for every compact } K \\subseteq G, \\text{ and }\\end{equation} \n\\[\n\\lim_{n \\rightarrow \\infty} \\|\\phi(x) - u_n(x \\otimes 1) u_n^*\\|_{\\tau \\otimes \\tau} = 0 \\text{ for all } x \\in \\mathcal{R}.\n\\]\nNote that \n\\[\\mathrm{Ad}(u_n^*) \\circ \\phi \\circ \\delta_g = \\mathrm{Ad}(u_n^*\\mathbbm{u}_g (\\delta \\otimes \\delta)_g(u_n)) \\circ (\\delta \\otimes \\delta)_g \\circ \\mathrm{Ad}(u_n^*) \\circ \\phi.\\]\nAs a consequence of \\eqref{eq:approx_coboundary}, then for every compact $K \\subseteq G$ one has \n\\begin{equation}\\label{eq:approximate_invariance_perturbation}\\lim_{n \\rightarrow \\infty} \\max_{g \\in K} \\sup_{x \\in (\\mathcal{R})_1} \\|(\\mathrm{Ad}(u_n^*) \\circ \\phi \\circ \\delta_g)(x) - ((\\delta \\otimes \\delta)_g \\circ \\mathrm{Ad}(u_n^*) \\circ \\phi)(x)\\|_{\\tau \\otimes \\tau} = 0.\\end{equation}\nIn particular, applying this to $x = \\phi^{-1}(u_n)$ and using that $(\\tau \\otimes \\tau) = \\tau \\circ \\phi^{-1}$ yields\n\\[\\lim_{n \\rightarrow \\infty} \\max_{g \\in K}\\|(\\mathrm{Ad}(\\phi^{-1}(u_n^*)) \\circ \\delta_g) (\\phi^{-1}(u_n)) -(\\phi^{-1}\\circ (\\delta \\otimes \\delta)_g)(u_n)\\|_\\tau = 0.\\]\nCombining this with \\eqref{eq:approx_coboundary} again, one gets\n\\begin{equation}\\label{eq:approx_coboundary_inverse}\n \\lim_{n \\rightarrow \\infty} \\max_{g \\in K} \\| \\phi^{-1}(u_n^*)\\, \\delta_g(\\phi^{-1}(u_n)) - \\phi^{-1}(\\mathbbm{u}^*_g)\\|_{\\tau \\otimes \\tau} = 0 \\text{ for every compact } K \\subseteq G.\n\\end{equation}\nFirst, we prove that $\\delta$ has approximately inner half-flip. \nDefine the cocyle morphism $(\\psi, \\mathbbm{v}) := (\\phi, \\mathbbm{u})^{-1}  \\circ (1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R})$. Note that\n\\begin{align*}\n1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R} \n&= (\\phi, \\mathbbm{u}) \\circ (\\psi, \\mathbbm{v})\\\\\n&\\approx_{\\mathrm{u}} (\\mathrm{id}_\\mathcal{R} \\otimes 1_\\mathcal{R}) \\circ (\\psi, \\mathbbm{v})\\\\\n&= (\\psi \\otimes 1_\\mathcal{R}, \\mathbbm{v} \\otimes 1).\n\\end{align*} \nApplying the equivariant flip automorphism to both sides of this equivalence, we get that\n\\begin{equation}\\label{eq:approx_equivalence_first_factor_embedding} \\mathrm{id}_\\mathcal{R} \\otimes 1_\\mathcal{R} \\approx_\\mathrm{u} (1_\\mathcal{R} \\otimes \\psi, 1 \\otimes \\mathbbm{v}) .\\end{equation}\nWe also get\n\\begin{align*}\n(\\psi \\otimes 1_\\mathcal{R}, \\mathbbm{v} \\otimes 1) &= (\\phi^{-1} \\otimes \\mathrm{id}_\\mathcal{R}, \\phi^{-1}(\\mathbbm{u})^* \\otimes 1) \\circ (1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R} \\otimes 1_\\mathcal{R})\\\\\n\\overset{\\eqref{eq:approx_equivalence_first_factor_embedding}}&{\\approx_{\\mathrm{u}}} (\\phi^{-1} \\otimes \\mathrm{id}_\\mathcal{R}, \\phi^{-1}(\\mathbbm{u})^* \\otimes 1) \\circ (1_\\mathcal{R} \\otimes 1_\\mathcal{R} \\otimes \\psi, 1\\otimes 1 \\otimes \\mathbbm{v})\\\\\n&=(1_\\mathcal{R} \\otimes \\psi, \\phi^{-1}(\\mathbbm{u})^* \\otimes \\mathbbm{v})\\\\\n\\overset{\\eqref{eq:approx_coboundary_inverse}}&{\\approx_{\\mathrm{u}}} (1_\\mathcal{R} \\otimes \\psi, 1 \\otimes \\mathbbm{v}).\n\\end{align*}\nBy transitivity we get that ${1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R}\\approx_\\mathrm{u} \\mathrm{id}_\\mathcal{R} \\otimes 1_\\mathcal{R}}$.\n\nNext we prove the existence of a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow (\\mathcal{R}_{\\omega,\\delta}, \\delta_\\omega)$. Define the sequence of trace-preserving $*$-homomorphisms \n\\[\\chi_n = \\phi^{-1} \\circ \\mathrm{Ad}(u_n) \\circ (1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R}).\\] \nWe conclude from \\eqref{eq:approximate_invariance_perturbation} that for all $x \\in \\mathcal{R}$\n\\[\\lim_{n \\rightarrow \\infty} \\max_{g \\in K} \\|\\delta_g(\\chi_n(x)) - \\chi_n(\\delta_g(x))\\|_{\\tau} =0.\\]\nFrom this and the fact that all $\\chi_n$ are trace-preserving it also follows that $(\\chi_n(x))_{n \\in \\mathbb{N}}$ belongs to $\\mathcal{E}^\\omega_\\alpha$.\nMoreover, for any $x,y \\in \\mathcal{R}$\n\\begin{align*}\n\\lim_{n \\rightarrow \\infty} \\|[x, \\chi_n(y)] \\|_\\tau\n&= \\lim_{n \\rightarrow \\infty}\\|[\\phi(x), u_n(1 \\otimes y) u_n^*]\\|_{\\tau \\otimes \\tau}\\\\\n&= \\lim_{n \\rightarrow \\infty}\\|u_n[x \\otimes 1, 1 \\otimes y] u_n^*\\|_{\\tau \\otimes \\tau}\\\\\n&=0. \n\\end{align*}\nSo the $\\chi_n$ induce a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow (\\mathcal{R}_{\\omega,\\delta}, \\delta_\\omega)$.\n\\end{proof}\n\nThe following can be seen as a direct generalization of the famous McDuff theorem \\cite{McDuff70} to actions on semi-finite von Neumann algebras.\n\n\\begin{corollary} \\label{cor:equivalence_equivariant_McDuff}\nLet $\\alpha:G \\curvearrowright M$ be an action on a semi-finite von Neumann algebra with separable predual and let $\\delta:G \\curvearrowright \\mathcal{R}$ be a strongly self-absorbing action on the hyperfinite II$_1$-factor.\nThen the following are equivalent:\n\\begin{enumerate}[leftmargin=*,label=\\textup{(\\arabic*)}]\n\\item There exists a cocycle conjugacy $(\\theta,\\mathbbm{v})\\colon (M,\\alpha) \\to (M\\bar{\\otimes}\\mathcal{R},\\alpha \\otimes \\delta)$ with $\\theta|_{\\mathcal{Z}(M)}=\\operatorname{id}_{\\mathcal{Z}(M)}\\otimes 1_{\\mathcal R}$; \\label{prop:McDuff:1}\n\\item $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\delta$; \\label{prop:McDuff:2}\n\\item There exists a unital equivariant $*$-homomorphism $(\\mathcal{R},\\delta) \\rightarrow (M_{\\omega,\\alpha}, \\alpha_\\omega)$. \\label{prop:McDuff:3}\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof}\nThe implication \\ref{prop:McDuff:1}$\\Rightarrow$\\ref{prop:McDuff:2} is tautological.\nSince strong self-absorption implies approximately inner half-flip by Proposition~\\ref{theorem:equivalence_ssa}, the implication \\ref{prop:McDuff:3}$\\Rightarrow$\\ref{prop:McDuff:1} follows from Proposition~\\ref{theorem:sufficient_criterium_strong_absorption}.\n\nIn order to prove \\ref{prop:McDuff:2}$\\Rightarrow$\\ref{prop:McDuff:3}, it is enough to show that there exists a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow ((M \\bar{\\otimes} \\mathcal{R})_{\\omega,\\alpha \\otimes \\delta}, (\\alpha \\otimes \\delta)_\\omega)$.\nWe know there exists a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow (\\mathcal{R}_{\\omega,\\delta}, \\delta_\\omega)$ by Proposition \\ref{theorem:equivalence_ssa}.\nSince the latter is unitally and equivariantly contained in $((M \\bar{\\otimes} \\mathcal{R})_{\\omega,\\alpha \\otimes \\delta}, (\\alpha \\otimes \\delta)_\\omega)$, this finishes the proof.\n\\end{proof}\n\nThe following lemma is a straightforward application of the noncommutative Rokhlin Theorem of Masuda \\cite[Theorem 4.8]{Masuda13}\\footnote{This Rokhlin Theorem is actually a variant of Ocneanu's noncommutative Rokhlin Theorem \\cite[Theorem 6.1]{Ocneanu85}, and the proof of Masuda's version is essentially the same as Ocneanu's proof. While it is possible to deduce what we need from Ocneanu's Theorem, here we cite Masuda's version for convenience of the reader, as it is directly applicable and there is no need to deal with $\\varepsilon$-paving families of $G$.}\n\n\\begin{lemma}\\label{lem:approx-central-embeddings}\nLet $\\alpha\\colon G \\curvearrowright M$ be an action of a countable discrete group on a McDuff factor (i.e., $M \\cong M \\bar{\\otimes} \\mathcal{R}$) with separable predual.\nLet $N\\subseteq G$ be the normal subgroup consisting of all elements $g\\in G$ such that $\\alpha_{\\omega,g} \\in\\operatorname{Aut}(M_\\omega)$ is trivial.\nSuppose that the quotient group $G_0=G\/N$ is amenable with quotient map $\\pi: G\\to G_0$.\nLet $\\delta: G_0\\curvearrowright\\mathcal R$ be an action with induced $G$-action $\\delta_\\pi=\\delta\\circ\\pi$. \nThen there exists an equivariant unital $*$-homomorphism $(\\mathcal R,\\delta_\\pi)\\to (M_\\omega,\\alpha_\\omega)$.\n\\end{lemma}\n\\begin{proof}\nConsider the induced faithful action $\\gamma: G_0\\curvearrowright M_\\omega$ via $\\gamma_{gN}=\\alpha_{\\omega,g}$.\nThen clearly the claim is equivalent to finding a $G_0$-equivariant unital $*$-homomorphism $(\\mathcal R,\\delta)\\to (M_\\omega,\\gamma)$.\nLet us introduce some notation.\nLet $(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M)$ be a sequence representing an element $X \\in M_\\omega$. Then we set $\\tau_\\omega(X) = \\lim_{n \\rightarrow \\omega} x_n$,  where the limit is taken in the $\\sigma$-weak topology.\nSince $M$ is a factor and $\\tau_\\omega(X)$ is central, this limit belongs to $\\mathbb{C}$. For any $\\phi \\in M_*$ we have\n\\[\\phi^\\omega(X) = \\lim_{n \\rightarrow \\omega}\\phi(x_n) = \\phi(\\tau_\\omega(X)) = \\tau_\\omega(X).\\]\nIn particular, $\\tau_\\omega$ defines a normal faithful tracial state on $M_\\omega$ and we denote $\\|X\\|_1 = \\tau_\\omega(|X|)$.\n\nSince $M$ is McDuff we can find a unital $*$-homomorphism $\\Phi: \\mathcal{R} \\rightarrow M_\\omega$. Fix $\\varepsilon >0$ and a symmetric finite subset $F \\subset\\joinrel\\subset G_0$ containing the neutral element.\nBy \\cite[Lemmas 5.6 and 5.7]{Ocneanu85} we are allowed to apply \\cite[Theorem 4.8]{Masuda13} to the action $\\gamma\\colon G_0\\curvearrowright M_\\omega$.\nSo if $S \\subset\\joinrel\\subset G_0$ is a finite $(F,\\varepsilon)$-invariant subset, then there exists a partition of unity of projections $\\{E_s\\}_{s \\in S} \\subset M_\\omega$ such that\n\\begin{align}\n\\sum_{s \\in g^{-1}S \\cap S} \\|\\gamma_{g}(E_s) - E_{gs}\\|_1 &< 4\\varepsilon^{1\/2} \\text{ for all } g \\in F; \\label{eq:r-lemma1}\\\\\n\\sum_{s \\in S\\setminus g^{-1}S} \\|E_s\\|_1 &< 3\\varepsilon^{1\/2} \\text{ for all } g \\in F;\\label{eq:r-lemma2}\\\\\n[E_s, \\gamma_h(X)] &= 0 \\text{ for all } s \\in S,\\ h\\in G_0,\\ X \\in \\Phi(\\mathcal{R}).\\label{eq:commutation_image}\n\\end{align}\nDefine \n\\[\n\\Psi: \\mathcal{R} \\rightarrow M_\\omega \\text{ via } \\Psi(x) = \\sum_{s \\in S} \\gamma_{s}(\\Phi(\\delta_{s}^{-1}(x))) E_s.\n\\]\nThis is a unital trace-preserving $*$-homomorphism because the projections $E_s$ form a partition of unity and condition \\eqref{eq:commutation_image}.\nFor $g \\in F$ and $x \\in \\mathcal{R}$ we use conditions \\eqref{eq:r-lemma1} and \\eqref{eq:r-lemma2} to observe\n\\begin{align*}\n\\|\\gamma_{g}(\\Psi(x)) - \\Psi(\\delta_g(x))\\|_1 &=  \\Big\\| \\sum_{s \\in S} \\gamma_{gs}(\\Phi(\\delta_s^{-1}(x))) \\gamma_{g}(E_s) - \\sum_{s \\in g^{-1}S} \\gamma_{gs}(\\Phi(\\delta_{s}^{-1}(x))) E_{gs} \\Big\\|_1\n\\\\\n&\\leq \\sum_{s \\in S \\cap g^{-1}S}  \\left\\| \\gamma_{gs}(\\Phi(\\delta_s^{-1}(x))) ( \\gamma_{g}(E_s) - E_{gs}) \\right\\|_1\\\\\n& \\qquad + \\sum_{s \\in S\\setminus g^{-1}S} \\|\\gamma_s(\\Phi(\\delta_s^{-1}(x))) E_s\\|_1 + \\sum_{s \\in S \\setminus gS} \\|\\gamma_{s}(\\Phi(\\delta_{g^{-1}s}^{-1}(x)))E_s\\|_1 \\\\\n&< 10\\varepsilon^{1\/2}\\|x\\|.\n\\end{align*}\nSince we can do this for arbitrary $\\varepsilon >0$ and $F \\subset\\joinrel\\subset G$, the claim follows via a standard reindexing trick.\n\\end{proof}\n\nThe following result recovers a famous result due to Ocneanu \\cite[Theorem 1.2 and following remark]{Ocneanu85} as well as his uniqueness theorem of outer actions of amenable groups on $\\mathcal R$.\nWe include this proof for the reader's benefit as it is comparably elementary with the methods established so far.\n\n\\begin{theorem} \\label{theorem:model-absorption}\nLet $G$ and $G_1$ be countable discrete groups with $G_1$ amenable.\nLet $\\delta: G_1\\curvearrowright\\mathcal R$ be an outer action and $\\alpha\\colon G \\curvearrowright M$ an action on a semi-finite McDuff factor (i.e.\\ $M \\cong M \\bar{\\otimes} \\mathcal{R}$) with separable predual.\nThen:\n\\begin{enumerate}[label=\\textup{(\\roman*)},leftmargin=*]\n\\item $\\delta$ is strongly self-absorbing and cocycle conjugate to any other outer action $G_1\\curvearrowright\\mathcal R$.\\label{theorem:model-absorption:1}\n\\item Suppose $H\\subseteq G$ is a normal subgroup containing all elements $g\\in G$ such that $\\alpha_{\\omega,g}$ is trivial.\nSuppose $G_1=G\/H$ with quotient map $\\pi: G\\to G_1$.\nThen $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\delta_\\pi$. \\label{theorem:model-absorption:2}\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\\ref{theorem:model-absorption:1}:\nLet $\\tau$ be the unique tracial state on $\\mathcal R$, which we may use to define the 1-norm $\\|\\cdot\\|_1=\\tau(|\\cdot|)$ on $\\mathcal R$.\nSet $\\delta^{(2)}=\\delta\\otimes\\delta: G_1\\curvearrowright \\mathcal R\\bar{\\otimes}\\mathcal R=:\\mathcal R^{(2)}$, which is also an outer action.\nSince the flip automorphism $\\sigma$ on $\\mathcal R^{(2)}$ is known to be approximately inner, we may pick a unitary $U\\in\\mathcal{U}(\\mathcal R^{(2)\\omega})$ with $UxU^*=\\sigma(x)$ for all $x\\in\\mathcal R^{(2)}$.\n\nBy \\cite[Theorem 3.2]{Connes77}, the induced action $\\delta^{(2)\\omega}: G_1\\curvearrowright\\mathcal R^{(2)}_\\omega$ is faithful.\nWe may hence argue exactly as in the proof of Lemma~\\ref{lem:approx-central-embeddings} and apply Masuda's noncommutative Rokhlin lemma.\nSo let $F\\subset\\joinrel\\subset G_1$ be a symmetric finite set and $\\varepsilon>0$.\nIf $S \\subset\\joinrel\\subset G_1$ is a finite $(F,\\varepsilon)$-invariant subset, then there exists a partition of unity of projections $\\{E_s\\}_{s \\in S} \\subset \\mathcal R^{(2)}_\\omega$ such that\n\\begin{align}\n\\sum_{s \\in g^{-1}S \\cap S} \\|\\delta^{(2)\\omega}_{g}(E_s) - E_{gs}\\|_1 &< 4\\varepsilon^{1\/2} \\text{ for all } g \\in F; \\label{eq:r-lemma1-}\\\\\n\\sum_{s \\in S\\setminus g^{-1}S} \\|E_s\\|_1 &< 3\\varepsilon^{1\/2} \\text{ for all } g \\in F;\\label{eq:r-lemma2-}\\\\\n[E_s, x] &= 0 \\text{ for all } s \\in S,\\ x\\in\\{ \\delta^{(2)\\omega}_h(U)\\}_{h\\in G_1}.\\label{eq:commutation_image-}\n\\end{align}\nDefine $W = \\sum_{s \\in S} \\delta^{(2)\\omega}_{s}(U) E_s$.\nThis is also a unitary in $\\mathcal R^{(2)\\omega}$ implementing the flip $\\sigma$ because the projections $E_s$ form a partition of unity and condition \\eqref{eq:commutation_image-}.\nFor $g \\in F$ we use conditions \\eqref{eq:r-lemma1-} and \\eqref{eq:r-lemma2-} to observe\n\\begin{align*}\n\\|\\delta^{(2)\\omega}_{g}(W) - W\\|_1 &=  \\Big\\| \\sum_{s \\in S} \\delta^{(2)\\omega}_{gs}(U) \\delta^{(2)\\omega}_{g}(E_s) - \\sum_{s \\in g^{-1}S} \\delta^{(2)\\omega}_{gs}(U) E_{gs} \\Big\\|_1\n\\\\\n&\\leq \\sum_{s \\in S \\cap g^{-1}S}  \\left\\| \\delta^{(2)\\omega}_{gs}(U) ( \\delta^{(2)\\omega}_{g}(E_s) - E_{gs}) \\right\\|_1\\\\ & \\qquad + \\sum_{s \\in S\\setminus g^{-1}S} \\| \\delta^{(2)\\omega}_{s}(U) E_s\\|_1 + \\sum_{s \\in S \\setminus gS} \\|\\delta^{(2)\\omega}_{s}(U) E_{s}\\|_1 \\\\\n&< 10\\varepsilon^{1\/2}.\n\\end{align*}\nSince we can do this for arbitrary $\\varepsilon >0$ and $F \\subset\\joinrel\\subset G_1$, we can use a reindexing trick to obtain a unitary $W\\in\\mathcal{U}((\\mathcal R^{(2)\\omega})^{\\delta^{(2)\\omega}})$ with $WxW^*=\\sigma(x)$ for all $x\\in\\mathcal R^{(2)}$.\nIn particular, $\\delta$ has approximately inner half-flip.\nIf we apply Lemma~\\ref{lem:approx-central-embeddings} for $G=G_1$, $N=\\{1\\}$ and $\\delta$ in place of $\\alpha$, it follows with Theorem~\\ref{theorem:equivalence_ssa} that $\\delta$ is strongly self-absorbing.\nIf $\\gamma\\colon G_1\\curvearrowright\\mathcal R$ is another outer action, then the same follows for $\\gamma$.\nBy applying Lemma~\\ref{lem:approx-central-embeddings} and Corollary~\\ref{cor:equivalence_equivariant_McDuff} twice, we obtain that $\\gamma$ and $\\delta$ absorb each other, hence they are cocycle conjugate.\n\n\\ref{theorem:model-absorption:2}:\nDefine $N$ to be the subgroup of all elements $g\\in G$ such that $\\alpha_{\\omega,g}$ is trivial, and set $G_0=G\/N$ with quotient map $\\pi^0: G\\to G_0$.\nBy assumption we have $N\\subseteq H$, hence $G_1$ can be viewed as a quotient of $G_0$ via a map $\\pi^{0\\to 1}: G_0\\to G_1$.\nThen $\\pi=\\pi^{0\\to 1}\\circ\\pi^0$ and the action $\\delta_{\\pi^{0\\to 1}}:=\\delta\\circ\\pi^{0\\to 1}$ is a $G_0$-action with $(\\delta_{\\pi^{0\\to 1}})_{\\pi^0}=\\delta_\\pi$.\nBy Lemma~\\ref{lem:approx-central-embeddings}, it follows that there exists an equivariant unital $*$-homomorphism $(\\mathcal R,\\delta_\\pi) \\to (M_\\omega,\\alpha_\\omega)$.\nSince $\\delta$ was strongly self-absorbing, so is $\\delta_\\pi$ as a $G$-action and the claim follows by Corollary~\\ref{cor:equivalence_equivariant_McDuff}.\n\\end{proof}\n\n\\section{Actions of discrete amenable groupoids}\n\nWe begin by recalling the definition of a discrete measured groupoid. This concept dates back to \\cite{Mackey63}. \n\n\\begin{definition}A discrete measured groupoid $\\mathcal{G}$ is a groupoid in the usual sense that carries the following additional structure:\n\\begin{enumerate}[label=$\\bullet$,leftmargin=*]\n\\item The groupoid $\\mathcal{G}$ is a standard Borel space and the units $\\mathcal{G}^{(0)} \\subset \\mathcal{G}$ form a Borel subset.\n\\item The source and target maps $s,t\\colon \\mathcal{G} \\rightarrow \\mathcal{G}^{(0)}$ are Borel and countable-to-one.\n\\item Define $\\mathcal{G}^{(2)}:= \\{(g,h) \\in \\mathcal{G} \\times \\mathcal{G} \\mid s(g) = t(h)\\}.$ The multiplication map $\\mathcal{G}^{(2)} \\rightarrow \\mathcal{G}\\colon (g,h) \\mapsto gh$ and the inverse map $\\mathcal{G}\\rightarrow \\mathcal{G}\\colon g \\mapsto g^{-1}$ are Borel.\n\\item $\\mathcal{G}^{(0)}$ is equipped with a measure $\\mu$ satisfying the following property.\nLet $\\mu_s$ and $\\mu_t$ denote the $\\sigma$-finite measures on $\\mathcal{G}$ obtained by integrating the counting measure over $s,t\\colon \\mathcal{G}\\rightarrow \\mathcal{G}^{(0)}$, respectively. Then $\\mu_s \\sim \\mu_t$. \n\\end{enumerate}\n\\end{definition}\n\\begin{example}\nAn important example of a discrete measured groupoid is the \\emph{transformation groupoid} associated to a non-singular action $G \\curvearrowright (X, \\mu)$ of a countable, discrete group $G$ on a standard measure space $(X, \\mu)$. In that case the unit space can be identified with $X$ and the measure $\\mu$ satisfies the necessary requirements. We denote this transformation groupoid by $G \\ltimes X$. \n\\end{example}\nWe assume the reader is familiar with the concept of amenability for discrete measured groupoids, see \\cite[Definition 3.2.8]{AnantharamanRenault00}. In particular, recall that a groupoid $\\mathcal{G}$ is amenable if and only if the associated equivalence relation\n\\[\n\\big\\{ \\big( s(g),t(g) \\big) \\mid g \\in \\mathcal{G} \\big\\}\n\\]\nand almost all associated isotropy groups\n\\[\\{g \\in \\mathcal{G} \\mid s(g) = t(g) = x\\} \\quad \\text{for } x \\in \\mathcal{G}^{(0)}\\]\nare amenable (see e.g.\\ \\cite[Corollary 5.3.33]{AnantharamanRenault00}). \nIn case of a non-singular action $G \\curvearrowright (X, \\mu)$, the associated transformation groupoid $G \\ltimes X$ is amenable if and only if the action is amenable in the sense of Zimmer (\\cite{Zimmer78, Zimmer77, Zimmer77b}).\n\n\\begin{remark}\nIn this paper we work with measurable fields of all kinds of separable structures, such as Polish spaces, Polish groups, von Neumann algebras with separable predual, and fields that can be derived from these.\nFor Polish groups the definition is explicitly given in \\cite{Sutherland85}, while the other notions can be defined in an analogous way.\nWe only consider the measurable setting and hence will often implicitly discard sets of measure zero whenever needed.\nThis means all measurable fields, groupoids and isomorphisms between measure spaces are defined up to sets of measure zero.\nBecause of this, all statements should be interpreted as holding only almost everywhere whenever appropriate.\nThis also means that we have no problem to apply the von Neumann measurable selection theorem (see e.g.\\ \\cite[Theorem 18.1]{Kechris95}) to obtain measurable sections after deletion of a suitable null set, and we will often omit the fine details related to such arguments.  \n\\end{remark}\n\n\\begin{definition}\nLet $\\mathcal{G}$ be a discrete measured groupoid with unit space $(X,\\mu)$. An \\emph{action} $\\alpha$ of $\\mathcal{G}$ on a measurable field $(B_x)_{x \\in X}$ of factors with separable predual is given by a measurable field of $*$-isomorphisms \\[\\mathcal{G} \\ni g \\mapsto \\alpha_g\\colon B_{s(g)} \\rightarrow B_{t(g)},\\] \nsatisfying $\\alpha_g \\circ \\alpha_h = \\alpha_{gh}$ for all $(g,h) \\in \\mathcal{G}^{(2)}$.\n\\end{definition}\n\n\\begin{definition}\\label{def:cc_groupoid_actions}\nLet $\\mathcal{G}$ be a discrete measured groupoid with unit space $(X,\\mu)$. Suppose that $\\alpha$ and $\\beta$ are actions of $\\mathcal{G}$ on the measurable fields of factors with separable predual $(B_x)_{x \\in X}$ and $(D_x)_{x \\in X}$, respectively. The actions are said to be \\emph{cocycle conjugate} if there exists a measurable field of $*$-isomorphisms $X \\ni x \\mapsto \\theta_x\\colon B_x \\rightarrow D_x$ and a measurable field of unitaries $\\mathcal{G} \\ni g \\mapsto w_g \\in \\mathcal{U}(D_{t(g)})$ satisfying  \n\\begin{align*}\n\\theta_{t(g)} \\circ \\alpha_g \\circ \\theta_{s(g)}^{-1} &= \\mathrm{Ad} w_g \\circ \\beta_g \\text{ for all } g \\in \\mathcal{G}\\\\\n  w_g \\beta_g(w_h) &= w_{gh} \\text{ for all } (g,h) \\in \\mathcal{G}^{(2)}.\n\\end{align*}\n\\end{definition}\n\n\\begin{example} \\label{ex:central_decomposition}\nLet $B$ be a von Neumann algebra acting on a separable Hilbert space $\\mathcal{H}$. Then we can centrally decompose $B$ as \n\\[ (B, \\mathcal{H}) = \\int_{X}^\\oplus (B_x, \\mathcal{H}_x)\\, d\\mu(x), \\]\nwhere $(X,\\mu)$ is a standard probability space such that $L^\\infty(X,\\mu) \\cong \\mathcal{Z}(B)$ (see e.g.\\ \\cite[Theorem IV.8.21]{Takesaki02}).\nIn this way we get a measurable field of factors $(B_x)_{x \\in X}$. When $B$ is of type I, II$_1$, II$_\\infty$ or III, every $B_x$ has the same type by \\cite[Corollary V.6.7]{Takesaki02}.\nWe claim that if $B \\cong B \\bar{\\otimes} \\mathcal{R}$, then every fibre $B_x$ is McDuff.\nPick a $*$-isomorphism $\\Phi\\colon B \\rightarrow B \\bar{\\otimes} \\mathcal{R}$.\nThen there exists (see for example \\cite[Theorem III.2.2.8]{Blackadar}) a unitary $U\\colon \\mathcal{H} \\otimes \\ell^2(\\mathbb{N}) \\rightarrow \\mathcal{H} \\otimes L^2(\\mathcal{R}, \\tau_\\mathcal{R}) \\otimes \\ell^2(\\mathbb{N})$ such that the amplification of $\\Phi$ is spatial, i.e.\\ $\\Phi(b) \\otimes 1 = U(x \\otimes 1)U^*$.\nWe have the decompositions\n\\[(B \\otimes \\mathbb{C}, \\mathcal{H} \\otimes \\ell^2(\\mathbb{N})) =\\int_X^\\oplus (B_x \\otimes \\mathbb{C}, \\mathcal{H}_x \\otimes \\ell^2(\\mathbb{N})) \\, d \\mu(x),\\text{ and}\\]\n\\[(B \\bar{\\otimes} \\mathcal{R} \\otimes \\mathbb{C},\\mathcal{H} \\otimes L^2(\\mathcal{R}, \\tau_\\mathcal{R}) \\otimes \\ell^2(\\mathbb{N}))  = \\int_X^\\oplus \\left(B_x \\bar{\\otimes}\\mathcal{R} \\otimes \\mathbb{C}, \\mathcal{H}_x \\otimes L^2(\\mathcal{R}, \\tau_\\mathcal{R}) \\otimes \\ell^2(\\mathbb{N})\\right) \\, d\\mu(x).\\]\nAs the amplification of $\\Phi$ necessarily maps the diagonal algebras (i.e.\\ the respective centers) to each other, we can use the fact that the disintegration is unique \\cite[Theorem 8.23]{Takesaki02}. In particular, this means every $B_x$ is isomorphic to some $B_y \\bar{\\otimes} \\mathcal{R}$ and hence, $B_x \\cong B_x \\bar{\\otimes} \\mathcal{R}$. \n\nNow suppose $\\alpha\\colon G \\curvearrowright B$ is an action of a countable discrete group. \nLet ${\\mathcal{G} = G \\ltimes X}$ denote the transformation groupoid associated to the action on $(X, \\mu)$ induced by $\\alpha$.\n Then $\\alpha$ can be disintegrated as an action $\\bar{\\alpha}$ of $\\mathcal{G}$ on the measurable field $(B_x)_{x \\in X}$ (see e.g.\\ \\cite[Corollary X.3.12]{Takesaki02}\\footnote{When the field of factors $(B_x)_{x \\in X}$ is constant (for example when $B$ is injective type II$_1$ and all $B_x$ are $\\mathcal{R}$), this construction dates back to \\cite{SutherlandTakesaki89}. \n There, the groupoid $\\mathcal{G}$ and action $\\bar{\\alpha}$ are also called the \\emph{ancillary groupoid} and \\emph{ancillary action} associated to $\\alpha$. }) such that given $b= \\int_x^\\oplus b_x\\,  d \\mu(x)$, we have\n\\[\\alpha_g(b)_{g \\cdot x} = \\bar{\\alpha}_{(g,x)}(b_{x}) \\text{ for } (g,x) \\in \\mathcal{G}.\\]\n\nAssume $\\beta\\colon G \\curvearrowright D$ is another action on a separably acting von Neumann algebra $(D, \\mathcal{K}) = \\int_X^\\oplus (D_x, \\mathcal{K}_x)\\, d\\mu(x)$, and assume that $\\beta$ induces the same action on $(X,\\mu)$ as $\\alpha$. \nLet $\\bar{\\beta}$ denote its decomposition as an action of $\\mathcal{G}$ on $(D_x)_{x \\in X}$. \nIf $\\bar{\\alpha}$ and $\\bar{\\beta}$ are cocycle conjugate in the sense of Definition~\\ref{def:cc_groupoid_actions}, then $\\alpha$ and $\\beta$ are cocycle conjugate as actions on von Neumann algebras.\nIndeed, let $X \\ni x \\mapsto \\theta_x\\colon A_x \\rightarrow B_x$ and $\\mathcal{G} \\ni (g,x) \\mapsto w_{(g,x)} \\in \\mathcal{U}(B_{g \\cdot x})$ denote the measurable fields of $*$-isomorphisms and unitaries realizing a cocycle conjugacy between $\\bar{\\alpha}$ and $\\bar{\\beta}$.\nThis gives rise to a $*$-isomorphism $\\theta\\colon A \\rightarrow B$ given by $\\theta(a)_x = \\theta_x(a_x)$ for $a = \\int_X^\\oplus a_x\\, \\mathrm{d} \\mu(x) \\in A$, and for each $g \\in G$ we get a unitary $\\mathbbm{v}_g \\in \\mathcal{U}(B)$ by\n$(\\mathbbm{v}_g)_x = w_{(g, g^{-1}\\cdot x)}$. The pair $(\\theta, \\mathbbm{v})$ is a cocycle conjugacy.\n\nConversely, one can show that every cocycle conjugacy $(\\theta, \\mathbbm{v}):(B, \\alpha) \\rightarrow (D, \\beta)$ with $\\theta\\big\\lvert_{L^\\infty(X)} =  \\mathrm{id}\\big\\lvert_{L^\\infty(X)}$ gives rise to a cocycle conjugacy in the sense of Definition~\\ref{def:cc_groupoid_actions}.\n\\end{example}\n\nWe will subsequently need the following lemma (albeit only for discrete groups) about strongly self-absorbing actions.\n\n\\begin{lemma} \\label{lem:special-cc}\nLet $G_j$, $j=1,2$, be two second-countable locally compact groups with a continuous group isomorphism $\\phi: G_1\\to G_2$.\nLet $\\delta^{(j)}\\colon G_j\\curvearrowright\\mathcal R$, $j=1,2$, be two strongly self-absorbing actions and choose a cocycle conjugacy $(\\Phi,\\mathbb{U})\\colon (\\mathcal R,\\delta^{(2)})\\to (\\mathcal R\\bar{\\otimes}\\mathcal R,\\delta^{(2)}\\otimes\\delta^{(2)})$ that is approximately unitarily equivalent to $\\operatorname{id}_{\\mathcal R}\\otimes 1_{\\mathcal R}$.\n(Note that $\\phi$ allows us to identify $(\\Phi,\\mathbb{U}\\circ\\phi)$ with a cocycle conjugacy between the $G_1$-action $\\delta_\\phi:=\\delta^{(2)}\\circ\\phi$ and its tensor square.)\nLet $\\alpha^{(j)}\\colon  G_j\\curvearrowright M_j$, $j=1,2$, be two actions on separably acting von Neumann algebras.\nGiven a cocycle conjugacy\n\\[\n(\\theta,\\mathbbm{v})\\colon (M_1,\\alpha^{(1)})\\to \\big( M_2\\bar{\\otimes}\\mathcal{R}, (\\alpha^{(2)}\\otimes\\delta^{(2)})\\circ\\phi \\big),\n\\]\nand a conjugacy $\\Delta\\colon (\\mathcal R, \\delta^{(2)} \\circ \\phi)\\to(\\mathcal R,\\delta^{(1)})$,\nconsider the cocycle conjugacy of $G_1$-actions\n\\[\n(\\Psi,\\mathbb{V})= \\big( (\\theta, \\mathbbm{v})^{-1} \\otimes \\Delta \\big) \\circ \\big( \\mathrm{id}_{M_2} \\otimes (\\Phi, \\mathbbm{U}\\circ\\phi) \\big) \\circ (\\theta, \\mathbbm{v})\n\\]\nbetween $(M_1,\\alpha^{(1)})$ and $(M_1\\bar{\\otimes}\\mathcal R,\\alpha^{(1)}\\otimes \\delta^{(1)} )$.\nThen there exists a sequence of unitaries $y_n\\in\\mathcal{U}(M_1\\otimes\\mathcal R)$ such that\n\\[\n\\operatorname{Ad}(y_n)\\circ (\\operatorname{id}_{M_1}\\otimes 1_{\\mathcal R})\\to \\Psi,\\quad \\operatorname{Ad}(y_n^*)\\circ\\Psi \\to \\operatorname{id}_{M_1}\\otimes 1_{\\mathcal R}\n\\]\npoint-strongly, and such that\n\\[\ny_n(\\alpha^{(1)}\\otimes\\delta^{(1)})_g(y_n)^* \\to \\mathbb{V}_g,\\quad y_n^*\\mathbb{V}_g(\\alpha^{(1)}\\otimes\\delta^{(1)})_g(y_n)\\to 1_{M_1\\otimes\\mathcal R}\n\\]\nin the strong operator topology for all $g\\in G$ and uniformly over compact sets.\n\\end{lemma}\n\\begin{proof}\nBy assumption, there is a sequence of unitaries $z_n\\in\\mathcal{U}(\\mathcal{R}\\bar{\\otimes}\\mathcal{R})$ such that \n\\begin{equation} \\label{eq:main-tech:ssa1}\n\\|\\Phi(x) - z_n(x\\otimes 1_{\\mathcal R})z_n^*\\|_2\\to 0\\quad\\text{for all } x\\in\\mathcal R.\n\\end{equation}\nand \n\\begin{equation} \\label{eq:main-tech:ssa2}\n\\max_{h\\in K} \\|\\mathbb{U}_h - z_n(\\delta^{(2)}\\otimes\\delta^{(2)})_h(z_n)^*\\|_2\\to 0 \\quad\\text{for every compact } K\\subseteq G_2.\n\\end{equation}\nBy definition, we have\n\\[\n\\Psi = (\\theta^{-1}\\otimes\\Delta)\\circ (\\operatorname{id}_{M_2}\\otimes\\Phi)\\circ\\theta\n\\]\nand\n\\[\n\\mathbb{V}_g = ({\\theta}^{-1}\\otimes\\Delta)\\Big( (\\operatorname{id}_{M_2}\\otimes\\Phi)(\\mathbbm{v}_g) \\cdot (1_{M_2}\\otimes \\mathbb{U}_{\\phi(g)})\\cdot (\\mathbbm{v}_g^* \\otimes 1_{\\mathcal R}) \\Big),\\quad g\\in G_1.\n\\]\nIf we consider the sequence of unitaries \n\\[\ny_n=(\\theta^{-1}\\otimes\\Delta)(1_{M_2}\\otimes z_n),\n\\]\nthen we can observe with \\eqref{eq:main-tech:ssa1} that\n\\[\n\\operatorname{Ad}(y_n^*)\\circ\\Psi \\to (\\theta^{-1}\\otimes\\Delta)\\circ (\\operatorname{id}_{M_2}\\otimes\\operatorname{id}_{\\mathcal R}\\otimes 1_{\\mathcal R})\\circ\\theta = \\operatorname{id}_{M_1}\\otimes 1_{\\mathcal R}\n\\]\nas well as\n\\[\n\\operatorname{Ad}(y_n)\\circ(\\operatorname{id}_{M_1}\\otimes 1_{\\mathcal R}) = \\operatorname{Ad}(y_n)\\circ(\\theta^{-1}\\otimes\\Delta)\\circ (\\operatorname{id}_{M_2}\\otimes\\operatorname{id}_{\\mathcal R}\\otimes 1_{\\mathcal R})\\circ\\theta \\to \\Psi\n\\]\npoint-strongly.\nMoreover, given $g\\in G_1$, the fact that $(\\theta^{-1}, \\big( \\theta^{-1}(\\mathbbm{v}^*_g) \\big)_{g\\in G_1} )$ is the inverse of $(\\theta,\\mathbbm{v})$ leads to the equation $\\alpha^{(1)}_g\\circ\\theta^{-1}=\\theta^{-1}\\circ\\operatorname{Ad}(\\mathbbm{v}_g)\\circ(\\alpha^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}$.\nIf we combine this with \\eqref{eq:main-tech:ssa1} and \\eqref{eq:main-tech:ssa2}, we can see that\n\\[\n\\begin{array}{cl}\n\\multicolumn{2}{l}{ y_n^*\\mathbb{V}_g(\\alpha^{(1)}\\otimes\\delta^{(1)})_g(y_n) } \\\\\n=& y_n^*\\mathbb{V}_g\\cdot (\\theta^{-1}\\otimes\\Delta)\\big( \\operatorname{Ad}(\\mathbbm{v}_g\\otimes 1_{\\mathcal R})(1_{M_2}\\otimes(\\delta^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}(z_n)) \\big)  \\\\\n=& (\\theta^{-1}\\otimes\\Delta)\\Big( (1_{M_2}\\otimes z_n)^*\\cdot (\\operatorname{id}_{M_2}\\otimes\\Phi)(\\mathbbm{v}_g) \\cdot (1_{M_2}\\otimes \\mathbb{U}_{\\phi(g)}) \\cdots \\\\\n& \\phantom{ (\\theta^{-1}\\otimes\\operatorname{id}_{\\mathcal{R}})\\Big( }\\cdots (1_{M_2}\\otimes(\\delta^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}(z_n))\\cdot (\\mathbbm{v}_g^* \\otimes 1_{\\mathcal R}) \\Big) \\\\\n=& (\\theta^{-1}\\otimes\\Delta)\\Big( \\underbrace{ \\big( \\operatorname{id}_{M_2}\\otimes (  \\operatorname{Ad}(z_n^*)\\circ\\Phi ) \\big) (\\mathbbm{v}_g) }_{\\to \\mathbbm{v_g}\\otimes 1_{\\mathcal R} } \n\\cdot \\big( 1_{M_2}\\otimes \\underbrace{ z_n^*\\mathbb{U}_{\\phi(g)}(\\delta^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}(z_n) }_{\\to 1_{\\mathcal{R}\\otimes\\mathcal{R}} } \\big)\\cdot (\\mathbbm{v}_g^* \\otimes 1_{\\mathcal R}) \\Big) \\\\\n\\to & 1_{M_1\\otimes\\mathcal R} \n\\end{array}\n\\]\nin the strong operator topology, uniformly over compact subsets.\nAnalogously, we observe the convergence\n\\[\n\\begin{array}{cl}\n\\multicolumn{2}{l}{ y_n(\\alpha^{(1)}\\otimes\\delta^{(1)})_g(y_n)^* }\\\\\n=& (\\theta^{-1}\\otimes\\Delta)\\Big( (1_{M_2}\\otimes z_n) \\cdot \\operatorname{Ad}(\\mathbbm{v}_g\\otimes 1_{\\mathcal R})(1_{M_2}\\otimes(\\delta^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}(z_n^*) ) \\Big) \\\\\n=& (\\theta^{-1}\\otimes\\Delta)\\Big( \\underbrace{ \\operatorname{Ad}(1_{M_2}\\otimes z_n)(\\mathbbm{v}_g\\otimes 1_{\\mathcal R})}_{\\to (\\operatorname{id}_{M_2}\\otimes\\Phi)(\\mathbbm{v}_g)}\n \\cdot \\big( 1_{M_2}\\otimes \\underbrace{ z_n(\\delta^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}(z_n^*) }_{\\to \\mathbb{U}_{\\phi(g)} } \\big) \\cdot (\\mathbbm{v}_g^*\\otimes 1_{\\mathcal R}) \\Big) \\\\\n\\to & \\mathbb{V}_g\n\\end{array}\n\\]\nuniformly over compact sets.\nThis finishes the proof.\n\\end{proof}\n\nIn the proof of the main technical result of this section we will make use of the following variant, due to Popa--Shlyakhtenko--Vaes, of the cohomology lemmas in \\cite[Appendix]{JonesTakesaki84} and \\cite[Theorem 5.5]{Sutherland85}.\n\n\\begin{lemma}[{\\cite[Theorem 3.5]{PopaShlyakhtenkoVaes20}}] \\label{lemma:cohomology_lemma}\nLet $\\mathcal{S}$ be an amenable countable nonsingular equivalence relation on the standard probability space $(X,\\mu)$.\nLet $(G_x \\curvearrowright P_x)_{x \\in X}$ be a measurable field of continuous actions of Polish groups on Polish spaces, on which $\\mathcal{S}$ is acting by conjugacies: we have measurable fields of group isomorphisms ${\\mathcal{S} \\ni (x,y) \\mapsto \\gamma_{(x,y)}\\colon G_y \\mapsto G_x}$ and homeomorphisms $\\mathcal{S} \\ni (x,y) \\mapsto \\beta_{(x,y)}\\colon P_y \\mapsto P_x$ satisfying \n\\[\n\\gamma_{(x,y)} \\circ \\gamma_{(y,z)} = \\gamma_{(x,z)}, \\quad \\beta_{(x,y)} \\circ \\beta_{(y,z)} = \\beta_{(x,z)}, \\quad \\beta_{(x,y)} (g \\cdot \\pi) = \\gamma_{(x,y)}(g) \\cdot \\beta_{(x,y)}(\\pi)\n\\]\nfor all $(x,y), (y,z) \\in \\mathcal{S}$ and $g \\in G_y, \\pi \\in P_y$.\nLet $X \\ni x \\mapsto \\sigma'(x) \\in P_x$ be a measurable section. Assume that for all $(x,y) \\in \\mathcal{S}$, the element $\\sigma'(x)$ belongs to the closure of ${G_x \\cdot \\beta_{(x,y)}(\\sigma'(y))}$. \nThen there exists a measurable family $\\mathcal{S} \\ni (x,y) \\mapsto v(x,y) \\in G_x$ and a section $X \\ni x \\mapsto \\sigma(x) \\in P_x$ satisfying:\n\\begin{itemize}\n\\item $v$ is a 1-cocycle: $v(x,y) \\gamma_{(x,y)}(v(y,z)) = v(x,z)$ for all $(x,y), (y,z)\\in \\mathcal{S}$;\n\\item $v(x,y)\\cdot \\beta_{(x,y)}(\\sigma(y)) = \\sigma(x)$ for all $(x,y) \\in \\mathcal{S}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{remark} \\label{rem:Polish-spaces}\nBefore we state and prove our main technical result in detail, we would like to outline for what kind of input data the lemma above will be used.\nIn the situation considered below, the typical Polish space $P$ will be the space of cocycle conjugacies $(M,\\alpha)\\to (N,\\beta)$, where $\\alpha: H\\curvearrowright M$ and $\\beta: H\\curvearrowright N$ are actions of a countable discrete group $H$ on separably acting von Neumann algebras.\nHere we consider the topology defined by declaring that one has a convergence of nets $(\\theta^{(\\lambda)},\\mathbbm{v}^{(\\lambda)})\\to(\\theta,\\mathbbm{v})$ if and only if $\\mathbbm{v}^{(\\lambda)}_g\\to\\mathbbm{v}_g$ in the strong operator topology for all $g\\in H$, and $\\|\\varphi\\circ\\theta-\\varphi\\circ\\theta^{(\\lambda)}\\|\\to 0$ for all $\\varphi\\in N_*$.\nThis generalizes the well-known topology on the space of isomorphisms $M\\to N$ that one usually called the ``$u$-topology''; cf.\\ \\cite{Haagerup75, Winslow98}.\nThe typical Polish group acting on this Polish space would be the unitary group $\\mathcal{U}(N)$, which is equipped with the strong operator topology, where the action is defined via composition with the inner cocycle conjugacy as per Example~\\ref{ex:inner-cc} and Remark~\\ref{rem:cocycle-category}.\nIn other words, a unitary $w\\in\\mathcal{U}(N)$ moves the cocycle conjugacy $(\\theta,\\mathbbm{v})$ to $\\operatorname{Ad}(w)\\circ(\\theta,\\mathbbm{v}) = \\big( \\operatorname{Ad}(w)\\circ\\theta, (w\\mathbbm{v}\\beta_g(w)^*)_{g\\in H} \\big)$.\n\nIf we assume in addition that $M$ and $N$ are semi-finite, we may pick a faithful normal semi-finite tracial weight $\\tau$ on $N$.\nAssume that $(\\Psi,\\mathbb{V})\\in P$ is a cocycle conjugacy.\nThen it follows from \\cite[Proposition 3.7]{Haagerup75} that on the space of all isomorphisms $\\Psi': M\\to N$ with $\\tau\\circ\\Psi'=\\tau\\circ\\Psi'$, the $u$-topology coincides with the topology of point-strong convergence.\nAs a direct consequence, we may conclude the following. \nIf $(\\Phi,\\mathbb{U})\\in P$ is another cocycle conjugacy and there exists a net of unitaries $w_\\lambda\\in\\mathcal{U}(N)$ such that $w_\\lambda\\mathbb{V}_g\\beta_g(w_\\lambda)^*\\to \\mathbb{U}_g$ for all $g\\in H$ in the strong operator topology and $\\operatorname{Ad}(w_\\lambda)\\circ\\Psi\\to\\Phi$ point-strongly, then $(\\Phi,\\mathbb{U})\\in\\overline{ \\mathcal{U}(N)\\cdot (\\Psi,\\mathbb{V}) }$.\n\\end{remark}\n\nThe following can be seen as the main technical result of this section, which we previously referred to as a kind of measurable local-to-global principle.\n\n\\begin{theorem} \\label{thm:main_technical}\nLet $G \\curvearrowright (X,\\mu)$ be an amenable action (in the sense of Zimmer) of a countable, discrete group on a standard probability space.\nLet $\\alpha$ be an action of ${\\mathcal{G}:= G \\ltimes X}$ on a measurable field of semi-finite factors with separable predual $(B_x)_{x \\in X}$. \nDenote by ${X \\ni x \\mapsto H_x}$ the measurable field of isotropy groups. \nFor any action ${\\delta\\colon G \\curvearrowright \\mathcal{R}}$ on the hyperfinite II$_1$-factor, we define a tensor product action $\\alpha \\otimes \\delta$ of $\\mathcal{G}$ on the field of factors $(B_x \\bar{\\otimes} \\mathcal{R})_{x \\in X}$ by $(\\alpha \\otimes \\delta)_{(g,x)} = \\alpha_{(g,x)} \\otimes \\delta_g$.\nIf $\\delta$ is strongly self-absorbing, then the following are equivalent:\n \\begin{enumerate}[leftmargin=*,label=\\textup{(\\arabic*)}]\n \\item $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\delta$; \\label{prop:main_technical:1}\n \\item For every $x \\in X$ we have $\\alpha{|_{H_x}} \\simeq_{\\mathrm{cc}} (\\alpha \\otimes \\delta){|_{H_x}}$ as actions of $H_x$ on $B_x$ and $B_x \\bar{\\otimes} \\mathcal{R}$. \\label{prop:main_technical:2}\n \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nWe note that by following the argument outlined in Example~\\ref{ex:central_decomposition} and by applying Corollary~\\ref{cor:equivalence_equivariant_McDuff}, we see that $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\delta$ implies that one can find a cocycle conjugacy between $\\alpha$ and $\\alpha\\otimes\\delta$ that induces the identity map on $X$.\nHence \\ref{prop:main_technical:1} implies \\ref{prop:main_technical:2}.\n\nIn order to prove the other implication, assume \\ref{prop:main_technical:2} holds. \nTo verify \\ref{prop:main_technical:1}, we will show the existence of a measurable field of $*$-isomorphisms ${X \\ni x \\mapsto \\theta_x \\colon B_x \\rightarrow B_x \\bar{\\otimes} \\mathcal{R}}$ and unitaries ${\\mathcal{G} \\ni (g,x) \\mapsto w_{(g,x)} \\in \\mathcal{U}(B_{g \\cdot x} \\bar{\\otimes} \\mathcal{R})}$ such that\n\\begin{align}\n\\theta_{g \\cdot x} \\circ \\alpha_{(g,x)} \\circ \\theta_{x}^{-1} &= \\mathrm{Ad} \\big( w_{(g,x)} \\big) \\circ (\\alpha \\otimes \\delta)_{(g,x)} \\text{ for all } (g,x) \\in \\mathcal{G}\\label{eq:required_cc}\\\\\n  w_{(g,h \\cdot x)} (\\alpha \\otimes \\delta)_{(g,h \\cdot x)}(w_{(h,x)}) &= w_{(gh,x)} \\text{ for all } g,h \\in G, x \\in X.\\label{eq:required_cocycle}\n\\end{align}\nFor every $x \\in X$, denote by $P_x$ the Polish space of cocycle conjugacies from $(B_x, \\alpha|_{H_x})$ to $(B_x \\bar{\\otimes} \\mathcal{R}, (\\alpha \\otimes \\delta)|_{H_x})$ as per Remark~\\ref{rem:Polish-spaces}. \nIn this way, we get a measurable field of Polish spaces $X \\ni x \\mapsto P_x$. Note that by assumption the sets $P_x$ are all non-empty and hence, there exists some measurable section \n$ X \\ni x \\mapsto (\\theta_x, \\mathbbm{v}_x) \\in P_x$.\nDefining $w_{(g,x)} := \\mathbbm{v}_{x}(g)$ for $g \\in H_x$, we get that --- although $w$ is not defined on all of $\\mathcal{G}$ yet --- the equations \\eqref{eq:required_cc}--\\eqref{eq:required_cocycle} are satisfied whenever they make sense.\nIn the rest of the proof we will show with the help of Lemma \\ref{lemma:cohomology_lemma} that there exists a (potentially different) section for which there exists a well-defined map $w$ on all of $\\mathcal{G}$ obeying conditions \\eqref{eq:required_cc}--\\eqref{eq:required_cocycle}.\n\nDenote by $\\mathcal{S}$ the countable non-singular orbit equivalence relation associated to $\\mathcal{G}$, i.e.,\n\\[\\mathcal{S}=\\{(g \\cdot x,x) \\mid (g,x) \\in \\mathcal{G}\\}.\\]\nAs $G \\curvearrowright (X,\\mu)$ is amenable, the relation $\\mathcal{S}$ is amenable and hence it follows by the Connes--Feldman--Weiss theorem \\cite{ConnesFeldmanWeiss81} that after neglecting a set of measure zero, there exists a partition of $X$ into $\\mathcal{S}$-invariant Borel subsets $X_0\\sqcup X_1$ such that the restriction of $\\mathcal{S}$ to $X_0$ has finite orbits and the restriction to $X_1$ is induced by a free non-singular action of $\\mathbb{Z}$. \nThis implies that the map $q \\colon \\mathcal{G} \\rightarrow \\mathcal{S}\\colon (g,x) \\mapsto (g \\cdot x,x)$ admits a measurable section, i.e., a measurable groupoid morphism $s \\colon \\mathcal{S} \\rightarrow \\mathcal{G}$ such that $q \\circ s = \\mathrm{id}_\\mathcal{S}$. \nTherefore, we can view $\\mathcal{G}$ as the semi-direct product of the field of groups $(H_x)_{x \\in X}$ and the measurable field of group isomorphisms $\\phi_{(x,y)}\\colon H_y \\rightarrow H_x$ given by $\\phi_{(x,y)}(g)= s(x,y) g s(x,y)^{-1}$.\nNote that $\\phi_{(x,y)} \\circ \\phi_{(y,z)} = \\phi_{(x,z)}$ for all $(x,y), (y,z) \\in \\mathcal{S}$.\n\nThis means that in order to define a measurable field $\\mathcal{G} \\ni (g,x) \\mapsto w_{(g,x)} \\in \\mathcal{U}(B_{g \\cdot x} \\bar{\\otimes} \\mathcal{R})$ satisfying \\eqref{eq:required_cocycle}, it suffices to find measurable families of unitaries \n${v{(x,y)} \\in \\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R})}$ for $(x,y) \\in \\mathcal{S}$ and \n$\\mathbbm{v}_x(g) \\in \\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R})$ for $x \\in X, g \\in H_x$ such that\n\\begin{itemize}\n\\item $v$ is a cocyle for the action of $\\mathcal{S}$ on the field of factors $(B_x\\bar{\\otimes} \\mathcal{R})_{x \\in X}$ induced by $s$, i.e.,\n$v{(x,y)} (\\alpha \\otimes \\delta)_{s(x,y)}(v{(y,z)})=v(x,z)$ for all $(x,y), (y,z) \\in \\mathcal{S}$;\n\\item for each $x \\in X$, the family $(\\mathbbm{v}_x(g))_{g \\in H_x}$ defines a cocycle for the action $H_x \\curvearrowright B_x \\bar{\\otimes} \\mathcal{R}$;\n\\item $\\mathbbm{v}_x(g) = v{(x,y)} (\\alpha \\otimes \\delta)_{s(x,y)}\\left(\\mathbbm{v}_y\\left({\\phi_{(y,x)}(g)}\\right)\\right) (\\alpha \\otimes \\delta)_g\\left(v(x,y)^*\\right) $ for all $(x,y) \\in \\mathcal{S}$ and $g \\in H_x$. \n\\end{itemize}\nIf these conditions are met, then setting $w_{g s(x,y)} := \\mathbbm{v}_x(g)(\\alpha \\otimes \\delta)_g (v(x,y))$ for $(x,y) \\in \\mathcal{S}$ and $g \\in H_x$ yields condition \\eqref{eq:required_cocycle}.\nMoreover, in order for a measurable field of $*$-isomorphisms ${X \\ni x \\mapsto \\theta_x \\colon B_x \\rightarrow B_x \\bar{\\otimes} \\mathcal{R}}$ to satisfy \\eqref{eq:required_cc}, it then suffices to check that \n\\begin{itemize}\n\\item for each $x \\in X$, the pair $(\\theta_x, \\mathbbm{v}_x)$ is a cocycle conjugacy in $P_x$;\n\\item for $(x,y) \\in \\mathcal{S}$ we have $\\theta_x \\circ \\alpha_{s(x,y)} \\circ \\theta_y^{-1} = \\mathrm{Ad}(v(x,y))\\circ (\\alpha \\otimes \\delta)_{s(y,z)}$.\n\\end{itemize}\nWe introduce some notation to rephrase this in terms of the terminology of Lemma~\\ref{lemma:cohomology_lemma}. Consider the natural action $\\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R}) \\curvearrowright P_x$ given by composing a cocycle conjugacy with the inner one given by $\\mathrm{Ad}(u)$ for $u \\in \\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R})$ as per Remark~\\ref{rem:Polish-spaces}.\nIn this way, we get a measurable field $(\\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R}) \\curvearrowright P_x)_{x \\in X}$ of continuous actions of Polish groups on Polish spaces.\nLet us convince ourselves that in the terminology of Lemma~\\ref{lemma:cohomology_lemma}, the equivalence relation $\\mathcal{S}$ acts by conjugacies on this field of actions.\nFirstly, we have a measurable field of group isomorphisms \n\\[\n{\\mathcal{S} \\ni (x,y) \\mapsto  \\gamma_{(x,y)} = (\\alpha \\otimes \\delta)_{s(x,y)}|_{\\mathcal{U}(B_y \\bar{\\otimes} \\mathcal{R})} \\colon \\mathcal{U}(B_y \\bar{\\otimes} \\mathcal{R})\\rightarrow \\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R})}\n\\]\nsuch that $\\gamma_{(x,y)} \\circ \\gamma_{(y,z)} = \\gamma_{(x,z)}$ for all $(x,y), (y,z) \\in \\mathcal{S}$.\nThe latter formula holds as $\\alpha\\otimes\\delta$ was a $\\mathcal G$-action and $s: \\mathcal{S}\\to\\mathcal{G}$ is a section.\nSecondly, we have an action of $\\mathcal{S}$ on $(P_x)_{x \\in X}$ as follows.\nGiven $(x,y) \\in \\mathcal{S}$ and $(\\theta, \\mathbbm{v}) \\in P_y$, we define $\\beta_{(x,y)}(\\theta, \\mathbbm{v}) := (\\tilde{\\theta}, \\tilde{\\mathbbm{v}})$, where\n\\[\n\\tilde{\\theta} = (\\alpha \\otimes \\delta)_{s(x,y)} \\circ \\theta \\circ \\alpha_{s(y,x)} \\quad\\text{and}\\quad\n\\tilde{\\mathbbm{v}}(h) = (\\alpha \\otimes \\delta)_{s(x,y)}(\\mathbbm{v}({\\phi_{(y,x)}(h)})) \\text{ for } h \\in H_x.\n\\]\nThis construction yields a well-defined cocycle conjugacy in $P_x$, and we get a well-defined map $\\beta_{(x,y)}\\colon P_y \\rightarrow P_x$.\nTogether these maps combine to form a measurable field of homeomorphisms $\\mathcal{S} \\ni (x,y) \\mapsto \\beta_{(x,y)}\\colon P_y \\to P_x$ such that $\\beta_{(x,y)} \\circ \\beta_{(y,z)} = \\beta_{(x,z)}$ for all $(x,y), (y,z) \\in \\mathcal{S}$. \nThis formula holds once again because $\\alpha\\otimes\\delta$ and $\\alpha$ are $\\mathcal G$-actions and $s\\colon \\mathcal{S}\\to\\mathcal{G}$ is a section.\nMoreover, the maps $\\beta$ and $\\gamma$ are compatible with the measurable field of actions ${(\\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R}) \\curvearrowright P_x)_{x \\in X}}$ (as required for Lemma~\\ref{lemma:cohomology_lemma}), since for any $(x,y) \\in \\mathcal{S}, u \\in \\mathcal{U}(B_y \\bar{\\otimes} \\mathcal{R})$ and $(\\theta,\\mathbbm{v}) \\in P_y$ we may simply compare definitions and observe\n\\[\n\\beta_{(x,y)}(u \\cdot (\\theta,\\mathbbm{v})) = \\gamma_{(x,y)}(u) \\cdot \\beta_{(x,y)}(\\theta,\\mathbbm{v}).\n\\]\nHaving introduced all this data, our previous discussion can be rephrased. \nIn order to complete the proof, it suffices to find a measurable section $X \\ni x \\mapsto \\sigma(x) \\in P_x$ and a measurable family $\\mathcal{S} \\ni (x,y) \\mapsto v(x,y) \\in \\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R})$ such that\n\\begin{itemize}\n\\item $v(x,y) \\gamma_{(x,y)}(v(y,z)) = v(x,z)$ for all $(x,y), (y,z) \\in \\mathcal{S}$;\n\\item $v(x,y) \\cdot \\beta_{(x,y)}(\\sigma(y)) = \\sigma(x)$ for all $(x,y) \\in \\mathcal{S}$.\n\\end{itemize}\nBy Lemma~\\ref{lemma:cohomology_lemma}, such maps exist if we can merely show that\nthere exists a measurable section $X \\ni x \\mapsto (\\theta_x, \\mathbbm{v}_x) \\in P_x$ such that for all $(x,y) \\in \\mathcal{S}$, the element $(\\theta_x, \\mathbbm{v}_x)$ belongs to the closure of $\\mathcal{U}(B_x \\bar{\\otimes}\\mathcal{R}) \\cdot \\beta_{(x,y)}(\\theta_y,\\mathbbm{v}_y)$.\nWe claim that this is indeed the case.\n\nConsider any measurable section $X \\ni x \\mapsto (\\theta'_x, \\mathbbm{v}'_x) \\in P_x$. \nAs $\\delta$ is strongly self-absorbing, we can fix a cocycle conjugacy $(\\Phi, \\mathbb{U})$ from $(\\mathcal{R}, \\delta)$ to $(\\mathcal{R}\\bar{\\otimes} \\mathcal{R}, \\delta \\otimes \\delta)$ that is approximately unitarily equivalent to $\\operatorname{id}_{\\mathcal R}\\otimes 1_{\\mathcal R}$.\nFor each $x \\in X$ we can define the map\n\\[\n\\Lambda_x\\colon P_x \\rightarrow P_x,\\quad (\\theta, \\mathbbm{v}) \\mapsto \\big( (\\theta, \\mathbbm{v})^{-1} \\otimes \\mathrm{id}_\\mathcal{R} \\big) \\circ \\big( \\mathrm{id}_{B_x} \\otimes (\\Phi, \\mathbb{U}) \\big) \\circ (\\theta, \\mathbbm{v})\n\\]\nThen we get a new measurable section\n\\[\nX \\ni x \\mapsto (\\theta_x, \\mathbbm{v}_x) := \\Lambda_x(\\theta'_x, \\mathbbm{v}'_x) \\in P_x.\n\\]\nWe claim that this section does the trick. Fix $(x,y) \\in \\mathcal{S}$. \nIf we denote $(\\tilde{\\theta}_x, \\tilde{\\mathbbm{v}}_x):= \\beta_{(x,y)}(\\theta_y, \\mathbbm{v}_y)$, we need to convince ourselves that the cocycle conjugacy $(\\theta_x, \\mathbbm{v}_x)$ is in the closure of $\\mathcal{U}(B_x \\bar{\\otimes}\\mathcal{R})\\cdot (\\tilde{\\theta}_x, \\tilde{\\mathbbm{v}}_x)$.\nFirst of all, we observe that the construction of $(\\theta_x, \\mathbbm{v}_x)=\\Lambda_x(\\theta_x', \\mathbbm{v}_x')$ can be seen as a special case of Lemma~\\ref{lem:special-cc} for $M_1=M_2=B_x$, $G=H_x$, $\\phi=\\operatorname{id}_{H_x}$ and $\\Delta=\\operatorname{id}_{\\mathcal R}$.\nHence we can find a sequence of unitaries $y_n\\in\\mathcal{U}(B_x\\otimes\\mathcal R)$ such that\n\\begin{equation} \\label{eq:main-tech:yn}\ny_n^*\\theta_x(b)y_n \\to b\\otimes 1_{\\mathcal R},\\quad y_n^*\\mathbbm{v}_x(h)(\\alpha\\otimes\\delta)_h(y_n) \\to 1_{B_x\\otimes\\mathcal R}\n\\end{equation}\nin the strong operator topology for all $b\\in B_x$ and $h\\in H_x$.\n\nNext, by our previous notation, the group isomorphism $\\phi_{(x,y)}\\colon H_y\\to H_x$ is exactly the one so that the isomorphism of von Neumann algebras $\\alpha_{s(x,y)}\\colon B_y\\to B_x$ can be viewed as a (genuine) conjugacy between the $H_x$-actions $(\\alpha|_{H_y})_{\\phi_{(y,x)}}$ and $\\alpha|_{H_x}$.\nMoreover if $s(x,y)=(k,y)$ for $k\\in G$, then $(\\alpha\\otimes\\delta)_{s(x,y)}=\\alpha_{s(x,y)}\\otimes\\delta_k$ and $\\delta_k$ can be seen as a conjugacy between the $H_x$-actions $(\\delta|_{H_y})_{\\phi_{(y,x)}}$ and $\\delta|_{H_x}$.\n\nBy definition of $\\beta_{(x,y)}$, we have\n\\[\n\\begin{array}{ccl}\n\\tilde{\\theta}_x &=& (\\alpha \\otimes \\delta)_{s(x,y)} \\circ \\theta_y \\circ \\alpha_{s(y,x)} \\\\\n&=&  (\\alpha \\otimes \\delta)_{s(x,y)} \\circ \\big( {\\theta_y'}^{-1} \\otimes \\mathrm{id}_\\mathcal{R} \\big) \\circ \\big( \\mathrm{id}_{B_y} \\otimes \\Phi \\big) \\circ \\theta_y' \\circ \\alpha_{s(y,x)} \\\\\n&=& \\big( ({\\theta_y'}\\circ\\alpha_{s(y,x)})^{-1} \\otimes \\delta_k \\big) \\circ \\big( \\mathrm{id}_{B_y} \\otimes \\Phi \\big) \\circ (\\theta_y' \\circ \\alpha_{s(y,x)})\n\\end{array}\n\\]\nand for all $h\\in H_x$ with $g=\\phi_{(y,x)}(h)$, one has\n\\[\n\\begin{array}{ccl}\n\\tilde{\\mathbbm{v}}_x(h) \n&=& (\\alpha \\otimes \\delta)_{s(x,y)}\\Big( ({\\theta_y'}^{-1}\\otimes\\operatorname{id}_{\\mathcal R})\\Big( (\\operatorname{id}_{B_y}\\otimes\\Phi)(\\mathbbm{v}_y'(g)) \\cdot (1_{B_y}\\otimes \\mathbb{U}_g)\\cdot (\\mathbbm{v}_y'(g)^* \\otimes1_{\\mathcal R}) \\Big) \\Big) \\\\\n&=& \\big( ({\\theta_y'}\\circ\\alpha_{s(y,x)})^{-1}\\otimes\\delta_k \\big)\\Big( (\\operatorname{id}_{B_y}\\otimes\\Phi)(\\mathbbm{v}_y'(g)) \\cdot (1_{B_y}\\otimes \\mathbb{U}_g)\\cdot (\\mathbbm{v}_y'(g)^* \\otimes1_{\\mathcal R}) \\Big)\n\\end{array}\n\\]\nWe conclude that Lemma~\\ref{lem:special-cc} is applicable to the cocycle conjugacy $(\\tilde{\\theta}_x,\\tilde{\\mathbbm{v}}_x)$, where we insert $G_1=H_x$, $G_2=H_y$, $\\phi=\\phi_{(y,x)}$, $M_1=B_x$, $M_2=B_y$, $\\Delta=\\delta_k$ and the cocycle conjugacy $(\\theta'_y\\circ\\alpha_{s(y,x)}, \\mathbbm{v}'_y\\circ\\phi_{(y,x)})$ in place of $(\\theta,\\mathbbm{v})$.\nThis allows us to find a sequence of unitaries $w_n\\in\\mathcal{U}(B_x\\otimes\\mathcal R)$ satisfying\n\\begin{equation} \\label{eq:main-tech:wn}\nw_n(b\\otimes 1_{\\mathcal R})w_n^* \\to \\tilde{\\theta}_x(b),\\quad w_n(\\alpha\\otimes\\delta)_h(w_n)^* \\to \\tilde{\\mathbbm{v}}_x(h)\n\\end{equation}\nin the strong operator topology for all $b\\in B_x$ and $h\\in H_x$.\nIf we consider both of the conditions \\eqref{eq:main-tech:yn} and \\eqref{eq:main-tech:wn} and keep in mind that $G$ is countable and $B_x$ is separable and semi-finite, with the help of Lemma \\ref{lemma:point_strong_dense_subset} it is possible to find an increasing sequence of natural numbers $m_n$ such that the resulting sequence of unitaries $z_n=w_ny_{m_n}^*$ satisfies\n\\[\nz_n\\theta_x(b)z_n^* \\to \\tilde{\\theta}_x(b),\\quad z_n\\mathbbm{v}_x(h)(\\alpha\\otimes\\delta)_h(z_n)^* \\to \\tilde{\\mathbbm{v}}_x(h)\n\\]\nin the strong operator topology for all $b\\in B_x$ and $h\\in H_x$.\nThen it follows from Remark~\\ref{rem:Polish-spaces} that $(\\theta_x, \\mathbbm{v}_x)$ indeed belongs to the closure of $\\mathcal{U}(B_x \\bar{\\otimes}\\mathcal{R})\\cdot (\\tilde{\\theta}_x, \\tilde{\\mathbbm{v}}_x)$.\nThis finishes the proof.\n\\end{proof}\n\n\\begin{definition}[see {\\cite[Definition~3.4]{Delaroche79}}]\nAn action $\\alpha \\colon G \\curvearrowright B$ of a countable discrete group on a von Neumann algebra is called amenable, if there exists an equivariant conditional expectation \n\\[\nP\\colon (\\ell^\\infty(G) \\bar{\\otimes} B, \\tau \\otimes \\alpha) \\rightarrow (B, \\alpha),\n\\]\nwhere $\\tau$ denotes the left translation action $G \\curvearrowright \\ell^\\infty(G)$.\n\\end{definition}\n\nBy \\cite{Delaroche82}, an action $\\alpha$ as above is amenable if and only if its restriction to $\\mathcal{Z}(B)$ is amenable, which is equivalent to the action on the measure-theoretic spectrum of the center being amenable in the sense of Zimmer.\nRecall that an automorphism $\\alpha\\in\\operatorname{Aut}(M)$ on a separably acting von Neumann algebra is properly centrally non-trivial, if for every non-zero projection $p\\in\\mathcal{Z}(M)$, the restriction of $\\alpha_\\omega$ on $pM_\\omega$ is non-trivial.\n\nThe following result contains Theorem~\\ref{theorem-A} for actions on semi-finite von Neumann algebras as a special case via $H=G$.\n\n\\begin{corollary} \\label{cor:McDuff-passes}\nLet $G$ be a countable discrete group and $B$ a semi-finite von Neumann algebra with separable predual such that $B \\cong B \\bar{\\otimes} \\mathcal{R}$. \nLet $\\alpha: G \\curvearrowright B$ be an amenable action.\nSuppose $H\\subseteq G$ is a normal subgroup such that for every $g\\in G\\setminus H$, the automorphism $\\alpha_g$ is properly centrally non-trivial.\nLet $G_1=G\/H$ with quotient map $\\pi: G\\to G_1$ and let $\\delta: G_1\\curvearrowright\\mathcal R$ be a strongly self-absorbing action.\nThen $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\delta_\\pi$.\n\\end{corollary}\n\\begin{proof}\nAdopt the notation from Example~\\ref{ex:central_decomposition} and Theorem~\\ref{thm:main_technical}.\nWe identify $\\alpha$ with an action of ${\\mathcal{G}:= G \\ltimes X}$ on a measurable field of semi-finite factors with separable predual $(B_x)_{x \\in X}$. \nDenote by ${X \\ni x \\mapsto H_x}$ the measurable field of isotropy groups.\nAmenability of the action $\\alpha$ implies that the action on $(X,\\mu)$ is amenable in the sense of Zimmer, \nwhich in turn implies amenability of the associated transformation groupoid. \nIn particular all isotropy groups $H_x$ are amenable. \n\nBy assumption on $H$ and \\cite[Theorem 9.14]{MasudaTomatsu16}, it follows for every $g\\in G\\setminus H$ and $\\mu$-almost all $x\\in X$ that either $g\\notin H_x$ or the automorphism $(\\alpha|_{H_x})_g$ on the McDuff factor $B_x$ is centrally non-trivial.\nIn other words, after discarding a null set from $X$, we may assume for all $x\\in X$ that for all $h\\in H_x\\setminus(H_x\\cap H)$, the automorphism $(\\alpha|_{H_x})_g$ on $B_x$ is centrally non-trivial.\nBy Theorem~\\ref{theorem:model-absorption}, we get that $(\\alpha|_{H_x})$ is cocycle conjugate to $(\\alpha\\otimes\\delta_\\pi)|_{H_x}$.\nThe claim then follows via Theorem~\\ref{thm:main_technical}.\n\\end{proof}\n\n\n\\section{Actions on arbitrary von Neumann algebras}\n\nIn this section we shall generalize some of the main results we obtained so far, namely Corollaries~\\ref{cor:equivalence_equivariant_McDuff} and \\ref{cor:McDuff-passes}, to the context of group actions on not necessarily semi-finite von Neumann algebras.\nThis uses standard results in Tomita--Takesaki theory, which allow us to reduce the generalize case to the semi-finite case considered in the previous sections.\nWe will henceforth assume that the reader is familiar with the basics of Tomita--Takesaki theory as well as the theory of crossed products (for a thorough treatment the reader should consult the book \\cite{Takesaki03}), although we are about to recall the specific points needed about the former for this section.\n\n\\begin{remark}[see {\\cite[Chapters VIII, XII]{Takesaki03}}] \\label{rem:tt-basics}\nLet $M$ be a separably acting von Neumann algebra.\nGiven a faithful normal state $\\varphi$ on $M$, we denote by $\\sigma^\\varphi: \\mathbb{R}\\curvearrowright M$ its associated \\emph{modular flow}.\nIf $\\psi$ is another faithful normal state on $M$, we denote by $(D\\psi: D\\varphi): \\mathbb{R}\\to\\mathcal{U}(M)$ the associated \\emph{Connes cocycle}, which is a $\\sigma^\\varphi$-cocycle satisfying $\\operatorname{Ad}(D\\psi: D\\varphi)_t\\circ\\sigma^\\varphi_t=\\sigma^\\psi_t$ for all $t\\in\\mathbb{R}$.\nThe crossed product von Neumann algebra $\\tilde{M}=M\\rtimes_{\\sigma^\\varphi}\\mathbb{R}$ is called the \\emph{continuous core} of $M$ and does not depend on the choice of $\\varphi$ up to canonical isomorphism.\nWith slight abuse of notation, we will consider $M$ as a von Neumann subalgebra in $\\tilde{M}$, and denote by $\\lambda^{\\sigma^\\varphi}: \\mathbb{R}\\to\\mathcal{U}(\\tilde{M})$ the unitary representation implementing the modular flow on $M$.\nThe continuous core $\\tilde{M}$ is always a semi-finite von Neumann algebra.\nGiven any automorphism $\\alpha\\in\\operatorname{Aut}(M)$, there is an induced \\emph{extended automorphism} $\\tilde{\\alpha}\\in\\operatorname{Aut}(\\tilde{M})$ uniquely determined by\n\\[\n\\tilde{\\alpha}|_M = \\alpha \\quad\\text{and}\\quad \\tilde{\\alpha}(\\lambda^{\\sigma^\\varphi}_t)=(D \\varphi\\circ\\alpha^{-1} : D\\varphi)_t\\cdot\\lambda^{\\sigma^\\varphi}_t,\\quad t\\in\\mathbb R.\n\\]\nThe assignment $\\alpha\\mapsto\\tilde{\\alpha}$ defines a continuous homomorphism of Polish groups.\nTherefore, given more generally a continuous action $\\alpha: G\\curvearrowright M$ of a second-countable locally compact group, we may induce the \\emph{extended action} $\\tilde{\\alpha}: G\\curvearrowright\\tilde{M}$.\nEvery extended automorphism on $\\tilde{M}$ has the property that it commutes with the dual flow $\\hat{\\sigma}^\\varphi: \\mathbb{R}\\curvearrowright\\tilde{M}$.\n\\end{remark}\n\n\n\\begin{prop} \\label{prop:cp-centralizer}\nLet $G$ be a second-countable locally compact group.\nLet $\\alpha: G\\curvearrowright M$ be an action on a separably acting von Neumann algebra.\nThen the normal inclusion $M\\subset M\\rtimes_\\alpha G$ induces (via componentwise application of representing sequences) a unital $*$-homomorphism\n$(M_{\\omega,\\alpha})^{\\alpha^\\omega} \\to (M\\rtimes_\\alpha G)_\\omega$.\n\\end{prop}\n\\begin{proof}\nAssume $M$ is represented faithfully on a Hilbert space $\\mathcal{H}$, and consider the canonical inclusion $\\pi\\colon M \\rtimes_\\alpha G \\rightarrow \\mathcal{B}(L^2(G)) \\bar{\\otimes} M \\subset \\mathcal{B}(L^2(G, \\mathcal{H}))$, which on $x \\in M \\subset M \\rtimes_\\alpha G$ is given by\n\\[ [\\pi(x) \\xi](g) = \\alpha_g^{-1}(x)\\xi(g), \\quad \\xi \\in L^2(G, \\mathcal{H}), g \\in G.\\]\nIf $(x_n)_{n \\in \\mathbb{N}}$ is any bounded sequence in $M$ representing an element $x \\in (M_{\\omega, \\alpha})^{\\alpha_\\omega}$, then the invariance of $x$ will guarantee that\n$\\pi(x_n) - (1 \\otimes x_n) \\rightarrow 0$ in the strong-$*$ operator topology as $n \\rightarrow \\omega$. Since $(1 \\otimes x_n)_{n \\in \\mathbb{N}}$ represents an element in $(\\mathcal{B}(L^2(G)) \\bar{\\otimes} M)_\\omega$, it follows that $(\\pi(x_n))_{n \\in \\mathbb{N}}$ represents an element in $(\\pi(M\\rtimes_\\alpha G))_\\omega$, so $x \\in (M\\rtimes_\\alpha G)_\\omega$. This finishes the proof.\n\\end{proof}\n\n\\begin{prop} \\label{prop:Takai-duality}\nLet $\\alpha: G \\curvearrowright M$ be an action on a von Neumann algebra with separable predual.\nLet $\\varphi$ be a faithful normal state on $M$.\nLet $\\tilde{\\alpha}: G\\curvearrowright\\tilde{M}=M\\rtimes_{\\sigma^\\varphi}\\mathbb R$ be the extended action on the continuous core as in Remark~\\ref{rem:tt-basics}.\nWith some abuse of notation, denote by $\\tilde{\\alpha}$ also the induced action $G\\curvearrowright\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R$ by acting trivially on the canonical unitary representation implementing $\\hat{\\sigma}^\\varphi$.\nThen under the Takesaki--Takai duality isomorphism $\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R \\cong \\mathcal B(L^2(\\mathbb R))\\bar{\\otimes} M$, the action $\\tilde{\\alpha}$ is cocycle conjugate to $\\operatorname{id}_{\\mathcal{B}(L^2(\\mathbb R))}\\otimes\\alpha$.\n\\end{prop}\n\\begin{proof}\nDenote $\\alpha'=\\operatorname{id}_{\\mathcal{B}(L^2(\\mathbb R))}\\otimes\\alpha$.\nWe apply Takesaki--Takai duality \\cite[Chapter X, Theorem 2.3(iii)]{Takesaki03} to understand the $G$-action $\\tilde{\\alpha}$.\nIf $M$ is represented faithfully on a Hilbert space $\\mathcal H$, then the natural isomorphism \n\\[\n\\Theta: \\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R = (M\\rtimes_{\\sigma^\\varphi}\\mathbb R)\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R \\to \\mathcal{B}(L^2(\\mathbb R)) \\bar{\\otimes} M\\subseteq \\mathcal{B}(L^2(\\mathbb R,\\mathcal H))\n\\]\nhas the following properties.\nLet $\\xi\\in L^2(\\mathbb R,\\mathcal H)$.\nFor $x\\in M$ and $s,t\\in\\mathbb R$ we have\n\\[\n[\\Theta(x)\\xi ](s) = \\sigma^\\varphi_{-s}(x)\\xi(s) \\quad\\text{and}\\quad [\\Theta(\\lambda^{\\sigma^\\varphi}_t)\\xi](s) = \\xi(s-t).\n\\]\nFurthermore, if the dual flow is given via the convention \n$\\hat{\\sigma}^\\varphi_t(\\lambda_s^{\\sigma^{\\varphi}}) = e^{its}\\lambda^{\\sigma^{\\varphi}}_s$, then we also have\n\\[\n[\\Theta(\\lambda^{\\hat{\\sigma}^\\varphi}_t)\\xi](s)=e^{its}\\xi(s).\n\\]\nThen we obtain an $\\alpha'$-cocycle $\\mathbb{W}$ via\n\\[\n[\\mathbb{W}_g\\xi](s) = (D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{-s}\\xi(s),\\quad \\xi\\in L^2(\\mathbb R,\\mathcal H).\n\\]\nGiven how $\\tilde{\\alpha}$ acts on the domain of $\\Theta$, we can observe (using \\cite[Corollary VIII.1.4]{Takesaki03}) for any $g\\in G$, $x \\in M$, $\\xi \\in L^2(\\mathbb{R}, \\mathcal{H})$ and $s \\in \\mathbb{R}$ that\n\\[\n\\begin{array}{ccl}\n[\\Theta(\\tilde{\\alpha}_g(x))\\xi ](s) &=& \\sigma^\\varphi_{-s}(\\alpha_g(x))\\xi(s) \\\\\n&=& \\operatorname{Ad}(D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{-s}\\Big( \\sigma_{-s}^{\\varphi\\circ\\alpha_g^{-1}}(\\alpha_g(x)) \\Big)\\xi(s) \\\\\n&=& \\Big(\\operatorname{Ad}(D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{-s}\\circ\\alpha_g\\Big) \\big( \\sigma^\\varphi_{-s}(x) \\big)\\xi(s) \\\\\n&=& \\Big[ \\Big( \\operatorname{Ad}(\\mathbb{W}_g)\\circ\\alpha'_g\\circ\\Theta\\Big)(x)\\, \\xi \\Big](s).\n\\end{array}\n\\]\nMoreover, using the cocycle identity and the chain rule for the Connes cocycle (\\cite[Theorem VIII.3.7]{Takesaki03}), we can see for any $g \\in G$, $\\xi \\in L^2(\\mathbb{R},\\mathcal{H})$ and $s,t \\in \\mathbb{R}$ that\n\\[\n\\begin{array}{ccl}\n[\\Theta(\\tilde{\\alpha}_g(\\lambda^{\\sigma^\\varphi}_t))\\xi](s) &=& [\\Theta\\big( (D \\varphi\\circ\\alpha_g^{-1} : D\\varphi)_t\\cdot\\lambda^{\\sigma^\\varphi}_t \\big)\\xi](s) \\\\\n&=& \\sigma^{\\varphi}_{-s}(D \\varphi\\circ\\alpha_g^{-1} : D\\varphi)_t [ \\Theta(\\lambda^{\\sigma^\\varphi}_t)\\xi ](s) \\\\\n&=& \\sigma^{\\varphi}_{-s}(D \\varphi\\circ\\alpha_g^{-1} : D\\varphi)_t \\xi(s-t) \\\\\n&=& (D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{-s} (D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{t-s}^* \\xi(s-t) \\\\\n&=& (D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{-s} \\Big[ \\Theta(\\lambda_t^{\\sigma^\\varphi}) \\mathbb{W}_g^* \\xi \\Big](s) \\\\\n&=& \\Big[ \\mathbb{W}_g \\Theta(\\lambda_t^{\\sigma^\\varphi}) \\mathbb{W}_g^* \\xi \\Big](s) \\\\\n&=& \\Big[ \\Big( \\operatorname{Ad}(\\mathbb{W}_g)\\circ\\alpha'_g\\circ\\Theta\\Big)(\\lambda_t^{\\sigma^\\varphi}) \\xi \\Big](s)\n\\end{array}\n\\]\nHere we used that $\\alpha'_g$ fixes the shift operator given by $\\Theta(\\lambda^{\\sigma^\\varphi}_t)$ for all $g\\in G$ and $t\\in\\mathbb{R}$.\nLastly, it is trivial to see that $\\alpha'$ fixes operators of the form $\\Theta(\\lambda^{\\hat{\\sigma}^\\varphi}_t)$, which in turn also commute pointwise with the cocycle $\\mathbb{W}$.\nIn conclusion, all of these observations culminate in the fact that the isomorphism $\\Theta$ extends to the cocycle conjugacy $(\\Theta,\\mathbb{W})$ between $\\tilde{\\alpha}$ and $\\alpha'$.\n\\end{proof}\n\nThe following represents our most general McDuff-type absorption theorem:\n\n\\begin{theorem} \\label{thm:general-McDuff}\nLet $G$ be a second-countable locally compact group.\nLet $\\alpha: G \\curvearrowright M$ be an action on a von Neumann algebra with separable predual and let $\\delta: G \\curvearrowright \\mathcal{R}$ be a strongly self-absorbing action on the hyperfinite II$_1$-factor.\nThen $\\alpha\\simeq_{\\mathrm{cc}}\\alpha\\otimes\\delta$ if and only if there exists a unital equivariant $*$-homomorphism $(\\mathcal R,\\delta) \\to (M_{\\omega,\\alpha},\\alpha_\\omega)$.\n\\end{theorem}\n\\begin{proof}\nThe ``only if'' part follows exactly as in the proof of Corollary~\\ref{cor:equivalence_equivariant_McDuff}, so we need to show the ``if'' part.\nSince Corollary~\\ref{cor:equivalence_equivariant_McDuff} already covers the case when $M$ is finite, we may split off the largest finite direct summand of $M$ and assume without loss of generality that $M$ is properly infinite.\n\nLet $\\varphi$ be a faithful normal state on $M$.\nAs in Remark~\\ref{rem:tt-basics}, we consider the (semi-finite) continuous core $\\tilde{M}$ and the extended $G$-action $\\tilde{\\alpha}: G\\curvearrowright\\tilde{M}$.\nSince the image of $\\tilde{\\alpha}$ commutes with the dual flow $\\hat{\\sigma}^{\\varphi}$, we have a continuous action $\\beta=\\tilde{\\alpha}\\times\\hat{\\sigma}^\\varphi: G\\times\\mathbb{R}\\curvearrowright\\tilde{M}$ via $\\beta_{(g,t)}=\\tilde{\\alpha}_g\\circ\\hat{\\sigma}^\\varphi_t$ for all $g\\in G$ and $t\\in\\mathbb{R}$.\nLet us also consider the action $\\delta^{\\mathbb R}: G\\times\\mathbb{R}\\curvearrowright\\mathcal R$ given by $\\delta^{\\mathbb R}_{(g,t)}=\\delta_g$ for all $g\\in G$ and $t\\in\\mathbb{R}$, which is evidently also strongly self-absorbing.\n\nWe apply Proposition~\\ref{prop:cp-centralizer} to $\\mathbb{R}$ in place of $G$ and to the modular flow in place of $\\alpha$.\nIn this context, recall that the ultrapower flow $(\\sigma^\\varphi)^\\omega$ already acts continuously on the ultrapower $M^\\omega$ (see \\cite[Theorem 4.1]{AndoHaagerup14}), and its restriction to $M_\\omega$ is trivial.\nSo $M_\\omega=(M_{\\omega,\\sigma^\\varphi})^{(\\sigma^\\varphi)^\\omega}$ and Proposition~\\ref{prop:cp-centralizer} implies that the inclusion $M\\subset\\tilde{M}$ induces an embedding $M_\\omega\\to \\tilde{M}_\\omega$.\nSince by definition, one has $\\tilde{\\alpha}|_M=\\alpha$ as $G$-actions, it is clear that bounded $(\\alpha,\\omega)$-equicontinuous sequences in $M$ become $(\\tilde{\\alpha},\\omega)$-equicontinuous sequences in $\\tilde{M}$.\nKeeping in mind that the dual flow $\\hat{\\sigma}^\\varphi$ acts trivially on $M$ by definition, the aforementioned inclusion therefore induces an equivariant unital $*$-homomorphism\n\\[\n(M_{\\omega,\\alpha},\\alpha_\\omega) \\to ( (\\tilde{M}_{\\omega,\\beta})^{(\\hat{\\sigma}^\\varphi)^\\omega},\\tilde{\\alpha}_\\omega ).\n\\]\nIf we compose this $*$-homomorphism with a given unital equivariant $*$-homomorphism $(\\mathcal R,\\delta)\\to (M_{\\omega,\\alpha},\\alpha_\\omega)$, we can view the resulting map as an $(G\\times\\mathbb{R})$-equivariant unital $*$-homomorphism $(\\mathcal R,\\delta^{\\mathbb R}) \\to (\\tilde{M}_{\\omega,\\beta},\\beta_\\omega)$.\nSince $\\tilde{M}$ is semi-finite, it follows from Corollary~\\ref{cor:equivalence_equivariant_McDuff} that $\\beta$ and $\\beta\\otimes\\delta^{\\mathbb R}$ are cocycle conjugate as $(G\\times\\mathbb R)$-actions, say via $(\\Psi,\\mathbb{V}): (\\tilde{M},\\beta)\\to (\\tilde{M}\\bar{\\otimes}\\mathcal{R},\\beta\\otimes\\delta^{\\mathbb R})$.\nRemembering $\\beta=\\tilde{\\alpha}\\times\\hat{\\sigma}^\\varphi$, we consider the $\\hat{\\sigma}^\\varphi\\otimes \\mathrm{id}_\\mathcal{R}$-cocycle $\\mathbbm{w}_t=\\mathbb{V}_{(1_G,t)}$ and the $\\tilde{\\alpha}\\otimes \\delta$-cocycle $\\mathbbm{v}_g=\\mathbb{V}_{(g,0)}$.\nThe cocycle identity for $\\mathbb{V}$ then implies the relation \n\\begin{equation} \\label{eq:cocycle-interaction}\n\\mathbbm{w}_t(\\hat{\\sigma}^\\varphi_t\\otimes\\operatorname{id}_{\\mathcal R})(\\mathbbm{v}_g)=\\mathbbm{v}_g(\\tilde{\\alpha}\\otimes\\delta)_g(\\mathbbm{w}_t)\n\\end{equation}\nfor all $g\\in G$ and $t\\in\\mathbb R$.\nThe cocycle conjugacy of flows $(\\Psi,\\mathbbm{w})$ induces an isomorphism\n\\[\n\\Lambda:=(\\Psi,\\mathbbm{w})\\rtimes\\mathbb R: \\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R \\to (\\tilde{M}\\bar{\\otimes}\\mathcal R)\\rtimes_{\\hat{\\sigma}^\\varphi\\otimes\\operatorname{id}_{\\mathcal R}}\\mathbb R \\cong \\big(\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R\\big)\\otimes\\mathcal R\n\\]\nvia\n\\[\n\\Lambda|_{\\tilde{M}} = \\Psi \\quad\\text{and}\\quad \\Lambda(\\lambda^{\\hat{\\sigma}^\\varphi}_t)=\\mathbbm{w}_t(\\lambda^{\\hat{\\sigma}^\\varphi}_t\\otimes 1_{\\mathcal R}),\\quad t\\in\\mathbb R.\n\\]\nWith slight abuse of notation (as in Proposition~\\ref{prop:Takai-duality}), we also denote by $\\tilde{\\alpha}$ the obvious induced $G$-action on the crossed product $\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R$.\nUsing that $(\\Psi,\\mathbb{V})$ was a cocycle conjugacy, we observe for all $g\\in G$ and $t\\in\\mathbb{R}$ that \n\\[\n\\operatorname{Ad}(\\mathbbm{v}_g)\\circ(\\tilde{\\alpha}\\otimes\\delta)_g\\circ\\Lambda|_{\\tilde{M}} = \\operatorname{Ad}(\\mathbb{V}_{(g,0)})\\circ(\\beta\\otimes\\delta^{\\mathbb R})_{(g,0)}\\circ\\Psi=\\Psi\\circ\\beta_{(g,0)}=\\Psi\\circ\\tilde{\\alpha}_g\n\\]\nand\n\\[\n\\begin{array}{ccl}\n\\big( \\operatorname{Ad}(\\mathbbm{v}_g)\\circ(\\tilde{\\alpha}\\otimes\\delta)_g\\circ\\Lambda \\big)(\\lambda^{\\hat{\\sigma}^\\varphi}_t) \n&=& \\mathbbm{v}_g \\big( (\\tilde{\\alpha}\\otimes\\delta)_g(\\mathbbm{w}_t) (\\lambda^{\\hat{\\sigma}^\\varphi}_t\\otimes 1_{\\mathcal R}) \\big) \\mathbbm{v}_g^* \\\\\n&\\stackrel{\\eqref{eq:cocycle-interaction}}{=}& \\mathbbm{w}_t(\\hat{\\sigma}^\\varphi_t\\otimes\\operatorname{id}_{\\mathcal R})(\\mathbbm{v}_g) (\\lambda^{\\hat{\\sigma}^\\varphi}_t\\otimes 1_{\\mathcal R}) \\mathbbm{v}_g^* \\\\\n&=& \\mathbbm{w}_t(\\lambda^{\\hat{\\sigma}^\\varphi}_t\\otimes 1_{\\mathcal R}) \\ = \\ \\Lambda(\\lambda^{\\hat{\\sigma}^\\varphi}_t).\n\\end{array}\n\\]\nIn conclusion, the pair $(\\Lambda,\\mathbbm{v})$ defines a cocycle conjugacy between the $G$-actions $\\tilde{\\alpha}$ on $\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R$ and $\\tilde{\\alpha}\\otimes\\delta$ on $\\big(\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R\\big)\\otimes\\mathcal R$.\nBy Proposition~\\ref{prop:Takai-duality}, the action $\\tilde{\\alpha}$ is cocycle conjugate to $\\operatorname{id}_{\\mathcal B(\\ell^2(\\mathbb{N}))}\\otimes\\alpha$.\nSince we assumed $M$ to be properly infinite, it furthermore follows that $\\operatorname{id}_{\\mathcal B(\\ell^2(\\mathbb{N}))}\\otimes\\alpha$ is cocycle conjugate to $\\alpha$.\\footnote{Although this appears to be well-known, we could not find a good literature source for this precise claim and in this generality. We note, however, that the recent proof of the analogous C$^*$-algebraic statement \\cite[Proposition 1.4]{GabeSzabo22kp} carries over in an obvious way to this setting.}\nCombining all these cocycle conjugacies yields one between $\\alpha$ and $\\alpha\\otimes\\delta$.\nThis finishes the proof.\n\\end{proof}\n\nThe following consequence is our last main result, which generalizes Corollary~\\ref{cor:McDuff-passes} to actions on arbitrary von Neumann algebras.\n\n\\begin{theorem} \\label{thm:general-amenable-McDuff}\nLet $G$ be a countable discrete group and $M$ a von Neumann algebra with separable predual such that $M \\cong M\\bar{\\otimes} \\mathcal{R}$. \nThen for every amenable action $\\alpha: G \\curvearrowright M$, one has $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\mathrm{id}_\\mathcal{R}$.\n\\end{theorem}\n\\begin{proof}\nChoose a faithful normal state $\\varphi$ on $M$.\nRecall that the induced faithful normal state $\\varphi^\\omega$ on $M^\\omega$ restricts to a tracial state on $M_\\omega$. We denote by $\\|\\cdot\\|_2=\\|\\cdot\\|_{\\varphi^\\omega}$ the induced tracial 2-norm on $M_\\omega$.\nSince we assumed $M$ to be McDuff, it follows that $M_\\omega$ is locally McDuff in the following sense:\nGiven any $\\|\\cdot\\|_2$-separable $*$-subalgebra $B\\subset M_\\omega$, there exists a unital $*$-homomorphism $\\mathcal R\\to M_\\omega\\cap B'$.\n\nNow we choose $N_1=\\mathcal{Z}(M)$ as a subalgebra of $M_\\omega$.\nWe may then choose a unital $*$-homomorphism $\\psi_1: \\mathcal{R}\\to M_\\omega$, and define $N_2$ to be the $\\|\\cdot\\|_2$-closed $*$-subalgebra generated by $N_1$ and the range of $\\alpha_{\\omega,g}\\circ\\psi_1$ for all $g\\in G$.\nAfter that, we may choose a unital $*$-homomorphism $\\psi_2: \\mathcal{R}\\to M_\\omega\\cap N_2'$, and define $N_3$ to be the $\\|\\cdot\\|_2$-closed $*$-subalgebra generated by $N_2$ and the range of $\\alpha_{\\omega,g}\\circ\\psi_2$ for all $g\\in G$.\nCarry on inductively like this and obtain an increasing sequence of $\\alpha_{\\omega}$-invariant separable von Neumann subalgebras $N_k\\subseteq M_\\omega$.\nThe $\\|\\cdot\\|_2$-closure of the union $\\bigcup_{k\\geq 1} N_k$ is then a separably acting finite von Neumann subalgebra, which is clearly McDuff and $\\alpha_{\\omega}$-invariant.\nFurthermore $N$ contains the center of $M$ equivariantly, which implies that the action $\\alpha_\\omega$ is amenable on $N$.\n\nBy Corollary~\\ref{cor:McDuff-passes} (with $H=G$), it follows that $\\alpha_{\\omega}|_N$ is cocycle conjugate to $(\\alpha_\\omega|_N)\\otimes\\operatorname{id}_{\\mathcal R}$.\nIn particular we may find some unital $*$-homomorphism $\\mathcal R\\to (N_\\omega)^{\\alpha_\\omega}$.\nApplying a standard reindexation trick, we may use this to obtain a unital $*$-homomorphism $\\mathcal R\\to (M_\\omega)^{\\alpha_\\omega}$, which finishes the proof by Theorem~\\ref{thm:general-McDuff}.\n\\end{proof}\n\n\n\\addcontentsline{toc}{section}{References}\n\\bibliographystyle{gabor}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nAutonomous, biomimetic robots can serve as tools in animal behavioural studies. Robots are used in ethology and behavioural studies to untangle the multimodal modes of interactions and communication between animals~\\cite{mondada2013general}. When they are socially integrated in a group of animals, they are capable of sending calibrated stimuli to test the animal responses in a social context~\\cite{halloy2007social}. Moreover, animal and autonomous robot interactions represent an interesting challenge for robotics. Confronting robots to animals is a difficult task because specific behavioural models have to be designed and the robots have to be socially accepted by the animals. The robots have to engage in social behaviour and convince somehow the animal that they can be social companions. In this context, the capabilities of the robots and their intelligence are put in harsh conditions and often demonstrate the huge gap that still exists between autonomous robots and animals not only considering motion and coping with the environment but also in terms of intelligence. It is a direct comparison of artificial and natural collective intelligence.\nMoreover, the design of such social robots is challenging as it involves both a luring capability including appropriate robot behaviours, and the social acceptation of the robots by the animals. We have shown that the social integration of robots into groups of fish can be improved by refining the behavioural models used to build their controllers~\\cite{cazenille2017acceptation}. The models have also to be calibrated to replicate accurately the animal collective behaviours in complex environments~\\cite{cazenille2017acceptation}.\n\nResearch on animal and robot interactions need also bio-mimetic formal models as behavioural controllers of the robots if the robots have to behave as congeners~\\cite{Bonnet2016IJARS,bonnet2017cats}. Robots controllers have to deal with a whole range of behaviours to allow them to take into account not only the other individuals but also the environment and in particular the walls \\cite{cazenille2017acceptation,cazenille2017automated}. However, most of biological collective behaviour models deal only with one sub-part at a time of fish behaviours in unbounded environments. Controllers based on neural networks, such as multilayer perceptron (MLP)~\\cite{king1989neural} or echo state networks (ESN)~\\cite{jaeger2007echo} have the advantage to be easier to implement and could deal with a larger range of events.\n\n\\subsection*{Objectives}\nWe aim at building models that generate accurately zebrafish trajectories of one individual within a small group of 5 agents. The trajectories are the result of social interactions in a bounded environment. Zebrafish are a classic animal model in the research fields of genetics and neurosciences of individual and collective behaviours. Building models that correctly reproduce the individual trajectories of fish within a group is still an open question~\\cite{herbert2015turing}. We explore MLP and ESN models, optimised by evolutionary computation, to generate individual trajectories. MLP and ESN are black-box models that need few \\textit{a priori} information provided by the modeller. They are optimised on the experimental data and as such represent a model of the complex experimental collective trajectories. However, they are difficult to calibrate on the zebrafish experimental data due to the complexity of the fish trajectories.\nHere, we consider the design and calibration by evolutionary computation of neural network models, MLP and ESN, that can become robot controllers. We test two evolutionary optimisation methods, CMA-ES~\\cite{auger2005restart} and NSGA-III~\\cite{yuan2014improved} and show that the latter gives better results. We show that such MLP and ESN behavioural models could be useful in animal robot interactions and could make the robots accepted by the animals by reproducing their behaviours and trajectories as in~\\cite{cazenille2017acceptation}. \n\n\n\\section{Materials and Methods} \\label{sec:methods}\n\n\\subsection{Experimental set-up} \\label{sec:setup}\nWe use the same experimental procedure, fish handling, and set-up as in~\\cite{cazenille2016automated,bonnet2017cats,seguret2017loose,cazenille2017acceptation,collignon2017collective,bonnet2018closed}. The experimental arena is a square white plexiglass aquarium of $1000\\times1000\\times100$~mm. An overhead camera captures frames at 15 FPS, with a $500\\times500$px resolution, that are then tracked to find the fish positions. We use 10 groups of 5 adults wild-type AB zebrafish (\\textit{Danio rerio}) in 10 trials lasting each one for 30-minutes as in~\\cite{cazenille2016automated,bonnet2017cats,seguret2017loose,cazenille2017acceptation,collignon2017collective,bonnet2018closed}.\nThe experiments performed in this study were conducted under the authorisation of the Buffon Ethical Committee (registered to the French National Ethical Committee for Animal Experiments \\#40) after submission to the French state ethical board for animal experiments.\n\n\n\\begin{figure*}[h]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{workflow3.pdf}\n\\caption{Methodology workflow. An evolutionary algorithm is used to evolve the weight of a MLP (1 hidden layer, 100 neurons) or an ESN (100 reservoir neurons) neural networks that serves as the controller of a simulated robot interacting with 4 fish described by the experimental data. Only the connections represented by dotted arrows are evolved (for MLP: all connections; for ESN: connections from inputs to reservoir, from reservoir to outputs and from outputs to outputs and to reservoir). The fitness function is computed through data-analysis of these simulations and represent the biomimetism metric of the simulated robot behaviour compared to the behaviour exhibited by real fish in experiments. Two evolutionary algorithms are tested: CMA-ES (mono-objective) and NSGA-III (multi-objective).}\n\\label{fig:workflow}\n\\end{center}\n\\end{figure*}\n\n\n\\subsection{Artificial neural network model} \\label{sec:model}\nBlack-box models, like artificial neural networks (ANN), can be used to model phenomena with few \\textit{a priori} information. Although they are not used yet to model fish collective behaviours based on experimental data, here we show that they are relevant to model zebrafish collective behaviour.\nWe propose a methodology (Fig.~\\ref{fig:workflow}) where either a multilayer perceptron (MLP)~\\cite{king1989neural} artificial neural network, or an echo state network (ESN)~\\cite{jaeger2007echo}, is calibrated through the use of evolutionary algorithms to model the behaviour of a simulated fish in a group of 5 individuals. The 4 other individuals are described by the experimental data obtained with 10 different groups of 5 fish for trials lasting 30 minutes.\n\nMLP are a type of feedforward artificial neural networks that are very popular in artificial intelligence to solve a large variety of real-world problems~\\cite{norgaard2000neural}. Their capability to universally approximate functions~\\cite{cybenko1989approximation} makes them suitable to model control and robotic problems~\\cite{norgaard2000neural}. We consider MLP with only one hidden layer of $100$ neurons (using a hyperbolic tangent function as activation function).\n\nESN are recurrent neural networks often used to model temporal processes, like time-series, or robot control tasks~\\cite{polydoros2015advantages}. They are sufficiently expressive to model complex non-linear temporal problems, that non-recurrent MLP cannot model.\n\nFor the considered focal agent, the neural network model takes the following parameters as input:\n    (i) the \\textit{direction vector} (angle and distance) from the focal agent towards each other agent;\n    (ii) the \\textit{angular distance} between the focal agent direction and each other agent direction (alignment measure);\n    (iii) the \\textit{direction vector} (angle and distance) from the focal agent towards the nearest wall;\n    (iv) the \\textit{instant linear speed} of the focal agent at the current time-step, and at the previous time-step;\n    (v) the \\textit{instant angular speed} of the focal agent at the current time-step, and at the previous time-step.\nThis set of inputs is typically used in multi-agent modelling of animal collective behaviour~\\cite{deutsch2012collective,sumpter2012modelling}. As a first step, we consider that it is sufficient to model fish behaviour with neural networks.\n\nThe neural network has two outputs corresponding to the change in linear and angular speeds to apply from the current time-step to the next time-step. Here, we limit our approach to modelling fish trajectories resulting from social interactions in a homogeneous environment but bounded by walls. Very few models of fish collective behaviours take into account the presence of walls~\\cite{collignon2016stochastic,calovi2018disentangling}.\n\n\n\\subsection{Data analysis} \\label{sec:dataAnalysis}\nFor each trial, $e$, and simulations, we compute several behavioural metrics using the tracked positions of agents: (i) the distribution of \\textit{inter-individual distances} between agents ($D_e$); (ii) the distributions of \\textit{instant linear speeds} ($L_e$); (iii) the distributions of \\textit{instant angular speeds} ($A_e$); (iv) the distribution of \\textit{polarisation} of the agents in the group ($P_e$) and (v) the distribution of \\textit{distances of agents to their nearest wall} ($W_e$).\nThe polarisation of an agent group measures how aligned the agents in a group are, and is defined as the absolute value of the mean agent heading: $P = \\frac{1}{N} \\bigl\\lvert  \\sum^{N}_{i=1} u_i \\bigr\\rvert$ where $u_i$ is the unit direction of agent $i$ and $N=5$ is the number of agents~\\cite{tunstrom2013collective}.\n\nWe define a similarity measure (ranging from $0.0$ to $1.0$) to measure the biomimetism of the simulated robot behaviour by comparing the behaviour of the group of agents in simulations where the robot is present (experiment $e_r$: four fish and one robot) to the behaviour of the experimental fish groups (experiment $e_c$: five fish):\n\\begin{equation}\nS(e_r, e_c) = \\sqrt[5]{I(D_{e_r}, D_{e_c}) I(L_{e_r}, W_{e_c}) I(A_{e_r}, O_{e_c}) I(P_{e_r}, T_{e_c}) I(W_{e_r}, T_{e_c})}\n\\end{equation}\nThe function $I(X, Y)$ is defined as such: $I(X, Y) = 1 - H(X, Y)$.\nThe $H(X, Y)$ function is the Hellinger distance between two histograms~\\cite{deza2006dictionary}. It is defined as: $H(X, Y) = \\frac{1}{\\sqrt{2}} \\sqrt{ \\sum_{i=1}^{d} (\\sqrt{X_i} - \\sqrt{Y_i}  )^2 }$ where $X_i$ and $Y_i$ are the bin frequencies.\n\nThis score measures the social acceptation of the robot by the fish, as defined in~\\cite{cazenille2017acceptation,cazenille2017automated}. Compared to the similarity measure defined in these articles, we added a measure of the polarisation of the agents. This was motivated by the tendency of our evolved neural models, without a polarisation factor, to generate agents with unnatural looping behaviour to catch up with the group.\n\n\n\\subsection{Optimisation} \\label{sec:optim}\nWe calibrate the ANN models presented here to match as close as possible the behaviour of one fish in a group of 5 individuals in 30-minute simulations (at $15$ time-steps per seconds, \\textit{i.e.}\\  $27000$ steps per simulation). This is achieved by optimising the connection weights of the ANN through evolutionary computation that iteratively perform global optimisation (inspired by biological evolution) on a defined fitness function so as to find its maxima~\\cite{salimans2017evolution,jiang2008supervised}. \n\nWe consider two optimisation methods (as in~\\cite{cazenille2017automated}), for MLP and ESN networks.\nIn the \\textbf{Sim-MonoObj-MLP} case, we use the CMA-ES~\\cite{auger2005restart} mono-objective evolutionary algorithm to optimise an MLP, with the task of maximising the $S_(e_1, e_2)$ function.\nIn the \\textbf{Sim-MultiObj-MLP} and \\textbf{Sim-MultiObj-ESN} cases, we use the NSGA-III~\\cite{yuan2014improved} multi-objective algorithm with three objectives to maximise. The first objective is a performance objective corresponding to the $S_(e_1, e_2)$ function. We also consider two other objectives used to guide the evolutionary process: one that promotes genotypic diversity~\\cite{mouret2012encouraging} (defined by the mean euclidean distance of the genome of an individual to the genomes of the other individuals of the current population), the other encouraging behavioural diversity (defined by the euclidean distance between the $D_{e}$, $L_{e}$, $A_{e}$, $P_{e}$ and $W_{e}$ scores of an individual). The NSGA-III algorithm was used with a $0.80\\%$ probability of crossovers and a $0.20\\%$ probability of mutations (we also tested this algorithm with only mutations and obtained similar results).\nThe NSGA-III algorithm~\\cite{yuan2014improved} is considered instead of the NSGA-II algorithm~\\cite{deb2002fast} employed in~\\cite{cazenille2017automated} because it is known to converge faster than NSGA-II on problems with more than two objectives~\\cite{ishibuchi2016performance}.\n\nIn both methods, we use populations of 60 individuals and 300 generations. Each case is repeated in 10 different trials.\nWe use a NSGA-III implementation based on the DEAP python library~\\cite{fortin2012deap}.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.99\\textwidth]{AllScores1.pdf}\n\\includegraphics[width=0.99\\textwidth]{AllScores2.pdf}\n\\caption{Similarity scores between the behaviour of the experimental fish groups (control) and the behaviour of the best-performing simulated individuals of the MLP models optimised by CMA-ES or NSGA-III.  Results are obtained over 10 different trials (experiments for fish-only groups, and simulations for NN models). We consider five behavioural features to characterise exhibited behaviours. \\textbf{Inter-individual distances} corresponds to the similarity in distribution of inter-individual distances between all agents and measures the capabilities of the agents to aggregate. \\textbf{Linear and Angular speeds distributions} correspond to the distributions of linear and angular speeds of the agents. \\textbf{Polarisation} measures how aligned the agents are in the group. \\textbf{Distances to nearest wall} corresponds to the similarity in distribution of agent distance to their nearest wall, and assess their capability to follow the walls. The \\textbf{Biomimetic score} corresponds to the geometric mean of the other scores.}\n\\label{fig:scores}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.90\\textwidth]{hists}\n\\caption{Comparison between 30-minutes trials involving 5 fish (control, biological data) and simulations involving 4 fish and 1 robot, over 10 trials and across 5 behavioural features: inter-individual distances (\\textbf{A}), linear (\\textbf{B}) and angular (\\textbf{C}) speeds distributions, polarisation (\\textbf{D}), and distances to nearest wall (\\textbf{E}).}\n\\label{fig:plotsHists}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Results} \\label{sec:results}\nWe analyse the behaviour of one simulated robot in a group of 4 fish. The robots are driven by ANN (either MLP or ESN) evolved with CMA-ES (\\textbf{Sim-MonoObj-MLP} case) or with NSGA-III (\\textbf{Sim-MultiObj-MLP} and \\textbf{Sim-MultiObj-ESN} cases) and compare it to the behaviour of fish-only groups (\\textbf{Control} case). We only consider the best-evolved ANN controllers. In the simulations, the simulated robot does not influence the fish because the fish are described by their experimental data that is replayed.\n\nExamples of agent trajectories obtained in the three tested cases are found in Fig.~\\ref{fig:plotsTraces}A. In the \\textbf{Sim-MonoObj-MLP} and \\textbf{Sim-MultiObj-*} cases, they correspond to the trajectory of the simulated robot agent.\nIn both case, we can see that the robot follow the walls like the fish, and are often part of the fish group as natural fish do. However, the robot trajectories can incorporate patterns not found in the fish trajectories. For example, small circular loop are done when the robot performs an U-turn to catch up with the fish group. This is particularly present in the \\textbf{Sim-MonoObj-MLP} case, and seldom appear in the \\textbf{Sim-MultiObj-*} cases.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.80\\textwidth]{traj}\n\\caption{Agent trajectories observed after 30-minute trials in a square ($1m$) aquarium, for the 4 considered cases: \\textbf{Control} reference experimental fish data obtained as in~\\cite{collignon2016stochastic,seguret2017loose}, \\textbf{Sim-MonoObj-MLP} MLP optimised by CMA-ES, \\textbf{Sim-MultiObj-MLP} MLP optimised by NSGA-III, \\textbf{Sim-MultiObj-ESN} ESN optimised by NSGA-III. \\textbf{A} Examples of an individual trajectory of one agent among the 5 making the group (fish or simulated robot) during 1-minute out of a 30-minute trial. \\textbf{B} Presence probability density of agents in the arena.}\n\\label{fig:plotsTraces}\n\\end{center}\n\\end{figure}\n\n\nWe compute the presence probability density of agents in the arena (Fig.~\\ref{fig:plotsTraces}B): it shows that the robot tend to follow the walls as the fish do naturally.\n\nFor the three tested cases, we compute the statistics presented in Sec.~\\ref{sec:dataAnalysis} (Fig.~\\ref{fig:plotsHists}). The corresponding similarity scores are shown in Fig.~\\ref{fig:scores}. The results of the \\textbf{Control} case shows sustained aggregative and wall-following behaviours of the fish group. Fish also seldom pass through the centre of the arena, possibly in small short-lived sub-groups. There is group behavioural variability, especially on aggregative tendencies (measured by inter-individual distances), and wall-following behaviour (measured by the distance to the nearest wall), because each one of the 10 groups is composed of different fish \\textit{i.e.} 50 fish in total.\n\nThe similarity scores of the \\textbf{Sim-MultiObj-*} cases are often within the variance domain of the \\textbf{Control} case, except for the inter-individual score. It suggests that groups incorporating the robot driven by an MLP evolved by NSGA-III exhibit relatively similar dynamics as a fish-only group, at least according to our proposed measures. However, it is still perfectible: the robot is sometimes at the tail of the group, possibly because of gap created between the robot and the fish group by small trajectories errors (\\textit{e.g.}\\  small loops shown in robot trajectories in Fig.~\\ref{fig:plotsTraces}A).\n\nThe \\textbf{Sim-MonoObj-MLP} case sacrifices biomimetism to focus mainly on group-following behaviour: this translated into a higher inter-individual score than in the \\textbf{Sim-MultiObj-*} cases, and robot tend to follow closely the fish group. With \\textbf{Sim-MonoObj-MLP}, the robot is going faster than the fish, and will fastly go back towards the centroid of the group if it is too far ahead of the group: this explains the large presence of loops in Fig.~\\ref{fig:plotsTraces}A. The \\textbf{Sim-MonoObj-MLP} does not take into account behavioural diversity like the \\textbf{Sim-MultiObj-*}, but focus on the one that is easier to find (namely the group-following behaviour) and stays stuck in this local optimum.\n\nThere are few differences between the results of the \\textbf{Sim-MultiObj-MLP} and the \\textbf{Sim-MultiObj-ESN} cases, the latter showing often slightly lower scores than the former. However, the \\textbf{Sim-MultiObj-ESN} displays a large variability of inter-individual scores, which could suggest that its expressivity could be sufficient to model agents with more biomimetic behaviours if the correct connection weights were found by the optimiser.\n\n\n\n\n\\section{Discussion and Conclusion} \\label{sec:conclusion}\nWe evolved artificial neural networks (ANN) to model the behaviour of a single fish in a group of 5 individuals. This ANN controller was used to drive the behaviour of a robot agent in simulations to integrate the group of fish by exhibiting biomimetic behavioural capabilities. Our methodology is similar to the calibration methodology developed in~\\cite{cazenille2017automated}, but employs artificial neural networks instead of an expert-designed behavioural model. Artificial neural networks are black-box models that require few \\textit{a-priori} information about the target tasks.\n\nWe design a biomimetism score from behavioural measures to assess the biomimetism of robot behaviour. In particular, we measure the aggregative tendencies of the agents (inter-individual distances), their disposition to follow walls, to be aligned with the rest of the group (polarisation), and their distribution of linear and angular speeds.\n\nHowever, finding ANN displaying behaviours of appropriate levels of biomimetism is a challenging issue, as fish behaviour is inherently multi-level (tail-beats as motor response vs individual trajectories vs collective dynamics), multi-modal (several kinds of behavioural patterns, and input\/output sources), context-dependent (different behaviours depending on the spatial position and proximity to other agents) and stochastic (leading to individual and collectives choices and action selection)~\\cite{collignon2016stochastic,sumpter2018using}. More specifically, fish dynamics involve trade-offs between social tendencies (aggregation, group formation), and response to the environment (wall-following, zone occupation); they also follow distinct movement patterns that allow them to move in a polarised group and react collectively to environmental and social cues.\n\nWe show that this artificial neural models can be optimised by using evolutionary algorithms, using the biomimetism score of robot behaviour as a fitness function. The best-performing evolved ANN controllers show competitive biomimetism scores compared to fish group behavioural variability. We demonstrate that taking into account genotypic and behavioural diversity in the optimisation process (through the use of the global multi-objective optimiser NSGA-III) improve the biomimetic scores of the evolved best-performing controllers.\nThe ANN models evolved through mono-objective optimisation tend to focus more on evolving a group-following behaviour rather than a biomimetic agent.\n\nOur approach is still perfectible, in particular, we only evolve the behaviour of a single agent in a group, rather than all agents of the group. This choice was motivated by the large increase in difficulty in evolving ANN models for the entire group, which would also involve additional behavioural trade-offs: \\textit{e.g.}\\  individual free-will and autonomous dynamics, individuals leaving or re-joining the group. However, it also means that here the fish do not react to the robot in simulations because the fish behaviour is a replay of fish experimental trajectories recorded without robot.\n\nAdditionally, it may be possible to improve the performance (in term of biomimetism) of the multi-objective optimisation process by combining additional selection pressures as objectives (\\textit{i.e.}\\  not just genotypic and behavioural diversity)~\\cite{doncieux2014beyond}. We already include behavioural and phenotypic diversities as selection pressures to guide the optimisation process; however, taking into account phenotypic diversity can bias the optimisation algorithm to explore rather than exploit, which can prevent some desired phenotypes to be considered by the optimisation algorithm. An alternative would be to use angular diversity instead~\\cite{szubert2016reducing}.\n\nThis study shows that ANN are good candidates to model individual and collective fish behaviours, in particular in the context of social bio-hybrid systems composed of animals and robots. By evolutionary computation, they can be calibrated on experimental data. This approach requires less \\textit{a priori} knowledge than equations or agent based modelling techniques. Although they are black box model, they could also produce interesting results from a biological point of view. Thus, ANN collective behaviour models can be an interesting approach to design animal and robot social interactions.\n\n\n\n\\section*{Acknowledgement}\n{\\small\nThis work was funded by EU-ICT project 'ASSISIbf', no 601074.}\n\n\n\\FloatBarrier\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{\\texorpdfstring{$k$}{k}-Fold Matroid Union} \\label{appendix:matroid-union-fold}\n\nIn this section, we study the special case of matroid union where we take the $k$-fold union of the same matroid.\nThat is, a basis of the $k$-fold union of $\\cM = (U, \\cI)$ is the largest subset $S \\subseteq U$ which can be partitioned into $k$ disjoint independent sets $S_1, \\ldots, S_k$ of $\\cI$.\nMany of the prominent applications of matroid union fall into this special case, particularly the $k$-disjoint spanning tree problem.\nAs a result, here we show an optimized version of the algorithm presented in \\cref{sec:matroid-union} with better and explicit dependence on $k$ that works in this regime. \n\n\\matroidunionfold*\n\nNote that in the breadth-first-search and blocking-flow algorithms in \\cref{sec:matroid-union}, there is an $O(k)$ overhead where we have to spend $O(k)$ time iterating through the $k$ binary search trees in order to explore the out-neighbors of the $O(kr)$ elements.\nOur goal in this section is thus to show that it is possible to further ``sparsify'' the exchange graphs to contain essentially only a basis, hence reducing its size from $\\Theta(kr)$ to $O(r)$.\nWe start with a slight modification to the BFS \\cref{alg:bfs-union} which reduces the running time by a factor of $O(k)$. The idea is that if we visit an element $u$ in the BFS which does not increase the rank of all visited elements so far, we can skip searching out-edges from $u$. Indeed, if $(u,v)$ is an edge of the exchange graph, then there must have been some element $u'$ visited earlier in the BFS which also has the edge $(u',v)$.\n\n\\begin{algorithm}\n  \\SetEndCharOfAlgoLine{}\n  \\SetKwInput{KwData}{Input}\n  \\SetKwInput{KwResult}{Output}\n  \n  \\caption{BFS in a $k$-fold union exchange graph}\n  \\label{alg:bfs-union-fold}\n  \\KwData{$S \\subseteq U$ with partition $S_1, \\ldots, S_k$ of independent sets and a basis $B$ of $U \\setminus S$}\n  \\KwResult{The $(s, v)$-distance $d(v)$ in $H(S)$ for each $v \\in S \\cup \\{t\\}$}\n\n  $\\mathsf{queue} \\gets B$ and $R \\gets \\emptyset$\\;\n  $d(v) \\gets \\infty$ for each $v \\in S \\cup \\{t\\}$, and $d(v) \\gets 1$ for each $v \\in B$\\;\n  $\\cT_i \\gets \\textsc{Initialize}(\\cM, S_i, S_i)$ (\\cref{thm:bst} with $\\beta = 1$)\\;\n  \\While{$\\mathsf{queue} \\neq \\emptyset$} {\n    $u \\gets \\mathsf{queue}.\\textsc{Pop}()$\\;\n    \\If{$R + u \\in \\cI$} {\n      \\For{$i \\in \\{1, 2, \\ldots, k\\}$} {\n        \\While{$v := \\cT_i.\\textsc{Find}(u) \\neq \\bot$} {\n          $d(v) \\gets d(u) + 1$ and $\\mathsf{queue}.\\textsc{Push}(v)$\\;\n          $\\cT_i.\\textsc{Delete}(v)$\\;\n        }\n        \\lIf{$S_i + u \\in \\cI$ and $d(t) = \\infty$} {\n          $d(t) \\gets d(u) + 1$\n        }\n      }\n      $R \\gets R + u$\\;\n    }\n  }\n  \\textbf{return} $d(v)$ for each $v \\in S \\cup \\{t\\}$.\n\\end{algorithm}\n\n\\begin{lemma}\n  Given $S \\in \\cI_\\text{part} \\cap \\hat{\\cI}$ and a basis $B$ of $U \\setminus S$, it takes $\\tO(kr)$ time to construct the distance layers $L_2, \\ldots, L_{d_t - 1}$ of $H(S)$.\n  \\label{lemma:bfs-union-fold}\n\\end{lemma}\n\n\\begin{proof}\n  The algorithm is presented as \\cref{alg:bfs-union-fold}.\n  It performs a BFS from a basis $B$ of the first layer and only explores out-edges from the first basis $R$ it found.\n  It takes\n  (i) $\\tO(|S|)$ time to construct the $\\cT_i$'s,\n  (ii) $\\tO(1)$ time to discover each of the $O(kr)$ element, and\n  (iii) an additional $\\tO(k \\cdot |R|)$ time to iterate through all $k$ binary search trees $\\cT_i$'s for each $u \\in R$.\n  The total running time is thus bounded by $\\tO(kr)$.\n  \n  We have shown in \\cref{lemma:bfs-union} that it is feasible to replace $U \\setminus S$ with simply $B$.\n  It remains to show that exploring only the out-neighbors of $u \\in R$ does not affect the correctness.\n  Consider a $v \\in S \\setminus R$ (we know that $B \\subseteq R$ so it suffices to consider elements in $S$) with an out-neighbor $x \\in S_i$, i.e., $S_i - x + v \\in \\cI$.\n  It then follows that $\\mathsf{rank}((S_i - x) + R_v) = \\mathsf{rank}((S_i - x) + (R_v + v)) \\geq \\mathsf{rank}(S_i)$ by \\cref{obs:exchange} and \\cref{lemma:basis-rank}, where $R_v$ is the set $R$ when $v$ is popped out of the queue (in other words, $R_v + v \\not\\in \\cI$).\n  This implies that there is a $u \\in R_v$ which is visited before $v$ that also has out-neighbor $x$.\n  The modification is therefore correct.\n\\end{proof}\n\nOur blocking-flow algorithm for $k$-fold matroid union is presented as \\cref{alg:blocking-flow-union-fold}.\nIt's essentially a specialization of \\cref{alg:blocking-flow-union} to the case where all the $k$ matroids are the same, except that we skip exploring the out-neighbors of $a_{\\ell}$ and remove it directly if it is ``spanned'' by the previous layers and the set $R_{\\ell} \\subseteq L_{\\ell}$ of elements that are \\emph{not} on any augmenting path of length $d_t$.\nWith this optimization, we obtain the following lemma analogous to \\cref{lemma:blocking-flow-union}.\n\n\\begin{algorithm}\n  \\SetEndCharOfAlgoLine{}\n  \\SetKwInput{KwData}{Input}\n  \\SetKwInput{KwResult}{Output}\n  \\SetKwInput{KwGuarantee}{Guarantee}\n  \n  \\caption{Blocking flow in a $k$-fold union exchange graph}\n  \\label{alg:blocking-flow-union-fold}\n  \\KwData{$S \\subseteq U$ which partitions into $S_1, \\ldots, S_k$ of independent sets and a dynamic-basis data structure $\\cD$ of $U \\setminus S$}\n  \\KwResult{$S^\\prime \\in \\cI_{\\text{part}} \\cap \\cI_k$ with $d_{H(S^\\prime)}(s, t) > d_{H(S)}(s, t)$}\n  \\KwGuarantee{$\\cD$ maintains a basis of $U \\setminus S^\\prime$ at the end of the algorithm}\n  \n  Build the distance layers $L_2, \\ldots, L_{d_t - 1}$ of $H(S)$ with \\cref{lemma:bfs-union-fold}\\;\n  $L_0 \\gets \\{s\\}$ and $L_{d_t} \\gets \\{t\\}$\\;\n  $B \\gets$ the basis maintained by $\\cD$ and $L_1 \\gets B$\\;\n  $A_\\ell \\gets L_\\ell$ for each $0 \\leq \\ell \\leq d_t$\\;\n  $\\cT_{\\ell}^{(i)} \\gets \\textsc{Initialize}(\\cM_i, S_i, Q_{S_i}, A_{\\ell} \\cap S_i)$ for each $2 \\leq \\ell < d_t$ and $1 \\leq i \\leq k$ (\\cref{thm:bst} with $\\beta = \\sqrt{r} \/ d_t$)\\;\n  $D_{\\ell} \\gets \\emptyset$ for each $1 \\leq \\ell < d_t$\\;\n  $R_{\\ell} \\gets \\emptyset$ for each $2 \\leq \\ell < d_t$\\;\n  $\\ell \\gets 0$ and $a_0 \\gets s$\\;\n  \\While{$\\ell \\geq 0$} {\n    \\If{$\\ell < d_t$} {\n      \\lIf{$A_{\\ell} = \\emptyset$} {\n        \\textbf{break}\n      }\n      \\lIf{$\\ell \\geq 2$ and $\\mathsf{rank}(L_1 \\cup \\cdots \\cup L_{\\ell - 1} \\cup R_{\\ell} \\cup \\{a_{\\ell}\\}) = \\mathsf{rank}(L_1 \\cup \\cdots \\cup L_{\\ell - 1} \\cup R_{\\ell})$} {\\label{line:check-not-spanned}\n        $A_{\\ell} \\gets A_{\\ell} - a_{\\ell}$ and \\textbf{continue}\n      }\n      \\lIf{$\\ell > 0$} {\n        Find an $a_{\\ell + 1} := \\cT_{\\ell + 1}^{(i)}.\\textsc{Find}(a_\\ell) \\neq \\bot$ for some $1 \\leq i \\leq k$\n      }\n      \\lElse {\n        $a_{\\ell + 1} \\gets$ an arbitrary element in $A_1$\n      }\n      \\If{such an $a_{\\ell + 1}$ does not exist} {\n        \\lIf{$\\ell \\geq 2$} {\n          $R_{\\ell} \\gets R_{\\ell} + a_{\\ell}$ and $\\cT_{\\ell}^{(j)}.\\textsc{Delete}(a_{\\ell})$ where $a_{\\ell} \\in S_j$\n        }\n        $A_\\ell \\gets A_\\ell - a_\\ell$ and $\\ell \\gets \\ell - 1$\\;\\label{line:remove}\n      }\n      \\lElse {\n        $\\ell \\gets \\ell + 1$\n      }\n    }\n    \\Else {\n    \\tcp{Found augmenting path $a_1, a_2, \\ldots a_\\ell$}\n      $B \\gets B - a_1$, $A_1 \\gets A_1 - a_1$, and $D_1 \\gets D_1 + a_1$\\;\n      \\If{$\\cD.\\textsc{Delete}(a_1)$ returns a replacement $x$} {\n        $B_i \\gets B_i + x$  and $A_i \\gets A_i + x$\\;\n      }\n      \\For{$i \\in \\{2, \\ldots, d_t - 1\\}$} {\n          $D_{i} \\gets D_{i} + a_{i}$ and $A_{i} \\gets A_{i} - a_{i}$\\;\n          $\\cT_{i}^{(j)}.\\textsc{Delete}(a_i)$ and $\\cT_{i}^{(j)}.\\textsc{Update}(\\{a_{i - 1}, a_{i}\\})$ where $a_i \\in S_j$\\;\n      }\n      Augment $S$ along $P = (s, a_1, \\ldots, a_{d_t - 1}, t)$\\;\n      $\\ell \\gets 0$\\;\n    }\n  }\n  \\textbf{return} $S$\\;\n\\end{algorithm}\n\n\\begin{lemma}\n  Given an $S \\in \\cI_{\\text{part}} \\cap \\hat{\\cI}$ with $d_{H(S)}(s, t) = d_t$ together with a data structure $\\cD$ of \\cref{thm:decremental-basis} that maintains a basis of $U \\setminus S$, it takes\n  \\begin{equation}\n    \\tO\\left(\\underbrace{kr}_{\\ref{item:term1}} + \\underbrace{\\left(|S^\\prime| - |S|\\right) \\cdot d_t\\sqrt{r}}_{\\ref{item:term2}} + \\underbrace{\\left((|S^\\prime| - |S|) \\cdot d_t + r\\right) \\cdot k}_{\\ref{item:term3}} + \\underbrace{\\left(kr + (|S^\\prime| - |S|)\\right) \\cdot \\frac{\\sqrt{r}}{d_t}}_{\\ref{item:term4}}\\right)\n    \\label{eq:complexity}\n  \\end{equation}\n  time to obtain an $S^\\prime \\in \\cI_{\\text{part}} \\cap \\hat{\\cI}$ with $d_{H}(S^\\prime)(s, t) > d_t$, with an additional guarantee that $\\cD$ now maintains a basis of $U \\setminus S^\\prime$.\n  \\label{lemma:blocking-flow-union-fold}\n\\end{lemma}\n\nWe need the following observation to bound the running time of \\cref{alg:blocking-flow-union-fold}.\n\n\\begin{observation}\n  In \\cref{alg:blocking-flow-union-fold}, it holds that $B \\cup R_2 \\cup R_{3} \\cup \\cdots \\cup R_{d_t - 1} \\in \\cI$.\n  \\label{claim:is-basis}\n\\end{observation}\n\n\n\\begin{proof}[Proof of \\cref{lemma:blocking-flow-union-fold}]\n  We analyze the running time of \\cref{alg:blocking-flow-union-fold} first.\n  In particular, there are four terms in \\cref{eq:complexity} which come from the following.\n  \\begin{enumerate}[label=(\\roman*)]\n    \\item\\label{item:term1} $\\tO(kr)$: It takes $\\tO(kr)$ time to compute the distance layers using \\cref{lemma:bfs-union-fold} and initialize all the binary search trees $\\cT_{\\ell}^{(i)}$'s. Computing the rank of $L_1 \\cup \\cdots \\cup L_{\\ell - 1} \\cup R_{\\ell}$ also takes $\\tO(kr)$ time in total since we can pre-compute query-sets of the form $L_1 \\cup \\cdots \\cup L_{k}$ for each $k$ in $\\tO(kr)$ time, and each insertion to $R_{\\ell}$ takes $\\tO(1)$ time.\n    \\item\\label{item:term2} $\\tO\\left(\\left(|S^\\prime| - |S|\\right) \\cdot d_t\\sqrt{r}\\right)$: For each of the $O(|S^\\prime| - |S|)$ augmentations, it takes $\\tO(r \\cdot \\frac{d_t}{\\sqrt{r}})$ time to update the binary search trees.\n    \\item\\label{item:term3} $\\tO(\\left((|S^\\prime| - |S|) \\cdot d_t + r\\right) \\cdot k)$: The number of elements whose out-edges are explored is bounded by $O\\left((|S^\\prime| - |S|) \\cdot d_t + r\\right)$. This is because for each such element $u$, either $u$ is included in an augmenting path of length $d_t$, or $u$ is removed in Line \\ref{line:remove}.\n    There are $O((|S^\\prime| - |S|) \\cdot d_t)$ such $u$'s in the augmenting paths.\n    For $u$ removed in Line \\ref{line:remove}, if $\\ell = 1$, then the number of such $u$'s is $O(|S^\\prime| - |S| + r)$ because there are initially $O(r)$ elements in $A_1$, and we add at most one to it every augmentation.\n    If $\\ell \\geq 2$, then we insert it into $R_{\\ell}$, and by Line \\ref{line:check-not-spanned}, the rank of $L_1 \\cup \\cdots \\cup L_{\\ell - 1} \\cup R_{\\ell}$ increases after including $u$ into $R_{\\ell}$.\n    By \\cref{claim:is-basis}, the number of such $u$'s is bounded by $O(r)$.\n    The term then comes from spending $O(k)$ time iterating through the $k$ binary search trees for each of the $O\\left((|S^\\prime| - |S|) \\cdot d_t + r\\right)$ elements whose out-neighbors are explored.\n    \\item\\label{item:term4} $\\tO(\\left(kr + (|S^\\prime| - |S|)\\right) \\cdot \\frac{\\sqrt{r}}{d_t})$:  The number of elements that are once in some $A_{\\ell}$ is bounded by $O(kr + |S^\\prime| - |S|)$. Initially, there are $O(kr)$ elements ($A_1$ plus all the $A_\\ell$ for $\\ell \\geq 2$), and each augmentation adds at most one element to $A_1$.\n    Each of these elements is discovered by $\\cT_{\\ell}^{(i)}.\\textsc{Find}(\\cdot)$ at most once, and thus we can charge the $\\tO(\\frac{\\sqrt{r}}{d_t})$ cost to it, resulting in the fourth term of \\cref{eq:complexity}.\n  \\end{enumerate}\n  Note that for each element whose out-neighbors are explored, any failed attempt of $\\cT_{\\ell}^{(i)}.\\textsc{Find}(\\cdot)$ costs only $\\tO(1)$ instead of $\\tO(\\frac{\\sqrt{r}}{d_t})$ according to \\cref{thm:bst}.\n  The $\\tO(\\frac{\\sqrt{r}}{d_t})$ cost of a successful search is charged to term \\ref{item:term4} instead of \\ref{item:term3}.\n  \n  As for correctness, it suffices to show that each of the $a_{\\ell}$ removed from $A_{\\ell}$ because it is spanned by $L_1 \\cup \\cdots \\cup L_{\\ell - 1} \\cup R_{\\ell}$ in Line \\ref{line:check-not-spanned} is not in any augmenting path of length $d_t$.\n  Consider its out-neighbor $a_{\\ell + 1}$ with respect to the current $S$, and we would like to argue that $a_{\\ell + 1}$ is not on any augmenting path of length $d_t$ anymore.\n  This is because we have already explored all the out-neighbors of elements in $R_{\\ell}$.\n  Since $a_{\\ell} \\in \\mathsf{span}(L_1 \\cup \\cdots \\cup L_{\\ell - 1} \\cup R_{\\ell})$, by \\cref{lemma:basis-rank}, there must exist some $u \\in L_1 \\cup \\cdots \\cup L_{\\ell - 1} \\cup R_{\\ell}$ with a directed edge $(u, a_{\\ell + 1})$.\n  We consider two cases:\n  \\begin{itemize}\n  \\item $u\\in R_{\\ell}$. This means that we have already explored $a_{\\ell+1}$, as we finished exploring all out-neighbors of $u$ already.\n  \\item $u\\in L_{1}\\cup \\cdots \\cup L_{\\ell-1}$. We know that by \\cref{lemma:monotone}, both $d_{H(S)}(s,v)$ and\n  $d_{H(S)}(v,t)$ can only increase after augmentations for all elements $v$.\n  Hence $a_{\\ell+1}$ cannot be part of an augmenting path of length $d_t$ anymore, since if it was its distance to $t$ must be $d-(\\ell+1)$, but then the distance from $u$ to $t$ must be at most $d-\\ell$ (which is smaller than its initial distance to $t$ at the beginning of the phase).\n  \\end{itemize}\n  \n  As a result, all of $u$'s out-neighbors have either already been explored or do not belong to any augmenting path of length $d_t$.\n  This implies that $u$ is not on any such path either, and thus it's correct to skip and remove it from $A_{\\ell}$.\n  This concludes the proof of \\cref{lemma:blocking-flow-union-fold}.\n\\end{proof}\n\n\n\\cref{thm:dynamic-matroid-union-fold} now follows from analyzing the total running time of $O(\\sqrt{\\min(n, kr)})$ runs of \\cref{lemma:blocking-flow-union-fold}.\n\n\\begin{proof}[Proof of \\cref{thm:dynamic-matroid-union-fold}]\n  We initialize the dynamic-basis data structure $\\cD$ of \\cref{thm:decremental-basis} on $U$ in $\\tO(n)$ time.\n  Let $p = \\min(n, kr)$ be the rank of the $k$-fold matroid union.\n  Using $\\cD$, we then run $O(\\sqrt{p})$ iterations of \\cref{lemma:blocking-flow-union-fold} until $d_{H(S)}(s, t) > \\sqrt{p}$.\n  Summing the first two terms of \\cref{eq:complexity} over these $O(\\sqrt{p})$ iterations gives (recall that \\cref{lemma:augmenting-path-lengths} guarantees that $\\sum_{d = 1}^{\\sqrt{p}}d \\cdot (|S_d| - |S_{d - 1}|) = \\tO(p)$)\n  \\[\n    \\tO\\left(kr\\sqrt{p} + \\sqrt{r} \\cdot \\sum_{d = 1}^{\\sqrt{p}}d \\cdot \\left(|S_d| - |S_{d - 1}|\\right)\\right) = \\tO\\left(kr\\sqrt{p}\\right)\n  \\]\n  since $p\\sqrt{r} \\leq kr\\sqrt{p}$.\n  The third term of \\cref{eq:complexity} contributes a total running time of\n  \\[\n    \\tO\\left(\\left(\\sum_{d = 1}^{\\sqrt{p}}dk \\cdot \\left(|S_d| - |S_{d - 1}|\\right)\\right) + kr\\sqrt{p}\\right) = \\tO\\left(kr\\sqrt{p} + kp\\right),\n  \\]\n  while the fourth term of \\cref{eq:complexity} sums up to\n  \\[\n    \\tO\\left(\\left(\\sum_{d = 1}^{\\sqrt{p}}kr\\frac{\\sqrt{r}}{d}\\right) + kr\\sqrt{r}\\right) = \\tO\\left(kr\\sqrt{r}\\right).\n  \\]\n  We finish the algorithm by finding the remaining $O(\\sqrt{p})$ augmenting paths one at a time with \\cref{lemma:bfs-union-fold} in a total of $\\tO(kr\\sqrt{p})$ time.\n  The $k$-fold matroid union algorithm thus indeed runs in $\\tO\\left(n + kr\\sqrt{\\min(n, kr)} + k\\min(n, kr)\\right)$ time, concluding the proof of \\cref{thm:dynamic-matroid-union-fold}.\n\\end{proof}\n\n\n\n\n\n\\section{Dynamic Oracles for Specific Matroids \\& Applications} \\label{appendix:applications}\n\n\nIn this appendix, we show how to leverage known dynamic algorithms to implement the dynamic rank oracle (\\cref{def:dyn-oracle}) efficiently for many important matroids.\nWhat we need are data structures that can maintain the rank of a set dynamically under insertions and deletions in \\emph{worst-case} update time (converting a \\emph{worst-case} data structure to \\emph{fully-persistent} can be done by the standard technique of \\cite{DriscollSST86,Dietz89}, paying an overhead of $O(\\log{n})$). Additionally, note that the data structures do not need to work against an adaptive adversary since we only ever use the \\emph{rank} of the queried sets, which is not affected by internal randomness.\n\nIn particular, for \\emph{partition, graphic, bicircular, convex transversal}, and \\emph{simple job scheduling matroids} it is possible to maintain the rank with polylog$(n)$ update-time, and for \\emph{linear matroids} in $O(n^{1.529})$ update-time.\n\nTogether with our matroid intersection (\\cref{sec:matroid-intersection}) and matroid union (\\cref{sec:matroid-union}) algorithms, this leads to a black-box approach to solving many different problems. In fact, we can solve matroid intersection and union on any combination of the above matroids, leading to improved or matching running times for many problems (see the introduction \\cref{sec:intro} with \\cref{tab:intro:dyn-oracle-implies-fast-algorithms,tab:intro:implications} for a more thorough discussion). For completeness, we define these problems in \\cref{appendix:problems}.\nThe same algorithms are powerful enough to also solve new problems which have not been studied before.\n\n\n\\paragraph{Example Application: Tree Scheduling (or Maximum Forest with Deadlines).} We give an example of a reasonably natural combinatorial scheduling problem, which---to our knowledge---has not been studied before. Suppose we are given a graph $G = (V,E)$ where each edge $e\\in E$ has two numbers associated with it: a release date $\\ell_e$ and a deadline $r_e$. Consider the problem where we want to for each day pick exactly one edge (say, to build\/construct), but we have constraints that edge $e$ can only be picked between days $\\ell_e$ and $r_e$. Now the task is to come up with a scheduling plan to build a spanning tree of the graph, if possible.\n\nThis problem is exactly a matroid intersection problem between a graphic matroid and a convex transversal matroid. Hence, by a black-box reduction, we know that we can solve this problem in $\\tO(|E|\\sqrt{|V|})$ time.\n\n\n\\subsection{Partition Matroids}\nIn a partition matroid $\\cM = (U,\\cI)$, each element $u\\in U$ is assigned a color $c_u$. We are also, for each color $c$, given a non-negative integer $d_c$, and we define a set of elements $S\\subseteq U$ to be independent if for each color $c$, $S$ includes at most $d_c$ elements of this color.\nImplementing the dynamic oracle for the partition matroid is easy:\n\n\\begin{lemma}\nOne can maintain the rank of a partition matroid in $O(1)$-update time.\n\\end{lemma}\n\\begin{proof}\nFor each color $c$ we maintain a counter $x_c$ of how many elements we have of color $c$. We also maintain $r = \\sum \\min_c(x_c, d_c)$, which is the rank of the current set.\n\\end{proof}\n\n\\begin{remark}\nBipartite matching can be modeled as a matroid intersection problem of two partition matroids. So our matroid intersection algorithm together with the above lemma match (up to poly-logarithmic factors induced by fully-persistence)---in a black-box fashion---the $O(|E|\\sqrt{|V|})$-time bound of the best \\emph{combinatorial} algorithm for bipartite matching \\cite{HopcroftK73}.\n\\end{remark}\n\n\\subsection{Graphic and Bicircular Matroids}\n\\label{appendix:applications-graphic}\nGiven a graph $G = (V,E)$, the \\emph{graphic} and \\emph{bicircular} matroids are matroids capturing the connectivity structure of the graph.\n\n\n\\paragraph{Graphic Matroid.}\nIn the graphic matroids $\\cM = (E,\\cI)$ a subset of edges $E'\\subseteq E$ are independent if and only if they do not contain a cycle.\nWe use the following result to implement the dynamic oracle for this matroid.\n\n\\begin{lemma}[\\cite{KapronKM13,GibbKKT15}]\n  There is a data structure that maintains an initially-empty graph $G = (V, E)$ and supports insertion\/deletion of edges $e$ into\/from $E$ in worst case $O(\\log^4{|V|})$ time and query of the connectivity between $u$ and $v$ in worst case $O(\\log{|V|}\/\\log{\\log{|V|}})$ time.\n  The data structure works with high probability against an oblivious adversary.\n  \\label{lemma:worst-case-dynamic-conn-appendix}\n\\end{lemma}\n\nWith a simple and standard extension, we can maintain the number of connected components as well, and hence also the rank (since $\\mathsf{rank}(E') = |V|- \\#\\text{connected components in }G[E']$).\n\n\\begin{corollary}\n  There is a data structure that maintains an initially-empty graph $G = (V, E)$ and supports insertion\/deletion of $e$ into\/from $E$ in worst-case $O(\\log^4{|V|})$ time.\n  After each operation, the data structure also returns the number of connected components in $G$.\n  The data structure works with high probability against an oblivious adversary.\n  \\label{cor:worst-case-dynamic-components-appendix}\n\\end{corollary}\n\n\\begin{proof}\n  We maintain the data structure $\\cC$ of \\cref{lemma:worst-case-dynamic-conn-appendix} and a counter $c := |V|$ representing the number of connected components.\n  For insertion of $e = (u, v)$, we first query the connectivity of $u$ and $v$ before inserting $e$ into $\\cC$.\n  If they are not connected before the insertion, decrease $c$ by one.\n  For deletion of $e = (u, v)$, after deleting $e$ from $\\cC$, we check if $u$ and $v$ are still connected.\n  If not, then we increase $c$ by one.\n\\end{proof}\n\n\n\\paragraph{Bicircular Matroid.}\nIn the bicircular matroid $\\cM = (E,\\cI)$, a subset of edges $E'\\subseteq E$ are independent if and only if each connected component in $G[E']$ has at most one cycle. Similar to the graphic matroid, dynamic connectivity algorithms can be used to implement the dynamic rank oracle for bicircular matroids too.\n\n\\begin{corollary}\n  There is a data structure that maintains an initially-empty graph $G = (V, E)$ and supports insertion\/deletion of $e$ into\/from $E$ in worst-case $O(\\log^4{|V|})$ time.\n  After each operation, the data structure also returns the rank of $E$ in the bicircular matroid.\n  The data structure works with high probability against an oblivious adversary.\n  \\label{cor:worst-case-dynamic-components-appendix-bicircular}\n\\end{corollary}\n\n\\begin{proof}\nThe dynamic connectivity data structure of \n\\cite{KapronKM13,GibbKKT15} (\\cref{lemma:worst-case-dynamic-conn-appendix}) can be adapted to also keep track of the number of edges and vertices in each connected component. Using this, the data structure can, for each connected component $c$ keep track of a number $x_c$ as the minimum of the number of edges in this component and the number of vertices in this component. Then the rank of the bicircular matroid is just the sum of $x_c$ (as in an independent set each component is either a tree or a tree with an extra edge). In each update two components can merge, a component can be split up into two, or the edge-count of a component may simply change. \n\\end{proof}\n\n\\begin{remark}[Deterministic Dynamic Connectivity]\nThe above dynamic connectivity data structures are randomized.\nThere are also deterministic connectivity data structures, but with slightly less efficient sub-polynomial $|V|^{o(1)}$ update time~\\cite{ChuzhoyGLNPS20}.\n\\end{remark}\n\n\\subsection{Convex Transversal and Scheduling Matroids}\n\nConvex transversal and  scheduling matroids are special cases of the \\emph{transversal} matroid, with applications in scheduling algorithms.\n\n\\begin{definition}[Transversal Matroid~\\cite{edmonds1965transversals}]\n  A \\emph{transversal matroid} with respect to a bipartite graph $G = (L, R, E)$ is defined over the ground set $L$, where each $S \\subseteq L$ is independent if and only if there is a perfect matching in $G$ between $S$ and a subset of $R$.\n\\end{definition}\n\nA bipartite graph $G = (L, R, E)$ is \\emph{convex} if $R$ has a linear order $R = \\{r_1, r_2, \\ldots, r_n\\}$ and each $\\ell \\in L$ corresponds to an interval $1 \\leq s(\\ell) \\leq t(\\ell) \\leq n$ such that $(\\ell, r_i) \\in E$ if and only if $s(\\ell) \\leq i \\leq t(\\ell)$, i.e., the neighbors of each $\\ell$ form an interval.\n\n\\begin{definition}[Convex Transversal Matroid and Simple Job Scheduling Matroid]\n  A \\emph{convex transversal matroid} is a transversal matroid with respect to a convex bipartite graph.\n  A \\emph{simple job scheduling matroid} is a special case of convex transversal matroids in which $s(\\ell) = 1$ for each $\\ell \\in L$.\n\\end{definition}\n\nOne intuitive way to think about the simple job scheduling matroid is that there is a machine capable of finishing one job per day.\nThe ground set of the matroid consists of $n$ jobs, where the $i$-th jobs must be done before its deadline $d_i$.\nA subset of jobs forms an independent set if it's possible to schedule these jobs on the machine so that every job is finished before its deadline.\n\n\\begin{lemma}[Dynamic Convex Bipartite Matching \\cite{BrodalGHK07}]\n  There is a data structure which, given a convex bipartite graph $G = (L, R, E)$, after $\\tO(|L|+|R|)$ initialization, maintains the size of the maximum matching of $G[A \\cup R]$ where $A \\subseteq L$ is a dynamically changing subset of $L$ that is initially empty.\n  The data structure supports insertion\/deletion of an $x \\in L$ to\/from $A$ in worst-case $O(\\log^{2}(|L|+|R|))$ update time.\n\\end{lemma}\n\n\\begin{remark}\n  The exact data structure presented in \\cite{BrodalGHK07} is different from the stated one.\n  In particular, they support insertion\/deletion of an \\emph{unknown} job, i.e., we do not know beforehand what the starting date and deadline of the job are, nor do we know its relative position among the current set of jobs.\n  As a result, they used a rebalancing-based or rebuilding-based binary search tree~\\cite{NievergeltR72,Andersson89,Andersson91}, resulting in their amortized bound.\n  For our use case, all the possible jobs are known and we are just activating\/deactivating them, hence a static binary tree with a worst-case guarantee over these jobs suffices.\n\\end{remark}\n\n\n\\subsection{Linear Matroid}\nIn a linear matroid $\\cM = (U, \\cI)$, $U$ is a set of $n$ vectors (of dimension $r$) in some vector space and the notion of independence is just that of \\emph{linear independence}. The dynamic algorithm to maintain the rank of a matrix of \\cite{BrandNS19} can be used without modification as the dynamic oracle.\n\n\\begin{lemma}[Dynamic Matrix Rank Maintenance~\\cite{BrandNS19}]\n  There is a data structure which, given an $n \\times n$ matrix $M$, maintains the rank of $M$ under row updates in worst-case $O(n^{1.529})$ update time.\n\\end{lemma}\n\n\\subsection{Problems} \\label{appendix:problems}\nFor completeness, here we define the problems we discuss in the introduction, and why they reduce to matroid union or intersection.\n\n\\paragraph{$k$-Forest.} In this problem we are given a graph $G = (V,E)$ and asked to find $k$ edge-disjoint forests of the graph, of the maximum total size.\nIt can be modeled as the $k$-fold matroids union over the graphic matroid of $G$.\n\n\\paragraph{$k$-Disjoint Spanning Trees.} This problem is a special case of the above $k$-forest problem where we ask to find $k$ edge-disjoint spanning trees of the graph. Clearly, if such exists, the $k$-forest problem will find them.\n\n\\paragraph{$k$-Pseudoforest.} Similar to above, in this problem we are given a graph $G = (V,E)$ and asked to find $k$ edge-disjoint \\emph{pseudoforests} of the graph, of the maximum total size. A pseudoforest is an undirected graph in which every component has at most one cycle.\nThe problem can be modeled as the $k$-fold matroids union over the bicircular matroid of $G$.\n\n\\paragraph{$(f,p)$-Mixed Forest-Pseudoforest.} Again, we are given a graph $G = (V,E)$ and asked to find $f$ forests and $p$ pseudoforest (all edge-disjoint), of the maximum total size. \nThe problem can be modeled as the matroids union over $f$ graphic matroids and $p$ bicircular matroids.\n\n\\paragraph{Tree Packing.} In the tree packing problem, we are given a graph $G = (V,E)$ and are asked to find the maximum $k$ such that we can find $k$-disjoint spanning trees in the graph. This number $k$ is sometimes called the \\emph{tree-pack-number} or \\emph{strength} of the graph. The problem can be solved with the $k$-disjoint spanning trees problem, by binary searching for $k$ in the range $[0,|E|\/(|V|-1)]$, and is an example of a \\emph{matroid packing} problem.\n\n\\paragraph{Arboricity and Pseudoarboricity.} The arboricity (respectively pseudoarboricity) of a graph $G = (V,E)$ is the least integer $k$ such that we can partition the edges into $k$ edge-disjoint forests (respectively pseudoforests). \nThis can be solved with the $k$-forest (respectively $k$-pseudoforest) problem with a binary search over $k$. It is well known that for a simple graph the (pseudo-)arboricity is at most $\\sqrt{|E|}$, so we need only search for $k$ in the range $[0,\\sqrt{|E|}]$. The problems are examples of \\emph{matroid covering} problems.\n\n\\paragraph{Shannon Switching Game.} The Shannon switching game is a game played on a graph $G = (V,E)$, between two players ``Short'' and ``Cut''. They alternate turns with Short playing first, and all edges are initially colored white. On Short's turn, he may color an edge of the graph black. On Cut's turn, he picks a remaining non-black edge and removes it from the graph. Short wins if he connects the full graph with black edges, and Cut if he manages to disconnect the graph. It can be shown that Short wins if and only if there exists two disjoint spanning trees in the graph (and these two spanning trees describes a winning strategy for Short). Hence solving this game is a special case of the $k$-disjoint spanning tree problem with $k = 2$.\n\n\\paragraph{Graph $k$-Irreducibility.} \nA (multi-)graph $G = (V,E)$ is called $k$-irreducible (\\cite{Whiteley88rigidity}) if and only if $|E| = k(|V|-1)$ and for any vertex-induced nonempty, proper subgraph $G[V']$ it holds that $|E(G[V'])| < k(|V'|-1)$.\nThe motivation behind this definition comes from the rigidity of \\emph{bar-and-body frameworks}. A bar-and-body framework where rigid bars are attached to rigid bodied with joints (represented by the graph $G$).\nThen any stress put on a $k$-irreducible structure will propagate to all the bars (i.e.\\ edges).\n\\cite{gabow1988forests} show how one can decide if a graph is $k$-irreducible by first determining if its edges can be partitioned into $k$ edge-disjoint trees, and then performing an additional $\\tO(k|V|)$ work.\n\n\\paragraph{Bipartite Matching.} In the bipartite matching problem, we are given a bipartite graph $G = (L\\cup R,E)$, and the goal is to find a \\emph{matching} (a set of edges which share no vertices) of maximum size. Bipartite matching can be modeled as a matroid intersection problem over two partition matroids $M_L = (E,\\cI_L)$ and $M_R = (E,\\cI_R)$. $M_L$ specifies that no two edges share the same vertex on the left $L$ (and $M_R$ is defined similarly on the right  set of vertices $R$).\n\n\\paragraph{Colorful Spanning Tree.} In this problem\\footnote{sometimes also called \\emph{rainbow spanning tree}.}, we are given a graph $G = (V,E)$ together with colors on the edges $c:E \\to \\bbZ$. We are tasked to find a spanning tree of $G$ such that no two edges in our spanning tree have the same color. This problem can be modeled by the matroid intersection of the graphic matroid of $G$ (ensuring we pick a forest),\nand a partition matroid of the coloring $c$ (ensuring that we pick no duplicate colors). We also note that this problem is more difficult than bipartite matching since any bipartite matching instance can be converted to a colorful spanning tree instance on a star-multi-graph.\n\n\\paragraph{Graphic Matroid Intersection.} In graphic matroid intersection we are given two graphs $G_1 = (V_1,E_1)$ and $G_2 = (V_2,E_2)$ and a bijection of the edges $\\phi : E_1 \\to E_2$. The task is to find a forest in $G_1$ of the maximum size, which also maps to a forest in $G_2$. By definition, this is a matroid intersection problem over two graphic matroids. Again, this problem is a further generalization of the colorful spanning tree problem.\n\n\\paragraph{Convex Transversal and Simple Job Scheduling Matroid Intersection.} In these problems, we are given a set of unit-size jobs $V$, where each job $v$ has two release times $\\ell_1(v)$, $\\ell_2(v) \\ge 1$ (in simple job scheduling $\\ell(v) = 1$) and two deadlines $r_1(v)$,$r_2(v)\\le \\mu$. The task is to find a set of jobs $S$ of the maximum size such that they can be scheduled on two machines as follows: each job needs to be scheduled at both machines, and at machine $i$ it must be scheduled at time $t \\in [\\ell_i(v), r_i(v)]$.\n\n\\paragraph{Linear Matroid Intersection}. In this problem, we are given two $n\\times r$ matrices $M_1$ and $M_2$ over some field.\nThe task is to find a set of indices $S\\subseteq \\{1,2,\\ldots,n\\}$ of maximum cardinality, such that the rows of $M_1$ (respectively $M_2$) indexed by $S$ are independent at the same time. This is a matroid intersection of two linear matroids defined by $M_1$ and $M_2$. We note that partition, graphic, and transversal matroids are special cases of linear matroids.\n\n\n\\section{Independence-Query Matroid Intersection Algorithm} \\label{appendix:dynamic-matroid-intersection-ind}\n\nIn this section, we show that we can obtain an $\\tO(nr^{3\/4})$ matroid intersection algorithm in the dynamic-independence-oracle model.\nThis matches the state-of-the-art traditional independence-query algorithm of Blikstad \\cite{blikstad21}.\nWe will only provide a proof sketch here because our algorithm is mostly an implementation of \\cite{blikstad21} in the new model with the help of (circuit) binary search trees.\n\nUsing the same construction as \\cref{thm:bst} and \\cref{obs:exchange}, circuit binary search trees work verbatim in the dynamic-independence-oracle model (however, co-circuit binary search trees do not).\nIn particular, \\cref{obs:exchange}\\ref{item:circuit-exchange} can be checked with a single independence query.\n\n\\begin{corollary}\n  For any integer $\\beta \\geq 1$, there exists a data structure that supports the following operations.\n  \\begin{itemize}\n    \\item $\\textsc{Initialize}(\\cM, S, Q_S, X)$: Given $S \\in \\cI$, the query-set $Q_S$ that corresponds to $S$, and $X \\subseteq S$ or $X = \\{t\\}$, initialize the data structure in $\\tO(|X|)$ time. The data structure also maintains $S$.\n    \\item $\\textsc{Find}(y)$: Given $y \\in \\bar{S}$,\n      \\begin{itemize}\n        \\item if $X \\subseteq S$, then return an $x \\in X$ such that $S - y + x \\in \\cI$, or\n        \\item if $X = \\{t\\}$, then return the only element $x = s$ or $x = t$ in $X$ if $S + y \\in \\cI$ and $\\bot$ otherwise.\n      \\end{itemize}\n      The procedure returns $\\bot$ if such an $x$ does not exist.\n      The procedure takes $\\tO(\\beta)$ time if the result is not $\\bot$, and $\\tO(1)$ time otherwise.\n    \\item $\\textsc{Delete}(x)$: Given $x \\in X$, if $x \\not\\in \\{s, t\\}$, delete $x$ from $X$ in $O(\\log{n})$ time.\n    \\item $\\textsc{Replace}(x, y)$: Given $x \\in X$ and $y \\not\\in X$, replace $x$ in $X$ by $y$ in $O(\\log{n})$ time.\n    \\item $\\textsc{Update}(\\Delta)$: Update $S$ to $S \\oplus (\\Delta \\setminus \\{s, t\\})$ in amortized $\\tO(\\frac{|X| \\cdot |\\Delta|}{\\beta})$ time.\n  \\end{itemize}\n  \\label{cor:bst-ind}\n\\end{corollary}\n\n\\paragraph{Framework.}\nThe algorithm of \\cite{blikstad21} consists of the following three phases.\n\n\\begin{enumerate}\n  \\item First, obtain an $(1 - \\epsilon)$-approximate solution $S$ using augmenting sets in $\\tO(\\frac{n\\sqrt{r}}{\\epsilon})$ time.\n  \\item Eliminate all augmenting paths in $G(S)$ of length at most $d$ using Cunningham's algorithm implemented by \\cite{chakrabarty2019faster} in $\\tO(nd + nr\\epsilon)$ time.\n  \\item Finding the remaining $O(r\/d)$ augmenting paths one at a time, using $\\tO(n\\sqrt{r})$ time each.\n\\end{enumerate}\n\nWith $\\epsilon = r^{-1\/4}$ and $d = r^{3\/4}$, the total running time is $\\tO(nr^{3\/4})$.\nWe briefly sketch how to implement the above three steps in the same running time also for the dynamic-independence-oracle model.\n\nNote that the primary difficulty independence-query algorithms face is that we are only capable of checking \\cref{obs:exchange}\\ref{item:circuit-exchange} (using \\cref{cor:bst-ind}), which means that we can only explore the neighbors of $u \\in \\bar{S}$.\nThe aforementioned rank-query algorithms for building distance layers and finding blocking-flow style augmenting paths are thus inapplicable in the dynamic-independence-oracle model.\n\n\\paragraph{Approximation Algorithm.}\nThe $O(n\\sqrt{r}\/\\epsilon)$-query $(1-\\epsilon)$-approximation algorithm of \\cite{blikstad21} needs to first compute distance layers up to distance $O(\\frac{1}{\\epsilon})$. This is done in a similar way as sketched below for ``Eliminating Short Augmenting Paths''.\n\nOtherwise, the approximation algorithm works through a series of ``refine'' operations\n(algorithms \\texttt{RefineAB}, \\texttt{RefineBA}, and \\texttt{RefineABA} in \\cite{chakrabarty2019faster,blikstad21}) to build a partial augmenting set. In each such operation, we only need to be able to do the following for some sets $(P,Q)$: start from some set $Q$ and find a maximal set $X\\subseteq P$ such that\n$Q+X$ is independent. This can be performed with a greedy algorithm in time (and dynamic query) $O(|P|)$, given that we already have queried set $Q$ before (which will be the case). \n\nFinally, the approximation algorithm falls back to finding a special type of augmenting paths \\emph{with respect to} the current augmenting set, in the \\texttt{RefinePath} algorithm of \\cite{blikstad21}, with $\\tO(n)$ queries for each such path. This algorithm can be implemented also in the dynamic-oracle model with the same query complexity. \\texttt{RefinePath} relies on the \\texttt{RefineAB} and \\texttt{RefineBA} algorithms (which we already covered), in addition to a binary search trick to find feasible exchange pairs. This binary search can be implemented with the circuit trees (\\cref{cor:bst-ind}), and it takes a total time $\\tO(n)$ to build them (since we can keep track of a queried set for $S$, and then we only need to build a circuit tree statically once for each layer in cost proportional to the size of the layer---which sums up to $\\tO(n)$).\n\n\\paragraph{Eliminating Short Augmenting Paths.}\nUsing \\cite{chakrabarty2019faster}'s implementation of Cunningham's algorithm, we can eliminate all augmenting paths of length $d$, thereby obtaining a $(1 - 1\/d)$-approximate solution.\nThe algorithm relies on \\cref{lemma:monotone} to ``fix'' the distance layers after each augmentation.\nInitially, all elements have distance $1$ or $2$ from $s$ depending on whether it belongs to $S$ (the common independent set obtained by the above approximation algorithm) or not.\nBefore the first and after each of the remaining $O(\\epsilon r)$ augmentation, we can fix the distance layers as follows.\n\n\\begin{itemize}\n  \\item For each $1 \\leq \\ell \\leq d$ and $u \\in L_{\\ell}$, if $u$ is not of distance $\\ell$ from $s$, i.e., there is no in-edge from $L_{\\ell - 1}$ to $u$ anymore, move $u$ from $L_{\\ell}$ to $L_{\\ell + 2}$. This check is done as follows, depending on the parity of $\\ell$.\n  \\begin{itemize}\n    \\item If $\\ell$ is even, then for each $v \\in L_{\\ell - 1}$, we find all the unmarked $u \\in L_{\\ell}$ that $v$ has an edge to and mark $u$.\n    In the end, all the unmarked $u \\in L_{\\ell}$ do not belong to $L_{\\ell}$ and should be moved to $L_{\\ell + 2}$.\n    \\item If $\\ell$ is odd, then we simply check if there is an in-edge from $L_{\\ell - 1}$ to decide whether $u$ should be moved to other layers.\n  \\end{itemize}\n\\end{itemize}\n\nBoth cases can be implemented efficiently with circuit binary search trees of \\cref{cor:bst-ind}: Each time we spend $\\tO(1)$ time to either confirm that $u$ has distance $\\ell$ from $s$ with respect to the current $S$ (in which case it will not be moved anymore in this iteration), or we increase the distance estimate of $u$.\nThe total running time is thus $\\tO(nd + nr\\epsilon)$, where $\\tO(nd)$ comes from increasing the distance estimate of each element to at most $d$, and $\\tO(nr\\epsilon)$ comes from confirming that each element belongs to its distance layer in the $O(\\epsilon r)$ iterations.\n\nA caveat here is that we need to support insertion\/deletion into the binary search trees.\nThis can be made efficient by doubling the size of a binary search tree (and re-initializing it) every time when there are not enough leaf nodes left in it.\nThe cost of re-building will be amortized to $\\tO(1)$ time per update (i.e., movement of an element to another layer).\n\n\\paragraph{Finding a Single Augmenting Path.}\nWith the $(1 - 1\/d)$-approximate solution obtained in the first two steps, \\cite{blikstad21} then finds the remaining $O(r\/d)$ augmenting paths one at a time, using the \\emph{reachability} algorithm of \\cite{quadratic2021}.\nThe reachability algorithm roughly goes as follows.\nFirst, we initialize two circuit binary search trees (\\cref{cor:bst-ind}) over the two matroids for discovering out-edges and in-edges of elements in $\\bar{S}$.\nWe then repeatedly run the following three steps until either an $(s, t)$-path is found (an arbitrary $(s, t)$-path suffices since such a path can be converted into a chordless one in $\\tO(r)$ time along which augmentation is valid) or we conclude that $t$ is unreachable from $s$.\nWe keep track of a set of visited vertices $F$ which we know are reachable from $s$.\n\n\\begin{enumerate}[label=(\\roman*)]\n  \\item Identify the set of unvisited \\emph{heavy} vertices in $\\bar{S}$ that have at least $\\sqrt{r}$ unvisited out-neighbors or has a direct edge toward $t$. This is done by sampling a set $R$ of unvisited vertices in $S$ and then computing for each vertex $u$ whether $R \\cap \\mathsf{OutNgh}(u) = \\emptyset$, or equivalently, whether $S - R + u \\in \\cI$.\n  Intuitively, vertices with more out-neighbors are more likely to fail the test.\n  This can be tested for a single $R$ and all $u$ in $O(n)$ time in the dynamic-independence-oracle model.\n  With $O(\\log{n})$ samples, heavy vertices can be successfully identified with high probability.\n  \\item Discover all the out-neighbors for each light vertex, taking a total of $\\tO(n\\sqrt{r})$ time using the circuit binary search tree over the whole run of the algorithm. (Each vertex turns from heavy to light at most once.)\n  \\item Perform a reversed breadth-first-search from all the heavy vertices simultaneously. We can assume that every vertex on the path is light (i.e., we find a ``closest'' heavy vertex reachable from $s$), and thus all its out-neighbors have already been discovered.\n  That is, going backward from $S$ to $\\bar{S}$, we use the out-edges of light vertices.\n  From $\\bar{S}$ to $S$, we use the circuit binary search tree.\n  This takes $\\tO(n)$ time, and we either find a heavy vertex reachable from $s$ (in which case we make progress by visiting at least $\\sqrt{r}$ vertices in $S$), or we conclude that all heavy vertices are unreachable from $s$ (in which case $t$ is unreachable either).\n\\end{enumerate}\n\nThe number of iterations is bounded by $O(\\sqrt{r})$ since we discover at least $\\sqrt{r}$ unvisited vertices in $S$ every time.\nThe total running time of finding a single augmenting path is thus $\\tO(n\\sqrt{r})$.\n\nUsing the same parameters $\\epsilon$ and $d$ as in \\cite{blikstad21} to combine the three phases, we obtain the following matroid intersection algorithm in the dynamic-independence-oracle model.\n\n\\begin{theorem}\n  For two matroids $\\cM_1 = (U, \\cI_1)$ and $\\cM_2 = (U, \\cI_2)$, it takes $\\tO(nr^{3\/4})$ time to obtain the largest $S \\in \\cI_1 \\cap \\cI_2$ with high probability in the dynamic-independence-oracle model.\n  \\label{thm:dynamic-matroid-intersection-ind-main}\n\\end{theorem}\n\n\\section{Binary Search Tree} \\label{sec:bst}\n\nIn this section, we give the core data structure of our algorithms which allows us to do binary searches and find free elements (elements $x$ such that $S + x \\in \\cI$) and exchange pairs (pairs $(x,y)$ such that $S-x+y\\in \\cI$, corresponding to edges in the exchange graph) efficiently.\nWe also support updating the common independent set $S$ that the exchange relationship is based upon.\nFor a matroid $\\cM = (U, \\cI)$, the data structure has the following guarantee ($s, t \\not\\in U$ denote the two distinguished vertices of the exchange graph as defined in \\cref{def:exchange-graph}).\n\n\\begin{theorem}\n  For any integer $\\beta \\geq 1$, there exists a data structure that supports the following operations.\n  \\begin{itemize}\n    \\item $\\textsc{Initialize}(\\cM, S, Q_S, X)$: Given $S \\in \\cI$, the query-set $Q_S$ that corresponds to $S$, and $X \\subseteq \\bar{S}$ (respectively, $X \\subseteq S$ or $X = \\{t\\}$), initialize the data structure in $\\tO(|X|)$ time. The data structure also maintains $S$.\n    \\item $\\textsc{Find}(y)$: Given $y \\in S \\cup \\{s\\}$ (respectively, $y \\in \\bar{S}$),\n      \\begin{itemize}\n        \\item if $y \\in S$ (respectively, $X \\subseteq S$), then return an $x \\in X$ such that $S - y + x \\in \\cI$ (respectively, $S - x + y \\in \\cI$), or\n        \\item if $y = s$ (respectively, $X = \\{t\\}$), then return an $x \\in X$ such that $S + x \\in \\cI$ (respectively, return the only element $x = s$ or $x = t$ in $X$ if $S + y \\in \\cI$ and $\\bot$ otherwise).\n      \\end{itemize}\n      The procedure returns $\\bot$ if such an $x$ does not exist.\n      The procedure takes $\\tO(\\beta)$ time if the result is not $\\bot$, and $\\tO(1)$ time otherwise.\n    \\item $\\textsc{Delete}(x)$: Given $x \\in X$, if $x \\not\\in \\{s, t\\}$, delete $x$ from $X$ in $O(\\log{n})$ time.\n    \\item $\\textsc{Replace}(x, y)$: Given $x \\in X$ and $y \\not\\in X$, replace $x$ in $X$ by $y$ in $O(\\log{n})$ time.\n    \\item $\\textsc{Update}(\\Delta)$: Update $S$ to $S \\oplus (\\Delta \\setminus \\{s, t\\})$ in amortized $\\tO(\\frac{|X| \\cdot |\\Delta|}{\\beta})$ time.\n  \\end{itemize}\n  \\label{thm:bst}\n\\end{theorem}\n\n\\begin{remark}\n  To make sense of the seemingly complicated input and casework of \\cref{thm:bst}, one should focus on the first item of $\\textsc{Find}(\\cdot)$.\n  We will use \\cref{thm:bst} to explore the exchange graphs, and thus we need to find an exchange element $x$ of $y$ as in the first case.\n  The additional complication is included solely because we also have to deal with edges incident to $s$ or $t$.\n  For instance, say $X \\subseteq \\bar{S}$, then $\\textsc{Find}(s)$ finds an edge in $G(S)$ directed from $s$ to $X$.\n  This will make our algorithms presented later cleaner (see \\cref{alg:blocking-flow} for example).\n  \\label{remark:bst}\n\\end{remark}\n\n\nSometimes, we will omit the $Q_S$ parameter of $\\textsc{Initialize}$, meaning that we explicitly build the query-set $Q_S$ from $S$ in $O(r)$ time before running the actual initialization.\nIn such cases, $|X|$ will be $\\Omega(r)$, and thus this incurs no overhead.\n\nWe will later refer to the case of $X \\subseteq \\bar{S}$ as the \\emph{co-circuit binary search tree} and the case of $X \\subseteq S$ as the \\emph{circuit binary search tree}.\nThe data structure follows from the binary search algorithm of \\cite[Lemma 10]{chakrabarty2019faster}, which is based on the following observation.\n\n\\begin{observation}[\\cite{chakrabarty2019faster}]\n  \\label{obs:exchange}\n  To find free elements and exchange pairs, we can use the following observations.\n\\begin{enumerate}[label=(\\roman*)]\n\\item\\label{item:free-element}\n\\emph{Free element}:\n  There exists an $x \\in X$ such that $S + x \\in \\cI$ if and only if $\\mathsf{rank}(S + X) > |S|$.\n\\item\\label{item:cocircuit-exchange}\n\\emph{Co-circuit exchange}:\n  Given $y\\in S$, there exists an $x \\in X$ such that $S - y + x \\in \\cI$ if and only if $\\mathsf{rank}(S - y + X) \\geq |S|$.\n  \\item\\label{item:circuit-exchange}\n\\emph{Circuit exchange}:\n  Given $y\\not\\in S$,\n  there exists an $x \\in X$ such that $S - x + y \\in \\cI$ if and only if $\\mathsf{rank}(S - X + y) = |S-X+y|$.\n  \\end{enumerate}\n\\end{observation}\n\nThe data structure of \\cref{thm:bst} is built upon the following similar data structure whose independent set $S$ is ``static'' in the sense that its update will be specified for each query.\nWe construct the data structure of \\cref{lemma:bst} first, and then use it for \\cref{thm:bst} in \\cref{sec:periodic-rebuild}.\n\n\\begin{lemma}\n  There exists a data structure that supports the following operations.\n  \\begin{itemize}\n    \\item $\\textsc{Initialize}(\\cM, S, Q_S, X)$: Given $S \\in \\cI$, a query-set $Q_S$ corresponding to $S$, and $X \\subseteq \\bar{S}$ (respectively, $X \\subseteq S$ or $X = \\{t\\}$), initialize the data structure in $\\tO(|X|)$ time.\n    \\item $\\textsc{Find}(y)$: Let $S^\\prime := S \\oplus \\Delta$. It is guaranteed that $S^\\prime \\in \\cI$. Given $y \\in S^\\prime \\cup \\{s\\}$ (respectively, $y \\in \\bar{S^\\prime}$),\n      \\begin{itemize}\n        \\item if $y \\in S^\\prime$ (respectively, $X \\subseteq S^\\prime$), then return an $x \\in X$ such that $S^\\prime - y + x \\in \\cI$ (respectively, $S^\\prime - x + y \\in \\cI$), otherwise\n        \\item if $y = s$ (respectively, $X = \\{t\\}$), then return an $x \\in X$ such that $S^\\prime + x \\in \\cI$ (respectively, return the only element $x = s$ or $x = t$ in $X$ if $S^\\prime + y \\in \\cI$ and $\\bot$ otherwise),\n      \\end{itemize}\n      in $\\tO(\\beta)$ time.\n      The procedure returns $\\bot$ if such an $x$ does not exist.\n    \\item $\\textsc{Delete}(x)$: Given $x \\in X$, delete $x$ from $X$ in $O(\\log{n})$ time.\n    \\item $\\textsc{Replace}(x, y)$: Given $x \\in X$ and $y \\not\\in X$, replace $x$ by $y$ in $X$ in $O(\\log{n})$ time.\n  \\end{itemize}  \n  \\label{lemma:bst}\n\\end{lemma}\n\nWe present the co-circuit version of the data structures as the circuit version is analogous (their difference is essentially stated in the two cases \\ref{item:cocircuit-exchange} and \\ref{item:circuit-exchange} of \\cref{obs:exchange}).\nThe data structure of \\cref{lemma:bst} is a balanced binary tree in which every node $v$ corresponds to a subset $X_v$ of $X$.\nThe subsets correspond to nodes at the same level form a disjoint partition of $X$.\nThere are $|X|$ leaf nodes, each of which corresponds to a single-element subset of $X$.\nAn internal node $v$ with children $u_1$ and $u_2$ has $X_v = X_{u_1} \\sqcup X_{u_2}$.\nEach node $v$ is also associated with a query-set $Q_v := S + X_v$, for which we have prepared a dynamic oracle (see \\cref{def:dyn-oracle}).\n\n\\paragraph{Initialization.}\nIn the initialization stage, we first compute the query-set of the root node $Q_r := S + X$ from $Q_S$ in $O(|X|)$ time.\nAs long as the current node $v$ has $|X_v| > 1$, we split $X_v$ into two equally-sized subsets $X_{u_1}, X_{u_2}$, compute $Q_{u_1}, Q_{u_2}$ from $Q_v$, and then recurse on the two newly created nodes $u_1$ and $u_2$.\nComputing $Q_{u_1}$ and $Q_{u_2}$ from $Q_v$ takes $O(|X_v|)$ time in total, and thus the overall running time for initialization is $\\tO(|X|)$.\n\n\\paragraph{Query.}\nTo find an exchange element of $y \\in S^\\prime$, we perform a binary search on the tree.\nFor each node $v$, we can test whether such an element exists in $X_v$ via \\cref{obs:exchange}\\ref{item:cocircuit-exchange} by computing the query-set $Q^\\prime_v := S^\\prime - y + X_v$ from $Q_v := S + X_v$ in $1+|\\Delta|$ dynamic-oracle queries.\nIf such an element does not exist for the root $X_r = X$, then we return $\\bot$.\nOtherwise, for node $v$ initially being $r$, there must exist one of the child nodes $u_i$ of $v$ where such an exchange $x$ exists in $X_{u_i}$.\nWe then recurse on $u_i$ until we reach a leaf node, at which point we simply return the corresponding element.\nSimilarly, to find a free element, we compute the rank of $Q^\\prime_v := S^\\prime + X_v$ instead (see \\cref{obs:exchange}\\ref{item:free-element}).\nSince we need to compute $Q^\\prime_v$ for each of the visited nodes, the running time is $O(|\\Delta|\\log{n})$.\n\n\\paragraph{Update.}\nFor deletion of $x$, we simply walk up from the leaf node corresponding to $x$ to the root node and remove $x$ from each of the $X_v$ and $Q_v$.\nThis takes time proportional to the depth of the tree, which is $O(\\log{n})$.\nReplacement of $x$ by $y$ follows similarly from deletion of $x$: instead of simply removing $x$ from $X_v$ and $Q_v$, we add $y$ to them as well.\n\n\n\\begin{remark}\n  Note that the above binary search tree is static in the sense that we only deactivate elements from a fixed initial set.\n  We can extend this data structure to support a dynamically changing input set $X$ by using a dynamic binary search tree based on partial rebuilding~\\cite{Andersson89,Andersson91} instead.\n  The amortized time complexity remains the same since rebuilding a subtree takes time proportional to the number of nodes of it.\n\\end{remark}\n\n\\subsection{Periodic Rebuilding} \\label{sec:periodic-rebuild}\n\nHere we extend \\cref{lemma:bst} to prove \\cref{thm:bst}. Recall that the difference between the two data structures is that we need to support a dynamically changing independent set in \\cref{thm:bst} (which we will need since $S$ changes after each augmentation in our matroid algorithms).\nHow we achieve this is to essentially employ a batch-and-rebuild approach to the binary search tree of \\cref{lemma:bst}.\n\n\n\\begin{proof}\n  We maintain a binary search tree $\\cT := \\textsc{Initialize}(\\cM, S, Q_S, X)$ of \\cref{lemma:bst} and a collection of ``batched'' updates $\\tilde{\\Delta}$ of size at most $\\beta$.\n  Throughout the updates, we also maintain the query-set corresponding to the current $S$ starting from the given $Q_S$ and the query-set corresponding to $S + X$, which initially can be computed from $Q_S$ in $O(|X|)$ time.\n  Each call to $\\textsc{Find}(y)$ is delegated to $\\cT.\\textsc{Find}(y, \\tilde{\\Delta})$, which runs in time $\\tO(|\\tilde{\\Delta}|) = \\tO(\\beta)$ time.\n  Note that we can test whether the result of $\\cT.\\textsc{Find}(\\cdot)$ will be $\\bot$ in $\\tO(1)$ time by simply checking if \\cref{obs:exchange}\\ref{item:cocircuit-exchange} (or \\ref{item:free-element} if $y = s$) holds with the query-set corresponding to $S + X$ we maintain.\n  \n  Each call to $\\textsc{Delete}(x)$ and $\\textsc{Replace}(x, y)$ translates simply to $\\cT.\\textsc{Delete}(x)$ and $\\cT.\\textsc{Replace}(x, y)$.\n  For an update to $S$ with $\\Delta$, we set $\\tilde{\\Delta} \\gets \\tilde{\\Delta} \\cup \\Delta$ and update $S$ and the query-sets accordingly.\n  If the size of $\\tilde{\\Delta}$ exceeds $B$, then we rebuild the binary search tree with the input common independent set being the up-to-date $S$ we maintain.\n  Note that we will pass query-set $Q_S$ to $\\textsc{Initialize}$ to not pay the extra $O(r)$ factor.\n  Finally, since the binary search tree is now up-to-date, we set $\\tilde{\\Delta}$ to be $\\emptyset$.\n  The rebuilding takes $\\tO(|X|)$ time and is amortized to $\\tO(\\frac{|X| \\cdot |\\Delta|}{\\beta})$ per update operation with $|\\Delta|$ changes.\n\\end{proof}\n\n\\section{Dynamically Maintaining a Basis of a Matroid} \\label{sec:decremental-basis}\n\nIn this section, we construct a data structure that allows us to maintain a basis of a matroid in a decremental set.\nThe data structure is used for obtaining an $\\tO_k(n + r\\sqrt{r})$ running time for matroid union, but it may be of independent interest as well.\nSpecifically, our data structure has the following guarantees.\n\n\\begin{theorem}\n  For a (weighted) matroid $\\cM = (U, \\cI)$, there exists a data structure supporting the following operations.\n  \\begin{itemize}\n    \\item $\\textsc{Initialize}(X)$: Given a set $X \\subseteq U$, initialize the data structure and return a (min-weight) basis $S$ of $X$ in $\\tO(n)$ time.\n    \\item $\\textsc{Delete}(x)$: Given $x \\in X$, remove $x$ from $X$ and return a new (min-weight) basis of $X$ in $\\tO(\\sqrt{r})$ time. Specifically, the new basis will contain at most one element (the replacement element of $x$) not in the old basis, and this procedure returns such an element if any.\n  \\end{itemize}\n  \\label{thm:decremental-basis}\n\\end{theorem}\n\n\n\n\nOur data structure for \\cref{thm:decremental-basis} will consist of two parts.\nThe first part, introduced in \\cref{sec:baseline}, is a baseline, unsparsified data structure that supports the $\\textsc{Delete}$ operation in $\\tO(\\sqrt{n})$ time, and the second one is a sparsification structure which brings the complexity down to $\\tO(\\sqrt{r})$, as presented in \\cref{sec:sparsification}.\n\nAs hinted by the statement of \\cref{thm:decremental-basis}, to make things simpler, we will assign an arbitrary but unique weight $w(x)$ to each $x \\in X$.\nNow, instead of maintaining an arbitrary basis of $X$, we maintain the \\emph{min-weight} basis instead.\nThe min-weight basis is well-known to be unique (as long as the weights are) and can be obtained greedily as shown in \\cref{alg:greedy} (see, e.g.,~\\cite{edmonds1971}).\n\n\\begin{algorithm}[!ht]\n  \\SetEndCharOfAlgoLine{}\n  \\SetKwInput{KwData}{Input}\n  \\SetKwInput{KwResult}{Output}\n  \n  \\caption{Greedy algorithm for computing the min-weight basis}\n  \\label{alg:greedy}\n  \\KwData{A set $X \\subseteq U$ of size $k$}\n  \\KwResult{The min-weight basis $S$ of $X$}\n  Order $X = (x_1, x_2, \\ldots, x_{k})$ so that $w(x_1) < w(x_2) < \\cdots < w(x_{k})$\\;\n  $S \\gets \\emptyset$\\;\n  \\For{$i \\in [1, k]$} {\n    \\If{$\\mathsf{rank}(S + x_i) > \\mathsf{rank}(S)$} {\\label{line:check}\n      $S \\gets S + x_i$\\;\n    }\n  }\n  \\textbf{return} $S$\\;\n\\end{algorithm}\n\nMoreover, suppose we remove $x \\in S$ from the set $X$. Then the new min-weight basis is either\n(i) $S - x + y$ where $y$ is the minimum weight element in $X - x$ that makes $S - x + y$ independent or\n(ii) simply $S - x$ if such a $y$ does not exist.\nIn case (i), $y$ is called the \\emph{replacement} element of $x$.\nNote that $w(y) > w(x)$ must hold.\n\nIt is useful to note that the $S$ in Line~\\ref{line:check} of \\cref{alg:greedy} is interchangeable with $X_{i - 1} = \\{x_1, \\ldots, x_{i - 1}\\}$, since\n$\\mathsf{span}(X_{i - 1}) = \\mathsf{span}(S \\cap X_{i - 1})$, so the sets $X_{i-1}$ and $S\\cap X_{i-1}$ have the same rank.\nIn other words, in each iteration $i$, we can imagine that \\cref{alg:greedy} has chosen every element before $x_i$.\n\n\\begin{observation}\n  In \\cref{alg:greedy}, $x_i \\in S$ if and only if $\\mathsf{rank}(X_{i}) > \\mathsf{rank}(X_{i - 1})$.\n  \\label{obs:greedy}\n\\end{observation}\n\n\n\\subsection{Baseline Data Structure} \\label{sec:baseline}\n\nOur baseline data structure supports the operations of \\cref{thm:decremental-basis}, except in time $\\tO(\\sqrt{k})$ where $k = |X|$ instead of $\\tO(\\sqrt{r})$.\n\n\\begin{lemma}\n  For a weighted matroid $\\cM = (U, \\cI)$, there exists a data structure supporting the following operations.\n  \\begin{itemize}\n    \\item $\\textsc{Initialize}(X)$: Given a set $X \\subseteq U$ with $|X| = k$, initialize the data structure and return the min-weight basis $S$ of $X$ in $\\tO(k)$ time.\n    \\item $\\textsc{Delete}(x)$: Given $x \\in X$, remove $x$ from $X$ and return the new min-weight basis of $X$ in $\\tO(\\sqrt{k})$ time. Specifically, the new basis will contain at most one element (the replacement element of $x$) not in the old basis, and this procedure returns such an element if any.\n    \\item $\\textsc{Insert}(x)$: Given $x \\not\\in X$, add $x$ to $X$. It's guaranteed that $x$ is not in the min-weight basis of the new $X$ and the size of $X$ does not exceed $2k$.\n  \\end{itemize}\n  \\label{lemma:decremental-basis_baseline}\n\\end{lemma}\n\n\\paragraph{Initialization.}\nIn the initialization stage, we order $X$ by the weights and split the sequence into $\\sqrt{k}$ blocks $X_1, X_2, \\ldots, X_{\\sqrt{k}}$ from left to right, where each block has roughly the same size $O(\\sqrt{k})$.\nThat is, $X_1$ contains the $\\sqrt{k}$ elements with the smallest weights while $X_{\\sqrt{k}}$ contains elements with the largest weights.\nWe also compute the basis $S$ of $X$ from left to right as in \\cref{alg:greedy} together with $\\sqrt{k}$ query-sets $Q_1, Q_2, \\ldots, Q_{\\sqrt{k}}$, where $Q_j = \\bigcup_{i = 1}^{j}X_i$ is the union of the first $j$ blocks.\nThis takes $\\tO(k)$ time in total.\n\n\\paragraph{Deletion.}\nFor each deletion of $x$ located in the block $X_i$, we first update the query-sets $Q_i, \\ldots, Q_{\\sqrt{k}}$ by removing $x$ from them.\nLet $Q_1^\\prime, \\ldots, Q_{\\sqrt{k}}^\\prime$ denote the old query-sets before removing $x$.\nIf $x$ is not in the basis $S$ we currently maintain, then $S$ remains the min-weight basis of the new $X$ and nothing further needs to be done.\nOtherwise, we would like to find the min-weight replacement element $y$ of~$x$.\nWe know that such a $y$, if it exists, can only be located in blocks $X_i, X_{i + 1}, \\ldots, X_{\\sqrt{k}}$.\nAs such, we find the first $j \\geq i$ with $\\mathsf{rank}(Q_j) = \\mathsf{rank}(Q_j^\\prime)$ and recompute the portion of $S$ inside $X_j$.\nThis can be done by running \\cref{alg:greedy} with the initial set $S$ being $Q_{i - 1}$, the union of the first $i - 1$ blocks (see \\cref{obs:greedy}).\nThus, the deletion takes $\\tO(\\sqrt{k})$ time.\n\n\\paragraph{Insertion.}\nFor insertion of $x$, we simply add $x$ to a block where it belongs (according to $w(x)$) and then update $Q_i$'s appropriately.\nThis takes $\\tO(\\sqrt{k})$ as well.\n\n\\paragraph{Rebalancing.}\nTo maintain an update time of $\\tO(\\sqrt{k})$, whenever the size of a block $X_i$ grows larger than $2\\sqrt{k}$, we split it into two blocks and recompute $Q_i$ and $Q_{i+1}$.\nSimilarly, to avoid having too many blocks, whenever the size of a block $X_i$ goes below $\\sqrt{k} \/ 2$, we merge it with an adjacent block and remove $Q_i$.\nEach of the above operations takes $\\tO(\\sqrt{k})$ time, which is subsumed by the cost of an update.\n\\\\\n\n\\noindent\nWe have shown how to implement each operation of \\cref{lemma:decremental-basis_baseline} in its desired running time, and the correctness of the data structure is manifest as we always follow the greedy basis algorithm (\\cref{alg:greedy}).\n\n\\subsection{Sparsification} \\label{sec:sparsification}\n\nIn this section, we prove \\cref{thm:decremental-basis} by ``sparsifying'' the input set of the data structure for \\cref{lemma:decremental-basis_baseline} in a recursive manner, similar to what \\cite{EppsteinGIN97} did to improve \\cite{Frederickson85}'s $O(\\sqrt{|E|})$ dynamic MST algorithm to $O(\\sqrt{|V|})$.\nThe following claim asserts that such sparsification is valid.\n\n\\begin{claim}\n  Let $S_X$ and $S_Y$ be the min-weight basis of $X$ and $Y$, respectively, where $w(x) < w(y)$ holds for each $x \\in X$ and $y \\in Y$.\n  Then, the min-weight basis of $S_X + S_Y$ is also the min-weight basis of $X + Y$.\n  \\label{claim:sparsification}\n\\end{claim}\n\n\\begin{proof}\n  Consider running the greedy \\cref{alg:greedy} on the set $X + Y$ to obtain the min-weight basis $S$ of it.\n  Clearly, we have $S_X \\subseteq S$ since $X$ contains the elements of smaller weights (in fact $S \\cap X = S_X$).\n  Assume for contradiction that $S \\cap Y \\not\\subseteq S_Y$, i.e., there exists a $y^{*} \\in S \\cap Y$ which does not belong to $S_Y$.\n  Then, it must be the case that there exists a $y \\in S_Y$ with $w(y) > w(y^{*})$, as otherwise (i.e., $y^{*}$ is ordered after everything in $S_Y$) by \\cref{lemma:basis-rank} the greedy algorithm stops before seeing $y^{*}$.\n  We claim that the greedy algorithm on $Y$ chooses $y^{*}$ before all such $y$'s, thereby contradicting the fact that $S_Y$ is the min-weight basis of $Y$.\n  This is true by the diminishing returns property\\footnote{The diminishing returns property of submodular functions states that $f(z+X)-f(X) \\geq f(z+Y)-f(Y)$ holds for each $Y \\subseteq X \\subseteq U$ and $z \\not\\in X$.} of the rank function: Let $Y^{*}$ be elements in $Y$ with weights smaller than $w(y^{*})$.\n  Since $y^{*} \\in S$, it follows that $\\mathsf{rank}(X + Y^{*} + y^{*}) > \\mathsf{rank}(X + Y^{*})$, implying $\\mathsf{rank}(Y^{*} + y^{*}) > \\mathsf{rank}(Y^{*})$ and the greedy algorithm run on $Y$ picks $y^{*}$.\n\\end{proof}\n\nWe are now ready to present our sparsification data structure.\n\n\\begin{proof}[Proof of \\cref{thm:decremental-basis}]\n  Our data structure is a balanced binary tree where the leaf nodes correspond to elements in $X$ and each internal node corresponds to the set consisting of elements in leaf nodes of this subtree.\n  We will abuse notation and use a node $v$ to also refer to the elements contained in the subtree rooted at $v$.\n  \n  We first build the binary tree top-down, starting with the root node containing $X$ and recursively splitting the current set into two subsets of roughly the same size and recursing on them.\\footnote{Note that unlike in \\cref{sec:bst}, we are not building query-sets here.}\n  We then build the min-weight basis of each node in a bottom-up manner, starting from the leaves.\n  For each node $v$ with children $u_1$ and $u_2$, we initialize the data structure $\\cD_v$ for \\cref{lemma:decremental-basis_baseline} with input set $S_{u_1} + S_{u_2}$, the min-weight basis of $u_1$ and $u_2$ which are obtained from $\\cD_{u_1}$ and $\\cD_{u_2}$.\n  By \\cref{claim:sparsification}, the basis $\\cD_v$ maintains is the min-weight basis of $v$.\n  Thus, by induction, the basis maintained in the root node is indeed the min-weight basis of the whole set $X$.\n  The data structure for \\cref{lemma:decremental-basis_baseline} takes time near-linear in the size of the input set to construct, and since the sparsified input is a subset of elements in the subtree, the initialization takes time near-linear in the sum of sizes of the subtrees, which is $\\tO(n)$ (indeed, every element occurs in at most $\\log n$ nodes).\n  \n  To delete an element $x \\in X$, we first identify the leaf node $v_x$ of the binary tree which corresponds to $x$.\n  Going upward, for each ancestor $p$ of $v_x$, we delete $x$ from $\\cD_p$.\n  If we find a replacement element $y$ for $x$, we insert $y$ into $\\cD_{q}$, where $q$ is $p$'s parent, before proceeding to $q$ ($y$ is not in $\\cD_p$ so such an insertion is valid by \\cref{claim:sparsification}).\n  Since $x$ will be removed from $\\cD_q$ shortly, the input set of $\\cD_q$ remains the union of the min-weight bases of $q$'s children.\n  This takes $\\tO(\\sqrt{r})$ time since $\\cD_p$ is of size $O(r)$.\n  Inductively, since the min-weight bases of the child nodes are updated, by \\cref{claim:sparsification}, the min-weight basis of each of the affect nodes (hence the min-weight basis of $X$) is correctly maintained.\n\\end{proof}\n\n\\section{Introduction} \\label{sec:intro}\n\n\n\nVia reductions to the max-flow and min-cost flow problems, exciting progress has been recently made for many graph problems such as maximum matching, vertex connectivity, directed cut, and Gomory-Hu trees \\cite{Madry13,LeeS14,Madry16,\nBrandLNPSSSW20,KathuriaLS20,LiP20,\nAbboudKT21stoc,BrandLLSS0W21-maxflow,LiNPSY21,\nAbboudKT21focs,AxiotisMV21,Cen0NPSQ21,0006PS21,GaoLP21,\nAbboudKT22,CenLP22,\nBrandGJLLPS22,\nAbboudK0PST22,ChenKLPGS22,\nCenHLP23}. \nHowever, many basic problems still witness no progress since a few decades ago. These problems include $k$-disjoint spanning trees \\cite{gabow1988forests,Gabow91}, colorful spanning tree \\cite{GabowS85}, arboricity \\cite{Gabow95}, spanning tree packing \\cite{gabow1988forests}, graphic matroid intersection \\cite{GabowS85,GabowX89}, and simple job scheduling matroid intersection \\cite{XuG94}. \nFor example, in the {\\em $k$-disjoint spanning trees problem} \\cite[Chapter 51]{schrijver2003}, \nwe want to find $k$ edge-disjoint spanning trees in a given input graph $G=(V,E)$. When $k=1$, this is the spanning tree problem and can be solved in linear time. \nFor higher values of $k$, the best runtime remains \n$\\tO(k^{3\/2}|V|\\sqrt{|E|})$-time algorithm from around 1990 \\cite{gabow1988forests,Gabow91}\\footnote{The stated bound was due to Gabow and Westermann \\cite{gabow1988forests}. Gabow \\cite{Gabow91} announced an improved bound of $O(kn\\sqrt{m+kn\\log(n)})$ but this bound was later removed from the journal version of the paper.}, which is also the best runtime for its applications such as Shannon Switching Game \\cite{Gardner61} and Graph $k$-Irreducibility \\cite{Whiteley88rigidity,graver1993combinatorial}. \nNo better runtime was known even for the special case of $k=2$.\n\n{\\em Can we improve the \nbounds of $k$-disjoint spanning trees and other problems?}\nMore importantly, since it is very unclear if these problems can be reduced to max-flow or min-cost flow\\footnote{For example, the best-known number of max-flow calls to decide whether there are $k$ disjoint spanning trees and to find the $k$ spanning trees are $O(n)$ and $O(n^2)$ respectively.}, \n{\\em is there an alternative approach to designing fast algorithms for many problems simultaneously?}\nFortunately, many of the above problems can be modeled as {\\em matroid problems}, giving hope that solving matroid problems would solve many of these problems in one shot. Unfortunately, this is not true in the traditional model for matroid problems---even the most efficient algorithm possible for a matroid problem does not \nnecessarily give a faster algorithm for any of its special cases. \nWe discuss this more below.\n\n\n\\paragraph{Matroid Problems.} A matroid $\\cM$ is a pair $(U, \\cI)$ where $U$ is a finite set (called the ground set) and $\\cI$ is a family of subsets of $U$ (called the independent sets) satisfying some constraints (see \\cref{def:matroid}; these constraints are not important in the following discussion).\nSince $\\cI$ can be very large, problems on matroid $\\cM$ are usually modeled with {\\em oracles} that answer {\\em queries}. Given a set $S\\subseteq U$, {\\em independence queries} ask if $S\\in \\cI$ and {\\em rank queries} ask for the value of $\\max_{I\\in \\cI, I\\subseteq S} |I|.$\nTwo textbook examples of matroid problems are {\\em matroid intersection} and {\\em union}\\footnote{\\emph{Matroid union} is also sometimes called \\emph{matroid sum}.} (e.g., \\cite[Chapters~41-42]{schrijver2003}).  \nWe will also consider the special case of matroid union called {\\em $k$-fold matroid union}.\n\n\\begin{definition}[Matroid intersection and ($k$-fold) matroid union]\\label{def:intro:matroid intersection union}\n(I) In matroid intersection, we are given two matroids $(U, \\cI_1)$ and $ (U, \\cI_2)$ and want to find a set of maximum size in $\\cI_1\\cap \\cI_2.$ \n(II) In matroid union, we are given $k$ matroids $(U_1, \\cI_1), (U_2, \\cI_2), \\ldots, (U_k, \\cI_k)$, and want to find the set $S_1\\cup S_2 \\cup \\cdots \\cup S_k$, where $S_i \\in \\cI_i$ for every $i$, of maximum size. \n(III) Matroid union in the special case where $U_1=U_2=\\cdots = U_k$ and $\\cI_1=\\cI_2=\\cdots = \\cI_k$ is called  $k$-fold matroid union.  \n\n{\\em Notations:} Throughout, for problems over matroids  $(U_1, \\cI_1), (U_2, \\cI_2), \\ldots, (U_k, \\cI_k)$, we define  $n:=\\max_i{|U_i|}$ and $r:=\\max_{i}\\max_{S \\in \\cI_i}|S|$. \\qed\n\\end{definition}\n\n\n\nMatroid problems are powerful abstractions that can model many fundamental problems. \nFor example, the $2$-disjoint spanning tree problem can be modeled as a $2$-fold matroid union problem: \n\\begin{tcolorbox}[breakable,boxrule=0pt,frame hidden,sharp corners,enhanced,borderline west={.5 pt}{0pt}{red},colback=white]\n\\vspace{-.2cm}\nGiven a graph $G=(V, E)$, let $\\cM=(U, \\cI)$ be the corresponding {\\em graphic matroids}, i.e. $U=E$ and $S\\subseteq E$ is in $\\cI$ if it is a forest in $G$. (It is a standard fact that such an $\\cM$ is a matroid.) \nThe $2$-fold matroid union problem with input $\\cM$ is a problem of finding two forests $F_1\\subseteq E$ and $F_2\\subseteq E$ in $G$ that maximizes $|F_1\\cup F_2|$. This is known as the {\\em $2$-forest} problem which clearly generalizes $2$-disjoint spanning tree  (a $2$-forest algorithm will return two disjoint spanning trees in $G$ if they exist). \n\\end{tcolorbox}\nObserve that this argument can be generalized to modeling the $k$-disjoint spanning trees problem by $k$-fold matroid union.  \nOther problems that can be modeled as matroid union (respectively, matroid intersection) include arboricity,  spanning tree packing, $k$-pseudoforest, and mixed $k$-forest-pseudoforest (respectively, bipartite matching and colorful spanning tree).\n\nThe above fact makes matroid problems a {\\em unified} approach for showing that many problems, including those mentioned above, can be solved in polynomial time. This is because (i) the matroid union, intersection, and other problems can be solved in polynomial time and rank\/independence queries, and (ii) for most problems  queries can be answered in polynomial time. For example, when we model $k$-disjoint spanning trees as $k$-fold graphic matroid union like above, the corresponding rank query is: given a set $S$ of edges, find the size of a spanning forest of $S$. This can be solved in $O(|S|)$ time. \n\nWhen it comes to more fine-grained time complexities, such as nearly linear and sub-quadratic time, matroid algorithms in the above model are not very helpful. This is because simulating a matroid algorithm in this model causes too much runtime blow-up. \nFor example, even if we can solve $k$-fold matroid union over $(U, \\cI)$ in {\\em linear} ($O(|U|)$) rank query complexity, it does not necessarily imply that we can solve its special case of $2$-disjoint spanning tree any faster.\nThis is because each query about a set $S$ of edges needs at least $O(|S|)$ time even to specify $S$, which can be as large as the number of edges in the input graph.\nIn other words, even a matroid union algorithm with linear complexities may only imply $O(|E|^2)$ time for solving 2-disjoint spanning trees on graphs $G=(V,E)$. \nThis is also the case for other problems that can be modeled as matroid union and intersection. \nBecause of this, previous works obtained improved bounds by simulating an algorithm for matroid problems and coming up with clever ideas to speed up the simulation for each of these problems one by one (e.g., \\cite{GabowT79,RoskindT85,GabowS85,gabow1988forests,FredericksonS89,GabowX89,Gabow91,XuG94}).\nIt cannot be guaranteed that recent and future improved algorithms for matroid problems (e.g., \\cite{chakrabarty2019faster,quadratic2021,blikstad21}) would imply improved bounds for any of these problems. \n\n\n\\paragraph{Dynamic Oracle.}\nThe main conceptual contribution of this paper is an introduction of a new matroid model called {\\em dynamic oracle}\nand an observation that, using dynamic algorithms, \nsolving a matroid problem efficiently in our model immediately implies efficient algorithms for many problems it can model. \nIn contrast to traditional matroids where a query can be made with an arbitrary set $S$, our model only allows queries made by slightly modifying the previous queries.\\footnote{The ``cost'' of a query in our dynamic model is the distance (size of the symmetric difference) from some (not necessarily the last) previous query.} More precisely, the dynamic-rank-oracle model, which is the focus of this work, is defined as follows.\\footnote{One can also define the dynamic-independence-oracle model where $\\textsc{Query}(i)$ returns only the independence of $S_i$.} \n\n\n\n\\begin{definition}[Dynamic-rank-oracle model]\n\\label{def:dyn-oracle}\n  For a matroid $\\cM = (U, \\cI)$, starting from $S_0 = \\emptyset$ and $k = 0$, the algorithm can access the oracle via the following three operations.\n  \\begin{itemize}[noitemsep]\n    \\item $\\textsc{Insert}(v, i)$: Create a new set $S_{k + 1} := S_i \\cup \\{v\\}$ and increment $k$ by one.\n    \\item $\\textsc{Delete}(v, i)$: Create a new set $S_{k + 1} := S_i \\setminus \\{v\\}$ and increment $k$ by one.\n    \\item $\\textsc{Query}(i)$: Return the rank of $S_i$, i.e., the size of the largest independent subset of $S_i$.\n  \\end{itemize}\n We say that a matroid algorithm takes $t$ time and dynamic-rank-query complexities if its time complexity and required number of operations are both at most $t$. \\qed\n \\end{definition}\n\nWe emphasize that a query can be obtained from \\emph{any} previous query, not just the last one.\n\n\\begin{table}[ht]\n    \\centering\n\n{\\small\n    \\begin{tabular}{c|l}\n        {\\bf matroid problems} &\n        \\begin{tabular}{l}\n        {\\bf special cases}\n        \\end{tabular}\n        \\\\\\hline\n        \\begin{tabular}{c}\n        $k$-fold matroid union in $T(n, r, k)$ \\\\\n        \\end{tabular} & \\begin{tabular}{l}\n             \\\\\n             $k$-forest in $\\tT(|E|, |V|, k))$ \\\\\n             $k$-pseudoforest in $\\tT(|E|, |V|, k))$ \\\\\n             $k$-disjoint spanning tree in $\\tT(|E|, |V|, k)$ (randomized) \\\\\n             \\phantom{$k$-disjoint spanning tree in} $\\hat{T}(|E|, |V|, k)$ (deterministic) \\\\\n             arboricity in $\\tT(|E|, |V|, \\sqrt{|E|}))$ \\\\\n             tree packing in $\\tT(|E|, |V|, |E|\/|V|))$ \\\\\n             Shannon Switching Game in $\\tT(|E|, |V|, 2))$ \\\\\n             graph $k$-irreducibility in $\\tT(|E|, |V|, k))$ \\\\\n             \\\\\n        \\end{tabular}\\\\\\hline\n        \\begin{tabular}{c}\n          matroid union in $T(n, r, k)$ \\\\\n        \\end{tabular}& \\begin{tabular}{l}\n          \\\\\n          $(f, p)$-mixed forest-pseudoforest in $\\tT(|E|, |V|, f + p))$ \\\\\n          \\\\\n        \\end{tabular}\\\\\\hline\n        \\begin{tabular}{c}\n          matroid intersection in $T(n, r)$ \\\\\n        \\end{tabular} & \\begin{tabular}{l}\n          \\\\\n          bipartite matching in $\\tT(|E|, |V|)$ \\\\\n          colorful spanning tree in $\\tT(|E|, |V|)$\\\\\n          graphic matroid intersection in $\\tT(|E|, |V|)$ \\\\\n          simple job scheduling matroid intersection in $\\tT(n, r)$ \\\\\n          convex transversal matroid intersection in $\\tT(|V|, \\mu)$ \\\\\n          \\\\\n        \\end{tabular}\\\\\\hline\n    \\end{tabular}\n}\n    \\caption{Examples of implications of dynamic-rank-oracle matroid algorithms. The complexities in the first column are in terms of time and dynamic-rank-query complexities. Notations $n$, $r$, and $k$ are as in \\Cref{def:intro:matroid intersection union}.\n    In the second column, the $\\mathrm{polylog}{n}$ factors are hidden in $\\tT$ and\n    subpolynomial factors are hidden in $\\hat{T}.$ Details are in  \\cref{sec:packing,appendix:applications}.}\n    \\label{tab:intro:dyn-oracle-implies-fast-algorithms}\n\\end{table}\n\n\n\n\n\\begin{observation}[Details in  \\cref{sec:packing,appendix:applications}]\\label{thm:intro:dyn-oracle implies fast alg}\nAlgorithms for the $k$-fold matroid union, matroid union, and matroid intersection problems imply algorithms for a number of problems with time complexities shown in \\Cref{tab:intro:dyn-oracle-implies-fast-algorithms}.\n\\end{observation}\n\\begin{proof}[Proof Idea]\nAs an example, we sketch the proof that if $k$-fold matroid union can be solved in $T(n, r, k)$ then $k$-disjoint spanning trees can be found in  $T(|E|, |V|, k)\\cdot\\mathrm{polylog}(|V|)$ time. \nRecall that in the traditional rank-oracle model, the algorithm can ask an oracle for the size of a spanning forest in an arbitrary set of edges $S$, causing $O(|S|)$ time to simulate. \nIn our dynamic-rank-oracle model, an algorithm needs to modify some set $S_i$ to the desired set $S$ using the {\\sc Insert} and {\\sc Delete} operations before asking for the size of a spanning forest in $S$. We can use a spanning forest data structure to keep track of the size of the spanning forest under edge insertions and deletions. This takes $\\mathrm{polylog}(|V|)$ time per operation \\cite{KapronKM13,GibbKKT15}.\\footnote{The dynamic spanning forest algorithms of \\cite{KapronKM13,GibbKKT15} are randomized and assume the so-called oblivious adversary (as opposed to, e.g.,   \\cite{NanongkaiS17,Wulff-Nilsen17} which work against adaptive adversaries). This is not a problem because we only need to report the size of the spanning forest and not an actual forest. We can also use a deterministic algorithm from \\cite{ChuzhoyGLNPS20,NanongkaiSW17} which requires $|V|^{o(1)}$ time per operation.}\nSo, if $k$-fold matroid union can be solved in $T(n, r, k)$ time and dynamic rank queries, then $k$-disjoint spanning trees can be solved in  $\\tO(T(n, r, k))=\\tO(T(|E|, |V|, k))$ time, where the equality is because the ground set size is the number of edges ($|U|=|E|$) and the rank $r$ is equal to the size of a spanning forest (thus at most $|V|$).\\footnote{Note that we also need a fully-persistent data structure \\cite{DriscollSST86,Dietz89} to maintain the whole change history in our argument.}\n\\end{proof}\n\nObserve that designing efficient algorithms in our dynamic-oracle model is {\\em not} easier than in the traditional model: a dynamic-oracle matroid algorithm can be simulated in the traditional model within the same time and query complexities. Naturally, the first challenge of the new model is this question: {\\em Can we get matroid algorithms in the new model whose performances match the state-of-the-art algorithms in the traditional model?}\nMoreover, for the new model to provide a unified approach to solve many problems simultaneously, one can further ask: {\\em Would these new matroid algorithms imply state-of-the-art bounds for many problems?}\n\n\n\n\\paragraph{Algorithms.}  \nIn this paper, we provide algorithms in the new model whose complexities not only match those in the traditional model but sometimes even improve them.  These lead to new bounds for some problems and, for other problems, a unified algorithm whose performance matches previous results developed in various papers for various problems. \n\n\nMore precisely, the best time and rank-query complexities for matroid intersection on input $(U, \\cI_1)$ and $(U, \\cI_2)$ were $\\tO(n\\sqrt{r})$ by Chakrabarty, Lee, Sidford, Singla, and Wong \\cite{chakrabarty2019faster} (improving the previous $\\tO(nr)$ bound based on Cunningham's classic algorithm \\cite{cunningham1986improved,LeeSW15,nguyen2019note}).\nDue to a known reduction, this implies $\\tO(k^2\\sqrt{k}n\\sqrt{r})$ bound for $k$-fold matroid union and matroid union. \nIn this paper, we present algorithms in the dynamic-oracle model that imply improved bounds in the traditional model for $k$-fold matroid union and matroid union and match the bounds for matroid intersection.\n\nHere, we only state our dynamic-rank-query complexities as they are the main focus of this paper, and for all the applications we have, answering (and maintaining dynamically) independence queries does not seem to be significantly easier.\nNote that we also obtain dynamic-independence-query algorithms that match the state-of-the-art traditional ones~\\cite{blikstad21} which we defer to \\cref{appendix:dynamic-matroid-intersection-ind}.\n\\begin{theorem}  \\label{thm:main}\n(I)  $k$-fold matroid union over input $(U, \\cI)$ can be solved in $\\tO(n + kr\\sqrt{\\min(n, kr)} + k \\min(n, kr))$ time and dynamic rank queries.\n(II) Matroid union over input $(U_1, \\cI_1), (U_2, \\cI_2), \\ldots, (U_k, \\cI_k)$ can be solved in $\\tO\\left(\\left(n + r\\sqrt{r}\\right) \\cdot \\mathrm{poly}(k)\\right)$ time and dynamic rank queries.\n(III) Matroid intersection over input $(U, \\cI_1)$ and $(U, \\cI_2)$ can be solved in $\\tO(n\\sqrt{r})$ time and dynamic rank queries.\n\\end{theorem}\n\n\n\n\n\\definecolor{applegreen}{rgb}{0.55, 0.71, 0.0}\n\n\\begin{savenotes}\n\\begin{table}[ht]\n    \\centering\n    \\small\n    \\begin{tabular}{c|l|l}\n      {\\bf problems} & {\\bf our bounds} & {\\bf state-of-the-art results} \\\\\\hline\n      {\\tiny (Via $k$-fold matroid union)}& & \\\\\n      $k$-forest\\footnote{For $k$-forest and its related graph problems in the table, we can assume that $k \\leq |V|$, and thus the $k^2r$ (where $r = \\Theta(|V|)$) term in \\cref{thm:main} is dominated by the $(kr)^{3\/2}$ term.} & $\\tO(|E| + (k|V|)^{3\/2})$ {\\color{green!50!black}\\cmark} & $\\tO(k^{3\/2}|V|\\sqrt{|E|})$ \\cite{gabow1988forests} \\\\\n      $k$-pseudoforest & $\\tO(|E| + (k|V|)^{3\/2})$ {\\color{red}\\xmark} & $|E|^{1 + o(1)}$ \\cite{ChenKLPGS22} \\\\\n      $k$-disjoint spanning trees & $\\tO(|E| + (k|V|)^{3\/2})$ {\\color{green!50!black}\\cmark} & $\\tO(k^{3\/2}|V|\\sqrt{|E|})$ \\cite{gabow1988forests} \\\\\n      arboricity\\footnote{Here we use the bound that $\\alpha \\leq \\sqrt{E}$~\\cite{Gabow95}.} & $\\tO(|E||V|)$ {\\color{red}\\xmark} & $\\tO(|E|^{3\/2})$ \\cite{Gabow95} \\\\\n      tree packing & $\\tO(|E|^{3\/2})$ & $\\tO(|E|^{3\/2})$ \\cite{gabow1988forests} \\\\\n      Shannon Switching Game & $\\tO(|E| + |V|^{3\/2})$ {\\color{green!50!black}\\cmark} & $\\tO(|V|\\sqrt{|E|})$ \\cite{gabow1988forests} \\\\\n      graph $k$-irreducibility & $\\tO(|E| + (k|V|)^{3\/2} + k^2|V|)$ {\\color{green!50!black}\\cmark} & $\\tO(k^{3\/2}|V|\\sqrt{|E|})$ \\cite{gabow1988forests} \\\\ & & \\\\ \\hline\n      {\\tiny (Via matroid union)} & & \\\\\n      $(f, p)$-mixed forest-pseudoforest & $\\tO_{f,p}(|E| + |V|\\sqrt{|V|})$ {\\color{green!50!black}\\cmark} & $\\tO((f + p)|V|\\sqrt{f|E|})$ \\cite{gabow1988forests} \\\\ & & \\\\\\hline\n      {\\tiny (Via matroid intersection)} & & \\\\\n      bipartite matching (combinatorial\\footnoteref{foot:combinatorial}) & $\\tO(|E|\\sqrt{|V|})$ & $O(|E|\\sqrt{|V|})$ \\cite{HopcroftK73} \\\\\n      bipartite matching (continuous) & $\\tO(|E|\\sqrt{|V|})$ {\\color{red}\\xmark} & $|E|^{1 + o(1)}$ \\cite{ChenKLPGS22} \\\\\n      graphic matroid intersection & $\\tO(|E|\\sqrt{|V|})$ & $\\tO(|E|\\sqrt{|V|})$ \\cite{GabowX89} \\\\\n      simple job scheduling matroid intersection & $\\tO(n\\sqrt{r})$ & $\\tO(n\\sqrt{r})$ \\cite{XuG94} \\\\\n      convex transversal matroid intersection & $\\tO(|V|\\sqrt{\\mu})$ & $\\tO(|V|\\sqrt{\\mu})$ \\cite{XuG94} \\\\\n      linear matroid intersection\\footnote{Our bound is with respect to the current value of $\\omega < 2.37286$~\\cite{AlmanW21}. If $\\omega = 2$, then our bound becomes $\\tO(n^{2.5}\\sqrt{r})$.} & $\\tO(n^{2.529}\\sqrt{r})$ {\\color{red}\\xmark} & $\\tO(nr^{\\omega - 1})$ \\cite{Harvey09} \\\\\n      colorful spanning tree & $\\tO(|E|\\sqrt{|V|})$ & $\\tO(|E|\\sqrt{|V|})$ \\cite{GabowS85} \\\\\n      maximum forest with deadlines & $\\tO(|E|\\sqrt{|V|})$ {\\color{green!50!black}\\cmark} & (no prior work) \\\\\n    \\end{tabular}\n    \\caption{Implications of our matroid algorithms stated in \\cref{thm:main} in comparison with previous results. Results marked with a {\\color{green!50!black}\\cmark} improve over the previous ones. Results marked with a {\\color{red}\\xmark} are worse than the best time bounds. Other results match the currently best-known algorithms up to poly-logarithmic factors. Details can be found in \\cref{appendix:applications}.}\n    \\label{tab:intro:implications}\n\\end{table}\n\\end{savenotes}\n\n\nCombined with \\Cref{thm:intro:dyn-oracle implies fast alg}, the above theorem immediately implies fast algorithms for many problems. \\Cref{tab:intro:implications} shows some of these problems. \nOne of our highlights is the improved bounds for $k$-forest and $k$-disjoint spanning trees. Even for $k=2$, there was no runtime better than the decades-old $\\tO(k^{3\/2}|V|\\sqrt{|E|})$ runtime \\cite{gabow1988forests,Gabow91}. Our result improves this to  $\\tO(|E| + (k|V|)^{3\/2})$.\nThis is nearly linear for dense input graphs and small $k$.\nThis also implies a faster runtime for, e.g., Shannon Switching Game (see \\cite{shannon1955game,gabow1988forests}) which is a special case of $2$-disjoint spanning trees. \n\n\nOur matroid intersection algorithm gives a unified approach to achieving time complexities that were previously obtained by various techniques in many papers. \nThus, improving this algorithm would imply breakthrough runtimes for many of these problems simultaneously.\nMoreover, in contrast to the previous approach where matroid algorithms have to be considered for each new problem one by one, our approach has the advantage that it can be easier to derive new bounds.\nFor example, say we are given a graph $G = (V, E)$, where the edge $e$ will stop functioning after day $d(e)$.\nEvery day we can ``repair'' one functioning edge. \nOur goal is to make the graph connected in the long run  (an edge will work forever once it has been repaired).\nThis is the \\emph{maximum forest with deadlines} problem.\nFormally speaking, the goal is to construct a spanning tree or a forest of the maximum size at the end, by selecting an edge $e$ with $d(e) \\geq t$ in the $t^{\\scriptsize \\mbox{{\\rm th}}}$ round.\\footnote{It is tempting to believe that we can use a greedy algorithm where we always select an edge $e$ with the smallest $d(e)$ to the solution. The following example shows why this does not work: There are three vertices $V=\\{a,b,c,d\\}$. Edges $e_1$ and $e_2$ between $a$ and $b$ have $d(e_1)=1$ and $d(e_2)=3$. Edges $e_3=(b,c)$ and $e_4=(c,d)$ have $d(e_3)=d(e_4)=2$.\n}\nOur result implies a runtime of  $\\tO(|E|\\sqrt{|V|})$ for this problem. The runtime holds even for the harder case where each edge is also associated with an {\\em arrival time} (edges cannot be selected before they arrive). \n\n\n\nWe also list some problems where our bounds cannot match the best bounds in \\Cref{tab:intro:implications}. Improving our matroid algorithms to match these bounds is a very interesting open problem. A particularly interesting case is the maximum bipartite matching problem. Our dynamic-oracle matroid intersection algorithm  implies a runtime that matches the runtime from the best combinatorial algorithm of Hopcroft and Karp \\cite{HopcroftK73} which has been recently improved via continuous optimization techniques (e.g., \\cite{Madry13,Madry16,CohenMSV17,AxiotisMV20,BrandLNPSSSW20,BrandLLSS0W21-maxflow,ChenKLPGS22}).\\footnote{\\label{foot:combinatorial}The term ``combinatorial'' is vague and varies in different contexts. Here, an algorithm is ``combinatorial'' if it does not use any of the continuous optimization techniques such as interior-point methods (IPMs).}\nThere are barriers to using continuous optimization even to solve some special cases of matroid intersection (e.g. colorful spanning tree's linear program requires exponentially many constraints). Thus, improving our matroid intersection algorithm requires either a new way to use continuous optimization techniques or a breakthrough idea in designing combinatorial algorithms that would improve Hopcroft-Karp algorithm.\n\n\n\n\n\n\n\n\\paragraph{Lower Bounds.} \nAnother advantage of our dynamic-oracle matroid model is that it can be easier to prove lower bounds than the traditional model. As a showcase, we show a simple super-linear rank-query lower bound in our new model. In fact, our argument also implies the first super-linear independence-query lower bound in the traditional model.\nThe latter result might be of independent interest. \n\\begin{theorem}\\label{thm:intro:lowerbound}\n(I) Any deterministic algorithms require $\\Omega(n\\log n)$ dynamic rank queries to solve the matroid union and matroid intersection problems.\n(II) Any deterministic algorithms require $\\Omega(n\\log n)$ (traditional) independence queries to solve the matroid union and matroid intersection problems.\n\\end{theorem}\nOur first lower bound suggests that the dynamic-oracle model might at best give nearly linear (and not linear) time algorithms.\nPrior to this paper, only a $\\log_2(3)n - o(n)$ independence-query lower bound for deterministic algorithms was known for (traditional) independence queries, due to Harvey \\cite{harvey2008matroid}.\\footnote{To the best of our knowledge, this lower bound does not hold for rank queries.}\nOur lower bound in the traditional model improves this decade-old bound. \nMoreover, showing super-linear independence-query lower bounds in the traditional model for matroid intersection is a long-standing open problem considered since 1976 (e.g. \\cite{Welsh1976matroid_book,chakrabarty2019faster}).\\footnote{As noted by Harvey, Welsh asked about the number of queries\nneeded to solve the matroid partition problem, which is equivalent to matroid union and intersection.} \nOur lower bound in the traditional model answers this open problem for deterministic algorithms. The case of randomized algorithms would be resolved too if an $\\omega(|V|)$ lower bound was proved for the communication complexity for computing connectivity of an input graph $G=(V,E)$. (It was conjectured to be $\\Omega(n\\log n)$ in \\cite{ApersEGLMN22}.)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Techniques}\nIn this section we briefly discuss our technical contributions. For a more in-depth overview of our algorithms, see the technical overview (\\cref{sec:overview}).\n\n\\paragraph{Exchange Graph \\& Blocking Flow.} Our algorithms and lower bounds are based on the notion of finding \\emph{augmenting paths} in the \\emph{exchange graph}, due to \\cite{edmonds1970submodular,Lawler75,aignerD}.\nGiven a common independent set $S\\in \\cI_1\\cap \\cI_2$, the exchange graph $G(S)$ is a directed graph where finding an $(s,t)$-path corresponds to increasing the size of $S$ by one. Starting with the work of Cunningham \\cite{cunningham1986improved}, modern matroid intersection algorithms (including the state-of-the-art \\cite{chakrabarty2019faster,blikstad21}) are based on a ``Blocking Flow'' idea inspired by the Hopcroft-Karp's \\cite{HopcroftK73} bipartite matching and Dinic's \\cite{dinic1970algorithm}  max-flow algorithms.\n\n\\paragraph{Matroid Intersection with Dynamic Oracle.} \nOur matroid intersection algorithms are implementations of the state-of-the-art $\\tO(n\\sqrt{r})$ rank-query algorithm of \\cite{chakrabarty2019faster} and\nthe $\\tO(n r^{3\/4})$ independence-query algorithm of \\cite{blikstad21}. Our contribution here is to show that versions of them can be implemented also in the \\emph{dynamic}-oracle model. \n\nThese algorithms explore the exchange graph efficiently in the classic non-dynamic models by performing binary searches with the oracle queries to find useful edges. However, such a binary search is very expensive in the dynamic-oracle model (as the queries differ by a lot): a single such binary search might cost up to $O(n)$ in the dynamic-oracle model instead of just $O(\\log n)$.\n\nOur contribution is to design a binary-tree data structure that supports finding these useful edges efficiently also in the dynamic-oracle model.\nNote that after each augmentation the underlying exchange graph changes, so the data structure must also support these dynamic updates efficiently. Some updates can just be propagated up the tree, while others we handle by batching them and rebuilding the tree periodically.\nWe also rely on a structural result ``Augmenting Sets'' by \\cite{chakrabarty2019faster} which states that the updates to the exchange graph are local, which helps us reduce the number of updates we need to make to our data structure, and achieve the final time bound.\n\n\\paragraph{Matroid Union with Dynamic Oracle.} \nOur $\\tO(n + r\\sqrt{r})$ matroid union algorithm with dynamic rank oracle is based on our $\\tO(n \\sqrt{r})$ matroid intersection algorithm (indeed, matroid union is a special case of matroid intersection).\nWe are able to obtain a more efficient algorithm by \ntaking advantage of the additional structure of the exchange graph in the case of matroid union.\nThe main idea is to run the blocking flow algorithm only on a dynamically-changing subgraph of size $\\Theta(r)$, instead of on the full exchange graph of size $\\Theta(n)$.\n\nA crucial observation is that all but $O(r)$ elements will be directly connected to the source vertex~$s$. To ``sparsify'' this first layer in the breadth-first-search tree, we argue that one only needs to consider a basis of it (this basis will have size at most $r$ as opposed to $n$). After an augmentation, this first layer changes, so we design a \\emph{dynamic algorithm to maintain a basis of a matroid}\\footnote{For example, maintaining a spanning forest in a dynamically changing graph.}, with $\\tO(\\sqrt{r})$ update time and $O(n)$ pre-computation.\nOur algorithm to maintain this basis dynamically is inspired by the\n dynamic minimum spanning tree algorithm of \\cite{Frederickson85} ($O(\\sqrt{|E|})$ update time), in combination with the sparsification trick of \\cite{EppsteinGIN97} ($\\tO(\\sqrt{|V|})$ update time).\n We believe that our dynamic algorithm to maintain a (min-weight) basis of a matroid might also be of independent interest.\n\n\n\n\\paragraph{Lower Bounds.} Our super-linear $\\Omega(n \\log n)$ query lower bound comes from studying the communication complexity of matroid intersection. The matroids $\\cM_1$ and $\\cM_2$ are given to two parties Alice and Bob respectively and they are asked to solve the matroid intersection problem using as few bits of communication between them. We show that even if Alice and Bob know some common independent set $S\\in \\cI_1\\cap \\cI_2$, they need to communicate $\\Omega(n\\log n)$ bits to see if $S$ is optimal. Essentially, they need to determine if there is an augmenting path in the exchange graph. Using a class of matroids called \\emph{gammoids} (see e.g.~\\cite{perfect1968applications,mason1972class}), we show a reduction from the $(s,t)$-connectivity problem which has a deterministic $\\Omega(n\\log n)$ communication lower bound \\cite{HajnalMT88}.\n\n\\subsection{Organization}\n\nThe rest of the paper is organized as follows.\nWe first give a high-level overview of how we obtain our algorithms in \\cref{sec:overview}.\nIn \\cref{sec:prelim}, we provide the necessary preliminaries.\nWe then construct the binary search tree data structure in \\cref{sec:bst}, followed in \\cref{sec:matroid-intersection} by how to use it to implement our $\\tO(n\\sqrt{r})$ matroid intersection algorithm in the new dynamic-rank-oracle model (the dynamic-\\emph{independence}-oracle algorithm is in \\cref{appendix:dynamic-matroid-intersection-ind}).\nIn \\cref{sec:decremental-basis} we describe our data structure to maintain a basis of a matroid dynamically, and then we use this in our $\\tO_k(n+r\\sqrt{r})$ matroid union algorithm in \\cref{sec:matroid-union} (the special case of $k$-fold matroid union is in \\cref{appendix:matroid-union-fold}).\nWe show our super-linear lower bound in \\cref{sec:lowerbound}. We end our paper with a discussion of open problems in \\cref{sec:openproblems}.\nIn \\cref{appendix:applications} we mention how to implement different matroids oracles in the dynamic-oracle model, and discuss some problems we can solve with our algorithms.\n\n\\section{Super-Linear Query Lower Bounds} \\label{sec:lowerbound}\n\nLower bounds for matroid intersection have been notoriously difficult to prove. The current highest lower bound is due to Harvey \\cite{harvey2008matroid} which says that $(\\log_2 3) n - o(n)$ queries are necessary for any deterministic independence-query algorithm solving matroid intersection.\nObtaining an $\\omega(n)$ lower bound has been called a challenging open question \\cite{chakrabarty2019faster}.\n\nIn this section, we show the first super-linear query lower bound for matroid intersection, both in our \\emph{new dynamic-rank-oracle} model (\\cref{def:dyn-oracle}), and also for the \\emph{traditional independence-oracle} model, thus answering the above-mentioned open question and improving on the bounds of \\cite{Harvey09}.\nWe obtain our lower bounds by studying the communication complexity for matroid intersection.\n\n\\begin{theorem}\nIf Alice is given a matroid $\\cM_1 = (U,\\cI_1)$\nand Bob a matroid $\\cM_2 = (U,\\cI_2)$, any deterministic communication protocol needs $\\Omega(n \\log n)$ bits of communication to solve the matroid intersection problem.\n\\label{thm:comm-lb}\n\\end{theorem}\n\nThe communication lower bound of \\cref{thm:comm-lb} implies a similar lower bound for the number of independence queries needed. \nWe argue that any independence-query algorithm can be simulated by Alice and Bob in the communication setting by exchanging a single bit per query asked. Whenever they want to ask an independence query ``Is $S \\in \\cI_{i}$?'', Alice or Bob will check this locally and share the answer with the other party by sending one bit of communication.\n\nUnfortunately, this argument does not extend to the traditional rank-oracle model (since each rank query can in fact reveal $\\Theta(\\log n)$ bits of information, which need to be sent to the other party). However, for the new \\emph{dynamic}-rank-oracle model, the $\\Omega(n\\log n)$ lower bound holds as now each new query only reveals constant bits of information: either the rank remains the same, increases by one, or decreases by one (and Alice or Bob can send which is the case to the other party with a constant number of bits). Our discussion proves the following corollaries, given \\cref{thm:comm-lb}.\n\n\n\\begin{corollary}\nAny deterministic (traditional) independence-query algorithm solving matroid intersection requires $\\Omega(n \\log n)$ queries.\n\\end{corollary}\n\n\\begin{corollary}\nAny deterministic dynamic-rank-query algorithm solving matroid intersection requires $\\Omega(n \\log n)$ queries.\n\\end{corollary}\n\n\\begin{remark}\nWe note that our lower bounds are also valid for the \\emph{matroid union} problem, due to the standard reductions\\footnote{See \\cref{sec:reduction} for a reduction from matroid union to matroid intersection. To reduce from matroid intersection to matroid union, consider $\\cM = \\cM_1 \\vee \\cM_2^{*}$, where $\\cM_2^{*}$ is the \\emph{dual matroid} of $\\cM_2$ ($S \\subseteq U$ is independent in $\\cM_2^{*}$ if and only if $U - S$ contains a basis). It's easy to show that the basis $B$ of $\\cM$ in $\\cM_1$ will be of the form $B = S \\cup (U \\setminus R)$, where $S$ is the solution to the intersection between $\\cM_1$ and $\\cM_2$ and $R$ is an arbitrary basis of $\\cM_2$ that contains $S$.} between matroid intersection and union.\n\\end{remark}\n\n\\subsection{Communication Setting}\n\nWe study the following communication game\nwhich we call \\textsf{Matroid-Intersection-with-Candidate}.\nAlice and Bob are given matroids $\\cM_1 = (U,\\cI_1)$ respectively $\\cM_2 = (U,\\cI_2)$.\nSuppose they are also both given a common independent set $S\\in \\cI_1\\cap \\cI_2$, and they wish to determine\nwhether $S$ is a maximum-cardinality independent\nset.\nClearly \n\\textsf{Matroid-Intersection-with-Candidate} is an easier version of\nthe matroid intersection problem, as Alice and Bob can just ignore the candidate $S$.\n\nOur idea is that in order to solve\n\\textsf{Matroid-Intersection-with-Candidate}, Alice and Bob need to determine if there exists an \\emph{augmenting path}---that is an $(s,t)$-path---in the \\emph{exchange graph} $G(S)$ (see \\cref{def:exchange-graph} and \\cref{lemma:augmenting-path}). It is known that \\textsf{$(s,t)$-connectivity} in a\ngraph requires $\\Omega(n\\log n)$ bits of communication (\\cref{lem:lb-conn}, \\cite{HajnalMT88}). Using \\emph{strict gammoids} as our matroids, we argue that we can choose exactly how the underlying exchange graph looks like, and hence that matroid intersection admits the same lower bound.\n\n\\begin{definition}[Strict Gammoid, see~\\cite{perfect1968applications,mason1972class}]\nLet $H = (V,E)$ be a directed graph and $X\\subseteq V$ a subset of vertices.\nThen $(H,X)$ defines a matroid $\\cM = (V,\\cI)$ called a \\emph{strict gammoid}, where a set of vertices $Y\\subseteq V$ is independent if and only if\n there exists a set of vertex-disjoint directed paths (some of which might just consist of single vertices) in $H$ whose starting points all belong to $X$ and whose ending points are exactly $Y$.\n\\end{definition}\n\n\\begin{claim}\n\\label{clm:lb-matroids}\nSuppose $G=(L,R,E)$ is a directed bipartite graph and $a, b \\in R$ are two unique vertices such that $a$ has zero in-degree and $b$ has zero out-degree.\nThen there exist two matroids $\\cM_1, \\cM_2$ over the ground set $L\\cup R$ such that\n$L$ is independent in both matroids and\nthe \\emph{exchange graph} $G(L)$ is exactly $G$ plus two extra vertices ($s$ and $t$) and two extra edges ($s\\to a$ and $b \\to t$).\n\\end{claim}\n\n\\begin{proof}\nLet $F_{1} = \\{(u,v) | (u,v)\\in E, u\\in L, v\\in R\\}$ be the directed edges from $L$ to $R$ in $G$,\nand\n$F_{2} = \\{(u,v) | (v,u)\\in E, u\\in L, v\\in R\\}$ be the (reversed) directed edges from $R$ to $L$ in $G$.\nAlso let $H_1 = (L\\cup R,F_1)$ and $H_2 = (L\\cup R,F_2)$ be the directed graphs with these edges respectively.\n\nWe let $\\cM_1$, respectively $\\cM_2$, be the strict gammoids defined by $(H_1, L+a)$ respectively $(H_2, L+b)$. Now $L$ is independent in both matroids. It is straightforward to verify that the exchange graph $G(L)$ is exactly as described in the claim. We certify that this is the case for the edges defined by $\\cM_1$ ($\\cM_2$ is similar):\n\n\n\\begin{enumerate}\n    \\item  $G(L)$ will have an edge from $s$ to $a$, since\n    $L+a$ is independent in $\\cM_1$. \n    Additionally note that $a$ has in-degree zero in $G$ (and hence is an isolated vertex in $H_1$).\n    \n    \\item\n    For any $x\\in L, y\\in R$, the edge $(x,y)$ exists in $G(L)$\n    if and only if $L - x + y$ is independent in $\\cM_1$. By definition this is if and only if there exists a vertex-disjoint path starting from $L$ and ending to $L-x+y$ in $H_1$, or equivalently if the edge $(x,y)$ exists in $H_1$ (indeed, all vertices in $L-x$ must be both starts and ends of paths, so the path to $y$ must have started in $x$).\n    \\qedhere{}\n    \n\\end{enumerate}\n\n\\end{proof}\n\nWe now proceed to reduce an instance of \\textsf{$(s,t)$-connectivity} to that of \\textsf{Matroid-Intersection-with-Candidate}, which concludes the proof of \\cref{thm:comm-lb}.\n\n\\begin{definition}[\\textsf{$(s,t)$-connectivity}]\nSuppose $G = (V, E_A \\cup E_B)$ is an undirected graphs on $n=|V|$ vertices,\nwhere Alice knows edges $E_A$ and Bob knows edges $E_B$.\nThey are also both given vertices $s$ and $t$, and want to determine\nif $s$ and $t$ are connected in $G$.\n\\label{def:st-conn}\n\\end{definition}\n\n\\begin{lemma}[\\cite{HajnalMT88}]\n\\label{lem:lb-conn}\nThe deterministic communication complexity of\n\\textsf{$st$-connectivity} is $\\Omega(n\\log n)$.\n\\end{lemma}\n\n\\begin{proof}[Proof of \\cref{thm:comm-lb}.]\nWe show that an instance of \\textsf{$(s,t)$-connectivity} can be converted to an instance\nof \\textsf{Matroid-Intersection-with-Candidate} of roughly the same size.\nSuppose the symbols are defined as in  Definition \\ref{def:st-conn}.\nLet $\\bar{V} = \\{\\bar{v} : v\\in V\\}$ be a copy of $V$.\nWe construct a directed bipartite graph $G' = (V, \\bar{V}, E'_A \\cup E'_B)$\nas follows:\n\n\\begin{itemize}\n    \\item $(v,\\bar{v}) \\in E'_A$ for all $v\\in V$.\n    \\item $(\\bar{v},v) \\in E'_B$ for all $v\\in V$.\n    \\item $(v,\\bar{u}), (u,\\bar{v}) \\in E'_A$ for all $\\{u,v\\}\\in E_A$.\n    \\item $(\\bar{v},u), (\\bar{u},v) \\in E'_A$ for all $\\{u,v\\}\\in E_B$.\n    \\item No other edges exist.\n\\end{itemize}\n\nAlice knows $E'_A$, and Bob knows $E'_B$. $G'$ has $2n$ vertices and $2n+2m$ edges.\n\nNow let $G''$ be $G'$ but removing all incoming edges from $\\bar{s}$ and\nall outgoing edges from $\\bar{t}$, in order to apply \\cref{clm:lb-matroids} on $G''$ with $a = \\bar{s}$ and $b = \\bar{t}$.\nSay we get matroids $\\cM_1$ and $\\cM_2$. Note that \nAlice knows $\\cM_1$ and Bob knows $\\cM_2$ by construction.\n\nNow $s$ and $t$ are connected in $G$ if and only if there is a directed $(\\bar{s},\\bar{t})$-path in $G''$. This happens if and only if $V$ is not a maximum-cardinality common independent set of $\\cM_1$ and $\\cM_2$ (i.e.\\ in the case we found an augmenting path for $V$).\n\nHence if there is a (deterministic) communication protocol for matroid intersection using $c$ bits of communication, there is also one for \\textsf{$(s,t)$-connectivity} using $O(c)$ bits of communication. \\cref{lem:lb-conn} then implies the\n$\\Omega(n\\log n)$ communication lower bound for matroid intersection.\n\\end{proof}\n\n\n\\section{Matroid Intersection} \\label{sec:matroid-intersection}\n\nIn this section, we present a matroid intersection algorithm in the dynamic-rank-oracle model that matches the state-of-the-art algorithm~\\cite{chakrabarty2019faster} in the traditional model.\n\n\\begin{restatable}{theorem}{matroidintersection}\n  For two matroids $\\cM_1 = (U, \\cI_1)$ and $\\cM_2 = (U, \\cI_2)$, it takes $\\tO(n\\sqrt{r})$ time to obtain the largest $S \\in \\cI_1 \\cap \\cI_2$ in the dynamic-rank-oracle model.\n  \\label{thm:dynamic-matroid-intersection-rank-main}\n\\end{restatable}\n\nThe algorithm follows the blocking-flow framework of \\cite{chakrabarty2019faster} similar to the Hopcroft-Karp algorithm for bipartite matching~\\cite{HopcroftK73}, which goes as follows.\nInitially, they start with $S = \\emptyset$.\n\n\\begin{enumerate}\n  \\item\\label{item:step1} First, they obtain a common independent set that is of size at least $(1 - \\epsilon)r$ by eliminating all augmenting paths of length $O(1\/\\epsilon)$. In each of the $O(1\/\\epsilon)$ iterations, they first compute the distance layers of $G(S)$ along which they find a maximal set of compatible shortest augmenting paths using an approach similar to a depth-first-search from $s$.\n  Augmenting paths are searched in a depth-first-search manner.\n  Whenever an element has no out-edge with respect to the current common independent set to the next layer, they argue that it can be safely removed as it will not be on a shortest augmenting path anymore in this iteration.\n  Augmenting along these augmenting paths increases the $(s, t)$-distance of $G(S)$ by at least one.\n  \\item\\label{item:step2} With the current solution which is only $\\epsilon$ fraction away from being optimal, they find the remaining $O(\\epsilon r)$ augmenting paths one at a time.\n\\end{enumerate}\n\nA proper choice of $\\epsilon$ (in this case it is $\\epsilon = 1\/\\sqrt{r}$) that balances the cost between the two steps results in their algorithm.\n\n\\subsection{Building Distance Layers}\n\nBuilding distance layers and finding a single augmenting path in Step~\\ref{item:step2} is immediate by replacing binary searches in \\cite[Algorithm 4]{chakrabarty2019faster} with the binary search trees of \\cref{thm:bst}.\n\n\\begin{lemma}\n  It takes $\\tO(n)$ time to compute the $(s, u)$-distance for each $u \\in U$ and find the shortest $(s, t)$-path in $G(S)$ or determine that $t$ is unreachable from $s$.\n  \\label{lemma:bfs}\n\\end{lemma}\n\n\\begin{proof}\n  First, we build two binary search trees via \\cref{thm:bst} with $\\beta = 1$, a circuit binary search tree ${\\mathcal{T}}_1 := \\textsc{Initialize}(\\cM_1, S, X_1)$ where $X_1 = S$ for the first matroid and a co-circuit binary search tree ${\\mathcal{T}}_2 := \\textsc{Initialize}(\\cM_2, S, X_2)$ where $X_2 = \\bar{S}$ for the second matroid. Initializing these takes $\\tO(n)$ time.\n  These two binary search trees allow us to explore the exchange graph efficiently.\n  \n  Then we run the usual BFS algorithm from the source $s$ (or equivalently, all $u \\in \\bar{S}$ with $S + u \\in \\cI_1$).\n  For each visited element $u$, if $u \\in S$, then we repeatedly find $x \\in X_2$ such that $S - u + x \\in \\cI_2$ using ${\\mathcal{T}}_2.\\textsc{Find}(u)$, mark $x$ as visited, and remove $x$ from $X_2$ via ${\\mathcal{T}}_2.\\textsc{Delete}(x)$ (until $\\bot$ is returned).\n  Similarly, for $u \\in \\bar{S}$, we find $x \\in X_1$ with $S - x + u \\in \\cI_1$, mark $x$ as visited, and remove $x$ from $X_1$ using ${\\mathcal{T}}_1$.\n  This explores all the unvisited out-neighbors of $u$ in $G(S)$.\n  Since each element will be visited at most once, the total running time is $\\tO(n)$.\n\\end{proof}\n\n\\subsection{Blocking Flow}\n\nIn this section, we prove the following lemma regarding a single phase of blocking-flow computation.\n\n\\begin{lemma}\n  Given an $S \\in \\cI_1 \\cap \\cI_2$ with $d_{G(S)}(s, t) = d$, it takes $\\tO\\left(n + \\frac{n\\sqrt{r}}{d} + \\frac{(|S^\\prime| - |S|) \\cdot nd}{\\sqrt{r}}\\right)$ time to obtain an $S^\\prime \\in \\cI_1 \\cap \\cI_2$ with $d_{G(S^\\prime)}(s, t) > d_{G(S)}(s, t)$.\n  \\label{lemma:blocking-flow}\n\\end{lemma}\n\nBefore proceeding to prove \\cref{lemma:blocking-flow}, we first use it to finish our matroid intersection algorithm.\nLike Hopcroft-Karp bipartite matching algorithm \\cite{HopcroftK73} and the matroid intersection algorithm of \\cite{chakrabarty2019faster}, we run several iterations of blocking-flow, and then keep augmenting until we get the optimal solution.\n\n\\begin{proof}[Proof of \\cref{thm:dynamic-matroid-intersection-rank-main}]\n  Starting from $S = \\emptyset$, we run the blocking-flow algorithm until $d_{G(S)}(s,t) \\geq \\sqrt{r}$.\n  This, by \\cref{lemma:blocking-flow}, takes\n  \\begin{equation}\n    \\tO\\left(n\\sqrt{r}\\right) + \\tO\\left(\\sum_{d = 1}^{\\sqrt{r}}\\frac{n\\sqrt{r}}{d}\\right) + \\tO\\left(\\frac{n}{\\sqrt{r}}\\cdot\\left(\\sum_{d = 1}^{\\sqrt{r}}{d \\cdot (|S_d| - |S_{d - 1}|)}\\right)\\right)\n    \\label{eq:timebound}\n  \\end{equation}\n  time, where $S_d$ is the size of the $S$ we get after augmenting along paths of length $d$.\n  Observe that $\\sum_{d = 1}^{\\sqrt{r}}{d \\cdot (|S_d| - |S_{d - 1}|)}$ is the sum of lengths of the augmenting paths that we use, and thus the third term in \\cref{eq:timebound} is $\\tO(\\frac{n}{\\sqrt{r}} \\cdot r) = \\tO(n\\sqrt{r})$ by \\cref{lemma:augmenting-path-lengths}.\n  The second term also sums up to $\\tO(n\\sqrt{r})$ (by a harmonic sum), and therefore the total running time of the blocking-flow phases is $\\tO(n\\sqrt{r})$.\n  The current common independent set $S$ has size at least $r - O(\\sqrt{r})$ by \\cref{lemma:approx}, and thus finding the remaining $O(\\sqrt{r})$ augmenting paths one at a time takes a total running time of $\\tO(n\\sqrt{r})$ via \\cref{lemma:bfs}.\n  This concludes the proof of \\cref{thm:dynamic-matroid-intersection-rank-main}.\n\\end{proof}\n\n\nThe rest of the section is to prove \\cref{lemma:blocking-flow}.\nOur blocking-flow algorithm is a slight modification to \\cite[Algorithm 5]{chakrabarty2019faster}, as shown in \\cref{alg:blocking-flow}.\nIt takes advantage of the data structure of \\cref{thm:bst} to explore an out-edge from the current element $a_{\\ell}$ to $A_{\\ell + 1}$---the set of ``alive'' elements in the next layers---while (approximately) keeping track of the current common independent set $S$.\nAn element $u$ is ``alive'' if it has not been included in the augmenting set $\\Pi := (D_1, \\ldots, D_{d_t - 1})$ yet, nor has the algorithm determines that there cannot be any shortest augmenting path through $u$.\n\n\\begin{algorithm}[!ht]\n  \\SetEndCharOfAlgoLine{}\n  \\SetKwInput{KwData}{Input}\n  \\SetKwInput{KwResult}{Output}\n  \n  \\caption{Blocking flow}\n  \\label{alg:blocking-flow}\n  \\KwData{$S \\in \\cI_1 \\cap \\cI_2$}\n  \\KwResult{$S^\\prime \\in \\cI_1 \\cap \\cI_2$ with $d_{G(S^\\prime)}(s, t) > d_{G(S)}(s, t)$}\n  \n  Build the distance layers $L_1, \\ldots, L_{d_t - 1}$ of $G(S)$ with \\cref{lemma:bfs}\\;\n  $L_0 \\gets \\{s\\}$ and $L_{d_t} \\gets \\{t\\}$\\;\n  $A_\\ell \\gets L_{\\ell}$ for each $0 \\leq \\ell \\leq d_t$\\;\n  $\\cT_{\\ell} \\gets \\textsc{Initialize}(\\cM_{1}, S, Q_S, L_{\\ell})$ for each \\emph{odd} $1 \\leq \\ell \\leq d_t$ by \\cref{thm:bst} with $\\beta = \\sqrt{r} \/ d$\\;\n  $\\cT_{\\ell} \\gets \\textsc{Initialize}(\\cM_{2}, S, Q_S, L_{\\ell})$ for each \\emph{even} $1 \\leq \\ell \\leq d_t$ by \\cref{thm:bst} with $\\beta = \\sqrt{r} \/ d$\\;\n  $\\ell \\gets 0$, $a_0 \\gets s$, and $D_{\\ell} \\gets \\emptyset$ for each $1 \\leq \\ell < d_t$\\;\n  \\While{$\\ell \\geq 0$} {\n    \\If{$\\ell < d_t$} {\n      \\lIf{$A_{\\ell} = \\emptyset$} {\n        \\textbf{break}\n      }\n      $a_{\\ell + 1} \\gets \\cT_{\\ell + 1}.\\textsc{Find}(a_\\ell)$\\;\\label{line:search}\n      \\If{$a_{\\ell + 1} = \\bot$} {\n        $\\cT_{\\ell}.\\textsc{Delete}(a_{\\ell})$\\;\n        $A_\\ell \\gets A_\\ell - a_\\ell$ and $\\ell \\gets \\ell - 1$\\;\n      }\n      \\Else {\n        $\\ell \\gets \\ell + 1$\n      }\n    }\n    \\Else {\n        \\tcp{Found augmenting path $a_1, a_2, \\ldots a_\\ell$}\n      \\For{$i \\in \\{1, 2, \\ldots, d_t - 1\\}$} {\n          $D_{i} \\gets D_{i} + a_{i}$ and $A_{i} \\gets A_{i} - a_{i}$\\;\n          $\\cT_{i}.\\textsc{Delete}(a_i)$ and $\\cT_{i}.\\textsc{Update}(\\{a_{i - 1}, a_{i}\\})$\\;\\label{line:update}\n      }\n      $\\ell \\gets 0$\\;\n    }\n  }\n  \\textbf{return} $S^\\prime := S \\oplus \\Pi$, where $\\Pi := (D_1, D_2, \\ldots, D_{d_t - 1})$\\;\n\\end{algorithm}\n\n\n\nWe emphasize that the difference between \\cref{alg:blocking-flow} and \\cite[Algorithm 5]{chakrabarty2019faster} is exactly in the replacement of binary searches with the data structure of \\cref{thm:bst}.\nNote that indeed by the specification stated in \\cref{thm:bst}, the binary search trees let us explore edges in the exchange graph (see \\cref{remark:bst}).\nAs a result, our proof will focus on showing that such a replacement does not affect the correctness.\nFor this, we need the concept of \\emph{augmenting sets} (see \\cref{def:augmenting-sets}) which characterizes a collection of ``mutually compatible'' augmenting paths---i.e.\\ a ``blocking flow''.\nThe structural results in \\cref{sec:prelim} culminate in the following lemma that is key to the correctness of our algorithm.\nIt models when we can safely ``remove'' an element since there will be no augmenting path through it in the future.\nThis is in particular required for us (as opposed to the simpler argument used in \\cite{chakrabarty2019faster}) because the set $S$ is not fully updated after each augmentation (at least in the binary search trees that we use to explore the exchange graphs).\n\n\\begin{lemma}\n  Let $\\Pi \\subseteq \\Pi^\\prime$ be augmenting sets in $G(S)$ with distance layers $L_1, \\ldots, L_{d_t - 1}$ where $d_{G(S)}(s, t) = d_t$.\n  For $x \\in L_\\ell$, if there is no $y \\in L_{\\ell + 1}$ such that\n  \\begin{equation}\n    (S \\oplus D_{\\ell} \\oplus D_{\\ell + 1}) \\oplus \\{x, y\\} \\in \\cI,\\;\\text{where $\\cI := \\cI_1$ if $\\ell$ is even and $\\cI := \\cI_2$ if $\\ell$ is odd},\n    \\label{eq:condition}\n  \\end{equation}\n  then there is no augmenting path of length $d_t$ through $x$ in $G(S \\oplus \\Pi^\\prime)$.\n  \\label{lemma:key}\n\\end{lemma}\n\n\\begin{proof}\n  We claim that there is no augmenting path of length $d_t$ through $x$ in $G(S \\oplus \\Pi)$: If there is such a $P$, then we can put $P$ into $\\Pi$ and get an augmenting set $\\tilde{\\Pi} := (\\tilde{D}_1, \\ldots, \\tilde{D}_{d_t - 1})$ by \\cref{claim:put-aug-path-in-aug-set}.\n  By definition of the augmenting set, this means that there is such a $y \\in \\tilde{D}_{k + 1} \\setminus D_{k + 1}$ satisfying \\eqref{eq:condition}, a contradiction to our assumption.\n  The lemma now follows from \\cref{claim:no-path}. \n\\end{proof}\n\nWe are now ready to prove \\Cref{lemma:blocking-flow}.\n\n\\begin{proof}[Proof of \\cref{lemma:blocking-flow}]\n  We analyze the running time of \\cref{alg:blocking-flow} first.\n  Similar to \\cite[Lemma 15]{chakrabarty2019faster}, in each iteration, we use $\\cT_\\ell.\\textsc{Find}(\\cdot)$ to find an out-edge of $a_{\\ell}$, taking $\\tO(\\beta) = \\tO(\\sqrt{r}\/d)$ time by \\cref{thm:bst}.\n  In each iteration, we either increase $\\ell$ and extend the current path by a new element, decrease $\\ell$ and remove one element, or find an $(s, t)$-path (then remove everything in it), and each element can participate in each of the event at most once.\n  Thus, there are only $O(n)$ iterations, and the total cost of $\\cT_{\\ell}.\\textsc{Find}(\\cdot)$ is consequently $\\tO(\\frac{n\\sqrt{r}}{d})$ by our choice of $\\beta$.\n  For each of the augmenting path, $\\cT_{\\ell}.\\textsc{Update}(\\cdot)$ takes $\\tO\\left(\\frac{|L_\\ell| \\cdot d}{\\sqrt{r}}\\right)$ time, contributing to a total running time of $\\tO\\left(\\frac{(|S^\\prime| - |S|) \\cdot nd}{\\sqrt{r}}\\right)$ since $L_\\ell$'s are disjoint.\n  \n  We then argue the correctness of the algorithm.\n  Observe that at any point in time, $\\cT_\\ell$ is a data structure capable of finding a replacement element with respect to the independent set $S \\oplus D_{\\ell - 1} \\oplus D_\\ell$, due to the updates that we gave it.\n  This means that the collection $\\Pi := (D_1, D_2, \\ldots, D_{d_t - 1})$ remains an augmenting set in $G(S)$ because $S \\oplus (D_{\\ell - 1} + a_{\\ell - 1}) \\oplus (D_{\\ell} + a_\\ell)$ is independent for each $\\ell$ whenever a path is found.\n  As a result, when the algorithm terminates, $S^\\prime := S \\oplus \\Pi$ is indeed a common independent set as guaranteed by \\cref{lemma:aug-set}.\n  \n  It remains to show that $d_{G(S^\\prime)}(s, t) > d_{G(S)}(s, t)$ by arguing that for each $a_{\\ell}$ not in $\\Pi$ but removed from $A_{\\ell}$ at time $t$, there is no shortest augmenting path in $G(S^\\prime)$ that passes through $a_{\\ell}$.\n  This is a direct consequence of \\cref{lemma:key} since $\\Pi^{(t)}$, the augmenting set obtained at time $t$, is contained in $\\Pi$.\n  The fact that $\\cT_{\\ell + 1}^{(t)}.\\textsc{Find}(a_{\\ell})$ returns nothing (equivalently, \\cref{eq:condition} is not satisfied) shows that $a_{\\ell}$ is not on any shortest augmenting path in $G(S^\\prime)$ since the set $X$ maintained in $\\cT_{\\ell + 1}^{(t)}$ (see \\cref{thm:bst}) is $A_{\\ell + 1}$ at all time.\n  We remark that $x$ might have an out-edge (with respect to $S \\oplus D_{\\ell}^{(t)} \\oplus D_{\\ell}^{(t + 1)}$) to a removed element with distance $\\ell + 1$ from $s$ (not in $A_{\\ell + 1}$), but such an element, by induction, is not on any augmenting path either.\n\\end{proof}\n\n\\section{Matroid Union} \\label{sec:matroid-union}\n\nIn this section, we present our improved algorithm for matroid union.\nOur main focus of this algorithm is on optimizing the $O(n\\sqrt{r})$ term to $O(r\\sqrt{r})$.\nThus, for simplicity of presentation, we will treat $k$ as a constant (the dependence on $k$ will be a small polynomial) and express our bounds using the $O_k(\\cdot)$ and $\\tO_k(\\cdot)$ notation.\n\n\\begin{theorem}\n  In the dynamic-rank-oracle model, given $k$ matroids $\\cM_i = (U_i, \\cI_i)$ for $1 \\leq i \\leq k$, it takes $\\tO_k(n + r\\sqrt{r})$ time to find a basis $S \\subseteq U_1 \\cup \\cdots \\cup U_k$ of $\\cM = \\cM_1 \\vee \\cdots \\vee \\cM_k$ together with a partition $S_1, \\ldots, S_k$ of $S$ in which $S_i \\in \\cI_i$ for each $1 \\leq i \\leq k$.\n  \\label{thm:dynamic-matroid-union}\n\\end{theorem}\n\nIn \\cref{appendix:matroid-union-fold}, we present an optimized (for the parameter $k$) version of the above algorithm which solves the important special case when all the $k$ matroids are the same---i.e.\\ \\emph{$k$-fold matroid union}---with applications in matroid packing problems. For example, the problem of finding $k$ disjoint spanning trees in a graph falls under this special case. In particular, in \\cref{appendix:matroid-union-fold}, we obtain the following \\cref{thm:dynamic-matroid-union-fold}, and we discuss some immediate consequences for the \\emph{matroid packing, matroid covering}, and \\emph{$k$-disjoint spanning trees problems} in \\cref{sec:packing,sec:spanning}.\n\n\\begin{restatable}{theorem}{matroidunionfold}\n  In the dynamic-rank-oracle model, given a matroid $\\cM = (U, \\cI)$ and an integer $k$, it takes $\\tO(n + kr\\sqrt{\\min(n, kr)} + k\\min(n, kr))$ time to find the largest $S \\subseteq U$ and a partition $S_1, \\ldots, S_k$ of $S$ in which $S_i \\in \\cI$ for each $1 \\leq i \\leq k$.\n  \\label{thm:dynamic-matroid-union-fold}\n\\end{restatable}\n\n\n\n\nThe rest of this section will focus on the proof of \n\\cref{thm:dynamic-matroid-union} (again, where the number of matroids $k$ is treated as a constant).\nOur algorithm is based on the matroid intersection algorithm in \\cref{sec:matroid-intersection}, in which we identify and optimize several components that lead to the improved time bound.\n\n\\subsection{Reduction to Matroid Intersection} \\label{sec:reduction}\n\nFor completeness, we provide a standard reduction from matroid union to matroid intersection.\nFor an in-depth discussion, see \\cite[Chapter 42]{schrijver2003}.\nLet $\\cM_i = (U_i, \\cI_i)$ be the given $k$ matroids and $U = U_1 \\cup \\cdots \\cup U_k$ be the ground set of the matroid union $\\cM = \\cM_1 \\vee \\cdots \\vee \\cM_k$.\nWe first relabel each element in the matroids with an identifier of its matroid, resulting in $\\hat{\\cM}_i = (\\hat{U}_i, \\cI_i)$, where $\\hat{U}_i = \\{(u, i) \\mid u \\in U_i\\}$.\nLet $\\hat{\\cM} = (\\hat{U}, \\hat{\\cI}) = \\hat{\\cM}_1 \\vee \\cdots \\vee \\hat{\\cM}_k$ be over the ground set $\\hat{U} = \\hat{U}_1 \\sqcup \\cdots \\sqcup \\hat{U}_k$.\n\nIn other words, in $\\hat{\\cM}$, we duplicate each element that is shared among multiple matroids into copies that are considered different, effectively making the ground sets of the $k$ matroids disjoint.\nAfter this modification, an independent set in $\\hat{\\cM}$ is now simply the union of $k$ independent sets, one from each matroid.\nHowever, that might not be what we want since these independent sets may overlap, i.e., contain copies that correspond to the same element. \nWe therefore intersection $\\hat{\\cM}$ with a partition matroid $\\cM_{\\text{part}} = (\\hat{U}, \\cI_{\\text{part}})$ given by\n\\[ \\cI_{\\text{part}} = \\{S \\subseteq \\hat{U} \\mid \\left|S \\cap \\{(u, i)\\;\\text{for $1 \\leq i \\leq k$} \\mid u \\in U_i\\}\\right| \\leq 1\\;\\text{holds for each $u \\in U$} \\} \\]\nto restrict different copies of the same element to be chosen at most once.\nThe matroid union problem is thus reducible to the matroid intersection problem in the sense that the intersection of $\\hat{\\cM}$ and $\\cM_{\\text{part}}$ maps exactly to the independent sets of the matroid union $\\cM$.\n\nNotation-wise, given the above mapping between the two worlds, whenever we write $S \\in \\cI_{\\text{part}} \\cap \\hat{\\cI}$, a subset set of $\\hat{U}$, we will equivalently regard $S$ as a subset of $U$ with an implicit partition $S_1, \\ldots, S_k$ where $S_i \\in \\cI_i$.\n\n\n\\subsection{Specialized Matroid Intersection Algorithm}\n\nGiven the reduction, to prove \\cref{thm:dynamic-matroid-union}, it suffices to compute the intersection of $\\hat{\\cM}$ and $\\cM_{\\text{part}}$ in the claimed time bound.\nIn the following, we will set $\\cM_1$ to be $\\cM_{\\text{part}}$ and $\\cM_2$ to be $\\hat{\\cM}$ when talking about exchange graphs and other data structures.\nOur main goal is to optimize the $O(n\\sqrt{r})$ term to $O(r\\sqrt{r})$, so it might be more intuitive to think of $r \\ll n$.\nWe first show that for an $S \\in \\cI_\\text{part} \\cap \\hat{\\cI}$, the exchange graph $G(S)$ is quite unbalanced in the sense that most elements appear in the first distance layer.\nIn fact, the first distance layer of $G(S)$ contains all duplicates of elements $u$ in $U$ that do not appear in $S$.\nThis is by definition of $G(S)$ and the fact that $\\cM_1 = \\cM_{\\text{part}}$ is the partition matroid.\nIn the following, when the context is clear, we let $d_t$ denote the $(s, t)$-distance of $G(S)$ and $L_1, \\ldots, L_{d_t - 1}$ denote the distance layers.\n\n\\begin{fact}\n  It holds that $L_1 = \\{(u, i) \\mid (u, i) \\in \\hat{U}\\;\\text{and}\\;(u, j) \\not\\in S\\;\\text{for any $1 \\leq j \\leq k$}\\}$.\n  \\label{fact:large-first-layer}\n\\end{fact}\n\nSimilarly, the odd layers of $G(S)$ (that corresponds to $\\bar{S}$) are well-structured in the sense that they consist of elements whose one of the duplicates appears in $S$.\nBy definition of $G(S)$, we also know that elements in odd layers have only a single in-edge, which is from their corresponding duplicate in $S$.\nThese elements thus all have the same distance from $s$.\n\n\\begin{fact}\n  It holds that $L_3 \\cup L_5 \\cup \\cdots \\cup L_{d_t - 1} = \\{(u, i) \\mid (u, i) \\in \\hat{U}\\;\\text{and}\\;(u, j) \\in S\\;\\text{for some}\\;i \\neq j\\}$,\n  and for each $(u, i) \\in L_3 \\cup \\cdots \\cup L_{d_t - 1}$, we have $d_{G(S)}(s, (u, i)) = d_{G(S)}(s, (u, j)) + 1$ where $(u, j) \\in S$.\n  \\label{fact:structured-odd-layers}\n\\end{fact}\n\n\n\\paragraph{Union Exchange Graph.} Given the above facts, we introduce another notion of exchange graphs which is commonly used for matroid union (see, e.g., \\cite{edmonds1968matroid,cunningham1986improved}).\nFor the given $k$ matroids $\\cM_i = (U_i, \\cI_i)$ and a subset $S \\subseteq U$ that can be partitioned into $k$ independent sets $S_1, \\ldots, S_k$ with $S_i \\in \\cI_i$, the \\emph{union exchange graph} is a directed graph $H(S) = (U \\cup \\{s, t\\}, E)$ with two distinguished vertices $s, t \\not\\in U$ and edge set $E = E_s \\cup E_t \\cup E_{\\text{ex}}$, where\n\n\\begin{align*}\n  E_s &= \\{(s, u) \\mid u \\not\\in S\\}, \\\\\n  E_t &= \\{(u, t) \\mid S_i + u \\in \\cI_i\\;\\text{for some $1 \\leq i \\leq k$}\\},\\;\\text{and} \\\\\n  E_{\\text{ex}} &= \\{(u, v) \\mid S_i - v + u \\in \\cI_i\\;\\text{where $v \\in S_i$}\\}. \\\\\n\\end{align*}\n\nWe can see that the exchange graph $G(S)$ with respect to $S \\in \\cI_{\\text{part}} \\cap \\hat{\\cI}$ (as a subset of $\\hat{U}$) and the union exchange graph $H(S)$ with respect to $S \\subseteq U$ is essentially the same in the sense that $H(S)$ can be obtained from $G(S)$ by contracting all copies of the same element in the first layers and skipping all other odd layers.\nIn particular, for each $(u, i) \\in S$, in $G(S)$, there might be a direct edge from $(u, i)$ to $(u, j)$ and an edge from $(u, j)$ to $(v, j)$, where $(v, j) \\in S_j$ and $S_j - v + u \\in \\cI_j$.\nCorrespondingly, in $H(S)$, we skip the intermediate vertex $(u, j)$ and meld the above two edges as one direct edge from $u \\in S_i$ to $v \\in S_j$.\nWe also merge all edges from $s$ to some $(u, i)$ of the same $u$ in the first layer to a single edge from $s$ to $u$ (\\cref{fact:large-first-layer}).\nThis simplification does not impact the distance layers of $H(S)$ since all such $(u, j)$ have the same distance from $s$ (\\cref{fact:structured-odd-layers}).\n\n\nFrom now on, for simplicity, our algorithms will run on the union exchange graphs $H(S)$, i.e., we will perform blocking-flow computation and augment $S$ along paths in $H(S)$.\nOn the other hand, to not repeat and specialize all the lemmas to the case of union exchange graphs, proofs and correctness will be argued implicitly in the perspective of the exchange graph $G(S)$ for matroid intersection.\nFor instance, for $P = (s, a_1, \\ldots, a_{d_t - 1}, t)$ a shortest $(s, t)$-path in $H(S)$, ``augmenting $S$ along $P$'' means moving $a_i$ to the independent set that originally contains $a_{i + 1}$ for each $i \\geq 1$, and thus effectively enlarge the size of $S$ by one via putting $a_1$ in it.\\footnote{One can show that the matroid union $\\cM$ is a matroid~\\cite[Chapter 42]{schrijver2003}. As such, a basis can be obtained by trying to include each element into $S$. From the union exchange graph perspective, the independence test of $S + x$ corresponds to asking whether ``there is a path in $H(S)$ from $x$ to $t$''.}\nOne can verify that this is indeed what happens if we map $P$ back to a path $P^\\prime$ in $G(S)$, and then perform the augmentation of $S$ (as a subset of $\\hat{U}$) along $P^\\prime$.\n\nOur main idea to speed up the matroid union algorithm to $\\tO_k(r\\sqrt r)$ (instead of $\\tO_k(n \\sqrt{r})$) is to ``sparsify'' the first layer of $H(S)$ by only considering a subset of elements contained in some basis.  We formalize this in the following \\cref{lemma:bfs-union,lemma:blocking-flow-union} together with \\cref{alg:bfs-union,alg:blocking-flow-union}.\n\n\n\\begin{lemma}\n  Given $S \\in \\cI_\\text{part} \\cap \\hat{\\cI}$ and $k$ bases $\\{B_i\\}_{i = 1}^{k}$ of $U_i \\setminus S$, it takes $\\tO_k(r)$ time to construct the distance layers $L_2, \\ldots, L_{d_t - 1}$ of $H(S)$.\n  \\label{lemma:bfs-union}\n\\end{lemma}\n\nNote that we know exactly what elements are in the first distance layer, so computing $L_2, \\ldots, L_{d_t - 1}$ suffices.\n\n\\begin{algorithm}\n  \\SetEndCharOfAlgoLine{}\n  \\SetKwInput{KwData}{Input}\n  \\SetKwInput{KwResult}{Output}\n  \n  \\caption{BFS in a union exchange graph}\n  \\label{alg:bfs-union}\n  \\KwData{$S \\subseteq U$ which partitions into $S_1, \\ldots, S_k$ of independent sets and $k$ bases $\\{B_i\\}_{i = 1}^{k}$ of $U_i \\setminus S$}\n  \\KwResult{The $(s, u)$-distance $d(u)$ in $H(S)$ for each $u \\in S \\cup \\{t\\}$}\n\n  $\\mathsf{queue} \\gets B_1 \\cup \\cdots \\cup B_k$\\;\n  $d(u) \\gets \\infty$ for each $u \\in S \\cup \\{t\\}$, and $d(u) \\gets 1$ for each $u \\in B_1 \\cup \\cdots \\cup B_k$\\;\n  $\\cT_i \\gets \\textsc{Initialize}(\\cM_i, S_i, S_i)$ (\\cref{thm:bst} with $\\beta = 1$)\\;\n  \\While{$\\mathsf{queue} \\neq \\emptyset$} {\n    $u \\gets \\mathsf{queue}.\\textsc{Pop}()$\\;\n    \\For{$i \\in \\{1, 2, \\ldots, k\\}$ where $u \\in U_i$ and $u \\not\\in S_i$} {\n      \\While{$v := \\cT_i.\\textsc{Find}(u) \\neq \\bot$} {\n        $d(v) \\gets d(u) + 1$ and $\\mathsf{queue}.\\textsc{Push}(v)$\\;\n        $\\cT_i.\\textsc{Delete}(v)$\\;\n      }\n      \\lIf{$S_i + u \\in \\cI$ and $d(t) = \\infty$} {\n        $d(t) \\gets d(u) + 1$\n      }\n    }\n  }\n  \\textbf{return} $d(u)$ for each $u \\in S \\cup \\{t\\}$\\;\n\\end{algorithm}\n\n\\begin{proof}\n  The algorithm is presented as \\cref{alg:bfs-union}, and it is essentially a breadth-first-search (BFS) starting from $B_1 \\cup \\cdots \\cup B_k$ instead of $s$.\n  Out-edges in $H(S)$ are explored via $k$ binary search trees $\\cT_1, \\cT_2, \\ldots, \\cT_k$ of \\cref{thm:bst}, one for each matroid $\\cM_i$ and independent set $S_i$.\n  Let's analyze the running time first.\n  Building $\\cT_i$ takes a total of $\\tO(|S|) = \\tO_k(r)$ time.\n  Exploring the graph takes $\\tO(|S \\cup B_1 \\cup \\cdots \\cup B_k| \\cdot k) = \\tO_k(r)$ time in total since each element in $S$ is found at most once by $\\cT_i.\\textsc{Find}(\\cdot)$ because $S_i$'s are disjoint, and we also spend $O(k)$ time for each element in $S \\cup B_1 \\cup \\cdots \\cup B_k$ iterating over $\\cT_i$.\n  \n  It remains to show that starting from $B_1 \\cup \\cdots \\cup B_k$ instead of $U \\setminus S$ does not affect the correctness of the BFS.\n  For this, it suffices to show that we successfully compute $d(u)$ for all $u \\in S$ with distance $2$ from $s$.\n  By definition, $u \\in S_i$ is of distance $2$ from $s$ if and only if there exists an $x \\in U_i \\setminus S$ such that $S_i - u + x \\in \\cI_i$.\n  This is equivalent to $\\mathsf{rank}_i(S_i - u + (U_i \\setminus S)) > \\mathsf{rank}_i(S_i)$ by \\cref{obs:exchange}.\n  But then by \\cref{lemma:basis-rank}, we have $\\mathsf{rank}_i(S_i - u + (U_i \\setminus S)) = \\mathsf{rank}_i(S_i - u + B_i)$, and so such an $x$ exists in $B_i$ as well.\n  This concludes the proof of \\cref{lemma:bfs-union}.\n\\end{proof}\n\n\n\\begin{lemma}\n  Given an $S \\in \\cI_{\\text{part}} \\cap \\hat{\\cI}$ with $d_{H(S)}(s, t) = d_t$ together with data structures $\\cD_i$ of \\cref{thm:decremental-basis} that maintains a basis of $U_i \\setminus S$ for each $1 \\leq i \\leq k$, it takes $\\tO_k(r + \\frac{r\\sqrt{r}}{d_t} + (|S^\\prime| - |S|) \\cdot d_t\\sqrt{r})$ time to obtain an $S^\\prime \\in \\cI_{\\text{part}} \\cap \\hat{\\cI}$ with $d_{H}(S^\\prime)(s, t) > d_t$, with an additional guarantee that $\\cD_i$ now maintains a basis of $U_i \\setminus S^\\prime$ for each $1 \\leq i \\leq k$.\n  \\label{lemma:blocking-flow-union}\n\\end{lemma}\n\n\\begin{algorithm}\n  \\SetEndCharOfAlgoLine{}\n  \\SetKwInput{KwData}{Input}\n  \\SetKwInput{KwResult}{Output}\n  \\SetKwInput{KwGuarantee}{Guarantee}\n  \n  \\caption{Blocking flow in a union exchange graph}\n  \\label{alg:blocking-flow-union}\n  \\KwData{$S \\subseteq U$ which partitions into $S_1, \\ldots, S_k$ of independent sets and a dynamic-basis data structure $\\cD_i$ of $U_i \\setminus S$ for each $1 \\leq i \\leq k$}\n  \\KwResult{$S^\\prime \\in \\cI_{\\text{part}} \\cap \\cI_k$ with $d_{H(S^\\prime)}(s, t) > d_{H(S)}(s, t)$}\n  \\KwGuarantee{$\\cD_i$ maintains a basis of $U_i \\setminus S^\\prime$ at the end of the algorithm for each $1 \\leq i \\leq k$}\n  \n  Build the distance layers $L_2, \\ldots, L_{d_t - 1}$ of $H(S)$ with \\cref{lemma:bfs-union}\\;\n  $L_0 \\gets \\{s\\}$ and $L_{d_t} \\gets \\{t\\}$\\;\n  $B_i \\gets$ the basis maintained by $\\cD_i$ and $L_1 \\gets B_1 \\cup \\cdots \\cup B_k$\\;\n  $A_\\ell \\gets L_\\ell$ for each $0 \\leq \\ell \\leq d_t$\\;\n  $\\cT_{\\ell}^{(i)} \\gets \\textsc{Initialize}(\\cM_i, S_i, Q_{S_i}, A_{\\ell} \\cap S_i)$ for each $2 \\leq \\ell < d_t$ and $1 \\leq i \\leq k$ (\\cref{thm:bst} with $\\beta = \\sqrt{r} \/ d_t$)\\;\n  $D_{\\ell} \\gets \\emptyset$ for each $1 \\leq \\ell < d_t$\\;\n  $\\ell \\gets 0$ and $a_0 \\gets s$\\;\n  \\While{$\\ell \\geq 0$} {\n    \\If{$\\ell < d_t$} {\n      \\lIf{$A_{\\ell} = \\emptyset$} {\n        \\textbf{break}\n      }\n      \\lIf{$\\ell = 0$} {\n        Find an $a_{\\ell + 1} := \\cT_{\\ell + 1}^{(i)}.\\textsc{Find}(a_\\ell) \\neq \\bot$ for some $1 \\leq i \\leq k$\n      }\n      \\lElse {\n        $a_{\\ell + 1} \\gets$ an arbitrary element in $A_1$\n      }\n      \\If{such an $a_{\\ell + 1}$ does not exist} {\n        \\lIf{$\\ell \\geq 2$} {\n          $\\cT_{\\ell}^{(j)}.\\textsc{Delete}(a_{\\ell})$ where $a_{\\ell} \\in S_j$\n        }\n        $A_\\ell \\gets A_\\ell - a_\\ell$ and $\\ell \\gets \\ell - 1$\\;\n      }\n      \\Else {\n        $\\ell \\gets \\ell + 1$\n      }\n    }\n    \\Else {\n    \\tcp{Found augmenting path $a_1, a_2, \\ldots a_\\ell$}\n      $D_1 \\gets D_1 + a_1$ and $A_1 \\gets A_1 - a_1$\\;\n      \\For{$i \\in \\{1, 2, \\ldots, k\\}$ where $a_1 \\in U_i$} {\n        $B_i \\gets B_i - a_1$\\;\n        \\If{$\\cD_i.\\textsc{Delete}(a_1)$ returns a replacement $x$} {\\label{line:delete}\n          $B_i \\gets B_i + x$ and $A_1 \\gets A_1 \\cup \\{x\\}$\\;\n        }\n      }\n      \\For{$i \\in \\{2, \\ldots, d_t - 1\\}$} {\n          $D_{i} \\gets D_{i} + a_{i}$ and $A_{i} \\gets A_{i} - a_{i}$\\;\n          $\\cT_{i}^{(j)}.\\textsc{Delete}(a_i)$ and $\\cT_{i}^{(j)}.\\textsc{Update}(\\{a_{i - 1}, a_{i}\\})$ where $a_i \\in S_j$\\;\n      }\n      Augment $S$ along $P = (s, a_1, \\ldots, a_{d_t - 1}, t)$\\;\n      $\\ell \\gets 0$\\;\n    }\n  }\n  \\textbf{return} $S$\\;\n\\end{algorithm}\n\n\\begin{proof}[Proof of \\Cref{lemma:blocking-flow-union}]\n  Our blocking-flow algorithm for matroid union is presented as \\cref{alg:blocking-flow-union}.\n  As it is equivalent to \\cref{alg:blocking-flow} running on $G(S)$ except that the first layer $L_1 := B_1 \\cup \\cdots \\cup B_k$ is now only a subset (which is updated after each augmentation) of $U \\setminus S$, we skip most parts of the proof and focus on discussing this difference.\n  That is, we need to show that if $A_1$ becomes empty, then there is no augmenting path of length $d_t$ in $H(S^\\prime)$ anymore.\n  Given how $A_1$ and $B_i$'s are maintained and \\cref{lemma:key} (note that the set $X$ maintained in $\\cT_{\\ell}^{(i)}$ is always $A_\\ell \\cap S_i$ with respect to the current $S_i$ and thus it lets us explore out-edges to $A_{\\ell} \\cap S_i$ satisfying \\cref{eq:condition}), $A_1$ is always the subset of $B_1 \\cup \\cdots \\cup B_k$ consisting of elements that still potentially admits augmenting path of length $d_t$ in $H(S^\\prime)$ through them.\n  That means if $A_1 = \\emptyset$, then there is no augmenting set of length $d_t$ in $G(S^\\prime)$, that starts from some $b \\in B_1 \\cup \\cdots \\cup B_k$.\n  This would imply that there is no such path even if we start from $x \\in (U_i \\setminus S) \\setminus D_1$ as $B_i$ is a basis of it: if $S_i + x - y \\in \\cI$ for some $x \\in (U_i \\setminus S) \\setminus D_1$ and $y \\in S_i$, then there is a $b \\in B$ with $S + b - y \\in \\cI$, and thus a path starting from $x$ can be converted into a path starting from $b$.\n  On the other hand, all elements in $D_1$ are not on a such path by \\cref{lemma:monotone} either.\n  This shows that indeed $d_{H(S^\\prime)}(s, t) > d_{H(S)}(s, t)$.\n  \n  The guarantee that $\\cD_i$ now operates on $U_i \\setminus S^\\prime$ is clear: Augmenting along $P = (s, a_1, \\ldots, a_{d_t - 1}, t)$ corresponds to adding $a_1$ into $S$, and since we call $\\cD_i.\\textsc{Delete}(a_1)$ in Line~\\ref{line:delete} after each such augmentation, $\\cD_i$ indeed stays up-to-date. \n  \n  It remains to analyze the running time of \\cref{alg:blocking-flow-union}.\n  Computing distance layers with \\cref{lemma:bfs-union} takes $\\tO_k(r)$ time.\n  The number of elements that have ever been in some $A_i$ is $O_k(r + |S^\\prime| - |S|)$ since\n  (i) $L_2 \\cup \\cdots \\cup L_{d_t - 1}$ has size $O_k(r)$,\n  (ii) the initial basis $B_i$ of $U_i \\setminus S$ for each $1 \\leq i \\leq k$ has total size $O_k(r)$, and\n  (iii) each of the $|S^\\prime| - |S|$ augmentations adds at most $O_k(1)$ elements to $A_1$.\n  Similar to \\cref{lemma:blocking-flow}, this means that there are at most $O_k(r)$ iterations, each taking $O_k(\\frac{\\sqrt{r}}{d_t})$ time in $\\cT_{\\ell + 1}^{(i)}.\\textsc{Find}(\\cdot)$ with our choice of $\\beta$.\n  The algorithm found $|S^\\prime| - |S|$ augmenting paths, taking $\\tO_k(d_t\\sqrt{r} \\cdot (|S^\\prime| - |S|))$ time in total to update the binary search trees.\n  Also, for each such augmentation, we need $\\tO_k(\\sqrt{r})$ time to update the basis $B_i$ for all $1 \\leq i \\leq k$, which is subsumed by the cost of updating $\\cT_{i}^{(j)}$.\n  These components sum up the total running time of\n  \\[ \\tO_k\\left(r + \\frac{r\\sqrt{r}}{d_t} + \\left(|S^\\prime| - |S|\\right) \\cdot d_t\\sqrt{r}\\right). \\]\n\\end{proof}\n\n\\cref{thm:dynamic-matroid-union} now follows easily.\n\n\\begin{proof}[Proof of \\cref{thm:dynamic-matroid-union}]\n  We initialize the dynamic-basis data structure $\\cD_i$ of \\cref{thm:decremental-basis} on $U_i$ for each of the matroid $\\cM_i$.\n  We then run \\cref{lemma:blocking-flow-union} for at most $\\sqrt{r}$ iterations with $\\{\\cD_i\\}_{i = 1}^{k}$ until $d_{H(S)}(s, t) \\geq \\sqrt{r}$ and get an $S \\in \\cI_{\\text{part}} \\cap \\hat{\\cI}$ with $\\cD_i$ now operating on $U_i \\setminus S$ for each $1 \\leq i \\leq k$.\n  This takes\n  \\[ \\tO_k\\left(r\\sqrt{r} + \\sum_{d = 1}^{\\sqrt{r}}\\frac{r\\sqrt{r}}{d} + \\sum_{d = 1}^{\\sqrt{r}}d \\cdot \\left(|S_d| - |S_{d - 1}|\\right)\\right) = \\tO_k(r\\sqrt{r})\\]\n  time.\n  By \\cref{lemma:approx}, $S$ is $O_k(\\sqrt{r})$ steps away from being optimal, and thus we find the remaining augmenting paths one at a time using \\cref{lemma:bfs-union} in $\\tO_k(r\\sqrt{r})$ time in total.\n  Note that since a single augmentation corresponds to adding an element to $S$ (hence removing it from $U \\setminus S$), we can maintain the basis of $U_i \\setminus S$ that \\cref{lemma:bfs-union} needs in $\\tO_k(\\sqrt{r} \\cdot \\sqrt{r})$ total update time, which is subsumed by other parts of the algorithm.\n\\end{proof}\n\n\\subsection{Matroid Packing and Covering}\n\\label{sec:packing}\n\n\nA direct consequence of our matroid union algorithm (\\cref{thm:dynamic-matroid-union-fold} in particular) is that we can solve the following packing and covering problem efficiently.\nAs a reminder, the exact dependence on $k$ of our algorithm is $\\tO(n + kr\\sqrt{\\min(n, kr)} + k \\min(n, kr))$ by \\cref{thm:dynamic-matroid-union-fold}.\n\n\\begin{restatable}[Packing]{corollary}{packingalgo}\n    For a matroid $\\cM = (U, \\cI)$, it takes $\\tO(n\\sqrt{n} + \\frac{n^2}{r})$ time to find the largest integer $k$ and a collection of disjoint subsets $\\cS = \\{S_1, S_2, \\ldots, S_k\\}$ of $U$ such that $S_i$ is a basis for each $1 \\leq i \\leq k$ under the dynamic-rank-query model.\n    \\label{cor:packing}\n\\end{restatable}\n\n\\begin{proof}\n  It's obvious that $k \\leq \\frac{n}{r}$ holds.\n  We do a binary search of $k$ in the range $[0, \\frac{n}{r}]$, and for each $k$, we can determine the largest subset $S$ of $U$ which can be partitioned into $k$ disjoint independent sets by \\cref{thm:dynamic-matroid-union-fold}.\n  If $|S| = kr$, then it means that there are at least $k$ disjoint bases.\n  Otherwise, there are less than $k$ disjoint bases.\n  The running time is $\\tO(n\\sqrt{n} + \\frac{n^2}{r})$.\n\\end{proof}\n\n\n\\begin{restatable}[Covering]{corollary}{coveringalgo}\n    For a matroid $\\cM = (U, \\cI)$, it takes $\\tO(\\alpha r\\sqrt{n} + \\alpha n)$ time to find the smallest integer $\\alpha$ and a partition $\\cS = \\{S_1, S_2, \\ldots, S_\\alpha\\}$ of $U$ such that $S_i \\in \\cI$ holds for each $1 \\leq i \\leq \\alpha$ under the dynamic-rank-query model.\n    \\label{cor:covering}\n\\end{restatable}\n\n\n\\begin{proof}\n  We first obtain a $2$-approximation $\\alpha^{\\prime}$ of $\\alpha$ (i.e., $\\alpha \\leq \\alpha^\\prime \\leq 2\\alpha$) by enumerating powers of $2$, running \\cref{thm:dynamic-matroid-union-fold} with $k = 2^i$, and checking if the returned $S$ has size $n$: If $|S| = n$, then we know $2^i$ independent sets suffice to cover $U$.\n  Note that the enumeration stops whenever we found a suitable value of $\\alpha^\\prime$.\n  The exact value of $\\alpha$ can then be found by a binary search in $[\\frac{\\alpha^\\prime}{2}, \\alpha^\\prime]$.\n  This takes $\\tO(\\alpha r\\sqrt{n} + \\alpha n)$ (note that $\\alpha r \\geq n$ must hold).\n\\end{proof}\n\n\\subsection{Application: Spanning Tree Packing}\n\\label{sec:spanning}\n\n\nWe demonstrate the applicability of our techniques by deriving an $\\tO(|E| + (k|V|)^{3\/2})$ algorithm for the $k$ disjoint spanning tree problem in a black-box manner.\nThis improves Gabow's specialized $\\tO(k^{3\/2}|V|\\sqrt{|E|})$ algorithm~\\cite{gabow1988forests}.\nSince all applications of our algorithms follow the same reduction, we only go through it once here.\nRefer to \\cref{appendix:applications} for other applications of both our matroid union and matroid intersection algorithms.\n\n\\begin{theorem}\n  Given an undirected graph $G = (V, E)$, it takes $\\tO(|E| + (k|V|)^{3\/2})$ time to find $k$ edge-disjoint spanning trees in $G$ or determine that such spanning trees do not exist with high probability\\footnote{We use \\emph{with high probability} to denote with probability at least $1 - |V|^{-c}$ for an arbitrarily large constant $c$.}.\n  \\label{thm:k-dst}\n\\end{theorem}\n\n\\begin{proof}\nBy \\cref{thm:dynamic-matroid-union-fold}, it suffices to provide a data structure that supports the three dynamic-oracle operations (\\cref{def:dyn-oracle}) in $\\mathrm{polylog}(|V|)$ time.\nOur black-box reduction makes use of the worst-case connectivity data structure of \\cite{KapronKM13,GibbKKT15}, which can be adapted to in $O(\\mathrm{polylog}(|V|))$ update time maintain the rank of a set of edges (see \\cref{appendix:applications-graphic} for a discussion on how this can be done).\n\n\n  Let $\\cM_G$ be the graphic matroid with respect to $G = (V, E)$.\n  $G$ admits $k$ edge-disjoint spanning trees if and only if $\\cM_G$ admits $k$ disjoint bases.\n  The theorem now follows from \\cref{thm:dynamic-matroid-union-fold} with $n = |E|$ and $r = |V| - 1$ since \\cref{thm:dynamic-matroid-union-fold} returns a union of $k$ disjoint bases if they exist (we note that $k \\le |E|\/(|V|-1) \\le O(|V|)$, and hence the $O(k^2 r)$ term is dominated by the $O((kr)^{3\/2})$ term).\n\\end{proof}\n\n\\section{Open Problems} \n\\label{sec:openproblems}\n\nOur dynamic-oracle model opens up a new path to achieve fast algorithms for many problems at once, where the ultimate goal is to achieve near-linear time and dynamic-rank-query complexities. This would imply near-linear time algorithms for many fundamental problems. We envision reaching this goal via a research program where the studies of algorithms and lower bounds in our and the traditional models \ncomplement each other. \nIn particular, a major open problem is to improve our algorithms further, which would imply improved algorithms for many problems simultaneously.\nA major step towards this goal is improved algorithms in the traditional model, which would already be a breakthrough. Moreover, failed lower bound attempts might lead to new algorithmic insights and vice versa,  \nwe leave improving our lower bounds as another major open problem. \nWe believe that the communication complexity of graph and matroid problems is an important component in this study since it plays a main role in our lower bound argument. Recently the communication and some query complexities of bipartite matching and related problems were resolved in \\cite{blikstadBEMN22}. How about the communication and query complexities of dynamic-oracle matroid problems and their special cases such as colorful spanning trees?\nIt is also fruitful to resolve some special cases as the solutions may shed more light on how to solve matroid problems in our model. Below are some examples.\n\n\\begin{itemize}\n  \\item \\textbf{Disjoint Spanning Trees.} Can we find $k$ edge-disjoint spanning trees in an undirected graph in near-linear time for constant $k$, or even do so for the case of $k = 2$ (which already has application in the Shannon Switching Game)? Our new $\\tO(|E|+|V|\\sqrt{|V|})$-time algorithm shows that it is possible for sufficiently dense graphs. For the closely related problem of finding $k$ edge-disjoint arborescences (rooted \\emph{directed} spanning trees) in a directed graph, the case of $k = 2$ has long been settled by Tarjan's linear time algorithm~\\cite{Tarjan76}, and the case of constant $k$ has also been resolved by~\\cite{BhalgatHKP08}. It is a very interesting question whether the directed case is actually computationally easier than the undirected case or not.\n\n  \\item \\textbf{Colorful Spanning Tree.} This problem generalizes the maximum bipartite matching problem, among others. Given the recent advances in max-flow algorithms which are heavily based on continuous optimization techniques, bipartite matching can now be solved in almost-linear time~\\cite{ChenKLPGS22} in general and nearly-linear time for dense input graphs \\cite{BrandLNPSSSW20}. \n  It is very unclear if continuous optimization can be used for colorful spanning tree since its linear program has exponentially many constraints. \n  This reflects the general challenge of using continuous optimization to solve matroid problems and many of their special cases. \n  Thus, improving  Hopcroft-Karp's $O(|E|\\sqrt{|V|})$ runtime \\cite{HopcroftK73} (which is matched by our dynamic-oracle matroid algorithm) may shed some light on either how to use continuous optimization for these problems or how combinatorial algorithms can break this runtime barrier for colorful spanning tree, bipartite matching, and matroid problems. \n\n\\end{itemize}\n\n\\paragraph{Other Problems with Dynamic Oracles.} It also makes sense to define dynamic oracles for problems like submodular function minimization (SFM), which asks to find the minimizer of a submodular function given an evaluation oracle. In this regime, similar to matroid intersection, we want to limit the symmetric difference from the current evaluation query to the previous ones. \nWe believe that the recent algorithms for submodular function minimization based on convex optimization and cutting-plane methods, particularly the work of \\cite{LeeSW15,JiangLSW20}, can be adapted to the dynamic-oracle setting.  \nHowever, \nwe are not aware of any applications of these dynamic-oracle algorithms. \nThe first step is thus improving the best bounds in the traditional oracle model. \nThe special case of the cut-query setting \\cite{RubinsteinSW18,MukhopadhyayN20,LeeSZ21,LeeLSZ21,ApersEGLMN22} is also very interesting; we leave getting algorithms for min $(s,t)$-cut \\cite{ChenKLPGS22} and directed global mincut \\cite{Cen0NPSQ21} with near-linear time and dynamic-query complexity as major open problems.\\footnote{Adapting the cut-query algorithm of \\cite{MukhopadhyayN20} to work with dynamic cut oracles and, even better, with a parallel algorithm \\cite{AndersonB21,Lopez-MartinezM21}, is also open; though, we suspect that these are not hard.}\nAnother interesting direction is the quantum setting. For example, can one define the notion of dynamic quantum cut query so that the quantum cut-query algorithm of \\cite{ApersEGLMN22} can imply a non-trivial quantum global mincut algorithm?\n\n\n\\paragraph{Improved Lower Bounds.} \nObtaining improved lower bounds for matroid intersection is also an important open problem. Getting $\\Omega(n\\log n)$ lower bound for traditional rank-query matroid intersection algorithms is particularly interesting since it would subsume our $\\Omega(n\\log n)$ lower bounds (traditional rank-query lower bound implies independence-query and dynamic-rank-query lower bounds) and the $\\Omega(n \\log n)$ SFM lower bound of \\cite{ChakrabartyGJS22}. \nFor the latter, \\cite{ChakrabartyGJS22} showed an $\\Omega(n\\log{n})$ lower bound for SFM against \\emph{strongly}-polynomial time algorithms. Since SFM generalizes matroid intersection in the traditional rank-oracle model (i.e., a rank query of a matroid corresponds to an evaluation of the submodular function), getting the same lower bound for traditional rank-query matroid intersection algorithms would further strengthen the result of \\cite{ChakrabartyGJS22} to hold against \\emph{weakly}-polynomial time algorithms. \n\nAdditionally, achieving a {\\em truly} super-linear lower bound (i.e. an $n^{1+\\Omega(1)}$ bound) for any of the above problems is extremely interesting. \n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Technical Overview of Algorithms}\n\\label{sec:overview}\n\n\n\\subsection{The Blocking-Flow Framework}\nIn this section, we give a high-level overview of our algorithms.\nWe will focus on the dynamic-rank-oracle model (\\cref{def:dyn-oracle}), and sketch how to efficiently implement the ``blocking flow''\\footnote{Similar to the Hopcroft-Karp's \\cite{HopcroftK73} bipartite matching and Dinic's \\cite{dinic1970algorithm} maximum flow algorithms.} matroid intersection algorithms of \\cite{GabowS85,cunningham1986improved,chakrabarty2019faster,nguyen2019note} in this model.\nAs such, we briefly recap how the $\\tO(n\\sqrt{r})$ rank-query (in the traditional oracle model) algorithm of \\cite{chakrabarty2019faster} works first, and then explain how to implement their framework in the new dynamic oracle model with the same cost.\n\nTheir algorithm, like most of the matroid intersection algorithms, is based on repeatedly finding \\emph{augmenting paths} in \\emph{exchange graphs} (see \\cref{sec:prelim} for a definition). Say we have already found some common independent set $S\\in \\cI_1\\cap \\cI_2$ (we start with $S = \\emptyset$). Then the \\emph{exchange graph} $G(S)$ is a directed bipartite graph in which finding an $(s,t)$-path exactly corresponds to increasing the size of $S$ by one.\nAccording to Cunningham's blocking-flow argument \\cite{cunningham1986improved}, if we always augment along the \\emph{shortest} augmenting path, the lengths of such augmenting paths are non-decreasing.\nMoreover, if the length of the shortest augmenting path in $G(S)$ is at least $1\/\\epsilon$, then the size of the current common independent set $S$ must be at least $\\left(1 - O(\\epsilon)\\right) r$ (i.e.\\ it is only $O(\\epsilon r)$ away from optimal).\nThus, the ``blocking flow''-style algorithms consists of two stages:\n\\begin{enumerate}\n\\item\nIn the first stage, they obtain a $(1 - \\epsilon)$-approximate solution by finding augmenting paths until their lengths become more than $O(1\/\\epsilon)$. This is done by running in phases, where in phase~$i$ they eliminate all augmenting paths of length $2i$ by finding a so-called ``blocking flow''---a maximal (not necessarily maximum) collection of compatible augmenting paths. Each such phase can be implemented using only $\\tO(n)$ rank queries, as shown in \\cite{chakrabarty2019faster}. This means that the first stage needs a total of $\\tO(n\/\\epsilon)$ rank-queries (in the classic non-dynamic model).\n\n\\item\nIn the second stage, they find the remaining $O(\\epsilon r)$ augmenting paths one at a time. Each such augmentation can be found in $\\tO(n)$ rank queries, for a total of $\\tO(\\epsilon nr)$ queries for this stage.\n\\end{enumerate}\nUsing $\\epsilon = 1\/\\sqrt{r}$, \\cite{chakrabarty2019faster} obtains their $\\tO(n \\sqrt{r})$ rank-query exact matroid intersection algorithm.\nThe crux of how to implement the stages efficiently is a binary search trick to explore useful edges of the exchange graph quickly (for e.g.\\ to implement a breadth-first-search on the graph). The exchange graph can have up to $\\Theta(nr)$ edges in total, but it is not necessary to find all of them. We will argue that this binary search trick (which issues queries far away from each other) can still be implemented in the \\emph{dynamic-oracle model}, with the use of some data structures.\n\n\n\\subsection{Matroid Intersection}\n\n\\paragraph{Binary Search Tree.}\nThe crux of why a breadth-first-search (BFS) and augmenting path searching can be implemented efficiently (in terms of the number of queries) in \\cite{chakrabarty2019faster} is that they show how to, for $S \\in \\cI$, $u \\in S$, and $X \\subseteq \\bar{S}$, discover an element $x \\in X$ with $(S \\setminus \\{u\\}) \\cup \\{x\\} \\in \\cI$ in $O(\\log{n})$ rank queries using binary search (such a pair $(u,x)$ is called an \\emph{exchange pair}, and corresponds to an edge in the exchange graph).\nThe idea is that such an $x$ exists in $X$ if and only if $\\mathsf{rank}((S \\setminus \\{u\\}) \\cup X) \\geq |S|$.\nThus, we can do a binary search over $X$: we split $X$ into two equally-sized subsets $X_1$ and $X_2$, and check if such an $x$ exists in $X_1$ via the above equation. If it does, then we recurse on $X_1$ to find $x$. Otherwise, such an $x$ must exist in $X_2$ (as it does in $X$), and so we recurse on $X_2$.\nTo make this process efficient in our new model, we pre-build a binary search tree over the elements of $X$, where the internal nodes contain all the query-sets we need. That is, in the root node we have the query-set for $S\\cup X$, and in its two children for $S\\cup X_1$ respectively $S\\cup X_2$.\n\nUsing this binary tree, one can simulate the binary search process as described above.\nSince what we need to do in a BFS is to (i) find a replacement element $x$ and (ii) mark $x$ as visited (thus effectively ``deactivate'' $x$ in $X$), each time we see $x$, we just need to remove $x$ from the $O(\\log{n})$ nodes on a root-to-leaf path, and thus the whole BFS algorithm runs in near-linear time as well.\n\n\\paragraph{Batching, Periodic Rebuilding, and Augmenting Sets.}\nThe above binary search tree is efficient when the common independent set $S$ is static. However, once we find an augmenting path, we need to update $S$. This means that every node in the binary search tree needs to be updated.\nIf done naively, this would need at least $\\Omega(nr)$ time, as there are up to $r$ augmentations, and rebuilding the tree takes $O(n)$ time.\nTherefore, we employ a batching approach here.\nThat is, we do not walk through every node and update them immediately when we see an update to $S$.\nInstead, we batch $k$ updates (for $k$ to be decided later) and pay an additional $O(k)$-factor every time we want to do a query in our tree. In other words, at some point, we might want to search for exchanges for a common independent set $S'$ (by doing queries like $(S' \\setminus \\{u\\}) \\cup X$ to find edges incident to $u$). Our binary tree might only have an outdated version $S$ (i.e.\\ store sets like $S\\cup X$). Then the cost of converting $S\\cup X$ to $(S'\\setminus \\{u\\}) \\cup X$ is $|S\\oplus S'| + 1$, which we assert is less than $k$.\nWhen this number exceeds $k$, we rebuild the binary search tree completely using the up-to-date $S'$ instead, in $\\tO(n)$ time.\n\nOver the whole run of the algorithm, there are only $O(r \\log r)$ updates to our common independent set $S$ (see, e.g., \\cite{cunningham1986improved,nguyen2019note}).\nHence, the total running time becomes\n\\[ \\underbrace{\\tO(nk\\epsilon^{-1})}_{\\text{Blocking-flow for $\\tO(\\epsilon^{-1})$ iterations}} + \\underbrace{\\tO(\\epsilon rn)}_{\\text{Remaining $\\tO(\\epsilon r)$ augmenting paths}} + \\underbrace{\\tO(nrk^{-1})}_{\\text{Rebuilding binary search trees}}, \\]\nwhich is $\\tO(nr^{2\/3})$ for $k = r^{1\/3}$ and $\\epsilon =r^{-1\/3}$.\n\nTo achieve the $\\tO(n\\sqrt{r})$ bound in our dynamic-rank-oracle model, there is one additional observation we need.\nBy the ``Augmenting Sets'' argument~\\cite{chakrabarty2019faster}, for each element $v$ that we want to query our tree, it suffices to consider changes to $S$ that are in the same distance layer as $v$ is (in a single blocking-flow phase).\nSince changes to $S$ are uniformly distributed among layers, when the $(s, t)$-distance in $G(S)$ is $d$, we only need to spend an additional $O(\\frac{k}{d})$-factor (instead of an $O(k)$-factor) when querying the binary search tree.\nThis brings our complexity down to\n\\[ \\tO\\left(n\\epsilon^{-1} + \\sum_{d = 1}^{1\/\\epsilon}\\frac{nk}{d}\\right) + \\tO(\\epsilon rn) + \\tO(nrk^{-1}), \\]\nwhere the first part is a harmonic sum which makes for $\\tO(n\\epsilon^{-1} + nk)$, and the total running time is $\\tO(n\\sqrt{r})$ for $k = r^{1\/2}$ and $\\epsilon = r^{-1\/2}$.\n\n\\subsection{Matroid Union}\n\nFor simplicity of the presentation in this overview, let's assume we are solving the $k$-fold matroid union problem and that $k$---the number of bases\\footnote{As an example, consider the problem of finding $k$ disjoint spanning trees of a graph.} we want to find---is constant.\nA standard black-box reduction from matroid intersection, combined with our algorithm outlined above, immediately gives us an $\\tO(n\\sqrt{r})$ bound in the dynamic-rank-oracle model.\nNevertheless, we show how to exploit certain properties of matroid union to speed this up to $\\tO(n + r\\sqrt{r})$, i.e.\\ \\emph{near-linear time} for sufficiently ``dense''\\footnote{We call matroids with $n \\gg r$ ``dense'' by analogy to the graphic matroids where $n$ denotes the number of edges and $r$ the number of vertices.} matroids.\n\nSuppose $\\cM = (U,\\cI)$ is the matroid we want to find $k$ disjoint bases for. The standard reduction to matroid intersection is that we create $k$ copies of all elements $u\\in U$. Then we define two matroids as follows:\n\\begin{itemize}\n\\item The first matroid $\\cM_1$ says that we only want to use one version of each element $u\\in U$. We set $\\cM_1 = (U\\times \\{1,\\ldots,k\\}, \\cI_\\text{part})$ to be the partition matroid defined as $S\\in \\cI_{\\text{part}}$ if and only if $|\\{(u,i)\\in S : i = x\\}| \\le 1$ for all $x$.\n\\item The second matroid $\\cM_2$ says that for each copy of the ground set $U$ we must pick an independent set according to $\\cM$. That is set $\\cM_2 = (U\\times \\{1,\\ldots,k\\},\\hat{\\cI})$ to be the \\emph{disjoint} $k$-fold union of $\\cM$, i.e.\\ $S\\in \\hat{\\cI}$ if and only if $\\{u : (u,i)\\in S\\}$ is independent in $\\cM$ for all $i$.\n\\end{itemize}\nFor a set $S$ which can be partitioned into $k$ disjoint independent sets, notice that in the exchange graph, the number of elements \\emph{not} in the first layer is bounded by $O(r)$.\nThis is because every $u\\in U$ who is not represented in $S$ will be in the first layer $L_1$ of the BFS tree.\nAs such, we can build the BFS layers starting from the second layer if we can identify all the elements in this second layer.\nThis can be done by checking for each $y$ \\emph{not} in the first layer whether $L_1$ contains an exchange element of $y$ (via computing the rank of $(S \\setminus \\{y\\}) \\cup L_1$; no need to do a binary search).\nAlthough binary search is not needed when identifying elements in the second layer, when going backward among layers to find an augmenting path $P$, we still have to find the exact element in the first layer which can be the first element of $P$ since it will decide which augmenting paths remain ``compatible'' later.\nThis inspires us to maintain two separate binary search trees: one, of size $O(r)$, for finding edges from the second layer and onward, and the other, of size $O(n)$, for finding the first elements of the augmenting paths.\nStill, doing a binary search for each element in the first layer results in a total number of $O(r\\epsilon^{-1})$ queries to the binary search tree, which is too much.\nTo reduce the number of queries down to $\\tO(r)$, we note that only binary searches which correspond to the actual augmenting paths will succeed, i.e., reach the leaf nodes of the binary search tree.\nSince there are at most $O(r\/d)$ augmenting paths when the $(s, t)$-distance in $G(S)$ is $d$, we only need to do $O(r\/d)$ queries to the binary search tree; other queries can be blocked by first checking if their corresponding exchange elements exist in the first layer.\nThis results in a running time of $\\tO(n + r\\sqrt{n})$ (note: $\\sqrt{n}$ and not $\\sqrt{r}$), which already matches Gabow's algorithm for $k$-disjoint spanning tree~\\cite{gabow1988forests}. \n\n\\paragraph{Toward $\\tO(n + r\\sqrt{r})$ for Matroid Union.}\nThe bottleneck of the above algorithm is that we need to do binary searches over (and hence rebuild periodically) the tree data structure for the first layer (of size $\\Omega(n)$).\nIf we can reduce the size of this tree down to $O(r)$, then the running time would be $\\tO(n + r\\sqrt{r})$.\nThis suggests that we might want to somehow ``sparsify'' the first layer.\nIndeed, for a single augmenting path, we only need a basis of the first layer.\nAs a concrete example, consider the case of a graphic matroid: Given a forest $S$, an edge $e \\in S$, and the set of non-tree edges $E \\setminus S$, we want to find a ``replacement'' edge $e^\\prime$ in $E \\setminus S$ for $e$ which ``restores'' the connectivity of $S - e + e^\\prime$.\nIn this case, it suffices to only consider a spanning forest (i.e.\\ ``basis'') $B$ of $E \\setminus S$, in the sense that such a replacement edge exists in $E \\setminus S$ if and only if it exists in this spanning forest $B \\subseteq E\\setminus S$.\n\nMoreover, note that after each augmentation a single element will be removed from the first layer.\nThus, if we can maintain a decremental basis of the first layer, we can build our binary search tree data structure dynamically on top of this basis and get the desired time bound.\n\n\\paragraph{Maintaining a Basis in a Matroid.}\nOur data structure for maintaining a basis is inspired by the dynamic minimum spanning tree algorithm of \\cite{Frederickson85}, in combination with the sparsification trick of \\cite{EppsteinGIN97}. It uses $\\tO(n)$ time to initialize, and then $\\tO(\\sqrt{r})$ dynamic rank queries\\footnote{In the application of our matroid union algorithm, there will only be $\\tO(r)$ updates, so this is efficient enough for our final $\\tO(n+r\\sqrt{r})$ algorithm.} per deletion.\nIt also supports maintaining a min-weight basis.\n\n\nLet $L \\subseteq U$ for $|L| = O(n)$ be the first layer in which we want to maintain a dynamic basis.\nIn the preprocessing stage, we split $L$ into $\\sqrt{n}$ blocks $L_1, L_2, \\ldots, L_{\\sqrt{n}}$ of size roughly $\\sqrt{n}$ and compute the basis of $L$ from left to right.\nWe also build the ``prefix sums'' of these $\\sqrt{n}$ blocks so that we can quickly access\/query sets of the form $L_1 \\cup L_2 \\cup \\cdots \\cup L_k$ for all values of $k$.\nWhen we remove an element $x$ from $L_i$, we first update the prefix sums in $O(\\sqrt{n})$ time.\nIf $x$ is not in the basis we currently maintain, then nothing additional needs to be done.\nOtherwise, we have to find the ``first'' replacement element, which is guaranteed to be located in blocks $L_i, \\ldots, L_{\\sqrt{n}}$.\nThe block $L_j$ in which the replacement element lies can be identified simply by inspecting the ranks of the prefix sums, and after that, we then go through elements in that block to find the exact element.\nNote that blocks after $L_j$ need not be updated, as for them it does not matter what basis we picked among blocks $L_1$ to $L_j$. \nThis gives us an $O(\\sqrt{n})$-update time algorithm for maintaining a basis of a matroid.\n\nTo get a complexity of $O(\\sqrt{r}\\log n)$, we show that a similar sparsification structure as that of \\cite{EppsteinGIN97} for dynamic graph algorithms also works for arbitrary matroids. The sparsification is a balanced binary tree over the $n$ elements, where in each node we have an instance of our (un-sparsified) underlying data structure to maintain a basis consisting of elements in the subtree rooted at the node. Only elements part of the basis of a node are propagated upwards to the parent node. This means that in each instance of our underlying data structure we work over a ground set of size at most $2r$.\nThus, each update corresponds to at most two updates (a single insertion and deletion) to at most $O(\\log n)$ (which is the height of the tree) nodes of the tree, each costing $O(\\sqrt{r})$ dynamic rank queries in order to maintain the basis at this node.\nThis results in the desired time bound.\n\n\n\n\\section{Preliminaries} \\label{sec:prelim}\n\n\n\\paragraph{Notation.}\nWe use standard set notation.\nIn addition to that, for two sets $X$ and $Y$, we use $X + Y$ to denote $X \\cup Y$ (when $X \\cap Y = \\emptyset$) and $X - Y$ to denote $X \\setminus Y$ (when $Y \\subseteq X$).\nFor an element $v$, $X + v$ and $X - v$ refer to $X + \\{v\\}$ and $X - \\{v\\}$, respectively.\nLet $X \\oplus Y$ denote the symmetric difference of $X$ and $Y$.\n\n\n\\paragraph{Matroid.}\nIn this paper, we use the standard notion of matroids which is defined as follows.\n\n\\begin{definition}\nA \\emph{matroid} $\\cM = (U, \\cI)$ is defined by a tuple consisting of a finite \\emph{ground set} $U$ and a non-empty family of \\emph{independent sets} $\\cI \\subseteq 2^U$ such that the following properties hold.\n\\begin{itemize}\n  \\item \\textbf{Downward closure:} If $S \\in \\cI$, then any subset $S^\\prime \\subseteq S$ is also in $\\cI$.\n  \\item \\textbf{Exchange property:} For any two sets $S_1, S_2 \\in \\cI$ with $|S_1| < |S_2|$, there exists an $x \\in S_2 \\setminus S_1$ such that $S_1 + x \\in \\cI$.\n\\end{itemize}\n\\label{def:matroid}\n\\end{definition}\n\nLet $U$ be the ground set of a matroid $\\cM$.\nFor $S \\subseteq U$, let $\\bar{S}$ denote $U \\setminus S$.\nFor $X \\subseteq U$, the \\emph{rank} of $X$, denoted by $\\mathsf{rank}(X)$, is the size of the largest independent set contained in $X$, i.e., $\\mathsf{rank}(X) = \\max_{S \\in \\cI}|X \\cap S|$.\nThe \\emph{rank} of a matroid $\\cM = (U, \\cI)$ is the rank of $U$.\nWe let $r \\leq n$ denote the rank of the input matriods.\nWhen the input consists of more than one matroid (e.g., in the matroid union problem), let $\\mathsf{rank}_i$ denote the rank function of the $i^{\\scriptsize \\mbox{{\\rm th}}}$ matroid.\nA \\emph{basis} of $X$ is an independent set $S \\subseteq X$ with $|S| = \\mathsf{rank}(X)$.\nA basis of $\\cM$ is a basis of $U$.\nA \\emph{circuit} is a minimal dependent set.\nThe \\emph{span} of $X$ contains elements whose addition to $X$ does not increase the rank of it, i.e., $\\mathsf{span}(X) = \\{u \\in U \\mid \\mathsf{rank}(X \\cup \\{u\\}) = \\mathsf{rank}(X)\\}$.\n\n\\begin{fact}\n  The rank function is submodular.\n  That is, $\\mathsf{rank}(X) + \\mathsf{rank}(Y) \\geq \\mathsf{rank}(X \\cap Y) + \\mathsf{rank}(X \\cup Y)$ holds for each $X, Y \\subseteq U$.\n\\end{fact}\n\n\\begin{fact}[see, e.g., {\\cite[Lemma 1.3.6]{price2015}}]\n  $\\mathsf{rank}(A) = \\mathsf{rank}(\\mathsf{span}(A))$ holds for every $A \\subseteq U$.\n  \\label{fact:span-does-not-increase-rank}\n\\end{fact}\n\n\\begin{lemma}\n  For two sets $X, Y$ and their bases $S_X, S_Y$, it holds that $\\mathsf{rank}(S_X + S_Y) = \\mathsf{rank}(X + Y)$.\n  \\label{lemma:basis-rank}\n\\end{lemma}\n\\begin{proof}\n  Since $X, Y \\subseteq \\mathsf{span}(S_X + S_Y)$, we have $S_X + S_Y \\subseteq X + Y \\subseteq \\mathsf{span}(S_X + S_Y)$.\n  The lemma then follows from $\\mathsf{rank}(S_X + S_Y) = \\mathsf{rank}(\\mathsf{span}(S_X + S_Y))$ using \\cref{fact:span-does-not-increase-rank}.\n\\end{proof}\n\n\\paragraph{Exchange Graph.}\nOur algorithms for matroid intersection and union will be heavily based on finding augmenting paths in exchange graphs.\n\n\\begin{definition}[Exchange Graph]\nFor two matroids $\\cM_1 = (U, \\cI_1)$ and $\\cM_2 = (U, \\cI_2)$ over the same ground set and an $S \\in \\cI_1 \\cap \\cI_2$, the \\emph{exchange graph} with respect to $S$ is a directed bipartite graph $G(S) = (U \\cup \\{s, t\\}, E_S)$ with $s, t \\not\\in U$ being two distinguished vertices and $E_S = E_1 \\cup E_2 \\cup E_s \\cup E_t$, where\n\\begin{align*}\n  E_1 &= \\{(x, y) \\mid x \\in S, y \\not\\in S,\\;\\text{and}\\;S - x + y \\in \\cI_1\\}, \\\\\n  E_2 &= \\{(y, x) \\mid x \\in S, y \\not\\in S,\\;\\text{and}\\;S - x + y \\in \\cI_2\\}, \\\\\n  E_s &= \\{(s, x) \\mid S + x \\in \\cI_1 \\},\\;\\text{and} \\\\\n  E_t &= \\{(x, t) \\mid S + x \\in \\cI_2 \\}.\n\\end{align*}\n\\label{def:exchange-graph}\n\\end{definition}\n\nThe \\emph{distance layers} of $G(S)$ is the sets $L_1, \\ldots, L_{d_{G(S)}(s, t) - 1}$, where $L_\\ell$ consists of elements in $U$ that are of distance $\\ell$ from $s$ in $G(S)$.\nMost matroid intersection algorithms including ours are based on augmenting a common independent set with an augmenting path in $G(S)$ until such a path does not exist.\nThe following lemma certifies the correctness of this approach.\n\n\\begin{lemma}[Augmenting Path]\n  Let $P$ be a shortest $(s, t)$-path\\footnote{In fact, $P$ only needs to be ``chordless''~\\cite{quadratic2021}, i.e., without shortcuts. Nonetheless, a shortest $(s, t)$-path suffices for our rank-query algorithms.} of $G(S)$.\n  Then, the set $S^\\prime := S \\oplus (V(P) \\setminus \\{s, t\\})$ is a common independent set with $|S^\\prime| = |S| + 1$.\n  On the other hand, if $t$ is unreachable from $s$ in $G(S)$, then $S$ is a largest common independent set.\n  \\label{lemma:augmenting-path}\n\\end{lemma}\n\nWe write $S \\oplus P$, where $P$ is an augmenting path in $G(S)$, for the common independent set $S^\\prime := S \\oplus (V(P) \\setminus \\{s, t\\})$ obtained by augmenting $S$ along $P$.\nLet $d_{G(S)}(u, v)$ denote the $(u, v)$-distance in $G(S)$.\nWhen $S$ is clear from context, let $d_t$ denote $d_{G(S)}(s, t)$.\nThe following lemma states that if $d_{G(S)}(s, t)$ is large, then $S$ is close to being optimal.\n\n\\begin{lemma}[\\cite{cunningham1986improved}]\n  If $S \\in \\cI_1 \\cap \\cI_2$ satisfies $d_{G(S)}(s, t) \\geq d$, then $|S|$ is at least $\\left(1 - O(\\frac{1}{d})\\right)r$.\n  \\label{lemma:approx}\n\\end{lemma}\n\nThe following bound on the total length of shortest augmenting paths will be useful for our analysis.\n\n\\begin{lemma}[\\cite{cunningham1986improved}]\n  If we solve matroid intersection by repeatedly finding the shortest augmenting paths, then the sum of the lengths of these augmenting paths is $O(r\\log r)$.\n  \\label{lemma:augmenting-path-lengths}\n\\end{lemma}\n\n\\begin{lemma}[\\cite{cunningham1986improved,price2015,chakrabarty2019faster}]\n  If we augment along a shortest $(s, t)$-path in $G(S)$ to obtain $S^\\prime$, then for each $u \\in U$, the following hold (let $d := d_{G(S)}$ and $d^\\prime := d_{G(S^\\prime)})$.\n  \\begin{enumerate}\n    \\item If $d(s, u) < d(s, t)$, then $d^\\prime(s, u) \\geq d(s, u)$. Similarly, if $d(u, t) < d(s, t)$, then $d^\\prime(u, t) \\geq d(u, t)$.\n    \\item If $d(s, u) \\geq d(s, t)$, then $d^\\prime(s, u) \\geq d^\\prime(s, t)$. Similarly, if $d(u, t) \\geq d(s, t)$, then $d^\\prime(u, t) \\geq d^\\prime(s, t)$.\n  \\end{enumerate}\n  \\label{lemma:monotone}\n\\end{lemma}\n\n\\paragraph{Augmenting Sets.}\nThe following notion of \\emph{augmenting sets}, introduced by \\cite{chakrabarty2019faster}, models a collection of ``mutually compatible'' augmenting paths, i.e., paths that can be augmented sequentially without interfering with each other.\n\n\\begin{definition}[Augmenting Set~{\\cite[Definition 24]{chakrabarty2019faster}}]\n  Let $S \\in \\cI_1 \\cap \\cI_2$ satisfy $d_{G(S)}(s, t) = d_t$ and let $L_1, L_2, \\ldots, L_{d_t - 1}$ be the distance layers of $G(S)$.\n  A collection $\\Pi := (D_1, D_2, \\cdots, D_{d_t - 1})$ is an \\emph{augmenting set} in $G(S)$ if\n  \\begin{enumerate}[noitemsep, label=(\\roman*)]\n    \\item $D_\\ell \\subseteq L_\\ell$ holds for each $1 \\leq \\ell < d_t$,\n    \\item $|D_1| = |D_2| = \\cdots = |D_{d_t - 1}|$,\n    \\item $S + D_1 \\in \\cI_1$,\n    \\item $S + D_{d_t - 1} \\in \\cI_2$,\n    \\item $S - D_\\ell + D_{\\ell + 1} \\in \\cI_1$ holds for each \\emph{even} $1 \\leq \\ell < d_t - 1$, and\n    \\item $S - D_{\\ell + 1} + D_{\\ell} \\in \\cI_2$ holds for each \\emph{odd} $1 \\leq \\ell < d_t - 1$.\n  \\end{enumerate}\n  \\label{def:augmenting-sets}\n\\end{definition}\n\nOne can think of the concept of augmenting sets as a generalization of augmenting paths.\nIndeed, an augmenting path is an augmenting set where $|D_1| = \\cdots = |D_{d_t - 1}| = 1$.\nThe term ``mutually compatible'' augmenting paths is formalized as follows.\n\n\\begin{definition}[Consecutive Shortest Paths~{\\cite[Definition 28]{chakrabarty2019faster}}]\n  A collection of vertex-disjoint shortest $(s, t)$-paths $\\cP = (P_1, \\ldots, P_k)$ in $G(S)$ is a collection of \\emph{consecutive shortest paths} if $P_i$ is a shortest augmenting path in $G(S \\oplus P_1 \\oplus \\cdots \\oplus P_{i - 1})$ for each $1 \\leq i \\leq k$.\n\\end{definition}\n\nThe following structural lemmas of \\cite{chakrabarty2019faster} will be useful for us, particularly in deriving \\cref{lemma:key} in \\cref{sec:matroid-intersection}.\n\n\\begin{lemma}[{\\cite[Theorem 25]{chakrabarty2019faster}}]\n  Let $\\Pi := (D_1, \\ldots, D_{d_t - 1})$ be an augmenting set in $G(S)$.\n  Then, $S^\\prime := S \\oplus \\Pi := S \\oplus D_1 \\oplus \\cdots \\oplus D_{d_t - 1}$ is a common independent set.\n  \\label{lemma:aug-set}\n\\end{lemma}\n\n\nFor two augmenting sets $\\Pi = (D_1, \\ldots, D_{d_t - 1})$ and $\\Pi^\\prime = (D_1^\\prime, \\ldots, D_{d_t - 1}^\\prime)$, we use $\\Pi \\subseteq \\Pi^\\prime$ to denote that $D_{\\ell} \\subseteq D_{\\ell}^\\prime$ hold for each $1 \\leq \\ell < d_t$.\nIn this case, let $\\Pi^\\prime \\setminus \\Pi := (D_1^\\prime \\setminus D_1, \\ldots, D_{d_t - 1}^\\prime \\setminus D_{d_t - 1})$.\nWe will hereafter abuse notation and let $\\Pi$ also denote the set of elements $D_1 \\cup \\cdots \\cup D_{d_t - 1}$ in it.\nIn particular, $U \\setminus \\Pi$ denotes $U \\setminus \\left(D_1 \\cup \\cdots \\cup D_{d_t - 1}\\right)$.\n\n\\begin{lemma}[{\\cite[Theorem 33]{chakrabarty2019faster}}]\n  For two augmenting sets $\\Pi \\subseteq \\Pi^\\prime$ in $G(S)$, $\\Pi^\\prime \\setminus \\Pi$ is an augmenting set in $G(S \\oplus \\Pi)$.\n  \\label{lemma:difference-is-aug-set}\n\\end{lemma}\n\n\\begin{lemma}[{\\cite[Theorem 29]{chakrabarty2019faster}}]\n  Given a collection of consecutive shortest paths $P_1, \\ldots, P_k$ in $G(S)$, where $P_i = (s, a_{i,1}, \\ldots, a_{i, d_t - 1}, t)$, the collection $\\Pi = (D_1, \\ldots, D_{d_t - 1})$, where $D_i = \\{a_{1,i}, \\ldots, a_{k, i}\\}$, is an augmenting set in $G(S)$.\n  \\label{lemma:aug-set-from-aug-paths}\n\\end{lemma}\n\nThe converse of \\cref{lemma:aug-set-from-aug-paths} also holds.\n\n\\begin{lemma}[{\\cite[Theorem 34]{chakrabarty2019faster}}]\n  Given an augmenting set $\\Pi$ in $G(S)$, there is a collection consecutive shortest paths $P_1, \\ldots, P_k$ in $G(S)$ where $P_i = (s, a_{i,1}, \\ldots, a_{i,d_t - 1}, t)$ such that $D_i = \\{a_{1,i}, \\ldots, a_{k,i}\\}$.\n  \\label{lemma:aug-paths-from-aug-set}\n\\end{lemma}\n\n\\begin{remark}\n  Note that \\cref{lemma:aug-set-from-aug-paths,lemma:aug-paths-from-aug-set} are not equivalent to the exact statements of {\\cite[Theorems 29 and 34]{chakrabarty2019faster}} (in particular, they did not specify how $\\Pi$ and $P_i$ are constructed), but our versions are clear from their proof.\n\\end{remark}\n\n\n\n\n\\begin{claim}\n  Let $S \\in \\cI_1 \\cap \\cI_2$ with $d_{G(S)}(s, t) = d_t$ and $\\Pi \\subseteq \\Pi^\\prime$ be two augmenting sets in $G(S)$.\n  Let $u \\not\\in \\Pi^\\prime$ be an element.\n  If $u$ is not on any augmenting path of length $d_t$ in $G(S \\oplus \\Pi)$, then $u$ is not on any augmenting path of length $d_t$ in $G(S \\oplus \\Pi^\\prime)$ either.\n  \\label{claim:no-path}\n\\end{claim}\n\n\\begin{proof}\n  Let $S^\\prime := S \\oplus \\Pi$ and $S^{\\prime\\prime} := S \\oplus \\Pi^\\prime$.\n  Since $S^{\\prime\\prime}$ can be obtained by augmenting $S^\\prime$ along a series of shortest augmenting paths (by \\cref{lemma:difference-is-aug-set,lemma:aug-paths-from-aug-set}), the claim follows from the fact that the $(s, u)$-distance and $(u, t)$-distance are monotonic (\\cref{lemma:monotone}).\n\\end{proof}\n\n\\begin{claim}\n  Let $\\Pi = (D_1, \\ldots, D_{d_t - 1})$ be an augmenting set in $G(S)$ and $P = (s, a_1, \\ldots, a_{d_t - 1}, t)$ be an augmenting path in $G(S \\oplus \\Pi)$.\n  Then, $\\Pi^\\prime = (D_1 + a_1, \\ldots, D_{d_t - 1} + a_{d_t - 1})$ is an augmenting set in $G(S)$.\n  \\label{claim:put-aug-path-in-aug-set}\n\\end{claim}\n\n\\begin{proof}\n  This directly follows from \\cref{lemma:aug-paths-from-aug-set,lemma:aug-set-from-aug-paths}.\n\\end{proof}\n\n\n\n\n\\paragraph{Using Dynamic Oracle.}\nIn the following sections except for \\cref{appendix:dynamic-matroid-intersection-ind} where we discuss independence-query algorithms, all algorithms and data structures will run in the \\emph{dynamic-rank-oracle} model (see \\cref{def:dyn-oracle}).\nIn other words, we will simply write ``in $t$ time'' for ``in $t$ time and dynamic rank queries''.\nWe will use the term \\emph{query-sets} to refer to the sets $S_i$ in \\cref{def:dyn-oracle}.\nIn particular, constructing a query-set means building the corresponding set from $S_0 = \\emptyset$ with the $\\textsc{Insert}(\\cdot)$ operation.\nInsertion\/Deletion of an element into\/from a query-set is done via the $\\textsc{Insert}$\/$\\textsc{Delete}$ operations.\nUsing the $\\textsc{Query}$ operation, we assume that we know the ranks of all the query-sets we construct in our algorithms.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\\label{sl:intro}\nDuring the past few decades, science has made great progress in understanding the biology of cancer \\cite{p53review_yu2014small,cancer_advances_2014}. The latest technological tools allow assaying tens of thousands of genes simultaneously, providing large volumes of data to search for cancer biomarkers \\cite{microarray_techn_2013,venter2001sequence}. Ideally, scientists would like to extract some qualitative signal from this data in the hope to better understand the underlying biological processes. At the same time it is desirable that the extracted signal is robust in the presence of noise and errors while effectively describing the dataset \\cite{valarmathi2014noise,pozhitkov2014revised}.\n\nOne suite of methods which have enjoyed increased level of success in recent years is based on concepts from the mathematical field of algebraic topology, in particular, {\\em persistent homology} \\cite{EdLeZo2002,Gh2008barcodes,CaZoCoGu2004}. The key benefits of using topological features to describe data include coordinate-free description of shape, robustness in the presence of noise and invariance under many transformations, as well as highly compressed representations of structures \\cite{Lumetal2013}. Analysis of the homology of data allows detection of high-dimensional features based on {\\em connectivity}, such as loops and voids, which could not be detected using traditional methods such as clustering \\cite{Ca2009,Gh2008barcodes}. Further, identifying the most {\\em persistent} of such features promises to pick up the significant shapes while ignoring noise \\cite{Gh2008barcodes}. Such analysis of the stable topological features of the data could provide helpful insights as demonstrated by several studies, including some recent ones on cancer data \\cite{dewoshkin2010,seeman2012}.\n\n\\subsection{Our contributions} \\label{contrib}\nWe present a new method of topological analysis of various cancer gene expression data sets. Our method belongs to the category of exploratory data analysis. In order to more efficiently handle the huge number of genes whose expressions are recorded in such data sets (typically in the order of tens of thousands), we transpose the data and analyze it in its {\\em dual space}, i.e., with each gene represented in the much lower dimensional (in the order of a few hundred)  sample space. We then sample critical genes as guided by the topological analysis. In particular, we choose a small subset (typically $120$--$200$) of genes as {\\em landmarks} \\cite{desilva2004}, and construct a family of nested simplicial complexes, indexed by a proximity parameter. We observe topological features (loops) in the first homology group ($\\homo_1$) that remain significant over a large range of values of the proximity parameter (we consider small loops as topological noise). By repeating the procedure for different numbers of landmarks, we select stable features that persist over large ranges of both the number of landmarks and the proximity parameter. We then further analyze these loops with respect to their membership, working under the hypothesis that their topological connectivity could reveal functional connectivity. Through the search of scientific literature, we establish that many loop members have been implicated in cancer biogenesis. We applied our methodology to five different data sets from a variety of cancers (brain, breast, ovarian, and acute myeloid leukemia (AML)), and observed that in each of the five cases, many members of the significant loops in $\\homo_1$ have been identified in the literature as having connections to cancer.\n\nOur method is capable of identifying geometric properties of the data that cannot be found by traditional algorithms such as clustering \\cite{cluster_genes_review2012,cluster_review2010}. By employing tools from algebraic topology, our method goes beyond clustering and detects connected components around holes (loops)  in the data space. The shown methodology is also different from techniques such as graph \\cite{cluster_massive_graphs_siam2013,graph_mining_book_cook2006} or manifold learning \\cite{Isomap_original2000,LLE_original2000,HessianLLE_original2003,laplacian_embedding_original2001,diffusion_maps2006}. Graph algorithms, while identifying connectivity, miss wealth of information beyond clustering. Manifold learning algorithms assume that the data comes from an intrinsically low-dimensional space and their goal is to find a low-dimensional embedding. We do not make such assumptions about the data.\n\n\n\n\\subsection{Related work}\\label{prev_work}\nSeveral applications of tools from algebraic topology to analyze complex biological data from the domain of cancer research have been reported recently. DeWoshkin et al.~\\cite{dewoshkin2010} used computational homology for analysis of comparative genomic hybridization (CGH) arrays of breast cancer patients. They analyzed DNA copy numbers by looking at the characteristics of \\(\\homo_0\\) group. Using $Betti_0$ (\\(\\beta_0\\)) numbers, which are the ranks of the zeroth homology groups ($\\homo_0$), their method was able to distinguish between recurrent and non-recurrent patient groups.\n\nLikewise, Seeman et al.~\\cite{seeman2012} applied persistent homology tools to analyze cancer data. Their algorithm starts with a set of genes that are preselected using the {\\it nondimensionalized standard deviation} metric \\cite{seeman2012}. Then, by applying persistent homology analysis to the \\(\\homo_0\\) group, the patient set is recursively subdivided to yield three subgroups with distinct cancer types. By inspecting the cluster membership, a core subset of genes is selected which allows sharper differentiation between the cancer subtypes.\n\nAnother example of topological data analysis is the work of Nicolau et al.~\\cite{nicolau2011}. Their method termed {\\em progression analysis of disease} (PAD) is applied to differentiate three subgroups of breast cancer patients. PAD is a combination of two algorithms -- disease-specific genome analysis (DSGA) \\cite{dsga2007} and the topology-based algorithm termed {\\it Mapper} \\cite{mapper2007}.  First, DSGA transforms the data by decomposing it into two components -- the disease component and the healthy state component, where the disease component is a vector of residuals from a {\\it Healthy State Model} (HSM) linear fit. A small subset of genes that show a significant deviation from the healthy state are retained and passed on to Mapper, which applies a specified filter function to reveal the topology of the data. Mapper identified three clusters corresponding to ER$+$, ER$-$, and normal-like subgroups of breast cancer. This work is somewhat different from the previous two papers mentioned above because it does not explicitly analyze features of any of the homology groups.\n\nAll studies mentioned above utilized $\\beta_0$ numbers, thus performing analyses that are topologically equivalent to clustering. In contrast, our method relies on $\\beta_1$ numbers (ranks of $\\homo_1$ groups). One can think of $\\beta_1$ numbers characterizing the loops constructed from connected components (genes) around ``holes'' in the data. The underlying idea is that connections around holes  may imply connections between the participating genes and biological functions. Also, most of other methods use some data preprocessing to limit the initial pool of candidate genes. Our method selects the optimal number of genes as part of the analysis itself.\n\n\\section{Mathematical background} \\label{sl:theory}\n\nWe review some basic definitions from algebraic topology used in our work. For details, refer one of the standard textbooks \\cite{Munkres1984,EdHa2009}. Illustrations of simplices, persistent homology, and identification of topological features from landmarks are available in the literature \\cite{Gh2008barcodes, desilva2004, javaplex2011}.\n\n\\subsection{Simplices and simplicial complexes} \\label{sl:simplicial_complexes}\nTopology represents the shape of point sets using combinatorial objects called simplicial complexes. Consider a finite set of points in \\(\\R^n\\). More generally, the space need not be Euclidean. We just need a unique pairwise distance be defined for every pair of points. The building blocks of the combinatorial objects are {\\em simplices}, which are made of collections of these points. \n\nFormally, the convex hull of $k+1$ affinely independent points $\\{v_{0}, v_{1}, \\dots, v_{k}\\}$ is a $k$-simplex. The dimension of the simplex is $k$, and $v_{j}$s are its vertices. Thus, a vertex is a $0$-simplex, a line segment connecting two vertices is a $1$-simplex, a triangle is a $2$-simplex, and so on. Observe that each $p$-simplex $\\sigma$ is made of lower dimensional simplices, i.e., $k$-simplices $\\tau$ with \\(k \\leq p\\). Here, $\\tau$ is called a {\\em face} of $\\sigma$, denoted $\\tau \\subset \\sigma$.  A collection of simplices satisfying two regularity properties forms a {\\em simplicial complex}. The first property is that each face of every simplex in a simplicial complex $K$ is also in $K$. Second, each pair of simplices in $K$ intersect in a face of both, or not at all. Due to these properties, algorithms to study shape and topology run much more efficiently on the simplicial complex than on the original point set.\n\nTo construct a simplicial complex on a given point set, one typically considers balls of a given diameter $\\epsilon$ (called $\\epsilon$-ball) centered at each point. The two widely studied complexes of this form are the {\\em \\v{C}ech} and the {\\em Vietoris-Rips} complexes. A $k$-simplex is included in the \\v{C}ech complex if there exists an $\\epsilon$-ball containing all its $k+1$ vertices. Such a simplex is included in the the Vietoris-Rips complex \\(R_{\\epsilon} \\) if each pair of its vertices is within a distance $\\epsilon$. As such, Vietoris-Rips complexes are somewhat easier to construct, since we only need to inspect pairwise, and not higher order, distances.\n\nHowever, both the \\v{C}ech and the Vietoris-Rips complexes have as vertex set all of the points in the data. Such complexes are computationally intensive for datasets of tens of thousands of points. The feasible option is to work with an approximation of the topological space of interest \\cite{desilva2004}. The key idea is to select only a small subset of points (landmarks), while the rest of points serve as {\\it witnesses} to the existence of simplices. Termed {\\em witness complexes}, such complexes have a number of advantages. They are easily computed,  adaptable to arbitrary metrics, and do not suffer from the curse of dimensionality. They also provide a less noisy picture of the topological space. We use the {\\it lazy Witness complex}, in which conditions for inclusion are checked only for pairs and not for higher order groups of points \\cite{desilva2004}, analogous to the distinction between the constructions of Vietoris-Rips and \\v{C}ech complexes.\n\nWe employ the heuristic landmark selection procedure called {\\it sequential maxmin} to select a representative set of landmark points \\cite{desilva2004,adams_carlsson_2009,carlsson_etal_2008}. The first landmark is selected randomly from the point set $S$. Then the algorithm proceeds inductively. If \\(L_{i-1}\\) is the set of the first \\(i-1\\) landmarks, then the $i$-th landmark is the point of $S$ which maximizes the function \\(d(x, L_{i-1})\\), the distance between the point \\(x\\) and the set \\(L_{i-1}\\). We vary the total number of landmarks, exploring each of the resulting lazy witness complexes. The final number of landmarks is chosen so that the resulting witness complex maximally exposes topological features.\n\n\\subsection{Persistent homology}\\label{persist_homo}\n\nHomology is the concept from algebraic topology which captures how space is {\\em connected}. Thus, homology can be used to characterize interesting features of a simplicial complex such as connected clusters, holes, enclosed voids, etc., which could reveal underlying relationships and behavior of the data set. Homology of a space can be described by its {\\em Betti numbers}. The $k$-th Betti number \\(\\beta_k\\) of a simplicial complex is the rank of its $k$-th homology group. For $k=0,1,2$, the $\\beta_k$ have intuitive interpretation. \\(\\beta_{0}\\) represents a number of connected components, \\(\\beta_{1}\\) the number of holes, and \\(\\beta_{2}\\) the number of enclosed voids. For example, a sphere has $\\beta_0=1, \\beta_1=0, \\beta_2=1$, as it has one component, no holes, and one enclosed void.\n\nConsider the formation of a simplicial complex using balls of diameter $\\epsilon$ centered on points in a set. For small \\(\\epsilon\\), the simplicial complex is just a set of disjoint vertices. For sufficiently large \\(\\epsilon\\), the simplicial complex becomes one big cluster. What value of \\(\\epsilon\\) reveals the ``correct'' structure? Persistent homology \\cite{EdLeZo2002,Gh2008barcodes} gives a rigorous response to this question. By increasing \\(\\epsilon\\), a sequence of nested simplicial complexes called a filtration is created, which is examined for attributes of connectivity and their robustness. Topological features appear and disappear as \\(\\epsilon\\) increases. The features which exist over a longer range of the parameter $\\epsilon$ are considered as signal, and short-lived features as noise \\cite{EdLeZo2002,zomo_carlsson_2005}. This formulation allows a visualization as a collection of barcodes (one in each dimension), with each feature represented by a bar. The longer the life span of a feature, the longer its bar. In the example barcodes in Figs.~\\ref{breast_evolve}--\\ref{aml170_dim25}, the x-axis represents the \\(\\epsilon\\) parameter, and the bars of persistent loops of interest are circled.\n\n\\section*{Research questions}\n\nOur approach could address several critical questions in the context of cancer data analysis. First, could we select a small subset of relevant genes while {\\em simultaneously} identifying robust nontrivial structure, i.e., topology, of the data? Most previous approaches require the selection of a subset of genes {\\em before} exploring the resulting structure, and hence limiting the generality. Second, could we elucidate higher order interactions (than clusters) between genes that could have potential implications for cancer biogenesis? Higher order structures such as loops could reveal critical subsets of genes with relevant nontrivial interactions, which together have implications to the cancer. Third, could this method work even when data is available from only a {\\em subset} of patients?\n\n\\section{Data}\\label{sl:data}\nWe analyzed five publicly available microarray datasets of gene expression from four different types of cancer -- breast, ovarian, brain, and acute myeloid leukemia (AML). Four of the datasets have the same protocol, GPL570 (HG\\_U133\\_Plus\\_2, Affymetrix Human Genome U133 Plus 2.0 Array). The fifth dataset has a different protocol, HG\\_U95Av2, which has a fewer number of genes (see Table~\\ref{sl:tbl_data}). By including data sets from different protocols, we could verify that the topological features identified are not just artifacts of a particular protocol. \n\n\\begin{wraptable}{r}{3.8in}\n\\tbl{Datasets used in the study.}\n{\n\\begin{tabular}{@{}lllcc@{}}\n\\toprule\nDataset & Series & Protocol & \\# Genes & \\# Samples \\\\ \\colrule\nBrain & GSE36245 & GPL570 & 46201 & 46\\\\\nBreast & GSE3744 & GPL570 & 54613 & 47\\\\ \nOvarian & GSE51373 & GPL570 & 54613 & 28\\\\\nAML188 & GSE10358 & GPL570 & 54613 & 188\\\\\nAML170 & willm-00119 & HG\\_U95Av2 & 12558 & 170\\\\ \\botrule\n\\end{tabular}\n} \\label{sl:tbl_data}\n\\end{wraptable}\nThe number of genes represents the number of unique gene id tags defined by a protocol excluding controls. While the brain dataset is of the same protocol as the breast and ovarian datasets, the former one has fewer genes -- 46201 vs.~54613. This variability, however, did not affect our procedure to find topological features.\n\nAll datasets, except for AML170, were obtained from NCBI Gene Expression Omnibus \\url{http:\/\/www.ncbi.nlm.nih.gov\/geo\/} in October 2013. AML170 was retrieved at the same time from the National Cancer Institute caArray Database at \\url{https:\/\/array.nci.nih.gov\/caarray\/project\/willm-00119}. \n\n\\section{Methods}\\label{sl:methods}\nWe work with the raw gene expression values. In particular, we do not log-transform them.\n\\subsection{Dual space of data}\nTraditionally, gene expression data is viewed in its gene space, i.e., the expression profile of patient $i$ with $m$ genes is a point in $\\R^m$, $\\vx_i= [ x_{i_1}~x_{i_2}~\\cdots~x_{i_m} ]$. Each \\(x_{i_j}\\) is an expression of gene $j$ in the patient $i$. For example, each patient in the Brain dataset is a point in $\\R^{46201}$.\n\nWe analyze expression data in its dual space, i.e., in its {\\it sample space}. Hence a gene $j$ is represented as a point in $\\R^n$, $\\vx_j = [ x_{j_1}~x_{j_2}~\\cdots~x_{j_n} ]$, where $n$ is the number of samples or patients. Each \\(x_{j_i}\\) is the expression of the gene $j$ in the patient $i$. For example, in the same Brain dataset, the points now sit in $\\R^{46}$ space. Hence we study gene expression across the span of all patients.\n\nThe key motivation for this approach is to handle the high dimensionality in a meaningful way for analyzing the {\\em shape} of data. Given the small size of patients, one could efficiently construct a Vietoris-Rips complex using the set of pairwise distances between patients in the gene space (no need to choose landmarks). But such distances become less discriminatory when the number of genes is large \\cite{meaningiful_nneighbor_beyer1999,concentration_ledoux2005}. Working in the dual space, we let our topological method select a manageable number of genes as landmarks to construct the witness complexes, which potentially capture interesting topology of the data. Hence we do not preselect a small number of genes before the topological analysis, as is done by some previous studies \\cite{nicolau2011,seeman2012}.\n\n\\subsection{Choosing the number of landmarks}\nFor construction of the witness complexes, the number of landmarks has to be defined {\\it a priori} (Sec.~\\ref{sl:simplicial_complexes}). Hence the question becomes how many landmarks do we select? We let the data itself guide the selection of genes used as landmarks. If there is a significant loop \n\\begin{wraptable}{r}{2in}\n\\vspace*{-0.1in}\n\\tbl{Numbers of landmarks selected in each dataset.}\n{\\begin{tabular}{@{}lc@{}}\\toprule\nDataset & \\# landmarks \\\\ \\colrule\nBrain & 120\\\\\nBreast & 110\\\\ \nOvarian & 200\\\\\nAML188 & 150\\\\\nAML170 & 130\\\\ \\botrule\n\\end{tabular}}\\label{sl:tbl_num_landmarks}\n\\end{wraptable}\nfeature in the data, it would persist through a range of landmarks in $\\homo_1$ of the complexes. We reconstruct the topological space incrementing the number of landmarks\nwhile observing appearance and disappearance of the topological features. Initially, there would be very few small, noisy features because of insufficient number of points. As the number of landmarks increases, some features stabilize, i.e., do not change much either in size or membership. Then they reach their maximal size, and start to diminish once some critical number of landmarks is exceeded (when the ``holes'' are all filled in). The ``optimal'' number of landmarks is chosen when the length of the bar representing the topological feature is maximal.\n\nA typical example of such behavior is seen in the Breast dataset (see Fig.~\\ref{breast_evolve}). A small loop appears when the number of landmarks is \\(L=50\\). It stabilizes around \\(L=90\\), reaches its maximum span at \\(L=110\\), and then decreases as \\(L\\) grows.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\textwidth]{Breast_evolution2_no_title}\n\\caption{Evolution of the loop of interest (circled) for varying number of landmarks from L=50 to L=130 in the breast dataset. Here and in Figs.~\\ref{brain_loop}--\\ref{aml170_dim25}, the x-axis represents the \\(\\epsilon\\) parameter.}\\label{breast_evolve}\n\\end{figure}\n\n\\subsection{Composition of loops}\n\nOne of the goals of our method is to determine the genes which participate in $\\homo_1$ features, which could indicate potential implications for cancer biogenesis. Since the first landmark is chosen randomly in the sequential maxmin procedure, the composition of the loops identified may differ based on this first choice. To circumvent this effect, we do $20$ different runs in each case to collect possible variations in loop formation. Members of the loops are then pooled together for further analysis. Due to the almost deterministic nature of sequential maxmin selection (apart from the first landmark being selected randomly), we observed very little variation over the $20$ runs in most cases. The recovered members of loops are then queried in scientific literature for cancer-related reports.\n\n\\medskip\n\\noindent We implemented our computations using the package JavaPlex \\cite{javaplex2011}. We explored the barcodes for $\\homo_0, \\homo_1,$ and $\\homo_2$, but interesting persistent features with members related to cancer biogenesis were detected only for $\\homo_1$.\n\n\\section{Results}\\label{sl:results}\n\nPersistent topological features in the homology group \\(\\homo_1\\) were observed in every cancer dataset we analyzed. Representative examples are shown in Figs.~\\ref{brain_loop}--\\ref{aml170_2loops}. The AML datasets both had two persistent loops, while the other datasets have one loop each. The Ovarian dataset had a few medium length bars in the $\\homo_1$ barcode, but we investigated only the longest loop. Once the persistent loops were identified, they were inspected with respect to their composition and relation to cancer through search of scientific literature. Below is a brief description of results for each of the datasets (the full list of all loop members is also available \\cite{suppl}).\n\n\\begin{table}[h]\n\\tbl{Selected representatives of loops in different datasets.}\n{\\begin{tabular}{@{}lllc@{}}\\toprule\nGene & Dataset & Description & References \\\\ \\colrule\nCAV1 & Brain & tumor suppressor gene & \\cite{gene_cav1}\\\\\nRPL36 & Brain & prognostic marker in hepatocellular carcinoma & \\cite{rpl36_2011}\\\\ \nRPS11 & Breast & downregulation in breast carcinoma cells & \\cite{rps11_2001}\\\\\nFTL & Breast & prognostic biomarkers in breast cancer & \\cite{ftl_2006}\\\\\nLDHA & Ovarian & overexpressed in tumors, important for cell growth & \\cite{ldha_2013}\\\\\nGNAS & Ovarian & biomarker for survival in ovarian cancer & \\cite{gnas_2010}\\\\\nLAMP1 & AML170 & regulation of melanoma metastasis & \\cite{lamp1_2014}\\\\\nPABPC1 & AML170 & correlation with tumor progression & \\cite{pabpc1_2006}\\\\\nHLF & AML188 & promotes resistance to cell death & \\cite{hlf_2013}\\\\\nDTNA & AML188 & induces apoptosis in leukemia cells & \\cite{dtna_2012}\\\\ \\botrule\n\\end{tabular}}\\label{sl:tb_some_members}\n\\end{table}\n\n\\vspace*{-0.2in}\n\\subsection{Brain dataset}\\label{sl:brain}\nThe lifespan of the longest loop in this dataset was about 820 (bar between $\\approx[480,1300]$). The loop was consistent over different choices of the first landmark. We identified \n\\begin{wrapfigure}{r}{3in}\n\\centering\n\\includegraphics[height=1.1in]{Brain_one_loop_no_title1}\n\\caption{Representative loop in brain dataset.}\\label{brain_loop}\n\\end{wrapfigure}\n13 loop members, out of which 9 were found in cancer literature. Some cancer-related members include EGR1 and CAV1, which have genes been characterized as cancer suppressor genes\\cite{gene_cav1, gene_egr1}, A2M, which has been identified as a predictor for bone metastases\\cite{a2m_2001}, and RPL36 which has been found to be a prognostic marker in hepatocellular carcinoma\\cite{rpl36_2011}.\n\n\n\\subsection{Breast dataset}\\label{sl:breast}\n\\begin{wrapfigure}{r}{3in}\n\\centering\n\\includegraphics[width=3in]{Breast_one_loop_no_title}\n\\caption{Representative loop in breast dataset.}\\label{breast_loop}\n\\end{wrapfigure}\nThe lifespan of the longest loop in this dataset is in the range $[10080.0, 16684.2]$. As with the brain dataset, this loop is very consistent. However, there were only 10 members of this loop, and 8 of which were found in cancer literature. An interesting feature of this loop is that it had five ribosomal proteins which are known to play a critical role in tightly coordinating p53 signaling with ribosomal biogenesis \\cite{ribo_prots_2011}.\n\n\\subsection{Ovarian dataset}\\label{sl:ovarian}\n\\begin{wrapfigure}{r}{3in}\n\\centering\n\\includegraphics[width=3in]{Ovarian_one_loop_no_title}\n\\caption{Representative loop in ovarian dataset.}\\label{ovarian_loop}\n\\end{wrapfigure}\nThe Ovarian dataset had the most variable features in \\(\\homo_1\\). However, we investigated the loop corresponding to the most consistent and longest bar, which ranged from about 4000 to over 7000. This loop consisted of 17 members, and 9 were mentioned in the cancer-related literature. Among cancer-related members were GNAS, which was identified as ``an independent, qualitative, and reproducible biomarker to predict progression-free survival in epithelial ovarian cancer'' \\cite{gnas_2010}, and HNRNPA1, which has been identified as a potential biomarker for colorectal cancer \\cite{hnrpra1_2009}. \n\n\\subsection{AML188 dataset}\\label{sl:aml188}\n\\begin{wrapfigure}{r}{3in}\n\\centering\n\\vspace*{-0.1in}\n\\includegraphics[width=3in]{AML188-representative-loops_no_title}\n\\caption{Representative loops in AML188 dataset.}\\label{aml188_2loops}\n\\end{wrapfigure}\nAcute myeloid leukemia 188 (AML188) had two significant loops (as did AML170). The first one occurred at $[25200.0, 102200.0]$ and the second one at $[78400.0, 146219.24]$. The first loop has 27 members while the second one only 6. Altogether, only 14 of these 33 genes were mentioned in cancer literature. Some cancer-related representatives were hepatic leukemia factor (HLF) which promotes resistance to cell death \\cite{hlf_2013}, RPL35A known for inhibition of cell death \\cite{rpl35a_2002}, and GRK5 which regulates tumor growth \\cite{grk5_2012}. A group of zinc finger proteins were present in the first loop, some of which have been reported as novel biomarkers for detection of squamous cell carcinoma \\cite{zink_finger_2011,zink_finger_1991}.\n\n\\subsection{AML170 dataset}\\label{sl:aml170}\n\\begin{wrapfigure}{r}{3in}\n\\centering\n\\vspace*{-0.1in}\n\\includegraphics[width=3in]{AML170-representative-loops_no_title}\n\\caption{Representative loops in AML170 dataset.}\\label{aml170_2loops}\n\\end{wrapfigure}\nAML170 comes from caArray database and its protocol HG\\_U95Av2 has only 12558 genes. Even though this protocol had a smaller (about $1\/4$) number of genes compared to the other data sets, we still detected two loops in this dataset. They were relatively shorter than the loops in AML188, and occurred at $[5800.0, 11400.0]$ and $[11000.0, 17600.0]$. The two loops comprised of 19 members, of which 10 were found in cancer-related literature. These relevant members included ubiquitin C (UBC), which was recently identified as a novel cancer-related protein \\cite{ubc_2014}, PABPC1 whose positive expression is correlated with tumor progression in esophageal cancer \\cite{pabpc1_2006}, and LAMP1 which facilitates lung metastasis \\cite{lamp1_2014}.\n\n\\section{Discussion}\\label{sl:discuss}\n\n\\begin{wrapfigure}{r}{3in}\n\\centering\n\\includegraphics[width=3in]{AML170_Dim25_L130_no_tilte}\n\\caption{Two persistent loops in the AML170 dataset detected using only $25$ dimensions at the number of landmarks L=130.} \\label{aml170_dim25}\n\\end{wrapfigure}\nThe Breast, Brain, and Ovarian datasets had only one persistent loop, while AML170 and AML188 had two. Also, the AML datasets had a higher number of patients (samples) than the other three sets (see Table~\\ref{sl:tbl_data}). Is this fact just a coincidence or, indeed, does the number of \\(\\homo_1\\) features (loops) correlate with the number of dimensions? To address this question, we chose samples of random  and progressively larger ($25$--$175$) subsets of patients from AML170 and AML188 while also increasing the number of landmarks, and studied the evolution of \\(\\homo_1\\) features. In other words, we repeated our method on smaller subsets of patients from these datasets. Both the AML datasets contained two loops even with only 25 dimensions (see Fig.~\\ref{aml170_dim25} for AML170), and continued to do so for the progressively larger subsets. Thus, the number of significant \\(\\homo_1\\) features appears to depend on intrinsic qualities of the data rather than the number of dimensions, demonstrating the robustness of our method to the number of patients in the dataset.\n\nAn important property of a loop is its lifespan \\cite{carlsson2006algebraic}. One may note that the life span of loops for different datasets vary significantly. For example, the lifespan of a significant \\(\\homo_1\\) feature in the brain dataset is only $820$, while for AML188 the lifespan of the first loop is $77,000$. This difference is not only due to the increase in the actual size of a loop as indicated by the number of points comprising the loop ($13$ vs.~$27$ in this case), but also in part because of the different absolute values in microarray expression data. The maximum  value for the brain dataset is \\(24 \\times 10^3\\), while for AML188 is \\(3 \\times 10^6\\). Therefore, the absolute length of an \\(\\homo_1\\) feature is not as important as its length relative to other \\(\\homo_1\\) features.\n\nThe crucial step of our method is the choice of landmarks. The goal here is the efficient inference of the topology of data, while selecting a small subset of potentially relevant genes. Landmarks chosen using sequential maxmin tend to cover the dataset better and are spread apart from each other, but the procedure is also known to pick outliers \\cite{desilva2004}. In our datasets, outliers are typically identified by extreme expression values \\cite{outlier_survey2004}. We examined the expression values of the chosen landmarks, and found that very few of them had extreme values (Figs.~\\ref{aml170_loop_hist} and \\ref{breast_loop_hist}). Similarly, the expressions of the genes implicated in cancer biogenesis (among the loop members) did not have any extreme values, and in fact appear to follow normal distributions. We infer that this observation results because sequential maxmin indeed picks points on the outskirts of the topological features. Further, the distribution of expressions of the loop members suggests that the group as a whole could have potential implications for the disease. More interestingly, the ``hole'' structure means such groups could not potentially be identified by traditional coexpression or even differential expression analyses \\cite{ChKe2009}. \n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=0.33]{hist_aml170_tripple}\n\\caption{Histograms for AML170 dataset, x-axis represents gene expression level of scale $10^4$. (a) distribution of gene expressions for the whole set; (b) distribution of gene expressions for only the loop members; and (c) distribution of gene expressions for cancer-related loop members.} \\label{aml170_loop_hist} \n\\end{figure}\n\\vspace*{-0.2in}\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[scale=0.33]{hist_breast_tripple}\n\\caption{\\label{breast_loop_hist} Histograms for Breast dataset, x-axis represents gene expression level of scale $10^5$. (a) distribution of gene expressions for the whole set, (b) distribution of gene expressions for the loop members only, (c) distribution of gene expressions for cancer-related loop members.}\n\\end{figure}\n\nThe computational complexity of our method is based on the current implementation of JavaPlex \\cite{javaplex2011}, where the two main steps are building and filtering simplicial complexes, and computing homology. Up to dimension $2$ (\\(\\beta_2\\)), homology could be computed in $O(n^3)$ time \\cite{EdHa2009}. However, building simplicial complexes relies on clique enumeration, which is NP-complete, and has a complexity of $O(3^{n\/3}) $\\cite{jholes2014,cliques65}. Further, JavaPlex requires explicit enumeration of simplicial facets appearing at each filtration step, implying the need for large memory resources\\cite{javaplex2011}.\n\nThe cancer-related loop members identified from the two AML datasets were distinct apart from the prominent group of ribosomal proteins. This observation could be explained by two main reasons. First, AML188 has four times the number of genes as AML170 (see Table~\\ref{sl:tbl_data}). Second, JavaPlex identifies only {\\em one} representative of each homology class. That is, if a significant topological feature (hole) exists, it will be identified, but only one loop will be found around that hole. There could be other relevant points in proximity to the hole, but are not guaranteed to be included in the loop. If we have prior knowledge of some genes being relevant, we could try to identify loops around the holes that include these genes as members. In this case, the other members of the identified loops could also have potential implications for the cancer biogenesis. Methods to find a member of a homology class that includes specific points could be of independent interest in the context of optimal homology problems \\cite{DeHiKr2011,DeSuWa2010}.\n\n\\section{Conclusion}\nWe have presented a method to look at cancer data from a different angle. Unlike previous methods, we look at characteristics of the first homology group ($\\homo_1$). We identify the persistent $\\homo_1$ features (which are loops, rather than connected components) and inspect their membership. Importantly, our approach finds potentially interesting connections among genes which cannot be found otherwise using traditional methods. This geometric connectedness may imply functional connectedness, however, this is yet to be investigated by oncologists. If such connections are indeed implied, then the genes in the loops could together form a characteristic ``signature'' for the cancer in question.\n\n\\paragraph{Acknowledgment:} \nKrishnamoorthy acknowledges support from the National Science Foundation through grant \\#1064600.\n\n\\vspace*{-0.1in}\n\\bibliographystyle{ws-procs11x85}\n\\input{BibLockwood_TopoFtrsCncrData}\n\n\n\\end{document}\n\n\n\n\\section{Using Other Packages}\\label{aba:sec1}\nThe class file loads the packages {\\tt amsfonts, amsmath, amssymb,\nchapterbib, cite, dcolumn, epsfig, rotating} and {\\tt url} at\nstartup. Please try to limit your use of additional packages as they\noften introduce incompatibilities. This problem is not specific to\nthe WSPC styles; it is a general \\LaTeX{} problem. 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+{"text":"\\section{Introduction and background}\n\\setcounter{equation}{0}\n\\setcounter{theo}{0}\nFor a fixed $ 0<\\alpha \\leq 2$ {\\it symmetric $\\alpha$-stable L\\'{e}vy motion} $\\{L_\\alpha (t), t\\geq 0\\}$   is a stochastic process characterized by having stationary independent increments with $L(0) = 0$ almost surely, and \n$L_\\alpha (t) - L_\\alpha (s)\\ (s>t)$ having the distribution of $S_\\alpha((t-s)^{1\/\\alpha},0,0)$,\nwhere $S_\\alpha(c, \\beta,\\mu)$ denotes a stable random variable with stability-index $\\alpha$, with  scale parameter $c$, skewness parameter $\\beta$ and shift $\\mu$.  A detailed account of such processes may be found in \\cite{Bk_Sam} but  we summarize here the  features we need.\nStable motion $L_\\alpha $  is $1\/\\alpha$-self-similar in the sense that $L_\\alpha (c t)$ and $c^{1\/\\alpha}L_\\alpha (t)$ are equal in distribution so in particular have the same finite-dimensional distributions. There is a version of  $L_\\alpha $ such that its sample paths are c\\`{a}dl\\`{a}g, that is right continuous with left limits. \n\nOne way of representing  symmetric $\\alpha$-stable L\\'{e}vy motion $L_\\alpha$ is as a sum over a plane point process.\n Throughout the paper we write\n$$\nr^{\\langle s\\rangle} = {\\rm sign}(r)|r|^s \\mbox{ for }r\\in \\mathbb{R}, s\\in \\mathbb{R}.\n$$\nThen\n\\begin{equation}\\label{sum}\nL_\\alpha  (t) = C_\\alpha \n\\sum_{(\\X,\\Y) \\in \\Pi} 1_{(0,t]}(\\X) \\Y^{\\langle-1\/\\alpha\\rangle},\n\\end{equation}\nwhere $C_\\alpha$ is a normalising constant given by\n$$\nC_\\alpha = \\Big(\\int_0^\\infty u^{-\\alpha} \\sin u\\, du \\Big)^{-1\/\\alpha}\n$$\nand where $\\Pi$  is a Poisson point process on $\\mathbb{R}^+ \\times\\mathbb{R}$  with plane Lebesgue measure $\\mathcal{L}^2$ as mean measure, so that for a Borel set $A \\subset \\mathbb{R}^+ \\times\\mathbb{R}$ the number of points of $\\Pi$ in $A$ has a Poisson distribution with  parameter $\\mathcal{L}^2(A)$, independently for disjoint $A$. \nThe sum \\eqref{sum} is almost surely absolutely uniformly convergent if $0<\\alpha<1$, but if $\\alpha\\geq 1$ then \\eqref{sum}  must be taken as the limit as $n\\to \\infty$ of symmetric partial sums\n$$\nL_{\\alpha ,n} (t) = C_\\alpha \n\\sum_{(\\X,\\Y) \\in \\Pi : |\\Y|\\leq n} 1_{(0,t]}(\\X) \\Y^{\\langle-1\/\\alpha\\rangle},\n$$\nin the sense that $\\|L_{\\alpha ,n}-L_\\alpha\\|_{\\infty}\\to 0$ almost surely.\n\nSeveral variants of $\\alpha$-stable motion have been considered. \nFor example, for {\\it multistable L\\'{e}vy motion} $\\{M_{\\alpha} (t), t\\geq 0\\}$ the stability index $\\alpha$ in  \\eqref{sum} can depend on  $\\X$ so that the local behaviour changes with $t$, see \\cite{FLL,FL,KFLL,XFL,LLA,LLL,LL}.  Thus given a continuous $\\alpha: \\mathbb{R}^+ \\to (0,2)$, \n$$\nM_{\\alpha}(t) =  \n\\sum_{(\\X,\\Y) \\in \\Pi} 1_{(0,t]}(\\X) C_{\\alpha(\\X)} \\Y^{\\langle-1\/\\alpha(\\X)\\rangle}.\n$$\nThen $M_{\\alpha}$ is a Markov process. Under certain conditions it is {\\it localisable}  with {\\it local form} $L_{\\alpha(t)}$, in the sense that near $t$ the process `looks like' an $\\alpha(t)$-stable process, that is for each $t>0$ and $u \\in \\mathbb{R}$,\n$$\\frac{M_{\\alpha}(t+ru) -M_{\\alpha}(t)}{r^{1\/\\alpha(t)}} \\tod L_{\\alpha(t)}(u)$$\nas $r\\searrow 0$, where convergence is in distribution with respect to the Skorohod metric and consequently  is convergent in finite dimensional distributions, see \\cite{FLL,FL}. \n\nThe local stability parameter of  multistable L\\'{e}vy motion depends on the time $t$ but in some contexts, for example in financial modelling,  it may be  appropriate for the local stability parameter to depend instead (or even as well) on the {\\it value} of the process at time $t$. Such a process might be called `self-stabilizing'. Thus, for suitable \n$\\alpha: \\mathbb{R} \\to (0,2)$, we seek a process $\\{Z (t), t\\geq 0\\}$ that is localisable with  local form $L^0_{\\alpha(Z (t))}$, in the sense that for each $t$ and $u>0$,\n\\begin{equation}\\label{locdef1}\n\\frac{Z(t+ru) -Z(t)}{r^{1\/\\alpha(Z (t))}}\\bigg|\\,  \\mathcal{F}_t \\  \\tod \\ L^0_{\\alpha(Z (t))}(u)\\end{equation}\nas $r\\searrow 0$,\nwhere convergence is in distribution and finite dimensional distributions and where $\\mathcal{F}_t$ indicates conditioning on the process up to time $t$. (For notational simplicity it is easier to construct  $Z_{\\alpha}$ with the non-normalised $\\alpha$-stable processes $L^0_{\\alpha} = C_\\alpha^{-1} L_{\\alpha}$ as its local form.)\n\nThroughout the paper we write $D[t_0,t_1)$ for the c\\`{a}dl\\`{a}g functions on the interval $[t_0,t_1)$, that is functions that are right continuous with left limits; this is the natural space for functions defined by sums over point sets. \n\nIn an earlier paper \\cite{FL2} we constructed self-stabilizing processes for  $\\alpha: \\mathbb{R}^+ \\to (0,1)$  by first showing that there exists a deterministic function $f\\in D[t_0,t_1)$ satisfying the relation\n$$f(t) = a_0 +\\sum_{(x,y) \\in \\Pi} 1_{(t_0,t]}(x) y^{\\langle-1\/\\alpha(f(x_-))\\rangle}$$\nfor a {\\it fixed} point set $\\Pi$, and then randomising to get a random function $Z$ such that\n\\begin{equation*}\nZ  (t) =  a_0 + \\sum_{(\\X,\\Y) \\in \\Pi} 1_{(t_0,t]}(\\X)\\, \\Y^{\\langle-1\/\\alpha(Z  (\\X_-))\\rangle} \\qquad (t_0\\leq t <t_1).\n\\end{equation*}\nThen, for all $t\\in [t_0,t_1)$ this random function satisfies \\eqref{locdef1} almost surely. However, this approach depends on the infinite sums being absolutely convergent, which need not be the case if $\\alpha(t) \\geq1$ for some $t$.\n\nHere we use an alternative approach to construct self-stabilizing processes where $\\alpha: \\mathbb{R}^+ \\to (0,2)$ and in general we cannot assume absolute convergence of the sums. We  show in Section \\ref{sec:signs}  that for a fixed point set $\\Pi^+\\subset (t_0,t_1)\\times \\mathbb{R}^+$ and independent random `signs' $S(x,y)=\\pm1$ there exists almost surely a random function $Z\\in D[t_0,t_1)$ satisfying\n$$ Z(t) = a_0 +\\sum_{(x,y) \\in \\Pi^+} 1_{(t_0,t]}(x) S(x,y)y^{-1\/\\alpha (Z(x_-))}\\qquad (t_0 \\leq t <t_1), $$\nsee Theorem \\ref{finlim2}. To achieve this we work with partial sums \n$$ Z_n(t) = a_0 +\\sum_{(x,y) \\in \\Pi^+:|y|\\leq n} 1_{(t_0,t]}(x) S(x,y)y^{-1\/\\alpha (Z(x_-))}\\qquad (t_0 \\leq t <t_1) $$\nand show that the limit as $n\\to \\infty$ exists in a norm given by $\\mathbb{E}\\big(\\|\\cdot\\|_\\infty^2\\big)^{1\/2}$, where $\\mathbb{E}$ denotes expectation. This is more awkward than it might seem at first sight since,  as $n$ increases, if a new  point $(x,y)\\in \\Pi^+$  enters the sum, then $Z_n(t)$ will change for $t\\geq x$ so for all  $(x',y')$ with $x'>x$ and  $y'<y$    the summands ${y'}^{\\langle-1\/\\alpha(Z_n({x'}_{-}))\\rangle}$ will change, with a knock on effect so that the change in $Z_n(t)$ may be considerably amplified as $t$ increases past further $x$ with $(x,y)\\in \\Pi^+$ and $y< n$.\n\nIn Section \\ref{sec:rand} we randomise the construction further by taking $\\Pi^+$ to be a Poisson point process on $ (t_0,t_1)\\times \\mathbb{R}^+$ with  mean measure $2{\\mathcal L}^2$ which, combined with the random signs, gives a point process with the same distribution as $\\Pi$ on $ (t_0,t_1)\\times \\mathbb{R}$.  We show that the resulting process $Z$ satisfies a H\\\"{o}lder continuity property and is self-stabilizing in the sense of \\eqref{locdef1}, see Theorem \\ref{rtloc}.\n\n\\subsection{Basic facts used throughout the paper}\\label{sec1.1}\nFor the rest of the paper we fix $a_0 \\in \\mathbb{R}$ and  $0<a <b<2 $ together with a function $\\alpha :\\mathbb{R} \\to [a,b]$ that is continuously differentiable with bounded derivative.\nBy the mean value theorem,\n$$ y^{-1\/\\alpha(v)} - y^{-1\/\\alpha(u)}\n\\ =\\ (v- u)\ny^{-1\/\\alpha(\\xi)}\\log y \\frac{\\alpha'(\\xi)}{\\alpha(\\xi)^2}\\qquad (y>0,\\ u,v \\in \\mathbb{R} ),$$\nwhere $\\xi \\in (u,v)$. In particular this gives the  estimate we will use frequently:\n\\be \n\\big| y^{-1\/\\alpha(v)} - y^{-1\/\\alpha(u)}\\big |\n\\  \\leq  \\ M\\  |v- u|\\, y^{-1\/(a,b)}\\qquad (y>0, \\ u,v \\in \\mathbb{R}), \\label{ydif4}\n \\ee\nwhere\n$$\nM\\  =\\  \\sup_{\\xi\\in\\mathbb{R}} \\frac{|\\alpha'(\\xi)|}{\\alpha(\\xi)^2},\n$$\nand for convenience we write\n$$\ny^{-1\/(a,b)} \\ = \\ \\max\\big\\{ y^{-1\/a}\\big(1+|\\log y| \\big), y^{-1\/b}\\big(1+|\\log y| \\big)\\big\\}\n\\qquad (y>0)\n$$\nand \n$$ y^{-2\/(a,b)} \\ =\\ (y^{-1\/(a,b)})^2.$$ \n\nFor $t_0< t_1$ and a suitable probability space $\\Omega$ (to be specified later), we will work with  functions \n$F: \\Omega\\times [t_0, t_1) \\to \\mathbb{R}\\cup \\{\\infty \\}$ which we assume to be measurable (taking  Lebesgue measure on  $[t_0, t_1)$).  Writing  $F_\\omega(t)$ for the value of $F$ at $\\omega \\in \\Omega$ and  $t\\in [t_0, t_1)$, we think of   $F_\\omega$ as a random function on  $[t_0, t_1)$ in the natural way (most of the time we will write $F$ instead of $F_\\omega$ when the underlying randomness is clear). In particular we will work in the space \n$${\\mathcal D} \\ = \\ \\big\\{F: F_\\omega \\in D[t_0, t_1) \\mbox{ for almost all }\n \\omega \\in \\Omega \\mbox{ with } \\mathbb{E}\\big(\\|F\\|_\\infty^2\\big)<\\infty\\big\\},$$\n where $\\mathbb{E}$ is expectation, $D[t_0,t_1)$ denotes the c\\`{a}dl\\`{a}g functions, and $\\|\\cdot\\|_\\infty$ is the usual supremum norm. By identifying $F$ and $F'$ if $F_\\omega = F'_\\omega$ for almost all $\\omega \\in \\Omega$, this becomes a normed space under the norm\n\\be\\label{norm}\n\\Big(\\mathbb{E}\\big(\\|F\\|_\\infty^2\\big)\\Big)^{1\/2}.\n\\ee\nA routine check shows that \\eqref{norm} defines a complete norm on ${\\mathcal D}$. \n\n\n\n\\section{Point sums with random signs}\\label{sec:signs}\n\\setcounter{equation}{0}\n\\setcounter{theo}{0}\n\nIn this section we fix a discrete point set $\\Pi^+ \\subset (t_0,t_1)\\times \\mathbb{R^+}$ \nand form sums over values at the points of $\\Pi^+$ with an independent random assignment of sign $+$ or $-$ at each point of $\\Pi^+$.\n\nWe will assume that the point set $\\Pi^+$ satisfies \n$$\n \\sum_{(x,y) \\in \\Pi^+}y^{-2\/(a,b)}< \\infty;\n$$\nthis will certainly be the case if $ \\sum_{(x,y) \\in \\Pi^+}y^{-2\/b'}< \\infty$ for some $b'$ with  $b<b'<2$.\n\n\n\nOur first aim is to show that if $\\{S(x,y) \\in \\{-1,1\\}: (x,y) \\in \\Pi^+\\}$ are random `signs', that is independent random variables taking the values $1$ and $-1$ with equal probability $\\frac{1}{2}$, then, almost surely, there exists  a random function $Z \\in D[t_0, t_1)$ satisfying\n\\begin{equation}\\label{identsigns}\nZ(t) = a_0 +\\sum_{(x,y) \\in \\Pi^+} 1_{(t_0,t]}(x) S(x,y)y^{-1\/\\alpha (Z(x_-))}\\qquad (t_0 \\leq t <t_1). \n\\end{equation}\nin an appropriate sense. This $Z$ will be the  limit in norm of the random functions obtained by restricting the sums to $y \\leq n$. Thus we define for $n\\in \\mathbb{N}$\n\\begin{equation}\\label{identsignsfin}\nZ_n(t) = a_0 +\\sum_{(x,y) \\in \\Pi^+:\\, y\\leq n} 1_{(t_0,t]}(x) S(x,y)y^{-1\/\\alpha (Z_n(x_-))}\\qquad (t_0 \\leq t <t_1).\n\\end{equation}\nHere, and throughout this section, our probability space has  $\\Omega$ as the set $ \\{-1,1\\}^{ \\Pi^+}\\equiv \\{S(x,y) \\in \\{-1,1\\}: (x,y) \\in \\Pi^+\\}$ of all assignments of signs $\\pm1$ to the points of $ \\Pi^+$, and  $\\sigma$-field  generated by the subsets of the form $S \\times \\{-1,1\\}^{ \\Pi^+\\setminus X}$ for each finite $X\\subset  \\Pi^+$ and each $S\\subset \\{-1,1\\}^X$. In particular, the probability that a given set of $k$ points of $ \\Pi^+$ having any particular assignment of signs is $2^{-k}$.\n\nNote that the sum in \\eqref{identsignsfin} is over the finite set $\\{(x,y)\\in \\Pi^+: y\\leq n\\}$ so, given the $S(x,y)$, the piecewise constant $Z_n(t)\\in D[t_0,t_1)$ can be evaluated inductively over increasing $x$ with $(x,y)\\in \\{\\Pi^+: y\\leq n\\}$  in a finite number of steps. Nevertheless, as has been remarked, we need to be careful about taking limits of $\\{Z_n\\}_n$ as $n \\to \\infty$ since on increasing $n$ the contributions from the summands with $0<y<n$  change as the values of $Z_n(x_-)$ change.   Note that  if $0<a<b<1$ the sum  \\eqref{identsigns} is absolutely convergent, and this case is considered in \\cite{FL2}, but if $0<a<b<2$ then $\\alpha(z)$ may take values greater than 1 and care is needed in defining $Z$.  We will show that there exists $Z\\in {\\mathcal D}$ such that $\\mathbb{E}\\big(\\|Z_n-Z\\|_\\infty^2\\big) \\to 0$  and $Z$ satisfies \\eqref{identsigns} in an appropriate sense. In particular  there is a sequence of integers $n_j\\nearrow\\infty$ such that almost surely $ \\|Z_{n_j}- Z\\|_\\infty \\to 0$, i.e. $Z_{n_j}$ converges uniformly to $Z$.\n\n\n\\subsection{Existence of functions defined by random signs}\\label{sec2.1}\n\n To obtain such a $Z$ we show that $\\{Z_n\\}_n$ is a Cauchy sequence in the complete norm \\eqref{norm}.\n$\\mathbb{E}\\big(\\|\\cdot\\|_\\infty^2\\big)^{1\/2}$ on ${\\mathcal D}$.\n\nIt is convenient to make a further assumption on $\\Pi^+$, that if $(x,y), (x',y') \\in\\Pi^+$ then $x\\neq x'$. Without this assumption the results remain valid with an essentially identical proof, but the notation becomes more cumbersome as the single terms added in \\eqref{mart} have to be replaced by sums over several terms corresponding to each point $(x_i,y_i)$ with a common value of $x_i$. In any case, when in Section 3 we let $\\Pi^+$ be a realisation of a Poisson point process, this assumption will hold almost surely.\n\n\\begin{prop}\\label{finlim1A}\nLet  $a_0, \\alpha $ and $\\Pi^+$ be as above  and  let  $Z_n$ be given by  \\eqref{identsignsfin}.  Then for $m\\geq n\\geq 1$,\n\\begin{equation}\n\\mathbb{E}\\big(\\|Z_m -Z_n\\|_\\infty^2\\big) \n\\  \\leq  \\ 4 \\prod_{(x,y)\\in \\Pi^+ : 0<y\\leq n} (1+M^2y^{-2\/(a,b)}) \\sum_{(x,y)\\in \\Pi ^+ : n<y\\leq m} y^{-2\/b}. \\label{cauchy0}\n \\end{equation}\nIn particular $\\{Z_n\\}_n$ is a Cauchy sequence in ${\\mathcal D}$ under the norm $\\mathbb{E}(\\|\\cdot\\|^2_\\infty)^{1\/2}$.\n\\end{prop}\n\n\\begin{proof}\nLet $m>n$. We list the points\n$$\\{(x,y) \\in \\Pi^+ : y\\leq m\\} = \\{(x_1,y_1), \\ldots, (x_N,y_N)\\},$$\nwith $t_0<x_1 < \\cdots < x_N <t_1$, where as mentioned we assume  that the $x_i$ are distinct. \nFor notational convenience we set $x_0 := t_0$ and $x_{N+1} := t_1$. \nWe write $i_1<i_2< \\cdots<i_K$ for the indices such that $y_{i_k} \\leq n$, and let $i_0 := 0$ and $i_{K+1} := N+1$\nWith this notation,  \\eqref{identsignsfin} restricts to the jump points $x_i$ as\n\\begin{equation}\\label{finalt}\nZ_m(x_i) = a_0 +\\sum_{0<j\\leq i} S(x_j,y_j)y_j^{-1\/\\alpha (Z_m(x_{j-1}))},\\quad \nZ_n(x_i) = a_0 +\\sum_{k: 0<i_k\\leq i} S(x_{i_k},y_{i_k})y_{i_k}^{-1\/\\alpha (Z_n(x_{{i_k}-1}))}. \n\\end{equation}\n\nWrite\n\\begin{equation}\\label{cki}\nc_k = M^2\\, y_{i_k}^{-2\/(a,b)} \\qquad (1\\leq k \\leq K)\n\\end{equation}\nand\n\\begin{equation}\\label{epki}\n\\epsilon_k = \\sum_{ i_{k-1}+1\\leq i \\leq i_k -1}  y_i^{-2\/b} \\qquad (1\\leq k \\leq K+1).\n\\end{equation}\nLet $\\mathcal{F}_i$ be the minimal $\\sigma$-field of subsets of $\\Omega$ such that the sign assignments $\\{S(x_j,y_j):1\\leq j\\leq i\\}$ are Borel measurable; thus conditioning on $\\mathcal{F}_i$ is equivalent to taking the finite set of values $\\{S(x_j,y_j): 1\\leq j\\leq i\\}$ as known.\nWe consider $\\big\\{Z_n(x_i) - Z_m(x_i),\\mathcal{F}_i\\big\\}_{i=0}^N$ which from \\eqref{finalt} and that $\\mathbb{E}(S(x_j,y_j)) = 0$ is a bounded martingale. Indeed, from \\eqref{finalt}, for $1\\leq i\\leq N$,\n \\begin{align}\\label{mart}\nZ_m(x_{i}) - & Z_n(x_{i})\\ =\\   Z_m(x_{i-1}) - Z_n(x_{i-1}) \\\\\n&+ \\left\\{\n\\begin{array}{ll}\n S(x_{i},y_{i})\\,y_{i}^{-1\/\\alpha (Z_m(x_{i-1}))}  &  \\mbox{ if } i\\neq i_k   \\mbox{ for all } k   \\\\\nS(x_{i},y_{i})\\big(y_{i}^{-1\/\\alpha (Z_m(x_{i-1}))} -   y_{i}^{-1\/\\alpha (Z_n(x_{i-1}))}\\big) & \\mbox{ if } i= i_k      \\mbox{ for some } k  \\nonumber \n\\end{array}.\n\\right.\n\\end{align}\n\nWe will\nshow by induction on $i$ that for all $1\\leq k\\leq K+1$ and all $i_{k-1} \\leq i <i_k$,\n\\begin{eqnarray}\n\\mathbb{E}\\big((Z_m(x_i) - Z_n(x_i))^2\\big)\n&\\leq& (1+c_1)\\cdots(1+c_{k-1}) ( \\epsilon_1+ \\epsilon_2+\\cdots + \\epsilon_{k-1})\\nonumber \\\\\n&&  +\\,  y_{i_{k-1}+1}^{-2\/b}  + y_{i_{k-1}+2}^{-2\/b} + \\cdots +  y_{i}^{-2\/b}\\label{ind1}\\\\\n&\\leq& (1+c_1)\\cdots(1+c_{k-1}) ( \\epsilon_1+ \\epsilon_2+\\cdots + \\epsilon_{k-1})\n \\,+\\,\\epsilon_k.\\label{ind2}\n\\end{eqnarray}\nNote that \\eqref{ind2} follows immediately from \\eqref{ind1} using \\eqref{epki}. Inequality  \\eqref{ind1} is trivially  true when $i=0$. Let $ 0< i \\leq N$ and assume inductively that \\eqref{ind1} holds with $i$ replaced by $i-1$.  There are two cases.\n\n(a) If $i\\neq i_k$  for all $k $ then from \\eqref{mart}\n \\begin{align*}\n\\mathbb{E}\\big((Z_m (x_i) - & Z_n(x_i))^2\\big| \\mathcal{F}_{i-1} \\big)\\\\\n&=\\ \\mathbb{E}\\Big(\\Big(Z_m(x_{i-1}) - Z_n(x_{i-1}) +  S(x_{i},y_{i})\\,y_{i}^{-1\/\\alpha (Z_m(x_{i-1}))}\\Big)^2\\ \\Big| \\mathcal{F}_{i-1}\\Big)\\\\\n&=\\ \\big(Z_m(x_{i-1}) - Z_n(x_{i-1})\\big)^2  +  y_{i}^{-2\/\\alpha (Z_m(x_{i-1}))}\\\\\n&\\leq \\ \\big(Z_m(x_{i-1}) - Z_n(x_{i-1})\\big)^2  +  y_{i}^{-2\/b}\n\\end{align*}\nas $y_i\\neq y_{i_k}$ for all $k$  implies  $y_i>n\\geq 1$. Thus  taking the unconditional expectation  and using \\eqref{ind1} for $i-1$ gives \\eqref{ind1} for $i$.\n\n(b) If $i=  i_k $ for some $1\\leq k\\leq K$, then from \\eqref{mart},\n \\begin{align*}\n\\mathbb{E}\\big((Z_m & (x_i)- Z_n(x_i))^2\\big| \\mathcal{F}_{i-1} \\big)\\\\\n&=\\ \\mathbb{E}\\Big(\\Big(Z_m(x_{i-1}) - Z_n(x_{i-1}) +  S(x_{i},y_{i})\\big(y_{i}^{-1\/\\alpha (Z_m(x_{i-1}))} -   y_{i}^{-1\/\\alpha (Z_n(x_{i-1}))}\\big)\\Big)^2\\Big| \\mathcal{F}_{i-1}\\Big)\\\\\n&=\\ \\big(Z_m(x_{i-1}) - Z_n(x_{i-1})\\big)^2  + \\big(y_{i}^{-1\/\\alpha (Z_m(x_{i-1}))} -   y_{i}^{-1\/\\alpha (Z_n(x_{i-1}))}\\big)^2\\\\\n&\\leq \\ \\big(Z_m(x_{i-1}) - Z_n(x_{i-1})\\big)^2\\big(1 +  M^2 y_{i_k}^{-2\/(a,b)}\\big)\\\\\n&\\leq \\ \\big(Z_m(x_{i-1}) - Z_n(x_{i-1})\\big)^2\\big(1 +  c_k\\big),\n\\end{align*}\n using \\eqref{ydif4} and \\eqref{cki}.\nAgain, taking the unconditional expectation and using \\eqref{ind2} for $i-1$ gives \\eqref{ind1} for $i$ with a vacuous sum of terms $y_j^{-2\/b}$, completing the induction.\n\nIt follows from  \\eqref{ind2} that for all $0\\leq i \\leq N$,\n\\begin{eqnarray}\n\\mathbb{E}\\big((Z_m(x_i) - Z_n(x_i))^2\\big)& \\leq &  \\prod_{k=1}^K (1+c_k) \\sum_{k= 1}^{K+1} \\epsilon_k\\nonumber\\\\ \n &  \\leq  &  \\prod_{(x,y)\\in \\Pi ^+ : y\\leq n} (1+M^2y^{-2\/(a,b)}) \\sum_{(x,y)\\in \\Pi ^+ : n<y\\leq m} y^{-2\/b} \\label{cauchy4}\n\\end{eqnarray}\nusing \\eqref{cki} and \\eqref{epki}.\n\nNoting that $Z_m(t) -Z_n(t)$ is constant except at the jump points $x_i$, and applying Doob's maximal inequality  \\cite{Wil} to the martingale \n$\\big\\{Z_m(x_i) - Z_n(x_i),\\mathcal{F}_i\\big\\}_{i=0}^N$,\n  we obtain\n\\begin{eqnarray*}\n\\mathbb{E}\\big(\\sup_{t_0\\leq t < t_1} (Z_m(t) - Z_n(t))^2\\big)\n&=& \\mathbb{E}\\big(\\max_{0\\leq i \\leq N} (Z_n(x_i) - Z_m(x_i))^2\\big)\\\\ \n&\\leq& 4\\, \\mathbb{E} \\big((Z_m(x_N) - Z_n(x_N))^2\\big) \n \\end{eqnarray*}\n for all $m\\geq n$. \n Combining with  \\eqref{cauchy4} gives \\eqref{cauchy0}.\nSince \n$$\\prod_{(x,y)\\in \\Pi ^+ } (1+M^2y^{-2\/(a,b)})\\ \\leq\\ \\exp\\Big(M^2\\sum_{(x,y) \\in \\Pi ^+ } y^{-2\/(a,b)}\\Big)\\ < \\infty$$\nand $\\sum_{(x,y) \\in \\Pi ^+ } y^{-2\/(a,b)}$ and $\\sum_{(x,y)\\in \\Pi ^+ } y^{-2\/b}$ are convergent by assumption, $\\{Z_n\\}_n$ is a Cauchy sequence.\n\\end{proof}\n\nWe now deduce the existence of $Z$ as the norm limit of the $Z_n$. \n \n\\begin{theo}\\label{finlim2}\nLet  $a_0, \\alpha $ and $\\Pi^+$ be as above. Then there exists $Z \\in {\\mathcal D}$ satisfying  \n\\eqref{identsigns},   \n with $\\lim_{n\\to \\infty}\\mathbb{E}\\big(\\|Z_n -Z\\|_\\infty^2\\big)=0$ where $Z_n$ as in  \\eqref{identsignsfin};  more specifically\n\\be \\label{diffbound}\n\\mathbb{E}\\big(\\|Z_n -Z\\|_\\infty^2\\big)\\ \\leq\\  4\\prod_{(x,y)\\in \\Pi ^+ } (1+M^2y^{-2\/(a,b)})\\sum_{(x,y)\\in \\Pi ^+ : y> n} y^{-2\/b}\\ \\to 0. \n\\ee\nMoreover, there exists a sequence $n_j \\nearrow \\infty$ such that almost surely $\\|Z_{n_j} -Z\\|_\\infty \\to 0$ i.e. $Z_{n_j} \\to Z$ uniformly.  If $0<b<1$ then almost surely $\\|Z_{n} -Z\\|_\\infty \\to 0$. \n\\end{theo}\n\n\\begin{proof}\nSince $\\{Z_n\\}_n$ is Cauchy in the complete norm $\\mathbb{E}\\big(\\|\\cdot\\|_\\infty^2\\big)^{1\/2}$ the desired limit $Z \\in {\\mathcal D}$ exists. Letting $m\\to \\infty$ in \\eqref{cauchy0} gives \\eqref{diffbound}. \n\nMoreover,  choosing  any increasing sequence $\\{{n_j}\\}_{j}$ such that\n\\be\\label{rapcon}\n \\sum_{(x,y)\\in \\Pi ^+ : y>n_j} y^{-2\/b}\\ < \\ 2^{-j}\n\\ee\nfor all sufficiently large $j$,\nthen by \\eqref{cauchy0} $\\mathbb{E}\\big(\\|Z_{n_{j+1}}-Z_{n_j}\\|_\\infty^2\\big)< c\\ 2^{-j}$ so almost surely\n $Z= Z_{n_1} + \\sum_{j=1}^\\infty (Z_{n_{j+1}}-Z_{n_j})$   is convergent in $\\|\\cdot\\|_\\infty$. If $0<b<1$ then\n\\begin{align*}\n\\mathbb{E}\\big(\\|Z_{n+1}&  -Z_n\\|_\\infty\\big)\\ \\leq\\ \\big(\\mathbb{E}\\big(\\|Z_{n+1} -Z_n\\|_\\infty^2\\big)\\big)^{1\/2} \\\\ \n& \\leq \\ c \\Big(\\sum_{(x,y)\\in \\Pi^+ : n<y\\leq n+1} y^{-2\/b}\\Big)^{1\/2} \n \\leq \\ c \\sum_{(x,y)\\in \\Pi^+ : n<y\\leq n+1} y^{-1\/b},\n \\end{align*}\nso $\\mathbb{E}\\big(\\sum_{n=1}^\\infty \\|Z_{n+1}  -Z_n\\|_\\infty\\big)<\\infty$, giving that\n$Z= Z_{1} + \\sum_{j=1}^\\infty (Z_{n+1}-Z_n) $  is  almost surely uniformly convergent.\n \\end{proof}\n\n\n\\subsection{Local properties of functions defined by random signs}\n\nWe next examine local properties  of the random function $Z$ constructed  in  Section \\ref{sec2.1}. We will show that for a given $t\\in [t_0,t_1)$, almost surely $Z$ satisfies a H\\\"{o}lder condition to the right of $t$   and also is locally approximable by a random function $L$ defined in a similar way to $Z$, but with a fixed exponent $\\alpha(Z(t))$.\nThroughout this section we fix $t\\in [t_0,t_1)$ and throughout this subsection we restrict $\\Omega$ to the subspace of full probability (by Theorem \\ref{finlim2}) such that there exists a sequence $n_j\\nearrow \\infty$ such that $Z_{n_j}(t) \\to Z(t)$.\nLet  $\\mathcal{F}_{t}$ be the $\\sigma$-field underlying the signs $\\{S(x,y)\\in \\{-1,1\\}: (x,y) \\in \\Pi^+, x\\leq t\\}$.\n\nLet $Z$ be given by the norm limit of the partial sums $Z_n$ in \\eqref{identsignsfin},  as in Section \\ref{sec2.1}. \nWe also construct $L_n, L \\in {\\mathcal D}$ restricted to $[t,t_1)$ in a similar way using the  point set $\\{(x,y) \\in \\Pi^+:  (t<x<t_1)\\}$ and a (conditional) fixed index  $\\alpha(Z(t)) \\in [a,b]$; thus  $L_n$ is the piecewise constant random function defined by\n$$\nL_n (u)\\  \\equiv\\ \\sum_{(x,y) \\in \\Pi^+\\,:\\, y\\leq n} 1_{(t ,u]}(x) S(x,y) y^{-1\/\\alpha(Z_n(t))}\n\\qquad (t\\leq u <t_1)\n$$\nwhere the $S(x,y)$ are independent random signs, and with $L\\in {\\mathcal D}$ defined by \n\\be\\label{limL}\n\\mathbb{E}\\big(\\|L_n - L\\|_\\infty^2\\big)\\to 0,\n\\ee\n as a particular case of Theorem \\ref{finlim2}, where here  $\\|\\cdot \\|_\\infty$ is the supremum norm on $[t,t_1)$. \n\n\\begin{prop}\\label{prop4.3}\nLet $\\alpha: \\mathbb{R} \\to [a,b]$, $\\Pi^+$ be as before and let  $Z_n, Z, L_n,L \\in {\\mathcal D}$ restricted to $[t,t_1)$ be  as above, taking the same realisations of $S(x,y)$ for $Z_n$ and $L_n$.\nThen, there are constants $c_1, c_2$ depending only on $\\Pi^+$ and $\\alpha$ such that, conditional on ${\\mathcal F}_{t}$, for all $0\\leq h <t_1-t$,\n\\be\\label{espest2}\n\\mathbb{E}\\Big(\\sup_{0\\leq h'\\leq h}\\big(Z(t+h')-Z(t)\\big)^2\\Big) \\ \\leq \\ \nc_1\\!\\!\\!\\!\\sum_{(x,y)\\in \\Pi^+\\, :\\,  t< x\\leq t+h}  y^{-2\/\\alpha(Z(t))} \n\\ee\nand\n\\begin{align}\n\\mathbb{E}\\Big(\\sup_{0\\leq h'\\leq h}\\big(&(Z(t+h')-Z(t)) -L(t+h')\\big)^2\\Big)\\nonumber\\\\\n& \\leq \\ \nc_2\\Big(\\sum_{(x,y)\\in \\Pi^+\\, :\\,  t< x\\leq t+h}  y^{-2\/(a,b)}\\Big)\n\\Big(\\sum_{(x,y)\\in \\Pi^+\\, :\\,  t< x\\leq t+h}  y^{-2\/\\alpha(Z(t))}\\Big).\\label{espest2L}\n\\end{align}\n\\end{prop}\n\n\\begin{proof} \nFor brevity write \n$$W_n(u) = Z_n(u)-Z_n(t)\\ \\ \\mbox{ and } \\ \\ D_n(u) = W_n(u)-L_n(u)    \\qquad (t\\leq u <t_1).$$\nLet $0 \\leq h \\leq t_1-t$.  \nFor each $n$, order the points $(x,y) \\in \\{\\Pi^+: t<x\\leq t+h, y\\leq n\\}$ as $(x_i,y_i)$ with  $x_1< x_2 < \\cdots < x_K$ and  let $x_0=t$ (as before we lose little other than awkward notation by assuming that the $x_i$ are distinct). Let $\\mathcal{F}_i$ be the $\\sigma$-field underlying the signs $\\{S(x_j,y_j): 1\\leq  j\\leq i\\}$.\nThen for $1\\leq i \\leq K$,\n$$W_n(x_i)\\  =\\  W_n(x_{i-1})   +  S(x_i,y_i)y_i^{-1\/\\alpha (Z_n(x_{i-1}))}$$\nand \n$$\nD_n(x_i)\\  =\\  D_n(x_{i-1})  +  S(x_i,y_i)\n\\big(y_i^{-1\/\\alpha (Z_n(x_{i-1}))}- y_i^{-1\/\\alpha (Z_n(t))}\\big)\n$$\nso  that $\\big\\{ W_n(x_i), \\mathcal{F}_{i}\\big\\}$ and  $\\big\\{ D_n(x_i), \\mathcal{F}_{i}\\big\\}$ are bounded martingales.\nFor each $1\\leq i \\leq K$, conditioning on $\\mathcal{F}_{i-1}$ gives\n\\begin{eqnarray}\n\\mathbb{E}\\big( W_n(x_i)^2\\big| \\mathcal{F}_{i-1}\\big) \n&=& W_n(x_{i-1})^2 \\ +\\  y_i^{-2\/\\alpha (Z_n(x_{i-1}))}\\nonumber\\\\\n&\\leq& W_n(x_{i-1})^2 \\ +\\   \\big(y_i^{-1\/\\alpha (Z_n(x_{i-1}))}- y_i^{-1\/\\alpha (Z_n(t))}+ y_i^{-1\/\\alpha (Z_n(t))}\\big)^2\\nonumber\\\\\n&\\leq& W_n(x_{i-1})^2 \\ +\\  2  \\big(y_i^{-1\/\\alpha (Z_n(x_{i-1}))}- y_i^{-1\/\\alpha (Z_n(t))}\\big)^2 \n\\ +\\  2  y_i^{-2\/\\alpha (Z_n(t))}, \\nonumber\\\\\n&\\leq& W_n(x_{i-1})^2 \\ +\\  2 M^2 y_i^{-2\/(a,b)}\\big(Z_n(x_{i-1}) - Z_n(t)\\big)^2 \n\\ +\\  2  y_i^{-2\/\\alpha (Z_n(t))}\\nonumber\\\\\n&\\leq& W_n(x_{i-1})^2 \\big(1 +\\  2 M^2 y_i^{-2\/(a,b)}\\big) \n\\ +\\  2  y_i^{-2\/\\alpha (Z_n(t))}\\label{step1} \n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\mathbb{E}\\big( D_n(x_i)^2\\big| \\mathcal{F}_{i-1}\\big) \n&=& D_n(x_{i-1})^2 \\ +\\big(y_i^{-1\/\\alpha (Z_n(x_{i-1}))}- y_i^{-1\/\\alpha (Z_n(t))}\\big)^2\\nonumber\\\\\n&\\leq& D_n(x_{i-1})^2  + M^2 y_i^{-2\/(a,b)} (Z_n(x_{i-1})-Z_n(t))^2\\nonumber\\\\\n&=& D_n(x_{i-1})^2  + M^2 y_i^{-2\/(a,b)} W_n(x_{i-1})^2\\label{step2} \n\\end{eqnarray}\nwhere we have used \\eqref{ydif4}.\nThen induction in decreasing $j$ using \\eqref{step1} gives\n$$\n\\mathbb{E}\\big( W_n(x_K)^2\\big| \\mathcal{F}_{j}\\big) \n\\ \\leq \\\n \\prod_{k=j+1}^{K} \\big(1+2M^2 y_k^{-2\/(a,b)}\\big) \n \\Big(W_n(x_{j})^2+2\\sum_{k=j+1}^K  y_k^{-2\/\\alpha(Z_n(t))} \\Big)\n$$\nfor  all $0\\leq j\\leq K-1$, and  induction in decreasing $j$ using \\eqref{step1} and \\eqref{step2} gives\n\\begin{eqnarray*}\n\\mathbb{E}\\big( D_n(x_K)^2\\big| \\mathcal{F}_{j}\\big) \n&\\leq&\\big(M^2 \\sum_{k=j+1}^K  y_k^{-2\/(a,b)} \\big)\n \\prod_{k=j+1}^{K-1} \\big(1+2M^2 y_k^{-2\/(a,b)}\\big) \\\\\n &&+\\  \\Big(W_n(x_{j})^2+2\\sum_{k=j+1}^{K-1} y_k^{-2\/\\alpha(Z_n(t))} \\Big)\n+ D_n(x_j)^2.\n\\end{eqnarray*}\nfor all $0\\leq j\\leq K-1$. Setting $j=0$ in these two estimates and noting that $W_n(x_0) = W_n(t)=0$ and $D_n(x_0) = D_n(t)=0$, we get  expectations conditioned only on $ \\mathcal{F}_{t}$:\n$$\n\\mathbb{E}\\big( W_n(x_K)^2\\big) \n\\ \\leq \\\n2 \\prod_{k=1}^{K} \\big(1+2M^2 y_k^{-2\/(a,b)}\\big) \n \\Big(\\sum_{k=1}^K  y_k^{-2\/\\alpha(Z_n(t))} \\Big)$$\nand \n$$\n\\mathbb{E}\\big( D_n(x_K)^2\\big) \n\\ \\leq\\ 2 M^2 \\big(\\sum_{k=1}^K  y_k^{-2\/(a,b)} \\big)\n \\prod_{k=1}^{K-1} \\big(1+2M^2 y_k^{-2\/(a,b)}\\big)\n  \\Big(\\sum_{k=1}^{K-1} y_k^{-2\/\\alpha(Z_n(t))} \\Big).\n$$\nNoting that $W_n$ and $D_n$ are constant between the $x_i$ and applying  Doob's inequality to the martingales $W_n(x_i)$ and $D_n(x_i)$,\n\\begin{align*}\n\\mathbb{E}\\big(\\sup_{0\\leq h'\\leq h} (& Z_n(t+h)-  Z_n(t))^2\\big)\\ = \\ \n\\mathbb{E}\\big(\\sup_{0\\leq h'\\leq h} W_n(t+h')^2\\big)\n\\ \\leq\\  \\mathbb{E}\\big(\\max_{0\\leq k \\leq K} W_n(x_k)^2\\big) \\\\\n&\\leq \\ 4\\mathbb{E}\\big( W(x_K)^2\\big) \\ \\leq \\ 8c_3  \\sum_{k=1}^K y_k^{-2\/\\alpha(Z_n(t))}\n\\ \\leq \\ 8c_3  \\sum_{(x,y) \\in \\Pi^+: t<x\\leq t+h}y^{-2\/\\alpha(Z_n(t))},\n\\end{align*}\nwhere $c_3 = \\prod_{k=1}^{K-1} \\big(1+2M^2 y_k^{-2\/(a,b)}\\big)$, and \n\\begin{align*}\n\\mathbb{E}\\Big(\\sup_{0\\leq h'\\leq h} \\big(&(Z_n(t+h')-Z_n(t))-L_n(t+h')\\big)^2\\Big)\\ = \\ \n\\mathbb{E}\\big(\\sup_{0\\leq h'\\leq h} D_n(t+h')^2\\big)\\\\\n& \\leq\\  \\mathbb{E}\\big(\\max_{0\\leq k \\leq K} D_n(x_k)^2\\big) \\ \\leq \\ 4\\mathbb{E}\\big( D(x_K)^2\\big) \\\\\n& \\leq \\ 8M^2 c_3  \\big(\\sum_{k=1}^K  y_k^{-2\/(a,b)} \\big)\\big(\\sum_{k=1}^K y_k^{-2\/\\alpha(Z_n(t))}\\big)\\\\\n& \\leq \\ 8M^2 c_3  \\big(\\sum_{(x,y) \\in \\Pi^+: t<x\\leq t+h} y_k^{-2\/(a,b)} \\big)\\big(\\sum_{(x,y) \\in \\Pi^+: t<x\\leq t+h} y_k^{-2\/\\alpha(Z_n(t))}\\big).\n\\end{align*}\nBy Theorem \\ref{finlim2}, $\\mathbb{E}\\big(\\|Z_n -Z\\|_\\infty^2\\big) \\to 0$ and $\\mathbb{E}\\big(\\|L_n -L\\|_\\infty^2\\big) \\to 0$ and there is a sequence $n_j\\nearrow \\infty$ such that $Z_{n_j}(t) \\to Z(t)$,  so we can take the limit of these inequalities along this subsequence using dominated convergence to get \\eqref{espest2} and  \\eqref{espest2L}  with $c_1 = 8c_3$ and $c_2=8M^2 c_3$. \n\\end{proof}\n\nWe remark that versions of \\eqref{espest2} and  \\eqref{espest2L} with\n$$\n4\\!\\!\\!\\!\\sum_{(x,y)\\in \\Pi^+\\, :\\,  t< x\\leq t+h}  y^{-2\/(a,b)} \\quad \\mbox{ and }\\quad\n4M^2\\Big(\\sum_{(x,y)\\in \\Pi^+\\, :\\,  t< x\\leq t+h}  y^{-2\/(a,b)}\\Big)^2 .\n$$\nrespectively as the right-hand side bounds  can be obtained using a simpler induction, but the  exponents are not so sharp.\n\n\nWe can immediately deduce a local right H\\\"{o}lder bound for $Z$ at $t$ as well getting a comparison with $L$. \n\n\\begin{prop}\\label{cty1}\nLet $Z\\in {\\mathcal D}$  be the random function given by Theorem \\ref{finlim2} and let $t\\in [t_0,t_1)$.  Suppose that for some $\\beta>0$,\n\\be\\label{Obound}\n\\sum_{(x,y)\\in \\Pi^+\\, :\\,  t< x\\leq t+h}  y^{-2\/\\alpha(Z(t))} \\ = \\ O(h^\\beta) \\qquad (0<h<t_1 - t).\n\\ee \nThen, conditional  on $\\mathcal{F}_{t}$, given $0<\\epsilon<\\beta$  there exist almost surely random numbers $C_1, C_2 <\\infty$  such that  for all $0\\leq h < t_1-t$,\n\\be\\label{holas}\n|Z(t+h) -Z(t)| \\ \\leq \\ C_1h^{(\\beta-\\epsilon)\/2}.\n\\ee\nIf, in addition to  \\eqref{Obound},\n$$\n\\sum_{(x,y)\\in \\Pi^+\\, :\\,  t< x\\leq t+h}  y^{-2\/(a,b)} \\ = \\ O(h^\\gamma) \\qquad (0<h< t_1 - t).\n$$ \nthen\n\\be\\label{locas}\n\\big|\\big(Z(t+h)-Z(t)\\big) -L(t+h)\\big|\\  \\leq \\ C_2 h^{(\\beta+\\gamma-\\epsilon)\/2}\\qquad (0<h< t_1 - t),\n\\ee\nwhere $L$ is as in  \\eqref{limL} and defined using the same realisation of $\\{S(x,y): (x,y) \\in \\Pi^+, t\\leq x< t_1\\}$ as $Z$.\n\\end{prop}\n\n\\begin{proof}\nSetting $h = 2^{-k}(t_1 - t)$ in  \\eqref{espest2}, multiplying by  $2^{k(\\beta- \\epsilon)}$ and summing, \n$$\n\\mathbb{E}\\Big(\\sum_{k=0}^\\infty 2^{k(\\beta- \\epsilon)}\\sup_{0\\leq h'\\leq 2^{-k}(t_1-t)}\\big(Z(t+h')-Z(t)\\big)^2\\Big) \\ \\leq \\ \nc\\sum_{k=0}^\\infty 2^{k(\\beta- \\epsilon)} 2^{-k\\beta} \\ < \\ \\infty\n$$\nfor some constant $c$, giving \\eqref{holas}. The bound \\eqref{locas} follows in a similar manner using \\eqref{espest2L}.\n\\end{proof}\n\n\n\n\n\\section{General Poisson point sums}\\label{sec:rand}\n\\setcounter{equation}{0}\n\\setcounter{theo}{0}\n\nWe apply the conclusions of Section \\ref{sec:signs} to random functions where $\\Pi_2^+$ is a realisation of  a Poisson point processes in the half-plane and show that this gives a self-stabilizing processes.\nThe key idea is that the distribution of the point sets $\\{(\\X,\\Y) \\in \\Pi\\}$ where $\\Pi \\subset (t_0,t_1)\\times \\mathbb{R}$ is  a Poisson point process with plane Lebesgue measure ${\\mathcal L}^2$ as mean measure, is identical to that of \n$\\{(\\X,S(\\X,\\Y)\\Y): (\\X,\\Y) \\in \\Pi_2^+, S(\\X,\\Y) = \\pm 1\\}$, where $\\Pi_2^+$ is a Poisson point process on $(t_0,t_1)\\times \\mathbb{R}^+$  with double Lebesgue measure $2{\\mathcal L}^2$ as mean measure and with the $S(\\X,\\Y)$ independently taking the values  $\\pm1$ with equal probability $\\frac{1}{2}$ for each $(\\X,\\Y) \\in \\Pi^+_2$; this follows from the superposition property of Poisson processes, see \\cite[Sections 2.2, 5.1]{King}. \nHence $\\Pi$ can be realised by first sampling $(\\X,\\Y)$ from $\\Pi_2^+$ and then assigning random signs to the $\\Y$ coordinates. \n\n\\subsection{Existence of random functions}\n\nGiven a Poisson process $\\Pi \\subset (t_0,t_1)\\times \\mathbb{R}$ and $\\alpha :\\mathbb{R} \\to [a,b]$   with  $0<a <b<2 $, we wish to show that there exist random functions ${\\mathcal D}$ satisfying \n\\begin{eqnarray}\\label{zedinfty}\nZ(t)\\ =\\  a_0 + \\sum_{(\\X,\\Y) \\in \\Pi } 1_{(t_0,t]}(\\X)\\, \\Y^{\\langle-1\/\\alpha (Z(\\X_-))\\rangle} \\qquad (t_0\\leq t <t_1)\n\\end{eqnarray}\nin an appropriate sense. If $0<a< b<1$ then almost surely $\\sum_{(\\X,\\Y) \\in \\Pi }  |\\Y|^{-1\/\\alpha (Z(\\X_-))}$ converges, in which case the sum in \\eqref{zedinfty} is almost surely absolutely convergent, but if $\\alpha(z)\\geq 1$ for some $z$ there is no a priori guarantee of convergence. In a similar way to Section \\ref{sec:signs} we define  $Z_n\\in {\\mathcal D}$ for $n\\in \\mathbb{N}$ by \n\\begin{eqnarray}\\label{zedn}\nZ_n (t)\\ =\\  a_0 + \\sum_{(\\X,\\Y) \\in \\Pi : |\\Y|\\leq n} 1_{(t_0,t]}(\\X)\\, \\Y^{\\langle-1\/\\alpha (Z_n(\\X_-))\\rangle} \\qquad (t_0\\leq t <t_1).\n\\end{eqnarray}\nAlmost surely this  sum is over a finite number of points and therefore, conditional on $\\Pi$, $Z_n$ is a well-defined piecewise-constant random function. We are interested in convergence of $Z_n$  to a limiting function $Z$ that satisfies \\eqref{zedinfty} in some sense. \n\nThe following result, which is part of Campbell's theorem, will be useful in bounding Poisson sums.\n\n\\begin{theo}[Campbell's theorem] \\label{camp}\nLet $\\Pi$ be a Poisson process on $S\\subset \\mathbb{R}^n$ with mean measure $\\mu$ and let $f:S \\to \\mathbb{R}$ be measurable. Then\n$$\\mathbb{E}\\bigg(\\sum_{\\Pt \\in \\Pi}f(\\Pt)\\bigg) = \\int_S f(u)d\\mu(u)$$\nand \n$$\\mathbb{E}\\bigg(\\exp\\sum_{\\Pt \\in \\Pi}f(\\Pt)\\bigg) = \\exp\\int_S \\big( \\exp f(u)-1\\big)d\\mu(u),$$\nprovided these integrals converge.\n\\end{theo}\n\\begin{proof}\nSee \\cite[Section 3.2]{King}.\n\\end{proof}\n\n\\begin{lem} \\label{sumest}\nLet $\\Pi \\subset (t_0,t_1)\\times \\mathbb{R}$ be a Poisson process  with mean measure ${\\mathcal L}^2$ and let $\\alpha :\\mathbb{R} \\to [a,b]$ where  $0<a <b<2 $.  Then for all $0< \\eta <2\/b-1$ there is almost surely a random $C<\\infty$ such that  \n\n\\be\\label{tailsum}\n\\sum_{(\\X,\\Y) \\in \\Pi: |\\Y|>n}  |\\Y|^{-2\/b}\\ <\\ C\\, n^{-\\eta}\\qquad (n \\in \\mathbb{N}).\n\\ee\n\\end{lem}\n\\begin{proof}\nBy Theorem \\ref{camp},\n$$\n\\mathbb{E}\\Big(\\sum_{(\\X,\\Y) \\in \\Pi: |\\Y|>n}  |\\Y|^{-2\/b}\\Big)\\ = \\ \\int_n^\\infty y^{-2\/b}\n\\ \\leq \\ \\frac{b}{2-b}n^{1-2\/b}.$$\nA Borel-Cantelli argument summing over $n=2^{-k}$ completes the proof.\n\\end{proof}\n\nWe can now obtain develop Theorem \\ref{finlim2} to sums over a Poisson point process. \n\n\n\\begin{theo}\\label{thmrand2}\nLet $\\Pi \\subset (t_0,t_1)\\times \\mathbb{R}$ \nbe a  Poisson point process with mean measure ${\\mathcal L}^2$, let $\\alpha :\\mathbb{R} \\to [a,b]$   where  $0<a <b<2 $ and let $a_0 \\in \\mathbb{R}$. \nThen there exists $Z \\in {\\mathcal D}$ satisfying  \n\\eqref{zedinfty} in the sense that   \n $\\lim_{n\\to \\infty}\\mathbb{E}\\big(\\|Z_n -Z\\|_\\infty^2\\big)=0$ where $Z_n$ is as in  \\eqref{zedn}. Moreover, there exists a sequence $n_j \\nearrow \\infty$ such that almost surely $\\|Z_{n_j} -Z\\|_\\infty \\to 0$.  If $0<b<1$ then almost surely $\\|Z_{n} -Z\\|_\\infty \\to 0$. \n\\end{theo}\n\n\\begin{proof}\nSince $0< b<2$,  the Poisson point set  $\\Pi$ is almost surely a countable set of isolated points with \n$$\n \\sum_{(\\X,\\Y) \\in \\Pi} 1_{(t_0,t_1]}(\\X)|\\Y|^{-2\/(a,b)}< \\infty\n$$\nand with the $\\X$ distinct.\nAs noted, $Z_n$ in \\eqref{zedn} has the same distribution as \n$$\nZ_n(t) = a_0+\\!\\!\\sum_{(\\X,\\Y) \\in \\Pi^+_2 : |\\Y|\\leq n} 1_{(t_0,t]}(\\X) S(\\X,\\Y)\\Y^{-1\/\\alpha (Z(\\X_-))}, \n$$\nwhere $\\Pi_2^+$ is a Poisson point process on $(t_0,t_1)\\times \\mathbb{R}^+$  with  $2{\\mathcal L}^2$ as mean measure and $S(\\X,\\Y)$ are random signs on $\\Pi_2^+$. Thus, by Theorem \\ref{finlim2}, for almost all realisations  of   $\\Pi_2^+$   there almost surely exists a random function  $Z \\in {\\mathcal D}$, such that  $\\mathbb{E}\\big(\\|Z_n-Z\\|_\\infty^2) \\to 0$, and also such that there exists a sequence $n_j \\nearrow \\infty$ with $\\|Z_{n_j} -Z\\|_\\infty \\to 0$; note that by \\eqref{tailsum} and \\eqref{rapcon} we can take the same sequence $n_j$ for all such realisations.\nThus the conclusion holds for almost all sign combinations for almost all realisations of $\\Pi_2^+$ and so  for almost all $\\{(\\X,S(\\X,\\Y)\\Y): (\\X,\\Y) \\in \\Pi_2^+, S(\\X,\\Y) =\\pm 1\\}$, that is for almost all $\\Pi$.\n\\end{proof}\n\nFor purposes of simulating these random functions we would like an estimate on how rapidly  $Z_n$ given by \\eqref{zedn} converges to $Z$. However we  cannot get useful estimates directly from  Theorem \\ref{finlim2} since allowing $(x,y) \\in \\Pi^+$ with $y$ arbitrarily small leaves the sum in  \\eqref{diffbound}  unbounded. However, we can get some concrete  estimates if we modify the setting slightly by assuming that there is  $y_0>0$ such that  $|\\Y|\\geq y_0$ if $(\\X,\\Y) \\in \\Pi$, which ensures that the right hand side of \\eqref{diffbound1} below converges. In practice this is a realistic assumption in that it  excludes the possibility of $Z$ having unboundedly large jumps.\n\n\n\\begin{theo}\\label{thmrand3}\nLet  $y_0>0$ and  let $\\Pi $ be a Poisson point process on $ (t_0,t_1)\\times (-\\infty, -y_0] \\cup [y_0,\\infty)$  \n with  mean measure ${\\mathcal L}^2$ restricted to this domain. Let $a_0\\in\\mathbb{R}$, let $0<a <b<2 $, let $\\alpha :\\mathbb{R} \\to [a,b]$ and  let $Z \\in {\\mathcal D}$ be the random function given by Theorem \\ref{thmrand2} using  this $\\Pi$ with $Z_n$ as in \\eqref{zedn}. Then as $n\\to \\infty$\n\\be\n\\mathbb{E}\\big(\\|Z_n -Z\\|_\\infty^2\\big)\\ \\leq \\ \n\\frac{8b(t_1-t_0)}{2-b}\\exp\\bigg(2M^2(t_1-t_0)\\int_{y_0}^\\infty  y^{-2\/(a,b)}\\ dy\\bigg) \n n^{-(2-b)\/b}\\ \\to 0. \n\\label{diffbound1}\n\\ee\n\n\\end{theo}\n\n\n\\begin{proof}\nWe proceed exactly as in the proofs of Proposition \\ref{finlim1A} and Theorem \\ref{finlim2}, except that we condition on  realisations  of a Poisson process $\\Pi_2^+$ with mean measure $2{\\mathcal L}$ on $(t_0,t_1)\\times [y_0,\\infty)$, to get  \\eqref{diffbound} in this setting and then use Theorem \\ref{thmrand2} to get the process on this $\\Pi$. \nThen for $n\\geq y_0$, using \\eqref{diffbound}, the independence of the  Poisson point process $\\Pi_2^+$ on  $(t_0,t_1)\\times [y_0,n]$ and on $(t_0,t_1)\\times (n,\\infty)$, and Theorem \\ref{camp},\t\n\\begin{align*}\n\\mathbb{E}\\big(\\|Z_n& -Z\\|_\\infty^2\\big) \\ =\\ \\mathbb{E}\\Big(\\mathbb{E}\\big(\\|Z_n -Z\\|_\\infty^2\\big| \\Pi_2^+ \\big)\\Big) \\\\\n&\\leq\\ \n4\\,\\mathbb{E}\\bigg( \\prod_{(\\X,\\Y) \\in \\Pi ^+_2: y_0\\leq \\Y\\leq n } (1+M^2\\Y^{-2\/(a,b)})\\sum_{(\\X,\\Y)\\in \\Pi ^+_2 : \\Y> n} |\\Y|^{-2\/b}\\bigg)\\\\\n&=\\ \n4\\,\\mathbb{E}\\bigg( \\exp\\Big( \\sum_{(\\X,\\Y) \\in \\Pi ^+_2: y_0\\leq \\Y\\leq n } \\log(1+M^2\\Y^{-2\/(a,b)})\\Big)\\bigg)\n\\mathbb{E}\\bigg( \\sum_{(\\X,\\Y)\\in \\Pi ^+_2 : \\Y> n} |\\Y|^{-2\/b}\\bigg)\\\\\n&=\\ \n4\\,\\exp\\bigg(\\int_{y_0}^n 2(t_1-t_0)\\ M^2 y^{-2\/(a,b)} dy\\bigg) \n(t_1-t_0) \\int_n^\\infty  2y^{-2\/b}  dy.\n \\end{align*}\nLetting $n\\to \\infty$ in the first integral and evaluating the second integral gives \\eqref{diffbound1}.\n\\end{proof}\n\nWe remark that Theorem \\ref{thmrand3} allows us to quantify the rate of convergence in probability of $\\|Z_n -Z\\|_\\infty\\to 0$ in Theorem \\ref{thmrand2}. By the Poisson distribution $\\P\\{\\Pi \\cap ((t_0,t_1)\\times [0,y_0] )= \\emptyset\\} = \\exp(-y_0(t_1-t_0))$. Given $\\epsilon >0$ we can choose $y_0$ to make this probability at most $\\epsilon\/2$, then using  \\eqref{diffbound1} and Markov's inequality it follows that if  $n$ is sufficiently large then   $\\P\\{ \\|Z_n -Z\\|_\\infty>\\epsilon\\} < \\epsilon$. In practice, this leads to an enormous value of $n$. \n\n\\subsection{Local properties and self-stabilising processes}\n\nWe next obtain local properties of the random functions defined by a Poisson point process as in Theorem \\ref{thmrand2}. Not only are the sample paths   right-continuous, but they satisfy a local H\\\"{o}lder continuity estimate and are self-stabilizing, that is locally they look like $\\alpha$-stable processes. \n\nWe will use a bound provided by the $\\alpha$-{\\it stable subordinator} which may be defined for each (constant) $0<\\alpha<1$ by\n$$\nS_\\alpha (t) := \\sum_{(\\X,\\Y) \\in \\Pi} 1_{(t_0 ,t]}(\\X)\\, |\\Y|^{-1\/\\alpha}\\qquad (t_0 \\leq t <t_1),\n$$\nwhere the sum, which is almost surely convergent, is over a plane Poisson point process $\\Pi$ with mean measure ${\\mathcal L}^2$.\nThen $S_\\alpha $ on $[t_0 ,t_1)$ has stationary increments and for each $0<\\epsilon<1\/\\alpha$ satisfies the H\\\"{o}lder property\n\\be\\label{sub2}\nS_\\alpha (t)\\   \\leq\\ C (t-t_0)^{(1\/\\alpha)\\,-\\,\\epsilon},\n\\ee\nwhere $C$ is almost surely finite; this may be established using Campbell's Theorem \\ref{camp} in a similar way to the proof of Lemma \\ref{sumest}, or see  \\cite[Section III.4]{Ber} or \\cite{Tak}.\n\nWe write $L^0_\\alpha $ for the non-normalized $\\alpha$-stable process, which has a representation\n\\begin{equation}\\label{sumnn}\nL^0_\\alpha  (t) =\\sum_{(\\X,\\Y) \\in \\Pi} 1_{(0,t]}(\\X) \\Y^{<-1\/\\alpha>}\n\\end{equation}\nwhere $\\Pi$ is a Poisson point process on $(t_1,t_2) \\times \\mathbb{R}$ with mean measure ${\\mathcal L}^2$.\nThis sum is almost surely absolutely convergent if $0<\\alpha<1$ but for general $0<\\alpha<2$\nit is the limit as $n \\to \\infty$ of\n$$L^0_{\\alpha,n}(t) =\\sum_{(\\X,\\Y) \\in \\Pi:|\\Y| \\leq n} 1_{(0,t]}(\\X) \\Y^{<-1\/\\alpha>}.$$ \nWhilst $\\mathbb{E}\\big(\\|L^0_{\\alpha,n}  -L^0_\\alpha\\|_\\infty^2\\big)\\to 0$ as a special case of Theorem \\ref{thmrand2}, the constant value of $\\alpha$ means that $L^0_{\\alpha,n}$ and $L^0_{\\alpha,m}-L^0_{\\alpha,n}$ are independent for $m>n$ and also that $\\{L^0_{\\alpha,n}\\}_n$ is a martingale, which ensures that $\\|L^0_{\\alpha,n}  -L^0_\\alpha\\|_\\infty\\to 0$ almost surely.  \n\nIn the same way to $Z_n$ we can think of $L^0_{\\alpha,n}(t)$ in terms of  $\\Pi^+_2$ and random signs, so that \n$$L^0_{\\alpha,n}(t) =\\sum_{(\\X,\\Y) \\in \\Pi^+_2:\\Y \\leq n} 1_{(0,t]}(\\X) S(\\X,\\Y)\\Y^{<-1\/\\alpha>}.$$ \n\n\nThe following proposition is the analogue of Proposition \\ref{cty1} in this context.\n\n\\begin{prop}\\label{cty3}\nLet $Z$  be the random function given by Theorem \\ref{thmrand2} and let $t\\in [t_0,t_1)$. Then, conditional on ${\\mathcal F}_{t}$, given $0< \\epsilon< 1\/b$ there exist almost surely random numbers $C_1, C_2 <\\infty$  such that  for all $0\\leq h <t_1-t$,\n\\be\\label{holas3}\n|Z(t+h) -Z(t)|  \\leq C_1 h^{1\/\\alpha(Z(t))\\,-\\,\\epsilon}.\n\\ee\nand\n\\be\\label{locas3}\n\\big|\\big(Z(t+h)-Z(t)\\big)   -\\big(L^0_{\\alpha(t)}(t+h)-L^0_{\\alpha(t)}(t)\\big)\\big|\n \\ \\leq \\ C_2 h^{1\/\\alpha(Z(t))+ 1\/b -\\,\\epsilon},\n\\ee\nwhere  $L^0_{\\alpha(t)}$ is the $\\alpha(t)$-stable process \\eqref{sumnn} defined using the same realisations of $\\Pi$ as $Z$.\n\\end{prop}\n\n\\begin{proof} \nLet $\\epsilon>0$. Let  $\\Pi^+_2$ be a Poisson point process on $\\mathbb{R}$ with mean measure $2{\\mathcal L}$.\nFrom \\eqref{sub2} \n$$\n\\sum_{(X,Y)\\in \\Pi^+_2\\, :\\,  t< x\\leq t+h}  \\Y^{-2\/\\alpha(Z(t))}\n\\ =\\ 2S_{\\alpha(Z(t))\/2}(h)\\ \\leq\\ C h^{2\/\\alpha(Z(t)) -\\epsilon}\n$$\nwhere $C<\\infty$ for almost all realisations of $\\Pi^+_2$.\nFor such $\\Pi^+_2$,  Proposition \\ref{cty1} gives on randomising the signs,\n$$\\big|Z(t+h) -Z(t)|\\Pi^+_2\\big|\\  \\leq\\ C_1 h^{1\/\\alpha(Z(t))-\\epsilon\/2-\\epsilon\/2}$$\nfor some random $C_1$, \nand hence \\eqref{holas3} holds almost surely.\n\nIn the same way,\n$$\\sum_{(X,Y)\\in \\Pi^+_2\\, :\\,  t< x\\leq t+h}  \\Y^{-2\/(a,b)}\\ \\leq\\ C' (h^{-1})^{-2\/(a,b) -\\epsilon\/2} \\ \\leq\\ C^{\\prime\\prime}h^{2\/b -\\epsilon} \n$$\nwhere $C', C^{\\prime\\prime}< \\infty$ for almost all realisations of $\\Pi^+_2$.\nThen in Proposition \\ref{cty1}, $L(t+h) = L^0_{\\alpha(t)}(t+h)-L^0_{\\alpha(t)}(t)$, so  for such $\\Pi^+_2$, on randomising the signs,\n$$\\Big|\\big(Z(t+h)-Z(t)\\big)   -\\big(L^0_{\\alpha(t)}(t+h)-L^0_{\\alpha(t)}(t)\\big)\\big|\\Pi^+_2\\Big| \\  \\leq\\ C_2 h^{1\/\\alpha(Z(t))+ 1\/b -\\,\\epsilon}$$\nfor random $C_2<\\infty$, so \\eqref{locas3} holds almost surely.\n\\end{proof}\n\n\nWe finally show that almost surely at each $t\\in [t_0,t_1)$ the random function $Z$ of Theorem \\ref{thmrand2} is right-localisable with local form  an $\\alpha(Z(t))$-stable process, so that $Z$ may indeed be thought of as self-stablizing. \n\n\\begin{theo}\\label{rtloc}\nLet $Z$ be the random function given by Theorem \\ref{thmrand2} and let $t\\in [t_0,t_1)$. Then, conditional on ${\\mathcal F}_{t}$, almost surely $Z$ is strongly right-localisable at $t$, in the sense that\n$$\\frac{Z(t+ru) -Z(t)}{r^{1\/\\alpha(Z(t))}}\\bigg|\\,  \\mathcal{F}_t \\  \\tod \\ L^0_{\\alpha(Z(t))}(u) \\qquad (0\\leq u \\leq 1)\n$$\nas $r\\searrow 0$, where convergence is in distribution with respect to $(D[0,1], \\rho_S)$, with $ \\rho_S$ is the Skorohod metric.\n\\end{theo}\n\n\n\\begin{proof}\nLet $0<\\epsilon< 1\/b$. For $u\\in [0,1]$   and $0<r<t_1-t$, almost surely \n\\begin{align*}\n\\big|\\big(Z(t+  ru) & -Z(t)\\big) -\\big(L^0_{\\alpha(t)} (t+ru)-L^0_{\\alpha(t)} (t)\\big)\\big| \\\\\n& \\leq\\  C_2 (ru)^{1\/\\alpha(Z(t))+ 1\/b -\\,\\epsilon}\n\\ \\leq\\ C_2 r^{1\/\\alpha(Z(t))+ 1\/b -\\,\\epsilon}\n\\end{align*}\nfor a random $C_2<\\infty$, by Proposition \\ref{cty3}. Thus\n\\begin{eqnarray*}\n\\bigg\\| \\frac{Z(t+  ru) - Z(t)}{r^{1\/\\alpha(Z(t))}}\\  -\\ \\frac{L^0_{\\alpha(Z (t))}(t+ru)-L^0_{\\alpha(Z (t))}(t)}{ r^{1\/\\alpha(Z(t))}}\\bigg\\|_\\infty \n&\\leq&\nC_2 \\frac{r^{1\/\\alpha(Z(t))+ 1\/b -\\,\\epsilon}}{ r^{1\/\\alpha(Z(t))}}\\\\\n&=& C_2  r^{1\/b-\\epsilon} \\to 0\n\\end{eqnarray*}\nalmost surely as $r\\searrow 0$. In particular, since $\\|\\cdot\\|_\\infty$ dominates $\\rho_S$ on $D[0,1]$,\n$$ \\rho_S \\bigg( \\frac{Z(t+  ru) - Z(t)}{r^{1\/\\alpha(Z(t))}}, \\frac{L^0_{\\alpha(Z (t))}(t+ru)-L^0_{\\alpha(Z (t))}(t)}{ r^{1\/\\alpha(Z(t))}}    \\bigg) \\topp 0$$\nalmost surely and in probability. Using that $\\alpha$-stable processes have stationary increments and scale with exponent $1\/\\alpha$,\n$$\\frac{L^0_{\\alpha(Z (t))}(t+ru)-L^0_{\\alpha(Z (t))}(t)}{ r^{1\/\\alpha(Z(t))}} \n\\ \\ed \\  L^0_{\\alpha(Z (t))}(u)-L^0_{\\alpha(Z (t))}(0) \\ed L^0_{\\alpha(Z (t))}(u),$$\nso we conclude, using \\cite[Theorem 4.1]{Bil} to combine convergence in probability and in distribution, that  \n$$\\frac{Z(t+  ru) - Z(t)}{r^{1\/\\alpha(Z(t))}}\\bigg|\\,  \\mathcal{F}_t \\  \\tod \\ L^0_{\\alpha(Z (t))}(u)$$\n as $r\\searrow 0$.\n\\end{proof}\n\n\\section*{Acknowledgements}\nThe authors thank the referee for some helpful comments. KJF gratefully acknowledges the hospitality of Institut Mittag-Leffler in  Sweden, where part of this work was carried out. JLV is grateful to SMABTP for financial support.\n\n\n\n\n\n\\bibliographystyle{plain}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCognitive radio technology provides an efficient means of utilizing the radio spectrum \\cite{zhao2007survey}. The basic idea is to allow secondary users to access the spectrum while providing certain guaranteed quality of service (QoS) performance measures for the high-priority licensed primary users (PUs). The secondary user is a battery-powered device in many practical situations and its operation, which involves spectrum sensing and access, is accompanied by energy consumption. Consequently an energy-constrained secondary user must optimize its sensing and access decisions to efficiently utilize the energy at its disposal. An emerging technology for energy-constrained terminals is energy harvesting which allows the terminal to gather energy from its environment. An overview of the different energy harvesting technologies is provided in \\cite{survey} and the references therein.\n\nData transmission by an energy harvesting transmitter with a rechargeable battery has got a lot of attention recently  \\cite{lei2009generic,sharma2010optimal,ho2010optimal,yang2010transmission,yang2010optimal,tutuncuoglu2010optimum,pappas2011optimal,krikidis2012stability}. The optimal online policy for controlling admissions into the data buffer\nis derived in \\cite{lei2009generic} using a dynamic programming framework. In \\cite{sharma2010optimal},\nenergy management policies which stabilize the data queue\nare proposed for single-user communication and some delay-optimal properties are derived. Throughput optimal\nenergy allocation is investigated in \\cite{ho2010optimal} for energy harvesting systems in\na time-constrained slotted setting. In \\cite{yang2010transmission,yang2010optimal}, minimization\nof the transmission completion time is considered in an\nenergy harvesting system and the optimal solution is obtained\nusing a geometric framework. In \\cite{tutuncuoglu2010optimum}, energy harvesting transmitters with\nbatteries of finite energy storage capacity are considered and\nthe problem of throughput maximization by a deadline is\nsolved for a static channel. The authors of \\cite{pappas2011optimal} consider the scenario in which a set of\nnodes shares a common channel. The PU has a rechargeable\nbattery and the secondary user is plugged to a reliable power supply. They obtain the maximum stable throughput region which describes the maximum arrival rates that maintain the stability of the network queues. In \\cite{krikidis2012stability}, the authors investigate the effects of network layer\ncooperation in a wireless three-node network with energy harvesting\nnodes and bursty traffic.\n\nIn this work, we develop spectrum sensing and transmission methods for an energy harvesting secondary user. We leverage the primary automatic repeat request (ARQ) feedback for secondary access. Due to the broadcast nature of the wireless channel, this feedback can be overheard and utilized by the secondary node assuming that it is unencrypted. The problem with depending on spectrum sensing only is that sensing does not inform the secondary terminal about its impact on the primary receiver. This issue has induced interest in utilizing the feedback from the primary receiver to the primary transmitter to optimize the secondary transmission\nstrategies. For instance, in \\cite{eswaran2007bits}, the secondary user observes the ARQ feedback from the primary\nreceiver as it reflects the PU's achieved packet rate. The secondary user's objective is to maximize its throughput while guaranteeing a certain packet rate for the PU. In \\cite{lapiccirella2010cognitive}, the authors use a partially observable Markov decision process (POMDP) to optimize the\nsecondary action on the basis of the spectrum sensing outcome and primary ARQ feedback. Secondary power control based on primary feedback is investigated in \\cite{huang2010distributed}. In \\cite{levorato2009cognitive}, the optimal transmission policy for the secondary user when the PU adopts a retransmission based error control scheme is investigated. The policy of the secondary user determines how often it transmits according to the retransmission state of the packet being served by the PU.\n\nOur contributions in this paper can be summarized as follows. We investigate the case of a secondary user equipped with an energy harvesting mechanism and a rechargeable battery. We propose a novel access and sensing scheme where the secondary user possibly senses the channel for a certain fraction of the time slot duration and accesses the channel with some access probability that depends on the sensing outcome. The secondary user may access the channel probabilistically without sensing in order to utilize the whole slot duration for transmission. We assume multipacket reception (MPR) capability added to the physical layer of the receiving nodes. We maximize the secondary throughput such that the primary and secondary queues are stable. The optimization problem to get the various probabilities also includes a constraint on the maximum tolerable primary queueing delay. Finally, we compare our system with the random access system. The numerical results show the gains of our proposed systems in terms of the secondary user throughput.\n\n\\begin{figure}\n \n  \\includegraphics[width=1.05\\columnwidth]{system_model_energy}\\\\\n  \\caption{Primary and secondary queues and links. The PU has data queue $Q_p$, whereas the secondary terminal has data queue $Q_s$ and energy queue $Q_e$. There is a feedback channel between the primary receiver (PR) and the PU to acknowledge the reception of data packets. This feedback channel is overheard by the secondary transmitter. Both the PR and the secondary receiver (SR) may suffer interference from the other link. }\\label{fig1}\n\\end{figure}\n\n\nThe rest of the paper is organized as follows. In the next section, we discuss secondary access without incorporating the primary feedback. In Section \\ref{sec3} we discuss the feedback-based scheme. We provide numerical results in Section \\ref{numerical} and conclude the paper in Section \\ref{secfinal}.\n\n\n\\section{Secondary Access Without Employing Primary Feedback}\\label{sec2}\n\\subsection{System Model}\nWe consider the system model shown in Fig. \\ref{fig1}. The model consists of one PU and\none secondary user. The channel is slotted in time and\na slot duration equals the packet transmission time. The PU and the secondary user have infinite buffer queues, $Q_p$ and $Q_s$, respectively, to store fixed-length data packets. If a terminal transmits during a time slot, it sends exactly one packet to its receiver. The\narrivals at $Q_p$ and $Q_s$ are independent and identically\ndistributed (i.i.d.) Bernoulli random variables from slot to slot with means $\\lambda_p$ and $\\lambda_s$, respectively.\n\nThe secondary user has an additional energy queue, $Q_e$, to store harvested energy from the environment. The arrival at the energy queue is also Bernoulli with mean $\\lambda_e$ and is independent from arrivals at the other queues. The Bernoulli model is simple, but it captures the random availability of ambient energy sources. It is assumed that the transmission of one data packet consumes one packet of energy. Adequate system operation requires that all the queues are stable. We employ the standard definition of stability for queue as in, e.g., \\cite{stabN}, that is, a queue is stable if and only if its probability of being empty does not vanish as time progresses.\n\nInstead of the collision channel model where simultaneous transmission by different terminals leads to sure packet loss, we assume that the receivers have multipacket reception (MPR) capability as in \\cite{ghez1988stability,ghez1989optimal,naware2005stability}. This means that transmitted data packets can survive the interference caused by concurrent\ntransmissions if the received signal to interference\nand noise ratio ({\\rm SINR}) exceeds the threshold required for successful decoding at the receiver. With MPR capability, the secondary user may use the channel simultaneously with the PU.\n\nThe PU accesses the channel whenever\nit has a packet to send. The secondary transmitter, given that it has energy, senses the channel or possibly transmits the packet at the head of its queue immediately at the beginning of the time slot without sensing the channel. We explain below why direct transmission can be beneficial for system performance.\n\nThe secondary user operation can be summarized as follows.\n\\begin{itemize}\n  \\item If the secondary terminal's energy and data queues are not empty, it senses the channel with probability $p_s$\n   from the beginning of the time slot for a duration of $\\tau$ seconds to detect the possible activity of the PU. If the slot duration is $T$, $\\tau < T$.\n    \\item If the channel is sensed to be free, the secondary transmitter accesses the channel with probability $p_f$. If the PU is detected to be active, it accesses the channel with probability $p_b$.\n    \\item If at the beginning of the time slot the secondary user decides not to sense the spectrum (which happens with probability $1-p_s$) it immediately decides whether to transmit with probability $p_t$ or to remain idle for the rest of the time slot with probability $1-p_t$.\n\\end{itemize}\n\\noindent This means that the transmission duration is $T$ seconds if the secondary user accesses the channel without spectrum sensing and $T-\\tau$ seconds if transmission is preceded by a sensing phase. We assume that the energy consumed in spectrum sensing is negligible, whereas data transmission dissipates exactly one unit of energy from $Q_e$.\n\nWe study now secondary access in detail to obtain the mean service rates of queues $Q_s$, $Q_e$ and $Q_p$. The meaning of the various relevant symbols are provided in Table \\ref{table1}. For the secondary terminal to be served, its energy queue must be nonempty. If the secondary user does not sense the channel, which happens with probability $1-p_s$, it transmits with probability $p_t$. If the PU's queue is empty and, hence, the PU is inactive, secondary transmission is successful with probability $\\overline{P}_{{\\rm out},0s}$, whose expression as a function of the secondary link parameters, transmission time $T$, and the data packet size is provided in Appendix A. If $Q_p \\neq 0$ and the PU is active, the success probability is $\\overline{P}_{{\\rm out},0s}^{\\left({\\rm c}\\right)}$ (see Appendix B).\n\\begin{table}\n\\renewcommand{\\arraystretch}{1}\n\\begin{center}\n\\begin{tabular}{ c |l  }\n    \\hline\\hline\n    $\\tau$ & {\\footnotesize Sensing duration} \\\\[5pt]\\hline\n  $T$ & {\\footnotesize Slot duration} \\\\[5pt]\\hline\n  $p_s$ & {\\footnotesize Probability of sensing the channel} \\\\[5pt]\\hline\n   $P_{\\rm MD}$ & {\\footnotesize Misdetection probability} \\\\[5pt]\\hline\n    $P_{\\rm FA}$ & {\\footnotesize False alarm probability} \\\\[5pt]\\hline\n  $p_t$ & {\\footnotesize Probability of direct channel access if the channel is } \\\\\n   & {\\footnotesize not sensed} \\\\[5pt]\\hline\n  $p_f$ & {\\footnotesize Probability of channel access if the channel is sensed} \\\\\n   & {\\footnotesize to be free} \\\\[5pt]\\hline\\\\\n    $p_b$ & {\\footnotesize Probability of channel access if the channel is sensed} \\\\\n   & {\\footnotesize to be busy} \\\\[5pt]\\hline\\\\\n  $\\overline{P}_{{\\rm out},p}$ & {\\footnotesize Probability of successful primary transmission to the } \\\\\n   & {\\footnotesize primary receiver if the secondary terminal is silent}\\\\[5pt]\\hline\n   $\\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}$ & {\\footnotesize Probability of successful primary transmission to the } \\\\\n   & {\\footnotesize primary receiver with concurrent secondary transmission}\\\\[5pt]\\hline\n $\\overline{P}_{{\\rm out},0s}$ & {\\footnotesize Probability of successful secondary transmission if the PU} \\\\\n   & {\\footnotesize is silent and transmission occurs over $T$ seconds}\\\\[5pt]\\hline\n   $\\overline{P}_{{\\rm out},1s}$ & {\\footnotesize Probability of successful secondary transmission if the PU} \\\\\n   & {\\footnotesize is silent and transmission occurs over $T-\\tau$ seconds}\\\\[5pt]\\hline\n   $\\overline{P}_{{\\rm out},0s}^{\\left({\\rm c}\\right)}$ & {\\footnotesize Probability of successful secondary transmission if the PU} \\\\\n   & {\\footnotesize is active and transmission occurs over $T$ seconds}\\\\[5pt]\\hline\n   $\\overline{P}_{{\\rm out},1s}^{\\left({\\rm c}\\right)}$ & {\\footnotesize Probability of successful secondary transmission if the PU} \\\\\n   & {\\footnotesize is active and transmission occurs over $T-\\tau$ seconds}\\\\[5pt]\\hline\n\\end{tabular}\n\\caption{List of symbols involved in the queues' mean service rates.}\n\\label{table1}\n\\end{center}\n\\end{table}\n\nIf the secondary user decides to sense the channel, there are four possibilities depending on the sensing outcome and the state of the primary queue. If the PU is sensed to be free, secondary transmission takes place with probability $p_f$. This takes place with probability $1-P_{\\rm FA}$ if the PU is actually silent. In this case, the probability of successful secondary transmission is $\\overline{P}_{{\\rm out},1s}$, which is lower than $\\overline{P}_{{\\rm out},0s}$ as proven in Appendix C. On the other hand, if the PU is on, the probability of detecting the channel to be free is $P_{\\rm MD}$ and the probability of successful secondary transmission is $\\overline{P}_{{\\rm out},1s}^{\\left({\\rm c}\\right)}$. If the channel is sensed to be busy, the secondary terminal transmits with probability $p_b$. Sensing the PU to be active occurs with probability $P_{\\rm FA}$ if the PU is actually inactive, or with probability $1-P_{\\rm MD}$ if the PU is actively transmitting. The probability of successful secondary transmission is $\\overline{P}_{{\\rm out},1s}$ when the PU is silent and $\\overline{P}_{{\\rm out},1s}^{\\left({\\rm c}\\right)}$ when the PU is active. Given these possibilities, we can write the following expression for the mean secondary service rate.\n\\begin{equation}\n\\begin{split}\n\\mu_s&=\\left(1-p_s\\right)p_t{\\rm Pr}\\{Q_p =0,Q_e \\ne 0\\} {\\overline{P}}_{{\\rm out},0s}\n\\\\&+\\left(1-p_s\\right)p_t{\\rm Pr}\\{Q_p \\ne 0,Q_e \\ne 0\\} {\\overline{P}}_{{\\rm out},0s}^{\\left({\\rm c}\\right)}\n\\\\& +p_s p_f {\\rm Pr}\\{Q_p =0,Q_e \\ne 0\\} \\left(1-P_{\\rm FA}\\right){\\overline{P}}_{{\\rm out},1s}\n\\\\&+p_s p_f {\\rm Pr}\\{Q_p \\ne0,Q_e \\ne 0\\} P_{\\rm MD} {\\overline{P}}_{{\\rm out},1s}^{\\left({\\rm c}\\right)}\n\\\\&+p_s p_b {\\rm Pr}\\{Q_p =0,Q_e \\ne 0\\}P_{\\rm FA} {\\overline{P}}_{{\\rm out},1s}\n\\\\&+p_s p_b  {\\rm Pr}\\{Q_p \\ne0,Q_e \\ne 0\\} \\left(1-P_{\\rm MD}\\right) {\\overline{P}}_{{\\rm out},1s}^{\\left({\\rm c}\\right)}\n\\end{split}\n\\end{equation}\n\n\\noindent Based on the above analysis, it can be shown that the mean service rate of the energy queue is\n\\begin{equation}\n\\begin{split}\n\\mu_e&=\\left(1-p_s \\right)p_t{\\rm Pr}\\{Q_s \\ne 0\\}\\\\& +p_s p_f \\bigg(P_{\\rm MD} {\\rm Pr}\\{Q_s \\ne 0,Q_p\\ne 0\\}\\\\&+(1-P_{\\rm FA}){\\rm Pr}\\{Q_s \\ne 0,Q_p=0\\}\\bigg) \\\\& +p_s p_b \\bigg(P_{\\rm FA} {\\rm Pr}\\{Q_s \\ne 0,Q_p=0\\}\\\\&+(1-P_{\\rm MD}){\\rm Pr}\\{Q_s \\ne 0,Q_p\\ne 0\\}\\bigg)\n\\end{split}\n\\end{equation}\n\n\nA packet from the primary queue can be served in either one of the following events. If the secondary user is silent because either of its data queue or energy queue is empty, the primary transmission is successful with probability $\\overline{P}_{{\\rm out},p}$. If both secondary queues are nonempty, secondary operation proceeds as explained above. In all cases, if the secondary user does not access the channel, the probability of successful primary transmission is $\\overline{P}_{{\\rm out},p}$, else it is $\\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}$. Therefore,\n\\begin{equation}\n\\begin{split}\n\\mu_p&=\\bigg(1-{\\rm Pr}\\{Q_s \\ne0,Q_e \\ne 0\\}\\bigg)\\overline{P}_{{\\rm out},p}\n\\\\&+{\\rm Pr}\\{Q_s \\ne0,Q_e \\ne 0\\}\\Bigg(\n\\\\&\\left(1-p_s\\right)\\bigg[p_t \\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}+\\left(1-p_t\\right)\\overline{P}_{{\\rm out},p}\\bigg]\n\\\\&+p_s P_{\\rm MD}\\bigg[p_f \\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}+\\left(1-p_f\\right)\\overline{P}_{{\\rm out},p}\\bigg]\n\\\\&+p_s \\left(1-P_{\\rm MD}\\right)\\bigg[p_b \\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}+\\left(1-p_b\\right)\\overline{P}_{{\\rm out},p}\\bigg]\\Bigg)\n\\end{split}\n\\end{equation}\n\n\n\\subsection{Backlogged Secondary User}\n\nSince the queues are interacting with each other, the exact analysis of the system is infeasible. In order to make the queueing analysis tractable, we make the following two approximations. We assume that a dummy data packet is transmitted from $Q_s$ when it is empty. That is, we assume a backlogged secondary user who always has data packets to transmit. In addition, we assume, as in \\cite{pappas2011optimal}, that one energy packet is consumed at each time slot regardless of actual secondary activity. The first approximation reduces the primary throughput as it eliminates the possibility of an empty secondary queue, which corresponds to an interference-free situation for the PU and a reduced outage probability. The second approximation means that the energy queue empties faster than in the actual system, thereby lowering the probability that the queue is nonempty. This reduces the secondary throughput by increasing the probability that the secondary node does not have energy. Therefore, our approximations, for each value of $\\lambda_p$, result in a {\\bf lower bound} on the secondary service rate or throughput.\n\n\n\n\\begin{figure}\n \n  \\includegraphics[width=1\\columnwidth]{DS_MC}\\\\\n  \\caption{Markov chain of the PU when the secondary user does not use primary feedback. State self-transitions are omitted for visual clarity. Probabilities $\\overline{\\mu_p}=1-\\mu_p$ and $\\overline{\\lambda_p}=1-\\lambda_p$.}\\label{fig22}\n\\end{figure}\n\nUnder the aforementioned assumption, the probability of the energy queue being nonempty is $\\lambda_e$ \\cite{pappas2011optimal}. This is because $\\mu_e=1$ and the probability of the energy queue being empty is $1-\\frac{\\lambda_e}{\\mu_e}$. The average data service rate of $Q_p$ is given by\n\n\\begin{equation}\n\\begin{split}\n\\mu_p&=\\bigg(1-\\lambda_e\\bigg)\\overline{P}_{{\\rm out},p}+\\lambda_e\\Bigg(\n\\\\&\\left(1-p_s\\right)\\bigg[p_t \\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}+\\left(1-p_t\\right)\\overline{P}_{{\\rm out},p}\\bigg]\n\\\\&+p_s P_{\\rm MD}\\bigg[p_f \\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}+\\left(1-p_f\\right)\\overline{P}_{{\\rm out},p}\\bigg]\n\\\\&+p_s \\left(1-P_{\\rm MD}\\right)\\bigg[p_b \\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}+\\left(1-p_b\\right)\\overline{P}_{{\\rm out},p}\\bigg]\\Bigg)\n\\end{split}\n\\label{mup}\n\\end{equation}\n\\noindent Since ${\\rm Pr}\\{Q_p=0\\}=1-\\frac{\\lambda_p}{\\mu_p}$,\n\\begin{equation}\n\\begin{split}\n\\mu_s&=\\lambda_e \\Bigg[\\bigg(1-\\frac{\\lambda_p}{\\mu_p}\\bigg) \\Bigg(\\left(1-p_s\\right)p_t {\\overline{P}}_{{\\rm out},0s}+\n\\\\&p_s p_b P_{\\rm FA} {\\overline{P}}_{{\\rm out},1s}+p_s p_f  \\left(1-P_{\\rm FA}\\right){\\overline{P}}_{{\\rm out},1s} \\Bigg)+\n\\\\&\\frac{\\lambda_p}{\\mu_p}\\Bigg(\\left(1-p_s\\right)p_t {\\overline{P}}_{{\\rm out},0s}^{\\left({\\rm c}\\right)} +p_s p_f  P_{\\rm MD} {\\overline{P}}_{{\\rm out},1s}^{\\left({\\rm c}\\right)}\n\\\\&+p_s p_b   \\left(1-P_{\\rm MD}\\right) {\\overline{P}}_{{\\rm out},1s}^{\\left({\\rm c}\\right)} \\Bigg)\\Bigg]\n\\end{split}\n\\label{mus}\n\\end{equation}\n\nThe Markov chain modeling the primary queue is provided in Fig. \\ref{fig22}. Solving the state balance equations, it is straightforward to show that the probability that the primary queue has $k$ packets is\n\\begin{equation}\n\\nu_k=\\nu_{\\circ}\\frac{1}{\\overline{\\mu_p}}\\Bigg[\\frac{\\lambda_p \\overline{\\mu_p}}{\\overline{\\lambda_p }\\mu_p}\\Bigg]^{k}\n\\label{nuk}\n\\end{equation}\n\\noindent where $\\overline{\\lambda_p}=1-\\lambda_p$ and $\\overline{\\mu_p}=1-\\mu_p$. Using the condition $\\sum_{k=0}^{\\infty}\\nu_k=1$,\n\\begin{equation}\n\\nu_{\\circ}=1-\\frac{\\lambda_p}{\\mu_p}\n\\end{equation}\n\\noindent For the sum $\\sum_{k=0}^{\\infty}\\nu_k$ to exist, we should have $\\lambda_p < \\mu_p$. This is equivalent to Loynes' theorem which states that if the arrival and service processes of a queue are strictly stationary processes, then the queue is stable if and only if the mean arrival rate is less than the mean service rate \\cite{loynes1962stability}.\n\n\\subsection{Primary User Packet Delay}\nLet $D_p$ be the average delay of the primary queue. Using Little's law and (\\ref{nuk}),\n\\begin{equation}\n\\begin{split}\nD_p=\\frac{1}{\\lambda_p}\\sum_{k=1}^{\\infty}k\\nu_k=\\frac{1-\\lambda_p}{\\mu_p-\\lambda_p}\n\\end{split}\n\\label{190990}\n\\end{equation}\n \n\\noindent For the optimal random access and sensing, we solve the following constrained optimization problem. We maximize the mean secondary service rate under the constraints that the primary queue is stable and that the primary packet delay is smaller than or equal a specified value $\\overline{D_p}$. The optimization problem with\n $\\mu_p$ given in (\\ref{mup}) and $\\mu_s$ in (\\ref{mus}) can be written as\n\\begin{equation}\n\\begin{split}\n& \\max_{p_s,p_f,p_b,p_t} \\,\\,\\mu_s\\\\\n&\\,\\,{\\rm s.t.} \\,\\,\\,\\,\\  0 \\le p_s,p_f,p_b,p_t \\le 1\\\\\n&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\lambda_p\\le \\mu_p\\\\& \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,D_p \\le \\overline{D_p}\n\\end{split}\n\\label{190990}\n\\end{equation}\nThis optimization problem and the others presented in this work are solved numerically\\footnote{Specifically, we use Matlab's fmincon. Since the problems are nonconvex, fmincon produces a locally optimum solution. To enhance the solution and increase the likelihood of obtaining the global optimum, the program is run many times with different initializations of the optimization variables.}.\n\n\\section{Feedback-based Access}\\label{sec3}\nIn this section we analyze the use of the primary feedback messages by the cognitive terminal. In the feedback-based access scheme, the secondary user utilizes the available primary feedback information for accessing the channel in addition to spectrum sensing. Leveraging the primary feedback is valid when it is available and unencrypted.\n\nIn the proposed scheme, the secondary user monitors the PU feedback\nchannel. It may overhear an acknowledgment (ACK) if the primary receiver correctly decodes the primary transmission, a negative acknowledgment (NACK) if decoding fails, or nothing if there is no primary transmission. We introduce the following modification to the protocol introduced earlier in the paper. If a NACK is overheard by the secondary user, it assumes that the PU will retransmit the lost packet during the next time slot \\cite{KarimSultan}. Being sure that the PU will be active, the secondary terminal does not need to sense the channel to ascertain the state of primary activity. Therefore, it just accesses the channel with some probability $p_r$. If an ACK is observed on the feedback channel or no primary feedback is overheard, the secondary user proceeds to operate as explained earlier in Section \\ref{sec2}. We assume the feedback packets are very short compared to $T$ and are always received correctly by both the primary and secondary terminals due to the use of strong channel codes. \n\nIt is important to emphasize here the benefit of employing primary feedback. By avoiding spectrum sensing, the secondary terminal does not have to waste $\\tau$ seconds for sensing. It can use the whole slot duration for transmission. As proven in Appendix C, this reduces the outage probability of the secondary link. Therefore, by differentiating between the primary states of transmission, i.e., whether they are following the reception of an ACK or not, the secondary user can potentially enhance its throughput by eliminating the need for spectrum sensing when the PU is about to retransmit a previously lost packet.\n\n\\subsection{Queueing Analysis with a Backlogged Secondary User}\n\n\\begin{figure}\n \n  \\includegraphics[width=1\\columnwidth]{MC}\\\\\n  \\caption{Markov chain of the PU for the feedback-based access scheme. Probabilities $\\overline{\\Gamma_p}=1-\\Gamma_p$ and $\\overline{\\alpha_p}=1-\\alpha_p$. State self-transitions are not depicted for visual clarity.}\\label{fig2}\n\\end{figure}\n\nThe PU's queue evolution\nMarkov chain is shown in Fig. \\ref{fig2}. The probability of the queue\nhaving $k$ packets and transmitting for the first time is $\\pi_k$, where\n$F$ in Fig. \\ref{fig2} denotes first transmission. The probability of the\nqueue having $k$ packets and retransmitting is $\\epsilon_k$, where $R$ in\nFig. \\ref{fig2} denotes retransmission. Define $\\alpha_p$ as the probability of successful transmission of the PU's packet in case of first transmission and $\\Gamma_p$ is the probability of successful transmission of the PU's packet in case of retransmission. It can be shown that both probabilities are given by:\n \\begin{equation}\\label{def}\n\\begin{split}\n   \\alpha_p&=\\bigg(1-\\lambda_e\\bigg)\\overline{P}_{{\\rm out},p}+\\lambda_e\\Bigg(\n\\\\&\\left(1-p_s\\right)\\bigg[p_t \\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}+\\left(1-p_t\\right)\\overline{P}_{{\\rm out},p}\\bigg]\n\\\\&+p_s P_{\\rm MD}\\bigg[p_f \\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}+\\left(1-p_f\\right)\\overline{P}_{{\\rm out},p}\\bigg]\n\\\\&+p_s \\left(1-P_{\\rm MD}\\right)\\bigg[p_b \\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}+\\left(1-p_b\\right)\\overline{P}_{{\\rm out},p}\\bigg]\\Bigg)\n    \\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n\\Gamma_p&=\\bigg(1-\\lambda_e\\bigg)\\overline{P}_{{\\rm out},p}+\\lambda_e \\bigg[p_r \\overline{P}_{{\\rm out},p}^{\\left({\\rm c}\\right)}+\\left(1-p_r\\right)\\overline{P}_{{\\rm out},p}\\bigg]\n\\end{split}\n\\end{equation}\nSolving the state balance equations, we can obtain the state probabilities which are provided in Table \\ref{table}. The probability $\\pi_0$ is obtained using the normalization condition $\\sum_{k=0}^{\\infty} (\\pi_k+\\epsilon_k)=1$.\n\nIt should be noticed that $\\lambda_p < \\eta$, where $\\eta$ is defined in Table \\ref{table}, is a condition for the sum $\\sum_{k=0}^{\\infty} (\\pi_k+\\epsilon_k)$ to exist. This condition ensures the existence of a stationary distribution for the Markov chain and guarantees the stability of the primary queue. The service rate of the secondary user is given by:\n\\begin{equation}\n\\begin{split}\n\\mu_s&=\\lambda_e \\Bigg[\\pi_0 \\Bigg(\\left(1-p_s\\right)p_t {\\overline{P}}_{{\\rm out},0s}+\n\\\\&p_s p_b P_{\\rm FA} {\\overline{P}}_{{\\rm out},1s}+p_s p_f  \\left(1-P_{\\rm FA}\\right){\\overline{P}}_{{\\rm out},1s} \\Bigg)+\n\\\\&\\Bigg(\\sum_{k=1}^{\\infty}\\pi_k\\Bigg)\\Bigg(\\left(1-p_s\\right)p_t {\\overline{P}}_{{\\rm out},0s}^{\\left({\\rm c}\\right)} +p_s p_f  P_{\\rm MD} {\\overline{P}}_{{\\rm out},1s}^{\\left({\\rm c}\\right)}\n\\\\&+p_s p_b   \\left(1-P_{\\rm MD}\\right) {\\overline{P}}_{{\\rm out},1s}^{\\left({\\rm c}\\right)} \\Bigg)+\\Bigg(\\sum_{k=1}^{\\infty}\\epsilon_k\\Bigg)p_r \\overline{P}_{{\\rm out},0s}^{\\left({\\rm c}\\right)}\\Bigg]\n\\end{split}\n\\label{muss}\n\\end{equation}\n\\noindent where the probability summations are given in Table \\ref{table}.\n\\begin{table}\n\\renewcommand{\\arraystretch}{2}\n\\begin{center}\n\\begin{tabular}{ c |c  }\n    \\hline\\hline\n    $\\eta$ & $\\lambda_p\\alpha_p+\\left(1-\\lambda_p\\right)\\Gamma_p$ \\\\[5pt]\\hline\n  $\\pi_{\\circ}$ & $\\frac{\\eta-\\lambda_p}{\\Gamma_p}$ \\\\[5pt]\\hline\n  $\\epsilon_{\\circ}$ & 0 \\\\[5pt]\\hline\n  $\\pi_1$& $\\pi_{\\circ}\\tfrac{\\lambda_p}{1-\\lambda_p}\\frac{\\lambda_p+\\left(1-\\lambda_p\\right)\\Gamma_p}{\\eta}$  \\\\[5pt]\\hline\n  $\\epsilon_1$ & $\\pi_{\\circ}\\frac{\\lambda_p}{\\eta}\\left(1-\\alpha_p\\right)$ \\\\ [5pt]\\hline\n  $\\pi_k,k\\geq 2$ & $\\pi_{\\circ}\\frac{\\lambda_p\\left(1-\\alpha_p\\right)}{\\left(1-\\eta\\right)^2} \\bigg[ \\frac{\\lambda_p \\left(1-\\eta\\right)}{\\left(1-\\lambda_p\\right) \\eta}  \\bigg]^k$ \\\\[5pt]\\hline\n  $\\epsilon_k,k\\geq 2$ & $\\pi_{\\circ}\\frac{\\left(1-\\lambda_p\\right)\\left(1-\\alpha_p\\right)}{\\left(1-\\eta\\right)^2} \\bigg[ \\frac{\\lambda_p \\left(1-\\eta\\right)}{\\left(1-\\lambda_p\\right) \\eta}  \\bigg]^k$ \\\\[5pt]\\hline\n  $\\sum_{k=1}^{\\infty}\\pi_k$ & $\\pi_{\\circ}\\frac{\\lambda_p\\Gamma_p}{\\eta-\\lambda_p}=\\lambda_p$\\\\[5pt]\\hline\n   $\\sum_{k=1}^{\\infty}\\epsilon_k$ & $\\pi_{\\circ}\\frac{\\lambda_p}{\\eta-\\lambda_p}\\left(1-\\alpha_p\\right)=\\frac{\\lambda_p }{\\Gamma_p}\\left(1-\\alpha_p\\right)$\n\\end{tabular}\n\\caption{State probabilities for the feedback-based access scheme.}\n\\label{table}\n\\end{center}\n\\end{table}\n\n\\subsection{Primary User Packet Delay}\nApplying Little's law, the primary queueing delay is given by\n\\begin{equation}\nD_p=\\frac{1}{\\lambda_p}\\sum_{k=1}^{\\infty} k \\left(\\pi_k+\\epsilon_k\\right)\n\\end{equation}\n\\noindent Using the state probabilities provided in Table \\ref{table},\n\\begin{equation}\n\\begin{split}\nD_p &=\\frac{(\\alpha_p-\\eta)(\\eta-\\lambda_p)^2+\\left(1-\\lambda_p\\right)^2 \\left(1-\\alpha_p\\right) \\eta}{(\\eta-\\lambda_p)\\left(1-\\lambda_p\\right) \\left(1-\\eta\\right) \\Gamma_p}\n  \\end{split}\n\\end{equation}\n For a fixed $\\lambda_p$, the maximum mean\nservice rate for the secondary user is given by solving the following\noptimization problem using expression (\\ref{muss}) for $\\mu_s$\n\\begin{equation}\n\\begin{split}\n& \\max_{p_s,p_f,p_t,p_b,p_r} \\,\\,\\mu_s\\\\\n&\\,\\,{\\rm s.t.} \\,\\,\\,\\,\\  0 \\le p_s,p_f,p_t,p_b,p_r \\le 1\\\\\n&\\,\\,\\,\\ \\lambda_p \\le \\eta\n\\\\\n&\\,\\,\\,\\ D_p \\le \\overline{D_p}\n\\end{split}\n\\label{190990}\n\\end{equation}\n\n\n\\section{Numerical Results}\\label{numerical}\nWe provide here some numerical results for the optimization problems presented in this paper and compare with a random access system without spectrum sensing. A random access scheme is simply obtained from the schemes described in this paper by setting $p_s$ to zero. We define here two variables $\\delta=\\frac{{\\overline{P}}_{{\\rm out},1s}}{{\\overline{P}}_{{\\rm out},0s}}$ and $\\delta^c=\\frac{{\\overline{P}}_{{\\rm out},1s}^{\\left({\\rm c}\\right)}}{{\\overline{P}}_{{\\rm out},0s}^{\\left({\\rm c}\\right)}}$, both of them are less than $1$ as shown in Appendix C. Figs.\\ \\ref{r1} and \\ref{r9} provide a comparison between the maximum secondary rate of our proposed systems and the random access systems. The difference between the two figures is that the delay of the PU is constrained by $\\overline{D_p}=2$ time slots and $\\overline{D_p}=200$ time slots in Figs. \\ref{r1} and \\ref{r9}, respectively. The figures show clearly the advantage of employing random sensing and access over a purely random access system. Moreover, the benefit of leveraging the primary feedback is evident from the figures. Fig.\\ \\ref{r5} provides the optimal sensing and access probabilities corresponding to the proposed scheme employing feedback in Fig.\\ \\ref{r1}.\n\nTo facilitate investigating the impact of the specified delay constraint, Fig.\\ \\ref{delay_fig} provides the effect of varying this constraint on the secondary service rate. As is clear from the figure, the secondary service rate is reduced when the primary queueing delay constraint is more strict. Fig.\\ \\ref{rx} shows the impact of the energy arrival rate. As expected, the secondary service rate is reduced when $\\lambda_e$ decreases. Finally, Fig.\\ \\ref{ry} shows the impact of the MPR capability. Without MPR capability, collisions are assumed to lead to sure packet loss. Therefore, a collision model without MPR corresponds to the case of the probabilities of correct reception being zero when there are simultaneous transmissions. As shown in  Fig.\\ \\ref{ry}, the secondary service rate is reduced when there is no MPR capability.\n\n\n\n\\section{conclusion} \\label{secfinal}\nIn this paper, we discussed an energy harvesting cognitive radio. We proposed a novel sensing\/access scheme to enable the secondary user efficiently use its own energy to access and sense the channel. We considered two schemes based on feedback of the primary receiver channel. We compared our results with the random access scheme with and without using the feedback information. The results have shown the benefit of employing the primary feedback in making secondary access decisions.\n\n\\section*{Appendix A}\nWe adopt a flat fading channel model and assume that the channel gains remain constant over the duration of the time slot. We do not assume the availability of transmit channel state information (CSI) at the transmitting terminals. The transmitters adjust their transmission rates depending on when they start transmission during the time slot.\nAssuming that the number of bits in a packet is $b$, the transmission rate is\n\\begin{equation}\nr_i=\\frac{b}{T\\left(1-i\\frac{\\tau}{T}\\right)}.\n\\label{r_i}\n\\end{equation}\n\\noindent where $\\tau$ is the amount of time spent to sense the channel, $\\tau < T$, $i=0$ if transmission proceeds at the very beginning of the time slot, and $i=1$ if transmission is preceded by a spectrum sensing period of $\\tau$ units of time. Outage for a secondary user occurs when the transmission rate exceeds the channel capacity\n\\begin{equation}\nP_{{\\rm out},is}={\\rm Pr}\\{r_i > W \\log_{2}\\left(1+\\gamma \\beta\\right)\\}\n\\end{equation}\n\\noindent where $W$ is the bandwidth of the channel, $\\gamma$ is the received SNR when the channel gain is equal to unity, and $\\beta$ is the channel gain, which is exponentially distributed in the case of Rayleigh fading.\n\nThe outage probability can be written as\n\\begin{equation}\nP_{{\\rm out},is}={\\rm Pr}\\Big\\{\\beta<\\frac{2^{\\frac{r_i}{W}}-1}{\\gamma}\\Big\\}\n\\end{equation}\n\\noindent Assuming that the mean value of $\\beta$ is $\\overline{\\beta}$,\n\\begin{equation}\nP_{{\\rm out},is}=1-\\exp\\bigg(-\\frac{2^{\\frac{r_i}{W}}-1}{\\gamma\\overline{\\beta}}\\bigg)\n\\end{equation}\n\\noindent Let $\\overline{P}_{{\\rm out},is}=1-P_{{\\rm out},is}$ be the probability of correct reception. It is therefore given by\n\\begin{equation}\\label{correctreception}\n\\overline{P}_{{\\rm out},is}=\\exp\\bigg(-\\frac{2^{\\frac{b}{TW\\left(1-i\\frac{\\tau}{T}\\right)}}-1}{\\gamma\\overline{\\beta}}\\bigg)\n\\end{equation}\n\\noindent The probability $\\overline{P}_{{\\rm out},p}$ for the PU is given by a similar expression with $i=0$ as the PU transmits starting from the beginning of the time slot and the relevant primary link parameters.\n\n\\begin{figure}\n \n  \\includegraphics[width=1\\columnwidth]{1}\\\\\n   \\caption{Maximum secondary service rate for the parameters: $\\lambda_e=0.8$, $\\overline{P}_{{\\rm out},p}=0.7$, ${\\overline{P}}_{{\\rm out},p}^{\\left({\\rm c}\\right)}=0.14$, $\\overline{P}_{{\\rm out},0s}=0.6065$, ${\\overline{P}}_{{\\rm out},0s}^{\\left({\\rm c}\\right)}=0.1820$, $\\delta=0.9782$,  $\\delta^{\\rm c}=0.8$, $P_{\\rm FA}=0.1$, $P_{\\rm MD}=0.08$, and $\\overline{D_p}=2$ time slot.}\\label{r1}\n\\end{figure}\n\\begin{figure}\n \n  \\includegraphics[width=1\\columnwidth]{5}\\\\\n   \\caption{Optimal values of the sensing and access probabilities for the feedback-based system depicted in Fig.\\ \\ref{r1}.}\\label{r5}\n\\end{figure}\n\n\\begin{figure}\n \n  \\includegraphics[width=1\\columnwidth]{9}\\\\\n   \\caption{Maximum secondary service rate for the parameters: $\\lambda_e=0.8$, $\\overline{P}_{{\\rm out},p}=0.7$, ${\\overline{P}}_{{\\rm out},p}^{\\left({\\rm c}\\right)}=0.14$, $\\overline{P}_{{\\rm out},0s}=0.6065$, ${\\overline{P}}_{{\\rm out},0s}^{\\left({\\rm c}\\right)}=0.1820$, $\\delta=0.9782$,  $\\delta^{\\rm c}=0.8$, $P_{\\rm FA}=0.1$, $P_{\\rm MD}=0.08$, and $\\overline{D_p}=200$ time slot.}\\label{r9}\n\\end{figure}\n\n\n \n \n  \n\n\n\\begin{figure}\n  \\includegraphics[width=1\\columnwidth]{delay_fig}\\\\\n   \\caption{Maximum secondary service rate for different values of the primary packet delay constraint. Parameters used to generate the figure are: $\\lambda_e=0.8$, $\\overline{P}_{{\\rm out},p}=0.7$, ${\\overline{P}}_{{\\rm out},p}^{\\left({\\rm c}\\right)}=0.14$, $\\overline{P}_{{\\rm out},0s}=0.6065$, ${\\overline{P}}_{{\\rm out},0s}^{\\left({\\rm c}\\right)}=0.1820$, $\\delta=0.9782$,  $\\delta^{\\rm c}=0.8$, $P_{\\rm FA}=0.1$, and $P_{\\rm MD}=0.08$}\\label{delay_fig}\n\\end{figure}\n\n\\begin{figure}\n \n  \\includegraphics[width=1\\columnwidth]{beta_curves2}\\\\\n   \\caption{Impact of the energy arrival rate $\\lambda_e$ on the maximum secondary service rate for the parameters: $\\overline{P}_{{\\rm out},p}=0.7$, ${\\overline{P}}_{{\\rm out},p}^{\\left({\\rm c}\\right)}=0.14$, $\\overline{P}_{{\\rm out},0s}=0.6065$, ${\\overline{P}}_{{\\rm out},0s}^{\\left({\\rm c}\\right)}=0.1820$, $\\delta=0.9782$,  $\\delta^{\\rm c}=0.8$, $P_{\\rm FA}=0.1$, $P_{\\rm MD}=0.08$, and $\\overline{D_p}=2$ time slot.}\\label{rx}\n\\end{figure}\n\n\\begin{figure}\n \n  \\includegraphics[width=1.09\\columnwidth]{MPR_versus_stability}\\\\\n   \\caption{Impact of the MPR capability on the maximum secondary service rate for the parameters: $\\lambda_e=0.5$, $\\overline{P}_{{\\rm out},p}=0.7$, $\\overline{P}_{{\\rm out},0s}=0.6065$, $\\delta=0.9782$,  $\\delta^{\\rm c}=0.8$, $P_{\\rm FA}=0.1$, $P_{\\rm MD}=0.08$, and $\\overline{D_p}=2$ time slot.}\\label{ry}\n\\end{figure}\n\n\n\\section*{Appendix B}\nWe derive here a generic expression for the outage probability at the receiver of link $j$ when there is concurrent transmission from the transmitter of link $v$. Outage occurs when the transmission rate $r_i$, given by (\\ref{r_i}), exceeds the channel capacity\n\\begin{equation}\nP_{{\\rm out},ij}^{\\left({\\rm c}\\right)}={\\rm Pr}\\biggr\\{r_i > W \\log_{2}\\left(1+\\frac{\\gamma_{j} \\beta_{j}}{\\gamma_{v} \\beta_{v}+1}\\right)\\biggr\\}\n\\end{equation}\n\\noindent where the superscript c denotes concurrent transmission, $\\gamma_j$ is the received SNR at the receiver of link $j$ without interference when the channel gain $\\beta_j$ is equal to unity, and $\\gamma_v$ is the received SNR at the receiver of link $j$ when it only receives a signal from the interfering transmitter of link $v$ given that the gain of the interference channel, $\\beta_v$, is equal to unity.\nThe outage probability can be written as\n\\begin{equation}\\label {1900}\nP_{{\\rm out},ij}^{\\left({\\rm c}\\right)}={\\rm Pr}\\Big\\{\\frac{\\gamma_{j} \\beta_{j}}{\\gamma_{v} \\beta_{v}+1}<{2^{\\frac{r_i}{W}}-1}\\Big\\}\n\\end{equation}\n\\noindent Assuming that $\\beta_j$ and $\\beta_v$ are exponentially distributed with means $\\overline{\\beta_{j}}$ and $\\overline{\\beta_{v}}$, respectively, we can use the probability density functions of these two random variables to obtain the outage as\n \\begin{eqnarray*}\\label{193}\n P_{{\\rm out},ij}^{\\left({\\rm c}\\right)}=1-\\frac{1}{1+\\Big( {2^{\\frac{r_i}{W}}-1} \\Big)\\frac{\\gamma_v\\overline{\\beta_v}}{ \\gamma_j\\overline{\\beta_j}}} {e^{-\\frac{{2^{\\frac{r_i}{W}}-1}}{\\gamma_j \\overline{\\beta_j}}}}\n\\end{eqnarray*}\n\\noindent The probability of correct reception $  \\overline{P}^{\\left({\\rm c}\\right)}_{{\\rm out},ij}=1-P^{\\left({\\rm c}\\right)}_{{\\rm out},ij}$ is thus given by\n\n  \\begin{eqnarray}\\label{conctra}\n \\overline{P}_{{\\rm out},ij}^{\\left({\\rm c}\\right)}=\\frac{\\overline{P}_{{\\rm out},ij}}{1+\\Big({2^{\\frac{b}{TW\\left(1-\\frac{i\\tau}{T}\\right)}}-1} \\Big)\\frac{\\gamma_v\\overline{ \\beta_v}}{\\gamma_j \\overline{\\beta_j}}}\n\\end{eqnarray}\n\\noindent As is obvious, the probability of correct reception is lowered in the case of interference.\n\\section*{Appendix C}\nWe prove here that $\\overline{P}_{{\\rm out},0j} > \\overline{P}_{{\\rm out},1j}$. As a function of $\\tau$, $\\overline{P}_{{\\rm out},1j}$ is given by (\\ref{correctreception}) with $i=1$. Let $x=1-\\frac{\\tau}{T}$ where\n$x \\in [0,1]$. Assuming that the energy unit used per slot is $e$, the transmit power is $\\frac{e}{T-\\tau}=\\frac{e}{Tx}$. This means that the received SNR $\\gamma$ is inversely proportional to $x$. The exponent in (\\ref{correctreception}) with $i=1$ is thus proportional to $\\boldsymbol{g(x)=x (\\exp(\\frac{a}{x})-1)}$ where $a =\\frac{b \\ln 2} {WT} > 0$. Differentiating $g(x)$ with respect to $x$, the derivative is\n\\begin{equation}\n\\begin{split}\n-1+\\Big[1-\\frac{a}{x}\\Big] \\exp(\\frac{a}{x})&=-1+\\Big[1-\\frac{a}{x}\\Big]\\sum_{k=0}^{\\infty}\\frac{1}{k!}\\Big(\\frac{a}{x}\\Big)^{k}\\\\&=\n-1+\\!\\sum_{k=0}^{\\infty}\\frac{1}{k!}\\Big(\\frac{a}{x}\\Big)^{k}\\!-\\!\\sum_{k=0}^{\\infty}\\frac{1}{k!}\\Big(\\frac{a}{x}\\Big)^{k+1}\\\\&=\n\\sum_{k=1}^{\\infty}\\frac{1}{k!}\\Big(\\frac{a}{x}\\Big)^{k}-\\sum_{k=1}^{\\infty}\\frac{1}{(k-1)!}\\Big(\\frac{a}{x}\\Big)^{k}\\\\&=\n\\sum_{k=1}^{\\infty}\\bigg[\\frac{1}{k!}-\\frac{1}{(k-1)!}\\bigg]\\Big(\\frac{a}{x}\\Big)^{k+1}<0\n\\end{split}\n\\end{equation}\nTherefore, the derivative is always negative. Since $x=1-\\frac{\\tau}{T}$, function $g(x)$ increases with $\\tau$. This means that $\\overline{P}_{{\\rm out},1j}$ decreases with $\\tau$ and its maximum value occurs when the transmission starts at the beginning of the time slot, i.e., $\\tau=0$. This proves that $\\overline{P}_{{\\rm out},0j} > \\overline{P}_{{\\rm out},1j}$ for $\\tau > 0$. For the case of concurrent transmission given in (\\ref{conctra}), the numerator is $\\overline{P}_{{\\rm out},1j}$, which decreases with $\\tau$. The denominator is proportional to $g(x)$ which has been shown to decrease with $x$ and, hence, increase with $\\tau$. This means that $\\overline{P}^{\\left({\\rm c}\\right)}_{{\\rm out},1j}$ decreases with $\\tau$, i.e.,  $\\overline{P}^{\\left({\\rm c}\\right)}_{{\\rm out},0j} > \\overline{P}^{\\left({\\rm c}\\right)}_{{\\rm out},1j}$.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nA set of pieces on a chessboard is said to be independent if no piece may attack another.  Independence problems on chessboards have long been studied; both in terms of maximum arrangements as well as the number of such arrangements.  For all traditional chess pieces, kings, queens, bishops, rooks, knights, and pawns, the maximum size of an independent set is known.  When enumerating maximum arrangements, some of the problems, for example in the case of rooks or bishops, have elementary solutions.  (See Dudeney~\\cite{Dudeney} for an early discussion of independence problems.)  For other pieces, such as in the case of queens, the number of maximum independent arrangements is unknown, or in the case of kings an asymptotic approximation is given by Larson~\\cite{Larson}, but an exact value is unknown.  Here we wish to enumerate the number of maximum arrangements of nonattacking pawns.  Arrangements of nonattacking pawns have been studied by Kitaev and Mansour~\\cite{Kitaev_Mansour} who provide upper and lower bounds on the number of arrangements of pawns on $m\\times n$ rectangles using Fibonacci numbers as well as an algorithm to generate an explicit formula.\n\nAs there are only two distinct arrangements for odd length chessboards, we focus on boards with even length.  Because we can divide a $2m\\times 2m$ chessboard into $m^2$ $2\\times 2$ squares each with at most two pawns, the maximum number of independent pawns is at most $2m^2$.  This value is easily achieved, and examples are illustrated in Figure~\\ref{pawns}.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics{EvenLengthArrangements}\n\\caption{Arrangements of nonattacking pawns for even length chessboards}\n\\label{pawns}\n\\end{figure}\n\nWe will provide a bijection between the set of maximum nonattacking arrangements of pawns on a $2m\\times 2m$ chessboard and the set of subsets of $m$ rows and $m$ columns of a $2m\\times 2m $ matrix.\n\n\\section{Bijection}\nInstead of considering full arrangements of nonattacking pawns on a $2m\\times 2m$ chessboard, we first consider arrangements on a $2\\times 2$ chessboard.  There are four possible arrangements labeled with A, B, C, and D, as illustrated in Figure~\\ref{2times2}.  We define a function $f$ on this set, where $f(A) = D$ and $f(B)=f(C)=f(D)=C$.  We use this function to define an $(m+1)\\times (m+1)$ matrix $M_{2m} = (m_{i,j})_{1\\leq i, j \\leq m+1}$ whose entries correspond to arrangements of $2m$ independent pawns on a $2 \\times 2m$ rectangular chessboard.\n   \n\\begin{figure}[ht]\n\\centering\n\\includegraphics{2x2Types}\n\\caption{The four maximum arrangements of 2 pawns on a $2\\times 2$ chessboard}\n\\label{2times2}\n\\end{figure}\n\n\\begin{definition}\\label{defM2m}\nLet $M_{2m}=(m_{i,j})_{1\\leq i,j \\leq m+1}$ be the matrix who entries consist of arrangements of $2m$ nonattacking pawns on a $2\\times 2m$ rectangular chessboard. We can think of each rectangle as a string of $m$ $2\\times 2$ squares, each with exactly two pawns.  The entries of $M_{2m}$ are defined as follows:\n\\begin{enumerate}[i.]\n\\item For $1\\leq j \\leq m+1$, let $m_{1,j}$ be the arrangement where the leftmost $(m+1-j)$ $2\\times 2$ squares of the rectangular chessboard are of Type A and the remaining rightmost $(j-1)$ squares are of Type B.   \n\\item For $1\\leq i \\leq m+1$, use $m_{1,j}$ to generate the arrangements $m_{i,j}$ by replacing the leftmost $(i-1)$ $2\\times 2$ squares of $m_{1,j}$, with their image under the function $f$ and leaving the rightmost $(m+1-i)$ $2\\times 2$ squares fixed. \n\\end{enumerate}\n\\end{definition}\n\n\nSee Figure~\\ref{entries} for an example of an entry in the first row and fifth row of $M_{14}$, and see Figure~\\ref{M6} for the entire matrix $M_6$.  We claim this matrix contains all possible nonattacking arrangements of pawns on a $2\\times 2m$ rectangular chessboard.\n \n\\begin{figure}[ht]\n\\centering\n\\includegraphics{ExampleEntries}\n\\caption{Entries from the matrix $M_{14}$}\n\\label{entries}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\scalebox{0.9}{\\includegraphics{M6}}\n\\caption{Entries in the matrix $M_6 = (m_{i,j})_{1\\leq i,j \\leq 4}$}\n\\label{M6}\n\\end{figure}\n\n\\begin{proposition}\\label{uniqueness}\nEvery nonattacking arrangement of $2m$ pawns on a $2\\times 2m$ rectangle appears exactly once in the matrix $M_{2m}$.\n\\end{proposition}\n\n\\begin{proof}\nTo begin, we show the number of distinct arrangements of pawns on a $2\\times 2m$ rectangle is $(m+1)^2$.  For $m=1$, a $2\\times 2$ square has the four distinct arrangements shown in Figure~\\ref{2times2}, so we induct on $m$.  The leftmost $2\\times 2$ square of a $2\\times 2m$ rectangle may have Type A, B, C, or D.  First, assume this leftmost square has Type D.  Any maximum independent arrangement of a $2\\times 2(m-1)$ rectangle may be appended to the Type D square creating $m^2$ distinct maximum nonattacking arrangements.  Next, if the leftmost square has Type A or C, it must be followed by a square of same type or of Type B.  But in any $2\\times 2m$ rectangle, when reading from left to right, as soon as a Type B square is introduced in the strip, all remaining squares to the right must also be of Type B.  Thus any $2\\times 2m$ strip beginning with a Type A or Type C square consists of $k$ squares of Type A or C followed by $m-k$ squares of Type B for $1\\leq k \\leq m$.  Finally there is one possible arrangement beginning with a Type B square.  Thus we have\n\\begin{equation*}\nm^2 +2m+1 = (m+1)^2\n\\end{equation*}\ndistinct arrangements as desired.\n\nFurther, no arrangement appears more than once in the matrix $M_{2m}$.  We continue to think of the entries of the matrix $M_{2m}$ as a string of $m$ $2\\times 2$ squares.  We observe, by construction, as one reads from top to bottom down a column of the matrix, the only actions on these $2\\times 2$ squares are:\n\\begin{enumerate}[i.]\n\\item Type A squares may be changed to Type D squares.\n\\item Type B squares may be changed to Type C squares.\n\\item Any type square may remain fixed.\n\\end{enumerate}\nSimilarly, as you read from left to right across a row of the matrix, the only actions are:\n\\begin{enumerate}[i.]\n\\item Type A squares may be changed to Type B squares.\n\\item Type D squares may be changed to Type C squares.\n\\item Any type square may remain fixed.\n\\end{enumerate}\nGiven any two arrangements in distinct positions in the matrix $M_{2m}$, at least one square has changed from the lower-indexed entry to the higher-indexed entry.  If that square was of Type B or D, respectively, it was changed into a Type C square and no action may change it back to a Type B or D square, respectively.  If the square was of Type A, then it was changed to a Type B, C, or D square, but in any case, may not return to Type A.  Because Type C squares cannot be changed, we have a matrix with unique elements whose size is equal to the size of the set, so therefore each independent maximum arrangement of pawns occurs exactly once in $M_{2m}$.\n\\end{proof}\n\n\nNow, we define a map from the set of subsets of $m$ rows and $m$ columns of a $2m\\times 2m$ matrix into the set of nonattacking arrangements of $2m^2$ pawns.\n\\begin{definition}\nSuppose the rows and columns of a $2m\\times 2m$ matrix are indexed by $[2m]$.  Set \n\\begin{equation*}\nA =\\{ C\\cup R : C,R\\subset [2m] \\hbox{ and } |C|=|R|=m\\},\n\\end{equation*}\nthat is, $A$ is the set of all subsets\nconsisting of $m$ rows $R=\\{r_1, r_2, \\ldots, r_{m}\\}\\subset [2m]$ and $m$ columns $C=\\{c_1,c_2, \\ldots, c_{m}\\}\\subset [2m]$.  Let $B$ be the set of all nonattacking arrangements of $2m^2$ pawns on a $2m\\times 2m$ chessboard.  Define the map $\\Phi: A \\longrightarrow B$ as follows:\n\nGiven a subset $C\\cup R$, assume without loss of generality that $r_1 < r_2< \\cdots <r_m$ and $c_1 < c_2 < \\cdots < c_m$.  Then set $S$ to be the set of $m$ ordered pairs where \n\\begin{equation*}\nS=\\{(a_i, b_i) : (a_i, b_i) = (r_i-i+1, c_i-i+1) \\hbox{ for } 1\\leq i \\leq m\\}.\n\\end{equation*}\nFor each ordered pair $(a_i, b_i)$, identify the $2\\times 2m$ chessboard arrangement $m_{a_i,b_i}$ from the matrix $M_{2m}$.  Concatenate these strips sequentially so that $m_{a_i,b_i}$ is directly above $m_{a_{i+1}, b_{i+1}}$ for $1\\leq i \\leq m-1$ to create an arrangement of $2m^2$ pawns on a $2m\\times 2m$ chessboard.  This arrangement is the image of the subset $C\\cup R$ under $\\Phi$.\n\\end{definition}\n\n\\begin{example}\nGiven $2m=6$, suppose $R=\\{1,4,5\\}$ and $C=\\{2,4,6\\}$.  Then $S=\\{(1,2), (3,3), (3,4)\\}$.  Thus, we concatenate the arrangements $m_{1,2}, m_{3,3}, m_{3,4}$ from Figure~\\ref{M6} to get the maximum $6\\times 6$ arrangement:\n\\begin{center}\n\\includegraphics{Example6x6}\n\\end{center}\n\\end{example}\n\n\\begin{example}\nGiven $2m=8$, suppose $R=\\{2,3,4,8\\}$ and $C=\\{1,6,7,8\\}$.  Then $S=\\{(2,1),(2,5),(2,5),(5,5)\\}$ and we have the following $8\\times 8$ arrangement:\n\\begin{center}\n\\includegraphics{Example8x8}\n\\end{center}\n\\end{example}\n\n\nWe check that the arrangements of pawns given by the function $\\Phi$ are nonattacking.\n\\begin{proposition}\\label{independent}\nEach arrangement of $2m^2$ pawns on a $2m\\times 2m$ chessboard in the image $\\Phi(A)$ is independent.\n\\end{proposition}\n\n\n\\begin{proof}\nBy construction, we know the the pawns may not attack within each $2\\times 2m$ rectangular chessboard, so it is left to show that the pawns may not attack from one rectangle to another.  We apply the restrictions on movement along rows and columns noted in the proof of Proposition~\\ref{uniqueness}.\n\n Let $a=m_{i,j}$ and $b=m_{i',j'}$ be any two nonattacking arrangements from the matrix $M_{2m}$ such that $i\\leq i'$ and $j\\leq j'$.  We assume $a$ lies directly above $b$ in a maximum arrangement of independent pawns on the $2m\\times 2m$ chessboard. Divide each $2\\times 2m$ rectangle into $2\\times 2$ squares and denote a $2\\times 2$ square of $a$, or $b$ respectively, at position $k$ where $1\\leq k \\leq m$ by $A_k$ or $B_k$, respectively.  First, if $A_k$ has Type A, then the pawns in $A_k$ may not attack any pawns in the arrangement $b$.  Next, suppose $A_k$ has Type B, so thus $B_k$ has Type B or Type C.  In either case the pawns in $A_k$ may not attack the pawns in $B_k$.  However the pawn in $A_k$ may also attack the upper left corner of $B_{k+1}$.  Because the square $A_{k+1}$ must also have Type B, we know $B_{k+1}$ has Type B or C.  In either case there is no pawn in the upper left corner, so pawns in $A_k$ may not attack pawns in $B_{k+1}$.  Similarly, if $A_k$ has Type D then $B_i$ also has Type D, and thus no attack is possible.  In this case a pawn in $A_i$ could also attack the upper right corner of $B_{i-1}$.  We see that $A_{i-1}$ also has Type D, so $B_{k-1}$ has Type D and thus no attack is possible from $A_k$ to $B_{k-1}$.  Finally, suppose $A_k$ is of Type C, so pawns in $A_k$ may attack squares $B_{k-1}, B_k$, and $B_{k+1}$.  We know $A_{k-1}$ is of Type C or Type D and $A_{k+1}$ is of Type C.  So we have that $B_{k-1}$ is of Type C  or D, thus not susceptible to an attack from $A_k$.  The squares $B_k$ and $B_{k+1}$ are both of Type C and also have no pawns that may be attacked by pawns in $A_k$.  Finally, we note in any case, if the squares $B_{k-1}$ or $B_{k+1}$ do not exist, then trivially there is no attacking pawn. Therefore, we have shown that any entry weakly to the left or above another entry in $M_{2m}$ may not attack when placed directly above the second entry, and thus have proven the claim.\n\\end{proof}\n\n\nWe have shown that each subset in $A$ provides exactly one maximum nonattacking arrangement of pawns on a $2m\\times 2m$ chessboard, thus $\\Phi(A) \\subseteq B$.  It is left to show that no other maximum independent arrangements are possible.\n\n\\begin{proposition}\nEvery nonattacking arrangement of $2m^2$ pawns on a $2m\\times 2m$ chessboard is the image of a subset $C\\cup R \\in A$ under the map $\\Phi$.\n\\end{proposition}\n\n\\begin{proof}\nAny arrangement of $2m^2$ pawns on a $2m\\times 2m$ chessboard may be divided into $m$ $2\\times 2m$ rectangular boards which correspond to the entries $(m_{a_1, b_1}, \\ldots, m_{a_m, b_m})$ in the matrix $M_{2m}$.  For all $i$, as long as $a_i\\leq a_{i+1}$ and $b_i \\leq b_{i+1}$, then the arrangement is an element of the image $\\Phi(A)$.  Suppose to the contrary $a_i > a_{i+1}$ for some $i$.  This implies the arrangement $m_{a_i, b_i}$ is in a lower row in $M_{2m}$ than arrangement $m_{a_{i+1},b_{i+1}}$, but appears directly above $m_{a_{i+1},b_{i+1}}$ in the $2m\\times 2m$ arrangement. We apply a similar argument to that used in Proposition~\\ref{independent}.  At least one square, say $A_k$ in $m_{a_i,b_i}$ is different from the square in the same position, $B_k$, in $m_{a_{i+1},b_{i+1}}$.  If $A_k$ is of type D, then $B_k$ is of type A or C, hence the pawn in the lower left corner of $A_k$ may attack the pawn in the upper right corner of $B_k$.  If $A_k$ is of Type C, then $B_k$ is of Type A or B, and the pawn in the lower left corner of $A_k$ may attack the pawn in the upper right corner of $B_k$.  Thus $a_i \\not> a_{i+1}$.  Similarly, if $b_i > b_{i+1}$, the arrangement $m_{a_i, b_i}$ is in column further to the right in $M_{2m}$ than arrangement $m_{a_{i+1},b_{i+1}}$, but appears directly above $m_{a_{i+1},b_{i+1}}$ in the $2m\\times 2m$ arrangement.  Again at least one square, say $A_k$ in $m_{a_i,b_i}$ is different from the square in the same position, $B_k$, in $m_{a_{i+1},b_{i+1}}$.  If $A_k$ of Type B, then $B_k$ is of Type A or D and the pawn in the lower right corner of $A_k$ may attack the pawn in the upper left corner of $B_k$.  Further if $A_k$ is of Type C, then $B_k$ is of Type A or D and the pawn in the lower right corner of $A_k$ may attack the pawn in the upper left corner of $B_k$.  Thus $b_i \\not> b_{i+1}$, and we have arrived at the contradiction.\n\\end{proof}\n\n Therefore we have the following corollary.\n\n\\begin{corollary}\nThe function $\\Phi: A \\longrightarrow B$ is a bijection.\n\\end{corollary}\n\nHence, because we may choose an $m$-subset of $[2m]$ in ${2m\\choose m}$ ways, we have our main result.\n\n\\begin{theorem}\nThe number of maximum nonattacking arrangements of pawns on a $2m\\times 2m$ chessboard is ${2m\\choose m}^2$.\n\\end{theorem}\n\n\nWe may generalized this result to maximum independent arrangements of pawns on $2n\\times 2m$ rectangles.\n\n\\begin{theorem}\nThe number of maximum nonattacking arrangements of pawns on a $2n\\times 2m$ chessboard is ${m+n\\choose n}^2$.\n\\end{theorem}\n\\begin{proof}\nAssume without loss of generality that $n\\leq m$.  We may utilize the bijection $\\Phi$ from above.  Given a nonattacking arrangement of $2mn$ pawns on a $2n\\times 2m$ chessboard, we may divide the arrangement into $n$ rectangles of size $2\\times 2m$.  These correspond to $n$ (not necessarily distinct) entries in the matrix $M_{2m}$.  Thus we have a set of indices from the matrix entries\n\\begin{equation*}\nS=\\{(a_i, b_i) | 1 \\leq a_1 \\leq a_2 \\leq \\cdots \\leq a_n\\leq m+1 \\hbox{ and }1 \\leq b_1 \\leq b_2 \\leq \\cdots \\leq b_n \\leq m+1 \\}.\n\\end{equation*}\nTwo create distinct column and row entries we have\n\\begin{equation*}\nC \\cup R = \\{a_1, a_2+1, a_3+2, \\ldots, a_n+n-1\\} \\cup \\{b_1, b_2+1, b_2+3, \\ldots, b_n+n-1\\}.\n\\end{equation*}\nWe note the maximum value of elements in $C$ or $R$ is $m+n$, thus $C, R \\subset [m+n]$.  Hence we are choosing an $n$-subset of rows from $[m+n]$ and an $n$-subset of columns from $[m+n]$, and the result follows.\n\\end{proof}\n\n\n\n\n\n\\newcommand{\\journal}[6]{{\\sc #1,} #2, {\\it #3}, {\\bf #4}, #6 (#5)}\n\\newcommand{\\book}[4]{{\\sc #1,} #2, #3 (#4)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $(x_n)\\in[0,1)$ be a sequence, $s>0$ be a real number and $N$ be a natural number. The pair correlation statistic of $(x_n)$ is defined as follows:\n\\[R_2(s,(x_n),N) :=\\frac{1}{N}\\#\\Big\\{1\\leq n\\neq m\\leq N: \\|x_n-x_m\\|\\leq \\frac{s}{N}\\Big\\},\\]\nwhere for any real $x$, $\\displaystyle\\|x\\|:=\\inf_{m\\in\\mathbb{Z}}|x+m|,$ the nearest integer distance. The sequence $(x_n)$ is said to have Poissonian pair correlation (PPC) if for all $s>0$,\\\\\n\\[\\lim_{N\\to\\infty}R_2(s,(x_n),N)= 2s .\\]\nThis concept originated from theoretical physics and it plays a crucial role in the Berry\u2013Tabor conjecture.\nRudnick and Sarnak~\\cite{rudnick1998pair} first studied this notion from a mathematical point of view. Since then this topic achieved wide attention~\\cite{RSZ2001, RZ1999}, several generalizations are known, see e,g.,~\\cite{hauke2021weak,hinrichs2019multi,marklof2020pair,steinerberger2020poissonian}.\n\nThe theory of uniform distribution (or equidistribution) of a sequence has a long history. An interesting connection between the pair correlation and uniform distribution of a sequence is known. This two properties of a sequence are different in nature but they are not independent. Infact, it is known that a Poissonian pair correlated sequence necessarily uniformly distributed(see ~\\cite{aistleitner2018pair,steinerberger2020poissonian,Grepstad2017Larcher}) but the converse is not true.\n\nOnly recently, a concept of pair correlation for the higher dimensional sequences have introduced in \\cite{hinrichs2019multi} with respect to sup-norm and in \\cite{steinerberger2020poissonian} with respect to $2$-norm. Let  $d \\geq 2$ and throughout the article we also assume the same. For $\\boldsymbol{x}=(x^{(1)},\\ldots,x^{(d)}) \\in \\mathbb{R}^d$, we denote $\\|\\boldsymbol{x}\\|_\\infty=\\max{\\{\\|x^{(1)}\\|,\\ldots,\\|x^{(d)}\\|\\}}$ and $\\|\\boldsymbol{x}\\|_2=(\\|x^{(1)}\\|^2+\\cdots +\\|x^{(d)}\\|^2)^{1\/2}$.\n\tLet $ (\\boldsymbol{x}_n)_{n\\geq 1}$ be a sequence in $[0,1)^d$. For $s>0$ we write\n\t\\[R_{2,\\infty}^{(d)}(s,(\\boldsymbol{x}_n),N):=\\displaystyle\\frac{1}{N}\\#\\Big\\{1\\leq m\\neq n\\leq N: \\|\\boldsymbol{x}_n-\\boldsymbol{x}_m\\|_\\infty\\leq \\frac{s}{N^{1\/d}}\\Big\\}\\] \n\tand, \\[R_{2,2}^{(d)}(s,(\\boldsymbol{x}_n),N):=\\displaystyle\\frac{1}{N}\\#\\Big\\{1\\leq n\\neq m\\leq N: \\|\\boldsymbol{x}_n-\\boldsymbol{x}_m\\|_2\\leq \\frac{s}{N^{1\/d}}\\Big\\}.\\]\n\n\t\\begin{definition}\n\tA sequence $(\\boldsymbol{x}_n)_{n\\geq 1}$ in $\\mathbb{R}^d$ is said to have $\\infty$-PPC if for all $s>0$, \n\t\\[R_{2,\\infty}^{(d)}(s,(\\boldsymbol{x}_n),N) \\to (2s)^d \\text{ as }N\\to\\infty,\\] \n\tand $2$-PPC if for all $s>0$,\n\t\\[R_{2,2}^{(d)}(s,(\\boldsymbol{x}_n),N) \\to w_ds^d \\text{ as }N\\to\\infty,\\]\n\twhere $w_d$ is the volume of unit ball of $\\mathbb{R}^d$ in $2$-norm.\n\\end{definition}\nSimilar to one dimensional case, it has been proved in \\cite{hinrichs2019multi} that  $\\infty$-PPC implies uniform distribution and in \\cite{steinerberger2020poissonian} that $2$-PPC implies uniform distribution for higher dimensional sequences.\n\nThe purpose of this article is to show $\\infty$-PPC and $2$-PPC for some higher dimensional sequences of the form $(\\{a_n^{(1)}\\alpha_1\\},\\ldots,\\{a_n^{(d)}\\alpha_d\\})$, where $(a_n^{(i)})$, for $i=1,2,\\ldots,d$ are sequences of natural numbers and  $\\boldsymbol{\\alpha}=(\\alpha_1,\\ldots,\\alpha_d)\\in\\mathbb{R}^d$. For simplicity, in this case we denote the respective  pair correlation statistic by $R_{2,\\infty}^{(d)}(s,\\boldsymbol{\\alpha},N)$ and $R_{2,2}^{(d)}(s,\\boldsymbol{\\alpha},N)$. \nWe say that $(a_n^{(1)},\\ldots,a_n^{(d)})$ has metric Poissonian pair correlation with respect to sup-norm ($\\infty$-MPPC) if \\[R_{2,\\infty}^{(d)}(s,\\boldsymbol{\\alpha},N) \\rightarrow (2s)^d \\mbox{ as } N\\rightarrow \\infty,\\] for almost all $\\boldsymbol{\\alpha}\\in\\mathbb{R}^d$. Moreover, $2$-MPPC is defined analogously.\n\n\nOur results depend on the notion of additive energy of integers sequences. For a finite subset $A$ of integers, the additive energy $E(A)$ of $A$ is defined by\n\\[E(A):=\\#\\{a,b,c,d\\in A: a+b=c+d\\}.\\]\n\nRecently, Aistleitner et al.~\\cite{aistleitner2017additive} proved that  for a strictly increasing sequence $(a_n)$ of natural numbers the sequence $(\\{\\alpha a_n\\})$ has Poissonian pair correlation for almost all $\\alpha,$ provided $E(A_N)\\ll N^{3-\\epsilon}$ for some $\\epsilon>0$, where $A_N$ denotes the set of first $N$ elements of $(a_n)$. In \\cite{Bloom2019GCDSA}, Bloom and Walker improved their result by relaxing the condition on the upper bound of additive energy.\nAnalogous result for some special higher dimensional sequences has been established in \\cite{hinrichs2019multi} with respect to sup-norm. Specifically, they proved the following theorem.\n\\begin{thm}\n\tLet $(a_n)$ be a strictly increasing sequence of natural numbers, $A_N$ denote the first $N$ elements of $(a_n)$ and suppose that\n\t\\[E(A_N)=\\operatorname{O}\\Big(\\frac{N^3}{(\\log N)^{1+\\epsilon}}\\Big), \\text{ for some }\\epsilon>0,\\]\n\tthen for almost all $\\boldsymbol{\\alpha}=(\\alpha_1,\\ldots,\\alpha_d)\\in\\mathbb{R}^d$, $(\\{a_n\\boldsymbol{\\alpha}\\})=(\\{a_n\\alpha_1\\},\\ldots,\\{a_n\\alpha_d\\})$ has $\\infty$-PPC.\n\\end{thm}\n\nWe consider more general sequences, namely, $(\\{a_n^{(1)}\\alpha_1\\},\\ldots,\\{a_n^{(d)}\\alpha_d\\})$ and want to show that they have $\\infty$-PPC and $2$-PPC. To state our results we introduce joint notion of additive energy for several increasing sequences of natural numbers. \n\\begin{definition}[Joint additive energy]\n\tLet $(a_n^{(1)}),(a_n^{(2)}),\\ldots,(a_n^{(d)})$ be strictly increasing sequences of natural numbers and $A_N^{(i)}$ denote the first $N$ elements of $(a_n^{(i)})$, for $1\\leq i\\leq d.$ \nThe joint additive energy $E(A_N^{(1)};\\ldots;A_N^{(d)})$ is given by\n\t\\[E\\Big(A_N^{(1)};\\ldots;A_N^{(d)}\\Big)=\\#\\Big\\{1\\leq n,m,k,l\\leq N: a_n^{(i)}+a_m^{(i)}=a_k^{(i)}+a_l^{(i)}, i=1,\\ldots,d\\Big\\}.\n\t\\]\n\\end{definition}\n\\noindent Note that joint additive energy is additive energy in higher dimension.\nIn section~\\ref{s2} we study joint additive energy in details and obtain an upper bound of it.\n \nNow, we state our results.\n\\begin{theorem}\\label{infinity-PPC thm}\n\tLet $(a_n^{(1)}),(a_n^{(2)}),\\ldots,(a_n^{(d)})$ be strictly increasing sequences of natural numbers and $A_N^{(i)}$ denote the first $N$ elements of $(a_n^{(i)})$, for $1\\leq i\\leq d.$ Suppose that for some $\\delta>0$,\n\t\\[E\\Big(A_N^{(1)};\\ldots;A_N^{(d)}\\Big) = \\operatorname{O}\\left( N^{3-\\delta}\\right).\\]\n\tThen $(a_n^{(1)},\\ldots,a_n^{(d)})$ has $\\infty$-MPPC.\n\\end{theorem}\n\\begin{corollary}\\label{infinity-PPC thm cor}\n\tSuppose that for some $\\delta>0$,\n\t\\[\\displaystyle \\min_{1\\leq i\\leq d}E(A_N^{(i)}) = \\operatorname{O}\\left( N^{3-\\delta}\\right).\\]\n\tThen $(a_n^{(1)},\\ldots,a_n^{(d)})$ has $\\infty$-MPPC.\n\\end{corollary}\n\n\n  \n  In \\cite{steinerberger2020poissonian}, Steinerberger introduced the notation of $2$-PPC but did not mention any sequences which satisfy $2$-PPC. In following theorems we study the $2$-PPC of certain sequences.\n\\begin{theorem}\\label{2-PPC thm1}\n\tLet $(a_n)$ be a strictly increasing sequence of natural numbers and $A_N$ denote the first $N$ elements of $(a_n).$\n\tAssume that \n\t\\[\\displaystyle E(A_N)=\\operatorname{O}\\Big(\\frac{N^3}{(\\log N)^{1+\\epsilon}}\\Big),\\text{ for some $\\epsilon>0$.}\\]\n\tThen for almost all $\\boldsymbol{\\alpha}=(\\alpha_1,\\ldots,\\alpha_d)\\in\\mathbb{R}^d$, the sequence $(\\{a_n\\boldsymbol{\\alpha}\\})=(\\{a_n\\alpha_1\\},\\ldots,\\{a_n\\alpha_d\\})$ has $2$-PPC.\n\\end{theorem}\n\\begin{theorem}\\label{2-PPC thm2}\n\tLet $(a_n),(b_n)$ be strictly increasing sequences of natural numbers and $A_N$, $B_N$ denote their first $N$ elements of $(a_n)$ and $(b_n)$ respectively. Suppose that\n\\begin{align*}\nE\\Big(A_N^{(1)};A_N^{(2)}\\Big)=\\operatorname{O}\\Big(N^{3}\\exp{\\big(-(\\log{N})^{\\frac{1}{2}+\\delta}\\big)}\\Big),\\,\\mbox{ for some } \\delta>0.\n\\end{align*}\n\tThen $(a_n,b_n)$ has $2$-MPPC.\n\\end{theorem}\n\n\\begin{remark}\nFor an arbitrary strictly increasing sequence $(a_n)$ of natural numbers and for any $l\\geq 2\\in\\mathbb{N},$\n$(a_n,n^l)$ has $\\infty$-MPPC and $2$-MPPC. \nIn particular, $(\\{n\\alpha\\},\\{n^2\\beta\\})$ has $\\infty$-PPC and $2$-PPC for almost all $(\\alpha,\\,\\beta)\\in\\mathbb{R}^2.$  It is known that $(\\{n\\alpha\\})_{n\\in \\mathbb{N}}$ does not have PPC for any $\\alpha\\in \\mathbb{R}$(for instance see ~\\cite{LS}).\nThus, the $\\infty$-PPC and $2$-PPC in higher dimension does not imply that each component has PPC.\n\\end{remark}\n\n\n\nNext, we obtain another such example in two dimension. \n\\begin{theorem}\\label{thm6}\n\tLet $A \\in [1,\\,2]$ be any real number. Then $(n,[n(\\log n)^A])$ has $\\infty$-MPPC and $2$-MPPC.\n\\end{theorem}\nCurrently we do not know any result showing that $(\\{[n(\\log n)^A]\\alpha_2\\})$ has PPC. But, from \\cite[Corollary 1]{Garaev2004} we obtain the associate additive energy $\\ll N^3(\\log N)^{1-A}$.\n\nWe fix some notations here to use later sections.\\\\\n\\textbf{Notations:}\nLet $h\\geq 2$ be an integer, $a$ be a real number and   $\\boldsymbol{x}=(x_1,\\ldots,x_h),\\boldsymbol{y}=(y_1,\\ldots,y_h)\\in\\mathbb{R}^h.$ \n\\begin{itemize}\n\t\\item Coordinate wise product $\\boldsymbol{x}\\boldsymbol{y}=(x_1y_1,\\ldots,x_hy_h)$\n\t\\item Inner product $\\boldsymbol{x}.\\boldsymbol{y}=\\sum_{1\\leq i\\leq h}x_iy_i$\n\t\\item Fractional part $\\{\\boldsymbol{x}\\}=(\\{x_1\\},\\ldots,\\{x_h\\}).$\n\t\\item Scaler product $a\\boldsymbol{x}=(ax_1,\\ldots,ax_h)$\n\t\\item $\\Sigma_{\\boldsymbol{x}\\in\\mathbb{Z}^h}^{'}$ means all components of $\\boldsymbol{x}$ are non-zero.\n\\end{itemize}\nFurther, for $a, b \\in \\mathbb{Z}$ their greatest common divisor(GCD)  is  $(a, b).$ Define $e(x):=e^{2\\pi ix}$ for $x \\in \\mathbb{R}$. \n\nLet $h\\geq 1$. If $a_n^{(i)}=a_n$ for all $i$ then we denote $A_N^{(i)}$'s by $A_N$ and $E(A_N^{(1)};\\ldots;A_N^{(h)})$ by $E(A_N).$\nFor $\\boldsymbol{v}=(v_1,\\cdots,v_h), v_i\\neq 0\\in\\mathbb{Z},$ we define representation function $\\mathcal{R}_N(\\boldsymbol{v})$ by,\n\\begin{align}\\label{R_N}\n\\mathcal{R}_N(\\boldsymbol{v}):=\\#\\{1\\leq m\\neq n\\leq N: a_n^{(i)}-a_m^{(i)}=v_i,\\:1\\leq i\\leq h\\}.\n\\end{align}\nWe use $\\mathcal{R}_N(\\boldsymbol{v})$ and $\\mathcal{R}_N(v_1,\\ldots,v_h)$ interchangeably.\nOne can see that, \n\\[\\displaystyle\\sum_{\\boldsymbol{v}\\in\\mathbb{Z}^h}\\mathcal{R}_N(\\boldsymbol{v})^2\\leq E(A_N^{(1)};\\ldots;A_N^{(d)}).\\]\n\n\n\\section{Joint additive energy}\\label{jadditive}\\label{s2}\nLet $A_N^{(i)}$ denote the first $N$ elements of $(a_n^{(i)})$, for $1\\leq i\\leq d$. It is easy to see that the joint additive energy satisfy the following trivial estimate \n\\[N^2\\leq E(A_N^{(1)};\\ldots;A_N^{(d)})\\leq\\min_{1\\leq i\\leq d}{E(A_N^{(i)})}\\leq N^3.\n\\]\nFrom this observation we conclude that for any strictly increasing sequence $A:=(a_n)_{1\\leq n\\leq N}$ of $N$ natural numbers and any fixed integer $l\\geq 2$, $B_l:=\\{n^l: 1\\leq n\\leq N\\}$ we have $E(A;B)\\ll N^{2+\\epsilon}$ for any $\\epsilon>0$ (since $E(B)\\ll N^{2+\\epsilon}$, see ~\\cite{aistleitner2017additive}).\n\n\nFor $1\\leq s\\leq d$, Vinogradov's mean value $J_{s,d}(N)$ is given by the number of solutions in $\\mathbb{N}$ of the system\n\\[x_1^{i}+\\cdots+x_s^i = y_1^{i}+\\cdots+y_s^i, \\quad 1\\leq i\\leq d.\\] For $B_l:=\\{n^l: 1\\leq n\\leq N\\}$, we see that $E(B_1;B_2;\\ldots;B_d)$ is equal to the Vinogradov's mean value $J_{2,d}(N)$. \nThen from the work of Ford and Wooley~\\cite{FW} we get\n\\begin{equation*}\nE(B_1;B_2;\\ldots;B_d)= J_{2,d}(N) \\ll N^{2+\\epsilon},\n\\end{equation*}\nfor any $\\epsilon>0$. In this section, we concentrate on obtaining the joint additive energy of certain sequences. A natural question is how small the joint additive energy can be, whenever all the components have considerably large additive energy. For example, \\cite[Theorem 1]{Garaev2001} says that the sequence $([n\\log n])$ has additive energy $\\gg \\frac{N^3}{\\log N}$, but we show that the joint additive energy of $(n)$ and \n$([n\\log n])$ is much smaller.\n\nLet $(f(n))$, $(g(n))$ be two strictly increasing sequences of natural numbers and $F_N$, $G_N$ be the set of their first $N$ elements respectively. For a given positive integer $l$ with $1\\leq l< N$ assume that  $J_l:=J_l(N)$ is  the number of solutions of the system of equations\n\\begin{align}\\label{jointeq1}\n\\begin{cases}\n\tf(x)+f(y)=f(x+l)+f(z)\\\\\n\tg(x)+g(y)=g(x+l)+g(z),\n\t\\end{cases}\n\t\\text{where } \\,1\\leq x<x+l\\leq z<y\\leq N.\n\\end{align}\nWe have the following upper bound for the joint additive energy.\n\\begin{proposition}\\label{bound thm}\n\tFor any $0<\\epsilon<1$ we have\n\t\\[\\frac{N^4}{(f(N)+1)(g(N)+1)} \\ll E(F_N;G_N)\\ll N^{2+\\epsilon}+N^\\epsilon\\displaystyle\\sum_{1\\leq l\\leq N^{1-\\epsilon}}J_l.\\]\n\\end{proposition}\n\\begin{proof} We adopt the idea of the proof of \\cite[Theorem 1]{Garaev2004} and \\cite[Theorem 1]{Garaev2001} for upper and lower bound respectively to obtain these estimates.\\\\\nFor any real $\\alpha_1, \\alpha_2$ we set \n\t\\[S(\\alpha_1,\\alpha_2)=\\sum_{1\\leq n\\leq N}e\\big(f(n)\\alpha_1+g(n)\\alpha_2\\big).\\] \nThen by orthogonality of the exponential function we obtain\n\\begin{equation}\\label{4th moment}\nE(F_N;G_N)=\\int_{[0,1]^2}|S(\\alpha_1,\\alpha_2)|^4d\\boldsymbol{\\alpha}.\n\\end{equation}\nfor integers $1\\leq s\\leq[N^\\epsilon],$ set\n $I_s:=\\{n\\in\\mathbb{Z}\\colon(s-1)N^{1-\\epsilon}<n\\leq sN^{1-\\epsilon}\\}$ and $I_{[N^\\epsilon]+1}:=\\{n\\in\\mathbb{Z}\\colon [N^\\epsilon]N^{1-\\epsilon}<n\\leq N\\}.$ Then, by separating diagonal and off-diagonal terms, applying triangular inequality and the partition of $[1,N]$ in to $I_s$'s we deduce\n\t\\begin{align*}\n\t\t|S(\\alpha_1,\\alpha_2)|^4 &\\ll N^2+\\Big|\\displaystyle\\sum_{1\\leq s\\leq [N^\\epsilon]+1}\\sum_{n\\in I_s}\\sum_{1\\leq n<m\\leq N}e\\big((f(n)+f(m))\\alpha_1+(g(n)+g(m))\\alpha_2\\big)\\Big|^2.\n\t\\end{align*}\nNow, by applying Cauchy-Schwarz inequality on the sum over $s$  we get\n\\begin{align*}\n\t\t|S(\\alpha_1,\\alpha_2)|^4 \\ll N^2+N^\\epsilon\\displaystyle\\sum_{1\\leq s\\leq [N^\\epsilon]+1}\\Big|\\sum_{n\\in I_s}\\sum_{1\\leq n<m\\leq N}e\\big((f(n)+f(m))\\alpha_1+(g(n)+g(m))\\alpha_2\\big)\\Big|^2.\n\t\\end{align*}\nWe apply the above bound of $|S(\\alpha_1,\\alpha_2)|^4$ in \\eqref{4th moment}, expand the squares and integrate to obtain \n\\begin{align}\\label{28\/6}\nE(F_N;G_N)\\ll N^2+N^\\epsilon\\displaystyle\\sum_{1\\leq s\\leq [N^\\epsilon]+1}\\sum_{n,n_1\\in I_s}\\sum_{\\substack{1\\leq n<m\\leq N\\\\1\\leq n_1<m_1\\leq N\\\\f(n)+f(m)=f(n_1)+f(m_1)\\\\g(n)+g(m)=g(n_1)+g(m_1)}}1.\n\\end{align}\nObserve that $n, n_1 \\in I_s$ imply that $|n-n_1|\\leq N^{1-\\epsilon}$. If $n_1=n$ then $m=m_1$ and in this case the contribution of the sums in the right hand side of \\eqref{28\/6} is $N^2$. Otherwise, by writing $|n-n_1|=l$ the contribution for such sums \n\\[ \\ll \\sum_{l\\leq N^{1-\\epsilon}} J_l.\\]\nCombining these estimates with \\eqref{28\/6} we obtain the upper bound.\n\nTo estimate the lower bound we start with the identity:\n\\begin{align*}\nN^2=\\int_{[0,1]^2}S^2(\\alpha_1,\\alpha_2) \\sum_{m=0}^{2f(N)}e(-m\\alpha_1)  \\sum_{n=0}^{2g(N)}e(-n\\alpha_2) d\\boldsymbol{\\alpha}\n\\end{align*}\nHence the lower bound follows from the above identity after applying Cauchy-Schwarz inequality to the right hand side of this identity.\n\\end{proof}\n\\begin{theorem}\\label{n and F(n) thm}\n\tLet $f(n)=n$ and $g(n)=[F(n)]$, where $F$ is a real valued function which is three times continuously differentiable on the segment $[1,N]$ with $F^{'}(n)>0$, $F^{''}(n)>0$ and $F^{'''}(n)<0$. For any real $0<\\epsilon<1$,\n\t\\[E(F_N;G_N)\\ll N^{2+\\epsilon}+\\frac{N^{1+\\epsilon}\\log N}{F^{''}(N)}.\\]\n\\end{theorem}\nThe following corollary is immediate.\n\\begin{corollary}\\label{coro2.3}\nLet $1\\leq A \\leq 2$ and $0<\\epsilon<1$. \n\tIf $F(n)=n(\\log n)^A$ then, \\[E(F_N;G_N)\\ll N^{2+\\epsilon}(\\log{N})^{2-A}.\\]\n\\end{corollary}\n\\begin{proof}[Proof of Theorem \\ref{n and F(n) thm}]\n\tHere we take  $f(n)=n$ and $g(n)=[F(n)]$, so $J_l$ in  \\eqref{jointeq1}  reduces to the  number of solutions of the equation\n\t\\begin{align*}\n\t\t[F(n)]+[F(m+l)]=[F(n+l)]+[F(m)], \\mbox{ with }1\\leq n<n+l\\leq m<m+l\\leq N.\n\t\\end{align*}\nNow, following the proof of \\cite[Theorem 1]{Garaev2001} we get\n\\[J_l\\ll N\\Big(\\frac{2}{lF^{''}(N)}+1\\Big).\n\\]\nCombining this bound with Proposition \\ref{bound thm} we obtain the required upper bound.\n\\end{proof}\n\n\\section{``Generalized\" GCD Sums}\nGiven a function $f:\\mathbb{N}\\rightarrow \\mathbb{C}$ with finite support and $\\alpha \\in (0, 1]$, the so-called GCD sum is defined as \n\\[S_f(\\alpha):=\\sum_{a, b} f(a)\\overline{f(b)} \\frac{(a, b)^{2\\alpha}}{(ab)^\\alpha}.\\] \nThis sum plays a key role in finding large value of Riemann zeta function (see \\cite{BS,BT} ). Also it is connected with the theory of equidistribution and pair correlation (for instance, see \\cite{aistleitner2017additive, Bloom2019GCDSA}). \nWe introduce here a new type of GCD sums in higher dimension. Let $f:\\mathbb{N}^d \\rightarrow \\mathbb{C}$ be a function with finite support and $\\alpha \\in (0, 1]$. Define $d$ dimensional GCD sum\n\\[S_f(d;\\alpha):= \\displaystyle\\sum_{\\boldsymbol{a},\\boldsymbol{b} \\in \\mathbb{N}^d}f(\\boldsymbol{a})\\overline{f(\\boldsymbol{b})}\\prod_{i=1}^d\\frac{(a_i,b_i)^{2\\alpha}}{(a_ib_i)^\\alpha}.\\]\nThe next result provides an upper bound for $S_f(d;\\alpha)$.\n\n\\begin{proposition}\\label{gen gcd lemma}\n\tLet $f:\\mathbb{N}^d\\to\\mathbb{C}$ be an arithmetic function with finite support of cardinality $K.$ Then we have the following estimates:\n\t\\begin{align*}\n\tS_f(d;\\alpha) \\ll \\begin{cases}\n\t(\\log\\log K)^{\\operatorname{O}(1)}\\|f\\|_2^2, & \\mbox{ if } \\,\\alpha =1,\\\\\n\t\\exp(C(\\alpha)(\\log K)^{1-\\alpha})(\\log\\log K)^{-\\alpha})\\|f\\|_2^2, & \\mbox{ if } \\,\\frac{1}{2}<\\alpha<1,\n\t\\end{cases}\n\t\\end{align*}\n\twhere $C(\\alpha)$ is an absolute positive constant depending on $\\alpha$.\n\t\nFurther, if $f$ is positive real valued satyisfying $\\|f\\|_1\\geq 3$ and $K\\geq\\log\\|f\\|_1$. Then \n\t\\[ S_{f}\\left(d; {1}\/{2}\\right) \\ll \\frac{\\exp(C(\\log K \\log\\log \\|f\\|_1)^{1\/2})}{(\\log \\|f\\|_1+O(1))^d}\\|f\\|_2^2,\\] \n\tfor some absolute constant $C$.\n\\end{proposition}\n\nFor simplicity we prove Proposition~\\ref{gen gcd lemma} for the case $d=2$ by extending an idea of the random model given by Lewko and Radziwi{\\l\\l} in \\cite{lewko2017refinements}. For $d\\geq 3$, the proof is straightforward extension of the method.\n \nLet $\\{X(p): p \\text{ prime}\\}$ and $\\{Y(p): p \\text{ prime}\\}$ be two collections of independent random variables uniformly distributed on $\\mathbb{S}^1$. Also, assume that $\\{X(p),Y(p): p\\text{ prime}\\}$ is an independent collection. For every $n\\in\\mathbb{N}$ we define $X(n):=\\prod_{p^a\\parallel n}X(p)^a$ and similarly $Y(n):=\\prod_{p^a\\parallel n}Y(p)^a$. The random zeta function associated to $X(n)$ is defined by \n\\[\\zeta_X(\\alpha):=\\displaystyle\\sum_{n\\geq 1}\\frac{X(n)}{n^\\alpha}, \\text{ where $\\alpha>1\/2,$} \\]\nand $\\zeta_Y(\\alpha)$ is defined similarly.\nFor fixed $\\alpha>1\/2$, these series converge a.e. by Kolmogorov three series theorem (see \\cite[Theorem ~15.51]{K}). \n\nNow, we recall moment estimates of $\\zeta_X(\\alpha)$ from Lemma~ 7 of \\cite{Bloom2019GCDSA} and we use it to prove Proposition~\\ref{gen gcd lemma}.\n\\begin{lemma}\\label{moment lemma} For a real number $l$,\n\t\\begin{equation*}\n\t\t\\log\\mathbb{E}\\big[|\\zeta_X(\\alpha)|^{2l}\\big]\\ll\\begin{cases}\n\t\t\tl\\log\\log l, \\quad & \\mbox{if }\\,\\alpha=1, \\\\\n\t\t\tC'(\\alpha)l^{1\/\\alpha}(\\log l)^{-1}, \\quad &\\mbox{if }\\,  1\/2<\\alpha<1,\\\\\n\t\t\tl^2\\log((\\alpha-1\/2)^{-1}), \\quad & \\mbox{if }\\, \\alpha\\rightarrow \\frac{1}{2},\n\t\t\\end{cases}\n\t\\end{equation*}\nwhere $C'(\\alpha)$ is a positive constant, $l\\geq 3$ for first two cases and $l\\geq 1$ for the final case.\n\\end{lemma}\n\n\\begin{proof}[Proof of Proposition~\\ref{gen gcd lemma}]\n\t Let us start with defining the double sum\n\t$$D(X,Y):=\\displaystyle\\sum_{a,b}f(a,b)X(a)Y(b).$$\n\tThen, we look at the expectation of $|\\zeta_X(\\alpha)\\zeta_Y(\\alpha)D(X,Y)|^2$. Squaring this product and considering the expectation to get \n\t\\begin{align*}\n\t\\mathbb{E}\\big[|\\zeta_X(\\alpha)\\zeta_Y(\\alpha)D(X,Y)|^2\\big]=& \\displaystyle\\sum_{m_1,n_1,m_2,n_2,a,b,c,d}\\frac{f(a,b)\\overline{f(c,d)}}{m_1^\\alpha n_1^\\alpha m_2^\\alpha n_2^\\alpha}\\mathbbm{1}_{n_1a=n_2c}\\mathbbm{1}_{m_1b=m_2d}.\n\t\\end{align*}\nNow, indicator functions allow us to write $n_1=\\frac{h_1c}{(a,\\, c)},\\, n_2=\\frac{h_1a}{(a,\\, c)}$ and $m_1=\\frac{h_2d}{(b,\\, d)}, \\,m_2=\\frac{h_2b}{(b,\\, d)}$ for positive integers $h_1, h_2$. Thus,\n\t\\begin{align}\\label{eq0}\n\t\t\\mathbb{E}\\big[|\\zeta_X(\\alpha)\\zeta_Y(\\alpha)D(X,Y)|^2\\big]\n\t\t=& \\sum_{h_1,h_2,a,b,c,d}f(a,b)\\overline{f(c,d)}\\frac{(a,c)^{2\\alpha}(b,d)^{2\\alpha}}{(acbd)^\\alpha}\\frac{1}{h_1^{2\\alpha}h_2^{2\\alpha}}\\\\\n\t\\nonumber\t= &  \\displaystyle\\zeta(2\\alpha)^2S_f(2;\\alpha).\n\t\\end{align}\n\tAlso, we notice that $\\mathbb{E}\\big[|D(X,Y)|^2\\big]=\\|f\\|_2^2$.\n\tLet $V$ and $l$ be positive real parameters to be chosen later. Consider the event $\\mathcal{A}=(|\\zeta_X(\\alpha)|<V,|\\zeta_Y(\\alpha)|<V)$. Now by splitting the expectation over $\\mathcal{A}$ and its complement we get\n\t\\begin{align}\\label{eq1}\n\t\t\\mathbb{E}\\big[|\\zeta_X(\\alpha)\\zeta_Y(\\alpha)D(X,Y)|^2\\big] \\leq&V^4\\|f\\|_2^2+V^{2-2l}\\mathbb{E}\\big[|\\zeta_Y(\\alpha)|^{2+2l}|D(X,Y)|^2\\big]\\\\\n\t\t&+V^{2-2l}\\mathbb{E}\\big[|\\zeta_X(\\alpha)|^{2+2l}|D(X,Y)|^2\\big]\\nonumber\\\\\n\t\t &+V^{-4l}\\mathbb{E}\\big[|\\zeta_X(\\alpha)|^{2+2l}|\\zeta_Y(\\alpha)|^{2+2l}|D(X,Y)|^2\\big]. \\nonumber\n\t\\end{align}\nBy Cauchy-Schwarz inequality $|D(X,Y)|^2\\leq \\|f\\|_2^2K$. Using above,\n\t\\begin{align}\\label{eq2}\n\t\t\\mathbb{E}\\big[|\\zeta_X(\\alpha)\\zeta_Y(\\alpha)D(X,Y)|^2\\big] \\leq \\|f\\|_2^2 \\,(V^4 &+V^{2-2l}K\\mathbb{E}\\big[|\\zeta_Y(\\alpha)|^{2+2l}\\big]+V^{2-2l}K\\mathbb{E}\\big[|\\zeta_X(\\alpha)|^{2+2l}\\big]\\\\\n\t\t&+V^{-4l}K\\mathbb{E}\\big[|\\zeta_X(\\alpha)|^{2+2l}|\\zeta_Y(\\alpha)|^{2+2l}\\big]. \\nonumber\n\t\\end{align}\n\t\\begin{case}{ $\\alpha=1.$} Using Lemma~\\ref{moment lemma} in (\\ref{eq2}) we get the upper bound,\n\t\\begin{align*}\n\t\t \\|f\\|_2^2(V^4+V^{2-2l}K\\exp(Cl\\log\\log l)).\n\t\\end{align*}\n\tWe choose $l=\\log K+3$ and $V=(\\log l)^C$, where $C$ is a large positive constant and get\n\t\\begin{align*}\n\t\\mathbb{E}\\big[|\\zeta_X(\\alpha)\\zeta_Y(\\alpha)D(X,Y)|^2\\big] \\leq \\|f\\|_2^2(\\log\\log K)^{O(1)}.\n\t\\end{align*}\n\t\\end{case}\n\t\n\t\\begin{case}{ $\\frac{1}{2} < \\alpha <1$}.  Applying Lemma~\\ref{moment lemma} in (\\ref{eq2}) we get,\n\t\\begin{align*}\n\t\t\\mathbb{E}\\big[|\\zeta_X(\\alpha)\\zeta_Y(\\alpha)D(X,Y)|^2\\big]\\leq \\|f\\|_2^2\\,(V^4+V^{2-2l}K\\exp(C'(\\alpha)l^{1\/\\alpha}(\\log l)^{-1})).\n\t\t\\end{align*}\n\tChoosing $l=(\\log K)^\\alpha(\\log\\log K)^\\alpha+3$ and $V= \\exp(C(\\alpha)l^{-1+1\/\\alpha}(\\log l)^{-1})$ we get the upper bound,\n\t\\begin{align*}\n\t\t\t\\|f\\|_2^2\\, \\exp{(C(\\alpha)(\\log K)^{1-\\alpha}(\\log\\log K)^{-\\alpha})}.\n\\end{align*}\n\t\\end{case}\n\t\n\t\\begin{case}{ $\\alpha \\rightarrow \\frac{1}{2}$.} Let $\\alpha=1\/2+1\/\\log \\|f\\|_1$. Then, (\\ref{eq2}) and Lemma \\ref{moment lemma} give us,\n\t\\begin{align}\\label{eq4}\n\t\t\\mathbb{E}\\big[|\\zeta_X(\\alpha)\\zeta_Y(\\alpha)D(X,Y)|^2\\big] \\leq&\\|f\\|_2^2\\big(V^4+V^{2-2l}K\\exp(C'(\\alpha)l^2\\log\\log\\|f\\|_1)\\big).\n\t\\end{align}\n\t\n\tBy H\\\"{o}lder's inequality we get,\n\t\\begin{align}\\label{eq3}\n\t\tS_f(2, 1\/2)\\leq \\Big(S_f(2,\\alpha)\\Big)^{\\frac{1}{2\\alpha}}\\|f\\|_1^{2(1-\\frac{1}{2\\alpha})}\n\t\t\\ll \\Big(S_f(2,\\alpha)\\Big)^{\\frac{1}{2\\alpha}}.\n\t\t\\end{align}\nSince $2\\alpha \\rightarrow 1$, we can use the estimate \\[\\zeta(2\\alpha)=\\frac{1}{2\\alpha-1}+O(1)\\] in \\eqref{eq0} and combining with \\eqref{eq4}) and \\eqref{eq3} we obtain the upper bound ,\n\t\\begin{align}\\label{eq5}\n\tS_f(2, 1\/2) \\ll\t(\\log\\|f\\|_1+O(1))^{-2}\\|f\\|_2^2\\big(V^4+V^{2-2l}K\\exp(C'(\\alpha)l^2\\log\\log\\|f\\|_1)\\big).\n\t\\end{align}\n\tHence we choose $l=(\\log K)^{1\/2}(\\log\\log\\|f\\|_1)^{-1\/2}$ and $V=\\exp(C''(\\alpha)(l\\log\\log \\|f\\|_1))$ in \\eqref{eq5} to get the upper bound\n\t\\[S_f(2, 1\/2) \\ll (\\log\\|f\\|_1+O(1))^{-2}\\|f\\|_2^2\\exp(C(\\alpha)(\\log K\\log\\log \\|f\\|_1)^{1\/2}).\\]\n\\end{case}\n\\noindent This completes the proof.\n\\end{proof}\n\t\n\\section{Proof of Theorem \\ref{infinity-PPC thm}}\n\\begin{proof}\n\tLet $s>0$ be fixed and $N\\geq (2s)^d$ be an integer. For $\\boldsymbol{\\alpha}\\in \\mathbb{R}^d $ and  $\\boldsymbol{a}_n\\in \\mathbb{N}^d$ consider the sequence $(\\boldsymbol{x_n})=(\\{\\boldsymbol{a}_n\\boldsymbol{\\alpha}\\})$. Then,\n\t\\begin{align*}\n\t\tR_{2,\\infty}^{(d)}(s,\\boldsymbol{\\alpha},N)=\\frac{1}{N}\\displaystyle\\sum_{1\\leq m\\neq n\\leq N}\\chi_{s,N}(\\boldsymbol{\\alpha}(\\boldsymbol{a}_m-\\boldsymbol{a}_n)),\n\t\\end{align*}\nwhere $\\chi_{s,N}$ is a characteristic function defined for $\\boldsymbol{x}\\in \\mathbb{R}^d$ as follows:\n\\begin{align*}\n\\chi_{s,N}(\\boldsymbol{x})=\\begin{cases}\n\t1\\quad& \\text{if } \\|\\boldsymbol{x}\\|_\\infty\\leq s\/N^{1\/d},\\\\\n\t0 \\quad& \\text{otherwise.}\n\\end{cases}\n\\end{align*}\nThe Fourier series expansion of $\\chi_{s,N}$ is given by\n\\begin{align}\\label{char}\n\t\\chi_{s,N}(\\boldsymbol{\\alpha})\\sim\\displaystyle\\sum_{\\substack{\\boldsymbol{r}\\in\\mathbb{Z}^d\\\\ \\boldsymbol{r}=(r_1,\\cdots,r_d)}}c_{\\boldsymbol{r}}e(\\boldsymbol{r}.\\boldsymbol{\\alpha}),\n\\end{align}\nwhere\n\\begin{align*}\n\tc_{\\boldsymbol{r}}=&\\displaystyle\\int_{-s\/N^{1\/d}}^{s\/N^{1\/d}}\\cdots\\int_{-s\/N^{1\/d}}^{s\/N^{1\/d}}e{\\Big(-\\sum_{i\\leq d}r_i\\alpha_i\\Big)}d\\alpha_1\\cdots d\\alpha_d\\nonumber\\\\\n\t=& c_{r_1}\\cdots c_{r_d},\n\\end{align*}\nand \n\\[c_{r_j}=\\displaystyle\\int_{-s\/N^{1\/d}}^{s\/N^{1\/d}}e{(-r_j\\alpha_j)}d\\alpha_j, \\quad j=1,2,\\ldots,d.\n\\]\n\n Further, one gets the following upper bound:\n\\begin{align*}\n\t|c_{r_j}|\\leq\\min{\\Big(2s N^{-\\frac{1}{d}},\\, |r_j|^{-1}\\Big)}.\n\\end{align*}\nA straightforward calculation gives the expectation:\n \\begin{align*}\n \\mathbb{E}\\big[R_{2,\\infty}^{(d)}(s,.,N)\\big]=\\displaystyle\\int_{[0,1)^d}R_{2,\\infty}^{(d)}(s,\\boldsymbol{\\alpha},N)d\\boldsymbol{\\alpha}=(2s)^d\\frac{N-1}{N}.\n \\end{align*}\n Now, the variance of $R_{2,\\infty}^{(d)}(s,\\boldsymbol{\\alpha},N) $ is defined as \n \\begin{align*}\n \\operatorname{Var}(R_{2,\\infty}^{(d)}(s,.,N)) :=\\int_{[0,1)^d}\\bigg(R_{2,\\infty}^{(d)}(s,\\boldsymbol{\\alpha},N)-\\frac{(2s)^d(N-1)}{N}\\bigg)^2d\\boldsymbol{\\alpha}.\n \\end{align*}\n By using Fourier series expansion of $\\chi_{s,N}$ from \\eqref{char}, we write\n \\begin{align*}\n \\operatorname{Var}(R_{2,\\infty}^{(d)}(s,.,N)) = \\frac{1}{N^2}\\displaystyle\\int_{[0,1)^d}\\Bigg(\\sum_{1\\leq m\\neq n\\leq N}\\sum_{\\substack{\\boldsymbol{r}\\in\\mathbb{Z}^d\\setminus \\{\\boldsymbol{0}\\}}}c_{\\boldsymbol{r}}e(\\boldsymbol{r}.(\\boldsymbol{\\alpha}(\\boldsymbol{a}_m-\\boldsymbol{a}_n)))\\Bigg)^2d\\boldsymbol{\\alpha}.\n \\end{align*}\n After squaring the integrand, we want to interchange the summations and integrations. Such a rearrangement can be justify by using the fact that the partial sums of a Fourier series of an indicator function are uniformly bounded and  hence the dominated convergence theorem is applicable ( for example, see,  \\cite[Chapter 3, Exercise 18]{SS})\n Thus, $\\operatorname{Var}(R_{2,\\infty}^{(d)}(s,.,N))$ equals to\n \\begin{align*}\n&\\frac{1}{N^2}\\displaystyle\\sum_{\\substack{1\\leq m\\neq n\\leq N\\\\1\\leq k\\neq l\\leq N}}\\sum_{\\substack{\\boldsymbol{r},\\boldsymbol{t}\\in\\mathbb{Z}^d\\setminus \\{\\boldsymbol{0}\\}}}c_{\\boldsymbol{r}}c_{\\boldsymbol{t}}\\int_{[0,1)^d}e(\\boldsymbol{r}.(\\boldsymbol{\\alpha}(\\boldsymbol{a}_m-\\boldsymbol{a}_n))-\\boldsymbol{t}.(\\boldsymbol{\\alpha}(\\boldsymbol{a}_k-\\boldsymbol{a}_l)))d\\boldsymbol{\\alpha}\\\\\n\t=& \\frac{1}{N^2}\\displaystyle\\sum_{\\substack{1\\leq m\\neq n\\leq N\\\\1\\leq k\\neq l\\leq N}}\\sum_{\\substack{\\boldsymbol{r},\\boldsymbol{t}\\in\\mathbb{Z}^d\\setminus \\{\\boldsymbol{0}\\}}}c_{\\boldsymbol{r}}c_{\\boldsymbol{t}}\\prod_{i\\leq d}\\int_{[0,1)}e\\Big(\\big(r_i(a_m^{(i)}-a_n^{(i)})-t_i(a_k^{(i)}-a_l^{(i)})\\big)\\alpha_i\\Big)d\\alpha_i.\n\\end{align*}\nBut, we have the inner most integral equals to $1$, whenever $r_i=t_i=0 $ or  $r_i(a_m^{(i)}-a_n^{(i)})=t_i(a_k^{(i)}-a_l^{(i)})$, otherwise the integral is zero.\nNote that when all components of $\\boldsymbol{r}, \\boldsymbol{t}$ are nonzero, the associated sums will produce the main contribution. Otherwise, when some of them are zero, we save powers of $N$. \nPrecisely, the variance equals to\n\\begin{align}\\label{eq6}\n\t\\frac{1}{N^2}\\displaystyle\\sum_{\\substack{j_1<\\cdots<j_q \\leq d\\\\1\\leq q\\leq d}}\\sideset{}{'}\\sum_{\\substack{\\boldsymbol{v}=(v_{j_1},\\ldots,v_{j_q}),\\\\\\boldsymbol{w}=(w_{j_1},\\dots,w_{j_q})\\in\\mathbb{Z}^q}}\\mathcal{R}_N(v_{j_1},\\ldots,v_{j_q})\\mathcal{R}_N(w_{j_1},\\ldots,w_{j_q})\\Big(\\frac{2s}{N^{1\/d}}\\Big)^{2(d-q)}\\sideset{}{'}\\sum_{\\substack{\\boldsymbol{r},\\boldsymbol{t}\\in\\mathbb{Z}^q\\\\ \\boldsymbol{r}\\boldsymbol{v}=\\boldsymbol{t}\\boldsymbol{w}}}c_{\\boldsymbol{r}} c_{\\boldsymbol{t}},\n\\end{align}\nwhere $\\mathcal{R}_N$ is defined in \\eqref{R_N}.\n\nThe relation $r_iv_i=t_iw_i$ allows us to write $r_i, t_i$ as\n\\begin{align}\n\tr_i=\\frac{h_iw_i}{\\gcd(v_i,w_i)}, \\text{ and } t_i=\\frac{h_iv_i}{\\gcd(v_i,w_i)},\\nonumber\n\\end{align}\nwhere $h_i$ is a nonzero integers. Now we separate the sum over $h_i$ into three sub-intervals of the real line given by\n\\begin{align}\n\t&|h_i|\\leq\\frac{N^{1\/d}\\gcd(v_i,w_i)}{s\\max(|v_i|,|w_i|)},\\nonumber\\\\\n\t&\\frac{N^{1\/d}\\gcd(v_i,w_i)}{s\\max(|v_i|,|w_i|)}<|h_i|\\leq\\frac{N^{1\/d}\\gcd(v_i,w_i)}{s\\min(|v_i|,|w_i|)},\\nonumber\\\\\n\t&|h_i|>\\frac{N^{1\/d}\\gcd(v_i,w_i)}{s\\min(|v_i|,|w_i|)}.\\nonumber\n\\end{align}\nFor any fixed $j_1<j_2<\\cdots<j_q$, we follow the arguments in page 344 of ~\\cite{hinrichs2019multi} and get,\n\\[\\sideset{}{'}\\sum_{\\substack{\\boldsymbol{r},\\,\\boldsymbol{t}\\in\\mathbb{Z}^q\\\\ \\boldsymbol{r}\\boldsymbol{v}=\\boldsymbol{t}\\boldsymbol{w}}}c_{\\boldsymbol{r}} c_{\\boldsymbol{t}}\\ll\\frac{s^q(\\log N)^q}{N^{q\/d}}\\displaystyle\\prod_{1\\leq \\beta\\leq q}\\frac{\\gcd(v_{j_\\beta},w_{j_\\beta})}{\\sqrt{v_{j_\\beta}w_{j_\\beta}}}.\n\\]\n\nReplace the above bound in the right hand side of \\eqref{eq6} to get\n\\begin{align}\n\t\\operatorname{Var}(R_2^{(d)}(s,.,N))&\\ll\\frac{s^d(\\log N)^d}{N^3}\\displaystyle\\sideset{}{'}\\sum_{\\boldsymbol{v},\\boldsymbol{w}\\in\\mathbb{Z}^d}\\mathcal{R}_N(\\boldsymbol{v})\\mathcal{R}_N(\\boldsymbol{w})\\prod_{1\\leq i\\leq d}\\frac{\\gcd(v_i,w_i)}{\\sqrt{v_iw_i}}\\nonumber\\\\\n\t+&\\sum_{\\substack{j_1<\\cdots<j_q\\in D\\\\1\\leq q<d}}\\frac{s^{2d-q}(\\log N)^q}{N^{4-q\/d}}\\sideset{}{'}\\sum_{\\boldsymbol{v},\\boldsymbol{w}\\in\\mathbb{Z}^q}\\mathcal{R}_N(\\boldsymbol{v})\\mathcal{R}_N(\\boldsymbol{w}))\\prod_{1\\leq \\beta\\leq q}\\frac{\\gcd(v_{j_\\beta},w_{j_\\beta})}{\\sqrt{v_{j_\\beta}w_{j_\\beta}}}. \\nonumber\n\\end{align}\n\nNow applying Proposition~\\ref{gen gcd lemma} to the above GCD sums, we get for any $\\gamma>0$,\n\\begin{align*}\n\\operatorname{Var}(R_2^{(d)}(s,.,N)) &\\ll \\frac{s^d(\\log N)^d}{N^3}E\\left(A_N^{(1)};\\ldots;A_N^{(d)}\\right)\\frac{\\exp(C\\sqrt{\\log N\\log\\log N})}{(\\log N+\\operatorname{O}(1))^d}+\\frac{1}{N^{1\/d-\\gamma}}\\\\\n&\\ll  \\frac{s^d}{N^3}E\\left(A_N^{(1)};\\ldots;A_N^{(d)}\\right)\\exp\\Big(C\\sqrt{\\log N\\log\\log N}\\Big).\n\\end{align*}\nThus, under the hypothesis of the theorem we get the variance is $\\operatorname{O}(N^{-\\delta\/2})$. This is the main part of the proof. The rest of the arguments follow a standard method of applying Chebyshev's inequality and Borel-cantelli lemma(see the proof of Theorem 1 in \\cite{aistleitner2017additive} and also \\cite{hinrichs2019multi, RZ1999}.)\n\\end{proof}\n\n\n\\section{Proof of theorem \\ref{2-PPC thm1} and \\ref{2-PPC thm2}.}\n\\subsection{Properties of Bessel functions}\nHere we state some properties of the Bessel function which are important  tools to prove Theorem \\ref{2-PPC thm1} and Theorem~\\ref{2-PPC thm2}. For complex number $\\nu$ with $\\Re(\\nu)>-1\/2$ and $t\\geq 0$ the Bessel function $J_\\nu$ of order $\\nu$ is defined by \n\\begin{align}\\label{bessel1}\nJ_\\nu(t) = \\frac{(t\/2)^\\nu}{\\Gamma(\\nu+1\/2)\\Gamma(1\/2)}\\int_{-1}^1e^{itx}(1-x^2)^{\\nu}\\frac{dx}{\\sqrt{1-x^2}}.\n\\end{align}\nThis definition is also valid for $t\\in\\mathbb{C}.$ The function $J_\\nu$ has many interesting properties. For our application we mention a few of them below and these are essentially given in \\cite[Apendix B]{loukasgrafakos}.\nFor $\\Re(\\nu)>-1\/2$ and $t\\geq 1$ we have a nice approximation which gives an asymptotic formula as $t\\rightarrow \\infty,$\n\\begin{align}\\label{p1}\nJ_\\nu(t) = \\sqrt{\\frac{2}{\\pi t}} \\cos{\\left(t-\\frac{\\pi \\nu}{2}-\\frac{\\pi}{4}\\right)} +\\operatorname{O}_\\nu(t^{-3\/2}).\n\\end{align}\nFor $0< t\\leq 1$ and $\\Re(\\nu)>-1\/2$, we have the following upper bound\n\\begin{align}\\label{p2}\nJ_\\nu(t) \\ll_\\nu \\exp\\big({\\max\\{(\\Re(\\nu)+1\/2)^{-2}, (\\Re(\\nu)+1\/2)^{-1} \\}|\\Im(\\nu)|^2}\\big) t^{\\Re(\\nu)}.\n\\end{align}\nWhenever $t>0$ and $\\nu \\in \\mathbb{N}$, the Bessel function reduces to a simple form:\n\\begin{align}\\label{bessel2}\nJ_\\nu(t) = \\frac{1}{2\\pi}\\int_0^{2\\pi}\\cos(t\\sin{\\theta}- \\nu \\theta) d\\theta.\n\\end{align}\n\n\\begin{lemma}\\label{bessel fn lemma}\n\tLet $N$, $\\nu \\geq 2$ be two integers and $s,r>0$ be real numbers. Then\n\t\\begin{itemize}\n\t\\item[(1)] $J_{\\nu\/2}(r)\\ll_\\nu 1$ if $r\\leq 1,$\\\\\n\t\\item[(2)] $J_{\\nu\/2}(r)\\ll_\\nu \\frac{1}{\\sqrt{r}}$ if $r > 1,$\\\\\n\t\\item[(3)] $\\frac{J_{\\nu\/2}( 2\\pi s r N^{-{1}\/{\\nu}})}{r^{\\nu\/2}}\\ll_\\nu \\frac{ s^{\\nu\/2}}{\\sqrt{N}}$, for any $r>0$\\\\\n\t\\item[(4)] $|J_\\mu(r)|\\leq 1$, for any $r>0$, and $\\mu\\in\\mathbb{N}.$\n\t\\end{itemize}\t\n\\end{lemma}\nProperties $1,\\,2,\\,3$ and $4$ follows from \\eqref{p2},\\,\\eqref{p1},\\, \\eqref{bessel1} and \\eqref{bessel2}, respectively.\n\\subsection{Preperation of the proofs of Theorem \\ref{2-PPC thm1} and Theorem \\ref{2-PPC thm2}}\nFor the sequence $(\\boldsymbol{x}_n)= (\\{\\boldsymbol{a}_n\\boldsymbol{\\alpha}\\})$, the $d$-dimensional pair correlation statistic in $2$-norm is,\n\\begin{align}\n\tR_{2,2}^{(d)}(s,\\boldsymbol{\\alpha},N)=\\frac{1}{N}\\displaystyle\\sum_{1\\leq m\\neq n\\leq N}I_{s,N}(\\boldsymbol{\\alpha}(\\boldsymbol{a}_m-\\boldsymbol{a}_n)),\\nonumber\n\\end{align}\nwhere $I_{s,N}$ is the indicator function for all $\\boldsymbol{x}\\in\\mathbb{R}^d$  which satisfy $\\|\\boldsymbol{x}\\|_2\\leq s\/N^{1\/d}$.\nThen, Fourier series expansion of $I_{s,N}$ is given by\n\\begin{align}\n\tI_{s,N}(\\boldsymbol{\\alpha})\\sim\\displaystyle\\sum_{\\substack{\\boldsymbol{r}\\in\\mathbb{Z}^d\\\\ \\boldsymbol{r}=(r_1,\\cdots,r_d)}}c_{\\boldsymbol{r}}e(\\boldsymbol{r}.\\boldsymbol{\\alpha}),\\nonumber\n\\end{align}\nwhere\n\\begin{align}\\label{c_r}\nc_{\\boldsymbol{r}}= \\begin{cases}\n\\omega_d\\frac{s^d}{N}, & \\mbox{ if } \\boldsymbol{r}=\\boldsymbol{0},\\\\\n\\frac{s^{d\/2}}{\\sqrt{N}\\|\\boldsymbol{r}\\|_2^{d\/2}}J_{d\/2}\\big(\\frac{2\\pi s}{N^{1\/d}}\\|\\boldsymbol{r}\\|_2\\big), & \\mbox{ if } \\boldsymbol{r}\\in\\mathbb{Z}^d\\setminus \\{\\boldsymbol{0}\\}.\n\\end{cases}\n\\end{align} \nFor the details of such formulas one can see~\\cite[Apendix B]{loukasgrafakos}. \n\nIt is not hard to verify that  \n\\begin{align*}\n\\mathbb{E}(R_{2,2}^{(d)}(s, . ,N))=\\omega_ds^d\\frac{N-1}{N}.\n\\end{align*}\nWe now proceed to calculate the variance of $R_{2,2}^{(d)}(s, . ,N)$, that is,\n\\begin{align*}\n\\operatorname{Var}(R_{2,2}^{(d)}(s,.,N)) &= \\displaystyle\\int_{[0,1)^d}\\bigg(R_{2,2}^{(d)}(s,\\boldsymbol{\\alpha},N)-\\frac{\\omega_ds^d(N-1)}{N}\\bigg)^2d\\boldsymbol{\\alpha}\\\\\n& = \\frac{1}{N^2}\\displaystyle\\int_{[0,1)^d}\\Bigg(\\sum_{1\\leq m\\neq n\\leq N}\\sum_{\\substack{\\boldsymbol{r}\\in\\mathbb{Z}^d\\setminus \\{\\boldsymbol{0}\\}}}c_{\\boldsymbol{r}}e(\\boldsymbol{r}.(\\boldsymbol{\\alpha}(\\boldsymbol{a}_m-\\boldsymbol{a}_n)))\\Bigg)^2d\\boldsymbol{\\alpha}.\n\\end{align*}\nBy interchanging summation and integration we obtain $\\operatorname{Var}(R_{2,2}^{(d)}(s,.,N))$ is bounded above by\n\\begin{align}\\label{eqppc2}\n \\frac{1}{N^2}\\displaystyle\\sum_{\\substack{1\\leq m\\neq n\\leq N\\\\1\\leq k\\neq l\\leq N}}\\sum_{\\substack{\\boldsymbol{r},\\boldsymbol{t}\\in\\mathbb{Z}^d\\setminus \\{\\boldsymbol{0}\\} }}c_{\\boldsymbol{r}}\\overline{c_{\\boldsymbol{t}}}\\prod_{i\\leq d}\\int_{[0,1)}e\\Big(\\big(r_i(a_m^{(i)}-a_n^{(i)})-t_i(a_k^{(i)}-a_l^{(i)})\\big)\\alpha_i\\Big)d\\alpha_i.\n\\end{align}\nSuch rearrangement can be justified as follows. If we split the Fourier series of the indicator function at some parameter $M$ and consider the partial sum upto $M$ and the tail part. Apply Cauchy-Schwarz inequality on the tail part to interchange sums and the integration, which leaves us with the square-integral of the tail. But the $2$-norm of tail goes to zero as $M\\rightarrow \\infty$. Thus, for arbitrarily large $M$ the partial sum upto $M$ remains for which the rearrangement is obvious.\n\n  As in Theorem~\\ref{infinity-PPC thm}, it is enough to show that the variance is arbitrarily small.\n\\begin{proof}[Proof of Theorem~\\ref{2-PPC thm1}]\n\tIn this theorem  we consider $a_n^{(i)}=a_n$ for all $1\\leq i\\leq d,$ then \\eqref{eqppc2} and the orthogonality of exponential function allow us to write,\n\\begin{align}\t\n\t\\operatorname{Var}(R_{2,2}^{(d)}(s,.,N))\\leq&\\frac{1}{N^2}\\displaystyle\\sum_{v,w\\in\\mathbb{Z}\\setminus \\{{0}\\}}\\mathcal{R}_N(v)\\mathcal{R}_N(w)\\sum_{\\substack{\\boldsymbol{r},\\boldsymbol{t}\\in\\mathbb{Z}^d\\setminus \\{\\boldsymbol{0}\\}\\\\ vr_i=wt_i}}|c_{\\boldsymbol{r}}c_{\\boldsymbol{t}}|. \\label{eq1ppc2}\n\\end{align}\nLet us fix $j_1<j_2<\\cdots<j_q\\leq d$ where $1\\leq q\\leq d$. We claim that  \n\\begin{align}\\label{eq3ppc2}\n\t\\sideset{}{''}\\sum_{\\substack{\\boldsymbol{r},\\boldsymbol{t}\\in\\mathbb{Z}^d\\setminus \\{\\boldsymbol{0}\\}\\\\ r_{j_\\beta}v=r_{j_\\beta}w}}|c_{\\boldsymbol{r}}c_{\\boldsymbol{t}}|\\ll_d\\frac{s^{2d-q}}{N^{2-q\/d}}\\frac{(v,w)^q}{\\sqrt{|vw|^q}},\n\\end{align}\nwhere the sum $\\Sigma^{''}$ is over $\\boldsymbol{r},\\boldsymbol{t}$ with all components zero other than $r_{j_\\beta}, t_{j_\\beta}$ for $1\\leq\\beta\\leq q.$ This imply for $q<d$ we will always save a power of $N$ and the main contribution to the variance comes from the case $q=d$. \n\nNow, from relations $r_{j_\\beta}v=r_{j_\\beta}w$ we can write,\n\\[r_{j_\\beta}=\\frac{wh_{j_\\beta}}{(v,w)}\\text{ and, }t_{j_\\beta}=\\frac{vh_{j_\\beta}}{(v,w)}\\text{ for }1\\leq\\beta\\leq q,\\]\nwhere $h_{j_\\beta}$ takes integer values. Let us denote the $q$-tuple $(h_{j_1},\\ldots,h_{j_q})$ by $\\boldsymbol{h}$ and from now onwards we use the notation $\\|\\boldsymbol{h}\\|_2$  for the usual $2$-norm.\n\nUsing the definition of $c_{\\boldsymbol{r}}$'s from \\eqref{c_r} the left hand side of \\eqref{eq3ppc2} is bounded by,\n\\begin{align}\\label{2nd eq}\n\t\\ll \\displaystyle\\frac{s^d}{N}\\sideset{}{'}\\sum_{\\boldsymbol{h}\\in\\mathbb{Z}^{q}}\\frac{(v,w)^d}{(|vw|)^{d\/2}}\\frac{|J_{d\/2}(\\frac{2\\pi s|w|}{N^{1\/d}(v,w)}\\|\\boldsymbol{h}\\|_2)|}{\\|\\boldsymbol{h}\\|_2^{d\/2}}\\frac{|J_{d\/2}(\\frac{2\\pi s|v|}{N^{1\/d}(v,w)}\\|\\boldsymbol{h}\\|_2)|}{\\|\\boldsymbol{h}\\|_2^{d\/2}}.\n\\end{align}\n Now we divide this sum into three parts \n $\\Sigma^{'}=\\Sigma_1^{'}+\\Sigma_2^{'}+\\Sigma_3^{'}$ depending on the following range of values of $\\|\\boldsymbol{h}\\|_2$, \n\\begin{align}\n\t1\\leq&\\|\\boldsymbol{h}\\|_2\\leq\\frac{N^{1\/d}(v,w)}{s\\max(|v|,|w|)}:=\\max \\boldsymbol{h},\\nonumber \\\\\n\t\\frac{N^{1\/d}(v,w)}{2\\pi s\\max(|v|,|w|)}\\leq&\\|\\boldsymbol{h}\\|_2\\leq\\frac{N^{1\/d}(v,w)}{2\\pi s\\min(|v|,|w|)}:=\\min \\boldsymbol{h},\\nonumber\\\\\n\t&\\|\\boldsymbol{h}\\|_2\\geq\\frac{N^{1\/d}(v,w)}{2\\pi s\\min(|v|,|w|)}.\\nonumber\n\\end{align}\nUsing (3) of Lemma~\\ref{bessel fn lemma} to get,\n\\begin{align*}\n\t\\Sigma_1^{'} &\\ll \\frac{s^d}{N}(\\max \\boldsymbol{h})^q\\ll\\frac{s^{d-q}(v,w)^q}{N^{1-q\/d}\\sqrt{|vw|^q}}.\n\\end{align*}\nBy using (1) and (2) of Lemma~\\ref{bessel fn lemma}, we obtain\n\\begin{align}\n\t\\Sigma_2^{'}&\\ll\\displaystyle\\frac{(v,w)^d}{(|vw|)^{d\/2}}\\sum_{\\max \\boldsymbol{h}\\leq\\|\\boldsymbol{h}\\|_2\\leq \\min \\boldsymbol{h}}\\frac{\\sqrt{\\max \\boldsymbol{h}}}{\\|\\boldsymbol{h}\\|_2^{d+1\/2}}\\nonumber\\\\\n\t&\\ll\\frac{s^{d-q}(v,w)^q\\max(|v|,|w|)^{d-q}}{N^{1-q\/d}|vw|^{d\/2}}\\ll\\frac{s^{d-q}(v,w)^q}{N^{1-q\/d}\\sqrt{|vw|^q}}.\\nonumber\n\\end{align}\nAgain, using case (2) of Lemma~\\ref{bessel fn lemma} we get,\n\\begin{align*}\n\t\\Sigma_3^{'}&\\ll \\frac{(v,w)^{d+1}N^{1\/d}}{s(|vw|)^{d\/2+1\/2}}\\displaystyle\\sum_{\\|\\boldsymbol{h}\\|_2> \\min \\boldsymbol{h}}\\frac{1}{\\|\\boldsymbol{h}\\|_2^{d+1}}\\\\\n\t&\\ll \\frac{(v,w)^{d+1}N^{1\/d}}{s(|vw|)^{d\/2+1\/2}}\\frac{1}{(\\min \\boldsymbol{h})^{d-q+1}}\\\\\n\t&\\ll \\frac{s^{d-q}(v,w)^q}{N^{1-q\/d}\\sqrt{|vw|^q}}.\n\\end{align*}\nCombining all three bounds with \\eqref{2nd eq} we obtain the claim \\eqref{eq3ppc2}. Then by using \\eqref{eq3ppc2} in \\eqref{eq1ppc2} we get,\n\\begin{align*}\n\t\\operatorname{Var}(R_{2,2}^{(d)}(s,.,N))\\ll &\\displaystyle\\sum_{1\\leq q\\leq d}\\frac{s^{2d-q}}{N^{4-q\/d}}\\sum_{v,w\\in\\mathbb{Z}\\setminus \\{0\\}}\\mathcal{R}_N(v)\\mathcal{R}_N(w)\\frac{(v,w)^q}{\\sqrt{|vw|^q}}\\\\\n\\end{align*}\nNow, using the result on GCD sums by G\\'al~\\cite{gel} we obtain\n\\begin{align*}\n\\operatorname{Var}(R_{2,2}^{(d)}(s,.,N)) \\ll \\begin{cases}\n\t\\frac{s^2}{N^3}E(A_N)(\\log\\log N)^2+ \\frac{s^2}{N^{1\/2 -\\gamma }},& \\mbox{ if } d=2,\\\\\n\t\\frac{s^d}{N^3}E(A_N) + \\frac{s^d}{N^{{1}\/{d}- \\gamma}}, & \\mbox{ if } d\\geq 3,\n\t\\end{cases}\n\\end{align*}\nfor some sufficiently small $\\gamma>0.$ Thus, under the hypothesis of this theorem the variance $\\operatorname{O}((\\log N)^{-1-\\epsilon\/2})$. The rest of the arguments follow from the proof of ~\\cite[Theorem 5]{Bloom2019GCDSA}.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{2-PPC thm2}]\n\tFor $d=2$, (\\ref{eqppc2}) gives us,\n\\begin{align}\\label{eq4ppc2}\n\t\\operatorname{Var}(R_{2,2}^{(2)}(s,.,N))&\\leq \\frac{1}{N^2}\\sum_{\\substack{v_1,v_2,w_1,w_2\\in\\mathbb{Z}\\setminus \\{0\\}}}\\mathcal{R}_N(v_1,v_2)\\mathcal{R}_N(w_1,w_2)\\sideset{}{'}\\sum_{\\substack{\\boldsymbol{r}, \\boldsymbol{t}\\in\\mathbb{Z}^2\\\\ r_iv_i=t_iw_i}}|c_{\\boldsymbol{r}}c_{\\boldsymbol{t}}| \\\\ \\nonumber\n& + \\frac{1}{N^2}\\sum_{\\substack{v_1, w_1\\in\\mathbb{Z}\\setminus \\{0\\}}}\\mathcal{R}_N(v_1)\\mathcal{R}_N(w_1) \\sum_{\\substack{r_1,t_1 \\in\\mathbb{Z}\\setminus \\{0\\}\\\\ r_1v_1=t_1w_1}}|c_{(r_1,0)}c_{(t_1,0)}|\\\\ \\nonumber\n& + \\frac{1}{N^2}\\sum_{\\substack{v_2, w_2\\in\\mathbb{Z}\\setminus \\{0\\}}}\\mathcal{R}_N(v_2)\\mathcal{R}_N(w_2)  \\sum_{\\substack{r_2,t_2\\in\\mathbb{Z}\\setminus \\{0\\}\\\\ r_2v_2=t_2w_2}}|c_{(0,r_2)}c_{(0,t_2)}|.\n\\end{align} \nNote that the second and third terms on the right hand side above are $\\operatorname{O}(N^{-1\/2+\\gamma})$ for some $\\gamma>0$, as it follows from \\eqref{eq3ppc2} with $d=2$ and $q=1$. \n\nLet us call the inner sum in first term as $I$.\nFrom the relation $r_iv_i=t_iw_i$ we have \n$r_i=\\frac{w_ih_i}{(v_i,w_i)}$ and $t_i=\\frac{v_ih_i}{(v_i,w_i)}$ for $i=1,2$ where $h_i$'s are nonzero integers. For simplicity we write,\n\\[A=\\frac{w_1}{(v_1,w_1)},\\, B=\\frac{w_2}{(v_2,w_2)},\\, C=\\frac{v_1}{(v_1,w_1)},\\, D=\\frac{v_2}{(v_2,w_2)}.\\]\nBy using \\eqref{c_r} we obtain\n\\begin{align*}\n\tI&\\ll\\frac{s^2}{N}\\sum_{h_1,h_2\\neq0}\\frac{\\Big|J_1\\Big(\\frac{2\\pi s}{\\sqrt{N}}(A^2h_1^2+B^2h_2^2)^{\\frac{1}{2}}\\Big)\\Big|}{(A^2h_1^2+B^2h_2^2)^{\\frac{1}{2}}}\\frac{\\Big|J_1\\Big(\\frac{2\\pi s}{\\sqrt{N}}(C^2h_1^2+D^2h_2^2)^{\\frac{1}{2}}\\Big)\\Big|}{(C^2h_1^2+D^2h_2^2)^{\\frac{1}{2}}}\\nonumber\\\\\n\t= &\\frac{s^2}{N|AC|}\\sum_{h_1\\neq 0}\\frac{1}{h_1^2}\\sum_{h_2\\neq 0}\\frac{\\Big|J_1\\Big(\\frac{2\\pi s|Ah_1|}{\\sqrt{N}}(1+\\frac{B^2}{A^2h_1^2}h_2^2)^{\\frac{1}{2}}\\Big) J_1\\Big(\\frac{2\\pi s|Ch_1|}{\\sqrt{N}}(1+\\frac{D^2}{C^2h_1^2}h_2^2)^{\\frac{1}{2}}\\Big)\\Big|}{\\Big(1+\\frac{B^2}{A^2h_1^2}h_2^2\\Big)^{\\frac{1}{2}} \\Big(1+\\frac{D^2}{C^2h_1^2}h_2^2\\Big)^{\\frac{1}{2}}}.\n\\end{align*}\nNow we divide the sum over $h_1$ into two parts,\n\\[|h_1|\\leq\\frac{\\sqrt{N}}{2\\pi s\\min(|A|,|C|)}\\text{ and, }|h_1|>\\frac{\\sqrt{N}}{2\\pi s\\min(|A|,|C|)}.\\]\nApplying (4) and (2) of Lemma \\ref{bessel fn lemma} respectively for small and large argument of Bessel function in the above inequality,\n\\begin{align*}\nI\t&\\ll \\frac{s^2}{N|AC|}\\Bigg(\\sum_{|h_1|\\leq\\frac{\\sqrt{N}}{2\\pi s\\min(|A|,|C|)}}\\frac{1}{h_1^2}\\sum_{h_2\\neq 0}\\Big(1+\\frac{B^2}{A^2h_1^2}h_2^2\\Big)^{-\\frac{1}{2}}\\Big(1+\\frac{D^2}{C^2h_1^2}h_2^2\\Big)^{-\\frac{1}{2}}\\\\\n\t&+\\sum_{|h_1|>\\frac{\\sqrt{N}}{2\\pi s\\min(|A|,|C|)}}\\frac{\\sqrt{N}}{s|AC|^{1\/2}|h_1|^3}\\sum_{h_2\\neq 0}\\Big(1+\\frac{B^2}{A^2h_1^2}h_2^2\\Big)^{-\\frac{1}{2}-\\frac{1}{4}}\\Big(1+\\frac{D^2}{C^2h_1^2}h_2^2\\Big)^{-\\frac{1}{2}-\\frac{1}{4}}\\Bigg).\n\\end{align*}\nNow observe that for $a_1,a_2\\in\\mathbb{R}$, $(1+a_1^2)(1+a_2^2)\\geq(1+a_1a_2)^2$. Then, sum on $h_2$ in the above two terms can be bounded by,\n\\[\\int_0^{\\infty}\\Big(1+\\Big|\\frac{BD}{AC}\\Big|\\frac{x^2}{h_1^2}\\Big)^{-1}dx\\,\n\\text{ and, } \\,\\int_0^{\\infty}\\Big(1+\\Big|\\frac{BD}{AC}\\Big|\\frac{x^2}{h_1^2}\\Big)^{-3\/2}dx\\]\nrespectively and both are bounded by\n\\[|h_1|\\Big|\\frac{AC}{BD}\\Big|^{1\/2}.\\]\nThus, we have \n\\begin{align*}\nI & \\ll \\frac{s^2}{N|AC|}\\Bigg(\\Big|\\frac{AC}{BD}\\Big|^{1\/2}\\sum_{|h_1|\\leq\\frac{\\sqrt{N}}{2\\pi s\\min(|A|,|C|)}}\\frac{1}{|h_1|}\n\t+ \\frac{\\sqrt{N}}{s \\sqrt{|BD|}}\\sum_{|h_1|>\\frac{\\sqrt{N}}{2\\pi s\\min(|A|,|C|)}}\\frac{1}{|h_1|^2}\\Bigg)\\\\\n\t& \\ll\\frac{s^2}{N|AC|}\\Bigg(\\Big|\\frac{AC}{BD}\\Big|^{1\/2}\\log N +\\frac{\\min(|A|,|C|)}{|BD|^{1\/2}}\\Bigg)\\ll \\frac{s^2}{N}\\frac{\\log{N}}{|ABCD|^{1\/2}}.\n\\end{align*}\nHence, we deduce the variance estimate \n\\begin{align}\\label{estimate1}\n\t\\operatorname{Var}(R_{2,2}^{(2)}(s,.,N))\\ll \\frac{s^2\\log N}{N^3}\\displaystyle\\sum_{\\substack{v_1,v_2,\\\\w_1,w_2\\in\\mathbb{Z}\\setminus \\{0\\}}}\\mathcal{R}_N(v_1,v_2)\\mathcal{R}_N(w_1,w_2)\\frac{(v_1,w_1)(v_2,w_2)}{\\sqrt{|v_1w_1v_2w_2|}}.\n\\end{align}\nNow applying Proposition~\\ref{gen gcd lemma} in \\eqref{estimate1} we conclude that \n\\[\\operatorname{Var}(R_{2,2}^{(2)}(s,.,N))\\ll \\frac{s^2}{N^3\\log{N}}E(A_N^{(1)};A_N^{(2)})\\exp(C\\sqrt{\\log N\\log\\log N}).\\]\nThis completes the proof.\n\\end{proof}\n\\section{Proof of Theorem~\\ref{thm6}}\nThis follows easily by combining Theorem \\ref{infinity-PPC thm} with Corollary~\\ref{coro2.3} and Theorem \\ref{2-PPC thm2} with Corollary~\\ref{coro2.3} .\n \\section{Acknowledgement}  The authors would like to thank Prof. Christoph Aistleitner for helpful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}First-level heading:\\protect\\\\ The line\n\n\\section{\\label{sec1}Introduction}\nIn atomic, molecular, or cluster physics, the interaction of a target system with a projectile typically leads to its excitation, ionization, or in case of bound systems to their fragmentation with corresponding emissions of a variety of particles such as electrons, ions, neutral species and photons.\nFor processes that produce excited states of the target, there is in general not a singular de-excitation path, but branching into competing channels.\nThe weaker processes are often difficult to identify and even harder to quantify, as their occurrence may be masked by other, more intense signals or they may barely emerge above the noise level at all.\\\\\nThe simultaneous detection of several or all reaction products belonging to a particular projectile-target interaction allows to disentangle individual decay pathways and to increase the signal to noise contrast for these weak processes significantly.\nThis powerful technique is called coincidence measurement and is widely used in experimental particle, nuclear, atomic, molecular, and cluster physics (Ref.~\\onlinecite{Arion2015} and references therein).\nThe most essential experimental parameters in a coincidence experiment are the detection probabilities $P_{i} (\\in [0,1])$ of the respective particles $i$, which may be separated into products of the accepted relative solid angles $\\Omega_{\\text{rel},i} (\\in [0,1])$ and the detector efficiencies $\\varepsilon_{i} (\\in [0,1])$.\nThe total probability of recording a coincident event $P_\\text{coinc}$ is the product of the individual detection probabilities of all involved particles:\n\\begin{eqnarray}\nP_\\text{coinc}=\\prod\\limits_{\\text{all particles}}{P_i}=\\prod\\limits_\\text{all particles}{\\Omega_{\\text{rel},i}\\varepsilon_i}\n\\label{pcoinc_eq}.\n\\end{eqnarray}\nThe detector efficiency $\\varepsilon_i$ is the probability that a particle reaching the detector is recorded as an actual event.\nIt is an intrinsic characteristic of the detector and typically depends on the particle type and its properties such as energy and mass.\nThe achievable solid angle, however, is given by the experimental geometry.\nIn general, it can be described by the active detector surface area $A_\\text{D}$ and its distance to the interaction volume $D$:\n\\begin{eqnarray}\n\\Omega_\\text{rel}\\approx\\frac{A_\\text{D}}{4\\pi D^2}\n\\label{photodet_eq}.\n\\end{eqnarray}\nIn the case of charged reaction products (electrons or ions), $\\Omega_\\text{rel}$ often can be increased to values close to unity by guiding these particles towards the active detector area via electric and\/or magnetic fields.\nA prominent example for the latter case are \"reaction microscope\" experiments in which charged fragments of atoms, molecules, or clusters are measured in multiple coincidences  \\cite{Doerner2000,Ullrich2003}.\nHowever, since guiding fields are not applicable to neutral fragments and photons, the solid angle for the detection of these particles is typically small and results in a low coincidence rate.\nBecause of the above mentioned challenge, reports of successful coincidence measurements with detection of these kind of particles are rare compared to ones restricted to charged particles.\nIn the following we will restrict the discussion to experiments in which a charged particle and a photon from the same ionization event are recorded.\\\\\nOne class of these experiments uses excitation by monochromatized emission of noble gas lamps to investigate fluorescing ionic fragments of various molecular systems ranging from diatomic N$_2$ or CO to fluorobenzene molecules \\cite{Bloch1975,Eland1976,Dujardin1981,Field1992}.\nAnother approach was the measurement of scattering angles in  electron energy loss spectroscopy in coincidence with an emitted photon to investigate doubly-excited states in H$_2$ and H$_2$O \\cite{Ishikawa2011,Tsuchida2011} as well as interferences in the He(3l,3l') excitation \\cite{Dogan1998}.\\\\\nHowever, these measurements do not use the opportunities offered by modern synchrotron radiation facilities, in particular the narrow bandwidth of the exciting radiation in combination with the high tunability of the exciting-photon energies.\nThis type of excitation was used for photon-threshold electron measurements which allow to determine the lifetime of radiative molecular states\\cite{Schlag1977}.\nPhotoion-photon coincidence techniques were also applied at synchrotron radiation facilities to investigate for example the dissociative photoionization of N$_2$ \\cite{Kitajima1995,Kitajima1996,Soderstroma2004} and the core-hole decay of argon atoms and clusters \\cite{Gejo2013}.\\\\\nA series of experiments showed that the coincident measurement of photons and electrons, combined with polarization analysis of the fluorescence, for atomic photoionization can be used to perform so-called complete experiments, in which all amplitudes and phases in a partial wave description of the process are determined. \\cite{Beyer1995,Beyer1996,West1998,Arp1996}.\nYet, all these studies suffer from low coincidence rates, compensated to some extent by using long acquisition times \\cite{Bloch1975}.\nHowever, if the experiment needs to make use of synchrotron radiation, for which allocated beamtime is limited, reasonable statistics might not be achievable or losses in data quality have to be accepted.\\\\\nThese challanges can be overcome according to the relation given in equation \\ref{pcoinc_eq}, if collecting optics are used to increase the solid angle of photon detection and with that increase the coincidence count rate as described in Ref.~\\onlinecite{Reiss2015} for photon-photon and in Ref.~\\onlinecite{Meyer1995} for ion-photon coincidence experiments.\\\\\nIn this paper, we show how to improve the usability of optical assemblies dedicated to the coincident detection of a photon and a charged particle.\nThese assemblies dramatically enhance the solid angle of photon detection and enable efficient coincidence experiments with at least one participating photon.\nWe present two configurations:\nA) A mirror assembly surrounding the interaction volume.\nThis design is optimized for maximum solid angle.\nB) A combination of flexible optical elements adaptable to a variety of experimental constraints, suited for experiments in which a direct view on the interaction volume is impossible. \nThe applicability of both designs is demonstrated by performing electron-photon coincidence measurements on fundamental processes after excitation of supersonic noble gas jets by synchrotron radiation.\n\n\\section{\\label{sec2}Coincident detection of electrons and photons}\nIn this section, we explain the timing scheme of our experiment, followed by other details of the set-up.\nIn many coincidence experiments, including the ones described in this work, either pulsed target delivery or pulsed excitation sources like pulsed lasers, ion bunches in ion storage facilities, or synchrotron radiation pulses with an appropriate reference clock for the measurements are used to ensure that all measured particles have originated in the same physical event.\nThe reference clock pulse is used as a start signal for a time-to-digital converter (TDC).\nIn the case of electron-photon coincidence experiments, a coincidence event occurs, if at least one electron and one photon are detected after a common start signal.\nIn general, coincidence events can result from into true or accidental (or false\\cite{Wehlitz1995}) coincidences.\nFor true coincidences, both detected particles originate in the same interaction process.\nAccidental coincidences occur, if the electron and the photon originate from two different physical processes or from two independent interactions at different sites of the sample.\nFor simplicity and without loss of generality, it is assumed in this work that for a true coincidence event, both the photon and the electron reach the respective detector within the time interval between two consecutive excitation pulses.\\\\\nOften, true and accidental coincidences bear no experimental signature that allows their separation on an event-by-event basis.\nA method to eliminate accidental coincidences uses data acquisition over several consecutive excitation pulses.\nIt is assumed that the rate of accidental coincidences is the same when both particles are detected coincidentally between one reference clock pulse and the successive one as compared to the accidental coincidence signal when the two particles are recorded individually with respect to different successive reference clock pulses.\nTherefore, the true coincidence spectrum can be obtained by the subtraction of the pure accidental coincidence spectrum from the total coincidence spectrum.\nThis is a purely statistical method and does not allow identification of individual true coincidences.\\\\  \nIn a typical time of flight spectrometer, the electron spectrum is obtained by collecting the arrival times of all electron events in a histogram and the time axis encodes the kinetic energy, while the respective histogram of the photon events yields the information about the lifetime of the radiative state.\nThe signal acquisition of an electron-photon-coincidence event is shown in Figure \\ref{fig_detscheme}.\n\\begin{figure}\n\\includegraphics{Figure1}\n\\caption{\\label{fig_detscheme}Detection scheme of electron-photon coincidences.\nThe TDC is triggered by the reference clock and records the arrival times of the electrons $t_\\text{el}$ and photons $t_\\text{ph}$ relative to the reference clock pulse $t_\\text{ref}$.\nAt synchrotron radiation facilities, the time of excitation $t_\\text{exc}$ typically has a constant offset to the provided reference clock. For the showcase experiments, the time of $800\\,\\text{ns}$ between two excitations corresponds to the circulation time of an electron bunch in BESSY II in single bunch mode.}\n\\end{figure}\nThe true coincident electron spectrum is explicitly obtained by separation of the total coincident electron spectrum into four cases:\n\\begin{itemize}\n\\item Both electron and photon were detected between the first and the second reference clock pulse ($t_\\text{el}, t_\\text{ph}<t^2_\\text{ref}$). These events are named $\\text{sp}_{11}$. \n\\item The electron was detected between the first and the second pulse of the reference clock and the photon was detected after the second reference clock signal ($t_\\text{el}<t^2_\\text{ref}$ and $t_\\text{ph}>t^2_\\text{ref}$), named $\\text{sp}_{12}$.\n\\item The electron was detected after the second reference clock pulse and the photon was detected between the first and the second reference clock signal ($t_\\text{el}>t^2_\\text{ref}$ and $t_\\text{ph}<t^2_\\text{ref}$). These events are named $\\text{sp}_{21}$.\n\\item Both electron and photon were detected after the second reference clock pulse ($t_\\text{el},t_\\text{ph}>t^2_\\text{ref}$), named $\\text{sp}_{22}$.\n\\end{itemize}\nThe spectrum of true coincidences is then obtained by:\n\\begin{eqnarray}\n\\text{True coincidences}=\\text{sp}_{11}+\\text{sp}_{22}-\\text{sp}_{12}-\\text{sp}_{21}\n\\label{coinc_eq}\n\\end{eqnarray}\nThis method of data acquisition and processing also allows the detection of multiple electrons with a photon in coincidence.\\\\\nProof-of-principle experiments were performed at the BESSY~II storage ring of the Helmholtz-Zentrum Berlin.\nIn all examples, the synchrotron was operated in single bunch mode with a circulation time of $800\\,\\text{ns}$, i.e. $800\\,\\text{ns}$ temporal spacing between two subsequent excitations.\nIn the presented examples, magnetic bottle type electron spectrometers were used for electron detection.\nDetails for one of the instruments can be found in Ref.~\\onlinecite{Mucke2012}.\nBriefly, the spectrometer uses the inhomogeneous field of a magnetic tip to collect and redirect the electrons from the interaction volume towards a drift tube.\nA magnetic field parallel to the tube axis prevents the loss of electrons due to lateral velocity components. At the end of the drift tube, electrons are amplified by a chevron stack of microchannel plates (MCPs) \\cite{Wiza1979}.\nDetection is carried out by an anode, where the corresponding voltage drop is retrieved from the high voltage potential using capacitive coupling \\cite{Mucke2012}.\nThe signal is processed by a constant fraction discriminator and its arrival time relative to the reference clock is recorded by a TDC.\\\\\nFor photon detection, a single-photon detector as described in Ref.~\\onlinecite{Hans2018} is used. The photons are passing through an MgF$_2$ window coated with a CsTe layer acting as a photocathode which converts photons into photoelectrons and allows the detection of photons with wavelengths in the range of about $120\\,\\text{nm}$ to $300\\,\\text{nm}$ ($4.1\\,\\text{eV}$ to $10.3\\,\\text{eV}$).\nThe electrons are amplified by an MCP chevron stack and the resulting electron cloud hits a delay line type position-sensitive anode \\cite{Jagutzki2002,Hans2018}.\nThe drop of the high voltage at the front MCP is measured using a capacitive coupling and used as the time signal.\nWhile the capability of position sensitive detection is not used for the exemplary measurements presented in this work, future experiments can incorporate these additional information.\nWhile the design of the magnetic bottle provides a relative solid angle $\\Omega_{\\text{rel},\\text{el}}$ close to unity, the relative solid angle $\\Omega_{\\text{rel},\\text{ph}}$ of the photon detection can be estimated to $\\Omega_{\\text{rel},\\text{ph}}\\approx\\frac{1.26\\cdot10^{-3}\\,\\text{m}^2}{4\\pi D^2}$  using Eq. \\ref{photodet_eq} with an active area diameter of $40\\,\\text{mm}$ of the used detector.\nIn the following, we illustrate two specific optics configurations in detail, which increase $\\Omega_{\\text{rel},\\text{ph}}$  in order to enable electron-photon coincidence experiments within a reasonable data acquisition time.\n\n\\section{\\label{sec3}Optics design and applications}\n\\subsection{\\label{sec_maxsolid}Configuration optimized for efficiency}\nFor this approach, the photon detector is attached to a chamber designed for electron coincidence spectroscopy of gaseous and cluster jets similar as in Ref.~\\onlinecite{Hans2016}.\nThe distance between the active detector surface and the interaction volume is $365\\,\\text{mm}$, resulting in a relative solid angle (without optics) of $\\Omega_{\\text{rel},\\text{ph}}=0.075\\,\\%$.\nA mirror system for photon detection was designed as illustrated in Figure \\ref{fig_hollowmirror} to maximize the solid angle.\nIt surrounds the complete interaction volume with apertures specifically designed for the used magnetic bottle spectrometer, target jet, and exciting-photon beam.\\\\\nThe mirror surfaces are made of polished aluminum to ensure a high reflectance for photons within the sensitivity range of the detector.\nThe system consists of a combination of a parabolic and two spherical mirrors which guide photons from the interaction volume towards the detector as shown by exemplary ray trajectories in Figure \\ref{fig_hollowmirror}:\nFirst, the paraboloid facing the detector parallelizes all photons emitted towards this hemisphere (ray path 1, blue).\nSecond, the inner spherical mirror possesses a radius such that all photons hitting this mirror from the interaction volume are reflected and also parallelized towards the detector (ray path 2, red). The size of the inner spherical mirror's area is equal to the entrance width of the parabolic part.\nThird, the outer spherical mirror opposite to the detector possesses a radius equal to the distance to the interaction volume, resulting in a reflection back into the interaction volume (ray path 3, violet).\nThose rays are then parallelized by the parabolic mirror.\n\n\\begin{figure}\n\\includegraphics{Figure2\n\\caption{\\label{fig_hollowmirror}Sketch of the basic rotational symmetric mirror system. The rotation axis is indicated by a grey dashed line.\nIn the direction of the detector, a parabolic mirror guides the photons onto its active area (path 1, blue).\nOn the opposite side of the detector, two spherical mirrors reflect photons towards it.\nThe radius of the inner spherical mirror $R_i$ is twice its distance to the interaction volume $A$ and photons are reflected in a collimated beam to the detector.\nThe radius of the outer mirror $R_o$ is equal to the distance to the interaction volume $A$ and therefore reflects the photons back into the interaction volume (path 3, violet). From here on, the path coincides with photons of path 1.}\n\\end{figure}\nThis configuration was tested in an experiment at the \\mbox{U49-2 PGM1} beamline (BESSY II, HZB)  \\cite{Kachel2016}.\nThe exciting-photon energy was set to $90\\,\\text{eV}$.\nUsing a $50\\,\\mu\\text{m}$ exit slit of the beamline monochromator, the resulting photon beam with a bandwidth of $9\\,\\text{meV}$ was crossed with a He gas jet.\\\\\nFor these experimental conditions, the dominant process is the photoionization of a single 1s electron, \\mbox{i.e. $\\text{He}(1\\text{s}^2)+h\\nu\\rightarrow\\text{He}^+(1\\text{s}^1)+\\text{e}^-$}.\n\\begin{figure}[t!]\n\\includegraphics{Figure3}\n\\caption{\\label{fig_He}Electron spectrum of a He gas jet after irradiation with a photon energy of 90 eV.\na) Total electron spectrum (red solid) and its 10-fold magnification (red dashed).\nb) All recorded one electron - one photon coincidences (true + accidental).\nc) True coincident electron signal after subtraction of accidental coincidences by the method described in section \\ref{sec2}.}\n\\end{figure}\nWith a comparably low cross section the second electron can be additionally promoted into an excited state during the photoionization process, \\mbox{$\\text{He}(1\\text{s}^2)+h\\nu\\rightarrow\\text{He}^+(1\\text{s}^0n\\text{p})+\\text{e}^-$}, which results in the appearance of so-called satellite lines in the photoelectron spectrum \\cite{Heimann1986}.\nAll satellite states subsequently decay by photon emission, but only the $3\\text{p}\\rightarrow2\\text{s}$ transition with about $7.6\\,\\text{eV}$ \\cite{Kramida2018} transition energy is within the sensitivity range of the employed detector.\nThe cross section of the $n=3$ satellite is $1.5(2)\\,\\%$ ,compared to the single 1s ionization \\cite{Lindle1987}.\nOf course, the 3p electron can also decay to the 1s level with a branching ratio of $\\frac{3\\text{p}\\rightarrow2\\text{s}}{3\\text{p}\\rightarrow1\\text{s}}= 0.112$.\nIf the cross sections for all other processes are neglected, about $0.17\\,\\%$ of the detected electrons should be in coincidence with a photon in the sensitivity range of the detector.\n\nThe $4\\text{p}\\rightarrow2\\text{s}$ transition from the $n=4$ satellite at $10.2\\,\\text{eV}$ above the ground state lies at the edge of the detector sensitivity and combined with the reduced cross section of the this satellite should lead to a negligible intensity compared to the $n=3$ case.\\\\ \nThe total non-coincident photoelectron spectrum of He is shown in Figure \\ref{fig_He}a. The 1s photoelectron line is the most prominent feature followed by the $n=2$ satellite and the suggested appearance of the $n=3$ satellite. In a magnified presentation, the satellite lines up to $n=5$ can be identified (not shown). The energy axis was calibrated using the corresponding energies from Ref.~\\onlinecite{Kikas1996}.\nIf only the coincident electrons are taken into account, the $n=3$ satellite increases in intensity relative to the other features as shown in Figure \\ref{fig_He}b.\nThe elimination of accidental coincidences as described in Section \\ref{sec2} yields the (true) photon-coincident electron spectrum shown in Figure \\ref{fig_He}c.\nHere, only the $n=3$ satellite remains as the radiative $n=3 \\rightarrow n=2$ transition is within the sensitivity range of the used detector.\nThe count rate of true coincidences in this measurement was approximately $0.5\\,\\text{Hz}$ with a total electron count rate of approximately $58\\,\\text{kHz}$.\n\\begin{figure}[t!]\n\\includegraphics{Figure4}\n\\caption{\\label{fig_Ar}Ar electron spectrum after irradiation with a photon energy of $449\\,\\text{eV}$.\na) Total electron spectrum, showing the two $2\\text{p}$ photoelectron lines and the four Ar Auger lines with the highest kinetic energies (Assignment according to Ref. \\onlinecite{McGuire1975}):\n(I) \\mbox{$\\text{Ar}^{+}\\,\\text{L}_3\\text{M}_{2,3}\\text{M}_{2,3}(^1\\text{D}_{2})$}.\n(II) \\mbox{$\\text{Ar}^{+}\\,\\text{L}_{3}\\text{M}_{2,3}\\text{M}_{2,3}(^3\\text{P}_{0,1,2})$} and \\mbox{$\\text{Ar}^{+}\\,\\text{L}_{2}\\text{M}_{2,3}\\text{M}_{2,3}(^1\\text{D}_{2})$} (not resolved).\n(III) \\mbox{$\\text{Ar}^{+}\\,\\text{L}_2\\text{M}_{2,3}\\text{M}_{2,3}(^3\\text{P}_{0,1,2})$}. \nb) True coincident electron spectrum. Details are discussed in the text.}\n\\end{figure}\nFor an estimate of the effective solid angle of photon detection achieved with this configuration, the ratio of the number of coincidence events to the total intensity of the $n=3$ satellite in the total electron spectrum, which is about $0.0015$, may be used.\nThis ratio has to be normalized by the $\\frac{3\\text{p}\\rightarrow2\\text{s}}{3\\text{p}\\rightarrow1\\text{s}}$ branching ratio and the quantum efficiency of the photon detector.\nSince the exact quantum efficiency in the spectral range of the $3\\text{p}\\rightarrow 2\\text{s}$ transition in He II ($165\\,\\text{nm}$) is not known, for a conservative estimation of the lower limit we use the peak quantum efficiency of $0.255$ at $254\\,\\text{nm}$\\cite{Photek2010}, resulting in an effective solid angle in the order of $\\geq5\\,\\%$.\nThe geometrical solid angle of the mirror assembly is about $41\\,\\%$.\nWe assign the deviation to imperfect reflection of the mirror and variations in the quantum efficiency of the detector. Nevertheless, this conservative estimate results in an increase of the solid angle by a factor of about $70$ compared to the case without optics.\\\\\nAs a second example, atomic Ar was photoionized with an exciting-photon energy of $449\\,\\text{eV}$.\nAt this photon energy, the 2p photoelectrons and the Ar$^+$ LMM Auger electrons have similar kinetic energies and can be resolved simultaneously by applying a retardation voltage to the drift tube of the magnetic bottle electron spectrometer.\nWhile Auger final states of the form $\\text{Ar}^{2+}(3p^{-2})$ cannot decay further, some of the Ar$^+$ LMM Auger channels will end in radiative satellite states of the configuration $\\text{Ar}^{2+}(3p^{-3}nl)$.\nIn Figure \\ref{fig_Ar}a, the electron spectrum composed of the 2p$_{1\/2}$ and 2p$_{3\/2}$ fine structure components and the four Auger channels of highest kinetic energies, corresponding to the $\\text{Ar}^{+}\\,\\text{L}_2\\text{M}_{2,3}\\text{M}_{2,3}$ and $\\text{Ar}^{+}\\,\\text{L}_3\\text{M}_{2,3}\\text{M}_{2,3} (^3\\text{P}_{0,1,2}\\;\\text{and}\\; ^1\\text{D}_2)$ final states, is shown \\cite{McGuire1975}.\nWhile the latter Auger electrons should not be accompanied by photon emission, the photoelectrons are, because some 2p vacancies lead to radiative Auger final states (of which the corresponding Auger electrons are not within the detected range).\nThis is indeed what is observed in Figure \\ref{fig_Ar}b, which shows the true photon-coincident electron spectrum.\nSurprisingly, one Auger channel is also present at about $203.2\\,\\text{eV}$ kinetic energy.\nWe suggest that this weak channel corresponds to radiative decay of the $\\text{Ar}^{+}\\,\\text{L}_2\\text{M}_{2,3}\\text{M}_{2,3}(^1\\text{S}_0)$ Auger final state\\cite{McGuire1975} to the $\\text{Ar}^{+}\\,\\text{L}_2\\text{M}_{2,3}\\text{M}_{2,3}(^3\\text{P}_{0,1,2})$ state via magnetic dipole or electronic quadrupole transitions, which are within the sensitivity range of the detector.\n\\subsection{\\label{sec_adapt}Configuration optimized for adaptability}\nIn certain cases, the application of configuration A might be hindered by experimental constraints.\nFor example, the layout of the vacuum chamber, target source, or electron spectrometer can interfere spatially with the mirror or the direct view towards the interaction volume might be blocked.\nThen, the solid angle can still be increased significantly using a combination of flexible optical elements.\\\\\nIn the presented experiment, a magnetic bottle electron spectrometer similar to configuration A was used with a different interaction chamber.\nHowever, no port with a direct view of the interaction volume was available for the photon detector as illustrated in Figure \\ref{fig_adaptset}.\nIn addition, a valve at the entrance of the electron spectrometer would intersect with parts in the close vicinity of the interaction volume thus preventing a mirror design as described above.\nWith these constraints, an assembly of three different optical elements is used for photon guidance.\nA combination of an UV enhanced Al coated spherical mirror and a fused silica spherical plano convex lens, positioned at opposite sides of the interaction volume, may achieve a relative solid angle of up to $7.5\\,\\%$.\nSimultaneously, the lens collimates the emitted photons onto a planar mirror, which is used to redirect the photons onto the detector.\n\\begin{figure}\n\\includegraphics{Figure5}\n\\caption{\\label{fig_adaptset}Isometric view of the adaptable set-up.\nA spherical mirror and a plano convex lens are used to increase the solid angle for the photon detection.\nThe plano convex lens distance and focal length are chosen such that the transmitted photons are parallelized while the plane mirror deflects the photons onto the detector axis.\nTherefore, an observation of the emitted photons without direct view is possible.}\n\\end{figure}\nThe functionality of this configuration was validated in an experiment conducted at the \\mbox{UE56\/2 PGM1} beamline at BESSY II in Berlin.\nNeon atoms were injected effusively into the interaction chamber through a $25\\,\\mu\\text{m}$ nozzle.\nThe exciting-photon energy was set to $867.1\\,\\text{eV}$ corresponding to the resonant $1\\text{s}^22\\text{s}^22\\text{p}^6 \\rightarrow1\\text{s}^12\\text{s}^22\\text{p}^63\\text{p}$ excitation of atomic Ne.\nHere, a variety of different de-excitation pathways are possible \\cite{Hayaishi1995}.\n\\begin{figure}\n\\includegraphics{Figure6}\n\\caption{\\label{fig_Ne}Electron spectra of Ne after irradiation with a photon energy of $867.1\\,\\text{eV}$.\na) Total electron spectrum.\nb) True coincident electron spectrum.\n}\n\\end{figure}\nThe total electron spectrum in Figure 6a consists of an intense peak at short times of flight, comprised of unresolved Auger and valence electrons.\nThe slower electrons are the result of further autoionizing Auger final states. However, the only relaxation channels observable by an electron-photon coincidence are spectator Auger final states of the form $1\\text{s}^22\\text{s}^22\\text{p}^4n\\text{p}$ with fast Auger electrons included in the fast electron peak.\nConsequently, the autoionizing Auger final states vanish in the true coincidence spectrum in Figure 6b.\nHere, the true coincidence rate was about $0.01\\,\\text{Hz}$, compared to a total electron count rate of about $60\\,\\text{kHz}$.\nDespite the low count rate, the experiment yields an interpretable result.\nThis illustrates that the coincidence measurement involves an extremely efficient noise reduction, allowing to detect signals orders of magnitude weaker than the non-coincident-noise.\n\\section{\\label{sec4}Conclusion}\nWe have developed two optical assemblies which can be combined with magnetic bottle electron spectrometers to result in a highly efficient set-up for electron-photon coincidence experiments.\nWhile configuration A focuses on the optimization of the solid angle of the photon detection, configuration B is adaptable to different experimental constraints.\\\\\nBoth configurations were tested successfully for exemplary physical processes that feature electron-photon coincidences after excitation of atomic noble gas targets with synchrotron radiation.\nWe demonstrated that this method is capable of identifying obscured physical processes and can circumvent the signal to noise ratio problem of very low count rate experiments.\nWe envision this method to be capable to unravel energy and charge transfer processes in dense media.\nHere, the additional insight of photon detection allows to further characterize ultra-fast phenomena taking place only in such dense media.\nExamples where photon emission plays an decisive role are resonant Interatomic Coulombic Decay\\cite{Knie2014}, Radiative Charge Transfer\\cite{Hans2018a}, or possibly ultra fast proton transfer in liquid samples\\cite{Hans2017,Thuermer2013}.\n\\begin{acknowledgments}\nWe thank Claudia Kolbeck, Sebastian Malerz, Marvin Pohl, Anne Stephansen (FHI Berlin, Germany) and Clara Saak (Uppsala University, Sweden) for their support during the synchrotron radiaton beamtime. \nWe gratefully acknowledge the HZB for the allocation of synchrotron radiation beamtime and the BESSY II staff for their support. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) \u2013 Projektnummer 328961117 \u2013 SFB 1319 ELCH and Research Unit FOR 1789.\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe nonlinear variational wave equation $\\psi_{tt}-c(\\psi)(c(\\psi)\\psi_x)_x=0$ was derived by Saxton \\cite{S} as a model of the director field of a nematic liquid crystal. The nonlinear variational wave equation has recieved wide attention \\cite{BCZ,BZ,GHZ,HR} due to mathematical challenges in the form of wavebreaking in finite time, and distinct ways to extend the solution to a global weak solution. In this context wavebreaking means that either the time or the space derivative becomes unbounded at certain points, while the solution remains H\\\" older continuous.\n\nFurthermore, Hunter and Saxton derived the equation $u_{tx}+(uu_x)_x=\\frac12u_x^2$ from the nonlinear variational wave equation \\cite{HS} as an asymptotic equation for small perturbations in the long time regime in a moving frame. The Hunter--Saxton equation share many of the features of the nonlinear variational wave equation such as wavebreaking, conservative and dissipative weak solutions, and H\\\" older continuity \\cite{BC,BHR,BZZ,CGH,D,HZ2,HZ3,ZZ,ZZ2}. In addition it exhibited novel features such as complete integrability and interpretation as a geodesic flow \\cite{HZ,KM}. It also proved  easier to work with due to there being only one family of characteristics and existence of explicit solutions.\n\nA two-component generalization for the Hunter--Saxton equation was derived from two-component Camassa--Holm \\cite{DP}, and independently form the Gurevich--Zybin system \\cite{P}. The two-component generalization is similar to the Hunter--Saxton equation since wavebreaking is also possible, there are both conservative and dissipative weak solutions, and $u$ is H\\\" older continuous \\cite{GN,N,W}. However when the second variable is nonzero almost everywhere initially, there will be no wave breaking \\cite{N}, and in that sense the introduction of a second variable regularizes the equation. The two-component Hunter--Saxton system has, however, not been shown to be related to the theory of nematic liquid crystals, and one of the aims of this paper is to establish that the two-component Hunter--Saxton system can indeed be derived from the theory of nematic liquid crystals, and that the new variable is related to the order parameter\n\nIn the Ericksen--Leslie theory of nematic liquid crystals the configuration is described by a director field $\\bn$ which gives the local orientation of the rods, and an order parameter field $s$ which gives the local degree of orientation \\cite{E,L}. When the nonlinear variational wave equation was first derived the local degree of orientation was assumed constant, so that  a scalar equation was obtained \\cite{S}. Here we will follow a similar route, but account for variable degree of orientation. Thus, in Section \\ref{section:derivation}, we will arrive at a novel two-component system of nonlinear wave equations. Moreover, we show that a similar asymptotic expansion to the one by Hunter and Saxton \\cite{HS} yields the two-component Hunter--Saxton equation. In Section \\ref{section:analysis} existence of global solutions is shown in the case of constant wave speed, and local solutions is shown to exist in general. The proofs rely on fixed point iterations and standard semigroup theory for evolution equations \\cite{K}. \n\n\\section{Derivation of the equations} \\label{section:derivation}\nIn Ericksen--Leslie theory a nematic liquid crystal in a domain $\\Omega$ is desribed by a director field $\\bn: [0,T]\\times\\Omega\\rightarrow \\R\\mathbb{P}^2$ and an order parameter $s:[0,T]\\times\\Omega\\rightarrow (-\\frac12,1)$ \\cite{E,L}. The Lagrangian of the system is then \\cite{E} given by\n\\begin{equation}\n\\mc L = -(s\\bn)_t^2+ W_2(s,\\nabla s,\\bn,\\nabla\\bn) + W_0(s),\n\\end{equation}\nwith the potential energy term $W_2$ \n\\begin{align}\nW_2(s,\\nabla s,\\bn,\\nabla\\bn) &= \\left(K_1+L_1s\\right)s^2\\left(\\nabla\\cdot\\bn\\right)^2 + \\left(K_2+L_2s\\right)s^2\\left(\\bn\\cdot\\nabla\\times\\bn\\right)^2\\non\n\t&\\quad+ \\left(K_3+L_3s\\right)s^2\\left|\\bn\\times\\nabla\\times\\bn\\right|^2 \\non \n\t&\\quad+ \\left((K_2+K_4)+(L_2+L_4)s\\right)s^2\\left[\\mathrm{tr}\\:\\nabla\\bn^2-(\\nabla\\cdot\\bn)^2\\right]\\non\n\t&\\quad+ \\left(\\kappa_1+\\lambda_1s\\right)\\left|\\nabla s\\right|^2 + \\left(\\kappa_2+\\lambda_2s\\right)\\left(\\nabla s\\cdot\\bn\\right)^2 \\non\n\t&\\quad+ \\left(\\kappa_3+\\lambda_3s\\right)\\left(\\nabla\\cdot\\bn\\right)\\left(\\nabla s\\cdot\\bn\\right) + \\left(\\kappa_4+\\lambda_4s\\right)\\nabla s\\cdot\\left((\\bn\\cdot\\nabla)\\bn\\right),\n\\end{align}\nand some function $W_0$ satisfying \n\\begin{equation}\n\\label{eq:W_0 cond}\n\t\\lim_{s\\downarrow-\\frac12}W_0(s)=+\\infty,\\quad \\lim_{s\\uparrow 1}W_0(s)=+\\infty, \\text{ and } \\lim_{s\\rightarrow 0}\\frac{W_0(s)}{s^2} \\in\\R.\n\\end{equation}\nWe are interested in the case where $\\bn=(\\cos\\psi,\\sin\\psi,0)$, and $s$ and $\\psi$ depends on $t$ and $x$ only.\nThen the expression for $W_2$ can be written\n\\begin{align}\nW_2 &= s^2\\psi_x^2\\left((K_1+L_1s)\\sin^2\\psi+(K_3+L_3s)\\cos^2\\psi\\right)\\non\n\t&\\quad + s_x^2\\left((\\kappa_1+\\lambda_1s)\\sin^2\\psi+(\\tilde\\kappa_2+\\tilde\\lambda_2s)\\cos^2\\psi\\right)\\non\n\t&\\quad -s_x\\psi_x\\sin 2\\psi\\left(\\frac{\\kappa_3+\\kappa_4}2+\\frac{\\lambda_3+\\lambda_4}2s\\right),\n\\end{align}\nwhere $\\tilde\\kappa_2=\\kappa_1+\\kappa_2$ and $\\tilde\\lambda_2 = \\lambda_1+\\lambda_2$. We simplify the problem by considering the case where\n\\begin{subequations}\n\\begin{align}\n\\kappa_3 &= \\kappa_4 = \\lambda_3 = \\lambda_4 = 0,\\\\\nL_1 &= L_3 = \\lambda_1 = \\tilde\\lambda_2 = 0,\\\\\nK_1&= \\kappa_1,\\\\\nK_3 &= \\tilde\\kappa_2.\n\\end{align}\n\\end{subequations}\nThen with the definition $c(\\psi)^2= K_1\\sin^2\\psi + K_3\\cos^2\\psi$, we get the Lagrangian density\n\\begin{align}\n\\label{eq:2NVW lagr}\n\\mc L^{2NVW} &= -\\frac 12s_t ^2 -\\frac 12s^2\\psi_t^2+\\frac12s^2c(\\psi)^2\\psi_x^2+\\frac 12c(\\psi)^2s_x^2 + W_0(s)\\non\n\t&= -\\frac 12(s\\bn)_t^2+\\frac12 c(\\bn)^2(s\\bn)_x^2 + W_0(s).\n\\end{align}\n\\begin{definition}\nWe define the two-component nonlinear variational wave system to be\n\\begin{subequations}\n\\label{eq:2NVW}\n\\begin{align}\n\ts^2\\left(\\psi_{tt}-c(\\psi)\\left(c(\\psi)\\psi_x\\right)_x\\right)+2s\\left(\\psi_ts_t-c(\\psi)^2\\psi_xs_x\\right)+c(\\psi)c'(\\psi)  s_x^2 &= 0,\\\\\n\ts_{tt} - c(\\psi)\\left(c(\\psi) s_x\\right)_x -c(\\psi)c'(\\psi)\\psi_xs_x -s\\left(\\psi_t^2-c(\\psi)^2\\psi_x^2\\right) + W_0'(s) &=0,\\\\\n\tc(\\psi)^2 - K_1\\sin^2\\psi-K_3\\cos^2\\psi &= 0,\\\\\n\tW_0 &\\in C^4\\left(\\left(-\\frac12,1\\right)\\right)\n\n\\end{align}\n\\end{subequations}\n\\end{definition}\n\\begin{remark}\nThe system \\eqref{eq:2NVW} is the Euler--Lagrange equations for \\eqref{eq:2NVW lagr}.\n\\end{remark}\nOne can define a conserved energy for classical solutions.\n\\begin{proposition}\nDefine the energy density \n\\begin{equation}\n\t\\label{def:E}\n\t\\mc E = \\frac12\\left(s^2(\\psi_t^2+c(\\psi)^2\\psi_x^2)+(s_t^2+c(\\psi)^2s_x^2)\\right)+W_0(s).\n\\end{equation}\nand the energy density flux\n\\begin{equation}\n\t\\label{def:F}\n\t\\mc F = \\left(s^2\\psi_t\\psi_x+s_ts_x\\right).\n\\end{equation}\nThen for classical solutions of \\eqref{eq:2NVW} the energy satisfy the equations\n\\begin{subequations}\n\\label{eq:cons E}\n\\begin{align}\n\t\\mc E_t-\\left(c(\\psi)^2\\mc F\\right)_x &= 0,\\\\\n\t\\mc F_t - \\left(\\mc E-2W_0(s)\\right)_x &= 0.\n\\end{align}\n\\end{subequations}\n\\end{proposition}\nWe will now derive the two-component Hunter--Saxton system from \\eqref{eq:2NVW lagr}. To follow the work of Hunter and Saxton \\cite{HS} we introduce $\\psi(t,x) = \\psi_0 + \\epsilon u(\\epsilon t,x-c_0t)$ and $s = s_0+\\epsilon r(\\epsilon t,x-c_0t)$ with $c_0^2 = K_1\\sin^2\\psi_0 + K_3\\cos^2\\psi_0$. Then expansion of \\eqref{eq:2NVW lagr} in powers of $\\epsilon$ gives\n\\begin{align}\n\\mc L^\\epsilon &= W_0(s_0) + \\epsilon W_0'(s_0)r + \\epsilon^2\\frac12W_0''(s_0)r^2\\non\n&\\quad + \\epsilon^3\\bigg(c_0r_tr_x+s_0^2c_0u_tu_x + s_0^2\\left(cc'\\right)_0 uu_x^2 + \\left(cc'\\right)_0 ur_x^2 + \\frac16W_0'''(s_0)r^3\\bigg)+ O(\\epsilon^4).\n\\end{align}\nIf we select $s_0$ such that $W_0'(s_0) = W_0''(s_0) = W'''(s_0) = 0$, the third order terms in the above Lagrangian gives (possibly by rescaling)\n\\begin{equation}\n\\label{eq:2HS lagr}\n\\mc L^{2HS} = u_tu_x +uu_x^2 + r_tr_x + ur_x^2.\n\\end{equation}\n\\begin{theorem}\n\tThe two-component Hunter--Saxton system\n\t\\begin{subequations}\n\t\\begin{align}\n\t(u_t+uu_x)_x &= \\frac 12u_x^2+\\frac 12\\rho^2,\\\\\n\t\\rho_t + (u\\rho)_x &= 0,\n\t\\end{align}\n\t\\end{subequations}\n\tis the Euler--Lagrange equations for the Lagrangian density \\eqref{eq:2HS lagr} with $\\rho=r_x$.\n\\end{theorem}\n\n\\section{The two-component nonlinear variational wave system} \\label{section:analysis}\nWe will first consider the case $K_1=K_3$, that is the wave speed $c$ is independent of $\\psi$. We will assume $s \\geq 0$ such that we can introduce the complex variable $\\zeta = se^{i\\psi}$, and \\eqref{eq:2NVW} reduces to\n\\begin{equation}\n\\label{eq:2NVWlinear}\n\\zeta_{tt}-c^2\\zeta_{xx} +\\frac{W_0'\\left(|\\zeta|\\right)}{|\\zeta|}\\zeta = 0,\n\\end{equation}\nwhich is a defocusing nonlinear wave equation. Recall that the physical interpretation of $\\psi$ is an angle, and thus any solutions $\\psi$ and $\\psi + n\\cdot 2\\pi$ should be considered equal from an application point of view. We have conservation laws for energy $\\mc E = \\frac12|\\zeta_t|^2+\\frac12c^2|\\zeta_x|^2+W_0(|\\zeta|)$ and energy flux $\\mc F = \\frac12\\left(\\bar\\zeta_t\\zeta_x+\\zeta_t\\bar\\zeta_x\\right)$ as follows\n\\begin{subequations}\n\\begin{align}\n\t\\frac{\\partial}{\\partial t}\\mc E - c^2\\frac{\\partial}{\\partial x}\\mc F &= 0,\\\\\n\t\\frac{\\partial}{\\partial t}\\mc F - \\frac{\\partial}{\\partial x}\\left(\\mc E-2W_0(|\\zeta|)\\right) &= 0.\n\\end{align}\n\\end{subequations}\nHence both $\\int_\\R\\mc E\\:\\d x$ and $\\int_\\R\\mc F\\:\\d x$ are conserved for any classical solution with bounded energy. To analyze solutions we will need some conditions on the function $W_0$. We will assume that $W_0(s)\\rightarrow\\infty$ rapidly enough as $s\\rightarrow1$ to be able to bound $\\|W_0(|\\zeta|)\\|_\\infty$ in terms of the total energy $E$. In addition we will assume that $W_0$ is non-negative and that $W_0$ is well behaved close to $s=0$, and also close to zeros of $W_0$. We will also relax \\eqref{eq:W_0 cond} slightly to allow for $s=0$ to be a local maximum or minimum instead of a global minimum, in accordance with \\cite{E}.\n\\begin{definition}\n\\label{def:W_0}\nWe will assume that $W_0\\in C^4([0,1))$ and non-negative, and that there exists a finite number of $s^*$ such that $W_0(s^*)=0$, and for all zeros $s^*$,\n\\begin{align}\n\t\\lim_{s\\rightarrow 0}\\frac{W_0(s)-W_0(0)}{s^2} &= \\frac 12W_0''(0)\\in\\R,\\\\\n\t\\lim_{s\\rightarrow s^*}\\frac{W_0(s)}{(s-s^*)^2} &= \\frac12W_0''(s^*)\\in [0,\\infty).\n\\end{align}\nMoreover, we let $W_0$ satisfy\n\\begin{equation}\n\t\\int_s^1W_0(u)(1-u)\\:d u = \\infty,\n\\end{equation}\nand that there exists $\\tilde s\\in[0,1)$ such that $W_0(s)>0$ and $W_0'(s)>0$ for all $s\\in (\\tilde s,1)$. \n\\end{definition}\nWe define the function spaces for the solutions in the next definition.\n\\begin{definition}\n\\label{def:spaces}\n\tLet $\\zeta^*\\in\\C$, $|\\zeta^*|<1$, and\n\t\\begin{equation}\t\n\t\tX_{\\zeta^*} = \\{(\\zeta,\\sigma)\\mid \\zeta \\in W^{1,\\infty}(\\R),\\zeta-\\zeta^*\\in H^1(\\R), \\sigma \\in L^\\infty(\\R)\\cap L^2(\\R)\\},\n\t\\end{equation}\t\n\tand define\n\t\\begin{align}\n\t\t\\mc E(\\zeta,\\sigma) &= \\frac12|\\sigma|^2+\\frac12|\\zeta_x|^2+W_0(|\\zeta|),\\\\\n\t\t\\mc F(\\zeta,\\sigma) &= \\frac12\\sigma\\bar\\zeta_x+\\frac12\\bar\\sigma\\zeta_x,\n\t\\end{align}\n\tand note that both $\\mc E$ and $\\mc F$ are real valued. Let now\n\t\\begin{equation}\n\t\tX_E = \\{(\\zeta,\\sigma)\\in X_{\\zeta^*}\\mid |\\mc E\\|_{L^1} \\leq E\\},\n\t\\end{equation}\n\tand denote\n\t\\begin{align}\n\t\t\\D_{T,E} &= \\left\\{\\zeta\\in C([0,T],W^{1,\\infty}(\\R,\\C))\\cap C^1([0,T],L^\\infty(\\R,\\C) \\mid (\\zeta(t),\\zeta_t(t))\\in X_E\\right\\}.\n\t\\end{align}\n\tIn the case $T<\\infty$ we equip $\\D_{T,E}$ with the metric $d_{\\D_{E,T}}$ induced from \n\t\\begin{align}\n\t\t\\|\\zeta\\| &= \\sup_{t\\in[0,T]}\\big(\\|\\zeta(t)\\|_{W^{1,\\infty}(\\R)} +\\| \\zeta(t)-\\zeta^*\\|_{H^1(\\R)}\\non &\\quad+ \\|\\zeta_t(t))\\|_{L^\\infty(\\R)} + \\| \\zeta_t(t))\\|_{L^2(\\R)} + \\|W_0(|\\zeta(t)|)\\|_{L^1(\\R)}\\big).\n\t\\end{align}\n\\end{definition}\nFor bounded energy we can now a priori put constraints on possible solutions.\n\\begin{proposition}\n\\label{prop:W_0}\nLet $W_0$ satisfy the conditions of Definition \\ref{def:W_0}. Then there exist positive constants $c_E, C_E$, depending on $W_0$ and $E$ only, such that for any $(\\zeta,\\sigma)\\in X_E$ we have\n\\begin{align}\n\t\\|W_0(|\\zeta|)\\|_{L^\\infty(\\R)} &\\leq C_E,\\non\n\t\\|\\zeta\\|_{L^\\infty(\\R)} &\\leq c_E < 1.\\nonumber\n\\end{align}\nMoreover for $s\\in [0,c_E]$ there is a positive constant $k_E$ such that\n\\begin{equation}\n\tW_0'(s)^2 \\leq k_E W_0(s).\n\\end{equation}\nFurthermore, we can define\n\\begin{align}\n\tL_E &= \\sup_{s\\in[0,c_E]} |W_0'(s)|,\\\\\n\tL_E' &= \\sup_{s\\in[0,c_E]} |\\frac{W_0'(s)}{s}|,\\\\\n\tL_E'' &= \\sup_{s\\in[0,c_E]} |W_0''(s)|.\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nSince $\\frac12\\|\\sigma\\|_2^2+\\frac12 c^2 \\|\\zeta_x\\|_2^2 + \\|W_0(|\\zeta|)\\|_1 \\leq E$ we have that $|\\zeta(x_1)-\\zeta(x_2)|\\leq\\frac{\\sqrt{2E_1}}{c}\\sqrt{|x_1-x_2|}$ where $E_1 = E-\\frac12\\|\\sigma\\|_2^2-\\|W_0(|\\sigma|)\\|_1 \\geq 0$. Since $\\zeta$ is continuous we have that there exists $\\hat x$ such that $|\\zeta(\\hat x)| = \\|\\zeta\\|_{L^\\infty(\\R)}\\leq 1$. Then we have that\n\\begin{equation}\n\t|\\zeta(x)| \\geq \n\t\t\\begin{cases}\n\t\t\t\\|\\zeta\\|_\\infty-\\frac{\\sqrt{2E_1}}c\\sqrt{|x-\\hat x|},&\\hat x-\\frac{\\|\\zeta\\|_\\infty^2c^2}{2E_1}<x<\\hat x+\\frac{\\|\\zeta\\|_\\infty^2c^2}{2E_1},\\\\\n\t\t\t0, &\\text{ else}.\n\t\t\\end{cases}\n\\end{equation}\nIn particular, we have that\n\\begin{align}\n\t\\int_\\R W_0(|\\zeta(x)|)\\:\\d x &\\geq \\int_{\\hat x-\\frac{c^2}{2E_1}(\\|\\zeta\\|_\\infty-\\tilde s)^2}^{\\hat x+\\frac{c^2}{2E_1}(\\|\\zeta\\|_\\infty-\\tilde s)^2} W_0(|\\zeta(x)|)\\:\\d x\\non\n\t\t&\\geq \\int_{\\hat x-\\frac{c^2}{2E_1}(\\|\\zeta\\|_\\infty-\\tilde s)^2}^{\\hat x+\\frac{c^2}{2E_1}(\\|\\zeta\\|_\\infty-\\tilde s)^2} W_0\\left(\\|\\zeta\\|_\\infty-\\frac{\\sqrt{2E_1}}{c}\\sqrt{|x-\\hat x|})\\right)\\:\\d x\\non\n\t\t&= \\int_{\\tilde s}^{\\|\\zeta\\|_\\infty}W_0(u)\\left(\\|\\zeta\\|_\\infty-u\\right)\\:\\d u\n\\end{align}\nBy the monotone convergence theorem we have that\n\\begin{equation}\n\\lim_{S\\rightarrow 1^-}\\int_s^S W_0(u)\\left(S-u\\right)\\:\\d u = \\int_s^1W_0(u)\\left(1-u\\right)\\:\\d u = \\infty.\n\\end{equation}\nHence we have the inequality\n\\begin{equation}\n\t\\int_{\\tilde s}^{\\|\\zeta\\|_\\infty}W_0(u)\\left(\\|\\zeta\\|_\\infty-u\\right)\\:\\d u \\leq E - E_1,\n\\end{equation}\nwhich proves the existence of $c_E, C_E$ such that $\\|\\zeta\\|_\\infty = c_E < 1$ and $\\|W_0(|\\zeta|)\\|_\\infty = W_0(\\|\\zeta\\|_\\infty) = C_E<\\infty$. The constant can be chosen to depend on $W_0$ and $E$ only.\n\nTo prove that $W_0'^2\\leq k_E W_0$, note that whenever $W_0$ tends to zero that\n\\begin{equation}\n\t\\lim_{s\\rightarrow s^*}\\frac{W_0'(s)^2}{W_0(s)} = 2W_0''(s^*)\\in[0,\\infty).\n\\end{equation} \nIn between zeros of $W_0$ the fraction $\\frac{W_0'^2}{W_0}$ is continuous. Since there is only a finite number of intervals where $W_0$ is zero and $[0,c_E]$ is bounded, $\\frac{W_0'^2}{W_0}$ has to be bounded as well by a constant dependent on $c_E$. Since $W_0$ is non-negative we get the desired inequality. \n\\end{proof}\n\\begin{remark}\n\\label{rmk:W_0}\nNote that $\\zeta_1$ and $\\zeta_2$ in $X_E$ we have that\n\\begin{align}\n\t\\int_\\R W_0(|\\zeta_1(x)|)\\:\\d x &= \\int_\\R\\frac12L_E''|\\zeta_1-\\zeta^*|^2\\:\\d x \\non\n\t&\\leq \\frac12L_E''\\|\\zeta_1-\\zeta^*\\|_2^2, \n\\end{align}\nand similarly, since $W_0(|\\zeta|) = \\int_{|\\zeta^*|}^{|\\zeta|}W_0''(u)(|\\zeta|-u)\\:\\d u$,\n\\begin{align}\n\t\\int_\\R|W_0(|\\zeta_1|)-W_0(|\\zeta_2|)|\\:\\d x &=\\int_\\R\\Bigg| \\int_{|\\zeta^*|}^{|\\zeta_1|}W_0''(u)\\left(|\\zeta_1|-|\\zeta_2|\\right)\\:\\d u\\non&\\qquad+\\int_{|\\zeta_2|}^{|\\zeta_1|}W_0''(u)\\left(|\\zeta_2|-u\\right)\\:\\d u\\Bigg|\\:\\d x \\non\n\t\t&\\leq L_E''\\|\\zeta_1-\\zeta_2\\|_2^2 + L_E''\\|\\zeta_1-\\zeta^*\\|_2\\|\\zeta_1-\\zeta_2\\|_2.\n\\end{align}\n\\end{remark}\n\\begin{theorem}\n\\label{thm:global}\nLet $(\\zeta_0,\\zeta_{t0})\\in X_E$, and assume that $W_0$ satisfies the conditions in Definition \\ref{def:W_0}. Then there exists a unique global strong energy conserving solution $(\\zeta,\\zeta_t)\\subseteq \\D_{\\infty,E}$ of \\eqref{eq:2NVWlinear} depending continuously on the initial data $(\\zeta_0,\\zeta_{t0})\\in X_E$.\n\\end{theorem}\n\\begin{proof}\nThe idea of the proof is to show that the solution in strong form\n\\begin{align}\n\t\\label{eq:strong form}\n\t\\zeta(t,x) &= \\frac12\\left(\\zeta_0(x-ct)+\\zeta_0(x+ct)\\right) + \\frac1{2c}\\int_{x-ct}^{x+ct}\\zeta_{t0}(y)\\:\\d y\\non&\\quad - \\frac1{2c}\\int_0^t\\int_{x-c(t-s)}^{x+c(t-s)}\\frac{W_0'(|\\zeta(s,y)|)}{|\t\\zeta(s,y)|}\\zeta(s,y)\\:\\d y\\d s,\n\\end{align}\nis the unique fixed point of a contraction on $\\D_{T,E}$, as is done for Picard iteration for a to be specified $T$. To be able to prove that energy is conserved by strong solutions we will at first assume that the initial data in $X_E$ is smooth, $(\\zeta_0,\\zeta_{t0})\\in C^2(\\R)\\times C^1(\\R)$. We will then prove that the strong solution with smooth initial data preserve energy, and that the solution operator taking initial data in $X_E$ to solutions in $\\D_{T,E}$ is continuous and can thus be extended to all of $X_E$. We prove that the energy is still conserved for these solutions. Finally we note that the existence time $T$ depends on $W_0$ and $E$ only, and thus that the existence time can be extended indefinitely.\n\nWe will now assume that $0<E<E'$ and that $(\\zeta_0,\\zeta_{t0})\\in X_{E}$ and $\\zeta_0\\in C^2(\\R), \\zeta_{t0}\\in C^1(\\R)$, and let $(\\hat\\zeta,\\hat\\zeta_t)\\in\\D_{T,E'}\\cap C^2([0,\\infty)\\times\\R)$ with $(\\hat\\zeta(0),\\hat\\zeta_t(0))=(\\zeta_0,\\zeta_{t0})$. Then let $(\\zeta,\\zeta_t)\\in C([0,T],X_{E'})$, $\\zeta\\in C^2([0,\\infty)\\times\\R)$, be the classical solution of\n\\begin{equation}\n\t\\label{eq:linear iterative}\n\t\\zeta_{tt}-c^2\\zeta_{xx}=-\\frac{W'_0(|\\hat\\zeta|)}{|\\hat\\zeta|}\\hat\\zeta,\\quad (\\zeta,\\zeta_t)|_{t=0}=(\\zeta_0,\\zeta_{t0}).\n\\end{equation}\nThen by Duhamel's principle we have\n\\begin{subequations}\n\\label{eq:iterative solutions}\n\\begin{align}\n\t\\zeta(t,x) &= \\frac12\\left(\\zeta_0(x-ct)+\\zeta_0(x+ct)\\right) + \\frac1{2c}\\int_{x-ct}^{x+ct}\\zeta_{t0}(y)\\:\\d y\\non &\\quad - \\frac1{2c}\\int_0^t\\int_{x-c(t-s)}^{x+c(t-s)}\\frac{W_0'(|\\hat\n\t\t\\zeta(s,y)|)}{|\\hat\\zeta(s,y)|}\\hat\\zeta(s,y)\\:\\d y\\:\\d s,\\\\\n\t\\zeta_t(t,x) &=  \\frac c2\\left(\\zeta_{x0}(x+ct)-\\zeta_{x0}(x-ct)\\right) + \\frac12\\left(\\zeta_{t0}(x+ct)+\\zeta_{t0}(x-ct)\\right)\\:\\d y\\non &\\quad - \\frac12\\int_0^t\\frac{W_0'(|\\hat\n\t\t\\zeta(s,x-c(t-s))|)}{|\\hat\\zeta(s,x-c(t-s))|}\\hat\\zeta(s,x-c(t-s))\\non &\\qquad\\qquad+\\frac{W_0'(|\\hat\\zeta(s,x+c(t-s))|)}{|\\hat\\zeta(s,x+c(t-s))|}\\hat\\zeta(s,x+c(t-s))\\:\\d s.\n\\end{align}\n\\end{subequations}\nTo prove that the strong solution \\eqref{eq:iterative solutions} is a member of $\\D_{T,E'}$ we will need norm estimates on the solutions. For $(\\zeta,\\zeta_t) \\in \\D_{T,E}$ we have that\n\\begin{align}\n\t\\left|\\frac1{2c}\\int_{x-c(t-s)}^{x+c(t-s)} \\frac{W_0'(|\\zeta(s,y)|)}{|\\zeta(s,y)|}\\zeta(s,y)\\:\\d y\\right| &\\leq L_E (t-s),\\\\\n\t\\left|\\frac{\\partial}{\\partial x} \\frac1{2c}\\int_{x-c(t-s)}^{x+c(t-s)} \\frac{W_0'(|\\zeta(s,y)|)}{|\\zeta(s,y)|}\\zeta(s,y)\\:\\d y\\right| &\\leq \\frac1c L_E.\n\\end{align}\nThe integrals can be interpreted as convolutions\n\\begin{equation}\n\t\\int_{x-ct}^{x+ct} f(y)\\:\\d y = \\mathbf{1}_{[-ct,ct]}*f (x),\n\\end{equation}\nand then by Young's inequality we get for any $p\\in[1,\\infty]$, \n\\begin{equation}\n\t\\label{eq:conv}\n\t\\left\\|\\int_{x-ct}^{x+ct} f(y)\\:\\d y\\right\\|_p \\leq 2ct\\|f\\|_p.\n\\end{equation}\nThus, the solution $(\\zeta,\\zeta_t)$ satisfies for $p=2,\\infty$ the estimates\n\\begin{align}\n\t\\|\\zeta(t)-\\zeta^*\\|_p &\\leq \\|\\zeta_0-\\zeta^*\\|_p + t\\|\\zeta_{t0}\\|_p + \\frac12t^2\\max\\{ L_{E'},\\sqrt{k_{E'}E'}\\},\\\\\n\t\\|\\zeta_x(t)\\|_p &\\leq \\|\\zeta_{x0}\\|_p + \\frac1c\\|\\zeta_{t0}\\|_p + \\frac2ct\\max\\{ L_{E'},\\sqrt{k_{E'}E'}\\},\\\\\n\t\\|\\zeta_t(t)\\|_p &\\leq c\\|\\zeta_{x0}\\|_p + \\|\\zeta_{t0}\\|_p + \\frac12t^2\\max\\{ L_{E'},\\sqrt{k_{E'}E'}\\}\n\\end{align}\nWe need to show that the energy is bounded. We have that\n\\begin{align}\n\\label{eq:iterative energy}\n\t&\\frac{\\partial}{\\partial t}\\left(\\frac12|\\zeta_t|^2+\\frac12c^2|\\zeta_x|^2+W_0(|\\zeta|)\\right) - \\frac{\\partial}{\\partial x}\\frac12c^2\\left(\\bar\\zeta_t\\zeta_x+\\zeta_t\\bar\\zeta_x\\right) =\\non &\\quad \\frac12\\zeta_t\\left(\\frac{W_0'(|\\zeta|)}{|\\zeta|}\\bar\\zeta - \\frac{W_0'(|\\hat\\zeta|)}{|\\hat\\zeta|}\\bar{\\hat\\zeta}\\right)+\\frac12\\bar\\zeta_t\\left(\\frac{W_0'(|\\zeta|)}{|\\zeta|}\\zeta - \\frac{W_0'(|\\hat\\zeta|)}{|\\hat\\zeta|}\\hat\\zeta\\right).\n\\end{align}\nIntegrating in space and using the above estimates gives, whenever $E(t)<E'$,\n\\begin{align}\n\t\\frac{\\d}{\\d t} E(t) &\\leq \\frac12\\int_\\R\\left|\\zeta_t\\right|\\left|\\frac{W_0'(|\\zeta|)}{|\\zeta|}\\bar\\zeta - \\frac{W_0'(|\\hat\\zeta|)}{|\\hat\\zeta|}\\bar{\\hat\\zeta}\\right|\\:\\d x\\non &\\quad+\\frac12\\int_\\R\\left|\\bar\\zeta_t\\right|\\left|\\frac{W_0'(|\\zeta|)}{|\\zeta|}\\zeta - \\frac{W_0'(|\\hat\\zeta|)}{|\\hat\\zeta|}\\hat\\zeta\\right|\\:\\d x\\no\n\t\t&\\leq \\|\\zeta_t(t)\\|_2\\left(\\|W_0'(|\\zeta(t)|)\\|_2 +\\|W_0'(|\\hat\\zeta(t)|)\\|_2\\right)\\non\n\t\t&\\leq \\sqrt{E(t)}\\left(\\sqrt{k_{E'}\\|W_0(|\\zeta(t)|)\\|_1}+\\sqrt{k_{E'}\\|W_0(|\\hat\\zeta(t)|)\\|_1}\\right)\\non\n\t\t&\\leq 2\\sqrt{k_{E'}}E'\n\\end{align}\nwhich yields\n\\begin{equation}\n\tE(t) \\leq E+2\\sqrt{k_{E'}}E't.\n\\end{equation}\nThus for $T\\leq \\frac1{2\\sqrt{k_{E'}}}\\left(1-\\frac{E}{E'}\\right)$ we have that the solution map $\\Phi:\\hat\\zeta \\mapsto \\zeta$ of \\eqref{eq:linear iterative} maps smooth functions in $\\D_{T,E'}$ to smooth functions in $\\D_{T,E'}$.\n\nTo prove that $\\Phi$ is a contraction we will need further estimates on the integrals in \\eqref{eq:iterative solutions}. For $\\zeta_1,\\zeta_2 \\in \\C$ with $0<|\\zeta_1|<|\\zeta_2|<1$ we have\n\\begin{align}\n\t\\left|\\frac{W'(|\\zeta_1|)}{|\\zeta_1|}\\zeta_1 -  \\frac{W'(|\\zeta_2|)}{|\\zeta_2|}\\zeta_2\\right| &\\leq \\left|\\frac{W'(|\\zeta_1|)}{|\\zeta_1|}\\left(\\zeta_1 - \\frac{|\\zeta_1|}{|\\zeta_2|}\\zeta_2\\right) + \t\t\\frac{\\zeta_2}{|\\zeta_2|}\\left(W'(|\\zeta_1|) - W'(|\\zeta_2|)\\right)\\right|\\non\n\t&\\leq \\left|\\frac{W'(|\\zeta_1|)}{|\\zeta_1|}\\right||\\zeta_1-\\zeta_2| + \\sup_{\\zeta\\in(\\zeta_1,\\zeta_2)}\\left|W''(\\zeta)\\right|\\left|\\zeta_1-\\zeta_2\\right|\n\\end{align}\nThen for $(\\zeta_1,\\zeta_{1t}),(\\zeta_2,\\zeta_{2t})\\in\\D_{T,E}$ we get\n\\begin{equation}\n\t\\label{eq:rhs diff}\n\t\\left|\\frac{W'(|\\zeta_1(t,x)|)}{|\\zeta_1(t,x)|}\\zeta_1(t,x) -  \\frac{W'(|\\zeta_2(t,x)|)}{|\\zeta_2(t,x)|}\\zeta_2(t,x)\\right| \\leq (L_E'+L_E'')|\\zeta_1(t,x)-\\zeta_2(t,x)|.\n\\end{equation}\nHence\n\\begin{align}\n\\label{eq:diff Linf}\n\t&\\left|\\frac1{2c}\\int_{x-c(t-s)}^{x+c(t-s)}\\frac{W_0'(|\\zeta_1(s,y)|)}{|\\zeta_1(s,y)|}\\zeta_1(s,y) -  \\frac{W_0'(|\\zeta_2(s,y)|)}{|\\zeta_2(s,y)|}\\zeta_2(s,y)\\:\\d y\\right|\\non &\\qquad\\leq (t-s)(L_E'+L_E'')\\|\\zeta_1(s)-\\zeta_2(s)\\|_{L^\\infty(\\R)},\\\\\n\\label{eq:diff der Linf}\n\t&\\left|\\frac{\\partial}{\\partial x}\\frac1{2c}\\int_{x-c(t-s)}^{x+c(t-s)}\\frac{W'(|\\zeta_1(s,y)|)}{|\\zeta_1(s,y)|}\\zeta_1(s,y) -  \\frac{W'(|\\zeta_2(s,y)|)}{|\\zeta_2(s,y)|}\\zeta_2(s,y)\\:\\d y\\right|\\non &\\qquad\\leq 2(L_E'+L_E'')\\|\\zeta_1(s)-\\zeta_2(s)\\|_{L^\\infty(\\R)}.\n\\end{align}\nWe will need $L^2(\\R)$-estimates in $x$ as well. For $(\\zeta_1,\\zeta_{1t}),(\\zeta_2,\\zeta_{2t})\\in\\D_{T,E}$ we get from \\eqref{eq:rhs diff} and \\eqref{eq:conv}\n\\begin{align}\n\\label{eq:L2 estimate}\n\t&\\quad\\left\\|\\int_{x-c(t-s)}^{x+c(t-s)}\\frac{W_0'(|\\zeta_1(s,y)|)}{|\\zeta_1(s,y)|}\\zeta_1(s,y) -  \\frac{W_0'(|\\zeta_2(s,y)|)}{|\\zeta_2(s,y)|}\\zeta_2(s,y)\\:\\d y\\right\\|_2\\non\n\t&\\leq (L_E'+L_E'')2c(t-s)\\|\\zeta_1(s,\\cdot)-\\zeta_2(s,\\cdot)\\|_2.\n\\end{align}\nTo prove that $\\Phi$ is a contraction let $\\hat\\zeta_1,\\hat\\zeta_2 \\in \\D_{T,E'}\\cap C^2([0,T]\\times\\R)$ with coinciding initial data $(\\zeta_0,\\zeta_{t0})$, and denote $\\zeta_1=\\Phi(\\hat\\zeta_1),\\zeta_2 = \\Phi(\\hat\\zeta_2)$. Then by \\eqref{eq:diff Linf}, \\eqref{eq:diff der Linf}, \\eqref{eq:L2 estimate}, and \\eqref{eq:iterative solutions}, we have for $p=2,\\infty$,\n\\begin{align}\n\t\\sup_{t\\in[0,T]}\\|\\zeta_1(t) -\\zeta_2(t)\\|_{L^p(\\R)} &\\leq \\left(L_{E'}'+L_{E'}''\\right)\\int_0^t(t-s)\\|\\hat\\zeta_1(s)-\\hat\\zeta_2(s)\\|_p\\:\\d s,\\\\\n\t\\sup_{t\\in[0,T]}\\|\\zeta_{1,t}(t) -\\zeta_{2,t}(t)\\|_{L^p(\\R)} &\\leq \\left(L_{E'}'+L_{E'}''\\right)\\int_0^t\\|\\hat\\zeta_1(s)-\\hat\\zeta_2(s)\\|_p\\:\\d s,\\\\\n\t\\sup_{t\\in[0,T]}\\|\\zeta_{1,x}(t) -\\zeta_{2,x}(t)\\|_{L^p(\\R)} &\\leq \\frac1c\\left(L_{E'}'+L_{E'}''\\right)\\int_0^t\\|\\hat\\zeta_1(s)-\\hat\\zeta_2(s)\\|_p\\:\\d s.\n\\end{align}\nWe have that\n\\begin{align}\n\t\\frac\\d{\\d t}\\left(W_0(|\\zeta_1|)-W_0(|\\zeta_2|)\\right) &= W_0'(|\\zeta_1|)\\frac{\\bar\\zeta_1\\zeta_{1,t} + \\zeta_1\\bar\\zeta_{1,t} }{2|\\zeta_1|}\\non &\\quad -  W_0'(|\\zeta_2|)\\frac{\\bar\\zeta_2\\zeta_{2,t} + \\zeta_2\\bar\\zeta_{2,t}}{2|\\zeta_2|}\\non\n\t&= \\left(W_0'(|\\zeta_1|)- W_0'(|\\zeta_2|)\\right)\\frac{\\bar\\zeta_1\\zeta_{1,t} + \\zeta_1\\bar\\zeta_{1,t}}{2|\\zeta_1|}\\non &\\quad  + W_0'(|\\zeta_2|)\\left(\\frac{\\bar\\zeta_1\\zeta_{1,t} + \\zeta_1\\bar\\zeta_{1,t}}{2|\\zeta_1|} - \\frac{\\bar\\zeta_2\\zeta_{2,t} + \\zeta_2\\bar\\zeta_{2,t}}{2|\\zeta_2|}\\right),\n\\end{align}\nFurther note that\n\\begin{align}\n\t \\left(\\frac{\\bar\\zeta_1\\zeta_{1,t} + \\zeta_1\\bar\\zeta_{1,t}}{2|\\zeta_1|} - \\frac{\\bar\\zeta_2\\zeta_{2,t} + \\zeta_2\\bar\\zeta_{2,t}}{2|\\zeta_2|}\\right) &= \\frac{1}{|\\zeta_2|}\\bigg[\\frac12\\left(|\\zeta_2|-|\\zeta_1|\\right)(\\frac{\\bar\\zeta_1}{|\\zeta_1|}\\zeta_{1,t}+\\frac{\\bar\\zeta_1}{|\\zeta_1|}\\bar\\zeta_{1,t}) \\non &\\quad + \\frac12\\big((\\bar\\zeta_1-\\bar\\zeta_2)\\zeta_{1,t}+\\bar\\zeta_2(\\zeta_{1,t}-\\zeta_{2,t})\\non &\\quad+(\\zeta_1-\\zeta_2)\\bar\\zeta_{1,t}+\\zeta_2(\\bar\\zeta_{1,t}-\\bar\\zeta_{2,t})\\big)\\bigg],\n\\end{align}\nand hence\n\\begin{align}\n\\label{eq:diff W_0}\n\t\\frac\\d{\\d t}\\left\\|W_0(|\\zeta_1|)-W_0(|\\zeta_2|)\\right\\|_1 &\\leq \\int_\\R\\left|W_0'(|\\zeta_1|)- W_0'(|\\zeta_2|)\\right|\\left|\\frac{\\bar\\zeta_1\\zeta_{1,t} + \\zeta_1\\bar\\zeta_{1,t}}{2|\\zeta_1|}\\right|\\:\\d x\\non &\\quad+2\\int_\\R\\left|\\frac{W_0'(|\\zeta_2|)}{|\\zeta_2|}\\right|\\left|\\zeta_1-\\zeta_2\\right|\\left|\\zeta_{1,t}\\right|\\:\\d x \\non &\\quad+  \\int_\\R\\left|\\frac{W_0'(|\\zeta_2|)}{|\\zeta_2|}\\right|\\left|\\zeta_{1,t}-\\zeta_{2,t}\\right||\\zeta_2|\\:\\d x\\non\n\t&\\leq \\left(L_{E'}''+2L_{E'}'\\right)\\sqrt{E'}\\|\\zeta_1 - \\zeta_2\\|_2 + \\sqrt{k_{E'}E'}\\|\\zeta_{1,t}-\\zeta_{2,t}\\|_2.\n\\end{align}\n\nHence for $T$ small enough $\\Phi$ is a contraction on the subspace of $\\D_{T,E'}$ consisting of $\\zeta\\in C^2([0,T]\\times\\R)$. Thus there is a unique fixed point in $\\D_{T,E'}$ which is a strong solution in the sense of \\eqref{eq:strong form}. The limit may not be $C^2$, however. We want to prove that energy is actually conserved. From the construction of the metric on $\\D_{T,E'}$ we see that the energy converges. A more careful investigation of \\eqref{eq:iterative energy} reveals that\n\\begin{align}\n\\label{eq:energy cons}\n\tE(t) - E(0)&= \\frac 12\\int_0^t \\int_\\R \\frac{W_0'(|\\hat\\zeta(s,y)|)}{|\\hat\\zeta(s,y)|}\\bar{\\hat\\zeta}(s,y)(\\hat\\zeta_t(s,y)-\\zeta_t(s,y)) \\non &\\qquad+\\frac{W_0'(|\\hat\\zeta(s,y)|)}{|\\hat\\zeta(s,y)|}\\hat\\zeta(s,y)(\\bar{\\hat\\zeta}_t(s,y)-\\bar\\zeta_t(s,y))\\:\\d y \\d s.\n\\end{align}\nThus the limit conserves energy since the right hand side of \\eqref{eq:energy cons} tends to zero as $\\zeta$ tends to the fixed point. Then we know that the solution $(\\zeta,\\zeta_t)$ is actually in $\\D_{T,E}\\subsetneq\\D_{T,E'}$, and hence, $(\\zeta(T),\\zeta_t(T))\\in X_E$.\n\nWe will now show that we can relax the smoothness conditions on the initial values. From the strong formulation of the equation \\eqref{eq:strong form} and estimate \\eqref{eq:rhs diff} we get, from the Gr\\\" onwall inequality, the estimates\n\\begin{align}\n\t\\|\\zeta_1(t)-\\zeta_2(t)\\|_p &\\leq \\left(\\|\\zeta_{1,0}-\\zeta_{2,0}\\|_p+t\\|\\zeta_{1,t,0}-\\zeta_{2,t,0}\\|_p\\right)e^{\\frac12(L_E'+L_E'')t^2},\\\\\n\t\\|\\zeta_{1,t}(t)-\\zeta_{2,t}(t)\\|_p &\\leq \\|\\zeta_{1,t,0}-\\zeta_{2,t,0}\\|_p + c\\|\\zeta_{1,x,0}-\\zeta_{2,x,0}\\|_p\\non &\\quad + (L_E'+L_E'')t\\left(\\|\\zeta_{1,0}-\\zeta_{2,0}\\|_p+t\\|\\zeta_{1,t,0}-\\zeta_{2,t,0}\\|_p\\right)e^{\\frac12(L_E'+L_E'')t^2},\\\\\n\t\\|\\zeta_{1,x}(t)-\\zeta_{2,x}(t)\\|_p &\\leq \\frac1c\\|\\zeta_{1,t,0}-\\zeta_{2,t,0}\\|_p + \\|\\zeta_{1,x,0}-\\zeta_{2,x,0}\\|_p\\non &\\quad  +\\frac{L_E'+L_E''}ct\\left(\\|\\zeta_{1,0}-\\zeta_{2,0}\\|_p+t\\|\\zeta_{1,t,0}-\\zeta_{2,t,0}\\|_p\\right)e^{\\frac12(L_E'+L_E'')t^2},\n\\end{align}\nfor $p=2,\\infty$. Thus the strong solution depends continuously on the initial data in the metric induced from $\\|(\\zeta_0,\\zeta_{t0})\\| = \\|\\zeta_0-\\zeta^*\\|_{H^1(\\R)} + \\|\\zeta_0\\|_{W^{1,\\infty}(\\R)}\\| + \\|\\zeta_{t0}\\|_{L^2(\\R)} + \\|\\zeta_{t0}\\|_{L^\\infty(\\R)} + \\|W_0(|\\zeta_0|)\\|_{L^1(\\R)}$. Hence we can extend the solution operator to all initial data in $X_E$. The argument can now be repeated to yield a solution in $[0,nT]$ for any $n\\in\\N$, and hence there exists a global strong solution that conserve energy.\n\\end{proof}\nIn the case where the initial data is smooth, $\\zeta_0\\in C^2(\\R)$ and $\\zeta_{t0}\\in C^1(\\R)$, the strong solution can be upgraded to a classical solution $\\zeta\\in C^2([0,\\infty)\\times\\R)$.\n\\begin{corollary}\nFor initial data $(\\zeta_0,\\zeta_{t0})\\in X_E\\cap C^2(\\R)\\times C^1(\\R)$ the strong solution $\\zeta$ in Theorem \\ref{thm:global} belongs to $C^2([0,\\infty)\\times\\R)$ and satisfies the equation \\eqref{eq:2NVWlinear} pointwise.\n\\end{corollary}\n\\begin{proof}\nWe now assume that $\\zeta$ is a strong solution given implicitly by \\eqref{eq:strong form}, and that $\\zeta\\in C([0,\\infty),W^{1,\\infty}(\\R))\\cap C^1([0,\\infty),L^\\infty(\\R))$. Note that we have\n\\begin{equation}\n\tW_0(s) = W_0(0) + \\frac12W_0^{(2)}(0)s^2 + \\frac16W_0^{(3)}(0)s^3 + \\int_0^s\\frac16W_0^{(4)}(u)(s-u)^3\\:\\d u,\\\\\n\\end{equation}\nand thus\n\\begin{equation}\n\t\\frac{\\d}{\\d s}\\frac{W_0'(s)}{s} = \\frac12W_0^{(3)}(0) - \\frac1{s^2}\\int_0^s\\frac12W_0^{(4)}(u)(s-u)^2\\:\\d u + \\frac1s\\int_0^sW_0^{(4)}(u)(s-u)\\:\\d u.\n\\end{equation}\nOne can then readily verify that\n\\begin{align}\n\t\\left|\\frac{\\d}{\\d s}\\frac{W_0'(s)}{s}(s)\\right| &\\leq \\frac12\\left|W_0^{(3)}(0)\\right| + \\frac32\\left\\|W_0^{(4)}\\right\\|_{L^\\infty([0,s])}s,\\\\\n\t\\left|\\frac{\\d}{\\d s}\\frac{W_0'(s)}{s}(s_1) - \\frac{\\d}{\\d s}\\frac{W_0'(s)}{s}(s_2)\\right| &\\leq \\frac52 \\left\\|W_0^{(4)}\\right\\|_{L^\\infty([0,\\max\\{s_1,s_2\\}])}\\left|s_1-s_2\\right|.\n\\end{align}\nThus $\\zeta\\mapsto \\frac{W_0'(|\\zeta|)}{|\\zeta|}\\zeta$ is $C^1$ in the open unit disk in two dimensions. Direct computation then yields that for\n\\begin{equation}\n\tI(t,x) = \\int_0^t\\int_{x-c(t-\\tau)}^{x+c(t-\\tau)}\\frac{W_0'(|\\zeta(\\tau,y)|)}{|\\zeta(\\tau,y)|}\\zeta(\\tau,y)\\:\\d y\\d\\tau\n\\end{equation}\nthe first derivatives are\n\\begin{align}\n\t\\frac{\\partial}{\\partial t} I(t,x) &= c\\int_0^t \\frac{W_0'(|\\zeta(\\tau,x+c(t-\\tau))|)}{|\\zeta(\\tau,x+c(t-\\tau))|}\\zeta(\\tau,x+c(t-\\tau))\\non &\\qquad + \\frac{W_0'(|\\zeta(\\tau,x-c(t-\\tau))|)}{|\\zeta(\\tau,x-c(t-\\tau))|}\\zeta(\\tau,x-c(t-\\tau))\\:\\d\\tau,\\\\\n\\frac{\\partial}{\\partial x} I(t,x) &= \\int_0^t \t\n\t\\frac{W_0'(|\\zeta(\\tau,x+c(t-\\tau))|)}{|\\zeta(\\tau,x+c(t-\\tau))|}\\zeta(\\tau,x+c(t-\\tau))\\non &\\qquad - \\frac{W_0'(|\\zeta(\\tau,x-c(t-\\tau))|)}{|\\zeta(\\tau,x-c(t-\\tau))|}\\zeta(\\tau,x-c(t-\\tau))\\:\\d\\tau,\n\\end{align}\nwhich are continuous since $\\zeta$ is continuous by assumption. Hence $\\zeta\\in C^1([0,\\infty)\\times\\R)$. Similarly the second derivatives are\n\\begin{align}\n\t\\frac{\\partial^2}{\\partial t^2} I(t,x) &= 2c\\frac{W_0'(|\\zeta(t,x)|)}{|\\zeta(t,x)|}\\zeta(t,x)\\non &\\quad+ c^2\\int_0^t\\frac{\\partial}{\\partial x}\\frac{W_0'(|\\zeta(\\tau,x+c(t-\\tau))|)}{|\\zeta(\\tau,x+c(t-\\tau))|}\\zeta(\\tau,x+c(t-\\tau))\\non &\\qquad- \\frac{\\partial}{\\partial x}\\frac{W_0'(|\\zeta(\\tau,x-c(t-\\tau))|)}{|\\zeta(\\tau,x-c(t-\\tau))|}\\zeta(\\tau,x-c(t-\\tau))\\:\\d\\tau,\\\\\n\t\\frac{\\partial^2}{\\partial t\\partial x}I(t,x) &= c\\int_0^t\\frac{\\partial}{\\partial x}\\frac{W_0'(|\\zeta(\\tau,x+c(t-\\tau))|)}{|\\zeta(\\tau,x+c(t-\\tau))|}\\zeta(\\tau,x+c(t-\\tau))\\non &\\qquad + \\frac{\\partial}{\\partial x}\\frac{W_0'(|\\zeta(\\tau,x-c(t-\\tau))|)}{|\\zeta(\\tau,x-c(t-\\tau))|}\\zeta(\\tau,x-c(t-\\tau))\\:\\d\\tau,\\\\\n\t\\frac{\\partial^2}{\\partial x^2}I(t,x) &=\\int_0^t\\frac{\\partial}{\\partial x}\\frac{W_0'(|\\zeta(\\tau,x+c(t-\\tau))|)}{|\\zeta(\\tau,x+c(t-\\tau))|}\\zeta(\\tau,x+c(t-\\tau))\\non &\\qquad - \\frac{\\partial}{\\partial x}\\frac{W_0'(|\\zeta(\\tau,x-c(t-\\tau))|)}{|\\zeta(\\tau,x-c(t-\\tau))|}\\zeta(\\tau,x-c(t-\\tau))\\:\\d\\tau,\n\\end{align}\nwhich are continuous. Thus $\\zeta$ given by \n\\begin{equation}\n\t\\zeta(t,x) = \\frac12\\left(\\zeta(t,x+ct)+\\zeta(t,x-ct)\\right) + \\frac1{2c}\\int_{x-ct}^{x+ct}\\zeta_{t0}(y)\\:\\d y -\\frac1{2c}I(t,x),\n\\end{equation}\nis a sum of $C^2([0,\\infty)\\times\\R)$ functions and is thus $C^2([0,\\infty)\\times\\R)$. We get\n\\begin{align}\n\t\\frac{\\partial^2}{\\partial t^2}\\zeta(t,x)-c^2\\frac{\\partial^2}{\\partial t^2}\\zeta(t,x) &= -\\frac{\\partial^2}{\\partial t^2}\\frac1{2c}I(t,x) + c^2\\frac{\\partial^2}{\\partial x^2}\\frac1{2c}I(t,x) \\non\n\t&= - \\frac{W_0'(|\\zeta(t,x)|)}{|\\zeta(t,x)|}\\zeta(t,x)\n\\end{align}\npointwise.\n\\end{proof}\nWe will only give a local existence theorem for \\eqref{eq:2NVW}. The main difficulty for \\eqref{eq:2NVW} compared to \\eqref{eq:2NVWlinear} is that \\eqref{eq:2NVW} is quasilinear, that is the terms $c(\\psi)^2\\psi_{xx}$ and $c(\\psi)^2s_{xx}$ are nonlinear. The method used in the proof of Theorem \\ref{thm:global} relied heavily on \\eqref{eq:2NVWlinear} being semilinear.\n\\begin{theorem}\n\\label{thm:local}\nThere exists a unique short time solution of \\eqref{eq:2NVW} for smooth initial data $(\\psi_0,\\psi_{t0},s_0,s_{t0})$, where $s_0$ is bounded away from $0$ and the energy is finite.\n\\end{theorem}\n\\begin{proof}\nLocal existence, uniqueness, and continuous dependence on initial data follows from the standard approach taken to semigroups of nonlinear evolution equations. Here we will diverge from the standard semigroup approach by considering linear operators on Banach spaces, and instead solve the linear equations by characteristics.\n\nIn particular, inspired by \\cite{K2}, the system \\eqref{eq:2NVW} can be rewritten as a quasilinear symmetric hyperbolic system and the results in \\cite{K} applied. Indeed, introduce the variables\n\\begin{subequations}\n\\begin{align}\n\t\\phi &= \\psi_t,\\\\\n\tv &= s_t,\\\\\n\t\\omega &= c(\\psi)\\psi_x,\\\\\n\tr &= c(\\psi)s_x,\n\\end{align}\n\\end{subequations}\nthe system \\eqref{eq:2NVW} can be written as a quasilinear symmetric hyperbolic system\n\\begin{subequations}\n\\label{eq:symhyp}\n\\begin{align}\n\t\\psi_t &= \\phi,\\\\\n\ts_t &= v,\\\\\n\t\\phi_t-c(\\psi)\\omega_x &= -\\frac2s\\left(\\phi v-\\omega r\\right)+\\frac{c'(\\psi)}{c(\\psi)}r^2,\\\\\n\tv_t-c(\\psi)r_x &=s\\left(\\phi^2-\\omega^2\\right)+\\frac{c'(\\psi)}{c(\\psi)}\\omega r-W_0'(s),\\\\\n\t\\omega_t-c(\\psi)\\phi_x &= \\frac{c'(\\psi)}{c(\\psi)}\\phi\\omega,\\\\\n\tr_t-c(\\psi)v_x &= \\frac{c'(\\psi)}{c(\\psi)}\\phi r.\n\\end{align}\n\\end{subequations}\nWe introduce the Banach space $X=W^{1,\\infty}(\\R,\\R^6)$, and let \n\\begin{align}\n\tX_{E,L} = \\bigg\\{&U=(\\psi,s,\\phi,v,\\omega,r)\\in X \\mid \\|U\\|_{W^{1,\\infty}}\\leq L, \\left\\|\\frac1s\\right\\|_{L^\\infty(\\R)}\\leq L,\\non &\\int_{\\R}\\frac12s^2\\left(\\phi^2+\\omega^2\\right)+\\frac12\\left(v^2+r^2\\right)+W_0(s)\\:\\d x \\leq E \\bigg\\}.\n\\end{align}\nNote that for elements of $X_{E_L}$ the variable $s$ is of definite sign since $s$ is continuous and  $\\frac1s$ is bounded. We define our solution space $\\D_{T,E,L}$ as follows\n\\begin{equation}\n\t\\D_{T,E,L} = C([0,T],X_{E,L})\\cap C^1([0,T],L^\\infty(\\R,\\R^6)).\n\\end{equation}\nWe equip $\\D_{T,E,L}$ with the metric\n\\begin{equation}\n\td_{\\D_{T,E,L}}(U_1,U_2) = \\sup_{0\\leq t\\leq T}\\|U_1(t)-U_2(t)\\|_{W^{1,\\infty}(\\R,\\R^6)}+ \\sup_{0\\leq t\\leq T}\\|U_{1,t}(t)-U_{2,t}(t)\\|_{L^\\infty(\\R,\\R^6)},\n\\end{equation}\nwhich renders $\\D_{T,E,L}$ a complete metric space. Formally, we can formulate \\eqref{eq:symhyp} as\n\\begin{equation}\n\tU_t-c(\\psi)AU_x = F(U),\n\\end{equation}\nwhere $A$ is a constant symmetric hyperbolic matrix with eigenvalues $-1,0,1$. We want to show existence of short time solutions from a fix point argument. To be able to make a contraction we will further restrict our space for approximate solutions to\n\\begin{equation}\n\t\\D_{T,E,L}^{\\mathrm{lin}} = \\left\\{U\\in\\D_{T,E,L}\\mid U(t)\\in W^{2,\\infty}(\\R,\\R^6), \\|U_{xx}(t)\\|_{L^\\infty(\\R,\\R^6)}\\leq L\\right\\}.\n\\end{equation}\n\nLet $0<E<E', 0<L<L'$ and $U_0\\in X_{E,L}$ with $\\|U_{0xx}\\|_\\infty\\leq L$ be the initial data, then for any $\\hat U\\in\\D_{T,E',L'}^{\\mathrm{lin}}$ with $\\hat U(0) = U_0$, the linear system of transport equations\n\\begin{equation}\nU_t -c(\\hat\\psi)AU_x = F(\\hat U)\n\\end{equation}\ncan be solved by characteristics. Define the backward characteristics $x_\\pm$ at time $\\tau$ from point $(t,x)$ by\n\\begin{equation}\n\t\\frac{\\d}{\\d\\tau}x_\\pm(\\tau;t,x) = \\mp c\\bigg(\\hat\\psi(\\tau,x_\\pm(\\tau;t,x))\\bigg), \n\\end{equation}\nand note that $x_\\pm(\\tau;t,x) = x_\\pm(\\tau;s,x_\\pm(s;t,x))$ and $x_\\pm(t;t,x))=x$. Then, by taking the derivative with respect to $s$, we get the following\n\\begin{equation}\n\t\\frac{\\partial}{\\partial t}x_\\pm(\\tau;t,x) \\mp c(\\hat\\psi(\\tau,x_\\pm(\\tau;t,x)))\\frac{\\partial}{\\partial x}x_\\pm(\\tau;t,x) = 0.\n\\end{equation}\nThus\n\\begin{align}\n\t|x_\\pm(\\tau_1;t,x)-x_\\pm(\\tau_2;t,x)| &\\leq \\|c\\|_\\infty|\\tau_2-\\tau_1|,\\\\\n\t|x_\\pm(\\tau;t,x_1)-x_\\pm(\\tau;t,x_2)| &\\leq \\mathrm{exp}\\left\\{\\left\\|\\frac1c\\right\\|_\\infty L'|t-\\tau|\\right\\}|x_1-x_2|,\\\\\n\t|x_+(\\tau;t,x)-x_-(\\tau;t,x)| &\\leq 2\\|c\\|_\\infty|t-\\tau|,\\\\\n\t\\left|\\frac{\\partial}{\\partial t}x_\\pm(\\tau;t,x)\\right| &\\leq \\|c\\|_\\infty\\:\\mathrm{exp}\\left\\{\\left\\|\\frac1c\\right\\|_\\infty L'|t-\\tau|\\right\\}.\n\\end{align}\nWe need to show that the Duhamel operator $DF(\\hat U)(t) = \\int_0^t T_\\tau(t-\\tau)F(\\hat U(\\tau))\\:\\d\\tau$, where $T_{\\tau}$ is the solution operator of $U_t(t)-c(\\psi(t+\\tau))AU_x(t) = 0$, satisfies $DF(\\hat U)\\in C([0,\\infty),W^{2,\\infty}(\\R,\\R^6))$, and $\\|DF(\\hat U_1)(t)-DF(\\hat U_2)(t)\\|_{W^{1,\\infty}(\\R)}\\leq C(t,E',L')\\|\\hat U_1-\\hat U_2\\|_{W^{1,\\infty}(\\R)}$ for $\\hat U_1,\\hat U_2\\in \\D_{T,E',L'}^{\\mathrm{lin}}$. Since\n\\begin{equation}\n\t\\frac{\\partial}{\\partial x}\\frac{c'(\\psi)}{c(\\psi)} = \\frac{c''(\\psi)c(\\psi)-c'(\\psi)^2}{c(\\psi)^2}\\psi_x,\n\\end{equation}\nand  $F$ is a smooth function, we have that for $\\hat U\\in \\D_{T,E',L'}^{\\mathrm{lin}}$ there is\n\\begin{equation}\n\t\\|F(\\hat U(t))\\|_{W^{2,\\infty}(\\R)} \\leq C_{E',L'}.\n\\end{equation}\nHence the Duhamel operator satisfies $\\|DF(\\hat U)(t)\\|_{W^{2,\\infty}(\\R)}\\leq C_{E',L'}t$. Note that\n\\begin{equation}\n\\frac\\partial{\\partial\\tau}\\left(x_\\pm^1(\\tau;t,x_1)-x_\\pm^2(\\tau;t,x_2)\\right)=\\mp\\left(c(\\hat\\psi_1(\\tau,x_\\pm^1(\\tau;t,x_1)))-c(\\hat\\psi_2(\\tau,x_\\pm^2(\\tau;t,x_2)))\\right),\n\\end{equation}\nand thus\n\\begin{align}\n\t|x_\\pm^1(\\tau;t,x_1)-x_\\pm^2(\\tau;t,x_2)| &\\leq |x_1-x_2|e^{\\|c'\\|_\\infty L'(t-\\tau)}\\non &\\quad+\\frac1{L'} \\left(e^{\\|c'\\|_\\infty L'(t-\\tau)}-1\\right)\\|\\hat\\psi_1-\\hat\\psi_2\\|_{L^\\infty([0,t],L^\\infty(\\R))},\\\\\n\t\\left|\\frac{\\partial}{\\partial x}x_\\pm^1(\\tau;t,x)-\\frac{\\partial}{\\partial x}x_\\pm^2(\\tau;t,x)\\right|  &\\leq C_{L'}\\mathrm{exp}\\left\\{k_{L'}(t-\\tau)\\right\\}\\sup_{0\\leq\\tau\\leq t}\\|\\hat\\psi_1-\\hat\\psi_2\\|_{W^{1,\\infty}(\\R)},\n\\end{align}\nwhere $C_{L'}$ and $k_{L'}$ depends on $c$ and $L'$ only. Thus, since $F$ is a smooth function and $\\|\\hat U_1\\|_{W^{2,\\infty}(\\R)}, \\|\\hat U_2\\|_{W^{2,\\infty}(\\R)} \\leq L'$, \n\\begin{align}\n\t&\\left|\\int_0^tF_j(\\hat U_1)(\\tau,x^1_\\pm(\\tau;t,x))-F_j(\\hat U_2)(\\tau,x^2_\\pm(\\tau;t,x))\\:\\d\\tau\\right|\\non\n\t&\\leq \\left|\\int_0^tF_j(\\hat U_1)(\\tau,x^1_\\pm(\\tau;t,x))-F_j(\\hat U_2)(\\tau,x^1_\\pm(\\tau;t,x))\\:\\d\\tau\\right| \\non &\\quad+ \\left|\\int_0^tF_j(\\hat U_2)(\\tau,x^1_\\pm(\\tau;t,x))-F_j(\\hat U_2)(\\tau,x^2_\\pm(\\tau;t,x))\\:\\d\\tau\\right|\\non\n\t&\\leq \\left|\\int_0^tF_j(\\hat U_1)(\\tau,x^1_\\pm(\\tau;t,x))-F_j(\\hat U_2)(\\tau,x^1_\\pm(\\tau;t,x))\\:\\d\\tau\\right| \\non &\\quad+ t \\bar C(L',E')\\|\\hat\\psi_1-\\hat\\psi_2\\|_{L^\\infty([0,t],L^\\infty(\\R)}\\non\n\t&\\leq t\\tilde C(L',E')\\|\\hat U_1 -\\hat U_2\\|_{L^\\infty([0,t],L^\\infty(\\R,\\R^6)}\n\\end{align}\nTo compute $\\frac\\partial{\\partial x}\\left(DF(\\hat U_1)(t)-DF(\\hat U_2)(t)\\right)$ we estimate\n\\begin{align}\n\t&\\quad\\left|\\frac\\partial{\\partial x}\\int_0^t F_j(\\hat U_1)(\\tau,x_\\pm^1(\\tau;t,x)) - F_j(\\hat U_2)(\\tau,x_\\pm^2(\\tau;t,x)) \\:\\d\\tau\\right|\\non &\\leq \\left|\\int_0^t\\left(\\nabla_UF_j(\\hat U_1)\\frac{\\partial \\hat U_1}{\\partial x}-\\nabla_UF_j(\\hat U_2)\\frac{\\partial \\hat U_2}{\\partial x}\\right)(\\tau,x_\\pm^1(\\tau;t,x)) \\frac{\\partial}{\\partial x}x_\\pm^1(\\tau;t,x)\\:\\d\\tau\\right| \\non\n\t&\\quad + \\left|\\int_0^t\\nabla_UF_j(\\hat U_2)\\frac{\\partial \\hat U_2}{\\partial x}(\\tau,x_\\pm^1(\\tau;t,x))\\frac{\\partial}{\\partial x}\\left(x_\\pm^1(\\tau;t,x)-x_\\pm^2(\\tau;t,x)\\right)\\:\\d\\tau \\right|\\non\n\t&\\quad +\\bigg|\\int_0^t\\left(\\nabla_UF_j(\\hat U_2)\\frac{\\partial \\hat U_2}{\\partial x}(\\tau,x_\\pm^1(\\tau;t,x))-\\nabla_UF_j(\\hat U_2)\\frac{\\partial \\hat U_2}{\\partial x}(\\tau,x_\\pm^2(\\tau;t,x))\\right)\\non &\\qquad\\qquad\\times\\frac\\partial{\\partial x}x_\\pm^2(\\tau;t,x)\\:\\d\\tau\\bigg|\\non\n&\\leq C_{E',L'}\\left(\\mathrm{exp}\\left\\{\\left\\|\\frac1c\\right\\|_\\infty L't\\right\\}-1\\right)\\sup_{0\\leq\\tau\\leq t}\\|\\hat U_1(\\tau)-\\hat U_2(\\tau)\\|_{W^{1,\\infty}(\\R)}\\non\n\t&\\quad + C_{E',L'}\\:(\\mathrm{exp}\\left\\{K_{L'}t\\right\\}-1)\\sup_{0\\leq\\tau\\leq t}\\|\\hat \\psi_1(\\tau)-\\hat \\psi_2(\\tau)\\|_{W^{1,\\infty}(\\R)}\\non\n\t&\\quad + C_{E',L'}\\:(\\mathrm{exp}\\left\\{K_{L'}t\\right\\}-1)\\sup_{0\\leq\\tau\\leq t}\\|\\hat \\psi_1(\\tau)-\\hat \\psi_2(\\tau)\\|_{L^\\infty(\\R)}.\n\\end{align}\nThat is, we have that\n\\begin{equation}\n\t\\|U_1(t)-U_2(t)\\|_{W^{1,\\infty}(\\R)} \\leq C_{L',E'}(\\mathrm{exp}\\{k_{L'}t\\}-1)\\|\\hat U_1-\\hat U_2\\|_{W^{1,\\infty}(\\R)},\n\\end{equation}\nand hence for $T$ small enough we have that \n\\begin{equation}\n\t\\sup_{0\\leq t\\leq T}\\|U_1(t)-U_2(t)\\|_{W^{1,\\infty}(\\R)} < \\sup_{0\\leq t\\leq T}\\|\\hat U_1(t)-\\hat U_2(t)\\|_{W^{1,\\infty}(\\R)}.\n\\end{equation}\nSince $U$ is continuously differentiable, we can do similar estimates for $\\sup_{0\\leq t\\leq T}\\|U_{1,t}(t)-U_{2,t}(t)\\|_{L^\\infty(\\R)}$ directly from the linear system. We need  that the energy of the solution is bounded by $E'$. Recall that $E(t)=\\int_\\R\\mc E(t)\\:\\d x = \\int_\\R s(t)^2(\\phi(t)^2+\\omega(t)^2)+v(t)^2 + r(t)^2 + W_0(s(t)) \\:\\d x$, and thus\n\\begin{align}\n\t\\frac\\partial{\\partial t}\\mc E - \\frac\\partial{\\partial x}c(\\hat\\psi)\\left(s^2\\phi\\omega+vr\\right) &= s\\phi^2\\hat v+v\\hat s\\hat\\phi^2-\\frac2{\\hat s}s^2\\phi\\hat\\phi\\hat v\\non\n\t&\\quad + s\\omega^2\\hat v - v\\hat s\\hat\\omega^2+ W_0'(s)\\hat v-W_0'(\\hat s)v\\non\n\t&\\quad +\\frac{c'(\\hat\\psi)}{c(\\hat\\psi)}\\left(s^2\\omega\\hat\\omega(\\hat\\phi-\\phi) +\\hat r(\\hat\\phi r-\\phi\\hat r) + v(\\hat\\omega\\hat r-\\omega r)\\right)\\non\n\t&\\quad +\\frac{c'(\\hat\\psi)}{c(\\hat\\psi)}2s\\phi\\left(\\frac s{\\hat s}\\hat\\omega\\hat r-\\omega c(\\hat\\psi)s_x\\right).\n\\end{align}\nIntegration yields\n\\begin{align}\n\t\\frac\\d{\\d t}E(t) &\\leq 2{L'}^2E(t)+ 2{L'}^2E' +2{L'}^2\\sqrt{E'}\\sqrt{E(t)} +\\|W_0'(s)\\hat v\\|_1 + \\|W_0'(\\hat s)v\\|_1\\non\n\t&\\quad + K\\left({L'}^3\\sqrt{E(t)}\\sqrt{E'} +  4{L'}\\sqrt{E(t)}\\sqrt{E'}\\right)\\non\n\t&\\quad + K\\left(2{L'}^3\\sqrt{E(t)}\\sqrt{E'}+2E(t)\\|c\\|_\\infty L'\\right)\n\\end{align}\nBy Definition \\ref{def:W_0} we have that\n\\begin{align}\n\t\\|W_0'(s)\\hat v\\|_1 &= \\left\\|W_0'(s)\\right\\|_2\\left\\|\\hat v\\right\\|_2 \\non\n\t&\\leq \\sqrt{k_{E(t)}\\|W_0(s)\\|_1}\\sqrt{E'}\\non\n\t&\\leq \\sqrt{k_{E'}E(t)}\\sqrt{E'}, \n\\end{align}\nand hence for $t$ small enough, depending on $E'$, $L'$, $W_0$, and $c$ only, $E(t)\\leq E'$.\n\nHence the sequence $U_n$, where $U_{n+1}$ is the solution to the linear equation with $U_n$ inserted, is a Cauchy sequence in $\\D_{T,E',L'}$, and hence there is a solution of the equations. Moreover, the limit will satisfy $\\mc E_t-(c(\\psi)^2\\mc F)_x = 0$, and thus conserves energy.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\def\\theequation{1.\\arabic{equation}}\\makeatother\n\\setcounter{equation}{0}\n\\def\\thetheorem{1.\\arabic{thm}}\\makeatother\n\\setcounter{thm}{0}\n\n\nIn this work, we analyze qualitative properties of non-negative solutions for the chemotaxis system in dimension $n \\ge 3$ with linear diffusion given by\n\\begin{align}\\label{star00}\n\\left\\{\n  \\begin{array}{ll}\n u_t=\\Delta u-\\nabla \\cdot (u^{\\sigma} \\nabla v)+u^\\alpha \\left( 1-\\int_{\\R^n} u^\\beta dx \\right),\\quad   &x \\in \\R^n, t>0,\n\\\\\n u(x,0)=u_0(x)\\geq 0,\\quad & x \\in \\R^n.\n  \\end{array}\n\\right.\n\\end{align}\nHere $v(x,t)$ expresses the chemical substance concentration and it is given by the fundamental solution\n\\begin{align}\nv(x,t)=K \\ast u(x,t) =c_n \\int_{\\R^n} \\frac{u(y,t)}{|x-y|^{n-2}}dy\n\\end{align}\nwhere\n\\begin{align}\nc_n=\\frac{1}{n(n-2)b_n},~~b_n=\\frac{\\pi^{n\/2}}{\\Gamma(n\/2+1)},\n\\end{align}\n$b_n$ is the volume of $n$-dimensional unit ball. This system without the reaction has been proposed as a model for chemotaxis driven cell movement \\cite{KS70}. Here $\\sigma \\ge 1$ in chemotaxis is to model the nonlinear aggregation, $u^\\alpha \\left( 1-\\int_{\\R^n} u^\\beta dx \\right)$ with $\\alpha>1, \\beta>1$ is the reaction term representing nonlinear growth under nonlocal resource consumption of the bacteria \\cite{F37,KPP37}.\n\nInitial data will be assumed\n\\begin{align}\\label{initial}\nu_0 \\ge 0,~~ u_0 \\in W^{2,n-1}(\\R^n) \\cap L^1(\\R^n).\n\\end{align}\nUnder the initial assumptions, we consider the case $\\sigma \\ge 1$ and prove that in the following cases,\n\\begin{itemize}\n  \\item[(1)] $\\sigma+1 \\le \\alpha$ and $\\alpha<1+2\\beta\/n$,\n  \\item[(2)] $\\sigma+1>\\alpha$ and $(\\sigma+1)(n+2)<2 \\beta+2 \\alpha+n$,\n\\end{itemize}\nthe solution of \\er{star00} is unique and global without any restriction on the initial data.\n\nIn the follows, we define the exponent $p$ arising from Sobolev inequality \\cite{Lieb202} and the notation $Q_T$\n\\begin{align}\np&:=\\frac{2n}{n-2},~~\\label{p}\\\\\nQ_T&:=\\R^n \\times (0,T) \\mbox{~~for~all~} T>0. \\nonumber\n\\end{align}\nThroughout this paper, we deal with a strong solution of \\er{star00} which is defined:\n\\begin{definition}\\label{def1}\nLet $\\sigma \\ge 1, \\alpha>1, \\beta>1$, $u_0$ satisfies \\er{initial}. $u(x,t)$ is called a strong solution on [0,T) if\n\\begin{itemize}\n  \\item[(1)] $u \\in W^{2,n-1}(Q_T),~u_t\\in L^{n-1}(Q_T)$,\n  \\item[(2)] $u \\in L^\\infty(0,T;L^1\\cap L^\\infty(\\R^n))$.\n \n\\end{itemize}\n\\end{definition}\nOur main result concerning the solution can be summarized as follows:\n\\begin{thm}\\label{thm1}\nLet $\\alpha>1, \\beta>1, \\sigma \\ge 1$. If either\n$$\\sigma+1 \\le \\alpha <1+\\frac{2 \\beta}{n}$$ or $$\\alpha<\\sigma+1<\\frac{2(\\beta+\\alpha)}{n+2}+\\frac{n}{n+2},$$ then for any initial data satisfying \\er{initial}, problem \\er{star00} possesses a unique global strong solution defined by Definition \\ref{def1} which is uniformly bounded, i.e., for any $t>0,$ then\n\\begin{align}\n\\|u(\\cdot,t)\\|_{L^q(\\R^n)} \\le C_1(\\|u_0\\|_{L^q(\\R^n)}),~~q \\in [\\beta+\\alpha-1,\\infty], \\\\\n\\|u(\\cdot,t)\\|_{L^q(\\R^n)} \\le C_2(\\|u_0\\|_{L^q(\\R^n)},t),~~q \\in [1,\\beta+\\alpha-1).\n\\end{align}\nHere $C_1$ is a positive constant only depending on $\\|u_0\\|_{L^q(\\R^n)}$ but not on $t$.\n\\end{thm}\n\nActually, problem \\er{star00} contains three terms, the diffusion term $\\Delta u$, the nonlocal aggregation term $\\nabla\\cdot(u^\\sigma \\nabla v)$ and the nonlinear growth term $u^\\alpha (1-\\int_{\\R^n} u^\\beta dx)$ (where $-u^\\alpha \\int_{\\R^n} u^\\beta dx$ can be viewed as the death contributing to the global existence), then the competition arises between the diffusion, the death and the aggregation, the growth. Indeed, \\er{star00} can be recast as\n\\begin{align}\\label{starstar00}\nu_t=\\Delta u-\\sigma u^{\\sigma-1}\\nabla u \\cdot \\nabla v+u^{\\sigma+1}+u^\\alpha\\left(1-\\int_{\\R^n} u^\\beta dx\\right),\n\\end{align}\nif $\\sigma+1=\\alpha,$ then the particular nonlinear reaction exponent\n$$\\alpha=2\\beta\/n+1$$\ngives the balance of diffusion and aggregation, reaction. In fact, plugging $u_\\lambda(x,t)=\\lambda^{\\frac{n}{\\beta}}u(\\lambda x, \\lambda^2 t)$ into \\er{star00}, it's easy to verify that $u_\\lambda(x,t)$ is also a solution of \\er{star00} and the scaling preserves the $L^\\beta$ norm in space, the diffusion term $\\lambda^{n\/\\beta+2}\\Delta u(\\lambda x, \\lambda^2 t)$ has the same scaling as the aggregation term $\\lambda^{(\\sigma+1)n\/\\beta}\\nabla \\cdot (u \\nabla (K\\ast u))(\\lambda x, \\lambda^2 t)$ and the reaction term $\\lambda^{n\\alpha\/\\beta} u^\\alpha (1-\\int_{\\R^n}u^\\beta dx)(\\lambda x, \\lambda^2 t)$ if and only if $\\alpha=2\\beta\/n+1$. From observing the rescaled equation we can see that when $n(\\alpha-1)\/\\beta<2,$ for low density (small $\\lambda$), the aggregation dominates the diffusion thus prevents spreading. While for high density (large $\\lambda$), the diffusion dominates the aggregation and thus blow-up is precluded. Hence, in this case, the solution will exist globally (Theorem \\ref{thm1}). On the other hand, if $n(\\alpha-1)\/\\beta>2,$ then the diffusion dominates for low density and the density had infinite-time spreading, the aggregation manipulates for high density and the density has finite time blow-up. Therefore, our conjecture is that there exists finite time blow-up for $\\alpha-1>2\\beta\/n.$ Moreover, for $\\alpha-1=2\\beta\/n$, similar to \\cite{Jose2009}, we guess that there is a critical value for the initial data sharply separating global existence and finite time blow-up.\n\nMoreover, noticing that \\er{starstar00} includes $u_t, \\Delta u, u^{\\sigma+1}, u^\\alpha, \\nabla \\cdot (u^\\sigma \\nabla v)$. Therefore, we detect the following equations are similar to \\er{starstar00}:\n\\begin{align}\\label{DR}\n\\left\\{\n  \\begin{array}{ll}\n    u_t=\\Delta u+u^\\alpha, & x \\in \\R^n, t>0, \\\\\n    u(x,0)=u_0(x), & x \\in \\R^n.\n  \\end{array}\n\\right.\n\\end{align}\nand\n\\begin{align}\\label{DA}\n\\left\\{\n  \\begin{array}{ll}\n    u_t=\\Delta u-\\nabla \\cdot (u^\\sigma \\nabla v), & x \\in \\R^n, t>0, \\\\\n    u(x,0)=u_0(x), & x \\in \\R^n.\n  \\end{array}\n\\right.\n\\end{align}\nIn order to compare \\er{DR}, \\er{DA} with \\er{starstar00}, we take $\\sigma+1=\\alpha$ (for simplicity) to find the effects of the nonlocal reaction $u^\\alpha \\left(1-\\int_{\\R^n} u^\\beta dx \\right)$. Concerning \\er{starstar00} and \\er{DR}, in this paper, we prove that for $\\alpha<1+2\\beta\/n,$ \\er{star00} admits a unique and global solution. While for the Fujita type equation \\er{DR}, it's known that for $\\alpha<1+2\/n,$ there is no global solution \\cite{Fu66} (For comparison, $\\lambda^{\\frac{n}{\\beta}}u(\\lambda x, \\lambda^2 t)$ in \\er{star00} is just the mass invariant scaling). As to \\er{DA} and \\er{starstar00}, the most remarkable difference is that the mass conservation holds for \\er{DA} but not for \\er{starstar00}, using this property it's been proved that the solution of \\er{DA} exists globally with small initial data \\cite{BL13,BDP06,JL92,P07,SK06}, while \\er{starstar00} has a unique and global solution without any restriction on the initial data. Thus we conclude that the reaction term can prevent blow-up.\n\nIn addition, Keller-Segel model with local reaction term in bounded domain has been widely studied by virtue of comparison principle and those estimates that are valid in bounded domain \\cite{GST16,HZ16,IS16,NT13,TW07,WL17,WMZ14,ZhLi15,Zh15}, that's\n\\begin{align} \\label{star11}\n\\left\\{\n      \\begin{array}{ll}\n       u_t=\\Delta u-\\chi \\nabla \\cdot (u^{\\sigma} \\nabla c)+f(u), &  x \\in \\Omega, t>0, \\\\\n       -\\Delta c+ c=u, & x \\in \\Omega, t>0, \\\\\n       \\nabla u \\cdot \\vec{n}=\\nabla v \\cdot \\vec{n}=0, &  x \\in \\partial \\Omega.\n      \\end{array}\\right.\n\\end{align}\nFor $\\sigma=1,$ \\cite{TW07} showed that model \\er{star11} with $f(u) \\le a-b u^2, u \\ge 0, a,b>0$ possesses a global bounded solution under the assumption $b>\\frac{n-2}{n}.$ The case $\\sigma>1$ was considered in \\cite{GST16} with $f(u)=\\mu u(1-u^\\alpha)$, if $\\alpha>\\sigma$ or $\\alpha=\\sigma$ and $\\mu>\\frac{n\\alpha -2}{n\\alpha+2(\\sigma-1)}$, then \\er{star11} admits a unique global solution. For other similar cases, one can refer to \\cite{HT17,NT13,WMZ14}.\n\nIn brief, comparing to the above models, the absence of mass conservation (model \\er{DA}) and the comparison principle (model \\er{DR},\\er{star11}) are two obstacles in our model \\er{star00} \\cite{Fu66,P07,SK06,WMZ14}. Besides, the nonlocal reaction makes the key energy estimates more difficult and many tools in the bounded domain  can't work in the whole space \\cite{GST16,NT13,TW07,WMZ14}. In our results, we will use analytical methods in the energy estimates and derive the conditions on $\\alpha,\\beta,\\sigma$ for global existence (Theorem \\ref{thm1}).\n\nThe main work is devoted to the global unique solution of model \\er{star00} for $\\alpha>1, \\beta>1, \\sigma \\ge 1$, with that aim Section \\ref{sec3} considers the local existence and uniqueness of the solution. In Section \\ref{sec4}, the a priori estimates are performed and show that $-u^\\alpha \\int_{\\R^n} u^\\beta dx$ plays a crucial role on the global existence, thus complete the proof of Theorem \\ref{thm1}. Here we split the arguments into several parts strongly depending on the exponents $\\alpha,\\beta$ and $\\sigma,$ consequently the uniformly boundedness is obtained by virtue of the Moser iterative method. Section \\ref{sec5} discusses some open questions of Eq. \\er{star00}.\n\n\\section{Preliminaries} \\label{sec2}\n\\def\\theequation{2.\\arabic{equation}}\\makeatother\n\\setcounter{equation}{0}\n\\def\\thetheorem{2.\\arabic{thm}}\\makeatother\n\nWe firstly state some lemmas which will be used in the proof of local existence and Theorem 2.\n\nConsider the Cauchy problem\n\\begin{align}\\label{Cauchy}\n\\left\\{\n  \\begin{array}{ll}\n    z_t=\\Delta z-\\nabla \\cdot (z H)+F,~~x \\in \\R^n, t>0, \\\\\n    z(x,0)=z_0(x).\n  \\end{array}\n\\right.\n\\end{align}\nThen the solution $z(x,t)$ can be expressed from semigroup theory \\cite{Pa83} as follows:\n\\begin{lemma}\\label{lem21}\nLet $X$ be a Banach space, $z_0 \\in X, H \\in L^\\infty(0,T;X)$ and $F\\in L^\\infty(0,T;X)$. $z(x,t)$ is given by \\cite{BDP06,SK06}\n\\begin{align}\nz(x,t)=G(\\cdot,t)\\ast z_0+\\int_0^t \\nabla G(\\cdot,t-s) \\ast [z(\\cdot,s)H(\\cdot,s)] ds+\\int_0^t G(\\cdot,t-s)\\ast F(\\cdot,s)ds,~~0 \\le t \\le T\n\\end{align}\nis the mild solution of \\er{Cauchy} on $[0,T].$ Here $G(x,t)=\\frac{1}{(4 \\pi t)^{\\frac{n}{2}}}e^{-\\frac{|x|^2}{4t}}$ is the Green function associated to the heat equation.\n\\end{lemma}\n\nThe following lemma is an immediate consequence from Sobolev inequality \\cite{Lieb202} which will play an important role in the proof of global existence of solutions for equation \\er{star00}.\n\\begin{lemma}[\\cite{BL16}]\\label{interpolation}\nLet $p$ is expressed by \\er{p}, $1 \\le r<q<p$ and $\\frac{q}{r}<\\frac{2}{r}+1-\\frac{2}{p}$, then for any $v\\in H^1(\\R^n)$ and $v \\in L^r(\\R^n)$, it holds\n\\begin{align}\n\\|v\\|_{L^q(\\R^n)}^q \\le C(n) C_0^{-\\frac{ \\lambda q}{2-\\lambda q}} \\|v\\|_{L^r(\\R^n)}^{\\gamma} + C_0 \\|\\nabla v\\|_{L^2(\\R^n)}^2,~~n \\ge 3, \\label{nge3}\n\\end{align}\nHere $C(n)$ is a constant depending on $n$, $C_0$ is an arbitrarily positive constant and\n\\begin{align}\n\\lambda=\\frac{\\frac{1}{r}-\\frac{1}{q}}{\\frac{1}{r}-\\frac{1}{p}} \\in (0,1),~~\n\\gamma=\\frac{2(1-\\lambda) q}{2-\\lambda q}=\\frac{2\\left(1-\\frac{q}{p}\\right)}{\\frac{2-q}{r}-\\frac{2}{p}+1}.\n\\end{align}\n\\end{lemma}\n\n\n\\section{Local existence and uniqueness} \\label{sec3}\n\\def\\theequation{3.\\arabic{equation}}\\makeatother\n\\setcounter{equation}{0}\n\\def\\thetheorem{3.\\arabic{thm}}\\makeatother\n\nThis part concerns local existence of the strong solution of \\er{star00}. The result is standard, more detailed arguments can be found in \\cite{BDP06,SK06,TW07}.\n\\begin{proposition}\\label{pro23}\nLet $\\alpha>1, \\beta>n\/2, \\sigma \\ge 1$. Assume that the initial data $u_0\\in W^{2,n-1}(\\R^n) \\cap L^1(\\R^n),$ then there exists a maximal existence time $T_{\\max} \\in (0,\\infty]$ such that $u(x,t) \\in W^{2,n-1}(Q_T) \\cap L^\\infty(0,T;L^1(\\R^n))$ is the unique non-negative strong solution of problem \\er{star00}. Furthermore, if $T_{\\max}<+\\infty,$ then\n\\begin{align}\\label{blowup}\n\\displaystyle \\lim_{t \\to T_{\\max}} \\|u(\\cdot,t)\\|_{L^\\infty(\\R^n)}=\\infty.\n\\end{align}\n\\end{proposition}\n\\begin{remark}\nHere $\\beta$ is chosen to be $\\beta>n\/2$ in order to prove the local existence and the a priori estimates Proposition \\ref{pro25}. By sobolev embedding theorem, $W^{2,n-1}(\\R^n)$ embedding into $L^\\infty(\\R^n)$ directly yields $u_0 \\in L^\\infty(\\R^n)$.\n\\end{remark}\n\\noindent\\textbf{Proof of Proposition \\ref{pro23}.} The proof can be divided into 2 steps. Step 1 investigates a semilinear parabolic equation and shows the local existence of the strong solution of Eqn. \\er{star00}. Step 2 gives the uniqueness of the strong solution. \\\\\n\n\\noindent\\textbf{Step 1 (Local existence).} In this step, we show the local existence of the strong solution, the proof is refined in spirit of \\cite{BDP06,SK06}. Here, we denote $X_T$ by\n\\begin{align}\nX_T:=\\{f \\in L^\\infty(0,T;W^{2,n-1}(\\R^n)), f_t \\in L^{n-1}(Q_T)\\big|~f \\ge 0, \\nonumber \\\\\n\\|f\\|_{L^\\infty(0,T;L^1 \\cap L^\\infty(\\R^n))} \\le C_1 \\|u_0\\|_{ L^\\infty(0,T;L^1 \\cap L^\\infty(\\R^n))}+C_2\\}\n\\end{align}\nfor some $C_1,C_2$ are constants only depending on $n,\\alpha,\\beta,\\sigma$ and $T>0$ to be determined later in Remark \\ref{Tguji}. We also define\n\\begin{align}\nW_u=\\{u \\in L^{n-1}(0,T;W^{2,n-1}(\\R^n)) \\mbox{~and~} u_t \\in L^{n-1}(0,T;L^{n-1}(\\R^n))\\}.\n\\end{align}\nFirstly, we consider\n\\begin{align}\nV=K \\ast f(x,t),\\qquad x \\in \\R^n,0<t<T,\n\\end{align}\nwhere $f \\in X_T.$ Then by the weak Young inequality \\cite{Lieb202}\n\\begin{align}\n\\nabla V \\in L^\\infty(0,T;L^{\\frac{n}{n-1}}\\cap L^\\infty(\\R^n)), \\\\\nV \\in L^\\infty(0,T;L^{\\frac{n}{n-2}}\\cap L^\\infty(\\R^n)).\n\\end{align}\nIn addition, by the Maximum principle \\cite{LSU} one has\n\\begin{align}\n0 \\le V(x,t) \\le \\|f\\|_{L^\\infty(Q_T)},~~~~x \\in \\R^n,~0<t<T.\n\\end{align}\nNow we introduce the following problem\n\\begin{align}\\label{E1}\n\\left\\{\n  \\begin{array}{ll}\nu_t=\\Delta u-\\sigma f^{\\sigma-1} \\nabla V\\cdot \\nabla u-f^{\\sigma-1}\\Delta V u+u^\\alpha \\left( 1-\\int_{\\R^n} f^\\beta dx \\right),\\quad x \\in \\R^n,~0<t<T,\\\\\nu\\big|_{t=0}=u_0(x) \\ge 0, \\quad x \\in \\R^n.\n  \\end{array}\n\\right.\n\\end{align}\nLet $u_0$ satisfies \\er{initial}. Assume $f \\in X_T,$ then $1-\\int_{\\R^n}f^\\beta dx$ is bounded by a constant $C(\\|u_0\\|_{L^1 \\cap L^\\infty(\\R^n)})$ which only depending on the initial data, then by virtue of \\cite[Theorem 9.1]{LSU} and \\cite{Fu66,H73,SK06}, equation \\er{E1} corresponding to the initial data $u_0$ has a strong solution $u^{f} \\in W_u$ and can be expressed by\n\\begin{align}\nu(\\cdot,t)&=G(\\cdot,t)\\ast u_0 - \\sigma \\int_0^t \\left(f^{\\sigma-1} \\nabla V \\cdot \\nabla u(\\cdot,s)\\right)\\ast G(\\cdot,t-s) ds-\\int_0^t \\left(f^{\\sigma-1} \\Delta V u(\\cdot,s)\\right)\\ast G(\\cdot,t-s) ds \\nonumber \\\\\n&+\\int_0^t \\left( 1-\\int_{\\R^n} f^\\beta dx \\right) u^\\alpha(\\cdot,s)\\ast G(\\cdot,t-s) ds,\n\\end{align}\nwhere $G(\\cdot,t)$ is the Green function as in Lemma \\ref{lem21}.\n\nNext we define a mapping $\\Phi$ by\n\\begin{align}\n\\Phi: f \\in X_T \\mapsto u^f \\in W_u,\n\\end{align}\nWe show the solution $u^f$ is nonnegative as follows. Multiplying \\er{E1} by $|u|^{k-2}u ~(k>1)$ and using Young's inequality we have\n\\begin{align}\n&\\frac{1}{k}\\frac{d}{dt}\\int_{\\R^n} |u|^k dx+ (k-1) \\int_{\\R^n} |u|^{k-2} |\\nabla u|^2dx \\nonumber \\\\\n=& -\\int_{\\R^n} \\sigma f^{\\sigma-1} \\nabla V \\cdot \\nabla u |u|^{k-2}u dx-\\int_{\\R^n} f^{\\sigma-1} \\Delta V |u|^kdx+\\int_{\\R^n} u^{\\alpha+1} |u|^{k-2} dx\\left( 1-\\int_{\\R^n} f^\\beta dx\\right)\\nonumber \\\\\n\\le &\\frac{k-1}{4}\\int_{\\R^n} |u|^{k-2} |\\nabla u|^2dx+\\frac{\\|\\sigma f^{\\sigma-1}\\nabla V\\|_{L^\\infty(Q_T)}^2}{k-1} \\int_{\\R^n} |u|^k dx \\nonumber \\\\\n&+\\|f^{\\sigma-1}\\Delta V\\|_{L^\\infty(Q_T)} \\int_{\\R^n} |u|^k dx +\\left( 1+\\|f\\|_{L^\\infty(0,T;L^1 \\cap L^\\infty(\\R^n))}^\\beta \\right)\\int_{\\R^n} |u|^{\\alpha+k-1} dx, \\label{es1}\n\\end{align}\nletting $F_0=1+\\|f\\|_{L^\\infty(0,T;L^1 \\cap L^\\infty(\\R^n))}^\\beta$, we apply\n\\begin{align}\nv=u^{k\/2}, q=\\frac{2(k+\\alpha-1)}{k}, r=2, C_0=\\frac{2(k-1)}{k^2}\n\\end{align}\nin Lemma \\ref{interpolation} for $k>\\frac{n(\\alpha-1)}{2}$ such that\n\\begin{align}\n&F_0 \\int_{\\R^n} |u|^{\\alpha+k-1} dx \\nonumber\\\\\n\\le & \\frac{2(k-1)}{k^2} \\int_{\\R^n} |\\nabla u^{k\/2}|^2dx+C\\left(n,F_0 \\right) \\left( \\frac{k^2}{2(k-1)} \\right)^{\\frac{\\lambda q}{2-\\lambda q}} \\left( \\int_{\\R^n} |u|^k dx \\right)^{\\delta },\n\\end{align}\nwhere\n$$\n\\lambda=\\frac{\\frac{1}{2}-\\frac{1}{q}}{\\frac{1}{2}-\\frac{1}{p}} \\in (0,1),~\\delta=\\frac{1-\\frac{q}{p}}{2-\\frac{q}{2}-\\frac{2}{p}}.\n$$\nPlugging it into \\er{es1} one has\n\\begin{align}\n&\\frac{d}{dt} \\|u\\|_{L^k(\\R^n)} \\nonumber \\\\\n\\le &\\left( \\frac{\\|\\sigma f^{\\sigma-1} \\nabla V\\|_{L^\\infty(Q_T)}^2}{k-1} + \\|f^{\\sigma-1} \\Delta V \\|_{L^\\infty(Q_T)} \\right) \\|u\\|_{L^k(\\R^n)}\n+C\\left(n,F_0 \\right) \\left( \\frac{k^2}{2(k-1)} \\right)^{C\\left(\\frac{1}{k}\\right)} \\|u\\|_{L^k(\\R^n)}^{k \\delta-k+1},\n\\end{align}\ntaking $k \\to \\infty$ we derive\n\\begin{align}\n\\frac{d}{dt} \\|u\\|_{L^\\infty(\\R^n)} \\le C\\left(n,F_0,\\|f^{\\sigma-1}\\Delta V\\|_{L^\\infty(Q_T)}\\right) \\|u\\|_{L^\\infty(\\R^n)},\n\\end{align}\nusing Gronwall's inequality it's obtained that\n\\begin{align}\\label{Linflocal}\n\\displaystyle \\sup_{0<t<T} \\|u(\\cdot,t)\\|_{L^\\infty(\\R^n)} \\le \\|u_0\\|_{L^\\infty(\\R^n)}e^{C\\left(n,F_0,\\|f^{\\sigma-1}\\Delta V\\|_{L^\\infty(Q_T)}\\right)T }.\n\\end{align}\nFurther integrating \\er{E1} over $\\R^n$ arrives at\n\\begin{align}\\label{L1local}\n\\displaystyle \\sup_{0<t<T} \\|u(\\cdot,t)\\|_{L^1(\\R^n)} \\le \\|u_0\\|_{L^1(\\R^n)} e^{C\\left(\\alpha,\\|u\\|_{L^\\infty(Q_T)}\\right)T}.\n\\end{align}\nThe nonnegativity of $u$ can be obtained by multiplying \\er{E1} with $u^-:=-\\min(u,0)$ that\n\\begin{align}\n&\\frac{1}{2}\\frac{d}{dt} \\int_{\\R^n} |u^-|^2 dx \\nonumber \\\\\n&= -\\int_{\\R^n} |\\nabla u^-|^2 dx-\\sigma \\int_{\\R^n} f^{\\sigma-1}\\nabla V \\cdot \\nabla u^- u^- dx -\\int_{\\R^n} f^{\\sigma-1} \\Delta V |u^-|^2 dx +\\left(1-\\int_{\\R^n} f^\\beta dx \\right)\\int_{\\R^n} u^\\alpha u^- dx \\nonumber \\\\\n&\\le -\\frac{1}{2} \\int_{\\R^n} |\\nabla u^-|^2 dx+ \\frac{1}{2} \\|\\sigma f^{\\sigma-1}\\nabla V\\|_{L^\\infty(Q_T)}^2 \\int_{\\R^n}|u^-|^2 dx \\nonumber \\\\\n&+ \\|f^{\\sigma-1}\\Delta V\\|_{L^\\infty(Q_T)} \\int_{\\R^n}|u^-|^2 dx+ F_0 \\|u\\|_{L^\\infty(Q_T)}^{\\alpha-1} \\int_{\\R^n} |u^-|^2 dx,\n\\end{align}\nit follows\n\\begin{align}\n\\displaystyle \\sup_{0<t<T} \\|u(\\cdot,t)\\|_{L^2(\\R^n)} \\le e^{(\\|\\sigma f^{\\sigma-1} \\nabla V\\|_{L^\\infty(Q_T)}^2+2\\|f^{\\sigma-1}\\Delta V\\|_{L^\\infty(Q_T)}+2F_0)T} \\|u_0^-(\\cdot,0)\\|_{L^2(\\R^n)}=0\n\\end{align}\ndirectly assures that for all $0 \\le t<T$\n$$\nu(x,t)\\ge0,~~a.e.~~x \\in \\R^n.\n$$\nFurthermore, $\\Phi$ is a contraction map in $L^\\infty(0,T;L^{n-1}(\\R^n))$. In fact, we consider the complete metric space $(X_T,d)$ where $d$ is defined by $d(f_1-f_2)=\\|f_1-f_2\\|_{L^\\infty(0,T;L^{n-1}(\\R^n))}$, we denote\n\\begin{align}\nu_1=u^{f_1},~~u_2=u^{f_2},~~w=u_1-u_2,\n\\end{align}\nfrom \\er{E1} one has\n\\begin{align}\\label{wstar}\n&(u_1-u_2)_t=\\Delta (u_1-u_2)-\\left( \\sigma f_1^{\\sigma-1}\\nabla V_1 \\cdot \\nabla u_1-\\sigma f_2^{\\sigma-1} \\nabla V_2 \\cdot \\nabla u_2 \\right) \\nonumber \\\\\n&-\\left( f_1^{\\sigma-1} \\Delta V_1 u_1-f_2^{\\sigma-1}\\Delta V_2 u_2 \\right)+\\left(u_1^\\alpha-u_2^\\alpha\\right) \\left( 1-\\int_{\\R^n}f_2^\\beta dx  \\right)-u_1^\\alpha \\int_{\\R^n} (f_1^\\beta-f_2^\\beta) dx,\n\\end{align}\nthe multiplication \\er{wstar} by $|w|^{n-3}w$ and using H\\\"{o}lder's inequality give that for $\\beta>n\/2$\n\\begin{align}\n&\\frac{1}{n-1} \\frac{d}{dt} \\int_{\\R^n} |w|^{n-1}dx +(n-2)\\int_{\\R^n} |\\nabla w|^2 |w|^{n-3} dx \\nonumber \\\\\n&~\\le \\frac{n-2}{4} \\int_{\\R^n} |w|^{n-3}|\\nabla w|^2 dx+\\sigma \\|f\\|_{L^\\infty(Q_T)}^{\\sigma-1}\\| u_1\\|_{W^{2,n-1}(Q_T)}\\int_{\\R^n} |f_1-f_2|\\cdot |w|^{n-2} dx \\nonumber \\\\\n&~~+\\sigma \\|f_2\\|_{L^\\infty(Q_T)}^\\sigma \\int_{\\R^n} |\\nabla w|\\cdot |w|^{n-2}dx +\\alpha \\|u\\|_{L^\\infty(Q_T)}^{\\alpha-1}F_0 \\int_{\\R^n} |w|^{n-1} dx+\\int_{\\R^n} \\beta |f|^{\\beta-1}|f_1-f_2| dx \\int_{\\R^n} u_1^\\alpha |w|^{n-2} dx \\nonumber \\\\\n&~\\le \\frac{n-2}{4} \\int_{\\R^n} |w|^{n-3}|\\nabla w|^2 dx+C\\left(\\|f\\|_{L^\\infty(Q_T)},\\|u_1\\|_{W^{2,n-1}(Q_T)}\\right) \\|f_1-f_2\\|_{L^{n-1}(\\R^n)} \\| w^{n-2}\\|_{L^{\\frac{n-1}{n-2}}(\\R^n)} \\nonumber \\\\\n&~~+\\frac{n-2}{4} \\int_{\\R^n} |w|^{n-3}|\\nabla w|^2 dx +C\\left( \\|f_2\\|_{L^\\infty(Q_T)},\\|u\\|_{L^\\infty(Q_T)},F_0 \\right) \\int_{\\R^n} |w|^{n-1} dx\\nonumber \\\\\n&~~+\\beta \\|f_1-f_2\\|_{L^{n-1}(\\R^n)} \\|f^{\\beta-1}\\|_{L^{\\frac{n-1}{n-2}}(\\R^n)} \\|w^{n-2}\\|_{L^{\\frac{n-1}{n-2}}(\\R^n)} \\|u_1^\\alpha\\|_{L^{n-1}(\\R^n)}. \\label{wcontraction}\n\\end{align}\nHere $u$ and $f$ satisfy $\\alpha u^{\\alpha-1}=u_1^\\alpha-u_2^\\alpha$ and $\\beta f^{\\beta-1}=f_1^\\beta-f_2^\\beta$ by mean value theorem, \\er{wcontraction} follows that\n\\begin{align}\n&\\frac{d}{dt} \\|w\\|_{L^{n-1}(\\R^n)} \\nonumber \\\\\n\\le & C_1\\left( \\|f\\|_{L^\\infty(Q_T)},\\|u\\|_{L^\\infty(Q_T)}\\right)\\|w\\|_{L^{n-1}(\\R^n)}+C_2\\left(\n\\|f\\|_{L^\\infty(Q_T)}, \\|u_1\\|_{W^{2,n-1}(Q_T)} \\right)\\|f_1-f_2\\|_{L^{n-1}(\\R^n)},\n\\end{align}\napplying Gronwall's inequality yields\n\\begin{align}\n\\displaystyle \\sup_{0<t<T} \\|w(t)\\|_{L^{n-1}(\\R^n)} \\le C_2 e^{C_1 T} \\|f_1-f_2\\|_{L^\\infty(0,T;L^{n-1}(\\R^n))},\n\\end{align}\nhence there exists $T_\\ast=T_\\ast\\left(\\|u_0\\|_{L^1 \\cap W^{2,n-1}(\\R^n)},\\|u_0\\|_{L^\\infty(Q_T)},T\\right)\\le T$ small such that\n\\begin{align}\n\\displaystyle \\sup_{0<t<T_\\ast} \\|w(t)\\|_{L^{n-1}(\\R^n)} \\le \\frac{1}{2} \\|f_1-f_2\\|_{L^\\infty(0,T_\\ast;L^{n-1}(\\R^n))}.\n\\end{align}\nUsing Banach's fixed point theorem, we have that $\\Phi$ has a fixed point $\\Phi(f)=u^f=f \\in X_{T_\\ast}$. Iterating the method we prove the existence of the strong solution $u$ of equation on $[0,T)$\n\\begin{align}\\label{232}\n\\left\\{\n  \\begin{array}{ll}\n u_t=\\Delta u-\\sigma u^{\\sigma-1} \\nabla V \\cdot \\nabla u-u^{\\sigma-1} \\Delta V u+u^\\alpha \\left( 1-\\int_{\\R^n} u^\\beta dx \\right),\\quad   &x \\in \\R^n, 0<t<T,\n\\\\\nv=K \\ast u, &x \\in \\R^n, 0<t<T,\n\\\\\n u(x,0)=u_0(x)\\geq 0,\\quad & x \\in \\R^n.\n  \\end{array}\n\\right.\n\\end{align}\n\\\\\n\n\\noindent\\textbf{Step 2 (Uniqueness).} The uniqueness of the strong solution can be shown in the follows. Assume $(u_1,v_1)$ and $(u_2,v_2)$ solve \\er{232} in $Q_{T}$ with initial data \\er{initial}, then\n\\begin{align}\n(u_1-u_2)_t=\\Delta(u_1-u_2)-\\nabla \\cdot \\left((u_1^\\sigma-u_2^\\sigma) \\nabla v_1+u_2^\\sigma \\nabla(v_1-v_2)\\right) \\nonumber\\\\\n+u_1^\\alpha-u_2^\\alpha+\\left(u_2^\\alpha-u_1^\\alpha\\right)\\int_{\\R^n} u_1^\\beta dx+u_2^\\alpha \\int_{\\R^n} \\left(u_2^\\beta-u_1^\\beta\\right) dx, \\label{233}\n\\end{align}\nmultiplying \\er{233} with $u_1-u_2$ we obtain\n\\begin{align}\\label{I1}\n&\\frac{1}{2}\\frac{d}{dt}\\int_{\\R^n} |u_1-u_2|^2 dx \\nonumber \\\\\n&=-\\int_{\\R^n} |\\nabla (u_1-u_2)|^2dx+\\int_{\\R^n} (u_1^\\sigma-u_2^\\sigma) \\nabla(u_1-u_2)\\cdot \\nabla v_1 dx+\\int_{\\R^n} u_2^\\sigma \\nabla(u_1-u_2)\\cdot\\nabla(v_1-v_2) dx\\nonumber \\\\\n&~~+\\left(1-\\int_{\\R^n}u_1^\\beta dx\\right)\\int_{\\R^n} (u_1^\\alpha-u_2^\\alpha)(u_1-u_2)dx+ \\int_{\\R^n} \\left(u_2^\\beta-u_1^\\beta\\right) dx \\int_{\\R^n} u_2^\\alpha (u_1-u_2) dx \\nonumber \\\\\n&\\le -\\int_{\\R^n} |\\nabla(u_1-u_2)|^2 dx+\\frac{1}{4}\\int_{\\R^n} |\\nabla(u_1-u_2)|^2 dx+\\|\\nabla v_1\\|_{L^\\infty(Q_{T})}^2 \\left( \\sigma \\|u\\|_{L^\\infty(Q_{T})}^{\\sigma-1}\\right)^2 \\int_{\\R^n} |u_1-u_2|^2dx \\nonumber \\\\\n&~~+\\frac{1}{4}\\int_{\\R^n} |\\nabla(u_1-u_2)|^2 dx+\\int_{\\R^n}|u_2^\\sigma \\nabla(v_1-v_2)|^2 dx+\\alpha \\left(1+\\|u_1\\|_{L^\\infty(0,T;L^\\beta(\\R^n))}^\\beta \\right)\\|u\\|_{L^\\infty(Q_{T})}^{\\alpha-1} \\int_{\\R^n} |(u_1-u_2)|^2 dx\\nonumber \\\\ &~~+\\int_{\\R^n}|u_2^\\beta-u_1^\\beta|dx \\int_{\\R^n}|u_2^\\alpha(u_1-u_2)|dx.\n\\end{align}\nHere $u$ comes from mean value theorem by $\\alpha u^{\\alpha-1}=u_1^\\alpha-u_2^\\alpha$. Therefore\n\\begin{align}\n&\\frac{d}{dt}\\int_{\\R^n} |u_1-u_2|^2 dx \\le C\\big(\\|\\nabla v_1\\|_{L^\\infty(Q_{T})},\\|u\\|_{L^\\infty(Q_{T})}, \\|u_1\\|_{L^\\infty(0,T;L^\\beta(\\R^n))} \\big)\\int_{\\R^n} |u_1-u_2|^2dx \\nonumber \\\\\n&+ \\int_{\\R^n}|u_2^\\sigma \\nabla(v_1-v_2)|^2dx+\\int_{\\R^n}|u_2^\\beta-u_1^\\beta|dx \\int_{\\R^n}|u_2^\\alpha(u_1-u_2)|dx\n=I_1+I_2+I_3,\n\\end{align}\nBy H\\\"{o}lder's inequality and weak Young's inequality \\cite{Lieb202} we have\n\\begin{align}\\label{I2}\nI_2 &\\le \\big\\||\\nabla(v_1-v_2)|^2\\big\\|_{L^{\\frac{n}{n-2}}(\\R^n)}\\|u_2^{2\\sigma}\\|_{L^{\\frac{n}{2}}(\\R^n)} \\nonumber\\\\\n&\\le C \\big\\|\\frac{x-y}{|x-y|^n} \\big\\|^2_{L_w^{\\frac{n}{n-1}}(\\R^n)} \\|u_1-u_2\\|_{L^2(\\R^n)}^2 \\|u_2^{2 \\sigma}\\|_{L^{\\frac{n}{2}}(\\R^n)},\n\\end{align}\nand\n\\begin{align}\\label{I3}\nI_3 &\\le \\int_{\\R^n} \\beta u^{\\beta-1}|u_1-u_2|dx \\int_{\\R^n} u_2^\\alpha |u_1-u_2|dx \\nonumber \\\\\n&\\le \\|u_1-u_2\\|_{L^2(\\R^n)} \\|\\beta u^{\\beta-1}\\|_{L^2(\\R^n)} \\|u_2^\\alpha \\|_{L^2(\\R^n)} \\|u_1-u_2\\|_{L^2(\\R^n)} \\nonumber \\\\\n&\\le C\\left(\\|u\\|_{L^1\\cap L^\\infty(\\R^n)}\\right) \\|u_1-u_2\\|_{L^2(\\R^n)}^2,  ~~(\\beta \\ge 3\/2)\n\\end{align}\ntaking \\er{I1}, \\er{I2} and \\er{I3} together one has\n\\begin{align}\n\\frac{d}{dt}\\int_{\\R^n} |u_1-u_2|^2 dx\\le C(\\|u\\|_{L^1\\cap L^\\infty(\\R^n)},n) \\int_{\\R^n} |u_1-u_2|^2 dx,\n\\end{align}\nthis yields $u_1=u_2$ in $Q_{T}$ which implies the uniqueness of solutions. Thus we complete the proof of local existence and uniqueness of the strong solution. $\\Box$\n\n\\begin{remark}\\label{Tguji}\nIn Proposition \\ref{pro23}, the bounded time $T$ in $X_T$ can be preestimated as follows\n\\begin{align}\nf_t&=\\Delta f-\\nabla \\cdot(f^\\sigma \\nabla v)+ f^\\alpha \\left(1-\\int_{\\R^n}f^\\beta dx\\right), \\label{333} \\\\\nv&=K \\ast f, \\nonumber \\\\\nf(0)&=u_0 \\ge 0. \\nonumber\n\\end{align}\nmultiplying \\er{333} by $rf^{r-1}(r>1)$ obtains that\n\\begin{align}\n\\frac{d}{dt}\\int_{\\R^n} f^r dx &=-\\frac{4(r-1)}{r} \\int_{\\R^n}|\\nabla f^{\\frac{r}{2}}|^2 dx +\\frac{r(r-1)}{\\sigma+r-1}\\int_{\\R^n} f^{\\sigma+r} dx+r (1-\\int_{\\R^n} f^\\beta dx) \\int_{\\R^n} f^{\\alpha+r-1} dx \\nonumber \\\\\n&\\le \\frac{r(r-1)}{\\sigma+r-1} \\|f\\|_{L^\\infty(\\R^n)}^\\sigma \\|f\\|_{L^r(\\R^n)}^r+r \\|f\\|_{L^\\infty(\\R^n)}^{\\alpha-1}\\|f\\|_{L^r(\\R^n)}^r,\n\\end{align}\nthus\n\\begin{align}\n\\frac{d}{dt} \\|f\\|_{L^r(\\R^n)} \\le \\frac{r-1}{\\sigma+r-1}\\|f\\|_{L^\\infty(\\R^n)}^\\sigma \\|f\\|_{L^r(\\R^n)}+ \\|f\\|_{L^\\infty(\\R^n)}^{\\alpha-1}\\|f\\|_{L^r(\\R^n)},\n\\end{align}\nletting $r \\to \\infty$ one has\n\\begin{align}\n\\|f\\|_{L^\\infty(\\R^n)} \\le \\|u_0\\|_{L^\\infty(\\R^n)}+\\int_0^t \\|f\\|_{L^\\infty(\\R^n)}^{\\sigma+1} ds+\\int_0^t \\|f\\|_{L^\\infty(\\R^n)}^\\alpha ds,\n\\end{align}\nhence from ODE inequality we have that there is a maximum existence time $T=T(\\|u_0\\|_{L^\\infty(\\R^n)})$ such that $f$ is bounded from above in $[0,T)$.\n\\end{remark}\n\n\\section{Proof of Theorem \\ref{thm1}}\\label{sec4}\n\\def\\theequation{4.\\arabic{equation}}\\makeatother\n\\setcounter{equation}{0}\n\\def\\thetheorem{4.\\arabic{thm}}\\makeatother\n\nIn this section, we derive the a priori estimates of the strong solution and complete the proof of Theorem \\ref{thm1}.\n\\begin{proposition}\\label{pro25}\nLet $\\sigma \\ge 1, \\beta>1, \\alpha>1,$ $u_0$ satisfies \\er{initial}. If\n\\begin{align}\n\\sigma+1\\le \\alpha \\mbox{ and } \\alpha<1+2\\beta\/n\n\\end{align}\nor\n\\begin{align}\n\\sigma+1>\\alpha \\mbox{ and } (\\sigma+1-\\alpha)(n+2)<2\\beta-n(\\alpha-1),\n\\end{align}\nthen for any $0<t<T_{\\max}$, the solution of \\er{star00} satisfies that \\\\\n\\noindent(1) For $\\beta+\\alpha-1 \\le k<\\infty,$\n\\begin{align}\n\\|u(\\cdot,t)\\|_{L^k(\\R^n)} \\le C(\\|u_0\\|_{L^k(\\R^n)},k).\n\\end{align}\n\\noindent(2) The uniformly boundedness of solution $$\\|u(\\cdot,t)\\|_{L^\\infty(\\R^n)}\\le C \\left(\\|u_0\\|_{L^{\\beta+\\alpha-1}(\\R^n)},\\|u_0\\|_{L^\\infty(\\R^n)}\\right),$$ where $C$ is a positive constant depending on $\\|u_0\\|_{L^{\\beta+\\alpha-1}(\\R^n)}$ and $\\|u_0\\|_{L^\\infty(\\R^n)}$ but not on $T_{\\max}$. \\\\\n\\noindent(3) For $1 \\le k<\\beta+\\alpha-1$,\n\\begin{align}\n\\|u(\\cdot,t)\\|_{L^k(\\R^n)} \\le C(\\|u_0\\|_{L^k(\\R^n)},T_{\\max})\n\\end{align}\nwhere $C$ is a positive constant depending on $\\|u_0\\|_{L^k(\\R^n)}$ and $T_{\\max}$. \\\\\n\nEspecially, when $\\sigma+1=\\alpha,$ we have that for $\\beta \\le k \\le \\infty$\n\\begin{align}\n\\|u(\\cdot,t)\\|_{L^k(\\R^n)} \\le C(\\|u_0\\|_{L^k(\\R^n)}),\n\\end{align}\nwhere $C$ only depends on $\\|u_0\\|_{L^k(\\R^n)}$ not on $T_{\\max}$.\n\\end{proposition}\n\\noindent{\\bf Proof of Proposition \\ref{pro25}.} For the rigorous proof, we should multiply \\er{star00} by $ku^{k-1}\\psi_l$, where $\\psi_l$ is a standard cut-off function. By the limiting process we can justify the following formal calculation. Throughout the proof, we suppose $\\sigma+1=\\eta.$ \\\\\n\n\n\\noindent\\textbf{Step 1 (A priori estimates).} Firstly multiplying \\er{star00} with $k u^{k-1}(k \\ge 1)$ one has\n\\begin{align}\\label{baseguji}\n& \\frac{d}{dt} \\int_{\\R^n}  u^k dx  + \\frac{4(k-1)}{k}\n\\int_{\\R^n} |\\nabla u^{\\frac{k}{2}} |^2 dx + k \\int_{\\R^n} u^{k+\\alpha-1} dx \\int_{\\R^n} u^\\beta dx \\nonumber\\\\\n=& ~k \\int_{\\R^n} u^{k+\\alpha-1} dx +\\frac{k(k-1)}{k+\\sigma-1} \\int_{\\R^n} u^{\\eta+k-1} dx.\n\\end{align}\nIn order to control the right hand side of \\er{baseguji} by using the two nonnegative terms in the left hand side of \\er{baseguji}, we apply\n$$~v=u^{k\/2}, q=\\frac{2(k+\\alpha-1)}{k}, r=\\frac{2k'}{k}, C_0=\\frac{k-1}{k^2} $$\nin Lemma \\ref{interpolation} with\n$k>\\frac{2(\\alpha-1)}{p-2}$ (which is $q<p$) and $\\frac{p(\\alpha-1)}{p-2}<k'<k+\\alpha-1$ (which is $\\frac{q}{r}<\\frac{2}{r}+1-\\frac{2}{p},r<q$)\nsuch that\n\\begin{align}\\label{1}\n\\int_{\\R^n} u^{\\alpha+k-1} dx \\le \\frac{k-1}{k^2} \\| \\nabla u^{\\frac{k}{2}} \\|_{L^2(\\R^n)}^2+C(k) \\|u\\|_{L^{k'}(\\R^n)}^{b_\\alpha},\n\\end{align}\nwhere\n$$ b_\\alpha=\\frac{(1-\\lambda_\\alpha)(\\alpha+k-1)}{1-\\frac{\\lambda_\\alpha(\\alpha+k-1)}{k}},~\\lambda_\\alpha=\\frac{\\frac{k}{2k'}-\\frac{k}{2(\\alpha+k-1)}}{\\frac{k}{2k'}-\\frac{1}{p}}\\in(0,1).\n$$\nSimilarly, taking\n$$~v=u^{k\/2}, q=\\frac{2(k+\\eta-1)}{k}, r=\\frac{2k'}{k}, C_0=\\frac{k+\\sigma-1}{k^2} $$\nin Lemma \\ref{interpolation} with $k>\\frac{2(\\eta-1)}{p-2}$ (which is $q<p$) and $\\frac{p(\\eta-1)}{p-2}<k'<k+\\eta-1$ (which is $\\frac{q}{r}<\\frac{2}{r}+1-\\frac{2}{p},r<q$) leads to\n\\begin{align}\\label{2}\n\\int_{\\R^n} u^{\\eta+k-1} dx \\le \\frac{k+\\sigma-1}{k^2} \\| \\nabla u^{\\frac{k}{2}} \\|_{L^2(\\R^n)}^2+C(k) \\|u\\|_{L^{k'}(\\R^n)}^{b_\\eta},\n\\end{align}\nwhere\n$$ b_\\eta=\\frac{(1-\\lambda_\\eta)(\\eta+k-1)}{1-\\frac{\\lambda_\\eta(\\eta+k-1)}{k}},~\\lambda_\\eta=\\frac{\\frac{k}{2k'}-\\frac{k}{2(\\eta+k-1)}}{\\frac{k}{2k'}-\\frac{1}{p}} \\in(0,1).$$\nHence taking \\er{1} and \\er{2} together we will conduct further estimates of \\er{baseguji} for\n\\begin{align}\\label{k}\nk>\\max \\left( \\frac{2(\\eta-1)}{p-2},\\frac{2(\\alpha-1)}{p-2},1 \\right)\n\\end{align}\nand\n\\begin{align}\\label{kprime}\n\\max \\left( \\frac{p(\\eta-1)}{p-2},\\frac{p(\\alpha-1)}{p-2},1,\\frac{k}{2} \\right)<k'<\\min\\left( k+\\alpha-1, k+\\eta-1 \\right)\n\\end{align}\nwhere we have used the fact that $1 \\le r$ is equivalent to $\\frac{k}{2} \\le k'.$\n\nCombining \\er{1} and \\er{2} we infer from \\er{baseguji} that\n\\begin{align}\\label{guji1}\n& \\frac{d}{dt}\\int_{R^n} u^k dx+ k \\int_{\\R^n} u^\\beta dx \\int_{\\R^n} u^{k+\\alpha-1} dx+\\frac{2(k-1)}{k} \\|\\nabla u^{\\frac{k}{2}} \\|_{L^2(\\R^n)}^2 \\nonumber \\\\\n\\le ~& C(k,\\alpha)  \\|u\\|_{L^{k'}(\\R^n)}^{b_\\alpha}+ C(k,\\eta) \\|u\\|_{L^{k'}(\\R^n)}^{b_\\eta}.\n\\end{align}\nWe further assume $\\beta<k'$ and use the following interpolation inequalities such that\n\\begin{align}\\label{guji2}\n\\|u\\|_{L^{k'}}^{b_\\alpha} \\le \\left( \\|u\\|_{L^{k+\\alpha-1}}^{k+\\alpha-1} \\|u\\|_{L^\\beta}^\\beta \\right)^{\\frac{b_\\alpha \\theta}{k+\\alpha-1}} \\|u\\|_{L^\\beta}^{b_\\alpha(1-\\theta-\\frac{\\theta \\beta}{k+\\alpha-1})}\n\\end{align}\nand\n\\begin{align}\\label{guji3}\n\\|u\\|_{L^{k'}}^{b_\\eta} \\le \\left( \\|u\\|_{L^{k+\\alpha-1}}^{k+\\alpha-1} \\|u\\|_{L^\\beta}^\\beta \\right)^{\\frac{b_\\eta \\theta}{k+\\alpha-1}} \\|u\\|_{L^\\beta}^{b_\\eta(1-\\theta-\\frac{\\theta \\beta}{k+\\alpha-1})}\n\\end{align}\nwhere\n$$\n\\theta=\\frac{\\frac{1}{\\beta}-\\frac{1}{k'}}{\\frac{1}{\\beta}-\\frac{1}{k+\\alpha-1}}.\n$$\nTo use Young's inequality, we need the following three conditions that\n\\begin{align}\\label{star10}\n1-\\theta-\\frac{\\theta \\beta}{k+\\alpha-1}=0\n\\end{align}\nand\n\\begin{align} \\label{star}\n\\frac{b_\\alpha \\theta }{k+\\alpha-1}<1\n\\end{align}\nas well as\n\\begin{align}\\label{starstar}\n\\frac{b_\\eta \\theta }{k+\\alpha-1}<1.\n\\end{align}\nFirstly for \\er{star10}, thanks to the arbitrariness of $k',$ we take\n$$\nk'=\\frac{k+\\alpha-1+\\beta}{2} \\in (\\beta,k+\\alpha-1)\n$$\nso that \\er{star10} holds true. Here combining \\er{k} and \\er{kprime}, $k$ satisfies\n\\begin{align}\\label{kxin}\nk>\\max \\left( \\frac{2(\\eta-1)}{p-2},~\\frac{2(\\alpha-1)}{p-2}, ~\\frac{2p(\\eta-1)}{p-2}-\\beta-(\\alpha-1),~ \\frac{2p(\\alpha-1)}{p-2}-\\beta-(\\alpha-1)\\right)=:K_0.\n\\end{align}\nNext recalling $b_{\\alpha}$ and $\\theta$, \\er{star} is equivalent to\n\\begin{align}\n(1-\\lambda_{\\alpha})\\left( \\frac{1}{\\beta}-\\frac{1}{k'}  \\right)< \\left( 1-\\frac{\\lambda_\\alpha (k+\\alpha-1)}{k} \\right)\\left( \\frac{1}{\\beta}-\\frac{1}{k+\\alpha-1}\\right),\n\\end{align}\nit also reads\n\\begin{align}\n\\lambda_\\alpha-1<\\frac{\\lambda_\\alpha}{k} k'-\\frac{k'}{k+\\alpha-1}-\\frac{\\lambda_\\alpha(\\alpha-1)}{k \\beta} k',\n\\end{align}\nsubstituting $\\lambda_\\alpha$ into the above formula one has that for\n\\begin{align}\\label{tiaojian1}\n1 \\le \\alpha<1+\\left( 1-\\frac{2}{p} \\right) \\beta,\n\\end{align}\n\\er{star} holds true for any $k$ satisfying \\er{kxin}.\nFor \\er{starstar}, under the condition \\er{tiaojian1} similar arguments arrive at\n\\begin{align}\n\\left( \\frac{k}{2}-\\frac{k+\\eta-1}{p} \\right) \\left( \\frac{k'}{\\beta}-1  \\right)<\\left( \\left(\\frac{1}{2}-\\frac{1}{p} \\right)k'-\\frac{\\eta-1}{2}  \\right) \\left( \\frac{\\alpha-1}{\\beta}-1+\\frac{k}{\\beta} \\right),\n\\end{align}\nit can be written as\n\\begin{align}\\label{starstarstar}\n\\frac{\\eta-\\alpha}{\\beta} ~\\left( \\frac{k}{2}-\\frac{k'}{p} \\right)<(\\eta+k-1-k') ~\\left( \\frac{1}{2}-\\frac{1}{p}-\\frac{\\alpha-1}{2 \\beta} \\right).\n\\end{align}\n\nDenote\n$$\nA_0=\\frac{1}{2}-\\frac{1}{p}-\\frac{\\alpha-1}{2 \\beta}>0,~A_1=\\frac{\\eta-\\alpha}{\\beta},\n$$\nrecalling \\er{k} and \\er{kprime}, if $\\eta \\le \\alpha,$ then $A_0>0$ is enough to guarantee that \\er{starstar} holds true. Otherwise if $\\eta > \\alpha,$ plugging\n$\nk'=\\frac{k+\\alpha-1+\\beta}{2}\n$\ninto \\er{starstarstar} yields\n$$\nA_0(\\eta-1)+\\frac{A_1}{2p}(\\alpha+\\beta-1)+\\Big[  \\frac{A_0}{2}-A_1 \\left( \\frac{1}{2}-\\frac{1}{2p} \\right)  \\Big] k~>~A_0 \\frac{\\alpha+\\beta-1}{2}.\n$$\nHence if\n$$\n\\frac{A_0}{2}-A_1 \\left( \\frac{1}{2}-\\frac{1}{2p} \\right)>0,\n$$\nor equivalently\n\\begin{align}\\label{tiaojian2}\n\\frac{\\eta-\\alpha}{\\beta} \\left(1-\\frac{1}{p} \\right)< \\frac{1}{2}-\\frac{1}{p}-\\frac{\\alpha-1}{2 \\beta},\n\\end{align}\nthen \\er{starstar} holds true for\n\\begin{align}\\label{DB}\nk>\\frac{A_0 \\left( \\frac{\\alpha+\\beta-1}{2}-(\\eta-1) \\right) -\\frac{A_1}{2p}(\\alpha+\\beta-1)}{\\frac{A_0}{2}-A_1\\left( \\frac{1}{2}-\\frac{1}{2p} \\right)}=:\\frac{D}{B}.\n\\end{align}\nIn the following, we prove $D>0$. In fact,\n\\begin{align}\nD&=A_0 \\left( \\frac{\\alpha+\\beta-1}{2}-(\\eta-1) \\right) -\\frac{A_1}{2p}(\\alpha+\\beta-1) \\nonumber\\\\\n& > A_0 \\left(\\frac{\\alpha+\\beta-1}{2}-\\frac{p-2}{p-1} \\frac{\\alpha+\\beta-1}{2} \\right)-\\frac{A_1}{2p}(\\alpha+\\beta-1) \\nonumber\\\\\n&= \\frac{A_0}{p-1}\\frac{\\alpha+\\beta-1}{2} -\\frac{A_1}{2p}(\\alpha+\\beta-1) \\nonumber \\\\\n&>0,\n\\end{align}\nwhere we have used \\er{tiaojian2} and its reformulation\n\\begin{align}\\label{tiaojian2bianxing}\n\\eta-1<\\frac{p-2}{2(p-1)} (\\alpha-1+\\beta).\n\\end{align}\nFurthermore, after some calculations we have\n\\begin{align}\n\\frac{D}{B} & = \\frac{\\left(-A_0-\\frac{\\alpha+\\beta-1}{2p\\beta}  \\right) \\eta + \\frac{\\alpha}{p \\beta}\\frac{\\alpha+\\beta-1}{2}+A_0\\frac{\\alpha+\\beta+1}{2} }{ -\\frac{1}{\\beta} \\left( \\frac{1}{2}-\\frac{1}{2p} \\right) \\eta+\\frac{A_0}{2}+\\frac{\\alpha}{\\beta} \\left( \\frac{1}{2}-\\frac{1}{2p} \\right)   }   \\nonumber \\\\[2mm]\n& = \\frac{-\\frac{1}{\\beta}(\\beta-(\\alpha-1)) \\left( \\frac{1}{2}-\\frac{1}{2p} \\right) \\eta+ \\big(\\beta-(\\alpha-1)\\big) \\left(\\frac{A_0}{2}+\\frac{\\alpha}{\\beta} \\left( \\frac{1}{2}-\\frac{1}{2p} \\right) \\right)}{ -\\frac{1}{\\beta} \\left( \\frac{1}{2}-\\frac{1}{2p} \\right) \\eta+\\frac{A_0}{2}+\\frac{\\alpha}{\\beta} \\left( \\frac{1}{2}-\\frac{1}{2p} \\right)   } \\nonumber\\\\\n&=\\beta-(\\alpha-1).\n\\end{align}\n\nNow coming back to \\er{guji2} and \\er{guji3}, we infer from \\er{star} and \\er{starstar} by using Young's inequality that\n\\begin{align}\nC(k,\\alpha)\\|u\\|_{L^{k'}}^{b_\\alpha} & \\le \\left( \\|u\\|_{L^{k+\\alpha-1}}^{k+\\alpha-1} \\|u\\|_{L^\\beta}^\\beta \\right)^{\\frac{b_\\alpha \\theta}{k+\\alpha-1}} \\nonumber\\\\\n& \\le \\frac{k}{4} \\|u\\|_{L^{k+\\alpha-1}(\\R^n)}^{k+\\alpha-1} \\|u\\|_{L^\\beta(\\R^n)}^\\beta+C_1(k)\n\\end{align}\nand\n\\begin{align}\nC(k,\\eta) \\|u\\|_{L^{k'}}^{b_\\eta} & \\le \\left( \\|u\\|_{L^{k+\\alpha-1}}^{k+\\alpha-1} \\|u\\|_{L^\\beta}^\\beta \\right)^{\\frac{b_\\eta \\theta}{k+\\alpha-1}} \\nonumber \\\\\n& \\le \\frac{k}{4} \\|u\\|_{L^{k+\\alpha-1}(\\R^n)}^{k+\\alpha-1} \\|u\\|_{L^\\beta(\\R^n)}^\\beta+C_2(k).\n\\end{align}\n\nTherefore, substituting the above arguments into \\er{guji1} we thus end up with for all $0<t<T_{\\max}$\n\\begin{align}\\label{guji}\n\\frac{d}{dt}\\int_{\\R^n} u^k dx+\\frac{k}{2} \\int_{\\R^n} u^\\beta dx \\int_{\\R^n} u^{k+\\alpha-1} dx+\\frac{2(k-1)}{k} \\|\\nabla u^{\\frac{k}{2}} \\|_{L^2(\\R^n)}^2 \\le C(k)\n\\end{align}\nfor $k>\\beta-(\\alpha-1)$ from \\er{kxin}, \\er{DB} and \\er{tiaojian1}, \\er{tiaojian2}. Precisely, for $\\eta>\\alpha$, \\er{tiaojian2} is equivalent to\n\\begin{align}\n\\frac{2(\\eta-1)}{p-2}(p-1)<\\beta+\\alpha-1,\n\\end{align}\nand then $k>\\max\\{K_0,\\beta-(\\alpha-1)\\}=\\beta-(\\alpha-1).$ Otherwise for $\\eta \\le \\alpha,$ by virtue of \\er{tiaojian1} it leads to\n$k>\\max\\{ K_0, \\beta-(\\alpha-1) \\}=\\beta-(\\alpha-1)$. \\\\\n\n\n\\noindent\\textbf{Step 2 ($L^k(\\R^n)$ estimates for $\\beta+\\alpha-1 \\le k \\le \\infty$).} Firstly we note that for $\\eta \\le \\alpha$, then $\\beta+\\alpha-1>\\frac{2(\\alpha-1)}{p-2}$ with the help of \\er{tiaojian1}, therefore we can take $k=\\beta+\\alpha-1$ in \\er{guji} and by H\\\"{o}lder inequality and Young's inequality on has\n\\begin{align}\n\\|u\\|_{L^{\\beta+\\alpha-1}(\\R^n)}^{\\beta+\\alpha-1} & \\le \\left( \\|u\\|_{L^{\\beta+2(\\alpha-1)}(\\R^n)}^{\\beta+2(\\alpha-1)} \\|u\\|_{L^\\beta(\\R^n)}^\\beta \\right)^{1\/2} \\nonumber \\\\\n& \\le \\frac{\\beta+\\alpha-1}{2} \\|u\\|_{L^{\\beta+2(\\alpha-1)}(\\R^n)}^{\\beta+2(\\alpha-1)} \\|u\\|_{L^\\beta(\\R^n)}^\\beta +\\frac{1}{2(\\beta+\\alpha-1)},\n\\end{align}\nplugging the above formula into \\er{guji} we have\n\\begin{align}\n\\frac{d}{dt} \\int_{\\R^n} u^{\\beta+\\alpha-1} dx+ \\int_{\\R^n} u^{\\beta+\\alpha-1} \\le C(\\beta+\\alpha-1)\n\\end{align}\nwhich follows the uniformly boundedness in time\n\\begin{align}\n\\int_{\\R^n} u^{\\beta+\\alpha-1} dx \\le \\|u_0\\|_{L^{\\beta+\\alpha-1}(\\R^n)}^{\\beta+\\alpha-1} +C(\\alpha,\\beta).\n\\end{align}\nOn the other hand, letting\n\\begin{align}\nv=u^{k\/2},~q=2,~1\\le r<2,~C_0=\\frac{2(k-1)}{k}\n\\end{align}\nin Lemma \\ref{interpolation} one has that for $n \\ge 1$\n\\begin{align}\\label{336}\n\\int_{\\R^n} u^k dx \\le \\frac{2(k-1)}{k} \\| \\nabla u^{\\frac{k}{2}} \\|_{L^2(\\R^n)}^2 +C(n,k) \\|u\\|_{L^{k_1}(\\R^n)}^k\n\\end{align}\nwhere $k_1=\\frac{kr}{2} <k$. Furthermore, for $\\beta<k_1<k+\\alpha-1$ we can take $k_1=\\frac{\\beta+\\alpha-1+k}{2}<k$ which is\n\\begin{align}\nk>\\beta+\\alpha-1\n\\end{align}\nso that\n\\begin{align}\\label{339}\n\\|u\\|_{L^{\\frac{\\beta+\\alpha-1+k}{2}}(\\R^n)}^k \\le \\left( \\|u\\|_{L^{k+\\alpha-1}(\\R^n)}^{k+\\alpha-1} \\|u\\|_{L^\\beta(\\R^n)}^\\beta \\right)^{\\frac{k}{\\beta+\\alpha-1+k}}.\n\\end{align}\nCombining \\er{336} and \\er{339} together yields\n\\begin{align}\\label{340}\n\\int_{\\R^n} u^k dx &\\le \\frac{2(k-1)}{k} \\| \\nabla u^{\\frac{k}{2}} \\|_{L^2(\\R^n)}^2+C(n,k) \\left( \\|u\\|_{L^{k+\\alpha-1}(\\R^n)}^{k+\\alpha-1} \\|u\\|_{L^\\beta(\\R^n)}^\\beta \\right)^{\\frac{k}{\\beta+\\alpha-1+k}} \\nonumber \\\\\n& \\le \\frac{2(k-1)}{k} \\| \\nabla u^{\\frac{k}{2}} \\|_{L^2(\\R^n)}^2+\\frac{k}{2}  \\|u\\|_{L^{k+\\alpha-1}(\\R^n)}^{k+\\alpha-1} \\|u\\|_{L^\\beta(\\R^n)}^\\beta +C(n,k).\n\\end{align}\nSubstituting \\er{340} into \\er{guji} one has\n\\begin{align}\n\\frac{d}{dt} \\int_{\\R^n} u^k dx + \\int_{\\R^n} u^k dx \\le C(n,k).\n\\end{align}\nIt can obtained that for any $\\beta+\\alpha-1<k<\\infty$\n\\begin{align}\n\\int_{\\R^n} u^k dx \\le C(\\|u_0\\|_{L^k(\\R^n)},k).\n\\end{align}\n\nFurthermore, for the $L^\\infty$ norm, according to \\er{baseguji},\nwe can conduct similar procedures as Step 4 in \\cite{BL16} in terms of $\\int_{\\R^n} u^{k+\\alpha-1} dx$ and $\\int_{\\R^n} u^{k+\\eta-1} dx$ together on the right hand side of \\er{baseguji} and get\n\\begin{align}\\label{inf}\n\\|u\\|_{L^\\infty(\\R^n)} \\le C(\\alpha,\\eta,\\|u_0\\|_{L^\\beta(\\R^n)},\\|u_0\\|_{L^\\infty(\\R^n)}).\n\\end{align}\n\\\\\n\n\\noindent\\textbf{Step 3 ($L^\\beta$ estimates).}\nWhen $\\sigma+1=\\alpha,$ letting $k=\\beta$ in \\er{baseguji} one has\n\\begin{align}\n\\frac{d}{dt}\\int_{\\R^n} u^\\beta dx \\le \\beta \\int_{\\R^n} u^{\\beta+\\alpha-1} dx \\left(1+\\frac{\\beta-1}{\\beta+\\sigma-1}-\\int_{\\R^n} u^\\beta dx \\right),\n\\end{align}\nit follows that for all $0<t<T_{\\max}$\n\\begin{align}\n\\int_{\\R^n} u(\\cdot,t)^\\beta dx \\le \\max \\left(\\int_{\\R^n}u_0^\\beta dx,1+\\frac{\\beta-1}{\\beta+\\sigma-1}\\right).\n\\end{align}\n\\\\\n\n\\noindent\\textbf{Step 4 ($L^k$ estimates for $1 \\le k<\\beta+\\alpha-1$).} By virtue of \\er{guji}, we have that for any $0<t<T_{\\max}$ and $\\beta-(\\alpha-1)<k<\\beta+\\alpha-1$,\n\\begin{align}\\label{beal}\n\\int_{\\R^n} u(\\cdot,t)^k dx\\le C(k)T_{\\max}+\\int_{\\R^n} u_0^k dx.\n\\end{align}\nFurthermore, integrating \\er{star00} over $\\R^n$ and using \\er{inf} and \\er{beal} get\n\\begin{align}\n\\frac{d}{dt} \\int_{\\R^n} u dx &= \\int_{\\R^n} u^\\alpha dx \\left( 1-\\int_{\\R^n} u^\\beta dx  \\right) \\nonumber \\\\\n& \\le \\|u\\|_{L^\\infty(\\R^n)}^{\\alpha-1}\\left(1+\\|u\\|_{L^\\beta(\\R^n)}\\right) \\int_{\\R^n} u dx \\nonumber \\\\\n& \\le C\\left(\\|u_0\\|_{L^\\infty(\\R^n)},\\|u_0\\|_{L^\\beta(\\R^n)},T_{\\max} \\right) \\int_{\\R^n} u dx,\n\\end{align}\nthus for any $0<t<T_{\\max}$\n\\begin{align}\n\\|u(\\cdot,t)\\|_{L^1(\\R^n)}\\le e^{C\\left(\\|u_0\\|_{L^\\infty(\\R^n)},\\|u_0\\|_{L^\\beta(\\R^n)},T_{\\max} \\right) T_{\\max}} \\|u_0\\|_{L^1(\\R^n)}.\n\\end{align}\nConsequently we conclude that for $1\\le k<\\beta+\\alpha-1$ and any $0<t<T_{\\max}$\n\\begin{align}\n\\|u(\\cdot,t)\\|_{L^k(\\R^n)} \\le C(\\|u_0\\|_{L^k(\\R^n)},\\beta,\\alpha,T_{\\max}).\n\\end{align}\nThis completes the a priori estimates. $\\Box$ \\\\\n\nProposition \\ref{pro25} together with the blow-up criterion \\er{blowup} allow us to state without further arguments\n\\noindent{\\bf Proof of Theorem \\ref{thm1}.} With the aid of the blow-up criterion \\er{blowup} and the uniformly boundedness of the solution in Proposition \\ref{pro25}, there exists a positive constant $C\\left(\\|u_0\\|_{L^{\\beta+\\alpha-1}(\\R^n)},\\|u_0\\|_{L^\\infty(\\R^n)}\\right)$ such that\n\\begin{align}\n\\|u(\\cdot,t)\\|_{L^\\infty(\\R^n)} \\le C~~\\mbox{ for all } 0<t<\\infty.\n\\end{align}\nBy Proposition \\ref{pro23} we obtain the desired result. The proof of Theorem \\ref{thm1} is completed. $\\Box$\n\n\\section{Conclusions}\\label{sec5}\n\\def\\theequation{5.\\arabic{equation}}\\makeatother\n\\setcounter{equation}{0}\n\\def\\thetheorem{5.\\arabic{thm}}\\makeatother\n\\setcounter{thm}{0}\n\nThis paper concerns Eq. \\er{star00} in terms of different reaction and aggregation exponents. If $\\alpha \\ge \\sigma+1,$ then the growth in reaction dominates and letting $\\alpha<1+2\\beta\/n$ in the death $u^\\alpha \\int_{\\R^n} u^\\beta dx$ can prevent blow-up. While for $\\alpha < \\sigma+1,$ the aggregation dominates and let the aggregation exponent $\\sigma+1<(2\\alpha+2\\beta+n)\/(n+2)$ thus the solution will exist globally. Moreover, if $\\sigma+1=\\alpha,~n(\\alpha-1)\/\\beta=2,$ then $u_\\lambda(x,t)=\\lambda^{\\frac{n}{\\beta}}u(\\lambda x, \\lambda^2 t)$ is also a solution of \\er{star00} and the scaling preserves the $L^\\beta$ norm in space. When $\\frac{n}{\\beta}(\\alpha-1)<2,$ for low density (small $\\lambda$), the aggregation dominates the diffusion thus prevents spreading. While for high density (large $\\lambda$), the diffusion dominates the aggregation and thus blow-up is precluded. Hence, in this case, the solution will exist globally (Theorem \\ref{thm1}) and we believe that the solution converges to the stationary solution as time goes to infinity. On the contrary, both global existence and finite time blow-up may occur for $\\frac{n}{\\beta}(\\alpha-1)>2,$ hence our conjecture is that there exists finite time blow-up for $\\alpha-1>2\\beta\/n.$ As to the case $\\alpha-1=2\\beta\/n$, similar to \\cite{Jose2009}, whether there is a critical value for the initial data sharply separating global existence and finite time blow-up is also unknown. Our result is to be considered as the first step to a more general theory of chemotaxis system with nonlocal nonlinear reaction. This will be a fertile area to explore.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\t\n\t\\noindent In this paper, we address the universality of the determinant of a class of random Hermitian matrices.\n\tBefore discussing results specific to this symmetry assumption, we give a brief history of results in the non-Hermitian setting. In both settings, a priori bounds preceded estimates on moments of determinants, and the distribution of\n\tdeterminants for integrable models of random matrices. The universality of such determinants has then been the \n\tsubject of recent active research.\t\n\t\n\t\n\t\\subsection{Non-Hermitian matrices.}\\ \n\tEarly papers on this topic treat non-Hermitian matrices with independent and identically distributed entries.\n\t\tMore specifically, Szekeres and Tur\\'an first studied an extremal problem on  the determinant of $\\pm 1$  matrices \\cite{SzeTur1937}. \n\tIn the 1950s, a series of papers \\cite{For1951,ForTuk1952, NyqRicRio1954, Tur1955, Pre1967} calculated \n\tmoments of the determinant of random matrices of fixed size (see also \\cite{Gir1980}). \n\tIn general, explicit formulae are unavailable for high order moments of the determinant except\n\twhen the entries of the matrix have particular distribution (see, for example, \\cite{Dem1989} and the references therein).\n\tEstimates for the moments and the Chebyshev inequality give upper bounds\n\ton the magnitude of the determinant. \\\\\n\t\n\t\\noindent Along a different line of research,\n\tfor an $N\\times N$ non-Hermitian random matrix $A$,\n\tErd\\H{o}s asked whether $\\det A$ is non-zero with probability\n\ttending to one as $N$ tends to infinity.  In \\cite{Kom1967, Kom1968}, Kolm\\'{o}s proved\n\tthat for random matrices with Bernoulli entries,\n\tindeed $ \\det A \\neq 0$ with probability converging to 1 with $N$. \n\tIn fact, this method works for more general models, and following \\cite{Kom1967},\n\t\\cite{KahKomSze1995, TaoVu2006, TaoVu2007, BouVuWoo2010} give improved, exponentially small bounds on the probability that $\\det A = 0$. \\\\\n\t\n\n\t\\noindent \n\tIn \\cite{TaoVu2006}, the authors made the first steps towards quantifying the typical size of $\\left| \\det A \\right|$,\n\tproving that for Bernoulli random matrices, with probability\n\ttending to 1 as $N$ tends to infinity,\n\t\t\\begin{equation}\n\t\t\\label{tao eq 1}\n\t\t\\sqrt{N!}\\exp\\left(-c \\sqrt{N\\log N}\\right) \\leq \\left| \\det A\\right| \\leq \\omega(N)\\sqrt{N!}, \n\t\t\\end{equation}\n\tfor any function $\\omega(N)$ tending to infinity with $N$. In particular, with overwhelming probability\n\t\t\\[ \\log \\left| \\det A \\right| = \\left( \\frac{1}{2} + \\oo(1) \\right) N\\log N. \\]\n\t\n\t\\noindent In \\cite{Goo1963}, Goodman considered $A$ with independent standard real Gaussian entries.\n\tIn this case, he was able to express $\\left| \\det A \\right|^2$ as the product of independent\n\tchi-square variables. This enables one to identify the asymptotic distribution\n\tof $\\log\\left| \\det A \\right|$. Indeed, one can prove that\n\t\t\\begin{equation}\n\t\t\\label{GOE result}\n\t\t \\frac{\\log \\left| \\det A \\right|  - \\frac{1}{2}\\log N! + \\frac{1}{2}\\log N}{\\sqrt{\\frac{1}{2}\\log N}} \n\t\t\t\\to \\mathscr{N}(0, 1),\n\t\t\\end{equation}\n\t(see \\cite{RemWes2005}). In the case of $A$ with independent complex Gaussian entries, a similar analysis yields \n\t\t\\[ \\frac{\\log \\left| \\det A \\right|  - \\frac{1}{2}\\log N! + \\frac{1}{4}\\log N}{\\sqrt{\\frac{1}{4}\\log N}} \n\t\t\t\\to \\mathscr{N}(0, 1). \\]\n\t\n\t\\noindent \n\tIn \\cite{NguVu2014}, the authors proved (\\ref{GOE result}) holds under just\n\tan exponential decay hypothesis on the entries. Their method yields an explicit  rate of convergence\n\tand extends to handle the complex case. Then in \\cite{BaoPanZho2015}, the authors extended \n\t(\\ref{GOE result}) to the case where the matrix entries only require bounded fourth moment.\\\\\n\t\n\t\\noindent The analysis of determinants of non-Hermitian random matrices relies crucially on the\n\tassumption that the rows of the random matrix are independent. The fact that this independence\n\tno longer holds for Hermitian random matrices forces one to look for new methods to prove similar\n\tresults to those of the non-Hermitian case. \n\tNevertheless, the history of this problem mirrors the history of the non-Hermitian case. \n\t\n\t\\subsection{Hermitian matrices.}\\ In the 1980s, Weiss posed the Hermitian analogs of \\cite{Kom1967, Kom1968} as an open problem.\n\tThis problem was solved, many years later in \\cite{CosTaoVu2006}, and \n\tthen in \\cite[Theorem 34]{TaoVu2011} the authors proved the Hermitian analog of \n\t(\\ref{tao eq 1}). This left open the question of describing the limiting\n\tdistribution of the determinant. \\\\\t\n\t\n\t\\noindent In \\cite{DelLeC2000}, Delannay and Le Ca\\\"{e}r used\n\tthe explicit formula for the joint distribution of the eigenvalues to prove\n\tthat for $H$ an $N \\times N$ matrix drawn from the GUE,\n\t\t\\begin{equation}\n\t\t\\label{GUE CLT}\n\t\t \\frac{\\log \\left| \\det H \\right|  - \\frac{1}{2}\\log N! + \\frac{1}{4}\\log N}{\\sqrt{\\frac{1}{2}\\log N}} \n\t\t\t\\to \\mathscr{N}(0, 1). \n\t\t\\end{equation}\n\tAnalogously, one has\n\t\t\\begin{equation}\n\t\t\\label{GOE CLT}\n\t\t\\frac{\\log \\left| \\det H \\right|  - \\frac{1}{2}\\log N! + \\frac{1}{4}\\log N}{\\sqrt{\\log N}}\n\t\t\t\\to \\mathscr{N}(0, 1)\n\t\t\\end{equation}\n\twhen $H$ is drawn from the GOE.\n\tProofs of these central limit theorems also appear in \\cite{TaoVu2012, CaiLiaZho2015, BorLaC2015,\n\tEdeLaC2015}. For related results concerning other models of random matrices, see \\cite{Rou2007} and the references therein.\\\\\n\n\t\n\t\\noindent While the authors of \\cite{TaoVu2012} give their own proof of (\\ref{GUE CLT}) and (\\ref{GOE CLT}), their main interest is\n\tto establish such a result in the more general setting of Wigner matrices. \n\tIndeed, they show that in (\\ref{GOE CLT}), we may replace $H$ by $W$, a Wigner matrix whose entries' first\n\tfour moments match those of $\\mathscr{N}(0,1)$.\n\tThey also prove the analogous result in the complex case.\n\tIn this paper, we will relax this four moment matching assumption to a \n\ttwo moment matching assumption (see Theorem \\ref{main theorem}). \\\\\n\t\n\t\\noindent Finally, we mention that new interest in averages of determinants of random (Hermitian) matrices \n\thas emerged from the study of complexity of high-dimensional landscapes \\cite{FyoWil2007,AufBenCer2013}.\n\t\n\t\n\t\\subsection{Statement of results: The determinant. }\nThis subsection gives our main result and suggests extensions in connection with the general class of log-correlated random fields. Our theorems apply to Wigner matrices as defined below. \n\t\n\n\t\n\t\\begin{definition}\n\t\\label{def wigner}\n        \tA complex Wigner matrix, $W = \\left(w_{ij}\\right)$, is an $N\\times N$ \n\tHermitian matrix with entries  \n\t\t\\[ W_{ii} = \\sqrt{\\frac{1}{N}}\\,x_{ii},\\, i = 1, \\dots, N, \\quad W_{ij} = \\frac{1}{\\sqrt{2N}} \\left(x_{ij} + {\\rm i} y_{ij}\\right), \\, \n\t\t1 \\leq i < j \\leq N.\\]\n\tHere \n\t$\\{x_{ii}\\}_{1\\leq i \\leq N}$, $\\{x_{ij}\\}_{1 \\leq i < j \\leq N}$, $\\{y_{ij}\\}_{1 \\leq i <  j \\leq N}$ are \n\tindependent identically distributed random variables satisfying\n\t\t$\n\t\t\\mathbb{E} \\left(x_{ij}\\right) = 0, \\mathbb{E}\\left(x_{ij}^2\\right) = \\mathbb{E}\\left(y_{ij}^2\\right) = 1. \n\t\t$\n\tWe assume further \n        \tthat the common distribution $\\nu$ of $\\{x_{ii}\\}_{1\\leq i \\leq N}$, $\\{x_{ij}\\}_{1 \\leq i < j \\leq N}$, $\\{y_{ij}\\}_{1 \\leq i <  j \\leq N}$, has subgaussian decay, i.e.\\ there exists \n        \t$\\delta_0 > 0$ such that\n        \t\t\\begin{equation}\n\t\t\\label{subgaussian} \n\t\t\\int_\\mathbb{R} e^{\\delta_0 x^2}{\\rm d}\\nu(x) < \\infty. \n\t\t\\end{equation}\n\tIn particular, this means that all the moments of the entries of the matrix are bounded.\n\tIn the special case $\\nu = \\mathscr{N}(0,1)$, \n\t$W$ is said to be drawn from the Gaussian Unitary Ensemble (GUE). \n\t\n\tSimilarly, we define a real Wigner matrix to have entries of the form\n\t\t$ W_{ii} = \\sqrt{\\frac{2}{N}} x_{ii}$, $W_{ij} = \\sqrt{\\frac{1}{N}} x_{ij}$, \n\twhere $\\left\\{x_{ij}\\right\\}_{1\\leq i, j \\leq N}$ are independent identically distributed random variables satisfying\n\t\t$ \\mathbb{E}\\left(x_{ij}\\right) = 0, \\mathbb{E}\\left(x^2_{ij}\\right) = 1.$\n\tAs in the complex case, we assume\n\tthe common distribution $\\nu$ satisfies (\\ref{subgaussian}).\n\tIn the special case $\\nu = \\mathscr{N}(0,1)$,\n\t$W$ is said to be drawn from the Gaussian Orthogonal Ensemble (GOE).\n\t\\end{definition}\n\t\t\t\n\t\\noindent Our main result extends (\\ref{GUE CLT}) and  (\\ref{GOE CLT}) to the above class of Wigner matrices.  In particular,\n\tthis answers a  conjecture from \\cite[Section 8]{TaoVu2014}, which asserts that the central limit theorem \n\t(\\ref{GOE CLT})\n\tholds for Bernoulli ($\\pm 1$) matrices. Note that in the following statement, our centering\n\tdiffers from (\\ref{GUE CLT}) and (\\ref{GOE CLT}) because we normalize our matrix entries\n\tto have variance of size $N^{-1}$. \n\t\t\t\n\n\t\\begin{theorem}\\label{main theorem}\n\t\tLet $W$ be a real Wigner matrix satisfying (\\ref{subgaussian}). Then\n\t\t\\begin{equation}\\label{main}\n\t\t\\frac{\\log\\left| \\det W\\right| + \\frac{N}{2} }{\\sqrt{\\log N}} \\to \\mathscr{N}(0, 1).  \n\t\t\\end{equation}\n\t\tIf $W$ is a complex Wigner matrix satisfying (\\ref{subgaussian}), then\n\t\t\\begin{equation}\n\t\t\t\\frac{\\log\\left| \\det W \\right| + \\frac{N}{2}}{\\sqrt{\\frac{1}{2}\\log N}} \n\t\t\t\\to \\mathscr{N}(0, 1). \n\t\t\\end{equation}\n\t\\end{theorem}\n\t\n\t \\noindent Assumption (\\ref{subgaussian}) may probably  be relaxed to a finite moment assumption, but we will not pursue this direction here. Similarly, it is likely that the matrix entries do not need to be identically distributed;\n\t only the first two moments need to match. However we consider the case of a \n\t unique $\\nu$ in this paper. \n\t\n\t\n\t\\begin{remark}\\label{rem:multidim}\n\tLet $H$ be drawn from the GUE normalized so that in the limit as $N\\to\\infty$, the distribution\n\tof its eigenvalues is supported on $[-1,1]$, and let\n\t\t\\[D_N(x) = -\\log\\left| \\det\\left( H - x\\right) \\right|. \\]\n\tIn \\cite{Kra2007}, Krasovsky proved that  for $x_k \\in (-1,1)$, $k =1, \\hdots, m$, \n\t$x_j \\neq x_k$, uniformly in $\\Re \\left(\\alpha_k\\right) > - \\frac{1}{2}$, $\\mathbb{E}\\left( e^{-\\sum_{k=1}^m \\alpha_k D_N\\left(x_k\\right)} \\right)$ is asymptotic to\n\t\t\\begin{align}\\label{krasovsky}\n\t\t\t\\prod_{k=1}^m \\left( C\\left(\\frac{\\alpha_k}{2}\\right) \\left(1-x_k^2\\right)^{\\frac{\\alpha_k^2}{8}} \n\t\t\t N^{\\frac{\\alpha_k^2}{4}} e^{\\frac{\\alpha_k N}{2}\\left( 2x_k^2 -1 - 2\\log 2 \\right)}\\right)\n\t\t\t &\\prod_{1\\leq \\nu < \\mu \\leq m} \\left(2\\left| x_\\nu - x_\\mu \\right| \\right)^{-\\frac{\\alpha_\\nu \\alpha_\\mu}{2}}\n\t\t\t \\left(1 + {\\rm O}\\left( \\frac{\\log N}{N} \\right) \\right),\n\t\t\\end{align}\n\tas $N\\to\\infty$. Here $C(\\cdot)$ is the Barnes function.\n\tSince the above estimate holds uniformly for $\\Re \\left(\\alpha_k\\right) > -\\frac{1}{2}$,  \n\t(\\ref{krasovsky}) shows that\n\tletting\n\t\t\\[\t\n\t\t\t\\widetilde D_N(x) = \\frac{D_N(x) - N\\left( x^2 - \\frac{1}{2} - \\log 2 \\right)}{\\sqrt{\\frac{1}{2} \\log N}},\n\t\t\\]\n\tthe vector $\\left( \\widetilde D_N\\left(x_1\\right), \\hdots, \\widetilde D_N\\left(x_m\\right) \\right)$ converges in distribution \n\tto a collection of $m$ independent standard Gaussians. Our proof of Theorem\n\t\\ref{main theorem} automatically extends this result to Hermitian Wigner matrices as defined above.\n\tIf one were to prove an analogous convergence for the GOE, our proof of Theorem\n\t\\ref{main theorem} would extend the result to real symmetric Wigner matrices as well. \n\t\\end{remark}\n\t\n\t\n\\begin{remark} \\label{rem:multidim2}\n\tWe note that (\\ref{krasovsky}) was proved for fixed, distinct $x_k$'s. If  (\\ref{krasovsky}) holds for collapsing\n\t$x_k$'s, this means that fluctuations of the log-characteristic polynomial of the GUE  become log-correlated for large dimension, \n\tas in the case of the Circular Unitary Ensemble \\cite{BourgadeZeta}. More specifically, let\n\t$\\widetilde D_N(\\cdot)$ be as above, and let $\\Delta$ denote the distance between two points $x, y$ in\n\t$(-1, 1)$. For $\\Delta \\geq 1\/N$, we expect the covariance between $\\widetilde D_N(x)$ and $\\widetilde D_N(y)$\n\tto behave like $\\frac{\\log\\left(1\/\\Delta\\right)}{\\log N}$, and for $\\Delta \\leq 1\/N$, we expect it to\n\tconverge to 1.\n\\end{remark}\n\n\n\\noindent \n\tOur method  automatically\n\t establishes the content of Remark \\ref{rem:multidim2} for Wigner matrices,\n\t conditional on the knowledge of GOE and GUE cases. \n\t The exact statement is as follows, and we omit the proof, strictly similar to Theorem \\ref{main theorem}. Denote\n\t \t$$\n\t \tL_N(z)=\\log|\\det(W-z)|-N\\int_{-2}^2\\log|x-z|\\,{\\rm d} \\rho_{\\rm sc}(x).\n\t \t$$\n\t \t\n\t \t\n\\begin{theorem}\\label{multldim}\n\t\tLet $W$ be a real Wigner matrix satisfying (\\ref{subgaussian}). Let \n\t\t$\\ell\\geq 1$, $\\kappa>0$ and let $E^{(1)}_N)_{N\\geq 1},\\dots$, $(E^{(\\ell)}_N)_{N\\geq 1}$\n\t\tbe energy levels in $[-2+\\kappa,2-\\kappa]$.\n\t\tAssume that for all $i\\neq j$, for some constants $c_{ij}$ we have\n$$\n\\frac{\\log |E_N^{(i)}-E_N^{(j)}|}{-\\log N}\\to c_{ij}\\in[0,\\infty]\n$$\t\t\nas $N\\to\\infty$. Then\n\\begin{equation}\\label{eqn:logcor}\n\\frac{1}{\\sqrt{\\frac{1}{2}\\log N}}\\left(\nL_N\\left( \\left(E^{(1)}_N\\right) \\right),\\dots,\nL_N\\left(\\left(E^{(\\ell)}_N\\right)\\right)\n\\right)\n\\end{equation}\nconverges in distribution to a Gaussian vector with covariance $ (\\min(1,c_{ij}))_{1\\leq i,j\\leq N}$ (with diagonal $1$ by convention), provided the same result holds for GOE. \\\\\n\n\\noindent The same result holds for Hermitian Wigner matrices, assuming it is true in the GUE case, up to a change in the normalization from $\\sqrt{\\frac{1}{2}\\log N}$ to $\\sqrt{\\log N}$ in (\\ref{eqn:logcor}).\n\\end{theorem}\t \t\n\t \t\n\t \t\\noindent Theorem \\ref{multldim} says $L_N$ converges to a log-correlated field, provided this result holds for the Gaussian ensembles. It therefore suggests that the universal limiting behavior of extrema and convergence to Gaussian multiplicative chaos conjectured for unitary matrices in \\cite{FyoHiaKea12} extends to the class of Wigner matrices. \nTowards these conjectures, \\cite{ArgBelBou15,ChaMadNaj16,PaqZei17,FyoSim2015,LamPaq16} proved asymptotics on the maximum of characteristic polynomials of circular unitary and invariant ensembles, and \\cite{BerWebWon2017,NikSakWeb2018,Web2015} established convergence to the Gaussian multiplicative chaos, for the same models.\nWe refer to  \\cite{Arg16} for a survey on log-correlated fields and their connections with random matrices, branching processes, the Gaussian free field, and analytic number theory. \n\n\n\n\n\\subsection{Statement of results: Fluctuations of Individual Eigenvalues. }\t\nWith minor modifications, the proof of Theorem \\ref{main theorem} also extends the results of \\cite{Gus2005} and \\cite{ORo2010} \nwhich describe the fluctuations of individual eigenvalues in the GUE and GOE cases, respectively. \nBy adapting the method of \\cite{TaoVu2011}, \\cite{ORo2010} proves the following theorem under\nthe assumption that the first four moments of the matrix entries match those of a standard Gaussian.\nIn Appendix B, we show that the individual eigenvalue fluctuations of the GOE (GUE) also hold for real (complex) Wigner matrices in the sense of Definition \\ref{def wigner}.\nIn particular, the fluctuations of eigenvalues of Bernoulli matrices are Gaussian in the large dimension limit, which answers a question from \\cite{TaoVu2014}. \\\\\n\n\n\\noindent To state the following theorem, we follow the notation of Gustavsson \\cite{Gus2005} and write \n$k(N) \\sim N^\\theta$ to mean that $k(N) = h(N)N^\\theta$ where $h$ is a function such that for \nall ${\\varepsilon} > 0$, for large enough $N$,\n\t\\begin{equation}\n\t\\label{gustavsson sim}\n\tN^{-{\\varepsilon}}\\leq h(N)\\leq N^{\\varepsilon}.\n\t\\end{equation}\n\tIn the following, $\\gamma_k$ denotes the $k$\\textsuperscript{th} quantile of the semicircle law,\n\t\t\t\\begin{equation}\\label{quantiles}\n\t\t\t \\frac{1}{2\\pi}\\int_{-2}^{\\gamma_k} \\sqrt{\\left(4 - x^2 \\right)_+}{\\rm d}x = \\frac{k}{N}.\n\t\t\t\\end{equation}\n\n\\begin{theorem}\n\\label{fluctuations of individual eigenvalues}\n\tLet $W$ be a Wigner matrix satisfying (\\ref{subgaussian}) with eigenvalues $\\lambda_1 < \\lambda_2 <\n\t\\dots < \\lambda_N$. Consider $\\left\\{ \\lambda_{k_i} \\right\\}_{i=1}^m$ such that  $0 < k_{i+1} - k_{i} \\sim N^{\\theta_i},\n\t0 < \\theta_i \\leq 1$, and $k_i \/N \\to a_i \\in (0,1)$ as $N \\to \\infty$.\n\tLet\n\t\\begin{equation}\n\t\\label{def: X_i}\n\t\tX_i = \\frac{\\lambda_{k_i} - \\gamma_{k_i}}{\\sqrt{ \\frac{4\\log N}{\\beta \\left(4 - \\gamma_{k_i}^2 \\right) N^2}  }},\n\t\t\\quad i = 1, \\dots, m,\n\t\\end{equation}\n\twith $\\beta =1$ for real Wigner matrices, and $\\beta = 2$ for complex Wigner matrices.\n\tThen as $N \\to \\infty$,\n\t\\[\n\t\t\\P\\left\\{ X_1 \\leq \\xi_1, \\dots, X_m \\leq \\xi_m \\right\\} \\to \\Phi_{\\Lambda}\\left(\\xi_1, \\dots, \\xi_m \\right),\n\t\\]\n\twhere $\\Phi_\\Lambda$ is the cumulative distribution function for the $m$-dimensional normal distribution with\n\tcovariance matrix $\\Lambda_{i,j} = 1 - \\max\\left\\{ \\theta_k: i \\leq k < j < m \\right\\}$ if \n\t$i < j$, and $\\Lambda_{i,i} = 1$.\n\\end{theorem}\n\n\t\n\\noindent The above theorem  has been known to follow from the  homogenization result in \\cite{BouErdYauYin2016} (this technique gives a simple expression for the relative {\\it individual} positions of coupled eigenvalues from GOE and Wigner matrices) and fluctuations of mesoscopic linear statistics; see \\cite{LanSos2018} for a proof of eigenvalue fluctuations for Wigner and invariant ensembles.  However, the technique from  \\cite{BouErdYauYin2016} is not enough for Theorem \\ref{main theorem}, as the determinant depends on the positions of {\\it all} eigenvalues.\n\t\n\t\n\t\\subsection{Outline of the proof. }\\  In this section, we give the main steps of\n\tthe proof of Theorem  \\ref{main theorem}. Our outline discusses the real case, but\n\tthe complex case follows the same scheme. \\\\\n\t\n\t\\noindent The main conceptual idea of the proof follows the three step strategy of \n\t\\cite{ErdPecRmSchYau2010,ErdSchYau2011}. With a priori localization of eigenvalues (step one, \\cite{ErdYauYin2012Rig, CacMalSch2015}), one can prove that the determinant has universal fluctuations after a adding a small Gaussian noise (this second step relies on a stochastic advection equation from \\cite{BourgadeExtreme}). The third step proves by a density argument that the Gaussian noise does not change the distribution of the determinant, thanks to a perturbative moment matching argument as in \\cite{TaoVu2011,ErdYauYin2012Bulk}. We include Figure \\ref{diagram of implications} below to \n\thelp summarize\n\tthe argument.\\\\\n\t\n\t\n\n\t\\noindent {\\it First step: small regularization.}\n\tIn Section \\ref{smoothing section}, \n\twith Theorems \\ref{schlein} and \\ref{fixed energy thm}, we reduce the proof of Theorem \\ref{main theorem} to showing the convergence\n\\begin{equation}\\label{eqn:probaconv}\n \\frac{\\log\\left| \\det (W+\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta_0)\\right| + c_N}{\\sqrt{\\log N}} \\to \\mathscr{N}(0, 1)\n\t\t\t\t\\end{equation}\n\twith  some explicit deterministic $c_N$, and the small regularization parameter\n\t\t\\begin{equation}\n\t\t\\label{def:eta0}\n\t\t\t\\eta_0 = \\frac{e^{ \\left(\\log N\\right)^{ \\frac{1}{4}  } }}{N}.\n\t\t\\end{equation}\n\t\n\n\t\n\t\n\t\n\t\n\t\n\t\n\\noindent {\\it Second step: universality after coupling.} \nLet $M$ be a symmetric matrix which serves as the initial condition for the\n\tmatrix Dyson's Brownian Motion (DBM) given by \n\t\t\\begin{equation}\n\t\t\t\\label{eq:DBM evolution} \n\t\t{\\rm d} M_t = \\frac{1}{\\sqrt{N}}{\\rm d}B^{(t)} - \\frac{1}{2}M_t{\\rm d}t.\n\t\t\\end{equation}\nHere $B^{(t)}$ is a symmetric $N \\times N$ matrix such that $B^{(t)}_{ij}\\left(i < j\\right)$ and $B_{ii}^{(t)}\/\\sqrt{2}$ are\nindependent standard Brownian motions.\n\tThe above matrix DBM induces \n\ta collection of independent standard Brownian motions (see \\cite{AndGuiZei2010}), $\\tilde{B}^{(k)}_t\/\\sqrt{2}$, $k = 1, \\dots, N$ such that the eigenvalues of $M$ satisfy  the system of stochastic differential equations\n\t\\begin{align}\n\t\t\\label{eqn:DBM}\n\t\t{\\rm d}x_k(t) &= \\frac{{\\rm d}\\tilde{B}_{t}^{(k)}}{\\sqrt{N}} + \\left( \\frac{1}{N} \\sum_{l \\neq k}\n\t\t\t\\frac{1}{x_k(t) - x_l(t)} - \\frac{1}{2} x_k(t)\\right){\\rm d}t \t\n\t\\end{align}\n\twith initial condition given by the eigenvalues of $M$.\n\tIt has been known since \\cite{McK1969} that the system \n\t(\\ref{eqn:DBM}) has a unique strong solution.\n\tWith this in mind, we follow \\cite{BouErdYauYin2016} and introduce the following\n\tcoupling scheme. First, run the matrix DBM taking $\\tilde{W}_0$, a Wigner matrix, as the initial condition. Using\n\tthe induced Brownian motions, run the dynamics given by (\\ref{eqn:DBM}) using the eigenvalues \n\t$y_1 < y_2 < \\dots < y_N$\n\tof $\\tilde{W}_0$ as the initial condition. Call the solution to this system ${\\bm y}(\\tau)$.\n\tUsing the very same (induced) Brownian motions, run\n\tthe dynamics given by (\\ref{eqn:DBM}) again, this time using the eigenvalues of a GOE matrix, ${\\bm x}(0)$,\n\tas the initial condition. Call the solution to this system ${\\bm x}(\\tau)$. \\\\\n\t\n\t\n\t\\noindent Now fix $\\epsilon > 0$ and let \n\t\\begin{equation}\n\t\t\\label{def:tau}\n\t\t\t\\tau = N^{-\\epsilon}.\n\t\t\\end{equation}\n\tUsing Lemma \\ref{extreme}, we show that \n\t\t\\begin{equation}\n\t\t\\label{coupling at tau}\n\t\t \\frac{\\sum_{k = 1}^N \\log \\left|x_k(\\tau) + {\\rm i}\\eta_0\\right| - \n\t\t\t\t\\sum_{k = 1}^N \\log \\left|y_k(\\tau) + {\\rm i}\\eta_0\\right| }{\\sqrt{\\log N}}\n\t\t\\end{equation}\n\tand\n\t\t\\begin{equation} \\label{smoothing t} \\frac{\\sum_{k = 1}^N \\log \\left|x_k(0) + z_\\tau\\right| - \n\t\t\t\t\\sum_{k = 1}^N \\log \\left|y_k(0) + z_\\tau\\right| }{\\sqrt{\\log N}}\n\t\t\\end{equation}\n\tare very close. Here $z_\\tau$ is as in (\\ref{def:ztau}) with $z = {\\rm i} \\eta_0$. \n\tThe  significance of this is that since\n\t$z_\\tau \\sim \\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\tau$, we can use\n\tLemma \\ref{expectation} and well-known central limit theorems which apply to nearly macroscopic scales \n\tto show that (\\ref{smoothing t}) has variance of order ${\\varepsilon}$. Consequently,\n\t(\\ref{coupling at tau}) is also small, and since ${\\bm x}(\\tau)$ is distributed as the eigenvalues of a GOE matrix,\n\twe have proved universality of the regularized determinant after coupling.\\\\\n\t\n\n\n\t\n\t\\noindent {\\it Third step: moment matching}. In Section \\ref{conclusion}, we conclude the proof \n\tof Theorem \\ref{main theorem}. First, we choose\n\t$\\tilde{W}_0$ so that $\\tilde{W}_\\tau$ and $W$ have entries whose first four moments are\n\tclose, as in \\cite{ErdYauYin2012Bulk}. With this approximate moment matching, we use a perturbative argument, as in \\cite{TaoVu2012},\n\tto prove that (\\ref{eqn:probaconv})\n\tholds for $W$ if and only if it holds for $\\tilde{W}_\\tau$. \t\n\tBut as (\\ref{coupling at tau}) is small, this means (\\ref{eqn:probaconv})\n\tholds for $W$ if and only if it holds for a GOE matrix. By (\\ref{GOE CLT}), this concludes the proof.\n\t\n\t\n\n\n\t\n\t\n\t\\begin{wrapfigure}[22]{l}{8cm}\n\t\\vspace{-1.2cm}\n\t\\begin{center}\n\t\\begin{tikzcd}[row sep = 3cm, column sep = 3cm] \n\t\t& W \\\\\n\t\t\\tilde{W}_0 \\arrow{r} \\arrow[rightarrow]{r} [anchor=center,yshift=2ex] \n\t\t\t{\\text{Matrix DBM } {\\rm d} B_{ij}} & \\tilde{W}_\\tau  \\arrow[leftrightarrow]{u} [anchor=center,rotate=-90,yshift=2ex]{\\text{Moment Matching (3)}\n\t\t\\\\ [-85pt]\n\t\t{\\bm y(0)} \\arrow{r} [anchor=center,yshift=-2ex]{\\text{Eigenvalues DBM } {\\rm d} \\tilde{B}_k}& {\\bm y(\\tau)} \\\\ \n\t\t[-30pt]\n\t\t{{\\bm x}(0)} \\arrow[rightarrow]{r}[anchor=center,yshift=2ex]{ \\text{Eigenvalues DBM }{\\rm d} \\tilde{B}_k}& {{\\bm x}(\\tau)}\n\t\t\\arrow{u} [anchor=center,rotate=-90,yshift=2ex]{\\text{Coupling (2)}} \t\n\t\\end{tikzcd} \n\t\\caption{\n       \t\tWe show (\\ref{main}) holds for $\\tilde{W}_\\tau$ if and only\n\t\tif it holds for $W$, and\n\t\twe prove that (\\ref{main}) holds for\n\t\t${\\bm x}(\\tau)$ if and only if (\\ref{main}) holds for $\\tilde{W}_\\tau$. \n\t\tSince ${\\bm x}(\\tau)$ is distributed as the eigenvalues of a GOE matrix, it satisfies\n\t\t(\\ref{GOE CLT}) and we conclude the proof.\n\t\tNote that $\\log| \\det \\tilde{W}_\\tau | = \\sum \\log \\left|y_k(\\tau) \\right|$ pathwise because\n\t\t$B$ induces $\\tilde{B}$. \n    } \\label{diagram of implications}\n\t\\end{center}\n\\end{wrapfigure}\n\n\t\n\t\\subsection{Notation. }\n\tWe shall make frequent use of the notations $s_W$ and $m_{sc}$ in the remainder of this paper. We \n\tstate their definitions here for easy reference.\n\tLet $W$ be a Wigner matrix with eigenvalues $\\lambda_1 < \\lambda_2 < \\dots < \\lambda_N$.\n\tFor $\\Im (z) > 0$, define\n\t\t\\begin{equation}\n\t\t\\label{def:stransform}\n\t\ts_W(z) = \\frac{1}{N} \\sum_{k=1}^N \\frac{1}{\\lambda_k - z},\n\t\t\\end{equation}\n\tthe Stieltjes transform of $W$. \n\tNext, let\n\t\t\\begin{equation}\n\t\t\\label{def:msc}\n\t\t\tm_{sc}(z) =  \\frac{-z + \\sqrt{z^2 -4}}{2},\n\t\t\\end{equation}\n\twhere the square root $\\sqrt{z^2 -4}$ is chosen with the branch cut in $[-2, 2]$ so that $\\sqrt{z^2-4} \\sim z$ \n\tas $z \\to \\infty$. Note that\n\t\t\\begin{equation}\n\t\t\t\\label{self consistent m}\n\t\t\tm_{sc}(z) + \\frac{1}{m_{sc}(z)} + z = 0.\n\t\t\\end{equation} \n\n\\noindent Finally, throughout this paper, unless indicated otherwise,\n\t$C$ ($c$) denotes a large (small) constant independent of all other parameters of the problem. \n\tIt may vary from line to line.\t\n\t\n\t\t\t\t\t\n\\section{Initial Regularization}\\label{smoothing section}\n\n\t\\noindent Let $y_1 < y_2 < \\dots <y_N$ denote \n\tthe eigenvalues of $W$, a real Wigner matrix\n\tsatisfying (\\ref{subgaussian}). \n\tWe first prove \n\twe only need to show Theorem \\ref{main theorem} for a slight regularization of the logarithm.\n\t\n\t\n\t\\begin{proposition}\\label{smoothing}\n\t\tSet\n\t\t\\begin{equation*}\n\t\tg(\\eta) = \\sum_{k} \\left(\\log \\left| y_k+\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta\\right| - \\log \\left| y_k\\right|\\right)-\\int_0^{\\eta}N\\Im \n\t\t\t\\left( m_{sc}(\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i} s)\\right) {\\rm d} s\n\t\t\\end{equation*}\n\t\tand recall\n\t\t$ \\eta_0 =\\frac{e^{ \\left(\\log N\\right)^{ \\frac{1}{4}  } }}{N} $\n\t\tas in (\\ref{def:eta0}).\n\t\tThen we have the convergence in probability\n\t$$\\frac{g(\\eta_0)}{\\sqrt{\\log N}}\\to 0.$$\n\t\\end{proposition}\n\t\\noindent To prove Proposition \\ref{smoothing}, we will use Theorems \\ref{schlein} and \\ref{fixed energy thm}\n\tas input. In \\cite{CacMalSch2015}, Theorem \\ref{schlein} is stated for complex Wigner matrices, however,\n\tthe argument there proves the same statement for real Wigner matrices.\n\t\\begin{theorem}[Theorem 1 in \\cite{CacMalSch2015}]\\label{schlein}\n\t\tLet $W$ be a Wigner matrix and\n\t\tfix $\\tilde{\\eta} > 0$. \n\t\tFor any $\\tilde{E} > 0$, there exist constants $M_0, N_0, C, c, c_0 >0$ such that \n\t\t\t\t\t\\[ \\mathbb{P}\\left( \\left| \\Im \\left(s_W\\left(E + {\\rm i}\\eta\\right)\\right) - \n\t\t\t\t\t\\Im \\left(m_{sc}\\left(E + {\\rm i}\\eta\\right)\\right) \\right| \n\t\t\t\t\t\t\\geq \\frac{K}{N\\eta} \\right) \n\t\t\t\t\t\t\\leq \\frac{\\left(Cq\\right)^{cq^2}}{K^q} \\]\n\t\tfor all $\\eta \\leq \\tilde{\\eta}$, $|E| \\leq \\tilde{E}$, $K > 0$, $N > N_0$ such that $N\\eta > M_0$,\n\t\tand\n\t\t$q \\in \\mathbb{N}$ with $q \\leq c_0\\left(N\\eta\\right)^{\\frac{1}{8}}$.\n\t\\end{theorem}\t\n\t\\begin{remark}\n\t\tIn  \\cite{ErdYauYin2012Rig}, the authors proved that for some positive constant $C_0$, and $N$ large enough,\n\t\t\t\\[ \\left| s_W\\left(E + {\\rm i}\\eta\\right) - m_{sc}\\left(E + {\\rm i}\\eta\\right) \\right| \\leq\n\t\t\t\t\\frac{e^{C_0(\\log\\log N)^2}}{N\\eta} \\] \n\t\tholds with high probability. Though this estimate is weaker than the estimate\n\t\tof Theorem \\ref{schlein}, it holds for a more general model of Wigner matrix in which\n\t\tthe entries of the matrix need not have identical variances. On the other hand, we require the  \n\t\tstronger estimate in Theorem \\ref{schlein} in our proof of Proposition\n\t\t\\ref{smoothing}, and so we restrict ourselves to Wigner matrices as defined\n\t\tin Definition \\ref{def wigner}.\n\t\tThe proof of Lemma \\ref{expectation} also relies on Definition \\ref{def wigner}. \n\t\\end{remark}\n\t\\begin{theorem}[Theorem 2.2 in \\cite{BouErdYauYin2016}]\\label{fixed energy thm}\n\t\tLet $\\rho_1$ denote the first correlation function for the eigenvalues of an $N\\times N$ Wigner matrix,\n\t\tand let $\\rho(x) = \\frac{1}{2\\pi} \\sqrt{\\left( 4 - v^2 \\right)_+}$. \n\t\tThen for any $F: \\mathbb{R} \\to \\mathbb{R}$ continuous and compactly supported, and for any $\\kappa > 0$,\n\t\twe have, \n\t\t\t\\begin{equation}\\label{FEU} \\lim_{N\\to\\infty} \\sup_{E \\in [-2 + \\kappa, 2 - \\kappa]} \\left |\\frac{1}{\\rho(E)} \n\t\t\t\\int F(v)\\rho_1\\left( E + \\frac{v}{N\\rho(E)}\\right){\\rm d}v -\n\t\t\t\\int F(v) \\rho(v)\\, {\\rm d}v \\right| = 0.\\end{equation}\n\t\\end{theorem}\n\t\\begin{remark}\n\t\tIn fact Theorem 2.2 in \\cite{BouErdYauYin2016} makes a much stronger statement, namely it states the\n\t\tanalogous convergence for all correlation functions in the case of generalized Wigner matrices.\n\t\\end{remark}\n\t\n\t\\begin{corollary}\\label{repulsion}\n\t\tFor any small fixed $\\kappa,\\gamma > 0$ there exists $C,N_0>0$ such that for any $N\\geq N_0$ and any interval $I \\subset [-2 + \\kappa, 2 - \\kappa]$ we have\n\t\t\t\\[ \\mathbb{E}\\left( \\left| \\left\\{ y_k: y_k \\in I \\right\\} \\right|\\right)\\leq CN |I| + \\gamma. \\]\n\t\t\t\t\\end{corollary}\n\t\\begin{proof}\n\t\tIn Theorem \\ref{fixed energy thm}, choosing $F$ to be an indicator of an interval of length\n\t\t$1$ gives an expected value ${\\rm O}(1)$.\n\t\tSince the statement of Theorem \\ref{fixed energy thm} holds uniformly in $E$, \n\t\twe may divide the interval $I$ into sub-intervals of length order $1\/N$\n\t\tto conclude. \n\t\\end{proof}\n\n\t\\begin{corollary}\\label{micro fixed energy}\n\t\tLet $E\\in[-2+\\kappa,2-\\kappa]$ be fixed and $I_\\beta = (E -  \\beta\/2, E + \\beta\/2)$ with $\\beta=\\oo(N^{-1})$. Then\n\t\t\t\\[ \\lim_{N\\to\\infty} \\mathbb{P}\\left( \\left| \\left\\{ y_k \\in I_\\beta \\right\\}\\right| = 0 \\right) = 1. \\]\n\t\\end{corollary}\n\t\\begin{proof}\n\t\tLet ${\\varepsilon}$ be any fixed small constant.\n\t\tLet $f$ be fixed, smooth, positive, equal to $1$ on $[-1,1]$ and $0$ on $[-2,2]^c$.\n\t\tThen\n\t\t$$\n\t\t\t\t\\mathbb{P}\\left( \\left| \\left\\{ y_k \\in I_\\beta \\right\\}\\right| \\geq 1 \\right) \\leq\n\t\t\t\t\\mathbb{E} \\left( \\left| \\left\\{ y_k \\in I_\\beta \\right\\}\\right| \\right) \\leq\n\t\t\t\t\\mathbb{E} \\left( \\sum_k f\\left(N(y_k-E)\/{\\varepsilon}\\right) \\right) \\leq 10{\\varepsilon},\n\t\t$$\n\t\twhere the last bound holds for large enough $N$ by Theorem \\ref{fixed energy thm}. \n\t\\end{proof}\n\t\\begin{proof}[Proof of Proposition \\ref{smoothing}]\n\tWe first choose $\\tilde{\\eta}<\\eta_0$ so that we can use Theorem \\ref{schlein} to estimate\n\t\t\\[\\mathbb{E}\\left( \\left| g\\left(\\eta_0\\right) - g\\left(\\tilde{\\eta}\\right) \\right| \\right),\\]\n\tand then take care of the remaining error using Corollaries \\ref{repulsion} and \\ref{micro fixed energy}.\n\tLet\n\t\t\\[ \\tilde{\\eta} = \\frac{d_N}{N}, \\ \\ {\\rm with} \\ \\ d_N = \\left( \\log N \\right)^{\\frac{1}{4}},\\]\n\tand observe that\n\t\t\\begin{equation}\\label{eqn:inter1} \\mathbb{E}\\left( \\left| g\\left(\\eta_0\\right) - g\\left(\\tilde{\\eta}\\right) \\right| \\right)= \n\t\t  \\mathbb{E}\\left( \\left|\\int_{\\tilde{\\eta}}^{\\eta_0} N \\Im \\left( s_W(\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i} t) -m_{sc}(\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i} t)\\right)\n\t\t  \t{\\rm d}t \\right| \\right)\n\t\t  \\leq \n\t\t  \\int_{\\tilde{\\eta}}^{\\eta_0} \\mathbb{E} (N\\left|\\Im \\left( s_{W_1}\\left({\\rm i}t\\right) - \n\t\t  \tm_{sc} ({\\rm i}t) \\right|\\right))\\, {\\rm d}t. \\end{equation}\n\tIn estimating the right hand side above, we will use the notation\n\t\t\\[ \\Delta(t) = \\left|\\Im \\left( s_{W_1}({\\rm i}t) - \n\t\t  \tm_{sc}({\\rm i}t \\right) \\right|. \\]\n\tFor $N$ sufficiently large, by Theorem \\ref{schlein} with $q =2$, we can write the right hand side of (\\ref{eqn:inter1}) as\n\t\t\\begin{multline}\n\t\t\t\\int_{\\tilde{\\eta}}^{\\eta_0} \\int_0^\\infty \\mathbb{P}\\left(N  \\Delta\\left( t \\right) > u\\right){\\rm d}u{\\rm d}t \n\t\t\t= \\int_{\\tilde{\\eta}}^{\\eta_0}\\left( \n\t\t\t\t\\int_0^1 \\mathbb{P}\\left( \\Delta\\left( t \\right) > \\frac{K}{Nt}\\right)\\,\\frac{{\\rm d}K}{t} +\n\t\t\t\t\\int_{1}^\\infty \\mathbb{P}\\left( \\Delta\\left( t \\right) > \\frac{K}{Nt}\\right)\\frac{{\\rm d}K}{t} \\right){\\rm d}t \\\\\n\t\t\t\\leq \\int_{\\tilde{\\eta}}^{\\eta_0} \\left( \\frac{1}{t} + \\int_1^\\infty \\frac{C}{K^2} \\frac{dK}{t} \\right){\\rm d}t\n\t\t\t\\leq\\left(1 + C\\right)\\log\\left( \\frac{\\eta_0}{\\tilde{\\eta}}\\right) \n\t\t\t={\\rm o}\\left( \\sqrt{\\log N} \\right).\\label{est1}\n\t\t\\end{multline}\n\tNext we estimate\n\t $\\sum_{k} \\left(\\log \\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log \\left| y_k\\right|\\right)$, and \n\tthis will give us a bound for $\\mathbb{E}\\left(\\left| g(\\tilde{\\eta}) \\right|\\right)$. \n\tTaylor expansion yields\n\t\t\\[ \\sum_{|y_k| > \\tilde{\\eta}}  \\left(\\log \\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log \\left| y_k\\right|\\right) \\leq\n\t\t\\sum_{|y_k| > \\tilde{\\eta}} \\frac{\\tilde{\\eta}^2}{y_k^2}.\n \\]\n\tDefine\n\t\t$N_1(u) = \\left| \\left\\{ y_k : \\tilde{\\eta} \\leq |y_k| \\leq u \\right\\}\\right|$.\n\tUsing integration by parts and Corollary \\ref{repulsion}, we have\n\t\t\\begin{equation} \\mathbb{E} \\left( \\sum_{|y_k| > \\tilde{\\eta}} \\frac{\\tilde{\\eta}^2}{y_k^2} \\right) =\n\t\t\t\\mathbb{E}\\left( \\int_{\\tilde{\\eta}}^\\infty \\frac{\\tilde{\\eta}^2}{y^2}\\,{\\rm d}N_1(y) \\right)\n\t\t\t= 2\\tilde{\\eta}^2 \\int_{\\tilde{\\eta}}^\\infty \\frac{\\mathbb{E}\\left(N_1(y)\\right)}{y^3}\\,{\\rm d}y = {\\rm O}\\left (d_N\\right).\\label{eqn:est2}\n\t\t\t\\end{equation}\n\tWe now estimate $\\sum_{|y_k| \\leq \\tilde{\\eta}}  \\left(\\log \\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log \\left| y_k\\right|\\right)$.\n\tWe consider two cases.\n\tFirst, let $A_N = b_N\/N$ for some very small $b_N$, for example \n\t\t\\[ b_N = e^{-\\left(\\log N\\right)^{\\frac{1}{4}}}.\\]\n\tFor $u > 0$ we denote\n\t\t$ N_2(u) = \\left| \\left\\{ y_k \\,:\\, A_N < | y_k |\\leq u \\right\\} \\right|$. Then again using integration by parts and Corollary \\ref{repulsion}  we obtain\n\t\t\\begin{multline*}\n\t\t\t\\mathbb{E}\\left(\n\t\t\t\\sum_{A_N <| y_k| < \\tilde{\\eta}}  \\left(\\log \\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log \\left| y_k\\right|\\right)\n\t\t\t\\right)\n\t\t\t= \\mathbb{E}\\left( \\int_{A_N}^{\\tilde{\\eta}} \n\t\t\t\t\\left( \\log \\left| y + {\\rm i}\\tilde{\\eta}\\right| - \\log \\left|y\\right|\\right){\\rm d}N_2(y)  \\right)\\\\\n\t\t\t\t\\leq \\log\\left(\\sqrt{2}\\right) \\mathbb{E}\\left( N_2\\left( \\tilde{\\eta} \\right) \\right) + \n\t\t\t\t\\int_{A_N}^{\\tilde{\\eta}} \\frac{\\mathbb{E}\\left(N_2(y)\\right)}{y}{\\rm d}y\n\t\t\t\t= {\\rm O}\\left(d_N + d_N\\log\\left( \\frac{d_N}{b_N} \\right)\\right) = \\oo\\left( \\sqrt{\\log N}\\right).\n\t\t\\end{multline*}\nIt remains\n\tto estimate $\\sum_{|y_k| < A_N}\\left( \\log\\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log\\left|y_k\\right|\\right)$. \n\tBy Corollary \\ref{micro fixed energy}, we have\n\t\t\\begin{equation}\\label{est3}\n\t\t \\mathbb{P}\\left(\\sum_{|y_k| < A_N}\\left( \\log\\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log\\left|y_k\\right| \\right) = 0\\right) \n\t\t \t\\geq \\mathbb{P}\\left( \\left|\\left\\{ y_k \\in [-A_N,A_N]\\right\\} \\right|= 0 \\right) \\to 1.\n\t\t\\end{equation}\n\t\\noindent \n\tThe estimates (\\ref{est1}) and (\\ref{eqn:est2}) along with Markov's inequality, and the bound (\\ref{est3}),\n\tconclude the proof.\n\t\\end{proof}\n\\section{Coupling of Determinants}\\label{DBM}\n\t\n\t\\noindent In this section, we use the coupled Dyson Brownian Motion introduced in \\cite{BouErdYauYin2016} to\n\tcompare (\\ref{smoothing t}) and (\\ref{coupling at tau}). \n\tDefine $\\tilde{W}_\\tau$ by running the matrix Dyson Brownian Motion\n\t(\\ref{eq:DBM evolution}) with initial condition $\\tilde{W}_0$\n\twhere $\\tilde{W}_0$ is a Wigner matrix with eigenvalues ${\\bm y}$.\n\tRecall that this induces a collection of Brownian motions\n\t$\\tilde{B}^{(k)}_t$ \n\tso that the system (\\ref{eqn:DBM}) with initial condition ${\\bm y}$\n\thas a (unique strong) solution \n\t${\\bm y}(\\cdot)$, and ${\\bm y}(\\tau)$ are the eigenvalues of $\\tilde{W}_\\tau$. Using the same (induced) Brownian motions as we used to define ${\\bm y}(\\tau)$, define\n\t${\\bm x}(\\tau)$ by running the dynamics (\\ref{eqn:DBM}) with initial condition given by the eigenvalues of a GOE matrix.\n\tUsing the result of Section \\ref{smoothing section} as an input to Lemma \\ref{extreme}, we now prove Proposition\n\t\\ref{prop:advection} which says that (\\ref{coupling at tau}) and (\\ref{smoothing t})  are asymptotically equal in law. \\\\\n\n\n\t\n\t\\noindent To study the coupled dynamics of ${\\bm x}(t)$ and ${\\bm y}(t)$, we\n\tfollow \\cite{LanSosYau2016, BourgadeExtreme}. For $\\nu \\in [0,1]$, let\n\t\t\\begin{equation}\n\t\t\\label{def:initialcondition}\n\t\t\t\\lambda^\\nu_k(0) = \\nu x_k + \\left(1 - \\nu\\right)y_k\n\t\t\\end{equation}\n\twhere ${\\bm x}$ is the spectrum of a GOE matrix, and ${\\bm y}$ is the spectrum of $\\tilde{W}_0$.\n\tWith this initial condition, we denote the (unique strong) solution to (\\ref{eqn:DBM}) by ${\\bm \\lambda}^{(\\nu)}(t)$.\n\tNote that\n$\n\t\t\t{\\bm \\lambda}^{(0)}(\\tau) = {\\bm y}(\\tau)$ and $\n\t\t\t{\\bm \\lambda}^{(1)}(\\tau) = {\\bm x}(\\tau).\n$ \nLet\n\t\t\\begin{equation}\n\t\t\\label{eqn:ft}\n\t\tf^{(\\nu)}_t(z) =e^{-\\frac{t}{2}} \\sum_{k=1}^N \\frac{u_k(t)}{\\lambda^{(\\nu)}_k(t)-z}, \n\t\t\\quad  u_k(t) = \\frac{{\\rm d}}{{\\rm d}\\nu}\\lambda^{(\\nu)}_k(t),\n\t\t\\end{equation}\n\t(see \\cite{LanSosYau2016} for existence of this derivative) and observe that\n\t\t\\begin{equation}\n\t\t\\label{main obs}\n\t\t\\frac{{\\rm d}}{{\\rm d}\\nu} \\sum_k \\log\\left| \\lambda_k^{(\\nu)}(t) - z \\right| = e^{\\frac{t}{2}}\\Re  \\left(f_t(z)\\right).\n\t\t\\end{equation}\n\tLemma \\ref{extreme} below from \\cite[Proposition 3.3]{BourgadeExtreme}, tells us that we may estimate\n\t$f_\\tau(z)$ by $f_0\\left(z_\\tau\\right)$, with $z_\\tau$ as in (\\ref{def:ztau}) and $\\tau$ as in (\\ref{def:tau}). \n\t\n\t\\begin{lemma}\n\t\t\\label{extreme}\n\t\tThere exists $C_0>0$ such that with $\\varphi=e^{C_0(\\log\\log N)^2}$, \n\t\tfor any $\\nu\\in[0,1]$, $\\kappa>0$ (small) and $D>0$ (large),  \n\t\t there exists $N_0(\\kappa,D)$ so that for any $N\\geq N_0$ we have\n\t\t\t\\[ \\mathbb{P}\\left(\\left| f^{(\\nu)}_t(z) - f^{(\\nu)}_0\\left(z_t\\right) \\right| <  \\frac{\\varphi}{N\\eta}\\ {\\rm for\\ all}\\ 0<t<1\\ {\\rm and}\\ z=E+\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i} \\eta, \\frac{\\varphi}{N}<\\eta<1,|E|<2-\\kappa\\right)\n\t\t\t\t\\geq 1-N^{-D}.\\]\n\t\tIn the above, $z_t$ is given by\n\t\t\t\\begin{equation}\n\t\t\t\\label{def:ztau} z_t = \\frac{1}{2}\\left( e^{\\frac{t}{2}}\\left(z + \\sqrt{z^2 - 4}\\right) + \n\t\t\t\te^{-\\frac{t}{2}}\\left(z - \\sqrt{z^2 - 4}\\right) \\right).\n\t\t\t\\end{equation}\n\t\\end{lemma}\t\n\t\\noindent For $z = {\\rm i}\\eta_0$, we have\n\t\t\\begin{equation}\n\t\t\\label{zt expansion}\n\t\t z_t = {\\rm i} \\left( \\eta_0 + \\frac{t \\sqrt{\\eta_0^2 + 4}}{2} \\right) + {\\rm O}\\left (t^2\\right),\n\t\t\\end{equation}\n\tand $\\eta_0$ is large enough to make use of Lemma \\ref{extreme}.\n\tTherefore,\n\tintegrating both sides of (\\ref{main obs}), we have by Lemma \\ref{extreme} that with overwhelming probability,\n\t\t\\begin{multline*}\n\t\t\t\\sum_k \\left( \\log \\left|x_k(\\tau) + {\\rm i}\\eta_0\\right| - \\log \\left|y_k(\\tau) + {\\rm i}\\eta_0\\right|  \\right) = \n\t\t\t\\int_0^1 \\frac{{\\rm d}}{{\\rm d}\\nu} \\sum_k \\log\\left| \\lambda_k^{(\\nu)}(\\tau) - z \\right| {\\rm d}\\nu \\\\\n\t\t\t= e^{\\frac{t}{2}} \\Re  \\int_0^1 f^{(\\nu)}_\\tau(z) {\\rm d}\\nu \n\t\t\t= e^{\\frac{t}{2}}\\Re  \\int_0^1 \n\t\t\t\\left( f_0^{(\\nu)}\\left(z_\\tau\\right) + {\\rm O}\\left ( \\frac{\\varphi}{N\\eta_0}\\right)\\right) {\\rm d}\\nu \n\t\t\t= e^{\\frac{t}{2}}\\Re  \\int_0^1 f_0^{(\\nu)}\\left(z_\\tau\\right){\\rm d}\\nu + {\\rm o}(1).\n\t\t\\end{multline*}\n\t\tMore precisely, the above estimates hold with probability $1-N^{-D}$ for large enough $N$, with rigorous justification by Markov's inequality based on the large moments \n\t\t$\\mathbb{E}((\\int_0^1 \\left( f_\\tau^{(\\nu)}\\left(z_0\\right)- f_0^{(\\nu)}\\left(z_\\tau\\right)  \\right) {\\rm d}\\nu)^{2p})$, which are bounded by Lemma \\ref{extreme}.\n\tAs a consequence, we have proved the following proposition.\n\t\\begin{proposition}\n\t\\label{prop:advection}\n\tLet $\\epsilon >0$, $\\tau = N^{-\\epsilon}$ and let $z_\\tau$ be as in (\\ref{def:ztau}) with $z = {\\rm i}\\eta_0$.\n\tThen for any $\\delta > 0$, \n\t\t\\[ \\lim_{N\\to\\infty} \\mathbb{P}\\left(\\left|\n\t\t\\sum_k \\left( \\log \\left|x_k(\\tau) + {\\rm i}\\eta_0\\right| - \\log \\left|y_k(\\tau) + {\\rm i}\\eta_0\\right|  \\right) \n\t\t - \\sum_k \\left( \\log \\left|x_k(0) + z_\\tau\\right| - \\log \\left|y_k(0) + z_\\tau\\right|  \\right)\\right| > \\delta  \\right) = 0.  \\]\n\t\\end{proposition}\n\n\\section{Conclusion of the Proof}\\label{conclusion}\n\n\t\\noindent We will conclude the proof of Theorem \\ref{main theorem} in the real symmetric case\n\tin two steps. The first step is to prove a Green's function comparison\n\ttheorem, and the second is to establish Theorem \\ref{main theorem} assuming Lemma\n\t\\ref{expectation}, proved  in the Appendix. \n\n\t\\subsection{Green's Function Comparison Theorem. }\\label{moment matching section}\n\tIn this section, we first use Lemma \\ref{moment matching} to \n\tchoose a $\\tilde{W}_0$ so that $\\tilde{W}_{\\tau}$ given by\n\t(\\ref{eq:DBM evolution})\n\tand initial condition $\\tilde{W}_0$,\n\tmatches $W$ closely up to fourth moment. We will then prove Theorem \\ref{4 moment matching theorem},\n\twhich by the result of Section \\ref{smoothing section}, says that\n\t$ \\log | \\det \\tilde{W}_\\tau |$ and \n\t$ \\log \\left| \\det W \\right|$ have the same law as $N\\to\\infty$.  \n\t\n\t\\begin{lemma}[Lemma 6.5 in \\cite{ErdYauYin2012Bulk}]\n\t\t\\label{moment matching}\n\t\tLet $m_3$ and $m_4$ be two real numbers such that\n\t\t\t\\begin{equation}\\label{moment condition}\n\t\t\t m_4 - m_3^2 - 1 \\geq 0, \\quad m_4 \\leq C_2 \n\t\t\t\\end{equation}\n\t\tfor some positive constant $C_2$. Let $\\xi^G$ be a Gaussian random variable with mean \n\t\t$0$ and variance $1$. Then for any sufficiently small $\\gamma > 0$ (depending on $C_2$),\n\t\tthere exists a real random variable $\\xi$, with subgaussian decay\n\t\tand independent of $\\xi^G$ such that the first four moments of\n\t\t\t\\[ \\xi' = \\left( 1 - \\gamma\\right)^{\\frac{1}{2}} \\xi_\\gamma + \\gamma^{\\frac{1}{2}}\\xi^G\\]\n\t\tare $m_1\\left(\\xi'\\right) = 0$, $m_2\\left(\\xi'\\right) = 1$, $m_3\\left(\\xi'\\right) = m_3$,\n\t\tand \n\t\t\t\\[ \\left|m_4\\left(\\xi'\\right) - m_4\\right| \\leq C\\gamma \\]\n\t\tfor some $C$ depending on $C_2$. \n\t\\end{lemma}\n\t\\noindent Now since $\\tilde{W}_\\tau$ is defined by independent Ornstein-Uhlenbeck processes in each entry, \n\tit has the same distribution as\n\t\t\\[ e^{-\\tau\/2} \\tilde{W}_0 + \\sqrt{1-e^{-\\tau}}W   \\]\n\twhere $W$ is a GOE matrix independent of $\\tilde{W}_0$.\n\tSo choosing $\\gamma = 1-e^{-\\tau}$,\n\tLemma \\ref{moment matching} says we can\n\tfind $\\tilde{W}_0$ so that the first three moments of the entries of $\\tilde{W}_\\tau$ \n\tmatch the first three moments of the entries of $W$, and the fourth moments of the entries\n\tof each differ by ${\\rm O}(\\tau)$.  \n\tOur next goal is to prove Theorem \\ref{4 moment matching theorem} which says that\n\twith $\\tilde{W}_\\tau$ constructed this way, if Theorem \\ref{main theorem} \n\tholds for $\\tilde{W}_\\tau$, then it holds for $W$. We first introduce stochastic domination and state\n\tTheorem \\ref{local law} which we will use in the proof.\n\t\n\t\\begin{definition}\n\t\tLet $X = \\left(X^N(u): N \\in \\mathbb{N}, u \\in U^N\\right), Y = \\left(Y^N(u): N \\in \\mathbb{N}, u \\in U^N\\right)$\n\t\tbe two families of nonnegative random variables, where $U^N$ is a possibly $N$-dependent parameter set.\n\t\tWe say that $X$ is stochastically dominated by $Y$, uniformly in $u$, if for every $\\epsilon > 0$ and $D > 0$, there exists $N_0(\\epsilon, D)$ such that\n\t\t\t\\[ \\sup_{u \\in U^N} \\mathbb{P}\\left[ X^N(u) > N^\\epsilon Y^N(u)\\right] \\leq N^{-D} \\]\n\t\tfor $N\\geq N_0$. Stochastic domination is always uniform in all parameters,\n\t\tsuch as matrix indices and spectral parameters, that are not explicitly fixed.\n\t\tWe will use the notation $X = O_\\prec(Y)$ or $X \\prec Y$ for the above property.\n\t\\end{definition}\n\t\n\t\\begin{theorem}[Theorem 2.1 in \\cite{ErdYauYin2012Rig}]\\label{local law}\n\tLet $W$ be a Wigner matrix satisfying (\\ref{subgaussian}). Fix $\\zeta > 0$ and define the domain\n\t\t\\[ S = S_N(\\zeta) := \\left\\{ E + {\\rm i}\\eta \\,:\\, |E| \\leq \\zeta^{-1}, \\,N^{-1+\\zeta} \\leq \\eta \\leq \\zeta^{-1} \\right\\}. \\]\n\tThen uniformly for $i,j = 1,\\hdots, N$ and $z \\in S$, we have\n\t\\begin{align*}\n\t\t s(z) &= m(z) +  {\\rm O}_\\prec\\left( \\frac{1}{N\\eta} \\right),\\\\\n\tG_{ij}(z)& = \\left(W -z\\right)^{-1}_{ij} = \n\t\tm(z)\\delta_{ij} + {\\rm O}_\\prec\\left( \\sqrt{\\frac{\\Im  \\left(m(z)\\right)}{N\\eta}} + \\frac{1}{N\\eta} \\right).\n\t\t\\end{align*}\n\t\\end{theorem}\n\n\t\\begin{theorem}\\label{4 moment matching theorem}\n\t\tLet $F: \\mathbb{R} \\to \\mathbb{R}$ be smooth with compact support, and\n\t\tlet $W$ and $V$ be two Wigner matrices \n\t\tsatisfying (\\ref{subgaussian}) such that for $1 \\leq i, j \\leq N$,\n\t\t\t\\begin{numcases}{\\mathbb{E}\\left(w_{ij}^a\\right) = }\n\t\t\t\t\\label{moment matching assumption 3}\n\t\t\t\t\\mathbb{E}\\left(v_{ij}^a\\right)  & $a \\leq 3$ \\\\\n\t\t\t\t\\label{moment matching assumption 4}\n\t\t\t\t\\mathbb{E}\\left(v_{ij}^a\\right)+ \\OO(\\tau) & $a = 4$,\n\t\t\t\\end{numcases}\n\t\twhere $\\tau$ is as in (\\ref{def:tau}). Further, let $c_N$ be any deterministic sequence and define\n\t\t\t\\[\n\t\t\t\tu_N(W) = \\frac{\\log | \\det \\left(W + \\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta_0\\right)| +c_N}{\\sqrt{\\log N}}.\n\t\t\t\\]\n\t\twhere $\\eta_0$ is as in (\\ref{def:eta0}). Then\n\t\t\t\\begin{equation}\n\t\t\t\t\\lim_{N\\to \\infty} \\mathbb{E} \\left(F\\left( u_N(W)\\right) - F\\left( u_N(V) \\right)\\right) = 0.\\label{eqn:enough}\n\t\t\t\\end{equation} \n\t\\end{theorem}\n\t\\begin{proof}\t\t\n\t\tAs in \\cite{TaoVu2012}, where the authors also used the following technique to analyze\n\t\tfluctuations of determinants, we show that the effect of substituting $W_{ij}$ in place of $V_{ij}$ in $V$ is\n\t\tnegligible enough that making $N^2$ replacements, we conclude the theorem. \\\\\n\n\t\t\n\t\t\\noindent Fix $(i,j)$ and let\n\t\t$E^{(ij)}$ be the matrix whose elements are $E^{(ij)}_{kl} = \\delta_{ik}\\delta_{jl}$.\n\t\tLet $W_1$ and $W_2$ be two adjacent matrices in the swapping process described above. \n\t\tSince $W_1, W_2$ differ in just the $(i,j)$ and $(j,i)$ coordinates, we may write\n\t\t\t$$\n\t\t\t\tW_1 = Q + \\frac{1}{\\sqrt{N}} U, \\ \\ \\  \\ \\\n\t\t\t\tW_2 = Q + \\frac{1}{\\sqrt{N}} \\tilde{U}\n\t\t\t$$\n\t\twhere $Q$ is a matrix with $Q_{ij} = Q_{ji} = 0$, and\n\t\t\t$$\n\t\t\t\tU = u_{ij}E^{(ij)} + u_{ji}E^{(ji)}\\ \\ \\ \\ \\ \\ \n\t\t\t\t\\tilde{U} = \\tilde{u}_{ij}E^{(ij)} + \\tilde{u}_{ji}E^{(ji)}.\n\t\t$$\n\t\tImportantly $U,\\tilde{U}$ satisfy the same moment matching conditions\n\t\twe have imposed on $\\tilde{W}_\\tau$ and $W$. \n\t\tNow by the fundamental theorem of calculus, we have for any symmetric matrix $W$,\n\t\t\t\\begin{equation}\n\t\t\t\\label{log via fundamental theorem}\n\t\t\t\\log\\left|\\det(W + {\\rm i}\\eta_0)\\right| = \n\t\t\t\t\\sum_{k=1}^N \\log\\left| x_k + {\\rm i}\\eta_0\\right| = \\log\\left|\\det(W + {\\rm i})\\right| - N\\,\n\t\t\t\t \\Im  \\int_{\\eta_0}^1\n\t\t\t\ts_W\\left({\\rm i}\\eta \\right){\\rm d}\\eta.  \n\t\t\t\\end{equation}\n\t\t\tFrom the central limit theorems for linear statistics of Wigner matrices on macroscopic scales \\cite{LytPas2009}, \n\t\t\t $(\\log\\left|\\det(W + {\\rm i})\\right|-\\mathbb{E}(\\log\\left|\\det(W + {\\rm i})\\right|))\/\\sqrt{\\log N}$ \n\t\t\tconverges to $0$ in probability (the same result holds with $W$ replaced with $V$), and from Lemma \\ref{expectation} (which clearly holds with $1$ in place of $\\tau$), \n\t\t\t$(\\mathbb{E}(\\log\\left|\\det(W + {\\rm i})\\right|)-\\mathbb{E}(\\log\\left|\\det(V + {\\rm i})\\right|))\/\\sqrt{\\log N}\\to 0$.\n\t\t\tTherefore (\\ref{eqn:enough}) is equivalent to\n\t\t\t\t\\begin{equation}\n\t\t\\label{key expression}\n\t\t\t\\lim_{N\\to \\infty} \\mathbb{E} \\left(\\widetilde F\\left(N\\,\n\t\t\t\t \\Im  \\int_{\\eta_0}^1\n\t\t\t\ts_W\\left({\\rm i}\\eta \\right){\\rm d}\\eta\\right) - \\widetilde F\\left(N\\,\n\t\t\t\t \\Im  \\int_{\\eta_0}^1\n\t\t\t\ts_V\\left({\\rm i}\\eta \\right){\\rm d}\\eta\\right)\\right)= 0,\n\t\t\t\\end{equation}\n\t\t\twhere\n\t\t\t$$\n\t\t\t\\widetilde F(x)=F\\left(\\frac{\\mathbb{E}(\\log\\left|\\det(W + {\\rm i})\\right|)+c_N-x}{\\sqrt{\\log N}}\\right).\n\t\t\t$$\n\n\t\tWe now expand $s_{W_1}$ and $s_{W_2}$ around $s_{Q}$, and then\n\t\tto Taylor expand $\\widetilde F$.\n\t\tSo let \n\t\t\t\\[ R =R(z)= \\left(Q-z\\right)^{-1} \\text{ and } S=S(z)=\\left(W_1 - z\\right)^{-1}. \\]\n\t\tBy the resolvent expansion\n\t\t\t\\[ \n\t\t\tS = R - N^{-1\/2}RUR + \\hdots + N^{-2}(RU)^4R - N^{-5\/2}(RU)^5 S,\n\t\t\t\\]\n\t\twe can write\n\t\t\t\\[N \\int_{\\eta_0}^1 s_{W_1}({\\rm i}\\eta){\\rm d}\\eta = \\int_{\\eta_0}^1\\text{Tr}\\left(S({\\rm i}\\eta)\\right){\\rm d}\\eta = \n\t\t\t\\int_{\\eta_0}^1 \\text{Tr}\\left(R({\\rm i}\\eta)\\right){\\rm d}\\eta +\n\t\t\t\\left( \\sum_{m=1}^4 N^{-m\/2}\\hat{R}^{(m)}({\\rm i}\\eta) - N^{-5\/2}\\Omega\\right)\n\t\t\t\t:= \\hat{R} + \\xi  \\]\n\t\twhere \n\t\t\t\\[ \\hat{R}^{(m)} = (-1)^m \\int_{\\eta_0}^1 \\text{Tr}\\left((R({\\rm i}\\eta)U)^mR({\\rm i}\\eta)\\right){\\rm d}\\eta \n\t\t\t\\quad \\text{and} \\quad\n\t\t\t\\Omega = \\int_{\\eta_0}^1 \\text{Tr} \\left( (R({\\rm i}\\eta)U)^5S({\\rm i}\\eta) \\right){\\rm d}\\eta.\\]\n\t\tThis gives us an expansion of $s_{W_1}$ around $s_{Q}$. \n\t\tNow Taylor expand $\\widetilde F(\\hat{R} + \\xi)$ as\n\t\t\t\\begin{equation} \\label{taylor expansion} \\widetilde F\\left(\\hat{R}+\\xi\\right) = \\widetilde F\\left(\\hat{R}\\right) + \n\t\t\t\t\\widetilde F'\\left(\\hat{R}\\right)\\xi + \\hdots + \\widetilde F^{(5)}\\left(\\hat{R}+\\xi'\\right)\\xi^5 = \\sum_{m=0}^5 N^{-m\/2}A^{(m)}\n\t\t\t\\end{equation}\n\t\twhere $0 < \\xi' < \\xi$, and we have introduced the notation $A^{(m)}$ in order to arrange terms according to\n\t\tpowers of $N$. For example\n\t\t\t$$\n\t\t\t\tA^{(0)} = \\widetilde F\\left(\\hat{R}\\right),\\ \\ \n\t\t\t\tA^{(1)} = \\widetilde F'\\left(\\hat{R}\\right)\\hat{R}^{(1)}, \\ \\ \n\t\t\t\tA^{(2)} = \\widetilde F'\\left(\\hat{R}\\right)\\hat{R}^{(2)} + \\widetilde F''\\left(\\hat{R}\\right)\\left(\\hat{R}^{(1)}\\right)^2.\n\t\t\t$$\n\t\tMaking the same expansion for $W_2$, we record our two expansions as\n\t\t\t\\[ \\widetilde F\\left(\\hat{R} + \\xi_i\\right) = \\sum_{m=0}^5 N^{-m\/2} A^{(m)}_i, \\quad i = 1, 2, \\]\n\t\twith $\\xi_i$ corresponding to $W_i$. With this notation, we have\n\t\t\t\\begin{align*} \n\t\t\t\t\\mathbb{E}\\left(\\widetilde F\\left(\\hat{R} + \\xi_1\\right)\\right) - \\mathbb{E} \n\t\t\t\t\\left(\\widetilde F\\left(\\hat{R} + \\xi_2\\right)\\right)\n\t\t\t\t&=  \\mathbb{E} \\left(\\sum_{m=0}^5 N^{-m\/2}\\left( A_1^{(m)} - A_2^{(m)} \\right)\\right) .\n\t\t\t\\end{align*}\n\t\tNow only the first three moments of $U,\\tilde{U}$ appear in the terms corresponding to $m=1, 2, 3$, so\n\t\tby the moment\n\t\tmatching assumption (\\ref{moment matching assumption 3}), all of these terms are all identically zero. \n\t\tNext, consider $m = 4$. Every term with first, second, and third moments of\n\t\t$U$ and $\\tilde{U}$ is again zero, and what remains is\n\t\t\t\\[ \\mathbb{E} \\left(\\widetilde F'(\\hat{R}) \\left( \\hat{R}_1^{(4)} - \\hat{R}_2^{(4)}\\right)\\right). \\]  \n\t\t\\noindent So we can discard $A^{(4)}$ if \n\t\t\t\\begin{equation}\\label{4th moment}\n\t\t\t \\int_{\\eta_0}^1 \\left|\\mathbb{E} \\left( \n\t\t\t \t{\\rm Tr}\\left( (RU)^4R \\right) - {\\rm Tr}\\left( (R\\tilde{U})^4R\\right) \\right)   \\right| {\\rm d}\\eta \n\t\t\t \\end{equation}\n\t\tis small. To see that this is in fact the case, we expand the traces, and apply Theorem \\ref{local law}\n\t\talong with our fourth moment matching assumption (\\ref{moment matching assumption 4}). \n\t\tSpecifically,\n\t\t\t\\[ \\text{Tr}\\left( (RU)^4R\\right) = \\sum_j \\left(\\sum_{i_1, \\hdots, i_8} \n\t\t\t\tR_{ji_1}U_{i_1i_2}R_{i_2i_3}\\hdots U_{i_7i_8}R_{i_8j}\\right). \\]\n\t\tWriting the corresponding Tr for $W_2$ and applying the moment matching assumption, we see that we can\n\t\tbound (\\ref{4th moment}) by\n\t\t\t\\[ {\\rm O}(\\tau) \\int_{\\eta_0}^1\\sum_j \\sum_{i_1, \\hdots, i_8} \\mathbb{E} \\left(\n\t\t\t\t\\left| R_{ji_1}R_{i_2i_3}R_{i_4i_5}R_{i_6i_7}R_{i_8j} \\right| \\right){\\rm d}\\eta. \\]\n\t\tTo bound the terms in the sum, we need to count the number of diagonal and off-diagonal terms in each product.\n\t\tTo do this, let us say $U_{pq}, \\tilde{U}_{pq}$ and $U_{qp}, \\tilde{U}_{qp}$ are the only non-zero entries of \n\t\t$U, \\tilde{U}$. Then\n\t\teach of the sums over $i_1, \\hdots, i_8$ are just sums over $p, q$, and when $j \\notin \\{p,q\\}$, \n\t\t$R_{ji_1}$ and $R_{i_8j}$ are certainly off-diagonal entries of $R$. This means we can apply Cauchy-Schwartz\n\t\tto write that for any $\\gamma > 0$,\n\t\t\t\\begin{align*}\n\t\t\t\t{\\rm O}(\\tau) \\int_{\\eta_0}^1\\sum_{j \\notin \\{p, q\\}} \\sum_{i_1, \\hdots, i_8} \\mathbb{E} \\left(\n\t\t\t\t\t\\left| R_{ji_1}R_{i_2i_3}R_{i_4i_5}R_{i_6i_7}R_{i_8j} \\right| \\right){\\rm d}\\eta \n\t\t\t\t= {\\rm O}\\left (\\tau N^{1 + 2\\gamma}  \\int_{\\eta_0}^1 \\frac{1}{N\\eta}{\\rm d}\\eta\\right) = \n\t\t\t\t\t{\\rm O}\\left (N^{2\\gamma - \\epsilon} \\log(N)\\right).\n\t\t\t\\end{align*}\n\t\tSimilarly, \n\t\t\t\\begin{align*}\n\t\t\t\t{\\rm O}(\\tau) \\int_{\\eta_0}^1\\sum_{j \\in \\{p, q\\}} \\sum_{i_1, \\hdots, i_8} \\mathbb{E} \\left(\n\t\t\t\t\t\\left| R_{ji_1}R_{i_2i_3}R_{i_4i_5}R_{i_6i_7}R_{i_8j} \\right| \\right){\\rm d}\\eta \n\t\t\t\t= {\\rm O}\\left (\\tau N^{\\epsilon \/2} \\right) = \n\t\t\t\t\t{\\rm O}\\left (N^{- \\epsilon\/2} \\right).\n\t\t\t\\end{align*}\n\t\tSince $A^{(4)}$ has a pre-factor of $N^{-2}$ in (\\ref{taylor expansion}), and the above holds\n\t\tfor every choice of $\\gamma > 0$, in our entire \n\t\tentry swapping scheme starting from $V$ and ending with $W$, the corresponding error\n\t\tis ${\\rm o}(1)$.\\\\\n\t\t\n\t\t\\noindent Lastly we comment on the error term $A^{(5)}$. All terms in $A^{(5)}$ not involving \n\t\t$\\Omega$ can be dealt\n\t\twith as above. The only term involving $\\Omega$ is $\\widetilde F'(\\hat{R})\\Omega$, and to deal with this,\n\t\twe can expand the expression for $\\Omega$ as above. We do not have any\n\t\tmoment matching condition for the fifth moments of $U, \\tilde{U}$, but (\\ref{subgaussian}) means\n\t\tthat their fifth\n\t\tmoments are bounded which is enough for our purpose since $A^{(5)}$ has a pre-factor of $N^{-5\/2}$ above. \n\t\t\\end{proof}\t\n\t\t\n\t\\subsection{Proof of Theorem \\ref{main theorem}. }\\label{sub:end}\n\tIn this section we first prove Proposition \\ref{variance} and, using Lemma \n\t\\ref{expectation}, we\n\tconclude the proof of Theorem \\ref{main theorem}.\n\t\n\t\n\t\n\t\n\t\\begin{proposition}\\label{prop:variance} Recall $\\tau=N^{-{\\varepsilon}}$.\n\t\\label{variance} There exist ${\\varepsilon}_0,C$ such that for any fixed $0<{\\varepsilon}<{\\varepsilon}_0$, for large enough $N$, we have\n\t\\[ {\\rm Var}\\left( \\sum_k\\log|x_k(0)+\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\tau|\\right) \\leq C (1+\\epsilon \\log N).\\]\n\\end{proposition}\t\n\\begin{proof}\nWe outline two proofs, which are trivial extensions of existing linear statistics asymptotics on global scales, to the case of almost macroscopic scales. The tool for this extension is the rigidity estimate from \\cite{ErdYauYin2012Rig}: for any $c,D>0$, there exists $N_0$ such that for any $N\\geq N_0$ and $k\\in\\llbracket 1,N\\rrbracket$ we have\n\\begin{equation}\\label{eqn:rigidity}\n\\mathbb{P}\\left(|x_k-\\gamma_k|>N^{-\\frac{2}{3}+c}\\min(k,N+1-k)^{-\\frac{1}{3}}\\right)\\leq N^{-D}.\n\\end{equation}\n\n\n\\noindent For the first proof, we use (\\ref{eqn:rigidity}) to bound all the error terms in the proof of \\cite[Theorem 3.6]{LytPas2009} (these error terms all depend on \\cite[Theorem 3.5]{LytPas2009}, which can be improved via \n(\\ref{eqn:rigidity}) to ${\\rm Var}(u_N(t))\\leq N^c(1+|t|)$ and ${\\rm Var}\\left(\\mathcal{N}_N(\\varphi)\\right)\\leq N^c\\|\\varphi\\|_{\\rm Lip}^2$). What we obtain is that if $\\varphi$ (possibly depending on $N$) satisfies $\\int|t|^{100}\\hat \\varphi(t)<N^{1\/100}$, then $\\sum\\varphi(x_k)-\\mathbb{E}(\\sum\\varphi(x_k))$ has limiting variance asymptotically equivalent to \n\t\t\\begin{equation}\n\\label{vwig} \n\t\tV_{\\rm Wig}[\\varphi] =  \\frac{1}{2\\pi^2} \\int_{(-2,2)^2}\n\t\t\t\\left( \\frac{\\Delta \\varphi}{\\Delta \\lambda}\\right)^2 \n\t\t\t\\frac{4 - \\lambda_1\\lambda_2}{\\sqrt{4 - \\lambda_1^2}\n\t\t\t\\sqrt{1- \\lambda_2^2}} \\,\n\t\t\t{\\rm d}\\lambda_1{\\rm d}\\lambda_2+ \\frac{\\kappa_4}{2\\pi^2}\n\t\t\t\\left( \\int_{-2}^{2} \\varphi(\\mu) \\frac{2 - \\mu^2}{\\sqrt{4 - \\mu^2}}\\,\n\t\t\t{\\rm d}\\mu\\right)^2,\n\t\t\\end{equation}\nwhere $\\Delta \\varphi = \\varphi\\left(\\lambda_1\\right) - \\varphi\\left(\\lambda_2\\right)$, $\\Delta \\lambda = \\lambda_1 - \\lambda_2$, $\\mu_4 = \\mathbb{E} \\left(W_{jk}^4\\right)$,\n\t$\\kappa_4 = \\mu_4 - 3$ is the fourth cumulant of the off-diagonal entries of $W$.\nWe choose $\\varphi(x)=\\varphi_N(x)=\\frac{1}{2}\\log(x^2+\\tau^2)\\chi(x)$ with $\\chi$ fixed, smooth, compactly supported, equal to 1 on $(-3,3)$. Note that for ${\\varepsilon}_0$ small enough, we have $\\int|t|^{100}\\hat \\varphi(t)<N^{1\/100}$. Then by (\\ref{eqn:rigidity}) and (\\ref{vwig}), \n$$\nV_{\\rm Wig}[\\log|\\cdot-\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\tau|]\\sim V_{\\rm Wig}[\\varphi]\\leq C \\iint \\left(\\frac{\\Delta\\varphi}{\\Delta\\lambda}\\right)^2\n\t\\, {\\rm d} \\lambda_1 {\\rm d} \\lambda_2\n\t= C \\int |\\xi|\\, \\left| \\hat{\\varphi}(\\xi) \\right|^2{\\rm d}\\xi,\n$$\nand the above right hand side can be bounded as follows.\n\tWe have\n\t\t\\begin{align*}\n\t\t \\left| \\hat{\\varphi}_N(\\xi) \\right| = \\left| \\frac{1}{2\\pi} \\int_{\\mathbb{R}}\n\t\t \t \\varphi_N(x)e^{-i\\xi x}\\, {\\rm d} x\\right|\n\t\t &\\leq \n\t\t C \\left| \\int_{-5}^{5} \\frac{x}{x^2 + \\tau^2} \\frac{e^{-i\\xi x}}{i\\xi}\\, {\\rm d} x\\right| \n\t\t =C\\left| \\frac{1}{\\xi} \\int_0^{5\/\\tau} \\frac{x}{x^2 + 1} \\sin(x \\xi \\tau) \\, {\\rm d} x \\right|.\n\t\t\\end{align*}\n\tFor $0 < \\xi < 5$, the inequality $|\\sin x| < x$ shows $\\left| \\hat{\\varphi}_N(\\xi) \\right| = {\\rm O}(1)$, and when\n\t$\\xi > 5\/\\tau$, integration by parts shows $\\left|\\hat{\\varphi}_N(\\xi)\\right| = {\\rm O}\\left (\\frac{1}{\\xi^2\\tau}\\right)$. \n\tWhen $5 < \\xi < 5\/\\tau$, first note\n\t\t\\begin{align*}\n\t\t\\int_0^{\\frac{5}{\\tau}} \\sin\\left(\\xi \\tau x\\right) \\frac{x}{x^2 + 1} \\, {\\rm d}x &= C + \n\t\t\t\\int_1^{\\frac{5}{\\tau}} \\frac{\\sin\\left(\\xi \\tau x\\right)}{x} \\,{\\rm d}x \n\t\t\t=  C + \\int_{\\xi \\tau}^{1} \\frac{\\sin y}{y} \\,{\\rm d}y + \\int_1^{5\\xi} \\frac{\\sin y}{y}\\,{\\rm d}y.\n\t\t\\end{align*}\n\tUsing $|\\sin y| < |y|$, we see that the first term is ${\\rm O}(1)$, and integrating by parts, we see\n\tthat the second term is ${\\rm O}(1)$ as well. This means \n\t\t\\[ \\int |\\xi| \\left| \\hat{\\varphi}_N(\\xi) \\right|^2{\\rm d}\\xi \\leq C +C \n\t\t\\int_{5}^{\\frac{5}{\\tau}} \\frac{1}{\\xi} \\,{\\rm d}\\xi = {\\rm O}\\left (1+|\\log \\tau|\\right), \n\t\t\\]\n\twhich concludes the proof.\\\\\n\t\n\\noindent The  second proof is similar but more direct. Theorem 3 in \\cite{KhoKhoPas} implies that for $z_1=\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta_1,z_2=\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta_2$ at macroscopic distance from the real axis, and $\\eta_1=\\im z_1>0,\\eta_2=\\im z_2<0$, we have\n\t$$\n\t\\left|{\\rm Cov}\\left(\\sum_k\\frac{1}{z_1-x_k},\\sum_k\\frac{1}{z_2-x_k}\\right)\\right|\\leq\\frac{C}{(\\eta_1-\\eta_2)^2}+f(z_1,z_2)+\\OO(N^{-1\/2}),\n\t$$\n\twhere $f$ is a function uniformly bounded on any compact subset of $\\mathbb{C}^2$.\n\tUsing (\\ref{eqn:rigidity}), one easily obtains that the formula above holds uniformly with $|\\im z_1|,|\\im z_2|>N^{-1\/10}$, and the deteriorated error term $\\OO(N^{-1\/10})$, for example.\t\nNote that\n$$\n\\log\\left|\\det(W + {\\rm i}\\eta)\\right| = \\log\\left|\\det(W + {\\rm i})\\right| - N\\,\n\t\t\t\t \\Im  \\int_{\\eta}^1\n\t\t\t\ts_W\\left({\\rm i}x \\right){\\rm d}x.  \n$$\nand $\\log\\left|\\det(W + {\\rm i})\\right|$ has fluctuations of order 1 due to the above macroscopic central limit theorems. \nFor for $\\eta>N^{-1\/10}$, the variance of the above integral can be bounded by \n$\n\\iint_{[\\eta,1]^2}\\frac{1}{|\\eta_1+\\eta_2|^2}\\,{\\rm d} \\eta_1{\\rm d} \\eta_2\\leq C|\\log \\eta|,\n$\nwhich concludes the proof.\n\t\\end{proof}\n\t\n\t\n\t\n\n\n\n\\noindent From (\\ref{GOE CLT}) and Proposition \\ref{smoothing}, for some explicit deterministic  $c_N$ we have\n\\begin{equation}\\label{eqn:1}\n\\frac{ \t\t\t\\sum_{k =1}^N \\log \\left|x_k(\\tau) + {\\rm i} \\eta_0\\right|+c_N}{\\sqrt{\\log N}}\\to\\mathscr{N}(0,1),\n\\end{equation}\nand Proposition \\ref{prop:advection} implies that\n$$\n\\frac{ \t\t\t\\sum_{k =1}^N \\log \\left|y_k(\\tau) + {\\rm i} \\eta_0\\right|+c_N}{\\sqrt{\\log N}}\n+\\frac{ \t\t\t\\sum_{k =1}^N \\log \\left|x_k(0) + z_\\tau\\right|-\\sum_{k =1}^N \\log \\left|y_k(0) + z_\\tau\\right|}{\\sqrt{\\log N}}\n\\to\\mathscr{N}(0,1).\n$$\n\n\n\t\\noindent Lemma \\ref{expectation} and Proposition \\ref{variance} show that the second term above, call it $X$, satisfies $\\mathbb{E}(X^2)<C{\\varepsilon}$, for some universal $C$. Thus for any fixed smooth and compactly supported function $F$,\n\t\\begin{align*}\n\t\\mathbb{E} \\left(F\\left(\\frac{ \t\t\t\\sum_{k =1}^N \\log \\left|y_k(\\tau) + {\\rm i} \\eta_0\\right|+c_N}{\\sqrt{\\log N}}\\right)\\right)&=\\mathbb{E} \\left(F\\left(\\frac{ \t\t\t\\sum_{k =1}^N \\log \\left|x_k(\\tau) + {\\rm i} \\eta_0\\right|+c_N}{\\sqrt{\\log N}}+X\\right)\\right)\n\t+\\OO\\left(\\|F\\|_{\\rm Lip}(\\mathbb{E}\\left(X^2\\right))^{1\/2}\\right)\\\\\n\t&=\\mathbb{E} \\left(F(\\mathscr{N}(0,1))\\right)+\\oo(1)+\\OO\\left({\\varepsilon}^{1\/2}\\right).\n\t\\end{align*}\nWith Theorem \\ref{4 moment matching theorem}, the above equation implies\n$$\n\\mathbb{E}\\left( F\\left(\\frac{\\log|\\det(W+\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta_0)|+c_N}{\\sqrt{\\log N}}\\right)\\right)=\\mathbb{E} \\left(F(\\mathscr{N}(0,1))\\right)+\\oo(1)+\\OO\\left({\\varepsilon}^{1\/2}\\right),\n$$\t\t\nand by Proposition \\ref{smoothing}, we obtain\n\\begin{equation}\\label{eqn:2}\n\\mathbb{E} \\left(F\\left(\\frac{\\log|\\det W|+\\frac{N}{2}}{\\sqrt{\\log N}}\\right)\\right)=\\mathbb{E} \\left(F(\\mathscr{N}(0,1))\\right)+\\oo(1)+\n\\OO\\left({\\varepsilon}^{1\/2}\\right).\n\\end{equation}\nSince ${\\varepsilon}$ is arbitrarily small, this concludes the proof.\n\\normalcolor\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\renewcommand{\\thetheorem}{A.\\arabic{theorem}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{IntroSec}\n\nThe \\emph{ping-pong argument} was first used in\n\\cite[\\S III,16]{Klein:Riemann} and \\cite[\\S II,3.8]{Fricke+Klein} to\nanalyze the actions of certain groups of linear fractional\ntransformations on the Riemann sphere.\nLater distillations and generalizations of the arguments\n(e.g., \\cite[Theorem 1]{MR0148731})\nwere used to establish that a given group is a free product.\nIn the current paper we will adapt the \\emph{ping-pong argument}\nto the setting of subgroups of \\({\\operatorname{Homeo}_+(I)}\\), the\ngroup of the orientation preserving homeomorphisms of the unit interval.\nOur main motivation is to develop a better understanding of the finitely generated\nsubgroups of the group \\({\\operatorname{PL}_+(I)}\\) of piecewise linear order-preserving homeomorphisms of\nthe unit interval.\nThe analysis in the current paper resembles the\noriginal \\emph{ping-pong argument} in that it establishes a tree structure on\ncertain orbits of a group action.\nHowever, the arguments in the current paper differ from the usual analysis\nas the generators of the group action have large sets of fixed points.\n\nThe focus of our attention in this article will be subgroups of \\({\\operatorname{Homeo}_+(I)}\\)\nwhich are specified by what we will term \\emph{geometrically fast} generating sets.\nOn one hand, our main result shows that the isomorphism types of the groups specified by\ngeometrically fast generating sets are determined by their \\emph{dynamical diagram} \nwhich encodes their qualitative dynamics;\nsee Theorem \\ref{CombToIso} below.\nThis allows us to show, for instance, that such sets generate groups which\nare always embeddable into Richard Thompson's group \\(F\\).\nOn the other hand, we will see that a broad class of subgroups\nof \\({\\operatorname{PL}_+(I)}\\) can be generated using such sets.\nThis is substantiated in part by Theorem \\ref{GeoProperGen} below.\n\nAt this point it is informative to consider an example.\nFirst recall the classical \\emph{Ping-Pong Lemma}\n(see \\cite[Prop. 1.1]{free_subgrp_linear}):\n\n\\begin{ping_pong_lem}\nLet \\(S\\) be a set and \\(A\\) be a set of permutations of \\(S\\)\nsuch that \\(a^{-1} \\not \\in A\\) for all \\(a \\in A\\).\nSuppose there is an assignment \\(a \\mapsto D_a \\subseteq S\\) of pairwise disjoint sets\nto each \\(a \\in A^\\pm := A \\cup A^{-1}\\) and an\n\\(x \\in S \\setminus \\bigcup_{a \\in A^\\pm} D_a\\) such that\nif \\(a \\ne b^{-1}\\) are in \\(A^\\pm\\), then\n\\[\n\\big(D_b \\cup \\{x\\}\\big)a \\subseteq D_a.\n\\] \nThen \\(A\\) freely generates \\(\\gen{A}\\).\n\\end{ping_pong_lem}\n\n\\noindent\n(We adopt the convention of writing permutations to the right of their arguments;\nother notational conventions and terminology will be reviewed in Sections \\ref{DefSec} and \\ref{FastBumpsSec}.)\nIn the current paper, we relax the hypothesis so that the\ncontainment \\(D_b a \\subseteq D_a\\)\nis required only when \\(D_b\\) is contained in the \\emph{support}\nof \\(a\\); similarly \\(x a \\in D_a\\) is only required when\n\\(x a \\ne x\\).\n\nConsider the three functions \\((b_i \\mid i < 3)\\) in\n\\({\\operatorname{Homeo}_+(I)}\\) whose graphs are shown in Figure \\ref{FastF3}.\n\\begin{figure}\n\\[\n\\xy\n(0,36); (0,0)**@{-}; (36,0)**@{-}; (0,0); (36,36)**@{-};\n(10,9.6)*{\\dvarbump{2}{14}{16}}; (5,15)*{b_0}; \n(26,25.6)*{\\dvarnbump{2}{14}{16}}; (31,21)*{b_2};\n(18,17.6)*{\\dvarbump{2}{14}{16}}; (15,24)*{b_1};\n\\endxy\n\\]\n\\caption{Three homeomorphisms}\\label{FastF3}\n\\end{figure}\n\\newcommand{\\varbump}[3]\n{\n\\xy\n(0,0); (#3,0)**\\crv{(#1,#1)&(#2,#1)};\n\\endxy\n}\n\\newcommand{\\varnbump}[3]\n{\n\\xy\n(0,0); (#3,0)**\\crv{(#1,-#1)&(#2,-#1)};\n\\endxy\n}\nA schematic diagram (think of the line \\(y=x\\) as drawn\nhorizontally) of these functions might be:\n\\[\n\\xy\n(12,2.6)*{\\varbump{12}{36}{48}};  (12,9)*{b_0};\n(36,2.6)*{\\varbump{12}{36}{48}};  (36,9)*{b_1};\n(60,-9)*{\\varnbump{12}{36}{48}}; (60,-10)*{b_2};\n(-12,-3); (-3.5,-3)**@{-};(-3,-3)*{\\circ}; (-8,-5)*{S_1}; (-3,-5)*{};\n(12,-3); (20.5,-3)**@{-};(21,-3)*{\\circ};  (16,-5)*{S_2}; (21,-5)*{};\n(36,-3); (27.5,-3)**@{-};(27,-3)*{\\bullet};  (32,-5)*{D_1}; (27,-5)*{};\n(36,-3); (44.5,-3)**@{-};(45,-3)*{\\circ};  (40, -1)*{D_3}; (45,-5)*{};\n(60,-3); (51.5,-3)**@{-};(51,-3)*{\\bullet};  (56, -5)*{D_2}; (51,-5)*{};\n(84,-3); (75.5,-3)**@{-};(75,-3)*{\\bullet};  (80,-1)*{S_3};  (75,-5)*{};\n\\endxy\n\\]\nIn this diagram, we have assigned intervals \\(S_i\\) and \\(D_i\\)\nto the ends of the support to each \\(b_i\\) so that\nthe entire collection of intervals is pairwise disjoint. \nOur system of homeomorphisms is assumed to have an additional\ndynamical property reminiscent of the hypothesis of the Ping-Pong Lemma:\n\\begin{itemize}\n\n\\item \\(S_i b_i \\cap D_i = \\emptyset\\);\n\n\\item \\(b_i\\) carries \\(\\operatorname{supt}(b_i) \\setminus S_i\\) into \\(D_i\\);\n\n\\item  \\(b_i^{-1}\\) carries \\(\\operatorname{supt}(b_i) \\setminus D_i\\) into \\(S_i\\) for each \\(i\\).\n\n\\end{itemize}\n\nA special case of our main result\nis that these dynamical requirements on the\n\\(b_i\\)'s are sufficient to characterize the isomorphism type of\nthe group \\(\\gen{b_i \\mid i < 3}\\):\nany triple \\((c_i \\mid i < 3)\\)\nwhich produces this same \\emph{dynamical diagram} and satisfies these dynamical\nrequirements will generate a group isomorphic to \\(\\gen{b_i \\mid  i < 3}\\).\nIn fact the map \\(b_i \\mapsto c_i\\) will extend to an isomorphism.\nIn particular,\n\\[\n\\gen{b_i \\mid  i < 3} \\cong\n\\gen{b_i^{k_i} \\mid  i < 3}\n\\]\nfor any choice of \\(k_i\\geq 1\\) for each \\(i < 3\\).\n\nWe will now return to the general discussion and be more precise.\nRecall that if \\(f\\) is in \\({\\operatorname{Homeo}_+(I)}\\), then its \\emph{support} is defined to\nbe \\(\\operatorname{supt}(f):=\\{ t \\in I \\mid tf \\ne t\\}\\).\nA left (right) transition point of \\(f\\) is a \\(t \\in I \\setminus \\operatorname{supt}(f)\\) such that for every \\(\\epsilon > 0\\),\n\\((t,t+\\epsilon) \\cap \\operatorname{supt}(f) \\ne \\emptyset\\)\n(respectively \\((t-\\epsilon,t) \\cap \\operatorname{supt}(f) \\ne \\emptyset\\)).  \nAn \\emph{orbital} of \\(f\\) is a component its support.\nAn orbital of \\(f\\) is \\emph{positive} if \\(f\\) moves elements of the orbital to\nthe right;\notherwise it is negative.\nIf \\(f\\) has only finitely many orbitals, then the left (right) transition points\nof \\(f\\) are precisely the left (right) end points of its orbitals.\n\nA precursor to the notion of a \\emph{geometrically fast} generating set is that of a\n\\emph{geometrically proper} generating set.\nA set \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is\n\\emph{geometrically proper} if there is no element of \\(I\\) which is\na left transition point of more than one element of \\(X\\)\nor a right transition point of more than one element of \\(X\\).\nObserve that any geometrically proper generating set with only finitely many transition points\nis itself finite.\nFurthermore, geometrically proper sets are equipped with a canonical ordering\ninduced by the usual ordering on the least transition points of its elements.\nWhile the precise definition of \\emph{geometrically fast}\nwill be postponed until Section \\ref{FastBumpsSec},\nthe following statements describe the key features of the definition:\n\\begin{itemize}\n\n\\item geometrically fast generating sets are geometrically proper.\n\n\\item if \\(\\{a_i \\mid  i < n\\}\\) is geometrically proper, then\nthere is a \\(k \\geq 1\\) such that \n\\(\\{a_i^k \\mid  i < n\\}\\) is geometrically fast.\n\n\\item if \\(\\{a_i^k \\mid  i < n\\}\\) is geometrically fast and \\(k \\leq k_i\\) for \\(i < n\\), then\n\\(\\{a_i^{k_i} \\mid  i < n\\}\\) is geometrically fast.\n\n\\end{itemize}\n\\noindent\nOur main result is that the isomorphism types of groups with geometrically fast generating sets\nare determined by their qualitative dynamics.\nSpecifically, we will associate a \\emph{dynamical diagram} to each geometrically fast set\n\\(\\{a_i \\mid i < n\\} \\subseteq {\\operatorname{Homeo}_+(I)}\\) which has finitely many transition points.\nRoughly speaking, this is a record of the relative order of the orbitals\nand transition points of the various \\(a_i\\), as well as the orientation of their orbitals.\nIn the following theorem \\(M_X\\) is a certain finite set of points chosen from the orbitals\nof elements of \\(X\\) and \\(M \\gen{X} = \\{t g \\mid  t \\in M_X \\textrm{ and } g \\in \\gen{X}\\}\\).\nThese points will be chosen such that any nonidentity element of \\(\\gen{X}\\) moves a point in \\(M\\gen{X}\\).\n\n\\begin{thm} \\label{CombToIso}\nIf two geometrically fast sets \\(X , Y \\subseteq {\\operatorname{Homeo}_+(I)}\\) have only finitely many transition points\nand have isomorphic dynamical diagrams, then the induced\nbijection between \\(X\\) and \\(Y\\) extends to an isomorphism of\n\\(\\gen{X}\\) and \\(\\gen{Y}\\) (i.e. \\(\\gen{X}\\) is \\emph{marked isomorphic} to \\(\\gen{Y}\\)).\nMoreover, there is an order preserving bijection\n\\(\\theta :  M \\gen{X} \\to M \\gen{Y}\\) such that\n\\(f \\mapsto f^\\theta\\) induces the isomorphism \n\\(\\gen{X} \\cong \\gen{Y}\\).\n\\end{thm}\n\nWe will also establish that under some circumstances the map \\(\\theta\\) can be extended to a\ncontinuous order preserving surjection \\(\\hat \\theta : I \\to I\\).\n\n\\begin{thm} \\label{SemiConjThm}\nFor each finite dynamical diagram \\(D\\), there is a geometrically fast \\(X_D \\subseteq {\\operatorname{PL}_+(I)}\\)\nsuch that if \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is geometrically fast and has dynamical diagram \\(D\\),\nthen there is a marked isomorphism \\(\\phi:  \\gen{X} \\to \\gen{X_D}\\) and a continuous\norder preserving surjection \\(\\hat \\theta : I \\to I\\) such that\n\\(f \\hat \\theta = \\hat \\theta \\phi (f)\\) for all \\(f \\in \\gen{X}\\).\n\\end{thm}\n\nTheorem \\ref{CombToIso} has two immediate consequences.\nThe first follows from the readily verifiable fact that any dynamical diagram can\nbe realized inside of \\(F\\) (see, e.g., \\cite[Lemma 4.2]{CFP}).\n\n\\begin{cor} \\label{EmbeddInF}\nAny finitely generated\nsubgroup of \\({\\operatorname{Homeo}_+(I)}\\) which admits a geometrically fast generating set\nembeds into Thompson's group \\(F\\).\n\\end{cor}\n\\noindent\nSince by \\cite{brin+squier} \\(F\\) does not contain nontrivial free produces of groups,\nsubgroups of \\({\\operatorname{Homeo}_+(I)}\\) which admit geometrically fast generating sets are not free products.\nIt should also be remarked that while our motivation comes from studying\nthe groups \\(F\\) and \\({\\operatorname{PL}_+(I)}\\), the conclusion of Corollary \\ref{EmbeddInF} remains valid if\n\\(F\\) is replaced by, e.g. \\(\\operatorname{Diff}^\\infty_+ (I)\\).\n\n\\begin{cor} \\label{AlgFast}\nIf \\(\\{f_i \\mid i < n\\}\\) is geometrically fast, then\n\\(\\gen{f_i \\mid i < n}\\) is marked isomorphic to\n\\(\\gen{f_i^{k_i} \\mid i < n}\\) for any choice of \\(k_i \\geq 1\\).\n\\end{cor}\n\nIt is natural to ask how restrictive having a geometrically fast or geometrically\nproper generating set is.\nThe next theorem shows that many finitely generated subgroups of \\({\\operatorname{PL}_+(I)}\\)\nin fact do have at least a geometrically proper generating set.\n\n\\begin{thm} \\label{GeoProperGen}\nEvery \\(n\\)-generated one orbital subgroup of \\({\\operatorname{PL}_+(I)}\\)\neither contains an isomorphic copy of \\(F\\) or else admits an\n\\(n\\)-element geometrically proper generating set.\n\\end{thm}\n\n\\noindent\nNotice that every subgroup of \\({\\operatorname{Homeo}_+(I)}\\) is contained in a direct product of\none-orbital subgroups of \\({\\operatorname{Homeo}_+(I)}\\). \nThus if one's interest lies in studying the structure of subgroups of\n\\({\\operatorname{PL}_+(I)}\\) which do not contain copies of \\(F\\), then\nit is typically possible to restrict one's attention to groups\nadmitting geometrically proper generating sets.\nThe hypothesis of not containing an isomorphic copy of \\(F\\) in Theorem \\ref{GeoProperGen}\ncan not be eliminated.\nThis is a consequence of the following theorem and the fact that there are finite\nindex subgroups of \\(F\\) which are not isomorphic to \\(F\\) (see \\cite{bleak+wassink}).\n\n\\begin{thm} \\label{NoGeoProperGen}\nIf a finite index subgroup of \\(F\\) is isomorphic to \\(\\gen{X}\\) for some  geometrically proper\n\\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\), then it is isomorphic to \\(F\\).\n\\end{thm}\n\\noindent\nWe conjecture, however, that every finitely generated subgroup of \\(F\\) is bi-embeddable\nwith a subgroup admitting a geometrically fast generating set.\n\nWhile the results of this paper do of course readily adapt to \\({\\operatorname{Homeo}_+(\\R)} \\cong {\\operatorname{Homeo}_+(I)}\\),\nit is important to keep in mind that \\(\\pm \\infty\\) must be allowed as possible\ntransition points when applying the definition of geometric properness and hence\ngeometric fastness.\nFor example, it is easy to establish that \\(\\gen{t + \\sin (t), t + \\cos ( t)}\\) contains\na free group using the \\emph{ping-pong lemma} stated above\n(the squares of the generators generate a free group).\nMoreover, once we define \\emph{geometrically fast} in Section \\ref{FastBumpsSec},\nit will be apparent that the squares of the generators satisfies all of the requirements\nof being geometrically fast except that it is not geometrically proper\n(since, e.g., \\(\\infty\\) is a right transition point of both functions).\nAs noted above, this group does not embed into \\(F\\) and thus\ndoes not admit a geometrically fast (or even a geometrically proper) generating set.\nSee Example \\ref{InfiniteBumpEx} below for a more detailed discussion of a related example.\n\nThe paper is organized as follows.\nWe first review some standard definitions, terminology and notation in Section \\ref{DefSec}.\nIn Section \\ref{FastBumpsSec}, we will give a formal definition of \\emph{geometrically fast}\nand a precise definition of what is meant by a \\emph{dynamical diagram}.\nSection \\ref{GeoFastCriteriaSec} gives a reformulation of \\emph{geometrically fast}\nfor finite subsets of \\({\\operatorname{Homeo}_+(I)}\\) which facilitates algorithmic checking.\nThe proof of Theorem \\ref{CombToIso} is then divided between Sections\n\\ref{PingPongSec} and \\ref{CombToIsoSec}.\nThe bulk of the work is in\nSection \\ref{PingPongSec}, which uses an analog of the \\emph{ping-pong argument}\nto study the dynamics of geometrically fast sets of one orbital homeomorphisms.\nSection \\ref{CombToIsoSec} shows how this analysis implies \nTheorem \\ref{CombToIso} and how to derive its corollaries.\nIn Section \\ref{SemiConjSec}, we will prove Theorem \\ref{SemiConjThm}.\nThe group \\(F_n\\), which is the \\(n\\)-ary analog of Thompson's group \\(F\\), is\nshown to have a geometrically fast generating set in Section \\ref{FnSec}.\nSection \\ref{ExcisionSec} examines when bumps in\ngeometrically fast generating sets are extraneous and\ncan be excised without affecting the marked isomorphism type.\nProofs of Theorems \\ref{GeoProperGen} and \\ref{NoGeoProperGen} are given in Section \\ref{GeoProperSec}.\nFinally, the concept of \\emph{geometrically fast}\nis abstracted in Section \\ref{AbstractPingPongSec},\nwhere a generalization of Theorem \\ref{CombToIso} is stated and proved, as well as corresponding embedding\ntheorems for Thompson's groups \\(F\\), \\(T\\), and \\(V\\).\nThis generalization in particular covers infinite geometrically fast subsets of \\({\\operatorname{Homeo}_+(I)}\\).\nEven in the context of geometrically fast sets \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) with only finitely many\ntransition points, this abstraction gives a new way of understanding \\(\\gen{X}\\)\nin terms of symbolic manipulation.\n\n\n\\section[Preliminaries]{Preliminary definitions, notation and conventions}\n\n\\label{DefSec}\n\nIn this section we collect a number of definitions and conventions\nwhich will be used extensively in later sections.\nThroughout this paper, the letters \\(i,j,k,m,n\\) will be assumed to range\nover the nonnegative integers unless otherwise stated.\nFor instance, we will write \\((a_i \\mid i < k)\\) to denote a sequence\nwith first entry \\(a_0\\) and last entry \\(a_{k-1}\\).\nIn particular, all counting and indexing starts at 0 unless stated otherwise.\nIf \\(f\\) is a function and \\(X\\) is a subset of the domain of \\(f\\), we will write \\(f \\restriction X\\) to\ndenote the restriction of \\(f\\) to \\(X\\).\n\nAs we have already mentioned,\n\\({\\operatorname{Homeo}_+(I)}\\) will be used to denote the\nset of all orientation preserving homeomorphisms of \\(I\\);\n\\({\\operatorname{PL}_+(I)}\\) will be used to denote the set of all piecewise linear elements\nof \\({\\operatorname{Homeo}_+(I)}\\). \nThese groups will act on the right.\nIn particular, \\(tg\\) will denote the result of applying a homeomorphism\n\\(g\\) to a point \\(t\\).\nIf \\(f\\) and \\(g\\) are elements of a group, we will\nwrite \\(f^g\\) to denote \\(g^{-1}fg\\).\n\nRecall that from the introduction that\nif \\(f\\) is in \\({\\operatorname{Homeo}_+(I)}\\), then its \\emph{support} is defined to\nbe \\(\\operatorname{supt}(f):=\\{ t \\in I \\mid tf \\ne t\\}\\).\nThe support of a subset of \\({\\operatorname{Homeo}_+(I)}\\) is the union of the supports of its\nelements. \nA left (right) transition point of \\(f\\) is a \\(t \\in I \\setminus \\operatorname{supt}(f)\\) such that for every \\(\\epsilon > 0\\),\n\\((t,t+\\epsilon) \\cap \\operatorname{supt}(f) \\ne \\emptyset\\)\n(respectively \\((t-\\epsilon,t) \\cap \\operatorname{supt}(f) \\ne \\emptyset\\)).  \nAn \\emph{orbital} of \\(f\\) is a component its support.\nAn orbital of \\(f\\) is \\emph{positive} if \\(f\\) moves elements of the orbital to\nthe right;\notherwise it is negative.\nIf \\(f\\) has only finitely many orbitals, then the left (right) transition points\nof \\(f\\) are precisely the left (right) end points of its orbitals.\nAn \\emph{orbital} of a subset of \\({\\operatorname{Homeo}_+(I)}\\)\nis a component of its support.\n\nAn element of \\({\\operatorname{Homeo}_+(I)}\\) with one orbital will be referred to as a \\emph{bump function}\n(or simply a \\emph{bump}).\nIf a bump \\(a\\) satisfies that \\(t a > t\\) on its support,\nthen we say that \\(a\\) is \\emph{positive};\notherwise we say that \\(a\\) is \\emph{negative}.\nIf \\(f \\in {\\operatorname{Homeo}_+(I)}\\), then \\(b \\in {\\operatorname{Homeo}_+(I)}\\) is a \\emph{signed bump of \\(f\\)} if\n\\(b\\) is a bump which agrees with \\(f\\) on its support.\nIf \\(X\\) is a subset of \\({\\operatorname{Homeo}_+(I)}\\), then a bump \\(a\\)\nis \\emph{used in} \\(X\\)\nif \\(a\\) is positive and\nthere is an \\(f\\) in \\(X\\) such that \\(f\\)\ncoincides with either \\(a\\) or \\(a^{-1}\\) on the support of \\(a\\).\nA bump \\(a\\) is used in \\(f\\) if it is used in \\(\\{f\\}\\).\nWe adhere to the convention that only positive\nbumps are used by functions to avoid ambiguities in some statements.\nObserve that if \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is such that\nthe set \\(A\\) of bumps used in \\(X\\) is finite, then\n\\(\\gen{X}\\) is a subgroup of \\(\\gen{A}\\).\n\nIf \\((g_i \\mid i < n)\\) and \\((h_i \\mid i < n)\\) are two generating sequences for groups,\nthen we will say that \\(\\gen{g_i \\mid i < n}\\) is \\emph{marked isomorphic} to\n\\(\\gen{h_i \\mid i < n}\\) if the map \\(g_i \\mapsto h_i\\) extends to an isomorphism\nof the respective groups.\nIf \\(X\\) is a finite geometrically proper subset of \\({\\operatorname{Homeo}_+(I)}\\), then\nwe will often identify \\(X\\) with its enumeration in which the minimum transition\npoints of its elements occur in increasing order.\nWhen we write \\(\\gen{X}\\) is marked isomorphic to \\(\\gen{Y}\\), we are\nmaking implicit reference to these canonical enumerations of \\(X\\) and \\(Y\\).\n\nAt a number of points in the paper it will be important to distinguish between\nformal syntax (for instance words) and objects (such as group elements) to which they refer.\nIf \\(A\\) is a set, then a \\emph{string}\nof elements of \\(A\\) is a finite sequence of elements of \\(A\\).\nThe length of a string \\(\\str{w}\\) will be denoted \\(|\\str{w}|\\).\nWe will use \\(\\varepsilon\\) to denote the string of length 0.\nIf \\(\\str{u}\\) and \\(\\str{v}\\) are two strings,\nwe will use \\(\\str{uv}\\) to denote\ntheir concatenation;\nwe will say that \\(\\str{u}\\) is a \\emph{prefix} of \\(\\str{uv}\\)\nand \\(\\str{v}\\) is a \\emph{suffix} of \\(\\str{uv}\\).\nIf \\(A\\) is a subset of a group, then a \\emph{word} (in \\(A\\))\nis a string of elements of\n\\(A^\\pm := A \\cup A^{-1}\\).\nA \\emph{subword} of a word \\(\\str {w}\\) must preserve the order from \\(\\str {w}\\),\nbut does not have to consist of consecutive symbols from \\(\\str {w}\\).\nWe write \\(\\str {w}^{-1}\\) for the formal inverse of \\(\\str {w}\\): the product of\nthe inverses of the symbols in \\(\\str {w}\\) in reverse order.\n\nOften strings have an associated evaluation (e.g. a word represents an element of a group).\nWhile the context will often dictate whether we are working with a string or\nits evaluation, we will generally use the typewriter font (e.g. \\(\\str{w}\\)) for strings\nand symbols in the associated alphabets and standard math font (e.g. \\(w\\)) for the\nassociated evaluations.\n\nIn Section \\ref{GeoProperSec}, we will use the notion of the \\emph{left (right) germ}\nof a function \\(f \\in {\\operatorname{Homeo}_+(I)}\\) at an \\(s \\in I\\) which is fixed by \\(f\\)\n(left germs are undefined at 0 and right germs are undefined at 1).\nIf \\(0 \\leq s < 1\\), then define the \\emph{right germ of \\(f\\) at \\(s\\)} to be the set of all \\(g \\in {\\operatorname{Homeo}_+(I)}\\) such\nthat for some \\(\\epsilon > 0\\), \\(f \\restriction (s,s+\\epsilon) = g \\restriction (s,s+\\epsilon)\\);\nthis will be denoted by \\(\\gamma_s^+(f)\\).\nSimilarly if \\(0 < s \\leq 1\\), then one defines the \\emph{left germ of \\(f\\) at \\(s\\)};\nthis will be denoted by \\(\\gamma_s^-(f)\\).\nThe collections \n\\[\n\\{\\gamma_s^+(f) \\mid f \\in {\\operatorname{Homeo}_+(I)} \\textrm{ and } sf = s\\}\n\\]\n\\[\n\\{\\gamma_s^-(f) \\mid f \\in {\\operatorname{Homeo}_+(I)} \\textrm{ and } sf = s\\}\n\\]\nform groups and\nthe functions \\(\\gamma_s^+\\) and \\(\\gamma_s^-\\) are homomorphisms defined\non the subgroup of \\({\\operatorname{Homeo}_+(I)}\\) consisting of those functions which fix \\(s\\).\n\n\n \n\\section[Fast collections of bumps]{Fast collection of bumps and their dynamical diagrams}\n\n\\label{FastBumpsSec}\n\nWe are now ready to turn to the definition of \\emph{geometrically fast} in the context of finite subsets of\n\\({\\operatorname{Homeo}_+(I)}\\).\nFirst we will need to develop some terminology.\nA \\emph{marking} of a geometrically proper collection of bumps \\(A\\)\nis an assignment of a \\emph{marker} \\(t \\in \\operatorname{supt}(a)\\) to each \\(a\\) in \\(A\\).\nIf \\(a\\) is a positive bump with orbital \\((x,y)\\) and marker \\(t\\), then we define\nits \\emph{source} to be the interval \\(\\operatorname{src}(a) := (x,t)\\)\nand its \\emph{destination} to be the interval \\(\\operatorname{dest}(a) := [t a,y)\\).\nWe also set \\(\\operatorname{src}(a^{-1}) := \\operatorname{dest}(a)\\) and \\(\\operatorname{dest} (a^{-1}) := \\operatorname{src}(a)\\).\nThe source and destination of a bump are collectively called its \\emph{feet}. \nNote that there is a deliberate asymmetry in this definition:\nthe source of a positive bump is an open interval whereas the destination is half open.\nThis choice is necessary so that for any \\(t \\in \\operatorname{supt} (a)\\), there is a unique \\(k\\) such that\n\\(t a^k\\) is not in the feet of \\(a\\), something which is a key feature of the definition.\n\nA collection \\(A\\) of bumps is \\emph{geometrically fast}\nif it there is a marking of \\(A\\) for which its feet form a pairwise disjoint family\n(in particular we require that \\(A\\) is geometrically proper).\nThis is illustrated in Figure \\ref{GeoFastFig}, where the feet of\n\\(a_0\\) are \\((p,q)\\) and \\([r,s)\\) and the feet of \\(a_1\\) are\n\\((q,r)\\) and \\([s,t)\\).\n\\newcommand{\\hashlabel}[1]\n{\n\\xy\n(0,0); (0,-2)**@{-}; (0,-4)*{\\scriptstyle #1};\n\\endxy\n}\n\\begin{figure}\n\\[\n\\xy\n(0,36); (0,0)**@{-}; (36,0)**@{-}; (0,0); (36,36)**@{-};\n(16,15.6)*{\\dvarbump{4}{16}{19}}; (9,19)*{a_0};\n(22,21.6)*{\\dvarbump{2}{14}{16}}; (17.5,27.5)*{a_1};\n(14,14);(14,20)**@{.};(20,20)**@{.};\n(20,25.5)**@{.};(25.5,25.5)**@{.};\n(6.57,4.1)*{\\hashlabel{p}};\n(14.02,11.55)*{\\hashlabel{q}};\n(20.0,17.55)*{\\hashlabel{r}};\n(25.55,23.15)*{\\hashlabel{s}};\n(30,27.55)*{\\hashlabel{t}};\n\\endxy\n\\]\n\\caption{A geometrically fast set of bumps}\\label{GeoFastFig}\n\\end{figure}\nBeing geometrically fast is precisely the set of dynamical requirements made on the set\n\\(\\{a_i \\mid i < 3\\}\\) of homeomorphisms mentioned in the introduction.\nWe do not require here that \\(A\\) is finite and we will explicitly state finiteness as\na hypothesis when it is needed.\nNotice however that, since pairwise disjoint families of intervals\nin \\(I\\) are at most countable, any geometrically fast set of bumps is\nat most countable.\nThe following are readily verified and can be used axiomatically to\nderive most of the lemmas in Section \\ref{PingPongSec}\n(specifically Lemmas \\ref{LocRedBasics}--\\ref{FellowTraveler3}):\n\\begin{itemize}\n\n\\item for all \\(a \\in A^\\pm\\), \\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(a)\\) and if\n\\(x \\in \\operatorname{supt}(a)\\) then there exists a \\(k\\) such that \\(xa^k \\in \\operatorname{dest}(a)\\);\n\n\\item if \\(a \\ne b \\in A^\\pm\\), then \\(\\operatorname{dest}(a) \\cap \\operatorname{dest}(b) = \\emptyset\\);\n\n\\item if \\(a \\in A^\\pm\\) and \\(x \\in \\operatorname{supt}(a)\\), then\n\\(x a \\in \\operatorname{dest}(a)\\) if and only if \\(x \\not \\in \\operatorname{src}(a) := \\operatorname{dest}(a^{-1})\\).\n\n\\item if \\(a,b \\in A^\\pm\\), then \\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(b)\\) or\n\\(\\operatorname{dest}(a) \\cap \\operatorname{supt}(b) = \\emptyset\\).\n\n\\end{itemize}\nThis axiomatic viewpoint will be discussed further in Section \\ref{AbstractPingPongSec}.\n\nA set \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is geometrically fast\nif it is geometrically proper and the set of bumps used in \\(X\\) is geometrically fast.\nNote that while geometric properness is a consequence of the disjointness of the feet\nif \\(X\\) uses only finitely many bumps, it is an additional requirement in general.\nThis is illustrated in the next example.\n\n\\begin{example} \\label{InfiniteBumpEx}\nConsider the following homeomorphism of \\({\\mathbf R}\\):\n\\[\nt \\gamma = \n\\begin{cases}\n3t & \\textrm{ if } 0 \\leq t \\leq 1\/2 \\\\\n(t+4)\/3 & \\textrm{ if } 1\/2 \\leq t \\leq 2 \\\\\nt & \\textrm{ otherwise}\n\\end{cases}\n\\]\nDefine \\(\\alpha , \\beta \\in {\\operatorname{Homeo}_+(\\R)}\\) by \n\\((t+2p) \\alpha = t \\gamma + 2p\\) and \\((t+2p+1) \\beta = t \\gamma + 2p+1\\)\nwhere \\(p \\in {\\mathbf Z}\\) and \\(t \\in [0,2]\\).\nThus the bumps used in \\(\\alpha\\) are obtained by translating \\(\\gamma\\) by even integers;\nthe bumps in \\(\\beta\\) are the translates of \\(\\gamma\\) by odd integers.\nIf we assign the marker \\(1\/2\\) to \\(\\gamma\\) and mark the translation of \\(\\gamma\\) by \\(p\\)\nwith \\(p + 1\/2\\), then it can be seen that the feet of \\(\\alpha\\) and \\(\\beta\\) are the intervals\n\\(\\{(p,p+1\/2) \\mid p \\in {\\mathbf Z}\\} \\cup \\{[p+1\/2,p+1) \\mid p \\in {\\mathbf Z}\\}\\),\nwhich is a pairwise disjoint family.\nThus the bumps used in \\(\\{\\alpha,\\beta\\}\\) are geometrically fast.\nSince \\(\\infty\\) is a right transition point of both \\(\\alpha\\) and \\(\\beta\\), \n\\(\\{\\alpha,\\beta\\}\\) is not geometrically proper and hence not fast.\nIn fact, it follows readily from the formulation of the classical \\emph{ping-pong lemma}\nin the introduction that \\(\\gen{\\alpha,\\beta}\\) is free.\n\\end{example}\n\nObserve that if \\(X\\) is geometrically proper, each of its elements uses only finitely many bumps,\nand the set of transition points of \\(X\\) is discrete, \nthen there is a map \\(f \\mapsto k(f)\\) of \\(X\\) into the positive integers such that\n\\(\\{f^{k(f)} \\mid  f \\in X\\}\\) is geometrically fast.\nTo see this, start with a marking such that the closures of the sources of the bumps used in\n\\(X\\) are disjoint;\npick \\(f \\mapsto k(f)\\) sufficiently large so that all of the feet become disjoint.\nAlso notice that if \\(\\{f^{k(f)} \\mid f \\in X\\}\\) is geometrically fast and if \\(k(f) \\leq l(f)\\) for \\(f \\in X\\),\nthen \\(\\{f^{l(f)} \\mid f \\in X \\}\\) is geometrically fast as well.\n\nIf \\(X\\) is a geometrically fast generating set with only finitely many transition points,\nthen the \\emph{dynamical diagram} \\(D_X\\) of \n\\(X\\) is the edge labeled vertex ordered directed graph defined as follows:\n\\begin{itemize}\n\n\\item the vertices of \\(D_X\\) are the feet of \\(X\\) with the order\ninduced from the order of the unit interval;\n\n\\item the edges of \\(D_X\\) are the signed bumps of \\(X\\) directed\nso that the source (destination) of the edge is the source (destination) of the\nbump;\n\n\\item the edges are labeled by the elements of \\(X\\) that they come from.\n\n\\end{itemize}\n\\noindent\nWe will adopt the convention that dynamical diagrams are necessarily finite.\nThe dynamical diagram of a generating set for the Brin-Navas group \\(B\\)\n\\cite{MR2160570} \\cite{MR2135961}\nis illustrated in the left half of Figure \\ref{BNFigure};\nthe generators are \\(f = a_0^{-1} a_2\\) and \\(g = a_1^{-1}\\),\nwhere the \\((a_i \\mid  i < 3)\\) is the geometrically fast generating sequence illustrated in\nFigure \\ref{TrackingPoint}.\nWe have found that when drawing dynamical diagram \\(D_X\\) of a given \\(X\\), it is more\n{\\ae}sthetic whilst being unambiguous to collapse\npairs of vertices \\(u\\) and \\(v\\) of \\(D_X\\) such that:\n\\begin{itemize}\n\n\\item  \\(v\\) is the immediate successor of \\(u\\) in the order on \\(D_X\\),\n\n\\item \\(u\\)'s neighbor is below \\(u\\), and \\(v\\)'s neighbor is above \\(v\\).\n\n\\end{itemize}\nAdditionally, arcs can be drawn as over or under arcs to indicate their direction,\neliminating the need for arrows.\nThis is illustrated in the right half of Figure \\ref{BNFigure}.\nThe result qualitatively resembles the graphs of the homeomorphisms rotated so that\nthe line \\(y =x\\) is horizontal.\n\nAn isomorphism between dynamical diagrams is a directed graph isomorphism\nwhich preserves the order of the vertices and\ninduces a bijection between the edge labels\n(i.e. two directed edges have equal labels before applying the isomorphism if and only if\nthey have equal labels after applying the isomorphism).\nNotice that such an isomorphism is unique if it exists --- there is at most one order preserving bijection\nbetween two finitely linear orders.\n\n\\begin{figure}\n\\[\n\\xy\n(0,5)*{\n\\xymatrix{\n{\\bullet} &\n{\\bullet} &\n{\\bullet} \\ar@\/^1.5pc\/[ll]^f  &\n{\\bullet} \\ar@\/^1.5pc\/[rr]^f &\n{\\bullet} \\ar@\/^1.5pc\/[lll]^g&\n{\\bullet}\n}};\n(-2,10); (63,10)**@{.};\n(40,27)*{}; \n(50,18)*{};\n\\endxy\n\\quad\n\\quad\n\\xy\n(0,5)*{\n\\xymatrix{\n{\\bullet} &\n{\\bullet} &\n{\\bullet} \\ar@{-}@\/^1.5pc\/[ll]^f \\ar@{-}@\/^1.5pc\/[rr]^f&\n{\\bullet} \\ar@{-}@\/^1.5pc\/[ll]^g&\n{\\bullet} \n}};\n(-2,10); (51,10)**@{.};\n(40,27)*{}; \n(50,18)*{};\n\\endxy\n\\]\n\\caption{The dynamical diagram for the Brin-Navas generators, with an illustration\nof the contraction convention \\label{BNFigure}}\n\\end{figure}\n\\begin{figure}\n\\[\n\\xy\n(0,36); (0,0)**@{-}; (36,0)**@{-}; (0,0); (36,36)**@{-};\n(10,9.6)*{\\dvarbump{2}{14}{16}}; (5,15)*{a_0};\n(18,17.6)*{\\dvarbump{2}{14}{16}}; (13.5,23.5)*{a_1};\n(26,25.6)*{\\dvarbump{2}{14}{16}}; (22,32)*{a_2};\n(10,10);(10,15.25)**@{.};(15.25,15.25)**@{.};(15.25,21)**@{.};(21,21)**@{.};(21,26.2)**@{.};(26.2,26.2)**@{.};\n\\endxy\n\\]\n\\caption{A point is tracked through a fast transition chain}\\label{TrackingPoint}\n\\end{figure}\nObserve that the (uncontracted) dynamical diagram of any geometrically\nfast \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) which has finitely many transition points has the property that\nall of its vertices have total degree 1.\nMoreover, any finite edge labeled vertex ordered directed graph in which each vertex has total\ndegree 1 is isomorphic to the dynamical diagram of some geometrically fast \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\)\nwhich has finitely many transition points (see the proof of Theorem \\ref{SemiConjThm} in Section\n\\ref{SemiConjSec}).\nThus we will write \\emph{dynamical diagram} to mean a finite edge labeled vertex ordered\ndirected graph in which each vertex has total degree 1.\nThe edges in a dynamical diagram will be referred to as \\emph{bumps} and the vertices\nin a dynamical diagram will be referred to as \\emph{feet}.\nTerms such as \\emph{source}, \\emph{destination}, \\emph{left\/right foot} will be given the obvious meaning\nin this context.\n\nNow let \\(A\\) be a geometrically fast set of positive bump functions.\nAn element of \\(A\\) is \\emph{isolated} (in \\(A\\)) if\nits support contains no transition points of \\(A\\).\nIn the dynamical diagram of \\(A\\), this corresponds to a bump whose source and destination are\nconsecutive feet.\nThe next proposition shows that we may always eliminate isolated bumps in \\(A\\) by\nadding new bumps to \\(A\\).\nThis will be used in Section \\ref{SemiConjSec}\n\n\\begin{figure}\n\\[\n\\xy\n\\xymatrix{\n{\\bullet} \\ar@\/^2.5pc\/[rrrrr]^a &\n{\\bullet} \\ar@\/^1.0pc\/[rr]^{b_0} &\n{\\bullet} \\ar@\/^1.0pc\/[rr]^{b_1} &\n{\\bullet} &\n{\\bullet} &\n{\\bullet} &\n}\n\\endxy\n\\]\n\\caption{The bump \\(a\\) is made nonisolated by the addition of bumps \\(b_0\\) and \\(b_1\\).\n\\label{IsolatedFixFig}}\n\\end{figure}\n\n\\begin{prop} \\label{IsolatedFixProp}\nIf \\(A \\subseteq {\\operatorname{Homeo}_+(I)}\\) is a geometrically fast set of positive bump functions,\nthen there is a geometrically fast \\(B \\subseteq {\\operatorname{Homeo}_+(I)}\\) such that \\(A \\subseteq B\\)\nand \\(B\\) has no isolated bumps.\nMoreover, if \\(A\\) is finite, then \\(B\\) can be taken to be finite as well.\n\\end{prop}\n\n\\begin{proof}\nIf \\(a \\in A\\) is isolated, let \\(b_0\\) and \\(b_1\\) be a geometrically fast pair of bumps with supports contained in\n\\(\\operatorname{supt}(a) \\setminus (\\operatorname{src}(a) \\cup \\operatorname{dest}(a))\\) such that neither \\(b_0\\) nor \\(b_1\\) is isolated in \\(\\{b_0,b_1\\}\\);\nsee Figure \\ref{IsolatedFixFig}.\nSince the feet of \\(A\\) are disjoint, so are the feet of \\(A \\cup \\{b_0,b_1\\}\\) and \\(a\\) is no longer isolated\nin \\(A \\cup \\{b_0,b_1\\}\\).\nLet \\(B\\) be the result of adding such a pair of bumps for each isolated bump in \\(A\\).\n\\end{proof}\n\n\n\\section[Criteria for geometric fastness]{An algorithmically check-able criteria for geometric fastness}\n\\label{GeoFastCriteriaSec}\n\nIn this section we will consider geometrically proper sets which have finitely many transition points and\ndevelop a characterization of when they are geometrically fast.\nThis characterization moreover allows one to determine algorithmically when such sets are geometrically fast.\nIt will also provide a canonical marking of geometrically fast sets with finitely many transition points.\nWe need the following refinement of the notion of a \\emph{transition chain}\nintroduced in \\cite{MR2466019}.\nLet \\(A \\subseteq {\\operatorname{Homeo}_+(I)}\\)\nbe a finite geometrically proper set of positive bump functions. \n\nA sequence \\((a_i \\mid i\\le k)\\) of nonisolated elements of \\(A\\)  is a\n\\emph{stretched transition chain of \\(A\\)} if:\n\\begin{enumerate}\n\n\\item \\label{TransitionChain}\nfor all \\(i<k\\), \\(x_i<x_{i+1}<y_i<y_{i+1}\\), where \n\\((x_i,y_i)\\) is the support of \\(a_i\\);\n\n\\item \\label{StretchedChain}\nno transition point of \\(A\\) is in any interval \\((x_{i+1},y_i)\\).\n\n\\end{enumerate}\nNotice that whether \\(C\\) is a stretched transition chain depends not only on the\nelements of \\(A\\) listed in \\(C\\) but on the entire collection \\(A\\).\nSince the left transition points in a stretched transition chain are strictly\nincreasing, we will often identify such sequences with their range.\nIn particular, a stretched transition chain \\(C\\) is \\emph{maximal} if is maximal with\nrespect to containment when\nregarded as a subset of \\(A\\).\nWe will use \\(\\prod C\\) to denote the composition \\(a_0 \\cdots a_k\\).\nAn element \\(a\\) of \\(A\\) is \\emph{initial} (in \\(A\\)) if either \\(a\\) is isolated\nor else the least transition point of \\(A\\) in the support of \\(a\\) is not\nthe right transition point of some element of \\(A\\).\nThese are precisely the elements of \\(A\\) which are the initial element of any\nstretched transition chain which contains them.\n\nFigure \\ref{ChainDecompFig} shows a dynamical diagram, along with a list of the\nmaximal stretched transition chains.\n\\begin{figure}\n\\[\n\\xy\n\\xymatrix{\n{\\bullet} \\ar@\/^1.5pc\/[rrr]^{a_0} &\n{\\bullet} \\ar@\/^2.5pc\/[rrrrr]^{a_1} &\n{\\bullet} \\ar@\/^3.0pc\/[rrrrrrr]^{a_2} &\n{\\bullet} \\ar@\/^2.0pc\/[rrrr]^{a_3} &\n{\\bullet} \\ar@\/^0.5pc\/[r]^{a_4} &\n{\\bullet} \\ar@\/^1.5pc\/[rrr]^{a_5} &\n{\\bullet} &\n{\\bullet} &\n{\\bullet} &\n{\\bullet} &\n}\n\\endxy\n\\]\n\\caption{The stretched transition chains in this dynamical diagram are\n\\(\\{a_0,a_2\\}\\), \\(\\{a_1,a_5\\}\\), and \\(\\{a_3\\}\\); \\(a_4\\) is isolated\n\\label{ChainDecompFig}}\n\\end{figure}\n\n\\begin{prop}\\label{StretchPartProp}\nIf \\(A\\) is a finite geometrically proper set of positive bump functions in \\({\\operatorname{Homeo}_+(I)}\\),\nthen the maximal stretched transition chains in \\(A\\) partition the nonisolated elements of \\(A\\).\n\\end{prop}\n\n\\begin{remark}\nWe will see in Section \\ref{FnSec} that geometrically fast stretched transition chains of length \\(n\\) generate\n\\(F_n\\), the \\(n\\)-ary analog of Thompson's group \\(F\\).\nThus if \\(A\\) is geometrically fast, then \\(\\gen{A}\\) is a sort of amalgam of copies of the\n\\(F_n\\) and copies of \\({\\mathbf Z}\\).\n\\end{remark}\n\n\\begin{proof}\nFirst observe that any sequence consisting of a single nonisolated element of \\(A\\)\nis a stretched transition chain and thus is a subsequence of\nsome maximal stretched transition chain in \\(A\\).\nNext  suppose that \\(a\\) and \\(b\\) are consecutive members of a stretched transition chain.\nIt follows that \\(b\\)'s left transition point is in the support of \\(a\\) and is\nthe greatest transition point of \\(A\\) in the support of \\(a\\).\nIn particular, \\(b\\) must immediately follow \\(a\\) in any maximal stretched transition chain.\nSimilarly, \\(a\\) must immediately precede \\(b\\) in any maximal stretched transition chain.\nThis shows that every nonisolated element of \\(A\\)\noccurs in a unique maximal stretched transition chain of \\(A\\).\n\\end{proof}\n\nIf \\(C\\) is a stretched transition chain, we define \\(C_{\\min}\\) as the\nleast transition point of \\(A\\) in the support \\(C\\) and\n\\(C_{\\max}\\) as the greatest transition point of \\(A\\) in the support \\(C\\).  \nSince \\(C\\) has no isolated bumps, both \\(C_{\\min}\\) and \\(C_{\\max}\\) are well defined.\n\n\\begin{prop}\\label{FastCriterion}\nIf \\(A \\subseteq {\\operatorname{Homeo}_+(I)}\\) is a finite geometrically\nproper set of positive bump functions, then the following are equivalent:\n\\begin{enumerate}\n\n\\item \\label{FastHyp}\n\\(A\\) is geometrically fast;\n\n\\item \\label{FastChainsHyp}\nevery stretched transition chain \\(C\\) of \\(A\\) satisfies\n\\(C_{\\max} \\leq C_{\\min} \\prod C\\).\n\n\\item \\label{MaxFastChainsHyp}\nevery maximal stretched transition chain \\(C\\) of \\(A\\) satisfies\n\\(C_{\\max} \\leq C_{\\min} \\prod C\\).\n\\end{enumerate}\n\\end{prop}\n\n\\begin{remark}\nThis criterion for being geometrically fast was our original motivation for\nthe choice of the terminology: \nthe dynamics of the homeomorphisms are such that\ntransition points can be moved to the right\nthrough transition chains as efficiently as possible. \nThis is illustrated in Figure \\ref{TrackingPoint}.\n\\end{remark}\n\n\\begin{remark}\nThe choice not to allow isolated bumps to be singleton stretched transition chains is somewhat arbitrary,\nalthough it would be necessary to make awkward adjustments to the definitions above if\nwe took the alternate approach.\nIt seems appropriate to omit them since they play no role in determining whether a group\nis fast other than contributing to the collective set of transition points of the set of bumps under consideration.\n\\end{remark}\n\n\n\n\\begin{proof}\nTo see that (\\ref{FastHyp}) implies (\\ref{FastChainsHyp}), let \\(A\\) be given\nequipped with a fixed marking witnessing that it is geometrically fast.\nLet \\(C = (a_k \\mid k \\leq m)\\) and\nlet \\(s_k\\) denote the marker of \\(a_k\\).\nNotice that if \\(t\\) is a transition point of \\(A\\) which is in the support of \\(a_k\\),\nthen \\(s_k \\leq t \\leq s_k a_k\\); otherwise\na foot associated to \\(t\\) would intersect a foot of \\(a_k\\).\nIt follows that \\(s_k a_k\\) is in the support of \\(a_{k+1}\\)\nsince the left transition point of \\(a_{k+1}\\) is in the support of \\(a_k\\) by\n(\\ref{TransitionChain}) in the definition of stretched transition chain.\nTherefore the left foot of \\(a_{k+1}\\) is to the left of the right foot of \\(a_k\\)\nand so \\(s_{k+1} \\leq s_k a_k\\).\nWe now have inductively that \\(s_m \\leq s_0 a_0 \\cdots a_{m-1}\\) and\nhence\n\\[\nC_{\\max} \\leq s_m a_m \\leq s_0 a_0 \\cdots  a_m \\leq C_{\\min}  a_0 \\cdots a_m.\n\\]\n\nIn order to see that (\\ref{FastChainsHyp}) implies (\\ref{FastHyp}),\nlet \\(( a_k \\mid  k \\leq n )\\) be the enumeration of \\(A\\)\nsuch that the left transition points of the \\(a_k\\)'s occur in increasing order.\nLet \\((x_k,y_k)\\) denote the support of \\(a_k\\) for \\(k \\leq n\\).\nBegin by setting \\(s_k\\) to be the least transition point of \\(A\\) in the support of \\(a_k\\);\nif \\(a_k\\) is isolated we let \\(s_k\\)\nbe the midpoint of the support of \\(a_k\\).\nDefine \\(t_k\\) by induction.\nIf \\(a_k\\) is initial, we define \\(t_k := s_k\\).\nIf \\(s_k\\) is the right transition point of \\(a_j\\) for some \\(j < k\\),\nthen define \\(t_k := t_j a_j\\).\nBy induction, \\(t_k < y_k\\).\nObserve that if \\(t_k = t_j a_j\\), then \\(t_k = t_j a_j < s_k\\)\nsince \\(s_k\\) is the right transition point of \\(a_j\\) and in particular is fixed by \\(a_j\\).\n\n\\begin{claim} \\label{ChainClaim}\nIf \\(s\\) is a transition point of \\(A\\) in the support of \\(a_k\\),\nthen \\(s \\leq t_k a_k\\).\n\\end{claim}\n\n\\begin{proof}\nLet \\(C\\) be the stretched transition chain which finishes with \\(a_k\\) and\nwhich is maximal with this property.\nIf \\(C\\) is \\(\\{a_k\\}\\), then \\(t_k = s_k = C_{\\min}\\) and the claim follows\nfrom our hypothesis that\n\\(s \\leq C_{\\max} \\leq C_{\\min} \\prod C \\leq t_k a_k\\).\nIf \\(a_j\\) is the immediate predecessor of \\(a_k\\) in \\(C\\)\nand \\(C\\) starts with \\(a_{i}\\),\nthen \\(C_{\\min} = t_i = s_i\\).\nMoreover, if \\(C'\\) is obtained from \\(C\\) by removing \\(a_k\\), then\nby an inductive argument we have that \\(C_{\\min} \\prod C' = t_i \\prod C' = t_k\\).\nBy hypothesis (\\ref{FastChainsHyp}), we have\nthat \\(C_{\\max} \\leq C_{\\min} \\prod C = t_k a_k\\).\nSince \\(s_k\\) is the right transition point of \\(a_j\\) which is the last entry\nof \\(C'\\), we must\nhave that \\(t_i \\prod C' = t_k < s_k\\).\nFurthermore, since any transition point of \\(A\\) in the support of \\(a_k\\)\nis between \\(s_k\\) and \\(C_{\\max}\\) we have our desired conclusion.\n\\end{proof}\n\nWe now claim that the assignment \\(a_k \\mapsto t_k\\) defines a marking which witnesses that \\(A\\) is geometrically fast.\nWe need to show that if \\(i < j \\leq n\\), then the feet of \\(a_i\\) are disjoint from\nthose of \\(a_j\\).\nBy our assumption on the enumeration, we have that \\(x_i < x_j\\).\nNote that if \\(y_i < x_j\\), the support of \\(a_i\\) is disjoint from the support of\n\\(a_j\\).\nIn particular the feet of \\(a_i\\) and \\(a_j\\) are disjoint.\n\nIf \\(x_i < x_j < y_j < y_i\\),\nthen \\(x_i < t_i \\leq x_j\\).\nIn particular, the left foot \\((x_i,t_i)\\)\nof \\(a_i\\) is disjoint from the support of\n\\(a_j\\) and hence from its feet.\nClaim \\ref{ChainClaim}\nimplies that \\(y_j \\leq t_i a_i \\) and hence that the right foot of \\(a_i\\), which\nis \\([t_i a_i,y_i)\\) is disjoint from the support of \\(a_j\\).\n\nFinally, suppose that \\(x_i < x_j < y_i < y_j\\).\nObserve that the left foot of \\(a_i\\) is disjoint from the support\nof \\(a_j\\) and, by a similar argument as in the previous case,\nthe right foot of \\(a_j\\) is disjoint from the support of \\(a_i\\).\nThus we only need to verify that the right foot of \\(a_i\\) is disjoint\nfrom the left foot of \\(a_j\\).\nBy Claim \\ref{ChainClaim} \n\\[\nx_j < t_j \\leq s_j \\leq t_i a_i < y_i\n\\]\nand hence \\((x_j,t_j)\\) is disjoint from \\([t_ia_i,y_i)\\), as desired.\n\nIt now remains to show that (\\ref{MaxFastChainsHyp}) implies (\\ref{FastChainsHyp});\nwe argue the contrapositive.\nSuppose that \\(C\\) is a stretched transition chain and that, as a sequence, \\(C\\) is the\nconcatenation of \\(C'\\) and \\(C''\\), each of which are stretched transition chains.\nSince we have seen above that every stretched transition chain is an interval within\na maximal stretched transition chain,\nthe desired implication will follow if we can show that\n\\(C_{\\max} > C_{\\min} \\prod C\\) holds if either\n\\(C'_{\\max} > C'_{\\min} \\prod C'\\) or \\(C''_{\\max} > C''_{\\min} \\prod C''\\).\nIf \\(C'_{\\max} > C'_{\\min} \\prod C'\\), then\n\\[\nC_{\\max} \\geq C'_{\\max} > C'_{\\min} \\prod C' =\nC_{\\min} \\prod C' \\prod C'' =C_{\\min} \\prod C\n\\]\nsince \\(C'_{\\max}\\) is the least transition point of \\(C''\\) and hence a lower bound for the support\nof \\(\\prod C''\\).\nIf  \\(C'_{\\max} \\leq C'_{\\min} \\prod C'\\) but \\(C''_{\\max} > C''_{\\min} \\prod C''\\), then\n\\[\nC_{\\max} = C''_{\\max} > C''_{\\min} \\prod C''  \\geq C'_{\\min} \\prod C' \\prod C'' = C_{\\min} \\prod C\n\\]\nsince \\(C''_{\\min}\\) is the greatest transition point of \\(C'\\) and thus\nan upper bound for the support of \\(\\prod C'\\).\n\\end{proof}\n\nObserve that the proof of Proposition \\ref{FastCriterion}\ngives an explicit construction\nof a marking of a family \\(A\\) of positive bumps.\nThis marking has the property that if \\(A\\) is geometrically fast,\nthen it is witnessed as such by the marking.\nWe will refer to this marking as the \\emph{canonical marking} of \\(A\\).\n\nFinally, let us note that Proposition \\ref{FastCriterion} gives us a means\nto algorithmically check whether a set of positive bumps \\(A\\) is fast.\nSpecifically, perform the following sequence of steps:\n\\begin{itemize}\n\n\\item determine whether \\(A\\) is geometrically proper;\n\n\\item if so, partition the non isolated elements of\n\\(A\\) into maximal stretched transition chains;\n\n\\item for each maximal stretched transition chain \\(C\\) of \\(A\\),\ndetermine whether \\(C_{\\max} \\leq C_{\\min} \\prod C\\).\n\n\\end{itemize}\nThis is possible provided we are able to perform the following basic queries:\n\\begin{itemize}\n\n\\item test for equality among the transition points of elements of \\(A\\);\n\n\\item determine the order of the transition points of elements of \\(A\\);\n\n\\item determine the truth of \\(C_{\\max} \\leq C_{\\min} \\prod C\\) whenever \\(C\\) is a stretched transition chain.\n\n\\end{itemize}\n\n\n\n\\section{The ping-pong analysis of geometrically fast sets of bumps}\n\\label{PingPongSec}\n\nIn this section, we adapt the ping-pong argument to the setting of fast families of\nbump functions.\nWhile the culmination will be Theorem \\ref{Faithful} below, the lemmas we will develop will\nbe used in subsequent sections.\nThey also readily adapt to the more abstract setting of Section \\ref{AbstractPingPongSec}.\n\nFix, until further notice, a (possibly infinite) geometrically fast collection \\(A\\) of positive bumps\nequipped with a marking; in particular we will write \\emph{word} to mean \\emph{\\(A\\)-word}.\nCentral to our analysis will be the notion of a \\emph{locally reduced word}.\nA word \\(\\str{w}\\) is \\emph{locally reduced} at \\(t\\) if it is freely reduced and whenever\n\\(\\str{ua}\\) is a prefix of \\(\\str{w}\\) for \\(a \\in A^\\pm\\),\n\\(tua \\ne tu\\).\nIf \\(\\str{w}\\) is locally reduced at every element of a set \\(J \\subseteq I\\), then we write that\n\\(\\str{w}\\) is \\emph{locally reduced on \\(J\\)}.\n\nThe next lemma collects a number of useful observations about locally\nreduced words; we omit the obvious proofs.\nRecall that if \\(\\str{u}\\) and \\(\\str{v}\\) are freely reduced, then the free\nreduction of \\(\\str{uv}\\) has the form \\(\\str{u}_0 \\str{v}_0\\) where \\(\\str{u}=\\str{u}_0 \\str{w}\\),\n\\(\\str{v}=\\str{w}^{-1}\\str{v}_0\\), and \\(\\str{w}\\) is the longest common suffix of\n\\(\\str{u}\\) and \\(\\str{v}^{-1}\\).\nIn particular, if \\(\\str{u}\\), \\(\\str{v}\\), and \\(\\str{w}\\) are freely reduced words and the\nfree reductions of \\(\\str{uv}\\) and \\(\\str{uw}\\) coincide, then \\(\\str{v} = \\str{w}\\).\n\n\\begin{lemma}\\label{LocRedBasics} All of the following are true:\n\\begin{itemize}\n\n\\item For all \\(x\\in I\\) and all words \\(\\str{w}\\), there is a subword\n\\(\\str{v}\\) of \\(\\str{w}\\) which is locally reduced at \\(x\\) so that\n\\(xv=xw\\).\n\n\\item For all \\(x\\in I\\) and words \\(\\str{u}\\) and \\(\\str{v}\\), if \\(\\str{u}\\) is\nlocally reduced at \\(x\\), and \\(\\str{v}\\) is locally reduced at \\(xu\\), then\nthe free reduction of \\(\\str{uv}\\) is locally reduced at \\(x\\).\n\n\\item For all \\(x\\in I\\) and words \\(\\str{w}\\), if \\(\\str{w}\\) is locally\nreduced at \\(x\\) and \\(\\str{w}=\\str{uv}\\), then \\(\\str{u}\\) is locally reduced at\n\\(x\\) and \\(\\str{v}\\) is locally reduced at \\(xu\\).\n\n\\item For all \\(x\\in I\\) and words \\(\\str{w}\\), if \\(\\str{w}\\) is locally\nreduced at \\(x\\), then \\(\\str{w}^{-1}\\) is locally reduced at \\(xw\\).\n\n\\end{itemize}\n\\end{lemma}\n\\noindent\n(Recall here our convention that a \\emph{subword} is not required to consist\nof consecutive symbols of the original word.)\nFor \\(x\\in I\\), we use \\(x\\gen{A}\\) to denote the orbit\nof \\(x\\) under the action of \\(\\gen{A}\\) and for\n\\(S\\subseteq I\\), we let \\(S\\gen{A}\\) be the union of\nthose \\(x\\gen{A}\\) for \\(x\\in S\\).\n\nA marker \\(t\\) of \\(A\\) is \\emph{initial} if whenever \\(s < t\\) is the marker of \\(a \\in A\\),\nthen \\(t \\ne sa\\).\nIf \\(A\\) is finite and we are working with the canonical marking, then the initial markers\nare precisely the markers of the initial intervals.\nLet \\(M_A\\) be the set of initial markers of \\(A\\).\nWe will generally suppress the subscript if the meaning is clear from the context;\nin particular we will write \\(M\\gen{A}\\) for \\(M_A\\gen{A}\\).\nNotice that every marker is contained in \\(M \\gen{A }\\).\n\nAside from developing lemmas for the next section,\nthe goal of this section is to prove that the action of\n\\(\\gen{A}\\) on \\(M\\gen{A}\\) is faithful.\nThe next lemma is the manifestation of the \\emph{ping-pong argument}\nin the context in which we are working.\nIf \\(\\str{w} \\ne \\varepsilon\\) is a word, the \\emph{source} (\\emph{destination}) of\n\\(\\str{w}\\) is the source of the first\n(destination of the last) symbol in \\(\\str{w}\\).\nThe source and destination of \\(\\varepsilon\\) are \\(\\emptyset\\).\n\n\\begin{lemma} \\label{PingPong}\nIf \\(x \\in I\\) and \\(\\str{w} \\ne \\varepsilon\\)\nis a word which is locally reduced at \\(x\\), then\neither \\(x \\in \\operatorname{src}(\\str{w})\\) or \\(xw \\in \\operatorname{dest}(\\str{w})\\).\n\\end{lemma}\n\n\\begin{proof}\nThe proof is by induction on the length of \\(\\str{w}\\).\nWe have already noted\nthat if \\(\\str{w} = \\str{a}\\) and \\(t \\in \\operatorname{supt}(a)\\),\nthen \\(ta \\in \\operatorname{dest}(a)= \\operatorname{dest}(\\str{w})\\) if and only if \\(t \\not \\in \\operatorname{src}(a) = \\operatorname{src}(\\str{w})\\).\nNext suppose that \\(\\str{w}\\) has length at least 2, \\(x \\not \\in \\operatorname{src}(\\str{w})\\),\nand let \\(\\str{v}\\) be a (possibly empty) word such that \\(\\str{w} = \\str{vab}\\)\nfor \\(a,b \\in A^\\pm\\).\nSince \\(\\str{w}\\) is locally reduced, \\(b \\ne a^{-1}\\)\nand thus the destination of \\(a\\) is not the source of \\(b\\).\nSince \\(A\\) is geometrically fast, the destination of \\(a\\) is disjoint from the source of\n\\(b\\).\nBy our inductive hypothesis, \\(y = x v a\\) is in\n\\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(b) \\setminus \\operatorname{src}(b)\\).\nThus \\(x w = x v ab  = y b\\) is in the destination of \\(b\\).\n\\end{proof}\n\nIf \\(\\str{w}\\) is a word, define \\(J(\\str{w}) :=\\operatorname{supt}(a)\\setminus \\operatorname{src}(a)\\)\nwhere \\(\\str{a}\\) is the first symbol of\n\\(\\str{w}\\).\nNotice that if \\(\\str{w}\\) is locally reduced at \\(x\\), then ``\\(x \\in J(\\str{w})\\)'' is equivalent to\n``\\(x \\not \\in \\operatorname{src}(\\str{w})\\)''.\nThe following lemma is easily established by induction on the length of \\(\\str{w}\\)\nusing Lemma \\ref{PingPong}.\n\n\\begin{lemma} \\label{threaded}\nIf \\(\\str{w}\\) is a word and \\(x \\in J(\\str{w})\\), then \\(\\str{w}\\) is locally reduced\nat \\(x\\) if and only if \\(\\str{w}\\) is freely reduced and \n\\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(b)\\) whenever \\(\\str{ab}\\) are consecutive symbols in \\(\\str{w}\\).\nIn particular, if \\(\\str{w}\\) is freely reduced, then \\(\\str{w}\\) is locally reduced on \\(J(\\str{w})\\)\nprovided \\(\\str{w}\\) is locally reduced at some element of \\(J(\\str{w})\\).\n\\end{lemma}\n\nWhen applying Lemma \\ref{PingPong}, it will be useful to be able to assume\nthat \\(x\\) is not in \\(\\operatorname{src}(\\str{w})\\).\nNotice that if \\(x\\) is not in the feet of any elements of \\(A\\), then this is automatically true\n(for instance this is true if \\(x \\in M\\)).\nThe next lemma captures an important consequence of Lemma \\ref{PingPong}.\n\n\\begin{lemma}\\label{Collision}\nSuppose \\(x_0,x_1 \\in I\\), \\(\\str{u}_i\\) is locally reduced at \\(x_i\\) and\n\\(x_i \\not \\in \\operatorname{src}(\\str{u}_i)\\).\nIf \\(|\\str{u}_0| \\leq |\\str{u}_1|\\) and \\(x_0 u_0 = x_1 u_1\\),\nthen \\(\\str{u}_0\\) is a suffix of \\(\\str{u}_1\\).\nIn particular if \\(t \\in I\\) is not in any of the feet of \\(A\\)\nand \\(\\str{u}\\) and \\(\\str{v}\\) are words\nthat are locally reduced at \\(t\\) with \\(tu=tv\\), then \\(\\str{u}=\\str{v}\\).\n\\end{lemma}\n\n\\begin{proof}\nThe main part of the lemma is proved by induction on \\(|\\str{u}_0|\\).\nIf \\(\\str{u}_0 = \\varepsilon\\), this is trivially true.\nNext suppose that \\(\\str{u}_i \\str{a}_i\\) is locally\nreduced at \\(x_i\\) and \\(x_i \\not \\in \\operatorname{src}(\\str{u}_i \\str{a}_i)\\).\nIf \\(x_0 u_0 a_0 = x_1 u_1 a_1\\),\nthen by Lemma \\ref{PingPong},\n\\(\\str{a}_0 = \\str{a}_1\\).\nWe are now finished by applying our\ninduction hypothesis to conclude that \\(\\str{u}_0\\) is a suffix of \\(\\str{u}_1\\).\n\nIn order to see the second conclusion, let \\(t\\), \\(\\str{u}\\) and \\(\\str{v}\\) be given such that\n\\(x:= tu = tv\\) and assume without loss of generality that \\(|\\str{u}| \\leq |\\str{v}|\\).\nBy the main assertion of the lemma, \\(\\str{v} = \\str{wu}\\) for some \\(\\str{w}\\).\nSince \\(tw = t\\), Lemma \\ref{PingPong} implies \\(\\str{w} = \\varepsilon\\).\n\\end{proof}\n\nIf \\(x \\in \\operatorname{src}(a)\\) for some \\(a \\in A^\\pm\\), then \\(x \\in \\operatorname{dest}(a^{-1})\\).\nThis suggests that we have ``arrived at'' \\(x\\) by applying a locally reduced word to some other point.\nMoreover  \\(a^{-1}\\) is the unique element \\(b\\) of \\(A^\\pm\\) such that \\(x \\in \\operatorname{dest}(b)\\).\nThus we may attempt to ``trace back'' to where \\(x\\) ``came from.''\nThis provides a recursive definition of a sequence which starts at\n\\(\\str{a}^{-1}\\) and grows to the left, possibly infinitely far.\nThis gives rise to the notion of a \\emph{history} of a point \\(x \\in I\\), which\nwill play an important role in the\nproof of Theorem \\ref{Faithful} below and also in Section \\ref{AbstractPingPongSec}.\nIf \\(t \\in I\\) is not in \\(\\operatorname{dest}(a)\\) for any \\(a \\in A^\\pm\\), then we say that\n\\(t\\) has \\emph{trivial history} and define \\(\\tilde t := \\{a \\in A : t \\in \\operatorname{supt}(a)\\}\\).\nIf \\(x \\in I\\), define \\(\\eta(x)\\) to be the set of all strings of the following form:\n\\begin{itemize}\n\n\\item words \\(\\str{u}\\) such that for some \\(t \\in I\\),\n\\(tu = x\\), \\(\\str{u}\\) is locally reduced at \\(t\\), and \\(t \\not \\in \\operatorname{src}(\\str{u})\\);\n\n\\item strings \\(\\str{\\tilde t u}\\) such that \\(t\\) has trivial history, \\(\\str{u}\\) is locally\nreduced at \\(t\\), and \\(tu = x\\).\n\n\\end{itemize}\n\\noindent\nNotice that if \\(\\str{w}\\) is a word, then \\(\\str{w}^{-1}\\) is in \\(\\eta(x)\\) if and only if\n\\(\\str{w}\\) is locally reduced at \\(x\\) and \\(xw \\not \\in \\operatorname{dest}(\\str{w})\\).\n\nWe will refer to elements of \\(\\eta(x)\\) as \\emph{histories} of \\(x\\).\nWe will say that \\(x\\) has \\emph{finite history} if \\(\\eta(x)\\) is finite.\nThe following easily established using Lemmas \\ref{LocRedBasics} and \\ref{Collision};\nthe proof is omitted.\n\n\\begin{lemma} \\label{BasicHistory}\nThe following are true for each \\(x \\in I\\):\n\\begin{itemize}\n\n\\item\n\\(\\eta(x)\\) is closed under taking suffixes;\n\n\\item\nfor each \\(n\\), \n\\(\\eta(x)\\) contains at most one sequence of length \\(n\\);\n\n\\item\nIf  \\(\\str{v}\\) is a word in \\(\\eta(x)\\), then \\(\\eta(xv^{-1}) = \\{\\str{u} : \\str{uv} \\in \\eta(x)\\}\\).\n\n\\end{itemize}\n\\end{lemma}\n\\noindent\nIt is useful to think of \\(\\eta(x)\\) as the suffixes of a single sequence\nwhich is either finite or grows infinitely to the left.\n\nIn what follows, we will typically use \\(s\\) and \\(t\\) to denote elements of \\(I\\)\nwith finite history and \\(x\\) and \\(y\\) for arbitrary elements of \\(I\\).\nThe following is a key property of having a trivial history.\n\n\\begin{lemma}\\label{DisjointOrbits}\nIf \\(s\\ne t\\) have trivial history, then\n\\(s\\gen{A}\\) and \\(t\\gen{A }\\) are disjoint.\n\\end{lemma}\n\n\\begin{proof}\nIf the orbits intersect, then \\(t=sw\\) for some word \\(\\str{w}\\).\nBy Lemma \\ref{LocRedBasics} we can take \\(\\str{w}\\) to be locally reduced\nat \\(s\\).\nBy Lemma \\ref{PingPong}, \\(sw\\) is in the destination of \\(\\str{w}\\).\nBut \\(t = sw\\) has trivial history, which is impossible.\n\\end{proof}\n\nRecall that the set of\nfreely reduced words in a given generating set has the structure of\na rooted tree with the empty word as root and where ``prefix of'' is\nsynonymous with ``ancestor of.''\nThe \\emph{ping-pong argument} discovers orbits that reflect this structure.\nDefine a labeled directed graph on \\(I\\) by putting an arc with label \\(a\\) from \\(x\\) to\n\\(xa\\) whenever \\(a \\in A^\\pm\\) and \\(xa \\ne x\\).\nThe second part of Lemma \\ref{Collision} asserts that if \\(x\\) is in the orbit of a point \\(t\\) with trivial\nhistory, then there is a unique path in this graph connecting \\(t\\) to \\(x\\).\nIt follows that if there is a path between two elements of \\(I\\) with finite history, it is unique,\nyielding the following lemma.\n\n\\begin{lemma}\\label{TreeAction} If\n\\(s,t \\in I\\) have finite histories,\nthen there is at most one word \\(\\str{w}\\) which is\nlocally reduced at \\(s\\) so that \\(sw=t\\).\n\\end{lemma}\n\nNotice that the assumption of finite history in this lemma is necessary.\nFor instance if we consider the positive bumps \\(a_0\\) and \\(a_1\\) Figure \\ref{TrackingPoint},\nthere must be an \\(x \\in \\operatorname{supt}(a_0) \\cap \\operatorname{supt}(a_1)\\) such that \\(xa_0 a_1^{-1} = x\\).\nThis follows from the observation that if \\(s < t\\) are,\nrespectively, the left transition point of \\(a_1\\)\nand the right transition point of \\(a_0\\), then\n\\[s a_0 a_1^{-1} > s a_1^{-1} =  s \\qquad \\textrm{ and } \\qquad  t a_0 a_1^{-1} = t a_1^{-1} < t\\]\nwhich implies the existence of the desired \\(x\\) by applying the Intermediate Value Theorem\nto \\(t \\mapsto t a_0 a_1^{-1} - t\\).\n\nGiven two points \\(x,y \\in I\\) and a word \\(\\str{w}\\), it will be useful to find a single\nword \\(\\str{w}'\\) which is locally reduced at \\(x\\) and \\(y\\) and which satisfies\n\\(xw' = xw\\) and \\(yw' = yw\\).\nThe goal of the next set of lemmas is to provide a set of sufficient conditions for the existence of such a \\(\\str{w}'\\).\nIt will be convenient to introduce some additional terminology at this point.\nIf \\(x \\in I\\), then\nwe say that \\(\\str{w}\\) is a \\emph{return word} for \\(x\\) if\n\\(xw = x\\) and \\(\\str{w} \\ne \\varepsilon\\);\na \\emph{return prefix} for \\(x\\) is a prefix which is a return word.\nWe will see that ``\\(\\str{w}\\) does not have a return prefix for \\(s\\)'' is a useful hypothesis.\nThe next lemma provides some circumstances under which this is true.\n\n\\begin{lemma} \\label{NoReturn}\nIf \\(s \\in I\\) has finite history,\n\\(\\str{u}\\) is locally reduced at \\(s\\),\nand \\(\\str{w}\\) is a word of length less than \\(\\str{u}\\),\nthen \\(\\str{uw}\\) has no return prefix for \\(s\\).\n\\end{lemma}\n\n\\begin{proof}\nNotice that it suffices to prove that \\(\\str{uw}\\)\nis not a return word for \\(s\\).\nIf it were, then there would be a locally reduced subword\n\\(\\str{v}\\) of \\(\\str{w}^{-1}\\) such that \\(su = sv\\).\nSince \\(|\\str{v}| \\leq |\\str{w}| < |\\str{u}|\\),\nthis would contradict Lemma \\ref{TreeAction}.\n\\end{proof}\n\n\\begin{lemma} \\label{FellowTraveler1}\nSuppose that \\(\\str{w} \\ne \\varepsilon\\) is a word and \\(x \\in J (\\str{w})\\).\nIf \\(\\str{w}\\) has no return prefix for \\(x\\) and \\(\\str{w'}\\) is locally reduced at\n\\(x\\) with \\(xw' = xw\\), then:\n\\begin{itemize}\n\n\\item \\(J(\\str{w}') = J(\\str{w})\\);\n\n\\item \\(\\str{w}'\\) is locally reduced on \\(J(\\str{w})\\);\n\n\\item if \\(y \\in J(\\str{w})\\), then \\(yw' = yw\\).\n\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nThe proof is by induction on the length of \\(\\str{w}\\).\nIf \\(\\str{w}\\) has length \\(1\\), then there is nothing to show.\nSuppose now that \\(\\str{w} = \\str{ub}\\) for some \\(b \\in A^\\pm\\) and \\(\\str{u} \\ne \\varepsilon\\).\nLet \\(\\str{u}'\\) be locally reduced at \\(x\\) such that \\(xu' = xu\\).\nBy our inductive assumption, \\(J:= J(\\str{u}') = J(\\str{u}) = J(\\str{w})\\),\n\\(\\str{u}'\\) is locally reduced on \\(J\\) and if \n\\(y \\in J\\), then \\(yu' = yu\\).\nBy our assumption, \\(xu' = xu \\ne x\\) and so \\(\\str{u}' \\ne \\varepsilon\\).\nIf \\(\\str{u}'\\str{b}\\) is not freely reduced, then its free reduction \\(\\str{w}'\\)\nsatisfies that \\(xw' = xu'b = xub = xw \\ne x\\).\nIn particular, \\(\\str{w}' \\ne \\varepsilon\\) and retains the first symbol of \\(\\str{u}'\\).\nFurthermore, since \\(\\str{u}'\\) is locally reduced on \\(J\\) and since\n\\(\\str{w}'\\) is a prefix of \\(\\str{u}'\\), \\(\\str{w}'\\) is also locally reduced on \\(J\\).\nAlso, if \\(y \\in J\\), then \\(yw = yub = yu'b = yw'\\).\n\nSuppose now that \\(\\str{u}'\\str{b}\\) is freely reduced. \nBy Lemma \\ref{PingPong}, \\(Ju = Ju' \\subseteq \\operatorname{dest}(\\str{u}')\\) for all \\(y \\in J\\).\nIf \\(\\operatorname{dest}(\\str{u}')\\) is disjoint from \\(\\operatorname{supt}(b)\\), then\n\\(yu' = yu'b = yub = yw\\) for all \\(y \\in J\\).\nSince \\(\\str{u}'\\) is locally reduced, we are again done in this case.\nIn the remaining case, \\(\\operatorname{dest}(\\str{u}') \\subseteq \\operatorname{supt}(b)\\) in which case\n\\(\\str{w}' = \\str{u}'\\str{b}\\) is locally reduced at all \\(y \\in J\\).\nSince  for all \\(y \\in J\\), \\(yu = yu' \\ne yu'b = yub = yw\\) we have that\n\\(\\str{w}'\\) is locally reduced on \\(J\\).\nClearly \\(J(\\str{w}') = J(\\str{u}') = J(\\str{w})\\) and we are finished.\n\\end{proof}\n\\noindent\nLemma \\ref{FellowTraveler1} has two immediate consequences which will be easier to\napply directly.\n\n\\begin{lemma} \\label{FellowTraveler2}\nIf \\(\\str{w}\\) is a word and\nthere is an \\(s \\in J:=J (\\str{w})\\) with finite history such that\n\\(\\str{w}\\) is a minimal return word for \\(s\\),\nthen \\(w\\) is the identity on \\(J\\).\n\\end{lemma} \n\n\\begin{proof}\nLet \\(\\str{w} = \\str{ua}\\) and let \\(\\str{u}'\\) be locally reduced at \\(s\\) with \\(su' = su\\).\nSince \\(sw = su'a = s\\), it must be that \\(\\str{u}' = \\str{a}^{-1}\\).\nBy Lemma \\ref{FellowTraveler1},\n\\(t u' = tu\\) whenever \\(t \\in J\\).\nThus \\(tw = tu' a = ta^{-1} a = t\\) for all \\(t \\in J\\).\n\\end{proof}\n\n\\begin{lemma} \\label{FellowTraveler3}\nIf \\(s,t \\in I\\) have finite histories and \\(\\eta(s) = \\eta(t)\\), then\nany return word for \\(s\\) is a return word for \\(t\\).\nIf moreover \\(s\\) and \\(t\\) have trivial history and\n\\(\\str{w} \\ne \\varepsilon\\) is not a return word for \\(s\\),\nthen there is an \\(a \\in A^\\pm\\) such that \\(\\{sw,tw\\} \\subseteq \\operatorname{dest}(a)\\).\n\\end{lemma}\n\n\\begin{proof}\nIf there is a minimal return prefix of \\(\\str{w}\\) for \\(s\\), then by Lemma \\ref{FellowTraveler2},\nit is also a return prefix for \\(t\\).\nBy iteratively removing minimal return prefixes for \\(s\\), \nwe may assume \\(\\str{w}\\) has no return prefixes for \\(s\\).\nObserve that since \\(\\eta(s) = \\eta(t)\\), \\(\\{s,t\\} \\subseteq J(\\str{w})\\).\nLemma \\ref{FellowTraveler1} now yields the desired conclusion.\n\\end{proof}\n\n\nThe following theorem shows that the restriction of the action of \\(\\gen{A}\\) to \\(M\\gen{A}\\) is faithful.\n\n\\begin{thm}\\label{Faithful}\nSuppose that \\(A \\subseteq {\\operatorname{Homeo}_+(I)}\\)\nis a (possibly infinite) geometrically fast set of positive bump functions, equipped with a marking.\nIf \\(g \\in \\gen{A}\\) is not the identity, then\nthere is a \\(y \\in M \\gen{A}\\) such that \\(y g \\ne y\\).\n\\end{thm}\n\n\\begin{remark}\nIf \\(A\\) is finite and equipped with the canonical marking, then\nthe cardinality of \\(M\\) is the sum of the number of maximal stretched transition chains in \\(A\\)\nand the number of isolated elements of \\(A\\).\n\\end{remark}\n\n\\begin{proof}\nFirst observe that if there is an \\(x \\in I\\) such that \\(xg \\ne x\\), then by continuity of \\(g\\),\nthere is an \\(x \\in I\\) such that \\(xg \\ne x\\) and \\(x\\) is not in the orbit of a transition point or\nmarker (orbits are countable and neighborhoods are uncountable).\nFix such an \\(x\\) and a word \\(\\str{w}\\) representing \\(g\\) for the duration of the proof.\nThe proof of the theorem breaks into two cases, depending on whether \\(x\\) has\nfinite history.\n\nWe will first handle the case in which \\(x\\) has trivial history;\nthis will readily yield the more general case in which \\(x\\) has finite history.\nSuppose that \\(x \\not \\in \\operatorname{src}(a)\\) for all \\(a \\in A\\).\nLet \\(y < x\\) be maximal such that \\(y\\) is either a transition point or a marker.\n\n\\begin{claim} \\label{NotSplit}\n\\(y\\) has trivial history and\n\\(\\tilde x = \\tilde y\\).\n\\end{claim}\n\n\\begin{proof}\nFirst suppose that \\(y\\) is the marker of some \\(a \\in A\\).\nNotice that by our assumption of maximality of \\(y\\), the right transition point\nof \\(a\\) is greater than \\(y\\).\nIn this case, both \\(x\\) and \\(y\\) are in the support of \\(a\\).\nFurthermore, observe that \\(y\\) is not in the foot of any \\(b \\in A\\).\nTo see this, notice that this would only be possible if \\(y\\) is in the right foot of some \\(b\\).\nHowever since \\(x\\) is not in the right foot of \\(b\\),\nthe right transition point of \\(b\\) would then be less than \\(x\\),\nwhich would contradict the maximal choice of \\(y\\).\nFinally, if \\(b \\in A \\setminus \\{a\\}\\), the maximal choice of \\(y\\) implies that\n\\(x\\) is in the support of \\(b\\) if and only if \\(y\\) is.\n\nIf \\(y\\) is a transition point of some \\(a \\in A\\), then \\(y\\) must be the right transition point\nof \\(a\\) since otherwise our maximality assumption on \\(y\\) would imply that\n\\(x\\) is in the left foot of \\(a\\), contrary to our assumption that \\(x\\) has trivial history.\nIn this case neither \\(x\\) nor \\(y\\) are in the support of \\(a\\).\nIf \\(b \\in A \\setminus \\{a\\}\\), then our maximality assumption on \\(y\\) implies\nthat \\(\\{x,y\\}\\) is either contained in or disjoint from the support of \\(b\\).\nTo see that \\(y\\) has trivial history, observe that the only way a transition point\ncan be in a foot is for it to be the left endpoint of a right foot.\nIf \\(y\\) is the left endpoint of a right foot,\nthen our maximal choice of \\(y\\) would mean that \\(x\\) is also in this foot, which is contrary\nto our assumption.\nThus \\(y\\) must have trivial history.\n\\end{proof}\n\nBy Claim \\ref{NotSplit} and Lemma \\ref{FellowTraveler3}, \\(yw \\ne y\\).\nIf \\(y\\) is a marker, we are done.\nIf \\(y\\) is a transition point of some \\(a \\in A\\), then as noted above it is the right transition point of \\(a\\).\nIf \\(s\\) is the marker of \\(a\\), then \\(sa^k \\to y\\) and by continuity \\(sa^k w \\to yw\\).\nThus for large enough \\(k\\), \\(sa^k w \\ne sa^k\\).\nSince \\(sa^k \\in M\\gen{A}\\), we are done in this case.\n\nNow suppose that \\(x\\) has finite history and let \\(\\str{\\tilde{t}u} \\in \\eta(x)\\) with \\(t \\in I\\) and \\(tu = x\\).\nBy definition of \\(\\eta(x)\\), \\(t\\) has trivial history.\nSince \\(xw \\ne x\\), we have that \\(tuw \\ne tu\\) and hence \\(tuwu^{-1} \\ne t\\).\nIt follows from the previous case\nthat there is an \\(s \\in M\\gen{A}\\) such that \\(s uwu^{-1} \\ne s\\).\nWe now have that \\(y :=su\\) is in \\(M\\gen{A}\\) and satisfies \\(yw \\ne y\\) as desired.\n\nFinally, suppose that \\(\\eta(x)\\) is infinite.\nLet \\(\\str{u} \\in \\eta(x)\\) be longer than \\(\\str{w}\\), let \\(s\\) be the marker\nfor the initial symbol of \\(\\str{u}\\), and set \\(y := xu^{-1}\\) and \\(t := su\\).\nSince \\(\\str{u}\\) is locally reduced at \\(y\\) by assumption,\nLemma \\ref{threaded} implies that \\(\\str{u}\\) is locally reduced at \\(s\\).\nBy Lemma \\ref{NoReturn}, \\(\\str{uw}\\) has no return prefix for \\(s\\).\nLet \\(\\str{v}\\) be locally reduced at \\(s\\) such that \\(suw = sv\\).\nApplying Lemma \\ref{FellowTraveler1} to \\(\\str{uw}\\), \\(s\\) and \\(\\str{v}\\), we\ncan conclude that \\(J(\\str{v}) = J(\\str{uw})\\), that\n\\(\\str{v}\\) is locally reduced at \\(y\\), and \\(yuw = yv\\).\nNotice that since \\(xw \\ne x\\), \\(yuw = yv \\ne yu\\) and in particular \\(\\str{v} \\ne \\str{u}\\).\nBy Lemma \\ref{TreeAction}, we have that \\(t = su \\ne sv = suw = tw\\).\nThis finishes the proof of Theorem \\ref{Faithful}.\n\\end{proof}\n\nWe finish this section with two lemmas which concern multi-orbital homeomorphisms but \nwhich otherwise fit the spirit of this section.\nThey will be needed in Section \\ref{ExcisionSec}.\n\n\\begin{lemma} \\label{FindPrefix}\nSuppose that \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is geometrically fast and equipped with a fixed marking.\nLet \\(A\\) be the set of bumps used in \\(X\\) and \\(s \\in M\\).\nIf \\(\\str{w}\\) is an \\(X\\)-word and \\(\\str{u}\\) is an \\(A\\)-word which is locally reduced at \\(s\\)\nand satisfies \\(su = sw\\), then for every prefix \\(\\str{u}'\\) of \\(\\str{u}\\), there is a prefix \\(\\str{w}'\\) of\n\\(\\str{w}\\) such that \\(su' = sw'\\). \n\\end{lemma}\n\n\\begin{proof}\nLet \\(\\str{w}_i\\) be the prefix of \\(\\str{w}\\) of length \\(i\\) for \\(i \\leq |\\str{w}|\\) and let\n\\(\\str{u}_i\\) be the unique word which is locally reduced at \\(s\\) such that\n\\(su_i = sw_i\\).\nNotice that if \\(\\str{u}_{i+1} \\ne \\str{u}_i\\), then \\(\\str{u}_{i+1}\\)\nis obtained by inserting or deleting a single symbol at\/from the end of \\(\\str{u}_i\\).\nIt follows that all prefixes of \\(\\str{u}\\) occur among the \\(\\str{u}_i\\)'s.\n\\end{proof}\n\nIf \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\), then an element \\(s\\) of \\(I\\) is defined to have finite history with respect to \\(X\\)\nif it has finite history with respect to the set of bumps used in \\(X\\).\nThe meaning of \\emph{return word} is unchanged in the context of \\(X\\)-words.\n\n\\begin{lemma} \\label{X_to_A}\nSuppose that \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is geometrically fast, equipped with a fixed marking, and that \\(A\\)\ndenotes the set of bumps used in \\(X\\).\nLet \\(\\str{w} \\ne \\varepsilon\\) be an \\(X\\)-word and \\(\\str{a} \\in A^\\pm\\) be a signed bump of\nthe first symbol of \\(\\str{w}\\).\nIf \\(s \\in J: = J(\\str{a})\\) and \\(\\str{w}\\) has no proper return prefix for \\(s\\),\nthen there is an \\(A\\)-word \\(\\str{v}\\) which begins with \\(\\str{a}\\) and is such that\n\\(v\\) and \\(w\\) coincide on \\(J\\).\n\\end{lemma}\n\n\\begin{proof}\nThe proof is by induction on the length of \\(\\str{w}\\).\nObserve that the lemma is trivially true if \\(\\str{w}\\) has length at most \\(1\\).\nTherefore suppose that \\(\\str{w} = \\str{u g}\\) with \\(g \\in X^\\pm\\) and \\(\\str{u} \\ne \\varepsilon\\).\nLet \\(\\str{v}'\\) be an \\(A\\)-word which begins with \\(\\str{a}\\) and is such that \\(v'\\) and \\(u\\) agree on \\(J\\).\nSince \\(\\str{u}\\) has no return prefix, Lemma \\ref{FindPrefix} implies that \\(\\str{v}'\\) has no return prefix.\nLet \\(\\str{v}''\\) be a subword of \\(\\str{v}'\\) which is locally reduced at \\(s\\) and satisfies\n\\(sv'' = sv'\\).\nBy Lemma \\ref{FellowTraveler1}, \\(\\str{v}''\\) is locally reduced on \\(J\\) and\n\\(J(\\str{v}'') = J(\\str{v}') = J\\).\nIn particular, \\(J u \\subseteq \\operatorname{dest}(\\str{v}'')\\).\nIf the support of \\(g\\) is disjoint from \\(\\operatorname{dest}(\\str{v}'')\\), set \\(\\str{v}:=\\str{v}'\\).\nOtherwise,\nlet \\(b \\in A^\\pm\\) be the signed bump of \\(g\\) such that \\(\\operatorname{dest}(\\str{v}'') \\subseteq \\operatorname{supt}(b)\\)\nand set \\(\\str{v}:=\\str{v}' \\str{b}\\).\nObserve that \\(\\str{v}\\) satisfies the conclusion of the lemma.\n\\end{proof}\n\n\n\n\\section[The isomorphism theorem]{The isomorphism theorem for geometrically fast generating sets}\n\n\\label{CombToIsoSec}\n\nAt this point we have developed all of the tools needed to prove Theorem \\ref{CombToIso},\nwhose statement we now recall.\n\n\\begin{customthm}{\\ref{CombToIso}}\nIf two geometrically fast sets \\(X , Y \\subseteq {\\operatorname{Homeo}_+(I)}\\) have only finitely many transition points\nand have isomorphic dynamical diagrams, then the induced\nbijection between \\(X\\) and \\(Y\\) extends to an isomorphism of\n\\(\\gen{X}\\) and \\(\\gen{Y}\\) (i.e. \\(\\gen{X}\\) is \\emph{marked isomorphic} to \\(\\gen{Y}\\)).\nMoreover, there is an order preserving bijection\n\\(\\theta :  M \\gen{X} \\to M \\gen{Y}\\) such that\n\\(f \\mapsto f^\\theta\\) induces the isomorphism \n\\(\\gen{X} \\cong \\gen{Y}\\).\n\\end{customthm}\n\nObserve that it is sufficient to prove this theorem in the special case when\n\\(X\\) and \\(Y\\) are finite geometrically fast collections of positive bumps:\nif \\(A\\) and \\(B\\) are the bumps used in \\(X\\) and \\(Y\\) respectively,\nthen the dynamical diagrams of \\(A\\) and \\(B\\) are isomorphic and\nthe isomorphism of \\(\\gen{A}\\) and \\(\\gen{B}\\) restricts to \na marked isomorphism of \\(\\gen{X}\\) and \\(\\gen{Y}\\).\n\nFix, for the moment, a finite geometrically fast set of positive bumps \\(X\\).\nAs we have noted, it is a trivial matter, given a word \\(\\str{w}\\) and a \\(t \\in M\\),\nto find a subword \\(\\str{w}'\\) which is locally reduced at \\(t\\) and satisfies \\(t w = tw'\\).\nTheorem \\ref{CombToIso} will fall out of an analysis of a question of\nindependent interest:\nhow does one determine \\(\\str{w}'\\) from \\(\\str{w}\\)\nand \\(t\\) using only the dynamical diagram of \\(X\\)?\nToward this end, we define \\(\\str{\\tilde{t}w}\\) to be a \\emph{local word} if \\(t \\in M\\) and \\(\\str{w}\\) is a word.\n(Notice that \\(t \\mapsto \\tilde t\\) is injective on \\(M\\);\nthe reason for working with \\(\\tilde t\\) is in anticipation\nof a more general definition in Section \\ref{AbstractPingPongSec}.)\nA local word \\(\\str{\\tilde{t}w}\\) is \\emph{freely reduced} if \\(\\str{w}\\) is.\nIt will be convenient to adopt the convention that \\(\\operatorname{dest}(\\str{\\tilde{t}}) = \\{t\\}\\) if\n\\(t \\in M\\).\nDefine \\(\\Lambda = \\Lambda_X\\) to be the set of all freely reduced local words \\(\\str{\\tilde{t}w}\\)\nsuch that if \\(\\str{ab}\\) are consecutive symbols in \\(\\str{\\tilde{t}w}\\), then\nthe destination of \\(a\\) is between \\(\\operatorname{src}(b)\\) and \\(\\operatorname{dest}(b)\\) in the diagram's ordering.\nNotice that the assertion that \\(\\str{\\tilde{t}w}\\) is in \\(\\Lambda\\) can be formulated as an\nassertion about \\(\\str{w}\\), the element of \\(X\\) which has \\(t\\) as a marker and\nthe dynamical diagram of \\(X\\).\n\nEvery local word \\(\\str{\\tilde{t}w}\\)\ncan be converted into an element of \\(\\Lambda\\) by iteratively\nremoving symbols by the following procedure:\nif \\(\\str{ab}\\) is the first consecutive pair in \\(\\str{\\tilde{t}w}\\) which witnesses that it\nis not in \\(\\Lambda\\), then:\n\\begin{itemize}\n\n\\item if \\(b = a^{-1}\\) then delete the pair \\(\\str{ab}\\);\n\n\\item if \\(b \\ne a^{-1}\\) then delete \\(\\str{b}\\).\n\n\\end{itemize}\nObserve that since the first symbol of a local word is not removed by this procedure,\nthe result is still a local word.\nThe \\emph{local reduction} of a local word \\(\\str{\\tilde{t}w}\\) is the result of applying\nthis procedure to \\(\\str{\\tilde{t}w}\\) until it terminates at an element of \\(\\Lambda\\).\nThe following lemma admits a routine proof by induction, which we omit.\n\n\\begin{lemma} \\label{LambdaLem}\nSuppose that \\(X\\) is a geometrically fast set of positive bumps.\nIf \\(t \\in M\\) and \\(\\str{w}\\) is a word,\nthen \\(\\str{w}\\) is locally reduced at \\(t\\) if and only if \\(\\str{\\tilde{t}w}\\)\nis in \\(\\Lambda\\).\nMoreover, if \\(\\str{w}'\\) is such that \\(\\str{\\tilde{t}w}'\\)\nis the local reduction of \\(\\str{\\tilde{t}w}\\),\nthen \\(tw'\\) and \\(tw\\) coincide.\n\\end{lemma} \n\nNext, order \\(A^\\pm\\) so that if \\(a,b \\in A^\\pm\\), then \\(a\\) is less than \\(b\\) if every element\nof \\(\\operatorname{dest}(a)\\) is less than every element of \\(\\operatorname{dest}(b)\\).\nOrder \\(\\Lambda\\) with the \\emph{reverse lexicographic order}:\nif \\(\\str{uw}\\) and \\(\\str{vw}\\) are in \\(\\Lambda\\) and the last symbol of\n\\(\\str{u}\\) is less than that of \\(\\str{v}\\), then we declare\n\\(\\str{uw}\\) less than \\(\\str{vw}\\).\nRecall that the \\emph{evaluation map} on \\(\\Lambda\\) is the function\nwhich assigns the value \\(tu\\) to each string \\(\\str{\\tilde{t}u} \\in \\Lambda\\).\n(This is well defined since \\(t \\mapsto \\str{\\tilde{t}}\\) is injective on \\(M\\).)\nThis order is chosen so that the following lemma is true.\n\n\\begin{lemma} \\label{RevLex}\nThe evaluation map defined on \\(\\Lambda\\) is order preserving.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that \\(\\str{uw}\\) and \\(\\str{vw}\\) are in \\(\\Lambda\\) and the last symbol of \n\\(\\str{u}\\) is less than the last symbol of \\(\\str{v}\\).\nObserve that by Lemma \\ref{PingPong}, the evaluation of \\(\\str{u}\\)\nis an element of its destination.\nThus if the destination of \\(\\str{u}\\) is less than the destination of \\(\\str{v}\\),\nthen this is true of their evaluations as well.\nSince \\(t \\mapsto tw\\) is order preserving, we are done.\n\\end{proof}\n\nNow we are ready to prove Theorem \\ref{CombToIso}.\n\\begin{proof}[Proof of Theorem \\ref{CombToIso}]\nAs noted above, we may assume that \\(X\\) and \\(Y\\) are geometrically fast families of\npositive bump functions with isomorphic dynamical diagrams.\nBy Theorem \\ref{Faithful}, we know that \\(\\gen{X} \\restriction \\big(M\\gen{X}\\big)\\)\nis marked isomorphic to \\(\\gen{X}\\); similarly \\(\\gen{Y} \\restriction \\big(M\\gen{Y}\\big)\\) is marked\nisomorphic to \\(\\gen{Y}\\).\nIt therefore suffices to define an order preserving bijection \\(\\theta: M\\gen{X} \\to M\\gen{Y}\\)\nsuch that \\(\\theta(sa) = \\theta(s) \\tau (a)\\), where \\(s \\in M\\gen{X}\\) and \\(a \\in X\\) and\nwhere \\(\\tau : X \\to Y\\) is the bijection induced by the isomorphism of the dynamical diagrams of \\(X\\) and \\(Y\\).\n\nDefine \\(\\mu : M_X \\to M_Y\\) by \\(\\mu(s) = t\\) if \\(s\\) is the marker for \\(a \\in X\\) and \\(t\\) is the marker\nfor \\(\\tau(a) \\in Y\\).\nLet \\(\\lambda\\) denote the translation of local \\(X\\)-words into local \\(Y\\)-words\ninduced by \\(\\mu\\) and \\(\\tau\\).\nDefine \\(\\theta : M\\gen{X} \\to M\\gen{Y}\\) so that \\(\\theta(tu)\\) is the evaluation of \\(\\lambda (\\str{tu})\\)\nfor \\(\\str{\\tilde{t}u} \\in \\Lambda_X\\).\nThis is well defined by Lemmas \\ref{DisjointOrbits}, \\ref{TreeAction}, and \\ref{LambdaLem}.\nBy Lemma \\ref{RevLex} and the fact that\n\\(\\lambda\\) preserves the reverse lexicographic order,\n\\(\\theta\\) is order preserving.\n\nNow suppose that \\(s \\in M\\gen{X}\\) and \\(a \\in X^\\pm\\).\nFix \\(\\str{\\tilde{t}u} \\in \\Lambda_X\\) such that \\(s = tu\\) and let\n\\(\\str{\\tilde{t}v} \\in \\Lambda_X\\) be the local reduction of \\(\\str{\\tilde{t}ua}\\).\nObserve that on one hand \\(\\theta(sa) = \\theta (tv)\\) is the evaluation of \\(\\lambda(\\str{\\tilde{t}v})\\).\nOn the other hand \\(\\theta(s)\\tau(a)\\) is the evaluation of \\(\\lambda(\\str{\\tilde{t}ua})\\).\nSince \\(\\lambda\\) is induced by an isomorphism of dynamical diagrams, it satisfies\nthat \\(\\str{w}'\\) is the local reduction of \\(\\str{w}\\) if and only if \\(\\lambda(\\str{w}')\\) is the local\nreduction of \\(\\lambda(\\str{w})\\).\nIn particular, \\(\\lambda(\\str{\\tilde{t}v})\\) is the local reduction of \\(\\lambda(\\str{\\tilde{t}ua})\\).\nBy Lemma \\ref{LambdaLem}, these local \\(Y\\)-words have the same evaluation and\ncoincide with \\(\\theta(sa)\\) and \\(\\theta(s) \\tau(a)\\), respectively.\nThis completes the proof of Theorem \\ref{CombToIso}.\n\\end{proof}\n\nAs we noted in the introduction Theorem \\ref{CombToIso} has two immediate consequences.\nFirst, geometrically fast sets \\(X = \\{f_i \\mid i < n\\} \\subseteq {\\operatorname{Homeo}_+(I)}\\) with finitely\nmany transition points are \\emph{algebraically fast}:\nif \\(1 \\leq k_i\\) for each \\(i < n\\), then\n\\(\\gen{f_i \\mid i < n}\\) is marked isomorphic to \\(\\gen{f^{k_i}_i \\mid i < n}\\).\nThe reason for this is that the dynamical diagrams associated to\n\\(\\{f_i \\mid i < n\\}\\) and \\(\\{f_i^{k_i} \\mid i < n\\}\\) are isomorphic.\nSecond, since the dynamical diagram of any geometrically fast set with finitely many transition points\ncan be realized by a geometrically fast subset of \\(F\\) (see, e.g., \\cite[Lemma 4.2]{CFP}),\nevery group admitting a finite geometrically fast generating set can be embedded into \\(F\\).\nThe notion of \\emph{history} in the previous section is revisited in Section \\ref{AbstractPingPongSec}\nwhere it is used to prove a relative  of Theorem \\ref{CombToIso}.\n\n\nMore evidence of the restrictive nature of geometrically fast generating sets\ncan be found in \\cite{kim+koberda+lodha} where groups generated by stretched\ntransition chains \\(C\\) as defined in Section \\ref{FastBumpsSec} are\nconsidered under the weaker assumption\n that consecutive pairs of elements in \\(C\\) are geometrically fast.\nGroups generated by such a \\(C\\) with \\(n\\) elements are called\n{\\itshape \\(n\\)-chain groups}.\nIt is proven in\n\\cite{kim+koberda+lodha} that every \\(n\\)-generated subgroup of\n\\({\\operatorname{Homeo}_+(I)}\\) is a subgroup of an \\((n+2)\\)-chain group.\nAnother result of \\cite{kim+koberda+lodha} is that for each\n\\(n\\ge3\\), there are uncountably many isomorphism types of\n\\(n\\)-chain groups. \nBy contrast, Theorem \\ref{CombToIso} (with Corollary \\ref{Excision} below) implies that\nthe number of isomorphism types of groups with finite, geometrically fast generating sets is\ncountable because the number of isomorphism types of dynamical\ndiagrams is countable.\n\n\n\n\\section[semi-conjugacy]{Minimal representations of\ngeometrically fast groups and topological semi-conjugacy}\n\n\\label{SemiConjSec}\n\nTheorem \\ref{CombToIso} partitions the subgroups of \\({\\operatorname{Homeo}_+(I)}\\)\ngenerated by geometrically fast sets with finitely many transition points: two such sets are\nconsidered equivalent if their dynamical diagrams are isomorphic.\nIn this section we show that each class contains a (nonunique)\nrepresentative \\(Y\\) so that for each \\(X\\) in the class there is a\nmarked isomorphism \\(\\phi : \\gen{X} \\to \\gen{Y}\\) which is\ninduced by a semi-conjugacy on \\(I\\).\nSpecifically, the bijection \\(\\theta : M\\gen{X} \\to M\\gen{Y}\\) of Theorem \\ref{CombToIso}\nextends to a continuous order preserving surjection \\(\\hat{\\theta}:I \\to I\\)\nso that for all \\(f\\in \\gen{X}\\) we have \\(f \\hat{\\theta} =\\hat{\\theta}\\phi(f)\\).\nNotice that in this situation, the graph of \\(\\phi(f)\\) is the image of the graph of \\(f\\) under the\ntransformation \\((x,y) \\mapsto (x\\hat\\theta,y\\hat\\theta)\\).\nWe will refer to such a \\(Y \\subseteq {\\operatorname{Homeo}_+(I)}\\) as \\emph{terminal}.\nTheorem \\ref{SemiConjThm} can now be stated as follows.\n\n\\begin{thm}\nEach dynamical diagram \\(D\\) can be realized by a terminal \\(X_D \\subseteq {\\operatorname{PL}_+(I)}\\).\n\\end{thm}\n\n\\begin{proof}\nAs in the proof of Theorem \\ref{CombToIso}, it suffices to prove the theorem\nunder the assumption that all bumps in \\(D\\) are positive and all labels are distinct.\nFurthermore, by Proposition \\ref{IsolatedFixProp}, we may assume that \\(D\\) has no isolated bumps.\nLet \\(n\\) denote the number of bumps of \\(D\\), set \\(\\ell := 1\/(2n)\\) and \n\\[\n\\mathcal{J} : = \\{ [i \\ell, (i+1) \\ell) \\mid 0 \\leq i < 2n \\},\n\\]\nobserving that \\(\\mathcal{J}\\) has the same cardinality as the set of feet of \\(D\\).\nOrder \\(\\mathcal{J}\\) by the order on the left endpoints of its elements.\nIf \\(i < 2n\\), we will say that the \\(i{}^{\\textrm{th}}\\) interval in \\(\\mathcal{J}\\) \\emph{corresponds} to the \\(i{}^{\\textrm{th}}\\) foot\nof \\(D\\).\n\nFor \\(i < n\\), let \\((x_i,s_i)\\) and \\((t_i,y_i)\\) be the intervals in \\(\\mathcal{J}\\) which correspond to the left and right feet of\nthe \\(i{}^{\\textrm{th}}\\) bump of \\(D\\), respectively.\nNote that since \\(D\\) has no isolated bumps, \\(s_i < t_i\\).\nDefine \\(b_i\\) to be the bump which has support \\((x_i,y_i)\\), maps \\(s_i\\) to \\(t_i\\) and is linear on\n\\((x_i,s_i)\\) and \\((s_i,y_i)\\) --- see Figure \\ref{PLBumpFig}.\n\\begin{figure}\n\\[\n\\xy\n(0,0); (30,30)**@{-};\n(3,3); (11,19)**@{-}; (27,27)**@{-}; (11,11); (11,19)**@{.};\n(19,19)**@{.};\n(3,3); (3,1)**@{-}; (3,-1)*{x_i};\n(11,11); (11,9)**@{-}; (11,7)*{s_i};\n(19,19); (19,17)**@{-}; (19,15)*{t_i};\n(27,27); (27,25)**@{-}; (27,23)*{y_i};\n\\endxy\n\\]\n\\caption{The function \\(b_i\\).\\label{PLBumpFig}}\n\\end{figure}\nIf we assign \\(b_i\\) the marker \\(s_i\\), then \nthe feet of \\(b_i\\) are either in \\(\\mathcal{J}\\) or are the interior of an element of \\(\\mathcal{J}\\);\nin particular, the feet of \\(X_D = \\{b_i \\mid i < n\\}\\) are disjoint.\nThus \\(X_D\\) is geometrically fast and has dynamical diagram isomorphic to \\(D\\).\n\nNotice that the feet \\((x_i,s_i)\\) and \\([t_i,y_i)\\) of \\(b_i\\)\nare each intervals of length \\(\\ell\\) contained in\n\\(I\\) while the middle interval \\([s_i,t_i)\\) is of length \\(m\\ell\\) for some positive integer \\(m\\).\nMoreover, since \\(D\\) has no isolated bumps,\nthere is an interval of \\(\\mathcal{J}\\) between \\(s_i\\) and \\(t_i\\);\nin particular, \\(t_i - s_i \\geq \\ell\\).\nIt follows that the slope of the graph of \\(b_i\\)\non its source \\((x_i,s_i)\\) and the slope of the graph of \\(b_i^{-1}\\)\non its source \\([t_i,y_i)\\) are both at least 2.\n\n\n\\begin{claim}\\label{OrbitDensity}\nIf \\(X_D\\) is the set of positive bumps constructed above, then \\(M\\gen{X_D}\\) is dense in \\(I\\).\n\\end{claim}\n\n\\begin{remark}\nNote that we are working under the assumption that \\(D\\) has no isolated elements.\nIf \\(D\\) has isolated bumps, then \\(M\\gen{X_D}\\) can not be dense.\n\\end{remark}\n\n\\begin{proof} \nSince every transition point of \\(X_D\\) is in the closure of \\(M\\gen{X_D}\\),\nit suffices to show that if \\(0 \\leq p < q \\leq 1\\), then \n\\((p,q)f\\) contains an endpoint of an interval in \\(\\mathcal{J}\\) for some \\(f \\in \\gen{X_D}\\).\nThe proof is by induction on the minimum \\(k \\geq 0\\) such that \\(\\ell 2^{-k} < q-p\\).\nObserve that if \\(k = 0\\), then \\(q-p > \\ell\\) and thus \\(q-p\\) must contain an endpoint of\nan interval in \\(\\mathcal{J}\\).\n\nNext observe that if \\((p,q)\\) does not contain an endpoint of an element of \\(\\mathcal{J}\\),\nthen \\((p,q)\\) is contained in the foot of some \\(b_i\\) for \\(i < n\\).\nIf \\((p,q) \\subseteq \\operatorname{src}(b_i)\\), then since the derivative of \\(b_i\\) is at least 2 on it source,\nit follows that  \\((p,q)b_i\\) is at least twice as long as \\((p,q)\\).\nBy our induction hypothesis, there is an \\(f \\in \\gen{X_D}\\) such that\n\\((p,q)b_i f\\) contains an endpoint of \\(\\mathcal{J}\\).\nSimilarly, if \\((p,q) \\subseteq \\operatorname{src}(b_i^{-1})\\), then \\((p,q)b_i^{-1}\\) is at least twice\nas long as \\((p,q)\\) and we can find an \\(f \\in \\gen{Y}\\) such that\n\\((p,q)b_i^{-1} f\\) contains an endpoint of \\(\\mathcal{J}\\).\n\\end{proof}\n\nIn order to see that \\(X_D\\) is terminal, let\n\\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) be geometrically fast, have finitely many transition points, and have a dynamical diagram\nisomorphic to \\(D\\).\nLet \\(\\theta : M\\gen{X} \\to M\\gen{X_D}\\) be order preserving and satisfy that\n\\(t f \\theta = t \\theta \\phi(f)\\) for all \\(f \\in M\\gen{X}\\),\nwhere \\(\\phi : \\gen{X} \\to \\gen{X_D}\\) is the marked isomorphism.\nDefine \\(\\hat \\theta : I \\to I\\) by\n\\[\nx \\hat \\theta := \\sup \\{t\\theta \\mid t \\in M\\gen{X} \\textrm{ and } t \\leq x\\}\n\\]\nwhere we adopt the convention that \\(\\sup \\emptyset = 0\\).\nClearly \\(\\hat \\theta : I \\to I\\) is order preserving and extends \\(\\theta\\).\nIn particular its range contains \\(M\\gen{X_D}\\), which by\nClaim \\ref{OrbitDensity} is dense in \\(I\\).\nIt follows that \\(\\hat \\theta\\) is a continuous surjection (\\emph{any} order preserving map from \\(I\\) to \\(I\\) with\ndense range is a continuous surjection).\nThat \\(x f \\hat \\theta = x \\hat \\theta \\phi (f)\\) follows from\nthe fact that this is true for \\(x \\in M\\gen{X}\\) and from the continuity of \\(f\\) and \\(\\phi(f)\\).\nThis completes the proof that \\(X_D\\) is terminal.\n\\end{proof}\n\n\\section{Fast generating sets for the groups \\(F_n\\)}\n\\label{FnSec}\n\nIn this section we will give explicit generating sets\nfor some well known variations of Thompson's group \\(F\\).\nFirst notice that since \\((0,1)\\) is homeomorphic to \\({\\mathbf R}\\) by an order preserving map,\nall of the analysis of geometrically fast subsets of \\({\\operatorname{Homeo}_+(I)}\\) transfers to \\({\\operatorname{Homeo}_+(\\R)}\\)\n(with the caveat that \\(\\pm \\infty\\) must be considered as possible\ntransition points of elements of \\({\\operatorname{Homeo}_+(\\R)}\\); see Example \\ref{x+1_x^3_ex} below).\nFix an integer \\(n\\ge2\\).\nFor \\(i < n\\), let \\(g_i\\) be a homeomorphism from \\({\\mathbf R}\\) to itself defined by:\n\\[\ntg_i := \\begin{cases}\nt & \\textrm{if } t\\le i \\\\\ni+n(t-i) & \\textrm{if } i\\le t\\le i+1 \\\\\nt+(n-1) & \\textrm{if } i+1 \\le t.\n\\end{cases}\n\\]\nIn words, \\(g_i\\) is the identity below \\(i\\), has constant slope\n\\(n\\) on the interval \\([i,i+1]\\), and is translation by \\(n-1\\)\nabove \\(i+1\\).\nWe will use \\(F_n\\) to denote \\(\\gen{g_i \\mid i < n}\\).\nThe group \\(F_2\\) is one of the standard\nrepresentations of Thompson's group \\(F\\).\n(The more common representation of \\(F\\) is as a set of piecewise linear\nhomeomorphisms of the unit interval \\cite[\\S1]{CFP}.)\n\nThe groups \\(F_n\\) \\((n \\ge 2)\\)  are discussed in\n\\cite[\\S4]{brown:finiteprop} where \\(F_n\\) is denoted\n\\(F_{n,\\infty}\\), and in \\cite[\\S2]{brin+fer} where \\(F_n\\) is\ndenoted \\(F_{n,0}\\).\nThe standard infinite presentation of \\(F_n\\) is given in \\cite[Cor. 2.1.5.1]{brin+fer}.\nIt follows easily from that\npresentation that the commutator quotient of \\(F_n\\) is a free abelian group of rank \\(n\\).\nIn particular the \\(F_n\\)'s are pairwise nonisomorphic.\n\nWe now describe an alternate generating set for \\(F_n\\) which \nconsists of \\(n\\) positive  bump functions and is geometrically fast.\nFor \\(i<n-1\\), set \\(h_i : = g_i g_{i+1}^{-1}\\) and let\n\\(h_{n-1}\\) denote \\(g_{n-1}\\).\nIt is clear that \\(C = \\{h_i \\mid i<n\\}\\) generates \\(F_n\\).\nWe claim that \\(C\\) is a geometrically fast stretched transition chain.\nIf \\(i < n-1\\), then \\(h_i\\) is the identity outside \\([i,i+2]\\).\nOn that interval it is a positive, one bump function since chain\nrule considerations show that \\(h_i\\) has slope \\(n\\) on\n\\([i,i+\\frac 1 n]\\), slope one on \\([i+\\frac 1 n, i+1]\\) and slope\n\\(1\/n\\) on \\([i+1, i+2]\\). \nThus the support of \\(h_i\\) is \\((i,i+2)\\) and\nthe support of \\(h_{n-1}\\) is \\((n-1, \\infty)\\).\nIn particular, \\(C\\) forms a stretched transition chain.\n\nNext we will show that \\(C\\) is geometrically fast.\nObserve that for \\(0 < j \\leq n\\) that we have:\n\\[ \n\\left( i+\\frac{j}{n} \\right) h_i\n= \n(i+1) + \\frac{j-1}{n}.\n\\]\n\nFor \\(0\\le i\\le n\\) set\n\\(t_i := i+ \\frac{n-i}{n}\\).\nIt follows from the above computation that for \\(0\\le i<n-1\\), we\nhave \\(t_ih_i = t_{i+1}\\) so that \\(t_0=1\\) and \\(t_n=n\\).  This\nmakes \\(t_0\\) the leftmost transition point in the support of\n\\(h_0\\) and \\(t_n\\) the rightmost transition point in the support of\n\\(h_{n-1}\\).\nBy Proposition \\ref{FastCriterion}, \\(C\\) is geometrically fast.\nCombining this with Theorem \\ref{CombToIso}, we now have that\nany geometrically fast stretched transition chain generates a copy of \\(F_n\\).\n\n\\begin{remark}\nThe above proof that \\(F_n\\) is isomorphic to the group generated by a geometrically fast\nstretched transition chain of length \\(n\\) is not an efficient\nway to reach this conclusion.\nThere is a more straightforward\nargument based on the standard presentation of \\(F_n\\) as indicated\nin the proof of \\cite[Proposition 1.11]{kim+koberda+lodha}.\nWe included this example as an introduction to the\nvariety of isomorphism classes in groups generated by geometrically fast sets.  \n\\end{remark}\n\n\\section{Excision of extraneous bumps in fast generating sets}\n\n\\label{ExcisionSec}\n\nSometimes fast generating sets use bumps which do not affect the marked isomorphism type\nof the resulting group.\nThis section gives a sufficient condition for when those bumps\ncan be excised while preserving the marked isomorphism type.\nWe make no finiteness assumptions in this section.\n\nIf \\(f \\in {\\operatorname{Homeo}_+(I)}\\) and \\(E\\) is a set of positive bumps, then we define \\(f\/E \\in {\\operatorname{Homeo}_+(I)}\\) to be the function\nwhich agrees with \\(f\\) on  \n\\[\nI \\setminus \\bigcup \\{\\operatorname{supt} (a) : a \\in E \\textrm{ and } a \\textrm{ is used in } f\\}\n\\]\nand is the identity elsewhere.\nIf \\(X\\) is a fast generating set and \\(E\\) is a set of positive bumps, we \ndefine \\(X\/E := \\{f\/E : f \\in X\\}\\).\n\nNow let \\(X\\) be a fast generating set and let \\(J \\subseteq I\\) be an interval.\nWe say a set \\(E\\) of bumps is \\emph{extraneous in \\(X\\) as witnessed by \\(J\\)} if\nfor some \\(f \\in X\\):\n\\begin{itemize}\n\n\\item every element of \\(E\\) is an isolated bump used in \\(f\\) whose\nsupport is contained in \\(J\\);\n\n\\item there is a bump \\(a\\) used in \\(f\\) not in \\(E\\) such that \\(J\\) contains a foot of \\(a\\);\n\n\\item \\(J\\) is disjoint from the feet of all \\(g \\in X \\setminus \\{f\\}\\).\n\n\\end{itemize}\nObserve that if \\(E\\) is extraneous in \\(X\\), then \\(g \\mapsto g\/E\\) preserves the canonical ordering of \\(X\\).\n\n\\begin{thm} \\label{Excision}\nIf \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is a (possibly infinite) geometrically fast set and\n\\(E\\) is an extraneous set of bumps used in \\(X\\),\nthen the map \\(g \\mapsto g\/E\\) extends to an isomorphism between \\(\\gen{X}\\) and \\(\\gen{X\/E}\\).\n\\end{thm}\n\n\n\\begin{proof}\nFix \\(X\\) and \\(E\\) as in the statement of the theorem and\nlet \\(J\\) be an interval witnessing that \\(E\\) is extraneous\nand \\(f \\in X\\) be the element of \\(X\\) such that the elements of \\(E\\) occur in \\(f\\).\nLet \\(A\\) denote the set of bumps used in \\(X\\),\n\\(M\\) denote \\(M_X\\), and \\(\\widetilde M\\) denote \\(M_{X\/E}\\).\nSet \\(S :=M \\gen{X} = M \\gen{A}\\) and\n\\(\\widetilde S := \\widetilde M \\gen{X\/E} = \\widetilde M \\gen{A \\setminus E}\\).\nObserve that since the elements of \\(E\\) are isolated in \\(A\\) and since\nevery element of \\(\\widetilde M \\gen{A \\setminus E} \\setminus \\widetilde M\\) is in \\(\\operatorname{dest}(b)\\) for some\n\\(b \\in (A \\setminus E)^{\\pm}\\), it follows that \\(\\widetilde M \\gen{A \\setminus E} = \\widetilde M \\gen{A}\\).\nIn particular, \\(\\widetilde S\\) is \\(\\gen{A}\\)-invariant and hence \\(g \\mapsto g \\restriction \\widetilde S\\)\ndefines a homomorphism of \\(\\gen{X}\\) to \\(\\gen{X} \\restriction \\tilde S\\).\n\nRecall that, by Theorem \\ref{Faithful}, \\(g \\mapsto g \\restriction S\\) defines an isomorphism\nbetween \\(\\gen{X}\\) and \\(\\gen{X} \\restriction S\\).\nSimilarly, \\(g\/E \\mapsto g\/E \\restriction \\widetilde S\\) defines an isomorphism between \\(\\gen{X\/E}\\)\nand \\(\\gen{X\/E} \\restriction \\widetilde S\\).\nSince \\(g \\restriction \\widetilde S = (g\/E) \\restriction \\widetilde S\\), \nit suffices to show that \\(g \\mapsto g \\restriction \\widetilde S\\) is injective on \\(\\gen{X}\\).\nThat is, if \\(\\str{w}\\) is an \\(X\\)-word, \\(s\\) is a marker for an \\(a \\in E\\),\nand \\(suw \\ne su\\) for some \\(X\\)-word \\(\\str{u}\\),\nthen there is a \\(t \\in \\widetilde S\\) such that \\(tuw \\ne tu\\).\nEquivalently we need to show that for every marker \\(s\\) of an \\(a \\in E\\) and \\(X\\)-word \\(\\str{w}\\),\nif  \\(sw \\ne s\\), then there is a \\(t \\in \\widetilde S\\) with \\(tw \\ne t\\).\nTo this end let such \\(s\\), \\(a\\), and \\(\\str{w}\\) be given with \\(sw \\ne s\\).\n\n\\begin{claim}\nIf there is an \\(x \\in \\operatorname{supt}(a)\\) such that \\(xw \\not \\in \\operatorname{supt}(a)\\), then there is a \\(t \\in \\widetilde S\\)\nsuch that \\(tw \\ne t\\).\n\\end{claim}\n\n\\begin{proof}\nSuppose that \\(xw \\not \\in \\operatorname{supt}(a)\\) for some \\(x \\in \\operatorname{supt}(a)\\).\nLet \\(\\str{uv}\\) be a locally reduced at \\(x\\) such that \\(xuv = xw\\) with \\(\\str{u}\\) maximal\nsuch that \\(xu \\in \\operatorname{supt}(a)\\);\nnotice that \\(\\str{v} \\ne \\varepsilon\\).\nSince \\(a\\) is isolated, \\(xu \\not \\in \\operatorname{src}(\\str{v})\\) and thus by Lemma \\ref{PingPong}\n\\(xuv \\in \\operatorname{dest}(\\str{v})\\).\nIt follows that \\(w\\) must move an endpoint of \\(\\operatorname{dest}(\\str{v})\\), both of which are in\nthe closure of \\(\\widetilde S \\cap \\operatorname{dest}(\\str{v})\\).\nHence \\(w\\) moves an element of \\(\\widetilde S\\). \n\\end{proof}\n\nWe may therefore assume that \\(sw \\ne s\\) but that \\(w\\) maps \\(\\operatorname{supt}(a)\\) to \\(\\operatorname{supt}(a)\\).\nLet \\(\\str{w} = \\prod_{i < n} \\str{w}_i\\) be a factorization of \\(\\str{w}\\) into minimal\npositive length words which define maps from \\(\\operatorname{supt}(a)\\) into \\(\\operatorname{supt}(a)\\).\nLet \\(b\\) be the bump used in \\(f\\) with \\(b \\not \\in E\\) and such that\n\\(J\\) contains a foot of \\(b\\).\nLet \\(t \\in J \\cap \\widetilde S\\) be sufficiently close to the transition point of \\(b\\) which is in \\(J\\)\nsuch that for all \\(k \\in {\\mathbf Z}\\) with \\(|k| \\leq |\\str{w}|\\), \\(t f^k \\in J\\).\n\n\\begin{claim}\nFor all \\(i < n\\), there is a \\(p \\in \\{1,0,-1\\}\\) such that\n \\(w_i \\restriction J = f^p \\restriction J\\).\n\\end{claim}\n\n\\begin{proof}\nIf \\(\\str{w}_i\\) begins with \\(f\\) or \\(f^{-1}\\), then it must have length \\(1\\) and there is nothing to\nshow.\nIf \\(\\str{w}_i\\) begins with \\(\\str{g} \\ne \\str{f}^{\\pm 1}\\),\nthen either \\(J\\) is contained in or disjoint from the support of\n\\(g\\) and it is disjoint from all of the feet of \\(g\\).\nIf \\(g \\restriction J\\) is the identity, then again \\(\\str{w}_i\\) has length 1.\n\nNow suppose that \\(J\\) is contained in the support of \\(g\\) where \\(g\\) is neither \\(f\\) nor \\(f^{-1}\\) and\nhas \\(s\\) in its support.\nNotice that since \\(J\\) is disjoint from the feet of \\(g\\),\n\\(J\\) is contained in the support of a single bump \\(c \\in A^\\pm\\) of \\(g\\).\n\nWe will now show that \\(sw_i = s\\).\nLet \\(\\str{gv}\\) be a minimal prefix of \\(\\str{w}_i\\) such that \\(sgv \\in \\operatorname{supt}(a)\\).\nBy Lemmas \\ref{FellowTraveler1} and \\ref{X_to_A}, there is an \\(A\\)-word \\(\\str{u}\\) which \nis locally reduced on \\(Jc = Jg\\) and such that \\(u\\) and \\(v\\) coincide on \\(Jg\\).\nSince \\(\\str{c}\\) is locally reduced at \\(s\\) and \\(\\str{u}\\) is locally reduced at \\(sc\\),\nthe free reduction of \\(\\str{cu}\\) is locally reduced at \\(s\\).\nIf this free reduction is not \\(\\varepsilon\\), then it must be that \\(\\str{cu}\\) was freely reduced and\n\\(scu \\in \\operatorname{dest}(\\str{cu})\\).\nSince \\(scu = sgv \\in \\operatorname{supt}(a)\\), \\(\\str{cu}\\) would have to end in \\(\\str{a}\\) or \\(\\str{a}^{-1}\\).\nBut then if \\(\\str{u}'\\) is the result of deleting the last symbol of \\(\\str{u}\\),\nthen \\(scu'\\) is in \\(\\operatorname{supt}(a)\\) and, by Lemma \\ref{FindPrefix}, coincides with\n\\(sgv'\\) for some proper prefix \\(\\str{v}'\\) of \\(\\str{v}\\).\nThis contradicts the minimal choice of \\(\\str{gv}\\).\nIt must therefore have been that \\(\\str{u} = \\str{c}^{-1}\\).\nIt follows that \\(c^{-1}\\) and \\(v\\) coincide on \\(Jg\\) and thus\n\\(gv\\) is the identity on \\(J\\).\nBy minimality of \\(\\str{w}_i\\), \\(\\str{w}_i = \\str{v}\\) and thus \\(\\str{w}_i\\) is the identity on \\(J\\). \n\\end{proof}\n\nApplying the claim, there are \\(p_i \\in \\{1,0,-1\\}\\) for \\(i < n\\) such that:\n\\[\nsw = s\\prod_{i < n} f^{p_i}\n\\qquad \\qquad \ntw = t \\prod_{i < n} f^{p_i}.\n\\]\nIn particular, since \\(sw \\ne w\\), it follows that \\(tw \\ne t\\).\n\\end{proof}\n\n\\begin{cor}\nIf \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is geometrically fast and finite, then there is a\ngeometrically fast \\(Y \\subseteq {\\operatorname{Homeo}_+(I)}\\) which has finitely many transition points such that\n\\(\\gen{X}\\) is marked isomorphic to \\(\\gen{Y}\\).\n\\end{cor}\n\n\\begin{proof}\nLet \\(X\\) be given and let \\(\\mathcal{J}\\)  consist\nof the maximal intervals \\(J \\subseteq I\\) such that for some \\(f \\in X\\) (dependent on \\(J\\)):\n\\begin{itemize}\n\n\\item \\(J\\) contains at least one transition point of \\(f\\);\n\n\\item \\(J\\) contains no transition points of \\(X \\setminus \\{f\\}\\);\n\n\\item \\(J\\) is disjoint from the feet of \\(X \\setminus \\{f\\}\\).\n\n\\end{itemize}\nObserve that since \\(X\\) is geometrically proper, \\(\\mathcal{J}\\) is finite.\nThe proof is by induction on the number of elements of \\(\\mathcal{J}\\) which contain infinitely many\ntransition points of \\(X\\).\nSuppose that \\(J \\in \\mathcal{J}\\) contains infinitely many transition points of some \\(f \\in X\\).\nLet \\(E\\) consist of all but one of the isolated bumps of \\(f\\) with support contained in \\(J\\).\nNotice that \\(J\\) witnesses that \\(E\\) is extraneous in \\(X\\).\nBy Theorem \\ref{Excision}, \\(\\gen{X\/E}\\) is marked isomorphic to \\(\\gen{X}\\).\nWe are now finished by our induction hypothesis.\n\\end{proof}\n\n\\section[Geometrically proper generating sets]{The existence of geometrically proper generating sets}\n\n\\label{GeoProperSec}\n\nIn this section we consider the question of when finitely generated subgroups of\n\\(F\\) admit geometrically proper generating sets.\nOur first task will be to prove:\n\n\\begin{customthm}{\\ref{GeoProperGen}}\nEvery \\(n\\)-generated one orbital subgroup of \\({\\operatorname{PL}_+(I)}\\)\neither contains an isomorphic copy of \\(F\\) or else admits an\n\\(n\\)-element geometrically proper generating set.\n\\end{customthm}\n\\noindent\nIn this section we will use the standard embedding of Thompson's group \\(F\\) in \\({\\operatorname{PL}_+(I)}\\)\nand we will make use of the homomorphism \\(\\pi:{\\operatorname{PL}_+(I)} \\rightarrow {\\mathbf R} \\times {\\mathbf R}\\)\ndefined by \\(\\pi(f) := \\big(\\log_2 \\left(f'(0)\\right),\\log_2\\left(f'(1)\\right)\\big)\\).\nIf \\(G\\) is a subgroup of \\({\\operatorname{PL}_+(I)}\\), we will write \\(\\pi_G\\) for \\(\\pi \\restriction G\\).\nIt is well known that \\(F'\\) is exactly\nthe kernel of \\(\\pi_F\\) and that \\(F'\\) is simple.\nWe will need the following lemma which combines the main result of \\cite{brin:ubiq} and\nLemma 3.11 of \\cite{MR2383051}.\n\n\\begin{lemma} \\label{CyclicGerm}\nLet \\(G\\) be a finitely generated subgroup of \\({\\operatorname{PL}_+(I)}\\) with connected support into which \n\\(F\\) does not embed.\nThe image of the homomorphism \\(\\pi_G\\) is either trivial or cyclic.\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Theorem \\ref{GeoProperGen}]\nLet \\(G\\) be a one orbital subgroup of \\({\\operatorname{PL}_+(I)}\\) which\nis generated by a finite set \\(X\\).\nBy conjugating by an appropriate affine homeomorphism of \\({\\mathbf R}\\),\nwe may assume that the support of \\(G\\) is \\((0,1)\\).\nWe assume that \\(F\\) does not embed in \\(G\\).\nBy Lemma \\ref{CyclicGerm}, the image of\n\\(\\pi_G\\) must then be isomorphic to \\({\\mathbf Z}\\).\nLet \\(\\varpi\\) be the composition of \\(\\pi_G\\) with this isomorphism.\nIf there is more than one element of \\(X\\) not in the\nkernel of \\(\\varpi\\), then we apply the Euclidean algorithm and\nrepeatedly reduce \\(\\sum \\left\\{ |\\varpi(f)| \\mid f \\in X\\right\\}\\) by replacing at each stage some\n\\(g\\in X\\) by \\(gh^\\epsilon\\), for some suitably chosen \\(h \\in X\\) and \\(\\epsilon\\in\\{-1,1\\}\\), where  \\(|\\varpi(h)|\\le |\\varpi(g)|\\).\nNote that this\nreplacement does not change the cardinality of \\(X\\).  Thus we can\nassume that there is only one element \\(f\\) of \\(X\\) not in the\nkernel of \\(\\varpi\\).\n\nIf \\(X = \\{g_i \\mid i < n\\}\\), define \\(N(X)\\) to be the number of\npairs \\((j,x)\\) such that for some \\(i < j\\) either \\(x\\) is left\ntransition point of both \\(g_i\\) and \\(g_j\\) or a right transition\npoint of both \\(g_i\\) and \\(g_j\\).\nObserve that \\(N(X)\\) is finite since \\(X\\) has only\nfinitely many transition points and that \\(X\\) is geometrically\nproper precisely if \\(N(X) = 0\\).\nNotice also that if \\((j,x)\\) is a pair counted by \\(N(X)\\),\nthen \\(x\\) can be neither 0 nor 1.\n\nAssume that \\(N(X) > 0\\), and let \\(x\\) be a point that contributes\nto \\(N(X)\\) with \\(g\\ne h\\) two functions in \\(X\\) with \\(x\\) a\ncommon left transition point or common right transition point of\nboth \\(g\\) and \\(h\\).\nSince \\(x\\in (0,1)\\), our assumption on the support of \\(G\\)\nimplies that there is an \\(f\\in X\\) with \\(xf \\ne x\\).\nLet \\(T\\) be the set of transition points of\n\\(g\\) that are moved by \\(f\\).\nSince each point in \\(T\\) is moved to infinitely\nmany places by powers of \\(f\\) and since the set of\ntransition points of elements of \\(X\\) is finite, we can find a\npower \\(f^k\\) of \\(f\\) so that no element of \\(T{f^k}\\) is a\ntransition point of an element of \\(X\\). \nLet \\(X'\\) be obtained from \\(X\\) by replacing \\(g\\) by \\(g^{f^k}\\).\nWe have arranged that \\(|X'|=|X|\\),\nthat \\(\\gen{X'} = \\gen{X}=G\\),\nthat there is only one element of \\(X'\\) outside the kernel of \\(\\varpi\\),\nand that \\(N(X')<N(X)\\).\nBy induction \\(\\gen{X'} = \\gen{X}\\)\nadmits a geometrically proper generating set.\n\\end{proof}\n\nThe assumption that the support of \\(G\\) be connected in Theorem \\ref{GeoProperGen}\nturns out to be necessary as the next example shows.\nBefore proceeding we will develop some notation which will be helpful in proving\nTheorem \\ref{NoGeoProperGen} as well.\nFor the remainder of the section, fix two elements \\(a\\) and \\(b\\) of \\(F\\) which are\none bump functions whose supports are, respectively, \\(\\left(0,\\frac{1}{2}\\right)\\)\nand \\(\\left(\\frac{1}{2}, 1\\right)\\) and which satisfy \\(a'(0)=b'(1)=2\\).\nNotice that \\(a\\) is a positive bump and \\(b\\) is a negative bump.\n\n\\begin{example}\nThe group\n\\(\\gen{ ab, a^{-1} b }\\) is isomorphic to \\({\\mathbf Z}\\times{\\mathbf Z}\\) but has\nno geometrically proper generating set.\nWe leave the details as an exercise for the reader.\n\\end{example}\n\n\\begin{example} \\label{x+1_x^3_ex}\nIt was shown in \\cite{x+1_x^p_free} that\nsubgroup \\(\\gen{t+1,t^3}\\) of \\({\\operatorname{Homeo}_+(\\R)}\\)\nis free.\nSince any minimal generating set for a free group is a free basis for the group,\nany minimal generating set is algebraically fast.\nOn the other hand this group is not embeddable\ninto \\(F\\) by \\cite{brin+squier} and hence by Corollary \\ref{EmbeddInF}\nis not isomorphic to a group with a geometrically proper generating set.\n\\end{example}\n\nNext we will prove Theorem \\ref{NoGeoProperGen},\nwhich shows that the ``\\(F\\)-less'' hypothesis in Theorem \\ref{GeoProperGen} can not be removed.\nNote that no assumption is made in this theorem concerning the number of bumps used in \nthe geometrically proper generating set.\n\n\\begin{customthm}{\\ref{NoGeoProperGen}}\nIf a finite index subgroup of \\(F\\) is isomorphic to \\(\\gen{X}\\) for some  geometrically proper\n\\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\), then it is isomorphic to \\(F\\).\n\\end{customthm}\n\n\\begin{proof}\nLet \\(G\\) be a subgroup of \\(F\\) of finite index such that\n\\(G \\cong \\gen{X}\\) for some finite geometrically proper \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\).\nLet \\(H\\) denote \\(\\gen{X}\\) and \\(\\phi : G \\to H\\) be a fixed isomorphism.\nWe must show that \\(G\\) is isomorphic to \\(F\\).\n\nBy \\cite{bleak+wassink} we know that, since \\(G \\leq F\\) is of\nfinite index, it is\nof the form \\(\\pi^{-1}(K)\\) for some finite\nindex subgroup \\(K\\) of \\({\\mathbf Z}\\times{\\mathbf Z}\\).\nFurthermore, also by \\cite{bleak+wassink},\n\\(\\pi^{-1}(K)\\) is isomorphic to \\(F\\) if and only if \\(K\\) admits a generating set of the form \\(\\left\\{\\left(p,0\\right),\\left(0,q\\right)\\right\\}\\) for\n\\(p,\\,q\\in{\\mathbf Z}\\setminus\\{0\\}\\).\n\nObserve in any case that \\(K\\) admits a two element generating\nset \\(\\{(p_i,q_i) \\mid  i < 2\\}\\), as it is finite index in \\({\\mathbf Z}\\times{\\mathbf Z}\\).\nWe will first derive some properties of \\(G\\) which come from viewing it\nas a subgroup of \\(F\\).\nDefine \\(f_i := a^{p_i} b^{q_i}\\).\nIt is shown in \\cite{bleak+wassink} that \\(G\\) can be generated by\n\\(\\{f_ i \\mid i < 2\\} \\cup F'\\).\nFor \\(0\\le x<y\\le 1\\) with both \\(x\\) and \\(y\\) in\n\\({\\mathbf Z}[1\/2]\\) we let \\(F_{[x,y]}\\) denote the subgroup of \\(F\\)\nconsisting of those elements of \\(F\\) whose support is contained in\n\\([x,y]\\).  It is standard that \\(F_{[x,y]}\\cong F\\) and that if\n\\(x<w<y<z\\), then \\(F_{[x,y]}\\cup F_{[w,z]}\\) generates\n\\(F_{[x,z]}\\).\n\nNext we claim that the kernel of \\(\\pi_G\\) is \\(G' = F'\\).\nIt is trivial that \\(F'\\subseteq G\\) and since \\(F'\\) is simple and not abelian\nit follows that we also have \\(F'\\subseteq G'\\).\nOn the other hand, since \\(G\\subseteq F\\) we have \\(G'\\subseteq F'\\).\nSince the image of \\(\\pi_G\\) is abelian, we have\n\\(G'\\) is contained in the kernel of \\(\\pi_G\\).\nLastly, \\(\\ker(\\pi_G) \\subseteq \\ker(\\pi) \\subseteq F'\\). \n\n\\begin{claim} \\label{BothGermsNontrivial}\nIf \\(g \\in G\\) is such that neither coordinate of \\(\\pi (g)\\)\nis \\(0\\), then there is a finite subset \\(T\\) of \\(G'\\) with\n\\(G' \\subseteq \\gen{T \\cup\\{g\\}}\\).\n\\end{claim}\n\n\\begin{proof}\nNotice that there is an\n\\(\\epsilon>0\\) so that \\(g\\) moves all points in \\((0,\\epsilon)\\)\nand \\((1-\\epsilon,1)\\).  It follows that for some \\(x<y\\) in\n\\({\\mathbf Z}[1\/2]\\) the images of \\((x,y)\\) under integral powers of \\(g\\)\ncover \\((0,1)\\).\nHence the conjugates of \\(F_{[x,y]}\\) under integral powers of \\(g\\) generate \\(F'=G'\\).\nThe claim now follows from the fact that\n\\(F_{[x,y]} \\cong F\\) is generated by two elements.\n\\end{proof}\n\nIn what follows, we will refer to the closure of the support of an \\(f\\in{\\operatorname{Homeo}_+(I)}\\)\nas the \\emph{extended support of \\(f\\)}.\n\n\\begin{claim} \\label{EquivCommute}\nSuppose \\(g \\in G \\setminus G'\\) has connected extended support and \\(h \\in G\\).\nIf \\(g\\) commutes with \\(g^h\\) then \\(g\\) commutes with \\(h\\).\n\\end{claim}\n\n\\begin{proof}\nWe know that 0 or 1 is in the extended support of \\(g\\) and both 0 and 1 are fixed by \\(h\\). \nSince \\(g\\) has only finitely many bumps, it has finitely many transition points.\nIf any of these are moved by \\(h\\), then at least one fixed point of\n\\(g^h\\) is moved by \\(g\\) or at least one fixed point of \\(g\\) is\nmoved by \\(g^h\\).\nIn either case this implies that \\(g\\) does not commute with \\(g^h\\).\nIf no transition point of \\(g\\) is moved by \\(h\\), then the orbitals of \\(g^h\\) are\nthose of \\(g\\).\nIt follows from the chain rule that on each\norbital \\(J\\) of \\(g\\), \\(g\\) and \\(g^h\\) agree on a small neighborhood of the\nendpoints of \\(J\\).\nBy \n\\cite[\\S 4]{MR1855112} two elements of \\({\\operatorname{PL}_+(I)}\\) commute over a\ncommon orbital of support only if they admit a common root over that orbital.\nIn particular, if \\(g \\restriction J\\) and \\(g^h \\restriction J\\)\nagree in a neighborhood of the endpoints of \\(J\\)\nand they commute, then they must be equal.  Hence \\(h\\) commutes with \\(g\\) over each orbital of support of \\(g\\), so \\(h\\) and \\(g\\) commute.\n\\end{proof}\n\nWe now turn our attention to our representation of \\(G\\) as\na subgroup \\(H=\\gen{X}\\) of \\({\\operatorname{Homeo}_+(I)}\\) where \\(X\\) is geometrically proper.\nIf \\(H\\) has more than one component of support, then the\nrestriction to each is a quotient of \\(H\\).\nSince no non-trivial element of \\(F\\) commutes with every element of \\(F'\\) and since \\(F'=G'\\) is simple,\nit follows that every\nnontrivial normal subgroup of \\(G\\) contains \\(G'\\).\nConsequently every proper quotient of \\(H\\cong G\\) is abelian.\nThus if no restriction were an isomorphism, \\(H\\) would be abelian (which is absurd).\nWe can thus replace \\(H\\) by its restriction to a component of its support on which the\nrestriction is faithful, and further, \nwe can conjugate by a homeomorphism of \\({\\mathbf R}\\)\nso that the support is \\((0,1)\\).  Note that this new embedding of \\(G\\) in \\({\\operatorname{Homeo}_+(I)}\\) is geometrically proper if the original embedding is geometrically proper.\n\nLet \\(\\pi_0\\) denote the restriction of the germ homomorphism \\(\\gamma_0^+\\) to \\(H\\) and \n\\(\\pi_1\\) denote the restriction of \\(\\gamma_1^-\\) to \\(H\\);\ndefine \\(\\pi_H(h) := (\\pi_0(h),\\pi_1(h))\\).\nObserve that since \\(X\\) is geometrically proper we have that\nfor each \\(i < 2\\), there is at most one\nelement of \\(X\\) which is not in the kernel of \\(\\pi_i\\).\nObserve that this implies the image of \\(\\pi_H\\) is abelian and hence\n\\(H' \\subseteq \\ker (\\pi_H)\\).\n\n\\begin{claim} \\label{NonCyclicImage}\nThe image of \\(\\pi_H\\) is not cyclic.\n\\end{claim}\n\n\\begin{proof}\nRecall that \\(G' = \\ker (\\pi_G)\\) and \\(H' \\subseteq \\ker(\\pi_H)\\).\nIn particular, \\(\\phi\\) induces a well defined homomorphism\nfrom \\(G\/G'\\) to \\(H\/\\ker (\\pi_H)\\).\nIf the image of \\(\\pi_H\\) were cyclic, then this homomorphism would\nhave a nontrivial kernel. \n\nPick \\(g:=f_0^m f_1^n \\in G\\setminus G'\\) in the kernel of \\(\\pi_H\\).\nThe element \\(g\\) will have connected extended support.\nAs \\(\\phi(g)\\in\\ker(\\pi_H)\\) we must have the support of \\(\\phi(g)\\) is contained in \\([x,y]\\) for\nsome \\(0 < x < y < 1\\).  As the support of \\(H\\) is \\((0,1)\\) there is an \\(h \\in H\\) so that \\(xh>y\\), and for this \\(h\\) we have \n\\(\\phi(g)\\) commutes with \\(\\phi(g)^h\\), but not with \\(h\\).\nNow, \\(g\\) commutes with \\(\\phi^{-1} (\\phi(g)^h)\\), but not with \\(\\phi^{-1}(h)\\), however, \\(\\phi^{-1}(\\phi(g)^h) = g^{\\phi^{-1}(h)}\\),\ncontradicting Claim \\ref{EquivCommute}.\n\\end{proof}\n\nAt this point we know that, by the geometric properness of \\(X\\), for each coordinate \\(i\\in\\{0,1\\}\\)\nthere is a unique element \\(h_i\\in X\\) so that\n\\(i\\) is in the extended support of \\(h_i\\) and so that \\(h_0\\ne h_1\\).\nThe fact that \\(h_0\\) and \\(h_1\\) are distinct and unique implies that\nthe image of \\(\\pi_H\\) is the product of the images of \\(\\pi_0\\) and \\(\\pi_1\\).\n\n\\begin{claim}\\label{claim_noFiniteGenSet}\nFor each element \\(h\\) of \\(X\\) whose extended support contains 0 or 1,\nthere is no finite subset \\(T\\) of the kernel of \\(\\pi_{H}\\) so that\n\\(\\ker(\\pi_{H}) \\subseteq \\gen{T\\cup \\{h\\}}\\).\n\\end{claim}\n\n\\begin{proof}\nLet \\(h\\) be given and suppose without loss of generality that \\(0\\) is\nin the extended support of \\(h\\).\nAs noted above, \\(1\\) can not be in the extended support of \\(h\\). \nSince \\(\\gen{X}'\\subseteq \\ker(\\pi_{H})\\),\nthere are nontrivial elements in \\(\\ker(\\pi_{H})\\).\nBecause the orbital of \\(H\\) is \\((0,1)\\),\nthere are points arbitrarily close to 0 and 1 moved by elements of\n\\(\\ker (\\pi_{H})\\).\nIt follows that if \\(T\\subseteq\n\\ker(\\pi_{H})\\) is finite, then there is a neighborhood of \\(1\\)\nfixed by all elements of \\(\\gen{T\\cup \\{h\\}}\\) and thus\n\\(\\gen{T\\cup \\{h\\}}\\) cannot contain all of \\(\\ker(\\pi_{H})\\).\n\\end{proof}\n\nIn order to finish the proof, it suffices to show that \\(\\pi_G\\left(\\phi^{-1}(h_0)\\right)\\) and\n\\(\\pi_G\\left(\\phi^{-1}(h_1)\\right)\\) generate the image of \\(\\pi_G\\) and that\neach have exactly one (necessarily different) nonzero coordinate. \nSince, by the proof of Claim \\ref{NonCyclicImage},\n\\(\\phi\\) induces an isomorphism \\(G\/G' \\cong H\/\\ker (\\pi_H)\\), it follows that\n\\(\\{\\pi_G(\\phi^{-1}(h_i)) \\mid i < 2\\}\\) must generate \\(G\/G'\\).\nOn the other hand, by Claims \\ref{BothGermsNontrivial} and \\ref{claim_noFiniteGenSet},\nit must be that one coordinate\nof \\(\\pi_G(\\phi^{-1}(h_i))\\) must be 0 for each \\(i \\in \\{0,1\\}\\).\nThis shows that these two elements, which generate the group \\(K\\), together make a set of the form \\(\\left\\{(p,0),(0,q)\\right\\}\\) for some non-zero \\(p\\) and \\(q\\), and therefore  \\(G \\cong F\\).\n\\end{proof}\n\n\\begin{remark} \nThe group \\(E=\\{(p,q)\\in{\\mathbf Z}\\times{\\mathbf Z}\\mid p+q \\equiv0 \\mod2\\}\\)\nis a subgroup of \\({\\mathbf Z} \\times {\\mathbf Z}\\) which is not of the form \\(P \\times Q\\) and hence\n\\(\\pi^{-1}_F (E)\\) is a finite index subgroup of \\(F\\) which is not isomorphic to \\(F\\).\nIn particular, there are finite index subgroups of \\(F\\) which do not admit\ngeometrically proper generating sets.\n\\end{remark}\n\n\\section{Abstract Ping-Pong Systems}\n\n\\label{AbstractPingPongSec}\n\nIn this section we will abstract the analysis of geometrically\nfast systems of bumps in previous sections to\nthe setting of permutations of a set \\(S\\).\n(By \\emph{permutation} of \\(S\\) we simply mean a bijection from \\(S\\) to \\(S\\).)\nOur goal will be to state the analog of Theorem \\ref{CombToIso} and its consequences.\nThe proofs are an exercise for the reader.\n\nSuppose now that \\(A\\) is a collection of permutations of a set \\(S\\) such that\n\\(A \\cap A^{-1} = \\emptyset\\).\nA \\emph{ping-pong system} on \\(A\\) is an assignment \\(a \\mapsto \\operatorname{dest}(a)\\) of sets\nto each element of \\(A^\\pm\\) such that whenever \\(a\\) and \\(b\\) are in \\(A^\\pm\\) and \\(s \\in S\\):\n\\begin{itemize}\n\n\\item \\label{basic_Dp}\n\\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(a)\\) and if \\(s \\in \\operatorname{supt}(a)\\), then\nthere is an integer \\(k\\) such that \\(s a^k \\in \\operatorname{dest} (a)\\);\n\n\\item \\label{ping-pong_cond}\nif \\(s \\in \\operatorname{supt}(a)\\), then \\(s a  \\in \\operatorname{dest}(a)\\) if and only if \\(s \\not \\in \\operatorname{src}(a) : = \\operatorname{dest}(a^{-1})\\);\n\n\\item if \\(a \\ne b\\), then \\(\\operatorname{dest}(a) \\cap \\operatorname{dest}(b)\\) is empty;\n\n\\item if \\(\\operatorname{dest}(a) \\cap \\operatorname{supt}(b) \\ne \\emptyset\\), then \\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(b)\\).\n\n\\end{itemize}\n\\noindent\nThe following lemma summarizes some immediate consequences of this definition.\n\\begin{lemma}\nGiven a set \\(S\\) and  a collection \\(A\\) of permutations of \\(S\\)\nequipped with a ping pong system, the following are true:\n\\begin{itemize}\n\n\\item if \\(a \\in A\\) and \\(s \\in \\operatorname{supt} (a)\\), then there is a unique \\(k \\in {\\mathbf Z}\\) such that:\n\\[\ns a^k \\in \\operatorname{supt}(a) \\setminus (\\operatorname{src}(a) \\cup \\operatorname{dest}(a))\\]\n\n\\item if \\(a \\in A\\), then \\(\\operatorname{dest}(a) a \\subseteq \\operatorname{dest}(a)\\);\n\n\\end{itemize}\nIn particular, all elements of \\(A\\) have infinite order.\n\\end{lemma}\nAs remarked in Section \\ref{FastBumpsSec}, geometrically fast sets of bumps admit\na ping-pong system.\nThe meanings of \\emph{source}, \\emph{destination}, and \\emph{locally reduced word}\nall readily adapt to this new context.\nFurthermore, the proofs of Lemmas \\ref{LocRedBasics}--\\ref{FellowTraveler3} given in Section \\ref{PingPongSec}\nuse only the axiomatic properties of a ping-pong system and thus these lemmas \nare valid in the present context.\nThe next example is simplistic, but it will serve to illustrate a number of points in this section.\n\n\\begin{example} \\label{PSL2Example}\nView the real projective line \\(\\mathbf{P}\\) as \\({\\mathbf R} \\cup \\{\\infty\\}\\) and \\(\\operatorname{PSL}_2({\\mathbf Z})\\) as a group of\nfractional linear transformations of \\(\\mathbf{P}\\).\nThe homeomorphisms \\(\\alpha\\) and \\(\\beta\\) of \\(\\mathbf{P}\\) defined by\n\\[\nt\\alpha := t + 1 \\qquad t \\beta := \\frac{t}{1-t}\n\\]\ngenerate \\(\\operatorname{PSL}_2({\\mathbf Z})\\).\nIf we take \\(A = \\{\\alpha^2,\\beta^2\\}\\), then\n\\[\n\\operatorname{src}(\\alpha^2) := (-\\infty,-1)  \\qquad \\qquad \\operatorname{dest}(\\alpha^2) := [1,\\infty)\n\\]\n\\[\n\\operatorname{src}(\\beta^2) := (0,1) \\qquad \\qquad \\operatorname{dest}(\\beta^2) := [-1,0) \n\\]\ndefines a ping-pong system.\nIt is well known that \\(\\gen{\\alpha^2,\\beta^2}\\) is free;\nin fact this is one of the classical applications of the Ping-Pong Lemma.\n\\end{example}\n\nIn order to better understand \\(\\gen{A}\\) when \\(A\\) is a set of permutations admitting a\nping-pong system, it will be helpful to represent \\(\\gen{A}\\) as a family of homeomorphisms\nof a certain space \\(K_A\\).\nThis compact space can be thought of as a space of \\emph{histories} in the sense\nof Section \\ref{PingPongSec}.  \nIf \\(S\\) is the underlying set which elements of \\(A\\) permute,\nlet \\(M = M_A\\) denote the collection of all sets of the form\n\\[\n\\tilde s: = \\{a \\in A \\mid s \\in \\operatorname{supt}(a)\\}\n\\]\nwhere \\(s \\in S \\setminus \\bigcup \\{\\operatorname{dest}(a) \\mid a \\in A^\\pm\\}\\).\nElements of \\(M\\) will play the same role as the \ninitial markers of a geometrically fast collection of bumps.\n\n\\begin{example} \\label{PSL2Marker}\nContinuing with the Example \\ref{PSL2Example}, \\(M\\) consists of two points:\n\\(\\widetilde 0 = \\{\\alpha\\}\\) and \\(\\widetilde \\infty = \\{\\beta\\}\\).\nWe can also restrict the action of \\(\\operatorname{PSL}_2({\\mathbf Z})\\) on \\(\\mathbf{P}\\) to the irrationals.\nIn this case \\(M\\) is empty.\n\\end{example}\n\nIt will be convenient to define \\(\\operatorname{dest}(\\tilde s) := \\bigcap \\{\\operatorname{supt}(a) \\setminus \\operatorname{src}(a) \\mid a \\in \\tilde s\\}\\) and\n\\(\\operatorname{supt}(\\tilde s) := \\emptyset\\).\nDefine \\(K_A\\) to be all \\(\\eta\\) such that:\n\\begin{itemize}\n\n\\item \\(\\eta\\) is a suffix closed family of finite \nstrings in the alphabet \\(A^\\pm \\cup M\\);\n\n\\item if \\(\\str{ab}\\) are consecutive symbols of an element of \\(\\eta\\), then\n\\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(b) \\setminus \\operatorname{src}(b)\\);\n\n\\item for each \\(n\\), there is at most one element of \\(\\eta\\) of length \\(n\\);\n\n\\item if \\(\\str{w} \\in \\eta\\) and \\(\\eta\\) does not contain a symbol from \\(M\\),\nthen \\(\\str{w}\\) is a proper suffix of an element of \\(\\eta\\).\n\n\\end{itemize}\n\\noindent\nThe second condition implies that elements of \\(\\eta\\) are freely reduced since if \\(b = a^{-1}\\),\nthen \\(\\operatorname{src}(b) = \\operatorname{dest}(a)\\).\nObserve that if \\(\\str{w}\\) is in \\(\\eta\\), the only occurrence of an element of \\(M\\) in \\(\\str{w}\\) must\nbe as the first symbol of \\(\\str{w}\\) (and there need not be any occurrence of an element of \\(M\\) in \\(\\str{w}\\)).\n\nNotice that every \\(\\eta \\in K_A\\) has at least one element other than \\(\\varepsilon\\) and that all elements\nof \\(\\eta\\) of positive length must have the same final symbol.\nWe define \\(\\operatorname{dest}(\\eta) := \\operatorname{dest}(a)\\) where \\(\\str{a}\\) is the final symbol of every\nelement of \\(\\eta\\) other than \\(\\varepsilon\\).\nWe topologize \\(K_A\\) by declaring that\n\\([\\str{w}] : = \\{\\eta \\in K_A \\mid \\str{w} \\in \\eta\\}\\) is closed and open.\nNotice that if \\(\\eta\\) is finite, it is an isolated point of \\(K_A\\).\n\n\\begin{prop}\n\\(K_A\\) is a Hausdorff space and if \\(A\\) is finite, then \\(K_A\\) is compact.\n\\end{prop}\n\n\\noindent\nEach \\(a \\in A^\\pm\\) defines a homeomorphism \\(\\hat a : K_A \\to K_A\\) by:\n\\[\n\\eta \\hat a := \n\\begin{cases}\n\\{\\str{ua} \\mid \\str{u} \\in \\eta\\} \\cup \\{\\varepsilon\\} & \\textrm{ if } \\operatorname{dest}(\\eta) \\subseteq \\operatorname{supt}(a) \\setminus \\operatorname{src}(a) \\\\\n\\{\\str{u} \\mid \\str{ua}^{-1} \\in \\eta\\}  & \\textrm{ if } \\operatorname{dest} (\\eta) = \\operatorname{src}(\\str{a}) \\\\\n\\eta & \\textrm{ if } \\operatorname{dest}(\\eta) \\cap \\operatorname{supt}(\\str{a}) = \\emptyset\n\\end{cases}\n\\]\nThus \\(\\eta \\hat a\\) is obtained by \nappending \n\\(\\str{a}\\) to the end of every element of \\(\\eta\\),\ncollecting the local reductions, and possibly including \\(\\varepsilon\\).\nSet \\(\\hat A = \\{\\hat a: a \\in A\\}\\).\n\nWe say that a ping-pong system on \\(A\\) is \\emph{faithful} if\n\\(\\Lambda_A : = \\{\\eta \\in K_A : \\eta \\textrm{ is finite}\\}\\) is dense in \\(K_A\\) (i.e.\nwhenever \\(\\str{w}\\) is in some \\(\\eta \\in K_A\\), there is a finite \\(\\eta' \\in K_A\\)\nwhich has \\(\\str{w}\\) as an element).\n\n\\begin{example} \\label{NonfaithfulExample}\nAs noted above, if we restrict the elements of \\(\\operatorname{PSL}_2({\\mathbf Z})\\) to the irrationals,\nthen  \\(M = \\emptyset\\) and in particular the system is not faithful.\nOn the other hand, \n\\[\n\\operatorname{dest}(\\alpha^4) := (2,\\infty) \\cap S \\qquad \\qquad \\operatorname{dest}(\\alpha^{-4}) := (-\\infty,-2) \\cap S\n\\]\n\\[\n\\operatorname{dest}(\\beta^4) := (-1\/2,0) \\cap S \\qquad \\qquad \\operatorname{dest}(\\beta^{-4}) := (0,1\/2) \\cap S.\n\\]\ndefines a ping-pong system in which \\(M\\) contains a single element \\(\\{\\alpha,\\beta\\}\\).\n\\end{example}\n\n\\noindent\nWhile not every ping-pong system is faithful,\nthe reader is invited to verify that if \\(A\\) admits a ping-pong system, then\n\\(\\{a^2 \\mid a \\in A\\}\\) admits a faithful ping-pong system.\n\n\\begin{thm}\nIf \\(A\\) is a set of permutations which admits a ping-pong system, then\n\\(a \\mapsto \\hat a\\) extends to an epimorphism of \\(\\gen{A}\\) onto \\(\\gen{\\hat A}\\).\nIf the ping-pong system is faithful, then the epimorphism is an isomorphism.\n\\end{thm}\n\nThe map \\(x \\mapsto \\eta(x)\\) defined in Section \\ref{PingPongSec} adapts \\emph{mutatis mutandis}\nto define a map \\(s \\mapsto \\eta(s)\\) from \\(S\\) into \\(K_A\\).\nIt is readily verified that if \\(a \\in A\\) and \\(s \\in S\\), then \\(\\eta(sa) = \\eta(s)\\hat a\\).\nThe existence of a faithful ping-pong system also has the following structural consequence\nwhich follows readily from the abstract form of Lemma \\ref{TreeAction}.\n\n\\begin{prop}\nIf \\(A\\) admits a faithful ping-pong system, then \\(\\gen{A}\\) is torsion free.\n\\end{prop}\n\\noindent\nThis shows in particular that \\(\\operatorname{PSL}_2({\\mathbf Z}) = \\gen{\\alpha,\\beta}\\) --- which contains elements of finite order such as\n\\(t \\mapsto -1\/t\\) --- does not admit a ping-pong system.\n\nA \\emph{blueprint} for a ping-pong system is a pair \\(\\mathfrak{B} = (\\str{B},\\str{supt})\\)\nsuch that:\n\\begin{itemize}\n\n\\item \\(\\str{B}\\) is a set and \\(\\str{supt}\\) is a binary relation on \\(\\str{B}\\)\nwhich is interpreted as a set-valued function:\n\\(\\str{b} \\in \\str{supt}(\\str{a})\\) if \\((\\str{a}, \\str{b}) \\in \\str{supt}\\);\n\n\\item if \\(\\str{a} \\in \\str{B}\\) and \\(\\str{supt}( \\str{a})\\) is nonempty, then \\(\\str{a} \\in \\str{supt}(\\str{a})\\);\n\n\\item if \\(\\str{a} \\in \\str{A}\\), then there is a unique \\(\\str{a}^{-1} \\in \\str{A} \\setminus \\{\\str{a}\\}\\) with\n\\(\\str{supt} (\\str{a}^{-1}) = \\str{supt} (\\str{a})\\).\n\n\\end{itemize}\nAdditionally, setting \n\\(\\str{A} : = \\{\\str{a} \\in \\str{B} \\mid \\str{supt}(\\str{a}) \\ne \\emptyset\\}\\) we require that:\n\\begin{itemize}\n\n\\item if \\(\\str{b} \\ne \\str{c} \\in \\str{B} \\setminus \\str{A}\\), then \\(\\str{\\tilde b} \\ne \\str{\\tilde c}\\) where\n\\({\\str{\\tilde b}} := \\{\\str{a} \\in \\str{A} \\mid \\str{b} \\in \\str{supt}(\\str{a}) \\}\\).\n \n\\end{itemize}\nTwo blueprints are isomorphic if they are isomorphic as structures.\nIf \\(A\\) is a set of permutations which admits a ping-pong system, then the blueprint\n\\(\\mathfrak{B}_A = (\\str{B}_A,\\str{supt}_A)\\)\nfor the system is defined by\n\\(\\str{B}_A := \\{\\str{a} \\mid a \\in A^\\pm\\} \\cup \\{{\\str{\\tilde  s}} \\mid \\tilde s \\in \\widetilde S\\}\\)\nwith \\(\\str{b} \\in \\str{supt}_A(\\str{a})\\) if \\(\\operatorname{dest}(b) \\subseteq \\operatorname{supt}(a)\\).\nAlso, if \\(\\mathfrak{B} = (\\str{A},\\str{supt})\\) is a blueprint for a ping-pong system, then\none defines \\(K_{\\mathfrak{B}}\\) and homeomorphisms \\({\\str{\\hat a}} : K_{\\mathfrak{B}} \\to K_{\\mathfrak{B}}\\)\nfor \\(\\str{a} \\in \\str{A}^{\\mathfrak{B}}\\) by a routine adaptation of the construction above.\nIn fact \\(K_A\\) is can be regarded as factoring though its blueprint in the sense that\n\\(K_A = K_{\\mathfrak{B}_A}\\) modulo identifying \\(a\\) and \\(\\str{a}\\).\nIt is routine to check that the following theorem holds as well.\n\n\\begin{thm} \\label{AbsCombToIso}\nIf \\(\\mathfrak{B}_0\\) and \\(\\mathfrak{B}_1\\) are isomorphic blueprints for ping-pong systems,\nthen the isomorphism induces an homeomorphism \\(\\theta:K_{\\mathfrak{B}_0} \\to K_{\\mathfrak{B}_1}\\)\nsuch that \\(g \\mapsto g^\\theta\\) defines an isomorphism between\n\\(\\gen{\\hat {\\str{a}} \\mid \\str{a} \\in \\str{A}_0}\\) and \\(\\gen{\\hat {\\str{a}} \\mid \\str{a} \\in \\str{A}_1}\\).\n\\end{thm}\n\nIf \\(A\\) is a set of permutations which admits a faithful ping-pong system,\nthen the blueprint for \\(A\\) and for \\(\\{a^{k(a)} \\mid a \\in A\\}\\) are canonically\nisomorphic whenever \\(a \\mapsto k(a)\\) is an assignment of a positive integer to each element of \\(A\\).\n\n\\begin{cor}\nAny set of permutations \\(A\\) which admits a faithful ping-pong system is an algebraically\nfast generating set for \\(\\gen{A}\\).\n\\end{cor}\n\nIf the system is not faithful, then \\(\\{a^2 \\mid a \\in A^2\\}\\) may contain new markers, as\nwas illustrated in Example \\ref{NonfaithfulExample}.\nNotice that in this example,\nboth \\(\\gen{\\alpha^2,\\beta^2}\\) and \\(\\gen{\\alpha^4,\\beta^4}\\) are free --- and hence marked isomorphic ---\neven though the blueprints associated to the ping-pong systems are not isomorphic.\n\nA blueprint \\(\\mathfrak{B}\\) is (cyclically) orderable if there is a (cyclic) ordering on \\(\\str{B}\\)\nsuch that for all \\(\\str{a} \\in \\str{A}\\), \n\\(\\str{supt}(\\str{a})\\) is an interval in the (cyclic) ordering with endpoints \\(\\operatorname{src}(\\str{a})\\) and \\(\\operatorname{dest}(\\str{a})\\).\nIt is readily checked that the (cyclic) order on \\(\\str{B}\\)\ninduces a reverse lexicographic (cyclic) order on \\(K_{\\mathfrak{B}}\\) which is preserved by the\nhomeomorphisms \\(\\hat {\\str{a}}\\) for \\(\\str{a} \\in \\str{A}\\).\n\n\\begin{cor}\nIf \\(A\\) is a finite set of permutations which admits a faithful ping-pong system, then\n\\(\\gen{A}\\) embeds into Thompson's group \\(V\\).\nIf the blueprint of \\(A\\) is cyclically orderable, then \\(\\gen{A}\\) embeds into\n\\(T\\).\nIf the blueprint of \\(A\\) is orderable, then \\(\\gen{A}\\) embeds into \\(F\\). \n\\end{cor}\n\n\\begin{proof}\nIt is readily verified that any finite blueprint can be realized as the blueprint of\na ping-pong system of a finite subset of \\(V\\).\nMoreover, if the blueprint is cyclically orderable (or orderable), then it can be realized\nby elements of \\(T\\) (respectively \\(F\\)). \nThe corollary follows by Theorem \\ref{AbsCombToIso}.\n\\end{proof}\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\n\\providecommand{\\MRhref}[2]{\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nHypertensive disorders of pregnancy are a class of high blood pressure disorders that occur during the second half of pregnancy, which include gestational hypertension, preeclampsia and severe preeclampsia. They are characterized by a diastolic blood pressure higher than 90 mm Hg and\/or a systolic blood pressure higher than 140 mm Hg and they are often accompanied by proteinuria. These disorders affect about 10\\% of pregnant women around the world, with preeclampsia occurring in 2--8\\% of all pregnancies \\citep{timokhina2019}. These disorders represent one of the leading causes \nof maternal and fetal morbidity and mortality, contributing to 7--8\\% of maternal death worldwide \\citep{dolea2003,shah2009,mcclure2009}. The World \nHealth Organization estimates that the incidence of preeclampsia is seven \ntimes higher in developing countries than in developed countries. However, the occurrence  of these diseases appears under-reported in low and middle income countries, implying that the true incidence is unknown \\citep{igberase2006,malik2018}. While there is evidence that hypertensive disorders of pregnancy are related with the development of cardiac dysfunctions both in the mother and in the child \\citep{bellamy2007,davis2012, ambrovzic2020dynamic, garcia2020maternal, aksu2021cardiac, demartelly2021long}, there is no common agreement on the relation between the severity of hypertension and cardiac dysfunction \\citep{tatapudi2017} and echocardiography is not included in baseline evaluation of hypertensive disorders of pregnancy. Further investigations on these disorders are needed, especially for developing countries, where women often give birth at a younger age with respect to developed countries. \n\nThe goal of this work is to detect which cardiac function is altered and \nunder which hypertensive disorders by relying on a principled Bayesian nonparametric approach. An interesting case-control study to explore the relation between cardiac dysfunction and hypertensive disorders is provided by \\citet{data}, where the measures of ten different cardiac function indexes were recorded in four groups of pregnant women in India. Groups of women are characterized by different hypertensive disorder diagnoses, that are naturally ordered based on the severity of the diagnosed disorder: healthy (C), gestational hypertension (G), mild preeclampsia (M) and severe preeclampsia (S). \nHypertensive diagnoses are used as identifiers for what we call populations of patients and we refer to cardiac function indexes also with the term response variables. For each response variable we want to determine a partition of the four populations of patients. This amounts to identifying similarities between different hypertensive disorders, with respect to each cardiac index. Supposing, for instance, that the selected partition assigns all the populations to the same cluster, one can conclude that no \nalteration is shown for the corresponding  cardiac index across different hypertensive diseases.\n\nOur goal of identifying a partition of the four patients' populations for each of the ten responses can be rephrased as a problem of multiple model selection: we want to select the most plausible partition for each cardiac index. \nFrequentist hypothesis testing does not allow to deal with more than two \npopulations in a straightforward way\nand pairwise comparisons may lead \nto conflicting conclusions. Conversely, a Bayesian approach yields the posterior distribution on the space of partitions, which can be used for simultaneous comparisons. \nMoreover, the presence of $M=10$ jointly tested cardiac indexes requires to perform model selection repeatedly ten times. Once again, a Bayesian approach seems to be preferred, because, as observed for instance by \\citet{scott2006}, it does not require the introduction of a penalty term for multiple comparison, thanks to the prior distribution build-in penalty. \n\nHere we design\na Bayesian nonparametric model, that is tailored to deal with both a collection of ordered populations and the multivariate information of the response variables, while preserving the typical flexibility of nonparametric models and producing easily interpretable results.\nWhen applied to the dataset on transthoracic echocardiography results for a cohort of Indian pregnant women in Section \\ref{s:results}, our model effectively identifies \nmodified cardiac functions in hypertensive patients compared to healthy subjects and progressively increased alterations with the severity of the disorder, in addition to other more subtle findings. The observed data $X_{i,j,m}$ represent the measurement of the $m$-th response variable (cardiac index) on the $i$-th individual (pregnant woman) in the $j$-th population (hypertensive disorder) \nand, as in standard univariate ANOVA models, they are assumed to be partially exchangeable across disorders.\nThis means that for every $m\\in\\{1,\\ldots,M\\}$,  \nthe law of $(\\,(X_{i,1,m})_{i\\ge 1},\\ldots,(X_{i,J,m})_{i\\ge 1})$ is invariant with respect to permutations within each sequence of random variables, namely for any positive integers $n_1,\\ldots,n_J$ \n\\[\n(\\,(X_{i,1,m})_{i=1}^{n_1},\\ldots,(X_{i,J,m})_{i=1}^{n_J})\\stackrel{\\mbox{\\scriptsize{d}}}{=} \n(\\,(X_{{\\sigma_1(i)},1,m})_{i=1}^{n_1},\\ldots,(X_{{\\sigma_J(i)},J,m})_{i=1}^{n_J})\n\\]\nfor all permutations $\\sigma_j$ of $(1,\\ldots,n_j)$, with $j=1,\\ldots,J$. This is a natural generalization of exchangeability to tackle heterogeneous data and, by de Finetti's representation theorem, it amounts to assuming the existence of a collection of (possibly dependent) random probability measures $\\{\\pi_{j,m}:\\:\\: j=1,\\ldots,J\\;\\: m=1,\\ldots,M\\}$ such that\n\\[\nX_{i,j,m} \\mid \\pi_{j,m} \\stackrel{\\mbox{\\scriptsize{\\rm iid}}}{\\sim} \\pi_{j,m} \\qquad i=1,\\ldots,n_j\n\\]  \nHence, for any two populations $j\\ne j'$, homogeneity corresponds to $\\pi_{j,m}=\\pi_{j',m}$ (almost surely). However, a reliable assessment of this type of homogeneity is troublesome when having just few patients per \ndiagnosis, as it happens in the mild preeclampsia subsample.  \nWithout relying on simplifying parametric assumptions, a small sub-sample size may not be sufficiently informative to infer equality of entire unknown distributions. \nTo overcome this issue, without introducing parametric assumptions, we resort to an alternative weaker notion of homogeneity between populations $j$ and $j'$: we only require the conditional means of the two populations to (almost surely) coincide\n\\begin{equation}\n\t\\label{eq:equality_means}\n\t\\mathbb{E}(X_{i,j,m}\\mid\\pi_{j,m})=\\mathbb{E}(X_{i,j',m}\\mid\\pi_{j',m}).\n\\end{equation}\nAccording to this definition, the detection of heterogeneities in cardiac function indexes amounts to inferring which cardiac indexes have means that differ \nacross diagnoses, as it is done in standard parametric ANOVA models.\nBesides clustering populations according to \\eqref{eq:equality_means}, it is also of interest to cluster patients, both within and across different groups, once the effect of the specific hypertensive disorder is taken \ninto account. This task may be achieved by assuming a model that decomposes the observations as \n\\begin{equation}\n\t\\label{eq:decomp_x}\n\tX_{i,j,m}=\\theta_{j,m}+\\varepsilon_{i,j,m}\\qquad\n\t\\varepsilon_{i,j,m}|(\\xi_{i,j,m},\\sigma_{i,j,m}^2)\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim} \\mbox{N}(\\xi_{i,j,m},\\sigma_{i,j,m}^2)\n\\end{equation} \nand the $\\xi_{i,j,m}$ have a symmetric distribution around the origin, in order to ensure $E(\\xi_{i,j,m})=0$. In view of this decomposition, we \nwill let $\\theta_{j,m}$ govern the clustering of populations while the  $(\\xi_{i,j,m},\\sigma^2_{i,j,m})$'s determine the clustering of individuals, namely patients, \nafter removing the effect of the specific hypertensive disorder. In order to pursue this, for each cardiac index $m$, we will specify a\nhierarchical process prior for $(\\xi_{i,j,m},\\sigma^2_{i,j,m})$ that is suited to infer the clustering structure both within and across different \nhypertensive disorders for a specific cardiac index. In particular, we will deploy a novel instance of hierarchical Dirichlet process, introduced in \\cite{teh2006}, that we name \\textit{symmetric}, to highlight its centering in $0$.\n\n\nEarly examples of Bayesian nonparametric models for ANOVA can be found in \\cite{cifreg78} and \\cite{mulierepetrone}, while the first popular proposal, due to \\citet{de2004}, uses the dependent Dirichlet process (DDP) \\citep{maceachern2000dependent} and is therefore termed ANOVA-DDP. This model is mainly tailored to estimate populations' probability distributions, while we draw inferences over clusters of populations' means and obtain estimates of the unknown distributions as a by-product. Moreover, the ANOVA-DDP of \\citet{de2004} was not introduced as a model selection procedure. A popular Bayesian nonparametric model, that does cluster \npopulations and can be used for model selection, is the nested Dirichlet process of \\citet*{rodriguez2008}. As shown in \\citet{camerlenghi2019}, such a prior is biased towards homogeneity, in the sense that even a single \ntie between populations $j$ and $j'$, namely $X_{i,j,m}=X_{i',j',m}$ for some $i$ and $i'$, entails $\\pi_{j,m}=\\pi_{j'm}$ (almost surely). In order to overcome such a drawback, a novel class of nested, and more flexible, priors has been proposed in   \n\\citet{camerlenghi2019}. See also \\cite{soriano_ma} for related work. Interesting alternatives that extend the analysis to more than two populations can be found in \\cite{ChristMa}, \\cite*{Rebaudo} and in \\cite*{Beraha}. Another similar proposal is the one by \\citet{gutierrez2019bayesian}, whose model identifies differences over cases' distributions and the control group. These models imply that two populations belong to the same cluster if they share the entire distribution. However, as already mentioned, distribution-based clustering is not ideal when dealing with scenarios as the one of hypertensive dataset. Further evidence will be provided in Section~\\ref{ss:sim}, through simulation \nstudies. In addition, note that all these contributions deal with only one response variable and would need to be suitably generalized to fit \nthe setup of this paper. As far as the contributions treating multiple response variables are concerned, uses of nonparametric priors for multiple \ntesting can be found, for instance, in \\citet{gopalan1998}, \\citet*{do2005bayesian}, \\citet{dahl2007multiple}, \\citet*{guindani2009}, \\citet{martin2012} and more recently in \\citet*{cipolli2016bayesian}, who propose an approximate finite P\\'olya tree multiple testing procedure to compare two-samples' locations, and in \\citet{dentitwo}. However, in all these contributions, models are developed directly over summaries of the original data (e.g. averages, z-scores) and, as such, do not allow to draw any inference on the entire distributions and clusters of subjects. \n\nThe outline of the paper is as follows. In Section~\\ref{s:model} we introduce the model, which makes use of an original hierarchical prior structure for symmetric distributions (Section~\\ref{ss:s-HDP}). In Section~\\ref{s:partition} we derive the prior law of the random partitions induced by \nthe model, key ingredient for the Gibbs sampling scheme devised in Section~\\ref{s:inference}. In Section~\\ref{s:results}, we first present a series of simulation studies that highlight the behaviour of the model before applying it to obtain our results on cardiac dysfunction in hypertensive disorders. Section~\\ref{s:conclusion} contains some concluding remarks. As Supplementary Material we provide the datasets and Python codes, some further background material and details about the derivation of \nthe posterior sampling scheme as well as additional simulation studies and results on the application, including an analysis of prior sensitivity.\n\n\\section{The Bayesian nonparametric model}\n\\label{s:model}\n\nThe use of discrete nonparametric priors for Bayesian model-based clustering has become standard practice. The Dirichlet process (DP) \\citep{ferguson1973} is the most popular instance, and clustering is typically addressed by resorting to a mixture model, which with our data structure amounts to\n\\begin{equation*}\n\t\\label{eq:mixture_kernel}\n\tX_{i,j,m}|\\psi_{i,j,m}\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim} k(X_{i,j,m};\\psi_{i,j,m}),\\qquad\n\t\\psi_{i,j,m}|\\tilde p_{j,m}\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim} \\tilde p_{j,m} \n\\end{equation*}\nfor $m=1,\\ldots,M$, $j=1,\\ldots,J$ and $i=1,\\ldots,n_j$. Here $k(\\,\\cdot\\,;\\,\\cdot\\,)$ is some kernel and the $\\tilde p_{j,m}$'s are discrete random probability measures. Hence, the $\\psi_{i,j,m}$'s may exhibit ties. The model specification for $\\tilde p_{j,m}$ will be tailored to address the following goals: (i) cluster the $J$ probability distributions based on their means; (ii) cluster the observations $X_{i,j,m}$ according to the ties induced on the $\\psi_{i,j,m}$'s by the $\\tilde p_{j,m}$'s for \na given fixed $j$ and across different $j$'s. These two issues will be targeted separately: we first design a clustering scheme for the populations, through the specification of a prior on the means of the $X_{i,j,m}$'s and, then, we cluster the data using a hierarchical DP having a specific invariance structure that is ideally suited to the application at hand. \n\n\\subsection{The prior on disease-specific locations}\n\\label{ss:like}\nAs a model for the observations we consider a nonparametric mixture of Gaussian distributions specified as\n\\begin{equation} \n\t\\label{eq:model}\n\tX_{i,j,m}\\,|\\,(\\bm{\\theta}_m,\\bm{\\xi}_m,\\bm{\\sigma}^2_m)\n\t\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim} \\mathcal{N}(\\theta_{j,m}+\\xi_{i,j,m},\\sigma_{i,j,m}^2)\n\\end{equation}\nwhere $\\bm{\\theta}_m=(\\theta_{1,m},\\ldots,\\theta_{J,m})$, $\\bm{\\xi}_m=(\\xi_{1,1,m},\\ldots,\\xi_{1,n_1,m},\\xi_{2,1,m},\\ldots,\n\\xi_{n_J,J,m})$, with a similar definition for the vector $\\bm{\\sigma}_m^2$, and $\\mathcal{N}(\\,\\mu,\\,\\sigma^2\\,)$ denotes a normal distribution with mean $\\mu$ and variance $\\sigma^2$. The assumption in \\eqref{eq:model} clearly reflects \\eqref{eq:decomp_x}. Moreover, in order to account for the two levels of clustering we are interested in, we will assume that \n\\begin{equation}\n\t\\label{eq:model_parameters}(\\bm{\\theta}_{1},\\ldots,\\bm{\\theta}_{M})\\sim P,\\qquad\n\t(\\xi_{i,j,m},\\sigma_{i,j,m}^2)\\,|\\,\\tilde{q}_{j,m}\\stackrel{\\mbox{\\scriptsize{\\rm iid}}}{\\sim} \\tilde q_{j,m}\\quad (i=1,\\ldots,n_j)\n\\end{equation}\nwhere $\\tilde q_{1,m},\\ldots,\\tilde q_{J,m}$ are discrete random probability measures independent from  $(\\bm{\\theta}_{1},\\ldots,\\bm{\\theta}_{M})$. Thus, the likelihood corresponds to\n\\begin{equation}\\label{eq1}\n\t\\prod_{m=1}^M\\,\\prod_{j=1}^J \\,\n\t\\prod_{i=1}^{n_j}\n\t\\frac{1}{\\sigma_{i,j,m}}\\:\n\t\\varphi\\Big(\\frac{x_{i,j,m}-\\theta_{j,m}-\\xi_{i,j,m}}{\\sigma_{i,j,m}}\\Big)\n\t\\:\\tilde{q}_{j,m}(\\mathrm{d}\\xi_{i,j,m},\n\t\\mathrm{d}\\sigma_{i,j,m})\n\\end{equation}\nwith $\\varphi$ denoting the standard Gaussian density. Relevant inferences can be carried out if one is able to marginalize this expression with respect to both $(\\bm{\\theta}_{1},\\ldots,\\bm{\\theta}_{M})$ and $(\\tilde q_{1,m},\\ldots,\\tilde{q}_{J,m})$ for each \n$m=1,\\ldots,M$. \n\nThis specification allows to address the model selection problem in the following way. If $\\mathcal{M}^m$ stands for the space of all partitions of the $J$ populations for the $m$-th cardiac function index, then $\\mathcal{M}^m=\\{M_b^m \\,:\\, b =1,\\ldots,\\mbox{card}(\\mathcal{P}_J)\\}$ where $\\mathcal{P}_J$ is the collection of all possible partitions of $[J]=\\{1,\\ldots,J\\}$. In our specific case, $J=4$ and card$(\\mathcal{P}_J)=15$, thus we have 15 competing models per cardiac index. Each competing model corresponds to a specific partition in $\\mathcal{M}^m$. In particular, the partition arises from ties between the population specific means in $\\bm{\\theta}_m$ and, hence, the distribution $P$ in \\eqref{eq:model_parameters} needs to associate positive probabilities to ties between the parameters within the vector $\\bm{\\theta}_m$, for each $m=1,\\ldots,M$. \n\n\nLet us start considering as distribution $P$ a well-known effective clustering prior, i.e. a mixture of DPs in the spirit of \\cite{antoniak1974mixtures}, namely\n\\begin{equation}\n\t\\label{eq2}\n\t\\begin{aligned}\n\t\t\\theta_{j,m}\\mid \\tilde p_m \\enskip&\\stackrel{\\mbox{\\scriptsize{\\rm iid}}}{\\sim}\\enskip \\tilde p_m \\qquad &j=1,\\ldots,J \\\\\n\t\t\\tilde p_m\\mid\\omega\\enskip &\\stackrel{\\mbox{\\scriptsize{\\rm iid}}}{\\sim} \\enskip \\mbox{DP}(\\omega,G_{m})\\qquad& m=1,\\ldots,M \\\\\n\t\t\\omega \\enskip &\\sim \\enskip p_\\omega&\n\t\\end{aligned}\n\\end{equation}\nwhere DP$(\\omega,G_m)$ denotes the DP with concentration parameter $\\omega$ and non-atomic baseline probability measure $G_m$ and $p_{\\omega}$ is a probability measure on $\\mathds{R} ^+$. The discreteness of the DP implies the presence (with positive probability) of ties within the vector of locations $\\bm{\\theta}_m$ associated to a certain cardiac index $m$, as desired. The ties give rise to a random partition: as shown in \\cite{antoniak1974mixtures}, the probability of observing a specific partition of the elements in $\\bm{\\theta}_m$ consisting of $k\\le J$ distinct values with respective frequencies $n_1,\\ldots,n_k$ coincides with\n\\begin{equation}\n\t\\label{eq:eppf_dir}\n\t\\Pi_k^{(J)}(n_1,\\ldots,n_k)=\\frac{\\omega^k}{(\\omega)_J}\\,\\prod_{i=1}^k (n_i-1)!\n\\end{equation}\nwhere $(\\omega)_J=\\Gamma(\\omega+J)\/\\Gamma(\\omega)$. The use of a shared concentration parameter over \\eqref{eq:eppf_dir} to address multiple model selection has been already successfully employed in \\cite{moser2021multiple}, where they cluster parameters in a probit model.  When there is no pre-experimental information available on competing partitions, the use of \\eqref{eq:eppf_dir} as a prior for model selection has some relevant benefits. Indeed, it induces borrowing of strength across diagnoses and, being $\\omega$ random, it generates borrowing of information also across cardiac indexes, thus improving the Bayesian learning mechanism. These two features can also be given a frequentist interpretation in terms of desirable penalties.\nAs a matter of fact, the procedure penalizes for the multiplicity of the model selections that are performed. The penalty has to be meant \nin the following way: while $J$ and\/or $M$ increase, the prior odds change in favor of less complex models. For more details on this, see \\cite{scott2010bayes}. Summing up, the mixture of DPs automatically induces a prior distribution on $\\{\\mathcal{M}^m:m=1,\\ldots,M\\}$, that arises from \\eqref{eq:eppf_dir} combined with the prior $p_\\omega$ on $\\omega$, and it presents desirable properties for model selection \nthat can be interpreted either in terms of borrowing of information or in terms of penalties. \n\nHowever, in the analysis of hypertensive disorders, some prior information on competing models is available, and this is not yet incorporated in \\eqref{eq:eppf_dir}. In fact, as already mentioned, there is a natural order of the diagnoses, which is given by the severity of the disorders, i.e.\\ C, G, M, S. Partitions that do not comply with this ordering, e.g. $\\{\\{C, S\\}\\{G\\},\\{M\\}\\}$, should be excluded from the support of the prior. Thus, we consider a prior over $\\mathcal{M}^m$ that associates zero probability to partitions that do not respect the natural order of the diagnoses and a probability proportional to that in \\eqref{eq:eppf_dir} for the remaining partitions, i.e.\\\n\\begin{equation}\\label{eq:prior2}\n\t\\mathbb{P}(M_b^m\\mid \\omega) \\propto \\begin{cases}\n\t\\Pi_k^{(J)}(n_1,\\ldots,n_k)&\\text{if $M_b^m$ is compatible with the natural order}\\\\\n0 & \\text{otherwise}\\end{cases}\n\\end{equation} \nThis amounts to a distribution $P$ for $(\\bm{\\theta}_{1},\\ldots,\\bm{\\theta}_{M})$ given by \n\\begin{equation}\n\t\\label{eqnew}\n\t\\begin{aligned}\n\t\t(\\theta_{1,m},\\ldots,\\theta_{J,m})\\mid \\omega \\enskip&\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim}\\enskip P_{\\omega, G_m} \\qquad m=1,\\ldots,M \\\\\n\t\t\\omega \\enskip &\\sim \\enskip p_\\omega&\n\t\\end{aligned}\n\\end{equation}\nwhere $P_{\\omega, G_m}$ is the distribution obtained sampling a partition according to \\eqref{eq:prior2} and associating to each cluster a unique value sampled from $G_m$. \nUsing \\eqref{eqnew} as a prior for the disease-specific locations, we preserve the desirable properties of the mixture of DPs mentioned before, while incorporating prior information on the severity of the diseases.\n\nAs detailed in the next section, we further define random probability measures $\\tilde q_{j,m}$ that satisfy the symmetry condition\n\\begin{equation}\n\t\\label{eq:symmetry}\n\t\\tilde q_{j,m}(A\\times B ) \n\t= \\tilde \n\tq_{j,m}((-A)\\times B )\\qquad\\qquad  \\text{a.s.}\n\\end{equation}\nfor any $A$ and $B$. This condition ensures that the parameters $\\theta_{j,m}$, for $j=1,\\ldots,J$ and $m=1,\\ldots,M$, in \\eqref{eq:model} are identified, namely $\\mathbb{E}(X_{i,j,m}\\mid \\boldsymbol{\\theta}_{m}, \\, \\tilde q_{j,m} ) = \\theta_{j,m}$ with probability one. This identifiability property is crucial to make inference over the location parameters $\\bm{\\theta}_m$'s. Similar model specifications for discrete \nexchangeable data have been proposed and studied in \\cite{dalal1979nonparametric},  \\cite{doss1984}, \\cite{diaconis1986} and \\cite*{ghosal1999}, of which \\eqref{eq1} represents a generalization to \ndensity functions and partially exchangeable data. \n\n\\subsection{The prior for the error terms}\n\\label{ss:s-HDP}\nWhile the clustering of populations is governed by \\eqref{eq:prior2}, we use a mixture of hierarchical discrete processes for the error terms. This has the advantage of modeling the clustering of the observations, both within and across different samples, once the disease-specific effects are account for. This clustering structure allows to model heterogeneity across patients in a much more realistic way with respect to standard ANOVA models based on assumption of normality. Cardiac indexes may be influenced by a number of factors that are not directly observed in the study, such as pre-existing conditions \\citep{hall2011heart} and psychosocial factors \\citep{pedersen2017psychosocial}. These unobserved relevant factors may be shared across patients with the same or a different diagnosis and may also result in outliers. To take into account this latent heterogeneity of the data, we introduce the hierarchical\nsymmetric DP that satisfies the symmetry condition in \\eqref{eq:symmetry} and, moreover, allows to model heterogeneous data similarly to the \nhugely popular hierarchical DP \\citep{teh2006}.\n\nThe basic building block of the proposed prior is the invariant Dirichlet \nprocess, which was introduced for a single population ($J=1$) in an exchangeable framework by \\cite{dalal1979}. Such a modification of the DP satisfies a symmetry condition, in the sense that it is a random probability measure that is invariant with respect to a chosen group of transformations $\\mathcal{G}$. A more formal definition and detailed description of the invariant DP can be found in Section A of the Supplement. For our purposes it is enough to consider the specific case of the symmetric Dirichlet process, \nwhich can be constructed through a symmetrization of a Dirichlet process. Consider a non-atomic probability measure $P_0$ on $\\mathbb{R}$\nand let $\\tilde Q_0 \\sim \\text{DP}(\\alpha,P_0)$. \nIf \n\\begin{equation}\n\t\\label{eq:inv_DP}\n\t\\tilde Q(A) = \\frac{\\tilde Q_0(A) + \\tilde Q_0(-A)}{2}\\quad \\quad \\forall A \\in \\mathcal{B}(\\mathbb{R})\n\\end{equation}\nwhere $-A=\\{x\\in \\mathbb{R}: -x \\in A\\}$, then $\\tilde Q$ is symmetric about 0 (almost surely) and termed symmetric DP, in symbols $\\tilde Q\\sim\\text{s-DP}(\\alpha\\,,\\,P_0)$. For convenience and without loss of generality, we assume that $P_0$ is symmetric: this implies that $P_0$ is the expected value of $\\tilde Q$ making it an interpretable parameter.\nThe random probability measure $\\tilde Q$ is the basic building block of the hierarchical process that we use to model the heterogeneity of the error terms across different populations, $j=1,\\ldots,J$, in such a way that clusters identified by the unique values can be shared within and across populations. \nThis prior is termed \\textit{symmetric hierarchical Dirichlet process} (s-HDP) and described as\n\\begin{equation}\n\t\\label{eq:s-hdp_def}\n\t\\begin{split}\n\t\t\\tilde q_{j,m}\\mid\\gamma_{j,m}, \\, \n\t\t\\tilde \n\t\tq_{0,m}\\enskip\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim}\\enskip\\text{s-DP}(\\gamma_{j,m},\\tilde q_{0,m})\\\\\n\t\t\\tilde q_{0,m}\\mid\\alpha_{m}\\enskip\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim}\\enskip\\text{s-DP}(\\alpha_{m},P_{0,m})\n\t\\end{split}\n\\end{equation}\nwhere $\\gamma_{j,m}$ and $\\alpha_{m}$ are positive parameters and $P_{0,m}$ is a non-atomic probability distribution symmetric about 0. We use the \nnotation $(\\tilde q_{1,m},\\ldots,\\tilde q_{J,m})\\sim \\mbox{s-HDP}(\\bm{\\gamma}_m,\\alpha_m,P_{0,m})$, where $\\bm{\\gamma}_m=(\\gamma_{1,m},\\ldots,\\gamma_{j,m})$. This definition clearly ensures the validity of \\eqref{eq:symmetry}. \nA graphical model representation of the over-all proposed model is displayed in Figure~\\ref{fig:figure1}. \n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\begin{tikzpicture}[-latex ,auto ,node distance =2 cm and 2cm ,on grid ,\n\t\t\tsemithick ,\n\t\t\tstate\/.style ={ circle ,top color =white , bottom color = white ,\n\t\t\t\tdraw,black , text=black , minimum width =1 cm},\n\t\t\ttransition\/.style = {rectangle, draw=black!50, thick, minimum width=6.1cm, minimum height = 5.8cm},\n\t\t\ttransition3\/.style = {rectangle, draw=black!50, thick, minimum width=4cm, minimum height = 3.8cm}, \n\t\t\ttransition2\/.style = {rectangle, draw=black!50, thick, minimum width=6.5cm, minimum height = 8cm}, scale=0.8]\n\t\t\t\\node[state] (C){$\\boldsymbol{\\theta_{m}}$};\n\t\t\t\\node[draw=none] (A) [below =of C] { };\n\t\t\t\\node[draw=none] (B) [below =of A] {};\n\t\t\t\\node[circle ,top color =white , bottom color = white ,\n\t\t\tdraw,black , text=black , minimum width =1 cm]  (D) [left =2 cm of C] {${\\omega}$};\n\t\t\t\\node[circle ,top color =white , bottom color = ashgrey ,\n\t\t\tdraw,black , text=black , minimum width =1 cm] (E) [below = of B] {${X_{i,j,m}}$};\n\t\t\t\\node[state] (F) [ right =of B] {$\\varepsilon_{i,j,m}$};\n\t\t\t\\node[state] (G) [above =of F] {${\\tilde q_{j,m}}$};\n\t\t\t\\node[state] (Z) [above =of G] {${\\tilde q_{0,m}}$};\n\t\t\t\\node[state] (X) [right =of Z] {${\\alpha_{m}}$};\n\t\t\t\\node[state] (Y) [right =of G] {${\\gamma_{j,m}}$};\n\t\t\t\\node[transition] (H)[ below =0.02cm of F] { };\n\t\t\t\\node[transition3] (J)[  above  right =1.5cm of E] { };\n\t\t\t\\node[transition2] (L)[ below =1cm of G] { };\n\t\t\n\t\t\t\\path (D) edge node[right] {} (C);\n\t\t\t\\path (C) edge node[below] {} (E);\n\t\t\t\\path (G) edge node[below] {} (F);\n\t\t\t\\path (F) edge node[below] {} (E);\n\t\t\t\\path (X) edge node[below] {} (Z);\n\t\t\t\\path (Y) edge node[below] {} (G);\n\t\t\t\\path (Z) edge node[below] {} (G);\n\t\t\\end{tikzpicture}\n\t\t\\caption{\\small Graphical representation of the model. Each node represents a random variable and each rectangle denotes conditional i.i.d. replications of the model within the rectangle.}\n\t\t\\label{fig:figure1}\n\t\\end{center}\n\t\\vspace{-0.5 cm}\n\\end{figure}\n\nStill referring to the decomposition of the observations into disease-specific locations and an error term, i.e. $X_{i,j,m}=\\theta_{j,m}+\\varepsilon_{i,j,m}$, it turns out that the $\\varepsilon_{i,j,m}$'s are from a symmetric hierarchical DP mixture (s-HDP mixture) with a normal kernel. Hence, the patients' clusters are identified through the $\\varepsilon_{i,j,m}$, which, according to \\eqref{eq:model}, are conditionally independent from a $\\mathcal{N}(\\xi_{i,j,m},\\sigma_{i,j,m}^2)$ given $(\\xi_{i,j,m},\\sigma_{i,j,m}^2)$. The choice of the specific invariant DP is aimed at ensuring that $\\mathbb{E}(\\varepsilon_{i,j,m}|\\tilde \nq_{j,m})=0$. \nThe clusters identified by the s-HDP mixture can be interpreted as representing common unobserved factors across patients, once the disease-specific locations have been accounted for. Indeed, for any pair of patients, we may consider the decomposition $X_{i,j,m}-X_{i',j',m}=\\,\\Delta^{(m)}_{\\theta} + \\Delta^{(m)}_{\\xi} + (e_{i,j,m}-e_{i',j',m})$ where $\\Delta^{(m)}_{\\theta}=\\theta_{j,m} - \\theta_{j',m}$, $\\Delta^{(m)}_{\\xi}=\\xi_{i,j,m} - \\xi_{i',j',m}$ and $e_{i,j,m}$ and $e_{i',j',m}$ are independent and normally distributed random variables with zero mean and variances $\\sigma_{i,j,m}^2$ and $\\sigma_{i',j',m}^2$, respectively. \n\nHence, patients' clustering reflects the residual heterogeneity that is not captured by the disease-specific component $\\Delta^{(m)}_{\\theta}$ and are related to the subject-specific locations $\\Delta^{(m)}_{\\xi}$ and to the zero-mean error component $(e_{i,j,m}-e_{i',j',m})$. \nIn view of this interpretation, using a s-HDP mixture over error terms offers a three-fold advantage. Firstly, the presence of clearly separated clusters of patients within and across populations will indicate the presence of unobserved relevant factors which affect the cardiac response variables. Secondly, single patients with very low probabilities of co-clustering \nwith all other subjects will have to be interpreted as outliers. \nFinally, the estimated clustering structure can also be used to check whether the relative effect of a certain disease (with respect to another) is fully explained by the corresponding $\\Delta^{(m)}_{\\theta}$. To clarify this last point consider two diseases: if the posterior co-clustering probabilities among patients sharing the same disease are different between the two populations, this will indicate that different diagnoses not only have an influence on disease-specific locations (which is measured by $\\Delta^{(m)}_{\\theta}$), but they also have \nan impact on the shape of the distribution of the corresponding cardiac index. More details on this can be found in Section D of the Supplement.\n\n\n\\section{Marginal distributions and random partitions}\n\\label{s:partition}\nAs emphasized in the previous sections, ties among the $\\theta_{j,m}$'s and the $(\\xi_{i,j,m},\\sigma_{i,j,m}^2)$'s are relevant for inferring the clustering structure both among the populations (hypertensive diseases) and among the individual units (patients). Indeed, for each $m$ (cardiac index) they induce a random partition that emerges as a composition of two partitions generated respectively by the prior in \\eqref{eqnew} and the s-HDP. The laws of these random partitions are not only crucial to understand the clustering mechanism, but also necessary in order to derive posterior sampling schemes. In this section such a law is derived and used to compute the predictive distributions that, jointly with the likelihood, determine the full conditionals of the Gibbs sampler devised in Section~\\ref{s:inference}. \nTo reduce the notational burden, in this and the following section, we remove the dependence of observations and parameters on the specific response variable $m$ and denote with $\\phi_{i,j}$ the pair $(\\xi_{i,j},\\sigma_{i,j}^2)$ and with $\\boldsymbol{\\phi}$ the collection $(\\phi_{1,1},\\ldots,\\phi_{1,n_1},\\phi_{2,1},\\ldots \\phi_{n_J,J})$.  \n \nConditionally on $\\omega$, the law of the partition in \\eqref{eq:prior2} leads to the following predictive distribution for the disease-specific locations\n\\[\n\\theta_j\\,|\\,\\omega,\\theta_1,\\ldots,\\theta_{j-1}\\:\\sim\\:\na_j(\\omega,\\theta_1,\\ldots,\\theta_{j-1})\\,\\delta_{\\theta_{j-1}}+\n\\left[1-a_j(\\omega,\\theta_1,\\ldots,\\theta_{j-1})\\right] G\n\\]\nwhere \n\\begin{equation}\n\\label{eq:a}\na_j(\\omega,\\theta_1,\\ldots,\\theta_{j-1}) = \\frac{\\sum_{(*_j)}\\Pi^{(J)}_{k}(n_1,\\ldots,n_k)}{\\sum_{(\\Delta_j)}\\Pi^{(J)}_{k}(n_1,\\ldots,n_k)}\n\\end{equation}\nwhere the sum at the denominator runs over the set of partitions consistent with the one generated by\n \t$(\\theta_1,\\ldots,\\theta_{j-1})$ and the one at the numerator runs over a subset of those \n \tpartitions where one further has $\\theta_j=\\theta_{j-1}$. For $j=4$, the predictive equals\n\\[\n\\theta_{4}\\mid \\omega,\\theta_{1}, \\theta_{2},\\theta_{3} \\begin{cases} \n\t\\frac{3}{\\omega+3}\\delta_{\\theta_{3}} + \n\t\\frac{\\omega}{\\omega+3}G &\\mbox{if }\\theta_{1}=\\theta_{2}=\\theta_{3}\\\\\n\t\\frac{2}{\\omega+2}\\delta_{\\theta_{3}} + \n\t\\frac{\\omega}{\\omega+2}G &\\mbox{if }\\theta_{1}\\neq\\theta_{2}=\\theta_{3}\\\\\n\t\\frac{1}{\\omega+1}\\delta_{\\theta_{3}} + \n\t\\frac{\\omega}{\\omega+1}G&\\mbox{otherwise}\n\t\\end{cases}\n\\]\nExplicit expressions for the function $a$, for $j=1,2,3$, can be easily computed using \\eqref{eq:a} and \\eqref{eq:prior2} and are provided in Section B of the Supplement.\n\nMoving to second-level partitions induced by the s-HDP, we recall that the key concept for studying random partitions on multi-sample data is the \\textit{partially exchangeable partition probability function} (pEPPF). See, e.g., \\cite*{lnp2014} and \\citet{camerlenghi2019b}. The pEPPF returns the probability of \na specific multi-sample partition and represents the appropriate generalization of the well-known single-sample EPPF, which in the DP case corresponds to \\eqref{eq:eppf_dir}.\nDiscreteness of the s-HDP $(\\tilde q_1,\\ldots,\\tilde q_m)$ in \\eqref{eq:s-hdp_def} induces a partition of the elements of \n$\\boldsymbol{\\phi}$ into equivalence classes identified by the distinct values. Taking into account the underlying partially exchangeable structure, such a random partition is characterized by the pEPPF\n\\begin{equation}\n\t\\label{eq:peppf}\n\t\\tilde \\Pi_k^{(N)}(\\bm{n}_1,\\ldots,\\bm{n}_J)=\\mathbb{E}\\left(\\int_{\\Phi^k}\n\t\\prod_{j=1}^J \\prod_{h=1}^k\\tilde q_{j,m}^{n_{j,h}}(\\mathrm{d}\\phi_i)\\right)\n\\end{equation}\nwhere $\\bm{n}_{j}=(n_{j,1},\\ldots,n_{j,k})$ are non-negative integers, for any $j=1,\\ldots,J$, such that $n_{j,h}$ is the number of elements in $\\bm{\\phi}$ corresponding to population $j$ and belonging to cluster $h$. Thus $\\sum_{j=1}^J n_{j,h}\\ge 1$ for any $h=1,\\ldots,k$, $\\sum_{h=1}^kn_{j,h}=n_j$ and $\\sum_{h=1}^k \\sum_{j=1}^J n_{j,h}=N$.\nThe determination of probability distributions of this type is challenging and only recently the first explicit instances have appeared in the literature. See e.g., \\citet{lnp2014}, \\citet{camerlenghi2019} and \\citet{camerlenghi2019b}. With respect to the hierarchical case considered in \\citet{camerlenghi2019b}, the main difference is that here we have to take into account the specific structure \\eqref{eq:inv_DP} of the $\\tilde q_{j,m}$. The almost sure symmetry of the process generates a natural random matching between sets in the induced partition. Therefore, instead of studying the marginal law in \\eqref{eq:peppf}, we derive the joint law of the partition and of the random matching. Formally, consider a specific partition $\\{A_1^{+},A_1^{-},\\ldots,A_k^{+},A_k^{-}\\}$ of $\\boldsymbol{\\phi}$, such that, for $h=1,\\ldots,k$, all the elements in $A_h^+$ belong to $\\mathbb{R}^+\\times\\mathbb{R}^+$, all the elements in $A_h^-$ belong to $\\mathbb{R}^-\\times\\mathbb{R}^+$ and, if $\\phi_{i,j}\\in A_h^+$ and $\\phi_{i',j'}\\in A_h^-$, then the element-wise absolute values of $\\phi_{i,j}$ and $\\phi_{i',j'}$ are equal. Denote with $n^+_{j,h}$ the number of elements in $A_h^{+}\\cap\\{\\phi_{i,j},i=1,\\ldots,n_j\\}$ and with $n^-_{j,h}$ the number of elements in $A_h^{-}\\cap\\{\\phi_{i,j},i=1,\\ldots,n_j\\}$. The probability of observing  $\\{A_1^{+},A_1^{-},\\ldots,A_k^{+},A_k^{-}\\}$ is\n\\begin{equation}\n\t\\label{eq:peppfsym}\n\t\\ddtilde{\\Pi}_{k}^{(N)}(\\boldsymbol{n_1}^+,\\boldsymbol{n_1}^-,\\ldots,\\boldsymbol{n_J}^+,\\boldsymbol{n_J}^-) = \\mathbb{E}\\left(\\int_{\\Phi^k}\\prod_{j=1}^J\\prod_{h=1}^k{\\tilde q_{j,m}}^ {n^{+}_{j,h}+n^{-}_{j,h}}(d\\phi)\\right)\n\\end{equation} \nwith $\\boldsymbol{n_j^+}=(n_{j,1}^+,\\ldots,n_{j,k}^+)$. As for the determination of \\eqref{eq:peppfsym}, a more intuitive understanding may be gained if one considers its corresponding Chinese restaurant franchise (CRF) metaphor, which displays a variation of both the standard Chinese restaurant franchise of \\cite{teh2006} and the skewed Chinese restaurant process of \\cite*{iglesias2009nonparametric}. Figure~\\ref{fig:figure2} provides a graphical representation. \n\\begin{figure}\n\t\\centering\n\t\\resizebox{14cm}{!}{\n\t\t\\begin{tikzpicture}[-latex, auto, node distance = 2.4 cm and 3.4cm, \n\t\ton grid,\n\t\tsemithick,\n\t\tstate\/.style ={circle, top color = white, bottom color = white,\n\t\t\tdraw, black, text=black, minimum width = 0.5 cm},\n\t\tfill fraction\/.style n args={2}{path picture={\n\t\t\t\t\\fill[#1] (path picture bounding box.south west) rectangle\n\t\t\t\t($(path picture bounding box.north west)!#2!(path picture bounding box.north east)$);}},\n\t\ttransition\/.style = {rectangle, draw=black!50, thick, minimum width=17cm, minimum height = 6.7cm} ]\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (A)\n\t\t{\\Large$\\boldsymbol{\\phi^{**}_1}$\\quad $-\\boldsymbol{\\phi^{**}_1}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (B) [right =of A]\n\t\t{\\Large$\\boldsymbol{\\phi^{**}_2}$\\quad $-\\boldsymbol{\\phi^{**}_2}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (C) [right =of B] \t{\\Large$\\boldsymbol{\\phi^{**}_3}$\\quad $-\\boldsymbol{\\phi^{**}_3}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (D) [right =of C] \t{\\Large$\\boldsymbol{\\phi^{**}_1}$\\quad $-\\boldsymbol{\\phi^{**}_1}$};\n\t\t\\node[circle, fill, draw, scale=0.3] (M) [right =1.5 cm of D] { };\n\t\t\\node[circle, fill, draw, scale=0.3] (N) [right =0.5 cm of M] { };\n\t\t\\node[circle, fill, draw, scale=0.3] (O) [right =0.5 cm of N] { };\n\t\t\\node[draw=none,fill=none] (E) [above left =1.5 cm of A] {\\large$\\phi_{1,1}$};\n\t\t\\node[draw=none,fill=none] (F) [above left =1.55 cm of B] {\\large$\\phi_{2,1}$};\n\t\t\\node[draw=none,fill=none] (G) [below left =0.60 cm of E] {\\large$\\phi_{3,1}$};\n\t\t\\node[draw=none,fill=none] (H) [left =1.56 cm of A] {\\large$\\phi_{4,1}$};\n\t\t\\node[draw=none,fill=none] (I) [above left =1.55 cm of C] {\\large$\\phi_{5,1}$};\n\t\t\\node[draw=none,fill=none] (L) [above right=1.55 cm of B] {\\large$\\phi_{6,1}$};\n\t\t\\node[draw=none,fill=none] (P) [below left =1.35 cm of A] {  };\n\t\t\\node[draw=none,fill=none] (Q) [above right=1.50 cm of A] {\\large$\\phi_{7,1}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (AA) \n\t\t[below =3.2cm of A]\n\t\t{\\Large$\\boldsymbol{\\phi^{**}_3}$\\quad $-\\boldsymbol{\\phi^{**}_3}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (BB) \n\t\t[below =3.2cm of B]\n\t\t{\\Large$\\boldsymbol{\\phi^{**}_1}$\\quad $-\\boldsymbol{\\phi^{**}_1}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (CC) \n\t\t[below =3.2cm of C] \t{\\Large$\\boldsymbol{\\phi^{**}_4}$\\quad $-\\boldsymbol{\\phi^{**}_4}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (DD) \n\t\t[below =3.2cm of D]\n\t\t{\\Large$\\boldsymbol{\\phi^{**}_5}$\\quad $-\\boldsymbol{\\phi^{**}_5}$};\n\t\t\\node[circle, fill, draw, scale=0.3] (MM) [right =1.5 cm of DD] { };\n\t\t\\node[circle, fill, draw, scale=0.3] (NN) [right =0.5 cm of MM] { };\n\t\t\\node[circle, fill, draw, scale=0.3] (OO) [right =0.5 cm of NN] { };\n\t\t\\node[draw=none,fill=none] (EE) [above left =1.5 cm of AA] {\\large$\\phi_{4,2}$};\n\t\t\\node[draw=none,fill=none] (FF) [above left =1.55 cm of BB] {\\large$\\phi_{2,2}$};\n\t\t\\node[draw=none,fill=none] (GG) [below left =0.60 cm of EE] {\\large$\\phi_{3,2}$};\n\t\n\t\t\\node[draw=none,fill=none] (II) [above left =1.55 cm of CC] {\\large$\\phi_{5,2}$};\n\t\t\\node[draw=none,fill=none] (LL) [above right=1.55 cm of CC] {\\large$\\phi_{6,2}$};\n\t\t\\node[draw=none,fill=none] (PP) [below left =1.35 cm of AA] {  };\n\t\t\\node[draw=none,fill=none] (QQ) [above right=1.5 cm of AA] {\\large$\\phi_{7,2}$};\n\t\t\\node[transition] (R)  [above  left=2.2 cm of CC]{ };\n\t\\end{tikzpicture}\n}\n\\caption{{Chinese restaurant franchise representation of the symmetric hierarchical DP for $J=2$ populations. Each circle represents a table.}}\n\\label{fig:figure2}\n\\end{figure}\nThe scheme is as follows: there are $J$ restaurants\nsharing the same menu and the customers are identified by their choice of \n$\\phi_{i,j}$ but, unlike in the usual CRF, at each table two \\textit{symmetric dishes} are served. Denote with $\\phi^*_{t,j}=(\\xi^*_{t,j},\\, \\sigma_{t,j}^{2*})$ and $-\\phi^*_{t,j}=(-\\xi^*_{t,j},\\, \\sigma_{t,j}^{2*})$ the two dishes served at table $t$ in restaurant $j$, with \n$\\phi^{**}_{h}=(\\xi^{**}_{h},\\, \\sigma^{**2}_{h})$ and $-\\phi^{**}_{h}=(-\\xi_{h},\\, \\sigma_{h}^{**2})$ the $h$-th pair of dishes in the menu and with $n_{j,h}^{+}$ and $n_{j,h}^{-}$ the number of customers in restaurant $j$ eating dish $\\phi^{**}_{h}$ and $-\\phi^{**}_{h}$, respectively. This means that two options are available to a customer entering restaurant \n$j$: she\/he will either sit at an already occupied table, with probability proportional to the number of customers at that table or will sit at a new table with probability proportional to the concentration parameter $\\gamma_j$. In the former case, the customer will choose the dish $\\phi^*_{t,j}$ with probability $1\/2$ and $-\\phi^*_{t,j}$ otherwise. In the latter \ncase, the customer will eat a dish served at another table of the franchise with probability proportional to half the number of tables that serve that dish, or will make a new order with probability proportional to the concentration parameter $\\alpha$. In view of this scheme, the probability in \\eqref{eq:peppfsym} turns out to be\n\\[\n\\ddtilde{\\Pi}_{k}^{(N)}(\\boldsymbol{n_1^+},\\ldots,\\boldsymbol{n_J^-}) = 2 ^ {-N}\\bar\\Pi_{k}^{(N)}(\\boldsymbol{n_1^+}+\\boldsymbol{n_1^-},\\ldots,\\boldsymbol{n_J^+}+\\boldsymbol{n_J^-}) \n\\]\nand $\\bar{\\Pi}_k^{(N)}$ on the right-hand-side is the pEPPF of the hierarchical DP derived in \\citet{camerlenghi2019b}, namely\n\\[\n\\bar{\\Pi}_k^{(N)}(\\bm{n}_1,\\ldots,\\bm{n}_k)=\\Biggl(\\prod_{j=1}^J \\frac{\\prod_{i=1}^k (\\gamma_j)_{n_{j,h}}}{(\\gamma_j)_{n_j}}\\Biggr)\\:\n\\sum\\limits_{\\boldsymbol{\\ell}}\\frac{\\alpha^k}{(\\alpha)_{|\\boldsymbol{\\ell}|}}\\:\n\\prod_{h=1}^k (\\ell_{\\bullet,h} -1)!\\prod_{j=1}^J P(K_{n_{j,h}}=\\ell_{j,h})\n\\] \nwhere each sums runs over all $\\ell_{j,h}$ in $\\{1,\\ldots,n_{j,h}\\}$, if $n_{j,h}\\ge 1$, and equals $1$ if $n_{j,h}=0$, whereas $\\ell_{\\bullet,h}=\\sum_{j=1}^J \\ell_{j,h}$ and $|\\boldsymbol{\\ell}| = \\sum_{j=1}^J \\sum_{h=1}^k \\ell_{j,h}$. Note that the latent variable $\\ell_{j,h}$ is the number of tables in restaurant $j$ serving the $h$-th pair of dishes. Moreover, $K_{n_{j,h}}$ is a random variable denoting the number of distinct clusters, out of $n_{j,h}$ observations generated by a DP with parameter $\\gamma_j$ and diffuse baseline $P_0$ and it is well-known that  \n\\[\n\\mathbb{P}(K_{n_{j,h}}=\\ell_{j,h})=\\frac{\\gamma_j^{\\ell_{j,h}}}\n{(\\gamma_j)_{n_{j,h}}}\\:\n|\\mathfrak{s}(n_{j,h},\\ell_{j,h})|\n\\] \nwhere $|\\mathfrak{s}(n_{j,h},\\ell_{j,h})|$ is the signless Stilring number of the first kind. \nIn view of this, one can deduce the predictive distribution\n\\begin{equation*}\n\\begin{split}\n\t\\mathbb{P}(\\phi_{n_j+1,j}\\in \\cdot&\\,|\\, \\boldsymbol{\\phi}) =\n\t\\frac{\\gamma_j}{i-1+\\gamma_j}\n\t\\sum\\limits_{\\boldsymbol{\\ell}}\n\t\\frac{\\alpha}{|\\boldsymbol{\\ell}| + \\alpha}\\pi(\\boldsymbol{\\ell}\\,|\\,\\boldsymbol{\\phi})P_0(\\cdot)\\\\\n\t&+\\sum\\limits_{h=1}^{k}\n\t\\left[\\frac{n_{j,h}^+ + n_{j,h}^-}{n_j+\\gamma_j} + \n\t\\frac{\\gamma_j}{n_j+\\gamma_j}\n\t\\sum\\limits_{\\boldsymbol{\\ell}}\n\t\\frac{\\ell_{\\bullet,h}}{|\\boldsymbol{\\ell}| + \\alpha}\\pi(\\boldsymbol{\\ell}\\,|\\,\\boldsymbol{\\phi})\\right]\n\t\\Bigg(\\frac{\\delta_{\\phi^{**}_{h}}(\\cdot)+\\delta_{-\\phi^{**}_{h}}(\\cdot)}{2}\t\\Bigg)\n\\end{split}\n\\end{equation*}\nwhere \n\\begin{equation*}\n\\pi(\\boldsymbol{\\ell}\\,|\\,\\boldsymbol{\\phi})\\propto\n\\frac{\\alpha^k}{(\\alpha)_{|\\boldsymbol{\\ell}|}}\n\\prod_{h=1}^k (\\ell_{\\bullet,h} -1)!\\prod_{j=1}^J\n\\frac{\\gamma_j^{\\ell_{j,h}}}\n{(\\gamma_j)_{n_{j,h}^+ + n_{j,h}^-}}\\:\n|\\mathfrak{s}(n_{j,h}^+ + n_{j,h}^-,\\ell_{j,h})|\\mathds{1}_{\\{1,\\ldots,n_{j,h}^+ + n_{j,h}^-\\}}(\\ell_{j,h})\n\\end{equation*}\nis the posterior distribution of the latent variables $\\ell_{j,h}$'s, and $\\mathds{1}_A$ is the indicator function of set $A$.\n\n\\section{Posterior inference}\n\\label{s:inference}\nThe findings of the previous section are the key ingredients to perform posterior inference with a marginal Gibbs sampler.\nThe output of the sampler is structured into three levels: the first produces posterior probabilities on partitions of disease-specific locations; the second generates density estimates; the third provides clusters of patients. \nFor notational simplicity, we omit\nthe dependence on $m$, except for the description of the sampling step that generates $\\omega$.\nRecall that ${\\boldsymbol{\\theta}}=(\\theta_{1},\\ldots,\\theta_{J}) $ and \n$\\boldsymbol{\\phi}=\\{(\\phi_{1,j},\\ldots,\\phi_{n_j,j}):\\,j=1,\\ldots,J\\}$, with $\\phi_{i,j}=(\\xi_{i,j},\\sigma_{i,j}^2)$. The target distribution of \nthe sampler is the joint distribution of $\\bm{\\theta}$, $\\bm{\\phi}$ and $\\omega$ conditionally on the observed data $\\bm{X}$.\n\n\\textbf{Sampling $\\boldsymbol{\\phi}$.} In view of the CRF representation of the \ns-HDP, $t_{i,j}$ stands for the label of the table where the $i$-th customer in restaurant $j$ sits \nand $h_{t,j}$ for the dish label served at table $t$ in restaurant $j$ and with $\\boldsymbol{t}$ and $\\boldsymbol{h}$ we denote the corresponding arrays. \nMoreover, define the assignment variable\n$s_{i,j}=\\mathds{1}(\\phi_{i,j}=\\phi^*_{t_{i,j},j})-\n\\mathds{1}(\\phi_{i,j}=-\\phi^*_{t_{i,j},j})$ and $\\boldsymbol{s}$ is the corresponding arrays. In order to generate $\\boldsymbol{\\phi}$, we need to sample \n\\begin{itemize}\n\\item[(i)]  $(t_{i,j}, s_{i,j})$ for $i=1,\\ldots,n_j$ and $j=1,\\ldots,J$;\n\\item[(ii)] $h_{t,j}$ for $t \\in \\boldsymbol{t}$ and $j=1,\\ldots,J$;\n\\item[(ii)] $\\phi^{**}_{h}$ for $h \\in \\boldsymbol{h}$. \n\\end{itemize}\n\nNote that, using the latent allocation indicators in $\\boldsymbol{t}$ and $\\boldsymbol{h}$, the sampling scheme is more efficient than sampling directly from the full conditional of each $\\phi_{i,j}$, since the algorithm can update more than one parameter simultaneously \\citep{neal2000}. Define $\\varepsilon_{i,j} = X_{i,j} - \\theta_{j}$ and denote with $h(\\varepsilon_{i,j,m}|\\phi^*)$ the conditional normal density of $\\varepsilon_{i,j}$\ngiven $\\phi^*=(\\xi^*,\\sigma^{2*})$, while the marginal density\nis\n\\[\n\\bar h (\\varepsilon_{i,j}) = \\int h(\\varepsilon_{i,j}|\\phi) P_{0}(d\\phi)\n\\]\n\nTo sample $(t_{i,j}, s_{i,j})$ from their joint full conditional, we first sample $t_{i,j}$ from\n\\[\nP(t_{i,j}=t\\mid \\boldsymbol{t}^{-(i,j)}, \\boldsymbol{h}^{-(i,j)}, \\boldsymbol{\\phi}^{*-({i,j})},\\boldsymbol{\\phi}^{**-({i,j})},\\varepsilon_{i,j}) \\propto\n\\begin{cases} n_{t,j}^{-(i,j)}\\,p_{\\mbox{\\footnotesize old}}(\\varepsilon_{i,j}|\\phi^*_{t,j})\n& \\mbox{if } t\\in \\boldsymbol{t}^{-(i,j)}\n\\\\ \n\\gamma_{j}\\,p_{\\mbox{\\footnotesize new}}(\\varepsilon_{i,j}|{\\boldsymbol{\\phi}^{**-({i,j})}})\n& \\mbox{if } t=t^{\\mbox{new}}\\end{cases}\n\\]\nwhere $\\boldsymbol{t}^{-(i,j)}$, $\\boldsymbol{h}^{-(i,j)}$ $\\boldsymbol{\\phi}^{*-({i,j})}$,$\\boldsymbol{\\phi}^{**-({i,j})}$ coincide with the arrays $\\boldsymbol{t}$, $\\boldsymbol{h}$ $\\boldsymbol{\\phi}^*$,$\\boldsymbol{\\phi}^{**}$ after having removed the entries corresponding to the $i$-th customer in restaurant $j$. Moreover\n\\[\np_{\\mbox{\\footnotesize old}}(\\varepsilon_{i,j} | \\phi^*_{t,j}) \n=\n\\frac{1}{2}h(\\varepsilon_{i,j}|\\phi^*_{t,j})  + \\frac{1}{2}h(\\varepsilon_{i,j}|-\\phi^*_{t,j})\n\\]\nand\n\\[\np_{\\mbox{\\footnotesize new}}(\\varepsilon_{i,j}|{\\boldsymbol{\\phi}^{**-({i,j})}}) \n= \\sum\\limits_{h=1}^{k^{-(i,j)}}\\frac{\\ell_{\\bullet,h}}{|\\boldsymbol{\\ell}|+ \\alpha}\\left\\{\\frac{1}{2}h(\\varepsilon_{i,j}|\\phi^{**}_h)+\\frac{1}{2}h(\\varepsilon_{i,j}|-\\phi^{**}_h)\t\\right\\}+\\frac{\\alpha}{|\\boldsymbol{\\ell}|+ \\alpha} \\bar h (\\varepsilon_{i,j})\n\\]\t\nThen we sample $s_{i,j}$ from its full conditional\n\\[\np(s_{i,j}=s\\mid\\boldsymbol{\\phi}^*, t_{i,j},\\epsilon_{i,j}) \\propto \\begin{cases} h(\\varepsilon_{i,j}|\\phi^*_{t_{i,j}}) & \\mbox{if } s=1 \\\\ h(\\varepsilon_{i,j}|-\\phi^*_{t_{i,j}}) & \\mbox{if } s=-1\\end{cases}\n\\]\nThe conditional distribution of $h_{t,j}$ is\n\\[\np(h_{t,j}=h\\mid \\boldsymbol{t},\\boldsymbol{h}^{-(t,j)},\\boldsymbol{\\phi}^{**-(t,j)},\\boldsymbol{s},\\boldsymbol{\\varepsilon}) \n\\propto\n\\begin{cases} \\ell_{\\bullet,h}^{-(t,j)}\\prod\\limits_{\\{(i,j):\\:t_{i,j}=t\\}}\nh(s_{i,j}\\,\\varepsilon_{i,j}|\\phi_{h})\n& \\mbox{if } h \\in \\boldsymbol{h}^{-(t,j)} \\\\ \n\\alpha\\,\\displaystyle\\int \\prod\\limits_{\\{(i,j):\\:t_{i,j}=t\\}} h(s_{i,j}\\,\\varepsilon_{i,j}|\\phi) P_{0}(d\\phi) & \\mbox{if } h=h^{new}\\end{cases}\n\\]\n\nFinally, when $P_{0}$ is conjugate with respect to the Gaussian kernel, the full conditional distribution of $\\phi_h^{**}$ is obtained in closed form as posterior distribution of a Gaussian model, using as observations the collection $\\{\\,(s_{i,j}\\,\\epsilon_{i,j}):\\: h\t_{t_{i,j},j} = h\\}$. \n\n\\textbf{Sampling $\\boldsymbol{\\theta}$.} When sampling the disease-specific location parameters, one can rely on \na Chinese restaurant process restricted to those partitions that are consistent with the ordering of the diseases.\nThus, in order to generate $\\boldsymbol{\\theta}$, we first sample the labels $\\boldsymbol{t}_{\\theta}=\\{t_{1},\\ldots,t_{J}\\}$, where $t_{j}$ is the label of the table where the $j$-th customer sits. Then, we sample the dish $\\theta^*_{t}$ associated to table $t$ for all $t\\in\\boldsymbol{t}_{\\theta}$. If $z_{i,j} = X_{i,j} - \\xi_{i,j}$, the conditional density of $\\bm{z}_{j}=(z_{1,j},\\ldots,z_{n_j,j})$ associated to the location parameter $\\theta^*$,\ngiven $\\bm{\\sigma}_{j}=(\\sigma_{1,j},\\ldots,\\sigma_{n_j,j})$, is\n\\[\nf_{\\theta^*}(\\bm{z}_{j}|\\bm{\\sigma}_{j})=\\frac{1}{\\sqrt{2\\pi}\\prod\\limits_{i=1}^{n_j} \\sigma_{i,j}}\\:\n\\exp\\left\\{-\\frac{1}{2}\\sum\\limits_{i=1}^{n_j}\\frac{(z_{i,j}-\\theta^*)^2}{\\sigma_{i,j}^2}\n\\right\\}\n\\]\nUnder the prior in \\eqref{eqnew}, the full conditional distribution of $\\boldsymbol{t}_{\\theta}$ is provided by\n\\[\n\\begin{split}\n\tp(t_{j}=t\\mid t_1,\\ldots,&t_{j-1},\\theta_{j-1},\\bm{z}_{j}, \\bm{\\sigma}_{j})\\\\\n\t&\\propto\n\t\\begin{cases} \n\t\ta(\\omega,\\theta_1,\\ldots,\\theta_{j-1})\\,f_{\\theta_{j-1}}(\\bm{z}_{j}|\\bm{\\sigma}_{j}) \n\t\t& \\mbox{if } t=t_j \\\\[4pt] \n\t\t[1 - a(\\omega,\\theta_1,\\ldots,\\theta_{j-1})] \\, \n\t\t\\displaystyle\\int f_{\\theta}(\\bm{z}_{j}|\\bm{\\sigma}_{j})\\, G (d\\theta)& \\mbox{if } t=t^{\\mbox{\\footnotesize new}}\\\\[4pt]\n\t\t0&\\mbox{otherwise}\n\t\\end{cases}\n\\end{split}\n\\]\nFinally, when $G$ is conjugate with respect to the Gaussian kernel, the full conditional distribution of $\\theta_{t}^{*}$, given $\\{\\bm{z}_{j}:\\: t_{j}=t\\}$, is obtained in closed form using conjugacy of the Normal-Normal model.\n\n\\textbf{Sampling the concentration parameter.} Finally, \nthe concentration parameter $\\omega$ can be sampled through an importance sampling step using as importance distribution the prior $p_{\\omega}$ over $\\omega$. Denoting with $M_m$ the selected partition for $\\bm{\\theta}_{m}$ and with $T_m$ the number of clusters in $M_m$, we have \n\\[\np(\\omega\\,|\\,M_m:m=1,\\ldots,M)\\propto p_{\\omega} (\\omega)  \\,\\frac{\\omega^ {\\sum_{m=1}^M T_m - M}}{(\\omega + 2)^{M} (\\omega^2 + \\omega + 3)^{M}}.\n\\]\n\n\\section{Results}\n\\label{s:results}\n\\subsection{Simulation studies}\n\\label{ss:sim}\nWe perform a series of simulation studies with two main goals. First, we aim to highlight the drawbacks of clustering based on the entire distribution with respect to our proposal in the context of small sample sizes. Second, we check the model's ability of detecting the presence of underlying relevant factors in the sense described in Section~\\ref{ss:s-HDP}. \n\nTo accomplish the first goal, we compare the results obtained using our model with the nested Dirichlet process (NDP) \\citep{rodriguez2008}, arguably the most popular Bayesian model to cluster populations. Mimicking the real hypertensive dataset, we simulate data for 4 samples, ideally corresponding to four diseases, with respective sample sizes \nof 50, 19, 9 and 22, which correspond to the sample sizes of the real data investigated in Section \\ref{ss:real}. Since the NDP does not allow to treat jointly multiple response variables, we consider only one response variable to ensure a fair comparison. The observations are sampled from the following distributions and 100 simulation studies are performed.\n\\begin{equation*}\n\\begin{split}\n\tX_{i,1}\\,\\overset{iid}{\\sim}0.5\\,\\mathcal{N}(\\,0,\\,0.5\\,)+0.5\\,\\mathcal{N}(\\,2,\\,0.5\\,) \\qquad&\\text{for}\\enskip i=1,\\ldots,n_1\\\\\n\tX_{i,2}\\,\\overset{iid}{\\sim}0.5\\,\\mathcal{N}(\\,2,\\,0.5\\,)+0.5\\,\\mathcal{N}(\\,4,\\,0.5\\,)\\qquad&\\text{for}\\enskip i=1,\\ldots,n_2\\\\\n\tX_{i,3}\\,\\overset{iid}{\\sim}0.5\\,\\mathcal{N}(\\,4,\\,0.5\\,)+0.5\\,\\mathcal{N}(\\,6,\\,0.5\\,) \\qquad&\\text{for}\\enskip i=1,\\ldots,n_3\\\\\n\tX_{i,4}\\,\\overset{iid}{\\sim}0.5\\,\\mathcal{N}(\\,6,\\,0.5\\,)+0.5\\,\\mathcal{N}(\\,8,\\,0.5\\,) \\qquad&\\text{for}\\enskip i=1,\\ldots,n_4\\\\\n\\end{split}\n\\end{equation*}\nNote that the true data generating process corresponds to samples from distinct distributions with pairwise sharing of a mixture component. Alternative scenarios are considered in the additional simulation studies that can be found in Section D of the Supplement.\n\nThe implementation of the NDP was carried out through the marginal sampling scheme proposed in \\cite{zuanetti2018clustering}, which is suitably extended \nto accommodate hyperpriors on the concentration parameters of the NDP. To simplify the choice of the hyperparameters, as suggested by \\citet[p.~535 and p.~551--554]{gelman2013bayesian} we estimate both models over standardized data. For our model, we set $G_m=\\mathcal{N}(0,\\,1)$ and $P_{0,m}=\\text{NIG}(\\mu=0, \\,\\tau=1,\\, \\alpha=2,\\,\\beta=4)$. Here, $\\text{NIG}(\\mu, \\,\\tau,\\, \\alpha,\\,\\beta)$ indicates a normal inverse gamma distribution. The base distribution for the NDP is $\\text{NIG}(\\mu=0, \\,\\tau=0.01,\\, \\alpha=3,\\,\\beta=3)$, as in \\cite{rodriguez2008}. Finally, we use gamma priors with shape 3 and rate 3 for all concentration parameters, which is a common choice.\nFor each simulation study, we perform 10,000 iterations of the MCMC algorithms with the first 5,000 used as burn-in.\n\n\\begin{table}[t]\n\t\t\\caption{Simulation studies summaries.}\n\t\\label{tab:table1}\n\\begin{center}\n\t\t\\begin{tabular}{lcccccc}\n\t\t\t&\\multicolumn{3}{c}{\\textbf{sHDP}}& \\multicolumn{3}{c}{\\textbf{NDP}}\\\\\\hline\\hline\n\t\t\t&MAP&Average&Median&MAP&Average&Median\\\\\n\t\t\tPartitions& count& post. prob.&post. prob.& count& post. prob.& post. prob.\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1},\\textcolor{burntorange}{2},\\textcolor{bostonuniversityred}{3},\\textcolor{burgundy}{4}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1}\\}\\{\\textcolor{burntorange}{2},\\textcolor{bostonuniversityred}{3},\\textcolor{burgundy}{4}\\}&0&0.000&0.000&2&0.020&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1},\\textcolor{burntorange}{2}\\}\\{\\textcolor{bostonuniversityred}{3},\\textcolor{burgundy}{4}\\}&0&0.000&0.000&\\textbf{72}&\\textbf{0.695}&\\textbf{0.860}\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{bostonuniversityred}{3},\\textcolor{burgundy}{4}\\}\\{\\textcolor{burntorange}{2}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1}\\}\\{\\textcolor{burntorange}{2}\\}\\{\\textcolor{bostonuniversityred}{3},\\textcolor{burgundy}{4}\\}&0&0.027&0.007&3&0.035&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1},\\textcolor{burntorange}{2},\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burgundy}{4}\\}&0&0.000&0.000&5&0.061&0.000\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{burgundy}{4}\\}\\{\\textcolor{burntorange}{2},\\textcolor{bostonuniversityred}{3}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1}\\}\\{\\textcolor{burntorange}{2},\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burgundy}{4}\\}&1&0.054&0.015&0&0.014&0.000\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burntorange}{2},\\textcolor{burgundy}{4}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{burntorange}{2},\\textcolor{burgundy}{4}\\}\\{\\textcolor{bostonuniversityred}{3}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1}\\}\\{\\textcolor{burntorange}{2},\\textcolor{burgundy}{4}\\}\\{\\textcolor{bostonuniversityred}{3}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1},\\textcolor{burntorange}{2}\\}\\{\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burgundy}{4}\\}&0&0.004&0.000&18&0.175&0.032\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burntorange}{2}\\}\\{\\textcolor{burgundy}{4}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{burgundy}{4}\\}\\{\\textcolor{burntorange}{2}\\}\\{\\textcolor{bostonuniversityred}{3}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1}\\}\\{\\textcolor{burntorange}{2}\\}\\{\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burgundy}{4}\\}&\\textbf{99}&\\textbf{0.915}&\\textbf{0.954}&0&0.000&0.000\n\t\\end{tabular}\n\\end{center}\n\\vspace{-\\baselineskip}\n\\end{table}\n\nTable~\\ref{tab:table1} displays summaries of the results on population clustering, darker rows correspond to partitions that are not consistent with the natural ordering of the diseases. The true clustering structure is given by the finest partition. \nAs already observed in \\cite{rodriguez2008}, the NDP tends to identify fewer, rather than more clusters, due to the presence of small sample sizes. Using the \\textit{maximum a posteriori} estimate, our model correctly \nidentifies the partition in 99 out of 100 simulation studies and a partition with three elements or more in 100 out of 100 simulation studies. The \nsame counts for the NDP are, respectively, 0 out of 100 and 21 out of 100. Analogous conclusions can be drawn looking at posterior probability averages and medians across the 100 simulation studies (see Table~\\ref{tab:table1}) leaving \nno doubt about the model to be preferred under this scenario. \n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{7cm}\n\t\t\\centering\n\t\t\\includegraphics[ width=7cm]{figures\/confmean26}\n\t\t\\caption{95\\% credible intervals for population-specific locations}\n\t\t\\label{fig:fig3a}\n\t\\end{subfigure}\\hspace{0.2cm}%\n\t\\begin{subfigure}{7cm}\n\t\n\t\t\\centering\n\t\t\\includegraphics[width=7cm]{figures\/K_secondlevel_scenario0_shdp26}\n\t\t\\caption{Number of second-level clusters.}\n\t\t\\label{fig:fig3b}\n\t\\end{subfigure}\n\t\\begin{subfigure}{7cm}\n\t\n\t\t\\centering\n\t\t\\includegraphics[ width=7cm, trim={2cm 0 2cm 0}]{figures\/cluster_scenario0_natord26}\n\t\t\\caption{Co-clustering.}\n\t\t\\label{fig:fig3c}\n\t\\end{subfigure}\\hspace{0.2cm}%\n\t\\begin{subfigure}{7cm}\n\t\n\t\t\\centering\n\t\t\\includegraphics[width=7cm,, trim={2cm 0 2cm 0}]{figures\/cluster_scenario0_reord26}\n\t\t\\caption{Co-clustering.}\n\t\t\\label{fig:fig3d}\n\t\\end{subfigure}\n\\caption{ Panel (a): Mean point estimates and 95\\% credible intervals for the four populations, vertical lines correspond to true values. Panel (b): Posterior distribution on the number of second-level clusters. Panels (c) and (d): heatmaps of second-level clustering, darker colors correspond to higher probability of co-clustering; in (c) patients are ordered based on the diagnosis and the four black squares highlight the within-sample probabilities and in (d) patients are reordered based on co-clustering probabilities.}\n\\end{figure}\n\nFinally, we randomly select three simulation studies among the 100 to better understand the performance in estimating the other model parameters. Here we comment on one of the studies, the other two leading to similar results are reported in Section D.1.1 of the Supplement. \nFigure~\\ref{fig:fig3a} shows point estimates and credible intervals for the population-specific location parameters $\\theta_1,\\theta_2,\\theta_3,\\theta_4$. The true means belong to the 95\\% credible intervals.\n\nMoreover, it turns out that the model is able to detect the presence of two clusters of subjects leading to a posterior distribution for the number of clusters that is \nrather concentrated on the true value, see Figure~\\ref{fig:fig3b}--\\ref{fig:fig3d}. Moreover, the point estimate for the subject partition, obtained minimizing the Binder loss function, also contains two clusters, proving the ability of the model to detect the underlying relevant factor. In Section D of the Supplement, a number of additional simulation studies are conducted, both using alternative specifications over the disorder-specific parameters and different data generating mechanisms: the results highlight a good performance of the model, which appears also able to detect outliers, to highlight non-location effects of the disorders and to produce reliable outputs even under deviation from symmetry.\n\n\\subsection{Impact of hypertensive disorders on maternal cardiac dysfunction}\n\\label{ss:real}\nOur analysis is based on the dataset of \\cite{data}, which can be obtained from https:\/\/data.mendeley.com\/datasets\/d72zr4xggx\/1. The dataset contains observations for $10$ cardiac function measurements collected through a prospective case-control study on women in the third semester of pregnancy divided in $n_1=50$ control cases (C), $n_2=19$ patients with gestational hypertension (G), $n_3=9$ patients with mild preeclampsia (M) and $n_4=22$ patients with severe preeclampsia (S). The cases are \nwomen admitted from 2012 to 2014 to King George Hospital in Visakhapatnam, India. The healthy sample is composed by normotensive pregnant women. All women with hypertension were on antihypertensive treatment with oral Labetalol or Nifedipine. Women with severe hypertension were treated with either oral nifedipine and parenteral labetalol or a combination. For more details on the dataset, we refer to \\cite{tatapudi2017}. The prior specification is the same as in the previous section. Sections E.2 and E.3 of the Supplement contain a prior-sensitivity analysis and show rather robust results w.r.t. different prior specifications. Inference is based on 10,000 MCMC iterations with the \nfirst half used as burn-in. \n\n\\begin{table}\n\\caption{Posterior probabilities over partitions of means. Maximum a posteriori probabilities are in \\textbf{bold}.}\n\\label{tab:table2}\n\\begin{center}\n\\resizebox{\\textwidth}{!}{\n\t\t\\begin{tabular}{lcccccccccc}\n\t\t\tpartitions&CI&CWI&LVMI&IVST&LVPW&EF&FS&EW&AW&E\/A\\\\\\hline\\hline\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.021&0.000&0.000&0.000&0.000&\\textbf{0.365}&\\textbf{0.303}&0.096&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.002&\\textbf{0.546}&0.001&0.083&0.016&0.078&0.190&0.021&0.036&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.002&0.000&0.001&0.000&0.000&0.037&0.038&0.072&0.076&0.049\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.001&0.139&0.001&0.019&0.024&0.028&0.078&0.042&0.232&0.055\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&\\textbf{0.463}&0.000&\\textbf{0.595}&0.000&0.000&0.276&0.045&\\textbf{0.498}&0.020&0.002\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.146&0.099&0.188&\\textbf{0.551}&\\textbf{0.672}&0.074&0.164&0.092&0.260&0.033\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.233&0.000&0.107&0.000&0.000&0.083&0.062&0.114&0.091&0.371\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.133&0.216&0.108&0.347&0.288&0.060&0.121&0.065&\\textbf{0.287}&\\textbf{0.491}\\\\\n\t\t\t\\hline\n\t\t\t$\\sum\\log_{15} \\left(p_i ^{-p_i}\\right)$ &0.501&0.430&0.415&0.361&0.289&0.632&0.688&0.598&0.613&0.424\n\t\\end{tabular}}\n\\end{center}\n\\end{table} \nTable \\ref{tab:table2} displays the posterior distributions for the partitions of unknown disease-specific means along with the corresponding entropy measurements, that can be used as measures of uncertainty. \nFirst note that if one takes also the ordering among distinct disease-specific locations into account, the posterior partition probabilities are, as desired, concentrated on specific orders of the associated unique values for all ten cardiac indexes. For instance, we have \n$\\mathbb{P}(\\{\\theta_{C,CI}=\\theta_{G,CI}=\\theta_{M,CI}\\}\\{\\theta_{S,CI}\\}\\mid X) =\\mathbb{P}(\\theta_{C,CI}=\\theta_{G,CI}=\\theta_{M,CI}>\\theta_{S,CI}\\mid X) = 0.463$. The ordered partitions with the highest posterior probability are displayed in Table~\\ref{tab:tableord}. \n\\begin{table}\n\\caption{Posterior probabilities over ordered partitions of means.}\n\\label{tab:tableord}\n\\begin{center}\n\t\\begin{tabular}{lcc} \n\t\t& ordered partition with& \\\\\n\t\tcardiac index& highest posterior probability&posterior prob\\\\\n\t\t\\hline\\hline\n\t\tCI&\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$>$\\{\\textcolor{burgundy}{S}\\}&0.463\\\\\n\t\tCWI&\\{\\textcolor{aoe}{C}\\}$<$\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}&0.546\\\\\n\t\tLVMI&\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$<$\\{\\textcolor{burgundy}{S}\\}&0.595\\\\\n\t\tIVST&\\{\\textcolor{aoe}{C}\\}$<$\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$<$\\{\\textcolor{burgundy}{S}\\}&0.548\\\\\n\t\tLVPW&\\{\\textcolor{aoe}{C}\\}$<$\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$<$\\{\\textcolor{burgundy}{S}\\}&0.671\\\\\n\t\tEF&\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}&0.365\\\\\n\t\tFS&\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}&0.303\\\\\n\t\tEW&\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$>$\\{\\textcolor{burgundy}{S}\\}&0.497\\\\\n\t\tAW&\\{\\textcolor{aoe}{C}\\}$<$\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$<$\\{\\textcolor{burgundy}{S}\\}&0.256\\\\\n\t\tE\/A&\\{\\textcolor{aoe}{C}\\}$>$\\{\\textcolor{burntorange}{G}\\}$>$\\{\\textcolor{bostonuniversityred}{M}\\}$>$\\{\\textcolor{burgundy}{S}\\}&0.466\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/confmean0}\n\t\\end{subfigure}\\hspace{0.05\\textwidth}%\n\t\\begin{subfigure}{.45\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figures\/confmean1}\n\t\\end{subfigure}\n\\caption{95\\% credible intervals for population-specific locations for CI and CWI}\n\\label{fig:fig5}\n\\end{figure}\n\nConsidering the posterior probabilities summarized in Table~\\ref{tab:table2} and in Table~\\ref{tab:tableord}, we find that the cardiac index (CI) is reduced in severe preeclampsia compared to all other patients, indicating reduced myocardial contractility in the presence of the most severe disorder. \nThe cardiac work index (CWI) is a good indicator to distinguish between cases and control, but not among cases. The left ventricular mass index (LVMI) is increased in severe preeclampsia patients compared to other pregnant women, indicating ventricular remodelling. While inter ventricular septal thickness (IVST) and left ventricular posterior wall thickness (LVPW) differ both between cases and controls and between severe preeclampsia and other disorders, indicating a progressive increase in the indexes with the severity of the disorder. The posterior probabilities associated to \nindexes of systolic function such as ejection fraction (EF) and fraction shortening (FS) are relatively concentrated on the partition of complete homogeneity, letting us to conclude that no differences are present among \npatients.\nAs for the parameters of the diastolic function, the posterior distribution for the E-wave indicator identifies a modified index in severe preeclampsia patients, while the mean E\/A ratio indicates a \ndecreasing diastolic function with the severity of the disorder. The posterior for the A-wave index is actually concentrated on three distinct partitions, leaving a relatively high uncertainty regarding the modifications of the index. However, considering jointly the three partitions with the highest posterior probability, differences are detected between control and cases with a total posterior probability equal to 0.779. Figure~\\ref{fig:fig5} shows point estimates and credible intervals for disorder-specific location parameters for the first two cardiac indexes. Analogous plots for all cardiac indexes can be found in Section E.1 of the Supplement.\n\nTable~\\ref{tab:table3} shows the results obtained using the prior in \\eqref{eq:eppf_dir}, instead of \\eqref{eq:prior2}. We remark that for all ten cardiac indexes, the posterior associates negligible probabilities to partitions that are in contrast with the natural order of the diagnoses. This is particularly reassuring in that the model, even without imposing such an order a priori, is able to single it out systematically across cardiac indexes.\nMoreover, we observe how the partitions identified by MAP are the same of Table~\\ref{tab:table2} for all cardiac index except AW. However, even under this alternative prior, the A-wave index is concentrated on the same three distinct partitions leading to the conclusion that there exists a difference between cases and control.\n\\begin{table}\n\t\\caption{Posterior probabilities over partitions of means. Maximum a posteriori probabilities are in \\textbf{bold}.}\n\t\\label{tab:table3}\n\t\\vspace{-0.3cm}\n\t\\begin{center}\n\\resizebox{\\textwidth}{!}{\n\t\t\\begin{tabular}{lcccccccccc}\n\t\t\tpartitions&CI&CWI&LVMI&IVST&LVPW&EF&FS&EW&AW&E\/A\\\\\\hline\\hline\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.019&0.000&0.000&0.000&0.000&\\textbf{0.332}&\\textbf{0.247}&0.078&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.002&\\textbf{0.643}&0.001&0.114&0.031&0.065&0.130&0.048&0.080&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.004&0.000&0.003&0.000&0.000&0.044&0.019&0.152&0.073&0.103\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G}\\}\n\t\t\t&0.004&0.000&0.000&0.000&0.000&0.037&0.105&0.013&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.002&0.065&0.002&0.047&0.078&0.027&0.036&0.063&\\textbf{0.424}&0.167\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&\\textbf{0.316}&0.000&\\textbf{0.527}&0.000&0.000&0.178&0.032&\\textbf{0.288}&0.002&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.023&0.000&0.000&0.000&0.000&0.019&0.103&0.006&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.173&0.089&0.124&\\textbf{0.472}&\\textbf{0.594}&0.033&0.054&0.064&0.140&0.042\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.002&0.000&0.001&0.003&0.000&0.044&0.031&0.017&0.000&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.018&0.000&0.000&0.000&0.000&0.061&0.067&0.016&0.000&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.005&0.163&0.001&0.095&0.006&0.028&0.040&0.015&0.016&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.213&0.000&0.124&0.000&0.000&0.052&0.014&0.121&0.036&0.241\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.074&0.000&0.137&0.003&0.000&0.041&0.022&0.055&0.001&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.014&0.000&0.000&0.000&0.000&0.011&0.067&0.004&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.133&0.040&0.079&0.265&0.291&0.029&0.033&0.059&0.229&\\textbf{0.448}\\\\\n\t\t\t\\hline\n\t\t\t$\\sum\\log_{15} \\left(p_i ^{-p_i}\\right)$&0.687&0.407&0.509&0.501&0.371&0.828&0.886&0.823&0.582&0.505\n\t\t\\end{tabular}}\n\t\\end{center}\n\\vspace{-0.1cm}\n\\end{table} \n\n\n\\begin{figure}[t]\n\t\\vspace{-0.1 cm}\n\t\\centering \n\t\\begin{subfigure}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[ width=\\linewidth]{figures\/EA}\n\t\t\\caption{density estimation}\n\t\\end{subfigure}\\hspace{0.05\\textwidth}%\n\t\\begin{subfigure}{.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={3cm 1cm 3cm 1cm}, clip,  width=\\linewidth]{figures\/cluster_EA}\n\t\t\\caption{co-clustering}\\label{fig:fig4b}\n\t\\end{subfigure}\\hspace{0.05\\textwidth}%\n\t\\begin{subfigure}{.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={3cm 1cm 3cm 1cm}, clip,  width=\\linewidth]{figures\/cluster_ord_EA}\n\t\t\\caption{co-clustering}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[ width=\\linewidth]{figures\/LVMI}\n\t\t\\caption{density estimation}\n\t\\end{subfigure}\\hspace{0.05\\textwidth}%\n\t\\begin{subfigure}{.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={3cm 1cm 3cm 1cm}, clip,  width=\\linewidth]{figures\/cluster_LVMI}\n\t\t\\caption{co-clustering}\\label{fig:fig4e}\n\t\\end{subfigure}\\hspace{0.05\\textwidth}%\n\t\\begin{subfigure}{.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={3cm 1cm 3cm 1cm}, clip, width=\\linewidth]{figures\/cluster_ord_LVMI}\n\t\t\\caption{co-clustering}\\label{fig:fig4f}\n\t\\end{subfigure}\n\\vspace{-0.1 cm}\n\\caption{ Panels (a) and (d): density estimates. Panels (b)--(c) and (e)--(f): heatmaps of the posterior probabilities of co-clustering; in (b) and (e) patients are ordered based on the diagnosis and the four black squares highlight the within-sample probabilities; in (c) and (f) patients are reordered based on co-clustering probabilities.}\n\\label{fig:fig4}\n\\vspace{-0.1 cm}\n\\end{figure} \n\nAs far as prediction and second-level clustering are concerned, Figure~\\ref{fig:fig4} displays the density estimates and the heatmap of co-clustering probabilities between pairs of patients for the E\/A ratio and LVMI. Figure~\\ref{fig:fig4b} shows that co-clustering probabilities are similar within and across diagnoses, indicating that the effect of the diseases on the distribution of the cardiac index is mostly explained through shifts between disease-specific locations. Moreover Figure~\\ref{fig:fig4b} suggests the presence of three outliers that have low probability of co-clustering with all the other subjects and that would be ignored by the model using a more traditional ANOVA structure. On the other hand, Figure~\\ref{fig:fig4e} shows a slightly different pattern for co-clustering probabilities in the fourth square, which suggests that the heterogeneity between severe preeclampsia patients and the other patients is not entirely explained by shifts in disease-specific locations. Finally, Figure~\\ref{fig:fig4f} suggests the presence of an underlying relevant factor. The corresponding figures for all ten response variables are reported in Section E.1 of the Supplement and can be used for prediction and for a graphical analysis aimed at controlling the presence of underlying relevant factors, outliers and differences across diseases distinct from shifts between disease-specific locations.\n\nOur results are coherent with almost all of the findings in \\cite{tatapudi2017}, where results were obtained through a series of independent frequentist tests. However, importantly, we are able to provide more insights thanks to the simultaneous comparison approach and the latent clustering of subjects. For instance, considering the response LVMI, \\cite{tatapudi2017} detected a significant increase in cases compared to controls and an increase in severe preeclampsia compared to gestational hypertensive and mild preeclampsia patients. Such results do not clarify whether a modification exists between the control group and gestational hypertensive patients or between the latter and mild preeclampsia patients. Moreover, in contrast to our analysis, their results do not provide any information concerning the presence of underlying common factors, outliers or distributional effects (different from shifts in locations).\n\n\n\n\n\n\n\\section{Concluding remarks}\n\\label{s:conclusion}\nWe designed a Bayesian nonparametric model to detect clusters of hypertensive disorders over different cardiac function indexes and found modified cardiac functions in hypertensive patients compared to healthy subjects as well as progressively increased alterations with the severity of the disorder. The proposed model has application potential also beyond the \nconsidered setup when the goal is to cluster populations according to multivariate information: it borrows strength across response variables, preserves the flexibility intrinsic to nonparametric models, and correctly detects partitions of populations even in presence of small sample sizes, \nwhen alternative distribution-based clustering models tend to underestimate the number of clusters. The key component of the model is the s-HDP, a hierarchical nonparametric structure for the error terms that offers flexibility and serves as a tool to investigate the presence of unobserved factors, outliers and effects other than changes in locations.  Interesting extensions of the model include generalizations to other types of invariances in order to accommodate identifiability in generalized linear models, for instance in presence of count data and a log link function, as well as generalizations to other types of processes, beyond the Dirichlet process.\n\n\\section*{Acknowledgements}\nMost of the paper was completed while B. Franzolini was a Ph.D. student at the Bocconi University, Milan.\n A. Lijoi and I. Pr\\\"unster are partially supported by MIUR, PRIN Project 2015SNS29B.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\nThe authors acknowledge K. Ishii and K. Tomiyasu for their useful remarks. The experiments at the Materials and Life Science Experimental Facility at J-PARC were performed under a user program (Proposal No. 2013A0052). M.F. is supported by a Grant-in-Aid for Scientific Research (A) (16H02125).\n\n\t\\begin{figure}[t]\n\t\\begin{center}\n\t\\includegraphics[width=75mm]{Fig3_dispersion_v1.pdf}\n\t\\caption{(Color online)~Momentum-dependence of (a) peak-position and (b) intensity in Sr$_3$Ir$_2$O$_7$. The gray lines are the results from RIXS [\\ref{Kim2012}]. The intensity is not corrected with $|{\\rm f}({\\bf Q})|^2$ and the absorption coefficient.}\n\t\\label{dispersion}\n\t\\end{center}\n\t\\end{figure}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction and Vocabulary }\n\\label{sec:vocabulary}\n\nOne of the important issues of contemporary physics is the\nunderstanding of strong interactions and in particular the study\nof the properties of \\textbf{strongly interacting matter} -- a system of strongly interacting particles in equilibrium.\nThe advent of the quark model of hadrons and the development of \nthe commonly accepted theory of strong interactions, \\textbf{quantum chromodynamics (QCD)},\nnaturally led to expectations that matter at very high densities\nmay exist in a state of quasi-free quarks and gluons, the \\textbf{quark-gluon\nplasma (QGP)}.\n\n\n\n\\begin{figure}[hbt]\n\n\\centering\n\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/v_qgp_hic.jpg}\n\n\\caption{\n\\textit{Left:} \n Artistic sketch of the two phases of strongly interaction matter, hadron-resonance gas and quark-gluon plasma.\n \\textit{Middle:}\nPhase diagram of QCD in temperature $T$ and baryon chemical potential $\\mu_B$, and the region covered by running or planned experiments~\\cite{Alemany:2019vsk}. The density range covered by LHC, LHC-FT and SPS experiments is indicated by the shaded areas in the figure. The lower boundary of the grey and blue shaded area follows the chemical freeze-out.\nThe upper boundary relates to the parameters at the early stage of the collisions. The potential deconfinement critical point is labelled with d-CP, the onset of deconfinement with OD. The black line at small temperatures and high densities shows the nuclear liquid-gas transition, also ending in a critical point n-CP. The density range of other experiments is indicated in the bar below the figure. This\nincludes RHIC at BNL, NICA at JINR, SIS100 at FAIR, J-PARC-HI at J-PARC, the Nuclotron at JINR (NUCL), and HIAF at HIRFL.\n \\textit{Right:} \n  Evolution of a heavy-ion collision at high energies. Successive snapshots of a central collision are shown versus time.  \n }\n \\label{fig:v_qgp}\n\\end{figure}\n\n\nDoes the QGP exist in nature?\nHow does the transition proceed from a low-density state of strongly interacting matter, \nin which quarks and gluons are confined in hadrons, to the QGP? \nIs it similar to the transition from liquid water to water vapour along\na first order transition line ending in a second order critical point and followed by a\ncross over transition, see illustration plots in Fig.~\\ref{fig:v_qgp}?\n\nThe study of high energy collisions of two atomic nuclei gives us\nthe unique possibility to address these issues in well controlled\nlaboratory experiments. \nThis is because it is observed that a system of strongly \ninteracting particles created in \\textbf{central heavy ion collisions} is close to (at least local) equilibrium.\nHow does the transition from a non-equilibrium system created in \\textbf{inelastic proton-proton interactions}\nto the equilibrium system in central heavy ion collisions look like?\n\nThese questions have motivated broad\nexperimental and theoretical efforts for about 50 years. \nSystematic measurements of particle production properties in nucleus-nucleus (A+A) collisions at different collision energies and for different masses of\ncolliding nuclei have been performed.\nBy changing collision energy and nuclear mass number \none changes macroscopic parameters of the created system -- its volume, energy, and net baryon number.\nThis allows to move across the phase diagram and look for the theoretically predicted boundaries of equilibration and matter phases, see  illustration plots in Fig.~\\ref{fig:v_qgp}.\nConsequently several physics phenomena might be \nobserved when studying experimentally nuclear collisions at high energies. \nThese are:\n\\begin{enumerate}[(i)]\n\\item \n\\textbf{onset of fireball} -- beginning of creation of large-volume ($\\gg 1$~fm$^3$) strongly interacting matter,\n\n\\item\n\\textbf{onset of deconfinement} -- beginning of QGP creation with increasing collision energy,\n\n\\item \n\\textbf{deconfinement critical point}  -- a hypothetical end point of the first order transition line to quark-gluon plasma that has properties of a second order phase transition. \n\\end{enumerate} \n\nThese phenomena are expected to lead to rapid changes of hadron production properties -- the \\textbf{critical structures} -- \nwhen changing collision energy and\/or nuclear mass number of the colliding nuclei. \n\n\n\n\n\n\\section{ Strongly interacting matter in heavy ion collisions}\n\\label{sec:strongly}\n\n\n\\textbf{Strongly interacting matter.}\nThe equation of state defines the macroscopic properties of matter in equilibrium. It  is a subject of statistical mechanics. The first step in this modelling is to clarify  the types of particle species and \ninter-particle interactions.\nOne should also choose an appropriate statistical ensemble\nwhich fixes the boundary conditions and conserves the corresponding global physical quantities, like energy and conserved charges.\nStrongly interacting matter at high energy density can be formed at the early stages of\nrelativistic \nA+A collisions.\nAs mentioned in the introduction the questions --\n\\begin{enumerate}[(i)]\n\\item\nWhat types of particles should be considered as fundamental?\n\\item\nWhat are the composite objects?\n\\item\nWhat are the fundamental forces between the matter constituents?\n\\item\nWhat are the conserved charges?\n\\end{enumerate}\n-- should be addressed.\nAnswers to these questions are changing with time as our knowledge about basic  properties of elementary particles and their interactions increases.\n\nThe first model of strongly interacting matter at high energy density was formulated in 1950 by Fermi~\\cite{fermi:1950jd}.\nIt assumes that a system created in high energy proton-proton (p+p) interactions emits  pions like black-body radiation, i.e., pions are treated as non-interacting particles,\nand the pion mass $m_\\pi\\cong 140$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace is neglected compared to the high temperature of the system.\nThe pressure $p$ and energy density $\\varepsilon$ can then be represented by the following functions of the temperature $T$ \n(the system of units with $h\/(2\\pi)=c=k_B=1$ will be used),\n\\eq{\\label{p}\np(T)~=~\\frac{\\sigma}{3}T^4~,~~~~~\\varepsilon(T)~\\equiv~T\\frac{dp}{dT}~-~p~=\\sigma T^4~,\n}\nwhere $\\sigma= \\pi^2g\/30$ is the the so-called Stephan-Boltzmann constant, with $g$ being the degeneracy factor (the number of spin and isospin states), and $g=3$ counting the three isospin states $\\pi^+,~\\pi^0,~\\pi^-$.  \n\n\\vspace{0.2cm}\n\\textbf{Hadrons and resonances.}\nThe study of particle production in high energy collisions started in the 1950s with discoveries\nof the lightest hadrons -- $\\pi$, $K$, and $\\Lambda$\n-- in cosmic-ray experiments. Soon after with the rapid advent of particle accelerators\nnew particles were discovered almost day--by--day.\nThe main feature of the strong interactions appears to be creation of new and new types of particle species -- hadrons and resonances -- when increasing the collision energy. \nA huge number (several hundreds) of different hadron and resonance species are known today. The simplest statistical model treats the hadron matter, i.e. a system of strongly interacting particles at not too high energy density, as a mixture of ideal gases of different hadron-resonance species.  \n\n\\vspace{0.2cm}\n\\textbf{Hadron-resonance gas.}\nIn the grand canonical ensemble the pressure \nfunction is then written as\n\\begin{eqnarray}\n\\label{p-id}\np^{\\rm id}(T,\\mu) \n= \\sum_i\\frac{g_i}{6\\pi^2}\\int d m\\, f_i(m)\\int_0^{\\infty} \\frac{k^4dk}{\\sqrt{k^2+m^2}}\n\\left[ \\exp\\left(\\frac{\\sqrt{k^2+m^2} - \\mu_i}{T}\\right)+\\eta_i\\right]^{-1}~,\n\\end{eqnarray}\nwhere $g_i$ is the degeneracy factor of the $i^{\\textrm{th}}$ particle and the normalized\nfunction $f_i(m)$\ntakes into account the Breit-Wigner shape of resonances with finite width\n$\\Gamma_i$ around their average mass $m_i$. For the stable hadrons,\n$f_i(m)=\\delta(m-m_i)$.\nThe sum over $i$ in Eq.~\\eqref{p-id} \nis taken over all\nnon-strange and strange hadrons listed in the Particle Data Tables.\nNote, that in the equation\n$\\eta_i =-1$ and $\\eta_i = 1$ for bosons\nand fermions, respectively, while $\\eta = 0$ corresponds to the Boltzmann approximation.\nThe chemical potential for the $i^{\\textrm{th}}$  hadron is given by\n\\begin{equation}\n\\mu_i\\ =\\ b_i\\,\\mu_B\\, +\\, s_i\\,\\mu_S\\, +\\, q_i\\,\\mu_Q\n\\label{eq:mui}\n\\end{equation}\nwith $b_i = 0,\\, \\pm 1$, $s_i = 0,\\, \\pm 1,\\, \\pm 2,\\, \\pm 3$, and\n$q_i = 0,\\, \\pm 1,\\, \\pm 2$\nbeing the corresponding baryonic number, strangeness, and electric charge of\nthe $i$th  hadron. Hadrons composed of charmed and beauty quarks are rather heavy and thus rare in the hadron-resonance gas, and their contribution to the thermodynamical functions are often neglected. \nChemical potentials are denoted as $\\mu\\equiv (\\mu_B,\\mu_S,\\mu_Q)$ and correspond to the conservation of net-baryon number, strangeness, and electric charge in the hadron-resonance gas.\nThe entropy density $s$,  net-charge densities $n_i$ (with $i=B,Q,S$), and energy density $\\varepsilon$  are calculated from the pressure function (\\ref{p-id}) according to the standard thermodynamic identities:\n\\eq{\\label{therm-id}\ns(T,\\mu)=\\left(\\frac{\\partial p}{\\partial T}\\right)_\\mu~,~~~\nn_i(T,\\mu)=\\left(\\frac{\\partial p}{\\partial \\mu_i}\\right)_T~,~~~~ \\varepsilon(T,\\mu)=Ts+\\sum_in_i\\mu_i-p~.\n}\n\nNote that only one chemical potential $\\mu_B$ is considered as independent variable in fits of the model to particle multiplicities produced in A+A reactions. Two others, $\\mu_S$ and $\\mu_Q$, should be found, at each pair of $T$ and $\\mu_B$, from the requirements that the net-strangeness density equals to zero, $n_S=0$, and the ratio of the net-electric charge density, $n_Q$, to $n_B$ equals to the ratio of the number of protons, $Z$, to the number of all nucleons, $A$ (protons and neutrons) in the colliding nuclei, $n_Q\/n_B=Z\/A$.  \nEquations (\\ref{p-id}-\\ref{therm-id}) define the ideal hadron-resonance gas  \nmodel. In spite of evident simplifications this model  rather successfully fits the rich data on mean multiplicities of hadrons measured in central A+A collisions at high energies.  \n\n\\vspace{0.2cm}\n\\textbf{Hagedorn model.} \nIs there an upper limit for the masses of mesonic and baryonic resonances? \nIn 1965 Hagedorn  formulated a statistical model assuming an exponentially increasing spectrum  of hadron-resonance states at large masses~\\cite{Hagedorn:1965st}:\n\\eq{\\label{H}\n\\rho(m)~\\cong~C\\,m^{-a}\\,\\exp\\left(\\frac{m}{T_H}\\right)~, \n}\nwhere $C$, $a$, and $T_H$ are the model parameters. \nAt that time the number of experimentally detected hadron-resonance states was much smaller than it is today. Nevertheless, Hagedorn made the brave assumption that these $m$-states interpolate the low-mass spectrum and extend to $m\\rightarrow \\infty$, and that their density at large $m$ behaves as in Eq.~(\\ref{H}).   \nThe pressure function at $\\mu=0$ then becomes\n\\eq{ \\label{pH}\np(T)~=~T\\int_0^\\infty dm\\,\\rho(m)\\,\\phi_m(T)~,\n}\nwith the function $\\phi_m(T)$ behaving at $m\/T \\gg 1$ as\n\\eq{ \\label{large-m}\n \\phi_m(T)~\\cong ~ g\\,\\left(\\frac{mT}{2\\pi}\\right)^{3\/2}\\,\\exp\\left(-\\,\\frac{m}{T}\\right)~.\n}\nThe result (\\ref{large-m}) can be found from Eq.~(\\ref{p-id}) after $k$-integration. At $m\/T\\gg1$ and $\\mu=0$\nboth quantum statistics and relativistic effects become negligible.\n\n\n There are two exponential functions in the integrand (\\ref{pH}):  $\\exp(-m\/T)$ defines the exponentially decreasing  contribution of each individual $m$-state, and $\\exp(m\/T_H)$ defines the exponentially increasing number of these $m$-states. It is clear that the $m$-integral\nin Eq.~(\\ref{pH}) exists only for $T \\le T_H$.\nTherefore, a new hypothetical\nphysical constant -- the limiting temperature\n$T_H$ -- was introduced. The numerical value of $T_H$ was estimated by Hagedorn from two sources: from the straightforward comparison of Eq.~(\\ref{H}) with the experimental mass spectrum of hadrons and resonances, $\\Delta N\/\\Delta m$, and  from the inverse slope parameter of the transverse momentum spectra of final state hadrons in p+p interactions at high energies. Both estimates gave similar values $T_H=150-160$~MeV.  \nThe hadron states with large $m$ in the Hagedorn model are named the Hagedorn fireballs. These states were defined in a democratic (bootstrap) way:\nthe Hagedorn fireball consists of an arbitrary number of non-interacting Hagedorn fireballs, each of which in turn consists of ... \n\n\nIn the 1960s it was not clear up to what masses the hadron-resonances spectrum can be extended.  \nThe answer to this question is still unclear today. The large (exponential) density of resonance states\n$\\rho(m)$ and the finite widths $\\Gamma(m)$ of these states  make their experimental observation very problematic.  \nMoreover, several conceptual problems of the Hagedorn  model were obvious from the very beginning. The lightest hadron species, e.g., pion, kaon and proton can not be considered as  composed of other (non-interacting) hadrons, and should therefore have their own (non-democratic) status. Besides, the fireballs are treated as point like non-interacting  objects. However, from nuclear physics it was already evident that at least protons and neutrons are (strongly) interacting particles: nucleons should have both attractive and repulsive interactions to be able to form stable nuclei.  Most probably, similar interactions exist between other types of baryons. Evident physical arguments suggest that the same type of repulsive and attractive interactions should exist between anti-baryon species.  \n\n\n\\vspace{0.2cm}\n\\textbf{Quark-gluon plasma.}\nThe quark model of hadron classification was proposed by \nGell--Mann~\\cite{GellMann:1964nj} and  Zweig~\\cite{Zweig:352337} in 1964. It was the alternative to the bootstrap approach. Only three types of objects -- $u$, $d$, $s$ quarks and their anti-quarks -- were needed to construct the quantum numbers of  all known hadrons and successfully predict several new ones.  A 15 years \nperiod then started in which the idea of the existence of sub-hadronic particles -- quarks and gluons -- \nwas transformed into the fundamental  theory of strong interactions,\nquantum chromodynamics (QCD). Soon after the discovery of the $J\/\\psi$-meson in 1974 three new types of quarks -- $c$, $b$, and $t$ -- were added to QCD.  \nIn parallel, an important conjecture was formulated~\\cite{Ivanenko:1965dg,Itoh:1970uw} --\nmatter at high energy density, as in super-dense star cores, may consist of quasi-free Gell-Mann-Zweig quarks instead of\ndensely packed hadrons. Some years later Shuryak investigated the properties of QCD matter and came to a qualitatively similar conclusion:  QCD matter at high temperature is best described by quark and gluon degrees of\nfreedom and the name quark-gluon plasma was coined~\\cite{Shuryak:1977ut,Shuryak:1980tp}.\n\nQuestions concerning QGP properties and properties of its transition\nto matter consisting of hadrons, have been considered since the late 1970s (see, e.g., Ref.~\\cite{Cabibbo:1975ig}). The Hagedorn model was still rather popular at that time  due to its successful phenomenological applications. For example, the temperature parameter $T_{\\rm ch}$ found from fitting the data on hadron multiplicities in p+p interactions and A+A collisions at high energies (the so-called chemical freeze-out temperature) was found to be close to the limiting Hagedorn temperature, $T_{\\rm ch} = 140-160~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace \\cong T_H $.  \nHagedorn and Rafelski~\\cite{Hagedorn:1980cv}\nas well as Gorenstein, Petrov, and Zinovjev~\\cite{Gorenstein:1981fa} \nsuggested that the \nupper limit of the hadron temperature, the Hagedorn\ntemperature $T_H$, is not the limiting temperature but the transition temperature\nto the QGP, $T_C = T_H \\approx 150$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace.  \nThe Hagedorn fireball was then interpreted\nas the quark-gluon bag formed in the early stage of the collision. It also had an exponential mass spectrum (\\ref{H}) like in the Hagedorn model,\nbut was not a point like object. The average volume of the quark-gluon bag increases linearly with its mass.\nThis causes the excluded volume effects in the system of bags and leads to the transition to the high temperature QGP phase.\nNote that the first QCD-inspired estimate of the transition temperature to the\nQGP gave $T_C \\approx 500~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace$~\\cite{Shuryak:1977ut}, the most recent QCD-based estimates obtain $T_C \\approx T_H \\approx 150~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace$~\\cite{Bazavov:2018mes}. \n\nMany physicists started to speculate that the QGP could be formed in\nA+A collisions at sufficiently high energies in which one expects that strongly interacting matter of high energy density will be created.\nTherefore the QGP might be discovered in laboratory experiments.\n\n\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/v_hic_events.jpg}\n\\caption{\nTracks produced in nucleus-nucleus collisions recorded by heavy-ion experiments\nlocated in JINR Dubna (the SKM-200 streamer chamber~\\cite{Aksinenko:1980nm}), LBL Berkeley (the LBL streamer chamber~\\cite{Sandoval:1982cn}), CERN SPS (the NA35 streamer chamber~\\cite{Stock:1987rn}) and CERN SPS (the NA49 time projection chambers~\\cite{Afanasev:1999iu}), from left to right, respectively. \n }\n\\label{fig:s_events}\n\\end{figure}\n\n\\vspace{0.2cm}\n\\textbf{The first experiments.}\nIn parallel to the theoretical ideas and models, experimental studies of A+A\ncollisions were initiated in 1970 at the Synchrophasotron (JINR Dubna)~\\cite{Issinsky:1994it,Malakhov:2013zjq} and in 1975 at the Bevelac (LBL Berkeley)~\\cite{Alonso:1985awy}. Figure~\\ref{fig:s_events} (\\textit{first} and \\textit{second from left}) shows two examples of recorded collisions.\n\nSeveral effects were observed which could be attributed to a collective behaviour of the created system of hadrons. These are anisotropic and radial flow of particles~\\cite{Lisa:1994yr,Nagamiya:1981sd,Gustafsson:1984nd}, enhanced production of strange particles~\\cite{Anikina:1984cu} and suppressed production of pions~\\cite{Sandoval:1980bm}. They could only be\nexplained by\nassuming that  strongly interacting matter is produced in the studied collisions~\\cite{Stock:1985xe,Danielewicz:1985hn}. \nIn what follows we use the term fireball as the notation for a large volume\n($\\gg 1$~fm$^3$) system consisting of strongly interacting particles close to at least local equilibrium. They can be either hadrons and resonances or quarks and gluons.   \n\n\n\\vspace{0.2cm}\n\\textbf{Initiating the hunt for QGP.}\nThe two findings,\n\\begin{enumerate}[(i)]\n\\item\ntheoretical: QCD matter at sufficiently high temperature is in the state of a QGP;\n\\item\nexperimental: strongly interacting matter is produced in heavy ion collisions at energies of several GeV per nucleon;\n\\end{enumerate}\nled activists of the field~\\cite{Satz:2016xba} to the important decision to collide heavy ions at the maximum possible energy with the aim to discover the QGP.\nIn the 1980s the maximum possible energy for heavy ion collisions was available at CERN, Geneva.\nThis is why heavy ion physics entered the Super Proton Synchrotron (SPS) program at CERN.\nThe \\textit{Workshop on future relativistic heavy ion experiments}, GSI Darmstadt, October 7-10, 1980, organized by Bock and Stock~\\cite{Bock:1981iyr}, with an opening talk by Willis and a summary talk by Specht led to the formulation of the new program. Moreover, it initiated a series of \\textit{Quark Matter} conferences~\\cite{Satz:2016xba}.\n\n\n\\section{Evidence for the quark-gluon plasma}\n\\label{sec:q}\n\n\\textbf{Predicted QGP signals.}\nThe experimental search for a quark-gluon plasma in heavy ion collisions at the CERN SPS was shaped by several model predictions of possible QGP signals:\n\\begin{enumerate}[(i)]\n  \\item\n    suppressed production of charmonium states, in particular $J\/\\psi$ mesons~\\cite{Matsui:1986dk},\n    \\item\n    enhanced production of strange and multi-strange hadrons from the QGP~\\cite{Rafelski:1982pu},\n     \\item \n    characteristic radiation of photons and dilepton pairs from the QGP~\\cite{Shuryak:1980tp}.\n\\end{enumerate}\n\n\n\\vspace{0.2cm}\n\\textbf{Measurements at the CERN SPS.}\nThe search for the QGP at the CERN SPS was performed in two steps:\n\\begin{enumerate}[(i)]\n    \\item \n    In 1986-1987 oxygen and sulphur nuclei were accelerated to 200\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace.\n    Data on collisions with various nuclear targets were recorded by seven experiments, NA34-2, NA35, NA36, NA38, WA80, WA85 and WA94. \n    \\item\n    In 1996-2003 lead and indium beams at 158\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace were collided with lead and indium targets. Data were recorded by nine experiments, NA44, NA45, NA49, NA50, NA52, NA57, NA60, WA97 and WA98. \n\\end{enumerate}\n\nFigure~\\ref{fig:s_events} shows a S+Au collision at 200\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace (\\textit{second from right}) and a Pb+Pb collision at 158\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace (\\textit{right}) recorded by the NA35 streamer chamber~\\cite{Stock:1987rn} and the NA49 time projection chambers~\\cite{Afanasev:1999iu}, respectively. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\linewidth,clip=true]{plots\/q_sm1.jpg}\n        \\caption{Transverse energy distributions in S+A collisions at the top CERN SPS energy measured by HELIOS\/NA34-2~\\cite{Akesson:1988yt}.}\n    \\label{fig:detadet}\n\\end{figure}\n\nAn estimate of the energy density in A+A\ncollisions can be obtained from measurement\nof the transverse energy production and the size of the collision system. Already in \nS collisions with heavy nuclei (see Fig.~\\ref{fig:detadet}) it was found that values above 1~GeV\/fm$^3$ were\nreached (NA34~\\cite{Akesson:1987kh,Akesson:1988yt}, NA35~\\cite{Heck:1988cm}, NA49~\\cite{Margetis:1994tt}).\nMoreover the fireball showed effective temperature increasing linearly with particle mass,\na characteristic of collective radial expansion (see Fig.~\\ref{fig:exp-stat} (\\textit{left})). Also mean\nmultiplicities of produced hadrons are well reproduced by the statistical model~\\cite{Becattini:2003wp} (see Fig.~\\ref{fig:exp-stat} (\\textit{right})).\nThus conditions in collisions of heavy nuclei at the top energy of the CERN SPS are promising for the production of the QGP.\n\n\\begin{figure}\n   \\centering\n    \\includegraphics[width=0.48\\linewidth,clip=true]{plots\/q_sm2.jpg}\n    \\includegraphics[width=0.51\\linewidth,clip=true]{plots\/q_sm3.jpg}\n    \\caption{\\textit{Left:} Inverse slope parameter (effective temperature) of the transverse mass distribution versus particle mass measured by WA97, NA44 and NA49~\\cite{Antinori:2000sb}. \\textit{Right:} Mean hadron multiplicities\n    measured by NA49 compared to the statistical model fit~\\cite{Becattini:2003wp}. Pb+Pb collisions at the top CERN SPS energy. }\n   \\label{fig:exp-stat}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\centering\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/q_signals1a.jpg}\n\\caption{\n\\textit{Left:} Ratio of $J\/\\psi$ meson to Drell-Yan muon pair production (data points) yields\ncompared to predictions (curves) of $J\/\\psi$ absorption by hadronic \nmatter~\\cite{Baglin:1990iv,Alessandro:2004ap} (NA38, NA50).\n\\textit{Center:} Comparison of \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace yield ratio in p+p, p+A and A+A collisions~\\cite{Bartke:1990cn,Appelshauser:1998vn}\n(NA35, NA49). \\textit{Right:} Comparison of the mid-rapidity ratios of\nof hyperon production to number of wounded nucleons in p+Be, p+Pb and Pb+Pb\ncollisions~\\cite{Antinori:2006ij} (NA57). Top CERN SPS energy.\n }\n\\label{fig:q_signalsa}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\centering\n \\includegraphics[width=0.4\\linewidth,clip=true]{plots\/q_sm4.jpg}\n \\includegraphics[width=0.4\\linewidth,clip=true]{plots\/q_sm5.jpg}\n\\caption{\n\\textit{Left:} Direct photon signal observed in central Pb+Pb collisions~\\cite{Manko:1999tgz} (WA98).\n\\textit{Right:} Effective temperature of directly produced dimouns in In+In collisions as function of dimuon mass~\\cite{Arnaldi:2008er} (NA60). Top CERN SPS energy.\n }\n\\label{fig:q_signalsb}\n\\end{figure}\n\n\\vspace{0.2cm}\n\\textbf{The QGP discovery.}\nResults on central collisions of medium size and heavy nuclei from the QGP search programme at the CERN SPS appeared to be consistent with the predictions for the QGP:\n\\begin{enumerate}[(i)]\n     \\item\n    The relative yield of $J\/\\psi$ mesons is significantly suppressed compared to that in p+p and p+A interactions\n    (NA38~\\cite{Baglin:1990iv}, NA50~\\cite{Alessandro:2004ap}, NA60~\\cite{Arnaldi:2007zz}) as expected for the $J\/\\psi$ melting in a QGP\n    (see Fig.~\\ref{fig:q_signalsa} (\\textit{left})).\n    \\item\n    The relative strangeness yield is consistent with the yield expected for the equilibrium QGP. Moreover, it is significantly enhanced compared to that in p+p and p+A \n    interactions (see Fig.~\\ref{fig:q_signalsa} (\\textit{center}))\n    (NA35~\\cite{Bartke:1990cn}, NA49~\\cite{Appelshauser:1998vn}). \n    Even larger enhancement is measured for the relative yield of multi-strange hyperons (see Fig.~\\ref{fig:q_signalsa} (\\textit{right})),\n    (WA97~\\cite{Andersen:1999ym}, NA57~\\cite{Antinori:2006ij}). \n    Note that QGP formation was not expected in p+p and p+A collisions.\n    \\item\n    Spectra of directly produced dimouns (virtual photons) and photons emerge above a dominant background at large mass respectively transverse momentum and show a thermal contribution with an effective temperature of about 200~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace. This is significantly larger than the expected transition temperature to QGP (see Fig.~\\ref{fig:q_signalsb})\n    (NA60~\\cite{Arnaldi:2008er} and WA98~\\cite{Manko:1999tgz}).\n    \\end{enumerate}\n\n\n\\vspace{0.2cm}\n\\textbf{Standard model of heavy-ion collisions.}\nThese and other results\nestablished the standard picture of heavy-ion collisions~\\cite{Florkowski:2010zz}:\n\\begin{enumerate}[(i)]\n    \\item \n    High density strongly interacting matter is created at the early stage of heavy-ion collisions. Starting at SPS collision energies it is in the QGP phase.\n    \\item\n    The high-density matter enters a hydrodynamic expansion, cools down and emits photons and dileptons.\n    \\item\n    At the phase transition temperature, $T_C \\approx 150~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace$, hadrons are created.\n    Statistical haronization models fit hadron yields at this stage quite well.\n    \\item \n    The hadronic matter after hadronization is still dense enough to modify the hadron composition and continue expansion.\n    \\item\n    At sufficiently low densities the hadron interaction rate drops to zero (freeze-out).\n    resonances decay and long-lived hadrons freely fly away e.g. towards particle detectors at the CERN SPS.\n\\end{enumerate}\n\n\\vspace{0.2cm}\n\\textbf{Conclusion from the QGP-search.}\nThese major achievements were compiled by the heavy-ion community~\\cite{Heinz:2000bk} and led to the CERN press release - \non February 10, 2000 the CERN Director General Luciano Maiani said:\n\\textit{\nThe combined data coming from the seven experiments on CERN's Heavy Ion programme have given a clear picture of a new state of matter. This result verifies an important prediction of the present theory of fundamental forces between quarks. It is also an important step forward in the understanding of the early evolution of the universe. We now have evidence of a new state of matter where quarks and gluons are not confined. There is still an entirely new territory to be explored concerning the physical properties of quark-gluon matter. The challenge now passes to the Relativistic Heavy Ion Collider at the Brookhaven National Laboratory and later to CERN's Large Hadron Collider.}\n\nThis was in fact the moment when the majority of heavy-ion physicists moved to\nstudy heavy-ion collisions at much higher energies at the Relativistic Heavy Ion Collider (RHIC) of Brookhaven National Laboratory (BNL).\nRich and precise results obtained during the period of 2000-2010 at RHIC provided extensive information on the properties of the QGP. There were already no doubts about QGP formation at the early stage of A+A collisions at the CERN SPS,  and all the more  at RHIC energies.  Two basic properties of the QGP were established at the RHIC BNL: jet quenching (deceleration  of high momentum partons in the hot QGP)  and a small ratio of  the shear viscosity $\\eta$ to the entropy density $s$. It was estimated that $\\eta\/s\\cong 0.1$, i.e the QGP appears to be an almost perfect liquid (see Ref.~\\cite{Adams:2005dq,Adcox:2004mh,Back:2004je,Arsene:2004fa} for details).  \n\nThe situation after the announcement of the QGP discovery in 2000 at CERN was however rather confusing. Many were pretty sure about its formation in central Pb+Pb collisions at the CERN SPS, but unambiguous evidence of the QGP state was still missing. \nNeedless to say that the Nobel prize for the QGP discovery was not yet awarded.  \nThis may be attributed to the difficulty of obtaining unique and quantitative predictions of\nthe expected QGP signals from QCD.\n\n\\vspace{0.2cm}\n\\textbf{Question marks.}\nLet us briefly discuss questions addressed to the two main signals of the QGP: the $J\/\\psi$ suppression and the strangeness enhancement.\n\n{\\bf The $J\/\\psi $ suppression.} The standard picture of $J\/\\psi$  production in collisions of hadrons and nuclei assumes\na two step process: \nthe creation of a $c\\overline{c}$ pair in hard parton collisions at the very early stage of the reaction and a subsequent formation of a bound charmonium state or two open charm hadrons. Further more it was assumed that the initial yield of $c\\overline{c}$ pairs is proportional to the yield Drell-Yan pairs. Then the $J\/\\psi $\/(Drell-Yan pairs) ratio is expected to be the same in p+p, p+A and A+A collisions providing there are no other processes which can lead to $J\/\\psi$\ndisintegration and\/or creation.\nThe measured suppression of the ratio in p+A collisions respectively to p+p interactions was interpreted as due to $J\/\\psi$ interactions with nucleons of the target nucleus and with hadronic secondaries (`co-movers'). \nIn central Pb+Pb collisions at 158\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace the suppression was found to be significantly stronger than expected in the models including nuclear and co-mover suppression. This anomalous $J\/\\psi$  suppression was interpreted as the evidence of the QGP creation in central Pb+Pb collisions at the top CERN SPS. \nHowever, the uncertainties related to the assumption $c\\overline{c} \\sim \\textrm{Drell-Yan pairs}$ and estimates of the nuclear and co-mover suppression\nlead to uncertainty in interpretation of the anomalous $J\/\\psi$  suppression as the QGP signal.\nMoreover, models of  $J\/\\psi$  production in the later stages of the collision process have been developed:\n\\begin{enumerate}[(i)]\n\\item\nthe statistical model of $J\/\\psi$ production at the hadronization~\\cite{Gazdzicki:1999rk},\n\\item\nthe dynamical and statistical models of $J\/\\psi$ production via coalescence of $c\\overline{c}$ quarks~\\mbox{\\cite{Thews:2000rj,Levai:2000ne,BraunMunzinger:2000px,Gorenstein:2000ck}}.\n\\end{enumerate}\nClearly, in order to distinguish between different effects and verify the $J\/\\psi$ signal of the QGP creation systematic data on open charm production is needed.\n\n\\vspace{0.2cm}\n\\textbf{Strangeness enhancement.}\nA fast equilibration of $s\\overline{s}$ pairs was predicted as a QGP signature~\\cite{Rafelski:1982pu,Rafelski:2019twp}. This is mainly because of the small mass of the strange quark, $m_s\\sim 100$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace compared to the QGP temperature: $T \\ge T_C > m_s \\approx 100$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace. The estimated strangeness equilibration time was found to similar to the life time of the QGP phase in heavy-ion collisions at high energies.\nIn fact the strangeness yield measured in A+A collisions at the top SPS energy and above corresponds to the QGP equilibrium yield, for recent review see Ref.~\\cite{Rafelski:2019twp}.\nMoreover, it was estimated that the strangeness equilibration time in the confined matter is about 10 times longer than the life time of the hadronic phase in A+A collisions. This is because masses of strange hadrons, starting form the lightest one, the kaon ($m_K\\sim 500$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace) are much larger than the maximum temperature of the hadron-resonance gas $T\\le T_C \\approx 150$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace.  Thus a small yield of strangeness was expected for reactions in which the QGP was not expected, p+p and p+A interactions and A+A collisions at low energies. Consequently the enhanced production of strangeness was predicted as the next QGP signal~\\cite{Glendenning:1984ta}.\n \nThe strangeness enhancement is quantified by comparing a strange-hadron to pion ratio in A+A collisions with that in p+p interactions.\nIn particular a double ratio is calculated:\n\\eq{\\label{str-pion}\nR(\\sqrt{s_{NN}})= \\frac{\\langle K^+\\rangle_{AA}\/\\langle \\pi^+\\rangle_{AA}}{\\langle K^+\\rangle_{pp}\/\\langle \\pi^+\\rangle_{pp}}~,\n}\nwhere $\\langle\\ldots \\rangle_{AA}$ and $\\langle \\ldots \\rangle_{pp}$ denote the event averages of $K^+$ and $\\pi^+$ yields in, respectively, A+A collisions and p+p interactions at the same center of mass collision energy $\\sqrt{s_{NN}}$ of the nucleon pair.  Ratios of different strange hadrons to pions  \nwere considered, e.g.,  $\\langle K+\\overline{K}\\rangle\/\\langle \\pi\\rangle$, $\\langle \\Lambda\\rangle\/\\langle \\pi\\rangle$, $\\ldots$, $\\langle\\Omega\\rangle \/\\langle \\pi \\rangle $,  \nand then analyzed by forming the double ratios $R$, similar to the one given by Eq.~(\\ref{str-pion}).\n The confrontation of this expectation with the data was for the\nfirst time possible in 1988 when results from the SPS and the AGS\nbecame available. NA35  reported~\\cite{Bartke:1990cn} that in central S+S collisions at 200\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace\nthe strangeness to pion ratio (\\ref{str-pion}) is indeed about two times higher than in nucleon-nucleon interactions at\nthe same energy per nucleon. But an even larger enhancement ($R= 14 - 5 $) \nwas measured at the AGS at 2$A -10A$~GeV~\\cite{Pinkenburg:2001fj,Abbott:1990mm} demonstrating that strangeness enhancement {\\it increases} with {\\it decreasing} collision energy.\nMoreover, the enhancement factor (\\ref{str-pion}) should evidently go to infinity at the threshold energy of strange hadron production in nucleon-nucleon interactions. \nNote also that the strangeness neutrality introduced to statistical models using the canonical ensemble leads to a suppression of the relative yield of strange particles in systems with a low multiplicity of strangeness carriers~\\cite{Rafelski:1980gk}, e.g. p+p interactions at SPS energies.  \nIn any case, the AGS measurements indicating a strangeness enhancement larger than that at the CERN SPS show clearly difficulties in interpreting the strangeness enhancement as the QGP signal. \n\n\\vspace{0.2cm}\n\\textbf{New strategy.}\nDifficulties in interpretation of the QGP signatures forced scientists to rethink the QGP-hunt strategy. \nThe emerging new strategy was similar to the one followed by physics studying molecular liquids and gases. In these essentially simpler and familiar cases it is also sometimes  difficult to distinguish the properties of a dense gas from those of a liquid. It is much easier to identify the effects of the liquid-gas transition.  \nThus, if one believes that the QGP is formed in central Pb+Pb collisions at the top SPS energy one should observe qualitative signals of the {\\it transition} to the QGP at a lower collision energy. \nSeveral such signals were predicted within the statistical model of the early stage\n~\\cite{Gazdzicki:1998vd}.\nTheir observation would  serve as strong evidence of QGP creation in heavy-ion collisions at high enough collision energies.\n\nThis idea  motivated some of us to propose the collision energy scan at the CERN SPS with the aim to search for the {\\it onset of deconfinement}. This was the beginning of the search for the critical structures in heavy-ion collisions, for detail see the next section and Ref.~\\cite{Gazdzicki:2003fj}.\n\n\n\n\\section{ Critical structures }\n\\label{sec:cs}\n\n\n\\subsection{Evidence for the onset of deconfinement}\n\\label{sec:cs_ood}\n\n\n\\textbf{Predicted signals of the onset of deconfinement.}\nThe experimental search for the onset of deconfinement in heavy ion collisions at the CERN SPS was shaped by several model predictions of possible measurable signals:\n\\begin{enumerate}[(i)]\n    \\item \n    characteristic enhanced production of pions and suppression of the strangeness to pion ratio ~\\cite{Gazdzicki:1998vd},\n    \\item\n    softening of collective flow of hadrons~\\cite{Gorenstein:2003cu,Csernai:1999nf,Bleicher:2000sx,Bleicher:2005tb}, which should be observed in hadron distributions in transverse~\\cite{Gorenstein:2003cu} and longitudinal momenta~\\cite{Bleicher:2005tb} and azimuth angle~\\cite{Csernai:1999nf,Bleicher:2000sx}.\n\\end{enumerate}\n\n\\vspace{0.2cm}\n\\textbf{Measurements at the CERN SPS and RHIC BES.}\nThe search for the onset of deconfinement at the CERN SPS started \nin 1999 with the data taking on Pb+Pb collisions at 40\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace.\nThe data were registered by NA49, NA45, NA50 and NA57. \nIn 2000 a beam at 80\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace was delivered to NA49 and NA45.\nThe program was completed in 2002 by runs of NA49 and NA60 at 20\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace and 30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace.\nThus, together with the previously recorded data at 158\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace, NA49 gathered data at  five collision energies. Other experiments collected data at two (NA50, NA57) or three (NA45, NA60) energies. Starting in 2010 the beam energy scan program BES was started at RHIC with the aim of covering the low\nenergy range overlapping with the CERN SPS and providing important\nconsistency checks on the measurements.\n\n\n\\begin{figure}[hbt]\n\\centering\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/cs_ood_signals.jpg}\n\\caption{\nExamples of results illustrating the observation of the onset-of-deconfinement signals in central Pb+Pb (Au+Au) collisions~\\cite{Aduszkiewicz:2019zsv}, see text for details and more references.\n }\n\\label{fig:cs_ood_signals}\n\\end{figure}\n\n\\vspace{0.2cm}\n\\textbf{Discovery of the onset of deconfinement.}\nResults on the collision energy dependence of hadron production in central Pb+Pb collisions from the onset-of-deconfinement search programme at the CERN SPS~\\cite{Afanasiev:2002mx,Alt:2007aa} appeared to be consistent with the predicted signals (for review see Ref.~\\cite{Gazdzicki:2010iv}):\n\\begin{enumerate}[(i)]\n\\item\nThe average number of pions per wounded nucleon, $\\langle N_\\pi \\rangle \/\\langle W \\rangle $, \nin low energy A+A collisions is smaller that this value in p+p reactions.  This relation is however changed to the opposite at  collision energies larger than $\\approx$~30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace, the so-called {\\bf kink} structure. \n     \\item\n    The collision energy dependence of the $\\langle K^+\\rangle_{AA}\/\\langle \\pi^+\\rangle_{AA}$ ratio shows the so-called \\textbf{horn} structure. Following a fast rise the ratio passes through a maximum in the SPS range, at approximately 30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace, and then decreases and settles to a plateau value at higher energies. This plateau was found to continue up to the RHIC and LHC energies.\n    \\item\n    The collision energy dependence of the inverse slope parameter of the transverse mass spectra, $T^*$, of charged kaons shows the so-called \\textbf{step} structure. Following a fast rise the $T^*$ parameter passes through a stationary region (or even a weak minimum for $K^-$), which starts at the low SPS energies, approximately 30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace, and then enters a domain of a steady increase above the top SPS energy.\n    \\end{enumerate}\n    \nFigure~\\ref{fig:cs_ood_signals} shows examples of the most recent plots~\\cite{Aduszkiewicz:2019zsv} illustrating the observation of the onset-of-deconfinement signals. As seen data from the RHIC BES~I programme\n(2010-2014) and LHC (see Ref.~\\cite{Aduszkiewicz:2019zsv} for references to original experimental papers) confirm the NA49 results and their interpretation. \n\n\nTwo comments are appropriate here.\nThe strangeness to pion ratio, e.g., \n$\\langle K^+\\rangle_{AA}\/\\langle \\pi^+\\rangle_{AA}$,\nstrongly increases with collision energy in the hadron phase. This happens because \n$m_K\/T\\gg 1$, whereas $m_\\pi\/T\\cong 1$.\nThus, a much stronger increase with increasing temperature is expected for the multiplicities of heavy strange hadrons than that of pions. The strangeness to pion ratio reaches its maximum inside the hadron phase at the onset of the deconfinement. \nThe plateau-like behavior  at high collision energies reflects the approximately constant value  of the \nstrangeness to entropy ratio in the QGP.  It equals to the ratio of the degeneracy factor of strange quarks, \n\\eq{\\label{s-quark}\ng_s=\\frac{7}{8}\\cdot 2\\cdot 2\\cdot 3 =10.5~,\n}\nto the total degeneracy factor of the quark-gluon constituents in the QGP,\n\\eq{\\label{qg}\ng= 2\\cdot 8+ \\frac{7}{8}\\cdot 2\\cdot 2\\cdot 3 \\cdot 3  = 47.5~.\n}\nThese degeneracy factors count 2 spin states of quarks and gluons, 3 flavor quark states, 8 colour states of gluons and 3 colour states of quarks, one more factor 2 appears due to antiquarks (the factor 7\/8 is due to the Fermi statistics of quarks).   \nThe strangeness to entropy ratio in the HRG at the largest  hadron temperature\n$T\\cong T_H\\cong 150$~MeV appears to be larger than this ratio in the QGP which is approximately constant at all QGP temperatures $T\\ge T_H$. Therefore, \nthe transition region from hadron matter to the QGP \nreveals itself as the {\\it suppression} of strangeness yield relative to pion yield.\n\nThe second comment concerns the inverse slope parameter $T^*$ of the transverse mass ($m_T=\\sqrt{m^2+p_T^2}$) spectrum \n\\eq{\\label{pT}\n\\frac{dN}{m_T dm_T}\\sim \\exp\\left(-\\,\\frac{m_T}{T^*}\\right)~.\n}\nThe parameter $T^*$ is sensitive to both the thermal and collective motion transverse to the collision axis and behaves as\n\\eq{\\label{T*}\nT^*\\cong T+ \\frac{1}{2}m v_T^2~,\n}\nwhere $T$ is the temperature and $v_T$ is the transverse collective (hydrodynamic) velocity of the hadronic matter at the kinetic freeze-out. The parameter $T^*$ increases strongly with collision energy \nup to the energy $\\approx$~30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace. This is because an increasing collision energy leads to an increase of both terms in Eq.~(\\ref{T*}) - the temperature  $T$ and velocity $v_T$ - in the hadron phase ($v_T$ increases due to the increase of the pressure). At collision energy larger than $\\approx$~30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace the parameter $T^*$ is approximately independent of the collision energy in the SPS energy range. In this region one expects the transition between confined and deconfined matter. In the transition region both values - $T$ and $v_T$ - remain approximately constant, and this leads to the plateau-like structure in the energy dependence of the $T^*$ parameter. \nAt  RHIC--LHC energies, the parameter $T^*$ again increases with collision energy. The early stage QGP pressure increases with collision energy, and thus $v_T$ in Eq.~(\\ref{T*}) increases too. \n\n\\vspace{0.1cm}\nThe workshop \\textit{Tracing the onset of deconfinement in nucleus-nucleus collisions}, ECT* Trento, April 24-29, 2004, \nsummarized the results from the energy scan programme at the CERN SPS and concluded that future measurements in the SPS energy range are needed~\\cite{Gazdzicki:2004gss}.\nThe goal is  to search for the deconfinement critical point and study system size dependence of the onset of deconfinement. Possibilities to perform  these measurements at the CERN SPS, FAIR SIS300 and RHIC were discussed. The event initiated a series of the \\textit{Critical point and onset of deconfinement} workshops.\n\n\n\\begin{figure}[hbt]\n\\centering\n \\includegraphics[width=0.45\\linewidth,clip=true]{plots\/cs_cp_hill.jpg}\n \\includegraphics[width=0.45\\linewidth,clip=true]{plots\/cs_cp_sigma.jpg}\n\\caption{\n \\textit{Left:}\n Sketch of the expected signal of the deconfinement critical point - a maximum of fluctuations in the (nuclear mass number)-(collision energy) plane.\n \\textit{Right:}\n     Results from the NA61\\slash SHINE\\xspace two dimensional scan of energy and system size for (pion multiplicity)-(transverse momentum) fluctuations in terms of the strongly intensive quantity $\\Sigma[P_T,N]$~\\cite{Gazdzicki:2017zrq}.\n }\n\\label{fig:cs_cp_signals}\n\\end{figure}\n\n\\subsection{Searching for deconfinement critical point}\n\\label{sec:cs_cp}\n\n\n\\textbf{Predicted d-CP signals.}\nThe possible existence and location of the deconfinement critical point (d-CP) is a subject of\nvivid theoretical discussion, for a recent review see Ref.~\\cite{Bzdak:2019pkr}. \nThe experimental search for the d-CP\nin A+A collisions at the CERN SPS was shaped by several model predictions (for detail see Ref.~\\cite{Bialas:1990xd}) of its potential signals:\n\\begin{enumerate}[(i)]\n    \\item \n    characteristic multiplicity fluctuations of hadrons~\\cite{Bialas:1990xd,Antoniou:2000ms,Hatta:2003wn,Antoniou:2006zb},\n    \\item\n    enhanced fluctuations of (pion multiplicity)-(transverse momentum)~\\cite{Stephanov:1999zu},\n\\end{enumerate}\n\n\nThe signals were expected to have a maximum in the parameter space of collision energy and nuclear mass number of colliding nuclei - \\textbf{the hill of fluctuations}~\\cite{Gazdzicki:2006fy}.\nThis motivated NA61\\slash SHINE\\xspace to perform a two dimensional scan at the CERN SPS~\\cite{Gazdzicki:995681} in these two parameters, which are well controlled in laboratory experiments. \n\n\n\n\\begin{figure}[hbt]\n\\centering\n\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/cs_cp_exp.jpg}\n\n\n\\caption{\n Summary of data recorded by NA61\\slash SHINE\\xspace at the CERN SPS~(\\textit{left}) and STAR at RHIC~(\\textit{right}) relevant for the search for the deconfinement-CP, see text for details. \n }\n\\label{fig:cs_cp_exp}\n\\end{figure}\n\n\\vspace{0.2cm}\n\\textbf{Measurements at SPS and RHIC.}\nThe systematic search for the d-CP of strongly interacting matter was started in 2009 with the NA61\\slash SHINE\\xspace data taking on p+p interactions at six beam momenta in the range from 13\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace to 158\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace.\nIn the following years data on Be+Be, Ar+Sc, Xe+La and Pb+Pb collisions were\nrecorded, see Fig.~\\ref{fig:cs_cp_exp}~(\\textit{left}) for an overview.\n\nIn 2010 the beam energy scan (BES-I and BES-II) with Au+Au collisions started at the BNL RHIC~\\cite{Odyniec:2019kfh}. Search for the deconfinement critical point has been the most important goal of this programme. Above the collision energy of $\\sqrt{s_{NN}} = 7.7~\\ensuremath{\\mbox{Ge\\kern-0.1em V}}\\xspace$ ($\\approx$~30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace) the scan was conducted in the collider mode, whereas below in the fixed target mode. The location of the recorded data in the phase diagram are shown in Fig.~\\ref{fig:cs_cp_exp}~(\\textit{right}).  \n\n\n\\vspace{0.2cm}\n\\textbf{Status of the d-CP search.}\nMany experimental results have already been obtained within the d-CP search programmes at SPS and RHIC, for a recent review see Ref.~\\cite{Czopowicz:2020twk}. Five of them were considered as possible indications of the d-CP and are presented and discussed in the following. \n\n\n\n\\begin{enumerate}[(i)]\n\\item\nA maximum of fluctuations is expected in a scan of the phase diagram (see Fig.~\\ref{fig:cs_cp_signals} (\\textit{left})). Measurements of (pion multiplicity)-(transverse momentum) fluctuations from NA61\\slash SHINE\\xspace shown in Fig.~\\ref{fig:cs_cp_signals} (\\textit{right}) do not show such a \nfeature~\\cite{Gazdzicki:2017zrq}.\n\\item\nThe energy dependence of fluctuations of conserved quantities such as the net baryon number is predicted to be sensitive to the presence of the d-CP. This holds in particular for higher moments. The scaled third and fourth moments of the net-proton multiplicity distribution in Au+Au collisions from the STAR experiment is plotted in Fig.~\\ref{fig:cs_cp_results1}~\\cite{Adam:2020unf}. The non-monotonic behaviour of the fourth moment in central collisions and its sign change around $\\sqrt{s_{NN}} \\approx 7$~\\ensuremath{\\mbox{Ge\\kern-0.1em V}}\\xspace is debated as possible indication of the d-CP.\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.7\\linewidth,clip=true]{plots\/cs_cp_results1.jpg}\n\\caption{\n The energy dependence of the scaled third (\\textit{left}) and fourth (\\textit{right}) moments of the net-proton multiplicity distribution in central and peripheral Au+Au collisions from the STAR experiment~~\\cite{Adam:2020unf}.\n }\n\\label{fig:cs_cp_results1}\n\\end{figure}\n\n\\item\nAt the d-CP the correlation length diverges and leads to power-law type fluctuations of the baryon number. These were investigated by the NA61\\slash SHINE\\xspace experiment by measuring the momentum bin size dependence of the scaled second factorial moment of the proton multiplicity distribution (intermittency study) in semi-central Ar+Sc collisions at  $\\sqrt{s_{NN}} \\approx 17~\\ensuremath{\\mbox{Ge\\kern-0.1em V}}\\xspace$~\\cite{Mackowiak:2019qm}. While previous measurements by the NA49 experiment in Si+Si collisions indicated a signal the new measurements \nshown in Fig.~\\ref{fig:cs_cp_results2} do not confirm the effect.\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.4\\linewidth,clip=true]{plots\/cs_cp_results2.jpg}\n\\caption{\n Scaled second factorial moment $\\Delta F_2$ (background subtracted) of the proton multiplicity distribution as function of the number of subdivisions M of transverse momentum space obtained in Ar+Sc collisions at $\\sqrt{s_{NN}} \\approx$ 17~\\ensuremath{\\mbox{Ge\\kern-0.1em V}}\\xspace~\\cite{Mackowiak:2019qm}.\n }\n\\label{fig:cs_cp_results2}\n\\end{figure}\n\n\\item\nThe ratio of yields of light nuclei production can be related to nucleon number fluctuations~\\cite{Liu:2019nii}. The measurements from STAR in central Au+Au collisions show strong collision energy dependence and peak at $\\sqrt{s_{NN}} \\approx 20 - 30~\\ensuremath{\\mbox{Ge\\kern-0.1em V}}\\xspace$~\\cite{Xu:2019qm}. These results are presented in Fig.~\\ref{fig:cs_cp_results3}. Such behaviour is not reproduced by model calculations without a d-CP and may thus be attributable\nto a critical point.\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.4\\linewidth,clip=true]{plots\/cs_cp_results3.jpg}\n\\caption{\n Energy dependence of the ratio of yields of light nuclei production in central Au+Au collisions~\\cite{Xu:2019qm} measured by the STAR experiment at RHIC. \n }\n\\label{fig:cs_cp_results3}\n\\end{figure}\n\n\\item\nThe energy and centrality dependence of short-range two-pion correlations as parameterized by source radius parameters determined from Bose-Einstein correlation analysis was used to search for indications of the d-CP~\\cite{Lacey:2014wqa,Adamczyk:2014mxp}.\nThe result for the difference $R^2_{out}$ - $R^2_{side}$ in Au+Au collisions at RHIC is shown in Fig.~\\ref{fig:cs_cp_results4}.\nA finite size scaling analysis of these results led to an estimate of the position of a d-CP at  $T \\approx$~165~MeV and \n$\\mu_B$ $\\approx$ 95~MeV.\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.4\\linewidth,clip=true]{plots\/cs_cp_results4.jpg}\n\\caption{\n Centrality and energy dependence of the difference $R^2_{out}$ - $R^2_{side}$\n of radius parameters obtained from Bose-Einstein two-pion correlation analysis in Au+Au collisions from the PHENIX experiment at RHIC\n ~\\cite{Lacey:2014wqa,Adamczyk:2014mxp}.\n }\n\\label{fig:cs_cp_results4}\n\\end{figure}\n\n\\end{enumerate}\n\nThese observations, when interpreted as due to the d-CP, yield different estimates of the d-CP location on the\nphase diagram of strongly interacting matter, see Fig.~\\ref{fig:cs_cp_results_pd}~\\cite{Czopowicz:2020twk}.\nThus, as for now, the experimental results concerning the d-CP are inconclusive. New results from NA61\\slash SHINE\\xspace and STAR BES-II are expected within the coming years. \n\\vspace{0.2cm}\n\n\n\\textbf{Nuclear and deconfinement critical points.}\nThe nuclear critical point (n-CP) corresponds to the liquid-gas phase transition in the system of interacting nucleons and is located at small temperature $T_C\\approx 19$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace and large baryonic chemical potential $\\mu_B\\approx 915$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace, see Fig.~\\ref{fig:v_qgp} (\\textit{middle}) and Fig.~\\ref{fig:cs_cp_results_pd} for illustration.\n\nThe effect of the n-CP on fluctuations of conserved charges,\nbaryon number (B), electric charge (Q), and strangeness (S),\nwas studied in Refs.~\\cite{Vovchenko:2017ayq, Poberezhnyuk:2019pxs} within the \nHRG\nmodel with van der Waals interactions between baryons and between anti-baryons. \nThe second, third, and fourth order cumulants \n(susceptibilities) are\ncalculated in the grand canonical ensemble from the pressure function by taking the derivatives over the corresponding chemical potentials:\n\\eq{\n\\chi^{i}_{n}& =\\frac{\\partial ^{n}\\left( p\/T^{4}\\right) }{\\partial \\left( \\mu_{i} \n\/T\\right) ^{n}}~,\n}\nwhere $i$ stands for $B, Q, S$ and $n$ is the moment order. \n\nThe obtained results show that\nthe n-CP may significantly impact event-by-event fluctuations in A+A collisions even at high energies.  \nThus, the nuclear-CP should be taken into account in  future searches for the deconfinement-CP.\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.6\\linewidth,clip=true]{plots\/cs_cp_results_pd.jpg}\n\\caption{\nCompilation of theoretical predictions~\\cite{Stephanov:2007fk} and experimental hints~\\cite{Czopowicz:2020twk} on the location of the deconfinement critical\npoint, d-CP, in the phase diagram of strongly interacting matter. The position \nof nuclear critical point, n-CP, as suggested by theoretical and experimental results is indicated for comparision.\n }\n\\label{fig:cs_cp_results_pd}\n\\end{figure}\n\n\n\\subsection{Indication of the onset of fireball}\n\\label{sec:cs_oof}\n\n\\textbf{Predictions of reference models on system-size dependence.}\nThere are two models often used to obtain reference predictions concerning the system-size dependence of hadron production properties~\\cite{Gazdzicki:2013lda} - the Wounded Nucleon Model (WNM)~\\cite{Bialas:1976ed} and the Statistical Model (SM)~\\cite{Rafelski:1980gk}. For the \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio at the CERN SPS energies they read:  \n\\begin{enumerate}[(i)]\n\\item\nThe WNM prediction: the \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio is independent of the system size (number of wounded nucleons).\n\\item\nThe SM prediction: in the canonical formulation incorporating global quantum number conservation the \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio increases monotonically with the system size and approaches the limit given\nby the grand canonical approximation of the model. The rate of this increase is the fastest for small systems.\n\\end{enumerate}\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.47\\linewidth,clip=true]{plots\/cs_oof_system-size.jpg}\n \\includegraphics[width=0.45\\linewidth,clip=true]{plots\/cs_oof_energy.jpg}\n\\caption{\n Measurements of the the \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio in p+p, Be+Be, Ar+Sc and Pb+Pb collisions: system size dependence at 150\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace~\\cite{Podlaski:2019} (\\textit{left}) and  collision energy dependence~\\cite{Aduszkiewicz:2019zsv} (\\textit{right}). \n }\n\\label{fig:cs_oof_results}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.47\\linewidth,clip=true]{plots\/cs_oof_system-size_o.jpg}\n \\includegraphics[width=0.45\\linewidth,clip=true]{plots\/cs_oof_energy_o.jpg}\n\\caption{\n Measurements of the scaled variance $\\omega$ of the multiplicity distribution of negatively charged hadrons in inelastic p+p interactions and central Be+Be and Ar+Sc collisions~\\cite{AndreySeryakovfortheNA61\/SHINE:2017ygz}: system size dependence at 150\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace (\\textit{left})  and  collision energy dependence (\\textit{right}). \n }\n\\label{fig:cs_oof_results_o}\n\\end{figure}\n\n\\vspace{0.2cm}\n\\textbf{Unexpected result of measurements.}\nMeasurements of the system size dependence of hadron production properties at different collision energies were carried out by NA61\\slash SHINE\\xspace, for detail see Sec.~\\ref{sec:cs_ood}. \nFigures~\\ref{fig:cs_oof_results} and~\\ref{fig:cs_oof_results_o} show the unexpected result~\\cite{Podlaski:2019,AndreySeryakovfortheNA61\/SHINE:2017ygz}.\nThe \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio in Fig.~\\ref{fig:cs_oof_results} and the scaled variance of the multiplicity distribution at 150\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace in Fig.~\\ref{fig:cs_oof_results_o} are similar in inelastic p+p interactions and in central Be+Be collisions, whereas they are very different in central Ar+Sc collisions which are close to central Pb+Pb collisions. Both reference models, WNM and SM, qualitatively disagree with the data. \nThe WNM seems to work in the collisions of light nuclei (up to Be+Be) and becomes qualitatively wrong for heavy nuclei (like Pb+Pb). On the contrary, the SM is approximately valid for collisions of heavy nuclei.  However, its predictions\ndisagree with the data on p+p to Be+Be collisions.  \n\n\nThe rapid change of hadron production properties when moving from Be+Be to Ar+Sc collisions is interpreted and referred to as the onset of fireball. \nFrom Fig.~\\ref{fig:cs_oof_results}~(\\textit{right}) follows that the increase of the \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio depends on the collision energy. On the other hand,\nthe scaled variance $\\omega$ of the multiplicity distribution shows only weak collision energy dependence \n(see Fig.~\\ref{fig:cs_oof_results_o}~(\\textit{right})).\nThe physics behind the onset of fireball is under discussion~\\cite{Motornenko:2018gdc}.\nWhereas one does not observe the formation of fireball in the collisions of light nuclei at the SPS energies, the large size system of strongly interacting matter is probably formed at LHC energies even in p+p high multiplicity events. \n\n\n\\vspace{0.3cm}\n\\textbf{In summary}, the scans in collision energy and nuclear mass number of colliding nuclei\nperformed at SPS and RHIC indicate four domains\nof hadron production separated by two thresholds: the onset of deconfinement and the\nonset of fireball. The sketch presented in Fig.~\\ref{fig:future}~(\\textit{left}) illustrates this conclusion.\n\n\n\\section{Plans for future measurements}\n\\label{sec:futures}\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/future_phases.jpg}\n \\caption{\\textit{Left:} \nTwo-dimensional scan conducted by NA61\\slash SHINE\\xspace varying collision energy and nuclear mass number of colliding nuclei indicates four domains of hadron\nproduction separated by two thresholds: the onset of deconfinement and the onset\nof fireball. The onset of deconfinement is well established in central Pb+Pb~(Au+Au)\ncollisions, its presence in collisions of low mass nuclei, in particular, inelastic p+p\ninteractions is questionable.\n\\textit{Right:} \nRegions in the phase diagram of strongly interacting matter studied by present (red) and future (green) heavy ion facilities.\n }\n\\label{fig:future}\n\\end{figure}\n\nLet us close by discussing possible future measurements which are suggested by this review and which should be considered as priorities: \n\\begin{enumerate}[(i)]\n\\item\nA collision energy scan in the onset of deconfinement region to measure open and hidden charm production in Pb+Pb collisions and establish the impact of the onset on the heavy quark sector. \nThis requires high statistics data collected with detectors optimized for open and hidden charm measurements. Detailed physics arguments and possible experimental set-ups are presented in Refs.~\\cite{Aduszkiewicz:2309890,Agnello:2018evr}.\n\\item\nA detailed study of the onset of fireball and its collision energy dependence in the onset of deconfinement region. The goal is to understand the underlying physics of this phenomenon, for details see Ref.~\\cite{Aduszkiewicz:2309890}. This requires a two dimensional scan in the nuclear mass number of the colliding nuclei and in collision energy performed with small steps in nuclear mass number.\n\\end{enumerate}\nConclusive results from the data recorded by NA61\\slash SHINE\\xspace and RHIC BES-II are needed to\nplan future measurements for the deconfinement-CP search.\n\nFigure~\\ref{fig:future}~(\\textit{right}) presents a compilation of present and future facilities and their region of coverage in the phase diagram of strongly interacting matter. \nCharm measurements are planned by NA61\\slash SHINE\\xspace~\\cite{Aduszkiewicz:2309890}, NA60+~\\cite{Agnello:2018evr} at the CERN SPS, they are considered by MPD~\\cite{Kekelidze:2018nyo} at NICA and J-PARC-HI~\\cite{Sako:2019hzh} at J-PARC.\nA detailed two dimensional scan is considered by NA61\\slash SHINE\\xspace at the CERN SPS~\\cite{Aduszkiewicz:2309890}.\n\n\n\\begin{acknowledgments} \nThe work was supported by the the National Science Centre Poland grant 2018\\slash 30\\slash A\\slash ST2\\slash 00226 and the German Research Foundation grant GA1480\\slash 8-1. The work of M.I.G. is supported by the Program of Fundamental Research of the Department of Physics and Astronomy of National Academy of Sciences of Ukraine.\n\\end{acknowledgments}\n\n\\bibliographystyle{ieeetr}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nWhile data-driven methods for system analysis and control have become increasingly popular over the recent years, only few such methods give theoretical guarantees on, e.g., stability or constraint satisfaction of system variables~\\cite{Hou13,Recht18}.\nA control method, which is naturally well-suited for achieving these objectives is model predictive control (MPC), which can handle nonlinear system dynamics, hard constraints on input, state and output, and it takes performance criteria into account~\\cite{Rawlings09}.\nIt centers around the repeated online solution of an optimization problem over predicted future system trajectories.\nThus, for the implementation of MPC, a model of the plant is required, which is usually obtained from first principles or from measured data via system identification~\\cite{Ljung87}.\nAn appealing alternative is to implement an MPC controller directly from measured data, without prior knowledge of an accurate model.\nIn various recent works, learning-based or adaptive MPC schemes have been proposed, which improve an inaccurate initial model using online measurements~\\cite{adetola2011robust,Aswani13,tanaskovic2014adaptive,Berkenkamp17,Zanon19}, while giving guarantees on the resulting closed loop.\nSimilarly, MPC based on Gaussian Processes has received increasing attraction~\\cite{Hewing18}, but proving desirable closed-loop properties remains an open issue.\nA different approach, which uses linear combinations of past trajectories to predict future trajectories, has been presented in~\\cite{salvador2018data}, but also no guarantees on, e.g., stability of the closed loop were given.\nThe design of purely data-driven MPC approaches with guarantees on stability and constraint satisfaction thus remains an open problem.\n\nIn this paper, we present a novel data-driven MPC scheme to control linear time-invariant (LTI) systems with stability and robustness guarantees for the closed loop.\nOur approach relies on a result from behavioral systems theory, which shows that the Hankel matrix consisting of a previously measured input-output trajectory spans the vector space of all trajectories of an LTI system, given that the input component is persistently exciting~\\cite{Willems05}.\nAlthough this result has found various applications in the field of system identification~\\cite{Katayama05,Markovsky05,Markovsky08}, it has only recently been used to develop data-driven methods for system analysis and control with theoretical guarantees.\nAn exposition of the main result of~\\cite{Willems05} in the classical state-space control framework and an extension to certain classes of nonlinear systems are provided in~\\cite{Berberich19}.\nFurther, the result is employed in~\\cite{Persis19} to design state- and output-feedback controllers and in~\\cite{Romer19} to verify dissipation inequalities from measured data, whereas~\\cite{Waarde19} investigates data-driven control without requiring persistently exciting data.\n\nMoreover, the recent contributions~\\cite{Yang15,Coulson19,Coulson19b} set up an MPC scheme based on~\\cite{Willems05}, but no guarantees on recursive feasibility or closed-loop stability can be given since neither terminal ingredients are included in the MPC scheme nor sufficient lower bounds on the prediction horizon are derived.\nIn the present paper, we propose a related MPC scheme, which utilizes terminal equality constraints, and we provide a theoretical analysis of various desirable properties of the closed loop.\nTo the best of our knowledge, this is the first analysis regarding recursive feasibility and stability\nof purely data-driven MPC.\nThe main advantage of the proposed MPC scheme over existing adaptive or learning-based methods such as~\\cite{adetola2011robust,Aswani13,tanaskovic2014adaptive,Berkenkamp17,Zanon19} is that it requires only an initially measured, persistently exciting data trajectory as well as an upper bound on the system order, but no (set-based) model description and no online estimation process.\nMoreover, since it relies on the data-driven system description from~\\cite{Willems05}, the presented scheme is inherently an output-feedback MPC scheme and does not require online state measurements.\\newpage\n\nAfter stating the required definitions and existing results in Section~\\ref{sec:setting}, we expand the nominal MPC scheme of~\\cite{Yang15,Coulson19} by terminal equality constraints in Section~\\ref{sec:tec}.\nUnder the assumption that the output of the plant can be measured exactly, we prove recursive feasibility, constraint satisfaction, and exponential stability of the scheme.\nIn Section~\\ref{sec:robust}, we propose a robust data-driven MPC scheme to account for bounded additive noise in both the initial data for prediction as well as the online measurements.\nUnder suitable assumptions on the system and design parameters, we prove that the closed loop under application of the scheme in a multi-step fashion leads to a practically exponentially stable closed loop.\nIn Section~\\ref{sec:example}, we illustrate the advantages of the proposed scheme over the scheme without terminal constraints from~\\cite{Yang15,Coulson19,Coulson19b} by means of a numerical example.\nThe paper is concluded in Section~\\ref{sec:conclusion}.\n\n\\section{Preliminaries}\\label{sec:setting}\n\nLet $\\mathbb{I}_{[a,b]}$ denote the set of integers in the interval $[a,b]$.\nFor a vector $x$ and a positive definite matrix $P=P^\\top\\succ0$, we write $\\lVert x\\rVert_P=\\sqrt{x^\\top Px}$.\nFurther, we denote the minimal and maximal eigenvalue of $P$ by $\\lambda_{\\min}(P)$ and $\\lambda_{\\max}(P)$, respectively.\nFor two matrices $P_1=P_1^\\top,P_2=P_2^\\top$, we write $\\lambda_{\\min}(P_1,P_2)=\\min\\{\\lambda_{\\min}(P_1),\\lambda_{\\min}(P_2)\\}$, and similarly for $\\lambda_{\\max}(P_1,P_2)$.\nMoreover, $\\lVert x\\rVert_2$, $\\lVert x\\rVert_1$, and $\\lVert x\\rVert_\\infty$ denote the Euclidean, $\\ell_1$-, and $\\ell_\\infty$-norm of $x$, respectively.\nIf the argument is matrix-valued, then we mean the corresponding induced norm.\nFor $\\delta>0$, we define $\\mathbb{B}_\\delta=\\left\\{x\\in\\mathbb{R}^n\\mid \\lVert x\\rVert_2\\leq\\delta\\right\\}$.\nA sequence $\\{x_k\\}_{k=0}^{N-1}$ induces the Hankel matrix\n\\begin{align*}\nH_L&(x)\\coloneqq\\begin{bmatrix}x_0 & x_1& \\dots & x_{N-L}\\\\\nx_1 & x_2 & \\dots & x_{N-L+1}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\nx_{L-1} & x_{L} & \\dots & x_{N-1}\n\\end{bmatrix}.\n\\end{align*}\nFor a stacked window of the sequence, we write\n\\begin{align*}\nx_{[a,b]}=\\begin{bmatrix}x_a\\\\\\vdots\\\\x_b\\end{bmatrix}.\n\\end{align*}\nWe denote by $x$ either the sequence itself or the stacked vector $x_{[0,N-1]}$ containing all of its components.\nWe consider the following standard definition of persistence of excitation.\n\n\\begin{definition}\nWe say that a sequence $\\{u_k\\}_{k=0}^{N-1}$ with $u_k\\in\\mathbb{R}^m$ is persistently exciting of order $L$ if $\\text{rank}(H_L(u))=mL$.\n\\end{definition}\n\nOur goal is to control an unknown LTI system, denoted by $G$, of order $n$ with $m$ inputs and $p$ outputs, using only measured input-output data.\n\n\\begin{definition}\\label{def:trajectory_of}\nWe say that an input-output sequence $\\{u_k,y_k\\}_{k=0}^{N-1}$ is a trajectory of an LTI system $G$, if there exists an initial condition $\\bar{x}\\in\\mathbb{R}^n$ as well as a state sequence $\\{x_k\\}_{k=0}^{N}$ such that\n\\begin{align*}\nx_{k+1}&=Ax_k+Bu_k,\\>\\>x_0=\\bar{x},\\\\\ny_k&=Cx_k+Du_k,\n\\end{align*}\nfor $k=0,\\dots,N-1$, where $(A,B,C,D)$ is a minimal realization of $G$.\n\\end{definition}\n\nNote that we define a trajectory of an LTI system as an input-output sequence that can be produced by a minimal realization, entailing controllability and observability of the system.\nExtending the results of this paper to systems whose input-output behavior cannot be explained via a minimal realization is an interesting issue for future research.\nThe following result lays the foundation of the present paper.\nIt shows that a Hankel matrix, involving a single persistently exciting trajectory, spans the vector space of all system trajectories of an LTI system.\nThe result originates from behavioral systems theory~\\cite{Willems05}, but we employ the formulation in the classical state-space control framework~\\cite{Berberich19}.\n\n\\begin{theorem}[\\cite{Berberich19}]\\label{thm:traj_rep}\nSuppose $\\{u_k^d,y_k^d\\}_{k=0}^{N-1}$ is a trajectory of an LTI system $G$, where $u^d$ is persistently exciting of order $L+n$.\nThen, $\\{\\bar{u}_k,\\bar{y}_k\\}_{k=0}^{L-1}$ is a trajectory of $G$ if and only if there exists $\\alpha\\in\\mathbb{R}^{N-L+1}$ such that\n\\begin{align}\\label{eq:thm_hankel}\n\\begin{bmatrix}H_L(u^d)\\\\H_L(y^d)\\end{bmatrix}\\alpha\n=\\begin{bmatrix}\\bar{u}\\\\\\bar{y}\\end{bmatrix}.\n\\end{align}\n\\end{theorem}\n\nRecently, Theorem~\\ref{thm:traj_rep} has received increasing attention to develop data-driven controllers~\\cite{Persis19}, verify dissipativity~\\cite{Romer19}, or to design MPC schemes~\\cite{Yang15,Coulson19,Coulson19b}.\nThis is due to the fact that~\\eqref{eq:thm_hankel} provides an appealing data-driven characterization of all trajectories of the unknown LTI system, without requiring any prior identification step.\nIn this paper, we use Theorem~\\ref{thm:traj_rep} to develop a data-driven MPC scheme with \\emph{provable}\nstability guarantees despite noisy measurements.\nNote that, if a sequence is persistently exciting of order $L$, then it is also persistently exciting of order $\\tilde{L}$ for any $\\tilde{L}\\leq L$.\nTherefore, Theorem~\\ref{thm:traj_rep} and hence all of our results hold true if $n$ is replaced by a (potentially rough) upper bound.\n\nAlthough we assume that only input-output data of the unknown system are available, we make extensive use of the fact that an input-output trajectory of length greater than or equal to $n$ induces a unique internal state in some minimal realization of the unknown system.\n\\begin{comment}\nThe following result shows that input and initial conditions suffice to fix an $\\alpha$, which predicts the corresponding output.\nThe following result is from data-driven simulation~\\cite{Markovsky08,Berberich19} and slightly reformulated.\n\\begin{proposition}\\label{prop:DD_sim}\nSuppose $\\{u_k^d,y_k^d\\}_{k=0}^{N-1}$ is a trajectory of an LTI system $G$, where $u$ is persistently exciting of order $L+n$.\nLet $\\{\\bar{u}_k,\\bar{y}_k\\}_{k=0}^{L-1}$ be an arbitrary trajectory of $G$.\nDenote by $\\{x_k^d\\}_{k=0}^{N-1}$ and $\\{\\bar{x}_k\\}_{k=0}^{L-1}$ the corresponding state trajectories in some minimal realization.\nIf $\\nu\\geq n$, then there exists an $\\alpha\\in\\mathbb{R}^{N-L+1}$ such that\n\\begin{align}\\label{eq:prop_DD_sim}\n\\begin{bmatrix}\nH_L\\left(u^d\\right)\\\\H_1\\left(x^d_{[0,N-L]}\\right)\n\\end{bmatrix}\\alpha=\n\\begin{bmatrix}\n\\bar{u}\\\\\\bar{x}_0\n\\end{bmatrix}.\n\\end{align}\nFurther, it holds that $\\bar{y}=H_L(y^d)\\alpha$.\n\\end{proposition}\n\\begin{proof}\nFollows directly from~\\cite[Proposition 1]{Markovsky08}.\n\\end{proof}\nSince the matrix on the left-hand-side of~\\eqref{eq:prop_DD_sim} will be essential for many of our arguments, we abbreviate it as \n\\begin{align}\\label{eq:Hux}\nH_{ux}\\coloneqq\\begin{bmatrix}\nH_L\\left(u^d\\right)\\\\H_1\\left(x^d_{[0,N-L]}\\right)\n\\end{bmatrix}.\n\\end{align}\nA simple choice to satisfy~\\eqref{eq:prop_DD_sim} is $\\alpha=H_{ux}^\\dagger\\begin{bmatrix}\\bar{u}\\\\\\bar{x}_0\\end{bmatrix}$, where $H_{ux}^\\dagger$ denotes the right-inverse of $H_{ux}$.y\n\\end{comment}\nWe employ MPC to stabilize a desired equilibrium of the system.\nSince a model of this system is not available, we define an equilibrium via input-output pairs.\n\n\\begin{definition}\\label{def:equil}\nWe say that an input-output pair $(u^s,y^s)\\in\\mathbb{R}^{m+p}$ is an equilibrium of an LTI system $G$, if the sequence $\\{\\bar{u}_k,\\bar{y}_k\\}_{k=0}^{n}$ with $(\\bar{u}_k,\\bar{y}_k)=(u^s,y^s)$ for all $k\\in\\mathbb{I}_{[0,n]}$ is a trajectory of $G$.\n\\end{definition}\n\nFor an equilibrium $(u^s,y^s)$, we define $u^s_n$ and $y^s_n$ as the column vectors containing $n$ times $u^s$ and $y^s$, respectively.\nWe assume that the system is subject to pointwise-in-time input and output constraints, i.e., $u_t\\in\\mathbb{U}\\subseteq\\mathbb{R}^m$, $y_t\\in\\mathbb{Y}\\subseteq\\mathbb{R}^p$ for all $t\\geq0$, and we assume $(u^s,y^s)\\in\\text{int}(\\mathbb{U}\\times\\mathbb{Y})$.\nThroughout this paper, $\\left\\{u_k^d,y_k^d\\right\\}_{k=0}^{N-1}$ denotes an a priori measured data trajectory of length $N$, which is used for prediction as in~\\eqref{eq:thm_hankel}.\nThe predicted input- and output-trajectories at time $t$ over some prediction horizon $L$ are written as $\\left\\{\\bar{u}_k(t),\\bar{y}_k(t)\\right\\}_{k=-n}^{L-1}$.\nNote that the time indices start at $k=-n$, since the last $n$ inputs and outputs will be used to invoke a unique initial state at time $t$.\nFurther, the closed-loop input, the state in some minimal realization, and the output at time $t$ are denoted by $u_t$, $x_t$, and $y_t$, respectively.\n\n\\section{Nominal data-driven MPC}\\label{sec:tec}\nIn this section, we propose a simple, nominal data-driven MPC scheme with terminal equality constraints. \nThe scheme relies on noise-free measurements to predict future trajectories using Theorem~\\ref{thm:traj_rep} and is described in Section~\\ref{sec:tec_scheme}.\nUnder mild assumptions, we prove recursive feasibility, constraint satisfaction, and exponential stability of the closed loop in Section~\\ref{sec:stab}.\n \n\\subsection{Nominal MPC scheme}\\label{sec:tec_scheme}\nCommonly, MPC relies on a model of the plant to predict future trajectories and to optimize over them.\nTheorem~\\ref{thm:traj_rep} provides an appealing alternative to a model since~\\eqref{eq:thm_hankel} suffices to capture all system trajectories.\nThus, to implement a data-driven MPC scheme, one can simply replace the system dynamics constraint by the constraint that the predicted input-output trajectories satisfy~\\eqref{eq:thm_hankel}.\nTo be more precise, the proposed data-driven MPC scheme minimizes, at time $t$, given the last $n$ input-output pairs, the following open-loop cost\\\\\n\\begin{subequations}\n\\begin{align}\nJ_L(u_{[t-n,t-1]},y_{[t-n,t-1]},&\\alpha(t))=\\sum_{k=0}^{L-1}\\ell\\left(\\bar{u}_k(t),\\bar{y}_k(t)\\right),\\\\\\label{eq:MPC_modela}\n\\begin{bmatrix}\\bar{u}_{[-n,L-1]}(t)\\\\\\bar{y}_{[-n,L-1]}(t)\\end{bmatrix}&=\\begin{bmatrix}H_{L+n}(u^d)\\\\H_{L+n}(y^d)\\end{bmatrix}\\alpha(t),\\\\\n\\label{eq:MPC_initial_conditionsa}\n\\begin{bmatrix}\\bar{u}_{[-n,-1]}(t)\\\\\\bar{y}_{[-n,-1]}(t)\\end{bmatrix}&=\\begin{bmatrix}u_{[t-n,t-1]}\\\\y_{[t-n,t-1]}\\end{bmatrix}.\n\\end{align}\n\\end{subequations}\nAs described above, the constraint~\\eqref{eq:MPC_modela} replaces the system dynamics compared to classical model-based MPC schemes.\nFurther,~\\eqref{eq:MPC_initial_conditionsa} ensures that the internal state of the true trajectory aligns with the internal state of the predicted trajectory at time $t$.\nNote that the overall length of the trajectory $(\\bar{u}(t),\\bar{y}(t))$ is $L+n$ since the past $n$ elements $\\{\\bar{u}_k(t),\\bar{y}_k(t)\\}_{k=-n}^{-1}$ are used to specify the initial conditions in~\\eqref{eq:MPC_initial_conditionsa}.\nThese initial conditions are specified until time step $t-1$, since the input at time $t$ might already influence the output at time $t$, in case of a feedthrough-element of the plant.\nThe open-loop cost depends only on the decision variable $\\alpha(t)$, since $\\bar{u}(t)$ and $\\bar{y}(t)$ are fixed implicitly through the dynamic constraint~\\eqref{eq:MPC_modela}.\nThroughout the paper, we consider quadratic stage costs, which penalize the distance w.r.t. a desired equilibrium $(u^s,y^s)$, i.e.,\n\\begin{align*}\n\\ell(\\bar{u},\\bar{y})=\\lVert\\bar{u}-u^s\\rVert_R^2+\\lVert\\bar{y}-y^s\\rVert_Q^2,\n\\end{align*}\nwhere $Q,R\\succ0$.\nIn~\\cite{Yang15,Coulson19}, it was suggested to directly minimize the above open-loop cost subject to constraints on input and output.\nIt is well-known that MPC without terminal constraints requires a sufficiently long prediction horizon to ensure stability and constraint satisfaction~\\cite{grune2017nonlinear,grune2012nmpc}.\nWithout such an assumption, the application of MPC can even destabilize an open-loop stable system.\nThere are two main approaches in the literature to guarantee stability:\na) providing bounds on the minimal required prediction horizon~\\cite{grune2012nmpc} and b) including terminal ingredients such as terminal cost functions or terminal region constraints~\\cite{Mayne00}.\nBoth approaches are usually based on model knowledge and thus, it is not straightforward to use them in the present, purely data-driven setting.\n\nIn this paper, we consider a simple terminal equality constraint, which can be directly included into the data-driven MPC framework, and which guarantees exponential stability of the closed loop.\nTo this end, we propose the following data-driven MPC scheme with a terminal equality constraint.\n\\begin{subequations}\\label{eq:term_eq_MPC}\n\\begin{align}\\nonumber\nJ_L^*(u_{[t-n,t-1]}&,y_{[t-n,t-1]})=\\\\\n\\underset{\\substack{\\alpha(t)\\\\\\bar{u}(t),\\bar{y}(t)}}{\\min}\\>\\>&\\sum_{k=0}^{L-1}\\ell\\left(\\bar{u}_k(t),\\bar{y}_k(t)\\right)\\\\\\label{eq:MPC_model}\ns.t.&\\begin{bmatrix}\\bar{u}_{[-n,L-1]}(t)\\\\\\bar{y}_{[-n,L-1]}(t)\\end{bmatrix}=\\begin{bmatrix}H_{L+n}(u^d)\\\\H_{L+n}(y^d)\\end{bmatrix}\\alpha(t),\\\\\n\\label{eq:MPC_initial_conditions}\n&\\begin{bmatrix}\\bar{u}_{[-n,-1]}(t)\\\\\\bar{y}_{[-n,-1]}(t)\\end{bmatrix}=\\begin{bmatrix}u_{[t-n,t-1]}\\\\y_{[t-n,t-1]}\\end{bmatrix},\\\\\\label{eq:term_eq_MPC2}\n&\\begin{bmatrix}\\bar{u}_{[L-n,L-1]}(t)\\\\\\bar{y}_{[L-n,L-1]}(t)\\end{bmatrix}=\\begin{bmatrix}u^s_n\\\\y^s_n\\end{bmatrix},\\\\\\label{eq:term_eq_MPC3}\n&\\bar{u}_k(t)\\in\\mathbb{U},\\>\\>\\bar{y}_k(t)\\in\\mathbb{Y},\\>\\>k\\in\\mathbb{I}_{[0,L-1]}.\n\\end{align}\n\\end{subequations}\nThe terminal equality constraint~\\eqref{eq:term_eq_MPC2} implies that $\\bar{x}_L(t)$, which is the internal state predicted $L$ steps ahead corresponding to the predicted input-output trajectory, aligns with the steady-state $x^s$ corresponding to $(u^s,y^s)$, i.e., $\\bar{x}_L(t)=x^s$ in any minimal realization.\nWhile Problem~\\eqref{eq:term_eq_MPC} requires that $(u^s,y^s)$ is an equilibrium of the unknown system in the sense of Definition~\\ref{def:equil}, this requirement can be dropped when $(u^s,y^s)$ is replaced by an artificial equilibrium, which is also optimized online (compare~\\cite{limon2008mpc}).\nThe recent paper~\\cite{berberich2020tracking} extends the above MPC scheme to such a setting, thereby leading to a significantly larger region of attraction for the closed loop without requiring knowledge of a reachable equilibrium of the unknown system.\nAs in standard MPC, Problem~\\eqref{eq:term_eq_MPC} is solved in a receding horizon fashion, which is summarized in Algorithm~\\ref{alg:MPC}.\n\n\\begin{algorithm}\n\\begin{Algorithm}\\label{alg:MPC}\n\\normalfont{\\textbf{Data-Driven MPC Scheme}}\n\\begin{enumerate}\n\\item At time $t$, take the past $n$ measurements $u_{[t-n,t-1]}$, $y_{[t-n,t-1]}$ and solve~\\eqref{eq:term_eq_MPC}.\n\\item Apply the input $u_t=\\bar{u}_0^*(t)$.\n\\item Set $t=t+1$ and go back to 1).\n\\end{enumerate}\n\\end{Algorithm}\n\\end{algorithm}\n\nWith slight abuse of notation, we will denote the open-loop cost and the optimal open-loop cost of~\\eqref{eq:term_eq_MPC} by $J_L(x_t,\\alpha(t))$ and $J_L^*(x_t)$, respectively,\nwhere $x_t$ is the state in some minimal realization, induced by $u_{[t-n,t-1]}$, $y_{[t-n,t-1]}$.\n\n\\subsection{Closed-loop guarantees}\\label{sec:stab}\nWithout loss of generality, we assume for the analysis that $u^s=0$, $y^s=0$, and thus $x^s=0$.\nFurther, we define the set of initial states, for which~\\eqref{eq:term_eq_MPC} is feasible, by $\\mathbb{X}_L=\\left\\{x\\in\\mathbb{R}^n\\mid J_L^*(x)<\\infty\\right\\}$.\nTo prove exponential stability of the proposed scheme, we assume that the optimal value function of~\\eqref{eq:term_eq_MPC} is quadratically upper bounded.\nThis is, e.g., satisfied in the present linear-quadratic setting if the constraints are polytopic\\footnote{While~\\cite{Bemporad02} considered model-based linear-quadratic MPC, the result applies similarly to the present data-driven MPC setting since~\\eqref{eq:MPC_model} (together with the initial conditions~\\eqref{eq:MPC_initial_conditions}) describes the input-output behavior of the system exactly and thus, both settings are equivalent in the nominal case.}~\\cite{Bemporad02}.\n\\begin{assumption}\\label{ass:quad_upper_bound}\nThe optimal value function $J_L^*(x)$ is quadratically upper bounded on $\\mathbb{X}_L$, i.e., there exists $c_u>0$ such that $J_L^*(x)\\leq c_u\\lVert x\\rVert_2^2$ for all $x\\in\\mathbb{X}_L$.\n\\end{assumption}\n\nMoreover, we assume that the input $u^d$ generating the data used for prediction is sufficiently rich in the following sense.\n\n\\begin{assumption}\\label{ass:pe}\nThe input $u^d$ of the data trajectory is persistently exciting of order $L+2n$.\n\\end{assumption}\n\nNote that we assume persistence of excitation of order $L+2n$, although Theorem~\\ref{thm:traj_rep} requires only an order of $L+n$.\nThis is due to the fact that the reconstructed trajectories in~\\eqref{eq:term_eq_MPC} are of length $L+n$ (compared to length $L$ in Theorem~\\ref{thm:traj_rep}), since $n$ components are used to fix the initial conditions.\nFurthermore, due to the terminal constraints~\\eqref{eq:term_eq_MPC2}, the prediction horizon needs to be at least as long as the system order $n$.\n\n\\begin{assumption}\\label{ass:length_n}\nThe prediction horizon satisfies $L\\geq n$.\n\\end{assumption}\n\nThe following result shows that the MPC scheme based on~\\eqref{eq:term_eq_MPC} is recursively feasible, ensures constraint satisfaction, and leads to an exponentially stable closed loop.\n\n\\begin{theorem}\\label{thm:mpc_tec}\nSuppose Assumptions~\\ref{ass:quad_upper_bound},~\\ref{ass:pe} and~\\ref{ass:length_n} are satisfied.\nIf the MPC problem~\\eqref{eq:term_eq_MPC} is feasible at initial time $t=0$, then \n\\begin{itemize}\n\\item[(i)] it is feasible at any $t\\in\\mathbb{N}$,\n\\item[(ii)] the closed loop satisfies the constraints, i.e., $u_t\\in\\mathbb{U}$ and $y_t\\in\\mathbb{Y}$ for all $t\\in\\mathbb{N}$,\n\\item[(iii)] the equilibrium $x^s=0$ is exponentially stable for the resulting closed loop.\n\\end{itemize}\n\\end{theorem}\n\\begin{proof}\nRecursive feasibility (i) and constraint satisfaction (ii) follow from standard MPC arguments, i.e., by defining a candidate solution as the shifted, previously optimal solution and appending zero (compare~\\cite{Rawlings09}).\\\\\n\\textbf{(iii). Exponential Stability}\\\\\nDenote the standard candidate solution mentioned above by $\\bar{u}'(t+1),\\bar{y}'(t+1),\\alpha'(t+1)$.\nThe cost of this solution is\n\\begin{align*}\n&J_L(x_{t+1},\\alpha'(t+1))\\\\\n&=\\sum_{k=0}^{L-1}\\ell\\left(\\bar{u}_k'(t+1),\\bar{y}_k'(t+1)\\right)\n=\\sum_{k=1}^{L-1}\\ell\\left(\\bar{u}^*_k(t),\\bar{y}^*_k(t)\\right)\\\\\n&=J_L^*(x_t)-\\ell\\left(\\bar{u}_0^*(t),\\bar{y}_0^*(t)\\right).\n\\end{align*}\nHence, it holds that\n\\begin{align}\\label{eq:thm_eq0_proof1}\nJ_L^*(x_{t+1})\\leq J_L^*(x_t)-\\ell\\left(\\bar{u}_0^*(t),\\bar{y}_0^*(t)\\right).\n\\end{align}\nSince $x$ is the state of an observable (and hence detectable) minimal realization, there exists a matrix $P\\succ0$ such that $W(x)=\\lVert x\\rVert_P^2$ is an input-output-to-state stability (IOSS) Lyapunov function\\footnote{Note that, in~\\cite[Section 3.2]{cai2008input}, only strictly proper systems with $y=Cx$ are considered, while we allow for more general systems with $y=Cx+Du$.\nThe result from~\\cite{cai2008input} can be extended to $y=Cx+Du$ by considering a modified $\\tilde{B}=B+LD$ in~\\cite[Inequality (12)]{cai2008input}.}, which satisfies\n\\begin{align}\\label{eq:thm_eq0_proof2}\nW(Ax+Bu)-W(x)\\leq-\\frac{1}{2}\\lVert x\\rVert_2^2+c_1\\lVert u\\rVert_2^2 +c_2\\lVert y\\rVert_2^2,\n\\end{align}\nfor all $x\\in\\mathbb{R}^n,u\\in\\mathbb{R}^m,y=Cx+Du$, and for suitable $c_1,c_2>0$~\\cite{cai2008input}.\nDefine the candidate Lyapunov function $V(x)=\\gamma W(x)+J_L^*(x)$ for some $\\gamma>0$.\nNote that $V$ is quadratically lower bounded, i.e., $V(x)\\geq\\gamma W(x)\\geq\\gamma\\lambda_{\\min}(P)\\lVert x\\rVert_2^2$ for all $x\\in\\mathbb{X}_L$.\nFurther, $J_L^*$ is quadratically upper bounded by Assumption~\\ref{ass:quad_upper_bound}, i.e., $J_L^*(x)\\leq c_u\\lVert x\\rVert_2^2$ for all $x\\in\\mathbb{X}_L$.\nHence, we have\n\\begin{align*}\nV(x)=J_L^*(x)+\\gamma W(x)\\leq\\left(c_u+\\gamma\\lambda_{\\max}(P)\\right)\\lVert x\\rVert_2^2,\n\\end{align*}\nfor all $x\\in\\mathbb{X}_L$, i.e., $V$ is quadratically upper bounded.\nWe consider now\n\\begin{align*}\n\\gamma=\\frac{\\lambda_{\\min}(Q,R)}{\\max\\{c_1,c_2\\}}>0.\n\\end{align*}\nAlong the closed-loop trajectories, using both~\\eqref{eq:thm_eq0_proof1} as well as~\\eqref{eq:thm_eq0_proof2}, it holds that\n\\begin{align*}\nV(x_{t+1})-V(x_t)\\leq&\\>\\gamma\\left(-\\frac{1}{2}\\lVert x_t\\rVert^2_2+c_1\\lVert u_t\\rVert^2_2 +c_2\\lVert y_t\\rVert^2_2\\right)\\\\\n&-\\lVert u_t\\rVert_R^2-\\lVert y_t\\rVert_Q^2\\\\\n\\leq&-\\frac{\\gamma}{2}\\lVert x_t\\rVert^2_2.\n\\end{align*}\nIt follows from standard Lyapunov arguments with Lyapunov function $V$ that the equilibrium $x^s=0$ is exponentially stable with region of attraction $\\mathbb{X}_L$.\n\\end{proof}\n\nThe proof of Theorem~\\ref{thm:mpc_tec} applies standard arguments from model-based MPC with terminal constraints (compare~\\cite{Rawlings09}) to the data-driven system description derived in~\\cite{Willems05}, similar to the approaches of~\\cite{Yang15,Coulson19} which did however not address closed-loop guarantees. \nTo handle the fact that the stage cost $\\ell$ is merely positive \\emph{semi}-definite in the state, detectability of the stage cost is exploited via an IOSS Lyapunov function~\\cite{cai2008input}, similar to~\\cite{grimm2005model}.\nAs we will see in Section~\\ref{sec:robust}, this analogy between model-based MPC and the proposed data-driven MPC scheme is only present in the nominal case, where the data is noise-free.\nFor the more realistic case of noisy output measurements, we develop a robust data-driven MPC scheme and we provide a novel theoretical analysis of the closed loop in Section~\\ref{sec:robust}, which is the main contribution of this paper.\n\n\\begin{remark}\\label{rk:complexity_nominal}\nWe would like to emphasize the simplicity of the proposed MPC scheme.\nWithout any prior identification step, a single measured data trajectory can be used directly to set up an MPC scheme for a linear system.\nCompared to other learning-based MPC approaches such as~\\cite{adetola2011robust,Aswani13,tanaskovic2014adaptive,Berkenkamp17,Zanon19}, which require initial model knowledge as well as an online estimation process, the complexity of~\\eqref{eq:term_eq_MPC} is similar to classical MPC schemes, which rely on full model knowledge.\nTo be more precise, the decision variables $\\bar{u}(t),\\bar{y}(t)$ can be replaced by $\\alpha(t)$ via~\\eqref{eq:MPC_model} (using a condensed formulation) and hence, since $\\alpha(t)\\in\\mathbb{R}^{N-L-n+1}$, Problem~\\eqref{eq:term_eq_MPC} contains in total $N-L-n+1$ decision variables.\nFor $u^d$ to be persistently exciting of order $L+2n$, it needs to hold that $N-L-2n+1\\geq m(L+2n)$.\nAssuming equality, Problem~\\eqref{eq:term_eq_MPC} hence has $m(L+2n)+n$ free parameters.\nOn the contrary, a condensed model-based MPC optimization problem contains $mL$ decision variables for the input trajectory (assuming that state measurements are available).\nThus, the online complexity of the proposed data-driven MPC approach is slightly larger ($2mn+n$ additional decision variables) than that of model-based MPC, but it does not require an a priori (offline) identification step.\nIt is worth noting that the difference in complexity is independent of the horizon $L$.\nMoreover, the proposed data-driven MPC is inherently an output-feedback controller since no state measurements are required for its implementation.\nFinally, as in model-based MPC, for convex polytopic (or quadratic) constraints $\\mathbb{U},\\mathbb{Y}$,~\\eqref{eq:term_eq_MPC} is a convex (quadratically constrained) quadratic program which can be solved efficiently.\n\\end{remark}\n\\section{Robust data-driven MPC}\\label{sec:robust}\nIn this section, we propose a multi-step robust data-driven MPC scheme and we prove practical exponential stability of the closed loop in the presence of bounded additive output measurement noise.\nThe scheme includes a slack variable, which is regularized in the cost and compensates noise both in the initial data $(u^d,y^d)$ used for prediction and in the online measurement updates $\\left(u_{[t-n,t-1]},y_{[t-n,t-1]}\\right)$. \nSection~\\ref{sec:scheme} contains the scheme, which is essentially a robust modification of the nominal scheme of Section~\\ref{sec:tec}, as well as detailed explanations of the key ingredients.\nIn Sections~\\ref{sec:Lyapunov_bound} and~\\ref{sec:prediction_error}, we prove two technical Lemmas, which will be required for our main theoretical results.\nRecursive feasibility of the closed loop is proven in Section~\\ref{sec:robust_feasibility}.\nIn Section~\\ref{sec:robust_stability}, we show that, under suitable assumptions, the closed loop resulting from the application of the multi-step MPC scheme leads to a practically exponentially stable closed loop.\nMoreover, if the noise bound tends to zero, then the region of attraction of the closed loop approaches the set of all initially feasible points.\nIn this section, we do not consider output constraints, i.e., $\\mathbb{Y}=\\mathbb{R}^p$.\nIn~\\cite{berberich2020constraints}, we recently extended the results of this section by incorporating tightened output constraints in order to guarantee closed-loop constraint satisfaction despite noisy data.\n\n\n\\subsection{Robust MPC scheme}\\label{sec:scheme}\n\nIn practice, the output of the unknown LTI system $G$ is usually not available exactly, but might be subject to measurement noise.\nThis implies that the stacked data-dependent Hankel matrices in~\\eqref{eq:thm_hankel} do not span the system's trajectory space exactly and thus, the output trajectories cannot be predicted accurately.\nMoreover, noisy output measurements enter the initial conditions in Problem~\\eqref{eq:term_eq_MPC}, which deteriorates the prediction accuracy even further.\nTherefore, a direct application of the MPC scheme of Section~\\ref{sec:tec} may lead to feasibility issues or it may render the closed loop unstable.\nIn this section, we tackle the issue of noisy measurements with a robust data-driven MPC scheme with terminal constraints.\nWe consider output measurements with bounded additive noise in the initially available data $\\tilde{y}_k^d=y_k^d+\\varepsilon_k^d$ as well as in the online measurements $\\tilde{y}_k=y_k+\\varepsilon_k$.\nWe make no assumptions on the nature of the noise, but we require that it is bounded as $\\lVert\\varepsilon_k^d\\rVert_{\\infty}\\leq\\bar{\\varepsilon}$ and $\\lVert\\varepsilon_k\\rVert_{\\infty}\\leq\\bar{\\varepsilon}$ for some $\\bar{\\varepsilon}>0$. \nThus, the present setting includes two types of noise.\nThe data used for the prediction via the Hankel matrices in~\\eqref{eq:thm_hankel} is perturbed by $\\varepsilon^d$, which can thus be interpreted as a multiplicative model uncertainty.\nOn the other hand, $\\varepsilon$ perturbs the online measurements and hence, the overall control goal is a noisy output-feedback problem.\n\nThe key idea to account for noisy measurements is to relax the equality constraint~\\eqref{eq:MPC_model}, where the relaxation parameter is penalized appropriately in the cost function.\nGiven a noisy initial input-output trajectory $\\left(u_{[t-n,t-1]},\\tilde{y}_{[t-n,t-1]}\\right)$ of length $n$, and noisy data $(u^d,\\tilde{y}^d)$, we propose the following robust modification of~\\eqref{eq:term_eq_MPC}.\n\\begin{subequations}\\label{eq:robust_MPC}\n\\begin{align}\\nonumber\nJ_L^*&\\big(u_{[t-n,t-1]},\\tilde{y}_{[t-n,t-1]}\\big)=\\\\\\nonumber\n\\underset{\\substack{\\alpha(t),\\sigma(t)\\\\\\bar{u}(t),\\bar{y}(t)}}{\\min}&\\sum_{k=0}^{L-1}\\ell\\left(\\bar{u}_k(t),\\bar{y}_k(t)\\right)+\\lambda_\\alpha\\bar{\\varepsilon}\\lVert\\alpha(t)\\rVert_2^2+\\lambda_\\sigma\\lVert\\sigma(t)\\rVert_2^2\\\\\n\\label{eq:robust_MPC1} s.t.\\>\\> &\\>\\begin{bmatrix}\n\\bar{u}(t)\\\\\\bar{y}(t)+\\sigma(t)\\end{bmatrix}=\\begin{bmatrix}H_{L+n}\\left(u^d\\right)\\\\H_{L+n}\\left(\\tilde{y}^d\\right)\\end{bmatrix}\\alpha(t),\\\\\\label{eq:robust_MPC2}\n&\\>\\begin{bmatrix}\\bar{u}_{[-n,-1]}(t)\\\\\\bar{y}_{[-n,-1]}(t)\\end{bmatrix}=\\begin{bmatrix}u_{[t-n,t-1]}\\\\\\tilde{y}_{[t-n,t-1]}\\end{bmatrix},\\\\\\label{eq:robust_MPC3}\n&\\>\\begin{bmatrix}\\bar{u}_{[L-n,L-1]}(t)\\\\\\bar{y}_{[L-n,L-1]}(t)\\end{bmatrix}=\\begin{bmatrix}u^s_n\\\\y^s_n\\end{bmatrix},\\>\\>\\bar{u}_k(t)\\in\\mathbb{U},\\\\\n\\label{eq:robust_MPC5}\n&\\>\\lVert\\sigma_k(t)\\rVert_\\infty\\leq\\bar{\\varepsilon}\\left(1+\\lVert\\alpha(t)\\rVert_1\\right),\\>\\>k\\in\\mathbb{I}_{[0,L-1]}.\n\\end{align}\n\\end{subequations}\nCompared to the nominal MPC problem~\\eqref{eq:term_eq_MPC}, the output data trajectory $\\tilde{y}^d$ as well as the initial output $\\tilde{y}_{[t-n,t-1]}$, which is obtained via online measurements, have been replaced by their noisy counterparts.\nFurther, the following ingredients have been added:\n\\begin{itemize}\n\\item[a)] A slack variable $\\sigma$, bounded by~\\eqref{eq:robust_MPC5}, to account for the noisy online measurements $\\tilde{y}_{[t-n,t-1]}$ and for the noisy data $\\tilde{y}^d$ used for prediction, which can be interpreted as a multiplicative model uncertainty,\n\\item[b)] Quadratic regularization (i.e., \\emph{ridge regularization}) of $\\alpha$ and $\\sigma$ with weights $\\lambda_\\alpha\\bar{\\varepsilon},\\lambda_\\sigma>0$, i.e., the regularization of $\\alpha$ depends on the noise level.\n\\end{itemize}\nThe above $\\ell_2$-norm regularization for $\\alpha(t)$ implies that small values of $\\lVert\\alpha(t)\\rVert_2^2$ are preferred.\nSince the noisy Hankel matrix $H_{L+n}\\left(\\tilde{y}^d\\right)$ is multiplied by $\\alpha(t)$ in~\\eqref{eq:robust_MPC1}, this implicitly reduces the influence of the noise on the prediction accuracy.\nIntuitively, for increasing $\\lambda_\\alpha$, the term $\\lambda_\\alpha\\bar{\\varepsilon}\\lVert\\alpha(t)\\rVert_2^2$ reduces the ``complexity'' of the data-driven system description~\\eqref{eq:robust_MPC1}, similar to regularization methods in linear regression, thus allowing for a tradeoff between tracking performance and the avoidance of overfitting.\nThe term $\\lambda_\\sigma\\lVert\\sigma(t)\\rVert_2^2$ yields small values for the slack variable $\\sigma(t)$, thus improving the prediction accuracy.\nFor our theoretical results, $\\lambda_\\sigma$ can be chosen to be zero since $\\sigma(t)$ is already rendered small by the constraint~\\eqref{eq:robust_MPC5}.\nHowever, as we discuss in more detail in Remark~\\ref{rk:sigma_bound}, the constraint~\\eqref{eq:robust_MPC5} is non-convex but can be neglected if $\\lambda_\\sigma$ is large enough.\n\nAn alternative to the present regularization terms are general quadratic regularization kernels, i.e., costs of the form $\\lVert\\alpha(t)\\rVert_{P_\\alpha}^2$, $\\lVert\\sigma(t)\\rVert_{P_\\sigma}^2$ for suitable matrices $P_\\alpha,P_\\sigma\\succ0$.\nFurther, in~\\cite{Coulson19,Coulson19b}, $\\ell_1$-regularizations of $\\alpha$ and $\\sigma$ were suggested and the resulting MPC scheme, without terminal equality constraints, was successfully applied to a nonlinear stochastic control problem.\nHowever, theoretical guarantees on closed-loop stability were not given.\nThroughout this paper, we consider simple quadratic penalty terms since this simplifies the arguments, but we conjecture that our theoretical results remain to hold for general norms $\\lVert\\alpha(t)\\rVert_p,\\lVert\\sigma(t)\\rVert_q$ with arbitrary $p,q=1,\\dots,\\infty$.\nAn interesting open question, which is beyond the scope of this paper, is to investigate the impact of particular choices of regularization norms on the practical performance of the presented MPC approach.\nThe choice of norms in the constraint~\\eqref{eq:robust_MPC5} is independent of the norms in the cost and essentially follows from the $\\ell_\\infty$-noise bound and the proofs of the value function upper bound (Lemma~\\ref{lem:value_fcn_upper_bound}) and recursive feasibility (Proposition~\\ref{prop:robust_rec_feas}).\n\nIn this section, we study the closed loop resulting from an application of~\\eqref{eq:robust_MPC} in an $n$-step MPC scheme (compare~\\cite{Gruene15,Worthmann17}).\nTo be more precise, we consider the scenario that, after solving~\\eqref{eq:robust_MPC} online, the first $n$ computed inputs are applied to the system.\nThereafter, the horizon is shifted by $n$ steps, before the whole scheme is repeated (compare Algorithm~\\ref{alg:MPC_n_step}).\n\n\\begin{algorithm}\n\\begin{Algorithm}\\label{alg:MPC_n_step}\n\\normalfont{\\textbf{$n$-Step Data-Driven MPC Scheme}}\n\\begin{enumerate}\n\\item At time $t$, take the past $n$ measurements $u_{[t-n,t-1]}$, $\\tilde{y}_{[t-n,t-1]}$ and solve~\\eqref{eq:robust_MPC}.\n\\item Apply the input sequence $u_{[t,t+n-1]}=\\bar{u}_{[0,n-1]}^*(t)$ over the next $n$ time steps.\n\\item Set $t=t+n$ and go back to 1).\n\\end{enumerate}\n\\end{Algorithm}\n\\end{algorithm}\n\nAs we will see in the remainder of this section, for the considered setting with output measurement noise, the multi-step MPC scheme described in Algorithm~\\ref{alg:MPC_n_step} has superior theoretical properties compared to its corresponding $1$-step version.\nThis is mainly due to the terminal equality constraints~\\eqref{eq:robust_MPC3}, which complicate the proof of recursive feasibility, similar as in model-based robust MPC with terminal equality constraints and model mismatch.\nIn particular, we show in this section that, for an $n$-step MPC scheme with a terminal equality constraint, practical exponential stability can be proven.\nOn the other hand, we comment on the differences for the corresponding $1$-step MPC scheme in Section~\\ref{sec:robust_feasibility} (Remark~\\ref{rk:one_step}).\nIn particular, for a $1$-step MPC scheme relying on~\\eqref{eq:robust_MPC}, recursive feasibility holds only locally around $(u^s,y^s)$ and thus, only local stability can be guaranteed.\nNevertheless, as we will see in Section~\\ref{sec:example} for a numerical example, the practical performance of the $n$-step scheme is almost indistinguishable from the $1$-step scheme.\n\n\\begin{remark}\nIn the nominal case of Section~\\ref{sec:tec}, i.e., for $\\bar{\\varepsilon}=0$,~\\eqref{eq:robust_MPC5} implies $\\sigma=0$.\nFurther, the regularization of $\\alpha$ vanishes for $\\bar{\\varepsilon}=0$, and the system dynamics~\\eqref{eq:robust_MPC1} as well as the initial conditions~\\eqref{eq:robust_MPC2} approach their nominal counterparts.\nThus, for $\\bar{\\varepsilon}=0$, Problem~\\eqref{eq:robust_MPC} reduces to the nominal Problem~\\eqref{eq:term_eq_MPC}.\n\\end{remark}\n\n\\begin{remark}\\label{rk:sigma_bound}\nIf the constraint~\\eqref{eq:robust_MPC5} is neglected and the input constraint set $\\mathbb{U}$ is a convex polytope, then Problem~\\eqref{eq:robust_MPC} is a strictly convex quadratic program and can be solved efficiently.\nHowever, the constraint on the slack variable $\\sigma$ in~\\eqref{eq:robust_MPC5} is non-convex due to the dependence of the right-hand side on $\\lVert\\alpha(t)\\rVert_1$, making it difficult to implement~\\eqref{eq:robust_MPC} in an efficient way.\nAs will become clear later in this section,~\\eqref{eq:robust_MPC5} is required to prove recursive feasibility and practical exponential stability.\nIt may, however, be replaced by the (convex) constraint $\\lVert\\sigma_k(t)\\rVert_\\infty\\leq c\\cdot\\bar{\\varepsilon}$ for a sufficiently large constant $c>0$, retaining the same theoretical guarantees.\nGenerally, a larger choice of $c$ increases the region of attraction, but also the size of the exponentially stable set to which the closed loop converges.\nFurthermore, the constraint~\\eqref{eq:robust_MPC5} can be enforced implicitly by choosing $\\lambda_\\sigma$ large enough.\nIn simulation examples, it was observed that the constraint~\\eqref{eq:robust_MPC5} is usually satisfied (for suitably large choices of $\\lambda_\\sigma$) without enforcing it explicitly in the optimization problem and thus, it may in most cases be neglected in the online optimization.\n\\end{remark}\n\nAs in the previous section, we require that the measured input $u^d$ is persistently exciting of order $L+2n$ (Assumption~\\ref{ass:pe}).\nFurther, to establish a local upper bound on the optimal cost of~\\eqref{eq:robust_MPC} and to prove recursive feasibility, we require that the horizon $L$ is not shorter than twice the system's order, as captured in the following assumption.\n\n\\begin{assumption}\\label{ass:length_2n}\nThe prediction horizon satisfies $L\\geq2n$.\n\\end{assumption}\n\nIn some minimal realization, we denote the state trajectory corresponding to $(u^d,y^d)$ by $x^d$.\nAccording to~\\cite[Corollary 2]{Willems05}, Assumption~\\ref{ass:pe} implies that the matrix\n\\begin{align}\\label{eq:ass_pe_matrix}\nH_{ux}=\\begin{bmatrix}H_{L+n}\\left(u^d\\right)\\\\H_1\\left(x^d_{[0,N-L-n]}\\right)\\end{bmatrix}\n\\end{align}\nhas full row rank and thus admits a right-inverse $H_{ux}^\\dagger=H_{ux}^\\top\\left(H_{ux}H_{ux}^\\top\\right)^{-1}$.\nDefine the quantity\n\\begin{align}\nc_{pe}\\coloneqq\\left\\lVert H_{ux}^\\dagger\\right\\rVert_2^2.\n\\end{align}\nFor our stability results, we will require that $c_{pe}\\bar{\\varepsilon}$ is bounded from above by a sufficiently small number.\nEssentially, this corresponds to a quantitative ``persistence-of-excitation-to-noise''-bound.\nTo be more precise, abbreviate in the following $U=H_{L+n}(u^d)$ and suppose that\n\\begin{align}\\label{eq:ass_pe_quantitative}\n\\rho I_{m(L+n)}\\preceq UU^\\top\\preceq\\nu I_{m(L+n)}\n\\end{align}\nfor scalar constants $\\rho,\\nu>0$.\nFurther, define the quantity $c_{pe}^u=\\lVert U^\\dagger\\rVert_2^2=\\lVert U^\\top(UU^\\top)^{-1}\\rVert_2^2$.\nThen, it holds that\n\\begin{align}\\label{eq:ass_pe_bound}\nc_{pe}^u&\\leq \\left\\lVert U^\\top\\right\\rVert_2^2\\left\\lVert \\left(UU^\\top\\right)^{-1}\\right\\rVert_2^2\\\\\\nonumber\n&=\\lambda_{\\max}(UU^\\top)\\cdot\\lambda_{\\max}\\left((UU^\\top)^{-1}(UU^\\top)^{-1}\\right)\\\\\\nonumber\n&\\leq\\frac{\\lambda_{\\max}(UU^\\top)}{\\lambda_{\\min}(UU^\\top)^2} \\stackrel{\\eqref{eq:ass_pe_quantitative}}{\\leq}\\frac{\\nu}{\\rho^2}.\n\\end{align}\nThus, if a persistently exciting input $u^d$ is multiplied by a constant $c>1$, then $c_{pe}^u$ decreases proportionally to $\\frac{1}{c^2}$.\nFurther, the constant $\\rho$ can typically be chosen larger if the data length $N$ increases.\nThe same arguments can be carried out when assuming a bound of the form~\\eqref{eq:ass_pe_quantitative} for the matrix~\\eqref{eq:ass_pe_matrix}, but finding a suitable input which generates data achieving such a bound is less obvious.\nIt is well-known for classical definitions of persistence of excitation that larger excitation of the input implies larger excitation of the state.\nTherefore, we conjecture (and we have observed for various practical simulation examples) that $c_{pe}$ decreases with increasing data horizons $N$ and with multiplications of a persistently exciting input data trajectory $u^d$ by a scalar constant greater than one.\nThis means that, for a given noise level $\\bar{\\varepsilon}$, robust stability as guaranteed in the following sections can be obtained by choosing a large enough persistently exciting input $u^d$ and\/or a sufficiently large data horizon $N$.\n\nSimilar to Section~\\ref{sec:tec}, we denote the open-loop cost of the robust MPC problem~\\eqref{eq:robust_MPC} by $J_L\\left(u_{[t-n,t-1]},\\tilde{y}_{[t-n,t-1]},\\alpha(t),\\sigma(t)\\right)$, and the optimal cost by $J_L^*\\left(u_{[t-n,t-1]},\\tilde{y}_{[t-n,t-1]}\\right)$.\nMoreover, we assume for the analysis that $(u^s,y^s)=(0,0)$.\nFor the presented robust data-driven MPC scheme, setpoints $(u^s,y^s)\\neq (0,0)$ change mainly one quantitative constant in Lemma~\\ref{lem:value_fcn_upper_bound}.\nWe comment on the main differences in the case $(u^s,y^s)\\neq(0,0)$ in Section~\\ref{sec:robust_feasibility} (Remark~\\ref{rk:wlog_zero}).\n\n\\subsection{Local upper bound of Lyapunov function}\\label{sec:Lyapunov_bound}\n\nIn this section, we show that the optimal cost of~\\eqref{eq:robust_MPC} admits a quadratic upper bound, similar to the nominal case (cf. Assumption~\\ref{ass:quad_upper_bound}).\nIt is straightforward to see that such an upper bound can not be quadratic in the state $x$ of some minimal realization:\nthe optimal cost $J_L^*$ depends explicitly on $\\alpha^*(t)$ via $\\lambda_\\alpha\\bar{\\varepsilon}\\lVert\\alpha^*(t)\\rVert_2^2$, which in turn depends on the past $n$ inputs and outputs $(u_{[t-n,t-1]},y_{[t-n,t-1]})$ through~\\eqref{eq:robust_MPC1} and~\\eqref{eq:robust_MPC2}. \nEven if the current state is zero, i.e., $x_t=0$, these may in general be arbitrarily large and hence, $\\alpha$ and therefore also $J_L^*$ may be arbitrarily large.\nThus, $J_L^*$ does not admit an upper bound in the state $x_t$ of a minimal realization.\nTo overcome this issue, we consider a different (not minimal) state of the system, defined as\n\\begin{align*}\n\\xi_t\\coloneqq\\begin{bmatrix}u_{[t-n,t-1]}\\\\y_{[t-n,t-1]}\\end{bmatrix}.\n\\end{align*}\nFurther, we define the noisy version of $\\xi$ as\n\\begin{align*}\n\\tilde{\\xi}_t\\coloneqq\\begin{bmatrix}u_{[t-n,t-1]}\\\\\\tilde{y}_{[t-n,t-1]}\\end{bmatrix}\n=\\begin{bmatrix}u_{[t-n,t-1]}\\\\y_{[t-n,t-1]}+\\varepsilon_{[t-n,t-1]}\\end{bmatrix}.\n\\end{align*}\nDenote the (not invertible) linear transformation from $\\xi$ to an arbitrary but fixed state $x$ in some minimal realization by $T$, i.e., $x_t=T\\xi_t$.\nClearly, this implies $\\lVert x_t\\rVert_2^2\\leq\\lVert T\\rVert_2^2\\lVert\\xi_t\\rVert_2^2\\eqqcolon\\Gamma_x\\lVert\\xi_t\\rVert_2^2$.\nNote that $\\xi$ is the state of a detectable state-space realization and thus, there exists an IOSS Lyapunov function $W(\\xi)=\\lVert \\xi\\rVert_P^2$, similar to the proof of Theorem~\\ref{thm:mpc_tec}.\nFor some $\\gamma>0$, define $V_t\\coloneqq J_L^*(\\tilde{\\xi}_t)+\\gamma W(\\xi_t)$.\nThe following result shows that, for the state $\\xi$, a meaningful quadratic upper bound on $V$ can be proven.\n\n\\begin{lemma}\\label{lem:value_fcn_upper_bound}\nSuppose Assumptions~\\ref{ass:pe} and~\\ref{ass:length_2n} hold.\nThen, there exists a constant $c_3>0$ as well as a $\\delta>0$ such that, for all $\\xi_t\\in \\mathbb{B}_\\delta$, Problem~\\eqref{eq:robust_MPC} is feasible and $V$ is bounded as\n\\begin{align}\\label{eq:lem:value_fcn_bound}\n\\gamma\\lambda_{\\min}(P)\\lVert \\xi_t\\rVert^2_2\\leq V_t\\leq c_3\\lVert \\xi_t\\rVert_2^2+c_4,\n\\end{align}\nwhere $c_4=2np\\bar{\\varepsilon}^2\\lambda_\\sigma$.\n\\end{lemma}\n\\begin{proof}\nThe lower bound is trivial.\nFor the upper bound, we construct a feasible candidate solution to Problem~\\eqref{eq:robust_MPC} which brings the state $x$ in some minimal realization (and thus the output $y$) to zero in $L$ steps.\nObviously, we have $\\bar{u}_{[-n,-1]}(t)=u_{[t-n,t-1]}$ as well as $\\bar{y}_{[-n,-1]}(t)=\\tilde{y}_{[t-n,t-1]}$ by~\\eqref{eq:robust_MPC2}.\nBy assumption, we have $L\\geq2n$ as well as $0\\in\\text{int}(\\mathbb{U})$.\nThus, by controllability, there exists a $\\delta>0$ such that for any $x_t$ with $\\frac{1}{\\Gamma_x}\\lVert x_t\\rVert_2\\leq\\lVert\\xi_t\\rVert_2\\leq\\delta$, there exists an input trajectory $u_{[t,t+L-1]}\\in\\mathbb{U}^L$, which brings the state $x_{[t,t+L-1]}$ and the corresponding output $y_{[t,t+L-1]}$ to the origin in $L-n$ steps while satisfying \n\\begin{align}\\label{eq:thm_robust_proof_control}\n\\left\\lVert\\begin{bmatrix}u_{[t,t+L-1]}\n\\\\y_{[t,t+L-1]}\\end{bmatrix}\n\\right\\rVert_2^2&\\leq \\Gamma_{uy}\\lVert x_t\\rVert_2^2\n\\end{align}\nfor a suitable constant $\\Gamma_{uy}>0$.\nAs candidate input-output trajectories for~\\eqref{eq:robust_MPC}, we choose these $u,y$, i.e., $\\bar{u}_{[0,L-1]}(t)=u_{[t,t+L-1]},\\bar{y}_{[0,L-1]}(t)=y_{[t,t+L-1]}$.\nMoreover, $\\alpha(t)$ is chosen as\n\\begin{align}\\label{eq:lem_value_fcn_bound_alpha}\n\\alpha(t)=H_{ux}^\\dagger\\begin{bmatrix}u_{[t-n,t+L-1]}\\\\x_{t-n}\n\\end{bmatrix},\n\\end{align}\nwhere $H_{ux}$ is defined in~\\eqref{eq:ass_pe_matrix}.\nAs is described in more detail in~\\cite{Markovsky08,Berberich19}, the output of an LTI system is a linear combination of its initial condition and the input, and therefore, the above choice of $\\alpha(t)$ implies\n\\begin{align*}\n\\begin{bmatrix}H_{L+n}\\left(u^d\\right)\\\\H_{L+n}\\left(y^d\\right)\\end{bmatrix}\\alpha(t)&=\n\\begin{bmatrix}\\bar{u}_{[-n,L-1]}(t)\\\\y_{[t-n,t+L-1]}\\end{bmatrix}\\\\\n&=\n\\begin{bmatrix}\\bar{u}_{[-n,L-1]}(t)\\\\\n\\bar{y}_{[-n,-1]}(t)-\\varepsilon_{[t-n,t-1]}\n\\\\\\bar{y}_{[0,L-1]}(t)\\end{bmatrix},\n\\end{align*}\nwhere $\\varepsilon_{[t-n,t-1]}$ is the true noise instance.\nFor the slack variable $\\sigma$, we choose \n\\begin{align}\\label{eq:lem_1_sigma}\n\\begin{split}\n\\sigma_{[-n,-1]}(t)&=H_n\\left(\\varepsilon^d_{[0,N-L-1]}\\right)\\alpha(t)-\\varepsilon_{[t-n,t-1]},\\\\\n\\sigma_{[0,L-1]}(t)&=H_L\\left(\\varepsilon^d_{[n,N-1]}\\right)\\alpha(t),\n\\end{split}\n\\end{align}\nwhich implies that~\\eqref{eq:robust_MPC1}-\\eqref{eq:robust_MPC3} are satisfied.\nFinally, writing $e_i$ for a row vector whose $i$-th component is equal to $1$ and which is zero otherwise, we obtain\n\\begin{align}\\label{eq:infty_bound}\n\\begin{split}\n\\lVert H_{L+n}(\\varepsilon^d)\\alpha(t)\\rVert_\\infty&=\\max_{i\\in\\mathbb{I}_{[1,p(L+n)]}}|e_iH_{L+n}(\\varepsilon^d)\\alpha(t)|\\\\\n&\\leq\\bar{\\varepsilon}\\lVert\\alpha(t)\\rVert_1.\n\\end{split}\n\\end{align}\nThis implies $\\lVert\\sigma(t)\\rVert_\\infty\\leq\\bar{\\varepsilon}\\left(\\lVert\\alpha(t)\\rVert_1+1\\right)$, which in turn proves that~\\eqref{eq:robust_MPC5} is satisfied.\n\nIn the following, we employ the above candidate solution to bound the optimal cost and thereby, the function $V$.\nDue to observability of the pair $(A,C)$, corresponding to the minimal realization with state $x$, it holds that\n\\begin{align}\\label{eq:lem_value_fcn_bound_ctrb}\n\\begin{bmatrix}\\bar{u}_{[-n,-1]}(t)\\\\x_{t-n}\\end{bmatrix}=\n\\underbrace{\\begin{bmatrix}\nI_{mn}&0\\\\M_1&\\Phi_\\dagger\n\\end{bmatrix}}_{M\\coloneqq}\\xi_t,\n\\end{align}\nwhere $\\Phi_\\dagger=(\\Phi^\\top\\Phi)^{-1}\\Phi^\\top$ is a left-inverse of the observability matrix $\\Phi$.\nThe lower block of~\\eqref{eq:lem_value_fcn_bound_ctrb} follows from observability and the linear system dynamics $x_{k+1}=Ax_k+Bu_k,\\>y_k=Cx_k+Du_k$ for $k\\in\\mathbb{I}_{[t-n,t-1]}$, which can be used to compute the matrix $M_1$ depending on $A,B,C,D$.\nHence, $\\alpha(t)$ can be bounded as\n\\begin{align}\\nonumber\n\\lVert&\\alpha(t)\\rVert_2^2\\stackrel{\\eqref{eq:lem_value_fcn_bound_alpha}}{\\leq}\\left\\lVert H_{ux}^\\dagger\\right\\rVert_2^2\n\\left(\\left\\lVert\\bar{u}_{[-n,L-1]}(t)\\right\\rVert_2^2\n+\\left\\lVert x_{t-n}\\right\\rVert_2^2\\right)\\\\\\nonumber\n&=\\left\\lVert H_{ux}^\\dagger\\right\\rVert_2^2\n\\left(\\left\\lVert\\bar{u}_{[0,L-1]}(t)\\right\\rVert_2^2\n+\\left\\lVert\\begin{bmatrix}\\bar{u}_{[-n,-1]}(t)\\\\x_{t-n}\\end{bmatrix}\\right\\rVert_2^2\\right)\\\\\\label{eq:lem_value_fcn_bound_3}\n&\\stackrel{\\eqref{eq:thm_robust_proof_control},\\eqref{eq:lem_value_fcn_bound_ctrb}}{\\leq}\\underbrace{\\left\\lVert H_{ux}^\\dagger\\right\\rVert_2^2}_{c_{pe}=}\n\\left(\\Gamma_{uy}\\lVert x_t\\rVert_2^2+\\lVert M\\rVert_2^2\\lVert\\xi_t\\rVert_2^2\\right).\n\\end{align}\nUsing standard norm equivalence properties, it holds for arbitrary $k\\in\\mathbb{N}$ that\n\\begin{align}\\label{eq:lem_constant}\n\\left\\lVert H_{k}\\left(\\varepsilon^d_{[0,N-L-n+k-1]}\\right)\\right\\rVert_2^2\n\\leq c_5k\\bar{\\varepsilon}^2,\n\\end{align}\nwhere $c_5\\coloneqq p(N-L-n+1)$.\nBased on the definition of $\\sigma(t)$ in~\\eqref{eq:lem_1_sigma}, and using~\\eqref{eq:lem_constant} as well as the inequality $(a+b)^2\\leq2(a^2+b^2)$, we can bound $\\sigma(t)$ in terms of $\\alpha(t)$ as\n\\begin{align}\\label{eq:lem_bound_sigma}\n\\lVert\\sigma(t)\\rVert_2^2&\\leq 2np\\bar{\\varepsilon}^2+c_5(L+2n)\\bar{\\varepsilon}^2\\lVert\\alpha(t)\\rVert_2^2.\n\\end{align}\nCombining the above inequalities, $V$ is upper bounded as\n\\begin{align}\n\\nonumber\n&V_t\\leq J_L(\\tilde{\\xi}_t,\\alpha(t),\\sigma(t))+\\gamma W(\\xi_t)\\\\\n\\nonumber\n&\\leq\n\\lambda_{\\max}(Q,R)\\Gamma_{uy}\\lVert x_t\\rVert_2^2+\\gamma\\lambda_{\\max}(P)\\lVert \\xi_t\\rVert_2^2\\\\\n\\nonumber\n&+\\left(\\lambda_\\alpha+c_5(L+2n)\\lambda_\\sigma\\bar{\\varepsilon}\\right)c_{pe}\\bar{\\varepsilon}\\left(\\Gamma_{uy}\\lVert x_t\\rVert_2^2+\\lVert M\\rVert_2^2\\lVert\\xi_t\\rVert_2^2\\right)\\\\\\nonumber\n&+2np\\bar{\\varepsilon}^2\\lambda_\\sigma.\n\\end{align}\nFinally, $x_t$ is bounded by $\\xi_t$ as $\\lVert x_t\\rVert_2^2\\leq\\Gamma_x\\lVert\\xi_t\\rVert_2^2$, which leads to $V_t\\leq c_3\\lVert \\xi_t\\rVert_2^2 +c_4$, where \n\\begin{align*}\nc_3&=\\lambda_{\\max}(Q,R)\\Gamma_{uy}\\Gamma_x+\\gamma\\lambda_{\\max}(P)\\\\\n&+\\left(\\lambda_\\alpha+c_5(L+2n)\\lambda_\\sigma\\bar{\\varepsilon}\\right)c_{pe}\\bar{\\varepsilon}\\left(\\Gamma_{uy}\\Gamma_x+\\lVert M\\rVert_2^2\\right),\\\\\nc_4&=2np\\bar{\\varepsilon}^2\\lambda_\\sigma.\n\\end{align*}\n\\end{proof}\nIn Section~\\ref{sec:tec}, we assumed that the optimal cost is quadratically upper bounded (cf. Assumption~\\ref{ass:quad_upper_bound}), which is not restrictive in the nominal linear-quadratic setting.\nLemma~\\ref{lem:value_fcn_upper_bound} proves that, under mild assumptions, the optimal cost of the robust MPC problem~\\eqref{eq:robust_MPC} admits (locally) a similar upper bound and can thus be seen as the robust counterpart of Assumption~\\ref{ass:quad_upper_bound}.\n\nThe term $c_4$ is solely due to the slack variable $\\sigma$.\nThis can be explained by noting that, for $\\xi_t=0$, $\\alpha(t)$, $\\bar{u}_{[0,L-1]}(t)$, $\\bar{y}_{[0,L-1]}(t)$ can all be chosen to be zero, as long as $\\sigma$ compensates the noise, i.e., $\\sigma_{[-n,-1]}(t)=-\\varepsilon_{[t-n,t-1]}$.\n\n\\subsection{Prediction error bound}\\label{sec:prediction_error}\nDenote the optimizers of~\\eqref{eq:robust_MPC} by $\\alpha^*(t),\\sigma^*(t),\\bar{u}^*(t),\\bar{y}^*(t)$, and the output trajectory resulting from an open-loop application of $\\bar{u}^*(t)$ by $\\hat{y}$.\nOne of the reasons why it is difficult to analyze the presented MPC scheme is the non-trivial relation between the predicted output $\\bar{y}^*(t)$ and the ``actual'' output $\\hat{y}$.\nIn the following, we derive a bound on the difference between the two quantities, which will play an important role in proving recursive feasibility\nand practical stabiliy of the proposed scheme.\nFor an integer $k$, define constants $\\rho_{2,k},\\rho_{\\infty,k}$ such that\n\\begin{align*}\n\\rho_{2,k}&\\geq\\left\\lVert CA^k\\Phi_\\dagger\\right\\rVert_2^2,\\\\\n\\rho_{\\infty,k}&\\geq\\left\\lVert CA^k\\Phi_\\dagger\\right\\rVert_\\infty, \n\\end{align*}\nwhere $\\Phi_\\dagger$ is a left-inverse of the observability matrix $\\Phi$.\n\n\\begin{lemma}\\label{lem:prediction_error}\nIf~\\eqref{eq:robust_MPC} is feasible at time $t$, then the following inequalities hold for all $k\\in\\mathbb{I}_{[0,L-1]}$\n\\begin{align}\n\\label{eq:lem_prediction_error1}\n\\lVert &\\hat{y}_{t+k}-\\bar{y}^*_k(t)\\rVert_2^2\\leq8c_5\\bar{\\varepsilon}^2\\lVert\\alpha^*(t)\\rVert_2^2+2\\lVert\\sigma_k^*(t)\\rVert_2^2\\\\\\nonumber\n&+\\rho_{2,n+k}\\left(16n\\bar{\\varepsilon}^2\\left(c_5\\lVert\\alpha^*(t)\\rVert_2^2+p\\right)+4\\lVert\\sigma_{[-n,-1]}^*(t)\\rVert_2^2\\right),\\\\\n\\label{eq:lem_prediction_error}\n\\lVert &\\hat{y}_{t+k}-\\bar{y}^*_k(t)\\rVert_\\infty\\leq\\bar{\\varepsilon}\\lVert\\alpha^*(t)\\rVert_1+\\lVert\\sigma_k^*(t)\\rVert_\\infty \\\\\\nonumber\n&+\\rho_{\\infty,n+k}\\left(\\bar{\\varepsilon}\\left(\\lVert\\alpha^*(t)\\rVert_1+1\\right)+\\left\\lVert\\sigma_{[-n,-1]}^*(t)\\right\\rVert_\\infty\\right),\n\\end{align}\nwith $c_5$ from~\\eqref{eq:lem_constant}.\n\\end{lemma}\n\\begin{proof}\nWe show only~\\eqref{eq:lem_prediction_error} and note that~\\eqref{eq:lem_prediction_error1} can be derived following the same steps, using~\\eqref{eq:lem_constant} as well as the inequality $(a+b)^2\\leq2a^2+2b^2$.\nAs written above, $\\hat{y}$ is the trajectory, resulting from an open-loop application of $\\bar{u}^*(t)$ and with initial conditions specified by $\\left(u_{[t-n,t-1]},\\hat{y}_{[t-n,t-1]}\\right)=\\left(u_{[t-n,t-1]},y_{[t-n,t-1]}\\right)$.\nOn the other hand, according to~\\eqref{eq:robust_MPC1}, $\\bar{y}^*(t)$ is comprised as \n\\begin{align*}\n\\bar{y}^*(t)=H_{L+n}\\left(\\varepsilon^d\\right)\\alpha^*(t)+H_{L+n}\\left(y^d\\right)\\alpha^*(t)-\\sigma^*(t).\n\\end{align*}\nIt follows directly from~\\eqref{eq:robust_MPC1} and~\\eqref{eq:robust_MPC2} that the second term on the right-hand side $H_{L+n}\\left(y^d\\right)\\alpha^*(t)$ is a trajectory of $G$, resulting from an open-loop application of $\\bar{u}^*(t)$ and with initial output conditions \n\\begin{align*}\n\\tilde{y}_{[t-n,t-1]}+\\sigma_{[-n,-1]}^*(t)-H_n\\left(\\varepsilon^d_{[0,N-L-1]}\\right)\\alpha^*(t).\n\\end{align*}\nDefine\n\\begin{align*}\ny^-_{[t-n,t+L-1]}=\\hat{y}_{[t-n,t+L-1]}-H_{L+n}\\left(y^d\\right)\\alpha^*(t).\n\\end{align*}\nSince $G$ is LTI and $y^-$ contains the difference between two trajectories with the same input, we can assume $\\bar{u}^*(t)=0$ for the following arguments without loss of generality.\nHence, $y^-$ is equal to the output component of a trajectory $\\left(u^-,y^-\\right)$ with zero input and with initial trajectory\n\\begin{align}\\label{eq:lem_2_yminus}\n&\\begin{bmatrix}u^-_{[t-n,t-1]}\\\\y^-_{[t-n,t-1]}\\end{bmatrix}=\\\\\\nonumber\n&\\qquad\\begin{bmatrix}0\\\\H_n\\left(\\varepsilon^d_{[0,N-L-1]}\\right)\\alpha^*(t)-\\varepsilon_{[t-n,t-1]}-\\sigma_{[-n,-1]}^*(t)\\end{bmatrix}.\n\\end{align}\nThe relation to the internal state $x^-$ can be derived as\n\\begin{align*}\ny^-_{[t-n,t-1]}=\\Phi x^-_{t-n},\n\\end{align*}\nwith the observability matrix $\\Phi$.\nThis leads to the corresponding output at time $t+k$\n\\begin{align*}\ny^-_{t+k}=CA^{n+k}\\Phi_\\dagger y^-_{[t-n,t-1]},\n\\end{align*}\nwhere $\\Phi_\\dagger$ is a left-inverse of $\\Phi$.\nUsing this fact, the expression for $y^-_{[t-n,t-1]}$ in~\\eqref{eq:lem_2_yminus}, and the inequality~\\eqref{eq:infty_bound}, $\\lVert y^-_{t+k}\\rVert_\\infty$ can be bounded as\n\\begin{align*}\n\\lVert y^-_{t+k}\\rVert_\\infty\\leq\\rho_{\\infty,n+k}\\left(\\bar{\\varepsilon}\\left(\\lVert\\alpha^*(t)\\rVert_1+1\\right)+\\left\\lVert\\sigma_{[-n,-1]}^*(t)\\right\\rVert_\\infty\\right).\n\\end{align*}\nNote that\n\\begin{align*}\n\\left\\lVert \\hat{y}_{t+k}-\\bar{y}_k^*(t)\\right\\rVert_\\infty\n\\leq \\lVert y^-_{t+k}\\rVert_\\infty+\\bar{\\varepsilon}\\lVert\\alpha^*(t)\\rVert_1+\\lVert\\sigma^*_k(t)\\rVert_\\infty,\n\\end{align*}\nwhich concludes the proof.\n\\end{proof}\n\nEssentially, Lemma~\\ref{lem:prediction_error} gives a bound on the mismatch between the predicted output and the actual output resulting from the open-loop application of $\\bar{u}^*(t)$, depending on the optimal solutions $\\alpha^*,\\sigma^*$, and on system parameters.\nIn model-based robust MPC schemes, similar bounds are typically used to propagate uncertainty, where the role of the weighting vector $\\alpha$ to account for multiplicative uncertainty is replaced by the state $x$ and a model-based uncertainty description (compare~\\cite{Kouvaritakis16} for details).\nThe main difference in the proposed MPC scheme is that the predicted trajectory $\\bar{y}^*(t)$ is in general not a trajectory of the system in the sense of Definition~\\ref{def:trajectory_of}, corresponding to the input $\\bar{u}^*(t)$.\nOn the contrary, in model-based robust MPC, the predicted trajectory usually satisfies the dynamics of a (nominal) model of the system.\n\n\\subsection{Recursive feasibility}\\label{sec:robust_feasibility}\nThe following result shows that, if the proposed robust MPC scheme is feasible at time $t$, then it is also feasible at time $t+n$,\nassuming that the noise level is sufficiently small.\n\n\\begin{proposition}\\label{prop:robust_rec_feas}\nSuppose Assumption~\\ref{ass:pe} and~\\ref{ass:length_2n} hold.\nThen, for any $V_{ROA}>0$, there exists an $\\bar{\\varepsilon}_0>0$ such that for all $\\bar{\\varepsilon}\\leq\\bar{\\varepsilon}_0$,\nif $V_t\\leq V_{ROA}$ for some $t\\geq 0$, then the optimization problem~\\eqref{eq:robust_MPC} is feasible at time $t+n$.\n\\begin{comment}\n\\begin{itemize}\n\\item[(i)] the $n$-step MPC scheme is feasible at any $t\\in\\mathbb{N}$,\n\\item[(ii)] the closed-loop output with the $n$-step MPC scheme satisfies $y_t\\in\\mathbb{Y}$ for all $t\\in\\mathbb{N}$.\n\\end{itemize}\n\\end{comment}\n\\end{proposition}\n\\begin{proof}\nSuppose the robust MPC problem~\\eqref{eq:robust_MPC} is feasible at time $t$ with $V_t\\leq V_{ROA}$ and denote the optimizers by $\\alpha^*(t),\\sigma^*(t),\\bar{u}^*(t),\\bar{y}^*(t)$.\nAs in Lemma~\\ref{lem:prediction_error}, the trajectory resulting from an open-loop application of $\\bar{u}^*(t)$ and with initial conditions specified by $\\left(u_{[t-n,t-1]},y_{[t-n,t-1]}\\right)$ is denoted by $\\hat{y}$.\nFor $k\\in\\mathbb{I}_{[-n,L-2n-1]}$, we choose for the candidate input the shifted previously optimal solution, i.e., $\\bar{u}_k'(t+n)=\\bar{u}_{k+n}^*(t)$.\nOver the first $n$ steps, the candidate output must satisfy $\\bar{y}_{[-n,-1]}'(t+n)=\\tilde{y}_{[t,t+n-1]}$ due to~\\eqref{eq:robust_MPC2}.\nFurther, for $k\\in\\mathbb{I}_{[0,L-2n-1]}$, the output is chosen as $\\bar{y}_k'(t+n)=\\hat{y}_{t+n+k}$.\nSince $\\bar{y}_{[L-n,L-1]}^*(t)=0$ by~\\eqref{eq:robust_MPC3}, the prediction error bound of Lemma~\\ref{lem:prediction_error} implies that, for any $k\\in\\mathbb{I}_{[L-n,L-1]}$, it holds that\n\\begin{align*}\n\\lVert\\hat{y}_{t+k}&\\rVert_\\infty\\leq\\bar{\\varepsilon}\\lVert\\alpha^*(t)\\rVert_1+\\lVert\\sigma^*(t)\\rVert_\\infty\\\\\n&+\\rho_{\\infty,n+k}\\left(\\bar{\\varepsilon}\\left(\\lVert\\alpha^*(t)\\rVert_1+1\\right)+\\lVert\\sigma_{[-n,-1]}^*(t)\\rVert_\\infty\\right).\n\\end{align*}\nFor $\\bar{\\varepsilon}_0$ sufficiently small, $\\lVert\\sigma^*(t)\\rVert_\\infty$ becomes arbitrarily small due to~\\eqref{eq:robust_MPC5}.\nFurther, using that $\\lambda_\\alpha\\bar{\\varepsilon}\\lVert\\alpha^*(t)\\rVert_2^2\\leq J_L^*(u_{t-n,t-1]},\\tilde{y}_{[t-n,t-1]})\\leq V_{ROA}$, we can bound $\\alpha^*(t)$ as\n\\begin{align*}\n\\lVert\\alpha^*(t)\\rVert_1&\\leq\\sqrt{N-L-n+1}\\lVert\\alpha^*(t)\\rVert_2\\\\\n&\\leq\\sqrt{N-L-n+1}\\sqrt{\\frac{V_{ROA}}{\\lambda_\\alpha\\bar{\\varepsilon}}}.\n\\end{align*} \nHence, if $\\bar{\\varepsilon}_0$ is sufficiently small, then $\\hat{y}_{t+k}$ becomes arbitrarily small at the above time instants.\nThis implies that the internal state in some minimal realization corresponding to the trajectory $(\\bar{u}^*(t),\\hat{y})$ at time $t+L-n$, i.e., $\\hat{x}_{t+L-n}=\\Phi_\\dagger\\hat{y}_{[t+L-n,t+L-1]}$, approaches zero for $\\bar{\\varepsilon}\\to0$.\nThus, similar to the proof of Lemma~\\ref{lem:value_fcn_upper_bound}, there exists an input trajectory $\\bar{u}_{[L-2n,L-n-1]}'(t+n)$, which brings the state and the corresponding output $\\bar{y}_{[L-2n,L-n-1]}'(t+n)$ to zero in $n$ steps, while satisfying\n\\begin{align}\\label{eq:prop_proof_rec_feas_ctrb}\n\\left\\lVert\\begin{bmatrix}\\bar{u}_{[L-2n,L-n-1]}'(t+n)\n\\\\\\bar{y}_{[L-2n,L-n-1]}'(t+n)\\end{bmatrix}\n\\right\\rVert_2^2&\\leq \\Gamma_{uy}\\lVert \\hat{x}_{t+L-n}\\rVert_2^2.\n\\end{align}\nMoreover, in the interval $\\mathbb{I}_{[L-n,L-1]}$, we choose $\\bar{u}_{[L-n,L-1]}'(t+n)=0$, $\\bar{y}_{[L-n,L-1]}'(t+n)=0$, i.e.,~\\eqref{eq:robust_MPC3} is satisfied.\nThe above arguments imply that\n\\begin{align*}\n\\left(\\bar{u}'(t+n),\\begin{bmatrix}\\hat{y}_{[t,t+n-1]}\\\\\\bar{y}_{[0,L-1]}'(t+n)\n\\end{bmatrix}\\right)\n\\end{align*}\nis a trajectory of the unknown LTI system in the sense of Definition~\\ref{def:trajectory_of}.\nDenote the corresponding internal state in some minimal realization by $\\bar{x}'(t+n)$.\nWe choose $\\alpha'(t+n)$ as a corresponding solution to~\\eqref{eq:thm_hankel}, i.e., as\n\\begin{align}\\label{eq:prop_proof_rec_feas_1}\n\\alpha'(t+n)=H_{ux}^\\dagger\\begin{bmatrix}\\bar{u}'_{[-n,L-1]}(t+n)\\\\x_t\\end{bmatrix}\n\\end{align}\nwith $H_{ux}$ from~\\eqref{eq:ass_pe_matrix}.\nFinally, we fix\n\\begin{align}\\label{eq:prop_1_sigma}\n\\sigma'(t+n)=H_{L+n}\\left(\\tilde{y}^d\\right)\\alpha'(t+n)-\\bar{y}'(t+n),\n\\end{align}\nwhich implies that~\\eqref{eq:robust_MPC1} holds.\nIt remains to show that the constraint~\\eqref{eq:robust_MPC5} is satisfied.\nOver the first $n$ time steps,~\\eqref{eq:robust_MPC5} holds since\n\\begin{align}\\nonumber\n&\\sigma_{[-n,-1]}'(t+n)=H_n\\left(\\tilde{y}^d_{[0,N-L-1]}\\right)\\alpha'(t+n)-\\tilde{y}_{[t,t+n-1]}\\\\\\nonumber\n&\\stackrel{\\eqref{eq:prop_proof_rec_feas_1}}{=}H_n\\left(\\varepsilon^d_{[0,N-L-1]}\\right)\\alpha'(t+n)+y_{[t,t+n-1]}-\\tilde{y}_{[t,t+n-1]}\\\\\\label{eq:prop_proof_rec_feas_sigma1}\n&=H_n\\left(\\varepsilon^d_{[0,N-L-1]}\\right)\\alpha'(t+n)-\\varepsilon_{[t,t+n-1]}.\n\\end{align}\nFurther, using the definition of $\\sigma'(t+n)$ in~\\eqref{eq:prop_1_sigma} and the bound~\\eqref{eq:infty_bound}, we obtain\n\\begin{align}\\label{eq:prop_proof_rec_feas_sigma2}\n\\lVert&\\sigma_{[0,L-1]}'(t+n)\\rVert_\\infty\\leq\\bar{\\varepsilon}\\lVert\\alpha'(t+n)\\rVert_1\\\\\\nonumber\n&+\\underbrace{\\left\\lVert H_{L}\\left(y^d_{[n,N-1]}\\right)\\alpha'(t+n)-\\bar{y}_{[0,L-1]}'(t+n)\\right\\rVert_\\infty}_{=0},\n\\end{align}\nand thus,~\\eqref{eq:robust_MPC5} holds.\n\\end{proof}\n\nProposition~\\ref{prop:robust_rec_feas} shows that, for any sublevel set of the Lyapunov function $V$, there exists a sufficiently small noise bound $\\bar{\\varepsilon}_0$ such that, for any $\\bar{\\varepsilon}\\leq\\bar{\\varepsilon}_0$ and any state starting in the sublevel set at time $t$, the $n$-step MPC scheme is feasible at time $t+n$.\nIn particular, the required noise bound decreases if the size of the sublevel set, i.e., $V_{ROA}$, increases and vice versa.\nThis can be explained by noting that the noise in~\\eqref{eq:robust_MPC1} corresponds to a multiplicative uncertainty, which affects the prediction accuracy more strongly if the current state is further away from the origin and hence the Lyapunov function $V_t$ is larger.\nWe note that this does not imply recursive feasibility of the $n$-step MPC scheme in the standard sense since it remains to be shown that the sublevel set $V_t\\leq V_{ROA}$ is invariant, which will be proven in Section~\\ref{sec:robust_stability}.\nIn our main result, the set of initial states for which $V_0\\leq V_{ROA}$ will play the role of the guaranteed region of attraction of the closed-loop system.\n\nThe input candidate solution used to prove recursive feasibility in Proposition~\\ref{prop:robust_rec_feas} is analogous to a candidate solution one would use to show robust recursive feasibility in model-based robust MPC with terminal equality constraints.\nThe output candidate solution is sketched in Figure~\\ref{fig:sketch_candidate}.\nUp to time $L-2n-1$, $\\bar{y}'(t+n)$ is equal to $\\hat{y}$ (shifted by $n$ times steps), which is the output, resulting from an open-loop application of $\\bar{u}^*(t)$.\nThis choice together with the prediction error bound of Lemma~\\ref{lem:prediction_error} implies that the internal state corresponding to $\\bar{y}'(t+n)$ at time $L-2n$ is close to zero.\nThus, by controllability, there exists an input trajectory satisfying the input constraints, which brings the state and the output to zero in $n$ steps.\nIn the interval $\\mathbb{I}_{[L-2n,L-n-1]}$, the candidate output is chosen as this trajectory.\nThis also implies that the choice $\\bar{y}_{[L-n,L-1]}'(t+n)=0$ makes the candidate solution between $0$ and $L-1$, i.e., $\\left(\\bar{u}_{[0,L-1]}'(t+n),\\bar{y}_{[0,L-1]}'(t+n)\\right)$, a trajectory\\footnote{\nIn most practical cases, $(\\bar{u}^*(t),\\bar{y}^*(t))$ are not trajectories of the system due to the slack variable $\\sigma$ and the noise.} of the unknown system $G$ in the sense of Definition~\\ref{def:trajectory_of}.\nFinally, the suggested candidate input is also similar to~\\cite{Yu14}, where inherent robustness of quasi-infinite horizon (model-based) MPC is shown.\n\n\\begin{figure}\n\t\t\\includegraphics[width=0.49\\textwidth]{sketch_candidate_tikz}\n\t\t\\caption{Sketch of the candidate output for recursive feasibility.\n\t\tDue to the terminal equality constraints~\\eqref{eq:robust_MPC3}, the last $n$ steps of the optimal predicted output $\\bar{y}^*(t)$ are equal to zero.\n\t\tAccording to the prediction error bound derived in Lemma~\\ref{lem:prediction_error}, this implies that the state resulting from an open-loop application of the optimal input $\\bar{u}^*(t)$ is small at time $L-2n$, provided that $\\bar{\\varepsilon}$ is sufficiently small.\n\t\tTherefore, a candidate solution $\\bar{y}'(t+n)$ can be constructed by appending the open-loop output $\\hat{y}$ by a local deadbeat controller, which steers the state to the origin in $n$ steps.\n\t\t}\t\\label{fig:sketch_candidate}\n\\end{figure}\n\n\n\\begin{remark}\\label{rk:one_step}\nFor a $1$-step MPC scheme, a similar argument to prove recursive feasibility can be applied, given that $\\bar{u}^*_{[L-2n,L-n-1]}(t)$ and $\\bar{y}^*_{[L-2n,L-n-1]}(t)$ (and hence $\\hat{y}_{[t+L-2n,t+L-n-1]}$) are close to zero.\nThis is required to construct a feasible input which steers the state and the corresponding output to zero, similar to the proof of Proposition~\\ref{prop:robust_rec_feas}, and it is, e.g., the case if the initial state $x_t$ is close to zero.\nThat is, the result of Proposition~\\ref{prop:robust_rec_feas} holds locally for a $1$-step MPC scheme, as expected based on model-based MPC with terminal equality constraints under disturbances using inherent robustness properties.\n\\end{remark}\n\n\\begin{remark}\n\\label{rk:wlog_zero}\nAs mentioned in Section~\\ref{sec:scheme}, all of our theoretical guarantees for the presented robust MPC scheme can be straightforwardly extended to the case $(u^s,y^s)\\neq0$, with the corresponding steady-state $\\xi^s\\neq0$.\nThe main difference lies in the bound~\\eqref{eq:lem:value_fcn_bound}, which becomes $V_t\\leq \\tilde{c}_3\\lVert \\xi_t-\\xi^s\\rVert_2^2+\\tilde{c}_4$ for constants $\\tilde{c}_3\\neq c_3,\\tilde{c}_4\\neq c_4$, where $\\tilde{c}_3$ can be made arbitrarily close to $c_3$.\nOn the other hand, $\\tilde{c}_4$ changes depending on $\\xi^s$, since the right-hand side of~\\eqref{eq:lem_value_fcn_bound_3} would need to be proportional to $\\lVert \\xi_t-\\xi^s\\Vert_2^2+\\lVert \\xi^s\\rVert_2^2$.\nThe same phenomenon can be observed in a bound of $\\alpha'(t+n)$ based on~\\eqref{eq:prop_proof_rec_feas_1}, which will be used in the stability proof.\nAs will become clear later in this section, such changes in the bound of $\\alpha'(t+n)$ as well as in the constant $\\tilde{c}_4$ do not affect our qualitative theoretical results, but they may potentially (quantitatively) deterioriate the robustness w.r.t. the noise level $\\bar{\\varepsilon}$.\nIntuitively, this can be explained by noting that~\\eqref{eq:robust_MPC1} corresponds to a multiplicative uncertainty and thus, stabilization of the origin is simpler than stabilization of any other equilibrium.\nSince equilibria with $(u^s,y^s)\\neq0$ require a significantly more involved notation, we omit this extension.\n\\end{remark}\n\n\\subsection{Practical exponential stability}\\label{sec:robust_stability}\n\nThe following is our main stability result.\nIt shows that, under Assumptions~\\ref{ass:pe} and~\\ref{ass:length_2n}, for a low noise amplitude and large persistence of excitation, and for suitable regularization parameters, the application of the scheme~\\eqref{eq:robust_MPC} as described in Algorithm~\\ref{alg:MPC_n_step} leads to a practically exponentially stable closed loop.\n\\begin{theorem}\\label{thm:robust}\nSuppose Assumptions~\\ref{ass:pe} and~\\ref{ass:length_2n} hold.\nThen, for any $V_{ROA}>0$, there exist constants $\\underline{\\lambda}_\\alpha,\\overline{\\lambda}_\\alpha,\\underline{\\lambda}_\\sigma,\\overline{\\lambda}_\\sigma>0$ such that, for all $\\lambda_\\alpha,\\lambda_\\sigma$ satisfying\n\\begin{align}\\label{eq:thm_bounds1}\n\\begin{split}\n\\underline{\\lambda}_\\alpha\\leq\\lambda_\\alpha\\leq\\overline{\\lambda}_\\alpha,\\quad\\underline{\\lambda}_\\sigma\\leq\\lambda_\\sigma\\leq\\overline{\\lambda}_\\sigma,\n\\end{split}\n\\end{align}\nthere exist constants $\\bar{\\varepsilon}_0,\\bar{c}_{pe}>0$, as well as a continuous, strictly increasing $\\beta:[0,\\bar{\\varepsilon}_0]\\to[0,V_{ROA}]$ with $\\beta(0)=0$, such that, for all $\\bar{\\varepsilon},c_{pe}$ satisfying\n\\begin{align}\\label{eq:thm_bounds2}\n\\bar{\\varepsilon}\\leq\\bar{\\varepsilon}_0,\\quad c_{pe}\\bar{\\varepsilon}\\leq\\overline{c}_{pe},\n\\end{align}\nthe sublevel set $V_t\\leq V_{ROA}$ is invariant and $V_t$ converges exponentially to $V_t\\leq\\beta(\\bar{\\varepsilon})$ in closed loop with the $n$-step MPC scheme for all initial conditions for which $V_0\\leq V_{ROA}$.\n\\end{theorem}\n\\begin{proof}\nThe proof consists of three parts:\nFirst, we bound the increase in the Lyapunov function $V$.\nThereafter, we prove that, for suitably chosen bounds on the parameters, there exists a function $\\beta$, which satisfies the above requirements.\nFinally, we show invariance of the sublevel set $V_t\\leq V_{ROA}$ and exponential convergence of $V_t$ to $V_t\\leq\\beta(\\bar{\\varepsilon})$.\\\\\n\\textbf{(i). Practical Stability}\\\\\nSuppose Problem~\\eqref{eq:robust_MPC} is feasible at time $t$ and let $V_{ROA}>0$ be arbitrary.\nFurther, let $\\bar{\\varepsilon}_0$ be sufficiently small such that Proposition~\\ref{prop:robust_rec_feas} is applicable.\nThe cost of the candidate solution derived in Proposition~\\ref{prop:robust_rec_feas} at time $t+n$ is\n\\begin{align*}\nJ_L&\\left(u_{[t,t+n-1]},\\tilde{y}_{[t,t+n-1]},\\alpha'(t+n),\\sigma'(t+n)\\right)\\\\\n=&\\sum_{k=0}^{L-1}\\ell\\left(\\bar{u}_k'(t+n),\\bar{y}_k'(t+n)\\right)+\\lambda_\\alpha\\bar{\\varepsilon}\\lVert\\alpha'(t+n)\\rVert_2^2\\\\\n&+\\lambda_\\sigma\\lVert\\sigma'(t+n)\\rVert_2^2.\n\\end{align*}\nThus, we obtain for the optimal cost\n\\begin{align}\\nonumber\n&J_L^*(u_{[t,t+n-1]},\\tilde{y}_{[t,t+n-1]})\\\\\n\\nonumber\n&\\leq J_L\\left(u_{[t,t+n-1]},\\tilde{y}_{[t,t+n-1]},\\alpha'(t+n),\\sigma'(t+n)\\right)\\\\\n\\label{eq:thm_proof_value_fcn_diff}\n&= J_L^*(u_{[t-n,t-1]},\\tilde{y}_{[t-n,t-1]})-\\sum_{k=0}^{L-1}\\ell\\left(\\bar{u}_k^*(t),\\bar{y}_k^*(t)\\right)\\\\\n\\nonumber\n&-\\lambda_\\alpha\\bar{\\varepsilon}\\lVert\\alpha^*(t)\\rVert_2^2-\\lambda_\\sigma\\lVert\\sigma^*(t)\\rVert_2^2+\\lambda_\\alpha\\bar{\\varepsilon}\\lVert\\alpha'(t+n)\\rVert_2^2\\\\\n\\nonumber\n&+\\lambda_\\sigma\\lVert\\sigma'(t+n)\\rVert_2^2+\\sum_{k=0}^{L-1}\\ell(\\bar{u}_k'(t+n),\\bar{y}_k'(t+n)).\n\\end{align}\nIn the following key technical part of the proof (Parts (i.i)-(i.iv)), we derive useful bounds for most terms on the right-hand side of~\\eqref{eq:thm_proof_value_fcn_diff}.\nThis will lead to a decay bound of the optimal cost which is then used to prove practical exponential stability of the closed loop.\\\\\n\\textbf{(i.i) Stage Cost Bounds}\\\\\nWe first bound those terms in~\\eqref{eq:thm_proof_value_fcn_diff}, which involve the stage cost.\nThe above difference can be decomposed as\n\\begin{align}\\label{eq:thm_proof_stage_cost_diff}\n&\\sum_{k=0}^{L-1}\\ell(\\bar{u}_k'(t+n),\\bar{y}_k'(t+n))-\\sum_{k=0}^{L-1}\\ell\\left(\\bar{u}_k^*(t),\\bar{y}_k^*(t)\\right)\\\\\\nonumber\n&=\\sum_{k=L-2n}^{L-n-1}\\ell\\left(\\bar{u}_k'(t+n),\\bar{y}_k'(t+n)\\right)-\\sum_{k=0}^{n-1}\\ell\\left(\\bar{u}_k^*(t),\\bar{y}_k^*(t)\\right)\\\\\\nonumber\n&+\\sum_{k=0}^{L-2n-1}\\ell\\left(\\bar{u}_k'(t+n),\\bar{y}_k'(t+n)\\right)-\\ell\\left(\\bar{u}_{k+n}^*(t),\\bar{y}_{k+n}^*(t)\\right),\n\\end{align}\nwhere we use that $\\bar{u}_k'(t+n),\\bar{y}_k'(t+n),\\bar{u}_k^*(t),\\bar{y}_k^*(t)$ are all zero for $k\\in\\mathbb{I}_{[L-n,L-1]}$ due to~\\eqref{eq:robust_MPC3}.\nTo bound the first term on the right-hand side of~\\eqref{eq:thm_proof_stage_cost_diff}, note that\n\\begin{align*}\n\\lVert \\hat{x}_{t+L-n}\\rVert_2^2\\leq\\lVert\\Phi_\\dagger\\rVert_2^2\\lVert\\hat{y}_{[t+L-n,t+L-1]}\\rVert_2^2,\n\\end{align*}\nwith $\\hat{x}_{t+L-n}$ as in the proof of Proposition~\\ref{prop:robust_rec_feas}.\nFurther, since $\\bar{y}_{[L-n,L-1]}^*(t)=0$, $\\hat{y}$ can be bounded in the considered time interval as in~\\eqref{eq:lem_prediction_error1}, i.e.,\n\\begin{align*}\n&\\lVert\\hat{y}_{[t+L-n,t+L-1]}\\rVert_2^2\\leq8c_5n\\bar{\\varepsilon}^2\\lVert\\alpha^*(t)\\rVert_2^2+2\\lVert\\sigma^*(t)\\rVert_2^2\\\\\n&+\\sum_{k=L-n}^{L-1}\\rho_{2,n+k}\\\\\n&\\cdot\\left(16n\\bar{\\varepsilon}^2\\left(c_5\\lVert\\alpha^*(t)\\rVert_2^2+p\\right)+4\\lVert\\sigma^*_{[-n,-1]}(t)\\rVert_2^2\\right).\n\\end{align*}\nHence, it holds that\n\\begin{align}\\label{eq:thm_proof_stage_cost_bound_1}\n&\\sum_{k=L-2n}^{L-n-1}\\ell\\left(\\bar{u}_k'(t+n),\\bar{y}_k'(t+n)\\right)\\\\\n\\nonumber\n&\\stackrel{\\eqref{eq:prop_proof_rec_feas_ctrb}}{\\leq}\\lambda_{\\max}(Q,R)\\Gamma_{uy}\\lVert\\hat{x}_{t+L-n}\\rVert_2^2\\\\\n\\nonumber\n&\\leq\\lambda_{\\max}(Q,R)\\Gamma_{uy}\\lVert\\Phi_\\dagger\\rVert_2^2\\Big(8c_5n\\bar{\\varepsilon}^2\\lVert\\alpha^*(t)\\rVert_2^2+2\\lVert\\sigma^*(t)\\rVert_2^2\\\\\n\\nonumber\n&+\\sum_{k=L-n}^{L-1}\\rho_{2,n+k}\\\\\n\\nonumber\n&\\cdot\\left(16n\\bar{\\varepsilon}^2\\left(c_5\\lVert\\alpha^*(t)\\rVert_2^2+p\\right)+4\\lVert\\sigma^*_{[-n,-1]}(t)\\rVert_2^2\\right)\\Big).\n\\end{align}\nNext, we bound the difference between the third and the fourth term on the right-hand side of~\\eqref{eq:thm_proof_stage_cost_diff}.\nThe following relations are readily derived:\n\\begin{align}\\nonumber\n&\\lVert\\bar{y}_k'(t+n)\\rVert_Q^2-\\lVert\\bar{y}_{k+n}^*(t)\\rVert_Q^2\\\\\\nonumber\n=&\\lVert\\bar{y}_k'(t+n)-\\bar{y}^*_{k+n}(t)+\\bar{y}_{k+n}^*(t)\\rVert_Q^2-\\lVert\\bar{y}_{k+n}^*(t)\\rVert_Q^2\\\\\\nonumber\n=&\\lVert\\bar{y}_k'(t+n)-\\bar{y}_{k+n}^*(t)\\rVert_Q^2\\\\\\label{eq:robust_thm_proof_bound5}\n&+2\\left(\\bar{y}_k'(t+n)-\\bar{y}_{k+n}^*(t)\\right)^\\top Q\\bar{y}_{k+n}^*(t)\\\\\\nonumber\n\\leq&\\lVert\\bar{y}_k'(t+n)-\\bar{y}_{k+n}^*(t)\\rVert_Q^2\\\\\\nonumber\n&+2\\lVert\\bar{y}_k'(t+n)-\\bar{y}_{k+n}^*(t)\\rVert_Q\\lVert\\bar{y}_{k+n}^*(t)\\rVert_Q.\n\\end{align}\nBy using $2\\lVert\\bar{y}_{k+n}^*(t)\\rVert_Q\\leq1+\\lVert\\bar{y}_{k+n}^*(t)\\rVert_Q^2$ as well as $\\lVert\\bar{y}_{k+n}^*(t)\\rVert_Q^2\\leq V_{ROA}$, we arrive at\n\\begin{align}\\label{eq:thm_proof_auxiliary_bound1}\n2\\lVert\\bar{y}_k'&(t+n)-\\bar{y}_{k+n}^*(t)\\rVert_Q\\lVert\\bar{y}_{k+n}^*(t)\\rVert_Q\\\\\n\\nonumber\n\\leq&\\lVert\\bar{y}_k'(t+n)-\\bar{y}_{k+n}^*(t)\\rVert_Q\\left(1+V_{ROA}\\right).\n\\end{align}\nTherefore, since the inputs coincide over the considered time interval, and due to~\\eqref{eq:robust_thm_proof_bound5} as well as~\\eqref{eq:thm_proof_auxiliary_bound1}, it holds that\n\\begin{align}\\nonumber\n&\\sum_{k=0}^{L-2n-1}\\ell\\left(\\bar{u}_k'(t+n),\\bar{y}_k'(t+n)\\right)-\\ell\\left(\\bar{u}_{k+n}^*(t),\\bar{y}_{k+n}^*(t)\\right)\\\\\n\\label{eq:thm_proof_stage_cost_bound_2}\n&\\leq\\sum_{k=0}^{L-2n-1}\\lVert\\bar{y}_k'(t+n)-\\bar{y}_{k+n}^*(t)\\rVert_Q^2\\\\\\nonumber\n&\\quad+\\lVert\\bar{y}_k'(t+n)-\\bar{y}_{k+n}^*(t)\\rVert_Q\\left(1+V_{ROA}\\right).\n\\end{align}\nThe difference $\\lVert\\bar{y}_k'(t+n)-\\bar{y}_{k+n}^*(t)\\rVert_Q$ can be bounded similar to Lemma~\\ref{lem:prediction_error}.\nUsing the constraint~\\eqref{eq:robust_MPC5} to bound $\\lVert\\sigma^*(t)\\rVert_2$, it can be shown that the bound is of the form $\\lVert\\bar{y}_k'(t+n)-\\bar{y}_{k+n}^*(t)\\rVert_Q\\leq \\tilde{C}_1\\lVert\\alpha^*(t)\\rVert_2+\\tilde{C}_2\\leq\\tilde{C}_1\\left(1+\\lVert\\alpha^*(t)\\rVert_2^2\\right)+\\tilde{C}_2$, where both $\\tilde{C}_1$ and $\\tilde{C}_2$ are proportional to $\\bar{\\varepsilon}$.\nHence, applying Lemma~\\ref{lem:prediction_error} to~\\eqref{eq:thm_proof_stage_cost_bound_2}, the sum of~\\eqref{eq:thm_proof_stage_cost_bound_1} and~\\eqref{eq:thm_proof_stage_cost_bound_2} can be bounded as $C_1\\lVert\\alpha^*(t)\\rVert_2^2+C_2\\lVert\\sigma^*(t)\\rVert_2^2+C_3$ for suitable $C_i>0$, where $C_1$ and $C_3$ are quadratic in $\\bar{\\varepsilon}$ and vanish for $\\bar{\\varepsilon}=0$.\nTherefore, if $\\underline{\\lambda}_\\alpha$ and $\\underline{\\lambda}_\\sigma$ are sufficiently large, then~\\eqref{eq:thm_proof_value_fcn_diff} implies\n\\begin{align}\n\\nonumber\n&J_L^*(u_{[t,t+n-1]},\\tilde{y}_{[t,t+n-1]})\\\\\n\\label{eq:thm_proof_value_fcn_diff2}\n&\\leq J_L^*(u_{[t-n,t-1]},\\tilde{y}_{[t-n,t-1]})-\\sum_{k=0}^{n-1}\\ell\\left(\\bar{u}_k^*(t),\\bar{y}_k^*(t)\\right)\\\\\n\\nonumber\n&+\\lambda_\\alpha\\bar{\\varepsilon}\\lVert\\alpha'(t+n)\\rVert_2^2+\\lambda_\\sigma\\lVert\\sigma'(t+n)\\rVert_2^2\n+c_6,\n\\end{align}\nfor a suitable constant $c_6>0$, which is quadratic in $\\bar{\\varepsilon}$ and vanishes for $\\bar{\\varepsilon}=0$.\n\\\\\n\\textbf{(i.ii) Bound of $\\mathbf{\\lVert\\sigma'(t+n)\\rVert_2^2}$}\\\\\nBy applying standard norm bounds to the slack variable candidate $\\sigma'(t+n)$ as defined in~\\eqref{eq:prop_1_sigma} (compare also~\\eqref{eq:prop_proof_rec_feas_sigma1} and~\\eqref{eq:prop_proof_rec_feas_sigma2}), we obtain\n\\begin{align}\\label{eq:thm_proof_sigma_bound}\n\\lVert\\sigma'(t+n)\\rVert_2^2&\\leq 2np\\bar{\\varepsilon}^2+c_5(L+2n)\\bar{\\varepsilon}^2\\lVert\\alpha'(t+n)\\rVert_2^2,\n\\end{align}\nwith $c_5=p(N-L-n+1)$ as in~\\eqref{eq:lem_constant}.\\\\\n\\textbf{(i.iii) Bound of $\\mathbf{\\lVert\\alpha'(t+n)\\rVert_2^2}$}\\\\\nFor the weighting vector $\\alpha'(t+n)$, it holds that\n\\begin{align*}\n&\\lVert\\alpha'(t+n)\\rVert_2^2\\stackrel{\\eqref{eq:prop_proof_rec_feas_1}}{\\leq} c_{pe}\\left\\lVert\\begin{bmatrix}\\bar{u}'_{[-n,L-1]}(t+n)\\\\\\bar{x}_{-n}'(t+n)\\end{bmatrix}\\right\\rVert_2^2\\\\\n&= c_{pe}\\left( \\lVert x_t\\rVert_2^2+\\lVert\\bar{u}_{[-n,L-1]}'(t+n)\\rVert_2^2\\right)=c_{pe}\\lVert x_t\\rVert_2^2\\\\\n&+c_{pe}\\left(\\lVert\\bar{u}_{[0,L-n-1]}^*(t)\\rVert_2^2+\\lVert\\bar{u}'_{[L-2n,L-n-1]}(t+n)\\rVert_2^2\\right).\n\\end{align*}\nSimilar to~\\eqref{eq:thm_proof_stage_cost_bound_1}, we can use~\\eqref{eq:prop_proof_rec_feas_ctrb} to bound the last term as\n\\begin{align}\\label{eq:thm_proof_alpha_bound}\n&\\lVert\\bar{u}'_{[L-2n,L-n-1]}(t+n)\\rVert_2^2\\\\\\nonumber\n&\\leq\\Gamma_{uy}\\lVert\\Phi_\\dagger\\rVert_2^2\\Big(8c_5n\\bar{\\varepsilon}^2\\lVert\\alpha^*(t)\\rVert_2^2+2\\lVert\\sigma^*(t)\\rVert_2^2+\\sum_{k=L-n}^{L-1}\\rho_{2,n+k}\\\\\n\\nonumber\n&\\cdot\\left(16n\\bar{\\varepsilon}^2\\left(c_5\\lVert\\alpha^*(t)\\rVert_2^2+p\\right)+4\\lVert\\sigma^*_{[-n,-1]}(t)\\rVert_2^2\\right)\\Big).\n\\end{align}\nThe bound~\\eqref{eq:thm_proof_alpha_bound} is of the same form as~\\eqref{eq:thm_proof_stage_cost_bound_1} and~\\eqref{eq:thm_proof_stage_cost_bound_2}.\nDue to this fact, using the bound~\\eqref{eq:thm_proof_sigma_bound} for $\\sigma'(t+n)$, and by potentially choosing $\\underline{\\lambda}_\\alpha$ and $\\underline{\\lambda}_\\sigma$ larger,~\\eqref{eq:thm_proof_value_fcn_diff} implies\n\\begin{align}\n\\nonumber\n&J_L^*(u_{[t,t+n-1]},\\tilde{y}_{[t,t+n-1]})\\\\\n\\label{eq:thm_proof_value_fcn_diff3}\n&\\leq J_L^*(u_{[t-n,t-1]},\\tilde{y}_{[t-n,t-1]})-\\sum_{k=0}^{n-1}\\ell\\left(\\bar{u}_k^*(t),\\bar{y}_k^*(t)\\right)\\\\\n\\nonumber\n&+\\left(\\lambda_\\alpha+\\lambda_\\sigma c_7\\right)\\left(\\lVert x_t\\rVert_2^2+\\lVert\\bar{u}_{[0,L-n-1]}^*(t)\\rVert_2^2\\right)c_{pe}\\bar{\\varepsilon}+c_8,\n\\end{align}\nfor suitable constants $c_7,c_8>0$, which vanish for $\\bar{\\varepsilon}=0$.\\\\\n\\textbf{(i.iv) IOSS Bound}\\\\\nAs in the proof of Theorem~\\ref{thm:mpc_tec}, we consider now $V_t=J_L^*(\\tilde{\\xi}_t)+\\gamma W(\\xi_t)$ with the IOSS Lyapunov function $W$ for some $\\gamma>0$.\nIt follows directly from~\\eqref{eq:thm_eq0_proof2},~\\eqref{eq:thm_proof_value_fcn_diff3}, and from $\\lVert x_t\\rVert_2^2\\leq\\Gamma_x\\lVert \\xi_t\\rVert_2^2$ that\n\\begin{align}\\label{eq:thm_proof_value_fcn_diff4}\n&V_{t+n}-V_t\\leq-\\sum_{k=0}^{n-1}\\ell\\left(\\bar{u}_k^*(t),\\bar{y}_k^*(t)\\right)\\\\\n\\nonumber\n&+\\gamma(-\\frac{1}{2}\\lVert \\xi_{[t,t+n-1]}\\rVert_2^2+c_1\\lVert u_{[t,t+n-1]}\\rVert_2^2+c_2\\lVert y_{[t,t+n-1]}\\rVert_2^2)\\\\\n\\nonumber\n&+\\left(\\lambda_\\alpha+\\lambda_\\sigma c_7\\right)\\left(\\Gamma_x\\lVert \\xi_t\\rVert_2^2+\\lVert\\bar{u}_{[0,L-n-1]}^*(t)\\rVert_2^2\\right)c_{pe}\\bar{\\varepsilon}+c_8.\n\\end{align}\nThe identity $(a+b)^2\\leq2(a^2+b^2)$ yields\n\\begin{align*}\n\\lVert y_{[t,t+n-1]}\\rVert_2^2&\\leq2\\lVert\\bar{y}_{[0,n-1]}^*(t)\\rVert_2^2\\\\\n&+2\\lVert y_{[t,t+n-1]}-\\bar{y}_{[0,n-1]}^*(t)\\rVert_2^2,\n\\end{align*}\nwhere the latter term can again be bounded using Lemma~\\ref{lem:prediction_error}.\nSimilar to the earlier steps of this proof, the components of the bound $\\lVert y_{[t,t+n-1]}-\\bar{y}_{[0,n-1]}^*(t)\\rVert_2^2$ vanish in~\\eqref{eq:thm_proof_value_fcn_diff4} if $\\underline{\\lambda}_\\sigma,\\underline{\\lambda}_\\alpha$ are chosen sufficiently large, except for an additive constant, which depends solely on the noise.\nMoreover, choosing $\\gamma=\\frac{\\lambda_{\\min}(Q,R)}{\\max\\{c_1,2c_2\\}}$, it holds that\n\\begin{align*}\n\\gamma(c_1\\lVert u_{[t,t+n-1]}\\rVert_2^2+2c_2\\lVert \\bar{y}_{[0,n-1]}^*&(t)\\rVert_2^2)\\\\\n\\leq&\\sum_{k=0}^{n-1}\\ell(\\bar{u}_k^*(t),\\bar{y}_k^*(t)).\n\\end{align*}\nCombining these facts, we arrive at\n\\begin{align*}\nV_{t+n}-V_t&\\leq\\left(\\left(\\lambda_\\alpha+\\lambda_\\sigma c_7\\right)\\Gamma_x c_{pe}\\bar{\\varepsilon}-\\frac{\\gamma}{2}\\right)\\lVert \\xi_t\\rVert_2^2\\\\\n&+\\left(\\lambda_\\alpha+\\lambda_\\sigma c_7\\right)c_{pe}\\bar{\\varepsilon}\\lVert\\bar{u}_{[0,L-n-1]}^*(t)\\rVert_2^2+c_9\n\\end{align*}\nfor a suitable constant $c_9$, which vanishes for $\\bar{\\varepsilon}=0$.\nFinally, note that $\\lambda_{\\min}(R)\\lVert\\bar{u}_{[0,L-n-1]}^*(t)\\rVert_2^2\\leq V_t$, which leads to\n\\begin{align}\\label{eq:thm_proof_value_fcn_diff5}\nV_{t+n}-V_t&\\leq\\left(\\left(\\lambda_\\alpha+\\lambda_\\sigma c_7\\right)\\Gamma_x c_{pe}\\bar{\\varepsilon}-\\frac{\\gamma}{2}\\right)\\lVert \\xi_t\\rVert_2^2\\\\\\nonumber\n&\\quad+\\frac{\\left(\\lambda_\\alpha+\\lambda_\\sigma c_7\\right)c_{pe}\\bar{\\varepsilon}}{\\lambda_{\\min}(R)}V_t+c_9\\\\\\nonumber\n&\\eqqcolon \\left(c_{10}-\\frac{\\gamma}{2}\\right)\\lVert \\xi_t\\rVert_2^2+c_{11}V_t+c_9.\n\\end{align}\n\\textbf{(ii). Construction of $\\mathbf{\\beta}$}\\\\\nThe local upper bound in Lemma~\\ref{lem:value_fcn_upper_bound}, which holds for any $\\xi_t\\in \\mathbb{B}_\\delta$, implies that the following holds for any $V_{ROA}>0$, and any $\\xi_t$ with $V_t\\leq V_{ROA}$:\n %\n\\begin{align}\n\\label{eq:robust_thm_proof_bound4}\nV_t\\leq\\underbrace{\\max\\left\\{ c_{3},\\frac{V_{ROA}-c_{4}}{\\delta^2}\\right\\}}_{c_{3,V_{ROA}}\\coloneqq}\\lVert \\xi_t\\rVert_2^2+c_{4}.\n\\end{align}\nWe first consider $V_{ROA}=\\delta^2c_3+c_4$, which implies $c_{3,V_{ROA}}=c_3$.\nFurther, we define $c_{12}\\coloneqq\\frac{\\gamma}{2}-c_{10}-c_{3}c_{11}$ as well as\n\\begin{align*}\n\\beta(\\bar{\\varepsilon})=\\frac{\\frac{\\gamma}{2}c_{4}+c_{3}c_9}{c_{12}}\n\\end{align*}\nfor any $\\bar{\\varepsilon}$ for which $c_{12}>0$.\nRecall that $c_3=a_1\\bar{\\varepsilon}^2+a_2\\bar{\\varepsilon}+a_3,c_4=a_4\\bar{\\varepsilon}^2,c_9=a_5\\bar{\\varepsilon}^2+a_6\\bar{\\varepsilon},c_{10}=a_7\\bar{\\varepsilon}^2+a_8\\bar{\\varepsilon},c_{11}=a_9\\bar{\\varepsilon}^2+a_{10}\\bar{\\varepsilon}$, for suitable constants $a_i>0$.\nThis implies $\\beta(0)=0$.\nNext, we show the existence of a constant $\\bar{\\varepsilon}_0$ such that $\\beta$ is strictly increasing on $[0,\\bar{\\varepsilon}_0]$.\nIf $c_{12}>0$, then $\\beta$ is strictly increasing since its numerator increases with $\\bar{\\varepsilon}$ whereas its denominator decreases with $\\bar{\\varepsilon}$. \nIn the following, we show that $c_{12}>0$.\nBy definition, we have\n\\begin{align*}\nc_{12}&=\\frac{\\gamma}{2}-\\left(\\lambda_\\alpha+\\lambda_\\sigma c_7\\right)\\Gamma_xc_{pe}\\bar{\\varepsilon}\\\\\n&-\\frac{\\left(\\lambda_\\alpha+\\lambda_\\sigma c_7\\right)c_{pe}\\bar{\\varepsilon}}{\\lambda_{\\min}(R)}\\Big(\\lambda_{\\max}(Q,R)\\Gamma_{uy}\\Gamma_x+\\gamma\\lambda_{max}(P)\\\\\n&+\\left(\\lambda_\\alpha+c_5(L+2n)\\lambda_\\sigma\\bar{\\varepsilon}\\right)c_{pe}\\bar{\\varepsilon}(\\Gamma_{uy}\\Gamma_x+\\lVert M\\rVert_2^2\\Big).\n\\end{align*}\nIt can be seen directly from this expression that, if $\\lambda_\\alpha\\leq\\overline{\\lambda}_{\\alpha},\\lambda_\\sigma\\leq\\overline{\\lambda}_\\sigma$, with arbitrary but fixed upper bounds $\\overline{\\lambda}_{\\alpha},\\overline{\\lambda}_\\sigma$, and $c_{pe}\\bar{\\varepsilon}$ is sufficiently small, then $c_{12}>0$.\nIt remains to show that $\\beta(\\bar{\\varepsilon}_0)\\leq V_{ROA}$, or, equivalently,\n\\begin{align*}\n\\frac{\\frac{\\gamma}{2}c_4+c_3c_9}{\\frac{\\gamma}{2}-c_{10}-c_3c_{11}}\\leq \\delta^2c_3+c_4,\n\\end{align*}\nwhich can be ensured by choosing $\\bar{\\varepsilon}_0$ sufficiently small.\\\\\n\\textbf{(iii). Invariance and Exponential Convergence}\\\\\nTake an arbitrary $\\xi_t$ with $V_t\\leq V_{ROA}$ and note that this implies that~\\eqref{eq:robust_MPC} is feasible and thus,~\\eqref{eq:thm_proof_value_fcn_diff5} and~\\eqref{eq:robust_thm_proof_bound4} hold.\nMoreover, $c_{12}>0$ implies $c_{10}<\\frac{\\gamma}{2}$.\nDefining $V_{\\beta,t}\\coloneqq V_t-\\beta(\\bar{\\varepsilon})$, we thus obtain\n\\begin{align*}\n&V_{t+n}\\stackrel{\\eqref{eq:thm_proof_value_fcn_diff5}}{\\leq}  \\left(1+c_{11}\\right)V_t+\\left(c_{10}-\\frac{\\gamma}{2}\\right)\\lVert \\xi_t\\rVert_2^2+c_9\\\\\n&\\stackrel{\\eqref{eq:robust_thm_proof_bound4}}{\\leq} \\left(1+c_{11}+\\frac{c_{10}-\\frac{\\gamma}{2}}{c_3}\\right) V_t+\\frac{c_4}{c_3}\\left(\\frac{\\gamma}{2}-c_{10}\\right)+c_9\\\\\n&\\leq\\left(1+c_{11}+\\frac{c_{10}-\\frac{\\gamma}{2}}{c_3}\\right)V_{\\beta,t}+\\beta(\\bar{\\varepsilon}),\n\\end{align*}\nwhere the last inequality follows from elementary computations.\nThis in turn implies the following contraction property\n\\begin{align}\\label{eq:thm_proof_contraction_property}\nV_{\\beta,t+n}\\leq\\underbrace{\\left(1+c_{11}+\\frac{c_{10}-\\frac{\\gamma}{2}}{c_3}\\right)}_{<1}V_{\\beta,t}.\n\\end{align}\nIf the noise bound $\\bar{\\varepsilon}_0$ is sufficiently small, then this implies invariance of the sublevel set $V_t\\leq V_{ROA}$ and hence, by Proposition~\\ref{prop:robust_rec_feas}, recursive feasibility of the $n$-step MPC scheme.\nApplying the contraction property~\\eqref{eq:thm_proof_contraction_property} recursively, we can thus conclude that $V_t$ converges exponentially to $V_t\\leq\\beta(\\bar{\\varepsilon})$.\n\nSo far, we have only considered the case $V_{ROA}=\\delta^2c_3+c_4$.\nIt remains to show that, \\emph{for any} $V_{ROA}>0$, there exist suitable parameter bounds such that\n\\begin{align*}\nc_{12,V_{ROA}}\\coloneqq \\frac{\\gamma}{2}-c_{10}-c_{3,V_{ROA}}c_{11}>0\n\\end{align*}\nwith $c_{3,V_{ROA}}$ from~\\eqref{eq:robust_thm_proof_bound4}.\nIt is easily seen from the above discussion that, for any \\emph{fixed} $V_{ROA}>0$ and for \\emph{fixed} bounds $\\overline{\\lambda}_\\alpha,\\overline{\\lambda}_\\sigma$, $c_{12,V_{ROA}}>0$ can always be ensured if $c_{pe}\\bar{\\varepsilon}$ is sufficiently small, i.e., if the bound $\\bar{c}_{pe}$ is sufficiently small.\n\\end{proof}\n\\begin{comment}\n\\begin{itemize}\n\\item Note that $\\lVert\\sigma'(t+n)\\rVert\\leq ...$ does not make use of $\\lVert\\sigma^*(t)\\rVert\\leq...$, i.e., the constraints do not really exploit recursive arguments - this is due to the fact that we do \/ can not relate $\\alpha'(t+n)$ and $\\alpha^*(t)$ $\\rightarrow$ big difference to standard robust MPC arguments.\n\\end{itemize}\n\\end{comment}\n\nTheorem~\\ref{thm:robust} shows that the closed loop of the proposed data-driven MPC scheme admits a (practical) Lyapunov function, which converges robustly and exponentially to a set, whose size shrinks with the noise level.\nSince $\\lVert\\xi_t\\rVert_2^2\\leq\\frac{1}{\\gamma\\lambda_{\\min}(P)}V_t$ due to~\\eqref{eq:lem:value_fcn_bound}, this implies practical exponential stability of the equilibrium $\\xi=0$.\nThe result requires that the noise level $\\bar{\\varepsilon}$ is small, the amount of persistence of excitation is large compared to the noise level (i.e., $c_{pe}\\bar{\\varepsilon}$ is small), and the regularization parameters are chosen suitably.\nConcerning the latter requirement, $\\lambda_\\alpha$ cannot be chosen arbitrarily large, which can be explained by noting that the optimal $\\alpha$ is usually not zero, even in the noise-free case.\nOn the other hand, $\\lambda_\\alpha$ cannot be too close to zero since solutions $\\alpha(t)$ of~\\eqref{eq:robust_MPC1} are not unique and large choices of $\\alpha(t)$ amplify the influence of the noise in $\\tilde{y}^d$ on the prediction accuracy.\nFurther, $\\lambda_\\sigma$ has to be chosen sufficiently large to ensure stability, but not arbitrarily large for a fixed noise level.\nTo be more precise, $\\lambda_\\alpha c_{pe}\\bar{\\varepsilon}$ and $\\lambda_\\sigma c_{pe}\\bar{\\varepsilon}^2$ have to be small, i.e., for a fixed $c_{pe}$, choosing the regularization parameters too large deteriorates the robustness of the scheme w.r.t. the noise level.\nOne can show that the theoretical properties in Theorem~\\ref{thm:robust} are also valid without imposing the lower bound in~\\eqref{eq:thm_bounds1} on $\\lambda_{\\sigma}$, by using the more conservative constraint~\\eqref{eq:robust_MPC5} in the proof.\nHowever,~\\eqref{eq:robust_MPC5} is non-convex (cf. Remark~\\ref{rk:sigma_bound}), but can typically be enforced implicitly if $\\lambda_\\sigma$ is chosen large enough.\n\nIn the proof of Theorem~\\ref{thm:robust}, a close connection between the region of attraction, i.e., the set of initial conditions with $V_0\\leq V_{ROA}$, and various parameters becomes apparent.\nFirst of all, the noise bound $\\bar{\\varepsilon}$ needs to be sufficiently small depending on $V_{ROA}$ to allow for an application of Proposition~\\ref{prop:robust_rec_feas}.\nMoreover, if $V_{ROA}$ increases, then also $c_{3,V_{ROA}}$ increases and hence, $c_{11}$ must decrease to ensure $c_{12,V_{ROA}}>0$ and thereby exponential stability.\nTo render $c_{11}$ small, $c_{pe}\\bar{\\varepsilon}$ must decrease, i.e., the amount of persistence of excitation compared to the noise level must increase.\nThus, for $c_{pe}\\bar{\\varepsilon}\\to0$ (and a sufficiently small noise bound $\\bar{\\varepsilon}$ due to Proposition~\\ref{prop:robust_rec_feas}), the region of attraction approaches the set of all initially feasible points.\nFor a fixed $c_{pe}$, the size of the region of attraction increases if the noise level decreases and vice versa.\nA similar connection between the maximal disturbance and the region of attraction can be found in~\\cite{Yu14}, which studies inherent robustness properties of quasi-infinite horizon MPC (but the result applies similarly to model-based $n$-step MPC with terminal equality constraints).\nFurther, if $c_{pe}$ decreases then so do $c_{10}$ as well as $c_{11}$ and hence also $\\beta(\\bar{\\varepsilon})$.\nThis implies that larger persistence of excitation (i.e., a lower $c_{pe}\\bar{\\varepsilon}$) does not only increase the region of attraction but it also reduces the tracking error.\n\n\\begin{remark}\\label{rk:RMPC_practical_application}\nTo apply the proposed data-driven MPC scheme in practice, the following ingredients are required.\nFirst of all, the design parameters in the cost, i.e., $Q,R,\\lambda_\\alpha,\\lambda_\\sigma$, have to be selected suitably.\nThe proof and discussion of Theorem~\\ref{thm:robust} give a qualitative guideline for choosing the regularization parameters.\nFurther, as in the nominal case (Section~\\ref{sec:tec}), measured data with a persistently exciting input as well as a (potentially rough) upper bound on the system's order need to be available.\nFinally, an upper bound on the noise level $\\bar{\\varepsilon}$ is required.\n\nWhile these ingredients suffice to apply the proposed scheme, computing bounds as in~\\eqref{eq:thm_bounds1} and~\\eqref{eq:thm_bounds2} is a difficult task in practice.\nTheorem~\\ref{thm:robust} should be interpreted as a qualitative result which illustrates a) the influence of the regularization parameters on stability and robustness of the presented MPC scheme and b) that large persistence of excitation (compared to the noise level) increases the region of attraction and reduces the tracking error.\nFurther, many of the employed bounds rely on conservative estimates such as $(a+b)^2\\leq2a^2+2b^2$.\nIn principle, it is possible to improve some of the quantitative estimates at the price of a more involved notation.\nNevertheless, such improved estimates may lead to meaningful, non-conservative, verifiable conditions on the noise level $\\bar{\\varepsilon}$ for closed-loop stability, and are therefore an interesting issue for future research.\n\\end{remark}\n\n\\begin{remark}\nIn the nominal MPC scheme~\\eqref{eq:term_eq_MPC} as well as in its robust modification~\\eqref{eq:robust_MPC}, the data $(u^d,y^d)$ used for prediction is fixed.\nAlternatively, one may update the data using online measurements, given that the closed loop is persistently exciting.\nIndeed, we believe that one of the main advantages of the proposed scheme is its ability to cope (locally) with nonlinear components of the unknown system.\nNonlinear dynamical systems are in general difficult to identify and thus, the proposed approach may be simpler than a model-based MPC scheme with prior system identification.\nAs illustrated in~\\cite{Coulson19} with an application of a similar MPC scheme to a nonlinear stochastic quadcopter system, the approach is already applicable in practice to time-varying or nonlinear dynamics without updating the data online.\nProviding theoretical guarantees for the application of the proposed scheme to a nonlinear system is an interesting and relevant problem for future research.\n\\end{remark}\n\nSimilar to the nominal MPC scheme, it is easy to see that the only free decision variables of Problem~\\eqref{eq:robust_MPC} are $\\alpha(t)$ and $\\sigma(t)$ with at least $m(L+2n)+n$ and $p(L+n)$ free parameters, respectively (cf. Remark~\\ref{rk:complexity_nominal}).\nOn the contrary, to implement a model-based MPC scheme (with state measurements), $mL$ parameters are required.\nWhen neglecting the constraint~\\eqref{eq:robust_MPC5} (cf. Remark~\\ref{rk:sigma_bound}), the slack variable $\\sigma(t)$ can be eliminated from~\\eqref{eq:robust_MPC} by directly penalizing the norm of the model mismatch $\\bar{y}(t)-H_{L+n}(\\tilde{y}^d)\\alpha(t)$ in the cost.\nHence, considering the minimal amount of data required for persistence of excitation, Problem~\\eqref{eq:robust_MPC} has roughly the same number of decision variables as a model-based MPC problem.\nIn contrast to the nominal case, however, Theorem~\\ref{thm:robust} implies that larger data horizons $N$ are beneficial for the theoretical properties of the proposed scheme as they typically decrease the constant $c_{pe}$.\nOn the other hand, increasing values for $N$ also lead to an increasing online complexity of~\\eqref{eq:robust_MPC} since $\\alpha(t)\\in\\mathbb{R}^{N-L+1}$, i.e., the presented MPC approach allows for a tradeoff between computational complexity and desired closed-loop performance by appropriately selecting $N$.\n\nOn the contrary, the performance of identification-based MPC typically improves if larger amounts of data are employed, whereas the online complexity is independent of $N$.\nHowever, while the scheme presented in this paper provides end-to-end guarantees for the closed loop using noisy data of finite length, the derivation of non-conservative estimation bounds on system parameters from such data, which would be required for guarantees in model-based MPC, is difficult in general and an active field of research~\\cite{Matni19,Matni19b}.\nAn extensive \\emph{quantitative} comparison of model-based MPC and the proposed data-driven MPC in theory and for practical examples is an interesting issue for future research.\n\n\\section{Example}\\label{sec:example}\nIn this section, we apply the robust data-driven MPC scheme of Section~\\ref{sec:robust} to a four tank system, which has been considered in~\\cite{raff2006nonlinear}.\nThis system is well-known as a real-world example, which is open-loop stable, but can be destabilized by an MPC without terminal constraints if the prediction horizon is too short.\nSimilarly, we show in this section that our proposed scheme is able to track a specified setpoint, whereas a scheme without terminal constraints as suggested in~\\cite{Yang15,Coulson19,Coulson19b} leads to an unstable closed loop, unless it is suitably modified.\n\nWe consider a linearized version of the system from~\\cite{raff2006nonlinear}, which takes the form\n\\begin{align*}\nx_{k+1}&=\\begin{bmatrix}0.921&0&0.041&0\\\\\n0&0.918&0&0.033\\\\\n0&0&0.924&0\\\\0&0&0&0.937\\end{bmatrix}x_k\\\\\n&+\\begin{bmatrix}\n0.017&0.001\\\\0.001&0.023\\\\0&0.061\\\\0.072&0\n\\end{bmatrix}u_k,\n\\end{align*}\n\\begin{align*}\ny_k&=\\begin{bmatrix}1&0&0&0\\\\0&1&0&0\n\\end{bmatrix}x_k.\n\\end{align*}\nFor the following application of the robust data-driven MPC scheme, the system matrices are \\emph{unknown} and only measured input-output data is available.\nThe control goal is tracking of the setpoint of the linearized system\n\\begin{align*}\n(u^s,y^s)=\\left(\\begin{bmatrix}1\\\\1\\end{bmatrix},\\begin{bmatrix}\n0.65\\\\0.77\\end{bmatrix}\\right),\n\\end{align*}\nwhich is readily shown to satisfy the dynamics.\nWe consider no constraints on the input or the output.\nIn an open-loop experiment, an input-output trajectory of length $N=400$ is measured, where the input is chosen randomly from the unit interval, i.e., $u^d_k\\in[-1,1]^2$, and the output is subject to \nuniformly distributed additive measurement noise with bound $\\bar{\\varepsilon}=0.002$.\nThe online measurements used to update the initial conditions~\\eqref{eq:robust_MPC2} in the MPC scheme are subject to the same type of noise.\n\nWe choose $L=30$ for the prediction horizon as well as the following design parameters\n\\begin{align*}\nQ=3\\cdot I_p,\\>R=10^{-4} I_m,\\>\\lambda_\\sigma=1000,\\>\\lambda_\\alpha\\bar{\\varepsilon}=0.1.\n\\end{align*}\nThe closed-loop output resulting from the application of Problem~\\eqref{eq:robust_MPC} in a $1$-step MPC scheme is displayed in Figure~\\ref{fig:four_tank}.\nIt can be seen that the control goal is fulfilled, with only slight deviations from the desired equilibrium.\nOn the other hand, if the same scheme without terminal constraints is applied to the system, then the closed loop is unstable and diverges with the chosen parameters for both a $1$-step and an $n$-step MPC scheme (cf. again Figure~\\ref{fig:four_tank}).\nThis confirms our initial motivation that rigorous guarantees are indeed desirable for data-driven MPC methods, in particular when they are applied to practical systems.\nFurthermore, it can also be observed in Figure~\\ref{fig:four_tank} that an $n$-step version of the proposed MPC scheme with terminal equality constraints yields slightly better tracking accuracy, compared to the $1$-step scheme.\nWe note that, with the above choice of parameters, the non-convex constraint~\\eqref{eq:robust_MPC5} is automatically satisfied without enforcing it explicitly (cf. Remark~\\ref{rk:sigma_bound}).\n\n\\begin{figure}\n\t\t\\begin{center}\n\t\n\t\n\t\n\t\n\t\t\\subfigure[Closed-loop output $y_1$]\n\t\t{\\includegraphics[width=0.49\\textwidth]{example_y1}}\n\t\t\\subfigure[Closed-loop output $y_2$]\n\t\t{\\includegraphics[width=0.49\\textwidth]{example_y2}}\n\t\t\\end{center}\n\t\t\\caption{Closed-loop output, resulting from the application of the robust data-driven MPC scheme with terminal equality constraints in a $1$-step fashion (TEC), in an $n$-step fashion (TEC, $n$-step), and without terminal equality constraints in a $1$-step fashion (UCON).}\t\\label{fig:four_tank}\n\\end{figure}\n\nTheorem~\\ref{thm:robust} gives qualitative guidelines for the tuning of the design parameters to guarantee robust stability.\nIn the following, we analyze the influence of various parameters on the closed-loop behavior.\nTheorem~\\ref{thm:robust} requires that the regularization parameters lie within specific bounds.\nThis is confirmed for the present example, where the MPC scheme achieves desirable closed-loop performance similar to Figure~\\ref{fig:four_tank} as long as $0.05\\leq\\lambda_\\alpha \\bar{\\varepsilon}\\leq0.5$.\nIf $\\lambda_\\alpha$ is chosen too low, then the closed loop is unstable since the norm of $\\alpha^*(t)$ and hence the amplification of the measurement noise in~\\eqref{eq:robust_MPC1} is too large.\nOn the contrary, if $\\lambda_\\alpha$ is chosen too large, then the asymptotic tracking error increases since the cost term $\\lambda_\\alpha\\bar{\\varepsilon}\\lVert\\alpha^*(t)\\rVert_2$ dominates over the tracking cost.\nSimilarly, if $\\lambda_\\sigma<500$, then the closed loop may be unstable since we did not consider the constraint~\\eqref{eq:robust_MPC5} and therefore the slack variable is too large, which has a negative impact on the prediction accuracy.\nAn upper bound on $\\lambda_\\sigma$ beyond which the closed-loop behavior is undesirable could not be observed for the present example.\nFurther, if the input weighting $R$ is chosen too low, then the robustness with respect to the noise deteriorates, which can be explained via the bound~\\eqref{eq:thm_proof_value_fcn_diff5}, which grows with $1\/\\lambda_{\\min}(R)$.\nIf the input weighting is chosen large enough, then also an MPC scheme without terminal constraints stabilizes the desired equilibrium.\n\nRegarding the knowledge of the system order $n=4$, it suffices if an upper bound on $n$ is available, i.e., if for instance $n=10$ is used in~\\eqref{eq:robust_MPC}.\nIf the system order is assumed lower than $n=4$, then the closed loop can be unstable.\nThe prediction horizon $L$ can be chosen (roughly) between $7\\leq L\\leq 70$.\nThe upper bound can be explained by noting that a larger $L$ implies that the constant $c_{pe}$ increases (compare the discussion after~\\eqref{eq:ass_pe_quantitative}) and therefore, the asymptotic tracking error increases.\nOn the other hand, the lower bound is due to the terminal equality constraints which require local controllability.\nMoreover, the steady-state tracking error, which can be seen e.g. in Figure~\\ref{fig:four_tank} (b), may increase or decrease, depending on the particular noise instance, and generally increases with the noise level $\\bar{\\varepsilon}$.\nThis confirms again the analysis of Section~\\ref{sec:robust}, which showed exponential stability of a set which grows with the noise level.\nFinally, if the norm of the data input $u^d$ increases (i.e., $c_{pe}$ decreases), then the tracking error decreases.\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn the present paper, we proposed and analyzed a novel MPC scheme with terminal equality constraints, which uses only past measured data for the prediction, without any prior system identification step.\nWe showed that, for a low noise amplitude, for a large ratio between persistence of excitation and the noise level, and for suitably tuned parameters, the closed loop in an $n$-step MPC scheme is recursively feasible and practically exponentially stable w.r.t. the noise level.\nTo the best of our knowledge, we have provided the first analysis regarding recursive feasibility and stability for a purely data-driven (model-free) MPC scheme.\nFurther, the analysis provides qualitative guidelines to choose the design parameters, and it illustrates the influence of other parameters, such as a persistence of excitation bound, on the region of attraction.\nWhile the MPC scheme is simple to implement, its analysis is challenging since we consider two sorts of noise: a) additive output noise and b) in the prediction model, similar to a multiplicative, parametric error in model-based MPC.\nIn an application to a practical example, we showed that the proposed MPC scheme guarantees stability, whereas an existing data-driven MPC scheme without terminal constraints leads to an unstable closed loop.\n\nSeveral topics for future research are left open.\nExtensions of the presented data-driven MPC approach to online optimization over artificial equilibria and robust output constraint satisfaction are provided in the recent works~\\cite{berberich2020tracking} and~\\cite{berberich2020constraints}, respectively.\nAnother extension, which would be highly interesting but also challenging, is the development of data-driven MPC schemes for \\emph{nonlinear} systems with meaningful closed-loop guarantees.\nFinally, many of the bounds employed in our proofs are conservative, and improving them may lead to less conservative, verifiable conditions on the admissible noise level for closed-loop stability.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMany modern applications encounter large data sets of very high dimension $d$, which requires new approaches for modeling as well as statistical inference, since classical methods usually fail. A particularly prominent example is the covariance matrix $\\Sigma$ of a random vector of $d$ variables, which has $d^2$ unknown parameters.\nIt is well-known that the sample covariance matrix $\\hat{\\Sigma}_n$ of sample size $n$ is singular if $n<d$, and consistency of $\\hat{\\Sigma}_n$ for growing dimension $d$ requires to control the $O(d^2) $ covariances and otherwise generally fails. Specifically, for $n$ i.i.d.\\ samples of $d$-dimensional random vectors with finite fourth moments, consistency in terms of the expected squared Frobenius norm of $\\hat{\\Sigma}_n- \\Sigma_n $ requires that $ \\mathrm{diag}( \\Sigma_n ) $ and the variance of the average of the squared variables to be of the order $ o(n\/d) $, \\citep{Ledoit2004}. In terms of eigenvalues and eigenvectors, classic inconsistency results for the largest eigenvalue and the top eigenvector under the regime $ d\/n \\to c \\in (0,1) $ are due to \\citep{Johnstone2001,Johnstone2009}, even when $ \\Sigma = I $. Since, however, it is generally believed that many high-dimensional data sets are governed by, say $r \\ll d$, strong signals, spiked covariance models with $r$ leading eigenvalues strictly larger than $1$, whereas the remaining bulk of the spectrum consists of unit eigenvalues, are quite extensively studied. Here the leading eigenvalues of $ \\hat{\\Sigma}_n $ are biased, thus requiring de-biasing procedures which, however, depend on the loss function, and the eigenvectors are inconsistent as well, see \\citep{DonohoGavinJohnstone2018} and the references therein. \nFurther potential remedies are to impose additional structural assumptions on $\\Sigma$, e.g.\\ sparsity \\citep{bickel2008a}, Toeplitz shape \\citep{cai2013}, or assuming the entries further from the diagonal to decay rapidly, also known as a bandable covariance matrix \\citep{bickel2008}.\nIn these frameworks, it is possible to derive consistent estimators of $\\Sigma$ even if $d\\gg n$, see \\cite{Cai2016} for a survey of suitable estimators and matching minimax bounds.\n\nThe structured covariance matrix estimators admit optimal rates of convergence within the specified class of matrices.\nHowever, if the true $\\Sigma$ does not satisfy the structural assumptions, the estimators might perform poorly.\nFor instance, if we use an estimator of Toeplitz form, but $\\Sigma$ is not a Toeplitz matrix, then this estimator will be inconsistent.\nIn order to strike a compromise between structured and model-free estimation, one may consider a shrinkage estimator of the form \n\\begin{align*}\n    \\hat{\\Sigma}^w_n = (1-w)\\,\\hat{\\Sigma}_n\\; + \\; w\\, \\tilde{\\Sigma}_n, \\qquad w\\in [0,1],\n\\end{align*}\nwhere $\\hat{\\Sigma}_n$ is the usual sample covariance matrix, and the shrinkage target $\\tilde{\\Sigma}_n$ is a structured estimator. \nThe weight $w$ may be interpreted as quantifying the confidence in the structural assumption underlying the estimator $\\tilde{\\Sigma}$. \nA particularly useful feature of shrinkage estimators is that $\\hat{\\Sigma}_n^w$ will be regular, as $\\tilde{\\Sigma}_n$ is usually chosen as a regular matrix. \nThis makes shrinkage estimation especially interesting for situations where the inverse of the covariance matrix is required, e.g.\\ in portfolio optimization.\n\\cite{Ledoit2004} suggest to use a multiple of the identity matrix, and \\cite{Steland2018a} studies more general diagonal matrices.\nLinear shrinking towards a structured target with respect to the scaled Frobenius loss has been studied in \\citep{LedoitWolf2003}, but the results therein on consistent estimators of the optimal shrinking weights consider classical fixed-$d$ asymptotics only. In the present paper, we go beyond a convex combination of the sample covariance matrix with a single shrinkage target. Instead, we study convex combinations with a structured covariance estimator with known minimax optimality properties for bandable $ \\Sigma $, and a structured covariance estimator which is minimax-optimal if $ \\Sigma $ is of Toeplitz type. \n\nTheoretical analysis of these structured covariance estimators typically focuses on their consistency and rate of convergence. Here, we contribute by showing that the estimators proposed when $ \\Sigma $ is bandable and Toeplitz, respectively, are still consistent under a general nonstationary  nonlinear time series model, whereas the known results assume iid $d$-dimensional random vectors. \nMoreover, to perform statistical inference, distributional approximations are required. \nHere, we consider inference about the high-dimensional covariance matrix in terms of projections $v^T\\Sigma v$ for some vector $v\\in\\mathbb{R}^d$.\nThe vector $v$ may either be determined by the application, or chosen at random, independently from the sample.\nFor example, in risk management, $v$ may represent weights of a portfolio of risky assets, and $v^T\\Sigma v$ corresponds to the variance of this portfolio. \nAs another example, the hypothesis $\\Sigma = \\Sigma_0$ may be studied via random projections: \nIf $v$ is drawn from some continuous distribution, then $\\Sigma\\neq \\Sigma_0$ implies $v^T\\Sigma v \\neq v^T\\Sigma_0 v$ almost surely.\nIt is thus reasonable to employ a test statistic of the form $v^T(S-\\Sigma_0) v$ for some estimator $S$, either structured, unstructured, or shrunken. Further applications (lasso prediction, sparse principal components) are discussed below. \nThe idea to study high-dimensional covariance matrices via bilinear forms has been suggested in \\cite{Steland2017, Steland2018}, and applied to changepoint testing \\citep{Steland2019} and to a K-sample problem \\cite{Mause2020}.\nIn the latter references, as well as in the present paper, the important assumption is that the projection vectors $v$ have a bounded $l_1$-norm. This condition is satisfied in many applications and enables the methodology to work also for very high dimensions $d$.\n\n\nIn this paper, we derive Gaussian approximation results for shrinkage estimators of the covariance matrix with structured shrinkage targets $\\tilde{\\Sigma}_n$. \nOur results are valid for nonstationary, nonlinear, high-dimensional time series $X_t$, and we demonstrate how to perform inference based on a suitable bootstrap scheme.\nNotably, we only impose assumptions on the marginal time series, while the dependency between the components may be arbitrary, and we need no restrictions on the dimension $d$. \nThis is a beneficial consequence of the projection technique, as described in Section \\ref{sec:projection}. \nAs we allow the dimension to grow with sample size, the considered models are arrays of time series.\nThus, classical central limit theory is not applicable because a limiting distribution in general does not exist.\nInstead, we formulate our results in terms of so-called strong approximations, or sequential couplings, in the sense of \\cite{komlos1975}.\nCompared to the existing work of \\cite{Steland2017, Steland2018}, we consider a much broader class of nonlinear time series models, which also allows us to study non-diagonal, structured shrinkage targets.\nUnderlying our mathematical results are recent Gaussian couplings for nonstationary time series established in \\cite{mies2022} for the regime $d\\ll n^\\frac{1}{3}$.\nWe extend these results to projections $v^T X_t$ of very high-dimensional time series $X_t$.\nIn particular, it is shown that for $l_1$-bounded projection vectors $v$, the Gaussian approximation is valid irrespective of the ambient dimension $d$, and of the dependence structure between the $d$ components.\nIt turns out that this general projection framework is also applicable to bilinear forms of sample covariance matrices, as demonstrated in Section \\ref{sec:coupling-cov}.\nThe presented distributional approximation also holds sequentially, and we demonstrate how to apply this to a test for structural breaks in the covariance matrix. \nAn attractive feature of the proposed test is that it is robust against general nonstationarity under the null hypothesis, and only detects changes in the target parameter, i.e.\\ the marginal covariance structure.\nFurthermore, we discuss the optimal choice of shrinkage weights and provide a data-driven criterion.\nSimulation results demonstrate that the asymptotic theory is also applicable in finite samples, and that shrinkage may improve the performance both for estimation and for change testing.\n\nThe rest of this paper is structured as follows. \nIn Section \\ref{sec:projection}, we present a general sequential Gaussian approximation result for projections of high-dimensional nonstationary time series, which is applied to bilinear forms of covariance matrices in Section \\ref{sec:coupling-cov}. \nThe data-driven choice of shrinkage weights is discussed in Section \\ref{sec:optimal}.\nSection \\ref{sec:changepoint} describes the application of our distributional approximations to tests for changes in the covariance structure of a high-dimensional time series, and presents a feasible bootstrap scheme.\nSimulation results are presented in Section \\ref{sec:simulation}.\n\n\n\n\n\n\\section{Gaussian approximations for projections}\\label{sec:projection}\n\nWe consider a $d$-variate time series $X_t$ which may be nonlinear and nonstationary, given by\n\\begin{align}\n\tX_t = G_t(\\epsilon_t,\\epsilon_{t-1},\\ldots), \\label{eqn:ts-model}\n\\end{align}\nfor measurable mappings $G_{t}:\\mathbb{R}^\\infty\\to\\mathbb{R}^{d}$, $t=1,\\ldots, n$, where we endow $\\mathbb{R}^\\infty$ with the $\\sigma$-algebra generated by all finite projections.\nThe $\\epsilon_i$ are iid $U[0,1]$ random variables, which may be regarded as random seeds for the time series.\nThe model formulation \\eqref{eqn:ts-model} allows for a convenient formulation of ergodicity conditions via the physical dependence measure introduced by \\cite{Wu2005}.\nTo formulate our assumptions on the kernels $G_t$, introduce a second sequence $\\tilde{\\epsilon}_i$ of iid $U[0,1]$ random variables, independent of the sequence $\\epsilon_i$.\nFor any $t\\in\\mathbb{N}$, denote \n\\begin{align*}\n\t\\boldsymbol{\\epsilon}_t &= (\\epsilon_t, \\epsilon_{t-1},\\ldots) \\in\\mathbb{R}^\\infty, \\\\\n\t\\tilde{\\boldsymbol{\\epsilon}}_{t,j} &= (\\epsilon_t,\\ldots, \\epsilon_{j+1}, \\tilde{\\epsilon}_j,  \\epsilon_{j-1},\\ldots) \\in\\mathbb{R}^\\infty, \n\\end{align*}\nsuch that $X_t = G_t(\\boldsymbol{\\epsilon}_t) = (G_{t,l}(\\boldsymbol{\\epsilon}_t))_{l=1}^d$.\nWe assume that the impact of past random seeds decays to zero polynomially, such that for some $\\beta>1$\n\\begin{align}\n\t\\begin{split}\n\t\t\\left(\\mathbb{E} |G_{t,l}(\\boldsymbol{\\epsilon}_t) - G_{t,l}(\\tilde{\\boldsymbol{\\epsilon}}_{t,t-j})|^q\\right)^\\frac{1}{q} &\\leq \\tilde{\\Theta} j^{-\\beta},\\qquad  j\\geq 0,\\quad l=1,\\ldots,d,\\\\\n\t\t\\left(\\mathbb{E} |G_{t,l}(\\boldsymbol{\\epsilon}_0)|^q\\right)^\\frac{1}{q} &\\leq \\tilde{\\Theta},\\qquad l=1,\\ldots,d. \n\t\\end{split}\\label{eqn:ass-ergodic-proj} \\tag{P.1}\n\\end{align}\n\nThe time series $X_t$ may be nonstationary, and the nonstationarity is explicit by making the kernel $G_t$ depend on $t$. \nNevertheless, to obtain stronger asymptotic results, we require some regularity in time. \nHere, we do not choose classical smoothness conditions, but rather formulate the regularity in terms of the total variation norm of the mapping $t\\mapsto G_t$, measured in $L_2(P)$.\nIn particular, we suppose that for some $\\tilde{\\Gamma}>0$,\n\\begin{align}\n\t\\sum_{t=2}^{n}\\left( \\mathbb{E} |G_{t,l}(\\boldsymbol{\\epsilon}_0) - G_{t-1,l}(\\boldsymbol{\\epsilon}_0)|^2 \\right)^\\frac{1}{2} \n\t&\\leq \\tilde{\\Gamma}\\cdot\\tilde{\\Theta},\\quad l=1,\\ldots,d. \\label{eqn:ass-BV-proj} \\tag{P.2}\n\\end{align}\n\nA few results require to assume\n\\[\n \\text{$ X_t $ has finite and stationary eighth moment and $ \\beta > 2 $.} \\tag{P.3}\n\\]\n\nWe highlight that \\eqref{eqn:ass-ergodic-proj} and \\eqref{eqn:ass-BV-proj} only impose conditions on the individual components $l=1,\\ldots,d$, and are thus rather easy to verify even for high-dimensional models. To emphasize, we impose no assumption on the dependency among the $d$ components.\n\nNow consider the projection of the $d$-variate time series $X_t$ to a $m$-dimensional time series $Z_t$, with $m\\ll d$.\nThat is, for a matrix $V\\in\\mathbb{R}^{m\\times d}$, we consider the time series $Z_t = V X_t$.\nWe measure the size of the matrix $V$ by the operator norm, $\\|V\\|_\\infty$, with respect to the maximum vector norm, i.e.\\\n\\begin{align*}\n\t\\|V\\|_\\infty = \\sup_{w\\in\\mathbb{R}^d} \\frac{\\|Vw\\|_\\infty}{\\|w\\|_\\infty} = \\max_{l=1,\\ldots,m} \\|V_{l,\\cdot}\\|_1,\n\\end{align*}\nwhere $V_{l,\\cdot}$ denotes the $l$-th row of the matrix $V$. This choice of norm uniformly controls the $ \\ell_1 $-norms of the projection vectors and is motivated by highdimensional statistical problems. Indeed, as well known, if $d$ is large compared to $n$, classical statistical methods usually fail, because even strong signals are overlayed by too many random noise sources (noise accumulation) and spurious correlations occur. To overcome the curse of high dimensionality, sparse methods typically use $ \\| \\cdot \\|_r$-norms with $ 1 \\le r < 2$, often leading to $s$-sparse solutions, i.e.\\ with only $s$ active (non-zero) coordinates.  For example, sparse PCA constructs $s$-sparse directions $ u $ on which the data vectors are projected by calculating $ u^T X_t$, and the variance of such a coefficients $ u^T X_t $, given by the quadratic form $ u^T \\Sigma_n u $ and coinciding with the associated eigenvalue when $ u $ is an eigenvector, provides information about the importance of that direction and is thus of interest. Similarly, when predicting a future response $\\tilde{Y}_t$ by a lasso regression on $ X_t $, one calculates $ \\hat{\\beta}_n^T \\tilde{X}_t $ for future regressors $ \\tilde{X}_t $, where the lasso estimate $ \\hat{\\beta}_n $ is $s-$sparse and thus selects $s \\ll d$ variables, so that it behaves like a low-dimensional statistic circumventing the issues arising for large dimension. The variance $ \\hat{\\beta}_n^T \\Sigma_n \\hat{\\beta}_n^T $ of the prediction $ \\hat{\\beta}_n^T \\tilde{X}_t$ is a natural measure of the prediction accuracy. When considering slightly more general vectors $ v $ with bounded $ \\| v \\|_1 $-norm (uniformly in $d$ and $n$), then we have the bound $ | v^T x_t | \\le \\| v \\|_1 \\max_{1 \\le j \\le d} | X_{tj} |$ at our disposal. Indeed, such projections have bounded moments under weak assumptions guaranteed by (P.1) as summarized in the following lemma, whose assertions no longer hold if $ \\| v \\|_2 $ is uniformly bounded but $ \\| v \\|_1 \\to \\infty $.\n\n\\begin{lemma}\\label{lem:jensen} Let $ q \\ge 1 $ and suppose that $ \\max_{1 \\le j \\le d} \\mathbb{E} |X_{tj}|^q \\le \\tilde{\\Theta} $ for some constant $C$ and all $d, n$, cf. (P.1). Then for any $ v \\in \\mathbb{R}^s $ with norm $ \\| v \\|_1 $ uniformly bounded by some $M$ \n\\[\n  \\left(\\mathbb{E} | v^T X_t |^q\\right)^{\\frac{1}{q}} \\le \\| v \\|_1 \\max_{1 \\le j \\le d} ( \\mathbb{E} |X_{tj}|^q )^{\\frac{1}{q}} \\le \\tilde{\\Theta} M\n\\]\n\\end{lemma}\n\n\n\n\nThe formulation of our first main result requires the two rates\n\\begin{align*}\n\t\\chi(q,\\beta) = \\begin{cases}\n\t\t\\frac{q-2}{6q-4}, & \\beta \\geq \\frac{3}{2}, \\\\\n\t\t\\frac{(\\beta-1)(q-2)}{q(4\\beta-3)-2}, &\\beta\\in(1,\\frac{3}{2}).\n\t\\end{cases},\n\t\\quad\n\t\\xi(q,\\beta) = \\begin{cases}\n\t\t\\frac{q-2}{6q-4}, & \\beta \\geq 3, \\\\\n\t\t\\frac{(\\beta-2)(q-2)}{(4\\beta-6)q-4},& \\frac{3+\\frac{2}{q}}{1+\\frac{2}{q}} < \\beta < 3, \\\\\n\t\t\\frac{1}{2}-\\frac{1}{\\beta}, & 2< \\beta \\leq \\frac{3+\\frac{2}{q}}{1+\\frac{2}{q}}.\\\\\n\t\\end{cases}\n\\end{align*}\n\n\\begin{theorem}\\label{thm:Gauss-ts-proj}\n\tLet $X_t=G_t(\\boldsymbol{\\epsilon}_t)$ with $\\mathbb{E}(X_t)=0$ be such that \\eqref{eqn:ass-ergodic-proj} holds for some $q>2$, $\\beta>1$, and let $Z_t = V X_t$ for some $V\\in\\mathbb{R}^{m\\times d}$ such that $m\\leq cn$ for some $c>0$.\n\tThen, on a potentially different probability space, there exist random vectors $(X_t')_{t=1}^n\\overset{d}{=} (X_t)_{t=1}^n$ and independent, mean zero, Gaussian random vectors $Y_t'$ such that\n\t\\begin{align}\n\t\t\\left(\\mathbb{E} \\max_{k\\leq n} \\left\\|\\frac{1}{\\sqrt{n}} \\sum_{t=1}^k  (V X_t' - Y_t') \\right\\|_2^2 \\right)^\\frac{1}{2}\n\t\t&\\leq C \\tilde{\\Theta} \\|V\\|_\\infty \\sqrt{m \\,\\log(n)}  \\left( \\frac{m}{n} \\right)^{\\chi(q,\\beta)}. \\label{eqn:Gauss-ts-1-proj}\n\t\\end{align} \n\tfor some universal constant $C=C(q,\\beta,c)$.\n\t\n\tIf $\\beta>2$, the local long-run variance $\\Xi_t(v) = \\sum_{h=-\\infty}^\\infty V\\operatorname{Cov}(G_t(\\boldsymbol{\\epsilon}_0),G_t(\\boldsymbol{\\epsilon}_h))V^T$ is well defined.\n\tIf \\eqref{eqn:ass-BV-proj} is satisfied aswell, then there exist random vectors $(X_t')_{t=1}^n\\overset{d}{=} (X_t)_{t=1}^n$ and independent, mean zero, Gaussian random variables $Y_t'\\sim\\mathcal{N}(0,\\Xi_t(v))$ such that\n\t\\begin{align}\n\t\t\\begin{split}\n\t\t\t\\left(\\mathbb{E} \\max_{k\\leq n} \\left\\| \\frac{1}{\\sqrt{n}}\\sum_{t=1}^k  (V X_t' - Y_t') \\right\\|_2^2 \\right)^\\frac{1}{2} \n\t\t\t&\\leq C \\tilde{\\Theta} \\|V\\|_\\infty \\sqrt{m \\,\\log(n)}  \\left( \\frac{m}{n} \\right)^{\\xi(q,\\beta)}.\n\t\t\\end{split}\n\t\t\\label{eqn:Gauss-ts-2-proj}\n\t\\end{align}\n\\end{theorem}\n\n\nThe rate of the approximation \\eqref{eqn:Gauss-ts-1-proj} is faster than \\eqref{eqn:Gauss-ts-2-proj}. \nThe difference is that in the first case, the covariance structure of the approximating Gaussian random vectors $Y_t'$ is not explicit.\nIn the second case, the distribution of the approximating random vectors is explicit, at the price of assuming some temporal regularity of the stochastic process $X_t$.\nIn Section \\ref{sec:changepoint}, we will describe a bootstrap scheme to perform inference based on Theorem \\ref{thm:Gauss-ts-proj}.\n\nNeglecting the high-dimensional context for a moment, the projection $Z_t=V X_t$ may be analyzed as a multivariate, nonstationary time series of fixed dimension $m$. \nIn this setting, \\cite{Wu2011} and \\cite{Karmakar2020} study sequential Gaussian approximations, the latter obtaining optimal rates in $n$ which are faster than the rates of Theorem \\ref{thm:Gauss-ts-proj}. \nHowever, they require a lower bound on the covariance matrices of $Z_t$, while our result does not need this assumption. \nThis is particularly relevant because we do not impose any conditions on the dependency among the components of $X_t$. \nAs an extreme example, if all $d$ components are independent, then a $l_1$-bounded projection will lead to some concentration due to averaging, such that a lower bound on the variance of the projection can in general not be guaranteed.\nThis prevents an application of the result of \\cite{Karmakar2020} in the present situation. \nMoreover, the covariance of the approximating random vectors in \\cite{Karmakar2020} is not explicit.\nInstead, we employ a result on Gaussian couplings presented in \\cite{mies2022}, which is in turn enabled by a recent result of \\cite{eldan2020}.\n\n\n\n\n\n\n\\section{Asymptotics of high-dimensional shrinkage covariance matrix estimators and their projections}\\label{sec:coupling-cov}\n\nFor a $d$-variate centered time series $X_t$ as above, denote the sample covariance matrix by $\\hat{\\Sigma}_{n,n} = \\frac{1}{n}\\sum_{t=1}^n X_t X_t^T$, the partial sums by $\\hat{\\Sigma}_{k,n} = \\frac{1}{n}\\sum_{t=1}^k X_t X_t^T$, $k=1,\\ldots, n$, and its mean by\n\\begin{align*}\n    \\Sigma_n \\;=\\; \\mathbb{E} \\hat{\\Sigma}_{n,n}  \\;=\\; \\frac{1}{n} \\sum_{t=1}^n \\operatorname{Cov}(X_t).\n\\end{align*}\nHence, we account for nonstationarity of the time series $X_t$ by averaging the covariance matrices at all times $t=1,\\ldots, n$.\n\nThe central observation enabling our subsequent analysis is that the matrix-valued time series $X_t X_t^T$ fits within the framework of Section \\ref{sec:projection}.\n\n\\begin{proposition}\\label{prop:matrix}\n\tIf the time series $X_t = G_t(\\boldsymbol{\\epsilon}_t)$ satisfies \\eqref{eqn:ass-ergodic-proj} resp.\\ \\eqref{eqn:ass-BV-proj} with power $2q$ and factor $\\tilde{\\Theta}$, then the $d^2$-variate time series $\\check{X}_t = X_t X_t^T = \\check{G}_t(\\boldsymbol{\\epsilon}_t)$ satisfies \\eqref{eqn:ass-ergodic-proj} resp.\\ \\eqref{eqn:ass-BV-proj} with power $q$ and factor $\\tilde{\\Theta}^2$.\n\\end{proposition}\n\n\nIt has been suggested by \\cite{Steland2017} to perform inference on the covariance matrix via the quadratic form $v^T \\hat{\\Sigma}_{n,n} v$ for some vector $v\\in\\mathbb{R}^d$, i.e., the variance of the projection $ v^T X_t$. As already discussed above, such projections are ubiquitous in statistical problems, for example arising in optimal portfolio selection, as linear predictors in regression models, or when reducing dimensionality by principal component analysis. The associated empirical version, $v^T \\hat{\\Sigma}_{n,n} v = \\frac{1}{n}\\sum_{t=1}^n |v^T X_t|^2$, may be interpreted as the estimated variance of the univariate projection $\\tilde{X}_t=v^TX_t$. The results of \\cite{Steland2017} and \\cite{Steland2018} on Gaussian approximations of $ v^T \\hat{\\Sigma}_n v $, related change-point procedures, and Shrinkage estimators, impose the assumption that the process $X_t$ is a linear time series with a single innovation process, which may be interpreted as a special type of single-factor model, although approximate vector autoregressions and spiked covariance models are included under regularity conditions, see \\cite{Steland2019}. Extensions to multi-factor models with infinitely many factors and general multivariate linear processes with finite-dimensional innovations are studied in \\cite{Bours2021}. Although such processes, especially factor models, provide good description for many application scenarios and are widely used, they nevertheless restrict the dependence structure of the $d$ component processes and rule out phenomena such as conditional heteroscedasticity and nonlinearity. Here, by relying on the results of the previous section, we allow for nonstationary nonlinear time series and impose much weaker assumptions on the dependence of the components.\n\n\n\nTo estimate $ \\Sigma_n $ and related quadractic forms $ v^T \\Sigma_n v $, we shall shrink the nonparametric estimator $ \\hat{\\Sigma}_{n,n} $ towards structured targets, in order to overcome\nvarious issues of the sample covariance matrix arising if $d\\gg n$.  A covariance matrix $\\Sigma$ is called bandable if $\\Sigma_{l,l'} \\leq g(|l-l'|)$ for some function $g(x)\\to 0$ as $x\\to\\infty$.\nIn this situation, optimal rates of convergence can be achieved via thresholding, see \\cite[Thm.\\ 7]{Cai2016} and references therein.\nIn particular, if $g(x) \\propto |x|^{-\\alpha-1}$ for some $\\alpha>0$, then a rate-optimal estimator is given by the tapering estimator\n\\begin{align*}\n\t\\hat{\\Sigma}_{k,n}^{\\dagger} &= ((\\hat{\\Sigma}_{k,n})_{l,l'} \\omega(|l-l'|))_{l,l'=1}^d, \\\\\n\t\\omega(x) &= \\begin{cases}\n\t\t1, & x\\leq \\frac{\\tau}{2}, \\\\\n\t\t2-\\frac{2x}{\\tau}, & \\frac{\\tau}{2}< x \\leq \\tau, \\\\\n\t\t0, & x>\\tau,\n\t\\end{cases}\n\\end{align*}\nThe optimal rate is achieved by the threshold $\\tau=\\min(n^\\frac{1}{2\\alpha+c},d)$ with $ c = 2 $ under the Frobenius norm $\\|\\cdot\\|_F$ and with $c=1$ under the spectral norm $\\|\\cdot\\|$, see \\cite{Cai2010}. \nIn particular, the optimal rates are\n\\begin{align*}\n    d^{-1}\\|\\widehat{\\Sigma}_n - \\Sigma\\|_F^2 \n    = \\mathcal{O}\\left( n^{-\\frac{2\\alpha+1}{2\\alpha+2}} + \\frac{d}{n} \\right), \\qquad\n   \n    \\|\\widehat{\\Sigma}_n - \\Sigma\\|^2 \n    = \\mathcal{O}\\left( n^{-\\frac{2\\alpha}{2\\alpha+1}} + \\frac{\\log d}{n} \\right).\n\\end{align*}\n\nAnother approach is to impose a Toeplitz structure, i.e.\\ $\\Sigma_{l,l'} = \\sigma_{|l-l'|}$, for some sequence $ \\sigma_0, \\sigma_1, \\ldots $. The corresponding tapered Toeplitz estimator of $\\Sigma$ is \n\\begin{align*}\n\t\\hat{\\Sigma}_{k,n}^\\diamond &= \\left( \\hat{\\sigma}_{|l-l'|} \\omega(|l-l'|) \\right)_{l,l'=1}^d, \\\\\n\t\\hat{\\sigma}_m &= \\frac{1}{d-m} \\sum_{l-l'=m} (\\hat{\\Sigma}_{k,n})_{l,l'}, \\quad 0 \\le m < d,\n\\end{align*}\nwith tapering weights $\\omega(x)$ as above.\nIf $|\\sigma_{m}|\\leq C m^{-\\alpha-1}$, then the optimal rate under the spectral norm is achieved by the threshold choice $\\tau = (nd \/ \\log(nd))^\\frac{1}{2\\alpha+1}$, see \\cite[Thm.\\ 9]{Cai2016} and \\cite{cai2013}.\n\nThe following theorem shows that the full-sample estimators $ \\hat{\\Sigma}_n^\\dagger = \\hat{\\Sigma}_{n,n}^\\dagger $ and $ \\hat{\\Sigma}_n^\\diamond = \\hat{\\Sigma}_{n,n}^\\diamond $ are consistent in the (rescaled) Frobenius norm, under mild regularity conditions. To the best of our knowledge, these estimators have not yet been studied under such a general nonstationary nonlinear time series model as given by (P.1) and (P.2). Moreover, $\\hat{\\Sigma}_n^\\dagger$ is rate-optimal for bandable covariance matrices. Note that the optimal rates and thresholds have been obtained under the assumption that $ X_t $ are iid random vectors with mean zero and a bandable or Toeplitz covariance matrix $ \\Sigma $ satisfying further regularity conditions. \n\nDenote by $ \\Sigma_n^\\dagger $ and $ \\Sigma_n^\\diamond $ the theoretical oracle estimators associated to the estimators $ \\hat{\\Sigma}_{n,n}^{\\dagger} $ and $ \\hat{\\Sigma}_{n,n}^\\diamond $, respectively, obtained by replacing in their definitions the estimates $ (\\hat{\\Sigma}_{n,n})_{i,j} $ by the true covariances $ (\\Sigma_{n,n})_{i,j} $.\nThat is, $\\Sigma_n^\\diamond = \\mathbb{E}( \\hat{\\Sigma}_n^\\diamond )$ and $\\Sigma_n^\\dagger = \\mathbb{E}( \\hat{\\Sigma}_n^\\dagger )$. \nRecall also that $ \\Sigma_n = \\mathbb{E}( \\hat{\\Sigma}_{n,n} )$. \nDenote for $A, B \\ge 0 $ the scaled inner product by $ \\langle A,B\\rangle_* = A \\cdot B \/ d $ and let $ \\| A \\|_{F*} = \\sqrt{\\langle A,A\\rangle_*} $ be the associated scaled Frobenius matrix norm. This scaling ensures that the unit matrix has norm $1$ whatever the dimension $d$. The following theorem studies the  above estimators under the general time series model given by (P.1) and (P.2) and provides sufficient conditions on the growth of dimension $ d $ relative to the sample size when measuring the estimation error in terms of the scaled Frobenius risk. \n\n\\begin{theorem} \n\\label{ConsistencyTapperToeplitz}\n\tSuppose that $ \\{ X_t \\} $ satisfies (P.1) with $q\\geq 4$ and $\\beta>1$, and that $ \\operatorname{Cov}(X_t) $ is bandable with exponent $\\alpha>0$ uniformly for all $ t $, i.e.\\ $\\operatorname{Cov}(X_t)_{i,j} \\leq C |i-j|^{-\\alpha-1}$.\n\t\n\t\\noindent(i)\n\tFor any threshold $\\tau$, it holds\n    \\[\n      \\mathbb{E} \\| \\hat{\\Sigma}_n^\\dagger - \\ \\Sigma_n \\|_{F*}^2 \n      \\;\\leq\\; C(\\Theta,\\beta)\\, \\left\\{ \\tau^{-2\\alpha-1} + \\frac{\\min(\\tau,d)}{n} \\right\\}.\n    \\]\n\tChoosing $\\tau = \\min(n^{\\frac{1}{2\\alpha+2}},\\, d)$, it holds\n\t\\[\n      \\mathbb{E} \\| \\hat{\\Sigma}_n - \\ \\Sigma_n^\\dagger \\|_{F*}^2 \n      \\;\\leq\\; C(\\Theta,\\beta)\\, \\min(n^{-\\frac{2\\alpha+1}{2\\alpha+2}},\\; \\tfrac{d}{n}).\n    \\]\n\n    \\noindent(ii)\n    If $\\Sigma_n$ is a Toeplitz matrix, then \n    \\[\n     \\mathbb{E} \\| \\hat{\\Sigma}_n^\\diamond - \\ \\Sigma_n \\|_{F*}^2  \n      \\;\\leq\\; C(\\Theta,\\beta)\\, \\left\\{ \\tau^{-2\\alpha-1} + \\frac{\\min(\\tau,d)}{n} \\right\\}.\n    \\]\n\tChoosing $\\tau = \\min(n^{\\frac{1}{2\\alpha+2}},\\, d)$, it holds\n\t\\[\n      \\mathbb{E} \\| \\hat{\\Sigma}_n^\\diamond - \\ \\Sigma_n \\|_{F*}^2  \n      \\;\\leq\\; C(\\Theta,\\beta)\\, \\min(n^{-\\frac{2\\alpha+1}{2\\alpha+2}},\\; \\tfrac{d}{n}).\n    \\]\n\\end{theorem}\n\nIn particular, the optimal rate of convergence of the tapering estimator $\\hat{\\Sigma}_n^\\dagger$ carries over to the dependent, nonstationary case, with identical threshold values.\n\nA shrinkage estimator may be defined for any weight $w\\in\\Delta = \\{(w_1, w_2,w_3)\\in[0,1]^3 : w_1+w_2+w_3=1\\}$, as \n\\begin{align*}\n\t\\hat{\\Sigma}_{k,n}^w &= w_1 \\hat{\\Sigma}_{k,n} + w_2 \\hat{\\Sigma}_{k,n}^\\dagger + w_3\\hat{\\Sigma}_{k,n}^\\diamond.\n\\end{align*}\nIt turns out that the theory presented in Section \\ref{sec:projection} may be applied to the quadratic form $v^T\\hat{\\Sigma}_n^w v$ for any shrinkage weight $w$, and hence also for the individual statistics $v^T\\hat{\\Sigma}_n v$, $v^T\\hat{\\Sigma}_n^\\dagger v$, and $v^T\\hat{\\Sigma}_n^\\diamond v$.\nThe crucial observation is that these quadratic forms can be regarded as linear projections of the $(d\\times d)$-variate time series $X_t X_t^T$ with respect to the Frobenius inner product $ A \\cdot B = \\sum_{i,j} a_{ij} b_{ij} $ defined for $ d \\times d $ matrices $ A = (a_{ij})_{i,j} $ and $ B = (b_{ij})_{i,j} $, which coincides with the usual inner product on $ \\mathbb{R}^{d^2} $ of the vectorized matrices. Indeed, we have the representations\n\\[\n  v^T \\hat{\\Sigma}_{k,n}^\\dagger v = \\nu^2 \\cdot \\hat{\\Sigma}_{k,n}, \\qquad v^T \\hat{\\Sigma}_{k,n}^\\diamond v = \\nu^3 \\cdot \\hat{\\Sigma}_{k,n},\n\\]\nfor the projection weighting matrices $\\nu^1,\\nu^2, \\nu^3\\in\\mathbb{R}^{d\\times d}\\equiv \\mathbb{R}^{d^2}$ given by\n\\begin{align*}\n\t\\nu^1 = (v_l v_{l'})_{l,l'=1}^d, &\\qquad \\|\\nu^1\\|_1 = \\sum_{l,l'=1}^d |v_l v_{l'}| = \\|v\\|_1^2, \\\\\n\t\\nu^2 = (v_l v_{l'} \\omega(|l-l'|))_{l,l'=1}^d, & \\qquad \\|\\nu^2\\|_1 \\leq \\|\\nu^1\\|_1 = \\|v\\|_1^2, \\\\\n\t\\nu^3 =  \\left( \\frac{\\omega(|l-l'|)}{d-|l-l'|}\\sum_{i-j=l-l'} v_iv_j\\right)_{l,l'=1}^d, \n\t& \\qquad \\|\\nu^3\\|_1 \\leq \\|\\nu^1\\|_1 = \\|v\\|_1^2\n\\end{align*}\nIn particular, we may write $v^T X_tX_t^T v = \\nu^1 \\cdot (X_t X_t^T)$ as the Frobenius inner product.\n\n\n\n\nWith Theorem \\ref{thm:Gauss-ts-proj}, we obtain the following result.\n\n\\begin{theorem}\\label{thm:Gauss-matrix}\n\tSuppose that the time series $X_t = G_t(\\boldsymbol{\\epsilon}_t)$ satisfies \\eqref{eqn:ass-ergodic-proj} resp.\\ \\eqref{eqn:ass-BV-proj} with power $2q$ and factor $\\tilde{\\Theta}$.\n\tThen, the time series $X_t$ may be defined on a different probability space, such that there exist independent, mean zero, Gaussian random vectors $\\eta_t\\in\\mathbb{R}^3$, such that\n\t\\begin{align*}\n\t\t\\left( \\mathbb{E} \\max_{k=1,\\ldots, n} \\max_{w\\in\\Delta} \\left|n v^T \\left[n\\hat{\\Sigma}_{k,n}^w - \\mathbb{E} \\hat{\\Sigma}_{k,n}^w\\right] v - w^T \\sum_{t=1}^k \\eta_t\\right|  \\right)^\\frac{1}{2} \n\t\t\\leq C \\tilde{\\Theta}^2 \\|v\\|_1^2 \\sqrt{\\log(n)}  n^{-\\xi(q,\\beta)}.\n\t\\end{align*}\n\tThe covariance of the Gaussian vectors $\\eta_t\\sim\\mathcal{N}(0,\\Xi_t)$ is given by\n\t\\begin{align*}\n\t\t(\\Xi_t)_{i,j} = \\sum_{h=-\\infty}^\\infty \\operatorname{Cov}\\left( \\nu^i \\cdot G_t(\\boldsymbol{\\epsilon}_0)G_t(\\boldsymbol{\\epsilon}_0)^T,\\; \\nu^j \\cdot G_t(\\boldsymbol{\\epsilon}_h)G_t(\\boldsymbol{\\epsilon}_h)^T \\right),\\quad i,j\\in \\{1,2,3\\}.\n\t\\end{align*}\n\\end{theorem}\n\nIt is worth mentioning that the bound in Theorem~\\ref{thm:Gauss-matrix} allows that $ \\| v \\|_1 $ increases as the dimension, $d$, or the sample size, $n$, grow, as long as $ \\| v \\|_1 = o( (\\log n)^{1\/4} n^{-\\xi(q,\\beta)\/2} ) $.\n\n\\section{Optimal shrinkage weights}\\label{sec:optimal}\n\nThe question arises, how the shrinking weights relate to the theoretical (oracle) performance of the resulting shrinkage estimator. To pursue such a study, we first consider the problem to combine the nonparametric estimator with oracles of the two targets within a framework proposed by \\cite{Ledoit2004} and aim at minimizing the squared Frobenius risk. This leads to a convex but box-constrained optimization problem allowing for an explicit interior solution (if it exists), which we briefly discuss. We propose consistent estimators of the unknown quantities determining the optimization problem and its solution(s) leading to data-adaptive bona fide estimates. \n\nFor simplicity of presentation, we elaborate this for the full sample of size $n$; the corresponding sequential estimates and oracles based on the first $k$ observations can then be derived easily. Write $ \\hat{\\Sigma}_n = \\hat{\\Sigma}_{n,n} $ and recall that  $ \\Sigma_n^\\dagger $ and $ \\Sigma_n^\\diamond $ are the theoretical oracle estimators associated to the estimators $ \\hat{\\Sigma}_{n,n}^{\\dagger} $ and $ \\hat{\\Sigma}_{n,n}^\\diamond $, respectively, obtained by replacing in their definitions the estimates $ (\\hat{\\Sigma}_{n,n})_{i,j} $ by the true covariances $ (\\Sigma_{n,n})_{i,j} $. Equivalently, $\\Sigma_n^\\diamond = \\mathbb{E} \\hat{\\Sigma}_n^\\diamond$ and $\\Sigma_n^\\dagger = \\mathbb{E} \\hat{\\Sigma}_n^\\dagger$.\n\nWe wish to determine the optimal ideal shrinking weights such that the resulting (scaled) Frobenius risk  is minimized when combining the nonparametric estimator  $ \\hat{\\Sigma}_n $ with the oracle estimators $ \\Sigma_n^\\dagger $ and $ \\Sigma_n^\\diamond $. The scaled mean squared error of the sample covariance matrix is given by $\\operatorname{MSE}( \\hat{\\Sigma}_n ) = \\operatorname{MSE}( \\hat{\\Sigma}_n; \\Sigma_n ) = E \\| \\hat{\\Sigma}_n - \\Sigma_n \\|_{F*}^2$, and we consider the risk minimization problem \n\\begin{align}    \n\\label{RMinP}\n    \\begin{split}\n  \\min_w& \\ \\ \\mathbb{E} \\| \\Sigma_n^w - \\Sigma_n \\|_{F*}^2 \\\\\n  \\text{where}\\quad&\\Sigma_n^w = w_1 \\hat{\\Sigma}_n + w_2 \\Sigma_n^\\dagger + w_3 \\Sigma_n^\\diamond,\\qquad \\ w_1+w_2+w_3=1.\n  \\end{split}\n\\end{align}\nFurther, let \n\\[\n  E_n^{\\dagger} = \\| \\Sigma_n^\\dagger - \\Sigma_n \\|_{F*}^2, \\qquad  E_n^{\\diamond} = \\| \\Sigma_n^\\diamond - \\Sigma_n \\|_{F*}^2\n\\]\nbe the squared approximation errors corresponding to the oracles, and \\[ D_n = \\langle \\Sigma_n^\\dagger - \\Sigma_n,  \\Sigma_n^\\diamond - \\Sigma_n \\rangle_*. \\] \nSubstituting $ w_1 = 1-w_2-w_3 $ and observing that $ \\mathbb{E} \\langle \\hat{\\Sigma}_n-\\Sigma_n, A \\rangle = 0$, for $ A \\in \\{ \\Sigma_n^\\dagger, \\Sigma_n^\\diamond \\} $, it suffices to determine minimizers of\n\\begin{equation}\n\\label{OptSimplified}\n  f( w_2, w_3 ) = (1-w_2-w_3)^2 \\mathbb{E} \\| \\hat{\\Sigma}_n - \\Sigma_n \\|_{F*}^2 + w_2^2 E_n^\\dagger + w_3^2 E_n^\\diamond + 2w_2w_3 D_n.\n\\end{equation}\nAdding the constraints $ 0 \\le w_1, w_2 \\le 1 $ leads to a quadratic programming problem under box constraints, which generally requires numerical algorithms for its solution. We discuss the special case of an interior solution, where explicit formulas can be derived, at the end of this section. The optimal weights depend on the unknown values $MSE(\\hat{\\Sigma_n}) = \\mathbb{E} \\|\\hat{\\Sigma_n}-\\Sigma_n\\|_{F*}^2$, $E_n^\\dagger$, $E_n^\\diamond$, and $D_n$. To obtain a data-driven solution for the optimal weights, we need consistent estimators of these quantities, which are also interesting in their own right.\n\nObserve that \n\\[\n  \\operatorname{MSE}(\\hat{\\Sigma}_n) = \\frac{1}{n d} \\sum_{i,j=1}^d \\operatorname{Var}( \\sqrt{n} ( \\hat{\\Sigma}_{n} )_{i,j} ). \n\\]\nUnder the assumptions of Proposition~\\ref{prop:matrix}  $ \\sqrt{n} (\\hat{\\Sigma}_{n} )_{i,j} = \\frac{1}{\\sqrt{n}} \\sum_{t=1}^n (X_t)_i (X_t)_j $ can be approximated in $ L_2 $ by $ \\frac{1}{\\sqrt{n}} \\sum_{t=1}^n Y_{t,i,j} $ for independent Gaussian random variables \n\\[ Y_{t,i,j} \\sim N( 0, \\xi_{t,i,j}^2 ) \\quad \\text{with}\\quad  \\xi_{t,i,j}^2 = \\sum_{h=-\\infty}^\\infty \\operatorname{Cov}( G_{t,i}(\\boldsymbol{\\epsilon}_0)G_{t,j}(\\boldsymbol{\\epsilon}_0), \\, G_{t,i}(\\boldsymbol{\\epsilon}_h )G_{t,j}(\\boldsymbol{\\epsilon}_h ) ). \\]\nThis yields \n\\[\n\\operatorname{MSE}(\\hat{\\Sigma}_n) = \\frac{1}{n d} \\sum_{i,j=1}^d \\frac{1}{n} \\sum_{t=1}^n \\xi_{t,i,j}^2 + o(1).\n\\]\n\n\n\\cite{Sancetta2008} proposed an estimator of $ \\operatorname{MSE}(\\hat{\\Sigma}_n) $ by estimating the long-run-variance parameters \n$\n\\operatorname{Var}( \\sqrt{n} ( \\hat{\\Sigma}_{n} )_{i,j} ) \n$ by the long-run-variance estimators \n\\[\n  \\widehat{\\operatorname{Var}}(  \\sqrt{n} ( \\hat{\\Sigma}_{n} )_{i,j} ) = \\hat{\\Gamma}_{n,i,j}(0) + 2 \\sum_{s=1}^{n-1} K(s\/b) \\hat{\\Gamma}_{n,i,j}(s),  \n\\]\nwhere $ K $ is a positive weighting function, $b>0$ is a bandwidth, and\n\\[\n    \\hat{\\Gamma}_{n,i,j}(s) =  \\frac{1}{n} \\sum_{t=1}^{n-s} [(X_t)_i (X_t)_j - (\\hat{\\Sigma}_n)_{i,j} ] \\cdot [ (X_{t+s})_i (X_{t+s})_j - (\\hat{\\Sigma}_n)_{i,j}  ].\n\\]\nThe mean square error may then be estimated as \n\\[\n\\widehat{MSE}(\\hat{\\Sigma}_n) = \\frac{1}{nd} \\sum_{i,j=1}^d \\widehat{\\operatorname{Var}}(\\sqrt{n} (\\hat{\\Sigma}_n)_{i,j}).\n\\] \nThe result is as follows, see \\cite[Lemma~1]{Sancetta2008}.\n\n\\begin{lemma}\\label{lem:MSE} If (P.1)-(P.3) hold, $ K : \\mathbb{R} \\to \\mathbb{R} $ is a decreasing positive c\\`adl\\`ag-function with $\\lim_{x\\to0+} K(x) = 1$ and $ \\int_0^\\infty K^2(x) dx < \\infty$, and $ b \\to \\infty $ with $ b = o( \\sqrt{n} ) $, then\n\\[\n  \\left| \\widehat{MSE}(\\hat{\\Sigma}_n) - MSE(\\hat{\\Sigma}_n) \\right| = o\\left( \\frac{d}{n} \\right).\n\\]\n\\end{lemma}\nIn practice, the bandwidth $b$ needs to be chosen in a data-driven way, e.g.\\ by the method described in \\cite{NeweyWest1994}.\n\n\nLet us now derive consistent estimators for the approximation errors $ E_n^\\dagger $ and $ E_n^\\diamond $ and the inner product $D_n$. The quantity $ E_n^\\diamond = \\| \\Sigma_n - \\Sigma_n^\\diamond \\|_{F*}^2$ can be estimated by plugging in the estimators $ \\hat{\\Sigma}_n^\\diamond $ and $\\hat{\\Sigma}_n^\\dagger $. Hence, define\n\\[\n  \\hat{E}_n^\\diamond = \\| \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger \\|_{F*}^2.\n\\]\nTo estimate $ E_n^\\dagger $ observe that $ \\Delta_n^\\dagger  = ( \\Delta_{n,ij}^\\dagger )_{i,j} = \\Sigma_n - \\Sigma_n^\\dagger $ is given by\n\\[\n  \\Delta_n^\\dagger = \\left\\{\n  \\begin{array}{ll}\n    0, & | i - j | \\le \\tau\/2, \\\\\n    (1-\\omega(|i-j|)) \\Sigma_{n,ij}, & \\tau\/2 < |i-j| \\le \\tau, \\\\\n    \\Sigma_{n,ij}, & |i-j| > \\tau.\n  \\end{array}\n  \\right.\n\\]\nEstimation of $\\Delta_n^\\dagger $ is delicate, since this involves estimating $\\sum_{m>\\tau} \\sum_{|i-j|=m} 1 = (d-\\tau)(d-\\tau+1)\/2 = O(d^2)$ off-diagonal terms which would lead to the unsatisfactory condition $ d^2 = o(n) $. For the class of bandable covariance matrices under consideration, we can proceed by introducing a second threshold and estimate only the covariances on the diagonals $ \\tau\/2+1, \\ldots, \\sigma $. This means, we estimate $ \\Delta_n^\\dagger $ by $ \\hat{\\Delta}_n^\\dagger = ( \\hat{\\Delta}_{n,ij}^\\dagger )_{i,j} $ with\n\\[\n  \\hat{\\Delta}_{n,ij}^\\dagger = \\left\\{\n  \\begin{array}{ll}\n    0, & | i - j | \\le \\tau\/2, \\\\\n    (1-\\omega(|i-j|)) \\hat{\\Sigma}_{n,ij}, & \\tau\/2 < |i-j| \\le \\tau, \\\\\n    \\hat{\\Sigma}_{n,ij}, & \\tau < |i-j| \\le \\sigma, \\\\\n    0, & |i-j| > \\sigma,\n  \\end{array}\n  \\right.\n\\]\nfor some threshold $ \\sigma $ with $ \\tau \\ll \\sigma \\ll d $, and put\n\\[\n  \\hat{E}_n^\\dagger = \\| \\hat{\\Delta}_n^\\dagger \\|_{F*}^2.\n\\]\nLastly, the estimation of $ D_n = \\langle \\Sigma_n - \\Sigma_n^\\dagger, \\Sigma_n - \\Sigma_n^\\diamond \\rangle_* $ can be based on the above estimators of $ \\Delta_n^\\dagger = \\Sigma_n - \\Sigma_n^\\dagger $ and $ \\Sigma_n - \\Sigma_n^\\diamond $,\n\\[\n  \\hat{D}_n = \\langle \\hat{\\Delta}_n^\\dagger, \\hat{\\Sigma}_n^\\dagger - \\hat{\\Sigma}_n^\\diamond \\rangle_*.\n\\]\nConsistency of $ \\hat{D}_n $, however, requires an extra condition compared to the estimators of the approximation errors. The reason is that $ \\Sigma_n - \\Sigma_n^\\dagger $ tends to zero in the $\\| \\cdot \\|_{F*} $-norm within the class of bandable matrices, whereas the error $ \\Sigma_n - \\Sigma_n^\\diamond $ may diverge. But within the the subclass of bandable matrices which are located in a neighborhood of their Toeplitz-approximation in the sense that \\[ \\| \\Sigma_n - \\Sigma_n^\\diamond \\|_{F*} = O(1) \\] consistency can be established, where $ \\Sigma_n^\\diamond $ is determined using the threshold $ n^{\\frac{1}{2\\alpha+c}} $; the threshold for the estimators are given below. We formulated the results for the highdimensional case $ d \\ge n^{\\frac{1}{2\\alpha+c}} $.\n\n\\begin{theorem}\n\\label{ConsistencyApproxErrors}\n\tSuppose that $ \\{ X_t \\} $ satisfies (P.1) and (P.2) and $ \\operatorname{Var}(X_t) $ is bandable for all $ t $.\nFurther, assume that $ d \\ge n^{\\frac{1}{2\\alpha+c}} $, and the estimators $ \\hat{\\Sigma}_n^\\dagger $, $ \\hat{\\Delta}_n^\\dagger $ and $ \\hat{\\Sigma}_n^\\diamond $ are calculated with thresholds $\\tau_n^\\dagger = n^{\\frac{1}{2\\alpha+c}} $, $\\sigma_n^\\dagger = n^{\\frac{1+s}{2\\alpha +c}} $ for some $ 0 < s < 2\\alpha+c-1$ and $ \\tau_n^\\diamond = \\left( \\frac{nd}{\\log nd} \\right)^{\\frac{1}{2\\alpha +1} }$, respectively. \n\\begin{itemize}\n\\item[(i)] If $ d = o( n^{\\frac{2\\alpha+2}{2\\alpha + 1}} ) $, then\n\\[\n \\mathbb{E} \\| \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger  - (\\Sigma_n^\\diamond - \\Sigma_n ) \\|_{F*}  = o(1).\n\\]\nand\n\\[\n  \\mathbb{E} \\left| \n    \\| \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger \\|_{F*} - \\| \\Sigma_n^\\diamond - \\Sigma_n \\|_{F*} \\right| = o(1),\n\\]\nwhich implies $ \\| \\hat{E}_n^\\diamond - E_n^\\diamond \\|_{F*} \\to 0 $ in probability.\n\n\\item[(ii)] We have\n\\[\n  \\mathbb{E} \\| \\hat{\\Delta}_n^\\dagger - \\Delta_n^\\dagger \\|_{F*}^2 = \n  \\left\\{\n  \\begin{array}{ll}\n    O\\left( n^{-\\frac{2\\alpha-1+c-s}{2\\alpha+c}} \\right), & \\qquad s > \\frac{c}{2\\alpha}, \\\\\n    O\\left( n^{-\\frac{(s+1)(2\\alpha-1)}{2\\alpha+c}} \\right), & \\qquad s \\le \\frac{c}{2\\alpha}, \\\\    \n  \\end{array}\n  \\right.\n\\]\nand\n\\[\n  \\mathbb{E} \\left|  \\| \\hat{\\Delta}_n^\\dagger \\|_{F*} - \\| \\Delta_n^\\dagger \\|_{F*} \n    \\right| = o(1),\n\\]\nsuch that $ \\| \\hat{E}_n^\\dagger - E_n^\\dagger \\|_{F*} \\to 0 $ in probability.\n\\item[(iii)] If $  \\| \\Sigma_n - \\Sigma_n^\\diamond \\|_{F*} = O(1) $ and $ d = o( n^{\\frac{2\\alpha+2}{2\\alpha + 1}} ) $, then\n\\[\n  | \\hat{D}_n - D_n | \\to 0,\n\\]\nin probability.\n\\end{itemize}\n\\end{theorem}\n\n\n\nIf, for example, $ \\alpha = 1 $, then the dimension needs to satisfy $ d^{3\/4}=o(n)$, i.e. it may grow almost as fast as $ n^{4\/3} $, and for $ \\alpha = 2 $ almost as fast as $ n^{6\/5} $.\n\nHaving consistent estimators at our disposal, we can propose the following bona fide estimators of the optimal weights, \n\\[ \n    \\left( \\hat{w}_1^*, \\hat{w}_2^*, \\hat{w}_3^*   \\right) = \\arg\\min_{w\\in \\Delta} w_1^2 \\widehat{MSE}(\\hat{\\Sigma}_n) + w_2^2 \\hat{E}_n^\\dagger + w_3^2 \\hat{E}_n^\\diamond + 2 w_2 w_3 \\hat{D}_n.\n\\]\nThis yields the feasible shrinkage estimator\n\\begin{align*}\n    \\hat{\\Sigma}_n^{w^*} = w_1^* \\hat{\\Sigma}_n + w_2^* \\hat{\\Sigma}_n^\\dagger + w_3^* \\hat{\\Sigma}_n^\\diamond.\n\\end{align*}\nThe performance of the corresponding shrinkage estimator $\\hat{\\Sigma}_n^{\\hat{w}^*}$ is assessed via simulations in the Section \\ref{sec:simulation}.\n\nLet us now briefly discuss the case of an interior solution of the minimization problem \\eqref{OptSimplified}. \nWe have\n\\begin{align*}\n  f( w_2, w_3 ) &= ( w_2,w_3) Q_n \\begin{pmatrix} w_2 \\\\ w_3 \\end{pmatrix} - a_n^\\top \\begin{pmatrix} w_2 \\\\ w_3 \\end{pmatrix} + \\operatorname{MSE}(\\hat{\\Sigma}_n),\n  \\end{align*}\nwhere\n\\[\nQ_n = \\begin{pmatrix} \\operatorname{MSE}(\\hat{\\Sigma}_n) + E^\\dagger & \\operatorname{MSE}(\\hat{\\Sigma}_n) + D_n \\\\ \\operatorname{MSE}(\\hat{\\Sigma}_n) + D_n & \\operatorname{MSE}(\\hat{\\Sigma}_n) + E_n^\\diamond \\end{pmatrix},\n\\quad a_n = \\begin{pmatrix} \\operatorname{MSE}(\\hat{\\Sigma}_n) \\\\ \\operatorname{MSE}(\\hat{\\Sigma}_n) \\end{pmatrix}.\n\\]\nHence, we are given a quadratic minimization problem under box constraints, and a natural condition is to assume that $ \\operatorname{det}(Q_n) > 0$. If an interior solution $ w^* \\in (0,1)^3$ exists (or the box constraint is omitted), the explicit solution is easily seen to be given by \n\\[\n  w_2^* = \\frac{ \\operatorname{MSE}(\\hat{\\Sigma}_n)( E_n^\\diamond - D_n) }{ F_n }, \\qquad w_3^* = \\frac{\\operatorname{MSE}(\\hat{\\Sigma}_n)( E_n^\\dagger - D_n )}{F_n}\n\\]\nwith\n\\[\n  F_n = \\operatorname{det}(Q_n) = E_n^\\dagger E_n^\\diamond + \\operatorname{MSE}( \\hat{\\Sigma}_n )( E_n^\\dagger + E_n^\\diamond - 2D_n ) - D_n^2.\n\\]\nThe resulting optimal weight for the sample covariance matrix $ \\hat{\\Sigma}_n $ is\n\\[\n  w_1^* = \\frac{E_n^\\dagger E_n^\\diamond - D_n^2}{F_n}. \n\\]\nWe see from the formula for $ w_1^* $ that the optimal solution prefers the nonparametric estimator $ \\hat{\\Sigma}_n $, if the truth $ \\Sigma_n $ cannot be well approximated by a banded or Toeplitz matrix. The banded estimator receives a large weight, if the distance between the Toeplitz oracle and $ \\Sigma_n $ is large. Analogously, the optimal solution assigns a large weight to the Toeplitz estimator, if $ \\Sigma_n $ is not well approximated by the banded oracle, so that $ \\| \\Sigma_n - \\Sigma_n^\\dagger \\|_F^2 $ is large.  As a result, it is optimal to distribute the weights across the estimators $ \\hat{\\Sigma}_n, \\hat{\\Sigma}_n^\\dagger $ and $ \\hat{\\Sigma}_n^\\diamond $ according to how well they approximate $ \\Sigma_n $, where the approximation accuracy is measured by the squared distances of the oracles to the truth and, in case of the nonparametric estimator, by the MSE. \n\nSince $ D_n^2 \\le E_n^\\dagger E_n^\\diamond $, we have $ w_1^* \\ge 0 $. If $D_n \\le \\min( E_n^\\dagger, E_n^\\diamond) $, then $ w_2^*, w_3^* \\ge 0 $ so that $ w \\in \\Delta $. \nIt is interesting to note that $ \\frac{w_2^*}{w_3^*} = \\frac{E_n^\\diamond - D_n}{E_n^\\dagger - D_n} $, i.e., the ratio of the oracle shrinking weights no longer dependents on $ \\operatorname{MSE}(\\hat{\\Sigma}_n) $. Further, $ w_{2}^* > w_3^* $ (banding preferable compared to Toeplitz) if and only if \n\\[ E_n^\\dagger < E_n^\\diamond \\Leftrightarrow \\| \\Sigma_n^\\dagger - \\Sigma_n \\|_{F*} < \\| \\Sigma_n^\\dagger - \\Sigma_n \\|_{F*}. \\]\nEven more is true: The above derivations remain formally true without any changes, if we use an arbitrary unbiased estimator $ \\tilde{\\Sigma}_n $ of $ \\Sigma_n $, i.e., an arbitrary measurable function $ \\tilde{\\Sigma}_n $ of the data with $ \\mathbb{E}( \\tilde{\\Sigma}_n ) = \\Sigma_n $. Therefore, we have the following interesting result: The ratio $\\frac{w_2^*}{w_3^*} $ is a universal characteristic of the risk minimization problem (\\ref{RMinP}) in the sense that it is independent of the nonparametric estimator and completely determined by $ \\Sigma_n $.\n\n\n\\section{Testing for structural breaks}\\label{sec:changepoint}\n\nThe sequential approximation result of Theorem \\ref{thm:Gauss-matrix} may be applied to test for changes in the covariance matrix $\\Sigma_t = \\operatorname{Cov}(X_t, X_t)\\in\\mathbb{R}^{d\\times d}$ of a time series $X_t$.\nTo be precise, we want to test the hypothesis\n\\begin{align*}\n\tH_0: \\Sigma_t \\equiv \\Sigma_0 \\quad\\leftrightarrow\\quad H_1: \\Sigma_t \\not\\equiv \\Sigma_0.\n\\end{align*}\nWe propose the following CUSUM-type statistics\n\\begin{align*}\n\tT_n(w) &= \\sqrt{n} \\max_{k\\leq n}  \\left|v^T \\left(\\hat{\\Sigma}^w_{k,n} - \\tfrac{k}{n} \\hat{\\Sigma}^w_{n,n}\\right) v \\right|,\\quad w\\in\\Delta, \\\\\n\tT_n^*(D) &= \\sup_{w\\in D}\\; T_n(w), \\qquad D \\subset \\Delta.\n\\end{align*}\n\nTo determine critical values for the statistic $T_n(w)$ resp.\\ $T_n^*$, we employ a bootstrap scheme suggested in \\cite{mies2022}.\nIn particular, let $\\eta_t\\sim \\mathcal{N}(0,A_t)$ be independent Gaussian random vectors in $\\mathbb{R}^3$, where\n\\begin{align*}\n\tA_t \t&= \\frac{1}{b} \\left(\\sum_{s=t-b+1}^t \\chi_s\\right)^{\\otimes 2} , \\qquad t=b,\\ldots, n, \\\\\n\t\\chi_s\t&= \\left[\\nu^j \\cdot (X_s X_s^T - \\hat{\\Sigma}_{n,n}) \\right]_{j=1}^3.\n\\end{align*}\nand $A_t=0$ for $t<b$. \nHere, $b$ is a block-length tuning parameter which needs to be chosen suitably, see below.\nWe now define the bootstrapped version of the changepoint statistics as\n\\begin{align*}\n\tR_n(w) &= \\sqrt{n}\\max_{k\\leq n} \\left|  \\frac{1}{n}\\sum_{t=b}^k w^T\\eta_t -  \\frac{k}{n} \\sum_{t=b}^n w^T \\eta_t  \\right|, \\\\\n\tR_n^*(D)&= \\sup_{w\\in D} R_n(w).\n\\end{align*}\nFinally, for $\\alpha\\in(0,1)$, introduce the conditional quantiles\n\\begin{align*}\n\ta_n(\\alpha,w) &= \\inf \\{a>0\\; :\\;  P(R_n(w)|X_1,\\ldots, X_n) \\leq \\alpha \\}, \\\\\n\ta_n^*(\\alpha,D) &= \\inf \\{a>0\\; :\\;  P(R_n^*(D)|X_1,\\ldots, X_n) \\leq \\alpha \\}.\n\\end{align*}\nWe suggest to reject the null hypothesis if\n\\begin{align*}\n\tT_n(w) > a_n(\\alpha,w)+\\delta_n, \\qquad \\text{resp.} \\qquad T_n^*(D) > a_n^*(\\alpha_n,D)+\\delta_n,\n\\end{align*}\nfor $\\delta_n = 1\/(\\log n)^p$ for some $p\\geq 1$.\n\n\\begin{theorem}\\label{thm:bootstrap}\n\tLet $X_{t} = X_{t,n} = G_{t}^n(\\boldsymbol{\\epsilon}_t)$ be an array of $d_n$-variate time series, such that each kernel $G_{t}^n$ satisfies (P.1) and (P.2), for some $q>8$, $\\beta>2$, and with factors $\\tilde{\\Theta}$ and $\\tilde{\\Gamma}$ not depending on $n$.\n\tChoose the block length $b$ such that $b=b_n \\asymp n^\\zeta$ for some $\\zeta\\in(0,1)$.\n\tIf $\\operatorname{Cov}(X_{t,n}) = \\Sigma_0$ for all $t=1,\\ldots, n$, and for some covariance matrix $\\Sigma_0$, then\n\t\\begin{align*}\n\t\t\\limsup_{n\\to\\infty} P\\left(  T_n(w) > a_n(\\alpha,w) + \\delta_n  \\right) &\\leq \\alpha, \\qquad w\\in\\Delta,\\\\\n\t\t\\limsup_{n\\to\\infty} P\\left(  T_n^*(D) > a_n^*(\\alpha,D) + \\delta_n  \\right) &\\leq \\alpha, \\qquad D\\subset\\Delta.\n\t\\end{align*}\n\\end{theorem}\n\n\nHence, the proposed bootstrap scheme indeed maintains the specified size for a change in covariance. \nA particular feature of this test is that it is robust against nuisance changes, because even under the null hypothesis the time series $X_t$ may be nonstationary, except for its stationary marginal covariance matrix. \nThe relevance of this kind of robustness has been first highlighted by \\cite{Zhou2013} for changes in mean, see also \\cite{Gorecki2018} and \\cite{Pesta2018}. Changes in the second moment structure in the presence of non-constant mean have been studied by \\cite{Dette2018} and \\cite{schmidt2021}, and a robust CUSUM test for a broad class of parameters is introduced in \\cite{mies2021}. Nevertheless, all these references focus on the low-dimensional case, whereas our new test is also applicable in high dimensions.\n\nThe power of the proposed test and bootstrap scheme is hard to investigate analytically, because the model framework is rather broad. Instead, we analyze the behavior of the proposed changepoint test via simulations in the subsequent section.\n\n\n\\section{Simulation}\\label{sec:simulation}\n\nTo demonstrate the implementation of our proposed changepoint test and the benefits of shrinkage, we assess our methodology for three examples of high-dimensional time series.\n\n\\subsection{Model A}\nFirst, we simulate a high-dimensional time series according to the vector ARMA(1,1) model\n\\begin{align*}\n\tX_t = a \\Pi X_{t-1} + b\\epsilon_t + \\epsilon_{t-1}.\n\\end{align*}\nThe matrix $\\Pi\\in\\mathbb{R}^{d\\times d}$ is a permutation matrix, such that the model is non-explosive.\nFurthermore, for any $t$, the $(\\epsilon_t)_i$, $i=1,\\ldots, d$, are chosen as independent, zero mean random variables having a symmetrized Gamma distribution with shape parameter $\\alpha(t\/n)$, standardized to unit variance. \nWe set $\\alpha(u) = 2+\\sin(2\\pi u)$, $u\\in[0,1]$. \nThus, the autocovariance structure is stationary, but the time-series is nonstationary. \nIn particular, the long run covariance matrices $\\Xi_t$ which occur in Theorem \\ref{thm:Gauss-matrix} are non-constant.\n\nWe assess our changepoint procedure under the null hypothesis of no change, and under the alternative where $b$ changes at time $t= \\lceil\\frac{n}{2}\\rceil$.\nThe projection vector $v$ is chosen randomly as $N\/\\|N\\|_1$ for a standard Gaussian random vector $N\\sim\\mathcal{N}(0,I_{d\\times d})$.\nThe threshold for the tapering estimator is chosen as $\\tau^{\\dagger}_n = n^{\\frac{1}{2\\alpha+1}} $, and $\\tau_n^{\\diamond} = (nd\/\\log(nd))^\\frac{1}{2\\alpha+1}$. \nNote that these thresholds are optimal for estimation under the spectral norm, in the class of bandable matrices with decay rate $\\alpha$. \nIn practice, $\\alpha$ is unknown, and we use the threshold for $\\alpha=2$ in our simulations.\nTo determine the shrinkage weights, we also use the threshold $\\sigma=\\sigma_n^\\dagger = n^{\\frac{1}{2\\alpha+1}} \\gg \\tau_n^{\\dagger}$.\nFor the bootstrap scheme, the additional offset is chosen as $\\delta_n = \\log(n)^{-4}$, and the block-length is $b_n=\\lceil 5n^{0.2} \\rceil$.\n\nTable \\ref{tab:power} reports the simulated size and power of the bootstrap test for different combinations of $n$ and $d=d_n$, and shrinkage weights $w^{(1)} = (1,0,0)$ and $w^{(2)}=(0.3, 0.3, 0.4)$.\nWe also consider the data-driven shrinkage weight $w^*$ as described in Section \\ref{sec:optimal}, which is not covered by our Gaussian approximation results.\nRegarding the choice of the bandwidth for the estimator $\\widehat{MSE}(\\hat{\\Sigma}_n)$, we employ the method of \\cite{NeweyWest1994} with Bartlett kernel as implemented in the R package \\textsc{sandwich} (see \\cite{sandwich2020}), and we determine a separate bandwidth for each component $\\widehat{\\operatorname{Var}}(\\sqrt{n}(\\hat{\\Sigma}_n)_{i,j})$.\n\nWhen simulating the process, we set $a=0.5$ and $b=0.5$ (resp.\\ $b=0.75$ post-change).\nBased on our simulation results, we find that the test maintains the specified size $10\\%$ and is indeed slightly conservative, as established theoretically in Section \\ref{sec:changepoint}. \nMoreover, the power of the changepoint test increases with sample size, as expected, but also with increasing dimension. \nThis may be explained by the fact that here, the change affects all coordinates, such that the additional information can be used to improve the detection performance. \nIt is also interesting to observe that the additional usage of the tapered and the Toeplitz estimator further improves the power of the test.\nThis is interesting because the shrinkage estimator has originally been proposed as a method to improve the performance of a point estimate. \nOur simulation results demonstrate that shrinkage can also be also useful for testing.\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{ll|cccc}\n         &  & $n=100$ & $n=500$ & $n=1000$ & $n=2000$ \\\\ \\hline\\hline\n         && \\multicolumn{4}{c}{Model A} \\\\ \\hline\\hline\n         $w^{(1)}$    & $d=4$    & 0.102 (0.09) & 0.220 (0.08) & 0.433 (0.09) & 0.727 (0.09) \\\\ \n         & $d=4n^{0.3}$   & 0.101 (0.09) & 0.289 (0.08) & 0.520 (0.08) & 0.824 (0.09) \\\\ \n         & $d=4n^{0.5}$  & 0.098 (0.07) & 0.282 (0.08) & 0.535 (0.08) & 0.822 (0.09) \\\\ \\hline\n         $w^{(2)}$   & $d=4$  & 0.107 (0.08) & 0.334 (0.08) & 0.594 (0.09) & 0.878 (0.10) \\\\ \n         & $d=4n^{0.3}$   & 0.155 (0.07) & 0.823 (0.08) & 0.990 (0.09) & 1.000 (0.09) \\\\ \n         & $d=4n^{0.5}$  & 0.201 (0.05) & 0.967 (0.07) & 1.000 (0.07) & 1.000 (0.08) \\\\ \\hline\n         $w^{*}$ & $d=4$  & 0.102 (0.08) & 0.252 (0.08) & 0.458 (0.09) & 0.751 (0.09) \\\\ \n         & $d=4n^{0.3}$   & 0.119 (0.08) & 0.530 (0.08) & 0.826 (0.08) & 0.985 (0.09) \\\\ \n         & $d=4n^{0.5}$  & 0.130 (0.06) & 0.697 (0.07) & 0.956 (0.07) & 1.000 (0.09) \\\\  \\hline\\hline\n   \n         && \\multicolumn{4}{c}{Model B} \\\\ \\hline\\hline\n    $w^{(1)}$    & $d=4$  & 0.108 (0.08) & 0.277 (0.08) & 0.503 (0.09) & 0.803 (0.09) \\\\ \n        & $d=4n^{0.3}$    & 0.102 (0.08) & 0.275 (0.07) & 0.502 (0.08) & 0.804 (0.09) \\\\ \n        & $d=4n^{0.5}$    & 0.101 (0.08) & 0.276 (0.07) & 0.505 (0.08) & 0.798 (0.09) \\\\ \\hline\n    $w^{(2)}$   & $d=4$   & 0.137 (0.08) & 0.438 (0.08) & 0.712 (0.09) & 0.938 (0.09) \\\\ \n        & $d=4n^{0.3}$    & 0.198 (0.08) & 0.861 (0.07) & 0.993 (0.08) & 1.000 (0.09) \\\\ \n        & $d=4n^{0.5}$    & 0.270 (0.07) & 0.966 (0.08) & 1.000 (0.08) & 1.000 (0.09) \\\\ \\hline\n        $w^{*}$ & $d=4$ & 0.115 (0.08) & 0.302 (0.08) & 0.531 (0.09) & 0.829 (0.09) \\\\ \n        & $d=4n^{0.3}$    & 0.133 (0.08) & 0.528 (0.07) & 0.837 (0.08) & 0.989 (0.09) \\\\ \n        & $d=4n^{0.5}$    & 0.157 (0.07) & 0.671 (0.07) & 0.941 (0.08) & 0.999 (0.09) \\\\\n        \\hline\\hline\n   \n        && \\multicolumn{4}{c}{Model C} \\\\ \\hline\\hline\n        $w^{(1)}$    & $d=4$  & 0.098 (0.08) & 0.273 (0.08) & 0.502 (0.08) & 0.806 (0.09) \\\\ \n        & $d=4n^{0.3}$        & 0.104 (0.08) & 0.277 (0.08) & 0.510 (0.08) & 0.806 (0.09) \\\\ \n        & $d=4n^{0.5}$        & 0.094 (0.08) & 0.269 (0.07) & 0.506 (0.08) & 0.799 (0.09) \\\\ \\hline\n    $w^{(2)}$   & $d=4$       & 0.124 (0.08) & 0.415 (0.08) & 0.692 (0.09) & 0.929 (0.09) \\\\ \n        & $d=4n^{0.3}$ & 0.170 (0.08) & 0.774 (0.08) & 0.971 (0.08) & 1.000 (0.09) \\\\ \n        & $d=4n^{0.5}$ & 0.225 (0.07) & 0.908 (0.08) & 0.997 (0.08) & 1.000 (0.09) \\\\ \n        \\hline\n     $w^{*}$   & $d=4$ & 0.108 (0.08) & 0.321 (0.08) & 0.573 (0.08) & 0.856 (0.09) \\\\ \n        & $d=4n^{0.3}$ & 0.132 (0.08) & 0.460 (0.08) & 0.733 (0.08) & 0.947 (0.09) \\\\ \n        & $d=4n^{0.5}$ & 0.143 (0.08) & 0.518 (0.07) & 0.786 (0.08) & 0.952 (0.09) \\\\ \\hline\n    \\end{tabular}\n\t\\caption{Power (and size), nominal level $10\\%$. p-values based on $10^3$ bootstrap samples, reported values based on $10^4$ Monte Carlo simulations.}\n    \\label{tab:power}\n\\end{table}\n\n\n\n\\subsection{Models B \\& C}\nAs a second example, we consider the vector ARMA(1,1) process $X_t$ given by\n\\begin{align*}\n    X_t = a X_{t-1} + b\\epsilon_t + \\epsilon_{t-1},\n\\end{align*}\nwith $a$ and $b$ as above, and $\\epsilon_t\\sim \\mathcal{N}(0, A)$ iid random vectors, and we initialize $X_0 \\sim \\mathcal{N}(0,A\\,(1+b^2)\/(1-a^2))$ to ensure weak stationarity. \nThe symmetric positive semidefinite matrix $A\\in\\mathbb{R}^{d\\times d}$ is chosen as $A_{i,j}=\\gamma(t_i-t_j)$, for $\\gamma(h) = |h+1|^{2H} + |h-1|^{2H} - 2|h|^{2H}$, which is the autocovariance function of fractional Gaussian noise. y\nFor the values $t_i$, we consider the scenarios (B) $t_i=i$, such that $A$ is a Toeplitz matrix, and (C) $t_i = \\sqrt{i}$, such that $A$ is not a Toeplitz matrix.\nFor the bootstrap scheme and for the determination of shrinkage weights, we use the same settings as in Model A.\n\nThe size and power of the changepoint test are presented in Table \\ref{tab:power}.\nAs for Model (A), the test turns out to be conservative, and gains power with increasing sample size, and higher dimension. \nIn our simulations, we also find the deterministic weight $w^{(2)}$ leading to higher power compared to the data-driven weight $w^*$.\nThus, future work could explore the optimal choice of shrinkage weights from a testing perspective.\n\nUnder the null hypothesis of no change, the marginal covariance matrix of $X_t$ is given by $\\operatorname{Cov}(X_t) = A \\frac{1+2ab + b^2}{1-a^2}$. \nThanks to this explicit formula, we are able to analyze the error of estimation of the proposed shrinkage estimator with data-driven shrinkage weights, see Table \\ref{tab:error-ARMA}. \nAs benchmarks, we evaluate the performance of the estimators $\\hat{\\Sigma}_n$, $\\hat{\\Sigma}_n^\\dagger$, and $\\hat{\\Sigma}_n^\\diamond$ individually. \nFor model B, the Toeplitz estimator $\\hat{\\Sigma}_n^\\diamond$ is found to perform best, which could be expected because the true covariance matrix is of Toeplitz type. \nOn the other hand, the shrinkage estimator always improves upon the sample covariance matrix $\\hat{\\Sigma}_n$, and is often close the performance of the tapered estimator $\\hat{\\Sigma}_n^\\dagger$.\nFor model C, the decay of the off-diagonal entries of the true covariance matrix is slower, hence the performance of $\\hat{\\Sigma}_n^\\dagger$ is also worse. \nIndeed, its estimation error is larger than the error of the sample covariance $\\hat{\\Sigma}_n$.\nYet, for $n\\geq 500$, the error of the shrinkage estimator is lower than that of all other estimators. \nThis demonstrates that our proposed shrinkage estimator not only chooses among the three estimators, but may indeed improve the performance even further due to the convex combination.\n\n\n\n\n\\begin{table}\n    \\centering\n    \\small\n    \\begin{tabular}{l|rrrr}\n         & \\multicolumn{1}{c}{$n=100$} & \\multicolumn{1}{c}{$n=500$} & \\multicolumn{1}{c}{$n=1000$} & \\multicolumn{1}{c}{$n=2000$} \\\\ \n        \n         \\hline\\hline\n         & \\multicolumn{4}{c}{Model B} \\\\ \\hline\\hline\n        \n         $d=4$ & 2.50$|$2.32$|$0.87$|$2.34 & 0.52$|$0.48$|$0.18$|$0.48 & 0.26$|$0.24$|$0.09$|$0.24 & 0.13$|$0.12$|$0.05$|$0.12 \\\\ \n        $d=4n^{0.3}$ & 6.38$|$3.84$|$0.78$|$4.01 & 2.55$|$1.29$|$0.16$|$1.42 & 1.49$|$0.76$|$0.08$|$0.82 & 0.95$|$0.49$|$0.04$|$0.53 \\\\ \n        $d=4n^{0.5}$  & 19.96$|$4.32$|$0.60$|$7.71 & 9.06$|$1.43$|$0.15$|$3.17 & 6.39$|$0.86$|$0.08$|$2.13 & 4.54$|$0.56$|$0.04$|$1.48 \\\\ \n       \n        \\hline\\hline\n        & \\multicolumn{4}{c}{Model C} \\\\ \\hline\\hline\n       \n        $d=4$ & 2.63$|$2.61$|$1.49$|$2.45 & 0.55$|$0.67$|$0.61$|$0.52 & 0.28$|$0.42$|$0.50$|$0.27 & 0.14$|$0.29$|$0.44$|$0.14 \\\\ \n        $d=4n^{0.3}$ & 6.85$|$5.17$|$3.91$|$4.80 & 2.73$|$2.55$|$5.21$|$1.98 & 1.58$|$1.79$|$5.69$|$1.23 & 1.01$|$1.26$|$6.80$|$0.82 \\\\ \n        $d=4n^{0.5}$ & 21.25$|$12.47$|$9.05$|$10.64 & 9.48$|$11.13$|$12.98$|$5.75 & 6.66$|$12.37$|$15.69$|$4.55 & 4.70$|$12.27$|$19.12$|$3.53 \\\\ \\hline\n    \\end{tabular}\n    \\caption{Mean square estimation error of the marginal covariance $\\Sigma=\\operatorname{Cov}(X_t)$ under stationarity, measured in the scaled Frobenius norm $\\|\\cdot\\|_{F*}$. Errors are reported, in this order, for (i) sample covariance $\\hat{\\Sigma}_n$, (ii) tapered estimator $\\hat{\\Sigma}_n^\\dagger$, (iii) Toeplitz-type estimator $\\hat{\\Sigma}_n^\\diamond$, (iv) the shrinkage estimator $\\hat{\\Sigma}_n^{w^*}$. Reported values based on $10^4$ Monte Carlo simulations.}\n    \\label{tab:error-ARMA}\n\\end{table}\n\n\\section{Proofs}\\label{sec:proofs}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:jensen}] The result follows from Jensen's inequality:\n\\begin{align*}\n\t\\mathbb{E} \\left| \\sum_{j=1}^d v_j X_{tj} \\right|^{q} \n\t\\le \\| v \\|_{1}^{q} \\mathbb{E} \\left( \\sum_{j=1}^d \\frac{ |v_j| }{\\| v \\|_{1}} | X_{tj} | \\right)^{q} \n\t\\le \\| v \\|_{1}^{q}  \\sum_{j=1}^d \\frac{ |v_j| }{\\| v \\|_{1}} \\mathbb{E} | X_{tj} |^{q} \n\t = \\| v \\|_{1}^{q} \\max_{1 \\le j \\le d} \\mathbb{E} | X_{tj} |^{q}\n\\end{align*}\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:Gauss-ts-proj}]\n\tThe time series $Z_t$ may be written as $Z_t = \\tilde{G}_t(\\boldsymbol{\\epsilon}_t)$, with kernel $\\tilde{G}_t = V G_t$, such that we may apply Theorem 3.1 of \\cite{mies2022}.\n\tIt can be verified that $\\tilde{G}_t$ satisfies conditions (G.1) and (G.2) therein, with $\\Gamma=\\tilde{\\Gamma}$ and $\\Theta = \\sqrt{m}\\tilde{\\Theta} \\|V\\|_\\infty$.\n\tTo see this, note that for any $m$-variate random variable $X$, we have\n\t\\begin{align*}\n\t\t(\\mathbb{E} \\|X\\|_2^q)^\\frac{1}{q} \n\t\t\\leq (\\mathbb{E} \\|X\\|_q^q)^\\frac{1}{q} m^{\\frac{1}{2}-\\frac{1}{q}} \n\t\t\\leq m^\\frac{1}{2} \\max_{l=1,\\ldots,m} (\\mathbb{E} |X_l|^q)^\\frac{1}{q},\n\t\\end{align*}\n\tbecause $\\|x\\|_2 \\leq m^{\\frac{1}{2}-\\frac{1}{q}} \\|x\\|_q$ for any vector $x\\in\\mathbb{R}^m$.\n\tMoreover, for any $l=1,\\ldots,m$,\n\t\\begin{align*}\n\t\t\\left( \\mathbb{E} |V_{l,\\cdot} G_t(\\boldsymbol{\\epsilon}_0)|^q\\right)^\\frac{1}{q} \n\t\t&= \\left( \\mathbb{E} \\left|\\sum_{r=1}^d V_{l,r} G_{t,r}(\\boldsymbol{\\epsilon}_0)\\right|^q\\right)^\\frac{1}{q} \\\\\n\t\t&\\leq  \\sum_{r=1}^d |V_{l,r}| \\, \\left(\\mathbb{E} |G_{t,r}(\\boldsymbol{\\epsilon}_0)|^q\\right)^\\frac{1}{q} \\qquad\n\t\t\\leq \\|V\\|_\\infty \\tilde{\\Theta},\n\t\\end{align*}\n\tsuch that\n\t\\begin{align*}\n\t\t\\left( \\mathbb{E} \\|V G_t(\\boldsymbol{\\epsilon}_0)\\|^q\\right)^\\frac{1}{q} \\leq \\sqrt{m} \\|V\\|_\\infty \\tilde{\\Theta}.\n\t\\end{align*}\n\tThe same argument can be used to establish (G.1) and (G.2).\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:matrix}]\n\tObserve that for four random variables $Y_1, Y_2, Z_1, Z_2 \\in L_{2q}$, it holds that\n\t\\begin{align*}\n\t\t(\\mathbb{E} |Y_1Z_1 - Y_2 Z_2|^{q})^\\frac{1}{q} \n\t\t&\\leq (\\mathbb{E} |(Y_1-Y_2)Z_1|^{q})^\\frac{1}{q} + (\\mathbb{E} |Y_2(Z_1- Z_2)|^{q})^\\frac{1}{q} \\\\\n\t\t&\\leq (\\mathbb{E} |Y_1-Y_2|^{2q})^\\frac{1}{2q} (\\mathbb{E} |Z_1|^{2q})^\\frac{1}{2q} + (\\mathbb{E} |Y_w|^{2q})^\\frac{1}{2q}(\\mathbb{E} |Z_1-Z_2|^{2q})^\\frac{1}{2q}.\n\t\\end{align*}\n\tInequality \\eqref{eqn:ass-ergodic-proj} may be obtained by setting $Y_1 = G_{t,l}(\\boldsymbol{\\epsilon}_t), Y_2 = G_{t,l}(\\tilde{\\boldsymbol{\\epsilon}}_{t,t-j})$ and $Z_1 = G_{t,l'}(\\boldsymbol{\\epsilon}_t), Z_2 = G_{t,l'}(\\tilde{\\boldsymbol{\\epsilon}}_{t,t-j})$.\n\tInequality \\eqref{eqn:ass-BV-proj} may be obtained by setting $Y_1 = G_{t,l}(\\boldsymbol{\\epsilon}_t), Y_2 = G_{t-1,l}(\\boldsymbol{\\epsilon}_{t})$ and $Z_1 = G_{t,l'}(\\boldsymbol{\\epsilon}_t)$, $Z_2 = G_{t-1,l'}(\\boldsymbol{\\epsilon}_{t})$. \n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{ConsistencyTapperToeplitz}]\n\t By \\cite[Th.~3.2]{mies2022} we have under Assumption (P.1) for all $d, n$\n\t \\[\n\t  \\left( \\mathbb{E} \\max_{k \\le n} \\left| \\sum_{t=1}^k X_{ti} X_{tj} - ( \\operatorname{Var}(X_t) )_{ij} \\right|^2 \\right)^{\\frac{1}{2}} \n\t  \\leq C(\\Theta,\\beta) n^\\frac{1}{2},\\quad 1\\leq i,j\\leq d,\n\t \\]\n\t such that $ \\hat{\\Sigma}_{n,ij} = ( \\hat{\\Sigma}_n)_{ij} $ satisfies \n\t \\begin{equation}\n\t    \\max_{1 \\le i, j \\le d} \\mathbb{E}( \\hat{\\Sigma}_{n,ij} -  \\ \\Sigma_{n,ij} )^2 \\leq C(\\Theta,\\beta) n^{-1}.\n\t \\end{equation}\n\t \n\t \\noindent\\underline{To show the bound for $\\hat{\\Sigma}_n^\\dagger$}, observe that\n\t \\[\n\t   \\mathbb{E} \\| \\hat{\\Sigma}_n^\\dagger - \\ \\Sigma_n \\|_{F*}^2 \\leq 4\\, ( A_n + B_n + C_n )\n\t \\]\n\t where\n\t \\begin{align*}\n\t \t  A_n & = d^{-1} \\sum_{m \\le \\tau\/2} \\sum_{|i-j|=m} \\mathbb{E}( \\hat{\\Sigma}_{n,ij} -  \\ \\Sigma_{n,ij} )^2, \\\\\n\t \t  B_n &= d^{-1} \\sum_{\\tau\/2 < m \\le \\tau} \\sum_{|i-j|=m} \\omega(|i-j|)^2 \\mathbb{E}\\left(  \\hat{\\Sigma}_{n,ij} -   \\Sigma_{n,ij} \\right)^2, \\\\\n\t \t  C_n &= d^{-1} \\sum_{\\tau\/2 < m \\le \\tau} \\sum_{|i-j|=m}  (1-\\omega(|i-j|))^2 \\Sigma_{n,ij}^2 , \\\\\n\t \t  D_n & =  d^{-1} \\sum_{m > \\tau} \\sum_{|i-j|=m} \\Sigma_{n,ij}^2.\n \t\\end{align*}\n    Clearly, $ |C_n + D_n| = O( \\tau^{-2\\alpha-1} ) $. \n    Furthermore, \n    \\begin{align*}\n     |A_n + B_n| \n     \\leq  \\frac{C}{n d} \\sum_{m\\leq \\tau} \\sum_{|i-j|=m} 1 \n     \\leq \\frac{C}{n} \\min(\\tau,d).\n     \\end{align*}\n    Thus, \n    \\begin{align*}\n        \\mathbb{E} \\| \\hat{\\Sigma}_n^\\dagger - \\ \\Sigma_n \\|_{F*}^2\n        \\leq C \\left\\{ \\tau^{-2\\alpha-1} + \\frac{\\min(\\tau,d)}{n} \\right\\}.\n    \\end{align*}\n    The optimal rate is attained for $\\tau = \\min( n^{\\frac{1}{2\\alpha+2}},\\, d)$. \n   \n   \\noindent\\underline{To show the bound for $\\hat{\\Sigma}_n^\\diamond$} observe that by Jensen's inequality\n   \\begin{align*}\n     \\mathbb{E}( \\hat{\\sigma}_m - \\ \\sigma_m )^2 \n     & =\\mathbb{E} \\left(  \\frac{1}{d-m} \\sum_{k-l=m} \\hat{\\Sigma}_{n,kl} - \\ \\Sigma_{n,kl} \\right)^2 \\\\\n     & \\le   \\frac{1}{d-m}  \\sum_{k-l=m}  \\mathbb{E}( \\hat{\\Sigma}_{n,kl} - \\ \\Sigma_{n,kl} )^2\n     \\\\\n     & \\leq C(\\Theta,\\beta) n^{-1},\n   \\end{align*}\nwhere $ \\ \\sigma_m = \\frac{1}{d-m} \\sum_{k-l=m} \\ \\Sigma_{n,kl} $ for all $i,j $ with $ i-j= m \\le \\tau$. We have\n\\begin{align*}\n\t\\mathbb{E} \\| \\hat{\\Sigma}_n^\\diamond - \\ \\Sigma_n \\|_{F,*}^2 \n\t& \\le \\frac{2}{d} \\sum_{m \\le \\tau} \\sum_{i-j=m} \\mathbb{E}(  \\hat{\\Sigma}_{n,ij}^\\diamond - \\ {\\Sigma}_{n,ij} )^2 + \\frac{2}{d} \\sum_{m>\\tau} \\sum_{i-j=m} \\ ({\\Sigma}_{n,ij})^2 \\\\\n\t& = \\frac{2}{d} \\sum_{m \\le \\tau} \\sum_{i-j=m} \\mathbb{E}( \\omega(i-j) ( \\hat{\\sigma}_m -  \\ \\sigma_m ) )^2 + \\frac{2}{d} \\sum_{m>\\tau} \\sum_{i-j=m} \\ ({\\Sigma}_{n,ij})^2 \\\\\n\t& = G_n + H_n.\n\\end{align*}\nSince $ \\ {\\Sigma}_{n,ij}^\\diamond \\leq C \\, m^{-\\alpha-1}$, we have $\\sigma_m \\leq C m^{-\\alpha-1}$ for $ i-j = m$.\nThus, $H_n = O(\\tau^{-2\\alpha-1})$.\nMoreover, $G_n = O(\\min(d,\\tau) \/ n)$.\nWe obtain the same bound as in case (i), i.e.\n    \\begin{align*}\n        \\mathbb{E} \\| \\hat{\\Sigma}_n^\\diamond - \\ \\Sigma_n \\|_{F*}^2\n        \\leq C \\left\\{ \\tau^{-2\\alpha-1} + \\frac{\\min(\\tau,d)}{n} \\right\\}.\n    \\end{align*}\n    Optimal choice of $\\tau$ yields the same rates as in case (i).\n \\end{proof}\n \n\\begin{proof}[Proof of Lemma \\ref{lem:MSE}] One only needs to verify Condition 2 of \\cite{Sancetta2008}. Essentially, this is specific Doukhan-Louhichi type weak dependence condition. It is known that Bernoulli shifts satisfy such weak dependence conditions, but the applying the results in the literature would require to impose stronger assumptions on $G_t$. Therefore, we give a direct proof working under (P.1). Let $ \\bar{\\epsilon}_{i,j} = (\\epsilon_i, \\ldots, \\epsilon_{j+1}, \\epsilon_j', \\epsilon_{j-1}', \\ldots) $ where $ \\{ \\epsilon_t' \\} $ is an independent copy of $\\{ \\epsilon_t \\} $. Fix $u, v \\in \\{1, \\ldots, 4\\} $, $ (i_1,\\ldots, i_v), (s_1,\\ldots, s_v) \\in \\mathbb{N}^v $ and $ (j_1,\\ldots, j_u), (t_1, \\ldots, t_u) \\in \\mathbb{N}^u$ such that $ s_1 \\le \\cdots \\le s_u + r \\le t_1 \\le \\cdots \\le t_v $ for some $ r \\in \\mathbb{N}$. Put $ U_{t_v}(\\boldsymbol{\\epsilon}_{t_v} ) = X_{t_1,j_1} \\cdots X_{t_v,j_v}$ and $ V_{s_u}(\\boldsymbol{\\epsilon}_{s_u} ) = Y_{s_1,i_1} \\cdots Y_{s_u,i_u} $. Then\n\\begin{align*}\n  \\operatorname{Cov}( U_{t_v}(\\boldsymbol{\\epsilon}_{t_v} ), V_{s_u}(\\boldsymbol{\\epsilon}_{s_u} ) ) &\n   = \\operatorname{Cov}( U_{t_v}( \\boldsymbol{\\epsilon}_{t_v-s_u} ), V_{s_u}(\\boldsymbol{\\epsilon}_0) ) \\\\\n  &= \\operatorname{Cov}( U_{t_v}( \\bar\\boldsymbol{\\epsilon}_{t_v-s_u,0} ), V_{s_u}(\\boldsymbol{\\epsilon}_0) ) + \\operatorname{Cov}( U_{t_v}( \\boldsymbol{\\epsilon}_{s_u,0} ) - U_{t_v}( \\bar\\boldsymbol{\\epsilon}_{t_v-s_u,0} ), V_{s_u}(\\boldsymbol{\\epsilon}_0) )  \\\\\n  &= \\operatorname{Cov}( U_{t_v}( \\boldsymbol{\\epsilon}_{s_u,0} ) - U_{t_v}( \\bar\\boldsymbol{\\epsilon}_{t_v-s_u,0} ), V_{s_u}(\\boldsymbol{\\epsilon}_0) )\n\\end{align*}\nBy telescoping over the random variables $ U_{t_v}(l) = U_{t_v}( \\epsilon_{t_v-s_u}, \\ldots, \\epsilon_{t_v-s_u-l+1}, \\bar\\boldsymbol{\\epsilon}_{t_v-s_u-l} ) $, so that $ \\sum_{l>t_v-s_u} U_{t_v}(l) - U_{t_v}(l-1) = U_{t_v}( \\boldsymbol{\\epsilon}_{s_u,0} ) - U_{t_v}( \\bar\\boldsymbol{\\epsilon}_{t_v-s_u,0} ) $, and noting that $ \\mathbb{E} | (U_{t_v}(l) - U_{t_v}(l-1) ) V_{s_u}(\\boldsymbol{\\epsilon}_0) | \\le \\Theta' l^{-\\beta} $ for some constant $ \\Theta' $ by (P.1), we obtain\n\\[\n\\operatorname{Cov}( U_{t_v}(\\boldsymbol{\\epsilon}_{t_v} ), V_{s_u}(\\boldsymbol{\\epsilon}_{s_u} ) ) = O( r^{-\\beta+1} ),\n\\]\nwhich establishes the Doukhan-Louhichi type weak dependence condition required in Condition 2 of \\cite{Sancetta2008}, since $ \\beta > 2$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{ConsistencyApproxErrors}] The first assertion follows from rearranging terms,\n\t\\[\n\t   \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger  - (\\Sigma_n^\\diamond - \\Sigma_n )  = (\\hat{\\Sigma}_n^\\diamond - \\Sigma_n^\\diamond ) - (\\hat{\\Sigma}_n^\\dagger - \\Sigma_n)\n\t\\]\n\tand the triangle inequality, so that\n\t\\begin{align*}\n\t\\mathbb{E} \\| \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger  - (\\Sigma_n^\\diamond - \\Sigma_n )  \\|_F &\\le \\mathbb{E} \\| \\hat{\\Sigma}_n^\\diamond - \\Sigma_n^\\diamond \\|_F + \\mathbb{E} \\| \\hat{\\Sigma}_n^\\dagger - \\Sigma_n \\|_F \\\\\n\t& \\le \\left(  \\mathbb{E} \\| \\hat{\\Sigma}_n^\\diamond - \\Sigma_n^\\diamond \\|_F^2 \\right)^\\frac{1}{2} + \\left( \\mathbb{E} \\| \\hat{\\Sigma}_n^\\diamond - \\Sigma_n^\\diamond \\|_F^2  \\right)^\\frac{1}{2}. \n\t\\end{align*}\n\tHence, the result follows from Theorem~\\ref{ConsistencyTapperToeplitz}.\n\t\n\tFor the second assertion, use the fact that if $ A_n - B_n \\le C_n $ on $ A_n \\ge B_n $ and $ B_n-A_n \\le D_n $ on $ A_n < B_n $, for random variables $A_n, B_n $ and bounds $ C_n, D_n $, then\n\t\\begin{align*}\n\t  \\mathbb{E} | A_n - B_n | &\\le \\mathbb{E} (A_n-B_n)1_{A_n\\ge B_n} + \\mathbb{E} (-A_n+B_n)1_{A_n<B_n} \\\\\n\t  & \\le \\mathbb{E} C_n + \\mathbb{E} D_n \n\t\\end{align*}\n    Apply this bound with $ A_n = \\| \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger \\|_{F*} $, $ B_n = \\| \\Sigma_n^\\diamond - \\Sigma_n^\\dagger \\|_{F*} $ and the bounds \n\\[\n     \\| \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger \\|_{F*} - \\| \\Sigma_n^\\diamond - \\Sigma_n\\|_{F*}  \\\\\n      \\le \\| \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger - ( \\Sigma_n^\\diamond - \\Sigma_n  ) \\|_{F*} = C_n\n\\]\nas well as\n\\[\n \\| \\Sigma_n^\\diamond - \\Sigma_n \\|_{F*} -\\| \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger \\|_{F*}  \\\\\n\\le \\| \\Sigma_n^\\diamond - \\Sigma_n  - ( \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger ) \\|_{F*} = D_n,\n\\]\nwhich follow from the triangle inequality. Therefore,\n\\[\n  \\mathbb{E} \\left| \\| \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger \\|_{F*} - \\| \\Sigma_n^\\diamond - \\Sigma_n\\|_{F*} \\right| = o(1),\n\\]\nwhich implies $ \\left| \\| \\hat{\\Sigma}_n^\\diamond - \\hat{\\Sigma}_n^\\dagger \\|_{F*} - \\| \\Sigma_n^\\diamond - \\Sigma_n\\|_{F*} \\right| \\to 0 $, in probability, which in turn gives $\\left| \\hat{E}_n^\\diamond - E_n^\\diamond \\right| \\to 0 $, in probability. \n\nTo show (ii) observe that we have, similarly as above, the bounds\n\\[\n \\pm ( \\| \\hat{\\Delta}_n^\\dagger \\|_{F*} - \\| \\Delta_n^\\dagger \\|_{F*} ) \\le \\| \\hat{\\Delta}_n^\\dagger - \\Delta_n^\\dagger \\|_{F*}\n\\]\nyielding\n\\[\n  \\mathbb{E} \\left| \\| \\hat{\\Delta}_n^\\dagger \\|_{F*} - \\| \\Delta_n^\\dagger \\|_{F*} \\right|\n  \\le 2 \\mathbb{E} \\| \\hat{\\Delta}_n^\\dagger - \\Delta_n^\\dagger \\|_{F*}\n  \\le 2 \\sqrt{ \\mathbb{E} \\| \\hat{\\Delta}_n^\\dagger - \\Delta_n^\\dagger \\|_{F*}^2 }.\n\\]\nThus, it suffices to study $ \\mathbb{E} \\| \\hat{\\Delta}_n^\\dagger - \\Delta_n^\\dagger \\|_{F*}^2 $. We have\n\\begin{align*}\n  \\mathbb{E} \\| \\hat{\\Delta}_n^\\dagger - \\Delta_n^\\dagger \\|_{F*}^2\n  & = d^{-1} \\sum_{\\tau^\\dagger\/2 \\le m \\le \\tau^\\dagger} \\sum_{|i-j|=m} (1-\\omega(|i-j|))^2 \\mathbb{E}( \\hat{\\Sigma}_{n,ij} - \\Sigma_{n,ij} )^2 \\\\\n  & \\qquad + d^{-1} \\sum_{\\tau^\\dagger<m\\le \\sigma^\\dagger} \\sum_{|i-j|=m} \\mathbb{E}( \\hat{\\Sigma}_{n,ij} - \\Sigma_{n,ij} )^2 \\\\\n  & \\qquad +  d^{-1} \\sum_{m>\\sigma^\\dagger} \\sum_{|i-j|=m} \\Sigma_{n,ij}^2 \\\\\n  & = O( \\sigma^\\dagger n^{-1} ) + O( (\\sigma^\\dagger)^{-2\\alpha+1} ) \\\\\n  & =O\\left( n^{-\\frac{2\\alpha-1+c-s}{2\\alpha+c}} \\right) + O\\left( n^{-\\frac{(s+1)(2\\alpha-1)}{2\\alpha+c}} \\right).\n\\end{align*}\nBoth terms are $o(1) $ and the first term dominates if and only if $s>c\/2\\alpha$. \n\n(iii) The claim follows directly from the fact that\n\\[\n \\langle \\hat{\\Delta}_n^\\dagger, \\hat{\\Sigma}_n^\\dagger - \\hat{\\Sigma}_n^\\diamond \\rangle_* \n - \\langle \\Delta_n^\\dagger, \\Sigma_n^\\dagger - \\Sigma_n^\\diamond \\rangle_* \n = \\langle \\hat{\\Delta}_n^\\dagger - \\Delta_n^\\dagger, \\hat{\\Sigma}_n^\\dagger - \\hat{\\Sigma}_n^\\diamond \\rangle_*\n -  \\langle \\Delta_n^\\dagger, \\Sigma_n^\\dagger - \\Sigma_n^\\diamond -\\hat{\\Sigma}_n^\\dagger - \\hat{\\Sigma}_n^\\diamond  \\rangle_* \n\\]\nso that\n\\begin{align*}\n\\left| \\langle \\hat{\\Delta}_n^\\dagger, \\hat{\\Sigma}_n^\\dagger - \\hat{\\Sigma}_n^\\diamond \\rangle_* \n - \\langle \\Delta_n^\\dagger, \\Sigma_n^\\dagger - \\Sigma_n^\\diamond \\rangle_*   \\right| \n & \\quad \\le \\| \\hat{\\Delta}_n^\\dagger - \\Delta_n^\\dagger \\|_{F*} \\| \\hat{\\Sigma}_n^\\dagger - \\hat{\\Sigma}_n^\\diamond  \\|_{F*} \\\\\n & \\quad + \\| \\Sigma_n - \\Sigma_n^\\dagger \\|_{F*} \\| \\Sigma_n^\\dagger - \\Sigma_n^\\diamond -\\hat{\\Sigma}_n^\\dagger - \\hat{\\Sigma}_n^\\diamond \\|_{F*} \\\\\n & = O( \\| \\hat{\\Delta}_n^\\dagger - \\Delta_n^\\dagger \\|_{F*} \\| ( \\| \\Sigma_n - \\Sigma_n^\\diamond \\|_{F*} + o_P(1) ) + o(1) \\\\\n & = o_P(1),\n\\end{align*}\nsince\n\\[\n  \\| \\hat{\\Sigma}_n^\\dagger - \\hat{\\Sigma}_n^\\diamond \\|_{F*} \n  = \\| \\Sigma_n - \\Sigma_n^\\diamond \\|_{F*} + o_P(1)\n\\]\nby (i),\n\\[\n  \\| \\hat{\\Delta}_n^\\dagger - \\Delta_n^\\dagger \\|_{F*} \\to 0,  \\quad \\text{in probability},\n\\]\nby (ii), \n\\[\n  \\| \\Sigma_n - \\Sigma_n^\\dagger \\|_{F*}^2 \\le d^{-1} \\sum_{m>\\tau\/2} \\sum_{|i-j|=m} \\Sigma_{n,ij}^2 = O\\left( (\\tau\/2)^{-2\\alpha+1} \\right)\n  = O\\left( n^{-\\frac{2\\alpha-1}{2\\alpha+c}} \\right)\n\\]\nand $ \\| \\Sigma_n - \\Sigma_n^\\diamond \\|_{F*} = O(1) $ by assumption. This completes the proof. \n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:bootstrap}]\n    Note that the statistics $T_n$ and the critical values $a_n(\\alpha,w)$ are linear in $\\|v\\|_1$, such that we may suppose without loss of generality that $\\|v\\|_1=1$.\n    If we replace the $\\chi_s$ by the centered random vectors $\\bar{\\chi}_s = [\\nu^j \\cdot (X_sX_s^T - \\Sigma_0)]_{j=1}^3$, then Theorem \\ref{thm:bootstrap} is a special case of \\cite[Prop.\\ 4.3]{mies2022}.\n    We briefly show that the estimation error of $\\hat{\\Sigma}_n$ versus the true covariance matrix is negligible.\n    To this end, denote the sequential estimators of the asymptotic covariance, for $k=b,\\ldots, n$, by \n    \\begin{align*}\n        \\hat{Q}(k) = \\sum_{t=b}^k \\frac{1}{b} \\left( \\sum_{s=t-b+1}^t \\chi_s \\right)^{\\otimes 2}, \\qquad\n        \\overline{Q}(k) = \\sum_{t=b}^k \\frac{1}{b} \\left( \\sum_{s=t-b+1}^t \\bar{\\chi}_s \\right)^{\\otimes 2} .\n    \\end{align*}\n    It suffices to show that\n    \\begin{align}\n        \\mathbb{E} \\max_{k=1,\\ldots, n} \\|\\hat{Q}(k) - \\overline{Q}(k)\\|_\\text{tr} = \\mathcal{O}(\\sqrt{nb}), \\label{eqn:boostrap-estimation}\n    \\end{align}\n    where $\\|A\\|_\\text{tr} = \\text{tr}((A A^T)^\\frac{1}{2})$ denotes the trace norm of a symmetric matrix.\n    Since the matrices $\\hat{Q}(k)$ and $\\overline{Q}(k)$ are always of fixed dimension $3\\times 3$, the choice of norm is in fact irrelevant as all norms are equivalent, and we may use an arbitrary matrix norm $\\|\\cdot\\|$.\n    If \\eqref{eqn:boostrap-estimation} holds, then both $\\hat{Q}(k)$ and $\\overline{Q}(k)$ satisfy Theorem 5.1 in \\cite{mies2022}, such that the bootstrap scheme based on $\\hat{Q}(k)$ is valid.\n    \n    To establish \\eqref{eqn:boostrap-estimation}, denote $\\Delta_n = \\left[\\nu^j \\cdot (\\Sigma_0 - \\hat{\\Sigma}_{n,n})\\right]_{j=1}^3 = \\chi_s - \\bar{\\chi}_s \\in \\mathbb{R}^3$, for all $s$. \n    Via quadratic expansion, we find that, for some universal $C$ which may change from line to line,\n    \\begin{align*}\n        &\\mathbb{E} \\max_{k}\\left\\| \\hat{Q}(k) - \\overline{Q}(k) \\right\\| \\\\\n        &= \\mathbb{E} \\max_k \\left\\| \\sum_{t=b}^k \\frac{1}{b} \\left[ \\left(\\sum_{s=t-b+1}^t (\\bar{\\chi}_s + \\Delta_n) \\right)^{\\otimes 2} -  \\left(\\sum_{s=t-b+1}^t \\bar{\\chi}_s \\right)^{\\otimes 2} \\right] \\right\\| \\\\\n        &= \\mathbb{E} \\max_k \\left\\| \\sum_{t=b}^k \\frac{1}{b} \\left[ \\left(\\sum_{s=t-b+1}^t (\\bar{\\chi}_s (b\\Delta_n)^T + (b\\Delta_n) \\bar{\\chi}_s^T + (b\\Delta_n)^{\\otimes 2})\\right)  \\right] \\right\\| \\\\\n        &\\leq C \\, \\mathbb{E} \\sum_{t=b}^n \\left[ 2\\left\\|\\sum_{s=t-b+1}^t \\bar{\\chi}_s  \\Delta_n^T \\right\\| + b\\left\\|\\Delta_n\\right\\|^2 \\right] \\\\\n        &\\leq C\\sum_{t=b}^n \\sqrt{\\mathbb{E} \\left\\| \\sum_{s=t-b+1}^t \\bar{\\chi}_s \\right\\|^2} \\sqrt{\\mathbb{E} \\|\\Delta_n\\|^2} + nb \\mathbb{E} \\|\\Delta_n\\|^2.\n    \\end{align*}\n    By virtue of Proposition \\ref{prop:matrix} and Lemma \\ref{lem:jensen}, the centered time series $\\bar{\\chi}_s$ satisfies condition (G.1) of \\cite{mies2022}, such that the Rosenthal-type inequality, Theorem 3.2 therein, is applicable. \n    This yields,\n    \\begin{align*}\n        \\sqrt{\\mathbb{E} \\left\\| \\sum_{s=t-b+1}^t \\bar{\\chi}_s \\right\\|^2}\n        &\\leq C \\sqrt{b} \\sum_{j=1}^\\infty \\tilde{\\Theta} j^{-\\beta} = \\mathcal{O}(\\sqrt{b}), \n    \\end{align*}\n    because $\\tilde{\\Theta}$ is fixed in the asymptotic regime of Theorem \\ref{thm:bootstrap}.\n    Analogously, \n    \\begin{align*}\n        \\sqrt{\\mathbb{E} \\|\\Delta_n\\|^2} \n        &= \\sqrt{ \\mathbb{E} \\left\\| \\tfrac{1}{n} \\sum_{t=1}^n \\bar{\\chi}_t \\right\\|^2 } = \\mathcal{O}(1\/\\sqrt{n}).\n    \\end{align*}\n    Hence, $\\mathbb{E} \\max_{k}\\left\\| \\hat{Q}(k) - \\overline{Q}(k) \\right\\| = \\mathcal{O}(\\sqrt{nb} + b) = \\mathcal{O}(\\sqrt{nb})$. \n    The last inequality holds because $b\\leq n$, and establishes \\eqref{eqn:boostrap-estimation}.\n\\end{proof}\n\n\\section*{Acknowledgements}\nThe authors acknowledge support from Deutsche Forschungsgemeinschaft (DFG, grant STE 1034\/11-2).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe interaction of light with vibrational modes close to the ground state leads to an asymmetry in the generated Stokes and anti-Stokes scattering. \nPhysically, this asymmetry originates from the fact that a mechanical oscillator in the ground state can only absorb energy.\nThe anti-Stokes scattering (resulting in blue-shifted radiation) rate scales with $\\bar n$, while the Stokes scattering rate scales with $\\bar n+1$, where $\\bar n$ denotes the average thermal occupation of the vibrational mode~\\cite{clerk_introduction_2010}.\nThe ratio of the two scattering rates is given by the Boltzmann factor $\\exp(-\\hbar\\Omega_m\/k_BT)$, where $\\Omega_m$ is the frequency of the oscillator and $T$ its temperature,\nwhich therefore allows for absolute and self-calibrated quantum noise thermometry.\nSuch motional sideband asymmetry has been observed in the quantized motion of laser-cooled trapped ions~\\cite{diedrich_laser_1989} and cold atoms in optical lattices~\\cite{jessen_observation_1992}.\nIt has been exploited for Raman noise thermometry in solids~\\cite{hart_temperature_1970, nemanich_raman_1984, cui_noncontact_1998} and molecules~\\cite{eckbreth_laser_2000}, and it is commercially used in fiber optical distributed temperature measurements~\\cite{ukil_distributed_2012}.\n\nCavity optomechanical techniques have enabled to prepare mechanical oscillators close to the ground state via optomechanical sideband cooling~\\cite{schliesser_resolved-sideband_2008,chan_laser_2011,verhagen_quantum-coherent_2012}, and therefore reach a regime where the interaction of mechanical oscillators and optical (or microwave) fields has to be treated quantum mechanically.\nIn this context, significant attention has been devoted to sideband asymmetry~\\cite{safavi-naeini_observation_2012,purdy_optomechanical_2015,underwood_measurement_2015,weinstein_observation_2014}, as a signature of the quantum-mechanical nature of the interaction between light and engineered mechanical oscillators.\n\n\nTo exploit sideband asymmetry for an absolute thermometer without the need for calibration~\\cite{purdy_quantum_2017}, understanding all noise contributions is central.\nFor example, if the laser exhibits excess classical noise in the phase or amplitude quadrature, sideband asymmetry is not a faithful measure of the mechanical thermal occupation, but is artificially increased~\\cite{jayich_cryogenic_2012, sudhir_appearance_2017, kippenberg_phase_2013}, resulting in incorrect temperature measurements.\n\nHere we observe experimentally and describe theoretically a novel mechanism that can lead to an artificially modified sideband asymmetry,\nwhich occurs in the presence of multiple laser drives due to the cavity nonlinear response, such as the Kerr effect or the photothermal cavity frequency shift.\nThermal effects are common for both optical~\\cite{verhagen_quantum-coherent_2012,barclay_nonlinear_2005} and microwave~\\cite{suh_thermally_2012} cavities.\nIn a cavity driven by multiple lasers, the cavity nonlinearity leads to a coupling of originally independent thermomechanical sidebands, modifying the sideband asymmetry measurements.\nMore generally, any optomechanical measurement protocol that utilizes multiple drive tones on a single cavity mode can be affected.\nThis therefore includes backaction-evading measurements~\\cite{suh_mechanically_2014}, mechanical squeezing~\\cite{kronwald_arbitrarily_2013, lecocq_quantum_2015,wollman_quantum_2015,pirkkalainen_squeezing_2015}, dissipative squeezing of the cavity field~\\cite{kronwald_dissipative_2014},  entanglement of two mechanical oscillators via reservoir engineering~\\cite{woolley_two-mode_2014,ockeloen-korppi_stabilized_2018}, and recently demonstrated non-reciprocal devices~\\cite{bernier_nonreciprocal_2017, peterson_demonstration_2017}.\n\nThis unwanted artificial asymmetry can be eliminated by sufficiently separating the driving tones in frequency, thus operating beyond the bandwidth of the Kerr-type nonlinearity, as verified by independent measurements.\nThe revealed intrinsic quantum motional sideband asymmetry then enables to perform self-calibrated thermometry.\nWe proceed to optomechanically cool a $5.3\\unit{GHz}$ breathing mode of a nanobeam optomechanical crystal, thermalized in a $^3$He buffer gas cryostat at $\\sim 4.5\\unit{K}$,\nself-calibrated at $1.50\\pm 0.2$ quanta (40\\% ground state probability).\n\n\\section{Experimental Results}\n\\label{sec:ExprResults}\n\n\\begin{figure*}\n  \\includegraphics[scale=1]{fig_experimental_combined_5.pdf}\n  \\caption{\\textbf{Optomechanical crystal and experimental setup.}\n  (a)~False-color SEM image of the silicon optomechanical crystal cavity with a waveguide for laser input coupling.\n  The path of the tapered fiber is indicated.\n  The inset shows the simulated mechanical breathing mode and optical mode.\n  (b)~SEM image of the cavity.\n  (c)~Scheme for motional sideband asymmetry measurement using two probe tones and a cooling tone.\n  (d)~Experimental setup. PM, phase modulator; VOA, variable optical attenuator;\nAOM, acousto-optical modulator; BHD, balanced heterodyne detector; SA, spectrum analyzer; NA, network analyzer; PLL, phase-locked loop.}\n\\label{fig:experimental}\n\\end{figure*}\n\nOur experimental system is a nanobeam optomechanical crystal~\\cite{chan_optimized_2012}, shown in Fig.~\\ref{fig:experimental}.\nOptically, the device functions as a single-sided cavity with a partially transmitting input mirror.\nLight is evanescently coupled from a tapered optical fiber into a waveguide that forms part of the nanobeam with efficiency exceeding 50\\%.\nThe optical resonance is at $1540\\unit{nm}$ with a linewidth of $\\kappa\/2\\pi=1.6\\unit{GHz}$, of which $\\kappa\\s{ex} = 0.3\\kappa$ are extrinsic losses to the input mirror.\nThe optical mode is optomechanically coupled to a mechanical breathing mode of frequency $\\Omega_m\/2\\pi=5.3\\unit{GHz}$, strongly confined due to a phononic bandgap, with an intrinsic linewidth of  $\\Gamma\\s{int}\/2\\pi=84\\unit{kHz}$.\nThis places the system in the resolved sideband regime~\\cite{schliesser_resolved-sideband_2008}, $\\Omega_m>\\kappa$, as required for ground state cooling~\\cite{wilson-rae_theory_2007,marquardt_quantum_2007}.\nThe optomechanical coupling parameter is $g_0\/2\\pi = 780\\unit{kHz}$.\nFull details of the device design, the setup and the system are given in Appendices~\\ref{sec:fabrication} and~\\ref{sec:expr_details}.\n\nThough optomechanical crystals, owing to their GHz-scale frequencies, have been put in their ground state by employing passive cooling in dilution refrigerators at $\\sim 10\\unit{mK}$, \nsevere heating due to optical absorption precluded continuous measurements with high optomechanical cooperativities~\\cite{meenehan_silicon_2014,meenehan_pulsed_2015,riedinger_non-classical_2016,hong_hanbury_2017,riedinger_remote_2018}.\nOur system is passively cooled to low initial thermal occupancy at $4.5\\unit{K}$ (thermal occupancy at bath temperature $\\bar n\\s{th}\\simeq 17$) using a $^3$He buffer gas cryostat.\nThe buffer gas provides additional mechanical damping, increasing $\\Gamma\\s{int}$ by several 10s kHz to the actual mechanical linewidth $\\Gamma_m$.\nHowever,\nas detailed in Appendix~\\ref{sec:sideband_cooling}\nand shown in the measurements described in this section, the buffer gas environment enables greatly enhanced thermalization of the oscillator and is necessary for cooling close to the ground state.\n\nTo measure motional sideband asymmetry, we employ two probe tones, around upper and lower motional sidebands, and in addition apply a strong tone near the lower sideband for sideband cooling (Fig.~\\ref{fig:experimental}c).\nSuch multi-tone probing has been applied in previous experiments in the microwave domain~\\cite{weinstein_observation_2014,lecocq_quantum_2015}.\nThe two weak equal-power probes (about $n_c=100$ mean intracavity photons each) are applied at frequencies $\\omega\\s{cav}\\pm(\\Omega_m+\\delta)$.\nThe cooling tone is blue-detuned from the lower probe by the frequency $\\Omega\\s{mod}$, where $\\Gamma_m\\ll\\Omega\\s{mod}\\ll\\kappa$.\nThe red-detuned probe will generate a resonantly enhanced anti-Stokes Raman process, where a probe photon is upconverted in frequency from $-\\Omega_m-\\delta$ to $-\\delta$, while destroying a phonon in the mechanical oscillator. \nThe converse occurs for the blue-detuned probe, where a probe photon is downconverted from $\\Omega_m+\\delta$ to $\\delta$ while creating a phonon, thus forming the resonant enhanced Stokes sideband.\nThe experimental setup is shown in Fig.~\\ref{fig:experimental}d.\nWe use balanced heterodyne detection for quantum-limited measurement of the scattered thermomechanical sidebands in the output spectrum.\nThe overall detection  efficiency is $\\eta\\simeq 4\\%$.\nWe measure the symmetrized autocorrelator of the photocurrent \\cite{weinstein_observation_2014,BowenMilburn2015}\n$\\bar S_I(\\omega)=\\frac12\\int_{-\\infty}^\\infty\\langle\\{\\overline{\\hat{I}_{\\text{out}}(t+t'),\\hat{I}_{\\text{out}}(t')}\\}\\rangle e^{i\\omega t}dt$, where the average over $t'$ is denoted by an overbar.\nThe one-sided heterodyne spectrum takes the form (in the resolved sideband limit)\n\\begin{equation}\n  \\begin{aligned}\n\t  {}&\\bar S_I(\\omega+\\Delta_{\\text{LO}})=1\n\t+\\eta\\Gamma_{\\text{tot}}\\left\\{\\frac{\\Gamma_m\\C_{\\text{red}}\\bar n}{\\Gamma_{\\text{tot}}^2\/4+(\\omega+\\delta)^2}\\right.\\\\\n\t&+\\left.\\frac{\\Gamma_m\\C_{\\text{cool}}\\bar n}{\\Gamma_{\\text{tot}}^2\/4+(\\omega+\\delta-\\Omega_{\\text{mod}})^2}\n\t+\\frac{\\Gamma_m\\C_{\\text{blue}}(\\bar n+1)}{\\Gamma_{\\text{tot}}^2\/4+(\\omega-\\delta)^2}\\right\\},\n  \\end{aligned}\n  \\label{eq:ideal_theory}\n\\end{equation}\nwhere we have introduced cooperativities $\\C_{\\text{red,blue,cool}}\\equiv 4g_{r,b,c}^2\/(\\kappa\\Gamma_m)$, the light-enhanced optomechanical coupling $g_{r,b,c}\\equiv g_0\\sqrt{n_{r,b,c}}$, reduced occupancy $\\bar n=\\Gamma_m\\bar n\\s{th}\/\\Gamma_{\\text{tot}}$, local oscillator detuning $\\Delta_{\\text{LO}}=\\omega_{\\text{LO}}-\\omega_{\\text{cav}}$, and incorporated the effect of sideband cooling into a broadened mechanical linewidth $\\Gamma_{\\text{tot}} \\simeq \\Gamma_m (1+\\C_{\\text{cool}})$.\nIn the last expression we assume weak probe tones of equal strength, such that they do not contribute to the total mechanical damping, and taken the good cavity limit $\\kappa\/\\Omega_m\\to\\infty$, thus neglecting the quantum backaction limit~\\cite{wilson-rae_theory_2007,marquardt_quantum_2007}.\nWe further assume that the cavity linewidth is much larger than the detuning and effective mechanical linewidth $\\kappa\\gg \\delta,\\Gamma_{\\text{tot}}$, such that the optical susceptibility can be evaluated on resonance, and neglect classical laser noise. \nEquation~\\eqref{eq:ideal_theory} is normalized to the shot noise floor, includes the overall detection efficiency $\\eta$, \nand we have chosen to show only sidebands close to resonance, the others are heavily suppressed due to the cavity resonance.\n\nFor our measurement, the red and blue cooperativities are chosen to be equal $\\C_{\\text{red}}=\\C_{\\text{blue}}\\equiv\\C$,\nsuch that the Lorentzian probe tone sidebands centered around $\\Delta_{\\text{LO}}\\pm\\delta$\nhave weights proportional to $\\C \\bar n$ and $\\C(\\bar n+1)$.\nThe asymmetry can be interpreted as a consequence of the commutation relations of either mechanical or optical creation and annihilation operators.\nWhich one of the two is responsible depends on the detection scheme used.\nFor heterodyne detection the sideband asymmetry arises from quantum backaction (i.e., the optical input vacuum fluctuations), while for photon counting of the sidebands (as done in Raman scattering experiments of solids) the asymmetry arises from the mechanical commutation relations \\cite{khalili_quantum_2012,weinstein_observation_2014,borkje_heterodyne_2016}.\nFrom the sidebands we extract $\\bar n$ either directly from the red-detuned probe sideband weight (with suitable calibration as outlined in Appendix~\\ref{sec:sideband_cooling})\nor from the ratio of the weights of the two sidebands, as in previous optomechanical experiments~\\cite{purdy_optomechanical_2015, underwood_measurement_2015}.\nRaman thermometry can be done using the cooling tone sideband instead of the red-detuned probe sideband, as long as one accounts for the difference in optomechanical coupling.\nClassical laser noise, which can be a source of artificially enhanced asymmetry~\\cite{jayich_cryogenic_2012,safavi-naeini_laser_2013,sudhir_appearance_2017} is not affecting our measurements, as is shown in Appendix~\\ref{sec:excess_noise}.\n\n\\begin{figure*}[ht]\n  \\includegraphics[scale=1]{fig_cooling_curve_true_fake_asymmetry.pdf}\n  \\caption{\\textbf{Artificial and quantum sideband asymmetry in optomechanical sideband cooling.} In (a), (b), the cooling tone is detuned $80\\unit{MHz}$ from red-detuned probe.\n  (a)~Inferred $\\bar n$ ($\\bar n+1$) from anti-Stokes (Stokes) mechanical sidebands, based on total thermomechanical noise.\n  Overall cooling is observed, however a clear discrepancy exists since the cooling tone and the red-detuned probe should yield the same and be equal to the occupancy $\\bar n$.\n  Inset: Example of an actual observed noise spectrum, colored for different tones as in the main panel.\n  Note that the choice of local oscillator frequency (Fig.~\\ref{fig:experimental}c) leads to the noise spectrum `folded over' and to the cooling tone sideband located close to the probe sidebands.\n  The $\\omega$-axis does not strictly correspond to Eq.~\\eqref{eq:ideal_theory} in this case.\n  (b)~The occupancy $\\bar n$ obtained from motional sideband asymmetry, obtained both from the ratio of red- and blue-detuned probes, and the ratio of cooling tone and blue-detuned probe, which shows strong disagreement.\n  Error bars represent $\\pm 20\\unit{MHz}$ tuning accuracy, see text for discussion.\n  (c), (d) show the data corresponding to (a), (b), respectively, only with the cooling tone detuned $220\\unit{MHz}$ from red-detuned probe, where the effect of the thermal Kerr-type nonlinearity is strongly diminished.\n  In~(c), the lower curve is a fit according to the model of Appendix~\\ref{sec:sideband_cooling} with the average asymmetry of the last two points, shown in (d), used for calibration; one quantum is added to result in the upper curve, coinciding with the Stokes sideband data.\n  Inset (c)~shows spectra of the last data point. \n}\n\\label{fig:fake_asymmetry}\n\\end{figure*}\n\nThroughout the experiment, the power of the cooling tone is increased, lowering the mechanical occupancy $\\bar{n}$ through sideband cooling, while the probe tones are held constant.\nFigure~\\ref{fig:fake_asymmetry}a,b we show thermometry results for a detuning of the red-probe from the cooling tone of $\\Omega\\s{mod}\/2\\pi=80\\unit{MHz}$.\nIn Fig.~\\ref{fig:fake_asymmetry}a we use the total sideband power to infer $\\bar n$ ($\\bar n+1$) from the anti-Stokes (Stokes) mechanical sidebands, plotted against the cooling tone intracavity photons.\nStrikingly, the curves of the cooling tone and red-detuned probe \\emph{do not coincide}, making it impossible to associate $\\bar n$ with either.\nThe discrepancy is also reflected in Fig.~\\ref{fig:fake_asymmetry}b, where  \\emph{quantum sideband asymmetries} of either of the two anti-Stokes sidebands compared to the Stokes sideband yield \\emph{different} $\\bar n$.\nWe next repeat the measurement with larger seperation between the red-detuned probe and the cooling tone, with\n$\\Omega\\s{mod}\/2\\pi=220\\unit{MHz}$ (Fig.~\\ref{fig:fake_asymmetry}c,d).\nThe inferred $\\bar n$ from both anti-Stokes sidebands \\emph{now show excellent agreement} (Fig.~\\ref{fig:fake_asymmetry}c), and $\\bar n$ inferred from motional sideband asymmetry also agree within experimental errors\n(Fig.~\\ref{fig:fake_asymmetry}d). Our measurements thus show that the presence of the cooling tone, when tuned closely to the red-detuned probe, modifies the quantum-mechanical motional sideband asymmetry.\n\nTo investigate this novel phenomenon further, we perform an auxiliary experiment, shown in Fig.~\\ref{fig:dual_tone_all}a.\nOmitting the blue-detuned probe, we apply two \\emph{equal power} tones near the lower mechanical sideband (still referred to as red-detuned probe and cooling tone).\nAs illustrated in Fig.~\\ref{fig:dual_tone_all}a, the observed anti-Stokes sidebands are not equal even for $\\Omega\\s{mod}\\ll\\kappa$, with the higher-frequency sideband stronger, in disagreement with standard optomechanical theory.\nKeeping the red-detuned probe fixed at $\\Delta=-\\Omega_m$ and scanning the cooling tone, Fig.~\\ref{fig:dual_tone_all}b shows the ratio of the two sidebands, normalized to the expected bare optomechanical response, vs.~$\\Omega\\s{mod}$.\nThe normalized ratio decreases with $\\Omega\\s{mod}$, only reaching the expected behavior at $\\Omega\\s{mod}\\gtrsim 200\\unit{MHz}$.\nNote that, depending on the power used, the ratio may be very large for small $\\Omega\\s{mod}$, e.g.~exceeding 2 for $\\Omega\\s{mod}\/2\\pi=20\\unit{MHz}$ in Fig.~\\ref{fig:dual_tone_all}b.\nFigure~\\ref{fig:dual_tone_all}c shows the noise spectra at low $\\Omega\\s{mod}\/2\\pi=4\\unit{MHz}$ for increasing tone power, where additional higher-order sidebands, spaced by $\\Omega\\s{mod}$, are observed.\nThese measurements point to the presence of a cavity nonlinearity, which couples the thermomechanical sidebands and modifies the observed asymmetry.\n\nThe existence of cavity nonlinearities due to e.g.~thermal effects is not unusual and has been observed in both bulk optical cavities as well as ultrasmall optical mode volume resonators~\\cite{ilchenko_thermal_1992,fomin_nonstationary_2005,rokhsari_observation_2005,barclay_nonlinear_2005}.\nWhile there are several potential sources for a ``Kerr-type'' effect, we consider the photothermorefractive frequency shift (PTRS) as the dominant mechanism.\nPhysically, photons circulating in the cavity are absorbed, leading to heating, and thus shifting of the cavity resonance (e.g., via the temperature dependent refractive index). The temperature deviation $\\delta T$ is governed in the simplest case by a single timescale and temperature-independent absorption\n\\begin{align}\n\t\\delta\\dot T(t)&=-\\gamma\\s{th}\\delta T(t)+g_{\\text{abs}}|\\bar a(t)|^2.\n\t\\label{eq:temperature_deviation}\n\\end{align}\nHere the absorption rate is given by $g_\\mathrm{abs}$, and the thermalization rate by $\\gamma_\\mathrm{th}$.\nIn the presence of two tones, the cavity field intensity beats $n_c(t)=|\\bar a(t)|^2\\propto\\text{const}+\\cos(\\Omega_{\\text{mod}}t)$, which, via the absorption heating, causes a periodic modulation of the cavity frequency, captured by the detuning\n\\begin{equation}\n\t\\Delta\\s{th}(t)=\\Delta_k \\exp(i\\Omega_{\\text{mod}}t)+\\text{c.c.}\n\t\\label{eq:PTRS}\n\\end{equation}\nwhere \n\\begin{equation}\n\t\\Delta_{k}=\\frac{g\\s{PT}g_{\\text{abs}}\\bar a_{c}\\bar a_{r}}{\\sqrt{\\gamma \\s{th}^2+\\Omega_{\\text{mod}}^2}}e^{-i\\phi_{\\text{th}}},\n\t\\quad \\phi_{\\text{th}}=\\tan^{-1}\\left(\\frac{\\Omega_{\\text{mod}}}{\\gamma_{\\text{th}}}\\right).\n\t \\label{eq:Delta_k}\n\\end{equation}\nHere $\\bar a_{c},\\bar a_{r}$ are the amplitudes of the intracavity fields produced by cooling tone and red-detuned probe, and\n$g_{\\text{th}}$ relates cavity frequency shift to temperature deviation via $\\Delta_{\\text{th}}=g\\s{PT}\\delta T$. Consequently the static thermal cavity shift per mean intracavity photon is given by $g\\s{0,th}=g\\s{PT}{g\\s{abs}}\/\\gamma\\s{th}$.\n\n\n\\begin{figure*}\n  \\includegraphics[width=\\linewidth]{fig_dual_tone_all_5.pdf}\n  \\caption{\\textbf{Observation of asymmetric noise spectra due to Kerr-type nonlinearity.}\n  (a)~Pumping scheme with two equal red-detuned pumps.\n  (b)~The peak ratio of the observed spectra, relative to the ratio expected from bare optomechanical response, vs.~$\\Omega\\s{mod}$ for constant power of $n\\s{c}=640$ intracavity photons, showing decreased effect for higher modulation frequencies.\n  The inset shows the spectra for data points of the same color in the main panel.\n  (c)~Increasing the pump power for a constant $\\Omega\\s{mod}\/2\\pi=4\\unit{MHz}$, showing  higher-order peaks.\n  Also shown are fits to the analytic model of Sec.~\\ref{sec:theory}, see text for details.\n  (d)~Calibration of sideband cooling using motional sideband asymmetry, with $\\Omega\\s{mod}\/2\\pi=220\\unit{MHz}$, free from Kerr-type artificial asymmetry. See text for more details. The oscillator is cooled down to $\\bar n=1.5$ phonons.\n}\n\\label{fig:dual_tone_all}\n\\end{figure*}\n\nThe cavity frequency modulation mediates processes in which photons are scattered from a frequency $\\omega$ to $\\omega\\pm\\Omega_{\\text{mod}}$, which causes coupling of the sidebands with strength $\\Delta_k$.\nIn Section~\\ref{sec:theory}, we incorporate the PTRS into standard optomechanical theory to model our experiments, and use a Floquet approach, based on writing the cavity and mechanical annihilation operators as sums of Fourier modes, to solve the time-dependent quantum Langevin equations.\n\nIn the first approximation, corresponding to retaining the dominant Fourier modes, the output spectrum takes the same form as above \\eqref{eq:ideal_theory} but with modified cooperativities\n\\begin{subequations}\n  \t\\begin{align}\n  \t\t\\tilde\\C\\s{red}&=\\C\\s{red}\\left|1-2ig_c\\Delta_k \/(g_r\\kappa)\\right|^2,\\\\\n\t\t\\tilde\\C\\s{cool}&=\\C\\s{cool}\\left|1-2ig_r\\Delta_k ^*\/(g_c\\kappa)\\right|^2,\n  \t\\end{align}\n  \t\\label{eq:modified_cooperativities}\n\\end{subequations}\nleaving the ideal theory [Eq.~\\eqref{eq:ideal_theory}] otherwise unchanged.\nImportantly, these expressions can lead to an asymmetry when $\\Delta_k$ is complex even when $g_r=g_c$.\nThis explains our observations in Fig.~\\ref{fig:dual_tone_all}b, where the cavity is driven by two equal-strength pumps, and asymmetric sidebands are observed.\nThe asymmetry diminishes as $\\Omega\\s{mod}$ is increased beyond the bandwidth $\\gamma\\s{th}$.\nA fit to this simple model shown in Fig.~\\ref{fig:dual_tone_all}b is in good agreement and captures this general behavior.\nIn this case of $\\Omega\\s{mod}\\gg\\gamma\\s{th}$ we find that the approximation~\\eqref{eq:modified_cooperativities} is sufficient to describe our data, with added Fourier modes producing no change to the  fitted curve.\nFrom the fit we obtain the two quantities characterizing $\\Delta_k$~\\eqref{eq:Delta_k}, $\\gamma\\s{th}\/2\\pi\\sim 6\\unit{MHz}$ and $g\\s{abs}g\\s{PT}\/4\\pi^2\\sim 10\\unit{MHz^2}$, with other parameters determined independently.\nDeviations from the data at high frequencies may be accounted for by considering additional Kerr-type effects of higher bandwidths or more complicated thermal behavior than that afforded by Eq.~\\eqref{eq:temperature_deviation}, however we keep the simplicity of our model and emphasize that for $\\Omega\\s{mod}\/2\\pi=220\\unit{MHz}$ used in Fig.~\\ref{fig:fake_asymmetry}c,d and all our subsequent motional sideband asymmetry measurements, we consistently observe the same normalized sideband power (i.e.~same $\\bar n$) for both cooling tone and red-detuned probe (within experimental errors of $\\sim 5\\%$), confirming the weakness of Kerr-type effects, and allowing to operate in a regime where the quantum sideband asymmetry arising from optomechancial quantum correlations can be observed.\nWe also note that the coupling of red and blue tones is negligible as their spacing is $2\\Omega_m\\gg\\gamma\\s{th}$.\n\n\nIn order to capture the physics observed in Fig.~\\ref{fig:dual_tone_all}c, where a series of thermomechanical sidebands are generated, more Fourier modes have to be included (six optical and five mechanical; see Fig.~\\ref{fig:fourier_modes}).\nThe relative weight of the higher order sidebands is in good agreement with the data.\nStrong thermal effects at $\\Omega\\s{mod}\/2\\pi=4\\unit{MHz}$ lead to distortion in the cavity linewidth measurement, introducing large detuning errors, leading to large uncertanties in cavity parameters.\nThe fits shown in Fig.~\\ref{fig:dual_tone_all}c yield $\\gamma\\s{th}\/2\\pi$ in the range 4--$8\\unit{MHz}$ and $g\\s{abs}g\\s{PT}\/4\\pi^2\\sim 6$--$17\\unit{MHz}^2$, in close agreement with Fig.~\\ref{fig:dual_tone_all}b.\nIn order to further confirm our model, we have carried out pump-probe response measurements of the cavity, detailed in Appendix~\\ref{sec:frequency_response}.\nWe found the bandwidth of power induced cavity frequency shifts (likely thermal in origin) $\\sim 10\\unit{MHz}$, in agreement with the results presented here.\n\nNext, we turn back to sideband measurements at $\\Omega\\s{mod}\/2\\pi=220\\unit{MHz}$, where only the quantum asymmetry is prominent.\nFigure~\\ref{fig:dual_tone_all}d shows an extended set of measurements done at cryostat temperature of $4.4\\unit{K}$ and buffer gas pressure of $140\\unit{mbar}$, including occupancies $\\bar n$ inferred from both motional sideband asymmetry and power in the cooling sideband.\nInferring $\\bar n$ from motional sideband asymmetry is straightforward, however the probes must be weak to avoid extraneous heating of the oscillator.\nThe much-larger signal-to-noise ratio of the cooling sideband is more suitable for determination of $\\bar n$ for the highest cooling powers.\nReferring to Eq.~\\eqref{eq:ideal_theory} and neglecting the weak probes, we see that apart from the easily-measured optomehcanical parameters, accurate knowledge of the quantity $\\eta\\,\\bar n\\s{th}$ is required.\nMoreover, extraneous heating due to optical absorption modifies the actual bath occupancy from that measured, $\\bar n\\s{th}\\rightarrow\\bar n\\s{th}+\\heating n_c$, where $\\heating$ is the added bath phonons per intracavity photon (see also Appendix~\\ref{sec:sideband_cooling}).\nThus, at least one of the parameters $\\eta,\\bar n\\s{th},\\heating$ needs to be independently determined.\nIt is often difficult to obtain accurate measurement of these parameters.\nHere we use the sideband asymmetry at an intermediate data point to compute $\\bar n$ and, unequivocally, $\\eta$, providing calibration for the entire cooling curve of Fig.~\\ref{fig:dual_tone_all}d.\n\nThe two main sources of measurement error are tuning accuracy, estimated at\n$\\pm 2\\pi\\times 20\\unit{MHz} = \\pm 0.0125\\kappa$, that leads to slightly different cavity response seen by the two probes (error bars in Figs.~\\ref{fig:fake_asymmetry} and~\\ref{fig:dual_tone_all}d), and error in estimate of Lorentzian peaks of the sidebands, that yield different $\\bar n$ values for the two flavors of asymmetry used in Fig.~\\ref{fig:fake_asymmetry}b,d (see inset of Fig.~\\ref{fig:fake_asymmetry}c).\nThe former error is dominant for large $\\bar n$ (small asymmetry), while the latter is dominant at strong cooling powers hence low signal-to-noise ratios (for example the last point in Fig.~\\ref{fig:fake_asymmetry}d).\nFor every data point in Fig.~\\ref{fig:dual_tone_all}d, we add these errors in quadrature.\nWe then compute $\\eta$ and its uncertainty using weighted average.\nThe final result, $\\eta = 0.044\\pm 0.005$ remains essentially the same if we take the last few, minimum error points.\nThis calibration gives $\\bar{n}=1.5\\pm 0.2$ (40\\% ground state occupation) for the minimum occupation in Fig.~\\ref{fig:dual_tone_all}d.\nIn addition, fitting the entire data set yields bath thermal occupation $\\bar n\\s{th}=17.5$ and extraneous heating of $\\heating=1.3\\,\\C_0$, in excellent agreement with the independently calibrated measurements of Appendix~\\ref{sec:sideband_cooling}.\n\n\\section{Theory\n}\n\\label{sec:theory}\n\nTo describe our experiment we take the standard optomechanical model in a rotating frame\n\\begin{equation}\n  \t\\hat{H}\\s{OM}\/\\hbar = [\\Delta_{\\text{th}}(t)-\\Delta]\\hat a\\dagg\\hat a\n  + \\Omega_m\\hat b\\dagg\\hat b\n  - g_0\\hat a\\dagg\\hat a(\\hat b+\\hat b\\dagg),\n  \\label{eq:HOM}\n\\end{equation}\nbut include the PTRS (via $\\Delta_{\\text{th}}$) as well as optical and mechanical baths. \nA standard calculation~\\cite{gardiner2004quantum,aspelmeyer_cavity_2014} gives quantum Langevin equations,\nwhich we linearize around the mean intracavity field $\\hat a(t)=\\bar a(t)+\\delta\\hat a(t)$ (and the same for the mechanical mode) to obtain the equations for linear optomechanics~\\cite{aspelmeyer_cavity_2014}\n\\begin{equation}\n  \\begin{aligned}\n\t\\delta \\dot{\\hat a} &= \\left\\{i\\left[\\Delta-\\Delta\\s{th}(t)\\right]-\\frac{\\kappa}{2}\\right\\}\\delta\\hat a\n\t+ig(t)(\\delta\\hat b+\\delta\\hat b\\dagg)+\\sqrt{\\kappa}\\delta\\hat a\\s{in},\\\\\n\t\\delta\\dot{\\hat b} &= \\left(-i\\Omega_m-\\frac{\\Gamma_m}{2} \\right)\\delta\\hat b\n\t+i[g(t)\\delta\\hat a\\dagg+g^*(t)\\delta\\hat a]+\\sqrt{\\Gamma_m}\\hat b\\s{in},\n  \\end{aligned}\n  \\label{eq:langevin_eoms}\n\\end{equation}\nwhere $\\Delta\\approx-\\Omega_m$ is the average detuning of the pumps from the cavity,\n$g(t)=g_0\\bar a(t)\\exp[i(\\omega\\s{cav}+\\Delta)t]$ is the modulated light-enhanced optomechanical coupling strength, \nand the input noises obey $\\langle\\delta\\hat a\\s{in}(t)\\delta\\hat a\\s{in}\\dagg(t')\\rangle=\\delta(t-t')$ and $\\langle\\hat b\\s{in}(t)\\hat b\\s{in}\\dagg(t')\\rangle=(\\bar n\\s{th}+1)\\delta(t-t')$ (as well as standard commutation relations).\nLangevin equations with periodic time-dependence can be analyzed with a recently developed method~\\cite{malz_floquet_2016}.\nNote that closely related models have been studied in the context of levitated optomechanics~\\cite{Aranas2016,Aranas2017}, where instead of the cavity, the mechanical resonator frequency is modulated.\n\nIn the experiment, we apply up to three tones to the cavity: a strong cooling tone, as well as weak red- and blue-detuned probes. Since the blue-detuned probe is very far detuned from the other two tones, it remains unaffected and we will not consider it in this section.\nCooling tone and red-detuned probe are close to the red sideband, separated in frequency by $\\Omega_{\\text{mod}}$, as is shown in Fig.~\\ref{fig:dual_tone_all}a.\nNeglecting all other effects, the resulting intracavity field $\\bar a(t)=\\bar a_{r}e^{-i(\\omega_{\\text{cav}}+\\Delta)t}+\\bar a_{c}e^{-i(\\omega_{\\text{cav}}+\\Delta+\\Omega_{\\text{mod}})t}$.\n\nWe model the PTRS~\\cite{verhagen_quantum-coherent_2012,Li2014} through Eq.~\\eqref{eq:temperature_deviation} in conjuction with $\\Delta_{\\text{th}}(t)=g_{\\text{th}}\\delta T(t)$ as discussed above.\nIn our level of approximation we neglect the effect of the PTRS on the mean intracavity field, such that we can solve directly for the temperature deviation.\nSince a constant temperature shift leads to a static frequency shift which can be absorbed into the overall detuning $\\Delta$,\nwe only consider the time-dependent part\n\\begin{equation}\n  \\begin{aligned}\n\t\\delta T(t)&=g_{\\text{abs}}\\int_{-\\infty}^t2\\bar a_{c}\\bar a_{r}e^{-\\gamma\\s{th}(t-t')}\\cos(\\Omega_{\\text{mod}}t')dt'\\\\\n\t&=\\frac{2g_{\\text{abs}}\\bar a_{c}\\bar a_{r}}{\\sqrt{\\gamma\\s{th}^2+\\Omega_{\\text{mod}}^2}}\\cos(\\Omega_{\\text{mod}}t-\\phi\\s{th}),\n  \\end{aligned}\n  \\label{eq:thermal_response}\n\\end{equation}\nwhere the phase lag \n\\begin{equation}\n  \\phi\\s{th}=\\tan^{-1}\\left(\\Omega\\s{mod}\/\\gamma\\s{th}\\right)\n  \\label{eq:phase_lag}\n\\end{equation}\narises due to the finite thermalization time.\nEquation \\eqref{eq:thermal_response} yields the PTRS displayed in Eqs.~\\eqref{eq:PTRS} and \\eqref{eq:Delta_k}.\nThe phase $\\phi\\s{th}$ plays a crucial role in the observed asymmetric response. Note that if $\\Delta\\s{th}$ is positive, the cavity resonance is blue-shifted.\n\nIf the two pumps lie close to the red sideband ($\\Delta\\approx-\\Omega_m$), and the modulation frequency is much smaller than $\\Omega_m$, as is the case here, we can neglect terms rotating at $2\\Omega_m$ in a rotating-wave approximation.\nHowever, the resulting equations still have an explicit time-dependence, which can be removed by splitting the fields $\\delta\\hat a$ and $\\delta\\hat b$ into Fourier components~\\cite{malz_floquet_2016}\n\\begin{equation}\n  \\begin{aligned}\n    \\delta\\hat a(t)&=\\sum_n\\exp(in\\Omega\\s{mod}t)\\delta\\hat a^{(n)}(t),\\\\\n    \\delta\\hat b(t)&=\\sum_n\\exp(in\\Omega\\s{mod}t)\\delta\\hat b^{(n)}(t),\n  \\end{aligned}\n  \\label{eq:fourier_cpts}\n\\end{equation}\nat the cost of introducing an infinite set of coupled equations of motion\n\\begin{equation}\n  \\begin{aligned}\n\t\\delta \\dot{\\hat a}^{(n)}&=\\left( i\\tilde\\Delta-in\\Omega_{\\text{mod}}-\\frac{\\kappa}{2} \\right)\\delta\\hat a^{(n)}+\\delta_{n,0}\\sqrt{\\kappa}\\delta\\hat a_{\\text{in}}\\\\\n\t&+i\\left(g_{c} \\delta\\hat b^{(n+1)}+g_{r}\\delta\\hat b^{(n)}-\\Delta_k ^*\\delta\\hat a^{(n+1)}-\\Delta_k \\delta\\hat a^{(n-1)} \\right),\\\\\n\t\\delta\\dot{\\hat b}^{(n)}&=\\left(-in\\Omega_{\\text{mod}}-\\frac{\\Gamma_m}{2}\\right)\\delta\\hat b^{(n)}\\\\\n\t&+i\\left( g_{c}\\delta\\hat a^{(n-1)}+g_{r}\\delta\\hat a^{(n)}\\right)+\\delta_{n,0}\\sqrt{\\Gamma_m}\\hat b_{\\text{in}},\n  \\end{aligned}\n  \\label{eq:eom}\n\\end{equation}\nwhich are depicted as lattice in Fig.~\\ref{fig:fourier_modes}.\nHere, $g_{r,c}=g_0\\bar a_{r,c}$ and we have written $\\Delta\\s{th}(t)=\\Delta_k \\exp(i\\Omega_{\\text{mod}}t)+\\text{c.c.}$, and defined the residual detuning $\\tilde\\Delta=\\Omega_m+\\Delta\\ll\\kappa$, which presents only a small correction and will thus be neglected in the following.\nIn principle, the cavity frequency shift could have a number of origins, which can be included in $\\Delta_k $. Here for simplicity we consider only the PTRS \\eqref{eq:PTRS}, such that $\\Delta_k $ is given by Eq.~\\eqref{eq:Delta_k}.\n\n\nThe Fourier components we have introduced are sometimes called auxiliary modes~\\cite{Peterson2017}, frequency-shifted operators \\cite{Aranas2016}, or sidebands. \nThe explicitly time-dependent terms in the linearized equations of motion~\\eqref{eq:langevin_eoms} couple the Fourier modes.\nThis off-diagonal coupling (in Fourier space) is suppressed by the response of the modes (especially the narrow mechanical mode), such that good approximations are obtained with only few Fourier modes.\n\n\\begin{figure}\n  \\includegraphics[width=\\linewidth]{unequal_sketch.pdf}\n  \\caption{\\textbf{An illustration of the infinite array of coupled Fourier modes.}\n  The map to Fourier components \\eqref{eq:fourier_cpts} introduces an infinite set of coupled Langevin equations, which has to be truncated at some order to obtain a solution.\n  Due to the two pumps (cooling and red-detuned probe), thermomechanical noise incident on mode $\\delta\\hat b^{(0)}$ is distributed onto $\\delta\\hat a_r$ and $\\delta\\hat a_{c}$, generating two sidebands.\n  The cavity Fourier modes are coupled due to a cavity Kerr-type nonlinearity [$\\Delta_k $~\\eqref{eq:Delta_k}].\n  This effect modifies the sideband weight, which to lowest order (considering only the bold three-mode system)\n  can be accounted for with new effective optomechanical couplings~\\eqref{eq:modified_cooperativities},\n  but in general leads to more sidebands (Fig.~\\ref{fig:dual_tone_all}c), modeled by including more Fourier modes in the description.\n  Notably, due to the finite response time, the coupling between the Fourier modes is \\emph{complex}, with the phase $\\phi_{\\text{th}}$ given in Eq.~\\eqref{eq:phase_lag}.}\n  \\label{fig:fourier_modes}\n\\end{figure}\n\nFor now, we assume that only the central modes $\\delta\\hat a_r\\equiv\\delta\\hat a^{(0)}$, $\\delta\\hat a_{c}\\equiv\\delta\\hat a^{(-1)},\\delta\\hat b^{(0)}$ are nonzero.\nThese modes are shown in bold colors in Fig.~\\ref{fig:fourier_modes} and are most relevant as they contain the most thermomechanical noise.\nThis reduces the equations of motion~\\eqref{eq:eom} to the matrix equation\n\\begin{equation}\n  \\begin{aligned}\n\t&\\mat{\\chi\\s{opt}^{-1}(\\omega)&-ig_{r}&i\\Delta_k \\\\\n\t-ig_{r}&\\chi_m^{-1}(\\omega)&-ig_{c}\\\\\n\ti\\Delta_k ^*&-ig_{c}&\\chi\\s{opt}^{-1}(\\omega+\\Omega\\s{mod})}\n\t\\mat{\\delta\\hat a_r\\\\\\delta\\hat b^{(0)}\\\\ \\delta\\hat a_{c}}\\\\\n\t&=\\mat{\\sqrt{\\kappa}\\delta\\hat a\\s{in}\\\\ \\sqrt{\\Gamma_m}\\hat b\\s{in}\\\\ 0},\n  \\end{aligned}\n  \\label{eq:matrix_equation}\n\\end{equation}\nwhere we have defined the susceptibilities\n\\begin{equation}\n  \\chi\\s{opt}^{-1}(\\omega)=\\kappa\/2-i(\\omega+\\tilde\\Delta), \\quad\\chi_m^{-1}(\\omega)=\\Gamma_m\/2-i\\omega.\n  \\label{eq:susceptibilities}\n\\end{equation}\nNote that the Kerr-type effect thus leads to a coupling of the Fourier modes $\\delta\\hat a_{c}$ and $\\delta\\hat a_r$, captured by $\\Delta_k $.\nThe noise spectrum contains mostly noise from the mechanical oscillators, such that we can neglect $\\delta\\hat a_{\\text{in}}$.\nApplying the red-detuned probe and the cooling tone introduces optical damping, yielding an effective susceptibility $\\chi\\s{m,eff}^{-1}(\\omega)=\\Gamma_m(1+\\C_{\\text{red}}+\\C_{\\text{cool}})\/2-i\\omega$.\nGiven the approximation \\eqref{eq:matrix_equation}, the spectrum reads\n\n\\begin{equation}\n  \\begin{aligned}\n  \t  \\bar S_I&(\\omega+\\Delta_{\\text{LO}})=1+\\eta\\kappa\\bar n\\Gamma_m\n  \t  \\left|\\frac{g_r-ig_c\\Delta_k\\chi\\s{opt}(\\Omega\\s{mod}+\\omega)}{\\chi\\s{opt}^{-1}(\\omega)\\chi\\s{m,eff}^{-1}(\\omega)}\\right|^2\\\\\n  \t  &+\\eta\\kappa\\bar n\\Gamma_m\\left|\\frac{g_c-ig_r\\Delta_k^*\\chi\\s{opt}(\\omega-\\Omega\\s{mod})}{\\chi\\s{opt}^{-1}(\\omega)\\chi\\s{m,eff}^{-1}(\\omega-\\Omega\\s{mod})}\\right|^2\n  \\end{aligned}\n  \\label{eq:three_mode_spectrum}\n\\end{equation}\nwhere we have neglected the frequency dependence of the cavity response, \nand introduced the overall detection efficiency $\\eta$~\\cite{gardiner2004quantum,BowenMilburn2015}.\nThis spectrum can be understood as follows: \nthermomechanical noise is filtered by the mechanical response $\\chi_{\\text{m,eff}}$ and is scattered to the optical modes, where it interferes with itself. In the Fourier mode corresponding to the red-detuned probe sideband, the amplitudes $g_r$ and $ig_c\\Delta_k \\chi\\s{opt}(\\Omega\\s{mod})$ add.\nThe rate $g_{r}$ comes from scattering directly into that Fourier mode, whereas $ig_{c}\\Delta_k \\chi\\s{opt}(\\Omega\\s{mod})$ has the clear interpretation of noise scattering first into cooling mode $\\delta\\hat a_{c}$ with amplitude $g_c$, where it picks up the optical susceptibility $\\chi\\s{opt}(\\Omega\\s{mod})$, and then hopping from there onto the red-detuned probe mode with amplitude $\\Delta_k$.\n\nWe find that even for equal pump strength ($g_{r}=g_{c}$) the thermomechanical sideband weights can differ, \\emph{as long as $\\Delta_k $ has a nonzero phase}.\nThis phase occurs only if $\\Omega\\s{mod}$ and $\\gamma\\s{th}$ are comparable, such that the phase lag of the thermal response behind the intracavity field [Eq.~\\eqref{eq:PTRS}] is non-trivial.\nThis phase is conjugated between the clockwise and counterclockwise process, allowing us to interpret it as a synthetic gauge flux $\\phi_{\\text{th}}$ threading the triangles in Fig.~\\ref{fig:fourier_modes}.\nFrom this point of view, the interpretation is similar to the nonreciprocal noise scattering observed in the optomechanical isolator~\\cite{Bernier2017}, except that here the phase arises dynamically, among the virtual Fourier modes, whereas in optomechanical nonreciprocal systems the phase is a direct consequence of the phase relation of drives. \nIn the more relevant case for thermometry, $g_{r}\\ll g_{c}$, we can neglect the backaction of the probe tones and Eq.~\\eqref{eq:three_mode_spectrum} reverts to the ideal theory \\eqref{eq:ideal_theory}, but now with the modified optomechanical cooperativities displayed above [Eq.~\\eqref{eq:modified_cooperativities}].\nThe three-mode approximation is sufficient for fitting our data in Fig.~\\ref{fig:dual_tone_all}b, where mostly $\\Omega\\s{mod}\\gg\\gamma\\s{th}$. \n\nFor strong driving and small $\\Omega\\s{mod}\\sim\\gamma\\s{th}$, the interaction $\\Delta_k $ between the optical Fourier modes is enhanced [Eq.~\\eqref{eq:Delta_k}],\nand thus higher-order Fourier modes are populated, and more sidebands appear in the output spectrum.\nThis is precisely what we observe in Fig.~\\ref{fig:dual_tone_all}c. \nThe higher-order sidebands can be modeled by including more Fourier components (faint color in Fig.~\\ref{fig:fourier_modes}).\nThe matrix in Eq.~\\eqref{eq:matrix_equation} is straightforwardly generalized to larger systems.\nIt can then be shown~\\cite{malz_floquet_2016} that the normal-ordered time-averaged noise spectrum is given by\n\\begin{equation}\n  S^N(\\omega)=\\sum_n\\int\\frac{d\\omega'}{2\\pi}\\langle \\delta\\hat a_{\\text{out}}^{(n) \\dagger}\n  (\\omega+n\\Omega\\s{mod})\\delta\\hat a_{\\text{out}}^{(-n)}(\\omega')\\rangle.\n  \\label{eq:measured_spectrum}\n\\end{equation}\nThe heterodyne spectrum we quote above is related through $\\bar S_I(\\omega+\\Delta_{\\text{LO}})=1+\\eta S^N(\\omega)$, which is shown explicitly in Appendix~\\ref{app:heterodyne_spectrum}.\nIn Fig.~\\ref{fig:dual_tone_all}c, we fit the data to this model including 6~optical Fourier modes and 5~mechanical Fourier modes\n[i.e.~$d^{(2)}\\ldots d^{(-3)}$, $b^{(2)}\\ldots b^{(-2)}$ in Fig.~\\ref{fig:fourier_modes}].\n\nFinally, we note that the instantaneous Kerr effect also leads to a cavity frequency shift $\\Delta\\s{Kerr}(t) = g\\s{Kerr}|\\bar a(t)|^2$ without a phase lag.\nThe coupling strength $g\\s{Kerr}$ can be estimated through \\cite{Matsko2005}\n\\begin{equation}\n  g\\s{Kerr} = -\\omega\\s{cav}\\frac{n_2}{n_0}\\frac{\\hbar\\omega\\s{cav}c}{V\\!\\s{mode}n_0},\n\\end{equation}\nwhere $n_0$ is the linear refractive index, $n_2$ the Kerr coefficient, $V\\!\\s{mode}$ the mode volume, and $c$ the speed of light. \nFor our system, we find $g\\s{Kerr}\/2\\pi\\sim 13\\unit{kHz}$, which is considerably weaker than the PTRS, which has a static, single intracavity photon coupling $g\\s{0,th}\/2\\pi=1.7\\unit{MHz}$.\nNevertheless, in precision thermometry the intrinsic Kerr effect might be important especially since it affects pumps with much larger spacing.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nRapid advances in cavity optomechanics over the last decade now enable quantum control of mechanical oscillators using electromagnetic radiation.\nHere we have shown that, similar to previously studied classical noise phenomena, quantum effects can be masked by classical nonlinearities, that lead to modification of the effective scattering rates between thermomechanical sidebands of closely-tuned applied drives.\nAlthough we have focused on motional sideband asymmetry measurements, our results apply to \nany experiment utilizing continuous monitoring of the mechanical oscillator with multiple tones~\\cite{wollman_quantum_2015,kronwald_dissipative_2014}, \nsuch as backaction-evading measurement~\\cite{suh_mechanically_2014}, \nmechanical quantum squeezing~\\cite{kronwald_arbitrarily_2013} and dissipative optical squeezing ~\\cite{kronwald_dissipative_2014}, entanglement of two mechanical oscillators using reservoir engineering\n~\\cite{woolley_two-mode_2014,ockeloen-korppi_stabilized_2018}, \nand recently demonstrated non-reciprocal devices~\\cite{bernier_nonreciprocal_2017, peterson_demonstration_2017}.\nMoreover, quantum optomechanics encompasses a wide range of electromagnetic cavities susceptible to thermal Kerr-type effects, such as photonic crystal~\\cite{barclay_nonlinear_2005}, whispering-gallery mode~\\cite{verhagen_quantum-coherent_2012}, as well as microwave cavities~\\cite{suh_thermally_2012}.\nIt is therefore necessary, in addition to the characterization of noise properties, to rule out such effects, by performing additional measurements, such as equal-tone red-detuned pump (Fig.~\\ref{fig:dual_tone_all}) and cavity frequency response measurements (Fig.~\\ref{fig:response}). \nBy comparing sideband asymmetry measurements with calibrated thermometry, we have shown that quantum mechanical effects are once again dominant when the system is operated outside the bandwidth of cavity Kerr-type effects.\n\n\n\\begin{acknowledgments}\n  LQ acknowledges support by Swiss National Science Foundation under grant No.~163387.\n  IS acknowledges support by the European Union's Horizon 2020 research and innovation programme under Marie Sko\\l{}odowka-Curie IF grant agreement No.~709147 (GeNoSOS).\n  DM acknowledges support by the UK Engineering and Physical Sciences Research Council (EPSRC) under Grant No.~EP\/M506485\/1.\n  AN acknowledges a University Research Fellowship from the Royal Society and support from the Winton Programme for the Physics of Sustainability.\n  TJK acknowledges financial support from an ERC AdG (QuREM).\n  This work was supported by the SNF, the NCCR Quantum Science and Technology (QSIT), and the European Union's Horizon 2020 research and innovation programme under grant agreement No.~732894 (FET Proactive HOT).\n  All samples were fabricated in the Center of MicroNanoTechnology (CMi) at EPFL.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nSpontaneous emergence of coordination between people or animals, without\nexternal control, is a remarkable phenomenon that can be crucial for\noptimal functioning or even survival of the population\n\\cite{king2009origins,conradt2009group,courchamp2002small}.\nIn some circumstances individuals are facing a choice between two\nor more possible actions, called strategies, they\ncan take. It often happens that the best outcome\nfor everyone can be obtained only if we choose the same strategy\nas our neighbours. In game theory such situation is referred to\nas coordination game\n\\cite{weidenholzer2010coordination,antonioni2013coordination,mazzoli2017equilibria,antonioni2014global}.\nAdditionally, it might matter under which strategy\nthe population coordinates.\nOne action can lead to higher prosperity than the other, what\nis modelled by different strategies having different payoffs.\nConditions required to coordinate have been scrutinised under various\nassumptions and for numerous environments, yet there are still unanswered\nquestions. Here, we study coordination and equilibrium selection in\ngames on multilayer networks.\n\nPeople interact in multiple contexts and through different media.\nOne of natural ways to represent it in a strict manner is by using a\nmultilayer network \\cite{boccaletti2014structure,kivela2014multilayer,battiston2014structural,battiston2017new,aleta2019multilayer}.\nEach layer is a separate network of interactions\nin a given context. For example, we interact with each other in work place,\nat home, online etc. In principle, the pattern of interactions can be\ndifferent in every layer resulting in a different network topology.\nAdditionally, some layers can be hidden \\cite{gajewski2021detecting}.\nNot every person must be active in every environment,\ntherefore some nodes are present only on a specific layer. If a node exists\nin many layers, however, it represents the same person, which often acts\nsimilarly in every context. It is therefore\nconnected between layers to itself\nvia inter-layer links, which provide coupling between the layers.\nIt is important to note that, if a system has a multilayer structure,\nit can not be simply reduced to a single-layer graph without\nchanging the dynamics \\cite{diakonova2016irreducibility}.\nHence, the scrutiny of layered systems is highly relevant.\n\nWe use the approach from evolutionary\ngame theory \\cite{sigmund1999evolutionary,axelrod1981evolution,nowak1992evolutionary,nowak2006five}\nto analyse synchronisation \nbetween the layers and equilibrium selection in coordination games.\nCoordination games have been studied in depth on single layer networks,\na comprehensive literature review can be found here\n\\cite{raducha2022coordination}. From previous results it is\nworth mentioning the KMR model which\nexplored the equilibrium selection in populations equivalent to\ncomplete graphs with the best response update rule \\cite{kandori1993learning}.\nThe risk-dominant equilibrium was always evolutionary\nfavoured in the model and several extensions did not find any deviation from\nthis behaviour \\cite{young1993evolution,youngindividual,ellison2000basins,peski2010generalized}. That outcome is preserved also on a circular network\n\\cite{ellison1993learning}, unless the unconditional imitation\nis used to update strategies \\cite{alos2006imitation}. In general,\nimitative update rules can favour Pareto-efficiency over risk dominance\n\\cite{raducha2022coordination,ohtsuki2006replicator,roca2009evolutionary}\n\n\nEvolutionary games were also extended to multilayer networks\n\\cite{wang2015evolutionary}.\nPrisoner's dilemma was studied on many layers with a possibility\nof using different strategies on different layers.\nThe strategy was updated according to replicators dynamics,\nbut using the collective payoff from all\nlayers \\cite{gomez2012evolution,matamalas2015strategical}.\nIt was also studied together with the stag hunt, the harmony game,\nand the snow drift on two-layer networks with the game being\nplayed on one layer and strategy imitation on the other\n\\cite{wang2014degree}. Additionally, the same games\non one layer were mixed with opinion dynamics and\nsocial influence on the second layer \\cite{amato2017interplay}.\nThe idea of separating the group in which we play the game\nfrom the one where we learn or imitate the strategy had been\nalready studied before within a single network\n\\cite{alos2014imitation,cui2016collaboration,khan2014coordination,alos2021efficient}.\nThe public goods game\n\\cite{tomassini2021computational,giardini2021gossip,maciel2021framing}\nwas considered on two \\cite{wang2013interdependent}\nand more layers \\cite{battiston2017determinants} with the game\nbeing played on each layer.\nInterestingly, in some of the mentioned research the multilayer structure\nwas said to enhance cooperation\n\\cite{gomez2012evolution,amato2017interplay,wang2013interdependent}.\nFinally, coordination games were also investigated on multilayer\nnetworks. The pure coordination game on one layer was\ncoupled with social dynamics and coevolution on the other, leading\nto a possible segregation \\cite{lipari2019investigating}.\nA particular version of the general coordination game\nwas studied on a two interconnected layers, with the strategy being imitated on the layers\nand the game played between the layers \\cite{lugo2015learning,lugo2020local,gonzalez2019coordination}.\nSimilarly to single-layer networks,\nthe unconditional imitation and smaller degree favoured\nthe Pareto-optimal equilibrium.\nHowever, the body of work on coordination games on multilayer\nnetworks is still very limited and consists of particular cases\nof more complex models mixed with opinion dynamics.\nMoreover, different works consider different update rules and it is\ndifficult to judge to which extent results are determined by\nthe multilayer structure, the particular payoff matrix,\nor the chosen update rule. Comparison between different\nupdate rules is necessary.\nFor these reasons, we provide a broader analysis\nof different payoff matrices laying within\nthe coordination games scope together with\nthree different update rules.\n\nWe focus on the two-player general coordination game\\cite{raducha2022coordination} described by a $2 \\times 2$ payoff matrix:\n\\begin{equation}\n\\begin{blockarray}{ccc}\n & $A$ & $B$  \\\\\n\\begin{block}{c(cc)}\n  $A$ & 1 & S  \\\\\n  $B$ & T & 0  \\\\\n\\end{block}\n\\end{blockarray}~,\n\\label{eqn:matrix_most_general}\n\\end{equation}\nwhere A and B are available strategies, while $T$ and $S$ are\nparameters defining payoffs. By definition, coordination games\nmust fulfil conditions $T<1$ and $S<0$.\nNote that the payoff matrix (\\ref{eqn:matrix_most_general})\nis equivalent to the most general one with all four payoffs\ndefined by a parameter, if only coordination on the strategy A is\nmore lucrative than on the strategy B.\nWithout loss of generality we can reduce it\nto the presented form. A game described\nby such payoff matrix contains a social\ndilemma. Obviously, the most rewarding outcome is obtained if both\nplayers choose the same strategy, but there is a hidden trade off\nbetween security and profit. Clearly, the highest possible profit is made\nwhen both play the strategy A, hence it is called the payoff-dominant\nor Pareto-optimal strategy. On the other hand, the risk-dominant strategy\nis the best choice in the lack of knowledge, i.e. it is the strategy that\nresults in the highest average payoff assuming that the opponent\nwill play either way with the same probability\n\\cite{harsanyi1988general}.\nIt is easy to check that for $T<S+1$ the strategy A is risk-dominant,\nand for $T>S+1$ the strategy B is be risk-dominant. This calculation\nprovides a theoretical line $T=S+1$ at which risk dominance changes.\nWhen all players coordinate on one of these strategies we \nrefer to such state as a payoff-dominant or risk-dominant equilibrium.\n\nIn the evolutionary game theory the game evolves because the players\nupdate their strategies after interacting and observing their peers.\nIt is well known that the update rule is as important as the payoff\nmatrix in defining the end result of the game\n\\cite{raducha2022coordination,ohtsuki2006replicator,roca2009evolutionary,xia2012role,szolnoki2018dynamic,danku2018imitate,poncela2016humans}.\nMultiple update\nrules have been proposed in the literature \n\\cite{szabo2007evolutionary,pangallo2021towards,blume1993statistical,traulsen2007pairwise}. We focus on three\nwell established ones: the replicator dynamics (RD)\n\\cite{schuster1983replicator,hammerstein1994game,nowak2004evolutionary}, the best response\n(BR)\n\\cite{kandori1993learning,young1993evolution,blume1995statistical,ellison1993learning,sandholm1998simple,buskens2008consent}, and the unconditional imitation (UI)\n\\cite{nowak1992evolutionary,vilone2012social,vilone2014social,lugo2015learning,gonzalez2019coordination,lugo2020local}. It is important to note\nthat RD and UI are imitative in nature, as players adapt the strategy\nof one of the neighbours. BR on the other hand is a strategical update rule\nwhich requires from the player knowledge abut the payoff matrix.\nAnother distinction between the update rules is their determinism\n-- BR and UI are deterministic, meaning that the same configuration\nwill always lead to the same strategy being chosen, while RD is a\nprobabilistic update rule. See Methods section for more details.\n\n\\begin{figure}[ht]\n\\centerline{\n\\includegraphics[scale=1,align=c]{figures\/multilayer_diagram.pdf}\n}\n\\caption{\n(a) Schematic representation of a miniature of multilayer network used in our\nsimulations. Both layers have the same topology of a random regular\ngraph with $N=8$ nodes of degree $k=3$ each\nand a fraction $q=5\/8$ of nodes\nis shared between the layers. Shared nodes are connected by inter-layer\nconnections (dashed lines). The node overlap $q$ is the number of\nshared nodes divided by $N$. White nodes play the strategy A\nand black ones play the strategy B. Shared nodes always have the same\nstate on both layers. Each layer has a specific payoff matrix\ngiven by $(S^I, T^I)$ and $(S^{II}, T^{II})$.\n(b) Diagram of the $S$-$T$ parameter space showing parametrisation\nof the layers. Each circle on the diagonal lines represents a game\nplayed on one of the layers. On layer I the strategy A is always\nrisk-dominant (yellow area),\nand on layer II the strategy B is always risk-dominant (purpule area).\nRisk-dominance changes at the line $T=S+1$. Examplary values\nof $(S^I, T^I)$ and $(S^{II}, T^{II})$ are highlighted in green\nwith $\\Delta S$ and $\\Delta T$ illustrated.}\n\\label{fig:multilayer_diagram}\n\\end{figure}\n\nIt was shown that on a single-layer network the risk aversion\nis usually stronger than the drive to profit. Therefore, on complete\ngraphs the risk-dominant equilibrium is always obtained. For sparse\nnetworks under unconditional imitation the system\ncan favour the Pareto-optimal\nequilibrium over the risk-dominant one, but only for a limited\nrange of parameters \\cite{raducha2022coordination}.\nFor RD and BR, however, the risk-dominant equilibrium is always selected.\nIn general, local effects were shown to be more important\nfor update rules\nwhich have an imitative nature, such as unconditional imitation.\n\\cite{alos2006imitation,ohtsuki2006replicator,roca2009evolutionary}.\nA natural question is which equilibrium, if any, will be chosen\nwhen the population is placed on a multilayer network\nwith two layers on opposite sides of the $T=S+1$ risk-dominance transition line.\nIn other words,\non layer I agents play a game where the strategy A is risk-dominant\nand on layer II a game where the strategy B is risk-dominant.\nWe investigate it by means of numerical simulations.\n\nWe study a population of players participating in two games on\na multilayer network with\ntwo inter-connected layers, as depicted in\nFigure~\\ref{fig:multilayer_diagram}.\nBoth layers have the same number of nodes $N$.\nIf a node is connected\nto itself between the layers via an inter-link, it plays the same\nstrategy everywhere. The fraction of nodes connected (or shared) between the layers\nis controlled by a parameter $q \\in [0,1]$, called node overlap\nor degree of multiplexity \\cite{diakonova2014,diakonova2016irreducibility}. There are $Nq$ inter-layer connections. For $q=0$ the two layers\nare effectively two independent networks, for $q=1$ the layers\nare equivalent to one network (every node is the same on each layer all the time)\nplaying each game half of the times. The edge overlap is kept constant\nwith both layers having the same topology, since we did not observe\nany change under varying edge overlap. We use random regular graphs\n\\cite{newmannetworks}. See Methods for more details on our simulations.\n\nPlayers on each layer are engaged in different games, i.e. parameters\n$S^\\beta$ and $T^\\beta$, $\\beta \\in \\{  \\textrm{I, II}\\}$, \ndefining the payoff matrix have different values\non each layer. In order to give the same relevance to both layers,\ntheir preferences towards one of the equilibria are set to be\nequally strong. This is achieved by choosing the points $(S^I, T^I)$\nand $(S^{II}, T^{II})$ equally distant from the $T=S+1$ line,\nas visible in Figure~\\ref{fig:multilayer_diagram}. Another choice to make\nis the angle between the $T=S+1$ line and the line created by\npoints $(S^I, T^I)$ and $(S^{II}, T^{II})$. We focus on cases\nwhere all points lay on a line $T^\\beta=-S^\\beta +C$,\nwhere $C$ is a constant\n(see Supplementary Material for other cases).\nThis is because only then the average payoffs $\\langle \\Pi^I \\rangle$\nand $\\langle \\Pi^{II} \\rangle$ of both layers are equal, therefore\ngames are truly symmetrical. We analyse the case of $T^\\beta=-S^\\beta -3$,\nwhich we call \\textit{diagonal}, and $T^\\beta=-S^\\beta$\nwhere all games are variants of\nthe well known \\textit{stag hunt} \\cite{skyrms2001stag,skyrms2004stag}.\nNote, that the stag hunt game can be obtained for different values of $C$ and that both cases\nare ,,diagonal'' in the sense that they have the same slope.\nNevertheless, we call the case of $C=-3$ \\textit{diagonal} and\n$C=0$ \\textit{stag hunt} to easily distinguish them in the discussion of results that\nfollows in the manuscript.\nIn both cases we cover with\nthe parameters $S$ and $T$ the whole width of the general\ncoordination game area (see Figure~\\ref{fig:multilayer_diagram}).\n\nSince the layers are placed symmetrically around the $T=S+1$ line,\nor more precisely around a point $(S_0, T_0)$ at this line,\nthe parameter $\\Delta S = S^I - S^{II}$ is sufficient to determine\nvalues of all four parameters $S^I, T^I, S^{II}, T^{II}$. Namely:\n\\begin{equation}\n\\begin{split}\n& S^I = S_0 + \\frac{\\Delta S}{2} ,\\\\\n& T^I = T_0 - \\frac{\\Delta T}{2} ,\\\\\n& S^{II} = S_0 - \\frac{\\Delta S}{2} ,\\\\\n& T^{II} = T_0 + \\frac{\\Delta T}{2} ,\n\\end{split}\n\\label{eqn:layers_params}\n\\end{equation}\nwhere $(S_0, T_0) = (-2,-1)$ for the diagonal case and\n$(S_0, T_0) = (-0.5,0.5)$ for the stag hunt case.\nNote also that for $T^\\beta=-S^\\beta+C$, that is both cases,\nwe have $\\Delta S = \\Delta T$.\nWe use the coordination rate $\\alpha \\in [0,1]$ to describe the state\nof the population. When $\\alpha^\\beta = 1$ every player on the\nlayer $\\beta$ chooses the strategy A, therefore layer $\\beta$\nis in the Pareto-optimal equilibrium.\nWhen $\\alpha^\\beta = 0$ the layer also coordinates,\nbut on the strategy B. For $\\alpha^\\beta = 0.5$ both strategies\nare mixed in equal amounts in the layer $\\beta$. We say that the\nlayers are synchronised when $\\alpha^I = \\alpha^{II}$ and\nthen we use just $\\alpha$ to describe both of them.\nNote that synchronisation does not require coordination within\nthe layers and vice versa,\nalthough they usually come together in our results.\n\n\n\\begin{figure}[h]\n\\centerline{\n\\includegraphics[scale=0.65]{figures\/fig2_alpha_long_res_repl_k8_.pdf}\n\\includegraphics[scale=0.65]{figures\/fig2_alpha_long_res_best_k8_.pdf}\n\\includegraphics[scale=0.65]{figures\/fig2_alpha_res_imit_k499_.pdf}\n}\n\\centerline{\n\\includegraphics[scale=0.65]{figures\/fig2_alpha_long_res_repl_k8_SH.pdf}\n\\includegraphics[scale=0.65]{figures\/fig2_alpha_long_res_best_k8_SH.pdf}\n\\includegraphics[scale=0.65]{figures\/fig2_alpha_long_res_imit_k499_SH.pdf}\n}\n\\caption{\nCoordination rate $\\alpha=\\alpha^I=\\alpha^{II}$\nvs gap size $\\Delta S$ for full node overlap $q=1$ (the multiplex case).\nThe upper row (a, b, c) presents the diagonal case\nand the bottom row (d, e, f) the stag hunt.\nFor RD and BR each layer\nhas $N=1000$ nodes with an intra-layer degree $k=8$, for\nUI it is a complete graph with $N=500$. Each circle represents\nthe value of $\\alpha$ (for both layers) in one of 400 realisations\nand solid lines show the average values.\n}\n\\label{fig:alpha}\n\\end{figure}\n\n\\section*{Results}\n\nWe study the synchronisation between the layers, coordination, and\nequilibrium selection under varying conditions.\nFor RD and BR update rules we set the connectivity\nat $k=8$, since it was shown that the degree does not\nchange the equilibrium selection in their case\n\\cite{raducha2022coordination}. However, for UI the line\n$T=S+1$, at which risk-dominance of strategies changes,\noverlaps with the actual transition in equilibrium selection\nonly for a complete graph\\cite{raducha2022coordination}.\nHence, we analyse the case of unconditional imitation\nalways with full connectivity in order to obtain true\nsymmetry between the layers.\n\nThe two main parameters whose influence we investigate\nare the node overlap $q$ and the distance between the games\n$\\Delta S$ or $\\Delta T$. For simplicity, we start with an analysis\nof the multiplex case, i.e. full node overlap, $q=1$.\nIn Figure~\\ref{fig:alpha} we present the coordination rate\n$\\alpha$ for synchronised layers at $q=1$ (layers are always synchronised\nat full node overlap, because all nodes have to be the same on both layers by definition).\nThe RD update rule clearly favours the payoff-dominant strategy A\nat the maximal level of multiplexity. In the diagonal case the difference\nis moderate with $0.6 \\lesssim \\alpha \\lesssim 0.8$, but in\nthe stag hunt case coordination rarely happens at the strategy B\nand $ \\alpha$ is close to 1.\n\nInterestingly, for $q=1$ the UI\nupdate rule does not favour the strategy A. As we\ncan see in the figure, for small size of the gap $\\Delta S$\nthe outcome is symmetrical with both strategies selected\nhalf of the time. But for increasing distance between the layers\nthe system starts to coordinate more often on the strategy B,\nto finally select exclusively the not Pareto-optimal equilibrium\nfor the maximal gap size. It has to be noted that\nthe maximal gap size results in payoff matrices that are on the\nborder between coordination games area and non-coordination games,\ntherefore this very point technically does not represent\na coordination game. Nonetheless, the decline in the payoff-dominant\nequilibrium selection is visible already before this limit value.\nThis result is especially surprising, since the UI is the only update rule\nthat on a single-layer network can lead to the Pareto-optimal equilibrium\neven though it is not risk-dominant\\cite{raducha2022coordination}. However,\nthe requirement for that was a sparse network.\n\nThe only truly symmetric update rule is the BR which does not\nreveal any preference towards one of the strategies for\nfull node overlap. Additionally, the diagonal case is \nidentical to the stag hunt case. For gaps $\\Delta S < \\Delta S_{max} \/ 2$\nand $q=1$ synchronised layers reach\neither equilibrium with equal probability,\nand for $\\Delta S > \\Delta S_{max} \/ 2$ the system does not coordinate\nstaying at $\\alpha = 0.5$. At the transition value of\n$\\Delta S = \\Delta S_{max} \/ 2$ both states -- the coordination on one of\nthe strategies and non-coordinated fully-mixing state -- are possible\n(see Figure~\\ref{fig:alpha}).\n\n\n\\begin{figure}[ht]\n\\centerline{\n\\includegraphics[scale=0.65]{figures\/fig3_repl_k8_gap1.6.pdf}\n\\includegraphics[scale=0.65]{figures\/fig3_best_k8_gap0.8.pdf}\n\\includegraphics[scale=0.65]{figures\/fig3_imit_k499_gap2.0.pdf}\n}\n\\centerline{\n\\includegraphics[scale=0.65]{figures\/fig3_repl_k8_gap0.4.pdf}\n\\includegraphics[scale=0.65]{figures\/fig3_best_k8_gap0.3.pdf}\n\\includegraphics[scale=0.65]{figures\/fig3_imit_k499_gap0.5.pdf}\n}\n\\caption{\nCoordination rates on layers $\\alpha^{I}$, $\\alpha^{II}$,\nand $\\Delta \\alpha$ vs node overlap $q$ for exemplary\nvalues of $\\Delta S$. The upper row (a, b, c) presents the diagonal case\nand the bottom row (d, e, f) the stag hunt.\nFor RD and BR each layer\nhas $N=1000$ nodes with an intra-layer degree $k=8$, for\nUI it is a complete graph with $N=500$. Each circle represents\none of 500 realisations and solid lines show the average values.\nFor each realisation there is one circle for layer I (yellow) and one for layer II (purple).\nNote, that when layers synchronise $\\alpha^{I} = \\alpha^{II}$,\n $\\Delta \\alpha = 0$, and both circles overlap looking like one of brownish colour,\n as well as the solid lines for $\\alpha^{I}$ and $\\alpha^{II}$ merge.\n The dashed line in (a, b, d, e) shows the function fitted to $\\Delta \\alpha$.\n}\n\\label{fig:alpha_vs_q_diag}\n\\end{figure}\n\nIn addition to the results showed in Figure~\\ref{fig:alpha} for $q=1$, we know that\nfor $q=0$ each layer will obtain full coordination on its preferred strategy --\nA for layer I and B for layer II \\cite{raducha2022coordination}.\nThe middle ground between those two extreme values of $q$  must therefore contain\nsome kind of transition. We investigate it in Figure~\\ref{fig:alpha_vs_q_diag},\nwhere we can see how the coordination rate $\\alpha$ changes at both layers\nwith increasing $q$. First thing to notice is that for any update\nrule and any parameter choice, but with $q=0$,\neach layer converges to a different limit value of $\\alpha$.\nThis means that both layers indeed obtain full coordination\non their preferred strategies. Consequently,\nthe difference between layers is maximal and $\\Delta \\alpha = 1$.\nSuch outcome was to be expected, because with no inter-connections\nthe two layers are effectively separate networks, as we\ndescribed. Therefore, each network selects the risk-dominant\nequilibrium. Similarly, for $q=1$ layers must fully overlap\nwith $\\Delta \\alpha = 0$, because each node is present on all\nlayers and the state of a shared node must be\nthe same across the layers. As visible in\nFigure~\\ref{fig:alpha_vs_q_diag} this is exactly what happens.\n\nThe above considerations lead to a conclusion that there must be\na certain point $q_c \\in [0,1]$ at which $\\Delta \\alpha$\nbecomes zero. In Figure~\\ref{fig:alpha_vs_q_diag}\nwe see that the value of $q_c$ can vary for replicator\ndynamics and best response update rules, but is close to\nzero in both cases for unconditional imitation. In fact,\n$q_c \\to 0$ for any configuration of the layers when\nplayers update their strategies according to UI\n(see Supplementary Materials for plots of different cases).\nIn other words, synchronisation between the layers is the strongest\nfor the UI update rule. One has to still bear in mind that for\nUI the network has a higher degree than in other cases.\nIt is a complete graph, whereas for RD and BR the networks are much\nsparser with $k=8$. Nevertheless, simulations for higher degree for BR\nindicate that synchronisation is weakened by increasing\nconnectivity (see Supplementary Materials), which makes the\nupdate rule a natural explanation of the observed differences.\n\nAnother surprising observation is that not all the results are\nsymmetrically placed around $\\alpha = 0.5$. Both layers\nhave equally strong preferences towards their natural equilibria\n-- payoff matrix parameters $(S^I, T^I)$ and $(S^{II}, T^{II})$\nare equally distant from the transition line $T=S+1$ and\naverage payoffs of the games on both layers are the same.\nThere is no reason, in principle, why the system as a whole\nshould choose one equilibrium over the other. Nevertheless,\nwe can see that in some cases with RD and UI synchronised layers\ncoordinate exclusively on the Pareto-optimal strategy A ($\\alpha = 1$),\nwhile it doesn't happen for the strategy B at any point\n(except for $q=1$ with the maximal gap $\\Delta S$ for UI, see Figure~\\ref{fig:alpha}).\nThis symmetry breaking is especially interesting, because\nit is driven by the level of multiplexity $q$ in a non-trivial way.\nIn examples shown in Figure~\\ref{fig:alpha_vs_q_diag}, and in \ngeneral, if the Pareto-optimal equilibrium is obtained on both layers\nit happens as soon as they synchronise, i.e. at $q_c$. When increasing\nthe node overlap further at some point $q_p$ the synchronised state\nwith coordination on the strategy $B$ starts to appear and\nthe average value of $\\alpha$ drops below 1. For $q>q_p$\nsynchronised layers can coordinate on either strategy,\nhowever $\\alpha>0.5$ most of the time meaning that the Pareto-optimal\nequilibrium is still dominant.\n\nThe fully symmetrical outcome that one would expect from the symmetrical\ndesign of the experiment is obtained solely for BR. We can\nsee in Figure~\\ref{fig:alpha_vs_q_diag}\nthat there are only two types of behaviour\nthat the system displays with BR update. The first one, for $q<q_c$\nis characterised by no synchronisation between the layers and each\nof them following a specific level of coordination, which is\n$\\alpha^I = -q\/2+1$ and $\\alpha^{II} = q\/2$. This calculation\ncomes from a simple assumption that all nodes that are not shared\nplay the dominant strategy of their layer and all shared nodes\nplay either strategy half of the time. Put differently,\nhalf of the shared nodes play the strategy A and half the strategy B.\nThis gives a fraction $(1-q)+q\/2=-q\/2+1$ of nodes playing \nthe strategy A on the layer I and the same fraction of nodes\nplaying the strategy B on layer II, so $1-(-q\/2+1)$ of them playing\nthe strategy A. As we can see in the figure this scenario\nis realised until reaching $q_c$. The second type of behaviour, for\n$q>q_c$, is coordination of both synchronised layers on one of\nthe strategies with equal probability of choosing either of them.\n\n\\begin{figure}[ht]\n\\centerline{\n\\includegraphics[scale=0.65]{figures\/fig4_sc_diag_repl.pdf}\n\\includegraphics[scale=0.65]{figures\/fig4_sc_diag_best.pdf}\n}\n\\centerline{\n\\includegraphics[scale=0.65]{figures\/fig4_q_c_repl_k8.pdf}\n\\includegraphics[scale=0.65]{figures\/fig4_q_c_best_k8.pdf}\n}\n\\caption{\n(a, b) Coordination rate difference between the layers $\\Delta \\alpha$\nvs node overlap $q$ for increasing value of $\\Delta S$ (given in the legend).\n(c, d) Critical value of $q_c$ and $q_c^{fit}$ vs gap size $\\Delta S$.\nResults for the diagonal case,\n$N=1000$ nodes on each layer with an intra-layer\ndegree $k=8$ averaged over 100 realisations.\n}\n\\label{fig:alpha_scaling_diag}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centerline{\n\\includegraphics[scale=0.65]{figures\/fig5_sc_diag_repl.pdf}\n\\includegraphics[scale=0.65]{figures\/fig5_sc_diag_best.pdf}\n}\n\\centerline{\n\\includegraphics[scale=0.65]{figures\/fig5_q_c_repl_k8.pdf}\n\\includegraphics[scale=0.65]{figures\/fig5_q_c_best_k8.pdf}\n}\n\\caption{\n(a, b) Coordination rate difference between the layers $\\Delta \\alpha$\nvs node overlap $q$ for increasing value of $\\Delta S$ (given in the legend).\n(c, d) Critical value of $q_c$ and $q_c^{fit}$ vs gap size $\\Delta S$.\nResults for the stag hunt case,\n$N=1000$ nodes on each layer with an intra-layer\ndegree $k=8$ averaged over 100 realisations.\n}\n\\label{fig:alpha_scaling_stag}\n\\end{figure}\n\nThe behaviour observed so far leads to a logical question\nabout the change, if any, we would observe when varying\nthe distance between the layers,\ni.e. for different values of the gap size $\\Delta S$.\nIn Figures~\\ref{fig:alpha_scaling_diag} and~\\ref{fig:alpha_scaling_stag} (a, b)\nwe present the dependence of $\\Delta \\alpha$ on the degree\nof multiplexity $q$ for values of $\\Delta S$ ranging from\n0.4 to 4 in the diagonal case, and from 0.1 to 1 for the stag hunt.\nThis range essentially covers the whole width of the general coordination game\narea, as presented in Figure~\\ref{fig:multilayer_diagram}.\nA bigger gap would result\nin payoff matrices out of the coordination game's scope.\nWhat we can see is that $\\Delta \\alpha$ drops to zero at higher\nnode overlap when increasing the gap size. More precisely,\nfor RD it roughly follows the line of $\\Delta \\alpha = -q +1$\nto diverge from it at some point and eventually reach the lowest\npossible value of 0. The line is followed for much longer in the\ndiagonal case than in the stag hunt case. For BR there is virtually\nno difference between those cases and the dependence on the gap\nsize is slightly different. Values of $\\Delta \\alpha$ are the same\nfor gap sizes equal 0.4 and 0.8, then again for 1.2 and 1.6, and from \n$\\Delta S = 2$ onwards  $\\Delta \\alpha = -q +1$ (these values are\nfor the diagonal case, for stag hunt the general picture is the same\nwith values rescaled by a factor of 1\/4).\n\nWe can clearly see that $q_c$\ndepends on the gap size $\\Delta S$ and this dependence\nis presented in Figures~\\ref{fig:alpha_scaling_diag}\nand~\\ref{fig:alpha_scaling_stag} (c, d). We use two approaches in order to\nestimate the value of $q_c$. The first one is simply taking the\nlowest value of $q$ at which $\\Delta \\alpha$ is equal 0\nfor the first time. This approach, however, is prone to numerical\nnoise and a tiny divergence from 0 will result in a change of the value.\nTo obtain the second one we fit a parabola with an exponential\ncutoff to the function $\\Delta \\alpha (q)$ (dashed line in Figure~\\ref{fig:alpha_vs_q_diag}) and we take the first\nvalue of $q$ at which $\\Delta \\alpha < 0.01$ as $q_c^{fit}$.\nAs we can see in the plots, it does not make a real difference\nfor BR, but can give different results for RD for higher values\nof $\\Delta S$. Regardless the approach, $q_c$ changes from approximately\n0.2 up to 1 for RD in the diagonal case (for the stag hunt values are slightly\nlower), and from 0.5 to 1 for BR with no visible difference between\nthe diagonal and the stag hunt case.\nWe similarly estimate the value of $q_p$, however without fitting a function,\nbecause the behaviour of  $\\alpha$ for synchronised layers\nis more complex than the one of $\\Delta \\alpha$.\nWe take as an approximation of $q_p$ the first value of $q$\nafter synchronisation for which the coordination rate\n$\\alpha$ drops below 0.95  (dashed lines in Figure~\\ref{fig:phase_diagram}).\n\nIn summary, for any gap $\\Delta S$ (or $\\Delta T$) between the layers\nat $q=0$ there is no synchronisation and each layer gravitates\ntowards its preferred equilibrium. Then, at $q=q_c$ layers start\nto synchronise. For RD and UI synchronised layers coordinate \non the Pareto-optimal strategy for $q_c<q<q_p$ and for $q>q_p$\nthey coordinate on either of the strategies. For some values\nof $\\Delta S$, however, as well as for BR in general, $q_p$\noverlaps with $q_c$ and the system goes from unsynchronised state\nstraight into coordination on any strategy, without the phase of\npure Pareto-optimal equilibrium\n(see Figure~\\ref{fig:phase_diagram}).\nAdditionally, there are two \nupdate-rule-specific phenomena.\nFor UI at the maximal\ngap between the layers ($\\Delta S_{max} = 4$ for the diagonal case and\n$\\Delta S_{max} = 1$ for the stag hunt) and for $q=1$ synchronised layers\ncoordinate only on the strategy B just at this point.\nAnd for BR for $\\Delta S > \\Delta S_{max} \/ 2$ at full node overlap\nwhen the layers get synchronised they do not reach coordination.\nInstead they both end up in a fully mixing state with\n$\\alpha^I=\\alpha^{II}=0.5$.\n\nWe can also see from\nFigure~\\ref{fig:phase_diagram}\nthat an increase in the absolute values of payoffs $S^\\beta$ and $T^\\beta$\non both layers,\ni.e. a shift from the diagonal to the stag hunt case, significantly\nenlarges the relative area of Pareto-optimal equilibrium for RD and UI.\nIt does not, however, change the relative size of the no-synchronisation\nphase and it seems not to influence the best response dynamics at all.\nOne explanation of the enlargement of the Pareto-optimal phase,\nat least for RD, could be the fact that in the stag hunt case\nthe layers are closer to each other -- the gap $\\Delta S$\n(and $\\Delta T$) is 4 times smaller on average. Games being more\nsimilar and closer to the transition line could justify\nwhy it is easier for the layer I to shift the layer II into\nits preferred equilibrium on the strategy A.\nNevertheless, for UI in the diagonal case there is a minimal value\n$\\Delta S \\approx 2$ below which the Pareto-optimal phase\ndoes not exist at all, hence here the proximity of layers can not be the \nexplanation of synchronisation in the payoff-dominant equilibrium.\nMoreover, there is an optimal size of the gap $\\Delta S$\nfor which the Pareto-optimal phase is the widest.\nFor UI it is approximately the maximal gap $\\Delta S_{max}$\nand for RD it is one of the middle values, but certainly not\nthe smallest gap. These considerations lead us to a conclusion\nthat synchronisation and equilibrium selection in\ncoordination games on multilayer networks are very complex\nphenomena where obtaining the most advantageous outcome\nrequires accurate parameter selection.\n\n\n\\begin{figure}[ht]\n\\centerline{\n\\includegraphics[scale=0.65]{figures\/fig6_diag_repl.pdf}\n\\includegraphics[scale=0.65]{figures\/fig6_diag_best.pdf}\n\\includegraphics[scale=0.65]{figures\/fig6_diag_imit.pdf}\n}\n\\centerline{\n\\includegraphics[scale=0.65]{figures\/fig7_stag_repl.pdf}\n\\includegraphics[scale=0.65]{figures\/fig7_stag_best_k8.pdf}\n\\includegraphics[scale=0.65]{figures\/fig7_stag_imit.pdf}\n}\n\\caption{\nPhase diagram of coordination rate $\\alpha=\\alpha^I=\\alpha^{II}$\nin the $q$-$\\Delta S$ space, for synchronised layers\nfor the diagonal (a, b, c) and the stag hunt (d, e, f) case.\nThe pink area represents the range of parameters where\nsynchronisation is not obtained and $\\alpha^I \\neq \\alpha^{II}$\n(for UI it happens only at $q=0$).\nThe solid lines show the critical value $q_c^{fit}$ and\nthe dashed lines $q_p$.\nFor RD and BR each layer\nhas $N=1000$ nodes with an intra-layer degree $k=8$, for\nUI it is a complete graph with $N=500$. \nResults are  averaged over 100 realisations.\n}\n\\label{fig:phase_diagram}\n\\end{figure}\n\n\n\n\\section*{Discussion}\n\nWe investigated synchronisation between layers\nand equilibrium selection in the general coordination game\non a multilayer network. The game played on each layer\nis described by a different payoff matrix, but both games are\nequally distant from the risk-dominance transition line $T=S+1$.\nThe layers are connected  by $Nq$ inter-links,\nwhere the parameter $q$ is the node overlap or degree of multiplexity. We studied the impact of\nof the value of  $q$ and the gap $\\Delta S$ between the layers\nfor three update rules: the replicator dynamics, the best response,\nand the unconditional imitation.\n\nThe most prominent outcome is the symmetry breaking\nin equilibrium selection. In neither of the cases,\ndiagonal and stag hunt, there is a difference in\naverage payoffs of games played on the layers. The strategies\npreferred by each layer are equally risk-dominant, i.e.\nthe distance from the transition line $T=S+1$ is the same.\nThe only difference, of course, is that the strategy A\ngives the highest possible payoff, hence it's the most\nprofitable one. A common-sense approach would lead us to\nbelieve that the payoff-dominant strategy A should be\nnaturally promoted by the population. This is however not\nthe case on single-layer networks, where the risk-dominant\nstrategy is always selected in the range of connectivities\nthat we considered \\cite{raducha2022coordination}.\nIn our multilayer model, which strategy is risk-dominant \ndepends on the layer, but coordination on the strategy A prevails\nin most of the parameters space or is at least favoured on average.\nIt is therefore clear that the multilayer structure\nenhances the Pareto-optimal outcome and it does so in \na complex manner.\n\nWe identified three main phases depending on the node overlap $q$\nand the gap size $\\Delta S$. The first one for lower values of $q$\nis a no-synchronisation phase with $\\alpha^I \\neq \\alpha^{II}$.\nEach layer obtains a certain level of\ncoordination close to its preferred equilibrium. The second phase\nbegins when $\\Delta \\alpha$ drops to zero, i.e. at $q_c$.\nHere, layers are synchronised and fully coordinate on the\nPareto-optimal strategy A. Finally, the third phase appears\nfor a higher node overlap $q>q_p$. In this phase layers are also\nsynchronised and they also coordinate, but not always on\nthe strategy A -- either equilibrium is possible, although\ndepending on the parameters one of them might be preferred on\naverage. In some cases $q_c = q_p$ and the second phase does not\nappear.\n\nThe Pareto-optimal phase is not a mere effect of high node overlap\nbetween layers or low gap size. It has a more complex shape that\ndepends on both parameters and on the update rule. For BR the\nPareto-optimal phase does not exist at all. For RD it is\nplaced, surprisingly, in the middle rage of the node overlap $q$,\nbut its position and width depends also on $\\Delta S$. Neither too\nlow nor too high degree of multiplexity helps in achieving the optimal\nequilibrium, and the same is true for the gap size.\nNevertheless, the value of $q_c$ grows with increasing distance $\\Delta S$.\nFor UI the Pareto-optimal phase might not even exist\nfor lower values of $\\Delta S$,\nand its usually wider for bigger gaps. If the phase exists, however,\nit appears already for any $q>0$, as the synchronisation is\nmuch faster for UI.\n\nOur work contributes to the understanding of the  necessary conditions\nfor the optimal outcome in coordination games. The study of evolutionary\ncoordination games is well established and\nhad its place in the literature for a long time.\nOur work also fits into more recent general studies of social\nsystems on multilayer networks. Since many of socio-technical\nsystems have multiple environments where people can interact,\nthe application of layered structures in their modelling is \na natural step forward. As we showed, this approach can be\nhighly relevant in analysis of coordination dilemmas, because\nit leads to non-trivial new effects that have not been observed\nin single-layer networks.\n\n\n\\section*{Methods}\n\nWe run numerical simulations of the general coordination game\ndefined by the payoff matrix~(\\ref{eqn:matrix_most_general})\non a multilayer graph.\nAgents are placed on two networks of $N$ nodes forming\ntwo layers of the multilayer network. Each layer is a random regular graph\nwith a degree $k$, generated using \\verb,K_Regular, algorithm form the\n\\textit{igraph} python package \\cite{csardi2006igraph,igraph}.\nThe coupling between layers can be adjusted using\ntwo parameters: node overlap $q$ and edge overlap. As we didn't observe\nany influence of varying edge overlap on the results we maintain\na perfect edge overlap, i.e. both layers have exactly\nthe same structure of connections.\nThe node overlap $q$ takes values from 0 to 1, defining\nthe fraction of nodes connected (or shared) between both layers.\nIf two nodes are shared, their state has to be the same\non both layers at all times.\nIn other words, it's the same node present on both layers.\nFor $q=0$ there is no connection between\nthe layers and their dynamics are fully separated,\nfor $q=1$ it's effectively a single-layer network\nwith each game played half of the time.\n\nThe game played on each layer\nis described by different values of $S^\\beta$ and\n$T^\\beta$ parameters of the payoff matrix, given in\nequation~(\\ref{eqn:layers_params}).\nWe use an asynchronous algorithm where at the beginning of each time step\na layer is randomly selected with equal probability for both layers. \nThen, the update is performed on the chosen layer\nas for a single-layer network and according\nto the game played on the layer.\nFirst, a random node is chosen with equal probability for all nodes\non the layer. We call it the active or focal node.\nThe active node then plays the game with all its $k$ neighbours\non the layer and receives a given payoff, which is saved.\nFinally, the strategy of the active node is updated according\nto one of the following three update rules:\n\\begin{itemize}\n    \\item {the Replicator Dynamics (RD)} (aka replicator rule, or proportional imitation rule) -- the active node compares the payoff with a random neighbour on the layer and copies its strategy with probability $p=(\\mathrm{payoff~diff.})\/\\phi$, if the neighbour's payoff is bigger. Normalisation $\\phi$ is the largest possible payoff difference allowed by the payoff matrix and network structure and it sets the probability $p$ within $[0,1]$ range,\n    \\item {the myopic Best Response (BR)} -- the active node chooses the best strategy given the current strategies of the neighbours on the layer, i.e. it compares all payoffs it would obtain playing each possible strategy against the current strategies of the neighbours and chooses the strategy resulting in the largest payoff,\n    \\item {the Unconditional Imitation (UI)} -- the active node copies the strategy of the most successful neighbour on the layer, i.e. the one with the highest payoff, if its payoff is larger.\n\\end{itemize}\nAt the end, the state of the focal node is copied onto\nthe other layer, if the updated node is connected (shared) between the layers.\nMore precisely, the new strategy selected by the node and\nthe last payoff are copied. The simulation\nruns until a stationary state is reached,\nor a frozen configuration is obtained on all layers.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:introduction}\n\nThe ``pre-train then fine-tune'' paradigm has become the new favorite in software intelligence~\\cite{niu2022deep}. Since pre-training tasks can be done with unlabeled data, a model can be pre-trained on large amounts of, easily accessible data, thus obtaining much common sense and linguistic knowledge~\\cite{de2021javabert,chirkova2021empirical,karmakar2021pre}. With this knowledge, pre-trained models can achieve better generalizability, which means that it is able to perform better than its ``no pre-training'' counterpart on a wide variety of software engineering (SE) tasks after being fine-tuned on the data of the target task~\\cite{feng2020codebert,guo2021graphcodebert,mastropaolo2021t5-learning,ahmad2021plbart,wang2021codet5,niu2022sptcode,guo2022unixcoder}. Therefore, rather than learning models from scratch, the adoption of pre-trained models as backbone for downstream tasks has become a common practice in the field of software intelligence~\\cite{guo2021graphcodebert,gotmare2021cascaded,wang2022bridging}.\n\nHowever, the fine-tuning stage requires updating the weights of a pre-trained model by training on thousands of supervised labels specific to the target task. Therefore, the ``pre-train then fine-tune'' paradigm still relies on the data from the target task. And the effectiveness of transfer learning of the pre-trained model to the target task depends heavily on the size and quality of the fine-tuning data~\\cite{niu2022sptcode}. Unfortunately, in practice, we often encounter situations where we need to apply a pre-trained model to a task with very low available data resources. In such cases, we are unable to fine-tune the pre-trained model on sufficient target task data to obtain a fine-tuned model that can be applied to the target task.\n\nFacing the same issue, pioneers in the field of Natural Language Processing (NLP) have proposed many ways to address this issue. Few-shot learning~\\cite{radford2019gpt2,yu2018diverse,gao2021making} skips the fine-tuning phase and applies the pre-trained model directly to the target task, without any weight updates. To bridge the gap between the pre-trained model and the target task, few-shot learning gives the model a task description and a few demonstrations (i.e., supervised examples) of the task at inference time. If there is only one positive example, it is also called one-shot learning~\\cite{vinyals2016matching}. Moreover, instead of any examples, zero-shot learning gives the model a natural language description of the task~\\cite{brown2020gpt3,raffel2020t5}. In the few\/one\/zero-shot settings, a large-scale pre-trained language model is able to show strong performance on many NLP tasks and benchmarks, in some cases nearly matching the performance of state-of-the-art fine-tuned systems~\\cite{radford2019gpt2,brown2020gpt3,raffel2020t5,gao2021making,ye2021crossfit}. In addition, learning from task instructions~\\cite{efrat2020turking,weller2020learning,mishra2022cross} adopts task definition, positive and negative examples, where task definition can be seen as a specification of solving the target task.\n\nThe SE community also does some explorations. Mastropaolo et al.~\\cite{mastropaolo2021t5-learning} and Wang et al.~\\cite{wang2021codet5} utilizes multi-task learning~\\cite{raffel2020t5,aghajanyan2021muppet} to achieve a better performance on the target task. By learning multiple related tasks simultaneously, multi-task learning aims to make models exploit both task-generic and task-specific information, thereby improving the model's performance on tasks with low available resources. However, multi-task learning favors tasks with significantly larger amounts of data than others, thus requiring sufficient supervised examples of the target task compared to other tasks to guarantee the availability of the model~\\cite{gu2018meta,dou2019investigating}. Rather, Prenner and Robbes~\\cite{prenner2021making} experiments with several other techniques that promised a possible benefit for small datasets, i.e., active learning, data augmentation, soft labels, self-training and intermediate-task fine-tuning~\\cite{phang2018sentence}. They find that soft labels to be more useful, while other methods are relatively more narrowly applicable, less effective, more costly, or inconclusive. Instead of using any data from some target tasks, Guo et al.~\\cite{guo2022unixcoder} directly apply the pre-trained model to the code-to-code retrieval task in order to evaluate the performance of code fragment embeddings. Given that there are many very large-scale pre-trained models of source code being proposed (e.g., GitHub Copilot, Codex~\\cite{chen2021codex} and AlphaCode~\\cite{li2022alphacode}), there is also a lot of work exploring the few\/zero-shot performance of these models on specific tasks and domains, such as program repair~\\cite{prenner2021automatic,kolak2022patch,pearce2022examining}, software security~\\cite{asare2022github,pearce2022asleep} and program synthesis~\\cite{austin2021program}.\n\nAll things considered, in the field of software intelligence, there is no systematic work to evaluate and explore the cross-task generalizability of code models. Therefore, in order to evaluate the cross-task capability of the model on code-related tasks in detail and comprehensively, we build a large-scale benchmark called CrossCodeBench. We start by collecting as many and as diverse code-related tasks as possible, and end up with 216 tasks across 28 categories, 7 types and 18 programming languages. Then, to make our benchmark available for multiple cross-task learning settings (e.g., few-shot learning, learning from task instructions), we manually label each of the 216 tasks with extensive meta-information such as task description, definition, positive\/negative examples, etc. Next, we create 10 training\/evaluation splits corresponding to different benchmark types.\n\nGiven the data splits, we perform experiments by adopting two types of pre-trained models: (1) off-the-shelf model, which are used directly on the evaluation set, and (2) models further fine-tuned on the training set. All models are applied to all splits by using all or suitable cross-task learning methods, such as few-shot learning, learning from instructions, etc. Last but not least, we carry out a number of scaling experiments, through which we find that (1) when the data of each task reaches a certain level (e.g., 10,000 instances), it is difficult for the model to maintain a high improving speed of performance as data instances increasing, and (2) larger models always lead to better performance. We hope that the benchmark, experimental results and analysis we provide will facilitate future research to more powerful cross-task approaches in SE literature. Furthermore, since our benchmark contains massive datasets (and is open to updates), not only our CrossCodeBench, but we hope that such a large-scale meta-dataset (i.e., dataset of datasets~\\cite{triantafillou2019meta,wang2022benchmarking}) will facilitate the construction of more benchmarks\\footnote{All datasets, tasks and their summaries are available at \\url{https:\/\/doi.org\/10.5281\/zenodo.7321934}. Source code is available at \\url{https:\/\/github.com\/NougatCA\/CrossCodeBench}.}.\n\n\\section{Related Work}\n\\label{section:related}\n\n\\subsection{Few-Shot, One-Shot, and Zero-Shot Learning}\n\\label{section:related_few_shot}\n\nAlbeit defeating human in many fields~\\cite{taigman2014deepface,silver2016alphago,he2016resnet,najberg2018alibaba,berner2019dota,brown2020gpt3}, current artificial intelligence (AI) techniques still rely on learning from large-scale task-specific data, and they are unable to rapidly generalize from a few examples. Rather, humans are able to learn new tasks quickly by using what they are born with, or what they have learned in the past.\n\nFew-shot learning (FSL) is therefore proposed in order to learn from a limited number of examples with supervised information. In the cross-task setup, it is an in-context learning approach where a pre-trained language model does not need any fine-tuning and weight updating~\\cite{xie2021explanation}\\footnote{Since our work only discusses the cross-task scenario, i.e., where no supervised training is performed on the data of the target task, we only introduce the FSL methods that can be applied in this scenario, for more FSL methods please refer to Wang et al.~\\cite{wang2020generalizing} and Yin~\\cite{yin2020meta}.}. The input can be divided into three parts, namely task description, examples, and prompt~\\cite{brown2020gpt3}. The task description is a typically short natural language description of the task, e.g. ``translate English to French''. Examples consist of $k$ canonical supervised examples (usually $10<k<100$, one-shot learning for $k=1$~\\cite{brown2020gpt3}), each of which includes the context (i.e., the input\/question of the example) and the desired completion (i.e., the output\/answer of the example). And the prompt is the context part of the example for which the model needs to make a prediction.\n\nFSL shows promising results compared to the supervised approaches. In the FSL settings, GPT-3~\\cite{brown2020gpt3}, a large pre-trained language model, significantly improves the state-of-the-art (SOTA) on various datasets across many task types, such as completion~\\cite{paperno2016lambada}, open-domain question answering~\\cite{joshi2017triviaqa} and translation~\\cite{durrani2014edinburgh}. In addition, Madotto et al.~\\cite{madotto2020language} demonstrate that in some task-oriented dialogue system tasks, language model priming FSL is able to achieve similar or better results than fine-tuning-based baseline. Chen et al.~\\cite{chen2020few} show that in the FSL settings, a language model can achieve very reasonable performances and outperforms the strongest baseline by an average of over 8.0 BLEU points improvement, across multiple domains. Under the multilingual translation setting, Winata et al.~\\cite{winata2021language} find the in-context few-shot cross-lingual prediction results of language models are comparative to the existing SOTA cross-lingual and translation models. In order to investigate whether and how cross-task generalization ability can be acquired, Ye et al.~\\cite{ye2021crossfit} propose CrossFit challenge, a task setup that standardizes the training pipeline, data access and evaluation protocol. As a complement, they present the NLP Few-shot Gym, a repository of 160 diverse few-shot NLP tasks. Experimental results show that the cross-task generalization ability can be obtained by using multi-task and meta learning, and can be affected by the selection of seen tasks.\n\nIn contrast to FSL, zero-shot learning (ZSL) is proposed to enable generalization to the target task without any examples. Alternatively stated, the language model is given only the task description and the prompt. As only a short task description is required, ZSL is able to provide maximum convenience, potential for robustness, and avoidance of spurious correlations for transferring language models to new tasks~\\cite{brown2020gpt3}. However, this presents huge challenges, such as the possibility of ambiguity in the task description, the absence of examples of output formats, etc. Even so, on some tasks, zero-shot GPT-3 can still outperform one-shot GPT-3 (e.g., completion~\\cite{paperno2016lambada,zellers2019hellaswag}), or even few-shot GPT-3 (common sense reasoning~\\cite{lai2017race}). Besides, on ANLI~\\cite{nie2020anli}, GPT-3 under the zero-shot setting scores higher than under the few-shot and one-shot settings for some parameter size settings (i.e., 0.1B, 2.6B and 6.7B).\n\n\\subsection{Learning from Task Instructions}\n\\label{section:related_instructions}\n\nIn addition to investigating on the number of examples, researchers do some exploration on task description as well. Recall that in the FSL and ZSL settings, task descriptions are often short (e.g., 12.6 tokens on average~\\cite{mishra2022cross}), sometimes causing ambiguity, sometimes missing necessary formatting instructions, etc. Therefore, researchers investigated the use of longer and more detailed task descriptions.\n\nInspired by current NLP datasets built using crowdsourcing, Efrat and Levy~\\cite{efrat2020turking} examine if language models can follow crowdsourcing instructions with no further training. Weller et al.~\\cite{weller2020learning} construct a crowdsourced dataset, called Zest, with question-like task descriptions. To study the ability of a model that learns a new task by understanding the human-readable instructions that define it, Mishra et al.~\\cite{mishra2022cross} introduce a dataset including 61 distinct tasks, their human-authored instructions and instances, which is named NatInst. Then they adopt existing pre-trained language models to encode task-specific instructions along with input and generate task output. Results show that the model can benefit from instructions and can improve performance by 19\\% when evaluated in generalization for unseen tasks. PromptSource~\\cite{bach2022promptsource}, FLAN~\\cite{sanh2021multitask}, and InstructGPT~\\cite{ouyang2022gpt-instruct} also study the cross-task generalization ability by following the provided in-context task instructions. As a subsequent work to NatInst, Wange et al.~\\cite{wang2022benchmarking} introduce NatInst$_{v2}$, a benchmark of over 1,600 diverse language tasks and their expert-written instructions, which covers 70+ distinct task types. They also propose T$k$-Instruct, a model trained to follow a variety of in-context instructions, which include plain language task definitions and $k$-shot examples.\n\nThe above methods can be collectively referred to as learning from task instructions (LTI). Although the above work differs in the content of task instructions, after reading a lot of recent related work, we believe that a typical task instruction can generally be divided into the following parts: task definition, positive (and negative) examples, and other elements (e.g., ``Things to Avoid''~\\cite{mishra2022cross}). Task definition is a detailed definition of the task, unlike task description in FSL, it details how to map the given input to the required output in the current task. As a result, task description is typically longer and more detailed than the task description in FSL\\footnote{For example, the task definition of the task description ``Translate English to French'' is ``Given a sentence in English, provide an equivalent paraphrased translation in French that retains the same meaning both through the translation and the paraphrase.''~\\cite{wang2022benchmarking}}. In addition, task instruction sometimes contains negative instances as opposed to only positive instances in FSL.\n\nBy learning in-context task instructions, models such as InstructGPT and NatInst$_{v2}$ are able to achieve better cross-task generalization performance on unseen tasks than current FSL-based models. For example, on the FLAN benchmark, InstructGPT has about 76.2\\% win rate compared to baselines such as FSL-based GPT-3~\\cite{brown2020gpt3}, and the 3B-parameter T$k$-Instruct outperforms 175B-parameter InstructGPT by 3.3 ROUGE-L points~\\cite{lin2002rouge} when evaluated on 119 unseen tasks.\n\n\\subsection{Code-Related Multi-Task Benchmarks}\n\\label{section:related_benchmark}\n\nThere are existing benchmarks that span multiple code-related tasks. CodeXGLUE~\\cite{lu2021codexglue}, a benchmark which includes 14 datasets for 10 diversified code-related tasks covering code-to-code, text-to-code, code-to-text and text-to-text scenarios. CodeXGLUE is now widely used to evaluate the performance of pre-trained models of source code on various downstream tasks~\\cite{wang2021codet5,guo2022unixcoder}. Elnaggar et al.~\\cite{elnaggar2021codetrans} collect a benchmark that contains 6 code-related tasks across 9 programming languages (PLs). Compared to these benchmarks, our work provides a larger scale and more diverse code-related tasks (216 tasks and more than 54M data instances in total). In addition, we also provide systematic and reasonable cross-task task splits, which can comprehensively evaluate cross-task capabilities in various scenarios.\n\nXLCoST~\\cite{zhu2022xlcost}, a benchmark for cross-lingual code intelligence proposed by Zhu et al., consists of fine-grained parallel data from 7 PLs and English. This parallel data in a total of 8 languages allow XLCoST to support 10 cross-lingual code-related tasks, for example, program synthesis, code summarization, cross-language code retrieval, etc. Puri et al.~\\cite{puri2021codenet} introduce a large-scale dataset CodeNet, aiming to benchmark a variety of critical coding tasks, including code similarity and classification, code translation between a large variety of PLs, and code performance (runtime and memory) improvement techniques. Although these efforts provide large-scale multi-task benchmark, the data for each of their tasks are extracted from the same collected dataset. Therefore, the data distribution between their different tasks is identical, and it is difficult for us to obtain rigorous and valid cross-task evaluation results by these benchmarks.\n\n\\input{tables\/benchmark_compare}\n\nTable~\\ref{table:benchmark_compare} compares our proposed CrossCodeBench with aforementioned benchmarks. We first compare whether each task in these benchmarks has a well-validated task description, definition, positive and negative examples, and whether it is off-the-shelf (i.e., the inputs and outputs are already preprocessed for each task). In addition, we compare their number of tasks, the number of task types (cf. Table~\\ref{table:task_type_statistics}, where the first number indicates the number of types in a total of 7 types, the next number indicates the types after considering sub-types, and ``+1'' indicates that there are task types other than those in Table~\\ref{table:task_type_statistics}\\footnote{CodeXGLUE and XLCoST have two retrieval tasks, namely natural language code search and code-to-code retrieval. In this work, we exclude the retrieval task type because our goal is to evaluate the cross-task capability of the unified model, however, retrieval tasks cannot be converted to a unified text-to-text form.}), the number of datasets included, the number of programming languages (PLs), and the approximate number of total instances.\n\n\\section{CrossCodeBench}\n\\label{section:benchmark}\n\n\\subsection{Collecting Tasks}\n\\label{section:benchmark_collect}\n\nIn order to collect as many code-related datasets as possible, we hired 5 Ph.D. student (including one author of this paper) and 6 M.S. students (whose research area is software intelligence) to provide all datasets they have used. Their research span the areas of automated program repair, defect localization, code completion, code summarization, code generation, etc. Then, we recursively expand the scope of collecting datasets by reading surveys and based on paper relationships such as citations, related work, etc. In addition, we conduct an exhaustive search on a number of websites widely used to publish or collect datasets, such as GitHub, HuggingFace Datasets, PapersWithCode Datasets, etc. Finally, we end up collecting a total of 66 code-related datasets.\n\n\nSince some datasets contain multiple subsets (e.g. CodeSearchNet~\\cite{husain2020codesearchnet} contains 6 subsets corresponding to different programming languages) or can support multiple tasks (e.g. the XLCoST dataset~\\cite{zhu2022xlcost} can support translation between multiple parallel elements), and they do not correspond to the same task descriptions, we split these datasets into different tasks. As a result, we end up with 216 tasks. Moreover, for the task instances, we directly use the input and output in the form of text sequences provided by the dataset without any further processing. Since we are proposing a unified benchmark for all types of tasks, we formalize the inputs and outputs of all tasks in text-to-text form with reference to T5~\\cite{raffel2020t5}. In particular, for the classification task, we convert all labels to corresponding task-relevant text\\footnote{For example, for some binary classification tasks, we convert the label ``0'' to text ``No'' and ``1'' to ``Yes''.}. \n\n\\subsection{Meta Information}\n\\label{section:benchmark_meta}\n\nSome basic information is already done when collecting datasets and tasks, such as the URL, BibTeX, and input\/output languages. Next, in order to make our benchmark support multiple cross-task in-context learning methods, namely FSL, ZSL, and LTI, we need to pair each task with some corresponding meta information. In this section, we first present the schema of the meta information we need to complete, then illustrate the open coding procedure~\\cite{sgier2012qualitative} that we follow to complete the meta information.\n\n\\subsubsection{Schema}\n\\label{section:benchmark_meta_schema}\n\nWith Sections~\\ref{section:related_few_shot} and \\ref{section:related_instructions}, we decide to match meta information for each task in the following fields:\n\n(a) \\textbf{Type}: the type of task to which the task belongs.\n\n(b) \\textbf{Description}: a short description of the task, e.g. ``Translate English to French'', ``Summarize'', etc.\n\n(c) \\textbf{Definition}: detailed instruction of the task, including a description of the input, and how to map input to output.\n\n(d) \\textbf{Positive\/Negative Examples}: canonical positive or negative examples.\n\nWith these fields, our benchmark can support all current cross-task learning approaches. For FSL, we use task descriptions and positive examples; for ZSL, we only use task descriptions; and for LTI, we use task definitions and positive\/negative examples.\n\n\\subsubsection{Coding Procedure}\n\\label{section:benchmark_meta_coding}\n\nOpen coding procedure is a widely used standard analytical process that can be utilized to label a dataset~\\cite{chaparro2017detecting,zhai2020cpc,chen2021my}. We invite 1 Ph.D. student and 4 M.S. students mentioned in Section~\\ref{section:benchmark_collect} to participate in the completion of the meta information, all of whom have more than 4 years of programming experience and have been working on code-related tasks for more than 2 years. As a prerequisite background knowledge, we ask all coders to first read the NLP-related work mentioned in Sections~\\ref{section:related_few_shot} and \\ref{section:related_instructions}, and to become familiar with the corresponding data as well. Below we show the steps of the coding procedure.\n\n\\textbf{(a) \\textit{Pilot Study}}. \nWith the intent of defining the coding framework, one of the coders conduct a pilot study on randomly selected NLP 30 tasks with meta information, which are equally derived from the work of Brown et al.~\\cite{brown2020gpt3}, Bach et al.~\\cite{bach2022promptsource}, and Wang et al.~\\cite{wang2022benchmarking}. The goal of this study is to identify the initial task type, analyze and identify the common pattern of the task description, definition, and the positive\/negative examples. The work ends with the definition of four initial classifications, in addition to the experience and specifications for writing task descriptions (14 patterns), task definitions (5 patterns), and positive\/negative examples. Here are details:\n\n\\textbf{$\\bullet$} Initial task type includes \\textit{classification}, \\textit{translation}, \\textit{generation}, \\textit{summarization} and \\textit{type prediction}.\n\n\\textbf{$\\bullet$} Task description is a short single sentence command that often appears before the input instance and connects it to the instructions. It briefly describes the intent of the task, usually with one or a few words. 14 initial patterns are identified, to name a few, \\textit{Translate A to B}, \\textit{Summarize A}, \\textit{Detect (defect\/clone\/variable misuse)}.\n\n\\textbf{$\\bullet$} Task definition is a detailed guide to solving the task and can be divided into three parts: (1) the description and interpretation of the input under the task, including the language and form of the input, e.g., \\textit{given a Java method with the method name masked by a special symbol `[MASK]'}, \\textit{given a natural language description}; (2) how to convert the given input into output, and the language and form of the output, e.g., \\textit{translate the given Java method to Python function with the same functionality}, etc.; (3) others, including the search space of the output (for classification tasks, e.g., \\textit{if ..., outputs `Yes', otherwise outputs `No'}), and the format of the output (e.g., \\textit{for each identifier, outputs the name of the identifier and its type, separated by a colon `:'}). 5 initial patterns are identified which connect these three elements into a complete, fluent, easy-to-understand natural language sentence.\n\n\\textbf{$\\bullet$} Four positive and four negative examples are required. Examples need to reflect the most typical situation in the task, explaining the main points of generating positive examples for positive examples, and pointing out the errors and giving the correct modifications for negative examples.\n\nBased on this, this coder organizes a 60-minute session for the other four coders for training and discussion.\n\n\\textbf{(b) \\textit{Completion Procedure}}. \nEach coder is assigned to all 216 tasks. For each task, coders are asked to learn about the task by referring to the websites and paper through the URL and BibTeX, then observe the data for that task, and finally identify the task type and fill in the rest of the meta information as required. \n\nSpecifically, for task types, coders are allowed to identify a task as a new task type despite the fact that an initial task type already exists. In addition, it is encouraged to identify the task type while also giving the sub-types of the task under this task type, if possible. All other coders are informed when a new task type\/sub-type is proposed, and coders will have a discussion for at least 10 minutes. If all the coders agree on the new task type\/sub-type, then the type\/sub-type list would be updated to include the new one.\n\nAs for the task description and definition, coders are asked to try to follow the 5\/14 initial patterns for writing, and if this is not possible, adding new patterns is also allowed. An online list of writing pattern is shared between all coders, including both the initial schema and the newly added schema. In this way, the latest schema list is visible to all coders and open for them to use and add. It is worth noting that a new pattern can only be added to the list after it has been verified by all coders and the disagreement is solved by discussion.\n\nIn addition, for task types\/sub-types, pattern lists of task descriptions and definitions, when adding new entries (i.e., new types\/sub-types or task description\/definition patterns), the similarity between the new entry and the existing entry is inspected. Similar entries are merged into a new, more general entry in due course, without disagreement, and the existing labels are updated accordingly as well. The process is fully iterative and includes continuous refinement of the entry and discussion of ambiguous cases. Each decision made during the entry extraction process represents the opinion of all coders.\n\nLastly, when it comes to positive\/negative examples we are not able to follow the procedure described above since they are all free-form labeled content. We first ask all coders to independently complete writing the task description, task definition, and 2 positive\/negative examples. After finishing, we gather all coders to discuss each task, and for the task description and definition, we put together what all coders had written, vote on the most concise and clear version, and revised it until everyone is satisfied. For the positive and negative examples, we first put together a total of 10 examples they written, vote on the most representative 4 until no disagreement remains.\n\n\\textbf{(c) \\textit{Agreement Measurement}}. \nTo evaluate the validity and reliability of our coding procedure, we use Cohen's Kappa value~\\cite{cohen1960coefficient} to measure the agreement among all coders. The results are 85.4\\%, 76.7\\%, 73.2\\% and 70.9\\% for task type, task description, task definition, selection of positive and negative examples, respectively. This shows that all coders have ``almost perfect agreement'' on the task type and ``substantial agreement'' on the other three meta information~\\cite{viera2005understanding}.\n\nFor fields that still have disagreements, we resolve them by applying a third person solution. We assign conflicting reports to 4 external coders (the 5 Ph.D. students mentioned in Section~\\ref{section:benchmark_collect} except the one internal coder) and let them judge and resolve these disagreements. Our analysis shows that the disagreements mainly focus on the second part of the task definition, where different coders have different understandings and representations of the input-to-output mapping, as well as positive and negative examples.\n\nFinally, in order to ensure the resulting meta information is sufficient for average software engineering researchers. We invite five undergraduate software engineering students with programming experience in non-software intelligence fields to read the meta information of all tasks. They are then asked to review the definition, intent, input and output formats, and other information for each task. Where they are unclear or incorrectly stated, we will feed back to the 4 external coders mentioned in the previous paragraph for revision until the task can be clearly defined and understood.\n\n\\subsubsection{Summary}\n\\label{section:benchmark_meta_summary}\n\nIn the end, we get 7 task types as follows.\n\n\\textbf{$\\bullet$ Classification}: output the corresponding labels based on the input. It is further divided into two sub-types, Binary and Multi-label, which correspond to tasks with only two labels and more than two labels, respectively.\n\n\\textbf{$\\bullet$ Fill in the blank}: predict the missing token or sequence in a given input.\n\n\\textbf{$\\bullet$ Translation}: translate code snippets written in one language into another preserving semantics and functionality.\n\n\\textbf{$\\bullet$ Generation}: generate a sequence based on the input. Three sub-types are (1) Rewrite: modify a part of the given code sequence and output the modified version; (2) Text-to-Code: input the natural language description and output the corresponding code sequence; (3) Code-to-Text: input as code, output the required natural language sequence as required.\n\n\\textbf{$\\bullet$ Summarization}: given a piece of code, output the functional description of that piece of code\\footnote{We separate this task from the Code-to-Text sub-type in Generation because this type of task is in a very important position in software intelligence research. And its research approach is different from other Code-to-Text approaches (e.g., commit message generation)}.\n\n\\textbf{$\\bullet$ Type Prediction}: predict the type of all identifiers in a given code snippet, which is a kind of sequence tagging tasks.\n\n\\textbf{$\\bullet$ Question Answering}: given a piece of code and a natural language question, output the answer to that question. \n\n\\input{tables\/benchmark_statistics}\n\\input{tables\/task_type_statistics}\n\nTable~\\ref{table:benchmark_statistics} provides statistical information about CrossCodeBench. In addition, Table~\\ref{table:task_type_statistics} shows the statistics of the final task types. In total, the benchmark includes 216 tasks, 28 task categories and over 54M instances.\n\nWe define a task \\textbf{category} as the set of tasks that have the same intent.\nWe define a \\textbf{task} as a \\textit{\\textless task category, dataset\\textgreater} pair, and a task \\textbf{type} depends on the input and output of the task, not on the intent. A task category is a superset of a task. But a task category is equal to a task if and only if the task category has only one dataset. A task category is a subset of a task type (e.g. Bug Fixing~\\cite{tufano2019bfp} and Mutant Generation~\\cite{tufano2019mutant} categories are both Rewrite sub-types), they are equal if and only if the task type has only one task category, e.g. the Translation type has only one category, Code Translation.\n\n\\subsubsection{Demonstration}\n\\label{section:benchmark_meta_demonstation}\n\nIn our benchmark, each task consists of two json files, one containing the meta information and the other containing the data instances. Both files have the same name in the first part, i.e., ``task\\_\\{task\\_id\\}\\_\\{dataset\\_name\\}\\_\\{task\\_type\\}\", followed by \".meta.json\" and \".data.json\", respectively.\n\nTo better illustrate, Figure~\\ref{figure:meta_example} shows the contents of a json file with the meta information of a task. We can see that in addition to the fields we mentioned in Section~\\ref{section:benchmark_meta_schema}, we also include some information about the task\/dataset, such as the language of input\/output, BibTeX and URL of the dataset, etc.\n\n\\begin{figure}\n    \\centering\n\\begin{tcolorbox}[title=task\\_006\\_swapped\\_operands\\_classification.meta.json,fontupper=\\footnotesize,left=1mm,right=1mm,top=1mm,bottom=1mm]\n    ``\\textbf{Type}'': [``Classification'', ``Binary''],\\\\\n    ``\\textbf{Description}\": ``Detect swapped operands'',\\\\\n    ``\\textbf{Definition}'': ``You are given a function, your task is to identify whether the operands of non-commutative binary operators are swapped. Construct an answer that is `Swapped operands' if such a swap occurs and `Correct' otherwise.'',\\\\\n    ``\\textbf{Input Language}'': ``Programming Language -\\textgreater~Python'',\\\\\n    ``\\textbf{Output Language}'': ``Natural Language -\\textgreater~English'',\\\\\n    ``\\textbf{Positive Examples}'': [\\{\n    \n        ~~~~``Input\": ``def \\_\\_contains\\_\\_(self, x): return x in self.columns'',\n        \n        ~~~~``Output\": ``Correct'',\n        \n        ~~~~...\\\\\n    ],\\\\\n    ``\\textbf{Negative Examples}'': [...],\\\\\n    ``\\textbf{BibTeX}'': ...,\\\\\n    ``\\textbf{URL}'': ...,\\\\\n    ...\n\\end{tcolorbox}\n\\vspace{-0.5em}\n    \\caption{An example of the task meta information.}\n    \\label{figure:meta_example}\n\\end{figure}\n\n\\subsection{Splits}\n\\label{section:benchmark_splits}\n\nAfter preparing the data, we need to create different training\/evaluation splits to evaluate the cross-task learning ability of the model in different application scenarios and difficulties. Table~\\ref{table:task_splits} lists the 10 training\/evaluation splits we create and use in this paper. We classify them in two dimensions: the level of the cross-task, and the scope of the training task. In addition, we list the name of each split, the number of tasks and instances of the training\/evaluation split in Table~\\ref{table:task_splits}. Next, we will introduce these splits in order from 3 cross-task levels, namely, cross-category, cross-sub-type and cross-type.\n\n\\input{tables\/task_splits}\n\n\\subsubsection{Cross-Category}\n\\label{section:benchmark_splits_category}\nDefine new task category is common in SE, but collecting and labeling a large amount of data for the newly defined task category can be very labor-intensive, even impossible. Instead, we can define the task category exactly and give several examples. At this point, we can consider using cross-task learning methods, where learning on a large number of existing tasks acquires generalization capabilities on this new task. Therefore, we first wish to explore whether, and to what extent, current cross-task learning methods will make models achieve cross task category ability. We choose a binary classification task, i.e., clone detection~\\cite{svajlenko2014bigclonebench} and a rewrite generation task, bug fixing~\\cite{tufano2019bfp} as evaluation task categories, respectively. In addition, since task categories are a subset of task types, we also want to explore whether more data from different task types can help improve the cross-task performance of the model on one task category. \n\nTherefore, for each of the two evaluation sets, we use two training sets for training: (1) the training set includes only tasks of the same task type as the evaluation task in the other tasks (Intra-Type); (2) the training set includes all other tasks (Inter-Type). Finally, we create 4 splits for the cross-category level, namely Cat-Intra-CD, Cat-Intra-BF, Cat-Inter-CD, and Cat-Inter-BF.\n\n\\subsubsection{Cross-Sub-Type}\n\\label{section:benchmark_splits_sub_type}\n\nRecall that we also have sub-types for Classification and Generation types in Table~\\ref{table:task_type_statistics}. We also want to explore to what extent the model can learn cross-task capabilities at the level of sub-type. Therefore, we select two sub-types for each of them, namely Multi-label Classification and Rewrite Generation. For multi-label classification sub-type, we want to investigate whether the model can generalize to multi-label tasks by learning only binary classification sub-type (and other types of tasks). For rewriting tasks, we wish to explore how much rewriting skill the model can acquire without having learned any rewriting task. As with the setup in the cross-category, we create 2 splits for each of the two selected sub-types under the different scopes of the training set. Consequently, we create 4 splits as well, i.e., Sub-Intra-ML, Sub-Intra-C2T, Sub-Inter-ML, and Sub-Inter-C2T.\n\n\\subsubsection{Cross-Type}\n\\label{section:benchmark_splits_type}\n\nThe last is the most challenging setup, the cross-type, where we use all tasks in the entire task type as the evaluation set and the other types of tasks as the training set. First, we want to use the Translation type as the evaluation set, and we want to explore whether the model can learn the correspondence between different languages by learning from other types of tasks. Second, we choose a type that contains the fewest tasks, i.e., Question Answering. This split is closest to the real practice situation, i.e., by learning on a well-resourced task type, we expect the model to have a better generalization ability on the new task type without resources. Therefore, we create two splits at the cross-type level, namely Type-Trans and Type-QA.\n\nBy now, we create 10 training\/evaluation splits of different difficulty levels and different application scenarios. These splits are used in the experiments. Besides, we are able to ensure that there is no data leakage in these splits, in another word, there is no task in the training set with the same resource dataset as any task in the evaluation set. \n\n\\section{Experimental Setup}\n\\label{section:setup}\n\nIn this section we describe the setup for preliminary experiments on CrossCodeBench, including cross-task learning methods, baselines, evaluation metrics and other settings.\n\n\\subsection{Cross Task Learning Methods}\n\\label{section:setup_learning_methods}\n\nHere we introduce the in-context cross-task learning method that we use for our experiments. In-context learning methods also determine the composition of the inputs to the model in our experiments. By referring to Section~\\ref{section:related_few_shot}, \\ref{section:related_instructions} and the meta information we introduced in Section~\\ref{section:benchmark_meta}, we propose the following learning methods.\n\n\\textbf{$\\bullet$} $k$-shot Learning ($k$-shot) : recall that the input of the few-shot learning consists of three parts: task description, $k$ examples and a prompt. Since each task in CrossCodeBench contains 4 positive examples, we randomly select $k (=4,3,2,1)$ positive examples as the examples of the input context, resulting four methods named \\textbf{4-shot}, \\textbf{3-shot}, \\textbf{2-shot} and \\textbf{1-shot}.\n\n\\textbf{$\\bullet$} Zero-shot learning (\\textbf{zero-shot}): same as $k$-shot, but without the second part of the input, i.e. the example.\n\n\\textbf{$\\bullet$} Learning from instructions with $m\/n$ positive\/negative examples ($m\/n$-instruct): use instructions items introduced in Wang et al.~\\cite{wang2022benchmarking}, with $m\/n$ positive\/negative examples. Specifically, we select the two combinations that worked best shown by Wang et al.~\\cite{wang2022benchmarking} (1) \\textbf{2\/0-instruct}: $m=2$, $n=0$ when training and $m=4$ when evaluating; (2) \\textbf{2\/2-instruct}: $m=2$ and $n=2$ for both training and evaluating.\n\nFinally, we derive 7 cross-task learning methods that will be used as input to the baseline model to be presented below.\n\n\\subsection{Models}\n\\label{section:setup_models}\n\n\\subsubsection{Shortcut Methods}\n\\label{section:setup_models_shortcut}\n\nThe feasibility of a proposed benchmark is important for the subsequent work, which determines whether the benchmark is practical and meaningful to study. Specifically for CrossCodeBench, it is whether the model can actually gain cross-task capability by using a particular cross-task learning method, rather than just behaving like it has gained such capability through some easy shortcuts. Therefore, we propose two such shortcut methods for each split, (1) \\textbf{Copy Ex-output}: randomly copying the output of one of the four positive examples and (2) \\textbf{Copy Ins-input}: copying the input of the current instance.\nWe use these two shortcut methods to evaluate the feasibility of CrossCodeBench, i.e., doing cross-task research in the field of software intelligence.\n\n\\subsubsection{Off-the-Shelf Models}\n\\label{section:setup_models_off}\n\nOff-the-shelf models are those that are evaluated directly on the evaluation set, without any further fine-tuning. We first wish to choose a model that is a ``few-shot learner'', such as GPT-3~\\cite{brown2020gpt3}. However, our request to use the GPT-3 API has not been approved. Instead, we turn to \\textbf{Tk-Instruct}~\\cite{wang2022benchmarking}, a T5-based model that is already trained to follow general language task instructions (including a portion of code-related tasks). Tk-Instruct is shown to have better cross-task performance than few-shot learners, such as GPT-3, and other task instruction learners, such as InstructGPT~\\cite{ouyang2022gpt-instruct}. Tk-Instruct is only used under the $m\/n$-Instruct learning settings. Specifically, we use the 3B parameter version of the model, and for the learning method in the 2\/0-instruct and 2\/2-instruct settings, we use the ``Tk-Instruct-3b-def-pos' and ``Tk-Instruct-3b-def-pos-neg'' versions of the model, respectively, in order to achieve the best performance~\\cite{wang2022benchmarking}. We use the Tk-Instruct model without any fine-tuning and directly evaluate them on the evaluation set, under the different learning methods.\n\n\\subsubsection{Fine-tuned Models}\n\\label{section:setup_models_tuned}\n\nWe evaluate pre-trained models that are fine-tuned on the training set with aforementioned in-context learning method as well. First, we choose two recent and widely used pre-trained models of source code, \\textbf{PLBART}~\\cite{ahmad2021plbart} and \\textbf{CodeT5}~\\cite{wang2021codet5}. In particular, PLBART is a sequence-to-sequence model based on BART~\\cite{lewis2020bart} and pre-trained on an extensive collection of Java and Python functions and associated NL text. CodeT5 is a T5~\\cite{raffel2020t5}-based model that is pre-trained on a large corpus containing 8 programming languages and natural language. We ensure that neither PLBART nor CodeT5 had supervised training on any evaluation task in any splits. We also ensure that there is no overlap between the dataset they use in pre-training and the source dataset of the evaluation set in all splits. Besides, we further fine-tune \\textbf{Tk-Instruct}~\\cite{wang2022benchmarking} by using the $m\/n$-instruct learning method. We hope that further fine-tuning will help Tk-Instruct to adapt to code-specific tasks while understanding generic task instruction. Consequently, we obtain three fine-tuned models, two pre-trained models of source code, PLBART and CodeT5, which will be fine-tuned under all cross-task learning methods. There is also an instruction learner, Tk-Instruct, which will be fine-tuned under only two $m\/n$-instruct methods. In order to balance efficiency and effectiveness, we use the official ``large'' version of all three models.\n\n\\subsubsection{Supervised Models}\n\\label{section:setup_models_supervised}\n\nWe estimate an upper bound performance of each split by supervised fine-tuning a CodeT5-large model on all task instances (except instances used for evaluation) on all evaluation set. This approach follows the classical ``pre-train then fine-tune'' paradigm, where we use all evaluation tasks in the fine-tuning and do not apply any cross-task learning methods. Since this approach allows the model to see the data of the target task and perform supervised fine-tuning, this approach is theoretically the upper limit of the cross-task approach on the corresponding split.\n\nFinally, we end up with 2 shortcut methods, 2 off-the-shelf models, 3 fine-tuned models and 1 supervised model. All model checkpoints are loaded from the official models published on HuggingFace Hub.\n\n\\subsection{Metrics}\n\\label{section:setup_metrics}\n\nSince outputs of all tasks are in text form (see Section~\\ref{section:benchmark_collect}), so text-specific metrics are used in our experiments. Specifically, for classification tasks whose output is short and limited, we adopt \\textbf{Exact Match (EM)} to measures the ratio of the instances for which a model produces exactly the same string as the gold labels. For the other tasks, their output is a longer sequence, so we employ \\textbf{BLEU (B.)}~\\cite{papineni2002bleu} and \\textbf{ROUGE-L (R.L)}~\\cite{lin2002rouge}. Both are widely adopted string overlap metrics that measure the similarity of between the text sequence generated by the model and the gold sequence. For all metrics, we report scores under percentages.\n\n\\subsection{Others}\n\\label{section:setup_others}\n\nIn order to avoid the data imbalance problem, we limit the number of instances in each training tasks to 10,000. These 10,000 instances are fixed across different running if the number of instances for a task exceeds 10,000. Similarly, to avoid the evaluation results being unevenly affected by the amount of task data and to make the evaluation more efficient, we select a fixed number of 500 evaluation instances from each task.\n\nAll experiments are conducted on 4$\\times$NIVDIA Tesla V100 32Gs with a total fine-tuning epochs of 3. We tune hyperparameters using grid search. We select learning rate from \\{1e-5, 3e-5, 5e-5, 1e-4\\}, warm-up steps from \\{500, 1000, 2000\\}, and batch size per device from \\{4, 8, 16\\}. As a result, for the fine-tuning phase, we use an initial learning rate of 5e-5 for CodeT5 and PLBART, and 1e-5 for Tk-Instruct, with a batch size per device of 16 and 1000 warmup steps for all. We run each experiments three times using different random seeds and report the mean.\n\n\\section{Results and Discussion}\n\\label{section:results}\n\nIn this section, we present and discuss the preliminary experimental results on CrossCodeBench.\n\n\\subsection{Overall Results}\n\\label{section:results_overall}\n\n\\input{tables\/results_overall}\n\nTable~\\ref{table:results_overall} shows the overall benchmarking results. Except scores of the supervised model, the best scores are bolded and those within 5\\% below the best scores are underlined, and if there are no scores within 5\\% below, the bolded best scores are further underlined. Based on these results, we have the following observations and discussions.\n\n\\subsubsection{Room for improvement exists}\n\nFirst of all, we can see that supervised fine-tuned CodeT5 following the classical ``pre-train then fine-tune'' paradigm has very significant advantages over any other methods. Since supervised CodeT5 performs supervised training on the target task, it can be seen as an upper bound for the performance of large language models of source code on the corresponding evaluation set. This suggests that there is still a lot of room for theoretical improvement in cross-task learning based on pre-trained models of source code.\n\n\\subsubsection{Shortcut wins on low-resource classification}\n\nWe find that the best fine-tuned model outperforms shortcut methods on all metrics for all splits except Cat-Intra-CD and Sub-Intra-ML. This illustrates that the cross-task learning approach allows the models to gain real cross-task capability, not just some simple shortcuts. And on the two splits where the shortcut method wins, we discover two points, one is that scopes of the training set of both splits are both ``Intra'', and the second is that only the evaluation sets of these two splits contain the classification tasks.\n\nFor the first point, recall that in Section~\\ref{section:benchmark_splits}, we create splits of ``Intra'' and``Inter'' by changing the scope of tasks in the training set, the former restricts the tasks in the training set to the same type as the verification tasks, while the latter has all other tasks in the training set. On the two ``Inter'' corresponding to these two ``Intra'', the fine-tuned method outperforms the shortcut method instead. Therefore, we can conclude that having more data for other types of tasks could improves the cross-task performance on tasks of a certain type.\n\nAs for the second point, we believe that the main reason is that the output of these tasks is short (usually one word) and fixed (corresponding to all labels), and our experiments do not restrict these conditions, relying only on giving some ``soft'' hints in the task definition and in the output of the examples. For example, in the classification task shown in Figure~\\ref{figure:meta_example}, we illustrate the restriction\/range of output only in two places, (1) ``Definition'': the output are ``Swapped Operands'' in the case of operands swapping only in definition, and ``Correct'' otherwise, (2) ``Positive\/Negative Examples''. These restrictions are simply entered into the model as plain text, and thus cannot pose any substantial restrictions on what the model generates. In this case, we can only expect the model to be trained to acquire the knowledge of how to find the output range on each classification task in the input. But the results show that this is far from the case, and the model often generates text outside the output range, leading to a low performance. This conclusion is further corroborated by browsing the output of the fine-tuned model on the classification task.\n\nThe performance of the shortcut method on the classification task suggests that further restrictions on the model's output on the classification task are needed if one wants a unified text-to-text model to have excellent cross-task performance on the classification task.\n\n\\subsubsection{Further fine-tuning matters}\n\nIt is clear that the fine-tuned model has a significant performance advantage over the off-the-shelf model. This is because fine-tuning on our training set can help the model understand the input representation of cross-task learning methods (for PLBART and CodeT5), or become more familiar with tasks in the software intelligence domain (for Tk-Instruct). Even though Tk-Instruct has included several code-related tasks, such as ``code-to-text'', in its original training set~\\cite{wang2022benchmarking}, we believe that these tasks are diluted in a huge number of NLP tasks, making off-the-shelf Tk-Instruct unable to obtain good power in code-related tasks.\n\n\\subsubsection{Domain-specific models are better}\n\nWe also notice that CodeT5, a pre-trained model of source code, is able to achieve better results than Tk-Instruct, a model trained to follow task instructions, in most tasks. This suggests that domain-specific pre-trained models are generally a better choice than cross-domain models. Moreover, we also notice that Tk-Instruct wins in the task of generating natural language, such as Code-to-Text and Question Answering. This is not difficult to understand, because training on huge amount of natural language corpus helps it generate more fluent and reasonable natural language text.\n\n\\subsubsection{Detailed in-context information generally helps}\n\nAmong all cross-task learning methods, LTI produces the best results for all tasks and metrics. Specifically, in most cases, using only two positive examples is more effective than using two positive and two negative examples. By comparing the input representations of FSL, ZSL and LTI, we can see that compared to FSL and ZSL, LTI extends the short task description to a detailed task definition, which allows the model to understand the task more comprehensively, including the input\/output format, the search space of outputs, etc., and thus learn the solution to the task. In addition, LSI further introduces negative examples, but it appears that the extra negative examples do not contribute positively to the performance of the model in most cases. This also confirms the results obtained by Wang et al.~\\cite{wang2022benchmarking}.\n\n\\subsection{Data Scaling}\n\\label{section:results_data}\n\nIn addition to more data on other task types, we also explore whether the size of the training instances could affect the cross-task performance of the model given the same tasks. Therefore, on Type-Trans, we vary the number of instances per task that are used for fine-tuning CodeT5, and the evaluation results are shown in Figure~\\ref{figure:data_scaling}.\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/data_scaling.pdf}\n    \\caption{Scaling trends of models performance as the maximum number of instances per task changes from 500 to 20,000. The error is indicated by light shading. The x-axis is on a log-scale in order to be more intuitive.}\n    \\label{figure:data_scaling}\n\\end{figure}\n\nIt can be seen that when the number of tasks in the training set is constant, increasing the maximum number of instances per task within a certain range can significantly improve the cross-task performance of the model. However, when the maximum number of instances per task increases from 10,000 to 20,000, the BLEU shows a significant decrease, so for the BLEU metric, the model achieves its best performance at 10,000. On the other hand, for ROUGE-L, although the average value has been increasing all the time, if the range of data errors is taken into account, the model may start to show a decrease in performance when increasing from 5,000 to 10,000. Thus, in aggregate, the model may achieve its best performance at a maximum number of instances of 10,000 per task, which, if continued to increase, leads to time and space costs that do not match the gains in model performance.\n\n\\subsection{Model Scaling}\n\\label{section:results_model}\n\nWe also study the effect of model scaling by initializing CodeT5 and Tk-Instruct from different sizes of checkpoints and the results are in Figure~\\ref{figure:model_scaling}.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/model_scaling.pdf}\n    \\caption{Scaling trends of models performance as the number of model parameter changes. The error is indicated by light shading. The x-axis is on a log-scale in order to be more intuitive.}\n    \\label{figure:model_scaling}\n\\end{figure}\n\nWe find that increasing the model size consistently delivers an improvement in the cross-task performance of the model over the range of model sizes we experimented with, and is roughly log-linear with parameter size. Combining the results in Figure~\\ref{figure:data_scaling}, we can see that models of a certain size have a fixed demand on the amount of data. For example, CodeT5-large reaches a performance bottleneck at a maximum of 10k instances per task. Conversely, a larger model always leads to better performance when the data scale is of a certain size (note that the data volume should be large enough to avoid overfitting).\n\n\\subsection{Scope Scaling}\n\\label{section:results_scope}\n\nFinally, we investigate a problem with practical applications, i.e., the case when we propose a new task class and obtaining the corresponding data is very costly or impossible. Considering the use of cross-task learning is a feasible approach at this point. Thus, there is a question, how the scope of the training set is selected is important for the cross-task capability of the model. Therefore, we select a target task category, Bug Fixing~\\cite{tufano2019bfp}, and then change various ranges of training sets to investigate under which scope the model would achieve the best cross-task performance for the Bug Fixing task category.\n\nWe also present two splits in Section~\\ref{section:benchmark_splits_category}, with experiments on two application scenarios where the target task is Bug Fixing. The training sets in both splits in Section~\\ref{section:benchmark_splits_category} include the sub-type (Rewrite) or type (Generation), to which Bug Fixing belongs. However, in practical, the proposed task category may be a new sub-type\/type, and other task data of the same sub-type and type is not available. What we further study here is whether in this case, the more data is still better when we can only select tasks of other sub-types or types.\n\nAfter selecting Bug Fixing as the target task category, we change the scope of the training set data from three dimensions: (1) In-Sub-Type: whether the training set contains task data of the same task sub-type as the target task, i.e., Rewrite; (2) In-Type: whether the training set has data of the same task type as the target task, i.e. Generation; (3) Out-Type: whether there are data in the training set that do not belong to the same task type as the target task, i.e., the other 6 types except the Generation type. We investigate the data in the training set under various combinations in these three dimensions and present the results in Table~\\ref{table:scope_scaling}.\n\\input{tables\/scope_scaling}\n\nComparing experiments 2 and 4, or 3 and 5 in Table~\\ref{table:scope_scaling}, we learn that the data within the sub-type plays an important role. But such a conclusion holds only when other task data are available. If we compare experiment 1 with 5 and 6, it can be found that the performance achieved by using only in-sub-type data is not as good as that achieved by using data outside the sub-type. \n\nBesides, in comparison with experiments 2 and 6, we can find that the performance using in-type data is worse than that using out-type data only. We believe the reason is that the amount of out-type data is larger than that of in-type data, which allows the model to learn more cross-task capabilities with more in-context information.\n\nTherefore, we can conclude that the amount of data still plays an important role. Its contribution to the cross-task performance of the model is more important than the data for the same type of task.\n\n\\subsection{Implications}\n\\label{section:results_implications}\n\nAfter our experiments and discussions above, we have some findings and implications that can facilitate subsequent researchers and practitioners.\n\nThe first thing we can see is that all models performs very poorly on the classification task in the cross-task setting. After adding the instruction containing the set of classification labels to the input, the performance of the model on the classification task, though improved, is still poor because it still often outputs words outside the output range. So, how to constrain the model to output valid classification labels requires further investigation.\n\nIn addition, our results show that the different combinations of SE tasks used for training a model would yield different performances on a target task. So, when given a new SE task that lacks data, how to quickly determine which existing tasks should be used to achieve the best results is an urgent issue to explore.\n\nFinally, we learn from Sections~\\ref{section:results_data}--\\ref{section:results_scope} that more data, larger model and more tasks can all benefit the cross-task generalization ability. Therefore, if there are limited resources, is it better to devote it to a larger model or to more data, and if to data, is it better to increase the data per task or to increase the diversity of tasks. That is, how to make better use of the limited resources is also an issue worth investigating.\n\n\\section{Threats to Validity}\n\\label{section:threats}\n\n\\paragraph*{Internal Validity}\n\nThreats to internal validity relate to the bias introduced when manually annotating some fields in meta information of the task. To mitigate this, we start with the open coding procedure to reduce the impact of individual bias. Then for remaining conflicts, we address these by employing a third person solution (see Section~\\ref{section:benchmark_meta_coding} for details). We believe that the approach we adopt minimizes the human impact on the meta information, but it still may not be completely avoidable.\nAnother factor for threatening internal validity is hyperparameters. Due to time constraints, we only perform a coarse-grained hyperparameter search. Therefore, other hyperparameter settings may lead to better results. But this does not affect the observations and conclusions presented in this paper, since all experiments are carried out with the same hyperparameter searching strategy. \n\n\\paragraph*{External Validity}\n\nThreats to external validity concern about the data imbalanced among different types of tasks. Even though we limit the maximum number of instances per task, the number of tasks contained in different task categories, types varies greatly (see Table~\\ref{table:task_type_statistics}). Therefore, the model may have different generalization performance on different types of unseen tasks. In addition, since we need to apply tasks to the unified sequence-to-sequence model, all tasks in our benchmark are formalized in a text-to-text form. Those code-related tasks that cannot be converted to this form are excluded. An example is the retrieval tasks, such as natural language code search and code-to-code retrieval~\\cite{guo2022unixcoder}. The retrieval task requires computing the similarity between the representation vectors generated by the model for different inputs and cannot be applied to the text-to-text framework.\n\n\\paragraph*{Construct Validity}\n\nThe major threat is the data or task overlap between the evaluation set and the data\/task the model has seen during self-supervised pre-training and fine-tuning on the training set. First, since the minimum cross-task granularity of our proposed split is task categories, there is no overlap of task categories between the training and evaluation sets. Then, to mitigate such data overlap threats, we use the tool provided by Allamanis~\\cite{allamanis2019adverse} to ensure that the data in the evaluation set of the proposed 10 splits do not have duplication with the data in the training set during our experiments. Therefore, we can guarantee that no overlap problem will be introduced in the fine-tuning phase, but since we use pre-trained models as the backbone of our baselines, such threats may be introduced in the pre-training phase.\n\n\\section{Conclusion}\n\\label{section:conclusion}\n\nTo study the problem of cross-task generalization in software intelligence, we collect a large-scale meta-dataset containing various types of tasks and label each task with rich meta information to support various cross-task learning methods. We then create a benchmark, CrossCodeBench, for studying the generalization ability of source code-related deep learning models. Through preliminary experiments, we demonstrate and analyze the feasibility and possible research directions for cross-task studies of software intelligence.\n\n\\section*{Acknowledgment}\n\nThis work was supported by National Natural Science Foundation of China (61802167), Cooperation Fund of Huawei-NJU Creative Laboratory for the Next Programming, and NSF award 2034508. We also thank the reviewers for their helpful comments. Chuanyi Li is the corresponding author.\n\n\\bibliographystyle{IEEEtran}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe notion of a microset is due to F\\\"{u}rstenberg \\cite{Furst}, it was introduced to understand the infinitesimal structure of a compact set $K\\subset \\mathbb{R}^d$. Microsets are obtained by `zooming' in on $K$ and taking the Hausdorff limit of what we see. For the definition of Hausdorff metric and other notions see the Preliminaries section. \n\n\\begin{definition} Let $d\\geq 1$ be a positive integer and let $K\\subset \\mathbb{R}^d$ be compact. We say that $E$ is a \\emph{microset} of $K$ if $E \\cap (0, 1)^d \\neq \\emptyset$ and there exist homotheties $S_n\\colon \\mathbb{R}^d\\to \\mathbb{R}^d$ defined as $S_n(z)=\\lambda_n z+u_n$ with $\\lambda_n\\geq 1$ and $u_n\\in \\mathbb{R}^d$ such that $S_n(K)\\cap [0,1]^d$ converges to $E$ in the Hausdorff metric. Then $\\{\\lambda_n\\}_{n\\geq 1}$ is called a \\emph{scaling sequence} of $E$. \n\\end{definition} \n\nIn order not to lose  information about the `thinnest' part of our set $K$, we added the non-standard assumption $E\\cap (0,1)^d \\neq \\emptyset$ to the definition. Without this property $\\{0\\}$ would be a microset of every Cantor set $K\\subset \\mathbb{R}$.  Besides this, the definition also varies across different sources. F\\\"{u}rstenberg \\cite{Furst} allows to converge to $E$ by subsets of $S_n(K)\\cap [0,1]^d$, which approach only provides information about the `thickest' part of $K$. Bishop and Peres \\cite{BP} assumes that $\\lambda_n\\to \\infty$. Fraser, Howroyd, K\\\"aenm\\\"aki, and Yu \\cite{microsets} define microsets without the restriction $E\\cap (0,1)^d\\neq \\emptyset$, they only add this extra condition to the notion of gallery. Some authors also consider weak tangents, which means that $S_n$ might be arbitrary expanding similarities in the definition instead of homotheties. As the orthogonal group of $\\mathbb{R}^d$ is compact, this concept would not significantly differ from ours.\n\n\\begin{notation} For a compact set $K\\subset \\mathbb{R}^d$ define \n\\begin{equation*} \\mathcal{M}_K=\\{E: E \\text{ is a microset of } K \\}.\n\\end{equation*}\n\\end{notation}\t\n\nLet $\\dim_H$, $\\underline{\\dim}_B$, $\\overline{\\dim}_B$, and $\\dim_P$ denote the Hausdorff, lower box, upper box, and packing dimensions, respectively. For the following inequalities see \\cite[Page~82]{Ma}.\n\\begin{fact} \\label{f:ineq} For any set $E\\subset \\mathbb{R}^d$ we have \n\\begin{equation*} \n\\dim_H E\\leq \\underline{\\dim}_B \\, E\\leq  \\overline{\\dim}_B \\, E \\quad \\text{and} \\quad \n\\dim_H E\\leq \\dim_P E\\leq  \\overline{\\dim}_B \\, E.\n\\end{equation*}\n\\end{fact}\n\nThe infimum part of the following theorem was proved by Fraser, Howroyd, K\u00e4enm\u00e4ki, and Yu \\cite[Theorem~1.1]{microsets}, while the supremum part is basically due to Furstenberg \\cite[Theorem~5.1]{Furst}, see also \\cite[Theorem~2.4]{microsets}. \n\n\\begin{theorem}[Fraser--Howroyd--K\\\"aenm\\\"aki--Yu and F\\\"urstenberg] \\label{t:Fu}\nLet $\\dim$ be one of $\\dim_H$, $\\underline{\\dim}_B$, $\\overline{\\dim}_B$, or $\\dim_P$. Assume that $d\\geq 1$ and $K\\subset \\mathbb{R}^d$ is a non-empty compact set. Then $\\{\\dim E: E\\in \\mathcal{M}_K\\}$ contains its infimum and supremum. \n\\end{theorem}\n\nFraser, Howroyd, K\u00e4enm\u00e4ki, and Yu \\cite[Theorem~1.3]{microsets} proved the following. Recall that a set is $F_\\sigma$ if it is a countable union of closed sets.\n\n\\begin{theorem}[Fraser--Howroyd--K\\\"aenm\\\"aki--Yu] \nLet $d\\geq 1$ and let $A\\subset [0,d]$ be an $F_{\\sigma}$ set which contains its infimum and supremum. Then there exists a compact set $K\\subset \\mathbb{R}^d$ such that \n\\begin{equation*} \\{\\dim_H E: E\\in \\mathcal{M}_K\\}=A.\n\\end{equation*}  \n\\end{theorem}\n\nThey asked the following question \\cite[Question~7.4]{microsets}, see also \\cite[Question~17.3.2]{F}.\n\n\\begin{question}[Fraser--Howroyd--K\\\"aenm\\\"aki--Yu] If $K\\subset \\mathbb{R}^d$ is compact, then is $\\{\\dim_H E: E\\in \\mathcal{M}_K\\}$ an $F_{\\sigma}$ set? If not, does it belong to a finite Borel class? \n\\end{question}\n\nAs the main result of this paper, in Section~\\ref{s:character} we answer the above questions in the negative. In fact, we prove a complete characterization as follows. \n\n\\begin{ch}[Main Theorem] Let $\\dim$ be one of $\\dim_H$, $\\underline{\\dim}_B$, or $\\overline{\\dim}_B$.\nLet $d\\geq 1$ and let $A \\subset [0, d]$ be a non-empty set. Then the following are equivalent:\n\\begin{enumerate}\n\\item There exists a compact set $K \\subset \\mathbb{R}^d$ such that $\\{\\dim E : E \\in \\mathcal{M}_K\\} = A$;\n\\item $A$ is an analytic set which contains its infimum and supremum;\n\\item There exists a compact set $K \\subset \\mathbb{R}^d$ such that $\\dim_H E=\\overline{\\dim}_B \\, E$ for all $E\\in \\mathcal{M}_K$ and $\\{\\dim_H E : E \\in \\mathcal{M}_K\\} = A$.\n\\end{enumerate} \n\\end{ch}\nFor packing dimension we can only prove one direction due to measurability problems, see Problem~\\ref{p:meas}. The following corollary is immediate. \n\\begin{corollary}\nLet $d\\geq 1$ and let $A \\subset [0, d]$ be a non-empty analytic \nset which contains its infimum and supremum. Then there exists a compact set $K \\subset \\mathbb{R}^d$ with \n\\begin{equation*}\n\\{\\dim_P E : E \\in \\mathcal{M}_K\\} = A.\n\\end{equation*}\n\\end{corollary}\n\nIn Section~\\ref{s:K} we consider the range of dimensions of compact families of compact subsets of a given compact set $K\\subset \\mathbb{R}^d$. Let $\\mathcal{K}(K)$ denote the set of non-empty compact subsets of $K$ endowed with the Hausdorff metric. Subsection~\\ref{ss:Haus} is dedicated to the proof of the following theorem, which relies on the stochastic co-dimension method applied to suitable fractal percolations. \n\n\\begin{co}\nLet $K \\subset \\mathbb{R}^d$ be a non-empty compact set and let $A \\subset [0, \\dim_H K]$. Then the following are equivalent: \n\\begin{enumerate}\n\\item There is a compact set $\\mathcal{C} \\subset \\mathcal{K}(K)$ with $\\{\\dim_H C : C \\in \\mathcal{C}\\} = A$;\n\\item $A$ is an analytic set. \n\\end{enumerate} \n\\end{co}\n\nIn Subsection~\\ref{ss:box} we consider the analogous problems for box and packing dimensions. In case of lower box dimension the analogue of Theorem~\\ref{t:compact family} does not hold. The following theorem of Feng, Wen, and Wu \\cite[Theorem~3]{FWW} demonstrates that $\\underline{\\dim}_B$ even fails to satisfy the Darboux property. \n\n\\begin{theorem}[Feng--Wen--Wu] There exists a compact set $K\\subset [0,1]$ such that $\\underline{\\dim}_B \\, K=1$ and any set $E\\subset K$ satisfies $\\underline{\\dim}_B \\, E\\in \\{0,1\\}$. \n\\end{theorem}\n\nFor upper box dimension an analogous version of Theorem~\\ref{t:compact family} holds. \n\n\\begin{cbox} Let $K$ be a non-empty compact metric space and $A \\subset [0, \\overline{\\dim}_B \\, K]$. The following statements are equivalent: \n\\begin{enumerate}\n\\item There is a compact set $\\mathcal{C} \\subset \\mathcal{K}(K)$ with $\\{\\overline{\\dim}_B \\, C : C \\in \\mathcal{C}\\} = A$;\n\\item $A$ is an analytic set. \n\\end{enumerate} \n\\end{cbox}\n\nFor packing dimension we will show the following.\n\n\\begin{pack}\nLet $K$ be a non-empty compact metric space and let $A \\subset [0, \\dim_P K]$ be analytic. Then there is a compact set $\\mathcal{C} \\subset \\mathcal{K}(K)$ with $\\{\\dim_P C : C \\in \\mathcal{C}\\} = A$.\n\\end{pack}\n\nFinally, we collect the open problems in Section~\\ref{s:open}.\n\n\\section{Preliminaries} \nLet $d\\geq 1$ be an integer. A set $A\\subset \\mathbb{R}^d$ is \\emph{analytic} if there exist a Polish space $X$ and a continuous onto map $f\\colon X\\to A$. Let $(X,\\rho)$ be a Polish space or compact metric space. Let $(\\mathcal{K}(X),d_{H})$ be the set of non-empty compact subsets of\n$X$ endowed with the \\emph{Hausdorff metric}, that is, for each $K_1,K_2\\in \\mathcal{K}(X)$ we have\n\\begin{equation*} \nd_{H}(K_1,K_2)=\\min \\left\\{r: K_1\\subset B(K_2,r) \\textrm{ and } K_2\\subset B(K_1,r)\\right\\},\n\\end{equation*}\nwhere $B(A,r)=\\{x\\in X: \\exists y\\in A \\textrm{ such that } \\rho(x,y)|\\leq r\\}$.\nThen $(\\mathcal{K}(X),d_{H})$ is a Polish space, and it is compact when $X$ is compact, see \\cite[Theorems~4.25,~4.26]{Ke}.\n\nLet $X$ be a metric space. For $x\\in X$ and $r>0$ let $B(x,r)$ be the closed ball of radius $r$ centered at $x$. For every $s\\geq 0$ the \\emph{$s$-Hausdorff content} of $X$ is defined as\n\\begin{equation*} \\mathcal{H}^{s}_{\\infty}(X)=\\inf \\left\\{ \\sum_{i=1}^\\infty (\\diam\nE_{i})^{s} : X \\subset \\bigcup_{i=1}^{\\infty} E_{i} \\right\\},\n\\end{equation*}\nwhere $\\diam E_i$ denotes the diameter of $E_i$. The \\emph{Hausdorff dimension} of $X$ is\n\\begin{equation*} \\dim_{H} X = \\inf\\{s \\ge 0: \\mathcal{H}_{\\infty}^{s}(X) =0\\}.\n\\end{equation*}\nLet $N_n(X)$ be the minimal number of closed balls of radius at most $2^{-n}$ needed to cover $X$. The \\emph{lower box dimension} and \\emph{upper box dimension} of $E$ are defined as\n\\begin{equation*}  \\underline{\\dim}_{B}\\, X=\\liminf_{n \\to \\infty} \\frac{\\log N_{n}(X)}{n\\log 2} \\quad \\text{and} \\quad \\overline{\\dim}_{B} \\, X=\\limsup_{n\\to \\infty} \\frac{\\log N_{n}(X)}{n\\log 2},\n\\end{equation*} \nrespectively. The \\emph{packing dimension} of $X$ is defined by\n\\begin{equation*} \\dim_P X=\\inf \\left\\{\\sup_{i} \\overline{\\dim}_{B} \\, E_i: X\\subset \\bigcup_{i=1}^{\\infty} E_i\\right\\}.\n\\end{equation*}\nFor the following lemma see \\cite[Lemma~2.8.1]{BP}.\n\\begin{lemma} \\label{l:packing}\nLet $X$ be a separable metric space. \n\\begin{enumerate}[(i)]\n\\item \\label{i:i} If $X$ is complete and $\\overline{\\dim}_B \\, U\\geq \\alpha$ for each open set $U\\subset X$, then $\\dim_P X\\geq \\alpha$.\n\\item \\label{i:ii} If $\\dim_P X>\\alpha$ then there is a closed subset $F\\subset X$ such that $\\dim_P (F\\cap U)>\\alpha$ for each open set $U$ intersecting $F$.\n\\end{enumerate}\n\\end{lemma}\n\nThe following fact easily follows from the finite stability of the dimensions in question. \n\n\\begin{fact} \\label{f:dimK}\nLet $\\dim$ be one of $\\dim_H$, $\\overline{\\dim}_B$, or $\\dim_P$. Let $K$ be a non-empty compact metric space. Then there exists $y\\in K$ such that $\\dim B(y,r)=\\dim K$ for all $r>0$. \n\\end{fact}\n\n\n\nFor the next claim see \\cite[Product formulas 7.2 and 7.5]{Fa}.\n\\begin{claim} \\label{c:product}\nLet $A\\subset \\mathbb{R}^d$ and $B\\subset \\mathbb{R}^m$ be compact sets. Then \n\\begin{equation*} \\dim_H A+\\dim_H B\\leq \\dim_H(A\\times B)\\leq  \\overline{\\dim}_B \\, (A\\times B) \\leq  \\overline{\\dim}_B \\, A+\\overline{\\dim}_B \\, B.\n\\end{equation*}  \n\\end{claim}\n\\noindent Consult~\\cite{Fa} or \\cite{Ma} for more on these concepts.\n\n\\bigskip\n\nFor any $x\\in 2^{\\omega}$ define a compact set $K(x)\\subset [0,1]$ as\n\\begin{equation*} \nK(x)=\\left\\{\\sum_{i=0}^{\\infty} a_i 2^{-i-1}: a_i=0 \\text{ if } x(i)=0 \\text{ and } a_i\\in \\{0,1\\} \\text{ if } x(i)=1\\right\\}.\n\\end{equation*}\nFor $n\\in \\omega$ let $x\\restriction n$ be the restriction of $x$ to its first $n$ coordinates. We endow $2^{\\omega}$ with a metric $d$ compatible with the product topology defined as \n\\begin{equation*} d(x,y)=2^{-\\min\\{i: x(i)\\neq y(i)\\}} \\quad \\text{for all } x,y\\in 2^{\\omega}.\n\\end{equation*}\nThe following fact is straightforward. \n\\begin{fact} \\label{f:cont} \nThe map $K\\colon 2^{\\omega}\\to \\mathcal{K}([0,1])$ mapping $x$ to $K(x)$ is continuous, more precisely, \n\\begin{equation*} d_H(K(x),K(y))\\leq d(x,y)  \\quad \\text{for all } x,y\\in 2^{\\omega}.\n\\end{equation*}\n\\end{fact} \nDefine the left shift $T\\colon 2^{\\omega} \\to  2^{\\omega}$ such that \n\\begin{equation*} \nT(x)(n)=x(n+1) \\text{ for all } n\\in \\omega.\n\\end{equation*} \nWe define the \\emph{lower density} and \\emph{upper density of $x$} by\n\\begin{equation*} \n\\underline{\\varrho}(x)=\\liminf_{n \\to \\infty} \\frac{\\sum_{i=0}^{n-1} x(i)}{n} \\quad \\text{and} \\quad \\overline{\\varrho}(x)=\\limsup_{n \\to \\infty} \\frac{\\sum_{i=0}^{n-1} x(i)}{n},\n\\end{equation*}    \nrespectively. If $\\underline{\\varrho}(x)=\\overline{\\varrho}(x)$ then the common value $\\varrho(x)$ is called the \\emph{density of $x$}. For the following claim see e.g.~\\cite[Example~3.2.3]{BP}.\n\\begin{claim} \\label{cl:1}\nFor any $x\\in 2^{\\omega}$ we have \n\\begin{equation*} \n\\dim_H K(x)=\\underline{\\dim}_B \\, K(x)=\\underline{\\varrho}(x) \\quad \\text{and} \\quad \\dim_P K(x)=\\overline{\\dim}_B \\, K(x)=\\overline{\\varrho}(x).\n\\end{equation*}\n\\end{claim}\nClaims~\\ref{c:product} and \\ref{cl:1} allow us to calculate the dimensions of the Cartesian product $K(x)^d$ as follows. \n\\begin{fact} \\label{f:dim}\nLet $\\dim$ be one of $\\dim_H$, $\\underline{\\dim}_B$, $\\overline{\\dim}_B$ or $\\dim_P$. Assume that $d\\in \\mathbb{N}^+$ and $x\\in 2^{\\omega}$ are given such that $\\varrho(x)$ exists. Then \n\\begin{equation*} \\dim K(x)^d=d\\varrho(x). \n\\end{equation*} \n\\end{fact}\n\nFor $s,t\\in 2^{<\\omega}$ let $s^\\frown t\\in 2^{<\\omega}$ denote the concatenation of $s$ and $t$.\n\n\n\n\n\n\t\n\\section{Range of dimensions of microsets} \\label{s:character}\n\nThe goal of this section is to prove Theorem~\\ref{t:characterization} after some preparation. \n\n\\subsection{A description of the microsets of $K(x)^d$} The goal of this subsection is to prove Theorem~\\ref{t:Cx}, which provides us with a good tool to work with the microsets of $K(x)^d$. For technical reasons we generalize $\\mathcal{M}_K$ as follows. \n\n\\begin{definition}\n\tFor $d\\geq 1$ and $\\mathcal{F}\\subset \\mathcal{K}(\\mathbb{R}^d)$ define $\\mathcal{M}(\\mathcal{F})$ as the set of compact sets $K\\subset [0,1]^d$ for which $K\\cap (0,1)^d\\neq \\emptyset$ and there exist $K_n\\in \\mathcal{F}$, $\\lambda_n\\geq 1$, and $u_n\\in \\mathbb{R}^d$ such that $(\\lambda_n K_n+u_n)\\cap [0,1]^d\\to K$. \n\\end{definition}\t\n\n\\begin{fact} \\label{f:subset} Let $d\\geq 1$ and assume that $C_n\\subset K_n\\subset \\mathbb{R}^d$ are compact sets such that $C_n\\to C$ and $K_n\\to K$. Then $C\\subset K$.\n\\end{fact}\n\n\\begin{definition}\n\tFor $x\\in 2^{\\omega}$ and $n\\in \\omega$ define the finite set\n\t\\begin{equation*} \n\tF_n(x)=\\left\\{\\sum_{i=0}^{n-1} a_i 2^{-i-1}: a_i=0 \\text{ if } x(i)=0 \\text{ and } a_i\\in \\{0,1\\} \\text{ if } x(i)=1\\right\\}\n\t\\end{equation*}\n\tand let\n\t\\begin{equation*} \n\t\\mathcal{D}_n(x)=\\{2^{-n}K(T^n(x))+u: u\\in F_n(x)\\}.\n\t\\end{equation*}\n\tClearly, $\\bigcup \\mathcal{D}_n(x)=K(x)$ and elements of $\\mathcal{D}_n(x)$ have pairwise non-overlapping convex hulls.\n\\end{definition}\n\n\\begin{lemma} \\label{l:main} \n\tLet $E\\in \\mathcal{M}(\\{K(x): x\\in 2^{\\omega}\\})$. Assume $(\\lambda_n K(x_n)+u_n)\\cap [0,1]\\to E$ for some $\\lambda_n\\geq 1$, $x_n\\in 2^{\\omega}$, and $u_n\\in \\mathbb{R}$. Then there exist $x\\in 2^{\\omega}$, $c\\in \\mathbb{R}^+$, $m_n\\in \\omega$ for all $n$, a subsequence of positive integers $k_n\\uparrow \\infty$, a similar copy $C(x)$ of $K(x)$, $w_0=0$ and $w_1,w_2,w_3\\in \\mathbb{R}$ such that \n\t\\begin{enumerate} \n\t\t\\item \\label{e:T1} $T^{m_n}(x_{k_n})\\to x$, \n\t\t\\item \\label{e:T2} $\\lambda_{k_n}2^{-m_n}\\to c$, \n\t\t\\item \\label{e:T3} $C(x)\\subset E\\subset \\bigcup_{i=0}^3 (C(x)+w_i)$.\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\tDefine $C^i_n=2^{-i} K(T^i(x_n))$ for all $i,n\\in \\omega$. Let $\\varepsilon\\in (0,1\/2)$ such that $E\\cap (\\varepsilon,1-\\varepsilon)\\neq \\emptyset$. For all $n$ let $\\varphi_n \\colon \\mathbb{R}  \\to \\mathbb{R}$ be defined as $\\varphi_n(z)=\\lambda_n z+u_n$ and let $p_n\\in \\omega$ be the minimal number such that there is a $v^0_n\\in F_{p_n}(x_n)$ such that \n\t\\begin{equation} \\label{e:vn} \\varphi_n (C_n^{p_n}+v^0_n) \\subset [0,1].\n\t\\end{equation} \n\tBy the minimality of $p_n$ for all $n$ there are at most $4$ translations $v\\in F_{p_n}(x_n)$ satisfying \n\t\\[\\varphi_n(C_n^{p_n}+v)\\cap [0,1] \\neq \\emptyset,\\] \n\tassume that they are $v^0_n,\\dots,v_n^{\\ell_n}$ for some $\\ell_n\\in\\{0,1,2,3\\}$. \n\t\n\tFirst we show $\\lambda_n 2^{-p_n}\\geq \\varepsilon\/2$ for all $n$ large enough. Indeed, if $n$ is large enough then $\\varphi_n (K(x_n))\\cap (\\varepsilon,1-\\varepsilon)\\neq \\emptyset$. Thus for $k=p_n-1$ there is a $D\\in \\mathcal{D}_{k}(K(x_n))$ with $\\varphi_n(D)\\cap (\\varepsilon,1-\\varepsilon)\\neq \\emptyset$. Assume to the contrary that $\\lambda_n 2^{-p_n}<\\varepsilon\/2$. Then $\\diam \\varphi_n(D)\\leq \\lambda_n 2^{-k}<\\varepsilon$,  so $\\varphi_n(D)\\subset [0,1]$, contradicting the minimality of $p_n$.\n\t\n\tNow let $q_n\\geq p_n$ be the minimal integer for which $\\lambda_n 2^{-q_n}\\leq 2$. We prove $x_n(i)=0$ for all $i\\in \\{p_n,\\dots,q_n-1\\}$. Assume to the contrary that this is not the case, and take the minimal $i$ such that $p_n\\leq i<q_n$ and $x_n(i)=1$. Then clearly $C_n^i=C_n^{p_n}$, so $\\varphi_n (C_n^{i} + v^0_n)\\subset [0,1]$. Since $x_n(i)=1$, we have $\\diam K(T^i(x_n))\\geq 1\/2$, so \n\t\\begin{equation*} \n\t\\lambda_n 2^{-i-1}\\leq \\lambda_n 2^{-i} \\diam K(T^i(x_n))=\\diam \\varphi_n (C_n^{i}) = \\diam \\varphi_n (C_n^{i} + v^0_n) \\leq 1.\n\t\\end{equation*} \n\tHence $\\lambda_n 2^{-i}\\leq 2$, which contradicts the minimality of $q_n$. As $x_n(i)=0$ for all $i\\in \\{p_n,\\dots,q_n-1\\}$, we obtain $C_n^{q_n}=C_n^{p_n}$. Since $\\lambda_n 2^{-q_n}\\in [\\varepsilon\/2,2]$ for all large enough $n$, we can choose $c\\in [\\varepsilon\/2,2]$ and a subsequence $k_n\\uparrow \\infty$ such that $\\ell_{k_n}=\\ell\\in \\{0,1,2,3\\}$ for all $n$ and $m_n=q_{k_n}$ satisfies \n\t\\begin{equation*}  \\lambda_{k_n} 2^{-m_n} \\to c \\text{ as } n\\to \\infty,\n\t\\end{equation*}\n\tso \\eqref{e:T2} holds. As $2^{\\omega}$ is compact, by choosing a subsequence we may assume that there exists $x\\in 2^{\\omega}$ such that \n\t\\begin{equation*}  T^{m_n}(x_{k_n})\\to x  \\text{ as } n\\to \\infty,\n\t\\end{equation*}\n\tso \\eqref{e:T1} holds as well. As $C_n^{q_n}=C_n^{p_n}$ for all $n$, by \\eqref{e:vn} and Fact \\ref{f:cont} we may assume by choosing a subsequence that \n\t\\begin{equation*} \n\t\\varphi_{k_n}(C^{m_n}_{k_n}+v^0_{k_n})\\to cK(x)+v\\subset [0,1] \\text{ as } n\\to \\infty \n\t\\end{equation*} \n\twith some $v\\in \\mathbb{R}$, and let $C(x)=cK(x)+v$. As $\\varphi_{k_n}(C^{m_n}_{k_n}+v^i_{k_n})\\subset [-1,2]$ is isometric to $\\varphi_{k_n}(C^{m_n}_{k_n}+v^0_{k_n})$ for every $1\\leq i\\leq \\ell$, we may suppose by choosing a subsequence that \n\t\\begin{equation*} \n\t\\varphi_{k_n}(C^{m_n}_{k_n}+v^i_{k_n})\\to C(x)+w_i \\text{ as } n\\to \\infty\n\t\\end{equation*} \n\tfor all $1\\leq i\\leq \\ell$ with some $w_1,\\dots,w_{\\ell}\\in \\mathbb{R}$. Let $w_0=0$, by Fact~\\ref{f:subset} we obtain \n\t\\begin{equation*}\n\tC(x)\\subset E\\subset \\bigcup_{i=0}^{\\ell} (C(x)+w_i),\n\t\\end{equation*}\n\tso \\eqref{e:T3} holds no matter how we define $w_i$ for $i>\\ell$. The proof is complete.\n\\end{proof}\n\n\\begin{theorem} \\label{t:Cx}\n\tLet $d\\geq 1$ and let $E\\in \\mathcal{M}(\\{K(x)^d: x\\in 2^{\\omega}\\})$. Assume that $(\\lambda_n K(x_n)^d+u_n)\\cap [0,1]^d \\to E$ for some $\\lambda_n\\geq 1$, $x_n\\in 2^{\\omega}$, and $u_n\\in \\mathbb{R}^d$. Then there exist $x\\in 2^{\\omega}$, $m_n\\in \\omega$ for all $n$, a subsequence of positive integers $k_n\\uparrow \\infty$, a similar copy $C(x)$ of $K(x)$, and $v_1,\\dots,v_\\ell\\in \\mathbb{R}^d$ such that \n\t\\begin{enumerate}[(i)]\n\t\t\\item \\label{e:d1} $T^{m_n}(x_{k_n})\\to x$,\n\t\t\\item \\label{e:d2} $C(x)^d+v_1\\subset E\\subset  \\bigcup_{i=1}^\\ell (C(x)^d+v_i)$. \n\t\\end{enumerate}\n\\end{theorem}\n\\begin{proof} Let $u_n=(u_n^1,\\dots, u_n^d)$ for all $n$. It easily follows that $E=E_1\\times\\dots \\times E_d$, where $E_i\\subset [0,1]$ are compact sets such that $E_i\\cap (0,1)\\neq \\emptyset$ and \n\t\\begin{equation*} \n\t(\\lambda_n K(x_n)+u^i_n)\\cap [0,1] \\to E_i \\text{ as } n\\to \\infty\n\t\\end{equation*}\n\tfor all $1\\leq i\\leq d$. Applying Lemma~\\ref{l:main} for all $1\\leq i\\leq d$ successively implies that there exist $z_i\\in 2^{\\omega}$, $c_i\\in \\mathbb{R}^+$, $m_{i,n}\\in \\omega$ for all $n$, a subsequence of positive integers $k_{n}\\uparrow \\infty$ (note that this does not depend on $i$), a similar copy $C(z_i)$ of $K(z_i)$, and $w_{i,j}\\in \\mathbb{R}$ for $0\\leq j\\leq 3$ such that \n\t\\begin{enumerate} \n\t\t\\item \\label{e:dd1} $T^{m_{i,n}}(x_{k_n})\\to z_i$, \n\t\t\\item \\label{e:dd2} $\\lambda_{k_n}2^{-m_{i,n}}\\to c_i$, \n\t\t\\item \\label{e:dd3} $C(z_i)\\subset E_i\\subset  \\bigcup_{j=0}^3 (C(z_i)+w_{i,j})$. \n\t\\end{enumerate}\n\tBy \\eqref{e:dd2} for large enough $n$ and for all $1\\leq i\\leq j\\leq d$ we obtain that $m_{i,n}-m_{j,n}$ is independent of $n$. We may assume that $m_{1,n}-m_{i,n}=\\ell_i\\in \\mathbb{N}$ for any $1\\leq i\\leq d$ and any large enough $n$. Let $m_n=m_{1,n}$ and $x=z_1$, then clearly \\eqref{e:d1} holds. By \\eqref{e:dd1} we obtain $x=T^{\\ell_i}(z_i)$ for all $1\\leq i\\leq d$. Therefore, $C(z_i)$ is a union of at most $2^{\\ell_i}$ many translates of $C(x)$ for all $1\\leq i\\leq d$, so taking the product of \\eqref{e:dd3} for all $i\\in \\{1,\\dots,d\\}$ implies \\eqref{e:d2} with $\\ell= \\prod_{i=1}^d 2^{\\ell_i+2}$, which finishes the proof.\n\\end{proof}\n\n\n\\subsection{Useful lemmas for analytic sets and balanced sequences} The goal of this subsection is to prove Lemmas~\\ref{l:varphi existence} and \\ref{l:balanced}.\n\n\\begin{fact} \\label{f:Gd}\nLet $A \\subset [0,\\infty)$ be a non-empty analytic set. Then there exist a $G_\\delta$ set $G \\subset 2^\\omega$ and a continuous map $f \\colon G \\to [0,\\infty)$ such that $f(G) = A$.\n\\end{fact} \n\\begin{proof} \nLet $g \\colon 2^\\omega \\to [0,1]$ be a continuous surjection, and let $\\psi \\colon [0, 1) \\to [0, \\infty)$ be a homeomorphism. Since $g^{-1}(\\psi^{-1}(A)) \\subset 2^\\omega$ is analytic, by \\cite[Exercise~14.3]{Ke} we can find a $G_\\delta$ set $H \\subset 2^\\omega \\times 2^\\omega$ such that $\\pi_1(H) = g^{-1}(\\psi^{-1}(A))$, where $\\pi_1$ denotes the projection onto the first coordinate. Let $h\\colon 2^{\\omega } \\to 2^\\omega \\times 2^\\omega$ be a homeomorphism. As $g(g^{-1}(\\psi^{-1}(A))) = \\psi^{-1}(A)$, taking $G=h^{-1}(H)$ and $f=\\psi \\circ g \\circ \\pi_1\\circ h|_{G}$ finishes the proof. \n\\end{proof} \n\n\\begin{definition} \nFor $s\\in 2^{<\\omega}$ we denote by $\\length(s)$ the number of coordinates of $s$, where $\\length(\\emptyset)=0$. Define \n\\begin{equation*} \n[s]=\\{x\\in 2^{\\omega}: x\\restriction \\length(s)=s\\}.\n\\end{equation*} \n\\end{definition} \n\n\\begin{lemma}\n\\label{l:varphi existence}\nLet $A \\subset [0,\\infty)$ be a non-empty analytic set. Then there exists a map $\\varphi \\colon 2^{<\\omega} \\to [0, \\infty)$ such that\n\\begin{equation*} \n\\overline{\\varphi}(x) = \\lim_{n \\to \\infty} \\varphi(x \\restriction n)\n\\end{equation*} \nexists for each $x \\in 2^\\omega$, and the resulting function $\\overline{\\varphi}$ satisfies $\\overline{\\varphi}(2^\\omega) = A$. \n\\end{lemma}\n\\begin{proof}\nAccording to Fact~\\ref{f:Gd} we can choose a $G_\\delta$ set $G \\subset 2^\\omega$ and a continuous map $f \\colon G \\to [0, \\infty)$ such that $f(G) = A$. The set $F = 2^\\omega \\setminus G$ is $F_{\\sigma}$, so it can be written as $F = \\bigcup_{n=1}^{\\infty} F_n$ where $F_n\\subset 2^{\\omega}$ are closed and $F_n \\subset F_{n+1}$ for all $n\\geq 1$. We define $\\varphi$ on an element $s \\in 2^{<\\omega}$ by induction on the length of $s$. \nFix $a_0\\in A$ arbitrarily and define $\\varphi(\\emptyset) = a_0$. Now suppose that $\\varphi$ is already defined on $s \\in 2^{<\\omega}$, our task is to define it on $s^\\frown c$ where $c \\in \\{0, 1\\}$. \nIf $[s^\\frown c] \\cap F = \\emptyset$ then let $\\varphi(s^\\frown c)$ be an arbitrary element of $f([s^\\frown c])$. If $[s^\\frown c] \\cap G = \\emptyset$ then let $\\varphi(s^\\frown c) = \\varphi(s)$. \n\t\nIt remains to define $\\varphi(s^\\frown  c)$ if $[s^\\frown  c]$ intersects both $G$ and $F$. Let $m(s^\\frown  c)$ and $m(s)$ be the smallest indices with $[s^\\frown  c] \\cap F_{m(s^\\frown  c)} \\neq \\emptyset$ and $[s] \\cap F_{m(s)} \\neq \\emptyset$, respectively. If $m(s^\\frown  c) = m(s)$ then let $\\varphi(s^\\frown  c) = \\varphi(s)$. Otherwise, let $\\varphi(s^\\frown  c)$ be an arbitrary element of $f([s^\\frown  c] \\cap G)$, concluding the definition of $\\varphi$. \n\t\nIt remains to check that $\\varphi$ satisfies the conditions of the lemma. First note that \n\\begin{equation} \\label{e:varphi s in A}\n\\varphi(s) \\in A \\text{ for each } s \\in 2^{<\\omega},\n\\end{equation}\na fact that can be quickly checked by induction. Let $x \\in 2^\\omega$ be fixed. It is enough to show that $\\overline{\\varphi} (x) = \\lim_{n \\to \\infty} \\varphi(x \\restriction n)$ exists, $\\overline{\\varphi}(x) \\in A$, and if $x \\in G$ then $\\overline{\\varphi}(x) = f(x)$. \n\t\nFirst assume that $[x \\restriction m] \\cap F = \\emptyset$ for some $m$. Then $\\varphi(x \\restriction n)$ is an element of $f([x \\restriction n])$ for each $n \\ge m$. Hence, using the continuity of $f$ and the fact that $x \\in G$, we obtain $\\varphi(x\\restriction n) \\to f(x) \\in A$. \n\t\nNow suppose that $[x \\restriction m] \\cap G = \\emptyset$ for some $m$. The definition of $\\varphi$ and \\eqref{e:varphi s in A} imply that there exists $a \\in A$ such that $\\varphi(x \\restriction n) = a$ for all $n\\geq m$. It follows that $\\varphi(x \\restriction n) \\to a \\in A$. Note that in this case $x \\in F$, so we do not have to check that $\\overline{\\varphi}(x) = f(x)$.\n\t\nFinally, assume that $[x \\restriction n]$ intersects both $G$ and $F$ for each $n$. Clearly $m(x \\restriction n)$ increases as $n \\to \\infty$. Suppose first that $m(x \\restriction n) \\to \\infty$. Then $x \\not \\in F_n$ for each $n$, hence $x \\in G$. Also, for infinitely many $n$, the value of $\\varphi(x \\restriction n)$ is chosen from $f([x \\restriction n])$, and when it is not, then $\\varphi(x \\restriction n) = \\varphi(x \\restriction (n - 1))$. It follows that $\\varphi(x \\restriction n) \\to f(x) \\in A$. Finally, assume that $m(x \\restriction n) $ converges, that is, there exists $m$ such that $m(x \\restriction n)= m$ for all large enough $n$. Then $[x \\restriction n] \\cap F_m \\neq \\emptyset$ for each $n$, hence $x \\in F_m \\subset F$, so we do not need to check $\\overline{\\varphi}(x) = f(x)$. Using \\eqref{e:varphi s in A} it also follows that there exists $a \\in A$ such that $\\varphi(x \\restriction n) = a$ if $n$ is large enough. The proof is complete.\n\\end{proof}\n\n\\begin{definition}\n We call $s\\in 2^{<\\omega}$ a \\emph{subsequence} of $x\\in 2^{\\omega}$ if \\begin{equation*} \n s = (x(k), x(k+1), \\dots, x(k+n-1))\n \\end{equation*} \nfor some $k, n\\in \\omega$. Such a relation is denoted by $s \\in x$. Let $I\\subset \\omega$ be a \\emph{discrete interval} if $I=\\{k,\\dots,k+n-1\\}$ for some $k,n\\in \\omega$, then define $x\\restriction I=(x(k),\\dots,x(k+n-1))$. For two discrete intervals $I,J\\subset \\omega$ we write $I<J$ if $\\max \\{i: i\\in I\\}<\\min \\{j: j\\in J\\}$. For $s\\neq \\emptyset$ let \n\\begin{equation*} \\sigma(s)=\\sum_{i=0}^{\\length(s)-1} s(i) \\quad \\text{and} \\quad  \\varrho(s)=\\frac{\\sigma(s)}{\\length(s)}.\n\\end{equation*}\nThe sequence $x\\in 2^{\\omega}$ is called \\emph{balanced} if for all $n$ for any two subsequences $s,t \\in x$ of length $n$ we have $|\\sigma (s) - \\sigma (t)|\\le 1$. It is known that for each $a \\in [0, 1]$ there exists a balanced sequence $x\\in 2^{\\omega}$ with $\\varrho(x) =a$, see e.g.~\\cite[Section~2.2]{balanced} for the details.\n\\end{definition}\n\n\\begin{lemma}\n\\label{l:balanced}\nLet $0\\leq a\\leq b \\leq 1$ and assume that $\\alpha, \\beta \\in 2^\\omega$ are balanced with $\\varrho(\\alpha) = a$ and $\\varrho(\\beta) = b$. For each $n$ let $x_n\\in 2^{\\omega}$ be defined as \n\\begin{equation*} x_n = {s^0_n}^\\frown {s^1_n} ^\\frown s^2_n \\dots, \n\\end{equation*}  \nwhere $s^k_n \\in 2^{<\\omega}$ is either a subsequence of $\\alpha$ or a subsequence of $\\beta$ for all $k\\in \\omega$. Suppose that $\\length(s^{1}_{n}) \\to \\infty$ and $x_n\\to x$ as $n \\to \\infty$. Then $\\varrho(x)\\in \\{\\alpha,\\beta\\}$. \n\\end{lemma}\n\\begin{proof}\nSince $\\alpha$ and $\\beta$ are balanced, it is easy to see that for any subsequences $s \\in \\alpha$ and $t \\in \\beta$ of length $n\\geq 1$ we have \n\\begin{equation}\n\\label{e:abdens}\n\\left| \\varrho(\\alpha)-  \\varrho(s) \\right|\\leq \\frac 1n  \\quad \\text{and} \\quad\t\\left| \\varrho(\\beta)- \\varrho(t)\\right|\\leq \\frac 1n.\n\\end{equation}\nFor $N\\in \\omega$ let $\\mathcal{I}(N)$ denote the set of discrete intervals $I$ satisfying $I\\subset [N,\\infty)$.  \nWe claim that it is enough to prove that there exists an $N\\in \\omega$ such that \n\\begin{equation}\n\\label{e:allN}\n\\text{ either $x \\restriction I \\in \\alpha$ for each $I \\in \\mathcal{I}(N)$, or $x \\restriction I \\in \\beta$ for each $I  \\in \\mathcal{I}(N)$}.\n\\end{equation}\nSuppose that \\eqref{e:allN} holds, we may assume by symmetry that $x \\restriction I \\in \\alpha$ for each $I \\in \\mathcal{I}(N)$. We want to show that $\\varrho(x) =\\varrho(\\alpha)$. Let $y\\in 2^{\\omega}$ such that $y(n)=x(n+N)$ for each $n\\in \\omega$. As $\\varrho(y)=\\varrho(x)$, it is enough to prove that $\\varrho(y)=\\varrho(\\alpha)$. By \\eqref{e:allN} for any $n\\geq 1$ we have $y\\restriction n \\in \\alpha$, so \\eqref{e:abdens} implies that \n\\begin{equation*} \n\\left| \\varrho(\\alpha)- \\varrho(y\\restriction n)\\right|\\leq \\frac 1n,\n\\end{equation*} \nthus $\\varrho(y)=\\varrho(\\alpha)$ follows. \n\t\nTherefore it remains to show \\eqref{e:allN}. Assume to the contrary that \\eqref{e:allN} fails for all $N$. Then one can find non-empty discrete intervals $I_1, I_2, I_3, I_4\\subset \\omega$ with $I_1 < I_2 < I_3 < I_4$ such that $x \\restriction I_1 \\not \\in \\alpha$, $x \\restriction I_2 \\not \\in \\beta$, $x \\restriction I_3 \\not\\in \\alpha$, and $x \\restriction I_4 \\not\\in\\beta$. Fix $k\\in \\omega$ large enough so that $I_1 \\cup I_2 \\cup I_3 \\cup I_4 \\subset [0, k)$ and then fix $n\\in \\omega$ large enough such that $x_n\\restriction k = x \\restriction k$ and $\\length(s_n^1)>k$. Then clearly $x_n \\restriction I_1 \\in s_n^0$ and $x_n \\restriction I_2 \\in s_n^0$, or $x_n\\restriction I_3 \\in s_n^1$ and $x_n\\restriction I_4 \\in s_n^1$. Using that both $s_n^0$ and $s_n^1$ are subsequences of $\\alpha$ or $\\beta$, and the fact that $x$ and $x_n$ coincide on these intervals, we obtain a contradiction.\n\\end{proof}\n\t\n\\subsection{The proof of the Main Theorem} Finally, in this subsection we are ready to prove Theorem~\\ref{t:characterization}. We need the following technical lemma which is implicitly contained in \\cite{microsets}.\n\n\\begin{lemma}\n\\label{l:building K}\nLet $K_n \\subset [0, 1]^d$ be compact sets such that $\\dim_H E=\\overline{\\dim}_B \\, E$ for all $E\\in \\mathcal{M}(\\{K_n\\}_{n\\geq 1})$ and let $\\gamma=\\sup\\{ \\dim_H K_n: n\\geq 1\\}$. Then there exists a compact set $K\\subset [0,1]^d$ such that \n\\begin{equation*}\n\\dim_H E=\\overline{\\dim}_B \\, E \\text{ for all } E\\in \\mathcal{M}_K    \n\\end{equation*}\nand \n\\begin{equation*} \\{\\dim_H E : E \\in \\mathcal{M}_K\\} =\\left\\{\\dim_H E : E \\in  \\mathcal{M}(\\{K_n\\}_{n\\geq 1})\\right\\} \\cup \\{\\gamma\\}.\n\\end{equation*} \n\\end{lemma}\n\\begin{proof}\nApply the proof of \\cite[Theorem~1.3]{microsets} to the multiset $\\Omega^0=\\{Q_i\\}_{i\\geq 1}$, where $\\Omega^0$ is an enumeration of $\\{K_n\\}_{n\\geq 1}$ such that each set $K_n$ is repeated infinitely often. \n\\end{proof}\n\t\n\\begin{theorem}[Main Theorem]\n\\label{t:characterization}\nLet $\\dim$ be one of $\\dim_H$, $\\underline{\\dim}_B$, or $\\overline{\\dim}_B$.\nLet $d\\geq 1$ and let $A \\subset [0, d]$ be a non-empty set. Then the following are equivalent:\n\\begin{enumerate}\n\\item \\label{i2} There exists a compact set $K \\subset \\mathbb{R}^d$ such that $\\{\\dim E : E \\in \\mathcal{M}_K\\} = A$;\n\\item \\label{i3} $A$ is an analytic set which contains its infimum and supremum;\n\\item \\label{i1} There exists a compact set $K \\subset \\mathbb{R}^d$ such that $\\dim_H E=\\overline{\\dim}_B \\, E$ for all $E\\in \\mathcal{M}_K$ and $\\{\\dim_H E : E \\in \\mathcal{M}_K\\} = A$.\n\\end{enumerate} \n\\end{theorem}\n\n\\begin{proof}  \nThe direction $\\eqref{i1} \\Rightarrow \\eqref{i2}$ is straightforward by Fact~\\ref{f:ineq}.\n\nNow we prove $\\eqref{i2} \\Rightarrow \\eqref{i3}$. Assume that $A = \\{\\dim E : E \\in \\mathcal{M}_K\\}$ for some compact set $K \\in \\mathcal{K}(\\mathbb{R}^d)$. Theorem~\\ref{t:Fu} yields that $A$ contains its infimum and supremum as well. To see that $A$ is analytic, note that the set $\\mathcal{M}_K$ is $F_\\sigma$, since the set $\\{E \\in \\mathcal{M}_K : E \\cap [\\varepsilon, 1 - \\varepsilon]^d \\neq \\emptyset\\}$ is closed for any $\\varepsilon \\in (0,1\/2)$. Mattila and Mauldin  \\cite{MM} proved that the mapping $\\dim \\colon \\mathcal{K}(\\mathbb{R}^d)\\to [0,d]$ is Borel measurable. Therefore, we obtain that $A$ is the image of a Borel set under a Borel map, hence it is analytic by \\cite[Proposition~14.4]{Ke}. \n\t\nFinally, we show $\\eqref{i3} \\Rightarrow \\eqref{i1}$. Fix an analytic set $A \\subset [0, d]$ which contains its infimum and supremum. Define $B=\\{z\/d: z\\in A\\}$, then $B\\subset [0,1]$ is analytic, and set $a = \\min B$ and $b = \\max B$. If $a = b = 0$, then $K$ can be a singleton. Hence we may assume that $b > 0$. Applying Lemma \\ref{l:varphi existence} for the analytic set $B$ yields a map $\\varphi \\colon 2^{<\\omega} \\to [0,\\infty)$ such that $\\overline{\\varphi}(x) = \\lim_{n \\to \\infty} \\varphi(x \\restriction n)$ exists for all $x\\in 2^{\\omega}$ and $\\overline{\\varphi}(2^{\\omega})=B$. We may assume that \n\\begin{equation}\n\\label{e:q3 a <= varphi <= b}\na \\le \\varphi(s) \\le b \\text{ for each $s \\in 2^{<\\omega}$.}\n\\end{equation}\nIndeed, let us replace a value $\\varphi(s)$ by $a$ if $\\varphi(s) < a$, and replace $\\varphi(s)$ by $b$ if $\\varphi(s) > b$. As $\\overline{\\varphi}(x) \\in B \\subset [a, b]$ for each $x \\in 2^\\omega$, the values of $\\overline{\\varphi}$ do not change by modifying $\\varphi$ in this way. \n\t\nWe now construct a continuous map $\\psi \\colon 2^\\omega \\to 2^\\omega$ and then use compact sets of the form $(K(\\psi(x)))^d$ to construct $K$. Let $\\alpha$ and $\\beta$ be the sequences provided by Lemma~\\ref{l:balanced} for $a$ and $b$. To construct $\\psi$, first we specify a mapping $\\phi \\colon  2^{<\\omega} \\to 2^{<\\omega}$ such that $\\phi(s)$ is a subsequence of either $\\alpha$ or $\\beta$ for all $s\\in 2^{<\\omega}$. Let $\\phi(\\emptyset) = \\emptyset$. For $s \\in 2^{<\\omega}$ with $\\length(s)=n\\geq 1$ define $\\phi(s) = (\\alpha \\restriction n) ^\\frown (\\beta \\restriction k)$, where $k=k(s)$ is a positive integer such that $ \\sqrt{n}-1<k< n \\sqrt{n}+1$ and \n\\begin{equation} \\label{e:q3  lambda close to varphi}\n\\left|\\frac{na + k b}{n+k} - \\varphi(s)\\right| \\le \\frac{2}{\\sqrt{n}}.\n\\end{equation}\nWe show that such $k$ exists. Indeed, \\eqref{e:q3 a <= varphi <= b} implies that $\\varphi(s) \\in [a, b]$, and consider the real function \\begin{equation*} \nr\\colon [\\sqrt{n}-1,n\\sqrt{n}+1] \\to [a,b], \\quad  r(z)=\\frac{an+bz}{n+z}.\n\\end{equation*}  \nThe inequalities \n\\begin{equation*} \n\\left|r\\left(\\sqrt{n}\\right)-a\\right|\\leq \\frac{2}{\\sqrt{n}}, \\quad  \\left|r\\left(n\\sqrt{n}\\right)-b \\right| \\leq \\frac{1}{\\sqrt{n}},\\text { and }  r(z+1)-r(z)\\leq \\frac{1}{n}\n\\end{equation*} \nfor all $\\sqrt{n}-1\\leq z\\leq n\\sqrt{n}$ easily imply the existence of $k$ satisfying \\eqref{e:q3  lambda close to varphi}. \n\t\nFor $x\\in 2^{\\omega}$ define \n\\begin{equation} \\label{e:psi}\n\\psi(x) = \\phi(x\\restriction 0) ^\\frown \\phi(x \\restriction 1) ^\\frown  \\phi(x \\restriction 2)   \\dots.\n\\end{equation}\nAs $\\phi(s)\\neq \\emptyset$ whenever $s\\neq \\emptyset$, the map $\\psi$ is continuous. Since $b>0$ and $\\beta$ is balanced with $\\varrho(\\beta)=b>0$, we obtain that $\\psi(x)\\neq \\mathbf{0}$ for all $x\\in 2^{\\omega}$, where $\\mathbf{0}\\in 2^{\\omega}$ is the zero sequence. We now claim that $\\psi$ satisfies \n\\begin{equation} \\label{e:dvarphi}\n\\varrho(\\psi(x)) = \\overline{\\varphi}(x)\\text{ for each $x \\in 2^\\omega$.} \n\\end{equation}\nLet us fix $x \\in 2^\\omega$, and let \n\\begin{equation*} \n\\psi_n(x) = \\phi(x \\restriction 0) ^\\frown \\phi(x \\restriction 1) ^\\frown \\dots ^\\frown  \\phi(x \\restriction n).\n\\end{equation*} \nAn elementary calculation using $k(s)=o(n^2)$ as $\\length(s)=n\\to \\infty$ shows that \n\\begin{equation*} \n\\frac{\\length(\\phi(x \\restriction n))}{\\length(\\psi_n(x))} \\to 0 \\text{ as } n \\to \\infty,\n\\end{equation*}  \nhence it is enough to show that $\\varrho(\\psi_n(x)) \\to \\overline{\\varphi}(x)$. Therefore, it is enough to show that $\\varrho(\\phi(x \\restriction n)) \\to \\overline{\\varphi}(x)$. Since $\\alpha$ and $\\beta$ are balanced, for $k=k(x \\restriction n)$ we obtain \n\\begin{equation} \\label{e:nk}\n|\\varrho(\\alpha \\restriction n) - a| \\le \\frac{1}{n} \\quad \\text{and} \\quad  \n|\\varrho(\\beta \\restriction k) - b| \\le \\frac{1}{k}.\n\\end{equation}  \nThen \\eqref{e:q3  lambda close to varphi}  and \\eqref{e:nk} imply that \n\\begin{align*} \n|\\varrho(\\phi(x \\restriction n)) - \\varphi(x \\restriction n) |&= \\left|\\frac{n \\varrho(\\alpha \\restriction n) +k \\varrho(\\beta \\restriction k)}{n + k} - \\varphi(x \\restriction n)\\right| \\\\\n&\\le \\frac{2}{n + k} + \\frac{2}{\\sqrt{n}},\n\\end{align*}\nwhich tends to $0$. This implies that $\\varrho(\\phi(x \\restriction n)) \\to \\overline{\\varphi}(x)$, so the proof of \\eqref{e:dvarphi} is complete.\n\t\nFinally, we can construct $K$. As $\\psi$ is continuous, Fact~\\ref{f:cont} implies that the map\n\\begin{equation} \\label{e:K_x}\nx \\mapsto K(\\psi(x)) \\text{ is also continuous.}\n\\end{equation}\nLet $\\{x_n\\}_{n\\geq 1}$ be a dense subset of $2^\\omega$ with $\\overline{\\varphi}(x_1) = b$. Let $K_n = K(\\psi(x_n))^d$ for all $n\\geq 1$. First we show that \n\\begin{equation} \\label{e:subs} A\\subset \\{\\dim_H E : E \\in \\mathcal{M}(\\{K_n\\}_{n\\geq 1})\\}.\n\\end{equation} \nLet $z \\in A$ be arbitrary. By the definition of $\\varphi$ there exists $x \\in 2^\\omega$ such that $\\overline{\\varphi}(x) = z\/d\\in B$. Fact~\\ref{f:dim} and \\eqref{e:dvarphi} imply\n\\begin{equation*} \n\\dim_H (K(\\psi(x))^d)=d \\varrho(\\psi(x))=z.\n\\end{equation*} \nAs $\\{x_n\\}_{n\\geq 1}$ is dense in $2^{\\omega}$, we can find a sequence $k_n$ such that $x_{k_n} \\to x$. By \\eqref{e:K_x} we obtain that \n\\begin{equation*} \nK_{k_n}=K(\\psi(x_{k_n}))^d \\to K(\\psi(x))^d \\quad \\text{as} \\quad  n\\to \\infty.\n\\end{equation*} \nAs $\\psi(x)\\neq \\mathbf{0}$, we obtain $K(\\psi(x))\\cap (0,1)\\neq \\emptyset$, hence $K(\\psi(x))^d\\cap (0,1)^d\\neq \\emptyset$. Thus $K(\\psi(x))^d \\in \\mathcal{M}(\\{K_n\\}_{n\\geq 1})$, so $z\\in \\{\\dim_H E : E \\in \\mathcal{M}(\\{K_n\\}_{n\\geq 1})\\}$ proving \\eqref{e:subs}.\n\nNow it is enough to prove that \n\\begin{equation} \\label{e:EiG} \n\\dim_H E=\\overline{\\dim}_B \\, E\\in A \\text{ for all } \nE \\in \\mathcal{M}(\\{K_n\\}_{n\\geq 1}).\n\\end{equation} \nIndeed, note that $\\sup\\{ \\dim_H K_n: n\\geq 1\\}=\\dim_H K_1$ and $K_1\\in \\mathcal{M}(\\{K_n\\}_{n\\geq 1})$. Then \\eqref{e:subs}, \\eqref{e:EiG}, and Lemma~\\ref{l:building K} will immediately imply \\eqref{i1}. \n\nFinally, we prove \\eqref{e:EiG}. Let $E \\in \\mathcal{M}(\\{K_n\\}_{n\\geq 1})$ and let $\\dim$ be one of $\\dim_H$ or $\\overline{\\dim}_B$, we will calculate $\\dim E$ independently of the choice of the dimension. Since we are only interested in $\\dim E$, by Theorem~\\ref{t:Cx} we may suppose that $E=K(y)^d$ for some $y\\in 2^{\\omega}$ for which there exists a subsequence of positive integers $k_n\\uparrow \\infty$ and $m_n\\in \\omega$ such that $T^{m_n}(\\psi(x_{k_n}))\\to y$. We may assume by choosing a subsequence that $x_{k_n}\\to x$ for some $x \\in 2^\\omega$. \n\nFirst suppose that $\\{m_n\\}_{n\\geq 1}$ is bounded. By choosing a subsequence we may assume that $m_n=m$ for all $n$. The continuity of $\\psi$ implies $y=T^m(\\psi(x))$. As $\\varrho(T^m(\\psi(x)))=\\varrho(\\psi(x))$, using \\eqref{e:dvarphi} and $\\overline{\\varphi}(x)\\in B$ we obtain  \n\\begin{equation*} \\dim E=\\dim (K(y)^d)=\\dim (K(\\psi(x))^d)=d\\varrho(\\psi(x))=d\\overline{\\varphi}(x)\\in A,\n\\end{equation*} \nand we are done. \n\nFinally, assume that $\\{m_n\\}_{n\\geq 1}$ is not bounded. We may suppose by choosing a subsequence that $m_n\\to \\infty$. Applying Lemma \\ref{l:balanced} for $T^{m_n}(\\psi(x_{k_n}))\\in 2^{\\omega}$ implies that $\\varrho(y)=a$ or $\\varrho(y)=b$, so \n\\begin{equation*}\n\\dim E=\\dim (K(y)^d)=d\\varrho(y)\\in \\{\\min A, \\max A\\}, \n\\end{equation*} \nwhich finishes the proof of \\eqref{e:EiG}. The proof of the theorem is complete.\n\\end{proof}\n\n\n\n\n\n\\section{Compact families of compact sets} \\label{s:K}\n\n\\subsection{The case of Hausdorff dimension} \\label{ss:Haus}\nThe goal of this subsection is to prove Theorem~\\ref{t:compact family} after some preparation. We define parametrized fractal percolations in axis parallel cubes $Q\\subset \\mathbb{R}^d$ as follows.\n\n\n\\begin{definition} \\label{d:p}\nLet $Q\\subset \\mathbb{R}^d$ be an axis parallel cube with side length $r$. For all $n\\in \\mathbb{N}^+$ we denote by $\\mathcal{D}_n$ the collection of compact dyadic subcubes of $Q$ of side length $r2^{-n}$. Given $\\alpha_n\\in [0,d]$ for all $n\\geq 1$, we construct a random compact set $\\Gamma(\\{\\alpha_n\\}_{n\\geq 1})\\subset Q$ as follows. We keep each of the $2^d$ cubes in $\\mathcal{D}_1$ with probability $2^{-\\alpha_1}$. Let $\\Delta_1\\subset \\mathcal{D}_1$ be the collection of kept cubes and let $S_1=\\bigcup \\Delta_1$ be their union. If $\\Delta_{n-1}\\subset \\mathcal{D}_{n-1}$ and $S_{n-1}=\\bigcup \\Delta_{n-1}$ are already defined, then we keep each cube in $D\\in \\mathcal{D}_{n}$ for which $D\\subset S_{n-1}$ independently with probability $2^{-\\alpha_{n}}$. Denote by $\\Delta_{n}\\subset \\mathcal{D}_{n}$ the collection of kept cubes and by $S_{n}=\\bigcup \\Delta_{n}$ their union. Define our \\emph{percolation limit set with generation dependent retention probabilities $2^{-\\alpha_n}$} as\n\\begin{equation*} \n\\Gamma(\\{\\alpha_n\\}_{n\\geq 1})=\\bigcap_{n=1}^{\\infty} S_n.\n\\end{equation*}\nIf $\\alpha_n=\\alpha$ for all $n\\geq 1$ then we simply use the notation $\\Gamma(\\alpha)$ instead of $\\Gamma(\\{\\alpha_n\\}_{n\\geq 1})$.\n\nWe say that a random set $X$ \\emph{stochastically dominates} $Y$ if they can be defined on a common probability space (coupled) such that $Y\\subset X$ almost surely. For two sets $A,B$ we write $A\\subset^{\\star} B$ if $A\\setminus B$ is countable. \n\\end{definition}\n\nThe following theorem is due to Hawkes \\cite[Theorem~6]{H} in the context of trees, see also \\cite[Theorem~9.5]{MP}.\n\n\\begin{theorem}[Hawkes] \\label{t:H} For every $\\beta\\in [0,d]$ and every compact set $K\\subset Q$ the following properties hold\n\\begin{enumerate}\n\\item \\label{e:H1} if $\\dim_H K<\\beta$, then almost surely, $K\\cap \\Gamma(\\beta)=\\emptyset$,\n\\item \\label{e:H2} if $\\dim_H K>\\beta$, then $K\\cap \\Gamma(\\beta)\\neq \\emptyset$ with positive probability,\n\\item \\label{e:H3} if $\\dim_H K>\\beta$, then almost surely, $\\dim_H(K\\cap \\Gamma(\\beta))\\leq \\dim_H K-\\beta$.\n\\end{enumerate} \n\\end{theorem}\n\n\\begin{lemma} \\label{l:H}\nLet $K\\subset Q$ be compact with $\\dim_H K=\\gamma$ and let $0<\\beta <\\gamma$. Then there exists a constant $c=c(Q,K,\\beta)>0$ such that the following holds. If $\\alpha_n \\in [0,\\beta]$ for all $n\\geq 1$ and $\\alpha_n\\to \\alpha $, then \n\\begin{equation} \\label{e:Kc}\n\\P(\\dim_H ( K\\cap \\Gamma(\\{ \\alpha_n \\}_{n\\geq 1}))\\geq \\beta-\\alpha)\\geq c.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} Define $c=\\P(K\\cap \\Gamma(\\beta)\\neq \\emptyset)$, we have $c>0$ by Theorem~\\ref{t:H}~\\eqref{e:H2}. For all $n\\geq 1$ let $\\delta_n=\\beta-\\alpha_n\\geq 0$  and consider $K\\cap \\Gamma(\\{\\alpha_n\\}_{n\\geq 1})\\cap \\Gamma(\\{\\delta_n\\}_{n\\geq 1})$, where $\\Gamma(\\{\\alpha_n\\}_{n\\geq 1})$ and $\\Gamma(\\{\\delta_n\\}_{n\\geq 1})$ are independent. Since  $\\Gamma(\\{\\alpha_n\\}_{n\\geq 1})\\cap \\Gamma(\\{\\delta_n\\}_{n\\geq 1})$ stochastically dominates $\\Gamma(\\beta)$, we obtain\n\t\\begin{equation} \\label{e:gamma1} \n\t\\P(K\\cap \\Gamma(\\{\\alpha_n\\}_{n\\geq 1})\\cap \\Gamma(\\{\\delta_n\\}_{n\\geq 1})\\neq \\emptyset)\\geq c.\n\t\\end{equation} \nAs $\\Gamma(\\{\\delta_n\\}_{n\\geq 1})$ is stochastically dominated by the union of finitely many affine copies of $\\Gamma(\\delta)$ for any given $\\delta<\\beta-\\alpha$, Theorem~\\ref{t:H}~\\eqref{e:H1}, the independence of $\\Gamma(\\{\\alpha_n\\}_{n\\geq 1})$ and $\\Gamma(\\{\\delta_n\\}_{n\\geq 1})$, and Fubini's theorem imply \n\t\\begin{equation*} \n\t\\P(K\\cap \\Gamma(\\{\\alpha_n\\}_{n\\geq 1})\\cap \\Gamma(\\{\\delta_n\\}_{n\\geq 1})\\neq \\emptyset \\text{ and } \\dim_H(K\\cap \\Gamma(\\{\\alpha_n\\}_{n\\geq 1}))<\\beta-\\alpha)=0.\n\t\\end{equation*}\nThis and \\eqref{e:gamma1} imply \\eqref{e:Kc}, and the proof is complete.\n\\end{proof}\n\n\n\n\\begin{theorem}\n\\label{t:compact family}\nLet $K \\subset \\mathbb{R}^d$ be a non-empty compact set and let $A \\subset [0, \\dim_H K]$. The following statements are equivalent: \n\\begin{enumerate}\n\\item \\label{ic1} There is a compact set $\\mathcal{C} \\subset \\mathcal{K}(K)$ with $\\{\\dim_H C : C \\in \\mathcal{C}\\} = A$;\n\\item \\label{ic2} $A$ is an analytic set. \n\\end{enumerate} \n\\end{theorem}\n\n\\begin{proof}  \nFirst we prove $\\eqref{ic1} \\Rightarrow \\eqref{ic2}$. Since $\\dim_H \\colon \\mathcal{K}(\\mathbb{R}^d)\\to [0,d]$ is Borel measurable by \\cite[Theorem 2.1]{MM}, if $\\{\\dim_H C : C \\in \\mathcal{C}\\} = A$ then $A$ must be analytic as the image of a compact set under a Borel map, see e.g.~\\cite[Proposition~14.4]{Ke}. \t\n\t\nNow we show $\\eqref{ic2} \\Rightarrow \\eqref{ic1}$. If $A=\\emptyset$ then $\\mathcal{C}=\\emptyset$ works, so we may suppose that $A\\subset [0,\\dim_H K]$ is non-empty and analytic. Let $\\gamma=\\dim_H K$. We may assume that $A \\subset(0, \\gamma]$, since if an appropriate family $\\mathcal{C}$ for $A \\cap (0, \\dim_H K]$ is constructed, then $\\mathcal{C}\\cup\\{\\{ y \\}\\}$ works for $A$ for any $y\\in K$. By Fact~\\ref{f:dimK} we can fix $y_0 \\in K$ such that $\\dim_H (K \\cap U)= \\dim_H K $ for any open neighborhood $U$ of $y_0$. \n\t\nFix a sequence of positive numbers $\\beta_k \\uparrow \\dim_H K $, and for all $k\\geq 1$ let $Q_k$ be a cube around $y_0$ of side length $1\/k$. Fix $k\\geq 1$ arbitrarily, and let $c_k>0$ be the constant we obtain by applying Lemma~\\ref{l:H} for $K\\cap Q_k\\subset Q_k$ and $\\beta_k$. Choose $i_k\\in \\mathbb{N}^+$ large enough so that \n\\begin{equation}\n\\label{e:choice of i_k}\n(1-c_k)^{i_k} < \\frac 12.\n\\end{equation}\nWe will run $i_k$ independent, parameterized family of percolations inside $Q_k$. For all $n\\geq 0$ let $\\mathcal{D}_n^k$ denote the collection of dyadic subcubes of $Q_k$ with side length $(1\/k)2^{-n}$ and let $\\mathcal{D}^k=\\bigcup_{n=1}^{\\infty} \\mathcal{D}_n^k$. For any $D\\in \\mathcal{D}^k$ let $u^k_i(D)$ be a random variable uniformly distributed in $[0, 1]$ such that the family $\\{u^k_i(D): k \\ge 1, \\, i \\leq  i_k, \\, D \\in \\mathcal{D}^k\\}$ is independent. Assume that a sequence $\\{\\alpha_n\\}_{n\\geq 1}$ is given such that $\\alpha_n\\in [0,\\gamma)$ for all $n\\geq 1$ and $\\alpha_n \\to \\alpha \\in [0,\\gamma)$. We define $\\Gamma_i^k(\\{ \\alpha_n \\}_{n\\geq 1})$ as follows. Let $S_0=Q_k$ and $\\mathcal{D}^k_0=\\{Q_k\\}$, and for each $1\\leq i\\leq i_k$ let \n\\begin{align*} \n&\\Delta_1=\\Delta_{i,1}^k(\\{\\alpha_n\\}_{n\\geq 1})=\\{D\\in \\mathcal{D}_1^k: u_i^k(D)\\leq 2^{-\\alpha_1} \\}, \\\\\n&S_1=S_{i,1}^k(\\{\\alpha_n\\}_{n\\geq 1})=\\bigcup \\Delta_1.\n\\end{align*} \nIf $\\Delta_{m-1}$ and $S_{m-1}$ are already defined, let \n\\begin{align*} \n&\\Delta_{m}=\\Delta_{i,m}^k(\\{\\alpha_n\\}_{n\\geq 1})=\\{D\\in \\mathcal{D}_m^k: D\\subset S_{m-1} \\text{ and } u_i^k(D)\\leq 2^{-\\alpha_m} \\}, \\\\\n&S_m=S_{i,m}^k(\\{\\alpha_n\\}_{n\\geq 1})=\\bigcup \\Delta_m.\n\\end{align*} \nFinally, we define \n\\begin{equation*} \n\\Gamma^k_i(\\{\\alpha_n\\}_{n \\geq 1}) = \\bigcap_{m=0}^{\\infty} S_m.\n\\end{equation*} \nSince for any $\\delta < \\alpha$ the percolation $\\Gamma^k_i(\\{\\alpha_n\\}_{n \\geq 1})$ is stochastically dominated by the union of finitely many affine copies of $\\Gamma(\\delta)$, by Theorem~\\ref{t:H}~\\eqref{e:H3} for all $1\\leq i\\leq i_k$ we obtain\n\\begin{equation}\n\\label{e:percolation probability}\n\\P\\left(\\dim_H \\left(K \\cap \\Gamma^k_i(\\{\\alpha_n\\}_{n \\geq 1})\\right) \n\\le \\gamma- \\alpha\\right) = 1.\n\\end{equation}\nMoreover, if $\\alpha_n \\in [0,\\beta_k]$ for all $n\\geq 1$, then Lemma~\\ref{l:H}, \\eqref{e:choice of i_k}, and the independence of the processes $\\Gamma^k_i(\\{\\alpha_n\\}_{n \\geq 1})$ for $1\\leq i\\leq i_k$ imply \n\\begin{equation}\n\\label{e:percolation probability 2}\n\\P\\left(\\dim_H \\left(K \\cap \\left(\\bigcup_{i=1 }^{i_k}\\Gamma^k_i(\\{\\alpha_n\\}_{n \\geq 1})\\right)\\right) \\ge \\beta_k - \\alpha \\right) \\geq \\frac 12.\n\\end{equation}\nFor each $k \\ge 1$ from each cube $D \\in \\mathcal{D}^k$ satisfying $D \\cap K \\neq \\emptyset$ we choose a point $z_D \\in D \\cap K$. For all \n$n\\geq 0$ let \n\\begin{equation*} \n\\mathcal{C}^k_n=\\{D\\in \\mathcal{D}^k_n: D\\cap K\\neq \\emptyset\\},\n\\end{equation*}\t\nand define the countable random set\n\\begin{equation*}\nF^k_i(\\{\\alpha_n\\}_{n \\geq 1}) = \\bigcup_{n=0}^{\\infty} \\big\\{z_D : D \\in \\mathcal{C}^k_n, \\, D \\subset S_n,\\text{ and } \\nexists C\\in \\mathcal{C}^k_{n+1}  \\text{ with } C\\subset S_{n+1} \\cap D \\big\\}.\n\\end{equation*}\nWe now claim that \n\\begin{equation}\\label{e:set is compact}\nF^k_i(\\{\\alpha_n\\}_{n \\geq 1}) \\cup \\Gamma^k_i(\\{\\alpha_n\\}_{n \\geq 1}) \\text{ is compact for each $k \\ge 1$ and $1\\leq i \\leq  i_k$}.\n\\end{equation} \nIndeed, $F^k_i(\\{\\alpha_n\\}_{n \\geq 1}) \\setminus S_m$ is finite for all $m\\geq 1$, hence $S_m^* = S_m \\cup F^k_i(\\{\\alpha_n\\}_{n \\geq 1})$ is compact. Therefore $F^k_i(\\{\\alpha_n\\}_{n \\geq 1}) \\cup \\Gamma^k_i(\\{\\alpha_n\\}_{n \\geq 1})  = \\bigcap_{m=1}^{\\infty} S_m^*$ is compact as well, which completes the proof of \\eqref{e:set is compact}. \n\t\nFor all $1\\leq i\\leq i_k$ let \n\\begin{equation*} \n\\Gamma^{*}_{k,i}(\\{\\alpha_n\\}_{n \\geq 1})=F^k_i(\\{\\alpha_n\\}_{n \\geq 1}) \\cup \\Gamma^k_i(\\{\\alpha_n\\}_{n \\geq 1}), \n\\end{equation*}\nand define\n\\begin{equation*} \n\\Gamma^*(\\{\\alpha_n\\}_{n \\geq 1}) = \\{x \\} \\cup \\bigcup_{k=1}^{\\infty} \\bigcup_{i=1}^{i_k} \\Gamma^{*}_{k,i}(\\{\\alpha_n\\}_{n \\geq 1}).\n\\end{equation*} \nUsing \\eqref{e:set is compact} and the fact that $\\Gamma^{*}_{k,i}(\\{\\alpha_n\\}_{n \\geq 1}) \\subset Q_k$ and $Q_k\\to \\{x\\}$ as $k\\to \\infty$, it is clear that $\\Gamma^*(\\{\\alpha_n\\}_{n \\geq 1})$ is compact. As $\\alpha<\\gamma$ and $\\alpha_n<\\gamma$ for all $n\\geq 1$,  we have $\\sup \\{\\alpha_n: n\\geq 1\\}\\leq \\beta_k$ for all large enough $k$, so we can apply \\eqref{e:percolation probability 2} for large values of $k$. Therefore  \\eqref{e:percolation probability}, \\eqref{e:percolation probability 2}, and the independence of the processes defining each $\\Gamma^k_i(\\{\\alpha_n\\}_{n \\geq 1})$ yield\n\\begin{equation*}\n\\P\\left(\\dim_H \\left(K \\cap \\Gamma^*(\\{\\alpha_n\\}_{n \\geq 1})\\right) = \\gamma - \\alpha\\right) = 1.\n\\end{equation*}\nOur coupling of percolations clearly implies the following monotonicity: Almost surely, for all sequences $\\{\\alpha_n\\}_{n \\geq 1}$ and $\\{ \\alpha'_n\\}_{n\\geq 1}$ we have\n\t\\begin{equation}\n\t\\label{e:monotonicity}\n\t\\Gamma^*(\\{\\alpha_n\\}_{n \\geq 1}) \\subset^{\\star} \\Gamma^*(\\{\\alpha'_n\\}_{n\\geq 1}) \\text{ if $\\alpha_n \\ge \\alpha'_n$ for each $n$}.\n\t\\end{equation}\nLet $Q =\\mathbb{Q} \\cap [0, \\gamma)$, and define the set \n\\begin{equation*} \nQ^* = \\{\\{\\alpha_n\\}_{n \\geq 1} : \\alpha_n\\in Q \\text{ for all $n$ and $\\alpha_n$ is eventually constant}\\}.\n\\end{equation*}  \nClearly, $Q^*$ is countable, therefore, with probability $1$ we have \n\\begin{equation}\n\\label{e:dimension prob for Q}\n\\dim_H (K \\cap \\Gamma^*(\\{\\alpha_n\\}_{n \\geq 1})) = \\gamma - \\alpha  \\text{ for all $\\{\\alpha_n\\}_{n \\geq 1} \\in Q^*$ with $\\alpha_n \\to \\alpha$}.\n\\end{equation}\n\t\nNow we are ready to define our family of compact sets $\\mathcal{C}$. By Lemma~\\ref{l:varphi existence} there exists a map $\\varphi \\colon 2^{<\\omega} \\to [0,\\infty)$ such that $\\overline{\\varphi}(x) = \\lim_{n \\to \\infty} \\varphi(x \\restriction n)$ exists for all $x \\in 2^\\omega$ and $\\overline{\\varphi}(2^\\omega) = A$. Since $A \\subset (0, \\gamma]$, we may assume that \n\\begin{equation} \\label{e:0gamma}\n0<\\varphi(s)\\leq \\gamma  \\text{ for all } s\\in 2^{<\\omega}.\n\\end{equation} \nIndeed, otherwise for all $s\\in 2^n$ we can replace $\\varphi(s)$ by $\\gamma 2^{-n}$ if $\\varphi(s)\\leq 0$ and by $\\gamma$ if $\\varphi(s)>\\gamma$, which modifications do not change the limit $\\overline{\\varphi}$. For each $x \\in 2^\\omega$ let $\\alpha(x)=\\{\\alpha(x)_n: n\\geq 1\\}$ such that\n\\begin{equation*} \n\\alpha(x)_n=\\gamma-\\varphi(x\\restriction n) \\text{ for all } n\\geq 1. \n\\end{equation*} \nLet us define $\\mathcal{C}$ as \n\\begin{equation*} \n\\mathcal{C} = \\{K \\cap \\Gamma^*(\\alpha(x)) : x \\in 2^\\omega\\}.\n\\end{equation*}\nIt is clear that $\\mathcal{C}$ is a random family of compact sets. Now we prove that, almost surely, $\\{\\dim_H C : C \\in \\mathcal{C}\\} = A$. Assume that the event of \\eqref{e:dimension prob for Q} holds, it is enough to show that $\\dim_H (K \\cap \\Gamma^*(\\alpha(x))) = \\overline{\\varphi}(x)$ for all $x \\in 2^\\omega$. Let $x \\in 2^\\omega$ be fixed. The definition of $\\varphi$ and \\eqref{e:0gamma} imply that $\\alpha(x)_n\\in [0,\\gamma)$ for all $n$, and $\\alpha(x)_n$ converges to $\\alpha_x=\\gamma-\\overline{\\varphi}(x)\\in  [0,\\gamma)$. Hence for any $\\varepsilon > 0$, we can find sequences $\\{\\alpha'_n\\}_{n \\geq 1}, \\{\\alpha''_n\\}_{n \\geq 1} \\in Q^*$ such that \n\\begin{equation*} \n\\alpha'_n \\le \\alpha(x)_n \\le \\alpha''_n \\text{ and } \\alpha''_n-\\alpha'_n\\leq \\varepsilon \\text{ for each } n.\n\\end{equation*} \nThen $\\alpha'_n\\to \\alpha'$ and $\\alpha''_n\\to \\alpha''$ such that \n\\begin{equation} \\label{e:alpha'} \n\\alpha'\\leq \\alpha_x\\leq \\alpha'' \\text{ and } \\alpha''-\\alpha'\\leq \\varepsilon.\n\\end{equation} \nBy \\eqref{e:dimension prob for Q} we have \n\\begin{equation} \\label{e:dimg}\n\\dim_H (K \\cap \\Gamma^*(\\{\\alpha'_n\\}_{n \\geq 1})) = \\gamma - \\alpha' \\text{ and } \\dim_H(K \\cap \\Gamma^*(\\{\\alpha''_n\\}_{n \\geq 1})) = \\gamma - \\alpha''.\n\\end{equation}\nMonotonicity \\eqref{e:monotonicity} yields \n\\begin{equation} \\label{e:mon}\n\\Gamma^{*} (\\{\\alpha''_n\\}_{n\\geq 1} )\\subset^{\\star} \\Gamma^{*}(\\alpha(x)) \\subset^{\\star} \n\\Gamma^{*} (\\{\\alpha'_n\\}_{n\\geq 1}).\n\\end{equation}\nAs $\\varepsilon>0$ was arbitrary, \\eqref{e:alpha'}, \\eqref{e:dimg}, and \\eqref{e:mon} imply that \n\\begin{equation*}\n\\dim_H(K \\cap \\Gamma^*(\\alpha(x))) =\\gamma-\\alpha_x=\\overline{\\varphi}(x),\n\\end{equation*}\nwhich proves that $\\{\\dim_H C : C \\in \\mathcal{C}\\} = A$ almost surely.\n\n\nFinally, we check that $\\mathcal{C}$ is compact with probability $1$. Since $2^\\omega$ is compact and $Q_k\\to \\{y_0\\}$ as $k\\to \\infty$, it is enough to prove that for an arbitrarily given $k\\geq 1$ and $1\\leq i\\leq i_k$ the map $x \\mapsto C_x\\stackrel{\\text{def}}{=} K\\cap \\Gamma_{k,i}^*(\\alpha(x))$ is continuous. Assume that $x$ and $y$ are two sequences with $x \\restriction n= y \\restriction n$ for some $n\\geq 1$, it is enough to show that \n\\begin{equation} \\label{e:dH} \nd_H(C_x, C_y)\\leq 2^{-n}\\diam Q_k.\n\\end{equation} \nThe construction implies that  $S^k_{i,n}(\\alpha(x))=S^k_{i,n}(\\alpha(y))\\stackrel{\\text{def}}{=} S_n$ coincide, and also $C_x \\setminus S_n=C_y\\setminus S_n$. Therefore, \n\\begin{equation} \\label{e:dH1}\nd_H(C_x, C_y)\\le d_H(C_x\\cap S_n, C_y\\cap S_n).\n\\end{equation} \nLet $D\\in \\mathcal{D}^k_n$ be arbitrary such that $D\\subset S_n$. By the construction we obtain that $C_x\\cap D\\neq \\emptyset$ iff $D\\in \\mathcal{C}^k_n$ iff $C_y\\cap D\\neq \\emptyset$, so $\\diam D=2^{-n}\\diam Q_k$ yields \n\\begin{equation} \\label{e:dH2}\nd_H(C_x\\cap S_n, C_y\\cap S_n)\\leq 2^{-n}\\diam Q_k.\n\\end{equation}\nSince \\eqref{e:dH1} and \\eqref{e:dH2} imply \\eqref{e:dH}, the proof is complete. \n\\end{proof}\n\n\\subsection{Packing and box dimensions} \\label{ss:box}\nWe prove Theorems~\\ref{t:compbox} and \\ref{t:comppack} in this subsection. For the following equivalent version of the upper box dimension and for more alternative definitions see \\cite[Chapter~3]{Fa}.\n\n\\begin{definition}\nIn a metric space $(X,\\rho)$ we say that $S\\subset X$ is a \\emph{$\\delta$-packing} if $\\rho(x,y)>\\delta$ for all distinct $x,y\\in S$. Let $P_n(X)$ be the cardinality of a maximal $2^{-n}$-packing in $X$. \n\\end{definition}\n\n\\begin{fact} \\label{f:equiv} \n\tFor any metric space $X$ we have $N_n(X)\\leq P_n(X)\\leq N_{n+1}(X)$, so \n\t\\begin{equation*}\n\t\\overline{\\dim}_B \\, X=\\limsup_{n \\to \\infty} \\frac{\\log P_n(X)}{n\\log 2}.\n\t\\end{equation*} \n\\end{fact} \n\n\n\n\\begin{theorem} \\label{t:compbox}\nLet $K$ be a non-empty compact metric space and $A \\subset [0, \\overline{\\dim}_B \\, K]$. The following statements are equivalent: \n\\begin{enumerate}\n\\item \\label{i01} There is a compact set $\\mathcal{C} \\subset \\mathcal{K}(K)$ with $\\{\\overline{\\dim}_B \\, C : C \\in \\mathcal{C}\\} = A$;\n\\item \\label{i02} $A$ is an analytic set. \n\\end{enumerate} \n\\end{theorem}\n\\begin{proof} \n\tThe direction $\\eqref{i01} \\Rightarrow \\eqref{i02}$ is analogous to the one in Theorem~\\ref{t:compact family}.\n\t\n\tNow we prove $\\eqref{i02} \\Rightarrow \\eqref{i01}$. We may assume that $A\\neq \\emptyset$, otherwise $\\mathcal{C}=\\emptyset$ works. Let $\\alpha=\\overline{\\dim}_B \\, K$. Choose a sequence $\\alpha_n \\uparrow \\alpha$. By Fact~\\ref{f:dimK} we can fix $y_0\\in K$ such that \n\\begin{equation*}\n\\overline{\\dim}_B \\, B(y_0,r)=\\alpha \\text{ for all } r>0.\n\\end{equation*}\nLet $g(n)=\\max\\{n+1,P_n(K)\\}$ for all $n\\in \\mathbb{N}$. We can choose a sequence $k_n\\uparrow \\infty$ with $k_0=0$ and a positive integer $j=j(n)$ such that $g(k_n)\\leq  j\\leq k_{n+1}-3$ and\n\\begin{equation} \\label{e:alpha}  P_{j}(B(y_0,2^{-g(k_n)}))\\geq 2^{\\alpha_n j}. \n\\end{equation}\nBy Lemma~\\ref{l:varphi existence} there is a map $\\varphi \\colon 2^{<\\omega} \\to [0, \\infty)$ such that\n\\begin{equation*} \n\\overline{\\varphi}(x) = \\lim_{n \\to \\infty} \\varphi(x \\restriction n) \n\\end{equation*} \n\texists for each $x \\in 2^\\omega$, and the resulting function $\\overline{\\varphi}$ satisfies $\\overline{\\varphi}(2^\\omega) = A$. We may assume that $\\varphi(s)\\leq \\alpha_n$ for all $s\\in 2^n$ and $n\\in \\mathbb{N}$, otherwise we can replace $\\varphi(s)$ by $\\alpha_n$ without changing $\\overline{\\varphi}$.\n\n\tLet $m(\\emptyset)=1$, $y_{\\emptyset}=y_0$, and $C(\\emptyset)=B(y_0,1)$. Assume that $s\\in 2^n$, a positive integer $m(s)$, and points $y_i(s)\\in K$ are given such that $y_0(s)=y_0$ and their pairwise distance is more than $2^{2-k_n}$, and  \n\t\\begin{equation*} \n\tC(s)=\\bigcup_{i=1}^{m(s)} B(y_i(s), 2^{-k_n}),\n\t\\end{equation*} \n\tso the distance between distinct balls of the form $B(y_i(s), 2^{-k_n})$ is bigger than $2^{-k_n}$.\n\t\n\tLet $c\\in \\{0,1\\}$ and $t=s^\\frown c$. Let $\\ell(t)$ be the minimal positive integer such that $g(k_n) \\leq \\ell(t) \\leq k_{n+1}-3$ and  \n\t\\begin{equation} \n\tP_{\\ell(t)}(B(y_0,2^{-g(k_n)}))\\geq 2^{\\varphi(t)\\ell(t)},\n\t\\end{equation} \n\tby \\eqref{e:alpha} the number $\\ell(t)$ is well-defined. Then we can choose a $2^{-\\ell_i}$-packing $S$ of size exactly $\\lfloor 2^{\\varphi(t)\\ell(t)} \\rfloor$ in $B(y_i(s),2^{-g(k_n)})$, where $\\lfloor \\cdot \\rfloor$ denotes the integer part. By replacing a suitable element of $S$ with $y_0$ we can obtain a $2^{-\\ell(t)-1}$-packing $T$ in $B(y_i(s),2^{-g(k_n)})$ such that $y_0\\in T$ and $\\#T=\\#S$. Indeed, this is straightforward if $S\\cup\\{y_0\\}$ is a $2^{-\\ell_i-1}$-packing. Otherwise $y_0\\in B(y,2^{-\\ell_i-1})$ for some $y\\in S$, and replacing $y$ with $y_0$ provides a suitable $T$. Let $m(t)=m(s)+(\\#T)-1$ and let \n\t\\begin{equation*} \n\ty_i(t)=y_i(s) \\text{ if } 1\\leq  i\\leq m(s) \\quad \\text{and} \\quad  \\{y_i(t)\\}_{m(s)<i\\leq m(t)}=T\\setminus \\{y_0\\}.\n\t\\end{equation*} \t\n\tDefine\n\t\\begin{equation*} C(t)=\\bigcup_{i=1}^{m(t)} B(y_i(s), 2^{-k_{n+1}}).\n\t\\end{equation*}\n\tso the distance between distinct balls $B(y_i(t), 2^{-k_{n+1}})$ is bigger than $2^{-k_{n+1}}$, and  clearly $C(t)\\subset C(s)$. Thus we defined $C(s)$ for all $s\\in 2^{<\\omega}$. \n\t\n\tFor all $x\\in 2^{\\omega}$ set\n\t\\begin{equation*} \n\tC(x)=\\bigcap_{n=1}^{\\infty} C(x\\restriction n).\n\t\\end{equation*}\n\tClearly $C(x)$ is compact. If $x\\restriction n=y\\restriction n=s$, then both $C(x)$ and $C(y)$ can be covered by the same union of balls of radius $2^{-k_n}$, and both intersect all such balls. Therefore \n\t\\begin{equation*} \n\td_H(C(x),C(y))\\leq 2^{1-k_n},\n\t\\end{equation*} \n\tso the map $x\\mapsto C(x)$ is continuous. Thus the definition \n\t\\begin{equation*} \\mathcal{C}=\\{C(x): x\\in 2^{\\omega}\\}\n\t\\end{equation*}\n\tyields a compact set $\\mathcal{C}\\subset \\mathcal{K}(K)$. In order to prove $\\{\\overline{\\dim}_B \\, C: C\\in \\mathcal{C}\\}=A$ let $x\\in 2^{\\omega}$ be arbitrarily fixed. It is enough to show that \n\t\\begin{equation} \\label{e:BC}\n\t\\overline{\\dim}_B \\, C(x)=\\overline{\\varphi}(x).\n\t\\end{equation} \n\tAs we never remove the centers of our balls during the construction, we obtain that $C(x)$ contains a $2^{-\\ell(x\\restriction n)-1}$-packing of size $\\lfloor 2^{\\varphi(x\\restriction n) \\ell(x\\restriction n)}\\rfloor$ for all $n$, so Fact~\\ref{f:equiv} yields that \n\t\\begin{equation} \\label{e:lower}\t\t\n\t\\overline{\\dim}_B \\, C(x)\\geq \\overline{\\varphi}(x).\n\t\\end{equation}\t\n\tFor the other direction we need \t\n\t\\begin{equation} \\label{e:upper}\n\t\\overline{\\dim}_B \\, C(x)\\leq \\overline{\\varphi}(x).\n\t\\end{equation}\n\tLet $\\gamma>\\overline{\\varphi}(x)$ be arbitrary, we use that $\\{y_i(x\\restriction n)\\}_{1\\leq i\\leq m(x\\restriction n)}$ forms a $2^{-k_n}$-packing in $K$, so $m(x\\restriction n)\\leq g(k_n)$. Then for all large enough $n$ and $g(k_n)\\leq \\ell \\leq k_{n+1}-3$ by Fact~\\ref{f:equiv} we have\n\t\\begin{align*}\n\tN_{\\ell}(C(x))&\\leq m(x\\restriction n)+N_{\\ell}\\left(C(x)\\cap B\\left(y_0, 2^{-g(k_n)}\\right)\\right) \\\\\n\t&\\leq  g(k_n)+ P_{\\ell}\\left(C(x)\\cap B\\left(y_0, 2^{-g(k_n)}\\right)\\right) \\\\\n\t&\\leq \\ell+2^{\\varphi(x\\restriction (n+1))\\ell } \\leq 2^{\\gamma \\ell}.\n\t\\end{align*} \n\tFor all large enough $n$ and $k_n-3< \\ell < g(k_n)$ the inequality \n\t$N_{\\ell}(C(x))\\leq 2^{\\gamma \\ell}$ clearly holds. As $\\gamma>\\overline{\\varphi}(x)$ was arbitrary, we obtain \\eqref{e:upper}. Then \\eqref{e:lower} and \\eqref{e:upper} imply \\eqref{e:BC}, so the proof is of the theorem is complete.\n\\end{proof} \n\n\nFor the packing dimension we prove the following theorem.\n\n\\begin{theorem} \\label{t:comppack}\n\tLet $K$ be a non-empty compact metric space and let $A \\subset [0, \\dim_P K]$ be analytic. Then there is a compact set $\\mathcal{C} \\subset \\mathcal{K}(K)$ with $\\{\\dim_P C : C \\in \\mathcal{C}\\} = A$.\n\\end{theorem}\n\n\\begin{proof}\n\tWe may assume that $A\\neq \\emptyset$, otherwise $\\mathcal{C}=\\emptyset$ works. Let $\\dim_P K=\\alpha$. First suppose that $A\\subset [0,\\beta]$ for some $\\beta<\\alpha$. By  Lemma~\\ref{l:packing}~\\eqref{i:ii} we may assume by shrinking $K$ if necessary that $\\overline{\\dim}_B \\,  U>\\beta$ for any non-empty open set $U\\subset K$. By Lemma~\\ref{l:varphi existence} there exists a map $\\varphi \\colon 2^{<\\omega} \\to [0, \\infty)$ such that\n\t\\begin{equation*} \\overline{\\varphi}(x) = \\lim_{n \\to \\infty} \\varphi(x \\restriction n)\n\t\\end{equation*} \n\texists for each $s \\in 2^\\omega$, and the resulting function $\\overline{\\varphi}$ satisfies $\\overline{\\varphi}(2^\\omega) = A$. We may assume that $\\varphi(s)\\leq \\beta$ for all $s\\in 2^{<\\omega}$. Let $g(n)=\\max\\{n+1,P_n(K)\\}$ for all $n\\in \\mathbb{N}$. \n\tBy compactness we can choose a sequence $k_n\\uparrow \\infty$ with $k_0=0$ such that for all $y\\in K$ and $n\\in \\mathbb{N}$ there exists a positive integer $j=j(n,y)$ such that $g(k_n)\\leq j\\leq k_{n+1}-2$ and\n\t\\begin{equation}  \\label{e:Pj} P_{j}(B(y,2^{-g(k_n)}))\\geq 2^{\\beta j}. \n\t\\end{equation}\n\tLet $m(\\emptyset)=1$, $y_{\\emptyset}\\in K$ be arbitrary, and $C(\\emptyset)=B(y_{\\emptyset},1)$. Assume that $s\\in 2^n$ and a positive integer $m(s)$, points $y_i(s)\\in K$ with pairwise distance more than $2^{2-k_n}$ are given, and  \n\t\\begin{equation*} \n\tC(s)=\\bigcup_{i=1}^{m(s)} B(y_i(s), 2^{-k_n}),\n\t\\end{equation*} \n\tso the distance between distinct balls of the form $B(y_i(s), 2^{-k_n})$ is bigger than $2^{-k_n}$.\n\t\n\tLet $c\\in \\{0,1\\}$ and $t=s^\\frown c$. For all $1\\leq i\\leq m(s)$ let $\\ell_i=\\ell_i(t)$ be the minimal number such that $g(k_n) \\leq \\ell_i \\leq k_{n+1}-2$ and  \n\t\\begin{equation} \\label{e:PL}\n\tP_{\\ell_i}(B(y_i(s),2^{-g(k_n)}))\\geq 2^{\\varphi(t)\\ell_i},\n\t\\end{equation} \n\tby \\eqref{e:Pj} the number $\\ell_i$ are well-defined.\n\tFor all $1\\leq i\\leq m(s)$ let us choose a $2^{-\\ell_i}$-packing $S_i$ of size exactly $\\lfloor 2^{\\varphi(t)\\ell_i} \\rfloor$ in $B(y_i(s),2^{-g(k_n)})$, where $\\lfloor \\cdot \\rfloor$ denotes the integer part.\n\tLet $S=\\bigcup_{i=1}^{m(s)} S_i$ and $m(t)=\\#S$, and let us define $y_i(t)\\in K$ such that $S=\\{y_i(t)\\}_{1\\leq i\\leq m(t)}$. Let \t\n\t\\begin{equation*} C(t)=\\bigcup_{i=1}^{m(t)} B(y_i(t), 2^{-k_{n+1}}),\n\t\\end{equation*}\n\tso the distance between distinct balls $B(y_i(t), 2^{-k_{n+1}})$ is bigger than $2^{-k_{n+1}}$, and  clearly $C(t)\\subset C(s)$. Thus we defined $C(s)$ for all $s\\in 2^{<\\omega}$. For $x\\in 2^{\\omega}$ let \n\t\\begin{equation*}\n\tC(x)=\\bigcap_{n=1}^{\\infty} C(x\\restriction n).\n\t\\end{equation*}\n\tThe construction clearly implies that $C(x)\\subset K$ is compact. If $x\\restriction n=y\\restriction n$, then $C(x)$ and $C(y)$ are covered by the same balls of radius $2^{-k_n}$ which they both intersect, so the map $x\\mapsto C(x)$ is continuous. Therefore, the definition \n\t\\begin{equation*} \\mathcal{C}=\\{C(x): x\\in 2^{\\omega}\\}\n\t\\end{equation*}\n\tyields a compact set $\\mathcal{C}\\subset \\mathcal{K}(K)$. In order to prove $\\{\\dim_P C: C\\in \\mathcal{C}\\}=A$ let $x\\in 2^{\\omega}$ be arbitrarily fixed, it is enough to show that \n\t\\begin{equation} \\label{e:PC}\n\t\\dim_P C(x)=\\overline{\\varphi}(x).\n\t\\end{equation} \n\tFirst we prove \n\t\\begin{equation} \\label{e:P>}\n\t\\dim_P C(x)\\geq \\overline{\\varphi}(x).\n\t\\end{equation}\n\tLet $U\\subset K$ be an arbitrary open set intersecting $C(x)$, by Lemma~\\ref{l:packing}~\\eqref{i:i} it is enough to show that $\\overline{\\dim}_B \\, (C(x)\\cap U)\\geq \\overline{\\varphi}(x)$. For all large enough $n$ we can fix $1\\leq i\\leq m(s)$ such that $B=B(y_i(x\\restriction n),2^{-k_n})\\subset U$. Let $\\ell(n)=\\ell_i(x\\restriction (n+1))$, by the construction and \\eqref{e:PL} the ball $B$ contains $\\lfloor 2^{\\varphi(x\\restriction (n+1))\\ell(n)} \\rfloor$ balls with pairwise distance greater than $2^{-\\ell(n)}$ such that all of them intersect $C(x)\\cap U$. Thus \n\t\\begin{equation*} P_{\\ell(n)}(C(x)\\cap U)\\geq \\lfloor 2^{\\varphi(x\\restriction (n+1))\\ell(n)}\\rfloor,\n\t\\end{equation*} \n\timplying \n\t\\begin{equation*}\n\t\\overline{\\dim}_B \\, (C(x)\\cap U)\\geq \\overline{\\varphi}(x).\n\t\\end{equation*}\n\tHence \\eqref{e:P>} holds. For the other direction it is enough to prove that\n\t\\begin{equation} \\label{e:PP}\n\t\\overline{\\dim}_B \\, C(x)\\leq \\overline{\\varphi}(x).\n\t\\end{equation}\n\tLet $\\gamma>\\overline{\\varphi}(x)$ be arbitrary, we use that $\\{y_i(x\\restriction n)\\}_{1\\leq i\\leq m(x\\restriction n)}$ form a $2^{-k_n}$-packing in $K$, so $m(x\\restriction n)\\leq g(k_n)$. Then for all large enough $n$ and $g(k_n)\\leq \\ell \\leq k_{n+1}-2$ by Fact~\\ref{f:equiv} we have\n\t\\begin{align*}\n\tN_{\\ell}(C(x))&\\leq \\sum_{i=1}^{m(x\\restriction n)} N_{\\ell}\\left(C(x)\\cap B\\left(y_i(s), 2^{-g(k_n)}\\right)\\right) \\\\\n\t&\\leq  \\sum_{i=1}^{m(x\\restriction n)} P_{\\ell}\\left(C(x)\\cap B\\left(y_i(s), 2^{-g(k_n)}\\right)\\right) \\\\\n\t&\\leq g(k_n) 2^{\\varphi(x\\restriction (n+1))\\ell } \\leq 2^{\\gamma \\ell}.\n\t\\end{align*} \n\tFor all large enough $n$ and $k_n-2< \\ell < g(k_n)$ the inequality \n\t$N_{\\ell}(C(x))\\leq 2^{\\gamma \\ell}$ clearly holds. As $\\gamma>\\overline{\\varphi}(x)$ was arbitrary, we obtain \\eqref{e:PP}. Inequalities \\eqref{e:P>} and \\eqref{e:PP} imply \\eqref{e:PC}, and the proof is complete if $A\\subset [0,\\beta]$ for some $\\beta<\\alpha$.  \n\t\nFinally, we prove the theorem in case of $A\\subset [0,\\alpha]$. If $A=\\{\\alpha\\}$ then $\\mathcal{C}=\\{K\\}$ works, so we may assume that there exists $\\beta_0\\in A\\cap [0,\\alpha)$. Take a sequence $\\beta_n\\uparrow \\alpha$, and define $A_n=A\\cap [0,\\beta_n]$ for all $n\\geq 0$. By Fact~\\ref{f:dimK} we can choose $y_0\\in K$ such that $\\dim_P B(y_0,r)=\\alpha$ for all $r>0$. By the first part we can choose a compact set $K_0\\subset K$ with $\\dim_P   K_0=\\beta_0$ and we may assume that $y_0\\in K_0$. By the first part we can also choose compact sets $\\mathcal{C}_n\\subset \\mathcal{K}(B(y_0,1\/n))$ such that \n\t\\begin{equation*}\n\t\\{\\dim_P  C: C\\in \\mathcal{C}_n\\}=A_n \\text{ for all } n\\geq 0.\n\t\\end{equation*} \n\tDefine\n\t\\begin{equation*}\n\t\\mathcal{C}=\\mathcal{C}_0\\cup \\{K\\} \\cup \\{K_0\\cup C: C\\in \\mathcal{C}_n,~n\\geq 1\\}.\n\t\\end{equation*}\t\n\tIt is easy to see that $\\mathcal{C}\\subset \\mathcal{K}(K)$ is compact and satisfies $\\{ \\dim_P  C: C\\in \\mathcal{C}\\}=A$, and the proof of the theorem is complete.\t\n\\end{proof}\n\n\n\\section{Open problems} \\label{s:open}\n\nOur first problem is the most ambitious one. For the sake of simplicity we only formalize it in case of the Hausdorff dimension.\n\n\\begin{problem} \\label{p:K} \nLet $K\\subset \\mathbb{R}^d$ be compact. Characterize the sets $A\\subset [0,\\dim_H K]$ for which there exists a compact set $C\\subset K$ such that $\\{\\dim_H E: E\\in \\mathcal{M}_C\\}=A$.\n\\end{problem}\n\n\nJ\\\"arvenp\\\"a\\\"a, J\\\"arvenp\\\"a\\\"a, Koivusalo, Li, Suomala, and Xiao \\cite[Lemma~2.3]{JJKLSX} proved an analogue of Hawkes' theorem in complete metric spaces satisfying a mild doubling condition. Therefore, the proof of Theorem~\\ref{t:compact family} possibly works in compact metric spaces with a suitable doubling condition as well. As the case of arbitrary compact metric spaces can be out of reach with this method, we ask the following.\n\n\\begin{problem}\nLet $K$ be a non-empty compact metric space and let $A \\subset [0, \\dim_H K]$ be an analytic set. Is there a compact set $\\mathcal{C} \\subset \\mathcal{K}(K)$ with $\\{\\dim_H C : C \\in \\mathcal{C}\\} = A$?\n\\end{problem}\n\nAs Mattila and Mauldin \\cite[Theorem~7.5]{MM} proved that the packing dimension $\\dim_P \\colon \\mathcal{K}(\\mathbb{R}^d)\\to [0,d]$ is not Borel measurable, we do not know whether analogue versions of Theorems~\\ref{t:characterization} and~\\ref{t:compact family} hold for the packing dimension. We state the measurability problems as follows. \n\n\\begin{problem} \\label{p:meas} If $K\\subset \\mathbb{R}^d$ is a compact set, is the set $\\{\\dim_P E: E\\in \\mathcal{M}_K\\}$ analytic? If $\\mathcal{C} \\subset \\mathcal{K}(\\mathbb{R}^d)$ is compact, is the set $\\{\\dim_P E: E\\in \\mathcal{C}\\}$ analytic?\n\\end{problem}\n\nIf the answer to the above problem is negative, we can ask for a characterization.  \n\\begin{problem} \\label{p:1} Let $d\\geq 1$. Characterize the sets $A \\subset [0, d]$ for which there exists a compact set $K \\subset \\mathbb{R}^d$ with $\\{\\dim_P E : E \\in \\mathcal{M}_K\\} = A$.\n\\end{problem}\n\n\\begin{problem} \\label{p:2} Let $K\\subset \\mathbb{R}^d$ be compact. Characterize the sets $A \\subset [0, \\dim_P K]$ for which there exists a compact set $\\mathcal{C}\\subset \\mathcal{K}(K)$ with $\\{\\dim_P C: C\\in \\mathcal{C}\\}=A$.\n\\end{problem}\n\n\\subsection*{Acknowledgments}\nWe are indebted to Jonathan M.~Fraser for some illuminating conversations and for providing us with the reference \\cite{FWW}. We thank Ville Suomala for pointing our attention to the paper \\cite{JJKLSX}. We also thank Ignacio Garc\\'ia for pointing out that a result stated in an earlier version of the manuscript was already known.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nAccording to the AdS\/CFT correspondence \\cite{Maldacena:1997re}, $\\mathcal{N}=4$ super Yang-Mills (SYM) theory on $\\mathbb{R} \\times S^3$ is dual to type IIB string theory on $\\mbox{AdS}_5\\times S^5$. This duality should in particular relate the phase transitions, critical behavior and thermal physics of the theories.  \n\nOne interesting example of a critical behavior is the Hagedorn temperature. In the planar limit of $\\mathcal{N}=4$ SYM theory on $\\mathbb{R} \\times S^3$, the origin of the Hagedorn temperature $T_{\\text{H}}$ is the confinement of the color degrees of freedom due to the theory being on a three-sphere. This enables the theory to have a phase transition that bears resemblance to the confinement\/deconfinement phase transition in QCD or pure Yang-Mills theory \\cite{Atick:1988si,Aharony:2003sx}.\n\nThe Hagedorn temperature is the lowest temperature for which the planar partition function $\\mathcal{Z}(T)$\ndiverges. Via the state\/operator correspondence, the partition function can be re-expressed in terms of the dilatation operator $D$ of $\\mathcal{N}=4$ SYM theory on $\\mathbb{R}^4$:\n\\begin{equation}\n \\mathcal{Z}(T)=\\tr_{\\mathbb{R}\\times S^3}[\\operatorname{e}^{-H\/T}]=\\tr_{\\mathbb{R}^4}[\\operatorname{e}^{-D\/T}]\\, , \n\\end{equation}\nwhere we have set the radius of $S^3$ to $1$.\nStates correspond to gauge-invariant operators consisting of one or more trace factors.\nThe energies correspond to the scaling dimensions of the operators, as measured by the dilatation operator.\nIn the planar limit, the scaling dimensions of multi-trace operators are entirely determined by those of their single-trace factors, and the latter can be enumerated via P\\'{o}lya theory to determine the partition function and thus the Hagedorn temperature in the free theory \\cite{Sundborg:1999ue}.\nThis procedure was later generalized to one-loop order and to the case of non-zero chemical potentials \\cite{Spradlin:2004pp,Yamada:2006rx,Harmark:2006di,Suzuki:2017ipd,GomezReino:2005bq}.\n\nOn the string-theory side, the Hagedorn temperature occurs due to the exponential growth of string states with the energy present in tree-level string theory. For interacting string theory, it is connected to the Hawking-Page phase transition \\cite{Witten:1998zw}. This suggests that the confinement\/deconfinement transition on the gauge-theory side is mapped on the string-theory side to a transition from a gas of gravitons (closed strings) for low temperatures to a black hole for high temperatures.\nIn particular, also the Hagedorn temperature on the gauge-theory and string-theory sides of the AdS\/CFT correspondence should be connected \n \\cite{Sundborg:1999ue,Aharony:2003sx}. \n \nOn the string-theory side, the Hagedorn temperature has been computed in pp-wave limits \\cite{PandoZayas:2002hh,Greene:2002cd,Brower:2002zx,Grignani:2003cs}. In \\cite{Harmark:2006ta}, the first quantitative interpolation of the Hagedorn temperature from the gauge-theory side to the string-theory side was made, exploiting a limit towards a critical point in the grand canonical ensemble \\cite{Harmark:2014mpa}.\nThis limit effectively reduces the gauge-theory side to the $\\mathfrak{su}(2)$ sector with only the one-loop dilatation operator surviving, which enables one to match the Hagedorn temperature of the gauge-theory side to that of string theory on a pp-wave background via the continuum limit of the free energy of the Heisenberg spin chain. \n\nA hitherto unrelated but very powerful property of planar $\\mathcal{N}=4$ SYM theory is integrability, see \\cite{Beisert:2010jr,Bombardelli:2016rwb} for reviews. It amounts to the existence of an underlying two-dimensional exactly solvable model, which reduces to an integrable sigma model at strong coupling and to an integrable spin chain at weak coupling. \nVia integrability, the planar scaling dimensions of all single-trace operators can in principle be calculated at any value of the 't Hooft coupling $\\lambda=g_{\\mathrm{\\scriptscriptstyle YM}}^2N$, allowing for a smooth interpolation between weak and strong coupling results.\nIn practice, however, the calculation for each operator is so involved that summing the results for all operators to obtain the partition function $\\mathcal{Z}(T)$ seems prohibitive.\n\nIn this letter, we show how to use integrability to compute the Hagedorn temperature at any value of the 't Hooft coupling.\nIn the spectral problem, the integrable model is solved \non a cylinder of finite circumference $L$, which accounts for wrapping contributions to the scaling dimension due to the finite length of the spin chain.\nIn order to calculate the partition function $\\mathcal{Z}(T)$, we would need to solve this model on the torus with circumferences $L$ and $1\/T$, an endeavor that has not been successful yet even for the Heisenberg spin chain.\nThe Hagedorn singularity, however, is driven by the contributions of spin chains with very high $L$, or rather very high classical scaling dimension, \nwhere the finite-size corrections play no role \\footnote{The fact that finite-size corrections are suppressed in $L$ is manifest. The suppression in the classical scaling dimension can for example be seen for so-called twist operators. Up to corrections in the classical scaling dimension, these operators are dual to Wilson loops, for which no finite-size effects occur \\cite{Korchemsky:1988si,Belitsky:2003ys}.}. \nThus, we can calculate it by solving the integrable model on a cylinder of circumference $1\/T$, a situation that is related to the one in the spectral problem via a double Wick rotation.\nIndeed, we find a direct relation between the continuum limit of the free energy of the spin chain associated with planar $\\mathcal{N}=4$ SYM theory and the Hagedorn temperature.\nUsing the integrability of the model, we derive thermodynamic Bethe ansatz (TBA) equations which determine the Hagedorn temperature at any value of the 't Hooft coupling. We present them in the form of a Y-system in \\eqref{eq: general Y system}--\\eqref{eq: theta n}.\nAs a first application, we solve them in the constant case as well as perturbatively at weak coupling, confirming the known tree-level and one-loop Hagedorn temperature. Moreover, we determine the previously unknown two-loop Hagedorn temperature:\n\\begin{equation}\n\\begin{aligned}\n T_{\\text{H}}&=\\frac{1}{2\\log(2+\\sqrt{3})}+\\frac{1}{\\log(2+\\sqrt{3})}g^2 \\\\\n &\\phaneq+\n \\left( -\\frac{86}{ \\sqrt{3}} + \\frac{24 \\log(12)}{\\log(2+\\sqrt{3})}\\right)\n g^4+\\mathcal{O}(g^6)\\,,\n \\end{aligned}\n\\end{equation}\nwhere $g^2=\\frac{\\lambda}{16\\pi^2}$.\n\n\\section{TBA equations for the Hagedorn temperature}\n\nIn the following, we relate the Hagedorn temperature to the spin-chain free energy and derive TBA equations for the latter. \n\n\\paragraph{The Hagedorn temperature from the free energy of the spin chain}\n\nIn the planar limit, the scaling dimensions of multi-trace operators are completely determined by the scaling dimensions of their single-trace factors. The partition function $\\mathcal{Z}(T)$ is then entirely determined by the single-trace partition function $Z(T)$.\nSplitting the dilatation operator into a classical and an anomalous part as $D=D_0+\\delta D$, \nwe can write \n\\begin{equation}\n Z(T)=\\sum_{m=2}^\\infty\\operatorname{e}^{-\\frac{m}{2}\\frac{1}{T}(1 + F_m(T))}\\, , \n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq: spin_chain_Z}\n F_m(T)=-T\\frac{2}{m}\\log \\left( \\tr_{\\text{spin-chain},D_0=\\frac{m}{2}}[\\operatorname{e}^{-\\delta D\/T}] \\right)    \n \\end{equation}\nis the spin-chain free energy per unit classical scaling dimension for fixed $D_0=\\frac{m}{2}$. \nThe multi-trace partition function $\\mathcal{Z}(T)$ is then given by \n\\begin{equation}\n\\label{eq: relation between partition function and free energy}\n \\mathcal{Z}(T)=\\exp\\sum_{n=1}^\\infty\\frac{1}{n}\\sum_{m=2}^\\infty(-1)^{m(n+1)}\\operatorname{e}^{-\\frac{m}{2T} (n + F_m(T\/n))}\\, , \n\\end{equation}\nwhere the alternating sign takes care of the correct statistics.\nThe Hagedorn singularity is the first singularity of $\\mathcal{Z}(T)$ encountered raising the temperature from zero.\nIt arises from the $n=1$ contribution to the sum over $n$, i.e.\\ from the infinite series\n\\begin{equation}\n\\label{n1contr}\n\\sum_{m=2}^\\infty \\operatorname{e}^{-\\frac{m}{2T} (1 + F_m(T))}\\, , \n\\end{equation}\nwhere each term in the series is finite as $F_m(T)$ only includes a finite number of states.\nWe can use Cauchy's root test to assess when this series diverges. To this end, we compute the $m$th root of the absolute value of the $m$th term and take the large $m$ limit, giving\n\\begin{equation}\nr = \\lim_{m\\rightarrow \\infty} \\operatorname{e}^{-\\frac{1}{2T}(1+F_m(T))} = \\operatorname{e}^{-\\frac{1}{2T} ( 1+ F(T))} \\, , \n\\end{equation}\nwhere\n\\begin{equation}\n F(T)= \\lim_{m\\rightarrow \\infty} F_m(T)\n\\end{equation}\nis the thermodynamic limit of the free energy.\nThe root test states that the series is convergent for $r<1$ and divergent for $r>1$. Thus, the Hagedorn temperature is  determined from $r=1$ or, equivalently, from\n\\begin{equation}\n\\label{eq: hagedorn temperature via F}\nF(T_{\\text{H}})=-1\\, . \n\\end{equation}\n\n\n\n\n\\paragraph{TBA equations}\n\nThe free energy $F$ of the spin chain can be calculated via the thermodynamic Bethe ansatz (TBA). The TBA equations for the Hagedorn temperature of $\\mathcal{N}=4$ SYM theory can be derived in analogy to the case of the spectral problem \\cite{Arutyunov:2009zu,Bombardelli:2009ns,Gromov:2009bc,Arutyunov:2009ur,Gromov:2009tv,Cavaglia:2010nm}.\nThe starting point are the all-loop asymptotic Bethe equations \\cite{Beisert:2004hm,Beisert:2006ez} for the \n$\\mathfrak{psu}(2,2|4)$ spin chain found in the spectral problem,\nwhich are written in terms of the length $L$ of the spin chain as well as the seven excitation numbers corresponding to the roots in the Dynkin diagram of $\\mathfrak{psu}(2,2|4)$. We then rewrite the Bethe equation so that the middle, momentum-carrying root is written in terms of $D_0$ instead of $L$, since it is $D_0$ that we keep fixed when calculating the free energy \\eqref{eq: spin_chain_Z}. We proceed by employing the string hypothesis, which  enables us to write the Bethe equations for many magnons. The next step is the continuum limit $D_0 \\rightarrow \\infty$, in which we can write the TBA equations in terms of the Y-functions defined from the densities of the strings. In particular, it allows us to write down the free energy. \nThe main difference compared to the TBA equations of the spectral problem is that we do not make a double Wick rotation i.e.\\ we consider the so-called direct theory and not the mirror theory. This means we use the Zhukovsky variable $x(u)$ with a short cut: \n\\begin{equation}\nx(u)=\\frac{u}{2}\\left(1+\\sqrt{1-\\frac{4g^2}{u^2}}\\right)\\, . \n\\end{equation}\nNote that TBA equations for the direct theory were also considered in \\cite{Cavaglia:2010nm,Arutynov:2014ota} but in different thermodynamic limits.\n\n\n\\paragraph{Y-system}\nThe TBA equations can be rephrased in terms of a Y-system consisting of the functions $\\mathcal{Y}_{a,s}$, where $(a,s)\\in M=\\{(a,s)\\in\\mathbb{N}_{\\geq0}\\times\\mathbb{N} \\,|\\,a=1 \\vee |s|\\leq 2 \\vee \\pm s=a=2\\}$.\nWith some exceptions, they satisfy the  equations\n\\begin{equation}\n\\label{eq: general Y system}\n \\log\\mathcal{Y}_{a,s}=\\log\\frac{(1+\\mathcal{Y}_{a,s-1})(1+\\mathcal{Y}_{a,s+1})}{(1+\\mathcal{Y}_{a-1,s}^{-1})(1+\\mathcal{Y}_{a+1,s}^{-1})}\\star s\\, , \n\\end{equation}\nwhere $\\star$ denotes the convolution with $s(u)=(2 \\cosh \\pi u)^{-1}$ on $\\mathbb{R}$ and the (inverse) Y-functions with shifted indices are assumed to be zero when the shifted indices are not in $M$.\nThe Y-functions are analytic in the strip with $|\\Im(u)|<\\frac{1}{2}|a-|s||$.\nFor the purpose of this letter, the chemical potentials are set to zero. Hence, the Y-system is symmetric, $\\mathcal{Y}_{a,s}=\\mathcal{Y}_{a,-s}$, with boundary conditions \n\\begin{equation}\n\\label{eq: Ybcs}\n\\lim_{a\\rightarrow \\infty} \\frac{\\mathcal{Y}_{a+1,s}}{\\mathcal{Y}_{a,s}} = 1 \\, ,  \\quad  \\lim_{n\\rightarrow \\infty} \\frac{\\mathcal{Y}_{1,n+1}}{\\mathcal{Y}_{1,n}} = 1 \\, , \n\\end{equation}\nfor $s=0,\\pm 1$.\nThe first of the aforementioned exceptions to the  equations \\eqref{eq: general Y system} then is \n\\begin{equation}\n\\label{eq: CY_10}\n \\log \\mathcal{Y}_{1,0}  = - \\rho\\hatstar s  + 2\\log (1+\\mathcal{Y}_{1,1})\\checkstar s-\\log(1+\\mathcal{Y}_{2,0}^{-1})\\star s\\,,\n\\end{equation}\nwhere we have defined $\\hatstar$ and $\\checkstar$ as the convolutions on $(-2g,2g)$ and $\\mathbb{R}\\setminus(-2g,2g)$, respectively.\nSimilarly, the convolution with $\\mathcal{Y}_{1,1}$ and $\\mathcal{Y}_{2,2}$ in \\eqref{eq: general Y system} for $(a,s)=(2,1),(1,2)$ is also understood to be $\\checkstar$.\nThe source term $\\rho(u)$ is defined as \n\\begin{equation}\n \\begin{aligned}\n\\rho &=  \\frac{\\epsilon_0}{T}    + 2 \\log (1+\\mathcal{Y}_{1,1}) (1+\\mathcal{Y}_{2,2}^{-1}) \\checkstar  H_0   \\\\ &\\phaneq  + 2 \\sum_{m=1}^\\infty \\log (1+\\mathcal{Y}_{m+1,1})\\star \n\\Big(H_m +H_{-m}\\Big)   \\\\&\\phaneq+ \\sum_{m=1}^\\infty \\log (1+\\mathcal{Y}_{m,0} ) \\star \\Sigma^{m} \\,,\n \\end{aligned}\n\\label{eq: rho}\n\\end{equation}\nwhere \n\\begin{equation}\n H_m(v,u) =\\frac{i}{2\\pi}\\partial_v\\log \\frac{x(u-i0)-\\frac{g^2}{x(v+\\frac{i}{2}m)}}{x(u+i0)-\\frac{g^2}{ x(v+\\frac{i}{2}m)}}\\,,\n\\end{equation}\n\\begin{equation}\n\\label{epsilon0}\n\\epsilon_0 (u) = \\begin{cases}\n                  0 &\\mbox{for }  | u | \\geq 2 g\\,,\\\\\n                  2\\sqrt{4g^2-u^2} & \\mbox{for }  |u| < 2g\\,,\n                 \\end{cases}\n\\end{equation}\nand the kernel\n\\begin{equation}\n\\begin{aligned}\n\\Sigma^{m} (v,u) =& \\frac{i}{2\\pi} \\partial_v  \\left( \\log \\frac{R^2(x(v+ \\frac{im}{2}),x(u+i0))}{R^2(x(v+ \\frac{im}{2}),x(u-i0))} \\right. \\\\ &+ \\left.  \\log \\frac{R^2(x(v- \\frac{im}{2}),x(u-i0))}{R^2(x(v- \\frac{im}{2}),x(u+i0))} \\right) \n\\end{aligned}\n\\end{equation}\nis given in terms of the dressing factor \\cite{Beisert:2006ez}\n\\begin{equation}\n\\begin{aligned}\n\\sigma^2 (u,v) =&  \\frac{R^2(x^+(u),x^+(v))R^2(x^-(u),x^-(v))}{R^2(x^+(u),x^-(v))R^2(x^-(u),x^+(v))}      \\,,\n\\end{aligned}\n\\end{equation}\nwith $x^\\pm(u) = x(u\\pm \\frac{i}{2})$.\nWhen applied to a function of two arguments such as $H_m(v,u)$, $\\star$, $\\hatstar$ and $\\checkstar$ are moreover understood as integrals over the respective intervals.\nThe other exceptions to the  equations \\eqref{eq: general Y system}\nare the non-local equations\n\\begin{equation}\n\\label{eq: Y11_times_Y22}\n\\log \\mathcal{Y}_{1,  1}\\mathcal{Y}_{2,2} (u) = \\sum_{m=1}^\\infty \\log (1+\\mathcal{Y}_{m,0} (v) ) \\star  \\Theta_{m} (v,u) \n\\end{equation}\nwith\n\\begin{equation}\n\\Theta_{m} (v,u) =\\frac{i}{2\\pi}\\partial_v\\log \\frac{x(u)-\\frac{g^2}{ x(v-\\frac{i m}{2})}}{x(u)-\\frac{g^2}{ x(v+\\frac{i m}{2})}}\\frac{x(u)-x(v+\\frac{i m}{2})}{x(u)-x\\left(v-\\frac{i m}{2}\\right)}\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq: Y11_divide_Y22}\n\\log \\frac{\\mathcal{Y}_{2,  2}}{\\mathcal{Y}_{1,1}}  = \\sum_{m=1}^\\infty a_m \\star \\log \\frac{ (1+ \\mathcal{Y}_{m+1,  1} )^2}{(   1+ \\mathcal{Y}_{1,m+1}^{-1} )^2(1+\\mathcal{Y}_{m,0}  )} \n\\end{equation}\nwith $a_n(u)=\\frac{n}{2\\pi(u^2+\\frac{n^2}{4})}$.\n\n\nThe free energy per unit scaling dimension is given by \n\\begin{equation}\n\\label{eq: free energy}\nF(T)= - T \\sum_{n=1}^\\infty  \\int_{-\\infty}^\\infty \\operatorname{d}\\! u \\, \\theta_n (u) \\log ( 1 + \\mathcal{Y}_{n,0} (u) )\\, , \n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq: theta n}\n\\theta_n(u) = \\frac{i}{2\\pi} \\partial_u \\log \\frac{x(u+\\frac{in}{2})}{x(u-\\frac{in}{2})}\\, . \n\\end{equation}\nThus, the TBA equations \\eqref{eq: general Y system}--\\eqref{eq: theta n} determine the Hagedorn temperature at any value of the 't Hooft coupling via \\eqref{eq: hagedorn temperature via F}.\n\n\n\\section{Solving the TBA equations}\n\nLet us now solve the TBA equations in the form of the Y-system.\n\n\\paragraph{Constant solution via T-system}\n\nAt large spectral parameter $u$, the Y-system approaches a constant value. This means we can find a constant Y-system that solves \\eqref{eq: general Y system} for all $(a,s)\\in M\\setminus \\{(1,1),(2,2)\\}$ as well as \\eqref{eq: Y11_divide_Y22}. \nNote that we cannot impose  \\eqref{eq: Y11_times_Y22} as it relates the behavior at finite and large $u$. \nThus, we find a one-parameter family of solutions with parameter $z$.\nThis solution is most easily expressed in terms of a T-system consisting of the functions $T_{a,s}$ with $(a,s)\\in \\hat{M}=\\{(a,s)\\in\\mathbb{Z}_{\\geq0}\\times\\mathbb{Z} \\,|\\,\\min(a,|s|)\\leq 2\\}$ and $T_{a,s}=0$ for $(a,s)\\notin \\hat{M}$.\nThe Y-functions are expressed in terms of the T-functions as\n\\begin{equation}\n \\mathcal{Y}_{a,s}=\\frac{T_{a,s+1}T_{a,s-1}}{T_{a+1,s}T_{a-1,s}}\\,.\n\\end{equation}\nIn the constant case, the  equations \\eqref{eq: general Y system} imply the following T-system (Hirota) equations for all $(a,s)\\in \\hat{M}$:\n\\begin{equation}\n\\label{eq: T-system equations}\n T_{a,s}^2 = T_{a+1,s} T_{a-1,s} + T_{a,s+1} T_{a,s-1}\\,.\n\\end{equation}\nThe latter are solved by\n\\begin{equation}\n\\label{consTbfgen1}\n\\begin{aligned}\nT_{a,0} &=  \\left(\\frac{1-\\noty}{1+\\noty}\\right)^{2a} \\frac{a + 2\\noty}{12 \\noty^4}  \\big( a^3 + 6 \\noty a^2 \\\\\n&\\hphantom{=  \\left(\\frac{1-\\noty}{1+\\noty}\\right)^{2a} \\frac{a + 2\\noty}{12 \\noty^4}  \\big(}+ ( 12\\noty^2-1) a + 6\\noty^3 \\big) \\,,\n\\\\\nT_{a,\\pm 1} &=  (-1)^{a} \\left(\\frac{1-\\noty}{1+\\noty}\\right)^{2a} \\frac{a+3\\noty}{6\\noty^4}  (a^2+3a\\noty + 3\\noty^2-1) \\,,\n\\\\\nT_{a,\\pm 2} &= \\frac{1}{\\noty^4}  \\left(\\frac{1-\\noty}{1+\\noty}\\right)^{2a}\\,,\n\\end{aligned}\n\\end{equation}\nfor $a \\geq |s|$, and\n\\begin{equation}\n\\label{consTbfgen2}\n\\begin{aligned}\nT_{0,s}&=1\\,,\\\\\nT_{1,s} &= \\frac{(-1)^s}{\\noty^2} \\left[ |s| + \\frac{1-3\\noty^2}{2\\noty} \\right] \\left(\\frac{1-\\noty}{1+\\noty}\\right)^{|s|}  \\,,\\\\\nT_{2,s} &= \\frac{1}{\\noty^4}  \\left(\\frac{1-\\noty}{1+\\noty}\\right)^{2|s|}\\,,\n\\end{aligned}\n\\end{equation}\nfor $|s| \\geq a$. \nThis solution is a special case of the most general, $\\mathfrak{psu}(2,2|4)$ character solution of \\eqref{eq: T-system equations} in \\cite{Gromov:2010vb}.\n\n\\paragraph{Solution at zero coupling}\n\nIn the limit of zero coupling, $g^2=0$, the source term $\\rho(u)$ in \\eqref{eq: CY_10} vanishes \\footnote{In particular, $\\text{constant}\\star H_m=\\text{constant}\\star\\Theta_m=0$.}, such that the functions $\\mathcal{Y}_{a,s}$ are constant for all $u$. Hence, the non-local equation \\eqref{eq: Y11_times_Y22} implies $\\mathcal{Y}_{1,1}\\mathcal{Y}_{2,2}=T_{1,0}=1$. We can use this to determine the parameter $z$ in the constant solution for the T-system above \nand thereby find the Y-system at zero coupling. Imposing $T_{1,0}=1$ is equivalent to $z = \\pm 1\/\\sqrt{3}$. The negative solution has to be discarded as it leads to a negative Hagedorn temperature. Thus, we conclude that to zeroth order $z = 1\/\\sqrt{3}$. Using \\eqref{eq: hagedorn temperature via F} and \\eqref{eq: free energy}, we find the zeroth-order Hagedorn temperature \n\\begin{equation}\n T_{\\text{H}}^{(0)}=\\frac{1}{2\\log(2+\\sqrt{3})}\\,,\n\\end{equation}\nwhich is in perfect agreement with \\cite{Sundborg:1999ue}.\n\n\\paragraph{Perturbative solution}\n\nWe can also solve the TBA equations in a perturbative expansion at weak coupling, expanding the Y-functions as\n\\begin{equation}\n\\mathcal{Y}_{a,s} (u) = \\mathcal{Y}_{a,s}^{(0)} \\left( 1+ \\sum_{\\ell=1}^\\infty g^{2\\ell} y_{a,s}^{(\\ell)} (u)\\right) \\,.\n\\end{equation}\n\nAt one-loop order, the solution takes the form\n\\begin{equation}\n\\label{eq: one-loop form}\ny_{a,s}^{(1)}(u) =  \\tilde{y}_{a,s}^{(1)}+  \\sum_{k=0}^\\infty c^{(1)}_{a,s,k} a_{2k+a+s}(u) \\,,\n\\end{equation}\nwhere $\\tilde{y}_{a,s}^{(1)}$ as well as $c^{(1)}_{a,s,k}$ are constants.\nThis follows from the expansions\n\\begin{equation}\n\\label{eq: expansions of kernels}\n \\begin{aligned}\n   \\epsilon_0\\hatstar s(u)&=4\\pi g^2 s(u) + 2\\pi g^4 s''(u)+ \\mathcal{O}(g^6)\\,,\\\\\n s(u)&=\\sum_{m=0}^\\infty (-1)^{m}a_{1+2m}(u)\\,, \\\\\n   \\theta_n(u)&=a_n(u)+g^2 a_n''(u)+\\mathcal{O}(g^4)\\,,\\\\\n \\Theta_m(v,u)&=a_m(u-v)-a_m(v)\\\\&\\phaneq+g^2\\left(\\frac{2}{u}a_m'(v)-a_m''(v)\\right)+\\mathcal{O}(g^4)\n \\end{aligned}\n\\end{equation}\nin combination with the convolution identity $a_n\\star a_m=a_{n+m}$ and the structure of the TBA equations. Inserting \\eqref{eq: one-loop form} into the expansion of the TBA equations, we can solve for the coefficients $c^{(1)}_{a,s,k}$.\nThe remaining one-loop parameter in the constant solution can be fixed from \n$\\left(\\mathcal{Y}_{1,1}\\mathcal{Y}_{2,2}\\right)^{(1)} (0)=0$, which follows from \\eqref{eq: Y11_times_Y22} and the last expansion in \\eqref{eq: expansions of kernels}.\nWe find for the one-loop Hagedorn temperature\n\\begin{equation}\n T_{\\text{H}}^{(1)}=\\frac{1}{\\log(2+\\sqrt{3})}\\,,\n\\end{equation}\nwhich perfectly agrees with the result of \\cite{Spradlin:2004pp}.\n\nAt two-loop order, $\\rho \\hatstar s$ in \\eqref{eq: CY_10} receives contributions from the one-loop solution\n$y_{a,s}^{(1)}(u)$ from the second and third term in \\eqref{eq: rho}. They can be calculated using \n\\begin{equation}\n (a_n\\star H_{m} \\hatstar s)(u) = g^2 \\frac{4}{(n+|m|)^2} s(u)  + \\mathcal{O}(g^4)\\,.\n\\end{equation}\nNote that the dressing kernel in the fourth term of \\eqref{eq: rho} vanishes at this loop order. The two-loop solution takes the form\n\\begin{equation}\n\\label{eq: two-loop form}\n\\begin{aligned}\ny_{a,s}^{(2)}(u) &=  \\tilde{y}_{a,s}^{(2)} + \\sum_{k=0}^\\infty c^{(2)}_{a,s,k,1} a_{2k+a+s}(u)\\\\ &\\phaneq+  \\sum_{k=0}^\\infty c^{(2)}_{a,s,k,2} a^2_{2k+a+s}(u) \\\\ &\\phaneq+  \\sum_{k=0}^\\infty c^{(2)}_{a,s,k,3} a^3_{2k+a+s}(u)   \\,,\n\\end{aligned}\n\\end{equation}\nas follows from simple reasoning paralleling the one at one-loop order.\nSolving for the coefficients $c^{(2)}_{a,s,k,1}$, $c^{(2)}_{a,s,k,2}$ and $c^{(2)}_{a,s,k,3}$ and fixing the two-loop parameter in the constant solution via \\eqref{eq: Y11_times_Y22}, we find the previously unknown two-loop Hagedorn temperature\n\\begin{equation}\n\\begin{aligned}\n T_{\\text{H}}^{(2)}= \n  -\\frac{86}{\\sqrt{3}} +  \\frac{24\\log(12)}{\\log(2+\\sqrt{3})}\\,.\n\\end{aligned}\n\\end{equation}\n\n\n\\paragraph{Solution at finite coupling}\n\nAt finite coupling, the infinite set of non-linear integral  equations \\eqref{eq: general Y system}--\\eqref{eq: theta n} can be solved numerically \nby iterating the equations and truncating to $a,s\\leq n_{\\max}$.\nThe convolutions are calculated for a finite number of sampling points from which the functions are recovered by interpolation and extrapolation at small and large $u$, respectively.\nWe have implemented this procedure in Mathematica following the strategy of \\cite{Bajnok:2013wsa}, where also \n$T_{\\text{H}}$ has to be iterated.\nWe will report on the resulting solution at finite coupling in our future publication \\cite{HW}.\n\n\\section{Outlook}\n\nIn this letter, we have derived integrability-based TBA equations \\eqref{eq: general Y system}--\\eqref{eq: theta n} that determine the Hagedorn temperature of planar $\\mathcal{N}=4$ SYM theory at any value of the 't Hooft coupling.\nAs an application, we have solved these equations perturbatively up to two-loop order.\nOur TBA equation can also be solved numerically at finite coupling, as was briefly discussed here but will be detailed on in a future publication \\cite{HW}.\nThus, they open up the door for an exact interpolation from weak to strong coupling, which, with the exception of \\cite{Harmark:2006ta},  \nwould be the first time for the case of thermal physics.\nPotentially, this could allow us to develop a better understanding of the phase structure of gauge theories and their dual gravitational theories in general.\n\nFor the spectral problem, the TBA equations have been recast into the form of the quantum spectral curve \\cite{Gromov:2013pga}, which allows to generate precision data at weak coupling \\cite{Marboe:2014gma} as well as at finite coupling \\cite{Gromov:2015wca}.\nWe will report on a similar reformulation of our equations in a future publication \\cite{HW}.\nMoreover, one can study the case of non-zero chemical potentials. We have generalized our method to this case as well, and we have solved the zeroth-order TBA equations for the case with chemical potentials turned on but corresponding still to a symmetric Y-system. We will report on this in a future publication as well \\cite{HW}.\n\nIn this letter, we have used the fact \\eqref{eq: hagedorn temperature via F} that the spin-chain free energy determines the Hagedorn temperature $T_{\\text{H}}$ at which the partition function diverges. The spin-chain free energy should however also determine the partition function in the vicinity of $T_{\\text{H}}$, which should allow to extract e.g.\\ critical exponents.\n\nThe partition function and Hagedorn temperature have also been studied in integrable deformations of $\\mathcal{N}=4$ SYM theory up to one-loop order \\cite{Fokken:2014moa}, where it was found that although $\\mathcal{Z}(T)$ is different, $T_{\\text{H}}$ is unchanged.\nIt would be interesting to see whether this statement continues to hold at higher loop orders. \nSimilarly, one might apply our framework to the three-dimensional $\\mathcal{N}=6$ superconformal Chern-Simons theory, which is known to be integrable as well.\n\n\n\\begin{acknowledgments}\n\\paragraph{Acknowledgements.}\nIt is a pleasure to thank Marta Orselli for collaboration in an earlier stage of the project.\nWe thank\nZoltan Bajnok,\nJohannes Br\\\"{o}del,\nSimon Caron-Huot,\nMarius de Leeuw,\nSergey Frolov,\nNikolay Gromov,\nSebastien Leurent,\nFedor Levkovich-Maslyuk,\nChristian Marboe,\nDavid McGady,\nRyo Suzuki,\nDmytro Volin\nand Konstantin Zarembo\nfor very useful discussions and \nRyo Suzuki for sharing his Mathematica implementation of the TBA equations used in \\cite{Bajnok:2013wsa}. \nT.H.\\ acknowledges support from FNU grant number DFF-6108-00340 and the Marie-Curie-CIG grant number 618284.\nM.W.\\ was supported in part by FNU through grants number DFF-4002-00037 and by the ERC advance grant 291092.\nM.W.\\ further acknowledges the kind hospitality of NORDITA during the program ``Holography and Dualities,'' where parts of this work were carried out.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe non-empty set $G$ together with an $n$-ary operation $f:G^n\\to\nG$ is called an {\\it $n$-ary groupoid} (or an {\\it $n$-ary\noperative}  -- in the Gluskin terminology, cf. \\cite{glu65}) and\nis denoted by $(G;f)$. We will assume that $n>2$.\n\nAccording to the general convention similar to that introduced in\nthe theory of $n$-ary systems by G. \\v{C}upona (cf. \\cite{Cup})\nthe sequence of elements $x_i,x_{i+1},\\ldots,x_j$ is denoted by\n$x_i^j$. In the case $j<i$ it is the empty symbol. If\n$x_{i+1}=x_{i+2}=\\ldots=x_{i+t}=x$, then instead of\n$x_{i+1}^{i+t}$ we write $\\stackrel{(t)}{x}$. In this convention\n$f(x_1,\\ldots,x_n)= f(x_1^n)$ and\n \\[\n f(x_1,\\ldots,x_i,\\underbrace{x,\\ldots,x}_{t},x_{i+t+1},\\ldots,x_n)=\n f(x_1^i,\\stackrel{(t)}{x},x_{i+t+1}^n) .\n \\]\nIf $\\,m=k(n-1)+1$, then the m-ary operation $\\,g\\,$ of the form\n \\[\n g(x_{1}^{k(n-1)+1}):=\\underbrace{f(f(...,f(f}_{k}\n(x_{1}^{n}),x_{n+1}^{2n-1}),...),x_{(k-1)(n-1)+2}^{k(n-1)+1})\n\\]\nis denoted by $\\,f_{(k)}$ and is called the {\\it simple iteration}\nof the operation $f$ (cf. \\cite{Mar'64}) or an $m$-ary operation\n{\\it derived} from $f$. In certain situations, when the arity of\n$\\,g\\,$ does not play a crucial role or when it will differ\ndepending on additional assumptions, we write $\\,f_{(.)}\\,$ to\nmean $\\,f_{(k)}\\,$ for some $\\,k=1,2,...$.\n\nAn $n$-ary groupoid $(G;f)$ is called {\\it $(i,j)$-associative} if\n \\begin{equation}\nf(x_1^{i-1},f(x_i^{n+i-1}),x_{n+i}^{2n-1})=\nf(x_1^{j-1},f(x_j^{n+j-1}),x_{n+j}^{2n-1})\\label{assolaw}\n \\end{equation}\nholds for all $x_1,\\ldots,x_{2n-1}\\in G$. If this identity holds\nfor all $1\\leqslant i<j\\leqslant n$, then we say that the\noperation $f$ is {\\it associative} and $(G;f)$ is called an {\\it\n$n$-ary semigroup} (or, in the Gluskin's terminology, an {\\it\n$n$-ary associative}). It is clear that an $n$-ary groupoid is\nassociative iff it is $(1,j)$-associative for all $j=2,\\ldots,n$.\nIn the binary case (i.e. for $n=2$) it is a usual semigroup.\n\nIf for all $x_0,x_1,\\ldots,x_n\\in G$ and fixed\n$i\\in\\{1,\\ldots,n\\}$ there exists an element $z\\in G$ such that\n\\begin{equation}                                                          \\label{solv}\nf(x_1^{i-1},z,x_{i+1}^n)=x_0 ,\n\\end{equation}\nthen we say that this equation is {\\it $i$-solvable} or {\\it\nsolvable at the place $i$}. If this solution is unique, then we\nsay that (\\ref{solv}) is {\\it uniquely $i$-solvable}.\n\nAn $n$-ary groupoid $(G;f)$ uniquely solvable for all\n$i=1,\\ldots,n$ is called an {\\it $n$-ary quasigroup}. An\nassociative $n$-ary quasigroup is called an {\\it $n$-ary group}.\nFor an $n$-ary quasigroup with the non-empty center to be an\n$n$-ary group it is sufficient to postulate the\n$(i,j)$-associativity for some fixed $1\\leqslant i<j\\leqslant n$\n(cf. \\cite{Du'86}). It is clear that for $n=2$ we obtain a usual\ngroup.\n\nNote by the way that in many papers $n$-ary semigroups ($n$-ary\ngroups) are called $n$-semigroups ($n$-groups, respectively).\nMoreover, in many papers, where the arity of the basic operation\ndoes not play a crucial role, we can find the term a {\\it polyadic\nsemigroup} ({\\it polyadic group}) (cf. \\cite{Busse}, \\cite{post},\n\\cite{Sokh}).\n\nNow such and similar $n$-ary systems have many applications in\ndifferent branches. For example, in the theory of automata\n\\cite{Busse} $n$-ary semigroups and $n$-ary groups are used, some\n$n$-ary groupoids are applied in the theory of quantum groups\n\\cite{Nik}. Different applications of ternary structures in\nphysics are described by R. Kerner in \\cite{Ker}. In physics there\nare used also such structures as $n$-ary Filippov algebras (see\n\\cite{Poj}) and $n$-Lie algebras (see \\cite{Vai}). Some $n$-ary\nstructures induced by hypercubes have application in\nerror-correcting and error-detecting coding theory, cryptology, as\nwell as in the theory of $(t,m,s)$-nets (see for example\n\\cite{LaMu} and \\cite{LaMuWhi}).\n\nThe idea of investigations of such groups seems to be going back\nto E. Kasner's lecture \\cite{kasner} at the fifty-third annual\nmeeting of the American Association for the Advancement of Science\nin 1904. But the first paper concerning the theory of $n$-ary\ngroups was written (under inspiration of Emmy Noether) by W.\nD\\\"ornte in 1928 (see \\cite{dor}). In this paper D\\\"ornte observed\nthat any $n$-ary groupoid $(G;f)$ of the form\n $\\, f(x_1^n)=x_1\\circ x_2\\circ\\ldots\\circ x_n$, where $(G;\\circ\n )$ is a group, is an $n$-ary group but for every $n>2$ there are\n$n$-ary groups which are not of this form. $n$-ary groups of the\nfirst form are called {\\it reducible} to the group $(G;\\circ )$ or\n{\\it derived} from the group $(G;\\circ )$, the second one are\ncalled {\\it irreducible}. Moreover, in some $n$-ary groups there\nexists an element $e$ (called an {\\it $n$-ary identity} or {\\it\nneutral element}) such that\n \\begin{equation}                                            \\label{n-id}\n f(\\stackrel{(i-1)}{e},x,\\stackrel{(n-i)}{e})=x\n \\end{equation}\nholds for all $x\\in G$ and for all $i=1,\\ldots,n$. It is\ninteresting that $n$-ary groups containing a neutral element are\nreducible (cf. \\cite{dor}). Irreducible $n$-ary groups do not\ncontain such elements. On the other hand, there are $n$-ary groups\nwith two, three and more neutral elements. The set\n$\\mathbb{Z}_{n-1}=\\{0,1,\\ldots,n-2\\}$ with the operation\n$f(x_1^{n})=(x_1+x_2+\\ldots +x_{n})({\\rm mod}\\,(n-1))$ is a simple\nexample of an $n$-ary group in which every element is neutral. All\n$n$-ary groups with this property are derived from the commutative\ngroup of the exponent $k|(n-1)$.\n\nIt is worthwhile to note that in the definition of an $n$-ary\ngroup, under the assumption of the associativity of $f$, it\nsuffices only to postulate the existence of a solution of\n(\\ref{solv}) at the places $i=1$ and $i=n$ or at one place $i$\nother than $1$ and $n$. Then one can prove the uniqueness of the\nsolution of (\\ref{solv}) for all $i=1,\\ldots,n$ (cf. \\cite{post},\np. $213^{17}$).\n\nThe above definition of $n$-ary groups is a generalization of the\nWeber's and Huntington formulation of axioms of a group as a\nsemigroup in which the equations $xa=b$, $ya=b$ have solutions.\nMany authors used the notion of $n$-ary groups as a generalization\nof Pierpont's definition of groups as a semigroup with neutral and\ninverse elements. Unfortunately, in this case we obtain only\n$n$-ary groups derived from groups.\n\nE.I. Sokolov proved in \\cite{sok} that in the case of $n$-ary\nquasigroups (i.e. in the case of the existence of a unique\nsolution of (\\ref{solv}) at any place $i=1,\\ldots,n$) it is\nsufficient to postulate the $(j,j+1)$-associativity for some fixed\n$j=1,\\ldots,n-1$.\n\nUsing the same method as Sokolov we can prove the following\nproposition (for details see \\cite{DGG}):\n\n\\begin{proposition}\\label{DGG1}\nAn $n$-ary groupoid $(G;f)$ is an $n$-ary group if and only if\n$($at least$)$ one of the following conditions is satisfied:\n\\begin{enumerate}\n\\item [$(a)$] the $(1,2)$-associative law holds and the\nequation $(\\ref{solv})$ is solvable for $\\,i=n\\,$ and uniquely\nsolvable for $\\,i=1$,\n\\item [$(b)$] the $(n-1,n)$-associative law holds and the\nequation $(\\ref{solv})$ is solvable for $\\,i=1\\,$ and uniquely\nsolvable for $\\,i=n$,\n\\item [$(c)$] the $(i,i+1)$-associative law holds for some\n$\\,i\\in \\{2,...,n-2\\}\\,$ and the equation $(\\ref{solv})$ is\nuniquely solvable for $\\,i\\,$ and some $j>i$.\n\\end{enumerate}\n\\end{proposition}\n\n\\medskip\n\nIn \\cite{DD80} (see also \\cite{cel77}) the following\ncharacterization of $n$-ary groups is given:\n\n\\begin{proposition}\nAn $n$-ary semigroup $(G;f)$ is an $n$-ary group if and only if\nfor some $k\\in\\{1,2,\\ldots,n-2\\}$ and all $a_1^k\\in G$ there are\nelements $x_{k+1}^{n-1},\\,y_{k+1}^{n-1}\\in G\\,$ such that\n\\begin{equation}\nf(a_1^k,x_{k+1}^{n-1},b)=f(b,y_{k+1}^{n-1},a_1^k)=b\\label{DG-80}\n\\end{equation}\nfor all $\\,b\\in G$.\n\\end{proposition}\n\n\n\\medskip\n\n\\begin{proposition}\nAn $n$-ary semigroup $(G;f)$ is an $n$-ary group if and only if\nfor some $i,j\\in\\{1,2,\\ldots,n-1\\}$ and all $a,b\\in G$ there are\n$x,y\\in G\\,$ such that\n\\begin{equation}\nf(x,\\stackrel{(i-1)}{b},\\stackrel{(n-i)}{a})=f(\\stackrel{(n-j)}{a},\\stackrel{(j-1)}{b},y)=b.\n\\label{gal-r1}\n\\end{equation}\n\\end{proposition}\n\n\n\\medskip\nPutting in the above proposition $i=j=1$ we obtain the following\nmain result of \\cite{Tyu85}.\n\n\\begin{corollary}\n{\\em An $n$-ary semigroup $(G;f)$ is an $n$-ary group if and only\nif for all $a,b\\in G$ there are $x,y\\in G\\,$ such that}\n\\[\nf(x,\\stackrel{(n-1)}{a})=f(\\stackrel{(n-1)}{a},y)=b.\n\\]\n\\end{corollary}\n\n\\bigskip\n\nFrom the definition of an $n$-ary group $(G;f)$ we can directly\nsee that for every $x\\in G$ there exists only one $z\\in G$\nsatisfying the equation\n \\begin{equation}                                               \\label{skew}\nf(\\stackrel{(n-1)}{x},z)=x .\n \\end{equation}\nThis element is called {\\it skew} to $x$ and is denoted by\n$\\overline{x}$. In a ternary group ($n=3$) derived from the binary\ngroup $(G;\\circ)$ the skew element coincides with the inverse\nelement in $(G;\\circ)$. Thus, in some sense, the skew element is a\ngeneralization of the inverse element in binary groups. This\nsuggests that for $n\\geqslant 3$ any $n$-ary group $(G;f)$ can be\nconsidered as an algebra $(G;f,\\bar{\\,}\\;)$ with two operations:\none $n$-ary $\\,f:G^n\\to G$ and one unary $\\;\\bar{\\,} :\nx\\to\\overline{x}$. D\\\"ornte proved (see \\cite{dor}) that in\nternary groups for all $x\\in G$ we have\n$\\overline{\\overline{x}}=x$, but for $n>3$ this is not true. For\n$n>3$ there are $n$-ary groups in which one fixed element is skew\nto all elements (cf. \\cite{D90}) and $n$-ary groups in which any\nelement is skew to itself. Then, in the second case, of course the\n$n$-ary group operation $f$ is idempotent. An $n$-ary group in\nwhich $f(\\stackrel{(n)}{x})=x$ for every $x\\in G$ is called {\\it\nidempotent}.\n\nNevertheless, the concept of skew elements plays a crucial role in\nthe theory of $n$-ary groups. Namely, as D\\\"ornte proved, the\nfollowing theorem is true.\n\n\\begin{theorem}\\label{dor-th}\n In any $n$-ary group $(G;f)$ the following identities:\n \\begin{eqnarray}\n  f(\\stackrel{(i-2)}{x},\\overline{x},\\stackrel{(n-i)}{x},y)=y, \\label{dor-r}\\\\\n  f(y,\\stackrel{(n-j)}{x},\\overline{x},\\stackrel{(j-2)}{x})=y, \\label{dor-l}\\\\\n  f(\\stackrel{(k-1)}{x},\\overline{x},\\stackrel{(n-k)}{x})=x\n  \\label{skew2}\n  \\end{eqnarray}\nhold for all $\\,x,y\\in G$, $\\,2\\leqslant i,j\\leqslant n\\,$ and\n$\\,1\\leqslant k\\leqslant n$.\n\\end{theorem}\n\n\n\\medskip\n\nThe first two identities, called now {\\it D\\\"ornte's identities},\nare used by many authors to describe the class of $n$-ary groups.\nFor example, in 1967 B. Gleichgewicht and K. G{\\l}azek proved in\n\\cite{GG67} (see also \\cite{Sio67}) that for fixed $n\\geqslant 3$\nthe class of all $n$-ary groups, considered as algebras of type\n$(n,1)$, forms a Mal'cev variety and found the system of\nidentities defining this variety. This means that all congruences\nof a given $n$-ary group commute and that the lattice of all\ncongruences of a fixed $n$-ary group is modular. But, as was\nobserved many years later, from the theorem on page 448 in\nGluskin's paper \\cite{glu65} it follows that the system of\nidentities given by B. Gleichgewicht and K. G\\l azek is not\nindependent. For similar axiom considerations, see also\n\\cite{cel77}, \\cite{rus79} and \\cite{rus81} (for other systems of\naxioms, see, e.g., \\cite{monk2}). The first independent system of\nidentities defining this variety was given in our paper\n\\cite{DGG}. Now we give the minimal system of such identities.\nThis is the main result of \\cite{Rem}.\n\n\\begin{theorem}\\label{DGG}\nThe class of $n$-ary groups coincides with the variety of $n$-ary\ngroupoids $(G;f,\\bar{}\\;)$ with a unary operation\n$\\,\\bar{}:x\\to\\overline{x}$ satisfying for some fixed\n$i,j\\in\\{2,\\ldots,n\\}$ the D\\\"ornte identities $(\\ref{dor-r})$,\n$(\\ref{dor-l})$ and the identity\n\\[\n f(f(x_1^{n}),x_{n+1}^{2n-1})=f(x_1,f(x_2^{n+1}),x_{n+2}^{2n-1}).\n \\]\n\\end{theorem}\n\n\\medskip\n\nTheorem \\ref{DGG} gives the minimal system of identities defining\n$n$-ary groups. In fact, for $n>3$ the set $Z$ of all integers\nwith the operation $f(x_1^n)=x_{n-1}+x_n$ is an example of a\n$(1,2)$-associative $n$-ary groupoid in which (\\ref{dor-r}) holds\nfor $\\overline{x}=0$ but (\\ref{dor-l}) is not satisfied.\nSimilarly, $(Z;f)$ with $f(x_1^n)=x_1$ is an example of a\n$(1,2)$-associative $n$-ary groupoid satisfying (\\ref{dor-l}) but\nnot (\\ref{dor-r}). It is clear that the $(1,2)$-associativity\ncannot be deleted.\n\nNote by the way that in some papers there are investigated\nso-called {\\it infinitary} semigroups and quasigroups, i.e.\ngroupoids $(G;f)$, where the number of variables in the operation\n$f:G^{\\infty}\\to G$ is infinite, but countable. Infinitary\nsemigroups are the infinitary groupoids $(G;f)$, where for all\nnatural $\\,i, j\\,$ the operation $f$ satisfies the identity\n \\[\nf(x_1^{i-1},f(x_i^{\\infty}),y_1^{\\infty})=\nf(x_1^{j-1},f(x_j^{\\infty}),y_1^{\\infty}).\n \\]\nInfinitary quasigroups are infinitary groupoids $(G;f)$ in which\nthe equation $\\,f(x_1^{k-1},z_k,x_{k+1}^{\\infty})=x_0\\,$ has a\nunique solution $z_k$ at any place $k$.\n\nFrom the general results obtained in \\cite{belz} and \\cite{MTC}\none can deduce that infinitary groups have only one element. Below\nwe present a simple proof of this fact.\n\nIf $(G;f)$ is an infinitary group, then, according to the\ndefinition, for any $y,z\\in G$ and $u=f(\\stackrel{(\\infty)}{y})$\nthere exists $x\\in G$ such that\n$z=f(u,y,x,\\stackrel{(\\infty)}{y})$. Thus\n\\[\\arraycolsep.5mm\n \\begin{array}{rl}\nf(z,\\stackrel{(\\infty)}{y})=&f(f(u,y,x,\\stackrel{(\\infty)}{y}),\\stackrel{(\\infty)}{y})=\n f(u,y,f(x,\\stackrel{(\\infty)}{y}),\\stackrel{(\\infty)}{y})\\\\[6pt]\n=&f(f(\\stackrel{(\\infty)}{y}),y,f(x,\\stackrel{(\\infty)}{y}),\\stackrel{(\\infty)}{y})\n=f(y,f(\\stackrel{(\\infty)}{y}),y,f(x,\\stackrel{(\\infty)}{y}),\\stackrel{(\\infty)}{y})\\\\[6pt]\n=&f(y,u,y,f(x,\\stackrel{(\\infty)}{y}),\\stackrel{(\\infty)}{y})\n=f(y,f(u,y,x,\\stackrel{(\\infty)}{y}),\\stackrel{(\\infty)}{y})=f(y,z,\\stackrel{(\\infty)}{y}),\n \\end{array}\n \\]\ni.e. for all $y,z\\in G\\,$ we have\n\\[\nf(z,\\stackrel{(\\infty)}{y})=f(y,z,\\stackrel{(\\infty)}{y}).\n\\]\nUsing this identity and the fact that for all $\\,x,y\\in G$ there\nexists $z\\in G$ such that $\\,x=f(z,\\stackrel{(\\infty)}{y}),\\,$ we\nobtain\n\\[\\arraycolsep.5mm\n \\begin{array}{rl}\nf(\\stackrel{(\\infty)}{x})=&f(x,f(z,\\stackrel{(\\infty)}{y}),\\stackrel{(\\infty)}{x})=\n f(x,f(y,z,\\stackrel{(\\infty)}{y}),\\stackrel{(\\infty)}{x})\\\\[6pt]\n=&f(x,y,f(z,\\stackrel{(\\infty)}{y}),\\stackrel{(\\infty)}{x})\n=f(x,y,\\stackrel{(\\infty)}{x}),\n \\end{array}\n \\]\nwhich together with the existence of only one solution at the\nsecond place implies $x=y$. Hence $G$ has only one element.\n\n\\medskip\n\nAccording to Theorem~\\ref{DGG}, the class of all $n$-groups (for\n$n>2$) can be considered as a variety of algebras\n$(G;f,\\bar{\\,}\\;)$ with one $n$-ary operation $f$ and one unary\n$\\,\\bar{}:x\\to\\overline{x}$. The class of $n$-ary groups can be\nalso considered as a variety of algebras of different types (cf.\n\\cite{filomat} and \\cite{jus'03}).\n\nTheorem~\\ref{DGG} is valid for $n>2$, but, as it was observed in\n\\cite{Rem}, this theorem can be extended to the case $n=2$.\nNamely, let \\ $\\hat{\\,}:x\\to\\hat{x}$ be a unary operation, where\n$\\hat{x}$ is defined as a solution of the equation\n$f_{(2)}(\\stackrel{(2n-2)}{x},\\hat{x})=x$. Then using the same\nmethod as in the proof of Theorem 2 in \\cite{DGG} we can prove:\n\n\\begin{theorem}\nLet $(G;f)$ be an $n$-ary $(n\\geqslant 2)$ semigroup with a unary\noperation $\\hat{}:x\\to\\hat{x}$. Then $(G;f,\\,\\hat{\\,}\\,)$ is an\n$n$-ary group if and only if for some $i,j\\in\\{2,\\ldots,2n-1\\}$\nthe following identities\n\\[\nf_{(2)}(y,\\stackrel{(i-2)}{x},\\hat{x},\\stackrel{(2n-1-i)}{x})=y=\nf_{(2)}(\\stackrel{(2n-1-j)}{x},\\hat{x},\\stackrel{(j-2)}{x},y)\n\\]\nhold.\n\\end{theorem}\n\nFrom this theorem we can deduce other definitions of $n$-ary\n$(n\\geqslant 2)$ groups presented in \\cite{Gal'95}, \\cite{Gal'03},\n\\cite{rus79} and \\cite{Sio67}.\n\n\\medskip\n\nAn $n$-ary group is said to be {\\em semiabelian} if the following\nidentity\n\\begin{equation}\nf(x_1^n)=f(x_n,x_2^{n-1},x_1)\n\\end{equation}\nis satisfied. In this case the operation $\\,\\bar{\\,}:x\\to\n\\overline{x}\\,$ is a homomorphism (cf. \\cite{GG'77}). Note by the\nway that the class of all semiabelian $n$-ary groups coincides\nwith the class of {\\it medial} $n$-ary groups (cf. \\cite{medial},\n[29]). (Some authors used also the name {\\it abelian} instead of\n{\\it semiabelian} (see, e.g., \\cite{Sio67}, [29]).) Such $n$-ary\ngroups are a special case of {\\it $\\sigma$-permutable} $n$-ary\ngroups (cf. \\cite{St-D'86}), i.e. $n$-ary groups in which\n$f(x_1^n)=f(x_{\\sigma(1)},x_{\\sigma(2)},\\ldots,x_{\\sigma(n)})$ for\nfixed $\\sigma\\in S_n$. An $n$-ary group which is\n$\\sigma$-permutable for every $\\sigma\\in S_n$ is usually called\n{\\it commutative}.\n\nAn {\\it $n$-ary power} of $x$ in an $n$-ary group $(G;f)$ is\ndefined in the following way: $x^{<0>}=x$, \\\n$x^{<1>}=f(\\stackrel{(n)}{x})\\;$ and\n$x^{<k+1>}=f(\\stackrel{(n-1)}{x},x^{<k>})\\;$ for all $k>0$. In\nthis convention $x^{<-k>}$ means an element $z$ such that\n$f(x^{<k-1>},\\stackrel{(n-2)}{x},z)=x^{<0>}=x$. Then\n$\\overline{x}=x^{<-1>}$ and\n\\[\n\\begin{array}{l}\nf(x^{<k_1>},\\ldots,x^{<k_n>})=x^{<k_1+\\ldots+k_n+1>}\\\\[4pt]\n(x^{<k>})^{<t>}=x^{<kt(n-1)+k+t>}\n\\end{array}\n\\]\n(cf. \\cite{post}, \\cite{Gl'82} or \\cite{auto}).\n\nNow, putting $x^{\\!\\!\\!\\!-(0)}=x$ and denoting by\n$x^{\\!\\!\\!\\!-(m+1)}$ the skew element to $x^{\\!\\!\\!\\!-(m)}$, from\nthe above two identities and results obtained by W. A. Dudek in\n\\cite{auto} and \\cite{medial} we deduce the following proposition.\n\\begin{proposition}\nIn any $n$-ary group \\ $x^{\\!\\!\\!\\!-(m)}=x^{<S_m>}$, where\n$S_m=\\frac{(2-n)^m-1}{n-1}$.\n\\end{proposition}\n\nThis means that for every $n>2$ we have\n$\\overline{\\overline{x}}=x^{<n-3>}$. In particular,\n$\\overline{\\overline{x}}=x^{<1>}$ in all $4$-ary groups, and\n$\\overline{\\overline{x}}=x^{<2>}$ in all $5$-ary groups (cf. \\cite{Gl'82})\n\n\\section{Hossz\\'u-Gluskin algebras}\n\nLet $(G;f)$ be an $n$-ary group. Fixing in $f(x_1^n)$ some $m<n$\nelements we obtain a new $(n-m)$-ary operation which in general is\nnot associative. It is associative only in the case when these\nfixed elements are located in some special places, for example, in\nthe case when this new operation has the form\n\\begin{equation}\ng(x_1^k)=f(x_1,a_1^r,x_2,a_1^r,x_3,a_1^r,\\ldots,a_1^r,x_{k-1},a_1^r,x_k),\n\\label{k-ary}\n \\end{equation}\nwhere $k+r(k-1)=n$ and $a_1,\\ldots,a_r\\in G$ are fixed. Of course,\nin this case $(G;g)$ is a $k$-ary group. It is denoted by\n$ret_{a_1^r}(G;f)$ and is called a {\\it $k$-ary retract of}\n$(G;f)$ (see, e.g., \\cite{DM2}).\n For different elements\n$a_1,\\ldots,a_r$ we obtain different $k$-ary retracts, but all\nthese $k$-ary retracts (for a fixed $n$-ary group) are isomorphic\n(cf. \\cite{DM2}).\nTherefore, we can consider only retracts for\n$a_1=\\ldots=a_r=a$. In such retracts the element skew to $x$ has\nthe form\n\\[\nf_{(\\cdot)}(\\stackrel{(n-r-2)}{a},\\overline{a},\\stackrel{(n-3)}{x},\\overline{x},\n\\stackrel{(n-r-2)}{a},\\overline{a},\\stackrel{(n-3)}{x},\\overline{x},\\ldots,\n\\stackrel{(n-r-2)}{a},\\overline{a},\\stackrel{(n-3)}{x},\\overline{x},\n\\stackrel{(n-r-2)}{a},\\overline{a}),\n\\]\nwhere $\\overline{a}$ and $\\overline{x}$ are skew in $(G;f)$. This\nmeans that the skew elements in this retract can be expressed by\nthe operations of an $n$-ary group $(G;f)$.\n\nA very important role play {\\it binary retracts}, especially\nretracts denoted by $ret_a(G;f)$, where $x\\circ\ny=f(x,\\stackrel{(n-2)}{a},y)$. The identity of the group\n$(G;\\circ)$ is $\\overline{a}$. One can verify that the inverse\nelement to $x$ has the form\n\\begin{equation}\nx^{-1}=f(\\overline{a},\\stackrel{(n-3)}{x},\\overline{x},\\overline{a}).\\label{inv}\n\\end{equation}\nThus, in this group\n \\begin{equation}\nx\\circ\ny^{-1}=f(x,\\stackrel{(n-3)}{y},\\overline{y},\\overline{a}).\\label{x-y}\n \\end{equation}\n\nBinary retracts of an $n$-ary group $(G;f)$ are commutative only\nin the case when $(G;f)$ is semiabelian (medial). So, $(G;f)$ is\nsemiabelian if and only if\n \\[\nf(x,\\stackrel{(n-2)}{a},y)=f(y,\\stackrel{(n-2)}{a},x)\n \\]\nholds for all $x,y\\in G$ and some fixed $a\\in G$.\n\nM. Hossz\\'u was first who observed a strong connection between\n$n$-ary groups and their binary retracts. He proved in\n\\cite{Hosszu} the following theorem:\n\n\\begin{theorem}\\label{thGH}\nAn $n$-ary groupoid $(G;f)$, $n>2$, is an $n$-ary group if and\nonly if\n\\begin{enumerate}\n\\item[$(i)$] on $G$ one can define a binary operation $\\cdot$ such\nthat $(G;\\cdot)$ is a group,\n\\item[$(ii)$] there exist an automorphism $\\varphi$ of $(G;\\cdot)$ and\n$b\\in G$ such that $\\varphi(b)=b$,\n\\item[$(iii)$] $\\varphi^{n-1}(x)=b\\cdot x\\cdot b^{-1}$ holds for every\n$x\\in G$,\n\\item[$(iv)$] $f(x_1^n)=x_1\\cdot\\varphi(x_2)\\cdot\\varphi^2(x_3)\\cdot\\varphi^3(x_4)\\cdot\\ldots\n\\cdot\\varphi^{n-1}(x_n)\\cdot b$ for all $x_1,\\ldots,x_n\\in G$.\n\\end{enumerate}\n\\end{theorem}\n\nTwo years later, this theorem was proved by L. M. Gluskin (see\n\\cite{glu65}) in a more general form (for so-called positional\noperatives). For a generalization to $n$-ary semigroups, see also\n\\cite{Monk} and \\cite{Zup}.  In another version this theorem was\nalso formulated by E. L. Post (cf. \\cite{post}, p. 246). An\nelegant short proof was given by E. I. Sokolov in \\cite{sok}. His\nproof is based on the observation that $(G;\\cdot)=ret_a(G;f)$.\nThen we have:\n  \\begin{equation}\n\\varphi(x)=f(\\overline{a},x,\\stackrel{(n-2)}{a})\\label{autom}\n \\end{equation}\nand\n \\begin{equation}\n b=f(\\stackrel{(n)}{\\overline{a}}).\\label{b}\n \\end{equation}\nFrom (14) and (7) or (8), we can deduce that commutative $n$-ary group\noperations have the form $f(x_1^n)=x_1\\cdot x_2\\cdot\\ldots\\cdot x_n\\cdot b$,\nwhere $(G;\\cdot)$ is a commutative group.\n\nNote that the last condition of Theorem~\\ref{thGH} can be\nrewritten in the form\n\\begin{equation}\nf(x_1^n)=x_1\\cdot\\varphi(x_2)\\cdot\\varphi^2(x_3)\\cdot\\varphi^3(x_4)\\cdot\\ldots\n\\cdot\\varphi^{n-2}(x_{n-1})\\cdot b\\cdot x_n. \\label{e10}\n\\end{equation}\n\nThe above theorem has the following generalization (cf.\n\\cite{DM1}):\n\\begin{theorem}\\label{genGH}\nAn $n$-ary groupoid $(G;f)$, $n>2$, is an $n$-ary group if and\nonly if\n\\begin{enumerate}\n\\item[$(i)$] on $G$ one can define a $k$-ary operation $g$ such\nthat $(G;g)$ is a $k$-ary group and $k-1$ divides $n-1$,\n\\item[$(ii)$] there exist an automorphism $\\varphi$ of $(G;g)$\nand elements $b_2,\\ldots,b_k\\in G$ such that $\\varphi(b_i)=b_i$\nfor $i=2,\\ldots,k$,\n\\item[$(iii)$] $g(\\varphi^{n-1}(x),b_2^k)=g(b_2^k,x)$ holds for every\n$x\\in G$,\n\\item[$(iv)$]\n$f(x_1^n)=g_{(\\cdot)}(x_1,\\varphi(x_2),\\varphi^2(x_3),\\ldots,\n\\varphi^{n-1}(x_n),b_2^k)$ for all $x_1,\\ldots,x_n\\in G$.\n\\end{enumerate}\n\\end{theorem}\n\nIn this theorem $(G;g)=ret_{a_1^r}(G;f)$, where $a_1=\\ldots=a_r=a$.\nIn this case, we get:\n\\begin{equation}\n\\varphi(x)=f(\\overline{a},\\stackrel{(n-r-2)}{a},x,\\stackrel{(r)}{a}),\n\\label{k-fi}\n\\end{equation}\n\\begin{equation}\nb_2=\nf_{(\\cdot)}(\\stackrel{(n-r-2)n}{a},\\stackrel{(n)}{\\overline{a}},\\stackrel{(k-2)(n-r-2)}{a}),\n \\ \\ b_3=\\ldots=b_k=\\overline{a}.\\label{b_2}\n\\end{equation}\n\nOther important generalizations can be found in \\cite{Hosszu2}\n(for heaps), \\cite{Mar-Jan} (for vector valued groups),\n\\cite{Sokh} (for partially associative $n$-ary quasigroups).\n\n\\medskip\n\nFollowing E. L. Post (see \\cite{post}, cf. [4], p. 36--40, and\n[28]) a binary group \\ $\\mathfrak{G}^{\\ast }=\\left(G^{\\ast };\\circ\n\\right)$ is said to be a {\\it covering group} for the $n$-ary\ngroup $(G;f)$ if there exists an embedding $\\tau :G\\rightarrow\nG^{\\ast }$ such that $\\tau(G)$ is a generating set of $G^{\\ast }$\nand \\ $\\tau (f(x_1^n)) = \\tau\n(x_1)\\circ\\tau(x_2)\\circ\\ldots\\circ\\tau(x_n)$ for every\n$x_1,\\ldots,x_n\\in G.$ \\ $\\mathfrak{G}^{\\ast }$ is a\n\\textit{universal covering group} (or a \\textit{free covering\ngroup}) if for any covering group $\\mathfrak{G}_{1}^{\\ast }$\\\nthere exists a homomorphism from $G^*$ onto $G^*_1$ such that the\nfollowing diagram is commutative (or compatible -- in another\nterminology):\n\n\\begin{center}\n\\begin{minipage}{6cm}\n\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $G$\n\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ {  \\ }$\\swarrow ${  \\ }$\n\\circlearrowleft ${  \\ }$\\searrow $\n\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $G^{\\ast}--\\rightarrow \\ G_{1}^{\\ast}$\n\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ {\\footnotesize onto}\n\\end{minipage}\n\\end{center}\n\nPost proved in \\cite{post} that for every $n$-ary group $(G;f)$\nthere exist a covering group $(G^{\\ast};\\circ)$ and its normal\nsubgroup $G_0$ such that $G^{\\ast}\\diagup G_0$ is a cyclic group\nof order $n-1$ and $f(x_1^n)=x_1\\circ x_2\\circ\\ldots\\circ x_n$ for\nall $x_1,\\ldots,x_n\\in G$. So, the theory of $n$-ary groups is\nclosely related to the theory of {\\it cyclic extensions of\ngroups}, but these theories are not equivalent.\n\nIndeed, the above theorems show that for any $n$-ary group $(G;f)$\nwe have the sequence\n$$\nO\\rightarrow (G_0;\\circ)\\rightarrow (G^{\\ast};\\circ )\n\\stackrel{\\zeta }{\\longrightarrow }\\mathcal{C}\\left( n\\right)\n\\rightarrow O,\n $$\nwhere $(G^{\\ast };\\circ)$ is the free covering group of $(G;f)$\nwith $G=\\zeta^{-1}\\left( 1\\right)$, and $1$ is a generator of the\ncyclic (additively writing) group $\\mathcal{C}\\left( n\\right)\n=(C_{n};+_{n}).$\n\nWe have\n\\[\n\\begin{array}{ccccc}\n&  & ( G_{1}^{\\ast };\\circ) &  &  \\\\\n& \\nearrow & \\uparrow & \\searrow &  \\\\\n(G_0;\\circ ) &  & \\circlearrowright \\;\\;\\; {\\vert}\n\\circlearrowleft\\circlearrowright &  & C\\left( n\\right) \\\\\n& \\searrow &\\downarrow & \\nearrow &  \\\\\n&  & (G_2^{\\ast};\\circ)&  &\n\\end{array}\n\\]\nwhere we use\n\n$\\circlearrowright $ \\ for the equivalence of extensions,\n\n$\\circlearrowleft $ \\ for the isomorphism of suitable $n$-ary\ngroups.\n\n\\medskip\n\nOf course, two $n$-ary groups determined in the above-mentioned\nsense by two equivalent cyclic extensions are isomorphic. However,\ntwo non-equivalent cyclic extensions can determine two isomorphic\n$n$-ary groups.\n\n\\begin{example} Consider two cyclic extensions of the cyclic group $%\n\\mathcal{C}(3)$ by $\\mathcal{C}(3)$:\n$$\n0\\rightarrow \\mathcal{C}(3)\\stackrel{\\alpha }{\\rightarrow\n}\\mathcal{C}(9)\\stackrel{\\beta _{1}}{\\rightarrow\n}\\mathcal{C}(3)\\rightarrow 0\n$$\nand\n$$\n0\\rightarrow \\mathcal{C}(3)\\stackrel{\\alpha }{\\rightarrow\n}\\mathcal{C}(9)\\stackrel{\\beta _{2}}{\\rightarrow\n}\\mathcal{C}(3)\\rightarrow 0,\n$$\nwhere the homomorphisms $\\alpha $, $\\beta _{1}$ and $\\beta _{2}$\nare given by:\n\\[\n\\begin{array}{lcl}\n\\;\\alpha (x)=3x&{\\rm for}& x\\in C_{3},\\\\[4pt]\n\\beta_{1}(x)\\equiv x({\\rm mod}\\,3)&{\\rm for}&x\\in C_{9},\\\\[4pt]\n\\beta_{2}(x)\\equiv 2x({\\rm mod}\\,3)&{\\rm for}&x\\in C_{9}.\n\\end{array}\n\\]\n\nIt is easy to verify that if $g(x,y,z,v)=(x+y+z+v)({\\rm mod}\\,9),$\nthen the $4$-ary groups $(\\beta _{1}^{-1}(1);g)$ and $(\\beta\n_{2}^{-1}(2);g)$, corresponding to those extensions are isomorphic\nto the $4$-ary groups $(C_{3};f_{1})$, $(C_{3};f_{2})$,\nrespectively, where\n $$\nf_{1}(x,y,z,v)\\equiv (x+y+z+v+1)({\\rm mod}\\,3)\n $$\nand\n$$\nf_{2}(x,y,z,v)\\equiv (x+y+z+v+2)({\\rm mod}\\,3).\n$$\nThese 4-groups are isomorphic. The isomorphism\n$\\varphi:(C_{3};f_{1})\\to(C_{3};f_{2})$ has the form\n$\\varphi(x)\\equiv 2x({\\rm mod}\\,3)$. Nevertheless, the\nabove-mentioned extensions are not equivalent (because there is no\nautomorphism $\\lambda $ of $\\mathcal{C}(9)$ such that $\\lambda\n\\circ \\alpha =\\alpha $ and \\ $\\beta _{2}\\circ \\lambda =\\beta\n_{1}$).\n\\end{example}\n\nThe algebra $(G;\\cdot,\\varphi,b)$ of the type $(2,1,0)$, where\n$(G;\\cdot)$ is a (binary) group, $b\\in G$ is fixed, $\\varphi\\in\nAut(G;\\cdot)$, $\\varphi(b)=b$ and $\\varphi^{n-1}(x)=b\\cdot x\\cdot\nb^{-1}$ for every $x\\in G$ is called a {\\it Hossz\\'u-Gluskin\nalgebra} (briefly: an {\\it $HG$-algebra}). We say that an\n$HG$-algebra $(G;\\cdot,\\varphi,b)$ is {\\it associated} with an\n$n$-group $(G;f)$ if the identity (\\ref{e10}) is satisfied. In\nthis case we say also that an $n$-ary group $(G;f)$ is {\\it\n$\\langle\\varphi,b\\rangle$-derived} from the group $(G;\\cdot)$. A\n$k$-ary $HG$-algebra $(G;g,\\varphi,b_2^k)$ can be defined\nsimilarly. Binary $HG$-algebras are studied in \\cite{jus'95a},\n\\cite{jus'95b} and \\cite{jus'03}.\n\nTheorems~\\ref{thGH} and \\ref{genGH} state that every $k$-ary\n$HG$-algebra is associated with some $n$-ary group. Any $n$-ary\ngroup is $\\langle\\varphi,b\\rangle$-derived from some binary group\nand $\\langle\\varphi,b_2^k\\rangle$-derived from some $k$-ary group.\n\n\\section{Calculation of $n$-ary groups on small sets}\n\nResults presented in the previous section give the possibility to\nevaluate the number of non-isomorphic $n$-ary groups. To calculate\nthese groups we must use the following result proved in\n\\cite{DM2}.\n\n\\begin{theorem}\\label{izoth}\nTwo $n$-ary groups $(G_1;f_1)$, $(G_2;f_2)$ are isomorphic if and\nonly if for every $c\\in G_1$ there exists an isomorphism\n$h:ret_c(G_1;f_1)\\to ret_d(G_2;f_2)$ such that $d=h(c)$, \\\n$h(f_1({\\overline{c}},\\ldots,{\\overline{c}}))=\nf_2(\\overline{d},\\ldots,\\overline{d}) $ and\n$h(f_1(\\overline{c},x,\\!\\stackrel{(n-2)}{c}))=\nf_2(\\overline{d},h(x),\\!\\stackrel{(n-2)}{d}).$\n\\end{theorem}\n\n\\begin{corollary}\\label{izoth2}\n{\\it Two commutative $n$-ary groups $(G_1;f_1)$, $(G_2;f_2)$ are\nisomorphic if and only if for every $c\\in G_1$ there exists an\nisomorphism $h:ret_c(G_1;f_1)\\to ret_d(G_2;f_2)$ such that\n$d=h(c)$ and $h(f_1({\\overline{c}},\\ldots,{\\overline{c}}))=\nf_2(\\overline{d},\\ldots,\\overline{d})$.}\n\\end{corollary}\n\nIf $(G,\\cdot)$ is an abelian group, then, of course, we can\nconsider the automorphism of the form $\\alpha(x)=x^{-1}$. Then $G$\nwith the operation\n\\begin{equation}\nf(x_1^n)=x_1\\cdot x_2^{-1}\\cdot x_3\\cdot\\ldots\\cdot\nx^{-1}_{n-1}\\cdot x_n\\label{-1}\n\\end{equation}\nis an $n$-ary group if $n$ is odd. Such $n$-ary groups are\ncharacterized by the following theorem proved in \\cite{Gl-Mi'84}.\n\n\\begin{theorem} Let $m$ be odd and let $(G;f)$ be an\n$n$-ary group. Then the operation $f$ has the form $(\\ref{-1})$,\nwhere $(G;\\cdot)$ is an abelian group, if and only if\n\\begin{enumerate}\n\\item[$(i)$] \\ $f\\left( x,\\ldots,x\\right) =x$,\n\\item[$(ii)$] \\ $f\\left(x_1^{i},y,y,x_{i+3}^{n}\\right)=\nf\\left(x_1^{i},z,z,x_{i+3}^{n}\\right)$ for all $\\;0\\leqslant\ni\\leqslant n-2.$\n\\end{enumerate}\nIn this case $(G;\\cdot)=ret_a(G;f)$ for some $a\\in G$.\n\\end{theorem}\n\nConsider an abelian group $(G;+)$. Then, as a special case of an $n$-ary\ngroup operation of form (19), one can obtain the ternary term operation\n$$\nf(x,y,z)=x-y+z\n$$\nwhich is a so-called Mal'tsev term in the group $G$. Of course, it is\nidempotent and medial ({\\it entropic} -- in another terminology). Such\nternary operations appear in several branches of mathematics. For example,\nthey play very important role in affine geometry and the theory of modes\n(because of idempotency and mediality), in the theory of congruences\nin general algebras (because existence of a Mal'tsev term in general algebras\nimplies permutability of congruences and then modularity of lattices of\ncongruences) and also in the theory of clones which is important in\nUniversal Algebra and as well in Multiple-valued Logics.\n\n\\medskip\nFrom results obtained in \\cite{Gl-Mi'84} (cf. also \\cite{Tim'72})\nwe can deduce:\n\n\\begin{proposition} Let $(G;\\cdot)$ be a group and let $t_1,\\ldots,t_n$\nbe fixed integers. Then $G$ with the operation\n\\[\nf(x_1^{m})=(x_1)^{t_{1}}\\cdot(x_{2})^{t_{2}}\\cdot\\ldots \\cdot\n(x_{n-1})^{t_{n-1}}\\cdot(x_{n})^{t_{n}},\n\\]\nis an $n$-ary group if and only if\n\\begin{enumerate}\n\\item \\ $x^{t_1}=x=x^{t_n}$,\n\\item \\ $t_j=k^{j}$ \\ for some integer $k$ and all $j=2,\\ldots,n-1$,\n\\item \\ $(x\\cdot y)^k=x^k\\cdot y^k.$\n\\end{enumerate}\n\\end{proposition}\n\nIn this case we say that $(G;f)$ is {\\em derived from the\n$k$-exponential group}.\n\\begin{proposition}\nAn $n$-ary group $(G;f)$ is derived from the $k$-exponential\n$(k>0)$ group $(G;\\cdot)$ if and only if there exists $a\\in G$\nsuch that\n\\begin{enumerate}\n\\item \\ $f(a,\\ldots,a)=a$,\n\\item \\ $f_{(k)}(\\stackrel{(n-2)}{a},x,\\stackrel{(n-2)}{a},x,\n\\ldots,\\stackrel{(n-2)}{a},x,a)=x$.\n\\end{enumerate}\nMoreover, $(G;\\cdot)=ret_a(G;f)$.\n\\end{proposition}\n\nUsing the above results we can describe all non-isomorphic $n$-ary\ngroups with small numbers of elements.\n\nFor this let $(\\mathbb{Z}_k;+)$ be the cyclic group modulo $k$.\nConsider the following $n$-ary operation:\n \\[\\arraycolsep=.5mm\n \\begin{array}{rl}\nf_{a}( x_1^n) &\\equiv (x_{1}+\\ldots+x_{n}+a)\\,({\\rm mod}\\,k) ,\\\\[4pt]\ng_{d}(x_1^n)&\\equiv (x_1+dx_2+\\ldots+d^{n-2}x_{n-1}+x_n)\\,({\\rm mod}\\,k),\\\\[4pt]\ng_{d,c}(x_1^n)&\\equiv\n(x_1+dx_2+\\ldots+d^{n-2}x_{n-1}+x_n+c)\\,({\\rm mod}\\,k),\n\\end{array}\n \\]\nwhere $a\\in\\mathbb{Z}_k$, \\ $c,d\\in\\mathbb{Z}_k\\setminus\\{0,1\\},$\n\\ $d^{\\,n-1}\\equiv 1\\,({\\rm mod}\\,k).$ Additionally, for the\noperation $g_{d,c}$ we assume that $dc\\equiv c\\,({\\rm mod}\\,k)$\nholds. By Theorem~\\ref{thGH}, $(\\mathbb{Z}_{k};f_{a})$,\n$(\\mathbb{Z}_{k};g_{d})$ and $(\\mathbb{Z}_{k};g_{d,c})$ are\n$n$-ary groups.\n\n\\bigskip\n\nIn \\cite{Gl-Mi-S} the following theorem is proved:\n\\begin{theorem} A $k$-element $n$-ary group $(G;f)$ is\n$\\langle\\varphi,b\\rangle$-derived from the cyclic group of order\n$k$ if and only if it is isomorphic to exactly one $n$-ary group\nof the form $(\\mathbb{Z}_{k};f_{a})$, $(\\mathbb{Z}_{k};g_{d})$ or\n$(\\mathbb{Z}_{k};g_{d,c})$, where $d|gcd(k,n-1)$ and $c|k.$\n\\end{theorem}\n\nAn infinite cyclic group can be identified with the group\n$(\\mathbb{Z};+)$. This group has only two automorphisms:\n$\\varphi(x)=x$ and $\\varphi(x)=-x$. So, according to\nTheorem~\\ref{thGH}, $n$-ary groups defined on $\\mathbb{Z}$ have\nthe form $(\\mathbb{Z};f_a)$ or $(\\mathbb{Z};g_{-1})$, where\n$$\ng_{-1}(x_1^n)=x_1-x_2+x_3-x_4+\\ldots+x_n, $$ and $n$ is odd. Since\n$\\varphi_k(x)=x+k$ is an isomorphism of $n$-ary groups\n$(\\mathbb{Z};f_a)$ and $(\\mathbb{Z};f_b)$, where $a=b+(n-1)k$, the\ncalculation of non-isomorphic $n$-ary groups of the form\n$(\\mathbb{Z};f_a)$ can be reduced to the case when\n$a=0,1,\\ldots,n-2$. From Corollary~\\ref{izoth2} it follows that\nthese $n$-ary groups are non-isomorphic.\n\nSo, we have proved\n\\begin{theorem}\nAn $n$-ary group $\\langle\\varphi,b\\rangle$-derived from the\ninfinite cyclic group $(\\mathbb{Z};+)$ is isomorphic to an $n$-ary\ngroup $(\\mathbb{Z};f_a)$, where $0\\leqslant a\\leqslant (n-2)$, or\nto $(\\mathbb{Z};g_{-1})$, where $n$ is odd.\n\\end{theorem}\n\nDenote by $Inn\\left(G;\\cdot\\right)$ the group of all inner\nautomorphisms of $(G;\\cdot)$, by $Out\\left( G;\\cdot \\right)$ the\nfactor group of $Aut\\left(G;\\cdot\\right)$ by\n$Inn\\left(G;\\cdot\\right)$, and by $Out_{n}\\left( G;\\cdot \\right) $\nthe set of all cosets $\\overline{\\gamma}\\in\nOut\\left(G;\\cdot\\right)$ containing $\\gamma$ such that\n$\\gamma^{n-1}\\in Inn\\left(G;\\cdot\\right)$. Then, as it is proved\nin \\cite{Gl-Mi-S}, for centerless groups, i.e. groups for which\n$\\,card(Cent\\left(G;\\cdot\\right))=1$, the following theorem is\ntrue.\n\n\\begin{theorem} Let $\\left( G;\\cdot \\right)$ be a centerless group such that\n$Out_{n}\\left(G;\\cdot\\right)$ is abelian, and let\n$\\left(G;f\\right)$ be $\\langle\\alpha,a\\rangle$-derived, and\n$\\left(G;g\\right)$ be $\\langle\\beta,b\\rangle$-derived from\n$\\left(G;\\cdot\\right)$. Then $\\left(G;f\\right)$ is isomorphic to\n$\\left( G;g\\right)$ if and only if $\\,\\alpha\\circ\\beta^{-1}\\in\nInn\\left( G;\\cdot\\right)$.\n\\end{theorem}\n\nThe number of pairwise non-isomorphic $n$-ary groups\n$\\langle\\varphi,b\\rangle$-derived from a centerless group\n$(G;\\cdot)$ is smaller or equal to $s=card(\nOut_{n}\\left(G;\\cdot\\right))$. It is equal to $s$ if and only if\n$Out\\left(G;\\cdot\\right)$ is abelian.\n\nFor every $n$ and $k\\neq 2,6$, there exists exactly one $n$-ary\ngroup which is $\\langle\\varphi,b\\rangle$-derived from $S_{k}$ (for\n$k=2$ and $k=6$ we have one or two such $n$-ary groups relatively\nto evenness of $n$).\n\nLet now $(G;\\cdot)$ be an arbitrary group, $c\\in G,$ $\\varphi \\in\nAut\\left(G;\\cdot \\right) $. Let us put\n\\[\n\\arraycolsep=.5mm \\begin{array}{rl}\n f_{c}^{(\\cdot)}(x_1^n)&=x_1\\cdot x_2\\cdot\\ldots\\cdot x_n\\cdot\n c,\\\\[4pt]\ng_{\\varphi }^{(\\cdot)}(x_1^n)& =x_1\\cdot\\varphi\\left(x_{1}\\right)\n\\cdot\\ldots\\cdot\\varphi^{n-1}( x_n) ,\\\\[4pt]\ng_{\\varphi ,c}^{( \\cdot)}(x_1^n)& =x_1\\cdot\\varphi\\left(\nx_{2}\\right)\\cdot\\ldots\\cdot\\varphi^{n-1}(x_n)\\cdot c.\n\\end{array}\n\\]\n\nFor example (for details see \\cite{Gl-Mi'87}), we have the\nfollowing:\n\n\\begin{theorem} Let $l=gcd\\left(n-1,12\\right),$ \\ $\\left(G_{4};\\ast\\right)$\nbe the Klein four-group $($with $0$ as the neutral element$)$, let\n$\\gamma ,\\varepsilon \\in Aut\\left( G_{4};\\ast \\right),$ where\n$\\gamma $ is of order $2$ and $\\varepsilon $ of order $3$, and let\n$c\\in G_{4}\\backslash \\{0\\}$ be the fix point of $\\gamma $. Then\nevery $n$-ary group $\\langle\\varphi,b\\rangle$-derived from\n$(G_{4};\\ast)$ is isomorphic to exactly one $(G_{4};f)$, where $f$\nis one of the following $n$-ary group operations:\n\\begin{enumerate}\n\\item[$(a)$] \\ $f_{0}^{\\left( \\ast \\right) },$ $f^{\\left( \\ast\n\\right) },g_{\\ \\gamma }^{\\left( \\ast \\right) },g_{\\ \\gamma\n,c}^{\\left( \\ast \\right) } $ or $g_{\\ \\varepsilon }^{\\left( \\ast\n\\right) }$ \\ \\ for \\ $l=12$,\n\\item[$(b)$] \\ $f_{0}^{\\left( \\ast \\right) },$ $f_{1}^{\\left(\n\\ast \\right) },g_{\\ \\gamma }^{\\left( \\ast \\right) }$ or $g_{\\\n\\varepsilon }^{\\left( \\ast \\right) }$ \\ \\ for \\ $l=6$,\n\\item[$(c)$] \\ $f_{0}^{\\left( \\ast \\right) },$ $f_{1}^{\\left( \\ast\n\\right) },g_{\\ \\gamma }^{\\left( \\ast \\right) }$ or $g_{\\ \\gamma\n,c}^{\\left( \\ast \\right) }$ \\ \\ for \\ $l=4$,\n\\item[$(d)$] \\ $f_{0}^{\\left( \\ast \\right) }$ or $g_{\\ \\varepsilon\n}^{\\left( \\ast \\right) }$ \\ \\ for \\ $l=3$,\n\\item[$(e)$] \\ $f_{0}^{\\left( \\ast \\right) },$ $f_{1}^{\\left( \\ast\n\\right) }$ or $g_{\\ \\gamma }^{\\left( \\ast \\right) }$ \\ \\ for \\\n$l=2$, \\item[$(f)$] \\ $f_{0}^{\\left( \\ast \\right) }$ \\ \\ for \\\n$l=1.$\n\\end{enumerate}\n\\end{theorem}\n\nComparing our results with results obtained in \\cite{Gl-Mi'87},\n\\cite{Gl-Mi'88} and \\cite{Gl-Mi-S} (cf. also \\cite{post} for\n$k=2,3$) we can tabularize the numbers of $n$-ary groups on\n$k$-element sets with $k<8$ in the following way (we use the\nabbreviations: commut. = commutative, idem. = idempotent):\n\n\\bigskip {\\small\\noindent\n\\begin{tabular}{|l|c|c|}\n\\hline $k=2$, \\ $l=gcd\\,(n-1,2)$ & $l=2$& $l=1\\rule{0mm}{3mm}$\n\\\\ \\hline\n$n\\equiv t({\\rm mod}\\,2)$&$t=1$&$t=0\\rule{0mm}{3mm}$\\\\\n\\hline all &2& 1\\\\ \\hline\n commutative&2 & 1 \\\\ \\hline\n commutative, idempotent & 1 & 0 \\\\ \\hline\n\\end{tabular}\n\n\\bigskip\\noindent\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline $k=3$, \\ $l=gcd\\,(n-1,6)$ &$l=6$ &$l=3$&$l=2$&$l=1\\rule{0mm}{3mm}$ \\\\\n\\hline $n\\equiv t\\,({\\rm mod}\\,6)$&$t=1$\n&$t=4$&$t=3,\\,5$&$t=0,\\,2\\rule{0mm}{3mm}$\\\\ \\hline\n {all} &3&2&2&1\\\\ \\hline\n{commutative} &2&2&1&1 \\\\ \\hline\n commutative, idempotent &1&1&0&0\\\\ \\hline\nnon-commut., medial, idempotent &1&0&1&0 \\\\ \\hline\n\\end{tabular}\n\n\\bigskip\\noindent\n\\begin{tabular}{|l|c|c|c|c|c|c|}\n\\hline $k=4$, \\ $l=gcd\\,(n-1,12)$ &$l=12$&$l=6$&$l=4$&$l=3$&$l=2$&$l=1\\rule{0mm}{3mm}$ \\\\\n\\hline $n\\equiv t\\,({\\rm mod}\\,12)$ & $t=1$\n&$t=7$&$t=5,\\,9$&$t=4,\\,10$&$t=3,11$&$t=t_0\\rule{0mm}{3mm}$\\\\\n\\hline {all} &10&8&9&3&7&2 \\\\ \\hline\n commutative &5&4&5&2&4&2 \\\\ \\hline\n {commutative, idempotent} &2&1&2&0&1&0 \\\\ \\hline\n non-commut., medial, idem. &3&2&1&1&1&0 \\\\ \\hline\n {non-commut., medial}, non-idem., &2&2&3&0&2&0 \\\\ \\hline\n\\end{tabular}\n\n\\smallskip $t_0=0,\\,2,\\,6,\\,8$.\n\n\\bigskip\\noindent\n\\begin{tabular}{|l|c|c|c|c|c|c|}\n\\hline $k=5$, \\ $l=gcd\\,(n-1,20)$ &$l=20$&$l=10$&$l=5$&$l=4$&$l=2$\n&$l=1\\rule{0mm}{3mm}$\n\\\\ \\hline $n\\equiv t\\,({\\rm mod}\\, 20)$&$t=1$ &$t=11$&$t=6,16$&$t=t_1$\n& $t=t_2$ & $t=t_3\\rule{0mm}{3mm}$\\\\ \\hline\n {all}&5&3&2&4&2&1 \\\\ \\hline\n {commutative}&2&2&2&1&1&1 \\\\ \\hline\n {commutative, idempotent} &1&1&1&0&0&0 \\\\ \\hline\n non-commut., idem., medial &3&1&0&3&1&0 \\\\ \\hline\n non-commut., non-idem., medial &0&0&0&0&0&0 \\\\ \\hline\n\\end{tabular}\n\n\\smallskip $t_1=5,9,13,17$, \\ \\\n\n$t_2=3,7,15,19$, \\ \\\n\n$t_3=0,2,4,8,10,12,14,18$.\n\n\\bigskip\\noindent\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline $k=6$, \\ $l=gcd\\,(n-1,6)$\n&$l=6$&$l=3$&$l=2$&$l=1\\rule{0mm}{3mm}$\n\\\\ \\hline\n$n\\equiv t\\,({\\rm mod}\\,6)$&$t=1$\n&$t=4$&$t=3$&$t=0,\\,2\\rule{0mm}{3mm}$\n\\\\ \\hline all&7&3&5&2\\\\ \\hline\ncommutative&4&2&2&1 \\\\\n\\hline {commutative, idempotent}&1&0&0&0\\\\ \\hline {medial,\nidempotent, non-commut.}&1&0&1&0 \\\\ \\hline\n non-commut., medial, non-idem.,&1&0&1&0 \\\\ \\hline\nnon-medial &1&1&1&1 \\\\\n\\hline\n\\end{tabular}\n\n\\bigskip\\noindent\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|}\\hline\n$k=7$, \\ $l=gcd\\,(n-1,42)$ &$l=42$&$l=21$&$l=14$&$l=7$ &$l=6$ &\n$l=3$&$l=2$ &$l=1\\rule{0mm}{3mm}$ \\\\\n\\hline $n\\equiv t\\,({\\rm mod}\\,42)$&$t=1$&$t=22$&$t=t_4$ &$t=t_5$ & $t=t_6$\n&$t=t_7$&$t=t_8$&$t=t_9\\rule{0mm}{3mm}$ \\\\\n\\hline all&7&4&3&2&6&3&2&1 \\\\\n\\hline {commutative}&2&2&2&2&1&1&1&1\\\\ \\hline\nnon-com., medial, idem.,& 5&2&1&0&5&2&1&0\\\\\n\\hline commutative, idempotent&1&1&1&1&0&0&0&0\\\\\n\\hline\n\\end{tabular}\n\n\\smallskip\n\n$t_4= 15, 29$,\n\n$t_5=8, 36$,\n\n$t_6=7,13,19,25,31,37$,\n\n$t_7=4,10,16,28,34,40$,\n\n$t_8= 3,5,9,11,17,21,23,33,35,39,41$,\n\n$t_9=0,2,6,12,14,18,20,24,26,30,32,38$.\n}\n\n\\section{Term equivalence of $n$-ary groups}\n\nFor any general algebra $\\frak{A}=(A;\\mathbb{F})$ one can define\nthe set $\\mathbb{T}^{(n)}(\\frak{A})$ of all {\\it $n$-ary term\noperations} as the smallest set of $n$-ary operations on $A$\ncontaining $n$-ary projections (or $n$-ary trivial operations, in\nanother terminology) and closed under compositions with\nfundamental operations. Then the set\n$\\mathbb{T}(\\frak{A})=\\bigcup\\limits_{n=1}^{\\infty}\\mathbb{T}(\\frak{A})$\nof all {\\it term operations} is the smallest set of operations on\nthe set $A$ containing the set $\\mathbb{F}$ of fundamental\noperations and all projections $e_i^{(n)}(x_1^n)=x_i$,\n($i=1,2,\\ldots,n$, $n=1,2,\\ldots$), and closed under (direct)\ncompositions. Of course, $\\mathbb{T}(\\frak{A})$ is a {\\it clone}\nin the sense of Ph. Hall (see, e.g., \\cite{Cohn}). It is worth\nmentioning that the term operations were also called {\\it\nalgebraic operations} by several authors (see, e.g.,\n\\cite{Mar'61}). Two algebras $\\frak{A}_1=(A;\\mathbb{F})$ and\n$\\frak{A}_2=(A;\\mathbb{G})$ are called {\\it term equivalent} if\n$\\mathbb{T}(\\frak{A}_1)=\\mathbb{T}(\\frak{A}_2)$ (see, e.g.,\n\\cite{Gb}, p. 32, 56). If elements from some subsets $A_1$ and\n$A_2$ of $A$ are treated as constant elements of algebras\n$\\frak{A}_1=(A;\\mathbb{F}\\cup A_1)$ and\n$\\frak{A}_2=(A;\\mathbb{G}\\cup A_2)$, respectively, and\n$\\mathbb{T}(\\frak{A}_1)=\\mathbb{T}(\\frak{A}_2)$, then $\\frak{A}_1$\nand $\\frak{A}_2$ are {\\it polynomially equivalent}. Two varieties\n$\\mathcal{V}_1$ and $\\mathcal{V}_2$ of algebras (perhaps of\ndifferent types) are term equivalent (polynomially equivalent,\nrespectively) if for every algebra $\\frak{A}_1\\in\\mathcal{V}_1$\nthere exists an algebra $\\frak{A}_2\\in\\mathcal{V}_2$ term\nequivalent (polynomially equivalent, resp.) to $\\frak{A}_1$, and\nvice versa.\n\n\\medskip\n\nUsing Theorem~\\ref{genGH} and taking into account formulas\n(\\ref{k-fi}) and (\\ref{b_2}), we have\n\n\\begin{theorem}\\label{teq}\nLet $\\mathfrak{G}=(G;f,\\bar{}\\;)$ be an $n$-ary group for a fixed\n$n>2$, an element $a$ belong to $G$, and let $k$ be such a natural\nnumber that $(k-1)$ divide $(n-1)$. Then the algebra\n$\\mathfrak{G}_a=(G;f,\\bar{}\\; ,a)$, with the additional constant\n$a\\in G$ is term equivalent to the algebra $(G;g,\\varphi,b_2^k)$,\nwhere $\\varphi$ is an automorphism of a $k$-ary group $(G;g)$,\n$(k-1)$ divides $(n-1)$, and $b_2,\\ldots,b_k$ are constant\nelements in $G$ such that $\\varphi(b_i)=b_i$ for $i=2,\\ldots,k$\nand $g(\\varphi^{n-1}(x),b_2^k)=g(b_2^k,x)$ for all $x\\in G $.\n\\end{theorem}\n\nIndeed, $f$ is determined by $g$, $\\varphi$ and $b_2,\\ldots,b_k$\nby the formula $(iv)$ from Theorem~\\ref{genGH}. The function \\\n$\\bar{}:x\\to\\overline{x}$ can be easily expressed by the operation\n$g$. Namely, if $f=g_{(t)}$, then $\\overline{x}=x^{<-t>}$, where\n$x^{<s>}$ is a $k$-ary power of $x$. According to\nTheorem~\\ref{genGH}, the element $\\overline{x}$ also can be\nexpressed by $g$, $\\varphi$ and $b_2,\\ldots,b_n$ as a solution $z$\nof the equation\n$$\nx=f(\\stackrel{(n-1)}{x},z)=g_{(\\cdot)}(x,\\varphi(x),\\varphi^2(x),\\ldots,\n\\varphi(x)^{n-2},b_2^k,z).\n$$\n\nConversely, the operations of $(G;g,\\varphi,b_2^k)$ are term\nderived from the operations of $(G;f,\\bar{}\\;)$ by (\\ref{k-fi})\nand (\\ref{b_2}). $(G;g)=ret_{a_1^r}(G;f)$, where\n$a_1=\\ldots=a_r=a$, which completes the proof of\nTheorem~\\ref{teq}.\n\n\\medskip\n\nBy Theorem~\\ref{thGH} and formulas (\\ref{autom}) and (\\ref{b}), we\nhave the following corollaries.\n\n\\begin{corollary}\\label{term1}{\\em\nLet $\\mathfrak{G}=(G;f,\\bar{}\\;)$ be an $n$-ary group for a fixed\n$n>2$, and let an element $a$ belong to $G$. Then the algebra\n$\\mathfrak{G}_a=(G;f,\\bar{}\\; ,a)$ is term equivalent to the\n$HG$-algebra $(G;\\cdot,\\varphi,b)$, where $(G;\\cdot)$ is a group,\n$\\varphi\\in Aut(G;\\cdot)$, $b\\in G$, $\\varphi(b)=b$,\n$\\varphi^{n-1}(x)=b\\cdot x\\cdot b^{-1}$ for all $x\\in G$.}\n\\end{corollary}\n\n\\begin{corollary}\\label{term2}\n{\\em For fixed $n>2$, the variety of $n$-ary groups $($as algebras\nof type $(n,1)\\,)$ is polynomially equivalent to the variety of the\ncorresponding $HG$-algebras $($as algebras of type $(2,1,1,0)\\,)$.\n}\n\\end{corollary}\n\nLet now $\\mathfrak{G}=(G;f,\\bar{}\\;)$ be a semiabelian $n$-ary\ngroup $(n>2)$. \\linebreak Then the $HG$-algebra associated with\n$\\mathfrak{G}$ has a commutative group operation denoted by $+$.\nLet $\\mathfrak{H}=(G;+,\\varphi,b)$ be associated with\n$\\mathfrak{G}$ and $\\mathfrak{G}_a=(G;f,\\bar{}\\;,a)$. Then\n$\\mathfrak{H}$ and $\\mathfrak{G}_a$ are term equivalent (see\nTheorems \\ref{thGH} and \\ref{genGH},  Corollary~\\ref{term1}, and\nformulas (\\ref{inv}) -- (\\ref{b_2})). In this case we have\n\\[\n\\arraycolsep=.5mm\n\\begin{array}{rl}\n-y&=f(\\overline{a},\\stackrel{(n-3)}{x},\\overline{x},\\overline{a}\\,),\\\\[4pt]\nx+y&=f(x,\\stackrel{(n-3)}{(-y)},\\overline{(-y)},\\overline{a}\\,),\\\\[4pt]\n\\varphi(x)&=f(\\overline{a},x,\\stackrel{(n-2)}{a}),\\\\[4pt]\n{\\rm and } \\ \\ \\ \\ b&=f(\\stackrel{(n)}{\\overline{a}}).\n\\end{array}\n\\]\n\nWe can describe of all term operations of $\\mathfrak{G}_a$ by using\nthe language of $HG$-algebras.\n\nAt first, we consider unary term operations. Denote by $g_i(x)$\nthe following operation\n\\begin{equation}\\label{g_i}\ng_i(x)=k_{i1}\\varphi^{l_{i1}}(x)+k_{i2}\\varphi^{l_{i2}}(x)+\\ldots\n+k_{it}\\varphi^{l_{it}}(x)\n\\end{equation}\nfor some $t,l_{i1},\\ldots,l_{it}$ non-negative integers and some\n$k_{i1},\\ldots,k_{it}\\in\\mathbb{Z}$. Then it is easily to verify\n\n\\begin{lemma}\nLet $\\,\\mathfrak{H}=(G;+,\\varphi,b)$ be the $HG$-algebra\nassociated with a semiabelian $n$-ary group $\\mathfrak{G}$. Then\nall unary term operations of $\\mathfrak{H}$ $($and of\n$\\;\\mathfrak{G}_a\\,)$ are of the form\n\\begin{equation}\\label{g}\ng(x)=g_i(x)+k_g b\n\\end{equation}\nfor some $g_i$ of the form $(\\ref{g_i})$ and $k_g\\in\\mathbb{Z}$.\n\\end{lemma}\n\nIndeed, it is enough to observe that\n$g\\in\\mathbb{T}^{(1)}(\\mathfrak{H})$, $\\varphi(g(x))$ is again of\nthe form (\\ref{g}), and the set of all such operations is closed\nunder addition.\n\n\\begin{theorem}\nLet $\\,\\mathfrak{H}=(G;+,\\varphi,b)$ be the $HG$-algebra\nassociated with a semiabelian $n$-ary group $\\mathfrak{G}$. Then\nall $m$-ary term operations of $\\mathfrak{H}$ $($and of\n$\\;\\mathfrak{G}_a\\,)$ are of the form\n\\begin{equation}\\label{sum}\nF(x_1,\\ldots,x_m)=\\sum\\limits_{i=1}^{m}g_i(x_i)+k_F b\n\\end{equation}\nfor some $g_i(x)$ of the form $(\\ref{g_i})$ and\n$k_F\\in\\mathbb{Z}$.\n\\end{theorem}\n\nA verification of this theorem can be done by induction with\nrespect to the complexity of term operations and we left it to\nreaders.\n\n\\section{$\\mathcal{Q}$-independent sets in $HG$-algebras}\n\nE. Marczewski observed at the end of the 1950s that there are\ncommon features of linear independence of vectors and\nset-theoretical independence, and proposed a general scheme of\nindependence called here {\\em $\\mathcal{M}$-independence}. Recall\nthat the notion of set-theoretical independence (or, more\ngenerally, independence in Boolean algebras, see, e.g.,\n\\cite{BalF82}, \\cite{BrRu81}, \\cite{Gla71}, \\cite{Mar60}) was\nintroduced at the mid-1930s by G. Fichtenholz and L. Kantorovich\n\\cite{FiKa34} and also, independently, by E. Marczewski himself,\nand this notion is very important in Measure Theory (see, e.g.,\n\\cite{FiKa34}, \\cite{Mar38}, \\cite{Mar48a}, \\cite {Myc68},\n\\linebreak and \\cite{Sik64}).\n\nLet $\\mathfrak{A}=(A;\\mathbb{F})$ be an algebra $\\emptyset \\neq\nX\\subseteq A$. The set $X$ is said to be $\\mathcal{M}$\\textit{-independent}\n(see \\cite{Mar'58}, \\cite{Mar'61}) $(X\\in Ind(\\mathfrak{A};\\mathcal{M})$,\nfor short$)$ if\n\n\\begin{enumerate}\n\\item[(a) ]  $(\\forall n\\in \\mathbb{N}$, $n\\leqslant card(X))$\n$(\\forall f,g\\in\\mathbb{T}^{(n)}(\\mathfrak{A}))$\n($\\forall\\underset{\\neq }{\\underbrace{a_{1},\\ldots ,a_{n}}}\\in X$)\n\n\\hspace*{3mm}$\\big[f(a_{1}^{n})=g(a_{1}^{n})\\Longrightarrow f=g \\\n({\\rm in }\\ A)\\big]$.\n\n\\bigskip\n\nThis condition is equivalent to each of the following ones\n\\medskip\n\n\\item[(b)]  $(\\forall n\\in \\mathbb{N}$, $n\\leqslant card(X))$\n$(\\forall f,g\\in\\mathbb{T}^{(n)}(\\mathfrak{A}))$ $(\\forall p:\nX\\rightarrow A)$ $(\\forall a_{1},\\ldots ,a_{n}\\in X)$\n$$\n\\big[f(a_{1}^{n})=g(a_{1}^{n})\\Longrightarrow f(p(a_{1}),\\ldots\n,p(a_{n}))=g(p(a_{1}),\\ldots ,p(a_{n}))\\big],\n$$\n\n\\item[(c)]  $(\\forall p\\in A^{X}) \\ (\\exists\\bar{p}\\in Hom(\\langle\nX\\rangle_{\\mathfrak{A}},\\mathfrak{A})) \\ \\bar{p}|_{X}=p$, where\n$\\langle X\\rangle_{\\mathfrak{A}}$ is a subalgebra of\n$\\mathfrak{A}$ generated by $X$,\n\n\\item[(d)]  $\\langle X\\rangle _{\\mathfrak{A}}$ is a $\\mathbb{K}$-{\\em free\nalgebra $\\mathbb{K}$-freely generated by} $X$, where\n$\\mathbb{K}=\\{\\mathfrak{A}\\}$ (or, by Birkhoff Theorem,\n$\\mathbb{K}=\\mathcal{H}\\mathcal{S}\\mathcal{P}\\{\\mathfrak{A}\\}$, a variety gene\\-rated\nby $\\mathfrak{A}$).\n\\end{enumerate}\n\n\\medskip\n\nBasic properties of $\\mathcal{M}$-independence are the following\nones:\n\\begin{itemize}\n\\item  (``\\textit{hereditarity}'') $X\\in Ind\\;(\\mathfrak{A},\\mathcal{M})$, \\\n$Y\\subseteq X\\Longrightarrow Y\\in\nInd\\;(\\mathfrak{A},\\mathcal{M})$,\\vspace{5pt}\n\n\\item  $(\\forall X\\subseteq A)$ $(\\forall\\,{\\rm finite} \\ Y\\subseteq\nX)\\;\\big( Y\\in Ind(\\mathfrak{A},\\mathcal{M})\\Longrightarrow X\\in\nInd(\\mathfrak{A},\\mathcal{M})\\big)$\\\\\n(i.e. the family $\\mathbb{J}=Ind(\\mathfrak{A},\\mathcal{M})$ is of finite\ncharacter).\n\\end{itemize}\n\nThe notion of $\\mathcal{M}$-independence is stronger than that of\nindependence with respect to the closure operator of such a kind\n$X\\mapsto\\langle X\\rangle_{\\mathfrak{A}}$ (for $X\\subseteq A$).\n\nThere are some notions of independence which are not special cases\nof $\\mathcal{M}$-independence, such as:\n\n\\noindent\n\\begin{tabular}{rl}\n$\\bullet $ & linear independence in abelian groups, \\\\\n$\\bullet $ & independence with respect to a closure operator\n$\\mathcal{C}$\n(i.e. $\\mathcal{C}$-independence), \\\\\n$\\bullet $ & stochastic independence, \\\\\n$\\bullet $ & ``independence-in-itself'' defined by J.~Schmidt (in 1962), \\\\\n$\\bullet $ & ``weak independence'' used by S.~\\thinspace\n\\'{S}wierczkowski (in 1964).\n\\end{tabular}\n\nFor this reason, a general notion of independence with respect to\na family of mappings was proposed by E.~Marczewski in 1966 (and\nstudied in \\cite{Mar'69} and \\cite{Gla71}). This notion is general\nenough to cover the above-mentioned kinds of independences.\n\nLet $\\emptyset\\neq X\\subseteq A$ and\n$$\n\\mathcal{Q}_X\\subseteq A^X=\\mathcal{M}_X=\\{p\\;|\\;p:X\\rightarrow A\\},\n$$\n$$\n\\mathcal{Q}(A)=\\mathcal{Q}=\\bigcup\\{\\mathcal{Q}_X \\;|\\;X\\subseteq A \\},\n$$\n$$\n\\mathcal{M}(A)=\\mathcal{M}=\\bigcup\\{A^X \\;|\\;X\\subseteq A\\}.\n$$\n\nFor an algebra $\\mathfrak{A} = (A, \\mathbb{F})$, a mapping \\ $p:X\n\\rightarrow A$ belongs to $\\mathcal{H}_X (\\mathfrak{A})$ if and only if\nthere exists a homomorphism \\ $\\bar{p}:\\langle X\\rangle\n_{\\mathfrak{A}}\\rightarrow A$ such that $\\bar{p}|_{X}=p$.\n\nThe set $X$ is said to be $\\mathcal{Q}$-{\\it independent} ($X\\in\nInd(\\mathfrak{A},\\mathcal{Q})$, for short) if\n\n\\begin{center}\n$\\mathcal{Q}_X \\subseteq \\mathcal{H}_X (\\mathfrak{A})$\n\\end{center}\n\n\\noindent or, equivalently,\n$$\n(\\forall p\\in \\mathcal{Q}_{X})\\;(\\forall\\;{\\rm finite }\\; n\\leqslant\ncard(X))\\;(\\forall f,g\\in \\mathbb{T}^{(n)}(\\mathfrak{A}))\\;\n(\\forall a_{1},\\ldots ,a_{n}\\in X)\\;$$\n$$\\big[f(a_{1}^{n})=g(a_{1}^{n})\\Longrightarrow f(p(a_{1}),\\ldots\n,p(a_{n}))=g(p(a_{1}),\\ldots ,p(a_{n}))\\big].$$\n\n\\medskip\n\\noindent{\\bf Examples.} (In the following examples we will use a\nterminology which differs from the original one.)\n\n\\begin{enumerate}\n\\item[1)] $\\mathcal{Q}=\\mathcal{M}=\\bigcup \\{A^{X}\\;|\\;X\\subseteq A\\};$ $\\mathcal{M}$-\\textit{independence}\n(E.~Marczewski: \\textit{general algebraic independence},\n\\cite{Mar'58}),\n\n\\item[2)] $\\mathcal{Q}=\\mathcal{G}=\\bigcup \\{p|_{X}\\;|\\;p\\in A^{A}$ is \\ diminishing,\n$X\\subseteq A\\};\\;\\mathcal{G}$-{\\it independence }(G.~Gr\\\"{a}tzer:\n\\textit{weak independence}, \\cite{Grat67}), where a mapping $p$ is\ncalled \\textit{diminishing} if\n$$\n(\\forall f,g\\in \\mathbb{T}^{(1)}(\\mathfrak{A}))\\;(\\forall a\\in A)\n\\;\\big[f(a)=g(a)\\Longrightarrow f(p(a))=g(p(a))\\big].$$\n\\end{enumerate}\nFor abelian groups, the notion of $\\mathcal{G}$-independence gives\nus the well-known \\textit{linear independence}.\n\nNow we can able to obtain some results on $\\mathcal{Q}$-independence (for\nspecial families $\\mathcal{Q}$ of mappings, e.g., for $\\mathcal{Q}=\\mathcal{M}$ and $\\mathcal{G}$ ) in\n$HG$-algebras of type $\\mathfrak{H}=(G;+,\\varphi,b)$, where $(G;+)$\nis an abelian group.\n\nIn this case, the equality\n\\begin{equation}\\label{F=G}\nF_1(x_1,\\ldots,x_m)=F_2(x_1,\\ldots,x_m)\n\\end{equation}\n(for two term operations of the form (\\ref{sum}) in $\\mathfrak{H}$)\nis equivalent to the equality\n\\begin{equation}\\label{H=0}\nH(x_1,\\ldots,x_m)=0,\n\\end{equation}\nwhere $H\\in\\mathbb{T}^{(m)}(\\mathfrak{H})$, i.e.\n$H(x_1,\\ldots,x_m)=\\sum\\limits_{i=1}^{m}h_i(x_i)+k_{_H} b$, and\n$0$ denotes the zero of the group $(G;+)$.\n\nConsider a subset $X$ of $G$. Let for $a_1,\\ldots,a_m\\in X$ the\nequality\n\\begin{equation}\\label{H_a}\nH(a_1,\\ldots,a_m)=0,\n\\end{equation}\nhold. Taking into account the mapping $p:X\\to\\langle\nX\\rangle_{\\mathfrak{A}}$ defined by $p(a_i)=0$ and $p(x)=x$ for\n$x\\in X\\setminus\\{a_1,\\ldots,a_m\\}$, we get $k_{_H}b=0$. (We\nobserve that such mapping $p$ belongs to families $\\mathcal{M}$ and $\\mathcal{G}$ .)\nTherefore we have\n$$\n\\sum\\limits_{i=1}^{m} h_i(a_i)=0.\n$$\nConsider the mapping $q_j:X\\to \\langle X\\rangle_{\\mathfrak{A}}$\ndefined for fixed $j\\in\\{1,\\ldots,m\\}$ as follows:\n\\[\nq_j(x)=\\left\\{\\begin{array}{ccl} a_j&{\\rm if }&x=a_j ,\\\\[4pt]\n0&{\\rm if }&x\\ne a_j.\\end{array}\\right.\n\\]\nWe obtain $h_j(a_j)=0$ for all $j=1,2,\\ldots,m$. (In the\nconsidered case all $q_j$ belong to $\\mathcal{M}$ and $\\mathcal{G}$ .)\n\nIn particular, we can easily observe, by similar considerations,\nthat the following result holds:\n\n\\begin{theorem}\nLet $X\\subseteq G$ be a subset of the $HG$-algebra\n$\\mathfrak{H}=(G;+,\\varphi,b)$. Then $X\\in Ind(\\mathfrak{H},\\mathcal{G})$ if and only if\nfor any $m\\leqslant card(X)$ for all $a_1,\\ldots,a_m\\in X$ and\nevery term operation\n$H(x_1,\\ldots,x_m)=\\sum\\limits_{i=1}^{m}h_i(x_i)+k_{_H}b$ the\nequality\n\\begin{equation}\n\\sum\\limits_{i=1}^{m}h_i(a_i)+k_{_H}b=0\n\\end{equation}\nis equivalent with\n\\[\n\\left(\\forall i\\in\\{1,\\ldots,m\\}\\right)\\,\\left(h_i(a)=0\\;\\&\\;\nk_{_H}b=0\\right).\n\\]\n\nMoreover, $X$ is $\\mathcal{M}$-independent in this $HG$-algebra\niff for all pairwise different elements $a_1,\\ldots,a_m$ from $X$\nequality $(26)$ implies $h_i(x)=0$ for all $i=1,2,\\ldots,m$ and\n$\\,k_Hb=0$.\n\\end{theorem}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe evaluation of the exponential of a square matrix $\\e^{\\bf A}$ is a classic problem of computational linear algebra.[1] A large number of methods have\nbeen proposed and used for its evaluation. None of these methods have produced a generally applicable method which is sufficiently accurate as well as being reasonably fast.\nThus this problem might be considered unsolved. It is clearly an important and pervasive problem which arises in a wide variety of contexts.[2,3] A method which is capable of producing\na solution to essentially arbitrary precision would thus be of great importance. The present work uses a variant of a method described in Ref. 1,  namely the introduction of an\nartificial  time-parameter\nwhich produces an initial-value problem.  Instead of calling a \\lq canned\\rq \\ solver, the present work uses a method introduced by several of the present authors [4] to solve quantum mechanical\ninitial-value problems. In this method, a finite element technique is used to propagate from an initial condition at $t=0$, which is the unit matrix, to the desired result at $t=1$. The time axis\nis broken up into an arbitrary number of time elements and the solution is propagated from element to element, using a special basis in time introduced here for the first time. \n\nThe next section presents the analysis of the problem and describes the solution algorithm. Then the algorithm is applied to evaluate the exponential of a number of test matrices. Finally, the\nconclusions are presented.\n\n\\section*{Analysis and Solution Algorithm}\n\nThe problem at hand is the evaluation of the exponential of a square (generally complex) matrix $e^{\\bf A}$. The present method introduces an\nartificial time parameter so as to transform the evaluation into the solution of an initial-value problem. For a given  $n\\times n$ square matrix $\\mathbf A$, consider the following parametrized function definition:\n\n\\begin{equation} \n   \\mathbf{\\Psi}(t) \\equiv \\rme^{\\mathbf{A}t}, \\\\\n\\end{equation}\n\n\\noindent\nwhere $\\bf \\Psi$ is also an $n\\times n$ square matrix and  the desired solution is $\\mathbf{\\Psi}(1)=\\rme^{\\mathbf{A}}$ which evolves\nfrom the initial value given by $\\mathbf{\\Psi}(0)=\\mathbf{1}$ ($\\mathbf{1}$ is the diagonal unit matrix). This is a solution of the following linear ordinary differential equation, written for each matrix element,\n\\begin{equation}\n\\dot{\\Psi}_{ij}(t)=\\sum_{k=1}^{n}A_{ik}\\Psi_{kj}(t),\n\\end{equation}\n\n\\noindent\nwhere the over-dot stands for the time-derivative. The time axis extends over the interval $[0, 1]$. Now break the time axis\nup into elements that extend between nodes $t_{\\rm{i}}$ and $t_{\\rm{i+1}}$, and define a local time $\\tau$ that spans $[-1, 1]$.\nThe local time transformation is defined by the relation, \n\\begin{equation}\n\\tau =qt -p,\n\\end{equation}\n\n\\noindent\nwhere, $q =  2\/(t_{\\rm{i+1}} - t_{\\rm{i}}) $ and $p= (t_{\\rm{i+1}} + t_{\\rm{i}})\/(t_{\\rm{i+1}} - t_{\\rm{i}})$.\nThus, for an arbitrary time element $e$, Eq. $(2)$ can be written in terms of local time $\\tau$ as\n\n\\begin{equation}\nq\\dot{\\Psi}^{(e)}_{ij}(\\tau)=\\sum_{k=1}^{n}A^{(e)}_{ik}\\Psi^{(e)}_{kj}(\\tau).\n\\end{equation}\n\nAt this point, we will use the following \\emph{ansatz} for $\\bf \\Psi^{(e)}$ to enforce continuity between two consecutive finite elements\n\n\\begin{equation}\n\\Psi^{(e)}_{ij}(\\tau) = f^{(e)}_{ij}(\\tau)+\\Psi^{(e-1)}_{ij}(+1), \\qquad f^{(e)}_{ij}(-1) = 0\n\\end{equation}\n\n\\noindent\nand expand $f^{(e)}_{ij}(\\tau)$ as\n\n\\begin{equation}\nf^{(e)}_{ij}(\\tau) = \\sum_{\\mu=0}^{m-1}  B_{\\mu}^{ij(e)} s_\\mu (\\tau)\n\\end{equation}\n\n\\noindent\nin a basis we define by\n\n\\begin{equation}\ns_{\\mu}(\\tau) = \\int_{-1}^\\tau T_{\\mu}(\\tau) \\, \\rm{d}\\tau\n\\end{equation}\n\n\\noindent\nwhere $T_{\\mu}(\\tau)$ are Chebyshev Polynomials of the first kind.[5] Note that these basis functions enforce the\ninitial condition on the $f$'s given in Eq. (5) since $s_{\\mu}(-1)=0$. The result for the decomposition of $f$ in $m$ basis functions is\n\n\\begin{eqnarray}\n\\Psi^{(e)}_{ij}(\\tau) = \\sum_{\\mu=0}^{m-1} B_{\\mu}^{ij(e)} s_\\mu (\\tau) +\\Psi_{ij}^{(e-1)}(+1) \\\\\n\\dot{\\Psi}^{(e)}_{ij} (\\tau) = \\sum_{\\mu=0}^{m-1} B_{\\mu}^{ij(e)} T_\\mu (\\tau).\n\\end{eqnarray}\n\n\\noindent\nNow, insert (8) and (9) into Eq. (4), and multiply from the left by $w(\\tau) s_{\\mu'}(\\tau)$ and integrate from $-1$ to $+1$\n(note that  $w(\\tau) = (1- \\tau^2)^{-1\/2}$ is the weighting function for Chebyshev polynomials). Rearranging terms we get,\n\n\\begin{equation}\n\\fl \\eqalign {q\\sum_{\\mu} [\\int_{-1}^{1} s_{\\mu'}(\\tau) \\omega (\\tau) T_{\\mu}(\\tau) \\, \\rm{d} \\tau] B_{\\mu}^{ij(e)} =\n\\sum_{k\\mu} A^{(e)}_{ik} [\\int_{-1}^{1} s_{\\mu'}(\\tau) \\omega (\\tau) s_{\\mu}(\\tau) \\, \\rm{d} \\tau] B_{\\mu}^{kj(e)} \\cr\n+ \\sum_{k} A^{(e)}_{ik} [\\int_{-1}^{1} s_{\\mu'}(\\tau) \\omega (\\tau) T_{0}(\\tau) \\, \\rm{d} \\tau] \\Psi_{kj}^{(e-1)}(+1)}\n\\end{equation}\n\n\\noindent\nwhere, $T_{0}(\\tau) = 1$. Defining, the integrals in the above equation as\n\n\\begin{eqnarray}\nC_{\\mu' \\mu} &\\equiv \\int_{-1}^{1} s_{\\mu'}(\\tau) \\omega (\\tau) T_{\\mu}(\\tau) \\, \\rm{d} \\tau \\\\\nD_{\\mu' \\mu} &\\equiv \\int_{-1}^{1} s_{\\mu'}(\\tau) \\omega (\\tau) s_{\\mu}(\\tau) \\, \\rm{d} \\tau \\\\\ng_{\\mu'} &\\equiv \\int_{-1}^{1} s_{\\mu'}(\\tau) \\omega (\\tau) T_{0}(\\tau) \\, \\rm{d} \\tau\n\\end{eqnarray}.\n\n\\noindent\nand substituting Eqs. (11-13) into Eq. (10) gives,\n\n\\begin{equation}\n{q\\sum_{\\mu} C_{\\mu' \\mu} B_{\\mu}^{ij(e)} =\n\\sum_{k\\mu} A^{(e)}_{ik} D_{\\mu' \\mu} B_{\\mu}^{kj(e)}\n+ g_{\\mu'} \\sum_{k} A^{(e)}_{ik} \\Psi_{kj}^{(e-1)}(+1)}\n\\end{equation}\n\n\\noindent\nor, rearranging\n\n\\begin{equation}\n\\sum_{\\mu k} (q C_{\\mu' \\mu} \\delta_{ik} - A^{(e)}_{ik} D_{\\mu' \\mu} )B_{\\mu}^{kj(e)}\n= g_{\\mu'} \\sum_{k} A^{(e)}_{ik} \\Psi_{kj}^{(e-1)}(+1)\n\\end{equation}\n\n\\noindent\nwhere $\\delta_{ik}$ is the usual Kronecker delta function. Then rewrite Eq. (15) as\n\n\\begin{equation}\n\\sum_{\\mu k} \\Omega^{(e)}_{(\\mu' i)(\\mu k)}B_{\\mu}^{kj(e)}\n= \\Gamma_{\\mu'}^{ij(e,e-1)}\n\\end{equation}\n\n\\noindent\nwhere\n\n\\begin{eqnarray}\n\\Omega^{(e)}_{(\\mu' i)(\\mu k)} &\\equiv (q C_{\\mu' \\mu} \\delta_{ik} - A^{(e)}_{ik} D_{\\mu' \\mu} ) \\\\\n\\Gamma_{\\mu'}^{ij(e,e-1)} &\\equiv g_{\\mu'} \\sum_{k} A^{(e)}_{ik} \\Phi_{kj}^{(e-1)}(+1).\n\\end{eqnarray}\n\nEquation (16) is a set of simultaneous equations of size $(n \\times m)$, which can be written in matrix form as, \n\n\\begin{equation}\n\\mathbf{\\Omega^{(e)} B}^{j (e)}= \\mathbf{\\Gamma}^{j(e,e-1)}\\qquad j = 1, 2,..., n.\n\\end{equation}\n\n\\noindent\nHere, $\\mathbf{\\Omega^{(e)}}$ is a (complex) matrix and for each $j$, $\\mathbf{\\Gamma^{j(e,e-1)}}$ and $\\mathbf{B^{j(e)}}$ are vectors. \nEq. (19) applies for each time element $e$. The solution is propagated from element to element, from $t=0$ to $t=1$. \nThe above equation can be solved numerically in many ways, but we have chosen the method of LU decomposition.[6]\nThe present method is ideally suited to high-performance computers where the solver of choice would probably\nbe iterative. In the present case, we\napply LU decomposition to \n$\\mathbf{\\Omega^{(e)}}$ and back substituting all of the $\\mathbf{\\Gamma}^{j(e,e-1)}$'s, we will have all the elements for the matrix (which can also be viewed as three dimensional) $\\mathbf{B^{j(e)}}$. This LU decomposition only needs to be done once since $\\mathbf{\\Omega^{(e)}}$ is independent of time. Thus, the propagation\njust involves a matrix vector multiply.\nThen, we employ Eq. $(8)$ to solve for $\\mathbf{\\Psi^{(e)}}(\\tau = 1)$ for the element e, which, in turn, will be used as $\\mathbf{\\Psi^{(e + 1)}}(\\tau = -1)$ for the next element e + 1. Starting off with a unit matrix for $\\mathbf{\\Psi^{(1)}}(t = 0)$, we continue this process till we calculate $ \\mathbf{\\Psi(t = 1)}$ at the last node, which is the exponential of the given matrix $\\mathbf{A}$.\n\n\\section*{Results}\n\nThe calculations presented below were done on a Macintosh Intel laptop using Gnu C++ which has machine accuracy limit of $2.22045\\times 10^{-16} $.\nAs an illustration, let's borrow a 'pathological' matrix from [1], which we have modified slightly to make it even worse. Consider a matrix $\\mathbf{M1}$ given by,\n\n\n\\begin{eqnarray}\n\\eqalign{\\mathbf{M1} &= \\left[\\begin{array}{cc}-73 & 36 \\\\-96 & 47\\end{array}\\right] \\\\\n&=\\left[\\begin{array}{cc}1 & 3 \\\\2 & 4\\end{array}\\right] \\left[\\begin{array}{cc}-1 & 0 \\\\0 & -25\\end{array}\\right]  {\\left[\\begin{array}{cc}1 & 3 \\\\2 & 4\\end{array}\\right]}^{-1}.}\n\\end{eqnarray}\n\n\\noindent\nThe exponent of $\\mathbf{M1}$ can be easily calculated as,\n\\begin{eqnarray*}\ne^\\mathbf{M1} &= \\left[\\begin{array}{cc}1 & 3 \\\\2 & 4\\end{array}\\right] \\left[\\begin{array}{cc}e^{-1} & 0 \\\\0 & e^{-25}\\end{array}\\right]  \\left[\\begin{array}{cc}-2 & 3\/2 \\\\1 & -1\/2\\end{array}\\right] \\\\\n&= \\left[\\begin{array}{cc}{-2e^{-1}+3e^{-25}} & {(3\/2)(e^{-1}-e^{-25})} \\\\{-4e^{-1}+4e^{-25}} & {3e^{-1}-2e^{-25}}\\end{array}\\right].\n\\end{eqnarray*}\n\n\\noindent\nThe above matrix, exact to $16$ decimal places, is given by\n\n\\begin{equation}\ne^\\mathbf{M1}\\cong \\left[\\begin{array}{cc}-0.7357588823012208 & 0.5518191617363316 \\\\-1.4715177646302175 & 1.1036383234865511\\end{array}\\right].\n\\end{equation}\n\n\\noindent\nThe result of our program is displayed below and we run it by using just $8$ time steps and $8$ basis functions. The result is accurate to $13$ decimal places already.\n\n\\begin{equation}\ne^\\mathbf{M1}\\cong \\left[\\begin{array}{cc}-0.7357588823012(181) & 0.5518191617363(358) \\\\-1.4715177646302(120) & 1.1036383234865(592)\\end{array}\\right].\n\\end{equation}\n\nAs an example of a non - diagonalizable matrix, consider the following matrix $\\mathbf{M2}$, with complex eigenvalues\n\n\\begin{equation}\n\\mathbf{M2} = \\left[\\begin{array}{cc}0 & -1 \\\\1 & 0\\end{array}\\right]\n\\end{equation}\n\n\\noindent\nIt can be shown that,\n\n\\begin{equation}\ne^\\mathbf{M2} = \\left[\\begin{array}{cc}cos(1) & -sin(1) \\\\sin(1) & cos(1)\\end{array}\\right]\n\\end{equation}\n\n\\noindent\nWe are able to achieve $14$ decimal digit accuracy with $8$ time steps and $8$ basis functions.\n\n\\begin{equation}\ne^\\mathbf{M2} \\cong \\left[\\begin{array}{cc} 0.54030230586814 & -0.84147098480790 \\\\ 0.84147098480790 & 0.54030230586814\\end{array}\\right]\n\\end{equation}\n\n\\noindent\n Table 1 shows the minimum number of basis functions, for a given number of time steps, which were required to achieve a precision of $\\pm 1\\times 10^{-14}$ on matrices whose exponential is known exactly. The matrices chosen are: the simplest possible matrix - a $2 \\times 2$ real unit matrix, for which the result is the constant $e$ on the diagonals, and the matrices $\\mathbf {M1}$ and $\\mathbf {M2}$.\n\nLet's check our program on matrices, which we picked randomly and for which we had no $\\it apriori$ knowledge as to the result of their exponentiation. We fixed 8 time steps and\/or 8 basis functions, and varied the other corresponding parameter from 5 to 40 and checked how the results of the program varied in accuracy. For the sake of saving space, we only displayed the result of the last element--the other elements of the matrix exponential behaved similarly. \nThe matrices chosen are a $5 \\times 5$ real matrix $\\mathbf{M3}$,\n\n\\begin{equation}\n\\mathbf{M3} = \\left[\\begin{array}{ccccc}-0.1 & -0.2 & -0.3 & -0.4 & -0.5 \\\\-0.6 & -0.7 & -0.8 & -0.9 & -1 \\\\0.1 & 0.2 & 0.3 & 0.4 & 0.5 \\\\0.6 & 0.7 & 0.8 & 0.9 & 1 \\\\1 & 2 & 3 & 4 & 0\\end{array}\\right]\n\\end{equation}\n\n\\noindent\nand a $3 \\times 3$ complex matrix $\\mathbf{M4}$\n\n\\begin{equation}\n\\mathbf{M4} = \\left[\\begin{array}{ccc}1+i & 1-i & i \\\\1 & 2i & 0 \\\\1+2i & -1+i & -1-i\\end{array}\\right].\n\\end{equation}\n\n\nFrom Table 2, one can see that the numbers up to $12$ decimal digits have saturated after $5$  time steps and\/or basis functions. Similarly, Table 3 shows $13$ digits of accuracy as we switch the two parameters from $5$ to $40$, except for the case of $5$ basis functions, which only shows $8$ accurate significant digits. This shows that for complex matrices, there is inherently more work for the program to handle because of the imaginary part of the matrix elements and there is apparently more sensitivity to the number of basis functions used than to the number of time steps.\n\n\n\\Table{\\label{tone}Minimum number of basis functions and time steps required for a precision of  $\\pm 1\\times 10^{-14}$ for a $2 \\times 2$ unit matrix, $\\mathbf{M1}$ and $\\mathbf{M2}$.}\n\\br\n\\centre{2}{$2\\times 2$ unit matrix}&\\centre{2}{$\\mathbf{M1}$}&\\centre{2}{$\\mathbf{M2}$}\\\\\n\n\\crule{2}&\\crule{2}&\\crule{2}\\\\\nTime steps & Basis functions& Time steps & Basis functions& Time steps & Basis functions\\\\\n\\mr\n\\01&\t11&\t\\0\\005&\t\t-&\t\t\\01&\t\t11\\\\\n\\02&\t\\09&\t\\0\\008&\t\t7&\t\\02&\t\t\\09\\\\\n\\04&\t\\08&\t\\016&\t\t6&\t\\04&\t\t\\08\\\\\n\\08&\t\\07&\t\\050&\t\t5&\t\\08&\t\t\\07\\\\\n16&\t\\06&\t256&\t\t\t4&\t15&\t\t\\06\\\\\n58&\t\\05&\t\\0\\0-&\t\t-&\t\t40&\t\t\\05\\\\\n\\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\n\\Table{\\label{ttwo}Results of matrix $e^{M3}_{5 5}$ for typical runs of 8 time steps and 8 basis functions.}\n\\br\nResult& Time steps & Basis functions\\\\\n\\mr\n\\underline{3.210309305973}118 &\t\t\\05&\t\t\\08\\\\\n\\underline{3.210309305973}288 &\t\t40&\t\t\\08\\\\\n\\underline{3.210309315373}377&\t\t\\08&\t\t\\05\\\\\n\\underline{3.210309305973}281&\t\t\\08&\t\t40\\\\\n\\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\n\\Table{\\label{tthree}Results of matrix $e^{M4}_{3 3}$ for typical runs of 8 time steps and 8 basis functions.}\n\\br\nResult& Time steps & Basis functions\\\\\n\\mr\n\\underline{-0.5119771222980}63 - i \\underline{0.0897728113135}12 &\t\t\\05&\t\t\\08\\\\\n\\underline{-0.5119771222980}81 - i \\underline{0.0897728113135}26 &\t\t40&\t\t\\08\\\\\n\\underline{-0.51197712}1264660 - i \\underline{0.08977281}0979965&\t\t\\08&\t\t\\05\\\\\n\\underline{-0.5119771222980}82 - i \\underline{0.0897728113135}26&\t\t\\08&\t\t40\\\\\n\\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\n\\section*{Conclusion}\n\nWe have presented a robust, easily used, and accurate algorithm for the evaluation of the exponential of a matrix. We did this by introducing an\nartificial time parameter and evaluating the matrix exponential as the solution of an initial-value problem in this artificial time. We solved the initial-value problem\nby using finite elements in time with a new time basis which we defined here so as to enforce the initial conditions on the solution at the beginning of each time finite element. This resulted in set of simultaneous equations for the expansion coefficients. \nThe actual algorithm employed here was an LU decomposition which was very fast and efficient. The relative efficiency of the method should be most\napparent when implemented on high-performance computers since the algorithm is highly parallel.\nThe method was applied to several matrices as a proof of the validity of the algorithm. \nThe results of our calculations show that we only need about $8$ basis functions and $8$ time steps for the matrices considered for accuracies as great as $13$ significant digits. We trust that this method of numerically  calculating the exponential of a matrix will be recognized  to be a \\emph{nondubious} one! \\\\\n\n\\section*{Acknowlegements}\nThis work was supported by the NSF CREST Center for Astrophysical Science and Technology under Cooperative Agreement \nHRD-0630370.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgment}\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Introduction} \\label{sec:intro}\n\n\\IEEEPARstart{D} {ynamic} systems which exhibit both continuous state evolution and discrete state transitions can typically be modeled as {\\em hybrid automata (HA)} (\\cite{henzinger:96, lynch:03}).\nComputing the reach set of a hybrid automaton from a given set of initial states is a problem of fundamental importance as it is related to safety verification and automated controller synthesis.\nEven though many systems can be so modeled, it is in general undecidable to compute the exact reach set \\cite{henzinger:95} except for classes of hybrid automata whose continuous dynamics are fairly simple, such as timed automata (TA) \\cite{alur:94} and initialized rectangular hybrid automata (IRHA) \\cite{henzinger:95}.\nNeither of these automata allow the standard linear systems dynamics which is widely used for control systems. \nTo broaden the class of systems that can be addressed, research in hybrid system verification in the recent years has focused on algorithms computing over-approximations of the reach set for various classes of hybrid automata (\\cite{henzinger:97, frehse:08, chutinan:03, girard:05, asarin:00, kurzh:00, clarke:03, tiwari:02}).\nHowever, even with this relaxation from exact reach set to over-approximations, it is still a challenging problem to compute an over-approximation of the reach set of hybrid automata with linear dynamics with arbitrarily small approximation error and a termination guarantee for the computation.\n\n\n\\subsection{Related Work}\nFor the computation of reach set of hybrid automata with linear dynamics, several tools and approaches have been proposed in the literature.\nAs an example, HyTech \\cite{henzinger:97} computes the reach set of hybrid automata whose continuous dynamics are more general than those of IRHA by translating the original model into an IRHA if the model is {\\em clock translatable}.\nOtherwise, an over-approximate reach set is computed through an approach, called {\\em linear phase-portrait approximation}, which approximates the original hybrid automaton by relaxing the continuous dynamics of the original automaton.\nPHAVer \\cite{frehse:08} can handle a class of systems called linear hybrid automata that have affine dynamics. \nIt computes a conservative over-approximation of the reach set of such hybrid automata through on-the-fly over-approximation of the phase portrait, which is a variation of the phase-portrait approximation in \\cite{henzinger:97}.\nRecently, another tool, SpaceEx, has been developed based on the algorithm called LeGuernic-Girard (LGG) algorithm \\cite{guernic:10} which allows the handling of hybrid automata with linear differential equations with a larger number of continuous variables compared to other approaches.\n\nIn \\cite{chutinan:03}, a class of hybrid automata, called {\\em polyhedral-invariant hybrid automata (PIHA)}, is defined and an algorithm is proposed to construct a finite state transition system, which is a conservative approximation of the original PIHA. \nDetermining a polyhedral approximation of each sampled segment of the continuous state evolution between switching planes is the underlying fundamental technique in the algorithm that is used.\nAnother approach proposed in \\cite{asarin:00} is also based on the idea of sampling and polyhedral over-approximation of continuous state evolution of a continuous linear dynamics. \nOn the other hand, in \\cite{kurzh:00} and \\cite{girard:05}, ellipsoids and zonotopes are used respectively for approximating continuous state evolution.\n\nHowever, while these algorithms and tools compute some over-approximation of the reach set of hybrid systems with linear dynamics, computation of an over-approximate reach set which is arbitrarily close to the exact reach set of such hybrid systems with guaranteed termination remains an open issue for further research. \n\n\n\n\\subsection{Challenges and Contributions}\nIn general, the key challenges in reach set computation of HA are \n\\begin{inparaenum}[(i)] \n\t\\item to over-approximate the exact continuous flow with arbitrarily small approximation error,\n\t\\item to determine when and where a discrete transition occurs, and\n\t\\item to develop a reach set computation algorithm with termination guarantee. \n\\end{inparaenum}\nIn this paper, we address the problem of computing an over-approximation of the reach set of a special class of hybrid automata, called {\\em Deterministic and Transversal Linear Hybrid Automaton (DTLHA)}, starting from an initial state over a finite time interval.\nWe call such an over-approximate reach set as a {\\em bounded $\\epsilon$-reach set}.\nOur approach can be related to other approaches that use sampling and polyhedral over-approximation as in \\cite{chutinan:03, asarin:00}.\nThe main contributions of our approach are as follows: \n\\begin{inparaenum}[(i)]\n\t\\item We show that an over-approximation of the reach set of a DTLHA can be computed arbitrarily closely to the exact reach set.\n\t\\item We also show that such computation is guaranteed to terminate under a deterministic and transversal restriction on the discrete dynamics. \n\t\\item Furthermore, to facilitate practical computation, we extend these theoretical results to consider the numerical calculation errors caused by finite precision calculation capabilities. \n\\end{inparaenum}\nBased on the theoretical results, we propose an algorithm to compute a bounded $\\epsilon$-reach set of a DTLHA, as well as a software architecture that is designed to improve the flexibility and the efficiency in computing such an over-approximation.\n\nThe paper is organized as follows.\nIn Section \\ref{sec:pre}, we introduce definitions and notations that are used throughout this paper.\nIn Section \\ref{sec:theory}, we show that, for arbitrarily small $\\epsilon > 0$, a bounded $\\epsilon$-reach set of a DTLHA starting from an initial state can be computed under the assumption of infinite precision numerical calculation capabilities. \nIn Section \\ref{sec:cond}, we first derive a set of conditions for computation of a bounded $\\epsilon$-reach set, and then extend these conditions to consider errors caused by finite precision numerical calculation capabilities.\nIn Section \\ref{sec:design}, we propose an algorithm for a bounded $\\epsilon$-reach set computation, as well as an architecture for software implementation of the proposed algorithm.\nFinally, we illustrate an example of bounded $\\epsilon$-reach set computation in Section \\ref{sec:imp}, followed by concluding remarks in Section \\ref{sec:con}.\n\n\n\n\n\n\\section{Preliminaries} \\label{sec:pre}\n\n\nLet $\\mathcal{X} \\subset \\mathbb{R}^n$  be a continuous state space over which a hybrid automaton is defined.\nFor a polyhedron $\\C \\subseteq \\mathbb{R}^n$, we denote its interior by $\\C^{\\circ}$, and its boundary by $\\partial \\C$.\nWe will also use the notation $\\B_r(x)$ to denote a closed ball of radius $r$ with center $x$, i.e., $\\B_r(x) := \\lbrace y \\in \\mathbb{R}^n : \\Vert y - x \\Vert \\leq r \\}$. \nThe specific norm that we use in the definition of $\\B_r(x)$ as well as the sequel is the $\\ell_{\\infty}$-norm. \nSince we are using the $\\ell_{\\infty}$-norm, $\\B_r(x)$ is a hypercubic neighborhood of $x$.\nOne of the advantages of using the $\\ell_{\\infty}$-norm is that the induced hypercubic neighborhood is easily computed. \nMore generally, a hypercube is a special case of a polyhedron, which is important since it is easy to propagate the image of this set under linear dynamics.\nThis is useful in Section \\ref{sec:theory} when we describe our approach for bounded $\\epsilon$-reach set computation.\n\n\nWe now describe the class of hybrid automata considered.\nWe assume that $\\mathcal{X}$ is a closed and bounded subset of Euclidean space, and is partitioned into a collection of polyhedral regions $\\mathcal{C} := \\{\\mathcal{C}_1, \\cdots, \\mathcal{C}_m \\}$ such that $\\C^{\\circ}_i \\ne \\emptyset$ for each $i \\in \\{1, \\cdots, m\\}$ and \n\n\n\\begin{equation}\\label{eq:pre:tess}\n\t\\bigcup_{i=1}^{m} \\mathcal{C}_{i} = \\mathcal{X}, \\quad \\mathcal{C}_{i}^{\\circ} \\cap \\mathcal{C}_{j}^{\\circ} = \\emptyset \\quad for~i \\ne j,\n\\end{equation}\nwhere $m$ is the size of the partition, and each $\\C_i$ is a polyhedron, called \\emph{cell}. \nTwo cells $\\C_i$ and $\\C_j$ are said to be \\emph{adjacent} if the affine dimension of $\\partial \\mathcal{C}_i \\cap \\partial \\mathcal{C}_j$ is $(n-1)$, or, equivalently, cells $\\C_i$ and $\\C_j$ intersect in an $(n-1)$-dimensional facet. \nTwo cells $\\C_i$ and $\\C_j$ are said to be \\emph{connected} if there exists a sequence of adjacent cells between $\\C_i$ and $\\C_j$.\n\n\n\\begin{defn} \\label{def:lha}\nAn $n$-dimensional \\emph{Linear Hybrid Automaton (LHA)},\\footnote{In the hybrid system literature \\cite{henzinger:97, alur:93} the word ``linear automaton'' has been used to denote a system where the differential equations and inequalities involved have constant right hand sides. This does not conform to the standard notion of linearity where the right hand side is allowed to be a function of state. In particular, it does not include the standard class of linear time-invariant systems that is of central interest in control systems design and analysis. We use the term ``linear'' in this latter more mathematically standard way that therefore encompasses a larger class of systems, and, more importantly, encompasses classes of switched linear systems that are of much interest.} \nis a tuple $(\\mathbb{L}, Inv, A, u, \\xrightarrow{G})$ satisfying the following properties: \n\\begin{enumerate}[(a)]\n\t\\item $\\mathbb{L}$ is a finite set of \\emph{locations} or \\emph{discrete states}. The state space is $\\mathbb{L} \\times \\mathbb{R}^n$, and an element $(l, x) \\in \\mathbb{L} \\times \\mathbb{R}^n$ is called a \\emph{state}.\n\t%\n\t\\item[(b)] $Inv: \\mathbb{L} \\rightarrow 2^{\\C}$ is a function that maps each location to a set of cells, called an \\emph{invariant set} of a location, such that \n\t \\begin{inparaenum}[(i)]\n\t\t\\item for each $l \\in \\mathbb{L}$, all the cells in $Inv(l)$ are connected, \n\t\t\\item for any two locations $l, l' \\in \\mathbb{L}$, $Inv(l)^{\\circ} \\cap Inv(l')^{\\circ} = \\emptyset$, and \n\t\t\\item $\\bigcup_{l \\in \\mathbb{L}} Inv(l) = \\X$.\n\t\\end{inparaenum}\n\t%\n\t\\item[(c)] $A: \\mathbb{L} \\rightarrow \\mathbb{R}^{n \\times n}$ is a function that maps each location to an $n \\times n$ real-valued matrix, and \n\t%\n\t\\item[(d)] $u: \\mathbb{L} \\rightarrow \\mathbb{R}^n$ is a function that maps each location to an $n$-dimensional real-valued vector.\n\t%\n\t\\item[(e)] $\\xrightarrow{G}: (\\mathbb{R}^n, \\mathbb{L}) \\times (\\mathbb{R}^n, \\mathbb{L})$ is a binary relation which defines a \\emph{discrete transition} from one state $(x_1, l_1)$ to another state $(x_2, l_2)$ such that $(x_1, l_1) \\xrightarrow{G} (x_2, l_2)$ when $G$ is satisfied and $x_2$ is set to $x_1$ after a discrete transition.\n\\end{enumerate}\n\\end{defn}\nIn the sequel, for each $l_i \\in \\mathbb{L}$, we use $A_i$, $u_i$, $Inv_i$ to denote $A(l_i)$, $u(l_i)$, and $Inv(l_i)$, respectively.\n\nAn example LHA which satisfies Definition \\ref{def:lha} is shown in Section \\ref{sec:imp:example}.\nNext, we define the behavior of LHA.\n\\begin{defn} \\label{def:traj}\nFor a location $l_i \\in \\mathbb{L}$, a \\emph{trajectory} of duration $t \\in \\mathbb{R}^{+}$ for an $n$-dimensional LHA $\\A$ is a continuous map $\\eta$ from $[0,t]$ to $\\mathbb{R}^n$, such that\n\\begin{enumerate}[(a)]\n    \\item $\\eta(\\tau)$ satisfies the differential equation\n            \\begin{equation}\\label{eq:pre:lti}\n                \\dot{\\eta}(\\tau) = A_i \\eta(\\tau) + u_i ,\n            \\end{equation}\n    \\item $\\eta(\\tau) \\in Inv_i$ for every $\\tau \\in [0,t]$.\n\\end{enumerate}\n\\end{defn}\n\n\n\\begin{defn} \\label{def:exec\nAn \\emph{execution} $\\alpha$ of an LHA $\\A$ from a starting state $(l_0, x_0) \\in \\mathbb{L} \\times \\mathbb{R}^n$ is defined to be the concatenation of a finite or infinite sequence of trajectories $\\alpha = \\eta_0 \\eta_1 \\eta_2 \\ldots$, such that\n\\begin{enumerate}[(a)]\n    \\item $\\eta_0(0) = x_0$,\n    \\item $\\eta_{k}(0) = \\eta_{k-1}(\\eta_{k-1}.dur)$ for $k \\ge 1$, \n\\end{enumerate}\nwhere $\\eta_k$ represents a trajectory defined at some location $l \\in \\mathbb{L}$ and $\\eta_k.dur$ denotes the duration of  $\\eta_k$.\nWe also define $\\alpha.dur := \\sum_k \\eta_k.dur$ where $\\alpha.dur$ denotes the duration of an execution $\\alpha$.\n\\end{defn}\n\n\nWe can represent an execution $\\alpha$ of an LHA $\\A$ from an initial condition $(l_0, x_0) \\in \\mathbb{L} \\times \\mathbb{R}^n$ for time $[0,t]$ as a continuous map $x:[0,t] \\rightarrow \\mathbb{R}^n$ such that\n\\begin{inparaenum}[(a)]\n    \\item $t = \\alpha.dur$,\n    \\item $x(0) = x_0 \\in Inv_0$,\n    \\item $x(\\tau_k) = \\eta_{k}(0)$, and\n    \\item $x(\\tau) = \\eta_{k-1}(\\tau - \\tau_{k-1})$ for $\\tau \\in [\\tau_{k-1}, \\tau_k]$,\n\\end{inparaenum}\nwhere $\\tau_0 = 0$, and $\\tau_k = \\sum_{i=0}^{k-1} \\eta_i.dur$ for $k \\ge 1$.\nNote that $\\tau_k$ for $k \\ge 1$ represents the time at the $k$-th discrete transition between locations and the continuous state is not reset during discrete transitions.\n\n\\begin{defn} \\label{def:trans}\nFor an execution $x(t)$ of an LHA, a discrete transition $(x_i, l_i) \\xrightarrow{G} (x_j, l_j)$ occurs if $x_i = x(\\tau')$ for some time $\\tau'$, \n$x(\\tau') \\in Inv_i \\cap Inv_j$ and $x(\\tau') = \\lim_{\\tau \\nearrow \\tau'} x(\\tau)$ where $x(\\tau) \\in (Inv_i)^{\\circ}$ for $\\tau \\in (\\tau'-\\delta,\\tau')$ for some $\\delta > 0$.\n\n\\end{defn}\n\\begin{defn} \\label{def:transversal} \nA discrete transition is called \\emph{deterministic} if there is only one location $l_j \\in \\mathbb{L}$ to which a discrete transition state $x(\\tau_k)$ can make a discrete transition from $l_i$. \nWe call a discrete transition a \\emph{transversal discrete transition} if there exists $\\epsilon > 0$ such that\n\\begin{equation}\\label{eq:pre:trans}\n    \\langle \\dot{x}_{i}(\\tau_k), \\vec{n}_i \\rangle \\ge \\epsilon ~~\\land~~ \\langle \\dot{x}_{j}(\\tau_k), \\vec{n}_i \\rangle \\ge \\epsilon ,\n\\end{equation}\nwhere $\\langle x, y \\rangle$ denotes the inner product between $x$ and $y$, $\\vec{n}_i$ is an outward normal vector of $\\partial Inv_i$ at $x(\\tau_k)$, and $\\dot{x}_{i}(\\tau_k) = A_i x(\\tau_k) + u_i$, and $\\dot{x}_{j}(\\tau_k) = A_j x(\\tau_k) + u_j$ are the vector fields at $x(\\tau_k)$ evaluated with respect to the continuous dynamics of location $l_i$ and $l_j$, respectively. \n\\end{defn}\nFig. \\ref{fig:transition} illustrates a case where $x(\\tau_k)$ satisfies such a deterministic and transversal discrete transition condition.\nNote that if $x(\\tau_k)$ satisfies a deterministic and transversal discrete transition condition, then $x(\\tau_k)$ must make a discrete transition from a location $l_i$ to the other location $l_j$, and $l_j$ has to be unique. \nFurthermore, the \\emph{Zeno behavior}, an infinite number of discrete transitions within a finite amount of time, does not occur if a discrete transition is a transversal discrete transition.\n\\begin{figure}\n\\begin{center}\n  \\includegraphics[width=7cm]{transition.png}\n\\caption{A deterministic and transversal discrete transition from a location $l_i$ to a location $l_j$ occurring at $x(\\tau_k) \\in \\partial Inv(l_i) \\cap \\partial Inv(l_j)$.}\n\\label{fig:transition}\n\\end{center}\n\\end{figure}\n\n\nWe now define a special class of LHA whose every discrete transition satisfies the deterministic and transversality conditions defined in Definition \\ref{def:transversal} as follows:\n\n\\begin{defn} \\label{pre:def:dtlha}\nGiven an LHA $\\mathcal{A}$, a starting state $(l_0, x_0) \\in  \\mathbb{L} \\times \\X$, a time bound $T$, and a jump bound $N$, we call an LHA $\\mathcal{A}$ as a \\emph{Deterministic and Transversal Linear Hybrid Automaton (DTLHA)}  if all discrete transitions in the execution starting from $x_0$ up to time $t_f := \\min \\{T, \\tau_N \\}$ are deterministic and transversal, where $\\tau_N$ is the time at the N-th discrete transition.\n\\end{defn}\n\n\nNext, we define the bounded reach set of a DTLHA and its over-approximation as follows:\n\n\\begin{defn}\nA continuous state in $\\X$ is \\emph{reachable} if there exists some time $t$ at which it is reached by some execution $x$.\n\\end{defn}\n\\begin{defn} \\label{def:pre:reach}\nGiven a state $x_0$ and a time $t$, the \\emph{bounded reach set} up to time $t$, denoted as $\\R_t(x_0)$, of a DTLHA $\\A$ is defined to be the set of continuous states that are reachable for some time $\\tau \\in [0, t]$ by some execution $x$ starting from $x_0 \\in Inv_0$.\n\\end{defn}\n\\begin{defn} \\label{def:pre:ereach}\nGiven $\\epsilon > 0$, a set of continuous states $S$ is called a \\emph{bounded $\\epsilon$-reach set} of a DTLHA $\\A$ over a time interval $[0,t]$ from an initial state $x_0$ if $\\R_t(x_0) \\subseteq S$ and\n\\begin{equation}\\label{eq:pre:ereach}\n\td_H(\\R_t(x_0), S) \\le \\epsilon,\n\\end{equation}\nwhere $d_H(\\P, \\Q)$ denotes the Hausdorff distance \nbetween two sets $\\P$ and $\\Q$\nthat is defined as $d_H(\\P, \\Q) := \\max \\{ \\sup_{p \\in \\P} \\inf_{q \\in \\Q} d(p, q), \\sup_{q \\in \\Q} \\inf_{p \\in \\P} d(p, q) \\}$ where $d(p,q) := \\Vert p - q \\Vert$.\n\\end{defn}\n\n\nIn the sequel, we use $\\D_t(\\P)$ to denote the set of states reached at time $t$ from a set $\\P$ at time $0$.\nSimilarly, for the set of reached states over a time interval $[t_1, t_2)$ from $\\P$, we use $\\D_{[t_1, t_2)}(\\P)$.\nWe also use $\\D_t(\\P, \\gamma)$ to denote an over-approximation of $\\D_t(\\P)$ with an approximation parameter $\\gamma > 0$, calling it a $\\gamma$-approximation of $\\D_t(\\P)$ if it satisfies \n\\begin{inparaenum}[(i)]\n\\item $\\D_t(\\P) \\subset \\D_t(\\P, \\gamma)$ and\n\\item $d_H(\\D_t(\\P),$ $\\D_t(\\P, \\gamma)) \\le \\gamma$.\n\\end{inparaenum}\nNote that $\\D_0(\\P, \\gamma)$ is simply a $\\gamma$-approximation of the set $\\P$.\n\n\n\n\\section{Bounded $\\epsilon$-Reachability of a DTLHA} \\label{sec:theory}\n\nIn this section, we consider the problem of a bounded $\\epsilon$-reach set computation of a DTLHA starting from an initial state over a finite time interval.\nMore precisely, we show that, for any given $\\epsilon > 0$, a DTLHA $\\A$, an initial condition $(l_0, x_0) \\in \\mathbb{L} \\times \\X$, a time upper bound $T \\in \\mathbb{R}^+$, and a discrete transition upper bound $N \\in \\mathbb{N}$, it is possible to compute a bounded $\\epsilon$-reach set of $\\A$ over a finite time interval $[0, t_f]$ under the assumptions that the following computations can be performed exactly:\n\\begin{inparaenum}[(i)]\n\t\\item  $x(t) = e^{At} x_0 + \\int_0^t e^{A(t-s)} u ds$,\n\t\\item the convex hull of a set of finite points in $\\mathbb{R}^n$, and\n\t\\item the intersection between a polyhedron and a hyperplane,\n\\end{inparaenum}\nwhere $t_f$ is as defined in Definition \\ref{pre:def:dtlha}, $A \\in \\mathbb{R}^{n \\times n}$, and $u \\in \\mathbb{R}^n$.\n\n\n\n\\subsection{Bounded $\\epsilon$-Reach Set of a DTLHA at Initial Location}  \\label{sec:theory:l0}\n\nWe first show how a trajectory of a DTLHA can be over-approximated through sampling and polyhedral over-approximation of each sampled state.\nThe basic approach for such over-approximation is shown in Fig. \\ref{fig:traj}. \nIt is necessary that, for a given size of over-approximation of each sampled state, a sampling period $h$ has to ensure that a trajectory $x(t)$ is contained in the computed set of polyhedra.\nFor a given value of $\\epsilon > 0$, we now show how we can determine a sampling period $h$ which guarantees that.\n\\begin{equation}\\label{eq:hcond}\n    \\max_{\\tau \\in [0, h]} \\Vert x(t+\\tau) - x(t) \\Vert < \\epsilon \\qquad \\forall x(t) \\in \\mathcal{X}.\n\\end{equation}\n\nTo determine a suitable value of $h$ which results in (\\ref{eq:hcond}), we suppose $x(s) \\in (Inv_i)^{\\circ}$ for all $s \\in [t, t+h]$ for some location $l_i \\in \\mathbb{L}$.\nThen for a given $\\Sigma_i$, $\\X$, and $x(s) \\in \\X$, we have \n\n\\begin{eqnarray} \\nonumber\n\\max_{s \\in [t, t+\\tau]} \\Vert \\dot{x}(s) \\Vert &=& \\max_{s \\in [t, t+ \\tau]} \\Vert A_i x(s) + u_i \\Vert \\\\ \\nonumber\n&\\le& \\max_{s \\in [t, t+\\tau]} \\{ \\Vert A_i \\Vert \\Vert x(s) \\Vert+ \\Vert u_i \\Vert \\} \\\\ \n&\\le& \\Vert A_i \\Vert \\bar{x}+ \\Vert u_i \\Vert,\n\\end{eqnarray}\nwhere $\\bar{x} = \\max_{x \\in \\mathcal{X}} \\Vert x \\Vert$. \n\nFor a fixed $\\tau \\in [0, h]$, we can compute an upper bound on $\\Vert x(t+\\tau) - x(t) \\Vert$ as follows:\n\\begin{eqnarray} \\label{eq:h:ineq:0} \\nonumber\n\\Vert x(t+\\tau) - x(t) \\Vert &\\le& \\int^{t+\\tau}_{t} \\Vert \\dot{x}(s) \\Vert ds \\nonumber \\\\\n&\\le& \\int^{t+\\tau}_{t} \\max_{s \\in [t, t+ \\tau]} \\Vert \\dot{x}(s) \\Vert ds \\nonumber \\\\\n&\\le& \\int^{t+\\tau}_{t} (\\Vert A_i \\Vert\\bar{x} + \\Vert u_i \\Vert) ds \\nonumber \\\\\n&=& (\\Vert A_i \\Vert \\bar{x} + \\Vert u_i \\Vert) \\tau.\n\\end{eqnarray}\n\nMaximization of both sides of (\\ref{eq:h:ineq:0}) over $\\tau \\in [0, h]$ gives us \n\\begin{eqnarray} \\label{eq:h:ineq:1} \\nonumber\n\t\\max_{\\tau \\in [0,h]} \\Vert x(t+\\tau) - x(t) \\Vert &\\le& (\\Vert A_i \\Vert \\bar{x} + \\Vert u_i \\Vert) h \\\\\n\t&\\le& \\max_{l_i \\in \\mathbb{L}} (\\Vert A_i \\Vert \\bar{x} + \\Vert u_i \\Vert) h.\n\\end{eqnarray}\n\nIf we upper bound the right hand side by $\\epsilon > 0$, then we can choose \n\\begin{equation} \\label{eq:h:ineq}\n\th < \\frac{\\epsilon}{\\bar{v}}.\n\\end{equation}\nwhere  $\\bar{v} := \\max_{l_i \\in \\mathbb{L}} (\\Vert A_i \\Vert \\bar{x} + \\Vert u_i \\Vert)$.\n\nSo, if we choose $h$ as \n\\begin{equation} \\label{eq:h:eq}\n    h = \\frac{\\epsilon\/2}{\\bar{v}} ,\n\\end{equation}\nthen it is clear that we can ensure (\\ref{eq:hcond}). \n\nWe now show that, for a given $\\epsilon > 0$, if a sampling period $h$ satisfies (\\ref{eq:h:eq}), then a set constructed as a union of $\\epsilon$-neighborhood of each sampled state along a trajectory is indeed a bounded $\\epsilon$-reach set at an initial location.  \nMoreover, such a bounded $\\epsilon$-reach set contains the bounded reach set not only from the initial state but also from the $(\\epsilon\/2)$-neighborhood of the initial state.\n\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[width=8cm]{traj.png}\n\t\\caption{An over-approximation of a trajectory $x(t)$ through sampling.}\\label{fig:traj}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{lem} \\label{lem:theory:l0}\nGiven $\\epsilon > 0$ and a time bound $T > 0$, a bounded $\\epsilon$-reach set $\\R_{t_f}(x_0, \\epsilon)$ of a DTLHA $\\A$ from an initial state $(x_0, l_0)$ can be determined as follows:\n\\begin{equation} \\label{eq:lem:theory:l0}\n    \\R_{t_f}(x_0, \\epsilon) := \\bigcup_{k=0}^{m-1} \\B_{\\epsilon}(x(k h)) ,\n\\end{equation}\nwhere $t_f := \\min \\{\\tau_1, T \\}$, $\\tau_1 := \\inf \\{ t \\in (0, T] : x(t) \\not\\in Inv_0 \\land x(0) = x_0 \\}$, $m := \\lceil t_f\/h \\rceil$ and $h = (\\epsilon\/2)\/\\max_{l_i \\in \\mathbb{L}} (\\Vert A_i \\Vert \\bar{x} + \\Vert u_i \\Vert)$.\nMoreover, this set has two additional properties:\n\\begin{enumerate}[(i)]\n    \\item $\\lim_{\\epsilon \\rightarrow 0} \\R_{t_f}(x_0, \\epsilon) = \\R_{t_f}(x_0)$, and\n    \\item It contains an $\\epsilon\/2$ neighborhood of $\\R_{t_f}(x_0)$, i.e.,\n    \\begin{equation}\\nonumber\n        \\bigcup_{z \\in \\R_{t_f}(x_0)} \\B_{\\epsilon\/2}(z) \\subseteq \\R_{t_f}(x_0, \\epsilon).\n    \\end{equation}\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nSince $h$ satisfies (\\ref{eq:h:ineq}), it is easy to see that $\\R_{t_f}(x_0) \\subset \\R_{t_f}(x_0, \\epsilon)$ from the construction of $\\R_{t_f}(x_0, \\epsilon)$.\nNext, by the relation between $\\epsilon$ and $h$ in (\\ref{eq:h:eq}), it is clear that $h \\rightarrow 0$ as $\\epsilon \\rightarrow 0$. \nThis implies that $\\R_{t_f}(x_0, \\epsilon) \\rightarrow \\R_{t_f}(x_0)$ as $\\epsilon \\rightarrow 0$, establishing (i). \nFor (ii), as noted above, (\\ref{eq:h:eq}) actually chooses half the sampling period that would have sufficed to make it a bounded $\\epsilon$-reach set over $[0,t_f]$. \nHence, replacing $\\epsilon$ by $\\epsilon\/2$ in the right hand side of (\\ref{eq:lem:theory:l0}) still yields a bounded $\\epsilon$-reach set. \nThus the over stringent choice of $h$ contains not just $\\R_{t_f}(x_0)$ but actually all points that are within a distance $\\epsilon\/2$ from it.\n\\end{proof}\n\n\n\n\n\\subsection{Continuity Property of DTLHA}  \\label{sec:theory:cont}\n\nNow let us consider the problem of computing a bounded $\\epsilon$-reach set of a DTLHA $\\A$ not from an initial state $x_0$ but from a $\\delta$-neighborhood of $x_0$.\nWe first show that there exists a $\\delta > 0$ such that the bounded reach set of a DTLHA $\\A$ from a set $\\B_{\\delta}(x_0)$ at an initial location $l_0$ is contained in a bounded $\\epsilon$-reach set of $\\A$ from $x_0$ defined in (\\ref{eq:lem:theory:l0}).\n\n\n\\begin{lem} \\label{lem:theory:cont:1}\nGiven $\\epsilon > 0$, a time bound $T > 0$, an initial state $x_0$, and a DTLHA $\\A$, there exists a $\\delta > 0$ such that \n\\begin{equation}\n   \\R_{t_f}(\\B_{\\delta}(x_0)) \\subseteq \\R_{t_f}(x_0,\\epsilon) ,\n\\end{equation}\nwhere $\\B_{\\delta}(x_0)$ is a $\\delta$-neighborhood around $x_0$ and $\\R_{t_f}(\\B_{\\delta}(x_0))$ is the bounded reach set of $\\A$ from $\\B_{\\delta}(x_0)$ up to time $t_f$ and $t_f$ is as defined in Lemma \\ref{lem:theory:l0}. \nIn particular, $\\R_{t_f}(\\B_{\\epsilon \/(2C)}(x_0)) \\subseteq \\R_{t_f}(x_0, \\epsilon)$ for an appropriate $C$.\n\\end{lem}\n\\begin{proof}\nNotice that $x(t) = e^{A_0t} x_0 + \\int_0^t e^{A_0(t-s)} u_0 ds$, where $A_0$ and $u_0$ define the linear dynamics in an initial location $l_0$.\nIf we consider two different initial states $x_0$ and $y_0$ in $\\B_{\\delta}(x_0)$, then their trajectories $x(t)$ and $y(t)$ satisfy $x(t) - y(t) = e^{At} (x_0 - y_0)$.\nHence $\\Vert x(t) - y(t) \\Vert \\leq c e^{\\lambda t} \\Vert x_0 - y_0 \\Vert$ for some positive constant $c$ and some constant $\\lambda$.\n\nLet $C := c \\cdot \\max_{0 \\leq t \\leq {t_f}}\\lbrace e^{\\lambda t}\\rbrace$.\nThen\n\\begin{equation} \\label{eq:cont:1}\n    \\Vert x(t) - y(t) \\Vert \\leq C \\Vert x_0 - y_0 \\Vert \\qquad \\mbox{for} \\quad t \\in [0, {t_f}].\n\\end{equation}\nSince $\\Vert x_0 - y_0 \\Vert \\leq \\delta$, $\\Vert x(t) - y(t) \\Vert \\leq C \\delta$ for all $t \\in [0, {t_f}]$.\nThis implies that any initial condition $y_0$ in $\\B_\\delta(x_0)$ results in a $y(t)$ that lies in a $C \\delta$ neighborhood of $\\R_{t_f}(x_0)$ for all $t \\in [0,{t_f}]$. \nIn particular, from property (ii) of Lemma \\ref{lem:theory:l0}, it also follows that $\\R_{t_f}(\\B_{\\delta}(x_0)) \\subseteq \\R_{t_f}(x_0, 2 C \\delta)$.\nIf we set $\\delta = \\epsilon \/(2C)$, then it is clear that $\\R_{t_f}(\\B_{\\delta}(x_0)) \\subseteq \\R_{t_f}(x_0,\\epsilon)$.\n\\end{proof}\n\n\nNext we extend the result in Lemma \\ref{lem:theory:cont:1} to show that there exist a $\\delta > 0$ and a $\\gamma > 0$ such that an over-approximation of the bounded reach set  $\\R_{t_f}(\\B_{\\delta}(x_0))$, denoted as $\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$, is also contained in $\\R_{t_f}(x_0,\\epsilon)$ that is defined in (\\ref{eq:lem:theory:l0}).\n\n\n\\begin{lem} \\label{lem:theory:cont:2}\nGiven $\\epsilon > 0$, a time bound $T > 0$, an initial state $x_0$, and a DTLHA $\\A$, there exist $\\delta > 0$ and $\\gamma > 0$ such that \n\\begin{equation}\n\t\\R_{t_f}(\\B_{\\delta}(x_0),\\gamma) \\subseteq \\R_{t_f}(x_0,\\epsilon),\n\\end{equation}\nwhere $\\R_{t_f}(\\B_{\\delta}(x_0),\\gamma)$ is a $\\gamma$-approximation of $\\R_{t_f}(\\B_{\\delta}(x_0))$, and $t_f$ is as defined in Lemma \\ref{lem:theory:l0}.\nIn particular, $\\R_{t_f}(x_0) \\subseteq \\R_{t_f}(\\B_{\\epsilon\/(4C)}(x_0), \\epsilon\/4) \\subseteq \\R_{t_f}(x_0, \\epsilon)$.\n\\end{lem}\n\\begin{proof}\nLet $x(t;z)$ denote the solution at time $t$ of the differential equation $\\dot{x}(t) = Ax(t)+u$ with initial condition $x(0) = z \\in \\B_\\delta(x_0)$.\nNow consider $w \\in \\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$. \nThen, by the definition of $\\R_{t_f}(\\B_{\\delta}(x_0))$ and $\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$, \n\\[ \\Vert w - x(t;z) \\Vert < \\gamma\\] \nfor some $t \\in [0,t_f]$ and $z \\in \\B_{\\delta}(x_0)$. \nHence\n\\begin{eqnarray} \\nonumber\n    \\Vert w - x(t;x_0) \\Vert &=& \\Vert w - x(t;z) + x(t;z) - x(t;x_0) \\Vert \\\\ \\nonumber\n    &\\le& \\Vert w - x(t;z) \\Vert + \\Vert x(t;z) - x(t;x_0) \\Vert \\\\ \\nonumber\n    &\\le& \\gamma + \\Vert x(t;z) - x(t;x_0) \\Vert . \\nonumber\n\\end{eqnarray}\nFrom (\\ref{eq:cont:1}), we know that \n\\[\n\t\\Vert x(t;z) - x(t;x_0) \\Vert \\le C \\Vert z - x_0 \\Vert \\le C \\delta.\n\\]\nHence \n\\[\n\t\\Vert w - x(t;x_0) \\Vert \\le \\gamma + C \\delta\n\\]\nwhich implies that $w$ lies in a $(\\gamma + C \\delta)$-neighborhood of $\\R_{t_f}(x_0)$.\nFrom the property (ii) in Lemma \\ref{lem:theory:l0}, if we replace $\\epsilon\/2$ with $(\\gamma + C \\delta)$, then we have $w \\in \\R_{t_f}(x_0, 2(\\gamma + C \\delta))$ which in turn implies that $\\R_{t_f}(\\B_{\\delta}(x_0),\\gamma) \\subseteq \\R_{t_f}(x_0, 2(\\gamma + C \\delta))$.\nSo, given $\\epsilon > 0$, we can choose $\\gamma = \\epsilon\/4$ and $\\delta = \\epsilon\/(4C)$, and then $\\R_{t_f}(\\B_{\\delta}(x_0),\\gamma) \\subseteq \\R_{t_f}(x_0, \\epsilon)$.\n\\end{proof}\n\n\n\n\n\\subsection{Decidability of Discrete Transition Event}  \\label{sec:theory:trans}\n\nRecall that $\\tau_1$ is the time $t$ when a reached state $x(t)$ of a DTLHA starting from an initial state first exits the invariant set of an initial location.\nWe now show that, for a given $T$, even though it is not known to be decidable to determine $\\tau_1$ exactly, we can still determine the event of exit of a reached state $x(t)$ from the invariant set of an initial location if $\\tau_1 < T$.\n\n\n\\begin{lem} \\label{lem:theory:trans:exit}\nGiven a time bound $T > 0$, an initial condition $(l_0, x_0) \\in \\mathbb{L} \\times \\mathbb{R}^n$, and a DTLHA $\\A$, if $\\tau_1 < T$, then for all small enough $\\delta > 0$ and for some small enough $h > 0$, $\\B_{\\delta}(x(nh)) \\subset (Inv_0)^{c}$ for some $n \\in \\mathbb{N}$ satisfying $nh \\le T$.\n\\end{lem}\n\\begin{proof}\nLet $\\vec{n}_1$ be an outward normal vector of $\\partial Inv_0$ at $x(\\tau_1)$.\nSince $\\langle \\dot{x}(\\tau_1), \\vec{n}_1 \\rangle > 0$ by assumption, then by the continuity of the vector field of a linear dynamics in $l_0$, there exists an $r > 0$ such that for all $z \\in \\B_{3 r}(x(\\tau_1)) \\cap \\partial Inv_0$, $\\langle \\dot{z}, \\vec{n}_1 \\rangle > 0$ where $\\dot{z} := A_0 z + u_0$.\nNotice that $\\Vert \\dot{z} \\Vert \\le \\bar{v}$ by the definition of $\\bar{v}$ in (\\ref{eq:h:ineq}).\nLet $x(t;z)$ denotes the solution at time $t$ of the differential equation $\\dot{x}(t) = A_0 x(t)+u_0$ with initial condition $x(0) = z$.\nThen for any $z \\in \\B_{r}(x(\\tau_1)) \\cap \\partial Inv_0$, it is guaranteed that $x(t;z) \\in (Inv_0)^C$ for $t \\in (0, 2 h)$ for any $h > 0$ satisfying $h < r \/ \\bar{v}$.\nThis implies that $x(nh) \\in (Inv_0)^{c}$ for some $n \\in \\mathbb{N}$.\nMoreover by compactness of $Inv_0$, there exists a $\\delta > 0$ such that $\\B_{\\delta}(x(nh)) \\subset (Inv_0)^C$.\n\\end{proof}\n\n\nNow suppose that $x(t) \\in Inv_0$ for all $0 \\le t \\le T+\\theta$ for some $\\theta > 0$.\nThen this fact can also be determined.\n\n\n\\begin{lem} \\label{lem:theory:trans:noexit}\nSuppose $x(t) \\in Inv_0$ for all $0 \\le t \\le T+\\theta$ for some $\\theta > 0$. \nThen for all small enough $\\delta > 0$ and $\\gamma > 0$,\n\\begin{equation}\n    \\R_{t_f}(\\B_{\\delta}(x_0), \\gamma) \\subseteq (Inv_0)^{\\circ}.\n\\end{equation}\nwhere $t_f := \\min \\{ \\tau_1, T \\} = T$.\n\\end{lem}\n\\begin{proof}\nSince $x(t) \\in (Inv_0)^{\\circ}$ for all $0 \\le t \\le T$, the result immediately follows from Lemma \\ref{lem:theory:cont:2}.\n\\end{proof}\n\n\n\n\\subsection{Over-approximation of Discrete Transition State}  \\label{sec:theory:over}\n\nFor a given time bound $T$, suppose that the event $\\tau_1 < T$ is determined for some $\\delta$ and $h$ as shown in Lemma \\ref{lem:theory:trans:exit}.\nThen, to continue to compute a bounded $\\epsilon$-reach set beyond an initial location, we need to determine \n\\begin{inparaenum}[(i)]\n\t\\item a new location to which a discrete transition is made from an initial location, and also\n\t\\item an over-approximation of a discrete transition state from which the bounded $\\epsilon$-reach set computation can be continued.\n\\end{inparaenum}\nWe now show that  these can be determined, if a discrete transition state $x(\\tau_1)$ is deterministic and, more importantly, transversal, as defined in Definition \\ref{def:transversal}.\n\n\n\\begin{lem} \\label{lem:theory:over}\nGiven $\\tau_1 < T$, if $x(\\tau_1) \\in \\partial Inv_0$ satisfies a deterministic and transversal discrete transition condition, then there exists a $\\delta > 0$ such that $\\B_{2\\delta}(x(\\tau_1)) \\subset (Inv_0 \\cup Inv_1)$ for some location $l_1$. Furthermore, there exists a $\\Delta > 0$ such that\n\\begin{enumerate}[(i)]\n    \\item $x(t) \\in (Inv_1)^{\\circ}$ for $t \\in (\\tau_1, \\tau_1 + \\Delta)$ , and\n    \\item\n         \\begin{equation} \\label{eq:lem:lha:6}\n                \\bigcup_{y \\in \\mathcal{J}_{0,1}} x(\\tau;y) \\subset (Inv_1)^{\\circ}  \\quad \\mbox{for } \\tau \\in (0, \\Delta),\n         \\end{equation}\n\\end{enumerate}\nwhere $x(\\tau;y)$ is the solution at time $\\tau$ of an LTI system for the location $l_1$ with an initial state $y$ and $\\mathcal{J}_{0,1} := \\B_{\\delta}(x(\\tau_1)) \\cap Inv_0 \\cap Inv_1$. \n\\end{lem}\n\\begin{proof}\nLet $Inv_1, Inv_2$ be invariant sets for some locations $l_1$ and $l_2$ such that $Inv_0 \\cap Inv_1 \\cap Inv_2 \\ne \\emptyset$.\nSince $x(\\tau_1)$ satisfies a deterministic discrete transition condition, if $x(\\tau_1) \\in Inv_0 \\cap Inv_1$, then $x(\\tau_1) \\notin Inv_0 \\cap Inv_2$.\nThis implies that $x(\\tau_1) \\not\\in Inv_2$.\nThen by compactness of $Inv_2$, we know that there exists a $\\delta' > 0$ such that $\\B_{\\delta'}(x(\\tau_1)) \\cap Inv_2 = \\emptyset$. Therefore, we conclude that $\\B_{\\delta'}(x(\\tau_1)) \\subset Inv_0 \\cup Inv_1$.\n\nLet $\\vec{n}_1$ be an outward normal vector of $\\partial Inv_0$ at $x(\\tau_1)$.\nSince $x(\\tau_1)$ satisfies a transversal discrete transition condition from the location $l_0$ to the other location $l_1$, we know that there exists a $\\delta'' > 0$ such that for all $x(t) \\in \\B_{\\delta''}(x(\\tau_1)) \\cap Inv_0 \\cap Inv_1$, $\\langle \\dot{x}(t), \\vec{n}_1 \\rangle > 0$, where $\\dot{x}(t)$ is taken as either $A_0 x(t) + u_0$ or as $A_1 x(t) + u_1$, by the continuity of vector fields of the LTI dynamics for $l_0$ and $l_1$.\n\nLet $\\delta = \\min \\lbrace \\delta'\/2, \\delta''\/2\\rbrace$, and $\\Delta := \\delta\/(2 \\bar{v})$ where $\\bar{v}$ is as defined in (\\ref{eq:h:ineq}).\nThen by the definition of $\\delta$ and $\\bar{v}$, it is clear that (i) and (ii) hold for these choices of $\\delta$ and $\\Delta$.\n\\end{proof}\n\n\n\nIn Lemma \\ref{lem:theory:over}, $\\mathcal{J}_{0,1}$ is an over-approximation of $x(\\tau_1)$ that is determined by taking a $\\delta$-ball around $x(\\tau_1)$ for suitably small $\\delta > 0$, and intersecting it with $Inv_0$ and $Inv_1$. Once such a suitably small $\\delta$ is known, then the following lemma shows that it is also possible to determine a $\\delta_0$-neighborhood of an initial state $x_0$ such that the reach set at time $\\tau_1$ of a DTLHA $\\A$ from $\\B_{\\delta_0}(x_0)$ is contained in $\\B_{\\delta}(x(\\tau_1))$.\n\n\\begin{lem} \\label{lem:theory:over:cont:1}\nGiven $\\delta$ determined by Lemma \\ref{lem:theory:over}, there exists a $\\delta_0$ such that\n\\begin{equation}\n       \\D_{\\tau_1}(\\B_{\\delta_0}(x_0)) \\subseteq \\B_{\\delta}(x(\\tau_1)),\n\\end{equation}\nand $\\D_{\\tau_1}(\\B_{\\delta_0}(x_0)) \\cap Inv_0 \\cap Inv_1$ is an over-approximation of $x(\\tau_1)$ determined by $\\delta_0$.\n\\end{lem}\n\\begin{proof}\nThis follows from the same argument used in the proof of Lemma \\ref{lem:theory:cont:1}, by choosing $\\delta_0 = \\delta\/C$.\n\\end{proof}\n\nThe next lemma shows that $\\delta_0$ for $\\B_{\\delta_0}(x_0)$ can be determined at each discrete transition time $\\tau_k$ for $k \\ge 1$.\n\n\\begin{lem} \\label{lem:theory:over:cont:2}\nLet $\\delta_{k}$ be the radius of a ball centered at $x(\\tau_k)$ intersecting only $Inv_{k-1}$ and $Inv_k$, where $\\tau_k$ is the $k$-th discrete transition time and $l_k$ is the location after the $k$-th discrete transition. Then for any $x(\\tau_k)$ satisfying a deterministic and transversal discrete transition condition, there exists a $\\delta_0$ such that\n\\begin{equation}\n       \\D_{\\tau_k}(\\B_{\\delta_0}(x_0)) \\subseteq \\B_{\\delta_{k}}(x(\\tau_k)) ,\n\\end{equation}\nwhere $\\D_{\\tau_k}(\\B_{\\delta_0}(x_0))$ is the reached states of a given DTLHA $\\A$ from $\\B_{\\delta_0}(x_0)$ at time $\\tau_k$.\n\\end{lem}\n\\begin{proof}\nFrom the continuity property shown in Lemma \\ref{lem:theory:cont:1}, there is a $\\delta_{k-1} > 0$ such that $\\D_{[0, \\tau_k - \\tau_{k-1}]}(\\B_{\\delta_{k-1}}(x(\\tau_{k-1})))$ $\\subseteq$ $\\D_{[0, \\tau_k - \\tau_{k-1}]}(x(\\tau_{k-1}), \\delta_{k})$ for a given $\\delta_{k}$ where $\\D_{[0, \\tau_k - \\tau_{k-1}]}(x(\\tau_{k-1}), \\delta_{k})$ denotes a $\\delta_k$-approximation of $\\D_{[0, \\tau_k - \\tau_{k-1}]}(x(\\tau_{k-1}))$.\nThen for this $\\delta_{k-1}$, it is clear that $\\D_{\\tau_k}(\\B_{\\delta_{k-1}}(x(\\tau_{k-1})))$ $\\subseteq$ $\\B_{\\delta_{k}}(x(\\tau_k))$.\nUsing the same argument, we can find $\\delta_{k-2}, \\delta_{k-3}, \\cdots, \\delta_1$.\nThen from Lemma \\ref{lem:theory:over:cont:1}, we know that there exists a $\\delta_0 > 0$ such that $\\D_{\\tau_1}(\\B_{\\delta_0}(x_0))$ $\\subseteq$ $\\B_{\\delta_1}(x(\\tau_1))$.\nSince $\\D_{\\tau_2 - \\tau_1}(\\B_{\\delta_1}(x(\\tau_1)))$ $\\subseteq$ $\\B_{\\delta_2}(x(\\tau_2))$, we have $\\D_{\\tau_2}(\\B_{\\delta_0}(x_0))$ $\\subseteq$ $\\B_{\\delta_2}(x(\\tau_2))$.\nThis relation holds for each $\\tau_i$ where $i = 1, 2, \\cdots, k$.\nTherefore, $\\D_{\\tau_k}(\\B_{\\delta_0}(x_0))$ $\\subseteq$ $\\B_{\\delta_k}(x(\\tau_k))$. \n\\end{proof}\n\n\nWe now present our main result for the bounded $\\epsilon$-reachability of a DTLHA.\n\n\\begin{thm} \\label{thm:theory:thm}\nGiven $\\epsilon > 0$, a time bound $T > 0$, a discrete transition bound $N \\in \\mathbb{N}$, and a DTLHA $\\A$ starting from an initial condition $(l_0, x_0) \\in \\mathbb{L} \\times \\mathbb{R}^n$, there exist $\\delta > 0$, $\\gamma > 0$, and a sampling period $h > 0$ satisfying $h < \\gamma\/\\bar{v}$ such that \n\\begin{equation} \\label{eq:theory:thm}\n\t\\R_{t_f}(x_0) \\subseteq \\R_{t_f}(\\B_{\\delta}(x_0), \\gamma) \\subseteq \\R_{t_f}(x_0, \\epsilon),\n\\end{equation}\nwhere $t_f := \\min \\{\\tau_N, T\\}$ and $\\tau_N$ is the time at the N-th discrete transition. \n\\end{thm}\n\\begin{proof}\nLet $C_i := \\max_{0 \\le t \\le t_f} \\{ e^{\\Vert A_i \\Vert t}\\}$ for a location $l_i \\in \\mathbb{L}$ and $C := \\max_{l_i \\in \\mathbb{L}} \\{ C_i \\}$.\nFor a given $\\epsilon > 0$, suppose $\\delta_k < \\epsilon\/(4 C)$ at each $\\tau_k$ up to $t_f$ where $\\delta_k$ is as defined in Lemma \\ref{lem:theory:over:cont:2}.\nThen, from Lemmas \\ref{lem:theory:over}, \\ref{lem:theory:over:cont:1}, and \\ref{lem:theory:over:cont:2}, we know that there exist a $\\delta' > 0$ such that $\\D_{\\tau_k}(\\B_{\\delta'}(x_0)) \\subseteq \\B_{\\delta_{k}}(x(\\tau_k))$ where $x(t)$ is the execution of a DTLHA $\\A$ starting from $x_0$ at time zero.\nFurthermore, from Lemmas \\ref{lem:theory:trans:exit} and \\ref{lem:theory:over}, there also exists $h > 0$ and $\\delta'' > 0$ such that \n\\begin{inparaenum}[(i)]\n\t\\item $h < \\Delta_k$ and\n\t\\item $h$ and $\\delta''$ satisfy Lemma \\ref{lem:theory:trans:exit}\n\\end{inparaenum}\nat every $\\tau_k$ up to $t_f$, where $\\Delta_k$ is the $\\Delta$ that is defined in Lemma \\ref{lem:theory:over} for the $k$-th deterministic and transversal discrete transition.\n\nLet $\\hat{\\delta} := \\min \\{ \\delta', \\delta'' \\}$.\nThen, with $\\hat{\\delta}$ and $h$, we can determine every discrete transition event and also construct an over-approximation of the discrete transition state as long as it is deterministic and transversal.\nSince $\\hat{\\delta} \\le \\delta'$, $\\D_{\\tau_k}(\\B_{\\hat{\\delta}}(x_0)) \\subseteq \\B_{\\delta_{k}}(x(\\tau_k))$ at each $\\tau_k$ up to $t_f$.\nThus, for any $\\gamma > 0$, \n\\[ \n\t\\D_{[0, \\tau_k^{k+1}]}(\\D_{\\tau_k}(\\B_{\\hat{\\delta}}(x_0)), \\gamma) \\subseteq \\D_{[0, \\tau_k^{k+1}]}(\\B_{\\delta_k}(x_{\\tau_k}), \\gamma)\n\\]\nwhere $ \\tau_k^{k+1} := \\tau_{k+1} - \\tau_k$.\n\nNow, we notice that if $\\gamma < \\epsilon\/4$, then from Lemma \\ref{lem:theory:cont:2},\n\\[\n\t\\D_{[0, \\tau_k^{k+1}]}(\\D_{\\tau_k}(\\B_{\\hat{\\delta}}(x_0)), \\gamma) \\subseteq \\D_{[0, \\tau_k^{k+1}]}(x(\\tau_k), \\epsilon),\n\\]\nfor each $\\tau_k$ up to $t_f$, where the left hand side is a segment of $\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$ for $[\\tau_k, \\tau_{k+1}]$, and the right hand side is a segment of $\\R_{t_f}(x_0, \\epsilon)$ for $[\\tau_k, \\tau_{k+1}]$ that is defined as $\\bigcup_{n = 0}^{N_k -1} \\B_{\\epsilon}(x(\\tau_k + n h))$ where $N_k := \\lceil (\\tau_{k+1} - \\tau_k)\/h\\rceil$.\n\nFurthermore, if $h < \\gamma\/\\bar{v}$, then from (\\ref{eq:h:ineq}) replaced with $\\epsilon$ by $\\gamma$, it is clear that \n\\[\n\t\\D_{[0, \\tau_k^{k+1}]}(\\D_{\\tau_k}(x_0)) \\subseteq \\D_{[0, \\tau_k^{k+1}]}(\\D_{\\tau_k}(\\B_{\\hat{\\delta}}(x_0)), \\gamma),\n\\]\nwhere the left hand side is a segment of $\\R_{t_f}(x_0)$ for $[\\tau_k, \\tau_{k+1}]$.\nTherefore, the result holds.\n\\end{proof}\n\n\n\n\n\n\n\\section{Computing a Bounded $\\epsilon$-Reach Set of a DTLHA}  \\label{sec:cond}\n\nFrom Theorem \\ref{thm:theory:thm}, we know that a set $\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$, a bounded $\\epsilon$-reach set of a DTLHA, can be computed for some $\\delta, \\gamma$, and $h$. \nIn this section, we discuss how to compute $\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$.\nMore precisely, we derive a set of conditions, based on the results in Section \\ref{sec:theory}, that are needed to correctly detect a deterministic and transversal discrete state transition event and also to determine whether the values for the parameters $\\delta, \\gamma$, and $h$ are appropriate so as to ensure that $\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$ is a correct bounded $\\epsilon$-reach set.\nFurthermore, later in this section, we extend these conditions to incorporate the numerical calculation errors caused by the finite precision numerical calculations capabilities. \n\n\n\n\\subsection{Conditions for Bounded $\\epsilon$-Reach Set Computation}  \\label{sec:cond:exact}\n\nWe first note some properties that a set $\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$ needs to satisfy so that it can be considered as a bounded $\\epsilon$-reach set of a DTLHA.\n\n\\begin{rem} \\label{rem:cond:exact}\nNotice that any $\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$ that can be determined by $\\delta, \\gamma$, and $h$ in Theorem \\ref{thm:theory:thm} for a given $\\epsilon > 0$ needs to satisfy the following properties.\n\\begin{enumerate}[(i)]\n\t\\item $d_H(\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma), \\R_{t_f}(x_0)) \\le \\epsilon$, \n\t\\item  $\\R_{t_f}(\\B_{\\delta}(x_0))) \\subset \\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$, and\n\t\\item $d_H(\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma), \\R_{t_f}(\\B_{\\delta}(x_0))) \\le \\gamma$.\t\n\\end{enumerate}\n\\end{rem}\n\n\nFor given $\\delta$ and $h$, the following lemma shows how we can detect a discrete state transition event if there is one.\n\n\n\\begin{lem} \\label{lem:cond:exact:trans}\nGiven a location $l_c$ and a DTLHA $\\A$, \nif $\\D_{t-h}(\\B_{\\delta}(x_0)) \\subset (Inv_c)^{\\circ}$ and $\\D_t(\\B_{\\delta}(x_0)) \\subset Inv_c^C$ for some $\\delta > 0$ and $h > 0$, where $\\B_{\\delta}(x_0)$ is a $\\delta$-neighborhood of the initial state $x_0$, then there is a discrete transition from the location $l_c$ to some other locations at some time in $(t-h, t)$.\n\\end{lem}\n\\begin{proof}\nRecall that $\\D_t(x_0)$ denotes the reached state of $\\A$ at time $t$ from $x_0$.\nThen it is clear that $\\D_t(x_0) \\in \\D_t(\\B_{\\delta}(x_0))$. \nSimilarly,  $\\D_{t-h}(x_0) \\in \\D_{t-h}(\\B_{\\delta}(x_0))$. \nHence, from the hypothesis, $\\D_t(x_0) \\in Inv_c^C$ and $\\D_{t-h}(x_0) \\in (Inv_c)^{\\circ}$.\nThis implies that there exists $\\tau \\in (t-h ,t)$ such that $\\D_{s}(x_0) \\in Inv_c^{\\circ}$ for $s \\in [t-h, \\tau)$ and  $\\D_{s}(x_0) \\in Inv_c^C$ for $s \\in (\\tau, t]$.\nTherefore, there is a discrete transition at some time $\\tau \\in (t-h, t)$.\n\\end{proof}\n\n\nOnce a discrete state transition is detected, then, by Lemma \\ref{lem:cond:exact:det}, we can check if it is deterministic or not.\n\n\n\\begin{lem} \\label{lem:cond:exact:det}\nGiven an initial state $x_0$ and a DTLHA $\\A$,\nsuppose that there is a discrete transition from a location $l_c$ to some other locations at time $t$, i.e., $\\D_{t-h}(\\B_{\\delta}(x_0)) \\subset (Inv_c)^{\\circ}$ and $\\D_t(\\B_{\\delta}(x_0)) \\subset Inv_c^C$ for some $\\delta > 0$ and $h > 0$. \nThen the discrete transition is deterministic if there exists a location $l_n$ such that $l_n \\ne l_c$ and $\\D_t(\\B_{\\delta}(x_0)) \\subset (Inv_n)^{\\circ}$.\n\\end{lem}%\n\\begin{proof}\nThis follows from the definition of a deterministic discrete transition in Definition \\ref{def:transversal}.\n\\end{proof}\n\n\nWe now present conditions to determine the transversality of a discrete state transition; this is more complicated than those in previous two lemmas.\nThe main idea of the conditions in the following Lemma \\ref{lem:cond:exact:transv} is that \n\\begin{inparaenum}[(i)]\n\t\\item $\\delta$ and $\\gamma$ have to be small enough so that every state in an over-approximation of a deterministic and transversal discrete transition state, which can be computed by $\\delta$ and $\\gamma$, is  also deterministic and transversal, and also\n\t\\item the sampling period $h$ should be small enough so that any reached states right after a discrete state transition can be captured correctly. \n\\end{inparaenum}\n\n\n\\begin{lem} \\label{lem:cond:exact:transv}\nGiven $\\gamma > 0$ and $h > 0$ satisfying $h < \\gamma\/\\bar{v}$,\nsuppose that there is a deterministic discrete transition from a location $l_c$ to another location $l_n$ at time $t$, i.e., $\\D_{t-h}(\\B_{\\delta}(x_0)) \\subset (Inv_c)^{\\circ}$ and $\\D_t(\\B_{\\delta}(x_0)) \\subset (Inv_n)^{\\circ}$ for some $\\delta > 0$ and $h > 0$. \nThen for any $\\epsilon > 0$, the discrete transition is transversal if the following conditions hold:\n\\begin{itemize}\n\\item[(i)] $h < (dia(\\mathcal{J}_{c,n})\/2)\/(2 \\bar{v})$, \n\\item[(ii)] $\\D_0(\\mathcal{J}_{c,n}, dia(\\mathcal{J}_{c,n})\/2) \\subset (Inv_c \\cup Inv_n)$, and\n\\item[(iii)] $\\langle \\dot{x}_c, \\vec{n}_c \\rangle \\ge \\epsilon \\land \\langle \\dot{x}_n, \\vec{n}_c \\rangle \\ge \\epsilon, ~~\\forall x \\in \\V(\\mathcal{J}_{c,n}')$,\n\\end{itemize}\nwhere $\\mathcal{J}_{c,n} := \\D_t(\\B_{\\delta}(x_0), \\gamma) \\cap Inv_c \\cap Inv_n$, $\\mathcal{J}_{c,n}' := \\D_0(\\mathcal{J}_{c,n},$ $dia(\\mathcal{J}_{c,n})\/2) \\cap Inv_c \\cap Inv_n$, $\\bar{v}$ is as defined in (\\ref{eq:h:ineq}), $\\V(\\P)$ is a set of vertices of a polyhedron $\\P$, $\\vec{n}_c$ is an outward normal vector of $\\partial Inv_c$, and $\\dot{x}_i$ is the vector flow evaluated with respect to the LTI dynamics of location $l_i \\in \\mathbb{L}$.\n\\end{lem}\n\\begin{proof}\nNotice that $\\D_{t-h}(\\B_{\\delta}(x_0)) \\subset  \\D_t(\\B_{\\delta}(x_0), \\gamma)$ since $\\gamma$ and $h$ satisfy $h < \\gamma\/\\bar{v}$.\nIn fact, $\\bigcup_{z \\in \\D_{t-h}(\\B_{\\delta}(x_0))} x(\\tau;z) \\subset \\D_{t}(\\B_{\\delta}(x_0), \\gamma)$ for $\\tau \\in [0, h]$ where $x(\\tau;z) := e^{A_c \\tau} z + \\int_{0}^{\\tau} e^{A_c s} u_c ds$ under the LTI dynamics of the location $l_c$.\nSince $\\D_{t-h}(x_0) \\in \\D_{t-h}(\\B_{\\delta}(x_0))$ and $\\D_t(x_0) \\in \\D_{t}(\\B_{\\delta}(x_0))$, $\\D_{\\tau'}(x_0) \\in \\mathcal{J}_{c,n}$ for some $\\tau' \\in (t-h, t)$ where $\\D_{\\tau'}(x_0)$ is a discrete transition state from $l_c$ to $l_n$ at time $\\tau'$.\nThus $\\mathcal{J}_{c,n} \\ne \\emptyset$ (more precisely, $\\mathcal{J}_{c,n}^{\\circ} \\ne \\emptyset$) and it is in fact an over-approximation of the deterministic discrete transition state $x_{\\tau'} \\in Inv_c \\cap Inv_n$.\n\nIf (ii) and (iii) hold, then it is easy to see that $z'$ satisfies the deterministic and transversal discrete transition condition in Definition \\ref{def:transversal} for any $z' \\in \\mathcal{J}'_{c,n}$.\nNow we suppose (i) holds \nand let $x(h;z)$ is the state reached from $z$ at time $h$ under the LTI dynamics of the location $l_n$, then,  for any $z \\in \\mathcal{J}_{c,n}$, \n\\[\n\t\\Vert x(h;z) - z \\Vert \\le \\bar{v}h < dia(\\mathcal{J}_{c,n})\/2.\n\\] \n\nIf we now consider the fact that $dia(\\mathcal{J}'_{c,n}) \\ge 2 \\cdot dia(\\mathcal{J}_{c,n})$, then it is easy to see that $x(\\tau;z) \\in Inv_n^{\\circ}$ for $\\tau \\in (0, h)$.\nSince $z \\in \\mathcal{J}_{c,n}$ is arbitrary, we conclude that \n\\[\n\t\\D_{\\tau}(\\mathcal{J}_{c,n}) \\in Inv_n^{\\circ}\n\\]\nfor all $\\tau \\in (0, h)$.\nThus, the discrete transition state $\\D_t(x_0) \\in \\mathcal{J}_{c,n}$ is transversal and it can be determined through $\\mathcal{J}_{c,n}$ with $h$ satisfying (i).\n\\end{proof}\n\n\n\n\\subsection{Finite Precision Basic Calculations}  \\label{sec:cond:basic}\n\nNotice that the results in Section \\ref{sec:cond:exact} are based on the assumption that the following quantities can be computed exactly:\n\\begin{itemize}\n\\item $x(t;x_0) = e^{At}x_0 + \\int_0^t e^{As} u ds$.\n\\item $\\H \\cap \\P$, where $\\H$ is a hyperplane and $\\P$ is a polyhedron.\n\\item $hull(\\mathcal{V})$, where $hull(\\mathcal{V})$ is the convex hull of $\\mathcal{V}$ that is a finite set of points in $\\mathbb{R}^n$.\n\\end{itemize}\nHowever, these exact computation assumptions cannot be satisfied in practice and we can only compute each of these with possibly arbitrarily small computation error. \nTherefore, instead of assuming exact computation capabilities for $x(t;x_0)$, $\\H \\cap \\P$, and $hull(\\mathcal{V})$, we now assume that the following basic calculation capabilities are available for approximately computing these quantities, and it only these that we can use to compute a bounded $\\epsilon$-reach set. \nMore precisely, we assume that for given $\\mu_c > 0$ and $\\mu_h > 0$, \n\\begin{itemize}\n\t\\item $a(\\mathcal{H} \\cap \\mathcal{P}, \\mu_c)$ and $a(hull(\\mathcal{V}), \\mu_h)$\n\\end{itemize}\nare available such that $d_H(x, a(x,y)) \\le y$, where $a(x, y)$ denotes an approximate computation of $x$, with $y > 0$ as an upper bound on the approximation error.\nWe also assume that for given $\\sigma_e > 0$ and $\\sigma_i > 0$,\n\\begin{itemize}\n\t\\item $a(e^{At}, \\sigma_e)$, and $a(\\int_0^t e^{A \\tau} d\\tau, \\sigma_i)$\n\\end{itemize}\nare available as an approximate computation of $x(t;x_0)$ such that $\\Vert x - a(x,y) \\Vert \\le y$.\nNotice that from these basic calculation capabilities for $x(t;x_0)$, we can compute $a(x(t;x_0), \\mu_x)$ with an approximation error  denoted as $\\mu_x$, which is upper bounded by a finite value as shown below.\n\n\nWe first note that, for all approximate computations $a(x,y)$ that are used for computing $x(t;x_0)$, we have \n\\begin{equation}\nx - y \\cdot {\\bf{1}}_{n \\times m} \\le a(x, y) \\le x + y \\cdot {\\bf{1}}_{n \\times m},\n\\end{equation}\nwhere $x \\in \\mathbb{R}^{n \\times m}$ and ${\\bf{1}}_{n \\times m}$ is an $n$ by $m$ matrix whose every element is $1$, and the inequalities hold elementwise.\nWith this, an upper bound of $\\mu_x$ can be derived as follows:\n\\begin{equation} \\nonumber\ne^{At} -\\sigma_e \\cdot {\\bf{1}}_{n \\times n} \\le a( e^{At}, \\sigma_e) \\le e^{At} + \\sigma_e \\cdot {\\bf{1}}_{n \\times n}.\n\\end{equation}\nSimilarly, \n\\begin{eqnarray} \\nonumber\n\\int_0^t e^{As}ds  - \\sigma_i \\cdot {\\bf{1}}_{n \\times n} \\le a(\\int_0^t e^{As}ds, \\sigma_i) \\\\ \\nonumber\n\\le \\int_0^t e^{As}ds + \\sigma_i \\cdot {\\bf{1}}_{n \\times n}.\n\\end{eqnarray}\nHence, we have \n\\begin{equation} \\nonumber\nx(t;x_0) - \\delta_x \\le a(x(t;x_0), \\delta_x)  \\le x(t;x_0) + \\delta_x,\n\\end{equation}\nwhere $\\delta_x := (\\sigma_e \\vert x_0 \\vert + \\sigma_i \\vert u \\vert) \\cdot {\\bf{1}}_{n \\times 1}$.\n\nNow, we know that $\\mu_x$ is upper bounded by the maximum of $\\vert \\delta_x \\vert$ over the continuous state space $\\X$ and the control input domain $\\U$,\n\\begin{equation} \\label{eq:mux}\n\t\\mu_x \\le \\max_{x \\in \\X, u \\in \\U} \\vert \\delta_x \\vert.\n\\end{equation}\n\n\n\n\\subsection{Conditions for Computation under Finite Precision Calculations}  \\label{sec:cond:inexact}\n\nIn this section, we extend the results in Section \\ref{sec:cond:exact} to derive a set of conditions for a bounded $\\epsilon$-reach set computation of the DTLHA under finite precision numerical calculation capabilities.\nThe following remark is an immediate extension of Remark \\ref{rem:cond:exact} in Section \\ref{sec:cond:exact}.\n\nIn the sequel, for simplicity of notation, we use $\\hat{x}$ to denote $a(x,\\rho)$ for a given approximation error bound $\\rho > 0$.\n\n\\begin{rem} \\label{rem:cond:inexact}\nLet $\\hat{\\R}_{t_f}(\\B_{\\delta}(x_0), \\gamma)$ be an approximation of $\\R_{t_f}(\\B_{\\delta}(x_0), \\gamma)$ that is determined by $\\delta, \\gamma$, and $h$ in Theorem \\ref{thm:theory:thm} and approximate calculations for $x(t;x_0)$, $\\H \\cap \\P$, and $hull(\\mathcal{V})$ defined in Section \\ref{sec:cond:basic}.\nThen, for a given $\\epsilon > 0$, it is sufficient for $\\hat{\\R}_{t_f}(\\B_{\\delta}(x_0), \\gamma)$ to be a bounded $\\epsilon$-reach set of a DTLHA $\\A$ if the following properties hold.\n\\begin{enumerate}[(i)]\n\t\\item $d_H(\\hat{\\R}_{t_f}(\\B_{\\delta}(x_0), \\gamma), \\R_{t_f}(x_0)) \\le \\epsilon$, \n\t\\item  $\\R_{t_f}(\\B_{\\delta}(x_0))) \\subset \\hat{\\R}_{t_f}(\\B_{\\delta}(x_0), \\gamma)$, and\n\t\\item $d_H(\\hat{\\R}_{t_f}(\\B_{\\delta}(x_0), \\gamma), \\R_{t_f}(\\B_{\\delta}(x_0))) \\le \\gamma$.\t\n\\end{enumerate}\n\\end{rem}\n\n\nNext, we discuss how the relation between $h$ and $\\gamma$ can be modified so as to satisfy (ii) and (iii) in Remark \\ref{rem:cond:inexact} when there is numerical calculation error in computing $x(t;x_0)$. \n\n\n\\begin{lem} \\label{lem:cond:inexact:isover}\nGiven a DTLHA $\\A$ and its reached state $x(t)$ at time $t$ starting from an initial condition $x(0)$, \nlet $\\rho > 0$ be an upper bound on the approximation errors such that $\\Vert x(t) - \\hat{x}(t) \\Vert \\le \\rho$.\nIf a given sampling period $h$ satisfies $h < (\\gamma - \\rho)\/\\bar{v}$ for a given $\\gamma$ satisfying $\\gamma > \\rho$,  where $\\bar{v}$ is as defined in (\\ref{eq:h:ineq}), then the following property holds at any location $l_i \\in \\mathbb{L}$ of $\\A$:\n\\begin{equation}\n\tx(t+\\tau) \\subset \\B_{\\gamma}(\\hat{x}(t)), ~~ \\forall \\tau \\in [0, h],\n\\end{equation}\nwhere $x(t+\\tau) = e^{A_i \\tau} x(t) + \\int_0^{\\tau} e^{A_i s} u_i ds$.\n\\end{lem}\n\\begin{proof}\nSince $\\Vert x(t) - \\hat{x}(t) \\Vert \\le \\rho$, $x(t) \\in \\B_{\\rho}(\\hat{x}(t))$.\nMoreover, from (\\ref{eq:h:ineq:1}), we know that for any $x(t) \\in \\X$,\n\\begin{eqnarray} \\nonumber\n\t\\max_{\\tau \\in [0,h]} \\Vert x(t+\\tau) - x(t) \\Vert &\\le& \\max_{\\tau \\in [0,h]}  \\int_t^{t+\\tau} \\Vert \\dot{x}(s) \\Vert ds \\\\ \\nonumber\n\t &\\le& \\bar{v} h. \n\\end{eqnarray}\nHence, if $h < (\\gamma - \\rho)\/\\bar{v}$, then, for any $x(t) \\in \\X$, \n\\[\n\t\\max_{\\tau \\in [0,h]} \\Vert x(t+\\tau) - x(t) \\Vert < \\gamma - \\rho. \n\\] \nThis means that $x(t+\\tau) \\in \\B_{\\gamma - \\rho}(x(t))$ for $\\tau \\in [0, h]$.\nTherefore, for $\\tau \\in [0, h]$, \n\\begin{eqnarray} \\nonumber\n\t\\Vert \\hat{x}(t) - x(t+\\tau)\\Vert &\\le& \\Vert \\hat{x}(t) - x(t) \\Vert + \\Vert x(t) -x(t+\\tau) \\Vert \\\\ \\nonumber\n\t&\\le& \\rho +  (\\gamma -\\rho).\n\\end{eqnarray}\nThus $\\Vert \\hat{x}(t) - x(t+\\tau)\\Vert  \\le \\gamma$.\n\\end{proof}\n\nNotice that Lemma \\ref{lem:cond:inexact:isover} says that if $h < (\\gamma - \\rho)\/\\bar{v}$ for a given $\\rho > 0$, then a $\\gamma$-neighborhood of a sampled state is indeed an over-approximation of a trajectory over the time interval $h$.\nWe now extend the result in Lemma \\ref{lem:cond:inexact:isover} to the case where we need to compute a $\\gamma$-approximation of a polyhedron.\n\n\\begin{lem} \\label{lem:cond:inexact:over}\nGiven a DTLHA $\\A$ and its reached states $\\D_t(\\B_{\\delta}(x_0))$ at some time $t$ from initial states in $\\B_{\\delta}(x_0)$, \nlet $\\rho > 0$ be an upper bound on the approximation errors such that $d_H(\\D_t(\\B_{\\delta}(x_0)), \\hat{\\D}_t(\\B_{\\delta}(x_0))) \\le \\rho$.\nIf a given sampling period $h$ satisfies the following inequality \n\\begin{equation} \\label{eq:lem:cond:inexact:over}\nh < \\frac{\\gamma - \\rho}{\\bar{v}} ,\n\\end{equation}\nthen, for a given $\\gamma$ satisfying $\\gamma > \\rho$, \n\\begin{equation}\n\t\\D_{t+ \\tau}(\\B_{\\delta}(x_0)) \\subset \\hat{\\D}_t(\\B_{\\delta}(x_0), \\gamma), \\quad \\forall \\tau \\in [0, h], \n\\end{equation}\n where $\\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma)$ is a $\\gamma$-approximation of $\\hat{\\D}_{t}(\\B_{\\delta}(x_0))$ that is  constructed as the convex hull of the set of extreme points of a polyhedral $\\gamma$-neighborhood of all vertices of $\\hat{\\D}_{t}(\\B_{\\delta}(x_0))$ and $\\bar{v}$ is as defined in (\\ref{eq:h:ineq}).\n\\end{lem}\n\\begin{proof}\nLet $\\V$ and $\\hat{\\V}$ be the set of extreme points of $\\D_t(\\B_{\\delta}(x_0))$ and $\\hat{\\D}_t(\\B_{\\delta}(x_0))$, respectively. \nSince $d_H(\\D_t(\\B_{\\delta}(x_0)), \\hat{\\D}_t(\\B_{\\delta}(x_0))) \\le \\rho$ and $\\gamma > \\rho$, it is clear that $\\D_{t}(\\B_{\\delta}(x_0)) \\subset \\hat{\\D}_t(\\B_{\\delta}(x_0), \\gamma)$.\nFrom Lemma \\ref{lem:cond:inexact:isover}, we know that for each $x(t) \\in \\V$, $x(t+\\tau) \\subset \\B_{\\gamma}(\\hat{x})$ for all $\\tau \\in [0, h]$ where $\\hat{x} \\in \\hat{\\V}$ corresponding to $x(t)$.\nLet $\\V_{t+\\tau}$ be the set of extreme points of $\\D_{t+ \\tau}(\\B_{\\delta}(x_0))$. \nThen $\\V_{t+\\tau} \\subset \\hat{\\D}_t(\\B_{\\delta}(x_0), \\gamma)$ for all $\\tau \\in [0, h]$ since \n\\begin{inparaenum}[(i)]\n\t\\item for each $x(t) \\in \\V$, $x(t+\\tau) \\subset \\B_{\\gamma}(\\hat{x})$ for all $\\tau \\in [0, h]$ and \n\t\\item from the construction of $\\hat{\\D}_t(\\B_{\\delta}(x_0), \\gamma)$, $\\B_{\\gamma}(\\hat{x}) \\subset \\hat{\\D}_t(\\B_{\\delta}(x_0), \\gamma)$ for each $\\hat{x} \\in \\hat{\\V}$.\n\\end{inparaenum}\nTherefore, the convex hull of $\\V_{t+\\tau}$, which is $\\D_{t+ \\tau}(\\B_{\\delta}(x_0))$, has to be contained in $\\hat{\\D}_t(\\B_{\\delta}(x_0), \\gamma)$ for all $\\tau \\in [0,h]$ since $\\hat{\\D}_t(\\B_{\\delta}(x_0), \\gamma)$ is convex and $\\V_{t+\\tau} \\subset \\hat{\\D}_t(\\B_{\\delta}(x_0), \\gamma)$ for all $\\tau \\in [0, h]$.\n\\end{proof}\n\n\nFor (i) in Remark \\ref{rem:cond:inexact}, Lemma \\ref{lem:cond:inexact:eps} below shows that the diameter of a set $\\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma)$ has to be smaller than a given $\\epsilon > 0$.\n\n\n\n\\begin{lem} \\label{lem:cond:inexact:eps}\nGiven $\\epsilon > 0$, $\\delta > 0$, $\\gamma >0$,  $\\rho > 0$, and a DTLHA $\\A$, \nsuppose a given sampling period $h > 0$ satisfies the inequality (\\ref{eq:lem:cond:inexact:over}).\nThen $\\D_{[t, t+h]}(x_0) \\subset \\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma)$ and $d_H(\\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma), \\D_{[t, t+h]}(x_0)) \\le \\epsilon$, if the following hold:\n\\begin{equation} \\label{eq:lem:cond:inexact:eps}\n\tdia(\\hat{\\D}_t(\\B_{\\delta}(x_0), \\gamma)) \\le \\epsilon,\n\\end{equation}\nwhere $\\D_{[t, t+h]}(x_0)$ is the set of reached states of $\\A$ starting from $x_0$ during the time interval $[t, t+h]$. \n\\end{lem}\n\\begin{proof}\nSince $h$ satisfies (\\ref{eq:lem:cond:inexact:over}), it is trivial to see that  $\\D_{[t, t+h]}(x_0) \\subset \\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma)$ holds from Lemma \\ref{lem:cond:inexact:over}.\nMoreover, if (\\ref{eq:lem:cond:inexact:eps}) is also true, then for any $z \\in \\D_{[t, t+h]}(x_0)$, $\\max_{y \\in \\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma)} \\Vert y - z \\Vert \\le \\epsilon$ since $z \\in \\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma)$.\nTherefore, it is clear that $d_H(\\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma), \\D_{[t, t+h]}(x_0)) \\le \\epsilon$ if (\\ref{eq:lem:cond:inexact:over}) and (\\ref{eq:lem:cond:inexact:eps}) hold.\n\\end{proof}\n\nNow we can extend the results of Lemmas \\ref{lem:cond:exact:trans}, \\ref{lem:cond:exact:det}, and \\ref{lem:cond:exact:transv} to incorporate a numerical calculation error $\\rho > 0$.\n\n\\begin{lem} \\label{lem:cond:inexact:istrans}\nGiven $\\rho > 0$,  a location $l_c$, and $\\hat{\\D}_t(\\B_{\\delta}(x_0))$ at time $t$,\nif\n\\begin{enumerate}[(i)]\n\\item $\\hat{\\D}_{t-h}(\\B_{\\delta}(x_0),$ $\\rho) \\subset (Inv_c)^{\\circ}$, and\n\\item $\\hat{\\D}_t(\\B_{\\delta}(x_0), \\rho) \\subset Inv_c^C$ \n\\end{enumerate} \nfor some $\\delta > 0$ and $h > 0$, then there is a discrete transition from the location $l_c$ to some other locations.\n\\end{lem}\n\\begin{proof}\nNotice that $d_H(\\D_t(\\B_{\\delta}(x_0)), \\hat{\\D}_t(\\B_{\\delta}(x_0))) \\le \\rho$, which implies $\\D_t(\\B_{\\delta}(x_0)) \\subset \\hat{\\D}_t(\\B_{\\delta}(x_0), \\rho)$.\nSimilarly, $\\D_{t-h}(\\B_{\\delta}(x_0))$ $\\subset \\hat{\\D}_{t-h}(\\B_{\\delta}(x_0), \\rho)$.\nHence if (i) and (ii) hold, then it is clear that $\\D_t(\\B_{\\delta}(x_0)) \\subset Inv_c^C$ and $\\D_{t-h}(\\B_{\\delta}(x_0)) \\subset (Inv_c)^{\\circ}$.\nThen the result follows from Lemma \\ref{lem:cond:exact:trans}. \n\\end{proof}\n\n\\begin{lem} \\label{lem:cond:inexact:det}\nGiven $\\rho > 0$, a location $l_c$, and $\\hat{\\D}_t(\\B_{\\delta}(x_0))$ at time $t$, suppose that a discrete transition from a location $l_c$ to some other locations is determined as in Lemma \\ref{lem:cond:inexact:istrans}.\nThen the discrete transition is a deterministic discrete transition from $l_c$ to $l_n$ if there exists a location $l_n$ such that $l_n \\ne l_c$ and $\\hat{\\D}_t(\\B_{\\delta}(x_0), \\rho) \\subset (Inv_n)^{\\circ}$.\n\\end{lem}\n\\begin{proof}\nNotice that $\\D_{t-h}(\\B_{\\delta}(x_0)) \\subset (Inv_c)^{\\circ}$ from the result in Lemma \\ref{lem:cond:inexact:istrans}.\nSince $\\D_t(\\B_{\\delta}(x_0)) \\subset \\hat{\\D}_t(\\B_{\\delta}(x_0), \\rho)$, if  $\\hat{\\D}_t(\\B_{\\delta}(x_0), \\rho) \\subset (Inv_n)^{\\circ}$, then  $\\D_t(\\B_{\\delta}(x_0)) \\subset (Inv_n)^{\\circ}$.\nThus by Lemma \\ref{lem:cond:exact:det}, the conclusion holds.\n\\end{proof}\n\n\n\n\n\\begin{lem} \\label{lem:cond:inexact:transv}\nGiven $\\rho > 0$, $\\gamma > 0$ and $h > 0$ satisfying (\\ref{eq:lem:cond:inexact:over}), suppose that a deterministic discrete transition from a location $l_c$ to another location $l_n$ is determined as in Lemma \\ref{lem:cond:inexact:istrans} and Lemma \\ref{lem:cond:inexact:det}, i.e., $\\hat{\\D}_{t-h}(\\B_{\\delta}(x_0), \\rho) \\subset (Inv_c)^{\\circ}$ and $\\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\rho) \\subset (Inv_n)^{\\circ}$.\nThen, for any $\\epsilon > 0$, the discrete transition is transversal if the following conditions hold:\n\\begin{itemize}\n\\item[(i)] $h < (dia(\\hat{\\mathcal{J}}_{c,n})\/2)\/(2 \\bar{v})$, \n \\item[(ii)] $\\D_0(\\hat{\\mathcal{J}}_{c,n}, dia(\\hat{\\mathcal{J}}_{c,n})\/2 + \\rho) \\subset (Inv_c \\cup Inv_n)$, and \n \\item[(iii)] $\\langle \\dot{x}_c, \\vec{n}_c \\rangle \\ge \\epsilon \\land \\langle \\dot{x}_n, \\vec{n}_c \\rangle \\ge \\epsilon,~~\\forall x \\in \\V(\\hat{\\mathcal{J}}_{c,n}')$,\n \\end{itemize}\n %\n where $\\hat{\\mathcal{J}}_{c,n} := \\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma + \\rho) \\cap Inv_c \\cap Inv_n$, $\\hat{\\mathcal{J}}_{c,n}' := \\D_0(\\hat{\\mathcal{J}}_{c,n},$ $dia(\\hat{\\mathcal{J}}_{c,n})\/2 + \\rho) \\cap Inv_c \\cap Inv_n$, $ \\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma + \\rho)$ is a $(\\gamma + \\rho)$-approximation of $ \\hat{\\D}_{t}(\\B_{\\delta}(x_0))$, and $\\dot{x}_i$ and $\\vec{n}_c$ are as defined in Lemma \\ref{lem:cond:exact:transv}. \n\\end{lem}\n\\begin{proof}\nNotice that $\\D_{t}(\\B_{\\delta}(x_0), \\gamma) \\subset \\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\gamma + \\rho)$.\nThen, by the definition of $\\mathcal{J}_{c,n}$ given in Lemma \\ref{lem:cond:exact:transv} and $\\hat{\\mathcal{J}}_{c,n}$, we know $\\mathcal{J}_{c,n} \\subset \\hat{\\mathcal{J}}_{c,n}$.\nHence, $\\hat{\\mathcal{J}}_{c,n} \\ne \\emptyset$ since $\\mathcal{J}_{c,n} \\ne \\emptyset$ by the construction of  $\\mathcal{J}_{c,n}$.\nNow if (i) holds, then it is easy to see that $\\D_{\\tau}(\\hat{\\mathcal{J}}_{c,n}) \\subset \\D_0(\\hat{\\mathcal{J}}_{c,n}, dia(\\hat{\\mathcal{J}}_{c,n})\/2)$ for $\\tau \\in (0, h)$.\nMoreover, (ii) and (iii) imply that $\\D_{\\tau}(\\hat{\\mathcal{J}}_{c,n})$ is in fact contained in $Inv_n^{\\circ}$ for $\\tau \\in (0, h)$.\n\\end{proof}\n\n\n\n\n\n\\section{Architecture and Algorithm for Bounded $\\epsilon$-Reach Set Computation of a DTLHA}  \\label{sec:design}\n\nWe are now in a position to propose an algorithm for bounded $\\epsilon$-reach set computation of a DTLHA.\nBefore proving its correctness, we first describe its architecture.\n\n\n\nFor \\emph{flexibility}, we decouple the higher levels of the algorithm, called \\emph{Policy}, from the component, called \\emph{Mechanisms}, where specific steps of calculations are performed through some numerical routines.\nThe proposed architecture of the algorithm, shown in Fig. \\ref{fig:arch}, consists of roughly five different components \\emph{Policy}, \\emph{Mechanism}, \\emph{System Description}, \\emph{Data}, and \\emph{Numerics}.\nA more detailed explanation of each of these modules is given below. \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width = 9cm]{arch4algo.png} \n\\caption{An architecture for bounded $\\epsilon$-reach set computation.}\n\\label{fig:arch}\n\\end{center}\n\\end{figure}\n\n\nThe System Description contains all information describing a problem of a bounded $\\epsilon$-reach set computation of a DTLHA.\nThis consists of $\\X$, the domain of continuous state space, a DTLHA $\\A$, and an initial condition $(l_0, x_0) \\in \\mathbb{L} \\times \\X$. \nAlso, an upper bound $T \\in \\mathbb{R}^+$ on terminal time, an upper bound $N \\in \\mathbb{N}$ on the total number of discrete transitions, and an approximation parameter $\\epsilon > 0$, are described.\nA bounded $\\epsilon$-reach set of a DTLHA $\\A$ is computed in the Mechanism component based on a given set of numerical calculation algorithms in Numerics, as well as a given Policy, which captures some of the higher level choices of the algorithm's outer loops. \nIn the Data component, all computation data that is relevant to a computed bounded $\\epsilon$-reach set, generated on-the-fly in the Mechanism part, are stored. \nEach of the functions in Numerics is in fact an implementation of some numerical computation algorithms.  \nAs an example, $e^{At}$ can be computed in many different ways as shown in \\cite{moler:78} and each of the different algorithms can compute the value with a certain accuracy. \nHere we assume that a set of such numerical computation algorithms for basic calculations are given\\footnote{In this way, we decouple the low-level numerical calculations from our bounded $\\epsilon$-reach set algorithm. This is the reason why the Numerics component is represented separately from the Mechanism component.} \nand the corresponding approximation error bounds, i.e., $\\sigma_e, \\sigma_i, \\mu_c$, and $\\mu_h$, are known a priori.\nThe Policy component represents a user-defined rules that choose appropriate values of the parameters, especially $\\delta > 0$, $\\gamma > 0$, and $h > 0$, which are needed to continue to compute a bounded $\\epsilon$-reach set of a DTLHA, when a bounded $\\epsilon$-reach set algorithm in Mechanism fails to determine some events or to satisfy some required properties, during its computation.\nThe Mechanism component represents the core of the bounded $\\epsilon$-reach set algorithm based on the theoretical results in Section \\ref{sec:theory} and \\ref{sec:cond}, and is detailed in Section \\ref{sec:design:algo}. \nGiven values for parameters $\\delta > 0$, $\\gamma > 0$, and $h > 0$, it computes a bounded $\\epsilon$-reach set of a DTLHA $\\A$ until it either successfully finishes its computation or cannot make further progress, which happens when some required conditions or properties are not met.\nNotice that,  as stated in Section \\ref{sec:cond}, there are a set of conditions and properties that a computed set needs to satisfy to be a correct bounded $\\epsilon$-reach set.\nIf the algorithm fails to resolve a computation, then it returns to Policy indicating the problems so that a user-defined rule in Policy can choose another set of values for the parameters to resolve the problems.\nEvery computation result is stored in the Data component to be possibly used later in Policy and Mechanism.\n\n\n\n\n\\subsection{Core Algorithm for Bounded $\\epsilon$-Reach Set of a DTLHA}  \\label{sec:design:algo}\n\n\n\n\n\nAn algorithm to compute a bounded $\\epsilon$-reach set of a DTLHA is proposed and shown in Algorithm \\ref{alg:main}.\nLet $k$ indicate a computation step of the algorithm from which the proposed algorithm starts its bounded $\\epsilon$-reach set computation.\nAll computation history up to the $(k-1)$-th computation step is stored as data, called {\\tt Reached}, in Data part.\nThen, given an input $(k, \\delta_k, \\gamma_k, h_k)$ from Policy, the algorithm first retrieves the computation data at the $(k-1)$-th computation step from {\\tt Reached} and starts its $k$-th computation step using this data.\nAs shown in Algorithm \\ref{alg:main}, the algorithm continues its computation until it either\n\\begin{inparaenum}[(i)]\n\t\\item returns {\\tt done} when it successfully finished to compute a bounded $\\epsilon$-reach set or\n\t\\item returns {\\tt error} when it encounters some erroneous situations during the execution of a function, called {\\tt Post()}.\n\\end{inparaenum}\nIf the algorithm returns an {\\tt error}, it also indicates the cause of the {\\tt error} so that a user-defined rule in Policy can choose appropriate values for the input parameters.\n\n\n\\begin{algorithm} \n\\SetAlgoLined \n\\BlankLine\n\\KwIn{$k, \\delta_k, \\gamma_k, h_k$ from Policy} \n\\BlankLine\n\\Compute $\\mu_x$ from $(\\sigma_e, \\sigma_i)$\\\\\n\\BlankLine\n\\While{\\True} {\n\\Get data at $(k-1)$-th step from \\Reached \\\\\n\\If{$\\delta_k \\ne \\delta_{k-1}$} {\n\t\\Compute $\\hat{\\D}_{t_{k-1}}(\\B_{\\delta_k}(x_0))$\\\\\n\t\\Update $\\rho_{k-1}$\n}\n$t_k \\leftarrow t_{k-1}+h_k$\\\\\n\\Call  \\Post  \\\\\n\\Store $k$-th computation data into \\Reached \\\\\n$k \\leftarrow k+1$\\\\\n\\lIf{$(t_k \\ge T) \\lor (\\Jump \\ge N)$}{\\Return \\Done}\n}\n\\BlankLine\n\\caption{An algorithm for bounded $\\epsilon$-reach set computation of a DTLHA.}\n\\label{alg:main}\n\\end{algorithm}\n\n\\begin{algorithm}\n\\SetAlgoLined \n\\BlankLine\n\\KwIn{$h_k, \\gamma_k, l_{k-1}, \\rho_{k-1}, \\hat{\\D}_{t_{k-1}}(\\B_{\\delta_k}(x_0))$} \n\\BlankLine\n\\Compute $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$ from $\\hat{\\D}_{t_{k-1}}(\\B_{\\delta_k}(x_0))$\\\\\n\\Compute $ \\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0),\\gamma_k)$ from $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$\\\\\n\\Update  $\\rho_k \\leftarrow \\rho_{k-1} + \\mu_x$\\\\\n\\BlankLine\n\\lIf{$h_k \\ge (\\gamma_k - \\rho_k)\/\\bar{v}$}{\\Return \\Err}\\\\\n\\lIf{$dia(\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k)) \\ge \\epsilon$} {\\Return \\Err} \\\\\n\\uIf{$\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0)) \\cap Inv(l_{k-1}) = \\emptyset$}{\n\t\\If{$\\hat{\\D}_{t_{k-1}}(\\B_{\\delta_k}(x_0)) \\subset Inv(l_{k-1})^{\\circ}$} {\n\t\t\\uIf{$\\Deterministic \\land \\Transversal$}{\n\t\t\t\\Update $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$ and $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k)$\n\t\t\t\\Update $\\rho_k \\leftarrow \\rho_k + \\mu_x + \\mu_c + \\mu_h$\\\\\n\t\t\t\\Update $l_k$\\\\\n\t\t\t \\Jump $\\leftarrow$ \\Jump + 1\n\t\t}\n\t\t\\lElse{\\Return \\Err}\n\t}\n}\n\\uElseIf{$\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0)) \\not\\subset Inv(l_{k-1})^{\\circ}$}{\\Return \\Err}\n\\lElse{$l_k \\leftarrow l_{k-1}$}\n\\BlankLine\n\\caption{A function {\\tt Post()}.}\n\\label{alg:post}\n\\end{algorithm}\n\n\nIn the proposed algorithm in {\\tt Post()}, $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$ is computed from $\\hat{\\D}_{t_{k-1}}(\\B_{\\delta_k}(x_0))$ as follows:\n\nGiven a polyhedron $\\hat{\\D}_{t_{k-1}}(\\B_{\\delta_k}(x_0))$, we first compute the set of the vertices of $\\hat{\\D}_{t_{k-1}}(\\B_{\\delta_k}(x_0))$ that is denoted as $\\V$.\nThen for each $v_i \\in \\V$, we compute \n\\[\n\tv_i(h_k) := e^{A_k h_k} v_i + \\int_0^{h_k} e^{A_k s} u_k ds\n\\] \nwhere $A_k$ and $u_k$ are given by the linear dynamics of a location $l_k$ on which the linear image of $\\hat{\\D}_{t_{k-1}}(\\B_{\\delta_k}(x_0))$ is computed at the $k$-the computation step in Algorithm \\ref{alg:main}. \nIf we let $\\V_h := \\{v_i(h_k) : v_i \\in \\V \\}$, then we can compute $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$ as follows: \n\\[\n\t\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0)) := hull(\\V_h)\n\\]\nwhere $hull(\\V_h)$ is the convex hull of $\\V_h$. \n\nOnce we have $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$, we compute $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k)$ in the following way.\nTo compute $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k)$ for a given $\\gamma_k$, we first construct a hypercubic $\\gamma_k$-neighborhood of $v_i(h_k)$ for each $v_i(h_k) \\in \\V_h$.\nLet $\\B_{\\gamma_k}(v_i(h_k))$ be such a $\\gamma_k$ hypercubic neighborhood of $v_i(h_k)$ and $\\V_{h}^{\\gamma}$ be the set of vertices of $\\B_{\\gamma_k}(v_i(h_k))$ for all $v_i(h_k) \\in \\V_h$.\nThen we can compute $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k)$ as follows:\n\\begin{equation} \\label{sec:design:algo:hull}\n\t\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k) := hull(\\V_h^{\\gamma}).\n\\end{equation}\n\nThis process of polyhedral image computation under a linear dynamics is illustrated in Fig. \\ref{fig:post:linear}.\nWe now show that $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k)$ that is computed as in (\\ref{sec:design:algo:hull}) is indeed a $\\gamma_k$-approximation of $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$ for a given $\\gamma_k$.\n\n\n\n\n\\begin{figure} [htbp]\n\\begin{center}\n\t\\includegraphics[width=8cm]{linear_image.png}\n\\caption{The image computation under a linear dynamics.}\n\\label{fig:post:linear}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{lem}\nLet $\\H$ be the convex hull of $\\V_h^{\\gamma}$. \nThen $\\H$ is exactly the closed $\\gamma$-neighborhood of the convex hull of $\\V_h$.\n\\end{lem}\n\\begin{proof}\nSuppose $\\bar{w} \\in \\H$ and $\\bar{w} \\not\\in hull(\\V_h)$.\nThen $\\bar{w} = \\lambda \\bar{y}_1 + (1-\\lambda)\\bar{y}_2$ for some $\\bar{y}_1$ and $\\bar{y}_2$ such that $\\Vert \\bar{y}_1 - v_1 \\Vert \\le \\gamma$ and $\\Vert \\bar{y}_2 - v_2 \\Vert \\le \\gamma$ for some $v_1, v_2 \\in \\V_h$ and $0 \\le \\lambda \\le 1$.\nThen there exists $v = \\lambda v_1 + (1-\\lambda)v_2 \\in hull(\\V_h)$ such that\n\\begin{eqnarray} \\nonumber\n\\Vert \\bar{w} -v \\Vert &=& \\Vert \\lambda(\\bar{y}_1 - v_1) + (1-\\lambda)(\\bar{y}_2 - v_2) \\Vert \\nonumber \\\\\n&\\le& \\lambda \\Vert \\bar{y}_1 - v_1 \\Vert + (1-\\lambda) \\Vert \\bar{y}_2 - v_2 \\Vert  \\nonumber \\\\\n&\\le& \\gamma. \\nonumber\n\\end{eqnarray}\nThus $\\bar{w}$ is in the $\\gamma$-neighborhood of the convex hull of $\\V_h$.\n\nFor the converse, consider $\\bar{z}$ in the $\\gamma$-neighborhood of the convex hull of $\\V_h$.\nThen for some $\\lambda_i \\ge 0$, $\\sum_i \\lambda_i = 1$, $\\Vert \\bar{z} - \\sum_i \\lambda_i v_i(h) \\Vert \\le \\gamma$, where $v_i(h) \\in \\V_h$.\nLet $s := \\bar{z} - \\sum_i \\lambda_i v_i(h)$.\nNow $\\bar{z} = \\sum_i \\lambda_i(v_i(h) + s)$.\nSo $\\bar{z}$ is in the convex hull of $\\lbrace v_i(h) + s \\rbrace$.\nHowever each $v_i(h) + s \\in \\B_{\\gamma}(v_i(h))$.\nHence each $v_i(h) + s$ is in the convex hull of the vertices of $\\B_{\\gamma}(v_i(h))$ which is $\\H$.\nThus $\\bar{z}$ is in $\\H$.\n\\end{proof}\n\n\n\nNotice that the first update of $\\rho_k$ in {\\tt Post()} is due to the computation of $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$ from $\\hat{\\D}_{t_{k-1}}(\\B_{\\delta_k}(x_0))$ over the time interval $h_k$ under the linear dynamics of $l_{k-1}$.\nThe second update after a deterministic and transversal discrete transition is due to a series of computations from $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$ that is used to determine such a discrete transition to a new $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$ that represents a reached states at time $t_k$ right after a deterministic and transversal discrete transition.\nAs described in Lemma \\ref{lem:cond:inexact:transv}, the steps involved during this discrete transition are to compute\n\\begin{inparaenum}[(i)]\n\t\\item $\\hat{\\mathcal{J}}_{c,n}$ from $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$ and\n\t\\item $\\D_{h_k}(\\hat{\\mathcal{J}}_{c,n})$ from $\\hat{\\mathcal{J}}_{c,n}$. \n\\end{inparaenum}\nNotice that (i) requires an intersection between a hyperplane and a polyhedron as well as a convex hull computation.\nMoreover, for (ii), we need to compute a polyhedral image under the linear dynamics of a new location that is determined in {\\tt Post()}.\nRecall that we have derived a set of conditions in Lemmas \\ref{lem:cond:inexact:istrans}, \\ref{lem:cond:inexact:det}, and \\ref{lem:cond:inexact:transv} to determine a deterministic and transversal discrete transition event.\nThese conditions are used in {\\tt Post()} to determine such an event.\nFurthermore, we also use conditions derived in Lemmas \\ref{lem:cond:inexact:over} and \\ref{lem:cond:inexact:eps}, to ensure that a set $\\hat{\\R}_{t_f}(\\B_{\\delta}(x_0), \\gamma)$, which can be constructed as a collection of $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k)$ as shown in the following theorem, satisfies the properties given in Remark \\ref{rem:cond:inexact}.\n\n\n\nNow, we present our main result for the problem of computing a bounded $\\epsilon$-reach set of a DTLHA.\n\n\n\\begin{thm}\nGiven input $(\\X, \\A, l_0, x_0, T,$ $N, \\epsilon)$ for a problem to compute a bounded $\\epsilon$-reach set of a DTLHA $\\A$, \nif Algorithm \\ref{alg:main} returns {\\tt done}, then a bounded $\\epsilon$-reach set of a DTLHA $\\A$ defined over the continuous state domain $\\X$ starting from an initial condition $(l_0, x_0) \\in \\mathbb{L} \\times \\mathbb{R}^n$, denoted as $\\hat{\\R}_{t_f}(x_0, \\epsilon)$, is the following:\n\\begin{equation}\n\\hat{\\R}_{t_f}(x_0, \\epsilon) := \\bigcup_{k = 1}^{K}  \\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0),\\gamma_k),\n\\end{equation}\nfor some $K \\in \\mathbb{N}$ where $t_f := \\min \\{T, \\tau_N \\}$ and $\\tau_N$ is the time at the $N$-th discrete transition. \n\\end{thm}\n\\begin{proof}\nFor each $k \\le K$, \n\\begin{inparaenum}\n\\item[(i)] $\\gamma_k, h_k, \\rho_k$ satisfies Lemma \\ref{lem:cond:inexact:over}, and \n\\item[(ii)] $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0),\\gamma_k)$ satisfies Lemma \\ref{lem:cond:inexact:eps}. \n\\end{inparaenum}\nHence $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k)$ is guaranteed to satisfy \n$\\D_{[t, t+h]}(x_0) \\subset \\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k)$ and $d_H(\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k), \\D_{[t, t+h]}(x_0)) \\le \\epsilon$.\nFurthermore, if a deterministic and transversal discrete transition is detected at the $k$-th step by $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$, then\n\\begin{inparaenum}\n\t\\item[(iii)] by Lemmas \\ref{lem:cond:inexact:istrans}, \\ref{lem:cond:inexact:det}, and \\ref{lem:cond:inexact:transv}, there is in fact a deterministic and transversal discrete transition in $(t_{k-1}, t_k)$. \n\\end{inparaenum}\nThis implies that a deterministic and transversal discrete transition event is correctly determined. \nFinally, if the proposed algorithm returns {\\tt done}, then this implies that \n\\begin{inparaenum}\n\t\\item[(iv)] either $t_k \\ge T$ or ${\\tt jump} \\ge N$. \n\\end{inparaenum}\nHence, $t_f$ is $\\min \\{ T, \\tau_N\\}$.\nTherefore, $\\R_{t_f}$ is a bounded $\\epsilon$-reach set of $\\A$ from $x_0$ by (i), (ii), (iii), and (iv).\n\\end{proof}\n\n\n\n\n\\section{Optimization and Implementation of the Proposed Algorithm}  \\label{sec:imp}\n\nA prototype software tool has been implemented, based on the architecture and the algorithm proposed in Section \\ref{sec:design}, to demonstrate the idea of a bounded $\\epsilon$-reach set computation. \nIn our implementation, we use the Multi-Parametric Toolbox \\cite{mpt} for polyhedral operations and also use some built-in Matlab functions for other calculations. \n\n\nNotice that the size of the $\\hat{\\D}_t(\\B_{\\delta}(x_0))$ right after a discrete transition increases roughly by the amount $\\gamma$ through the computation of $\\hat{\\mathcal{J}}_{c,n}$. \nThis can potentially affect the capability to determine a discrete transition event.\nHence, we determine a smaller value of $\\gamma$ to construct a tighter over-approximation of a discrete transition state. \nSuppose that a discrete transition from a location $l_i$ to some other location $l_j$ has already been determined by the proposed algorithm for given $h > 0$, $\\hat{\\D}_{t-h}(\\B_{\\delta}(x_0), \\rho)$, and $\\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\rho)$ at some time $t$.\nThen the procedure for construction of a tight over-approximation of a discrete transition state $x(\\tau_k)$ for some $\\tau_k \\in (t-h, t)$ can be improved shown in Algorithm \\ref{alg:opt}.\n\n\n\\begin{algorithm}\n\\SetAlgoLined \n\\BlankLine\n1. Partition the time interval $[t-h, t]$ into a finite sequence of $\\{ I_m \\}_{m=1}^M$ for some $M \\in \\mathbb{N}$, where\n\\[ I_m : = [t-h +(m-1) \\cdot \\Delta h, t-h + m \\cdot \\Delta h] \\]\nfor some $\\Delta h \\ll h$. \\\\\n\\BlankLine\n2. Find a time $\\tau := t-h + m \\cdot \\Delta h \\in (t-h, t)$ such that \n\t\\begin{itemize}\n\t\t\\item $volume(\\P_{\\tau}^i) > volume(\\P_{\\tau}^j)$ and\n\t\t\\item $volume(\\P_{\\tau + \\Delta h}^i) < volume(\\P_{\\tau + \\Delta h}^j)$, \n\t\\end{itemize}\n\twhere $\\P_t^k := Inv_k \\cap  \\hat{\\D}_{t}(\\B_{\\delta}(x_0), \\rho)$. \\\\\n\\BlankLine\n3. Construct $\\hat{\\D}_{\\tau+\\Delta h}(\\B_{\\delta}(x_0), \\gamma' + \\rho)$ where $\\gamma' > \\Delta h \\cdot \\bar{v}$. \\\\\n\\BlankLine\n4. Compute an over-approximate discrete transition state\n\\[ \\hat{\\mathcal{J}}_{i,j} := \\hat{\\D}_{\\tau+\\Delta h}(\\B_{\\delta}(x_0), \\gamma' + \\rho) \\cap Inv_i \\cap Inv_j. \\]\n\\caption{A procedure to compute a tight over-approximation of discrete transition state.}\n\\label{alg:opt}\n\\end{algorithm}\n\n\n\n\\subsection{An Example of Bounded $\\epsilon$-Reach Set Computation}  \\label{sec:imp:example}\n\nAs an example to evaluate the proposed algorithm for a bounded $\\epsilon$-reach set computation of a DTLHA $\\A$, we consider an LHA $\\A := (\\mathbb{L}, Inv, A, u, \\xrightarrow{G})$ over a  continuous state space $\\X := [-8, 8] \\times [-8, 8] \\subset \\mathbb{R}^2$ where\n\\begin{enumerate}[(i)]\n\t\\item $\\mathbb{L} = \\{Up, Down,$ $Left, Right \\}$,\n\t\\item $A(l)$ and $u(l)$ for each location $l \\in \\mathbb{L}$ are defined as shown in Table \\ref{tab:ex:Au},  \n\t\\item The invariant set for each location $l \\in \\mathbb{L}$, $Inv(l)$, is defined as shown in Fig. \\ref{fig:eg:rho}, and\n\t\\item $\\xrightarrow{G}$ holds at the intersection between invariant sets of different locations. \n\\end{enumerate}\n\nNotice that all the LTI dynamics defined in the given LHA $\\A$ are asymptotically stable.\nMoreover, from the the vector fields determined by $A(l)$ and $u(l)$ for each $l \\in \\mathbb{L}$, every discrete transition which occurs along the boundary of the invariant set between different locations is deterministic and transversal. \nHence the given LHA $\\A$ is in fact a DTLHA.\n\n\n\n\\begin{table}[htdp]\n\\caption{$A(l)$ and $u(l)$ for each $l \\in \\mathbb{L}$ of $\\A$}\n\\begin{center}\n{\\small\n\\begin{tabular}{c|p{0.15\\textwidth}|c}\n\\toprule\n$l$ & \\centering{$A(l)$} & $u(l)$ \\\\ \n\\midrule\n$UP$ & \\centering{$\\begin{pmatrix}  -0.2 & -1 \\\\ 3 & -0.2 \\end{pmatrix}$} & $\\begin{pmatrix}  0.1 \\\\ 0.1 \\end{pmatrix}$ \\\\ \n$DOWN$ & \\centering{$\\begin{pmatrix}  -0.2 & -1 \\\\ 3 & -0.2 \\end{pmatrix}$} & $\\begin{pmatrix}  -0.2 \\\\ -0.2 \\end{pmatrix}$ \\\\ \n$LEFT$ & \\centering{$\\begin{pmatrix}  -0.2 & -3 \\\\ 1 & -0.2 \\end{pmatrix}$} & $\\begin{pmatrix} 0.15 \\\\ 0.15 \\end{pmatrix}$ \\\\ \n$RIGHT$ & \\centering{$\\begin{pmatrix}  -0.2 & -3 \\\\ 1 & -0.2 \\end{pmatrix}$} & $\\begin{pmatrix} 0.3 \\\\ 0.3 \\end{pmatrix}$ \\\\ \n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\label{tab:ex:Au}\n\\end{table}%\n\nThe bounded $\\epsilon$-reach set computation problem is specified by $(\\A, l_0, x_0, T, N, \\epsilon)$ where $l_0 = Up$, $x_0 = (2.5, 6)^T$, $T = 20$ sec., $N = 10$, and $\\epsilon = 0.5$.\n\nIn this example, we also assume that numerical calculation algorithms are available for basic calculations defined in Section \\ref{sec:cond:basic} such that $a(e^{At}, \\rho)$, $a(\\int_{0}^{t} e^{A \\tau} d \\tau, \\rho)$, $a(\\H \\cap \\P, \\rho)$, and $a(hull(\\V), \\rho)$ where $\\rho$ is specified as $10^{-15}$.\n\n\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[width=8.5cm]{case_02_loc.png}\n\\caption{A bounded $\\epsilon$-reach set of a DTLHA $\\A$ starting from $(Up, [2.5, 6]^T)$.}\t\n\\label{fig:eg:rho}\n\\end{center}\n\\end{figure}\n\n\n\nA policy that is used to choose values for $(k, \\delta_k, \\gamma_k, h_k)$ is as follows:\n\\begin{enumerate}[(i)]\n\t\\item $k$ is chosen in non-decreasing manner, \n\t\\item $\\delta_k := 10^{-5}$ to define a fixed sufficiently small $\\B_{\\delta}(x_0)$,\n\t\\item $\\gamma_k := (\\epsilon - dia(\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\rho_k)))\/2$, and\n\t\\item $h_k := (\\gamma_k\/2)\/ \\bar{v}$ where $\\bar{v}$ is as defined in (\\ref{eq:h:ineq}).\n\\end{enumerate}\nNotice that (i) means that whenever the proposed $\\epsilon$-reach set algorithm fails to continue its computation at the $k$-th computation step, then the policy decides to restart the computation from the $k$-th step with different values of the other parameters.\nRecall that $\\rho_k$ denotes the approximation error of $\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0))$ when the algorithm computes $\\D_{t_k}(\\B_{\\delta_k}(x_0))$ at time $t_k$.\nAs shown in (iii), for a given $\\epsilon$, the policy chooses the largest value of $\\gamma_k$ at each computation step.\nThe equation for $\\gamma_k$ given in (iii) can easily be derived by considering \n\\begin{equation} \\nonumber\n\tdia(\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\gamma_k + \\rho_k)) \\le dia(\\hat{\\D}_{t_k}(\\B_{\\delta_k}(x_0), \\rho_k)) + 2 \\gamma_k.\n\\end{equation}\nIf we upper bound the right hand side by $\\epsilon$, then we have (iii).\n\n\nFig. \\ref{fig:eg:rho} shows the computation result.\nAs shown in Fig. \\ref{fig:eg:rho}, a bounded $\\epsilon$-reach set is successfully computed.\nIn this example, the algorithm terminates at the computation step $k = 2259$ right after the algorithm makes the tenth discrete transition from locations $Left$ to $Down$ at the time $t = 12.1415$ sec. and {\\tt jump $= 10$}.\nFor given $\\rho := 10^{-15}$, the accumulated numerical calculation error $\\rho_k$ for $\\D_{t_k}(\\B_{\\delta_k}(x_0))$ at this termination time is $2.5638 \\times 10^{-11}$.\n\n\n\n\n\\section{Conclusion} \\label{sec:con}\nWe have defined a special class of hybrid automata, called Deterministic and Transversal Linear Hybrid Automata (DTLHA), for which we can address the problem of bounded $\\epsilon$-reach set computation starting from an initial state.\nFor this class, we can also incorporate the impact of numerical calculation errors caused by finite precision numerical computation.\n\nIt is of importance to determine more  general and useful models of hybrid systems that permit computational verification of safety properties. Hybrid linear systems that incorporate linear models widely employed in control systems are a natural candidate around which to build such a theory of verification and validation.\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\emph{This Game Is Not Going To Load Itself} (TGINGTLI{}) \\cite{tgingtli} is a free game created in 2015 by Roger ``atiaxi'' Ostrander for the Loading Screen Jam, a game jam hosted on \\protect\\url{itch.io},\nwhere it finished $7$th overall out of $46$ entries.\nThis game jam was a celebration of the expiration of US Patent 5,718,632 \\cite{US5718632}, which covered the act of including mini-games during video game loading screens.\nIn this spirit, TGINGTLI{} is a real-time puzzle game themed around the player helping a game load three different resources of itself --- save data, gameplay, and music, colored red, green, and blue --- by placing arrows on the grid cells to route data entering the grid to a corresponding sink cell.\nFigure~\\ref{fig:realplay} shows an example play-through.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.63]{figs\/realplay_input}\\hfill\n  \\includegraphics[scale=0.63]{figs\/realplay}\n  \\caption{%\n    Left: The (eventual) input for a real-world Level 16 in TGINGTLI{}.\n    Right: A successful play-through that\n    routes every packet to its corresponding sink.}\n   \n  \\label{fig:realplay}\n\\end{figure}\n\nWe formalize TGINGTLI{} as follows.\nYou are given an $m \\times n$ grid where each unit-square cell is either empty,\ncontains a data sink, or contains an arrow pointing\nin one of the four cardinal directions.\n(In the implemented game, $m=n=12$ and no arrows are placed initially.)\nEach data sink and arrow has a color (resource) of red, green, or blue;\nand there is exactly one data sink of each color in the grid.\nIn the online version (as implemented), sources appear throughout the game;\nin the offline version considered here, all sources are known a priori.\nNote that an outer edge of the grid may have multiple sources of different colors.\nFinally, there is an loading bar that starts at an integer $k_0$\nand has a goal integer~$k^*$.\n\nDuring the game,\neach source periodically produces data packets of its color,\nwhich travel at a constant speed into the grid.\nIf a packet enters the cell of an arrow of the same color,\nthen the packet will turn in that direction.\n(Arrows of other colors are ignored.)\nIf a packet reaches the sink of its color,\nthen the packet disappears and the loading bar increases by one unit of data.\nIf a packet reaches a sink of the wrong color, or exits the grid entirely,\nthen the packet disappears and the loading bar decreases by one unit of data, referred to as taking damage.\nPackets may also remain in the grid indefinitely\nby going around a cycle of arrows;\nthis does not increase or decrease the loading bar.\nThe player may at any time permanently fill an empty cell with an arrow, which may be of any color and pointing in any of the four directions.\nIf the loading bar hits the target amount $k^*$, then the player wins; but if the loading bar goes below zero, then the player loses.\n\nIn Section~\\ref{sec:NP-hardness},\nwe prove NP-hardness of the {TGINGTLI} decision problem:\ngiven a description of the grid\n(including sources, sinks, and preplaced arrows),\ncan the player place arrows to win?\nThis reduction works even for just six sources and three colors;\nit introduces a new problem, \\defn{3DSAT}, where variables have three\ndifferent colors and each clause mixes variables of all three colors.\nIn Section~\\ref{sec:sigma-2}, we introduce more detailed models for the\nperiodic behavior of sources, and show that many sources of differing periods\nenable both NP- and coNP-hardness of winning the game,\n\\emph{even without player input} (just simulating the game).\nOn the positive side, we prove that these problems are in $\\Sigma_2^P$;\nand in NP when the source periods are all equal,\nas in our first NP-hardness proof, so this case is in fact NP-complete.\n\nIn Section~\\ref{sec:perfect-layouts}, we consider how levels start in\nthe implemented game: a grid with placed sinks but no preplaced arrows.\nWe give a full characterization of when there is a \\defn{perfect layout}\nof arrows, where all packets are guaranteed to route to the correct sink,\n\\emph{no matter where sources get placed}.\nIn particular, this result provides a winning strategy for most\nsink arrangements in the implemented game.\nNotably, because this solution works independent of the sources,\nit works in the online setting.\n\n\\xxx{We should perhaps point to some of the other hardness of videogames literature and if there is literature on solving small instances of puzzle games through math and exhaustive search.}\n\n\\section{NP-Hardness for Three Colors and Six Sources}\n\\label{sec:NP-hardness}\n\nWe first prove that TGINGTLI{} is NP-hard by reducing from a new problem called\n\\defn{3-Dimensional SAT (3DSAT)},\ndefined by analogy to 3-Dimensional Matching (3DM).\n3DSAT is a variation of 3SAT where, in addition to a 3CNF formula,\nthe input specifies one of three colors (red, green, or blue) to each variable\nof the CNF formula, and the CNF formula is constrained to have trichromatic\nclauses, i.e., to have exactly one variable (possibly negated) of each color.\n\n\\begin{lemma}\n  3DSAT is NP-complete.\n\\end{lemma}\n\n\\begin{proof}\n  We reduce from 3SAT to 3DSAT by converting a 3CNF formula $F$\n  into a 3D CNF formula~$F'$.\n  For each variable $x$ of~$F$,\n  we create three variables $x^{(1)}, x^{(2)}, x^{(3)}$ in $F'$\n  (intended to be equal copies of $x$ of the three different colors)\n  and add six clauses to $F'$ to force $x^{(1)} = x^{(2)} = x^{(3)}$:\n  %\n  \\def\\halfup#1{\\raisebox{2.5ex}{\\smash{$#1$}}}\n  \\begin{align*}\n    \\lnot x^{(1)} \\lor x^{(2)} \\lor x^{(3)}       &\\iff (x^{(1)} \\to x^{(2)}) \\lor x^{(3)} \\\\\n    \\lnot x^{(1)} \\lor x^{(2)} \\lor \\lnot x^{(3)} &\\iff (x^{(1)} \\to x^{(2)}) \\lor \\lnot x^{(3)} ~\\halfup{\\Bigg\\rbrace \\iff x^{(1)} \\to x^{(2)}} \\\\\n    x^{(1)} \\lor \\lnot x^{(2)} \\lor x^{(3)}       &\\iff (x^{(2)} \\to x^{(3)}) \\lor x^{(1)} \\\\\n    \\lnot x^{(1)} \\lor \\lnot x^{(2)} \\lor x^{(3)} &\\iff (x^{(2)} \\to x^{(3)}) \\lor \\lnot x^{(1)} ~\\halfup{\\Bigg\\rbrace \\iff x^{(2)} \\to x^{(3)}} \\\\\n    x^{(1)} \\lor x^{(2)} \\lor \\lnot x^{(3)}       &\\iff (x^{(3)} \\to x^{(1)}) \\lor x^{(2)}\\\\\n    x^{(1)} \\lor \\lnot x^{(2)} \\lor \\lnot x^{(3)} &\\iff (x^{(3)} \\to x^{(1)}) \\lor \\lnot x^{(2)} ~\\halfup{\\Bigg\\rbrace \\iff x^{(3)} \\to x^{(1)}}\n  \\end{align*}\n  %\n  Thus the clauses on the left are equivalent to the implication loop\n  $x^{(1)} \\implies x^{(2)} \\implies x^{(3)} \\implies x^{(1)}$,\n  which is equivalent to $x^{(1)} = x^{(2)} = x^{(3)}$.\n  \n  For each clause $c$ of $F$ using variables $x$ in the first literal,\n  $y$ in the second literal, and $z$ in the third literal,\n  we create a corresponding clause $c'$ in $F'$ using\n  $x^{(1)}$, $y^{(2)}$, and $z^{(3)}$ (with the same negations as in~$c$).\n  All clauses in $F'$ (including the variable duplication clauses above)\n  thus use a variable of the form $x^{(i)}$ in the\n  $i$th literal for $i \\in \\{1,2,3\\}$,\n  so we can 3-color the variables accordingly.\n\\end{proof}\n\n\\begin{theorem} \\label{thm:NP-hard36}\n  {TGINGTLI} is NP-hard, even with three colors and six sources\n  of equal period.\n\\end{theorem}\n\n\\begin{proof}\n  Our reduction is from 3DSAT.\n  Figure~\\ref{fig:reduction-sketch} gives a high-level sketch:\n  each variable gadget has two possible routes\n  for a packet stream of the corresponding color,\n  and each clause gadget allows at most two colors of packets\n  to successfully route through.\n  When a clause is satisfied by at least one variable,\n  the clause gadget allows the other variables to successfully pass\n  through; otherwise, at least one of the packet streams enters a cycle.\n  Variables of the same color are chained together\n  to re-use the packet stream of that color.\n\n  \\begin{figure}\n    \\centering\n    \\subcaptionbox{Satisfied clause}{\\includegraphics[scale=0.5]{figs\/reduction-sketch-sat}}\\hfil\n    \\subcaptionbox{Unsatisfied clause}{\\includegraphics[scale=0.5]{figs\/reduction-sketch-unsat}}\n    \\caption{Sketch of our NP-hardness reduction.}\n    \\label{fig:reduction-sketch}\n  \\end{figure}\n  \n  In detail, most cells of the game board\n  will be prefilled, leaving only a few empty cells\n  (denoted by question marks in our figures) that the player can fill.\n \n \n \n\n  For each color, say red, we place a red source gadget\n  on the left edge of the construction.\n  Then, for each red variable $x$ in sequence,\n  we place a variable gadget of Figure~\\ref{fig:gadget:variable}\n  at the end of the red stream.\n  To prevent the packets from entering a loop,\n  the player must choose between sending the stream upward or downward,\n  which results in it following one of the two rightward paths,\n  representing the literals $x$ and $\\overline x$ respectively.\n  The path followed by the packet stream is viewed as \\emph{false};\n  an empty path is viewed as \\emph{true}.\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{minipage}[b]{0.51\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.9]{figs\/clause.pdf}\n    \\caption{Clause gadget. At most two streams of data, representing false literals, can pass through the gadget (by placing upwards arrows in the ``?'' cells) without entering a cycle.\n      Placing any other direction of arrow also puts the stream into a cycle.}\n    \\label{fig:gadget:clause}\n   \n  \\end{minipage}\\hfill\n  \\begin{minipage}[b]{0.22\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.9]{figs\/variable.pdf}\n    \\caption{\\centering Variable gadget lets the player route packets from a source to one of two literal paths.}\n    \\label{fig:gadget:variable}\n  \\end{minipage}\\hfill\n  \\begin{minipage}[b]{0.22\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.9]{figs\/merge.pdf}\n    \\caption{\\centering Merge gadget combines two literal paths back into one.\\linebreak~}\n    \\label{fig:gadget:merge}\n  \\end{minipage}\n  \n  \\bigskip\n  \n  \\begin{minipage}[b]{0.5\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.9]{figs\/crossover.pdf}\n    \\caption{Crossover gadget between two literal paths of same or different colors. (The center cell is colored different from both paths.)}\n    \\label{fig:gadget:crossover}\n  \\end{minipage}\\hfill\n \n \n \n \n \n \n  \\begin{minipage}[b]{0.46\\linewidth}\n    \\includegraphics[scale=0.9]{figs\/death.pdf}\n    \\caption{Damage gadget forces damage at a unit rate after a desired start delay.\\hspace{\\fill}\\linebreak~}\n    \\label{fig:gadget:damage}\n  \\end{minipage}\n\\end{figure}\n\n  Then we route each literal path to sequentially visit\n  every clause containing it.\n  Figure~\\ref{fig:gadget:crossover} shows a crossover gadget\n  to enable such routing.\n  (Note that it still works if both lines are the same color.)\n\n  Figure~\\ref{fig:gadget:clause} shows the clause gadget,\n  which allows at most two packet streams to pass through it.\n  If all three literals are false then at least\n  one stream of data must be placed into a cycle.\n  On the other hand, if the clause is satisfied, then the literal paths\n  carrying data can pass their data on to the next clause, and so on.\n  Note that the length of the diverted path through the clause\n  is the same for all three colors.\n\n  After all uses of the red variable \\(x\\),\n  we lengthen the two red literal paths\n  corresponding to \\(x\\) and \\(\\overline{x}\\) to have the same length,\n  then combine them back together\n  using the merge gadget of Figure~\\ref{fig:gadget:merge}.\n  We then route this red path into the variable gadget\n  for the next red variable, and so on.\n  Finally, after all red variables, we connect the red stream\n  to a red sink.\n\n  We lengthen the red, green, and blue streams\n  to all have the same length~$\\ell$.\n  If the player successfully satisfies all clauses, then\n  they will increase the loading bar by $3$ units ($1$~per color)\n  after an initial delay of~$\\ell$.\n  We set the parameters so that the player wins in this case:\n  $k^* - k_0 = 3$.\n  Otherwise, the loading rate is at most~$2$.\n  To ensure that the player loses in this case, we add\n  $3$ damage gadgets of Figure~\\ref{fig:gadget:damage},\n  each incurring damage at a rate of $1$ after an initial delay of $\\ell+1$.\n  Thus we obtain a net of $-1$ per period,\n  so the player eventually loses even if $k_0$ is large.\n\\end{proof}\n\n\nThis NP-hardness result does not need a very specific model of sources and\nhow they emit packets.\nTo understand whether the problem is in NP, we need a more specific model,\nwhich is addressed in the next section.\n\n\n\\section{Membership in $\\Sigma_2^P$ and Hardness from Source Periodicity}\n\\label{sec:sigma-2}\n\n\\iffalse\nmembership in sigma 2 - algorithm\n    check if you didn't lose\n    check if you didn't win\n    skip to the next event\n\nco-np hard:\n    checking if you didn't win\/lose\n\\fi\n\nIn this section, we consider the effect of potentially differing periods\nfor different sources emitting packets.\nSpecifically, we show that carefully setting periods together with\nthe unbounded length of the game results in both NP- and coNP-hardness\nof determining the outcome of {TGINGTLI},\n\\emph{even when the player is not making moves}.\nConversely, we prove that the problem is in~$\\Sigma_2^P$,\neven allowing player input.\n\n\\subsection{Model and Problems}\n\nMore precisely, we model each source $s$ as emitting data packets of its color\ninto the grid with its own period $p_s$, after a small warmup time $w_s$\nduring which the source may emit a more specific pattern of packets.\n{TGINGTLI} as implemented has a warmup behavior of each source\ninitially (upon creation) waiting 5 seconds before the first emitted packet,\nthen emitting a packet after 2 seconds, after 1.999 seconds,\nafter 1.998 seconds, and so on,\nuntil reaching a fixed period of 0.5 seconds.\nThis is technically a warmup period of 1881.25 seconds with\n1500 emitted packets, followed by a period of 0.5 seconds.\n\nIn the \\defn{simulation} problem, we are given the initial state of the grid,\na list of timestamped \\defn{events} for\nwhen each source emits a packet during its warmup period,\nwhen each source starts periodic behavior,\nand when the player will place each arrow.\nWe assume that timestamps are encoded in binary but\n(to keep things relatively simple)\nperiods and warmup times are encoded in unary.\nThe problem then asks to predict whether the player wins;\nthat is, the loading bar reaches~$k^*$ before a loss occurs.\n\nIn the \\defn{game} problem, we are not given the player's arrow placements.\nIf we allow nondeterministic algorithms, the game problem reduces to the\nsimulation problem: just guess what arrows we place, where, and at what times.\n\nA natural approach to solving the simulation problem is to simulate the game\nfrom the initial state to each successive event.\nSpecifically, given a state of the game (a grid with sinks, sources of varying periods and offsets, placed arrows, and the number of in-flight packets at each location) and a future timestamp \\(t\\),\nwe wish to determine the state of the game at time \\(t\\).\nUsing this computation, we can compute future states of the game quickly by\n``skipping ahead'' over the time between events.\nOn an $m \\times n$ grid, there are $O(m n)$ events, so we can determine the state of the game at any time \\(t\\) by simulating polynomially many phases between events.\n\nThis computation is easy to do.\nGiven the time \\(t\\), we can divide by each source's period and the period of each cycle of arrows to determine\nhow many packets each source produces and where the arrows route them ---\neither to a sink which affects loading, off the grid, stuck in a cycle,\nor in-flight outside a cycle ---\nand then sum up the effects to obtain the new amount loaded and the number of packets at each location.\n\\xxx{Be more specific?}\n\nHowever, being able to compute future states\ndoes not suffice to solve the simulation and game problems because there might be an intermediate time\nwhere the loading amount drops below $0$ or reaches the target amount~$k^*$.\nNonetheless, this suffices to show that the problems are in \\(\\Sigma_2^P\\),\nby guessing a win time and verifying there are no earlier loss times:\n\n\\begin{lemma}\n  The simulation and game problems are in \\(\\Sigma_2^P\\).\n\\end{lemma}\n\\begin{proof}\n  The player wins if there exists a time with a win such that all smaller times are not losses.\n  To solve the simulation problem,\n  nondeterministically guess the winning time and verify that it is a win\n  by computing the state at that time.\n  Then check using a coNP query that there was no loss before that time,\n  again using the ability to quickly compute states at individual timestamps.\n\n  To solve the game problem, we first existentially guess the details of the arrow placements,\n  then solve the resulting simulation problem as before.\n\\end{proof}\n\nAn easier case is when the source periods are all the same\nafter warmup, as implemented in the real game.\nTheorem~\\ref{thm:NP-hard36} proved this version of the game NP-hard,\nand we can now show that it is NP-complete:\n\n\\begin{lemma}\n  If all sources have the same polynomial-length period after a\n  polynomial number $t_p$ of time steps,\n  then the simulation problem is in P and the game problem is in NP.\n\\end{lemma}\n\n\\begin{proof}\n  In this case, we can check for wins or losses in each phase between events by explicitly simulating longer than all packet paths,\n  at which point the loading bar value becomes periodic with the common source period.\n  (Cycles of arrows may have different periods but these do not affect the loading bar, and thus do not matter when checking for wins and losses.)\n  We skip over each phase, checking for win or loss along the way.\n  If the game continues past the last event, we measure the sign of the net score change over the period.\n  If it is positive, the player will eventually win;\n  if it is negative, the player will eventually lose; and\n  if it is zero, the game will go on forever.\n\\end{proof}\n\n\nIn the remainder of this section, we consider the case where each source can be assigned any integer period, and the period does not change over time.\n\n\\subsection{Periodic Sum Threshold Problem}\n\nWith varying source periods, the challenge is that the overall periodic\nbehavior of the game can have an extremely large (exponential) period.\n\nWe can model this difficulty via the\n\\defn{Periodic Sum Threshold Problem}, defined as follows.\nWe are given a function $f(x) = \\sum_{i} g_i(x)$\nwhere each $g_i$ has unary integer period $T_i$\nand unary maximum absolute value $M_i$.\nIn addition, we are given a unary integer $\\tau > 0$\nand a binary integer time $x^*$.\nThe goal is to determine whether there exists an integer $x$\nin $[0, x^*)$ such that $f(x) \\geq \\tau$.\n(Intuitively, reaching $\\tau$ corresponds to winning.)\n\n\\begin{theorem}\n\\label{thm:pstp-np-complete}\n  The Periodic Sum Threshold Problem is NP-complete,\n  even under the following restrictions:\n  \\begin{enumerate}\n  \\item \\label{prop:one-hot}\n    Each $|g_i|$ is a one-hot function, i.e.,\n    $g_i(x) = 0$ everywhere except for exactly one $x$ in its period\n    where $g_i(x) = \\pm 1$.\n  \\item \\label{prop:lambda}\n    We are given a unary integer $\\lambda < 0$ such that\n    $f(x) > \\lambda$ for all $0 \\leq x < x^*$ and $f(x^*) \\leq \\lambda$.\n    (Intuitively, dipping down to $\\lambda$ corresponds to losing.)\n  \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n  First, the problem is in NP: we can guess $x \\in [0,x^*)$\n  and then evaluate whether $f(x) \\leq c$ in polynomial time.\n\n  For NP-hardness, we reduce from 3SAT.\n  We map each variable $v_i$ to the $i$th prime number $p_i$ excluding~$2$.\n  Using the Chinese Remainder Theorem,\n  we can represent a Boolean assignment $\\phi$ as a single integer $0 \\le x < \\prod_i p_i$\n  where $x \\equiv 1 \\mod p_i$ when $\\phi$ sets $v_i$ to true,\n  and $x \\equiv 0 \\mod p_i$ when $\\phi$ sets $v_i$ to false.\n  (This mapping does not use other values of $x$ modulo~$p_i$.\n  In particular, it leaves $x \\equiv -1 \\mod p_i$ unused,\n  because $p_i \\geq 3$.)\n\n  Next we map each clause such as $C = (v_i \\vee v_j \\vee \\neg v_k)$\n  to the function\n  %\n  $$g_C(x) = \\max\\{[x \\equiv 1 \\mod p_i], [x \\equiv 1 \\mod p_j], [x \\equiv 0 \\mod p_k]\\},$$\n  %\n  i.e., positive literals check for $x \\equiv 1$\n  and negated literals check for $x \\equiv 0$.\n  This function is $1$ exactly when $x$ corresponds\n  to a Boolean assignment that satisfies~$C$.\n \n \n \n  This function has period $p_i p_j p_k$,\n  whose unary value is bounded by a polynomial.\n  Setting $\\tau$ to the number of clauses,\n  there is a value $x$ where the sum is $\\tau$ if and only if\n  there is a satisfying assignment for the 3SAT formula.\n  (Setting $\\tau$ smaller, we could reduce from Max 3SAT.)\n\n  To achieve Property~\\ref{prop:one-hot}, we split each $g_C$ function\n  into a sum of polynomially many one-hot functions\n  (bounded by the period).  In fact, seven functions per clause suffice,\n  one for each satisfying assignment of the clause.\n\n  To achieve Property~\\ref{prop:lambda},\n  for each prime $p_i$, we add the function\n  $h_i(x) = -[x \\equiv -1 \\mod p_i]$.\n  This function is $-1$ only for unused values of \\(x\\)\n  which do not correspond to any assignment \\(\\phi\\),\n  so it does not affect the argument above.\n  Setting $-\\lambda$ to the number of primes (variables)\n  and $x^* = \\prod_i p_i - 1$,\n  we have $\\sum_i h_i(x^*) = \\lambda$\n  because $h_i(x^*) \\equiv -1 \\mod p_i$ for all~$i$,\n  while $\\sum_i h_i(x) > \\lambda$ for all $0 \\leq x < x^*$.\n  All used values \\(x\\) are smaller than \\(x^*\\).\n\n  In total, $f(x)$ is the sum of the constructed functions\n  and we obtain the desired properties.\n\\end{proof}\n\n\n\\subsection{Simulation Hardness for Two Colors}\n\nWe can use our hardness of the Periodic Sum Threshold Problem\nto prove hardness of simulating {TGINGTLI}, even without player input.\n\n\\begin{theorem} \\label{thm:sim NP-hard}\n  Simulating {TGINGTLI} and determining whether the player wins\n  is NP-hard, even with just two colors.\n\\end{theorem}\n\\begin{proof}\n  We reduce from the Periodic Sum Threshold Problem proved NP-complete\n  by Theorem~\\ref{thm:pstp-np-complete}.\n\n  For each function $g_i$ with one-hot value $g_i(x_i) = 1$ and period~$T_i$,\n  we create a blue source $b_i$ and a red sources $r_i$,\n  of the same emitting period~$T_i$,\n  and route red and blue packets from these sources to the blue sink.\n  By adjusting the path lengths and\/or the warmup times of the sources,\n  we arrange for a red packet to arrive one time unit after each blue packet\n  which happens at times $\\equiv x_i \\mod T_i$.\n  Thus the net effect on the loading bar value is $+1$ at time $x_i$\n  but returns to $0$ at time $x_i + 1$.\n  Similarly, for each function $g_i$ with one-hot value $g_i(x_i) = -1$,\n  we perform the same construction but swapping the roles of red and blue.\n\n  Setting $k_0 = -\\lambda-1 \\geq 0$, the loading bar goes negative\n  (and the player loses)\n  exactly when the sum of the functions $g_i$ goes down to~$\\lambda$.\n  Setting $k^* = k_0 + \\tau$, the loading bar reaches $k^*$\n  (and the player wins)\n  exactly when the sum of the functions $g_i$ goes up to~$\\tau$.\n\\end{proof}\n\nThis NP-hardness proof relies on completely different aspects of the game\nfrom the proof in Section~\\ref{sec:NP-hardness}: instead of using player input,\nit relies on varying (but small in unary) periods for different sources.\nMore interesting is that we can also prove the same problem coNP-hard:\n\n\\begin{theorem}\n  Simulating {TGINGTLI} and determining whether the player wins\n  is coNP-hard, even with just two colors.\n\\end{theorem}\n\\begin{proof}\n  We reduce from the complement of the Periodic Sum Threshold Problem,\n  which is coNP-complete by Theorem~\\ref{thm:pstp-np-complete}.\n  The goal in the complement problem is to determine whether\n  there is \\emph{no} integer $x$ in $[0, x^*)$ such that $f(x) \\geq \\tau$.\n  The idea is to negate all the values to flip the roles of winning and losing.\n  \n  For each function $g_i$, we construct two sources and wire them to a sink\n  in the same way as Theorem~\\ref{thm:sim NP-hard}, but negated:\n  if $g_i(x_i) = \\pm 1$, then we design the packets to have a net effect\n  of $\\mp 1$ at time $x_i$ and $0$ otherwise.\n\n  Setting $k_0 = \\tau-1$, the loading bar goes negative\n  (and the player loses)\n  exactly when the sum of the functions $g_i$ goes up to~$\\tau$, i.e.,\n  the Periodic Sum Threshold Problem has a ``yes'' answer.\n  Setting $k^* = k_0 - \\lambda$, the loading bar reaches $k^*$\n  (and the player wins)\n  exactly when the sum of the functions $g_i$ goes down to $\\lambda$, i.e.,\n  the Periodic Sum Threshold Problem has a ``no'' answer.\n\\end{proof}\n\n\n\\section{Characterizing Perfect Layouts}\n\\label{sec:perfect-layouts}\n\nSuppose we are given a board which is empty except for the location of the\nthree data sinks.\nIs it possible to place arrows such that all possible input packets\nget routed to the correct sink?\nWe call such a configuration of arrows a \\defn{perfect layout}.\nIn particular, such a layout guarantees victory,\nregardless of the data sources.\nIn this section, we give a full characterization of boards\nand sink placements that admit a perfect layout.\nSome of our results work for a general number $c$ of colors,\nbut the full characterization relies on $c=3$.\n\n\\subsection{Colors Not Arrows}\n\nWe begin by showing that we do not need to consider the directions of the arrows, only their colors and locations in the grid.\n\nLet \\(B\\) be a board with specified locations of sinks,\nand let \\(\\partial B\\) be the set of edges on the boundary of \\(B\\).\nSuppose we are given an assignment of colors to the cells of \\(B\\)\nthat agrees with the colors of the sinks;\nlet \\(C_i\\) be the set of grid cells colored with color \\(i\\).\nWe call two cells of \\(C_i\\), or a cell of \\(C_i\\) and a boundary edge\n$e \\in \\partial B$, \\defn{visible} to each other if and only if\nthey are in the same row or the same column\nand no sink of a color other than \\(i\\) is between them.\nLet \\(G_i\\) be the graph whose vertex set is \\(C_i \\cup \\partial B\\),\nwith edges between pairs of vertices that are visible to each other.\n\n\\begin{lemma}\n  \\label{lem:perfect-colors}\n  Let \\(B\\) be a board with specified locations of sinks.\n  Then \\(B\\) admits a perfect layout if and only if it is possible to choose\n  colors for the remaining cells of the grid such that, for each color~\\(i\\),\n  the graph \\(G_i\\) is connected.\n\\end{lemma}\n\\begin{proof}\n  ($\\Longrightarrow$)\n  Without loss of generality, assume that\n  the perfect layout has the minimum possible number of arrows.\n  Color the cells of the board with the same colors as the sinks and arrows in the perfect layout.\n  (If a cell is empty in the perfect layout,\n  then give it the same color as an adjacent cell;\n  this does not affect connectivity.)\n  Fix a color \\(i\\).\n  Every boundary edge is connected to the sink of color \\(i\\)\n  by the path a packet of color \\(i\\) follows when entering from that edge.\n  (In particular, the path cannot go through a sink of a different color.)\n  By minimality of the number of arrows in the perfect layout,\n  every arrow of color \\(i\\) is included in such a path.\n  Therefore \\(G_i\\) is connected.\n\n  ($\\Longleftarrow$)\n  We will replace each cell by an arrow of the same color to form a perfect layout.\n  Namely, for each color~\\(i\\), choose a spanning tree of \\(G_i\\)\n  rooted at the sink of color~\\(i\\),\n  and direct arrows from children to parents in this tree.\n  By connectivity, any packet entering from a boundary edge\n  will be routed to the correct sink, walking up the tree to its root.\n\\end{proof}\n\n\\subsection{Impossible Boards}\n\\label{sec:impossible-boards}\n\nNext we show that certain boards cannot have perfect layouts.\nFirst we give arguments about boards containing sinks\ntoo close to the boundary or each other.\nThen we give an area-based constraint on board size.\n\n\\begin{lemma}\n  \\label{lem:sink-distance}\n  If there are fewer than $c-1$ blank cells in a row or column\n  between a sink and a boundary of the grid,\n  then there is no perfect layout.\n\\end{lemma}\n\\begin{proof}\n  A perfect layout must prevent packets of the other \\(c-1\\) colors\n  entering at this boundary from reaching this sink;\n  this requires enough space for \\(c-1\\) arrows.\n\\end{proof}\n\n\\begin{lemma}\n  \\label{lem:impossible-c3}\n  For $c=3$, a board has no perfect layout if either\n  (as shown in Figure~\\ref{fig:impossible})\n  \\begin{enumerate}[(a)]\n    \\item a data sink is two cells away from three boundaries and adjacent to another sink;\n    \\item a data sink is two cells away from two incident boundaries and is adjacent to two other sinks;\n    \\item a data sink is two cells away from two opposite boundaries and is adjacent to two other sinks; or\n    \\item a data sink is two cells away from three boundaries and is one blank cell away from a pair of adjacent sinks.\n  \\end{enumerate}\n\\end{lemma}\n\n\\begin{figure}\n  \\centering\n  \\subcaptionbox{}{\\includegraphics[scale=0.75]{figs\/impossible1}}\\hfil\n  \\subcaptionbox{}{\\includegraphics[scale=0.75]{figs\/impossible2}}\\hfil\n  \\subcaptionbox{}{\\includegraphics[scale=0.75]{figs\/impossible3}}\\hfil\n  \\subcaptionbox{}{\\includegraphics[scale=0.75]{figs\/impossible4}}\n  \\caption{Sink configurations with no perfect layout.\n    Dots indicate arrows of forced colors\n    (up to permutation within a row or column).}\n  \\label{fig:impossible}\n\\end{figure}\n\n\\begin{proof}\n  Assume by symmetry that, in each case, the first mentioned sink is red.\n\n  Cases (a), (b), and (c):\n  The pairs of cells between the red sink and the boundary\n  (marked with dots in the figure) must contain a green arrow and a blue arrow\n  to ensure those packets do not reach the red sink.\n  Thus there are no available places to place a red arrow\n  in the same row or column as the red sink,\n  so red packets from other rows or columns cannot reach the red sink.\n  \n  Case (d): The pairs of cells between the red sink and the boundary\n  (marked with green and blue dots in the figure)\n  must contain a green arrow and a blue arrow to ensure those packets\n  do not collide with the red sink.\n  Thus the blank square between the red sink and the other pair of sinks\n  must be a red arrow pointing toward the red sink,\n  to allow packets from other rows and columns to reach the red sink.\n  Assume by symmetry that the sink nearest the red sink is green.\n  As in the other cases, the pairs of cells between the green sink and the\n  boundary must be filled with red and blue arrows.\n  Thus there are no green arrows to route green packets from other\n  rows or columns to the green sink.\n \n \n \n\\end{proof}\n\nWe now prove a constraint on the sizes of boards that admit a perfect layout.\n\n\\begin{lemma}\n  \\label{lem:size-constraint}\n  Let \\(c\\) be the number of colors.\n  Suppose there is a perfect layout on a board where \\(m\\) and \\(n\\) are respectively the number of rows and columns, and \\(p\\) and \\(q\\) are respectively the number of rows and columns that contain at least one sink.\n  Then\n  \\begin{equation}\n    \\label{eqn:size-constraint}\n    c(m + n) + (c-2)(p + q) \\le m n - c.\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\n  Each of the \\(m - p\\) unoccupied rows must contain \\(c\\) vertical arrows in order to redirect packets of each color out of the row.\n  Each of the \\(p\\) occupied rows must contain \\(c-1\\) vertical arrows to the\n  left of the leftmost sink in order to redirect incorrectly colored packets\n  from the left boundary edge away from that sink;\n  similarly, there must be \\(c-1\\) vertical arrows to the right of the\n  rightmost sink.\n  Thus we require \\(c(m - p) + 2(c - 1)p = c m + (c - 2)p\\)\n  vertical arrows overall.\n  By the same argument, we must have \\(c n + (c - 2)q\\) horizontal arrows,\n  for a total of\n  \\(c(m + n) + (c - 2)(p + q)\\)\n  arrows.\n  There are \\(m n - c\\) cells available for arrows, which proves the claim.\n\\end{proof}\n\nUp to insertion of empty rows or columns, rotations, reflections, and\nrecolorings, there are six different configurations that $c=3$ sinks\nmay have with respect to each other, shown in Figure~\\ref{fig:3-sink-configs}.\nWe define a board's \\defn{type} according to this configuration of its sinks\n(C, I, J, L, Y, or \/).\n\n\\begin{figure}[htbp]\n  \\centering\n  \\hfil\n  \\subcaptionbox{C}{\\includegraphics[scale=0.9]{figs\/config-C}}\\hfil\n  \\subcaptionbox{I}{\\includegraphics[scale=0.9]{figs\/config-I}}\\hfil\n  \\subcaptionbox{J}{\\includegraphics[scale=0.9]{figs\/config-J}}\\hfil\n  \\subcaptionbox{L}{\\includegraphics[scale=0.9]{figs\/config-L}}\\hfil\n  \\subcaptionbox{Y}{\\includegraphics[scale=0.9]{figs\/config-Y}}\\hfil\n  \\subcaptionbox{\/}{\\includegraphics[scale=0.9]{figs\/config-slash}}%\n  \\hfil\n  \\caption{The six possible configurations of three sinks up to rotations, reflections, recolorings, and removal of empty rows.}\n  \\label{fig:3-sink-configs}\n\\end{figure}\n\nA board's type determines the values of \\(p\\) and \\(q\\)\nand thus the minimal board sizes as follows.\nDefine a board to have size \\defn{at least} $m \\times n$ if\nit has at least $m$ rows and at least $n$ columns, or vice versa.\n\n\\begin{lemma}\n  \\label{lem:size-constraint-c3}\n  For a perfect layout to exist with \\(c=3\\), it is necessary that:\n  \\begin{itemize}\n  \\item Boards of type Y or \/ have size at least $7\\times8$.\n  \\item Boards of type C or J have size at least $7\\times8$ or $6\\times9$.\n  \\item Boards of type L have size at least $7\\times7$ or $6\\times9$.\n  \\item Boards of type I have size at least $7\\times7$, $6\\times9$, or $5\\times11$.\n  \\end{itemize}\n\\end{lemma}\n\\begin{proof}\n  These bounds follow from Lemma~\\ref{lem:size-constraint} together with the requirement from Lemma~\\ref{lem:sink-distance} that it be possible to place sinks at least two cells away from the boundary.\n\\end{proof}\n\n\\subsection{Constructing Perfect Layouts}\n\nIn this section, we complete our characterization of boards\nwith perfect layouts for \\(c=3\\).\nWe show that Lemmas \\ref{lem:sink-distance}, \\ref{lem:impossible-c3}, and\n\\ref{lem:size-constraint-c3} are the only obstacles to a perfect layout:\n\n\\begin{theorem}\n  \\label{thm:characterization-c3}\n  A board with $c=3$ sinks has a perfect layout if and only if the following conditions all hold:\n  \\begin{enumerate}\n  \\item All sinks are at least two cells away from the boundary\n  (Lemma~\\ref{lem:sink-distance}).\n  \\item The board does not contain any of the four unsolvable configurations in Figure~\\ref{fig:impossible} (Lemma~\\ref{lem:impossible-c3}).\n  \\item The board obeys the size bounds of Lemma~\\ref{lem:size-constraint-c3}.\n  \\end{enumerate}\n\\end{theorem}\n\nWe call a board \\defn{minimal} if it has one of the minimal dimensions\nfor its type as defined in Lemma~\\ref{lem:size-constraint-c3}.\nOur strategy for proving Theorem~\\ref{thm:characterization-c3}\nwill be to reduce the problem to the finite set of minimal boards,\nwhich we then verify by computer.\nWe will accomplish this by removing empty rows and columns\nfrom non-minimal boards to reduce their size,\nwhich we show can always be done while preserving the above conditions.\n\n\\begin{lemma}\n  \\label{lem:characterization-c3-minimal}\n  All minimal boards satisfying the three conditions of\n  Theorem~\\ref{thm:characterization-c3} have a perfect layout.\n\\end{lemma}\n\\begin{proof}\n  \\renewcommand{\\qedsymbol}{\\(\\blacksquare\\)}\n  The proof is by exhaustive computer search of all such minimal boards.\n  We wrote a Python program to generate all possible board patterns,\n  reduce each perfect layout problem to Satisfiability Modulo Theories\n  (SMT), and then solve it using Z3 \\cite{z3}.\n  The results of this search are in Appendix~\\ref{apx:computer-solutions}.\n\\end{proof}\n\nIf \\(B_0\\) and \\(B_1\\) are boards, then we define \\defn{\\(B_0 \\pmb{\\lessdot} B_1\\)}\nto mean that \\(B_0\\) can be obtained by removing a single empty row or column\nfrom \\(B_1\\).\n\n\\begin{lemma}\n  \\label{lem:add-row}\n  If \\(B_0 \\lessdot B_1\\) and \\(B_0\\) has a perfect layout,\n  then \\(B_1\\) also has a perfect layout.\n\\end{lemma}\n\\begin{proof}\n  By symmetry, consider the case where $B_1$ has an added row.\n  By Lemma~\\ref{lem:perfect-colors}, it suffices to show that we can color\n  the cells of the new row while preserving connectivity in each color.\n  We do so by duplicating the colors of the cells (including sinks)\n  in an adjacent row.\n  Connectivity of the resulting coloring follows from that of the original.\n\\end{proof}\n\n\\begin{lemma}\n  \\label{lem:characterization-c3-nonminimal}\n  Let \\(B_1\\) be a non-minimal board satisfying the three conditions\n  of Theorem~\\ref{thm:characterization-c3}.\n  Then there exists a board \\(B_0\\) that also satisfies all three conditions\n  and such that \\(B_0 \\lessdot B_1\\).\n\\end{lemma}\n\\begin{proof}\n  By symmetry, suppose $B_1$ is non-minimal in its number $m$ of rows.\n  By removing a row from $B_1$ that is not among the first or last two rows\n  and does not contain a sink, we obtain a board \\(B'_0\\) satisfying conditions\n  (1) and (3) such that \\(B'_0 \\lessdot B_1\\).\n  If \\(B'_0\\) also satisfies condition (2), then we are done,\n  so we may assume that it does not.\n\n  Then \\(B'_0\\) must contain one of the four unsolvable configurations,\n  and \\(B_1\\) is obtained by inserting a single empty row or column to\n  remove the unsolvable configuration.\n  Figure~\\ref{fig:perfect-reductions} shows all possibilities for \\(B'_0\\),\n  as well as the locations where rows or columns may be inserted to yield\n  a corresponding possibility for \\(B_1\\).\n  (\\(B'_0\\) may have additional empty rows and columns beyond those shown,\n  but this does not affect the proof.)\n  For each such possibility, Figure~\\ref{fig:perfect-reductions} highlights\n  another row or column which may be deleted from \\(B_1\\) to yield\n  \\(B_0 \\lessdot B_1\\) where \\(B_0\\) satisfies all three conditions.\n\\end{proof}\n\n\\begin{figure}\n  \\centering\n  \\hfil\n  \\subcaptionbox{$L$, $7\\times7$}{\\includegraphics[scale=0.6]{solver\/reductions\/L7_7_reduction}}\\hfill\n  \\subcaptionbox{$L$, $6\\times9$}{\\includegraphics[scale=0.6]{solver\/reductions\/L6_9_reduction}}\\hfill\n  \\subcaptionbox{\\label{fig:perfect-reductions-row-choice}$I$, $5\\times11$}{\\includegraphics[scale=0.6]{solver\/reductions\/I5_11_0_reduction}}\\hfill\n  \\subcaptionbox{$I$, $5\\times11$}{\\includegraphics[scale=0.6]{solver\/reductions\/I5_11_1_reduction}\\vspace{0.2in}}\\hfill\n  \\subcaptionbox{$I$, $5\\times11$}{\\includegraphics[scale=0.6]{solver\/reductions\/I5_11_2_reduction}}%\n  \\hfil\n  \\caption{\n    All boards satisfying conditions (1) and (3) but not (2),\n    up to rotations, reflections, and recolorings.\n    An empty row or column may be inserted in any of the locations\n    marked ``$+$'' to yield a board satisfying all three conditions.\n    Removing the row or column marked ``$-$'' then preserves the conditions.\n    In case~(c),\n   \n    remove a row that does not contain the blue sink.\n    In case~(d),\n    $\\vcenter{\\hbox{\\includegraphics[scale=0.6]{figs\/dotdotdot}}}$\n    denotes zero or more rows.\n  }\n  \\label{fig:perfect-reductions}\n\\end{figure}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:characterization-c3}]\n  It follows from Lemmas \\ref{lem:sink-distance}, \\ref{lem:impossible-c3},\n  and \\ref{lem:size-constraint-c3} that all boards with perfect layouts\n  must obey the three properties of the theorem.\n  We prove that the properties are also sufficient\n  by induction on the size of the board.\n  As a base case, the claim holds for minimal boards\n  by Lemma~\\ref{lem:characterization-c3-minimal}.\n  For non-minimal boards \\(B_1\\),\n  Lemma~\\ref{lem:characterization-c3-nonminimal} shows that\n  there is a smaller board \\(B_0\\)\n  that satisfies all three conditions and such that \\(B_0 \\lessdot B_1\\).\n  By the inductive hypothesis, \\(B_0\\) has a perfect layout.\n  Lemma~\\ref{lem:add-row} shows that \\(B_1\\) also has a perfect layout.\n\\end{proof}\n\n\n\\section{Open Questions}\n\nThe main complexity open question is whether {TGINGTLI} is $\\Sigma_2^P$-complete.\nGiven our NP- and coNP-hardness results, we suspect that this is true.\n\nOne could also ask complexity questions of more restrictive versions of the game.\nFor example, what if the board has a constant number of rows?\n\nWhen characterizing perfect layouts, we saw many of our lemmas generalized to different numbers of colors. It may be interesting to further explore the game and try to characterize perfect layouts with more than three colors.\n\nA related problem is which boards and configurations of sinks\nadmit a \\defn{damage-free} layout, where any packet entering from\nthe boundary either reaches the sink of the correct color\nor ends up in an infinite loop.  Such a layout avoids losing,\nand in the game as implemented, such a layout actually wins the game\n(because the player wins if there is ever insufficient room\nfor a new source to be placed).\nCan we characterize such layouts like we did for perfect layouts?\n\nPerfect and damage-free layouts are robust to any possible sources.\nHowever, for those boards that do not admit a perfect or damage-free layout,\nit would be nice to have an algorithm that determines whether a given\nset of sources or sequence of packets still has a placement of arrows\nthat will win on that board.\nBecause the board starts empty except for the sinks,\nour hardness results do not apply.\n\nHaving a unique solution is often a desirable property of puzzles. Thus it is natural to ask about ASP-hardness and whether counting the number of solutions is \\#P-hard.\n\n\n\n\\section*{Acknowledgments}\n\nThis work was initiated during open problem solving in the MIT class on\nAlgorithmic Lower Bounds: Fun with Hardness Proofs (6.892)\ntaught by Erik Demaine in Spring 2019.\nWe thank the other participants of that class\nfor related discussions and providing an inspiring atmosphere.\nIn particular, we thank Quanquan C. Liu for helpful discussions\nand contributions to early results.\n\nMost figures of this paper were drawn using SVG Tiler\n[\\url{https:\/\/github.com\/edemaine\/svgtiler}].\nIcons (which match the game) are by looneybits and\nreleased in the public domain\n[\\url{https:\/\/opengameart.org\/content\/gui-buttons-vol1}].\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\nThere are insufficiently large or no datasets for research questions on camera-based AI systems from the field of medical operating rooms. Use cases for AI systems in this context include sterility of health professionals, whether and where certain medical devices are located or action recognition of health professionals. Furthermore, datasets are, if at all, only institution-related and not publicly available due to data protection regulations and ethics requirements.\n\nThe successes of deep learning in recent years are among others, due to the availability of large datasets such as Imagenet \\cite{ILSVRC15} for Image Classification, or MS COCO \\cite{DBLP:journals\/corr\/LinMBHPRDZ14} for Object Bounding Box Detection. In addition, the research of new methods or architectures like \\cite{He2016}\\cite{Lin2020}\\cite{Tan_2020}\\cite{Bochkovskiy2020} and the research of high performance hardware for parallel computing are to be mentioned. The authors of \\cite{Sze2018}\\cite{LeCun2019} analyze the hardware topic in depth. In this work however, special focus is put on the availability of datasets and methods for dataset generation in order to reduce the necessary amount of real data from the target domain. \n\nSeveral works already deal with the generation of synthetic data and with the goal of reducing the reality gap between synthetic and real data. Among these are the work on Domain Randomization (DR) and Structured Domain Randomization (SDR) \\cite{DrTobin}\\cite{DrTremblay}\\cite{Prakash_2019}. In addition to other work, it has already been shown that the use of synthetic image data can decrease the required amount of real data \\cite{DrTremblay}. Likewise, synthetic data of persons exist \\cite{h36m_pami}. However, the challenge lies in the specific characteristics of a medical intervention space. Health professionals wear special clothing with sometimes multiple layers, wear sterile gloves, masks and hairnets. The differences between conventional human data and data from the medical field are large. Nevertheless, Domain Randomization techniques sound promising for use in research questions around medical interventions.\n\nThis work presents a comparison in terms of detection accuracy and generalizability of different methods for synthetic clothing generation using either 3D clothing scans (SCANS) or designed CAD clothing (CAD) with SMPL models \\cite{SMPL:2015}. The comparison is performed using the example of medical clothing object detection. Both methods (SCANS, CAD) are incorporated into a domain randomization environment called NDDS \\cite{to2018ndds} and a Structured Domain Randomization environment implemented in Unity, based on \\cite{Technologies2017}, to generate synthetic training data. Likewise, the aim of the presented methodology is to explore a pipeline for the generation of synthetic data for the medical field, so that further research questions from the intervention space can be explored. In addition to synthetic data, real data of different persons are recorded in front of a green screen and in a clinical environment. All data are split into train\/val\/test sets while the clinical dataset serves as a test set for all methods.\n\n\n\\section{Related Work}\n\\label{sec:relatedWork}\nWith the rise of synthetic data generation methods, for example Domain Randomization \\cite{DrTobin}, it has already been shown that synthetic data can reduce the amount of real data required \\cite{DrTremblay}. However, one focus of research is the reduction of the reality gap between the synthetic data and the target domain.\n\nHere, the aforementioned Domain Randomization (DR) has turned out to be one way to reduce the gap. One idea of DR is that if enough variance can be generated in the synthetic data, reality represents another variance of the target domain \\cite{DrTobin}.\n\nThe work of \\textit{Tobin et al.} \\cite{DrTobin} and \\textit{Tremblay et al.} \\cite{DrTremblay} showed, that an object detection network for robot grasping or car detection can be trained from synthetic images with random positioning, random lighting, random backgrounds, distractor objects and non-realisitc textures alone. In addition, the work of \\textit{Tremblay et al.} showed, that the necessary amount of real target domain data can be reduced while maintaining adequate accuracy, when pretrained with DR generated images.\n\nAlso the work of \\textit{Borkman et al.}\\cite{Borkman2021} showed that when using Unity Perception for synthetic data generation, the amount of real-world data could be reduced to 10\\% when used together with the synthetic data, while achieving better AP score as with all real-world data alone. \n\nDomain Randomization has already been successfully applied in various fields. In addition to the mentioned areas of car detection and robot grasping, the work of Sadeghi et al. \\cite{Sadeghi2017} for flying a quadrocopter through indoor environments, \\textit{Zhang et al.} \\cite{DBLP:journals\/corr\/abs-1709-05746} for a table-top object reaching task through clutter or \\textit{James et al.} \\cite{DBLP:journals\/corr\/JamesDJ17} for grasping a cube and placing it in a basket can be named.\n\nThis leads us to believe that DR is a suitable approach for the medical intervention room domain, where no real data is largely available and access to that domain is widely restricted.\n\nAblation studies of \\cite{DrTremblay} and \\cite{DrTobin} showed, that high resolution textures and higher numbers of unique textures in the scene improve performance.\nAlso, \\cite{DBLP:journals\/corr\/abs-1709-05746} come to the conclusion, after testing their hypothesis, that using complex textures yields better performance than using random colors.\n\nIn contrast to the domain randomization approach is the photorealistic rendering of the scene and objects. A number of datasets have been created for this purpose in recent years. Here the works of \\cite{Tremblay2018a} \\cite{h36m_pami} \\cite{Varol2017LearningFS} or \\cite{GraspingDRphoto} are to be mentioned. Some of these works combine real image data with domain randomization and photorealistic rendered image data.\n\nIn \\cite{Tremblay2018a} a photorealistic rendered dataset was created for 21 objects of the YCB dataset. Here, the objects are rendered in different scenes with collision properties when falling down. The dataset is intended to accelerate progress in the area of object detection and pose estimation. \n\nIn \\cite{GraspingDRphoto}, domain randomization is combined with photo realistically rendered image data, for robotic grasping of household objects. Using the data generated in this way, the authors have managed to explore a real-time system for object detection and robot grasping with sufficient accuracy. They also showed that the combination of both domains improved performance as opposed to just one alone.\n\nIn the field of human pose estimation, the works of \\cite{h36m_pami} and \\cite{Varol2017LearningFS} need to be mentioned. Both works were able to show that the performance of networks can be increased by using synthetic and animated persons, respectively.\n\nThe work of \\cite{Varol2017LearningFS} generates photo-realistic synthetic image data and their ground truth for body part classifications.\n\nIn \\cite{h36m_pami}, animated persons are integrated into mixed reality environments. The movements were recorded by actors in a motion capture scenario and transferred to 3D scanned meshes. In their experiments, they were able to achieve a 20\\% increase in performance compared to the largest training set available in this domain.\n\nState of the art models for realistic human body shapes are the SMPL models introduced by \\cite{SMPL:2015} and improved by STAR in \\cite{Osman2020}. SMPL stands for Skinned Multi-Person Liner Model (SMPL) which according to the authors is a skinned vertex-based model which represents human shapes in a wide variety. In their work they learn male and female body shape from the CAESAR dataset \\cite{CAESAR}. Their model is compatible with a wide variety of rendering engines like Unity or Unreal and therefore highly suited to be used in synthetic data generation for humans. There also exist extensions to the SMPL model like MANO and SMPL-H which introduce a deformable hand model into the framework. MANO \\cite{MANO:SIGGRAPHASIA:2017} is learned from 1000 high resolution 3D scans of various hand poses.\n\n\\section{Method}\n\\label{sec:methods}\nAs previously mentioned, real-world data collection in medical intervention rooms is complex, costly, and requires approval from an ethics board and the persons involved. \nAs shown in the previous, DR can help train an object detection network with sufficient performance in real world applications.\n \nHowever, one challenge in dataset generation for the medical intervention space is domain-specific clothing. We argue, that randomizing the clothing textures with random textures would help improve detection rates of the clothing types, but when applied in real world applications, for example a colored T-shirt would not be distinguishable from the targeted blue colored specific area clothing. For the general detection of cars as in \\cite{DrTremblay} the randomization technique makes sense, but for the domain specific use case presented here something else should be used in our opinion.\n\n\nThe questions we try to address in this work are:\n\\begin{enumerate}\n\\item How can health professionals be modeled for synthetic data generation?\n\\item Which techniques are best suited for SDR\/DR clothing generation?\n\\item Can we close the reality gap further by including greenscreen data (Mixed Reality, MR)?\n\\item Can the required amount of real data be reduced by using SDR\/DR\/MR ?\n\\end{enumerate}\n\nFor point 1) we argue to use a deformable human shape model like the SMPL models. This provides sufficient variance of different human shapes and sizes. For point 2) we explore two different methods of clothing generation. First we 3D scan various persons wearing medical clothing and generate a database of different medical clothing scans for each clothing type, which we call SCANS. Second we commission a professional graphics designer to create assets based on the area clothing, which we call CAD. Regarding point 3) we take images in front of a greenscreen of different persons wearing medical clothing which we label by hand. For point 4) we explore wether and by how much the required amount of real data can be reduced.\n\nTo address the named questions further, we set up experiments where we want to detect the following classes with the help of the Scaled Yolov4 object detector \\cite{Wang_2021_CVPR}\n\nThe classes to be detected are:\n\\begin{itemize}\n\t\\item humans\n\t\\item area clothing shirt\n\t\\item area clothing pants\n\t\\item sterile gown\n\t\\item medical face mask\n\t\\item medical hairnet\n\t\\item medical gloves\n\\end{itemize}\n\nExamples of the medical clothing are given in figure \\ref{fig:expClothing}.\n\n\n\\begin{figure}\n\t\\centering\n\t\\begin{tabular}{l c}\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/clothingexp\/shirt}\n\t\t&\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/clothingexp\/glove} \\\\\n\t\t\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/clothingexp\/pants}\n\t\t&\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/clothingexp\/mask} \\\\\n\t\\end{tabular}\t\n\t\\caption{Medical clothing examples like area shirt, pants, mask and glove. These clothing types among others represent the target clothing for our object detection network.}\n\t\\label{fig:expClothing}\n\\end{figure}\n\n\n\\subsection{Character Creation}\n\\label{sec:charCreation}\nThe medical characters we use in SDR\/DR are built through a combination of SMPL body models, textures, animations and clothing assets. Within the following sections each of the components are described in detail. \n\n\\textbf{Body Model}\\\\\nAs the base of our characters we use the male and female model of the SMPL+H model from \\cite{MANO:SIGGRAPHASIA:2017}. The models cover a huge variety of realistic human shapes, which can be randomized through ten blend shapes. We decide to use the extended SMPL+H model instead of the original SMPL model \\cite{SMPL:2015}. This is because one of our cloth items are gloves and through the hand rig of the SMPL+H model we will be able to create more deformations of the glove asset. \n\n\\textbf{Human Texture}\\\\\nTo add more variation and realism to the appearance of the characters, the texture maps from \\cite{Varol2017LearningFS} are used.\nOut of the 930 textures, only 138 (69 of every gender) have been used. This is, as we created our own cloth assets, only the textures of people in undergarments were relevant. Those texture maps were created out of 3D body scans from the CAESAR dataset \\cite{CAESAR} and cover a variety of skin colors and identities, however all of the faces have been anonymized \\cite{Varol2017LearningFS}. \n\n\\textbf{Pose} \\\\\nTo provide a variety of realistic body poses, the models were animated through Motion Capture (MoCap) data, which has been captured within our laboratory. We track the movement of 74 joints down to the fingertips. We use an intrinsic Motion Capture suite with the Hand gloves Add-on called Perception Neuron Studio \\footnote{Perception Neuron Studio suite and gloves addon: https:\/\/neuronmocap.com\/perception-neuron-studio-system}. In order to keep the dataset simple, we only used one animation in our experiments. The potential to add more varying animations is given however.\\\\\n\n\\textbf{3D-scanned Cloth Assets (SCANS)} \\\\\nA 3D Scanner called Artec Leo\\footnote{3D Scanner Artec Leo: https:\/\/www.artec3d.com\/portable-3d-scanners\/artec-leo } with a 3D resolution of 0.2mm was used to capture the medical cloths. For our synthetic training dataset we used clothing scans of 4 male and 4 female models. In this way, variations of the real world textures, including reflections, wrinkles, colors and surface texture information are collected. After building an initial model from the 3D Scanner, we adapt the cloths to fit the standard male and female SMPL+H Character using 3D Modelling techniques. As the gloves should match the Characters fingers exactly, to ensure correct animations of the fingers, they have been modeled instead of scanned. Afterwards the textures of scanned gloves have been applied to the models. When the cloth assets are fitted, we create cloth blend shapes, which match the blend shapes of the SMPL+H model, in order to make them adaptable with the character. Additionally, the cloth meshes are bound to the same rig as the character, by transferring the skin weights of the SMPL+H models. Like this the cloth assets are just as adaptable in shape and pose as the body model. \nAccording to our research, medical cloths usually come in the colors blue, green and light pink. To cover this variation in our dataset, without scanning more cloth or performing augmentations on the whole image, we augmented the texture maps. Examples of the scanned and rigged clothing assets can be seen in figure \\ref{fig:expClothingScans}.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{tabular}{l c}\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/pics_3dmodels\/mask-and-net}\n\t\t&\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/pics_3dmodels\/mask-and-net-1} \\\\\n\t\t\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/pics_3dmodels\/shirt}\n\t\t&\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/pics_3dmodels\/gown} \\\\\n\t\\end{tabular}\t\n\t\\caption{Examples of our 3D-scanned clothing assets with color augmentation.}\n\t\\label{fig:expClothingScans}\n\\end{figure}\n\n\n\\textbf{Designed Cloth Assets (CAD)} \\\\\nTo evaluate the performance of 3D-scanned cloth assets we compare them to hand designed clothing assets. Therefore we have asked a designer\\footnote{azeemdesigns: https:\/\/www.fiverr.com\/azeemdesigns?\\\\source=order\\_page\\_user\\_message\\_link} on Fiver to model the clothes. Examples of those assets can be seen in figure \\ref{fig:expClothingCad}. We first evaluated to what extent freely available assets from the assets stores can be used for this purpose. However, there are no assets available that match our specific clothing in total. Therefore, we have decided to have the assets designed. The designed assets have been processed in the same way as our scanned assets. They are also deformable and are bound to the same rig.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{tabular}{l c}\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/pics_cadModels\/mask-and-hat-1}\n\t\t&\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/pics_cadModels\/mask-and-hat-2} \\\\\n\t\t\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/pics_cadModels\/shirt-1}\n\t\t&\n\t\t\\includegraphics[width=.22\\textwidth]{figures\/pics_cadModels\/gown-1} \\\\\n\t\\end{tabular}\t\n\t\\caption{Examples of the designed clothing assets with color augmentation.}\n\t\\label{fig:expClothingCad}\n\\end{figure}\n\n\\textbf{Modular Character DR, NDDS} \\\\\nTo create the datasets from the body models, animations, cloth assets and all other components are combined in a character blueprint in Unreal Engine 4.22.3(DR). Here we have created a modular character, who is able to take on random shapes, textures and combinations of clothing. Also the 3D scanned cloths and designed cloths will vary in shape with the character and move with the animation. When the data capturing begins, the character first iterates over 1700 body shapes. The SMPL shape parameters for these body shapes have been taken from \\cite{Varol2017LearningFS} and represent 1700 male and female body types taken from the CEASAR Dataset. Next, one randomly sampled human texture is applied to the body model. Afterwards one cloth item out of each category is randomly sampled, textured and added to the body model. After the blend shapes of all cloth assets are adapted to the current body shape, one animation is chosen. When the animation finished playing, the next body shape will be chosen and the process will repeat itself. We create two seperate datasets, one with 3D-scanned (SCANS) assets and another with designed (CAD) assets for DR. An activity diagram, which represents the blueprint for Modular Character creation in NDDS, is given in figure \\ref{fig:flowChartBlueprint}.\\\\\n\n\\textbf{Modular Character SDR, Unity}\\\\\nTo be able to use SMPL models in unity, the male and female models are converted to a Humanoid character and an avatar is created from the model. Variation of clothing in unity is made possible using a custom-made component which selects random pieces of clothing from each category and applies them to a randomly selected SMPL model. In order to enable the clothing to animate together with the model, the bones of each piece of clothing are changed to match with those of the model using said component.  Afterwards a random animation can be assigned to the model using the Animator component with a custom Animator Controller by setting the Motion parameter. Finally the model including all selected clothing is instantiated and randomly placed inside the scene. The animation is then moved along during data generation by setting the Motion Time parameter of the Animator Controller to a random value. The character itself is adapted using the custom-made randomization components by varying the texture of the models as well as the material of the clothing using predefined color variations. Models as well as clothing and position are varied each frame. The shape of the models is not modified as it resulted in clipping of the clothing and other unrealistic behavior. An activity diagram is shown in figure \\ref{fig:flowChartUnity}\n\n\\begin{figure}\n\t\\centering\t\n\t\\includegraphics[width=.49\\textwidth]{figures\/activity} \\\\\t\n\t\\caption{Activity Diagram of the modular character blueprint used in Unreal with NDDS.}\n\t\\label{fig:flowChartBlueprint}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\t\n\t\\includegraphics[width=.49\\textwidth]{figures\/activityUnity} \\\\\t\n\t\\caption{Activity Diagram of the character creation used in Unity for SDR.}\n\t\\label{fig:flowChartUnity}\n\\end{figure}\n\n\n\n\n\\textbf{Domain Randomization}\nFor the synthetic data generation of DR, an Unreal Engine 4 plugin called NDDS\\cite{to2018ndds} was used. This allows the generation of RGB images at rates similar to real cameras, as well as depth image data and segementation masks of the scene within Unreal Engine 4 (UE4). The plugin also creates bounding box labeling data for each object in the scene in 2D and 3D. The tool was specifically developed for Domain Randomization and therefore provides tools for scene randomization like object or camera position, lighting and distractor objects, among others. Using the aforementioned Modular Character blueprint, NDDS enables the generation of synthetic datasets for sterile clothing using 3D scanned clothing or designed based clothing. Example images are given in figure \\ref{fig:expSynth} on the top row.\n\n\\textbf{Structured Domain Randomization}\nFor dataset generation of SDR, we used a Unity plugin called ML-ImageSynthesis \\cite{Technologies2017}  as a base and adapted it to work with the universal rendering pipeline (URP) for quality improvement. Using Unity 2020.3.32f1, additional components have been added to enable an export of additional metadata regarding each generated image such as camera parameters, bounding boxes and world position in .json format. SDR is made possible by making use of a variety of custom-made components which allow the randomization of parameters such as lighting, material, texture, position. The plugin ProBuilder provided by Unity was used to build an intervention room based on the target domain of the real dataset (Klinikum). Scene randomization is achieved by utilizing the aforementioned randomization components.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{tabular}{l c}\n\t\t\\includegraphics[width=.23\\textwidth, trim={5cm 2cm 25cm 2cm},clip]{figures\/NDDS_exp\/000014.jpg}\n\t\t&\n\t\t\\includegraphics[width=.23\\textwidth, trim={25cm 2cm 15cm 9.5cm},clip]{figures\/NDDS_exp\/000024.jpg} \\\\\n\t\t\n\t\t\\includegraphics[width=.23\\textwidth, trim={25cm 2cm 2.5cm 9.5cm},clip]{figures\/SDR_exp\/scans.jpg}\n\t\t&\n\t\t\\includegraphics[width=.23\\textwidth, trim={25cm 2cm 2.5cm 4.5cm},clip]{figures\/SDR_exp\/cad.jpg} \\\\\n\t\\end{tabular}\t\n\t\\caption{Examples of synthetic RGB image data from DR and SDR datasets (top: DR, bottom: SDR, left: SCANS, right: CAD).}\n\t\\label{fig:expSynth}\n\\end{figure}\n\n\n\\subsection{Datasets}\n\\label{sec:dataset}\nTo investigate the potential accuracy difference between 3D scanned clothing (SCANS) and designed clothing (CAD) and the amount of necessary real data, different datasets were generated.\n\nFirst, synthetic datasets of DR and SDR were generated for both scanned and designed clothing, using the presented pipelines in Unreal-Engine und Unity. Second, a dataset in front of a greenscreen was collected (MR-DR). It consists of 8 persons in the training dataset and 2 persons in the validation dataset. The recorded persons move in front of the green screen with a certain grasping motion, which is also used as motion animation for the synthetic data. Finally a dataset of the target domain was recorded (Klinikum). It serves as a baseline comparison for all models and also presents the test data. All datasets are divided into training and validation data. \n\nExamples of real data in front of the green screen with exchanged background can be seen in figure \\ref{fig:realDataExps}. Examples of the synthetic data can be seen in figure \\ref{fig:expSynth} and finally examples from the clinical test data can be seen in figure \\ref{fig:expKlinikumTest}.\n\nTable \\ref{tab:datasetDistr} shows a breakdown of the sizes and distributions of the datasets.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{tabular}{l c}\n\t\t\\includegraphics[width=.23\\textwidth,trim={15cm 5cm 25cm 2cm},clip]{figures\/gsExps\/i-1617957374561}\n\t\t&\n\t\t\\includegraphics[width=.23\\textwidth, trim={20cm 2cm 20cm 5cm}, clip]{figures\/gsExps\/i-1617958913261}\n\t\\end{tabular}\t\n\t\\caption{Examples of the greenscreen dataset with exchanged backgrounds.}\n\t\\label{fig:realDataExps}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{tabular}{l c}\n\t\t\\includegraphics[width=.23\\textwidth,trim={20cm 5cm 20cm 5cm},clip]{figures\/expKlinikum\/i-1630600026553}\n\t\t&\n\t\t\\includegraphics[width=.23\\textwidth, trim={20cm 5cm 20cm 5cm}, clip]{figures\/expKlinikum\/i-1630600245053}\n\t\\end{tabular}\t\n\t\\caption{Examples of the Klinikum test dataset.}\n\t\\label{fig:expKlinikumTest}\n\\end{figure}\n\n\\begin{table}\n\t\\caption{This table provides data distributions in the different datasets.}\n\t\\centering\n\t\\begin{tabular}{llll}\n\t\t\\toprule\n\t\t\\multicolumn{4}{c}{dataset distributions}                   \\\\\n\t\t\\midrule\n\t\tdataset name \t& train \t& validation \t& test \\\\\n\t\t\\midrule\n\t\tDRscans \t\t& 10000\t\t& 1000\t\t\t& \/ \t\\\\\n\t\tDRcad  \t\t\t& 10000\t\t& 1000\t\t\t& \/\t\t\\\\\n\t\tSDRscans\t\t& 8000\t\t& 2000\t\t\t& \/\t \t\\\\\n\t\tSDRcad\t\t\t& 8000\t\t& 2000\t\t\t& \/\t \t\\\\\n\t\tMR-DR(GS) \t\t& 6443\t\t& 1536\t\t\t& \/\t\\\\\n\t\tKlinikum\t\t& 660\t\t& 110\t\t\t& 331\t\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:datasetDistr}\n\\end{table}\n\n\\section{Experiment Results}\n\\label{sec:experiments}\n\n\nFor the experimental investigation of whether and how well 3D scanned clothing compares to designed CAD clothing for detection in the medical environment and how much addition real data is needed to achieve sufficient accuracy, various tests were carried out.\n\nFor our experiments we used Scaled-Yolov4 implementation from GitHub\\cite{Wang_2021_CVPR}.\nAt first, 6 different baseline networks were trained to show a basic comparison of the different methods. These baseline models include trainings with synthetic (DRscans, DRcad, SDRscans, SDRcad), mixed-reality (MR-DR) and real data from the clinic domain (Klinikum train).\n\nTraining was conducted with default finetune parameters provided from Scaled Yolo-V4\\footnote{https:\/\/github.com\/WongKinYiu\/ScaledYOLOv4}. Only Mosaic Augmentation ratio parameters $\\alpha$ and $\\beta$ were increased from $8.0$ to $20.0$. Additionally a green channel augmentation was used when MR data was present in the training dataset in order to reduce the greenscreen reflection influence which we had troubles with in some classes. We experimentally found out that this gives the best results. All networks were trained for 300 epochs and achieved convergence. Tests were performed with IoU-threshold: 0.5 and confidence-threshold: 0.2. The used Yolov4 network was yolov4-p5, image size setting was 896 in training and test and the pretrained weights provided were used.\n\nAll trained models were tested on the Klinikum test-set. The results of the baseline models are displayed in table \\ref{tab:baselinesTableResults}.\n\nThe results show, that CAD based synthetic data generally give better results than SCAN based data on this experiment. This is why we used SDRcad dataset as a baseline for all follow up experiments.\n\nTo investigate by how much the amount of real data can be reduced when used together with synthetic or MR data, while maintaining sufficient accuracy, experiments were conducted with a percentage distribution of the Klinikum training data. We decided to use the mosaic augmentation during these experiments as well and use all datasets as training data instead of a finetune experiment. We argue that the network can better learn relevant features while maintaining the advantages of the additional synthetic data when seeing a variation of all used datasets mixed together with mosaic augmentation as when only finetuning. During these experiments we decided to include the aforementioned green-channel augmentation on all trainings. Additionally the real data runs are trained with equal step size.\n\nThe follow up results with SDRcad, MR-DR as well as real data is shown in table \\ref{tab:followUpResults}. While generally reaching worse results than the SDR based datasets, the MR dataset provides an additional boost in accuracy when used together with the SDRcad dataset.\n\nThese results show that when using synthetic data, MR data and all real data or only 15\\% real data, the mAP accuracy could be improved further compared to when only using real data. Additionally it shows that when only using 15\\% of real data the gap between the synthetic, MR and real data could be closed.\n\n\n\\begin{table}[ht]\n\t\\small\n\t\\caption{Results on Klinikum test-set for baseline trainings. *additional green-channel augmentation used}\n\t\\centering\n\t\\begin{tabular}{llllll}\n\t\t\\multicolumn{5}{c}{\\textbf{all}}\\\\\n\t\t\\toprule\t    \n\t\tExperiment     \t\t& $mAP$  \t\t\t& $mAP50$  \t\t\t&  P \t\t\t\t&  R                   \\\\\n\t\t\\toprule\n\t\tKlinikum train \t   \t& 81.60\t\t\t\t& 98.28 \t\t\t& 88.99\t\t\t\t& 98.73 \\\\\n\t\t\\midrule\n\t\tDRscans      \t\t& 46.06\t\t\t\t& 75.51 \t\t\t& 76.89\t\t\t\t& 77.83 \\\\\n\t\tDRcad    \t   \t\t& 51.50\t\t\t\t& 77.85 \t\t\t& 76.40\t\t\t\t& 81.56 \\\\\n\t\tSDRscans\t\t\t& 65.52\t\t\t\t& 87.12 \t\t\t& 77.92\t\t\t\t& 89.31 \\\\\n\t\tSDRcad  \t\t\t& \\textbf{67.44}\t& \\textbf{90.72} \t& \\textbf{81.98}\t& \\textbf{92.02} \\\\\n\t\tMR-DR*  \t\t\t& 60.94\t\t\t\t& 82.29 \t\t\t& 79.54\t\t\t\t& 84.04 \\\\\n\t\t\\bottomrule\t    \n\t\\end{tabular}\n\t\\label{tab:baselinesTableResults}\n\\end{table}\n\n\\begin{table}\n\t\\caption{Results on Klinikum test-set for follow up experiments. For these experiment we used the additional green-channel augmentation on all reported trainings.}\n\t\\centering\n\t\\begin{tabular}{llllll}\n\t\t\\multicolumn{5}{c}{\\textbf{all}}\\\\\n\t\t\\toprule\t    \n\t\tExperiment     \t\t& $mAP$  \t\t\t& $mAP50$  \t\t\t&  P \t\t\t\t&  R                 \\\\\n\t\t\\toprule\n\t\t\\midrule\n\t\tKlinikum(100) \t   \t& 81.95\t\t\t\t& \\textbf{98.57} \t& \\textbf{88.88}\t& \\textbf{98.85} \\\\\n\t\t\\midrule\n\t\tSDR+MR+\\\\real(100)  & \\textbf{83.35}\t& 98.14 \t\t\t& 88.41\t\t\t\t& 98.59 \\\\\n\t\t\\midrule\n\t\tKlinikum(15)   \t\t& 77.52\t\t\t\t& 96.67 \t\t\t& 87.83\t\t\t\t& 97.37 \\\\\n\t\t\\midrule\n\t\tSDR+MR+\\\\real(15)\t& 80.05\t\t\t\t& 96.92 \t\t\t& 87.06\t\t\t\t& 97.64 \\\\\n\t\t\\midrule\n\t\tSDR+MR  \t\t\t& 72.00\t\t\t\t& 92.27 \t\t\t& 83.81\t\t\t\t& 94.07 \\\\\n\t\t\\bottomrule\t    \n\t\\end{tabular}\n\t\\label{tab:followUpResults}\n\\end{table}\n\n\nInference result images of SDR+MR+real(100) can be seen in figure \\ref{fig:bestScoreKlinikExp}. Here we used a slightly higher confidence threshold of 0.4 and and IoU-threshold of 0.5.\n\n\n\\begin{figure}\n\t\\centering\n\t\\begin{tabular}{l c}\n\t\t\\includegraphics[width=.22\\textwidth,trim={15cm 5cm 25cm 0cm},clip]{figures\/resultsKlinikumTest\/i_1630599989803.jpg}\n\t\t&\n\t\t\\includegraphics[width=.22\\textwidth,trim={20cm 8cm 25cm 2cm},clip]{figures\/resultsKlinikumTest\/i_1630600036303.jpg} \\\\\n\t\t\n\t\t\\includegraphics[width=.22\\textwidth,trim={25cm 5cm 25cm 10cm},clip]{figures\/resultsKlinikumTest\/i_1630600109803.jpg}\n\t\t&\n\t\t\\includegraphics[width=.22\\textwidth,trim={20cm 5cm 25cm 5cm},clip]{figures\/resultsKlinikumTest\/i_1630600252803.jpg} \\\\\n\t\\end{tabular}\t\n\t\\caption{Inference results with SDR+MR+real(100) trained net. Confidence threshold of 0.4 and IoU-threshold of 0.5}\n\t\\label{fig:bestScoreKlinikExp}\n\\end{figure}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe were able to show that the use of SMPL models together with scanned or designed medical clothing is a suitable method for modeling heathcare professionals for AI questions in the intervention space using the example of medical clothing detection.\nDuring our experiments we found out that the designed clothing generally performed better on our test dataset than the 3D scanned cloths. This result surprised us, as we expected the potentially more accurate textures of the 3D scan to have a positive impact on detection rates. However, according to the results, it cannot be ruled out that artifacts in the rendering pipeline or pre-processing pipeline that we did not detect have an influence on this. In order to make a final statement about the potential of 3D scanned clothing for the modeling of health professionals, further experiments should be conducted.\nUsing Mixed-Reality data together with the synthetic data closed the gap further and while the margin is quite small, we could show that when using synthetic, mixed reality and 15\\% real data the remaining gap towards 100\\% real data could be nearly closed.\n\nFinally, it can be stated that the presented modeling of health professionals is a promising methodology to address the challenge of missing datasets from medical intervention rooms. We will further investigate it on various task around the medical field.\n\n\\section{Acknowledgements}\nThis research project is part of the Research Campus M2OLIE and funded by the German Federal Ministry of Education and Research (BMBF) within the Framework \"Forschungscampus \u2013 public-private partnership for Innovations\" und the funding code 13GW0389C.\n\nDisclaimer: The methods and information presented in this work are based on research and are not commercially available.\n\n\n\\bibliographystyle{unsrt}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet the domain of interest be $D=(0,1)^d$ where $d=2,3$. The Helmholtz\nequation is\n\\begin{equation}\n  \\Delta u(x)+\\dfrac{\\omega^2}{c^2(x)}u(x)=f(x),\\quad \\forall x\\in D,\n  \\label{eq:helm}\n\\end{equation}\nwhere $u(x)$ is the time-independent wave field generated by the\ntime-independent force $f(x)$, $\\omega$ is the angular frequency and\n$c(x)$ is the velocity field. Commonly used boundary conditions are\nthe approximations of the Sommerfeld radiation condition. By rescaling the system, we assume $c_{\\min}\\le c(x)\\le c_{\\max}$ where $c_{\\min}$\nand $c_{\\max}$ are of $\\Theta(1)$. Then $\\omega\/(2\\pi)$ is the typical\nwave number and $\\lambda=2\\pi\/\\omega$ is the typical wavelength.\n\nSolving the equation numerically is challenging in high frequency\nsettings for two reasons. First, in most applications, the equation is\ndiscretized with at least a constant number of points per wavelength,\nwhich makes the number of points in each direction $n=\\Omega(\\omega)$\nand the total degree of freedom $N=n^d=\\Omega(\\omega^d)$ very\nlarge. Second, the system is highly indefinite and has a very\noscillatory Green's function, which makes most of the classical\niterative methods no longer effective.\n\nThere has been a sequence of papers on developing iterative methods\nfor solving \\eqref{eq:helm}. The AILU method by Gander and Nataf\n\\cite{ailu} is the first to use the incomplete LU factorization to\nprecondition the equation. Engquist and Ying \\cite{sweephmf,sweeppml}\ndeveloped a series of sweeping preconditioners based on approximating\nthe inverse of the Schur complements in the LDU factorization and\nobtained essentially $\\omega$-independent iteration numbers. In\n\\cite{stolk2013domaindecomp}, Stolk proposed a domain decomposition\nmethod based on the PML which constructs delicate transmission\nconditions between the subdomains by considering the ``pulses''\ngenerated by the intermediate waves. In \\cite{vion2014doublesweep},\nVion and Geuzaine proposed a double sweep preconditioner based on the\nDirichlet-to-Neumann (DtN) map and several numerical simulations of\nthe DtN map were compared. In\n\\cite{chen2013sourcetrans,chen2013sourcetrans2}, Chen and Xiang\nintroduced a source transfer domain decomposition method which\nemphasizes on transferring the sources between the subdomains. In\n\\cite{demanet}, Zepeda-N{\\'u}{\\~n}ez and Demanet developed a novel\ndomain decomposition method for the 2D case by pairing up the waves\nand their normal derivatives at the boundary of the subdomains and\nsplitting the transmission of the waves into two directions. Most\nrecently in \\cite{Liu2015}, Liu and Ying proposed a recursive sweeping\npreconditioner for 3D Helmholtz problems. Other progresses includes\n\\cite{parallelsweep,sweepem,sweepemfem,sweepspectral} and we refer to\n\\cite{advances} by Erlangga and \\cite{why} by Ernst and Gander for a\ncomplete discussion.\n\nInspired by \\cite{stolk2013domaindecomp} and these previous\napproaches, we propose a new domain decomposition method in this paper\nwhich shares some similarities with\n\\cite{sweeppml,stolk2013domaindecomp}. The novelty of this new\napproach is that the transmission conditions are built with the\nboundary values of the intermediate waves directly. For each wave\nfield on the subdomains, we divide it into three parts -- the waves\ngenerated by the force to the left of the subdomain, to the right of\nthe subdomain, and within the subdomain itself. This corresponds to an\n$L+D+U$ decomposition of the Green's matrix $G$ as the sum of its\nlower triangular part, upper triangular part and diagonal part. This\nis why we call this new preconditioner the additive sweeping\npreconditioner.\n\nThe rest of this paper is organized as follows. First in Section\n\\ref{sec:1D} we use the 1D case to illustrate the idea of the\nmethod. Then in Section \\ref{sec:2D} we introduce the preconditioner\nin 2D and present the 2D numerical results. Section\n\\ref{sec:3D} discusses the 3D case.  Conclusions and some future\ndirections are provided in Section \\ref{sec:Conclusion}.\n\n\\section{1D Illustration}\n\\label{sec:1D}\n\nWe use the PML\\cite{berenger1994pml,chew1994pml,johnson2008pmlnotes}\nto simulate the Sommerfeld condition. The PML introduces the auxiliary\nfunctions\n\\begin{align*}\n\\sigma(x) &:=\n\\begin{dcases}\n\\dfrac{C}{\\eta}\\left(\\dfrac{x-\\eta}{\\eta}\\right)^2,\\quad &x\\in[0,\\eta),\\\\\n0,\\quad &x\\in[\\eta,1-\\eta],\\\\\n\\dfrac{C}{\\eta}\\left(\\dfrac{x-1+\\eta}{\\eta}\\right)^2,\\quad &x\\in(1-\\eta,1],\n\\end{dcases}\\\\\ns(x) &:= \\left(1+\\ii\\dfrac{\\sigma(x)}{\\omega}\\right)^{-1},\n\\end{align*}\nwhere C is an appropriate positive constant independent of $\\omega$,\nand $\\eta$ is the PML width which is typically around one wavelength.\n\nThe Helmholtz equation with PML in 1D is\n\\begin{equation*}\n\\begin{dcases}\n\\left((s(x)\\dfrac{\\dd}{\\dd x})^2+\\dfrac{\\omega^2}{c^2(x)}\\right)u(x)=f(x),\\quad \\forall x\\in (0,1),\\\\\nu(0)=0,\\\\\nu(1)=0. \n\\end{dcases}\n\\end{equation*}\nWe discretize the system with step size $h=1\/(n+1)$, then $n$\nis the degree of freedom. With the standard central difference\nnumerical scheme the discretized equation is\n\\begin{equation}\n\\label{eqn:1D}\n\\dfrac{s_{i}}{h}\\left(\\dfrac{s_{i+1\/2}}{h}(u_{i+1}-u_{i})-\\dfrac{s_{i-1\/2}}{h}(u_{i}-u_{i-1})\\right)+\\dfrac{\\omega^2}{c_i^2}u_{i}=f_{i}, \\quad \\forall 1\\le i\\le n,\n\\end{equation}\nwhere the subscript $i$ means that the corresponding function is evaluated at $x=ih$.\n\nWe denote Equation \\eqref{eqn:1D} as $A\\pmb u=\\pmb f$, where $\\pmb u$ and $\\pmb f$ are the discrete array of the wave field and the\nforce\n\\begin{align*}\n\\pmb u:=[u_1,\\dots,u_n]^T,\\quad \\pmb f:=[f_1,\\dots,f_n]^T. \n\\end{align*}\nIn 1D, $A$ is tridiagonal and Equation \\eqref{eqn:1D} can be solved\nwithout any difficulty. However, here we are aiming at an approach\nwhich can be generalized to higher dimensions so the rest of this\nsection takes another point of view to solve \\eqref{eqn:1D} instead of\nexploiting the sparsity structure of $A$ directly.\n\nWith the Green's matrix $G=A^{-1}$, $\\pmb u$ can be written\nas $\\pmb u=G\\pmb f$. Now let us divide the discrete grid into $m$ parts. We\nassume that $\\eta=\\gamma h$ and $n=2\\gamma+mb-2$ where $\\gamma$ and $b$ are some\nsmall constants and $m$ is comparable to $n$, and we define\n\\begin{align*}\nX_1 &:= \\{ih:1 \\le i \\le \\gamma + b-1 \\}, \\\\\nX_p &:= \\{ih:\\gamma + (p-1) b \\le i \\le \\gamma + pb-1 \\},\\quad p=2,\\dots, m-1, \\\\\nX_m &:= \\{ih:\\gamma + (m-1) b \\le i \\le 2 \\gamma + mb-2 \\}, \n\\end{align*}\nwhich means, $X_1$ is the leftmost part containing the left PML of the\noriginal problem and a small piece of grid with $b$ points, $X_m$ is\nthe rightmost part containing the right PML and a grid of $b$ points,\nand $X_p, p=2,\\dots,m-1$ are the middle parts each of which contains\n$b$ points. $\\pmb u_p$ and $\\pmb f_p$ are defined as the restrictions\nof $\\pmb u$ and $\\pmb f$ on $X_p$ for $p=1,\\dots,m$, respectively,\n\\begin{align*}\n\\pmb u_1 &:=  [u_1,\\dots,u_{\\gamma+b-1}]^T,\\\\\n\\pmb u_p &:=  [u_{\\gamma+(p-1)b},\\dots,u_{\\gamma + pb-1}]^T,\\quad p=2,\\dots,m-1,\\\\\n\\pmb u_m &:=  [u_{\\gamma + (m-1) b},\\dots,u_{2 \\gamma + m b -2}]^T,\\\\\n\\pmb f_1 &:=  [f_1,\\dots,f_{\\gamma+b-1}]^T,\\\\\n\\pmb f_p &:=  [f_{\\gamma+(p-1)b},\\dots,f_{\\gamma + pb-1}]^T,\\quad p=2,\\dots,m-1,\\\\\n\\pmb f_m &:=  [f_{\\gamma + (m-1) b},\\dots,f_{2 \\gamma + m b -2}]^T.\n\\end{align*}\nThen $u=Gf$ can be written as\n\\begin{align*}\n\\begin{bmatrix}\n\\pmb u_1\\\\\\pmb u_2\\\\ \\vdots\\\\\\pmb u_m\n\\end{bmatrix}\n=\n\\begin{bmatrix}\nG_{1,1}&G_{1,2}&\\ldots&G_{1,m}\\\\\nG_{2,1}&G_{2,2}&\\ldots&G_{2,m}\\\\\n\\vdots&\\vdots&&\\vdots\\\\\nG_{m,1}&G_{m,2}&\\ldots&G_{m,m}\n\\end{bmatrix}\n\\begin{bmatrix}\n\\pmb f_1\\\\\\pmb f_2\\\\ \\vdots\\\\\\pmb f_m\n\\end{bmatrix}.\n\\end{align*}\n\nBy introducing $\\pmb u_{p,q}:=G_{p,q}\\pmb f_q$ for $1\\le p,q\\le m$,\none can write $\\pmb u_p=\\sum_{q=1}^m \\pmb u_{p,q}$. The physical\nmeaning of $\\pmb u_{p,q}$ is the contribution of the force $\\pmb f_q$\ndefined on the grid $X_q$ acting upon the grid $X_p$. If we know the\nmatrix $G$, the computation of $\\pmb u_{p,q}$ can be carried out\ndirectly. However, computing $G$, or even applying $G$ to the vector\n$\\pmb f$, is computationally expensive. The additive sweeping method\ncircumvent this difficulty by approximating the blocks of $G$\nsequentially and the idea works in higher dimensions. In what follows,\nwe shall use $\\td{\\pmb u}_{p,q}$ to denote the approximations of $\\pmb\nu_{p,q}$.\n\n  \n\\subsection{Approximating $\\pmb u_{p,q}$ with auxiliary PMLs}\n\n\\subsubsection{Wave generated by $\\pmb f_1$}\n\nThe components ${\\pmb u}_{p,1}$ for $p=1,\\dots,m$ can be regarded as a\nsequence of right-going waves generated by $\\pmb f_1$. Note that the\nboundary condition of the system is the approximated Sommerfeld\ncondition. If we assume that the reflection during the transmission of\nthe wave is negligible, then, to approximate $\\pmb u_{1,1}$, we can\nsimply put an artificial PML on the right of the grid $X_1$ to solve a\nmuch smaller problem, since the domain of interest here is only $X_1$\n(see Figure \\ref{fig:approxU11}). To be precise, we define\n\\begin{align*}\n\\sigma_1^{M}(x) &:=  \n\\begin{dcases}\n\\dfrac{C}{\\eta}\\left(\\dfrac{x-\\eta}{\\eta}\\right)^2,\\quad &x\\in[0,\\eta),\\\\\n0,\\quad &x\\in[\\eta,\\eta+(b-1)h],\\\\\n\\dfrac{C}{\\eta}\\left(\\dfrac{x-(\\eta+(b-1)h)}{\\eta}\\right)^2,\\quad &x\\in(\\eta+(b-1)h,2\\eta+(b-1)h],\n\\end{dcases}\\\\\ns_1^{M}(x) &:=  \\left(1+\\ii\\dfrac{\\sigma_1^{M}(x)}{\\omega}\\right)^{-1}. \n\\end{align*}\nWe consider a subproblem on the auxiliary domain $D_1^M := (0,2\\eta + (b - 1)h)$\n\\begin{align*}\n\\begin{dcases}\n\\left((s_1^{M}(x)\\dfrac{\\dd}{\\dd x})^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=g(x),\\quad &\\forall x\\in D_1^M,\\\\\nv(x)=0, \\quad & \\forall x\\in \\partial D_1^M.\n\\end{dcases}\n\\end{align*}\nWith the same discrete numerical scheme and step size $h$, we have the\ncorresponding discrete system $H_1^{M} \\pmb v=\\pmb g$ on the extended grid\n\\begin{align*}\nX_1^{M}:=\\{ih :1\\le i\\le 2\\gamma+b-2\\}. \n\\end{align*}\nFigure \\ref{fig:Xp} shows a graphical view of $X_1^M$, as well as other extended grids which we will see later.\n\nWith the discrete system $H_1^M \\pmb v = \\pmb g$, we can define an operator $\\td{G}_{1}^{M}:\\pmb y\\to \\pmb z$, which is an approximation of\n$G_{1,1}$, by the following:\n\\begin{enumerate}\n\\item \n  Introduce a vector $\\pmb g$ defined on $X_1^{M}$ by setting\n  $\\pmb y$ to $X_1$ and zero everywhere else.\n\\item \n  Solve $H_1^{M} \\pmb v=\\pmb g$ on $X_1^{M}$.\n\\item \n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_1$.\n\\end{enumerate}\nThen $\\td{\\pmb u}_{1,1}$ can be set as\n\\begin{align*}\n\\td{\\pmb u}_{1,1}:=\\td{G}_{1}^{M}\\pmb f_1.\n\\end{align*}\n\n\\begin{figure}[h!]\n  \\centering\n    \n    \\begin{tikzpicture}\n      [x=2cm,y=2cm,>=latex]\n      \\draw\n      (0.0 , 0.0)--(7.0 , 0.0)\n      \n      (0.0 , 1.5)--(2.0 , 1.5)\n      (2.5 , 1.5)--(4.5 , 1.5)\n      (5.0 , 1.5)--(7.0 , 1.5)\n      \n      (0.0 , 1.0)--(1.5 , 1.0)\n      (2.5 , 1.0)--(4.0 , 1.0)\n            \n      (3.0 , 2.0)--(4.5 , 2.0)\n      (5.5 , 2.0)--(7.0 , 2.0)\n      ;\n      \n      \\draw\n      (0.0 , -0.1)--(0.0 , 0.1)\n      (1.5 , -0.1)--(1.5 , 0.1)\n      (3.0 , -0.1)--(3.0 , 0.1)\n      (4.0 , -0.1)--(4.0 , 0.1)\n      (5.5 , -0.1)--(5.5 , 0.1)\n      (7.0 , -0.1)--(7.0 , 0.1)\n      \n      (0.0 , 1.0 - 0.1)--(0.0 , 1.0 + 0.1)\n      (1.5 , 1.0 - 0.1)--(1.5 , 1.0 + 0.1)\n      (2.5 , 1.0 - 0.1)--(2.5 , 1.0 + 0.1)\n      (4.0 , 1.0 - 0.1)--(4.0 , 1.0 + 0.1)\n      \n      (3.0 , 2.0 - 0.1)--(3.0 , 2.0 + 0.1)\n      (4.5 , 2.0 - 0.1)--(4.5 , 2.0 + 0.1)\n      (5.5 , 2.0 - 0.1)--(5.5 , 2.0 + 0.1)\n      (7.0 , 2.0 - 0.1)--(7.0 , 2.0 + 0.1)\n      \n      (0.0 , 1.5 - 0.1)--(0.0 , 1.5 + 0.1)\n      (2.0 , 1.5 - 0.1)--(2.0 , 1.5 + 0.1)\n      (2.5 , 1.5 - 0.1)--(2.5 , 1.5 + 0.1)\n      (4.5 , 1.5 - 0.1)--(4.5 , 1.5 + 0.1)\n      (5.0 , 1.5 - 0.1)--(5.0 , 1.5 + 0.1)\n      (7.0 , 1.5 - 0.1)--(7.0 , 1.5 + 0.1)\n      ;\n      \n      \\draw\n      (0.0 , 0.0)node[anchor=south west]{PML}\n      (0.0 , 1.0)node[anchor=south west]{PML}\n      (0.0 , 1.5)node[anchor=south west]{PML}\n      (2.5 , 1.0)node[anchor=south west]{PML}\n      (2.5 , 1.5)node[anchor=south west]{PML}\n      (5.0 , 1.5)node[anchor=south west]{PML}\n      (7.0 , 0.0)node[anchor=south east]{PML}\n      (7.0 , 1.5)node[anchor=south east]{PML}\n      (7.0 , 2.0)node[anchor=south east]{PML}\n      (4.5 , 1.5)node[anchor=south east]{PML}\n      (4.5 , 2.0)node[anchor=south east]{PML}\n      (2.0 , 1.5)node[anchor=south east]{PML}\n      ;\n      \n      \\draw\n      (1.0 , 0.0)node[anchor = north]{$X_1$}\n      (1.0 , 1.0)node[anchor = north]{$X_1^L$}\n      (1.0 , 1.5)node[anchor = north]{$X_1^M$}\n      \n      (6.0 , 0.0)node[anchor = north]{$X_m$}\n      (6.0 , 2.0)node[anchor = north]{$X_m^R$}\n      (6.0 , 1.5)node[anchor = north]{$X_m^M$}\n      \n      (3.5 , 0.0)node[anchor = north]{$X_p$}\n      (3.5 , 1.0)node[anchor = north]{$X_p^L$}\n      (3.5 , 1.5)node[anchor = north]{$X_p^M$}\n      (3.5 , 2.0)node[anchor = north]{$X_p^R$}\n      ;\n      \n      \\draw\n      (2.25 , 0.0)node[fill=white]{$\\ldots$}\n     \n      (4.75 , 0.0)node[fill=white]{$\\ldots$}\n     \n      ;\n    \\end{tikzpicture}\n  \n  \\caption{This figure shows how the grids $X_p$ are extended with auxiliary PMLs.}\n  \\label{fig:Xp}\n\\end{figure}\n\nOnce we have computed $\\td{\\pmb u}_{1,1}$, we can use the right\nboundary value of $\\td{\\pmb u}_{1,1}$ to compute $\\td{\\pmb u}_{2,1}$\nby introducing an auxiliary PML on the right of $X_2$ and solving the\nboundary value problem with the left boundary value at\n$x=(\\gamma+b-1)h$ equal to the right boundary value of $\\td{\\pmb\n  u}_{1,1}$. The same process can be repeated to compute $\\td{\\pmb\n  u}_{p+1,1}$ by exploiting the right boundary value of $\\td{\\pmb\n  u}_{p,1}$ recursively for $p=2,\\dots,m-1$ (see Figure\n\\ref{fig:approxUp1}). In the following context of this section, we\nintroduce notations $g^{L}, g^{R}$ for a vector array $\\pmb\ng=[g_1,\\dots,g_s]^T$ by\n\\begin{align*}\ng^{L}:=g_1,\\quad g^{R}:=g_s,\n\\end{align*}\nwhere $g^{L}$ and $g^{R}$ should be interpreted as the\nleftmost and the rightmost element of the array $\\pmb g$.\n\n\n\\begin{figure}[h!]\n  \\centering\n  \\subfigure[The wave $\\pmb u_1$ (shown as the gray arrow) generated by $\\pmb f_1$.]{\n    \\begin{tikzpicture}\n      [x=3cm,y=3cm,>=latex]\n      \\draw[step=1](0,-0.1)grid(4,0.1);\n      \\draw(0.5,0)node[below=3]{$X_1$}(1.5,0)node[below=3]{$X_2$}(3.5,0)node[below=3]{$X_m$};\n      \\draw[<->,gray,very thick](0,0.5)--(4,0.5);\n      \\draw(0.5,0.5)node[fill=white]{$\\pmb f_1$};\n      \\draw(2.0,0.5)node[below=3]{$\\pmb u_1$};\n      \\draw(2.5,0.5)node[fill=white]{$\\ldots$}(2.5,0)node[fill=white]{$\\ldots$};\n      \\draw(0,0)node[anchor=south west]{PML}(4,0)node[anchor=south east]{PML};\n    \\end{tikzpicture}\n  }\n  \\subfigure[$\\td{\\pmb u}_{1,1}$ is computed by introducing an auxiliary PML on the right of $X_1$. The dotted gray arrow stands for the restriction of ${\\pmb u}_1$ on $X_2\\cup\\dots \\cup X_m$, which is to be approximated.]{\n    \\label{fig:approxU11}\n    \\begin{tikzpicture}\n      [x=3cm,y=3cm,>=latex]\n      \\draw[step=1](0,-0.1)grid(4,0.1);\n      \\draw(0.5,0)node[below=3]{$X_1$}(1.5,0)node[below=3]{$X_2$}(3.5,0)node[below=3]{$X_m$};\n      \\draw[<->,gray,very thick](0,0.5)--(1,0.5);\n      \\draw[->,dotted,gray,very thick](1,0.5)--(4,0.5);\n      \\draw(0.5,0.5)node[fill=white]{$\\pmb f_1$};\n      \\draw(0.5,0.5)node[below=3]{$\\td{\\pmb u}_{1,1}$};\n      \\draw(2.5,0.5)node[fill=white]{$\\ldots$}(2.5,0)node[fill=white]{$\\ldots$};\n      \\draw(0,0)node[anchor=south west]{PML}(1,0)node[anchor=south west]{PML};\n    \\end{tikzpicture}\n  }\n  \\subfigure[$\\td {\\pmb u}_{p,1}$ for $p=2,\\dots,m$ are computed sequentially.]{\n    \\label{fig:approxUp1}\n    \\begin{tikzpicture}\n      [x=3cm,y=3cm,>=latex]\n      \\draw[step=1](0,-0.1)grid(4,0.1);\n      \\draw(0.5,0)node[below=3]{$X_1$}(1.5,0)node[below=3]{$X_2$}(3.5,0)node[below=3]{$X_m$};\n      \\draw[<->,gray,very thick](0,0.5)--(1,0.5);\n      \\foreach \\x in{1,2,3}\n               {\n                 \\draw[->,gray,very thick](\\x,0.5)--(\\x+1,0.5);\n               }\n               \\draw(0.5,0.5)node[fill=white]{$\\pmb f_1$};\n               \\draw(0.5,0.5)node[below=3]{$\\td{\\pmb u}_{1,1}$}(1.5,0.5)node[below=3]{$\\td {\\pmb u}_{2,1}$}(3.5,0.5)node[below=3]{$\\td {\\pmb u}_{m,1}$};\n               \\draw(2.5,0.5)node[fill=white]{$\\ldots$}(2.5,0)node[fill=white]{$\\ldots$};\n    \\end{tikzpicture}\n  }\n  \\subfigure[$\\td {\\pmb u}_{p,m}$ for $p=m,\\dots,1$ are computed sequentially.]\n  {\n  \\label{fig:approxUpm}\n  \\begin{tikzpicture}\n    [x=3cm,y=3cm,>=latex]\n    \\draw[step=1](0,-0.1)grid(4,0.1);\n    \\draw(0.5,0)node[below=3]{$X_1$}(2.5,0)node[below=3]{$X_{m-1}$}(3.5,0)node[below=3]{$X_m$};\n    \\draw[<->,gray,very thick](3,0.5)--(4,0.5);\n    \\foreach \\x in{3,2,1}\n             {\n               \\draw[->,gray,very thick](\\x,0.5)--(\\x-1,0.5);\n             }\n             \\draw(3.5,0.5)node[fill=white]{$\\pmb f_m$};\n             \\draw(0.5,0.5)node[below=3]{$\\td {\\pmb u}_{1,m}$}(2.5,0.5)node[below=3]{$\\td {\\pmb u}_{m-1,m}$}(3.5,0.5)node[below=3]{$\\td {\\pmb u}_{m,m}$};\n             \\draw(1.5,0.5)node[fill=white]{$\\ldots$}(1.5,0)node[fill=white]{$\\ldots$};\n  \\end{tikzpicture}\n  }\n  \\subfigure[$\\td {\\pmb u}_{p,q}$ are computed for $p=q$ first, and then for $p=q+1,\\dots,m$ and for $p=q-1,\\dots,1$ sequentially.]\n  {\n  \\label{fig:approxUpq}\n  \\begin{tikzpicture}\n    [x=2cm,y=2cm,>=latex]\n    \\draw[step=1](0,-0.1)grid(7,0.1);\n    \\draw(0.5,0)node[below=3]{$X_1$}(2.5,0)node[below=3]{$X_{q-1}$}(3.5,0)node[below=3]{$X_q$}(4.5,0)node[below=3]{$X_{q+1}$}(6.5,0)node[below=3]{$X_m$};\n    \\draw[<->,gray,very thick](3,0.5)--(4,0.5);\n    \\foreach \\x in{3,2,1}\n             {\n               \\draw[->,gray,very thick](\\x,0.5)--(\\x-1,0.5);\n               \\draw[->,gray,very thick](\\x+3,0.5)--(\\x+4,0.5);\n             }\n             \\draw(3.5,0.5)node[fill=white]{$\\pmb f_q$};\n             \\draw(0.5,0.5)node[below=3]{$\\td {\\pmb u}_{1,q}$}(2.5,0.5)node[below=3]{$\\td {\\pmb u}_{q-1,q}$}(3.5,0.5)node[below=3]{$\\td {\\pmb u}_{q,q}$}(4.5,0.5)node[below=3]{$\\td {\\pmb u}_{q+1,q}$}(6.5,0.5)node[below=3]{$\\td {\\pmb u}_{m,q}$};\n             \\draw(1.5,0.5)node[fill=white]{$\\ldots$}(1.5,0)node[fill=white]{$\\ldots$}(5.5,0.5)node[fill=white]{$\\ldots$}(5.5,0)node[fill=white]{$\\ldots$};\n  \\end{tikzpicture}\n  }\n  \\caption{This figure shows how $\\td {\\pmb u}_{p,q}$ are generated. The direction of the arrows indicates the computing orders of the approximating waves.}\n\\end{figure}\n\n\nTo formalize the definition of $\\td{\\pmb u}_{p,1}$ for each\n$p=2,\\dots,m$, we introduce the auxiliary domain $D_p^R$, which will be defined below, to simulate the right-transmission of the waves. The superscript $R$ means that the auxiliary domain is intended for approximating the right-going waves. The left boundary of $D_p^R$ will be denoted as $\\partial^L D_p^R$, on which the boundary value will be used to approximate the wave transmission as we shall see. We also extend $X_p$ with an auxiliary PML on the right to form an extended grid $X_p^R$ (see Figure \\ref{fig:Xp}), which corresponds the discretization of $D_p^R$. To be specific, we define\n\\begin{align*}\nD_p^R &:=   (\\eta+((p-1)b-1)h,2\\eta+(pb-1)h), \\\\\n\\partial^L D_p^R  &:=  \\{\\eta+((p-1)b-1)h\\}, \\\\\n  X_p^{R} &:=  \\{ih:\\gamma+(p-1)b\\le i\\le 2\\gamma+pb-2\\}.\n\\end{align*}\nNote that the grid $X_m^{R}$ is $X_m$ itself since\n$X_m$ already contains the original right PML region. The purpose to use the notation $X_m^{R}$ is to simplify the description of the algorithm.\n\nFor the PML on $D_p^{R}$, we define\n\\begin{align*}\n  \\sigma_p^{R}(x) &:=  \n  \\begin{dcases}\n    0,\\quad &x\\in[\\eta+((p-1)b-1)h,\\eta+(pb-1)h],\\\\\n    \\dfrac{C}{\\eta}\\left(\\dfrac{x-(\\eta+(pb-1)h)}{\\eta}\\right)^2,\\quad &x\\in(\\eta+(pb-1)h,2\\eta+(pb-1)h],\n  \\end{dcases}\\\\\n  s_p^{R} &:=  \\left(1+\\ii\\dfrac{\\sigma_p^{R}(x)}{\\omega}\\right)^{-1}.\n\\end{align*} \nWe consider the following subproblem\n\\begin{align*}\n  \\begin{dcases}\n    \\left((s_p^{R}(x)\\dfrac{\\dd}{\\dd x})^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=0,\\quad &\\forall x\\in D_p^R,\\\\\n    v(x)=w,\\quad & \\forall x \\in \\partial^L D_p^R,\\\\\n    v(x)=0, \\quad & \\forall x \\in \\partial D_p^R \\setminus\\partial^L D_p^R,\n  \\end{dcases}\n\\end{align*}\nwhere $w$ is the left boundary value of the unknown $v(x)$. We define $H_p^{R} \\pmb v=\\pmb g$ as the discretization of this\nproblem on $X_p^{R}$ where the right-hand side $\\pmb g$ is given by\n$\\pmb g:= (-1\/h^2)[w,0,\\dots,0]^T$ as a result of the central\ndiscretization. The subproblem $H_p^{R} \\pmb v=\\pmb g$ for each\n$p=2,\\dots,m$ induces the approximation operator $\\td{G}_{p}^{R}:w \\to\n\\pmb z$ by the following procedure:\n\\begin{enumerate}\n\\item\n  Set $\\pmb g=(-1\/h^2)[w,0,\\dots,0]^T$. \n\\item\n  Solve $H_p^{R} \\pmb v =\\pmb g$ on $X_p^{R}$. \n\\item\n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_p$.\n\\end{enumerate}\nThen $\\td{\\pmb u}_{p,1}$ can be defined recursively for $p=2,\\dots,m$ by\n\\begin{align*}\n  \\td{\\pmb u}_{p,1}:=\\td{G}_{p}^{R}\\td{\\pmb u}_{p-1,1}^{R}.\n\\end{align*}\nNote that, the operator $\\td{G}_p^{R}$ is not an approximation of the\nmatrix block $G_{p,1}$, since $\\td{G}_p^R$ maps the right boundary\nvalue of $\\td{\\pmb u}_{p-1,1}$ to $\\td{\\pmb u}_{p,1}$ while $G_{p,1}$\nmaps $\\pmb f_1$ to $\\pmb u_{p,1}$.\n\n\\subsubsection{Wave generated by $\\pmb f_m$}\n\nThe components ${\\pmb u}_{p,m}$ for $p=1,\\dots,m$ can be regarded as a\nsequence of left-going waves generated by $\\pmb f_m$. The method for\napproximating them is similar to what was done for $\\pmb f_1$ (see\nFigure \\ref{fig:approxUpm}). More specifically, for $\\td{\\pmb\n  u}_{m,m}$ we define\n\\begin{align*}\n  D_m^M  &:=   (1-2\\eta-(b-1)h,1),\\\\\n  X_m^{M} &:=  \\{ih:(m-1)b+1\\le i \\le 2\\gamma+mb-2\\},\\\\\n  \\sigma_m^{M}(x) &:=  \n  \\begin{dcases}\n    \\dfrac{C}{\\eta}\\left(\\dfrac{x-(1-\\eta-(b-1)h)}{\\eta}\\right)^2,\\quad &x\\in[1-2\\eta-(b-1)h,1-\\eta-(b-1)h),\\\\\n      0,\\quad &x\\in[1-\\eta-(b-1)h,1-\\eta],\\\\\n      \\dfrac{C}{\\eta}\\left(\\dfrac{x-(1-\\eta)}{\\eta}\\right)^2,\\quad &x\\in(1-\\eta,1],\n  \\end{dcases}\\\\\n  s_m^{M}(x) &:=  \\left(1+\\ii\\dfrac{\\sigma_m^{M}(x)}{\\omega}\\right)^{-1}. \n\\end{align*}\nWe consider the continuous problem\n\\begin{align*}\n  \\begin{dcases}\n    \\left((s_m^{M}(x)\\dfrac{\\dd}{\\dd x})^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=g(x),\\quad & \\forall x\\in D_m^M,\\\\\n    v(x)=0,\\quad & \\forall x\\in \\partial D_m^M,\n  \\end{dcases}\n\\end{align*}\nand define $H_m^{M}\\pmb v=\\pmb g$ as its discretization on $X_m^{M}$.\nThe operator $\\td{G}_{m}^{M}: \\pmb y\\to \\pmb z$ can be defined as:\n\\begin{enumerate}\n\\item\n  Introduce a vector $\\pmb g$ defined on $X_m^{M}$ by setting $\\pmb y$ to\n  $X_m$ and zero everywhere else.\n\\item\n  Solve $H_m^{M} \\pmb v=\\pmb g$ on $X_m^{M}$. \n\\item\n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_m$. \n\\end{enumerate}\nThen\n\\begin{align*}\n  \\td{\\pmb u}_{m,m}:=\\td{G}_{m}^{M}\\pmb f_m. \n\\end{align*}\n\nFor each $\\td{\\pmb u}_{p,m},p=1,\\dots,m-1$, we introduce the auxiliary domain $D_p^L$, the right boundary $\\partial ^R D_p^L$, the extended grid $X_p^L$, and the corresponding PML functions $\\sigma_p^L(x), s_p^L(x)$ as follows\n\\begin{align*}\n  D_p^L  &:=   ((p-1)bh,\\eta+pbh),\\\\\n  \\partial ^R D_p^L  &:=  \\{\\eta + pbh\\},\\\\\n  X_p^{L} &:=  \\{x_i:(p-1)b+1\\le i\\le \\gamma+pb-1\\},\\\\\n  \\sigma_p^{L}(x) &:=  \n  \\begin{dcases}\n    \\dfrac{C}{\\eta}\\left(\\dfrac{x-(\\eta+(p-1)bh)}{\\eta}\\right)^2,\\quad &x\\in[(p-1)bh,\\eta+(p-1)bh),\\\\\n      0,\\quad &x\\in[\\eta+(p-1)bh,\\eta+pbh],\n  \\end{dcases}\\\\\n  s_p^{L}(x) &:=  \\left(1+\\ii\\dfrac{\\sigma_p^{L}(x)}{\\omega}\\right)^{-1}, \n\\end{align*}\nand we consider the continuous problem\n\\begin{align*}\n  \\begin{dcases}\n    \\left((s_p^{L}(x)\\dfrac{\\dd}{\\dd\n      x})^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=0,\\quad & \\forall x\\in D_p^L,\\\\\n      v(x) = w,\\quad & \\forall x \\in \\partial^R D_p^L,\\\\\n      v(x) = 0,\\quad & \\forall x \\in \\partial D_p^L \\setminus \\partial^R D_p^L,\n  \\end{dcases}\n\\end{align*}\nwhere $y$ is the right boundary value of $v(x)$. Let $H_p^{L}\\pmb v=\\pmb g$ be its discretization on $X_p^{L}$ with\n$\\pmb g:=(-1\/h^2)[0,\\dots,0,w]^T$. We introduce the operator\n$\\td{G}_{p}^{L}:w\\mapsto \\pmb z$ by:\n\\begin{enumerate}\n\\item\n  Set $\\pmb g=(-1\/h^2)[0,\\dots,0,w]^T$. \n\\item\n  Solve $H_p^{L} \\pmb v =\\pmb g$ on $X_p^{L}$. \n\\item\n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_p$. \n\\end{enumerate}\nThen $\\td{\\pmb u}_{p,m}$ can be defined recursively for\n$p=m-1,\\dots,1$ by\n\\begin{align*}\n\\td{\\pmb u}_{p,m}:=\\td{G}_{p}^{L}\\td{\\pmb u}_{p+1,m}^{L}. \n\\end{align*}\n\n\\subsubsection{Wave generated by $\\pmb f_q$ for $q=2,\\dots,m-1$}\n\nFor each $q$, the components ${\\pmb u}_{p,q}$ for $p=1,\\dots,m$ can\nbe regarded as a sequence of left- and right-going waves generated by\n$\\pmb f_q$ (see Figure \\ref{fig:approxUpq}). For $\\td{\\pmb u}_{q,q}$,\nwe introduce\n\\begin{align*}\nD_q^M &:=  ((q-1)bh,2\\eta+(qb-1)h),\\\\\nX_q^{M} &:=  \\{x_i:(q-1)b+1\\le i \\le 2\\gamma+qb-2\\},\\\\\n\\sigma_q^{M}(x) &:=  \n\\begin{dcases}\n\\dfrac{C}{\\eta}\\left(\\dfrac{x-(\\eta+(q-1)bh)}{\\eta}\\right)^2,\\quad &x\\in[(q-1)bh,\\eta+(q-1)bh),\\\\\n0,\\quad &x\\in[\\eta+(q-1)bh,\\eta+(qb-1)h],\\\\\n\\dfrac{C}{\\eta}\\left(\\dfrac{x-(\\eta+(qb-1)h)}{\\eta}\\right)^2,\\quad &x\\in(\\eta+(qb-1)h,2\\eta+(qb-1)h],\n\\end{dcases}\\\\\ns_q^{M}(x) &:=  \\left(1+\\ii\\dfrac{\\sigma_q^{M}(x)}{\\omega}\\right)^{-1}, \n\\end{align*}\nand define $H_q^{M} \\pmb v=\\pmb g$ as the discrete problem of the continuous problem\n\\begin{align*}\n\\begin{dcases}\n\\left((s_q^{M}(x)\\dfrac{\\dd}{\\dd x})^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=g(x),\\quad & \\forall x\\in D_q^M,\\\\\nv(x)=0, \\quad & \\forall x \\in \\partial D_q^M.\n\\end{dcases}\n\\end{align*}\nWe introduce the operator $\\td{G}_q^{M}: \\pmb y\\to \\pmb z$ as:\n\\begin{enumerate}\n\\item\n  Introduce a vector $\\pmb g$ defined on $X_q^{M}$ by setting $\\pmb y$ to $X_q$\n  and zero everywhere else.\n\\item\n  Solve $H_q^{M} \\pmb v=\\pmb g$ on $X_q^{M}$. \n\\item\n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_q$. \n\\end{enumerate}\nThen\n\\begin{align*}\n  \\td{\\pmb u}_{q,q}:= \\td{G}_q^{M} \\pmb f_q. \n\\end{align*}\nFollowing the above discussion, the remaining components $\\td{\\pmb\n  u}_{p,q}$ are defined recursively as\n\\begin{align*}\n\\td{\\pmb u}_{p,q} &:=  \\td{G}_{p}^{R} \\td{\\pmb u}_{p-1,q}^{R}, \\quad \\text{for }p=q+1,\\dots,m,\\\\\n\\td{\\pmb u}_{p,q} &:=  \\td{G}_{p}^{L}\\td{\\pmb u}_{p+1,q}^{L},\\quad \\text{for } p=q-1,\\dots,1. \n\\end{align*}\n\n\\subsection{Accumulating the boundary values}\nAfter all the above are done, an approximation of $\\pmb u_p$ is\ngiven by (see Figure \\ref{fig:approxUseparate})\n\\begin{align*}\n\\td{\\pmb u}_p:=\\sum_{q=1}^m\\td{\\pmb u}_{p,q},\\quad p=1,\\dots,m.\n\\end{align*}\n\n\n\\begin{figure}[h!]\n  \\centering\n  \\subfigure[$\\td {\\pmb u}_p$ is a superposition of $\\td {\\pmb u}_{p,q},q=1,\\dots,m$.]{            \n    \\label{fig:approxUseparate}\n    \\begin{tikzpicture}\n      [x=2cm,y=2cm,>=latex]\n      \\draw[step=1](0,-0.1)grid(7,0.1);\n      \\draw(0.5,0)node[below=3]{$X_1$}(1.5,0)node[below=3]{$X_2$}(2.5,0)node[below=3]{$X_3$}(4.5,0)node[below=3]{$X_{m-2}$}(5.5,0)node[below=3]{$X_{m-1}$}(6.5,0)node[below=3]{$X_m$};\n      \\foreach \\x in{0,...,5}\n               {\n                 \\draw[->,gray,very thick](\\x+1,2.5)--(\\x+2,2.5);\n                 \\draw[<-,gray,very thick](\\x,0.5)--(\\x+1,0.5);\n               }\n               \\draw[<->,gray,very thick](6,0.5)--(7,0.5);\n               \\draw[<->,gray,very thick](0,2.5)--(1,2.5);\n               \n               \\foreach \\x in{4,5,6}\n               \\draw[->,gray,very thick](\\x,1.5)--(\\x+1,1.5);\n               \\foreach \\x in{0,1,2}\n               \\draw[<-,gray,very thick](\\x,1.5)--(\\x+1,1.5);\n               \\draw[<->,gray,very thick](3,1.5)--(4,1.5);\n               \n               \\draw(6.5,0.5)node[fill=white]{$\\pmb f_m$};\n               \\draw(0.5,2.5)node[fill=white]{$\\pmb f_1$};\n               \n               \\foreach \\x in{0.5,1.5,2.5}\n               \\draw(3.5,\\x)node[fill=white]{$\\ldots$};\n               \n               \\foreach \\x in{0.5,1.5,2.5,4.5,5.5,6.5}\n               \\foreach \\y in{1.0,2.0}\n               \\draw(\\x,\\y)node{$\\vdots$};\n               \n               \\draw(0.5,0.5)node[below=3]{$\\td {\\pmb u}_{1,m}$}(1.5,0.5)node[below=3]{$\\td {\\pmb u}_{2,m}$}(2.5,0.5)node[below=3]{$\\td {\\pmb u}_{3,m}$}(4.5,0.5)node[below=3]{$\\td {\\pmb u}_{m-2,m}$}(5.5,0.5)node[below=3]{$\\td {\\pmb u}_{m-1,m}$}(6.5,0.5)node[below=3]{$\\td {\\pmb u}_{m,m}$};\n               \n               \\draw(3.5,0)node[fill=white]{$\\ldots$};\n               \\draw(0.5,1.5)node[below=3]{$\\td {\\pmb u}_{1,q}$}(1.5,1.5)node[below=3]{$\\td {\\pmb u}_{2,q}$}(2.5,1.5)node[below=3]{$\\td {\\pmb u}_{3,q}$}(4.5,1.5)node[below=3]{$\\td {\\pmb u}_{m-2,q}$}(5.5,1.5)node[below=3]{$\\td {\\pmb u}_{m-1,q}$}(6.5,1.5)node[below=3]{$\\td {\\pmb u}_{m,q}$};\n               \n               \\draw(3.5,0)node[fill=white]{$\\ldots$};\n               \\draw(0.5,2.5)node[below=3]{$\\td {\\pmb u}_{1,1}$}(1.5,2.5)node[below=3]{$\\td {\\pmb u}_{2,1}$}(2.5,2.5)node[below=3]{$\\td {\\pmb u}_{3,1}$}(4.5,2.5)node[below=3]{$\\td {\\pmb u}_{m-2,1}$}(5.5,2.5)node[below=3]{$\\td {\\pmb u}_{m-1,1}$}(6.5,2.5)node[below=3]{$\\td {\\pmb u}_{m,1}$};\n               \n               \\foreach \\x in{0.5,1.5,2.5,4.5,5.5,6.5}\n               \\draw[->,>=latex,line width=3 pt](\\x,2.7)--(\\x,3.0);\n               \n               \\draw(0.5,3.0)node[above]{$\\td {\\pmb u}_1$}(1.5,3.0)node[above]{$\\td {\\pmb u}_2$}(2.5,3.0)node[above]{$\\td {\\pmb u}_3$}(4.5,3.0)node[above]{$\\td {\\pmb u}_{m-2}$}(5.5,3.0)node[above]{$\\td {\\pmb u}_{m-1}$}(6.5,3.0)node[above]{$\\td {\\pmb u}_m$};\n    \\end{tikzpicture}\n  }\n  \n  \\subfigure[$\\td {\\pmb u}_p$ is a superposition of $\\td {\\pmb u}_{p,1:p-1}$, $\\td {\\pmb u}_{p,p}$ and $\\td {\\pmb u}_{p,p+1:m}$.]{\n    \\label{fig:approxUaccumulate}\n    \\begin{tikzpicture}\n      [x=2cm,y=2cm,>=latex]\n      \\draw[step=1](0,-0.1)grid(7,0.1);\n      \\draw(0.5,0)node[below=3]{$X_1$}(1.5,0)node[below=3]{$X_2$}(2.5,0)node[below=3]{$X_3$}(4.5,0)node[below=3]{$X_{m-2}$}(5.5,0)node[below=3]{$X_{m-1}$}(6.5,0)node[below=3]{$X_m$};\n      \n      \\foreach \\x in{0,...,5}\n               {\n                 \\draw[->,gray,very thick](\\x+1,1.5)--(\\x+2,1.5);\n                 \\draw[<-,gray,very thick](\\x,0.5)--(\\x+1,0.5);\n               }\n               \\draw[<->,gray,very thick](6,0.5)--(7,0.5);\n               \\draw[<->,gray,very thick](0,1.5)--(1,1.5);\n               \n               \\draw(6.5,0.5)node[fill=white]{$\\pmb f_m$};\n               \\draw(0.5,1.5)node[fill=white]{$\\pmb f_1$};\n               \n               \\foreach \\x in{0.5,1.5}\n               \\draw(3.5,\\x)node[fill=white]{$\\ldots$};\n               \n               \\foreach \\x in{1,...,5}\n                        {\n                          \\draw[<->,gray,very thick](\\x,0.5)--(\\x+1,1.5);\n                          \\fill[white](3.35,0.7)rectangle(3.65,1.3);\n                        }\n                        \n                        \\draw(1.5,1.0)node[fill=white]{$\\pmb f_2$}(2.5,1.0)node[fill=white]{$\\pmb f_3$}(3.5,1.0)node[fill=white]{$\\ldots$}(4.5,1.0)node[fill=white]{$\\pmb f_{m-2}$}(5.5,1.0)node[fill=white]{$\\pmb f_{m-1}$};\n                        \n                        \\draw(1.5,1.0)node[below=3]{$\\td {\\pmb u}_{2,2}$}(2.5,1.0)node[below=3]{$\\td {\\pmb u}_{3,3}$}(4.5,1.0)node[below=3]{$\\td {\\pmb u}_{m-2,m-2}$}(5.5,1.0)node[below=3]{$\\td {\\pmb u}_{m-1,m-1}$};\n                        \n                        \\draw(0.5,0.5)node[below=3]{$\\td {\\pmb u}_{1,2:m}$}(1.5,0.5)node[below=3]{$\\td {\\pmb u}_{2,3:m}$}(2.5,0.5)node[below=3]{$\\td {\\pmb u}_{3,4:m}$}(4.5,0.5)node[below=3]{$\\td {\\pmb u}_{m-2,m-1:m}$}(5.5,0.5)node[below=3]{$\\td {\\pmb u}_{m-1,m}$}(6.5,0.5)node[below=3]{$\\td {\\pmb u}_{m,m}$};\n                        \n                        \\draw(3.5,0)node[fill=white]{$\\ldots$};\n                        \\draw(0.5,1.5)node[below=3]{$\\td {\\pmb u}_{1,1}$}(1.5,1.5)node[below=3]{$\\td {\\pmb u}_{2,1}$}(2.5,1.5)node[below=3]{$\\td {\\pmb u}_{3,1:2}$}(4.5,1.5)node[below=3]{$\\td {\\pmb u}_{m-2,1:m-3}$}(5.5,1.5)node[below=3]{$\\td {\\pmb u}_{m-1,1:m-2}$}(6.5,1.5)node[below=3]{$\\td {\\pmb u}_{m,1:m-1}$};\n                        \n                        \\foreach \\x in{0.5,1.5,2.5,4.5,5.5,6.5}\n                        \\draw[->,>=latex,line width=3 pt](\\x,1.7)--(\\x,2.0);\n                        \n                        \\draw(0.5,2.0)node[above]{$\\td {\\pmb u}_1$}(1.5,2.0)node[above]{$\\td {\\pmb u}_2$}(2.5,2.0)node[above]{$\\td {\\pmb u}_3$}(4.5,2.0)node[above]{$\\td {\\pmb u}_{m-2}$}(5.5,2.0)node[above]{$\\td {\\pmb u}_{m-1}$}(6.5,2.0)node[above]{$\\td {\\pmb u}_m$};\n                        \n    \\end{tikzpicture}\n  }\n  \\caption{This figure shows how the boundary values are accumulated after each step. The thin arrows indicate the transmission directions of the waves. The bold, up-pointing arrows symbolizes that summing up the corresponding waves on $X_p$ gives the superposition wave $\\td {\\pmb u}_p$.}\n\\end{figure}\n\n\nIn the algorithm described above, the computation of each component\n$\\td{\\pmb u}_{p,q}$ requires a separate solution of a problem of form\n$H_p^{R} \\pmb v =\\pmb g$ or $H_p^{L} \\pmb v =\\pmb g$. Since there are $O(m^2)$ such\ncomponents, the algorithm is computationally expensive. A key\nobservation is that the computation associated with each $p$ can be\ncombined in one single shot by accumulating the boundary values of the waves.  More precisely, we define\n\\begin{align*}\n\\td{\\pmb u}_{p,q_1:q_2}:=\\sum_{t=q_1}^{q_2} \\td{\\pmb u}_{p,t}, \n\\end{align*}\nwhich is the total contribution of the waves generated by $\\pmb\nf_{q_1},\\dots,\\pmb f_{q_2}$ restricted to the grid $X_{p}$. The\nquantity $\\td{\\pmb u}_{p,1:p-1}$, which is the total right-going wave\ngenerated by $\\pmb f_1,\\dots,\\pmb f_{p-1}$ upon $X_p$, can be computed\nsequentially for $p=2,\\dots,m$ without computing each component and\nthen adding them together as we described above, as long as we\naccumulate the boundary values after each intermediate\nstep. Specifically, we first compute $\\td{\\pmb\n  u}_{q,q}=\\td{G}_q^{M}\\pmb f_{q}$ for $q=1,\\dots,m$.  This step is similar to what we did above. Then, to compute $\\td{\\pmb\n  u}_{p,1:p-1}$ we carry out the following steps\n\\begin{align*}\n  \\td{\\pmb u}_{p,1:p-1}=\\td{G}_{p}^{R}\\td{\\pmb u}_{p-1,1:p-1}^{R},\\quad \\td{\\pmb u}_{p,1:p}^{R}=\\td{\\pmb u}_{p,1:p-1}^{R}+\\td{\\pmb u}_{p,p}^{R}, \\quad \\text{for } p=2,\\dots,m.  \n\\end{align*}\nThis means, before computing the total right-going wave $\\td{\\pmb u}_{p+1,1:p}$ on subdomain $X_{p+1}$, the boundary values of the previous right-going\nwaves, $\\td{\\pmb u}_{p,1:p-1}^{R}$ and $\\td{\\pmb u}_{p,p}^{R}$, are added together, so\nthat the the current right-going wave $\\td{\\pmb u}_{p+1,1:p}$ can be computed in one shot, eliminating the\ntrouble of solving the subproblems for many times and adding the\nresults together (see Figure \\ref{fig:approxUaccumulate}).\n\nFor the left going waves $\\td{\\pmb u}_{p,p+1:m}$, a\nsimilar process gives rise to the recursive formula\n\\begin{align*}\n\\td{\\pmb u}_{p,p+1:m}=\\td{G}_p^{L}\\td{\\pmb u}_{p+1,p+1:m}^{L},\\quad \\td{\\pmb u}_{p,p:m}^{L}=\\td{\\pmb u}_{p,p}^{L}+\\td{\\pmb u}_{p,p+1:m}^{L}, \\quad \\text{for } p=m-1,\\dots,1.\n\\end{align*}\n\nFinally, each $\\td{\\pmb u}_p$ can be computed by summing $\\td{\\pmb\n  u}_{p,1:p-1}$, $\\td{\\pmb u}_{p,p}$ and $\\td{\\pmb u}_{p,p+1:m}$\ntogether (for the leftmost and the rightmost one, $\\td{\\pmb u}_1$ and\n$\\td{\\pmb u}_m$, only two terms need to be summed), i.e.,\n\\begin{align*}\n\\td{\\pmb u}_1 &= \\td{\\pmb u}_{1,1}+\\td{\\pmb u}_{1,2:m},\\\\\n\\td{\\pmb u}_p &= \\td{\\pmb u}_{p,1:p-1}+\\td{\\pmb u}_{p,p}+\\td{\\pmb u}_{p,p+1:m}, \\quad p=2,\\dots,m-1,\\\\\n\\td{\\pmb u}_m &= \\td{\\pmb u}_{m,1:m-1}+\\td{\\pmb u}_{m,m}. \n\\end{align*}\n\nWe see that, by accumulating the boundary values after each intermediate step, we only need to solve $O(m)$ subproblems instead of $O(m^2)$.\n\nIn this algorithm, the approximation $\\td{\\pmb u}_{p}$ on each small\nsubdomain is divided into three parts. From a matrix point of view,\nthis is analogous to splitting the block matrix $G$ into its lower\ntriangular part, diagonal part and upper triangular part, and then\napproximating each part as an operator to get the intermediate waves and then summing the intermediate results\ntogether. This is why we call it the additive sweeping method. \n\nEquation \\eqref{eqn:LDUsplit} shows an analogy of this procedure, where the matrix $G$ is split into $3m - 2$ blocks, each of which corresponds to a subproblem solving process:\n\\begin{align*}\n\\td{\\pmb u}_{q,q} &\\approx  {\\pmb u}_{q,q} = G_{q,q} {\\pmb f}_q,\\quad q = 1,\\dots, m,\\\\\n\\td{\\pmb u}_{p,1:p-1} &\\approx  {\\pmb u}_{p,1:p-1} = \\sum_{q=1}^{p-1} G_{p,q} {\\pmb f}_q,\\quad p = 2,\\dots, m,\\\\\n\\td{\\pmb u}_{p,p+1:m} &\\approx  {\\pmb u}_{p,p+1:m} = \\sum_{q=p+1}^{m} G_{p,q} {\\pmb f}_q,\\quad p = 1,\\dots, m-1.\\\\\n\\end{align*}\n\n\\begin{equation}\n  \\label{eqn:LDUsplit}\n  \\begin{bmatrix}\n    \\pmb u_1\\\\\\pmb u_2\\\\ \\ldots\\\\\\pmb u_m\n  \\end{bmatrix}\n  =\n  \\begin{bmatrix}\n    \\pmb u_{1,1}+\\pmb u_{1,2:m}\\\\\\pmb u_{2,1}+\\pmb u_{2,2}+\\pmb u_{2,3:m}\\\\ \\ldots\\\\\\pmb u_{m,1:m-1}+\\pmb u_{m,m}\n  \\end{bmatrix}\n  =\n  \\left[\n    \\begin{array}{ccccc}\n      \\multicolumn{1}{c|}{G_{1,1}} & G_{1,2} & \\dots & & G_{1,m} \\\\ \n      \\hline\n      G_{2,1} & \\multicolumn{1}{|c|}{G_{2,2}} & G_{2,3} & \\ldots & G_{2,m} \\\\ \n      \\hline\n      & & \\ldots & & \\\\ \n      \\hline\n      G_{m,1} & \\ldots & \u2022 & G_{m,m-1} & \\multicolumn{1}{|c}{G_{m,m}}\n    \\end{array} \n    \\right]\n  \\begin{bmatrix}\n    \\pmb f_1\\\\\\pmb f_2\\\\ \\ldots\\\\\\pmb f_m\n  \\end{bmatrix}\n\\end{equation}\n\nWhen combined with standard iterative solvers, the approximation\nalgorithm serves as a preconditioner for Equation \\eqref{eqn:1D} and it can be easily generalized to higher dimensions. In the following sections, we will discuss the details of the algorithm in 2D and 3D. To be structurally consistent,  we will keep the notations for 2D and 3D the same with the 1D case without causing ambiguity. Some of the key notations and concepts are listed below as a reminder to the reader:\n\\begin{itemize}\n\\item\n$\\{X_p\\}_{p=1}^m$\\quad The sliced partition of the discrete grid.\n\\item\n$\\{D_q^M\\}_{q=1}^m$\\quad The auxiliary domains with two-sided PML padding.\n\\item\n$\\{D_p^R\\}_{p=2}^m$\\quad The auxiliary domains with right-side PML padding.\n\\item\n$\\{D_p^L\\}_{p=1}^{m-1}$\\quad The auxiliary domains with left-side PML padding.\n\\item\n$\\{X_q^M\\}_{q=1}^m$\\quad $X_q$ with two-sided PML padding, the discretization of $D_q^M$.\n\\item\n$\\{X_p^R\\}_{p=2}^m$\\quad $X_p$ with right-side PML padding, the discretization of $D_p^R$.\n\\item\n$\\{X_p^L\\}_{p=1}^{m-1}$\\quad $X_p$ with left-side PML padding, the discretization of $D_p^L$.\n\\item\n$\\{\\tilde{G}_q^M\\}_{q=1}^m$ \\quad The auxiliary Green's operators each of which maps the force on $X_q$ to the approximation of the wave field restricted to $X_q$.\n\\item\n$\\{\\tilde{G}_p^R\\}_{p=2}^m$ \\quad The auxiliary Green's operators each of which maps the left boundary value to the approximated wave field restricted to $X_p$, which simulates the right-transmission of the waves.\n\\item\n$\\{\\tilde{G}_p^L\\}_{p=1}^{m-1}$ \\quad The auxiliary Green's operators each of which maps the right boundary value to the approximated wave field restricted to $X_p$, which simulates the left-transmission of the waves.\n\\end{itemize}\n\n\\section{Preconditioner in 2D}\n\\label{sec:2D}\n\n\\subsection{Algorithm}\n\nThe domain of interest is $D=(0,1)^2$. We put PML on the two\nopposite sides of the boundary, $x_2=0$ and $x_2=1$, to illustrate the\nidea. The resulting equation is\n\\begin{align*}\n  \\begin{dcases}\n    \\left(\\partial_1^2+(s(x_2)\\partial_2)^2+\\dfrac{\\omega^2}{c^2(x)}\\right)u(x)=f(x),&\\quad \\forall x=(x_1,x_2)\\in D,\\\\\n    u(x)=0,&\\quad \\forall x\\in \\partial D, \n  \\end{dcases}\n\\end{align*}\nWe discretize $D$ with step size $h=1\/(n+1)$ in each direction, which\nresults the Cartesian grid\n\\begin{align*}\n  X:=\\{(i_1h,i_2h):1\\le i_1,i_2\\le n\\},  \n\\end{align*} \nand the discrete equation\n\\begin{equation}\n  \\label{eqn:2D}\n  \\begin{gathered}\n    \\dfrac{s_{i_2}}{h}\\left(\\dfrac{s_{i_2+1\/2}}{h}(u_{i_1,i_2+1}-u_{i_1,i_2})-\\dfrac{s_{i_2-1\/2}}{h}(u_{i_1,i_2}-u_{i_1,i_2-1})\\right)\\\\\n    +\\dfrac{u_{i_1+1,i_2}-2u_{i_1,i_2}+u_{i_1-1,i_2}}{h^2}+\\dfrac{\\omega^2}{c_{i_1,i_2}^2}u_{i_1,i_2}=f_{i_1,i_2}, \\quad \\forall 1\\le i_1,i_2\\le n,\n  \\end{gathered}\n\\end{equation}\nwhere the subscript $(i_1,i_2)$ means that the corresponding function\nis evaluated at $(i_1h,i_2h)$, and since $s(x_2)$ is a function of\n$x_2$ only, we omit the $i_1$ subscript. $\\pmb u$ and $\\pmb f$ are\ndefined to be the column-major ordering of the discrete array $u$ and\n$f$ on the grid $X$\n\\begin{align*}\n  \\pmb u:=[u_{1,1},\\dots,u_{n,1},\\dots,u_{n,n}]^T,\\quad \\pmb f:=[f_{1,1},\\dots,f_{n,1},\\dots,f_{n,n}]^T.\n\\end{align*}\nNow \\eqref{eqn:2D} can be written as $A\\pmb u=\\pmb f$.\n\nWe divide the grid into $m$ parts along the $x_2$ direction\n\\begin{align*}\n  X_1 &:= \\{(i_1h,i_2h):1\\le i_1\\le n,1 \\le i_2 \\le \\gamma + b-1 \\}, \\\\\n  X_p &:= \\{(i_1h,i_2h):1\\le i_1\\le n,\\gamma + (p-1) b \\le i_2 \\le \\gamma + pb-1 \\},\\quad p=2,\\dots, m-1, \\\\\n  X_m &:= \\{(i_1h,i_2h):1\\le i_1\\le n,\\gamma + (m-1) b \\le i_2 \\le 2 \\gamma + mb-2 \\}, \n\\end{align*}\nand we define $\\pmb u_p$ and $\\pmb f_p$ as the column-major ordering\nrestriction of $u$ and $f$ on $X_p$\n\\begin{align*}\n  \\pmb u_1 &:= [u_{1,1},\\dots,u_{n,1},\\dots,u_{n,\\gamma+b-1}]^T,\\\\\n  \\pmb u_p &:= [u_{1,\\gamma+(p-1)b},\\dots,u_{n,\\gamma+(p-1)b},\\dots,u_{n,\\gamma + pb-1}]^T, \\quad p=2,\\dots,m-1,\\\\\n  \\pmb u_m &:= [u_{1,\\gamma + (m-1) b},\\dots,u_{n,\\gamma + (m-1) b},\\dots,u_{n,2 \\gamma + mb-2}]^T,\\\\\n  \\pmb f_1 &:= [f_{1,1},\\dots,f_{n,1},\\dots,f_{n,\\gamma+b-1}]^T,\\\\\n  \\pmb f_p &:= [f_{1,\\gamma+(p-1)b},\\dots,f_{n,\\gamma+(p-1)b},\\dots,f_{n,\\gamma + pb-1}]^T, \\quad p=2,\\dots,m-1,\\\\\n  \\pmb f_m &:= [f_{1,\\gamma + (m-1) b},\\dots,f_{n,\\gamma + (m-1) b},\\dots,f_{n,2 \\gamma + mb-2}]^T,\n\\end{align*}\nthen $\\pmb u=G\\pmb f$ for $G=A^{-1}$ can be written as\n\\begin{align*}\n\\begin{bmatrix}\n\\pmb u_1\\\\\\pmb u_2\\\\ \\vdots\\\\\\pmb u_m\n\\end{bmatrix}\n=\n\\begin{bmatrix}\nG_{1,1}&G_{1,2}&\\ldots&G_{1,m}\\\\\nG_{2,1}&G_{2,2}&\\ldots&G_{2,m}\\\\\n\\vdots&\\vdots&&\\vdots\\\\\nG_{m,1}&G_{m,2}&\\ldots&G_{m,m}\n\\end{bmatrix}\n\\begin{bmatrix}\n\\pmb f_1\\\\\\pmb f_2\\\\ \\vdots\\\\\\pmb f_m\n\\end{bmatrix}.\n\\end{align*}\n\n\\paragraph{Auxiliary domains.}\nFollowing to the 1D case, the extended subdomains and the\ncorresponding left and right boundaries are defined by\n\\begin{align*}\n  D_q^{M} &= (0,1)\\times((q-1)bh,2\\eta+(qb-1)h),\\quad q=1,\\dots,m,\\\\\n  D_p^{R} &= (0,1)\\times(\\eta+((p-1)b-1)h,2\\eta+(pb-1)h),\\quad p=2,\\dots,m,\\\\\n  D_p^{L} &= (0,1)\\times((p-1)bh,\\eta+pbh),\\quad p=1,\\dots,m-1,\\\\\n  \\partial^{L} D_p^{R} &= (0,1)\\times\\{\\eta+((p-1)b-1)h\\},\\quad p=2,\\dots,m,\\\\\n  \\partial^{R} D_p^{L} &= (0,1)\\times\\{\\eta+pbh\\},\\quad p=1,\\dots,m-1.\n\\end{align*}\nThe extended grid for these domains are\n\\begin{align*}\n  X_q^{M} &:= \\{(i_1h,i_2h):1\\le i_1\\le n,(q-1)b+1\\le i_2 \\le 2\\gamma+qb-1\\}, \\quad q=1,\\dots,m, \\\\\n  X_p^{R} &:= \\{(i_1h,i_2h):1\\le i_1\\le n,\\gamma+(p-1)b\\le i_2\\le 2\\gamma+pb-2\\}, \\quad p=2,\\dots,m, \\\\\n  X_p^{L} &:= \\{(i_1h,i_2h):1\\le i_1\\le n,(p-1)b+1\\le i_2\\le \\gamma+pb-1\\}, \\quad p=1,\\dots,m-1. \n\\end{align*}\n\n\\paragraph{Auxiliary problems.}\nFor $q=1,\\dots,m$, we define $H_q^{M} \\pmb v=\\pmb g$ to be the\ndiscretization on $X_q^{M}$ of the problem\n\\begin{align*}\n  &\\begin{dcases}\n     \\left(\\partial_1^2+(s_q^{M}(x_2)\\partial_2)^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=g(x),\\quad &\\forall x\\in D_q^{M},\\\\\n     v(x)=0,\\quad &\\forall x\\in \\partial D_q^{M}. \n   \\end{dcases}\n\\end{align*}\nFor $p=2,\\dots,m$, $H_p^{R} \\pmb v=\\pmb g$ is the discretization on\n$X_p^{R}$ of the problem\n\\begin{align*}\n  &\\begin{dcases}\n     \\left(\\partial_1^2+(s_p^{R}(x_2)\\partial_2)^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=0,\\quad &\\forall x\\in D_p^{R},\\\\\n     v(x)=w(x_1),\\quad &\\forall x\\in \\partial^{L} D_p^{R}, \\\\\n     v(x)=0,\\quad &\\forall x\\in \\partial D_p^{R} \\setminus \\partial^{L} D_p^{R},\n   \\end{dcases}\n\\end{align*}\nwhere $\\pmb g:=(-1\/h^2)[\\pmb w^T,0,\\dots,0]^T$ and $\\pmb w:= [w_1,\\dots,w_n]^T$ is the discrete value of $w(x_1)$. Finally, for\n$p=1,\\dots,m-1$, $H_p^{L} \\pmb v=\\pmb g$ is the discretization on\n$X_p^{L}$ of the problem\n\\begin{align*}\n  &\\begin{dcases}\n     \\left(\\partial_1^2+(s_p^{L}(x_2)\\partial_2)^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=0,\\quad &\\forall x\\in D_p^{L},\\\\\n     v(x)=w(x_1),\\quad &\\forall x\\in \\partial^{R} D_p^{L}, \\\\\n     v(x)=0,\\quad &\\forall x\\in \\partial D_p^{L} \\setminus \\partial^{R} D_p^{L},\n   \\end{dcases}\n\\end{align*}\nwhere $\\pmb g:=(-1\/h^2)[0,\\dots,0,\\pmb w^T]^T$ and $\\pmb w:= [w_1,\\dots,w_n]^T$.\n\n\\paragraph{Auxiliary Green's operators.}\nFor $q=1,\\dots,m$, we define $\\td{G}_q^{M}:\\pmb y\\mapsto \\pmb z$ to be\nthe operator defined by the following operations:\n\\begin{enumerate}\n\\item\n  Introduce a vector $\\pmb g$ defined on $X_q^{M}$ by setting $\\pmb y$ to $X_q$ and zero everywhere else. \n\\item\n  Solve $H_q^{M} \\pmb v=\\pmb g$ on $X_q^{M}$. \n\\item\n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_q$. \n\\end{enumerate}\nFor $p=2,\\dots,m$, the operators $\\td{G}_p^{R}:\\pmb w\\mapsto \\pmb z$\nis given by:\n\\begin{enumerate}\n\\item\n  Set $\\pmb g=(-1\/h^2)[\\pmb w^T,0,\\dots,0]^T$. \n\\item\n  Solve $H_p^{R} \\pmb v =\\pmb g$ on $X_p^{R}$. \n\\item\n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_p$. \n\\end{enumerate}\nFinally, for $p=1,\\dots,m-1$, $\\td{G}_p^{L}:\\pmb w\\mapsto \\pmb z$ is\ndefined as:\n\\begin{enumerate}\n\\item\n  Set $\\pmb g=(-1\/h^2)[0,\\dots,0,\\pmb w^T]^T$. \n\\item\n  Solve $H_p^{L} \\pmb v =\\pmb g$ on $X_p^{L}$. \n\\item\n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_p$. \n\\end{enumerate}\n\n\\paragraph{Putting together.}\nSimilar to the previous section, we introduce the left boundary value\n$\\pmb g^{L}$ and the right boundary value $\\pmb g^{R}$ for a\ncolumn-major ordering array\n$\\pmb g=[g_{1,1},\\dots,g_{s_1,1},\\dots,g_{s_1,s_2}]^T$\ninduced from some grid with size $s_1\\times s_2$ by\n\\begin{align*}\n  \\pmb g^{L}:=[g_{1,1},\\dots,g_{s_1,1}]^T, \\quad \\pmb g^{R}:=[g_{1,s_2},\\dots,g_{s_1,s_2}]^T. \\\\\n\\end{align*}\nThen the approximations for $\\pmb u_p,p=1,\\dots,m$, can be defined step by step as\n\\begin{align*}\n  \\td{\\pmb u}_{q,q} &:= \\td{G}_q^{M} \\pmb f_q,\\quad q=1,\\dots,m,\\\\\n  \\td{\\pmb u}_{p,1:p-1} &:= \\td{G}_{p}^{R}\\td{\\pmb u}_{p-1,1:p-1}^{R},\\quad \\td{\\pmb u}_{p,1:p}^{R}:=\\td{\\pmb u}_{p,1:p-1}^{R}+\\td{\\pmb u}_{p,p}^{R},\\quad \\text{for } p=2,\\dots,m,\\\\\n  \\td{\\pmb u}_{p,p+1:m} &:= \\td{G}_p^{L}\\td{\\pmb u}_{p+1,p+1:m}^{L},\\quad \\td{\\pmb u}_{p,p:m}^{L}:=\\td{\\pmb u}_{p,p}^{L}+\\td{\\pmb u}_{p,p+1:m}^{L}, \\quad \\text{for } p=m-1,\\dots,1,\\\\\n  \\td{\\pmb u}_1 &:= \\td{\\pmb u}_{1,1}+\\td{\\pmb u}_{1,2:m},\\\\\n  \\td{\\pmb u}_p &:= \\td{\\pmb u}_{p,1:p-1}+\\td{\\pmb u}_{p,p}+\\td{\\pmb u}_{p,p+1:m},\\quad p=2,\\dots,m-1,\\\\\n  \\td{\\pmb u}_m &:= \\td{\\pmb u}_{m,1:m-1}+\\td{\\pmb u}_{m,m}. \n\\end{align*}\n\nTo solve the subproblems $H_q^{M} \\pmb v=\\pmb g$, $H_p^{R}\\pmb v=\\pmb\ng$ and $H_p^{L}\\pmb v=\\pmb g$, we notice that they are indeed quasi-1D\nproblems since $\\gamma$ and $b$ are some small constants. Therefore,\nfor each one of them, we can reorder the system by grouping\nthe elements along dimension 2 first and then dimension 1, which\nresults a banded linear system that can be solved by the LU\nfactorization efficiently. These factorization processes induce the\nfactorizations for the operators $\\td{G}_q^{M}$, $\\td{G}_p^{R}$ and\n$\\td{G}_p^{L}$ symbolically, which leads to our setup algorithm of the\npreconditioner in 2D as described in Algorithm \\ref{alg:2dsetup} and\nthe application algorithm as described in Algorithm \\ref{alg:2dapp}.\n\n\\begin{algorithm}[h!]\n  \\caption{Construction of the 2D additive sweeping preconditioner of\n    the Equation \\eqref{eqn:2D}. Complexity\n    $=O(n^2(b+\\gamma)^3\/b)=O(N(b+\\gamma)^3\/b)$.}\n  \\label{alg:2dsetup}\n  \\begin{algorithmic}\n    \\FOR {$q=1,\\dots,m$}\n    \\STATE Construct the LU factorization of $H_q^{M}$, which defines $\\td{G}_q^{M}$.\n    \\ENDFOR\n    \\FOR {$p=2,\\dots,m$}\n    \\STATE Construct the LU factorization of $H_p^{R}$, which defines $\\td{G}_p^{R}$.\n    \\ENDFOR\n    \\FOR {$p=1,\\dots,m-1$}\n    \\STATE Construct the LU factorization of $H_p^{L}$, which defines $\\td{G}_p^{L}$.\n    \\ENDFOR\n  \\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[h!]\n  \\caption{Computation of $\\td{\\pmb u}\\approx G \\pmb f$ using the\n    preconditioner from Algorithm \\ref{alg:2dsetup}. Complexity\n    $=O(n^2(b+\\gamma)^2\/b)=O(N(b+\\gamma)^2\/b)$.}\n  \\label{alg:2dapp}\n  \\begin{algorithmic}\n    \\FOR {$q=1,\\dots,m$}\n    \\STATE\n    $\\td{\\pmb u}_{q,q}=\\td{G}_q^{M} \\pmb f_q$\n    \\ENDFOR\n    \\FOR {$p=2,\\dots,m$}\n    \\STATE\n    $\\td{\\pmb u}_{p,1:p-1}=\\td{G}_{p}^{R}\\td{\\pmb u}_{p-1,1:p-1}^{R}$\\\\\n    $\\td{\\pmb u}_{p,1:p}^{R}=\\td{\\pmb u}_{p,1:p-1}^{R}+\\td{\\pmb u}_{p,p}^{R}$\n    \\ENDFOR\n    \\FOR {$p=m-1,\\dots,1$}\n    \\STATE\n    $\\td{\\pmb u}_{p,p+1:m}=\\td{G}_p^{L}\\td{\\pmb u}_{p+1,p+1:m}^{L}$\\\\\n    $\\td{\\pmb u}_{p,p:m}^{L}=\\td{\\pmb u}_{p,p}^{L}+\\td{\\pmb u}_{p,p+1:m}^{L}$\n    \\ENDFOR\n    \\STATE\n    $\\td{\\pmb u}_1=\\td{\\pmb u}_{1,1}+\\td{\\pmb u}_{1,2:m}$\\\\\n    \\FOR {$p=2,\\dots,m-1$}\n    \\STATE\n    $\\td{\\pmb u}_p=\\td{\\pmb u}_{p,1:p-1}+\\td{\\pmb u}_{p,p}+\\td{\\pmb u}_{p,p+1:m}$\n    \\ENDFOR\n    \\STATE\n    $\\td{\\pmb u}_m=\\td{\\pmb u}_{m,1:m-1}+\\td{\\pmb u}_{m,m}$ \n  \\end{algorithmic}\n\\end{algorithm}\n\nTo analyze the complexity, we note that, in the setup process, there\nare $O(n\/b)$ subproblems, each of which is a quasi-1D problem with\n$O(\\gamma+b)$ layers along the second dimension. Therefore, the setup\ncost of each subproblem by the LU factorization is $O(n(\\gamma+b)^3)$\nand the application cost is $O(n(\\gamma+b)^2)$. So the total setup\ncost is $O(n^2(\\gamma+b)^3\/b)$. Besides, one needs to solve each\nsubproblem once during the application process so the total\napplication cost is $O(n^2(\\gamma+b)^2\/b)$.\n\nThere are some differences when implementing the method practically:\n\\begin{enumerate}\n\\item\n  In the above setting, PMLs are put only on two opposite sides of the\n  unit square for illustration purpose. In reality, PMLs can be put on\n  other sides of the domain if needed. As long as there are two\n  opposite sides with PML boundary condition, the method can be\n  implemented.\n\\item\n  The thickness of the auxiliary PMLs introduced in the interior part\n  of the domain needs not to be the same with the thickness of the PML\n  at the boundary. In fact, the thickness of the auxiliary PML is\n  typically thinner in order to improve efficiency.\n\\item\n  The widths of the subdomains are completely arbitrary and they need\n  not to be the same. Practically, the widths can be chosen to be\n  larger for subdomains where the velocity field varies heavily.\n\\item\n  The symmetric version of the equation can be adopted to save memory\n  and computational cost.\n\\end{enumerate}\n\n\\subsection{Numerical results}\n\\label{sec:2Dnumerical}\n\nHere, we present some numerical results in 2D to illustrate the\nefficiency of the algorithm. The proposed method is implemented in\nMATLAB and the tests are performed on a 2.0 GHz computer with 256 GB\nmemory. GMRES is used as the iterative solver with relative residual\nequal to $10^{-3}$ and restart value equal to $40$. PMLs are put on\nall sides of the unit square. The velocity fields tested are given in\nFigure \\ref{fig:2D}:\n\\begin{enumerate}[(a)]\n\\item\n  A converging lens with a Gaussian profile at the center of the domain.\n\\item\n  A vertical waveguide with a Gaussian cross-section.\n\\item\n  A random velocity field.\n\\end{enumerate}\n\n\\begin{figure}[h!]\n  \\centering\n  \\subfigure[]\n  {\\includegraphics[width=0.32\\textwidth]{2Da.pdf}}\n  \\subfigure[]\n  {\\includegraphics[width=0.32\\textwidth]{2Db.pdf}}\n  \\subfigure[]\n  {\\includegraphics[width=0.32\\textwidth]{2Dc.pdf}}\n  \\caption{The three velocity fields tested in 2D.}\n  \\label{fig:2D}\n\\end{figure}\n\nFor each velocity field, two external forces are tested:\n\\begin{enumerate}[(a)]\n\\item\n  A Gaussian point source centered at $(1\/2,1\/8)$.\n\\item\n  A Gaussian wave packet with wavelength comparable to the typical\n  wavelength of the domain. The packet centers at $(1\/8,1\/8)$ and\n  points to the direction $(1\/\\sqrt{2},1\/\\sqrt{2})$.\n\\end{enumerate}\n\n\\begin{table}[h!]\n  \\centering\n  \\begin{overpic}\n    [width=0.45\\textwidth]{2Daa.pdf}\n    \\put(40,6){force (a)}\n  \\end{overpic}\n  \\begin{overpic}\n    [width=0.45\\textwidth]{2Dab.pdf}\n    \\put(40,6){force (b)}\n  \\end{overpic}\n  \\begin{tabular}{ccc|cc|cc}\n    \\hline \n    \\multicolumn{3}{c|}{velocity field (a)} & \\multicolumn{2}{c|}{force (a)} & \\multicolumn{2}{c}{force (b)} \\\\ \n    \\hline \n    $\\omega\/(2\\pi)$ & $N$ & $T_{\\text{setup}}$ & $N_\\text{iter}$ & $T_\\text{solve}$ & $N_\\text{iter}$ & $T_\\text{solve}$ \\\\ \n    \\hline \n    16 & $127^2$ & 8.1669e$-$01 &  4 & 5.3199e$-$01 &  4 & 2.5647e$-$01 \\\\\n    32 & $255^2$ & 3.4570e$+$00 &4& 7.3428e$-$01 &4&  7.2807e$-$01\\\\ \n    64 & $511^2$ & 1.5150e$+$01  &5&  3.6698e$+$00  &4& 3.7239e$+$00\\\\ \n    128 & $1023^2$ & 6.2713e$+$01 &5&  1.6812e$+$01 &4&  1.6430e$+$01\\\\\n    256 & $2047^2$ & 2.6504e$+$02 &6&   7.8148e$+$01 &4& 5.6936e$+$01\\\\\n    \\hline \n  \\end{tabular} \n  \\caption{Results for velocity field (a) in 2D. Solutions with $\\omega\/(2\\pi)=32$ are presented.}\n  \\label{tab:2domegaa}\n\\end{table}\n\\begin{table}[h!]\n  \\centering\n  \\begin{overpic}\n    [width=0.45\\textwidth]{2Dba.pdf}\n    \\put(40,6){force (a)}\n  \\end{overpic}\n  \\begin{overpic}\n    [width=0.45\\textwidth]{2Dbb.pdf}\n    \\put(40,6){force (b)}\n  \\end{overpic}\n  \\begin{tabular}{ccc|cc|cc}\n    \\hline \n    \\multicolumn{3}{c|}{velocity field (b)} & \\multicolumn{2}{c|}{force (a)} & \\multicolumn{2}{c}{force (b)} \\\\ \n    \\hline \n    $\\omega\/(2\\pi)$ & $N$ & $T_{\\text{setup}}$ & $N_\\text{iter}$ & $T_\\text{solve}$ & $N_\\text{iter}$ & $T_\\text{solve}$ \\\\ \n    \\hline \n    16 & $127^2$ & 7.0834e$-$01 &6& 2.9189e$-$01 &4& 1.9408e$-$01\\\\\n    32 & $255^2$ & 3.2047e$+$00 &8& 1.6147e$+$00  &4&  7.9303e$-$01\\\\ \n    64 & $511^2$ & 1.4079e$+$01 &8& 6.3057e$+$00   &4&  3.9008e$+$00\\\\ \n    128 & $1023^2$ & 6.0951e$+$01 &8&  2.9097e$+$01  &4&  1.5287e$+$01\\\\\n    256 & $2047^2$ & 2.6025e$+$02  &8&  1.1105e$+$02  &5& 7.2544e$+$01\\\\\n    \\hline \n  \\end{tabular} \n  \\caption{Results for velocity field (b) in 2D. Solutions with $\\omega\/(2\\pi)=32$ are presented.}\n  \\label{tab:2domegab}\n\\end{table}\n\\begin{table}[h!]\n\\centering\n\\begin{overpic}\n  [width=0.45\\textwidth]{2Dca.pdf}\n  \\put(40,6){force (a)}\n\\end{overpic}\n\\begin{overpic}\n  [width=0.45\\textwidth]{2Dcb.pdf}\n  \\put(40,6){force (b)}\n\\end{overpic}\n\\begin{tabular}{ccc|cc|cc}\n  \\hline \n  \\multicolumn{3}{c|}{velocity field (c)} & \\multicolumn{2}{c|}{force (a)} & \\multicolumn{2}{c}{force (b)} \\\\ \n  \\hline \n  $\\omega\/(2\\pi)$ & $N$ & $T_{\\text{setup}}$ & $N_\\text{iter}$ & $T_\\text{solve}$ & $N_\\text{iter}$ & $T_\\text{solve}$ \\\\ \n  \\hline \n  16 & $127^2$ & 7.0495e$-$01  &5&  2.4058e$-$01  &6& 2.8347e$-$01\\\\\n  32 & $255^2$ & 3.1760e$+$00 &5&  1.0506e$+$00  &5&  9.9551e$-$01\\\\ \n  64 & $511^2$ & 1.4041e$+$01   &6&  4.7083e$+$00  &7&  6.7852e$+$00\\\\\n  128 & $1023^2$ & 6.1217e$+$01  &6&  1.8652e$+$01  &6&   1.9792e$+$01\\\\\n  256 & $2047^2$ & 2.5762e$+$02  &8&  1.1214e$+$02  &6&  8.6936e$+$01\\\\\n  \\hline \n\\end{tabular} \n\\caption{Results for velocity field (c) in 2D. Solutions with $\\omega\/(2\\pi)=32$ are presented.}\n\\label{tab:2domegac}\n\\end{table}\n\nIn these tests, each typical wavelength is discretized with 8\npoints. The width of the PML at the boundary and the one of the PMLs introduced\nin the interior parts of the domain are both $9h$, i.e., $\\gamma=9$. The number of layers in each interior subdomain is $b=8$,\nthe number of layers in the leftmost subdomain is $b+\\gamma-1=16$ and\nthe one in the rightmost is $b+\\gamma-2=15$. \n\nWe vary the typical wave number $\\omega\/(2\\pi)$ and test the behavior\nof the algorithm. The test results are presented in Tables\n\\ref{tab:2domegaa}, \\ref{tab:2domegab} and\n\\ref{tab:2domegac}. $T_\\text{setup}$ is the setup time of the\nalgorithm in seconds. $T_{\\text{solve}}$ is the total solve time in\nseconds and $N_{\\text{iter}}$ is the iteration number. From these\ntests we see that the setup time scales like $O(N)$ as well as the\nsolve time per iteration, which is consistent with the algorithm\ncomplexity analysis. The iteration number remains constant or grows at\nmost logarithmically, which shows the efficiency of the\npreconditioner.\n\n\\section{Preconditioner in 3D}\n\\label{sec:3D}\n\n\\subsection{Algorithm}\n\nIn this section we briefly state the preconditioner in 3D case. The\ndomain of interest is $D=(0,1)^3$. PMLs are put on two opposite faces\nof the unit cube, $x_3=0$ and $x_3=1$, which results the equation\n\\begin{align*}\n  \\begin{dcases}\n    \\left(\\partial_1^2+\\partial_2^2+(s(x_3)\\partial_3)^2+\\dfrac{\\omega^2}{c^2(x)}\\right)u(x)=f(x),&\\quad \\forall x=(x_1,x_2,x_3)\\in D,\\\\\n    u(x)=0,&\\quad \\forall x\\in \\partial D, \n  \\end{dcases}\n\\end{align*}\nDiscretizing $D$ with step size $h=1\/(n+1)$ gives the grid\n\\begin{align*}\n  X:=\\{(i_1h,i_2h,i_3h):1\\le i_1,i_2,i_3\\le n\\},  \n\\end{align*} \nand the discrete equation\n\\begin{equation}\n  \\label{eqn:3D}\n  \\begin{gathered}\n    \\dfrac{s_{i_3}}{h}\\left(\\dfrac{s_{i_3+1\/2}}{h}(u_{i_1,i_2,i_3+1}-u_{i_1,i_2,i_3})-\\dfrac{s_{i_3-1\/2}}{h}(u_{i_1,i_2,i_3}-u_{i_1,i_2,i_3-1})\\right)\\\\\n    +\\dfrac{u_{i_1+1,i_2,i_3}-2u_{i_1,i_2,i_3}+u_{i_1-1,i_2,i_3}}{h^2}+\\dfrac{u_{i_1,i_2+1,i_3}-2u_{i_1,i_2,i_3}+u_{i_1,i_2-1,i_3}}{h^2}\\\\\n    +\\dfrac{\\omega^2}{c_{i_1,i_2,i_3}^2}u_{i_1,i_2,i_3}=f_{i_1,i_2,i_3}, \\quad \\forall 1\\le i_1,i_2\\le n. \n  \\end{gathered}\n\\end{equation}\n$\\pmb u$ and $\\pmb f$ are defined as the column-major ordering of $u$ and $f$ on the grid $X$\n\\begin{align*}\n  \\pmb u:=[u_{1,1,1},\\dots,u_{n,1,1},\\dots,u_{n,n,1},\\dots,u_{n,n,n}]^T,\\quad \\pmb f:=[f_{1,1,1},\\dots,f_{n,1,1},\\dots,f_{n,n,1},\\dots,f_{n,n,n}]. \n\\end{align*}\n$X$ is divided into $m$ parts along the $x_3$ direction\n\\begin{align*}\n  X_1 &:= \\{(i_1h,i_2h,i_3h):1\\le i_1\\le n,1\\le i_2\\le n,1 \\le i_3 \\le \\gamma + b-1 \\}, \\\\\n  X_p &:= \\{(i_1h,i_2h,i_3h):1\\le i_1\\le n,1\\le i_2\\le n,\\gamma + (p-1) b \\le i_3 \\le \\gamma + pb-1 \\}, \\quad p=2,\\dots, m-1, \\\\\n  X_m &:= \\{(i_1h,i_2h,i_3h):1\\le i_1\\le n,1\\le i_2\\le n,\\gamma + (m-1) b \\le i_3 \\le 2 \\gamma + mb-2 \\}. \n\\end{align*}\n$\\pmb u_p$ and $\\pmb f_p$ are the column-major ordering restrictions of  $u$ and $f$ on $X_p$ \n\\begin{align*}\n\\pmb u_1 &:= [u_{1,1,1},\\dots,u_{n,1,1},\\dots,u_{n,n,1},\\dots,u_{n,n,\\gamma+b-1}]^T,\\\\\n\\pmb u_p &:= [u_{1,1,\\gamma+(p-1)b},\\dots,u_{n,1,\\gamma+(p-1)b},\\dots,u_{n,n,\\gamma+(p-1)b},\\dots,u_{n,n,\\gamma + pb-1}]^T, \\quad p=2,\\dots,m-1,\\\\\n\\pmb u_m &:= [u_{1,1,\\gamma + (m-1) b},\\dots,u_{n,1,\\gamma + (m-1) b},\\dots,u_{n,n,\\gamma + (m-1) b},\\dots,u_{n,n,2 \\gamma + mb-2}]^T,\\\\\n\\pmb f_1 &:= [f_{1,1,1},\\dots,f_{n,1,1},\\dots,f_{n,n,1},\\dots,f_{n,n,\\gamma+b-1}]^T,\\\\\n\\pmb f_p &:= [f_{1,1,\\gamma+(p-1)b},\\dots,f_{n,1,\\gamma+(p-1)b},\\dots,f_{n,n,\\gamma+(p-1)b},\\dots,f_{n,n,\\gamma + pb-1}]^T, \\quad p=2,\\dots,m-1,\\\\\n\\pmb f_m &:= [f_{1,1,\\gamma + (m-1) b},\\dots,f_{n,1,\\gamma + (m-1) b},\\dots,f_{n,n,\\gamma + (m-1) b},\\dots,f_{n,n,2 \\gamma + mb-2}]^T. \n\\end{align*}\n\n\\paragraph{Auxiliary domains.}\nThe extended subdomains, the extended grids, and the corresponding\nleft and right boundaries are defined by\n\\begin{align*}\nD_q^{M} &:= (0,1)\\times(0,1)\\times((q-1)bh,2\\eta+(qb-1)h), \\quad q=1,\\dots,m,\\\\\nD_p^{R} &:= (0,1)\\times(0,1)\\times(\\eta+((p-1)b-1)h,2\\eta+(pb-1)h), \\quad p=2,\\dots,m,\\\\\nD_p^{L} &:= (0,1)\\times(0,1)\\times((p-1)bh,\\eta+pbh), \\quad p=1,\\dots,m-1,\\\\\n\\partial^{L} D_p^{R} &:= (0,1)\\times(0,1)\\times\\{\\eta+((p-1)b-1)h\\}, \\quad p=2,\\dots,m,\\\\\n\\partial^{R} D_p^{L} &:= (0,1)\\times(0,1)\\times\\{\\eta+pbh\\}, \\quad  p=1,\\dots,m-1, \\\\\nX_q^{M} &:= \\{(i_1h,i_2h,i_3h):1\\le i_1\\le n,1\\le i_2\\le n,(q-1)b+1\\le i_3 \\le 2\\gamma+qb-1\\}, \\quad q=1,\\dots,m, \\\\\nX_p^{R} &:= \\{(i_1h,i_2h,i_3h):1\\le i_1\\le n,1\\le i_2\\le n,\\gamma+(p-1)b\\le i_3\\le 2\\gamma+pb-2\\}, \\quad p=2,\\dots,m, \\\\\nX_p^{L} &:= \\{(i_1h,i_2h,i_3h):1\\le i_1\\le n,1\\le i_2\\le n,(p-1)b+1\\le i_3\\le \\gamma+pb-1\\}, \\quad p=1,\\dots,m-1. \n\\end{align*}\n\n\\paragraph{Auxiliary problems.}\nFor each $q=1,\\dots,m$, $H_q^{M} \\pmb v=\\pmb g$ is defined as the\ndiscretization on $X_q^{M}$ of\n\\begin{align*}\n  &\\begin{dcases}\n     \\left(\\partial_1^2+\\partial_2^2+(s_q^{M}(x_3)\\partial_3)^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=g(x),&\\quad \\forall x\\in D_q^{M},\\\\\n     v(x)=0,&\\quad \\forall x\\in \\partial D_q^{M}, \n   \\end{dcases}\n\\end{align*}\nFor $p=2,\\dots,m$, $H_p^{R} \\pmb v=\\pmb g$ is defined as the\ndiscretization on $X_p^{R}$ of\n\\begin{align*}\n  &\\begin{dcases}\n     \\left(\\partial_1^2+\\partial_2^2+(s_p^{R}(x_3)\\partial_3)^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=0,\\quad &\\forall x\\in D_p^{R},\\\\\n     v(x)=w(x_1,x_2),\\quad &\\forall x\\in \\partial^{L} D_p^{R}, \\\\\n     v(x)=0,\\quad &\\forall x\\in \\partial D_p^{R} \\setminus \\partial^{L} D_p^{R}, \n   \\end{dcases}\n\\end{align*}\nwhere $\\pmb\ng:=(-1\/h^2)[\\pmb w^T,0,\\dots,0]^T$ and $\\pmb w := [w_{1,1},\\dots,w_{n,1},\\dots,w_{n,n}]$ is the discrete boundary value. Finally,\nfor $p=1,\\dots,m-1$, $H_p^{L} \\pmb v=\\pmb g$ is the discretization on\n$X_p^{L}$ of\n\\begin{align*}\n  &\\begin{dcases}\n     \\left(\\partial_1^2+\\partial_2^2+(s_p^{L}(x_3)\\partial_3)^2+\\dfrac{\\omega^2}{c^2(x)}\\right)v(x)=0,\\quad &\\forall x\\in D_p^{L},\\\\\n     v(x)=w(x_1,x_2),\\quad &\\forall x\\in \\partial^{R} D_p^{L}, \\\\\n     v(x)=0,\\quad &\\forall x\\in \\partial D_p^{L} \\setminus \\partial^{R} D_p^{L},\n   \\end{dcases}\n\\end{align*}\nwhere $\\pmb g:=(-1\/h^2)[0,\\dots,0,\\pmb w^T]^T$ and $\\pmb w := [w_{1,1},\\dots,w_{n,1},\\dots,w_{n,n}]$. \n\n\\paragraph{Auxiliary Green's operators.}\nFor $q=1,\\dots,m$, $\\td{G}_q^{M}:\\pmb y\\mapsto \\pmb z$ is defined\nusing the following operations:\n\\begin{enumerate}\n\\item\n  Introduce a vector $\\pmb g$ defined on $X_q^{M}$ by setting $\\pmb y$ to $X_q$ and zero everywhere else. \n\\item\n  Solve $H_q^{M} \\pmb v=\\pmb g$ on $X_q^{M}$. \n\\item\n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_q$. \n\\end{enumerate}\nFor $p=2,\\dots,m$, $\\td{G}_p^{R}:\\pmb w\\mapsto \\pmb z$ is given by:\n\\begin{enumerate}\n\\item\n  Set $\\pmb g=(-1\/h^2)[\\pmb w^T,0,\\dots,0]^T$. \n\\item\n  Solve $H_p^{R} \\pmb v =\\pmb g$ on $X_p^{R}$. \n\\item\n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_p$. \n\\end{enumerate}\nFinally, for $p=1,\\dots,m-1$, the operators $\\td{G}_p^{L}:\\pmb\nw\\mapsto \\pmb z$ is introduced to be:\n\\begin{enumerate}\n\\item\n  Set $\\pmb g=(-1\/h^2)[0,\\dots,0,\\pmb w^T]^T$. \n\\item\n  Solve $H_p^{L} \\pmb v =\\pmb g$ on $X_p^{L}$. \n\\item\n  Set $\\pmb z$ as the restriction of $\\pmb v$ on $X_p$. \n\\end{enumerate}\n\n\\paragraph{Putting together.}\nIn the 3D case, $\\pmb g^{L}$ and $\\pmb g^{R}$ for the column-major ordering array\\\\\n$\\pmb\ng=[g_{1,1,1},\\dots,g_{s_1,1,1},\\dots,g_{s_1,s_2,1},\\dots,g_{s_1,s_2,s_3}]^T$\ninduced from some 3D grid with size\\\\\n$s_1\\times s_2 \\times s_3$ are\ngiven by\n\\begin{align*}\n  \\pmb g^{L}:=[g_{1,1,1},\\dots,g_{s_1,1,1},\\dots,g_{s_1,s_2,1}]^T, \\quad \\pmb g^{R}:=[g_{1,1,s_3},\\dots,g_{s_1,1,s_3},\\dots,g_{s_1,s_2,s_3}]^T. \n\\end{align*}\n\nThe subproblems $H_q^{M} \\pmb v=\\pmb g$, $H_p^{R} \\pmb v=\\pmb g$ and\n$H_p^{L} \\pmb v=\\pmb g$ are quasi-2D. To solve them, we group the\nelements along dimension 3 first, and then apply the nested dissection\nmethod\\cite{george1973nested,duff1983multifrontal} to them, as in \\cite{sweeppml}. This gives the setup process of the 3D\npreconditioner in Algorithm \\ref{alg:3dsetup} and the application\nprocess in Algorithm \\ref{alg:3dapp}.\n\\begin{algorithm}[h!]\n  \\caption{Construction of the 3D additive sweeping preconditioner of\n    the system \\eqref{eqn:3D}. Complexity\n    $=O(n^4(b+\\gamma)^3\/b)=O(N^{4\/3}(b+\\gamma)^3\/b)$.}\n  \\label{alg:3dsetup}\n  \\begin{algorithmic}\n    \\FOR {$q=1,\\dots,m$}\n    \\STATE Construct the nested dissection factorization of $H_q^{M}$, which defines $\\td{G}_q^{M}$.\n    \\ENDFOR\n    \\FOR {$p=2,\\dots,m$}\n    \\STATE Construct the the nested dissection factorization of $H_p^{R}$, which defines $\\td{G}_p^{R}$.\n    \\ENDFOR\n    \\FOR {$p=1,\\dots,m-1$}\n    \\STATE Construct the the nested dissection factorization of $H_p^{L}$, which defines $\\td{G}_p^{L}$.\n    \\ENDFOR\n  \\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[h!]\n  \\caption{Computation of $\\td{\\pmb u}\\approx G \\pmb f$ using the\n    preconditioner from Algorithm \\ref{alg:3dsetup}. Complexity\n    $=O(n^3\\log n (b+\\gamma)^2\/b)=O(N\\log N(b+\\gamma)^2\/b)$.}\n  \\label{alg:3dapp}\n  \\begin{algorithmic}\n    \\FOR {$q=1,\\dots,m$}\n    \\STATE\n    $\\td{\\pmb u}_{q,q}=\\td{G}_q^{M} \\pmb f_q$\n    \\ENDFOR\n    \\FOR {$p=2,\\dots,m$}\n    \\STATE\n    $\\td{\\pmb u}_{p,1:p-1}=\\td{G}_{p}^{R}\\td{\\pmb u}_{p-1,1:p-1}^{R}$\\\\\n    $\\td{\\pmb u}_{p,1:p}^{R}=\\td{\\pmb u}_{p,1:p-1}^{R}+\\td{\\pmb u}_{p,p}^{R}$\n    \\ENDFOR\n    \\FOR {$p=m-1,\\dots,1$}\n    \\STATE\n    $\\td{\\pmb u}_{p,p+1:m}=\\td{G}_p^{L}\\td{\\pmb u}_{p+1,p+1:m}^{L}$\\\\\n    $\\td{\\pmb u}_{p,p:m}^{L}=\\td{\\pmb u}_{p,p}^{L}+\\td{\\pmb u}_{p,p+1:m}^{L}$\n    \\ENDFOR\n    \\STATE\n    $\\td{\\pmb u}_1=\\td{\\pmb u}_{1,1}+\\td{\\pmb u}_{1,2:m}$\\\\\n    \\FOR {$p=2,\\dots,m-1$}\n    \\STATE\n    $\\td{\\pmb u}_p=\\td{\\pmb u}_{p,1:p-1}+\\td{\\pmb u}_{p,p}+\\td{\\pmb u}_{p,p+1:m}$\n    \\ENDFOR\n    \\STATE\n    $\\td{\\pmb u}_m=\\td{\\pmb u}_{m,1:m-1}+\\td{\\pmb u}_{m,m}$ \n  \\end{algorithmic}\n\\end{algorithm}\n\nFor the algorithm analysis, we notice that each quasi-2D subproblem\nhas $O(\\gamma+b)$ layers along the third dimension. Therefore, the\nsetup cost for each subproblem is $O((\\gamma+b)^3n^3)$ and the\napplication cost is $O((\\gamma+b)^2n^2\\log n)$. Taking the total\nnumber of subproblems into account, the total setup cost for the 3D\npreconditioner is $O(n^4(b+\\gamma)^3\/b)$ and the total application\ncost is $O(n^3\\log n (b+\\gamma)^2\/b)$.\n\n\\subsection{Numerical results}\n\\label{sec:3Dnumerical}\n\nHere we present the numerical results in 3D. All the settings and\nnotations are kept the same with Section \\ref{sec:2Dnumerical}\nunless otherwise stated. The PMLs are put on all sides of the boundary\nand the symmetric version of the equation is adopted to save memory\ncost. The PML width is $\\eta= 9h$ for the boundary and is\n$\\eta_\\text{aux}=5h$ for the interior auxiliary ones. The number of\nlayers in each subdomain is $b=4$ for the interior ones,\n$b+\\gamma-1=12$ for the leftmost one and $b+\\gamma-2=11$ for the\nrightmost one.\n\nThe velocity fields tested are (see Figure \\ref{fig:3D}):  \n\\begin{enumerate}[(a)]\n\\item\n  A converging lens with a Gaussian profile at the center of the domain.\n\\item\n  A vertical waveguide with a Gaussian cross-section.\n\\item\n  A random velocity field.\n\\end{enumerate}\n\n\\begin{figure}[h!]\n  \\centering\n  \\subfigure[]\n  {\\includegraphics[width=0.32\\textwidth]{3Da.pdf}}\n  \\subfigure[]\n  {\\includegraphics[width=0.32\\textwidth]{3Db.pdf}}\n  \\subfigure[]\n  {\\includegraphics[width=0.32\\textwidth]{3Dc.pdf}}\n  \\caption{The three velocity fields tested in 3D.}\n  \\label{fig:3D}\n\\end{figure}\n\nThe forces tested for each velocity field are: \n\\begin{enumerate}[(a)]\n\\item\n  A Gaussian point source centered at $(1\/2,1\/2,1\/4)$.\n\\item\n  A Gaussian wave packet with wavelength comparable to the typical\n  wavelength of the domain. The packet centers at $(1\/2,1\/4,1\/4)$ and\n  points to the direction $(0,1\/\\sqrt{2},1\/\\sqrt{2})$.\n\\end{enumerate}\n\n\\begin{table}[h!]\n\\centering\n\\begin{overpic}\n[width=0.45\\textwidth]{3Daa.pdf}\n\\put(40,6){force (a)}\n\\end{overpic}\n\\begin{overpic}\n[width=0.45\\textwidth]{3Dab.pdf}\n\\put(40,6){force (b)}\n\\end{overpic}\n\\begin{tabular}{ccc|cc|cc}\n\\hline \n\\multicolumn{3}{c|}{velocity field (a)} & \\multicolumn{2}{c|}{force (a)} & \\multicolumn{2}{c}{force (b)} \\\\ \n\\hline \n$\\omega\/(2\\pi)$ & $N$ & $T_{\\text{setup}}$ & $N_\\text{iter}$ & $T_\\text{solve}$ & $N_\\text{iter}$ & $T_\\text{solve}$ \\\\ \n\\hline \n5 & $39^3$ & 2.3304e$+$01  &3& 2.9307e$+$00  &4&  3.7770e$+$00\\\\\n10 & $79^3$ & 3.2935e$+$02 &3&  3.6898e$+$01  &4&  4.6176e$+$01\\\\ \n20 & $159^2$ & 4.2280e$+$03  &4&  4.3999e$+$02  &4&  4.6941e$+$02\\\\\n\\hline \n\\end{tabular} \n\\caption{Results for velocity field (a) in 3D. Solutions with $\\omega\/(2\\pi)=10$ at $x_1=0.5$ are presented.}\n\\label{tab:3domegaa}\n\\end{table}\n\\begin{table}[h!]\n\\centering\n\\begin{overpic}\n[width=0.45\\textwidth]{3Dba.pdf}\n\\put(40,6){force (a)}\n\\end{overpic}\n\\begin{overpic}\n[width=0.45\\textwidth]{3Dbb.pdf}\n\\put(40,6){force (b)}\n\\end{overpic}\n\\begin{tabular}{ccc|cc|cc}\n\\hline \n\\multicolumn{3}{c|}{velocity field (b)} & \\multicolumn{2}{c|}{force (a)} & \\multicolumn{2}{c}{force (b)} \\\\ \n\\hline \n$\\omega\/(2\\pi)$ & $N$ & $T_{\\text{setup}}$ & $N_\\text{iter}$ & $T_\\text{solve}$ & $N_\\text{iter}$ & $T_\\text{solve}$ \\\\ \n\\hline \n5 & $39^3$ & 2.1315e$+$01  &3&  2.7740e$+$00  &3&  2.7718e$+$00\\\\\n10 & $79^3$ & 3.4256e$+$02 &4&  4.4286e$+$01  &3&  3.4500e$+$01\\\\ \n20 & $159^2$ & 4.3167e$+$03  &5&  5.7845e$+$02  &4& 4.6462e$+$02\\\\\n\\hline \n\\end{tabular} \n\\caption{Results for velocity field (b) in 3D. Solutions with $\\omega\/(2\\pi)=10$ at $x_1=0.5$ are presented.}\n\\label{tab:3domegab}\n\\end{table}\n\\begin{table}[h!]\n\\centering\n\\begin{overpic}\n[width=0.45\\textwidth]{3Dca.pdf}\n\\put(40,6){force (a)}\n\\end{overpic}\n\\begin{overpic}\n[width=0.45\\textwidth]{3Dcb.pdf}\n\\put(40,6){force (b)}\n\\end{overpic}\n\\begin{tabular}{ccc|cc|cc}\n\\hline \n\\multicolumn{3}{c|}{velocity field (c)} & \\multicolumn{2}{c|}{force (a)} & \\multicolumn{2}{c}{force (b)} \\\\ \n\\hline \n$\\omega\/(2\\pi)$ & $N$ & $T_{\\text{setup}}$ & $N_\\text{iter}$ & $T_\\text{solve}$ & $N_\\text{iter}$ & $T_\\text{solve}$ \\\\ \n\\hline \n5 & $39^3$ & 2.1063e$+$01 &4& 3.8074e$+$00  &4&  3.7975e$+$00\\\\\n10 & $79^3$ & 3.4735e$+$02  &4&  4.4550e$+$01  &4&  4.5039e$+$01\\\\ \n20 & $159^2$ & 4.3391e$+$03  &4&  4.4361e$+$02  &5&  5.8090e$+$02\\\\\n\\hline \n\\end{tabular} \n\\caption{Results for velocity field (c) in 3D. Solutions with $\\omega\/(2\\pi)=10$ at $x_1=0.5$ are presented.}\n\\label{tab:3domegac}\n\\end{table}\n\nThe results are given in Tables \\ref{tab:3domegaa}, \\ref{tab:3domegab}\nand \\ref{tab:3domegac}. From these tests we see that the iteration\nnumber grows mildly as the problem size grows. We also notice that the\nsetup cost scales even better than $O(N^{4\/3})$, mainly because MATLAB\nperforms dense linear algebra operations in a parallel way, which\ngives some extra advantages to the nested dissection algorithm as the\nproblem size grows.\n\n\\section{Conclusion}\n\\label{sec:Conclusion}\nIn this paper, we proposed a new additive sweeping preconditioner for\nthe Helmholtz equation based on the PML. When combined with the\nstandard GMRES solver, the iteration number grows mildly as the\nproblem size grows. The novelty of this approach is that the unknowns\nare split in an additive way and the boundary values of the\nintermediate results are utilized directly. The disadvantage is that,\nfor each subdomains, three subproblems need to be built up, which is\ntime consuming compared to \\cite{sweeppml} and\n\\cite{stolk2013domaindecomp}. However, the costly parts of the\nalgorithm, i.e. the whole setup process and the solve processes of the\nsubproblems $H_q^{M} \\pmb v=\\pmb g$, can be done in parallel. The only\nparts that must be implemented sequentially are the accumulations of\nthe left-going and right-going waves, where only the solve processes\nof the subproblems $H_p^{L} \\pmb v=\\pmb g$ and $H_p^{R} \\pmb v=\\pmb g$\nare involved, which are the cheapest parts of the algorithm. Besides,\nwe think that the whole approximation process is simple and\nstructurally clear from a physics point of view and the idea might be\neasy to be generalized to other equations.\n\nThere are also some other directions to make potential\nimprovements. First, other numerical schemes of the equation and other\napproximations of the Sommerfeld radiation condition can be used to\ndevelop more efficient versions of this additive\npreconditioner. Second, the parallel version of the nested dissection\nalgorithm can be combined to solve large scale problems. Last, in the\n3D case, the quasi-2D subproblems can be solved recursively by\nsweeping along the $x_2$ direction with the same technique, which\nreduces the theoretical setup cost to $O(N)$ and the application cost\nto $O(N)$. However, compared to \\cite{sweeppml}, the coefficient of\nthe complexity in this new method is larger, so it is not clear\nwhether or not the recursive approach will be more efficient\npractically. Nevertheless, it is of great theoretical interest to look\ninto it.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec: intro}\nInterstellar dust grains participate in many important physical and chemical processes in the interstellar medium (ISM). For example, the surface of dust is the catalyst for formation of some molecules, especially $\\rm H_2$ \\citep{GOULD63, CAZAUX04}. Dust also shields gas from the interstellar radiation field (ISRF), and allows the low temperatures crucial to star formation to emerge deep within molecular clouds \\citep{KRUMHOLZ11, YAMASAWA11, GLOVER12}. Dust plays an important role in the observed spectral energy distribution (SED) of galaxies: it absorbs and scatters starlight, and reemits the absorbed energy at infrared (IR) wavelengths \\citep{CALZETTI01, BUAT12}. Thus, it is important to understand the properties of dust before we can fully understand the ISM and the observed SED from galaxies.\n\nThe amount of interstellar dust depends on the balance between dust formation and dust destruction. The mechanisms of dust destruction include supernovae (SNe) shocks, thermal evaporation, cosmic rays, and dust incorporated into newly formed stars \\citep{DWEK98, HIRASHITA99}. The dust formation mechanisms include accretion of metals in the ISM onto existing dust grains, formation of new dust grains in the winds of AGB stars, and dust formation in type II SNe \\citep{DWEK98, ASANO13}. Different dominant dust destruction and formation mechanisms would result in a different dust-to-gas mass ratio (DGR):\n\\begin{equation}\n{\\rm DGR} \\equiv \\Sigma_d \/ \\Sigma_{\\rm gas}\n\\end{equation}\nand dust-to-metals ratio (DTM):\n\\begin{equation}\n    {\\rm DTM} \\equiv {\\rm DGR} \/ Z,\n\\end{equation}\nwhere $\\Sigma_d$ is the dust mass surface density, $\\Sigma_{\\rm gas}$ is the total gas mass surface density, which includes the contribution from \\textsc{HI}, H$_2$ and He, and $Z$ is the metallicity. Note that some authors replace $\\Sigma_{\\rm gas}$ with hydrogen mass surface density in the definition of DGR, e.g., \\citet{DRAINE14, GORDON14}. Other than the formation and destruction mechanisms affecting DGR and DTM, the DTM itself can directly impact the ISM dust accretion rate \\citep{DWEK98}. Thus, studying DGR and DTM provides key insights into the dust life cycle.\n\nTheoretical dust life cycle models yield varying predictions for the DTM as a function of metallicity and local environment. Models in \\citet{SODROSKI97, DWEK98} show that the DGR gradient scales linearly with the metallicity gradient, and the DTM is nearly a constant. This can be achieved by a constant rate of dust formation and destruction, which results in a constant fraction of metal incorporated into dust, and thus DTM at all chemical evolution stages is a constant \\citep{GALLIANO08}. Other studies show that DTM is not always a constant, but a multi-stage variable as metallicity increases. At low metallicity, ISM accretion is less effective and the dust production rate is dominated by stellar ejecta, which could result in a locally constant DTM in this low metallicity regime \\citep{HIRASHITA11}. Above a certain critical metallicity, the efficiency of dust accretion may increase, which would result in a DTM increasing with metallicity \\citep{ZHUKOVSKA08, HIRASHITA11, FELDMANN15}. The critical metallicity depends on model and choices of parameters, and usually falls in the range of $\\rm 12+\\logt({\\rm O\/H}) = 7.5$ and $8.5$ \\citep{HIRASHITA99, ZHUKOVSKA08, HIRASHITA11, ASANO13, ZHUKOVSKA16}.\n\nSeveral observational studies support a constant DTM. In \\citet{ISSA90}, the authors collated the DGR gradients and metallicity gradients from previous studies in M31, M33, and M51, and reached the conclusion that the slopes of DGR and metallicity with galactic radius are consistent with each other. In \\citet{LEROY11}, the authors followed the approaches in \\citet{DRAINE07} to derive the dust masses in local group galaxies. They showed that DTM is a constant across $8.0 \\la \\rm 12+\\logt({\\rm O\/H}) \\la 9.0$. In \\citet{DRAINE14}, the authors fit the IR SED in M31 to a renormalized version of dust model described in \\citet{DRAINE07}. The authors showed that their derived DGR scales linearly with metallicity where metallicity measurements are reported by \\citet{ZURITA12}. Importantly, the relation between dust and metallicity is consistent with $M_d\/M_H \\sim 0.0091 Z\/Z_\\sun$, a prediction from depletion conditions in the cloud toward $\\zeta$Oph in the Milky Way (MW) \\citep{DRAINE11, DRAINE14}.\n\nThere are also observational results supporting a varying DTM. In \\citet{LISENFELD98}, the authors studied the DTM in 44 dwarf galaxies, and found a varying DTM. In \\citet{HIRASHITA02}, the authors study 16 blue compact dwarf (BCD) galaxies, and found that $\\log_{10}(\\rm DGR)$ spreads from $-3.3$ to $-4.6$ within $\\rm 7.9 < \\rm 12+\\logt({\\rm O\/H}) < 8.6$, indicating an variable DTM because the slope between DGR and metallicity is not unity. The authors hypothesized that this phenomenon is the result of the variation in dust destruction efficiency by SNe, which depends on the star formation history of the region. \\citet{HUNT05} also showed a 2 dex spread of DGR at $8 \\leq \\rm 12+\\logt({\\rm O\/H}) \\leq 9$. They also reported that the BCD SBS 0335$-$052, which has a metallicity $\\rm 12+\\logt({\\rm O\/H}) = 7.32$, has an extremely low dust mass, two orders of magnitude below a linear trend with metallicity. Similarly, \\citet{HERRERA-CAMUS12} and \\citet{FISHER14} showed that the local dwarf galaxy I Zw 18 has a DGR two orders of magnitude below the linear trend derived from local galaxies. In \\citet{REMY-RUYER14}, the authors compiled DGR measurements for 126 galaxies, with 30\\% of their sample having $\\rm 12+\\logt({\\rm O\/H})\\leq 8.0$. They showed that there might be a discontinuity of the linear DTM at oxygen abundance $\\rm 12+\\logt({\\rm O\/H}) = 8$, and the galaxies below that metallicity have $\\rm DGR \\propto Z^{3.1}$. That is, instead of a simple linear relation between DGR and $Z$, the authors suggest a broken power-law. In \\citet{ROMAN-DUVAL17}, the authors showed that the DGR changes by factors of 3 to 7 in the Magellanic Clouds, where metallicity is considered to be constant. This result also indicates a variable DTM. In \\citet{GIANNETTI17}, the authors found a ${\\rm DGR}(Z) \\propto Z^{1.4}$ in a sample set composed by 23 massive and dense star-forming regions in the far outer MW.\n\nIn this work, we revisit the possible variation of DTM in a single galaxy, M101. There are several benefits to studying DTM within a single galaxy. First, metallicity measurements are calibrated more uniformly within one galaxy than across galaxies, which is crucial for studying DTM variation \\citep{REMY-RUYER14, BERG15, CROXALL16}. Moreover, focusing on one galaxy can avoid the problem in galaxy-integrated results that DTM can be underestimated by integrating over dust-poor HI in outer disks \\citep{DRAINE07}. By comparing the DTM within one galaxy and across galaxies, we will also be able to determine whether the possible variation in DTM depends more on local physical properties or galactic properties. Lastly, observations within one galaxy would have the minimum differences of MW foreground, calibration, and background level estimation, which means the data are more uniform.\n\nM101 is an ideal target for this study for four reasons: 1) M101 has one of the most detailed studies of its metallicity from the Chemical Abundances Of Spirals survey \\citep[CHAOS,][]{BERG15, CROXALL16}, based on electron temperature ($T_e$) derived from auroral line measurements. 2) M101 has the largest metallicity gradient among those galaxies where direct $T_e$-based metallicity measurements are available, ranging $7.5 \\la \\rm 12+\\logt({\\rm O\/H}) \\la 8.8$ \\citep{CROXALL16}. This range covers both as high as the solar neighborhood and as low as the turning point in \\citet{REMY-RUYER14} broken-power law. 3) M101 has a good radial resolution even at far-infrared (FIR) observations because it is nearby (distance $\\sim \\rm 6.7~Mpc$), physically large (the 25th magnitude isophote in B band, or r$_{25}$, is $0.2^\\circ=23.4~{\\rm kpc}$ at distance $\\rm 6.7~Mpc$), and relatively face on \\citep[inclination $\\approx 16^\\circ$,][]{FREEMAN01, MAKAROV14}. 4) M101 also has high sensitivity \\textsc{Hi} and CO maps \\citep{WALTER08, LEROY09}, which let us map the total gas distribution.\n\nThis paper is presented as follows. \\secref{sec: observations} presents FIR, \\textsc{Hi}, CO, and other supporting data used in this study, with our data processing procedures. The five modified blackbody (MBB) model variants and the fitting methodologies are described in \\secref{Sec: methods}. We present our fitting results in \\secref{sec: results}, and compare them with known physical limitations and statistical properties. In \\secref{sec: discussions}, we discuss the implication of our results, and the relation between our DTM and previous findings. Finally, we give our conclusions in \\secref{sec: conclusions}.\n\n\\section{Observations} \\label{sec: observations}\n\\subsection{Data}\\label{sec: data}\nIn this section, we introduce the multi-wavelength measurements of M101 from several surveys and their uncertainties, which we adopted for this study. The physical properties (position, distance and orientation) of M101 adopted for this study are listed in Table \\ref{tab: samples}.\n\\begin{deluxetable}{lll}\n\\tablecaption{Properties of M101. \\label{tab: samples}}\n\\tablehead{\\colhead{Property} & \\colhead{Value} & \\colhead{Reference}}\n\\startdata\nR.A. (2000.0)\t& 14h~03m~12.6s & (1) \\\\\nDec (2000.0)\t& +54d~20m~57s & (1) \\\\\nDistance\t\t& 6.7~Mpc\\tablenotemark{$\\dagger$} &(2) \\\\\nr$_{25}$\t\t\t& $0^\\circ.19990$ & (1) \\\\\nInclination\t\t& $16^\\circ$ & (1) \\\\\nP.A.\t\t\t& $38^\\circ$ &(3) \\\\\n$\\alpha_{{\\rm CO}~J=(2-1)}$\\tablenotemark{*}\t& $\\rm (2.9\/R_{21})~M_\\sun~pc^{-2} (K~km~s^{-1})^{-1}$ & (4) \\\\\n$R_{21}$ & 0.7 & (4) \\\\\n\\enddata\n\\tablenotetext{\\dagger}{Consistent with the value in \\citet{SHAPPEE11}.}\n\\tablenotetext{*}{See \\secref{subsec: CO} for discussion of the $\\alpha_{\\rm CO}$ factor we use.}\n\\tablerefs{(1) HyperLeda database (\\url{http:\/\/leda.univ-lyon1.fr\/}), \\citet{MAKAROV14}; (2) \\citet{FREEMAN01}; (3) \\citet{SOFUE99}; (4) \\citet{SANDSTROM13}.}\n\\end{deluxetable}\n\n\\subsubsection{Infrared Imaging} \\label{subsec: IR}\nWe use FIR images from the ``Key Insights on Nearby Galaxies: A Far-Infrared Survey with \\textit{Herschel}'' survey \\citep[KINGFISH,][]{KENNICUTT11} to fit dust surface densities in M101. KINGFISH imaged 61 nearby galaxies in the FIR with the \\textit{Herschel Space Observatory} \\citep{PILBRATT10}, covering $70~\\micron$, $100~\\micron$, and $160~\\micron$ from Photoconductor Array Camera and Spectrometer \\citep[PACS,][]{POGLITSCH10}, and $250~\\micron$, $350~\\micron$, and $500~\\micron$ from Spectral and Photometric Imaging Receiver \\citep[SPIRE,][]{GRIFFIN10}. We do not include the $70~\\micron$ flux in our SED modeling because stochastic heating from small dust grains makes non-negligible contribution in that spectral range \\citep{DRAINE07}, which is not accounted for by the simple SED models we employ in this study. The PACS images were processed from level 1 with \\texttt{Scanamorphos v16.9} \\citep{ROUSSEL13} by the KINGFISH team. The SPIRE images were processed with \\texttt{HIPE} \\citep{OTT10} version \\texttt{spire-8.0.3287} and from level 1 to final maps with \\texttt{Scanamorphos v17.0} \\citep{ROUSSEL13} by the KINGFISH team. According to the KINGFISH DR3 user guide \\citep{KINGFISH13}, the SPIRE images have been multiplied by correction factors of 0.9282, 0.9351, and 0.9195 for SPIRE250, SPIRE350, and SPIRE500, respectively, due to improved effective beam size estimation. The FWHMs are approximately $7\\arcsec.0 = 0.23~\\rm kpc$, $11\\arcsec.2 = 0.36~\\rm kpc$, $18\\arcsec.2 = 0.59~\\rm kpc$, $24\\arcsec.9 = 0.81~\\rm kpc$, and $36\\arcsec.1 = 1.17~\\rm kpc$ for the 100\\micron, 160\\micron, 250\\micron, 350\\micron, and 500\\micron\\ band images, respectively.\n\n\\subsubsection{\\textsc{Hi}} \\label{subsec: HI}\nWe obtain \\textsc{Hi} 21 cm line data from ``The \\textsc{Hi} Nearby Galaxy Survey'' \\citep[THINGS,][]{WALTER08}. The images were obtained at the Very Large Array (VLA)\\footnote{The VLA is operated by the National Radio Astronomy Observatory (NRAO), which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.}. The M101 dataset in this survey has (10\\arcsec.8, 10\\arcsec.2)$\\sim$(0.35~kpc, 0.33~kpc) angular resolution and $\\rm 5.2~km~s^{-1}$ velocity resolution with natural weighting. The observed 21 cm emission can be converted to \\textsc{Hi} column density ($N_{\\rm HI}$) via Eq. (1) and Eq. (5) in \\citet{WALTER08} assuming it is optically thin, and then further converted to surface density $\\Sigma_{\\rm HI}$ by multiplying by the atomic weight of hydrogen. The uncertainty in the THINGS survey is dominated by the estimated zero-point uncertainty in \\textsc{Hi}, which is around $1~\\rm M_\\sun\/pc^2$, corresponding to 0.04 to 0.17 dex in the center of M101 (molecular gas dominated region), 0.03 to 0.04 dex for most atomic gas dominated region, and goes above 0.08 dex for the outer most pixels.\n\n\\subsubsection{CO and Total Gas} \\label{subsec: CO}\nWe obtain CO emission line measurements from the ``HERA CO Line Extragalactic Survey'' \\citep[HERACLES,][]{LEROY09, SCHRUBA11, SCHRUBA12, LEROY13}, a survey mapping the ${\\rm ^{12}CO}~J=(2-1)$ rotational line at $230.538~\\rm GHz$ of 48 nearby galaxies, including M101. The observation was carried out with Heterodyne Receiver Array \\citep[HERA,][]{SCHUSTER04} on the IRAM 30-m telescope\\footnote{IRAM is supported by CNRS\/INSU (France), the MPG (Germany) and the IGN (Spain).}. The survey has 13\\arcsec\\ angular resolution and $\\rm 2.6~km~s^{-1}$ velocity resolution. The CO line integrated intensity can be converted to surface density of $\\rm H_2$ plus He ($\\Sigma_{\\rm mol}$) by:\n\\begin{equation}\n    \\Sigma_{\\rm mol} = \\alpha_{\\rm CO}\\frac{I_{{\\rm CO}~J=(2-1)}}{R_{21}},\n\\end{equation}\nwhere $\\alpha_{\\rm CO}$ is the CO-to-$\\rm H_2$ conversion factor, see Table \\ref{tab: samples}. The standard $\\alpha_{\\rm CO}$ is quoted for $I_{{\\rm CO}~J=(1-0)}$, thus, we convert the $I_{{\\rm CO}~J=(2-1)}$ with a fixed line ratio\\footnote{We adopt the $\\alpha_{\\rm CO}$ value from \\citet{SANDSTROM13}, which the authors originally derived with $I_{{\\rm CO}~J=(2-1)}$ data and convert with $R_{21}=0.7$. Thus we need to use the same $R_{21}$ for consistency.} $R_{21}=(2-1)\/(1-0)=0.7$ \\citep{SANDSTROM13}.\n\nWith $\\Sigma_{\\rm HI}$ and $\\Sigma_{\\rm mol}$, we calculate the total gas mass surface density ($\\Sigma_{\\rm gas}$) with Eq. \\ref{eq: total gas}. A multiplier of value 1.36 is included in $\\Sigma_{\\rm mol}$ for helium mass \\citep{SANDSTROM13}. We multiply the $\\Sigma_{\\rm HI}$ by this factor to calculate the total gas surface density correctly.:\n\\begin{equation} \\label{eq: total gas}\n\\Sigma_{\\rm gas} = 1.36~\\Sigma_{\\rm HI} + \\alpha_{\\rm CO}\\frac{I_{{\\rm CO}~J=(2-1)}}{R_{21}}\n\\end{equation}\n\nWe have checked that a metallicity dependent $\\alpha_{\\rm CO}$ \\citep{WOLFIRE10, BOLATTO13} would make no significant difference in $\\Sigma_{\\rm gas}$ because in the region where H$_2$ is important in M101, the metallicity is still relatively high. See more discussion in \\secref{sec: alpha_CO discussion}.\n\n\\subsubsection{Metallicity\\label{subsec: metal_data}}\nWe obtained metallicity measurements from CHAOS survey \\citep{CROXALL16}. Measurements were taken in 109 \\textsc{Hii} regions by the Multi-Object Double Spectrographs (MODS) on the Large Binocular Telescope \\citep[LBT,][]{POGGE10}. They derived $T_e$ from a three-zone model with \\textsc{[Oiii]}, \\textsc{[Siii]}, and \\textsc{[Nii]} line ratios. The electron densities are derived from \\textsc{[Sii]} line ratios. This gives us gas phase oxygen abundances in 74 \\textsc{Hii} regions inside M101, and also an average metallicity gradient spread over the galactocentric radius considered in this study. We will compare our derived DGR with their derived metallicity gradient \\citep[Eq. 10 in][second line\\footnote{Instead of the 7.4 Mpc distance quoted in \\citet{CROXALL16}, we used a galaxy distance of 6.7 Mpc, thus we multiplied the slope in their Eq. 10 by $\\frac{7.4}{6.7}$ to account for the difference.}]{CROXALL16}.  The uncertainty in $\\rm 12+\\logt({\\rm O\/H})$ from the average metallicity gradient is $\\sim0.02$ dex in the center and $\\sim0.07$ dex in the outer most part.\n\n\\subsubsection{Star formation rate and stellar mass\\label{subsec: other}}\nWe calculate star formation rate surface density ($\\Sigma_{\\rm SFR}$) from the Galaxy Evolution Explorer (GALEX) FUV \\citep{MARTIN05} and \\textit{Spitzer} Multiband Imaging Photometer (MIPS) 24 \\micron\\ data \\citep{WERNER04, RIEKE04}, and stellar mass surface density ($\\Sigma_\\star$) from \\textit{Spitzer} Infrared Array Camera (IRAC) 3.6 \\micron. These data are from the Local Volume Legacy survey \\citep[LVL,][]{DALE09}.\n\nWe use the following equation to convert observed FUV and IR emission to $\\Sigma_{\\rm SFR}$:\n\\begin{equation} \\label{eq: SFR}\n\\Sigma_{\\rm SFR}=(8.1\\times 10^{-2} I_{\\rm FUV} + 3.2 \\times 10^{-3} I_{24})\\cos i,\n\\end{equation}\nwhere $i$ is the inclination of M101. $\\Sigma_{\\rm SFR}$ is in $M_{\\sun}~\\rm kpc^{-2}~yr^{-1}$, and both $I_{\\rm FUV}$ and $I_{24}$ are in $\\rm MJy~sr^{-1}$. Eq. \\ref{eq: SFR} is adopted from \\cite{LEROY08}, and it is functionally similar to the prescription in \\citet{KENNICUTT12}.\n\nFor converting 3.6 \\micron\\ SED to $\\Sigma_\\star$, we use the relation:\n\\begin{equation}\n    \\Sigma_\\star = 350I_{3.6}\\cos i,\n\\end{equation}\nwhere $\\Sigma_\\star$ is in $M_{\\sun}~\\rm pc^{-2}$, and $I_{3.6}$ is in $\\rm MJy~sr^{-1}$. Note that the appropriate mass to light ratio ($\\Upsilon_\\star^{3.6}$) remains a topic of research \\citep{MCGAUGH14, MEIDT14}. Here, we assume the $\\Upsilon_\\star^{3.6}=0.5$ \\citep{MCGAUGH14}, see discussions in \\citet{LEROY08} and A. K. Leroy et al. (2018, in preparation).\n\n\\subsection{Data processing\\label{sec: data proc}}\n\\subsubsection{Background subtraction} \\label{subsec: background error}\nThe IR and GALEX images that we use include contributions from various backgrounds and foregrounds. Throughout this study, we will neglect the structure in MW foreground over the relatively small angular ($r_{25}=0^\\circ.2$) extent of M101. To estimate the foreground\/background (hereafter referred to as background) level for each image, we need a uniform definition of background region. We define our background region as where $N_{\\rm HI} < 1.0\\times \\rm 10^{18}~cm^{-2}$. For the GALEX map, we take the mean value in the background region as recommended due to the Poisson statistics of the GALEX counts. For the IR images, we fit a tilted plane and iteratively reject outliers. This includes several steps: we fit a tilted plane to all the background pixels. We then subtract the tilted plane from the data and calculate the absolute deviation (AD) from the median for all pixels and derive median absolute deviation (MAD). Finally, we use only the pixels with AD smaller than three times MAD to fit a tilted plane, and iterate over step two and three for five times, keeping the last fitted tilted plane as the background to be removed.\n\nAfter background subtraction and convolution (\\secref{subsec: convolution}), we calculate the covariance matrix\\footnote{A matrix with its i-j element as the i-band to j-band covariance. Our covariance matrix has a dimension of 5x5, corresponding to the 100$-$500 \\micron\\ bands in \\textit{Herschel}.} in the background region of the five \\textit{Herschel} bands. This covariance matrix ($\\mathcal{C}_{\\rm bkg}$) will play an important role in calculation of likelihood in our fitting procedure because it incorporates the observed band-to-band correlation in the noise due to confusion and other astronomical sources into our fitting (\\secref{subsec: fitting}). \n\n\\subsubsection{Convolution}\\label{subsec: convolution}\nMaps obtained from different surveys do not have the same pixel scale and point spread function (PSF). In order to compare them pixel-by-pixel, we first convolve all the maps to match the PSF of SPIRE500 using the \\texttt{convolve\\_fft} function in \\texttt{astropy.convolution} \\citep{ASTROPY13}. Most kernels in this study were adapted from \\citet{ANIANO11}, except the Gaussian kernels for THINGS and HERACLES surveys. For these two surveys, we built elliptical or circular Gaussian kernels according to their beam sizes \\citep{WALTER08, LEROY09} to convolve them to match a Gaussian PSF with 25\\arcsec\\ FWHM. Then, we convolve the images with a second kernel from \\citet{ANIANO11}, which convolves Gaussian PSF with 25\\arcsec\\ FWHM to SPIRE500 PSF.\n\n\\subsubsection{Alignment\\label{subsec: alignment}}\nAfter convolution, we align the coordinates of all the images with the SPIRE500 image and its pixel scale using the function \\texttt{reproject\\_exact} in \\texttt{reproject}, an \\texttt{astropy} affiliated package. The final pixel scale is 14.0\\arcsec, or $\\rm \\sim 0.45~kpc$, which is smaller than half of SPIRE500 PSF FWHM, 36\\arcsec, thus enough for properly sampling the PSF. In the final images, one resolution element contains $\\sim 5.2$ pixels, therefore, neighboring pixels are not independent.\n\n\\subsubsection{Binning\\label{subsec: voronoi}}\nOne of our main interests is to analyze DTM in regions with $\\rm 12+\\logt({\\rm O\/H}) \\la 8.0$, where the relation of DTM with metallicity is expected to change \\citep{HIRASHITA99, HIRASHITA11, REMY-RUYER14}. However, individual pixels in the low metallicity region, or outer disk, tend to have insufficient signal-to-noise ratio (SNR) for analysis. One way we can solve this problem is to bin neighboring pixels together and average the measured quantities in those pixels to increase SNR according to:\n\n\\begin{equation}\n    {\\rm SNR_{avg}} = \\frac{(\\sum_i {\\rm Signal}_i)\/n}{\\sqrt{(\\sum_i {\\rm Noise}_i^2)\/n^2}},\n\\end{equation}\n\nwhere the summation is over resolution elements inside the binned and $n$ is the number of resolution elements. As a consequence, uniform binning requires all regions on the map to sacrifice their spatial resolution in order to recover the regions with lower SNR, which means some structures that could have been resolved would be smoothed out in the binning process. To optimize the resolution and extend to the outer disk simultaneously, we choose to use adaptive binning: binning more pixels together in the low SNR region, while binning fewer pixels together or leaving pixels as individuals in the high SNR region.\n\nThe adaptive binning method we choose is the \\texttt{voronoi\\_2d\\_binning} function \\citep{CAPPELLARI03}. Instead of directly apply the algorithm to the entire SED, we execute some extra procedures listed below in order to preserve radial information:\n\\begin{enumerate}\n    \\item We calculate SNR map for all five \\textit{Herschel} bands using the square root of diagonal terms in the covariance matrix ($\\mathcal{C}_{\\rm bkg}$), which is the variance of each band, as the noise of each band.\n    \\item For each pixel, we select the lowest SNR among five bands at that pixel to build the worst SNR map, which is plotted in Figure \\ref{fig: voronoi} (a). This worst SNR map is used for the subsequent binning process in order to make sure all five bands will reach the target SNR with the same binned regions. 58\\% of pixels have their worst SNR from PACS100.\n    \\item We cut the target galaxy into concentric rings with the same radial spacing, which is set to be the same as the FWHM of the SPIRE500 PSF. This initial radial cut is shown in Figure \\ref{fig: voronoi} (b).\n    \\item Starting from the outermost ring, if the average SNR of all pixels within a ring is lower than target SNR, we combine it with one ring inside until target SNR is achieved. This final radial cut is shown in Figure \\ref{fig: voronoi} (c). The target SNR is set to be 5. However, since the pixels are oversampled with the SPIRE500 PSF (see \\secref{subsec: alignment}), the effective target SNR is $ 5 \/ \\sqrt{5.2} \\sim 2.2 $.\n    \\item We apply \\texttt{voronoi\\_2d\\_binning} with \\texttt{targetSN} set to 5, to each ring from Step 4 and worst SNR map from Step 2 to generate the final binned regions, as shown in Figure \\ref{fig: voronoi} (d).\n\\end{enumerate}\nNote that we discard the \\texttt{roundness} threshold in the original function \\citep{CAPPELLARI03}. This \\texttt{roundness} threshold makes sure all binned region are nearly circular, which will result in malfunctions when we cut the image into concentric circles at the beginning. All pixels within radius 7.4 kpc ($0.3~\\rm r_{25}$) have high enough SNR thus remain unbinned.\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{_Voronoi_NGC5457.pdf}\n\\caption{Voronoi binning process in this study. (a) The worst-SNR map. Among the 16,403 points, 58\\% have their worst SNR in PACS100. Both PACS160 and SPIRE500 take around 18\\%. (b) The initial radial cut. (c) The final radial cut after grouping rings according to target SNR. (d) The final binned regions. The white circles in panel (a) and (d) show the radius 7.4 kpc. All pixels within 7.4 kpc remain unbinned.\\label{fig: voronoi}}\n\\end{figure}\n\n\\section{Methods\\label{Sec: methods}}\n\\subsection{Models\\label{Sec: Model}}\nIn this work, we focus on the FIR part of the dust emission SED. It is reasonable to assume that emission from dust grains in thermal equilibrium dominates the FIR range \\citep{LI01, BLAIN02, GORDON14}, therefore we start with fitting the FIR emission with a modified blackbody (MBB) model:\n\\begin{equation}\nI_\\nu = \\kappa_{\\nu}\\Sigma_d B_\\nu(T_d),\n\\end{equation}\nwhere $I_\\nu$ is the specific intensity, $\\kappa_{\\nu}$ is the wavelength-dependent emissivity, $\\Sigma_d$ is the dust surface density, and $B_\\nu(T_d)$ is the blackbody spectral radiance at dust temperature $T_d$. An empirical power law emissivity is often assumed, that is, $\\kappa_\\nu = \\kappa_{\\nu_0}(\\nu\/\\nu_0)^\\beta$, where the emissivity index $\\beta$ is a constant and $\\nu_0=c\/\\lambda_0$. Throughout this study, $\\lambda_0=160~\\micron$ is used.\n\nThere are a few possible drawbacks to this simple model, some of them are physical, and the others are inherent to the process of fitting the model. The physical drawbacks include: 1) The simple model above does not allow for wavelength or environmental dependence of $\\beta$, which might exist \\citep{REACH95, FINKBEINER99, LI01, GORDON14}. 2) The model does not include stochastic heating \\citep{DRAINE07}, which might contribute to our shortest wavelength observation due to the width of the response functions of the PACS instruments. 3) The model does not include the broadening in the SED due to multiple heating conditions involved in one resolution element \\citep{DALE01}. The fitting process drawbacks include: 1) $\\kappa_{\\nu_0}$ and $\\Sigma_d$ are completely degenerate, thus there will be an inherent uncertainty in $\\Sigma_d$ from how we determine the $\\kappa_{\\nu_0}$ value. 2) Due to the nature of this model, $\\beta$ and $T_d$ are covariant, since they both shift the peak wavelength of the SED. Thus, there might be artificial correlation between them. \\citet{KELLY12} demonstrated this artificial correlation with traditional $\\chi^2$-minimization fitting.\n\nWe calibrate $\\kappa_{\\nu_0}$ with high-latitude MW diffuse ISM following the approach in \\citet{GORDON14} (see \\secref{Sec: calibration}). It is possible that this calibration is not appropriate at all local environmental conditions and it would result in a systematic uncertainty in our results (see \\secref{subsec: discuss_emissivity} for further discussion). We also use a probabilistic fitting procedure following \\citet{GORDON14} that lets us assess the correlations between fit parameters and properly marginalize over the degeneracy between $\\beta$ and $T_d$. Still, there is no simple way to solve all the physical drawbacks of the MBB model. In order to address the physical shortcomings of the MBB model, we construct five variant models. These each address a shortcoming of the MBB. They are not all mutually exclusive, and a full model \\citep[e.g.,][]{DRAINE07} might incorporate several of these. Our goal here is to identify the simplest possible modifications that yield a good fit to the IR SED. These variants are listed below:\n\n\\subsubsection{Simple emissivity (SE)\\label{subsubsec: SE}}\nHere, we assume a simple power-law emissivity, which gives a dust emission SED described by the following equation:\n\\begin{equation} \\label{eq: SE}\nI_\\nu = \\kappa_{\\nu_0}(\\frac{\\nu}{\\nu_0})^\\beta \\Sigma_d B_\\nu(T_d).\n\\end{equation}\nThe free parameters in this model are $\\Sigma_d$, $T_d$ and $\\beta$. This method allows $\\beta$ to vary spatially, thus could partially avoid the environmental-dependent $\\beta$ drawback. However, it is also heavily affected by the possible artificial correlation between $\\beta$ and $T_d$.\n\n\\subsubsection{Fixing $\\beta$ (FB)\\label{subsubsec: FB}}\nUsing the same functional form as Eq. \\ref{eq: SE}, we can also fix the $\\beta$ value. This is one way to remove the inherent covariance between $T_d$ and $\\beta$ based on what is expected for the optical properties of ISM dust grain materials. In some previous studies \\citep{MENNELLA98, BOUDET05, GALLIANO17} and our preliminary test of SE method, there are fitting results with anti-correlated $T_d$ and $\\beta$. This could mean that $\\beta$ is a function of $T_d$, however, due to the degeneracy of $T_d$ and $\\beta$ in the model, it is also possible that this anti-correlation is all, or partially, artificial \\citep{SHETTY09, SHETTY09B, KELLY12}. In the latter case, fixing $\\beta$ can improve the accuracy of fitted $T_d$ \\citep{SHETTY09B}. Thus, we adapted $\\beta=2$ from previous studies \\citep{REACH95, DUNNE01, DRAINE07} as a variation of MBB spectrum. We also tested $\\beta$ values of 1.6, 1.8, and 2.2 and the difference in $\\Sigma_d$ and chi-square values between them and $\\beta=2$ results are insignificant. The insensitivity of the resulting $\\Sigma_d$ to our choice of $\\beta$ results from the fact that we calibrate the emissivity for each $\\beta$ value accordingly. The process of emissivity calibration is described in \\secref{Sec: calibration}. It is also true for the other methods where we also have $\\beta$ fixed at 2 at short wavelength or the whole spectral range.\n\n\\subsubsection{Broken Emissivity (BE)}\nIt is possible that the dust emissivity is not a simple power law, but varies with wavelength. Previous studies have shown that the emissivity in the long wavelength end tends to be flatter than the short wavelength end. Thus, many authors including \\citet{REACH95} and \\citet{GORDON14} have tried to build more complicated forms of emissivity as a function of wavelength. Here, we adapted the BEMBB model in \\citet{GORDON14}: assuming $\\beta$ as a step function in wavelength, which makes the emissivity a broken-power law (Eq. \\ref{eq: BE}).\n\\begin{equation}\\label{eq: BE}\n\\kappa_\\nu=\\left\\{\\begin{array}{ll}\n    \\kappa_{\\nu_0}(\\frac{\\nu}{\\nu_0})^{\\beta} & \\lambda < \\lambda_c \\\\\n    \\kappa_{\\nu_0}(\\frac{\\nu_c}{\\nu_0})^{\\beta}(\\frac{\\nu}{\\nu_c})^{\\beta_2} & \\lambda \\geq \\lambda_c\n    \\end{array}\\right.\n\\end{equation}\n$\\lambda_c$ is the critical wavelength corresponding to the break, and $\\nu_c$ is the frequency corresponding to $\\lambda_c$. $\\lambda_c$ is fixed at 300~\\micron\\ in this study. We explored varying the break wavelength with the spectral range of 50 to 600 \\micron and found it had no major impact on the results. $\\beta_2$ is the dust emissivity index at long wavelength. The short wavelength dust emissivity index $\\beta$ is fixed at 2 in this study.\n\n\\subsubsection{Warm dust component (WD)}\nIn the spectral region below $100~\\micron$, it is possible that the SED is affected by stochastic emission from small grains \\citep{DRAINE07}, which is within the effective bandpass of the PACS100 response function (around $80$ to $120~\\micron$). In this model, we add a second MBB component with $T_d=\\rm 40~K$ to our SED, called ``warm dust'', to simulate the contribution from stochastically heated dust. We made this choice of $T_d$ to have the peak of warm dust SED at the boundary of PACS100 response function. The fraction of warm dust relative to total dust is symbolized as $f_W$. The fitting model in this method becomes (Note that both components have power-law emissivity with $\\beta=2$):\n\\begin{equation}\nI_\\nu = \\kappa_{\\nu_0}(\\frac{\\nu}{\\nu_0})^\\beta\\Sigma_d \\Big((1-f_W) B_\\nu(T_d) + f_W B_\\nu(40K) \\Big).\n\\end{equation}\n\nTo properly take this effect into account, one would need to adopt a complete physical dust model. However, among the dust properties, we are mainly interested in $\\Sigma_d$, which is necessary for calculating DGR and DTM, and which does not require adopting a full dust model. This is because within our current understanding of dust heating and the dust grain size distribution, only a small fraction of the dust mass is stochastically heated \\citep{DRAINE07}. Our preliminary test confirms this: the mass fraction of stochastically heated dust in the WD modeling is usually under 1\\%. This means that we can still acquire reasonable accuracy in $\\Sigma_d$ even when the SED of stochastically heated dust is not modeled with high accuracy. \n\n\\subsubsection{Power Law distribution (PL)}\\label{sec: PL}\nAt the SPIRE500 resolution, the FWHM of PSF would have a large physical size ($\\sim 1.22~{\\rm kpc}$). Thus, it is likely that there are various dust heating conditions within one resolution element. To attempt to model such a distribution of heating conditions, we adopt a model wherein a fraction ($1-\\gamma$) of the dust mass is heated by a single value ISRF $U_{\\rm min}$, while the other $\\gamma$ fraction is heated by a distribution of ISRF between $U_{\\rm min}$ and $U_{\\rm max}$ with $\\frac{d\\Sigma_d}{dU}\\propto U^{-\\alpha}$ \\citep{DALE01, DRAINE07}. Each mass fraction emits a FB MBB spectrum, which makes the total emission\\footnote{The normalization factor $\\frac{1-\\alpha}{U_{\\rm max}^{1-\\alpha} - U_{\\rm min}^{1-\\alpha}}$ in Eq. \\ref{eq: PL} only works when $\\alpha \\neq 1$. For $\\alpha = 1$ (which is excluded in this study), one should use $\\frac{1}{\\ln(U_{\\rm max}\/U_{\\rm min})}$ instead.}:\n\\begin{equation} \\label{eq: PL}\n\\begin{array}{ll}\nI_\\nu = \\kappa_{\\nu_0}(\\frac{\\nu}{\\nu_0})^\\beta\\Sigma_d & \\Big((1-\\gamma) B_\\nu(U_{\\rm min}) + \\\\\n& \\gamma \\frac{1-\\alpha}{U_{\\rm max}^{1-\\alpha} - U_{\\rm min}^{1-\\alpha}}\\int^{U_{\\rm max}}_{U_{\\rm min}}U^{-\\alpha}B_\\nu(U)dU \\Big).\n\\end{array}\n\\end{equation}\nTo calculate the equivalent MBB temperature, we convert $U$ to $T_d$ as $U \\propto T_d^{\\beta + 4}$, with a normalization of $U=1$ corresponding to $T_d=\\rm 18~K$ \\citep{DRAINE14}. This approach adds several free parameters, however, since we do not have good constraints for all of them, we fix some parameters before fitting: $U_{\\rm max}$ is fixed at $10^7$ \\citep[following][]{ANIANO12}, and $\\beta$ is fixed at 2. Thus, the number of free parameters is 4, which is not a major difference from the other models.\n\n\\subsection{Fitting techniques} \\label{subsec: fitting}\n\\begin{deluxetable}{lllll}\n\\tablecaption{Grid parameters for fitting.\\label{tab: grid space}}\n\\tablecolumns{4}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Parameter} &\n\\colhead{Range} &\n\\colhead{Spacing} &\n\\colhead{Range$_c$\\tablenotemark{f}} &\n\\colhead{Spacing$_c$}}\n\\startdata\n$\\log_{10}\\Sigma_d$ & -4 to 1\\tablenotemark{a} & 0.025 & $\\pm0.2$ & 0.002\\\\\n$T_d$ & 5 to 50\\tablenotemark{b} & 0.5 & $\\pm1.5$ & 0.1\\\\\n$\\beta$ & -1.0 to 4.0\\tablenotemark{c} & 0.1 & $\\pm0.3$ & 0.02\\\\\n$\\lambda_c$ & 300\\tablenotemark{d} & N\/A & 300 & N\/A\\\\\n$\\beta_2$ & -1.0 to 4.0 & 0.25 & $\\pm0.3$ & 0.02\\\\\n$f_W$ & 0.0 to 0.05 & 0.002 & $\\pm0.006\\tablenotemark{g}$ & 0.0005\\\\\n$\\alpha$ & 1.1 to 3.0 & 0.1 & $\\pm0.3$ & 0.01\\\\\n$\\log_{10}\\gamma$ & -4.0 to 0.0 & 0.2 & $\\pm0.3$ & 0.1\\\\\n$\\log_{10} U_{\\rm min}$ & -2.0 to 1.5\\tablenotemark{e} & 0.1 & $\\pm0.1$ & 0.01\\\\\n$\\log_{10} U_{\\rm max}$ & 7 & N\/A & 7 & N\/A \\\\\n\\enddata\n\\tablecomments{(a) $\\Sigma_d$ in $M_\\sun ~{\\rm pc}^{-2}$. (b) In K. (c) For SE only. All the others are fixed at $\\beta=2$. (d) In \\micron. (e) $9.3\\leq T_d \\leq 35.6~\\rm K$ under our conversion. (f) Range for second iteration during calibration. (g) While none negative.}\n\\end{deluxetable}\nWe follow the fitting techniques in \\citet{GORDON14}: we build model SEDs on discrete grids in parameter space, and then calculate the likelihood for all models given the SED in each binned region. The multi-dimensional (3 dimensional for SE, BE and WD methods, 2 for FB and 4 for PL) grids have axes defined in \\secref{Sec: Model}, and grid spacing defined in Table \\ref{tab: grid space}.\n\nFor each grid point, we can generate a model SED $M_{ij...d}(\\nu)$, where the subscript represents a unique combination of parameters in the grid with $d$ dimensions. The calculated model is a continuous function of frequency $\\nu$. To compare with the real observation, we integrated $M_{ij...d}(\\nu)$ over the response function $R^n(\\nu)$ of each band $n$ in PACS and SPIRE with the following integral:\n\\begin{equation}\n\\overline{M^n_{ij...d}} = \\frac{\\int^\\infty_0 R^n(\\nu)M_{ij...d}(\\nu)d\\nu}{\\int^\\infty_0 R^n(\\nu)(\\nu_n\/\\nu)d\\nu}\n\\end{equation}\nNote that the denominator is added to account for the fact that \\textit{Herschel} intensities are quoted assuming a spectrum with $S(\\nu) \\propto \\nu^{-1}$ within the response function. The $\\nu_n$ values are the frequencies corresponding to the representative wavelength at each band, that is, 100, 160, 250, 350, and 500 \\micron.\n\nNext, in each binned region, we calculate the relative likelihood ($\\mathcal{L}$) of the model SED ($\\overline{M_{ij...d}}$) given the observed SED ($I_{\\rm obs}$) assuming Gaussian errors\\footnote{See \\citet{GORDON14} for discussion about statistical advantages of this matrix form definition}, that is:\n\\begin{equation}\\label{eq: likelihood}\n\\mathcal{L}(\\overline{M_{ij...d}}|I_{\\rm obs}) = \\exp\\big( -\\frac{1}{2} \\chi^2_{ij...d} \\big),\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq: chi2}\n\\chi_{ij...d}^2 \\equiv  (\\overline{M_{ij...d}}-I_{\\rm obs})^T \\mathcal{C}^{-1} (\\overline{M_{ij...d}}-I_{\\rm obs})\n\\end{equation}\nand\n\\begin{equation}\\label{eq: covariance_sum}\n\\mathcal{C} = \\mathcal{C}_{\\rm bkg} + \\mathcal{C}_{\\rm cal}.\n\\end{equation}\nThe $^T$ sign represents the transpose matrix, and $^{-1}$ sign represents the inverse matrix. $\\mathcal{C}_{\\rm bkg}$ is the background covariance matrix discussed in \\secref{subsec: background error} with values:\n\\begin{eqnarray}\n\\mathcal{C}_{\\rm bkg} = \n \\left[\n\\arraycolsep=3.0pt\n\\begin{array}{ccccc}\n    1.548 & 0.09 & 0.057 & 0.025 & 0.01 \\\\\n    0.09 & 0.765 & 0.116 & 0.079 & 0.04 \\\\\n    0.057 & 0.116 & 0.098 & 0.071 & 0.037 \\\\\n    0.025 & 0.079 & 0.071 & 0.063 & 0.033 \\\\\n    0.01 & 0.04 & 0.037 & 0.033 & 0.028 \\\\\n\\end{array}\\right].\n\\end{eqnarray}\nAs described in \\secref{subsec: voronoi}, $\\mathcal{C}_{\\rm bkg}$ will be lower for resolution elements binned together. For a binned region with a number of pixels greater than one resolution element (5.2 pixels, see \\secref{subsec: alignment}), $\\mathcal{C}_{\\rm bkg}$ is divided by number of resolution elements in the region.\n\n$\\mathcal{C}_{\\rm cal} = I^T \\mathcal{M}_{\\rm fit} I$ is the covariance matrix generated from calibration error, where $\\mathcal{M}_{\\rm fit}$ is the percentage calibration errors and $I$ is the observed SED at the binned region. There are two kinds of errors from calibration. The first one is absolute calibration uncertainty, estimated from the systematic uncertainty by comparing the calibrator to model \\citep{BENDO17}. We assume this absolute calibration uncertainty will affect all the bands calibrated together at the same time, thus we will fill this uncertainty both in the diagonal terms and the band-to-band off diagonal terms in $\\mathcal{M}_{\\rm fit}$. The second one is the relative uncertainty, or random uncertainty, which is estimated from the ability of an instrument to reproduce the same measurement \\citep{BENDO17}. We assume this noise is band-independent thus we only put it in diagonal terms in $\\mathcal{M}_{\\rm fit}$.\n\nAmong the \\textit{Herschel} observations, the SPIRE instruments were calibrated with Neptune, and were estimated to have 4\\% absolute calibration and 1.5\\% relative calibration uncertainty. The PACS instruments were calibrated with 5 stars. and the result gave a 5\\% absolute uncertainty and 2\\% relative uncertainty \\citep{PACS13, BALOG14}. In the diagonal terms in $\\mathcal{M}_{\\rm fit}$, where we need to consider both kinds of uncertainties, it is recommended that we should take the direct sum of the two errors instead of quadratic sum \\citep{BALOG14, BENDO17}. Since our object is an extended source, we must also take the uncertainty in the beam shape into account when calculating calibration errors \\citep{BENDO17}. It is recommended that we double the absolute uncertainties for this \\citep{GORDON14}. The final $\\mathcal{M}_{\\rm fit}$ is:\n\\begin{eqnarray}\\label{Eq: Mcal}\n\\mathcal{M}_{\\rm fit} = \n \\left[\n\\arraycolsep=3.0pt\n\\begin{array}{ccccc}\n    0.12^2 & 0.1^2 & 0 & 0  & 0 \\\\\n    0.1^2 & 0.12^2 & 0 & 0  & 0 \\\\\n    0 & 0 & 0.095^2 & 0.08^2 & 0.08^2 \\\\\n    0 & 0 & 0.08^2 & 0.095^2 & 0.08^2 \\\\\n    0 & 0 & 0.08^2 & 0.08^2 & 0.095^2 \\\\\n\\end{array}\\right].\n\\end{eqnarray}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{_Model_5555_NGC5457_z_merged.pdf}\n\\caption{An example of observed SED versus fitted SED from a single binned region. Red: The observed SED and error used in the fit. The error bars only include the square root of diagonal terms from the complete covariance matrix $\\mathcal{C}$. Green dot: The SED convolved with response function. Orange dashed line: The model SED generated from expectation values in the fit. Gray lines: Some selected models with transparency proportional to $\\mathcal{L}$. For each method, we randomly select 50 models from the subset $\\mathcal{L}(M^n_{ij...d}|I^n) \\geq max\\big(\\mathcal{L}(M^n_{ij...d}|I^n)\\big)\/1000$ for plotting. Note that both WD and PL methods allow FB components with peak wavelength below 100 \\micron\\, where we do not include observational constraint in this study. Therefore, the unusual shape in SED at short wavelength will not affect the fitting qualities of those models. However, we can still get similar expectation values in $\\Sigma_d$ from these methods.\\label{fig: example_model_merged}}\n\\end{figure*}\nWith the relative likelihood $\\mathcal{L}(\\overline{M_{ij...d}}|I_{\\rm obs})$ calculated, we can construct the full probability distribution function (PDF) for each parameter by summing over all other dimensions in parameter space. For example, if the index $i$ corresponds to $\\Sigma_d$, then the PDF of $\\Sigma_d$ with observed $I^n$ would be $P_{\\Sigma_{d,i}} = \\sum_{j...d} \\mathcal{L}(\\overline{M_{ij...d}}|I_{\\rm obs})$. We can then calculate the expectation value\\footnote{When calculating the expectation values, we use logarithmic scales for variables with logarithmic spacing in the grid.}, and the probability weighted 16\\% and 84\\% values, which represent the 1-$\\sigma$ confidence interval and are sampled to represent the uncertainty of the fit. An example of observed SED versus fitted models with all methods is shown in Figure \\ref{fig: example_model_merged}. An example of the log-scale likelihood distribution and correlation between fitting parameters is shown in Figure \\ref{fig: example_model_corner}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{_Corner_5555_NGC5457_BE.pdf}\n\\caption{Likelihood distribution in the parameter space from results of BE method at the same binned region in Figure \\ref{fig: example_model_merged}. Both the histograms and 2-dimensional histograms are shown in log scale. The figure does not include the whole parameter space. It is magnified to emphasize the region with $\\chi^2 \\leq \\big(min(\\chi^2) + 6\\big)$.\\label{fig: example_model_corner}}\n\\end{figure}\n\n\\subsubsection{Calibrating $\\kappa_{160}$ \\label{Sec: calibration}}\nWe use the procedure and integrated dust SED of the MW diffuse ISM from \\citet{GORDON14} to calibrate $\\kappa_{160}$ in our models. The SED was originally measured with Cosmic Background Explorer (COBE), where the $\\lambda \\geq 127~\\micron$ measurements are from Far Infrared Absolute Spectrophotometer (FIRAS) and the 100 \\micron\\ measurement is from Diffuse Infrared Background Experiment (DIRBE). The resulting SED is 0.6887, 1.4841, 1.0476, 0.5432, and 0.2425 ${\\rm MJy~sr^{-1}~(10^{20}~H~atom)^{-1}}$ for the 100, 160, 250, 350, and 500 \\micron\\ bands. These values differ from those given by \\citet{GORDON14} because we include a factor of 0.97 for the molecular cloud correction \\citep{COMPIEGNE11}. The ionized gas factor in \\citet{COMPIEGNE11} is excluded because we do not include ionized gas through out this study, including the calculation of average DGR in the MW diffuse ISM \\citep{JENKINS09, GORDON14}. The dust-to-Hydrogen mass ratio appropriate for this high-latitude diffuse region is calculated by averaging the depletion strength factor $\\rm F_\\star$ value over sightlines in \\citet{JENKINS09} with similar hydrogen column densities as the observed region. The resulting $\\rm F_\\star$ is 0.36, and the dust-to-Hydrogen mass ratio is $1\/150$, which corresponds to a dust surface density to H column density ratio of $5.30\\times10^{-3}~{\\rm M_\\sun~ pc^{-2}}~(10^{20}~{\\rm H~atom})^{-1}$.\n\nDuring calibration, it is important to use the same models and fitting methods as the real fitting \\citep{GORDON14}. We follow the same steps of our fitting techniques except four necessary differences: 1) We replace the original $\\mathcal{M}_{\\rm fit}$ with $\\mathcal{M}_{\\rm cali}$ (Eq. \\ref{eq: M_cal cali}) for calibration since the calibration data came from COBE instead of \\textit{Herschel}. Following \\citet{FIXSEN97}, we assume 0.5\\% relative uncertainty and 2\\% absolute uncertainty for FIRAS (calibrating PACS160 and SPIRE bands), and 1\\% relative uncertainty and 10\\% absolute uncertainty for DIRBE (calibrating PACS100 \\micron).\n\\begin{eqnarray}\\label{eq: M_cal cali}\n\\mathcal{M}_{\\rm cali} = \n \\left[\n\\arraycolsep=3.0pt\n\\begin{array}{ccccc}\n    0.11^2 & 0 & 0 & 0  & 0 \\\\\n    0 & \\cdf & 0.02^2 & 0.02^2 & 0.02^2 \\\\\n    0 & 0.02^2 & \\cdf & 0.02^2 & 0.02^2 \\\\\n    0 & 0.02^2 & 0.02^2 & \\cdf & 0.02^2 \\\\\n    0 & 0.02^2 & 0.02^2 & 0.02^2 & \\cdf \\\\\n\\end{array}\\right]\n\\end{eqnarray}\n2) No $\\mathcal{C}_{\\rm bkg}$ term is applied. $\\mathcal{C}_{\\rm cal}$ is the only variance term considered. 3) Due to the small uncertainty of COBE data, the normal parameter spacing is not finely-sampled enough to resolve the PDF for all the parameters. Thus, we use a two-step calibration: first, we fit with the normal parameter space; then reduce the parameter range to a smaller region near the peak with a finer spacing (see ``Range$_c$'' and ``Spacing$_c$'' columns in Table \\ref{tab: grid space}); last, we fit with this new parameter spacing and report the results. 4) Our SED per hydrogen atom of the MW diffuse ISM is weaker than the one in \\citet{GORDON14} by a factor of 0.97 due to the molecular cloud fraction.\n\n\\begin{deluxetable*}{llcc}\n\\tablecaption{Results of calibrating emissivity to the MW high latitude SED.\\label{tab: cali result}}\n\\tablecolumns{4}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Model} &\n\\colhead{$\\kappa_{160}~({\\rm cm^2~g^{-1}})$} &\n\\colhead{Other parameters} &\n\\colhead{Expectation values}}\n\\startdata\nSE & $10.10\\pm1.42$ & ($T_d$, $\\beta$) & ($20.90\\pm0.62$~K, $1.44\\pm0.08$) \\\\\nFB & $25.83\\pm0.86$ & ($T_d$) & ($17.13\\pm0.12$~K) \\\\\nBE & $20.73\\pm0.97$ & ($T_d$, $\\beta_2$) & ($18.02\\pm0.18$~K, $1.55\\pm0.06$) \\\\\nWD & $27.46\\pm1.14$ & ($T_d$, $f_W$) & ($16.60\\pm0.25$~K, $0.00343\\pm0.00143$) \\\\\nPL & $26.60\\pm0.98$ & ($\\alpha$, $\\log_{10}\\gamma$, $\\log_{10} U_{\\rm min}$) & ($1.69\\pm0.19$, $-1.84\\pm0.21$, $-0.16\\pm0.03$)\\\\\n\\enddata\n\\end{deluxetable*}\nThe calibrated $\\kappa_{160}$ values range from 10.48 to 21.16$~\\rm cm^2~g^{-1}$, see complete results in Table \\ref{tab: cali result}. This is a fairly large range, which indicates that the choice of model does affect the measurement of dust properties. Our results are comparable with calculated $\\kappa_{160}$ values in literature, e.g., the widely used \\citet{DRAINE07} model, with updates in \\citet{DRAINE14}, gives $\\kappa_{160}$ equal to 13.11$~\\rm cm^2~g^{-1}$ for silicates and 10.69$~\\rm cm^2~g^{-1}$ for carbonaceous grains, and 12.51$~\\rm cm^2~g^{-1}$ in the combined model. The standard model in \\citet{GALLIANO11} gives a value of 14$~\\rm cm^2~g^{-1}$, and 16$~\\rm cm^2~g^{-1}$ after replacing graphite with amorphous carbons. A recent calculation by \\citet{RELANO18}, following the \\citet{DESERT90} dust model, gives an equivalent $\\kappa_{160}=22.97~\\rm cm^2~g^{-1}$.\n\nIn the MBB model calibration process in \\citet{GORDON14} and \\citet{GORDON17}, the resulting $\\kappa_{160}$ falls between 30.2 and 36.4$~\\rm cm^2~g^{-1}$, depending on the model used. The common model between us is the SMBB in \\citet{GORDON14}, where they have $\\kappa_{160}=30.2~\\rm cm^2~g^{-1}$, and our SE, where we have $\\kappa_{160}=10.1~\\rm cm^2~g^{-1}$. Our calibration method differs from \\citet{GORDON14} in four ways: 1) With the values of COBE uncertainty we quote, we are allowed to have more deviation at 100 \\micron\\ than the other bands. On the other hand, \\citet{GORDON14} have both correlated and uncorrelated uncertainty values uniform for all bands. 2) We use a $M_{\\rm cali}$ which assumes 100 \\micron\\ calibration independent of the other bands since DIRBE and FIRAS were calibrated independently. \\citet{GORDON14} assumed that all bands are correlated with the same absolute uncertainties. 3) We use a two-step fitting to increase the accuracy only for calibration, while \\citet{GORDON14} used exactly the same methods for calibration and fitting. 4) Our SED per hydrogen atom of the MW diffuse ISM is weaker by a factor of 0.97 due to the molecular cloud fraction. In Section~\\ref{sec:sensitivity} we discuss the sensitivity of the results to choices in the SED fitting and calibration in more detail.\n\n\\section{Results}\\label{sec: results}\nWe fit the SEDs from all binned regions with all five MBB variants introduced in \\secref{Sec: Model}. We calculate the DGR in each bin from the observed $\\Sigma_{\\rm gas}$ and the fitting results of $\\Sigma_d$. Here, we look at the DGR and dust temperature radial gradients for each model, and at the residuals and reduced chi-square values about the best fit. Doing so, we will be interested in which models meet our physically motivated expectations and which models provide good fits to the SED. The complete fitting results are shown in Appendix \\ref{app: fitting}, along with their correlations in Appendix \\ref{app: corner}.\n\n\\subsection{DGR-metallicity relation\\label{sec: max DGR}}\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{_DGR-vs-Metallicity_NGC5457_z_merged.pdf}\n\\caption{DGR expectation values versus radius and metallicity. The shaded regions show the intrinsic scatter of DGR from $\\Sigma_d$ fitting results and the zero-point fluctuation of $\\Sigma_{\\rm gas}$ (\\secref{subsec: HI}). MAX is the maximum possible DGR calculated as a function of metallicity. The range is set by the difference between \\citet{LODDERS03} and \\citet{ASPLUND09} chemical composition, which is small at this plotting scale.\\label{fig: DGR_grad_models_merged}}\n\\end{figure}\nIn Figure \\ref{fig: DGR_grad_models_merged}, we plot the DGR-metallicity relation from all fitting methods. The metallicity-radius relation is calculated with Eq. 10 in \\citet{CROXALL16}. We first separate M101 into 20 radial regions, and, at each region with $r_i\\leq r < r_j$, we take the sum of the expectation value of dust mass divided by the total gas mass as the expectation value of DGR (${\\rm < DGR >}$) in that region, that is:\n\\begin{equation}\n{\\rm <DGR>}_{ij} = \\frac{\\sum_{r_i\\leq r_k < r_j}<\\Sigma_d>_k A_k}{\\sum_{r_i\\leq r_k < r_j}M_{\\rm gas,k}},\n\\end{equation}\nwhere $<\\Sigma_d>_k$ and $A_k$ are the expectation value of $\\Sigma_d$ and area at the k-th binned region, respectively. We estimate the uncertainties of these expectation values of DGR with the ``realize'' method \\citep{GORDON14}, and the uncertainties are $\\sim$ 0.02 dex in the high metallicity region, $\\sim$ 0.09 dex at $\\rm 12+\\logt({\\rm O\/H})\\sim8.2$, and $\\sim$ 0.6 dex in the lowest metallicity region, which are reasonably small. However, there is also intrinsic scatter of DGR in each radial region, which would be larger than the uncertainties. To estimate this intrinsic scatter of DGR per $M_{\\rm gas}$ within one radial region, we calculate the distribution by summing up the PDFs of DGR from each bin in that radial region, weighted by their $M_{\\rm gas}$. Next, we take the region between the 16th and 84th percentile of the distribution as the range of the intrinsic scatter. This intrinsic scatter is included in Figure \\ref{fig: DGR_grad_models_merged}, along with the zero-point uncertainty in $\\Sigma_{\\rm gas}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{_DTM_C18_NGC5457_BE.pdf}\n\\caption{Our DGR and DTM versus metallicity from the BE method results. (a) The DGR expectation values fitted by the BE model from each binned region are shown with error bars. Shaded region: The scatter of DGR. The definition is described in Figure \\ref{fig: DGR_grad_models_merged}. (b) The DGR from the BE model with power-law (${\\rm DGR} \\propto Z^x$) fitting as listed in Table \\ref{tab: D2M}. Blue: The expectation values calculated from the combined PDF (same for figures in \\secref{sec: discussions}). Orange: The power-law result with whole data range. Green: Fitting with only $\\rm 12+\\logt({\\rm O\/H}) > 8.2$, where we have a more concentrated data point distribution. (c) The DTM from the BE model. The DTM scatter includes DGR scatter, the $\\rm 12+\\logt({\\rm O\/H})$ uncertainty \\citep{CROXALL16}, and $M_{\\rm O}\/M_{\\rm Z}$ uncertainty (\\secref{sec: MOMZ unc}). The horizontal lines are the DTM=0.1, 0.2, ......1.0 locations.\\label{fig: DGR_grad_models_BE}}\n\\end{figure}\nThe distribution of our original data points is denser in the region with $\\rm 12+\\logt({\\rm O\/H}) \\ga 8.2$, where the original SNR is high. This is illustrated in Figure \\ref{fig: DGR_grad_models_BE} (a) with the results from the BE model. Within this range, all models except SE have their DGR dropping by nearly 1 dex, which is around twice faster than the metallicity gradient. The SE has its DGR dropping by 1.5 dex. At $\\rm 12+\\logt({\\rm O\/H}) < 8.0$, the scatter in PDF is large (generally with $\\sigma \\ga 1~\\rm dex$), which makes determining a trend difficult. By treating metallicity as an independent variable, we fit our DGR versus metallicity with a linear equation ${\\log_{10} \\rm DGR} = a\\times(\\rm 12+\\logt({\\rm O\/H})) +b$ in both the full metallicity range and only $\\rm 12+\\logt({\\rm O\/H}) \\geq 8.2$ region. An example showing results from the BE model is shown in Figure \\ref{fig: DGR_grad_models_BE} (b). The results are listed in Table \\ref{tab: D2M}. All the fitting results indicate a $\\log_{10}{\\rm DGR}$ variation steeper than $\\rm 12+\\logt({\\rm O\/H})$. The three methods with $\\beta$ fixed over the whole spectral range, FB, WD, and PL, have fitted slopes closer to one.\n\\begin{table}\n    \\centering\n    \\caption{$\\log_{10}$DGR versus $\\rm 12+\\logt({\\rm O\/H})$ linear fitting results.\\label{tab: D2M}}\n    \\begin{tabular}{ccccc}\n        \\hline\n        \\hline\n        Model & \\multicolumn{2}{c}{Full range} & \\multicolumn{2}{c}{$\\rm 12+\\logt({\\rm O\/H})\\geq 8.2$} \\\\\n        & a & b & a & b \\\\\n        \\hline\n        SE & $2.7\\pm0.3$ & $-25.3\\pm 2.1$ & $3.2\\pm0.2$ & $-29.4\\pm 1.8$ \\\\\n        FB & $1.5\\pm0.1$ & $-14.9\\pm0.9$ & $1.5\\pm0.1$ & $-15.3\\pm 0.7$ \\\\\n        BE & $1.7\\pm0.1$ & $-16.9\\pm 1.0$ & $1.9\\pm 0.1$ & $-18.1\\pm 0.7$ \\\\\n        WD & $1.5\\pm0.2$ & $-14.9\\pm1.4$ & $1.3\\pm0.1$ & $-13.1\\pm0.5$ \\\\\n        PL & $1.3\\pm0.1$ & $-13.3\\pm 0.9$ & $1.2\\pm0.1$ & $-12.8\\pm0.5$ \\\\\n        \\hline\n    \\end{tabular}\n    \\raggedright Note: Data are fitted with ${\\log_{10} \\rm DGR} = a\\times(\\rm 12+\\logt({\\rm O\/H})) +b$.\n\\end{table}\n\n\\subsubsection{Physical limitations to DGR}\\label{sec: MOMZ unc}\nDust grains are built from metals. Thus, we can calculate the theoretical upper limit to the DGR by calculating the DGR for the case when all available metals are in dust. If the fitted DGR exceeds the calculated upper limit, we would consider the fitting result physically less plausible. To convert to total metallicity from oxygen abundance, we need to assume the ISM chemical composition. We calculate the mass ratio of oxygen to total metal from two literature of solar chemical composition: 1) \\citet{LODDERS03}, which gives $M_{\\rm O}\/M_Z = 51\\%$ where $M_Z$ is the mass of all metals. This is the composition used in \\citet{JENKINS09}, which we will discuss in \\secref{sec: J09}. 2) A later version in \\citet{ASPLUND09}, which gives $M_{\\rm O}\/M_Z = 44.5\\%$. The conversion from $\\rm 12+\\logt({\\rm O\/H})$ to metallicity is given by:\n\\begin{equation}\\label{eq: DGR_limit}\n    \\frac{M_Z}{M_{\\rm gas}} = \\frac{M_Z}{M_{\\rm O}}\\frac{M_{\\rm O}}{1.36M_{\\rm H}} = \\frac{\\frac{m_{\\rm O}}{m_{\\rm H}}10^{\\big(\\rm 12+\\logt({\\rm O\/H})\\big)-12}}{\\frac{M_{\\rm O}}{M_Z} \\times 1.36},\n\\end{equation}\nwhere $m_{\\rm O}$ and $m_{\\rm H}$ are the atomic weights of oxygen and hydrogen. The solar $\\rm 12+\\logt({\\rm O\/H})$ adopted in this study is $8.69\\pm0.05$ \\citep{ASPLUND09}. This estimation of the DGR upper limit can be incorrect if the actual chemical composition deviates from this range. For example, \\citet{CROXALL16} showed there is a trend that $\\log_{10}({\\rm N\/O})$ goes from $-0.4$ to $-1.4$ as radius increases in M101, which means we can overestimate the upper limit in the outer disk if other major elements have similar trends.\n\nWe overlay the DGR upper limit calculated between $M_{\\rm O}\/M_Z = 44.5\\%$ and $51\\%$ with our results in Figure \\ref{fig: DGR_grad_models_merged}. We find that in the highest metallicity region, the DGR given by SE method is greater than the upper limit by a factor of 3, which is outside the 16-84 percentile of intrinsic scatter. This is unlikely being a result of $\\alpha_{\\rm CO}$ variation because we will need to have $\\alpha_{\\rm CO}\\sim 9$ in the center of M101 to explain this apparent DGR. This $\\alpha_{\\rm CO}$ value is unlikely to be true with our knowledge of $\\alpha_{\\rm CO}$ in M101 \\citep{SANDSTROM13} and metallicity-dependency of $\\alpha_{\\rm CO}$ \\citep{BOLATTO13}. We thus consider the results from SE method less physically plausible.\n\nWe also notice that for all methods listed, there is a DGR spike in expectation value exceeding the upper limit near $\\rm 12+\\logt({\\rm O\/H}) \\sim 7.9$. Nevertheless, all the others still have their 16-84 percentile scatter falling under the DGR upper limit. Thus, we consider all methods except SE still reasonable under DGR upper limit test. Note that the scatter in the regions with $\\rm 12+\\logt({\\rm O\/H}) < 8.2$ reach the order of 1 dex, which means the fit values are less reliable.\n\n\\subsection{Temperature profiles}\\label{sec: T grad}\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{_T-profile_NGC5457_z_merged.pdf}\n\\caption{Radial profiles of dust temperature, $\\Sigma_{\\rm SFR}$ and $\\Sigma_\\star$. All the profiles are plotted in gas-mass-weighted average. Top panel: temperature profiles from all fitting methods. 16-84 percentile scatter from the fitting is shown in shaded areas. Bottom panel: $\\Sigma_{\\rm SFR}$ and $\\Sigma_\\star$ profiles. See \\secref{subsec: other} for data source and calculation. A 10\\% uncertainty is plotted in shaded region, which is an uncertainty suggested in \\citet{DALE09}.\\label{fig: T_prof_new}}\n\\end{figure}\nIn the top panel of Figure \\ref{fig: T_prof_new}, we plot the $M_{\\rm gas}$-weighted dust temperature as a function of radius for each method. Within a small radial range, we assume that the DGR variation is small, thus the $M_{\\rm gas}$-weighted dust temperature would be a representative $T_d$ in the corresponding radial region. For the PL method, temperature is not a directly fitted variable. Thus, we calculate the dust mass-weighted average $U$, and convert it to temperature according to \\secref{sec: PL}.\n\nThe equilibrium dust temperature depends on the heating radiation field, which should be related to a combination of $\\Sigma_\\star$ and $\\Sigma_{\\rm SFR}$ here, shown in bottom panel of Figure \\ref{fig: T_prof_new}. By comparing to the radial trend of heating sources, the one model that stands out is SE: it has a temperature profile rising from the galaxy center to $0.8R_{25}$. It is possible to change the relationship between heating sources and dust temperature if the geometry and\/or the opacity of the ISM changes with radius. However, with both heating source tracers having intensity decreasing by more than one dex within $0.8R_{25}$, we expect that a decreasing $T_d$ with radius to be the dominant trend. Thus, we also reach the conclusion as previous section that results from SE method are less physically plausible.\n\n\\subsection{Residual distributions}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{_Residual_NGC5457_merged.pdf}\n\\caption{2-dimensional histograms of relative residual versus observed SED at each band. The x-axes have unit in $\\rm MJy~sr^{-1}$. The zero relative residual line is marked in gray.\\label{fig: Residual_merged}}\n\\end{figure*}\nThe residual distribution is one of the most straightforward ways to check the goodness of fit. For each method, we plotted the 2-dimensional histogram of relative surface brightness residuals in Figure \\ref{fig: Residual_merged}. We expect that a good fit will give a residual distribution that is symmetric about zero (The gray lines in all panels in Figure \\ref{fig: Residual_merged}) and has no trend with the measured surface brightness. An example of well-behaved residual distribution can be seen for the BE model at the SPIRE 250 band. Otherwise, there may be a underlying systematic effect which tells us that the model is flawed or an additional free parameter is needed.\n\nThere are two features occurring for all MBB methods: 1) At the high intensity end, all of our methods underestimate PACS160. 2) In general, the relative residuals are smaller at the low intensity end (see more discussion in \\secref{sec: compare-chi2}). The SE method gives the most compact residual distributions. This means that letting both $T_d$ and $\\beta$ free provides the highest flexibility to fit the SED among all models here. However, we should bear in mind that the SE model yields DGR and temperature gradients distinct from the other models and that we consider these results less physically plausible, as previously shown in \\secref{sec: max DGR} and \\ref{sec: T grad}. The FB method yields the residual distribution least consistent with random scatter about the model. It shows the least compact residual distribution with long tails in positive residuals, especially in PACS100, SPIRE350 and SPIRE500. These positive residuals mainly come from low intensity regions. These indicate the need for $\\beta$ to change between high and low intensity regions. Among the remaining methods, both the WD and PL improve the residuals the short wavelengths covered by PACS100. This reflects the expected presence of warm, possibly out-of-equilibrium dust at these short wavelengths. The BE method has the second most compact residual distribution, and shows a better fit to the long wavelength bands that are crucial to accurately trace $\\Sigma_d$.\n\n\\subsection{The reduced chi-square values\\label{sec: compare-chi2}}\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{_Residual_Chi2_NGC5457.pdf}\n\\caption{The $\\tilde{\\chi}^2$ distributions for all fitting methods. Left: 2-dimensional histograms of $\\tilde{\\chi}^2$ with PACS100. The x-axes have unit in $\\rm MJy~sr^{-1}$. Note that the 2-dimensional histograms of $\\tilde{\\chi}^2$ with all five bands demonstrate similar information, thus we only plot the ones from PACS100. Right: the horizontal histograms of $\\tilde{\\chi}^2$. The orange lines show the expected distribution according to DoF.\\label{fig: residuals_chi2}}\n\\end{figure}\nThe reduced chi-square value is defined as $\\tilde{\\chi}^2 \\equiv \\chi^2\/(n - m)$, where $n$ is the number of observations (which is $5$ in our study) and $m$ is the number of fitting parameters (3 for SE, 2 for FB, 3 for BE, 3 for WD and 4 for PL). This value takes both uncertainties in the observations and the degrees of freedom (DoF) of the models into account. The $\\tilde{\\chi}^2$ value gives the information of how good the fitting is and how much an extra fitting parameter improves the fitting quality. We plot the $\\tilde{\\chi}^2$ distribution versus observation in the left panels in Figure \\ref{fig: residuals_chi2}. As we have seen in residual maps, the FB and WD methods have long tails in the low luminosity region. FB and WD methods have $\\tilde{\\chi}^2 \\geq 1$ in the high luminosity region, mainly due to the residuals in long wavelength, where the corresponding uncertainties are much smaller. The PL method has relatively large $\\tilde{\\chi}^2$ everywhere, which means the extra DoF does not offer an improvement in the quality of the fitting. Note that this result does not imply the physical correctness of single temperature over ISRF distribution, but indicates that the DoF from ISRF distribution is less effective in improving the quality of FIR SED fitting.\n\nAll the methods have a gradually rising $\\tilde{\\chi}^2$ toward the high luminosity region. By calculating the contribution to $\\tilde{\\chi}^2$ from each band, the most important contributor to this phenomenon is the PACS160 band. There is in general a $\\sim$20\\% systematic underestimation by the model fits in PACS160 in the center of M101. One possible explanation is that the contribution from \\textsc{[Cii]} 158 \\micron\\ line is integrated into the PACS160 SED, which makes the PACS160 SED brighter than what is predicted by dust emission models. This effect is shown to be minor by \\citet{GALAMETZ14}, where the authors demonstrated that \\textsc{[Cii]} contributes only around 0.4\\% to integrated 160~\\micron\\ emission. Another possible explanation is an unknown systematic uncertainty in PACS160. Previous work by \\citep{ANIANO12} found that PACS160 was $\\sim$20\\% higher than {\\em Spitzer} MIPS160 measurements in the bright regions of some nearby galaxies.\n\nWe also examine the histograms of $\\tilde{\\chi}^2$ (Figure \\ref{fig: residuals_chi2} right panels) with two features: 1) The mean value, which is expected to be one. 2) The shape of the histogram, which should resemble the $\\chi^2$-distribution with $k$ DoF\\footnote{We normalized the $\\chi^2$-distribution to a mean value of one, i.e., $k\\times f(k\\tilde{\\chi}^2, k)$.}. The SE method has mean $\\tilde{\\chi}^2$ of 0.77. The histogram is more compact than a $\\chi^2$-distribution with $k=2$. Both indicate that we might be overestimating the uncertainties in the SE method. FB and WD have mean values of 1.5 and 1.64, respectively, and flatter histograms than expected. BE has a mean value of 0.97 and a distribution resembling what we expected. PL has a mean value of 3.16, which means the extra parameters in the PL model do not help it making a more precise fit corresponding to its DoF.\n\n\\subsection{Summary of model comparison}\nAmong the MBB variants we have tested, we consider the SE method physically less plausible because the resulting temperature and DGR gradient do not match our physically-motivated expectations. The DGR results from the other four MBB variants are consistent with each other in regions with $\\rm 12+\\logt({\\rm O\/H}) \\leq 8.5$, as illustrated in Figure \\ref{fig: DGR_grad_models_merged}. This implies that the dust masses measure from the MBB fitting is mostly insensitive to the specific choices about the radiation field distribution. According to the residual distribution and $\\tilde{\\chi}^2$ values, the BE model gives the statistical best fit, which means that the most important first-order correction to the basic MBB is to allow $\\beta$ vary in the long wavelength region. We will consider BE as the preferred model based on these tests.\n\n\\section{Discussion}\\label{sec: discussions}\n\\subsection{Is DTM Constant in M101?}\\label{sec: dicuss_variable_DTM}\nAll of our models indicate that DGR falls off steeper than metallicity, showing a variable DTM ratio. Our preferred model (BE) has ${\\rm DGR} \\propto Z^{1.7}$, which is equivalent to DTM changing from 0.25 at $\\rm 12+\\logt({\\rm O\/H})\\sim$7.8 to 1 above $\\rm 12+\\logt({\\rm O\/H})\\sim$8.5. Models with $\\beta$ fixed have smaller power-law indices, specifically the FB and WD models show ${\\rm DGR} \\propto Z^{1.4}$, and PL model shows ${\\rm DGR} \\propto Z^{1.2}$. Even if we only consider region with $\\rm 12+\\logt({\\rm O\/H}) \\geq 8.2$, where the majority of our data points reside, we still obtain a DGR trend steeper than metallicity gradient. These results are based on direct-$T_e$ method metallicity measurements \\citep{CROXALL16} with uncertainties in $\\rm 12+\\logt({\\rm O\/H})$ around $0.04-0.08$ dex.\n\nIn order to understand what aspects of the dust life cycle could result in a variable DTM, we look for mechanisms that affect dust mass and metals in the ISM with different rates. The five most important mechanisms of this kind are: 1) Accretion of metals in the ISM onto existing dust grains, which raises DTM. 2) ISM enrichment from stellar sources (e.g. AGB stars, SNe), which have DTM characteristic of the particular stellar source instead of DTM in the current ISM. 3) Dust destruction by SNe, which lowers DTM. 4) Infall of circumgalactic medium (CGM) into the galaxy, which dilutes the ISM DTM with the lower DTM in the CGM \\citep{DWEK98, HIRASHITA99, ZHUKOVSKA16}. 5) Outflows of dust and metals into CGM, which increases the ISM DTM because the outflow is less dusty than the ISM \\citep{LISENFELD98}.\n\nAmong these mechanisms, ISM accretion has a rate that increases with ISM density, especially in cold clouds \\citep{DWEK98, ASANO13}. Observationally, ISM density can be roughly traced by the mass fraction of molecular hydrogen ($\\rm f_{H_2}$)\\footnote{Without knowing the three-dimensional ISM geometry, $\\rm f_{H_2}$ would be a better indicator of ISM density than $\\Sigma_{\\rm gas}$.}. The rate of enrichment from stellar sources should follow the stellar mass surface density ($\\Sigma_\\star$) modulo stellar age effects. The effects of production and destruction of dust by SNe should track both the massive star formation rate ($\\Sigma_{\\rm SFR}$) and the older stellar populations ($\\Sigma_\\star$).\n\\begin{table}\n    \\centering\n    \\caption{Correlation between $\\log_{10}$DTM and physical quantities $\\log_{10}\\rm f_{H_2}$, $\\log_{10}\\Sigma_\\star$ and $\\log_{10}\\Sigma_{\\rm SFR}$.\\label{tab: Residual trend}}\n    \\begin{tabular}{ccccc}\n        \\hline\n        \\hline\n        Quantity & \\multicolumn{2}{c}{Direct} & \\multicolumn{2}{c}{Residual} \\\\\n        & $\\rho_S$ & $p$-value\\footnote{$p$-value is the probability that we get a $\\rho_S$ greater or equal to the calculated value from the given data when null hypothesis is true. In other words, $p$-value goes from 0 to 1, and a smaller $p$-value implies a more significant correlation.} & $\\rho_S$ & $p$-value \\\\\n        \\hline\n        $\\log_{10}\\rm f_{H_2}$ & 0.80 & $\\ll 1$ & 0.26 & $\\ll 1$ \\\\\n        $\\log_{10}\\Sigma_\\star$ & 0.72 & $\\ll 1$ & -0.05 & 0.12 \\\\\n        $\\log_{10}\\Sigma_{\\rm SFR}$ & 0.22 & $\\ll 1$ & -0.08 & 0.007 \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{_residual_trends_NGC5457_BE.pdf}\n\\caption{Relation between DTM and the three physical quantities: $\\rm f_{H_2}$ (a, d), $\\Sigma_\\star$ (b, e) and $\\Sigma_{\\rm SFR}$ (c, f). (a, b, c): Relations in the raw data. (d, e, f): Relations after removing the radial trends in all four quantities: $\\log_{10}$DTM, $\\log_{10}\\rm f_{H_2}$, $\\log_{10}\\Sigma_\\star$ and $\\log_{10}\\Sigma_{\\rm SFR}$. The radial trend removal is done by first fitting the quantities versus radius with linear regression, and then subtracting the regression results from the original data. The discussion of radial trend removal is described in \\secref{sec: dicuss_variable_DTM}. The mean uncertainty in $\\Sigma_d$ is 0.1 dex. $\\Sigma_\\star$ has unit in $M_{\\sun}~\\rm pc^{-2}$ and $\\Sigma_{\\rm SFR}$ has unit in $M_{\\sun}~\\rm kpc^{-2}~yr^{-1}$.\\label{fig: residual trend}}\n\\end{figure*}\nTo test these potential correlations of DTM with environmental characteristics, we calculate the Spearman's rank correlation coefficient ($\\rho_S$) and $p$-value between $\\log_{10}$DTM and these three quantities. Note that we only include the region with $\\rm f_{H_2}\\geq 5\\%$ for all four quantities, namely DTM, $\\rm f_{H_2}$, $\\Sigma_\\star$ and $\\Sigma_{\\rm SFR}$, due to the detection limit of HERACLES. $\\log_{10}$DTM correlates strongly and significantly with both $\\log_{10} \\rm f_{H_2}$ and $\\log_{10} \\Sigma_\\star$, while it shows a weaker but significant correlation with $\\log_{10}\\Sigma_{\\rm SFR}$. This is shown in the ``direct'' columns in Table \\ref{tab: Residual trend} and top panels in Figure \\ref{fig: residual trend}.\n\nWhile there are significant correlations between DTM and these environmental characteristics, all the quantities here (DTM, $\\rm f_{H_2}$, $\\Sigma_\\star$ and $\\Sigma_{\\rm SFR}$) to first order have major trends that vary with radius. $\\rm f_{H_2}$, $\\Sigma_\\star$ and $\\Sigma_{\\rm SFR}$ all have $\\rho_S$ with radius greater than the $\\rho_S$ with $\\log_{10}$DTM. $\\log_{10}$DTM also has a higher $\\rho_S$ with radius than with other quantities. The results of calculating the $\\rho_S$ and $p$-value directly will therefore be dominated by this major radial trend. In order to investigate what drives the DTM variation, we need to remove these dominant radial trends. This removal is done by first fitting $\\log_{10}$DTM, $\\log_{10} \\rm f_{H_2}$, $\\log_{10}\\Sigma_\\star$ and $\\log_{10}\\Sigma_{\\rm SFR}$ versus radius with linear regression, and then subtracting the regression results from the original data points to get the residuals. The correlations between $\\log_{10}$DTM and $\\log_{10} \\rm f_{H_2}$, $\\log_{10}\\Sigma_\\star$ and $\\log_{10}\\Sigma_{\\rm SFR}$ after radial trend removal are shown in the bottom panels in Figure \\ref{fig: residual trend} and the ``Residual'' columns in Table \\ref{tab: Residual trend}.\n\nThe resulting $\\rho_S$ between residual $\\log_{10}$DTM and residual $\\log_{10}\\rm f_{H_2}$ is 0.26, with a $p$-value $\\ll 1$. This indicates that the correlation between them is weak compared to the scatter in the data but significant. The null hypothesis, that the two variables (residual DTM and $\\rm f_{H_2}$) are unrelated, is extremely unlikely to be true. Residual $\\log_{10}\\Sigma_\\star$ and residual $\\log_{10}\\Sigma_{\\rm SFR}$, on the other hand, have their $\\rho_S$ drop relative to the direct correlation and the residual $\\rho_S$ of them show extremely weak correlations, and thus considered negligible.\n\nBased on this calculation, we suggest that ISM density may be the most important environmental factor that affects DTM in M101. This would explain the correlation between variations of DTM at a fixed radius and variations in $\\rm f_{H_2}$. The stellar sources, traced by $\\Sigma_\\star$ and $\\Sigma_{\\rm SFR}$, do not correlate significantly with the variations of DTM at a fixed radius.\n\n\\subsubsection{Variable emissivity coefficient}\\label{subsec: discuss_emissivity}\nAlthough we have thus far interpreted our results as changes in DTM, an alternative possibility is that $\\kappa_{160}$ varies with environment instead. As discussed in \\secref{Sec: Model} all of our MBB variants are subject to the degeneracy between $\\Sigma_d$ and $\\kappa_{160}$. The way we deal with it is by calibrating $\\kappa_{160}$ with the MW diffuse ISM SED (\\secref{Sec: calibration}) and assuming all the variation in temperature-corrected SED amplitude is due to $\\Sigma_d$ only. However, this assumption might fail if we observe environments that differ from the high-latitude MW diffuse ISM we used for calibration and if $\\kappa_{160}$ varies with local environment. In general, our DGR($Z$) does not follow the DGR($Z$) calculated from ${\\rm F_\\star}=0.36$, which have been used for our calibration. This leaves the possibility that the changes we see in DTM are still degenerate with the changes in $\\kappa_{160}$.\n\n$\\kappa_{160}$ can be a function of dust size, temperature, and composition, which may change as gas transitions from diffuse to dense phases. The calculations in \\citet{OSSENKOPF94, KOHLER11} show an enhanced dust emissivity due to coagulation of dust particles in dense ISM regions. This phenomenon is also observed by \\citet{PLANCK14} and \\citet{PLANCK15} in the MW, where the authors show an increase in total opacity with increasing ISM density and decreasing $T_d$. However, we note that both \\citet{PLANCK14} and \\citet{PLANCK15} \\textit{assumed} a constant DGR, and explained their observations with a change in the composition and structure of the dust particles.\n\nWe will focus on the dense regions in M101 for discussing emissivity variation with coagulation, where coagulation is more likely to happen. \nWe use the constant DTM in MW \\citep{DRAINE11} as our reference true DTM and calculate how our DTM deviates from the reference as a function of ISM density, traced by $\\rm f_{\\rm H_2}$, plotted Figure \\ref{fig: rDGR_vs_metal_BE}. Note that the figure only includes the region with significant detection from HERACLES ($f_{\\rm H_2} \\ga 5\\%$, or $\\rm 12+\\logt({\\rm O\/H}) \\ga 8.4$), not the full range of our DGR-to-metallicity figures.\n\nWe calculate the Pearson's correlation coefficient of all four combinations of log\/linear $\\frac{DTM}{DTM_{MW}}$-to-$ f_{\\rm H_2}$ relation, i.e., $\\rm \\frac{DTM}{DTM_{MW}}$-to-$f_{\\rm H_2}$, $\\rm \\frac{DTM}{DTM_{MW}}$-to-$\\log_{10} f_{\\rm H_2}$, $\\log_{10} \\rm \\frac{DTM}{DTM_{MW}}$-to-$f_{\\rm H_2}$, and $\\log_{10} \\rm \\frac{DTM}{DTM_{MW}}$-to-$\\log_{10} f_{\\rm H_2}$. The result shows 0.712, 0.790, 0.694, and 0.795, respectively. Thus we continue our analysis with $\\log_{10} \\rm \\frac{DTM}{DTM_{MW}}$-to-$\\log_{10} f_{\\rm H_2}$ relation. By fitting $\\log_{10} \\rm \\frac{DTM}{DTM_{MW}}$ to $\\log_{10} f_{\\rm H_2}$, our $\\rm \\frac{DTM}{DTM_{MW}}$ varies from 0.9 to 2.0 in this region. If we attribute this change to the increase in emissivity, then $\\kappa_{160}$ will go from 19 to $41~\\rm cm^2~g^{-1}$ in this region, with a relation of $\\kappa_{160} \\propto f_{\\rm H_2}^{0.2}$. This is comparable to the emissivity changes inferred by \\citet{PLANCK14} using similar reasoning in MW clouds and well within the range allowed by theoretical grain coagulation models \\citep{OSSENKOPF94, KOHLER11}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{_relDGR_vs_afH2_NGC5457_.pdf}\n\\caption{Our DTM normalized by the MW DTM \\citep{DRAINE11} plotted as a function of H$_2$ mass fraction ($f_{\\rm H_2}$). The original distribution is shown in blue. A representative error bar in cyan, which only include the uncertainties in DGR, is shown at top-left. Another error bar including extra uncertainty in $\\rm 12+\\logt({\\rm O\/H})$, which is considered systematic, is shown in green at top-left. The linear regression of $\\log_{10} \\rm DTM\/DTM_{MW}$-to-$\\log_{10} f_{\\rm H_2}$ is shown in red. Note that this plot only includes data with $f_{\\rm H_2} \\ga 5\\%$ ($\\rm 12+\\logt({\\rm O\/H}) \\ga 8.4$), and that the y-axis is in log scale.\\label{fig: rDGR_vs_metal_BE}}\n\\end{figure}\n\n\\subsubsection{Variable conversion factor}\\label{sec: alpha_CO discussion}\nAnother potential explanation of the change in DGR (and thereby DTM) is that the conversion factor $\\alpha_{\\rm CO}$ is not a constant, therefore, we could be wrong in estimating $\\Sigma_{\\rm H_2}$. There are two major observed trends in $\\alpha_{\\rm CO}$ \\citep{BOLATTO13}. The first trend is a metallicity dependent $\\alpha_{\\rm CO}$. In the model derived in \\citet{WOLFIRE10}, among others, $\\alpha_{\\rm CO}$ increases as metallicity decreases, which means we could be overestimating DGR in the outer part of M101. Recovering this overestimation would increase the variation in DTM and make the observed trends stronger. Moreover, since $\\rm f_{H_2}$ traced by a fixed $\\alpha_{\\rm CO}$ drops steeply with increasing radius in M101, any modification from metallicity dependent $\\alpha_{\\rm CO}$ that can affect DGR in the disk must posit a large and almost totally invisible reservoir of CO-dark molecular gas. It is suggested by \\citet{BOLATTO13} to use a constant $\\alpha_{\\rm CO}$ in regions with $\\rm 12+\\logt({\\rm O\/H})\\ge 0.5Z_\\sun$. When we test the total gas mass from a constant $\\alpha_{\\rm CO}$ against the one calculated with \\citet{WOLFIRE10} metallicity-dependent $\\alpha_{\\rm CO}$, the difference between them is at most 0.12 dex. This small change is due to the fact that in the radial region of M101 where H$_2$ makes a substantial contribution to the total gas mass, the metallicity is greater than $\\rm 12+\\logt({\\rm O\/H})=8.4$, where $\\alpha_{\\rm CO}$ only changes by a small amount. Considering the unknown uncertainties caused by the constant DTM assumption in the metallicity-dependent model \\citep{BOLATTO13}, we decide to only present the results with a fixed $\\alpha_{\\rm CO}$.\n\nThe second trend is the decrease of $\\alpha_{\\rm CO}$ in the very center of some nearby galaxies, shown by \\citet{SANDSTROM13}. It is worth noting that the \\citet{SANDSTROM13} analysis assumed DGR was locally independent of $\\rm f_{H2}$ to simultaneously solve for $\\alpha_{\\rm CO}$ and DGR in their solution pixels. Over most of M101, however, the average $\\alpha_{\\rm CO}$ they find is similar to the standard MW conversion factor, so using the \\citet{SANDSTROM13} values or making the standard assumption of a MW $\\alpha_{\\rm CO}$ will not greatly impact our results. \\citet{SANDSTROM13} found that M101 has one of the largest observed central decreases in $\\alpha_{\\rm CO}$, showing $\\alpha_{\\rm CO}=0.35^{+0.21}_{-0.13}$ in the central solution pixel, which is far lower than the galaxy-average value. Adopting the galaxy average value of $\\alpha_{\\rm CO}$ therefore causes us to overestimate the amount of gas in the center and subsequently underestimate the DGR and DTM. As shown in Figure \\ref{fig: DGR_grad_models_BE} (c), we do observe a decrease in the DGR and DTM in the central $\\sim$kpc of M101, which is likely the result of an incorrect conversion factor assumption there. However, since the affected region is small compared to our full M101 maps, we can neglect this effect in the DTM discussion.\n\nBeyond radial trends that alter $\\alpha_{\\rm CO}$ relative to what we have assumed, it is also possible that $\\alpha_{\\rm CO}$ varies from cloud-to-cloud at a fixed radius. If we overestimate $\\alpha_{\\rm CO}$ for a cloud, the DTM would be underestimated and $\\rm f_{H_2}$ would be overestimated. If we underestimate $\\alpha_{\\rm CO}$, we would underestimate $\\rm f_{H_2}$ and overestimate DTM. Both overestimation and underestimation work in the opposite sense of the correlation we observe in the residual DTM and $\\rm f_{H_2}$ and, if corrected for, would therefore strengthen our conclusions. Thus, the positive correlation between DTM and $\\rm f_{H_2}$ we calculate previously is not a result of $\\alpha_{\\rm CO}$ variation.\n\n\\subsubsection{Summary of DTM Measurements}\nTo summarize, we can explain our fitting results from all our MBB variants except the SE model with a variable DTM, where ${\\rm DGR} \\propto Z^{1.7}$ in the BE model. The maximum DGR is still within the available total metal abundance limits. By comparing the correlation between DTM and physical quantities $\\rm f_{H_2}$, $\\Sigma_\\star$ and $\\Sigma_{SFR}$, we conclude that the strongest environmental correlation of DTM is with $\\rm f_{H_2}$, which we take to be a reasonable observational indicator of ISM density and thus a tracer for accretion process. We see no clear trends that indicate correlations of DTM with stellar sources or massive star formation.\n\nOn the other hand, we could also explain the DTM results with enhanced dust emissivity in dense regions due to coagulation. The increase in $\\kappa_{160}$ is at most twice of the originally calibrated value, which is within the findings in \\citet{PLANCK14}. A non-extreme metallicity-dependent $\\alpha_{\\rm CO}$ does not affect our DGR trend much due to the low $\\rm f_{H_2}$ in most regions, however, the change of $\\alpha_{\\rm CO}$ in the center is related to our observed decrease of DGR in the central kpc. Variability of $\\alpha_{\\rm CO}$ from cloud to cloud at fixed radius would lead to a negative correlation between residual DTM and residual $\\rm f_{H_2}$, which is opposite what we observe.\n\nBoth explanations of variable DTM and variable emissivity are within the physically plausible range, thus we cannot definitively conclude if the variations we see are mainly due to changes in DGR or changes in the emissivity. However, given the observation that elemental depletions in the Milky Way are a function of ISM density and $\\rm f_{H_2}$ \\citep[][see further discussion below]{JENKINS09}, which is equivalent to a variable DTM, we argue that attributing all variation to emissivity is unlikely. To break the degeneracy between emissivity and $\\Sigma_d$, one future path is to calculate emissivity from dust models according to physical properties of local ISM. Another is to build an observational database of $\\Sigma_d$-to-SED, with known metallicity and ISM density, for future calibration. \nAnother powerful test available in the near future will be to measure the properties of the UV\/optical extinction curve, like $\\rm R_V$, as a tracer for coagulation and processes that can change the IR emissivity in the Local Group, and correlate this extinction curve tracer with quantities observable outside the Local Group.\n\n\\subsection{Comparison with previous DTM studies}\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{_DTM_D14_NGC5457_BE.pdf}\n\\caption{Top: We compare our DGR($Z$) with that from M31 measured by \\citet{DRAINE14}. The solid line is where \\citet{DRAINE14} presents their DGR fitting in M31 with observed metallicity, and the dashed line is extrapolation of their linear DGR($Z$). Within this metallicity region, our M101 results suggest a DTM 2 times higher than M31. However, if we instead select the range of radii where the M31 $\\rm f_{H_2}$ matches what we see in M101 (red region), we find a much better agreement between our observed DTM and extrapolation of \\citet{DRAINE14}. Bottom: Demonstration of how we select the green and red zones. Grey zone: \\citet{DRAINE14} $\\rm f_{H_2}$ range corresponding to the presented $\\rm 12+\\logt({\\rm O\/H})$ range. Blue: $\\rm f_{H_2}$-metallicity relation in M101. Green zone: Region with the same metallicity as \\citet{DRAINE14} data range. Red zone: Region where M101 $\\rm f_{H_2}$ corresponds to \\citet{DRAINE14} $\\rm f_{H_2}$.\\label{fig: DTM_D14}}\n\\end{figure}\nIn Figure \\ref{fig: DTM_D14}, we plot our results compared to the linear DGR($Z$) relation discussed in \\citet{DRAINE14}. \\citet{DRAINE14} show that the M31 DTM matches very well with the DTM predicted from depletions along the line of sight to $\\zeta$Oph in the MW \\citep[${\\rm F_\\star}=1$ line of sight in][]{JENKINS09}. In the corresponding metallicity range, our DGR is larger than the one in \\citet{DRAINE14}. This is illustrated in Figure \\ref{fig: DTM_D14} green zone. The derived $\\kappa_{160}$ value in \\citet{DRAINE14} is 12.51, which is around 0.75 times of our $\\kappa_{160}$ value. Thus, the DGR discrepancy at high metallicity end is not a result of our choice of $\\kappa_{160}$. Moreover, \\citet{DALCANTON15, PLANCK16} indicates that the \\citet{DRAINE07} model might overestimate $\\Sigma_d$ by $\\sim 2$ times, which also makes the difference larger. Thus, The difference between \\citet{DRAINE14} and our results in high metallicity region is not due to parameter selection, but due to physical differences between M101 and M31, or differences in the modeling.\n\nInstead of comparing region with the same metallicity, we can also compare the DTM between regions in M31 and M101 with similar ISM density, traced by $\\rm f_{H_2}$ here. According to \\citet{NIETEN06}, the region in M31 where \\citet{DRAINE14} gives the direct metallicity measurements has $\\rm f_{H_2}$ below 0.2, marked by the horizontal dashed line in Figure \\ref{fig: DTM_D14}. This $\\rm f_{H_2}=0.2$ upper limit meets our M101 data at $\\rm 12+\\logt({\\rm O\/H})=8.44$, indicated at where the horizontal dashed line meets the blue curve in Figure \\ref{fig: DTM_D14}. We pick the region between $\\rm 12+\\logt({\\rm O\/H})=8.44$ and where we have minimum $\\rm f_{H_2}$, shown in red in Figure \\ref{fig: DTM_D14}, as the region that has similar ISM density with M31 data in \\citet{DRAINE14}. Within this region, our DTM is consistent with the extrapolation of \\citet{DRAINE14} DTM. This suggests that the difference in DTM between our results and \\citet{DRAINE14} may be a consequence of M101 having a higher $\\rm f_{H_2}$ and therefore enhanced depletion (e.g. larger DTM) at the metallicity of M31.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{_DTM_R14_NGC5457_BE.pdf}\n\\caption{Our DGR versus metallicity with \\citet{REMY-RUYER14} results (data points in blue). The power law (orange dahsed line) and broken power law (green dotted line) fitting are quoted with MW conversion factors.\\label{fig: DTM_R14}}\n\\end{figure}\n\\citet{REMY-RUYER14} has compiled integrated DGR($Z$) for a large set of galaxies observed by \\textit{Herschel}. In Figure \\ref{fig: DTM_R14} we compare our measured DGR($Z$) with theirs. At the high metallicity end, our slope is shallower than their power law fitting, but within 1-$\\sigma$ confidence level of each other ($2.02 \\pm 0.28$ from \\citet{REMY-RUYER14}). Unfortunately, the turnover point of broken power law derived in \\citet{REMY-RUYER14} is at $\\rm 12+\\logt({\\rm O\/H}) = 8.10 \\pm 0.43$, and we do not have enough reliable DGR fitting results below that metallicity to compare with. It is hard to draw a conclusion whether a broken power law with turnover point around $\\rm 12+\\logt({\\rm O\/H})=8.0$ would fit our results better than a power law. The \\citet{REMY-RUYER14} broken power law in high metallicity region is basically identical to the \\citep{DRAINE11} power law.\n\n\\subsection{Comparison with MW depletion\\label{sec: J09}}\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{_DTM_J09_NGC5457_BE.pdf}\n\\caption{The comparison of our results with the DTM corresponding to various MW $\\rm F_\\star$ values described in \\citet{JENKINS09}. Most MW measurable regions have $0 \\la {\\rm F_\\star} \\la 1$. ${\\rm F_\\star}=0.36$ represents the average property of our $\\kappa_{160}$ calibration, and ${\\rm F_\\star}=\\rm inf$ means total depletion. The 40\\% $\\rm H_2$ location is marked because all \\citet{JENKINS09} data points have $f_{\\rm H_2} \\la 0.4$.\\label{fig: DTM_J09}}\n\\end{figure}\nStudies of the depletion of heavy elements in the MW \\citep{JENKINS09} also found a dependence of DTM on average ISM density and $\\rm f_{H_2}$. In Figure \\ref{fig: DTM_J09}, we display DTM corresponding to various MW $F_\\star$ regions described in \\citet{JENKINS09}. All of their original data points have $f_{\\rm H_2} \\la 0.4$ and $17.4 \\la \\log_{10} (N_{\\rm HI}) \\la 21.8$. Regions with ${\\rm F_\\star}=1$ and ${\\rm F_\\star}=0$ are by definition the representative regions of high and low depletion in the diffuse ISM of the MW, respectively. Thus, the region between these two lines corresponds to a DTM similar to the MW range extending to lower metallicity. Most points with $\\rm 12+\\logt({\\rm O\/H}) \\leq 8.4$ fall inside this range. The high-latitude diffuse ISM in the MW used to calibrate our $\\kappa_{160}$ has an ${\\rm F_\\star}$ of 0.36, thus it was selected for DGR calculation in calibrating our $\\kappa_{160}$, see \\secref{Sec: calibration}. The ${\\rm F_\\star}=\\rm inf$ line means total depletion, which is physically the same as the DGR upper limit discussed in \\secref{sec: max DGR}. All our DGR fitting results are within this limit. It is interesting to note that the point where the DGR trend falls below the maximum depletion is at the boundary of molecular gas dominant region and atomic gas dominant region (f$_{\\rm H_2}\\sim$0.4).\n\n\\subsection{Sensitivity of results to fitting methods}\\label{sec:sensitivity}\nIt is worth noting that given the same dust emission SED, the fitting results are sensitive to methods and parameters in the fitting process. Thus, it is important to be clear and self-consistent about the choices we make for calibration and fitting, as demonstrated by \\citet{GORDON14}. We also need to be careful when comparing cross-study results. Here, we use the process of $\\kappa_{160}$ calibration with SE model, which gives $\\kappa_{160}=10.48\\pm1.48~\\rm cm^2~g^{-1}$ with the SED of the MW diffuse ISM from \\citet{GORDON14}, to illustrate the possible variations in results due to different choices. Note that we want to focus only on the methods, thus we use the MW diffuse ISM from \\citet{GORDON14} in this section instead of ours described in \\secref{Sec: calibration} to eliminate the simple offset.\n\\begin{itemize}\n    \\item By changing to different models, $\\kappa_{160}$ can go up to 21.16 (PL model), which is a 100\\% change. Thus, the choice of fitting model strongly affects fitting results.\n    \\item By making the fitting grid spacing coarser, from the original 0.002 spacing to a 0.1 spacing in $\\log_{10}\\kappa_{160}$, the resulting $\\kappa_{160}$ becomes 11.7, which is a 10\\% change. This has a mild effect on fitting results, and is especially important when the grid spacing is larger than the adopted uncertainties.\n    \\item The matrix form and values of the covariance matrix can affect the fitting results. By changing the covariance matrix from ours to the one in \\citet{GORDON14} and keeping all other factors the same, the resulting $\\kappa_{160}$ goes to 17.9, which is a 70\\% change. This also affects the results strongly.\n    \\item The covariance matrix can also change the fitting residuals. For example, \\citet{GORDON14} assumes a flat uncertainty across the five bands and equal correlation, which results in similar residuals among the five bands. On the other hand, we assume different values and correlation between DIRBE and FIRAS bands, which results in better residuals in FIRAS bands and worse residual in the DIRBE band.\n\\end{itemize}\n\n\\section{Conclusions} \\label{sec: conclusions}\nWe present dust SED fitting results from five MBB variants in M101 with kpc scale spatial resolution. We compare the resulting $\\Sigma_d$ and $T_d$ with known physical limitations, and conclude the results from a simple, variable emissivity, modified blackbody model are not physically plausible. The other four models have results consistent with each other at $\\rm 12+\\logt({\\rm O\/H}) \\leq 8.5$, which demonstrates the robustness of modified blackbody model under many conditions. Among the four models, the one with a single temperature blackbody modified by a broken power-law emissivity has the highest fitting quality in residuals and $\\tilde{\\chi}^2$ distribution. Thus, the first order correction to the MBB, necessitated by our observed SEDs in M101, is to add flexibility in the emissivity spectral index at long wavelengths.\n\nThe resulting DTM, derived from our dust and gas surface densities and direct $\\rm T_e$-based metallicities, is not constant with radius or metallicity in M101 from all five models. From the preferred BE model, a relation of $\\rm DGR \\propto Z^{1.7}$ is observed overall, and $\\rm DGR \\propto Z^{1.9}$ in region with $\\rm 12+\\logt({\\rm O\/H}) \\geq 8.2$. We try to explain this variable DTM by searching for correlations between tracers of formation and destruction mechanisms of dust and metallicity to the observed physical quantities. By comparing the correlation between DTM and physical quantities ($\\rm f_{H_2}$, $\\Sigma_\\star$ and $\\Sigma_{\\rm SFR}$) after removing the major radial trend, we argue that the accretion of metals in ISM onto existing dust grains could be a cause of this variable DTM, while we do not see evidence for correlations with stellar or SNe related production and destruction.\n\nIt is also possible that the change in DTM is actually the enhancement of emissivity due to coagulation. In the center of M101, if we assume the \\citet{DRAINE14} DTM and calculate the possible change in emissivity, the resulting $\\kappa_{160}$ would be $\\sim$19 to $41~\\rm cm^2~g^{-1}$, which are 0.9 to 2.0 larger than the originally calibrated value of $16.52~\\rm cm^2~g^{-1}$ in the high latitude diffuse ISM in the MW. This change is still within the range of previous observational and theoretical calculations. Both changes in DTM and in emissivity are possible according to our current knowledge.\n\nWhen comparing with previous DTM studies, our DTM is 2 times larger than the \\citet{DRAINE14} results in the same metallicity region, but our DTM are consistent with their DTM extrapolated to the region with similar $\\rm f_{H_2}$. Comparing with \\citet{REMY-RUYER14}, our DTM has a slope consistent with their power-law fitting slope. Unfortunately, we do not have enough low-metallicity data to compare with their broken-power law. When comparing with known depletion relations from the MW and the amount of available metals in the central 5 kpc of M101, our DTM suggests essentially all available heavy elements are in dust, which is consistent with ${\\rm F_\\star}={\\rm inf}$ line from extrapolating the \\citet{JENKINS09} calculations, and also larger than most of the previous studies. Our DTM results in the lower metallicity region would fall between ${\\rm F_\\star}=1$ and ${\\rm F_\\star}=0$ in the MW. This suggests that even in the lowest metallicity regime of our study, we have not yet probed conditions where the dust life cycle differs in major ways from that in the Milky Way.\n\nDuring the fitting process, we found that the fitting results from the likelihood calculated with a multi-dimensional Gaussian distribution and a complete covariance matrix are sensitive to the choice of model and covariance matrix. Therefore, it is important to be self-consistent between calibration and fitting processes. It is also important to note the covariance matrix adopted when comparing fitting results across studies because the fitting results could change by 70\\% with different covariance matrices.\n\n\\acknowledgments\nWe thank the referee for useful comments that helped to improve the quality of the manuscript. We gratefully acknowledge the hard work of the KINGFISH, THINGS, HERACLES, LVL, and CHAOS teams and thank them for making their data publicly available. We acknowledge the usage of the HyperLeda database (http:\/\/leda.univ-lyon1.fr). IC thanks K. Gordon for helpful conversations regarding calibration and fitting. IC thanks Y.-C. Chen for helpful conversations. The work of KS, IC, AKL, DU and JC is supported by National Science Foundation grant No. 1615728 and NASA ADAP grants NNX16AF48G and NNX17AF39G. The work of AKL and DU is partially supported by the National Science Foundation under Grants No. 1615105, 1615109, and 1653300.\n\nThis work uses observations made with \\textit{Herschel}. \\textit{Herschel} is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. \nPACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-IFSI\/OAA\/OAP\/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA\/CNES (France), DLR (Germany), ASI\/INAF (Italy), and CICYT\/MCYT (Spain). \nSPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA).\n\nThis work uses observations based on National Radio Astronomy Observatory (NRAO) Karl G. Jansky Very Large Array. The NRAO is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\nThis work uses observations based on HERA on IRAM 30-m telescope. IRAM is supported by CNRS\/INSU (France), the MPG (Ger-\nmany) and the IGN (Spain). \nThis work uses observations made with the \\textit{Spitzer} Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. \nThis research has made use of NASA's Astrophysics Data System. This research has made use of the NASA\/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.\n\n\\vspace{5mm}\n\\facilities{Herschel(PACS and SPIRE), VLA, GALEX, IRAM(HERA), Spitzer(MIPS and IRAC), LBT(MODS)}\n\\software{astropy \\citep{ASTROPY13},\n          matplotlib \\citep{HUNTER07},\n          numpy \\& scipy \\citep{VANDERWALT11},\n          pandas \\citep{MCKINNEY10},\n          voronoi\\_2d\\_binning \\citep{CAPPELLARI03},\n          corner \\citep{Foreman-Mackey16}, \n          Scanamorphos \\citep[v16.9 and v17.0;][]{ROUSSEL13}, \n          HIPE \\citep[vspire-8.0.3287;][]{OTT10}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{}\n\\vspace{-1cm}\n\n\n\n\n\\footnotetext{\\textit{$^{a}$~Laboratory of Nonlinear Chemical Dynamics, Institute of Chemistry, ELTE E\\\"otv\\\"os Lor\\'and University, Budapest, Hungary.}}\n\\footnotetext{\\textit{$^{b}$~Budapest University of Technology and Economics, Department of Analysis, Budapest, Hungary. Fax: +361 463 3172; Tel: +361 463 2314; E-mail: jtoth@math.bme.hu}}\n\\footnotetext{\\textit{$^{c}$~Chemical Kinetics Laboratory, Institute of Chemistry, ELTE E\\\"otv\\\"os Lor\\'and University, Budapest, Hungary.}}\n\n\\footnotetext{\\dag~Electronic Supplementary Information (ESI) available: [details of any supplementary information available should be included here]. See DOI: 10.1039\/cXCP00000x\/}\n\n\\footnotetext{\\ddag~Based on the talk given at the 2nd International Conference on Reaction Kinetics, Mechanisms and Catalysis. 20--22 May 2021, Budapest, Hungary}\n\n\\section{Introduction} \nThe concept of reaction extent (most often denoted by \\(\\xi\\)) is more than 100 years old \\cite{dedondervanrysselberghe}. \nIts importance is emphasized by the mere fact that it has been included in the IUPAC Green Book\\cite{millscvitashomankallaykuchitsu} (see page 43). \nTwo definitions are given that are equivalent in the treated very simple special case of a\nsingle reaction step:\n\\begin{equation}\\label{eq:cm}\n\\ce{$\\sum_{m=1}^M\\alpha_{m}$X($m$) = $\\sum_{m=1}^M\\beta_{m}$X($m$)}; \n\\end{equation}\nwhere \\(M\\) is the number of chemical species \\ce{X(1)}, \\ce{X(2)}, \\dots, \\ce{X($M$)};\nand the integers \\(\\alpha_m\\) and \\(\\beta_m\\) are the corresponding stoichiometric coefficients of the reactant and product species, respectively.\nThe first definition is:\n\\begin{equation}\\label{eq:iupac1def}\n n_{\\ce{X($m$)}}=n^0_{\\ce{X($m$)}}+ \\nu_{m}\\xi,  \n\\end{equation}\nwhere \\ce{n_{\\ce{X($m$)}}} and $n^0_{\\ce{X($m$)}}$ are the actual and initial quantities (number of moles) of the species \\ce{X($m$)}, respectively. The symbol \\(\\nu_m\\) is the generalized stoichiometric number. It is negative for a reactant and positive for a product species.\nThe second definition is\n\\begin{equation}\\label{eq:iupac2def}\n\\Delta\\xi=\\frac{\\Delta n_{\\ce{X($m$)}}}{\\nu_m}=\n\\frac{n_{\\ce{X($m$)}}-n^0_{\\ce{X($m$)}}}{\\nu_m}.\n\\end{equation}\nA slightly different version is given in Ref.\\cite{goldloeningmcnaughtsehmi} and by the electronic version \\url{https:\/\/goldbook.iupac.org\/terms\/view\/E02283} called IUPAC Gold Book\\cite{mcnaughtwilkinson}:\n\\begin{equation}\\label{eq:iupac3def}\n\\mathrm{d}\\xi=\\frac{\\mathrm{d} n_{\\ce{X($m$)}}}{\\nu_m}.\n\\end{equation}\nThe above-cited definitions have been summarized in the book by Stepanov et al.\\cite{stepanoverlikinafilippov}. The authors also give a good introduction to the methods of linear algebra applied in Reaction Kinetics.\n\nWith an eye on the applicability of the concept in modern formal reaction kinetics (or, chemical reaction network theory) \nas exposed by Feinberg\\cite{feinbergbook} and T\u00f3th et al.\\cite{tothnagypapp} the following points seem crucial:\n\\begin{enumerate}\n\\item\nStarting from the original definition by De Donder and Van Rysselberghe\\cite{dedondervanrysselberghe}, we extend the definition to an \\emp{arbitrary} number of reaction steps.\n\\item\nWe do not restrict ourselves to reversible steps.\n\\item\nWe do not require linear independence of the reaction steps.\n\\item\nWe do not \"order the steps to one side\" which would result in hiding the difference between the steps like \\ce{X -> Y} and \\ce{X + Y -> 2Y}.\n\\item\nWe do not consider and take into account the atomic (or molecular) structure of the species.\n\\item \nWe do not use differentials when introducing the concept\n(cf. p. 61. of Ref.\\cite{truesdell}).\n\\item\nWe shall give an explicit definition more similar to Eq. \\eqref{eq:iupac2def} rather than to Eq. \\eqref{eq:iupac1def}.\n\\item\nWe take into consideration the volume of the reacting mixture to be able to calculate the number of individual reaction events.\n\\end{enumerate}\n\nThe structure of our paper is as follows. \nSection \\ref{sec:concept} introduces the concept for reaction networks of arbitrary complexity: for any number of reaction steps and species, mass action kinetics is not assumed. \nAs it is a usual requirement that the reaction extent tends to 1 when \"the reaction tends to its end\", we try to find quantities derived from our reaction extent having this property in Section \\ref{sec:what}.\nIt will turn out in many examples that the reaction extents do not tend to 1 in any sense. We show, however, that they contain quite relevant information about the time evolution of the reactions: they measure (or give) the number of the occurrences of the individual reaction events.\nThese examples will also reflect the fact that the reaction events do not cease  during equilibrium, and this can be seen without referring to fluctuations.\nAs the closure of our paper, we show applications of the concept\nto more complicated cases: those with multiple stationary states, oscillation, and chaos. \n\n\nIn this part, first, we analyze the classical multi-stationary example\nby Horn and Jackson\\cite{hornjackson}. \nAs to oscillatory reactions, we start with the irreversible Lotka--Volterra reaction, and we also study the reversible Lotka--Volterra reaction both in the detailed balanced and not detailed balanced cases. \nOur following oscillatory example will be an experimental system studied by R\u00e1bai\\cite{rabai}. \nAs a chaotic example, we shall take a slightly modified version of that oscillatory system. \nDiscussion of the Conclusions and a list of Notations come last.\nThe proofs of the statements and Theorems are relegated to an Appendix so as to improve the logical flow of the manuscript without getting side-tracked.\nSupporting Information is given in a PDF file; upon request, the corresponding Wolfram Language notebook---the source of the PDF file---will be provided to the interested reader.\n\\section{The concept of reaction extent}\\label{sec:concept}\nStarting from the classical works\\cite{dedondervanrysselberghe,arisprolegomena1,croce}\nand relying on the consensus of the chemists' community as formulated by Laidler\\cite{laidlerglossary} our aim is to present a treatment more general than any of the definitions introduced and applied up to now.\n\\subsection{Motivation and fundamental definitions}\nWe are going to use the following concepts.\n\\subsubsection{Fundamental notations and definitions: The framework.}\nFollowing the books by Feinberg\\cite{feinbergbook} and by T\\'oth et al.\\cite{tothnagypapp} we consider a \\emp{complex chemical reaction}, simply \\emp{reaction}, or \\emp{reaction network} as a set consisting of \\emp{reaction steps} as follows: \n\\begin{equation}\\label{eq:ccr}\n\\left\\{\\ce{$\\sum_{m=1}^M\\alpha_{m,r}$X($m$) -> $\\sum_{m=1}^M\\beta_{m,r}$X($m$)}\\quad (r=1,2,\\dots,R)\\right\\}; \n\\end{equation}\nwhere \n\\begin{enumerate}\n\\item \nthe chemical \\emp{species} are \\ce{X($1$)}, \\ce{X($2$)}, \\dots, \\ce{X($M$)}---take note that their quantities \\(N_{\\ce{X($m$)}}\\) or \\(N_m\\) will be applied interchangeably;\n\\item\nthe \\emp{reaction steps} are numbered from 1 to \\(R;\\) \n\\item\nhere \\(M\\) and \\(R\\) are positive integers;\n\\item\n\\(\\boldsymbol{\\alpha}:=[\\alpha_{m,r}]\\) and \n\\(\\boldsymbol{\\beta}:=[\\beta_{m,r}]\\) are \\(M\\times R\\) matrices of non-negative integer components called \\emp{stoichiometric coefficients}, with the properties that all the species take part in at least one reaction step (\\(\\forall m \\exists r: \\beta_{m,r}\\neq\\alpha_{m,r}\\)), and all the reaction steps do have some effect (\\(\\forall r \\exists m: \\beta_{m,r}\\neq\\alpha_{m,r}\\)), and finally\n\\item\n\\(\\boldsymbol{\\gamma}:=\\boldsymbol{\\beta}-\\boldsymbol{\\alpha}\\) is the \\emp{stoichiometric matrix} of  \\emp{stoichiometric numbers}.\n\\end{enumerate}\nInstead of Eq. \\eqref{eq:ccr} some authors prefer writing this:\n\\begin{equation}\\label{eq:ccrreduced}\n\\ce{$\\sum_{m=1}^M\\gamma_{m,r}$X($m$) = 0}\\quad (r=1,2,\\dots,R).\n\\end{equation}\nThis formulation immediately excludes reaction steps like \\ce{X + Y -> 2Y} (used e.g. to describe a step in the Lotka--Volterra reaction), or reduces it to \\ce{X -> Y}, \nchanging the stoichiometric coefficients used to formulate mass-action type kinetics. Similarly, an autocatalytic step that may be worth studying, see e.g. pp. 63 and 66 in the book by Aris\\cite{arisintro}, like \\ce{X -> 2X} appears oversimplified as \\ce{0 -> X}. \nAnother possibility is to exclude the \\emp{empty complex}, implying involuntarily that we get rid of the possibility to simply represent in- and outflow with \\ce{0 -> X} and \\ce{X -> 0}, respectively. \nThese last two examples evidently mean mass-creation and \nmass-destruction. \nIf one does not like these one should explicitly say that one is only interested in mass-conserving reactions. \nSometimes mass creation and mass destruction are slightly less obvious than above, \nsee the reaction network\n\\begin{equation*}\n\\ce{X -> Y + U},\\quad \\ce{Z -> X + U},\\quad \\ce{Z -> U},\n\\end{equation*}\nwhich is mass-producing.\n\nIt may happen that one would like to exclude reaction steps with more than two particles on the left side, such as \n\\[\n\\ce{2MnO4- + 6H+ + 5H2C2O4 = 2Mn^2+ + 8H2O + 10CO2}. \n\\]\nSuch steps do occur e.g. on page 1236 of Kov\\'acs et al.\\cite{kovacsvizvaririedeltoth} when dealing with \\emp{overall reactions}. The theory and applications of decomposition of overall reactions into elementary steps\\cite{kovacsvizvaririedeltoth,pappvizvari} would have been impossible without the framework of formal reaction kinetics. Someone may be interested in complex chemical reactions  consisting of reversible steps only. Then, they have to write down all the forward and the corresponding backward reaction steps.\n\nTaking into consideration restrictions of the above kind usually \ndoes not make the mathematical treatment easier. \nSometimes it needs hard work to figure out how they can be checked, \nas it is in the case of mass conservation of models containing species without atomic structure,\\cite{deaktothvizvari,tothnagypapp} or in relation to the existence of oscillatory reactions.\\cite{potalotka}\nTo sum up: an author has the right to make any restriction \nthought to be chemically important, but these restrictions should be declared at the outset. \nFinally, we mention our main assumption: all the steps in Eq. \\eqref{eq:ccr} are really present, i.e. they proceed with a positive rate whenever the species on the left side are present.\n\n\nWe now provide a simple example to make the understanding easier.\n\\subsubsection{A simple example.}\nLet us take an example that may be deemed chemically oversimplified but not too trivial, still simple enough so as not to be lost in the details. \nAssume that water formation follows the reversible reaction step \n\\ce{2H2 + O2 <=> 2H2O}.\nThis means that we do not take into consideration either the atomic structure of the species, or the realistic details of water formation.\nLet the forward step be represented in a more abstract way: \\ce{2X + Y -> 2Z}.\n\nThe number of species, denoted as above by \\(M\\), is 3, and the number of (irreversible) reaction steps, denoted as above by \\(R\\), is 1. \nThe \\(3\\times1\\) stoichiometric matrix \\(\\boldsymbol{\\gamma}\\) consisting of the  stoichiometric numbers is:\n\\(\\begin{bmatrix}-2\\\\-1\\\\2\\end{bmatrix}\\). \nIn case this step occurs five times, the vector of the \n\\emp{number of individual species}\nwill change as follows:\n\\begin{equation*}\n\\begin{bmatrix}\nN_{\\ce{X}}-N_{\\ce{X}}^0\\\\\nN_{\\ce{Y}}-N_{\\ce{Y}}^0\\\\\nN_{\\ce{Z}}-N_{\\ce{Z}}^0\n\\end{bmatrix}=\n5\\begin{bmatrix}\n-2\\\\\n-1\\\\\n2\n\\end{bmatrix},\n\\end{equation*}\nwhere \\(N_{\\ce{X}}^0\\) is the number of molecules of species \\ce{X} at the beginning, and \\(N_{\\ce{X}}\\) is the number of molecules of species \\ce{X} after five reaction events, and so on. \nIf one considers the reversible reaction \n\\ce{2X + Y <=> 2Z}, and assumes that the backward reaction step takes place three times then the total change is\n\\begin{equation}\n\\begin{bmatrix}\nN_{\\ce{X}}\\\\mathbb{N}_{\\ce{Y}}\\\\mathbb{N}_{\\ce{Z}}\n\\end{bmatrix}-\n\\begin{bmatrix}\nN_{\\ce{X}}^0\\\\mathbb{N}_{\\ce{Y}}^0\\\\mathbb{N}_{\\ce{Z}}^0\n\\end{bmatrix}=\n5\\begin{bmatrix}\n-2\\\\-1\\\\2\n\\end{bmatrix}+\n3\\begin{bmatrix}\n2\\\\1\\\\-2\n\\end{bmatrix}.\\label{eq:spec}\n\\end{equation}\nNote that both the number of molecules and the number of the occurrence of reaction events are positive integers. \n\nEq. (2.2) of Ref.\\cite{kurtz} is of the same form as our Eq. \\eqref{eq:spec}. Kurtz is interested mainly in reversible and detailed balanced complex chemical reactions, and, more importantly, in the relationship of their deterministic and stochastic models. This is the reason why he formulates his Eq. (2.2) for the slightly restricted case only. As to the relationship between discrete and continuous descriptions, we follow here more or less Kurtz\\cite{kurtz} and T\\'oth et al.\\cite{tothnagypapp} We cannot rely on a discrete state deterministic model of reaction kinetics---that would be desirable---because such a model does not exist as far as we know. \n\n\\subsubsection{The general case.}\nBefore providing general definitions, we mention that Dumon et al.\\cite{dumonlichanotpoquet} formulated a series of requirements that---according to them---should be obeyed by a well-defined reaction extent.\nUnfortunately, we are unable to accept most of these requirements. \nLet us mention only one: the reaction extent should be independent of the choice of stoichiometric coefficients (invariant under multiplication), i.e. it should have the same value for the reaction \\ce{2H2 + O2 -> 2 H2O} and for the reaction \\ce{H2 + $\\frac{1}{2}$O2 -> H2O}. Our point of view is that the reaction extent is strongly connected to kinetics, and it is not a tool to describe stoichiometry as some other authors\\cite{garst} also think. The only requirement that we accept will be mentioned later, in the discussion of Definition \\ref{def:extent}.\n\nWe assume throughout that the volume (\\(V\\)) is constant, and\none can write the generalized form of Eq. \\eqref{eq:spec} as\n\\begin{equation*}\n\\begin{bmatrix}\nN_{\\ce{1}}\\\\\nN_{\\ce{2}}\\\\\n\\dots\\\\\nN_{\\ce{M}}\n\\end{bmatrix}-\\begin{bmatrix}\nN_{\\ce{1}}^0\\\\\nN_{\\ce{2}}^0\\\\\n\\dots\\\\\nN_{\\ce{M}}^0\n\\end{bmatrix}=\n\\sum_{r=1}^{R}\n\\begin{bmatrix}\n\\gamma_{1,r}\\\\\n\\gamma_{2,r}\\\\\n\\dots\\\\\n\\gamma_{M,r}\n\\end{bmatrix}W_r, \n\\end{equation*}\nor shortly\n\\begin{equation*}\n\\mathbf{N}-\\mathbf{N}^0=\\boldsymbol{\\gamma} \\mathbf{W},\n\\end{equation*}\nwhere component \\(W_r\\) of the vector \\(\\mathbf{W}=\\begin{bmatrix}\nW_1&W_2&\\dots&W_R\n\\end{bmatrix}^{\\top}\\) gives the number of occurrences of the \\(r^{\\mathrm{th}}\\) reaction step. \nNote that we do not speak about \\emp{infinitesimal} changes. \n\nWith a slight abuse of notation let \\(\\mathbf{W}(t)\\), the vector of the numbers of occurrences of reaction events, a step function in the interval \\([0,t]\\).\nThen:\n\\begin{equation*}\n\\mathbf{N}(t)-\\mathbf{N}^0=\\boldsymbol{\\gamma} \\mathbf{W}(t),\n\\end{equation*}\nor turning to moles\n\\begin{equation}\\label{def:moles}\n\\mathbf{n}(t)-\\mathbf{n}^0=\\frac{\\mathbf{N}(t)-\\mathbf{N}^0}{L}=\\boldsymbol{\\gamma} \\frac{\\mathbf{W}(t)}{L}=\\boldsymbol{\\gamma}\\boldsymbol{\\xi}(t),\n\\end{equation}\nwhere \\(L\\)\\emp{ is the Avogadro constant} having the unit \\({\\,\\mathrm{mol}}^{-1}\\), and\n\\begin{equation*}\n\\mathbf{n}(t):=\\frac{\\mathbf{N}(t)}{L},\n\\mathbf{n}^0:=\\frac{\\mathbf{N}^0}{L},\n\\boldsymbol{\\xi}(t):=\\frac{\\mathbf{W}(t)}{L}.\n\\end{equation*}\nHere we had to choose the less often used notation \\(L\\) (\\url{https:\/\/goldbook.iupac.org\/terms\/view\/A00543}) to avoid mixing up with other notations.\n\nThe relationship \\eqref{def:moles} can be expressed in concentrations as\n\\begin{equation}\n\\mathbf{c}(t)-\\mathbf{c}^0=\\frac{\\mathbf{n}(t)-\\mathbf{n}^0}{V}=\\frac{\\boldsymbol{\\gamma}}{V}\\boldsymbol{\\xi}(t),\\label{eq:implicit}\n\\end{equation}\nwhere \\(V\\in\\mathbb{R}^+\\), the volume of the reaction vessel is assumed to be constant, \\(\\mathbf{c}(t):=\\frac{\\mathbf{n}(t)}{V}\\) and\n\\(\\mathbf{c}^0:=\\frac{\\mathbf{n}^0}{V}\\). \nThe component \\(c_{\\ce{X($m$)}}\\) or \\(c_m\\) of \\(\\mathbf{c}\\) is traditionally denoted in chemical textbooks as \\([\\ce{X($m$)}],\\) see e.g. Section 1.2 of Ref.\\cite{pillingseakins}.\n\nThe concentration in Eq. \\eqref{eq:implicit} is again a step function; however, if the number of particles (molecules, radicals, electrons, etc.) is very large, as very often it is, it may be considered to be a continuous, even differentiable function. Remember that the components of \\(\\boldsymbol{\\xi}(t)\\) have the dimension of the amount of substance, measured in moles.\n\nLet us now give a general, formal, and \\emp{explicit} definition of reaction extent valid \\emp{for an arbitrary number of species and reaction steps, and not restricted to mass action type kinetics}. \n(Few qualitative---mainly technical---restrictions are usually made on the function \n\\(\\mathbf{rate}\\)\\cite{feinbergbook,tothnagypapp,volperthudyaev},\nbut we now mention the continuous differentiability only.)\nWe start with rewriting the induced kinetic differential equation \\begin{equation}\\label{eq:ikdegen}\n\\dot{\\mathbf{c}}(t)=\\boldsymbol{\\gamma}\\mathbf{rate}(\\mathbf{c}(t))\n\\end{equation} \ntogether with the initial condition \\(\\mathbf{c}(0)=\\mathbf{c}^0\\) into an (equivalent) integral equation:\n\\begin{equation}\\label{eq:int}\n\\mathbf{c}(t)-\\mathbf{c}^0=\\boldsymbol{\\gamma}\\int_0^t\\mathbf{rate}(\\mathbf{c}(\\overline{t}))\n{\\;\\mathrm{d}\\overline{t}}.\n\\end{equation}\n\nThe component \\(rate_r\\) of the vector \\(\\mathbf{rate}\\) provides the reaction rate of the \\(r^{\\mathrm{th}}\\) reaction step.\nNote that in the mass action case Eq. \\eqref{eq:ikdegen} specializes into\n\\begin{equation}\\label{eq:ikdemass}\n\\dot{\\mathbf{c}}=\\boldsymbol{\\gamma}\\mathbf{k}\\odot\\mathbf{c}^{\\boldsymbol{\\alpha}} \n\\end{equation} \nor, in coordinates\n\\begin{equation*\n\\dot{c}_m(t)=\\sum_{r=1}^{R}\\gamma_{mr}k_r\\prod_{p=1}^{M}c_p^{\\alpha_{p,r}}\\quad(m=1,2,\\cdots,M),\n\\end{equation*}\nwhere \\(\\mathbf{k}\\) is the vector of (positive) reaction rate coefficients \\(k_r.\\)\n(We prefer using the expression \\emp{reaction rate coefficients} to \\emp{reaction rate constants},\nas these numbers do depend on many factors---except species concentrations.)\nIn Eq. \\eqref{eq:ikdemass} we used the usual vectorial operations, see e.g. Section 13.2 in T\\'oth et al.\\cite{tothnagypapp}\nTheir use in formal reaction kinetics has been initiated by Horn and Jackson.\\cite{hornjackson}\n\nIn accordance with what has been said up to now, we can introduce the explicit definition of reaction extent by combining Eqs. \\eqref{eq:implicit} and \\eqref{eq:int}.\n\n\\begin{definition}\\label{def:extent}\nThe \\emp{reaction extent}\nof a complex chemical reaction or reaction network defined by Eq. \\eqref{eq:ccr} is the scalar variable, vector-valued function given by the formula\n\\begin{equation}\\label{eq:specre}\n\\boxed{\n\\boldsymbol{\\xi}(t):=V\\int_0^t\n\\mathbf{rate}(\\mathbf{c}(\\overline{t}))\n{\\;\\mathrm{d}\\overline{t}}}.\n\\end{equation}\nIts time derivative \\(\\dot{\\boldsymbol{\\xi}}(t)=V\\mathbf{rate}(\\mathbf{c}(t))\\) is usually called the \\emp{rate of conversion} or \\emp{reaction flux}\\cite{polettiniesposito}.\n\\end{definition}\nNote that Eq. \\eqref{eq:specre} shows that the reaction extent, in general, depends on the whole history\n(past and present) of the vector of concentrations, as if it had a \\emp{memory}.\\label{page:memory}\n\nDefinition \\ref{def:extent} of the reaction extent has been derived from the number of reaction events in order to reveal its connection to changes in the concentrations. \nAssuming here also that \\(V\\) is constant, \none can formulate the following trivial (equivalent) consequences of the definition:\n\\begin{equation}\\label{eq:trivicons}\n\\dot{\\mathbf{n}}=\\boldsymbol{\\gamma}\\dot{\\boldsymbol{\\xi}},\\quad\\dot{\\mathbf{c}}=\\frac{1}{V}\\boldsymbol{\\gamma}\\dot{\\boldsymbol{\\xi}},\\quad\n\\mathbf{c}=\\mathbf{c}^0+\\boldsymbol{\\gamma}\\frac{\\boldsymbol{\\xi}}{V}\n\\end{equation}\nmentioned also by Laidler\\cite{laidlerglossary},\nsometimes as definitions, sometimes as statements.\n\nNote that neither the rate of the reaction: \n\\(\\mathbf{rate}(\\mathbf{c}(t))=\\frac{\\dot{\\boldsymbol{\\xi}}(t)}{V},\\)\nnor the reaction extent \\({\\boldsymbol{\\xi}},\\) nor the rate of conversion \\(\\dot{\\boldsymbol{\\xi}}\\) depends on the stoichiometric matrix \\(\\boldsymbol{\\gamma}\\), thereby this\none of the requirements formulated by Dumont et al.\\cite{dumonlichanotpoquet}\nis fulfilled.\n\nWhat is wrong with the almost ubiquitous implicit \"definition\" Eq. \\eqref{eq:implicit}? \nWe show an example to enlighten this.\n\\begin{example}\nConsider the reaction steps\n\\begin{equation*}\n\\ce{X ->[$k_1$] Y}, \\quad\\ce{X + Y ->[$k_2$] 2Y},\n\\end{equation*}\nexpressing the fact that \\ce{X} is transformed into \\ce{Y} directly and also via autocatalysis.\nAlthough the reaction steps \n\\begin{equation*}\n\\ce{X ->[$k_1$] Y + P},\\quad \\ce{X + Y ->[$k_2$] 2Y + P}\n\\end{equation*} with the external species \\ce{P}  is a more realistic description of genuine chemical reactions,\ne.g. the acid autocatalysis in ester hydrolysis\\cite{ostwald,bansagitaylor}, they lead to the same kinetic differential equations for \\ce{X} and \\ce{Y}. Therefore, we shall analyze the simpler scheme.\nNow the stoichiometric matrix $\\gamma$ is as follows:\n\\begin{equation*}\n\\boldsymbol{\\gamma}=\\begin{bmatrix}\n-1&-1\\\\\n1&1\n\\end{bmatrix}.\n\\end{equation*}\nThen Eq. \\eqref{eq:int} specializes into \n\\begin{align*}\nc_{\\ce{X}}(t)-c_{\\ce{X}}(0)&=\n-\\int_{0}^{t}k_1c_{\\ce{X}}(\\overline{t}){\\;\\mathrm{d}\\overline{t}}\n-\\int_{0}^{t}k_2c_{\\ce{X}}(\\overline{t})c_{\\ce{Y}}(\\overline{t}){\\;\\mathrm{d}\\overline{t}}\n=\n-\\frac{\\xi_1(t)+\\xi_2(t)}{V}\\\\\nc_{\\ce{Y}}(t)-c_{\\ce{Y}}(0)&=\n\\int_{0}^{t}k_1c_{\\ce{X}}(\\overline{t}){\\;\\mathrm{d}\\overline{t}}+\n\\int_{0}^{t}k_2c_{\\ce{X}}(\\overline{t})c_{\\ce{Y}}(\\overline{t}){\\;\\mathrm{d}\\overline{t}}\n=\n\\frac{\\xi_1(t)+\\xi_2(t)}{V}.\n\\end{align*}\nThese two relations do not determine \\(\\xi_1\\) and \\(\\xi_2\\) individually but only their sum (even if one utilizes \\(c_{\\ce{Y}}(t)=c_{\\ce{X}}(0)+c_{\\ce{Y}}(0)-c_{\\ce{X}}(t)\\)).\nThe problem originates from the fact that the reaction steps are not linearly independent as reflected in the singularity of the matrix \\(\\boldsymbol{\\gamma}.\\)\n\\end{example}\n\n\nIf the reaction steps of a complex chemical reaction are independent, the situation is better.\n\\begin{example}\\label{ex:indep}\nIn some special cases, there is a way of making the \"definition\" Eq. \\eqref{eq:implicit} into a real, explicit definition.\nAssume that \\(R\\le M\\), and that the stoichiometric matrix \\(\\boldsymbol{\\gamma}\\) is of the full rank, i.e. the reaction steps are independent. \nThen, one can rewrite Eq. \\eqref{eq:implicit} in two steps as follows:\n\\begin{align}\n\\boldsymbol{\\gamma}^{\\top}(\\mathbf{c}(t)-\\mathbf{c}^0)&=\\frac{1}{V}\\boldsymbol{\\gamma}^{\\top}\\boldsymbol{\\gamma}\\boldsymbol{\\xi}(t)\\nonumber\\\\\n\\boldsymbol{\\xi}(t)&=V(\\boldsymbol{\\gamma}^{\\top}\\boldsymbol{\\gamma})^{-1}\\boldsymbol{\\gamma}^{\\top}(\\mathbf{c}(t)-\\mathbf{c}^0).\\label{eq:memoryless}\n\\end{align}\nNow one can accept Eq. \\eqref{eq:memoryless} as a definition for the reaction extent.\nNevertheless, in this special case \nEq. \\eqref{eq:int} \nimplies\n\\begin{equation*}\n\\boldsymbol{\\gamma}^{\\top}(\\mathbf{c}(t)-\\mathbf{c}^0)=\\boldsymbol{\\gamma}^{\\top}\\boldsymbol{\\gamma}\\int_0^t\\mathbf{rate}(\\mathbf{c}(\\overline{t})){\\;\\mathrm{d}\\overline{t}}\n\\end{equation*}\nand\n\\begin{equation*}\n(\\boldsymbol{\\gamma}^{\\top}\\boldsymbol{\\gamma})^{-1}\\boldsymbol{\\gamma}^{\\top}(\\mathbf{c}(t)-\\mathbf{c}^0)=\n\\frac{1}{V}\\boldsymbol{\\xi}(t)=\n\\int_0^t\\mathbf{rate}(\\mathbf{c}(\\overline{t})){\\;\\mathrm{d}\\overline{t}},\n\\end{equation*}\nthus this definition is the same as the one in Eq. \\eqref{eq:specre}. \nThis derivation can always be done if \\(R=1\\) that is, in a not-so-interesting trivial case.\nUnfortunately, the case \\(R\\le M\\) does not happen very often. On the contrary, for example, in\ncase of combustion reactions, Law's law\\cite{law} (see page 11) states that \\(R\\approx 5 M.\\)\n\nNote also, that Eq. \\eqref{eq:memoryless} shows the  following:  in these cases, i.e. when the stoichiometric matrix is of the full rank---as opposed to the general case, see page \\pageref{page:memory}---the reaction extents do not depend on the whole history of the concentration vector, it only depends on its instantaneous value.\n\\end{example}\nLet us make a trivial remark on the independence of reaction steps. \nIf the complex chemical reaction consists of a single irreversible step, then the reaction steps(!) are independent. \nIf any of the reaction steps are reversible, then the reaction steps are not independent. \n\\subsection{Properties of the reaction extent}\nThe usual assumptions \\label{pg:assump} on the vector-valued function \\(\\mathbf{rate}\\) are as follows, see Refs.\\cite{tothnagypapp,volperthudyaev}.\n\\begin{enumerate}\n\\item\nAll of its components are continuously differentiable functions defined on \\(\\mathbb{R}^M\\) taking only non-negative values. This is usual, e.g. in the case of mass action kinetics, but---with some restrictions---also in the case when the reaction rates are rational functions as in the case of Michaelis--Menten or Holling type kinetics, see e.g. Refs.\\cite{polczkulcsarszederkenyi,polczpeniszederkenyi,kisstoth,laidlerglossary}\n\\item\nThe value of \\(rate_r(\\mathbf{c})\\)  is zero if and only if some of the species needed for the \\(r^{\\mathrm{th}}\\) reaction step is missing, i.e. for some \\(m: \\alpha_{m,r}>0\\) and \\(c_m=0\\) (see p. 613, Condition 1 in Ref.\\cite{volperthudyaev}). \nWe shall say in this case that reaction step \\(r\\) \\emp{cannot start} from the concentration vector  \\(\\mathbf{c}.\\)\n\\end{enumerate}\nThe second assumption implies---even in the general case, i.e. without restriction to the mass action type kinetics---that \\(rate_r(\\mathbf{c})>0\\) if all the necessary species (\\emp{reactants}, see below) are present initially: \\(\\alpha_{m,r}>0\\Longrightarrow c_m>0.\\)\n\nLet us sum up the relevant qualitative characteristics of the reaction extent.\n(Remember that the proof can be found in the Appendix.)\n\\begin{theorem}\\label{thm:basics}\n\\begin{enumerate}\n\\item[\\nonumber]\n\\item\nThe domain of the function \\(t\\mapsto\\boldsymbol{\\xi}(t)\\) is the same as that of \\(\\mathbf{c}\\).\n\\item\nBoth \\(\\mathbf{c}\\) and \\(\\boldsymbol{\\xi}\\in\\mathcal{C}^2(J,\\mathbb{R}^R);\\) with some open interval \\(J\\subset\\mathbb{R}\\) such that \\(0\\in J.\\)\n \\item\n \\(\\boldsymbol{\\xi}\\) obeys the following initial value problem:\n\\begin{equation}\\label{eq:rmdiffegy}\n\\dot{\\boldsymbol{\\xi}}(t)=V\\mathbf{rate}(\\mathbf{c}^0+\\frac{1}{V}\\boldsymbol{\\gamma}\\boldsymbol{\\xi}(t)),\\quad \\boldsymbol{\\xi}(0)=\\boldsymbol{0}.\n\\end{equation}\n\\item\nAt the beginning, the velocity vector of the reaction extent (also called the \\emp{rate of conversion}) points into the closed first orthant, and this property is kept for all times in the domain of the function \\(t\\mapsto\\boldsymbol{\\xi}(t).\\) \n\\item\nThe components of \\(\\boldsymbol{\\xi}\\) are either positive, strictly monotonously increasing functions or constant zero. \nIf for some positive time \\(t\\) we find that \\(\\xi_r(t)=0\\) then, obviously, the reaction step \\(r\\) did not start at all at the beginning.\n\\end{enumerate}\n\\end{theorem}\n\n\n\nLet us make a few remarks:\n\\begin{itemize}\n\\item \nThe last property (positivity) mentioned in the Theorem can be realized with \\(\\lim_{t\\to+\\infty}\\xi(t)=+\\infty\\)\n(the simplest example for this being \\(\\ce{X -> 2X}, c_{\\ce{X}}^0>0\\)),  or with a finite positive value of \\(\\lim_{t\\to+\\infty}\\xi(t),\\) see the example \\ce{X -> Y -> Z} below.\n\\item\nEq. \\eqref{eq:rmdiffegy} shows that we would have got simpler formulas if we used\n\\(\\frac{\\xi(t)}{V}\\) as proposed by Aris\\cite{arisintro} on p. 44, \nbut this form is valid only if \\(V\\) is constant. \n\\item\nIn the mass action case both \\(\\mathbf{c}\\) and \\(\\boldsymbol{\\xi}\\) are infinitely many times differentiable.\n\\item\nIf one uses a kinetics different from the mass action type\nnot fulfilling assumptions 1 and 2 on page \\pageref{pg:assump}, \nor---as P\\'ota\\cite{potarabai} has shown---if one applies an approximation, \nthen it may happen that some of the initially positive concentrations turn to zero.\n\\end{itemize}\n\nIn order to proceed, we need to make a technical remark on the figures shown hereinafter. We label the first axis (usually: horizontal) in the figures with \\(t\\mathrm{\/s}\\), where \\(\\mathrm{s}\\) is the time unit second. Labels of other axes are formed in a similar way. With this procedure we want to emphasize that the figures show the relationship between pure numbers and not between physical quantities. \n\nThe condition in part 5 of Theorem \\ref{thm:basics} is only necessary but not sufficient as the example below shows.  \n\\begin{example}\\label{ex:convexconcave}\nLet us start the consecutive reaction \\ce{X ->[$k_1$] Y ->[$k_2$] Z} from the vector of the initial concentrations:\n\\(\\begin{bmatrix}\nc^0_{\\ce{X}}&0&0\n\\end{bmatrix}^{\\top}\\), and suppose \\(k_1\\neq k_2.\\) \nAlthough, the second step cannot start at the beginning, yet the second reaction extent is positive for all positive times as the solution of the evolution equations\n\\begin{equation}\\label{eq:velo}\n\\dot{\\xi}_1=Vk_1\\left(c^0_{\\ce{X}}-\\frac{\\xi_1}{V}\\right),\\quad\n\\dot{\\xi}_2=Vk_2\\left(0+\\frac{\\xi_1-\\xi_2}{V}\\right)\n\\end{equation}\nare as follows\n\\begin{align*}\n&\\xi_1(t)=Vc^0_{\\ce{X}}(1-e^{-k_1t}),\\\\\n&\\xi_2(t)=\\frac{Vc^0_{\\ce{X}}}{k_2-k_1}\\left(k_2(1-e^{-k_1t})-k_1(1-e^{-k_2t})\\right).\n\\end{align*}\nPositivity also follows without any calculations from the fact that the velocity vector of the differential equations in \\eqref{eq:velo} point inward, into the interior of the first quadrant, or using the fact that Eqs. \\eqref{eq:velo} are also kinetic type differential equations.\n\nNote that \\(\\lim_{t\\to+\\infty}\\xi_1(t)=\\lim_{t\\to+\\infty}\\xi_2(t)=Vc^0_{\\ce{X}}.\\)\nIt means that the number of occurrences of the reaction events for both reactions, and thus the reaction extents, are exactly the same at the end of the whole process. \nMoreover, it does not depend on the reaction rate coefficients.\n\nEasy calculations show the following facts. \nThe function \\(\\xi_2\\,\/\\,\\mathrm{mol}\\) in Fig. \\ref{fig:convexconcave} has an inflection point, because its second derivative is zero at some positive time \\(t_{\\mathrm{infl}}\\) for all choices of the reaction rate coefficients, and the third derivative is not zero at \\(t_{\\mathrm{infl}}\\). \nThe function \\(\\xi_1\\,\/\\,\\mathrm{mol}\\) in Fig. \\ref{fig:convexconcave} is concave from below no matter what the reaction rate coefficients are. \n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{convexconcave}\n\\caption{Reaction extents in the consecutive reaction \\ce{X ->[$k_1$] Y ->[$k_2$] Z} when \n\\(k_1~=~1~{\\mathrm{s}}^{-1}\\), \n\\(k_2=~2~{\\mathrm{s}}^{-1}\\), \n\\(c_{X}^0=~1~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \n\\(c_{\\ce{Y}}^0=c_{\\ce{Z}}^0=~0~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \n\\(V=~1~\\mathrm{dm}^3.\\) \nThe limit 1 here is only a consequence of the choice of the parameters. \nThe large dot is the inflection point of the curve \\(\\xi_2\\,\/\\,\\mathrm{mol}\\).}\n\\label{fig:convexconcave}\n\\end{figure}\n\\end{example}\n\nTo characterize the convexity of the reaction extents in the general case is an open problem. \nOne should take into consideration that although in the practically interesting cases the number of equations in \\eqref{eq:rmdiffegy} is larger than those in Eq.\n\\eqref{eq:ikdegen}, that is \\(R > M\\), the  equations for the reaction extents are of a simpler structure. \n\nMonotonicity mentioned in Theorem \\ref{thm:basics} implies that all the components of the reaction extent do have a finite or infinite limit as \\(t\\) tends to \\(\\sup(J)=:t^*,\\) where \\(J:=\\Dom(\\mathbf{c}),\\)\nand \\(t^*\\) is a finite or infinite time. \nIt is an interesting open question: \nwhen does a coordinate of the reaction extent vector tend to infinity?\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{graphical1}\n\\caption{\n\\(c_{\\ce{X}}^0~=~3.0~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(c_{\\ce{Y}}^0~=~1.0~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(c_{\\ce{Z}}^0~=~1.0~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(k_1~=~1.0~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\),\n\\(k_{-1}~=~1.0~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(V~=~1~\\mathrm{dm}^3.\\)\nWhile the concentrations tend to and are becoming very close to the equilibrium values, the reaction extents tend to infinity in a monotonously increasing way in the reaction: \n\\ce{2X + Y <=>[$k_1$][$k_{-1}$] 2Z}.}\n\\label{fig:dynamiceq}\n\\end{figure}\nAs the \\emp{emphatic} closure of this series of remarks, we mention that the strictly monotonous increase of the number of the occurrence of reaction events shows that the reaction events never stop, see \\ref{fig:dynamiceq}. This important fact is independent on the form of kinetics, and it is a property of the deterministic models of reaction kinetics.\nThis sheds light on the meaning of \\emp{dynamic equilibrium} as generally taught. \nNote that \\emp{no reference to thermodynamics or statistical physics} has been invoked here,\nanalyzing the connections are left to the reader.\n\\begin{example}\\label{ex:blowup}\nIn case when the domain of the function \\(t\\mapsto\\mathbf{c}(t)\\) is a proper subset of the non-negative real numbers, \\(\\boldsymbol{\\xi}\\) has the same property. \nLet us consider the induced kinetic differential equation\\ \\(\\dot{c}=kc^2\\) of the quadratic auto-catalytic reaction \\ce{2X ->[k] 3X} with the initial condition \\(c(0)=c^0>0.\\) Then, \n\\(\nc(t)=\\frac{c^0}{1-kc^0t},  \\left(t\\in\\left[0,\\frac{1}{kc^0}\\right[\n\\,\\subsetneq\\,[0,+\\infty[\\right).\n\\) \nNow Eq. \\eqref{eq:rmdiffegy} specializes into\n\\(\\dot{\\xi}=Vk(c^0+\\xi\/V)^2,{\\ }\\xi(0)=0\\)\nhaving the solution\n\\(\n\\xi(t)=V(c^0)^2\\frac{kt}{1-kc^0t},  \\left(t\\in\\left[0,\\frac{1}{kc^0}\\right[\\right),\n\\)\nthus \\(\\xi\\) blows up at the same time (\\(t^*:=\\frac{1}{kc^0}\\)) when \\(c\\) does.\nUp to the blow-up, the reaction event occurs infinitely many times:\n\\(\\lim_{t\\to+t^*}\\xi(t)=+\\infty.\\)\nDefinitions and a few statements about blow-up in kinetic differential equations are given in the works by Csikja et al.\\cite{csikjapapptoth,csikjatoth}\n\\end{example}\n\n\n\\section{What is it that tends to 1?}\\label{sec:what}\nOur interest up to this point was the number of occurrences of reaction events. \nHowever, many authors think it is useful and visually attractive that the \"reaction extent tends to 1 when the reaction tends to its end,\" see e.g. Fig. 1 of Glasser\\cite{glasser}.\nBorge\\cite{borge} and Peckham\\cite{peckham} also argue for [0,1].\n\nAnother approach is given by Moretti\\cite{moretti}, Dumon et al.\\cite{dumonlichanotpoquet} and others via introducing the \\emp{reaction advancement ratio} \\(\\frac{\\xi}{\\xi_{\\max}}\\), and stating that this ratio is always between 0 and 1. \nPeckham\\cite{peckham} noticed that Atkins\\cite{atkins5} (pp. 272--276) shows a figure of the free energy \\(G\\) of the reacting system versus \\(\\xi,\\) where the first axis is labeled from zero to one. \nHowever, in the next edition\\cite{atkins6} (pp. 216--217), the graph has been changed, and it now shows the first axis without 1 as an upper bound.\nBeing loyal to the usual belief\\cite{peckham,treptow}, \nwe are looking for quantities (pure numbers) tending to 1 as e.g. \\(t\\to+\\infty\\). \nScaling might help to find such quantities.\n\\subsection{Scaling by the initial concentration: One reaction step}\nNow we are descending from the height of generality by considering a single irreversible reaction step (\\(R=1\\)), assuming that the kinetics is of the mass action type. Thus, the reaction is\n\\begin{equation}\\label{eq:onestep}\n\\ce{$\\sum_{m=1}^M\\alpha_m$X($m$) ->[k] $\\sum_{m=1}^M\\beta_m$X($m$)}.\n\\end{equation}\nTherefore, one has the reaction extent\n\\begin{equation}\\label{eq:ximone}\n\\dot{\\xi}=Vk\\left(\\begin{bmatrix}\nc_1^0\\\\c_2^0\\\\\\dots\\\\c_M^0\n\\end{bmatrix}+\n\\begin{bmatrix}\n\\gamma_1\\\\\\gamma_2\\\\\\dots\\\\\\gamma_M\n\\end{bmatrix}\\frac{\\xi}{V}\\right)^{\\boldsymbol{\\alpha}}\n=\nVk\\prod_{m=1}^{M}\\left(c_m^0+\\gamma_m\\frac{\\xi}{V}\\right)^{\\alpha_m},\\quad \\xi(0)=0;\n\\end{equation}\nwith \\(\\gamma_m:=\\beta_m-\\alpha_m.\\)\n\n\\begin{theorem}\\label{thm:singlestep}\n\\begin{enumerate}\n\\item[\\nonumber]\n\\item \nIf the reaction in Eq. \\eqref{eq:onestep} cannot start, then\n\\(\\xi(t)=0\\) for all non-negative real times \\(t\\):\n\\begin{equation}\\label{eq:cannot}\n\\exists m:(\\alpha_m\\neq0\\ \\&\\ c_m^0=0)\\Longrightarrow\\forall t\\in\\mathbb{R}^+_0:\\xi(t)=0.\n\\end{equation}\n\\item\nIf the reaction in Eq. \\eqref{eq:onestep} does start and all the species are produced (i.e. for all \\(m: \\gamma_m>0\\)), then \\(\\xi(t)\\)\ntends to infinity (blow-up included):\n\\begin{equation}\\label{eq:produced}\n\\forall m:\\gamma_m:=\\beta_m-\\alpha_m>0\\Longrightarrow\\lim_{t\\to t^*}\\xi(t)=+\\infty,\n\\end{equation} \nwhere \\(t^*:=\\sup(J)\\) with \\(J:=\\Dom(\\xi)\\).\n\\item \nIf some of the species is consumed, that is \\(\\exists m: \\gamma_m<0\\), then \n\\begin{equation}\\label{eq:decreases}\n\\lim_{t\\to +\\infty}\\xi(t)=\\min\\left\\{-\\frac{Vc_m^0}{\\gamma_m};\\gamma_m<0\\right\\}.\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{example}\n\\begin{enumerate}\n\\item [\\nonumber]\n\\item \nReaction \\ce{X ->[$k$] Y} with \\(c^0_{\\ce{X}}=0\\) and \\(c^0_{\\ce{Y}}\\)\narbitrary, illustrates the first case as here \\(\\dot{\\xi}=-k\\xi, \\xi(0)=0\\)\nimplies \\(\\forall t\\in\\mathbb{R}:\\xi(t)=0.\\)\n\\item \nReaction \\ce{X ->[$k$] 2X} with \\(c^0_{\\ce{X}}>0\\) is an illustration for the second case \n\\begin{equation*}\n\\forall t\\in\\mathbb{R}:\\xi(t)=V c^0_{\\ce{X}}(e^{k t}-1 )\n\\end{equation*}\nwith \n\\(\n\\lim_{t\\to+\\infty}\\xi(t)=+\\infty.   \n\\)\n\\item \nReaction \\ce{2X ->[$k$] 3X} with \\(c^0_{\\ce{X}}>0\\) (Example \\ref{ex:blowup}) is another illustration for the second case with \n\\begin{equation*}\n\\forall t\\in[-\\infty,\\frac{1}{kc^0_{\\ce{X}}}[ :\\xi(t)=\\frac{k t V (c^0_{\\ce{X}})^2}{1-ktc^0_{\\ce{X}}}\n\\end{equation*} with\n\\(\\lim_{t\\to\\frac{1}{kc^0_{\\ce{X}}}}\\xi(t)=+\\infty.\\)\n\\item \nReaction \\ce{X + Y ->[$k$] 2X}\nis an illustration for the third case \n\\begin{equation*}\n\\forall t\\in\\mathbb{R}:\\xi(t)=\n\\frac{(-1 + e^{k t (c^0_{\\ce{X}} + c^0_{\\ce{Y}})}) V c^0_{\\ce{X}} c^0_{\\ce{Y}}}{(c^0_{\\ce{Y}} + e^{k t (c^0_{\\ce{X}} + c^0_{\\ce{Y}})} c^0_{\\ce{X}})}\n\\end{equation*} \nwith \\(\\lim_{t\\to+\\infty}\\xi(t)=Vc^0_{\\ce{Y}},\\) if \\(c^0_{\\ce{X}},c^0_{\\ce{Y}}\\neq0\\). If either \\(c^0_{\\ce{X}}=0,\\) or \\(c^0_{\\ce{Y}}=0,\\)\nthen \\(\\forall t\\in \\mathbb{R}:\\xi(t)=0.\\)\n\\item\nThe example \\ce{X -> Y} with \\(c_{\\ce{X}}^0=0, c_{\\ce{Y}}^0>0\\) shows that\na species (here \\ce{Y}) can have positive concentration for all positive times in a reaction where \"none of the steps\" can start.\n\\end{enumerate}\n\\end{example}\n\nThe table below shows a series of examples illustrating different types of single irreversible reaction steps.\n\n\\begin{table}\n  \\caption{Reaction extent for various reaction types}\n  \\label{table:cases}\n\\begin{tabular}{lllll}\n\\hline\nStep                      &\\(\\mathbf{c}^0\\)&\\(\\dot{\\xi}=\\)                              &\\({\\xi}(t)=\\)                                                &Case\\\\ \n\\hline\\ce{X ->[$k$] 2X}     & 0       &\\(k\\xi\\) & 0             &\\eqref{eq:cannot}\\\\\n\\hline\\ce{X ->[$k$] 2X}     & 1       &\\(Vk(1+\\xi\/V)\\)& \\(V(e^{kt}-1)\\)&\\eqref{eq:produced}\\\\\n\\hline\\ce{2X ->[$k$] 3X}    & 1       &\\(Vk(1+\\xi\/V)^2\\)                               &\\(\\frac{Vkt}{1-kt}\\)                                           &\\eqref{eq:produced}\\\\\n\\hline\\ce{2X + Y ->[$k$] 2Z}&         &\\(Vk(c_{\\ce{X}}^0-2\\xi\/V)^2*\\)&\\\\\n&&\\((c_{\\ce{Y}}^0-\\xi\/V)\\)&            &\\eqref{eq:decreases}\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{example}\nHere we analyze the last example of Table \\ref{table:cases}.\nIn the case of the reaction \\ce{2X + Y ->[$k$] 2Z} mimicking water formation one has the following quantities:\n\\(R:=1,M:=3,\\ce{X}:=\\ce{H2}, \\ce{Y}:=\\ce{O2},\\ce{Z}:=\\ce{H2O}.\\)\nFurthermore,\n\\[\n\\boldsymbol{\\alpha}=\n\\begin{bmatrix}\n2\\\\1\\\\0\n\\end{bmatrix}{\\!}{\\!},\\quad\n\\boldsymbol{\\beta}=\\begin{bmatrix}\n0\\\\0\\\\2\n\\end{bmatrix}{\\!}{\\!},\\quad \n\\boldsymbol{\\gamma}=\\begin{bmatrix}\n-2\\\\-1\\\\2\n\\end{bmatrix}{\\!}{\\!}.\n\\]\nThe initial value problem to describe the time evolution of reaction extent is\n\\begin{equation}\\label{eq:wateru}\n\\dot{\\xi}=Vk\\left(\n\\begin{bmatrix}\nc_{\\ce{X}}^0\\\\c_{\\ce{Y}}^0\\\\c_{\\ce{Z}}^0\n\\end{bmatrix}+\n\\begin{bmatrix}\n-2\\\\-1\\\\2\\\\\n\\end{bmatrix}\\frac{\\xi}{V}\\right)^{\\begin{bmatrix}\n2\\\\1\\\\0\n\\end{bmatrix}}\n=\nVk(c_{\\ce{X}}^0-2\\frac{\\xi}{V})^2(c_{\\ce{Y}}^0-\\frac{\\xi}{V}),\\quad \\xi(0)=0.\n\\end{equation}\nWe can provide only the inverse of the solution to Eq. \\eqref{eq:wateru}. However, one can state that the reaction extent tends strictly monotonously to its limit (independent on the value of the reaction rate coefficient): \\(\\lim_{t\\to+\\infty}\\xi(t)=\\min\\{\\frac{Vc_{\\ce{X}}^0}{2},Vc_{\\ce{Y}}^0\\}.\\)\nDifferent initial conditions lead to different results, see Figs. \\ref{fig:rm11}--\\ref{fig:rm13}. Obviously,  the third point of Theorem \\ref{thm:singlestep} is of main practical use here. For this case one has the following statement.\n\\end{example}\n\n\\begin{corollary}\\label{corr:pure}\nIn the third case of Theorem \\ref{thm:singlestep}, \ndividing \\(\\xi\\) by the initial concentration and scaled by the quantity \\(-\\gamma_m\/V\\), \nwe obtain a (pure) number tending to 1 as \\(t\\) tends to infinity:\n\\fbox{\\(\\lim_{t\\to +\\infty}\n\\frac{\\xi(t)}{Vc_m^0\/(-\\gamma_m)}=1.\\)}\n\\end{corollary}\n\\subsection{Stoichiometric initial condition, excess and deficit}\nBefore studying the above-mentioned figures, we need some definitions in order to avoid the sin of using a concept without having defined it. The concepts of \\emp{stoichiometric initial condition} and \\emp{initial stoichiometric excess} are often used elsewhere but never defined.\n\\begin{definition\nConsider the induced kinetic differential equation \n\\eqref{eq:ikdegen} of the reaction \\eqref{eq:ccr} with the initial condition\n\\(\\mathbf{c}(0)=\\mathbf{c}^0\\neq\\boldsymbol{0}.\\)\nThis initial condition is said to be a \\emp{stoichiometric initial condition} (and \\(\\mathbf{c}^0\\) is a stoichiometric initial concentration), if for all such\n\\(m=1,2,\\dots,M;{\\ }r=1,2,\\dots,R\\) for which \\(\\gamma_{m,r}<0\\) the ratios \\(\\frac{c_m^0}{-\\gamma_{m,r}}\\)\nare independent from \\(m\\) and \\(r.\\) \nIf the ratios are independent of \\(r\\), but for some \n\\(p=1,2,\\dots,M\\) the ratio \\(\\frac{c_p^0}{-\\gamma_{p,r}}\\) is larger than the others, then \\ce{X(p)} is said to be in \\emp{initial stoichiometric excess}, or it is in excess initially.\nIf the ratios are independent of \\(r\\), but for some \n\\(p=1,2,\\dots,M\\) the ratio \\(\\frac{c_p^0}{-\\gamma_{p,r}}\\) is smaller than the others, then \\ce{X(p)} is said to be in \\emp{initial stoichiometric deficit}, or it is in deficit initially.\nThe last notion is mathematically valid, but in such cases, one prefers saying that all the other species are in excess.\nIn combustion theory the expressions \\emp{stoichiometric}, \\emp{fuel lean} and \\emp{fuel rich} are used in the same sense, see page 115 of the book by Tur\\'anyi and Tomlin\\cite{turanyitomlin}.\n\\end{definition}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{p1}\n\\caption{\n\\(c_{\\ce{X}}^0\/2~=~0.7~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}<c_{\\ce{Y}}^0~=~1.2~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(c_{\\ce{Z}}^0~=~3~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(k~=~1.0~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\),\n\\(V~=~1~\\mathrm{dm}^3.\\)\n\\ce{Y} is in stoichiometric excess in the reaction \\ce{2X + Y -> 2Z}. \nThe limiting value is denoted with the dashed line.} \n\\label{fig:rm11}\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{p2}\n\\caption{\n\\(c_{\\ce{X}}^0\/2~=c_{\\ce{Y}}^0=~0.6~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(c_{\\ce{Z}}^0~=~3~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \n\\(k~=~1.0~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\),\n\\(V~=~1~\\mathrm{dm}^3.\\) Stoichiometric initial condition in the reaction \\ce{2X + Y -> 2Z}; slow convergence. The limiting value is denoted with the dashed line.}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{p3}\n\\caption{\n\\(c_{\\ce{X}}^0\/2~=~0.6~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}>c_{\\ce{Y}}^0~=~0.5~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(c_{\\ce{Z}}^0~=~3~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \n\\(k~=~1.0~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\),\n\\(V~=~1~\\mathrm{dm}^3.\\)\n\\ce{X} is in stoichiometric excess in the reaction \\ce{2X + Y -> 2Z}. \nThe limiting value is denoted with the dashed line.}\n\\label{fig:rm13}\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.3\\linewidth]{q1}\\quad\n\\includegraphics[width=0.3\\linewidth]{q2}\\quad\n\\includegraphics[width=0.3\\linewidth]{q3}\n\\caption{Scaled reaction extents tend to 1. \nThe reaction rate coefficient and the initial data are the same as in Figs. \\ref{fig:rm11}--\\ref{fig:rm13}.}\n\\label{fig:rm1skalazott}\n\\end{figure*}\nWe suggest that instead of saying that the scaling factor is some initial concentration as in the third case of Theorem \\ref{thm:singlestep},\none can equally well say that the divisor is the limiting value of the reaction extent, as it is in the cases in figures \\ref{fig:rm1skalazott}: \\(Vc_{\\ce{X}}^0\/2, Vc_{\\ce{X}}^0\/2=Vc_{\\ce{Y}}^0,\nVc_{\\ce{Y}}^0,\\) respectively.\nThis result will come in handy below.\n\nUnder stoichiometric initial conditions Peckham\\cite{peckham} gives a definition of \\(\\xi_{\\max}\\) and proposes to use \\(\\frac{\\xi(t)}{\\xi_{\\max}}\\) in extremely special cases. The domain of this ratio is \\([0,1].\\)\n\nAt this point it may not be obvious how to generalize Corollary \\ref{corr:pure}. In order to treat more complicated cases we shall choose another way.\n\\subsection{Scaling by the \"maximum\"}\nIn cases when \\(\\xi^*:=\\lim_{t\\to+\\infty}\\xi(t)\\) is finite, then \n\\(\\xi^*=\\sup\\{\\xi(t);t\\in\\mathbb{R}\\}\\), thus  \n \\(\\xi^*\\) may be identified (mathematically incorrectly) with \\(\\xi_{\\max},\\) and surely \\(\\lim_{t\\to+\\infty}\\frac{\\xi(t)}{\\xi^*}=1.\\)\n That is the procedure applied by most authors\\cite{dumonlichanotpoquet,vandezandevandergrienddekock,moretti}.\n\\subsection{Detailed balanced reactions}\n\\begin{definition}\nThe complex chemical reaction\n\\begin{equation}\\label{eq:revccr}\n\\ce{$\\sum_{m=1}^M\\alpha_{m,r}$X($m$) <=>[$k_r$][$k_{-r}$] $\\sum_{m=1}^M\\beta_{m,r}$X($m$)}\\quad (r=1,2,\\dots,R);\n\\end{equation}\nendowed with mass action kinetics is said to be \\emp{conditionally detailed balanced} at the positive stationary point \\(\\mathbf{c}^*\\) if\n\\begin{equation}\\label{eq:dbgeneralcond}\nk_r(\\mathbf{c}^*)^{\\boldsymbol{\\alpha}_{.,r}}=k_{-r}(\\mathbf{c}^*)^{\\boldsymbol{\\beta}_{.,r}}\n\\end{equation}\nholds. It is unconditionally \\emp{detailed balanced} if\nEq. \\eqref{eq:dbgeneralcond} holds for any choice of (positive) reaction rate coefficients.\n\\end{definition}\nNote that all the steps in \\eqref{eq:revccr} are reversible. Furthermore, in such cases the reaction steps are indexed\nby \\(r\\) and \\(-r,\\). It is always our choice in which order the forward and backward steps are written, expressing the fact that \"forward\" and \"backward\" has no true physical meaning.\n\\subsubsection{Ratio of two reaction extents.}\nSuppose we have a reversible reaction\n\\begin{equation}\\label{eq:singledb}\n\\ce{$\\sum_{m=1}^M\\alpha_m$X($m$) <=>[$k_1$][$k_{-1}$] $\\sum_{m=1}^M\\beta_m$X($m$)}\n\\end{equation}\nbeing unconditionally detailed balanced because the number of the forward and backward reaction pairs is 1.\nThen the initial value problem for the reaction extents is as follows.\n\\begin{align*}\n&\\dot{\\xi}_1=Vk_1\\prod_{m=1}^{M}(c_m^0+\\gamma_m(\\xi_1-\\xi_{-1})\/V)^{\\alpha_m},&\\xi_1(0)=0,\\\\\n&\\dot{\\xi}_{-1}=Vk_{-1}\\prod_{m=1}^{M}(c_m^0+\\gamma_m(\\xi_1-\\xi_{-1})\/V)^{\\beta_m},&\\xi_{-1}(0)=0,\n\\end{align*}\nwhere \\(\\gamma_m:=\\beta_m-\\alpha_m.\\)\n\\begin{proposition}\\label{prop:specdb}\nUnder the above conditions, one has\n\\fbox{\\(\\lim_{t \\to +\\infty}\n\\frac{\\xi_1(t)}{\\xi_{-1}(t)}=1.\\)}\n\\end{proposition}\nNote that initially one only knows that it is the \\emp{derivative}s of the reaction extents that have the same value at equilibrium.\n\n\\begin{example}\\label{ex:waterform}\nConsider the reversible reaction\n\\ce{2X + Y <=>[$k_1$][$k_{-1}$] 2Z} for water formation with the data \n\\( k_1~=~1~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1},\\) \n\\(k_{-1}~=~1~\\mathrm{dm}^3\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1},\\)\n\\(c_{\\ce{Y}}^0~=~1~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}, c_{\\ce{Z}}^0~=~1~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}.\\)\n\\begin{itemize}\n\\item \nIf \\(c_{\\ce{X}}^0~=~3~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}, \\) then \\ce{X} is in excess initially (a);\n\\item\nif \\(c_{\\ce{X}}^0~=~2~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}, \\) then one has a stoichiometric initial condition (b);\n\\item\nif \\(c_{\\ce{X}}^0~=~1~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}, \\) or\n\\(c_{\\ce{X}}^0~=~1\/2~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3},\\) then \\ce{Y} is in excess initially (c or d).\n\\end{itemize}\nNote that it is not the excess or deficit that is relevant, see Conjecture \\ref{conj:iniratio} below.\nThe initial rates of the forward and backward reactions are as follows:\n\\begin{itemize}\n\\item \n\\(1\\cdot9\\cdot1 > 1\\cdot1,\\)\n\\item \n\\(1\\cdot4\\cdot1 > 1\\cdot1,\\)\n\\item \n\\(1\\cdot1\\cdot1 = 1\\cdot1,\\)\n\\item \n\\(1\\cdot1\/4\\cdot1 <1\\cdot1.\\)\n\\end{itemize}\nThe results are in accordance with Conjecture \\ref{conj:iniratio} below and can be seen in Figs. \\ref{fig:rm21} and \\ref{fig:rm22}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\linewidth]{xexcess1}\\\\\n\\includegraphics[width=0.7\\linewidth]{xexcess2}\\\\\n\\caption{The ratio \\(\\frac{\\xi_{1}(t)}{\\xi_{-1}(t)}\\) is tending to 1 from above: \\ce{X} is in excess at the top figure (case a of Example \\ref{ex:waterform}) and the initial condition is stoichiometric at the bottom figure (case b) in case of the reaction \\ce{2X + Y <=>[$k_1$][$k_{-1}$] 2Z}. \\(V~=~1~\\mathrm{dm}^3,\\) and other data are given in the text.}\n\\label{fig:rm21}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\linewidth]{stoichio}\\\\\n\\includegraphics[width=0.7\\linewidth]{yexcess}\\\\\n\\caption{The ratio \\(\\frac{\\xi_{1}(t)}{\\xi_{-1}(t)}\\) is constant at the top figure (case c of Example \\ref{ex:waterform}) and is tending to 1 from below at the bottom figure (case d) in case of the reaction \\ce{2X + Y <=>[$k_1$][$k_{-1}$] 2Z}. \\ce{Y} is in excess in both cases. \\(V~=~1~\\mathrm{dm}^3,\\) and other data are given in the text.}\n\\label{fig:rm22}\n\\end{figure}\n\\end{example}\n\nNow we formulate our experience collected on several models. \nConsider reaction \\eqref{eq:singledb}.\n\\begin{conjecture}\\label{conj:iniratio}\nThe sign of the difference \\(k_{-1}(\\mathbf{c}^0)^{\\beta}-k_1(\\mathbf{c}^0)^{\\alpha}\\)\nand the sign of \\(\\lim_{t\\to0}\\frac{\\xi_1(t)}{\\xi_{-1}(t)}\\)\nis the same.\n\\end{conjecture}\n\n\nConvergence has been proved above.\nThe ratio at \\(t=0\\) is not defined, but  the limit of the ratio when \\(t \\to +0\\)\ncan be calculated using the l'Hospital Rule as\n\\begin{equation*}\n\\lim_{t\\to+0}\\frac{\\xi_{1}(t)}{\\xi_{-1}(t)}=\n\\lim_{t\\to+0}\\frac{\\dot{\\xi}_{1}(t)}{\\dot{\\xi}_{-1}(t)}=\n\\frac{k_1}{k_{-1}}(\\mathbf{c}^0)^{\\boldsymbol{\\alpha}-\\boldsymbol{\\beta}},\n\\end{equation*}\nand \\(\\frac{k_1}{k_{-1}}(\\mathbf{c}^0)^{\\boldsymbol{\\alpha}-\\boldsymbol{\\beta}}<1\\) is equivalent to saying that \n\\(k_1(\\mathbf{c}^0)^{\\boldsymbol{\\alpha}}<k_{-1}(\\mathbf{c}^0)^{\\boldsymbol{\\beta}}.\\)\n\nThe meaning of the above conjecture is quite obvious: \nif the forward reaction proceeds slower at the beginning than the backward reaction,\nthen limit 1 of the ratio is approached from below etc.\nThe concept of stoichiometric initial condition seems to play no role here.\n\nInstead of studying other simple reactions, we generalize the above result.\n\\subsubsection{Multiplication of the ratios.}\n\\begin{theorem}\\label{thm:product}\nIf the complex chemical reaction \\eqref{eq:revccr} is detailed balanced, i.e. \\eqref{eq:dbgeneralcond} is fulfilled, then\n\\fbox{\\(\n\\lim_{t\\to +\\infty}\n\\prod_{r=1}^R\n\\frac{\\xi_{r}(t)}{\\xi_{-r}(t)}=1.\n\\)}\n\\end{theorem}\n\\Pf{\nIt is similar to that of Proposition \\ref{prop:specdb}: \nall the separate factors tend to 1 as \\(t\\to+\\infty.\\)}\n\n\\begin{example}\nConsider the example in Fig. \\ref{fig:rm3} that is not unconditionally detailed balanced.\n\\begin{figure}\n\\centering\n\\includegraphics[height=0.5\\textheight]{bonyifhj}\n\\caption{A triangle coupled with a Wegscheider-type reaction\\cite{wegscheider}}\n\\label{fig:rm3}\\end{figure}\nIn this case, the condition of detailed balancing to hold is\n\\[\nk_{-1} k_{-2} k_{-3}=k_1 k_2 k_3,\\quad k_3 k_5=k_{-3} k_{-5},\\quad k_3 k_4=k_{-3} k_{-4}\n\\]\nas applying either the \\emp{circuit conditions} and the \\emp{spanning forest conditions} \\cite{feinbergdb} or use the algebraic condition coming from the \\emp{Fredholm alternative theorem} (see p. 133 of\\cite{tothnagypapp}) gives. \nThe calculations leading to similar equalities in more complicated cases\ncan be carried out using the function \\mma{DetailedBalanced} of the package \\mma{ReactionKinetics}, supplement to the book by T\\'oth et al.\\cite{tothnagypapp}. See an example on p. 136 of the cited book.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\linewidth]{megbonyolultabb}\n\\caption{Convergence of the product of ratios from above in case of reaction in Fig. \\ref{fig:rm3} with parameters implying \\emp{detailed balancing} as follows:\n\\(k_1~=~1~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\), \n\\(k_{-1}~=~2~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\), \n\\(k_2~=~3~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\), \n\\(k_{-2}~=~1~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\), \n\\(\\mathbf{k_3}~=~\\frac{1}{3}~ \\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(\\mathbf{k_{-3}}~=~\\frac{1}{2}~ \\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\),\n\\(k_4~=~3~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_{-4}~=~2~{\\mathrm{s}}^{-1}\\), \n\\(k_5~=~3~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\),\n\\(k_{-5}~=~2~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\), \n\\(c_{\\ce{X}}^0~=~c_{\\ce{Y}}^0~=~c_{\\ce{Z}}^0~=~1~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \n\\(V~=~1~\\mathrm{dm}^3.\\)}\n\\label{fig:rm30}\n\\end{figure}\nThus, Fig. \\ref{fig:rm30} shows that the ratio tends to 1. \n\\end{example}\n\\begin{example}\nEven if the reversible reaction is \\emp{not detailed balanced}, it will not blow up\\cite{boroshofbauer}. \nThe stationary point exists and is unique, and the product of the ratios will converge; although the limit will be different from 1, see Fig. \\ref{fig:rm30notdb}. \nLet us use l'Hospital's Rule to show this, in the case of a single factor of the product:\n\\begin{equation*}\n\\lim_{t\\to+\\infty}\\frac{\\xi_r(t)}{\\xi_{-r}(t)}=    \n\\lim_{t\\to+\\infty}\\frac{\\dot{\\xi}_r(t)}{\\dot{\\xi}_{-r}(t)}=\n\\frac{\\dot{\\xi}_r(+\\infty)}{\\dot{\\xi}_{-r}(+\\infty)}=\n\\frac{k_r}{k_{-r}}\\frac{(\\mathbf{c}^*)^{\\boldsymbol{\\alpha}(.,r)}}{(\\mathbf{c}^*)^{\\boldsymbol{\\beta}(.,r)}},\n\\end{equation*}\nwhich is not necessarily equal to 1.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\linewidth]{megbonyolultabbnemdb}\n\\caption{Convergence of the product of ratios in the case of reaction in Fig. \\ref{fig:rm3} with parameters \\emp{not fulfilling detailed balance} as follows:\n\\(\nk_1~=~1~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\), \n\\(k_{-1}~=~2~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\), \n\\(k_2~=~3~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\), \n\\(k_{-2}~=~1~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(\\boldsymbol{k_3}~=~1~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\), \n\\(\\boldsymbol{k_{-3}}~=~1~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\), \n\\(k_4~=~3~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_{-4}~=~2~{\\mathrm{s}}^{-1}\\), \n\\(k_5~=~3~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\),\n\\(k_{-5}~=~2~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\), \n\\(c_{\\ce{X}}^0~=~c_{\\ce{Y}}^0~=~c_{\\ce{Z}}^0~=~1~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3},\\) \n\\(V~=~1~\\mathrm{dm}^3.\\)}\n\\label{fig:rm30notdb}\n\\end{figure}\n\\end{example}\n\\subsection{Reactions with an attractive stationary point}\nAll our observations can be summarized in the trivial proposition below. Before stating it, we need a definition for general ordinary differential equations.\n\nLet \\(M\\in\\mathbb{N},\\)  \\(\\mathbf{f}\\in\\mathcal{C}^1(\\mathbb{R}^M,\\mathbb{R}^M)\\) and consider the initial value problem\n\\begin{equation}\\label{eq:ivp}\n\\dot{\\mathbf{x}}(t)=\\mathbf{f}(\\mathbf{x}(t)), \\mathbf{x}(0)=\\mathbf{x}^0\\, (\\in\\mathbb{R}^M).\n\\end{equation}\n\\begin{definition}\nThe stationary point \\(\\mathbf{x}^*\\) of \\eqref{eq:ivp} is said to be \\emp{attractive}, if all the solutions starting from a neighbourhood of \\(\\mathbf{x}^*\\) are defined for all positive time, and tend to it as \\(t\\to+\\infty\\).\n\\end{definition}\nNote that attractiveness is less then being asymptotically stable and different from being stable. The neighbourhood mentioned in the definition is the \\emp{domain of attraction} of the point \\(\\mathbf{x}^*.\\)\n\nAs a trivial reformulation of the definitions, we arrive at our general statement.\n\\begin{proposition}\\label{prop:main}\nLet \\(\\mathbf{x}^*\\) be an attractive stationary point of the initial value problem Eq. \\eqref{eq:ivp}, and assume that \\(\\mathbf{x}^0\\) is located in the attracting domain of \\(\\mathbf{x}^*\\).\nAssume furthermore, that for some\n\\(g\\in\\mathcal{C}(\\mathbb{R}^M,\\mathbb{R}):g(\\mathbf{x}^*)\\neq0,\\) then \\(\\lim_{t\\to+\\infty}\\frac{g(\\mathbf{x}(t))}{g(\\mathbf{x}^*)}=1.\\)\n\\end{proposition}\nThis proposition is not useful enough. \nThe reason is that in most cases the derivatives of the reaction extents are positive,\ntherefore, the reaction extents themselves tend to infinity, that is they have no (finite) stationary values. \nThey can only have it when the derivative tends to zero like in the consecutive Example \\ref{ex:convexconcave} above: we have seen that \\(\\xi_1(t),\\xi_2(t)\\to Vc^0_{\\ce{X}}.\\)\n\nNow the problem arises that one has to find cases when some of the reaction rates tend to zero,\nas this is a necessary (although not necessarily sufficient!) condition to have a finite reaction extent if \\(t\\to +\\infty.\\) \n\nNote that we did not use Proposition \\ref{prop:main} when calculating the limit of the ratios of reaction extents.\n\\begin{comment}\n\\section{Preliminaries}\nTo make the present text easier to read, here we repeat the most important formulas. Our object of study is\nthe \\emp{complex chemical reaction}, simply \\emp{reaction}, or \\emp{reaction network} as a set consisting of \\emp{reaction steps} as follows: \n\\begin{equation}\\label{eq:ccr}\n\\left\\{\\ce{$\\sum_{m=1}^M\\alpha_{m,r}$X($m$) -> $\\sum_{m=1}^M\\beta_{m,r}$X($m$)}\\quad (r=1,2,\\dots,R)\\right\\} \n\\end{equation}\nwith the usual notations: \n\\(\\alpha_{m,r}\\) and \\(\\beta_{m,r}\\)\nare the stoichiometric coefficients, \\ce{X($m$)} are the species. \nThe induced kinetic differential equation describing the time evolution of the quantities of the species is in general \\begin{equation}\\label{eq:ikdegen}\n\\dot{\\mathbf{c}}(t)=\\boldsymbol{\\gamma}\\mathbf{rate}(\\mathbf{c}(t))\n\\end{equation} \ntogether with the initial condition \\(\\mathbf{c}(0)=\\mathbf{c}^0.\\) The component \\(rate_r\\) of the vector \\(\\mathbf{rate}\\) provides the reaction rate of the \\(r^{\\mathrm{th}}\\) reaction step.\nNote that in the mass action case Eq. \\eqref{eq:ikdegen} specializes into\n\\begin{equation}\\label{eq:ikdemass}\n\\dot{\\mathbf{c}}=\\boldsymbol{\\gamma}\\mathbf{k}\\odot\\mathbf{c}^{\\boldsymbol{\\alpha}} \n\\end{equation} \nor, in coordinates\n\\begin{equation*\n\\dot{c}_m(t)=\\sum_{r=1}^{R}\\gamma_{mr}k_r\\prod_{p=1}^{M}c_p^{\\alpha_{p,r}}\\quad(m=1,2,\\cdots,M),\n\\end{equation*}\nwhere \\(\\mathbf{k}\\) is the vector of (positive) reaction rate coefficients \\(k_r.\\)\n\nThe \\emp{reaction extent}\nof a complex chemical reaction or reaction network defined by Eq. \\eqref{eq:ccr} is the scalar variable, vector-valued function given by the formula\n\\begin{equation}\\label{eq:specre}\n\\boxed{\n\\boldsymbol{\\xi}(t):=V\\int_0^t\n\\mathbf{rate}(\\mathbf{c}(\\overline{t}))\n{\\;\\mathrm{d}\\overline{t}}}.\n\\end{equation}\nIts time derivative \\(\\dot{\\boldsymbol{\\xi}}(t)=V\\mathbf{rate}(\\mathbf{c}(t))\\) is usually called the \\emp{rate of conversion}.\nWe shall calculate the reaction extent for a few exotic reactions and see how they reflect the special properties of the reactions.\n\\end{comment}\n\\section{What if the conditions are not fulfilled?}\nIn the first part of the paper, we calculated the reaction extents for the reaction steps of simple reactions, for detailed balanced reactions, for reactions with a kinetic differential equation having an attracting stationary point, etc. \nOur main question in the present part is:\nWhat happens if one takes an exotic reaction that has multiple stationary point(s), and shows oscillations or even chaos?\n\\subsection{Multistationarity}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\linewidth]{multist1}\n\\caption{The Horn--Jackson reaction network\nwith \\(k_1=k_3=1~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\) and \\(k_2=k_4=k\\). \nThe value of \\(k\\) is varied as described in the text, \nits unit is the same as that of the other rate coefficients.}\n\\label{fig:hj0}\n\\end{figure}\nHorn and Jackson\\cite{hornjackson}, (see p. 110) has shown that the complex chemical reaction in Fig. \\ref{fig:hj0} has three (positive) stationary points in every stoichiometric compatibility class if \nthe numerical value of \\(k\\) lies between 0 and \\(\\frac{1}{6}.\\)\nTo be more specific, let us choose \n\\(\nk~=~\\frac{1}{10}~\\mathrm{dm}^6~\\,\\mathrm{mol}^{-2}~\\mathrm{s}^{-1}\\), and  \\(c^0_{\\ce{X}}~=~c^0_{\\ce{Y}}~=~\\frac{1}{2}~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}.\\) \nThen, easy calculation--neglecting the units for simplicity--shows that in the \\emp{stoichiometric compatibility} class\n\\(\\{[c_{\\ce{X}}\\quad c_{\\ce{Y}}]; c_{\\ce{X}}+c_{\\ce{Y}}=1\\}\\)\n(or, for cases when the total concentration is unity) there are three stationary points:\n\\begin{enumerate}\n\\item \nthe stationary point\n\\(\n\\mathbf{c}^*_1:=\\begin{bmatrix}\n\\frac{1}{3 + \\sqrt{3}}&\\frac{3 + \\sqrt{3}}{6}\n\\end{bmatrix}\n\\) is  globally asymptotically stable (i.e. attracting)  with the attracting domain \n\\(\\{[c_{\\ce{X}}\\quad c_{\\ce{Y}}]: 0\\le c_{\\ce{X}}<\\frac{1}{3 + \\sqrt{3}}, c_{\\ce{X}}+c_{\\ce{Y}}=1\\},\n\\)  and\n\\item \nthe stationary point \\(\\mathbf{c}^*_2:=[\\frac{1}{2}\\quad\\frac{1}{2}]\\)\nis unstable (i.e. non-attracting),  and\n\\item \nthe stationary point\n\\(\n\\mathbf{c}^*_3:=\\begin{bmatrix}\n\\frac{1}{3 - \\sqrt{3}}&\\frac{3 - \\sqrt{3}}{6}\n\\end{bmatrix}\n\\) \nis globally asymptotically stable (i.e. attracting)\n with the attracting domain \n\\(\n\\{[c_{\\ce{X}}\\quad c_{\\ce{Y}}];\\frac{1}{3 - \\sqrt{3}}<c_{\\ce{X}}\\le1, c_{\\ce{X}}+c_{\\ce{Y}}=1\\}\n\\).\n\\end{enumerate}\nThe character of the stationary points can be determined by using standard linear stability analysis.\nFig. \\ref{fig:hj1} summarizes the behavior of the trajectories in the neighborhood of the stationary points. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\linewidth]{multist2}\n\\caption{Phase plane of the reaction network in Fig. \\ref{fig:hj0}.\nThe black line is one of the compatibility classes, where the total concentration is unity. \\(E_1, E_2\\), and \\(E_3\\) are the sets of stationary points, and their intersections with the line of the given compatibility class results in the actual stationary points \\(\\mathbf{c}^*_1, \\mathbf{c}^*_2\\), and \\(\\mathbf{c}^*_3\\). Blue denotes asymptotic stability, red denotes instability. The black arrows show the direction of the motion along the trajectories in the neighborhood of the attracting stationary points.}\n\\label{fig:hj1}\n\\end{figure}\n\nThe reaction extents tend to infinity in all cases,\nbut they are ordered differently for different initial conditions, see in Fig. \\ref{fig:hj}.\nThis reflects the fact that not the same reactions are the fastest or slowest in the two cases.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\linewidth]{multist4}\\\\\n\\includegraphics[width=0.85\\linewidth]{multist5}\n\\caption{Reaction extents starting from the attracting domain of \\(\\mathbf{c}^*_1\\) (above) and \\(\\mathbf{c}^*_3\\) (below). As there are large differences in the values,\nthe smallest one seems to be zero, but it is not.\nThe reaction steps in  Fig. \\ref{fig:hj0} are numbered clockwise, starting with the reaction \\ce{X + 2Y -> 3Y}. \nNote that the ranking of the reaction extents is different in the two cases.}\n\\label{fig:hj}\n\\end{figure}\n\\subsection{Oscillation}\nWe shall study here two oscillatory reactions. \nFirst, the often used Lotka--Volterra reaction\\cite{lotka,volterra} comes that is not only theoretically interesting because it can be used to describe the oscillations in cold flames\\cite{frankkamenetskii} or see Ref.\\cite{frankkamenetskiiPUP}.\nThe experimentally based R\\'abai reaction\\cite{rabai} aimed at describing pH oscillations follows as the second. \nOne may say that the Brusselator model\\cite{prigoginelefever} would be a more realistic choice, as it results in  limit cycle solutions. However, it has a third-order step that makes the calculations more tedious.\nThe type of calculations shown below would give almost the same kind of results with the Brusselator, too.\n\\subsubsection{The Lotka--Volterra reaction.}\nThe irreversible and reversible cases behave in qualitatively different ways.\n\n\\paragraph{Irreversible case:}\n\nIt is known\\cite{potalotka,schumantoth} that under some mild conditions the only two-species reaction to show oscillations is the (irreversible) Lotka--Volterra reaction \\ce{X ->[$k_1$] 2X}, \\ce{X + Y ->[$k_2$] 2Y}, \\ce{Y ->[$k_3$] 0}. (Cf. also the paper by T\\'oth and H\\'ars\\cite{tothhars} and that by Banaji and Boros\\cite{banajiboros}.) \nIt has a single positive stationary point that is stable but not attractive, therefore, one cannot apply Proposition \\ref{prop:main} above.\nNote that the individual reaction extents are not oscillating; they are \"pulsating\" while monotonously increasing to infinity. \nThey have an oscillatory derivative, and the zeros of their second derivative clearly show the endpoints of the periods, \nsee Fig. \\ref{fig:pulsating}. \nIt may be a good idea to calculate any kind of reaction extent \\emp{for a period} in case of oscillatory reactions.\nWe are going to study this point later.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{lotkaxi0}\\\\\n\\includegraphics[width=0.85\\linewidth]{lotkaxi1}\\\\\n\\includegraphics[width=0.85\\linewidth]{lotkaxi2}\n\\caption{The individual reaction extents and their first and second derivatives in case of the irreversible Lotka--Volterra reaction with \n\\(k_1~=~3~{\\mathrm{s}}^{-1}\\),\n\\(k_2~=~4~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\), \n\\(k_3~=~5~{\\mathrm{s}}^{-1}\\),\n\\(c^0_{\\ce{X}}~=~1~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}, c^0_{\\ce{Y}}~=~2~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \n\\(V~=~1~\\mathrm{dm}^3.\\)}\n\\label{fig:pulsating}\n\\end{figure}\nIt is interesting to have a look at the ratios of the reaction extents, as they seem to tend 1, see Fig. \\ref{fig:lotkaratios}.\nWe assume that this phenomenon is related to the fact that the oscillatory solution results in a closed curve in the phase plane of the irreversible Lotka--Volterra reaction.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{lotkaratio12}\\\\\n\\includegraphics[width=0.75\\linewidth]{lotkaratio23}\\\\\n\\includegraphics[width=0.75\\linewidth]{lotkaratio31}\n\\caption{The ratios of the reaction extents in case of the irreversible Lotka--Volterra reaction with the parameters as in Fig. \\ref{fig:pulsating}.}\n\\label{fig:lotkaratios}\n\\end{figure}\n\n\\paragraph{Reversible case, detailed balanced:}\n\nThe reversible Lotka--Volterra reaction\n\\ce{X <=>[$k_1$][$k_{-1}$] 2X}, \n\\ce{X + Y <=>[$k_2$][$k_{-2}$] 2Y}, \n\\ce{Y <=>[$k_3$][$k_{-3}$] 0}\nis also worth studying.\nFirst, let us note that for all values of the reaction rate coefficients it has a single, positive stationary point because the reaction steps are reversible. \nTherefore, the system is permanent\\cite{simon,borosexistence}, i.e. the trajectories remain in a compact set.  \nIf the trajectories remain in a compact set, then they are either tending to a limit cycle, or the stationary point is asymptotically stable. \nThe first possibility is excluded by the above-mentioned theorem by P\\'ota\\cite{potalotka}, thus it is only the second possibility that remains. \nFig. \\ref{fig:lvrevdb} shows the behavior of the individual reaction extents.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{lotkarevxi0}\\\\\n\\includegraphics[width=0.85\\linewidth]{lotkarevxi1}\\\\\n\\includegraphics[width=0.85\\linewidth]{lotkarevxi2}\n\\caption{The individual reaction extents and their first and second derivatives in case of the reversible, detailed balanced Lotka--Volterra reaction with \n\\(k_1~=~1~{\\mathrm{s}}^{-1}\\),\n\\(k_{-1}~=~2~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_2~=~3~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_{-2}~=~4~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_3=~5~{\\mathrm{s}}^{-1}\\),\n\\(k_{-3}~=~\\frac{15}{8}~\\,\\mathrm{mol}~\\mathrm{dm}^{-3}~{\\mathrm{s}}^{-1}\\),\n\\(c^0_{\\ce{X}}~=~1~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \\(c^0_{\\ce{Y}}~=~2~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \\(V~=~1~\\mathrm{dm}^3.\\)}\n\\label{fig:lvrevdb}\n\\end{figure}\nLet us note that both the existence and uniqueness of the stationary state also follow from the Deficiency One Theorem (see p. 106 in Feinberg\\cite{feinbergbook}, or p. 176 in T\u00f3th et al.\\cite{tothnagypapp})\n\nIf the reaction is detailed balanced that holds if and only if\n\\begin{equation}\\label{eq:lvdb}\nk_1k_2k_3=k_{-1}k_{-2}k_{-3}\n\\end{equation}\nis true, then our Proposition 2 of the previous paper implies that\nthe product of the ratios of the reaction extents tends to 1, see Fig \\ref{fig:lotkarevdbratios}. \nThis follows also from our Theorem 3 there.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{lotkarevdbratios}\\\\\n\\includegraphics[width=0.85\\linewidth]{lotkarevdbprod}\n\\caption{\nAbove: Time evolution of the ratios of the individual reaction extents---blue for Reaction (1), orange for Reaction (2), and green for Reaction (3)---in case of the reversible, detailed balanced Lotka--Volterra reaction with\n\\(k_1~=~1~{\\mathrm{s}}^{-1}\\),\n\\(k_{-1}~=~2~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\), \n\\(k_2~=~3~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_{-2}~=~4~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_3=~5~{\\mathrm{s}}^{-1}\\),\n\\(k_{-3}~=~\\frac{15}{8}~\\,\\mathrm{mol}~\\mathrm{dm}^{-3}~{\\mathrm{s}}^{-1}\\),\n\\(c^0_{\\ce{X}}~=~1~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \\(c^0_{\\ce{Y}}~=~2~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \\(V~=~1~\\mathrm{dm}^3.\\)\nBelow: Time evolution of the product of the ratios tending to 1.}\n\\label{fig:lotkarevdbratios}\n\\end{figure}\n\n\\paragraph{Reversible case, not detailed balanced:}\n\nIf Condition \\eqref{eq:lvdb} does not hold, the reaction still has an \\emp{attracting stationary point}. What is more, it has an asymptotically stable stationary point. \nFig. \\ref{fig:lotkanotdbxi} shows the behavior of the individual reaction extents.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{lotkanotdbxi0}\\\\\n\\includegraphics[width=0.85\\linewidth]{lotkanotdbxi1}\\\\\n\\includegraphics[width=0.85\\linewidth]{lotkanotdbxi2}\n\\caption{The individual reaction extents and their first and second derivatives in case of the reversible, \\emp{not} detailed balanced Lotka--Volterra reaction with \\(k_1~=~1~{\\mathrm{s}}^{-1},\\)\n\\(k_{-1}~=~2~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_2~=~3~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_{-2}~=~4~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_3=~5~{\\mathrm{s}}^{-1}\\),\n\\(k_{-3}~=~6~\\,\\mathrm{mol}~\\mathrm{dm}^{-3}~{\\mathrm{s}}^{-1}\\),\n\\(c^0_{\\ce{X}}~=~1~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \\(c^0_{\\ce{Y}}~=~2~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\), \\(V~=~1~\\mathrm{dm}^3.\\)}\n\\label{fig:lotkanotdbxi}\n\\end{figure}\n\n\\subsubsection{The R\\'abai reaction of pH oscillation.}\nHere we include a reaction proposed by R\\'abai\\cite{rabai} to describe pH oscillations. \nThis reaction has much more direct contact with chemical kinetic experiments, and it is much more challenging---from the point of view of numerical mathematics---than the celebrated Lotka--Volterra reaction.\n\nR\\'abai\\cite{rabai} starts with the steps\n\\begin{align*}\n&\\ce{A- + H+ <=>[$k_1$][$k_-1$] AH},\\\\\n&\\ce{AH + H+ +\\{B\\} ->[$k_2$] 2H+ + P-}\n\\end{align*}\nwhere \\ce{\\{B\\}} is an external species with a constant concentration. \nThis reaction has a single stationary point\n\\begin{equation*}\nc^*_{\\ce{A-}}=0,\\quad\nc^*_{\\ce{H+}}=c^0_{\\ce{H+}}+c^0_{\\ce{AH}},\\quad\nc^*_{\\ce{AH}}=0,\\quad\nc^*_{\\ce{P}}=c^0_{\\ce{A-}}+c^0_{\\ce{AH}}+c^0_{\\ce{P}}\n\\end{equation*}\nspecializing into\n\\(\nc^*_{\\ce{A-}}=0,\nc^*_{\\ce{H+}}=c^0_{\\ce{H+}},\nc^*_{\\ce{AH}}=0,\nc^*_{\\ce{P}}=c^0_{\\ce{A-}}\n\\)\nwith the natural restriction on the initial condition\n\\(\nc^0_{\\ce{AH}}=0,\nc^0_{\\ce{P}}=0.\n\\)\n\nPutting the reaction into a CSTR (continuously stirred flow-through tank reactor) \nmeans in the terms of formal reaction  kinetics that\nall the species can flow out and some of the species may flow in, \nso that in the meantime the volume is maintained constant.\nIn the present case, the following steps are added\n\\begin{align}\n&\\ce{A- ->[$k_0$] 0},\\label{rabaiout1}\\\\\n&\\ce{0 ->[$k_0c^0_{\\mathrm{A}^-}$]A-},\\label{rabaiin1}\\\\\n&\\ce{H+ ->[$k_0$] 0},\\label{rabaiout2}\\\\\n&\\ce{0 ->[$k_0c^0_{\\mathrm{H}^+}$] H+},\\label{rabaiin2}\\\\\n&\\ce{AH ->[$k_0$] 0},\\label{rabaiout3}\n\\end{align}\nwhere $k_0$ is the volumetric flow rate normalized to the volume of the reactor (often called the reciprocal of the residence time) measured in unit \\(\\mathrm{s}^{-1}\\). \nAs a result of adding these steps, multistability may occur with appropriately chosen values of the parameters. When the reaction step\n\\begin{equation}\\label{rabai3}\n\\ce{H+ + \\{C$^-$\\} ->[$k_3$] CH}\n\\end{equation}\nis also added, one may obtain periodic solutions having appropriate parameter values, see Fig. \\ref{fig:rabaiosc}. Let us remark that neither the R\\'abai reaction nor the Lotka--Volterra reaction is mass-conserving.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{rabaioscsol}\\\\\n\\includegraphics[width=0.85\\linewidth]{rabaiosctraj}\n\\caption{Time evolution of the pH and the projection of the negative logarithm of the first three coordinates of the trajectory in case of the oscillating R\\'abai reaction with \n\\(k_1~=~10^{10}~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_{-1}~=~10^3~{\\mathrm{s}}^{-1}\\),\n\\(k_2~=~10^6~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_3~=~1~\\mathrm{dm}^3~\\,\\mathrm{mol}^{-1}~\\mathrm{s}^{-1}\\),\n\\(k_0~=~10^{-3}~{\\mathrm{s}}^{-1}\\),\n\\(c^0_{\\ce{A-}}~=~5~\\times~10^{-3}~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(c^0_{\\ce{H+}}~=~10^{-3}~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(c^0_{\\ce{AH}}~=~0~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(c^0_{\\ce{P}}~=~0~\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\),\n\\(V~=~1~\\mathrm{dm}^3.\\)}\n\\label{fig:rabaiosc}\n\\end{figure}\n\nIt is instructive to cast a glance to the reaction extents in such a complex system.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{rabaioscext01forward}\\\\\n\\includegraphics[width=0.85\\linewidth]{rabaioscext01backward}\n\\caption{Reaction extents of the forward (above) and backward (below) steps of the fast equilibrium reaction \\ce{A- + H+ <=>[$k_1$][$k_-1$] AH} of the oscillating R\\'abai reaction (in the same time window as that of in Fig. \\ref{fig:rabaiosc}.)}\n\\label{fig:rabaioscext12} \n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{rabaioscext02}\\\\\n\\includegraphics[width=0.85\\linewidth]{rabaioscext03}\n\\caption{Reaction extents of the reaction steps\n\\ce{AH + H+ +\\{B\\} ->[$k_2$] 2H+ + P-} (left) and \n\\ce{H+ + \\{C$^-$\\} ->[$k_3$] CH} (right) of the oscillating R\\'abai reaction (in the same time window as that of in Fig. \\ref{fig:rabaiosc}.)}\n\\label{fig:rabaioscext23} \n\\end{figure}\nNote that the reaction extents of the fast equilibrium reaction \\ce{A- + H+ <=>[$k_1$][$k_-1$] AH} shown in Fig. \\ref{fig:rabaioscext12} are practically the same, or: their ratio tends to 1, as if no other steps were present! \nThey are also four-five orders of magnitude higher than those of the auto-catalytic production \\ce{AH + H+ +\\{B\\} ->[$k_2$] 2H+ + P-} and the slow pseudo first-order chemical removal \\ce{H+ + \\{C$^-$\\} ->[$k_3$] CH} of \\ce{H+} ion shown in Fig. \\ref{fig:rabaioscext23}.\nNote also the step-wise increase of reaction extent \\(\\xi_3(t)\\).\n\\subsection{Chaos}\nHere we use a version of the R\\'abai reaction that \\emp{can numerically be shown} to exhibit chaotic behavior, see Fig. \\ref{fig:rabaichaos}. \nThis is a good model for experimental pH oscillators also showing behavior that seems to be chaotic according to the usual standards.\n\nWhen the reaction step \\eqref{rabai3} is made reversible  \n\\begin{equation}\\label{rabai5}\n\\ce{H+ + \\{C$^-$\\} <=>[$k_3$][$k_{-3}$] CH},\n\\end{equation}\nand one also introduces both the chemical \"removal\" and the outflow of \\ce{CH} \n\\begin{equation}\\label{rabai6}\n\\ce{CH ->[$k_{4}$] Q},\n\\end{equation}\n\\begin{equation}\\label{rabai7}\n\\ce{CH ->[$k_{0}$] 0},\n\\end{equation}\nchaotic solutions are obtained by using appropriate parameters and favorable input concentrations. Fig. \\ref{fig:rabaichaos} illustrates this fact. The reaction extents tend to \\(+\\infty\\) (see for example Fig. \\ref{fig:rabaichaosextent}) in such a way that their derivative is chaotically oscillating (not shown), as expected.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{rabaichaossol}\\\\\n\\includegraphics[width=0.85\\linewidth]{rabaichaostraj}\n\\caption{Time evolution of the pH and the projection of the negative logarithm of the first two coordinates of the trajectory in case of the chaotically oscillating R\\'abai reaction with \nthe same parameters as in Fig. \\ref{fig:rabaiosc} and\n\\(k_{-3}~=~1.5~\\times~10^{-2}~{\\mathrm{s}}^{-1}\\),\n\\(k_{4}~=~~5~\\times~10^{-2}~{\\mathrm{s}}^{-1}.\\) \n}\n\\label{fig:rabaichaos} \n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{rabaichaosextent}\n\\caption{Time evolution of the reaction extent \nof the forward step of reaction \\eqref{rabai5}\nin case of the chaotic R\\'abai reaction with the same parameters as in Fig. \\ref{fig:rabaichaos}.}\n\\label{fig:rabaichaosextent}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{Conclusions}\n\nA generalized definition for the reaction extent has been given by Bowen\\cite{bowen} (included in Chapter 6 of the book edited and partially written by Truesdell\\cite{truesdell}).\nAnother definition that turned out to be much better fitting into the framework of modern formal reaction kinetics in the last 50 years was given by Aris\\cite{arisprolegomena1}. \nStill, neither of them became popular among chemists and chemical engineers. Our goal here is to further generalize the definition (essentially by Aris) to make it compatible with the present theory of reaction kinetics. The result will reveal that there existed a kind of sleeping definition with no use in chemical kinetics, and we show that this should not be the case.\n\nWe have introduced the concept of reaction extent for reaction networks of arbitrary complexity (any number of reaction steps and species) without assuming mass action kinetics. \nThe newly defined reaction extent gives the advancement of each individual irreversible reaction step; \nin case of reversible reactions, we have a pair of reaction extents. \nIn all the practically important cases, the  fact that the reaction extent is strictly monotonously increasing, implies that the reaction events never stop. \nThis observation sheds new light onto the concept of \\emp{dynamic equilibrium},\nwithout alluding to either thermodynamics or statistical mechanics.\n\nAfter a few statements on the qualitative behaviour of the reaction extent we made efforts to connect the notion with the traditional ones. \nThus, we have shown that if the number of reaction steps is one, \nthe reaction extent in the long run (as \\(t\\to+\\infty\\))\ntends to 1 if appropriately scaled.\nWe have not used the expression \\emp{progress of reaction}, \nand even less the \\emp{reaction coordinate}. \nWe agree that it is convenient to accept the proposal by Aris\\cite{aris} to work with \\(\\frac{1}{V}\\boldsymbol{\\xi}\\), that is usually called the \\emp{degree of advancement}. \nWe have also shown that for an arbitrary number of reversible detailed balanced reaction steps the product of the ratios of the individual reaction extents also tends to 1 as \\(t\\to+\\infty.\\)\n\nOur most general statement follows for arbitrary reactions having an attracting stationary point\nand with a function not vanishing on the stationary point: in this case the value of the chosen function \nalong the time dependent concentrations divided by the value of the given function at the stationary concentration tends to 1. \nThus, this statement is true not only for the reaction extent but also for any appropriate functions.\n\nOne should take into consideration that although in the practically interesting cases, when the number of equations \\(R\\) in \\eqref{eq:rmdiffegy} for the reaction extents is larger than \\(M\\), i.e. the number of the kinetic differential equations in \\eqref{eq:ikdegen}: \\(R > M\\), then\nthe equations for the reaction extents are of a much simpler structure, \nas the right hand side of the differential equations \\eqref{eq:rmdiffegy} describing them consist only of a single term. \nDuring calculations, we had the experience that it was numerically less demanding to solve the system of differential equations of the reaction extents than those of the concentrations. \n\nThe main advantage of the new definition of reaction extent is that by knowing the kinetic model of a reacting system one can now calculate not only the time evolution of the concentration of each reacting species but also the number of occurrences of the individual reaction events.\n\nAs a by-product we have given an exact definition of the stoichiometric initial concentrations and the initial concentration in excess. \nOne can  say that the concept of the newly defined reaction extent can be usefully applied to a larger class of reactions than usual, but in some (exotic) cases its use needs further investigations,\nthis we will start in the forthcoming paper.\nIt is for the reader to decide if we succeeded in avoiding all the traps mentioned in the Introduction. \nQuite a few authors treat the methodology of teaching the concept\\cite{garst,moretti,vandezandevandergrienddekock};\nwe think this approach will only have its \\textit{raison d'\\^{e}tre} when the scientific background will have been clarified and agreed on. \n\n\nLet us mention a few limitations and future directions of research. We have assumed throughout that volume (together with temperature and pressure) is constant. Tacitly, we assumed that we deal with homogeneous kinetics; heterogeneous systems are not taken into consideration. Also, we have not dealt with reaction-diffusion systems. \nWe mention that recently, Peka\\v{r}\\cite{pekar} and Rodrigues et al.\\cite{rodriguesbilleterbonvin} have applied the concept of reaction extent to the case when diffusion is also present. \nWe have also mentioned a few mathematical conjectures that are to be investigated later. \n\n\\section*{Supporting Information}\nThe file FiguresandCalculationGasparToth.pdf contains all the calculations and drawings made using the Wolfram language. \nInterested readers may request from the authors the .nb file usable for calculations.\n\n\\section*{Author Contributions}\nThe authors equally participated in all parts of the paper.\n\n\\section*{Conflicts of interest}\nThere are no conflicts to declare.\n\n\\section*{Acknowledgements}\nThe present work has been supported by the National Office for Research and Development  (2018-2.1.11-T\\'ET-SI-2018-00007 and FK-134332). \nJT is grateful to Dr. J. Karsai (Bolyai Institute, Szeged University) and for Daniel Lichtblau (Wolfram Research) for their help. \nMembers of the Working Committee for Reaction Kinetics and Photochemistry, \nespecially Profs. T. Tur\\'anyi and G. Lente, furthermore Drs. Gy. P\\'ota and T. Nagy, made a number of useful critical remarks.\n\n\\section*{Notations}\nSome of the readers may appreciate that we have collected the used notations.\n\n\\begin{table*}\n\\caption{Notations}\n\\label{tbl:notations1}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}clcc}\n\\hline\nNotation & Meaning & Unit & Typical value \\\\\n\\hline\\([a, b[\\)&left-closed, right-open interval&&\\\\\n\\hline\\(\\Longrightarrow\\)&implies&&\\\\\n\\hline\\(\\in\\)&belongs to&&\\\\\n\\hline\\(\\forall\\)&universal quantifier&&\"for all\"\\\\\n\\hline\\(\\exists\\)&existential quantifier&&\"there is\"\\\\\n\\hline\\(\\odot\\)&coordinate-wise product of vectors&&\\\\\n\\hline\\(\\mathbf{A}^{\\top}\\)&transpose of the matrix \\(\\mathbf{A}\\)&&\\\\\n\\hline\\(c_m, c_{\\ce{X}}\\) & concentration of \\ce{X($m$)} and \\ce{X} & \\(\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\)& \\\\\n\\hline\\(\\mathbf{c}\\) &vector of concentrations  & & \\\\\n\\hline\\(c_m^0\\) & initial concentration of \\ce{X($m$)} & \\(\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\)& \\\\\n\\hline\\(c_m^*\\) & stationary concentration of \\ce{X($m$)} & \\(\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}\\)& \\\\\n\\hline\\(\\mathcal{C}^i(A,B) \\) & \\(i\\) times continuously differentiable \\\\\n  & functions from \\(A\\) into \\(B\\) & & \\\\\n\\hline\\(\\Dom(u)\\) & the domain of the function \\(t\\mapsto u(t)\\)  & &\\\\\n\\hline \\(J\\subset\\mathbb{R}\\) & an open interval &&\\\\ \n\\hline \\(k, k_r,k_{-r}\\) & reaction rate coefficient &\\((\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3})^{1-\\sum_{m=1}^{M}\\alpha_{m,r}}~{\\mathrm{s}}^{-1}\\)&\\\\\n\\hline \\(k_{0}\\) & normalized volumetric flow rate& \\({\\mathrm{s}}^{-1}\\) &\\\\\n\\hline\\(M\\) & the number of chemical species & & \\(\\in\\mathbb{N}\\)\\\\\n\\hline\\(n_m\\) & the quantity of species \\ce{X($m$)}&\\(\\,\\mathrm{mol}\\) & \\(\\in\\mathbb{N}\\)\\\\\n\\hline\\(n_m^0\\) & the initial quantity of species \\ce{X($m$)}  &\\(\\,\\mathrm{mol}\\) & \\(\\in\\mathbb{N}\\)\\\\\n\\hline\\(\\mathbf{n}\\) & the vector of the quantity of species & &\\\\\n\\hline\\(\\mathbf{n}^0\\) & the vector of the initial  quantity of species\\\\\n\\hline \\(\\mathbb{N}\\) & the set of positive integers &&\\\\\n\\hline \\(\\mathbb{N}_0\\) & the set of non-negative integers &&\\\\\n\\hline\\(rate_r\\) & rate of the \\(r^{\\mathrm{th}}\\) reaction step &\\(\\,\\mathrm{mol}\\;\\mathrm{dm}^{-3}~\\mathrm{s}^{-1}\\) & \\\\\n\\hline\\(\\mathbf{rate}\\) & vector of reaction rates & & \\\\\n\\hline \\(\\mathbb{R}\\) & the set of  real numbers&&\\\\\n\\hline \\(\\mathbb{R}^+\\) & the set of positive real numbers &&\\\\\n\\hline \\(\\mathbb{R}^+_0\\) & the set of non-negative real numbers &&\\\\\n\\hline\\(t\\) & time & s &  \\(\\in\\mathbb{R}\\)\\\\\n\\hline \\(V\\) & volume & \\(\\mathrm{dm}^3\\) & \\\\\n\\hline \\(W_r\\) & number of occurrences of the \\(r^{\\mathrm{th}}\\) step & &\\(\\in\\mathbb{N}_0\\) \\\\\n\\hline \\(\\mathbf{W}\\) & vector of number of occurrences &&\\\\\n\\hline \\(x_m\\) & \\(m^{\\mathrm{th}}\\) dependent variable in a &  &\\\\\n& differential equation&&\\\\\n\\hline\\(\\mathbf{x}\\) & vector of variables in a &  &\\\\\n& differential equation &&\\\\\n\\hline\\(\\ce{X},\\ce{Y},\\ce{X($m$)}\\) &chemical species  & & \\\\\n\\hline\\(\\alpha_{m,r},\\alpha_{m}\\) & stoichiometric coefficient & 1 & \\(0,1,2,3\\) \\\\\n& in the reactant complex &  & \\\\\n\\hline \\(\\boldsymbol{\\alpha}\\)  & matrix of reactant complex vectors  & & \\\\\n\\hline \\(\\beta_{m,r},\\beta_{m}\\) & stoichiometric coefficient & 1 & \\(0,1,2,3\\) \\\\\n& in the product complex &  & \\\\\n\\hline \\(\\boldsymbol{\\beta}\\)  & matrix of  product complex vectors & & \\\\\n\\hline \\(\\gamma_{m,r},\\gamma_{m}\\) &  stoichiometric number & 1 & \\(-3,-2,\\dots,2,3\\) \\\\\n\\hline \\(\\boldsymbol{\\gamma}\\) & stoichiometric matrix &  &\\\\\n\\hline \\(\\xi_r\\) & reaction extent of the \\(r^{\\mathrm{th}}\\) step &\\(\\,\\mathrm{mol}\\)&\\\\\n\\hline \\(\\boldsymbol{\\xi}\\) &vector of reaction extents & & \\\\\n\\hline\n\\end{tabular*}\n\\end{table*}\n\n\n\n\n\\renewcommand\\refname{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn 1992 Sedykh \\cite{sedykh:originalvertex, sedykh:vertex} generalized the classical four vertex theorem of planar curves by showing that the torsion of any closed space curve  vanishes at least $4$ times, if it lies on a convex surface, see Note \\ref{note:4vertex}. Recently the author \\cite{ghomi:rosenberg} extended Sedykh's theorem to curves which lie on a \\emph{locally} convex simply connected surface. In this work we prove another generalization of Sedykh's theorem which does not require the existence of any underlying surface for the curve. \n\nTo state our result, let us recall that \na set $X$ in Euclidean space $\\mathbf{R}^3$ is \\emph{star-shaped} with respect to a point $o$ if no ray emanating from $o$ intersects $X$ in more than one point. Further let us say that $X$ is \\emph{locally convex} with respect to $o$ if through  every point $p$ of $X$ there passes a plane $H$, not containing $o$, such that a neighborhood of $p$ in $X$ lies on the same side of $H$ as does $o$. For instance, the boundary of any convex body in $\\mathbf{R}^3$ is both star-shaped and locally convex with respect to any of its interior points. In this paper we show:\n\n\\begin{thm}\\label{thm:main}\nLet $\\Gamma\\subset\\mathbf{R}^3$ be a simple closed $\\mathcal{C}^3$ immersed curve with nonvanishing curvature. Suppose that $\\Gamma$ is star-shaped and locally convex with respect to a point  $o$ in the interior of its convex hull. Then the torsion of $\\Gamma$ changes sign at least $4$ times.\n\\end{thm}\n\nIn particular,  if $\\Gamma$ lies on  the boundary of a convex body, then it immediately follows that $\\Gamma$ has at least $4$ points of vanishing torsion, which is Sedykh's result. The above theorem also generalizes a similar result of Thorbergsson and Umehara \\cite[Thm. 0.2]{thorbergsson&umehara}, see Note \\ref{note:TU2}.\n\n\nThe general strategy for proving Theorem \\ref{thm:main} hinges on the fact that at a point of nonvanishing torsion $p$, the torsion $\\tau$ of $\\Gamma$ is positive (resp. negative) if and only if $\\Gamma$ crosses its osculating plane at $p$ in the direction (resp. opposite direction) of the binormal vector $B(p)$. To exploit this phenomenon, we project $\\Gamma$ into a sphere $S$ centered at $o$ to obtain a simple closed  curve $\\overline\\Gamma$ which contains $o$ in its convex hull. By a result of Segre \\cite{segre:tangents, weiner:inflection, ghomi:verticesC}, also known as Arnold's tennis ball theorem, $\\overline\\Gamma$ has at least $4$ inflections $\\overline p_i$. It turns out that  the osculating planes $\\overline\\Pi_{\\overline p_i}$ of $\\overline\\Gamma$ coincide with the osculating planes $\\Pi_{p_i}$ of $\\Gamma$ at $p_i$, where $p_i$ are the preimages of $\\overline p_i$, see Figure \\ref{fig:projection}(a). Further, the local convexity assumption will  ensure that the binormal vectors of these planes are parallel, i.e., $\\overline B(\\overline p_i)=B(p_i)$. So the local position of $\\Gamma$ with respect to $\\Pi_{p_i}$ mirrors that of $\\overline\\Gamma$ with respect to $\\overline\\Pi_{\\overline p_i}$. After a perturbation of $o$, we may assume that $\\tau(p_i)\\neq 0$ and $\\overline p_i$ are \\emph{genuine} inflections, i.e.,  the geodesic curvature of $\\overline\\Gamma$ changes sign at $\\overline p_i$. Further, it is easy to see that at every pair of consecutive genuine inflections, $\\overline\\Gamma$ crosses its osculating planes in opposite directions with respect to $\\overline B$, because $\\overline B$ can never be orthogonal to $S$, see Figure \\ref{fig:projection}(b).\nThus it follows that $\\tau$ changes sign at least once within each of the $4$ intervals of $\\Gamma$ determined by $p_i$.\n \\begin{figure}[h]\n   \\centering\n    \\begin{overpic}[height=1.75in]{projection.pdf}\n         \\put(13.5,12.5){\\small $p_i$}\n         \\put(13,20){\\small $B(p_i)$}\n         \\put(22,16){\\small $\\Gamma$}\n          \\put(34.5,19){\\small $\\overline p_i$}\n         \\put(33,27){\\small $\\overline B(\\overline{p}_i)$}\n         \\put(38.5,23){\\small $\\overline\\Gamma$}\n         \\put(30,2){\\small (a)}\n         \\put(81,2){\\small (b)}\n         \\put(13.5,6){$\\Pi_{p_i}=\\overline\\Pi_{\\overline{p}_i}$}\n                    \\put(81,25.5){\\small $\\overline\\Gamma$}\n            \\put(94,23){\\small $\\overline{B}$}\n           \\put(68,14){\\small $\\overline{B}$}\n\n    \\end{overpic}\n    \\caption{}\\label{fig:projection}\n\\end{figure}\n\n\n\nThe above approach for studying the sign of torsion is due to Thorbergsson and Umehara \\cite[p. 240]{thorbergsson&umehara}, who in turn attribute the spherical projection technique to Segre \\cite[p. 258]{segre1968alcune}; however, the method of Thorbergsson and Umehara does not quite lead to a generalization of Sedykh's theorem to a class of nonconvex curves, as we describe in Note \\ref{note:TU} below. \n\nNext we discuss some examples which validate Theorem \\ref{thm:main}.  Figure \\ref{fig:curls}(a) shows a curve which is star-shaped and locally convex, but is not convex. Thus Theorem \\ref{thm:main} is  indeed a nontrivial generalization of Sedykh's result.  Figure \\ref{fig:curls}(b), which shows a torus curve of type $(1, 7)$, demonstrates that the star-shaped assumption by itself would not be sufficient in Theorem \\ref{thm:main}. Indeed all torus curves of type $(1,n)$ are star-shaped with respect to any point on their axis of symmetry, and Costa \\cite{costa:twisted} has shown that, for $n\\geq 2$, they may be realized with nonvanishing torsion if the underlying torus is sufficiently thin. Finally Figure \\ref{fig:curls}(c) shows that the local convexity by itself is not sufficient either. This figure depicts a spherical curve with only two extrema of geodesic curvature and hence only two sign changes of torsion (the torsion of a spherical curve vanishes precisely when its geodesic curvature has a local extremum).\n\n \\begin{figure}[h]\n   \\centering\n    \\begin{overpic}[height=1in]{curls.jpg}\n    \\put(10,-2){\\small (a)}\n    \\put(45,-2){\\small (b)}\n    \\put(78,-2){\\small (c)}\n        \\end{overpic}\n    \\caption{}\\label{fig:curls}\n\\end{figure}\n\nFour vertex theorems have had a multifaceted and interesting history, with unexpected applications, since Mukhopadhyaya proved the first version of this phenomenon in 1909. For extensive background and more references see \\cite{ghomi:rosenberg, gluck:notices, ovsienko&tabachnikov}. In particular see \\cite{bray&jauregui} for some recent applications in General Relativity, and \\cite{sedykh1996, sedykh1997} for discrete versions.\n\n\\begin{note}\\label{note:4vertex}\nThe classical four vertex theorem  states that the curvature of any simple closed curve in $\\mathbf{R}^2$ has at least $4$ critical points, which are called vertices.  A point of the curve is a vertex if and only if the osculating circle at that point has contact of order $3$ with the curve. Consequently, the geodesic curvature of simple closed curves on the sphere also satisfies the four vertex theorem, because the stereographic projection preserves circles. Further note that the plane which contains the osculating circle of a spherical curve is actually the osculating plane of the curve. Thus at a vertex, a spherical curve has contact of order $3$ with its osculating plane, which means that the torsion at that point vanishes. Hence all simple closed spherical curves have at least $4$ points of vanishing torsion. Sedykh generalized this result to curves lying on any convex surface. It is in this sense that Sedykh's theorem, and more generally our Theorem \\ref{thm:main}, are extensions of the classical four vertex theorem.\n\\end{note}\n\n\n\n\\begin{note}\\label{note:TU}\nFor convex space curves, a refinement of Sedykh's four vertex theorem is proved by  Thorbegsson and Umehara  in \\cite[Thm. 0.1]{thorbergsson&umehara}. Furthermore,  these authors \\cite[Thm. 0.2]{thorbergsson&umehara} obtain  a $4$ vertex result for space curves $\\Gamma\\subset \\mathbf{R}^3$ which are star-shaped with respect to a point $o$ in the interior of their convex hull, have no tangent line passing through $o$, and further satisfy the property that for all points $p\\in\\Gamma$ the angle between the principal normal $N(p)$ of $\\Gamma$ and the position vector $p-o$ is obtuse, i.e., \n\\begin{equation}\\label{eq:N}\n\\langle p-o, N(p) \\rangle <0.\n\\end{equation}\nThey claim that this result implies Sedykh's theorem, because a ``convex space curve $\\gamma$ satisfies the conditions in Theorem 0.2\" \\cite[p. 230]{thorbergsson&umehara}; however, this is not the case. Indeed there exists a simple closed curve  which lies on the boundary of a convex body,  but does not satisfy the condition \\eqref{eq:N} for any point $o$; see Figure \\ref{fig:apple}.\n  \\begin{figure}[h]\n   \\centering\n    \\begin{overpic}[height=1.1in]{apple.jpg}\n    \\put(71.5,26){\\small$p_1$}\n    \\put(80,26){\\small$p_2$}\n     \\put(57,1.5){\\small$H_1^+$}\n      \\put(94,1.5){\\small$H_2^+$}\n    \\end{overpic}\n    \\caption{}\\label{fig:apple}\n\\end{figure}\nThe left diagram here depicts the curve,  and the right diagram shows its projection into its plane of symmetry. Let $p_i$, $i=1$, $2$ be the intersections of the curve with it symmetry plane. Note that at these points the principal normals $N(p_i)$ are antiparallel. Let $H_i$  be the planes orthogonal to $N(p_i)$ which pass through $p_i$, and $H_i^+$ be the corresponding (closed) half-spaces into which $N(p_i)$ point. Note that each $H_i^+$ consists of the set of all points $o$ such that \n$\\l p_i-o, N(p_i) \\r \\leq 0$. But $H_1$ and $H_2$ are disjoint. So there exists no point $o$ with respect to which $\\Gamma$ can satisfy condition \\eqref{eq:N}. Thus Theorem 0.2 in \\cite{thorbergsson&umehara} does not imply Sedykh's theorem.\n\\end{note}\n\n\\begin{note}\\label{note:TU2}\nThe result of Thorbergsson and Umehara  \\cite[Thm. 0.2]{thorbergsson&umehara}  mentioned in the previous note is a special case of Theorem \\ref{thm:main}. Indeed let $H_p$ be the rectifying plane of $\\Gamma$ at $p$, i.e., the plane which passes through $p$ and is orthogonal to the principal normal $N(p)$. Then \\eqref{eq:N} implies that $N(p)$ points to the side of $H$ which contains $o$. Consequently a neighborhood of $p$ in $\\Gamma$ lies on the same side as well, which establishes the local convexity of $\\Gamma$.\n\\begin{comment}\n (let $\\gamma\\colon (-\\epsilon,\\epsilon)\\to\\Gamma$ be a local unit speed parametrization with $\\gamma(0)=p$, and $f(t):=\\langle \\gamma(t)-p,N(p)\\rangle$; by Taylor's theorem, $f(t)=\\|\\gamma''(s)\\|\\langle N(s), N(0)\\rangle t^2\/2$ for some $s\\in(0,t)$, which yields $f\\geq 0$ for small $t$)\n\\end{comment}\n\\end{note}\n\n\n\n\n\n\\section{Basic Notation and Terminology}\nThroughout this work we assume that $\\Gamma$, $o$ are as in Theorem \\ref{thm:main}. In particular $\\Gamma$ has nonvanishing curvature (so that its torsion is well defined). For convenience we also assume that $o$ is the origin of $\\mathbf{R}^3$. Further we let $\\overline\\Gamma$ denote the radial projection of $\\Gamma$ into the unit sphere $\\S^2$ centered at $o$. For every point $p\\in\\Gamma$, $\\overline p:=p\/\\|p\\|$ denotes the corresponding point of $\\overline\\Gamma$. We assume that $\\Gamma$ is oriented, and let $T$, $N$, $B:=T\\times N$, denote the corresponding unit tangent, principal normal, and the binormal vectors of $\\Gamma$ respectively. For every point $p\\in\\Gamma$ there exists a $(\\mathcal{C}^3)$ unit speed parametrization $\\gamma\\colon (-\\epsilon,\\epsilon)\\to \\Gamma$ with $\\gamma(0)=p$ such that $\\gamma'(0)=T(p)$. Then $N(p):=\\gamma''(0)\/\\|\\gamma''(0)\\|$, and  the torsion of $\\Gamma$ is given by \n$$\n\\tau(p):=\\frac{\\langle \\gamma'''(0),B(p)\\rangle}{\\|\\gamma''(0)\\|}.\n$$\n The osculating plane $\\Pi_p$ of $\\Gamma$ at $p$ is the plane which passes through $p$ and is orthogonal to $B(p)$. We let $\\overline\\gamma$, $\\overline T$, $\\overline N$, $\\overline B$,   $\\overline \\Pi$ denote the corresponding quantities for $\\overline\\Gamma$. More specifically, $\\overline\\gamma:=\\gamma\/\\|\\gamma\\|$, $\\overline T:=\\overline\\gamma'\/\\|\\overline\\gamma'\\|$, $\\overline B:=\\overline\\gamma'\\times\\overline\\gamma''\/\\|\\overline\\gamma'\\times\\overline\\gamma''\\|$, and $\\overline N:=\\overline B\\times \\overline T$. In particular note that these quantities are well defined, since by assumption through each point of $\\Gamma$ there passes a local support plane of $H$ not containing $o$. Consequently the tangent lines of $\\Gamma$ do not pass through $o$, since they  lie in $H$. So $\\|\\overline\\gamma'\\|\\neq 0$.\n Further note that $\\overline\\Gamma$ inherits its orientation from $\\Gamma$.\nAn \\emph{inflection point} of $\\overline\\Gamma$ is a point where the geodesic curvature $\\overline k$ of $\\overline\\Gamma$ in $\\S^2$ vanishes. Here we define $\\overline k$ with respect to the conormal vector $\\overline n(\\overline p):=\\overline p\\times \\overline T(\\overline p)$:\n$$\n\\overline k(\\overline p):=\\langle \\overline N(\\overline p),\\overline n(\\overline p)\\rangle.\n$$\nWe say that an inflection point $\\overline p$ is genuine if $\\overline k$ changes sign at $\\overline p$.\n\n\\section{Osculating Planes and Inflections}\nA key part of the proof of Theorem \\ref{thm:main}, which facilitates its reduction to Segre's tennis ball theorem, is the following observation. The first part of this lemma is trivial, the second part goes back to Segre, and the third part is a consequence of our local convexity assumption. \n\\begin{lem}\\label{lem:osculate}\nLet $\\overline p$ be an inflection point of $\\overline\\Gamma$. Then \n\\begin{enumerate}\n\\item[(i)] $o\\in\\overline\\Pi_{\\overline p}$, \n\\item[(ii)] $\\overline\\Pi_{\\overline p}=\\Pi_p$, \n\\item[(iii)] $B(p)= \\overline B(\\overline p)$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nThe argument is presented in three parts corresponding to each of the items enumerated above. Here for any pair of vector $v$, $w$, we use the notation $v\\parallel w$ to indicate that $v=\\lambda w$ for some $\\lambda>0$.\n\n(\\emph{i}) \nThe point $\\overline p$ is an inflection if and only if $\\overline N(\\overline p)$ is orthogonal to $\\S^2$, or\n\\begin{equation}\\label{eq:olN}\n\\overline N(\\overline p)=-\\overline p.\n\\end{equation}\n Thus $o=\\overline p+\\overline N(\\overline p)\\in\\overline \\Pi_{\\overline p}$, as claimed. \n\n(\\emph{ii}) \nSince $\\overline\\Pi_{\\overline p}$ passes through $\\overline p$ and $o$, it contains $p$ as well. So it suffices to check that $\\overline\\Pi_{\\overline p}$ and $\\Pi_p$ are parallel, or that $\\overline B(\\overline p)$ is orthogonal to $\\Pi_p$. To this end let $\\gamma\\colon (-\\epsilon, \\epsilon)\\to\\Gamma$ be a  local parametrization with $\\gamma(0)=p$ and $\\gamma'(0)\\parallel T(p)$. Then $\\overline\\gamma:=\\gamma\/\\|\\gamma\\|$ yields a local parametrization for $\\overline\\Gamma$ with $\\overline\\gamma(0)=\\overline p$ and $\\overline\\gamma'(0)\\parallel \\overline T(\\overline p)$.  We need to check that $\\overline B(\\overline p)$ is orthogonal to both $\\gamma'(0)$ and $\\gamma''(0)$. Simple computations show that\n\\begin{equation}\\label{eq:olgamma1}\n\\overline\\gamma\\times \\overline\\gamma'=\\frac{\\gamma\\times \\gamma'}{\\|\\gamma\\|^2},\n\\end{equation}\n\\begin{equation}\\label{eq:olgamma2}\n\\overline\\gamma\\times\\overline\\gamma''=\\frac{(\\gamma\\times\\gamma'')\\|\\gamma\\|^2-2(\\gamma\\times\\gamma')\\l\\gamma,\\gamma'\\r}{\\|\\gamma\\|^4}.\n\\end{equation}\nUsing \\eqref{eq:olN} and \\eqref{eq:olgamma1}, we have\n\\begin{equation}\\label{eq:olB}\n\\overline B(\\overline p)=\\overline T(\\overline p)\\times \\overline N(\\overline p)=\\overline p\\times \\overline T(\\overline p)=\\overline\\gamma(0)\\times \\overline\\gamma'(0)\n=\\frac{\\gamma(0)\\times \\gamma'(0)}{\\|\\gamma(0)\\|^2}.\n\\end{equation}\nThus\n$\n\\langle \\gamma'(0), \\overline B(\\overline p)\\rangle=0.\n$\nNext, to compute $\\l\\gamma''(0), \\overline B(\\overline p)\\r$, we may assume that $\\overline\\gamma$ has unit speed. Then $\\overline N(\\overline p)\\parallel \\overline\\gamma''(0)$ and \\eqref{eq:olN} yields that \n$$\n\\overline\\gamma(0)\\times \\overline\\gamma''(0)\\parallel  \\overline p\\times \\overline N(\\overline p)\\parallel \\overline p\\times(-\\overline p)=0.\n$$\nConsequently, \\eqref{eq:olgamma2} yields that $\\gamma(0)\\times\\gamma''(0)=\\alpha\\; \\gamma(0)\\times\\gamma'(0)$, for some constant $\\alpha$. Now\nusing \\eqref{eq:olB} we have, \n$$\n\\|\\gamma(0)\\|^2\\l\\gamma''(0), \\overline B(\\overline p)\\r=\\l\\gamma'(0),\\gamma''(0)\\times\\gamma(0)\\r=\n\\alpha\\l\\gamma'(0),\\gamma'(0)\\times\\gamma(0)\\r=0,\n$$\nas desired. \n\n(\\emph{iii})\nWe may assume that $\\gamma$ has unit speed. Then $N(p)\\parallel \\gamma''(0)$.\nConsequently,\n$$\nB(p)=T(p)\\times N(p)\\parallel \\gamma'(0)\\times \\gamma''(0).\n$$\nThis together with \\eqref{eq:olB} shows that $B(p)\\parallel\\overline B(\\overline p)$ if and only if\n$$\n\\gamma'(0)\\times\\gamma''(0)\\parallel \\gamma(0)\\times \\gamma'(0).\n$$\nBy assumption there exists a local support plane $H$ of $\\Gamma$ passing through $p$.   Let $L$ be the line given by $H\\cap \\Pi_p$ ($L$ is well defined because $o\\in \\Pi_p$ by parts $(i)$ and $(ii)$ above, but $o\\not\\in H$ by assumption; thus $H\\neq\\Pi_p$). Let $L^+$ denote the side of $L$ in $\\Pi_p$ which contains $o$. Then $N(p)$ points into $L^+$, see Figure \\ref{fig:planes}; because by assumption $\\Gamma$ lies locally on the side of $H$ which contains $o$, and thus $N(p)$ must point into this side as well. \n  \\begin{figure}[h]\n   \\centering\n    \\begin{overpic}[height=1.8in]{planes.pdf}\n    \\put(52.5,16){\\small$o$}\n    \\put(42,14){\\small$\\overline{p}$}\n    \\put(21,14){\\small$p$}\n    \\put(11,3){\\small$L$}\n     \\put(6,23){\\small$H$}\n     \\put(63,11){\\small$L^+$}\n     \\put(76,29){\\small$L$}\n      \\put(19,23.5){\\small$T(p)$}\n       \\put(43,24){\\small$\\overline T(\\overline p)$}\n       \\put(27,11.5){\\small$N(p)$}\n       \\put(50,13){\\small$\\overline N(\\overline p)$}\n       \\put(0,11){\\small$\\Pi_{p}=\\overline\\Pi_{\\overline{p}}$}\n       \\put(89, 6.5){\\small$o$}\n        \\put(89, 16.5){\\small$\\overline p$}\n         \\put(89, 28.5){\\small$p$}\n         \\put(79, 1){\\small$L^+$}\n         \\put(93.5, 26.5){\\small$T(p)$}\n         \\put(94, 16){\\small$\\overline T(p)$}\n         \\put(80.5, 22){\\small$N(p)$}\n         \\put(82, 9){\\small$\\overline N(p)$}\n       \\end{overpic}\n    \\caption{}\\label{fig:planes}\n\\end{figure}\n Further, $L$ is tangent to $\\Gamma$ since $H$ is tangent to $\\Gamma$. Thus $N(p)\\perp L$. Consequently \n $$\n L^+=\\{ x\\in \\Pi_p\\mid \\l\\ x-p,N(p)\\r\\geq 0\\}.\n $$  \n So since $o$ lies in the interior of $L^+$, $\\l o-p, N(p)\\r>0$, or\n $\n\\langle N(p), p\\rangle <0,\n$\n which  yields\n\\begin{equation}\\label{eq:a}\n \\langle \\gamma''(0), \\gamma(0)\\rangle <0.\n\\end{equation}\n Since $\\Pi_p$ passes through $o$,  $\\{\\gamma(0),\\gamma'(0),\\gamma''(0)\\}$ is linearly dependent.\n Further, since we are assuming that $\\gamma$ has unit speed, $\\gamma'\\perp\\gamma''$. Thus \n $$\n \\gamma(0)=a \\gamma'(0)+b\\gamma''(0),\n $$\n where $b=\\langle \\gamma''(0), \\gamma(0)\\rangle\/\\|\\gamma''(0)\\|^2<0$ according to \\eqref{eq:a}.\n So\n $$\n \\gamma(0)\\times\\gamma'(0)=-b\\;\\gamma'(0)\\times\\gamma''(0),\n $$\n which completes the proof.\n\\end{proof}\n\n\\section{Maximum Principles for Torsion and Geodesic Curvature}\n\nHere we collect the facts we need concerning the relation between the sign of torsion of a space curve, and its relative position with respect to its osculating plane. Further we discuss the corresponding facts for the geodesic curvature of spherical curves, which will be used in the proof of our main result in the next section.\n\n\nWe start with torsion. Here by the region \\emph{above} the osculating plane we mean the (closed) half-space into which the binormal vector $B$ points, and the region \\emph{below} will be the other half-space. Since we assume that $\\Gamma$ is oriented, for every pair of points $p$, $q$ of $\\Gamma$, there is a unique choice of a segment with initial point $p$ and final point $q$ which we denote by $[p,q]$. The interior of this segment will be denoted by $(p,q)$.\n\n\\begin{lem}[Lem. 6.12, \\cite{ghomi:rosenberg}]\\label{lem:maxprincipletorsion}\nSuppose that $\\tau\\geq  0$ (resp. $\\tau\\leq 0$) on a segment $[p, q]$ of $\\Gamma$. Then, near $p$,  the segment lies above (resp. below) the osculating plane of $\\Gamma$ at $p$.\\qed\n\\end{lem}\n\nThe above lemma quickly yields the following converse:\n\n\\begin{cor}\\label{cor:maxprincipletorsion}\nSuppose that a segment $[p,q]$ of $\\Gamma$ lies above its osculating  plane at $p$ (resp. $q$) and does not lie completely in the osculating plane. Then $\\tau >0$ (resp. $\\tau<0$) at some point of $[p,q]$.\n\\end{cor}\n\\begin{proof}\nFirst suppose that $[p,q]$ lies above its osculating plane at $p$ and assume, towards a contradiction, that $\\tau\\leq 0$ on $[p,q]$. Then $[p,q]$ lies below $\\Pi_p$ by Lemma \\ref{lem:maxprincipletorsion}. Thus $[p,q]$ must lie entirely in $\\Pi_p$ which is a contradiction. The case where $[p,q]$ lies above its osculating plane at $q$, also follows from Lemma \\ref{lem:maxprincipletorsion}, once we switch the orientation of $[p,q]$ and observe that this switches the direction of $B$, but does not effect the sign of $\\tau$.\n\\end{proof}\n\nSimilarly, here are the  facts concerning geodesic curvature which we need. For every $\\overline p\\in\\overline\\Gamma$, let $C_{\\overline p}$ denote the great circle in $\\S^2$ which is tangent to $\\overline\\Gamma$ at $\\overline p$. By the region \\emph{above} $C_{\\overline p}$ we mean the (closed) hemisphere into which the conormal vector $\\overline n(\\overline p):=\\overline p\\times \\overline T(\\overline p)$ points, and the other hemisphere will be referred to as the region \\emph{below} $C_{\\overline p}$.\n\n\\begin{lem}[Lem. 2.1, \\cite{ghomi:verticesC}]\\label{lem:maxprinciplecurvature}\nSuppose that  a segment $[\\overline p, \\overline q]$ of $\\overline\\Gamma$ lies above (resp. below) the tangent great circle $C_{\\overline p}$. Then either  $[\\overline p, \\overline q]$ is a part of $C_{\\overline p}$, or else $\\overline k > 0$ (resp. $< 0$) at some point of $[\\overline p, \\overline q]$.\\qed\n\\end{lem}\n\nNow we quickly obtain:\n\n\\begin{cor}\\label{cor:maxprinciplecurvature}\nSuppose that $\\overline k>0$ on the interior of a segment $[\\overline p,\\overline q]$ of $\\overline\\Gamma$. Then, near $\\overline p$ and $\\overline q$,  $[\\overline p, \\overline q]$ lies above $C_{\\overline p}$ and $C_{\\overline q}$ respectively, and does not coincide with them.\n\\end{cor}\n\\begin{proof}\nIf $[\\overline p,\\overline q]$ lies locally below  $C_{\\overline p}$, then $[\\overline p, \\overline q]$  coincides with $C_{\\overline p}$ near $\\overline p$, by Lemma \\ref{lem:maxprinciplecurvature}. So $[\\overline p, \\overline q]$ lies locally above $C_{\\overline p}$ as claimed. The same argument may also be applied to $\\overline q$ after switching the orientation of $\\overline\\Gamma$. Finally, since $\\overline k\\neq 0$ on $(p,q)$, these circles cannot coincide with $\\overline\\Gamma$ near $\\overline p$ and $\\overline q$.\n\\end{proof}\n\nNext, we link the last two corollaries together by recording that if $\\overline p$ is an inflection point of $\\overline\\Gamma$, then the region above $C_{\\overline p}$ corresponds to the region above $\\Pi_{p}$. Indeed recall that if $\\overline p$ is an inflection, then $\\overline N(\\overline p)=-\\overline p$ as we discussed in the proof of Lemma \\ref{lem:osculate}. Thus\n$$\n\\overline n(\\overline p)=\\overline p\\times \\overline T(\\overline p)=-\\overline N(\\overline p)\\times \\overline T(\\overline p)=\\overline T(\\overline p)\\times \\overline N(\\overline p)=\\overline B(\\overline p).\n$$\n\nSo Lemma \\ref{lem:osculate} quickly yields:\n\n\n\\begin{lem}\\label{lem:above}\nAt an inflection point  $\\overline p$ of $\\overline\\Gamma$, the region above the great circle $C_{\\overline p}$ in $\\S^2$ coincides with the hemisphere which lies above the osculating plane $\\Pi_p$.\\qed\n\\end{lem}\n\n\n\\section{Proof of the Main Result}\n\nBefore proving our main theorem, we require the following technical fact which shows that the local convexity and star-shaped  properties of $\\Gamma$ are stable under small perturbations of $o$.\n\n\\begin{lem}\\label{lem:oprime}\nThere exists an open neighborhood $U$ of $o$ such that $\\Gamma$ is star-shaped and locally convex with respect to every point $o'$ in $U$.\nFurther, we may choose $o'$ so that only finitely many osculating planes of $\\Gamma$ pass through $o'$.\n \\end{lem}\n\\begin{proof}\nThe local convexity assumption ensures that the tangent lines of $\\Gamma$ do not pass through $o$. Thus $\\overline\\Gamma$ is a $\\mathcal{C}^3$ immersed curve. Further, the star-shaped assumption ensures that $\\overline\\Gamma$ is embedded. Now since embeddings of compact manifold are open in the space of $\\mathcal{C}^1$ mappings, it follows that projections of $\\Gamma$ into unit spheres centered at $o'\\in U$ are embedded  as well, for some open neighborhood $U$ of $o$. So the star-shaped property is preserved under small perturbations of $o$. Next we  check that the local convexity assumption is preserved as well. To this end it suffices to show that the local support planes of $\\Gamma$ may be chosen continuously, for then the support planes cannot get arbitrarily close to $o$, and hence $\\Gamma$ remains locally convex with respect to all points of $U$, assuming $U$ is sufficiently small. To see that the local support planes may be chosen continuously, see \\cite[Sec 3.1]{ghomi:stconvex}.\nFinally, using Sard's theorem, we may choose a point $o'$ in $U$ such that only finitely many osculating planes of $\\Gamma$ pass through $o'$: consider  $f\\colon\\Gamma\\times\\mathbf{R}^2\\to\\mathbf{R}^3$ given by $f(p, t, s)= p+t\\, T(p) +s\\, N(p)$, and let $o'$ be a regular value of $f$. \n\\end{proof}\n\nWe also need the following refinement of the tennis ball theorem. Recall that an inflection point $\\overline p$ of $\\overline\\Gamma$ is \\emph{genuine}, provided that the geodesic curvature $\\overline k$ of $\\overline\\Gamma$ changes sign at $\\overline p$.\n\n\\begin{lem}[Thm. 1.2, \\cite{ghomi:verticesC}]\\label{lem:tennisball}\nSuppose that $\\overline\\Gamma$ has at most  finitely many inflections. Then at least $4$ of these inflections must be genuine.\\qed\n\\end{lem}\n \n\nFinally we are ready to establish our main result:\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nBy Lemma \\ref{lem:oprime}, after replacing $o$ by a nearby point, we may assume that at most finitely many osculating planes of $\\Gamma$ pass through $o$. By Lemma \\ref{lem:osculate}, osculating planes of $\\overline\\Gamma$ at its inflections pass through $o$ and coincide with the osculating planes of $\\Gamma$. Thus  $\\overline\\Gamma$ has now at most finitely many inflections. \nConsequently, by the refinement of the tennis ball theorem, Lemma \\ref{lem:tennisball},  $\\overline\\Gamma$ has at least $4$ genuine inflections. \n\nLet $\\overline p_0$, $\\overline p_1$ be a pair of  genuine inflections of $\\overline\\Gamma$ such that the oriented segment \n$[\\overline p_0, \\overline p_1]$   has no genuine inflections in its interior $(\\overline p_0, \\overline p_1)$. There are at least $4$ such intervals with pairwise disjoint interiors. Thus, to complete the proof, it suffices to show that $\\tau$ changes sign on $(p_0, p_1)$. \n\nWe may assume that $\\overline k\\geq 0$ on $(\\overline p_0,\\overline p_1)$, after switching the orientations of $\\Gamma$ and $\\overline\\Gamma$ if necessary. Then $\\overline k>0$ near $\\overline p_i$ on $(\\overline p_0,\\overline p_1)$, since  $\\overline\\Gamma$ has only finitely many inflections. Consequently,  by the maximum principle for geodesic curvature, Corollary \\ref{cor:maxprinciplecurvature}, $[\\overline p_0, \\overline p_1]$ lies locally above its tangent great circles at $\\overline p_i$ and does not coincide with them near $\\overline p_i$.\nThis in turn implies, by Lemma \\ref{lem:above},\n that \n$[p_0, p_1]$ lies locally above its osculating planes at $p_i$, and does not coincide with them near $p_i$. Thus, by  the maximum principle for torsion, Corollary \\ref{cor:maxprincipletorsion}, $\\tau$ changes sign on $(p_0, p_1)$ as desired.\n\\end{proof}\n\n\\section*{Acknowledgment}\nThe author is grateful to Slava Sedykh for correcting some computations in the proof of Lemma \\ref{lem:osculate}, and several other comments to improve the exposition of this work. Thanks also to Masaaki Umehara for helpful communications.\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCurrently there is a large body of data coming from cosmological and astrophysical observations that is mostly consistent with the existence of dark matter.  Such observations also suggest that the hypothesized particles that constitute dark matter have very small cross section and travel at speeds much lower than light. These lead to the cold dark matter framework, which is one of the pillars of the current standard cosmological model $\\Lambda$CDM.\n\n\nIt is not only tempting, but mandatory to check if such dark matter particles exist (by detecting them in laboratory based experiments, for instance) and also to check if the gravitational effects that lead to the dark matter hypothesis could follow from a more detailed and complete approach to gravity. The effects of pure classical General Relativity at galaxies have been studied for a long time and, considering galaxy rotation curves, the differences between General Relativity and Newtonian gravity are negligible, since in a galaxy matter moves at speeds much lower than that of light and is typically subject to weak gravitational fields ($\\Phi \\ll c^2$), which leads to the Newtonian limit of General Relativity \\footnote{There are some proposals that consider General Relativity  in the context of galaxies which do not lead to Newtonian gravity, see e.g. \\cite{Cooperstock:2006dt, Vogt:2005hn, Cooperstock:2007sc, Vogt:2005va, Vogt:2005hv}.  It is not impossible that a reasonable explanation for galaxy rotation curves may rely on similar approaches, nevertheless up to now none of such proposals have found a baryonic mass distribution that is in conformity with astrophysical expectations.}. \n\nThere is however a newer approach that may change considerably the role of dark matter, while following standard physical principles. Namely, the investigation of the running of the gravitational coupling parameter $G$ on large scales as induced by the renormalization group framework.\n\nThe running of coupling constants is a well known phenomenon within \nQuantum Field Theory. It is well-known that the renormalization group \nmethod can be extended to quantum field theory on curved space time\nand to some models of quantum gravity (see, e.g., \\cite{Buchbinder:1992rb}), such that  the beta functions can be interpreted in this framework. Concerning \nthe high energy (UV) behavior, there is hope that the running of $G$ \nin quantum gravity may converge to a non-Gaussian fixed point \n(asymptotic safety) \\cite{Niedermaier:2006wt,Weinberg:2009bg}. Our present concern is, however, not about \nthe UV completeness, but with the behavior of $G$ in the far infrared \nregime (IR). In the electromagnetic case the IR behavior corresponds \nto the Appelquist-Carazzone decoupling \\cite{Appelquist:1974tg} (see e.g., \\cite{Goncalves:2009sk} for a recent derivation of this theorem). In the case of gravity the same effect of decoupling has been obtained in \\cite{Gorbar:2002pw,Gorbar:2003yt},\nbut only for the higher derivative terms in the gravitational action. \nIt remains unclear whether such decoupling takes place for the other terms. This possibility was studied on phenomenological grounds a number of times before, e.g. \\cite{Shapiro:2004ch,Reuter:2004nx}.\n\nIn  \\cite{Rodrigues:2009vf} we presented new results on the application of renormalization group (RG) corrections to General Relativity (GR) in the astrophysical domain. Previous attempts to apply this picture to galaxies have considered for simplicity point-like galaxies. We extended previous considerations by identifying the proper renormalization group energy scale $\\mu$ and by evaluating the consequences considering the observational data of disk galaxies. Also we propose a natural choice for the identification of $\\mu$, linking it to the local value of the Newtonian gravitational potential. With this choice, the renormalization group-based approach is capable to mimic dark matter effects with great precision. This picture induces a very small variation on the gravitational coupling parameter $G$, namely a variation of about $10^{-7}$ of its value across $10^5$ light-years. We call our model RGGR, in reference to renormalization group effects in General Relativity.\n\nIn order to evaluate the observational consequences of the RGGR model and to compare it to other proposals, \nrecent high quality observational data \\cite{2008AJ....136.2648D,Gentile:2004tb} from nine regular spiral galaxies were mass-modelled using the standard procedures for the baryonic part, and four different models for the ``dark'' component: i) the RGGR model; ii) one of the most phenomenological successful dark matter profiles, the Isothermal profile \\cite{1991MNRAS.249..523B}; and two alternative models which were built to avoid the need for the dark matter: iii) the Modified Newtonian Dynamics (MOND) \\cite{1983ApJ...270..365M,1983ApJ...270..371M} and iv) the Scalar-Tensor-Vector Gravity (STVG) \\cite{Moffat:2005si}. The latter is a recent relativistic proposal that is capable of dealing with galaxy rotation curves and other phenomena usually attributed to dark matter. For galaxy rotation curves phenomena, STVG becomes equivalent to a similar proposal called MSTG \\cite{Moffat:2004bm, Brownstein:2005zz, Brownstein:2009gz}. The model parameters that we use to fit galaxies in the STVG framework can be found in ref. \\cite{Brownstein:2005zz}. While for MOND we use the $a_0$ value as given in \\cite{Sanders:2002pf}.\n\nThe quality of the rotation curve fits and total stellar mass as inferred from the RGGR model is perfectly satisfactory considering both the general behavior of the model and its results when applied to nine particular galaxies, as analyzed in \\cite{Rodrigues:2009vf}. It is about the same of the Isothermal profile quality, while it seems significantly better than the quality of MOND and STVG. In the case of MOND, we did numerical experiences with $a_0$ as a free parameter and found that, albeit the concordance with the shape of the rotation curve can considerably increase in this case, the concordance with the expected stellar mass-to-light ratios remains unsatisfactory (similar conclusions have also appeared in some recent papers, e.g. \\cite{Gentile:2004tb,Gentile:2010xt,Famaey:2005fd}, and it seems that the concordance can only be improved by adjusting the MOND's $\\mu(x)$ function in an {\\it ad-hoc} way).\n\n\\section{The running of $G$}\n\n\nThe $\\beta$-function for  the gravitational coupling parameter $G$ has been discussed in the framework of different approaches to Quantum Gravity and Quantum Field Theory in curved space-time. In \\cite{Rodrigues:2009vf} we followed the derivation used previously in \\cite{Shapiro:2004ch}.  If  $G$ does not behave as  a constant  in the far  IR limit, it was argued in \\cite{Shapiro:2004ch} (and recently in more details in \\cite{Farina:2011me}) that the logarithmic running of $G$  is a direct consequence of covariance and must hold in all loop orders.  As far as direct derivation of the physical running of $G$ is not available, it is worthwhile to explore the possibility of a logarithmically running $G$ at the phenomenological level.\n\n\nConsider the following infrared $\\beta$-function for General Relativity,\n\\begin{equation}\n\t \\beta_{G^{-1}} \\equiv \\mu \\frac{dG^{-1}}{d \\mu} = 2 \\nu \\,  \\frac{M_{\\mbox{\\tiny Planck}}^{2}}{c \\, \\hbar} = 2 \\nu G_0^{-1}.\n\t\\label{betaG}\n\\end{equation}\n\nEquation (\\ref{betaG}) leads to the logarithmically  varying $G(\\mu)$ function,\n\\begin{equation}\n\t\\label{gmu}\n\tG(\\mu) = \\frac {G_0}{ 1 + \\nu \\,\\mbox{ln}\\,(\\mu^2\/\\mu_0^2)},\n\\end{equation}\nwhere $\\mu_0$ is a reference scale introduced such that $G(\\mu_0) =G_0 $. The constant $G_0$ is the gravitational constant as measured in the Solar System  (actually, there is no need to be very precise on where $G$ assumes the value of $G_0$, due to the smallness of the variation of $G$). The dimensionless constant $\\nu$ is a phenomenological parameter which depends on the details of the quantum theory leading to eq. (\\ref{gmu}). Since we have no means to compute the latter from first principles, its value should be fixed from observations. It will be shown that even a very small $\\nu$ can lead to observational consequences at galactic scales.\n\n\nThe action for this model is simply the Einstein-Hilbert one in which $G$ appears inside the integral, namely,\n\\begin{equation}\n\tS_{\\mbox{\\tiny RGGR}}[g] = \\frac {c^3}{16 \\pi }\\int \\frac {R  } G \\, \\sqrt{-g} \\,  d^4x.\n\t\\label{rggraction}\n\\end{equation}\nIn the above, $G$ should be understood as an external scalar field that satisfies (\\ref{gmu}).  Since for the problem of galaxy rotation curves  the cosmological constant effects are negligible, we have not written the $\\Lambda$ term above. Nevertheless, for a complete cosmological picture, $\\Lambda$ is necessary and it also runs covariantly with the RG flow of $G$ (see e.g.,\\cite{Shapiro:2004ch}). \n\nThere is a simple procedure to map solutions from the Einstein equations with the gravitational constant $G_0$  into RGGR solutions. One need not to follow this route, one may find all the dynamics from the RGGR equations of motion, which can be found by a direct variation of the action (\\ref{rggraction}) in respect to the metric, leading to equations of motion that have the same form of those of a scalar-tensor gravity\\footnote{We stress that it is only the from since RGGR is not a type of scalar-tensor gravity, and $G$ is not a fundamental field of the model.}. In this review, we will proceed to find RGGR solutions via a conformal transformation of the Einstein-Hilbert action, and to this end first we write \n\\begin{equation}\n\tG = G_0 + \\delta G,\n\\end{equation} \nand we assume $\\delta G \/ G_0 \\ll 1$, which will be justified latter. Introducing the conformally related metric\n\\begin{equation}\n\t\\bar g_{\\mu \\nu} \\equiv \\frac {G_0}{G} g_{\\mu \\nu}, \n\t\\label{ct}\n\\end{equation}\nthe RGGR action can be written as\n\\begin{equation}\n\tS_{\\mbox{\\tiny RGGR}}[g] = S_{\\mbox{\\tiny EH}}[\\bar g] + O(\\delta G^2),\n\\end{equation}\nwhere $S_{\\mbox{\\tiny EH}}$ is the Einstein-Hilbert action with $G_0$ as the gravitational constant. The above suggest that the RGGR solutions can be generated from the Einstein equations solutions via the conformal transformation (\\ref{ct}). Indeed, within a good approximation, one can check that this relation persists when comparing the RGGR equations of motion to the Einstein equations even in the presence of matter \\cite{Rodrigues:2009vf}.\n\nIn the context of rotation curves of galaxies, standard General Relativity gives essentially the same predictions of Newtonian gravity. The Newtonian potential $ \\Phi_{\\mbox{\\tiny Newt}}$ is related to the metric by \n\\begin{equation}\n\t\\bar g_{00} = - \\left ( 1  + \\frac {2 \\Phi_{\\mbox{\\tiny Newt}}}{c^2} \\right ).\n\\end{equation}\nHence, using eq. (\\ref{ct}), the effective RGGR potential $\\Phi$ in the non-relativistic limit is given by\n\\begin{equation}\n\t\\Phi = \\Phi_{\\mbox{\\tiny Newt}} + \\frac {c^2}2 \\frac{\\delta G}{G_0}.\n\\end{equation}\nAn equivalent result can also be found from the evaluation of a test particle geodesics \\cite{Rodrigues:2009vf}. In the context of weak gravitational fields $\\Phi_{\\mbox{\\tiny Newt}}\/ c^2 \\ll 1$ (with $\\Phi_{\\mbox{\\tiny Newt}} = 0$ at spatial infinity) holds, and hence the term $\\delta G\/G_0$ should not be neglected.\n\nIn order to derive a test particle acceleration, we have to specify the proper energy scale $\\mu$ for the problem setting in question, which is a time-independent gravitational phenomena in the weak field limit. This is a recent area of exploration of the renormalization group application, where the usual procedures for high energy scattering of particles cannot be applied straightforwardly. Previously to \\cite{Rodrigues:2009vf} the selection of $\\mu \\propto 1\/r$, where $r$ is the distance from a massive point, was repeatedly used, e.g. \\cite{Reuter:2004nv,Dalvit:1994gf,Bertolami:1993mh,Goldman:1992qs, Shapiro:2004ch}. This identification adds a constant velocity proportional to $\\nu$ to any rotation curve. Although it was pointed as an advantage due to the generation of ``flat rotation curves'' for galaxies, it introduced difficulties with the Tully-Fisher law, the Newtonian limit, and the behavior of the galaxy rotation curve close to the galactic center, since there the behavior is closer to the expected one without dark matter. In \\cite{Rodrigues:2009vf} we introduced a $\\mu$ identification that seems better justified both from the theoretical and observational points of view. The characteristic weak-field gravitational energy does not comes from the geometric scaling $1\/r$, but from the Newtonian potential $\\Phi_{\\mbox{\\tiny Newt}}$. However, the straight relation $ \\mu \\propto \\Phi_{\\mbox{\\tiny Newt}}$ leads to $\\mu \\propto 1\/r$ in the large $r$ limit; which is unsatisfactory on observational grounds (bad Newtonian limit and correspondence to the Tully-Fisher law). One way to recover the Newtonian limit is to impose a suitable cut-off, but this does not solves the Tully-Fisher issues \\cite{Shapiro:2004ch}. Another one is to use \\cite{Rodrigues:2009vf}\n\\begin{equation}\n\t\\frac {\\mu}{\\mu_0} =\\left( \\frac{\\Phi_{\\mbox{\\tiny Newt}}}{\\Phi_0} \\right)^\\alpha,\n\t\\label{muphi}\n\\end{equation}\nwhere $\\Phi_0$ and $\\alpha$ are constants. Apart from the condition $0 < \\Phi_0 < c^2$ (i.e., essentially $\\Phi_0$ is a reference Newtonian potential), the precise value of $\\Phi_0$ is largely irrelevant for the problem of rotation curves. The relevant parameter is $\\alpha$. It is a phenomenological parameter that depends on the mass of the system, and it must go to zero when the mass of the system goes to zero. This is necessary to have a good Newtonian limit. From the Tully-Fisher law, it is expected to increase monotonically with the increase of the mass. Such behavior is indeed found from the galaxy fits done in \\cite{Rodrigues:2009vf}. In a recent paper, an upper bound on $\\nu \\alpha$ in the Solar System was derived \\cite{Farina:2011me}. In galaxy systems, $\\nu \\alpha|_{\\mbox{\\tiny Galaxy}} \\sim 10^{-7}$, while for the Solar System, whose mass is about $10^{-10}$ of that of a galaxy,  $\\nu \\alpha|_{\\mbox{\\tiny Solar System}} \\lesssim 10^{-17}$. It shows that a linear decrease on $ \\alpha$ with the mass is sufficient to satisfy both the current upper bound from the Solar System and the results from galaxies.\n\nWe also point that the above energy scale setting (\\ref{muphi}) was recently re-obtained from a more theoretical perspective \\cite{Domazet:2010bk}.\n\nOnce the $\\mu$ identification is set, it is straightforward to find the rotation velocity for a static gravitational system sustained by its centripetal acceleration,\n\\begin{equation}\n\tV^2_{\\mbox{\\tiny RGGR}} \\approx V^2_{\\mbox{\\tiny Newt}} \\left ( 1 - \\frac {\\nu \\, \\alpha  \\, c^2} {\\Phi_{\\mbox{\\tiny Newt}}} \\right ).\n\t\\label{v2rggr}\n\\end{equation}\n\nContrary to Newtonian gravity, the value of the Newtonian \npotential at a given point does play a significant role \nin this approach. This sounds odd from the perspective of \nNewtonian gravity, but this is not so \nfrom the General Relativity viewpoint, since the latter has no \nfree zero point of energy. In particular, the Schwarzschild \nsolution is not invariant under a constant shift of the \npotential.\n\nIn the following, we will comment on the effect of the relation (\\ref{v2rggr}) to galaxy rotation curves. First from a more general perspective, and then to the modeling of individual galaxies.\n\n \n\n\n\n\n\\section{Galaxy rotation curves}\n\nBefore proceeding to specific galaxy rotation curves modeling, it is more instructive to analyze general features of the relation (\\ref{v2rggr}), and to compare it to the standard approach. In \\cite{Rodrigues:2009vf} we analyze some general aspects and scaling laws of the RGGR  model, with no dark matter, in comparison to the isothermal profile; both of them, at this step, without gas and with an exponential stellar disk. In particular, it was pointed that the RGGR rotation curves have a reasonable shape to fit galaxies (i.e., no clear problems like increasing or decreasing too fast, oscillations...),  and that they effectively behave similarly to cored dark matter profiles at inner radii, whose effective core radius scales with the galaxy disk scale length. Further details in our paper.\n\nWe have also extended the previous analysis by adding a gas-like contribution (a re-scaled version of the NGC 3198 gaseous part). In particular, this numerically evaluates how the model behaves on the presence of density perturbations at large radii.  In the first plot of fig. (\\ref{mus}) it is displayed the result for RGGR, which is remarkably good (a similar plot can be found in our original paper), while in the others plots in fig. (\\ref{mus}) (presented at the Conference, but not in \\cite{Rodrigues:2009vf}) one sees the results for the same mass distribution but different choices for the energy scale\\footnote{These other choices are also unsatisfactory from the theoretical perspective, since they have no direct relation to the local energy of the gravitational field in the weak field regime ($\\Phi_{\\mbox{\\tiny Newt}} \\ll c^2 $). } $\\mu$. \n\n\n\n\\begin{figure}[ht]\n\t  \\includegraphics[width=150mm]{IC2Cosmo3mus2.pdf}\n\\caption{The additional circular velocity squared induced by different  choices of $\\mu$. The first plot refers to RGGR, the two others to different identifications of $\\mu$: one depends on the Newtonian acceleration (a variation inspired on MOND) and the other on the (baryonic) matter density. All the plots above display the additional squared velocity of each model divided by $V^2_\\infty \\equiv \\nu \\alpha c^2$ and  as a function of $R\/R_D$, where  $R$ is the the radial cylindrical coordinate and  $R_D$ is the stellar disk scale length. Black lines depict the additional velocity due to a pure exponential stellar disk, while the gray solid lines  take into account the gas mass $M_{\\mbox{gas}}$ for different values of $f \\equiv M_{\\mbox{gas}}\/M_{\\mbox{stars}}$, with $f = 0.2, 0.7, 1.2,..., 9.7$ (i.e., the black lines stand for $f=0$). See \\cite{Rodrigues:2009vf} for further details.}\n\\label{mus}\n\\end{figure}\n\nFrom fig. (\\ref{mus}), the two other proposals different from RGGR are seen to be unsuited as replacements for dark matter. In particular, both are too sensitive to the gas presence, and both eventually add negative contributions to the total circular velocity at large radii.\n\n\nIn \\cite{Rodrigues:2009vf} we used a sample of nine high quality and regular rotation curves of disk galaxies from \\cite{2008AJ....136.2648D, Gentile:2004tb}. In figs. (\\ref{ngc2403}, \\ref{masstolightplot}) we show one of ours results (see \\cite{Rodrigues:2009vf} for the complete set and further details) in comparison to the results of three other models: a cored dark matter profile (Isothermal profile), the  Modified Newtonian Dynamics (MOND) and the recently proposed Scalar-Tensor-Vector Gravity (STVG). \n\n\\begin{figure}[ht]\n\t  \\includegraphics[width=100mm]{NGC2403PlotsIC2Models.pdf}\n\\caption{NGC 2403 rotation curve fits. The red dots and its error bars are the rotation curve observational data, the gray ones close to the abscissa are the residues of the fit. The solid black line for each model is its best fit rotation curve, the dashed yellow curves are the stellar rotation curves from the bulge and disk components,  the dotted purple curve is the gas rotation curve, and the dot-dashed green curve is the resulting Newtonian, with no dark matter, rotation curve. }\n\\label{ngc2403}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\t  \\includegraphics[width=100mm]{ColorVsYIC2Models}\n\\caption{Stellar disk mass-to-light ratio ($Y_{*D}$) in the $3.6  \\mu m$ band  as a function of the color $J - K$.  Each galactic disk is represented above by an open circle, with a {\\it reference} error bar of 50$\\%$ of the $Y_{*D}$ value. The black open squares display the $Y_{*D}$ values and their associated 1$\\sigma$ errors for each galaxy as inferred from the rotation curve fits for each model. The highlighted square and circle correspond to the NGC 2403 galactic disk mass-to-light ratios. See \\cite{Rodrigues:2009vf} for further details.}\n\\label{masstolightplot}\n\\end{figure}\n\n\nDue to the considerably large uncertainty in the total stellar mass of each stellar component (disk and bulge), we first use the total stellar mass as a free parameter for the fittings (achieved from a $\\chi^2$ minimization considering the errors). At a second stage, we compare the resulting value with stellar population expectations, following the  standard approach.\n\nOn the free parameters of each model, we remark that besides the total stellar mass, the Isothermal profile has two additional free parameters, the RGGR model has a single free parameter ($\\alpha$) while MOND and STVG have no free parameters that can vary from galaxy to galaxy. On the other hand, both of the latter depend on constants whose values are calibrated considering its best fit in a large sample of galaxies. We remark that the $\\nu$ parameter in RGGR cannot vary from galaxy to galaxy, but $\\alpha $ can, and galaxy rotation curves are sensible to the combination $\\nu \\alpha$, whose value is about the order of $10^{-7}$. The best fit for NGC 2403 yields $ \\nu \\alpha = (1.66 \\pm0.01) \\times 10^{-7}$.\n\n\n\n\\section{Conclusions}\n\nWe  presented a model, motivated by renormalization group corrections to the Einstein-Hilbert action, that introduces small inhomogeneities in the gravitational coupling across a galaxy (of about 1 part in $10^7$) and can generate galaxy rotation curves in agreement with the observational data, without the introduction of dark matter as a new kind of matter. Both High and Low Surface Brightness galaxies were tested \\cite{Rodrigues:2009vf} . Considering the samples of  galaxies evaluated in \\cite{Rodrigues:2009vf}, the quality of the RGGR rotation curves,  together with the corresponding mass-to-light ratios, is about the same than  the Isothermal profile quality, but with one less free parameter. We expect that similar results would hold in regard to other cored dark matter profiles, while our results seem better than those achieved by the NFW profile \\cite{Rodrigues:2009vf}. We also compared the results of our model with MOND and STVG, and at face value our model yielded clearly better results.   \n\nOur results can be seen as a next step compared to the \nprevious models motivated by renormalization group effects in \ngravity, e.g. \\cite{Shapiro:2004ch, Reuter:2007de}. Their original analyses \ncould only yield a rough estimate on the galaxy rotation curves, \nsince they were restricted to modeling a galaxy as a single \npoint. Trying to extend this approach to real galaxies, we \nhave shown that the proper scale \nfor the renormalization group phenomenology is not of a \ngeometric type, like the inverse of the distance, but is\nrelated to the Newtonian potential with null boundary \ncondition at infinity. \n\n\nThe essential feature for the RGGR rotation curves fits is the formula (\\ref{v2rggr}), which is by itself a  simple formula that provides a very efficient description of galaxy rotation curves. \n\nThere are several tests and implications of this model yet to be evaluated. In particular we are working on applying the RGGR framework to a larger sample of galaxies (including elliptical galaxies) \\cite{FabrisGalaxies} and galaxy-galaxy strong lensing \\cite{rodrigueslentes}. Related work on CMB, BAO and LSS in search for a new cosmological concordance model is also a work in progress \\cite{toribioCMB}.  \n\n\n\n\\bigskip\n\n\\noindent\n{ \\it \\bf Acknowledgements}\n\n\\noindent\nDCR thanks FAPESP and PRPPG-UFES for partial financial support. PSL thanks CNPq and FAPESP for partial financial support. The work of I.Sh. was partially supported by CNPq, FAPEMIG, FAPES and ICTP. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper we establish the structured ring spectra analogs of Goodwillie's widely exploited and powerful cubical diagram results \\cite{Goodwillie_calc2} for spaces. These cubical diagram results are a key ingredient in the authors' homotopic descent results \\cite{Ching_Harper} on a structured ring spectra analog of Quillen-Sullivan theory \\cite{Quillen_rational, Sullivan_MIT_notes, Sullivan_genetics}. They also establish an important part of the foundations for the theory of Goodwillie calculus in the context of structured ring spectra; see, for instance, Arone and Ching \\cite{Arone_Ching}, Bauer, Johnson, and McCarthy \\cite{Bauer_Johnson_McCarthy}, Ching \\cite{Ching_duality}, Harper and Hess \\cite[1.14]{Harper_Hess}, Kuhn \\cite{Kuhn_survey}, and Pereira \\cite{Pereira_general_context, Pereira_spectral_operad}. For example, it follows from our results that the identity functor on a category of structured ring spectra is analytic in the sense of Goodwillie \\cite{Goodwillie_calc2}.\n\n\\begin{assumption}\n\\label{assumption:commutative_ring_spectrum}\nFrom now on in this paper, we assume that ${ \\mathcal{R} }$ is any commutative ring spectrum; i.e., we assume that ${ \\mathcal{R} }$ is any commutative monoid object in the category $({ \\mathsf{Sp}^\\Sigma },{ \\otimes }_S,S)$ of symmetric spectra \\cite{Hovey_Shipley_Smith, Schwede_book_project}. We work mostly in the category of ${ \\mathcal{R} }$-modules which we denote by ${ \\mathsf{Mod}_{ \\mathcal{R} } }$.\n\\end{assumption}\n\n\\begin{rem}\nOur results apply to many different types of algebraic structures on spectra including (i) associative ring spectra, which we simply call ring spectra, (ii) commutative ring spectra, (iii) all of the $E_n$ ring spectra for $1\\leq n\\leq \\infty$ that interpolate  between these two extremes of non-commutativity and commutativity. These structures, and many others, are examples of algebras over operads. We therefore work in the following general context: throughout this paper, ${ \\mathcal{O} }$ is an operad in the category of ${ \\mathcal{R} }$-modules (unless otherwise stated), ${ \\Alg_{ \\mathcal{O} } }$ is the category of ${ \\mathcal{O} }$-algebras, and ${ \\Lt_{ \\mathcal{O} } }$ is the category of left ${ \\mathcal{O} }$-modules. \n\nWhile ${ \\mathcal{O} }$-algebras are the main objects of interest for most readers, our results also apply in the more general case of left modules over the operad ${ \\mathcal{O} }$; that generalization will be needed elsewhere.\n\\end{rem}\n\n\\begin{rem}\nIn this paper, we say that a symmetric sequence $X$ of ${ \\mathcal{R} }$-modules is $n$-connected if each ${ \\mathcal{R} }$-module $X[\\mathbf{t}]$ is an $n$-connected spectrum. We say that an algebra (resp. left module) over an operad is $n$-connected if the underlying ${ \\mathcal{R} }$-module (resp. symmetric sequence of ${ \\mathcal{R} }$-modules) is $n$-connected, and similarly for operads. Similarly, we say that a map $X { \\rightarrow } Y$ of symmetric sequences is $n$-connected if each map $X[\\mathbf{t}] { \\rightarrow } Y[\\mathbf{t}]$ is an $n$-connected map of ${ \\mathcal{R} }$-modules, and a map of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) is $n$-connected if the underlying map of spectra (resp. symmetric sequences) is $n$-connected.\n\\end{rem}\n\nThe main results of this paper are Theorems~\\ref{thm:higher_blakers_massey} and \\ref{thm:higher_dual_blakers_massey}, which are the analogs of Goodwillie's higher Blakers-Massey theorems \\cite[2.4 and 2.6]{Goodwillie_calc2}. These results include various interesting special cases which we now highlight.\n\nOne such case is given by the homotopy excision result of Theorem~\\ref{thm:homotopy_excision} below. Goerss and Hopkins \\cite[2.3.13]{Goerss_Hopkins_moduli} prove a closely related homotopy excision result in the special case of simplicial algebras over an $E_\\infty$ operad, and remark that it is true more generally for any simplicial operad \\cite[2.3.14]{Goerss_Hopkins_moduli}. In a closely related setting, Baues \\cite[I.C.4]{Baues_combinatorial} proves a homotopy excision result in an algebraic setting that includes simplicial associative algebras, and closely related is a result of  Schwede \\cite[3.6]{Schwede_algebraic} that is very nearly a homotopy excision result in the context of algebras over a simplicial theory. Our result also recovers Dugger and Shipley's \\cite[2.3]{Dugger_Shipley} homotopy excision result for associative ring spectra as a very special case.\n\n\\begin{thm}[Homotopy excision for structured ring spectra]\n\\label{thm:homotopy_excision}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. Let ${ \\mathcal{X} }$ be a homotopy pushout square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow } { \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i=1,2$), then\n\\begin{itemize}\n\\item[(a)] ${ \\mathcal{X} }$ is $l$-cocartesian in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) with $l=k_1+k_2 +1$,\n\\item[(b)] ${ \\mathcal{X} }$ is $k$-cartesian with $k=k_1+k_2$.\n\\end{itemize}\n\\end{thm}\n\nRelaxing the assumption in Theorem~\\ref{thm:homotopy_excision} that ${ \\mathcal{X} }$ is a homotopy pushout square, we obtain the following result which is the direct analog for structured ring spectra of the original Blakers-Massey Theorem for spaces.\n\n\\begin{thm}[Blakers-Massey theorem for structured ring spectra]\n\\label{thm:blakers_massey}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. Let ${ \\mathcal{X} }$ be a commutative square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$, and  $k_{12}\\in{ \\mathbb{Z} }$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow } { \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i=1,2$) and ${ \\mathcal{X} }$ is $k_{12}$-cocartesian, then ${ \\mathcal{X} }$ is $k$-cartesian, where $k$ is the minimum of $k_{12}-1$ and $k_{1}+k_{2}$.\n\\end{thm}\n\nThe following higher homotopy excision result lies at the heart of this paper. It can be thought of as a structured ring spectra analog of higher homotopy excision (see Goodwillie \\cite[2.3]{Goodwillie_calc2}) in the context of spaces. This result also implies that the identity functors for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$ are $0$-analytic in the sense of \\cite[4.2]{Goodwillie_calc2}.\n\n\\begin{thm}[Higher homotopy excision for structured ring spectra]\n\\label{thm:higher_homotopy_excision}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a strongly $\\infty$-cocartesian $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Let $k_i\\geq -1$ for each $i\\in W$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow }{ \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i\\in W$),  then\n\\begin{itemize}\n\\item[(a)] ${ \\mathcal{X} }$ is $l$-cocartesian in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) with $l=|W|-1+\\sum_{i\\in W}k_i$,\n\\item[(b)] ${ \\mathcal{X} }$ is $k$-cartesian with $k=\\sum_{i\\in W}k_i$.\n\\end{itemize}\n\\end{thm}\n\nThe preceding results are all special cases of the following theorem which relaxes the assumption in Theorem \\ref{thm:higher_homotopy_excision} that ${ \\mathcal{X} }$ is strongly $\\infty$-cocartesian. This result is a structured ring spectra analog of Goodwillie's higher Blakers-Massey theorem for spaces \\cite[2.4]{Goodwillie_calc2}. \n\n\\begin{thm}[Higher Blakers-Massey theorem for structured ring spectra]\n\\label{thm:higher_blakers_massey}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_\\emptyset$ and ${ \\mathcal{X} }_V$) is $k_V$-cocartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen ${ \\mathcal{X} }$ is $k$-cartesian, where $k$ is the minimum of $-|W|+\\sum_{V\\in\\lambda}(k_V+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets.\n\\end{thm}\n\nFor instance, when $n=3$, $k$ is the minimum of\n\\begin{align*}\n  k_{\\{1,2,3\\}}-2,\\quad\n  &k_{\\{1,2\\}}+k_{\\{3\\}}-1,\\\\\n  &k_{\\{1,3\\}}+k_{\\{2\\}}-1,\\\\\n  &k_{\\{2,3\\}}+k_{\\{1\\}}-1,\\quad\n  k_{\\{1\\}}+k_{\\{2\\}}+k_{\\{3\\}}.\n\\end{align*}\n\nOur other results are dual versions of Theorems~\\ref{thm:homotopy_excision}, \\ref{thm:blakers_massey}, \\ref{thm:higher_homotopy_excision} and \\ref{thm:higher_blakers_massey}.\n\n\\begin{thm}[Dual homotopy excision for structured ring spectra]\n\\label{thm:dual_homotopy_excision}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. Let ${ \\mathcal{X} }$ be a homotopy pullback square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$. If ${ \\mathcal{X} }_{\\{2\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_1$-connected and ${ \\mathcal{X} }_{\\{1\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_2$-connected, then ${ \\mathcal{X} }$ is $k$-cocartesian with $k=k_{1}+k_{2}+2$.\n\\end{thm}\n\nThe following result relaxes the assumption that ${ \\mathcal{X} }$ is a homotopy pullback square.\n\n\\begin{thm}[Dual Blakers-Massey theorem for structured ring spectra]\n\\label{thm:dual_blakers_massey}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. Let ${ \\mathcal{X} }$ be a commutative square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2,k_{12}\\geq -1$ with $k_1\\leq k_{12}$ and $k_2\\leq k_{12}$. If ${ \\mathcal{X} }_{\\{2\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_1$-connected, ${ \\mathcal{X} }_{\\{1\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_2$-connected, and ${ \\mathcal{X} }$ is $k_{12}$-cartesian, then ${ \\mathcal{X} }$ is $k$-cocartesian, where $k$ is the minimum of $k_{12}+1$ and $k_{1}+k_{2}+2$.\n\\end{thm}\n\n\\begin{thm}[Higher dual homotopy excision for structured ring spectra]\n\\label{thm:higher_dual_homotopy_excision}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a finite set with $|W|\\geq 2$. Let ${ \\mathcal{X} }$ be a strongly $\\infty$-cartesian $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Let $k_i\\geq -1$ for each $i\\in W$. If each ${ \\mathcal{X} }_{W-\\{i\\}}{ \\rightarrow }{ \\mathcal{X} }_W$ is $k_i$-connected ($i\\in W$),  then ${ \\mathcal{X} }$ is $k$-cocartesian with $k=|W|+\\sum_{i\\in W}k_i$.\n\\end{thm}\n\nThe last three results are all special cases of the following theorem which is a structured ring spectra analog of Goodwillie's higher dual Blakers-Massey theorem for spaces \\cite[2.6]{Goodwillie_calc2}. This specializes to the higher dual homotopy excision result (Theorem \\ref{thm:higher_dual_homotopy_excision}) in the special case that ${ \\mathcal{X} }$ is strongly $\\infty$-cartesian, and to Theorem~\\ref{thm:dual_blakers_massey} in the case $|W| = 2$.\n\n\\begin{thm}[Higher dual Blakers-Massey theorem for structured ring spectra]\n\\label{thm:higher_dual_blakers_massey}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_{W-V}$ and ${ \\mathcal{X} }_W$) is $k_V$-cartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen ${ \\mathcal{X} }$ is $k$-cocartesian, where $k$ is the minimum of $k_W+|W|-1$ and $|W|+\\sum_{V\\in\\lambda}k_V$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$.\n\\end{thm}\n\n\n\\subsection{Organization of the paper}\n\nIn Section \\ref{sec:preliminaries} we recall some preliminaries on algebras and modules over operads. In Section \\ref{sec:cubical_diagrams} we prove our main results. Much of the work is concerned with proving higher homotopy excision (Theorem~\\ref{thm:higher_homotopy_excision}) which we obtain as a special case of a more general result, Theorem \\ref{thm:pushout_cofibration_cube_homotopical_analysis}. We then use an induction argument due to Goodwillie to pass from this to the higher Blakers-Massey result (Theorem~\\ref{thm:higher_blakers_massey}). We can then use higher Blakers-Massey to deduce, first, higher dual homotopy excision (Theorem~\\ref{thm:higher_dual_homotopy_excision}) and then higher dual Blakers-Massey (Theorem~\\ref{thm:higher_dual_blakers_massey}). Finally, in Section \\ref{sec:chain_complexes_over_a_commutative_ring}, we observe that the analogs of the main theorems stated above remain true in the context of unbounded chain complexes over a commutative ring.\n\n\n\\subsection*{Acknowledgments}\n\nThe second author would like to thank Greg Arone, Kristine Bauer, Bjorn Dundas, Bill Dwyer, Brenda Johnson, Nick Kuhn, Ib Madsen, Jim McClure, and Donald Yau for useful remarks. The second author is grateful to Dmitri Pavlov and Jakob Scholbach for helpful comments that directly led to \\cite{Harper_Spectra_Correction}, and to Mark Behrens and Haynes Miller for a stimulating and enjoyable visit to the Department of Mathematics at the Massachusetts Institute of Technology in summer 2011, and for their invitation which made this possible. The first author was partially supported by National Science Foundation Grant DMS-1144149.\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nThe purpose of this section is to recall various preliminaries on algebras and left modules over operads needed in this paper. Define the sets $\\mathbf{n}:=\\{1,\\dots,n\\}$ for each $n\\geq 0$, where $\\mathbf{0}:=\\emptyset$ denotes the empty set. If $W$ is a finite set, we denote by $|W|$ the number of elements in $W$. For a more detailed development of the material in this section, see \\cite{Harper_Spectra, Harper_Modules}.\n\n\\begin{defn}\n\\label{defn:symmetric_sequences}\nLet ${ \\mathsf{M} }$ be a category and $n\\geq 0$.\n\\begin{itemize}\n\\item $\\Sigma$ is the category of finite sets and their bijections.\n\\item $({ \\mathsf{Mod}_{ \\mathcal{R} } },{ \\,\\wedge\\, },{ \\mathcal{R} })$ is the closed symmetric monoidal category of ${ \\mathcal{R} }$-modules.\n\\item A \\emph{symmetric sequence} in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{M} }$) is a functor $\\functor{A}{\\Sigma^{{ \\mathrm{op} }}}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}$ (resp. $\\functor{A}{\\Sigma^{{ \\mathrm{op} }}}{{ \\mathsf{M} }}$). Denote by ${ \\mathsf{SymSeq} }$ the category of symmetric sequences in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ and their natural transformations.\n\\item A symmetric sequence $A$ is \\emph{concentrated at $n$} if $A[\\mathbf{r}]=\\emptyset$ for all $r\\neq n$.\n\\end{itemize}\n\\end{defn}\n\n\\begin{defn} Let $A_1,\\dotsc,A_t\\in{ \\mathsf{SymSeq} }$. Their \\emph{tensor product} $A_1{ \\check{\\tensor} }\\dotsb{ \\check{\\tensor} } A_t\\in{ \\mathsf{SymSeq} }$ is the left Kan extension of objectwise smash along coproduct of sets\n\\begin{align*}\n\\xymatrix{\n  (\\Sigma^{{ \\mathrm{op} }})^{\\times t}\n  \\ar[rr]^-{A_1\\times\\dotsb\\times A_t}\\ar[d]^{\\coprod} & &\n  ({ \\mathsf{Mod}_{ \\mathcal{R} } })^{\\times t}\\ar[r]^-{{ \\,\\wedge\\, }} & { \\mathsf{Mod}_{ \\mathcal{R} } } \\\\\n  \\Sigma^{{ \\mathrm{op} }}\\ar[rrr]^{A_1{ \\check{\\tensor} }\\dotsb{ \\check{\\tensor} }\n  A_t}_{\\text{left Kan extension}} & & & { \\mathsf{Mod}_{ \\mathcal{R} } }\n}\n\\end{align*}\n\\end{defn}\n\nIf $X$ is a finite set and $A$ is an object in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$, we use the usual dot notation $A\\cdot X$ (see Mac Lane \\cite{MacLane_categories} or \\cite[2.3]{Harper_Modules}) to denote the copower $A\\cdot X$ defined by\n$\n  A\\cdot X := \\coprod_X A\n$,\nthe coproduct in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ of $|X|$ copies of $A$. Recall the following useful calculations for tensor products.\n\n\\begin{prop}\nLet $A_1,\\dotsc,A_t\\in{ \\mathsf{SymSeq} }$ and $R\\in\\Sigma$, with $r:=|R|$. There are natural isomorphisms\n\\begin{align}\n  \\notag\n  (A_1{ \\check{\\tensor} }\\dotsb{ \\check{\\tensor} } A_t)[R]&{  \\ \\cong \\ }\\\n  \\coprod_{\\substack{\\function{\\pi}{R}{\\mathbf{t}}\\\\ \\text{in ${ \\mathsf{Set} }$}}}\n  A_1[\\pi^{-1}(1)]{ \\,\\wedge\\, }\\dotsb{ \\,\\wedge\\, }\n  A_t[\\pi^{-1}(t)],\\\\\n  \\label{eq:tensor_check_calc}\n  &{  \\ \\cong \\ }\n  \\coprod_{r_1+\\dotsb +r_t=r}A_1[\\mathbf{r_1}]{ \\,\\wedge\\, }\\dotsb{ \\,\\wedge\\, }\n  A_t[\\mathbf{r_t}]\\underset{{\\Sigma_{r_1}\\times\\dotsb\\times\n  \\Sigma_{r_t}}}{\\cdot}\\Sigma_{r}\n\\end{align}\n\\end{prop}\n\nHere, ${ \\mathsf{Set} }$ is the category of sets and their maps, and \\eqref{eq:tensor_check_calc} displays the tensor product $(A_1{ \\check{\\tensor} }\\dotsb{ \\check{\\tensor} } A_t)[R]$ as a coproduct of $\\Sigma_{r_1}\\times\\dotsb\\times\\Sigma_{r_t}$-orbits. It will be conceptually useful to extend the definition of tensor powers $A^{{ \\check{\\tensor} } t}$ to situations in which the integers $t$ are replaced by a finite set $T$.\n\n\\begin{defn}\nLet $A\\in{ \\mathsf{SymSeq} }$ and $R,T\\in\\Sigma$. The \\emph{tensor powers} $A^{{ \\check{\\tensor} } T}\\in{ \\mathsf{SymSeq} }$ are defined objectwise by\n\\begin{align*}\n  (A^{{ \\check{\\tensor} }\\emptyset})[R]:=\n  \\coprod_{\\substack{\\function{\\pi}{R}{\\emptyset}\\\\ \\text{in ${ \\mathsf{Set} }$}}}\n  S,\\quad\\quad\n  &(A^{{ \\check{\\tensor} } T})[R]:=\n  \\coprod_{\\substack{\\function{\\pi}{R}{T}\\\\ \\text{in ${ \\mathsf{Set} }$}}}\n  \\bigwedge_{t\\in T} A[\\pi^{-1}(t)]\\quad(T\\neq\\emptyset).\n\\end{align*}\nNote that there are no functions $\\function{\\pi}{R}{\\emptyset}$ in ${ \\mathsf{Set} }$ unless $R=\\emptyset$. We will use the abbreviation $A^{{ \\check{\\tensor} } 0}:=A^{{ \\check{\\tensor} }\\emptyset}$.\n\\end{defn}\n\n\\begin{defn}\\label{defn:circle_product}\nLet $A,B,C\\in{ \\mathsf{SymSeq} }$, and $r,t\\geq 0$. The \\emph{circle product} (or composition product) $A\\circ B\\in{ \\mathsf{SymSeq} }$ is defined objectwise by the coend\n\\begin{align}\n  \\label{eq:circle_product_calc}\n  (A\\circ B)[\\mathbf{r}] := A{ \\,\\wedge\\, }_\\Sigma (B^{{ \\check{\\tensor} }-})[\\mathbf{r}]\n  &{  \\ \\cong \\ }\n  \\coprod_{t\\geq 0}A[\\mathbf{t}]{ \\,\\wedge\\, }_{\\Sigma_t}\n  (B^{{ \\check{\\tensor} } t})[\\mathbf{r}].\n\\end{align}\n\\end{defn}\n\n\n\\begin{prop}\n\\label{prop:closed_monoidal_on_symmetric_sequences}\n\\\n\\begin{itemize}\n\\item [(a)] $({ \\mathsf{SymSeq} },{ \\check{\\tensor} },1)$ has the structure of a closed symmetric monoidal category with all small limits and colimits. The unit for ${ \\check{\\tensor} }$ denoted ``$1$'' is the symmetric sequence concentrated at $0$ with value ${ \\mathcal{R} }$.\n\\item [(b)] $({ \\mathsf{SymSeq} },\\circ,I)$ has the structure of a closed monoidal category with all small limits and colimits. The unit for $\\circ$ denoted ``$I$'' is the symmetric sequence concentrated at $1$ with value ${ \\mathcal{R} }$. Circle product is not symmetric.\n\\end{itemize}\n\\end{prop}\n\n\\begin{defn}\n\\label{defn:hat_construction_embed_at_zero}\nLet $Z\\in{ \\mathsf{Mod}_{ \\mathcal{R} } }$. Define $\\hat{Z}\\in{ \\mathsf{SymSeq} }$ to be the symmetric sequence concentrated at $0$ with value $Z$.\n\\end{defn}\n\nThe functor $\\function{\\hat{-}}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}{{ \\mathsf{SymSeq} }}$ fits into the adjunction\n\\begin{align*}\n\\xymatrix{\n  { \\mathsf{Mod}_{ \\mathcal{R} } }\\ar@<0.5ex>[r]^-{\\hat{-}} &\n  { \\mathsf{SymSeq} }\\ar@<0.5ex>[l]^-{{ \\mathrm{Ev} }_0}\n}\n\\end{align*}\nwith left adjoint on top and ${ \\mathrm{Ev} }_0$ the \\emph{evaluation} functor defined objectwise by ${ \\mathrm{Ev} }_0(B):=B[\\mathbf{0}]$. Note that $\\hat{-}$ embeds ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ in ${ \\mathsf{SymSeq} }$ as the full subcategory of symmetric sequences concentrated at $0$.\n\n\\begin{defn}\\label{defn:corresponding_functor}\nLet ${ \\mathcal{O} }$ be a symmetric sequence and $Z\\in{ \\mathsf{Mod}_{ \\mathcal{R} } }$. The corresponding functor $\\functor{{ \\mathcal{O} }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}$ is defined objectwise by\n$\n  { \\mathcal{O} }(Z):={ \\mathcal{O} }\\circ(Z):=\\amalg_{t\\geq 0}{ \\mathcal{O} }[\\mathbf{t}]\n  { \\,\\wedge\\, }_{\\Sigma_t}Z^{\\wedge t}.\n$\n\\end{defn}\n\n\\begin{prop}\nLet ${ \\mathcal{O} },A\\in{ \\mathsf{SymSeq} }$ and $Z\\in{ \\mathsf{Mod}_{ \\mathcal{R} } }$. There are natural isomorphisms\n\\begin{align}\n\\label{eq:circ_product_and_evaluate_at_zero}\n  \\widehat{{ \\mathcal{O} }(Z)}=\n  \\widehat{{ \\mathcal{O} }\\circ(Z)}{  \\ \\cong \\ }{ \\mathcal{O} }\\circ\\hat{Z},\\quad\\quad\n  { \\mathrm{Ev} }_0({ \\mathcal{O} }\\circ A){  \\ \\cong \\ } { \\mathcal{O} }\\circ\\bigl({ \\mathrm{Ev} }_0(A)\\bigr).\n\\end{align}\n\\end{prop}\n\n\\begin{proof}\nThis follows from \\eqref{eq:circle_product_calc} and \\eqref{eq:tensor_check_calc}.\n\\end{proof}\n\n\\begin{defn}\n\\label{defn:operad}\nAn \\emph{operad} in ${ \\mathcal{R} }$-modules is a monoid object in $({ \\mathsf{SymSeq} },\\circ,I)$ and a \\emph{morphism of operads} is a morphism of monoid objects in $({ \\mathsf{SymSeq} },\\circ,I)$.\n\\end{defn}\n\n\\begin{rem} If ${ \\mathcal{O} }$ is an operad, then the associated functor $\\function{{ \\mathcal{O} }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}$ is a monad.\n\\end{rem}\n\n\\begin{defn}\n\\label{defn:algebras_and_modules}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules.\n\\begin{itemize}\n\\item A \\emph{left ${ \\mathcal{O} }$-module} is an object in $({ \\mathsf{SymSeq} },\\circ,I)$ with a left action of ${ \\mathcal{O} }$ and a \\emph{morphism of left ${ \\mathcal{O} }$-modules} is a map that respects the left ${ \\mathcal{O} }$-module structure. Denote by ${ \\Lt_{ \\mathcal{O} } }$ the category of left ${ \\mathcal{O} }$-modules and their morphisms.\n\\item An \\emph{${ \\mathcal{O} }$-algebra} is an algebra for the monad $\\functor{{ \\mathcal{O} }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }}$ and a \\emph{morphism of ${ \\mathcal{O} }$-algebras} is a map in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ that respects the ${ \\mathcal{O} }$-algebra structure. Denote by ${ \\Alg_{ \\mathcal{O} } }$ the category of ${ \\mathcal{O} }$-algebras and their morphisms.\n\\end{itemize}\n\\end{defn}\n\nIt follows easily from \\eqref{eq:circ_product_and_evaluate_at_zero} that an ${ \\mathcal{O} }$-algebra is the same as an ${ \\mathcal{R} }$-module $Z$ with a left ${ \\mathcal{O} }$-module structure on $\\hat{Z}$, and if $Z$ and $Z'$ are ${ \\mathcal{O} }$-algebras, then a morphism of ${ \\mathcal{O} }$-algebras is the same as a map $\\function{f}{Z}{Z'}$ in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ such that $\\function{\\hat{f}}{\\hat{Z}}{\\hat{Z'}}$ is a morphism of left ${ \\mathcal{O} }$-modules. In other words, an algebra over an operad ${ \\mathcal{O} }$ is the same as a left ${ \\mathcal{O} }$-module that is concentrated at $0$, and ${ \\Alg_{ \\mathcal{O} } }$ embeds in ${ \\Lt_{ \\mathcal{O} } }$ as the full subcategory of left ${ \\mathcal{O} }$-modules concentrated at $0$, via the functor $\\function{\\hat{-}}{{ \\Alg_{ \\mathcal{O} } }}{{ \\Lt_{ \\mathcal{O} } }}$, $Z\\longmapsto \\hat{Z}$. Define the \\emph{evaluation} functor $\\function{{ \\mathrm{Ev} }_0}{{ \\Lt_{ \\mathcal{O} } }}{{ \\Alg_{ \\mathcal{O} } }}$ objectwise by ${ \\mathrm{Ev} }_0(B):=B[\\mathbf{0}]$.\n\n\\begin{prop}\n\\label{prop:basic_properties_LTO}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{C} }$. There are adjunctions\n\\begin{align}\n\\label{eq:free_forgetful_adjunction}\n\\xymatrix{\n  { \\mathsf{Mod}_{ \\mathcal{R} } }\\ar@<0.5ex>[r]^-{{ \\mathcal{O} }\\circ(-)} & { \\Alg_{ \\mathcal{O} } },\\ar@<0.5ex>[l]^-{U}\n}\\quad\\quad\n\\xymatrix{\n  { \\mathsf{SymSeq} }\\ar@<0.5ex>[r]^-{{ \\mathcal{O} }\\circ-} & { \\Lt_{ \\mathcal{O} } },\\ar@<0.5ex>[l]^-{U}\n}\\quad\\quad\n\\xymatrix{\n  { \\Alg_{ \\mathcal{O} } }\\ar@<0.5ex>[r]^-{\\hat{-}} & { \\Lt_{ \\mathcal{O} } },\\ar@<0.5ex>[l]^-{{ \\mathrm{Ev} }_0}\n}\n\\end{align}\nwith left adjoints on top and $U$ the forgetful functor. All small colimits exist in ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$, and both reflexive coequalizers and filtered colimits are preserved (and created) by the forgetful functors. All small limits exist in ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$, and are preserved (and created) by the forgetful functors.\n\\end{prop}\n\nThroughout this paper, we use the following model structures on the categories of ${ \\mathcal{O} }$-algebras and left ${ \\mathcal{O} }$-modules.\n\n\\begin{defn}\n\\label{defn:stable_flat_positive_model_structures}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. The \\emph{positive flat stable model structure} on ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) has as weak equivalences the stable equivalences (resp. objectwise stable equivalences) and  as fibrations the positive flat stable fibrations (resp. objectwise positive flat stable fibrations).\n\\end{defn}\n\nThe model structures in Definition \\ref{defn:stable_flat_positive_model_structures} are established in \\cite{Harper_Spectra, Harper_Spectra_Correction, Harper_Hess}. For a description of the cofibrations, see \\cite[Section 4]{Harper_Spectra} and \\cite[Section 7]{Harper_Hess}. For ease of notation, we have followed Schwede \\cite{Schwede_book_project} in using the term \\emph{flat} (e.g., flat stable model structure) for what is called $S$ (e.g., stable $S$-model structure) in \\cite{Hovey_Shipley_Smith, Schwede, Shipley_comm_ring}. For some of the good properties of the flat stable model structure, see \\cite[5.3.7 and 5.3.10]{Hovey_Shipley_Smith}.\n\n\n\\section{Homotopical Analysis of Cubical Diagrams}\n\\label{sec:cubical_diagrams}\n\nIn this section we prove the main results of the paper. The following definitions and constructions appear in Goodwillie \\cite{Goodwillie_calc2} in the context of spaces, and will also be useful in our context of structured ring spectra.\n\n\\begin{defn}[Indexing categories for cubical diagrams]\nLet $W$ be a finite set and ${ \\mathsf{M} }$ a category.\n\\begin{itemize}\n\\item Denote by $\\powerset(W)$ the poset of all subsets of $W$, ordered by inclusion $\\subset$ of sets. We will often regard $\\powerset(W)$ as the category associated to this partial order in the usual way; the objects are the elements of $\\powerset(W)$, and there is a morphism $U{ \\rightarrow } V$ if and only if $U\\subset V$.\n\\item Denote by $\\powerset_0(W)\\subset\\powerset(W)$ the poset of all nonempty subsets of $W$; it is the full subcategory of $\\powerset(W)$ containing all objects except the initial object $\\emptyset$.\n\\item Denote by $\\powerset_1(W)\\subset\\powerset(W)$ the poset of all subsets of $W$ not equal to $W$; it is the full subcategory of $\\powerset(W)$ containing all objects except the terminal object $W$.\n\\item A \\emph{$W$-cube} ${ \\mathcal{X} }$ in ${ \\mathsf{M} }$ is a $\\powerset(W)$-shaped diagram ${ \\mathcal{X} }$ in ${ \\mathsf{M} }$; in other words, a functor $\\function{{ \\mathcal{X} }}{\\powerset(W)}{{ \\mathsf{M} }}$.\n\\end{itemize}\n\\end{defn}\n\n\\begin{rem}\nIf $n=|W|$ and ${ \\mathcal{X} }$ is a $W$-cube in ${ \\mathsf{M} }$, we will sometimes refer to ${ \\mathcal{X} }$ simply as an \\emph{$n$-cube} in ${ \\mathsf{M} }$. In particular, a $0$-cube is an object in ${ \\mathsf{M} }$ and a $1$-cube is a morphism in ${ \\mathsf{M} }$.\n\\end{rem}\n\n\\begin{defn}[Faces of cubical diagrams]\nLet $W$ be a finite set and ${ \\mathsf{M} }$ a category. Let ${ \\mathcal{X} }$ be a $W$-cube in ${ \\mathsf{M} }$ and consider any subsets $U\\subset V\\subset W$. Denote by $\\partial_U^V{ \\mathcal{X} }$ the $(V-U)$-cube defined objectwise by\n\\begin{align*}\n  T\\mapsto(\\partial_U^V{ \\mathcal{X} })_T:={ \\mathcal{X} }_{T\\cup U},\\quad\\quad T\\subset V-U.\n\\end{align*}\nIn other words, $\\partial_U^V{ \\mathcal{X} }$ is the $(V-U)$-cube formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_U$ and ${ \\mathcal{X} }_V$. We say that $\\partial_U^V{ \\mathcal{X} }$ is a \\emph{face} of ${ \\mathcal{X} }$ of \\emph{dimension} $|V-U|$.\n\\end{defn}\n\n\\begin{defn}\n\\label{defn:cofibration_cubes_etc}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a finite set. Let ${ \\mathcal{X} }$ be a $W$-cube in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) or ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) and $k\\in{ \\mathbb{Z} }$.\n\\begin{itemize}\n\\item ${ \\mathcal{X} }$ is a \\emph{cofibration cube} if the map\n$\\colim_{\\powerset_1(V)}{ \\mathcal{X} }{ \\rightarrow }\\colim_{\\powerset(V)}{ \\mathcal{X} }{  \\ \\cong \\ }{ \\mathcal{X} }_V$ is a cofibration for each $V\\subset W$; in particular, each ${ \\mathcal{X} }_V$ is cofibrant.\n\\item ${ \\mathcal{X} }$ is \\emph{$k$-cocartesian} if the map\n$\\hocolim_{\\powerset_1(W)}{ \\mathcal{X} }{ \\rightarrow }\\hocolim_{\\powerset(W)}{ \\mathcal{X} }{ \\ \\simeq \\ }{ \\mathcal{X} }_W$ is $k$-connected.\n\\item ${ \\mathcal{X} }$ is \\emph{$\\infty$-cocartesian} if the map\n$\\hocolim_{\\powerset_1(W)}{ \\mathcal{X} }{ \\rightarrow }\\hocolim_{\\powerset(W)}{ \\mathcal{X} }{ \\ \\simeq \\ }{ \\mathcal{X} }_W$ is a weak equivalence.\n\\item ${ \\mathcal{X} }$ is \\emph{strongly $\\infty$-cocartesian} if each face of dimension $\\geq 2$ is $\\infty$-cocartesian.\n\\item ${ \\mathcal{X} }$ is a \\emph{pushout cube} if the map\n$\\colim_{\\powerset_1(V)}{ \\mathcal{X} }{ \\rightarrow }\\colim_{\\powerset(V)}{ \\mathcal{X} }{  \\ \\cong \\ }{ \\mathcal{X} }_V$  is an isomorphism for each $V\\subset W$ with $|V| \\geq 2$; i.e., if it is built by colimits in the usual way out of the maps ${ \\mathcal{X} }_\\emptyset{ \\rightarrow }{ \\mathcal{X} }_V$, $V\\subset W$, $|V|=1$.\n\\end{itemize}\n\\end{defn}\n\nThese definitions and constructions dualize as follows. Note that when looking for the appropriate dual construction, it is useful to observe that ${ \\mathcal{X} }=\\partial_\\emptyset^V{ \\mathcal{X} }$ when restricted to $\\powerset(V)$; for instance, $\\colim_{\\powerset_1(V)}{ \\mathcal{X} }=\\colim_{\\powerset_1(V)}\\partial_\\emptyset^V{ \\mathcal{X} }$.\n\n\\begin{defn}\n\\label{defn:fibration_cubes_etc}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a finite set. Let ${ \\mathcal{X} }$ be a $W$-cube in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) or ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) and $k\\in{ \\mathbb{Z} }$.\n\\begin{itemize}\n\\item ${ \\mathcal{X} }$ is a \\emph{fibration cube} if the map ${ \\mathcal{X} }_V{  \\ \\cong \\ }\\lim_{\\powerset(W-V)}\\partial_V^W{ \\mathcal{X} }{ \\rightarrow }\\lim_{\\powerset_0(W-V)}\\partial_V^W{ \\mathcal{X} }$ is a fibration for each $V\\subset W$; in particular, each ${ \\mathcal{X} }_V$ is fibrant.\n\\item ${ \\mathcal{X} }$ is \\emph{$k$-cartesian} if the map ${ \\mathcal{X} }_\\emptyset{ \\ \\simeq \\ }\\holim_{\\powerset(W)}{ \\mathcal{X} }{ \\rightarrow }\\holim_{\\powerset_0(W)}{ \\mathcal{X} }$ is $k$-connected.\n\\item ${ \\mathcal{X} }$ is \\emph{$\\infty$-cartesian} if the map ${ \\mathcal{X} }_\\emptyset{ \\ \\simeq \\ }\\holim_{\\powerset(W)}{ \\mathcal{X} }{ \\rightarrow }\\holim_{\\powerset_0(W)}{ \\mathcal{X} }$ is a weak equivalence.\n\\item ${ \\mathcal{X} }$ is \\emph{strongly $\\infty$-cartesian} if each face of dimension $\\geq 2$ is $\\infty$-cartesian.\n\\item ${ \\mathcal{X} }$ is a \\emph{pullback cube} if the map ${ \\mathcal{X} }_V{  \\ \\cong \\ }\\lim_{\\powerset(W-V)}\\partial_V^W{ \\mathcal{X} }{ \\rightarrow }\\lim_{\\powerset_0(W-V)}\\partial_V^W{ \\mathcal{X} }$ is an isomorphism for each $V\\subset W$ with $|W-V| \\geq 2$; i.e., if it is built by limits in the usual way out of the maps ${ \\mathcal{X} }_V{ \\rightarrow }{ \\mathcal{X} }_W$, $V\\subset W$, $|W-V|=1$.\n\\end{itemize}\n\\end{defn}\n\n\\begin{rem}\nIt is important to note that every $1$-cube in ${ \\Alg_{ \\mathcal{O} } }$, ${ \\Lt_{ \\mathcal{O} } }$, ${ \\mathsf{Mod}_{ \\mathcal{R} } }$, or ${ \\mathsf{SymSeq} }$ is strongly $\\infty$-cocartesian (resp. strongly $\\infty$-cartesian), since there are no faces of dimension $\\geq 2$, but only the $1$-cubes that are weak equivalences are $\\infty$-cocartesian (resp. $\\infty$-cartesian).\n\\end{rem}\n\nThe following is an exercise left to the reader.\n\\begin{prop}\n\\label{prop:connectivity_estimates_for_composition_of_maps}\nLet $k\\in{ \\mathbb{Z} }$. Consider any maps $X{ \\rightarrow } Y{ \\rightarrow } Z$ in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) or ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$).\n\\begin{itemize}\n\\item[(a)] If $X{ \\rightarrow } Y$ and $Y{ \\rightarrow } Z$ are $k$-connected, then $X{ \\rightarrow } Z$ is $k$-connected.\n\\item[(b)] If $X{ \\rightarrow } Y$ is $(k-1)$-connected and $X{ \\rightarrow } Z$ is $k$-connected, then $Y{ \\rightarrow } Z$ is $k$-connected.\n\\item[(c)] If $X{ \\rightarrow } Z$ is $k$-connected and $Y{ \\rightarrow } Z$ is $(k+1)$-connected, then $X{ \\rightarrow } Y$ is $k$-connected.\n\n\\end{itemize}\n\\end{prop}\n\nVersions of the following connectivity estimates are proved in Goodwillie \\cite[1.6--1.8]{Goodwillie_calc2} in the context of spaces, and exactly the same arguments give a proof of Propositions \\ref{prop:map_of_cubical_diagrams} and \\ref{prop:composed_map_of_cubical_diagrams} below in the context of structured ring spectra; this is an exercise left to the reader.\n\n\\begin{prop}\n\\label{prop:map_of_cubical_diagrams}\nLet $W$ be a finite set and $k\\in{ \\mathbb{Z} }$. Consider any map ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ of $W$-cubes in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) or ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$).\n\\begin{itemize}\n\\item[(a)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ and ${ \\mathcal{X} }$ are $k$-cocartesian, then ${ \\mathcal{Y} }$ is $k$-cocartesian.\n\\item[(b)] If ${ \\mathcal{X} }$ is $(k-1)$-cocartesian and ${ \\mathcal{Y} }$ is $k$-cocartesian, then ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ is $k$-cocartesian.\n\\item[(c)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ and ${ \\mathcal{Y} }$ are $k$-cartesian, then ${ \\mathcal{X} }$ is $k$-cartesian.\n\\item[(d)] If ${ \\mathcal{X} }$ is $k$-cartesian and ${ \\mathcal{Y} }$ is $(k+1)$-cartesian, then ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ is $k$-cartesian.\n\n\\end{itemize}\n\\end{prop}\n\n\\begin{prop}\n\\label{prop:composed_map_of_cubical_diagrams}\nLet $W$ be a finite set and $k\\in{ \\mathbb{Z} }$. Consider any map ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }{ \\rightarrow }{ \\mathcal{Z} }$ of $W$-cubes in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) or ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$).\n\\begin{itemize}\n\\item[(a)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ and ${ \\mathcal{Y} }{ \\rightarrow }{ \\mathcal{Z} }$ are $k$-cocartesian, then ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Z} }$ is $k$-cocartesian.\n\\item[(b)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ is $(k-1)$-cocartesian and ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Z} }$ is $k$-cocartesian, then ${ \\mathcal{Y} }{ \\rightarrow }{ \\mathcal{Z} }$ is $k$-cocartesian.\n\\item[(c)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ and ${ \\mathcal{Y} }{ \\rightarrow }{ \\mathcal{Z} }$ are $k$-cartesian, then ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Z} }$ is $k$-cartesian.\n\\item[(d)] If ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Z} }$ is $k$-cartesian and ${ \\mathcal{Y} }{ \\rightarrow }{ \\mathcal{Z} }$ is $(k+1)$-cartesian, then ${ \\mathcal{X} }{ \\rightarrow }{ \\mathcal{Y} }$ is $k$-cartesian.\n\\end{itemize}\n\\end{prop}\n\nThe following results depend on the fact that the model structures on ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ and ${ \\mathsf{SymSeq} }$ are stable, so that fibration and cofibration sequences coincide. Note that these do not hold, in general, for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$.\n\n\\begin{prop}\n\\label{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR}\nLet $W$ be a finite set and $k\\in{ \\mathbb{Z} }$. Let ${ \\mathcal{X} }$ be a $W$-cube in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$).\n\\begin{itemize}\n\\item[(a)] ${ \\mathcal{X} }$ is $k$-cocartesian if and only if ${ \\mathcal{X} }$ is $(k-|W|+1)$-cartesian.\n\\item[(b)] ${ \\mathcal{X} }$ is $k$-cartesian if and only if ${ \\mathcal{X} }$ is $(k+|W|-1)$-cocartesian.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nThis is because the total homotopy cofiber of ${ \\mathcal{X} }$ (see Goodwillie \\cite[1.4]{Goodwillie_calc2}) is weakly equivalent to the $|W|$-th suspension, usually denoted $\\Sigma^{|W|}$, of the total homotopy fiber of ${ \\mathcal{X} }$ (see \\cite[1.1a]{Goodwillie_calc2}).\n\\end{proof}\n\n\n\\subsection{Proof of higher homotopy excision for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$}\n\nThe purpose of this section is to prove Theorem~\\ref{thm:higher_homotopy_excision}. At the heart of our proof is a homotopical analysis of the construction ${ \\mathcal{O} }_A$ described in Proposition~\\ref{prop:coproduct_modules}. We deduce Theorem~\\ref{thm:higher_homotopy_excision} from a more general result about the effect of the construction $A \\mapsto { \\mathcal{O} }_A$ on strongly $\\infty$-cocartesian cubes. \n\n\\begin{defn}\\label{def:symmetric_array}\nConsider symmetric sequences in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$. A \\emph{symmetric array} in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ is a symmetric sequence in ${ \\mathsf{SymSeq} }$; i.e., a functor $\\functor{A}{\\Sigma^{ \\mathrm{op} }}{{ \\mathsf{SymSeq} }}$. Denote by ${ \\mathsf{SymArray} }:={ \\mathsf{SymSeq} }^{\\Sigma^{ \\mathrm{op} }}$ the category of symmetric arrays in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ and their natural transformations.\n\\end{defn}\n\nThe following ${ \\mathcal{O} }_A$ construction is crucial to our arguments; a proof of the following proposition is given in \\cite[4.7]{Harper_Spectra}.\n\n\\begin{prop}\n\\label{prop:coproduct_modules}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$, $A\\in{ \\Alg_{ \\mathcal{O} } }$ (resp. $A\\in{ \\Lt_{ \\mathcal{O} } }$), and $Y\\in{ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. $Y\\in{ \\mathsf{SymSeq} }$). Consider any coproduct in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) of the form $A\\amalg{ \\mathcal{O} }\\circ(Y)$ (resp. $A\\amalg({ \\mathcal{O} }\\circ Y)$). There exists a symmetric sequence ${ \\mathcal{O} }_A$ (resp. symmetric array ${ \\mathcal{O} }_A$) and natural isomorphisms\n\\begin{align*}\n  A\\amalg{ \\mathcal{O} }\\circ(Y) {  \\ \\cong \\ }\n  \\coprod\\limits_{q\\geq 0}{ \\mathcal{O} }_A[\\mathbf{q}]\n  { \\,\\wedge\\, }_{\\Sigma_q}Y^{\\wedge q}\\quad\n  \\Bigl(\\text{resp.}\\quad\n  A\\amalg({ \\mathcal{O} }\\circ Y) {  \\ \\cong \\ }\n  \\coprod\\limits_{q\\geq 0}{ \\mathcal{O} }_A[\\mathbf{q}]\n  { \\check{\\tensor} }_{\\Sigma_q}Y^{{ \\check{\\tensor} } q}\n    \\Bigr)\n\\end{align*}\nin the underlying category ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$). If $q\\geq 0$, then ${ \\mathcal{O} }_A[\\mathbf{q}]$ is naturally isomorphic to a colimit of the form\n\\begin{align*}\n  { \\mathcal{O} }_A[\\mathbf{q}]&{  \\ \\cong \\ }\n  \\colim\\biggl(\n  \\xymatrix{\n    \\coprod\\limits_{p\\geq 0}{ \\mathcal{O} }[\\mathbf{p}\\boldsymbol{+}\\mathbf{q}]\n    { \\,\\wedge\\, }_{\\Sigma_p}A^{\\wedge p} &\n    \\coprod\\limits_{p\\geq 0}{ \\mathcal{O} }[\\mathbf{p}\\boldsymbol{+}\\mathbf{q}]\n    { \\,\\wedge\\, }_{\\Sigma_p}({ \\mathcal{O} }\\circ (A))^{\\wedge p}\\ar@<-0.5ex>[l]^-{d_1}\n    \\ar@<-1.5ex>[l]_-{d_0}\n  }\n  \\biggl),\\\\\n  \\text{resp.}\\quad\n  { \\mathcal{O} }_A[\\mathbf{q}]&{  \\ \\cong \\ }\n  \\colim\\biggl(\n  \\xymatrix{\n    \\coprod\\limits_{p\\geq 0}{ \\mathcal{O} }[\\mathbf{p}\\boldsymbol{+}\\mathbf{q}]\n    { \\,\\wedge\\, }_{\\Sigma_p}A^{{ \\check{\\tensor} } p} &\n    \\coprod\\limits_{p\\geq 0}{ \\mathcal{O} }[\\mathbf{p}\\boldsymbol{+}\\mathbf{q}]\n    { \\,\\wedge\\, }_{\\Sigma_p}({ \\mathcal{O} }\\circ A)^{{ \\check{\\tensor} } p}\\ar@<-0.5ex>[l]^-{d_1}\n    \\ar@<-1.5ex>[l]_-{d_0}\n  }\n  \\biggl),\n\\end{align*}\nin ${ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_q^{ \\mathrm{op} }}$ (resp. ${ \\mathsf{SymSeq} }^{\\Sigma_q^{ \\mathrm{op} }}$), with $d_0$ induced by operad multiplication and $d_1$ induced by the left ${ \\mathcal{O} }$-action map $\\function{m}{{ \\mathcal{O} }\\circ (A)}{A}$ (resp. $\\function{m}{{ \\mathcal{O} }\\circ A}{A}$).\n\\end{prop}\n\nRecall from \\cite{Harper_Hess} the following proposition.\n\n\\begin{prop}\n\\label{prop:OA_commutes_with_certain_colimits}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ and let $q\\geq 0$. Then the functor\n$\n  \\function{{ \\mathcal{O} }_{(-)}[\\mathbf{q}]}{{ \\Alg_{ \\mathcal{O} } }}{{ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_q^{ \\mathrm{op} }}}\n$ (resp.\n$\n  \\function{{ \\mathcal{O} }_{(-)}[\\mathbf{q}]}{{ \\Lt_{ \\mathcal{O} } }}{{ \\mathsf{SymSeq} }^{\\Sigma_q^{ \\mathrm{op} }}}\n$) preserves reflexive coequalizers and filtered colimits.\n\\end{prop}\n\n\\begin{defn}\\label{def:filtration_setup_modules}\nLet $\\function{i}{X}{Y}$ be a morphism in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) and $t\\geq 1$. Define $Q_0^t:=X^{\\wedge t}$ (resp. $Q_0^t:=X^{{ \\check{\\tensor} } t}$) and $Q_t^t:=Y^{\\wedge t}$ (resp. $Q_t^t:=Y^{{ \\check{\\tensor} } t}$). For $0<q<t$ define $Q_q^t$ inductively by the left-hand (resp. right-hand) pushout diagrams\n\\begin{align*}\n\\xymatrix{\n  \\Sigma_t\\cdot_{\\Sigma_{t-q}\\times\\Sigma_{q}}X^{\\wedge(t-q)}\n  { \\,\\wedge\\, } Q_{q-1}^q\\ar[d]^{i_*}\\ar[r]^-{{ \\mathrm{pr} }_*} & Q_{q-1}^t\\ar[d]\\\\\n  \\Sigma_t\\cdot_{\\Sigma_{t-q}\\times\\Sigma_{q}}X^{\\wedge(t-q)}\n  { \\,\\wedge\\, } Y^{\\wedge q}\\ar[r] & Q_q^t\n}\\quad\n\\xymatrix{\n  \\Sigma_t\\cdot_{\\Sigma_{t-q}\\times\\Sigma_{q}}X^{{ \\check{\\tensor} }(t-q)}\n  { \\check{\\tensor} } Q_{q-1}^q\\ar[d]^{i_*}\\ar[r]^-{{ \\mathrm{pr} }_*} & Q_{q-1}^t\\ar[d]\\\\\n  \\Sigma_t\\cdot_{\\Sigma_{t-q}\\times\\Sigma_{q}}X^{{ \\check{\\tensor} }(t-q)}\n  { \\check{\\tensor} } Y^{{ \\check{\\tensor} } q}\\ar[r] & Q_q^t\n}\n\\end{align*}\nin ${ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_t}$ (resp. ${ \\mathsf{SymSeq} }^{\\Sigma_t}$). We sometimes denote $Q_q^t$ by $Q_q^t(i)$ to emphasize in the notation the map $\\function{i}{X}{Y}$. The maps ${ \\mathrm{pr} }_*$ and $i_*$ are the obvious maps induced by $i$ and the appropriate projection maps.\n\\end{defn}\n\nRecall from \\cite{Harper_Hess} the following proposition.\n\n\\begin{prop}\n\\label{prop:filtering_OA}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$, $A\\in{ \\Alg_{ \\mathcal{O} } }$ (resp. $A\\in{ \\Lt_{ \\mathcal{O} } }$), and $\\function{i}{X}{Y}$ in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$). Consider any pushout diagram in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) of the form\n\\begin{align}\n\\label{eq:small_arg_pushout_modules}\n\\xymatrix{\n  { \\mathcal{O} }\\circ (X)\\ar[r]^-{f}\\ar[d]^{{ \\mathrm{id} }\\circ (i)} & A\\ar[d]^{j}\\\\\n  { \\mathcal{O} }\\circ (Y)\\ar[r] & B\n}\\quad\\quad\\text{resp.}\\quad\n\\xymatrix{\n  { \\mathcal{O} }\\circ X\\ar[r]^-{f}\\ar[d]^{{ \\mathrm{id} }\\circ i} & A\\ar[d]^{j}\\\\\n  { \\mathcal{O} }\\circ Y\\ar[r] & B\n}\n\\end{align}\nFor each $r\\geq 0$, ${ \\mathcal{O} }_B[\\mathbf{r}]$ is naturally isomorphic to a filtered colimit of the form\n\\begin{align}\n\\label{eq:filtered_colimit_modules_refined}\n  { \\mathcal{O} }_B[\\mathbf{r}]{  \\ \\cong \\ }\n  \\colim\\Bigl(\n  \\xymatrix{\n    { \\mathcal{O} }_A^0[\\mathbf{r}]\\ar[r]^{j_1} &\n    { \\mathcal{O} }_A^1[\\mathbf{r}]\\ar[r]^{j_2} &\n    { \\mathcal{O} }_A^2[\\mathbf{r}]\\ar[r]^{j_3} & \\dotsb\n  }\n  \\Bigr)\n\\end{align}\nin ${ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_r^{ \\mathrm{op} }}$ (resp. ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$), with ${ \\mathcal{O} }_A^0[\\mathbf{r}]:={ \\mathcal{O} }_A[\\mathbf{r}]$ and ${ \\mathcal{O} }_A^t[\\mathbf{r}]$ defined inductively by pushout diagrams in ${ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_r^{ \\mathrm{op} }}$ (resp. ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$) of the form\n\\begin{align}\n\\label{eq:good_filtration_modules_refined}\n\\xymatrix{\n  { \\mathcal{O} }_A[\\mathbf{t}\\boldsymbol{+}\\mathbf{r}]{ \\,\\wedge\\, }_{\\Sigma_t}Q_{t-1}^t\\ar[d]^{{ \\mathrm{id} }\\wedge_{\\Sigma_t}i_*}\n  \\ar[r]^-{f_*} & { \\mathcal{O} }_A^{t-1}[\\mathbf{r}]\\ar[d]^{j_t}\\\\\n  { \\mathcal{O} }_A[\\mathbf{t}\\boldsymbol{+}\\mathbf{r}]{ \\,\\wedge\\, }_{\\Sigma_t}Y^{\\wedge t}\\ar[r]^-{\\xi_t} & { \\mathcal{O} }_A^t[\\mathbf{r}]\n}\\quad\\quad\n\\text{resp.}\\quad\n\\xymatrix{\n  { \\mathcal{O} }_A[\\mathbf{t}\\boldsymbol{+}\\mathbf{r}]{ \\check{\\tensor} }_{\\Sigma_t}Q_{t-1}^t\\ar[d]^{{ \\mathrm{id} }{ \\check{\\tensor} }_{\\Sigma_t}i_*}\n  \\ar[r]^-{f_*} & { \\mathcal{O} }_A^{t-1}[\\mathbf{r}]\\ar[d]^{j_t}\\\\\n  { \\mathcal{O} }_A[\\mathbf{t}\\boldsymbol{+}\\mathbf{r}]{ \\check{\\tensor} }_{\\Sigma_t}Y^{{ \\check{\\tensor} } t}\\ar[r]^-{\\xi_t} & { \\mathcal{O} }_A^t[\\mathbf{r}]\n}\n\\end{align}\n\\end{prop}\n\n\\begin{rem}\nIt is important to note (see \\cite{Harper_Hess}) that for $r=0$ the filtration \\eqref{eq:filtered_colimit_modules_refined} specializes to a filtered colimit of the pushout in \\eqref{eq:small_arg_pushout_modules} of the form\n\\begin{align}\n\\label{eq:good_filtration_modules}\n  B{  \\ \\cong \\ }\n  { \\mathcal{O} }_B[\\mathbf{0}]{  \\ \\cong \\ }\n  \\colim\\bigl(\n  \\xymatrix@1{\n    A_0\\ar[r]^{j_1} & A_1\\ar[r]^{j_2} & A_2\\ar[r]^{j_3} & \\dotsb\n  }\n  \\bigr)\n\\end{align}\nin the underlying category ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$), with $A_0:={ \\mathcal{O} }_A[\\mathbf{0}]{  \\ \\cong \\ } A$ and $A_t:={ \\mathcal{O} }_A^t[\\mathbf{0}]$.\n\\end{rem}\n\n\\begin{prop}\\label{prop:pushout_of_n_connected_cofibration}\nLet $n\\geq -1$. If the map $\\function{i}{X}{Y}$ in Proposition \\ref{prop:filtering_OA} is an $n$-connected generating cofibration or generating acyclic cofibration in ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$) with the positive flat stable model structure, and ${ \\mathcal{R} },{ \\mathcal{O} }_A$ are $(-1)$-connected, then each map $j_t$ in \\eqref{eq:good_filtration_modules} and \\eqref{eq:good_filtration_modules_refined} is an $n$-connected monomorphism. In particular, the map $j$ in \\eqref{eq:small_arg_pushout_modules} is an $n$-connected monomorphism in the underlying category ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$).\n\\end{prop}\n\n\\begin{proof}\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The generating cofibrations and acyclic cofibrations in ${ \\mathsf{SymSeq} }$ have cofibrant domains. Hence by \\cite[4.28*]{Harper_Spectra_Correction}, each $j_t$ in \\eqref{eq:good_filtration_modules} is a monomorphism. We know that $A_t\/A_{t-1}{  \\ \\cong \\ } { \\mathcal{O} }_A[\\mathbf{t}]{ \\check{\\tensor} }_{\\Sigma_t}(Y\/X)^{{ \\check{\\tensor} } t}$ and $*{ \\rightarrow } Y\/X$ is an $n$-connected cofibration in ${ \\mathsf{SymSeq} }$. It follows from \\cite[4.40]{Harper_Hess} that each $j_t$ in \\eqref{eq:good_filtration_modules} is $n$-connected. The case for each map $j_t$ in \\eqref{eq:good_filtration_modules_refined} is similar.\n\\end{proof}\n\nThe following proposition is closely related to Dugger-Shipley \\cite[A.3, A.4]{Dugger_Shipley}.\n\n\\begin{prop}\n\\label{prop:factorization_into_n_connected_cofibration}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $n\\geq -1$. If $\\function{f}{A}{C}$ is an $n$-connected map in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$), $C$ is fibrant,  and ${ \\mathcal{R} },{ \\mathcal{O} }_A$ are $(-1)$-connected, then $f$ factors in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) as\n\\begin{align}\n\\label{eq:factoring_n_connected_maps}\n\\xymatrix{\n  A\\ar[r]^-{j} & B\\ar[r]^-{p} & C\n}\n\\end{align}\na nice $n$-connected cofibration followed by an acyclic fibration. Here, ``nice'' means that $j$ is a (possibly transfinite) composition of pushouts of $n$-connected generating cofibrations and generating acyclic cofibrations in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$). This factorization is functorial in all such $f$.\n\\end{prop}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:factorization_into_n_connected_cofibration}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. Let $\\function{i}{X}{Y}$ be an $n$-connected generating cofibration or generating acyclic cofibration in ${ \\mathsf{SymSeq} }$ with the positive flat stable model structure, and consider the pushout diagram\n\\begin{align}\n\\label{eq:gluing_on_cells}\n\\xymatrix{\n  { \\mathcal{O} }\\circ X\\ar[r]\\ar[d] & Z_0\\ar[d]^{i_0}\\\\\n  { \\mathcal{O} }\\circ Y\\ar[r] & Z_1\n}\n\\end{align}\nin ${ \\Lt_{ \\mathcal{O} } }$. Assume ${ \\mathcal{O} }_{Z_0}$ is $(-1)$-connected; let's verify that ${ \\mathcal{O} }_{Z_1}$ is $(-1)$-connected and $i_0$ is an $n$-connected monomorphism in ${ \\mathsf{SymSeq} }$. Let $A:=Z_0$. By Proposition \\ref{prop:filtering_OA}, we know ${ \\mathcal{O} }_{Z_1}[\\mathbf{r}]$ is naturally isomorphic to a filtered colimit of the form\n\\begin{align*}\n  { \\mathcal{O} }_{Z_1}[\\mathbf{r}]&{  \\ \\cong \\ }\n  \\colim\\bigl(\n  \\xymatrix{\n    { \\mathcal{O} }_A^0[\\mathbf{r}]\\ar[r]^{j_1} &\n    { \\mathcal{O} }_A^1[\\mathbf{r}]\\ar[r]^{j_2} &\n    { \\mathcal{O} }_A^2[\\mathbf{r}]\\ar[r]^{j_3} & \\dotsb\n  }\n  \\bigr)\n\\end{align*}\nin ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$, and Proposition \\ref{prop:pushout_of_n_connected_cofibration} verifies that each $j_t$ is an $n$-connected monomorphism. Since ${ \\mathcal{O} }_A^0={ \\mathcal{O} }_{Z_0}$ is $(-1)$-connected by assumption, it follows that ${ \\mathcal{O} }_{Z_1}$ is $(-1)$-connected, and taking $r=0$ (or using Proposition \\ref{prop:pushout_of_n_connected_cofibration} again) finishes the argument that $i_0$ is an $n$-connected monomorphism in ${ \\mathsf{SymSeq} }$.\n\nConsider a sequence\n$\n\\xymatrix@1{\n  Z_0\\ar[r] & Z_1\\ar[r] & Z_2\\ar[r] & \\dotsb\n}\n$\nof pushouts of maps as in \\eqref{eq:gluing_on_cells}, and let $Z_\\infty:=\\colim_k Z_k$. Consider the naturally occurring map $Z_0{ \\rightarrow } Z_\\infty$, and assume ${ \\mathcal{O} }_{Z_0}$ is $(-1)$-connected. By the argument above we know that\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{Z_1}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{Z_2}[\\mathbf{r}]\\ar[r] & \\dotsb\n}\n\\end{align*}\nis a sequence of $n$-connected monomorphisms, hence ${ \\mathcal{O} }_{Z_\\infty}$ is $(-1)$-connected, and taking $r=0$ verifies that $Z_0{ \\rightarrow } Z_\\infty$ is an $n$-connected monomorphism in ${ \\mathsf{SymSeq} }$.\n\nThe small object argument (see \\cite[7.12]{Dwyer_Spalinski} for a useful introduction) produces a factorization \\eqref{eq:factoring_n_connected_maps} of $f$ such that $p$ has the right lifting property with respect to the $n$-connected generating cofibrations and generating acyclic cofibrations in ${ \\Lt_{ \\mathcal{O} } }$, and $j$ is a (possibly transfinite) composition of pushouts of maps as in \\eqref{eq:gluing_on_cells}, starting with $Z_0=A$. Since $C$ is fibrant in ${ \\Lt_{ \\mathcal{O} } }$ by assumption, it follows from the latter lifting property that $p$ is a fibration between fibrant objects in ${ \\Lt_{ \\mathcal{O} } }$. By the argument above, it follows that $j$ is $n$-connected. Since $f$ is $n$-connected by assumption, it follows that $p$ is $n$-connected, and since $p$ furthermore has the right lifting property with respect to the $n$-connected generating cofibrations, it follows that $p$ is a weak equivalence which completes the proof.\n\\end{proof}\n\n\\begin{rem}\nIt is important to note that an $n$-connected fibration between fibrant objects in symmetric spectra with the positive flat stable model structure (resp. positive stable model structure) is a positive levelwise $n$-connected fibration; this property has been exploited in the argument above.\n\\end{rem}\n\n\\begin{rem}\nTo keep the statement of Proposition \\ref{prop:factorization_into_n_connected_cofibration} as simple and non-technical as possible, we have been conservative in our choice for the set of maps used in the small object argument. In other words, running the small object argument with the set of $n$-connected generating cofibrations and generating acyclic cofibrations in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) is sufficient for our purposes and makes for an attractive and simple statement, but one can obtain the desired factorizations using a smaller set of maps; this is an exercise left to the reader.\n\\end{rem}\n\n\\begin{prop}\n\\label{prop:OA_construction_on_n_connected_cofibrations}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $n\\geq -1$.  Let $\\function{j}{A}{B}$ be a cofibration in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$). Assume that ${ \\mathcal{R} },{ \\mathcal{O} }_A$ are $(-1)$-connected. If $j$ is $n$-connected, then ${ \\mathcal{O} }_A[\\mathbf{r}]{ \\rightarrow }{ \\mathcal{O} }_B[\\mathbf{r}]$ is an $n$-connected monomorphism for each $r\\geq 0$.\n\\end{prop}\n\n\\begin{proof}\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The proof is in two parts; in part (a) we assume that $B$ is fibrant in ${ \\Lt_{ \\mathcal{O} } }$, and in part (b) we do not assume that $B$ is fibrant in ${ \\Lt_{ \\mathcal{O} } }$. Consider part (a). Proceed exactly as in the proof of Proposition \\ref{prop:factorization_into_n_connected_cofibration}, and consider the pushout diagram \\eqref{eq:gluing_on_cells}. Assume ${ \\mathcal{O} }_{Z_0}$ is $(-1)$-connected; let's verify that ${ \\mathcal{O} }_{Z_0}[\\mathbf{r}]{ \\rightarrow }{ \\mathcal{O} }_{Z_1}[\\mathbf{r}]$ is an $n$-connected monomorphism for each $r\\geq 0$. This follows by arguing exactly as in the proof of Proposition \\ref{prop:factorization_into_n_connected_cofibration}. Consider a sequence\n\\begin{align*}\n\\xymatrix@1{\n  Z_0\\ar[r] & Z_1\\ar[r] & Z_2\\ar[r] & \\dotsb\n}\n\\end{align*}\nof pushouts of maps as in \\eqref{eq:gluing_on_cells}, and let $Z_\\infty:=\\colim_k Z_k$. Consider the naturally occurring map $Z_0{ \\rightarrow } Z_\\infty$, and assume ${ \\mathcal{O} }_{Z_0}$ is $(-1)$-connected. By the argument above we know that\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{Z_1}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{Z_2}[\\mathbf{r}]\\ar[r] & \\dotsb\n}\n\\end{align*}\nis a sequence of $n$-connected monomorphisms, hence ${ \\mathcal{O} }_{Z_0}[\\mathbf{r}]{ \\rightarrow } { \\mathcal{O} }_{Z_\\infty}[\\mathbf{r}]$ is an $n$-connected monomorphism. Noting that every $n$-connected cofibration of the form $A{ \\rightarrow } B$ in ${ \\Lt_{ \\mathcal{O} } }$ is a retract of a (possibly transfinite) composition of pushouts of maps as in \\eqref{eq:gluing_on_cells}, starting with $Z_0=A$, finishes the proof of part (a).\n\nConsider part (b). The map $B{ \\rightarrow } *$ in ${ \\Lt_{ \\mathcal{O} } }$ factors as $B{ \\rightarrow } C{ \\rightarrow } *$, an acyclic cofibration followed by a fibration. Since the map $B{ \\rightarrow } C$ is an acyclic cofibration and the composite map $A{ \\rightarrow } C$ is an $n$-connected cofibration, we know from part (b) that the composite map\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{O} }_{A}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{B}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{C}[\\mathbf{r}]\n}\n\\end{align*}\nis an $n$-connected monomorphism and the right-hand map is a monomorphism and a weak equivalence. Hence the left-hand map is an $n$-connected monomorphism which finishes the proof of part (b).\n\\end{proof}\n\n\\begin{prop}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules. If $A$ is a cofibrant ${ \\mathcal{O} }$-algebra (resp. left ${ \\mathcal{O} }$-module) and ${ \\mathcal{R} },{ \\mathcal{O} },A$ are $(-1)$-connected, then ${ \\mathcal{O} }_A$ is $(-1)$-connected.\n\\end{prop}\n\n\\begin{proof}\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. This follows from Proposition \\ref{prop:OA_construction_on_n_connected_cofibrations} by considering the map ${ \\mathcal{O} }\\circ\\emptyset{ \\rightarrow } A$ in ${ \\Lt_{ \\mathcal{O} } }$, together with the natural isomorphisms ${ \\mathcal{O} }\\circ\\emptyset{  \\ \\cong \\ }\\widehat{{ \\mathcal{O} }[\\mathbf{0}]}$ and ${ \\mathcal{O} }_{{ \\mathcal{O} }\\circ\\emptyset}[\\mathbf{r}]{  \\ \\cong \\ }\\widehat{{ \\mathcal{O} }[\\mathbf{r}]}$ for each $r\\geq 0$ (see \\cite[5.31]{Harper_Hess}).\n\\end{proof}\n\n\\begin{rem}\n\\label{rem:map_of_n_cubes}\nAny $3$-cube $\\mathcal{X}$ of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) may be regarded as a map of $2$-cubes $A{ \\rightarrow } B$ with $A=\\partial_\\emptyset^{\\{1,2\\}}{ \\mathcal{X} }$ (the top face of ${ \\mathcal{X} }$) and $B=\\partial_{\\{3\\}}^{\\{1,2,3\\}}{ \\mathcal{X} }$ (the bottom face of ${ \\mathcal{X} }$) as follows:\n\\begin{align*}\n\\xymatrix@!0{\n{ \\mathcal{X} }_\\emptyset\\ar[rr]\\ar[dd]\\ar[dr] &&\n{ \\mathcal{X} }_{\\{1\\}}\\ar[dr]\\ar'[d][dd]\\\\\n&{ \\mathcal{X} }_{\\{2\\}}\\ar[rr]\\ar[dd] &&\n{ \\mathcal{X} }_{\\{1,2\\}}\\ar[dd]\\\\\n{ \\mathcal{X} }_{\\{3\\}}\\ar[dr]\\ar'[r][rr] &&\n{ \\mathcal{X} }_{\\{1,3\\}}\\ar[dr]\\\\\n&{ \\mathcal{X} }_{\\{2,3\\}}\\ar[rr] &&\n{ \\mathcal{X} }_{\\{1,2,3\\}}\n}\n\\quad\\quad\n\\xymatrix@!0{\nA_\\emptyset\\ar[rr]\\ar[dd]\\ar[dr] &&\nA_{\\{1\\}}\\ar[dr]\\ar'[d][dd]\\\\\n&A_{\\{2\\}}\\ar[rr]\\ar[dd] &&\nA_{\\{1,2\\}}\\ar[dd]\\\\\nB_\\emptyset\\ar[dr]\\ar'[r][rr] &&\nB_{\\{1\\}}\\ar[dr]\\\\\n&B_{\\{2\\}}\\ar[rr] &&\nB_{\\{1,2\\}}\n}\n\\end{align*}\nMore generally, we may regard an $(n+1)$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) as a map of $n$-cubes $A{ \\rightarrow } B$ with $A=\\partial_\\emptyset^{\\{1,\\dots,n\\}}{ \\mathcal{X} }$ and $B=\\partial_{\\{n+1\\}}^{\\{1,\\dots,n+1\\}}{ \\mathcal{X} }$, for each $n\\geq 0$. In particular, the map ${ \\mathcal{X} }_\\emptyset{ \\rightarrow }{ \\mathcal{X} }_{\\{n+1\\}}$ in ${ \\mathcal{X} }$ is the map $A_\\emptyset{ \\rightarrow } B_\\emptyset$ in $A{ \\rightarrow } B$.\n\\end{rem}\n\nWe now prove the following result of which Theorem~\\ref{thm:higher_homotopy_excision} is the special case $r = 0$. \n\n\\begin{thm}[Homotopical analysis of ${ \\mathcal{O} }_{ \\mathcal{X} }$ for a pushout cofibration cube ${ \\mathcal{X} }$]\n\\label{thm:pushout_cofibration_cube_homotopical_analysis}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $n\\geq 1$. Let ${ \\mathcal{X} }$ be a pushout $(\\mathbf{n}\\boldsymbol{+}\\mathbf{1})$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) regarded as a map of pushout $\\mathbf{n}$-cubes $A{ \\rightarrow } B$ as in Remark \\ref{rem:map_of_n_cubes}. Assume that ${ \\mathcal{R} },{ \\mathcal{O} }_{A_\\emptyset}$ are $(-1)$-connected. Let $k_1,\\dots,k_{n+1}\\geq -1$. Assume that each $A_\\emptyset{ \\rightarrow } A_{\\{i\\}}$ and $A_\\emptyset{ \\rightarrow } B_\\emptyset$ are cofibrations between cofibrant objects in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) $(1\\leq i\\leq n)$. Consider the associated left-hand diagram of the form\n\\begin{align}\n\\label{eq:colim_punctured_cube_associated_diagrams}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}A\\ar[d]\\ar[r] & \\tilde{A}\\ar[d]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}B\\ar[r] & \\tilde{B}\n}\\quad\\quad\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A[\\mathbf{r}]\\ar[d]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{A}}[\\mathbf{r}]\\ar[d]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_B[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{B}}[\\mathbf{r}]\n}\n\\end{align}\nin the underlying category ${ \\mathsf{Mod}_{ \\mathcal{R} } }$ (resp. ${ \\mathsf{SymSeq} }$), and more generally, the associated right-hand diagrams $(r\\geq 0)$ in ${ \\mathsf{Mod}_{ \\mathcal{R} } }^{\\Sigma_r^{ \\mathrm{op} }}$ (resp. ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$). If each $A_\\emptyset{ \\rightarrow } A_{\\{i\\}}$ is $k_i$-connected $(1\\leq i\\leq n)$ and $A_\\emptyset{ \\rightarrow } B_\\emptyset$ is $k_{n+1}$-connected, then the diagrams \\eqref{eq:colim_punctured_cube_associated_diagrams} are $(k_1+\\dots + k_{n+1}+n)$-cocartesian; here, $\\tilde{A}:=A_\\mathbf{n}$ and $\\tilde{B}:=B_{\\mathbf{n}}$.\n\\end{thm}\n\n\\begin{rem}\nIn other words, this theorem shows that the $(\\mathbf{n}\\boldsymbol{+}\\mathbf{1})$-cube ${ \\mathcal{O} }_{ \\mathcal{X} }[\\mathbf{r}]$ is $(k_1+\\dots + k_{n+1}+n)$-cocartesian for each $r\\geq 0$, or equivalently, it shows that the $(\\mathbf{n}\\boldsymbol{+}\\mathbf{1})$-cube ${ \\mathcal{O} }_{ \\mathcal{X} }$ is $(k_1+\\dots + k_{n+1}+n)$-cocartesian. The left-hand diagram in (\\ref{eq:colim_punctured_cube_associated_diagrams}) is the case $r = 0$ and the result here is precisely that needed for Theorem~\\ref{thm:higher_homotopy_excision}.\n\\end{rem}\n\n\\begin{proof}\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The proof is in two parts; in part (a) we assume that $B_\\emptyset$ is fibrant in ${ \\Lt_{ \\mathcal{O} } }$, and in part (b) we do not assume that $B_\\emptyset$ is fibrant in ${ \\Lt_{ \\mathcal{O} } }$. The argument is by induction on $n$. It is convenient to start the induction at $n=0$ in which case the diagrams in \\eqref{eq:colim_punctured_cube_associated_diagrams} are maps (i.e., $1$-cubes) of the form $\\tilde{A}{ \\rightarrow }\\tilde{B}$ and ${ \\mathcal{O} }_{\\tilde{A}}[\\mathbf{r}]{ \\rightarrow }{ \\mathcal{O} }_{\\tilde{B}}[\\mathbf{r}]$. Hence the case $n=0$ is verified by Proposition \\ref{prop:OA_construction_on_n_connected_cofibrations}. Let $N\\geq 1$ and assume the proposition is true for each $0\\leq n\\leq N-1$. Consider part (a); let's verify it remains true for $n=N$. Let $\\function{i}{X}{Y}$ be a $k_{n+1}$-connected generating cofibration or generating acyclic cofibration in ${ \\mathsf{SymSeq} }$ with the positive flat stable model structure, $Z_0$ a pushout $n$-cube in ${ \\Lt_{ \\mathcal{O} } }$, and consider any left-hand pushout diagram of the\nform\n\\begin{align}\n\\label{eq:gluing_on_cells_higher_cube}\n\\xymatrix{\n  { \\mathcal{O} }\\circ X\\ar[d]_-{{ \\mathrm{id} }\\circ i}\\ar[r] &\n  {Z_0}_\\emptyset\\ar[d]\\\\\n  { \\mathcal{O} }\\circ Y\\ar[r] &\n  {Z_1}_\\emptyset\n}\\quad\\quad\n\\xymatrix{\n  Z_0\\ar[d] \\\\\n  Z_1\n}\\quad\\quad\n\\xymatrix{\n  { \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[d]\\\\\n  { \\mathcal{O} }_{Z_1}[\\mathbf{r}]\n}\n\\end{align}\nin ${ \\Lt_{ \\mathcal{O} } }$ with the middle map of pushout $n$-cubes the associated pushout $(n+1)$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Assume each ${Z_0}_\\emptyset{ \\rightarrow } {Z_0}_{\\{i\\}}$ is a $k_i$-connected cofibration between cofibrant objects in ${ \\Lt_{ \\mathcal{O} } }$ $(1\\leq i\\leq n)$ and ${ \\mathcal{O} }_{{Z_0}_\\emptyset}$ is $(-1)$-connected; let's verify that the associated right-hand maps  of $n$-cubes $(r\\geq 0)$, each regarded as an $(n+1)$-cube in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$, are $(k_1+\\dots+k_{n+1}+n)$-cocartesian. If $A:=Z_0$ and $\\tilde{A}:=A_{\\{1,\\dots,n\\}}$, then by Proposition \\ref{prop:filtering_OA} there are corresponding filtrations\n\\begin{align}\n\\label{eq:induced_filtration_diagram_for_studying_induced_map_higher_cubes}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^0[\\mathbf{r}]\\ar[dd]^{\\xi_0}\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^1[\\mathbf{r}]\\ar@{.>}[d]^{\\xi_1}\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^2[\\mathbf{r}]\\ar@{.>}[d]^{\\xi_2}\\ar[r] &\n  \\dots\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^\\infty[\\mathbf{r}]\\ar[d]^{\\xi_\\infty}\\\\\n  &\n  \\cdot\\ar@{.>}[d]^{(*)_1}\\ar[r] &\n  \\cdot\\ar@{.>}[d]^{(*)_2}\\ar[r] &\n  \\dots\\ar[r] &\n  \\cdot\\ar[d]^{(*)_\\infty}\\\\\n  { \\mathcal{O} }_{\\tilde{A}}^0[\\mathbf{r}]\\ar[r]\\ar@\/_0.5pc\/[ur] &\n  { \\mathcal{O} }_{\\tilde{A}}^1[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{A}}^2[\\mathbf{r}]\\ar[r] &\n  \\dots\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{A}}^\\infty[\\mathbf{r}]\n}\n\\end{align}\ntogether with induced maps $\\xi_t$ and $(*)_t$ ($t\\geq 1$) that make the diagram in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$ commute; here, the upper diagrams are pushout diagrams and $\\xi_\\infty:=\\colim_t\\xi_t$, the maps $(*)_t$ are the obvious induced maps and $(*)_\\infty:=\\colim_t(*)_t$, the left-hand vertical map is naturally isomorphic to\n\\begin{align*}\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\longrightarrow\n  { \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}],\n\\end{align*}\nand the right-hand vertical maps are naturally isomorphic to the diagram\n\\begin{align}\n\\label{eq:right_hand_vertical_maps_higher_cubes}\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_{Z_1}[\\mathbf{r}]\\longrightarrow\n  \\bigl(\\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_{Z_1}[\\mathbf{r}]\\bigr)\\cup\n  { \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\longrightarrow\n  { \\mathcal{O} }_{\\tilde{Z}_1}[\\mathbf{r}];\n\\end{align}\nhere, $\\tilde{Z}_0:={Z_0}_{\\{1,\\dots,n\\}}$ and $\\tilde{Z}_1:={Z_1}_{\\{1,\\dots,n\\}}$. We want to show that the right-hand map in \\eqref{eq:right_hand_vertical_maps_higher_cubes} is $(k_1+\\dots+k_{n+1}+n)$-connected; since the horizontal maps in \\eqref{eq:induced_filtration_diagram_for_studying_induced_map_higher_cubes} are monomorphisms, it suffices to verify each map $(*)_t$ is $(k_1+\\dots+k_{n+1}+n)$-connected. The argument is by induction on $t$. The map $\\xi_0$ factors as\n\\begin{align*}\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^0[\\mathbf{r}]\\longrightarrow\n  \\bigl(\\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A^0[\\mathbf{r}]\\bigr)\\cup{ \\mathcal{O} }_{\\tilde{A}}^0[\\mathbf{r}]\\xrightarrow[{  \\ \\cong \\ }]{(*)_0}\n  { \\mathcal{O} }_{\\tilde{A}}^0[\\mathbf{r}]\n\\end{align*}\nand since the right-hand map $(*)_0$ is an isomorphism, it is $(k_1+\\dots+k_{n+1}+n)$-connected. Consider the commutative diagram\n\\begin{align}\n\\label{eq:filtration_quotients_diagram_for_analyzing_connectivity_higher_cubes}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_A^{t-1}[\\mathbf{r}]\\ar[d]^{\\xi_{t-1}}\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_A^t[\\mathbf{r}]\\ar[d]^{\\xi_t}\\ar[r] &\n  \\bigl(\\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_A[\\mathbf{t+r}]\\bigr){ \\check{\\tensor} }_{\\Sigma_t}(Y\/X)^{{ \\check{\\tensor} } t}\\ar[d]_{{  \\ \\cong \\ }}\\ar@\/^2pc\/[dd]^{(\\#)}\\\\\n  \\cdot\\ar[d]^{(*)_{t-1}}\\ar[r] &\n  \\cdot\\ar[d]^{(*)_t}\\ar[r] &\n  \\cdot\\ar[d]_{(\\#\\#)}\\\\\n  { \\mathcal{O} }_{\\tilde{A}}^{t-1}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{A}}^t[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{A}}[\\mathbf{t+r}]{ \\check{\\tensor} }_{\\Sigma_t}(Y\/X)^{{ \\check{\\tensor} } t}\n}\n\\end{align}\nwith rows cofiber sequences. Since we know $(Y\/X)^{{ \\check{\\tensor} } t}$ is at least $k_{n+1}$-connected and\n$\n  \\colim_{\\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_A[\\mathbf{t+r}]\\longrightarrow\n  { \\mathcal{O} }_{\\tilde{A}}[\\mathbf{t+r}]\n$\nis $(k_1+\\dots+k_{n}+n-1)$-connected by the induction hypothesis, it follows that $(\\#)$ is $(k_1+\\dots+k_{n+1}+n)$-connected, and hence $(\\#\\#)$ is also. Since the rows in \\eqref{eq:filtration_quotients_diagram_for_analyzing_connectivity_higher_cubes} are cofiber sequences, it follows by induction on $t$ that $(*)_t$ is $(k_1+\\dots+k_{n+1}+n)$-connected for each $t\\geq 1$. This finishes the argument that the the right-hand maps of $n$-cubes $(r\\geq 0)$ in \\eqref{eq:gluing_on_cells_higher_cube}, each regarded as an $(n+1)$-cube in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$, are $(k_1+\\dots+k_{n+1}+n)$-cocartesian.\n\nConsider a sequence $Z_0{ \\rightarrow } Z_1{ \\rightarrow } Z_2{ \\rightarrow }\\cdots$ of pushout $n$-cubes in ${ \\Lt_{ \\mathcal{O} } }$ as in \\eqref{eq:gluing_on_cells_higher_cube}, define $\\tilde{Z}_n:={Z_n}_{\\{1,\\dots,n\\}}$, $Z_\\infty:=\\colim_nZ_n$, and $\\tilde{Z}_\\infty:=\\colim_n\\tilde{Z}_n$, and consider the naturally occurring map $Z_0{ \\rightarrow } Z_\\infty$ of pushout $n$-cubes, regarded as a pushout $(n+1)$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the associated left-hand diagram of the form\n\\begin{align}\n\\label{eq:gluing_on_cells_filtration_sequence_O_construction_higher_cubes}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}Z_0\\ar[d]\\ar[r] & \\tilde{Z}_0\\ar[d]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}Z_\\infty\\ar[r] & \\tilde{Z}_\\infty\n}\\quad\\quad\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[d]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[d]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}{ \\mathcal{O} }_{Z_\\infty}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_\\infty}[\\mathbf{r}]\n}\n\\end{align}\nin the underlying category ${ \\mathsf{SymSeq} }$ and the associated right-hand diagrams ($r\\geq 0$) in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$. Assume each ${Z_0}_\\emptyset{ \\rightarrow } {Z_0}_{\\{i\\}}$ is a $k_i$-connected cofibration between cofibrant objects in ${ \\Lt_{ \\mathcal{O} } }$ $(1\\leq i\\leq n)$ and ${ \\mathcal{O} }_{{Z_0}_\\emptyset}$ is $(-1)$-connected. We want to show that the right-hand diagrams in \\eqref{eq:gluing_on_cells_filtration_sequence_O_construction_higher_cubes} are $(k_1+\\dots + k_{n+1}+n)$-cocartesian. Consider the associated commutative diagram\n\\begin{align}\n\\label{eq:filtration_quotients_diagram_for_analyzing_connectivity_for_Z_higher_cubes}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[dd]\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_1}[\\mathbf{r}]\\ar@{.>}[d]^{\\eta_1}\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_2}[\\mathbf{r}]\\ar@{.>}[d]^{\\eta_2}\\ar[r] &\n  \\cdots\\ar[r] &\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_\\infty}[\\mathbf{r}]\\ar[d]^{\\eta_\\infty}\\\\\n  &\n  \\cdot\n  \\ar@{.>}[d]^{(\\#)_1}\\ar[r] &\n  \\cdot\n  \\ar@{.>}[d]^{(\\#)_2}\\ar[r] &\n  \\cdots\\ar[r] &\n  \\cdot\\ar[d]^{(\\#)_\\infty}\\\\\n  { \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[r]\\ar@\/_0.5pc\/[ur] &\n  { \\mathcal{O} }_{\\tilde{Z}_1}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_2}[\\mathbf{r}]\\ar[r] &\n  \\cdots\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_\\infty}[\\mathbf{r}]\n}\n\\end{align}\nin ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$ and induced maps $\\eta_t$ and $(\\#)_t$ $(t\\geq 1)$; here, the upper diagrams are pushout diagrams and $\\eta_\\infty:=\\colim_t\\eta_t$, the maps $(\\#)_t$ are the obvious induced maps and $(\\#)_\\infty:=\\colim_t(\\#)_t$, and the right-hand vertical maps are naturally isomorphic to the diagram\n\\begin{align}\n\\label{eq:right_hand_vertical_maps_sequential_diagram_higher_cubes}\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_\\infty}[\\mathbf{r}]\n  \\longrightarrow\n  \\bigl(\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_\\infty}[\\mathbf{r}]\n  \\bigr)\n  \\cup{ \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\n  \\longrightarrow\n  { \\mathcal{O} }_{\\tilde{Z}_\\infty}[\\mathbf{r}]\n\\end{align}\nWe want to show that the right-hand map in \\eqref{eq:right_hand_vertical_maps_sequential_diagram_higher_cubes} is $(k_1+\\dots + k_{n+1}+n)$-connected; since the horizontal maps in \\eqref{eq:filtration_quotients_diagram_for_analyzing_connectivity_for_Z_higher_cubes} are monomorphisms, it suffices to verify each map $(\\#)_t$ is $(k_1+\\dots + k_{n+1}+n)$-connected. The argument is by induction on $t$. The map $(\\#)_t$ factors as\n\\begin{align}\n\\label{eq:factorization_of_maps_to_analyze_higher_cubes}\n\\xymatrix{\n  \\bigl(\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_t}[\\mathbf{r}]\n  \\bigr)\n  \\cup{ \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[r] &\n  \\bigl(\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_t}[\\mathbf{r}]\n  \\bigr)\n  \\cup{ \\mathcal{O} }_{\\tilde{Z}_{t-1}}[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_t}[\\mathbf{r}]\n}\n\\end{align}\nWe know from above that $(\\#)_1$ and the right-hand map in \\eqref{eq:factorization_of_maps_to_analyze_higher_cubes} are $(k_1+\\dots + k_{n+1}+n)$-connected for each $t\\geq 1$, hence it follows by induction on $t$ that $(\\#)_t$ is $(k_1+\\dots + k_{n+1}+n)$-connected for each $t\\geq 1$. This finishes the argument that the right-hand diagrams $(r\\geq 0)$ in \\eqref{eq:gluing_on_cells_filtration_sequence_O_construction_higher_cubes} are $(k_1+\\dots + k_{n+1}+n)$-cocartesian in  ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$.\n\nIt follows from Proposition \\ref{prop:factorization_into_n_connected_cofibration} that the pushout $(n+1)$-cube $A{ \\rightarrow } B$ factors as $Z_0\\xrightarrow{i_\\lambda} Z_\\lambda \\xrightarrow{p} B$, a composition of pushout $(n+1)$-cubes in ${ \\Lt_{ \\mathcal{O} } }$, starting with $Z_0=A$, where $i_\\lambda$ is a (possibly transfinite) composition of pushout $n$-cubes as in \\eqref{eq:gluing_on_cells_higher_cube} and $p$ is an objectwise weak equivalence. Consider the associated diagram\n\\begin{align*}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_0}[\\mathbf{r}]\\ar[d]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[d]\\ar@\/^0.5pc\/[dr]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_\\lambda}[\\mathbf{r}]\\ar[d]^{{ \\ \\simeq \\ }}\\ar[r] &\n  \\bigl(\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_{Z_\\lambda}[\\mathbf{r}]\n  \\bigr)\n  \\cup{ \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[d]^{{ \\ \\simeq \\ }}\\ar[r]^-{(**)} &\n  { \\mathcal{O} }_{\\tilde{Z}_\\lambda}[\\mathbf{r}]\\ar[d]^{{ \\ \\simeq \\ }}\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_B[\\mathbf{r}]\\ar[r] &\n  \\bigl(\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_B[\\mathbf{r}]\n  \\bigr)\n  \\cup{ \\mathcal{O} }_{\\tilde{Z}_0}[\\mathbf{r}]\\ar[r]^-{(*)} &\n  { \\mathcal{O} }_{\\tilde{B}}[\\mathbf{r}]\n}\n\\end{align*}\nNoting that the bottom vertical arrows are weak equivalences, it follows that $(*)$ has the same connectivity as $(**)$, which finishes the proof of part (a) that the right-hand diagrams $(r\\geq 0)$ in \\eqref{eq:colim_punctured_cube_associated_diagrams} are $(k_1+\\dots + k_{n+1}+n)$-cocartesian in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$.  In particular, taking $r=0$ verifies that the left-hand diagram in \\eqref{eq:colim_punctured_cube_associated_diagrams} is $(k_1+\\dots + k_{n+1}+n)$-cocartesian in ${ \\mathsf{SymSeq} }$.\n\nConsider part (b). The map $B_\\emptyset{ \\rightarrow } *$ in ${ \\Lt_{ \\mathcal{O} } }$ factors as $B_\\emptyset{ \\rightarrow } C_\\emptyset{ \\rightarrow } *$, an acyclic cofibration followed by a fibration. Consider the associated pushout $(n+1)$-cube $B{ \\rightarrow } C$ in ${ \\Lt_{ \\mathcal{O} } }$ and the associated diagram of pushout squares of the form\n\\begin{align*}\n\\xymatrix{\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_A[\\mathbf{r}]\\ar[d]\\ar[r] & { \\mathcal{O} }_{\\tilde{A}}[\\mathbf{r}]\\ar[d]\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_B[\\mathbf{r}]\\ar[d]^{{ \\ \\simeq \\ }}\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{B}}[\\mathbf{r}]\\ar[d]^{{ \\ \\simeq \\ }}\\\\\n  \\colim\\limits_{\\ \\ \\powerset_1(\\mathbf{n})}\n  { \\mathcal{O} }_C[\\mathbf{r}]\\ar[r] &\n  { \\mathcal{O} }_{\\tilde{C}}[\\mathbf{r}]\n}\n\\end{align*}\nin ${ \\Lt_{ \\mathcal{O} } }$. Since the map $B_\\emptyset{ \\rightarrow } C_\\emptyset$ is an acyclic cofibration and the composite map $A_\\emptyset{ \\rightarrow } C_\\emptyset$ is a $k_{n+1}$-connected cofibration, we know from part (a) that the outer diagram is $(k_1+\\dots+k_{n+1}+n)$-cocartesian in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$. Since the vertical maps in the bottom square are weak equivalences, it follows that the upper square is $(k_1+\\dots+k_{n+1}+n)$-cocartesian in ${ \\mathsf{SymSeq} }^{\\Sigma_r^{ \\mathrm{op} }}$ which finishes the proof of part (b).\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:higher_homotopy_excision}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. It is enough to treat the special case where ${ \\mathcal{X} }$ is a pushout cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. The case $|W|=1$ is trivial and the case $|W|\\geq 2$ follows from Theorem \\ref{thm:pushout_cofibration_cube_homotopical_analysis}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:homotopy_excision}]\nThis is the special case $|W| = 2$ of Theorem \\ref{thm:higher_homotopy_excision}.\n\\end{proof}\n\n\n\\subsection{Proof of the higher Blakers-Massey theorem for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$}\n\nThe purpose of this section is to prove the Blakers-Massey theorems \\ref{thm:blakers_massey} and \\ref{thm:higher_blakers_massey}. We first show that Blakers-Massey for square diagrams (Theorem \\ref{thm:blakers_massey}) follows fairly easily from the higher homotopy excision result proved in the previous section.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:blakers_massey}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. Let $W:=\\{1,2\\}$. It is enough to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the induced maps\n\\begin{align*}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(*)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(**)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset(W)}{  \\ \\cong \\ }{ \\mathcal{X} }_W\n}\n\\end{align*}\nWe know that $(*)$ is $(k_1+k_2+1)$-connected by homotopy excision (Theorem \\ref{thm:homotopy_excision}) and $(**)$ is $k_{12}$-connected by assumption. Hence by Proposition \\ref{prop:connectivity_estimates_for_composition_of_maps}(a) the composition is $l$-connected, where $l$ is the minimum of $k_1+k_2+1$ and $k_{12}$; in other words, we have verified that ${ \\mathcal{X} }$ is $l$-cocartesian in ${ \\mathsf{SymSeq} }$, and Proposition \\ref{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR}(a) finishes the proof.\n\n\\end{proof}\n\nWe now turn to the proof of the higher Blakers-Massey result (Theorem~\\ref{thm:higher_blakers_massey}). Our approach follows that used by Goodwillie at the corresponding point in \\cite{Goodwillie_calc2}.\n\nThe following is an important warm-up calculation for Proposition \\ref{prop:cocartesian_estimates_for_induction_argument}.\n\n\\begin{prop}\n\\label{prop:warmup_cocartesian_estimates_for_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a cofibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_\\emptyset$ and ${ \\mathcal{X} }_V$) is $k_V$-cocartesian,\n\\item[(ii)] $k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen, for every $U\\subsetneqq V\\subset W$, the $(V-U)$-cube $\\partial_U^V{ \\mathcal{X} }$ is $k_{V-U}$-cocartesian.\n\\end{prop}\n\n\\begin{proof}\nThe argument is by induction on $|U|$. The case $|U|=0$ is true by assumption. Let $n\\geq 1$ and assume the proposition is true for each $0\\leq|U|<n$. Let's verify it remains true for $|U|=n$. Let $u\\in U$ and note that $\\partial_{U-\\{u\\}}^V{ \\mathcal{X} }$ can be written as the composition of cubes\n\\begin{align*}\n\\xymatrix{\n  \\partial_{U-\\{u\\}}^{V-\\{u\\}}{ \\mathcal{X} }\\ar[r] &\n  \\partial_U^V{ \\mathcal{X} }\n}\n\\end{align*}\nWe know by the induction assumption that the composition of cubes is $k_{(V-U)\\cup\\{u\\}}$-cocartesian and the left-hand cube is $k_{V-U}$-cocartesian. Since $k_{V-U}\\leq k_{(V-U)\\cup\\{u\\}}$ by assumption, it follows from Proposition \\ref{prop:map_of_cubical_diagrams}(a) that the right-hand cube is $k_{V-U}$-cocartesian which finishes the proof.\n\\end{proof}\n\n\\begin{defn}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any subset ${ \\mathcal{B} }\\subset\\powerset(W)$.\n\\begin{itemize}\n\\item A subset ${ \\mathcal{A} }\\subset{ \\mathcal{B} }$ is \\emph{convex} if every element of ${ \\mathcal{B} }$ which is less than an element of ${ \\mathcal{A} }$ is in ${ \\mathcal{A} }$.\n\\item Define ${ \\mathcal{A} }^W_\\mathrm{min}:=\\{V\\subset W: |V|\\leq 1\\}$ and ${ \\mathcal{A} }^W_\\mathrm{max}:=\\powerset(W)$.\n\\item For each convex subset ${ \\mathcal{A} }\\subset\\powerset(W)$, the $W$-cube ${ \\mathcal{X} }_{ \\mathcal{A} }$ is defined objectwise by $({ \\mathcal{X} }_{ \\mathcal{A} })_U := \\colim_{{ \\mathcal{A} }\\cap\\powerset(U)}{ \\mathcal{X} }{  \\ \\cong \\ }\\colim_{T\\in{ \\mathcal{A} },\\, T\\subset U}{ \\mathcal{X} }_T$.\n\\end{itemize}\n\\end{defn}\n\nThe following proposition will be needed in the proof of Proposition \\ref{prop:properties_of_the_X_sub_A_construction} below.\n\n\\begin{prop}\n\\label{prop:colim_over_punctured_cube_of_X_sub_A_construction}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any convex subset ${ \\mathcal{A} }\\subset\\powerset(W)$. Then for each nonempty subset $V\\subset W$, there is a natural isomorphism\n\\begin{align*}\n  \\colim_{\\powerset_1(V)}{ \\mathcal{X} }_{ \\mathcal{A} }{  \\ \\cong \\ }\n  \\colim_{{ \\mathcal{A} }\\cap\\powerset_1(V)}{ \\mathcal{X} }=\n  \\colim_{T\\in{ \\mathcal{A} },\\,T\\subsetneqq V}{ \\mathcal{X} }_T\n\\end{align*}\n\\end{prop}\n\n\\begin{proof}\nThis is because the indexing sets $\\{T\\in{ \\mathcal{A} }:T\\subset U,\\, U\\subsetneqq V\\}$ and $\\{T\\in{ \\mathcal{A} }:T\\subsetneqq V\\}$ are the same.\n\\end{proof}\n\nThe following proposition explains the key properties of the ${ \\mathcal{X} }_{ \\mathcal{A} }$ construction and its relationship to ${ \\mathcal{X} }$; it is through these properties that the ${ \\mathcal{X} }_{ \\mathcal{A} }$ construction is useful and meaningful.\n\n\\begin{prop}\n\\label{prop:properties_of_the_X_sub_A_construction}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any convex subset ${ \\mathcal{A} }^W_\\mathrm{min}\\subset{ \\mathcal{A} }\\subset{ \\mathcal{A} }^W_\\mathrm{max}$.\n\\begin{itemize}\n\\item[(a)] There are natural isomorphisms $\\partial_\\emptyset^V{ \\mathcal{X} }_{ \\mathcal{A} }{  \\ \\cong \\ }\\partial_\\emptyset^V{ \\mathcal{X} }$ if $V\\in{ \\mathcal{A} }$,\n\\item[(b)] The $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }_{ \\mathcal{A} }$ is $\\infty$-cocartesian if $V\\notin{ \\mathcal{A} }$ and ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Alg_{ \\mathcal{O} } }$.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nConsider part (a). Let $V\\in{ \\mathcal{A} }$. Then we know that $\\powerset(V)\\subset{ \\mathcal{A} }$, by ${ \\mathcal{A} }$ convex, and hence ${ \\mathcal{A} }\\cap\\powerset(V)=\\powerset(V)$. It follows that $({ \\mathcal{X} }_{ \\mathcal{A} })_V=\\colim_{\\powerset(V)}{ \\mathcal{X} }{  \\ \\cong \\ }{ \\mathcal{X} }_V$. The same argument shows that $({ \\mathcal{X} }_{ \\mathcal{A} })_{V'}{  \\ \\cong \\ }{ \\mathcal{X} }_{V'}$, for each $V'\\subset V$. Consider part (b). Let $V\\subset W$ and $V\\notin{ \\mathcal{A} }$. Then ${ \\mathcal{A} }\\cap\\powerset(V)={ \\mathcal{A} }\\cap\\powerset_1(V)$ and hence the composition\n\\begin{align*}\n  \\colim_{\\powerset_1(V)}\\partial_\\emptyset^V{ \\mathcal{X} }_{ \\mathcal{A} }=\n  \\colim_{\\powerset_1(V)}{ \\mathcal{X} }_{ \\mathcal{A} }{  \\ \\cong \\ }\n  \\colim_{{ \\mathcal{A} }\\cap\\powerset_1(V)}{ \\mathcal{X} }=\n  \\colim_{{ \\mathcal{A} }\\cap\\powerset(V)}{ \\mathcal{X} }=\n  ({ \\mathcal{X} }_{ \\mathcal{A} })_V=(\\partial_\\emptyset^V{ \\mathcal{X} }_{ \\mathcal{A} })_V\n\\end{align*}\nis an isomorphism which finishes the proof of part (b).\n\\end{proof}\n\nThe following proposition shows that ${ \\mathcal{X} }_{ \\mathcal{A} }$ inherits several of the homotopical properties of ${ \\mathcal{X} }$.\n\n\\begin{prop}\n\\label{prop:cocartesian_estimates_for_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a cofibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any convex subset ${ \\mathcal{A} }^W_\\mathrm{min}\\subset{ \\mathcal{A} }\\subset{ \\mathcal{A} }^W_\\mathrm{max}$. Assume that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_\\emptyset$ and ${ \\mathcal{X} }_V$) is $k_V$-cocartesian,\n\\item[(ii)] $k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen, for each $U\\subsetneqq V\\subset W$, the $(V-U)$-cube $\\partial_U^V{ \\mathcal{X} }_{ \\mathcal{A} }$ is $k_{V-U}$-cocartesian.\n\\end{prop}\n\n\\begin{proof}\nBy the induction argument in the proof of Proposition \\ref{prop:warmup_cocartesian_estimates_for_induction_argument}, it suffices to verify that the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }_{ \\mathcal{A} }$ is $k_{V}$-cocartesian, for each nonempty subset $V\\subset W$, and this follows immediately from Proposition \\ref{prop:properties_of_the_X_sub_A_construction}.\n\\end{proof}\n\nThe following proposition is proved in Goodwillie \\cite[2.8]{Goodwillie_calc2} in the context of spaces, and exactly the same argument gives a proof in the context of structured ring spectra; this is an exercise left to the reader.\n\n\\begin{prop}\n\\label{prop:cofibration_cube_properties}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a cofibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any convex subset ${ \\mathcal{A} }\\subset\\powerset(W)$.\n\\begin{itemize}\n\\item[(a)] For each inclusion ${ \\mathcal{A} }'\\subset{ \\mathcal{A} }$ of convex subsets of $\\powerset(W)$, the induced map $\\colim_{{ \\mathcal{A} }'}{ \\mathcal{X} }{ \\rightarrow }\\colim_{{ \\mathcal{A} }}{ \\mathcal{X} }$ is a cofibration in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$),\n\\item[(b)] For any convex subsets ${ \\mathcal{A} },{ \\mathcal{B} }$ of $\\powerset(W)$, the diagram\n\\begin{align*}\n\\xymatrix{\n  \\colim_{{ \\mathcal{A} }\\cap{ \\mathcal{B} }}{ \\mathcal{X} }\\ar[r]\\ar[d] & \\colim_{{ \\mathcal{B} }}{ \\mathcal{X} }\\ar[d]\\\\\n  \\colim_{{ \\mathcal{A} }}{ \\mathcal{X} }\\ar[r] & \\colim_{{ \\mathcal{A} }\\cup{ \\mathcal{B} }}{ \\mathcal{X} }\n}\n\\end{align*}\nis a pushout diagram of cofibrations in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$).\n\\item[(c)] If $A\\in{ \\mathcal{A} }$ is maximal and the cofibration\n$\n  \\colim_{\\powerset_1(A)}{ \\mathcal{X} }{ \\rightarrow }\\colim_{\\powerset(A)}{ \\mathcal{X} }{  \\ \\cong \\ }{ \\mathcal{X} }_A\n$\nis $k_A$-connected, then the cofibration\n$$\n  \\colim_{{ \\mathcal{A} }-\\{A\\}}{ \\mathcal{X} }{ \\rightarrow }\\colim_{{ \\mathcal{A} }}{ \\mathcal{X} }\n$$\nis $k_A$-connected.\n\\end{itemize}\n\\end{prop}\n\nThe purpose of the following induction argument is to leverage the higher homotopy excision result (Theorem \\ref{thm:higher_homotopy_excision}) for structured ring spectra into a proof of the first main theorem in this paper (Theorem \\ref{thm:higher_blakers_massey})---the higher Blakers-Massey theorem for structured ring spectra. Proposition \\ref{prop:cubical_induction_argument} is motivated by Goodwillie \\cite[2.12]{Goodwillie_calc2}; it is essentially Goodwillie's cubical induction argument, appropriately modified to our situation.\n\n\\begin{prop}[Cubical induction argument]\n\\label{prop:cubical_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Suppose ${ \\mathcal{A} }\\subset{ \\mathcal{A} }_\\mathrm{max}^W$ is convex, $A\\in{ \\mathcal{A} }$ maximal, $|A|\\geq 2$, and ${ \\mathcal{A} }':={ \\mathcal{A} }-\\{A\\}$. Let ${ \\mathcal{X} }$ be a cofibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_\\emptyset$ and ${ \\mathcal{X} }_V$) is $k_V$-cocartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nIf $X_{{ \\mathcal{A} }'}$ is $k$-cartesian, then ${ \\mathcal{X} }_{ \\mathcal{A} }$ is $k$-cartesian, where $k$ is the minimum of $-|W|+\\sum_{V\\in\\lambda}(k_V+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets.\n\\end{prop}\n\n\\begin{rem}\n\\label{rem:3_cube_example_of_cubical_induction}\nIn the case that ${ \\mathcal{X} }$ is a $3$-cube, the following diagram illustrates one of the cubical decompositions covered by Proposition \\ref{prop:cubical_induction_argument}. It corresponds to the sequence of maximal elements: $\\{1,2\\}$, $\\{1,3\\}$, $\\{2,3\\}$, $\\{1,2,3\\}$; we benefitted from the discussion in Munson-Voli\\'c \\cite{Munson_Volic_book_project} where this cubical decomposition is used, in the context of spaces, to illustrate Goodwillie's cubical induction argument.\n\n\\begin{align*}\n\\xymatrix@!0{\n{ \\mathcal{X} }_\\emptyset\\ar[rr]\\ar[dd]\\ar[dr] &&\n{ \\mathcal{X} }_{\\{1\\}}\\ar[dr]\\ar'[d][dd]\\ar@{=}[rr] &&\n{ \\mathcal{X} }_{\\{1\\}}\\ar@{=}[rr]\\ar[dr]\\ar'[d][dd] &&\n{ \\mathcal{X} }_{\\{1\\}}\\ar[dr]\\ar[dd]|(0.5)\\hole|(0.53)\\hole\\\\\n&{ \\mathcal{X} }_{\\{2\\}}\\ar[rr]\\ar[dd] &&\n\\cdot\\ar[dd]\\ar[rr] &&\n{ \\mathcal{X} }_{\\{1,2\\}}\\ar@{=}[rr]\\ar[dd] &&\n{ \\mathcal{X} }_{\\{1,2\\}}\\ar[dd]\\\\\n{ \\mathcal{X} }_{\\{3\\}}\\ar[dr]\\ar'[r][rr]\\ar@{=}[dd] &&\n\\cdot\\ar[dr]\\ar@{=}'[r][rr] &&\n\\cdot\\ar[dr]\\ar'[r][rr] &&\n{ \\mathcal{X} }_{\\{1,3\\}}\\ar[dr]\\ar@{=}'[d][dd]\\\\\n&\\cdot\\ar[rr]\\ar[dd] &&\n\\cdot\\ar[rr] &&\n\\cdot\\ar[rr] &&\n\\cdot\\ar[dd]\\\\\n{ \\mathcal{X} }_{\\{3\\}}\\ar[dr]\\ar'[r][rrrrrr]\\ar@{=}[dd]&&&&&&\n{ \\mathcal{X} }_{\\{1,3\\}}\\ar[dr]\\ar@{=}'[d][dd]\\\\\n&{ \\mathcal{X} }_{\\{2,3\\}}\\ar[rrrrrr]\\ar@{=}[dd]&&&&&&\n\\cdot\\ar[dd]\\\\\n{ \\mathcal{X} }_{\\{3\\}}\\ar[dr]\\ar'[r][rrrrrr] &&&&&&\n{ \\mathcal{X} }_{\\{1,3\\}}\\ar[dr]\\\\\n&{ \\mathcal{X} }_{\\{2,3\\}}\\ar[rrrrrr]&&&&&&{ \\mathcal{X} }_{\\{1,2,3\\}}\n}\n\\end{align*}\n\\end{rem}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:cubical_induction_argument}]\nThe argument is by induction on $|W|$. The case $|W|=2$ is verified by the proof of Theorem \\ref{thm:blakers_massey}. Let $n\\geq 3$ and assume the proposition is true for each $2\\leq |W|<n$. Let's verify it remains true for $|W|=n$.\n\nWe know by assumption that ${ \\mathcal{X} }_{{ \\mathcal{A} }'}$ is $k$-cartesian. We want to verify that ${ \\mathcal{X} }_{ \\mathcal{A} }$ is $k$-cartesian (it might be helpful at this point to look ahead to  \\eqref{eq:final_decomposition_for_induction_argument} for the decomposition of ${ \\mathcal{X} }_{ \\mathcal{A} }$ that we will use to finish the proof). Consider the induced map of $W$-cubes ${ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }{ \\mathcal{X} }_{ \\mathcal{A} }$. Note that if $V\\not\\supset A$, then $\\powerset(V)\\not\\ni A$ and hence ${ \\mathcal{A} }'\\cap\\powerset(V)={ \\mathcal{A} }\\cap\\powerset(V)$; in particular, each of the maps\n\\begin{align}\n\\label{eq:subdiagram_of_identity_maps}\n\\xymatrix{\n  ({ \\mathcal{X} }_{{ \\mathcal{A} }'})_V\\ar[r]^-{{ \\mathrm{id} }} & ({ \\mathcal{X} }_{ \\mathcal{A} })_V,\n  & V\\not\\supset A\n}\n\\end{align}\nin ${ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }{ \\mathcal{X} }_{ \\mathcal{A} }$ is the identity map. Note that if $V\\supset U\\supset A$, then the diagram\n\\begin{align}\n\\label{eq:pushout_diagram_in_induction_step}\n\\xymatrix{\n  ({ \\mathcal{X} }_{{ \\mathcal{A} }'})_U\\ar[r]\\ar[d] & ({ \\mathcal{X} }_{ \\mathcal{A} })_U\\ar[d]\\\\\n  ({ \\mathcal{X} }_{{ \\mathcal{A} }'})_V\\ar[r] & ({ \\mathcal{X} }_{ \\mathcal{A} })_V,&\n  V\\supset U\\supset A\n}\n\\end{align}\nis a pushout diagram by Proposition \\ref{prop:cofibration_cube_properties}(b); in particular, focusing on this case is the same as focusing on the subdiagram $\\partial_A^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} }$ of ${ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }{ \\mathcal{X} }_{ \\mathcal{A} }$.\n\nLet's first verify that $\\partial_A^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} }$ is $(k+|A|-1)$-cartesian. Let ${ \\mathcal{Y} }$ denote $\\partial_A^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} }$ regarded as a $((W-A)\\cup\\{*\\})$-cube as follows:\n\\begin{align*}\n  { \\mathcal{Y} }_V&:=(\\partial_A^W{ \\mathcal{X} }_{{ \\mathcal{A} }'})_V=({ \\mathcal{X} }_{{ \\mathcal{A} }'})_{V\\cup A},\\quad\\ V\\subset W-A,\\\\\n  { \\mathcal{Y} }_V&:=(\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} })_U=({ \\mathcal{X} }_{ \\mathcal{A} })_{U\\cup A},\\quad\\quad U\\subset W-A,\\quad V=U\\cup\\{*\\}.\n\\end{align*}\nSince $|(W-A)\\cup\\{*\\}|<|W|$, our induction assumption can be applied to ${ \\mathcal{Y} }$, provided that the appropriate $k'_V$-cocartesian estimates are satisfied. We claim that with the following definitions\n\\begin{align*}\n  k'_V&:=k_V,\\quad\\quad \\emptyset\\neq V\\subset W-A,\\\\\n  k'_V&:=k_A,\\quad\\quad V=\\{*\\}\\\\\n  k'_V&:=\\infty,\\quad\\quad \\emptyset\\neq U\\subset W-A,\\quad V=U\\cup\\{*\\},\n\\end{align*}\nthe cube $\\partial_\\emptyset^V{ \\mathcal{Y} }$ (formed by all maps in ${ \\mathcal{Y} }$ between ${ \\mathcal{Y} }_\\emptyset$ and ${ \\mathcal{Y} }_V$) is $k'_V$-cocartesian, for each nonempty subset $V\\subset (W-A)\\cup\\{*\\}$. Let's verify this now: If $V\\subset W-A$ is nonempty, then $\\partial_\\emptyset^V{ \\mathcal{Y} }$ is the cube $\\partial_A^{A\\cup V}{ \\mathcal{X} }_{{ \\mathcal{A} }'}$ which is $k_V$-cocartesian by Proposition \\ref{prop:cocartesian_estimates_for_induction_argument}. If $V=\\{*\\}$, then $\\partial_\\emptyset^V{ \\mathcal{Y} }$ is the map $({ \\mathcal{X} }_{{ \\mathcal{A} }'})_A{ \\rightarrow }({ \\mathcal{X} }_{ \\mathcal{A} })_A$ which is $k_A$-connected by Proposition \\ref{prop:cofibration_cube_properties}(c). Finally, if $V=U\\cup\\{*\\}$ for any nonempty $U\\subset W-A$, then $\\partial_\\emptyset^V{ \\mathcal{Y} }$ is the cube $\\partial_A^{A\\cup U}{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^{A\\cup U}{ \\mathcal{X} }_{ \\mathcal{A} }$ which is $\\infty$-cocartesian by\n\\eqref{eq:pushout_diagram_in_induction_step} and Proposition \\ref{prop:map_of_cubical_diagrams}(b). Hence by our induction hypothesis applied to ${ \\mathcal{Y} }$: since the sum $\\sum_{V\\in\\lambda'}(k'_V+1)$ for a partition $\\lambda'$ of $(W-A)\\cup\\{*\\}$ by nonempty sets is always either $\\infty$, or the sum $\\sum_{U\\in\\lambda}(k_U+1)$ for a partition $\\lambda$ of $W$ by nonempty sets in which some $U$ is $A$, we know that ${ \\mathcal{Y} }$ is $k'$-cartesian, where $k'$ is the minimum of $-|(W-A)\\cup\\{*\\}|+\\sum_{U\\in\\lambda}(k_U+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets in which some $U$ is $A$. In particular, this implies that ${ \\mathcal{Y} }$, and hence $\\partial_A^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} }$, is $(k+|A|-1)$-cartesian.\n\nWe next want to verify that\n\\begin{align}\n\\label{eq:induction_argument_cartesian_estimate_for_certain_subdiagrams}\n  \\partial_{A'}^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_A^W{ \\mathcal{X} }_{ \\mathcal{A} }\n  \\quad \\text{is $(k+|A'|-1)$-cartesian}\n\\end{align}\nfor each $A'\\subset A$. We know that \\eqref{eq:induction_argument_cartesian_estimate_for_certain_subdiagrams} is true for $A'=A$ by above. We will argue by downward induction on $|A'|$. Suppose that \\eqref{eq:induction_argument_cartesian_estimate_for_certain_subdiagrams} is true for some nonempty $A'\\subset A$. Let $a\\in A'$ and note that the cube $\\partial_{A'-\\{a\\}}^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\\partial_{A'-\\{a\\}}^W{ \\mathcal{X} }_{ \\mathcal{A} }$ can be written as the diagram of cubes\n\\begin{align}\n\\label{eq:downward_induction_cube_factorization}\n\\xymatrix{\n  \\partial_{A'-\\{a\\}}^{W-\\{a\\}}{ \\mathcal{X} }_{{ \\mathcal{A} }'}\\ar[r]^{{ \\mathrm{id} }}\\ar[d] &\n  \\partial_{A'-\\{a\\}}^{W-\\{a\\}}{ \\mathcal{X} }_{ \\mathcal{A} }\\ar[d]\\\\\n  \\partial_{A'}^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}\\ar[r] &\n  \\partial_{A'}^W{ \\mathcal{X} }_{{ \\mathcal{A} }}\n}\n\\end{align}\nWe know the top arrow is the identity by \\eqref{eq:subdiagram_of_identity_maps}, and the bottom arrow is $(k+|A'|-1)$-cartesian by assumption. It follows from  Proposition \\ref{prop:map_of_cubical_diagrams}(d) that  \\eqref{eq:downward_induction_cube_factorization} is $(k+|A'|-2)$-cartesian, which finishes the argument that \\eqref{eq:induction_argument_cartesian_estimate_for_certain_subdiagrams} is true for each $A'\\subset A$.\n\nTo finish off the proof, let $a\\in A$ and note by \\eqref{eq:subdiagram_of_identity_maps} that ${ \\mathcal{X} }_{ \\mathcal{A} }$ can be written as the composition of cubes\n\\begin{align}\n\\label{eq:final_decomposition_for_induction_argument}\n  \\partial_\\emptyset^{W-\\{a\\}}{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\n  \\partial_{\\{a\\}}^W{ \\mathcal{X} }_{{ \\mathcal{A} }'}{ \\rightarrow }\n  \\partial_{\\{a\\}}^W{ \\mathcal{X} }_{ \\mathcal{A} }.\n\\end{align}\nThe right-hand arrow is $k$-cartesian by \\eqref{eq:induction_argument_cartesian_estimate_for_certain_subdiagrams} and the left-hand arrow is ${ \\mathcal{X} }_{{ \\mathcal{A} }'}$ which is $k$-cartesian by assumption, hence by Proposition \\ref{prop:composed_map_of_cubical_diagrams}(c) it follows that ${ \\mathcal{X} }_{ \\mathcal{A} }$ is $k$-cartesian which finishes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:higher_blakers_massey}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. It is enough to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Let ${ \\mathcal{A} }:={ \\mathcal{A} }_{\\mathrm{max}}^W$, ${ \\mathcal{A} }':={ \\mathcal{A} }_{\\mathrm{max}}^W-\\{W\\}$, and note that ${ \\mathcal{X} }_{ \\mathcal{A} }$ is equal to ${ \\mathcal{X} }$. Then it follows by induction from Proposition \\ref{prop:cubical_induction_argument}, together with Theorem \\ref{thm:higher_homotopy_excision} (to start the induction using ${ \\mathcal{A} }'={ \\mathcal{A} }^W_{\\mathrm{min}}$) that ${ \\mathcal{X} }$ is $k$-cartesian.\n\\end{proof}\n\n\n\\subsection{Proof of higher dual homotopy excision for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$}\n\nWe now turn to the dual versions of our main results. In this section we prove the dual homotopy excision results (Theorems \\ref{thm:dual_homotopy_excision} and \\ref{thm:higher_dual_homotopy_excision}). Notice that here we are leveraging the fact that cartesian-ness in the categories ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$ is detected in the underlying categories of ${ \\mathcal{R} }$-modules and symmetric sequences, and that, in those underlying categories, there is a close relationship between cartesian-ness and cocartesian-ness, given by Proposition~\\ref{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:dual_homotopy_excision}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. Let $W:=\\{1,2\\}$. It is enough to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the induced maps\n\\begin{align*}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(*)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(**)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset(W)}{  \\ \\cong \\ }{ \\mathcal{X} }_W\n}\n\\end{align*}\nWe know that $(*)$ is $(k_1+k_2+1)$-connected by homotopy excision (Theorem \\ref{thm:homotopy_excision}), and since ${ \\mathcal{X} }$ is $\\infty$-cocartesian in the underlying category ${ \\mathsf{SymSeq} }$, the composition is $\\infty$-connected. Hence by Proposition \\ref{prop:connectivity_estimates_for_composition_of_maps}(b) the map $(**)$ is $(k_1+k_2+2)$-connected which finishes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:higher_dual_homotopy_excision}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The argument is by induction on $|W|$. The case $|W|=2$ is verified by Theorem \\ref{thm:dual_homotopy_excision}. Let $n\\geq 3$ and assume the theorem is true for each $2\\leq |W|< n$. Let's verify it remains true for $|W|=n$.\n\nIt suffices to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the induced maps\n\\begin{align}\n\\label{eq:composition_of_maps_for_determining_cocartesian_estimate}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(*)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(**)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset(W)}{  \\ \\cong \\ }{ \\mathcal{X} }_W\n}\n\\end{align}\nWe want to show that $(**)$ is $k$-connected. Consider the $W$-cube ${ \\mathcal{X} }':={ \\mathcal{X} }_{{ \\mathcal{A} }_\\mathrm{max}-\\{W\\}}$ in ${ \\Lt_{ \\mathcal{O} } }$ and note that $(*)$ is isomorphic to the map\n\\begin{align*}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }'\\ar[r]^-{(*)'} &\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset(W)}{ \\mathcal{X} }'{  \\ \\cong \\ }{ \\mathcal{X} }'_W\n}\n\\end{align*}\nWe know that ${ \\mathcal{X} }'$ is $\\infty$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$, hence by higher Blakers-Massey (Theorem \\ref{thm:higher_blakers_massey}) applied to ${ \\mathcal{X} }'$, together with the induction hypothesis, it follows that ${ \\mathcal{X} }'$ is $k'$-cartesian with $k'=\\sum_{i\\in W}k_i$. Hence by Proposition \\ref{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR} we know that ${ \\mathcal{X} }'$ is $(k'+|W|-1)$-cocartesian in ${ \\mathsf{SymSeq} }$, and therefore $(*)$ is $(k'+|W|-1)$-connected. Since the composition in \\eqref{eq:composition_of_maps_for_determining_cocartesian_estimate} is $\\infty$-connected, it follows from Proposition \\ref{prop:connectivity_estimates_for_composition_of_maps}(b) that $(**)$ is $(k'+|W|)$-connected which finishes the proof.\n\\end{proof}\n\n\\subsection{Proof of the higher dual Blakers-Massey theorem for ${ \\Alg_{ \\mathcal{O} } }$ and ${ \\Lt_{ \\mathcal{O} } }$}\n\nThe purpose of this section is to prove Theorems \\ref{thm:dual_blakers_massey} and \\ref{thm:higher_dual_blakers_massey}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:dual_blakers_massey}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. Let $W:=\\{1,2\\}$. It is enough to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the induced maps\n\\begin{align}\n\\label{eq:composition_of_maps_in_dual_blakers_massey}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(*)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(**)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset(W)}{  \\ \\cong \\ }{ \\mathcal{X} }_W\n}\n\\end{align}\nSince ${ \\mathcal{X} }$ is $k_{12}$-cartesian, we know by Proposition \\ref{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR} that ${ \\mathcal{X} }$ is $(k_{12}+1)$-cocartesian in ${ \\mathsf{SymSeq} }$, and hence the composition in \\eqref{eq:composition_of_maps_in_dual_blakers_massey} is $(k_{12}+1)$-connected. Since we know the map $(*)$ is $(k_1+k_2+1)$-connected by homotopy excision (Theorem \\ref{thm:homotopy_excision}), it follows that $(**)$ is $k$-connected by Proposition \\ref{prop:connectivity_estimates_for_composition_of_maps}(b) which finishes the proof.\n\\end{proof}\n\n\n\n\\begin{defn}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any subset ${ \\mathcal{B} }\\subset\\powerset(W)$.\n\\begin{itemize}\n\\item A subset ${ \\mathcal{A} }\\subset{ \\mathcal{B} }$ is \\emph{concave} if every element of ${ \\mathcal{B} }$ which is greater than an element of ${ \\mathcal{A} }$ is in ${ \\mathcal{A} }$.\n\\item Define ${ \\mathcal{A} }_W^\\mathrm{min}:=\\{V\\subset W: |W-V|\\leq 1\\}$ and ${ \\mathcal{A} }_W^\\mathrm{max}:=\\powerset(W)$.\n\\item For each concave subset ${ \\mathcal{A} }\\subset\\powerset(W)$, the $W$-cube ${ \\mathcal{X} }^{ \\mathcal{A} }$ is defined objectwise by $({ \\mathcal{X} }^{ \\mathcal{A} })_U := \\lim_{T\\in{ \\mathcal{A} },\\, T\\supset U}{ \\mathcal{X} }_T$.\n\\end{itemize}\n\\end{defn}\n\n\n\\begin{prop}\n\\label{prop:warmup_cartesian_estimates_for_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a fibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }$  (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_{W-V}$ and ${ \\mathcal{X} }_W$) is $k_V$-cartesian,\n\\item[(ii)] $k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen, for every $U\\subsetneqq V\\subset W$, the $(V-U)$-cube $\\partial_U^V{ \\mathcal{X} }$ is $k_{V-U}$-cartesian.\n\\end{prop}\n\n\\begin{proof}\nIt will be useful to note that assumptions (i) and (ii) are equivalent to the following assumptions:\n\\begin{itemize}\n\\item[(1)] for each subset $U\\subsetneqq W$, the $(W-U)$-cube $\\partial_{U}^W{ \\mathcal{X} }$  (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_U$ and ${ \\mathcal{X} }_W$) is $k_{W-U}$-cartesian,\n\\item[(2)] $k_{W-V}\\leq k_{W-U}$ for each $U\\subset V$.\n\\end{itemize}\nWe want to verify that $\\partial_U^V{ \\mathcal{X} }$ is $k_{V-U}$-cartesian for each $U\\subsetneqq V\\subset W$. The argument is by downward induction on $|V|$. The case $|V|=|W|$ is true by assumption. Assume the proposition is true for some nonempty $V\\subset W$ and consider any $U\\subsetneqq V$. Let $v\\in V$ and note that $\\partial_{U-\\{u\\}}^V{ \\mathcal{X} }$ can be written as the composition of cubes\n\\begin{align*}\n\\xymatrix{\n  \\partial_{U-\\{u\\}}^{V-\\{u\\}}{ \\mathcal{X} }\\ar[r] &\n  \\partial_U^V{ \\mathcal{X} }\n}\n\\end{align*}\nWe know by the induction assumption that the composition of cubes is $k_{(V-U)\\cup\\{u\\}}$-cartesian and the right-hand cube is $k_{V-U}$-cartesian. Since $k_{V-U}\\leq k_{(V-U)\\cup\\{u\\}}$ by assumption, it follows from Proposition \\ref{prop:map_of_cubical_diagrams}(c) that the left-hand cube is $k_{V-U}$-cartesian which finishes the proof; note that the sets $(V-\\{u\\})-(U-\\{u\\})$ and $V-U$ are the same.\n\\end{proof}\n\nThe following proposition will be needed in the proof of Proposition \\ref{prop:properties_of_the_X_sup_A_construction} below.\n\n\\begin{prop}\n\\label{prop:lim_over_punctured_cube_of_X_sup_A_construction}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any concave subset ${ \\mathcal{A} }\\subset\\powerset(W)$. Then for each nonempty subset $V\\subset W$, there is a natural isomorphism\n\\begin{align*}\n  \\lim_{\\powerset_0(W-V)}\\partial_V^W{ \\mathcal{X} }^{ \\mathcal{A} }{  \\ \\cong \\ }\n  \\lim_{T\\in{ \\mathcal{A} },\\,T\\supsetneqq V}{ \\mathcal{X} }_T\n\\end{align*}\n\\end{prop}\n\n\\begin{proof}\nThis is because the indexing sets $\\{T\\in{ \\mathcal{A} }:T\\supset U,\\, U\\supsetneqq V\\}$ and $\\{T\\in{ \\mathcal{A} }:T\\supsetneqq V\\}$ are the same.\n\\end{proof}\n\nThe following proposition explains the key properties of the ${ \\mathcal{X} }^{ \\mathcal{A} }$ construction and its relationship to ${ \\mathcal{X} }$; it is through these properties that the ${ \\mathcal{X} }^{ \\mathcal{A} }$ construction is useful and meaningful.\n\n\\begin{prop}\n\\label{prop:properties_of_the_X_sup_A_construction}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any concave subset ${ \\mathcal{A} }_W^\\mathrm{min}\\subset{ \\mathcal{A} }\\subset{ \\mathcal{A} }_W^\\mathrm{max}$.\n\\begin{itemize}\n\\item[(a)] There are natural isomorphisms $\\partial_V^W{ \\mathcal{X} }^{ \\mathcal{A} }{  \\ \\cong \\ }\\partial_V^W{ \\mathcal{X} }$ if $V\\in{ \\mathcal{A} }$,\n\\item[(b)] The $(W-V)$-cube $\\partial_V^W{ \\mathcal{X} }^{ \\mathcal{A} }$ is $\\infty$-cartesian if $V\\notin{ \\mathcal{A} }$ and ${ \\mathcal{X} }$ is a fibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules).\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nConsider part (a). Let $V\\in{ \\mathcal{A} }$. Then we know that the indexing sets $\\{T\\in{ \\mathcal{A} }:T\\supset V\\}$ and $\\{T:W\\supset T\\supset V\\}$ are the same, by ${ \\mathcal{A} }\\subset{ \\mathcal{A} }_W^\\mathrm{max}$ concave. It follows that ${ \\mathcal{X} }_V{  \\ \\cong \\ }\\lim_{W\\supset T\\supset V}{ \\mathcal{X} }_T=\\lim_{T\\in{ \\mathcal{A} },\\,T\\supset V}=({ \\mathcal{X} }^{ \\mathcal{A} })_V$. The same argument shows that ${ \\mathcal{X} }_{V'}{  \\ \\cong \\ }({ \\mathcal{X} }^{ \\mathcal{A} })_{V'}$, for each $V'\\supset V$. Consider part (b). Let $V\\subset W$ and $V\\notin{ \\mathcal{A} }$. Then the indexing sets $\\{T\\in{ \\mathcal{A} }:T\\supset V\\}$ and $\\{T\\in{ \\mathcal{A} }:T\\supsetneqq V\\}$ are the same, and hence the composition\n\\begin{align*}\n  (\\partial_V^W{ \\mathcal{X} }^{ \\mathcal{A} })_\\emptyset=({ \\mathcal{X} }^{ \\mathcal{A} })_V=\\lim_{T\\in{ \\mathcal{A} },\\,T\\supset V}X_T=\\lim_{T\\in{ \\mathcal{A} },\\,T\\supsetneqq V}X_T{  \\ \\cong \\ }\\lim_{\\powerset_0(W-V)}\\partial_V^W{ \\mathcal{X} }^{ \\mathcal{A} }\n\\end{align*}\nis an isomorphism which finishes the proof of part (b).\n\\end{proof}\n\nThe following proposition shows that ${ \\mathcal{X} }^{ \\mathcal{A} }$ inherits several of the homotopical properties of ${ \\mathcal{X} }$.\n\n\n\\begin{prop}\n\\label{prop:cartesian_estimates_for_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a fibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any concave subset ${ \\mathcal{A} }_W^\\mathrm{min}\\subset{ \\mathcal{A} }\\subset{ \\mathcal{A} }_W^\\mathrm{max}$. Assume that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_{W-V}$ and ${ \\mathcal{X} }_W$) is $k_V$-cartesian,\n\\item[(ii)] $k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen, for each $U\\subsetneqq V\\subset W$, the $(V-U)$-cube $\\partial_U^V{ \\mathcal{X} }^{ \\mathcal{A} }$ is $k_{V-U}$-cartesian.\n\\end{prop}\n\n\\begin{proof}\nBy the downward induction argument in the proof of Proposition \\ref{prop:warmup_cartesian_estimates_for_induction_argument}, it suffices to verify that the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }^{ \\mathcal{A} }$ is $k_V$-cartesian, for each nonempty subset $V\\subset W$, and this follows immediately from Proposition \\ref{prop:properties_of_the_X_sup_A_construction}.\n\\end{proof}\n\nThe following proposition appears in Goodwillie \\cite[2.8]{Goodwillie_calc2} in the context of spaces, and exactly the same argument gives a proof in the context of structured ring spectra; this is an exercise left to the reader.\n\n\\begin{prop}\n\\label{prop:fibration_cube_properties}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Let ${ \\mathcal{X} }$ be a fibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) and consider any concave subset ${ \\mathcal{A} }\\subset\\powerset(W)$.\n\\begin{itemize}\n\\item[(a)] For each inclusion ${ \\mathcal{A} }'\\subset{ \\mathcal{A} }$ of concave subsets of $\\powerset(W)$, the induced map $\\lim_{{ \\mathcal{A} }}{ \\mathcal{X} }{ \\rightarrow }\\lim_{{ \\mathcal{A} }'}{ \\mathcal{X} }$ is a fibration in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$),\n\\item[(b)] For any concave subsets ${ \\mathcal{A} },{ \\mathcal{B} }$ of $\\powerset(W)$, the diagram\n\\begin{align*}\n\\xymatrix{\n  \\lim_{{ \\mathcal{A} }\\cup{ \\mathcal{B} }}{ \\mathcal{X} }\\ar[r]\\ar[d] & \\lim_{{ \\mathcal{B} }}{ \\mathcal{X} }\\ar[d]\\\\\n  \\lim_{{ \\mathcal{A} }}{ \\mathcal{X} }\\ar[r] & \\lim_{{ \\mathcal{A} }\\cap{ \\mathcal{B} }}{ \\mathcal{X} }\n}\n\\end{align*}\nis a pullback diagram of fibrations in ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$).\n\\item[(c)] If $A\\in{ \\mathcal{A} }$ is minimal and the fibration $${ \\mathcal{X} }_A{  \\ \\cong \\ }\\lim_{\\powerset(W-A)}\\partial_A^W{ \\mathcal{X} }{ \\rightarrow }\\lim_{\\powerset_0(W-A)}\\partial_A^W{ \\mathcal{X} }$$\nis $k_{W-A}$-connected, then the fibration $$\\lim_{ \\mathcal{A} }{ \\mathcal{X} }{ \\rightarrow }\\lim_{{ \\mathcal{A} }-\\{A\\}}{ \\mathcal{X} }$$ is $k_{W-A}$-connected.\n\\end{itemize}\n\\end{prop}\n\nThe purpose of the following induction argument is to leverage the higher dual homotopy excision result (Theorem \\ref{thm:higher_dual_homotopy_excision}) for structured ring spectra into a conceptual proof of the second main theorem in this paper (Theorem \\ref{thm:higher_dual_blakers_massey})---the higher dual Blakers-Massey theorem for structured ring spectra. Proposition \\ref{prop:dual_cubical_induction_argument} is motivated by Goodwillie \\cite[proof of (2.6)]{Goodwillie_calc2}; it is essentially Goodwillie's dual cubical induction argument, appropriately modified to our situation. The reader who is interested in an alternate proof of Theorem \\ref{thm:higher_dual_blakers_massey}, which is more efficient, but requires a little extra calculation at the end, may skip directly to Remark \\ref{rem:alt_proof_of_higher_dual_blakers_massey}.\n\n\\begin{prop}[Dual cubical induction argument]\n\\label{prop:dual_cubical_induction_argument}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathcal{R} }$-modules and $W$ a nonempty finite set. Suppose ${ \\mathcal{A} }\\subsetneqq{ \\mathcal{A} }_W^\\mathrm{max}$ is concave, $A\\in{ \\mathcal{A} }$ minimal, $|W-A|\\geq 2$, and ${ \\mathcal{A} }':={ \\mathcal{A} }-\\{A\\}$. Let ${ \\mathcal{X} }$ be a fibration $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{R} },{ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_{W-V}$ and ${ \\mathcal{X} }_W$) is $k_V$-cartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nIf $X^{{ \\mathcal{A} }'}$ is $j$-cocartesian, then ${ \\mathcal{X} }^{ \\mathcal{A} }$ is $j$-cocartesian, where $j$ is the minimum of $|W|+\\sum_{V\\in\\lambda}k_V$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$.\n\\end{prop}\n\n\n\\begin{rem}\nIn the case that ${ \\mathcal{X} }$ is a $3$-cube, the following diagram illustrates one of the cubical decompositions covered by Proposition \\ref{prop:dual_cubical_induction_argument}. It corresponds to the sequence of minimal elements: $\\{3\\}$, $\\{2\\}$, $\\{1\\}$, $\\emptyset$.\n\n\\begin{align*}\n\\xymatrix@!0{\n{ \\mathcal{X} }_{\\{1,2,3\\}} &&\n{ \\mathcal{X} }_{\\{2,3\\}}\\ar[ll]\\ar@{=}[rr] &&\n{ \\mathcal{X} }_{\\{2,3\\}}\\ar@{=}[rr] &&\n{ \\mathcal{X} }_{\\{2,3\\}}\\\\\n&{ \\mathcal{X} }_{\\{1,3\\}}\\ar[ul] &&\n\\cdot\\ar[ll]\\ar[ul] &&\n{ \\mathcal{X} }_{\\{3\\}}\\ar@{=}[rr]\\ar[ll]\\ar[ul] &&\n{ \\mathcal{X} }_{\\{3\\}}\\ar[ul]\\\\\n{ \\mathcal{X} }_{\\{1,2\\}}\\ar@{=}[dd]\\ar[uu] &&\n\\cdot\\ar@{=}'[r][rr]\\ar'[l][ll]\\ar'[u][uu] &&\n\\cdot\\ar'[u][uu] &&\n{ \\mathcal{X} }_{\\{2\\}}\\ar@{=}'[d][dd]\\ar'[l][ll]\\ar[uu]|(0.47)\\hole|(0.5)\\hole\\\\\n&\\cdot\\ar[dd]\\ar[ul]\\ar[uu] &&\n\\cdot\\ar[ll]\\ar[ul]\\ar[uu] &&\n\\cdot\\ar[ll]\\ar[ul]\\ar[uu] &&\n\\cdot\\ar[ll]\\ar[ul]\\ar[uu]\\\\\n{ \\mathcal{X} }_{\\{1,2\\}}\\ar@{=}[dd]&&&&&&\n{ \\mathcal{X} }_{\\{2\\}}\\ar@{=}'[d][dd]\\ar'[lllll][llllll]\\\\\n&{ \\mathcal{X} }_{\\{1\\}}\\ar@{=}[dd]\\ar[ul]\\ar[uu]&&&&&&\n\\cdot\\ar[llllll]\\ar[ul]\\ar[uu]\\\\\n{ \\mathcal{X} }_{\\{1,2\\}} &&&&&&\n{ \\mathcal{X} }_{\\{2\\}}\\ar'[lllll][llllll]\\\\\n&{ \\mathcal{X} }_{\\{1\\}}\\ar[ul]&&&&&&{ \\mathcal{X} }_\\emptyset\\ar[llllll]\\ar[ul]\\ar[uu]\n}\n\\end{align*}\n\\end{rem}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:dual_cubical_induction_argument}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The argument is by induction on $|W|$. The case $|W|=2$ is verified by the proof of Theorem \\ref{thm:dual_blakers_massey}. Let $n\\geq 3$ and assume the proposition is true for each $2\\leq|W|<n$. Let's verify it remains true for $|W|=n$.\n\nWe know by assumption that ${ \\mathcal{X} }^{{ \\mathcal{A} }'}$ is $j$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$. We want to verify that ${ \\mathcal{X} }^{ \\mathcal{A} }$ is $j$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ (it might be helpful at this point to look ahead to \\eqref{eq:dual_final_decomposition_for_induction_argument} for the decomposition of ${ \\mathcal{X} }^{ \\mathcal{A} }$ that we will use to finish the proof). Consider the induced map of $W$-cubes ${ \\mathcal{X} }^{{ \\mathcal{A} }}{ \\rightarrow }{ \\mathcal{X} }^{{ \\mathcal{A} }'}$. Note that if $U\\not\\subset A$, then $\\{T\\in{ \\mathcal{A} }:T\\supset U\\}=\\{T\\in{ \\mathcal{A} }':T\\supset U\\}$; in particular, each of the maps\n\\begin{align}\n\\label{eq:dual_subdiagram_of_identity_maps}\n\\xymatrix{\n  ({ \\mathcal{X} }^{{ \\mathcal{A} }})_U\\ar[r]^-{{ \\mathrm{id} }} & ({ \\mathcal{X} }^{{ \\mathcal{A} }'})_U,\n  & U\\not\\subset A\n}\n\\end{align}\nin ${ \\mathcal{X} }^{{ \\mathcal{A} }}{ \\rightarrow }{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ is the identity map. Note that if $U\\subset V\\subset A$, then the diagram\n\\begin{align}\n\\label{eq:pullback_diagram_in_induction_step}\n\\xymatrix{\n  ({ \\mathcal{X} }^{{ \\mathcal{A} }})_U\\ar[r]\\ar[d] & ({ \\mathcal{X} }^{{ \\mathcal{A} }'})_U\\ar[d]\\\\\n  ({ \\mathcal{X} }^{{ \\mathcal{A} }})_V\\ar[r] & ({ \\mathcal{X} }^{{ \\mathcal{A} }'})_V, &\n  U\\subset V\\subset A\n}\n\\end{align}\nis a pullback diagram by Proposition \\ref{prop:fibration_cube_properties}(b); in particular, focusing on this case is the same as focusing on the subdiagram $\\partial_\\emptyset^A{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ of ${ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }{ \\mathcal{X} }^{{ \\mathcal{A} }'}$.\n\nLet's first verify that $\\partial_\\emptyset^A{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ is $(j+|A|+1-|W|)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$. Let ${ \\mathcal{Y} }$ denote $\\partial_\\emptyset^A{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ regarded as an $(A\\cup\\{*\\})$-cube as follows:\n\\begin{align*}\n  { \\mathcal{Y} }_V&:=(\\partial_\\emptyset^A{ \\mathcal{X} }^{ \\mathcal{A} })_V=({ \\mathcal{X} }^{ \\mathcal{A} })_V,\\quad\\ V\\subset A,\\\\\n  { \\mathcal{Y} }_V&:=(\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }'})_U=({ \\mathcal{X} }^{{ \\mathcal{A} }'})_U,\\quad U\\subset A,\\quad V=U\\cup\\{*\\}.\n\\end{align*}\nSince $|A\\cup\\{*\\}|<|W|$, our induction assumption can be applied to ${ \\mathcal{Y} }$, provided that the appropriate $k'_V$-cartesian estimates are satisfied. We claim that with the following definitions\n\\begin{align*}\n  k'_V&:=k_V,\\quad\\quad \\emptyset\\neq V\\subset A,\\\\\n  k'_V&:=k_{W-A},\\quad\\quad V=\\{*\\}\\\\\n  k'_V&:=\\infty,\\quad\\quad \\emptyset\\neq U\\subset A,\\quad V=U\\cup\\{*\\},\n\\end{align*}\nthe cube $\\partial_{(A\\cup\\{*\\})-V}^{A\\cup\\{*\\}}{ \\mathcal{Y} }$ (formed by all maps in ${ \\mathcal{Y} }$ between ${ \\mathcal{Y} }_{(A\\cup\\{*\\})-V}$ and ${ \\mathcal{Y} }_{A\\cup\\{*\\}}$) is $k'_V$-cartesian, for each nonempty subset $V\\subset A\\cup\\{*\\}$. Let's verify this now: If $V\\subset A$ is nonempty, then $\\partial_{(A\\cup\\{*\\})-V}^{A\\cup\\{*\\}}{ \\mathcal{Y} }$ is the cube $\\partial_{A-V}^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ which is $k_V$-cartesian by Proposition \\ref{prop:cartesian_estimates_for_induction_argument}. If $V=\\{*\\}$, then $\\partial_{(A\\cup\\{*\\})-V}^{A\\cup\\{*\\}}{ \\mathcal{Y} }$ is the map $({ \\mathcal{X} }^{ \\mathcal{A} })_A{ \\rightarrow }({ \\mathcal{X} }^{{ \\mathcal{A} }'})_A$ which is $k_{W-A}$-connected by Proposition \\ref{prop:fibration_cube_properties}(c). Finally, if $V=U\\cup\\{*\\}$ for any nonempty $U\\subset A$, then $\\partial_{(A\\cup\\{*\\})-V}^{A\\cup\\{*\\}}{ \\mathcal{Y} }$ is the cube $\\partial_{A-U}^A{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_{A-U}^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ which is $\\infty$-cartesian by\n\\eqref{eq:pullback_diagram_in_induction_step} and Proposition \\ref{prop:map_of_cubical_diagrams}(d). Hence by our induction hypothesis applied to ${ \\mathcal{Y} }$: since the sum $\\sum_{V\\in\\lambda'}k'_V$ for a partition $\\lambda'$ of $A\\cup\\{*\\}$ by nonempty sets is always either $\\infty$, or the sum $\\sum_{U\\in\\lambda}k_U$ for a partition $\\lambda$ of $W$ by nonempty sets in which some $U$ is $W-A$, we know that ${ \\mathcal{Y} }$ is $j'$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$, where $j'$ is the minimum of $|A\\cup\\{*\\}|+\\sum_{U\\in\\lambda}k_U$ over all partitions $\\lambda$ of $W$ by nonempty sets in which some $U$ is $W-A$. In particular, this implies that ${ \\mathcal{Y} }$, and hence $\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }}{ \\rightarrow }\\partial_\\emptyset^A{ \\mathcal{X} }^{{ \\mathcal{A} }'}$, is $(j+|A|+1-|W|)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$.\n\n\nWe next want to verify that\n\\begin{align}\n\\label{eq:induction_argument_cocartesian_estimate_for_certain_subdiagrams}\n  \\partial_\\emptyset^{A'}{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_\\emptyset^{A'}{ \\mathcal{X} }^{{ \\mathcal{A} }'}\n  \\quad \\text{is $(j+|A'|+1-|W|)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$}\n\\end{align}\nfor each $A'\\supset A$. We know that \\eqref{eq:induction_argument_cocartesian_estimate_for_certain_subdiagrams} is true for $A'=A$ by above. We will argue by upward induction on $|A'|$. Suppose that \\eqref{eq:induction_argument_cocartesian_estimate_for_certain_subdiagrams} is true for some $A'\\supset A$. Let $a\\in W-A'$ and note that the cube $\\partial_\\emptyset^{A'\\cup\\{a\\}}{ \\mathcal{X} }^{{ \\mathcal{A} }}{ \\rightarrow }\\partial_\\emptyset^{A'\\cup\\{a\\}}{ \\mathcal{X} }^{{ \\mathcal{A} }'}$ can be written as the diagram of cubes\n\\begin{align}\n\\label{eq:upward_induction_cube_factorization}\n\\xymatrix{\n  \\partial_\\emptyset^{A'}{ \\mathcal{X} }^{ \\mathcal{A} }\\ar[r]\\ar[d] &\n  \\partial_\\emptyset^{A'}{ \\mathcal{X} }^{{ \\mathcal{A} }'}\\ar[d]\\\\\n  \\partial_{\\{a\\}}^{A'\\cup\\{a\\}}{ \\mathcal{X} }^{ \\mathcal{A} }\\ar[r]^-{{ \\mathrm{id} }} &\n  \\partial_{\\{a\\}}^{A'\\cup\\{a\\}}{ \\mathcal{X} }^{{ \\mathcal{A} }'}\n}\n\\end{align}\nWe know the bottom arrow is the identity by \\eqref{eq:dual_subdiagram_of_identity_maps}, and the top arrow is $(j+|A'|+1-|W|)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ by assumption. It follows from  Proposition \\ref{prop:map_of_cubical_diagrams}(b) that  \\eqref{eq:upward_induction_cube_factorization} is $(j+|A'|+2-|W|)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$, which finishes the argument that \\eqref{eq:induction_argument_cocartesian_estimate_for_certain_subdiagrams} is true for each $A'\\supset A$.\n\n\nTo finish off the proof, let $a\\in W-A$ and note by \\eqref{eq:dual_subdiagram_of_identity_maps} that ${ \\mathcal{X} }^{ \\mathcal{A} }$ can be written as the composition of cubes\n\\begin{align}\n\\label{eq:dual_final_decomposition_for_induction_argument}\n  \\partial_\\emptyset^{W-\\{a\\}}{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\n  \\partial_\\emptyset^{W-\\{a\\}}{ \\mathcal{X} }^{{ \\mathcal{A} }'}{ \\rightarrow }\n  \\partial_{\\{a\\}}^W{ \\mathcal{X} }^{{ \\mathcal{A} }'}.\n\\end{align}\nThe left-hand arrow is $j$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ by \\eqref{eq:induction_argument_cocartesian_estimate_for_certain_subdiagrams} and the right-hand arrow is ${ \\mathcal{X} }^{{ \\mathcal{A} }'}$ which is $j$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ by assumption, hence by Proposition \\ref{prop:composed_map_of_cubical_diagrams}(a) it follows that ${ \\mathcal{X} }^{ \\mathcal{A} }$ is $j$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ which finishes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:higher_dual_blakers_massey}]\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. It is enough to consider the special case where ${ \\mathcal{X} }$ is a fibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Let ${ \\mathcal{A} }:={ \\mathcal{A} }^{\\mathrm{max}}_W$, ${ \\mathcal{A} }':={ \\mathcal{A} }^{\\mathrm{max}}_W-\\emptyset$, and note by \\eqref{eq:dual_subdiagram_of_identity_maps} that ${ \\mathcal{X} }^{ \\mathcal{A} }$, which is equal to ${ \\mathcal{X} }$, can be written as the composition of cubes\n\\begin{align}\n\\label{eq:dual_final_decomposition_dual_higher_homotopy_excision}\n  \\partial_\\emptyset^{W-\\{a\\}}{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\n  \\partial_\\emptyset^{W-\\{a\\}}{ \\mathcal{X} }^{{ \\mathcal{A} }'}{ \\rightarrow }\n  \\partial_{\\{a\\}}^W{ \\mathcal{X} }^{{ \\mathcal{A} }'}.\n\\end{align}\nWe know by Proposition \\ref{prop:dual_cubical_induction_argument}, together with Theorem \\ref{thm:higher_dual_homotopy_excision} (to start the induction using ${ \\mathcal{A} }'={ \\mathcal{A} }_W^{\\mathrm{min}}$) that the right-hand arrow, which is ${ \\mathcal{X} }^{{ \\mathcal{A} }'}$, is $j$-cocartesian in ${ \\Alg_{ \\mathcal{O} } }$, where $j$ is the minimum of $|W|+\\sum_{V\\in\\lambda}k_V$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$. We claim that the left-hand arrow is $(k_W+|W|-1)$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$; this follows from upward induction by arguing exactly as in \\eqref{eq:upward_induction_cube_factorization}, but by starting with the observation (see the $k'_V$ estimates in the proof of Proposition \\ref{prop:dual_cubical_induction_argument}) that the map $\\partial_\\emptyset^\\emptyset{ \\mathcal{X} }^{ \\mathcal{A} }{ \\rightarrow }\\partial_\\emptyset^\\emptyset{ \\mathcal{X} }^{{ \\mathcal{A} }'}$, which is the map $({ \\mathcal{X} }^{ \\mathcal{A} })_\\emptyset{ \\rightarrow }({ \\mathcal{X} }^{{ \\mathcal{A} }'})_\\emptyset$, is $k_W$-connected. Hence it follows from Proposition \\ref{prop:composed_map_of_cubical_diagrams} that the composition,\nwhich is ${ \\mathcal{X} }$, is $k$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$ which finishes the proof.\n\\end{proof}\n\n\\begin{rem}\n\\label{rem:alt_proof_of_higher_dual_blakers_massey}\nAn alternate proof of Theorem \\ref{thm:higher_dual_blakers_massey} can be obtained from the higher Blakers-Massey theorem (Theorem \\ref{thm:higher_blakers_massey}) by arguing as in the proof of Theorem \\ref{thm:higher_dual_homotopy_excision}. Since the dual cubical induction argument is conceptually very useful and will be needed elsewhere, we include both approaches for the interested reader, with only a few details of the alternate proof below left to the reader.\n\\end{rem}\n\n\n\\begin{proof}\nIt suffices to consider the case of left ${ \\mathcal{O} }$-modules. The argument is by induction on $|W|$. The case $|W|$=1 is trivial and the case $|W|=2$ is verified by Theorem \\ref{thm:dual_blakers_massey}. Let $n\\geq 3$ and assume the theorem is true for each $2\\leq |W|< n$. Let's verify it remains true for $|W|=n$.\n\nIt suffices to consider the special case where ${ \\mathcal{X} }$ is a cofibration $W$-cube in ${ \\Lt_{ \\mathcal{O} } }$. Consider the induced maps\n\\begin{align}\n\\label{eq:alt_composition_of_maps_for_determining_cocartesian_estimate}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(*)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset_1(W)}{ \\mathcal{X} }\\ar[r]^-{(**)} &\n  \\colim\\nolimits^{{ \\Lt_{ \\mathcal{O} } }}_{\\powerset(W)}{  \\ \\cong \\ }{ \\mathcal{X} }_W\n}\n\\end{align}\nWe want to show that $(**)$ is $k$-connected. Consider the $W$-cube ${ \\mathcal{X} }':={ \\mathcal{X} }_{{ \\mathcal{A} }^W_\\mathrm{max}-\\{W\\}}$ in ${ \\Lt_{ \\mathcal{O} } }$ and note that $(*)$ is isomorphic to the map\n\\begin{align*}\n\\xymatrix{\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset_1(W)}{ \\mathcal{X} }'\\ar[r]^-{(*)'} &\n  \\colim\\nolimits^{{ \\mathsf{SymSeq} }}_{\\powerset(W)}{ \\mathcal{X} }'{  \\ \\cong \\ }{ \\mathcal{X} }'_W\n}\n\\end{align*}\nWe know that ${ \\mathcal{X} }'$ is $\\infty$-cocartesian in ${ \\Lt_{ \\mathcal{O} } }$. By Proposition \\ref{prop:warmup_cartesian_estimates_for_induction_argument} we know that the $V$-cube $\\partial_U^{U\\cup V}{ \\mathcal{X} }$ is $k_V$-cartesian for each disjoint $U$ and $V$. Hence by the induction hypothesis, for each $V\\subsetneqq W$ the cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ is $k'_V$-cocartesian, where $k'_V$ is the minimum of $k_V+|V|-1$ and $|V|+\\sum_{U\\in\\lambda'}k_U$ over all partitions $\\lambda'$ of $V$ by nonempty sets not equal to $V$. In particular, the $W$-cube ${ \\mathcal{X} }'$ satisfies the conditions of the higher Blakers-Massey theorem (Theorem \\ref{thm:higher_blakers_massey}) with $k'_W=\\infty$ and the $k'_V$ above, and hence it follows that ${ \\mathcal{X} }'$ is $k'$-cartesian, where $k'$ is the minimum of $-|W|+\\sum_{V\\in\\lambda}(k'_V+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$.\n\nHence by Proposition \\ref{prop:comparing_cocartesian_and_cartesian_estimates_in_ModR} we know that ${ \\mathcal{X} }'$ is $(k'+|W|-1)$-cocartesian in ${ \\mathsf{SymSeq} }$, and therefore $(*)$ is $(k'+|W|-1)$-connected. Since the composition in \\eqref{eq:alt_composition_of_maps_for_determining_cocartesian_estimate} is $(k_W+|W|-1)$-connected, it follows from Proposition \\ref{prop:connectivity_estimates_for_composition_of_maps}(b) that $(**)$ is $k$-connected, where $k$ is the minimum of $k_W+|W|-1$ and $\\sum_{V\\in\\lambda}(k'_V+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$; it is an exercise left to the reader to verify that this description of $k$ agrees with Theorem \\ref{thm:higher_dual_blakers_massey}.\n\\end{proof}\n\n\n\n\\section{Operads in chain complexes over a commutative ring}\n\\label{sec:chain_complexes_over_a_commutative_ring}\n\nThe purpose of this section is to observe that the main results of this paper remain true in the context of unbounded chain complexes over a commutative ring, provided essentially that the desired model category structures exist on algebras (resp. left modules) over the operad ${ \\mathcal{O} }$. Since the constructions and proofs of the theorems are essentially identical to the arguments above in the context of ${ \\mathcal{R} }$-modules, modulo the obvious changes, the arguments are left to the reader; see \\cite[5.17]{Harper_Hess} and Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes} below.\n\n\\begin{assumption}\n\\label{assumption:commutative_ring_notation}\nFrom now on in this section, we assume that ${ {\\mathcal{K}} }$ is any commutative ring.\\end{assumption}\n\nDenote by $({ \\mathsf{Ch} }_{ {\\mathcal{K}} },{ \\otimes },{ {\\mathcal{K}} })$ the closed symmetric monoidal category of unbounded chain complexes over ${ {\\mathcal{K}} }$ \\cite{Hovey, MacLane_homology}.\n\n\\begin{homotopical_assumption}\n\\label{HomotopicalAssumption_chain_complexes}\nIf ${ \\mathcal{O} }$ is an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$, assume that the following model structure exists on ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$): the model structure on ${ \\Alg_{ \\mathcal{O} } }$ (resp. ${ \\Lt_{ \\mathcal{O} } }$) has weak equivalences the homology isomorphisms (resp. objectwise homology isomorphisms) and fibrations the maps that are dimensionwise surjections (resp. objectwise dimensionwise surjections). Furthermore, assume that ${ \\mathcal{O} }[\\mathbf{r}]$ is cofibrant in $({ \\mathsf{Ch} }_{ {\\mathcal{K}} })^{\\Sigma_r^{ \\mathrm{op} }}$ with respect to the projective model structure for each $r\\geq 0$.\n\\end{homotopical_assumption}\n\nIf ${ {\\mathcal{K}} }$ is any field of characteristic zero, then Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes} is satisfied by every operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ (see, for instance, \\cite{Harper_Modules}).\n\nThe following is a commutative rings version of Theorem \\ref{thm:homotopy_excision}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a homotopy pushout square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow } { \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i=1,2$), then\n\\begin{itemize}\n\\item[(a)] ${ \\mathcal{X} }$ is $l$-cocartesian in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ (resp. ${ \\mathsf{SymSeq} }$) with $l=k_1+k_2 +1$,\n\\item[(b)] ${ \\mathcal{X} }$ is $k$-cartesian with $k=k_1+k_2$.\n\\end{itemize}\n\\end{thm}\n\nThe following is a commutative rings version of Theorem \\ref{thm:blakers_massey}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a commutative square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$, and  $k_{12}\\in{ \\mathbb{Z} }$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow } { \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i=1,2$) and ${ \\mathcal{X} }$ is $k_{12}$-cocartesian, then ${ \\mathcal{X} }$ is $k$-cartesian, where $k$ is the minimum of $k_{12}-1$ and $k_{1}+k_{2}$.\n\\end{thm}\n\nThe following is a commutative rings version of Theorem \\ref{thm:higher_homotopy_excision}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ and $W$ a nonempty finite set. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a strongly $\\infty$-cocartesian $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Let $k_i\\geq -1$ for each $i\\in W$. If each ${ \\mathcal{X} }_\\emptyset{ \\rightarrow }{ \\mathcal{X} }_{\\{i\\}}$ is $k_i$-connected ($i\\in W$),  then\n\\begin{itemize}\n\\item[(a)] ${ \\mathcal{X} }$ is $l$-cocartesian in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ (resp. ${ \\mathsf{SymSeq} }$) with $l=|W|-1+\\sum_{i\\in W}k_i$,\n\\item[(b)] ${ \\mathcal{X} }$ is $k$-cartesian with $k=\\sum_{i\\in W}k_i$.\n\\end{itemize}\n\\end{thm}\n\nThe following is a commutative rings version of the first main theorem of this paper (Theorem \\ref{thm:higher_blakers_massey}).\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ and $W$ a nonempty finite set. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_\\emptyset^V{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_\\emptyset$ and ${ \\mathcal{X} }_V$) is $k_V$-cocartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen ${ \\mathcal{X} }$ is $k$-cartesian, where $k$ is the minimum of $-|W|+\\sum_{V\\in\\lambda}(k_V+1)$ over all partitions $\\lambda$ of $W$ by nonempty sets.\n\\end{thm}\n\nThe following is a commutative rings version of Theorem \\ref{thm:dual_homotopy_excision}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a homotopy pullback square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2\\geq -1$. If ${ \\mathcal{X} }_{\\{2\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_1$-connected and ${ \\mathcal{X} }_{\\{1\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_2$-connected, then ${ \\mathcal{X} }$ is $k$-cocartesian with $k=k_{1}+k_{2}+2$.\n\\end{thm}\n\nThe following is a commutative rings version of Theorem \\ref{thm:dual_blakers_massey}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a commutative square of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules) of the form\n\\begin{align*}\n\\xymatrix{\n  { \\mathcal{X} }_\\emptyset\\ar[r]\\ar[d] & { \\mathcal{X} }_{\\{1\\}}\\ar[d]\\\\\n  { \\mathcal{X} }_{\\{2\\}}\\ar[r] & { \\mathcal{X} }_{\\{1,2\\}}\n}\n\\end{align*}\nAssume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Consider any $k_1,k_2,k_{12}\\geq -1$ with $k_1\\leq k_{12}$ and $k_2\\leq k_{12}$. If ${ \\mathcal{X} }_{\\{2\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_1$-connected, ${ \\mathcal{X} }_{\\{1\\}}{ \\rightarrow } { \\mathcal{X} }_{\\{1,2\\}}$ is $k_2$-connected, and ${ \\mathcal{X} }$ is $k_{12}$-cartesian, then ${ \\mathcal{X} }$ is $k$-cocartesian, where $k$ is the minimum of $k_{12}+1$ and $k_{1}+k_{2}+2$.\n\\end{thm}\n\nThe following is a commutative rings version of Theorem \\ref{thm:higher_dual_homotopy_excision}.\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ and $W$ a finite set with $|W|\\geq 2$. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a strongly $\\infty$-cartesian $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected. Let $k_i\\geq -1$ for each $i\\in W$. If each ${ \\mathcal{X} }_{W-\\{i\\}}{ \\rightarrow }{ \\mathcal{X} }_W$ is $k_i$-connected ($i\\in W$),  then ${ \\mathcal{X} }$ is $k$-cocartesian with $k=|W|+\\sum_{i\\in W}k_i$.\n\\end{thm}\n\nThe following is a commutative rings version of the second main theorem of this paper (Theorem \\ref{thm:higher_dual_blakers_massey}).\n\n\\begin{thm}\nLet ${ \\mathcal{O} }$ be an operad in ${ \\mathsf{Ch} }_{ {\\mathcal{K}} }$ and $W$ a nonempty finite set. Assume that ${ \\mathcal{O} }$ satisfies Homotopical Assumption \\ref{HomotopicalAssumption_chain_complexes}. Let ${ \\mathcal{X} }$ be a $W$-cube of ${ \\mathcal{O} }$-algebras (resp. left ${ \\mathcal{O} }$-modules). Assume that ${ \\mathcal{O} },{ \\mathcal{X} }_\\emptyset$ are $(-1)$-connected, and suppose that\n\\begin{itemize}\n\\item[(i)] for each nonempty subset $V\\subset W$, the $V$-cube $\\partial_{W-V}^W{ \\mathcal{X} }$ (formed by all maps in ${ \\mathcal{X} }$ between ${ \\mathcal{X} }_{W-V}$ and ${ \\mathcal{X} }_W$) is $k_V$-cartesian,\n\\item[(ii)] $-1\\leq k_{U}\\leq k_V$ for each $U\\subset V$.\n\\end{itemize}\nThen ${ \\mathcal{X} }$ is $k$-cocartesian, where $k$ is the minimum of $k_W+|W|-1$ and $|W|+\\sum_{V\\in\\lambda}k_V$ over all partitions $\\lambda$ of $W$ by nonempty sets not equal to $W$.\n\\end{thm}\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe optical properties of thin metal films secure their wide use in different fields \u2013 from mirrors to supersensitive detectors \\cite{Maier} capable of detecting refraction coefficient changes as small as ${10^{ - 7}}$ \\cite{Mikaelian}. These properties are considered to be stable to external actions and changes of the absorption spectrum are detected only in going from macro samples to nano samples. The latter effect is caused by collective oscillations of electrons (so-called plasmon oscillations) in nanoparticles  with  sizes smaller than  the electron mean free  path  in massive samples. It was revealed that the static polarizability of metal clusters containing  several tens of atoms, which determines the frequency of collective oscillations, differs sufficiently from  macroscopic polarizability \\cite{Knight}. Besides, calculations show that in $Ag$ clusters $1 - 2 nm$ in diameter an extra charge of $1$ to $3$ electron charges should shift the plasmon absorption line by about $5 nm$, while for even $10 - nm$ clusters this spectral shift is negligibly small \\cite{Quinten}. It sounds reasonable because the spectral shift is caused by a change of the charge density, which is noticeable in small clusters and negligibly small in large clusters. It is however known that plasmon frequencies are strongly dependent on the form and size of nanoparticles. The examples show that noticeable changes of optical properties of metal nano-size samples are hard to cause without changing their geometry.\n\n\tIn the present communication we give the experimental data that clearly show a considerable change of the effective relative permittivity of $Ag$ layers in the MDM nanostructure.\n\n\\section{ Experiment}\nHigh-vacuum electron-beam deposition was used to make plasmon 4-nanolayer MDM samples ($A{l_2}{O_3}\/Ag\/A{l_2}{O_3}\/Ag$). Figure~\\ref{fig:fig01} illustrates a 4-layer nanostructure whose optical properties are studied in the experiment. The layers of the structure are $12,\\,49,\\,177\\;{\\textrm{and}}\\;36\\,nm$ thick. A quartz substrate was kept at a relatively low temperature (about ${150^ \\circ }C$) in the process of layer deposition which allows obtaining smaller crystal grains in the layers  and was necessary to ensure the surface smothness of the film. The outer $A{l_2}{O_3}$ layer ($d = 12nm$) serves as antioxidation coating. No adhesive sublayers were used. A laser with wavelength $\\lambda  = 632.8nm$ and $p$-polarization was used to generate SPR. It should be noted that this structure can be considered as a metal-dielectric plasmonic waveguide with two SPR angles ${\\theta _1} = {43^0}$ and ${\\theta _2} = {56.3^0}$.\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig01}\n\\caption{\\label{fig:fig01}MDM plasmonic structure}\n\\end{figure}\n\n\tIn deposition we used masks to form a thin-film capacitor structure. The diameter of the outer silver electrode was 3 mm. A constant electric field was applied to the capacitor. The structures could stand up to $30V$ without electric breakdown. This corresponds to the field strength of $1.7 \\cdot {10^6}V\/cm$ in the dielectric layer, given the corundum permittivity $\\varepsilon  = 10$. However, when the air humidity was high, a lower breakdown voltage and greater leakage currents were observed.\n\n\tIn the experiment we  measured the SPR spectrum (the angular reflection spectrum of the MDM structure) using the Kretschmann configuration \\cite{Kretschmann}. We used an automatic semicylinder $\\theta - 2\\theta $ goniometer with additional plane-wavefront-forming optics to measure the angular spectrum of total reflection.\n\n\tThe experimental results are given in Figure~\\ref{fig:fig02} which shows how the reflection spectrum changes with the electric field changing from $0$ to $30$ volts. The curves allow us to assert that the reflection coefficient changes noticeably when the voltage increases above $4V$, at $U > 16V$ it reaches  saturation. The resonance peak width grows considerably with $U$, yet the shift from the resonance angles ${\\theta _1} = {43^0}$ and ${\\theta _2} = {56.3^0}$ does not occur. The reflection coefficient changes significantly: when $U$ approaches $20 V$, it decreases fourfold near the first resonance angle and increases fivefold near the second SPR angle. In Figure~\\ref{fig:fig02} we can select three groups of curves: the first group corresponds to $U \\in [0,\\;6V]$, the second to $U \\in [8 V,\\;16 V]$, and the third to $U \\in [18 V,\\;32 V]$. The grouping will become clear in the next paragraph where we examine the voltage dependence of optical parameters. Besides, there are two points where the reflection coefficient is independent of the applied voltage: $R \\approx 11.6\\% $ at ${\\theta} = {53.95^0}$ and  $R \\approx 11.8\\% $ at  ${\\theta} = {57.39^0}$ for any $U$. It is hard to explain the existence of these points, however they can be used for testing the results: the theoretical curves should intersect at the points.\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig02}\n\\caption{\\label{fig:fig02}The reflection spectrum versus the voltage $U$ across the MDM structure. The upper curves corresponds to $U$changing from $0 V$ to $28 V$ in $4 - V$ steps. The lowest curve corresponds to near-breakdown voltage $U = 30 V$.}\n\\end{figure}\n\n\tThe curves of reflection coefficient $R$  versus angle of incidence $\\theta $ allowed us to compute the optical parameters of all layers of the MDM structure (optical thickness, refraction coefficients $n$ and absorption coefficients $k$) as function of voltage $U \\in [0,\\;30 V]$. The effective optical parameters of the layers were determined by the best agreement between experimental and theoretical curves.\n\n\tThe conventional methods and software were used in computation of theoretical angular spectrum (the method is described in \\cite{Palagushkin1,Palagushkin2} in detail). The optical constants of silver and corundum from the SOPRA database and layer thicknesses measured during the deposition process were taken as initial values for the theoretical model in the absence of the external electric field. Then we computed the best agreement between the theoretical and experimental curves $R = R(\\theta )$ to correct the layer thicknesses to use them for further computations. The results of simulation are shown in Figure~\\ref{fig:fig03} and in Table~\\ref{table:table1} for $U = 0$.\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig03}\n\\caption{\\label{fig:fig03}SPR spectrum when there is no electric field (the solid line corresponds to the theory, marks to the experiments).}\n\\end{figure}\n\n\\begin{table}\n\\caption{\\label{table:table1} The initial optical parameters of the layers ($U=0$)}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\nNo & material \t\t& $n$ \t& $k$ \t& thickness, ($nm$) \\\\\n1  & ${Al_2}{O_3}$\t& 1.659\t& 0\t\t& 12.02\\\\\n2  & $Ag$\t\t\t& 0.136\t& 4.011\t& 49.20\\\\\n3  & ${Al_2}{O_3}$\t& 1.659\t& 0\t\t& 177.36\\\\\n4  & $Ag$\t\t\t& 0.136\t& 4.011\t& 36.46\\\\\n5  & substrate\t\t& 1.525\t& 0\t\t& -\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\tGiven non-zero external electric field ($U > 0$), the thicknesses of the layers were considered fixed (Table~\\ref{table:table1}) in computations and effective optical parameters of the $Ag$ and $A{l_2}{O_3}$ layers (refraction $n$ and absorption $k$ coefficients) were varied to reach the best agreement with the experimental spectrum. All parameters of the outer protective layer ($d = 12nm$) were also fixed according to Table~\\ref{table:table1} because it should have no electric field in it.\n\n\tComparing the experimental and theoretical curves allows the following conclusions. With no external field (Fig.~\\ref{fig:fig03}) the model is nearly perfect: the original values of optical parameters of the layers are identical to those given in the SOPRA data base. If there is an external field, the simulation needs closer consideration. If we assume that the optical properties of both $Ag$ layers keep the same in the presence of the electric field, the difference between theoretical and experimental data will increase with the field strength and we can't eliminate it by varying $n$ and $k$. Indeed, the theoretical curves in Fig.~\\ref{fig:fig04} built from this assumption agrees with the experiment poorly. It is clearly caused by the fact that the computations do not take into account nonlinear effects which can take place when the electric field strength is high. Surface charges caused by dielectric polarization and changes of  the numbers of electrons  in the metal layers should also be taken into consideration. The theory and experiment agree better if we assume that the properties of $Ag$ of the cathode and anode differ dramatically. Validation of such approach and corresponding simulation results are given in the next section.\n\n\\begin{figure*}\n\\includegraphics{fig04}\n\\caption{\\label{fig:fig04}SPR spectra for $U = 16 V$ and $U = 30 V$ (the marks are the experimental data, the solid line are the computation results). The theoretic curve  are built under assumption that the optical parameters of the both $Ag$ layers are the same. It is seen that these curves do not agree well with the experiment.}\n\\end{figure*}\n\n\\section{ The results of the simulation}\nThe difference between the theory and experiment can be eliminated if we suppose that the optical parameters of the upper (anode) and lower (cathode) $Ag$ layers can be different and must be optimized  separately. The separate optimization is possible under the assumption that charges accumulate at the dielectric-metal interface and the electron plasma density changes across the metal layers in a strong electric field.\n\n\tIndeed, when the dielectric layer is about $d_0\\sim200\\,nm$ thick and the applied voltage is nearly $30\\,V$, the surface charge density is $\\sigma  = U\/4\\pi \\varepsilon {d_0} \\sim 40CGSE$ (the permittivity of corundum $\\varepsilon \\sim 10$). If the layer area is $1\\;c{m^2}$, the number of excess electrons on one of the $Ag$ layer (and deficient ones on the other) is  $\\sim{10^{11}}$, while the full number of conduction electrons is $\\sim{10^{18}}$.\n\n\tLet us ask ourselves what the deficiency of electrons in the layer can result in. It is clear that the depth of the potential well for all electrons of the anode increases by $30\\;eV$. At the same time, we should take into account the changed band structure. The matter is that the additional attraction force from the uncompensated positive charge brings about a shift of the conduction band and, of course, the Fermi level ${E_F}$ down to a lower energy. Correspondingly, this results in the interband  absorption  threshold  ($\\sim4\\,eV$)  lowering. However, the shift of the d-electron band is much smaller because of  stronger binding to the nuclei.\n\nThe change of the interband absorption level changes the dielectric permittivity of the metal and, therefore, parameters $n$ and $k$. For instance, the model function $\\varepsilon (\\omega )$  is built for gold (see \\cite{Maier}) which allows for two interband transitions. The eleven  fitting parameters (resonance frequencies, resonance widths, complex amplitudes, asymptotic parameter, etc.) of the function allowed to reach a good agreement with the data \\cite{Etchegoin}. In the case of silver a similar function has not been built so far, so we limit ourselves to the rough estimation using formulae for gold \\cite{Maier}. A $0.1 - eV$ change of threshold $\\Delta $ results in an about unit change of the dielectric permittivity, which can be detected experimentally. Note, that the formula presented  in \\cite{Etchegoin2} describes well the permittivity of $Ag$ for the photon energies less than $3\\,eV$, which does not allow to analyze the influence of  the shift of  the interband  transition threshold on the permittivity value at $\\lambda  = 632.8nm$ ($1.96\\,eV$).\n\n\tAt the same time, the excess of electrons on the cathode raises the interband absorption level. Excess electrons do not leave the metal as long as the attraction caused by image charges exceeds the repulsion caused  by excess electrons. The condition of equilibrium of  these forces can be written as ${{e}^{2}}\/{{l}^{2}}=eU\/2d_0$, where $U\/2d_0$ is the field strength on the cathode (it is twice greater in the capacitor itself), $l$ is the electron-cathode distance. Then the saturation voltage is ${U_s} = 6 \\times {10^{ - 12}}\/{l^2}$. Let us suppose that slow electrons are scattered by defects whose concentration is about ${10^{18}}\\;c{m^{ - 3}}$. The average free path is about $10nm$ and saturation voltage is a few volts in this case. Further increase of $U$ leads to dynamic equilibrium, that is, the number of electrons arriving to the cathode in unit time is equal to the number of electrons leaking through the dielectric per unit time.\n\n\tThe above considerations suggest the necessity to optimize optical parameters of the upper (anode) and lower (cathode) $Ag$ layer independently. Indeed, the modelling using independent optimization of anode and cathode layers gives good agreement between theoretic and experimental curves. Figure~\\ref{fig:fig05} gives the results of simulation of an SPR angular spectrum for $U = 16\\,V$ and $U = 30\\,V$. Comparing Figure~\\ref{fig:fig04} and Figure~\\ref{fig:fig05} shows that the independent optimization allows a significantly better theory-experiment agreement. The computed values of optical properties for this kind of modelling are given in Table~\\ref{table:table2}. The first thing that draws attention is that the refractive index of the anode layer proves to be smaller than that of the cathode. This is an expected observation which agrees with measurements made during the deposition process and is explained by the fact that the deposition on a smooth substrate always gives a better quality of the $Ag$ layer than in deposition on a relatively loose intermediate layer. Secondly, with the growing voltage the $A{l_2}{O_3}$ layer starts exhibiting slight absorption and its refractive index decreases. Probably, it is caused by electrons arriving to this layer from the cathode. To avoid misunderstanding, we should point out that in modelling it is impossible to take into account surface irregularities, interpenetration of layers and non-linear effects caused by a strong electric field. That is why the values of $n$ and $k$ given in Table~\\ref{table:table2} should be regarded as certain effective values that allows for the drawbacks of this theoretical approach in a way.\n\n\\begin{figure*}\n\\includegraphics{fig05}\n\\caption{\\label{fig:fig05}SPR spectra for $U = 16 V$  and  $U = 30 V$ (marks \u2013 experiment, line \u2013 theory). The theoretic curves are built given separate optimizations of $Ag$  parameters on the anode and cathode.}\n\\end{figure*}\n\n\\begin{table*}\n\\caption{\\label{table:table2} The optical parameters of the MDM structure}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccc}\n   & \n\\multicolumn{2}{c}{$Ag$ (anode), $d=49 nm$} &\n\\multicolumn{2}{c}{${Al_2}{O_3}$, $d=177 nm$} &\n\\multicolumn{2}{c}{$Ag$ (cathode), $d=36 nm$} \\\\\n$U(V)$ \t& $n$ \t\t& $k$ \t& $n$ \t& $k$ \t& $n$ \t& $k$ \\\\\n0  \t\t& 0.1360\t&4.0110\t&1.6908\t&0.0000\t&0.1341\t&4.0100\\\\\t\n4  \t\t& 0.1341\t&4.0109\t&1.6592\t&0.0022\t&0.1340\t&4.0100\\\\\n8  \t\t& 0.5928\t&4.2967\t&1.6593\t&0.0087\t&0.0881\t&3.4853\\\\\n12  \t& 0.5931\t&4.2969\t&1.6533\t&0.0116\t&0.0882\t&3.4852\\\\\n16  \t& 0.5932\t&4.2970\t&1.6531\t&0.0113\t&0.0882\t&3.4852\\\\\n20  \t& 0.3752\t&4.6909\t&1.5059\t&0.0583\t&0.0000\t&1.0263\\\\\n24  \t& 0.3717\t&4.6840\t&1.5064\t&0.0589\t&0.0000\t&1.0308\\\\\n28  \t& 0.3828\t&4.6885\t&1.5122\t&0.0622\t&0.0000\t&1.0619\\\\\n30  \t& 0.3778\t&4.7022\t&1.5161\t&0.0642\t&0.0000\t&1.0816\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\tThe voltage dependence of the optical parameters of MDM layers are presented in Figure~\\ref{fig:fig06}. It is seen that all the optical parameters of the $Ag$ layers experience noticeable changes at $U\\sim8V$ \u0438 and $U\\sim16V$. Similar irregular behaviour of the parameters is observed for the $A{l_2}{O_3}$ layers. Between these two voltage points parameters $n$ and $k$ keep fairly stable for all layers. The exclusion is the absorption coefficient of the $A{l_2}{O_3}$ layer which grows slightly on the interval  $U =20-30V$. It is interesting that the refractive index of the cathode $Ag$ layer falls to zero when $U > 16V$, which means that the dielectric permittivity becomes strictly negative and loses its imaginary component.\n\t\n\t\t\\begin{figure*}\n\\includegraphics{fig06}\n\\caption{\\label{fig:fig06}The refractive indices and absorptions coefficients versus voltage $U$. The top panels show the behaviour of the parameters of the cathode and anode $Ag$ layers, the bottom ones refer to the $A{l_2}{O_3}$ layers.}\n\\end{figure*}\n\n\tFollowing the changes of parameters $n$ and $k$, the reflection coefficient also changes sharply.  The most pronounced changes in the MDM structure reflection occur when the voltage exceeds $16V$. An experiment on this MDM structure showed  that the voltage increase  up to $U =20-30V$ results in a $4.2$-times decrease of the reflection coefficient at ${\\theta} = {42.9^0}$ and $5.4$-times increase at ${\\theta} = {55.8^0}$. Calculations show that the contrast of the reflectivity can be increased dramatically by varying the layer thicknesses slightly. For instance, if we increase the anode layer thickness to $54\\,nm$, we get a structure which at fixed angle ${\\theta} = {60.5^0}$ may be considered as a switcher: with zero reflection at $U=0$ and  with $20\\% $ reflectivity when $U$ increases to $30V$.\n\n\\section{Conclusions}\nWe have discovered significant changes of the dielectric permittivity of silver when a constant voltage of up to $30V$ is applied to a MDM-structure waveguide. It is shown that we can modulate the reflection coefficient widely by varying the voltage. Looking forward, the effect may be used in development of electrically controlled optical valves.\n\n\tWe could not so far find an acceptable theory that can explain the abrupt change of optical parameters of the MDM structure. This will be the goal of later research.\n\n\\begin{acknowledgments} \nThe work was supported by the project 2.1 of the RAS Presidium and RFBR grant 11-07-92470.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\pagestyle{plain}\n\nDuring the last two years new connections between integrable\nmodels and quantum groups were established. The representation\ntheory of quantum groups at roots of unit provides new solutions\nto the start--triangle equation increasing in this way the\nfamily of lattice integrable models. Most of these new solutions\ncan be obtained by applying the ``descendent''--procedure [1] to\nwell known models as for instance the 6--vertex model.\n\nGiven an $R$ matrix satisfying the Yang--Baxter equation the first\nstep in the descendent procedure is to find all possible\nsolutions to:\n\n\\begin{equation}\nR (\\lambda, \\mu) \\left[ L (\\lambda) \\otimes L (\\mu) \\right]  =\n\\left[ L (\\mu) \\otimes L (\\lambda) \\right] R (\\lambda, \\mu)\n\\label{1}\n\\end{equation}\n\nIf the $R$--matrix is, for instance, the quantum $R$ matrix of\n$\\widehat{SU (2)}_q$ in the 1\/2--representation then we should\nexpect the solutions of (1) to be in one to one correspondence with the\ndifferent irreps of the Hopf algebra $\\widehat{SU (2)}_q$.\nIt is in this way that the representation theory plays an\nimportant role in the discovery of new solutions to thestar--triangle equations\nby the descendent procedure [2]. The second\nstep of this method is to find for two given solutions\n$L^{\\rho_1}, L^{\\rho_2}$ of (1) a new $R$--matrix, solution to\nthe Yang--Baxter equation, and satisfying:\n\n\\begin{equation}\n{\\cal R}^{\\rho_1 \\rho_2} \\left( L^{\\rho_1} \\otimes L^{\\rho_2}\n\\right) = \\left( L^{\\rho_2} \\otimes L^{\\rho_1} \\right) {\\cal\nR}^{\\rho_1 \\rho_2}\n\\label{2}\n\\end{equation}\n\nIf we interpret $L^{\\rho_1}, L^{\\rho_2}$ as associated with two\ndifferent irreps $(\\rho_1, \\rho_2)$ of $\\widehat{SU (2)}_q$ the\nnew ${\\cal R}$--matrix will define the intertwiner between these\ntwo representations. The two steps of the descendent--procedure\nare graphically represented in fig. 1.\n\n\\vspace{4cm}\n\\begin{center}\nFig. 1. \\underline{The descendent--equations}\n\\end{center}\n\n\\vspace{0.3cm}\nThe simplest example of this procedure is the one known as\nfusion where starting, for instance, with the $R^{1\/2,\n1\/2}$--matrix of $\\widehat{SU (2)}_q$ we obtain the intertwiner\nfor regular representations with higher spins [3].\n\nThe case of quantum groups at roots of unity is a very\ninteresting example for the application of this technique. In\nthis case, the number of different finite dimensional irreps is\nmuch bigger as a consequence of the existence of new central\nelements in the Hopf algebra [4]. Hence for the $SU(2)_q$ case, the\ncentral Hopf subalgebra for $q = \\varepsilon$ an Nth--root of unit,\ncontains, in addition to the standard casimir, the new elements\n$E^N, F^N, K^N$. The different irreps can be divided into two\nmain sets: i) regular representations and ii)\ngeneric--representation. The last ones are classified into\ncyclic, with non vanishing eigenvalue for $E^N$ and $F^N$,\nsemicyclic with vanishing eigenvalue only for $E^N$ or $F^N$ but\nnot both and nilpotent for which $E^N = F^N = 0$ but $K^N$ having\ngeneric values. Starting with the $R$--matrix of $\\widehat{SU\n(2)}_q$ it is possible to get solutions to the descendent\nequations (1) and (2) associated with cyclic, semicyclic and\nnilpotent representations. The first case i.e. cyclic, was first\ndiscovered by Bazhanov and Stroganov [1] and the Kyoto group [5]. Surprisingly\nenough, the solution obtained coincides with the one\ncorresponding to the chiral Potts model [6]. These kinds of models are\nspecially important from a mathematical point of view. Their\nspectral manifold is a higher genus curve:\n\n\\begin{equation}\n\\Gamma_k \\; : \\; x^N + y^N = k (1 + x^N y^N)\n\\label{3}\\end{equation}\n\n\\noindent\nwith $k$ a parameter of the model (for instance $k=0$\ncorresponds to Fateev--Zamolodchikov model). Given two irreps\n$(\\xi_1, \\xi_2)$ of $\\widehat{SU (2)}_q$ with $q^N =1$ and\n$\\xi \\in Spec Z_q$, where $Z_q$ is the central Hopf subalgebra,\nthen the intertwining condition defines a submanifold in $Spec Z_q$\nwhich can be shown to be isomorphic to the product of two copies\nof the spectral curve $\\Gamma_k$ of the chiral Potts model.\nMoreover the solution $R(\\xi_1 \\xi_2)$ to equation (2)\nfactorizes into four pieces which can be represented in terms of\nthe Boltzman weights $W, \\bar{W}$ of the chiral Potts model [5].\n\nThe new solution which we would like to discuss in this lecture\ncorresponds to the case of semicyclic and nilpotent\nrepresentations. The model defined in this way shares some\nsimilarities with chiral Potts and with the Heisenberg--Ising models with\nhigher spin. However, it presents some new features which we now move\non to present.\n\n\\section{The semi--cyclic descendent}\n\nIn a previous paper [7] we have presented the $R$ matrix solution\nto the Yang--Baxter and the intertwiner conditions:\n\n$$\n(1 \\otimes R (\\xi_1 \\xi_2)) (R (\\xi_1 \\xi_3) \\otimes 1) (1\n\\otimes R (\\xi_2 \\xi_3)) =\n\\eqno{(3.a)}\n$$\n\n\\[\n= (R (\\xi_2 \\xi_3) \\otimes 1) (1 \\otimes R(\\xi_1 \\xi_3)) (R\n(\\xi_1 \\xi_2) \\otimes 1)\n\\]\n\n$$\nR (\\xi_1 \\xi_2) \\Delta_{\\xi_1 \\xi_2} (a) = \\Delta_{\\xi_2 \\xi_1}\n(a) R (\\xi_1 \\xi_2)\n\\eqno{(3.b)}\n$$\n\n\\[\na \\in SU (2)_q \\; \\; \\; q^N = 1\n\\]\n\n\\noindent\nfor $\\xi_1, \\xi_2$ semicyclic representations of $SU(2)_q$.\nDenoting by $x, y, \\lambda^N$ the eigenvalues of $E^N, F^N$ and\n$K^N$ respectively the semicyclic representations are\nparametrized by $(y_1, \\lambda_1)$, $(y_2, \\lambda_2)$. Using (3.b)\nfor $a$ in the center, we get the intertwining constraints on\nthe values of $\\xi_1, \\xi_2$:\n\\begin{equation}\n\\frac{y_1}{1 - \\lambda^N_1} = \\frac{y_2}{1 - \\lambda^N_2} = \\chi\n\\label{4}\n\\end{equation}\n\n\\noindent\nwith $\\chi$ an arbitrary complex number. The solution $R (\\xi_1\n\\xi_2)$ satisfies:\n\n\\begin{equation}\n\\begin{array}{rl}\ni) & R (\\xi \\xi) = 1 \\\\\nii) & R (\\xi_1 \\xi_2) R (\\xi_2 \\xi_1) = 1 \\otimes 1 \\\\\niii) & R (\\xi_1 \\xi_2) = P R (\\xi_2 \\xi_1) P\n\\end{array}\n\\label{5}\n\\end{equation}\n\n\\noindent\nwhere $P$ is the permutation operator.\n\nFor more details on this solution see reference [7].\n\nFollowing the spirit of the descendent--technology we proceed\nnow to find solutions to equation (1), for $R$ the quantum\n$R$--matrix of $\\widehat{SU(2)}_q$ in the 1\/2--representation\ni.e. the six vertex model, which can be associated with\nsemicyclic representations of $\\widehat{SU(2)}_q$ at $q =\n\\varepsilon$ an Nth--root of unit (we shall consider $N$ to be\nan odd integer). We will first consider the simplest\nnon trivial case:  $x = y = 0$ and $\\lambda$ generic (the\nregular representations correspond to $\\lambda =\n\\varepsilon^{2s}$ with $s$ integer or half integer spin).\n\nThe 6--vertex $R$--matrix is given by [3]:\n\n\\begin{equation}\nR^{1\/2, 1\/2} (u) = sh \\left[ u + i\\gamma \\frac{1}{2} (1 + \\sigma^3\n\\otimes \\sigma^3) \\right] + i \\; {\\rm sin\\gamma} \\; (\\sigma^+ \\otimes\n\\sigma^- + \\sigma^- \\otimes \\sigma^+)\n\\label{6}\n\\end{equation}\n\nEquation (1) reads now:\n\n\\begin{equation}\nR^{1\/2, 1\/2} (u-v) (L (u) \\otimes L(v)) = (L (v) \\otimes L(u))\nR^{1\/2, 1\/2} (u-v)\n\\label{7}\n\\end{equation}\n\nThe solutions, we are interested in, are $L^{(\\lambda)} (u)$--matrices\nacting on $V^{1\/2} \\otimes V^\\lambda$ such that:\n\n\\begin{equation}\nL^{(\\lambda)} (u) : V^{1\/2} \\otimes V^\\lambda \\rightarrowV^{1\/2} \\otimes\nV^{\\lambda}\n\\label{8}\n\\end{equation}\n\n\\noindent\nwhere $V^\\lambda$ is the irrep of $\\widehat{SU\n(2)}_\\varepsilon$ with eigenvalue of $K^N = \\lambda^N$ and where\nthe tensor product in (7) is defined with respect to the\n$V^{1\/2}$--indices. Notice that the $L^{(\\lambda)} (u)$--matrices we\nare looking for define the intertwiners between the irreps 1\/2\nand $\\lambda$. In matrix notation the solution is given by:\n\n$$\nL^{(\\lambda)} (u) = \\frac{1}{e^u \\varepsilon^{1\/2} \\lambda^{1\/2}\n- e^{-u} \\varepsilon^{-1\/2} \\lambda^{-1\/2}} \\times\n$$\n\n\\begin{equation}\n\\times \\left( \\begin{array}{cc}\ne^u \\varepsilon^{1\/2} K^{1\/2}\n- e^{-u} \\varepsilon^{-1\/2} K^{-1\/2} & (\\varepsilon -\n\\varepsilon^{-1}) \\varepsilon^{-1\/2} FK^{1\/2} \\\\\n(\\varepsilon- \\varepsilon^{-1}) \\varepsilon^{-1\/2} E K^{-1\/2} &\ne^u \\varepsilon^{1\/2} K^{-1\/2}\n- e^{-u} \\varepsilon^{-1\/2} K^{1\/2} \\end{array} \\right)\n\\label{9}\n\\end{equation}\n\n\n\\noindent\nwith  $E, K, F$ the generators of $SU(2)_q$ which satisfy the relations:\n\n\\[\nEK = \\varepsilon^2 K E \\; \\; , \\; \\; FK = \\varepsilon^{-2} KF\n\\]\n\n\\begin{equation}\n[E, K] = \\frac{K - K^{-1}}{\\varepsilon - \\varepsilon^{-1}}\n\\label{10}\n\\end{equation}\n\nThe representation $V^\\lambda$ is defined in the basis $\\{ |r> \\}_{r=0}^{N-1}$\n\n\\begin{equation}\n\\begin{array}{l}\nE | r> = d_{r-1} | r-1> \\\\\nF | r> = d_{r} | r+1> \\\\\nK | r> = \\lambda \\varepsilon^{-2r} | r>\n\\end{array}\n\\label{11}\n\\end{equation}\n\n\\[\nd^2_j (\\lambda) = [j+1] \\frac{\\lambda \\varepsilon^{-j} -\n\\lambda^{-1} \\varepsilon^j}{\\varepsilon - \\varepsilon^{-1}}\n\\]\n\\[\n[ x] = \\frac{\\varepsilon^x - \\varepsilon^{-x}}{\\varepsilon - \\varepsilon^{-1}}\n\\]\n\nFor a graphic representation of the matrix $L^{(\\lambda)}(u)$ see\nFig. 2. Notice that each entry in the 2 x 2 matrix (9)\nrepresents an operator acting on the space $V^\\lambda$.\n\n\\vspace{5cm}\n\\begin{center}\nFig. 2.  Graphic representation of the $L^{(\\lambda)}(u)$--matrix\n\\end{center}\n\n\\vspace{1cm}\nThe second step in the descendent procedure corresponds to\nfinding the $R$--matrix solution to the equation (2):\n\n\\begin{equation}\n{\\cal R}^{\\lambda_1 \\lambda_2} (u-v) \\left( L^{(\\lambda_1)} (u)\n\\otimes L^{(\\lambda_2)} (v) \\right) = \\left( L^{(\\lambda_2)} (v)\n\\otimes L^{(\\lambda_1)} (u) \\right) R^{\\lambda_1 \\lambda_2} (u-v)\n\\label{12}\n\\end{equation}\n\n\\noindent\nwhere now the tensor product is defined with respect\nto the \\underline{$V^{\\lambda_1}$ - $V^{\\lambda_2}$} \\underline{indices} of the\n$L^{(\\lambda)}$ (u) - matrices (see fig. 1 B). The solution to\n(12) defines the intertwiner between the representations\n$\\lambda_1, \\lambda_2$ and it is given by:\n\n\\[\nR^{\\lambda_1 \\lambda_2} (u)^{l, r_1 + r_2 - l}_{r_1 r_2} =\n\\frac{\\epsilon^{(r_1 + r_2 - l) l - r_1\nr_2}}{\\prod^{r_1+r_2-1}_{j=0} (e^u \\lambda_1 \\lambda_2\n\\epsilon^{-j} - e^{-u} \\epsilon^j)} \\times\n\\]\n\n\\[\n\\times {\\sum^{r_1}_{l_1 = 0}\n\\sum^{r_2}_{l_2 = 0}} \\left[ \\begin{array}{c} r_1\n\\\\ l_1 \\end{array} \\right] \\left[ \\begin{array}{c} r_2 \\\\ l_2\n\\end{array} \\right] \\frac{[l] ! [r_2 - l_2] !}{[r_1 + l_2] !\n[r_2]!} (\\epsilon - \\epsilon^{-1})^{r_1 - l_1 +l_2}\n\\]\n\n\\[\n\\times \\prod^{r_1 + l_2 -1}_{j= r_1} d_j (\\lambda_1) \\prod^{r_1\n+l_2-1}_{j=l_1 +l_2} d_j (\\lambda_1) \\prod^{r_2\n- 1}_{j=r_2 - l_2} d_j (\\lambda_2) \\prod^{r_1+r_2\n-l-1}_{j=r_2 -l_2}\n\\]\n\n\\begin{equation}\\times \\lambda^{l_2}_1 \\lambda^{r_1-l_1}_2 \\prod^{r_2 -l_2\n-1}_{j=0} (e^u \\lambda_2 \\epsilon^{-j} - e^{-u} \\lambda_1\n\\epsilon^j) \\prod^{l_1 -1}_{j=0} (e^u \\lambda_1 \\epsilon^{-j+r_2-l_2}\n-e^{-u} \\lambda_2\n\\epsilon^{j+l_2 -r_2})\n\\label{13}\n\\end{equation}\n\n\\noindent\nwith the following conventions: a) whenever in above products the upper index\nis less than the lower index the result is one.\nb) the constraint $l_1 + l_2 = l$ must be used to carry out summation.\n\nIt is easy to check that this solution coincides in the case of\n$u=0$ with the $R^{\\xi_1 \\xi_2}$--matrix of reference [7] for\n$\\xi_1, \\xi_2$ two semi--cyclic irreps with $x=y=0$ and $K^N =\n\\lambda^N_1, \\lambda^N_2$. Moreover  for $\\lambda_1 = \\lambda_2\n= \\epsilon^{2s}$ eqn. (13) gives us the spin $s - R$ matrix.\n\nSummarizing, we have obtained starting with the $R$--matrix of\nthe six vertex model for $q$ an Nth-root of unit a class of\ndescendent models characterized by the quantum $R$--matrix (13).\nThe transfer matrix of these models can be defined for periodic\nboundary conditions as follows (see fig. 3):\n\n\\begin{equation}\nT_{\\lambda_0} (u, \\lambda) : \\otimes^L V^{\\lambda_0}\n\\rightarrow\n\\otimes^L V^{\\lambda_0}\n\\label{14}\n\\end{equation}\n\n\\begin{equation}\n\\langle r'_1 \\cdots r'_N | T_{\\lambda_0} (u, \\lambda) | r_1,\nr_2 \\cdots r_N \\rangle = \\left( \\sum_{l's} R^{\\lambda,\n\\lambda_0} (u)_{l_1 r_1}^{l_2 r'_1} R^{\\lambda \\lambda_0}\n(u)^{l_3 r'_2}_{l_2 r_2} \\cdots \\right)\n\\label{15}\n\\end{equation}\n\n\\noindent\nwhere $\\lambda_0$ appears as a parameter characterizing the model\nand $u$ and $\\lambda$ as spectral variables. The Yang-Baxter\nrelation for the $R$--matrix (eq. 13) implies the integrability equation:\n\n\\begin{equation}\n\\left[ T_{\\lambda_0} (u, \\lambda'), T_{\\lambda_0} (u', \\lambda'') \\right]\n= 0\n\\label{16}\n\\end{equation}\n\nEquation (16) is the standard integrability condition for\nsoluble models with the important peculiarity that the spectral\nvariables are now living on a manifold of complex dimension\nequal 2. Notice that for the kind of irreps we are considering\n(i.e. with vanishing eigenvalues of $E^N$ and $F^N$ and generic\neigenvalue of K) the intertwining condition is not imposing any\nconstraint on the allowed values of $\\lambda$.\n\n\\[(\\lambda'_1 u) \\begin{array}{cc|cc|cc|cc|cc|ccc}\n& & r'_1 & & r'_2 & & r'_3 & & r'_4 \\cdots & & r'_L & & \\\\\n\\hline\nl_1 & & &l_2  & & l_3  & & l_4  & & l_5 \\cdots  & e 1  & \\\\\n& & r_1  & & r_2  & &  r_3 & & r_4 \\cdots & & r_L & | r_1 \\cdots\nr_L> \\\\\n\\end{array}\n\\overbrace{\\otimes^L V^{\\lambda_0}}\n\\]\n\n\\begin{center}\n$L = \\#$ sites in the row.\n\nFig. 3: The transfer matrix $T_{\\lambda_0} (\\lambda, u)$\n\\end{center}\n\n\\noindent\n\n\\section*{3. The associated 1D--chain}\n\nGiven the transfer matrix (14-15) we can define a local 1D\nhamiltonian as follows:\n\n\\begin{equation}\n\\left. H^{(\\lambda_0)} = i \\frac{\\partial}{\\partial u}  l n\nT_{\\lambda_0} (u, \\lambda_0) \\right|_{u= 0}\n\\label{17}\n\\end{equation}\n\n\\noindent\nwhich will be hermitian for certain allowed regions of\n$\\lambda_0$. This hamiltonian defines a one dimensional\nspin chain with the Hilbert space given by:\n\n\\begin{equation}\n{\\cal H} = \\otimes^L V^{\\lambda_0}\n\\label{18}\n\\end{equation}\n\nThe most natural interpretation of this spin--chain is as a\ngeneralization to continuous ``spin\" of the higher spin quantum\nchains defined in references [8]. In fact, as we have\nalready mentioned for the special case $\\lambda_0 =\n\\epsilon^{2S_{max}} (2S_{max} +1 = N)$ the hamiltonian (16)\ncoincides with the hamiltonian $H^{S_{max}}_\\epsilon$ of spin\n$S_{max}$ and anisotropy depending of $\\epsilon$.\n\nThe two main new features of the model defined by the transfer\nmatrix (14) are:\n\n\\begin{itemize}\n\\item[i)] All the $V^{\\lambda_0}$ representations are of the\nsame dimensions $N$. This is specific of working with the\nquantum group at roots of unit.\n\\item[ii)] The integrability condition (16) with two independent\nspectral parameters.\n\\end{itemize}\n\nThe main consequence of (16) for the spin chain defined by (16)\nis the existence of a local operator $Q^{(\\lambda_0)}$ defined by:\n\n\n\\begin{equation}\nQ^{(\\lambda_0)} = \\left. 2 \\lambda \\frac{\\partial}{\\partial \\lambda} l n\nT_{\\lambda_0} (u = 0,\\lambda) \\right|_{\\lambda = \\lambda_0}\n\\label{19}\n\\end{equation}\n\n\\noindent\nsuch that:\n\n\\begin{equation}\n\\left[ H^{\\lambda_0}, Q^{\\lambda_0} \\right] = 0\n\\label{20}\n\\end{equation}\n\nDenoting $\\xi \\equiv (u, \\lambda)$, the transfer matrix (14) can\nbe written as:\n\n\\begin{equation}\nT_{\\xi, \\xi'} = T_{(u, \\lambda)|(u'\\lambda_0)} \\equiv\nT_{\\lambda_0} (u - u', \\lambda)\n\\label{21}\n\\end{equation}\n\n\\noindent\nwhere we have used the difference property with respect to the\nspectral variable $u$. The most general local hamiltonian we can\ndefine is:\n\n\\begin{equation}\n{\\cal H} = \\frac{d}{dt} \\left. ln T_{\\xi, \\xi'(t)}\n\\right|_{t= 0}\n\\label{22}\n\\end{equation}\n\n\\noindent\nwith $\\lim_{t\n\\rightarrow 0} \\xi'(t) =  \\xi$. This definition of the hamiltonian would\nbe, in spirit, very close to the way a 1D hamiltonian is defined\nfor the chiral Potts model. However, it is clear that in our case\nany Hamiltonian obtained as indicated by eqn. (22) will be a\nlinear combination of $H^{(\\lambda_0)}$ and $Q^{(\\lambda_0)}$.\nBefore entering into the detailed study of the 1D hamiltonians, it\nis worthwhile to discuss the differences between the models we\nare defining and the chiral Potts. The model which we can compare\nwith is the Fateev--Zamolodchikov model. The way the cyclicity\nof the irreps, entering into the definition of this model, is reflected\nin the physics is through the $Z(N)$--invariance. In fact, the spectrum of\nthe hamiltonian decomposes in different sectors with well\ndefined $Z(N)$--charge.\n\nIn the noncyclic case, we are presenting here, there exists a well\ndefined ``reference state\" $| \\Omega_0 > \\in \\otimes^L V^{\\lambda_0}$:\n\n\\begin{equation}\n| \\Omega_0 > = | 0,0 \\cdots 0 >\n\\label{23}\n\\end{equation}\n\nRecall that the states in $\\otimes^L V^\\lambda$ are given by $|r_1\nr_2 \\cdots r_L>$ with $r_i = 0,\\cdots,N-1$. The reference state\nis the natural generalization of the ferromagnetic state with\nall the spins up. The conserved number, as in the case of the\nHeisenberg chains, is the total number of spins ``down'' which in our\ncase is given by:\n\n\\begin{equation}\n\\sum^L_{i=1} r_i = \\rho.\n\\label{24}\n\\end{equation}\n\nDifferent sectors correspond to different values $\\rho$. In the\nchiral Potts model, i.e. cyclic case, there is no good\ndefinition of ``reference state\" and the conserved quantum\nnumber is given by the $Z(N)$ charge defined by:\n\n\\begin{equation}\nX = e^{2 i\\pi Q\/N}.\n\\label{25}\n\\end{equation}\n\n\\noindent\nwhere\n\n\\begin{equation}\nX | r_1 \\cdots r_L >= |r_1 +1 \\cdots r_2 +1>.\n\\label{26}\n\\end{equation}\n\nIn the next section and using the fact that we have a good\nreference state we will proceed to diagonalize the hamiltonian\n$H^{\\lambda_0}$ by using the Bethe Ansatz technique.\n\n\\section*{4. Bethe Ansatz Equations}\n\nIn terms of the $L^{\\lambda_0} (u)$--matrices we define the\nmonodromy matrix by:\n\n\\begin{equation}\nt_{\\lambda_0} (u) = L^{\\lambda_0}_L (u) \\cdots L^{\\lambda_0}_2\nL^{\\lambda_0}_1 (u)\\label{27}\n\\end{equation}\n\n\\noindent\nwith $L$ the length of the row in lattice units. The monodromy\nmatrix $t_{\\lambda_0}(u)$ satisfies:\n\n\\begin{equation}\nR^{1\/2, 1\/2} (u - v) t_{\\lambda_0} (u) t_{\\lambda_0} (v)\n= t_{\\lambda_0} (v) t_{\\lambda_0} (u) R^{1\/2, 1\/2} (u - v)\n\\label{28}\n\\end{equation}\n\nRepresenting it as a 2x2 matrix:\n\n\\begin{equation}\nt_{\\lambda_0} (u) = \\left[ \\begin{array}{cc} A_{\\lambda_0} (u) &\nB_{\\lambda_0} (u) \\\\\nC_{\\lambda_0} (u) & D_{\\lambda_0} (u) \\end{array} \\right]\n\\label{29}\n\\end{equation}\n\n\\noindent\nwe obtain the operators $B_{\\lambda_0} (u) : \\otimes^L\nV^{\\lambda_0} \\rightarrow \\otimes^L V^{\\lambda_0}$ which can be\ninterpreted as creating on the reference state (23) an\n``elementary excitation\". Notice that:\n\n\\begin{equation}\nB_{\\lambda_0} (u_1) B_{\\lambda_0} (u_2) = B_{\\lambda_0} (u_2)\nB_{\\lambda_0} (u_1)\n\\label{30}\n\\end{equation}\n\n$$\nC_{\\lambda_0} (u) | \\Omega_0 > = 0.\n\\eqno{(30;b)}\n$$\n\nTo diagonalize the transfer matrix (13) we use the standard\nalgebraic Bethe Ansatz:\n\n\\begin{equation}\n|\\psi> = \\prod^M_{i=1} B_{\\lambda_0} (u_i) | \\Omega_0 >\n\\label{31}\n\\end{equation}\n\nThe number $M$ of ``excitations\" is for our model a conserved\nquantum number and therefore we can diagonalize the transfer\nmatrix in each sector.\n\nUsing equation (9) it is easy to find the eigenvalues $\\Lambda\n(\\lambda_0, \\lambda, u; \\{ u_i \\}_{i=1, \\cdots, M})$ of the\ntransfer matrix T in each sector:\n\n\\begin{equation}T_{\\lambda_0} (u, \\lambda) |\\psi> = \\Lambda (\\lambda_0,\n\\lambda,\nu; \\{u_i \\}) |\\psi > +\\;unwanted\\;terms.\n\\label{32}\n\\end{equation}\n\n\\noindent\nwith\n\n\\begin{equation}\n\\Lambda (\\lambda_0, \\lambda,\nu; \\{u_i \\}) = \\sum^{N-1}_{r=0} (R^{r0}_{r0} (\\lambda,\n\\lambda', u))^L \\prod^M_{i=1} {\\cal L}^{\\lambda'}_r (u - u_i)\n\\label{33}\n\\end{equation}\n\n\\noindent\nwhere the $R$ matrix is the one given in (12) and\n${\\cal L}^{\\lambda'}_r (u-u_i)$ is:\n\n\\begin{equation}\n{\\cal L}^{\\lambda'}_r (u)= \\frac{(\\epsilon^u \\epsilon^{1\/2}\n\\lambda^{\\prime 1\/2} - e^{-u} \\epsilon^{-1\/2} \\lambda'^{-1\/2}) (e^u\n\\epsilon^{-1\/2} \\lambda'^{-1\/2} - e^{-u} \\epsilon^{1\/2}\n\\lambda'^{1\/2})}{(e^u \\epsilon^{1\/2 - r} \\lambda'^{1\/2} - e^{-u}\n\\epsilon^{r-1\/2} \\lambda'^{-1\/2}) (e^u \\epsilon^{-r-1\/2}\n\\lambda'^{1\/2} - e^{-u} \\epsilon^{r+1\/2} \\lambda'^{-1\/2})}\n\\label{34}\n\\end{equation}\n\nTo fix the values of the ``rapidities'' $u_i$ we must eliminate\nthe unwanted terms in (32). This can be done using equation (28)\nor more easily imposing that the residues at the poles both in\n$u$ and $\\lambda$ of (33) vanish. The result is:\n\n\\begin{equation}\n\\left( \\frac{e^{u_j} \\epsilon^{1\/2} \\lambda^{1\/2}_0 - e^{-u_j}\n\\epsilon^{-1\/2} \\lambda^{-1\/2}_0}{e^{u_j} \\epsilon^{1\/2}\n\\lambda^{-1\/2}_0 - e^{-u_j} \\epsilon^{-1\/2} \\lambda^{1\/2}_0}\n\\right)^L = \\prod^{M}_{\\begin{array}{c} k=1 \\\\ k \\neq j\n\\end{array}} \\frac{sh (u_j - u_k + i \\gamma)}{sh (u_j - u_k - i \\gamma)}\n\\label{35}\n\\end{equation}\n\n\\noindent\nwhere $\\epsilon = e^{i \\gamma}$. In the case $\\lambda_0 =\n\\epsilon^{2s}$ with 2s being an integer, i.e. regular representations of spin\ns, equations (35) become the Bethe Ansatz equations for the higher spin\nHeisenberg--Ising chains of references [8].\n\n{}From (33) we can find the eigenvalues of the operators\n$H^{\\lambda_0}$ and $Q^{\\lambda_0}$ defined in the previous section:\n\n\\begin{equation}\nE^{\\lambda_0} = 2i \\sum^M_{j=1} \\frac{\\lambda_0 -\n\\lambda^{-1}_0}{(e^{u_j} \\varepsilon^{1\/2} \\lambda^{1\/2}_0 -e^{-u_j}\n\\varepsilon^{-1\/2} \\lambda^{-1\/2}_0 ) (e^{u_j}\n\\varepsilon^{1\/2} \\lambda^{-1\/2}_0 - e^{-u_j} \\varepsilon^{-1\/2}\n\\lambda^{1\/2}_0)}\n\\label{36}\n\\end{equation}\n\n\\begin{equation}\nQ^{\\lambda_0} = 2 \\sum^M_{j=1}\n\\frac{e^{2u_j} \\varepsilon - e^{-2u_j} \\varepsilon^{-1}}{(e^{u_j}\n\\varepsilon^{1\/2} \\lambda^{1\/2}_0 -\ne^{-u_j} \\varepsilon^{-1\/2} \\lambda^{-1\/2}_0 ) (e^{u_j}\n\\varepsilon^{1\/2} \\lambda^{-1\/2}_0 - e^{-u_j} \\varepsilon^{-1\/2}\n\\lambda^{1\/2}_0)}\n\\label{37}\n\\end{equation}\n\nFor the intermediate case corresponding to $\\lambda_0 =\n\\varepsilon^{2s}$ with arbitrary $s$, the equations (36) (37) become:\n\n\\begin{equation}\nE^s = - \\sum^M_{j=1} \\frac{\\sin (2 \\gamma s)}{sh [\n\\frac{\\gamma}{2} (\\alpha_j + 2 is)] sh [ \\frac{\\gamma}{2} (\\alpha_j -\n2 is)]}\n\\label{38}\n\\end{equation}\n\n\\begin{equation}\nQ^s =  \\sum^M_{j=1} \\frac{s h ( \\gamma \\alpha_j)}{sh [\n\\frac{\\gamma}{2} (\\alpha_j + 2 is)] sh [ \\frac{\\gamma}{2} (d_j -\n2 is)]}\n\\label{39}\n\\end{equation}\n\n\\noindent\nwhere we have introduced the new variable $\\alpha$ defined by\n\n\\begin{equation}\nu + \\frac{i}{2} \\gamma = \\frac{\\gamma}{2} \\alpha\n\\label{40}\n\\end{equation}\n\nNow we briefly discuss the case of the nonvanishing $\\chi$, defined by eqn.(4).\nThe first observation to be made is that the number of excitations\n``$M$'', introduced in (31), is no longer ``good quantum number\".\nWhat replaces it is an eigenvalue of $\\Delta K$ (which is $\\lambda^L_0\ne^{i\\frac{2 \\pi Q}{N}}$ for $Q=0, 1, \\cdots N-1)$ since this\noperator commutes with both transfer matrices, namely\n\n\\[\n\\Delta K = K^{\\lambda_0}_L \\cdots K_2^{\\lambda_0} K_1^{\\lambda_0}\n\\]\n\n\\begin{equation}[ \\Delta K, tr t_{\\lambda_0} (u)] = [\\Delta K, T_{\\lambda_0}\n(u,\n\\lambda) ] =0\n\\label{41}\n\\end{equation}\n\nSurprisingly enough, some nice features of $\\chi =0$ case\nsurvive. In particular inspecting formula (13), one concludes\nthat R--matrix preserves its low--triangular form while acting\non the ``reference state\" (23). Therefore, this state remains an\neigenvector of $T_{\\lambda_0} (u_1 \\lambda)$. More precisely\n\n\\begin{equation}\nT_{\\lambda_0} (u, \\lambda) | \\Omega_0 > = \\sum^{N-1}_{r=0} (R^{r0}_{r0}\n(\\lambda_0, \\lambda, u))^L | \\Omega_0 >\n\\label{42}\n\\end{equation}\n\n\\noindent\nWhat makes $\\chi \\neq 0$ situation somewhat more complicated is\nthe fact that the commutation relations for $A_{\\lambda_0},\nB_{\\lambda_0}, C_{\\lambda_0}, D_{\\lambda_0}$ and the elements of\n$T_{\\lambda_0}(u,\\lambda)$ no longer have simple form and\ntherefore, one does not expect for the state (31) to be an\neigenvector of $T_{\\lambda_0} (u, \\lambda)$. Consequently,\nthe appropriate generalization of Algebraic Bethe Ansatz is\ncalled for in this case.\n\nTo get the feeling of what transpires, one should note that the\nstate (31) is still an eigenvector of $tr t_{\\lambda_0} (u)$\n\n\\begin{equation}\ntr t_{\\lambda_0} (u) | \\psi_M > = \\sum^1_{r=0} \\left( L^{\\lambda_0}\n(u)^{r0}_{r0} \\right)^L\n\\prod^M_{i=1} R^{1\/2, 1\/2} ( (-1)^r (u_i - u) )^{01}_{10}\n| \\psi_M >\n\\label{43}\n\\end{equation}\n\nIndeed, in deriving eqn.(43) above, one only uses 6--vertex\ncommutation relations (28) and the degeneracy eqn.(30;b) for the\noperator $C_{\\lambda_0} (u)$. Since neither eqn.(28) nor eqn.(30;b)\nare affected by turning on nonzero $\\chi$, one could infer that\neqn.(43) should hold true. Thus, it appears that going from $\\chi\n=0$ to $\\chi \\neq0$ case produces no visible effect on the\nspectrum of the transfer matrix $tr t_{\\lambda_0} (u)$. This\nconclusion, however, may be a bit premature. To see that, let us\nrecall, that not all the solutions of (35), generally speaking,\ncorrespond to non-zero vectors of the form (31).\n\nQuite frequently an additional investigation (involving delicate\nlimiting procedures) is required to determine the fate of the\nparticular solution. In general, the answer to {\\it ``To Be Or\nNot To Be}\" question may depend on whether or not $\\chi$ takes\non the zero value.\nLet us now generate \\underline{finite--dimensional} vector space\n$V_M (Q = M - int [ \\frac{M}{N}])$ by premultiplying non-zero\nvector $| \\psi_M >$ by any sum of products of operators\n$T_{\\lambda_0} (u, \\lambda), T_{\\lambda_0} (u', \\lambda')\n\\cdots$ for any $u, \\lambda; u', \\lambda'; \\cdots$ (but all\nwith the same $\\lambda_0$ and $\\chi$). It is clear, that in this\nspace one can diagonalize simultaneously the whole family of\ncommuting transfer--matrices $T_{\\lambda_0} (u, \\lambda)$.\nRecalling that $tr t_{\\lambda_0} (u)$ commutes with\n$T_{\\lambda_0} (u, \\lambda)$, one immediately arrives to the conclusion\nthat the eigenvectors of family $T_{\\lambda_0} (u, \\lambda)$\ncan be constructed as linear combinations of vectors (31), all\nof the same length (modulo N) and eigenvalue of transfer--matrix\n$tr t_{\\lambda_0} (u)$. Apparently, what seems to be going on is\nas follows: The eigenvalues of transfer matrices $T_{\\lambda_0}\n(u, \\lambda)$ and  $tr t_{\\lambda_0} (u)$ are highly degenerate\nwhen $\\chi = 0$. Turning on finite $\\chi$ results in lifting this\ndegeneracy for $T_{\\lambda_0} (u, \\lambda)$ and changing\nmultiplicities of eigenvalues of $tr t_{\\lambda_0} (u)$. This is\nvery intriguing phenomena and certainly warrants further\ninvestigations. To summarize, we can still characterize\neigenstates of commuting family $T_{\\lambda_0} (u, \\lambda)$\nby the set of Bethe Ansatz roots (35), abandoning, however, the\nsimple representation for Bethe vector given by formula (31). The\ngeneralization of algebraic Bethe Ansatz, briefly sketched\nabove, is similar in spirit to Tarasov's proposal for\nsuperintegrable chiral Potts Model [9].\n\nThis similarity is, by no means, accidental. Indeed, chiral Potts\nmodel on superintegrable line shares an important property of\nsemi--cyclic case: The L--matrix which intertwines spin 1\/2 and\ncyclic repr. has ``top\" reference state but no bottom state.\nWhether this similitude has a deeper implication remains to be seen.\n\nWhen algebraic Bethe Ansatz does not work (or does not provide\nthe shortest route), one may resort to the alternative procedure\nin order to solve for eigenvalues of Hamiltonian and transfer\nmatrix. This procedure, known as ``Functional Relations\"\nmethod, was exploited quite successfully in recent years to\nfind the spectrum of RSOS [10] and chiral Potts models [11].\nThe gist of this approach can be described in a few words as\nfollows: One keeps on fusing various R--matrices (related to the\nmodel under investigation) throwing in some obvious (and not so\nobvious) symmetries along the way, until one comes back to where\njourney began. The result is the system of functional equations\nfor the transfer matrices. In a sense, one can regard this\nprocedure as a generalization of Zamolodchikov--Karowski\nBootstrap program [12] to determine S-matrices of exactly\nintegrable models.\n\nIn order to apply this technique to our model, let us recall\nthat the tensor product of a spin $j$ representation with a\nsemi--cyclic one is completely reducible. In particular,\nfor $j= \\frac{1}{2}$, we have (see reference [14])\n\\begin{equation}\n( \\frac{1}{2} ) \\otimes (y, \\lambda) = ( y, \\epsilon \\lambda)_+\n+ (y, \\epsilon^{-1} \\lambda)_-\n\\label{44}\n\\end{equation}\n\nIn each subspace one can find the highest weight vector $V^+_0$\nand $V^-_0$ such that\n\n\\begin{equation}\n\\Delta(E) V^{\\pm}_0 = 0 \\;\\;and\\;\\;\\Delta (K) V^\\pm_0 =\n\\epsilon^{\\pm1} \\lambda V^\\pm_0\n\\label{45}\n\\end{equation}\n\nMaking use of (11) we obtain for each sign a semi--cyclic\nrepresentation with the basis $[ V^\\pm_0 , \\cdots, V^\\pm_{N-1} ].$\n\nLet us now define two projection points $u_\\pm$ as follows:\n\n\\begin{equation}\nL^{\\lambda_0} (u_\\mp) V^\\pm_i = 0\n\\label{46}\n\\end{equation}\n\nAt these points the operators $L^{\\lambda_0}$ becomes\nessentially the projector $P^\\pm$ onto corresponding $\\pm$ subspace.\n\n\\begin{equation}\nL^{\\lambda_0} (u_{\\pm}) \\sim P^\\pm;\\;\\;P^+ + P^- = 1\\;\\;and\\;\\;P^+P^-\n= P^- P^+ = 0\n\\label{47}\n\\end{equation}\n\nIt follows then from eqn. (12) that\n\n\\[\nP^- [ L^{\\lambda_0} (u) \\otimes R^{\\lambda_1, \\lambda_0} (u\n- u_-)] = \\cdots [ R^{\\lambda_1, \\lambda_0} (u - u_-) \\otimes\nL^{\\lambda_0} (u)] \\cdots P^-\n\\]\n\n\\noindent\nand\n\n\\begin{equation}\nP^- [ L^{\\lambda_0} (u) \\otimes R^{\\lambda_1, \\lambda_0} (u - u_-)\n] P^+ = 0\n\\label{48}\n\\end{equation}\n\nThe relation (48) above reveals a block--triangular structure of\nthe product\n\n\\[O^{-1} [ L^{\\lambda_0} (u) \\otimes R^{\\lambda_1, \\lambda_0} (u -\nu_-) ] O =\n\\left(\n\\begin{array}{cc}\nP^+ LRP^+ & * \\\\\n0 & P^-LRP^-\n\\end{array}\n\\right)\n\\]\n\nRemarkably, $P^+LRP^+$ and $P^-LRP^-$ turn out to be\n\n\\[\nP^+ L^{\\lambda_0} (u) \\otimes R^{\\lambda_1, \\lambda_0} (u - u_-)\nP^+ =\n\\]\n\n\\begin{equation}\n= L^{\\lambda_0} (u)^{00}_{00} R^{\\epsilon \\lambda_1, \\lambda_0}\n(u - u_- + i \\frac{2\\pi}{N})\n\\label{49}\n\\end{equation}\n\n\\[\nP^- L^{\\lambda_0} (u) \\otimes R^{\\lambda_1, \\lambda_0} (u - u_-)\nP^- =\n\\]\n\\begin{equation}\n= L^{\\lambda_0} (u)^{10}_{10} R^{\\epsilon^{-1} \\lambda_1,\n\\lambda_0} (u - u_- - i \\frac{2\\pi}{N})\n\\label{50}\n\\end{equation}\n\nIn the product of block--triangular matrices the diagonal blocks\nare multiplied independently. Thus, we have for transfer--matrices\n\n\\[\ntr t_{\\lambda_0} (u) T_{\\lambda_0} (u - u_-, \\lambda_1) =\n[L^{\\lambda_0} (u)^{00}_{00}]^L T_{\\lambda_0} (u-u_- + i\n\\frac{2\\pi}{N}, \\epsilon^{\\lambda_1}) +\n\\]\n\n\\begin{equation}\n+ (L^{\\lambda_0} (u)^{10}_{10} )^L T_{\\lambda_0} (u - u_- - i\n\\frac{2\\pi}{N}, \\epsilon^{-1} \\lambda_1)\n\\label{51}\n\\end{equation}\n\n\nThe use of the second projection point $u_+$ leads to the\nsimilar equation. Note, that $T_{\\lambda_0} (u, \\lambda)$\ndepends essentially on two parameters $u$ and $\\lambda$.\nTherefore, another functional relation is needed in order to\ndetermine eigenvalues of $T_{\\lambda_0} (u, \\lambda)$ interms of known\neigenvalues of $tr t_{\\lambda_0} (u)$. This\nadditional relation (along with eigenvalues of $T_{\\lambda_0}\n(u, \\lambda))$ will be presented in the forthcoming publication [13].\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nIt is well established that Type Ia supernovae (SNe Ia) originate from \nexplosive thermonuclear runaways in a C-O white dwarf \\citep[WD;][]{hf60},\nusually attributed to mass accretion from a non-degenerate binary companion\nor a merger\/collision of two WDs. \nThe former is known as the single-degenerate \\citep[e.g.,][]{wi73}\nscenario; the latter as the double-degenerate scenario \\citep[e.g.,][]{it84,web84}.\nAlthough the overall similarities in observed light curves of \\snia\\\nare indicative of the presence of shared underlying core mechanisms,\nthere exists substantial diversity in other observed properties,\nincluding uncertainties in the inferred accretion processes, explosion mechanisms, \nnature of progenitors as well as host galaxy environment. \nAccording to \\citet{liet01}, for example, about one third of nearby \\snia\\ are peculiar. \n\\moon{The cause of this diversity is not well understood and under considerable debate\n\\citep[see][for a recent review on peculiar \\snia]{tau17}.}\n\nAmong these stands out the long-lasting question about the origin of \nthe group of \\snia\\ exhibiting rapid post-peak evolution\nidentified with a large decline rate in light curves. \nThis group of \\snia\\ is called ``91bg-like\" following its archetype SN 1991bg \\citep{fet92,leiet93}, \nwhile the main population are ``Branch Normal.\"\nWhen viewed on the Phillips relation \\citep{phi93,phiet99},\nwhich compares the peak luminosity and the decline rate of \\snia\\\nusing the parameter \\dm15\\ (= Phillips parameter) that \nmeasures the change in $B$-band magnitude during the first 15 days after peak,\nthe 91bg-like rapid decliners appear as\na distinctive group with \\dm15\\ $\\simgt$ 1.6 mag \\citep[e.g.,][]{tauet08, buret14}.\nIt has been reported that a significant portion, i.e., 15--20~\\%, of the entire \n\\snia\\ population belongs to the group of rapid decliners \\citep[e.g.,][]{let11,sriet17}\naccompanied by a low peak luminosity in most cases \\citep[e.g.,][]{howell01,tauet08}.\nWhile normal \\snia\\ typically show double-peaked light curves in the $I$ and\nand near-infrared bands\nwhere the primary first peak precedes that of the $B$ band,\nthe rapid decliners show a single $I$-band peak after they reach \nthe maximum brightness in the $B$ band \\citep[see][and references therein]{tauet08,dhaet17}. \nThis behaviour is known to be related to \\ni56\\ production and\nthe evolution of temperature and opacity in \\snia,\nwith the 91bg-like group being driven by a smaller amount of \\ni56.\nThese faint and rapidly-declining \\snia\\ have mostly \nbeen found in large elliptical galaxies \nand are therefore thought to be associated with old stellar populations\n\\citep{howell01,sulet06,get11}. \n\nThe distinction of this group of rapid decliners from the normal \\snia\\ \nin the Phillips relation, however, is less apparent when the decline rate\nis represented by the color-stretch parameter (= \\sbv) \ndefined as the time span between the $B$-band maximum\nand that of the $B-V$ color normalized by 30 days \\citep{buret14}.\nThe rapid decliners tend to transition to the post-peak decline phase \ndominated by \\co56\\ decay in less than 15 days,\nwhich leads to a discontinuity in the slope of the Phillips relation near\n\\dm15\\ $\\simeq$ 1.6 mag and makes it\ninapplicable for \\dm15\\ $\\simgt$ 1.6 mag.\nThe disappearance of the clear distinction between Branch Normal and  91bg-like \nbased on the \\sbv\\ parameterization of the decline rate\nsuggests that the \\snia\\ in the two groups \nmay in fact be similar kinds with a continuous distribution of observables\nthat appear to be separated by a gap in the number of observed samples between them. \n\nThere is no clear consensus yet on whether or not \nthe rapidly-declining \\snia, or at least part of them, \nindeed have different origins from the rest.\nSome favor different origins for the rapid decliners,\nusually relying on the double-degenerate sub-Chandrasekhar (sub-Ch) mass explosions\nor the deflagrations in rotating WDs \\citep[see][and references therein]{tauet08,dhaet17},\nwhile others have argued that Chandrasekhar-mass explosions can explain at least part of them,\noften associated with delayed detonations \\citep[e.g.,][]{mrbh07, ashet18}.\nBased on the results of extensive radiative transport simulations,\n\\citet{gk18} reported that \n\\snia\\ with a large \\dm15, i.e., $\\simgt$ 1.55 mag, can only be produced by sub-Ch-mass explosions.\nThis is more or less consistent with the slightly \nbroader criterion \\dm15\\ $\\simgt$ 1.4 mag \nobtained by using non-local thermodynamic equilibrium calculations of light curves \\citep{bet17}.\nIn contrast, \\citet{het17} reported that a uniform explosion based on Chandrasekhar-mass spherical delayed\ndetonations can explain most of the observed light curves of \\snia\\\nwithout the need for a separate mechanism for (extreme) rapid decliners.\n\nA key clue to understanding the origin of the division between\nthe two groups of Branch Normal and 91bg-like \\snia\\ lies \nin the nature of the so-called transitional \\snia\\ straddling them. \nThe identification of a statistically significant number of \nsuch transitional SNe with a continuous distribution of observed properties\ncould support the common-origin hypothesis.\nThe light curves of the transitional \\snia\\ are usually featured with a large Phillips parameter\nand a $B$-band peak between two $I$-band peaks.\nTheir prototypical case is SN~1986G \\citep{phiet87} observed \nwith \\dm15\\ $\\simeq$ 1.81 mag, \nwhich is substantially larger than \\dm15\\ $\\sim$ 1.1 mag of Branch Normal,\nand a low explosion energy \\citep{phiet87,ashet16},\nshowing a very rapid decay in post-peak evolution.\nSpectroscopically, however, it appears to be very similar to normal \\snia,\nthereby revealing its intermediate nature.\nThe Phillips parameter of the transitional types lies roughly within the range of\n\\dm15\\ $\\simeq$ 1.5--1.8 mag \\citep{priet06, ashet16, sriet17, galet18}.\nHowever, transitional types have also been identified in \\snia\\ beyond this range,\nwhile some \\snia\\ within the range have been observed with no clear signs \nof intermediate properties \\citep[e.g.,][]{tauet08,hsiet15}. \nAccording to \\citet{tauet08}, \\snia\\ with \\dm15\\ in the range of 1.75--1.85 mag \nshow an inconspicuous secondary $I$-band peak and can be considered as the intermediate \ngroup between Branch Normal and 91bg-like, \nwhereas those in the range of \\dm15\\ = 1.5--1.75 mag \nfeatured with a double-peaked $I$-band light curve, \na relatively small luminosity and a rapid decay\nare similar to normal types.\n\\citet{hsiet15} suggested that \\rapd\\ \\snia\\ are \nof transitional nature if they show two long-waveband (i.e., $iYJHK$) peaks in the light curves\nwith the first and primary peak preceding the $B$-band peak. \nThe 10 transitional \\snia\\ compiled in their work show\n\\dm15\\ $\\simeq$ 1.30--1.80 mag and \\sbv\\  $\\simeq$ 0.46--0.86\nwith average values of 1.62 mag (\\dm15) and 0.70 (\\sbv).\nThese transitional \\snia\\ have been observed mostly \nin non-star-forming galaxies \\citep{ashet16, mrbh07},\nsimilar to 91bg-like events (see above).\n\nIn this paper, we present multi-color, high-cadence observations of \nthe \\sni\\ \\kspn\\ (or \\iaun) which appears to be a \\rapd\\ \\sni\\ \nof transitional nature that is more similar to Branch Normal than 91bg-like,\nproviding a rare opportunity to investigate\nthe early photometric evolution of rapid decliners.\nIn addition, the absence of any apparent host galaxy for this SN\nin our deep stack images reaching $\\mu_{BVI}$ $\\sim$ 28 \\mpas\\ implies that \nsuch \\snia\\ can occur in host galaxy environments that are quite different \nfrom what have been previously reported.\nIn \\S\\ref{sec:obs} below we provide the details of our discovery and monitoring observations of \\kspn,\nfollowed by the analyses of the observed light curves and color evolution \nas well as template fitting in \\S\\ref{sec:lc}.\nWe compare the observations and model predictions in \\S\\ref{sec:bol} and \\S\\ref{sec:pro}:\ncomparisons of the bolometric light curve with model predictions \nbased on different types of \\ni56\\ distribution (\\S\\ref{sec:bol}) and\nthe observed early light curves with model light curves\nexpected from the interactions between ejecta and companion in \\snia\\ (\\S\\ref{sec:pro}).\nWe show that there is no apparent host galaxy of \\kspn\\ in \\S\\ref{sec:host}\nand provide the summary and conclusion in \\S\\ref{sec:sum}. \n\n\\section{Observations}\\label{sec:obs}\n\nWe conducted high-cadence, multi-color ($BVI$) monitoring observations of \na 2\\degr\\ $\\times$ 2\\degr\\ field\ncontaining the galaxy NGC~300 in the direction of the South Galactic pole \nusing the Korea Microlensing Telescope Network \\citep[KMTNet,][]{ket16} as part of its\nKMTNet Supernova Program \\citep[KSP,][]{met16,afset19}.\nThe initial phase of the monitoring started in 2015 July during the commissioning \ntest period of the KMTNet and continued until 2017 August 9.\nThe KMTNet operates three wide-field 1.6-m telescopes located in \nthe Cerro-Tololo Inter-American Observatory (CTIO; Chile), \nthe South African Astronomical Observatory (SAAO) and the Siding Spring Observatory (SSO; Australia).\nAll of the three telescopes are equipped with a CCD camera of \n2\\degr\\ $\\times$ 2\\degr\\ field-of-view (FoV) at 0\\farcs4 pixel sampling with Johnson-Cousins $BVRI$ filters \\citep{ket16}.\nWe obtained about 2460 images of the field with 60-s exposure time for \neach of the $BVI$ bands,\nand the mean cadence of our observations within the monitoring period\nwas approximately 3.5 hours for each band.\nThe typical limiting magnitude for a point source detected with a signal-to-noise (S\/N) ratio\ngreater than 2 in the individual images \nwas $\\sim$ 22 mag\nwhen seeing was better than 2\\arcsec.\n\nA new point source was detected in our $B$-band image obtained \nwith the SAAO telescope at 21.56 h on 2015 September 24 (UT), \nor MJD 57289.89825,\nwith a magnitude of 21.29 $\\pm$ 0.37 mag at the coordinate\n(RA, decl.) = ($\\rm 00^h57^m03.19^s, -37\\degr02\\arcmin23\\farcs6$) (J2000).\nThe source was subsequently detected in the $V$ band two minutes later with 21.45 $\\pm$ 0.33 mag\nand in the $I$ band 28 minutes later with 21.59 $\\pm$ 0.32 mag, \nwhile it was not detected in a $B$-band image obtained 7.18 hours before the first detection.\nWe name this source \\kspn.\nIt remained above the detection limit over the ensuing period of four months at the same location,\nreaching observed peak magnitudes of 18.59 $\\pm$ 0.02 mag ($B$), \n18.49 $\\pm$ 0.02 mag ($V$) and 18.91 $\\pm$ 0.05 ($I$) mag over $\\sim$ 18 days.\nFigure~\\ref{fig:det} presents images of the field centered on the location of \\kspn.\nIn the figure, (a) is a deep $I$-band image, \nwhich reaches a sensitivity limit $\\mu_{I}$ $\\simeq$ 27.8 \\mpas, \ncreated by stacking 1167 individual images obtained either \nbefore the first detection of the source or after its disappearance. \n\\moon{The image stacking is made using the SWARP package \\citep{bet02}.\nEach individual frame is subtracted by its own background estimated with the\nbackground mesh size of 512 pixels, and then resampled and median combined\nwith other frames to create the final stack image.\nOnly the individual frames whose seeing are better than 2\\arcsec\\\nare used, which can help mitigate the effects of the confusion limit \nin image stacking \\citep[e.g.][]{ashcet18}.\nThere exists no underlying source at the location of \\kspn\\ in the stack image.}\nThe three other figures (b)--(d) are the individual 60-s $B$-band images for\nthe last non-detection exposure obtained 7.18 hours prior to the first detection (b),\nthe first detection (c), \nand the image from MJD 57304.04163 when the source reached the peak brightness (d). \n\nThe photometric calibration is carried out using more than 10 nearby \nstandard reference stars within 15\\arcmin\\ distance from \\kspn\\ available in the AAVSO Photometric All-Sky \nSurvey\\footnote{The AAVSO Photometric All-Sky Survey: Data Release 9, https:\/\/www.aavso.org\/apass} database. \nOnly the reference stars whose apparent magnitudes are in the range of 15--16 mag are used \nto secure high S\/N ratios in their flux measurement while avoiding CCD saturation and non-linearity effects.\nA local point spread function (PSF), which is obtained by fitting \na Moffat function \\citep{m69, tet01} and sky background emission \nto the reference stars, is fitted for the measurements of fluxes of \\kspn\\ and the reference stars. \nThe AAVSO photometric system consists of the standard Johnson $BV$ and the Sloan $i$ band, i.e., $BVi$.\nThe calibration of the KMTNet $B$-band data against \nthe Johnson $B$-band magnitudes of the AAVSO system\nrequires a color correction given by\n$\\Delta B$ $\\simeq$ 0.27 ($B-V$) + offset \\citep{pet17, pet19}, \nwhere $\\Delta B$ is the $B$-band magnitude differences\nbetween the magnitudes obtained in KMTNet images before color correction\nand the standard magnitudes of AAVSO reference stars in the database. \nThe same $B$-band color dependence in our data is identified, and\nthe $B$-band magnitudes of the AAVSO reference stars used in our photometric calibration\nare corrected with their known $B-V$ colors.\nThe final $B$-band magnitudes of \\kspn\\ are obtained after applying $S$-correction for SNe\nby following the procedure described in \\citet{stret02}\nusing template spectra from the \\snp\\ package (see \\S\\ref{subsec:temp}). \nNo such color dependence has been identified in the KMTNet $V$ and $I$ bands\nwhen calibrated against the AAVSO $V$ and $i$ bands, respectively \\citep{pet17, pet19},\nwhich is also confirmed in the data for \\kspn.\nThus, the photometric calibration of our KMTNet $VI$-band data are made \nagainst the AAVSO $Vi$ bands without any color correction. \nAs a result, the $BV$-band magnitudes of \\kspn\\ presented in this paper are\nin the Vega magnitude system, while its $I$-band magnitudes are in the AB system.\nTable~\\ref{tab:mag} contains a sample of the observed magnitudes of \\kspn.\n\nAccording to the dust map of \\citet{sfd98}, the total Galactic reddening in\nthe direction of KSP-OT-201509b is E(\\bv) $\\simeq$ 0.013 mag. \nUsing the updated $R_V$ = 3.1 dust model of \\citet{sf11},\nwe find corresponding extinction of 0.046, 0.034 and 0.021 mag in\nthe $B$, $V$ and $i$ band, respectively, \nThe small extinction is compatible with the location of \\kspn\\ near the Galactic pole.\nSince no host galaxy is identified for the source (see below and \\S\\ref{sec:host}), \nonly this small Galactic extinction is taken into account in our extinction correction.\n\nOne notable feature in Figure~\\ref{fig:det}(a) is the absence of \nany underlying host galaxy candidate for \\kspn\\\nat its location or in the immediate vicinity of the source in our deep $I$-band image.\nThe absence is also confirmed in the $B$ and $V$ bands.\nThe most conspicuously extended nearby object of \\kspn\\ is\nlocated at (RA, decl.) $\\simeq$ ($\\rm 00^h57^m01.23^s, -37\\degr02\\arcmin37\\farcs5$) (J2000),\n$\\sim$ 27\\arcsec\\ away in the southwest. \nWe call this object \\galgg\\ hereafter as we denote in the figure.\nIn order to measure the redshift and understand the nature of \\galgg, we conducted \nspectroscopic observations of the source using\nthe Low Dispersion Survey Spectrograph 3 (LDSS3)\\footnote{http:\/\/www.lco.cl\/telescopes-information\/magellan\/instruments\/ldss-3}\non the 6.5-m Magellan Clay telescope on 2016 August 11.\nWe used the VHP-All grism coupled with 1\\arcsec\\ slit for dispersion\nand obtained a single 600~s exposure.\nSee \\S\\ref{sec:host} for the spectrum of \\galgg\\ and its analysis.\n\n\\medskip\n\\section{Light Curves and Classification of \\kspn\\ as a Rapidly-Declining Transitional \\sni}\\label{sec:lc}\n\n\n\\subsection{Light Curves and Epoch of the First Light}\\label{subsec:lc}\n\nFigure \\ref{fig:lc} shows $BVI$-band light curves of \\kspn\\ \nobtained with the KMTNet for about two months.\nOver approximately the first two week, \nthe source gradually ascends to its peak brightness more than three \nmagnitudes brighter than that of the first detection.\nThe observed peak brightnesses are 18.59, 18.49 and 18.91 mag for\n$B$, $V$, and $I$ band, respectively.\n(See Table~\\ref{tab:par} for the observed and estimated parameters of \\kspn.)\nThere exist clear band-dependent differences \nin the manner of how post-peak decay evolves in Figure~\\ref{fig:lc}.\nThe $B$-band decay is much faster than those of the other bands, \nwhile the $I$-band light curve is featured with a secondary peak about two \nweeks after the first and primary peak, which precedes the $B$-band peak.\nWe estimate the average strength of the secondary $I$-band peak \nwithin 20--40 days after the epoch of the $B$-band peak \nto be $<I>_{\\rm 20-40}$ = 0.309 $\\pm$ 0.002\nwith respect to the maximum brightness of the first and primary\npeak of the $I$ band \\citep[][]{kriet01}. \nThe overall evolution of the observed light curves of \\kspn\\ -- \nwhich includes \nthe initial ascent in brightness,\nthe presence of a secondary peak in the $I$ band,\nthe absence of any apparent post-peak plateau or linear phase, and\nthe slow decay -- \nidentifies \\kspn\\ as a \\sni\\ powered by radioactive decay \nof \\ni56\\ and \\co56\\ \\citep[e.g.,][]{het96}. \nWe compare in Figure~\\ref{fig:comp} the $I$-band light curve (black) of \\kspn\\\nwith those of four well-sampled \\snia: \nSN2011fe (red), SN1994D (blue), SN2005ke (cyan) and SN2005bl (green).\nThe first two are Branch Normal with \\dm15\\ $\\simeq$ 1.1 and 1.4 mag, respectively, whereas \nthe last two are 91bg-like with more rapid post-peak decline rates \n\\dm15\\ $\\simeq$ 1.7 (SN2005ke) and 1.9 (SN2005bl) mag.\nThe post-peak decline rate of \\kspn\\ in Figure~\\ref{fig:comp}\nappears to be intermediate between the two groups. \n\nFigure~\\ref{fig:epo} presents the observed early light curves of \\kspn\\ \nprior to 8 days before the peak, normalized by its peak brightness.\nPower laws have been adopted to describe the evolution of early light curves of \\snia\\\nwhen intensities are substantially \nlower \\citep[e.g., $<$ 40~\\%, see][]{ollet15} than\nthe peak intensity since the brightness of a \\ni56-powered expanding homologous sphere\nmediated by photon diffussion process is expected to be $\\propto$ $t^2$ where $t$ is time\n\\citep[e.g.,][]{nuget11}.\nWe fit the early light curves of \\kspn\\ during the first $\\sim$ 10 days \nof its detection in Figure~\\ref{fig:epo} \nwith a power law $F(t)$ $\\propto$ $(t-t_0)^{\\alpha}$,\nwhere $F$ and $t_0$ represent the observed brightness \nand the epoch of the first light, respectively.\nThe best-fit power-law indices are \n$\\alpha$ = 2.0 $\\pm$ 0.2 ($B$), 1.9 $\\pm$ 0.2 ($V$) \nand 2.1 $\\pm$ 0.2 ($I$) \nand the epoch of first light is \\t0\\ =  --18.4 $\\pm$ 0.6 days \nsince the peak in the observer frame, \nor --17.2 $\\pm$ 0.5 days in the rest frame.\n(See \\S\\ref{subsec:temp} for the estimation of the epoch of the peak brightness \nand the redshift of the source.)\nThe fitted power-law indices are $\\simeq$ 2 as expected for a homologous expansion, \nsimilar to what have been found\nin other \\snia\\ \\citep[e.g.,][]{nuget11,ollet15,dimitriadis19}.\n\\moon{ We also estimate the epoch of the first light using\na Gaussian process extrapolation which can provide \na less model-dependent inspection of the epoch.\nThe Gaussian process extrapolation-based epoch of the first light \nis $-$16.4 $\\pm$ 0.5 days (rest frame) for the $BVI$ light curves,\nwhich is slightly smaller but consistent with $-$17.2 $\\pm$ 0.5 days  \nfrom our power-law fitting. We adopt $-$17.2 $\\pm$ 0.5 days as the\nepoch of the explosion of \\kspn\\ in its rest frame.}\n\n\n\n\\subsection{Color Evolution}\\label{subsec:color}\n\nFigure~\\ref{fig:color} shows the evolution of the $B-V$ and $V-I$ colors of\n\\kspn\\ aligned with that of the $I$-band light curve.\nThe three vertical dotted lines mark three color epochs over which\nthe colors show a notable phase transition in their evolution.\nThe first and third color epochs are --4.5 and +18 days, respectively, from\nthe epoch of the $B$-band peak brightness estimated using SN template fitting of the \nobserved $BVI$-band light curves (see below for the details).\nThese two color epochs roughly correspond to those of the primary and secondary $I$-band peaks, respectively. \nThe second color epoch is +7 days which is close to the midpoint\nbetween the first and third epochs,\nnear the onset of the $I$-band secondary rise.\n\nWe summarize the evolutions of the \\bv\\ and \\vi\\ colors,\nwhich are largely synchronous with that of the $I$ band, as follow.\n\\begin{itemize}\n\\itemsep0em \n\\item  Prior to the first color epoch of --4.5 days,\nthe \\bv\\ color starts at $\\simeq$ 0.2 mag around --14 days,\nor $\\sim$ 3.3 days in the rest frame since the epoch of first light (Table~\\ref{tab:par}), \nand slowly becomes bluer to $\\simeq$ 0.0 mag in about 10 days\nnear the first color epoch. \nThe blueward evolution of the \\bv\\ during this early phase of \\kspn\\ \nappears to be consistent with that of the early red population of \\snia,\nwhich comprises predominantly Branch Normal types \\citep{stret18}.\nIt is thought to be due mainly to the heating  from \\ni56\\ radioactive decay as \nthe deeper layer of the ejecta is revealed \\citep[see][]{het17}.\nIndeed, we show in \\S\\ref{sec:bol} that the observed early light curves of \\kspn\\ \ncan be explained by a shallow \\ni56\\ distribution  \\citep[e.g.,][]{pm16}\nleading to an early increase in color temperature.\nThe \\vi\\ color of \\kspn\\ during this phase, \non the other hand, shows a redward evolution\nfrom approximately --0.5 to --0.1 mag within the first two days \nbefore it stalls relatively flat until the first color epoch.\nSNe in general have rarely been observed with \\vi\\ colors at these early epochs, \nand the origin of the early redward evolution of \\kspn\\ is unclear. \nIt is conceivable, however, that the evolution of spectral features \n-- e.g., Ca II features in the $I$ band \\citep[][see Figure~1 therein]{paret12} --\nare responsible for it.\n\\item Across the first color epoch of --4.5 days, \nwhen the $I$-band light curve nears its primary peak, \nboth colors change the direction of their evolution \nin that \\bv\\ becomes redder while \\vi\\ becomes bluer.\nWe attribute this redward evolution in \\bv\\ to the absorption \nby iron peak elements \\citep[e.g.][]{het17},\nwhereas the blueward evolution in \\vi\\ \nto temperature increase.\n\\item The \\bv\\ color evolves redward relatively monotonically between\nthe first (i.e., --4.5 days) and third (i.e., +18 days) color epochs,\nwhich is largely equivalent to the period between the two $I$-band peaks,\nby $\\simeq$ 1.2 mag,\nwhereas the \\vi\\ color changes its direction of evolution again around the \nsecond color epoch (i.e., +7 days) from blueward to redward until\nit reaches the third color epoch. \nThis is due to the recombination process of Fe III leading to \na rebrightening in the $I$ band by redistribution of blue\/UV radiation\n\\citep[e.g.,][see also \\S\\ref{sec:intro}]{kas06}.\n\\moon{We note that the Fe III recombination also plays a critical role\nin determining the color evolution of \\snia\\ in the near-infrared wavebands\nwith respect to their light curves \\citep{dhaet15}, similar to what\nwe report here.}\n\n\\item After the third color epoch, both colors \ngradually evolve blueward as the SN enters the Lira law phase known\nto have a linear blueward evolution for an extended period \\citep{phiet99}.\n\\item The \\bv\\ and \\vi\\ colors at the peak epoch\nin Figure~\\ref{fig:color} are 0.056 and --0.34 mag, respectively.\nAccording to \\citet{tauet08},\nthe observed \\bv\\ color of \\kspn\\ at the peak epoch \nis significantly bluer than \nthe 0.4--0.7 mag range known for 91bg-like,\nwhile it is consistent with the value\nexpected for a \\sni\\ with \\dm15\\ $\\simlt$ 1.75 mag.\n\\item \\snia\\ with \\dm15\\ in the range of 1.5--1.75 mag\nare known to show an initial blueward evolution between --10 and 10 days \nin \\vi\\ followed by a rapid redward evolution \\citep{tauet08} as observed in \\kspn,\nwhile this initial blueward evolution is absent in 91bg-like.\nIt is quite noteworthy, however, that the \\vi\\ color evolution of \\kspn\\ \nin Figure~\\ref{fig:color} also reveals that the initial blueward evolution \n(between --10 and 10 days) is in fact preceded by an earlier \nredward evolution at $<$ --10 days as explained above.\nA similar \\vi\\ color pattern  at comparably early epochs has been observed \nin SN~1994D which is a normal but relatively over-luminous  \\sni\\ \\citep{patet96}.\n\\item Overall, the observed color evolutions of \\kspn\\ are \ncompatible with what can be expected from a \\sni\\ \nwith \\dm15\\ = 1.62 mag as measured for the source (see \\S\\ref{subsec:temp})\nand appear to be more similar to Branch Normal than 91bg-like.\nAlso, the similarity of its color evolutions to those observed\nin other \\snia\\ indicates that there is an insignificant extinction \nby its potential host galaxy as we assumed (\\S\\ref{sec:obs}).\n\\end{itemize}\n\n\\subsection{Template Fitting and Parameters of \\kspn}\\label{subsec:temp}\n\nIn order to estimate key physical parameters of \\kspn\\ as a \\sni, \nwe conduct template fitting of its observed light curves  (Figure~\\ref{fig:lc})  \nusing two \\sni\\ light curve templates from the SN template fitting package \\snp\\ \\citep[v181212;][]{buret11},\none for Branch Normal and the other for 91bg-like.\nBecause no spectroscopic observations were made for \\kspn\\ \n(due to its detection during the commissioning test period of the KMTNet)\nand its host galaxy is unknown (see \\S\\ref{sec:host}),\nwe estimate its distance using an iterative template model fitting technique.\nFor this, we adopt a set of 100 trial redshifts in the range  $z$ = 0.03--0.1\nfor potential redshifts of the source.\nThis corresponds to the distance modulus (DM) of $\\simeq$ 35.6--38.6 mag\nbased on the cosmological model of \\citet{rieet16} with parameters \n$H_0$ $\\simeq$ 73.24 \\kms\\ Mpc$^{-1}$,  \n$\\Omega_{\\rm M}$ $\\simeq$ 0.27 and $\\Omega_{\\rm \\Lambda}$ $\\simeq$ 0.73.\n\\moon{Note that these cosmological parameters are more recently updated values \nthat are slightly different from those adopted in the \\snp\\ package internally, \nwhich are  $H_0$ $\\simeq$ 72 \\kms\\ Mpc$^{-1}$, $\\Omega_{\\rm M}$ $\\simeq$ 0.28 \nand $\\Omega_{\\rm \\Lambda}$ $\\simeq$ 0.73 \\citep{speet07}.} \nGiven $B$ = 18.59 $\\pm$ 0.02 peak apparent magnitude of \\kspn\\ (Table~\\ref{tab:par}),\nthis redshift range ensures us to investigate the full range of \npeak absolute magnitudes typically displayed by \\snia,\nwhich is between --17 mag and --20 mag as shown in Figure~\\ref{fig:phil},\nfor conceivable peculiar motion of the source (see below).\nWe use \\snp\\ \nto conduct SN template fitting and $K$-corrections with the input redshifts, assuming no host galaxy extinction.\n\nFigures~\\ref{fig:LCtemplate} and \\ref{fig:LCzfitVi} show the results\nof our template fitting to the light curves of \\kspn. \nIn Figure~\\ref{fig:LCtemplate} we compare the observed $V$-band (top panel)\nand $I$-band (bottom panel) light curves of the source\nwith the best-fit template light curves for Branch Normal (solid line) and 91bg-like (dashed line).\nWe exclude the $B$-band light curve in this initial template fitting\nstage to avoid uncertainties involved in the \n$S$-correction process (see \\S\\ref{sec:obs}).\nWhile the best-fit $V$-band light curve from the Branch Normal template \nin the figure appears to give a good match to the observed light curve, \nthat from the 91bg-like template slightly overpredicts the decline rate\nimmediately after the peak. \nThis discrepancy is more obvious in the $I$ band where \nthe double-peaked light curve of \\kspn\\ is apparently \nincompatible with the best-fit 91bg-like template,\nshowing that the source is more similar to Branch Normal than 91bg-like.\n\nFigure~\\ref{fig:LCzfitVi} presents the distribution of the fitted \nparameters from the template\nfitting as a function of the trial redshifts (bottom abscissa),\nor corresponding cosmological DM (top abscissa).\nThe top panel shows $\\Delta$(DM) (left ordinate), \nwhich is the offset between the \\snp-fitted DM at a given redshift and the cosmological DM from the top abscissa, \nrepresenting the peculiar motion (as calculated in the right ordinate) of \\kspn\\\nif the fitted DM and assumed redshift are indeed the real values for the source.\nAs in the figure, no peculiar motion of \\kspn\\ is required if the source is \nlocated at $z$ = 0.072 for Branch Normal (solid line) or 0.077 for 91bg-like (dashed line).\n\\moon{At redshifts smaller than 0.06 or larger than 0.08, \nin order for the input redshift and the cosmological redshift \nfrom the distance modulus obtained in the SNooPy fitting to be comparable,\nthe required peculiar motions\nbecome significantly larger, i.e., $>$ 1000 \\kms, than what can be expected from \\kspn\\ (see below)\nfor the Branch Normal template fitting. This also applies to the 91bg-like template fitting,\nbut at slightly larger redshift values.}\n\nIn the bottom panel of Figure~\\ref{fig:LCzfitVi},\nthe distribution of the best-fit \\csp\\ \\sbv\\ (left ordinate) \nincreases along the trial redshifts as the quality of the fitting becomes poorer.\nThe growing incompatibility between the template and observed light curves  at a higher redshift,\ncaused mainly by the secondary $I$-band peak,\nresults in larger best-fit values of \\csp\\ for higher trial redshifts. \nThe Phillips parameters (\\dm15, right ordinate) in the panel \nare obtained using the relation between \\sbv\\ and \\dm15\\ in \\citet{buret14}.\nAt $z$ = 0.072, the best-fit \\sbv\\ is 0.75 (or \\dm15\\ = 1.54 mag) for Branch Normal,\nwhile it is 0.82 (or \\dm15\\ = 1.39 mag) for 91bg-like at $z$ = 0.077.\nThe \\chisqr\\ values of the best-fit templates at these redshifts are\n$\\simeq$ 2 for Branch Normal ($z$ = 0.072) \nand $\\simeq$ 3 for 91bg-like ($z$ = 0.077).\nOverall, the best-fit Branch Normal templates give \nsystematically, although slightly,\nsmaller \\chisqr\\ values of \\simlt\\ 3\nfor all the trial redshifts.\nAt higher redshifts $z$ \\simgt\\ 0.08, however, \n\\chisqr\\ values of the best-fit 91bg-like templates\nbecome increasingly greater than 3.\n\nBased on the results shown in Figures~\\ref{fig:LCtemplate} \nand \\ref{fig:LCzfitVi} from the \\snp\\ template fitting,\nwe estimate the redshift of \\kspn\\ to be $z$ = 0.072 $\\pm$ 0.003 and \nconclude that the source is closer to Branch Normal than 91bg-like  as follows. \nFirst of all, the comparison between the observed and best-fit templates (Figure~\\ref{fig:LCtemplate}),\nespecially the $I$-band comparison (bottom panel),\ngives clear preference for Branch Normal for \\kspn.\nAlso, the best-fit color stretch parameter \\sbv\\ $\\simeq$ 0.82 for \nthe 91bg-like template is unacceptably larger than those found in 91bg-like SNe,\nwhile the best-fit \\sbv\\ $\\simeq$ 0.75 for Branch Normal is largely acceptable, \nalthough somewhat small, for a Branch Normal SN.\nThis shows that the 91bg-like template is apparently incompatible with \nthe observed light curves of \\kspn\\ at an admissible redshift range. \nThe estimated redshift of \\kspn\\ is largely derived from the conceivable range \nof reasonable proper motions for its progenitor. \nAs explained in \\S\\ref{sec:host}, \nit is highly likely that the host of \\kspn\\ is an unidentified \ndwarf galaxy whose brightness is fainter than $\\sim$ 25 mag in $BVI$.\nIn the local environment around the Milky Way, the peculiar velocities of dwarf galaxies are at the level of 200 \\kms\\ \\citep{let07}, \nwith a few occasional outliers showing higher velocities\nin special environments \\citep[e.g.,][]{dgc01}. \nFor a galaxy like the Milky Way, $\\sim$ 500 \\kms\\ is the escape velocity,\nwhile rare objects with faster velocities of up to $\\sim$ 1000 \\kms\\ are considered \nhypervelocity stars \\citep{brown15, beei17}.\n\\moon{Adopting 500 \\kms\\ as a conservative, or upper-limit, assessment of \nthe potential peculiar motion of \\kspn,\nwe obtain $\\Delta z$ = 0.003 as the uncertainty of the best-fit redshift \nfrom the Branch Normal template fitting \nafter taking the uncertainty in the DM measurement\nfrom the \\snp\\ fitting, which includes\nthe uncertainty contributions  from the \\snp\\ template calibration\nas well as from adopting the more recently updated cosmological\nparameters than \\snp\\ (see above),\nand the uncertainty corresponding to 500 \\kms\\ peculiar motion into account together.\nThe $\\sim$ 4\\% level of the photometric redshift uncertainty is \ncomparable with what have been found in some other \nphotometric studies of \\snia\\ \\citep[e.g.,][]{palet10}.}\nThis gives $z$ = 0.072 $\\pm$ 0.003 and DM = 37.47 $\\pm$ 0.10 mag, \nor luminosity distance of 311.9 Mpc,\nfor the source (Figure~\\ref{fig:LCzfitVi}).\nThe small difference between the best-fit redshifts of Branch Normal and 91bg-like\nindicates that any potential systematic uncertainties resulting from applying the\nBranch Normal template to KSP-OT-201509b, which shows transitional nature (\\S\\ref{subsec:tran}), \nfor its redshift determination is small.\n\nWe now conduct $S$-correction of the $B$-band light curve \nusing this redshift and the spectral time series of the best-fit \nBranch Normal template shown in Figure~\\ref{fig:LCtemplate}\nas identified in the $VI$-band light curve analyses above.\n({\\it Note that the $B$-band light curve in Figure~\\ref{fig:lc} is the one \nobtained after $S$-correction.})\nWe then use SNooPy with the entire $S$-corrected $BVI$ light curves \nto obtain the best-fit Branch Normal template for \\kspn\\\n(Figure~\\ref{fig:lc}, black lines) \nfrom which we derive $K$-corrections. \nWe adopt the epoch of the $B$-band peak brightness \\tp\\ = 57305.08 $\\pm$ 0.03 (MJD) of the best-fit template,\nwhich is 15.18 $\\pm$ 0.03 days after the epoch of the first detection\nand $\\sim$18.4 days after the epoch of first light, \nas the peak epoch of \\kspn\\ from the template.\nUsing the $S$- and $K$-corrected $BVI$ curves of the SN in the rest frame, \nwe estimate absolute peak magnitudes  of \n--18.94 $\\pm$ 0.01 ($B$), --18.93 $\\pm$ 0.01($V$) and --18.38 $\\pm$ 0.01 ($I$) mag \nwith their respective epochs. \nWe finally estimate the \\csp\\ and \\pp\\ of \\kspn\\ to be \n\\sbv\\ = 0.54 $\\pm$ 0.05 using the \n$B$-band light curve and \n$B-V$ color evolution of the source as defined in \\citet[][]{buret14} and\n\\dm15\\ = 1.62 $\\pm$ 0.03 mag by conducting polynomial fitting \nof the $B$-band light curve in the rest frame of $z$ = 0.072.\nThese final \\sbv\\ and \\dm15\\ values are different from those obtained \nfrom the $VI$-band template fitting above\n(i.e., \\sbv\\ $\\simeq$ 0.75 and \\dm15\\ $\\simeq$ 1.54 mag for Branch Normal;\n\\sbv\\ $\\simeq$ 0.82 and \\dm15\\ $\\simeq$ 1.39 mag for 91bg-like),\nand we attribute the discrepancy to the transitional nature of \\kspn\\ \n(see below)\nsince neither template is an intrinsically good match to the observed light curves of the source.\nSee Table~\\ref{tab:par} for the compilation of the physical parameters of \\kspn.\n\n\\moon{As described above, the redshift of \\kspn\\ is estimated using its observed peak brightness\nby allowing reasonable peculiar motion of the source in the frame of the \\snp\\ template fitting, rather than purely based on the goodness of the light-curve fitting.\nThe minimum \\chisqr\\ values of the \\snp\\ template fittings change only slightly, \ni.e., mostly between 2 and 3,\nover the trial redshift range of 0.01--0.1, making the parameter \ninadequate to discern the redshift of the source alone.\nIn order to examine the validity of our redshift estimation from the \\snp-based analyses,\nwe apply two other \\snia\\ light curve fitting packages to the observed light curves of \\kspn.\nWe first use the SALT2 fitting package \\citep{guyet07} with the same trial redshift range\nof 0.01--0.1, which results in only a slight variation, i.e., between 4 and 5,  of the minimum \\chisqr\\ values,\nas we find in the \\snp\\ fitting.\nBoth the best-fit redshifts from the SALT2 fitting of the $B$- and $V$-band \nlight curves determined by the minimum \\chisqr\\ value are consistently $\\sim$ 0.07, \nsimilar to what we obtain for \\kspn\\ from the \\snp-based analyses above, \nwhile it is $\\sim$ 0.02 for the $I$ band.\nThis significant discrepancy between the $BV$ and $I$ band might be due to \nincompatibility between the SALT2 package and the transitional nature of \\kspn,\nwhich is expected to affect the $I$-band fitting more than the other bands,\nalthough it is important to note that these best-fit redshifts are \npoorly constrained given the small variations in the minimum \\chisqr\\ values.\nFigure~\\ref{fig:SLAT2}, which shows the distribution of the best-fit shape parameter (= \\x1)\nfrom our SALT2 fitting of the \\kspn\\ light curves over the trial redshift range of 0.01--0.1,\nprovides a consistency check between the \\snp- and SALT2-based fitting results.\nIn the figure, the two shaded areas represent the ranges of the redshift and \\x1\\ \nexpected from the \\snp\\ fitting for \\kspn: the vertically shaded area is $z$ = 0.072 $\\pm$ 0.003\nthat we estimate as the redshift of the source,\nwhile the horizontally shaded area is \\x1\\ = $-$1.89 $\\pm$ 0.05 converted from\nthe fitted \\sbv\\ values in Figure~\\ref{fig:LCzfitVi} (bottom panel) for the redshift\nusing the known relationship between \\x1\\ and \\sbv\\ in \\citet{buret14}.\nThe SALT2-based \\x1\\ parameter intersects with the two shaded areas, \nshowing that the results of the \\snp\\ fitting are consistent with \nthose of the SALT2 fitting on the known relation between \\sbv\\ and \\x1.\nIn addition to the SALT2 fitting, we also apply the SiFTO SN fitting package \\citep{conet08} \nto the observed light curves of \\kspn\\ over the trial redshift range of 0.01--0.1. \nAlthough the best-fit redshift from the SiFTO fitting is poorly constrained again \ndue to the small variation in the minimum \\chisqr\\ values, we obtain $\\sim$ 0.07 \nfor the best-fit redshift, compatible with \\snp\\ and SALT2 fitting results.}\n\n\n\\subsection{Transitional Nature of \\kspn}\\label{subsec:tran}\n\nThe observed characteristics and measured parameters \nof \\kspn\\ identify the source as a \\rapd\\ \\sni\\ of transitional nature\nthat is more similar to Branch Normal than 91bg-like as follows.\nThe \\rapd\\ nature of the source is easily confirmed in \nFigure~\\ref{fig:phil} (left panel) where we compare the $B$-band peak \nabsolute magnitude and \\sbv\\ of \\kspn\\ (filled yellow star)\nwith those of the group of \\snia\\ (crosses) from \\citet{buret18}.\nThe number of SNe gradually decreases as the \\sbv\\ \nvalues decrease (or the peak $B$-band luminosities become smaller) in the figure.\nAfter the location of \\kspn\\ at \\sbv\\ $\\simeq$ 0.54,\nthe gradual distribution of the SNe identified with larger \\sbv\\ values\ndisappears, and there exists only a small number of SNe as\n\\sbv\\ decreases in this range.\nThe right panel of Figure~\\ref{fig:phil}, \nwhere \\dm15\\ is used instead of \\sbv\\ for the post-peak decline rate\nfor the same \\snia\\ in the left panel,\nshows the mean \\dm15\\ values of the \nfour major subtypes of \\snia\\ \\citep[see][]{pfp14}:\n91T-like (filled red circle),\nCore-Normal (filled blue circle), Broad-Line (filled orange circle), \nand 91bg-like (filled green circle) \nas well as that (\\dm15\\ $\\simeq$ 1.62 mag) of \\kspn\\ (yellow star).\nNote that \\snia\\ in the 91T-like group are slowly-evolving and overluminous,\nwhereas Core-Normal and Broad-Line constitute Branch Normal. \n\\kspn\\ clearly appears to be associated with a small number of \\snia\\ \nbridging the gap between Branch Normal and 91bg-like in the figure, \nshowing its transitional nature. \n\n\\moon{In \\citet{buret14}, \\snia\\ with \\sbv\\ $\\simlt$ 0.5 show an $I$-band peak \n$\\sim$3--4 days later than $B$-band peak, while those with \\sbv\\ $\\simgt$ 0.7 do\n$\\sim$2--3 days earlier. The $I$-band peak of \\kspn\\ precedes that of the $B$ band by\nabout 1.5 days, which, together with its \\sbv\\ $\\simeq$ 0.54, places\nthe source in the gap connecting the two populations.\nThe relative strength of the secondary $I$-band peak is \n$<I>_{20-40}$ = 0.309 $\\pm$ 0.002 for \\kspn\\ (\\S\\ref{subsec:lc}). \nThe estimated values of \\sbv\\ and $<I>_{20-40}$ of \\kspn\\ \nfollow the overall correlation between the two parameters compiled for \\snia\\ in the paper.\nThe value $\\simeq$ 0.309 is low for Branch Normal but is high for 91bg-like \\citep{kriet01,buret14,sriet17}, \nconsistent with the interpretation that \\kspn\\ is of transitional nature. \nComparison of the source with other \\snia\\ in terms of \\sbv, $<I>_{20-40}$ and the time\ndelay between $B$- and $I$-band peaks shows that it overlaps with \"Cool\" (CL) objects\n\\citep[][see Figure~6 therein]{buret14} which are mostly rapid decliners, \nbut close to the boundary to the group of slow decliners, \nsupporting the rapidly-evolving and transitional nature of \\kspn.}\n\nThe \\bv\\ color of \\kspn\\ at the peak epoch is $\\simeq$ 0.08 mag, \nand this is consistent with what have been found in transitional \n\\snia\\ \\citep{galet18}.\nWhile the source is transitional, \nthe measured values of \\dm15\\ and \\sbv, the presence of the \nsecondary $I$-band peak, the overall color evolution, \nand the results of the template fitting all show\nthat it is more similar to Branch Normal than 91bg-like.\nAccording to \\citet{dhaet17}, fast-declining \\snia\\ more luminous than\n5 $\\times$ 10$^{42}$ \\ergs\\ make a smooth connection to normal \\snia\\ \nas we find in \\kspn\\ whose peak bolometric luminosity is\n(9.0 $\\pm$ 0.3) $\\times$ 10$^{42}$ \\ergs\\ (Table~\\ref{tab:par} and see \\S\\ref{sec:bol}).\n\n\\section{Bolometric Light Curve and \\ni56\\ Distribution}\\label{sec:bol}\n\nThe radioactive decay of \\ni56\\ followed by that of \\co56\\ \ndrives the light curves of \\snia\\ after explosion. \nWe compare here the bolometric light curve of \n\\kspn\\ with those expected from two different types of model \\ni56\\ distribution\n-- centrally concentrated and stratified -- in order to investigate\nhow \\ni56\\ was distributed in the progenitor of the source as well \nas its explosion parameters of ejecta mass and kinetic energy.\nDuring the photospheric phase of a \\sni\\ when the expansion is \ndriven by homologous spherical shocks within the first $\\sim$ 30 \ndays after the explosion,\nthe majority of its emission falls within \nthe UVOIR (ultraviolet-optical-infrared) waveband \\citep[][]{conet00}.\nWe integrate the best-fit \\sni\\ template of \\kspn\\ that we obtained\nfrom the \\snp\\ template fitting (\\S\\ref{subsec:temp})\nover the rest-frame UVOIR waveband, or $\\lambda$ = 3075--23763 \\AA, \nto construct its bolometric light curve.\\footnote{\\moon{We note that a bolometric light curve obtained this way is sometimes \ncalled quasi-bolometric light curve \\citep[e.g.][]{bet17} given the limited wavelength\nrange over which the luminosity is calculated. Since the difference is expected\nto be very small, we will call it bolometric luminosity light curve\nthroughout this paper for simplicity.}}\nFigure~\\ref{fig:arn}  shows the bolometric light curve of \\kspn\\  (filled black circles),\nwherein the peak bolometric luminosity \nand epoch are  (9.0 $\\pm$ 0.3) $\\times$ 10$^{42}$ erg s$^{-1}$ \nand --0.5 $\\pm$ 0.6 days, respectively.\n\n\\subsection{Centrally Concentrated \\ni56\\ Distribution}\\label{sec:cenni}\n\n\nIn the case that the distribution of \\ni56\\ is strongly peaked \ntowards the center of the ejected mass and that\nthe ejecta opacity is constant during SN explosion, \nthe radioactively-powered luminosity of a \\sni\\ \nduring the photospheric phase can be described as \\citep{arn82,arn96,valet08} \n\\begin{equation}\n    L(x) = M_{\\rm Ni} \\; e^{-x^{2}} \\times [(\\epsilon_{\\rm Ni}-\\epsilon_{\\rm Co}) \\; C(x) + \\epsilon_{\\rm Co} \\; D(x)]\n\\label{eq:lum}\n\\end{equation}\n\\noindent\nwhere $M_{\\rm Ni}$ is the total mass of \\ni56.\nThe parameter $x$ represents a scaled time dimension of SN explosion as  \n\\begin{equation}\n    x = \\frac{t}{\\tau_m}\\ \\ , \\ \\ \\ \\ \\tau_m = \\left(\\frac{\\kappa }{\\beta_{\\rm A} c}\\right)^{1\/2}\\left(\\frac{6 M_{\\rm ej}^3}{5 E_{\\rm ej}}\\right)^{1\/4}\n\\label{eq:time}\n\\end{equation}\n\\noindent\nwhere $t$ is time since explosion, $\\tau_m$ is the geometric mean of diffusion and expansion time scales, \n$\\kappa$ is the opacity, $\\beta_{\\rm A}=13.8$ is a model constant for SN density distribution, \n$c$ is the speed of light, and \\mej\\ and \\eej\\ are mass and kinetic energy of the ejecta, respectively.\nWe adopt $\\kappa$ = 0.1 cm$^2$~g$^{-1}$ dominated by the line transitions \nof \\ni56\\ during the photospheric phase \\citep{pe00, pn14}.\nIn Eqn.~\\ref{eq:lum}, $\\epsilon_{\\rm Ni} = 3.90\\times10^{10}$ erg~s$^{-1}$~g$^{-1}$ and\n$\\epsilon_{\\rm Co} = 6.78\\times10^{9}$ erg~s$^{-1}$~g$^{-1}$\nare the energy production rates per gram of \\ni56\\ and \\co56, respectively.\nThe term $C$($x$) is related to the luminosity produced by the nuclear reaction dacay of \\ni56,\nand so is $D$($x$) but to \\co56. \nThe equation shows that the luminosity of a \\sni\\ in this model is mainly determined by two parameters:\nthe mass of \\ni56\\ ($M_{\\rm Ni}$) and the mean time scale ($\\tau_m$). \n\n\\moon{If we simply assume that the onset of the model \nis the same as that of the first light obtained from \nthe power-law fit (\\S\\ref{subsec:lc}),\nwhich gives 16.7 days as the time interval between the epochs of \nthe first light and peak bolometric luminosity (or ``rise time''),\nwe obtain $M_{\\rm Ni}$ = 0.44 $\\pm$ 0.01 \\msol\\ \nand $\\tau_m$ = 14.07 $\\pm$ 0.67 days for the source\nby applying Eqn.~\\ref{eq:lum} at the bolometric peak.\nThis is equivalent to fixing Eqn.~\\ref{eq:lum} using only the two parameters: the peak bolometric luminosity and the rise time,\nregardless of the shape of the light curve.}\nFigure~\\ref{fig:arn} compares the model light curve (blue dotted line)\npredicted by the two obtained parameters, i.e., $M_{\\rm Ni}$ and $\\tau_m$,\nwith the bolometric light curve (black circles) of \\kspn,\nwhere we can clearly identify over-prediction of the bolometric luminosity \nby the model across the peak, \nshowing that this method is inappropriate.\nThis incompatibility is also confirmed with the inferred ejecta mass \nand kinetic energy of the source,\nwhich are \\mej\\ = 1.86 $\\pm$ 0.24 \\msol\\ and \\eej\\ = 1.35 $\\pm$ 0.30 $\\times$ 10$^{51}$ erg, respectively,\nobtained under the assumption of \na typical ejecta velocity \\vej\\ = 11000 $\\pm$ 1000 km~s$^{-1}$ \\citep{Scalet19}\nand $E_{\\rm ej}$ = $\\frac{3}{10}M_{\\rm ej}\\varv_{\\rm ej}^2$ for a \\sni.\nThe inferred ejecta mass and kinetic energy are \nunacceptably large for a fast-evolving \\sni\\ \\citep{Scalet19}.\nThis discrepancy is mainly due to  the long rise time (16.7 days) and \nthe fast post-peak decline rate (\\dm15\\ $\\simeq$ 1.62 mag) of the source\nthat are largely incompatible with  the model of Eqn.~\\ref{eq:lum}.\nIn other words, while this model may accurately estimate\nthe total \\ni56\\ mass based on the peak luminosity of \\kspn\\ (see below), \nit is likely insufficient to reproduce the detailed evolution of \nthe ascending and\/or declining phases.\n\nIn order to obtain more reliable SN explosion parameters\nthat match the evolution of the luminosity around the peak of \\kspn,\nwe conduct fitting of Eqn.~\\ref{eq:lum} to the \nbolometric light curve of \\kspn\\  (Figure~\\ref{fig:arn}), \nbut limiting the period  between --10 and +10 days across the peak in the fitting. \nThis method, which excludes the early and late part of the light curve, \nshould provide more reliable explosion parameters \nsince it is less affected by the assumed \\ni56\\ \ndistribution in Eqn.~\\ref{eq:lum}\nwhile more dependent on the bulk properties of the ejecta. \nThe best-fit parameters obtained this way are\n$M_{\\rm Ni}$ = 0.32 $\\pm$ 0.01 \\msol\\ and $\\tau_m$ = 9.45 $\\pm$ 0.52 days \nwhich provide a significantly improved match to the observed bolometric\nlight curve around the peak with \\chisqr\\ $\\simeq$ 0.42\n(Figure~\\ref{fig:arn}, black solid line).\n\\moon{The inferred ejecta mass and kinetic energy are \\mej\\ = 0.84 $\\pm$ 0.12 \\msol\\ \nand \\eej\\ = (0.61 $\\pm$ 0.14) $\\times$ 10$^{51}$ erg, respectively, \nconsistent with what have been observed in other fast-evolving SNe \\citep{dhaet18,Scalet19,wyget19}.}\nWe, therefore, adopt these values as the explosion parameters of \\kspn.\nIn this model fit, we also note that --13.5 $\\pm$ 0.4 days in the SN rest frame \nis the onset of the model light curves powered by centrally concentrated \\ni56 distribution\nwhich is approximately 3.7 days after the epoch of first light (\\t0, see \\S\\ref{subsec:lc}).\nThe difference of 3.7 days implies again that the centrally concentrated \\ni56\\ distribution \nin Eqn.~\\ref{eq:lum} is inadequate to properly model the observed early light curves of \\kspn,\nalthough it is capable of providing reliable SN explosion parameters \nwhen fitting is limited to the light curves around the peak.\nIn Figure~\\ref{fig:arn}, the black dotted line shows the\nextrapolated model prediction of Eqn.~\\ref{eq:lum} \nfor early ($<$ --10 days) and late ($>$ 10 days) epochs\nusing the explosion parameters obtained in the fitting above.\nThe best-fit model apparently underpredicts bolometric luminosities at early epochs.\nWe attribute this underprediction and the aforementioned difference between \\t0\\\nand the onset of the model light curves \nto a shallow \\ni56\\ distribution in \\kspn\\ \nas we detail in \\S\\ref{sec:strni} below.\n\nThe ejecta mass \\mej\\ = 0.84 $\\pm$ 0.12 \\msol\\ places \\kspn\\ \nin the group of sub-Ch-mass \\snia,\nconsistent with the results from previous studies of \\snia\\ that\nrapid decliners are from sub-Ch-mass explosions \\citep[e.g.,][]{Scalet19}. \nNote that, \nby assuming a smaller opaicty $\\kappa$ = 0.08 cm$^2$ g$^{-1}$  \\citep[e.g.,][]{arn82, Li19},\nwe obtain ejecta mass and kinetic energy \n\\mej\\ = 1.05 $\\pm$ 0.15 \\msol\\ and \\eej\\ = (0.76 $\\pm$ 0.18) $\\times$ 10$^{51}$ erg, respectively,\nstill within the limit of the sub-Ch-mass explosion.\nRecent results from extensive radiation transport simulations also \nindicate that rapid decliners are highly unlikely to be from \nChandrasekhar or super-Chandrasekhar mass explosions \\citep{gk18}.\n\n\\subsection{Stratified \\ni56\\ Distribution}\\label{sec:strni}\n\nAs shown above, the presence of early excess emission to what is expected \nby Eqn~\\ref{eq:lum} (Figure~\\ref{fig:arn}) and the difference of 3.7 days \nbetween the epoch of the first light \\t0\\ \nand the onset of the model light curves of Eqn.~\\ref{eq:lum} \nindicate that the real \\ni56\\ distribution \nof \\kspn\\ is different from the simple central concentration \nassumed in \\citet{arn82}.\nIt is conceivable, as recently suggested \\citep{pn14,pm16,conet18,maget20,mm20}, \nthat there exists \\ni56\\ distributed close to the progenitor surface responsible \nfor the early excess and the difference of 3.7 days. \nWe investigate this possibility below based on these models.\n\n\\subsubsection{Analytic Model}\\label{subsec:analytic}\nAccording to \\citet[][PN14, hereafter]{pn14}, \na stratified \\ni56\\ distribution extended more towards the surface\ncan adequately model the bolometric light curves of other \\snia\\ \nsuch as SN~2009ig, SN~2011fe, and SN~2012cg.\nPN14 modelled the local mass fraction of \\ni56\\ in the ejecta \nfollowing the spherically-symmetric logistic distribution of \n\\begin{equation}\n    X_{56}(x) \\propto \\frac{1}{1+\\exp{[-\\beta(x-x_{1\/2})]}}\n\\label{eq:ni}\n\\end{equation}\n\\noindent\nwhere $x= t\/t_{\\rm diff}$ is a scaled depth, measured from the surface to the center, \nin unit of diffusion time ($t_{\\rm diff}$) of the ejecta optical depth, while\n$\\beta$ and $x_{1\/2}$ are shape parameters representing \nthe radial decline rate of the \\ni56\\ distribution \nand the scaled depth at which\nthe distribution reaches half maximum, respectively.\nDuring a SN explosion, a diffusion wave travels backwards \ninto the expanding ejecta and SN luminosities are determined by \nthe amount of \\ni56\\ probed by the diffusion wave which reaches the \ncenter of the ejecta, or $x$ = 1, some time after the SN reaches the peak luminosity.\n\nFigure~\\ref{fig:SNpnmod} (top panel) shows the early bolometric \nlight curve (solid line) of \\kspn\\\npredicted by the best-fit stratified \\ni56\\ distribution (Eqn.~\\ref{eq:ni}),\nconfirming that the stratified \\ni56\\ distribution can match \nthe observed early bolometric luminosities (black circles) almost perfectly\nwith the best-fit parameters  \n$x_{1\/2}$ = 1.0, $\\beta$ = 2.4, and $t_{\\rm diff}$ = 23.1 days \n(see Figure~\\ref{fig:SNpnchi} \nbelow for the details of the fit).\nIn contrast, the extrapolation of the model light curve (dotted line)  \nbased on the centrally-distributed \\ni56\\ distribution from Eqn.~\\ref{eq:lum}\nshows a clear underprediction.\nThe bottom panel of the figure presents the early evolution \nof the estimated mass of ejecta (M$_{\\rm diff}$ for solid line)\nand \\ni56\\ (M$_{56}$ for dashed line) above the diffusion wave\n(or close to the progenitor surface)\nwhere we can identify the presence of about 0.0075 \\msol\\ \\ni56\\ mass,\nwhich corresponds to 3.4~\\% of the ejecta,\nlying above the diffusion depth  at 4 days.\nThis tells us that a small amount of excess \\ni56\\ distributed shallowly\ncan account for the early excess emission in \\kspn,\nrevealing that the \\ni56\\ distribution in the \nSN ejecta is likely more stratified towards the outer layers.\n\nThe distribution of \\chisqr\\ values from our model fitting \nof the stratified \\ni56\\ distribution, \nwhich is presented in Figure~\\ref{fig:SNpnchi} \nas a function of $x_{1\/2}$ and $\\beta$,\nshows that the model is reasonably well-fitted within a curved strip of\n$x_{1\/2}$ $\\simeq$ 0.4--1.0 and $\\beta$ $\\simeq$ 2--5\nwhere large $x_{1\/2}$ values are paired with small $\\beta$ values,\nor vice versa.\nThis distribution pattern of $x_{1\/2}$ and $\\beta$ describes\neither a more gradually extended \\ni56\\ distribution\nthat drops off closer to the center (= large $x_{1\/2}$ and small $\\beta$) or a more \nrapidly decaying distribution that drops off closer to the surface\n(= small $x_{1\/2}$ and large $\\beta$),\nboth of which are consistent with a shallow \\ni56\\ distribution.\nThe bottom panel of the figure shows the distribution of the local mass fraction of \\ni56\\ \n(= $X_{56}$) in two extreme cases marked in the top panel --\ni.e., circle for $x_{1\/2}$ = 1.0 and $\\beta$ = 2.4; \ndiamond for $x_{1\/2}$ = 0.4 and $\\beta$ = 4.6 -- \nwhere we can confirm that both distributions require nearly the same \\ni56\\ \nfraction at $\\sim$4-6 days after explosion.\n\n\\subsubsection{Radiative Transfer Model}\\label{subsec:radiative}\n\n\\moon{\n\\citet{maget20} recently provided radiative transfer-based \nmodel light curves of \\snia\\ with a logistic \\ni56\\ distribution  \nusing the following function\n\\begin{equation}\n    X_{56}(m) = \\frac{1}{1+\\exp{[-s \\; (m-M_{\\rm Ni})\/ {\\rm M}_\\odot ]}}\n\\label{eq:ni_mm}\n\\end{equation}\n\\noindent\nfor \\ni56\\ distribution, which is very similar to what is adopted in PN14 \n(or Eqn.~\\ref{eq:ni} in \\S~\\ref{subsec:analytic}). \nIn this distribution, $m$ is the mass coordinate from the ejecta surface,\n$M_{\\rm Ni}$ is the total \\ni56\\ mass, \nand $s$ describes how fast the \\ni56\\ distribution declines. \nWe compare in Figure~\\ref{fig:mm_Ni56} the observed colors (black circles) of \\kspn,\nwhich are binned to 1 day interval to increase S\/N ratios,\nduring the first 10 days post-explosion to\nthe best-fit radiative transfer models (black dashed curves,\n$M_{\\rm Ni}$ = 0.4 \\msol, $s$ = 4.4, and $E_{\\rm ej}$ = 0.78 $\\times$ 10$^{51}$ ergs)\nof \\snia\\ from \\citet{maget20} with Chandrasekhar-mass ejecta and exponential density distributions.\nThe shaded grey regions in the figure represent \nthe range of predicted colors for a set of radiative transfer models \nwith kinetic energies  in the range of (0.5--2.2) $\\times$ 10$^{51}$ ergs,\nwhile fixing $M_{\\rm Ni}$ and $s$ to those from the best-fit, \nwhich are 0.4 \\msol\\ and 4.4, respectively.\nWe identify in the figure that the observed early color evolution of \\kspn\\\nis largely consistent with radiative transfer-based predictions of \na centrally concentrated and monotonically stratified \\ni56\\ distribution.}\n\n\\moon{\n\\citet{mm20} extended the work of \\citet{maget20} to compute \nradiative transfer model light curves of \nChandrasekhar-mass \\snia\\ for a limited set of logistic \\ni56\\ distributions\nwith an external shell component in the outer layers of the ejecta. \nIn Figure~\\ref{fig:mm_clump}, we compare the observed colors of \\kspn\\\n(as in Figure~\\ref{fig:mm_Ni56}) to the predicted light curves of\nthe shell-added \\ni56\\ distributions of \\citet{mm20}.\nThe black curve represents the best-fit model from the limited set  \nof Chandrasekhar-mass \\snia\\ \nwith  $M_{\\rm Ni}$  = 0.6 \\msol, $s$ = 9.7, and $E_{\\rm ej}$ = 1.68 $\\times$ 10$^{51}$ ergs\nfor the case without any shell, \nwhile the blue, green, red curves are for the cases of \nan added shell of 0.01, 0.02, and 0.03 \\msol, respectively.\nThe \\ni56\\ distribution within these shells is assumed to be a Gaussian\ncentered at $m$ = 1.35 \\msol\\ from the center of the ejecta with \nwidths of 0.18 \\msol\\ (solid curves) and 0.06 \\msol\\ (dotted curves).\nAs seen in the figure, the observed colors are less consistent \nwith the presence of a \\ni56\\ shell of \\simgt\\ 0.01 \\msol\\\nthan the case of a centrally concentrated and monotonically \nstratified distribution of \\ni56\\ alone (Figure~\\ref{fig:mm_Ni56}),\nalthough we cannot rule out the possibility of a thinner \\ni56\\ shell of \\simlt\\ 0.01 \\msol\\  \nproducing colors more consistent with the observations.}\n\n\\moon{\nOverall, as shown above, the observed early color evolution of \\kspn\\ \nwithin the first 10 days post-explosion is consistent with\nwhat is expected from stratified, but still centrally concentrated, \\ni56\\ \nthat extends to the shallow layers of the ejecta near surface. \nThe color evolution is, however, largely incompatible with the presence of\na thick (\\simgt\\ 0.01 \\msol), external shell component of \\ni56\\ in addition to the logistic distribution. \nThis indicates that if \\kspn\\ originated in a \\subch\\ explosion triggered by \na He-shell detonation process \\citep[e.g.,][]{kroet10},\na thinner shell would have been required such as recently shown in simulations\nof He-shell detonations with thin, enriched He shells \\citep[e.g.][]{towet19}.}\n\n\\section{Constraint on the Progenitor of \\kspn}\\label{sec:pro}\n\nThe high-cadence, multi-color light curves of \\kspn\\ (Figure~\\ref{fig:lc}) provide\na rare opportunity to place thorough constraints on the progenitors of \nrapidly-declining \\snia\\ of transitional nature.\nFor a \\sni\\ from a single-degenerate progenitor system composed of\na white dwarf and \neither a main-sequence (MS) subgiant or a red giant companion,\n\\citet[][see also \\citealt{boehner17}]{kas10} calculated model\nluminosities from the shock interactions between  the SN ejecta and the companion \nwhich are mainly determined by the mass, kinetic energy and opacity of the ejecta\nas well as by the progenitor binary separation distance.\n\\moon{Such emission has been discussed as the source of early flashes within\nroughly 5 days post-explosion in \\snia\\ \\citep[e.g.,][]{miller18,miller20}.}\nThe observable luminosity from the interaction can be approximated by \n$L_{\\rm int}(t) f \\rm {(\\theta)}$, \nwhere $L_{\\rm int}(t)$ is the intrinsic luminosity from the interaction at time $t$\nand $f \\rm {(\\theta)}$ = 0.982 exp[--($\\theta$\/99.7)$^2$] + 0.018\nis the distribution of the observed luminosity as a function of the viewing angle $\\theta$ \\citep{ollet15}.\nThe maximum observable luminosity  occurs when a SN is viewed \nalong the interaction axis from the side of the companion, or $\\theta$ = 0$^\\circ$. \n\nFigure~\\ref{fig:kas} compares the observed $BVI$ magnitudes \n(black crosses) as well as the limiting magnitudes \n(black inverted triangles) of the \\kspn\\ light curves \nfrom its early phase\nwith those expected by the ejecta and companion \ninteraction model \\citep{kas10} for three\nparticular companion cases of\n1RG (blue solid line),  6MS (orange) and  2MS (green) \nwhen viewed along the interaction axis from the companion side.\nThese three cases are for progenitor systems where \nthe Roche Lobe-filling companion is \na red giant of 1 \\msol\\ (1RG),\na MS subgiant of 6 \\msol\\ (6MS)\nand 2 \\msol (2MS) located at \n2 $\\times$ 10$^{13}$ cm,\n2 $\\times$ 10$^{12}$ cm and\n5 $\\times$ 10$^{11}$ cm, respectively, from the white dwarf.\nWe use the estimated parameters of \\kspn\\ in Table~\\ref{tab:par}\nand ejecta opacity $\\kappa$ = 0.2 cm$^2$~g$^{-1}$ attributed to \nelectron scattering \\citep{kas10} to estimate luminosities \nfrom the interaction ($L_{\\rm int}$) between the ejecta and companion.\nAs in the figure, the observed $BVI$-band brightnesses \n(including the upper limits) of \\kspn\\ are lower than\nwhat are predicted by the interactions between \nthe companion and ejecta \nin the 1RG case (blue line) in most of the observed epochs, \nand this is also true for the case of 6MS (orange line) at the epochs \nearlier than day 3.\nThis comparison shows that both the 1RG and 6MS models are \nincompatible with the observations\nsince the observed fluxes need to be larger than \nthe model predictions to allow\nfor the presence of emission from the ejecta-companion interactions.\nThe 2MS case (green line) is different from the 1RG and 6MS\ncases because only the $B$-band brightness obtained around 0.6 day \nis lower than the model prediction while \nall the other observed brightnesses (including the upper limits)\nare higher.\nThe 2MS case, therefore, still appears to be incompatible\nwith the model prediction,\nbut with less confidence than the 1RG and 6MS cases. \nIn conclusion, if the interactions between the ejecta and \nthe binary companions are \nindeed viewed along the interaction axis from the companion side in \\kspn,\nour comparisons show that the companion of the source\nwas located in closer proximity to the white dwarf \nthan the 1RG and 6MS cases,\nand likely than the 2MS case too,\nindicating that the size of the progenitor companion \nof \\kspn\\ is smaller than those of these three stars.\nWe now provide a much more thorough and general investigation\ninto the presence of potential emission originating from the\nejecta-companion interactions and \nconclude that it is highly likely that the companion of \\kspn\\\nwas a white dwarf, supporting the double-degenerate scenario.\n\n\nWe first expand our search for the signal from the potential \ninteractions between the ejecta and companion in \\kspn\\ \nby including an extensive set of companion types,\nfar beyond the 1RG, 2MS and 6MS cases, \nfor all possible oblique viewing angles,\ni.e., $\\theta$ $>$ 0\\degr, \nand also by fully accounting for the uncertainties in our photometric measurements.\nFor this, we choose a set of 60 distances \nin the range of  (0.001--10) $\\times$ 10$^{13}$~cm \nin logarithmic scale as trial binary separation distances of its progenitor.\nThis range of the separation distances corresponds to that of the Roche-radius separations of stars \nspanning from the smallest red dwarfs to the largest supergiants. \nWe then calculate predicted brightnesses from the \nejecta-companion interaction in $BVI$ with these \nseparation distances using the model of \\citet{kas10}\nfor a set of 100 viewing angles equally sampled \nin the range of $\\theta$ = 0\\degr--180\\degr.\nIn this process, \nwe compare the predicted brightnesses to the observed light curves \nat various confidence levels by adopting a Gaussian distribution \nof the observed $BVI$ magnitudes\n(including the upper limits, Figure~\\ref{fig:kas}) of the source\nwith their photometric uncertainties. \nFigure~\\ref{fig:kassep} shows our results\nwhere the abscissa is the binary separation distance\nand the ordinate represents\nthe lower limit of acceptable viewing angles \nfor the interaction between the ejecta and its companion in \\kspn\\ based \non the model of \\citet{kas10}\nat the confidence level of 68~\\%\\ (solid curve) and 95~\\%\\ (dashed curve).\nAt the 95~\\%\\ confidence level (dashed curve), \nas in the figure,\nonly highly-oblique viewing angles $\\theta$ $\\simgt$ 130\\degr\\ are allowed\nfor the separation distances larger than 2 $\\times$ 10$^{13}$~cm,\nwhereas all of the small viewing angles are ruled out.\nFor the small separation distances of $<$ 0.03 $\\times$ 10$^{13}$~cm,\non the other hand, we cannot rule out any viewing angles.\nIn the separation distance range of \n(0.03--2) $\\times$ 10$^{13}$~cm, \nthe lower limit of acceptable viewing angles increases along the\nseparation distance, ruling out more viewing angles for larger separation distances. \nAt the 68~\\% confidence level (solid curve), \nthey are naturally more constrained\nthan the 95~\\% case (dashed curve) --\nalmost all viewing angles are excluded for large separation distances,\nwhile we cannot rule out any angle only for the separation distances\nsmaller than 0.007 $\\times$ 10$^{13}$~cm.\n\nThe results shown in Figure~\\ref{fig:kassep} accommodate the \nphotometric uncertainties of the light curves of \\kspn, \nbut not those of the estimated values of the redshift, epoch of the first light, and ejecta mass and kinetic energy\nneeded to compute the model luminosities from the ejecta-companion interactions.\nIn order to reflect the uncertainties of those parameters \nin our analyses, we conduct 40000 \nMonte Carlo simulations of the light curves of the ejecta-companion \ninteractions predicted by \\citet{kas10}\nby randomly selecting the values of these parameters \nunder the assumption that they follow a Gaussian distribution \nwith the measured uncertainties.\nWe use the same set of the separation distances \nand viewing angles that \nwe used above for Figure~\\ref{fig:kassep}\nand then conduct comparison between the observed light curves \nof \\kspn\\ with the model light curves. \nFigure~\\ref{fig:kasepo} shows the results of our comparison,\nwherein all the viewing angles are allowed for the separation distances \nup to $\\simeq$ 0.045 $\\times$ 10$^{13}$ cm at the confidence level of 95~\\%\nwhen all the uncertainties are statistically accounted for. \nThe separation distance of 0.045 $\\times$ 10$^{13}$ cm\nis about 90~\\% of that of 2MS after which the lower limit of acceptable viewing angle increases.\nAt the separation distances larger than that of 1RG, \nonly large viewing angles, i.e., $\\theta$ $>$ 125\\degr, are allowed.\nThe constraints on the viewing angles become more stringent at the 68~\\% confidence level\nwith even small viewing angles being ruled out at very short separation distances.\n\nOur analyses above apparently show  that \nthere is not much chance for the companion of \\kspn\\ to have been a red giant\nconsidering that the lower limit of acceptable viewing angles\nrules out most of the viewing angles\nfor such a large star.\nNote that it is highly unlikely the companion was \nat a different location from the Roche-radius separation where\nit can provide the stable accretion needed to trigger the explosion.\nA MS subgiant appears to be more probable \nthan the red giant in our analyses, \nespecially at highly-oblique viewing angles;\nhowever, sustaining substantial accretion from such a MS companion \nfor a \\sni\\ detonation is more challenging.\nThese results indicate that a double-degenerate system is much more \nlikely to have been the progenitor system of \\kspn,\nwhich agrees well with the results of the statistical \nsearches for ejecta-companion shock interactions\nin the surveys of SDSS, SNLS and TESS \\citep{hayden10,bianco11,fausnaugh19}\nand also with those of individual SN studies of \nSN2011fe \\citep{nuget11,liet11}, \nSN2012ht \\citep{yamanaka14}, \nSN2013gy \\citep{holmbo19}, \nASASSN-14lp \\citep{shapet16} \nand SN2012cg \\citep[][although see \\citealt{maret16}]{shapet18},\nstrengthening the case of small companions for the majority \nof \\snia\\ progenitors \\citep[e.g.,][]{ollet15}. \nWe, however, also note that some SNe, e.g.,\nSN2014J \\citep{goobar15}, \nSN2017cbv \\citep{hosseinzadeh17} and \nSN2018oh \\citep{shapet19,dimitriadis19}, \ndo show indications of early excess emission that could come \nfrom ejecta-companion shock interactions, \nalthough it is possible that some other processes---such as circumstellar interactions or a shallow layers of \\ni56---are responsible for it.\n\n\n\n\\section{Missing Host Galaxy for \\kspn}\\label{sec:host}\n\nAs shown in Figure~\\ref{fig:det} (see also \\S~\\ref{sec:obs}),\nno host galaxy underlying \\kspn\\ \nis detected in our deep stack images reaching the sensitivity\nlimits of $\\simeq$ 27.8 ($B$), 28.5 ($V$) and 28.2 ($I$) \\mpas, while the source\n\\galgg, which is the only apparently extended source  near \\kspn, \nis located $\\sim$27\\arcsec\\ away in the southwestern direction.\nIn order to understand the nature of \\galgg\\ and its potential \nconnection to \\kspn,\nwe fit its $V$-band surface brightness\nwith the S{\\'e}rsic profile\n$\\mu$ = $\\mu_0$ +1.0857 $b_n(r\/r_{\\rm e})^{1\/n}$\n(where $\\mu_0$, $r$, $r_{\\rm e}$ and $n$ are the central \nsurface brightness, radius, effective radius and S\\'ersic \ncurvature index, respectively, and \n$b_n$ = 1.9992$n$--0.3271, see \\citealt{graham05})\nto obtain $\\mu_{0}$ = 19.04 $\\pm$ 0.14 \\mpas,\n$r_{\\rm e}$ = 3\\farcs63 $\\pm$ 0\\farcs03 and $n$ = 2.05 $\\pm$ 0.06.\nThe apparent $V$-band magnitude and the \\bv\\ color of \\galgg\\ \nare $V$ $\\simeq$ 17.38 mag and \\bv\\ $\\simeq$ 1.4 mag, respectively.\nThese fitted parameters and color of \\galgg\\ are similar to those \nfound in early-type galaxies \\citep{baset13, valet11}, \nwhich is consistent with its spectroscopic properties.\nFigure~\\ref{fig:gal} shows our Magellan spectrum of \\galgg\\ (see \\S~\\ref{sec:obs} \nfor the details of the observations)\nwith clear detections of Ca H+K, G and Na ID absorption lines, \ntypical of early-type galaxies. \nWe determine the redshift of \\galgg\\ to be \n$z$ $\\simeq$ 0.167 $\\pm$ 0.001\nusing the measured wavelengths of the absorption lines,\nwhere the uncertainty is due to the rms wavelength \nsolution error of 0.135 \\AA.\nThe measured redshift is much larger than $z$ = 0.072 $\\pm$ 0.003 inferred for \\kspn\\ by\nfitting to \\sni\\ template (\\S\\ref{subsec:temp}).\nFigure~\\ref{fig:z0163fit} compares the best-fit \\snp\\ Branch Normal template (dotted curve) \nobtained at $z$ = 0.167 with those in Figure~\\ref{fig:LCtemplate}, \nshowing clearly that it gives significantly \nworse results than the fit obtained at $z$ = 0.072.\nIn addition, in order for \\kspn\\ to be a \\sni\\ with a high redshift $z$ = 0.167,\nits observed $B$-band peak brightness requires the SN to either have a\npeculiar motion greater than 14000 \\kms\\ or peak absolute magnitude\nmore luminous than --21.8 mag with \\dm15\\ $\\simeq$ 1.96 mag.\nBoth of these are unrealistic since the required peculiar motion is more than\n20 times greater than what can be expected for the source (see \\S\\ref{subsec:temp})\nand the required peak luminosity and \\dm15\\ make the source an extremely luminous \n\\sni\\ with an exceptionally large decline rate.\nWe, therefore, conclude that \\galgg\\ is unrelated to \\kspn\\\nand is an early-type galaxy located at a much larger redshift.\nThis leaves \\kspn\\ still hostless. \n\nIn order to investigate whether another nearby detected source\nmay be the host galaxy of \\kspn, we then \ncalculate probabilities that 22 sources identified within\nabout 35\\arcsec\\ from \\kspn\\ in our deep $BVI$ images\nwith S\/N $>$ 2\nare random galaxies coincidentally located in the field\nby adopting the methodology in \\citet{bkd02} using their \nmagnitudes and distances from \\kspn\\ \n\\citep[see also][and the details therein]{berger10}.\nIn Figure~\\ref{fig:maria}, which shows \nthe calculated probabilities of \nchance coincidence as a function of distance,\nall the sources have high ($>$ 0.4) chance coincidence probabilities \nwith the majority near 1. \nThis tells us that all these sources near \\kspn\\\nare likely to be coincident by chance\nand are unlikely related to the SN.\n\n\nThe absence of any host galaxy candidate of \\kspn\\ \nin our deep stack images hints to the nature of its \nhost galaxy since,\nat the luminosity distance of $\\simeq$ 310 Mpc (or $z$ $\\simeq$ 0.072), \nregular galaxies should be easily identifiable as an extended object \\citep[see][for example]{aet14} in our images.\nConsidering that there is no such extended object \nother than \\galgg, which is at a much higher redshift, \nin the vicinity of \\kspn, \nit is highly likely that the host galaxy of \\kspn\\ is \na faint dwarf galaxy.\nThe limiting magnitudes of an unresolved source in our images \nare $B\\simeq 25.83$, $V\\simeq 25.30$, and $I\\simeq 24.74$ mag.\nThis corresponds to an absolute magnitude limit of $\\simeq$ --12 mag \nin the $V$ band at $z$ = 0.072 and it is \ncompatible with the previously-known $V$-band absolute magnitude range\nof dwarf galaxies \\citep{tol09}.\nThe nearest unresolved source to \\kspn\\ in Figure~\\ref{fig:det} \nis about 5\\arcsec\\ away\nfrom \\kspn\\ with $V$- and $I$-band magnitudes of \n24.14 $\\pm$ 0.15 and 23.99 $\\pm$ 0.17 mag, respectively. \nIf this unresolved source is a dwarf galaxy hosting \\kspn,  \nthe SN is located $\\sim$ 6.6 kpc away from the center of the host galaxy.\nIts $V$-band absolute magnitude is --13.23 mag at $z$ = 0.072, \nand a dwarf galaxy with such magnitude \nis expected to have $\\simeq$ 25 \\mpas\\ effective surface brightness\nand $\\simeq$ 1 kpc effective radius \\citep{cb15,tol09}.\nThis requires that \\kspn\\ exploded at a location away \nfrom the center of a dwarf galaxy more than six times \nof its effective radius. \nAccording to \\citet{k13}, the stellar density in a dwarf galaxy \nat such a location drops significantly more, i.e., $\\gg$ 50 times, \nthan the central part.\nWe, therefore, conclude again that it is highly unlikely that \nany source identified in our deep stack images is \nthe host galaxy of \\kspn\\ and that the host galaxy of the SN\nis most likely a dwarf galaxy fainter than our detection limit.\nThe inferred insignificant host galaxy extinction of \\kspn\\  \n(\\S\\ref{sec:obs} and \\S\\ref{subsec:color}) points to \nthe outskirts of its potential host galaxy as the explosion location.\n\nRecently there has been a growing number of \\snia\\ detected \nin low-luminosity dwarf galaxies as well as those that remained hostless. \nThese include\n(1) SN1999aw, a luminous and slow-decaying SN from a \nvery faint galaxy with $M_V=-12.4$ mag \\citep{Strolger2002};\n(2) SN2007qc from an extremely faint host galaxy with $M_B \\sim -11$ mag \\citep{Quimby2012};\n(3) SN2007if, a luminous super-Chandrasekhar-mass \\sni\\\ndetected in a host galaxy with $M_g=-14.45$ mag \\citep{Childress2011};\nand (4) PTF10ops, a peculiar type SN with subluminous spectral \nproperties but with a normal light-curve width \nwhile remaining hostless to the detection \nlimit of $r$ $\\gtrsim$ --12 mag \\citep{maguire11}. \nIt is also noteworthy that the group of Ca-rich transients, \nwhose thermonuclear origin is still under debate, \nhave consistently been found in the outskirts of their host galaxies \nwhere there is no apparent stellar population,\npreferring high-velocity progenitors to low-metallicity environment \\citep{yuan13,foley15,lyman16,lunnan17}.\n\\cite{Graham2015} identified the host of a \\sni\\ with \n$M_V$ $\\simeq$ --8.4 mag which may be either\na dwarf galaxy or a globular cluster from a nearby elliptical \ngalaxy along with two other cases where no host galaxy \nis identified to the limit of $M_R$ $>$ --9.2 mag, \nsuggesting that their progenitors most likely belong to the intracluster \nstellar population.\nAlthough \\snia\\ from a faint host,\nincluding those from intra-cluster environment and dwarf galaxies, \ncan potentially bring us an important insight into the connection between\ntheir progenitors and stellar population and be used to trace missing dwarf galaxies in the \nlocal universe \\citep[e.g.,][]{Graham2015,cb15},\nour understanding of such \\snia\\ is still \nhighly incomplete, mainly due to the lack of statistically meaningful sample size. \nThe identification of \\kspn\\ as a rapidly-declining hostless\n\\sni\\ of transitional nature indicates that this type of  \n\\snia\\ can be produced\nin faint dwarf galaxies and that the coupling between\nspecific types of \\snia\\ and host galaxy environment may not be\nas strong as previously thought considering that\nthe transitional \\snia\\ have largely been detected\nin early-type galaxies \\citep[e.g.,][]{ashet16, mrbh07}.\n\n\\section{Summary and Conclusion}\\label{sec:sum}\n\nIn this paper, we report the discovery and \nidentification of \\kspn\\ \nas a rapidly-declining hostless \\sni\\ of transitional nature\nlikely originating from a sub-Ch explosion in a double-degenerate progenitor\nbased on high-cadence, multi-color observations made with the KMTNet.\nWe summarize our results and conclusion as follows.\n\n\\begin{itemize}\n\\item The observed light curves and colors of \\kspn\\ \nare compatible with a \\rapd\\ \\sni\\ at $z$ $\\simeq$ 0.072\nwhose properties are intermediate between Branch Normal \nand 91bg-like, but much closer to the former, \nwith clear signs of transitional nature.\nWhile the evolution of its early light curves is well fitted \nwith a power law representing a homologous expansion\npowered by \\ni56\\ radioactive decay which is mediated by photon diffusion processes,\nthe overall \\bv\\ and \\vi\\ color evolution of \\kspn\\ is largely\nsynchronous with that of the $I$ band as found in other \\snia.\nWe identify the presence of an early redward evolution\nin the \\vi\\ color prior to --10 days since peak in the SN\nbefore it enters the previously-known phase of blueward evolution.\nThis early redward evolution in \\vi\\ \nhas not been much studied but may bear an important clue\nto understanding the physical conditions of SN explosions.\n\n\\item The \\pp\\ and \\csp\\ of \\kspn\\ are \n\\dm15\\ = 1.62 $\\pm$ 0.03 mag and \\sbv\\ = 0.54 $\\pm$ 0.05, respectively,\nwhich place the source in the gap between the two groups of \nBranch Normal and 91bg-like with the peak luminosity of\n(9.0 $\\pm$ 0.3) $\\times$ 10$^{42}$ \\ergs. \nThe transitional nature of the source is also confirmed with \nthe relative strength ($\\simeq$ 0.309) \nof the secondary $I$-band peak of the source \nand its \\bv\\ $\\simeq$ 0.08 mag color at the peak epoch.\nWe obtain $M_{\\rm Ni}$ =  0.32 $\\pm$ 0.01 \\msol, \n\\mej\\ =  0.84 $\\pm$ 0.12 \\msol\\ and \n\\eej\\ = (0.61 $\\pm$ 0.14) $\\times$ 10$^{51}$ erg,\nwhich make \\kspn\\ a \\sni\\ explosion with\na sub-Ch mass.\n\n\\item The bolometric light curve of \\kspn\\ shows the presence of\nan early excess emission to what is expected from a centrally-concentrated\n\\ni56\\ distribution. \n\\moon{We find that a stratified \\ni56\\ distribution \nextended more shallowly to the surface of the progenitor provides \na good  match to the observed bolometric light curve,\nwhile the presence of a thick, \\simgt\\ 0.01 \\msol, external shell\nis largely incompatible with the observed early color evolution.}\nThorough comparisons between the observed light curves \nand those predicted from the ejecta-companion interactions \nclearly prefer a small binary separation distance for the progenitor, \nfavoring the double-degenerate scenario for its origin.\n\n\\item Even in our deep stack images reaching the sensitivity limit\n$\\mu_{BVI}$ $\\simeq$ 28 \\mpas,\n\\kspn\\ remains hostless, suggesting that its host galaxy \nis a faint dwarf galaxy. This contradicts to what has been previously\nthought for the types of host galaxies that produce transitional \\snia.\nIt will be worthwhile to investigate the nature of the host galaxy \nof \\kspn\\ with deeper imaging observations than presented in this paper\nthat can shed new insight into the relationship between \nhost galaxies and types of \\snia.\n\n\\end{itemize}\n\n\n\n\n\n\\acknowledgments\nThis research has made use of the KMTNet facility operated by the Korea Astronomy and Space Science Institute and the data were obtained at three host sites of CTIO\nin Chile, SAAO in South Africa, and SSO in Australia. \nWe acknowledge with thanks the variable star observations\nfrom the AAVSO International Database contributed by\nobservers worldwide and used in this research. \nDSM was supported in part by a Leading Edge Fund from the Canadian Foundation \nfor Innovation (project No. 30951) and a Discovery\nGrant (RGPIN-2019-06524) from the Natural Sciences and Engineering Research Council (NSERC) of Canada. \nMRD acknowledges support from NSERC through grant RGPIN-2019-06186, \nthe Canada Research Chairs Program, \nthe Canadian Institute for Advanced Research, \nand the Dunlap Institute at the University of Toronto. \nHSP was supported in part by the National Research Foundation of Korea (NRF) grant\nfunded by the Korea government (MSIT, Ministry of Science and ICT; No. NRF-2019R1F1A1058228).\nJA is supported by the Stavros Niarchos Foundation (SNF) and the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the 2nd Call of \"Science and Society\" Action Always strive for excellence Theodoros Papazoglou\" (Project Number: 01431)\n\n\n\n\n\\vspace{5mm}\n\\facilities{KMTNet, Magellan, AAVSO}\n\\software{\nAstropy \\citep{astropy13,astropy18},\n\\snp\\ \\citep{buret11},\nSCAMP \\citep{bertin06},\nSWARP \\citep{bet02},\nSiFTO \\citep{conet08}\n}\n\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Data Extraction}\nAt each gate setting we measure the qubit transition frequency over at least one period in offset charge $n_g$. From this we extract the qubit frequency $f_{0n}$ and charge dispersion $\\delta f_{0n}$ by applying a peak-finding algorithm to the raw two-tone spectroscopy data. The algorithm first smooths the data in frequency axis in order to combat noise, after which peaks are identified and fit with Lorentzian lineshapes in order to obtain their center frequency. For the gate voltage ranges in which two peaks are identified we take $\\delta f_{0n}$ to be the local maximum in peak separation. Conversely, $f_{0n}$ is obtained from the regions where only a single peak is identified. In the regions of parameter space where $\\delta f_{0n}$ is smaller than the qubit linewidth $\\gamma_{0n}$ such that only a single peak can be discerned for any $n_g$ (open markers in Fig. 4 of the main text), we take the center frequency to be $f_{0n}$ and use the extracted linewidth of the Lorentzian lineshape as an upper bound for $\\delta f_{0n}$.\n\n\\section{Modelling of the qubit}\n\\label{sec:supmodel}\nIn order to model the measured data we study the Hamiltonian of a capacitively shunted junction given by \\begin{equation}\n\\hat{H} = 4 E_{\\mathrm{c}} \\left(\\hat{n} - n_{\\mathrm{g}}\\right)^2 + U(\\hat{\\phi})\n\\label{eq:suphamiltonian}\n\\end{equation}\nwhere $U(\\hat{\\phi})$ is the junction potential, $E_{\\mathrm{c}}$ is the charging energy, $\\hat{n}$ is the number of Cooper pairs that have traversed the junction, $n_{\\mathrm{g}}$ is a dimensionless offset charge and $\\hat{\\phi}$ is the phase difference between the superconductors on either side of the junction. We obtain the qubit energy levels $E_n(n_g)$ and the corresponding qubit transitions $f_{ij}(n_g) = E_j(n_g) - E_i(n_g)$ through numerical diagonalization of the Hamiltonian, from which $f_{0n}$ and $\\delta f_{0n}$ are calculated by evaluating the transitions at the appropriate offset charges.\n\nWe perform this procedure for three possible models for the junction potential: a sinusoidal potential as encountered in tunnel junctions, a potential considering only occupation of the $E_-$ ABS branch of the resonant level model, and a potential including ITLZ tunneling between the ABS of the resonant level model. In the case of the sinusoidal model we take $U(\\hat{\\phi}) = -E_{\\mathrm{J}} \\cos{\\hat{\\phi}}$, where we define an effective $E_{\\mathrm{J}} \\equiv \\widetilde{\\Delta} \\widetilde{D} \/4$ in order to compare the models on equal footing. For the model including only the $E_{-}(\\hat{\\phi})$ ABS branch we use $U(\\hat{\\phi}) = -\\widetilde{\\Delta} \\sqrt{1-\\widetilde{D} \\sin^2 \\hat{\\phi}\/2}$, while in order to include ITLZ tunneling between the $E_{\\pm}(\\hat{\\phi})$ branches we follow the work of Ivanov and Feigel'man \\cite{Feigelman1999a} and approximate the many-body superconducting system by an effective two-level system. This results in a junction potential given by \n\\begin{equation}\nU(\\hat{\\phi})    =\n\\widetilde{\\Delta} \\begin{pmatrix}\n    \\cos{\\frac{\\hat{\\phi}}{2}} & \\sqrt{1-\\widetilde{D}}\\sin{\\frac{\\hat{\\phi}}{2}} \\\\[4pt]\n    \\sqrt{1-\\widetilde{D}}\\sin{\\frac{\\hat{\\phi}}{2}}  & -\\cos{\\frac{\\hat{\\phi}}{2}} \n\\end{pmatrix}\n\\label{eq:feigelman}\n\\end{equation}\nIn the above $\\widetilde{\\Delta}$ and $\\widetilde{D}$ are effective parameters resulting from the underlying quantum dot parameters discussed in section \\ref{sec:QD}.\n\nThe results of this procedure are demonstrated in Fig. \\ref{fig:supaverin}, which shows how $f_{01}$ and $\\delta f_{01}$ depend on $\\widetilde{D}$ for the three models. The sinusoidal model reproduces the expected results of the conventional tunnel junction transmon, exhibiting exponential suppression of $\\delta f_{01}$ for large values of $\\widetilde{D}$ and thus $E_{\\mathrm{J}}\/E_{\\mathrm{c}}$. The $E_{-}(\\phi)$ model exhibits similar behaviour up to moderate values of $\\widetilde{D}$, after which an enhanced suppression of $\\delta f_{01}$ takes place due to the increased height of the potential compared to the sinusoidal model. Finally, the model including ITLZ tunneling shows comparable behaviour to $E_{-}(\\phi)$ up to large values of $\\widetilde{D}$, after which a much more rapid decrease in charge dispersion takes place. This reproduces the scaling law predicted for ballistic Josephson junctions \\cite{Averin1999}. We note that the cross-over value of $\\widetilde{D}$ between the adiabatic regime well-described by only $E_-$ and the diabatic regime including ITLZ tunneling is approximately given by $\\widetilde{D} = 1-E_c\/\\widetilde{\\Delta}$, where the rate of phase evolution becomes comparable to the energy gap between the ABS \\cite{Feigelman1999}. \n\n\\begin{figure}[t!]\n\\includegraphics[width=12.9cm]{supp_averin_v1.pdf\n\\caption{\\label{fig:supaverin}  Qubit frequency (panel a) and charge dispersion (panel b) evaluated for three different models: a sinusoidal potential, the negative energy ABS branch of the resonant level model, and a potential considering ITLZ tunneling between both ABS branches. All models are evaluated for fixed parameter values $\\widetilde{\\Delta} = 14$~GHz and $E_c = 715$~MHz. The dashed line indicates the crossover value $1-E_c\/\\widetilde{\\Delta}$. Inset of panel b: Zoom in of the behaviour near $D=1$, where the ITLZ model exhibits rapid suppression in charge dispersion down to a small value set by tunneling through a potential barrier $\\widetilde{\\Delta} \\cos \\phi\/2$.}\n\\end{figure}\n\n\\section{Resonant Level Model}\n\\label{sec:QD}\nIn order to develop a quantitative understanding of our device we study a simplified model of a quantum dot between two superconducting leads known as the resonant level model \\cite{Beenakker2001}. Depicted in Fig. 4a of the main text, it considers the presence of a single spin-degenerate level in the junction with an energy $\\epsilon_{\\mathrm{0}}$ relative to the Fermi level, coupled to two identical superconductors with superconducting gap $\\Delta$ via the spin-degenerate tunnel rates $\\Gamma_{\\mathrm{l}}$ and $\\Gamma_{\\mathrm{r}}$. Its discrete energy spectrum follows from the solutions $\\epsilon \\in (0,\\Delta)$ of the equation\n\\begin{equation}\n\\left(\\Delta^2-\\epsilon^2)(\\epsilon^2-\\epsilon_0^2-\\Gamma^2\\right) + 4\\Delta^2\\Gamma_{\\mathrm{l}}\\Gamma_{\\mathrm{r}} \\sin{}^2(\\phi\/2) + 2\\Gamma \\epsilon^2(\\Delta^2-\\epsilon^2)^{1\/2} = 0\n\\label{eq:supbeenakker}\n\\end{equation}\nwhere $\\Gamma = \\Gamma_{\\mathrm{l}}+\\Gamma_{\\mathrm{r}}$. \nThis equation can be solved numerically for general parameter values, resulting in a single pair of spin degenerate ABS $E_{\\pm}(\\phi)$. Furthermore, in certain limits its solution can be recovered analytically (given in Eq. 1 of the main text), which coincides with the eigenvalues of Eq. \\ref{eq:feigelman}. However, we found these limits too constraining for the model to accurately describe our data. We therefore construct an approximate solution to Eq. \\ref{eq:supbeenakker} based on the ABS energies $E_{\\pm}(\\phi)$ and transparency $\\widetilde{D}$ given by Eq. 1 of the main text, whereas for $\\widetilde{\\Delta}$ we do not use the limiting values but instead solve Eq. \\ref{eq:supbeenakker} for the bound state energy at $\\phi = 0$. Shown in Fig. \\ref{fig:QDerr}, we tested the validity of this approximation for a wide range of parameters by explicit comparison to the solutions of Eq. \\ref{eq:supbeenakker}. The effective spectrum closely resembles the exact solutions over a wide range of parameters, with relative errors on the order of a few percent even in the regime $\\Gamma \\approx \\Delta$. We therefore argue that the effective model of Eq. \\ref{eq:feigelman} should accurately describe the phenomenology of the resonant level junction. A more detailed description of the model and its derivation can be found in \\cite{Kringhoj2019a}.\n\n\\begin{figure}[t!]\n\\includegraphics[width=12.9cm]{Supp_QD_v1.pdf\n\\caption{\\label{fig:QDerr} Comparison between the exact and approximate solutions for the ABS of the resonant level model. Top half shows the explicit energy levels for both solutions while the bottom half shows the relative error of the approximate model. Panel a shows the comparison in the regime of weak coupling ($\\Gamma \\ll \\Delta)$ on resonance ($\\epsilon_0 = 0$) with symmetric barriers ($\\delta\\Gamma = 0$). Panel b shows the regime of moderate coupling in which $\\Gamma_{\\mathrm{l}}=\\Gamma_{\\mathrm{r}} = \\Delta$ and where the level is off-resonant ($\\epsilon_0 \\neq 0$). Panel c shows the regime of far off-resonant strong coupling with asymmetric tunneling rates, in which $\\Gamma, \\epsilon_0 \\gg \\Delta$ and $\\delta\\Gamma \\neq 0$.}\n\\end{figure}\n\nHaving constructed the effective parameters $\\widetilde{D}$ and $\\widetilde{\\Delta}$, we now show how the ITLZ model introduced in section \\ref{sec:supmodel} behaves as a function of the underlying quantum dot parameters. We do this for the parameter values that resulted in the best fits to the measured data, shown in Fig. 4 of the main text. In Fig. \\ref{fig:fig4d} we study the effect of varying $\\epsilon_0$ at $\\Gamma_{\\mathrm{l}} = \\Gamma_{\\mathrm{r}}$. It results in a weak dependence of $\\widetilde{\\Delta}$, which is minimal when $\\epsilon_0 = 0$. This coincides coincides with a maximum in $\\widetilde{D}$, as given by Eq. 1 of the main text. Shown in panel b this translates into a qubit frequency that is maximal at $\\epsilon_0 = 0$, coinciding with a minimum in charge dispersion. Panels c and d in turn show the dependence on $\\Gamma_{\\mathrm{l}}$ at fixed $\\Gamma_{\\mathrm{r}}$ with $\\epsilon_0 = 0$. We find that $\\widetilde{\\Delta}$ is a monotonically increasing function of $\\Gamma_{\\mathrm{l}}$, whereas $\\widetilde{D}$ is maximal when the asymmetry $\\delta\\Gamma = \\vert \\Gamma_{\\mathrm{l}}-\\Gamma_{\\mathrm{r}} \\vert$ is minimized. Panel d shows that the relative rapidity at which $\\widetilde{\\Delta}$ and $\\widetilde{D}$ change around $\\delta\\Gamma = 0$ can result in a situation where the maximal qubit frequency does not coincide with minimal charge dispersion. We believe this effect to be the origin of the non-monotonic dependence between the qubit frequency and the charge dispersion seen in figures 3 and 4d of the main text.\n\n\\begin{figure}[t!]\n\\includegraphics[width=12.9cm]{supp_4d_v1.pdf\n\\caption{\\label{fig:fig4d} Effective junction and qubit parameters as a function of resonant level parameters. Panel a (b) shows how $\\widetilde{D}$ and $\\widetilde{\\Delta}$ ($f_{01}$ and $\\delta f_{01}$) behave as a function of $\\epsilon_0$ for $\\delta\\Gamma = 0$, panel c (d) shows how these parameters behave as a function of $\\Gamma_{\\mathrm{l}}$ for $\\epsilon_0 =0$ and $\\delta \\Gamma \\neq 0$.}\n\\end{figure}\n\n\\section{Fitting Routine}\nWe fit the measured relationships between $\\{f_{0n}, \\delta f_{0n}\\}$ using the models developed in section \\ref{sec:supmodel}. For the data measured as a function of $V_{\\mathrm{j}}$, shown in Fig. 4c of the main text, we assume that only $\\epsilon_{\\mathrm{0}}$ is varied. The remaining remaining parameters $\\Delta$, $E_{\\mathrm{c}}$, $\\Gamma_{\\mathrm{l}}$, and $\\Gamma_{\\mathrm{r}}$ are taken to be independent. Additionally, we fix $\\Delta = 53 \\ \\mathrm{GHz}$ based on DC transport experiments on similar nanowires \\cite{Deng2016} and we assume that $\\Gamma_{\\mathrm{l}} = \\Gamma_{\\mathrm{r}}$. We then apply a fitting routine in which for each set of parameter values a range of $\\{\\widetilde{\\Delta}, \\widetilde{{D}}\\}$ is generated from the effective resonant level ABS potential for a large range of $\\epsilon_{\\mathrm{0}}$. These effective parameters are then used in the three different junction potentials of section \\ref{sec:supmodel}, resulting in a set of calculated values for $\\{f_{01}, \\delta f_{01}\\}$ that can be compared to the measured values. The best fit to the data for each model is obtained through the standard method of least-squares.\n\nFor the extracted relationships between $\\{f_{0n}, \\delta f_{0n}\\}$ as a function of $V_{\\mathrm{g}}$, shown in Fig. 4d of the main text, this procedure is slightly modified. We now assume that only $\\Gamma_{\\mathrm{l}}$ is a function of $V_{\\mathrm{g}}$, with $\\Delta$, $E_{\\mathrm{c}}$, $\\epsilon_{\\mathrm{0}}$, and $\\Gamma_{\\mathrm{r}}$ taken to be independent. For simplicity we fix $\\epsilon_{\\mathrm{0}} = 0$. We note that, in order to obtain a good fit for the data versus $V_{\\mathrm{g}}$, $\\Delta$ needed to enter as a free parameter. In addition, a good fit could not be found simultaneously for all three measured transitions with a single set of parameters. Instead we only fit the $\\{f_{01}, \\delta f_{01}\\}$ data to the model, resulting in a qualitatively satisfying fit for the $01$ transition. However, the fit suggests a value of superconducting gap $\\Delta = 18.6$~GHz, much smaller than the value measured in DC experiments \\cite{Deng2016}. Moreover, the fit does not manage to capture the position of the higher order transitions. As discussed in the main text, we attribute this parameter discrepancy as well as the inability to fit all three transitions to possible modifications in the shape of the potential originating from the lack of electron-electron interactions in the model.\n\n\\section{Estimating transparencies}\nAn estimate for the transparencies $\\widetilde{D}$ realized the in the experiment can be obtained from the fits to the data. As illustrated in Fig. \\ref{fig:fig4d}, each numerically calculated value of $\\delta f_{0n}$ corresponds to a value of $\\widetilde{D}$, and one can therefore infer the values of $\\widetilde{D}$ by matching the measured values of $\\delta f_{0n}$ to the numerical values. We emphasize that these values are model and parameter dependent, and are therefore only an estimate. Shown in Fig. \\ref{fig:ts}a, we find that by varying $V_{\\mathrm{j}}$ transparencies between 0.5 and 1 are attained, with the largest transparency based on a distinguishable charge dispersion (filled markers) being 0.998 and the largest value based on the qubit linewidth $\\gamma_{01}$ (open markers) being 0.9996. Finally in panel b we show the asymptotic probability $p$ of the ABS remaining in the ground state as calculated from the extracted $\\widetilde{D}$ and $\\widetilde{\\Delta}$ \\cite{Averin1999}. This illustrates that the suppression of charge dispersion coincides with the vanishing of $p$. Furthermore, it shows that sizeable ITLZ probabilities are obtained over a wide range of the measured values, robust to small changes in fit parameters. We do not repeat this procedure for the data obtained by varying $V_{\\mathrm{g}}$, given the unsatisfactory fit to the data.\n\n\\begin{figure}[t!]\n\\includegraphics[width=12.9cm]{supp_fig_Ts.pdf\n\\caption{\\label{fig:ts} (a) Extracted $\\widetilde{D}$ for the values of $\\delta f_{01}$ measured as a function of $V_{\\mathrm{j}}$, with the inset showing the behaviour near $\\widetilde{D}=1$. (b) Asymptotic probability for the ABS to remain in the $E_-$ branch as calculated from the extracted $\\widetilde{D}$ and $\\widetilde{\\Delta}$.}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nVideo Object Segmentation (VOS) is a fundamental task in computer vision with many potential applications, including augmented reality~\\cite{ngan2011video} and self-driving cars~\\cite{zhang2016instance}. In this paper, we focus on semi-supervised VOS, which targets on segmenting a particular object across the entire video sequence based on the object mask given at the first frame. \n\\zongxin{The development of semi-supervised VOS can benefit many related tasks, such as video instance segmentation~\\cite{vis,Feng_2019_ICCV} and interactive video object segmentation~\\cite{oh2019fast,miao2020memory,liangmemory}.}\n\nEarly VOS works~\\cite{osvos,onavos,premvos} rely on fine-tuning with the first frame in evaluation, which heavily slows down the inference speed. Recent works (\\emph{e.g.},~\\cite{osmn,feelvos,spacetime}) aim to avoid fine-tuning and achieve better run-time. \nIn these works, STMVOS~\\cite{spacetime} introduces memory networks to learn to read sequence information and outperforms all the fine-tuning based methods. However, STMVOS relies on simulating extensive frame sequences using large image datasets~\\cite{voc,coco,cheng2014global,shi2015hierarchical,semantic} for training. The simulated data significantly boosts the performance of STMVOS but makes the training procedure elaborate. Without simulated data, FEELVOS~\\cite{feelvos} adopts a semantic pixel-wise embedding together with a global (between the first and current frames) and a local (between the previous and current frames) matching mechanism to guide the prediction. The matching mechanism is simple and fast, but the performance is not comparable with STMVOS.\n\n\n\n\\begin{wrapfigure}[17]{R}{0.55\\textwidth}\n\\centering \n\n\\includegraphics[width=0.98\\linewidth]{figs\/CI.pdf}\n\n\\caption{CI means collaborative integration. There are two foreground sheep (pink and blue). In the top line, the contempt of background matching leads to a confusion of sheep's prediction. In the bottom line, we relieve the confusion problem by introducing background matching (dot-line arrow).}\\label{fig:cl}\n\n\\end{wrapfigure}\n\n\n\nEven though the efforts mentioned above have made significant progress, current state-of-the-art works pay little attention to the feature embedding of background region in videos and only focus on exploring robust matching strategies for the foreground object (s). Intuitively, it is easy to extract the foreground region from a video when precisely removing all the background. Moreover, modern video scenes commonly focus on many similar objects, such as the cars in car racing, the people in a conference, and the animals on a farm. For these cases, the contempt of integrating foreground and background embeddings traps VOS in an unexpected background confusion problem. As shown in Fig.~\\ref{fig:cl}, if we focus on only the foreground matching like FEELVOS, a similar and same kind of object (sheep here) in the background is easy to confuse the prediction of the foreground object. Such an observation motivates us that the background should be equally treated compared with the foreground so that better feature embedding can be learned to relieve the background confusion and promote the accuracy of VOS.\n\n\n\nWe propose a novel framework for Collaborative video object segmentation by Foreground-Background Integration (CFBI) based on the above motivation. \nDifferent from the above methods, we not only extract the embedding and do match for the foreground target in the reference frame, but also for the background region to relieve the background confusion.\nBesides, our framework extracts two types of embedding (\\emph{i.e.}, pixel-level, and instance-level embedding) for each video frame to cover different scales of features. Like FEELVOS, we employ pixel-level embedding to match all the objects' details with the same global \\& local mechanism. However, the pixel-level matching is not sufficient and robust to match those objects with larger scales and may bring unexpected noises due to the pixel-wise diversity. Thus we introduce instance-level embedding to help the segmentation of large-scale objects by using attention mechanisms.\nMoreover, we propose a collaborative ensembler to aggregate the foreground \\& background and pixel-level \\& instance-level information and learn the collaborative relationship among them implicitly.\n\\zongxin{For better convergence, we take a balanced random-crop scheme in training to avoid learned attributes being biased to the background attributes.}\nAll these proposed strategies can significantly improve the quality of the learned collaborative embeddings for conducting VOS while keeping the network simple yet effective simultaneously.\n\n\n\n\nWe perform extensive experiments on DAVIS~\\cite{davis2016,davis2017}, and YouTube-VOS~\\cite{youtubevos} to validate the effectiveness of the proposed CFBI approach. Without any bells and whistles (such as the use of simulated data, fine-tuning or post-processing), CFBI outperforms all other state-of-the-art methods on the validation splits of DAVIS 2016 (ours, $\\mathcal{J}\\&\\mathcal{F}$ $\\mathbf{89.4\\%}$), DAVIS 2017 ($\\mathbf{81.9\\%}$) and YouTube-VOS ($\\mathbf{81.4\\%}$) while keeping a competitive single-object inference speed of about 5 FPS. By additionally applying multi-scale \\& flip augmentation at the testing stage, the accuracy can be further boosted to $\\mathbf{90.1\\%}$, $\\mathbf{83.3\\%}$ and $\\mathbf{82.7\\%}$, respectively. \nWe hope our simple yet effective CFBI will serve as a solid baseline and help ease VOS's future research.\n\n\n\\section{Related Work}\n\n\\noindent\\textbf{Semi-supervised Video Object Segmentation.}\nMany previous methods for semi-supervised VOS rely on fine-tuning at test time. Among them, OSVOS~\\cite{osvos} and MoNet~\\cite{xiao2018monet} fine-tune the network on the first-frame ground-truth at test time. OnAVOS~\\cite{onavos} extends the first-frame fine-tuning by an online adaptation mechanism, \\emph{i.e.}, online fine-tuning. MaskTrack~\\cite{masktrack} uses optical flow to propagate the segmentation mask from one frame to the next. PReMVOS~\\cite{premvos} combines four different neural networks (including an optical flow network~\\cite{flownet}) using extensive fine-tuning and a merging algorithm. Despite achieving promising results, all these methods are seriously slowed down by fine-tuning during inference.\n\nSome other recent works (\\emph{e.g.},~\\cite{osmn,favos}) aim to avoid fine-tuning and achieve a better run-time. OSMN~\\cite{osmn} employs two networks to extract the instance-level information and make segmentation predictions, respectively.  PML~\\cite{pml} learns a pixel-wise embedding with the nearest neighbor classifier. Similar to PML, VideoMatch~\\cite{videomatch} uses a soft matching layer that maps the pixels of the current frame to the first frame in a learned embedding space. Following PML and VideoMatch, FEELVOS~\\cite{feelvos} extends the pixel-level matching mechanism by additionally matching between the current frame and the previous frame. Compared to the methods with fine-tuning, FEELVOS achieves a much higher speed, but there is still a gap inaccuracy. Like FEELVOS, RGMP~\\cite{rgmp} and STMVOS~\\cite{spacetime} does not require any fine-tuning. STMVOS, which leverages a memory network to store and read the information from past frames, outperforms all the previous methods. However, STMVOS relies on an elaborate training procedure using extensive simulated data generated from multiple datasets. Moreover, the above methods do not focus on background matching.\n\nOur CFBI utilizes both the pixel-level and instance-level embeddings to guide prediction. Furthermore, we propose a collaborative integration method by additionally learning background embedding. \n\n\\noindent\\textbf{Attention Mechanisms.}\nRecent works introduce the attention mechanism into convolutional networks (\\emph{e.g.}, ~\\cite{attention_conv1,attention_conv2}). \nFollowing them, SE-Nets~\\cite{senet} introduced a lightweight gating mechanism that focuses on enhancing the representational power of the convolutional network by modeling channel attention. Inspired by SE-Nets, CFBI uses an instance-level average pooling method to embed collaborative instance information from pixel-level embeddings. After that, we conduct a channel-wise attention mechanism to help guide prediction. Compared to OSMN, which employs an additional convolutional network to extract instance-level embedding, our instance-level attention method is more efficient and lightweight.\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figs\/overview.pdf}\n    \\caption{An \\textbf{overview} of CFBI. F-G denotes Foreground-Background. We use \\textcolor{red}{red} and \\textcolor{blue}{blue} to indicate foreground and background separately. The deeper the red or blue color, the higher the confidence. Given the first frame ($t=1$), previous frame ($t=T-1$), and current frame ($t=T$), we firstly extract their pixel-wise embedding by using a backbone network. Second, we separate the first and previous frame embeddings into the foreground and background pixels based on their masks. After that, we use F-G pixel-level matching and instance-level attention to guide our collaborative ensembler network to generate a prediction.}\n    \\label{fig:overview}\n\n\\end{figure}\n\n\n\\section{Method}\\label{sec:model}\n\n\\noindent\\textbf{Overview.} Learning foreground feature embedding has been well explored by previous practices (\\emph{e.g.},~\\cite{osmn,feelvos}). OSMN proposed to conduct an instance-level matching, but such a matching scheme fails to consider the feature diversity among the details of the target's appearance and results in coarse predictions. PML and FEELVOS alternatively adopt the pixel-level matching by matching each pixel of the target, which effectively takes the feature diversity into account and achieves promising performance. Nevertheless, performing pixel-level matching may bring unexpected noises in the case of some pixels from the background are with a similar appearance to the ones from the foreground (Fig.~\\ref{fig:cl}).\n\nTo overcome the problems raised by the above methods and promote the foreground objects from the background, we present Collaborative video object segmentation by Foreground-Background Integration (CFBI), as shown in Figure~\\ref{fig:overview}. We use red and blue to indicate foreground and background separately. First, beyond learning feature embedding from foreground pixels, our CFBI also considers embedding learning from background pixels for collaboration. Such a learning scheme will encourage the feature embedding from the target object and its corresponding background to be contrastive, promoting the segmentation results accordingly. Second, we further conduct the embedding matching from both pixel-level and instance-level with the collaboration of pixels from the foreground and background. For the pixel-level matching, \nwe improve the robustness of the local matching under various object moving rates. For the instance-level matching, we design an instance-level attention mechanism to augment the pixel-level matching efficiently. Moreover, to implicitly aggregate the learned foreground \\& background and pixel-level \\& instance-level information, we employ a collaborative ensembler to construct large receptive fields and make precise predictions.\n\n\n\n\\subsection{Collaborative Pixel-level Matching}\n\nFor the pixel-level matching, we adopt a global and local matching mechanism similar to FEELVOS for introducing the guided information from the first and previous frames, respectively. Unlike previous methods~\\cite{pml,feelvos}, we additionally incorporate background information and apply multiple windows in the local matching, which is shown in the middle of Fig.~\\ref{fig:overview}. \n\nFor incorporating background information, we firstly redesign the pixel distance of~\\cite{feelvos} to further distinguish the foreground and background.\nLet $B_t$ and $F_t$ denote the pixel sets of background and all the foreground objects of frame $t$, respectively. We define a new distance between pixel $p$ of the current frame $T$ and pixel $q$ of frame $t$ in terms of their corresponding embedding, $e_p$ and $e_q$, by\n\\begin{equation} \\label{equ:distance}\n    D_t(p,q)=\n        \\begin{cases}\n            1-\\frac{2}{1+exp(||e_p-e_q||^2+b_B)} & \\text{if } q \\in B_t\\\\\n            1-\\frac{2}{1+exp(||e_p-e_q||^2+b_F)} & \\text{if } q \\in F_t\n        \\end{cases},\n\\end{equation}\nwhere $b_B$ and $b_F$ are trainable background bias and foreground bias. We introduce these two biases to make our model be able further to learn the difference between foreground distance and background distance.\n\n\n\\noindent\\textbf{Foreground-Background Global Matching.} Let $\\mathcal{P}_t$ denote the set of all pixels (with a stride of 4) at time $t$ and $\\mathcal{P}_{t,o}\\subseteq \\mathcal{P}_{t}$ is the set of pixels at time $t$ which belongs to the foreground object $o$. The global foreground matching between one pixel $p$ of the current frame $T$ and the pixels of the first reference frame (\\emph{i.e.}, $t=1$) is,\n\\begin{equation} \\label{equ:global_f}\n    G_{T,o}(p)=\\min_{q\\in\\mathcal{P}_{1,o}} D_1(p,q).\n\\end{equation}\nSimilarly, let $\\mathcal{\\overline{P}}_{t,o} =\\mathcal{P}_t \\backslash \\mathcal{P}_{t,o}$ denote the set of relative background pixels of object $o$ at time $t$, and the global background matching is,\n\\begin{equation} \\label{equ:global_b}\n    \\overline{G}_{T,o}(p)=\\min_{q\\in\\mathcal{\\overline{P}}_{1,o}} D_{1}(p,q).\n\\end{equation}\n\n\\noindent\\textbf{Foreground-Background Multi-Local Matching.}\n\n\\setlength{\\intextsep}{-10pt}\n\\begin{wrapfigure}[20]{R}{0.37\\textwidth}\n\\center\n\n\\subfloat[Slow moving rate]{\n\\label{fig:slow}\n\\includegraphics[width=0.9\\linewidth]{figs\/slow.pdf}\n}\n\n\\subfloat[Fast moving rate]{\n\\label{fig:fast}\n\\includegraphics[width=0.9\\linewidth]{figs\/fast.pdf}\n}\n\n\\caption{The moving rate of objects across two adjacent frames is largely variable for different sequences. Examples are from YouTube-VOS~\\cite{youtubevos}.}\\label{fig:offset}\n\\end{wrapfigure}\n\n\n\\noindent In FEELVOS, the local matching is limited in only one fixed extent of neighboring pixels, but the offset of objects across two adjacent frames in VOS is variable, as shown in Fig.~\\ref{fig:offset}. Thus, we propose to apply the local matching mechanism on different scales and let the network learn how to select an appropriate local scale, which makes our framework more robust to various moving rates of objects. Notably, we use the intermediate results of the local matching with the largest window to calculate on other windows. Thus, the increase of computational resources of our multi-local matching is negligible.\n\n\\setlength{\\intextsep}{0pt}\n\nFormally, let $K=\\{k_1,k_2,...,k_n\\}$ denote all the neighborhood sizes and $H(p,k)$ denote the neighborhood set of pixels that are at most $k$ pixels away from $p$ in both $x$ and $y$ directions, our foreground multi-local matching between the current frame $T$ and its previous frame $T-1$ is\n\\begin{equation} \n    ML_{T,o}(p,K)=\\{L_{T,o}(p,k_1),L_{T,o}(p,k_2),...,L_{T,o}(p,k_n)\\},\n\\end{equation}\nwhere\n\\begin{equation} \\label{equ:local_f}\n    L_{T,o}(p,k)=\n        \\begin{cases}\n            \\min_{q\\in\\mathcal{P}^{p,k}_{T-1,o}} D_{T-1}(p,q) & \\text{if }\\mathcal{P}^{p,k}_{T-1,o}\\neq\\emptyset \\\\\n            1 & \\text{otherwise}\n        \\end{cases}.\n\\end{equation}\nHere, $\\mathcal{P}^{p,k}_{T-1,o}:=\\mathcal{P}_{T-1,o}\\cap H(p,k)$ denotes the pixels in the local window (or neighborhood). And our background multi-local matching is\n\\begin{equation} \n    \\overline{ML}_{T,o}(p,K)=\\{\\overline{L}_{T,o}(p,k_1),\\overline{L}_{T,o}(p,k_2),...,\\overline{L}_{T,o}(p,k_n)\\},\n\\end{equation}\nwhere\n\\begin{equation} \\label{equ:local_b}\n    \\overline{L}_{T,o}(p,k)=\n        \\begin{cases}\n            \\min_{q\\in\\mathcal{\\overline{P}}_{T-1,o}^{p,k}} D_{T-1}(p,q) & \\text{if }\\mathcal{\\overline{P}}_{T-1,o}^{p,k}\\neq\\emptyset \\\\\n            1 & \\text{otherwise}\n        \\end{cases}.\n\\end{equation}\nHere similarly, $\\mathcal{\\overline{P}}^{p,k}_{T-1,o}:=\\mathcal{\\overline{P}}_{T-1,o}\\cap H(p,k)$.\n\n\nIn addition to the global and multi-local matching maps, we concatenate the pixel-level embedding feature and mask of the previous frame with the current frame feature. FEELVOS demonstrates the effectiveness of concatenating the previous mask. Following this, we empirically find that introducing the previous embedding can further improve the performance ($\\mathcal{J}$\\&$\\mathcal{F}$) by about $0.5\\%$.\n\nIn summary, the output of our collaborative pixel-level matching is a concatenation of (1) the pixel-level embedding of the current frame, (2) the pixel-level embedding and mask of the previous frame, (3) the multi-local matching map and (4) the global matching map, as shown in the bottom box of Fig.~\\ref{fig:overview}. \n\n\\setlength{\\intextsep}{-10pt}\n\\begin{wrapfigure}[22]{r}{0.3\\textwidth}\n\\center\n\n\\includegraphics[width=0.98\\linewidth]{figs\/ins_a.pdf}\n\n\\caption{The trainable part of the instance-level attention. $C_e$ denotes the channel dimension of pixel-wise embedding. $H$, $W$, $C$ denote the height, width, channel dimension of CE features.}\n\\label{fig:instance}\n\n\\end{wrapfigure}\n\n\\subsection{Collaborative Instance-level Attention}\n\nAs shown in the right of Fig~\\ref{fig:overview}, we further design a Collaborative instance-level attention mechanism to guide the segmentation for large-scale objects. \n\nAfter getting the pixel-level embeddings of the first and previous frames, we separate them into foreground and background pixels (\\emph{i.e.}, $\\mathcal{P}_{1,o}$, $\\mathcal{\\overline{P}}_{1,o}$, $\\mathcal{P}_{T-1,o}$, and $\\mathcal{\\overline{P}}_{T-1,o}$) according to their masks. Then, we apply channel-wise average pooling on each group of pixels to generate a total of four instance-level embedding vectors and concatenate these vectors into one collaborative instance-level guidance vector. Thus, the guidance vector contains the information from both the first and previous frames, and both the foreground and background regions.\n\n\\setlength{\\intextsep}{0pt}\n\n\n\nIn order to efficiently utilize the instance-level information, we employ an attention mechanism to adjust our Collaborative Ensembler (CE). We show a detailed illustration in Fig.~\\ref{fig:instance}. Inspired by SE-Nets~\\cite{senet}, we leverage a fully-connected (FC) layer (we found this setting is better than using two FC layers as adopted by SE-Net) and a non-linear activation function to construct a gate for the input of each Res-Block in the CE. The gate will adjust the scale of the input feature channel-wisely.\n\n\n\n\n\nBy introducing collaborative instance-level attention, we can leverage a full scale of foreground-background information to guide the prediction further. The information with a large (instance-level) receptive field is useful to relieve local ambiguities~\\cite{torralba2003contextual}, which is inevitable with a small (pixel-wise) receptive field.\n\n\n\n\n\n\\subsection{Collaborative Ensembler (CE)}\n\nIn the lower right of Fig.~\\ref{fig:overview}, we design a collaborative ensembler for making large receptive fields to aggregate pixel-level and instance-level information and implicitly learn the collaborative relationship between foreground and background. \n\nInspired by ResNets~\\cite{resnet} and Deeplabs~\\cite{deeplab,deeplabv3p}, which both have shown significant representational power in image segmentation tasks, our CE uses a downsample-upsample structure, which contains three stages of Res-Blocks~\\cite{resnet} and an Atrous Spatial Pyramid Pooling (ASPP)~\\cite{deeplabv3p} module. The number of Res-Blocks in Stage 1, 2, and 3 are $2$, $3$, $3$ in order. Besides, we employ dilated convolutional layers to improve the receptive fields efficiently. The dilated rates of the $3\\times3$ convolutional layer of Res-Blocks in one stage are separately $1$, $2$, $4$ ( or $1$, $2$ for Stage 1). At the beginning of Stage 2 and Stage 3, the feature maps will be downsampled by the first Res-Block with a stride of 2. After these three stages, we employ an ASPP and a Decoder~\\cite{deeplabv3p} module to increase the receptive fields further, upsample the scale of feature and fine-tune the prediction collaborated with the low-level backbone features.\n\n\n\n\\section{Implementation Details}\n\\setlength{\\intextsep}{-10pt}\n\\begin{wrapfigure}[11]{r}{0.5\\textwidth}\n\\center\\vspace{-9mm}\n\n\\subfloat[Normal]{\n\\label{fig:normal_crop}\n\\includegraphics[width=0.426\\linewidth]{figs\/normal_crop.pdf}\n}\n\\subfloat[Balanced]{\n\\label{fig:balanced_crop}\n\\includegraphics[width=0.52\\linewidth]{figs\/balanced_crop.pdf}\n}\n\n\\caption{When using normal random-crop, some red windows contain few or no foreground pixels. For reliving this problem, we propose balanced random-crop.}\\label{fig:crop}\n\n\\end{wrapfigure}\n\nFor better convergence, we modify the random-crop augmentation and the training method in previous methods~\\cite{spacetime,feelvos}.\n\n\\setlength{\\intextsep}{0pt}\n\n\\noindent\\textbf{Balanced Random-Crop.}\nAs shown in Fig.~\\ref{fig:crop}, there is an apparent imbalance between the foreground and the background pixel number on VOS datasets. Such an issue usually makes the models easier to be biased to background attributes. \n\nIn order to relieve this problem, we take a balanced random-crop scheme, which crops a sequence of frames (\\emph{i.e.}, the first frame, the previous frame, and the current frame) by using a same cropped window and restricts the cropped region of the first frame to contain enough foreground information. The restriction method is simple yet effective. To be specific, the balanced random-crop will decide on whether the randomly cropped frame contains enough pixels from foreground objects or not. If not, the method will continually take the cropping operation until we obtain an expected one.\n\n\\iffalse\n\\begin{figure}[!t]\n    \\center\n        \\includegraphics[width=0.8\\linewidth]{figs\/sequence_training.pdf}\n        \\caption{An illustration of the sequential training. In each step, the previous mask comes from the previous prediction (the green lines) except for the first step, whose previous mask comes from the ground-truth mask (the blue line).}\n        \\label{fig:sequence_training}\n\\end{figure}\n\\fi\n\n\\noindent\\textbf{Sequential Training.} \\zongxin{In the training stage, FEELVOS predicts only one step in one iteration, and the guidance masks come from the ground-truth data. RGMP and STMVOS uses previous guidance information (mask or feature memory) in training, which is more consistent with the inference stage and performs better. In the evaluation stage, the previous guidance masks are always generated by the network in the previous inference steps.}\n\n\\zongxin{Following RGMP, we train the network using a sequence of consecutive frames in each SGD iteration. \nIn each iteration, we randomly sample a batch of video sequences. For each video sequence, we randomly sample a frame as the reference frame and a continuous $N+1$ frames as the previous frame and current frame sequence with $N$ frames. When predicting the first frame, we use the ground-truth of the previous frame as the previous mask. When predicting the following frames, we use the latest prediction as the previous mask. \n}\n\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figs\/comparison.pdf}\n\n    \\caption{Qualitative comparison with STMVOS on DAVIS 2017. In the first video, STMVOS fails in tracking the gun after occlusion and blur. In the second video, STMVOS is easier to partly confuse with bicycle and person.}\n    \\label{fig:comparison}\n\n\\end{figure}\n\n\\noindent\\textbf{Training Details.}\nFollowing FEELVOS, we use the DeepLabv3+~\\cite{deeplabv3p} architecture as the backbone for our network. However, our backbone is based on the dilated Resnet-101~\\cite{deeplabv3p} instead of Xception-65~\\cite{xception} for saving computational resources. We apply batch normalization (BN)~\\cite{bn} in our backbone and pre-train it on ImageNet~\\cite{deng2009imagenet} and COCO~\\cite{coco}. The backbone is followed by one depth-wise separable convolution for extracting pixel-wise embedding with a stride of 4.\n\nWe initialize $b_B$ and $b_F$ to $0$. For the multi-local matching, we further downsample the embedding feature to a half size using bi-linear interpolation for saving GPU memory. Besides, the window sizes in our setting are $K=\\{2, 4, 6, 8, 10, 12\\}$. For the collaborative ensembler, we apply group normalization (GN)~\\cite{gn} and gated channel transformation~\\cite{gct} to improving training stability and performance when using a small batch size. For sequential training, the current sequence's length is $N=3$, which makes a better balance between computational resources and network performance.\n\n\n\nWe use the DAVIS 2017~\\cite{davis2017} training set (60 videos) and the YouTube-VOS~\\cite{youtubevos} training set (3471 videos) as the training data. \\zongxin{We downsample all the videos to 480P resolution, which is same as the default setting in DAVIS.} We adopt SGD with a momentum of $0.9$ and apply a bootstrapped cross-entropy loss, which only considers the $15\\%$ hardest pixels. During the training stage, we freeze the parameters of BN in the backbone. For the experiments on YouTube-VOS, we use a learning rate of $0.01$ for $100,000$ steps with a batch size of 4 videos (\\emph{i.e.}, 20 frames in total) per GPU using $2$ Tesla V100 GPUs. The training time on YouTube-VOS is about 5 days. For DAVIS, we use a learning rate of $0.006$ for $50,000$ steps with a batch size of 3 videos (\\emph{i.e.}, 15 frames in total) per GPU using $2$ GPUs.\nWe apply flipping, scaling, and balanced random-crop as data augmentations. The cropped window size is $465\\times 465$. For the multi-scale testing, we apply the scales of $\\{1.0, 1.15, 1.3, 1.5\\}$ and $\\{2.0, 2.15, 2.3\\}$ for YouTube-VOS and DAVIS, respectively. CFBI achieves similar results in PyTorch~\\cite{pytorch} and PaddlePaddle~\\cite{paddlepaddle}.\n\n\n\n\n\n\n\\setlength{\\intextsep}{-3pt}\n\\begin{wraptable}[28]{r}{0.55\\textwidth}\n\t\\centering\\vspace{-9mm}\n\t\\caption{The quantitative evaluation on YouTube-VOS~\\cite{youtubevos}. F, S, and $^*$ separately denote fine-tuning at test time, using simulated data in the training process and performing model ensemble in evaluation. CFBI$^{MS}$ denotes using a multi-scale and flip strategy in evaluation.}\\label{tab:youtubevos}\n\t\\begin{tabular}{lccccccc}\n\\toprule[1.5pt]\n            &  &  &  &  \\multicolumn{2}{c}{Seen}  &    \\multicolumn{2}{c}{Unseen} \\\\\n\\midrule[1pt]\n Methods & F & S  & Avg & $\\mathcal{J}$ & $\\mathcal{F}$ & $\\mathcal{J}$ & $\\mathcal{F}$ \\\\\n\\midrule[1pt]\n\\multicolumn{8}{c}{\\textit{Validation 2018 Split}} \\\\\n\\midrule[1pt]\nAG~\\cite{agame} &   &   &  66.1  &  67.8  &  -  &  60.8  &  - \\\\\nPReM~\\cite{premvos} & \\checkmark &   &  66.9  &  71.4  &  75.9  &  56.5  &  63.7 \\\\\nBoLT~\\cite{boltvos} & \\checkmark &   &  71.1  &  71.6  &  -  &  64.3  &  - \\\\\nSTM$^-$~\\cite{spacetime} &  &   &  68.2  &  -  &  -  &  -  &  -  \\\\\nSTM~\\cite{spacetime} &  & \\checkmark  &  79.4  &  79.7  &  84.2  &  72.8  &  80.9  \\\\\n\\hline\nCFBI &  &   &  \\textbf{81.4}  &  \\textbf{81.1}  & \\textbf{85.8}  & \\textbf{75.3}  & \\textbf{83.4}  \\\\\nCFBI$^{MS}$ &  &   &  \\textbf{82.7}  &  \\textbf{82.2}  & \\textbf{86.8}  & \\textbf{76.9}  & \\textbf{85.0}  \\\\\n\\midrule[1pt]\n\\multicolumn{8}{c}{\\textit{Validation 2019 Split}} \\\\\n\\midrule[1pt]\nCFBI &  &   &  \\textbf{81.0}  &  \\textbf{80.6}  & \\textbf{85.1}  & \\textbf{75.2}  & \\textbf{83.0}  \\\\\nCFBI$^{MS}$ &  &   &  \\textbf{82.4}  &  \\textbf{81.8}  & \\textbf{86.1}  & \\textbf{76.9}  & \\textbf{84.8}  \\\\\n\\bottomrule[1.5pt]\n\\multicolumn{8}{c}{\\textit{Testing 2019 Split}} \\\\\n\\midrule[1pt]\nMST$^*$~\\cite{mst} & & \\checkmark &  81.7  &  80.0  &  83.3  &  \\textbf{77.9}  &  85.5 \\\\\nEMN$^*$~\\cite{emn} & & \\checkmark  &  81.8  &  \\textbf{80.7}  &  \\textbf{84.7}  &  77.3  &  84.7 \\\\\n\\hline\nCFBI & & &  81.5  &  79.6  & 84.0 & 77.3  & 85.3  \\\\\nCFBI$^{MS}$ & & &  \\textbf{82.2}  &  80.4  & \\textbf{84.7}  & \\textbf{77.9}  & \\textbf{85.7}  \\\\\n\\bottomrule[1.5pt]\n\\end{tabular}\n\\end{wraptable}\n\n\n\\section{Experiments}\n\nFollowing the previous state-of-the-art method~\\cite{spacetime},\nwe evaluate our method on YouTube-VOS~\\cite{youtubevos}, DAVIS 2016~\\cite{davis2016} and DAVIS 2017~\\cite{davis2017}. For the evaluation on YouTube-VOS, we train our model on the YouTube-VOS training set~\\cite{youtubevos} (3471 videos). For DAVIS, we train our model on the DAVIS-2017 training set~\\cite{davis2017} (60 videos). Both DAVIS 2016 and 2017 are evaluated using an identical model trained on DAVIS 2017 for a fair comparison with the previous works~\\cite{feelvos,spacetime}. Furthermore, we provide DAVIS results using both DAVIS 2017 and YouTube-VOS for training following some latest works~\\cite{feelvos,spacetime}.\n\n\\setlength{\\intextsep}{0pt}\n\nThe evaluation metric is the $\\mathcal{J}$ score, calculated as the average IoU between the prediction and the ground truth mask, and the $\\mathcal{F}$ score, calculated as an average boundary similarity measure between the boundary of the prediction and the ground truth, and their average value ($\\mathcal{J}$\\&$\\mathcal{F}$). We evaluate our results on the official evaluation server or use the official tools.\n\n\n\n\n\n\n\\subsection{Compare with the State-of-the-art Methods}\n\n\n\n\\noindent \\textbf{YouTube-VOS}~\\cite{youtubevos} is the latest large-scale dataset for multi-object video segmentation. Compared to the popular DAVIS benchmark that consists of $120$ videos, YouTube-VOS is about 37 times larger. In detail, the dataset contains 3471 videos in the training set (65 categories), 507 videos in the validation set (additional 26 unseen categories), and 541 videos in the test set (additional 29 unseen categories). Due to the existence of unseen object categories, the YouTube-VOS validation set is much suitable for measuring the generalization ability of different methods. \n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figs\/quality.pdf}\n    \\caption{Qualitative results on DAVIS 2017 and YouTube-VOS. In the first video, we succeed in tracking many similar-looking sheep. In the second video, our CFBI tracks the person and the dog with a red mask after occlusion well. In the last video, CFBI fails to segment one hand of the right person (the white box). A possible reason is that the two persons are too similar and close.}\n    \\label{fig:quality}\n\\end{figure}\n\n\n\\begin{wraptable}[21]{r}{0.58\\textwidth}\n\\centering\n\\caption{The quantitative evaluation on DAVIS 2016~\\cite{davis2016} validation set. (\\textbf{Y}) denotes using YouTube-VOS for training.}\\label{tab:davis2016}\n\\begin{tabular}{l c c c c c c}\n\\toprule[1.5pt]\nMethods  & F & S  & Avg & $\\mathcal{J}$ & $\\mathcal{F}$ & t\/s \\\\\n\\midrule[1pt]\nOSMN~\\cite{osmn} &  &   &  - & 74.0  &   & 0.14 \\\\\nPML~\\cite{pml} &  &   &  77.4 & 75.5  & 79.3  & 0.28 \\\\\nVideoMatch~\\cite{videomatch} &  &  &   80.9  & 81.0  & 80.8  & 0.32 \\\\\nRGMP$^-$~\\cite{rgmp} &  &   & 68.8  & 68.6  & 68.9  & 0.14 \\\\\nRGMP~\\cite{rgmp} &  & \\checkmark  & 81.8 &  81.5 & 82.0  & 0.14 \\\\\nA-GAME~\\cite{agame} (\\textbf{Y}) &  &   &  82.1 & 82.2  & 82.0  & \\textbf{0.07} \\\\\nFEELVOS~\\cite{feelvos} (\\textbf{Y}) &  &   & 81.7  &  81.1 &  82.2 & 0.45 \\\\\nOnAVOS~\\cite{onavos}{} & \\checkmark &   & 85.0  & 85.7  & 84.2  & 13 \\\\\nPReMVOS~\\cite{premvos} & \\checkmark &   & 86.8  & 84.9  & 88.6  & 32.8 \\\\\nSTMVOS~\\cite{spacetime} &  & \\checkmark  & 86.5  & 84.8  & 88.1  & 0.16 \\\\\nSTMVOS~\\cite{spacetime} (\\textbf{Y}) &  & \\checkmark  &  \\textbf{89.3} & \\textbf{88.7}  & 89.9  & 0.16 \\\\\n\\hline\nCFBI  &   &   & 86.1  & 85.3 &  86.9 & 0.18 \\\\\nCFBI (\\textbf{Y})  &   &   & \\textbf{89.4}  & 88.3 & \\textbf{90.5}  & 0.18 \\\\\nCFBI$^{MS}$ (\\textbf{Y}) &   &   &  \\textbf{90.7}  & \\textbf{89.6}  & \\textbf{91.7} & 9 \\\\\n\\bottomrule[1.5pt]\n\\end{tabular}\n\\end{wraptable}\n\n\nAs shown in Table~\\ref{tab:youtubevos}, we compare our method to existing methods on both Validation 2018 and Testing 2019 splits. Without using any bells and whistles, like fine-tuning at test time~\\cite{osvos,onavos} or pre-training on larger augmented simulated data~\\cite{rgmp,spacetime}, our method achieves an average score of $\\mathbf{81.4\\%}$, which significantly outperforms all other methods in every evaluation metric. Particularly, the $81.4\\%$ result is $2.0\\%$ higher than the previous state-of-the-art method, STMVOS, which uses extensive simulated data from~\\cite{coco,voc,cheng2014global,semantic,shi2015hierarchical} for training. Without simulated data, the performance of STMVOS will drop from $79.4\\%$ to $68.2\\%$. Moreover, we further boost our performance to $\\mathbf{82.7\\%}$ by applying a multi-scale and flip strategy during the evaluation. \n\n\nWe also compare our method with two of the best results on the Testing 2019 split, \\emph{i.e.}, \\textit{Rank 1} (EMN~\\cite{emn}) and \\textit{Rank 2} (MST~\\cite{mst}) results in the 2nd Large-scale Video Object Segmentation Challenge. Without applying model ensemble, our single-model result ($\\mathbf{82.2\\%}$) outperforms the \\textit{Rank 1} result ($81.8\\%$) in the unseen and average metrics, which further demonstrates our generalization ability and effectiveness.\n\n\n\n\n\n\n\\noindent \\textbf{DAVIS 2016}~\\cite{davis2016} contains 20 videos annotated with high-quality masks each for a single target object. We compare our CFBI method with state-of-the-art methods in Table~\\ref{tab:davis2016}. On the DAVIS-2016 validation set, our method trained with an additional YouTube-VOS training set achieves an average score of $\\mathbf{89.4\\%}$, which is slightly better than STMVOS ($89.3\\%$), a method using simulated data as mentioned before. The accuracy gap between CFBI and STMVOS on DAVIS is smaller than the gap on YouTube-VOS. A possible reason is that DAVIS is too small and easy to over-fit.\nCompare to a much fair baseline (\\emph{i.e.}, FEELVOS) whose setting is same to ours, the proposed CFBI not only achieves much better accuracy ($\\mathbf{89.4\\%}$ \\emph{vs.}\\hspace{-0.8mm} $81.7\\%$) but also maintains a comparable fast inference speed ($0.18s$ \\emph{vs.}\\hspace{-0.8mm} $0.45s$). After applying multi-scale and flip for evaluation, we can improve the performance from $\\mathbf{89.4\\%}$ to $\\mathbf{90.1\\%}$. However, this strategy will cost much more inference time ($9s$).\n\n\n\n\\setlength{\\intextsep}{-3pt}\n\\begin{wraptable}[29]{r}{0.52\\textwidth}\n\n\\caption{The quantitative evaluation on DAVIS-2017~\\cite{davis2017}.}\\label{tab:davis2017}\n\\begin{center}\n\\begin{tabular}{l c c c c c}\n\\toprule[1.5pt]\n Methods & F & S  & Avg & $\\mathcal{J}$ & $\\mathcal{F}$ \\\\\n\\midrule[1pt]\n\\multicolumn{6}{c}{\\textit{Validation Split}} \\\\\n\\midrule[1pt]\nOSMN~\\cite{osmn} &   &   &  54.8  & 52.5  & 57.1 \\\\\nVideoMatch~\\cite{videomatch} &   &  & 62.4  & 56.5 & 68.2 \\\\\nOnAVOS~\\cite{onavos} & \\checkmark  &   &  63.6  & 61.0  & 66.1 \\\\\nRGMP~\\cite{rgmp} &   & \\checkmark   & 66.7 & 64.8   & 68.6 \\\\\nA-GAME~\\cite{agame} (\\textbf{Y}) &   &   & 70.0  &  67.2  & 72.7 \\\\\nFEELVOS~\\cite{feelvos} (\\textbf{Y}) &   &   &  71.5  &  69.1  & 74.0 \\\\\nPReMVOS~\\cite{premvos} & \\checkmark  &   &  77.8  &  73.9  & 81.7 \\\\\nSTMVOS~\\cite{spacetime}  &   & \\checkmark  &  71.6  & 69.2  & 74.0 \\\\\nSTMVOS~\\cite{spacetime} (\\textbf{Y}) &   & \\checkmark  &  \\textbf{81.8}  & \\textbf{79.2}  & 84.3 \\\\\n\\hline\nCFBI  &   &   &  74.9  & 72.1  & 77.7 \\\\\nCFBI (\\textbf{Y}) &   &   &  \\textbf{81.9}  & \\textbf{79.1}  & \\textbf{84.6} \\\\\nCFBI$^{MS}$ (\\textbf{Y}) &   &   &  \\textbf{83.3}  & \\textbf{80.5}  & \\textbf{86.0} \\\\\n\\bottomrule[1.5pt]\n\\multicolumn{6}{c}{\\textit{Testing Split}} \\\\\n\\midrule[1pt]\nOSMN~\\cite{osmn} &   &   &  41.3  & 37.7  & 44.9 \\\\\nOnAVOS~\\cite{onavos} & \\checkmark  &  &  56.5  & 53.4  & 59.6 \\\\\nRGMP~\\cite{rgmp} &   &  \\checkmark & 52.9 & 51.3   & 54.4 \\\\\nFEELVOS~\\cite{feelvos} (\\textbf{Y}) &   &  &  57.8  &  55.2  & 60.5 \\\\\nPReMVOS~\\cite{premvos}  & \\checkmark  &   &  71.6  &  67.5  & 75.7 \\\\\nSTMVOS~\\cite{spacetime} (\\textbf{Y}) &   & \\checkmark  &  72.2  & 69.3  & 75.2 \\\\\n\\hline\nCFBI (\\textbf{Y})&   &  &  \\textbf{74.8}  & \\textbf{71.1}  & \\textbf{78.5} \\\\\nCFBI$^{MS}$ (\\textbf{Y}) &   &  &  \\textbf{77.5}  & \\textbf{73.8}  & \\textbf{81.1} \\\\\n\\bottomrule[1.5pt]\n\\end{tabular}\n\\end{center}\n\n\\end{wraptable}\n\n\n\n\\noindent \\textbf{DAVIS 2017}~\\cite{davis2017} is a multi-object extension of DAVIS 2016. The validation set of DAVIS 2017 consists of 59 objects in 30 videos.\nNext, we evaluate the generalization ability of our model on the popular DAVIS-2017 benchmark. \n\n\nAs shown in Table~\\ref{tab:davis2017}, our CFBI makes significantly improvement over FEELVOS ($\\mathbf{81.9\\%}$ \\emph{vs.}\\hspace{-0.8mm} $71.5\\%$). Besides, our CFBI without using simulated data is slightly better than the previous state-of-the-art method, STMVOS ($\\mathbf{81.9\\%}$ \\emph{vs.}\\hspace{-0.8mm} $81.8\\%$). We show some examples compared with STMVOS in Fig.~\\ref{fig:comparison}. Same as previous experiments, the augmentation in evaluation can further boost the results to a higher score of $\\mathbf{83.3\\%}$. We also evaluate our method on the testing split of DAVIS 2017, which is much more challenging than the validation split. As shown in Table~\\ref{tab:davis2017}, we significantly outperforms STMVOS ($72.2\\%$) by $\\textbf{2.6\\%}$. By applying augmentation, we can further boost the result to $\\textbf{77.5\\%}$. The strong results prove that our method has the best generalization ability among the latest methods.\n\n\n\n\n\n\\noindent \\textbf{Qualitative Results} We show more results of CFBI on the validation set of DAVIS 2017 ($\\mathbf{81.9\\%}$) and YouTube-VOS ($\\mathbf{81.4\\%}$) in Fig.~\\ref{fig:quality}. It can be seen that CFBI is capable of producing accurate segmentation under challenging situations, such as large motion, occlusion, blur, and similar objects. In the \\emph{sheep} video, CFBI succeeds in tracking five selected sheep inside a crowded flock. In the \\emph{judo} video, CFBI fails to segment one hand of the right person. A possible reason is that the two persons are too similar in appearance and too close in position. Besides, their hands are with blur appearance due to the fast motion.\n\n\n\n\n\\subsection{Ablation Study}\n\n\\setlength{\\intextsep}{-2pt}\n\\begin{wraptable}[15]{r}{0.4\\textwidth}\n\n\\centering\n\\caption{Ablation of background embedding. P and I separately denote the pixel-level matching and instance-level attention. $^*$ denotes removing the foreground and background bias.}\\label{tab:ablation_a}\n\\setlength{\\tabcolsep}{6.5pt}\n\\begin{tabular}{l c c c c}\n    \\toprule[1.5pt]\n         P & I & Avg & $\\mathcal{J}$ & $\\mathcal{F}$ \\\\\n    \\midrule[1pt]\n        \\checkmark  & \\checkmark & 74.9 & 72.1 & 77.7 \\\\\n    \\hline\n        \\checkmark$^*$  & \\checkmark & 72.8 & 69.5 & 76.1 \\\\\n        \\checkmark  &  & 73.0 & 69.9 & 76.0 \\\\\n          & \\checkmark & 72.3 & 69.1 & 75.4 \\\\\n          &  & 70.9 & 68.2 & 73.6 \\\\\n    \\bottomrule[1.5pt]\n\\end{tabular}\n\\end{wraptable}\n\nWe analyze the ablation effect of each component proposed in CFBI on the DAVIS-2017 validation set. Following FEELVOS, we only use the DAVIS-2017 training set as training data for these experiments. \n\n\n\n\\noindent \\textbf{Background Embedding.} As shown in Table~\\ref{tab:ablation_a}, we first analyze the influence of removing the background embedding while keeping the foreground only as~\\cite{feelvos,osmn}. Without any background mechanisms, the result of our method heavily drops from $74.9\\%$ to $70.9\\%$.\nThis result shows that it is significant to embed both foreground and background features collaboratively. Besides, the missing of background information in the pixel-level matching or the instance-level attention will decrease the result to $73.0\\%$ or $72.3\\%$ separately. \nThus, compared to instance-level attention, the pixel-level matching performance is more sensitive to the effect of background embedding. A possible reason for this phenomenon is that the possibility of existing some background pixels similar to the foreground is higher than some background instances. Finally, we remove the foreground and background bias, $b_F$ and $b_B$, from the distance metric and the result drops to $72.8\\%$, which further shows that the distance between foreground pixels and the distance between background pixels should be separately considered.\n\n\\setlength{\\intextsep}{-3pt}\n\\begin{wraptable}[11]{r}{0.55\\textwidth}\n\n\\centering\n\\caption{Ablation of other components.}\\label{tab:ablation_b}\n\\begin{tabular}{l c c c c}\n    \\toprule[1.5pt]\n          & Ablation & Avg & $\\mathcal{J}$ & $\\mathcal{F}$ \\\\\n    \\midrule[1pt]\n        0  & Ours (CFBI) & 74.9 & 72.1 & 77.7 \\\\\n    \\hline\n        1  & w\/o multi-local windows & 73.8 & 70.8 & 76.8 \\\\\n        2  & w\/o sequential training & 73.3 & 70.8 & 75.7 \\\\\n        3  & w\/o collaborative ensembler & 73.3 & 70.5 & 76.1 \\\\\n       \n        4  & w\/o balanced random-crop & 72.8 & 69.8 & 75.8 \\\\\n        5  & w\/o instance-level attention & 72.7 & 69.8 & 75.5 \\\\\n       \n    \\hline\n        6  & baseline (FEELVOS) & 68.3 & 65.6 & 70.9 \\\\\n    \\bottomrule[1.5pt]\n\\end{tabular}\n\\end{wraptable}\n\n\\noindent \\textbf{Other Components.} The ablation study of other proposed components is shown in Table~\\ref{tab:ablation_b}. Line 0 ($74.9\\%$) is the result of proposed CFBI, and Line 6 ($68.3\\%$) is our baseline method reproduced by us. Under the same setting, our CFBI significantly outperforms the baseline.\n\n\n\nIn line 1, we use only one local neighborhood window to conduct the local matching following the setting of FEELVOS, which degrades the result from $74.9\\%$ to $73.8\\%$. It demonstrates that our multi-local matching module is more robust and effective than the single-local matching module of FEELVOS. Notably, the computational complexity of multi-local matching dominantly depends on the biggest local window size because we use the intermediate results of the local matching of the biggest window to calculate on smaller windows.\n\n\n\nIn line 2, we replace our sequential training by using ground-truth masks instead of network predictions as the previous mask. By doing this, the performance of CFBI drops from $74.9\\%$ to $73.3\\%$, which shows the effectiveness of our sequential training under the same setting.\n\n\n\n\n\nIn line 3, we replace our collaborative ensembler with 4 depth-wise separable convolutional layers. This architecture is the same as the dynamic segmentation head of~\\cite{feelvos}. Compared to our collaborative ensembler, the dynamic segmentation head has much smaller receptive fields and performs $1.6\\%$ worse.\n\n\nIn line 4, we use normal random-crop instead of our balanced random-crop during the training process. In this situation, the performance drops by $2.1\\%$ to $72.8\\%$ as well. As expected, our balanced random-crop is successful in relieving the model form biasing to background attributes.\n\nIn line 5, we disable the use of instance-level attention as guidance information to the collaborative ensembler, which means we only use pixel-level information to guide the prediction. In this case, the result deteriorates even further to $72.7$, which proves that instance-level information can further help the segmentation with pixel-level information.\n\n\nIn summary, we explain the effectiveness of each proposed component of CFBI. For VOS, it is necessary to embed both foreground and background features. Besides, the model will be more robust by combining pixel-level information and instance-level information, and by using more local windows in the matching between two continuous frames. Apart from this, the proposed balanced random-crop and sequential training are useful but straightforward in improving training performance. \n\n\n\n\\section{Conclusion}\n\nThis paper proposes a novel framework for video object segmentation by introducing collaborative foreground-background integration and achieves new state-of-the-art results on three popular benchmarks. Specifically, we impose the feature embedding from the foreground target and its corresponding background to be contrastive. Moreover, we integrate both pixel-level and instance-level embeddings to make our framework robust to various object scales while keeping the network simple and fast. We hope CFBI will serve as a solid baseline and help ease the future research of VOS and related areas, such as video object tracking and interactive video editing.\n\n\\noindent \\textbf{Acknowledgements.} This work is partly supported by ARC DP200100938 and ARC DECRA DE190101315.\n\n\n\n\n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nFeSi is a non-magnetic \\cite{wyi63}, narrow-gap semiconductor\n\\cite{jwwwa67,ktidv68} at low temperatures.  Its magnetic\nsusceptibility $\\chi(T)$ increases with temperature and passes through\na maximum at $T \\sim 500$ K \\cite{jwwwa67}.\nFeSi becomes metallic above 300 K\\cite{ktidv68,wwh65}.  The\nsubstitution of Co for Fe (about 10\\% Co) yields a magnet with a\nhelical spin order \\cite{bvr83}.  Local density\nfunctional\\cite{hohe64} band structure calculations\n\\cite{mh93,fkd94,gpb94} give the correct value of the semiconducting\ngap (about 0.1 eV), but can not explain the large magnitude of\n$\\chi(T)$. According to infrared and optical measurements\n\\cite{sfzmt93}, the gap of 50 meV is gradually filled with increasing\ntemperature, with new spectral weight which can not be explained\nwithin the conventional band structure picture.  In connection with a\ntemperature-induced local moment, a model based on unified\nspin-fluctuation theory was proposed in Ref.~\\onlinecite{tm79} which explains\n$\\chi(T)$ using a model density of states (DOS).\n\nIn spite of the large number of publications devoted to measurements\nof X-ray photoelectron spectra (XPS) and ultraviolet photoelectron\nspectra (UPS) of FeSi\\cite{slfsk89,kmmos92,cymty94,oal87,ksnik82,ssmfi94},\nmost measurements were performed using polycrystalline samples which do not\nallow precise measurements on clean surfaces free of contamination.\nIn this paper, we present a full set of precise spectral measurements,\ndata, including XPS of the valence band and X-ray emission valence spectra\nfor both components of FeSi, which were obtained on the same single\ncrystal, and providing experimental information about the distribution of\ntotal and partial DOS in the valence band.\n\nThe already published information on the calculated\nelectronic structure of FeSi presented in Ref.\\cite{mh93,fkd94,gpb94}\nreveals only total DOS and Fe $3d$, Si $3s$ partial DOS distributions\nin Ref. \\onlinecite{fkd94}, and Fe $3d$, Si $3p$\nDOS for the CsCl-type structure (which is rather\ndifferent from that known for the bulk FeSi\\cite{wyi63}) in\nRef. \\onlinecite{gpb94}.  Because of this we performed a new set of\nband structure calculations of FeSi by two independent methods --\nlinearized muffin-tin orbitals (LMTO) and linearized augmented plane\nwave (LAPW) -- which give more detailed information about the total\nand the Fe $3d$, Fe $4p$, Si $3s$, Si $3d$ and Si $3p$ partial\nDOS distributions.\n\n\\section{Experimental}\n\\label{sec:exp}\n\nThe Fe $L_{\\alpha}$ ($2p$--$3d4s$ transition) X-ray emission spectrum\nwas measured on the RSM-500 type X-ray vacuum spectrometer with a\ndiffraction grating ($N=600$~lines\/mm and $R=6$~m) and electron\nexcitation. The spectra were recorded in the second order of\nreflection by a secondary electron multiplier with a CsI\nphotocathode. The energy resolution was about 0.35--0.40 eV.  The\nX-ray tube was operated at $V=4.6$~keV, $I=0.4$~mA.\n\nThe Si $K_{\\beta_{1,3}}$ ($1s$--$3p$ transition) X-ray emission\nspectrum was measured using a fluorescent Johan-type vacuum\nspectrometer with a position-sensitive detector \\cite{dckgo84}. The Pd\n$L$-X-ray radiation from a special sealed X-ray tube was used for the\nexcitation of the fluorescent Si$K_{\\beta_{1,3}}$ spectra. A quartz\n$(10\\bar{1}0)$ single crystal curved to $R=1400$~mm served as a\ncrystal-analyzer. The spectra were measured with an energy resolution\nof approximately 0.2--0.3~eV.  The X-ray tube was operated at $V=25$~keV,\n$I=50$~mA.\n\nThe Si $L_{2,3}$ ($2p$--$3s3d$ transition) X-ray emission spectra of\nFeSi were taken from Ref. \\onlinecite{kvswk92}, and the Fe\n$K_{\\beta_5}$ ($1s$--$4p$ transition) X-ray emission spectrum was\nreproduced from Ref.~\\onlinecite{kolo70}.\n\nThe XPS valence band spectra of FeSi were measured using a\nPerkin-Elmer ESCA spectrometer (PHI 5600 ci, monochromatized Al\n$K_{\\alpha}$ radiation). The FeSi single crystal was cleaved in high\nvacuum prior to the XPS measurements.  The XPS spectra were calibrated\nbased on the Au $4f$-spectra of Au metal ($E_b$=84.0~eV).\n\nX-ray emission spectra have been brought to the scale of binding\nenergies with respect to the Fermi level using the binding energies of\nthe relevant initial (core level) states of the X-ray transitions as\nmeasured by the XPS technique.  Corresponding binding energies are\n$E_b({\\rm Fe~}2p)=706.7$~eV, $E_b({\\rm Si~}2p)=99.3$~ eV. The values\nof $E({\\rm Fe~}K_{\\alpha_1})=6403.86$~ eV and $E({\\rm\nSi~}K_{\\alpha_1})=1740.1$~eV were taken for comparison of Fe\n$L_{\\alpha}$ and Fe $K_{\\beta_5}$, Si $L_{2,3}$ and Si\n$K_{\\beta_{1,3}}$ X-ray emission spectra of FeSi.\n\nThe measured XPS and X-ray emission spectra are shown in\nFig.~\\ref{spec}.\n\n\\section{Details of calculation}\n\\label{sec:calc}\n\nElectronic structure calculations have been performed for the cubic FeSi\nstructure (8 atoms\/cell, space group $P2_{1}3$) as determined in\nRef.~\\onlinecite{wyi63} and discussed in detail in\nRef.~\\onlinecite{mh93}.  We have used the cubic lattice constant of\n$a=4.493$ \\AA, with Fe atoms occupying the $(0.1358,0.1358,0.1358)a$ and\nequivalent positions of the $B20$ structure, while Si atoms occupy the\n$(0.844,0.844,0.844)a$ and equivalent positions.\n\nIn the calculations using the tight-binding LMTO method \\cite{tblmto},\nwe used space-filling atomic spheres of equal size on Fe and Si sites\n($R=1.394$ \\AA) and no empty spheres were introduced.  The\nexchange-correlation potential as proposed by von Barth and\nHedin \\cite{vbh} was used. DOS calculated by the tetrahedron\nmethod over 470 {\\bf k}-points in the irreducible part of the\nBrillouin zone are shown in Fig.~\\ref{spec}, and compared with the\nexperimental spectra.  Our calculated electron bands are very close to\nthose obtained by Fu {\\it et al.}\\cite{fkd94} using the augmented\nspherical wave method, and our calculated DOS agree\nwell with those of Ref.~\\onlinecite{fkd94,cfu}.  We found a direct gap of\n0.14~eV at the $X$ point of the Brillouin zone and an indirect gap of\n0.05--0.08~eV, in agreement with the\nresistivity measurement data reported in Ref.~\\onlinecite{fkd94}.\n\nSome small deviations from the bands calculated in\nRef.~\\onlinecite{mh93} using the LAPW method without any shape\napproximation imposed on the potential seem to have quite negligible\neffect on the densities of states, in the present context of making a\ncomparison with X-ray spectra using the LMTO method.\nWe also carried out an independent LAPW\\cite{ande75,sing94}\ncalculation, in which we used the local density approximation\n\\cite{hohe64} form for exchange and correlation given by Ceperley and\nAlder \\cite{cepe80}, as parameterized by Perdew and\nZunger\\cite{perd81}.  In the expansion of the charge density and\npotential inside the muffin-tin spheres ($R_{\\rm Fe}=R_{\\rm Si}=1.111$\n\\AA), lattice harmonics up to angular momentum of $l=8$ have been\nused, while the interstitial region was described by a plane wave\nexpansion.  Core orbitals were treated self-consistently, retaining,\nhowever, only the spherically-symmetric part of their density. Scalar\nrelativistic effects (neglecting spin-orbit coupling) were included\nfor the valence states, and full relativistic effects were included\nfor the core states.  A large basis set consisting of both real-space\norbitals (inside the muffin-tin regions) and plane waves was used. The\ntotal number of basis functions used was $\\sim$ 570 per unit cell of 8\natoms.  The Brillouin zone summations were performed using 24 {\\bf\nk}-vectors in the irreducible section of the Brillouin zone in the\nself-consistency loop.  The densities of states calculated using\nthe LAPW method, with the tetrahedron integration\nover 176 {\\bf k}-points in the irreducible\npart of the Brillouin zone, are shown in\nFigs.~\\ref{dospart} and \\ref{dostot}.\nNote that the total DOS calculated in both LMTO\nand LAPW methods per unit cell (i.e. 4 formula units) are in good\nabsolute agreement, whereas the partial DOS are somehow smaller in the\nLAPW calculation (Fig.~\\ref{dospart} because they are attributed\nto non-overlapping muffin-tin spheres rather than space-filling\natomic spheres as in LMTO.\nThe expanded part of the total\ndensities of states plot in the vicinity of the gap is shown at the\ninset in Fig.~\\ref{dostot}. Two sharp peaks formed mostly by Fe $d$\nstates are separated by the indirect gap of about 0.065 eV.\n\nThere seems to be generally good agreement between the several LDA-based\nband structure calculation schemes in reproducing the bands, total\ndensities of states, and bandwidths in FeSi. Below, we concentrate on\ndiscussing particular features of partial densities of states of the\nconstituents as revealed in the X-ray emission spectra and the present\ncalculations.\n\n\\section{Experimental results and discussion}\n\\label{sec:disc}\n\nThe X-ray photoelectron valence band spectrum\nand X-ray emission valence band spectra\nof Fe (Fe $L_{\\alpha}$, Fe $K_{\\beta_5}$) and Si (Si $L_{2,3}$, Si\n$K_{\\beta_{1,3}}$) are presented in Fig.~\\ref{spec}. As is known,\nthe XPS valence band spectrum gives an information about the total\nDOS distribution (accurate up to the weight function depending\non the atomic photoemission cross-sections, see Ref.~\\onlinecite{yl85}),\nwhereas the Fe $L_{\\alpha}$, Fe $K_{\\beta_5}$, Si $L_{2,3}$,\nSi $K_{\\beta_{1,3}}$ X-ray emission spectra (in accordance to dipole\nselection rules) -- give information about partial Fe $3d4s$, Fe $4p$,\nSi $3s3d$ and Si $3p$ densities of states, respectively.\nThese spectra are compared in Fig.~\\ref{spec} with the results\nof the LMTO band structure calculations by aligning the calculated\nand experimental positions of the Fermi level. We note that\nthe experimental XPS valence band spectrum reproduces the total DOS\ndistribution of FeSi quite well.  Especially, it is very important\nthat in our experiments we have found the splitting of the main peak\nof XPS spectrum in the range 0--2 eV which corresponds to the\nFe $3d$ band splitting.  This distinct splitting was not found before\nin XPS\\cite{slfsk89} and UPS\\cite{kmmos92,cymty94,ksnik82,ssmfi94}\nmeasurements on FeSi, and its absence was attributed in\nRef.~\\onlinecite{ssmfi94} to a hole lifetime broadening which increases\nwith binding energies. The similar splitting of the XPS valence\nband spectrum of FeSi was found in Ref.~\\onlinecite{oal87} only\nin the measurements done at low temperatures ($T$=120 K).\nIn our opinion, this discrepancy between fine structure in the present\nand previous XPS (UPS) valence band spectra is due to the high quality\nof the FeSi single crystal which was used for the present measurements.\nThe low-energy Si $3p$ and Si $3s$ subbands are reflected\nin the XPS valence band spectrum as low-intensity\n(due to low values of respective photoinization\ncross-sections \\cite{yl85}) features located at binding energies of\napproximately 4.5 and 9 eV, respectively.\n\nThe experimental Fe $L_{\\alpha}$ X-ray emission spectrum reproduces\nfairly well the position of the center of gravity of calculated Fe\n$3d$ DOS distribution in the valence band of FeSi, but\nnot the splitting of the Fe $3d$ band. This is due to the large total\ndistortion of the Fe $L_{\\alpha}$ X-ray emission spectrum which\nincludes the instrumental distortion (about 0.4 eV) and the width of\nthe inner (core) Fe $2p$ level (about 0.8-1.0 eV) which is determined\nby the lifetime of the core-level vacancy under an X-ray transition.\n\nThe experimental Fe $K_{\\beta_5}$ spectrum shows two maxima located at\nbinding energies of approximately 3.5 and 9.0 eV that is in accordance\nwith the theoretical distribution of the Fe $4p$ partial DOS.\nIt is also seen from both the theoretical and experimental\nspectra that Fe $4p$ states are hybridized mostly with Si $3p$ and Si\n$3d$ states.\n\nThe experimental Si $L_{2,3}$ X-ray emission spectra corresponds to\nthe $2p$--$3s3d$ transition. Usually it is considered that the Si $3d$\nstates do not take part in the chemical bonding in $3d$ transition\nmetal silicides. As was emphasized in Ref.~\\onlinecite{kvswk92}, the\nhigh-energy subband of Si$L_{2,3}$ X-ray emission spectra of\ntransition metal silicides FeSi, MnSi and NiSi, as well as that of\ndisilicides FeSi$_2$, MnSi$_2$ and NiSi$_2$, cannot be explained\nwithout an assumption that Si $3d$ states contribute to the chemical\nbonding. Subsequently, the same conclusion was drawn for Pt silicides\n(Pt$_2$Si, PtSi) in Ref.~\\onlinecite{yhkin94} based on an analysis of\nthe Si $L_{2,3}$ X-ray emission spectra of these compounds. This\nconclusion is again confirmed now for FeSi by two independent sets of\nband structure calculations (see Figs.~\\ref{spec} and \\ref{dospart}).\nOne can see from Fig.~\\ref{spec} that the Si $3s$ DOS is\nconcentrated in the range of binding energies about 6-13 eV where the\nmost intensive low-energy subband of the Si $L_{2,3}$ X-ray emission\nspectrum is located. The center of gravity of the Si $3d$ states\ndistribution corresponds to a binding energy about 2 eV where the\nhigh-energy maximum of the Si $L_{2,3}$ spectrum is situated.  The Si\n$3d$ states are strongly hybridized with Fe3$d$ states, and the same\nsplitting of the Si $3d$ band (about 2 eV) as that in the Fe $3d$\nsubband is observed in the band structure calculations.\n\nThe energy position and fine structure of the Si $K_{\\beta_{1,3}}$\nX-ray emission spectrum is in a good accord with the Si3 $p$ partial\nDOS distribution (see Fig.~\\ref{spec}).\n\n\\section{Conclusion}\n\\label{sec:conclu}\n\nThe results of high-energy spectroscopy measurements of FeSi single\ncrystals, including the XPS valence band spectrum and the X-ray emission\nvalence band spectra of both constituents, are presented. They are\ncompared with two independent {\\it ab initio} band structure\ncalculations of FeSi, performed using the LMTO and LAPW methods, and\ngood agreement between the experimental and theoretical spectra is\nfound.\n\n\\acknowledgements\n\nThe authors are grateful to Castor Fu for sending his data on the\ncalculated density of states of FeSi.  AVP, StU and MN appreciate the\nfinancial support of the Deutsche Forschungsgemeinschaft (SFB~225,\nGraduate College).  This work was supported by Russian Foundation for\nFundamental Research (grant No.94-03-08040) and NATO International\nScientific Exchange Program (Project HTECH LG 940861). ZWL, BMK, and\nZPS thank the support by the University Research Funds of the\nUniversity of California at Davis.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\nAt the level of the thermodynamical description of out of equilibrium\ntransformations, the irreversibility of macroscopic processes is\nencoded in the second law expressed through strict inequalities\nsatisfied by various thermodynamic observables, like entropy and free\nenergies. At the more refined level of the statistical mechanical\ndescription of the same processes, the irreversibility ought to\nmanifest as an asymmetry in the probability distributions of the\nfluctuations of these same quantities.  The central and yet unsolved\nproblem is to find general methods, comparable to those available in\nequilibrium, to characterize these probability distributions. A major\nadvance in this direction has been made in the last two decades with\nthe development of Fluctuations Theorems (FT), which allow to\nconstrain the form of the distributions with a certain degree of\nuniversality and in conditions arbitrarily far from\nequilibrium~\\cite{Seifert}. This theoretical approach turned out to be\nvery useful, both in experimental and numerical studies, in the\ncharacterization of several nonequilibrium systems, such as colloidal\nparticles in harmonic traps~\\cite{Wang,Zon,imparato07,kim}, vibrated\ngranular media~\\cite{Sarracino,Naert,Gnoli}, models of coupled\nLangevin equations~\\cite{Farago,Joubaud,Crisanti}, driven stochastic\nLorentz gases~\\cite{Gradenigo,Gradenigo2}, and active\nmatter~\\cite{Kumar}, just to name a few examples.\n\nAn important development in the field has been achieved with the understanding\nof the impact of symmetry and symmetry breaking on the FT of currents\nin stationary diffusive systems~\\cite{Hurtado}.\nMore recently, it has been shown that relations in the form of FT are of\ngeneral occurrence also in the probability distributions of order\nparameters in equilibrium statistical mechanics, in connection with\nsymmetry breaking~\\cite{Gaspard}. Here we show that these equilibrium\nresults do feedback in the understanding of the nonequilibrium FT,\nshedding new light on the conditions under which the FT in the\nstandard form (namely, with a linear asymmetry function) is expected to\nhold~\\cite{Puglisi,Baiesi,Harris,Park}. In particular, we study the\nfluctuations of the heat exchanged with the environment by one or more\nBrownian oscillators in an interval of time $[t_w,t]$, during the\nrelaxation following the instantaneous quench from high to low\ntemperature. This protocol induces a \\emph{nonstationary dynamics}\nthat is noninvariant under time translation and therefore the heat\nexchanged along a given trajectory explicitly depends on the two times\n$(t_w,t)$. The study of the FT in similar processes has been addressed\nin~\\cite{Ritort,Picco,Zamponi}. Here, we derive exact expressions for\nthe heat probability density functions, for one, two and a large\nnumber of independent oscillators. Our analysis shows how, in the case\nof more than one oscillator, the heat probability obeys an FT which\ndeviates from the standard form and whose physical meaning can be\nrationalized by resorting to the analogy with the equilibrium problem\nin the presence of a nonuniform external field.\n\nIn the case of a large number of degrees of freedom, arising, for\ninstance, in the normal mode decomposition of an extended system, we\npresent a computation based on the steepest descent method, which\nallows us to obtain an accurate description of the large deviation\nfunction of the exchanged heat. Here, we will use\n  large deviation theory with a large number of degrees of freedom\n  (and not for long time intervals) \\cite{hugo,ld}. This kind of\nproblem was considered previously in Refs.~\\cite{Gonnella,Jo,Pechino},\nshowing that fluctuations undergo a condensation transition. Briefly,\nby this is meant that fluctuations in a multi-component system do\ncondense if there exists a critical threshold above which the\nfluctuation is feeded by {\\it just one} of the components (or degrees\nof freedom). Here, we analyse in detail the crossover region between\nthe normal and the condensed phase, providing an explicit expression\nfor the large deviation function. We also reconstruct numerically the\nphase diagram at finite temperature, in the parameter space\n$(t_w,\\tau=t-t_w)$, which shows a nontrivial reentrant behavior.\n\nThe paper is organized as follows. In Section~\\ref{zero} we present\nthe general framework within which various FT forms are derived and we\nargue on physical grounds for the particular form we adopt.  We also\nrecall in some detail how an FT arises in statics as a consequence of\nsymmetry breaking, highlighting its relevance for the dynamical\nproblem.  Section~\\ref{One} is devoted to the simplest case of a\nsingle oscillator. We compute exactly the probability distribution of\nheat fluctuations, we introduce the basic concept of time-dependent\neffective temperature and we derive the FT in the ``Gallavotti-Cohen''\nform~\\cite{Gallavotti}. In Section~\\ref{twobrownian} we consider the\ncase of two oscillators, and we analyse the modifications arising in\nthe FT form due to the presence of more than one degree of freedom.\nIn Section~\\ref{extended} we consider the case of a large number of\noscillators. We discuss the conditions leading to condensation and we\nmap out numerically the phase diagram in the parameter space. The\nconsequences on the FT are analysed.  Finally, conclusions are drawn\nin Section~\\ref{conclusions}.\n\n\n\n\n\n\\section{General setup}\n\\label{zero}\n\nThere exist many variants of FT whose derivation~\\cite{Maes,Seifert,Gawedzki}\ncan be unified into a single master theorem. Let us first outline the general setting\nwhich applies to all fluctuation problems, in or out of equilibrium. \nConsider a sample space $\\Omega$ with elements $\\sigma$.\nLet $\\mu(\\sigma)$ and $\\mu^{\\prime}(\\sigma)$ be two arbitrary probability measures\nover $\\Omega$ and let ${\\cal W}(\\sigma)$ be defined by\n\\be\n\\frac{\\mu^{\\prime}(\\sigma^*)}{\\mu(\\sigma)} = e^{-{\\cal W}(\\sigma)},\n\\label{I.1}\n\\ee\nwhere the $*$ operation denotes an involutory transformation of $\\Omega$ onto itself, \ni.e. $(\\sigma^*)^*=\\sigma$.\nIn the following this will be taken as the representation of the inversion element of\nthe $\\mathbb{Z}_2 =(\\mathbb{I},*)$ group, where $\\mathbb{I}$ is the identity operator.\nIntroducing an arbitrary function ${\\cal F}(\\sigma)$, from the above relation there follows\nthe identity\n\\be\n\\langle {\\cal F}(\\sigma)e^{-{\\cal W}(\\sigma)} \\rangle = \\langle {\\cal F}(\\sigma^*) \\rangle^{\\prime},\n\\label{I.2}\n\\ee\nwhere $\\langle \\cdot \\rangle$ and $\\langle \\cdot \\rangle^{\\prime}$ denote expectations with respect\nto $\\mu(\\sigma)$ and $\\mu^{\\prime}(\\sigma)$, respectively. Taking for ${\\cal F}(\\sigma)$ the characteristic \nfunction $\\theta_{{\\cal M}}(\\sigma|M)$ of a certain observable ${\\cal M}(\\sigma)$, that is\n\\be\n\\theta_{{\\cal M}}(\\sigma|M)  = \\left \\{ \\begin{array}{ll}\n        1,\\;\\; $if$ \\;\\; {\\cal M}(\\sigma) = M,\\\\\n        0,\\;\\; $if$ \\;\\; {\\cal M}(\\sigma) \\neq M,\n        \\end{array}\n        \\right .\n        \\label{I.2bis}\n        \\ee\nfrom Eq.~(\\ref{I.2}) follows\n\\be\n\\frac{P^{\\prime}({\\cal M}(\\sigma^*) = M)}{P({\\cal M}(\\sigma) = M)} = \n\\langle e^{-{\\cal W}(\\sigma)} | {\\cal M}(\\sigma) = M \\rangle,\n\\label{I.3}\n\\ee\nwhere $P^{\\prime}$ and $P$ are the probabilities of the events in the arguments induced\nby $\\mu^{\\prime}(\\sigma)$ and $\\mu(\\sigma)$, respectively, and\nin the right hand side there appears the expectation with respect to $\\mu(\\sigma)$,\nconditioned to ${\\cal M}(\\sigma) = M$. Defining the $*$ transformation on the\nset of random variables by ${\\cal M}(\\sigma) \\mapsto {\\cal M}^*(\\sigma) = {\\cal M}(\\sigma^*)$,\nwhich is also an involution, the above relation can be\nrecast in the form\n\\be\n\\frac{P^{\\prime}({\\cal M}^* = M)}{P({\\cal M} = M)} = \ne^{-{\\cal K}(M)},\n\\label{I.03}\n\\ee\nwith\n\\be\n{\\cal K}(M) = -\\ln \\langle e^{-{\\cal W}(\\sigma)} | {\\cal M}(\\sigma) = M \\rangle.\n\\label{I.04}\n\\ee\nIt appears, then, that Eq.~(\\ref{I.03}) is the transposition to the level of the observable\n${\\cal M}$ of the underlying basic relation~(\\ref{I.1}). In particular, regarding ${\\cal W}$\nas the bias which is necessary to apply on $\\mu$ in order to construct $\\mu^{\\prime}$ and ${\\cal K}$ as \nthe analogous quantity relating $P$ and $P^{\\prime}$, Eq.~(\\ref{I.04}) tells how these biases\nare related one to the other. \n\nAs long as $\\mu(\\sigma)$ and $\\mu^{\\prime}(\\sigma)$ are arbitrary, the above result does not\nhave predictive power. \nIt becomes the master FT when $\\mu^{\\prime}(\\sigma)$ is taken with a definite relation to $\\mu(\\sigma)$,\nwhich constrains the form\nof $P(M)$ if the right hand side of Eq.~(\\ref{I.03}) is accessible without having\nto actually compute the expectation, possibly via symmetry arguments.\nIn the simplest case $\\mu^{\\prime}(\\sigma)$ is taken to be the same as $\\mu(\\sigma)$. \nThen, from Eq.~(\\ref{I.1}) one has\n\\be\n\\frac{\\mu(\\sigma^*)}{\\mu(\\sigma)} = e^{-{\\cal W}(\\sigma)},\n\\label{I.1bis}\n\\ee\nshowing that ${\\cal W}$ is odd ${\\cal W}(\\sigma^*) = -{\\cal W}(\\sigma)$ and\ncharacterizes the asymmetry of the probability measure under inversion. \nNow, the question is to what extent this\nasymmetry is preserved or distorted  as the description is moved up\nfrom the microscopic level to the higher one of random variables. The answer from\nEq.~(\\ref{I.03}) is given by\n\\be\n\\frac{P({\\cal M}^* = M)}{P({\\cal M} = M)} = e^{-{\\cal K}(M)}.\n\\label{I.3bis}\n\\ee\nTaking ${\\cal M}$ to have definite parity, if it is even no information about the symmetry of\n$P$ is obtained and Eq.~(\\ref{I.3bis}) yields the conditional integral FT\n\\be\n\\langle e^{-{\\cal W}(\\sigma)} | {\\cal M}(\\sigma) = M \\rangle = 1.\n\\label{I.3tris}\n\\ee\nConversely, if ${\\cal M}$ is odd, one has\n\\be\n\\frac{P({\\cal M} = -M)}{P({\\cal M} = M)} = e^{-{\\cal K}(M)},\n\\label{I.3quater}\n\\ee\nwhose usefulness depends on the possibility of assessing the form of ${\\cal K}(M)$,\nwhich is called the asymmetry function (AF).\nIn particular, if ${\\cal M} \\propto {\\cal W}$, there follows\nan FT of the Gallavotti-Cohen~\\cite{Gallavotti} form with a linear AF\n\\be\n{\\cal K}(M) \\propto M.\n\\label{I.3pente}\n\\ee\nIf, instead, ${\\cal W}$ and ${\\cal M}$ are not simply related,\nthe meaning of ${\\cal K}(M)$ in general is not immediately transparent.\nFor an overview of the variety of the different FT forms arising in the general case see \nRef.~\\cite{Seifert}.\n\n\\subsection{FT and symmetry breaking in equilibrium}\n\nThe latter remarks are well clarified in the equilibrium context, used\nin Ref.~\\cite{Gaspard} to investigate the relation between FT and\nsymmetry breaking.  Let $\\Omega$ and $\\sigma$ be the system's phase\nspace and configurations, respectively. For definiteness, let $\\sigma\n= (s_1,...,s_N)$, with $s_i=\\pm 1$, be a spin configuration of a\nmagnetic system on the lattice in the presence of an external\nsite dependent field $\\mathbb{B}=\\{B_i\\}$, whose equilibrium state is described\nby the probability measure \\be \\mu(\\sigma) = \\mu_0(\\sigma)e^{\\beta\n  \\sum_i B_i s_i},\n\\label{equi.1}\n\\ee\nwhere $\\mu_0(\\sigma)$ is symmetric under spin inversion $\\sigma^* = (-s_1,...,-s_N)$,\nwhile the exponential term breaks explicitly the $\\mathbb{Z}_2$ symmetry.\nWe are interested in the fluctuations of the global magnetization ${\\cal M}(\\sigma) = \\sum_i s_i$.\nFrom Eq.~(\\ref{equi.1}) follows\n\\be\n{\\cal W}(\\sigma) = 2 \\beta \\sum_i B_i s_i,\n\\label{equi.01}\n\\ee\nand\n\\be\n{\\cal K}(M) = -\\ln \\langle e^{-2\\beta \\sum_i B_i s_i}|{\\cal M} = M \\rangle,\n\\label{equi.02}\n\\ee\nwhich takes a simple form only if the external field\nis uniform $B_i=B, \\forall i$, yielding an FT with the linear AF~\\cite{Gaspard}\n\\be\n{\\cal K}(M) = 2\\beta B M.\n\\label{equi.3}\n\\ee\nInstead, if $\\mathbb{B}$ is not uniform, {\\it deviations} from the FT arise.\nIn order to take a closer look, let the number of spins\nto become large and consider the case of the ideal paramagnet, in which\n$\\mu_0(\\sigma)$ in Eq.~(\\ref{equi.1}) is the uniform measure $Z^{-1} = [\\prod_i 2\\cosh (\\beta B_i)]^{-1}$.\nThen, by a straightforward saddle point\ncomputation one obtains the large deviation principle\n\\be\nP(M) \\sim e^{-NI(m)},\n\\label{G.1}\n\\ee\nwhere $m=M\/N$ is the magnetization per spin and the large deviation function is given by\n\\be\nI(m) = x^*(m) m + \\beta \\bigl[ f\\bigl(\\mathbb{B}^*(m)\\bigr) - f(\\mathbb{B})\\bigr].\n\\label{G.2}\n\\ee\nHere,\n\\be\nf(\\mathbb{B}) = \\lim_{N \\to \\infty} - \\frac{1}{N\\beta} \\sum_i \\ln [2\\cosh (\\beta B_i)],\n\\label{G.3}\n\\ee\nis the Helmotz free energy density, which depends on the field configuration $\\mathbb{B}$,\nwe have defined $\\mathbb{B}^*(m)=\\{B^*_i(m)\\}$ with  \n\\be\nB^*_i(m) = B_i + \\beta^{-1} x^*(m),\n\\label{G.3bis}\n\\ee\nand $x^*(m)$ is obtained by solving with respect to $x$ the\nequation of state\n\\bea\nm & = & - \\frac{\\partial}{\\partial x} f\\bigl(\\{B_i + \\beta^{-1} x\\}\\bigr)  \\nonumber \\\\\n& = & \\frac{1}{N} \\sum_i \\tanh (B_i + \\beta^{-1} x).\n\\label{G.4}\n\\eea\nConsequently, $\\beta^{-1} x^*(m)$ is the shift to be applied to the external field on each site in order\nto produce $m$ as the {\\it average} magnetization per spin.\nFrom the definition~(\\ref{I.3quater}) follows \n\\begin{eqnarray}\n& & \\frac{1}{N}{\\cal K}(M) = I(-m) - I(m) \\nonumber \\\\\n& = & - \\bigl[x^*(-m) + x^*(m)\\bigr]m + \\beta \\bigl[ f\\bigl(\\mathbb{B}^*(-m)\\bigr) - \nf\\bigl(\\mathbb{B}^*(m)\\bigr)\\bigr]. \\nonumber \\\\\n\\label{G.5}\n\\end{eqnarray} \nAveraging Eq.~(\\ref{G.3bis}) over $i$, we can write\n\\be\n\\beta^{-1} x^*(m) = \\overline{\\mathbb{B}^*(m)} - \\overline{\\mathbb{B}},\n\\label{G.01}\n\\ee\nwith\n\\be\n\\overline{\\mathbb{B}^*(m)} = \\frac{1}{N} \\sum_i B^*_i(m), \\,\\,\\, \\overline{\\mathbb{B}} = \\frac{1}{N} \\sum_i B_i,\n\\label{G.02}\n\\ee\nand Eq.~(\\ref{G.5}) can be put in the form\n\\be\n\\frac{1}{N}{\\cal K}(M) =  2 \\beta \\overline{\\mathbb{B}} m + \\beta \\bigl[g(-m) - g(m)\\bigr],\n\\label{G.03}\n\\ee\nwhere we have defined\n\\be\ng(m) = f\\bigl(\\mathbb{B}^*(m)\\bigr) + \\overline{\\mathbb{B}^*(m)} m.\n\\label{G.04}\n\\ee\nIn the particular case of the uniform external field \n\\begin{eqnarray}\n& & \\overline{\\mathbb{B}} = B, \\nonumber \\\\\n& & B^*_i(m) = B^*(m) = B + \\beta^{-1} x^*(m), \\,\\,\\,\\forall i, \\nonumber \\\\\n& & \\overline{\\mathbb{B}^*(m)} = B^*(m), \n\\label{G.05}\n\\end{eqnarray}\nand\n\\be\ng(m) = f(B^*) + B^*m\n\\label{G.06}\n\\ee\nis the Legendre transform of $f(B)$, which is even under $m$ reversal,\nsince $B^*(m)$ is odd. Hence, in this case Eq.~(\\ref{G.03}) reproduces the result~(\\ref{equi.3}).\nIn the nonuniform case $g(m)$ is not the Legendre transform of $f(\\mathbb{B})$ and\nin general does not have a definite parity, leading to a nonlinear AF function.\nWe will come back on this point in Section~\\ref{twobrownian}.\nWhat the above exercise shows is that the FT, in the sense of a linear AF, holds as\nlong as the macrovariable ${\\cal M}$, whose fluctuations are considered, is conjugate\nto the symmetry breaking field. Instead, if ${\\cal M}$ is not a conjugate variable, as it is the case\nwith a site dependent $\\mathbb{B}$, the FT in the form~(\\ref{equi.3}) does not hold.\n\n\\subsection{FT out of equilibrium}\n\nLet us, next, consider the nonequilibrium context. Assuming stochastic\nevolution, take for $\\Omega$ the space of stochastic trajectories\nand for $\\sigma$ an individual\ntrajectory. Then, $\\sigma^*$ stands for the time reversed trajectory, while\n$\\mu(\\sigma)$ and $\\mu^{\\prime}(\\sigma)$ are the probability measures associated with\ntwo different evolutions, whose relation is specified from case to case.\nHere, as in the equilibrium problem, we shall be concerned with $\\mu^{\\prime}(\\sigma) = \\mu(\\sigma)$,\nwhich in the dynamical context arises when there are \nno time dependent external parameters and the system evolves\nin contact with a single thermal reservoir at the final temperature $T$.\nThen, only heat\nis exchanged with the environment and one has~\\cite{Seifert}\n\\be\n{\\cal W}(\\sigma) = \\ln \\frac{P_0(x_0)}{P_0(x_t)} - \\beta {\\cal Q}(\\sigma),\n\\label{I.001}\n\\ee\nwhere $P_0$ is the initial probability distribution, $x_0$ and $x_t$ are the\ninitial and final entries in the trajectory $\\sigma$, $\\beta = 1\/T$ and ${\\cal Q}(\\sigma)$\nis the heat exchanged along the trajectory, which we take as negative if released\nto the environment. Using the above form of ${\\cal W}$ and the definition~(\\ref{I.04}),\nthe heat AF is given by\n\\be\n{\\cal K}(Q) = -\\beta Q -\\ln \\langle e^{\\ln P_0(x_t) - \\ln P_0(x_0)}|{\\cal Q} = Q \\rangle,\n\\label{I.002}\n\\ee \nwhose understanding requires some clue on the role of the boundary\nterms. This can be gained from the work of Puglisi {\\it et al.}~\\cite{Puglisi}. \n\nIn this paper, as anticipated\nin the Introduction, we shall be interested in the heat exchanged by a system \nof Brownian oscillators with the environment in an interval of time $[t_w,t]$. Eventually,\nthis will lead to recast the above equation in the form\n\\be\n{\\cal K}(Q) = - \\ln \\bigl\\langle e^{-\\sum_{\\mathbf{k}} \\Delta \\beta_{\\mathbf{k}} Q_{\\mathbf{k}}} |{\\cal Q} \n= Q \\bigr\\rangle,\n\\label{I.003}\n\\ee\nwhere $\\mathbf{k}$ are single oscillators labels and $Q_{\\mathbf{k}}$ the heat exchanged by each one of \nthem.\nThe above expression is clearly analogous to Eq.~(\\ref{equi.02}), where the role of\nthe nonuniform external field $\\mathbb{B}$ is played by the set of \naffinities $\\{\\Delta \\beta_{\\mathbf{k}} \\}$,\nwhich are $\\mathbf{k}$-dependent differences of inverse temperatures. The correspondence \nbetween the two problems helps to understand the deviations or modifications of the FT in terms\nof a collection of nonuniform degrees of freedom.\n\nIt should be emphasized that the choice of taking $\\mu^{\\prime} =\n\\mu$, and therefore $P^{\\prime} = P$, is dictated by the particular\nphysical setting of interest, since we consider the relaxation\nfollowing a temperature quench and we want to compare the probability\nof exchanging the heat $Q$ with that of exchanging $-Q$, in the {\\it\n  same quench process}, that is {\\it without time reversal}.  We focus\non the asymmetry of the heat distribution in the given process, as it\nwas done, for instance, in the experimental work of Gomez-Solano et\nal.~\\cite{gomez11}.\n\n\n\n\n\n\n\\section{Brownian oscillator}\n\\label{One}\n\n\\noindent \nThe equation of motion for the single overdamped Brownian oscillator is of the\nLangevin type \n\\begin{equation}\n\\dot{x}=-\\omega x + \\eta,\n\\label{process}\n\\end{equation}\nwhere $\\omega$ is the frequency and $\\eta$ is the white noise, modeling the\ninteraction with the thermal bath at the temperature $T$, with expectations\n\\begin{eqnarray}\n\\langle \\eta(t)\\rangle &=&0 \\\\\n\\langle \\eta(t)\\eta(t')\\rangle&=&2T\\delta(t-t').\n\\end{eqnarray}\nThe Boltzmann constant will be taken $k_B=1$ throughout.\nInitially the system is in equilibrium at the temperature $T_0$, with the\nposition probability distribution\n\\be\nP_0(x) = \\sqrt{\\frac{\\beta_0 \\omega}{2\\pi}} e^{-\\beta_0 {\\cal H}(x)}, \n\\label{HO.1}\n\\ee\nwhere ${\\cal H}(x) = \\frac{1}{2}\\omega x^2$ is the energy of the oscillator.\nInstantaneous cooling (quenches) or heating processes are realized by putting, at the time $t=0$, the system\nin contact with the thermal bath at the temperature $T < T_0$ or $T > T_0$, respectively.\nIn the following, we shall be mainly interested in the case of\nthe temperature quench.\n\n\n\n\\subsection{Fluctuations of exchanged heat}\n\nLet us focus on the fluctuations of\nthe heat exchanged by the oscillator with the thermal bath in the\ntime interval $(t_w \\geq 0,t>t_w)$ after the temperature step. \nSince no work can be carried out on or by the system,\ndue to $\\omega$ constant, the heat exchanged in a single realization of the dynamical evolution\ncoincides with the energy difference\n\\be\n\\mathcal{Q}(t,t_w)={\\cal H}\\bigl(x(t)\\bigr)- {\\cal H}\\bigl(x(t_w)\\bigr),\n\\label{Phex.1}\n\\ee\nwhich is positive if heat is absorbed from the bath and negative if\nit is released to the bath. Then,\nthe probability of exchanging the amount $Q$ of heat is given by\n\\be\nP(Q) = \\int_{-\\infty}^{\\infty} dx dx_w \\, P(x, t;x_w, t_w) \\delta (\\mathcal{Q} - Q),\n\\label{Phex.2}\n\\ee\nwhere $P(x, t;x_w, t_w)$ is the joint probability of the two events $(x,t)$ and $(x_w,t_w)$,\ngiven by\n\\begin{eqnarray}\\label{HO.9}\nP(x, t;x_w, t_w) \n&=&  \\frac{1}{\\sqrt{2 \\pi \\Delta(\\tau) \\nu(t_w)}} \\\\ \\nonumber\n&\\times & e^{-\\left \\{ \\frac{1}{2\\Delta(\\tau)}x^2  - \\frac{G(\\tau)}{\\Delta(\\tau)} xx_w\n+\\frac{1}{2}\\left [ \\frac{G^2(\\tau)}{\\Delta(\\tau)} + \\frac{1}{\\nu(t_w)} \\right ] x_w^2 \\right \\} },\n\\end{eqnarray}\nwhere $G(\\tau) = e^{-\\omega \\tau}$\nis the response function dependent on the time difference $\\tau=t - t_w$,\n$\\Delta(\\tau) = \\frac{T}{\\omega}[1-G^2(\\tau)]$\nand $\\nu(t_w) = G^2(t_w) \\nu_0 + {\\Delta(t_w)}$ is the position variance\nat the time $t_w$, whose initial value is given by $\\nu_0=T_0\/\\omega$.\n\nThe position probability distribution at the generic time $t$, obtained by integrating the\nabove quantity over $x_w$, maintains the equilibrium form~(\\ref{HO.1})\n\\be\nP(x,t) = \\sqrt{\\frac{\\beta_{\\rm eff}(t) \\omega}{2\\pi}} e^{-\\beta_{\\rm eff}(t){\\cal H}(x)},\n\\label{HO.6}\n\\ee\nwhere $\\beta_{\\rm eff}(t)$ is the time dependent inverse effective temperature\ndefined by the equipartition-like statement~\\cite{gomez11} \n\\be\n\\langle {\\cal H} \\rangle_t = \\frac{1}{2}T_{\\rm eff}(t),\n\\label{HO.6bis}\n\\ee\nwhich yields\n\\be\nT_{\\rm eff}(t) =  \\omega \\nu(t) =  (T_0-T)G^2(t) + T.\n\\label{HO.10}\n\\ee\nFor this quantity, which will play an important role in the following,\nwe shall use the short hand notation $T_w = T_{\\rm eff}(t_w)$ or $T_t = T_{\\rm eff}(t)$\nand similarly for the inverse temperature $\\beta_w = T_w^{-1}$ or $\\beta_t = T_t^{-1}$.\n\nReturning to Eq.~(\\ref{Phex.2}) and\nintroducing the integral representation of the $\\delta$ function\n\\be\n\\delta (\\mathcal{Q} - Q) = \\int_{-i \\infty}^{i \\infty} \\frac{d \\lambda}{2\\pi i} \\,\ne^{-\\lambda (Q - \\mathcal{Q})},\n\\label{Phex.3}\n\\ee\nwe obtain\n\\be\nP(Q) =  \\int_{-i \\infty}^{i \\infty} \\frac{d \\lambda}{2\\pi i} \\, e^{-\\lambda Q}\n\\int_{-\\infty}^{\\infty} dx dx_w \\, \\frac{e^{-f(\\lambda,x,x_w)}}{2 \\pi \\sqrt{\\Delta(\\tau)\\nu(t_w)}},\n\\label{Phex.4}\n\\ee\nwhere\n\\be\nf(\\lambda,x,x_w) = \\frac{1}{2 \\Delta(\\tau)} [x-G(\\tau) x_w]^2 + \\frac{x_w^2}{2\\nu(t_w)}\n-\\frac{\\lambda \\omega}{2}(x^2 - x_w^2).\n\\label{Phex.5}\n\\ee\nCarrying out the $x$ and $x_w$ integrations and rotating the $\\lambda$ integration from the\nimaginary to the real axis, this can be rewritten as\n\\be\nP(Q) =  \\int_{-\\infty}^{\\infty} \\frac{d \\lambda}{2\\pi \\sqrt{a}} \\,\n\\frac{e^{-i \\lambda Q}}{\\sqrt{(\\lambda -i\\lambda_-)(\\lambda + i\\lambda_+)}},\n\\label{Phex.16bis}\n\\ee\nwhere\n\\be\n\\lambda_\\pm = \\frac{1}{2 a} \\left [\\sqrt{b^2 + 4 a} \\pm b \\right ],\\,\\,\\,\\,\\lambda_+\\lambda_- = \n\\frac{1}{a},\n\\label{Phex.14}\n\\ee\nwith\n\\be\na =  TT_w(1-G^2), \\,\\,\\,\\, b =   \\Delta T G^2_w(1-G^2),\n\\label{Phex.11}\n\\ee\nafter using the notation\n\\be\n\\Delta T = T_0 - T,\\,\\,\\,\\, G_w=G(t_w),\\,\\,\\,\\,G = G(\\tau),\n\\label{Phex.10bis}\n\\ee which will be adopted from now on. Let us briefly comment on the\nparameters $a$ and $b$ introduced above. The asymmetry\nbetween $\\lambda_+$ and $\\lambda_-$ is controlled by $b$, while $a$ controls the symmetric\npart.  The sign of $b$ depends on that of $\\Delta T$, implying\n$\\lambda_+ > \\lambda_-$ in the quench process and $\\lambda_+ < \\lambda_-$ in the heating process.  \nThe most asymmetrical situation is\nobtained for $a=0$, which is realized either setting $T=0$, or $T_0=0$\nand $t_w=0$, yielding \n\\be \\lambda_+ = \\left \\{ \\begin{array}{ll}\n  \\infty,\\;\\; $for$ \\;\\; b > 0,\\\\ 1\/|b| ,\\;\\; $for$ \\;\\; b < 0,\n        \\end{array}\n        \\right .\n        \\label{eig.2}\n        \\ee\n\\be\n\\lambda_-  = \\left \\{ \\begin{array}{ll}\n        1\/b,\\;\\; $for$ \\;\\; b > 0,\\\\\n        \\infty,\\;\\; $for$ \\;\\; b < 0.\n        \\end{array}\n        \\right .\n        \\label{eig.3}\n\\ee \nInstead, the symmetrical situation with\n\\be\n\\lambda_+ = \\lambda_-= \\frac{1}{\\sqrt{a}},\n\\label{eig.1}\n\\ee\nis obtained when the system is equilibrated ($b=0$) for $\\Delta T=0$ or\nfor $t_w \\rightarrow \\infty$.\n\nThe integral in Eq.~(\\ref{Phex.16bis}) is evaluated by closing the contour either on the upper \nhalf plane or on the lower half plane around the branch cuts\nrunning along the imaginary axis from $\\pm i \\lambda_{\\mp}$ up to $ \\pm i \\infty$\n(see Fig.~\\ref{fig_int}), depending on $Q < 0$ or $Q > 0$, and obtaining\n\\be\nP(Q) = \\frac{\\sqrt{\\lambda_+\\lambda_-}}{\\pi} e^{\\frac{1}{2}\\Delta \\beta Q} \nK_0\\left (\\frac{\\lambda_+ +\\lambda_-}{2}|Q| \\right ),\n\\label{Phex.16}\n\\ee\nwhere \n\\be\n\\Delta \\beta = \\lambda_- -\\lambda_+ = \\beta_w - \\beta,\n\\label{aff.1}\n\\ee\nwhile $K_0$ is the modified Bessel function of the second kind. \n\n\\begin{figure}[!tb]\n\\includegraphics[width=0.6\\columnwidth,clip=true]{drawing.eps}\n\\caption{Contours of integration, for $Q<0$ and $Q>0$, in the case of a single oscillator.}\n\\label{fig_int}\n\\end{figure}\n\nIn the limit $t_w \\to \\infty$ one has $\\lambda_+ = \\lambda_- = \\beta$, recovering\nthe equilibrium result\n\\be\nP(Q) = \\frac{\\beta}{\\pi}K_0 (\\beta|Q|),\n\\label{Phex.16ter}\n\\ee\nwhich was derived in Refs.~\\cite{imparato07,chatterjee10} for a Brownian particle\noptically trapped in a stationary harmonic potential.  \nThe complementary case of a Brownian oscillator in the strongly underdamped limit \nhas been studied in the recent work of Salazar and Lira~\\cite{salazar16}, where a similar\nresult for the heat distribution is derived. The dependence on $t_w$ of the asymmetry\nof the distribution, for a quench to a small but finite temperature $T=0.1$, is illustrated in\nFig.~\\ref{fig1}, where the analytical expression~(\\ref{Phex.16}) is compared \nwith the numerical simulation of the process~(\\ref{process}). \n\n\\begin{figure}[!tb]\n\\includegraphics[width=0.9\\columnwidth,clip=true]{pq.eps}\n\\caption{Heat probability distribution for the single oscillator, with $\\omega=10^{-1}$, $T_0=100$ and $T=10^{-1}$, for $t=200$\nand different values of $t_w$.\nAnalytical form from Eq.~(\\ref{Phex.16}) (continuous lines) and\nnumerical simulations (symbols).}\n\\label{fig1}\n\\end{figure}\n\n\n\\subsection{Decomposition into cooling and heating contributions}\n\\label{realization}\n\n\nThe two quantities $\\lambda_{\\pm}$ turn out to be the basic building blocks in all\nwhat follows.\nThe physical meaning can be readily understood\nby rewriting $P(Q)$ as the convolution of the two distributions arising\nin the purely cooling and in the purely heating process. Defining\n\\be\nP_\\pm(Q) = \\sqrt{\\pm i\\lambda_\\pm} \\int_{-\\infty}^{\\infty} \\frac{d \\lambda}{2\\pi}\n\\frac{e^{-i \\lambda Q}}{\\sqrt{\\lambda \\pm i\\lambda_\\pm}}\n\\label{Decomp.1}\n\\ee\nand carrying out the integral, which involves only one of the branch points, one finds\n\\be\nP_-(Q) = \\left \\{ \\begin{array}{ll}\n        \\sqrt{\\frac{\\lambda_-}{\\pi |Q|}} e^{\\lambda_-Q},\\;\\; $for$ \\;\\; Q < 0,\\\\\n        0 ,\\;\\; $for$ \\;\\; Q > 0,\n        \\end{array}\n        \\right .\n        \\label{Decomp.3}\n        \\ee\nand\n\\be\nP_+(Q)  = \\left \\{ \\begin{array}{ll}\n        \\sqrt{\\frac{\\lambda_+}{\\pi Q}} e^{-\\lambda_+Q},\\;\\; $for$ \\;\\; Q > 0,\\\\\n        0 ,\\;\\; $for$ \\;\\; Q < 0,\n        \\end{array}\n        \\right .\n        \\label{Decomp.4}\n        \\ee\nfrom which follows\n\\be\n\\lambda_\\pm = \\pm \\frac{1}{2 \\langle {\\cal Q} \\rangle_\\pm},\n\\label{Decomp.5}\n\\ee\nwhere $\\langle {\\cal Q} \\rangle_\\pm$ are the average heats \nexchanged in the processes in which heat can only be absorbed or released.\nThen, the probability~(\\ref{Phex.16bis}) can be rewritten as \n\\be\nP(Q) =  \\int_{-\\infty}^{\\min(0,Q)} dQ^{\\prime}P_+(Q-Q^{\\prime})P_-(Q^{\\prime}),\n\\label{1ho}\n\\ee\nwhich reproduces the result of Eq.~(\\ref{Phex.16}) and\nthe total average heat exchanged is clearly given by\n\\bea\n\\langle {\\cal Q} \\rangle & = &  \\langle {\\cal Q} \\rangle_+ + \\langle {\\cal Q} \\rangle_- \\nonumber \\\\\n& = &\n\\frac{1}{2} \\left ( \\frac{1}{\\lambda_+} - \\frac{1}{\\lambda_-} \\right ) \n= - \\frac{b}{2}.\n\\label{Decomp.6}\n\\eea\n\nThe physical conditions for $P(Q) = P_-(Q)$ or $P(Q) = P_+(Q)$ are obtained \nputting the oscillator in contact with the thermal reservoir at $T=0$,\nor by arranging a purely heating process with $T_0=0, T > 0$ and $t_w=0$.\nIn the first case, one has\n$T_w=T_0 G^2_w, a=0, b=(T_w-T_t)$\nfrom which follows, according to Eqs.~(\\ref{eig.2}) and~(\\ref{eig.3}),\n$\\lambda_+=\\infty, \\lambda_-=(T_w-T_t)^{-1}$, implying\n$P_+(Q)= \\delta(Q)$ and, therefore, according to Eq.~(\\ref{Decomp.3})\n\\be\nP(Q) = P_-(Q) = \\left \\{ \\begin{array}{ll}\n        \\sqrt{\\frac{1}{\\pi |Q|(T_w-T_t)}} e^{Q\/(T_w-T_t)},\\;\\; $for$ \\;\\; Q < 0,\\\\\n        0 ,\\;\\; $for$ \\;\\; Q > 0.\n        \\end{array}\n        \\right .\n        \\label{Asy.4}\n        \\ee\nSimilarly, in the second case, with $T_0=0$ and $t_w=0$, one has \n$T_w=0, a=0, b=-T_t$, which yield $\\lambda_+=1\/T_t, \\lambda_-=\\infty$, \nimplying $P_-(Q)= \\delta(Q)$ and\n\\be\nP(Q) = P_+(Q) = \\left \\{ \\begin{array}{ll}\n        \\sqrt{\\frac{\\beta_t}{\\pi Q}} e^{-\\beta_tQ},\\;\\; $for$ \\;\\; Q > 0,\\\\\n        0 ,\\;\\; $for$ \\;\\; Q < 0.\n        \\end{array}\n        \\right .\n        \\label{Asy.8}\n        \\ee\n\n\n\n\n\n\\subsection{FT and time reversal symmetry breaking}\n\nIn this subsection we discuss the symmetry properties of the heat distribution.\nFrom Eq.~(\\ref{Phex.16}) follows immediately the FT in the standard form \n\\be\n\\frac{P(-Q)}{P(Q)} = e^{-\\Delta \\beta Q}.\n\\label{Phex.21}\n\\ee\nA similar result was derived in Ref.~\\cite{salazar16}\nwith $t_w=0$ and $\\Delta \\beta = \\beta_0 - \\beta$, while in Ref.~\\cite{gomez11}\nthere appears $\\Delta \\beta = \\beta_w - \\beta_t$, which holds true only in the limit of large\n$\\tau$ when the system is time decorrelated and $\\beta_t \\sim \\beta$.\nHence, as anticipated in section~\\ref{zero}, the AF is linear\n\\be\n{\\cal K}(Q) = \\Delta \\beta Q,\n\\label{aff.2}\n\\ee\nsince the role of the initial probability $P_0$\nnow is played by $P(x,t_w)$, which after Eqs.~(\\ref{Phex.1}) and~(\\ref{HO.6})\nyields\n\\be\n\\ln \\frac{P(x_w,t_w)}{P(x,t_w)} = \\beta_w {\\cal Q}.\n\\label{aff.3}\n\\ee\n\n\nHere, we want to highlight the connection between the above result and the\nbreaking of the time reversal symmetry, by adapting to the present context\nthe approach of Gaspard~\\cite{Gaspard} sketched in Section~\\ref{zero}.\nTaking for $\\Omega$ the set of ordered pairs $\\sigma = (x_w,x)$,\nthe time reversal symmetry operation is represented by the involution\n$\\sigma^* = (x,x_w)$ which exchanges the order of $x_w$ and $x$. \nThe task is to rewrite the joint probability Eq.~(\\ref{HO.9}) in the form of Eq.~(\\ref{equi.1}),    \nthat is as the product of a $\\mathbb{Z}_2$ symmetric\nfactor $\\mu_0(\\sigma)$ times an exponential where there appears the\nheat ${\\cal Q}$ in the role of the observable explicitly\nbreaking the $\\mathbb{Z}_2$ symmetry. In order to do this, let us first\ncast the joint probability Eq.~(\\ref{HO.9}) in the form\n\\be\n\\mu(\\sigma) = \\frac{1}{Z}e^{-{\\cal A}(\\sigma)},\n\\label{TRSB.1}\n\\ee\nwhere the ``action'' ${\\cal A}(\\sigma)$ can be read out from Eq.~(\\ref{HO.9}) as\n\\be\n{\\cal A}(\\sigma) = \\frac{\\beta \\omega}{2(1-G^2)} \\left [x^2 -2Gxx_w + G^2x_w^2 \\right ] \n+ \\beta_w \\frac{1}{2} \\omega x_w^2,\n\\label{TRSB.2}\n\\ee\nand the normalization factor is given by $Z=\\sqrt{\\frac{2\\pi \\beta \\beta_w \\omega^2}{(1-G^2)}}$. \nDecomposing ${\\cal A}(\\sigma)$ into the sum of an even and an odd part\n\\be\n{\\cal A}(\\sigma) = {\\cal E}(\\sigma) + {\\cal O}(\\sigma),\n\\label{TRSB.4}\n\\ee\nwith\n\\bea\n{\\cal E}(\\sigma) & = & \\frac{1}{2} \\left [ {\\cal A}(\\sigma) + {\\cal A}(\\sigma^*) \\right ] \\nonumber \\\\\n& = & \\frac{\\beta \\omega}{4(1-G^2)} \\left [(1+G^2)x^2 -4Gxx_w  \\right . \\nonumber \\\\\n&+&\\left . (1+G^2)x_w^2 \\right ]\n+ \\beta_w \\frac{1}{4} \\omega (x^2 + x_w^2),\n\\label{TRSB.5}\n\\eea\nand\n\\be\n{\\cal O}(\\sigma) =  \\frac{1}{2} \\left [ {\\cal A}(\\sigma) - {\\cal A}(\\sigma^*) \\right ] \n= -\\frac{1}{2} \\Delta \\beta {\\cal Q}(\\sigma),\n\\label{TRSB.6}\n\\ee\nwhere ${\\cal Q}(\\sigma)$ is defined by Eq.~(\\ref{Phex.1}), \nthe form~(\\ref{equi.1}) of $\\mu(\\sigma)$ is obtained\n\\be\n\\mu(\\sigma) = \\mu_0(\\sigma) e^{\\frac{1}{2} \\Delta\\beta {\\cal Q}(\\sigma)},\n\\label{TRSB.7}\n\\ee\nwhere\n\\be\n\\mu_0(\\sigma) = \\frac{1}{Z}e^{-{\\cal E}(\\sigma)},\n\\label{TRSB.8}\n\\ee\nis $\\mathbb{Z}_2$ invariant. Therefore, the invariance under time reversal in\n$\\mu(\\sigma)$ is explicitly broken by the exponential\nfactor involving ${\\cal Q}(\\sigma)$ in Eq.~(\\ref{TRSB.7}).\nOnce this is established, the FT in the form of Eq.~(\\ref{Phex.21}) \nis recovered straightforwardly. The following\nremarks are in order:\n\n\\begin{enumerate}\n\n\n\\item  the affinity $\\Delta \\beta$\ndriving the heat flow plays the role of the external field\ncausing the explicit breaking of the $\\mathbb{Z}_2$ invariance.\n\n\\item  The $\\mathbb{Z}_2$\ninvariant measure $\\mu_0(\\sigma)$ of Eq.~(\\ref{TRSB.8}) is not time translation\ninvariant, as it could have been naively expected, since the action \n${\\cal E}(\\sigma)$ in Eq.~(\\ref{TRSB.5}) \ndepends both on $\\tau$, through $G$, and on $t_w$, through $\\beta_w$.\nComputing $P(Q)$ from Eq.~(\\ref{TRSB.7}) one has\n\\be\nP(Q) = \\sum_{\\sigma} \\mu(\\sigma) \\delta({\\cal Q} - Q) = \ne^{\\frac{1}{2} \\Delta\\beta Q} \\sum_{\\sigma} \\mu_0(\\sigma) \\delta({\\cal Q} - Q),\n\\label{TRSB.9}\n\\ee\nand, comparing with Eq.~(\\ref{Phex.16}), one obtains\n\\be\n\\sum_{\\sigma} \\mu_0(\\sigma) \\delta({\\cal Q} - Q) =\n\\frac{\\sqrt{\\lambda_+\\lambda_-}}{\\pi} \nK_0\\left (\\frac{\\lambda_+ +\\lambda_-}{2}|Q| \\right ).\n\\label{TRSB.10}\n\\ee\n\n\\item  From Eq.~(\\ref{Phex.16bis}) it is straightforward to derive the\nmoment generating function\n\\be\n\\langle e^{\\vartheta {\\cal Q}} \\rangle =\n\\sqrt {\\frac{\\lambda_+\\lambda_-}{(\\vartheta + \\lambda_-)(\\lambda_+ - \\vartheta)}},\n\\label{CGF.2}\n\\ee\nimplying that the cumulant generating function \n$\\psi(\\vartheta) = \\ln \\langle e^{\\vartheta {\\cal Q}} \\rangle$\nsatisfies the same symmetry\n\\be\n\\psi(\\vartheta) = \\psi(-\\vartheta + \\Delta \\beta),\n\\label{CGF.2bis}\n\\ee\nwhich was found for a particle coupled to two thermostats at different temperatures~\\cite{Farago}.\n\n\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\\section{Two Brownian oscillators}\n\\label{twobrownian}\n\n\nLet us, next, see how the overall heat fluctuations are modified when\nthere are internal degrees of freedom. We consider first the case of\ntwo uncoupled oscillators with frequencies $\\omega_1$ and $\\omega_2$\nand, in the next section, we consider the limit of a large number of\noscillators. \n\nAs before, initially the system is in equilibrium at the temperature $T_0$ and at the time $t=0$\nis put in contact with the reservoir at the temperature $T$. Denoting by\n$Q_1$ and $Q_2$ the amounts of heat exchanged by each oscillator,\nthe probability that the system as a whole exchanges\nthe heat $Q$ is given by \n\\be\nP(Q) = \\int_{-\\infty}^{\\infty} dQ_1\\, P_1(Q_1)P_2(Q -Q_1),\n\\label{2osc.1}\n\\ee\nwhere $P_i(Q_i)$, with $i=1,2$, are the probabilities \npertaining to each component. \nInserting the expression~(\\ref{Phex.16}) and changing the integration variable from\n$Q_1$ to $y = Q_1 - Q\/2$, this can be put in the form analogous to Eq.~(\\ref{Phex.16})\n\\be\nP(Q) =  e^{\\frac{1}{2}\\overline{\\Delta \\beta}Q} R(Q),\n\\label{moz.1}\n\\ee\nwhere\n\\be\n\\overline{\\Delta \\beta} = \\frac{1}{2}(\\Delta \\beta_1 + \\Delta \\beta_2),\n\\label{moz.01}\n\\ee\nand \n\\be\nR(Q) = \\int_{-\\infty}^{\\infty} dy \\, W_1(y + Q\/2) W_2(y - Q\/2) \ne^{\\frac{1}{2}(\\Delta \\beta_1 - \\Delta \\beta_2)y},\n\\label{moz.2}\n\\ee\nwith\n\\be\nW_i(x) = \\frac{\\sqrt{\\lambda_{+,i}\\lambda_{-,i}}}{\\pi} \nK_0\\left (\\frac{\\lambda_{+,i} +\\lambda_{-,i}}{2}|x| \\right ).\n\\label{moz.3}\n\\ee\nHere, $\\lambda_{+,i}$ are defined in Eq.~(\\ref{Phex.14}), the index $i=1,2$\ndenoting the oscillator frequencies, $\\omega_1$ and $\\omega_2$, respectively\nand $\\Delta \\beta_i$ is defined by Eqs.~(\\ref{HO.6bis}) and~(\\ref{aff.1})\nwith frequency $\\omega_i$.\nThe resulting AF is given by\n\\be\n{\\cal K}(Q) = \\overline{\\Delta \\beta}Q\n+ \\ln \\left (\\frac{R(Q)}{R(-Q)} \\right ).\n\\label{moz.4}\n\\ee \nHence, if the two oscillators are equal ($\\omega_1 = \\omega_2$)\nthe second term disappears, since $R(Q)$ becomes an even function,\nand the linear result~(\\ref{aff.2}) is recovered.  If they are\nunequal, $R(Q)$ is not even under the change of sign of $Q$ and the second term in \nthe above equation gives a nonlinear contribution.\nThe modifications introduced by this second contribution are\nillustrated by Fig.~\\ref{fig_AF} obtained with parameters\n$\\omega_1=0.5$ and $\\omega_2=0.2$, $T_0=10$, $T=1$ at times\n$t=12,t_w=10$.  Around $Q=0$ one observes a linear behavior with slope\n$\\overline{\\Delta \\beta}$ (see dashed line in the\ninset), while for large values of $Q$ the AF shows a \nnonmonotonic behavior.\n\n\\begin{figure}[!tb]\n\\includegraphics[width=0.9\\columnwidth,clip=true]{AF_2oscil.eps}\n\\caption{Asymmetry function for two Brownian oscillators with\n  frequencies $\\omega_1=0.5$ and $\\omega_2=0.2$, for $T_0=10$, $T=1$,\n  $t_w=10$ and $t_w=12$. The line represents the analytical expression\n  Eq.~(\\ref{moz.4}), while the points are numerical simulations. In\n  the inset a zoom of the region at small $Q$ is shown.}\n\\label{fig_AF}\n\\end{figure}\n\n\n\nIn order to expose the connection between the presence of internal structure\nand the equilibrium problem mentioned in the Introduction, let us go back\nto the microscopic measure.\nWith two oscillators the phase space enlarges to the set of pairs $\\Omega = \\{ \\sigma_1,\\sigma_2 \\}$\nand the corresponding action, due to the absence of coupling, is additive\n\\bea\n{\\cal A}(\\sigma_1,\\sigma_2) = {\\cal A}_1(\\sigma_1) + {\\cal A}_2(\\sigma_2).\n\\label{FT2.1}\n\\eea\nThe two terms in the right hand side are given by Eq.~(\\ref{TRSB.2}), keeping into\naccount that the frequencies $\\omega_1$ and $\\omega_2$ are different, that is\n\\bea\n{\\cal A}_i(\\sigma_i) &=& \\frac{\\beta \\omega_i}{2(1-G_i^2)} \n\\left [x_i^2 -2G_ix_ix_{w,i} + G_i^2x_{w,i}^2 \\right ] \n\\nonumber \\\\\n&+& \\beta_{w,i} \\frac{1}{2} \\omega_i x_{w,i}^2.\n\\label{FT2.2}\n\\eea\nCarrying out the decomposition into even and odd parts, as in Eq.~(\\ref{TRSB.4}),\n\\be\n{\\cal A}(\\sigma_1,\\sigma_2) = {\\cal E}(\\sigma_1,\\sigma_2) + {\\cal O}(\\sigma_1,\\sigma_2),\n\\label{FT2.3}\n\\ee\nthe measure can be written as  \n\\be\n\\mu(\\sigma_1,\\sigma_2) = \\mu_0(\\sigma_1,\\sigma_2) e^{-{\\cal O}(\\sigma_1,\\sigma_2)},\n\\label{FT2.4}\n\\ee\nwhere \n\\be\n\\mu_0(\\sigma_1,\\sigma_2) = \\frac{1}{Z} e^{-{\\cal E}(\\sigma_1,\\sigma_2)},\n\\label{FT2.5}\n\\ee\nis even under $* : (\\sigma_1,\\sigma_2) \\mapsto (\\sigma^*_1,\\sigma^*_2)$, while\n\\be\n{\\cal O}(\\sigma_1,\\sigma_2) = -\\frac{1}{2} [ \\Delta \\beta_1 {\\cal Q}_1(\\sigma_1) +\n\\Delta \\beta_2  {\\cal Q}_2(\\sigma_2) ]\n\\label{FT2.6}\n\\ee\nis odd. Hence, \n${\\cal W}(\\sigma_1,\\sigma_2) = - 2{\\cal O}(\\sigma_1,\\sigma_2)$ and from Eq.~(\\ref{I.3quater})\nthere follows\n\\be\n\\frac{P(-Q)}{P(Q)} = e^{-{\\cal K}(Q)},\n\\label{flu.1}\n\\ee\nwith\n\\be\n{\\cal K}(Q) = -\\ln \\langle e^{-\\sum_i \\Delta \\beta_i {\\cal Q}_i} | {\\cal Q} = Q \\rangle.\n\\label{flu.2}\n\\ee \nThe formal\nsimilarity with Eq.~(\\ref{equi.02}) is evident, with the affinities $\\Delta \\beta_i$ playing the role\nof the site dependent external field $\\{B_i\\}$ and the partial heat ${\\cal Q}_i$ that of the local\norder parameter $s_i$. Accordingly, the above expression simplifies in the particular case \nakin to that of the uniform external field, that is for $\\omega_1=\\omega_2$ or\nfor $t_w=0$, yielding\n$\\Delta \\beta_i = \\beta_0 - \\beta$ for all $i$.\n\n\n\n\n\n\\section{Extended system}\n\\label{extended}\n\n\n\nIn this section we consider the case of a large number of oscillators.\nThe aim is to analyse important qualitatively new features arising in\nthe behavior of fluctuations when the size of the system becomes\nlarge.  As it is usually the case for large systems, fluctuations of\nextensive quantities, such as heat, obey a large deviation\nprinciple~\\cite{hugo,ld}. The non trivial feature, unexpected for a\nlinear system, is that in the large deviation function there appears a\nsingularity corresponding to a condensation\ntransition~\\cite{Gonnella,Jo,Pechino}. This means, as briefly\nanticipated in the Introduction, that there exists a critical\nthreshold $Q_c$ of the exchanged heat, such that fluctuations above\nthreshold are due to a {\\it single} oscillator.  We give a careful\ntreatment of the saddle point computation involved in the computation\nof the large deviation function and we study the phase diagram of the\ntransition. Mathematical details are collected in\nAppendix~\\ref{app:crit}.\n\n\nWe consider an extended system of linear size $L$ and volume $V=L^d$, where $d$ is\nthe space dimensionality. We suppose that this system is linear and that the normal\nmodes decomposition with periodic boundary conditions gives rise to a set of harmonic \ncomponents for the low lying modes \nwith frequencies obeying the dispersion relation~\\cite{note} \n\\be\n\\omega(\\mathbf{k}) = k^{\\alpha} + \\omega_0,\n\\label{dis.1}\n\\ee\nwhere $\\omega_0 > 0$, $\\mathbf{k} = \\frac{2\\pi}{L} \\mathbf{n}, \\,\\,n_i=0,\\pm 1,\\pm 2,...$ and $\\alpha >0$.\nJust to fix ideas, a dispersion relation of this type arises in the Gaussian model\nor Van Hove theory of critical phenomena~\\cite{Ma}.\nThe $\\mathbf{k}$-space volume per mode is $(2 \\pi)^d\/V$. Assuming that there exists \nan underlying lattice structure with lattice spacing $a_0$, the first Brillouin zone is bounded by\n$\\Lambda = 2\\pi\/a_0$ and the total number of modes is given by $N=V\/a_0^d$.\nDue to modes independence, the\nprobability of a generic microscopic state at the time $t_w$ is given by the product\nmeasure\n\\be\nP(\\{x_{\\mathbf{k}}\\},t_w) = \\prod_{\\mathbf{k}} \\sqrt{\\frac{\\beta_{\\mathbf{k},w} \\omega(\\mathbf{k})}{2 \\pi}}\ne^{-\\beta_{\\mathbf{k},w}{\\cal H}_{\\mathbf{k}}(x_{\\mathbf{k}})},\n\\label{dis.1bis}\n\\ee\nwhere, according to Eq.~(\\ref{HO.10}),\n\\be\n\\beta_{\\mathbf{k},w} =  [(T_0-T)e^{-2\\omega(\\mathbf{k})t_w} + T ]^{-1}.\n\\label{HO.10bis}\n\\ee\nis the inverse effective temperature of the mode $\\mathbf{k}$\nand ${\\cal H}_{\\mathbf{k}}(x_{\\mathbf{k}}) = (1\/2)\\omega(\\mathbf{k})x^2_{\\mathbf{k}}$ is the single mode\nenergy. Hence, Eq.~(\\ref{aff.3}) generalizes to\n\\be\n\\frac{P(\\{x_{\\mathbf{k}}\\},t_w)}{P(\\{x_{\\mathbf{k},w}\\},t_w)} = e^{-\\sum_{\\mathbf{k}} \\beta_{\\mathbf{k},w} {\\cal Q}_{\\mathbf{k}}},\n\\label{dis.1tris}\n\\ee\nwhere ${\\cal Q}_{\\mathbf{k}} = {\\cal H}_{\\mathbf{k}}(x_{\\mathbf{k}}) - {\\cal H}_{\\mathbf{k}}(x_{\\mathbf{k},w})$ is the heat\nexchanged by the mode $\\mathbf{k}$, and the sum is restricted to the first Brillouin zone. \nInserting into Eq.~(\\ref{I.002}) we recover Eq.~(\\ref{I.003})\n\\be\n{\\cal K}(Q) = - \\ln \\bigl\\langle e^{-\\sum_{\\mathbf{k}} \\Delta \\beta_{\\mathbf{k}} {\\cal Q}_{\\mathbf{k}}} |{\\cal Q} = Q \\bigr\\rangle.\n\\label{dis.001}\n\\ee\nComparing with Eq.~(\\ref{equi.02}), \nwe see that the differences $\\Delta \\beta_{\\mathbf{k}}=\\beta_{\\mathbf{k},w}-\\beta$\nplay the role of the site dependent external field in the equilibrium problem,\nas remarked in section~\\ref{twobrownian}, and that the AF linear form\nis recovered when these differences become independent\nof $\\mathbf{k}$, as for instance for $t_w=0$, or when the oscillators are\nidentical, i.e. $\\alpha = 0$ (see Appendix~\\ref{app:alpha0}).\n\nLet us, next, see what form takes the AF by letting $N$ to become large.\nGeneralizing Eq.~(\\ref{2osc.1}) and keeping $V$ finite,\nthe probability that the system exchanges the amount of heat $Q$\nis given by \n\\bea\n& & P(Q,V) =  \\int_{-\\infty}^{\\infty} \\prod_{\\mathbf{k}} dQ_{\\mathbf{k}}\\, P_{\\mathbf{k}}(Q_{\\mathbf{k}})\\,\n\\delta(Q - \\sum_{\\mathbf{k}}Q_{\\mathbf{k}}) \\nonumber \\\\\n& = & \\int_{-i\\infty}^{i\\infty}\\frac{d\\lambda}{2\\pi i} \\, e^{-\\lambda Q} \n\\prod_{\\mathbf{k}} \\sqrt{\\frac{\\lambda_{+}(k)\\lambda_{-}(k)}{[\\lambda_{+}(k) -\\lambda][\\lambda_{-}(k) +\\lambda]}}, \\nonumber \\\\\n\\label{dis.3}\n\\eea\nwhere  $\\lambda_{\\pm}(k)$ are defined in Eq.~(\\ref{Phex.14}),\nevaluated for $\\omega(\\mathbf{k})$ as in Eq.~(\\ref{dis.1}). \nDenoting by $\\lambda^+_0$ and $\\lambda^-_0$ the upper and lower edges of the two branches\nof the spectrum, that is \n\\be\n\\lambda^+_0 = \\min_{0\\leq k\\leq \\Lambda} \\{ \\lambda_{+}(k) \\}, \\,\\,\\, \n\\lambda^-_0 = \\min_{0\\leq k\\leq \\Lambda} \\{ \\lambda_{-}(k) \\},\n\\label{dis.5}\n\\ee\nthe contour of integration can be deformed to an arbitrary line $\\Gamma$ on the complex plane, provided\nit crosses the real axis within the gap $[-\\lambda^-_0,\\lambda^+_0]$. \nThis allows to rewrite the above integral as\n\\be\nP(Q,V) = \\int_{\\Gamma}\\frac{d z}{2\\pi i} \\, e^{-V[ zq + L(z;V)]},\n\\label{dis.6}\n\\ee\nwhere $q=Q\/V$ is the density of the exchanged heat and\n\\be\nL(z;V) = \\frac{1}{2V}\\sum_{0\\leq\\mathbf{k}\\leq\\Lambda}\n\\ln \\left[ \\frac{[\\lambda_{+}(k) - z][z + \\lambda_{-}(k)]}{\\lambda_{+}(k) \\lambda_{-}(k)} \\right].\n\\label{dis.4bis}\n\\ee\nThe large $V$ behavior of $P(Q,V)$ can be obtained by \nusing the steepest descent method to \nestimate the integral in Eq. \\eqref{dis.6}  \\cite{nota}.  \nWhen $V\\gg1$ \nthe wave vector $\\mathbf{k}$ can be assumed continuous and \nthe sum in Eq. \\eqref{dis.4bis}  approximated by an \nintegral with an error of $O(1\/V)$. \nThe neglected $O(1\/V)$ terms in the exponent, however, give an\n$O(1)$ contribution to $P(Q;V)$,\nresulting in an undetermined normalization of $P(Q;V)$,\nmemory of the discrete nature of $\\mathbf{k}$.\n\nThe function $L(z;V)$ contains the dangerous boundary terms \n$(1\/V)\\ln[\\lambda_0^{+} - z]$ and\n$(1\/V)\\ln[\\lambda_0^{-}+ z]$.\nIf the steepest descent path $\\Gamma$ does not come too close to either edges \n$z = \\pm \\lambda_0^{\\pm}$ of the gap\nthese terms just give an $O(1\/V)$ correction which can be  included into the normalization factor.\nConsequently, in this case the function\n\\be\n\\label{dis.8}\n   L(z) =  \\frac{1}{2} \\int_0^{\\Lambda} d\\mu(k) \\,\n\\ln \\left[ \\frac{[\\lambda_{+}(k) - z][z + \\lambda_{-}(k)]}{\\lambda_{+}(k) \\lambda_{-}(k)} \\right],\n\\ee\nwith \n$d\\mu(k) = \\Upsilon_d k^{d-1}dk$ and\n$\\Upsilon_d = [2^{d-1}\\pi^{d\/2} \\Gamma(d\/2)]^{-1}$  coming from the\nangular integration,\nis an uniform approximation to the function $L(z;V)$ as $V\\gg 1$.\nHere, $\\Gamma(x)$ denotes the Euler gamma function.\n\n\nDenoting by $\\phi(z;q) = zq + L(z)$ and by primes derivatives with respect to $z$, \nit is straightforward to verify\nthat $\\mathbb{I}m\\, \\phi^{\\prime}(z;q) \\sim \\mathbb{I}m\\, z$, \nimplying that the stationary point $z^*$ of $\\phi(z;q)$ is on the real\naxis. Thus, setting $z = x + iy$, the coordinate $x^*$ of the saddle point is \nobtained solving the equation $\\phi^{\\prime}(x^*;q) = 0$,\nwhich explicitly reads\n\\begin{eqnarray}\nq & = &  - L'(x^*) \\nonumber \\\\\n& = & \\frac{1}{2} \\int_0^{\\Lambda} d\\mu(k) \\, \\left [ \\frac{1}{\\lambda_{+}(k) - x^*}\n- \\frac{1}{\\lambda_{-}(k) + x^*} \\right ].\n\\label{Sdlpt.7bis}\n\\end{eqnarray}\nThe function $-L'(x)$  is  monotonically increasing in the interval\n$x\\in [-\\lambda_0^-,\\lambda_0^+]$, so Eq. \\eqref{Sdlpt.7bis} admits solution \non the condition that $q$ lies between the limiting values:\n\\be\nq_c^{\\pm} = -L'(\\pm \\lambda_0^{\\pm}),\n\\label{Sdlpt.01bis}\n\\ee\nwith $q_c^{-} < 0 < q_c^{+}$. \nIn this case the steepest descent path in the neighborhood of the saddle point is parallel to the \nimaginary axis\nand a straightforward calculation leads to the asymptotic result for $V\\gg 1$\n\\be\n\\label{eq:PQ_norm}\n  P(Q,V) = \\frac{1}{\\sqrt{-2\\pi V L''(x^*)}}\\, e^{-V [qx^* + L(x^*)]}.\n\\ee\nIf both $q_c^{\\pm}$ diverge  this result is valid\nfor arbitrary finite values of $q$ because $x^* $ is always inside the gap $[-\\lambda^-_0,\\lambda^+_0]$,\nand far from the edges.\nIf, instead, one or both $q_c^{\\pm}$ are finite, the saddle point $x^*$ may reach \nthe boundary of the  gap $[-\\lambda^-_0,\\lambda^+_0]$,\nand leave it for $q < q_c^{-}$ or $q > q_c^{+}$.\nWhen this occurs the asymptotic approximation in Eq. \\eqref{eq:PQ_norm} is  no more valid.\n\n\nThe boundary values $q_c^{\\pm}$ are finite or infinite depending on whether the singularities\nof the integrand function in $L'(x)$ are integrable or not.\nMore specifically, let us denote by $k^{\\pm}_0$ the wave vectors at which \n$\\lambda_{\\pm}(k)$ attain the minimum value,\nthat is\n$\\lambda_{\\pm}(k^{\\pm}_0) = \\lambda^{\\pm}_0$.\nThen, in the neighborhoods of $k^{\\pm}_0$, \n\\be\n\\lambda_{{\\pm}}(k) - \\lambda^{\\pm}_0 \\sim \\begin{cases}\n         C\\, (k - k^{\\pm}_0)^{2n}, & \\text{for}\\  k^{\\pm}_0 > 0;  \\\\\n          C\\, k^{\\alpha},         & \\text{for}\\  k^{\\pm}_0 = 0,\n      \\end{cases}\n        \\label{Sdlpt.10}\n\\ee\nwhere $C$ is a positive constant,  $n=1,2,\\dotsc$ and $\\alpha > 0$.\nTherefore, $q_c^{\\pm}$  diverge if $k^{\\pm}_0 > 0$ or if $k^{\\pm}_0=0$ and $d \\leq \\alpha$, while\nthey are finite if $k^{\\pm}_0=0$ and $d > \\alpha$. \nWhich is the case depends on the parameters of the quench $T_0,T,\\tau,t_w$. So, in general,\nthis manifold of parameters is partitioned into phases\ndistinguished by $q_c^{\\pm}$ being finite or infinite. \n\nFor what follows, and before  discussing how the steepest descent calculation must be modified, it \nis useful to get some insight into the meaning of the two different cases.\nRecalling that $\\lambda_+(k)$ and\n$-\\lambda_-(k)$ are the inverse average heat absorbed or released, it is evident\nfrom Eq.~(\\ref{Sdlpt.7bis}) that\nif $q$ coincides with the average heat density $\\langle q \\rangle$, then\n$x^*(\\langle q \\rangle) = 0$, implying $-L'(0) = \\langle q \\rangle$. \nConsequently, if $q$ differs\nfrom $\\langle q \\rangle$ then $x^*(q) \\neq 0$ and \n\\be\n\\lambda^*_{\\pm}(k,q) = \\lambda_{\\pm}(k) \\mp x^*(q)\n\\label{0.02}\n\\ee\nacquire the meaning of the inverse average heat absorbed or released in new conditions, such that\n$q$ would be the new average total heat density, exactly as the $\\{B_i^*\\}$ of section~\\ref{zero}\nhave been recognized to be the shifted external fields necessary to render the fluctuation\n$m$ equal to the average magnetization per spin.\nIn other words, a deviation of $q$ \nfrom the average $\\langle q \\rangle$\nbrings in a shift by $x^*(q)$ of the inverse heats exchanged by the single modes. \nAccordingly, when $x^*(q)$ approaches the edges of the gap the above defined $\\lambda_0^{\\pm *}$ vanish,\nwhich means that the corresponding edge modes give an {\\it infinite} contribution to the exchanged heat.\nThis may happen either because the fluctuation $q$ itself is infinite, or because the contributions\nof all the modes, other than the edge ones, can sum up at most to the finite amounts $q_c^{\\pm}$. In the\nlatter case, if $|q|$ goes above the thresholds $|q_c^{\\pm}|$, in order to make up\nfor the finite difference $|q| - |q_c^{\\pm}|$ a spike contribution must come from the edge modes\nat $k_0^{\\pm}$. This phenomenon is the condensation of fluctuations\nin the edge mode, since as anticipated in the Introduction and at the beginning of\nthis section, the entire amount of\nthe fluctuation above the critical threshold comes from a single degree of freedom. \nThe mechanism of the transition can be recognized to be the same as that of \nBose-Eintein condensation~\\cite{Huang}.\nMore in general, condensation of fluctuations\nhas been found in widely different contexts such as information theory~\\cite{Merhav},\nfinance~\\cite{Filiasi} and statistical mechanics encompassing\nboth equilibirum and out of equilibrium situations~\\cite{JoEPL}.\n\nWhen one or both $q_c^{\\pm}$ are finite, the steepest descent calculation must be \nmodified.\nThis case will be dealt with in the next subsection. Returning to the case in which\nEq.~(\\ref{eq:PQ_norm}) holds, neglecting subdominant terms and taking the ratio $P(-Q)\/P(Q)$, \none obtains the following explicit form of the AF\n\\be\n\\frac{1}{V}{\\cal K}(Q) = -[x^*(-q) + x^*(q)]q + [L(x^*(-q)) - L(x^*(q))],\n\\label{condns.7}\n\\ee\nwhich is clearly reminiscent of Eq.~(\\ref{G.5}) in sect.~\\ref{zero}. \nIn order to expose the analogy, let us\nsubtract equations~(\\ref{0.02}) one from the other and take the average over $k$,\nobtaining\n\\be\nx^*(q) = \\frac{1}{2} \\left (\\overline{\\lambda_+} - \\overline{\\lambda_-} \\right )\n+ \\frac{1}{2} \\left [\\overline{\\lambda_-^*(q)} - \\overline{\\lambda_+^*(q)} \\right ],\n\\label{0.03}\n\\ee\nwhere \n\\be\n\\overline{\\lambda_{\\pm}} = \\frac{1}{{\\cal N}}\\int_0^{\\Lambda} d\\mu(k) \\, \\lambda_{\\pm}(k), \\,\\,\\,\n{\\cal N} = \\int_0^{\\Lambda} d\\mu(k) \\, 1,\n\\label{0.04}\n\\ee\nand similar expressions for $\\overline{\\lambda_{\\pm}^*(q)}$. Inserting into Eq.~(\\ref{condns.7}),\nwe obtain\n\\be\n\\frac{1}{V}{\\cal K}(Q) = \\left (\\overline{\\lambda_-} - \\overline{\\lambda_+} \\right )q\n+ \\left [\\Psi(-q) - \\Psi(q) \\right ],\n\\label{0.05}\n\\ee \nwhere\n\\be\n\\Psi(q) = L(x^*(q)) + \n\\frac{1}{2} \\left [ \\overline{\\lambda_-^*(q)} - \\overline{\\lambda_+^*(q)} \\right ]q.\n\\label{0.06}\n\\ee \nHence, comparing with Eqs.~(\\ref{G.03}) and~(\\ref{moz.4}), we can recognize the same structure: \nthe prefactor $\\left (\\overline{\\lambda_-} - \\overline{\\lambda_+} \\right )$ \nof the linear contribution (the same appearing in the single oscillator case)\nplays the role of the external average field $\\mathbb{B}$, while the nonlinear\nterm $\\left [\\Psi(-q) - \\Psi(q) \\right ]$, arises from ``free energy'' contributions.\nAs remarked above, the presence of this latter term is due to the $k$-dependence of $\\Delta \\beta_k$,\nwhich plays the same role as the $i$-dependence of $B_i$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Condensation transition}\n\n\nOn physical grounds one can argue that condensation at finite\n$q_c^{-}$ can occur only in cooling experiments, where the final\ntemperature is lower than the initial one, while condensation at\nfinite $q_c^{+}$ can occur only in heating experiment.  We do not have\na proof of this, but numerical analysis of the conditions\nfor condensation confirm this conjecture.\n\nLet us thus concentrate on the case with $q_c^{+} = \\infty$ and\n$q_c^{-}$ finite, which occurs for $k_0^- = 0$ and $d > \\alpha$.  The\nanalysis of the opposite case in which $q_c^{+}$ is finite and\n$q_c^{-} = -\\infty$, (or both finite, if such a case exists) is\nstraightforward.\n\\begin{figure}[!tb]\n\\includegraphics[width=0.9\\columnwidth,clip=true]{path_cusp2.eps}\n\\caption{Schematic representation of the steepest descent path of\n  integration. If $x^*=x_1^*\\ge -\\lambda_0^-$, the integration contour\n  goes through $x_1^*$, where $\\phi'=0$, ($\\Gamma_1$ curve).\n  If $x^*=x_2^*< -\\lambda_0^-$,\n  the integration contour develops a cusp and sticks\n  to $-\\lambda_0^-$, ($\\Gamma_2$ curve).}\n\\label{cusp}\n\\end{figure}\nWhen $q_c^-$ is finite,  Eq.~(\\ref{Sdlpt.7bis}) \ndoes not admit a solution with  $x^*\\in[-\\lambda_0^{-}, \\lambda_0^{+}]$ whenever $q < q_c^-$.\nThe problem is well known from the theory of Bose-Einstein\ncondensation of an ideal gas of bosons~\\cite{Huang} or from\nthe ``sticking'' of the saddle point to a singularity\nin the solution of the spherical model of ferromagnetism~\\cite{Berlin}. \nIt arises because \nwhen $q < q_c^{-} = -L'(\\lambda_0^{-})$ the steepest descent path cannot pass through the \nstationary point $x^*< -\\lambda_0^{-}$ of $\\phi(z;q) = zq + L(z)$ \nand must traverse the real axis at the gap edge $z=-\\lambda_0^{-}$. \nSince $\\phi'(-\\lambda_0^-;q) = q - q_c^{-}$\nis negative for $q < q_c^{-}$,  the steepest descent path at $z=-\\lambda_0^{-}$ bends \ntoward the negative real axis forming  \nthe cusp characteristic of saddle point sticking at the gap edge (see Fig. \\ref{cusp}).\nThe large $V$ behavior of $P(Q,V)$ is dominated by the \nneighborhood of the gap edge because $\\phi(-\\lambda_0^{-};q) < \\phi(x^*;q)$\nleading to  the asymptotic behavior for $V\\gg 1$:\n\\be\n\\label{eq:PQ_cond1}\n  P(Q,V) \n  = \\frac{1}{\\sqrt{-\\pi V \\bigl(q - q_c^{-}\\bigr)}}\\,\n         e^{-V[ -\\lambda_0^{-} q + L(-\\lambda_0^{-})]},\n\\ee\nvalid for $q < q_c^{-}$.\n\nAs a matter of fact, this expression and Eq. \\eqref{eq:PQ_norm} valid\nfor $q > q_c^{-}$, hold for $x^*$ not to close to the gap edge\n$-\\lambda_0^{-}$.  That is, respectively, in the condensed phase and\nnormal phase for $q$ not too close to $q_c^{-}$.  The analysis of the\nasymptotic behavior of $P(Q,V)$ as $V\\gg 1$ for all values of $q$,\nincluding close to $q_c^{-}$, requires some care.  Details can be\nfound in Appendix \\ref{app:crit}.\n\n\\begin{figure}[!tb]\n\\includegraphics[width=0.9\\columnwidth,clip=true]{pq_new.eps}\n\\caption{Comparison of the saddle point computation of $P(q)$ with the\n  numerical computation of the same quantity for $\\omega_0=0.1$, $T_0=100$, $T=0.1$,\n  $t_w=3$, $\\tau=7$, for a system with $L=41$ in $d=2$ ($N=1681$\n  oscillators), and with $\\alpha=1$. (The quantity $q_\\delta^-$ is defined in Eq.~(\\ref{eq:q_d})).}\n\\label{fig20}\n\\end{figure}\n\nAdding and subtracting $\\lambda_0^{-}q_c^-$ in the exponent of Eq. \\eqref{eq:PQ_cond1}, \n$P(Q;V)$ in the condensed phase far from the critical point can be written as\n\\be\n  P(Q,V) =   \\frac{e^{\\lambda_0^- (Q-Q_c^-)}}{\\sqrt{\\pi \\bigl|Q - Q_c^{-}\\bigr|}}  \\,\ne^{V[\\lambda_0^- q_c^-  - L(-\\lambda_0^-)]},\n\\label{Img.21}\n\\ee \nwhere $Q_c^- = Vq_c^-$.\nComparing the first term with the expression~(\\ref{Decomp.3})\nfor the distribution in the purely cooling process we have\n\\be\n   P(Q,V) =  P_{k=0,-}(Q-Q^-_c)\\, P(Q^-_c,V),\n\\label{Img.22}\n\\ee\nwhere, up to normalization factors, \n\\be\n  P(Q_c^-,V)  \\sim    e^{V [\\lambda_0^{-} q_c^{-} - L(-\\lambda_0^{-})]}.\n\\ee\nThis means that negative fluctuations below the critical lower threshold $Q^-_c$\ncondense into the cooling contribution of the $\\mathbf{k} =0$ mode for the exceeding\npart $(Q - Q^-_c)$, while the contribution of all the other modes is locked onto\n$Q^-_c$.\nThe comparison of the saddle point estimates of $P(Q,V)$, in the normal and in the condensed\nphase, with the ``exact'' numerical computation is illustrated in Fig.~\\ref{fig20}.\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\\subsection{Phase diagram}\n\nIn order to establish the ``phase'' structure, it is necessary to analyse\nthe behavior of $k^{\\pm}_0 > 0$. Let us first recall the results \nof Ref.~\\cite{Pechino}, where the problem was analysed for the quench\nto $T=0$. This is the simplest case because,\nas specified in section~\\ref{realization}, heat can only be released\nand we have to deal only with the $\\lambda_-(k)$ branch of the spectrum.\nDifferentiating with respect to $k$ the expression for $\\lambda_-(k)$, given by the \nfirst line of Eq.~(\\ref{eig.3}), we get\n\\be\n\\frac{\\partial \\lambda_-}{\\partial k} = C(k,\\tau,t_w)A(k,\\tau,t_w),\n\\label{Part.01}\n\\ee\nwhere $C(k,\\tau,t_w) = b^{-2} \\Delta T e^{-2\\omega(\\vec k)(t_w + \\tau)}\\alpha k^{\\alpha -1}$\nis a positive quantity, while \n\\be\nA(k,\\tau,t_w) = t_w e^{2\\omega(\\vec k)\\tau}[1-e^{2(E-k^{\\alpha})\\tau}],\n\\label{Part.02}\n\\ee\nhas the sign of $(k^{\\alpha}-E)$, with \n\\be\nE(\\tau,t_w) = \\frac{1}{2\\tau} \\ln \\left (1 + \\frac{\\tau}{t_w} \\right ) - \\omega_0,\n\\label{Part.03}\n\\ee\nand vanishes at $k^{\\alpha} = E$.\nTherefore, imposing the condition $E(\\tau,t_w) = 0$, in the $(\\tau,t_w)$ plane\nthere remains defined the critical line, given by\n\\begin{equation}\nt_w = \\frac{\\tau}{e^{2\\omega_0 \\tau} - 1},\n\\label{Part.6}\n\\end{equation}\nsuch that above it $k_0^- =0$, while below $k_0^- > 0$. Thus, $k_0^-$ or, equivalently,\n$q_c^-$ act as order parameters and below the critical line the system is in the \nnormal phase, corresponding to $q_c^-=-\\infty$, while above it is in the \ncondensed phase, corresponding to $q_c^-$ finite,\nas illustrated in Fig.~\\ref{Phasediagr} (black line). \n\n\n\n\nIf $T > 0$, we must take into account both branches $\\lambda_{\\pm}(k)$ \nand keep track of the two order parameters $q_c^{\\pm}$. The\nanalytical search of the critical lines turns out to be quite complicated. \nSo, we have looked for the absolute minima of $\\lambda_{\\pm}(k)$ numerically.\nThe resulting phase diagram for $q_c^-$ is \ndepicted in Fig.~\\ref{Phasediagr}  for a few values of $T$, ranging from very low to\nalmost equal to $T_0$. When the system is equilibrated ($T=T_0$), there is no\ncondensed phase. Fig.~\\ref{Phasediagr} shows that upon lowering $T$ there appears a condensed\nphase which, starting from the far right, pronges toward the left eventually filling the entire\nregion above the $T=0$ critical line. The prominent qualitative difference between the $T=0$\nand the $T>0$ phase diagrams for $q_c^-$ is that in the latter case the transition driven by an increasing\n$t_w$, for fixed $\\tau$, manifests reentrant behavior. The same computation for\n$q_c^+$ does not show the existence of a condensed phase, namely $q_c^+ = \\infty$ all\nover the explored $(\\tau,t_w)$ plane.\n\nThe reentrant behavior of the phase diagram can be interpreted as follows. Let us consider a quench at finite\ntemperature, and let us fix $\\tau$. Then,\nfor small enough $t_w$, all oscillators are out of equilibrium and can exchange an arbitrary amount\nof heat with the bath. Therefore no condensation can take place in this region. Upon increasing $t_w$,\nall oscillators do equilibrate but the slowest one, corresponding to the mode $k=0$, \nthat can account for large exchange of heat (above the threshold), leading to the condensation of fluctuations. Eventually,\nfor large $t_w$, all oscillators are equilibrated and again the normal phase is recovered.\n\n\n\\begin{figure}[!tb]\n\\includegraphics[width=0.9\\columnwidth,clip=true]{phase3.eps}\n\\caption{$q_c^-$ phase diagram for various final temperatures $T$ and\nfor $T_0=1$. The normal phase corresponds to $q_c^- = \\infty$, while\nthe condensed phase to $q_c^-$ finite. }\n\\label{Phasediagr}\n\\end{figure}\n\n\n\\section{Conclusions}\n\\label{conclusions}\n\n\nIn this paper we have studied the fluctuations of the heat exchanged with the environment\nby a system of oscillators relaxing after a temperature quench. Focusing on the\nFT, we have investigated the relation between the deviations from linearity of the AF\nand the internal structure of the system, building on the analogy with the behavior\nof fluctuations in equilibrium when a symmetry is broken by nonuniform external\nperturbations.\n\nWe have first analysed in great detail the case of a single\noscillator, reducing it to the convolution of the two elementary\nprocesses in which heat can be only released or only absorbed. The\ninverse average heats released $(\\lambda_-)$ and absorbed\n$(\\lambda_+)$ in these processes turn out to be the basic objects\nunderlying the behavior of the quantities of interest.  In the\none-oscillator case the AF is linear with slope given by the\ndifference $\\Delta \\beta = \\lambda_- - \\lambda_+$ and, in the\nframework of the above mentioned analogy with the equilibrium problem,\nthis quantity acts like an external field explicitly breaking the time\nreversal symmetry. By adding a second oscillator it starts to surface\nthe role of the dishomogeneity of the external perturbation in\ndetermining deviations from linearity in the AF.  It should be pointed\nout that in the dynamical problem the lack of homogeneity is due to\nthe existence of different relaxation rates, related to the presence\nof degrees of freedom evolving on different time scales.  This\nproduces a temporal dishomogeneity, which induces a differentiation of\nthe affinities $\\Delta \\beta_1$ and $\\Delta \\beta_2$, analogous to the\nspatial heterogeneity generated by the external field $\\{B_i\\}$.\n\nThis picture emerges most clearly in the case of a large number of\noscillators. In this case, the system, although diagonal, presents\nnontrivial features, the most notable of which is the possible\ncondensation of fluctuations~\\cite{Gonnella,Jo,Pechino}. Here, we have\npresented a detailed analytical study of the large deviation function\nof the heat distribution, with an accurate treatment of the crossover\nfrom the normal to the condensed phase, pointing out some interesting\ntechnical points involved in the application of the steepest descent\nmethod in the presence of a transition. In addition, we have mapped\nout the phase diagram of the condensation transition, discovering an\nunexpected reentrant behavior, in the $(t_w,\\tau)$ plane.\n\nOur analysis allowed us to establish a close correspondence with the\nstructure of the AF for the paramagnet in equilibrium under the action\nof a nonuniform external field.  In the equilibrium case it seems\nreasonable to make the statement that the linearity of the AF depends\non whether the observable of interest ${\\cal M}(\\sigma)$ is conjugate\nto the external perturbation, the implication of which being that\ndeviations from the FT are due to the lack of conjugation. Clearly,\nthe interesting and challenging issue is to understand whether this\nway of looking at the AF and its deviations from linearity can be extended to the\nnonequilibrium case. The study we have presented in this paper is a\nfirst step in this direction, paving the way to futures investigations\nwithin the context of more complex interacting systems, showing slow\nrelaxation and aging phenomena.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nFirst notions of \\textit{artificial neural networks (ANNs)} -- in the context of supervised learning -- date back to the early 1940s~\\cite{McCulloch1943}. Although considerable research has been conducted in the succeeding decades, people had to wait for the emergence of powerful computers, and the accompanying (significant) drop in costs for storing and processing data, until ANNs turned into highly competitive machine learning algorithms~\\cite{GoodfellowBengio2016,hastie01statisticallearning}. Nowadays, ANNs represent the state of the art in a variety of supervised learning tasks, such as image, sound or object recognition \\cite{HeZRS16,SimonyanZ14,AbdelHamidMJDPY14}.\n\n\nHowever, while the strength of ANNs undoubtedly results from their huge amount of parameters, their enormous complexity also makes them rather incomprehensible. \nAlthough numerous works have introduced methods for interpreting the decisions made by ANNs \\cite{nguyen2016synthesizing,erhan2009visualizing,zhou2016learning}, these methods lack explanations for malicious and intentional behaviour against ANNs.\n\nOne possibility for fooling ANNs is the addition of carefully crafted noise, so-called \\textit{adversarial perturbations} (cf. Figure~\\ref{fig:adv_example}), to the model's input \\cite{SzegedyZSBEGF13,goodfellow2014,GuR14,TabacofV15,FawziMF16,CarliniW16,Carlini017,KurakinGB17a}. It should be noted that these perturbations do not need to be crafted individually per network; instead, ANNs that are trained for a similar task often can be fooled by the same adversaries \\cite{SzegedyZSBEGF13,goodfellow2014,PapernotMGJCS16}.\n\n\\newpage\nWhile a plethora of works have shown the advantages of (deep learning) neural networks, their black-box characteristic makes them very vulnerable. Hence, in order to enable a more frequent and, even more importantly, more secure integration of neural networks in our daily lives, we need to ensure their robustness against adversarial attacks. \n\nThe issue of a network's vulnerability and how to defend it against adversarial attacks will be investigated in this work. As such, our contributions can be summarized as follows:\n\\begin{compactenum}\n    \\item We present a \\textbf{novel method to craft transferable adversaries}, which in turn helps to improve the understanding of a network's vulnerability against adversarial attacks.\n    \\item We introduce a novel approach to regularize decision boundaries and, thus, \\textbf{enhance the resilience of neural networks against adversarial attacks}. Compared to previous attempts, our approach does not require additional backward-passes, which can decrease training speed significantly (cf. Table \\ref{tab:training_time}).\n\\end{compactenum}\n\\begin{figure}[t!]\n\\centering\n    \\begin{overpic}[width=0.95\\linewidth, trim = 2mm 0mm 0mm 0mm, clip]{figures\/example_adver.png}\n        \\put (31.5,11) {\\large{$+~\\varepsilon~\\cdot$}}\n        \\put (70,11) {\\large{$=$}}\n    \\end{overpic}\n   \n    \\caption{Example for an adversary (from left to right): the original image (classified as \\textit{great panda} with a confidence of 99.99\\%), adversarial noise with $\\varepsilon=3$ and the resulting adversarial image (classified as \\textit{saxophone} with a confidence of 83.8\\%).}\n  \\label{fig:adv_example}\n\\end{figure}\n\nThe remainder of this manuscript is structured as follows. In Section~\\ref{sec:related} we set our work into context. We then introduce a method for crafting transferable adversaries in Section~\\ref{sec:transfer}, and list possible defense strategies against adversarial attacks in Section~\\ref{sec:counteractions}. Thereafter, we provide our experimental setup in Section~\\ref{sec:experiments} and discuss our findings in Section~\\ref{sec:results}. At last, Section~\\ref{sec:conclusion} concludes our work.\n\n\\section{Related Work}\n\\label{sec:related}\n\n\nSeveral methods to craft adversarial examples have been published in recent years \\cite{goodfellow2014,MoosaviDezfooli15,Carlini017,madry2018towards}. In principle, one can categorize these methods into \\textit{single-} and \\textit{multi-step} attacks: the former only require a single gradient while the latter calculate several gradients. Kurakin et al.~\\cite{KurakinGB16a} found in their work that multi-step adversaries are less transferable than single-step ones and they concluded that models are indirectly robust against multi-step adversaries. Yet, in line with the findings of \\cite{dong2018boosting}, we observed that the transfer of adversaries between models is rather easy when using a so-called \\emph{ensemble attack}, i.e., a multi-step attack, which is executed on an ensemble of models simultaneously \\cite{dong2018boosting}.\n\nAs generating adversarial perturbations is rather simple, we will at first address the characteristics of these perturbations along with their accompanying risks of black-box attacks (see Section~\\ref{sec:charact}). Note that in contrast to white-box methods, black-box approaches have no access to the model's gradient. Thereafter, in Section~\\ref{sec:resil}, we briefly outline selected strategies to enhance resilience.\n\n\\subsection{Characteristics of Adversaries}\n\\label{sec:charact}\nSzegedy et al.~\\cite{SzegedyZSBEGF13} crafted artificial noise, which causes ANNs to misclassify images whose original versions they classified correctly. As a result, it could be demonstrated that already slight changes, which are hardly recognizable to the human eye, are sufficient to fool the network. As further demonstrated by \\cite{SzegedyZSBEGF13,goodfellow2014}, and also in this work, adversarial perturbations are not model-specific, but instead generalize well over networks with different hyper-parameters. In addition, even networks trained on disjoint data sets, yet fulfilling a similar classification task, are likely vulnerable to the same adversarial image \\cite{goodfellow2014,PapernotMGJCS16}. Thus, two models with different architectures, hyper-parameters, and trained on different data sets, often misclassify the same images. This property can be used to execute black-box attacks as demonstrated by \\cite{PapernotMGJCS16}. \n\nSeveral works have been conducted to examine the effects of adversaries. The neighborhoods of adversarial examples and their original counterparts were investigated by \\cite{TabacofV15}. The authors found that while already small and random noise applied to adversaries often caused it to be shifted back to their original class, the classification of the original images did not change (even when rather large random noise was applied). Therefore, the authors concluded that adversarial examples likely inhabit small sub-spaces within the decision space.\n\nNext, \\cite{tramer2017space} investigated these subspaces by measuring the number of orthogonal attack directions and found that the number of these directions is rather large. \nThey concluded that the more orthogonal attack directions exist at a given point, the larger these adversarial subspaces are and the more likely adversaries transfer between models (since larger subspaces are more likely to intersect between different models).\nThis may be the explanation on \\textit{why} adversarial perturbations are often universal across different model designs and often universal across datasets.\n\n\\subsection{Enhancing Resilience through Adversarial Training}\n\\label{sec:counteractions}\n\\begin{figure}[t!]\n\\flushright\n    \\begin{overpic}[width=0.92\\columnwidth]{figures\/example_alpha.png}\n        \\put (-8,11) {{$(1-\\alpha)~\\cdot$}}\n        \\put (29.5,11) {{$+~\\alpha~\\cdot$}}\n        \\put (70,11) {{$=$}}\n    \\end{overpic}\n   \n\t\\caption{Idea of our proposed method: linear combination of the original image (\\textit{great panda}) with a randomly selected image (\\textit{saxophone}) using weight $\\alpha \\in (0, 0.5)$ (here: $\\alpha = 0.3$). The resulting training image (right) inherits the class of the original image. This allows us to generate, on the one hand, more training examples and, on the other one, training examples which lie in close proximity to the decision boundaries.}\n  \\label{fig:alpha_example}\n\\end{figure}\n\\label{sec:resil}\n\nDefense strategies against adversarial attacks can mainly be distinguished into two categories: approaches that either change (i) a model's topology, or (ii) the way a model is trained. Gu and Rigazio~\\cite{GuR14} stated that a model's robustness \\enquote{is more related to intrinsic deficiencies in the training procedure and objective function than to [its] topology}.\n\n\n\nUnder the concept of \\textit{Adversarial Training}, several works have introduced regularizers which decrease a model's vulnerability to adversarial perturbations. The first one was introduced by Goodfellow et al.~\\cite{goodfellow2014}. Here, the objective function of the \\textit{Fast Gradient Sign Method} (FGSM) \\cite{goodfellow2014} is added to the classification loss.\nFor every training image, a single-step adversary is calculated and the network's sensitivity to this adversary is minimized. Thus, \\textit{Adversarial Training} acts as a regularizer to improve resilience \\cite{goodfellow2014,KurakinGB16a}.\n\n\nThe training method of \\cite{goodfellow2014} only uses a single back-propagation step to calculate adversarial perturbations. In contrast to the aforementioned approach, other works introduced a min-max optimization method \\cite{Huang2015LearningWA,shaham2015}. It prepares the network for the worst-case scenario under a limited perturbation and minimizes its sensitivity to it. The algorithm works in two steps. First, it maximizes the loss of the input image by adding a small adversarial perturbation. However, the network trained with a min-max regularizer performed worse on the classification tasks \\cite{Huang2015LearningWA} -- even when trained with rather small perturbations. Due to the additional maximization problem, the algorithm is even more complex to execute, which further increases training time.\n\nSimilar approaches were introduced by \\cite{Miyato2015DistributionalSW,madry2018towards}. The approach of \\cite{Miyato2015DistributionalSW}, denoted \\textit{Virtual Adversarial Training} (VAT), works in a similar way as the min-max optimization, yet it maximizes the Kullback-Leibler divergence \\cite{Kullback51klDivergence} between the probability vectors of the training image and an adversary. The KL-divergence between the original and an adversary can be interpreted as the local smoothness of the decision space. Thus, the network not only learns to classify the training images correctly, but also smoothens the decision boundaries around each of the training points. Miyato et al.~\\cite{Miyato2015DistributionalSW} used \\textit{Projected Gradient Descent} (PGD) to solve the inner maximization problem. The authors could demonstrate a guaranteed robustness even against multi-step adversaries under small $l_\\infty$-norm -- with $l_\\infty = max\\{|x~-~x^{adv}|\\}$ -- where $x$ is the unchanged input image while $x^{adv}$ is the image with additional adversarial perturbation. Multi-step black- and white-box attacks have been conducted under a $l_\\infty\\leq8$-constraint. Even under multi-step white-box attacks, the accuracy of their model is 45\\%, whereas the accuracy of a non-robustified, and thus defenseless, model would be close to zero. However, as there are additional forward-backward passes required, training time takes accordingly longer.\n\n\\section{Crafting Strong and Transferable Adversaries}\n\\label{sec:transfer}\n\\begin{figure}[t!]\n\\centering\n\t\\includegraphics[width=\\columnwidth, trim = 0mm 8mm 0mm 0mm, clip]{figures\/example_pic.png}\n\t\\caption[Random Images from the Dataset]{Exemplary images from the modified \\textit{Large Scale Visual Recognition Challenge 2017} dataset: Seal, Bear, Dragonfly and Rabbit (left to right).}\n\t\\label{fig:example_pic}\n\\end{figure}\n\n\nGoodfellow et al.~\\cite{goodfellow2014} introduced the \\textit{Fast Gradient Sign Method} (FGSM):\n\\begin{align}\n    x^{adv} &= x + \\varepsilon \\cdot \\text{sign} \\left(\\nabla_{x}~L(\\Theta, x, y)\\right) \\label{eq:fgsm}\n\\end{align}\nThe loss function\\footnote{In this work, we refer to the cross-entropy as loss function. Yet, any other function can most likely be used, too.} $L$ of the model $\\Theta$ is differentiated with respect to the input image $x$ and its true class $y$. The sign of the resulting gradient $\\nabla_{x}~L$ is used to calculate adversarial noise which, when applied to an image, increases the model's loss and, thereby, could shift its classification decision. The parameter $\\varepsilon$ is used to control the amount of perturbations. Note that a larger $\\varepsilon$ does not necessarily lead to a higher chance of fooling a model as demonstrated by Figure \\ref{fig:loss}. Instead, tuning $\\varepsilon$ is highly important. The resulting adversary is a single-step one.\n\nOne can execute the FGSM multiple times with a small $\\varepsilon$-value to converge towards the adversarial sub-space iteratively. The resulting adversaries are then referred to as multi-step ones. In order to find multi-step adversaries under limited amount of perturbations and close to the original image, several methods such as \\cite{MoosaviDezfooli15,Carlini017,madry2018towards,KurakinGB16a} have been introduced. When the perturbation exceeds a certain limit -- usually the $l_\\infty$- or $l_2$-norm of the adversarial perturbation is used -- the perturbation is projected back into the search space, e.g., by using pixel-wise clipping.\n\nAs pointed out by \\cite{TabacofV15,tramer2017space}, adversaries lie in sub-spaces which are in close proximity to true training examples and which often intersect between models. One can argue that when crafting adversaries with multi-step methods, these adversaries often do not lie in intersecting sub-spaces because the resulting adversary may be overfitted to the particular model as its loss is maximized. To overcome this issue, \\cite{dong2018boosting} introduced the (i) \\textit{Momentum Iterative Gradient Based Method} (MI-FGSM) and (ii) the \\textit{ensemble in logits}. The former method uses a momentum to overcome local maxima and the latter one combined the logits of an ensemble of models to find a common and intrinsic direction. Attacking an ensemble of models increases the likelihood of transferable adversaries as these adversaries are not optimized for a single model but for several models simultaneously. Hence, the adversaries lie more likely in intersecting adversarial sub-spaces. \n\nIn contrast to \\cite{dong2018boosting}, we used the combined gradient of an ensemble of models to craft adversaries which lie in intersecting sub-spaces. We call our method \\textit{gradient ensemble attack}. We found that these adversaries are likely to transfer between models of the same architecture -- and frequently transfer to other architectures as well (see Section~\\ref{sec:results}). To ensure that every model's gradient has the same impact on the resulting adversarial perturbation, we normalize each gradient to length one:\n\\begin{align}\n    \\hat{\\nabla}_{x}~L(\\Theta, x, y) &= \\frac{\\nabla_{x}~L(\\Theta, x, y)}{||\\nabla_{x}~L(\\Theta, x, y)||_2}. \\label{eq:grad_norm}\n\\end{align}\nAfterwards, the different gradients are summed up to find a common direction of intersecting adversarial sub-spaces. Similar to \\cite{KurakinGB16a}, we approximated the true direction by using the sign-method of the summed and normalized gradients to the image in an iterative process:\n\\begin{eqnarray}\n x^{adv}_{i} &=& x_{i-1} + \\lambda \\cdot \\text{sign} \\left(\\sum_{n=0}^N~\\hat{\\nabla}_{x}~L(\\Theta_n, x_{i-1}, y) \\right) \\label{eq:adv_methode}\\\\\n  \\text{s.t.} && ||x^{adv}_{i} - x||_{\\infty}~\\leq~\\varepsilon \\quad \\text{and} \\quad x_0 = x.\\nonumber\n\\end{eqnarray}\nTo ensure that the magnitude of the perturbations stays within a given limit, we used pixel-wise clipping. Further, we used a learning rate of $\\lambda=1$ (to slowly convert towards the adversarial spot) and set the number of steps to $I=\\min(\\varepsilon + 4, 1.25\\cdot\\varepsilon)$ as proposed by \\cite{KurakinGB16a} to craft adversarial images. In addition, we chose $\\varepsilon\\in\\{4,8,16\\}$ to limit the amount of perturbations in order to test and compare different magnitudes of adversarial perturbations.\n\n\\section{Proposed Defense Strategies}\nAs demonstrated by \\cite{TramerKPGBM18} and \\cite{athalye18a}, adversarial training may lead to \\emph{gradient masking}. This term describes a phenomenon in which the decision boundaries are roughly the same, yet the corresponding gradient is damaged or obfuscated in a way that white-box attacks fail. However, the model does not become more resilient against adversarial examples in black-box attack scenarios or against transferable adversaries as these attacks are based on surrogate models.\n\nIn order to avoid the risk of gradient masking, we propose a method that does not require gradient calculations and still flattens the decision space. In addition, we avoid the expensive optimization of an inner maximization problem as done in VAT or PGD. We found that while random noise is mostly ignored by the model, superimposing two pictures does distract a model. Therefore, as illustrated in Figure~\\ref{fig:alpha_example}, we designed training images $\\tilde{x}_i$ by placing a randomly selected image $x_r$ on top of the original training image $x_i$:\n\\begin{align}\n    \\tilde{x}_i = (1-\\alpha) \\cdot x_i + \\alpha \\cdot x_r. \\label{eq:alpha_method}\n\\end{align}\nThe parameter $\\alpha \\in (0, 0.5)$ controls the impact of image $x_r$ on the generated image $\\tilde{x}_i$. As the majority of the image originates from the original image $x_i$, the generated image $\\tilde{x}_i$ will inherit the class label $y_i$ of the original image. Our proposed approach thus allows to (i) generate \\emph{more training examples}, and (ii) create images that are closer to the decision boundaries of at least two classes -- and thus \\emph{harder to distinguish} -- as $\\tilde{x}_i$ contains properties of two image classes. Thereby, the space between the different classes is flattened and the boundaries are regularized. Thereafter, the networks will be trained by minimizing the loss $L(\\Theta,\\tilde{x}_i, y_i)$.\nAs the results depend on the choice of $\\alpha$, we considered three different approaches:\n\\begin{compactenum}\n\\item Using a fixed $\\alpha$-value.\n\\item Following the idea of \\cite{Miyato2015DistributionalSW}, i.e., first predicting the classes $y_i$ and $\\tilde{y}_i$ based on $x_i$ and $\\tilde{x}_i$ (and using a fixed $\\alpha$ as in 1.), and then minimizing the loss of the unmodified training examples \\emph{and} the KL-Divergence between the predicted classes of the unmodified and modified training examples:\n\\[\\underset{\\Theta}{\\text{minimize}} \\quad L(\\Theta, x_i, y_i) + \\lambda \\cdot KL(\\hat{y}_i~||~\\tilde{y}_i).\\]\n\\item Similarly to 2., but this time drawing $\\alpha$ from a beta-distribution instead of a fixed $\\alpha$-value; $\\alpha \\sim B(p,q)$ with $p=2$ and $q \\in \\{4,6,10\\}$.\n\\end{compactenum}\nNote that while the first method does not require any additional passes, the latter two methods require a second forward-pass to calculate the KL-Divergence -- but no additional backward-pass. As stated by \\cite{Miyato2015DistributionalSW}, the KL-divergence can be interpreted as the local smoothness of the decision space. If the divergence is high, the predictions of both input images are dissimilar, implying that the decision space between the two activations is not flattened. By minimizing the divergence, the models learn to find similar activations for both images and, thereby, flatten the decision space between the two activations.\n\nAs a side effect, our method allows to generate a multitude of training images (using Equation~(\\ref{eq:alpha_method})), which in turn allows to train a larger area.\n\\begin{figure}[t!]\n\\centering\n\t\\includegraphics[width=0.95\\columnwidth]{figures\/acc_over_time.png}\n\t\\caption[Validation Accuracy during training]{Visualization of the validation accuracy during training. The \\textit{Projected Gradient Descent} \\cite{madry2018towards} substantially reduces the overall accuracy compared to the \\textit{Base Model}, our proposed method, or the \\textit{Virtual Adversarial Training} \\cite{Miyato2015DistributionalSW}. Furthermore, the model $\\alpha \\sim B(2, 4)$, $KL_{\\lambda=10}$ achieves an accuracy of more than 70\\% with fewer methods than the \\textit{Base Model}.}\n\t\\label{fig:acc_druing_training}\n\\end{figure}\n\n\\section{Methodology and Experiments}\n\\label{sec:experiments}\nIn the following, we briefly outline the models' topologies and used training data.\n\n\\subsection{Dataset}\nWithin our experiments, we used the data from the second challenge of the \\textit{Large Scale Visual Recognition Challenge 2017}\\footnote{\\url{http:\/\/image-net.org\/challenges\/LSVRC\/2017\/}} \\cite{ILSVRC15} for training and validating our models. \nYet, as down-scaling the whole images to the required size was not reasonable, we first used bounding boxes to cut out the objects within each of the images. Then, as the bounding boxes were of different sizes, the cropped images were down-scaled to the desired resolution of 128 $\\times$ 128 pixels (cf. Figure \\ref{fig:example_pic}).\n\nAs a result, our dataset contains roughly 400,000 images. Unfortunately, the images are not uniformly distributed across all 200 classes. As the mismatch between the classes is too large to use oversampling, we selected 100 classes with an identical number of images per class. For training and validation purposes, we performed holdout with a 80-20-split. Thereby, we created a dataset which is larger and has a higher resolution than CIFAR10\/100~\\cite{krizhevsky2009learning} while (at the same time) being significantly smaller and less complex than ImageNet~\\cite{ILSVRC15}. In addition, by using balanced classes, we eliminate any side-effects which may occur from unbalanced datasets and may influence our results.\n\n\\subsection{Models}\n\\begin{table}[!t]\n    \\centering\n    \\caption{Measured training Time for different models (all models were trained on a single Nvidia Quadro RTX 6000). Training a single epoch with our proposed method takes only slightly longer than the \\textit{Base Model} without any defense. In addition, $\\alpha \\sim B(2, 4)$, $KL_{\\lambda=10}$ reaches 60\\% and 70\\% Validation Accuracy about 30 minutes earlier than the \\textit{Base Model} as it converges faster (cf. Figure \\ref{fig:acc_druing_training}). Training models using the \\textit{Projected Gradient Descent} \\cite{madry2018towards} or \\textit{Virtual Adversarial Training} \\cite{Miyato2015DistributionalSW} takes noticeably longer per epoch, and they take even longer to converge.}\n    \\label{tab:training_time}\n   \n   \n        \\begin{tabular}{l|cccc}\n                       & Median Time & Time until & Time until & Time until \\\\\n            Model      & p. Epoch    & 50\\% Acc.  & 60\\% Acc.  & 70\\% Acc. \\\\\n            \\midrule\n            \\textit{Base} &  \\cellcolor{gray!25}\\textbf{6:43m} & \\cellcolor{gray!25}\\textbf{00:59h} & 02:15h & 03:17h \\\\\n            $\\alpha=0.4$   & 6:55m & 02:18h & 03:14h & -\/- \\\\\n            $\\alpha \\sim B(2, 4)$ & 6:51m & 01:01h & \\cellcolor{gray!25}\\textbf{01:42h} & \\cellcolor{gray!25}\\textbf{02:44h} \\\\\n            VAT ($\\varepsilon=15$) & 14:45m & 02:12h & 03:55h & 08:05h \\\\\n            PGD ($I=7$) & 22:40m & 11:46h & 15:56h & -\/- \\\\\n            PGD ($I=3$) & 11:37m & 03:53h & 07:45h & -\/- \n        \\end{tabular}\n   \n   \n\\end{table}\nTo investigate the effects of adversarial perturbations, we trained multiple ANNs. First, an ``unprotected'' model -- denoted \\textit{base model} in the following -- is trained. For this purpose, we took a VGGNet~\\cite{SimonyanZ14}, as it provides a straightforward and uncluttered design of convolutions, pooling and dense layers. In addition, we modified the design by using a \\textit{Global-Average-Pooling} layer \\cite{LinCY13} and extended each CNN layer with a \\textit{Batch Normalization} layer \\cite{Ioffe15}. Afterwards, we compared our proposed methods to different ResNets \\cite{HeZRS16} to verify our findings.\n\nIn order to compare different methods to defend against adversarial attacks, we trained several models with and without different defense methods (see Section~\\ref{sec:results} for more details). All models were trained on a single Nvidia RTX 6000 or a single Nvidia V100 GPU. We found, that by using VAT \\cite{Miyato2015DistributionalSW} or PGD \\cite{madry2018towards} the training speed is reduced, significantly. \nEach training epoch not only takes longer, but the models also converge more slowly towards the optimum. \nIn contrast, our proposed method is more time efficient as it only requires an additional forward pass and it even converges faster (cf. Figure \\ref{fig:acc_druing_training} and Table \\ref{tab:training_time}).\n\n\\subsection{Assessing a Network's Resilience against Adversarial Perturbations}\n\\label{sec:assessment}\nTo assess the robustness of a network against adversaries, we trained six models per considered network architecture. We then used an ensemble of one, three or five of the six ANNs\\footnote{Note that our method applied to an ensemble of one model is identical to the i-FGSM.} (see Section~\\ref{sec:results} for details) to extract a set of adversarial images using Equation~(\\ref{eq:adv_methode}). More precisely, an image is considered being adversarial, if it is misclassified by \\emph{all} of the ensemble's networks -- note that for simplicity the networks do not have to agree on the same wrong class. The set of the extracted adversarial images is then classified by the sixth model and its accuracy is taken as quality indicator of the respective network architecture.\n\n\\section{Results and Discussion}\n\\label{sec:results}\n\\begin{table*}[!t]\n    \\centering\n    \\caption{Accuracy of different networks on the sets of original and adversarial images. All adversarial datasets are crafted using our proposed \\textit{gradient ensemble attack} approach with one, three or five (vulnerable) VGGNet13 models. Our method is equivalent to \\cite{KurakinGB16a,madry2018towards} if only one model is used. The shown results are for $\\varepsilon=16$. When training a model with PGD, the performance on the original images is significantly lower in comparison to the \\textit{Base Model}. In contrast, VAT and our proposed method slightly increase the validation accuracy. Across all adversarial perturbations, VAT models achieved the highest accuracy. Noteworthy values are highlighted.\n    }\n    \\label{tab:results}\n   \n   \n        \\begin{tabular}{r|cc|cc|cc|cc}\n        &\\multicolumn{2}{c|}{\\bf Accuracy on} & \\multicolumn{6}{c}{\\bf Accuracy on Adversarial Dataset based on} \\\\\n        &\\multicolumn{2}{c|}{\\bf Original Data} & \\multicolumn{2}{c|}{\\bf One Model} & \\multicolumn{2}{c|}{\\bf Three Models} & \\multicolumn{2}{c}{\\bf Five Models} \\\\\n        Parametrization \\, & \\, Train &Val. \\, & \\, True& Adv. \\, & \\, True \\, & \\, Adv. \\, & \\, True \\, & \\, Adv.\\\\\n        \\midrule\n        \\multicolumn{9}{l}{\\textit{Base Model}}\\\\\n         \\, & \\, 91.2\\% \\, & \\, \\cellcolor{gray!25}\\textbf{73.4}\\% \\, & \\, 89.9\\% \\, & \\, \\cellcolor{gray!25}\\textbf{23.8}\\% \\, & \\, 95.3\\% \\, & \\, \\cellcolor{gray!25}\\textbf{4.1}\\% \\, & \\, 96.7\\% \\, & \\, \\cellcolor{gray!25}\\textbf{0.9}\\% \\\\\n        \\midrule\n        \\multicolumn{9}{l}{\\textit{Projected Gradient Descent (PGD) with} $\\varepsilon = 8$ \\textit{and} $\\lambda = 2$}\\\\\n        $I = 3$ \\, & \\, 86.7\\% \\, & \\, \\cellcolor{gray!25}\\textbf{66.2}\\% \\, & \\, 79.6\\% \\, & \\, 79.0\\% \\, & \\, 84.6\\% \\, & \\, 83.6\\% \\, & \\, 86.6\\% \\, & \\, 86.5\\% \\\\\n        $I = 7$ \\, & \\, 77.8\\% \\, & \\, \\cellcolor{gray!25}\\textbf{62.4}\\% \\, & \\, 75.6\\% \\, & \\, 75.1\\% \\, & \\, 80.7\\% \\, & \\, 80.0\\% \\, & \\, 82.8\\% \\, & \\, 81.8\\% \\\\\n        \\midrule\n        \\multicolumn{9}{l}{\\textit{Virtual Adversarial Training (VAT) with} $I = 3$ \\textit{and} $\\lambda = 1$}\\\\\n        $\\varepsilon = \\phantom{1}5$ \\, & \\, 97.7\\% \\, & \\, 76.2\\% \\, & \\, 90.6\\% \\, & \\, 85.1\\% \\, & \\, 95.2\\% \\, & \\, \\cellcolor{gray!25}\\textbf{87.2}\\% \\, & \\, 96.6\\% \\, & \\, 86.9\\% \\\\\n        $\\varepsilon = 15$ \\, & \\, 99.0\\% \\, & \\, 74.6\\% \\, & \\, 88.5\\% \\, & \\, \\cellcolor{gray!25}\\textbf{86.0}\\% \\, & \\, 93.0\\% \\, & \\, 86.0\\% \\, & \\, 94.5\\% \\, & \\, \\cellcolor{gray!25}\\textbf{90.3}\\% \\\\\n        $\\varepsilon = 25$ \\, & \\, 96.9\\% \\, & \\, 76.5\\% \\, & \\, 91.1\\% \\, & \\, 83.1\\% \\, & \\, 95.5\\% \\, & \\, 82.8\\% \\, & \\, 96.8\\% \\, & \\, 80.2\\% \\\\\n        \\midrule\n        \\multicolumn{9}{l}{\\textit{Our Proposed Method} with $KL_{\\lambda=10}$}\\\\\n        $\\alpha \\sim B(2, \\phantom{1}4)$ \\, & \\, 90.6\\% \\, & \\, 74.3\\% \\, & \\, 88.8\\% \\, & \\, 71.9\\% \\, & \\, 93.5\\% \\, & \\, 62.1\\% \\, & \\, 95.1\\% \\, & \\, 53.7\\% \\\\\n        $\\alpha \\sim B(2, \\phantom{1}6)$ \\, & \\, 84.7\\% \\, & \\, 72.4\\% \\, & \\, 87.1\\% \\, & \\, 74.2\\% \\, & \\, 91.8\\% \\, & \\, 70.0\\% \\, & \\, 93.7\\% \\, & \\, 66.0\\% \\\\\n        $\\alpha \\sim B(2, 10)$ \\, & \\, 86.9\\% \\, & \\, 71.4\\% \\, & \\, 85.8\\% \\, & \\, 76.9\\% \\, & \\, 90.4\\% \\, & \\, 75.0\\% \\, & \\, 92.4\\% \\, & \\, 72.4\\% \\\\\n        \\end{tabular}\n   \n       \n\\end{table*}\n\nAt first, adversaries have been generated as described in Section~\\ref{sec:assessment}. For the gradient alignment, we considered an ensemble of one, three and five models, respectively. In a first analysis, we investigated the impact of the perturbation parameter for the values $\\varepsilon \\in\\{4,8,16\\}$. Interestingly, we observed that the success rate for crafting adversaries is not sensitive to the tested values for $\\varepsilon$. Therefore, we are only referring to the adversaries with $\\varepsilon=16$ as they are the most powerful ones. If a single unprotected model -- which is fully identical to the \\textit{Base Model} in terms of topology and training execution -- is used to calculate multi-step adversaries, the \\textit{Base Model's} classification accuracy is still $23.8\\%$ as shown in Table~\\ref{tab:results}. However, aligning the gradients of an ensemble of three or five models, the \\textit{Base Model's} accuracy on these adversaries decreases to $4\\%$ and $0.9\\%$, respectively.\n\nNext, we trained multiple models with \\textit{Virtual Adversarial Training (VAT)}, \\textit{Projected Gradient Descent Training (PGD)} and the three different defense methods proposed in this work (see Section~\\ref{sec:counteractions}). For VAT we used $I=3$, $\\lambda=1$ and $\\varepsilon\\in\\{5,10,15,20,25\\}$. Miyato et al. \\cite{Miyato2015DistributionalSW} recommended\nusing $I=1$ and $\\lambda=1$ as they found it to be sufficient. We increased the \\textit{power of iterations} to $I=3$ to ensure a better conversion (cf. Miyato et al. \\cite{Miyato2015DistributionalSW} for more details). As tuning $\\varepsilon$ is most important, we tried several different values and compared them to each other. Next, we adapted the default parameter for PGD as proposed by \\cite{madry2018towards}: $\\varepsilon=8$, $\\lambda=2$ and $I\\in\\{3, 7\\}$ as the number of iterations. $\\varepsilon=8$, $\\lambda=2$ and $I=\\{7\\}$ are the default settings used for on CIFAR10 by \\cite{madry2018towards}. In addition, we used $I=3$ to speed up training.\n\nTable \\ref{tab:results} shows the results of different methods based on their accuracy on our crafted adversarial images. As indicated by the base model's accuracy values on the adversarial data, the more models are used for our proposed \\textit{gradient ensemble attack}, the higher is the success rate of transferring the adversaries to other models. This demonstrates that our adversaries, crafted from an ensemble of models, are likely transferable to other networks. Moreover, the VAT models seem to perform best on adversarial images.\n\nTo test the generalizability of our approach, we additionally assessed our adversaries on ResNet32 and Res\\-Net50~\\cite{HeZRS16}. As shown in Table~\\ref{tab:transfer}, when applying a \\textit{gradient ensemble attack} on VGGNet13 and the ResNet models together, the resulting adversaries likely transfer between both topologies. The accuracy of unprotected (base) models on our combined adversarial dataset is 0.01\\% for the VGGNet13 network and about 26 to 27\\% for both ResNet models -- although all three models have an accuracy of over 90\\% on the original images. Even adversaries that were originally crafted for a different topology reduce a model's accuracy noticeably. \n    \nWe further tested our method on different ResNets. As shown in the bottom half of Table~\\ref{tab:transfer}, we found that not only the originally considered VGGNet13 models, but also ResNet32 and ResNet50 became more resilient against the transferable adversaries.\n\nHowever, comparing the performance of adversarial defense methods merely based on the model's accuracy or on the success rate for crafting adversaries is problematic. Adversarial sub-spaces may occur a little aside of the original ones or gradient masking could prevent gradient-based attack methods from being successful. Therefore, we do not only refer to a model's accuracy on strong and transferable adversaries, but also investigate the surrounding decision space, as well as its gradient.\n\n\\begin{figure*}[p]\n    \\centering\n    \\centering\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/gradient_base.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (a) \\textit{Base Model}\n    \\end{minipage}\n    \\hfill\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/gradient_vat.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (b) VAT ($\\varepsilon=15$) \\cite{Miyato2015DistributionalSW}\n    \\end{minipage}\n    \\hfill\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/gradient_pgd_07.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (c) PGD ($I=7$) \\cite{madry2018towards}\n    \\end{minipage}\\\\\n    \\medskip\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/gradient_vgg_alpha_04.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (d) $\\alpha=0.4$\n    \\end{minipage}\n    \\hfill\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/vgg_vialpha_4_10.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (e) $\\alpha=0.4$, $KL_{\\lambda=10}$\n    \\end{minipage}\n    \\hfill\n    \\begin{minipage}[b]{0.325\\textwidth}\n        \\begin{overpic}[width=\\textwidth]{figures\/gradients\/gradient_vibeta_204_10.png}\n                \\put (40,20) {\\tiny{$X$}}\n                \\put (30,2) {$\\varepsilon_1$}\n                \\put (80,5) {$\\varepsilon_2$}\n            \\end{overpic}\\\\\n            (f) $\\alpha \\sim B(2, 4)$, $KL_{\\lambda=10}$\n    \\end{minipage}\n   \n    \\caption{Visualization of the decision space of six different VGGNet13 models in two adversarial directions of the same input image $X$. The loss is plotted using $x^* = x + \\varepsilon_1 \\cdot \\text{sign} \\left(\\nabla_{x}~L_1\\right) + \\varepsilon_2 \\cdot \\text{sign} \\left(\\nabla_{x}~L_2\\right)$ (cf. \\cite{TramerKPGBM18}). The red peak is an adversarial spot, which corresponds to a very high loss. The magnitude of the related model's gradients $\\text{sign}\\left(\\nabla_{x}~L_2\\right)$ is controlled by $\\varepsilon_2\\in[-8,32]$, whereas $\\varepsilon_1\\in[-16,16]$ controls the magnitude of an unprotected model's gradient $\\text{sign}\\left(\\nabla_{x}~L_1\\right)$. The six images visualize how the loss values change when the input $x$ is moved in one of these two directions. The images in (d) to (f) correspond to our proposed models.}\n    \\label{fig:gradient}\n    \n    \\bigskip\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_vgg_base.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (a) \\textit{Base Model}\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_vat_15.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (b) VAT ($\\varepsilon=15$) \\cite{Miyato2015DistributionalSW}\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_pgd_07.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (c) PGD ($I=7$) \\cite{madry2018towards}\n        \\end{minipage}\\\\\n        \\medskip\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_vibeta_204_10.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (d) VGGNet13 (Ours)\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_resnet32.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (e) ResNet32 (Ours)\n        \\end{minipage}\n        \\hfill\n        \\begin{minipage}[b]{0.325\\textwidth}\n            \\begin{overpic}[width=\\textwidth]{figures\/losses\/loss_resnet50.png}\n                    \\put (30,1) {$X$}\n                    \\put (80,5) {$\\varepsilon$}\n                \\end{overpic}\\\\\n                (f) ResNet50 (Ours)\n        \\end{minipage}\n       \n        \\caption{Impact of different images $X$ with additional perturbation on the loss value. The magnitude of the loss is controlled by varying $\\varepsilon$. In comparison to the models from the literature (top row), our approach flattens the decision space significantly as demonstrated by the low loss values -- even for very large values of $\\varepsilon$. Note that our proposed models (bottom row) were trained using $\\alpha \\sim B(2, 4)$ and $KL_{\\lambda=10}$.}\n        \\label{fig:loss}\n\\end{figure*}\n\n\\begin{table*}[!t]\n    \\caption{Accuracy of different networks on the sets of original and adversarial images. All adversarial datasets are crafted using our \\textit{gradient ensemble attack} approach with five unprotected VGGNet13, ResNet32 and ResNet50 models. For the last dataset (last column), we combined three VGG13, ResNet32 and ResNet50 models. The table shows the accuracy of three unprotected models on different adversarial datasets. \n    As one can see, multi-step adversaries do transfer between topologies. Further, our proposed method provides resilience against these transferable adversaries.}\n    \\label{tab:transfer}\n   \n    \\centering\n    \\begin{small}\n   \n        \\begin{tabular}{r|cc|cc|cc|cc|cc}\n        &\\multicolumn{2}{c|}{\\bf Accuracy on} & \\multicolumn{8}{c}{\\bf Accuracy on Adversarial Dataset}\\\\\n        &\\multicolumn{2}{c|}{\\bf Original Data} & \\multicolumn{2}{c|}{\\bf VGGNet13} & \\multicolumn{2}{c|}{\\bf ResNet32} & \\multicolumn{2}{c|}{\\bf ResNet50} & \\multicolumn{2}{c}{\\bf Combined} \\\\\n        &Train &Val. & True& Adv. & True & Adv. & True & Adv. & True & Adv.\\\\\n        \\midrule \\multicolumn{11}{l}{\\textit{Base Models}}\\\\\n        VGG13 \\, & \\, 91.2\\% \\, & \\, \\cellcolor{gray!25}\\textbf{73.4}\\% \\, & \\, 96.7\\% \\, & \\, \\cellcolor{gray!25}\\textbf{0.9}\\% \\, & \\, 96.2\\% \\, & \\, 28.3\\% \\, & \\, 96.9\\% \\, & \\, 50.2\\% \\, & \\, 98.4\\% \\, & \\, \\cellcolor{gray!25}\\textbf{0.01}\\% \\\\\n        ResNet32 \\, & \\, 89.4\\% \\, & \\, 66.7\\% \\, & \\, 87.9\\% \\, & \\, 67.6\\% \\, & \\, 95.5\\% \\, & \\, \\cellcolor{gray!25}\\textbf{16.0}\\% \\, & \\, 95.3\\% \\, & \\, 44.0\\% \\, & \\, 96.1\\% \\, & \\, \\cellcolor{gray!25}\\textbf{26.1}\\% \\\\\n        ResNet50 \\, & \\, 87.5\\% \\, & \\, 62.8\\% \\, & \\, 84.1\\% \\, & \\, 64.9\\% \\, & \\, 92.2\\% \\, & \\, 24.6\\% \\, & \\, 94.7\\% \\, & \\, \\cellcolor{gray!25}\\textbf{30.9}\\% \\, & \\, 94.5\\% \\, & \\, \\cellcolor{gray!25}\\textbf{26.7}\\% \\\\\n        \\midrule \\multicolumn{11}{l}{Our Proposed Method with $\\alpha \\sim B(2, 4)$ and $KL_{\\lambda=10}$}\\\\\n        VGG13 \\, & \\, 90.6\\% \\, & \\, \\cellcolor{gray!25}\\textbf{74.3}\\% \\, & \\, 95.1\\% \\, & \\, \\cellcolor{gray!25}\\textbf{53.7}\\% \\, & \\, 96.6\\% \\, & \\, 69.6\\% \\, & \\, 97.1\\% \\, & \\, 81.5\\% \\, & \\, 98.1\\% \\, & \\, \\cellcolor{gray!25}\\textbf{32.4}\\% \\\\\n        ResNet32 \\, & \\, 83.1\\% \\, & \\, 59.8\\% \\, & \\, 80.3\\% \\, & \\, 67.8\\% \\, & \\, 88.5\\% \\, & \\, \\cellcolor{gray!25}\\textbf{38.5}\\% \\, & \\, 89.5\\% \\, & \\, 56.9\\% \\, & \\, 90.2\\% \\, & \\, \\cellcolor{gray!25}\\textbf{45.1}\\% \\\\\n        ResNet50 \\, & \\, 80.4\\% \\, & \\, 62.4\\% \\, & \\, 83.8\\% \\, & \\, 70.3\\% \\, & \\, 91.6\\% \\, & \\, 44.1\\% \\, & \\, 92.9\\% \\, & \\, \\cellcolor{gray!25}\\textbf{56.4}\\% \\, & \\, 93.6\\% \\, & \\, \\cellcolor{gray!25}\\textbf{46.9}\\% \\\\\n        \\end{tabular}\n   \n   \n    \\end{small}\n\\end{table*}\n\nFigure \\ref{fig:gradient} illustrates the loss of six different models based on the decision space of a single input image in two adversarial directions -- one taken from an unprotected model and the other one in direction of the related model. As depicted by Figure~\\ref{fig:gradient}~(a), i.e., the image of the \\textit{Base Model}, one can see an adversarial sub-space in close proximity of the input image as indicated by the high loss value (shown in red). Thus, it only takes a small change in  $\\varepsilon$ ($\\varepsilon=6$ is optimal in this particular case) to push the input image $X$ deep into this adversarial sub-space. A similar adversarial sub-space occurs in case of the VAT model as shown in Figure~\\ref{fig:gradient}~(b). Here, the distance is greater, but the sub-space still exists. So, it takes a larger $\\varepsilon$ to fool the VAT model.\n\nIn case of our proposed methods, using a fixed $\\alpha$-value is not sufficient either as the blind-spot still exists, as illustrated in Figures~\\ref{fig:gradient}~(d) and (e). Although our proposed methods with a fixed $\\alpha$-value weaken the adversarial spot, they do not eliminate it. Probably, the $\\alpha$-value is chosen too high in this case as there is another adversarial spot behind the first one. We found that sampling $\\alpha \\sim B(p,q)$ is essential to flatten the decision space. As demonstrated by Figure~\\ref{fig:gradient}~(f) the decision space of our method with varying $\\alpha$-values is significantly more flat than all the others. In fact, it is even more flat than the one of the PGD model -- especially for large $\\varepsilon$-values as illustrated in Figures~\\ref{fig:gradient}~(c) and (f), respectively. \n\nTo verify these findings, we generated 128 adversarial images $X$ (using the FGSM) for each defense method and compared the loss values based on varying $\\varepsilon\\in[0,128]$ (c.f. Figure \\ref{fig:loss}). As for the VAT model, adversarial attacks can be highly successful. In fact, $\\varepsilon=26$ provides the greatest success-rate for single-step adversaries while for the \\textit{Base Model} $\\varepsilon=8$ works best (see Figures \\ref{fig:loss}~(a) and (b)). This proves once again, that the adversarial spots are just a little further away compared to the \\textit{Base Model}. This may be an explanation, on why the VAT model performs rather good on our generated adversaries. However, the VAT model is not substantially more robust against adversarial attacks than the unprotected base model -- it only requires a larger $\\varepsilon$ to fool it.\n\nIn contrast, the PGD model and our proposed method clearly flatten the decision space and thereby strongly reduce the risk of adversarial spots (cf. Figure \\ref{fig:loss}~(c) and (d)). As one can see, adversarial examples occur in greater distance compared to the \\textit{Base Model} and occur significantly less frequent as the lower loss values demonstrate. However, there are adversarial sub-spaces which cause a high distraction of the PGD models. In contrast to the PGD model, our model (Virtual Alpha with $p=2$, $q=4$ and $\\lambda=10$) provides a significantly more flattened space, i.e., high loss values rarely occur at all.  \n\nNoticeably, for $\\varepsilon<16$ the loss values of our model are significantly higher than the ones of the PGD model. In addition, loss values grow rapidly for small $\\varepsilon$-values. The high average loss values for small $\\varepsilon$-values may explain \\textit{why} our model has a lower accuracy on our generated adversarial datasets than the PGD model as the loss values do not need do be maximal in order to indicate misclassification.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nThis work investigates the effects of adversarial attacks on deep learning networks. It analyzes different strategies for increasing a model's resilience and, thus, countervailing malicious attacks. The performance of the different defense strategies is compared across large sets of transferable, carefully generated adversaries. Next, three new approaches to improve resilience against such perturbations were first introduced and then compared against the state-of-the-art techniques \\textit{Virtual Adversarial Training} and \\textit{Projected Gradient Descent}. In addition, a novel adversarial attack method called \\textit{gradient ensemble attack} has been introduced. Further, this work has demonstrated the transferability of adversaries, which have been crafted using our proposed method.\n\nWithin our investigations we have observed that VAT does not provide substantial resilience against adversarial perturbations as the adversarial sub-spaces are just pushed a little further away. However, the incidence of these spaces is similar to an unprotected model. Further, PGD trained models reduce the frequency of adversarial sub-spaces and strongly increase the distance to them. Yet, these sub-spaces still occur. Our proposed method, superimposing two images\nand minimizing the KL-Divergence between the two activations, reduces the risk of adversarial sub-spaces with high loss. In fact, our results demonstrate that these spaces rarely occur. However, the average loss value is significantly higher which explains why our models performed worse on our adversarial test sets. Nevertheless, our proposed method is very promising as it (i) is easily executable (it only requires an additional forward pass), and (ii) still provides a noticeable regularized decision space. \n\n\n\nOur ideas for future work are two-fold: (i) we will compare additional methods to further decrease the overall loss of our proposed method and thereby improve its performance on adversaries; (ii) we will investigate the effects of our \\textit{gradient ensemble attack} for crafting strong and transferable adversaries in a wider context -- especially applying it to different white- and black-box attack scenarios.\n\n\\section*{Acknowledgment}\n\nAll three authors acknowledge support by the \\href{https:\/\/www.ercis.org}{\\emph{European Research Center for Information Systems (ERCIS)}}.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction: Prediction of Hemispheric Asymmetry}\n\nIt is well known, starting already with the researches of Rudolf Wolf in the\nnineteenth century, that the sunspot cycle is not strictly periodic. Significant\ncycle-to-cycle variability is observed in both the amplitude and duration\nof the cycle. The hemispheric asymmetry in sunspot coverage\nwas already noticed by \\cite{Waldmeier1955}, who also pointed out that such\nasymmetry could be sustained for several years.\n\\cite{Babcock1959} reported that this asymmetry appears in the case of the polar field reversal as well.\nBased on observational data for cycles 12-23, \\cite{Norton2010} analyzed the asymmetry and proposed 20\\% as upper limit of sunspot area asymmetry. Regarding the phase lag asymmetry for the same period of time, \\cite{McIntosh2013} found a roughly four cycles long periodicity. \\cite{Zolotova2010} investigated the phase difference of sunspot cycles in the hemispheres for a longer duration, back to the Maunder Minimum and found secular variation. On the other hand, this periodicity appears only in the sign of the phase lag but not in its magnitude \\citep{Norton2014}.\n\\cite{Hathaway2016} predicted the hemispheric asymmetry of cycle 25 by extrapolating the polar fields of cycle 24 using their Advective Flux Transport model.\nAccording to their results, the Southern hemisphere should dominate the North.\nHowever, from a purely statistical point of view, the available polar field data (eg. \\citealt{Munoz2012}) is insufficient to infer a significant\ncorrelation from past cycles.\n\n\\cite{Belucz2013} investigated the hemispheric asymmetry generated\nin a 2D flux transport dynamo model by inserting a second meridional circulation cell on the southern hemisphere with different amplitude, latitude and depth. They found significant hemispheric asymmetry depending the properties of the second cell. Using the same model, \\cite{Shetye2015} focused on the effects of the meridional inflow towards the activity belts. According to their results, the intense inflow in one hemisphere leads to stronger toroidal fields and this asymmetry is sustained for more than one cycle.\n\n\\cite{Karak2017}, using their 3D surface flux transport and Babcock-Leighton solar dynamo model (\\citealt{Miesch2014}; \\citealt{Miesch2016}),\ninvestigated the influence of the tilt angle distribution by adding random scatter on Joy's law and a tilt-angle saturation was also added to the model (on this point\nsee also \\citealt{Lemerle2017}).\nThe SpotMaker algorithm they use places new BMRs in each hemispheres, with a set time\ndelay in order to avoid artificially imposing hemispheric symmetry. One of their results is the hemispheric asymmetry appearing in the polar fluxes\nwas only weakly correlated to the toroidal flux of the subsequent cycle. Due to the\nstrong diffusive coupling between the hemispheres in the model, the asymmetry reduces quickly.\n\n\nIn the present paper \\fix{we extend the work of \\citep{Nagy2017} on\nthe hemispheric asymmetry triggered by rogue BMRs. We present a detailed analysis of how rogue BMRs affect the hemispheric asymmetry of the polar cap flux, including prediction of the asymmetry level of the subsequent cycle based on the polar field asymmetry. We also investigate whether the asymmetry in the model shows periodicity or temporal persistence in its characterizing parameters.\nFollowing \\citep{Nagy2017}, our analysis is based on simulations carried out\nusing the recent $2\\times2$D dynamo model of \\citealt{Lemerle2015, Lemerle2017}.}\n\n\n\\section{Rogue BMRs in the $2\\times2$D Dynamo Model}\n\nThe \\cite{Lemerle2017} solar cycle model invokes differential rotation shear\nand the regeneration of the solar dipole via surface decay of active regions\n(the so-called Babcock-Leighton mechanism) as its primary inductive mechanisms.\nThis mean-field-like kinematic model\ncouples a 2D surface flux transport module\n(SFT) with a\n2D axisymmetric flux transport dynamo (FTD). The SFT component provides the azimutally averaged radial field\nserving as the upper boundary condition for the FTD\nsimulation, while the FTD module drives the SFT through the emergence\nof new bipolar magnetic region (BMR). This step is based\non a semi-empirical emergence function that sets the\nprobability of a BMR emerging (radially)\nas a function of toroidal magnetic field $B_t$ at\nthe bottom of the convective zone in the FTD module.\n\\fix{Motivated by the modelling of the destabilization and buoyant rise of thin\nmagnetic flux ropes initially located immediately beneath the base of the\nsolar convection zone \\citep[see, e.g.,][and references therein]{Fan2009},\na lower threshold\non $B_t$ is introduced, below which the emergence probability\nvanishes. The presence of a threshold implies that the dynamo is not self-excited,\nas the internal magnetic field must remain above threshold for regeneration of\nthe surface dipole to take place. Above this threshold, the proportionality constant\n$K$ between emergence probability and\n$B_t^{1.5}$ acts as the dynamo number for the model, since it effectively\nsets the mean surface dipole growth rate for a given internal toroidal\nfield strength.}\nProperties of the\nnew BMR --- emergence latitude, longitude, flux, angular separation, tilt --- are randomly drawn\nfrom distribution functions for these quantities built from observed\nstatistics of solar active regions during solar cycle 21, as described in Appendix A of\n\\citealt{Lemerle2015}.\n\nBecause the model is kinematic and includes a steady quadrupole-like meridional\nflow in the FTD module, the only physical mechanism available to couple\nthe two hemispheres is diffusive transport, operating in both the SFT and FTD modules.\n\nThe only amplitude-limiting nonlinearity included in the model\nis a\nreduction of the average BMR tilt angle $\\alpha$ as a function of the deep-seated\ntoroidal field strength $B_t$, parametrized\naccording to the following ad hoc algebraic formula:\n\\begin{equation}\\label{eq:tiltquenching}\n    \\alpha_q = \\frac{\\alpha(\\theta)}{1 + (B_t\/B_q)^2},\n\\end{equation}\nwhere $B_q$ is the quenching field amplitude, and $\\alpha$ is the reference\ntilt variation with latitude, i.e., Joy's law \\citep{McClintock2013}.\n\\fix{Lacking any reliable information on the manner in which the emerging flux ropes\nproducing BMRs disconnect from the underlying toroidal magnetic flux system,\nwe do not implement any flux reduction in the FTD module when emergences\nare introduced in the SFT module.}\n\nThe main advantages of the $2\\times2$D model are its high numerical efficiency\nand the fact that it is calibrated to follow accurately the\nstatistical properties of the real Sun. The complete\nlatitude--longitude representation of the simulated solar surface in\nthe SFT component further makes it possible to achieve high spatial\nresolution and account for the effect of individual active region\nemergences.\n\nThe reference solar cycle solution presented in \\citet{Lemerle2017} is\ndefined by 11 adjustable parameters, for which optimal values were obtained\nvia a formal optimization by a\ngenetic algorithm. The algorithm was designed to minimize the differences between the\nspatiotemporal distribution of emergences produced by the model, and\nthe observed sunspot butterfly diagram during cycle 21.\n\nFigure \\ref{fig:refsolution} shows a portion of the reference dynamo solutions\nused for the analyses presented in what follows. The solid lines on the\ntop and middle panels\nshow time series of hemispheric pseudo-sunspot number and polar cap flux, \\fix{which is calculated here as the surface integral of the radial magnetic field over a latitudinal extent of $20^{\\circ}$ from the poles.}\nThe bottom panel shows the corresponding time-latitude ``butterfly''\ndiagram for the spatial density of emerging BMRs, encoded as a grayscale.\nThe red dot indicates the time and latitude at which a large ``rogue''\nBMR emerged in this simulation,\nits properties being listed in the first row of Table \\ref{tab:BMRs}.\nArtificially removing this single BMR from the simulation leads to\na markedly different subsequent evolution of the dynamo solution,\nas shown by the dashed time series on panels (a) and (b)\nof Figure \\ref{fig:refsolution}.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{Refcase_SSN.pdf}\\\\\n  \\includegraphics[width=\\textwidth]{Refcase_PC.pdf}\\\\\n  \\includegraphics[width=\\textwidth]{Refcase_Bfly.pdf}\\\\\n  \\caption{A short segment of the reference dynamo solution used for the analyses\ndiscussed in the present paper.\nPanel (a) shows the pseudo sunspot number time series separately for the hemispheres.\nThe solid lines shows the reference simulation run.\nOn panel (b) hemispheric time series of the polar cap flux are plotted, in\nred and blue for the Northern and Southern hemispheres, respectively.\nOn both of these panels dashed lines\npertain to a experiment\nin which a single large ``rogue'' BMR was removed from the simulation at the time indicated\nby the vertical black dashed line.\nPanel (c) shows the pseudo-sunspot butterfly diagram of the reference simulation plotted\nas solid lines in panels (a) and (b).\nThe gray scale encodes the density of emerging BMRs, and\nthe red dot indicates the position of the rogue BMR removed from the simulation\nto yield the dashed time series in (a) and (b).\n}\\label{fig:refsolution}\n\\end{figure}\n\nThe reference solution plotted on Figure \\ref{fig:refsolution}\nis the same as adopted in the numerical experiments of\n\\citet{Nagy2017} where the impact of individual ``rogue'' BMRs were analyzed. These peculiar active regions were identified based on their contribution to the global dipole moment defined as follows:\n\\begin{equation}\n \\delta D_{\\mathrm{BMR}} \\approx F \\,d \\, \\sin\\alpha \\,  \\sin\\theta,\n \\label{eq:thenumber}\n\\end{equation}\nwhere $F$ is magnetic flux, $d$ is the\nangular separation of the two polarities, $\\alpha$ is the tilt angle, $\\theta$\nis the colatitude. According to this expression, BMRs with high flux content, tilt angle and angular separation, close to the equator influence the most the building up dipole moment, and therefore the strength of the subsequent cycle \\fix{as suggested by \\citet{Jiang2015} as an explanation for the low amplitude of Cycle 24}.\n\n\\begin{figure}[t!]\n  \\centering\n  \\begin{minipage}{0.55\\textwidth}\n  \\includegraphics[width=0.95\\linewidth]{DeltaTheta_andothers.pdf}\n  \\end{minipage}\n  \\begin{minipage}{0.44\\textwidth}\n  \\caption{Average effect of varying the properties of a BMR (2nd data\nrow of Table \\ref{tab:BMRs}), inserted in the simulations at cycle\nmaximum, on the amplitude of the subsequent cycle. Variations in BMR\nflux (green), tilt angle (blue) and  polarity separation (orange) are\nconverted to the contribution to the dipole moment according to\nEquation (\\ref{eq:thenumber}), while the varying colatitudes (red) are\nshown on the top axis (adapted from Figure 11 in \\citealt{Nagy2017}).}\\label{fig:summarize}\n  \\end{minipage}\n\\end{figure}\n\n\\citet{Nagy2017} carried out\nseveral numerical experiments aiming to study how the parameters of BMRs affect the next, or even the ongoing cycle. A selected ``test'' BMR\nwith specified properties ($F$, $d$, etc.)\nwas inserted manually into simulations while the parameters of the active region were varied separately during each experimental series. The test-BMR emerged spontaneously during the reference simulation with parameters listed in the second row of Table \\ref{tab:BMRs}. The experiments were performed for three cycles with average, below average and above average amplitudes, respectively.\nIn each case two series of experiments were carried out with Hale (anti-Hale) test-BMR in order to increase (decrease) the dipole moment of the examined cycle. The characteristics of the\ntest-BMR -- emergence time and latitude, flux, tilt angle and angular separation -- were changed one by one in order to map the impact of each property on the subsequent simulated cycle.\nThe results of these experiments are summarized in Figure \\ref{fig:summarize}.\nBy jointly varying the flux, tilt angle or separation of the test-BMR, similar results were obtained for the amplitude of the upcoming cycle. This is because\nthese quantities have a combined effect in the form of Equation (\\ref{eq:thenumber}). The impact of the emergence latitude is also shown in Figure \\ref{fig:summarize} (red curve, top axis).\nThe effect of the BMR decreases as a function of the emergence latitude, but is still significant $20^{\\circ}$ away from the Equator. The emergence epoch is also an important factor; the strongest impact on the subsequent cycle is expected when the test-BMR appears at cycle maximum,\nand diminishes gradually as the rogue BMR is forced to emerge later and later in the descending\nphase of the cycle.\nWhen the emergence occurs during the rising phase of the perturbed cycle, it modifies the ongoing cycle as well.\n\n\\begin{table}[t!]\n    \\center\n  \\caption{Parameters of active regions discussed in the\npaper.  Colatitudes $\\theta_{\\mathrm{lead}}$ and\n$\\theta_{\\mathrm{trail}}$ are the\n  latitudinal positions of leading and trailing polarities;  $F$ is\nthe flux of the trailing polarity ($F_{\\mathrm{trail}} =\n-F_{\\mathrm{lead}}$); $\\alpha$ is the tilt angle and $d$ is the\nangular separation of leading and trailing polarities. $\\delta\nD_{\\mathrm{BMR}}$, the contribution of the BMR to the global dipole\nmoment, is defined according to Equation (\\ref{eq:thenumber}). J\/H\nindicates whether the active region is (anti-)Joy\/(anti-)Hale. In the\ncase of the second row a J\/H (J\/a-H) test-BMR increases (decreases)\nthe dipole moment during the experiments detailed in Section 5 of\n\\citep{Nagy2017}. }\\label{tab:BMRs}\n  \\begin{tabular}{ c  c  c  c  c  c  c }\n  \\hline\n  $\\theta_{\\mathrm{lead}}$ & $\\theta_{\\mathrm{trail}}$ & $F $ [$10^{23}$ Mx] & $\\alpha$  & $d$ & $\\delta D_{\\mathrm{BMR}}$[$10^{23}$ Mx] & J\/H  \\\\\n  \\hline\n  112$^{\\circ}$           & 118.6$^{\\circ}$           & 4.39              & $-11.08^{\\circ}$ & 34.08$^{\\circ}$   & $-0.5124$ &  J\/H\\\\\n  89.5$^{\\circ}$           & 82.1$^{\\circ}$            & --1.39              & 13.98$^{\\circ}$  & 30.97$^{\\circ}$   & --0.1810 & J\/H  \\\\\n                           &                           &                     &                  &                   &          & J\/a-H \\\\\n  \\hline\n  \\end{tabular}\n\n\\end{table}\n\n\n\\section{Hemispheric Asymmetry due to Rogue BMRs}\n\nAs proposed by \\cite{Hathaway2016} in the context of variation in the surface\nmeridional flow, the hemispheric asymmetry of a solar cycle originates from the polar cap flux asymmetry during the preceding cycle. \\citet{Nagy2017} analyzed whether the polar cap flux asymmetry is a good indicator of the upcoming simulated cycle's asymmetry in the $2\\times2$D model.\n\nComparison of the solid and dashed lines on Figure \\ref{fig:refsolution}(a) and (b) indicates that a single, large BMR with unusual characteristics\ncan have a large and persistent impact on\nhemispheric asymmetry in the resulting dynamo solution.\n\\fixx{The top panel on Figure \\ref{fig:synopticRogue} shows a synoptic magnetogram extracted from a simulation in which the rogue BMR listed in the first row of Table \\ref{tab:BMRs} was inserted at simulation time $t = 1992.72$, one cycle before the strongly asymmetric cycle that we will discuss in the next section.\nThis snapshot is extracted six months after emergence of the rogue BMR.\nThis\nis an anti-Hale BMR, with polarity ordering opposite to that which should\ncharacterize its hemisphere, so that the emergence impedes the build up\nof the Southern polar fields. The poleward surge of positive polarity (red)\ncan be seen quite clearly.\nFor comparison, the bottom\npanel shows a second synoptic magnetogram, extracted at the same time in\na parent simulation without artificial insertion of the rogue BMR.\nComparing the two snapshots reveals how\nthe positive trailing flux of the decaying rogue has strongly\naltered\nthe pattern of magnetic flux transport to the Southern polar regions.\nThis pattern is qualitatively similar to that highlighted by\n\\citet{Upton2018}, with the large\nactive region AR12192\nemerging in October 2014 and producing a poleward stream of positive\nmagnetic polarity which weakened the buildup of the southern polar cap\nnegative magnetic field.\n}\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{FrameT-400rogue5952-6950.pdf}\n  \\includegraphics[width=\\textwidth]{FrameTnorogue5952-6950.pdf}\n  \\caption{\\fixx{On the top we plot the synoptic magnetogram of the rogue BMR listed in the first row of Table \\ref{tab:BMRs} at central longitude $\\phi\\sim360^{\\circ}$ and simulation time $t = 1992.72$, six months after the time of emergence (see the corresponding time series in Figure \\ref{fig:refsolution} and \\ref{fig:magnetogr}).\nThe bottom panel shows the magnetic field distribution without the emergence of this active region at the same epoch.  Here (and on Fig.~\\ref{fig:magnetogr} following), the color scale is strongly saturated to make the weaker magnetic fields visible.\n}}\\label{fig:synopticRogue}\n\\end{figure}\n\nFigure \\ref{fig:magnetogr}\noffers another example of this effect, for a different rogue event\nand now in the form of time-latitude maps\nof the zonally-averaged\nsurface radial magnetic field component. The top panel corresponds to the\nreference solution, while the bottom panel shows the magnetogram resulting from the\nartificial removal of a single large BMR at the time indicated by the vertical dashed line.\nNote in particular how the reversal of the polar field occurs almost\nsimultaneously in both hemispheres when the rogue BMR is removed, while in the\noriginal reference solution the southern polar cap reverses polarity some two years\nprior to the northern hemisphere.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{Refcase_magnetogr.pdf}\\\\\n  \\includegraphics[width=\\textwidth]{Refcase_magnetogr_norogureBMR.pdf}\\\\\n  \\caption{The top panel shows a synoptic magnetogram of the surface radial magnetic\nfield component in the reference solution of Figure \\ref{fig:refsolution}. The bottom panel\nshows the synoptic magnetogram resulting from removal of the rogue BMR at the\ntime indicated by the vertical dashed line, corresponding to the dashed time\nseries on panels (a) and (b) of Figure \\ref{fig:refsolution}.}\\label{fig:magnetogr}\n\\end{figure}\n\\fix{Note that here the polar cap flux peaks 2--3 years prior to SSN minimum,\nwhile in\nthe case of the sun this peak usually occurs somewhat closer to cycle minimum.\nHowever, in our model as in the sun,\nthe peak polar cap flux does turn out\nto be a good predictor of the SSN amplitude of the subsequent cycle, so we retain\nit as a measure of cycle dipole strenth in all analyses that follow.}\n\nTo quantify the level of asymmetry, the normalized asymmetry of the peak polar cap flux ($\\Delta_{\\Phi}$) produced during the cycles is\ncompared to two asymmetry measures characterizing the subsequent cycles. These measures are the asymmetry of the total number of emergences in each hemisphere ($\\Delta_{\\mathrm{SSN}}$), and the time delay between the epochs when the new cycle BMRs first\nstart to emerge in the North and the South ($\\Delta_{T}$).\n\nThe asymmetry of the polar cap flux at a given cycle is defined as follows:\n        \\begin{equation}\n        \\Delta_{\\Phi} = \\frac{|\\Phi_{\\mathrm{N},\\mathrm{max}}| - |\\Phi_{\\mathrm{S},\\mathrm{max}}|} {( |\\Phi_{\\mathrm{N},\\mathrm{max}}| + |\\Phi_{\\mathrm{S},\\mathrm{max}}| )\/2},\n        \\end{equation}\n    \\noindent where $\\Phi_{\\mathrm{N},\\mathrm{max}}$ ($\\Phi_{\\mathrm{S},\\mathrm{max}}$) is the northern (southern) polar cap flux maximum.\nThe asymmetry of the activity level is defined similarly, but in\nterms of the pseudo-sunspot number constructed from the model output:\n        \\begin{equation}\n        \\Delta_{\\mathrm{SSN}} = \\frac{\\Sigma \\mathrm{SSN}_{\\mathrm{N}} - \\Sigma \\mathrm{SSN}_{\\mathrm{S}} } {( \\Sigma \\mathrm{SSN}_{\\mathrm{N}} + \\Sigma \\mathrm{SSN}_{\\mathrm{S}} )\/2},\n        \\end{equation}\n    \\noindent where $\\Sigma \\mathrm{SSN}_{\\mathrm{N}}$ ($\\Sigma \\mathrm{SSN}_{\\mathrm{S}}$) is the total number of emergences in the northern (southern) hemisphere.\nFinally, the time lag between the hemispheres is defined as:\n        \\begin{equation}\n          \\Delta_{T} = \\frac{t_{\\mathrm{N}} - t_{\\mathrm{S}}} {( T_{\\mathrm{N}} + T_{\\mathrm{S}} )\/2},\n        \\end{equation}\n    \\noindent where $t_{\\mathrm{N}}$ ($t_{\\mathrm{S}}$) is the beginning epoch of the cycle, while $T_{\\mathrm{N}}$ ($T_{\\mathrm{S}}$) is the duration of the cycle on the North (South).\n\nUpon calculating these asymmetry measures for 540 simulated cycles, \\citet{Nagy2017} found strong anticorrelations between the polar cap flux asymmetry of cycle $i$ and time delay, $\\Delta_{\\mathrm{T}}$ ($r = -0.7174$)\nduring cycle $i+1$.\nIn the case of asymmetry in number of emergences of cycle $i+1$, $\\Delta_{\\mathrm{SSN}}$  the correlation coefficient is $r = 0.7430$ as it is shown in Figure \\ref{fig:asymmetry500cycles}. This result shows that in the model the asymmetry of a cycle can be predicted via the asymmetry of the polar cap flux built up during the previous cycle.\n\n\n    \\begin{figure}[t!]\n      \\centering\n      \\begin{minipage}[b]{1\\textwidth}\n          \\includegraphics[width=0.5\\textwidth]{HemisphericAsymm_crossSSN.pdf}\n          \\includegraphics[width=0.5\\textwidth]{HemisphericAsymm_crossT.pdf}\n      \\end{minipage}\n      \\begin{minipage}[b]{1\\textwidth}\n        \\caption{\\small{ Two-dimensional histograms of the asymmetry of the hemispheric total\n     pseudo-SSN (\\textbf{left}) and the time lag between North and South (\\textbf{right})\n     in pseudo-solar cycle $i$ against the polar cap flux asymmetry during the\n     previous cycle for 540 simulated cycles. Some outlier data have been removed.\n     The number of cases (cycles) in each bin are\n     indicated by the colour codes. The correlation coefficient is\n     $r_{\\Delta SSN} = 0.7430$ and $r_{\\Delta T} = -0.7174$, respectively (adapted from Figure 6 in \\citealt{Nagy2017}).\n     }}\\label{fig:asymmetry500cycles}\n      \\end{minipage}\n    \\end{figure}\n\nIn order to assess the persistence of hemispheric asymetry, we separate the\nsimulated cycles in two groups according to hemispheric dominance, as measured\nby the quantity $\\Delta_{\\rm SSN}$ introduced above. For each group, we then\nconstruct the histograms of $\\Delta_{\\rm SSN}$ values characterising the\ncycle following each member of the groups. The resulting histograms are plotted\nin Figure\n\\ref{fig:PeriodHist}. Both are very well fit with Gaussian centered on\n$\\Delta_{\\rm SSN}=0$, with deviation from zero at the $10^{-2}$ level\nand standard deviation $\\sim 0.4$. This indicates that the probability of\nfinding a given hemispheric dominance in cycle $n+1$ is independent of\nhemispheric dominance in cycle $n$, and thus that hemispheric dominance is\ndetermined by processes operating on inter-cycle timescales.\n\\begin{figure}[t!]\n  \\centering\n  \\begin{minipage}{\\textwidth}\n  \\centering\n  \\includegraphics[width=0.4\\linewidth]{hist_SSN_afterN.pdf}\n  \\includegraphics[width=0.4\\linewidth]{hist_SSN_afterS.pdf}\n  \\end{minipage}\n  \\begin{minipage}{\\textwidth}\n  \\caption{ Strength asymmetry histograms of cycles following a North (South) dominated cycle on the left (right) panel. The histograms show the distribution of asymmetry\nparameters for all cycles following a Northern-dominated cycle (left) or\nSouthern-dominated cycle (right). The distributions are approximately Gaussian,\nwith means and standard deviations are respectively\n$0.012$ and $0.36$ in the left panel, and $-0.008$ and $0.41$ for the right panel.\nThis indicates that hemispheric dominance shows no significant\npersistence from one cycle to the next, at least in the (optimal) parameter regime\nof this simulation run.\n}\\label{fig:PeriodHist}\n  \\end{minipage}\n\\end{figure}\nWe repeated this exercise, this time constructing the distribution of asymetry\nparameters two cycles in the future instead of one, in order to possibly detect\npersistence of hemispheric asymmetry associated with one magnetic polarity\ndominating over the other for a few subsequent cycles, a features sometimes\nobserved in the \\cite{Lemerle2017} dynamo solutions. Once again the mean\nof the distribution are very close to zero, with standard deviations $\\sim 0.4$,\nindicating that hemispheric asymmetry does not persist beyond\none cycle in this model.\n\nBased on their numerical experiments, \\citet{Nagy2017} identified another interesting effect triggered by the emergence of a rogue BMRs. After such emergences in one cycle, the next cycle tends to be strongly asymmetric. This phenomenon was analyzed using a new test-BMR described in the first line of Table \\ref{tab:BMRs}, emerged on the southern hemisphere at cycle maximum, indicated by the vertical dashed line in Figure \\ref{fig:casestudy}. According to the previous results, the AR's impact on the upcoming cycle is the strongest at this epoch. On the other hand, the original position of the BMR is a bit far from the Equator.\nFor this reason, during the experimental runs the test region was replaced to emerge closer, about $15^{\\circ}$ far from the equator, within the region where significant effect was observed during the first experimental series.\nAt this position the active region's flux was decreased from about $4\\cdot10^{23}$ Mx down to $2.19\\cdot10^{23}$ Mx.\nThe black solid line in the top panel of Figure \\ref{fig:casestudy} shows the reference case when the BMR emerged at the original position, while black dashed line shows the case when the BMR was removed from the simulation. The coloured dashed lines show how the asymmetry changed for various values of the test-BMR's flux, as color-coded.\nOne can see that the asymmetry is changing according to the flux of the test region. There are slight changes in the amplitude of the northern hemisphere as well due to the\ndiffusive hemispheric coupling in the model.\nThe bottom panel of Figure \\ref{fig:casestudy} shows that the hemispheric asymmetry already appears in the form of polar cap flux asymmetry during the cycle within which\nthe test-BMR emerges.\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=\\textwidth]{cycle_03_hemSSN_BMRonS.pdf}\\\\\n  \\includegraphics[width=\\textwidth]{cycle_03_PC_BMRonS.pdf}\\\\\n  \\caption{The \\textbf{top} panel shows how a test-BMR can modify the amplitude of the subsequent cycle, separately in each hemisphere. The properties of this region are listed in the first row in Table \\ref{tab:BMRs}.\n  Black solid curves indicate the reference solution, with a ``rogue'' BMR emerging\n$25^{\\circ}$ away from the Equator, at the time indicated by the vertical dashed line.\nThe black dashed curves refer to a\nmodified simulation in which this ``rogue'' BMR is removed. Colored curves indicate the results with test-BMRs with different flux, $15^{\\circ}$ far from the Equator, as labeled.\nOn the \\textbf{bottom} panel the polar cap flux is shown. Solid and dot-dashed lines indicates the southern polar cap flux while dashed lines correspond to the northern polar cap flux.\n}\\label{fig:casestudy\n\\end{figure}\n\nThe correlation between the polar cap flux asymmetry triggered by the test-BMR and the asymmetry parameters of the subsequent cycle is plotted in Figure \\ref{fig:asymmetry_seed512}. Besides the results for both hemispheres of the reference cycle, we plot results for five more cycles that were studied using the same test-BMR emerging $15^{\\circ}$ far from the equator at cycle maximum.\nWhen the BMR is inserted in the northern hemisphere, its tilt and polarity are set to obey Joy's and Hale's law. Considering all the six experimental runs, the correlation coefficient between the polar cap flux asymmetry during the perturbed cycle ($\\Delta_{\\Phi,i-1}$) and the asymmetry of the number of emerged BMRs during the next cycle ($\\Delta_{\\mathrm{SSN},i}$) is 0.8431. In the case of the time lag between the hemispheres ($\\Delta_{T,i}$) this correlation is $-0.8029$.\n\n     \\begin{figure}[t!]\n      \\centering\n      \\begin{minipage}[b]{1\\textwidth}\n          \\includegraphics[width=0.51\\textwidth]{crossplot_AsymmCorr_SSN.pdf}\n          \\includegraphics[width=0.51\\textwidth]{crossplot_AsymmCorr_TL.pdf}\n      \\end{minipage}\n      \\begin{minipage}[b]{1\\textwidth}\n        \\caption{\\small{ Correlation plots between the asymmetry of the hemispheric total\n     pseudo-SSN (\\textbf{left}) and the time lag between North and South (\\textbf{right}) in\n     pseudo-solar cycle $i$ against the polar cap flux asymmetry during the\n     previous cycle. Colored markers correspond to the example shown in Figure \\ref{fig:casestudy} and also for the experimental series for the northern hemisphere (red markers). Gray markers show the same series for five more cycles in the same simulation. The correlation coefficients are\n     $0.9463$ and $-0.9158$, respectively, and $0.8431$ and $-0.8029$ when using\ndata from the combined six experiments.  }}\\label{fig:asymmetry_seed512}\n      \\end{minipage}\n    \\end{figure}\n\n\nDiffusive transport is the only cross-equatorial coupling mechanism operating\nin the \\cite{Lemerle2017} dynamo model used to carry out the various experiments\ndescribed above. The leading member of a BMR emerging $15^\\circ$ from the equator\nwill diffuse to the equator on a timescale $\\tau=(\\pi R\/12)^2\/\\eta_R\\simeq 2\\,$yr\nfor the surface diffusivity value $\\eta_R=6\\cdot 10^{12}\\,$cm$^2\\,$s$^{-1}$\nused in the SFT module. On the other hand, the internal toroidal field\nat 15 degrees will diffuse to the equator on a timescale controlled\nby the internal magnetic diffusivity $\\eta_t=10^{12}\\,$cm$^2\\,$s$^{-1}$ of the\nFTD module,\nleading to a timescale $\\tau\\simeq 12\\,$yr. The first timescale\nindicates that a rogue\nBMR emerging close to the equator can induce a polar cap asymmetry in the ongoing\ncycle, in agreement with the experimental results displayed\non Fig.~\\ref{fig:asymmetry_seed512};\nwhile the second timescale reveals that this asymmetry,\nonce it has built up in the internal toroidal field, can persist over\na full cycle before being diffusively balanced. Periodicity in hemispheric\nasymmetry cannot be driven or sustained by such diffusive coupling alone, and\nindeed is not observed in our dynamo simulations. Dynamical backreaction on\nlarge-scale flows, namely meridional circulation and differential rotation\nwould be the most likely candidate mechanism that could lead to such behavior.\n\n\\section{Conclusion}\n\nThe hemispheric asymmetry triggered by rogue active regions was studied using simulated data of the $2\\times2$D solar dynamo model. The flux of a selected test-BMR was changed while its position was fixed to $15^{\\circ}$ far from the Equator either\non the northern or southern hemispheres. The emergence epoch was the cycle maximum, while the polarity was set to increase the building up dipole moment. Experimental series were carried out in the case of six simulated cycles of varying amplitudes.\n\nIn contrast to the results of \\cite{Karak2017} we found strong correlation between the hemispheric asymmetry of the polar cap flux of cycle $i$ and the asymmetry of hemispheric activity levels\nduring the subsequent cycle $i+1$. The time lag between the hemispheric\nlag in the onset of cycle $i+1$ is also strongly correlated to the asymmetry of the\npolar cap flux of the preceding cycle. These results can be understood in terms\nof diffusive coupling of the magnetic field across the equatorial plane.\n\nOur results thus demonstrate that\nthe polar cap flux asymmetry at the end of a cycle can be determined\nby a single peculiar active region, emerging relatively close to the equator.\nThis offers an alternate scenario to that suggested by \\cite{Hathaway2016},\nbased on hemispheric variations in the surface meridional flow, which,\nin our kinematic dynamo model, remains strictly constant.\nIn view of the relatively strong\nhemispheric asymmetry observed in cycle 24, the unfolding of cycle 25 may\nallow to discriminate between these two scenarios.\n\n\\section*{Acknowledgements}\nM.N.'s research is currently supported by the \\'UNKP-18-3 New National Excellence Program of the Ministry of Human Capacities.\nP.C. and A.L. are supported through the Discovery Grant Program of the\nNatural Sciences and Engineering Research Council of Canada.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{&Intro}}\n\nThe symmetries of the Euler-Lagrange equations of motion\nwere recently used to study the constrained dynamics of singular\nLagrangians \\cite{ADS2020}. The focus was on almost regular\nLagrangians \\cite{Got1978, Got1979, Got1980, Car1990a}, and it was\nfound that for these Lagrangians the Euler-Lagrange equations of motion  \nadmit a generalized Lie symmetry (also known as a local gauge\nsymmetry). The generators $\\mathcal{S}\\hbox{ym}$ of this symmetry group\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ were determined in the Lagrangian\nphase space approach to Lagrangian mechanics, and were found to lie\nin the kernel of the Lagrangian two-form $\\mathbf{\\Omega}_L$. While it\nis well-known that the solutions $\\mathbf{X}_E$ of the energy equation,  \n\\begin{equation}\n  0=\\mathbf{d}E-i_{\\mathbf{X}_E}\\mathbf{\\Omega}_L,\n  \\label{EnergyE}\n\\end{equation}\nis not unique for almost regular Lagrangians, it was shown in \\cite{ADS2020} that\nthe action of $\\mathcal{S}\\hbox{ym}$ on a general solution  \nto this equation\\textemdash and in particular, on the \\textbf{second-order,\n  Lagrangian vector field} (SOLVF)\\textemdash will result in a vector\nfield that is no longer a solution of Eq.~$(\\ref{EnergyE})$. Thus, not all\nsolutions of the energy equation have\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ as a symmetry group. It is,\nhowever, possible to construct \nsolutions to Eq.~$(\\ref{EnergyE})$ for whom $\\mathcal{S}\\hbox{ym}$\ndoes generate a group of symmetry transformations\n\\cite{ADS2020}. These vector fields are called \\textbf{second-order,\n  Euler-Lagrange vector fields} (SOELVFs). As the \nevolution of the dynamical system for singular Lagrangians must lie on\nLagrangian constraint surfaces \\cite{Car1990a}, a Lagrangian constraint\nalgorithm for SOELVFs was also introduced in \\cite{ADS2020} to\nconstruct such solutions to the energy equation. It was\nthen shown that these SOELVFs, along with the dynamical structures\nin the Lagrangian phase space needed \nto describe and determine the motion of the dynamical system, are projectable to the\nHamiltonian phase space. In particular, the primary\nHamiltonian constraints can be constructed from vectors that lie in the kernel of\n$\\mathbf{\\Omega}_L$, and the Lagrangian constraint\nalgorithm for the SOELVF is equivalent to the stability analysis of \nthe total Hamiltonian (we follow the terminology found in\n\\cite{Hen1992}; see also \\cite{Dir1950, Mun1989, Lus2018}) obtained\nusing constrained Hamiltonian mechanics. Importantly, the end result\nof this stability analysis gives a Hamiltonian vector field that is the \nprojection of the SOELVF obtained from the Lagrangian constraint\nalgorithm. The Lagrangian and Hamiltonian formulations of mechanics\nfor almost regular Lagrangians were thereby shown to be equivalent. \n\nWhile \\cite{ADS2020} focused on the generalized Lie symmetries of the\nEuler-Lagrange equations of motion and whether the dynamical\nstructures constructed in the Lagrangian phase space are projectable to\nthe Hamiltonian phase space, in this paper the focus is on the symmetries\nof the action itself and the impact these symmetries have on the\nevolution of dynamical systems. This impact is found to be quite broad,\nsurprisingly restrictive, and unexpectedly subtle. Indeed, even the\nseemingly reasonable expectation that any generalized Lie symmetry of\nthe Euler-Lagrange equations of motion should be a reflection of the\nsymmetries of the action itself is not borne out.\n\nWe find that if the action has a generalized Lie\nsymmetry, then its Lagrangian is necessarily singular; the converse\nneed not be true, as we show through a specific example. We also find\nthat the generators of the generalized Lie symmetry of the action form\na Lie sub-algebra of the generators of the\ngeneralized Lie symmetry of the Euler-Lagrange equation of motion;\nonce again, the converse is not true. We give an example of a dynamical\nsystem for which the Euler-Lagrange equations of motion has a\ngeneralized Lie symmetry, while its action does not. Most importantly,\nfor systems where the Lagrangian is almost regular and for \nwhich the two-form $\\mathbf{\\Omega}_L$ has constant rank, we show that\neach generalized Lie symmetry of the action contributes one\narbitrary constant to the SOELVF. The dimensionality of the space of\nsolutions to the energy equation that have\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ as a symmetry group is thus at\nleast as large as the number of generalized Lie symmetries of the\naction. Moreover, if second- or higher-order Lagrangian\nconstraints are introduced during the application of the Lagrangian\nconstraint algorithm, these additional constraints cannot be due to the\ngeneralized Lie symmetry of the action.      \n      \nSymmetries of Lagrangian systems have been studied before. However,\nsuch analyses have been focused on time-dependent Lagrangians \\cite{Pri1983, Pri1985, \nCra1983, Car1991, Car1988b, Car1992, Car1993, Car2003}; on systems of\nfirst-order evolution equations \\cite{Car1990b, Mar1992, Gra2002, Gra2005,\n  Pop2017}; or on general solutions of Eq.~$(\\ref{EnergyE})$ \n\\cite{deL1995} (see also \\cite{Dim2016}). Importantly, the great majority of these\nstudies have been done using first-order prolongations on\nfirst-order jet bundles with a focus on the Lie symmetries of first-order\nevolution equations. Our interest is in the symmetries of the action, which\nnaturally leads us to consider generalized Lie symmetries and\nsecond-order prolongations. To our knowledge, such symmetry analysis\nof the action has not been done before. (The framework for\n$k^{\\hbox{\\textit{th}}}$-order prolongations on\n$k^{\\hbox{\\textit{th}}}$-order jet bundles have been introduced before\n\\cite{Car1993, deL1995, Car2003, Pop2009, Pop2011}, but they were not applied\nto the action or to the Euler-Lagrange equations of motion.)\n\nThe rest of the paper is arranged as follows. In \\textbf{Section\n  \\ref{&Sym}} the conditions under which the action for a dynamical\nsystem, and the conditions under which the Euler-Lagrange equations of\nmotion for this action, have a generalized Lie symmetry are\ndetermined. To compare the conditions for each, the analysis for the\ntwo are done separately, with each self-contained. In\n\\textbf{Section \\ref{&review}} properties of the Lagrangian phase\nspace are reviewed, and the notation used here established. The\ngenerators of the generalized Lie symmetry group for the\nEuler-Lagrange equations of motion were determined in \\cite{ADS2020},\nand a summary of the results found therein that are needed here is\ngiven. In \\textbf{Section \\ref{&A-S}} the generators of the\ngeneralized Lie symmetry group for the action is found within the\nLagrangian phase space approach, and their relation to the\ngenerators for the symmetry group of the Euler-Lagrange equations of\nmotion is determined. The impact of the symmetries of the action on\nthe SOELVF is then analyzed by applying the \nLagrangian constraint algorithm introduced in \\cite{ADS2020} to these\nSOELVF. The results obtained in this paper is then applied to three different\ndynamical systems in \\textbf{Section \\ref{&Exam}}. In particular, an\nexample of a dynamical system that has no generalized Lie symmetries\nand yet is still singular, and another example where the action has no\nsymmetries and yet the Euler-Lagrange equations of motion do, are\ngiven. Concluding remarks can be found in \\textbf{Section \\ref{&Conc}}.      \n\n\\section{Generalized Lie symmetries and Lagrangian Mechanics\\label{&Sym}}\n\nIn this section we determine the conditions under which the action of\na dynamical system, and the conditions under which the Euler-Lagrange\nequations of motion for this system, has a generalized Lie\nsymmetry. While the determination for both is done within Lagrangian\nmechanics, the analysis for the action is completed separately from that of\nthe equations of motion\\textemdash with each self-contained\\textemdash so \nthat the two conditions can be compared. We will later show that every\ngenerator of the generalized Lie symmetry of the action is a\ngenerator of a generalized Lie symmetry of the Euler-Lagrange\nequations of motion. Interestingly, the converse is not true. \n\n\\subsection{Symmetries of the Action \\label{&A-Sym}}\n\nWe begin with Lagrangian mechanics, and an analysis of the\ngeneralized Lie symmetry \\cite{Olv1993} of the action \n\\begin{equation}\n  S := \\int_{t_1}^{t_2}L\\left(q(t),\\dot{q}(t)\\right)dt,\n  \\nonumber\n\\end{equation}\nfor a dynamical\nsystem on a $D$-dimensional configuration space $\\mathbb{Q}$. Here,\n$L\\left(q(t), \\dot{q}(t)\\right)$ is the Lagrangian along a path $q(t)\n= \\left(q^1(t), \\dots, q^D(t)\\right)$ on $\\mathbb{Q}$ with end\npoints given by $Q_1:=q(t_1), Q_2:= q(t_2)$. These points are \nchosen at the same time the choice of $S$ is made, and are fixed.\n\nAs $L\\left(q(t),\\dot{q}(t)\\right)$ depends on both the position $q(t)$\nand the velocity $\\dot{q}(t)$ of the path, we consider a generalized\nLie symmetry that is generated by \n\\begin{equation}\n  \\mathbf{g}_L := \\rho_L(q, \\dot{q})\\cdot \\frac{\\mathbf{\\partial}\n    \\>\\>\\,}{\\mathbf{\\partial} q}, \n  \\nonumber\n\\end{equation}\nwhere $\\rho_L(q,\\dot{q})$ does not depend explicitly on\ntime. Evolution along the path gives the total time derivative \n\\begin{equation}\n  \\frac{\\mathbf{d}\\>\\>\\>}{\\mathbf{d}t} := \\dot{q}\\cdot\n  \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q} +\\ddot{q}\\cdot\n  \\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial} \\dot{q}}.\n  \\label{Dt}\n\\end{equation}\nThis in turn gives $\\dot{\\rho}_L:=\\mathbf{d}\\rho_L\/\\mathbf{d}t$, and the second-order\nprolongation vector \\cite{Olv1993}, \n\\begin{equation}\n  \\hbox{\\textbf{pr }}\\mathbf{g}_L := \\rho_L \\cdot\n  \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q} \n  + \\dot{\\rho}_L \\cdot \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} \\dot{q}} +\n  \\ddot{\\rho}_L\\cdot \\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial}\n    \\ddot{q}},\n  \\label{prog}\n\\end{equation}\non the second-order jet space $\\mathbb{M}^{(2)}=\\{(t, q,\n\\dot{q},\\ddot{q})\\}$ where this $\\hbox{\\textbf{pr\n}}\\mathbf{g}_L\\in\\mathbf{T}\\mathbb{M}^{(2)}$.\n\nUnder this generalized Lie symmetry, the action varies by\n\\begin{equation}\n  \\delta S=\\int_{t_1}^{t_2}\\hbox{\\textbf{pr }}\\mathbf{g}_L \\Big[L(q(t),\n  \\dot{q}(t))\\Big]dt,\n  \\nonumber\n\\end{equation}\nwith the requirement that \n$\\rho_L(q(t_1), \\dot{q}(t_1))=0=\\rho_L(q(t_2),\n\\dot{q}(t_2))$. Then after an integration by parts,\n\\begin{equation}\n  \\delta S=\\int_{t_1}^{t_2} \\rho_L\\cdot\\left[\\frac{\\partial L}{\\partial\n    q} - \\frac{d\\>\\>\\>}{dt}\\left(\\frac{\\partial L}{\\partial\n    \\dot{q}}\\right)\\right]dt.\n  \\label{Ae1}\n\\end{equation}\nIt is important to realize that the action may be evaluated along any path on\n$\\mathbb{Q}$. As such, if $\\mathbf{g}_L$ generates a symmetry of the action, then\nEq.~$(\\ref{Ae1})$ must vanish for \\textit{all} paths $q(t)$ on\n$\\mathbb{Q}$, and not just for those that minimize the action.\n\nTo make connection with the Lagrangian phase space approach used in\nthe rest of the paper, we make use of\n\\begin{equation}\nE\\left(  q,\\dot{q}\\right)  :=\\dot{q}^{a}\\frac{\\partial L\\left(  q,\\dot\n  {q}\\right)  }{\\partial\\dot{q}^{a}}-L\\left(  q,\\dot{q}\\right)  ,\n\\nonumber\n\\end{equation\nalong with\n\\begin{equation}\nM_{ab}\\left(  q,\\dot{q}\\right)  :=\\frac{\\partial^{2}L\\left(  q,\\dot{q}\\right)\n}{\\partial\\dot{q}^{a}\\partial\\dot{q}^{b}}, \\quad \\hbox{and }\\quad \nF_{ab}\\left(  q,\\dot{q}\\right)  := \\frac{\\partial^{2}L\\left(\n  q,\\dot{q}\\right)  }{\\partial\\dot{q}^{a}\\partial q^{b}} -\n\\frac{\\partial^{2}L\\left(  q,\\dot{q}\\right)}{\\partial\\dot{q}^{b}\\partial \n  q^{a}},\n\\nonumber\n\\end{equation}\nto express Eq.~$(\\ref{Ae1})$ as\n\\begin{equation}\n  \\delta S=-\\int_{t_1}^{t_2} \\rho^a_L\\left(\\frac{\\partial E}{\\partial\n    q^a} +F_{ab}(q,\n  \\dot{q})\\dot{q}^b+M_{ab}(q,\\dot{q})\\ddot{q}^b\\right)dt.\n  \\label{e2}\n\\end{equation}\nHere, Latin indices run from $1$ to $D$, and Einstein's summation\nconvention is used. We then arrive at our first result.\n\n\\begin{lemma} \\label{Action-Sym} An action $S$ of a dynamical\n  system has a generalized Lie symmetry generated by $\\mathbf{g}_L$ if\n  and only if there exists a $\\rho_L\\in \\hbox{ker }M_{ab}$ such that \n  \\begin{equation}\n    0=\\rho_L^a(q,\\dot{q})\\left(\\frac{\\partial E}{\\partial\n      q^a}+F_{ab}(q,\\dot{q})\\dot{q}^b\\right),\n    \\label{e3}\n  \\end{equation}\n  on $\\mathbf{T}\\mathbb{Q}$.\n  \n  \\begin{proof}\n    If $\\mathbf{g}_L$ generates a generalized Lie symmetry of $S$, then\n    Eq. $(\\ref{e2})$ must vanish for all paths on $\\mathbb{Q}$. For an\n    arbitrary path on $\\mathbb{Q}$ the\n    curvature of the path $\\ddot{q}$ will not depend on\n    either the $q(t)$ or the $\\dot{q}(t)$ for the path, however. As such, \n    for $\\delta S=0$, it must be that $\\rho^a_LM_{ab}\\ddot{q}^b=0$ for any choice of\n    $\\ddot{q}$, and thus $\\rho_L^a\\in\\hbox{ker }M_{ab}$. The remaining\n    terms in Eq.~$(\\ref{e2})$ gives the condition\n    Eq.~$(\\ref{e3})$. \n  \\end{proof}\n\\end{lemma}\n\nThe set of all vector fields $\\mathbf{g}_L$ that satisfy \\textbf{Lemma\n  $\\mathbf{\\ref{Action-Sym}}$} is denoted by  $\\mathfrak{g}_L$, while \n$\\hbox{\\textbf{pr }}\\mathfrak{g}_L := \\{\\hbox{\\textbf{pr\n}}\\mathbf{g}_L\\ \\vert \\ \\ \\mathbf{g}_L\\in \\mathfrak{g}_L\\}$ is the set of\ntheir prolongations. This $\\hbox{\\textbf{pr }}\\mathfrak{g}_L$ is involutive\n\\cite{Olv1993}, and the conditions under which $\\hbox{\\textbf{pr }}\\mathfrak{g}_L$\ngenerates a generalized Lie symmetry group are given in\n\\cite{Olv1993}.\n\nWe see from \\textbf{Lemma \\ref{Action-Sym}} that if the\naction has a generalized Lie symmetry, then the Lagrangian is\nnecessarily singular, and as such the Lagrangian two-form\n$\\mathbf{\\Omega}_L$ will not have maximum rank. It is also important\nto note that while equations of the form\nEq.~$(\\ref{e3})$ often appear in the Lagrangian phase space description of \nmechanics \\cite{ADS2020}, they appear as Lagrangian\nconstraints, conditions that must be imposed for evolution under the Euler-Lagrange\nequations to be well defined. Here, Eq.~$(\\ref{e3})$ is not a \nconstraint. Rather, because the action must have this symmetry for\n\\textit{all} possible paths on $\\mathbb{Q}$, and since the set of all possible \npaths cover $\\mathbb{Q}$, Eq.~$(\\ref{e3})$ is a condition on $\\rho_L$\nthat must be satisfied identically on \\textit{all} of\n$\\mathbf{T}\\mathbb{Q}$\\textemdash and \nthus, on the Lagrangian phase space\\textemdash for $\\mathbf{g}_L$ to\nbe a generator of the symmetry group. We will see that not all the vectors in\n$\\hbox{ker } M_{ab}$ satisfy the identity Eq.~$(\\ref{e3})$,\nhowever, and thus not all of these vectors will generate a generalized Lie\nsymmetry of the action.   \n  \n\\subsection{Symmetries of the Euler-Lagrange\n  Equations of Motion\\label{&EL-Sym}}\n\nWhile in \\textbf{Section \\ref{&A-Sym}} the focus was on arbitrary paths on the\nconfiguration space $\\mathbb{Q}$ and the symmetries of the action,\nin this section the focus is on the trajectories that minimizes the\naction and the generalized Lie symmetries of them. These trajectories\nare solutions of the Euler-Lagrange equations of motion, and for almost\nregular Lagrangians such solutions form a family of curves. It is, in fact, the\npresence of this family of curves that gives rise to the generalized\nLie symmetry. The treatment here follows closely to that given in \\cite{ADS2020}. \n\nFor almost regular Lagrangians the solutions of the Euler-Lagrange\nequations of motion \n\\begin{equation}\nM_{ab}(q, \\dot{q})\\ddot{q}^{b}=-\\frac{\\partial E}{\\partial\n  q^{a}}-F_{ab}(q, \\dot{q})\\dot{q}^{b}, \n\\label{2ndEL1\n\\end{equation}\nare not unique. While for these Lagrangians the rank of\n$M_{ab}\\left(q, \\dot{q} \\right) =D-N_{0}$\\textemdash with $N_0=\\hbox{dim\n}\\left(\\hbox{ker }M_{ab}(q, \\dot{q})\\right)$\\textemdash is constant, this rank\nis not maximal, and thus Eq.~$(\\ref{2ndEL1})$ does not have a unique\nsolution for $\\ddot{q}$. Instead, for a chosen set of initial data\n$\\left(q_{0}=q(t_0),\\dot{q}_{0}=\\dot{q}(t_0)\\right)$, the solution to\nEq.~$(\\ref{2ndEL1})$ results in a family of solutions that evolve from\nthis $(q_0, \\dot{q}_0)$. As with the paths in \\textbf{Section \\ref{&A-Sym}}, these\nsolutions are related to one another through a generalized Lie \nsymmetry \\cite{Olv1993}.\n\nFollowing \\cite{Olv1993}, the collection of functions\n\\begin{equation}\n  \\Delta_a(q,\\dot{q}, \\ddot{q}) := \\frac{\\partial E(q, \\dot{q})}{\\partial q^a} +\n  F_{ab}(q, \\dot{q})\\dot{q}^b + M_{ab}(q, \\dot{q}) \\ddot{q}^b,  \n\\label{delta}\n\\end{equation}\ndefines a set of  surfaces $\\Delta_a(q,\\dot{q}, \\ddot{q})=0$ on\n$\\mathbb{M}^{(2)}$, while the family of solutions to Eq.~$(\\ref{2ndEL1})$ \n\\begin{equation}\n\\mathcal{O}\\left(q_0, \\dot{q}_0\\right)  :=\\big\\{q\\left(  t\\right)\n\\ \\vert \\ \\\n\\Delta_a(q,\\dot{q}, \\ddot{q}) =0 \\hbox{ with }\nq\\left(  t_{0}\\right)\n=q_{0},\\ \\dot{q}\\left(  t_{0}\\right)  =\\dot{q}_{0}\\big\\}  ,\n\\nonumber\n\\end{equation}\nthat evolve from the same initial data $(q_0, \\dot{q}_0)$ gives the collection\nof trajectories that lie on these surfaces. Indeed, for any two such \nsolutions $q^a(t)$ and $Q^a(t)$ there exists a \n$\\mathfrak{z}(q, \\dot{q})\\in\\hbox{ker } \nM_{ab}(q, \\dot{q})$ such that $\\ddot{Q}^a-\\ddot{q}^a =\n\\mathfrak{z}^a$. Importantly, because $\\mathfrak{z}^a$ depends on both $q$\nand $\\dot{q}$, the symmetry group that maps one member of\n$\\mathcal{O}$ to another must be a generalized Lie symmetry. We\ntherefore take the generator of this symmetry group to be \n\\begin{equation}\n  \\mathbf{g} := \\rho(q, \\dot{q})\\cdot \\frac{\\mathbf{\\partial}\n    \\>\\>\\>}{\\mathbf{\\partial} q},\n  \\nonumber\n\\end{equation}\nwith the corresponding the second-order prolongation vector\nfor $\\mathbf{g}$ being,\n\\begin{equation}\n  \\hbox{\\textbf{pr }}\\mathbf{g} := \\rho \\cdot\n  \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q} \n  + \\dot{\\rho} \\cdot \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} \\dot{q}} +\n  \\ddot{\\rho}\\cdot \\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial} \\ddot{q}},\n  \\nonumber\n\\end{equation}\nwith this $\\hbox{\\textbf{pr }}\\mathbf{g}\\in\\mathbf{T}\\mathbb{M}^{(2)}$. As with\nthe above, the total time \nderivative is given by Eq.~$(\\ref{Dt})$, but unlike the analysis in\n\\textbf{Section \\ref{&A-Sym}}, the evolution of the path\\textemdash and indeed,\nfor all the trajectories in $\\mathcal{O}(q_0,\\dot{q}_0)$\\textemdash here is\ngiven by the Euler-Lagrange equations of motion.\n\nThe action of this\nprolongation on $\\Delta_a$ on the $\\Delta_a=0$ surface gives,   \n\\begin{equation}\n  \\hbox{\\textbf{pr }}\\mathbf{g}\\left[\\Delta_a(q,\n    \\dot{q},\\ddot{q})\\right] = -\\frac{\\partial \n    \\ddot{q}^b}{\\partial q^a}M_{bc}(q, \\dot{q})\\rho^c +\n  \\frac{d\\>\\>\\>}{dt}\\left[F_{ab}(q, \\dot{q})\\rho^b + M_{ab}(q,\n    \\dot{q})\\dot{\\rho}^b\\right]. \n  \\nonumber   \n\\end{equation}\nSince $N_0>0$, $\\ddot{q}$ is not unique on this surface, and yet \n$\\mathbf{g}$ must generate the same symmetry group for all the\ntrajectories in $\\mathcal{O}(q_0, \\dot{q}_0)$. Necessarily, \n$\\rho(q, \\dot{q})\\in\\hbox{ker }M_{ab}(q, \\dot{q})$. It then follows\nthat $\\hbox{\\textbf{pr g}}[\\Delta_a(q,\\dot{q}, \\ddot{q})] =0$\nif and only if (iff) there are constants $b_a$ such that $b_a = \nF_{ab}\\rho^b + M_{ab}\\dot{\\rho}^b$. The solutions in $\\mathcal{O}(q_0,\n\\dot{q}_0)$ all have the same initial data, however, and thus\nnecessarily $\\rho(q_0, \\dot{q}_0)=0=\\dot{\\rho}(q_0,\\dot{q}_0)$. We\nconclude that $b_a=0$. The following result, first proved in\n\\cite{ADS2020}, then follows.  \n\n\\begin{lemma} \\label{GS} If $\\mathbf{g}$ is\n  a generalized infinitesimal symmetry of $\\Delta_a$, then\n  $\\rho^a(q, \\dot{q})\\in\\hbox{ker } M_{ab}(q, \\dot{q})$, and\n  $\\dot{\\rho}^a(q, \\dot{q})$ is a solution of\n    \\begin{equation}\n    0=F_{ab}(q, \\dot{q})\\rho^b(q, \\dot{q}) +\n    M_{ab}(q, \\dot{q})\\dot{\\rho}^b(q, \\dot{q}).\n    \\label{sol}\n   \\end{equation}\n\\end{lemma}\n\nAs before, we denote the set of all vector fields $\\mathbf{g}$ that satisfy \\textbf{Lemma\n  $\\mathbf{\\ref{GS}}$} by  $\\mathfrak{g}$, while \n$\\hbox{\\textbf{pr }}\\mathfrak{g} := \\{\\hbox{\\textbf{pr\n}}\\mathbf{g}\\ \\vert \\ \\ \\mathbf{g}\\in \\mathfrak{g}\\}$ is the set of\ntheir prolongations. Once again $\\hbox{\\textbf{pr }}\\mathfrak{g}$ is\ninvolutive, and the conditions under which $\\hbox{\\textbf{pr\n}}\\mathfrak{g}$ \ngenerates a generalized Lie symmetry group are given in\n\\cite{Olv1993}. Note, however, that while $\\rho=0$ and\n$\\dot{\\rho}=\\mathfrak{z}$ for any $\\mathfrak{z}\\in\\hbox{ker\n}M_{ab}(q, \\dot{q})$ is a solution of \n  Eq.~$(\\ref{sol})$, we require that $\\dot{\\rho} =\n  \\mathbf{d}\\rho\/\\mathbf{d}t$; these solutions cannot be generators of the\n  generalized Lie symmetry. Next, if $\\dot{\\rho}$ is a solution of\n  Eq.~$(\\ref{sol})$, then \n$\\dot{\\rho}^a+\\mathfrak{z}$ is a solution of Eq.~$(\\ref{sol})$ as\n  well, and thus these solutions are not unique. This, along with the\n  previous observation, leads us to generators that are constructed from\n  equivalence classes of prolongations. Finally, Eq.~$(\\ref{delta})$\n  gives for any $\\mathfrak{z} \\in\\hbox{ker }M_{ab}(q, \\dot{q})$,\n\\begin{equation}\n  0=\\mathfrak{z}^a\\left(\\frac{\\partial E}{\\partial\n    q^a}+F_{ab}(q,\\dot{q})\\dot{q}^b\\right),\n  \\label{e}\n\\end{equation}\non the solution surface $\\Delta_a(q,\\dot{q},\\ddot{q})=0$. If\nEq.~(\\ref{e}) does not hold identically, it must be\nimposed, leading to Lagrangian constraints\n\\cite{Car1990a}. More importantly, because each\n$q(t)\\in\\mathcal{O}(\\mathfrak{u}_0)$   \nmust lie on the Lagrangian constraint submanifold, any symmetry transformation of\n$q(t)$ generated by $\\mathbf{pr }\\>\\> \\mathbf{g}$ must give a path\n$Q(t)$ that also lies on the constraint submanifold.\n\nNot all vectors in $\\mathbf{pr}\\> \\mathfrak{g}$ will be generators of\nthe generalized Lie symmetry group for\n$\\mathcal{O}(\\mathfrak{u}_0)$. Determining which of these vectors are,\nand the relationship between the generators of symmetries of\nthe Euler-Lagrange equations of motion and those of \nthe action, is best done within the Lagrangian phase space\nframework. To accomplish this, we will need the following generalization of\n\\textbf{Lemma \\ref{GS}}.\n\nConsider the vector\n\\begin{equation}\n  \\mathbf{k}:=c\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q}\n  +\\dot{c}\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}\n    \\dot{q}},\n    \\nonumber\n\\end{equation}\nwith a $c\\in\\hbox{ker }M_{ab}(q, \\dot{q})$ along with the quantity\n\\begin{equation}\n  l_a:=F_{ab}c^b(q, \\dot{q})+M_{ab}\\dot{c}^b(q,\\dot{q}).\n  \\nonumber\n\\end{equation}\nAfter an integration by parts,\n\\begin{eqnarray}\n    l_a&=& c^b(q,\n    \\dot{q})\\left\\{F_{ab}(q, \\dot{q})-\\frac{\\mathbf{d}\\>\\>\\,}{\\mathbf{d}t}\n    \\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial}\n        \\dot{q}^a\\mathbf{\\partial} \\dot{q}^b}\\right\\},\n      \\nonumber\n      \\\\\n      &=& c^b(q,\n    \\dot{q})\\left\\{F_{ab}(q, \\dot{q})-\\left[\\frac{\\mathbf{d}\\>\\>\\,}{\\mathbf{d}t}, \\frac{\\partial\n        \\>\\>\\,}{\\mathbf{\\partial} \\dot{q}^a}\\right]\\frac{\\mathbf{\\partial} L}{\\mathbf{\\partial}\n      \\dot{q}^b}- \\frac{\\mathbf{\\partial}\\>\\>\\,}{\\mathbf{\\partial}\n      \\dot{q}^a} \\left(\\frac{\\mathbf{d}\\>\\>\\,}{\\mathbf{d}t}\n    \\frac{\\mathbf{\\partial} L}{\\mathbf{\\partial}\n      \\dot{q}^b}\\right)\\right\\}. \n    \\nonumber\n\\end{eqnarray}\nUsing Eq.~$(\\ref{Dt})$ we have\n\\begin{equation}\n  \\left[\\frac{\\mathbf{d}\\>\\>\\,}{\\mathbf{d}t}, \\frac{\\mathbf{\\partial}\n      \\>\\>\\,}{\\mathbf{\\partial} \\dot{q}^a}\\right]\\frac{\\partial L}{\\partial \\dot{q}^b} =\n  -\\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial} q^a\\mathbf{\\partial}\n    \\dot{q}^b}-\n  \\frac{\\mathbf{\\partial}\n    \\ddot{q}^c}{\\mathbf{\\partial}\n    \\dot{q}^a}\\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial}\n    \\dot{q}^c\\mathbf{\\partial} \\dot{q}^b}.\n  \\nonumber\n\\end{equation}\nAs $q(t)$ is  a solution of the Euler-Lagrange equations of\nmotion, we find that\n\\begin{equation}\nl_a= c^b(q, \\dot{q})\\left\\{F_{ab}(q, \\dot{q})\n+\\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial} q^a\\mathbf{\\partial}\n    \\dot{q}^b}-\\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial}\n      \\dot{q}^a\\mathbf{\\partial}q_b} +\n  \\frac{\\mathbf{\\partial}\n    \\ddot{q}^c}{\\mathbf{\\partial}\n    \\dot{q}^a}\\frac{\\mathbf{\\partial}^2L}{\\mathbf{\\partial}\n    \\dot{q}^c\\mathbf{\\partial} \\dot{q}^b}\n  \\right\\}.\n    \\nonumber\n\\end{equation}\nThis last expression vanishes after the definition of $F_{ab}(q,\n\\dot{q})$ is used along with the requirement that $c\\in\\hbox{ker\n}M_{ab}(q,\\dot{q})$. We then have the following result. \n\n\\begin{lemma} \\label{AllS} For any vector \n\\begin{equation}\n  \\mathbf{k}=c\\cdot\\frac{{\\partial}\\>\\>\\>}{{\\partial} q}\n  +\\dot{c}\\cdot\\frac{{\\partial}\\>\\>\\>}{{\\partial}\n    \\dot{q}},\n  \\nonumber\n\\end{equation}\nsuch that $c\\in\\hbox{ker }M_{ab}$, \n\\begin{equation}\n  0=F_{ab}c^b(q, \\dot{q})+M_{ab}\\dot{c}^b(q,\\dot{q}).\n  \\nonumber\n\\end{equation}.\n\\end{lemma}\n\n\\section{Generators of the Generalized Lie Symmetry for the\n  Euler-Lagrange Equations of Motion\\label{&review}}\n\nThe generators of the generalized Lie symmetry for both the\nEuler-Lagrange equations of motion and the action are best found using\nthe Lagrangian phase space approach to mechanics. This phase space\nand its concomitant mathematical structure provide the tools needed to\ndetermine both the generators of the symmetry and the solutions to the\nenergy equation on which they act. For the Euler-Lagrange equations of \nmotion this determination was done in \\cite{ADS2020}. In this\nsection we will review the Lagrangian phase space approach, establish\nthe notation used in this paper, and summarize the results obtained in\n\\cite{ADS2020} that are needed here. (We will also take the\nopportunity to correct typographical errors made in \\cite{ADS2020}.)\nProofs of the majority of the assertions listed in this section will\nnot be given; the reader is instead referred to \\cite{ADS2020} where the\nproofs and the context of their development can be found.\n\n\\subsection{The Lagrangian Phase space\\label{&phase}}\n\nFor a configuration space $\\mathbb{Q}$ the \\textbf{Lagrangian phase\n  space} $\\mathbb{P}_L$ is the tangent space\n$\\mathbb{P}_L=\\mathbf{T}\\mathbb{Q}$, with the coordinates on\n$\\mathbb{P}_L$ denoted as $\\mathfrak{u}=(q^1, \\dots, q^D, v^1, \\dots \nv^D)$. Integral flows on $\\mathbb{P}_L$,\n$t\\in[t_0,\\infty)\\to\\mathfrak{u}(t)\\in\\mathbb{P}_L$ \n\\cite{Abr1978}, for a set of initial data $\\mathfrak{u}_0=(q_0,\nv_0)$ are given as solutions to  \n\\begin{equation}\n\\frac{d\\mathfrak{u}}{dt}:=\\mathbf{X}  (\\mathfrak{u}),\n\\nonumber\n\\end{equation}\nwhere $\\mathbf{X}$ is a smooth vector field in \n$\\mathbf{T}\\mathbb{P}_L=\\mathbf{T}(\\mathbf{T}\\mathbb{Q})$. The two\ntangent spaces $\\mathbf{T}\\mathbb{Q}$ and $\\mathbf{T}\\mathbb{P}_L$\nhave the bundle projections: $\\tau_{\\mathbb{Q}}:\\mathbf{T}\\mathbb{Q}\\to  \n\\mathbb{Q}$ and \n$\\tau_{\\mathbf{T}\\mathbb{Q}}:\\mathbf{T}(\\mathbf{T}\\mathbb{Q})\n\\to\\mathbf{T}\\mathbb{Q}$. They can be used to\nconstruct two other projection maps: $\\tau_{\\mathbb{Q}}\\circ \n\\tau_{\\mathbf{T}\\mathbb{Q}}:\\mathbf{T}(\\mathbf{T}\\mathbb{Q})  \n\\to\\mathbb{Q}$ and the prolongation of $\\tau_{\\mathbf{T}\\mathbb{Q}}$\nto $\\mathbf{T}(\\mathbf{T}\\mathbb{Q})$ (see \\cite{Got1979} and\n\\cite{Abr1978}). This prolongation is the map \n$\\mathbf{T}\\tau_{\\mathbb{Q}}:\\mathbf{T}(\\mathbf{T}\\mathbb{Q})\\to\\mathbf{T}\\mathbb{Q}$,\nand is defined by requiring that the two maps\n$\\tau_{\\mathbb{Q}}\\circ \\tau_{\\mathbf{T}\\mathbb{Q}}$ and \n$\\tau_{\\mathbb{Q}}\\circ \\mathbf{T}\\tau_{\\mathbb{Q}}$ map any point in\n$\\mathbf{T}(\\mathbf{T}\\mathbb{Q})$ to the same point in\n  $\\mathbb{Q}$. The \\textbf{vertical subbundle}\n$[\\mathbf{T}\\mathbb{P}_L]^v$ of $\\mathbf{T}(\\mathbf{T}\\mathbb{Q})$\nis $[\\mathbf{T}\\mathbb{P}_L]^v = \\hbox{ker\n}\\mathbf{T}\\tau_{\\mathbb{Q}}$ \\cite{Got1979}; a $\\mathbf{X}^v\\in\n[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L]^v$ above a point\n$\\mathfrak{u}\\in\\mathbb{P}_L$ is called a \\textbf{vertical vector\n  field}. The \\textbf{horizontal subbundle} \n$[\\mathbf{T}\\mathbb{P}_L]^q$ of\n$\\mathbf{T}(\\mathbf{T}\\mathbb{Q})$ is \n$[\\mathbf{T}\\mathbb{P}_L]^q = \\hbox{Image\n}\\mathbf{T}\\tau_{\\mathbb{Q}}$; a\n$\\mathbf{X}^q\\in[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L]^q$ is called a \n\\textbf{horizontal vector field}. Consequently, each  \n$\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L$ consists of a\n$\\mathbf{X}^q \\in\n\\left[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^q$ and a\n$\\mathbf{X}^v \\in\n\\left[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^v$ with $\\mathbf{X}\n= \\mathbf{X}^q + \\mathbf{X}^v$. In terms of local coordinates,  \n\\begin{equation}\n\\mathbf{X}^{q}\n:=X^{qa}\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}q^{a}} ,\\quad\n\\hbox{and}\\quad\\mathbf{X}^{v} :=X^{va}\n\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v^{a}}.\n\\nonumber\n\\end{equation}\nOf special interest is the second order Lagrangian vector field\n$\\mathbf{X}_L$. This vector field is the particular solution of\nEq.~$(\\ref{EnergyE})$ for which \n$\\mathbf{T}\\tau_{\\mathbb{Q}}\\circ\\mathbf{X}_L$ is the identity on\n$\\mathbf{T}\\mathbb{Q}$ (see \\cite{Abr1978}). In terms of local\ncoordinates\n\\begin{equation}\n  \\mathbf{X}_L = v^a \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q^a} +\n  X^{va} \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v^a}.\n  \\nonumber\n\\end{equation}\n\nThe space of one-forms on\n$\\mathbb{P}_L$ is the cotangent space\n$\\mathbf{T}^{*}\\mathbb{P}_L$. For a one-form $\\mathbf{\\alpha}\\in \n\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L$, and a vector\nfield $\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L$, the\ndual prolongation map $\\mathbf{T}^{*}\\tau_{\\mathbb{Q}}$ is defined as\n\\begin{equation}\n  \\langle \\mathbf{\\alpha}\\vert\n  \\mathbf{T}\\tau_{\\mathbb{Q}}\\mathbf{X}\\rangle = \\langle\n  \\mathbf{T}^{*}\\tau_{\\mathbb{Q}} \\mathbf{\\alpha}\\vert\n  \\mathbf{X}\\rangle,\n  \\nonumber\n\\end{equation}\nafter a useful adaptation of Dirac's bra and ket notation. In\naddition, for a general $k$-form \\textbf{$\\mathbf{\\omega}$} in the\n$k$-form bundle $\\mathbf{\\Lambda}^{k}\\left(\\mathbb{P}_L\\right)$, \n\\begin{equation}\n\\mathbf{\\omega}\\left(  x\\right)  :\\mathbf{Y}_{1}\\otimes\\cdots\\otimes\n\\mathbf{Y}_{k}\\rightarrow\\left\\langle \\left.  \\mathbf{\\omega}\\left(  x\\right)\n\\right\\vert \\mathbf{Y}_{1}\\otimes\\cdots\\otimes\\mathbf{Y}_{k}\\right\\rangle\n\\in\\mathbb{R},\n\\nonumber\n\\end{equation}\nwith\n$\\mathbf{Y}_{j}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L$ for $j = 1,\n\\dots, k$. The \\textbf{vertical one-form subbundle}\n$[\\mathbf{T}^{*}\\mathbb{P}_L]^v$ of $\\mathbf{T}^{*}\\mathbb{P}_L$ is \n$[\\mathbf{T}^{*}\\mathbb{P}_L]^v :=\\hbox{ker }\n\\mathbf{T}^{*}\\tau_{\\mathbb{Q}}$; a\n$\\mathbf{\\alpha}_v\\in [\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L]^v$ is\ncalled a \\textbf{vertical one-form}. The \\textbf{horizontal\none-form subbundle} $[\\mathbf{T}^{*}\\mathbb{P}_L]^q$ of\n$\\mathbf{T}^{*}\\mathbb{P}_L$ is \n$[\\mathbf{T}^{*}\\mathbb{P}_L]^q=\\hbox{Image }\n\\mathbf{T}^{*}\\mathbb{P}_L$; a\n$\\mathbf{\\alpha}_q\\in[\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{Q}]^q$ is\ncalled a \\textbf{horizontal one-form}. Each one-form $\\mathbf{\\varphi}\\in\n\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L$ consists of a\n$\\mathbf{\\varphi}_{q} \\in \\left[\\mathbf{T}_{\\mathfrak{u}}^{*}\\mathbb{P}_L\\right]^q$ and\na $\\mathbf{\\varphi}_{v} \\in \n\\left[\\mathbf{T}_{\\mathfrak{u}}^{*}\\mathbb{P}_L\\right]^q$\nsuch that $\\mathbf{\\varphi}=\\mathbf{\\varphi}_{q}\n+\\mathbf{\\varphi}_{v}$. In terms\nof local coordinates \n$\\mathbf{\\varphi}_{q} :=\\varphi_{qa} \\ \\mathbf{d}q^{a}$ and \n$\\mathbf{\\varphi}_{v} :=\\varphi_{va} \\mathbf{d}v^{a}$. \n\nFollowing \\cite{Got1979, Got1980}, the Lagrangian two-form is defined\nas $\\mathbf{\\Omega}_L := -\\mathbf{d}\\mathbf{d}_JL$, \nwhere $\\mathbf{d}_J$ is the vertical derivative (see\n\\cite{Got1979}). This two-form can be expressed as\n$\\mathbf{\\Omega}_{L}:=\\mathbf{\\Omega}_{F}+\\mathbf{\\Omega}_{M}$ such\nthat for any $\\mathbf{X},\n\\mathbf{Y}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L$. \n\\begin{equation}\n\\mathbf{\\Omega}_F(\\mathbf{X},\\mathbf{Y}) :=\n\\mathbf{\\Omega}_L(\\mathbf{T}\\tau_{\\mathbb{Q}}\\mathbf{X},\n\\mathbf{T}\\tau_{\\mathbb{Q}}\\mathbf{Y}),\n\\nonumber\n\\end{equation}\nand is thus the \\textbf{horizontal two-form} of $\\mathbf{\\Omega}_L$. As\n$\\mathbf{\\Omega}_M(\\mathbf{X},\\mathbf{Y})=\\mathbf{\\Omega}_L(\\mathbf{X},\\mathbf{Y})- \n\\mathbf{\\Omega}_F(\\mathbf{X},\\mathbf{Y})$, $\\mathbf{\\Omega}_M$ is then\na \\textbf{mixed two-form} of \n$\\mathbf{\\Omega}_L$. In terms of local coordinates,  \n\\begin{equation}\n\\mathbf{\\Omega}_{L}=-\\mathbf{d{\\theta}}_{L},\\quad \\hbox{where} \\quad\n\\mathbf{\\theta}_{L}:=\\frac{\\partial L}{\\partial v^{a\n}\\mathbf{d}q^{a},\n\\nonumber\n\\end{equation}\nwhile\n\\begin{equation}\n\\mathbf{\\Omega\n}_{F}:=\\frac{1}{2}F_{ab}\\mathbf{d}q^{a}\\wedge\\mathbf{d}q^{b\n,\\ \\hbox{and} \\ \\mathbf{\\Omega}_{M}:=M_{ab}\\mathbf{d}q^{a}\\wedge\\mathbf{d}v^{b}.\n\\nonumber\n\\end{equation}\n\nFor regular Lagrangians $\\mathbf{X}_L$ is the unique solution of\nEq.~$(\\ref{EnergyE})$. For almost regular Lagrangians, on the other\nhand, this solution is not unique, but instead depends on \n\\begin{equation}\n\\ker\\>  \\mathbf{\\Omega}_{L}\\left(  \\mathfrak{u}\\right) \n:=\\left\\{  \\mathbf{K}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_{L\n\\ \\vert\\ \\ i_{\\mathbf{K}}\\mathbf{\\Omega}_{L}=0\\right\\}.\n\\nonumber\n\\end{equation}\nFrom \\textbf{Section \\ref{&Sym}} we expect this kernel to\nplay a role in determining the generators of the generalized Lie \nsymmetry of both the Euler-Lagrange equations of motion and the\naction. Indeed, consider the natural isomorphism $iso: (t, q, \\dot{q}, \n\\ddot{q}) \\in \\mathbb{M}^{(2)} \\to (t, q, v, X^{va}_{L})$ defined in\n\\cite{ADS2020}, and the prolongation $\\hbox{\\textbf{pr} }\\mathbf{g}$ \nof a generator $\\mathbf{g}\\in\\mathbf{\\mathfrak{g}}$ of a \ngeneralized Lie symmetry of the Euler-Lagrange equations of\nmotion. This $\\hbox{\\textbf{pr }}\\mathbf{g}$ contains the vector  \n\\begin{equation}\n  \\mathbf{k}=\\rho\\cdot\\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial} q} +\n  \\dot{\\rho}\\cdot\\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial}\\dot{q}}.\n  \\nonumber\n\\end{equation}\nThe collection of all such vectors has been shown to be\ninvolutive (see \\cite{ADS2020}). The isomorphism maps $iso:\n\\mathbf{k}\\to\\mathbf{k}'$ where \n\\begin{equation}\n  \\mathbf{k}'=\\rho\\cdot\\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial} q} +\n  \\dot{\\rho}\\cdot\\frac{\\mathbf{\\partial} \\>\\>\\>}{\\mathbf{\\partial} v}.\n  \\nonumber\n\\end{equation}\nThen $\\mathbf{k}'\\in\\mathbf{T}\\mathbb{P}_L$, and from\n\\textbf{Lemma \\ref{GS}}, $\\mathbf{k}'\\in\\hbox{ker\n}\\mathbf{\\Omega}_L(\\mathfrak{u})$ as well. A similar result holds for\nthe generators in $\\mathbf{\\mathfrak{g}}_L$ after \n\\textbf{Lemma \\ref{Action-Sym}} and \\textbf{Lemma \\ref{AllS}} are used.\n\nThe two-form $\\mathbf{\\Omega}_L$ gives the lowering map\n$\\mathbf{\\Omega}_L^{\\flat}:\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_{L}\\rightarrow \n\\mathbf{T}_{\\mathfrak{u}}^{\\ast}\\mathbb{P}_{L}$, with\n$\\Omega_L^{\\flat}\\mathbf{X}:=i_{\\mathbf{X}}\\mathbf{\\Omega}_L$.\nThis map consists of\n$\\Omega_{L}^{\\flat}=\\Omega_{F}^{\\flat}+\\Omega_{M}^{v\\flat}+\\Omega_{M}^{q\\flat}$, \nwith $\\Omega_{F}^{\\flat}:\n\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L \\to\n\\left[\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^q$; \n$\\Omega_{M}^{q\\flat}:\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L \n\\to\\left[\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^q$; and\n$\\Omega_{M}^{v\\flat}:\n\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L \\to\n\\left[\\mathbf{T}^{*}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^v$. In terms of local\ncoordinates, \n$\\Omega_{F}^{\\flat}\\mathbf{X} = F_{ab}X^{qa}\\mathbf{d}q^b$, \n$\\Omega_{M}^{q\\flat}\\mathbf{X}= -M_{ab}X^{va}{}\\mathbf{d}q^{b}$, and\n$\\Omega_{M}^{v\\flat}\\mathbf{X}= M_{ab}X^{qa}\\mathbf{d}v^{b}$.\n\nFor almost regular Lagrangians $\\ker\\>\n  \\Omega_{M}^{v\\flat} = \\mathcal{C}\\oplus\n  \\left[ \\mathbf{T}_{\\mathfrak{u}} \\mathbb{P}_{L}\\right]  ^{v}$ \n  while $\\ker\\>  \\Omega_{M}^{q\\flat}  =\\left[\n    \\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_{L}\\right]  ^{q}\\oplus  \n  \\mathcal{G}$. Here\n  \\begin{equation}\n    \\mathcal{C}:=\\left\\{\\mathbf{C}\\in[\\mathbf{T}_q\\mathbb{P}_L]^q\n      \\ \\vert \\ i_{\\mathbf{C}}\\mathbf{\\Omega}_M =0\\right\\},\n      \\nonumber\n  \\end{equation}\n  and\n  \\begin{equation}\n    \\mathcal{G}:=\\left\\{\\mathbf{G}\\in\n    [\\mathbf{T}_q\\mathbb{P}_L]^v \\ \\vert\n    \\ i_{\\mathbf{G}}\\mathbf{\\Omega}_M =0\\right\\}.\n    \\nonumber\n  \\end{equation}\nAs $M_{ab}(\\mathfrak{u})$ has constant rank on\n$\\mathbb{P}_L$, there exists a basis,  \n\\begin{equation}\n\\Big\\{\n\\mathbf{\\mathfrak{z}}_{\\left(  n\\right)  }\\left(  \\mathfrak{u}\\right) =\\left(  \\mathfrak{z\n_{\\left(  n\\right)  }^{1}\\left(  \\mathfrak{u}\\right)  ,\\ldots,\\mathfrak{z\n_{\\left(  n\\right)  }^{D}\\left(  \\mathfrak{u}\\right)  \\right)  \\vert\n\\ M_{ab}\\left(  \\mathfrak{u}\\right)  \\mathfrak{z}_{\\left(  n\\right)  \n^{b}\\left(  \\mathfrak{u}\\right)  =0,\nn=1,\\ldots,N_{0} \\Big\\}  ,\n\\nonumber\n\\end{equation}\nfor $\\ker  M_{ab}\\left(  \\mathfrak{u}\\right) $ at each\n$\\mathfrak{u}\\in\\mathbb{P}_{L}$. Spans of both\n\\begin{eqnarray}\n  \\mathcal{C}&=& \\hbox{span }\n  \\left\\{\\mathbf{U}^q_{(n)}=\\mathbf{\\mathfrak{z}}_{(n)}\\cdot\n    \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q}, n=1,\n    \\dots, N_0\\right\\}, \\hbox{and}\\>\n    \\nonumber\n    \\\\\n  \\mathcal{G}&=& \\hbox{span }\n  \\left\\{\\mathbf{U}^v_{(n)}=\\mathbf{\\mathfrak{z}}_{(n)}\\cdot\n    \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v}, n=1,\n    \\dots, N_0\\right\\},\n    \\nonumber\n\\end{eqnarray}\ncan then be constructed. Importantly, $\\mathcal{G}$ is involutive\n\\cite{Car1990a}, and when the rank of\n$\\mathbf{\\Omega}_L(\\mathfrak{u})$ is constant on $\\mathbb{P}_L$,\n$\\hbox{ker } \\mathbf{\\Omega}_L(\\mathfrak{u})$ is involutive as well.\n\nCorresponding to $\\mathbf{U}^q_{(n)}$ and $\\mathbf{U}^v_{(n)}$ we have\nthe one-forms  \n$\\mathbf{\\Theta}^{(m)}_q$ and $\\mathbf{\\Theta}^{(m)}_v$\nwhere $\\langle\\mathbf{\\Theta}^{(m)}_q\\vert\n\\mathbf{U}^q_{(n)} \\rangle= \\delta_{(n)}^{(m)}$ and\n$\\langle\\mathbf{\\Theta}^{(m)}_v\\vert \\mathbf{U}^v_{(n)} \\rangle=\n\\delta_{(n)}^{(m)}$. Then $\\left[  \\mathbf{T\n_{\\mathfrak{u}}\\mathbb{P}_{L}\\right]  ^{q}\n\\mathcal{C\n\\oplu\n\\mathcal{C\n_{\\perp}$ and $\\left[  \\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_{L}\\right]  ^{v}\n\\mathcal{G\n\\oplu\n\\mathcal{G\n_{\\perp}$, where \n\\begin{eqnarray\n\\mathcal{C\n_{\\perp}:=\\bigg\\{  \\mathbf{X}\\in\\left[  \\mathbf{T}_{\\mathfrak{u}\n  \\mathbb{P}_{L}\\right]  ^{q}\\ &\\vert&\\\n\\left\\langle \\left.  \\mathbf{\\Theta}_{q}^{(n)}\\right\\vert \\mathbf{X\n\\right\\rangle =0,\\>\\>  n=1,\\dots,N_{0}  \\ \\bigg\\}  ,\n\\hbox{and\n\\nonumber\n\\\\\n\\mathcal{G}_{\\perp}:=\\bigg\\{  \\mathbf{X}\\in\\left[\n  \\mathbf{T}_{\\mathfrak{u}}\n  \\mathbb{P}_{L}\\right]  ^{v}\\ &\\vert&\\\n\\left\\langle \\left.  \\mathbf{\\Theta}_{v}^{(n)}\\right\\vert \\mathbf{X\n\\right\\rangle =0, \\>\\>   n=1,\\dots,N_{0}  \\ \\bigg\\}.\n\\nonumber\n\\end{eqnarray}\nThe vectors that lie in $\\hbox{ker\n}\\mathbf{\\Omega}_L(\\mathfrak{u})$ can be determined by using the reduced\nmatrix $ {F}_{nm}:=\\mathfrak{z}_{\\left(  n\\right)\n}^{a}F_{ab}\\mathfrak{z}_{\\left( \nm\\right)  }^{b}$ to define\n\\begin{equation}\n\\overline\n\\mathcal{C\n}:=\\left\\{  \\overline{\\mathbf{C}}\\i\n\\mathcal{C\n\\ \\bigg\\vert\\ \\sum_{m=1}^{N_{0}}\\bar{F}_{nm}\\overline{C}^{\\left(  m\\right)\n}=0\\right\\}  \\subse\n\\mathcal{C}\n\\nonumber\n\\end{equation}\nThen,\n\n\\begin{theorem}\n\\label{@NVGen}The vectors $\\mathbf{K=K}^{q}+\\mathbf{K}^{v}\\in\\ker\n\\mathbf{\\Omega}_{L}$ are given by\n\\begin{equation}\n\\mathbf{K}^{q}=\\overline{\\mathbf{C}}\\mathbf{,\\ \\ K}^{v}=\\mathbf{G+\n\\widehat{\\mathbf{C}},\n\\nonumber\n\\end{equation}\nwhere, $\\overline{\\mathbf{C}}\\in\\overline\n\\mathcal{C\n}$, $\\mathbf{G}\\i\n\\mathcal{G\n$, and $\\widehat{\\mathbf{C}}\\i\n\\mathcal{G\n_{\\perp}$ is the unique solution of $M_{ab}\\widehat{C}^{b}=-F_{ab}\\overline\n{C}^{b}$.\n\\end{theorem}\nWe found in \\cite{ADS2020} that $\\dim\\>  \\left(\\ker\n\\mathbf{\\Omega}_{L}\\left(  \\mathfrak{u}\\right)\\right) \n  =N_{0}+\\bar{D}, $ where $\n\\ \\bar{D}  :=\\dim \\>\\overline{\\mathcal{C}}\\le N_0$ (see \\cite{ADS2020}\nfor proof). However, the results of \\textbf{Lemma \\ref{AllS}} show\nthat we can construct from any vector $\\mathbf{U}^q\\in \\mathcal{C}$ a\nvector that lies in the $\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})$,\nand as dim $(\\mathcal{C})=N_0$, it follows that dim $(\\hbox{ker\n}\\mathbf{\\Omega}_L(\\mathfrak{u}))=2N_0$. \n\n\\subsection{First-order Lagrangian constraints\\label{&LCon}}\n\nFor singular Lagrangians solutions of the energy equation\n$\\mathbf{X}_E$ are not unique. It is well known that they also do not,\nin general, exist throughout $\\mathbb{P}_L$, but are instead confined\nto a submanifold of the space given by Lagrangian constraints. \n\nWith $\\mathbf{X}_{E}=\\mathbf{X}^q_{E}+\\mathbf{X}^v_{E}$, it is\nconvenient to use the one form $\\mathbf{\\Psi}$\n\\begin{equation}\n  \\Omega_{M}^{q\\flat}\\mathbf{X}_{E}^{v}=\\mathbf{\\Psi}.\n\\nonumber\n\\end{equation}\nconstructed from the energy equation. The \\textbf{first-order\n  constraint functions} are then $\\gamma_{n}^{\\left[\n    1\\right]  }:=\\left\\langle \n\\left.  \\mathbf{\\Psi}\\right\\vert \\mathbf{U}_{\\left(  n\\right)\n}^{q}\\right\\rangle=0$ for $n=1,\\ldots,N_{0}$. In terms of local coordinates,\n\\begin{equation}\n\\gamma_{n}^{\\left[  1\\right]  }= U_{\\left(  n\\right)  \n^{qa}\\left(  \\frac{\\partial E}{\\partial q^{a}}+F_{ab}v^{b}\\right).\n\\nonumber\n\\end{equation}\nThey may also be expressed \\cite{Got1979,\n  Got1980} as $\\gamma^{[1]}_n = \\langle\n\\mathbf{d}E\\vert\\mathbf{P}_{(n)}\\rangle=\\mathbf{P}_{(n)}E$ for any\nbasis $\\{\\mathbf{P}_{(n)}\\}$ of ker $\\mathbf{\\Omega}_L(\\mathfrak{u})$\nfor which $\\langle\\mathbf{\\Theta}^{(m)}_q\\vert\n\\mathbf{P}_{(n)}\\rangle=\\delta^{(m)}_{(n)}$. \nIn general, $\\gamma_{n}^{\\left[  1\\right]  }\\ne0$ on\n$\\mathbb{P}_L$. Instead, the condition $\\gamma_{n}^{\\left[  1\\right]\n}=0$ must be imposed, and this in turn defines a set of submanifolds\nof $\\mathbb{P}_L$ given by the collection $\\hbox{C}_{L}^{\\left[  1\\right]  }:=\\left\\{\n\\gamma_{1}^{\\left[  1\\right]  },\\ldots,\\gamma_{N_{0}}^{\\left[  1\\right]\n}\\right\\}  $. The collection of these\nsurfaces, \n$\\mathbb{P}_{L}^{\\left[  1\\right]  }:=\\left\\{ \n\\mathfrak{u}\\in\\mathbb{P}_{L}\\ \\vert\\ \\ \\gamma_{n}^{\\left[  1\\right]\n}\\left(  \\mathfrak{u}\\right)  =0\\, ,  n=1,\\ldots,N_{0}\\ \\right\\}$ is\ncalled the \\textbf{first-order Lagrangian constraint submanifold}, and\nhas $\\dim \\mathbb{P}_{L}^{\\left[1\\right]  } =2D-I_{\\left[\n    1\\right]}$. Here $I_{\\left[1\\right]  }$ is the\nnumber of independent functions in $\\hbox{C}_{L}^{\\left[  1\\right]\n}$ with $I_{\\left[ 1\\right]  }=\\hbox{rank }  \\left\\{\\mathbf{d}\\gamma_{n}^{\\left[\n      1\\right]  }\\right\\} \\leq N_{0}$. \n\nThe \\textbf{constraint\n  one-form}  \n\\begin{equation}\n  \\mathbf{\\beta}[\\mathbf{X}_E] :=\n  \\mathbf{d}E-i_{\\mathbf{X}_E}\\mathbf{\\Omega}_L,\n\\nonumber\n\\end{equation}\nwas introduced in \\cite{ADS2020} with the condition \n$\\mathbf{\\beta}[\\mathbf{X}_E]=0$ giving both the  \nsolution of the energy equation and the submanifold\n$\\mathbb{P}_L^{[1]}$. As $\\langle\\mathbf{\\beta} \\vert \\mathbf{U}^q_{(n)}\\rangle=\n\\gamma_n^{[1]}$, this $\\mathbf{\\beta}[\\mathbf{X}_E]$ can also\nbe expressed as \n\\begin{equation}\n  \\mathbf{\\beta}[\\mathbf{X}_E]=\\sum_{n=1}^{N_0}\n  \\gamma_n^{[1]}\\mathbf{\\Theta}^{(n)}_q.\n  \\label{beta}\n\\end{equation}\n\n\\subsection{The Generalized Lie Symmetry Group for the Euler-Lagrange\n  Equations of motion\\label{&GenEL-Sym}} \n\nThe generalized Lie symmetry group for $\\mathcal{O}(\\mathfrak{u}_0)$\nis determined using\n\\begin{equation}\n  \\overline{\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})} :=\n  \\{\\mathbf{P} \\in \\hbox{ker } \\mathbf{\\Omega}_L(\\mathfrak{u})\\\n  \\vert \\ [\\mathbf{G},\\mathbf{P}]\n  \\in\\left[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^v \\>\\>\\forall\n  \\>\\>\\mathbf{G}\\in\\mathcal{G}\\},  \n\\end{equation}\nalong with the following collection of functions on $\\mathbb{P}_L$,  \n\\begin{equation}\n  \\overline{\\mathcal{F}} := \\{f\\in C^\\infty \\hbox{on } \\mathbb{P}_L\n  \\ \\vert\\ \\ \\mathbf{G}f = 0 \\>\\>\\forall\\>\\> \\mathbf{G} \\in\n  \\mathcal{G}\\}.\n\\nonumber\n\\end{equation}\nThis $\\overline{\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ is also\ninvolutive.\n\nThe following results were proved in \\cite{ADS2020}.\n\n\\begin{lemma} \\label{basic} Let  \n  $\\mathbf{X}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L$ \n  and $\\mathbf{G}\\in\\mathcal{G}$ such that $[\\mathbf{G},\n    \\mathbf{X}]\\in\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})$. Then \n  $[\\mathbf{G},\n    \\mathbf{X}]\\in\\left[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^v$\n  iff $[\\mathbf{G}, \\mathbf{X}]\\in\\mathcal{G}$.\n\\end{lemma}\nIt then follows that $[\\mathbf{G},\\mathbf{P}]\\in\\mathcal{G}$ for all\n$\\mathbf{P}\\in \\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$. As $\\mathcal{G}$ is involutive\nand as $\\mathcal{G}\\subset \\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})$,\n$\\mathcal{G}\\subset \\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ as well, and thus   \n$\\mathcal{G}$ is an ideal of $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$.  \n\n\\begin{lemma}\n  \\label{basis}\n  There exists a choice of basis for ker\n  $\\mathbf{\\Omega}_L(\\mathfrak{u})$ that is \n  also a basis of $\\overline{\\hbox{ker\n    }\\mathbf{\\Omega}_L(\\mathfrak{u})}$.\n\\end{lemma}\n\nAs $\\mathcal{G}$ is an ideal of $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$, we may define for\nany $\\mathbf{P}_1, \\mathbf{P}_2\\in\\overline{\\hbox{ker \n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ the equivalence relation:\n$\\mathbf{P}_1\\sim\\mathbf{P}_2$ iff\n$\\mathbf{P}_1-\\mathbf{P}_2\\in\\mathcal{G}$. The equivalence class,\n\\begin{equation}\n  \\left[\\mathbf{P}\\right]:=\n  \\{\\mathbf{Y}\\in \\overline{\\hbox{ker\n    }\\mathbf{\\Omega}_L(\\mathfrak{u})}\\ \\vert \n  \\ \\mathbf{Y}\\sim\\mathbf{P}\\},  \n\\end{equation}\ncan be constructed along with the quotient space\n$\\overline{\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}$.   \n(For the sake of notational clarity  we will suppress\nthe square brackets for equivalence classes when there\nis no risk of confusion.) This space is a collection of\nvectors that lie in the kernel of $\\mathbf{\\Omega}_L$, but with the\nvectors in $\\mathcal{G}$ removed; $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}$ thereby addresses the\nfirst two observations listed at the end of \\textbf{Section\n  \\ref{&EL-Sym}}. \n\nWe now turn our attention to the third observation. Because the\nintegral flow $\\mathfrak{u}_{\\mathbf{X}}(t)$ of any solution \n$\\mathbf{X}$ of the energy equation must lie on $\\mathbb{P}_L^{[1]}$, a \nsymmetry transformation of $\\mathfrak{u}_{\\mathbf{X}}(t)$ must result\nin an integral flow $\\mathfrak{u}_{\\mathbf{Y}}(t)$ of another solution\n$\\mathbf{Y}$ of the energy equation, which must also lie on\n$\\mathbb{P}_L^{[1]}$. Implementing this condition is done through\n$\\mathbf{\\beta}[\\mathbf{X}_E]$. \n\nAs $\\langle\\mathbf{\\beta}[\\mathbf{X}_E]\\vert \\mathbf{G}\\rangle=\\langle\n\\mathbf{d}E\\vert\\mathbf{G}\\rangle= \\mathbf{G}E=0$ for all  \n$\\mathbf{G}\\in\\mathcal{G}$ on $\\mathbb{P}_L^{[1]}$, the Lie derivative\n$\\mathfrak{L}_{\\mathbf{G}}$ of $\\mathbf{\\beta}$ along $\\mathbf{G}$ is, \n\\begin{equation}\n  \\mathfrak{L}_{\\mathbf{G}}\\mathbf{\\beta}[\\mathbf{X}_E]= \\sum_{n=1}^{N_0}\n  \\left(\\mathbf{G}\\gamma_n^{[1]}\\right) \\mathbf{\\Theta}^{(n)}_q.\n  \\nonumber\n\\end{equation}\nGiven a\n$\\mathbf{P}_{(n)} \\in\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ such that $\\mathbf{P}_{(n)} = \n\\mathbf{U}_{(n)}^q + \\widehat{\\mathbf{U}}_{(n)} +\\mathbf{G}'$ with\n$\\mathbf{G}'\\in\\mathcal{G}$, $\\mathbf{G}\\gamma_n^{[1]}=\n       [\\mathbf{G}, \\mathbf{P}_{(n)}]E + \n\\mathbf{P}_{(n)}\\mathbf{G}E$. But $\\mathcal{G}$ is an ideal of\n$\\overline{\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})}$, and thus\n$\\mathbf{G}\\gamma_{n}^{[1]}=0$ on the first-order constraint\nmanifold. It follows that $\\mathfrak{L}_{\\mathbf{G}}\\mathbf{\\beta}\n=0$ on $\\mathbb{P}^{[1]}_L$. The collection of vectors,     \n\\begin{equation}\n  \\mathcal{S}\\hbox{ym} := \\big\\{\\mathbf{P}\\in \\overline{\\hbox{ker\n    }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}\n  \\ \\vert\\ \\\n  \\mathfrak{L}_{\\mathbf{P}}\\mathbf{\\beta}[\\mathbf{X}_E] =\n  \\mathbf{d}\\langle \\mathbf{\\beta}[\\mathbf{X}_E]\\vert \\mathbf{P}\\rangle \\hbox{ on }\n  \\mathbb{P}_L^{[1]}\\big\\}, \n  \\nonumber\n\\end{equation}\nis therefore well defined, and is involutive. It follows that\n$\\mathbf{P}\\in\\mathcal{S}\\hbox{ym}$ iff    \n$\\langle \\mathbf{d}\\mathbf{\\beta}[\\mathbf{X}_E]\\vert \\mathbf{P}\\otimes\n\\mathbf{X}\\rangle=0$ for all $\\mathbf{X}\\in\n\\mathbf{T}\\mathcal{P}_L$. We are then able to construct from each\n$\\mathbf{P}\\in\\mathcal{S}\\hbox{ym}$ a one-parameter subgroup\n$\\mathbf{\\sigma}_{\\mathbf{P}}(\\epsilon,x)$ defined as the solution\nto  \n\\begin{equation}\n  \\frac{d\\mathbf{\\sigma}_{\\mathbf{P}}}{d\\epsilon} :=\n  \\mathbf{P}\\left(\\mathbf{\\sigma}_{\\mathbf{P}}\\right), \n  \\nonumber\n\\end{equation}\nwhere $\\sigma_{\\mathbf{P}}(0,\\mathfrak{u}) = \\mathfrak{u}$ for\n$\\mathfrak{u}\\in\\mathbb{P}_L$. The collection of such \nsubgroups with give the Lie group\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$.\n\n\\subsection{Euler-Lagrange Solutions of the Energy\n  Equation\\label{&Sol}}\n\nWe denote the set of \\textbf{general solutions} to the energy equation as\n\\begin{equation}\n\\mathcal{S}\\hbox{ol}\n:=\\{\\mathbf{X}_{E}\\in\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\n\\ \\vert\\ \\ i_{\\mathbf{X}_{E}}\\mathbf{\\Omega}_L = \\mathbf{d}E\n\\hbox{ on } \\mathbb{P}_L^{[1]}\\}.\n\\nonumber\n\\end{equation}\nIf $\\mathfrak{u}(t)$ is the integral flow of a vector in\n$\\mathcal{S}\\hbox{ol}$ whose projection onto\n$\\mathbb{Q}$ corresponds to a solution of\nthe Euler-Lagrange equations of motion, then\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ must map one of\nsuch flows into another one. However, while\n$\\mathfrak{L}_{\\mathbf{G}}\\mathbf{X}_{L} = \n[\\mathbf{G}, \\mathbf{X}_{L}]\\in \\hbox{ker\n}\\mathbf{\\Omega}_L(\\mathfrak{u})$, in general\n$\\mathfrak{L}_{\\mathbf{G}}\\mathbf{X}_{L} \\notin \\mathcal{G}$. The \naction of $\\sigma_{\\mathbf{P}}$ on the flow\n$\\mathfrak{u}_{\\mathbf{X}_{L}}$ will in general result in a flow\n$\\mathfrak{u}_{\\mathbf{Y}}$ generated by a $\\mathbf{Y}$ that is\n\\textit{not} a SOLVF. It need not even be a solution of the energy\nequation. By necessity, general solutions of the energy equation must\nbe considered, leading us to consider the collection of solutions\n\\begin{equation}\n\\overline{\\mathcal{S}\\hbox{ol}}\n:=\\{\\overline{\\mathbf{X}}_{EL}\\in\\mathcal{S}\\hbox{ol}  \n\\ \\vert \\ [\\mathbf{G},\n  \\overline{\\mathbf{X}}_{EL}]\\in\n\\left[\\mathbf{T}_{\\mathfrak{u}}\\mathbb{P}_L\\right]^v \\>\\>\n\\>\\> \\forall \\mathbf{G}\\in\n\\mathcal{G}\\}.  \n\\nonumber\n\\end{equation}\nThis collection generates the family of integral flows  \n\\begin{equation}\n  \\mathcal{O}_{EL}(\\mathfrak{u}_0) := \\bigg\\{\\mathfrak{u}(t) \\ \\bigg\\vert \\\n  \\frac{d\\mathfrak{u}}{dt}=\\overline{\\mathbf{X}}_{EL}(\\mathfrak{u}),\n  \\overline{\\mathbf{X}}_{EL}\\in\\overline{\\mathcal{S}\\hbox{ol}},\n  \\>\\>\\hbox{and }\n  \\mathfrak{u}(t_0)=\\mathfrak{u}_0\\bigg\\}.\n  \\nonumber\n\\end{equation}\nImportantly, if $\\mathbf{P}\\in\\mathcal{S}\\hbox{ym}$, then \n\\begin{equation}\n  i_{[\\mathbf{X}_E,\n      \\mathbf{P}]}\\mathbf{\\Omega}_L=i_{\\mathbf{P}}\\mathbf{d}\\mathbf{\\beta}[\\mathbf{X}_E]=0.\n  \\nonumber\n\\end{equation}\nAs such, we find that \n\n\\begin{lemma} \\label{X_EL} $[\\overline{\\mathbf{X}}_{EL}, \\mathbf{P}] \\in\n  \\overline{\\hbox{ker } \\mathbf{\\Omega}_L(\\mathfrak{u})}$ for all\n  $\\mathbf{P}\\in \\mathcal{S}\\hbox{ym}$.  \n\\end{lemma}\n\n\\noindent It then follows that\n\n\\begin{theorem}\\label{Group-EL} $\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ forms a group of\n  symmetry transformations of $\\mathcal{O}_{EL}(\\mathfrak{u}_0)$.\n\\end{theorem}\n\n\\noindent Proof of both assertions can be found in \\cite{ADS2020}.\n\nThe generators of the generalized Lie symmetry for\n$\\mathcal{O}_{EL}(\\mathfrak{u}_0)$ are \nthus given by $\\mathcal{S}\\hbox{ym}$. The corresponding solutions\nto the Euler-Lagrange equations that have this symmetry are given by\n$\\overline{\\mathcal{S}\\hbox{ol}}$, and a vector\n$\\overline{\\mathbf{X}}_{EL} \\in \\overline{\\mathcal{S}\\hbox{ol}}$ is\ncalled a \\textbf{second-order, Euler-Lagrange vector field (SOELVF)}. It has\nthe general form,\n\\begin{equation}\n  \\overline{\\mathbf{X}}_{EL}=\\overline{\\mathbf{X}}_{L} +\n  \\sum_{m=1}^{N_0} u^m(\\mathfrak{u}) \\left[\\mathbf{P}_{(m)}\\right],\n  \\label{EL}\n\\end{equation}\nwhere $u^m(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$ and\n$\\{[\\mathbf{P}_{(n)}], n=1, \\dots, N_0\\}$ is a choice of basis \nfor $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}$. The vector field\n$\\overline{\\mathbf{X}}_L$ is constructed from the second order\nLagrangian vector field $\\mathbf{X}_L$ and vectors in $\\hbox{ker\n}\\mathbf{\\Omega}_L(\\mathfrak{u})$ by requiring\n$\\overline{\\mathbf{X}}_L\\in\\overline{\\mathcal{S}\\hbox{ol}}$. This\nconstruction is described in \\cite{ADS2020}; we will only need the\nexistence of such a vector field in this paper.\n\n\\section{Generalized Lie Symmetries of the Action and its Impact on\n  Dynamics\\label{&A-S}} \n\nWe now turn our attention to the generators of the\ngeneralized Lie symmetry of the action, and the impact this symmetry\nhas on the evolution of dynamical systems.\n\n\\subsection{The Generalized Lie Symmetry of the Action\\label{SandA}}\n\nIn determining the conditions (as listed in \\textbf{Lemma \\ref{Action-Sym}}) under\nwhich the action admits a generalized Lie symmetry, the understanding\nthat the action must have this symmetry for all possible paths on\n$\\mathbb{Q}$ played an essential role. By necessity, these\nconditions could only be placed on $\\rho_L$, and not on \n$\\dot{\\rho}_L$; unlike $\\rho_L$, $\\dot{\\rho}_L$ depends explicitly on the\nevolution of a particular path, while the symmetry must hold\nfor all paths. We note, however, that the family $\\mathcal{O}_{EL}$ of\ntrajectories determined by the Euler-Lagrange equations of motion also consists of\npaths on $\\mathbb{Q}$, and as such the generalized Lie symmetry of the\naction is also a symmetry of $\\mathcal{O}_{EL}$. Importantly, how these\ntrajectories evolve with time is known, and as such, the $\\dot{\\rho}_L$ for a\ngiven $\\rho_L$ is also known for these trajectories. With this\nunderstanding, and after comparing \\textbf{Lemma \\ref{Action-Sym}} and\nthe results of \\textbf{Lemma \\ref{AllS}} with \\textbf{Lemma \\ref{GS}},\nwe conclude that the generators of the generalized Lie symmetry of the\naction must also be generators of the generalized Lie symmetry of the\nEuler-Lagrange equations of motion. This leads us to consider the\nfollowing collection of vectors.  \n\\begin{equation}\n  \\mathcal{S}\\hbox{ym}\\mathcal{L} = \\{\\mathbf{P}\\in \\overline{\\hbox{ker\n    }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}\\ \\vert\\\n  \\gamma_{\\mathbf{P}}^{[1]}=\\langle \\mathbf{\\beta}\\vert\\mathbf{P}\\rangle\n    =0 \\hbox{ on }\\mathbb{P}_L \\}.\n    \\nonumber\n\\end{equation}\nWe will also need  $N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}=\\hbox{dim\n}(\\mathcal{S}\\hbox{ym}\\mathcal{L})$ in the following. \n\n\\begin{lemma}\\label{subset}\n  $\\mathcal{S}\\hbox{ym}\\mathcal{L}\\subset\\mathcal{S}\\hbox{ym}$.\n  \n  \\begin{proof}\n    Let $\\{\\mathbf{P}_{(l)}, l = 1, \\dots, N_0 \\}$ be a basis of\n    $\\overline{\\hbox{ker\n      }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}$ such that\n    $\\mathbf{P}_{(l)}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$ for\n    $l= 1, \\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$. We may choose the basis of\n    $\\mathcal{C}$ such that $\\langle \\mathbf{\\Theta}^{(m)}_q\\vert\n    \\mathbf{P}_{(l)}\\rangle=\\delta^{(m)}_{(l)}$. Then for any\n    $\\mathbf{P}_{(n)}\\in\\mathcal{S}\\hbox{ym}$, we see from\n    Eq.~$(\\ref{beta})$ that, \n    \\begin{equation}\n      \\langle\\mathbf{d}\\mathbf{\\beta}\\vert \\mathbf{P}_{(n)}\\otimes\\mathbf{Y}\\rangle= \\sum_{m=1}^{N_0}\n      \\left(\n      \\langle\\mathbf{d}\\gamma_m^{[1]}\\vert\n      \\mathbf{P}_{(n)}\\rangle \\langle \\mathbf{\\Theta}^{(m)}_q\\vert\n      \\mathbf{Y}\\rangle-\n      \\langle\\mathbf{d}\\gamma_m^{[1]}\\vert\\mathbf{Y}\n      \\rangle \\langle \\mathbf{\\Theta}^{(m)}_q\\vert\n      \\mathbf{P}_{(n)}\\rangle+\n      \\gamma_m^{[1]}\\langle\\mathbf{d}\\mathbf{\\Theta}^{(m)}_q\\vert\n      \\mathbf{P}_{(n)}\\otimes\\mathbf{Y}\\rangle \n      \\right),\n      \\nonumber\n    \\end{equation}\n    for any $\\mathbf{Y}\\in \\mathbf{T}\\mathbb{P}_L$. The last term\n    vanishes on the first-order constraint manifold\n    $\\mathbb{P}_L^{[1]}$, while for the second term, \n    $\\langle\\mathbf{d}\\gamma_m^{[1]}\\vert\\mathbf{Y} \\rangle\n    \\langle \\mathbf{\\Theta}^{(m)}_q\\vert \\mathbf{P}_{(n)}\\rangle=\n    \\langle\\mathbf{d}\\gamma_n^{[1]}\\vert\\mathbf{Y} \\rangle\n    \\delta ^{(m)}_{(n)}$. But as\n    $\\mathbf{P}_{(n)}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$,\n    $\\gamma_n^{[1]}=0$ on $\\mathbb{P}_L$, and this term\n    vanishes as well. Finally, for the first term,\n    $\\langle\\mathbf{d}\\mathbf{\\gamma}_m^{[1]}\\vert\n    \\mathbf{P}_{(n)}\\rangle= \\mathbf{P}_{(n)}\\mathbf{P}_{(m)}E=\n    [\\mathbf{P}_{(n)},\\mathbf{P}_{(m)}]E+\\mathbf{P}_{(m)}\\mathbf{P}_{(n)}E$. But\n    $\\gamma_n^{[1]}=\\mathbf{P}_{(n)}E=0$ on $\\mathbb{P}_L$, while\n    $\\overline{\\hbox{ker }\\mathbf{\\Omega}(\\mathfrak{u})}$ is\n    involutive. There then exists a $\\mathbf{P}_{(nm)}\\in\n    \\overline{\\hbox{ker }\\mathbf{\\Omega}(\\mathfrak{u})}$ such that\n    $\\mathbf{P}_{(nm)}=[\\mathbf{P}_{(n)}, \\mathbf{P}_{(m)}]$. As\n    $\\mathbf{P}_{(nm)}E:=\\gamma_{(nm)}^{[1]}$, this\n    $\\gamma_{(nm)}^{[1]}$ must be a linear\n    combination of first-order constraint functions, and they also vanish\n    on $\\mathbb{P}_L^{[1]}$. It then follows that\n    $\\langle\\mathbf{d}\\mathbf{\\beta}\\vert\n    \\mathbf{P}_{(n)}\\otimes\\mathbf{Y}\\rangle=0$ on\n    $\\mathbb{P}_L^{[1]}$, and\n    $\\mathbf{P}_{(n)}\\in\\mathcal{S}\\hbox{ym}$.\n  \\end{proof}\n\\end{lemma}\n\nIf $\\mathbf{P}_1, \\mathbf{P}_2 \\in \\mathcal{S}\\hbox{ym}\\mathcal{L}$, then\n$\\gamma^{[1]}_{[\\mathbf{P}_1,\\mathbf{P}_2]} =\\mathbf{P}_1\\mathbf{P}_2E-\n\\mathbf{P}_2\\mathbf{P}_1E=\\mathbf{P_1}\\gamma_{\\mathbf{P}_2}-\n\\mathbf{P_2}\\gamma_{\\mathbf{P}_1}=0$, and thus\n$\\mathcal{S}\\hbox{ym}\\mathcal{L}$ is involutive. Then for  \neach $\\mathbf{P}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$ we once again have the\none-parameter subgroup\n$\\sigma^{\\mathcal{S}\\hbox{ym}\\mathcal{L}}_{\\mathbf{P}}(\\epsilon,\\mathfrak{u})$\ndefine as the integral flow of\n\\begin{equation}\n  \\frac{\\mathbf{d}\\sigma^{\\mathcal{S}\\hbox{ym}\\mathcal{L}}_{\\mathbf{P}}}{\\mathbf{d}\\epsilon}\n  :=\\mathbf{P},\n  \\nonumber\n\\end{equation}\nwith\n$\\sigma^{\\mathcal{S}\\hbox{ym}\\mathcal{L}}_{\\mathbf{P}}(0,\\mathfrak{u})=\\mathfrak{u}$\nfor $\\mathfrak{u}\\in\\mathbb{P}_L$. The collection of such subgroups\ngives the Lie group $\\hbox{Gr}_{\\mathcal{S}{ym}\\mathcal{L}}$. As\n$\\mathcal{S}{ym}\\mathcal{L}\\subset \\mathcal{S}{ym}$,\n$\\hbox{Gr}_{\\mathcal{S}{ym}\\mathcal{L}}$ is a Lie subgroup of\n$\\hbox{Gr}_{\\mathcal{S}{ym}}$. It then follows from \\textbf{Theorem \\ref{Group-EL}}\nthat $\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ also forms a group of\n  symmetry transformations of $\\mathcal{O}_{EL}(\\mathfrak{u}_0)$. As\n  the family $\\mathcal{O}_{EL}(\\mathfrak{u}_0)$ of trajectories are\n  paths on $\\mathbb{Q}$, and as the symmetry transformation of the\n  action must be the same for all paths on $\\mathbb{Q}$, it also follows that, \n\n\\begin{theorem}\\label{Group-L}\n  $\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ forms the group of\n  symmetry transformations of the action $S$.\n\\end{theorem}\n\n\\subsection{Symmetries and Dynamics\\label{SandD}}\n\nWhile $\\mathcal{O}_{EL}(\\mathfrak{u}_0)$ gives the family of integral\nflows on which both $\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}}$ and\n$\\hbox{Gr}_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ act, a general flow in\n$\\mathcal{O}_{EL}(\\mathfrak{u}_0)$ need not be confined to $\\mathbb{P}_{L}^{\\left[1\\right]\n}$, and yet this is  the submanifold on which the solutions \n$\\overline{\\mathbf{X}}_{EL}\\in mathcal{S}{\\mathrm{ol}}$ of the energy\nequations exist. In such cases it is necessary to jointly choose a\nSOELVF $\\overline{\\mathbf{X}}_{EL}$ and a submanifold \nof $\\mathbb{P}_{L}^{\\left[  1\\right]  }$ on which the resultant flow\n$\\mathfrak{u}_{\\overline{\\mathbf{X}}_{EL}}$ will be confined. This is\ndone through the implementation of a constraint algorithm, one of\nwhich was proposed in \\cite{ADS2020}. In that paper the product of this\nalgorithm was the most that could be said about the general structure \nof SOELVFs that have integral flow fields which lie on\n$\\mathbb{P}_L^{[1]}$. Here, with the results obtained in\n\\textbf{Section \\ref{SandA}}, we can say much more, and we will see\nthat the presence of a generalized Lie symmetry of the action\ngreatly restricts the structure of the SOELVFs that such systems can have.\n\nFollowing \\cite{ADS2020}, we introduce for a \n$\\overline{\\mathbf{X}}_{EL}\\in\\overline{\\mathcal{S}\\hbox{ol}}$ the\nnotation \n\\begin{equation}\n\\overline{\\mathbf{X}}_{EL}^{[1]} := \\overline{\\mathbf{X}}_{EL}, \\>\n\\overline{\\mathbf{X}}_{L}^{[1]} := \\overline{\\mathbf{X}}_{L}, \\>\n\\mathbf{P}^{[1]}_{(n)} := \\mathbf{P}_{(n)}, \\> u_{[1]}^m := u^m, \\>\nN_0^{[1]} := N_0,\n\\nonumber\n\\end{equation}\nwhen the constraint algorithm is implemented, with the superscript $[1]$ denoting\nthe first iteration of this algorithm. (This notation is only used in\nthis section.) In addition, we choose \n$\\mathbf{P}^{[1]}_{(n)}\\in \\mathcal{S}\\hbox{ym}\\mathcal{L}$ for\n$n=1,\\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$.\n\nFor the integral flow field of $\\overline{\\mathbf{X}}_{EL}$ to lie on \n$\\mathbb{P}_L^{[1]}$,\n\\begin{equation}\n  \\mathfrak{L}_{\\overline{\\mathbf{X}}_{EL}}\\mathbf{\\beta}=0,\n  \\label{stable}\n\\end{equation}\nwhich reduces to\n$\\mathfrak{L}_{\\overline{\\mathbf{X}}_{EL}}\\gamma_n^{[1]}=0$ on $\\mathbb{P}_L^{[1]}$.\nThis is called the \\textbf{constraint condition}. \nAs both $u_{[1]}^n,\n\\gamma^{[1]}_n\\in \\overline{\\mathcal{F}}$,\n$\\left[\\mathbf{P}_{(n)}^{[1]}\\right]\\gamma^{[1]}_m = \n\\mathbf{P}_{(n)}\\gamma^{[1]}_m$, and after making use of the general\nform of a SOELVF given in Eq.~$(\\ref{EL})$, Eq.~$(\\ref{stable})$ reduces to\n\\begin{equation}\n  \\sum_{m=1}^{N_0} \\Gamma^{[1]}_{nm} u^m_{[1]} =\n  -\\left\\langle \\mathbf{d} \\gamma^{[1]}_n\\Big\\vert\n  \\overline{\\mathbf{X}}^{[1]}_{L}\\right\\rangle, \\>\\hbox{with }\n  \\Gamma^{[1]}_{nm} := \\left\\langle\n  \\mathbf{d}\\gamma^{[1]}_n\\Big\\vert \\mathbf{P}^{[1]}_{(m)}\\right\\rangle.\n  \\label{first-order}\n\\end{equation}\nSince $\\left\\langle\n\\mathbf{d}\\gamma^{[1]}_n\\Big\\vert\n\\mathbf{P}^{[1]}_{(m)}\\right\\rangle=\\mathbf{P}^{[1]}_{(m)}\\mathbf{P}^{[1]}_{(n)}E=\n       [\\mathbf{P}^{[1]}_{(m)}, \\mathbf{P}^{[1]}_{(n)}]E\n+\\Gamma^{[1]}_{mn}$.\nBut $\\overline{\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ is\ninvolutive, and thus $[\\mathbf{P}^{[1]}_{(m)},\n  \\mathbf{P}^{[1]}_{(n)}]E$ is a linear combination of first-order\nLagrangian constraints. As these constraints vanishes on\n$\\mathbb{P}_L^{[1]}$, \n$\\Gamma^{[1]}_{nm}=\\Gamma^{[1]}_{mn}$ on the first-order constraint\nmanifold.\n\nNext, when $n=1, \\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$,\n$\\mathbf{P}_{(n)}^{[1]}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$, and\n$\\gamma_n^{[1]}=0$. Thus, $\\Gamma^{[1]}_{nm}=0$ when $n=1, \\dots,\nN_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$, and as $\\Gamma^{[1]}_{nm}$ is a\nsymmetric matrix on $\\mathbb{P}_L^{[1]}$, $\\Gamma^{[1]}_{mn}=0$ for\nthese values of $n$ as well. Thus while\n$\\Gamma^{[1]}_{nm}$ is a $N_0\\times N_0$ matrix, \nthe only nonzero components of this matrix lie in the\n$\\left(N_0-N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}\\right)\\times\n\\left(N_0-N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}\\right)$ submatrix\n$\\bar{\\Gamma}^{[1]}_{\\overline{n}\\,\\overline{m}}:=\n\\left\\langle\\mathbf{d}\\gamma^{[1]}_{\\overline{n}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\\bigg\\vert\n  \\mathbf{P}^{[1]}_{\\overline{m}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\\right\\rangle$\nwhere $\\overline{n}, \\overline{m} = 1, \\dots,\nN_0-N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$. As $\\left\\langle \\mathbf{d}\n\\gamma^{[1]}_n\\Big\\vert\n\\overline{\\mathbf{X}}^{[1]}_{L}\\right\\rangle=0$ as well when $n=1,\n\\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$, Eq.~$(\\ref{first-order})$\n  reduces to  \n\\begin{equation}\n  \\sum_{\\overline{m}=1}^{N_0-N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\n  \\bar{\\Gamma}^{[1]}_{\\overline{n}\\bar{m}}\n  u^{\\overline{m}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}_{[1]} = -\\left\\langle \\mathbf{d}\n  \\gamma^{[1]}_{\\overline{n}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\\Big\\vert \n  \\overline{\\mathbf{X}}^{[1]}_{L}\\right\\rangle.\n  \\label{red-first-order}\n\\end{equation}\nIt is then readily apparent that the\n$N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ arbitrary functions $u^m_{[1]}$\nfor $m=1, \\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ are not\ndetermined at this iteration, while $r^{[1]} =\\hbox{rank }\n\\bar{\\Gamma}^{[1]}_{\\overline{n}\\,\\overline{m}}$ of the $u^m_{[1]}$ for\n$m>N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ are. There\nare then $N_0^{[2]} :=N_0^{[1]}-r^{[1]}$\n  \\textbf{second-order Lagrangian constraint functions} \n\\begin{equation}\n  \\gamma^{[2]}_{n_{[2]}} := \\left\\langle\n  \\mathbf{d}\\gamma^{[1]}_{n_{[2]}}\\Big\\vert \n  \\overline{\\mathbf{X}}_{L}^{[1]}\\right\\rangle, n_{[2]}=1, \\cdots,\n  N_0^{[2]},\n  \\nonumber\n\\end{equation}\nwith the conditions $\\gamma^{[2]}_{n_{[2]}}=0$ imposed if necessary. In general\nthere will be $I_{[2]}:=  \n\\hbox{rank }\\left\\{\\mathbf{d}\\gamma^{[1]}_{n_{[1]}},\n  \\mathbf{d}\\gamma^{[2]}_{n_{[2]}}\\right\\}$\nindependent functions in $\\hbox{C}^{[2]}_L :=\n\\hbox{C}^{[1]}\\cup \\left\\{\\gamma^{[2]}_{n_{[2]}}\\ \\vert\n\\ n_{[2]} = 1, \\dots, N_0^{[2]}\\right\\}$, and $\\mathbb{P}_L^{[1]}$ is\nreduced to the \\textbf{second-order constraint submanifold}, \n\\begin{equation}\n  \\mathbb{P}_L^{[2]} := \\left\\{\\mathfrak{u}\\in\\mathbb{P}_L^{[1]}\n  \\ \\Big\\vert \\ \\gamma^{[2]}_{[n_2]}(\\mathfrak{u})=0, n_{[2]} = 1, \\dots,\n  N_0^{[2]} \\right\\}, \n  \\nonumber\n\\end{equation}\nwhere dim $\\mathbb{P}^{[2]}_L = 2D-I_{[2]}$. At this point, there are two\npossibilities. If $I_{[2]}=I_{[1]}$ or\n$I_{[2]}=2D$, the iterative process stops, and no new Lagrangian\nconstraints are introduced. If not, the process continues. \n\nFor the second iteration in the constraint algorithm, we choose a basis\n$\\left\\{\\mathbf{P}_{(n)}^{[2]}\\right\\}$ for $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G}$ and the arbitrary functions\n$\\left\\{u_{[2]}^m\\right\\}$ such that for $m=1, \\dots, N_0^{[2]}$,\n$u_{[2]}^m$ are linear combinations of $u_{[1]}^m$ that lie in the kernel\n$\\Gamma^{[1]}_{nm}$. We once again require that\n$\\mathbf{P}_{(n)}^{[2]}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$\nfor $n=1, \\dots, N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$.  \nThen\n\\begin{equation}\n\\overline{\\mathbf{X}}_{EL}^{[2]} = \\overline{\\mathbf{X}}_{L}^{[2]} +\n\\sum_{m=1}^{N_0^{[2]}} u_{[2]}^m \\left[\\mathbf{P}_{(m)}^{[2]}\\right],\n\\nonumber\n\\end{equation}\nwith\n\\begin{equation}\n  \\overline{\\mathbf{X}}_{L}^{[2]} = \\overline{\\mathbf{X}}_{L}^{[1]} +\n  \\sum_{m=N_0^{[2]}+1} ^{N_0^{[1]}}u^m_{[2]}\\left[\\mathbf{P}^{[2]}_{(m)}\\right].\n  \\nonumber\n\\end{equation}\nHere, the functions $u^m_{[2]}$ for $m = N_0^{[2]}+1, \\dots, N_0^{[1]}$ have been\ndetermined through the constraint analysis of $\\gamma^{[1]}_n$. \n\nAs shown in \\cite{ADS2020}, $\\mathbf{G}u_{[1]}^m=0$. Similarly,\n  $\\mathbf{G}\\gamma^{[2]}_n = \\mathfrak{L}_{[\\mathbf{G},\n      \\overline{\\mathbf{X}}_{EL}]}\\mathbf{d}\\gamma^{[2]}_n=0$. Clearly\n  $\\gamma^{[2]}_n\\in\\overline{\\mathcal{F}}$ and we may require\n  $u^m_{[2]}\\in\\overline{\\mathcal{F}}$ as well. It \n  then follows that $\\left[\\mathbf{P}^{[2]}_{(n)}\\right]\\gamma^{[2]}_m \n=\\mathbf{P}^{[2]}_{(n)}\\gamma^{[2]}_m$, and imposing Eq.~$(\\ref{stable})$\non $\\gamma^{[2]}_n $, gives\n\\begin{equation}\n  \\sum_{m=1}^{N_0^{[2]}}\\Gamma^{[2]}_{nm} u^m_{[2]} =\n  -\\left\\langle \\mathbf{d}\\gamma^{[2]}_n\\Big\\vert\n  \\overline{\\mathbf{X}}^{[2]}_{L}\\right\\rangle,\\>\\hbox{where }\n  \\Gamma^{[2]}_{nm} := \\left\\langle\n  \\mathbf{d}\\gamma^{[2]}_n\\Big\\vert \\mathbf{P}^{[2]}_{(m)}\\right\\rangle,\n\\>\\> n=1, \\dots,\nN_0^{[2]}.\n\\label{second-order}\n\\end{equation}\nOnce again, $\\Gamma^{[2]}_{nm}=\\Gamma^{[2]}_{mn}$, but now on the constraint\nmanifold $\\mathbb{P}^{[2]}_L$. Moreover, since\n$\\gamma^{[2]}_n=\\gamma^{[1]}_n=0$ for $n=1, \\dots,\nN_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$,  \n$\\Gamma^{[2]}_{nm}=0=\\Gamma^{[2]}_{mn}$, and $\\left\\langle\n\\mathbf{d}\\gamma_n^{[2]}\\Big\\vert\\overline{\\mathbf{X}}^{[2]}_L\\right\\rangle\n=0$. There is once again a reduction of Eq.~$(\\ref{second-order})$,\nand we are left with \n\\begin{equation}\n  \\sum_{\\overline{m}=1}^{N_0^{[2]}-N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\n    \\overline{\\Gamma}^{[2]}_{\\overline{n}\\,\\overline{m}} u^{\\overline{m}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}_{[2]} =\n    -\n    \\langle \\mathbf{d}\\gamma^{[2]}_{\\overline{n}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\\vert\n  \\overline{\\mathbf{X}}^{[2]}_{L} \\rangle.\n\\nonumber\n\\end{equation}\nwhere\n$\\bar{\\Gamma}^{[2]}_{\\overline{n}\\,\\overline{m}}:=\n\\left\\langle\\mathbf{d}\\gamma^{[2]}_{\\overline{n}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}}\n\\bigg\\vert\\mathbf{P}^{[2]}_{(\\overline{m}+N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}})}\\right\\rangle$. As\nbefore, the $N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ arbitrary functions\n$u^{m}_{[2]}$ are not determined, while $r^{[2]} := \\hbox{rank\n}\\bar{\\Gamma}^{[2]}_{\\overline{n}\\,\\overline{m}}$ of the\nremaining $u^m_{[2]}$ for $m>N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$\nare. There are now \n$N^{[3]}_0=N_0^{[2]}-r^{[2]}$ \\textbf{third-order Lagrangian\n  constraint functions},\n\\begin{equation}\n  \\gamma^{[3]}_{n_{[3]}} =\n  \\left\\langle\\mathbf{d}\\gamma^{[2]}_{n_{[3]}}\\Big\\vert\n  \\overline{\\mathbf{X}}^{[2]}_{L}\\right\\rangle, \\> n_{[3]} = 1, \\dots,\n  N_0^{[3]},\n  \\nonumber\n\\end{equation}\nwith the conditions $\\gamma^{[3]}_{n_{[3]}}=0$ \nimposed if necessary. With \n\\begin{equation}\nI_{[3]} := \\hbox{rank } \\left\\{ \n  \\mathbf{d}\\gamma_{n_{[1]}}^{[1]},\n \\mathbf{d}\\gamma_{n_{[2]}}^{[2]},\n \\mathbf{d}\\gamma_{n_{[3]}}^{[3]}\n \\right\\},\n \\nonumber\n\\end{equation}\nindependent functions in $\\hbox{C}^{[3]}_L :=\n\\hbox{C}_L^{[2]}\\cup\\left\\{\\gamma_{n_{[3]}}^{[3]}, n_{[3]}=1,\n\\dots, N_0^{[3]}\\right\\}$, we now have the \\textbf{third-order\n  constraint submanifold},  \n\\begin{equation}\n  \\mathbb{P}_L^{[3]}:=\\left\\{\\mathfrak{u}\\in \\mathbb{P}^{[2]}_L\n  \\ \\Big\\vert \\ \\gamma^{[3]}_{n_{[3]}}(\\mathfrak{u})=0, n_{[3]}=1,\n  \\dots, N_0^{[3]}\\right\\}.\n  \\nonumber\n\\end{equation}\nOnce again, the process stops when\n$I_{[3]}=I_{[2]}$ or $I_{[3]}=2D$. However, if $I_{[2]}<I_{[3]}<2D$, the\nprocess continues until at the $n_F$-iteration when either\n$I_{[n_F]}=I_{[n_F]-1}$ or $I_{[n_F]}=2D$. \n\nFollowing \\cite{ADS2020}, the end result of this algorithm is \n\\begin{enumerate}\n\\item{A submanifold $\\mathbb{P}^{[n_F]}_L\\subset \\mathbb{P}_L$ on which\n  dynamics takes place.} \n  \\item{A collection\n    $\\hbox{C}_L^{[n_F]}\\subset\\overline{\\mathcal{F}}$ of constraint\n    functions of order $1$ to $n_F$.} \n\\item{A second-order, Euler-Lagrange vector field\n  \\begin{equation}\n    \\overline{\\mathbf{X}}_{EL}^{[n_F]}=\\overline{\\mathbf{X}}_{L}^{[n_F]} +\n    \\sum_{m=1}^{N^{[n_F]}_0}u^m_{[n_F]}(\\mathfrak{u})\\left[\\mathbf{P}_{(m)}^{[n_F]}\\right],  \n    \\nonumber\n  \\end{equation}\n  with $N_0^{[n_F]}\\ge N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ arbitrary functions\n  $u^m_{[n_F]}(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$ for $m=1,\n  \\dots, N_0^{[n_F]}$, and \n  \\begin{equation}\n    \\overline{\\mathbf{X}}_{L}^{[n_F]} =\n    \\overline{\\mathbf{X}}_{L}^{[1]}+\n    \\sum_{m=N_0^{[n_F]}+1}^{N_0^{[1]}}u^m_{[n_F]}(\\mathfrak{u})\n    \\left[\\mathbf{P}_{(m)}^{[n_F]}\\right],  \n    \\nonumber\n  \\end{equation}\n  where the $N_0^{[1]}-N_0^{[n_F]}$ functions\n  $u^m_{[n_F]}(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$, \n  $m=N_0^{[n_F]}+1, \\dots, N_0^{[1]}$, have been uniquely\n  determined through the constraint algorithm.} \n\\end{enumerate}\nWe assume that the rank of $\\Gamma^{[l]}_{nm}$ is constant on\n$\\mathbb{P}_L$ for each $l=1, \\dots, n_F$, and that\n$\\mathbb{P}^{[n_F]}_L$ is non-empty. \n\nThe end result of the constraint algorithm\n$\\overline{\\mathbf{X}}_{EL}^{[n_F]}$ is still a SOELVF, and we define\nthe collection of such vector fields as\n\\begin{equation}\n  \\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L} :=\n  \\{\\overline{\\mathbf{X}}_{EL}\\in\\overline{\\mathcal{S}\\hbox{ol}} \\>\\>\\vert\\>\n  \\mathfrak{L}_{\\overline{\\mathbf{X}}_{EL}}\\mathbf{\\beta}=0\\}.\n  \\nonumber\n\\end{equation}\nImportantly, $\\hbox{dim }\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}\\ge\nN_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$. \n\n\\section{The Generalized Lie Symmetries of Three Dynamical\n  Systems\\label{&Exam}}\n\nThree examples of dynamical systems with almost regular Lagrangians\nwere introduced in \\cite{ADS2020}. In that paper the focus of these\nexamples was on the explicit construction of the dynamical structures \nneeded to describe and predict motion in the Lagrangian phase space,\nand to show that these structures are projectable to \nthe Hamiltonian phase space. We return to these examples here, but\nwith the focus now being on the generalized Lie symmetries of each, and the\napplication of the results we have found in this paper. In particular,\nwe are in interested in the dimensionality of the symmetry groups for\neach of the systems as compared to the dimensionality of\n$\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}$ of each. A\nsummary of our results can be found in Table $\\ref{Table}$\n\n\\subsection{A Lagrangian With and Without a Generalized Lie Symmetry\\label{&special}}  \n\nWhether the action\n\\begin{equation}\n  S_{1}:=\\int\\left[\\frac{1}{2}m\\left(\\frac{d\\widehat{q}}{dt}\\right)^2\n    -V(q^a)\\right]dt, \n  \\nonumber\n\\end{equation}\nwith $\\vert q\\vert=\\sqrt{q^a q_a}$ and $\\widehat{q}^a := q^a\/\\vert\nq\\vert$, $a=1, \\dots, D$, has a generalized Lie symmetry depends on the\nchoice of potential $V(q)$. With one choice both the Lagrangian and\nthe Euler-Lagrange equations of \nmotion have a generalized gauge symmetry; with a second choice the equations\nof motion has a generalized Lie symmetry while the Lagrangian\ndoes not; and with a third choice neither the action nor the equations of motion have\na symmetry. Irrespective of the choice of $V(q)$, however, $L$ is\nsingular, demonstrating that while all actions with a \ngeneralized Lie symmetry have a singular Lagrangian, not all singular\nLagrangians have a generalized Lie symmetry. \n\nDefining $\\Pi_{ab}(q):= \\delta_{ab} - \\widehat{q}_a\\widehat{q}_b$, we find\n\\begin{equation}\n  \\mathbf{\\Omega}_M = \\frac{m}{\\vert q\\vert^2}\\Pi_{ab}(q)\n  \\mathbf{d}q^a\\wedge\\mathbf{d}v^b,\\qquad\n  \\mathbf{\\Omega}_F=\\frac{m}{\\vert q\\vert^3}\n  \\left(\\widehat{q}\\cdot\\mathbf{d}q\\right)\\wedge\n  \\left(v\\cdot\\Pi(q)\\cdot\\mathbf{d}q\\right).\n\\nonumber\\end{equation}\nThen $\\mathcal{C}$ and $\\mathcal{G}$ are spanned by $\\mathbf{U}^q_{(1)}\n= \\widehat{q}\\cdot\\mathbf{\\partial}\/\\mathbf{\\partial} q$ and  \n  $\\mathbf{U}^v_{(1)} =\n  \\widehat{q}\\cdot\\mathbf{\\partial}\/\\mathbf{\\partial} v$, \nrespectively, while $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ is spanned by\n$\\mathbf{U}^v_{(1)}$ and  \n\\begin{equation}\n  P_{(1)}=\\widehat{q}\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q} +\n    \\frac{1}{\\vert q\\vert}v\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}\n      v}.\n\\nonumber\\end{equation}\nThat $\\hbox{dim }(\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}\/\\mathcal{G})=1$ then follows.\n\nThe energy is\n\\begin{equation}\n  E=\\frac{1}{2}\\frac{m}{\\vert q\\vert^2}v\\cdot\\Pi(q)\\cdot v + V(q),\n  \\nonumber\n\\end{equation}\nand there is only one first-order Lagrangian constraint,\n\\begin{equation}\n  \\gamma^{[1]}=\\mathbf{U}^q_{(1)} V,\n  \\label{g-con}\n\\end{equation}\nso that $\\mathbf{\\beta}[\\mathbf{X}_{EL}] =\n\\gamma^{[1]}\\mathbf{\\Theta}^{(1)}_q$, where $\\mathbf{\\Theta}^{(1)}_q =\n\\widehat{q}\\cdot \\mathbf{d} q$. Using Eq.~$(\\ref{g-con})$,\n\\begin{equation}\n  \\mathfrak{L}_{\\mathbf{P}_{(1)}} \\mathbf{\\beta} =\n  \\mathbf{d}\\left[\\mathbf{U}^q_{(1)}V\n    \\right]-\\frac{1}{\\vert\n    q\\vert^2}\\widehat{q}\\cdot\\frac{\\partial\\>\\>\\>}{\\partial \n    q}\\left(\\Pi_a^{\\>\\>b}(q)\\frac{\\partial V}{\\partial\\widehat{q}^b}\\right)\n  \\mathbf{d}q^a.\n  \\label{e84}\n\\end{equation}\nWhether or not $\\mathcal{S}\\hbox{ym}$ or\n$\\mathcal{S}\\hbox{ym}\\mathcal{L}$ is empty therefore depends on the\nsymmetries of $V(q)$, as we would expect.  \n\nIt was found in \\cite{ADS2020} that\n\\begin{equation}\n  \\overline{\\mathbf{X}}_{L} = v\\cdot{\\Pi(q)}\\cdot\n  \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q} +\n  \\frac{(\\widehat{q}\\cdot v)}{\\vert q\\vert}\n  v\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\partial v} -\n  \\frac{\\vert q\\vert^2}{m}\\frac{\\partial V}{\\partial\n    q}\\cdot\\Pi(q)\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v},\n\\nonumber\\end{equation}\nand a general SOELVF is given by\n$\\overline{\\mathbf{X}}_{EL}=\\overline{\\mathbf{X}}_{L} + u(\\mathfrak{u})\n\\left[\\mathbf{P}_{(1)}\\right]$, where\n$u(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$.\nAs the constraint \nalgorithm gives \n\\begin{equation}\n  \\mathfrak{L}_{\\overline{\\mathbf{X}}_{EL}}\\gamma^{[1]} = v\\cdot\\Pi\n  \\cdot\\frac{\\partial \\gamma^{[1]}}{\\partial q} + u(\\mathfrak{u})\n  \\mathbf{U}^q_{(1)}\\gamma^{[1]},\n  \\label{stab}\n\\end{equation}\nwhether or not $u(\\mathfrak{u})$ (which in turn determines the\ndimensionality of\n$\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}_L^{[n_f]}}$) is determined\nby the constraint condition also depends on the symmetries of $V(q)$. \n\nThere are three cases to consider.\n\n\\bigskip\n\\noindent{\\textit{The symmetric potential}}\n\\bigskip\n    \nFor $\\mathbf{P}_{(1)}$ to generate a generalized Lie symmetry of the\nEuler-Lagrange equations of motion, \n\\begin{equation}\n  0=\\frac{1}{\\vert q\\vert^2} \\widehat{q}\\cdot\\frac{\\partial \\>\\>\\>}{\\partial q} \\left(\\Pi_a^{\\>\\>b}(q)\\frac{\\partial\n    V}{\\partial \\widehat{q}^b}\\right),\n\\nonumber\\end{equation}\nand as such the potential must satisfy\n\\begin{equation}\n  \\frac{\\partial V}{\\partial\\widehat{q}^a} = \\frac{\\partial\n    V_{AS}(\\widehat{q}^a)}{\\partial \\widehat{q}^a},\n\\nonumber\\end{equation}\nwhere $V_{AS}$ is a function of $\\widehat{q}^a$ only. It follows that\n$\\mathbf{P}_{(1)}$ generates a generalized Lie symmetry iff $V(q^a) =\nV_{Sph}(\\vert q\\vert)+V_{AS}(\\widehat{q}^a)$, where $V_{Sph}$ is a function of\n$\\vert q\\vert$ only. For this potential, $\\mathcal{S}\\hbox{ym}$ is \none-dimensional, and is spanned by $\\mathbf{P}_{(1)}$. \n\nThe constraint condition Eq.~$(\\ref{stab})$ for this potential\nreduces to \n\\begin{equation}\n  0=u(\\mathfrak{u})\\frac{d^2V_{Sph}(q)}{d\\vert q\\vert^2},\n\\nonumber\\end{equation}\nwhich must be satisfied on $\\mathbb{P}_L^{[1]}$. There are two possibilities.\n\n\\bigskip\n\\textit{Case 1:} $\\frac{d^2V_{Sph}}{d\\vert q\\vert^2}=0$.\n\\bigskip\n\n\\noindent Then $V_{Sph}(\\vert q\\vert) = a\\vert q\\vert+b$, but since \n\\begin{equation}\n  \\gamma^{[1]}=\\frac{dV_{Sph}}{d\\vert q\\vert} =a,\n\\nonumber\\end{equation}\nthe condition $\\gamma^{[1]}=0$ requires $a=0$. It then follows that\n$\\gamma^{[1]} =0$ on $\\mathbb{P}_L$, and thus\n$\\mathcal{S}\\hbox{ym}\\mathcal{L}$ is one-dimensional; it also is\nspanned by $\\mathbf{P}_{(1)}$. The potential is then $V(q) =\nb+V_{AS}(\\widehat{q}^a)$, and the Lagrangian is   \ninvariant under the transformation $q^a\\to \\alpha q^a$,\nwhere $\\alpha$ is an arbitrary, nonvanishing function on\n$\\mathbb{P}_L$. This Lagrangian therefore has a local conformal symmetry.\nImportantly, the function $u(\\mathfrak{u})$ is\nnot determined, and thus the dynamics of the \nparticle is given only up to an arbitrary function. Then  \n$\\hbox{dim\n}(\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L})=1$ as well,\nand is also spanned by $\\mathbf{P}_{(1)}$.\n\n\\bigskip\n\\textit{Case 2:} $\\frac{d^2V_{Sph}}{d\\vert q\\vert^2}\\ne0$.\n\\bigskip\n\n\\noindent In this case $u(\\mathfrak{u})=0$, and the dynamics of the\nparticle is completely determined by its initial data;\n$\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}=\\{\\overline{\\mathbf{X}}_{L}\\}$.\nThe first-order Lagrangian constraint $\\gamma^{[1]}$ does not vanish\nautomatically, but instead defines a surface\non $\\mathbf{P}_L$, and it follows that\n$\\mathcal{S}\\hbox{ym}\\mathcal{L}=\\emptyset$. Indeed, the action's lack\nof a local gauge symmetry in this case can be seen explicitly.\n\n\nEquation $(\\ref{g-con})$ reduces to\n\\begin{equation}\n  0=\\widehat{q}\\cdot\\frac{\\mathbf{\\partial}V_{sph}}{\\mathbf{\\partial}q},\n\\nonumber\\end{equation}\nand for dynamics to be possible the set of solutions \n\\begin{equation}\n    \\left\\{R_i\\in\\mathbb{R} \\ \\Bigg\\vert\n    \\ \\frac{dV_{Sph}}{d\\vert q\\vert}\\Bigg\\vert_{R_i} =0\\right\\},\n\\nonumber\\end{equation}\nmust be non-empty. Dynamics are on the surfaces $\\vert q\\vert\n-R_i =0$ where the potential reduces to $V(q)=V_{Sph}(R_i) +\nV_{AS}(\\widehat{q}^a)$. This reduced potential has the same symmetry\nas the potential $V_{AS}(\\widehat{q}^a)$ in \\textit{Case 1}, and it is for\nthis reason that the Euler-Lagrange equations of motion have the same\ngeneralized Lie symmetry for the two cases. This is explicitly shown\nin the appendix.\n\nIn \\textit{Case 1} the action has a local conformal symmetry, while in\n\\textit{Case 2} it does not. (In \\cite{ADS2020} it was\nerroneously stated that in this case the action has a global rotational\nsymmetry.) The Lagrangian for the two cases do not have the same \ninvariances, resulting in one case dynamics that are determined\nonly up to \nan arbitrary $u(\\mathfrak{u})$, and in the other case to a\n$u(\\mathfrak{u})=0$ and dynamics\nthat are instead completely determined by the choice of initial data. \n\n\\bigskip\n\\noindent{\\textit{The asymmetric potential}}\n\\bigskip\n\nFor a general $V$ the second term in\nEq.~$(\\ref{e84})$ does not vanish, $\\mathbf{P}_{(1)}$ does not\ngenerate a symmetry of the equations of motion, and \n$\\mathcal{S}\\hbox{ym}=\\{\\emptyset\\}$. As before, $\\gamma^{[1]}$ does\nnot vanish, and thus $\\mathcal{S}\\hbox{ym}\\mathcal{L}=\\{\\emptyset\\}$\nas well. Furthermore, as Eq.$(\\ref{stab})$ results in\n\\begin{equation}\n  \\overline{\\mathbf{X}}_E =\\overline{\\mathbf{X}}_L-\\frac{v\\cdot\\Pi\n  \\cdot\\frac{\\partial \\gamma^{[1]}}{\\partial q}}\n  {q^2\\mathbf{U}^q_{(1)}\\gamma^{[1]}}[\\mathbf{P}_{(1)}],\n\\nonumber\\end{equation}\nthe dynamics of the particle is uniquely determined by its initial\ndata, and\n$\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}=\\{\\overline{X}_{EL}\\}$\nonce again consists of a single point.\n\n\\subsection{A Lagrangian with Local\n  Conformal Symmetry}\n\nThe action,\n\\begin{equation}\n  S_{2} := \\int\\Bigg\\{\\frac{1}{2}m\n  \\left(\\frac{d\\widehat{q}_1}{dt}\\right)^2+\\frac{1}{2}m\n  \\left(\\frac{d\\widehat{q}_2}{dt}\\right)^2+\n  \\frac{\\lambda}{2}\\left[\\frac{q_1^a}{\\vert q_2\\vert}\n  \\frac{d\\>\\>}{dt}\\left(\\frac{q_{2a}}{\\vert q_1\\vert}\\right) -\n  \\frac{q_2^a}{\\vert q_1\\vert}\n  \\frac{d\\>\\>}{dt}\\left(\\frac{q_{1a}}{\\vert q_2\\vert}\\right)\n  \\right]\n  \\Bigg\\} dt,\n\\nonumber\\end{equation}\nwhere $a=1, \\dots, d$, $D=2d$, describes an interacting, two particle\nsystem that is invariant under the local conformal transformation\n$q_1^a \\to \\alpha(\\mathfrak{u}) q^a_1$ and $q_2^a \\to\n\\alpha(\\mathfrak{u}) q^a_2$. \n\nWith\n\\begin{widetext}\n\\begin{eqnarray}\n  \\mathbf{\\Omega}_M &=&  \\frac{m}{\\vert q_1\\vert^2} \\Pi_{ab}(q_1)\n  \\mathbf{d}q_1^a\\wedge \\mathbf{d} v_1^b + \\frac{m}{\\vert q_2\\vert^2}\n  \\Pi_{ab}(q_2) \\mathbf{d}q_2^a\\wedge \\mathbf{d} v_2^b, \\hbox{  and}\n  \\nonumber\n  \\\\\n  \\mathbf{\\Omega}_F &=& \\frac{m}{\\vert q_1\\vert^3}\n  \\left(\\widehat{q}_1\\cdot\\mathbf{d}\n  q_1\\right)\\wedge\\left(v_1\\cdot\\Pi(q_1)\\cdot \\mathbf{d} q_1\\right) + \n  \\frac{m}{\\vert q_2\\vert^3} \\left(\\widehat{q}_2\\cdot\\mathbf{d} \n  q_2\\right)\\wedge\\left(v_2\\cdot\\Pi(q_2)\\cdot \\mathbf{d} q_2\\right)- \n  \\nonumber\n  \\\\\n  &{}&\n  \\frac{\\lambda}{\\vert q_1\\vert \\vert\n    q_2\\vert}\\left[\\mathbf{d}q_1^a\\wedge\\left(\\Pi(q_2)\\cdot \n  \\mathbf{d} q_2\\right)_a+ \\left(\\Pi(q_1)\\cdot\n  \\mathbf{d} q_1\\right)_a\\wedge \\mathbf{d}q_2^a -\\left(\\Pi(q_1)\\cdot\n  \\mathbf{d} q_1\\right)^a\\wedge \\left(\\Pi(q_2)\\cdot\n  \\mathbf{d} q_2\\right)_a\\right]-\n  \\nonumber\n  \\\\\n  &{}& \\frac{\\lambda}{\\vert q_1\\vert^2}\n  \\left(\\widehat{q}_1\\cdot\\mathbf{d}q_1\\right)\\wedge\n  \\left(\\widehat{q}_2\\cdot\\Pi(q_1) \\cdot \\mathbf{d}q_1\\right) +\n  \\frac{\\lambda}{\\vert q_2\\vert^2} \n  \\left(\\widehat{q}_2\\cdot\\mathbf{d}q_2\\right)\\wedge\n  \\left(\\widehat{q}_1\\cdot\\Pi(q_2) \\cdot \\mathbf{d}q_2\\right),\n  \\nonumber\n\\end{eqnarray}\n\\end{widetext}\n$\\mathcal{C}$ and $\\mathcal{G}$ are two-dimensional, and\nare spanned by\n\\begin{equation}\n  \\mathbf{U}^q_{(1)} =\n  \\widehat{q}_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_1},\n    \\quad\n  \\mathbf{U}^q_{(2)} =\n  \\widehat{q}_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_2},\n  \\quad \\hbox{and}\\quad \n  \\mathbf{U}^v_{(1)} =\n  \\widehat{q}_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v_1},\n    \\quad\n  \\mathbf{U}^v_{(2)} =\n  \\widehat{q}_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v_2},\n\\nonumber\\end{equation}\nrespectively. The reduced $\\bar{F}=0$, and $\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L(\\mathfrak{u})}$ is spanned by $\\mathbf{U}^v_{(1)},\n\\mathbf{U}^v_{(2)}$, \n\\begin{eqnarray}\n  \\mathbf{P}_{(+)} &=&\n  q_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_1}+q_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_2}\n  +\n  v_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v_1} \n  +\n  v_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v_2},\n  \\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n  \\mathbf{P}_{(-)} &=&\n  q_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_1}-q_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q_2}\n  +\n  v_1\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v_1} \n  -\n  v_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}v_2} \n  -\n  \\nonumber\n  \\\\\n  &{}&\n  2\\frac{\\lambda}{m}\n  \\left[\\frac{\\vert q_1\\vert}{\\vert\n      q_2\\vert}q_2\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} \n      v_1}\n    +\\frac{\\vert q_2\\vert}{\\vert q_1\\vert}q_1\\cdot\n    \\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v_2}\\right].\n  \\nonumber\n\\end{eqnarray}\nAs such, $\\hbox{dim }\\overline{(\\hbox{ker }\\mathbf{\\Omega}_L)}\/\\mathcal{G}=2$. \n\nThe energy is\n\\begin{equation}\n  E=\\frac{1}{2}\\frac{m}{\\vert q_1\\vert^2}v_1\\cdot\\Pi(q_1)\\cdot v_1 +\n  \\frac{1}{2}\\frac{m}{\\vert q_2\\vert^2}v_2\\cdot\\Pi(q_2)\\cdot v_2.\n\\nonumber\n\\end{equation}\nWe find that $\\gamma^{[1]}_{(+)} =0$ while\n\\begin{equation}\n  \\gamma^{[1]}_{(-)} =-\\frac{2\\lambda}{\\vert q_1\\vert \\vert q_2\\vert}\\left(\n  q_2\\cdot \\Pi(q_1)\\cdot v_1 +\n  q_1\\cdot \\Pi(q_2) \\cdot v_2\\right),\n\\nonumber\\end{equation}\ngiving,\n\\begin{eqnarray}\n  \\mathbf{\\beta}[\\mathbf{X}_{EL}] &=& \\frac{1}{2}\\gamma^{[1]}_{(-)}\n  \\left(\\frac{\\mathbf{\\Theta}_q^{(1)}}{\\vert q_1\\vert} -\n  \\frac{\\mathbf{\\Theta}_q^{(2)}}{\\vert q_2\\vert}\\right).\n  \\nonumber\n\\end{eqnarray}\nThen $\\mathcal{S}\\hbox{ym}\\mathcal{L}$ is\none-dimensional and spanned by $\\mathbf{P}_{(+)}$. \nAs expected, $\\mathfrak{L}_{\\mathbf{P}_{(+)}}\n\\mathbf{\\beta} = 0$. Because\n\\begin{equation}\n  \\mathfrak{L}_{\\mathbf{P}_{(-)}} \\mathbf{\\beta} =\n  -\\frac{4\\lambda}{m}\\left[1-(\\widehat{q}_1\\cdot\\widehat{q}_2)^2\\right]\\left(\\frac{\\mathbf{\\Theta}_q^{(1)}}{\\vert q_1\\vert} -\n  \\frac{\\mathbf{\\Theta}_q^{(2)}}{\\vert q_2\\vert}\\right),\n  \\nonumber\n\\end{equation}\n$\\mathcal{S}\\hbox{ym}$ is also one-dimensional, and\nis also spanned by $\\mathbf{P}_{(+)}$. \n\nA general SOELVF is \n\\begin{equation}\n  \\overline{\\mathbf{X}}_{EL} = \\overline{\\mathbf{X}}_{L} -\n  \\frac{m}{8\\lambda^2}\\frac{\\overline{\\mathbf{X}}_{L}\\gamma^{[1]}_{(-)}}{ \n    \\left[1-(\\widehat{q}_1\\cdot\\widehat{q}_2)\\right]}\\left[\\mathbf{P}_{(-)}\\right] +\n  u^{(+)}(\\mathfrak{u}) \\left[\\mathbf{P}_{(+)}\\right],\n  \\label{103}\n\\end{equation}\nwhere $u^{(+)}(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$, and from \\cite{ADS2020},\n\\begin{eqnarray}\n\\overline{\\mathbf{X}}_{L} &=&\nv_1\\cdot\\Pi(q_1)\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}\n  q_1} + v_2\\cdot\\Pi(q_2)\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial}\n  q_2} +\n\\nonumber\n\\\\\n&{}&\n\\left(\\frac{\\widehat{q}_1\\cdot v_1}{\\vert q_1\\vert}\\right)\nv_1\\cdot\\Pi(q_1)\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v_1} +\n\\left(\\frac{\\widehat{q}_2\\cdot v_2}{\\vert q_2\\vert }\\right)\nv_2\\cdot\\Pi(q_2)\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v_2} \n+\n\\nonumber\n\\\\\n&{}&\n\\frac{\\lambda}{m} \\left(\\frac{\\vert q_1\\vert}{\\vert q_2\\vert}\n  v_2\\cdot\\Pi(q_2)\\cdot\\Pi(q_1)\\cdot \\frac{\\mathbf{\\partial}\n    \\>\\>\\>}{\\mathbf{\\partial} v_1}-\\frac{\\vert q_2\\vert}{\\vert q_1\\vert}\n  v_1\\cdot\\Pi(q_1)\\cdot\\Pi(q_2)\\cdot \\frac{\\mathbf{\\partial}\n    \\>\\>\\>}{\\mathbf{\\partial} v_2} \\right),\n  \\nonumber\n\\end{eqnarray}\nafter the constraint algorithm is applied. Equation $(\\ref{103})$ is\na consequence of the identity\n$\\langle\\mathbf{d}\\gamma^{[1]}_{(+)}\\vert \\overline{\\mathbf{X}}_{L}\\rangle=0$ and \n\\begin{eqnarray}\n -\\frac{1}{2\\lambda} \\langle\\mathbf{d}\\gamma^{[1]}_{(-)}\\vert \\overline{\\mathbf{X}}_{L}\\rangle&=& -\n  2(\\widehat{q}_1\\cdot\\widehat{q}_2)\\frac{E}{m}+ \\frac{2}{\\vert\n    q_1\\vert \\vert q_2\\vert}\n  v_1\\cdot\\Pi(q_1)\\cdot\\Pi(q_2)\\cdot v_2\n  -\n  \\nonumber\n  \\\\\n  &{}&\n  \\frac{\\lambda}{m}(\\widehat{q}_1\\cdot\\widehat{q}_2)\\left[v_2\\cdot\\Pi(q_2)\\cdot\\widehat{q}_1\n    - v_1\\cdot\\Pi(q_1)\\cdot \\widehat{q}_2\\right].\n\\nonumber\n\\end{eqnarray}\nWe see that $\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}$ is also\none-dimensional, and is also spanned by $\\mathbf{P}_{(+)}$. \n\n\\subsection{A Lagrangian with Local Conformal and\n  Time-reparametization Invariance}\n\nThe action\n\\begin{equation}\n  S_{3} := sm\\int\\left[s\\left(\\frac{d\\widehat{q}}{dt}\\right)^2\\right]^{1\/2} dt,\n\\nonumber\\end{equation}\nwhere $s=\\pm1$, is invariant under both the local conformal\ntransformations, $q^a\\to \\alpha(\\mathfrak{u}) q^a$, and the\nreparametization of time $t\\to \\tau(t)$ where $\\tau$ is a\nmonotonically increasing function of $t$. Then\n\\begin{equation}\n  \\mathbf{\\Omega}_L = \\frac{m}{\\vert q\\vert}\n  \\frac{P_{ab}(u)}{\\sqrt{sv\\cdot \\Pi(q)\\cdot\n      v}}\\mathbf{d}q^a\\wedge\\mathbf{d}v^b,\n\\nonumber\\end{equation}\nand $\\mathbf{\\Omega}_F=0$. Here, $a=1. \\dots, D$,\n\\begin{equation}\n  u_a=\\frac{\\Pi_{ab}(q)v^b}{\\sqrt{sv\\cdot\\Pi(q)\\cdot v}},\n\\nonumber\\end{equation}\nso that $u^2 = s$, while $P_{ab}(u) = \\Pi_{ab}(q) -su_a u_b$. As such,\n$\\hbox{ker }\\mathbf{\\Omega}_L(\\mathfrak{u}) = $ ker \n$\\mathbf{\\Omega}_M(\\mathfrak{u})$. Both $\\mathcal{C}$ and\n$\\mathcal{G}$ are two-dimensional, and are spanned by\n\\begin{equation}\n  \\mathbf{U}^q_{(1)} = \n  \\widehat{q}\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q}, \\quad\n    \\mathbf{U}^q_{(2)} =\n  u\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} q}, \\quad\n  \\hbox{and}\\quad \n    \\mathbf{U}^v_{(1)} =\n  \\widehat{q}\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v}, \\quad\n    \\mathbf{U}^v_{(2)} =\n    u\\cdot\\frac{\\mathbf{\\partial}\\>\\>\\>}{\\mathbf{\\partial} v},\n\\nonumber\\end{equation}\nrespectively. It follows that $\\hbox{dim }(\\overline{\\hbox{ker\n  }\\mathbf{\\Omega}_L}\/\\mathcal{G})=2$. \n\nBecause this system is fully constrained, $E=0$. As\n$\\mathbf{\\Omega}_F=0$ as well, there are no\nLagrangian constraints. It follows that $\\mathcal{S}\\hbox{ym}\\mathcal{L}$ is two\ndimensional and spanned by $\\mathbf{U}_{(1)}^q$ and\n$\\mathbf{U}_{(2)}^q$. As $\\mathbf{\\beta}=0$ as well, $\\mathcal{S}\\hbox{ym}$\nis also two dimensional, and is also spanned by  \n$\\mathbf{U}^q_{(1)}$ and $\\mathbf{U}^q_{(2)}$.   \n\nWe found in \\cite{ADS2020} that $\\overline{\\mathbf{X}}_{L} = 0$. A\ngeneral SOELVF is then $\\overline{\\mathbf{X}}_{EL} = u^{1}(\\mathfrak{u}) \n\\left[\\mathbf{U}^q_{(1)}\\right] + u^{2}(\\mathfrak{u})\n\\left[\\mathbf{U}^q_{(2)}\\right]$, with\n$u^{n}(\\mathfrak{u})\\in\\overline{\\mathcal{F}}$ for $n=1,2$. It follows\nthat $\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L}$ \nis also two-dimensional, and is spanned by $\\mathbf{U}^{q}_{(1)}$ and\n$\\mathbf{U}^q_{(2)}$ as well.  \n\n\\begin{table}\n  \\begin{tabular}{c|c|ccccc}\n    \\hline\n    Action & Potential & $\\>\\>\\>\\overline{\\hbox{ker }\\mathbf{\\Omega}_L}\/\\mathcal{G}\\>\\>\\>$ &\n    $\\>\\>\\>\\mathcal{S}\\hbox{ym}\\>\\>\\>$ & $\\>\\>\\>\\mathcal{S}\\hbox{ym}\\mathcal{L}\\>\\>\\>$\n    & $\\>\\>\\> I_{\\left[1\\right]  }\\>\\>\\>$ &\n    $\\>\\>\\>\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}^{[n_F]}_L} \\>\\>\\>$ \\\\\n    \\hline\n    {} & $V_{AS}(\\hat{q}^a)$ & 1 & 1 & 1 & 0 & 1 \\\\\n   \n    $S_{1}$ & $V_{sph}(\\vert q\\vert)+V_{AS}(\\hat{q}^a)$ & 1 & 1 & 0 & 1 & 0 \\\\\n   \n    {} & $V(q^a)$ & 1 & 0 & 0& 1& 0 \\\\\n    \\hline\n    $S_{2}$ & $\\frac{\\lambda}{2}\\left[\\frac{q_1^a}{\\vert q_2\\vert}\n  \\frac{d\\>\\>}{dt}\\left(\\frac{q_{2a}}{\\vert q_1\\vert}\\right) -\n  \\frac{q_2^a}{\\vert q_1\\vert}\n  \\frac{d\\>\\>}{dt}\\left(\\frac{q_{1a}}{\\vert q_2\\vert}\\right)\n  \\right]$ & 2 & 1 & 1& 1& 1 \\\\\n    \\hline\n    $S_{3}$ & $0$ & 2 & 2& 2& 0& 2\\\\\n  \\hline\n  \\end{tabular}\n  \\caption{A summary of the symmetries of the three examples\n    considered in this paper. With the \n    exception of the $I_{[1]}$ column, the numerical entries are the\n    dimensionality of the vector spaces listed along the first row. Notice\n    the case where the Euler-Lagrange equations of motion has a\n    generalized Lie symmetry while the action itself does not. In all\n    three examples, $\\hbox{dim\n    }(\\mathcal{S}\\hbox{ym}\\mathcal{L})=\\hbox{dim\n    }(\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}_L^{[n_f]}})$. \n  }\n  \\label{Table}\n\\end{table}\n\n\n\\section{Concluding Remarks\\label{&Conc}}\n\nThat each generalized Lie symmetry of the action\ncontributes one arbitrary function to the SOELVF for a dynamical\nsystem is known anecdotally, and is a result expected on physical\ngrounds. For almost regular Lagrangians, the\nappearance in physics of a generalized Lie symmetry is due to a local\ngauge symmetry of the \ndynamical system, and thus to the absence of a gauge\\textemdash the\nlength of vectors for local conformal invariance, or a measure\nfor time for time-reparametization invariance\\textemdash for some\ndynamical property of the system. As the generalized Lie symmetries of\nthe action for an almost regular Lagrangian would have\n$N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ of these gauge freedoms, it is\nreasonable that the absence of these gauges will result in an \nequal number of arbitrary functions in the SOELVF. An equal\nnumber of terms to fix these gauges would then be needed to determine the\ndynamics of the system uniquely. But while these expectations are\nreasonable, up to now they have been fulfilled only on a case-by-case\nbasis. This is in great part because the analysis of dynamical systems\nwith a local gauge symmetry has traditionally been done using\nconstrained Hamiltonian mechanics. Such analysis relies on the\ncanonical Hamiltonian, however, and the connection\nbetween the canonical Hamiltonian and the symmetries of the Lagrangian\nis indirect at best, in contrast to the Lagrangian approach followed\nhere. Moreover, the process of determining the total Hamiltonian\nfor the system is often prescriptive, with results that are specific to\nthe system at hand. By focusing on the Lagrangian and on the Lagrangian\nphase space, we have been able to show for all systems with an almost\nregular Lagrangian that has a constant rank Lagrangian two-form,\na direct link between local gauge symmetries and its dynamics. In\nparticular, it establishes a link between the number of gauge\nsymmetries of the action and the number of arbitrary functions that\nnaturally appear in the evolution of such dynamical systems.  \n\nAs $\\gamma_{\\mathbf{P}}^{[1]}=0$ for any choice of\n$\\mathbf{P}\\in\\mathcal{S}\\hbox{ym}\\mathcal{L}$, the vectors in\n$\\mathcal{S}\\hbox{ym}\\mathcal{L}$ do not contribute to the\nfirst-order constraint manifold $\\mathbb{P}_L^{[1]}$, and as such do\nnot contribute to the Lagrangian constraint algorithm at this order, or at\nany higher orders. It is for this reason that the\n$N_{\\mathcal{S}\\hbox{ym}\\mathcal{L}}$ arbitrary functions $u^m_{[1]}$\nare not determined by the algorithm, and why these functions will\nstill contribute to $\\overline{\\mathbf{X}}_{EL}$ even after the algorithm has been\ncompleted. It also means that if second- and higher-order Lagrangian constraints\nare introduced, they are accidental and cannot be due to the local\ngauge symmetries of the action. Interestingly, we have yet to find a dynamical system\nwith a Lagrangian that is both almost-regular and has a Lagrangian\ntwo-form with constant rank where second- or higher-order Lagrangian\nconstraints are introduced. \n\nThis impact of generalized Lie symmetries on the dynamics of particles\nillustrates the inherent differences between the analysis of the\nsymmetries of regular Lagrangians and that of almost regular\nLagrangians. For regular Lagragians, the generator of the generalized\nLie symmetry (at times referred to as a global symmetry) gives rise to\na prolongation vector, and the action of this prolongation on the Lagrangian gives\nthe variation of the action, $\\delta S$, under this symmetry. When the\nEuler-Lagrange equations of motion are thenimposed, the conserved\nquantity for this symmetry along the path given by the solution of these equations of\nmotion is then obtained. While the generator of the generalized \nLie symmetry for the almost regular Lagrangian $\\mathbf{g}_L$ does\ngive a prolongation vector \\textbf{pr} $\\mathbf{g}_L$ Eq.~$(\\ref{prog})$, and\nwhile the action of \\textbf{pr} $\\mathbf{g}_L$ on $L$ does give\n$\\delta S$, imposing the Euler-Lagrange equations of motion on $\\delta\nS$ in Eq.~$(\\ref{Ae1})$ gives the vacuous statement $\\delta\nS=0$. Instead, the requirement that $\\delta S=0$ for all paths on\n$\\mathbb{Q}$ gives the conditions that the  \ngenerators of the symmetry must satisfy. This in turn shows that the\nexistence of these generators is due solely to the Lagrangian being\nsingular. These conditions then \naffect the dynamics of the system through\n$\\gamma_{\\mathbf{P}}^{[1]}=0$, and in doing so, sets a lower bound to\nthe dimensionality of \n$\\overline{\\mathcal{S}\\hbox{ol}}_{\\mathbb{P}_L^{[n_f]}}$.  \n\nWe have found it quite difficult to construct more than one example of a\ndynamical system that has an almost regular Lagrangian with both a\ngeneralized Lie symmetry and a \nLagrangian two-form with constant rank on $\\mathbb{P}_L$. We have,\non the other hand, found it quite easy to construct examples of dynamical\nsystems that have an almost regular Lagrangian with a generalized Lie\nsymmetry and a Lagrangian two-form whose rank varies across\n$\\mathbb{P}_L$. Indeed, it is the latter case that is the more\nprevalent one, and yet much of the results of this \npaper and a good portion of the results of our previous one \\cite{ADS2020} relies\non the condition that the rank of the Lagrangian two-form be constant on\n$\\mathbb{P}_L$. This is even more concerning when we realize that these\nmore prevalent systems are expected, by their nature, to have much\nricher dynamics and mathematical structures (indeed, we have found that\nsuch systems often require the introduction of second- or higher-order\nLagrangian constraints), and yet it is not known\nwhich of the results that have been shown to hold for systems with\nconstant rank Lagrangian two-forms will still hold when the rank varies\nacross $\\mathbb{P}_L$. Determining the generalized Lie symmetries of\nthese systems; showing that the passage from the Lagrangian to the\nHamiltonian phase space is possible; and finding the links between\nsymmetry and dynamics is a necessity for future research.  \n\n\\begin{acknowledgments}\n  This paper would not have been possible without the contributions by\n  John Garrison, who provided much of the underlying symmetry analysis\n  of the action used in \\textbf{Section \\ref{&A-Sym}}, and most\n  of the essential mathematics in \\textbf{Section\n    \\ref{&review}}. Publication made possible in \n  part by support from the Berkeley Research Impact Initiative (BRII)\n  sponsored by the UC Berkeley Library. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe simple loop conjecture, proved by Gabai in \\cite{Ga}, states that any non-injective homomorphism from a closed surface group to another closed surface group has an element represented by a simple closed curve in the kernel.  It has been conjectured that the result still holds if the target is replaced by the fundamental group of an orientable 3-manifold (see Kirby's problem list in \\cite{Ki}).  Although special cases have been proved (e.g. \\cite{Ha}, \\cite{RW}), the general hyperbolic case is still open. \n\nRecently, Cooper and Manning showed that if instead of a 3-manifold group the target group is $\\SL(2, \\mathbb{C})$, then the conjecture is false.  Precisely, they show: \n\n\\begin{theorem}[Cooper-Manning \\cite{CM}]  \\label{cmthm} Let $\\Sigma$ be a closed orientable surface of genus $g \\geq 4$.  Then there is a homomorphism $\\rho: \\pi_1(\\Sigma) \\to \\SL(2, \\mathbb{C})$ such that \n\\begin{enumerate} \n\\item $\\rho$ is not injective\n\\item If $\\rho(\\alpha) = \\pm I$, then $\\alpha$ is not represented by a simple closed curve\n\\item If $\\rho(\\alpha)$ has finite order, then $\\rho(\\alpha) = I$\n\\end{enumerate}\nThe third condition implies in particular that no \\emph{power} of a simple closed curve lies in the kernel.  \n\\end{theorem}\n\nInspired by this, we asked whether a similar result holds for $\\PSL(2,\\mathbb{R})$, this being an intermediate case between Gabai's result for surface groups and Cooper and Manning's for $\\SL(2,\\mathbb{C})$.  Cooper and Manning's proof uses a dimension count on the $\\SL(2,\\mathbb{C})$ character variety and a proof that a specific subvariety is irreducible and smooth on a dense subset, much of which does not carry over to the $\\PSL(2, \\mathbb{R})$ case.  In general, complex varieties and their real points can behave quite differently.   However, we show here with different methods that an analogous result does hold.  \n\nWhile this note was in progress, we learned of work of Louder and Calegari (independently in \\cite{Lo} and \\cite{Ca}) that can also be applied to answer our question in the affirmative. Louder shows the simple loop conjecture is false for representations into limit groups, and Calegari gives a practical way of verifying no simple closed curves lie in the kernel of a non-injective representation using stable commutator length and the Gromov norm.  \n\nThe difference here is that our construction is entirely elementary.  We use an explicit representation from DeBlois and Kent in \\cite{DK} and verify that this representation it is non injective and kills no simple closed curve by elementary means.   Our end result parallels that of Cooper and Manning but also include surfaces with boundary and all genera at least 1: \n\n\\begin{theorem}\\label{main}\nLet $\\Sigma$ a surface of negative Euler characteristic and of genus $g \\geq 1$\n, possibly with boundary.  \nThen there is a homomorphism $\\rho: \\pi_1(\\Sigma) \\to \\SL(2, \\mathbb{R})$ such that \n\\begin{enumerate} \n\\item $\\rho$ is not injective\n\\item If $\\rho(\\alpha) = \\pm I$, then $\\alpha$ is not represented by a simple closed curve\n\\item In fact, if $\\alpha$ is represented by a simple closed curve, then $\\rho(\\alpha^k) \\neq 1$ for any $k \\in \\mathbb{Z}$.  \n\\end{enumerate}\nMoreover, there are uncountably many non-conjugate representations satisfying 1. through 3.  \n\\end{theorem}\n\n\n\n\n\n\n\n\n \n\n\n\n\n\\section{Proof of theorem \\ref{main}} \n\nWe first present a construction of a (non-injective) representation from DeBlois and Kent in \\cite{DK}, and then show that no power of a simple closed curve lies in the kernel of this representation.  The full construction appears in \\cite{DK}, we describe it here for convenience. \n\nLet $\\Sigma$ be a surface of genus $g \\geq 1$ and negative Euler characteristic, possibly with boundary. Assume for the moment that $\\Sigma$ is not the once-puntured torus -- Theorem \\ref{main} for this case will follow easily later on.  \n\nLet $c \\subset \\Sigma$ be a simple closed curve separating $\\Sigma$ into a genus 1 subsurface with single boundary component $c$, and a genus ($g-1$) subsurface with one or more boundary components.  Let $\\Sigma_A$ denote the genus $(g-1)$ subsurface and $\\Sigma_B$ the genus 1 subsurface.  See Figure \\ref{setup} below.  Finally, we let $A= \\pi_1(\\Sigma_A)$ and $B = \\pi_1(\\Sigma_B)$, so that $\\pi_1(\\Sigma) = A \\ast_C B$, where $C$ is the $\\mathbb{Z}$-subgroup generated by the element $[c]$ represented by the curve $c$.  We assume that the basepoint for $\\pi_1(\\Sigma)$ lies on $c$.   \n\n  \\begin{figure*}[h]\n  \\centerline{\n    \\mbox{\\includegraphics[width=2.5in]{simpleloopsetup.pdf}}}\n \\caption{The setup: decomposition of $\\Sigma$ and generators $x$ and $y$ for $B$}\n  \\label{setup}\n  \\end{figure*}\n\nLet $x \\in B$ and $y \\in B$ be generators such that $B = \\langle x, y \\rangle$, and that $c$ represents the commutator $[x,y]$.   Fix $\\alpha$ and $\\beta$ in $\\mathbb{R} \\setminus \\{0,\\pm1\\}$, and following \\cite{DK} define $\\phi_B: B \\to \\SL(2,\\mathbb{R})$ by \n\n$$\\phi_B(x) = \n \\begin{pmatrix}\n  \\alpha & 0 \\\\\n  0 & \\alpha\n \\end{pmatrix}$$\n  $$\\phi_B(y) = \n  \\begin{pmatrix}\n  \\beta & 1 \\\\\n  0 & \\beta^{-1} \n \\end{pmatrix}$$  \n  We have then \n  $$\\phi_B([x, y]) =  \\begin{pmatrix}\n 1 & \\beta(\\alpha^2 - 1)  \\\\\n  0 & 1  \n \\end{pmatrix}$$\nso that $\\phi_B([x, y])$ is invariant under conjugation by the matrix $\\lambda_t := \\bigl( \\begin{smallmatrix} \n 1 & t  \\\\\n  0 & 1  \n\\end{smallmatrix} \\bigr)$.  \n  \nProjecting this representation to $\\PSL(2,\\mathbb{R})$ gives a representation which is upper triangular, hence solvable and therefore non-injective.  Abusing notation, let $\\phi_B$ denote the representation to $\\PSL(2,\\mathbb{R})$.\n \nNow let $\\phi_A : A \\to \\PSL(2, \\mathbb{R})$ be Fuchsian and such that the image of the boundary curve $c$ under $\\phi_A$ agrees with $\\phi_B([x,y])$.  Such a representation exists for the following reasons.  First, if $\\Sigma$ has negative Euler characteristic, genus $g>1$, and is not the once punctured torus, then $\\Sigma_A$ will have negative Euler characteristic as well and admit a hyperbolic structure.  Secondly, the Fuchsian representation coming from the hyperbolic structure will send the element $[c]$ representing the boundary curve to a parabolic, so after conjugation we may assume that it is equal to $\\phi_B([x,y])$, provided $\\phi_B([x,y])$ is parabolic, i.e. $\\beta(\\alpha^2 - 1) \\neq 0$.  \n \nFinally, combine $\\phi_A$ and $\\phi_B$ to get a one-parameter family of representations $\\phi_t$ of $\\pi_1(\\Sigma) = A \\ast_C B$ to $\\PSL(2,\\mathbb{R})$ as follows.  For $t \\in \\mathbb{R}$ and $g \\in A \\ast_C B$, let\n\n$$\\phi_t(g) = \\left\\{ \\begin{array}{rcl} \\phi_A(g) & \\mbox{if} & g \\in A \\\\ \\lambda_t \\circ \\phi_B(g) \\circ (\\lambda_t)^{-1} & \\mbox{if} & g \\in B\n\\end{array}\\right.$$\n\nThis representation is well defined because $\\phi_B([x, y]) = \\phi_A([x,y])$ and $\\phi_B([x, y])$ is invariant under conjugation by $\\lambda_t$.  \n\nOur next goal is to show that for appropriate choice of $\\alpha, \\beta$ and $t$, the representation $\\phi_t$  satisfies the criteria in Theorem \\ref{main}.  The main difficulty will be checking that no element representing a simple closed curve is of finite order.  To do so, we employ a stronger form of Lemma 2 from \\cite{DK}.  This is:\n\n\\begin{lemma} \\label{transcendental}\nSuppose $w \\in A \\ast_C B$ is a word of the form $w = a_1b_1a_2b_2 ... a_lb_l$ with $a_i \\in A$ and $b_i \\in B$ for $1 \\leq i \\leq l$. Assume that for each $i$, the matrix $\\phi_t(a_i)$ has a nonzero 2,1 entry and $\\phi_t(b_i)$ is hyperbolic. If $t$ is transcendental over the entry field of $\\phi_0(A \\ast_C B)$, then $\\phi_t(w)$ is not finite order.  \n\\end{lemma}\n\n\\noindent By \\emph{entry field} of a group $\\Gamma$ of matrices, we mean the field generated over $\\mathbb{Q}$ by the collection of all entries of matrices in $\\Gamma$.\n\n\\begin{remark} \nLemma 2 of \\cite{DK} is a proof that $\\phi_t(w)$ is not the \\emph{identity}, under the assumptions of Lemma \\ref{transcendental}.    We use some of their work in our proof.  \n\\end{remark}\n\n\\begin{proof}[Proof of Lemma \\ref{transcendental}]\nIn \\cite{DK}, DeBlois and Kent show by a straightforward induction that the entries of $\\phi_t(w)$ are polynomials in $t$, where the degree of the 2,2 entry is $l$, the degree of the 1,2 entry is at most $l$, and the other entries have degree at most $l-1$.  Now suppose that $\\phi_t(w)$ is finite order.  Then it is conjugate to a matrix of the form \n$ \\bigl( \\begin{smallmatrix} \n u & v  \\\\\n  -v & u  \n\\end{smallmatrix} \\bigr)$. \nwhere $u = \\cos(\\theta)$ and $v = \\sin(\\theta)$ for some rational angle $\\theta$.  In particular, it follows from the deMoivre formula for sine and cosine that $u$ and $v$ are algebraic.  \n\nNow suppose that the matrix conjugating $\\phi_t(w)$ to \n$ \\bigl( \\begin{smallmatrix} \n u & v  \\\\\n  -v & u  \n\\end{smallmatrix} \\bigr)$ has entries $a_{ij}$.  Then we have\n$$ \\phi_t(w) =  \\begin{pmatrix}\nu -  (a_{12}a_{22} -  a_{11}a_{21})v   &  (a_{12}^2  a_{11}^2) v \\\\\n -(a_{22}^2  a_{21}^2)v &   u + (a_{12}a_{22} +  a_{11}a_{21})v\n \\end{pmatrix}$$\nLooking at the 2,2 entry we see that $a_{12}a_{22} +  a_{11}a_{21}$ must be a polynomial in $t$ of degree $l$.  But this means that the 1,1 entry is also a polynomial in $t$ of degree $l$, contradicting Deblois and Kent's calculation.  This proves the lemma.\n\\end{proof} \n\n\nTo complete our construction, choose $t$ to be transcendental over the entry field of $\\phi_0(A\\ast_C B)$.  We want to show that no power of an element representing a simple closed curve lies in the kernel of $\\phi_t$.  To this end, consider any word $w$ in $A \\ast_C B$ that has a simple closed curve as a representative.  There are three cases to check.  First, if $w$ is a word in $A$ alone, then $\\phi_t(w)$ is not finite order, since $\\phi_t(A)$ is Fuchsian and therefore injective.  Secondly, if $w$ is a word in $B$, then an elementary geometric argument shows that $w$ can only be represented by a simple closed curve if it has one of the following forms: \n\\begin{enumerate} \n\\item $w=x ^{\\pm 1}$ or $w= y^{\\pm 1}$\n\\item $w = [x^{\\pm1},y^{\\pm1}]$\n\\item Up to replacing $x$ with $x^{-1}$, $y$ with $y^{-1}$ and interchanging $x$ and $y$, there is some $n \\in \\mathbb{Z}^+$ such that $w = x^{n_1}y x^{n_2} y ... x^{n_s}y$ where $n_i \\in \\{n, n+1\\}$.  \n\\end{enumerate}\nWe leave this as an exercise for the reader.  \nThis classification of words representing simple closed curves in $\\Sigma_B$ also follows from a much more general theorem in \\cite{BS}.\n\nBy construction, no word of type 1, 2 or 3 is finite order provided that $\\alpha^s \\beta^k \\neq 1$ for any integers $s$ and $k$ other than zero -- indeed, we only need to check words of type 3, and these necessarily have trace $\\alpha^s\\beta^k + \\alpha^{-s}\\beta^{-k}$ for some $s, k \\neq 0$.  Note that in particular, under the condition that $\\alpha^s \\beta^k \\neq 1$ for $s, k \\neq 0$, all type 3 words are hyperbolic.  We will use this fact again later on.  \n\nFor the remaining case where $w$ is a word with letters in both $A$ and $B$, we claim that it can be written in a form where Lemma \\ref{transcendental} applies.  To write it this way, use the following procedure: First take a simple representative $\\gamma$ for $w$ and apply an isotopy so that each crossing of $\\gamma$ with $c$ occurs in some small neighborhood of the basepoint $p$.  This gives us a well defined shortest path along $c$ to $p$ from each crossing.  After further isotopy, we may assume additionally that no segment of $\\gamma$ together with the shortest path along $c$ from its endpoints to $p$ bounds a disc, and that $\\gamma$ is transverse to $c$.  All this can be done without introducing any self-crossings in $\\gamma$.  Now $\\gamma$ is of the form $\\gamma_1 \\delta_1 \\gamma_2 \\delta_2 ... \\gamma_l \\delta_l$ where $\\gamma_i$ is a simple arc in $\\Sigma_A$ and $\\delta_i$ a simple arc in $\\Sigma_B$. Close each $\\gamma_i$ and $\\delta_i$ into a simple loop by connecting its endpoints to $p$ using the shortest path along $c$ and let $a_i \\in A$ (respectively $b_i \\in B$) be the corresponding element of the fundamental group.  See Figure \\ref{fig1}.  This gives us a word $a_1b_1a_2b_2 ... a_lb_l$ equivalent to $w$ after cyclic reduction, and each $a_i$ is represented by a simple closed curve in $\\Sigma_A$ and each $b_i$ by a simple closed curve in $\\Sigma_B$.  The elimination of discs bounded between segments of $\\gamma$ and short segments of $c$ ensures that each $a_i$ and $b_i$ is nontrivial.  \n\n  \\begin{figure*}\n  \\centerline{\n    \\mbox{\\includegraphics[width=3.3in]{simpleloop.pdf}}}\n \\caption{$a_1$ and $b_1$ in $w$, represented by $\\gamma_i$ and $\\delta_i$ joined to $p$}\n  \\label{fig1}\n  \\end{figure*}\n\nWe can also show that each $a_i$ either has a non-zero 2,1 entry or is represented by the curve $c$ or its inverse.  This is because $\\phi_A$ is Fuchsian, so the only elements fixing infinity -- that is, with 2,1 entry equal to zero -- are powers of $c$, and no powers of $c$ other than $c^{\\pm1}$ have a simple closed curve representative.  Similarly, the classification of words representing simple closed curves in $\\Sigma_B$ shows that each $b_i$ is either hyperbolic or represented by $c$ or $c^{-1}$.  We claim that we may now rewrite $w$ to eliminate all appearances of $c$, keeping each $a_i$ with a non-zero 2,1 entry and each $b_i$ hyperbolic.  After doing so, we will have $w$ in a form where we can apply Lemma \\ref{transcendental}.  \n\nTo rewrite $w$ in the desired form, first note that all $\\gamma_i$ such that $a_i$ is represented by $c$ may be homotoped (simultaneously) into $\\Sigma_b$ without introducing any self intersections of $\\gamma$.  Thus, we can replace each such $\\delta_{i-1} \\gamma_i \\delta_i$ with a simple loop $\\delta'_i$ in $\\Sigma_B$ alone, and rewrite $w = a_1b_1... a_{i-1} b'_{i} a_{i+1} ... a_lb_l$.  Reindex so that $w = a_1b_1a_2b_2 ... a_kb_k$ for $k < l$, and reindex the corresponding $\\delta_i$ and $\\gamma_i$ as well.  \nNow repeat the procedure on this new word with each $b_i$: homotope all $\\delta_i$ such that $b_i$ is represented by $c$ over to $\\Sigma_A$ without introducing any self intersections of $\\gamma$, and then replace each such $\\gamma_i \\delta_i \\gamma_{i-1}$ with a simple loop $\\gamma_i'$ in $\\Sigma_B$ alone. Then rewrite $w$ so that, after reindexing, $w = a_1b_1a_2b_2 ... a_mb_m$ with $m<k$ and each $a_i$ and $b_i$ is a simple closed curve.  Repeat the process again with the $a_i$ of this new word.  The procedure ends when either no $a_i$ or $b_i$ is represented by $c$, or when $w$ is a word in $A$ or $B$ alone, represented by a simple loop in $\\Sigma_A$ or $\\Sigma_B$.  In the first case, Lemma \\ref{transcendental} applies to show that $\\phi_t(w)$ is not finite order.  In the second case, we have already shown that a word in $A$ or $B$ represented by a simple loop in $\\Sigma_A$ or $\\Sigma_B$ cannot be finite order.  \n\nIt remains only to remark that the representation $\\phi_t$ is non-injective and that, by choosing appropriate parameters, we can produce uncountably many nonconjugate representations.   Non-injectivity follows immediately since $\\phi_t(B)$ is solvable so the restriction of $\\phi_t$ to $B$ is non-injective.  Now for any fixed $\\alpha$ and $\\beta$  (satisfying the requirement that $\\alpha^s \\beta^k \\neq 1$ for all integers $s, k$), varying $t$ among transcendentals over the entry field of $\\phi_0(A \\ast_C B)$ produces uncountably many non-conjugate representations that are all non-injective, but have no power of a simple closed curve in the kernel.  This concludes the proof of Theorem \\ref{main}, assuming that the surface was not the punctured torus.  \n\nThe punctured torus case is now immediate: any representation of the form of $\\phi_B$ where $\\alpha^s \\beta^k \\neq 1$ for any integers $s$ and $k$ is non-injective and our work above shows that no element represented by a simple closed curve has finite order.  Fixing $\\alpha$ and varying $\\beta$ produces uncountably many non-conjugate representations.  \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis is a simple template for authors to write new MNRAS papers.\nSee \\texttt{mnras\\_sample.tex} for a more complex example, and \\texttt{mnras\\_guide.tex}\nfor a full user guide.\n\nAll papers should start with an Introduction section, which sets the work\nin context, cites relevant earlier studies in the field by \\citet{Others2013},\nand describes the problem the authors aim to solve \\citep[e.g.][]{Author2012}.\n\n\\section{Methods, Observations, Simulations etc.}\n\nNormally the next section describes the techniques the authors used.\nIt is frequently split into subsections, such as Section~\\ref{sec:maths} below.\n\n\\subsection{Maths}\n\\label{sec:maths}\n\nSimple mathematics can be inserted into the flow of the text e.g. $2\\times3=6$\nor $v=220$\\,km\\,s$^{-1}$, but more complicated expressions should be entered\nas a numbered equation:\n\n\\begin{equation}\n    x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}.\n\t\\label{eq:quadratic}\n\\end{equation}\n\nRefer back to them as e.g. equation~(\\ref{eq:quadratic}).\n\n\\subsection{Figures and tables}\n\nFigures and tables should be placed at logical positions in the text. Don't\nworry about the exact layout, which will be handled by the publishers.\n\nFigures are referred to as e.g. Fig.~\\ref{fig:example_figure}, and tables as\ne.g. Table~\\ref{tab:example_table}.\n\n\\begin{figure}\n\n\n\n\t\\includegraphics[width=\\columnwidth]{example}\n    \\caption{This is an example figure. Captions appear below each figure.\n\tGive enough detail for the reader to understand what they're looking at,\n\tbut leave detailed discussion to the main body of the text.}\n    \\label{fig:example_figure}\n\\end{figure}\n\n\\begin{table}\n\t\\centering\n\t\\caption{This is an example table. Captions appear above each table.\n\tRemember to define the quantities, symbols and units used.}\n\t\\label{tab:example_table}\n\t\\begin{tabular}{lccr}\n\t\t\\hline\n\t\tA & B & C & D\\\\\n\t\t\\hline\n\t\t1 & 2 & 3 & 4\\\\\n\t\t2 & 4 & 6 & 8\\\\\n\t\t3 & 5 & 7 & 9\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\\section{Conclusions}\n\nThe last numbered section should briefly summarise what has been done, and describe\nthe final conclusions which the authors draw from their work.\n\n\\section*{Acknowledgements}\n\nThe Acknowledgements section is not numbered. Here you can thank helpful\ncolleagues, acknowledge funding agencies, telescopes and facilities used etc.\nTry to keep it short.\n\n\n\n\n\n\n\n\\section{Introduction}\\label{sec:intro}\n\nThe first gravitational wave (GW) detection of a binary neutron star (BNS) merger, GW170817, by the Advanced\nLIGO and Virgo (aLIGO\/Virgo) facilities, was an eagerly awaited event \\citep{abbott17a}. Such a merger was predicted to produce a multi-wavelength electromagnetic afterglow radiating from radio to gamma-rays, and GW170817 did not disappoint \\citep{abbott17b,andreoni17}. The most conspicuous predicted counterpart was the prompt ejection of collimated, short-lived gamma-ray emitting jets, similar to the observed \nshort gamma-ray burst \\citep[SGRB, gamma-ray durations $<2$\\,s;][]{narayan92} phenomenon, which is one of the two main classes of gamma-ray burst \\citep[GRB;][]{norris84,dezalay92,kouveliotou93} detected by the \\textit{Neil Gehrels Swift Observatory} \\citep[hereafter {\\em Swift};][]{gehrels04} Burst Alert Telescope (BAT) and the \\textit{Fermi Gamma-ray Space Telescope} (hereafter {\\em Fermi}{}) Gamma-ray Burst Monitor \\citep[GBM;][]{meegan09}. \nThe other dominant population known as long GRBs (LGRBs) typically have durations $>2$\\,s and are attributed to core collapse supernovae \\citep[e.g.][]{galama98,bloom98}.\nIt was therefore the near-simultaneous detection of GW170817 and GRB 170817A \\citep[the latter of which was a SGRB detected by {\\em Fermi}{};][]{abbott17b} and the late-time radio and X-ray follow-up confirming the presence of an off-axis jet \\citep[e.g.][]{mooley18b,ghirlanda19,troja19gw} that strongly supported the link between BNS mergers and SGRBs.\n\nThe detection of the electromagnetic counterpart to a aLIGO\\slash Virgo-detected BNS merger is of great importance as it enables the localisation of the source, along with providing complementary information such as an independent distance measurement, insight into the central engine, the energy released, and the final merger remnant. However, the initial localisation of a GW event by aLIGO\/Virgo is tens to hundreds of square degrees, making it difficult to search for counterparts. \nWe therefore introduce a method designed to exploit the established link between GW-detected BNS mergers and SGRBs by using the Australia Telescope Compact Array (ATCA) rapid-response mode to trigger on {\\em Swift}{}-detected SGRBs. \n\nWhile the radio emission from SGRBs is usually short-lived \\citep[$\\lesssim2$\\,days;][]{fong15}, the ATCA rapid-response mode is capable of being on-source within $10$\\,minutes. \nBy rapidly responding to {\\em Swift}{} SGRB triggers, ATCA can become a new diagnostic tool for\nuncovering the range of radio behaviour shown by SGRBs to help interpret what to look for from GW events that have off-axis gamma-ray jets. \nAs targeted observations can usually reach deeper sensitivities than wide-field surveys, ATCA observations can provide\na template of the radio brightness and timing properties of BNS mergers, which will \nin-turn inform the follow-up strategies of the next era of aLIGO\/Virgo GW events by wide-field radio telescopes, such as Australian instruments like the Murchison Widefield Array \\citep[MWA;][]{tingay13} and the Australian Square Kilometre Array Pathfinder \\citep[ASKAP;][]{johnston08}.\n\nThe jet launched during an SGRB is expected to produce a radio afterglow as predicted by the fireball model \\citep{cavallo78,rees92}. \nIn this model, the relativistic ejecta interact with the circumstellar medium (CSM) producing a forward shock that accelerates electrons and generates synchrotron emission. Reverse shock synchrotron emission, produced by the shock that propagates back into the post-shock ejecta, may also be observed  \ndepending on the density of the CSM and the ejecta. \nThe broadband spectrum produced by the jet interactions in the GRB afterglow is described by the peak flux and 3 characteristic frequencies ($\\nu_m$, the minimum electron energy frequency; $\\nu_{sa}$, the synchrotron self-absorption frequency; and $\\nu_c$, the electron cooling frequency), which evolve over time \\citep{sari98,wijers99,granot99}. Only early-time radio observations are able to properly constrain 2 of these 3 frequencies ($\\nu_m$ and $\\nu_{sa}$), and also disentangle the reverse and forward shock components. \nBy combining ATCA observations with \nmulti-wavelength observations to perform SED modelling, these parameters can be derived, thus providing information about the blast wave kinetic energy, the CSM density, the magnetic field energy density and the power law electron energy distribution \\citep{sari98,wijers99,granot99}. \nLimits on the linear polarisation of the reverse shock can also provide information on the jet magnetic field structure \\citep{granot14}.\nEarly-time radio observations of SGRBs are also sensitive to temporal steepening from the jet-break \\citep{sari99},\nwhich constrains the jet opening-angle used to calculate the true energy released \\citep[and therefore merger BNS\/GW event rates, e.g.][]{fong14,fong15,deugartepostigo14}. \nEven early-time non-detections in the radio band can allow us to make predictions about when the forward-shock emission may peak, which can inform the cadence and duration of follow-up radio observations, potentially optimising the success of a late-time detection as we demonstrate in this paper.  \nIn addition, sensitive, multi-frequency, high-cadence radio observations may allow us to distinguish between more exotic emission models caused by the ejection of neutron star material or the propagation of shocks caused by the merger event, which may produce non- to ultra-relativistic omnidirectional radio emission \\citep[e.g.][]{nakar11,kyutoku14}.\nIt is therefore crucial to obtain early-time radio observations (within minutes to days) of a larger sample of SGRBs to better characterise the timescales and frequencies necessary for understanding the range of behaviours we might expect from GW radio counterparts. \n\nThere are also several BNS merger models that suggest \na short-lived, supramassive and highly magnetised neutron star (NS) or ``magnetar'', supported by rotation, can exist for a short time ($<10^{4}$\\,s) before finally forming a stable magnetar or further collapsing into a black hole \\citep[BH, e.g.][]{usov92,zhang01,falcke14,zhang14}.  \nEvidence for such merger products comes from the detection of a ``plateau phase'' in some SGRB X-ray light curves between $\\sim10^{2}-10^{4}$\\,s post-burst, where this departure from power-law decay indicates ongoing energy injection \\citep{rowlinson13}. \nSuch merger remnant scenarios may be sources of prompt, coherent radio emission \\citep[see][for a review]{rowlinson19}. However, no continuous monitoring of the radio behaviour has yet been performed at GHz frequencies during the plateau phase. Such detections or upper limits could constrain different central engine models as has been done at late-times \\citep[e.g.][]{fong16}. \n\nOnly eight SGRBs have published detections in the radio band to date: GRB 050724A, 051221A, 130603B, 140903A, 141212A, 150424A, 160821B and 200522A\n\\citep{berger05,soderberg06,fong14,fong15,fong17,troja16,zhang17,troja19,lamb19,fong21}.\nNote that this does not include GW170817 as it had a far more off-axis outflow \nthan standard cosmological SGRBs so the corresponding radio afterglow was detected much later when the ejecta had moved into our line-of-sight \\citep{mooley18b}. \nOut of a sample of $>70$ radio-observed SGRBs, only $\\sim10$\\% have been detected in the radio band at GHz frequencies \\citep{fong21}.\nThis low detection rate may be due to an observed fast rise in radio emission with a potentially short radio afterglow lifetime. \nFor example, 7 of the 8 radio-detected SGRBs were detected within 1\\,day post-burst, at least half of which faded below detectability within $\\sim2$ days (see Figure~\\ref{fig:lc}). \nGiven these short timescales, it is possible the radio emission is frequently dominated by the reverse-shock \\citep[as was the case for GRB 051221A;][]{soderberg06} since simulations of BNS mergers demonstrate forward shock radio emission may evolve over days to weeks \\citep{hotokezaka16} as is also the case for many LGRBs \\citep[e.g.][]{vanderhorst08,vanderhorst14}. \nIf we instead compare the radio-detected sample to those SGRBs that were initially observed at radio wavelengths $<1$\\,day post-burst, this gives a much higher radio detection rate of $\\sim30$\\% \\citep{fong15}. \nHowever, while the first four radio-detected SGRBs showed initial flux densities of $>0.1$ mJy\/beam at GHz frequencies, few of the other $<1$\\,day post-burst pre-2016 observations had sufficient sensitivity to detect a predicted peak flux density of $\\sim40\\mu$\\,Jy\/beam at 10\\,GHz for an SGRB at an average redshfit of $z=0.5$ with an expected CSM density of $n_{0} \\sim0.1$\\,cm$^{-3}$ \\citep{berger14}. In fact, the four most recent radio-detected SGRBs peak at $\\lesssim40\\mu$\\,Jy\/beam.\n\nThe small sample of radio detected SGRBs therefore provides limited knowledge of their radio afterglow brightnesses and timescales, and is insufficient for deriving the energy outputs and environmental properties of the population through multi-wavelength modelling. It is therefore vital to perform both rapid and sensitive radio follow-up observations of SGRBs to capture these short-lived and faint events. The key to achieving this is through the use of rapid-response (also known as triggering) systems, where a telescope has the ability to automatically respond to a transient alert, and either repoint at the event or update its observing schedule to begin observations when the source has risen above the horizon. Rapid-response radio telescopes have been in use since the 1990's \\citep[for example see][]{green95,dessenne96,bannister12,palaniswamy14,kaplan15} \nbut predominantly at low radio frequencies (100\\,MHz to 2.3\\,GHz), with the majority of experiments being designed to search for prompt, coherent radio emission. However, until recently, the only high frequency ($>5$\\,GHz) rapid-response program designed to target incoherent (synchrotron) radio emission from GRBs has been run on the Arcminute Microkelvin Imager (AMI) Large Array (LA), known as ALARRM (the AMI-LA Rapid Response Mode), which has been active since 2012 \\citep[][]{staley13,anderson18}. It was only through ALARRM that it was possible to be on-source fast enough to detect the rise and peak in the reverse-shock radio emission at 15\\,GHz from GRB 130427A within 1\\,day post-burst, which also represents one of the earliest radio detections of a GRB to date \\citep{anderson14}. In addition, the radio catalogue of AMI observations of 139 GRBs (12 were short GRBs but non-detections), the majority of which were automatically triggered on using the rapid-response mode within $1$\\,day post-burst, was the first representative sample of GRB radio properties that was unbiased by multi-wavelength selection criteria \\citep{anderson18}. This work revealed that possibly up to $\\sim44-56$\\% of {\\em Swift}{}-detected LGRBs have a radio counterpart (down to $\\sim0.1-0.15$\\,mJy\/beam), with the increase in detection rate from previous studies \\citep[$\\sim30$\\%;][]{chandra12} likely being due to the AMI rapid-response mode, which allows observations to begin while the reverse-shock is contributing to the radio afterglow. This program has motivated the installation of a rapid-response mode on the ATCA. \n\nHere we present the first triggered observation of a SGRB using the new ATCA rapid-response mode. \nATCA is an ideal instrument for performing triggered radio follow-up of {\\em Swift}{} SGRBs due to its high sensitivity and broadband receivers that provide simultaneous multi-frequency coverage. The ATCA response times (which can be as short as minutes) \nhave the potential to be \nmuch faster than the current median SGRB response of the Karl G. Jansky Very Large Array (VLA; $\\sim24.7$\\,hrs), which rely on manually scheduling target-of-opportunity observations \\citep{fong15}.\nIn Section 2, we describe the ATCA rapid-response system from the observer interaction (front-end) level and the observatory (back-end) level.\nIn Section 3, we describe the triggered ATCA observation and data reduction of GRB 181123B, and corresponding results. \nThis is followed by a comparison of our radio limits for GRB 181123B to the sample of radio-detected SGRBs and a discussion of the parameter space that the triggered ATCA observations are probing in Section 4. \nFinally, we perform modelling of the GRB 181123B afterglow \nand thus demonstrate the usefulness of obtaining early-time (within 1 day) radio observations of an SGRB (regardless of whether or not there is a detection) to place constraints on the GRB physics.  \n\n\\section{ATCA rapid-response mode}\n\nATCA is a six element, 22\\,m dish, East-West interferometer based in New South Wales in Australia. Its maximum baseline length is 6\\,km and it is capable of observing in multiple, broad frequency bands  \nwith full polarisation, and in a variety of array configurations. \nATCA is currently equipped with the Compact Array Broadband Backend \\citep[CABB;][]{wilson11}, which has a 2\\,GHz bandwidth that is capable of observing in two frequency bands simultaneously with tunable receivers that operate between 1.1-105\\,GHz. \n\nSince 2017 April 18, ATCA has been capable of rapidly responding to transient alerts. \nThe rapid-response mode can trigger using the 16\\,cm, 4\\,cm and 15\\,mm receivers, corresponding to a usable frequency range of $1.1-25$\\,GHz, and can observe in any CABB mode. \nIn the following, we describe both the observer front-end and the observatory back-end of this new triggering system.\n\n\\subsection{VOEvent parsing\/front-end}\\label{sec:front-end}\n\nThe front-end software we use to interface with the ATCA rapid-response system ({\\sc vo\\_atca})\\footnote{https:\/\/github.com\/mebell\/vo\\_atca} is designed to trigger on Virtual Observatory Events (VOEvent; \\citealt{seaman11}), which are the standard format for broadcasting machine readable astronomical alerts related to transient events. \nA VOEvent package contains all the required data (in {\\sc xml} format) that allow automated decisions to be made in real-time given certain keywords and parameters. \nVOEvents are brokered via the 4 Pi Sky VOEvent Broker \\citep{staley16pp} and the {\\sc comet} VOEvent client \\citep{swinbank14}. \nThese packages allow us to listen to multiple VOEvent streams, including those broadcast by {\\em Swift}{}. \nWe use the {\\sc Python} package {\\sc vovent-parse} \\citep{staley_voevent-parse_2014} as the main tool to read the VOEvents \nand to extract the required information to be assessed by the triggering algorithm. \n\nUpon receiving a {\\em Swift}{} VOEvent, the ATCA VOEvent parser uses the keyword {\\sc grb\\_identified = true} to initially identify a GRB packet. \nPackets containing {\\sc startrack\\_lost\\_lock=true} are ignored as it means that {\\em Swift}{} has lost its celestial position-lock so such an alert is unlikely to be from a real transient.  \nWhile the observatory back-end prevents the telescope from overriding for sources that are too far north (see Section~\\ref{sec:back-end}), we impose an additional declination cut-off for all SGRBs north of +15$^{\\circ}$ to \nensure the potential for $>8$\\,hr integrations for the triggered observations. \n\nOn passing these stages, the parser then assesses the duration of the trigger so that SGRB candidates can be identified. However, on the short timescales following the alert, and with growing uncertainty as the GRB burst duration increases, it is difficult to classify {\\em Swift}{} GRBs as short or long in an automated way.\nA rigorous classification of the GRB requires human inspection of the data, which is only published online on the Gamma-ray Coordinates Network Circulars (GCN)  Archive,\\footnote{https:\/\/gcn.gsfc.nasa.gov\/gcn3\\_archive.html} usually between 10\\,mins and 1\\,hr post-burst and therefore not via a VOEvent. \nTo account for this, we implemented a three-tiered system to flexibly respond to different GRB durations and therefore filter for those events more likely to be SGRBs. \nThe keyword {\\sc integ\\_time} (the length of time for the transient signal to reach a significant threshold) parameter is used as an estimator of the incoming GRB's true duration. \n\n\\begin{itemize}\n\\item GRBs with {\\sc integ\\_time}$<$0.257\\,s have a high probability of being SGRBs so the VOEvent parser will automatically submit these triggers to the observatory and alert team members via text and email of the override observation. \n\n\\item With durations $0.257$\\,s\\,$<${\\sc integ\\_time}\\,$<1.025$\\,s, we have implemented a \"wait-to-proceed\" algorithm as the probability of the GRB being a SGRB decreases with increasing {\\sc integ\\_time}. In this case, we issue email and text alerts so that team members can check the GCN Archive for adequate verification of the GRB classification. If the GRB is confirmed to be short, then the duty team member responds \"YES\" to the detection email, and this email reply is read by an algorithm (via the Google email Application Programming Interface\\footnote{https:\/\/developers.google.com\/gmail\/api}) \nthat then proceeds with submitting the trigger to ATCA, resulting in an override observation. This provides an easy interface to assess and submit triggers via a mobile phone, which can receive SMS alerts and allow responding to emails away from a computer. \n\n\\item If {\\sc integ\\_time}$>$1.025\\,s then we presume that the GRB is long and we do not proceed with submitting a trigger to override the telescope. \n\\end{itemize}\n\nAfter the parser (or duty team member) has successfully identified the event as an SGRB, \nour algorithm then searches the ATCA calibrator database for a nearby and suitable phase calibrator.  \nIt then automatically builds a schedule file (we use the ATCA scheduler software {\\sc cabb-schedule-api})\\footnote{https:\/\/github.com\/ste616\/cabb-schedule-api} for a 12-hour observation of the GRB in the requested frequency band (for GRB triggering we currently use the 4\\,cm receiver), which has interleaved \nphase calibrator observations every 20 minutes. \nNote that the total exposure time is also limited by how far the GRB is above the horizon at the time of the trigger. \nThe schedule file and override request is then submitted to the observatory where it is assessed for submission to the observing queue by the ATCA back-end.\n\n\\subsection{Observatory back-end}\\label{sec:back-end}\n\nTime on the ATCA is scheduled into two 6-month long semesters, and the order of observations in each semester is set months in advance. This is done to allow the project investigators, who are also responsible for conducting the observations, to plan their activities. A rapid-response system is not easily compatible with this mode of operation.\n\nNevertheless, demand for the telescope to quickly respond to events has been steadily rising. In 2016, roughly 10\\% of telescope time was given to  \nNAPA (Non A-priori Assignable) or ToO (Target of Opportunity) projects, while in 2019 this figure had risen to 19\\%. \nFor a NAPA project, a science case is given to the time assignment committee (TAC), which ranks its significance against the other projects for that semester. Provided the science is considered compelling, these projects are allowed to displace time from other projects during the semester, with the philosophy being that were we to know during the scheduling process when an event would happen, a compelling project would have been scheduled to observe it.\n\nRapid-response NAPAs operate in the same way. A scientific justification must be supplied to the TAC, who must agree that rapid response is warranted. The observatory then supplies an authentication Javascript Web Token (JWT) to the project, and assists the investigators to test their automatic triggering system.\n\nA web service is provided so that the trigger to start observations can be sent from any internet-connected device. A Python library ({\\sc atca-rapid-resonse-api}) is also available to make it easier to send requests to this service.\\footnote{https:\/\/github.com\/ste616\/atca-rapid-response-api} All requests must contain a valid schedule file, and must nominate the target coordinates and a nearby phase calibrator.\n\nUpon receipt of a trigger, the web service tries to schedule the observation as soon as possible. If the source is above the horizon and the user-nominated minimum useful observing time can be obtained before the source sets, the current and subsequent observations can be displaced  \nand the system can start the observations within 2 seconds of the trigger's arrival. Within that time, emails are sent to the projects that will be displaced, and to the triggering team, describing the new order of the scheduling. The schedule is also altered as necessary to add a scan on a flux density calibrator at an opportune time, and potentially to shorten the observations to fit the available time. At all times, the emphasis is to move the telescope to the target coordinates as quickly as possible.\n\nThe service can also provide an immediate rejection should no suitable time be found for the observation. For example, if no available time can be found up to 100 hours in the future (generally because the request was made during a time when the array is shutdown for maintenance or participating in VLBI observing), the observations are rejected and the proposal team are notified. \nWhile no explicit limit is set for the source declination, sources too far north may not be available for the user-nominated minimum useful observing time, and will thus be rejected.\n\nIf the web service can schedule the observations, a separate service then takes over, and takes control of the observing control software. Some more checks are made to see if the array can be used for observing, and will delay the start of the observations if the weather conditions are unsuitable. This service also monitors the observations for interruptions due to weather, equipment failure and human intervention. Rudimentary responses are pre-programmed for any such eventuality. The service stops once the observations have finished, the target sets, or the observations are cancelled, whichever comes first. Control of the telescope then goes to the investigators whose project was scheduled to be running at this end time.\n\nA more complete guide to the operation of the rapid-response system is provided in the ATCA Users Guide.\\footnote{https:\/\/www.narrabri.atnf.csiro.au\/observing\/users\\_guide\/html\/atug.html}\n\n\\subsection{Triggering performance} \\label{sec:trig_per}\n\nSince the commencement of the program, we have worked with the observatory to improve the success of SGRB triggered observations with ATCA, which involved extensive system and software debugging. Many SGRBs were missed due to the telescope being in uninterruptible modes such as maintenance, reconfiguration, participating in VLBI or operating in an incompatible correlator mode (the latter has since been resolved). \n\nOur original override strategy involved triggering on all {\\em Swift}{} GRBs with {\\sc integ\\_time}$<1.025$\\,s as SGRBs have been detected with {\\sc integ\\_time} up to $1.024$\\,s. However, as mentioned in Section~\\ref{sec:front-end}, the majority of events within this timescale are LGRBs. \n{\\em Swift}{} data requires a human in the loop to classify the event as long or short, which is usually based on the duration and the hardness of the event (note that SGRBs often produce higher energy prompt emission than LGRBs) and are only published on the GCN Archive up to an hour post-burst \\citep[also note that the distinction between events with durations between $1-2$\\,s can be tenuous and has led to discussions regarding intermediate GRB classes; e.g.][]{mukherjee98,horvath98,huja09,deugartepostigo11}. \nThis original strategy therefore resulted in several false ATCA triggers, most of which were identified and cancelled before the telescope was overridden as there was additional lead time before the event in question had risen above the horizon. However, \nthere were a few instances where some data was collected on LGRBs. \nRecent edits to the VOEvent parsing of event timescales using the keyword {\\sc integ\\_time}, which are described in Section~\\ref{sec:front-end} have resulted in a significant reduction in ATCA triggers of LGRB contaminants. \n\nWhen ATCA receives a trigger of an event that is above the horizon, the main limitation to the response time is the telescope slew speed. On receiving the VOEvent via the parsing code, it takes $2-3$\\,s for the observation to be queued and the subsequent maximum observing time calculated. Following a {\\em Swift}{} alert on the long GRB 190519A \\citep{ukwatta19}, ATCA was on target and observing the event in 2\\,min and 39\\,s. Other response times range between $3-6$\\,min post-burst, which make the ATCA rapid-response system competitive with other triggering facilities such as AMI \\citep[e.g.][]{anderson18} yet is also based in the Southern Hemisphere, has more collecting area, a larger number of frequency bands, polarisation capabilities, and (in some configurations) better angular resolution.\n\n\\subsection{Short GRB experimental design}\n\nThe majority of GRBs detected by {\\em Swift}-BAT are LGRBs, with SGRBs (in this case events with $T_{90} \\leq 2$\\,s including those found in ground analysis) only accounting for $\\sim7-8$\\% \n(this is based on event numbers between 2017 and 2019 using the {\\em Swift}{} GRB Look-up Table,\\footnote{https:\/\/swift.gsfc.nasa.gov\/archive\/grb\\_table\/} where $T_{90}$ is the time between 5 and 95\\% of the fluence being emitted). We therefore expect $\\sim5-10$ SGRBs to be detected by {\\em Swift}{} per year, and therefore predict $\\lesssim2$ will be observable with ATCA (below a Declination cut-off of +15\\,deg) during an observing semester. \n\nOur rapid-response observations are performed using the $4$\\,cm receiver, which has dual 2\\,GHz windows that are usually centered at 5.5 and 9\\,GHz, which is the most sensitive ATCA band. This choice is based on several factors: the full-width half-maximum of the primary beam encompasses the initial {\\em Swift}{}-BAT positional uncertainty of newly detected GRBs \\citep[1-4 arcmin;][]{barthelmy05}, it is largely immune to atmospheric instabilities, and is less disrupted by RFI than other ATCA bands.\nIn addition, as synchrotron emission from GRB reverse and forward shocks peaks earlier and brighter with increasing frequency, the 4\\,cm band (5.5\/9 GHz) is optimal for ensuring the source will be bright but not peaking before the telescope is on-target.\n\nAs mentioned in Section~\\ref{sec:intro}, the radio afterglows from SGRBs are usually detected within 1\\,day post-burst \\citep[e.g.][]{fong15}, which strongly motivates our need for the ATCA rapid-response mode. The triggered observations are designed to  observe between $2-12$\\,h (depending on how long the source is above the horizon following the trigger). \nAs previous SGRB radio studies have shown that the radio afterglow has already switched-on within $4-16$\\,h post-burst \\citep[e.g.][]{anderson18}, a $\\leq12$\\,hr observation allows us to track the rapid rise in emission with a sensitivity of $\\sim60 \\, \\mu$Jy ($3\\sigma$) on one hour timescales.\\footnote{https:\/\/www.narrabri.atnf.csiro.au\/myatca\/interactive\\_senscalc.html} This means that any delays of $\\leq1$\\,hr related to waiting for the GRB classification does not affect the rapid-response science goal (see Section~\\ref{sec:front-end}). \nA $\\leq12$\\,hr track also ensures some periods of simultaneous {\\em Swift}{} X-ray Telescope \\citep[XRT, observing band between $0.3-10$\\,keV;][]{lien18} observations, \nwhich is essential for modelling the spectral energy distribution (SED), and for exploring the radio properties associated with the plateau phase (e.g. see our modelling in Section~\\ref{sec:mod}). \n\nFollowing the triggered, rapid-response observation, we also request three $\\sim4$\\,hr follow-up observations in the $4$\\,cm band to occur between $1-3$, $4-6$, and $8-12$\\,days post-burst, which can reach a sensitivity of $30\\,\\mu$Jy ($3\\sigma$). \nWhile 3 of the previous radio-detected SGRBs faded below detectability within 2 days post-burst, the other 2 were detected up to 10\\,days post-burst (see Figure~\\ref{fig:lc}), thus motivating this more long-term monitoring of any triggered candidate. \n\n\\begin{figure*}\n  \\begin{center}\n  \\includegraphics[width=0.49\\textwidth]{figures\/SGRBs_flux_lc_v3.pdf}\\label{fig:flux}\n  \\includegraphics[width=0.49\\textwidth]{figures\/SGRBs_lum_lc_v3.pdf}\\label{fig:lum}\n  \\caption{Radio light curves of SGRB radio detections ($1\\sigma$ error bars) and $3\\sigma$ upper-limits observed at frequencies between 6 and 10\\,GHz. \n  Left: Radio flux density vs days post-burst and Right: $k$-corrected spectral luminosity vs days post-burst in the rest-frame. The 9\\,GHz upper-limit of GRB 181123B is depicted as a large white triangle. For those GRBs without a known redshift we assume $z=0.5$. The $3\\sigma$ upper limits of those SGRBs that were observed but not detected in the radio band are depicted as grey triangles.  \n  References for all radio flux densities and redshifts for radio-detected SGRBs: \\citet{berger05}, \\citet{fox05}, \\citet{prochaska05}, \\citet{soderberg06}, \\citet{cucchiara13}, \\citet{deugartepostigo14}, \\citet{cucchiara14}, \\citet{chornock14}, \\citet{fong14}, \\citet{fong15}, \\citet{troja19}, \\citet{lamb19}, \\citet{paterson20}, \\citet{fong21}. All radio upper limits shown in grey were taken from \\citet[][see references therein]{fong15}.}\n  \\label{fig:lc}\n  \\end{center}\n\\end{figure*}\n\n\\section{ATCA observations of GRB 181123B}\n\n{\\em Swift}{}-BAT detected the short GRB 181123B at 05:33:03 UT (trigger=873186), which was rapidly detected in the X-rays by the {\\em Swift}{}-XRT  \nand localised to the position $\\alpha \\mathrm{(J2000.0)} =  12^{\\mathrm{h}}17^{\\mathrm{m}}28\\overset{\\mathrm{s}}{.}05$ and $\\delta (\\mathrm{J2000.0}) = +14^{\\circ}35'52\\overset{''}{.}4~$ with a 90\\% confidence of $1\\overset{''}{.}8$ \\citep{osborne18}. Further optical and near-infrared follow-up detected a source coincident with the {\\em Swift}{}-XRT position \\citep{fong18,paterson18a,paterson18b}, \nresulting in the identification of the host galaxy at redshift $z=1.754$ and the detection of the optical afterglow to GRB 181123B \\citep[$i=25.1$\\,mag at $9.1$\\,h post-burst;][]{paterson20}. This makes GRB 181123B one of only three SGRBs at $z>1.5$ \\citep{paterson20}. \n\nOn receiving the VOEvent trigger, ATCA was automatically scheduled to begin observations on 2018 Nov 23 at 18:07:24.9 UT (12.6\\,h post-burst) for 8.3\\,h  \\citep{anderson18gcn}, when the GRB had risen above the horizon (minimum elevation of 12\\,deg). On this date, ATCA was in the 6B array configuration, and the triggered observations were taken in the 4\\,cm band, with the dual 2\\,GHz bandwidth windows centered at 5.5 and 9\\,GHz. The observation pointing was at the initial BAT position, which was $1.2$\\,arcmin offset from the final {\\em Swift}{}-XRT position of GRB 181123B. \nNote that we requested no follow-up ATCA observations due to the imminent reconfiguration and correlator reprogramming, with many subsequent programmes having priority.\n\nThe ATCA rapid-response observation was reduced and analysed with the radio reduction software {\\sc MIRIAD} \\citep{sault95} using standard techniques. Flux and bandpass calibration were conducted using PKS 1934-638 and phase calibration with PKS 1222+216. Several rounds of phase and amplitude self calibration were also applied \\citep[this was possible due to the nearby bright field source FIRST~J121731.7+143953;][]{helfand15}. In order to obtain the most robust flux density upper limits at the position of the GRB, we used {\\sc mfclean} to create a clean model of the sources in the field (manually drawing clean boxes) and subtracted this model from the visibilities. A primary beam correction was then applied due to the 1.2\\,arcmin offset between the pointing centre and the best known GRB position from the {\\em Swift}{}-XRT. GRB 181123B was not detected, and the final $3\\sigma$ upper-limits can be found in Table~\\ref{tab:obs}. \n\nAs we know the precise location of GRB 181123B to within the ATCA beam, we also report the peak force-fitted flux density at both 5.5 and 9\\,GHz in Table~\\ref{tab:obs}.\nThese were calculated using the task {\\sc imfit} to force-fit a Gaussian to the beam that was fixed at the {\\em Swift}{}-XRT position of the GRB (errors are the $1\\sigma$ rms).\nThe advantage of quoting the force-fitted flux density over an upper-limit is that such a measurement also accounts for the presence of nearby sources, as well as variations in the noise across the image.\nThe data were also divided into 3\\,h and 1\\,h timescales and then re-imaged to search for evidence of emission that may have switched on nearer the end of the observation; however none was detected. \n\n\\begin{center}\n\\begin{table}\n\\caption{ATCA observations of GRB 181123B at 5.5 and 9 GHz, which began on 2018 Nov 23 at 18:07:24.9 UT (12.6\\,h post-burst) for 8.3\\,h.}\n\\label{tab:obs}\n\\begin{tabular}{lcc}\n\\\\\n\\hline\nFrequency & $3\\sigma$ Upper-limit & Forced-fit flux density\\\\\n(GHz) & ($\\mu$Jy\/beam) & ($\\mu$Jy\/beam)\\\\\n\\hline\n 5.5 & 34 & $7 \\pm 12$ \\\\\n 9.0 & 32 & $15 \\pm 11$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{center}\n\n\\section{Discussion}\n\nIn this section, we first demonstrate that our radio flux density limits for GRB 181123B are consistent and competitive with previous studies of the radio-detected SGRB population. This is followed by afterglow modelling to demonstrate the importance of obtaining early-time radio observations (regardless of whether there is a detection) to better constrain the properties of the blast wave.\n\nIn Figure~\\ref{fig:lc}, we show the light curves of SGRBs observed in the radio band between 6 and 10\\,GHz. The 8 radio-detected SGRBs are colour-coded with $3\\sigma$ upper-limits represented by triangles. The $3\\sigma$ upper limits of those SGRBs observed but not detected in the radio band have been plotted as grey triangles. The ATCA 9\\,GHz $3\\sigma$ upper-limit of GRB 181123B is shown as a large white triangle. In the left panel of Figure~\\ref{fig:lc}, we have plotted the observed radio flux density vs days post-burst, whereas in the right panel we have plotted the spectral luminosity vs days post-burst in the rest frame, assuming a redshift of $z=0.5$ \\citep{berger14} for those events with no known redshift.  \nWhen converting the flux ($F$) to luminosity ($L$), a $k$-correction was also applied such that $L=4 \\pi F d_{L}^{2} (1+z)^{\\alpha - \\beta -1}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$, where $d_{L}$ is the luminosity distance for the redshift $z$ \\citep[assuming $\\Lambda$CDM cosmology with $H_{0}=68$\\,km\\,s$^{-1}$\\,Mpc$^{-1}$ and $\\Omega_m=0.3$;][]{planck16}, and $\\alpha$ and $\\beta$ are the temporal and spectral indices defined as $F \\propto t^{\\alpha}\\,\\nu^{\\beta}$ \\citep{bloom01}. We assume $\\alpha=0$ and $\\beta=1\/3$, which are appropriate for an optically thin, post-jet-break light curve \\citep[see][]{chandra12}.\n\n\nFrom Figure~\\ref{fig:lc} we can see that the ATCA flux limit for GRB 181123B is extremely competitive and consistent with the most constraining lower-limits. \nUsing formalism by \\citet{granot02}, \\citet{berger14} showed that if we assume fiducial parameters for SGRBs, along with typical microphysical parameters for LGRBs, the expected peak flux density at a redshift of $z=0.5$ is $F_{\\nu}\\sim40\\mu$\\,Jy at $\\sim10$\\,GHz for an ambient medium density of $n_{0}=0.1$\\,cm$^{-3}$. \nOur $3\\sigma$ sensitivity at 9\\,GHz was 32$\\mu$\\,Jy, and therefore sensitive enough to detect emission from a GRB with the above properties, however, it is important to note that some GRB microphysical and macrophysical parameters like the kinetic energy and the CSM density can vary by several orders of magnitude \\citep{granot14}.\n\nThe luminosity light curves in Figure~\\ref{fig:lc} show the $3\\sigma$ upper-limit for GRB 181123B at $\\sim1$\\,day post-burst (in the rest frame). \nGiven the very high redshift of GRB 181123B, even these sensitive ATCA observations would not have detected the radio counterpart to the seven SGRBs detected at early times (within a day post-burst in the rest frame) if they were placed at $z=1.754$.\nWe therefore cannot draw any further comparisons between the physical properties of the radio-detected GRB sample and GRB 181123B based on luminosity alone and require more detailed multi-wavelength light curve modelling (see Section~\\ref{sec:mod}).\n\n\\subsection{Modelling constraints} \\label{sec:mod}\n\nIn this section, we model the afterglow of GRB 181123B in order to explore how early-time ($<1$\\,day) radio observations of SGRBs can help to constrain the dynamical and microphysical parameters of such blast waves in the context of the fireball model. \nUsing the redshift derived from the identification of the host galaxy of GRB 181123B \\citep[$z=1.754$;][]{paterson20}, \nwe model the force-fitted flux density values at the {\\em Swift}{}-XRT position of the GRB from our ATCA observations together with the {\\em Swift}-XRT light curve \\citep{evans09,evans10}. \nWe have chosen to use the force-fitted flux measurements plus errors in our modelling as it allows us to assign a likelihood to a predicted model flux for a set of model parameters, which is not possible with an upper limit \n\\citep[for some examples of where radio force-fitted flux measurements are quoted and used in afterglow modelling see][]{galamawijers98,kulkarni99,vanderhorst11,vanderhorst15}. \n\nFor this modelling, we have chosen to only consider the forward-shock component to minimise complexity, particularly as we are dealing with a small number of data points. \nAs previously mentioned, \nthe reverse-shock could be dominant at early times ($\\lesssim1$\\,day) in the radio band as has been observed for some SGRBs \\citep[e.g.][]{soderberg06,lamb19,troja19}. \nGiven that the reverse-shock evolves to lower frequencies more rapidly than the forward-shock and we have no radio detection, our modelling depends primarily on the X-ray detections, which are always dominated by the forward-shock, thus motivating our model choice. \nOur afterglow fitting also does not rule out a reverse shock contribution.  \nWe therefore assume a spherical, relativistic, blast wave interacting with the circumburst medium and generating synchrotron emission. Since SGRBs are known to occur in homogeneous, low density environments \\citep[median densities of $n_{0}\\approx (3-15) \\times 10^{-3}$\\,cm$^{-3}$ with $\\approx80-95\\%$ of events being situated in environments of $n_{0}<1$\\,cm$^{-3}$;][]{fong15}, \nwe assume a constant density circumburst medium. \n\nWe use the \\texttt{boxfit} code to model the afterglow emission \\citep{vanEerten2012}. \\texttt{boxfit} makes use of pre-calculated hydrodynamics data to calculate the dynamics of the blast wave, and solves radiative transfer equations on the go. \nSince in this work we assume a spherical blast wave, we fix the opening angle ($\\theta_0$) to $\\pi \/ 2$. \nWe then use the C++ implementation of the \\texttt{MultiNest} nested sampling algorithm, which is a Bayesian inference tool, to determine the posterior distributions of the free parameters \\citep{feroz09}.  The free parameters of our model are defined as:\n\n\\begin{itemize}\n  \\item $E_{K, \\mathrm{iso}}$: Isotropic equivalent kinetic energy in units of erg.\n  \\item $n_0$: Circumburst medium number density in units of $\\mathrm{cm}^{-3}$.\n  \\item $p$: Power-law index of the accelerated electron distribution, such that $N(\\gamma) \\propto \\gamma^{-p}$, with some minimum Lorentz factor $\\gamma_{m}$ \\citep{wijers99}.\n  \\item $\\epsilon_B$: Fraction of thermal energy in the magnetic fields.\n  \\item $\\epsilon_e$: Fraction of thermal energy in the electrons.\n\\end{itemize}\n\nIn order to demonstrate how the inclusion of early-time radio data helps to further constrain \nthe dynamical and microphysical parameters (when combined with {\\em Swift}{}-XRT observations and regardless of whether or not there is a radio detection), we model the afterglow of GRB 181123B with and without the ATCA force-fitted fluxes and compare the posterior distributions of the free parameters. \nIn both fits, we use the same prior for the free parameters (Table~\\ref{tab:prior}) and the resulting best fit values in Table~\\ref{tab:post_param} are set with the lowest chi-squared value in the posterior. \n\nLight curves for the posterior predictive distribution when the ATCA force-fitted flux density values are included in the modelling, together with the best fit, can be found in Figure \\ref{fig:lc_post}. \nGiven that the modelling of the X-ray detections of GRB 181123B alone suggests an energetic solution, the inclusion of radio information aids to pull down the overall fit so that at both 5.5 and 9\\,GHz, the best fit light curves are clustered around the ATCA force-fitted flux densities. \nWhile the resulting model is consistent with the {\\em Swift}{} Ultraviolet\/Optical Telescope \\citep[UVOT;][]{roming05} upper-limits \\citep{oates18}, it over-predicts the Galactic extinction corrected $i$-band flux reported by \\citep{paterson20} by a factor of $\\sim3$ or 1.2 magnitudes. \nAt this high redshift, an $i$-band detection indicates the afterglow emission was produced at ultraviolet wavelengths in the rest frame, and would therefore be quite prone to extinction by dust.\nGiven our model does not consider extinction, intrinsic or otherwise, this over-prediction may therefore not be unreasonable. \nHowever, our $i$-band prediction is much higher than the host optical extinction calculated by \\citet{paterson20} from photometric observations ($A_{V}$=0.23) or calculated from their observed excess hydrogen column density ($N_{H}$; derived from X-ray afterglow spectral modelling), which is known to scale to optical extinction \\citep{guver09}, predicting $A_{V}$=0.38.\nThere are also other potential sources of optical and infrared emission from SGRBs such as a kilonova from r-process radiative decay \\citep[e.g.][]{metzger10}, which our model does not include. However, such emission usually does not dominate over the afterglow until $>1$\\,d post-burst \\citep[e.g.][]{tanvir13}. \n\nAs can be seen in Table~\\ref{tab:post_param}, the inclusion of the ATCA force-fitted fluxes in our modelling allows for much better constraints to be placed on $\\epsilon_e$ (see also Figure~\\ref{fig:marge_comp}, which shows a comparison between the marginal distributions of the parameters for both cases - modelling with and without the ATCA data). \nThe rest of the parameters are consistent between both modelling experiments but the $E_{K,\\mathrm{iso}}$ is on the brighter end of known SGRBs \\citep{fong15}. \nOur findings are consistent with those by \\cite{beniamini17}, who have shown that the flux density and time of the GRB radio light curve peak can be used to particularly constrain $\\epsilon_e$. We also note that our constraint on $\\epsilon_e$ is also consistent (within the 95\\% credible interval) with the distribution of $\\epsilon_e$ ($0.13-0.15$) found through the analysis of 36 GRB radio afterglows performed by \\citet{beniamini17}.\nThe predicted radio peak also suggests that at later times ($\\gtrsim3-4$\\,days post-burst), the forward shock radio emission from GRB 181123B may have been detectable at 5.5 and 9\\,GHz with $\\geq4$\\,hr ATCA integrations (see Figure~\\ref{fig:lc_post}). \nTherefore, the inclusion of early-time radio data in GRB afterglow modelling (regardless of whether or not there is a detection), \ntogether with an X-ray light curve, allows us to predict the forward shock peak radio flux density, \nthus constraining the fraction of shock energy in the relativistic electrons  ($\\epsilon_e$). \n\\citet{paterson20} also derived these same afterglow parameters for GRB 181123B but assumed fixed values of $\\epsilon_e=0.1$ and $\\epsilon_B=0.1$ or 0.01. While our parameters are far less constrained, our values for $E_{K,iso}$ and $n_{0}$ (as well as $\\epsilon_e$ and $\\epsilon_B$)  are consistent with \\citet{paterson20} within the 95\\% credible intervals. However, our value range for $p$ was higher and did not overlap with the range derived by \\citet{paterson20}. Note that our value range for $p$ is more consistent with those calculated for radio-detected SGRBs (see Section~\\ref{sec:comp}).\n\n\\begin{table}\n    \\setlength{\\tabcolsep}{5pt}\n    \\def1.25{1.25}\n    \\caption{Assumed priors for the free parameters for all modelling efforts.}\n    \\begin{tabular}{l|r}\n    \\hline\n    \\multicolumn{1}{l}{Parameter range} & \\multicolumn{1}{r}{Prior distribution} \\\\\n    \\hline\n    $10^{49}<E_{K, \\mathrm{iso}}~\\mathrm{(erg)}<10^{54}$ & log-uniform\\\\\n    $10^{-4}<n_{0}~\\mathrm{(cm^{-3})}<10$ &log-uniform\\\\\n    $2.0<p<3.5$& uniform\\\\\n    $10^{-7}<\\epsilon_B<0.50$& log-uniform\\\\\n    $10^{-4}<\\epsilon_e<0.50$& log-uniform\\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:prior}\n\\end{table}\n\n\\begin{figure}\n  \\includegraphics[width=.45\\textwidth]{figures\/lc_post-z.pdf}\n  \\caption{Fit result for the afterglow light curves of GRB 181123B at ATCA observing frequencies 5.5\\,GHz (top panel) and 9.0\\,GHz (middle panel), and with the {\\em Swift}{}-XRT ($0.3-10$\\,keV; bottom panel) when the ATCA data are included (in this case the force-fitted flux densities; red data point and error bar plotted on the 5.5 and 9.0\\,GHz light curves). The plotted values in the {\\em Swift}{}-XRT light curve (also red data points) were downloaded via the {\\em Swift}{} Burst Analyser \\citep{evans10}. For each of the three frequency bands, 200 light curves are drawn by sampling the inferred posterior distribution of the parameters. The solid line represents the best fit model. The horizontal dashed lines show the 3-$\\sigma$ detection limit for various ATCA integration times.\n  }\n  \\label{fig:lc_post}\n\\end{figure}\n\n\\begin{table*}\n    \\setlength{\\tabcolsep}{5pt}\n    \\def1.25{1.25}\n    \\caption{Fit results for GRB 181123B for when the ATCA force-fitted data are excluded, included or the ATCA force-fitted mean is lowered by an order of magnitude. The errors on the parameters represent the 95\\% credible interval.\n    It can be seen that when the ATCA data are included, $\\epsilon_e$ is better constrained. \n    }\n    \\begin{tabular}{l|r|r|r}\n    \\hline\n    Parameter name &  \\multicolumn{1}{c}{ATCA data excluded} & \\multicolumn{1}{c}{ATCA data included} &  \\multicolumn{1}{c}{Lower ATCA forced fit} \\\\\n    \\hline\n    {\\boldmath$\\log_{10}E_{K, \\mathrm{iso}}$} & $52.4^{+1.4}_{-1.6}        $ & $52.0^{+1.5}_{-1.2}        $ & $51.5^{+1.1}_{-0.85}       $\\\\\n    \n    {\\boldmath$\\log_{10}n_{0}$} & $-0.4^{+1.4}_{-1.5}        $ & $-0.5^{+1.4}_{-1.4}        $ & $-1.1^{+1.4}_{-1.3}        $\\\\\n    \n    {\\boldmath$p$} & $2.92^{+0.42}_{-0.37}      $ & $2.90^{+0.42}_{-0.38}      $ & $2.97^{+0.38}_{-0.39}      $\\\\\n    \n    {\\boldmath$\\log_{10}\\epsilon_B$} & $-2.9^{+2.5}_{-3.2}        $ & $-3.0^{+2.7}_{-3.5}        $ & $-2.5^{+2.1}_{-2.3}        $\\\\\n    \n    {\\boldmath$\\log_{10}\\epsilon_e$} & $-1.13^{+0.82}_{-1.2}      $ & $-0.75^{+0.39}_{-0.40}     $ & $-0.60^{+0.30}_{-0.35}     $\\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:post_param}\n\\end{table*}\n\n\\subsubsection{Robustness of the results for more complicated models} \\label{sec:robust}\n\nThe aim of the modelling analysis presented in Section~\\ref{sec:mod} is to demonstrate how early-time radio data (even a non-detection) can help to constrain physical parameters in the framework of the fireball model.\nHowever, the presence of a plateau feature in many GRB X-ray light curves indicates that energy injection and more complex emission mechanisms are at play beyond a simple forward and reverse shock. \nOne of the main interpretations of X-ray plateaus observed from SGRBs \nis likely an energy injection signature from a potentially short-lived, supramassive, highly magnetised, rapidly rotating neutron star remnant, often referred to as a `magnetar' \\citep[e.g.][]{zhang01,yu07,rowlinson13}. In fact, \\citet{gompertz15} has performed broad-band modelling of SGRBs that includes energy injection from the spin-down of such a magnetar. Alternatively, \\cite{leventis14} were able to demonstrate that X-ray plateaus could be explained by the combined emission from the reverse and forward shock, provided that the blast wave is in the thick shell regime, and such a reverse shock would also lead to an additional emission component in radio and optical wavelengths.\nIn the case of GRB 181123B, the X-ray light curve of the afterglow shows evidence of a plateau phase at early times that \\citet{rowlinson20pp} have interpreted as energy injection from an unstable magnetar that collapsed a few hundred seconds following its formation. \nWhile many models have been proposed in the literature to describe the X-ray light curve behaviour of GRBs, our limited radio data-set means that an exhaustive analysis of these complex models is beyond the scope of this paper. \nNonetheless, it is worth exploring whether the constraints we have derived from our simple forward shock model are still meaningful if we introduce additional free parameters. \n\nIn the following we investigate how the inclusion of the ATCA data affects the posterior of modelling efforts that also include energy injection. We therefore \nincorporate energy injection into \\texttt{boxfit}, \nwhich is modelled by varying the isotropic equivalent kinetic energy ($E_{K, \\mathrm{iso}}$) as a power-law in time. In this case, $E_{K, \\mathrm{iso}}$ is described as:\n\n\\begin{equation}\n    E = \n    \\begin{cases} \n      E_0 & t \\leq t_{\\mathrm{inj}} \\\\\n      E_0 (t \/ t_{\\mathrm{inj}})^\\alpha &  t_{\\mathrm{inj}} < t \\leq t_{\\mathrm{inj}} + dt_{\\mathrm{inj}} \\\\\n      E_0 (1 + dt_{\\mathrm{inj}} \/ t_{\\mathrm{inj}})^\\alpha & t_{\\mathrm{inj}} + dt_{\\mathrm{inj}} < t \n   \\end{cases}\n\\end{equation}\n\n\\noindent where the three additional parameters are defined as:\n\n\\begin{itemize}\n  \\item $t_{\\mathrm{inj}}$: Start time of the energy injection in seconds (s).\n  \\item $dt_{\\mathrm{inj}}$: Duration of energy injection in seconds (s).\n  \\item $\\alpha$: Power-law index of the energy injection.\n\\end{itemize}\n\n\\noindent The assumed prior distributions for the energy injection parameters can be seen in Table~\\ref{tab:prior_ei}. We use the same priors as before for all other burst parameters (Table~\\ref{tab:prior}).\n\nThe resulting best-fit parameters for the afterglow modelling of GRB 181123B that includes energy injection for the cases when the ATCA force-fitted flux densities are included and excluded can be found in Table~\\ref{tab:post_param_ei}, with the resulting light curves for the posterior predictive distribution, together with the best fit model plotted in Figure \\ref{fig:lc_post_ei}. \nDue to the limited multi-wavelength coverage of the afterglow, we are not able to place tight constraints on the energy injection parameters. \nHowever, note that $\\epsilon_e$ continues to be well constrained when ATCA data are included, even with a more complex model with additional parameters (see Figure~\\ref{fig:marge_comp_ei}, which shows the marginal distributions of the parameters for the energy injection cases when ATCA data are included and excluded). \nIn addition, Figure~\\ref{fig:marge_comp_classic_ei} compares the obtained marginalized distribution for the GRB parameters (omitting the energy injection parameters) for the two modelling cases that include the ATCA data: with and without including energy injection. It can be seen that the resulting parameter values common between both fits are consistent within the 95\\% credible intervals despite different model complexities.\nHowever, while the model is consistent with the {\\em Swift}{}-UVOT upper-limits, it over-predicts the $i$-band detection \\citep{paterson20}, which demonstrates the limitations of our modelling.\n\nThe modelling of GRB 181123B shows that early-time radio observations, \nregardless of whether they are detections or non-detections,\nare able to constrain the fraction of thermal energy in the accelerated electrons, $\\epsilon_e$, beyond what is possible with just the {\\em Swift}-XRT X-ray light curve data. \nOur modelling also predicted that observations at later times (1-10 days) may have resulted in a detection of the forward shock, which would further constrain the GRB parameters.\n\n\\begin{table}\n    \\setlength{\\tabcolsep}{5pt}\n    \\def1.25{1.25}\n    \\caption{Assumed priors for the energy injection parameters for all modelling efforts.}\n    \\begin{tabular}{l|r}\n    \\hline\n    \\multicolumn{1}{l}{Parameter range} & \\multicolumn{1}{r}{Prior distribution} \\\\\n    \\hline\n    $8.64<t_{\\mathrm{inj}}~\\mathrm{(s)}<4.32 \\times 10^5$ & log-uniform\\\\\n    \n    $8.64<dt_{\\mathrm{inj}}~\\mathrm{(s)}<4.32 \\times 10^5$ & log-uniform\\\\\n    \n    $0.0<\\alpha<2.0$& uniform \\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:prior_ei}\n\\end{table}\n\n\\begin{table}\n    \\setlength{\\tabcolsep}{5pt}\n    \\def1.25{1.25}\n    \\caption{Fit results for GRB 181123B for when the ATCA force-fitted data are excluded and included, with \n    energy injection also included in our modelling. The errors on the parameters represent the 95\\% credible interval. It can be seen that when the ATCA data are included, $\\epsilon_e$ is better constrained. \n    }\n    \\begin{tabular}{l|r|r}\n    \\hline\n    Parameter name &  \\multicolumn{1}{c}{ATCA data excluded} & \\multicolumn{1}{c}{ATCA data included} \\\\\n    \\hline\n    {\\boldmath$\\log_{10}E_{K, \\mathrm{iso}}$} & $51.8^{+1.9}_{-1.3}        $   & $51.4^{+1.7}_{-0.91}       $\\\\\n    \n    {\\boldmath$\\log_{10}n_{0}$}    & $-0.7^{+1.6}_{-1.7}        $ & $-0.9^{+1.7}_{-1.8}        $ \\\\\n    \n    {\\boldmath$p              $} &$3.08^{+0.42}_{-0.47}      $ & $3.16^{+0.35}_{-0.50}      $\\\\\n    \n    {\\boldmath$\\log_{10}\\epsilon_{B}$} &$-1.9^{+1.6}_{-3.5}        $ & $-2.1^{+1.9}_{-4.0}        $\\\\\n    \n    {\\boldmath$\\log_{10}\\epsilon_{e}$} &$-1.01^{+0.70}_{-1.2}      $ & $-0.66^{+0.35}_{-0.39}     $\\\\\n    \n    {\\boldmath$\\log_{10}t_{\\mathrm{inj}}$ (days)} &$-1.23^{+1.7}_{-0.91}      $ & $-1.32^{+1.8}_{-0.84}      $\\\\\n    \n    {\\boldmath$\\log_{10}dt_{\\mathrm{inj}}$ (days)} & $-1.2^{+1.8}_{-2.3}        $ & $-1.1^{+1.8}_{-2.3}        $\\\\\n    \n    {\\boldmath$\\alpha         $} &\n    $0.86^{+0.94}_{-0.77}      $ & $0.88^{+0.92}_{-0.76}      $\\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:post_param_ei}\n\\end{table}\n\n\\begin{figure}\n  \\includegraphics[width=.45\\textwidth]{figures\/lc_post-ei-z.pdf}\n  \\caption{As for Figure~\\ref{fig:lc_post} but with energy injection incorporated into our modelling. \n  } \n  \\label{fig:lc_post_ei}\n\\end{figure}\n\nTo test how the dependencies between the parameters in our model are affected by \nthe mean of the forced fitted values, we ran an additional fit using \\texttt{boxfit} (without energy injection) where we lowered the mean of the ATCA force-fitted flux density values by an order of magnitude. Table~\\ref{tab:post_param} shows the resulting inferred parameters with the best fit light curves shown in Figure~\\ref{fig:lc_post_mean}. \nThe main effect of lowering the mean value of the force-fitted ATCA flux densities on the parameters was to decrease the mean value of the circumburst medium density ($n_0$) and increase the fraction of thermal energy in the  magnetic fields ($\\epsilon_B$) by a similar amount (factor of $\\sim3-4$) when compared to the results from fitting the original force-fitted ATCA flux densities.\nHowever, note that all parameters presented in Table~\\ref{tab:post_param} that were derived from including ATCA force-fitted flux densities in our  modelling still agree within the 95\\% credible intervals of the modelling performed without the ATCA information.\nThe reduction in the ATCA force-fitted mean flux densities also indicate that the predicted peak forward shock emission in the radio band would be delayed and also drop below the ATCA $3\\sigma$ detection limits (see Figure~\\ref{fig:lc_post_mean}) when compared to the fit performed with the measured force-fitted flux densities shown in Figure~\\ref{fig:lc_post}.\n\n\\begin{figure}\n  \\includegraphics[width=.45\\textwidth]{figures\/lc_post-lower-z.pdf}\n  \\caption{As for Figure~\\ref{fig:lc_post} but with the ATCA forced-fitted flux densities lowered by an order of magnitude. \n  }\n  \\label{fig:lc_post_mean}\n\\end{figure}\n\n\nOverall, our analysis demonstrates the importance of quoting force-fitted flux density values for radio transients over just reporting $3\\sigma$ upper-limits, which is traditional in most fields of astrophysics. If only upper-limits are reported then it is not always possible to incorporate this information into some modelling analyses (e.g. like our GRB afterglow modelling), which means we are throwing away important data that could further constrain the physics of an event or source. \nFinally, the inclusion of early-time radio force-fitted flux densities allows us to make predictions about the time and brightness of the forward shock peak in the radio band, which can inform late-time radio follow-up strategies. \n\n\\subsection{Comparisons of GRB 181123B to radio-detected SGRBs}\\label{sec:comp}\n\nWe now compare our parameter constraints on the micro- and macro-physical properties of GRB 181123B resulting from our modelling to those obtained for other radio-detected GRBs. For this comparison, we only focus on the parameters derived from our forward-shock modelling using \\texttt{boxfit} with the inclusion of the ATCA data (see Table~\\ref{tab:post_param}). \nSix of the eight radio-detected SGRBs have constraints on the same parameters (e.g. Table~\\ref{tab:prior}) through afterglow modelling \n\\citep[GRB 050724A, 051221A, 130603B, 140903A, 160821B, GRB 200522A;][]{berger05,soderberg06,fong14,pandey19,troja16,zhang17,troja19,lamb19,fong21}. \nThe advantage of these modelling efforts was that these events had extensive multi-wavelength data (radio, infrared, optical, X-ray) so the afterglow analysis \nled to stringent constraints on these parameters and in many cases, an estimate of \nthe gamma-ray jet opening angle, which has important implications for rate calculations. \nWhile the parameters we derived for GRB 181123B were far less constrained, our 95\\% credible intervals agree with those values derived for the six  SGRBs mentioned above, but the upper limits in our prediction for both $n_0$ and $p$ are much higher overall. \nIn fact, our derived accelerated electron distribution is steeper than that usually expected for GRBs \\textbf{($p=2.90^{+0.42}_{-0.38}$)} but not unreasonably so.\nOverall, as previously mentioned, the inclusion of ATCA force-fitted flux densities within 1 day post-burst have allowed us to place \nreasonable constraints on $\\epsilon_{e}$. \n\n\\section{Conclusions}\n\nIn this paper, we introduce the ATCA rapid-response observing mode by presenting results from the first successful SGRB trigger on GRB 181123B. This new mode of operations allows the telescope to automatically and rapidly respond to transient alerts broadcast via VOEvents, causing the active observing programme to be interrupted to allow for time-critical observations of transient phenomena. Successful triggers on LGRBs (see Section~\\ref{sec:trig_per}) have demonstrated that if the source is above the horizon, the ATCA can be on target and observing the event as fast as 3 minutes post-burst, allowing us to probe this early-time radio regime over a wide range of frequencies ($1.1-25$\\,GHz) with full polarisation, and in a variety of array configurations.\n\nThe ATCA rapid-response observations of GRB 181123B began 12.6\\,hr post-burst, as soon as the target had risen above the horizon, collecting 8.3\\,h of data at 5.5 and 9\\,GHz. While no radio emission was detected from GRB 181123B, we quote force-fitted flux densities, which enabled more constraining GRB afterglow modelling to be performed then would usually be possible with just X-ray data from {\\em Swift}{}-XRT. The addition of early-time radio data in the modelling allowed us to obtain more stringent constraints on the fraction of thermal energy in the electrons behind the shock wave ($\\epsilon_e$), which in turn allowed us to predict the peak in the forward shock radio afterglow emission around $\\sim10$\\,days post-burst.  \nThis modelling indicates that $\\gtrsim3-4$\\,days post-burst, the radio afterglow of GRB 181123B may have been detectable with a $\\geq4$\\,hr ATCA integration. \n\nOur results demonstrate the importance of including early-time radio observations in afterglow modelling efforts for constraining GRB blast wave properties and predicting the late-time peak in the radio forward shock, regardless of whether or not there is a detection, provided that force-fitted flux densities are quoted rather than upper-limits in the case of a non-detection. \n\nThis project also demonstrates the importance of implementing rapid-response observing systems on radio telescopes to probe a new parameter space in transient science. \nEarly-time radio observations of SGRBs can allow us to distinguish between different sources of early-time synchrotron afterglow emission \\citep[e.g.][]{kyutoku14} and even detect prompt and persistent coherent signals predicted to be produced during the compact merger and from the merger remnant \\citep[e.g.][]{rowlinsonanderson19}.\nDetections from rapid-response radio observations will provide crucial insight into the radio brightness and timescales we might expect from aLIGO\/Virgo-detected merging BNSs or NS-BH systems, which will aid in our search for electromagnetic counterparts in the large GW positional error regions.\nOther science cases include \nLGRBs, which may show bright, early-time radio emission from the reverse shock \\citep[e.g.][]{anderson14} and flare stars, which have shown simultaneous high-energy and radio flaring behaviour \\citep[e.g.][]{fender15}. \n\n\nFinally, our efforts running rapid-response programs on ATCA act as an excellent test for transient observing strategies with the SKA. \nIt is only through the utilisation of a rapid-response system that we can exploit the SKA to study early-time BNS and BH-NS merger physics down to sub-micro-Jansky levels. \n\n\\section*{acknowledgements}\nWe thank the referee for their careful reading of the manuscript and recommendations.\nGEA is the recipient of an Australian Research Council Discovery Early Career Researcher Award (project number DE180100346) and JCAM-J is the recipient of Australian Research Council Future Fellowship (project number FT140101082) funded by the Australian Government. \n\nThe Australia Telescope Compact Array (ATCA) is part of the Australia Telescope National Facility, which is funded by the Australian Government for operation as a National Facility managed by CSIRO. \nThis work made use of data supplied by the UK {\\it Swift} Science Data Centre at the University of Leicester and the {\\it Swift} satellite. {\\it Swift}, launched in November 2004, is a NASA mission in partnership with the Italian Space Agency and the UK Space Agency. {\\it Swift} is managed by NASA Goddard. Penn State University controls science and flight operations from the Mission Operations Center in University Park, Pennsylvania. Los Alamos National Laboratory provides gamma-ray imaging analysis.\n\nThe ATCA rapid-response front-end software makes use of {\\sc comet} \\citep{swinbank14} and\n{\\sc voevent-parse} \\citep{staley16pp} to process the incoming VOEvents.\nBoth the front-end ({\\sc vo\\_atca})\\footnote{https:\/\/github.com\/mebell\/vo\\_atca} and back-end ({\\sc atca-rapid-resonse-api})\\footnote{https:\/\/github.com\/ste616\/atca-rapid-response-api} software for ATCA rapid-response triggering rely on the {\\sc Astropy}, a community-developed core Python package for Astronomy \\citep{TheAstropyCollaboration2013,TheAstropyCollaboration2018}, {\\sc numpy} \\citep{vanderWalt_numpy_2011} and {\\sc scipy} \\citep{Jones_scipy_2001} python modules. This research also makes use of {\\sc matplotlib} \\citep{hunter07}. \nThis research has made use of NASA's Astrophysics Data System. \nThis research has made use of SAOImage DS9, developed by Smithsonian Astrophysical Observatory.\nThis research has made use of the VizieR catalogue access tool \\citep{ochsenbein00} and the SIMBAD database \\citep{wenger00}, operated at CDS, Strasbourg, France.\n\n\\section*{Data Availability Statement}\nThe unprocessed ATCA visibility datasets are public and can be accessed via the ATCA online archive (\\href{https:\/\/atoa.atnf.csiro.au\/}{https:\/\/atoa.atnf.csiro.au\/}) under project code C3204 and PI G Anderson.\nThe reduced images of these observations have been uploaded to Zenodo \\citep{anderson21zen}.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nDeep neural networks are a powerful way to extract information from noisy, raw sensor data that is collected in an unconstrained real-world environment. This approach has been used successfully in computer vision \\cite{howard2017mobilenets}, computer audition \\cite{hannun2014deep} and physiological monitoring \\cite{ravi2017deep}. Since deep neural networks are generally memory- and computationally-intensive, they are typically implemented on high-end cloud compute servers. However, transmitting raw data from the sensor to the cloud has negative implications for battery life \\cite{miettinen2010energy}, real-time responsiveness \\cite{dillon2010cloud} and data privacy \\cite{takabi2010security}. It also demands a reliable communications connection to the cloud which is not always possible. These challenges can be avoided by implementing the deep neural networks directly onto sensors.\n\nTwo approaches to realizing neural networks on sensors, or \"edge\" devices \\cite{teerapittayanon2017distributed} are (1) customized hardware processors like the Nvidia Drive Px2 and the A11 Bionic Chip \\cite{nvidia_drive,apple_2017}, and (2) light-weight software libraries like uTensor and CMSIS-NN \\cite{Tan2017,Lai2018}. Unfortunately, these approaches are inadequate for wearable applications like the recently-introduced cough-detection device (see Fig. \\ref{fig:devices})  \\cite{amoh2016deep,amoh2015deepcough,amoh2013technologies,amohmobile}. To promote long-term use, wearable devices have stringent size requirements that translate to a small battery (often the single largest component) and a correspondingly small power budget, on the order of 10 mW. Custom neural network processors like the Jetson Tx2 would drain such a battery in less than a minute. An ultra-low-power micro controller unit (MCU) \\cite{ti:msp430,stm:STM32L010C6,siliconlabs:C8051F96x,microchip:PIC16LF1509} could stretch the limited power budget further, but it only provides a limited amount of memory and computational resources; even light-weight libraries have some minimum hardware requirements that cannot be met by the sparse resources of an ultra-low-power MCU.\n\nTo address these challenges, we developed a novel recurrent neural network that is specifically adapted for implementation on the Arm M0+ \\cite{Armltd} class of ultra-low-power MCUs. In particular, we introduce a new architecture for the recurrent unit which is optimized for keyword spotting tasks like wearable cough detection. This paper presents details on the architecture, training scheme and hardware implementation of the proposed recurrent cell. We also present experimental results that show our architecture requires $12\\times$ less memory and $60\\times$ less computational time than comparable conventional recurrent units.\n\n\n\\begin{figure}[t]\n    \\begin{subfigure}[b]{0.32\\textwidth}\n        \\includegraphics[width=\\textwidth]{sensor2.png}\n        \\caption{Our wearable cough detection device.}\n        \\label{fig:ausculto}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}[b]{0.65\\textwidth}\n        {\\renewcommand{\\arraystretch}{1.5}\n        \\begin{tabular}{|l|r|r|r|r|}\n        \\hline\n        \\multicolumn{1}{|c|}{\\textbf{Device}} & \\multicolumn{1}{c|}{\\textbf{Size}} & \\multicolumn{1}{c|}{\\textbf{Power}} & \\multicolumn{1}{c|}{\\textbf{Computations}} & \\multicolumn{1}{c|}{\\textbf{RAM}} \\\\ \\hline\n        Drive Px2     & $\\sim$9.6$\\times$9.2\"  & 30,000 mW      & 20 TFLOPs             & 16 GB        \\\\\n        Jetson Tx2    & $\\sim$3.5$\\times$2.5\"  & 7,500 mW       & 2 TFLOPs              & 8 GB         \\\\\n        iPhone X      & 5.6$\\times$2.8\"  & 1,000 mW       & 1 GFLOPs              & 3 GB         \\\\\n        Freedom K64F  & 3.2$\\times$2.1\"  & 500 mW         & 1 MFLOPs              & 256 kB       \\\\\n        \\textbf{Ausculto (ours)}   & \\textbf{1.1$\\times$0.7\"}  & \\textbf{40 mW}          & \\textbf{0.1 MFLOPs}            & \\textbf{32 kB}        \\\\ \\hline\n        \\end{tabular}}\n        \\vspace{10pt}\n        \\caption{Table of computational resources available on various portable systems.}\n        \\label{fig:resources}\n    \\end{subfigure}\n    \\caption{\\ref{fig:ausculto} Our wearable device for detecting coughs and adventitious sounds is about the size of a thumb drive. It features the Atmel ATSAMD21E18 \\cite{ATSAMD21}, an ultra-low power Arm Cortex M0+ micro-controller. Embedding a neural network to this device is difficult because of its size and power constraints. \\ref{fig:resources} Compared to our device, portable systems that run neural networks like Nvidia's Drive and the iPhone X are larger and have enormous power requirements. Even devices like Freedom K64F recommended for uTensor light-weight library still need $10\\times$ more power than available on our ultra-low power system.}\n    \\label{fig:devices}\n\\end{figure}\n\n\n\\section{Related Work}\n\\label{sec:related}\nNeural network optimizations similar to those proposed in this work have previously been considered in isolation by other researchers. For instance, recent efforts to optimize the GRU architecture include the removal of one of its gates. \\citet{Zhou2016} achieved this single gate architecture by coupling the update and reset gates into a single forget gate. The resulting `minimal' gated unit (MGU) directly effects both cell update and reset using the same gate. \\citet{Ravanelli2017} discarded the reset gate altogether, highlighting that it is potentially redundant in speech recognition where input signals evolve slowly. Our work investigates this further in AED tasks where events are sudden and infrequent, in sharp contrast to speech recognition. Furthermore, our proposed cell architecture replaces the sigmoid and hyperbolic tangent activation functions with more efficient softsign variants.\n\nAnother common area of optimization is weight quantization. In previous works \\cite{Han2015,Wu2016}, 8-bit weight quantization was shown to drastically reduce network size without significant loss in accuracy. However, these approaches focused solely on convolutional and fully-connected networks. They also required the storage of a codebook for decoding weights at run-time. With regards to recurrent networks, \\cite{Ott2016} demonstrated that 2-bit weight quantization is possible in GRU, with only a slight loss in accuracy. However, the combined effect of such a low precision quantization and other optimizations like a single gate architecture remained unknown.  \n\nUsing solely integer arithmetic in neural networks has also been studied in prior works \\cite{Han2015,Gupta2015,Chen2015}. \\citet{Courbariaux2014} found that in a fully connected network, a dynamic point numeric format yields much better results than 20-bit fixed point. Another work demonstrates that 32-bit fixed point formats are effective in convolutional networks \\cite{Chen2015}. Once again, all these efforts consider fully-connected and convolutional architectures. Since recurrent architectures are fundamentally different and much more difficult to train than other architectures \\cite{pascanu2013difficulty}, it is worthwhile to investigate fixed point arithmetic in recurrent neural networks.\n\nTo date, no one has ever simultaneously applied all of these modifications to a GRU and validated that they yield a usable network that is implementable on a low power, low resource MCU like the Arm Cortex M0+. This work bridges that gap.\n\n\n\\section{Embedded Gated Recurrent Unit}\n\\label{sec:eGRU}\nPrevious work showed that a network of gated recurrent units (GRU) \\cite{Cho2014} outperforms classical approaches like Hidden Markov Models on an acoustic event detection task when evaluated in a noisy, non-ideal environment \\cite{amoh2016deep}. But the GRU network described in \\cite{amoh2016deep} is too memory- and computationally-expensive to be implemented on a low-resource MCU like the Arm M0+.\n\nTo meet the resource constraints of a low power MCU like the M0+, we propose a new recurrent architecture: the embedded Gated Recurrent Unit (eGRU). It is based on the traditional GRU, but with four major modifications: (1) a single gate mechanism, (2) faster activation functions, (3) 3-bit exponential weight quantization and (4) fixed point arithmetic in all network operations. These modifications lead to massive reductions in memory and computations in the recurrent cell, making it feasible to run eGRU networks on our target device. Below, we discuss the features of the eGRU in detail.\n\n\n\\subsection{Single Gate Mechanism}\nA practical idea for optimizing recurrent units is the removal of gates. Modern recurrent cell architectures like the Long Short-Term Memory (LSTM) unit \\cite{Hochreiter1997} and GRU are characterized by gating mechanisms that regulate information flow in and out of the cell's memory. For instance, the LSTM cell has 3 gates: two for controlling the cell's input and output, and a third for forgetting or resetting the cell's internal state. Since gates are implemented using weights and activations, omitting a gate reduces the required memory and computations in a recurrent cell. Accordingly, GRU was introduced as an optimized form of LSTM by reducing the number of gates from 3 to 2: the update and reset gates. \n\nIn the same vein, GRU can also be optimized further by discarding yet another gate. \\citet{Zhou2016} accomplished this very `minimal' gated unit by using a single gate for both resetting and updating the cell's internal state. \\citet{Ravanelli2017} extended that work further by highlighting a redundancy between the two gates. They deduced that in applications like speech recognition where signals change slowly, reset gates are unnecessary and can be omitted altogether. However, in applications where events of interest are abrupt and isolated (eg. detecting cough sounds), the assumption by \\citet{Ravanelli2017} that state resets are irrelevant does not hold. In fact, we found that without state resets, recurrent units in our application are unable to recover from large impulse signals. Thus, for keyword spotting applications, we can eliminate the reset gate and rely on only the update gate.\n\n\\begin{figure}[t]\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\includegraphics[width=\\textwidth]{grucell.pdf}\n        \\caption{GRU cell.}\n        \\label{fig:grucell}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\includegraphics[width=\\textwidth]{egrucell.pdf}\n        \\caption{eGRU cell.}\n        \\label{fig:egrucell}\n    \\end{subfigure}\n    \\caption{ Circuit diagrams illustrating GRU (\\ref{fig:grucell}) and eGRU (\\ref{fig:egrucell}) cell architectures. Compared to GRU, eGRU omits the reset gate $r$ and does not require the weights, $w_r$. Additionally, it replaces sigmoid and $\\tanh$ activation functions with softsign variants.}\n    \\label{fig:cellcircuit}\n\\end{figure}  \n\n\n\\subsection{Activation Functions}\nThe original GRU cell utilized two kinds of activations; sigmoid functions ($\\sigma$) in the update and reset gate equations, and the hyperbolic tangent function ($\\tanh$) in the state update equation. However, since the M0+ lacks a dedicated floating point unit and DSP instruction set, executing either sigmoid or $\\tanh$ functions are quite slow (see Table \\ref{tab:activations}). For a more efficient recurrent unit, it was suggested in \\cite{Ravanelli2017} that the $\\tanh$ be replaced with a rectified linear unit (ReLU). Unfortunately, when combined with heavily quantized weights, recurrent cells with ReLU activations are too lossy to learn well. And since quantization is an essential part of our proposed eGRU architecture, ReLU activations are not an option.\n\nA desirable alternative to the activation functions above is the softsign function. In \\cite{glorot2010understanding}, softsign was shown to perform comparably with $\\tanh$ and sigmoid in feed-forward networks. Although not as cheap (computationally) as ReLU, softsign is much cheaper than sigmoid or $\\tanh$. A floating point implementation of the softsign function is more than 10x faster than either sigmoid or $\\tanh$ functions on the M0+ (Table \\ref{tab:activations}). Furthermore, the simplicity of the softsign function permits an even faster fixed point implementation. For these reasons, we adopt the softsign and a shifted version of it as activation functions for our proposed eGRU recurrent architecture.\n\n\n\\begin{table}\n  \\begin{subfigure}[b]{0.55\\textwidth}\n    \\centering\n    \\begin{tabular}{|c|c|}\n    \\hline\n    \\textbf{GRU} & \\textbf{eGRU} (this work)\\\\\n    \\hline\n    $\\begin{aligned}[t]\n        z_t &= \\sigma(W_z \\odot [h_{t-1},x_t]) \\\\\n        r_t &= \\sigma(W_r \\odot [h_{t-1},x_t]) \\\\\n        \\tilde{h_t} &= \\tanh(W_h \\odot [ r_t * h_{t-1}, x_t]) \\\\\n        h_t &= (1-z_t)*h_{t-1} + z_t*\\tilde{h_t}\n    \\end{aligned}$ & \n    $\\begin{aligned}[t]\n        z_t &= (\\varsigma(W_z \\odot [h_{t-1},x_t])+1)\/ 2 \\\\ \\\\\n        \\tilde{h_t} &= \\varsigma(W_h \\odot [ h_{t-1}, x_t]) \\\\\n        h_t &= (1-z_t)*h_{t-1} + z_t*\\tilde{h_t}\n    \\end{aligned}$\\\\\n    \\hline\n    \\end{tabular}\n    \\caption{State equations of traditional GRU and proposed eGRU cells.}\n    \\label{tab:equations}\n  \\end{subfigure}\\hfill\n  \\begin{subfigure}[b]{0.37\\textwidth}\n    {\\renewcommand{\\arraystretch}{1.18}\n    \\centering\n    \\begin{tabular}{|l|l|r|}\n    \\hline\n    \\multicolumn{1}{|c|}{Function} & \\multicolumn{1}{c|}{Definition} & \\multicolumn{1}{c|}{Latency} \\\\ \\hline\n    TanH                           & $f(x) = \\tanh(x)$               & 328 us    \\\\\n    Sigmoid                        & $f(x) = \\frac{1}{1+e^{-x}}$     & 230 us    \\\\\n    ReLU                           & $f(x) = \\max(0,x)$              &  11 us    \\\\\n    Softsign                       & $f(x) = \\frac{x}{1 + |x|}$      &  24 us    \\\\ \\hline\n    \\end{tabular}}\n    \\caption{Cost of activation functions on Arm M0+.}\n    \\label{tab:activations}\n  \\end{subfigure}\n\n  \\caption{\\ref{tab:equations} Equations for GRU and eGRU highlights architectural differences between the two cells. In a single cell, gate weights $W_r$, $W_z$ are vectors of size $N+1$, where $N$ is the dimension of the input $x$. eGRU omits the reset gate ($r_t$) and uses softsign ($\\varsigma$) activations instead of the $\\tanh$ and sigmoid ($\\sigma$) functions. \\ref{tab:activations} Latency of standard activation functions executed on the M0+ MCU. Sigmoid and $\\tanh$ functions are $10-30\\times$ slower than ReLU and softsign functions. However, softsign is ideal since ReLU does not work well with extremely quantized weights.} \n  \\vspace{-10pt}\n  \\label{tab:cellarchitecture}\n\\end{table}\n\n\n\\subsection{Weight Quantization}\nEfforts to reduce neural network memory footprint often involve weight quantization. As networks are purely defined by learned parameters or weights, a reduction in memory for each weight through quantization results in tremendous shrinkage in the overall network size. Several studies have shown that neural networks are still effective even after weights are quantized to only 8-bits, leading to $20 - 49\\times$ memory reduction \\cite{Han2015,Wu2016}. Specifically in GRU architectures, ternarization (2-bit weights) has been proposed as feasible for recurrent neural networks at a small cost in performance \\cite{Ott2016}.\n\nHowever, we discovered that 2-bit quantization does not work well for our application. When combined with the single gate and activation function optimizations, ternarized weights in eGRU lead to poor performance. On the other hand, 3-bit quantization with septenary weights (7 levels) proved to be quite effective. Thus, 3-bit quantization was adopted for eGRU.\n\nBesides the reduction in bits, our quantization scheme also ensures that quantized levels are negative integer exponents of two, similar to an approach in \\cite{Ott2016}. This exponential quantization enables the replacement of weight multiplications with bit shifting, which in turn drastically reduces the computation time of an eGRU cell. The septenary weights used are: $[0,\\pm0.25,\\pm0.5,\\pm1]$ and they are encoded using the mapping in Table \\ref{tab:wtencoding} such that no external look-up tables are required at run-time. To multiply an input by a quantized weight, a simple, fast procedure featuring only bitwise operations is employed (Algorithm \\ref{alg:qwtransform}).\n\n\\begin{table}\n    \\begin{minipage}{0.45\\textwidth}\n        \\centering\n        \\begin{tabular}{|c|c|c|}\n        \\hline\n        \\textbf{Weight} & \\textbf{Binary} & \\textbf{Decimal} \\\\ \\hline\n        \\textbf{+1.00}  & 000             & 0                \\\\\n        \\textbf{+0.50}  & 001             & 1                \\\\\n        \\textbf{+0.25}  & 010             & 2                \\\\\n        \\textbf{0}      & 111             & 7                \\\\\n        \\textbf{-0.25}  & 110             & 6                \\\\\n        \\textbf{-0.50}  & 101             & 5                \\\\\n        \\textbf{-1.0}   & 100             & 4                \\\\ \\hline\n        \\end{tabular}\n        \\caption{Encoding of quantized weights. Encoding scheme is chosen for fast computation of weight transformations using only bitwise operations in Algorithm \\ref{alg:qwtransform}. No additional look-up table is required at run-time.}\n        \\label{tab:wtencoding}\n    \\end{minipage} \n    \\hfill\n    \\begin{minipage}{0.5\\textwidth}\n    \\vspace{-25pt}\n        \\begin{algorithm}[H]\n        \\caption{Transformation by Quantized Weight}\n        \\label{alg:qwtransform}\n        \\KwIn{$x$, value to be scaled and, $w$, quantized weight.}\n        \\KwOut{Result of weight transformation, $y$.}\n        $y \\gets 0$\\;\n        \\eIf {$w = 7$}{\n           \\Return $y$\\;\n           }{\n           $y\\gets x >>$ last 2 bits of $w$\\;\n           \\lIf{bit $3$ of $w = 1$ }{$y\\gets - y$}\n           }\n        \\Return $y$\n        \\end{algorithm}\n    \\end{minipage}\n\\end{table}\n\n\n\\subsection{Fixed Point Arithmetic}\nThe final area of optimization in eGRU is the numeric format used for all math operations in the neural network. On better equipped processing units, single or double precision floating point formats are typically used. Unfortunately, since M0+ lacks an FPU, floating point operations are very costly (see Table \\ref{tab:operations}). Hence, using solely integer operations in eGRU is desirable. \n\nConsidering the M0+ has a 32-bit architecture, optimal execution of operations is achieved when operands are contained within 32-bits. Hence, we adopt the Q15 16-bit fixed point format for all arithmetic within eGRU. In Q15 format, 16-bit signed integers from 32,767 to 32,768 are used to represent decimals in the range of [-1,1) at intervals of $2^{-15}$. Necessary precautions ought to be taken during basic operations to prevent overflow and ensure closure under Q15. For instance, to divide two Q15 numbers, it is necessary to left-shift the dividend by 16 bits (yielding a 32 bit value) before undertaking the division to avoid losing precision. Hence, all eGRU operations (including activation functions) need to be translated to Q15 versions.\n\nWeight multiplication is the most frequent operation in a neural network. Fortunately, by virtue of our exponential quantization, affine transformations for all layers in the network can be implemented by right-shift operations, which remains the same in Q15 format. The summation of all transformed inputs can exceed 16 bits and is thus accumulated in a 32 bit register. However, since the activation function is bounded by [-1,1), the output of the recurrent node remains a 16-bit Q15 number which can then be fed into yet another node. From simulations, we discovered that all inputs to an eGRU network will flow through the entire model in Q15 format and result in an output that is precise to at least 2 decimal places compared to those from an equivalent full precision (floating point) network.\n\nAn interesting modification worth mention pertains to the Q15 implementation of the softsign activation function. Inputs to activations, in the scheme described above, is a 32-bit accumulation of all transformed inputs. Since such inputs are already in 32-bit, undertaking a Q15 division would be impractical as it would require left-shifting, resulting in an overflow. One way to circumvent this is to clip the accumulated value to a certain domain to prevent overflow. We found that clipping to the domain (64,-64] resulted in a fast and accurate integer softsign approximation (see Table \\ref{tab:operations}). A simple C++ definition of our softsign function is provided in Listing \\ref{lst:softsign}. \n\n\n\\begin{table}\n    \\begin{minipage}[b]{0.50\\textwidth}\n    \\centering\n        \\begin{tabular}{|c|c|c|}\n        \\hline\n        \\textbf{Operation} & \\textbf{int32 (ns)} & \\textbf{float32 (ns)} \\\\ \\hline\n        $\\pm$              & 252                 & 4,853                 \\\\\n        $<<,>>$            & 273                 & -                     \\\\\n        $\\times$           & 253                 & 6,578                 \\\\\n        $\\div$             & 1,923               & 12,942                \\\\ \\hline\n        softsign           & 4,740               & 24,910                \\\\ \\hline\n        \\end{tabular}\n        \\caption{Comparison of time cost of integer vs floating point arithmetic on M0+. For basic maths, integer operations are beyond $10\\times$ faster than floating point ones. Also, our fixed point approximation of the softsign is $5\\times$ faster than the floating point implementation.}\n        \\label{tab:operations}\n    \\end{minipage}\n    \\hspace{10pt}\n   \n    \\begin{minipage}[b]{0.40\\textwidth}\n    \\vspace{-30pt}\n    \\centering\n        \\begin{lstlisting}[language=C++]\n  int16 softsign(int32 x) {\n      int32 a = x << 10;\n      int32 b = (abs(x) + 32768) << 5; \n      int16 y = a \/ b;        \n      return y;\n  }\n      \\end{lstlisting}\n   \n    \\captionof{lstlisting}{C++ routine for softsign function in Q15. Only integer operations are used. With this routine, softsign function is correctly defined for domain of (64.0,-64.0].}\n    \\label{lst:softsign}\n    \\end{minipage}\n    \\vspace{-15pt}\n\\end{table}\n\n\n\\subsection{Summary of Optimizations}\nA summary of the proposed optimizations and their corresponding reductions in computational resource requirements is provided in Table \\ref{tab:optsummary}. Table \\ref{tab:cellrequirements} presents a breakdown of the resource requirements for an eGRU cell is compared with that for a traditional GRU cell. The overall number of operations are fewer for eGRU (12) than for GRU (17). Furthermore, eGRU trades off most multiplications for cheaper and faster bit shift operations. Also, note that all eGRU operations are integer operations (INOPs) compared to the floating point operations (FLOPs) in GRU. \n\nMemory and latency reduction scale factors for each optimization is also listed in Table \\ref{tab:cellrequirements}. Quantization mostly accounts for the miniaturization, reducing the cell size by a factor of 10. On the other hand, using solely Q15 integer operations rather than floating point operations drastically speeds up cell execution and yields a 20$\\times$ reduction in latency. Analytically, softsign activations also account for significant reduction in latency (10$\\times$), even without accounting for the additional gains in speed afforded by the faster integer softsign approximation in Listing \\ref{lst:softsign}. \n\n\n\n\\begin{table}\n\\begin{subfigure}[t]{0.36\\textwidth}\n    \\centering\n    \\begin{tabular}{|l|c|c|c|}\n    \\hline\n    \\textbf{Requirement} & \\textbf{eGRU} & \\textbf{GRU} \\\\ \\hline\n    \\# of Parameters      & 6                 & 9         \\\\    \n    Additions           & 6                 & 8         \\\\\n    Multiplications     & 2                 & 9         \\\\\n    Bit Shifts          & 4                 & -         \\\\ \\hline\n    \\end{tabular}\n    \\caption{Comparison of cell requirements.}\n    \\label{tab:cellrequirements}\n\\end{subfigure}\\hspace{1pt}\n\\begin{subfigure}[t]{0.62\\textwidth}\n    \\centering\n    \\begin{tabular}{|l|c|c|c|}\n    \\hline\n    \\textbf{Optimization} & \\textbf{Memory} & \\textbf{Latency} & \\textbf{Modification}                     \\\\ \\hline\n    Single Gate Unit      & 1.5$\\times$     & 1.5$\\times$             & $\\,\\,$3 gates $\\Rightarrow$ 2 gates              \\\\\n    Quantization          & $\\,$10$\\times$      & -                & 32 bits $\\Rightarrow$  3 bits              \\\\\n    Softsign Activations  & -               & $\\,$10$\\times$              & $\\sigma$ \\& $\\tanh$ $\\Rightarrow$ softsign \\\\\n    Q15 Arithmetic        & -               & $\\,$20$\\times$              & $\\,\\,$FLOPs $\\Rightarrow$ INOPs                  \\\\ \\hline\n    \\end{tabular}\n    \\caption{Reduction scale factors for individual optimizations.}\n    \\label{tab:reductions}\n\\end{subfigure}\n    \\caption{Summary of computational requirements and savings of an eGRU cell compared to GRU. \\ref{tab:cellrequirements} Comparison of number of parameters and operations required for each cell type. eGRU's fewer parameters decreases required memory storage and its fewer operations reduces execution latency. \\ref{tab:reductions} Tabulation of memory and latency reductions afforded by proposed optimizations. Quantization achieves the most memory reduction by a factor of 20, whereas Q15 integer implementation attains the most speed up at a 10$\\times$ reduction in latency.}\n    \\label{tab:optsummary}\n\\end{table}\n\n\n\n\n\\subsection{Training Setup}\nSince the sole interest is inference (not learning) on target devices, training of eGRU cells can be undertaken on high-end computers with enormous resources,  using cutting-edge deep learning libraries like PyTorch \\cite{paszke2017pytorch} and Tensorflow \\cite{abadi2016tensorflow}. However, due to the modifications proposed above, additional steps are still necessary to effectively train eGRU cells in a neural network. Specifically, the effects of quantization and fixed point arithmetic in eGRU need to be simulated during the training process as well. \n\nOne common approach to quantize neural network weights is by means of post-training step. The network is first trained with full-precision weights (32- or 64-bit floating point numbers), then learned weights are quantized to the desired precision. This approach requires no adjustments to the training process since quantization is applied afterwards. While this method is suitable for reducing model sizes, it is not robust especially with heavy quantization like ours. That is because during training, the network is oblivious to the post-training quantization and therefore cannot learn to accommodate it.\n\nA more desirable approach is to implement the quantization process as part of the network's computational graph so it can be incorporated into the training process. Yet, since quantization is discontinuous, it cannot be incorporated in backpropagation training without adjustments. To address this, we adopt an approach similar to that used by \\citet{Hubara2016}. During forward pass of backpropagation, weights are first cached and passed through a quantization function before they are used in the dot product. Then during backward pass, gradients are computed with respect to the stored full-precision weights.\n\nAn illustration for this is provided in Fig. \\ref{fig:trainingsetup} and the quantization function used is given by: \n\\[ W_Q = \\begin{cases} \n      +1, & W \\geq +1 \\\\\n      \\;\\;\\, 0,  &\\!\\!\\! |W| \\leq 0.25\\\\\n      -1, & W \\leq -1 \\\\\n      \\text{sgn}(W) \\cdot 2^{\\nint{(\\text{log}_2|W|)}} & \\text{otherwise}  \n   \\end{cases}\n\\]\nwhere $W$ is the full-precision weight, and $W_Q$ is the quantized weight. In a sense, quantization is realized just like weight clipping (where weights are restricted to a defined range), and can thus be similarly considered a regularization \\cite{merolla2016deep}. Note though, that once the network is trained, full-precision weights are discarded and only quantized weights are used in inferencing. \n\nBesides quantization, simulating eGRU's fixed point arithmetic also requires training adjustments. The Q15 fixed point format used can only represent decimal numbers in the range (1,-1]. Therefore, during training, all network operations have to be constrained to fit within these Q15 bounds, in order to avoid overflow on the target device. In our work, we achieve this by clipping all dot products and activations to the (1,-1] range. The weight clipping here, as alluded to above, serves as yet another regularizer for eGRU networks.  \n\n\n\\begin{figure}[t]\n    \\begin{subfigure}[b]{0.36\\textwidth}\n        \\includegraphics[width=\\textwidth]{quantization.pdf}\n        \\caption{Exponential quantization function.}\n        \\label{fig:quantization}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}[b]{0.60\\textwidth}\n        \\includegraphics[width=\\textwidth]{backpropagation.pdf}\n        \\caption{Quantization in backpropagation.}\n        \\label{fig:backprop}\n    \\end{subfigure}\n    \\caption{Modifications to include quantization in training process. \\ref{fig:quantization} Quantization function for yielding septenary weights that are negative integer exponents of 2. \\ref{fig:backprop} During forward pass, quantization is applied to weights before they are used in any transformation. In backward pass, gradients are computed with respect to stored full precision weights.}\n    \\label{fig:trainingsetup}\n\\end{figure}\n\n\n\\section{Experimentation}\nTo evaluate the proposed features of eGRU, four experiments are undertaken. The first experiment investigates the benefits of our architectural choices, mainly the single gate mechanism and the activation functions. The second experiment analyzes the isolated impact of the various optimizations on the performance of a recurrent network. The third experiment benchmarks the computational speed of an eGRU cell on an M0+ in contrast to the GRU and RNN. Finally, the last experiment evaluates the performance of a fully optimized eGRU neural network embedded onto the target low power device for acoustic event detection tasks. The tasks, network architecture and experimental setups are discussed below in detail.\n\n\\subsection{Task Description}\nTo evaluate the various recurrent cells, three Acoustic Event Detection (AED) tasks of varying complexity are considered. They are as follows:\n\n\\begin{itemize}[itemsep=4pt,topsep=6pt]\n    \\item \\textbf{Cough Detection:} The first and least complex of the three task is the classification of short acoustic events as cough, speech or other non-cough sounds. Other non-cough sounds consists of voiced non-speech sounds, ambient sounds and noise (white and pink). This task is actually the application of interest for our wearable device. Since there are only three possible classes, the problem is much easier than the other tasks. Furthermore, because cough sounds have a relatively short duration (350 ms on average), this problem would not require that much long-term memory compared to the others. The dataset used here is from a prior work \\cite{amoh2016deep}, and consists of 2,518 audio sounds. The dataset is not exactly balanced and contains more speech and other sounds than actual cough events.\n\n    \\item \\textbf{Spoken Digits:} The next task involves distinguishing between ten spoken digits (numbers 0 to 9). The audio data is obtained from Google's Speech Commands Dataset \\cite{Warden2018} which comprises of audio recordings covering 20 simple commands. In this work, we use a subset of that dataset, consisting of 23,666 one-second audio clips. There are ten digits and the dataset is well balanced, with roughly 2,400 examples per digit. On one hand, this task can be considered more complex than the cough detection problem because there more classes and audio samples are roughly 3$\\times$ longer. On the other hand, this dataset is 10$\\times$ larger and thus should be easier to learn.  \n\n    \\item \\textbf{Urban Sounds:} The third and most challenging of the tasks considered involves identifying different urban sounds. The UrbanSound8k dataset is used \\cite{salamon2014dataset} and it includes 8,732 sounds recorded from 10 urban environments. Thus, there are ten different classes in this problem as well, and some examples are: \\textit{children\\_playing}, \\textit{dog\\_barking}, \\textit{car\\_horn}, \\textit{siren} and \\textit{street\\_music}. Certain classes sound very similar and it can be difficult for even a human to accurately identify some entries. Moreover, entries have a maximum duration of 4 seconds, which is much longer than the other two tasks and would therefore require larger long-term memory capability.\n\\end{itemize}\n\n\\begin{figure}\n    \\begin{subfigure}[b]{0.60\\textwidth}\n    \\centering\n        \\includegraphics[width=\\textwidth]{aed_datasets.png} \n        \\caption{Dataset and tasks for training neural networks.}\n        \\label{fig:aed_datasets}  \n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.38\\textwidth}\n    \\centering\n        \\includegraphics[width=\\textwidth]{network.pdf}\n        \\caption{Recurrent neural network architecture.}\n        \\label{fig:network}  \n    \\end{subfigure}\n    \\caption{\\ref{fig:aed_datasets} Summary of the three audio event detection (AED) datasets and tasks investigated in experiments. Cough detection is easiest because samples are relatively short and it is a 3-class problem. Urban sounds are 10$\\times$ longer and consist of 10 different classes. \\ref{fig:network} An illustration of the neural network architecture used for all tasks. Recurrent units in the first two layers are RNN, eGRU or GRU cells. The input is a sequence of FFT vectors.}         \n\\end{figure}\n\nA summary of the three tasks is provided in Fig. \\ref{fig:aed_datasets}. Audio from all datasets are pre-processed in a similar manner. First, recordings are downsampled from their original sampling rates to 8 KHz, then further processed into spectrograms using the Short-Time Fourier Transform (STFT). For the STFT, a window length of 128 samples is used, with no zero-padding or overlap. These analysis configuration are motivated by what can feasibly be computed on the M0+ MCU in a real-time application. For each entry in the corresponding dataset, the spectral analysis yields a 64$\\times$24 spectral chunk (64 frequency bins and 24 time steps) for the Cough task, 64$\\times$64 for Spoken Digits and 64$\\times$250 for Urban Sounds.\n\n\\subsection{Network Architecture}\nFor all three tasks, the same recurrent neural network architecture is used. Though better performing architectures can be used for each task, fixing the architecture makes for a fair comparison across tasks and experiments. Also, it is also common practice to use similar network architectures when introducing new recurrent units to enable direct comparison of architectural benefits \\cite{chung2014empirical,collins2016capacity}. In our case, we are primarily interested in the differences in performance for a drop-in replacement of all GRU cells with eGRU. Thus, it is preferred to keep the same network architecture for both eGRU and GRU cells. A four layer network is used, featuring two recurrent layers with 30 and 20 corresponding recurrent units (eGRU, GRU, or RNN cells). This is preceded by a 16 node linear layer with ReLU activations. The fourth and final layer is the classification layer with task specific number of output nodes and preceded by a softmax function. An illustration of this neural network architecture is provided in Fig. \\ref{fig:network}.\n\nThe neural networks are implemented and trained using PyTorch on a computer with 64GB RAM, a 3.8 GHz Core i5 CPU and an Nvidia GTX 1080 Ti GPU. The ADAM optimizer is used, with mini-batch sizes of 128. Training is undertaken for a maximum of 200 epochs with checkpointing at each epoch to retain only the best performing model state. In checkpointing, if the network's loss on a validation set is the minimum seen so far, the network's state is saved as the current best model. Model checkpointing therefore enables training for much longer epochs without the risk of overfitting and ensures the best model state is used eventually for evaluation.\n\n\\subsection{Experiment 1}\nIn the first experiment, we seek to answer the following question: how does the proposed eGRU architecture compare with traditional recurrent architectures on acoustic event detection tasks? Of particular interest here is the performance of the single gate mechanism together with the softsign activation functions because those are distinctive features of the eGRU. Quantization and integer arithmetic, on the other hand, can be applied to any recurrent cell and are therefore not considered in this experiment.\n\nTo evaluate eGRU's architecture, RNN, GRU and eGRU networks are trained on the three datasets using the network architecture in Fig. \\ref{fig:network}. While eGRU is a single-gate version of the GRU, the vanilla RNN is in a sense an even simpler, ``zero-gate'' GRU. We include the RNN in this experiment to explore whether gated models like the GRU or eGRU are even necessary for our application. A 10-fold cross-validation scheme is adopted for a thorough evaluation. On each task, classification accuracies on test sets are computed for the different cell types and aggregated across all folds. In all, it takes about 5 days to train all models across the ten folds.\n\n\n\\subsection{Experiment 2}\nNext, the second experiment explores how the proposed optimizations, in isolation, impact performance of a GRU network. To that end, four variants of the network architecture in \\ref{fig:network} are implemented featuring: (1) full precision original GRU cells, (2) single gated recurrent cells (3) 3-bit quantized weights in GRU cells and linear layers and (4) softsign activations in GRU cells. These networks are to illustrate the performance costs of individual optimizations. However, integer arithmetic is not included in this analysis because it mainly affects run-time speed on the embedded system and has little effect on network performance in terms of accuracy.\n\nAll four networks are trained on the above mentioned AED datasets. Also, a 70\\% : 30\\% training and test set split is used rather than the 10-fold cross-validation scheme used in the first experiment. From the 70\\% subset of the dataset used for training, 25\\% is further set aside as the validation set to determine when training converges.\n\n\n\\subsection{Experiment 3}\nThe third experiment aims to measure the latency of the each recurrent cell type on the Arm M0+ MCU. Measuring latency will reveal and quantify the gains in speed and memory that eGRU cell has over the GRU cell. Consequently, single eGRU and GRU cells are implemented and benchmarked on the target M0+ device. The eGRU cell here features all optimizations proposed including quantization and fixed point arithmetic. \n\nFor each recurrent cell type, a sequence of 1000 one dimensional inputs ranging from -1.0 to +1.0 are iteratively fed to the cell. The time taken for each cell to process the sequence on the M0+ is measured and reported as the latency benchmark. Note that while the GRU cell and its activations are implemented in 32-bit floating point format, eGRU is implemented solely in 16-bit integer data types. The latency benchmarks provide a concrete measure of how much faster and smaller eGRU is compared to GRU on an M0+ MCU.\n\n\n\\subsection{Experiment 4}\nIn the fourth and final experiment, the goal is to evaluate the performance of eGRU models trained on the three datasets and deployed to the M0+ for inference. For each dataset, a fully optimized eGRU model with the same architecture as in Fig. \\ref{fig:aed_datasets} is trained. Since deployment is the aim, eGRU cells used here feature all optimizations: single gate architecture, 3-bit quantization, softsign activations and a Q15 implementation. The same 70\\% : 30\\% cross-validation scheme used in experiment 2 is also adopted here. \n \nOnce the network is trained, quantized network weights are included in the firmware of the target M0+ device. At test time, test audio samples are fed to the M0+ device via a USB serial communication for processing and classification. The entire audio classification by the eGRU model takes place on the device, which then returns a prediction afterwards. Predictions are aggregated and analyzed to obtain performance metrics. Also, the memory, computations and speed of the embedded model are analyzed. For comparison, an equivalent GRU network is implemented on a workstation computer and evaluated using the same train and test sets. \n\n\n\\begin{figure}[b]\n   \n    \\centering\n\n    \\begin{minipage}[b]{0.58\\textwidth}\n    \\centering\n    {\\renewcommand{\\arraystretch}{1.3}\n    \\begin{tabular}{|l|c|c|c|}\n    \\hline\n    \\multicolumn{1}{|c|}{\\textbf{Cell}} & \\textbf{CoughDetect}   & \\textbf{SpokenDigits}  & \\textbf{UrbanSounds}     \\\\ \\hline\n    RNN  & 88.0 $\\pm$ 2.1  & 34.2 $\\pm$ 5.7  & 51.0 $\\pm$ 2.4   \\\\\n    eGRU & \\textbf{92.1 $\\pm$ 1.7}  & \\textbf{92.3 $\\pm$ 0.5}  & \\textbf{74.3 $\\pm$ 2.1}  \\\\\n    GRU  & \\textbf{92.5 $\\pm$ 1.9}  & \\textbf{93.1 $\\pm$ 0.6}  & \\textbf{76.2 $\\pm$ 2.7}  \\\\ \\hline\n    \\end{tabular}}\n    \\captionof{table}{Experiment 1 results reporting average classification accuracies (\\%) for the recurrent cells across 10-folds in three AED tasks. The eGRU architecture performs comparably with GRU across all tasks.}\n    \\label{tab:expt1res}\n    \\end{minipage} \\hfill\n    \\begin{minipage}[b]{0.4\\textwidth}\n    \\centering\n        \\includegraphics[width=\\textwidth]{expt1_summary.pdf}\n        \\vspace{-20pt}\n        \\captionof{figure}{Plot visualizing accuracies from Table \\ref{tab:expt1res}. Tasks have varying complexities. Specialized cells perform better than RNN.}\n        \\label{fig:expt1res}\n    \\end{minipage}    \n\\end{figure}\n\n\\section{Results}\n\\subsection{Experiment 1}\nTable \\ref{tab:expt1res} and Fig. \\ref{fig:expt1res} report the classification accuracies across 10-folds for the various recurrent cells in the first experiment. As expected, performance in general drops as the sequence length and task complexity increases across tasks. Cough detection, with a short sequence length of 24, is shown to be the easiest task where even RNN does well. On the other hand, in urban sound identification where sequences are 10$\\times$ longer, even the gold standard GRU suffers a notable decrease in performance. \n\nIn all tasks, the baseline RNN cell significantly under-performs both GRU and eGRU cells, especially in the more complex tasks. This trend is observed in the accuracy scores in Table \\ref{tab:expt1res} and in the per-dataset box-plots and validation curves in Fig. \\ref{fig:coughdetect}-\\ref{fig:urbansound}. Counter to expectations, RNN actually performed worse on spoken digits classification rather than the longer, seemingly more complex urban sounds identification. A possible explanation for this observation is that although audio entries from the spoken digit dataset are one second long, the actual event contained is often much shorter than that as evident in Fig. \\ref{fig:aed_datasets}. Since recordings begin and end with long periods of silence or background sounds, traditional RNN cells lacking advanced memory capacity will struggle to identify the buried event. The validation curves in Fig. \\ref{fig:spokendigit} attests to this, showing the RNN cells' inability to learn much in spoken digits task. Such behavior is not observed in the more advanced GRU and eGRU cells. In all, the under-performance of RNN cells confirms that gated recurrent architectures like eGRU and GRU are indeed beneficial for the AED tasks in question. \n\nThe eGRU model performs consistently well across all tasks, closely following GRU's performance. Even in the difficult urban sounds task, eGRU only suffers a 2\\% loss in accuracy relative to GRU. The same trend is observed in the training curves in Fig. \\ref{fig:coughdetect}-\\ref{fig:urbansound}, where eGRU follows closely behind GRU. That eGRU performs decently even on longer sequences is interesting because one would expect the lack of a reset gate to greatly inhibit the cell's long-term memory. However, it turns out that with full precision weights, eGRU can indeed accomodate longer sequences.\n\nFrom the observations mentioned above, the proposed eGRU architecture, mainly the single gating mechanism, appears to be suitable for AED tasks. eGRU is certainly preferable to RNN and a good alternative to GRU in keyword-spotting applications.\n\n\n   \n\\begin{figure}[t]\n    \\vspace{-5pt}\n    \\includegraphics[width=\\textwidth]{coughdetect_plots.pdf}\n    \\caption{Cough detection task performance and training curves.}\n    \\label{fig:coughdetect}            \n\n    \\includegraphics[width=\\textwidth]{spokendigit_plots.pdf}\n    \\caption{Spoken digits task performance and training curves.}\n    \\label{fig:spokendigit}            \n\n    \\includegraphics[width=\\textwidth]{urbansound_plots.pdf}\n    \\caption{Urban sounds task performance and training curves.}\n    \\label{fig:urbansound}              \n\\end{figure}\n\n\n\n\\subsection{Experiment 2}\nThe ablation study results are presented in Fig. \\ref{fig:ablation}. In Fig. \\ref{fig:ablation_barplot}, classification accuracies for isolated optimizations are compared to that of a baseline GRU without any optimization. In all three AED tasks, 3-bit quantization impacts network performance the most, resulting in 1-20\\% reduction in accuracy. The effects of the single gate and softsign activations are not as pronounced except for the harder urban sounds task. In the longer urban sounds task, softsign activations evidently hurt performance (by $\\sim$6\\%) whereas single gate architecture appears to slightly help. \n\nThe validation curves on the spoken digits tasks provided in \\ref{fig:ablation_curves} also reveal how the individual optimizations influence the network training. From the curves, all optimizations are seen to make training more difficult, taking longer time to converge than the original GRU. Once again, 3-bit weight quantization has the most adverse effect on training, taking much longer to converge and exhibiting noisy performance due to the extremely low precision weights. Softsign activations also makes it harder to train, requiring 20 more epochs to catch up to the original GRU in accuracy. The single gate optimization impacts training the least. \n\n\n\n\\begin{figure}\n    \\begin{subfigure}[b]{0.58\\textwidth}\n    \\centering\n        \\includegraphics[width=\\textwidth]{relative_ablation_barplot.pdf} \n        \\caption{Bar chart comparing performance of individual optimizations.}\n        \\label{fig:ablation_barplot}  \n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.40\\textwidth}\n    \\centering\n        \\includegraphics[width=\\textwidth]{ablation_curves.pdf}\n        \\caption{Validation curves for spoken digit task.}\n        \\label{fig:ablation_curves}  \n    \\end{subfigure}\n    \\caption{Experiment 2 results analyzing the isolated impacts of proposed eGRU optimizations on performance. \\ref{fig:ablation_barplot} Relative classification accuracies for optimizations with respect to original GRU, across three AED tasks. 3-bit quantization affects accuracy the most, resulting in up to 19\\% reduction in accuracy. \\ref{fig:ablation_curves} Corresponding training curves for different optimizations in spoken digit task. All proposed optimizations make networks harder to training, requiring more epochs to converge compared to original GRU.}  \n    \\label{fig:ablation}       \n\\end{figure}\n\n\n\\subsection{Experiment 3}\nThe results for the third experiments are provided in Table \\ref{tab:expt3res} and it informs on the computational costs of the different recurrent cells. Evidently, eGRU's use of solely integer operations result in much faster execution on the M0+ compared to GRU. On the M0+, an eGRU cell takes only 13 ns to execute, which is more than 60$\\times$ faster than GRU. Besides speed, the extreme 3-bit quantization of eGRU results in a 12$\\times$ reduction in memory compared to GRU (with 32-bit floats). A single eGRU cell is only 3 bytes large. \n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{|c|r|r|r|}\n    \\hline\n    \\textbf{Cell} & \\textbf{Operations} & \\textbf{Memory} & {\\textbf{Latency}} \\\\ \\hline\n    eGRU  & 12 INOPs    &  \\textbf{3 bytes} &\\textbf{13 ns}   \\\\\n    GRU   & 17 FLOPs    & 36 bytes          & 794 ns          \\\\ \\hline\n    \\end{tabular}\n    \\caption{Experiment 3 results showing benchmark of M0+ latency for eGRU versus GRU cells. An eGRU cell requires only integer operations (INOPs) instead of the more expensive floating point operations (FLOPs) in traditional GRU. On the M0+, eGRU is \\textbf{60$\\times$ faster} and \\textbf{12$\\times$ smaller} than GRU.}\n    \\label{tab:expt3res}\n\\end{table}\n\n\n\n\\subsection{Experiment 4}\nFinally, in Fig. \\ref{fig:expt4res}, the performance of fully optimized eGRU models trained for the different tasks and embedded on the target M0+ device is presented. Note that while eGRU models here feature all optimizations proposed including quantization and fixed point operations, GRU models have full precision weights and employ floating point operations. Also, because a different validation scheme is used, the results for the GRU models here are a bit different than those in the first experiment.\n\nOn the embedded system, the eGRU model attains an accuracy of 95.5\\% on the cough detection task, which is only 2\\% less than the accuracy of a corresponding GRU model on a fully equipped desktop computer. This is impressive considering the eGRU model is only of 3kB in size, 10$\\times$ smaller than its GRU equivalent. From latency benchmarks, the embedded eGRU model is measured to take only 9.3 ms to analyze and classify a 350 ms (64 x 24 spectrogram) cough event.\n\nOn the spoken digits identification, the eGRU model running on the M0+ still compares well with the GRU. It attains an accuracy of 87.8\\% compared to 91.8\\% of GRU. That said, the more complex spoken digits task with longer sequences widens the performance difference between eGRU and GRU at 4\\%.\n\nSimilar to the first experiment, both recurrent cell types perform the worst on the urban sounds dataset. The deployed eGRU model achieves a 61.3\\% accuracy, compared to 72.1\\% of GRU. Here, the eGRU model under-performs GRU by a significant 11\\%. However, that is not very surprising considering the input sequences in this task are 4-10$\\times$ longer than in the other two tasks.\n\n\n\\begin{figure}\n    \\begin{subfigure}[b]{0.38\\textwidth}\n    \\centering\n\n    \\begin{tabular}{|l|c|c|}\n    \\hline\n    \\multicolumn{1}{|c|}{\\textbf{}} & \\textbf{eGRU} & \\textbf{GRU}  \\\\ \\hline\n    \\textbf{CoughDetect}            & 95.3\\%        & 97.3\\%        \\\\\n    \\textbf{SpokenDigits}           & 87.8\\%        & 91.8\\%        \\\\\n    \\textbf{UrbanSounds}            & 61.3\\%        & 72.1\\%        \\\\ \\hline\n    Model Size                      & 3 kB          & 34 kB         \\\\\n    Device                          & Arm M0+       & Intel Core i5 \\\\\n    Speed                           & 48 MHz        & 3,800 MHz     \\\\ \\hline\n    \\end{tabular}\n\n    \\vspace{5pt}\n    \\caption{eGRU deployed on Arm M0+ versus GRU running on Intel workstation.}\n    \\label{fig:egruperf}\n    \\end{subfigure}\\hspace*{\\fill}\n    \\begin{subfigure}[b]{0.28\\textwidth}\n    \\centering\n        \\includegraphics[width=\\textwidth]{val_acc.pdf}\n        \\captionof{figure}{Validation accuracy curves for eGRU versus GRU models.}\n        \\label{fig:valacc}\n    \\end{subfigure}\\hspace*{5pt}\n    \\begin{subfigure}[b]{0.28\\textwidth}\n    \\centering\n        \\includegraphics[width=\\textwidth]{val_loss.pdf}\n        \\captionof{figure}{Validation loss curves for eGRU versus GRU models.}\n        \\label{fig:valloss}\n    \\end{subfigure}    \n    \\caption{Experiment 4 results showing performance of eGRU model embedded unto the Atmel ATSAMD21 \\cite{ATSAMD21}, an ultra low-power Arm M0+ processor. \\ref{fig:egruperf} The eGRU model on the M0+ compares well with its full precision GRU counterpart on a computer, across the different task. Yet, the embedded eGRU model is highly efficient, taking only 10\\% the size of GRU and requiring inexpensive integer operations exclusively. \\ref{fig:valacc} Validation accuracy curves for eGRU and GRU models across all task. eGRU takes longer epochs to train than GRU. \\ref{fig:valloss} Corresponding validation loss curves. eGRU models approach similar loss in short-duration tasks.}\n    \\label{fig:expt4res}\n\\end{figure}\n\n\n\\section{Discussion}\nThe results from the first experiment (Fig. \\ref{fig:expt1res} and Fig. \\ref{fig:urbansound}) confirm that the proposed architectural modifications in eGRU are indeed effective for short acoustic event detection or keyword spotting tasks. In particular, single gate architecture with softsign activations is seen to be effective for short sequences, and even for longer ones. From a purely architectural stand-point, eGRU is seen to be certainly better than traditional RNN and an efficient candidate for replacing GRU cells. \n\nWith quantization and fixed point arithmetic, eGRU is demonstrated to be efficient enough to fit and run successfully on a resource constrained MCU like the M0+. From the second and third experiments, eGRU is shown to be 10$\\times$ smaller and 60$\\times$ faster than GRU. Yet, in short sequence tasks, eGRU only suffers 2-4\\% reduction in accuracy relative to GRU. Indeed, on more complex AED tasks with much longer sequences such as the urban sounds identification task, fully optimized eGRU under-performs GRU even more. The combination of extreme quantization as well as the single gate architecture appears to heavily impair eGRU's ability to handle long sequences. Thus, eGRU is most suitable for short-duration acoustic event detection applications such as keyword spotting. \n\nOne way to make eGRU effective even for long sequences is to use a less aggressive quantization scheme. As noted in experiment 1 results, full precision eGRU performs well on long sequences even with the single gate architecture. Thus, increasing weight precision from 3 to 4 or even 8 bits will considerably improve accuracy on long events. Another idea for tackling for lengthy sequences is to use eGRU in a sequence-to-sequence network architecture \\cite{sutskever2014sequence} with a sequential loss function such as the connectionist temporal classifier (CTC) \\cite{graves2006connectionist}. In the sequence-to-sequence with CTC setup, rather than reading an entire input sequence and outputting a single prediction as done above, recurrent networks yield sequential outputs that are aligned with the input. Sequence-to-sequence network architectures are more suitable for long sequences and is typically used in speech recognition tasks. \n\nHowever, the significance of eGRU is crucial in ultra-low power wearable applications where resources are limited. For instance, for our wearable cough detector application, the 3 kB eGRU model can easily fit within the 32 kB RAM available on the M0+, with a lot of space left over for other processes. Also, the 9 ms latency can be easily managed on the MCU since cough events are about 350 ms on average. In contrast, the equivalent GRU model requires 34 kB memory and up to 500 ms latency, both of which will not be feasible on the M0+. In such applications, eGRU is the only practical choice. \n\n\n\\section{Conclusion}\nIn summary, our work explores a means to realize recurrent neural network inferencing on ultra-low power wearable devices. We discussed the constraints of ultra-low-power devices in terms of size, power, memory and computation specifications. We then presented a new recurrent unit architecture, the embedded Gated Recurrent Unit (eGRU), specifically adapted for ultra-low power micro-controller implementation. In particular, eGRU featured a single gate mechanism, faster activation functions, 3-bit weight quantization and utilized exclusively fixed point arithmetic. From our experiments, we demonstrated that while eGRU can be feasibly implemented on target ultra-low power devices, it still performs comparably with GRU on short-duration acoustic event detection applications. \n\nTwo main areas for further investigation remain. First, since final eGRU network only utilized a fraction (10\\%) of the available RAM on the M0+, better, bigger and even more complex network architectures can be explored. It will be interesting and beneficial to incorporate recent neural network innovations such as batch normalization, attention mechanisms, and skip\/residual connections in the models.\n\nA more pressing next step is to prepare the entire system to run effectively in a real life scenario, untethered and robust to completely unexpected events. Currently, since the datasets used were all created under controlled environments, the model does not perform quite as well on events that are very different from those in the datasets. For the system to be completely untethered, the neural network model has to be combined with other real-time processing steps such as signal acquisition, voice activity detection and spectral decomposition, all running on the ultra-low power micro-controller. \n\n\n\n\\bibliographystyle{plainnat}\n\\input{preprint.bbl}\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe efficiency of any heat engine is limited by the Carnot efficiency defined as:\n\\begin{equation}\n\\eta_C=1-T_{C}\/T_H,\n\\end{equation}\n where $T_{H}$ and $T_C$ denote the temperatures of the hot and cold contacts respectively. Typically, the Carnot efficiency is achieved at zero power output and vanishing lattice thermal conductivity ($\\kappa_{ph}$). Modern thermoelectrc engineering aims at enhancing the  \\emph{figure of merit} ($zT$) defined as:\n\\begin{equation}\nzT=\\frac{S^2\\sigma}{\\kappa}T,\n\\label{eq:zt}\n\\end{equation}\nwhere $S$, $\\sigma$ and $\\kappa$ are the Seebeck coefficient, the electrical conductivity and the thermal conductivity of the material respectively, and $T$ is the average temperature between the hot and cold contacts.  In the linear response regime, the operation efficiency of a thermoelectric generator is closely related to the \\emph{figure of merit} $zT$. In  attempts towards enhancing $zT$, the two approaches commonly followed are $(i)$ limiting the thermal conductivity $\\kappa$  and $(ii)$ enhancing the power factor ($S^2\\sigma$). The thermal conductivity $\\kappa$ captures the heat flow due to lattice thermal conductivity ($\\kappa_{ph}$) as well as  due to electronic thermal conductivity ($\\kappa_{el}$). To enhance the efficiency, a lot of engineering effort has been dedicated towards reducing  $\\kappa_{ph}$. Such efforts include  nanostructuring, nano-inclusions,  embedded interfaces and heterostructures \\cite{supressk1,supressk2,phonon1,phonon2,phonon3,phonon4,phonon5,chen3}. An engineering of electronic density of states, on the other hand, is an independent and alternative route that also aims at enhancing the power factor ($S^2\\sigma$), while simultaneously reducing $\\kappa_{el}$ as far as possible \\cite{sothmann1,sothmann2,staircase_qdot,aniket1,aniket,aniket2,aniket3,rev_sofomahan,Humphrey,Humphrey2,yamamoto2,yamamoto1}. \\\\\n\\indent In thermoelectric generators, the overall generation efficiency is generally limited by the lattice thermal conductivity ($\\kappa_{ph}$). The heat energy flowing from the hot  contact via lattice thermal conductivity,  either dissipates to the environment from the channel region or flows towards the cold contact and can't be re-extracted back to the hot contact. This results in a deterioration of the overall efficiency. The overall Seebeck coefficient, hence, the power factor and efficiency, could be improved, if by clever engineering of electronic density of states, the heat flowing into channel region via lattice heat conductivity, could be absorbed by the electrons. In other words, the overall Seebeck coefficient can be improved, while still limiting the electronic heat conductivity, via inelastic processes in the channel region. A few proposals regarding the enhancement of the Seebeck coefficient or power factor  via inelastic scattering in the channel  region \\cite{kim1,kim2,neo} has  been put forward  in recent years. However, we are yet to see a powerful, but   general and compact design strategy to enhance the heat engine performance via inelastic processes. In this paper, I construct a general, but compact  design strategy for maximum enhancement in thermoelectric performance via absorption of thermal energy from  the channel. The optimal design features of the proposal were derived via rigorous mathematical calculations. Along with optimizing the design features, I also present numerical simulation results to assess the enhancement in thermoelectric performance via my proposed strategy. \\\\\n\\indent The absorption of waste heat from the channel region can be facilitated by restricting the energy resolved flow of electrons from the hot contact, which in other words is known as energy filtering \\cite{kim1,kim2,aniket,aniket3}. Energy filtering of incoming electrons in the channel region can be accomplished via  control of the transmission coefficient at the hot contact to channel junction. This  results in the injection of electrons with an effective temperature  much lower than the channel temperature. It should be noted that absorption of heat from the channel doesnot impact the heat flowing  from the hot contact via lattice heat conductivity, since the direction of heat flow is from the channel towards the cold contact and not vice-versa. In this context, it might be stated that the channel temperature is intermediate between the hot and cold contacts. It is worth mentioning that, unlike the given Refs. \\cite{kim1,kim2,neo}, where much of the enhancement in generated power stems from an enhanced  electrical conductivity $\\sigma(\\epsilon)$ at higher energy, the proposed concept in this paper is independent of such constraints. Rather, in this case, I aim towards finding out the optimal bounded transmission functions   for maximizing the efficiency at a  generated power. While doing so, I show that the efficiency of the heat engine with respect to the electronic heat flow, can surpass the ballistic limit at finite power output, provided that the transmission functions are optimized to achieve the best performance. \\\\\n\\indent This paper is organized as follows. In Sec.~\\ref{model}, I highlight the model used to validate the impact of the proposed design strategy. Sec \\ref{calculation} consists of detailed calculation to optimize the design features of the model, proposed in Sec. \\ref{model}, to achieve maximum enhancement in generated power. In Sec. \\ref{results}, I demonstrate result of numerical simulations for the proposed design strategy, considering the model discussed in  Sec.~\\ref{model}. I, finally, conclude this paper with a general discussion in Sec.~\\ref{conclusion}.\n \n\\section{Model.}\\label{model}\nTo validate the proposed concept, I employ a toy model as shown in Fig.~\\ref{fig:schem}. The model  consists  of a left hot contact at temperature $T_H$ and a right cold contact at temperature $T_C$. The contacts are connected to the middle channel region labeled `CH' via electronic filters with energy dependent transmission function $\\Gamma_H$ and $\\Gamma_C$. The channel, in this case, is assumed to be a macroscopic electronic bath with well defined quasi-Fermi energy $\\mu_{CH}$. Such an assumption is valid  when the resistance of the channel is much less compared to that of the energy filters $\\Gamma_H$ and $\\Gamma_C$. This basically means that the electronic flow between the hot (left) and cold (right) contacts is practically dictated by the two filters at the channel-to-contact junctions.  For ease of analysis, I also assume that there is no spatial variation of the channel temperature, labeled as $T_{CH}$, which  is valid when the thermal conductivity of the filters are much lower compared to that of the channel \\cite{kim1,kim2}.  In addition to dictating the electronic flow, the left filter ($\\Gamma_H$) also governs the electronic heat current that flows from the hot contact. By a suitable choice of the energy resolved transmission coefficient $\\Gamma_H$ of the filter at the hot contact to channel junction, electrons can be injected at an effective temperature that is lower compared to the channel. This in turn facilitates an absorption of heat energy from the channel via inelastic processes. \\\\\n\\begin{figure}[!tb]\n{\\includegraphics[scale=.25]{schem}}\n\\caption{Schematic diagram illustrating the model employed to validate the proposed design strategy. The model consists of a left (hot) contact at temperature $T_H$ and  a right (cold) contact at temperature $T_C$ connected by a middle channel region at temperature $T_{CH}$. Two electron filters with transmission function $\\Gamma_H$ and $\\Gamma_C$ are connected at the hot contact to channel and cold contact to channel interface.  The filters dictate the energy resolved electronic current flowing through the interfaces. Inelastic processes within the channel may be used to facilitate heat absorption by the incoming electrons, provided that the effective temperature of the injected electrons is lower compared to the channel.}\n\\label{fig:schem}\n\\end{figure}\n\\indent In the linear response regime,  the power factor $S^2G$ and  the figure of merit $zT$ can be used to assess the performance of a thermoelectric generator. However, these parameters are not of much ultility when the corresponding generator is operating in the non-linear regime \\cite{nakpathomkun}. In addition, these two parameters hardly give any information about the regimes of operation other than the point of maximum power generation, and hence, cannot facilitate a clear understanding of the physics of heat flow \\cite{Humphrey,jordan1,jordan2,sothmann,bm,whitney,whitney2}.     Hence, an analysis of power generation at a given  efficiency and operating point \\cite{whitney,whitney2,sothmann,bm,agarwal,BD,nakpathomkun,jordan2,bm,zimb,leij} is essential for a detailed analysis of the advantage gained via the proposed concept. In this paper, I hence, carry out an analysis of generated power at a given efficiency.\\\\\n\\indent Here, a direct calculation of maximum  power at a given efficiency  is performed by optimizing the filter transmission functions $\\Gamma_H$ and $\\Gamma_C$. The voltage drop across an external load, where the power dissipation takes place, is emulated by a potential bias across the generator \\cite{nakpathomkun}. The generated power $(P)$ and the efficiency ($\\eta$) for a  a given temperature  and voltage bias across two contacts can be calculated as:\n\\begin{equation}\nP=I \\times V\n\\label{eq:pow}\n\\end{equation} \n\\begin{equation}\n\\eta=\\frac{P}{I_Q},\n\\label{eq:eff}\n\\end{equation}\nwhere $I$ is the electronic charge current, $I_Q$ is the total heat current at the hot contact and the applied bias $V$ emulates the potential drop across the external load.   Since lattice thermal conductivity remains almost unchanged regardless of the electronic heat flow, it is logical to simplify our calculations by neglecting the degradation in efficiency due to phonon heat conductivity. In addition, reduction of lattice heat conductivity via engineering of nano-heterostructures is a different and independent path towards improving the overall efficiency of waste heat harvesting. So, the efficiency  that I use to gauge the effectiveness of the proposed concept   is  defined  by the equation:\n\\begin{equation}\n\\eta=\\frac{P}{I_{Qe}},\n\\label{eq:ef}\n\\end{equation}\nwhere $P$ is  defined in \\eqref{eq:pow} and $I_{Qe}$ is the electronic heat current at the hot contact.   As stated earlier, I assume that the resistance of the  channel `CH' is negligible compared to the energy filters. Hence, the electronic current between the contacts is effectively limited by the energy filters. This also means that the total voltage drop across the channel is negligible compared to that across the filters. I denote the respective electrochemical potentials of the hot contact,  cold contact and  the macroscopic channel by  $\\mu_H$, $\\mu_C$ and $\\mu_{CH}$ respectively. Without loss of generality, I assume that $\\mu_{CH}=0$. Assuming a voltage drop of $V_H$ and $V_C$  entirely across the left and right energy filters respectively, the quasi-Fermi levels of the hot and the cold contacts are  given by $\\mu_H=-eV_H$ and $\\mu_C=eV_C$ (assuming no spacial variation in $\\mu_{CH}$). The total voltage drop across the generator is hence given by $V=V_H+V_C$. Under the condition of ballistic electronic transport  through the two filters, the electronic charge currents through the left and right filters  can be calculated from Landauer's scattering theory via the following equations: \n\\begin{eqnarray}\nI_L=\\frac{e}{h} \\int \\Gamma _H(\\epsilon) \\{f_H(\\epsilon)-f_{CH}(\\epsilon)\\} d\\epsilon \\nonumber \\\\\nI_R=\\frac{e}{h} \\int \\Gamma _C(\\epsilon) \\{f_{CH}(\\epsilon)-f_C(\\epsilon)\\} d\\epsilon\n\\label{eq:curr}\n\\end{eqnarray}\nIn steady state, $I_L=I_R=I$.  The net irreversible heat flow within the system, on the other hand, occurs from the hot contact to the channel and subsequently towards the cold contact. Hence, the net irreversible heat flow can be accounted for by considering only the heat lost from the hot contact (since the heat lost from the channel to the cold contact ultimately comes from the hot contact via lattice heat conduction or electronic heat conduction). Hence, for the purpose of my calculations, the net irreversible  heat current  flow in the system  is defined as the net heat flow between the left (hot) contact to channel  and depends only on  the transmission function $\\Gamma_H$.  As stated before, for the purpose of calculations, I only consider the heat flow due to the electronic conduction. The net irreversible heat flow from the left (hot) contact can, hence, be defined by the equation:\n\\begin{equation}\nI_{Qe}=\\frac{1}{h} \\int (\\epsilon-\\mu_H)\\Gamma _H(\\epsilon) \\{f_H(\\epsilon)-f_{CH}(\\epsilon)\\} d\\epsilon\n\\label{eq:heat}\n\\end{equation}\nIn the above equations, $f_H(\\epsilon)$, $f_C(\\epsilon)$ and $f_{CH}(\\epsilon)$ denote the quasi Fermi electronic distribution at the hot contact, the cold contact and the macroscopic electronic channel `CH' respectively.\n\\[\nf_i=\\left[1+exp\\left(\\frac{\\epsilon-\\mu_i}{k_B T_i}\\right)\\right]^{-1}\n\\] \n\\indent The set of equations are written  under the assumption that the electrons injected from the left contact completely equilibriate with the channel temperature $T_{CH}$ via inelastic scattering while transport through the channel. It should be noted that the absorption of heat from the channel hardly impacts the electronic or lattice heat flow from the left contact. This is because both electronic and lattice heat flows from the hot contact towards the channel and subsequently cold contact, and not in the reverse direction. Hence,  absorption of heat via inelastic processes within the channel simply reduces the lattice heat flux into the cold contact. Throughout the discussion, I will assume  n-type channel, though the discussion is  valid for p-type channel as well, with a slight modification of the equations used in the following sections.\n\\section{Calculation and optimization of $\\Gamma_H$ and $\\Gamma_C$.}\\label{calculation}\nIn this section, I present analytical calculations for  $\\Gamma_H$ and $\\Gamma_C$ to maximize the generated power at a given efficiency. Since, the transmission function of any electronic filters can be complicated, I fix the upper bounds on the functions $\\Gamma_H$ and $ \\Gamma_C$ at a given energy $\\epsilon$, assuming that by clever engineering the filters can be manipulated to acquire the desired transmission characteristics. The  upperbound to the transmission functions are set such that:\n\\begin{equation}\n0\\leq \\Gamma_H(\\epsilon)\\leq 1;  ~0 \\leq \\Gamma_C(\\epsilon) \\leq 1,\n\\end{equation}\nfor any energy $\\epsilon$ in the range of electronic transport. My intention here is to derive the mathematical form  of the  optimal transmission function for  the filters $\\Gamma_H$ and $\\Gamma_C$. It should be noted in this regard that the derived concepts and results still hold if one multiplies the optimized transmission function by a constant.\n\\\\\n\\indent For  analysis of the problem,  I define a variable, \n\\begin{equation}\nI_C=I_L-I_R\n\\end{equation}\nIn steady state, $I_C=0$.  I  assume that the transmission function of each of the filters $\\Gamma _H(\\epsilon)$ and $\\Gamma _C(\\epsilon)$ can take any value from $0$ to a maximum value of $1$ and proceed towards calculating the maximum power at a given efficiency. My approach to the problem entails finding out the maximum value of the power $P$ at a given value of the electronic heat current at the left (hot) contact $I_{Qe}$. The generated power $P$ and the electronic heat current $I_{Qe}$ are functions of $V_H$, $V_C$, $\\Gamma_H$ and $\\Gamma_C$. Hence, a small variation in $P$ and $I_{Qe}$ due to small variation in this parameter space can be written as: \n\\indent \n\\begin{widetext}\n\\begin{equation}\n\\delta P= \\sum_i\\delta \\Gamma _i(\\epsilon)\\frac{\\partial P}{\\partial \\Gamma_i(\\epsilon)}\\Big|_{V_H, V_C}+\\delta V_H\\frac{\\partial P}{\\partial V_H}\\Big|_{V_C, \\Gamma}+\\delta V_C\\frac{\\partial P}{\\partial V_C}\\Big|_{V_H, \\Gamma},\n\\label{eq:power_differential}\n\\end{equation}\n\n\\begin{equation}\n\\delta I_{Qe}= \\sum_i \\delta \\Gamma _i(\\epsilon)\\frac{\\partial I_{Qe}}{\\partial \\Gamma_i(\\epsilon)}\\Big|_{V_H, V_C}+\\delta V_H\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C, \\Gamma}+\\delta V_C\\frac{\\partial I_{Qe}}{\\partial V_C}\\Big|_{V_H, \\Gamma},\n\\label{eq:heat_differential}\n\\end{equation}\n\\end{widetext}\nwhere  the symbol ~~`$|_{x}$'~~ implies that the partial differentiation is carried out at constant values of $x$ and $i \\in (H,~C)$, $\\Gamma=(\\Gamma_H, \\Gamma_C)$. Since $I_{Qe}$, in \\ref{eq:heat} doesn't depend explicitly on $V_C$ and $\\Gamma_C$, I put $\\frac{\\partial I_{Qe}}{\\partial V_C}\\Big|_{V_H, \\Gamma}=\\frac{\\partial I_{Qe}}{\\partial \\Gamma_C}\\Big|_{V_H, V_C}=0$. Hence,\n\n\\begin{equation}\n\\delta I_{Qe}= \\delta \\Gamma _H(\\epsilon)\\frac{\\partial I_{Qe}}{\\partial \\Gamma_H(\\epsilon)}\\Big|_{V_H, V_C}+\\delta V_H\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C, \\Gamma}.\n\\label{eq:heat_differential_rewrite}\n\\end{equation}\nSimilarly,\n\\begin{eqnarray}\n\\delta I_C=\\sum_i \\delta \\Gamma _i(\\epsilon)\\frac{\\partial I_C}{\\partial \\Gamma_i(\\epsilon)}\\Big|_{V_H, V_C}+\\delta V_H\\frac{\\partial I_C}{\\partial V_H}\\Big|_{V_C, \\Gamma} \\nonumber \\\\ +\\delta V_C\\frac{\\partial I_C}{\\partial V_C}\\Big|_{V_H, \\Gamma}~~~~~~~~\n\\label{eq:curr_differential_rewrite}\n\\end{eqnarray}\nAt first, I assume $\\Gamma_H$ to be some arbitrary function and proceed towards finding out the optimal bounded function for $\\Gamma_C$. While $\\Gamma_C$ varies, I try to maximize the generated power for a given heat current. Hence, I  fix $I_{Qe}$ and vary the parameters $\\Gamma _C(\\epsilon)$, $V_H$ and $V_C$ so that the output power  varies along a line where $I_{Qe}$ is kept constant in the parameter ($\\Gamma _C(\\epsilon),~V_C,~V_H$) space. In this case, I can rewrite Eqs.  \\eqref{eq:heat_differential_rewrite} and \\eqref{eq:curr_differential_rewrite} as:\n\\begin{equation}\n\\delta I_{Qe}= \\delta V_H\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C, \\Gamma}=0 \n\\label{eq:right_solve_1}\n\\end{equation} \n\\begin{eqnarray}\n\\delta I_C= \\delta \\Gamma _C(\\epsilon)\\frac{\\partial I_C}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C}+\\delta V_H\\frac{\\partial I_C}{\\partial V_H}\\Big|_{V_C, \\Gamma} \\nonumber \\\\ +~~\\delta V_C\\frac{\\partial I_C}{\\partial V_C}\\Big|_{V_H, \\Gamma}=0,~~~~~\n\\label{eq:right_solve_2}\n\\end{eqnarray}\nSolving equations \\eqref{eq:right_solve_1} and \\eqref{eq:right_solve_2} I get the values of $\\delta V_H$ and $\\delta V_C$ as:\n\\begin{equation}\n\\delta V_C=-\\delta \\Gamma _C(\\epsilon) \\frac{\\frac{\\partial I_C}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C}}{\\frac{\\partial I_C}{\\partial V_C}\\Big|_{V_H, \\Gamma}},~~\\delta V_H=0.\n\\label{eq:delta_1}\n\\end{equation}\nSubstituting  the values of $\\delta V_H$ and $\\delta V_C$ in \\eqref{eq:power_differential} from \\eqref{eq:delta_1} I get,\n\\begin{equation}\n\\delta P=\\delta \\Gamma _C(\\epsilon)\\frac{\\partial P}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C}-\\delta \\Gamma _C(\\epsilon) \\frac{\\frac{\\partial I_C}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C}}{\\frac{\\partial I_C}{\\partial V_C}\\Big|_{V_H, \\Gamma}}\\frac{\\partial P}{\\partial V_C}\\Big|_{V_H, \\Gamma},\n\\label{eq:delta_2}\n\\end{equation}\nTo proceed from the above equation, I substitute $\\frac{\\partial I_C}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C}$ in \\eqref{eq:delta_2} by the formulas :\n\\begin{equation}\n\\frac{\\partial I_C}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C}=-\\frac{\\partial I_R}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C}=-\\frac{1}{V_C}\\frac{\\partial P}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C}\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial P}{\\partial V_C}=I_R+V_C \\frac{\\partial I_R}{\\partial V_C}=I_R-V_C\\frac{\\partial I_C}{\\partial V_C} \n\\end{equation}\nI hence get:\n\\begin{align}\n\\delta P&=\\left\\{1+ \\frac{1}{V_C}\\frac{\\frac{\\partial P}{\\partial V_C}\\Big|_{V_H, V_C}}{\\frac{\\partial I_C}{\\partial V_C}\\Big|_{V_H, \\Gamma}}\\right\\}\\delta \\Gamma _C(\\epsilon)\\frac{\\partial P}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C} \\nonumber \\\\\n&=\\left\\{1+ \\frac{I_R}{V_C {\\frac{\\partial I_C}{\\partial V_C}}}-1\\right\\} \\delta \\Gamma _C(\\epsilon)\\frac{\\partial P}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C}\\nonumber \\\\\n&=\\frac{I_R}{V_C {\\frac{\\partial I_C}{\\partial V_C}}}\\delta \\Gamma _C(\\epsilon)\\frac{\\partial P}{\\partial \\Gamma_C(\\epsilon)}\\Big|_{V_H, V_C}\n\\end{align}\nNow, $\\frac{\\partial I_C}{\\partial V_C}=-\\frac{\\partial I_R}{\\partial V_C}>0$ (from Eq. \\ref{eq:curr}). Assuming an n-type channel and $\\mu_{CH}=0$, it can be readily found out that $\\frac{\\partial P}{\\partial \\Gamma_C (\\epsilon)}\\geq 0$ for $\\epsilon\\geq \\epsilon_C$, with $\\epsilon_C=\\mu_C \\left(1-\\frac{T_C}{T_{CH}}\\right)^{-1}$. Hence, the generated power increases for a small increase in $\\Gamma_C(\\epsilon)$ by  $\\delta \\Gamma _C(\\epsilon)$, when $\\epsilon \\geq \\epsilon_C$. For $\\epsilon<\\epsilon_C$, the generated power decreases with a small increase in $\\Gamma_C(\\epsilon)$. Hence, to generate maximum power for a given value of the $I_{Qe}$, the transmission function $\\Gamma_C$ should to kept to its maximum and minimum possible value for $\\epsilon \\geq \\epsilon_C$ and $\\epsilon < \\epsilon_C$ respectively.\n Hence, in this particular case, where $0 \\leq \\Gamma_C(\\epsilon) \\leq 1$, the mathematical expression for $\\Gamma_C$ to generate maximum power is:\n\\begin{equation}\n\\Gamma_C=\\theta(\\epsilon-\\epsilon_C),\n\\label{eq:final_gamma_c}\n\\end{equation}\n$\\theta$ being the unit step function.\\\\\n\\indent Next, I proceed towards finding out the optimized transmission function for the left filter. We first note that the transmission function $\\Gamma_C$ given by Eq. \\eqref{eq:final_gamma_c} is independent for $\\Gamma_H$ for any value of $\\epsilon$.  Hence, I  assume that the transmission function for the right filter has already been fixed to its optimal value. If the transmission function $\\Gamma_H$, at energy $\\epsilon$, changes by a small amount  $\\delta \\Gamma_H (\\epsilon)$, then for a fixed value of $I_{Qe}$, Eq. \\eqref{eq:heat_differential} becomes:\n\n\\begin{equation}\n\\delta I_{Qe}= \\delta \\Gamma _H(\\epsilon)\\frac{\\partial I_{Qe}}{\\partial \\Gamma_H(\\epsilon)}\\Big|_{V_H, V_C}+\\delta V_H\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C, \\Gamma}=0,\n\\label{eq:left_solve_1}\n\\end{equation}\n\\begin{eqnarray}\n\\delta I_C= \\delta \\Gamma _H(\\epsilon)\\frac{\\partial I_C}{\\partial \\Gamma_H(\\epsilon)}\\Big|_{V_H, V_C}+\\delta V_H\\frac{\\partial I_C}{\\partial V_H}\\Big|_{V_C, \\Gamma}\\nonumber \\\\+\\delta V_C\\frac{\\partial I_C}{\\partial V_C}\\Big|_{V_H, \\Gamma} =0\n\\label{eq:left_solve_2}\n\\end{eqnarray}\n\nSolving \\eqref{eq:left_solve_1} and \\eqref{eq:left_solve_2}, I get \n\\begin{equation}\n\\delta V_H=-\\frac{\\frac{\\partial I_{Qe}}{\\partial \\Gamma_H(\\epsilon)}\\Big|_{V_H,~V_C}}{\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{\\Gamma, V_C}}\\delta \\Gamma _H (\\epsilon) \\\\\n\\label{eq:left_solve_3}\n\\end{equation}\n\\begin{equation}\n\\delta V_C=\\left(\\frac{\\frac{\\partial I_{Qe} }{\\partial \\Gamma _H(\\epsilon)}\\Big|_{V_H,V_C} \\frac{\\partial I_C}{\\partial V_H}\\Big|_{V_C,\\Gamma_H}}{\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C,\\Gamma_H} \\frac{\\partial I_C}{\\partial V_C}\\Big|_{V_H,\\Gamma_H}}-\\frac{\\frac{\\partial I_C}{\\partial \\Gamma _H}\\Big|_{V_H,V_C}}{\\frac{\\partial I_C}{\\partial V_C}\\Big|_{V_H,\\Gamma_H}}\\right)\\delta \\Gamma_H(\\epsilon)\n\\label{eq:left_solve_4}\n\\end{equation}\n\\indent Note that $\\delta V_H$ in Eq. \\eqref{eq:left_solve_3} depends only on the parameters $V_H$ and $\\Gamma_H(\\epsilon)$. This basically means that the optimization of $\\Gamma_H(\\epsilon)$, and hence $V_H$, at a given value of $I_{Qe}$ are independent of $V_C$.\nI hence independently optimize the power $P_H$ given by $P_H=I_L \\times V_H$. The change in power $P_H$ due to a small change in $\\Gamma_H(\\epsilon)$ and $V_H$ is given by:\n\\begin{equation}\n\\delta P_H=\\delta \\Gamma_H(\\epsilon)\\frac{\\partial P_H}{\\partial \\Gamma_H(\\epsilon)}\\Big|_{V_H, V_C}+\\delta V_H\\frac{\\partial P_H}{\\partial V_H}\\Big|_{V_C, \\Gamma}\n\\label{eq:delta_P_H}\n\\end{equation}\nSubstituting the value of $\\delta V_H$ in the above equation from \\eqref{eq:left_solve_3} and replacing $\\frac{\\partial I_{Qe}}{\\partial \\Gamma_H(\\epsilon)}$ using the formula,\n\\begin{equation}\n\\frac{\\partial I_{Qe}}{\\partial \\Gamma_H(\\epsilon)}\\Big|_{V_H, V_C}=\\frac{\\epsilon-\\mu_H}{eV_H}\\frac{\\partial P_H}{\\partial \\Gamma_H(\\epsilon)}\\Big|_{V_H, V_C},\n\\end{equation}\nEq. \\eqref{eq:delta_P_H} can be re-written as,\n\\begin{equation}\n\\delta P_H=\\delta \\Gamma_H(\\epsilon)\\frac{\\partial P_H}{\\partial \\Gamma_H(\\epsilon)}\\Big|_{V_H, V_C}\\left\\{ 1-\\frac{\\epsilon-\\mu_H}{eV_H}\\frac{\\frac{\\partial P_H}{\\partial V_H}\\Big|_{V_C, \\Gamma}}{\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C, \\Gamma}}\\right\\}\n\\label{eq:delta_P_H_2}\n\\end{equation}\n\nIn Eq. \\eqref{eq:delta_P_H_2}, I observe that $\\frac{\\partial P_H}{\\partial \\Gamma_H(\\epsilon)}>0$ for $\\epsilon>\\epsilon_H$, where,\n\\begin{equation}\n \\epsilon_H=-\\mu_H\\left(\\frac{T_H}{T_{CH}}-1\\right)^{-1}.\n \\label{eq:jye_h}\n \\end{equation}\n \n  In addition, $I_{Qe}$ is a decreasing function of the applied voltage $V_H$, and hence  $\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C, \\Gamma}<0$\t (from Eq. \\ref{eq:heat}). $P_H$ on the other hand is a non-monotonic function of the applied voltage $V_H$. $P_H$ increases from zero when $V_H$ increases from $V_H=0$ and then reaches a point when $\\frac{\\partial P_H}{\\partial V_H}\\Big|_{V_C, \\Gamma}=0$ and then again decreases making $\\frac{\\partial P_H}{\\partial V_H}\\Big|_{V_C, \\Gamma}<0$. Eq. \\eqref{eq:delta_P_H_2} demonstrates that for $\\frac{\\partial P_H}{\\partial V_H}\\Big|_{V_C, \\Gamma}>0$, $\\delta P_H$ is positive for any  $\\epsilon>\\epsilon_H$. The challenge for optimization of $\\Gamma_H(\\epsilon)$ sets in when finally $\\frac{\\partial P_H}{\\partial V_H}\\Big|_{V_C, \\Gamma}$ becomes negative. In this case, an increase in the power $\\delta P_H$ is positive only if\n\\begin{widetext}\n\n\\begin{gather}\n\\left\\{ 1-\\frac{\\epsilon-\\mu_H}{eV_H}\\frac{\\frac{\\partial P_H}{\\partial V_H}\\Big|_{V_C, \\Gamma}}{\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C, \\Gamma}}\\right\\}=\\left\\{1-\\frac{\\frac{\\partial P_H}{\\partial V_H}\\Big|_{V_C, \\Gamma}}{\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C, \\Gamma}}-\\frac{\\epsilon}{eV_H}\\frac{\\frac{\\partial P_H}{\\partial V_H}\\Big|_{V_C, \\Gamma}}{\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C, \\Gamma}}\\right\\}>0,\\nonumber \\\\\n\\Rightarrow \\epsilon <\\epsilon_H'= eV_H\\left\\{\\frac{\\frac{\\partial I_{Qe}}{\\partial V_H}\\Big|_{V_C, \\Gamma}}{\\frac{\\partial P_H}{\\partial V_H}\\Big|_{V_C, \\Gamma}}-1\\right\\}\n\\label{eq:jye_h_d}\n\\end{gather}\n\n\\end{widetext}\nHence, from Eq.~\\eqref{eq:jye_h} and \\eqref{eq:jye_h_d},  we note that an increase in $\\Gamma_H$ at energy $\\epsilon$ by an infinitesimal quantity $\\delta \\Gamma_H(\\epsilon)$, results in an enhancement in power generation only when\n\\begin{equation}\n\\epsilon_H\\leq \\epsilon \\leq \\epsilon_H' \\nonumber\n\\end{equation}\nHence, the maximum generated power for a given heat current is achieved when the transmission function is set to its maximum possible value in the  the  energy range between $\\epsilon_H$ and $\\epsilon_H'$. For other energy range in the transport window, the transmission function should be set to the minimum possible value. In this particular case, where $0 \\leq \\Gamma_H(\\epsilon) \\leq 1$, the transmission function in the energy range between $\\epsilon_H$ and $\\epsilon_H'$ should be set to 1. For other energy range $\\Gamma_H$ should be set to 0. Hence , the optimal function for $\\Gamma_H$ can be given by:\n\\begin{equation}\n\\Gamma_H=\\theta(\\epsilon-\\epsilon_H)-\\theta(\\epsilon-\\epsilon_H'),\n\\end{equation}\nwhere $\\theta$ is the unit step function. \n\\begin{figure}[!htb]\n\\includegraphics[scale=1.6]{result}\n\\caption{(a) Maximum power ($P_{MAX}(\\eta)$) vs efficiency ($\\frac{\\eta}{\\eta_C}$) for our proposed design strategy, optimized single filter design proposed in Refs. \\cite{whitney,whitney2} and ballistic nanowire filters. (b) Logarithm of the enhancement factor ($\\chi$) with our proposed design strategy compared to a a single  optimized  filter design (boxcar transmission) proposed in Refs. \\cite{whitney,whitney2} and ballistic nanowire filters. The efficiency plotted in the above figures are normalized with respect to the Carnot efficiency. For comparison of the optimal single filter design with the proposed design employing dual filter, the maximum value transmission function at any energy $\\epsilon$ for the optimized single filter is limited to $0.5$. }\n\\label{fig:result}\n\\end{figure}\n\\section{Results}\\label{results}\nIn this section, I demonstrate numerical simulation results for the case when $T_H=360K$, $T_{CH}=330K$ and $T_C=300K$. In this case, the values of the quantities $\\epsilon_C$,~$\\epsilon_H$ and $\\epsilon_H'$ were numerically computed, followed by a numerical computation of the maximum generated power $P_{MAX}$ for a given efficiency $\\eta$. For comparison, I normalize the efficiency with respect to the Carnot efficiency. To assess the performance enhancement via the proposed design strategy, I also demonstrate (in Fig.~\\ref{fig:result}a) the results for (i) an optimal single filter design with boxcar transmission function (proposed in Refs. \\cite{whitney,whitney2}) and (ii)  the filters $\\Gamma_H$ and $\\Gamma_C$ replaced by ballistic nanowires with transmission functions $\\Gamma_H=\\theta(\\epsilon-\\epsilon_H)$ and $\\Gamma_C=\\theta(\\epsilon-\\epsilon_C)$.  Fig.~\\ref{fig:result}(a) demonstrates the maximum power generated at a given efficiency for these three cases.  We note that the advantage gained by the proposed design strategy is quite prominent in the high efficiency regime of operation, when the output power is low compared to the quantum bound \\cite{whitney,whitney2}. In fact, when the desired output power is low compared to the quantum bound, the efficiency of operation, in the proposed design strategy, can be driven beyond the ballistic limit. However, such advantage gradually decreases as we  approach the point of maximum power generation. In particular, it should be noted that  all the three cases have identical magnitude of maximum generated power, in addition to identical efficiency at the point of maximum power. This is  expected, since $\\epsilon_H'$ in our proposed model increase with the increase in generated power. Hence, the transmission function $\\Gamma_H$ gradually approaches that of a ballistic nanowire when the generated power gradually approaches the quantum bound.\\\\\n\\indent To assess the enhancement in generated power at a given efficiency, I define a metric termed the \\textit{advantage factor} ($\\chi$), which is the ratio of the maximum generated power via our proposed design to the maximum  generated power via other designs.\n\\[ \n\\chi(\\eta)=\\frac{P_{{MAX}}^{opt}(\\eta)}{P_{MAX}^{other}(\\eta)},\n\\]\nwhere $P_{{MAX}}^{opt}(\\eta)$ is the maximum power generated via the proposed design with optimal filters and $P_{MAX}^{other}(\\eta)$ is the maximum generated power via other designs or strategies at efficiency $\\eta$.  It should be noted that $\\chi$ is a function of efficiency and is high in the regime of high efficiency.  Fig.~\\ref{fig:result}(b) demonstrates the logarithm $\\chi$ for two different cases, where the enhancement of generated power in our proposed design is compared with  (i) optimized single filter with boxcar transmission function as proposed in Refs.~\\cite{whitney,whitney2}, and (ii) the same design shown in Fig.~\\ref{fig:schem}, with the two filters replaced by ballistic nanowires, with $\\Gamma_H=\\theta(\\epsilon-\\epsilon_H)$ and $\\Gamma_C=\\theta(\\epsilon-\\epsilon_C)$. It should be noted that the enhancement of generated power near Carnot efficiency in our case is nearly $10^2$ times compared to the design proposed in Refs. \\cite{whitney,whitney2}. The enhancement in generated power compared to ballistic nanowire filters is about $10^3$ times at $80\\%$ of the Carnot efficiency, which is the maximum efficiency achieved via ballistic nanowire filters. It should, however, be noted that in this case the efficiency is defined with respect to electronic heat flow. In practice, however, the overall efficiency, defined with respect to the total heat flow due to electronic transport and lattice heat conductivity, should be much lower than the calculation result.\n\\section{Conclusion}\\label{conclusion}\nIn this paper, I have  proposed a design strategy to enhance the performance of heat engines  via inelastic processes within the channel. In particular, I have proposed energy filters at the contact-to-channel interfaces for the said purpose. The energy filter at the hot contact to channel junction inject a stream of electrons with effective temperature much lower than the channel region. These electrons then absorb  thermal energy from the channel and flow towards the cold contact. Via rigorous mathematical derivations, it was shown that the heat engine reaches its peak performance (in case of electron-like conduction or n-type channel), when the filter at the hot-contact to channel junction  implements a box-car shaped transmission function, while that at the cold-contact to channel junction implements a unit-step shaped transmission function. Then, with the help of numerical simulation, I have  demonstrated that the principal utility of the proposed design strategy lies in the high efficiency regime of operation, where the efficiency of the engine, with respect to the electronic heat conduction, can be driven beyond the ballistic limit. Although, I have demonstrated the results for a particular case with   $T_H=360K$, $T_{CH}=330K$ and $T_C=300K$, the proposed concepts remain valid for other temperature range as well. In this paper, however, I have not considered an optimization of channel temperature for an optimal performance of the heat engine. It remains an interesting direction to explore the optimal channel temperature with respect to the contact temperatures for the proposed strategy.\nThe proposed concept in this paper can lead to the development of efficient heat engines, in cases where the source of usable thermal energy is limited.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and summary} \n\\label{sec:intro} \n\n\\subsection{Context: resonant excitation of stellar normal modes during binary inspirals} \n\nThe near-simultaneous measurement of a neutron-star binary merger in gravitational waves (GW170817 \\cite{GW170817:17}), gamma rays \\cite{GW170817fermi:18, GW170817integral:17}, and other parts of the electromagnetic-wave spectrum \\cite{GW170817multi:17} marked the beginning of a new era of multimessenger astronomy that promises, in the fullness of time, to reveal the intimate details of neutron-star interiors. This effort is reinforced with the recent deployment of the NICER instrument \\cite{gendreau-etal:16} on board the International Space Station, which measures X-ray emissions from pulsars. Of primary interest is to elucidate the nature of nuclear matter at very high densities, which determines the stellar structure through the equation of state. The measurement of the mass and radius of neutron stars, through all conceivable means, provides constraints on the equation of state, and this in turn constrains the physics of nuclear matter at supernuclear densities. \n\nFlanagan and Hinderer \\cite{flanagan-hinderer:08} recognized that details of the neutron-star interior could be revealed by gravitational-wave measurements of the star's tidal polarizability, which varies sensitively with the equation of state. This observation initiated a large effort \\cite{flanagan-hinderer:08, hinderer-etal:10, vines-flanagan-hinderer:11, baiotti-etal:11, pannarale-etal:11, ferrari-gualtieri-maselli:12, lackey-etal:12, damour-nagar-villain:12, maselli-etal:12, maselli-etal:13, vines-flanagan:13, read-etal:13, maselli-gualtieri-ferrari:13, favata:14, yagi-yunes:14, lackey-etal:14, wade-etal:14, bernuzzi-etal:15, lackey-etal:17, cullen-etal:17, harry-hinderer:18, piekarewicz-fattoyev:19} to determine just how well this measurement can be made, and what information it can deliver on the star's internal structure. Such a measurement was attempted with GW170817 \\cite{GW170817:18}, and it delivered an astrophysically interesting upper bound. On the other hand, the NICER measurement of the mass and radius of the pulsar PSR J0030+0451 has already provided useful constraints on the nuclear equation of state \\cite{miller-etal:19}.\n\nAll means of measurement of neutron-star properties are subjected to systematic errors than can be difficult to quantify and control. It is therefore important to identify many independent ways of performing these measurements, in the hope that a consistent picture will emerge, in spite of the systematic errors. Other ways to gain access to the internal physics of neutron stars through gravitational waves include exploiting post-merger information at high frequencies \\cite{bauswein-janka:12, hotokezaka-etal:13, takami-rezzolla-baiotti:14, bauswein-stergioulas:15, martynov-etal:19}, and accurate modeling of dynamical tides, which implicate the $f$-mode of vibration of neutron stars \\cite{hinderer-etal:16, steinhoff-etal:16, schmidt-hinderer:19, ma-yu-chen:20}. A recent study by Pan {\\it et al.}\\ \\cite{pan-etal:20} reveals that interface modes \\cite{mcdermott-vanhorn-hansen:88, pan-etal:20}, which result from an interaction between the fluid core and solid crust of a neutron star, can be manifested during a binary inspiral, leading to phasing effects that can be measured by an interferometric detector in the LIGO A+ configuration \\cite{LIGOAplus}. \n\nYet another way, the focus of this paper, is to rely on the resonant excitation of a star's normal modes of vibration. The main idea is this: The tidal forces exerted by the neutron star's companion serve to perturb the star, and the perturbation can be decomposed into normal modes, which behave as a collection of driven harmonic oscillators. At some time during the inspiral, the frequency of the tidal field (either the orbital frequency or twice the orbital frequency) becomes momentarily equal to a mode's eigenfrequency, and this provokes the development of a resonance. The rapid growth of the mode during resonance takes energy away from the orbital motion\\footnote{As we shall see below, in some circumstances the resonance can provide energy to the orbital motion, at the expense of the star's rotational energy.}, and this affects the inspiral in a way that can be measured in the phasing of the emitted gravitational waves. \n\nEarly studies of this phenomenon \\cite{lai:94, reisenegger-goldreich:94, shibata:94, kokkotas-schafer:95, lai:97, ho-lai:99} concluded that $f$-modes cannot become resonant (because the mode frequency is too large, beyond the LIGO\/Virgo frequency band), and that while $g$-modes can be resonantly excited (because their frequencies are sufficiently low), they lead to orbital changes that are too small to be detected (because the coupling between tidal forces and $g$-modes is very small). The interface modes studied by Pan {\\it et al.}\\ \\cite{pan-etal:20} do not typically achieve resonance. \n\nRotating neutron stars offer a wider spectrum of normal modes, and therefore provide a larger set of possibilities for resonant tidal interactions. The focus of this paper is with the {\\it inertial modes} first identified by Lockitch and Friedman \\cite{lockitch-friedman:99}  (see also Refs.~\\cite{yoshida-lee:00, wu:05, passamonti-etal:09} for additional studies). These modes are predominantly perturbations of the star's velocity field, governed by a restoring force supplied by the stellar rotation (the Coriolis force, when viewed in the star's corotating frame). The property that makes these modes interesting for resonant tidal interactions is that their frequencies are of the same order of magnitude as the star's rotational angular velocity. For a star spinning with a frequency comparable to $100\\ \\mbox{Hz}$, around the peak frequency of the LIGO\/Virgo interferometers, the inertial modes permit the development of resonances that have a measurable impact on the phasing of the gravitational waves. The inertial modes include $r$-modes \\cite{papaloizou-pringle:78, provost-berthomieu-rocca:81, saio:82, smeyers-martens:83, lindblom-mendell-owen:99} as special members; while an $r$-mode gives rise to a velocity perturbation of pure axial parity, the decomposition of a generic inertial mode in vector harmonics involves both polar and axial terms.   \n\nThe coupling of inertial modes to a Newtonian tidal field was first investigated by Lai and Wu \\cite{lai-wu:06} (see also Ref.~\\cite{xu-lai:17} for a recent refinement that also includes $g$-modes), who concluded that extremely high rotational frequencies (beyond $100\\ \\mbox{Hz}$) would be required to achieve a measurable phase shift in the gravitational-wave signal. Flanagan and Racine \\cite{flanagan-racine:07}, however, pointed out that inertial modes couple more strongly to a post-Newtonian tidal field. While the Newtonian field is associated with second derivatives of the gravitational potential $U$, generated by the mass density of the orbiting companion, the post-Newtonian field is given by second derivatives of a vector potential $U_j$, produced by the mass currents associated with the companion's orbital motion. Drawing an analogy with the vector potential of Maxwell's electrodynamics, $U_j$ is often called a gravitomagnetic potential, and the resulting tidal field described as a gravitomagnetic field. The stronger coupling of inertial modes with a gravitomagnetic tidal field led Flanagan and Racine to a more optimistic conclusion regarding the measurability of the resonant interaction.  \n\n\\subsection{This work: gravitomagnetic tidal resonances from inertial modes} \n\nFlanagan and Racine \\cite{flanagan-racine:07} calculated the dynamical impact on a binary inspiral, and on the phasing of the emitted gravitational waves, of a resonant gravitomagnetic tidal interaction. In their setup, the tidal field couples to an inertial mode of the rotating neutron star, and a resonance occurs when the orbital frequency momentarily becomes equal to the mode's eigenfrequency. During resonance the velocity perturbation within the star gives rise to a nonvanishing mass-current quadrupole moment\\footnote{This is essentially the integral of the current density $\\rho\\, \\delta \\bm{v}$, where $\\rho$ is the mass density and $\\delta \\bm{v}$ the velocity perturbation, multiplied by two powers of $\\bm{x}$, the position vector relative to the star's center of mass.}  $J^{jk}$ which, in turn, produces another contribution to the gravitomagnetic vector potential that affects the binary's orbital motion. So while the coupling between orbit and mode is mediated by the gravitomagnetic tidal field created by the companion, the coupling between mode and orbit is mediated by the current quadrupole moment developed within the neutron star.  \n\nFlanagan and Racine, however, examined only the coupling with $r$-modes, in spite of the fact that other inertial modes do participate in the tidal interaction. My purpose with this paper is to incorporate these additional modes in the analysis, and therefore to produce a more complete understanding of the resonant tidal interaction. As we shall see presently, proper inclusion of all relevant inertial modes produces a qualitatively different picture of the tidal encounter.\n\nInertial modes of a rotating star are labelled by two integers. The first is $m$, which determines the $e^{im\\phi}$ dependence of the velocity perturbation on the azimuthal angle $\\phi$; because a mode with a negative value of $m$ is the complex conjugate of a mode with the corresponding positive value, it is sufficient to take $m \\geq 0$. The second integer is $n$, which sequences the infinity of overtones for each value of $m$. Each inertial mode comes with a distinct eigenfrequency $\\omega_n^m = w_n^m \\Omega$, where $\\Omega$ is the star's rotational angular velocity, and \n$w_n^m$ is a number of order unity.\n\n\\begin{figure} \n\\includegraphics[width=0.7\\linewidth]{fig1}\n\\caption{Resonant tidal interaction during a binary inspiral. The thick black curve represents the sweep in orbital frequency that occurs during an inspiral. The binary parameters are given by the fiducial values adopted in Eq.~(\\ref{DeltaPhi_intro}); in particular, the star's rotational frequency is chosen to be 100 Hz. The zero of time corresponds to a matching orbital frequency of 100 Hz. The thin horizontal lines mark the frequencies of the four inertial modes discussed in the text; a resonance takes place when the orbital frequency momentarily becomes equal to one of these frequencies. For the stellar model adopted in this plot --- Eq.~(\\ref{rho_kmodel_intro}) with $k=1$ --- the lowest resonant frequency (41 Hz) corresponds to the mode labelled $(m = 1, n = \\mbox{II})$, the second one (103 Hz) corresponds to $(m = 0, n = \\mbox{I})$, the third (133 Hz) corresponds to the $r$-mode $(m=2, n = \\bullet)$, and the highest frequency (140 Hz) corresponds to $(m=1, n=\\mbox{I})$.}  \n\\label{fig:inspiral} \n\\end{figure} \n\nI show in the technical sections of the paper that the resonant tidal interaction implicates four dominant modes, those that possess the strongest couplings with the gravitomagnetic tidal field; the situation is illustrated in Fig.~\\ref{fig:inspiral}. The first is an $r$-mode with $m=2$, which is labelled $(m = 2, n = \\bullet)$. The second is a $m=1$ mode with a positive frequency, to which I assign the label $(m = 1, n = \\mbox{I})$. The third is another $m=1$ mode with a negative frequency, labelled $(m = 1, n = \\mbox{II})$. The fourth is actually a complex-conjugate pair of $m = 0$ modes, with positive and negative frequencies of equal magnitude; this pair of modes is labelled $(m = 0, n = \\mbox{I})$. Each mode gives rise to its own resonant tidal interaction, so that the encounter is actually a succession of four resonances; the total accumulated gravitational-wave phase shift is the sum of four contributions. I find that this phase shift is given by   \n\\begin{equation} \n\\Delta\\Phi_{\\rm GW} = \\gamma(\\iota) \\biggl( \\frac{R}{10\\ \\mbox{km}} \\biggr)^4 \n\\biggl( \\frac{1.4\\ M_\\odot}{M} \\biggr)^2 \n\\biggl( \\frac{1.4\\ M_\\odot}{M'} \\biggr) \n\\biggl( \\frac{2.8\\ M_\\odot}{M_{\\rm tot}} \\biggr)^{1\/3} \n\\biggl( \\frac{\\Omega\/2\\pi}{100\\ \\mbox{Hz}} \\biggr)^{2\/3}, \n\\label{DeltaPhi_intro} \n\\end{equation} \nwhere\n\\begin{equation}\n\\gamma(\\iota) := \\mathfrak{u}_\\bullet^2\\, \\sin^2\\iota(\\cos\\iota + 1)^2 \n+ \\mathfrak{u}_{\\rm I}^1\\, (\\cos\\iota + 1)^2 (2\\cos\\iota - 1)^2 \n+ \\mathfrak{u}_{\\rm II}^1\\, (\\cos\\iota - 1)^2 (2\\cos\\iota + 1)^2\n+ \\mathfrak{u}_{\\rm I}^0\\, \\sin^2\\iota \\cos^2\\iota, \n\\label{gamma_intro} \n\\end{equation} \nand where each number $\\mathfrak{u}_n^m$, defined by Eq.~(\\ref{unm_def}) in the main text, is essentially a normalized overlap integral between the mode $(m , n)$ and the gravitomagnetic tidal field.  \n\nIn Eq.~(\\ref{DeltaPhi_intro}), $R$ is the radius of the rotating neutron star, normalized by a fiducial value of $10\\ \\mbox{km}$, $M$ is its mass, normalized by a fiducial value of $1.4\\ M_\\odot$, and $\\Omega$ is the rotational angular velocity, normalized with a corresponding frequency of $100\\ \\mbox{Hz}$; the second mass $M'$, also normalized with $1.4\\ M_\\odot$, is the companion's mass, and $M_{\\rm tot} = M + M'$ is the binary's total mass. In Eq.~(\\ref{gamma_intro}), $\\iota$ is the orbit's inclination, the angle between the star's spin vector and the orbital angular-momentum vector. The numbers $\\mathfrak{u}_n^m$ are listed in Table~\\ref{tab:uvalues_intro} for stellar models defined by the density function\n\\begin{equation}\n\\rho = C \\biggl[ \\frac{\\sin(\\pi r\/R)}{(\\pi r\/R)} \\biggr]^k,\n\\label{rho_kmodel_intro} \n\\end{equation}\nwhere $r$ is the radial coordinate inside the star, and $C$, $k$ are constants; low values of $k$ correspond to stars of nearly uniform density, while high values correspond to centrally dense stars. The model with $k=1$ results from the polytropic equation of state $p = K\\rho^2$, where $p$ is the pressure and $K$ a constant. This makes a crude but serviceable model for a realistic equation of state for neutron-star matter. The stellar model with $k=1$ can therefore be considered to be the most realistic one.   \n\n\\begin{table} \n\\caption{\\label{tab:uvalues_intro} Numerical results for the coefficients $\\mathfrak{u}_n^m$, defined by Eq.~(\\ref{unm_def}) in the main text, obtained for the density models of Eq.~(\\ref{rho_kmodel_intro}). The first column gives $\\gamma$ for $\\iota = \\pi$, while the second column lists $\\gamma$ for $\\iota = 0$; these are defined in Eq.~(\\ref{gamma_intro}).} \n\\begin{ruledtabular} \n\\begin{tabular}{ccccccc}\n$k$ & $\\gamma(\\pi)$ & $\\gamma(0)$ & $\\mathfrak{u}_\\bullet^2$ & $\\mathfrak{u}_{\\rm I}^1$ & $\\mathfrak{u}_{\\rm II}^1$ & $\\mathfrak{u}_{\\rm I}^0$ \\\\ \n\\hline\n  0 & 2.3141e-1 & 4.7264e-2 & -6.1322e-2 & 1.1816e-2 & 5.7853e-2 & 2.8195e-1 \\\\\n  1 & 1.4877e-1 & 4.1400e-2 & -2.9403e-2 & 1.0350e-2 & 3.7192e-2 & 1.4807e-1 \\\\\n  2 & 9.0848e-2 & 2.6154e-2 & -1.6323e-2 & 6.5384e-3 & 2.2712e-2 & 8.5073e-2 \\\\\n  3 & 5.8932e-2 & 1.7123e-2 & -1.0156e-2 & 4.2809e-3 & 1.4733e-2 & 5.3653e-2 \n\\end{tabular} \n\\end{ruledtabular} \n\\end{table} \n\nThe first term on the right of Eq.~(\\ref{gamma_intro}) is the $r$-mode contribution to the total accumulated phase shift, the one that was previously calculated by Flanagan and Racine \\cite{flanagan-racine:07}. This term is proportional to $\\sin^2\\iota(\\cos\\iota+1)^2$, which is maximized when $\\iota = \\pi\/3$. As Table~\\ref{tab:uvalues_intro} indicates, the associated number $\\mathfrak{u}^2_\\bullet$ is negative, which implies that the $r$-mode makes a negative contribution to the phase shift. With the sign convention used here\\footnote{The sign convention is the one of Flanagan and Racine \\cite{flanagan-racine:07}, for whom a positive $\\Delta \\Phi_{\\rm GW}$ corresponds to a shorter inspiral. This convention is opposite to the one adopted by Lai in Ref.~\\cite{lai:94}.}, this means that the resonant tidal interaction provides energy to the orbital motion, thereby increasing the orbital radius and prolonging the inspiral. This behavior has to do with the well-known fact that the $r$-mode is subjected to the Chandrasekhar-Friedman-Schutz instability \\cite{chandrasekhar:70, friedman-schutz:78b, andersson:98, friedman-morsink:98}; the increase in orbital energy comes at the expense of the star's rotational energy.\n\nThe other terms on the right of Eq.~(\\ref{gamma_intro}) are the contributions from the remaining inertial modes. They are all positive, which means that the resonant tidal interaction removes energy from the orbital motion, thereby decreasing the orbital radius and shortening the inspiral. The contribution from the $(m=1, n = \\mbox{I})$ mode is proportional to $(\\cos\\iota + 1)^2(2\\cos\\iota - 1)^2$, which is maximized when $\\iota = 0$, that is, when the angular-momentum vectors are aligned. The contribution from the $(m=1, n = \\mbox{II})$ mode is proportional to $(\\cos\\iota - 1)^2(2\\cos\\iota + 1)^2$; this is maximized when $\\iota = \\pi$, for anti-aligned angular-momentum vectors. Finally, the contribution from the $(m=0, n = \\mbox{I})$ pair of modes is proportional to $\\sin^2\\iota \\cos^2\\iota$, which is maximized when $\\iota = \\pi\/4$ or $\\iota = 3\\pi\/4$. \n\nPlots of $\\gamma(\\iota)$, the sum of all four contributions to the gravitational-wave phase shift, are displayed in Fig.~\\ref{fig:phaseshift_intro} for selected values of $k$. The main observation is that the phase shift is maximized when $\\iota = \\pi$, that is, when the spin is anti-aligned with orbital angular momentum. In this situation the only contributing mode is the one labelled by $(m = 1, n = {\\rm II})$ --- the inertial mode with a negative frequency --- and the phase shift becomes $\\gamma(\\iota = \\pi) = 4 \\mathfrak{u}^1_{\\rm II}$. On the other hand, when the spin is aligned with the orbital angular momentum, so that $\\iota = 0$, the only contributing mode is $(m = 1, n = {\\rm I}$), and we have that $\\gamma(\\iota = 0) = 4 \\mathfrak{u}^1_{\\rm I}$. The values of $\\gamma$ at $\\iota = \\pi$ and $\\iota = 0$ are listed in Table~\\ref{tab:uvalues_intro}. \n\nIt can also be observed from Table~\\ref{tab:uvalues_intro} that $\\mathfrak{u}_n^m$ decreases (in absolute value) with increasing $k$: stellar models that are more centrally dense produce a smaller phase shift. This behavior is to be expected, on the grounds that everything else being equal, a star that is more centrally dense develops a smaller current quadrupole moment $J^{jk}$, and therefore undergoes a weaker tidal interaction. \n\nWith realistic values for $M$, $M'$, $\\Omega$, and $R$, and for density models that are neither too uniform nor centrally dense, $\\Delta \\Phi_{\\rm GW}$ can be expected to be in an interval between $0.02$ and $0.04$ radians when the spin is approximately aligned with the orbital angular momentum, or in an interval between $0.08$ and $0.16$ radians when the spin is approximately anti-aligned. If the companion also is a rotating neutron star, then both stars will participate in the resonant tidal interaction, and the total accumulated phase shift will be multiplied by two (assuming that the spins are comparable). The chosen fiducial value for $R$ --- $10\\ \\mbox{km}$ --- corresponds to a fairly soft equation of state that produces a relatively small star; a stiffer equation of state would return a larger stellar radius. A more optimistic choice of $13\\ \\mbox{km}$ for the fiducial value leads to an enhancement of the phase shift by a factor of $2.9$ --- thanks to the scaling with $R^4$, a little optimism goes a long way. On the other hand, the selection of $100\\ \\mbox{Hz}$ as a fiducial value for $\\Omega\/(2\\pi)$ might be overly optimistic, given that the fastest known pulsar rotates at a rate of about $44\\  \\mbox{Hz}$\\cite{burgay-etal:03}. A reduction of the fiducial value to $40\\ \\mbox{Hz}$ would reduce the phase shift by a factor of $0.54$ --- because of the scaling with $\\Omega^{2\/3}$, strong pessimism comes with a relatively small impact.   \n\n\\begin{figure} \n\\includegraphics[width=0.7\\linewidth]{fig2}\n\\caption{Gravitational-wave phase shift $\\gamma$ as a function of inclination angle $\\iota$, for selected values of the density model parameter $k$. As stated in the text, the density model with $k=1$ gives a crude but effective description of a neutron star with a realistic equation of state.}   \n\\label{fig:phaseshift_intro} \n\\end{figure} \n \nWith good constraints on the masses and spins (and therefore on the inclination angle $\\iota$) gathered from the inspiral waveform, $\\Delta \\Phi_{\\rm GW}$ provides information regarding the star's internal structure through the quantity $\\gamma(\\iota) R^4$, which depends on the equation of state. Just how well this information can be extracted from gravitational-wave measurements, and precisely what can be inferred about the equation of state, are important questions that lie beyond the scope of the paper. For the time being I shall limit myself to a statement of optimism regarding future measurements of the gravitomagnetic tidal resonances: \n\nIn their recent study of the excitation of interface modes during binary inspirals, Pan {\\it et al.}\\ \\cite{pan-etal:20} estimate that for an event comparable to GW170817 \\cite{GW170817:17}, the gravitational-wave phase shift $\\Delta \\Phi_{\\rm GW}$ can be measured by the current LIGO\/Virgo facility within an uncertainty comparable to $0.1$ radians. This, unfortunately, is about the size of the effect in the most favorable circumstances. But these authors also estimate that for a similar event, measured with a signal-to-noise ratio of 100 by LIGO A+ \\cite{LIGOAplus}, the uncertainty in the phase shift will decrease by an order of magnitude, a sufficient improvement to reveal the phenomenon. A similar conclusion was reached by Yu {\\it et al.}\\ \\cite{yu-etal:18} in their proposal for a detector upgrade (LIGO-LF) that dramatically improves the sensitivity at low frequencies. I take this as a strong indication that the gravitational-wave phase shift accumulated during a gravitomagnetic tidal resonance will be within reach in the current decade, at the cost of an incremental improvement of the LIGO\/Virgo instruments. Thinking ahead, the resonant tidal interaction will have to be incorporated in templates when the waves are measured by the next generation of detectors, such as Cosmic Explorer \\cite{CosmicExplorer, reitze-etal:19} and the Einstein Telescope \\cite{EinsteinTelescope, punturo:10}. \n\n\\subsection{Some fine print: approximations made in this paper}\n\nThe results summarized in the preceding subsection were obtained by formulating various approximations to the correct and complete physics of the problem. I state and discuss these assumptions here. \n\nFirst, the gravitomagnetic tidal field acting on the neutron star is described to leading order only in a post-Newtonian expansion in powers of $v'\/c$, where $v'$ is the orbital velocity and $c$ is the speed of light. At this leading, first post-Newtonian order, it derives from a vector potential $U_j$ created by the mass current associated with the companion's orbital motion. The vector potential is essentially the time-space part of the metric tensor. \n\nSecond, the fluid dynamics is described within a purely Newtonian framework. While the external forces have a post-Newtonian origin, and the resulting perturbation of the fluid configuration is also of the first post-Newtonian order, the perturbation is nevertheless calculated with the Newtonian fluid equations. There is no inconsistency with this approach, because the focus is on gravitomagnetic tidal effects; other post-Newtonian corrections are ignored. \n\nThird, the unperturbed ``neutron star'' is actually a Newtonian configuration of perfect fluid, self-gravitating and rotating rigidly with an angular velocity $\\Omega$, governed by a barotropic equation of state of the form $p = p(\\rho)$, where $p$ is the pressure and $\\rho$ the mass density. The perturbed star is assumed to obey the same equation of state. Deviations from the barotropic form, associated with composition gradients and\/or a nonzero temperature, would lead to buoyancy effects in the inertial modes, and these are neglected. Similarly, no attempt is made to account for the influence of the superfluid core on the modes (for a detailed study on $r$-modes, see Ref.~\\cite{lindblom-mendell:00}). \n\nFourth, it is assumed that the star is rotating slowly, in the sense that $\\Omega^2 \\ll GM\/R^3$, where $G$ is the gravitational constant. All centrifugal corrections to the stellar structure are neglected in this approximation; the surfaces of constant density and pressure are spheres. The inertial modes and their frequencies are calculated within this approximation. \n\nFifth, the binary's inspiral is described to leading order only in a post-Newtonian expansion of the radiation reaction force. At this order, the orbital evolution is a second-and-a-half post-Newtonian effect, associated with the loss of orbital energy and angular momentum to gravitational waves. The orbit is taken to be circular.\n\nSixth, the treatment provided here of the development of gravitomagnetic tidal resonances is based on the assumption that the transit time through a resonance is short compared with the radiation-reaction time scale. The ratio of time scales is denoted $\\epsilon$, and many expressions below neglect fractional corrections of order $\\epsilon$. Because $\\epsilon \\sim 10^{-2}$ for binaries of interest [see Eq.~(\\ref{epsilon_def}) below], this provides a perfectly adequate approximation. A consequence of this hierarchy of time scales is that the four resonances are well separated in time, since they each have a distinct frequency.  \n\nSeventh, dissipation of the velocity perturbation through the coupling of an inertial mode to the star's shear and bulk viscosity, and through the mode's own emission of gravitational waves, is neglected. In this treatment, the only source of dissipation in the system is the production of gravitational waves by the binary's orbital motion.   \n\nAll these approximations imply that the numbers listed in Table~\\ref{tab:uvalues_intro} can be expected to be accurate only up to about 10 or 20 percent. The dominant source of error is the Newtonian treatment of the neutron-star interior and the Newtonian calculation of inertial modes. Promotion to general relativity will have to await future work.  \n\n\\subsection{Organization of this paper}\n\nIn the first part of the paper --- Secs.~\\ref{sec:governing} to \\ref{sec:solution} --- I calculate the response of a slowly rotating star to an applied gravitomagnetic tidal field, over a time interval that is long compared with the orbital period, but short compared with the transit time through a resonance (and therefore also short compared with the radiation-reaction time scale). On this short-term view the binary's orbital radius can be taken to be fixed, and the frequency of the tidal field is approximately constant. The equations that govern the dynamics of a barotropic perfect fluid are reviewed in Sec.~\\ref{sec:governing}, and the gravitomagnetic tidal force is introduced in Sec.~\\ref{sec:gravitomagnetic}. The explicit form of the perturbation equations is obtained in Sec.~\\ref{sec:perturbation}, and solutions to these equations are constructed in Sec.~\\ref{sec:solution}. The main goal in this first part of the paper is to establish that the star's tidal response can be described in terms of dimensionless current quadrupole moments $\\gothj_m(\\omega)$, one for each value of $m = \\{0, 1, 2\\}$, which are functions of the orbital frequency $\\omega$. These quantities are defined in Eq.~(\\ref{j_components}) below, they are plotted in Figs.~\\ref{fig:j_m2}, \\ref{fig:j_m1}, and \\ref{fig:j_m0}, and the current quadrupole moment tensor $J^{jk}$ is related to them via Eqs.~(\\ref{J_decomposed}) and (\\ref{J_components}).  \n\nThe second part of the paper --- Sec.~\\ref{sec:oscillator} --- is also concerned with a short-term view of the gravitomagnetic tidal interaction. It is devoted to a representation of the velocity perturbation in terms of a sum over normal modes. While an exact expression would require a sum over an infinite number of modes, the main purpose of Sec.~\\ref{sec:oscillator} is to establish that an excellent approximation results from the inclusion of just four modes, namely the inertial modes introduced previously. The four-mode approximation is at once suggested and validated by Figs.~\\ref{fig:j_m2}, \\ref{fig:j_m1}, and \\ref{fig:j_m0}. The suggestion comes from the fact that the reduced current quadrupole moments $\\gothj_m(\\omega)$ feature simple poles at eigenfrequencies of the inertial modes; the validation comes from the agreement between the solid curves (the outcome of the four-mode approximation) and the open diamonds (the result of the direct calculation presented in Sec.~\\ref{sec:solution}).    \n\nIn the third part of the paper --- Sec.~\\ref{sec:dynamics} --- I turn to a long-term view in which the relevant time interval is of the order of the radiation-reaction time. In this view, the orbital radius decreases steadily in response to the radiation-reaction force, and at some point in the inspiral the orbital frequency momentarily becomes equal to a mode's  eigenfrequency. A succession of four resonances takes place, during which the orbital motion is significantly affected by its coupling with each mode; this coupling is mediated by the current quadrupole moment $J^{jk}$. I describe the inspiral, calculate how each mode grows during the development of a resonance, show how this affects the orbital motion, and obtain the gravitational-wave phase shift associated with each resonance. The final result was presented in  Eqs.~(\\ref{DeltaPhi_intro}) and (\\ref{gamma_intro}).    \n\nThe paper also contains an Appendix (Sec.~\\ref{sec:phaseshift}), in which some (nonessential) technical matters are relegated.  \n\nThe developments in the main text rely heavily on the Lagrangian theory of fluid perturbations supplied by Friedman and Schutz \\cite{friedman-schutz:78a}, as well as the extensions contributed by Schenk {\\it et al.}\\ \\cite{schenk-etal:01}. I also made use of the methods introduced by Lockitch and Friedman \\cite{lockitch-friedman:99} to calculate the inertial modes of slowly rotating, barotropic stars, and those devised by Flanagan and Racine \\cite{flanagan-racine:07} to determine the dynamical impact of a gravitomagnetic tidal resonance.   \n\n\\section{Fluid equations} \n\\label{sec:governing} \n\nWe (you and I, working together) consider a star of mass $M$ and radius $R$, rotating slowly and rigidly with an angular velocity $\\Omega$, perturbed by a gravitomagnetic tidal field described by a post-Newtonian vector potential $U_a$. The slow-rotation assumption implies that the unperturbed star is approximately spherically symmetric, with all centrifugal deformations neglected. We take the star to be described by a perfect fluid with a barotropic equation of state $p = p(\\rho)$, where $p$ is the pressure and $\\rho$ the mass density. The physics of the unperturbed stellar interior is described within the framework of Newtonian fluid mechanics and gravitation. The first governing equation is Euler's equation    \n\\begin{equation} \n0 = E_a := \\partial_t v_a + v^b \\nabla_b v_a - \\nabla_a (U - h), \n\\label{euler} \n\\end{equation} \nwhere $v^a$ is the velocity field, $U$ the Newtonian gravitational potential, and $h$ the specific enthalpy, defined by $dh = \\rho^{-1}\\, dp$. Euler's equation is written in covariant form, and can therefore be formulated in any coordinate system; the same remark applies to all equations in this section. For the unperturbed state we have that $v^a = \\Omega\\, \\phi^a$, where $\\phi^a$ is the azimuthal Killing vector, the time-derivative term vanishes in Eq.~(\\ref{euler}), and the term quadratic in $\\Omega$ is neglected within the slow-rotation approximation; the equation of hydrostatic equilibrium reduces to $\\nabla_a (U - h) = 0$. The second governing equation is the continuity equation \n\\begin{equation} \n0 = C := \\partial_t \\rho + \\nabla_a (\\rho v^a),  \n\\label{continuity} \n\\end{equation} \nwhich is trivially satisfied for a static and axisymmetric distribution of mass. A third governing equation is Poisson's equation, \n\\begin{equation} \n0 = P := \\nabla^2 U +  4\\pi G \\rho,   \n\\label{poisson} \n\\end{equation} \nwhich determines the gravitational potential.  \n\nThe perturbation takes its origin in post-Newtonian gravity. We suppose that the star is placed within a time-dependent gravitomagnetic tidal field described by a vector potential $U_a$, and that the fluid is therefore subjected to a perturbing force density given by $\\rho f_a$, where  \n\\begin{equation} \nf_a = \\frac{4}{c^2} \\bigl[ \\partial_t U_a + v^b (\\nabla_b U_a - \\nabla_a U_b) \\bigr]. \n\\label{force_density} \n\\end{equation} \nThis is Eq.~(8.119) of Ref.~\\cite{poisson-will:14}, after discarding all terms that are irrelevant for our purposes, because they do not involve the vector potential.  \n\nThe tidal force creates a perturbation in the fluid, described by the Eulerian changes $\\{ \\delta v^a, \\delta\\rho, \\delta p, \\delta h, \\delta U \\}$. The perturbation of Euler's equation is \n\\begin{equation} \n\\delta E_a = f_a, \n\\label{dEuler} \n\\end{equation} \nwhere \n\\begin{equation} \n\\delta E_a := \\partial_t \\delta v_a + \\delta v^b\\, \\nabla_b v_a + v^b \\nabla_b \\delta v_a \n- \\nabla_a (\\delta U - \\delta h). \n\\label{dEa} \n\\end{equation} \nThe velocity perturbation can be related to a Lagrangian displacement vector $\\xi^a$ by \n\\begin{equation} \n\\delta v^a = \\partial_t \\xi^a - \\Lie_\\xi v^a = (\\partial_t + \\Lie_v) \\xi^a  \n= (\\partial_t + \\Omega\\, \\partial_\\phi) \\xi^a,  \n\\label{xi_vs_dv} \n\\end{equation}  \nwhere $\\Lie$ is the Lie-derivative operator. The perturbed continuity equation implies that \n\\begin{equation} \n\\delta \\rho = -\\nabla_a (\\rho \\xi^a), \n\\label{drho} \n\\end{equation} \nand $\\delta p$, $\\delta h$ can be obtained from $\\delta \\rho$ and the equation of state. The perturbed Poisson equation is \n\\begin{equation} \n\\nabla^2 \\delta U + 4\\pi G \\delta\\rho = 0, \n\\label{dPoisson} \n\\end{equation} \nand it determines $\\delta U$. \n\nThe perturbation equations come with the requirement that $\\Delta h = 0$ at the stellar surface, where $\\Delta h = \\delta h + \\xi^a \\nabla_a h$ is the Lagrangian change in the specific enthalpy. With $dh\/dr = -GM\/R^2$ at $r=R$, the boundary condition becomes \n\\begin{equation} \n\\delta h(r = R) = \\frac{GM}{R^2} r_a \\xi^a (r=R), \n\\label{surface_condition} \n\\end{equation} \nwhere $r_a := \\nabla_a r$ is the radial unit vector. \n\n\\section{Gravitomagnetic tidal force} \n\\label{sec:gravitomagnetic} \n\n\\subsection{Tidal quadrupole moment} \n\nThe gravitomagnetic tidal field is characterized by a symmetric-tracefree quadrupole-moment tensor ${\\cal B}_{jk}(t)$, which gives rise to the vector potential \n\\begin{equation} \nU_j = -\\frac{1}{6} \\epsilon_{jkp} {\\cal B}^p_{\\ q} x^k x^q, \n\\label{U_vs_B} \n\\end{equation} \nwhere $\\epsilon_{jkp}$ is the permutation symbol, and $x^j := (x,y,z)$ are Cartesian coordinates attached to the star's center-of-mass; the $z$-direction is aligned with the rotation axis.\\footnote{While all equations in Sec.~\\ref{sec:governing} were written in covariant form, the expression for the vector potential is restricted to Cartesian coordinates. To distinguish covariant equations from equations valid only in Cartesian coordinates, we use indices $a, b, c, \\ldots$ in covariant equations, and indices $j, k, p, \\ldots$ in Cartesian equations.} Equation (\\ref{U_vs_B}) follows from the requirements that the vector potential must satisfy Laplace's equation $\\nabla^2 U_j = 0$ and the gauge condition $\\partial_j U^j = 0$. The potential is substituted within Eq.~(\\ref{force_density}) to obtain the tidal force density. \n\nThe tidal field is created by the orbital motion of a companion body. This companion has a mass $M'$, and the orbital radius is denoted $p$ (the standard symbol for semi-latus rectum). The orbital angular velocity is\n\\begin{equation} \n\\omega = \\sqrt{\\frac{GM_{\\rm tot}}{p^3}}, \n\\label{orbital} \n\\end{equation} \nwhere $M_{\\rm tot} := M + M'$ is the binary's total mass. The orbital velocity is $v' = p\\, \\omega$. \n\nIn a generic orientation, the normal to the orbital plane is directed along the vector $\\bm{l}$, the direction of the companion (with respect to the origin of the coordinate system) is $\\bm{n}$, and the direction of the orbital velocity vector is $\\bm{\\lambda}$. These vectors have components $\\bm{l} = [0,-\\sin\\iota,\\cos\\iota]$, $\\bm{n} = [\\cos\\omega t, \\cos\\iota\\, \\sin\\omega t, \\sin\\iota\\, \\sin\\omega t]$, and $\\bm{\\lambda} = [-\\sin\\omega t, \\cos\\iota\\, \\cos\\omega t, \\sin\\iota\\, \\sin\\omega t]$, where $\\iota$ is the inclination angle between the normal to the orbital plane and the star's rotation axis. The vectors have a unit length, they are mutually orthogonal, and $\\bm{l} = \\bm{n} \\times \\bm{\\lambda}$.\n\nThe tidal quadrupole moment is given by \\cite{taylor-poisson:08} \n\\begin{equation} \n{\\cal B}_{jk} = 3 {\\cal B}_0 \\bigl( l_j n_k + n_j l_k \\bigr), \\qquad \n{\\cal B}_0 := \\frac{GM'v'}{p^3} = \\frac{GM'}{(GM_{\\rm tot})^{2\/3}}\\, \\omega^{7\/3}.   \n\\label{Bjk} \n\\end{equation} \nAn explicit listing of components is \n\\begin{subequations} \n\\label{B_listing} \n\\begin{align} \n{\\cal B}_{xx} &= 0, \\\\ \n{\\cal B}_{xy} &= -3 {\\cal B}_0 \\sin\\iota\\, \\cos(\\omega t), \\\\\n{\\cal B}_{xz} &= 3 {\\cal B}_0 \\cos\\iota\\, \\cos(\\omega t), \\\\\n{\\cal B}_{yy} &= -6 {\\cal B}_0 \\sin\\iota\\cos\\iota\\, \\sin(\\omega t), \\\\\n{\\cal B}_{yz} &= 3 {\\cal B}_0 (2\\cos^2\\iota-1)\\, \\sin(\\omega t), \\\\\n{\\cal B}_{zz} &= 6 {\\cal B}_0 \\sin\\iota\\cos\\iota\\, \\sin(\\omega t). \n\\end{align} \n\\end{subequations} \n\n\\subsection{Force density} \n\nWe substitute Eqs.~(\\ref{B_listing}) within Eq.~(\\ref{U_vs_B}), and this within the force density of Eq.~(\\ref{force_density}). For our purposes below it is useful to express the result in spherical coordinates, as\n\\begin{equation}\nf_a = \\sum_{m=-2}^2 f_a^m e^{im\\phi} e^{-i\\omega t} + \\mbox{cc}, \n\\label{fa_decomposed} \n\\end{equation}\nwhere ``cc'' denotes the complex conjugate of the preceding expression. The coefficients are given by \n\\begin{subequations}\n\\label{fa_listing_m2} \n\\begin{align} \nf^{\\pm 2}_r &= \\frac{3i}{2} \\frac{{\\cal B}_0}{c^2} \\sin\\iota (\\cos\\iota \\pm 1) \\Omega \\sin^2\\theta\\cos\\theta\\, r^2, \\\\\nf^{\\pm 2}_\\theta &= \\frac{i}{2} \\frac{{\\cal B}_0}{c^2} \\sin\\iota (\\cos\\iota \\pm 1) (-3\\Omega \\sin^2\\theta\n\\pm \\omega) \\sin\\theta\\, r^3, \\\\\nf^{\\pm 2}_\\phi &= -\\frac{1}{2} \\frac{{\\cal B}_0}{c^2} \\sin\\iota (\\cos\\iota \\pm 1) \\omega \\sin^2\\theta\\cos\\theta\\, r^3, \n\\end{align}\n\\end{subequations} \n\\begin{subequations}\n\\label{fa_listing_m1} \n\\begin{align} \nf^{\\pm 1}_r &= \\pm \\frac{3}{2} \\frac{{\\cal B}_0}{c^2} (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) \\Omega (2\\cos^2\\theta - 1)\\sin\\theta\\, r^2, \\\\\nf^{\\pm 1}_\\theta &= \\pm \\frac{1}{2} \\frac{{\\cal B}_0}{c^2} (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) (-6\\Omega\\sin^2\\theta \\pm \\omega ) \\cos\\theta\\, r^3, \\\\\nf^{\\pm 1}_\\phi &= \\pm \\frac{i}{2} \\frac{{\\cal B}_0}{c^2} (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) \\omega \\sin\\theta\n(2\\cos^2\\theta - 1) r^3,\n\\end{align}\n\\end{subequations} \nand\n\\begin{subequations}\n\\label{fa_listing_m0} \n\\begin{align} \nf^0_r &= -9i\\frac{{\\cal B}_0}{c^2} \\sin\\iota \\cos\\iota\\, \\Omega \\sin^2\\theta \\cos\\theta\\, r^2, \\\\\nf^0_\\theta &= -3i\\frac{{\\cal B}_0}{c^2} \\sin\\iota \\cos\\iota\\, \\Omega \\sin\\theta (3\\cos^2\\theta-1)\\, r^3, \\\\\nf^0_\\phi &= 3\\frac{{\\cal B}_0}{c^2} \\sin\\iota \\cos\\iota\\, \\omega \\sin^2\\theta \\cos\\theta\\, r^3. \n\\end{align}\n\\end{subequations} \n\n\\subsection{Curl of the force density} \n\nIn subsequent developments we shall also need the curl of the force density, \n\\begin{equation} \nq^a := \\epsilon^{abc} \\nabla_b f_c,  \n\\label{q_def} \n\\end{equation}    \nwhere $\\epsilon_{abc}$ is the completely antisymmetric Levi-Civita tensor; in spherical coordinates we have that $\\epsilon_{r\\theta\\phi} = r^2\\sin\\theta$. We decompose $q^a$ as\n\\begin{equation}\nq^a = \\sum_{m=-2}^2 q^a_m e^{im\\phi} e^{-i\\omega t} + \\mbox{cc},\n\\label{qa_decomposed} \n\\end{equation}\nwith\n\\begin{subequations}\n\\label{qa_listing_m2}\n\\begin{align}\nq^r_{\\pm 2} &= \\mp \\frac{3}{2} \\frac{{\\cal B}_0}{c^2} \\sin\\iota (\\cos\\iota \\pm 1)(2\\Omega \\mp \\omega) \\sin^2\\theta\\, r, \\\\\nq^\\theta_{\\pm 2} &= \\mp \\frac{3}{2} \\frac{{\\cal B}_0}{c^2} \\sin\\iota (\\cos\\iota \\pm 1)(2\\Omega \\mp \\omega) \\sin\\theta\\cos\\theta, \\\\\nq^\\phi_{\\pm 2} &= -\\frac{3i}{2} \\frac{{\\cal B}_0}{c^2} \\sin\\iota (\\cos\\iota \\pm 1)(2\\Omega \\mp \\omega),\n\\end{align}\n\\end{subequations}\n\\begin{subequations}\n\\label{qa_listing_m1}\n\\begin{align}\nq^r_{\\pm 1} &= 3i \\frac{{\\cal B}_0}{c^2}  (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) (\\Omega \\mp \\omega) \\sin\\theta\\cos\\theta\\, r, \\\\\nq^\\theta_{\\pm 1} &= \\frac{3i}{2} \\frac{{\\cal B}_0}{c^2}  (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) (\\Omega \\mp \\omega) (2\\cos^2\\theta-1), \\\\\nq^\\phi_{\\pm 1} &= \\mp \\frac{3}{2} \\frac{{\\cal B}_0}{c^2}  (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) (\\Omega \\mp \\omega) \\cot\\theta, \n\\end{align}\n\\end{subequations}\nand \n\\begin{subequations}\n\\label{qa_listing_m0}\n\\begin{align}\nq^r_0 &= 3 \\frac{{\\cal B}_0}{c^2} \\sin\\iota \\cos\\iota\\, \\omega (3\\cos^2\\theta - 1) r, \\\\\nq^\\theta_0 &= -9 \\frac{{\\cal B}_0}{c^2} \\sin\\iota \\cos\\iota\\, \\omega \\sin\\theta\\cos\\theta, \\\\\nq^\\phi_0 &= 0.\n\\end{align}\n\\end{subequations}\n\n\\subsection{Spherical-harmonic decompositions}\n\\label{subsec:harmonics}\n\nSubsequent developments will require a set of vectorial harmonics to decompose vector fields such as $\\delta v^a$ and $f^a$. The first member of the set consists of the radial harmonics\n\\begin{equation}\nr^a Y_\\ell^m,\n\\end{equation}\nthe union of the usual scalar harmonics $Y_\\ell^m$ with the unit radial vector $r^a := \\nabla^a r$. In Cartesian coordinates, this vector is given by $r^j = x^j\/r$; in spherical coordinates, its only nonvanishing component is $r^r = 1$. The second member of the set consists of the polar harmonics\n\\begin{equation}\n(Y_\\ell^m)^a := \\nabla^a Y_\\ell^m.\n\\end{equation}\nAnd the third member of the set consists of the axial harmonics\n\\begin{equation}\n(X_\\ell^m)^a := \\epsilon^{abc} (\\nabla_b Y_\\ell^m) r_c, \n\\end{equation}\nthe cross product between the polar harmonics and the unit radial vector. \n\nThe scalar harmonics are normalized according to \n\\begin{equation}\n\\int \\bar{Y}_\\ell^m Y_{\\ell'}^{m'}\\, d\\Omega = \\delta_{\\ell\\ell'} \\delta_{mm'}, \n\\label{ortho1} \n\\end{equation} \nwhere an overbar indicates complex conjugation, and $d\\Omega := \\sin\\theta\\, d\\theta d\\phi$ is the element of solid angle. It follows from this and the definitions of the polar and axial harmonics that \n\\begin{subequations} \n\\label{ortho2} \n\\begin{align} \n\\int (\\bar{Y}_\\ell^m)_a (Y_{\\ell'}^{m'})^a\\, d\\Omega &= \\frac{\\ell(\\ell+1)}{r^2} \\delta_{\\ell\\ell'} \\delta_{mm'}, \\\\ \n\\int (\\bar{X}_\\ell^m)_a (X_{\\ell'}^{m'})^a\\, d\\Omega &= \\frac{\\ell(\\ell+1)}{r^2} \\delta_{\\ell\\ell'} \\delta_{mm'}.  \n\\end{align} \n\\end{subequations} \nWe also have that the polar harmonics are orthogonal to the axial harmonics, and both are orthogonal to the radial harmonics. We adhere to the convention $Y_\\ell^{-m} = (-1)^m \\bar{Y}_\\ell^m$. \n\nThe decomposition of $f_a$ in vector harmonics is accomplished by\n\\begin{equation}\nf_a = \\sum_{\\ell m} \\biggl[ \\frac{1}{r} (f_\\ell^m)^{\\rm R}\\, r_a Y_\\ell^m\n+ (f_\\ell^m)^{\\rm P} (Y_\\ell^m)_a  + (f_\\ell^m)^{\\rm A} (X_\\ell^m)_a \\biggr] e^{-i\\omega t} + \\mbox{cc},\n\\label{fa_vharm}\n\\end{equation}\nwhere the nonvanishing coefficients are\n\\begin{subequations} \n\\label{fa_radial} \n\\begin{align} \n(f_3^{\\rm \\pm 2})^{\\rm R} &= \\frac{2i}{35} \\sqrt{210\\pi} \\frac{{\\cal B}_0}{c^2} \\sin\\iota(\\cos\\iota \\pm 1) \\Omega\\, r^3, \\\\\n(f_3^{\\rm \\pm 1})^{\\rm R} &= -\\frac{8}{35} \\sqrt{21\\pi} \\frac{{\\cal B}_0}{c^2} (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) \\Omega\\, r^3, \\\\\n(f_1^{\\pm 1})^{\\rm R} &= \\frac{3}{5} \\sqrt{6\\pi} \\frac{{\\cal B}_0}{c^2} (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) \\Omega\\, r^3, \\\\\n(f_3^0)^{\\rm R} &= \\frac{36i}{35} \\sqrt{7\\pi} \\frac{{\\cal B}_0}{c^2} \\sin\\iota\\cos\\iota\\, \\Omega\\, r^3, \\\\\n(f_1^0)^{\\rm R} &= -\\frac{12i}{5} \\sqrt{3\\pi} \\frac{{\\cal B}_0}{c^2} \\sin\\iota\\cos\\iota\\, \\Omega\\, r^3, \n\\end{align}\n\\end{subequations}\n\\begin{subequations} \n\\label{fa_polar} \n\\begin{align} \n(f_3^{\\rm \\pm 2})^{\\rm P} &= \\frac{2i}{105} \\sqrt{210\\pi} \\frac{{\\cal B}_0}{c^2} \\sin\\iota(\\cos\\iota \\pm 1) \\Omega\\, r^3, \\\\\n(f_3^{\\rm \\pm 1})^{\\rm P} &= -\\frac{8}{105} \\sqrt{21\\pi} \\frac{{\\cal B}_0}{c^2} (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) \\Omega\\, r^3, \\\\\n(f_1^{\\pm 1})^{\\rm P} &= \\frac{1}{5} \\sqrt{6\\pi} \\frac{{\\cal B}_0}{c^2} (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) \\Omega\\, r^3, \\\\\n(f_3^0)^{\\rm P} &= \\frac{12i}{35} \\sqrt{7\\pi} \\frac{{\\cal B}_0}{c^2} \\sin\\iota\\cos\\iota\\, \\Omega\\, r^3, \\\\\n(f_1^0)^{\\rm P} &= -\\frac{4i}{5} \\sqrt{3\\pi} \\frac{{\\cal B}_0}{c^2} \\sin\\iota\\cos\\iota\\, \\Omega\\, r^3,\n\\end{align}\n\\end{subequations}\nand\n\\begin{subequations} \n\\label{fa_axial} \n\\begin{align} \n(f_2^{\\pm 2})^{\\rm A} &= \\mp \\frac{1}{15} \\sqrt{30\\pi} \\frac{{\\cal B}_0}{c^2} \\sin\\iota (\\cos\\iota \\pm 1) (2\\Omega \\mp \\omega)\\, r^3, \\\\\n(f_2^{\\pm 1})^{\\rm A} &= \\mp \\frac{i}{15} \\sqrt{30\\pi} \\frac{{\\cal B}_0}{c^2} (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) (\\Omega \\mp \\omega)\\, r^3, \\\\\n(f_2^0)^{\\rm A} &= \\frac{2}{5} \\sqrt{5\\pi} \\frac{{\\cal B}_0}{c^2} \\sin\\iota\\cos\\iota\\, \\omega\\, r^3.  \n\\end{align}\n\\end{subequations}\nThe superscripts on $(f_\\ell^m)$ stand for ``Radial'', ``Polar'', and ``Axial''.\n\nAn observation that will be exploited in the sequel is that \n\\begin{equation}\n(f_\\ell^m)^{\\rm R} = 3 k_\\ell^m\\, r^3, \\qquad\n(f_\\ell^m)^{\\rm P} = k_\\ell^m\\, r^3 \n\\label{RP-proportionality}\n\\end{equation}\nfor all relevant values of $\\ell$ and $m$, with the same constant $k_\\ell^m$ in both expressions. \n\nWe shall not require a decomposition of $q^a$ in vectorial harmonics. \n \n\\section{Perturbation equations} \n\\label{sec:perturbation} \n\nWe now examine the system of perturbation equations for the variables $\\{ \\delta v^a, \\delta \\rho, \\delta p, \\delta h, \\delta U\\}$. This consists of Euler's equation (\\ref{dEuler}), the continuity equation (\\ref{drho}), Poisson's equation (\\ref{dPoisson}), and the surface condition of Eq.~(\\ref{surface_condition}). Following the general strategy outlined by Lockitch and Friedman \\cite{lockitch-friedman:99}, we shall show that the perturbation is dominated by $\\delta v^a$. We continue to assume that $\\Omega$ is small, so that all centrifugal effects can be neglected, and we further assume that $\\omega$ and $\\Omega$ are of the same order of magnitude.   \n\n\\subsection{Velocity perturbation} \n\nFollowing Lockitch and Friedman \\cite{lockitch-friedman:99}, we eliminate $\\delta U$ and $\\delta h$ from Euler's equation (\\ref{dEuler}) by taking the curl of both sides. With the definition of Eq.~(\\ref{q_def}), this gives\n\\begin{equation}\nQ^a = q^a,\n\\label{curlEuler} \n\\end{equation} \nwhere $Q^a := \\epsilon^{abc} \\nabla_b \\delta E_c$. Equation (\\ref{curlEuler}) provides only two independent equations, because $\\nabla_a Q^a = 0 = \\nabla_a q^a$ (the divergence of a curl is always zero). \n\nThe explicit form of Eq.~(\\ref{curlEuler}) and the decomposition of Eq.~(\\ref{qa_decomposed}) imply that $\\delta v^a$ admits the decomposition\n\\begin{equation}\n\\delta v^a = \\sum_{m=-2}^2 \\delta v^a_m e^{-i\\omega t} + \\mbox{cc},\n\\label{va_decomposed1} \n\\end{equation}\nwhere each $\\delta v^a_m$ can be further decomposed in vectorial harmonics. We incorporate the scaling and numerical factors implied by Eqs.~(\\ref{qa_listing_m2}), (\\ref{qa_listing_m1}), and (\\ref{qa_listing_m0}), and write\n\\begin{subequations}\n\\label{va_decomposed2}\n\\begin{align}\n\\delta v^a_{\\pm 2} &= \\frac{{\\cal B}_0}{c^2} R^3 \\sin\\iota(\\cos\\iota \\pm 1)(2 \\mp \\omega\/\\Omega) \\Biggl[\n\\mp \\sum_{\\ell=3}^{\\rm odd} \\frac{1}{r} A_\\ell^{\\pm 2}\\, r^a Y_\\ell^{\\pm 2}\n\\mp \\sum_{\\ell=3}^{\\rm odd} B_\\ell^{\\pm 2}\\, (Y_\\ell^{\\pm 2})^a\n- i \\sum_{\\ell=2}^{\\rm even} C_\\ell^{\\pm 2}\\, (X_\\ell^{\\pm 2})^a \\Biggr], \\\\\n\\delta v^a_{\\pm 1} &= \\frac{{\\cal B}_0}{c^2} R^3 (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1)(1 \\mp \\omega\/\\Omega) \\Biggl[\ni \\sum_{\\ell=1}^{\\rm odd} \\frac{1}{r} A_\\ell^{\\pm 1}\\, r^a Y_\\ell^{\\pm 1}\n+ i\\sum_{\\ell=1}^{\\rm odd} B_\\ell^{\\pm 1}\\, (Y_\\ell^{\\pm 1})^a\n\\mp \\sum_{\\ell=2}^{\\rm even} C_\\ell^{\\pm 1}\\, (X_\\ell^{\\pm 1})^a \\Biggr], \\\\\n\\delta v^a_{0} &= \\frac{{\\cal B}_0}{c^2} R^3 \\sin\\iota\\cos\\iota\\, (\\omega\/\\Omega) \\Biggl[\n\\sum_{\\ell=1}^{\\rm odd} \\frac{1}{r} A_\\ell^{0}\\, r^a Y_\\ell^{0}\n+ \\sum_{\\ell=1}^{\\rm odd} B_\\ell^{0}\\, (Y_\\ell^{0})^a\n- i \\sum_{\\ell=2}^{\\rm even} C_\\ell^{0}\\, (X_\\ell^{0})^a \\Biggr],\n\\end{align}\n\\end{subequations}\nwhere the radial functions $A_\\ell^m(r)$, $B_\\ell^m(r)$, and $C_\\ell^m(r)$ are all dimensionless. The restrictions on the sums over $\\ell$ (even or odd terms only) will be justified below on the basis of the explicit form of Eq.~(\\ref{curlEuler}). The various factors of $\\pm$ and $i$ in front of these sums are inserted to simplify the perturbation equations.  \n\nEquation (\\ref{curlEuler}) implies that the radial functions satisfy the identities\n\\begin{equation}\nA_\\ell^{-2}(\\omega) = A_\\ell^2(-\\omega), \\qquad\nB_\\ell^{-2}(\\omega) = B_\\ell^2(-\\omega), \\qquad\nC_\\ell^{-2}(\\omega) = C_\\ell^2(-\\omega), \n\\label{ABCidentities1}\n\\end{equation}\n\\begin{equation}\nA_\\ell^{-1}(\\omega) = -A_\\ell^1(-\\omega), \\qquad\nB_\\ell^{-1}(\\omega) = -B_\\ell^1(-\\omega), \\qquad\nC_\\ell^{-1}(\\omega) = -C_\\ell^1(-\\omega),  \n\\label{ABCidentities2}\n\\end{equation}\nand \n\\begin{equation}\nA_\\ell^0(\\omega) = A_\\ell^0(-\\omega), \\qquad\nB_\\ell^0(\\omega) = B_\\ell^0(-\\omega), \\qquad\nC_\\ell^0(\\omega) = -C_\\ell^0(-\\omega). \n\\label{ABCidentities3}\n\\end{equation}\nIt also implies that all radial functions are real. \n\n\\subsection{Dominance of the velocity perturbation} \n\nWhen we insert within Eq.~(\\ref{dEuler}) the scaling factors for $\\delta v^a$ indicated in Eq.~(\\ref{va_decomposed2}),  we find that $\\delta h\/|\\delta v^a|$ and $\\delta U\/|\\delta v^a|$ must both be of the order of $\\omega R$ or $\\Omega R$, making $\\delta h$ and $\\delta U$ (and therefore $\\delta \\rho$ and $\\delta p$) of relative order $\\Omega$ compared with $\\delta v^a$. Stated in a more meaningful way, we have that \n\\begin{equation} \n\\frac{\\delta h\/h}{\\delta v\/v} \\sim \\frac{\\delta U\/U}{\\delta v\/v}\n\\sim \\frac{\\omega\\Omega}{GM\/R^3} \\ \n\\mbox{or} \\ \\frac{\\Omega^2}{GM\/R^3}, \n\\end{equation} \nwhere $\\delta v := |\\delta v^a|$ and $v := |v^a| \\sim \\Omega R$. The small-$\\Omega$ assumption ensures that these ratios are small. This relative scaling between $\\delta v^a$ and the remaining perturbation variables is the same as for the Lockitch-Friedman inertial modes \\cite{lockitch-friedman:99}, and this motivates an expectation that the gravitomagnetic tidal field acts as a driving force for these modes. This expectation will be verified in Sec.~\\ref{sec:oscillator}.      \n\n\\subsection{Continuity equation and surface condition} \n\nAs was stated previously, Eq.~(\\ref{curlEuler}) provides only two independent equations for the three components of $\\delta v^a$. A third equation is provided by the continuity equation (\\ref{drho}). We express the Lagrangian displacement as\n\\begin{equation}\n\\xi^a = \\sum_{m=-2}^2 \\xi^a_m e^{-i\\omega t} + \\mbox{cc},\n\\label{xi_decomposed}\n\\end{equation}\nand use Eq.~(\\ref{xi_vs_dv}) to relate $\\xi^a_m$ to $\\delta v^a_m$; we obtain\n\\begin{equation}\n\\xi^a_m = i(\\omega - m\\Omega)^{-1}\\, \\delta v^a_m.\n\\end{equation}\nMaking the substitutions within Eq.~(\\ref{drho}), taking $\\omega$ to be of the same order of magnitude as $\\Omega$, and recalling that $\\delta \\rho$ is formally of order $\\Omega$, we find that the continuity equation reduces to  \n\\begin{equation} \n\\nabla_a (\\rho \\delta v^a) = O(\\Omega^2). \n\\label{div_rhov} \n\\end{equation} \nThe third equation is therefore $\\nabla_a (\\rho \\delta v^a) = 0$, and after substituting Eqs.~(\\ref{va_decomposed1}) and (\\ref{va_decomposed2}) we eventually obtain\\footnote{To arrive at this we made use of the relations $\\nabla_a r^a = 2r^{-1}$, $\\nabla_a (Y_\\ell^m)^a = -\\ell(\\ell+1) r^{-2}\\, Y_\\ell^m$, and $\\nabla_a (X_\\ell^m)^a = 0$.}\n\\begin{equation} \nr \\frac{dA_\\ell^m}{dr} + \\biggl( 1 + \\frac{r}{\\rho} \\frac{d\\rho}{dr} \\biggr) A_\\ell^m - \\ell(\\ell+1) B_\\ell^m = 0.   \n\\label{Aprime} \n\\end{equation} \nEquation (\\ref{Aprime}) is the only equation that provides information regarding the star's internal structure, through the function $1 + (r\/\\rho)(d\\rho\/dr)$ involving the mass density $\\rho(r)$.\n\nBecause $\\delta h$ is formally of order $\\Omega$, the boundary condition of Eq.~(\\ref{surface_condition}) implies that $(GM\/R^2) r_a \\delta v^a(r=R) = O(\\Omega^2)$. Inserting Eqs.~(\\ref{va_decomposed1}) and (\\ref{va_decomposed2}), we find that this condition implies \n\\begin{equation} \nA_\\ell^m(r=R) = 0. \n\\label{BC_A} \n\\end{equation} \nThe radial component of the velocity field must therefore vanish on the stellar surface. \n\n\\subsection{Truncation and projections} \n\nInsertion of Eqs.~(\\ref{va_decomposed1}) and (\\ref{va_decomposed2}) within Eq.~(\\ref{curlEuler}) leads to an infinite sequence of equations for the radial functions $A_\\ell^m(r)$, $B_\\ell^m(r)$, and $C_\\ell^m(r)$; the sequence is such that a function with a given value of $\\ell$ is coupled to an infinite number of functions with distinct values of $\\ell$. Any hope of finding a solution to the perturbation equations must be based on a truncation of the system; the truncation implies that the solution can only be approximate. Fortunately, numerical exploration reveals that except for isolated values of $\\omega$ (which correspond to eigenfrequencies of subdominant Lockitch-Friedman inertial modes\\footnote{The dominant modes are those listed in Sec.~\\ref{sec:intro}. All other modes are subdominant.}), radial functions with the lowest values of $\\ell$ are much larger than functions with larger values of $\\ell$. An excellent representation of the solution can therefore be obtained with handfuls of terms. \n\nFor $m=\\pm 2$ we include the set of radial functions\n\\begin{equation}\n\\{ C_2^{\\pm 2}, A_3^{\\pm 2}, B_3^{\\pm 2}, C_4^{\\pm 2}, A_5^{\\pm 2}, B_5^{\\pm 2}, C_6^{\\pm 2}, A_7^{\\pm 2}, B_7^{\\pm 2} \\}\n\\label{set_m2} \n\\end{equation}\nin our truncated system of perturbation equations. For $m=\\pm 1$ we include the set\n\\begin{equation} \n\\{ A_1^{\\pm 1}, B_1^{\\pm 1}, C_2^{\\pm 1}, A_3^{\\pm 1}, B_3^{\\pm 1}, C_4^{\\pm 1}, A_5^{\\pm 1}, B_5^{\\pm 1}, C_6^{\\pm 1} \\}.\n\\label{set_m1} \n\\end{equation}\nAnd for $m=0$ we include\n\\begin{equation} \n\\{ A_1^{0}, B_1^{0}, C_2^{0}, A_3^{0}, B_3^{0}, C_4^{0}, A_5^{0}, B_5^{0}, C_6^{0} \\}.\n\\label{set_m0} \n\\end{equation}\nFor $m=\\pm 1$ and $m=0$ we find that $A_1^m$, $B_1^m$, and $C_2^m$ dominate the description of the solution by at least two orders of magnitude (except for isolated frequencies, as was mentioned previously). For $m=\\pm 2$ we shall find that the only nonvanishing radial function is $C_2^{\\pm 2}(r)$.  \n\nTo obtain an explicit listing of equations for the radial functions, we take the components of Eq.~(\\ref{curlEuler}) and project them into spherical-harmonic components. We thus define \n\\begin{subequations} \n\\label{Zdef} \n\\begin{align} \n(Z_\\ell^m)^r &:= \\int (Q^r - q^r) \\bar{Y}_\\ell^m\\, d\\Omega, \n\\label{Zr_def} \\\\\n(Z_\\ell^m)^\\theta &:= \\int \\sin\\theta (Q^\\theta - q^\\theta) \\bar{Y}_\\ell^m\\, d\\Omega, \n\\label{Ztheta_def} \\\\ \n(Z_\\ell^m)^\\phi &:= \\int \\sin^2\\theta (Q^\\phi - q^\\phi) \\bar{Y}_\\ell^m\\, d\\Omega, \n\\label{Zphi_def}\n\\end{align} \n\\end{subequations} \nand the perturbation equations become $(Z_\\ell^m)^a = 0$. As was pointed out, only two of these equations are independent. They are supplemented with Eq.~(\\ref{Aprime}) and the surface condition of Eq.~(\\ref{BC_A}).   \n\nAs we shall see in detail below, we find that the equations $(Z_\\ell^m)^r = 0$ provide algebraic relations between the radial functions, while $(Z_\\ell^m)^\\theta = 0$ and $(Z_\\ell^m)^\\phi = 0$ give rise to equations involving the functions and their first derivatives. The general strategy to obtain a workable system of equations is to use the radial equations to obtain $C_\\ell^m$ algebraically in terms of $A_\\ell^m$ and $B_\\ell^m$, invoke Eq.~(\\ref{Aprime}) to express $dA_\\ell^m\/dr$ also in terms of $A_\\ell^m$ and $B_\\ell^m$, and finally, to solve the angular equations for $dB_\\ell^m\/dr$. We shall now go through these steps for each value of $m$.      \n\n\\subsection{Equations for $m=\\pm 2$} \n\\label{sec:eqns_m2} \n\nSetting $w := \\omega\/\\Omega$, the listing of $(Z_\\ell^m)^a$ for $m = 2$ is  \n\\begin{subequations} \n\\label{Z_m2} \n\\begin{align} \n(Z_2^2)^r &= -(6w - 8)\\,  C_2^2 + \\frac{4}{7}\\sqrt{7}\\, A_3^2 + \\frac{16}{7}\\sqrt{7}\\,  B_3^2 \n+ \\frac{2}{5}\\sqrt{30\\pi}\\,  (r\/R)^3, \\\\\n(Z_3^2)^\\theta &= \\frac{4}{3}\\,  r \\frac{dA_3^2}{dr} - \\frac{4}{33}\\sqrt{11}\\,  r \\frac{d A_5^2}{dr} \n+ 2(w-2)\\,  r \\frac{d B_3^2}{dr} - \\frac{8}{11}\\sqrt{11}\\,  r \\frac{d B_5^2}{dr} \n- \\frac{2}{7}\\sqrt{7} w\\,  r \\frac{d C_2^2}{dr} + \\frac{2}{21}\\sqrt{21} (5w-14)\\,  r \\frac{d C_4^2}{dr} \n\\nonumber \\\\ & \\quad \\mbox{}\n+ \\frac{8}{7}\\sqrt{7}\\,  C_2^2 - (2w-4)\\,  A_3^2 - 8\\,  B_3^2 - \\frac{40}{21}\\sqrt{21}\\,  C_4^2 \n+ \\frac{2}{35}\\sqrt{210\\pi}\\,  (r\/R)^3, \\\\\n(Z_4^2)^r &= -\\frac{20}{21}\\sqrt{21}\\,  A_3^2 + \\frac{20}{7}\\sqrt{21}\\,  B_3^2 - (20w-36)\\,  C_4^2 \n+ \\frac{8}{33}\\sqrt{231}\\, A_5^2 + \\frac{16}{11}\\sqrt{231}\\, B_5^2, \\\\\n(Z_5^2)^\\theta &= -\\frac{4}{33}\\sqrt{11}\\, r \\frac{d A_3^2}{dr} + \\frac{44}{39}\\, r \\frac{dA_5^2}{dr} \n- \\frac{8}{143}\\sqrt{66}\\, r \\frac{d A_7^2}{dr}  + \\frac{4}{11}\\sqrt{11}\\, r \\frac{d B_3^2}{dr} \n+ \\frac{2}{13}(13w-28) \\, r \\frac{d B_5^2}{dr} - \\frac{64}{143}\\sqrt{66}\\, r \\frac{d B_7^2}{dr} \n\\nonumber \\\\ & \\quad \\mbox{}\n- \\frac{4}{33}\\sqrt{231} (w-1) \\, r \\frac{dC_4^2}{dr} + \\frac{4}{143}\\sqrt{286} (7w-18) \\, r \\frac{d C_6^2}{dr} \n+ \\frac{16}{33}\\sqrt{231}\\, C_4^2 - (2w-4) \\, A_5^2 - \\frac{112}{143}\\sqrt{286}\\, C_6^2 - 8 \\, B_5^2, \\\\ \n(Z_6^2)^r &= -\\frac{56}{143}\\sqrt{286}\\, A_5^2 + \\frac{280}{143}\\sqrt{286}\\, B_5^2 - (42w-80) \\, C_6^2 \n+ \\frac{12}{13}\\sqrt{39}\\, A_7^2 + \\frac{96}{13}\\sqrt{39}\\, B_7^2, \\\\ \n(Z_7^2)^\\theta &= -\\frac{8}{143}\\sqrt{66}\\, r \\frac{dA_5^2}{dr} + \\frac{236}{221}\\, r \\frac{dA_7^2}{dr} \n+ \\frac{40}{143}\\sqrt{66}\\, r \\frac{dB_5^2}{dr} + \\frac{2}{221} (221 w - 486) \\, r \\frac{dB_7^2}{dr} \n- \\frac{2}{13}\\sqrt{39} (3w-4) \\, r \\frac{d C_6^2}{dr} \n\\nonumber \\\\ & \\quad \\mbox{}\n+ \\frac{24}{13}\\sqrt{39}\\, C_6^2 - (2w-4) \\, A_7^2 - 8 \\, B_7^2. \n\\end{align} \n\\end{subequations} \nThe expressions for $m=-2$ can be obtained from these by applying the identities of Eq.~(\\ref{ABCidentities1}). \n\nThe equations $(Z_\\ell^2)^r = 0$ for $\\ell = 2, 4, 6$ provide algebraic solutions for $C_\\ell^2$ in terms of $A_\\ell^2$ and $B_\\ell^2$. Making the substitutions in $(Z_\\ell^2)^\\theta = 0$ for $\\ell = 3, 5, 7$, and making use of Eq.~(\\ref{Aprime}) to eliminate derivatives of $A_\\ell^2$, we obtain for $B_\\ell^2$ a system of equations of the schematic form \n\\begin{equation} \nr \\frac{dB_\\ell^2}{dr} = \\sum_{\\ell' = 3, 5, 7} \\bigl( {\\cal P}_\\ell^{\\ell'}\\, A_{\\ell'}^2 \n+ {\\cal Q}_\\ell^{\\ell'}\\, B_{\\ell'}^2 \\bigr) + F_\\ell^2, \n\\label{Bprime_m2} \n\\end{equation} \nwhere ${\\cal P}_\\ell^{\\ell'}$ and ${\\cal Q}_\\ell^{\\ell'}$ are coefficients that depend on $w$ and $r$, and $F_\\ell^2 \\propto (r\/R)^3$ are driving terms that originate from $q^a$, the curl of the tidal force density. The radial functions are therefore determined by the system of differential equations provided by Eqs.~(\\ref{Aprime}) and (\\ref{Bprime_m2}), the algebraic equations for $C_\\ell^2$, and the boundary conditions of Eq.~(\\ref{surface_condition}). \n\n\\subsection{Equations for $m=\\pm 1$} \n\\label{sec:eqns_m1} \n\nThe listing of $(Z_\\ell^m)^a$ for $m = 1$ is  \n\\begin{subequations} \n\\label{Z_m1} \n\\begin{align} \n(Z_1^1)^\\theta &= -\\frac{8}{5}\\, r \\frac{dA_1^1}{dr} + \\frac{4}{35}\\sqrt{14}\\, r \\frac{dA_3^1}{dr} \n- \\frac{1}{5}(5w - 3)\\, r \\frac{dB_1^1}{dr} + \\frac{16}{35} \\sqrt{14}\\, r \\frac{dB_3^1}{dr} \n- \\frac{1}{5} \\sqrt{5} (3w-5)\\, r \\frac{dC_2^1}{dr} \n\\nonumber \\\\ & \\quad \\mbox{}\n+ (w-1)\\, A_1^1 + 2\\, B_1^1 + \\frac{6}{5}\\sqrt{5}\\, C_2^1 - \\frac{3}{5}\\sqrt{6\\pi}\\, (r\/R)^3, \\\\ \n(Z_2^1)^r &= \\frac{6}{5}\\sqrt{5}\\, A_1^1 - \\frac{6}{5}\\sqrt{5}\\, B_1^1 + (6w-4)\\, C_2^1 \n- \\frac{8}{35}\\sqrt{70}\\, A_3^1 - \\frac{32}{35}\\sqrt{70}\\, B_3^1 + \\frac{2}{5}\\sqrt{30\\pi} (r\/R)^3, \\\\ \n(Z_3^1)^\\theta &= \\frac{4}{35}\\sqrt{14}\\, r \\frac{dA_1^1}{dr} - \\frac{16}{15}\\, r \\frac{d A_3^1}{dr} \n+ \\frac{4}{231}\\sqrt{770}\\, r \\frac{dA_5^1}{dr} - \\frac{4}{35}\\sqrt{14}\\, r \\frac{d B_1^1}{dr} \n- \\frac{1}{5}(5w-7)\\, r \\frac{d B_3^1}{dr} + \\frac{8}{77}\\sqrt{770}\\, r \\frac{dB_5^1}{dr} \n\\nonumber \\\\ & \\quad \\mbox{}\n+ \\frac{4}{35}\\sqrt{70}w\\, r \\frac{d C_2^1}{dr} - \\frac{1}{21}\\sqrt{105}(5w-7)\\, r \\frac{d C_4^1}{dr} \n- \\frac{8}{35}\\sqrt{70}\\, C_2^1 + (w-1)\\, A_3^1 + 2\\, B_3^1 + \\frac{10}{21}\\sqrt{105}\\, C_4^1 \n\\nonumber \\\\ & \\quad \\mbox{}\n+ \\frac{8}{35}\\sqrt{21\\pi} (r\/R)^3, \\\\ \n(Z_4^1)^r &= \\frac{10}{21}\\sqrt{105}\\, A_3^1 - \\frac{10}{7}\\sqrt{105}\\, B_3^1 + (20w-18)\\, C_4^1 \n- \\frac{16}{33}\\sqrt{66}\\, A_5^1 - \\frac{32}{11}\\sqrt{66}\\, B_5^1, \\\\ \n(Z_5^1)^\\theta &= \\frac{4}{231}\\sqrt{770}\\, r \\frac{d A_3^1}{dr} - \\frac{40}{39}\\, r \\frac{d A_5^1}{dr} \n- \\frac{4}{77}\\sqrt{770}\\, r \\frac{d B_3^1}{dr} - \\frac{1}{13} (13w - 19)\\, r \\frac{d B_5^1}{dr} \n+ \\frac{4}{33}\\sqrt{66}(2w-1)\\, r \\frac{d C_4^1}{dr} \n\\nonumber \\\\ & \\quad \\mbox{}\n- \\frac{1}{143}\\sqrt{5005}(7w-9)\\, r \\frac{d C_6^1}{dr} \n- \\frac{16}{33}\\sqrt{66}\\, C_4^1 + (w-1)\\, A_5^1 + 2\\, B_5^1 + \\frac{14}{143}\\sqrt{5005}\\, C_6^1, \\\\ \n(Z_6^1)^r &= \\frac{14}{143}\\sqrt{5005}\\, A_5^1 - \\frac{70}{143}\\sqrt{5005}\\, B_5^1 \n+ (42w-40)\\, C_6^1. \n\\end{align} \n\\end{subequations} \nExpressions for $m=-1$ are obtained from these by making use of Eq.~(\\ref{ABCidentities2}). \n\nThe equations $(Z_\\ell^1)^r = 0$ for $\\ell = 2, 4, 6$ determine $C_\\ell^1$ in terms of $A_\\ell^1$ and $B_\\ell^1$. Substituting these in $(Z_\\ell^1)^\\theta = 0$ for $\\ell = 1, 3, 5$, and eliminating derivatives of $A_\\ell^1$ with Eq.~(\\ref{Aprime}), we obtain for $B_\\ell^1$ a system of equations of the schematic form \n\\begin{equation} \nr \\frac{dB_\\ell^1}{dr} = \\sum_{\\ell' = 1, 3, 5} \\bigl( {\\cal P}_\\ell^{\\ell'}\\, A_{\\ell'}^1 \n+ {\\cal Q}_\\ell^{\\ell'}\\, B_{\\ell'}^1 \\bigr) + F_\\ell^1, \n\\label{Bprime_m1} \n\\end{equation} \nwhere ${\\cal P}_\\ell^{\\ell'}$ and ${\\cal Q}_\\ell^{\\ell'}$ are coefficients --- distinct from those appearing in Eq.~(\\ref{Bprime_m2}) --- that depend on $w$ and $r$, and $F_\\ell^1 \\propto (r\/R)^3$ are driving terms that originate from $q^a$. The radial functions are therefore determined by the system of differential equations provided by Eqs.~(\\ref{Aprime}) and (\\ref{Bprime_m1}), the algebraic equations for $C_\\ell^1$, and the boundary conditions of Eq.~(\\ref{surface_condition}).    \n\n\\subsection{Equations for $m=0$} \n\\label{sec:eqns_m0} \n\nFor $m=0$ we have \n\\begin{subequations} \n\\label{Z_m0} \n\\begin{align} \n(Z_2^0)^r &= \\frac{4}{5}\\sqrt{15}\\, A_1^0 - \\frac{4}{5}\\sqrt{15}\\, B_1^0 - 6w\\, C_2^0 \n- \\frac{12}{35}\\sqrt{35}\\, A_3^0 - \\frac{48}{35}\\sqrt{35}\\, B_3^0 - \\frac{12}{5}\\sqrt{5\\pi}\\, (r\/R)^3, \\\\ \n(Z_2^0)^\\phi &= -\\frac{2}{15}\\sqrt{15}w\\, r \\frac{dB_1^0}{dr} + \\frac{12}{35}\\sqrt{35}w\\, r \\frac{dB_3^0}{dr} \n+ \\frac{4}{7}\\, r \\frac{dC_2^0}{dr} + \\frac{8}{7}\\sqrt{5}\\, r \\frac{d C_4^0}{dr} \n\\nonumber \\\\ & \\quad \\mbox{}\n+ \\frac{2}{15}\\sqrt{15}w\\, A_1^0 - \\frac{40}{7}\\, C_2^0 - \\frac{12}{35}\\sqrt{35}w\\, A_3^0 \n+ \\frac{32}{7}\\sqrt{5}\\, C_4^0, \\\\ \n(Z_4^0)^r &= \\frac{40}{21}\\sqrt{7}\\, A_3^0 - \\frac{40}{7}\\sqrt{7}\\, B_3^0 - 20w\\, C_4^0 \n- \\frac{40}{33}\\sqrt{11}\\, A_5^0 - \\frac{80}{11}\\sqrt{11}\\, B_5^0, \\\\ \n(Z_4^0)^\\phi &= -\\frac{4}{7}\\sqrt{7}w\\, r \\frac{dB_3^0}{dr} + \\frac{10}{11}\\sqrt{11}w\\, r \\frac{dB_5^0}{dr} \n- \\frac{16}{35}\\sqrt{5}\\, r \\frac{dC_2^0}{dr} + \\frac{40}{77}\\, r \\frac{d C_4^0}{dr} \n+ \\frac{140}{143}\\sqrt{13}\\, r \\frac{dC_6^0}{dr} \n\\nonumber \\\\ & \\quad \\mbox{}\n+ \\frac{48}{35}\\sqrt{5}\\, C_2^0 + \\frac{4}{7}\\sqrt{7}w\\, A_3^0 \n- \\frac{1520}{77}\\, C_4^0 - \\frac{10}{11}\\sqrt{11}w\\, A_5^0 + \\frac{840}{143}\\sqrt{13}\\, C_6^0, \\\\ \n(Z_6^0)^r &=  \\frac{84}{143}\\sqrt{143}\\, A_5^0 - \\frac{420}{143}\\sqrt{143}\\, B_5^0 - 42w\\, C_6^0, \\\\ \n(Z_6^0)^\\phi &= -\\frac{30}{143}\\sqrt{143}w\\, r\\frac{dB_5^0}{dr} - \\frac{80}{143}\\sqrt{13}\\, r \\frac{dC_4^0}{dr} \n+ \\frac{28}{55}\\, r \\frac{d C_6^0}{dr} + \\frac{400}{143}\\sqrt{13}\\, C_4^0 + \\frac{30}{143}\\sqrt{143}w\\, A_5^0 \n- \\frac{2296}{55}\\, C_6^0. \n\\end{align} \n\\end{subequations} \nIt might be noted that in this case, the independent equations are provided by $(Z_\\ell^0)^r$ and $(Z_\\ell^0)^\\phi$, with $\\ell = 2, 4, 6$. For $m = \\pm 2$ and $\\pm 1$, the independent equations came from $(Z_\\ell^m)^r$ and $(Z_\\ell^m)^\\theta$, with both even and odd values of $\\ell$.  \n\nThe equations $(Z_\\ell^0)^r = 0$ for $\\ell = 2, 4, 6$ give $C_\\ell^0$ algebraically in terms of $A_\\ell^0$ and $B_\\ell^0$. Substituting these relations in $(Z_\\ell^0)^\\phi = 0$ for $\\ell = 2, 4, 6$, and using Eq.~(\\ref{Aprime}) to replace the derivatives of $A_\\ell^0$, we obtain for $B_\\ell^0$ the system of equations \n\\begin{equation} \nr \\frac{dB_\\ell^0}{dr} = \\sum_{\\ell' = 1, 3, 5} \\bigl( {\\cal P}_\\ell^{\\ell'}\\, A_{\\ell'}^0 \n+ {\\cal Q}_\\ell^{\\ell'}\\, B_{\\ell'}^0 \\bigr) + F_\\ell^0, \n\\label{Bprime_m0} \n\\end{equation} \nwhere ${\\cal P}_\\ell^{\\ell'}$ and ${\\cal Q}_\\ell^{\\ell'}$ are again coefficients that depend on $w$ and $r$, distinct from those appearing in Eqs.~(\\ref{Bprime_m2}) and (\\ref{Bprime_m1}), and $F_\\ell^0 \\propto (r\/R)^3$ are driving terms. The radial functions are therefore determined by the system of differential equations provided by Eqs.~(\\ref{Aprime}) and (\\ref{Bprime_m0}), the algebraic equations for $C_\\ell^0$, and the boundary conditions of Eq.~(\\ref{surface_condition}).    \n\n\\section{Solution to the perturbation equations} \n\\label{sec:solution} \n\nIn this section we shall find solutions to the perturbation equations developed in Sec.~\\ref{sec:perturbation}. For $m=\\pm 2$ the solution will be exact, and it will be obtained analytically. For $m=\\pm 1$ and $m=0$ the solutions will be approximate, and obtained numerically. \n\n\\subsection{Numerical methods} \n\nFor $m=\\pm 1$ and $m = 0$ we have the set of variables displayed in Eqs.~(\\ref{set_m1}) and (\\ref{set_m0}), the system of first-order differential equations provided by Eqs.~(\\ref{Aprime}), (\\ref{Bprime_m1}), and (\\ref{Bprime_m0}), and the surface conditions of Eq.~(\\ref{BC_A}). The various coefficients ${\\cal P}_\\ell^{\\ell'}$ and ${\\cal Q}_\\ell^{\\ell'}$ depend on $w := \\omega\/\\Omega$, the dimensionless frequency of the gravitomagnetic tidal field, and they depend on $r$ through the function $1 + (r\/\\rho) (d\\rho\/dr)$, which encodes the relevant details of the star's internal structure. For concreteness and simplicity in this section, we adopt for the stellar fluid a polytropic equation of state $p = K \\rho^2$, where $K$ is a constant. For this model we have that $K = 2GR^2\/\\pi$ and \n\\begin{equation} \n\\rho = \\frac{M}{4R^2 r} \\sin(\\pi r \/R),   \n\\label{density} \n\\end{equation} \nfrom which it follows that \n\\begin{equation} \n1 + \\frac{r}{\\rho} \\frac{d\\rho}{dr} = (\\pi r\/R) \\cot(\\pi r\/R).   \n\\end{equation} \nThis density model corresponds to $k = 1$ in Eq.~(\\ref{rho_kmodel_intro}). \n\nA local analysis of the differential equations (\\ref{Aprime}), (\\ref{Bprime_m1}), and (\\ref{Bprime_m0}) near $r=0$ reveals that for $m = \\pm 1$ and $m=0$, $A_\\ell^m$ and $B_\\ell^m$ admit the expansions \n\\begin{equation} \nA_\\ell^m = \\sum_{k=\\ell}^\\infty a_\\ell^k (r\/R)^k, \\qquad \nB_\\ell^m = \\sum_{k=\\ell}^\\infty b_\\ell^k (r\/R)^k,\n\\end{equation} \nwhere $a_\\ell^k$ and $b_\\ell^k$ are constant coefficients. The analysis reveals also that $a_1^1$, $a_3^3$, and $a_5^5$ can be specified freely, and that all remaining coefficients are determined by the differential equations. \n\nA local analysis of the differential equations near $r=R$, which takes into account the boundary conditions of Eq.~(\\ref{BC_A}), proceeds on the basis of the expansions \n\\begin{equation} \nA_\\ell^m = \\sum_{k=1}^\\infty \\alpha_\\ell^k (r\/R - 1)^k, \\qquad \nB_\\ell^m = \\sum_{k=0}^\\infty \\beta_\\ell^k (r\/R - 1)^k,\n\\end{equation} \nwhere $\\alpha_\\ell^k$ and $\\beta_\\ell^k$ are constant coefficients. The analysis shows that $\\alpha_1^1$, $\\alpha_3^1$, and $\\alpha_5^1$ can be specified freely, and that all remaining coefficients are determined by the differential equations. \n\nTo find the solutions to Eqs.~(\\ref{Aprime}), (\\ref{Bprime_m1}), and (\\ref{Bprime_m0}) we construct an inner solution in the interval $0 \\leq r \\leq r^\\sharp$, an outer solution in the interval $r^\\sharp \\leq r \\leq R$, and demand that the inner and outer solutions agree at $r = r^\\sharp$, with $r^\\sharp$ denoting an arbitrary middle point. To describe the method we collectively denote by $\\bm{e}$ the set of radial functions $A_\\ell^m$ and $B_\\ell^m$. For the inner solution we construct a particular solution $\\bm{e}^{\\rm in}_{\\rm part}$ to the differential equations by making a random selection of the constants $\\{ a_1^1, a_3^3, a_5^5 \\}$. We also form a basis of functions $\\bm{e}^{\\rm in}_n$ (with $n = \\{1, 2, 3\\}$), defined to be solutions to the homogeneous version of the differential equations, with all driving terms $F_\\ell^m$ switched off; each member of the basis is constructed with a different random selection of the constants $\\{ a_1^1, a_3^3, a_5^5 \\}$. The correct inner solution to the differential equations is \n\\begin{equation} \n\\bm{e}^{\\rm in} = \\bm{e}^{\\rm in}_{\\rm part} + p_1\\, \\bm{e}^{\\rm in}_1 + p_2\\, \\bm{e}^{\\rm in}_2\n+ p_3\\, \\bm{e}^{\\rm in}_3, \n\\end{equation} \nwhere $\\{ p_1, p_2, p_3 \\}$ are unknown coefficients. Moving on to the outer solution, we construct a particular solution $\\bm{e}^{\\rm out}_{\\rm part}$ to the differential equations by making a random selection of the constants $\\{ \\alpha_1^1, \\alpha_3^1, \\alpha_5^1 \\}$. The functions $\\bm{e}^{\\rm out}_n$ are solutions to the homogeneous version of the differential equations, with each member associated with a different random selection of the constants $\\{ \\alpha_1^1, \\alpha_3^1, \\alpha_5^1 \\}$. The correct outer solution to the differential equations is\n\\begin{equation} \n\\bm{e}^{\\rm out} = \\bm{e}^{\\rm out}_{\\rm part} + q_1\\, \\bm{e}^{\\rm out}_1 + q_2\\, \\bm{e}^{\\rm out}_2\n+ q_3\\, \\bm{e}^{\\rm out}_3, \n\\end{equation} \nwhere $\\{ q_1, q_2, q_3 \\}$ are unknown coefficients. The constants $\\{ p_n, q_n \\}$ are determined by the requirement that the inner and outer solutions agree at $r = r^\\sharp$:  \n\\begin{equation} \n\\bm{e}^{\\rm in}(r=r^\\sharp) = \\bm{e}^{\\rm out}(r=r^\\sharp). \n\\end{equation} \nThis gives us six equations for six unknowns, and we finally have our solution in the complete interval $0 \\leq r \\leq R$. \n\n\\subsection{Solution for $m=\\pm 2$} \n\nWe observe from the explicit form of Eq.~(\\ref{Bprime_m2}) that $F_\\ell^{\\pm 2} = 0$; there is no driving term for the radial functions $A_\\ell^{\\pm 2}$ and $B_\\ell^{\\pm 2}$. This implies that they may be set equal to zero, and we find that the remaining perturbation equations produce \n\\begin{equation} \nC_2^{\\pm 2} = \\pm \\frac{1}{5}\\sqrt{30\\pi}\\, \\frac{(r\/R)^3}{3w \\mp 4}, \n\\label{C22_solution} \n\\end{equation} \ntogether with $C_4^{\\pm 2} = C_6^{\\pm 2} = 0$. While this provides a solution to our truncated system of equations,  involving only the set of Eq.~(\\ref{set_m2}), it was verified that  \n\\begin{equation} \n\\delta v^a = \\mp \\frac{i}{5}\\sqrt{30\\pi}\\, \\frac{{\\cal B}_0}{c^2} R^3 \\sin\\iota(\\cos\\iota \\pm 1)\\,\n\\frac{2 \\mp w}{3w \\mp 4}\\, (r\/R)^3\\, (X_2^{\\pm 2})^a\\, e^{-i\\omega t} + \\mbox{cc} \n\\label{dv_m2} \n\\end{equation} \nis an {\\it exact solution} to the perturbation equations for $m = \\pm 2$; we recall that $w := \\omega\/\\Omega$. We notice that the solution is formally infinite when $w = \\pm 4\/3$. These frequencies coincide with the eigenfrequencies of $r$-modes for a barotropic star. The precise connection with $r$-modes will be made below in Sec.~\\ref{sec:oscillator}.  \n\n\\subsection{Solution for $m=\\pm 1$ and $m=0$} \n\n\\begin{figure} \n\\includegraphics[width=0.7\\linewidth]{fig3}\n\\caption{Radial functions $C^m_\\ell(r)$ for $m = 1$, computed for $w = 1.0$. The solid black line corresponds to $\\ell = 2$, the dash-dotted blue line corresponds to $\\ell=4$, and the dashed red line corresponds to $\\ell = 6$.}   \n\\label{fig:Cm1} \n\\end{figure} \n\n\\begin{figure} \n\\includegraphics[width=0.7\\linewidth]{fig4}\n\\caption{Radial functions $C^m_\\ell(r)$ for $m = 0$, computed for $w = 1.0$. The solid black line corresponds to $\\ell = 2$, the dash-dotted blue line corresponds to $\\ell=4$, and the dashed red line corresponds to $\\ell = 6$.} \n\\label{fig:Cm0} \n\\end{figure} \n\nNext we present a very small sampling of our numerical results for $m = \\pm 1$ and $m = 0$. In Fig.~\\ref{fig:Cm1} we display the functions $C_\\ell^1(r)$ for $\\ell = 2, 4, 6$, computed for $w = 1.0$. In Fig.~\\ref{fig:Cm0} we show $C_\\ell^0(r)$ for $\\ell = 2, 4, 6$, also computed for $w = 1.0$. As advertised, in both cases $|C_4^m|$ is smaller than $C_2^m$ by about two orders of magnitude, and $|C_6^m|$ is smaller still. This behavior is shared by the other radial functions: we have that $|A_5^m| < |A_3^m| \\ll |A_1^m|$ and $|B_5^m| < |B_3^m| \\ll |B_1^m|$. As was pointed out, this hierarchy is satisfied for all values of $w$, except for eigenfrequencies of subdominant inertial modes; the hierarchy continues to be respected when $w$ is an eigenfrequency of a dominant mode.   \n\n\\subsection{Current quadrupole moment} \n\\label{sec:quadmoment}  \n\nA meaningful measure of the velocity perturbation is provided by the current quadrupole moment $J^{jk}$, defined by \n\\begin{equation} \nJ^{jk} = \\int \\epsilon^{(j}_{\\ \\, pq} x^{k)} x^p \\rho\\, \\delta v^q\\, dV, \n\\label{J_def} \n\\end{equation} \nwhere $\\delta v^q$ is the velocity field of Eqs.~(\\ref{va_decomposed1}) and (\\ref{va_decomposed2}), and $dV$ is the volume element. As in Sec.~\\ref{sec:gravitomagnetic}, we use indices $j,k,p,\\ldots$ in all equations formulated in Cartesian coordinates. \n\nTo calculate $J^{jk}$ we rely on the identities \n\\begin{subequations} \n\\label{iden} \n\\begin{align} \n\\epsilon^{(j}_{\\ pq} r^{k)} r^p\\, r^q\\, Y_\\ell^m &= 0, \n\\label{iden1} \\\\ \n\\int \\epsilon^{(j}_{\\ pq} r^{k)} r^p\\, (Y_\\ell^m)^q\\, d\\Omega &= 0, \n\\label{iden2} \\\\ \n\\int \\epsilon^{(j}_{\\ pq} r^{k)} r^p\\, (X_\\ell^m)^q\\, d\\Omega&= \n\\frac{8\\pi}{5 r}\\, (\\bar{\\scrpt Y}_2^m)^{jk}\\, \\delta_{\\ell,2} \n\\label{iden3} \n\\end{align} \n\\end{subequations} \ninvolving the radial, polar, and axial harmonics of Sec.~\\ref{subsec:harmonics}, as well as the unit radial vector $r^j = x^j\/r$; in Eq.~(\\ref{iden3}) we have the constant tensors (see Box~1.5 of Ref.~\\cite{poisson-will:14}) \n\\begin{equation} \n(\\bar{\\scrpt Y}_2^{\\pm 2})^{jk} = \\frac{1}{8} \\sqrt{\\frac{30}{\\pi}} \\left( \n\\begin{array}{ccc} \n1 & \\pm i & 0 \\\\ \n\\pm i & -1 & 0 \\\\ \n0 & 0 & 0 \n\\end{array} \\right), \\qquad \n(\\bar{\\scrpt Y}_2^{\\pm 1})^{jk} = \\frac{1}{8} \\sqrt{\\frac{30}{\\pi}} \\left( \n\\begin{array}{ccc} \n0 & 0 & \\mp 1 \\\\ \n0 & 0 & -i \\\\ \n\\mp 1 & -i & 0 \n\\end{array} \\right), \\qquad \n(\\bar{\\scrpt Y}_2^{0})^{jk} = \\frac{1}{4} \\sqrt{\\frac{5}{\\pi}} \\left( \n\\begin{array}{ccc} \n-1 & 0 & 0 \\\\ \n0 & -1 & 0 \\\\ \n0 & 0 & 2 \n\\end{array} \\right), \n\\label{STFtensors} \n\\end{equation} \nwhich are all symmetric and tracefree. \n\nThe result of Eq.~(\\ref{iden1}) follows simply from the antisymmetry of $\\epsilon^j_{\\ pq}$ with respect to the indices $p$ and $q$ and the symmetry of $r^p r^q$; the presence of the spherical harmonics is irrelevant. The identity of Eq.~(\\ref{iden2}) is derived by importing the definition of the polar harmonics, $(Y_\\ell^m)^q = \\nabla^q Y_\\ell^m$, and performing an integration by parts; symmetrization of the $jk$ indices ensures that the result vanishes. To establish the third equation we insert the definition of the axial harmonics, $(X_\\ell^m)^q = \\epsilon^{qrs} \\nabla_r Y_\\ell^m r_s$, combine the permutation symbols, and perform an integration by parts to obtain \n\\begin{equation} \n\\int \\epsilon^{(j}_{\\ pq} r^{k)} r^p\\, (X_\\ell^m)^q\\, d\\Omega \n= \\frac{3}{r} \\int (r^j r^k - \\tfrac{1}{3} \\delta^{jk}) Y_\\ell^m\\, d\\Omega.\n\\end{equation} \nIn the next step we rely on the fact that with the help of the constant tensors introduced in Eq.~(\\ref{STFtensors}), the symmetric, tracefree tensor within the integral can be decomposed in $\\ell=2$ spherical harmonics, according to [Eq.~(1.164) of Ref.~\\cite{poisson-will:14}] \n\\begin{equation} \nr^j r^k - \\frac{1}{3} \\delta^{jk} = \\frac{8\\pi}{15} \\sum_{m=-2}^2  \n(\\bar{\\scrpt Y}_2^m)^{jk}\\, \\bar{Y}_2^m(\\theta,\\phi). \n\\end{equation} \nFor the final step we integrate over the angles and make use of the orthonormality of spherical harmonics.  \n\n\\begin{figure} \n\\includegraphics[width=0.7\\linewidth]{fig5}\n\\caption{Plot of $\\gothj_m$ for $m=2$ as a function of $w := \\omega\/\\Omega$. The vertical line at $w = 4\/3 \\simeq 1.3333$ corresponds to the eigenfrequency of an $r$-mode.} \n\\label{fig:j_m2} \n\\end{figure} \n\n\\begin{figure} \n\\includegraphics[width=0.7\\linewidth]{fig6}\n\\caption{Plot of $\\gothj_m$ for $m=1$ as a function of $w$. The black diamonds result from the numerical evaluation of Eq.~(\\ref{j_components}). The solid red curve is the mode-sum representation constructed in Sec.~\\ref{sec:oscillator}. The vertical lines at $w \\simeq -0.4130$ and $w \\simeq 1.4014$ correspond to the eigenfrequencies of dominant inertial modes (polar-led modes with $m=1$ and $\\ell_0 = 2$, calculated for a $p = K\\rho^2$ polytrope).}   \n\\label{fig:j_m1} \n\\end{figure} \n\nThe identities of Eqs.~(\\ref{iden}) imply that when Eqs.~(\\ref{va_decomposed1}) and (\\ref{va_decomposed2}) are inserted within Eq.~(\\ref{J_def}), the volume integral reduces to a radial integration involving the mass density $\\rho$, the functions $C_2^m(r)$, and powers of $r$. For $m = \\pm 2$ we can make further use of the exact solution of Eq.~(\\ref{C22_solution}). After working through the details, we arrive at \n\\begin{equation} \nJ^{jk} = \\sum_{m=-2}^2 J^{jk}_m,  \n\\label{J_decomposed} \n\\end{equation} \nwhere \n\\begin{subequations} \n\\label{J_components} \n\\begin{align} \nJ^{jk}_{\\pm 2} &= \\frac{{\\cal B}_0}{c^2} M R^4 \\sin\\iota (\\cos\\iota \\pm 1)\\, \\gothj_{\\pm 2}(\\omega) \n\\left( \\begin{array}{ccc} \n\\sin\\omega t & \\mp \\cos\\omega t & 0 \\\\ \n\\mp \\cos\\omega t & -\\sin\\omega t & 0 \\\\ \n0 & 0 & 0 \n\\end{array} \\right), \\\\ \nJ^{jk}_{\\pm 1} &= -\\frac{{\\cal B}_0}{c^2} M R^4 (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1)\\, \n\\gothj_{\\pm 1}(\\omega) \n\\left( \\begin{array}{ccc} \n0 & 0 & \\cos\\omega t \\\\ \n0 & 0 & \\pm \\sin\\omega t \\\\ \n\\cos\\omega t & \\pm \\sin\\omega t & 0 \n\\end{array} \\right), \\\\ \nJ^{jk}_{0} &= \\frac{{\\cal B}_0}{c^2} M R^4 \\sin\\iota \\cos\\iota\\, \\gothj_{0}(\\omega) \n\\left( \\begin{array}{ccc} \n\\sin\\omega t & 0 & 0 \\\\ \n0 & \\sin\\omega t & 0 \\\\ \n0 & 0 & -2\\sin\\omega t \n\\end{array} \\right), \n\\end{align} \n\\end{subequations} \nwith \n\\begin{subequations} \n\\label{j_components} \n\\begin{align} \n\\gothj_{\\pm 2}(\\omega) &:= \\frac{9}{5} \\frac{2 \\mp w}{4 \\mp 3w} \n\\int_0^1 \\hat{\\rho}\\, \\hat{r}^6\\, d\\hat{r}, \n\\label{j_components_m2} \\\\  \n\\gothj_{\\pm 1}(\\omega) &:= -3 \\sqrt{\\frac{3}{10\\pi}} (1 \\mp w) \n\\int_0^1 \\hat{\\rho}\\, C_2^{\\pm 1}\\, \\hat{r}^3\\, d\\hat{r}, \\\\ \n\\gothj_0(\\omega) &:= \\frac{3}{\\sqrt{5\\pi}} w \n\\int_0^1 \\hat{\\rho}\\, C_2^{0}\\, \\hat{r}^3\\, d\\hat{r}. \n\\end{align} \n\\end{subequations} \nWe introduced $\\hat{\\rho} := (4\\pi R^3\/3M) \\rho$ as a dimensionless density, and $\\hat{r} := r\/R$ as a dimensionless radial coordinate. By virtue of the definitions and Eqs.~(\\ref{ABCidentities1}), (\\ref{ABCidentities2}), and (\\ref{ABCidentities3}), we have that \n\\begin{equation} \n\\gothj_{-2}(\\omega) = \\gothj_{2}(-\\omega), \\qquad \n\\gothj_{-1}(\\omega) = -\\gothj_{1}(-\\omega) \\qquad \n\\gothj_0(\\omega) = \\gothj_0(-\\omega).  \n\\label{j_identities} \n\\end{equation} \nA minus sign was incorporated in the definition of $\\gothj_{\\pm 1}$ to enforce a convention that each  $\\gothj_m(\\omega)$ for $m \\geq 0$ is positive in a neighborhood of $\\omega = 0$. \n\n\\begin{figure} \n\\includegraphics[width=0.7\\linewidth]{fig7}\n\\caption{Plot of $\\gothj_m$ for $m=0$ as a function of $w$. The black diamonds result from the numerical evaluation of Eq.~(\\ref{j_components}). The solid red curve is the mode-sum representation constructed in Sec.~\\ref{sec:oscillator}. The vertical lines at $w \\simeq \\pm 1.0282$ correspond to the eigenfrequencies of dominant inertial modes (polar-led modes with $m=0$ and $\\ell_0 = 2$, calculated for a $p = K\\rho^2$ polytrope.)}  \n\\label{fig:j_m0} \n\\end{figure} \n\nIn Figs.~\\ref{fig:j_m2}, \\ref{fig:j_m1}, and \\ref{fig:j_m0} we display plots of the dimensionless quadrupole moments $\\gothj_m(\\omega)$ for $m = 2$, $m = 1$, and $m=0$, respectively. The black diamonds correspond to a numerical evaluation of the integrals in Eqs.~(\\ref{j_components}), calculated for the $p = K\\rho^2$ polytropic model, for which $\\hat{\\rho} = (\\pi\/3) \\sin(\\pi \\hat{r})\/\\hat{r}$. The solid red curves correspond to the construction detailed in Sec.~\\ref{sec:oscillator}, based on a mode-sum representation of the tidal perturbation; the point is to demonstrate the excellent agreement between these different calculations. There are no black diamonds for $m=2$, because in this case the mode sum is exact and agrees precisely with Eq.~(\\ref{j_components}). For $m=2$ the radial integration can be carried out analytically:\n\\begin{equation} \n\\int_0^1 \\hat{\\rho}\\, \\hat{r}^6\\, d\\hat{r} = \\frac{\\pi^4 - 20\\pi^2 + 120}{3\\pi^4}. \n\\label{rho-r6} \n\\end{equation} \n\nThe plots of $\\gothj_m(\\omega)$ feature simple poles when $\\omega$ matches an eigenfrequency of a dominant inertial mode. For $m=2$ the pole occurs at $w = 4\/3$, which corresponds to the eigenfrequency of an $r$-mode. For $m=1$ we have poles at $w \\simeq -0.4130$ and $w \\simeq 1.4014$; these are the eigenfrequencies of the polar-led inertial modes labelled $\\ell_0 = 2$ by Lockitch and Friedman \\cite{lockitch-friedman:99}. For $m=0$ the poles occur at $w \\simeq \\pm 1.0282$, the eigenfrequencies of the polar-led inertial modes also labelled $\\ell_0 = 2$. A finer sampling of $\\gothj_m(\\omega)$ would reveal weaker poles associated with higher-order, subdominant inertial modes.  \n \n\\section{Mode-sum representation of the perturbation} \n\\label{sec:oscillator} \n\nIn this section we construct an alternative representation of the velocity perturbation in terms of a sum over normal modes. Our goals here are twofold. First, we demonstrate that a set of just four modes provides a very accurate approximation to the perturbation. Second, we introduce some of the essential ingredients required in a discussion of the dynamical impact of the resonances on a binary inspiral; this discussion will be taken up in Sec.~\\ref{sec:dynamics}.  \n\n\\subsection{Formalism} \n\nMethods to represent a driven perturbation of a rotating fluid as a sum over normal modes are developed in Sec.~II and Appendix A of Ref.~\\cite{schenk-etal:01}. We begin with a very brief summary of these techniques.  \n\nThe formalism makes central use of the Lagrangian displacement vector $\\xi^a$, which is related to the velocity field $\\delta v^a$ by Eq.~(\\ref{xi_vs_dv}). It is useful to introduce a linear-algebra notation in which $\\xi^a$ is mapped to an abstract vector $\\xi$ in a Hilbert space, and the operation $\\KK^a_{\\ b} \\xi^b:= v^b \\nabla_b \\xi^a$ is mapped to $\\KK \\xi$. We also introduce an inner product $\\bkt{\\eta}{\\xi}$ between two vectors $\\eta$ and $\\xi$, defined by \n\\begin{equation}\n\\bkt{\\eta}{\\xi} := \\int_V \\rho\\, \\bar{\\eta}_a \\xi^a\\, dV, \n\\end{equation}\nwhere an overbar indicates complex conjugation. The inner product satisfies the usual property that $\\bkt{\\xi}{\\eta}$ is the complex conjugate of $\\bkt{\\eta}{\\xi}$. \n\nA normal mode is a solution to the perturbation equations (without an external force) of the specific form \n\\begin{equation}\n\\xi^a(t,\\bm{x}) = \\zeta^a(\\bm{x}) e^{-i\\omega t}. \n\\label{mode_def}\n\\end{equation} \nThe perturbation equations become an eigenvalue problem for the frequency $\\omega$ and mode functions $\\zeta^a$. There is an infinity of solutions, and we label each solution with a mode index $K$; for simplicity we assume that the mode spectrum is discrete and nondegenerate, so that there is a unique mode $\\xi_K$ for each eigenfrequency $\\omega_K$.  \n\nNext we consider the perturbation equations with an external force, represented by the abstract vector $f$. We wish to represent the solution $\\xi$ in terms of a sum over modes. We adopt a ``phase-space'' representation and write \n\\begin{equation} \n\\xi  = \\sum_K q_K(t)\\, \\zeta_K, \\qquad  \n\\partial_t \\xi = \\sum_K (-i\\omega_K) q_K(t)\\, \\zeta_K, \n\\label{mode-sum} \n\\end{equation} \nfor some amplitudes $q_K(t)$. The perturbation equations imply that these are determined by \n\\begin{equation} \n\\dot{q}_K + i\\omega_K q_K = i f_K := -\\frac{\\bkt{\\zeta_K}{f}}{2i\\omega_K N_K}, \n\\label{q_eq} \n\\end{equation} \nwhere \n\\begin{equation} \nN_K := \\bkt{\\zeta_K}{\\zeta_K} - \\frac{1}{i\\omega_K} \\bkt{\\zeta_K}{\\KK \\zeta_K} \n\\label{Ndef} \n\\end{equation} \nprovides a notion of mode norm. (This is to be interpreted with caution: while $N_K$ is real, it is not positive-definite.) For $f = 0$ we would have that $q_K \\propto \\exp(-i\\omega_K t)$, and the expression for $\\partial_t \\xi$ would follow from the one for $\\xi$ by direct differentiation. The expressions, however, are independent when $f \\neq 0$.  \n\nThe normal modes that are implicated in a gravitomagnetic tidal interaction are the Lockitch-Friedman inertial modes \\cite{lockitch-friedman:99}, which are labelled by the azimuthal integer $m$ and an additional mode index $n$. Inertial modes with negative values of $m$ are related to those with positive values by \n\\begin{equation} \n\\omega_n^{-m} = -\\omega_n^m, \\qquad \n\\zeta_n^{-m} = \\bar{\\zeta}_n^m. \n\\label{m_negative} \n\\end{equation} \nThese properties allow us to fold the sum over negative values of $m$ into the positive values. To achieve this folding, we note that since $f$ is a real vector, $\\bkt{\\zeta_n^{-m}}{f}$ is the complex conjugate of $\\bkt{\\zeta_n^m}{f}$. We also note that $N_n^{-m} = N_n^{m}$ is a direct consequence of the definition of Eq.~(\\ref{Ndef}). From all this it follows that $q_n^{-m} = \\bar{q}_n^m$. We further take into account the fact that the $m=0$ modes always come in complex-conjugate pairs; one member has a positive frequency, the second member has a negative frequency of the same absolute value, and the mode functions are complex conjugates of each other. These observations imply that Eqs.~(\\ref{mode-sum}) can be written as     \n\\begin{equation}\n\\xi = \\sum_{m \\geq 0} \\sideset{}{'}\\sum_n \\bigl[ q_n^m(t)\\, \\zeta_n^m + \\mbox{cc} \\bigr], \\qquad\n\\partial_t \\xi = \\sum_{m \\geq 0} \\sideset{}{'}\\sum_n \\bigl[ -i\\omega_n^m q_n^m(t)\\, \\zeta_n^m + \\mbox{cc} \\bigr]; \n\\label{mode_sum_inertial} \n\\end{equation} \nthe prime on the sum over $n$ reminds us that for $m = 0$, the sum is to include positive-frequency modes only. Inserting Eqs.~(\\ref{mode_sum_inertial}) within Eq.~(\\ref{xi_vs_dv}), we find that \n\\begin{equation}\n\\delta v^a = \\sum_{m \\geq 0} \\sideset{}{'}\\sum_n \\bigl[ i(m\\Omega - \\omega_n^m) q_n^m(t)\\, (\\zeta_n^m)^a\n+ \\mbox{cc} \\bigr] \n\\label{dv_inertialmodesum} \n\\end{equation}\nis the required representation of the velocity perturbation in terms of a sum over inertial modes. \n\nIn the same notation specialized to inertial modes, Eq.~(\\ref{q_eq}) becomes\n\\begin{equation} \n\\dot{q}_n^m + i\\omega_n^m q_n^m = i f_n^m := -\\frac{\\bkt{\\zeta_n^m}{f}}{2i\\omega_n^m N_n^m}, \n\\label{q_eq_inertial} \n\\end{equation} \nand the mode norm is now\n\\begin{equation} \nN_n^m := \\bkt{\\zeta_n^m}{\\zeta_n^m} - \\frac{1}{i\\omega_n^m} \\bkt{\\zeta_n^m}{\\KK \\zeta_n^m}.  \n\\label{Ndef_inertial} \n\\end{equation} \n\n\\subsection{Description of inertial modes} \n\\label{sec:description} \n\nThe decomposition of the external force $f$ given in Eq.~(\\ref{fa_vharm}) implies that the only relevant modes are those with $m = \\{0, 1, 2\\}$. The corresponding mode functions are \n\\begin{subequations} \n\\label{mode_functions} \n\\begin{align} \n(\\zeta_n^2)^a &= R^2 \\sum_\\ell \\biggl[ \\frac{i}{r} (a_\\ell^2)_n\\, r^a Y_\\ell^2 \n+ i (b_\\ell^2)_n\\, (Y_\\ell^2)^a + (c_\\ell^2)_n (X_\\ell^2)^a \\biggr], \\\\\n(\\zeta_n^1)^a &= R^2 \\sum_\\ell \\biggl[ \\frac{1}{r} (a_\\ell^1)_n\\, r^a Y_\\ell^1 \n+ (b_\\ell^1)_n\\, (Y_\\ell^1)^a - i (c_\\ell^1)_n (X_\\ell^1)^a \\biggr], \\\\  \n(\\zeta_n^0)^a &= R^2 \\sum_\\ell \\biggl[ \\frac{i}{r} (a_\\ell^0)_n\\, r^a Y_\\ell^0 \n+ i (b_\\ell^0)_n\\, (Y_\\ell^0)^a + (c_\\ell^0)_n (X_\\ell^0)^a \\biggr],\n\\end{align} \n\\end{subequations} \nwhere $(a_\\ell^m)_n$, $(b_\\ell^m)_n$, $(c_\\ell^m)_n$ are functions of the radial coordinate. These are all real and dimensionless, and for $m > 0$ they satisfy \n\\begin{equation} \n(a_\\ell^{-m})_n = -(a_\\ell^m)_n, \\qquad \n(b_\\ell^{-m})_n = -(b_\\ell^m)_n, \\qquad \n(c_\\ell^{-m})_n = (c_\\ell^m)_n\n\\end{equation} \nin order to enforce Eq.~(\\ref{m_negative}). Factors of $i$ were inserted in Eqs.~(\\ref{mode_functions}) to ensure that the overlap integrals $\\bkt{\\zeta_n^m}{f}$ are real. \n\nDifferential equations for the radial functions are obtained by following the same strategy as in Sec.~\\ref{sec:perturbation}. Here also we rely on a truncation of the system of equations, justified by the fact that for the dominant inertial modes, radial functions with the lowest value of $\\ell$ are much larger than functions with higher values of $\\ell$. More precisely stated, for the implicated modes we have that $|(a_5^m)_n| < |(a_3^m)_n| \\ll |(a_1^m)_n|$, $|(b_5^m)_n| < |(b_3^m)_n| \\ll |(b_1^m)_n|$, and $|(c_6^m)_n| < |(c_4^m)_n| \\ll |(c_2^m)_n|$. The equations take the same form as those listed in Secs.~\\ref{sec:eqns_m2}, \\ref{sec:eqns_m1}, and \\ref{sec:eqns_m0}, with all driving terms --- proportional to $(r\/R)^3$ --- set to zero.   \n\nOne of the mode equations is identical to Eq.~(\\ref{Aprime}), \n\\begin{equation} \nr \\frac{d(a_\\ell^m)_n}{dr} + \\biggl( 1 + \\frac{r}{\\rho} \\frac{d\\rho}{dr} \\biggr) (a_\\ell^m)_n \n- \\ell(\\ell+1) (b_\\ell^m)_n = 0,    \n\\label{aprime} \n\\end{equation} \nand it is easy to show that it implies \n\\begin{equation} \n\\rho r^3 \\bigl[ 3 (a_\\ell^m)_n + \\ell(\\ell+1) (b_\\ell^m)_n \\bigr] \n= \\frac{d}{dr} \\bigl[ \\rho r^4 (a_\\ell^m)_n \\bigr]. \n\\end{equation} \nIntegrating both sides from $r=0$ to $r=R$, we arrive at the identity \n\\begin{equation} \n\\int_0^R \\rho r^3 \\bigl[ 3 (a_\\ell^m)_n + \\ell(\\ell+1) (b_\\ell^m)_n \\bigr]\\, dr = 0, \n\\label{ab_identity} \n\\end{equation} \nwhich will be required below. On the right-hand side, the boundary term at $r=0$ vanishes because of the factor of $r^4$, and the one at $r=R$ vanishes because $\\rho = 0$ for our stellar models, and also because $(a_\\ell^m)_n = 0$, by virtue of the mode version of Eq.~(\\ref{BC_A}). \n\nNumerical solutions to the eigenvalue problem for the mode frequencies $\\omega_n^m$ and mode functions $(\\zeta_n^m)^a$ are obtained with the computational methods described in Sec.~\\ref{sec:solution}. We develop series expansions near $r=0$ and $r=R$, and we integrate outward from $r \\simeq 0$ and inward from $r \\simeq R$. To determine the eigenfrequencies and the free coefficients that appear in the series expansions, we search for solutions that match smoothly at a middle point $r^\\sharp$. Our results are in excellent agreement with those of Lockitch and Friedman \\cite{lockitch-friedman:99}.  \n\n\\subsection{Overlap integrals} \n\\label{sec:overlap} \n\nWe insert Eqs.~(\\ref{fa_vharm}) and (\\ref{mode_functions}) within the overlap integrals $\\bkt{\\zeta_n^m}{f}$, and perform  the angular integration with the help of the orthogonality relations of Eq.~(\\ref{ortho1}) and (\\ref{ortho2}). We find \n\\begin{align} \n\\bkt{\\zeta_n^m}{f} &=\\alpha^m \\frac{{\\cal B}_0}{c^2} R^2 \\sum_\\ell \\int \\rho \\biggl\\{ \n(a_\\ell^m)_n \\Bigl[ (f_\\ell^m)^{\\rm R} e^{-i\\omega t} + (-1)^m (\\bar{f}_\\ell^{-m})^{\\rm R} e^{i\\omega t} \\Bigr] \n\\nonumber \\\\ & \\quad \\mbox{} \n+ \\ell(\\ell+1) (b_\\ell^m)_n \\Bigl[ (f_\\ell^m)^{\\rm P} e^{-i\\omega t} \n+ (-1)^m (\\bar{f}_\\ell^{-m})^{\\rm P} e^{i\\omega t} \\Bigr] \n\\nonumber \\\\ & \\quad \\mbox{} \n+ i\\ell(\\ell+1) (c_\\ell^m)_n \\Bigl[ (f_\\ell^m)^{\\rm A} e^{-i\\omega t} \n+ (-1)^m (\\bar{f}_\\ell^{-m})^{\\rm A} e^{i\\omega t} \\Bigr] \\biggr\\}\\, dr, \n\\end{align} \nwhere $\\alpha^m = -i$ when $m=0, 2$, and $\\alpha^m = 1$ when $m=1$. Next we exploit the observation of Eq.~(\\ref{RP-proportionality}) and the identity of Eq.~(\\ref{ab_identity}) to eliminate the integrals involving the radial (R) and polar (P) components of the external force. And because the axial (A) components come with $\\ell = 2$ only, the overlap integrals reduce to \n\\begin{equation} \n\\bkt{\\zeta_n^m}{f} = 6i\\alpha^m \\frac{{\\cal B}_0}{c^2} R^2 \\int \\rho (c_2^m)_n \n\\Bigl[ (f_2^m)^{\\rm A} e^{-i\\omega t} \n+ (-1)^m (\\bar{f}_2^{-m})^{\\rm A} e^{i\\omega t} \\Bigr]\\, dr. \n\\end{equation} \nMaking the substitutions from Eq.~(\\ref{fa_axial}), we arrive at \n\\begin{subequations} \n\\label{overlap1} \n\\begin{align} \n\\bkt{\\zeta_n^2}{f} &= \\frac{{\\cal B}_0}{c^2} M R^3\\, \\gothp_n^2 \\bigl[ \n-\\sin\\iota(\\cos\\iota+1)(2\\Omega-\\omega) e^{-i\\omega t} \n+ \\sin\\iota(\\cos\\iota-1)(2\\Omega+\\omega) e^{i\\omega t} \\bigr], \\\\\n\\bkt{\\zeta_n^1}{f} &= \\frac{{\\cal B}_0}{c^2} M R^3\\, \\gothp_n^1 \\bigl[ \n(\\cos\\iota+1)(2\\cos\\iota-1)(\\Omega-\\omega) e^{-i\\omega t} \n- (\\cos\\iota-1)(2\\cos\\iota+1)(\\Omega+\\omega) e^{i\\omega t} \\bigr], \\\\ \n\\bkt{\\zeta_n^0}{f} &= \\frac{{\\cal B}_0}{c^2} M R^3\\, \\gothp_n^0\\, \\sin\\iota\\cos\\iota\\, \\omega \n(e^{-i\\omega t} + e^{i\\omega t}) \n\\end{align} \n\\end{subequations} \nwith \n\\begin{subequations} \n\\label{overlap2} \n\\begin{align} \n\\gothp_n^2 &:= 3 \\sqrt{\\frac{3}{10\\pi}} \\int \\hat{\\rho}\\, (c_2^2)_n\\, \\hat{r}^3\\, d\\hat{r}, \\\\ \n\\gothp_n^1 &:= 3 \\sqrt{\\frac{3}{10\\pi}} \\int \\hat{\\rho}\\, (c_2^1)_n\\, \\hat{r}^3\\, d\\hat{r}, \\\\ \n\\gothp_n^0 &:= \\frac{9}{\\sqrt{5\\pi}} \\int \\hat{\\rho}\\, (c_2^0)_n\\, \\hat{r}^3\\, d\\hat{r}, \n\\label{gothp2_def} \n\\end{align}\n\\end{subequations} \nwhere $\\hat{\\rho} := (4\\pi R^3\/3M) \\rho$ and $\\hat{r} := r\/R$. The reduced overlap integrals $\\gothp_n^m$ are dimensionless.  \n\n\\subsection{Implicated modes} \n\\label{sec:implicated} \n\nEquations (\\ref{overlap2}) indicate that the inertial modes implicated in the overlap integrals are those for which $(c_2^m)_n \\neq 0$. Adopting the terminology of Lockitch and Friedman \\cite{lockitch-friedman:99}, this restriction selects the following set of modes: \n\\begin{itemize} \n\\item For $m = 2$ we have axial-led modes with nonvanishing $(a_3^2)_n$, $(a_5^2)_n$, $(a_7^3)_n$ and so on, $(b_3^2)_n$, $(b_5^2)_n$, $(b_7^2)_n$ and so on, and $(c_2^2)_n$, $(c_4^2)_n$, $(c_6^2)_n$ and so on. But as we are about to show, there is only one mode that produces a nonvanishing overlap integral with the external force: the $r$-mode with frequency $\\omega_\\bullet^2 = (4\/3)\\Omega$ and sole nonvanishing radial function $(c_2^2)_\\bullet = (r\/R)^3$. For a barotropic star, the $r$-mode is independent of the equation of state.   \n\\item For $m = 1$ we have polar-led modes with nonvanishing $(a_1^1)_n$, $(a_3^1)_n$, $(a_5^1)_n$ and so on, $(b_1^1)_n$, $(b_3^1)_n$, $(b_5^1)_n$ and so on, and $(c_2^1)_n$, $(c_4^1)_n$, $(c_6^1)_n$ and so on. The two dominant modes are those designated by $\\ell_0 = 2$, and for the polytropic model $p = K\\rho^2$, they have the eigenfrequencies $\\omega_{\\rm I}^1 \\simeq 1.4014\\, \\Omega$ and  $\\omega_{\\rm II}^1 \\simeq -0.4130\\, \\Omega$.  \n\\item For $m = 0$ we have polar-led modes with nonvanishing $(a_1^0)_n$, $(a_3^0)_n$, $(a_5^0)_n$ and so on, $(b_1^0)_n$, $(b_3^0)_n$, $(b_5^0)_n$ and so on, and $(c_2^0)_n$, $(c_4^0)_n$, $(c_6^0)_n$ and so on. The two dominant modes are also those with $\\ell_0 = 2$, and for the polytropic model they have frequencies $\\omega_{\\rm I}^0 \\simeq 1.0282\\, \\Omega$ and $\\omega_{\\rm II}^0 \\simeq -1.0282\\, \\Omega$. As was discussed previously, only the positive-frequency mode is included in the mode sum. \n\\end{itemize} \nThe dominant modes for $m=1$ and $m=0$ are stable; the $r$-mode for $m=2$ is subjected to the Chandrasekhar-Friedman-Schutz instability \\cite{chandrasekhar:70, friedman-schutz:78b, andersson:98, friedman-morsink:98}. \n\nWe shall now establish that  \n\\begin{equation} \n\\int \\rho\\, (c_2^2)_n\\, r^3\\, dr = 0 \n\\label{rmode_only} \n\\end{equation} \nfor all inertial modes except for the $r$-mode. As was stated previously, this implies that for $m=2$, only the $r$-mode is implicated in the decomposition of the velocity perturbation. The property follows from the orthogonality of normal modes with distinct frequencies. \n\nThe statement of orthogonality for modes $\\zeta_K$ and $\\zeta_{K'}$ with $\\omega_K \\neq \\omega_{K'}$ is [Eq.~(2.32) of Ref.~\\cite{schenk-etal:01}]\n\\begin{equation} \n(\\omega_K + \\omega_{K'}) \\bkt{\\zeta_K}{\\zeta_{K'}} + 2i \\bkt{\\zeta_K}{\\KK \\zeta_{K'}} = 0. \n\\label{ortho_KK} \n\\end{equation}  \nIn this we insert for $\\zeta_K$ a generic inertial mode described by \n\\begin{equation} \n\\zeta_K^a = R^2 \\sum_\\ell \\biggl[ \\frac{i}{r} (a_\\ell^2)_n\\, r^a Y_\\ell^2 \n+ i (b_\\ell^2)_n\\, (Y_\\ell^2)^a + (c_\\ell^2)_n\\, (X_\\ell^2)^a \\biggr],  \n\\end{equation} \nand for $\\zeta_{K'}$ the $r$-mode with eigenfrequency $\\omega_{K'} = (4\/3)\\Omega$ and mode function $\\zeta^a_{K'} = R^2 (r\/R)^3 (X_2^2)^a$. For this mode we have that \n\\begin{equation} \n(\\KK \\zeta_{K'})^a = v^b \\nabla_b \\zeta_{K'}^b = \\Omega R^2 (r\/R)^3 \\biggl[ \\frac{2\\sqrt{7}}{7} \\frac{1}{r}\\, r^a Y_3^2 \n+ \\frac{2\\sqrt{7}}{21} (Y_3^2)^a + \\frac{5i}{3} (X_2^2)^a \\biggr]. \n\\end{equation} \nUsing the orthogonality relations of Eq.~(\\ref{ortho1}) and (\\ref{ortho2}), we find that \n\\begin{equation} \n\\bkt{\\zeta_K}{\\zeta_{K'}} = 6R \\int \\rho\\, (c_2^2)_n\\, r^3\\, dr\n\\end{equation} \nand \n\\begin{equation} \n\\bkt{\\zeta_K}{\\KK \\zeta_{K'}} = \n-\\frac{2i\\sqrt{7}}{21} \\Omega R \\int \\rho r^3 \\bigl[ 3(a_3^2)_n + 12(b_3^2)_n \\bigr]\\, dr \n+ 10 i \\Omega R \\int \\rho\\, (c_2^2)_n\\, r^3\\, dr. \n\\end{equation} \nThis simplifies to \n\\begin{equation} \n\\bkt{\\zeta_K}{\\KK \\zeta_{K'}} = 10 i \\Omega R \\int \\rho\\, (c_2^2)_n\\, r^3\\, dr \n\\end{equation} \nafter invoking Eq.~(\\ref{ab_identity}). The statement of orthogonality therefore becomes \n\\begin{equation} \n(\\omega_K - 2\\Omega) \\int \\rho\\, (c_2^2)_n\\, r^3\\, dr = 0, \n\\end{equation} \nand we arrive at Eq.~(\\ref{rmode_only}) whenever $\\omega_K \\neq 2\\Omega$. \n\nWith Eq.~(\\ref{rmode_only}) established, we conclude that for $m = 2$, the sums over $n$ in Eqs.~(\\ref{mode_sum_inertial}) reduce to a single term, which we denote $q_\\bullet^2\\, \\zeta_\\bullet^2 + \\mbox{cc}$ and $-i\\omega_\\bullet^2\\, q_\\bullet^2\\, \\zeta_\\bullet^2 + \\mbox{cc}$, respectively. Equation (\\ref{gothp2_def}) becomes  \n\\begin{equation} \n\\gothp_\\bullet^2 = 3 \\sqrt{\\frac{3}{10\\pi}} \\int \\hat{\\rho}\\, \\hat{r}^6\\, d\\hat{r};\n\\label{gothp2_result} \n\\end{equation} \nthe integral was evaluated in Eq.~(\\ref{rho-r6}) for the $p = K\\rho^2$ polytropic model. \n\n\\subsection{Mode norm} \n\\label{sec:norm} \n\nThe mode norm $N_n^m$ is defined by Eq.~(\\ref{Ndef_inertial}), and to calculate it we introduce the notation \n\\begin{equation} \n\\bkt{\\zeta_n^m}{\\zeta_n^m} = \\frac{3}{4\\pi} M R^2\\, \\gothn_n^m, \\qquad \n\\bkt{\\zeta_n^m}{\\KK\\zeta_n^m} = \\frac{3i}{4\\pi} M R^2\\Omega\\, \\mathfrak{m}_n^m,\n\\end{equation} \nin which $\\gothn_m^n$ and $\\gothn_m^n$ are dimensionless. With this we have that \n\\begin{equation} \nN_n^m = \\frac{3}{4\\pi} M R^2\\, \\hat{N}_n^m,  \n\\end{equation} \nwhere \n\\begin{equation} \n\\hat{N}_n^m := \\gothn_n^m - \\frac{\\mathfrak{m}_n^m}{w_n^m}, \\qquad \nw_n^m := \\omega_n^m\/\\Omega. \n\\label{Nhat_def} \n\\end{equation} \n\nFor $m=2$ we have the exact expressions \n\\begin{equation} \n\\gothn_\\bullet^2 = 6 \\int_0^1 \\hat{\\rho}\\, \\hat{r}^6\\, d\\hat{r}, \\qquad \n\\mathfrak{m}_\\bullet^2 = 10 \\int_0^1 \\hat{\\rho}\\, \\hat{r}^6\\, d\\hat{r}, \\qquad \n\\hat{N}_\\bullet^2 = -\\frac{3}{2} \\int_0^1 \\hat{\\rho}\\, \\hat{r}^6\\, d\\hat{r}. \n\\label{gothnm_m2} \n\\end{equation}  \nTo calculate $\\gothn_n^m$ and $\\mathfrak{m}_n^m$ for $m=0$ and $m=1$ we truncate the description of the mode to include only the dominant functions $a_1^m$, $b_1^m$, and $c_2^m$. This gives \n\\begin{subequations}\n\\label{gothnm_m1} \n\\begin{align} \n\\gothn_n^1 &= \\int \\hat{\\rho} \\bigl[ (a_1^1)_n^2 + 2 (b_1^1)_n^2 + 6 (c_2^1)_n^2 \\bigr]\\, d\\hat{r}, \\\\ \n\\mathfrak{m}_n^1 &= \\int \\hat{\\rho} \\Bigl\\{ (a_1^1)_n^2 - 2 (a_1^1)_n (b_1^1)_n + (b_1^1)_n^2 \n+ \\frac{6\\sqrt{5}}{5} \\bigl[ (a_1^1)_n  - (b_1^1)_n \\bigr] (c_2^1)_n + 5(c_2^1)_n^2 \\Bigr\\}\\,  d\\hat{r} \n\\end{align} \n\\end{subequations} \nand \n\\begin{subequations}\n\\label{gothnm_m0} \n\\begin{align} \n\\gothn_n^0 &= \\int \\hat{\\rho} \\bigl[ (a_1^0)_n^2 + 2 (b_1^0)_n^2 + 6 (c_2^0)_n^2 \\bigr]\\, d\\hat{r}, \\\\ \n\\mathfrak{m}_n^0 &= \\frac{4\\sqrt{15}}{5} \\int \\hat{\\rho} \\bigl[ (a_1^0)_n - (b_1^0)_n \\bigr] (c_2^0)_n\\, d\\hat{r}. \n\\end{align} \n\\end{subequations} \n\n\\subsection{Stellar models}\n\nWe compute eigenfrequencies $w_n^m := \\omega_n^m\/\\Omega$, overlap integrals $\\gothp_n^m$, and mode norms $\\gothn_n^m$, $\\mathfrak{m}_n^m$, and $\\hat{N}_n^m$ for a variety of stellar models. These are defined by the density function \n\\begin{equation}\n\\rho = C \\biggl[ \\frac{\\sin(\\pi r\/R)}{(\\pi r\/R)} \\biggr]^k,\n\\label{rho_kmodel} \n\\end{equation}\nin which $C$ is a constant (irrelevant for our purposes) and $k$ is an integer in the set $\\{ 0, 1, 2, 3 \\}$. With $k=0$ we have a star of constant density, and $k = 1$ corresponds to a polytrope with $p = K\\rho^2$. To the remaining values of $k$ we can assign an equation of state by obtaining $p(r)$ from the structure equations and expressing the result as $p(r(\\rho))$, where $r(\\rho)$ is the inversion of Eq.~(\\ref{rho_kmodel}). The stellar models described by Eq.~(\\ref{rho_kmodel}) range from being nearly uniform (for small values of $k$) to being centrally dense (for large values of $k$). We recall that $\\rho$ appears explicitly in Eqs.~(\\ref{overlap2}), (\\ref{gothnm_m1}), and (\\ref{gothnm_m0}), but that it also enters --- via the function $1 + (r\/\\rho) (d\\rho\/dr)$ --- in the mode equations.   \n\n\\begin{table} \n\\caption{\\label{tab:overlaps} Eigenfrequencies, overlap integrals, and norms for the relevant modes, calculated for the stellar models described by Eq.~(\\ref{rho_kmodel}).}  \n\\begin{ruledtabular} \n\\begin{tabular}{cccccccc}\n$k$ & $m$ & mode & $w_n^m$ & $\\gothp_n^m$ & $\\gothn_n^m$ & $\\mathfrak{m}_n^m$ & $\\hat{N}_n^m$ \\\\ \n\\hline\n  0 & 2 & $\\bullet$ & 1.3333 & 1.3244e-1 & 8.5714e-1 & 1.4286 & -2.1429e-1 \\\\\n    & 1 & I & 1.1766 & 1.3244e-1 & 1.2286 & 1.3371 & 9.2208e-2 \\\\\n    & 1 & II & -5.0994e-1 & 1.3244e-1 & 2.8349 & 6.9463e-1 & 4.1970 \\\\\n    & 0 & I &  8.9443e-1 & 3.2440e-1 & 1.7143 & 7.6665e-1 & 8.5714e-1 \\\\\n\\hline\n  1 & 2 & $\\bullet$ & 1.3333 & 6.3502e-2 & 4.1099e-1 & 6.8498e-1 & -1.0275e-1 \\\\\n    & 1 & I & 1.4014 & -4.1974e-2 & 3.0282e-1 & 3.6375e-1 & 4.3258e-2 \\\\\n    & 1 & II & -4.1300e-1 & 2.6417e-2 & 1.7609e-1 & 5.1716e-2 & 3.0131e-1 \\\\\n    & 0 & I & 1.0282 & -7.9947e-2 & 2.1767e-1 & 1.1196e-1 & 1.0879e-1 \\\\\n\\hline\n  2 & 2 & $\\bullet$ & 1.3333 & 3.5252e-2 & 2.2816e-1 & 3.8026e-1 & -5.7039e-2 \\\\\n    & 1 & I & 1.4979 & -3.1348e-2 & 3.2385e-1 & 4.0454e-1 & 5.3785e-2 \\\\\n    & 1 & II & -3.8059e-1 & 2.2777e-2 & 2.1528e-1 & 6.6684e-2 & 3.9049e-1 \\\\\n    & 0 & I & 1.0763 & -6.3472e-2 & 2.4632e-1 & 1.3269e-1 & 1.2303e-1 \\\\\n\\hline\n  3 & 2 & $\\bullet$ & 1.3333 & 2.1933e-2 & 1.4195e-1 & 2.3659e-1 & -3.5489e-2 \\\\\n    & 1 & I & 1.5466 & -2.4483e-2 & 3.2772e-1 & 4.1736e-1 & 5.7869e-2 \\\\\n    & 1 & II & -3.6583e-1 & 1.9210e-2 & 2.3666e-1 & 7.5054e-2 & 4.4182e-1 \\\\\n    & 0 & I & 1.1003 & -5.2790e-2 & 2.7400e-1 & 1.5080e-1 & 1.1369e-1\n\\end{tabular} \n\\end{ruledtabular} \n\\end{table} \n\nOur numerical results are displayed in Table~\\ref{tab:overlaps}. It is interesting to see how the mode frequency varies with the parameter $k$: while the $r$-mode frequency is a universal $w^2_\\bullet = 4\/3$, all other frequencies increase with $k$, that is, increase as the star becomes centrally dense. \n\nThe entries for $k = 0$ in Table~\\ref{tab:overlaps} can be obtained analytically, using results first presented by Lockitch and Friedman \\cite{lockitch-friedman:99}. For $m=1$ the mode frequencies are solutions to $15w^2 - 10w - 9 = 0$; we have  $w^1_{\\rm I} = 1\/3 + 4\\sqrt{10}\/15 \\simeq 1.1766$ and $w^1_{\\rm II} = 1\/3 - 4\\sqrt{10}\/15 \\simeq -0.5099$, and the corresponding mode functions are ($n = \\{ \\mbox{I}, \\mbox{II} \\}$) \n\\begin{equation}\n(a_1^1)_n = -\\frac{\\sqrt{5}}{3} (3 w_n^1 - 2) \\frac{r(r^2-R^2)}{R^3}, \\qquad\n(b_1^1)_n = -\\frac{\\sqrt{5}}{3} (3 w_n^1 - 2) \\frac{r(2r^2-R^2)}{R^3}, \\qquad\n(c_2^1)_n = (r\/R)^3.\n\\end{equation}\nFor $m=0$ the mode frequencies are solutions to $5w^2 - 4 = 0$, and the positive solution is $w_{\\rm I}^0 = 2\/\\sqrt{5} \\simeq 0.8944$. The corresponding mode functions are ($n = \\{ \\mbox{I}, \\mbox{II} \\}$, with $n = \\mbox{II}$ designating the negative-frequency mode) \n\\begin{equation}\n(a_1^0)_n = -\\frac{\\sqrt{15}}{2} w_n^0 \\frac{r(r^2-R^2)}{R^3}, \\qquad\n(b_1^0)_n = -\\frac{\\sqrt{15}}{2} w_n^0 \\frac{r(2r^2-R^2)}{R^3}, \\qquad\n(c_2^0)_n = (r\/R)^3.\n\\end{equation}\n\nThe entries that pertain to the $r$-mode can also be calculated analytically, for all the selected values of $k$. For all other modes, and for $k \\neq 0$, the eigenfrequencies and mode functions must be obtained numerically, as was described in Sec.~\\ref{sec:description}. The results for $\\gothp_n^m$, $\\gothn_n^m$, and $\\mathfrak{m}_n^m$ also require a numerical computation, and their values reflect a choice of normalization for the mode. We chose to normalize them with the condition $(b_1^m)_n(r=R) = \\frac{1}{2}(1+k)$.  \n\n\\subsection{Amplitudes, velocity field, and current quadrupole moment} \n\nCollecting the foregoing results, we have that the forcing functions $f_n^m$ defined by Eq.~(\\ref{q_eq_inertial}) can be expressed as\n\\begin{equation}\nf_n^m = A^m_{+n} e^{-i\\omega t} + A^m_{-n} e^{i\\omega t}\n\\label{f_nm_fixedorbit}\n\\end{equation}\nwith\n\\begin{subequations}\n\\label{Apm}\n\\begin{align}\nA^2_{\\pm \\bullet} &= \\mp \\frac{{\\cal B}_0 R}{c^2} \\frac{2\\pi \\gothp^2_\\bullet}{3 \\hat{N}^2_\\bullet \\omega^2_\\bullet}\n\\sin\\iota (\\cos\\iota \\pm 1) (2\\Omega \\mp \\omega), \\\\\nA^1_{\\pm n} &= \\pm \\frac{{\\cal B}_0 R}{c^2} \\frac{2\\pi \\gothp^1_n}{3 \\hat{N}^1_n \\omega^1_n}\n(\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1) (\\Omega \\mp \\omega), \\\\\nA^0_{\\pm n} &= \\frac{{\\cal B}_0 R}{c^2} \\frac{2\\pi \\gothp^0_n}{3 \\hat{N}^0_n \\omega^0_n}\n\\sin\\iota \\cos\\iota\\, \\omega. \n\\end{align}\n\\end{subequations} \nThe solution to Eq.~(\\ref{q_eq_inertial}) is then\n\\begin{equation}\nq_n^m = \\frac{A_{+n}^m}{\\omega_n^m - \\omega} e^{-i\\omega t}\n+ \\frac{A_{-n}^m}{\\omega_n^m + \\omega} e^{i\\omega t}.\n\\label{oscillators_fixed} \n\\end{equation}\n\nWe insert Eq.~(\\ref{oscillators_fixed}) within Eq.~(\\ref{dv_inertialmodesum}) for the velocity perturbation, and obtain \n\\begin{equation} \n\\delta v^a = (\\delta v^a)_{m=0} + (\\delta v^a)_{m=1} + (\\delta v^a)_{m=2}\n\\label{dv_modesum} \n\\end{equation} \nwith \n\\begin{subequations} \n\\label{dv_m} \n\\begin{align} \n(\\delta v^a)_{m=2} &= i\\frac{{\\cal B}_0}{c^2} R^3 \n\\frac{2\\pi \\gothp_\\bullet^2(2-w_\\bullet^2)}{3 w_\\bullet^2 \\hat{N}_\\bullet^2} \n\\biggl[ -\\sin\\iota (\\cos\\iota+1) \\frac{2-w}{w_\\bullet^2 - w} e^{-i\\omega t} \n+ \\sin\\iota (\\cos\\iota-1) \\frac{2+w}{w_\\bullet^2 + w} e^{i\\omega t} \\biggr] \n\\nonumber \\\\ & \\quad \\mbox{} \n\\times (r\/R)^3\\, (X_2^2)^a + \\mbox{cc}, \\\\ \n (\\delta v^a)_{m=1} &= \\frac{{\\cal B}_0}{c^2} R^3 \\sum_{n= {\\rm I}, {\\rm II}} \n\\frac{2\\pi \\gothp_n^1 (1-w_n^1)}{3 w_n^1 \\hat{N}_n^1} \n\\biggl[ (\\cos\\iota+1)(2\\cos\\iota-1) \\frac{1-w}{w_n^1 - w} e^{-i\\omega t} \n- (\\cos\\iota-1)(2\\cos\\iota+1) \\frac{1+w}{w_n^1 + w} e^{i\\omega t} \\biggr] \n\\nonumber \\\\ & \\quad \\mbox{} \\times \n\\sum_\\ell \\biggl[ \\frac{i}{r} (a_\\ell^1)_n\\, r^a Y_\\ell^1 \n+ i (b_\\ell^1)_n\\, (Y_\\ell^1)^a + (c_\\ell^1)_n\\, (X_\\ell^1)^a \\biggr] + \\mbox{cc}, \\\\ \n(\\delta v^a)_{m=0} &= \\frac{{\\cal B}_0}{c^2} R^3 \\frac{2\\pi \\gothp_{\\rm I}^0}{3 \\hat{N}_{\\rm I}^0} \n\\biggl[ \\sin\\iota\\cos\\iota \\frac{w}{w_{\\rm I}^0 - w} e^{-i\\omega t} \n+ \\sin\\iota\\cos\\iota \\frac{w}{w_{\\rm I}^0 + w} e^{i\\omega t} \\biggr] \n\\nonumber \\\\ & \\quad \\mbox{} \\times \n\\sum_\\ell \\biggl[ \\frac{1}{r} (a_\\ell^0)_{\\rm I}\\, r^a Y_\\ell^0 \n+ (b_\\ell^0)_{\\rm I}\\, (Y_\\ell^0)^a - i (c_\\ell^0)_{\\rm I}\\, (X_\\ell^0)^a \\biggr] + \\mbox{cc},\n\\end{align} \n\\end{subequations} \nwhere $w := \\omega\/\\Omega$ and $w_n^m := \\omega_n^m\/\\Omega$.  \n\nNext we involve the velocity field of Eq.~(\\ref{dv_m}) in a calculation of the current quadrupole moment $J^{jk}$, just as we did back in Sec.~\\ref{sec:quadmoment}. First we return to Eq.~(\\ref{dv_inertialmodesum}), which we insert within Eq.~(\\ref{J_def}). We make use of the identities of Eqs.~(\\ref{iden}), and observe that the surviving integrals are those displayed in Eq.~(\\ref{overlap2}). We arrive at\n\\begin{equation}\nJ^{jk} = \\sum_{m\\geq 0} \\sideset{}{'}\\sum_n (J^{jk})_n^m\n\\label{Jjk_mode1} \n\\end{equation}\nwith\n\\begin{equation}\n(J^{jk})_n^m = \\frac{6i}{5} \\frac{\\alpha^m}{\\beta^m} M R^3 (m\\Omega - \\omega_n^m) \\gothp_n^m q_n^m \n(\\bar{\\scrpt Y}_2^m)^{jk} + \\mbox{cc}, \n\\label{Jjk_mode2}\n\\end{equation} \nwhere $\\alpha^m = 1$ when $m=0, 2$ and $\\alpha^m = -i$ when $m = 1$, and where $\\beta^m$ denotes the numerical coefficients in front of the integrals in Eq.~(\\ref{overlap2}). More explicitly, we have that\n\\begin{subequations}\n\\label{Jjk_mode3}\n\\begin{align}\n(J^{jk})_\\bullet^2 &= \\frac{1}{2} M R^3 (2\\Omega - \\omega^2_\\bullet) \\gothp^2_\\bullet \\left[\nq_\\bullet^2 \\left( \n\\begin{array}{ccc} \ni  & -1 & 0 \\\\\n-1 & -i & 0 \\\\ \n0 & 0 & 0 \n\\end{array} \\right)\n+ \\mbox{cc} \\right], \\\\\n(J^{jk})_n^1 &= \\frac{1}{2} M R^3 (\\Omega - \\omega^1_n) \\gothp^1_n \\left[\nq_n^1 \\left( \n\\begin{array}{ccc} \n0  & 0 & -1 \\\\\n0 & 0 & -i \\\\ \n-1 & -i & 0 \n\\end{array} \\right)\n+ \\mbox{cc} \\right], \\\\\n(J^{jk})_n^0 &= \\frac{1}{6} M R^3 (-\\omega^0_n) \\gothp^0_n \\left[\nq_n^0 \\left( \n\\begin{array}{ccc} \n-i  & 0 & 0 \\\\\n0 & -i & 0 \\\\ \n0 & 0 & 2i \n\\end{array} \\right)\n+ \\mbox{cc} \\right].\n\\end{align}\n\\end{subequations} \nIf we now insert Eq.~(\\ref{oscillators_fixed}) with Eqs.~(\\ref{Apm}) and re-express Eq.~(\\ref{Jjk_mode1}) in the form of \nEq.~(\\ref{J_decomposed}), we arrive at Eq.~(\\ref{J_components}) with \n\\begin{subequations} \n\\label{j_modesum} \n\\begin{align} \n\\gothj_{\\pm 2}(\\omega) &= -\\frac{2\\pi}{3} \n\\frac{(\\gothp_\\bullet^2)^2}{\\hat{N}_\\bullet^2} \\frac{2-w_\\bullet^2}{w_\\bullet^2} \n\\frac{2 \\mp w}{w_\\bullet^2 \\mp w}, \n\\label{gothj_modesum_m2} \\\\\n\\gothj_{\\pm 1}(\\omega) &= \\pm \\sum_{n = {\\rm I}, {\\rm II}} \\frac{2\\pi}{3} \n\\frac{(\\gothp_n^1)^2}{\\hat{N}_n^1} \\frac{1-w_n^1}{w_n^1} \n\\frac{1 \\mp w}{w_n^1 \\mp w}, \\\\  \n\\gothj_{0}(\\omega) &= \\frac{4\\pi}{9} \n\\frac{(\\gothp_{\\rm I}^0)^2}{\\hat{N}_{\\rm I}^0} \n\\frac{w^2}{(w_{\\rm I}^0-w) (w_{\\rm I}^0+w)}. \n\\end{align} \n\\end{subequations} \nThese alternative expressions for $\\gothj_1(\\omega)$ and $\\gothj_0(\\omega)$ are plotted in Figs.~\\ref{fig:j_m1} and \\ref{fig:j_m0} for $k=1$ (the polytrope with $p = K\\rho^2$), along with the results obtained in Sec.~\\ref{sec:quadmoment}. The previous results are shown as black diamonds, while the mode-sum representations are plotted as solid red lines. We see that the numerical agreement is excellent, and conclude that even with just a handful of dominant modes, the mode-sum representation of the velocity perturbation is very accurate. For $\\gothj_{\\pm 2}(\\omega)$ the agreement is actually exact: with $w_\\bullet^2 = 4\/3$, $\\gothp_\\bullet^2$ given by Eq.~(\\ref{gothp2_result}), and $\\hat{N}_\\bullet^2$ by Eq.~(\\ref{gothnm_m2}), we find that Eq.~(\\ref{gothj_modesum_m2}) becomes precisely identical to Eq.~(\\ref{j_components_m2}).  \n\n\\section{Dynamical impact of gravitomagnetic tidal resonances} \n\\label{sec:dynamics} \n\nIn this section we calculate the impact on a binary inspiral of the gravitomagnetic tidal resonances identified in Sec.~\\ref{sec:solution}. Our end goal is to obtain the accumulated gravitational-wave phase shift produced by the resonances. We rely heavily on the methods developed by Flanagan and Racine \\cite{flanagan-racine:07} in their exploration of the $r$-mode resonance.  \n\n\\subsection{Inspiral}\n\\label{subsec:inspiral} \n\nWe consider a binary system on a circular orbit, driven into an inspiral by gravitational radiation reaction. We have that the mass of the reference star is $M$, the mass of the companion is $M'$, the orbital angular velocity is $\\omega$, and the orbital radius is $p =(GM_{\\rm tot})^{1\/3} \\omega^{-2\/3}$, where $M_{\\rm tot} = M + M'$ is the total mass. At leading order in a post-Newtonian description of the inspiral, the rate of change of the orbital radius is given by [Eq.~(12.251) of Ref.~\\cite{poisson-will:14}]\n\\begin{equation}\n\\dot{p} = -\\frac{64}{5} c^{-5} (G {\\cal M})^{5\/3} (G M_{\\rm tot})^{4\/3} p^{-3},\n\\label{pdot}\n\\end{equation}\nwhere an overdot indicates differentiation with respect to time, and\n\\begin{equation} \n{\\cal M} := \\frac{ (M M')^{3\/5} }{(M + M')^{1\/5} }\n\\label{chirp_mass}\n\\end{equation}\nis the binary's chirp mass. The corresponding rate of change of the angular velocity is \n\\begin{equation}\n\\dot{\\omega} = \\frac{96}{5} c^{-5} (G {\\cal M})^{5\/3} \\omega^{11\/3}. \n\\label{omegadot}\n\\end{equation}\n\nWe introduce a reference frequency $\\omega_0$ (later to be identified with one of the resonant frequencies) and a radiation-reaction time scale defined by $t_{\\rm rr} := (\\omega\/\\dot{\\omega})_{\\omega=\\omega_0}$. This is \n\\begin{equation}\nt_{\\rm rr} = \\frac{5}{96} c^5 (G {\\cal M})^{-5\/3} \\omega_0^{-8\/3}\n= 9.1 \\times 10^{-1} \\biggl( \\frac{1.2\\ M_\\odot}{\\cal M} \\biggr)^{5\/3}\n\\biggl( \\frac{100\\ \\mbox{Hz}}{\\omega_0\/2\\pi} \\biggr)^{8\/3}\\ \\mbox{s};\n\\label{trr_def} \n\\end{equation}\nin the second expression we inserted fiducial values for the binary parameters; a chirp mass of $1.2\\ M_\\odot$ corresponds to an equal-mass binary with $M = M' = 1.4\\ M_\\odot$.\n\nThe solutions to Eqs.~(\\ref{pdot}) and (\\ref{omegadot}) are\n\\begin{equation}\np(t) = p_0 (1 - \\tfrac{8}{3} \\tau)^{1\/4}, \\qquad\n\\omega(t) = \\omega_0 (1 - \\tfrac{8}{3} \\tau)^{-3\/8},\n\\label{p_omega}\n\\end{equation}\nwhere $\\tau := (t-t_0)\/t_{\\rm rr}$ and $p_0 := (GM_{\\rm tot})^{1\/3} \\omega_0^{-2\/3}$; $t_0$ is the time at which $p = p_0$ and $\\omega = \\omega_0$. The orbital phase $\\Phi$ is the integral of $\\omega$ with respect to time, and this is given by\n\\begin{equation}\n\\Phi(t) = \\Phi_0 + \\frac{3}{5} \\omega_0 t_{\\rm rr} \\bigl[ 1 - (1 - \\tfrac{8}{3} \\tau)^{5\/8} \\bigr],\n\\label{orb_phase}\n\\end{equation}\nwhere $\\Phi_0$ is the phase at time $t = t_0$.\n\nFor future reference we introduce a resonance-transit time scale $t_{\\rm res} := (\\pi t_{\\rm rr}\/\\omega_0)^{1\/2}$, the geometrical mean between the orbital period at resonance and the radiation-reaction time. This is \n\\begin{equation}\nt_{\\rm res} = \\sqrt{\\frac{5\\pi}{96}} c^{5\/2} (G {\\cal M})^{-5\/6} \\omega_0^{-11\/6}\n= 6.7 \\times 10^{-2} \\biggl( \\frac{1.2\\ M_\\odot}{\\cal M} \\biggr)^{5\/6}\n\\biggl( \\frac{100\\ \\mbox{Hz}}{\\omega_0\/2\\pi} \\biggr)^{11\/6}\\ \\mbox{s}. \n\\label{tres_def}\n\\end{equation}\nAs the name indicates, $t_{\\rm res}$ provides a measure of the time required for the inspiral to transit through a resonance at frequency $\\omega_0$. The ratio of time scales is $\\epsilon := t_{\\rm res}\/t_{\\rm rr}$, or\n\\begin{equation}\n\\epsilon = \\sqrt{\\frac{96\\pi}{5}} c^{-5\/2} (G {\\cal M})^{5\/6} \\omega_0^{5\/6}\n= 7.4 \\times 10^{-2} \\biggl( \\frac{\\cal M}{1.2\\ M_\\odot} \\biggr)^{5\/6}\n\\biggl( \\frac{\\omega_0\/2\\pi}{100\\ \\mbox{Hz}} \\biggr)^{5\/6}. \n\\label{epsilon_def}\n\\end{equation}\nWe see that $\\epsilon \\ll 1$ for binaries of interest, and we shall rely on this property in subsequent developments. It is useful to record the relations $\\omega_0 t_{\\rm rr} = \\pi\/\\epsilon^2$ and $\\omega_0 t_{\\rm res} = \\pi\/\\epsilon$.\n\nWhile $\\tau$ is a convenient time variable to describe the entirety of an inspiral, the transit through a resonance is best described in terms of the rescaled variable $\\lambda := (t-t_0)\/t_{\\rm res} = \\tau\/\\epsilon$. Over a time interval around $t_0$ that is much shorter than $t_{\\rm rr}$, we have that Eqs.~(\\ref{p_omega}) and (\\ref{orb_phase}) are usefully approximated by\n\\begin{equation}\np = p_0 \\bigl[ 1 + O(\\epsilon \\lambda) \\bigr], \\qquad\n\\omega = \\omega_0\\bigl[ 1 + O(\\epsilon \\lambda) \\bigr], \\qquad \n\\Phi = \\Phi_0 + \\frac{\\pi}{\\epsilon} \\lambda + \\frac{\\pi}{2} \\lambda^2 + O(\\epsilon \\lambda^3).\n\\label{orbit_short}\n\\end{equation}\nIt is useful to note that $\\pi \\lambda \/\\epsilon = \\omega_0(t-t_0)$.\n\n\\subsection{Mode amplitude across a resonance}\n\\label{subsec:across} \n\nWe return to the mode equation (\\ref{q_eq_inertial}),\n\\begin{equation} \n\\dot{q}_n^m + i\\omega_n^m q_n^m = i f_n^m, \n\\end{equation} \nand the forcing term of Eq.~(\\ref{f_nm_fixedorbit}), which we re-express as \n\\begin{equation}\nf_n^m = A^m_{+n} e^{-i\\Phi} + A^m_{-n} e^{i\\Phi}\n\\end{equation}\nto account for the inspiral; the coefficients $A^m_{\\pm n}$ are listed in Eq.~(\\ref{Apm}), and they are now slowly-varying functions of time. The solution to the mode equation, assuming an unperturbed state in the remote past, is\n\\begin{equation}\nq_n^m(t) = i e^{-i\\omega_n^m t} \\biggl\\{\n\\int_{-\\infty}^t A_{+n}^m(t') e^{i[\\omega_n^m t' - \\Phi(t')]}\\, dt' \n+ \\int_{-\\infty}^t A_{-n}^m(t') e^{i[\\omega_n^m t' + \\Phi(t')]}\\, dt' \\biggr\\}. \n\\label{q_sol} \n\\end{equation} \nWe are interested in the development of a resonance, which occurs when $\\omega(t')$ momentarily becomes equal to  $|\\omega_n^m|$. When this happens, one of the exponential factors inside the integrals stops oscillating (the phase becomes stationary), and this allows the integral to grow significantly. When $\\omega_n^m > 0$, only the first integral participates in the resonance; the second integral does not grow. Only the second integral participates when $\\omega_n^m < 0$.  \n\nWe wish to construct the solution to the mode equation before, during, and after the development of a resonance. To keep the following discussion simple we shall examine the specific case of a positive-frequency mode; the calculations for a negative-frequency mode require very few alterations, and our final results will apply to both cases. We simplify the notation by setting $q_n^m = q$, $\\omega_n^m = \\omega_0 > 0$, and $A_{+n}^m = A$. Because the second integral is now irrelevant in Eq.~(\\ref{q_sol}), the mode equation can be simplified to\n\\begin{equation} \n\\dot{q} + i\\omega_0 q = i A e^{-i\\Phi}.\n\\label{modeeq_simple} \n\\end{equation}\n\nThe solution to Eq.~(\\ref{modeeq_simple}) at early times, before resonance, is\n\\begin{equation}\nq(t) = \\frac{A(t) e^{-i\\Phi(t)}}{\\omega_0 - \\omega(t)} \\biggl[ 1\n  + O\\biggl( \\frac{1}{(\\omega_0-\\omega) t_{\\rm rr}} \\biggr) \\biggr].\n\\label{q_early1} \n\\end{equation}\nThis is obtained by writing down the exact solution as in Eq.~(\\ref{q_sol}), decomposing $\\Phi(t')$ into a rapidly-varying piece $\\omega(t)(t'-t)$ and a slowly-varying piece $\\Phi_{\\rm slow}(t')$, and integrating by parts to generate derivatives of $A$ and $\\Phi_{\\rm slow}$; the error term in Eq.~(\\ref{q_early1}) comes from these derivatives, if we note that $A$ and $\\Phi_{\\rm slow}$ vary over a time scale comparable to $t_{\\rm rr}$.\n\nWe may examine Eq.~(\\ref{q_early1}) at times $t$ such that $t_0 - t$ is small compared with $t_{\\rm rr}$, but still large enough to keep the error term small. We switch to the variable $\\lambda$ defined near the end of Sec.~\\ref{subsec:inspiral}, and we expand $A(t)$ and $\\Phi(t)$ in powers of $\\epsilon \\ll 1$, as in Eq.~(\\ref{orbit_short}). We arrive at\n\\begin{equation}\nq(t) = -\\frac{A(t_0)}{\\epsilon \\omega_0 \\lambda}\ne^{-i[\\Phi_0 + \\frac{\\pi}{\\epsilon} \\lambda + \\frac{\\pi}{2} \\lambda^2]}\n\\Bigl[ 1 + O(\\epsilon \\lambda, \\lambda\/\\epsilon) \\Bigr].\n\\label{q_early2} \n\\end{equation}\nThis provides an accurate description of the solution for times $\\lambda$ such that $\\epsilon \\ll |\\lambda| \\ll \\epsilon^{-1}$.\n\nNext we construct a solution to Eq.~(\\ref{modeeq_simple}) that is valid when $|\\lambda| \\ll \\epsilon^{-1}$; this will describe the transit through the resonance. We once again expand $A$ and $\\Phi$ in powers of $\\epsilon \\ll 1$, and rewrite the mode equation as\n\\begin{equation}\n\\frac{dq}{d\\lambda} + i \\frac{\\pi}{\\epsilon} q = i t_{\\rm res} A(t_0) \ne^{-i[\\Phi_0 + \\frac{\\pi}{\\epsilon} \\lambda + \\frac{\\pi}{2} \\lambda^2]}.\n\\end{equation}\nThe solution to this equation that is compatible with Eq.~(\\ref{q_early2}) at early times is\n\\begin{equation}\nq(t) = i t_{\\rm res} A(t_0) e^{-i[\\Phi_0 + \\frac{\\pi}{\\epsilon} \\lambda]} \\Bigl\\{\n\\bigl[ C(\\lambda) + \\tfrac{1}{2} \\bigr] - i \\bigl[ S(\\lambda) + \\tfrac{1}{2} \\bigr] \\Bigr\\},\n\\label{q_during}\n\\end{equation}\nwhere $C(\\lambda)$ and $S(\\lambda)$ are the Fresnel integrals [see, for example, Chapter 7 of Ref.~\\cite{NIST:10}] \n\\begin{equation}\nC(\\lambda) := \\int_0^\\lambda \\cos\\Bigl( \\frac{\\pi}{2} \\lambda^{\\prime 2} \\Bigr)\\, d\\lambda' ,\n\\qquad \nS(\\lambda) := \\int_0^\\lambda \\sin\\Bigl( \\frac{\\pi}{2} \\lambda^{\\prime 2} \\Bigr)\\, d\\lambda'. \n\\end{equation}\nThat Eq.~(\\ref{q_during}) reproduces the behavior required by Eq.~(\\ref{q_early2}) is verified with the asymptotic relations \n\\begin{equation}\nC(\\lambda) = \\frac{1}{2} \\mbox{sign}(\\lambda)\n+ \\frac{1}{\\pi\\lambda} \\sin\\Bigl( \\frac{\\pi}{2} \\lambda^2 \\Bigr) + O(\\lambda^{-3}),\n\\qquad\nS(\\lambda) = \\frac{1}{2} \\mbox{sign}(\\lambda)\n- \\frac{1}{\\pi\\lambda} \\cos\\Bigl( \\frac{\\pi}{2} \\lambda^2 \\Bigr) + O(\\lambda^{-3}).\n\\label{Fresnel_asymp} \n\\end{equation}\nIn this context it is useful to recall that $t_{\\rm res} = \\pi\/(\\epsilon \\omega_0)$.\n\nEquations (\\ref{q_during}) and (\\ref{Fresnel_asymp}) further reveal that beyond the resonance, when $\\lambda \\gg 1$, the mode amplitude is well approximated by\n\\begin{equation}\nq(t) = \\sqrt{2} t_{\\rm res} A(t_0) e^{-i[\\Phi_0 + \\frac{\\pi}{\\epsilon}\\lambda - \\frac{\\pi}{4}]}\n- \\frac{t_{\\rm res} A(t_0)}{\\pi \\lambda}\ne^{-i[\\Phi_0 + \\frac{\\pi}{\\epsilon} \\lambda + \\frac{\\pi}{2} \\lambda^2]}. \n\\label{q_late2} \n\\end{equation}\nThe second term is the same as in Eq.~(\\ref{q_early2}), and it represents a particular solution to the mode equation. The first term is a solution to the homogeneous version of the mode equation, and it describes a free oscillation at frequency $\\pi\/\\epsilon$ in terms of $\\lambda$, or $\\omega_0$ in terms of $t$. This free oscillation is the result of a resonant excitation of the mode associated with the amplitude $q$.\n\nEquation (\\ref{q_early1}) is a valid solution to Eq.~(\\ref{modeeq_simple}) after the resonance has occurred, just as it was prior to its occurrence. But this particular solution must be augmented by a solution to the homogeneous equation to properly account for the resonant excitation of the mode. The solution that matches the behavior required by Eq.~(\\ref{q_late2}) is\n\\begin{equation}\nq(t) = \\sqrt{2} t_{\\rm res} A(t_0) e^{-i[\\Phi_0 + \\omega_0(t-t_0) - \\frac{\\pi}{4}]} \n- \\frac{A(t) e^{-i\\Phi(t)}}{\\omega(t) - \\omega_0} \\biggl[ 1\n  + O\\biggl( \\frac{1}{(\\omega-\\omega_0) t_{\\rm rr}} \\biggr) \\biggr].\n\\label{q_late1} \n\\end{equation}\nBy patching together Eqs.~(\\ref{q_early1}), (\\ref{q_during}), and (\\ref{q_late1}) we obtain a complete description of the mode amplitude before, during, and after the onset of resonance.\n\nRestoring the original notation for the mode amplitudes, and allowing for the existence of negative-frequency modes, we have found that Eq.~(\\ref{q_sol}) is well approximated by \n\\begin{equation}\nq_n^m(t) = i t_{\\rm res} A_{\\pm n}^m(t_n^m) e^{\\mp i[\\Phi_n^m + \\frac{\\pi}{\\epsilon} \\lambda]} \\Bigl\\{\n\\bigl[ C(\\lambda) + \\tfrac{1}{2} \\bigr] \\mp i \\bigl[ S(\\lambda) + \\tfrac{1}{2} \\bigr] \\Bigr\\}\n\\label{q_during_final}\n\\end{equation}\nwhen $|\\lambda| \\ll \\epsilon^{-1}$. Here, $\\lambda := (t-t_n^m)\/t_{\\rm res}$ is the time variable introduced near the end of Sec.~\\ref{subsec:inspiral}, scaled in terms of the resonance-transit time of Eq.~(\\ref{tres_def}), in which we insert $\\omega_0 = |\\omega_n^m|$. In addition, $t_n^m$ (formally denoted $t_0$) is the time at which $\\omega(t) = |\\omega_n^m|$, and $\\Phi_n^m$ (formally denoted $\\Phi_0$) is the orbital phase at time $t = t_n^m$. Equation (\\ref{q_during_final}) involves a choice of coefficients $A_{\\pm n}^m$, as listed in Eq.~(\\ref{Apm}), a choice of sign in the exponential factor, and another choice of sign within the curly brackets; in all cases we select the upper sign for a positive-frequency mode, and the lower sign for a negative-frequency mode. \n\nA consequence of Eq.~(\\ref{q_during_final}) is that\n\\begin{equation}\n\\dot{q}_n^m = -i \\omega_n^m q_n^m \\bigl[ 1 + O(\\epsilon) \\bigr]\n\\label{qdot_vs_q}\n\\end{equation} \nwhen $|\\lambda| \\ll \\epsilon^{-1}$. This relation will be put to good use in subsequent developments.\n\n\\subsection{Mode-orbit coupling}\n\\label{subsec:mode-orbit}\n\nThe growth of a mode during the development of a resonance has a dynamical impact on the orbit. In the case of a stable mode, the energy given to the mode is taken away from the orbital motion; this produces a faster inspiral. In the case of an $r$-mode subjected to the Chandrasekhar-Friedman-Schutz instability, the star's rotational energy is given in part to the mode and in part to the orbital motion; this produces a slower inspiral. The mode-orbit interaction is mediated by the star's current quadrupole moment $J^{jk}$, which produces a post-Newtonian perturbing force on the companion. In this subsection we describe the effect of this force on the orbital motion.\n\nWe re-introduce the orbit-based vectorial basis of Sec.~\\ref{sec:gravitomagnetic}, suitably modified to account for the inspiral. We have $\\bm{n} = [\\cos\\Phi, \\cos\\iota\\, \\sin\\Phi, \\sin\\iota\\, \\sin\\Phi]$ pointing in the direction of the companion, $\\bm{\\lambda} = [-\\sin\\Phi, \\cos\\iota\\, \\cos\\Phi, \\sin\\iota\\, \\sin\\Phi]$ pointing in the direction of the orbital velocity, and $\\bm{l} = \\bm{n} \\times \\bm{\\lambda} = [0, -\\sin\\iota\\, \\cos\\iota]$ denoting the normal to the orbital plane.\n\nWhen we account for a perturbing acceleration $\\bm{a}$ in addition to the radiation-reaction force, the rate of change of the orbital radius is modified from Eq.~(\\ref{pdot}) and becomes [see, for example, Eq.~(3.64a) of Ref.~\\cite{poisson-will:14}] \n\\begin{equation}\n\\dot{p} = -\\frac{64}{5} c^{-5} (G {\\cal M})^{5\/3} (G M_{\\rm tot})^{4\/3} p^{-3}\n+ \\frac{2}{\\omega}\\,  {\\cal S},\n\\label{pdot_pert}\n\\end{equation}\nwhere ${\\cal S} := \\bm{a} \\cdot \\bm{\\lambda}$ is the tangential component of the perturbing acceleration. To integrate Eq.~(\\ref{pdot_pert}) we write $p = p_{\\rm ins} + \\delta p$, where $p_{\\rm ins}$ is the inspiral motion described by Eq.~(\\ref{p_omega}), and $\\delta p$ is the perturbation. Linearizing with respect to $\\delta p$, the differential equation becomes\n\\begin{equation}\n\\delta \\dot{p} = \\frac{2}{t_{\\rm rr}} (1 - \\tfrac{8}{3} \\tau)^{-1}\\, \\delta p\n+ \\frac{2}{\\omega_0} (1 - \\tfrac{8}{3} \\tau)^{3\/8}\\, {\\cal S},\n\\end{equation}\nand its solution is\n\\begin{equation}\n\\delta p(t) = \\frac{2}{\\omega_0} (1 - \\tfrac{8}{3} \\tau)^{-3\/4}\n\\int_{-\\infty}^t (1 - \\tfrac{8}{3} \\tau')^{9\/8}\\, {\\cal S}(t')\\, dt'.\n\\label{deltap_formal}\n\\end{equation}\nIn this equation, $\\omega_0$ is the reference frequency introduced in Sec.~\\ref{subsec:inspiral}, which will eventually be identified with one of the resonant frequencies.\n\nThe integral in Eq.~(\\ref{deltap_formal}) accumulates mostly during the resonance. After the resonance, $\\delta p(t)$ is well approximated by letting the upper bound of integration go to infinity. The expression then implies that $p = p_{\\rm ins} + \\delta p$ again corresponds to an inspiral, but one that is shifted in time relative to the original description. We have that\n\\begin{equation}\np(t) = p_{\\rm ins}(t) + \\delta p(t) = p_{\\rm ins}(t + \\Delta t),\n\\label{p_timeshift} \n\\end{equation}\nwhere\n\\begin{equation}\n\\Delta t = -\\frac{3 t_{\\rm rr}}{p_0 \\omega_0} \n\\int_{-\\infty}^\\infty (1 - \\tfrac{8}{3} \\tau)^{9\/8}\\, {\\cal S}(t)\\, dt. \n\\label{Deltat_def}\n\\end{equation}\nThe meaning of this result is that the perturbation has essentially no influence on the orbit before and after the resonance; in both cases the evolution of the orbital radius is described by Eq.~(\\ref{p_omega}). But the resonance produces a sudden shift in orbital radius, which is manifested by the time shift $\\Delta t$ in Eq.~(\\ref{p_timeshift}). Another way of stating this is that the inspirals before and after the resonance are both described by a solution to Eq.~(\\ref{pdot}), but that the solutions come with different boundary conditions.\n\nA crude description of the transit through a resonance, justified on the grounds that $t_{\\rm res} \\ll t_{\\rm rr}$, is one in which $p$ and $\\omega$ are taken to change discontinuously at $t = t_0$, but are otherwise not affected by the mode-orbit coupling. We write \n\\begin{equation}\np(t) = \\left\\{ \\begin{array}{ll}\np_{\\rm ins}(t) & \\qquad t < t_0 \\\\\np_{\\rm ins}(t + \\Delta t) & \\qquad t > t_0\n\\end{array} \\right., \n\\end{equation}\nwhere $\\Delta t$ is the time shift of Eq.~(\\ref{Deltat_def}). According to this, the instantaneous jump in orbital radius at resonance is given by $\\Delta t\\, \\dot{p}_{\\rm ins}(t_0)$; a positive $\\Delta t$ gives rise to a decrease in orbital radius. \n\nThe orbital phase is taken to change continuously across the resonance, and for this we write \n\\begin{equation}\n\\Phi(t) = \\left\\{ \\begin{array}{ll}\n\\Phi_{\\rm ins}(t) & \\qquad t < t_0 \\\\\n\\Phi_{\\rm ins}(t + \\Delta t) - \\Delta \\Phi & \\qquad t > t_0\n\\end{array} \\right., \n\\label{phase_model} \n\\end{equation}\nwhere \n\\begin{equation}\n\\Delta \\Phi = \\omega_0 \\Delta t\n\\label{Deltaphi_def}\n\\end{equation}\nis a constant introduced to ensure that $\\Phi(t)$ is indeed continuous at $t = t_0$. This is an important quantity: Flanagan and Racine \\cite{flanagan-racine:07} showed that in a measurement of gravitational waves emitted by a compact binary, $2\\Delta \\Phi$ is the accumulated phase difference between a model signal that properly incorporates the resonance and another model that does not; the factor of two accounts for the fact that the gravitational-wave frequency is twice the orbital frequency. A sizable phase shift implies that resonance effects can be detected in the gravitational-wave signal.   \n\nEquation Eq.~(\\ref{Deltaphi_def}) was derived on the basis of our crude description of the transit through a resonance. In Appendix~\\ref{sec:phaseshift} we show that it also follows from a more elaborate treatment. \n\n\\subsection{Calculation of the orbital phase shifts} \n\nThe perturbing acceleration $\\bm{a}$ created by a body with current quadrupole moment $J^{jk}$ was obtained by Flanagan and Racine \\cite{flanagan-racine:07}, as a special case of their post-Newtonian equations of motion for arbitrarily shaped bodies \\cite{racine-flanagan:05}. \nIt is given by\n\\begin{equation}\na^j = \\frac{4G}{c^2} \\frac{M_{\\rm tot}}{M} \\bigg( -\\epsilon^j_{\\ kp} \\dot{J}^p_{\\ q} \\frac{n^\\stf{kq}}{p^3}\n+ 5 \\epsilon^j_{\\ kp} J^p_{\\ q} v'_s \\frac{n^\\stf{kqs}}{p^4}\n- 5 \\epsilon^k_{\\ pq} J^q_{\\ s} v'_k \\frac{n^\\stf{jps}}{p^4} \\biggr),\n\\end{equation}\nwhere $v'_k$ are the components of the orbital velocity vector, and $n^\\stf{jk} := n^j n^k - \\frac{1}{3} \\delta^{jk}$, $n^\\stf{jkp} := n^j n^k n^p - \\frac{1}{5}(\\delta^{jk} n^p + \\delta^{jp} n^k + \\delta^{kp} n^j)$ are symmetric-tracefree combinations of the vector $n^j$. For a circular orbit we have that $v'_k = p\\omega \\lambda_k$, and we find that the second and third terms in $a^j$ cancel out when we calculate ${\\cal S} = a^j \\lambda_j$. We get\n\\begin{equation}\n{\\cal S} = \\frac{4G}{c^2} \\frac{M_{\\rm tot}}{M} \\frac{1}{p^3} \\dot{J}_{jk} l^j n^k\n\\label{S_result}\n\\end{equation}\nafter simplification.\n\nThe current quadrupole moment was expressed in terms of the mode amplitudes $q_n^m$ in Eqs.~(\\ref{Jjk_mode1}) and (\\ref{Jjk_mode3}). We substitute this in Eq.~(\\ref{S_result}), make use of\nEq.~(\\ref{qdot_vs_q}) to approximately equate $\\dot{q}_n^m$ to $-i\\omega_n^m q_n^m$, and invoke   \nEq.~(\\ref{p_omega}) to express $p \\simeq p_{\\rm ins}$ as a function of time. We obtain   \n\\begin{equation}\n{\\cal S} = \\sum_{m\\geq 0} \\sideset{}{'}\\sum_n {\\cal S}_n^m\n\\label{S_mode1} \n\\end{equation}\nwith\n\\begin{equation} \n{\\cal S}_n^m = \\frac{2R^3}{c^2} (m\\Omega - \\omega_n^m) (\\omega_n^m)^3 \\gothp_n^m\n(1 - \\tfrac{8}{3} \\tau)^{-3\/4} \\bigl[ a_+^m \\mbox{Re}(i q_n^m e^{i\\Phi})\n+ a_-^m \\mbox{Re}(i q_n^m e^{-i\\Phi}) \\bigr],\n\\label{S_mode2}\n\\end{equation}\nwhere\n\\begin{equation}\na_\\pm^2 = \\mp \\sin\\iota (\\cos\\iota \\pm 1), \\qquad\na_\\pm^1 = \\pm (\\cos\\iota \\pm 1)(2\\cos\\iota \\mp 1), \\qquad\na_\\pm^0 = \\mp \\sin\\iota \\cos\\iota.\n\\label{a_pm}\n\\end{equation}\nIn Eq.~(\\ref{S_mode2}), $\\tau := (t-t_n^m)\/t_{\\rm rr}$ is the time variable introduced in Sec.~\\ref{subsec:inspiral}, refined so that it refers specifically to the resonance time $t_n^m$ at which $\\omega(t) = |\\omega_n^m|$; the radiation-reaction time $t_{\\rm rr}$ is defined by Eq.~(\\ref{trr_def}), with $\\omega_0 = |\\omega_n^m|$. We recall that the prime on the sum over $n$ in Eq.~(\\ref{S_mode1}) is an instruction for $m = 0$ to include positive-frequency modes only.\n\nTo calculate the time shift $(\\Delta t)_n^m$ for a mode labelled by $m$ and $n$, we substitute  Eq.~(\\ref{q_during_final}) for $q_n^m$ within Eq.~(\\ref{S_mode2}), which we then insert within Eq.~(\\ref{Deltat_def}). For a positive-frequency mode, only the term involving $i q_n^m e^{i\\Phi}$ in Eq.~(\\ref{S_mode2}) produces a significant contribution to the integral; for a negative-frequency mode, the only relevant term is the one involving $i q_n^m e^{-i\\Phi}$. In either case we obtain\n\\begin{align}\n(\\Delta t)_n^m &= \\pm \\frac{6R^3}{c^2} (G M_{\\rm tot})^{-1\/3} t_{\\rm rr} t_{\\rm res}^2 a_\\pm^m\nA_{\\pm n}^m(t_n^m) \\gothp_n^m (m\\Omega - \\omega_n^m) |\\omega_n^m|^{8\/3}\n\\nonumber \\\\ & \\quad \\mbox{} \\times \n\\int_{-\\infty}^\\infty (1 - \\tfrac{8}{3} \\tau)^{3\/8} \\Bigl\\{\n\\bigl[ C(\\lambda) + \\tfrac{1}{2} \\bigr] \\cos\\frac{\\pi}{2} \\lambda^2\n+ \\bigl[ S(\\lambda) + \\tfrac{1}{2} \\bigr] \\sin\\frac{\\pi}{2} \\lambda^2 \\Bigr\\}\\, d\\lambda.\n\\label{Deltat_intermediate} \n\\end{align}\nTo evaluate the integral we observe that the factor within curly brackets is very small\\footnote{It behaves as $-\\pi^{-2} \\lambda^{-3}$.} when $\\lambda \\ll -1$, increases markedly in the interval $|\\lambda| \\lesssim 1$, and oscillates increasingly rapidly when $\\lambda \\gg 1$. The integral is therefore dominated by the second interval, in which we can set $(1 - \\frac{8}{3} \\tau)^{3\/8} = (1 - \\frac{8}{3} \\epsilon \\lambda)^{3\/8} = 1 + O(\\epsilon)$. Evaluation of the integral is then straightforward, and we obtain a value of unity.\n\nThe time shift for a mode labelled by $m$ and $n$ is therefore given by\n\\begin{equation}\n(\\Delta t)_n^m = \\pm \\frac{6R^3}{c^2} (G M_{\\rm tot})^{-1\/3} t_{\\rm rr} t_{\\rm res}^2 a_\\pm^m\nA_{\\pm n}^m(t_n^m) \\gothp_n^m (m\\Omega - \\omega_n^m) |\\omega_n^m|^{8\/3}. \n\\end{equation}\nWe next incorporate Eq.~(\\ref{trr_def}) for $t_{\\rm rr}$ and (\\ref{tres_def}) for $t_{\\rm res}$, replacing $\\omega_0$ in these expressions with $|\\omega_n^m|$. We also insert Eqs.~(\\ref{Apm}) for $A_{\\pm n}^m$, Eq.~(\\ref{Bjk}) for ${\\cal B}_0$, and Eq.~(\\ref{chirp_mass}) for the chirp mass ${\\cal M}$. Writing $\\omega_n^m = w_n^m \\Omega$ for the mode frequencies, we finally arrive at\n\\begin{equation}\n(\\Delta t)_n^m = \\frac{25\\pi^2}{2304} \\frac{(\\gothp_n^m)^2}{\\hat{N}_n^m}\n\\frac{(m-w_n^m)^2}{|w_n^m|^{7\/3}} (a_\\pm^m)^2 \n\\frac{c^6 R^4}{(G M)^2 (GM') (GM_{\\rm tot})^{1\/3} \\Omega^{1\/3}}.\n\\label{Deltat_final} \n\\end{equation}\n\nThe orbital phase shift $(\\Delta \\Phi)_n^m$ associated with a mode labelled by $m$ and $n$ is obtained by inserting Eq.~(\\ref{Deltat_final}) within Eq.~(\\ref{Deltaphi_def}), once again replacing $\\omega_0$ with $|\\omega_n^m|$. We get\n\\begin{equation}\n(\\Delta \\Phi)_n^m = \\frac{25\\pi^2}{2304} \\frac{(\\gothp_n^m)^2}{\\hat{N}_n^m}\n\\frac{(m-w_n^m)^2}{|w_n^m|^{4\/3}} (a_\\pm^m)^2 \n\\frac{c^6 R^4 \\Omega^{2\/3}}{(G M)^2 (GM') (GM_{\\rm tot})^{1\/3}}.\n\\label{Deltaphi_final} \n\\end{equation}\nWe recall that the overlap integrals $\\gothp_n^m$ were defined in Eq.~(\\ref{overlap2}), and that the mode norms $\\hat{N}_n^m$ were introduced in Eq.~(\\ref{Nhat_def}); these constants are displayed in Table~\\ref{tab:overlaps} for various stellar models. In addition, we recall that the constants $a_\\pm^m$ are listed in Eq.~(\\ref{a_pm}).  \n\n\\subsection{Gravitational-wave phase shift} \n\nAs the inspiral proceeds, the binary goes through a succession of four gravitomagnetic tidal resonances, one for each of the dominant Lockitch-Friedman inertial modes (including the $r$-mode). At each resonance the orbital phase undergoes a shift given by Eq.~(\\ref{Deltaphi_final}), and the total accumulated phase shift in the gravitational-wave signal is \n\\begin{equation} \n\\Delta\\Phi_{\\rm GW} = \\gamma(\\iota) \\biggl( \\frac{R}{10\\ \\mbox{km}} \\biggr)^4 \n\\biggl( \\frac{1.4\\ M_\\odot}{M} \\biggr)^2 \n\\biggl( \\frac{1.4\\ M_\\odot}{M'} \\biggr) \n\\biggl( \\frac{2.8\\ M_\\odot}{M_{\\rm tot}} \\biggr)^{1\/3} \n\\biggl( \\frac{\\Omega\/2\\pi}{100\\ \\mbox{Hz}} \\biggr)^{2\/3}, \n\\label{DeltaPhi_GW1} \n\\end{equation} \nwhere\n\\begin{equation}\n\\gamma(\\iota) := \\mathfrak{u}_\\bullet^2\\, \\sin^2\\iota(\\cos\\iota + 1)^2 \n+ \\mathfrak{u}_{\\rm I}^1\\, (\\cos\\iota + 1)^2 (2\\cos\\iota - 1)^2 \n+ \\mathfrak{u}_{\\rm II}^1\\, (\\cos\\iota - 1)^2 (2\\cos\\iota + 1)^2\n+ \\mathfrak{u}_{\\rm I}^0\\, \\sin^2\\iota \\cos^2\\iota\n\\label{DeltaPhi_GW2} \n\\end{equation} \nwith \n\\begin{equation} \n\\mathfrak{u}_n^m := 2 \\times 11.550 \\times \\frac{25\\pi^2}{2304} \\frac{(\\gothp_n^m)^2}{\\hat{N}_n^m}\n\\frac{(m-w_n^m)^2}{|w_n^m|^{4\/3}}. \n\\label{unm_def} \n\\end{equation} \nIn this expression, the factor of 2 accounts for the fact that the gravitational-wave frequency is twice the orbital frequency, the numerical factor of $11.550$ corresponds to the choice of fiducial values for $R$, $M$, $M'$, and $\\Omega$, and the remaining factors come from Eq.~(\\ref{Deltaphi_final}). \n\nEquations (\\ref{DeltaPhi_GW1}) and (\\ref{DeltaPhi_GW2}) as the same as Eqs.~(\\ref{DeltaPhi_intro}) and (\\ref{gamma_intro}), respectively. The numbers $\\mathfrak{u}_n^m$ are tabulated for selected values of the density-model parameter $k$ in Table~\\ref{tab:uvalues_intro}, and plots of $\\gamma$ as a function of the inclination angle $\\iota$ are displayed in Fig.~\\ref{fig:phaseshift_intro}. An in-depth discussion of these results was provided back in Sec.~\\ref{sec:intro}, and there is no need to repeat it here. \n\n\\begin{acknowledgments} \nI am grateful to John Friedman and Huan Yang for essential discussions that shaped the development of this work. I also thank Simon Alexandre Pekar, who verified some of my numerical results with independent computations, and Eanna Flanagan for helpful comments on an earlier draft of the paper. This work was supported by the Natural Sciences and Engineering Research Council of Canada.  \n\\end{acknowledgments} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{hexint}\n\n\nDiscrete breathers are time-periodic and spatially localised exact\nsolutions of translationally invariant nonlinear lattices.  For a\nbrief review of some general properties of breathers in\nhigher-dimensional systems, see our earlier work \\citep{buts06}.  \nIn particular, it is known that while some fundamental properties\nsuch as the existence of breathers are not affected by lattice\ndimension (see Flach \\etal \\citep{flat94} and Mackay and Aubry\n\\citep{mac94}), other properties are affected profoundly, for\ninstance, the energy properties of breathers (see Flach \\etal\n\\citep{fla97}, Kastner \\citep{kas04} and Weinstein \\citep{wei99}).\nIn this paper we investigate how the symmetry of the lattice \ninfluences the properties of discrete breathers found therein.  \n\n\nIn Hamiltonian systems, stationary breathers occur in one-parameter \nfamilies.  For a certain class of Hamiltonian systems, a critical spatial \ndimension $d_c$ exists such that for systems with $d \\geq d_c$, \nthere exists a positive lower bound on the energy of a breather \nfamily, and the breather energies do {\\em not} approach zero even \nas the amplitude tends to zero.   For lattices with dimension \n$d < d_c$, there is no positive lower bound on the energy of \nbreathers.  In other words, the energy of a family of breathers goes \nto zero with amplitude, and breathers of arbitrarily small energy can \nbe found.    The critical dimension $d_c$ is typically two.  A small \namplitude expansion yields the NLS reduction to ${\\mathrm i} F_T+\\Delta F \n+ \\kappa |F|^{2\\sigma}F=0$ which has a critical dimension of  \n$d_c=2\/\\sigma$, with blow-up in NLS occurring when $\\kappa>0$ \nand $d>d_c$.   For typical lattice potentials, $\\sigma=1$ again \nconfirming the critical dimension of $d_c=2$. \n\n\nMarin, Eilbeck and Russell have performed extensive numerical \ninvestigations of breather dynamics in two-dimensional lattices \n\\citep{mar98,mar00,mar01}.  Their results suggest that moving \nbreather modes exist (or are at least extremely long-lived), and \nthat the lattice exhibits a strong directional preference whereby \nbreathers can only move along symmetries of the lattice, and in \nno other direction.  Such quasi-one-dimensional behaviour is also \nobserved in higher-dimensional lattices.\n\n\nThe work in this paper follows on from our earlier study \n\\citep{buts06} of breathers in a two-dimensional lattice with square \nrotational symmetry;  hereupon referred to as the ``square'' lattice.  \nThis lattice has $C_4$ symmetry, by which we mean that a rotation \nthrough any multiple of $2\\pi\/4 = \\pi\/2$ about an axis perpendicular to \nthe lattice plane through a lattice site maps the lattice onto itself.  \nIn \\cite{buts06}, using the semi-discrete multiple-scale method \n(see Remoissenet \\citep{rem85}), we determined an approximate \nform for (as well as the properties of) small \namplitude breathers in a two-dimensional square Fermi-Pasta-Ulam \n(FPU) lattice \\citep{fer40}.  We found a third-order analysis to be \ninadequate, since the partial differential equation obtained at this order \ndescribing the breather envelope exhibits \\emph{blow-up}.  To\novercome this, we incorporated higher-order effects in the model,\nthereby obtaining a modified partial differential equation which\nincludes known stabilising terms. From this, we determined regions\nof parameter space where breather solutions are expected.\nNumerical simulations supported the results of our analysis, and\nsuggested that, in contrast to the two-component lattices studied\nby Marin \\etal\\ \\citep{mar98,mar00,mar01} (that is, with two\ndegrees of freedom at each lattice site), there is no restriction upon \nthe permitted directions of travel within the one-component square \nFPU lattice. We also found asymptotic estimates for the breather \nenergy which confirmed the existence of a minimum threshold \nenergy, in agreement with the work of Flach \\etal\\ \\citep{fla97}.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\resizebox{5in}{!}{\\includegraphics{hexlatsmen.eps}}\n\\caption{The 2D hexagonal electrical transmission lattice (HETL).}\n\\label{hexetl}\n\\end{center}\n\\end{figure}\n \n\nIn this paper, we consider a hexagonal electrical transmission lattice \n(HETL).  This two-dimensional network possesses $C_6$ (or hexagonal) \nrotational symmetry.  That is, rotation through any multiple of the angle \n$2\\pi\/6 = \\pi\/3$ about a lattice site maps the lattice onto itself.  The HETL \nis shown in \\Fref{hexetl}, pictured from a point vertically above the plane \nof the lattice.  We note from \\Fref{hexetl} that geometrically, the HETL is \nan arrangement of tessellating triangles (not hexagons).   Nevertheless, the \narrangement in \\Fref{hexetl} is referred to as ``hexagonal'' rather than \n``triangular'' since these descriptions refer to its symmetry properties and \nnot to the geometrical shapes which comprise the array (not all authors \nfollow this convention).  One might expect the analysis for the hexagonal \nlattice to be more involved, since it is geometrically more complicated in \nhaving more links emanating from each node.  However, the hexagonal \nsymmetry results in greater isotropy and hence simpler equations than \nthose obtained for the square lattice.\n\n\nWe derive the equations of motion and demonstrate a Hamiltonian\nformalism in \\Sref{hexder}.  In Sections \\ref{hexsym} and\n\\ref{hexasymm}, we present two cases for which the hexagonal FPU\nlattice equations can be reduced to a two-dimensional nonlinear\nSchr{\\\"o}dinger (NLS) equation with cubic nonlinearity.  We\nconsider lattices with a symmetric interaction potential, in which\ncase reduction to a cubic NLS equation can be performed for moving\nbreathers.  We find an ellipticity criterion for the wavenumbers\nof the carrier wave in \\Sref{hexdell}.  A reduction to the cubic NLS\nequation can also be carried out for lattices with an asymmetric\npotential, provided we consider only stationary breathers.  \n\n\nAs expected we find a minimum energy below which breathers cannot \nexist in the hexagonal FPU lattice.  We find that the\nenergy threshold is dependent upon the wavevector of the carrier\nwave. It is maximised for stationary breathers, and becomes\narbitrarily small near the boundary of the elliptic domain.   \n\n\nThe cubic NLS equation admits only unstable Townes solitons.  Hence, \nin \\Sref{hex5}, we extend our asymptotic analysis to higher order and \nfind an isotropic generalised NLS equation which incorporates known \nstabilising terms.  Our analytic work is supplemented by numerical simulations \npresented in \\Sref{hexnumerics}, which suggest that long-lived stationary\nand moving breather modes are supported by the system.\nIn \\Sref{hdisc}, we discuss the results obtained in this paper. \n\n\n\\section{A two-dimensional hexagonal Fermi-Pasta-Ulam lattice}\n\\label{hexfpulat}\n\n\n\\subsection{Preliminaries} \\label{hexpre}\n\n\nBefore we derive the equations governing the HETL, we describe our \nscheme for indexing the lattice nodes and our choice of basis vectors.  We \nintroduce a rectangular lattice with basis vectors $\\mathcal{B}~=~\\{ \\mathbf{i'} \\, , \n\\, \\mathbf{j'} \\}$, where $\\mathbf{i'} = \\mathbf{i} = [1,0]^T$ and $\\mathbf{j'} = [0,h]^T$ \n($\\mathbf{j}$ being $[0,1]^T$), illustrated in \\Fref{sqbase}.  We use only half of \nthe $(m,n)$ indices, namely, those for which the sum $m+n$ is even.  We \nchoose an origin with coordinates $(0,0)$;  the position of the site $(m,n)$ \nis $m\\mathbf{i'} + n\\mathbf{j'}$.  In order for the hexagonal lattice to be regular, \nwe specify $h=\\sqrt{3}$. \n\n \n\\begin{figure}[htbp]\n\\begin{center}\n\\vspace{-1.0cm}\n\\psfrag{0}{{\\scriptsize $(m,n)$}}\n\\psfrag{1}{{\\scriptsize $(m+2,n)$}}\n\\psfrag{2}[B][]{{\\scriptsize $(m+1,n+1)$}}\n\\psfrag{3}[B][]{{\\scriptsize $(m-1,n+1)$}}\n\\psfrag{4}[B][r]{{\\scriptsize $(m-2,n)$}}\n\\psfrag{5}[B][b]{{\\scriptsize $(m-1,n-1)$}}\n\\psfrag{6}[B][b]{{\\scriptsize $(m+1,n-1)$}}\n\\psfrag{i}{{\\footnotesize $\\mathbf{i'}$}}\n\\psfrag{j}{{\\footnotesize $\\mathbf{j'}$}}\n\\resizebox{4in}{!}{\\includegraphics{sqbasecorr2.eps}}\n\\caption{Labelling of nodes in the HETL with basis \n$ \\mathcal{B} = \\{ \\mathbf{i'} \\, , \\, \\mathbf{j'} \\}$.}\n\\label{sqbase}\n\\end{center}\n\\end{figure}\n\n\nAt every node of the HETL lies a nonlinear capacitor (not shown in \n\\Fref{hexetl}), and between every node and each of its six nearest \nneighbours is a linear inductor.  An enlarged view of the area surrounding \nthe capacitor at $(m,n)$ is shown in \\Fref{hexunitdir}, where the capacitor \nis visible.  The variable $V_{m,n}$ denotes the voltage across the \ncapacitor $(m,n)$ and $Q_{m,n}$ denotes the total charge stored on the \ncapacitor at $(m,n)$.  Also, $I_{m,n}$, $J_{m,n}$ and $K_{m,n}$ are the \ncurrents through the inductors immediately to the right of site $(m,n)$ in \nthe directions $\\mathbf{e_i} = [2,0]^T$, $\\mathbf{e_j} = [1,-\\sqrt{3}]^T$ and \n$\\mathbf{e_k}=[1,\\sqrt{3}]^T$ respectively, as illustrated in \\Fref{hexunitdir}. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\vspace{1.3cm}\n\\psfrag{a}{{\\large $V_{m,n}$}}\n\\psfrag{b}[B][]{{\\large $V_{m+1,n-1}$}}\n\\psfrag{c}[B][]{{\\large $V_{m+2,n}$}}\n\\psfrag{d}{{\\large $V_{m+1,n+1}$}}\n\\psfrag{e}{{\\large $I_{m,n}$}}\n\\psfrag{f}{{\\large $J_{m,n}$}}\n\\psfrag{g}{{\\large $K_{m,n}$}}\n\\psfrag{i}{{\\normalsize $\\mathbf{e_i}$}}\n\\psfrag{j}{{\\normalsize $\\mathbf{e_j}$}}\n\\psfrag{k}{{\\normalsize $\\mathbf{e_k}$}}\n\\resizebox{5.5in}{!}{\\includegraphics{hexundir.eps}}\n\\caption{Enlarged view of the HETL at site $(m,n)$.}\n\\label{hexunitdir}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Derivation of model equations} \\label{hexder}\n\n\nTo derive the equations relating current, charge and voltage in the lattice\nwe apply Kirchoff's law \n\\begin{eqnarray}\nV_{m+2,n} - V_{m,n} & = -L \\frac{{\\mathrm d} I_{m,n}}{{\\mathrm d} t}, \\label{hki} \\\\[1ex]\nV_{m+1,n-1} - V_{m,n} & = -L \\frac{{\\mathrm d} J_{m,n}}{{\\mathrm d} t}, \\label{hkj} \\\\[1ex]\nV_{m+1,n+1} - V_{m,n} & = -L \\frac{{\\mathrm d} K_{m,n}}{{\\mathrm d} t}, \\label{hkk}\n\\end{eqnarray}\nwhere the inductance $L$ is constant.  Conservation of charge gives\n\\begin{eqn} \\label{hcc}\nI_{m-2,n}-I_{m,n} + J_{m-1,n+1}-J_{m,n} + K_{m-1,n-1}-K_{m,n} =\n\\frac{{\\mathrm d} Q_{m,n}}{{\\mathrm d} t}.\n\\end{eqn}\nDifferentiating \\eref{hcc} with respect to time, and then using \n\\eref{hki}--\\eref{hkk} to find $\\dot{I}_{m-2,n}$, $\\dot{J}_{m-1,n+1}$ \nand $\\dot{K}_{m-1,n-1}$, we have  \n\\begin{eqnarray}\nL \\frac{{\\mathrm d}^2 Q_{m,n}}{{\\mathrm d} t^2} & = & \\quad (V_{m+2,n} - 2 V_{m,n} + \nV_{m-2,n}) + (V_{m+1,n-1} - 2 V_{m,n} + V_{m-1,n+1}) \\nonumber \\\\ & & \n+ (V_{m+1,n+1} - 2 V_{m,n} + V_{m-1,n-1}). \n\\label{heqvqfull} \\end{eqnarray}\nEquation \\eref{heqvqfull} may be written in the abbreviated form\n$L \\ddot{Q}_{m,n} = (\\delta^2_I + \\delta^2_J + \\delta^2_K)V_{m,n}$,\nwhere the centred second-difference operators are defined by\n\\begin{eqnarray}\n\\label{hdi} \\delta^2_I A_{m,n} & = A_{m+2,n}   - 2 A_{m,n} + A_{m-2,n},\\\\[1ex]\n\\label{hdj} \\delta^2_J A_{m,n} & = A_{m+1,n-1} - 2 A_{m,n} + A_{m-1,n+1},\\\\[1ex]\n\\label{hdk} \\delta^2_K A_{m,n} & = A_{m+1,n+1} - 2 A_{m,n} + A_{m-1,n-1} . \n\\end{eqnarray}\nHere, $A_{m,n}$ is an arbitrary quantity referenced by two indices; $\\delta^2_I$, \n$\\delta^2_J$ and $\\delta^2_K$ are centred second-difference operators in \nthe directions of $\\mathbf{e_i}$, $\\mathbf{e_j}$ and $\\mathbf{e_k}$ respectively.   \nSince the voltage $V_{m,n}$ is known in terms of the charge, we reformulate \n\\eref{heqvqfull} in terms of the single quantity $Q_{m,n}$.  We invert the \ncapacitor's nonlinear charge-voltage relationship $Q=VC(V)$ (see \nequations (2.9)--(2.14) of \\cite{buts06} for details), to obtain \n\\begin{equation} \nV(Q) = ( Q + a Q^2 + b Q^3 + c Q^4 + d Q^5 ) \/ C_0 , \n\\label{hqvexp} \\end{equation}\nwhere $C_0=C(0)$.  Hence, the HETL equations \\eref{heqvqfull} can \nbe written as \n\\begin{eqn} \\label{heqq}\n\\frac{{\\mathrm d}^2 Q_{m,n}}{{\\mathrm d} t^2} = (\\delta^2_I + \\delta^2_J + \\delta^2_K) \n\\left[Q_{m,n}+aQ_{m,n}^2+bQ_{m,n}^3+cQ_{m,n}^4+dQ_{m,n}^5\\right], \n\\end{eqn}\nwhere $m,n \\in {\\bbf{Z}}$ and, by rescaling the time variable, we set \n$LC_0=1$ without loss of generality.   Thus we have shown that the \nequation governing charge in the HETL \\eref{heqq} is a two-dimensional \nanalogue of the Fermi-Pasta-Ulam equation \n\\begin{equation}\n\\frac{{\\mathrm d}^2 Q_j}{{\\mathrm d} t^2} = W'(Q_{j+1}) - 2 W'(Q_j) + W'(Q_{j-1}) , \n\\end{equation}\nwhich is a Hamiltonian system that can be derived from both \n\\begin{equation}\nH = \\sum_j \\sfr{1}{2} \\pi_j^2 + W(\\phi_{j+1}-\\phi_j) , \\;\\;\\; {\\rm and} \\;\\;\\; \n\\widetilde H = \\sum_j \\sfr{1}{2} (P_{j+1}-P_j)^2 + W(Q_j) , \n\\end{equation}\nwhere $Q_j=\\phi_{j+1}-\\phi_j$.  \n\n\nThe lattice equations \\eref{heqq} can be derived from the Hamiltonian \n\\begin{eqnarray}\n\\widetilde{H} & =\\sum_{m,n} & \n\\mbox{$\\frac{1}{2}$} (P_{m+2,n}-P_{m,n})^2 + \n\\mbox{$\\frac{1}{2}$} (P_{m+1,n-1} - P_{m,n})^2 \\nonumber \\\\ & & + \n\\mbox{$\\frac{1}{2}$} (P_{m+1,n+1} - P_{m,n})^2 + \\Upsilon (Q_{m,n}), \n\\label{hham} \\end{eqnarray}\nwhere $\\Upsilon (Q_{m,n})$ satisfies $\\Upsilon\\!\\ '(Q_{m,n}) =\nV(Q_{m,n})$ given in \\eref{hqvexp}, hence   \n\\begin{eqn}\n\\Upsilon(Q) = \\mbox{$\\frac{1}{2}$} Q^2 + \n\\mbox{$\\frac{1}{3}$} a Q^3 + \\mbox{$\\frac{1}{4}$} b Q^4 + \n\\mbox{$\\frac{1}{5}$} c Q^5 + \\mbox{$\\frac{1}{6}$} d Q^6 .\n\\end{eqn} \nWe describe potentials which satisfy $\\Upsilon(-Q)=\\Upsilon(Q)$ (that is, \n$a=c=0$) as `symmetric'.  The variables $P_{m,n}$ and $Q_{m,n}$ are \nconjugate momenta and displacement variables of the system; and \neliminating $P_{m,n}$ from the equations \n\\begin{eqn} \\label{hhameq}\n\\frac{{\\mathrm d} Q_{m,n}}{{\\mathrm d} t} = -(\\delta^2_I + \\delta^2_J  +\n\\delta^2_K)P_{m,n}, \\qquad\n\\frac{{\\mathrm d} P_{m,n}}{{\\mathrm d} t} = -\\Upsilon\\!\\ '(Q_{m,n}) , \n\\end{eqn}\nyields (\\ref{heqq}). In \\Sref{heeh}, we derive expressions for \nthe energy of breathers given the small amplitude solutions \nwhich are obtained in the next section. \n\n\n\\subsection{Asymptotic analysis} \\label{haa}\n\n\nWe apply the method of multiple-scales to determine an approximate \nanalytic form for small amplitude breather solutions of \\eref{heqq}, with \nslowly varying envelope.  We introduce new variables defined by\n \\begin{eqn} \\label{hexvars}\n X=\\varepsilon m,\\quad Y=\\varepsilon h n,\\quad\\tau=\\varepsilon t\\quad\\textrm{and}\\quad T=\\varepsilon^2 t;\n \\end{eqn}\nnote the presence of the scaling factor $h$ in the definition of $Y$.  \nWe seek solutions of \\eref{heqq} of the form\n\\begin{eqnarray} \\fl \nQ_{m,n}(t) = \\hspace*{-3mm} & \\!\\!\\!\\! \\varepsilon {\\mathrm e}^{{\\mathrm i} \\psi} F(X,Y,\\tau,T) + \n\\varepsilon^2 G_0(X,Y,\\tau,T) +\n\\varepsilon^2 {\\mathrm e}^{{\\mathrm i} \\psi} G_1(X,Y,\\tau,T) + \\nonumber \\\\ & \\!\\!\\!\\! \n\\varepsilon^2 {\\mathrm e}^{2{\\mathrm i} \\psi} G_2(X,Y,\\tau,T) + \\varepsilon^3 H_0 (X,Y,\\tau,T) +\n\\varepsilon^3 {\\mathrm e}^{{\\mathrm i} \\psi} H_1(X,Y,\\tau,T) + \\nonumber \\\\\n& \\!\\!\\!\\!  \\varepsilon^3 {\\mathrm e}^{2{\\mathrm i} \\psi} H_2(X,Y,\\tau,T)\n+ \\varepsilon^3{\\mathrm e}^{3{\\mathrm i} \\psi}H_3(X,Y,\\tau,T) + \\varepsilon^4{\\mathrm e}^{{\\mathrm i}\\psi}I_1(X,Y,\\tau,T)\n+ \\nonumber\\\\ & \\!\\!\\!\\!  \n\\varepsilon^4 {\\mathrm e}^{2{\\mathrm i} \\psi} I_2(X,Y,\\tau,T) + \\varepsilon^4 {\\mathrm e}^{3{\\mathrm i}\\psi} I_3(X,Y,\\tau,T)\n+ \\varepsilon^4{\\mathrm e}^{4{\\mathrm i} \\psi} I_4(X,Y,\\tau,T) + \\nonumber \\\\\n& \\!\\!\\!\\! \\varepsilon^5{\\mathrm e}^{{\\mathrm i} \\psi} J_1(X,Y,\\tau,T)+ \\cdots + \\mathrm{c.c.},\n\\label{hexanz} \\end{eqnarray}\nwhere the phase $\\psi$ of the carrier wave is given by $km+lhn+\\omega t$ (once \nagain noting the extra factor $h$), and $\\mathbf{k}=[k,l]^T$ and $\\omega(\\mathbf{k})$ \nare its wavevector and temporal frequency respectively.  We substitute the \nansatz \\eref{hexanz} into the lattice equations \\eref{heqq} and equate \ncoefficients of each harmonic frequency at each order of $\\varepsilon$.  After \nmuch simplification, this yields the following system of equations: \\\\\n\\\\\n$\\mathcal{O}(\\varepsilon {\\mathrm e}^{{\\mathrm i}\\psi})$:\n\\begin{eqn} \\label{hexdisp}\n\\omega^2 F = 4\\sin^2(k) F + 4\\sin^2 \\left( \\frac{k+lh}{2} \\right) F + \n4\\sin^2 \\left( \\frac{k-lh}{2} \\right) F,\n\\end{eqn}\n$\\mathcal{O}(\\varepsilon^2 {\\mathrm e}^{{\\mathrm i}\\psi})$:\n\\begin{eqn} \\label{hexvel}\n\\omega F_{\\tau} = 2\\sin k [2\\cos k + \\cos(lh)]F_X + 2h\\cos k \\sin(lh) F_Y,\n\\end{eqn}\n$\\mathcal{O}(\\varepsilon^2 {\\mathrm e}^{2{\\mathrm i}\\psi})$:\n\\begin{eqn} \\label{hexg2}\n\\omega^2 G_2 = [\\sin^2(2k) + \\sin^2(k+lh) + \\sin^2(k-lh)](G_2 + a F^2),\n\\end{eqn}\n$\\mathcal{O}(\\varepsilon^3 {\\mathrm e}^{{\\mathrm i}\\psi})$:\n\\begin{eqnarray}\n\\fl 2{\\mathrm i} \\omega F_T + F_{\\tau\\tau}  & \n= & [4\\cos(2k) + 2\\cos k \\cos(lh)] F_{XX} + 2h^2\\cos k \\cos(lh)\nF_{YY} \\nonumber \\\\\n& & - 4h\\sin k \\sin(lh)F_{XY} \\nonumber \\\\\n& & - \\ 8a \\left[ \\sin^2(k) + \\sin^2 \\left( \\frac{k+lh}{2} \\right)\n+ \\sin^2 \\left( \\frac{k-lh}{2} \\right) \\right] [F(G_0+\\overline{G}_0)+\\overline{F}G_2] \n\\nonumber\\\\\n& & - \\ 12 b \\left[ \\sin^2(k) + \\sin^2 \\left( \\frac{k+lh}{2} \\right)\n+ \\sin^2 \\left( \\frac{k-lh}{2} \\right) \\right] |F|^2 F,  \\label{hexnls}\n\\end{eqnarray}\n$\\mathcal{O}(\\varepsilon^3 {\\mathrm e}^{3{\\mathrm i}\\psi})$:\n\\begin{eqn} \\fl \n9\\omega^2H_3 =  4 \\left[ \\sin^2(3k) + \\sin^2 \\left( \\frac{3k+3lh}{2} \\right) + \n\\sin^2 \\left( \\frac{3k-3lh}{2} \\right) \\right] \\left( H_3 + 2aFG_2+bF^3 \\right), \n\\label{hexh3} \\end{eqn}\n$\\mathcal{O}(\\varepsilon^4 {\\mathrm e}^0)$:\n\\begin{eqn} \\label{hexfo}\nG_{0\\tau\\tau} = 6\\, G_{0XX} + 2h^2 G_{0YY} + \na \\! \\left[ 6 \\left( |F|^2 \\right)_{XX} + 2h^2 \\left(|F|^2 \\right)_{YY} \\right] .\n\\end{eqn}\n\n\nThough each equation plays a similar role to its counterpart in the square \nlattice, equations \\eref{hexdisp}--\\eref{hexfo} are more complicated. \nEquation \\eref{hexdisp} is the dispersion relation for the system \\eref{heqq}.  \nSince we are interested only in solutions for which $F \\neq 0$, \\eref{hexdisp} yields \n\\begin{eqn} \\label{hexdisp2} \n\\omega^2  = 4\\sin^2(k)  + 4\\sin^2 \\left( \\frac{k+lh}{2} \\right) + \n4\\sin^2 \\left( \\frac{k-lh}{2} \\right),\n\\end{eqn}\nwhich does not simplify significantly.  From equation \\eref{hexvel}, \nwe determine the velocity of the travelling wave $F$, finding \n\\begin{eqn}  \\label{hexvels}\nF(X,Y,\\tau,T) \\equiv F(Z,W,T),\n\\end{eqn}\nwhere $Z=X-u\\tau$ and $W=Y-v\\tau$, and the horizontal and vertical\nvelocity components ($u$ and $v$) are found to be\n\\begin{eqn}  \\label{velsuv}\nu = \\frac{-2\\sin (k)[2\\cos(k)+\\cos(lh)]}{\\omega} \\quad\\textrm{and}\\quad\nv = \\frac{-2h\\cos(k)\\sin(lh)}{\\omega}.\n\\end{eqn}\nEquation \\eref{velsuv}, along with \\eref{hexdisp2} enables the elimination \nof terms involving $G_1$ from \\eref{hexnls} which are not shown.  We \ndenote the angle at which the envelope $F$ propagates through the \nlattice by $\\Psi$, which is measured from the direction of the basis vector \n$\\mathbf{e_i}$ to the line of travel and hence is given by $\\tan^{-1}(v\/u)$, \nwhich in turn depends upon the wavevector  $\\mathbf{k}$.   For both the cases \nthat we consider (namely, symmetric and asymmetric interaction potentials), \nwe find constraints upon the wavenumbers $k$ and $l$ which affect the \nvelocity components $u$ and $v$.   By taking $k=\\pi\/3$ and  $l=\\pi\/h$ we \nfind $u=v=0$, which corresponds to a static breather; and by choosing \n$k,l$ values which circle this point, we find breathers which can propagate \nin any direction ($0\\leq\\Psi<2\\pi$); in other words, our analysis suggests \nthat there is no restriction upon the direction of travel through the lattice.\n\n\nOur aim is to reduce \\eref{hexnls} to a nonlinear\nSchr{\\\"o}dinger (NLS) equation for $F$.  Before this\ncan be done, the quantities $G_0$ and $G_2$ in \\eref{hexnls} must\nbe found in terms of $F$.  As for the square lattice, it is straightforward\nto determine $G_2$ from the algebraic equation \\eref{hexg2}.\nHowever, the partial differential equation \\eref{hexfo} for $G_0$\ncan be solved for two special cases only; namely, \nsymmetric potentials, in which case the reduction of \\eref{hexnls}\nto an NLS equation can be completed even for moving breathers, and\nalso asymmetric potentials, provided we confine our attention to\nstationary breathers.  These two cases are considered in Sections \n\\ref{hexsym} and \\ref{hexasymm} respectively, where we also use \nour breather formulae to generate estimates for the breather energy.\n\n\n\\subsection{Asymptotic estimates for breather energy} \\label{heeh}\n\n\nThe HETL is a lossless network, meaning that the total electrical energy \nis conserved.  This quantity is related to the Hamiltonian \n(\\ref{hham}) by $E=\\widetilde{H}\/C_0$ and so is given by \n\\begin{eqn} \\label{hexentot}\nE = \\sum_{m,n} e_{m,n} =  \\sum_{m,n} \\frac{\\Upsilon(Q_{m,n})}{C_0} \n+ \\mbox{$\\frac{1}{2}$}L \\left( I_{m,n}^2 + J_{m,n}^2 + K_{m,n}^2 \\right) . \n\\end{eqn}\nThe electrical energy $e_{m,n}$ associated with a unit of the lattice is \n(see \\Fref{hexunitdir})\n\\begin{eqn} \\label{hexenun} \ne_{m,n} = \\frac{\\Upsilon(Q_{m,n})}{C_0} + \n\\mbox{$\\frac{1}{2}$}L ( I_{m,n}^2 + J_{m,n}^2 + K_{m,n}^2 ) . \n\\end{eqn}\nTo derive a leading-order estimate for the electrical energy, defined by \n\\begin{eqn}  \\label{hentotlo}\nE_0 = \\sum_{m,n} e_{m,n}^{(0)}=\\sum_{m,n}\\frac{Q_{m,n}^2}{2C_0}\n+ \\frac{L}{2} \\left( I_{m,n}^2 + J_{m,n}^2 + K_{m,n}^2 \\right) , \n\\end{eqn}\nwe use leading-order expressions for each of the terms in the terms in \nthe summand of \\eref{hexentot}.  The first term is $Q_{m,n}^2\/(2C_0)$;  \nfrom \\eref{hqvexp}, it follows that $V_{m,n} \\sim Q_{m,n}\/C_0$, and \nso to leading order, (\\ref{hexentot}) agrees with the linear approximation \nto the energy in the capacitor being $QV\/2$.   To find leading-order \nexpressions for the currents  $I_{m,n}$, $J_{m,n}$ and $K_{m,n}$, we \nuse equations \\eref{hki}--\\eref{hkk}, which imply \n\\begin{eqnarray}\nQ_{m+2,n} - Q_{m,n} & = - \\frac{{\\mathrm d} I_{m,n}}{{\\mathrm d} t}, \\label{hkiq} \\\\[1ex]\nQ_{m+1,n-1} - Q_{m,n} & = - \\frac{{\\mathrm d} J_{m,n}}{{\\mathrm d} t}, \\label{hkjq} \\\\[1ex]\nQ_{m+1,n+1} - Q_{m,n} & = - \\frac{{\\mathrm d} K_{m,n}}{{\\mathrm d} t}, \\label{hkkq}\n\\end{eqnarray}\nwhere $LC_0=1$.  The currents are determined by substituting the\nexpression for the breather $Q_{m,n}$ into \\eref{hkiq}--\\eref{hkkq} \nand then integrating with respect to time.  We obtain leading-order \nestimates for the energy of moving breathers in systems with symmetric \npotentials Section \\ref{hemv}, and in Section \\ref{hest}, the energies \nof stationary breathers with asymmetric potentials.\n\n\n\\subsection{The dispersion relation for the HETL} \\label{wplots}\n\n\nIn this section, we analyse the dispersion relation \\eref{hexdisp2} \nfor the system \\eref{heqq}.  A contour plot of $\\omega$ against $k$ and \n$l$ is shown in \\Fref{wcont}.  Since $w$ is periodic in both $k$ and \n$l$, with period $2\\pi$ along the $k$-direction, and $2\\pi\/h$ in the \n$l$-direction, we consider only $k$ and $l$ such that \n$(k,l) \\in \\mathcal{T}^2 = [0,2\\pi] \\times [0,2\\pi\/h]$.\n\n\nThe function $\\omega$ is minimised, and assumes the value zero, at the\ncentre of the circular patterns in \\Fref{wcont}.  The minima are located \nat $(0,0)$, $(2\\pi,0)$, $(2\\pi,2\\pi\/h)$, $(0,2\\pi\/h)$ and $(\\pi,\\pi\/h)$.  \nPlus signs (`$+$') mark the points in $(k,l)$-space at which $\\omega$ is \nmaximised and takes the value $\\omega=3$.  The maxima lie at the centres \nof the equilateral triangles, the wavevectors corresponding to these \npoints are denoted $\\mathbf{k_1}, \\ldots, \\mathbf{k_6}$, where\n\\begin{equation} \\begin{array}{ lclcl}\n\\mathbf{k_1}=[\\pi\/3,\\pi\/h]^T,  && \n\\mathbf{k_2}=[2\\pi\/3,0]^T,  && \n\\mathbf{k_3}=[4\\pi\/3,0]^T,    \\\\ \n\\mathbf{k_4}=[5\\pi\/3,\\pi\/h]^T, && \n\\mathbf{k_5}=[4\\pi\/3,2\\pi\/h]^T, &&\n\\mathbf{k_6}=[2\\pi\/3,2\\pi\/h]^T.\n\\end{array} \\end{equation} \nThe arrangement of the points corresponding to these wavevectors\nin $(k,l)$-space reflects the hexagonal symmetry properties of the\nlattice \\eref{heqq}.  It may be verified using \\eref{velsuv} that\nthe velocity components $u$ and $v$ are both zero for each of the\nwavevectors $\\{ \\mathbf{k_1},\\ldots, \\mathbf{k_6} \\}$.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\resizebox{4.5in}{!}{\\includegraphics{wcont.ps}}\n\\put(-250,103){\\footnotesize $\\mathbf{k_1}$}\n\\put(-200,44){\\footnotesize $\\mathbf{k_2}$}\n\\put(-100,44){\\footnotesize $\\mathbf{k_3}$}\n\\put(-48,103){\\footnotesize $\\mathbf{k_4}$}\n\\put(-100,183){\\footnotesize $\\mathbf{k_5}$}\n\\put(-200,183){\\footnotesize $\\mathbf{k_6}$}\n\\caption{Contour plot of $\\omega(\\mathbf{k})$; $\\omega$ attains its maximum \nvalue of 3 at the points $\\mathbf{k_1},\\ldots,\\mathbf{k_6}$, and \nis minimised at the points marked `o', where $\\omega=0$. }\n\\label{wcont}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Lattices with a symmetric potential} \\label{hexsym}\n\n\nIn this section, we consider lattices with a symmetric interaction potential, \nwhere $\\Upsilon (Q)$ is even and $\\Upsilon\\!\\ '(Q)$  has odd symmetry. \nThis corresponds to $a=c=0$ in \\eref{heqq}.  Since there are no even \nharmonics for vibrations controlled by symmetric potentials, it follows \nthat $G_0$ and $G_2$ are both zero.  In \\eref{hexnls}, the term \n$F_{\\tau\\tau}$ is eliminated using $F_{\\tau\\tau} = u^2F_{ZZ} + 2uvF_{ZW} \n+ v^2F_{WW}$ which is derived from \\eref{hexvels}.  This \nleads to (\\ref{hexnls}) being reduced to an NLS equation \n\\begin{eqnarray}\n\\fl 2{\\mathrm i}\\omega F_T + \\left[ u^2 - 4\\cos(2k) - 2\\cos k\\cos(lh) \\right] F_{ZZ}\n+ \\left[ v^2 - 2h^2\\cos k \\cos(lh) \\right] F_{WW} \\nonumber \\\\\n+ \\left[ 2uv + 4h \\sin k \\sin(lh) \\right] F_{ZW} + 3b\\omega^2|F|^2F = 0, \n\\label{hexrnls} \\end{eqnarray}\nwhere the velocities $u$ and $v$ are given by \\eref{velsuv}.  By\napplying an appropriate change of variables, we remove \nthe mixed derivative term from  \\eref{hexrnls}, and reduce the\nequation to a standard form.  To simplify the appearance of\nsubsequent expressions, we denote the coefficients of $F_{ZZ}$,\n$F_{WW}$ and $F_{ZW}$ by $D_1 = u^2 - 4\\cos(2k) - 2\\cos\nk\\cos(lh)$, $D_2 =  v^2 - 2h^2\\cos k \\cos(lh)$ and $D_3 = 2uv + 4h\n\\sin k \\sin(lh)$ respectively.  A suitable transformation is thus\n\\begin{eqn} \\label{hextr}\n\\xi = \\frac{hZ}{\\sqrt{D_1}} \\quad \\textrm{and} \\quad\n\\eta = \\frac{h(2D_1 W - D_3 Z)}{\\sqrt{D_1 (4D_1 D_2 - D_3^2)}},\n\\end{eqn}\nwhich implies \\eref{hexrnls} becomes\n\\begin{eqn} \\label{hexcan}\n2{\\mathrm i}\\omega F_T + 3\\nabla^2 F + 3b\\omega^2|F|^2F = 0,\n\\end{eqn}\nwhere the differential operator $\\nabla^2 F \\equiv F_{\\xi\\xi} +\nF_{\\eta\\eta}$ is isotropic in the $(\\xi,\\eta)$ variables.   An approximation \nto the Townes soliton solution of (\\ref{hexcan}) is given by equation \n(\\ref{vartown}) in the appendix.  Substituting the resulting expression for $F$ \ninto \\eref{hexanz} yields a leading-order expression for the breather\n\\begin{eqn} \\label{hexlo}\nQ_{m,n}(t) = 2 \\varepsilon \\alpha \\cos[km+lhn+(\\omega+\\varepsilon^2\\lambda)t] \n\\ \\mathrm{sech}(\\beta r) + \\mathcal{O}(\\varepsilon^3),\n\\end{eqn}\nwhere $\\alpha$ and $\\beta$ are determined as described by equation \n(\\ref{alpbet}) in the appendix, with $D=3\/2\\omega$, $B=3b\\omega\/2$.  Further,\n$r=\\sqrt{\\xi^2+\\eta^2}$ is found in terms of the physical discrete\nvariables $m$ and $n$ by inverting the transformations\n\\eref{hextr} and reverting back to the variables $Z$ and $W$, using\n\\begin{eqnarray} \nr^2 &=& \\xi^2 + \\eta^2 = \\frac{ 4 h^2 \\varepsilon^2 \n( D_2(m\\!-\\!ut)^2 + D_1(hn\\!-\\!vt)^2 - D_3(m\\!-\\!ut)(hn\\!-\\!vt) )}\n{4D_1D_2 -D_3^2}.  \\nonumber \\\\ && \\label{hexr}\n\\end{eqnarray}\nThe terms $D_1$, $D_2$ and $D_3$ are known from \\eref{hexrnls}, the \nvelocities $u$ and $v$ are given by \\eref{velsuv} and $\\omega$ is given in \n\\eref{hexdisp2}.\n\n\n\\subsubsection{Determining the domain of ellipticity. } \\label{hexdell}\n\n\nWe confine our attention to elliptic NLS equations.  We seek to determine \nthe region $\\mathcal{D}$ of $(k,l)$-parameter space (that is, the two-torus \n$\\mathcal{T}^2 = [(0,2\\pi)]\\times [0,2\\pi\/h]$) where the NLS equation \n\\eref{hexrnls} is elliptic. By definition, this equation is elliptic when \n$D_3^2 < 4 D_1 D_2$, where $D_1$, $D_2$ and $D_3$ are given in \n\\eref{hexrnls}.  Whilst the region $\\mathcal{D}$ cannot be specified \nexplicitly,  it is simple to find numerically, and is illustrated in  \\Fref{econt}.  \nDefining the function $e(k,l) = 4D_1(k,l) \\! \\cdot \\! D_2(k,l) - D_3(k,l)^2$, \nwe are concerned with the region where $e(k,l)>0$.  The subdomains \nhave been labelled $\\{ \\mathcal{D}_1 ,\\ldots, \\mathcal{D}_6\\}$ in \\Fref{econt}, \nwhere $\\mathcal{D} = \\mathcal{D}_1 \\cup \\ldots \\cup \\mathcal{D}_6$.  \n\n\n\\addtolength{\\captionmargin}{-15pt}\n\\begin{figure}[ht]\n\\begin{center}\n\\resizebox{4.5in}{!}{\\includegraphics{econt.ps}}\n\\put(-250,130){\\footnotesize $\\mathcal{D}_1$}\n\\put(-200,44){\\footnotesize $\\mathcal{D}_2$}\n\\put(-100,44){\\footnotesize $\\mathcal{D}_3$}\n\\put(-48,130){\\footnotesize $\\mathcal{D}_4$}\n\\put(-100,220){\\footnotesize $\\mathcal{D}_5$}\n\\put(-200,220){\\footnotesize $\\mathcal{D}_6$}\n\\caption{The domain $\\mathcal{D} = \\mathcal{D}_1 \\cup \\ldots \\cup \n\\mathcal{D}_6$ in which the NLS equation \\eref{hexrnls} is elliptic.}\n\\label{econt}\n\\end{center}\n\\end{figure}\n\\addtolength{\\captionmargin}{+15pt}\n\n\nAgain, the hexagonal symmetry properties of the HETL are reflected clearly \nin the function $e(k,l)$, which has six maxima (at which $e(k,l)=36$), each\nlying at the centre of one of the closed curves in \\Fref{econt}.  The maxima \nof $e(k,l)$ coincide with the six maxima of $\\omega(k,l)$ shown in \\Fref{wcont}, \nnamely, at the wavevectors $\\{\\mathbf{k_1},\\ldots,\\mathbf{k_6}\\}$ in $\\mathcal{T}^2$; \n$e(k,l)$ is minimised ($e(k,l)=-48$) at the six midpoints of the line \nsegments which connect adjacent maxima. \n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\resizebox{2.5in}{!}{\\includegraphics{WD1.eps}} \\quad \n\\resizebox{2.5in}{!}{\\includegraphics{SPD1.eps}}\n\\end{center}\n\\caption{Contour plots of $\\omega(k,l)$ (left) and speed \n($\\sqrt{u^2+v^2}$, right) for wavevectors $\\mathbf{k}\\in\\mathcal{D}_1$. \nContours are for $2.82\\leq\\omega\\leq3$ in steps of $0.02$ and speeds from \nzero to 0.6 in steps of 0.1.  The central point corresponds to $\\omega=3$ and \nzero speed, as one considers wave vectors nearer to the edge of \n$\\mathcal{D}_1$, the frequency $\\omega$ decreases and the speed increases. }\n\\label{wspcont}\n\\end{figure}\n\n\nFigure \\ref{wspcont} shows a contour plot for the frequency $\\omega$ in \n$\\mathcal{D}_1$: the ellipticity constraint only permits breathers with a \nrelatively high frequency, that is, with $\\omega>2.82$.  This constraint in turn \nimplies that not all breather envelope velocities are attainable, only breathers \nwith speeds upto about $0.7$ lattice sites per second are permitted; the plot \non the right of Figure \\ref{wspcont} shows that only breathers with speeds \nupto about $0.3$ sites per second can move in any direction. There is then an \nintermediate range of speeds, between $0.3$ and $0.7$ sites per second where \nbreathers can only move in certain directions, these correspond to the lattice \ndirections. The larger the speed, the more restricted is the direction of motion. \n\n\n\\subsubsection{Breather energy. } \\label{hemv}\n\n\nTo calculate the leading-order energy $E_0$ of moving breathers in \nlattices with a symmetric potential we use \\eref{hexlo} which we write as\n\\begin{eqn} \\label{hxs2}\nQ_{m,n}(t) \\sim 2 \\varepsilon \\alpha \\cos \\Phi \\ \\mathrm{sech}(\\beta r),\n\\end{eqn}\nwhere $\\Phi = km + lhn + \\Omega t$ is the phase of the carrier wave,\n$(k,l) \\in \\mathcal{D}$, $\\Omega = \\omega + \\varepsilon^2\\lambda$ is the breather\nfrequency including the first correction term,  ($\\psi=km+lhn+\\omega t$ \nis only the leading-order expression).  $\\lambda$ parameterises the \nbreather amplitude, $\\alpha=\\alpha(\\lambda)$, $\\beta=\\beta(\\lambda)$ \nand $r^2$ are as described in equations \\eref{hexlo} and \\eref{hexr}.  \n\n\nWe now find expressions for the currents $I_{m,n}$, $J_{m,n}$ and\n$K_{m,n}$: the current $I_{m,n}$ is obtained by substituting the \nexpression for $Q_{m,n}$ \\eref{hxs2} into \\eref{hkiq} and integrating \nwith respect to time.   Owing to the complexity of the expression for $r$ \ngiven by \\eref{hexr}, the left-hand side of \\eref{hkiq} can not be integrated \nwith respect to time. However, the variable $r$ varies more slowly in time \nthan $\\Phi$. Hence,  integration by parts (using $\\int f'(t) g(\\varepsilon t) {\\mathrm d} t = \n[f(t) g(\\varepsilon t)] - \\varepsilon \\int f(t) g'(\\varepsilon t){\\mathrm d} t$) gives, to leading-order, \n\\begin{eqn} \\label{himov} \\fl\nI_{m,n}\\sim\\frac{2\\varepsilon\\alpha}{\\omega}\\left[1-\\cos(2k)\\right]\\sin\\Phi\\,\\mathrm{sech}(\\beta r)\n- \\frac{2\\varepsilon\\alpha}{\\omega} \\sin(2k) \\cos \\Phi \\,\\mathrm{sech}(\\beta r),\n\\end{eqn}\nwhere we have taken the constant of integration to be zero, and\n$\\Omega \\sim \\omega$ to leading order.  Similarly, substituting for $Q_{m,n}$ in \nequations \\eref{hkjq} and \\eref{hkkq} and integrating, we find \n\\begin{eqnarray} \\fl\nJ_{m,n} & \\sim & \\frac{2\\varepsilon\\alpha}{\\omega} \\left[ 1-\\cos(k-lh) \\right] \n\\sin \\Phi\\, \\mathrm{sech}(\\beta r) - \n\\frac{2\\varepsilon\\alpha}{\\omega}\\sin(k-lh)\\cos\\Phi\\,\\mathrm{sech}(\\beta r), \\label{hjmov}\\\\[1ex] \n\\fl  K_{m,n} & \\sim & \\frac{2\\varepsilon\\alpha}{\\omega} \\left[ 1-\\cos(k+lh) \\right] \n\\sin \\Phi\\, \\mathrm{sech}(\\beta r)\n- \\frac{2\\varepsilon\\alpha}{\\omega} \\sin(k+lh) \\cos \\Phi \\,\\mathrm{sech}(\\beta r). \\label{hkmov}\n\\end{eqnarray}\nSubstituting these expressions into \\eref{hentotlo}, we obtain \n\\begin{eqnarray}\nE_0 & \\sim \\sum_{m,n} & \\frac{2\\varepsilon^2\\alpha^2}{C_0} \n\\cos^2 \\Phi \\ \\mathrm{sech}^2(\\beta r) \\nonumber \\\\ & & + \n\\frac{2L\\varepsilon^2\\alpha^2}{\\omega^2}\\mathrm{sech}^2(\\beta r) \\left\\{ \\left[ \n(1-\\cos(2k))\\sin\\Phi - \\sin(2k)\\cos\\Phi  \\right]^2 \\right. \\nonumber \\\\\n& & + \\left[ (1-\\cos(k-lh))\\sin\\Phi - \\sin(k-lh)\\cos\\Phi \\right]^2 \\nonumber \\\\ \n& & + \\left. \\left[ (1-\\cos(k+lh))\\sin\\Phi - \\sin(k+lh)\\cos\\Phi \\right]^2 \\right\\} .  \n\\label{hmvan} \\end{eqnarray}\nWe replace the sum by an integral since the variables $Z=\\varepsilon (m-ut)$ \nand $W=\\varepsilon (hn-v t)$  vary slowly with $m$ and $n$.  To simplify the \nresulting integral, we approximate the terms $\\cos^2\\Phi$, $\\sin^2\\Phi$ \nand $\\sin\\Phi\\cos\\Phi$ by their average values of $\\sfr{1}{2}$, $\\sfr{1}{2}$ \nand $0$ respectively.  Hence \\eref{hmvan} becomes \n\\begin{eqn} \\label{henmvint}\nE_0 \\sim \\sum_{m,n} \\frac{2\\varepsilon^2\\alpha^2}{C_0} \\mathrm{sech}^2(\\beta r).\n\\end{eqn}\n\n\n{}From the definition of $r$ (\\ref{hexr}) we note the function \n$\\mathrm{sech}(\\beta r)$ is not in general radially symmetric in $m$, $n$.  Hence \nwe work in $(\\xi,\\eta)$-space to facilitate evaluation of the double integral \nwhich approximates the double sum in (\\ref{henmvint}).   Evaluating the \nJacobian associated with the transformation from $(m,n)$ to $(\\xi,\\eta)$ \ncoordinates, we find \n\\begin{eqn}\nE_0 \\sim \\frac{\\alpha^2}{h^3C_0}\\sqrt{4D_1D_2 - D_3^2} \\int \\!\\!\\! \\int\n\\mathrm{sech}(\\beta\\sqrt{\\xi^2+\\eta^2}) \\ {\\mathrm d} \\xi {\\mathrm d} \\eta. \n\\label{hexemtri} \\end{eqn}\nEvaluating this and substituting for $\\alpha$ and $\\beta$ in terms of \n$D=3\/2\\omega(k,l)$ and $B=3b\\omega(k,l)\/2$ (see the appendix) we find \n\\begin{eqn} \\label{henar}\nE_0 \\sim \\frac{4\\pi\\log 2(2\\log 2 + 1)}\n{3\\, h^3 \\, C_0 \\, b \\, \\omega^2(4\\log 2 - 1)} \\sqrt{4D_1D_2 - D_3^2}.\n\\end{eqn}\nIt is evident from \\eref{henar} that the leading-order energy $E_0$ is \nindependent of the breather amplitude, again confirming the existence \nof a minimum energy of moving breathers in the two-dimensional HETL \nwith symmetric potential.  However, the threshold energy does depend \nupon the wavenumbers $k$ and $l$ hence moving breathers have a \ndifferent threshold energy.   A plot of the expression \\eref{henar} is \nshown in \\Fref{henmov}.  We see that $E_0$, given by \\eref{henar}, is \nstrictly positive in the region of ellipticity $\\mathcal{D}$, and is maximised \n(attaining the same value) at each of the points corresponding to \nwavevectors $\\{ \\mathbf{k_1},\\ldots, \\mathbf{k_6} \\}$, that is, at the points which\ncorrespond to stationary breathers.  The threshold energy (\\ref{henar}) \ndecays to zero towards the boundary of the elliptic domain $\\mathcal{D}$ (see \n\\Fref{econt}).   Hence, as for breathers in the square lattice, the energy \nthreshold for moving breathers is {\\em lower} than that for stationary \nbreathers.  The threshold becomes arbitrarily small at the boundary of the \ndomain of ellipticity.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\resizebox{3.5in}{!}{\\includegraphics{henmov.ps}}\n\\caption{Plot of $E_0(k,l)$ for lattices with a symmetric potential.}\n\\label{henmov}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Lattices with an asymmetric potential} \\label{hexasymm}\n\n\nIn this section, we consider the more general scenario where the potential \n$\\Upsilon\\!\\ '(Q)$ is asymmetric.  In this case, the terms $a$ and $c$ in \n\\eref{hqvexp} and \\eref{heqq} are not both zero.   If \\eref{hexnls} is to \nbe reduced to an NLS equation in  $F$, both $G_0$ and $G_2$ must be \nfound in terms of $F$.  Whilst $G_2$ is given by the simple algebraic \nequation \\eref{hexg2}; in order to find $G_0$, the partial differential \nequation \\eref{hexfo} must be solved.  We  assume that $G_0$ travels \nat the same velocity as $F$, that is, $G_0(X,Y,\\tau,T) \\equiv G_0(Z,W,T)$.  \nEliminating $G_{0\\tau\\tau}$,  \\eref{hexfo} becomes\n\\begin{eqn} \\label{hexg0}\n(u^2 - 6)G_{0ZZ} + (v^2 - 2h^2)G_{0WW} + 2uvG_{0ZW} =\n6a|F|_{ZZ}^2 + 2ah^2|F|_{WW}^2.\n\\end{eqn}\nFor general $k$ and $l$, it is difficult to solve for $G_0$ explicitly.  \nHowever, for any of the wavevectors $\\{\\mathbf{k_1},\\ldots,\\mathbf{k_6}\\}$, \nthe velocities $u$ and $v$ become zero, and so \\eref{hexg0} becomes \n$\\nabla^2 G_0 = -a\\nabla^2|F|^2$, where the operator $\\nabla^2$, \ndefined by $\\nabla_{(Z,W)}^2 \\equiv \\partial_Z^2 + \\partial_W^2$, is \nequivalent to $\\nabla_{(X,Y)}^2 \\equiv \\partial_X^2 + \\partial_Y^2$. It \nfollows that $G_0 = -a|F|^2$ for each of these wavevectors.\n\n\nEquations \\eref{hexdisp}--\\eref{hexfo} are the same whichever \nof the wavevectors $\\{\\mathbf{k_1},\\ldots,\\mathbf{k_6}\\}$ is used.  Thus, \n\\eref{hexdisp} gives $\\omega=3$, and  \\eref{hexg2} implies $G_2 = aF^2\/3$.  \nSubstituting these expressions for $G_0$ and $G_2$ in  \\eref{hexnls} \ngives the following NLS equation for asymmetric potentials for any \nof the wavevectors $\\{\\mathbf{k_1},\\ldots,\\mathbf{k_6}\\}$\n\\begin{eqn} \\label{hnlsasy}\n2{\\mathrm i}\\omega F_T + 3 \\nabla^2 F + \\omega^2(3b-\\mbox{$\\frac{10}{3}$}a^2)|F|^2F=0,\n\\end{eqn}\nThe anomalous dispersion regime thus corresponds to $b>10a^2\/9$.  \nSoliton solutions are derived in the appendix;   \\eref{hnlsasy} \ncorresponds to $D = 1\/2$ and $B =(9b-10a^2)\/2$, and also \n$r^2 = X^2 + Y^2$.   Substituting the solution for $F$ into the lattice \nansatz \\eref{hexanz}, along with the known expressions for $G_0$ and \n$G_2$ gives the following second-order formula for stationary \nbreathers in lattices with an asymmetric potential \n\\begin{eqnarray} \n\\fl Q_{m,n}(t) = 2\\ \\! \\varepsilon  \\ \\! \\alpha\\cos[km+lhn+(\\omega+\\varepsilon^2\\lambda)t]\\ \n\\mathrm{sech}(\\beta r) \\nonumber \\\\ \n+ \\mbox{$\\frac{2}{3}$}\\ \\! a \\ \\! \\varepsilon^2 \\ \\! \\alpha^2 \\mathrm{sech}^2(\\beta r)\\!\\left\\{ \n\\cos[2km+2lhn+(2\\omega+2\\varepsilon^2\\lambda)t] - 3 \n\\right\\} + \\mathcal{O}(\\varepsilon^3), \\label{hexqasym} \n\\end{eqnarray} \nwhere $\\alpha$ and $\\beta$ are determined in the appendix, \n$r=\\sqrt{X^2+Y^2}$, and $X=\\varepsilon m$ and $Y=\\varepsilon h n$ in terms of the \noriginal discrete variables $m$ and $n$.\n\n\n\\subsubsection{Breather energy. } \\label{hest}\n\n\nWe calculate the leading-order energy $E_0$ of stationary breathers in \nlattices with an asymmetric potential.  In this case, stationary breathers \nare given by \\eref{hexqasym}, from which we take only the leading order term \n\\begin{eqn} \\label{hxa2}\nQ_{m,n}(t) \\sim 2 \\varepsilon \\alpha \\cos \\Theta \\ \\mathrm{sech}(\\beta r),\n\\end{eqn}\nwhere $\\Theta = km + lhn + \\Omega t$ is the phase of the carrier wave \nwith $(k,l)$ corresponding to one of $\\{ \\mathbf{k_1},\\ldots, \\mathbf{k_6}\\}$, \n$\\Omega = \\omega+\\varepsilon^2\\lambda$ is the breather frequency, and $w=3$. \n\n\nThe currents $I_{m,n}$, $J_{m,n}$ and $K_{m,n}$ are obtained \nby substituting \\eref{hxa2} into equations \\eref{hkiq}--\\eref{hkkq} \nand integrating with respect to time, taking the constant of \nintegration to be zero.   Thus, to leading order, we find\n\\begin{eqnarray}\nI_{m,n} & \\sim & \\frac{3\\varepsilon\\alpha}{\\omega} \\sin \\Theta\\, \\mathrm{sech}(\\beta r)\n- \\frac{\\sqrt{3}\\varepsilon\\alpha}{\\omega} \\cos \\Theta \\,\\mathrm{sech}(\\beta r),\\label{heni} \\\\[1ex]\nJ_{m,n} & \\sim & \\frac{3\\varepsilon\\alpha}{\\omega} \\sin \\Theta\\, \\mathrm{sech}(\\beta r)\n+ \\frac{\\sqrt{3}\\varepsilon\\alpha}{\\omega} \\cos \\Theta \\,\\mathrm{sech}(\\beta r), \\label{henj} \\\\[1ex]\nK_{m,n} & \\sim & \\frac{3\\varepsilon\\alpha}{\\omega} \\sin \\Theta\\, \\mathrm{sech}(\\beta r)\n+ \\frac{\\sqrt{3}\\varepsilon\\alpha}{\\omega} \\cos \\Theta \\,\\mathrm{sech}(\\beta r).  \\label{henk}\n\\end{eqnarray}\nSubstituting these expressions into  \\eref{hentotlo} \ngives the leading-order energy \n\\begin{eqnarray} \\fl\nE_0 \\sim \\sum_{m,n} \\frac{2\\varepsilon^2\\alpha^2}{C_0} \n\\cos^2 \\Theta \\ \\mathrm{sech}^2(\\beta r) \\nonumber \\\\\n+  \\frac{Lh^2}{2\\omega^2}\\,\\varepsilon^2\\alpha^2  \\left[  3h^2\\sin^2\\Theta + \n2h\\sin\\Theta\\cos\\Theta + 3\\cos^2\\Theta  \\right]  \\mathrm{sech}^2(\\beta r)  . \n\\label{henas}\n\\end{eqnarray}\nWe approximate the term in square brackets by taking the average\nvalues of $\\cos^2 \\Theta$, $\\sin^2\\Theta$, $\\sin\\Theta\\cos\\Theta$ \nas \\mbox{$\\frac{1}{2}$}, \\mbox{$\\frac{1}{2}$} and $0$ respectively. \n\n\nWe replace the double sum by an integral giving\n\\begin{eqn} \\label{henint}\nE_0 \\sim  \\frac{2\\alpha^2}{hC_0} \\int \\!\\!\\! \\int\n\\mathrm{sech}^2(\\beta \\sqrt{X^2+Y^2}) \\ {\\mathrm d} X {\\mathrm d} Y.\n\\end{eqn}\nThis can be evaluated to \n\\begin{eqn} \\label{henex}\nE_0 \\sim \\frac{4\\pi\\log 2}{h C_0} \\frac{\\alpha^2}{\\beta^2}\n= \\frac{8 \\pi \\log 2(2\\log 2+1)}{C_0h(9b-10a^2)(4\\log 2-1)} . \n\\end{eqn}\nAgain, this estimate for energy is independent of the breather\namplitude $\\lambda$, demonstrating the energy threshold properties\nof the two-dimensional HETL; namely, the activation energy\nrequired to create a breather in the HETL is an $\\mathcal{O}(1)$\nquantity, irrespective of its amplitude ($2\\varepsilon\\alpha\\ll 1$).\n\n\n\\section{Higher-order asymptotic analysis}  \\label{hex5}\n\n\nThe cubic NLS equation exhibits blow-up, which cannot occur in a discrete \nsystem since energy is conserved, and even if all the energy were localised \nat a single site, the amplitude would still be finite.   The two-dimensional \ncubic NLS equation description of the breather envelope is lacking.   \nHigher-order dispersive and nonlinear effects play an important role in the \ndynamics of such discrete systems and therefore must be incorporated.  In \nthis section we extend our analysis of the lattice equations \\eref{heqq} to \nfifth-order, and derive a generalised NLS equation which includes higher-order \ndispersive and nonlinear terms.  It then remains to determine whether this \ngeneralised NLS equation supports stable soliton solutions.  \n\n\nDue to the complexity of a fifth-order analysis of  a general asymmetric \npotential, we consider only lattices with a symmetric potential, that is, \nthose for which $a=c=0$ in \\eref{heqq}. Since no second or fourth harmonic\nterms are generated by the nonlinearity, we use a much simpler ansatz, namely \n\\begin{eqn} \\label{h5anzgen}\nQ_{m,n}(t) = \\varepsilon {\\mathrm e}^{{\\mathrm i} \\psi} F(X,Y,\\tau,T)\n+ \\varepsilon^3 {\\mathrm e}^{3{\\mathrm i} \\psi}H_3(X,Y,\\tau,T) + \\cdots + \\mathrm{c.c.,}\n\\end{eqn}\nwhere the phase $\\psi = km+lhn+\\omega t$.  In this case, in addition to the \nequations \\eref{hexdisp2}--\\eref{velsuv}, \\eref{hexrnls} and \\eref{hexh3}, \nwe also have \\\\ \n\\\\\n$\\mathcal{O}(\\varepsilon^5 {\\mathrm e}^{{\\mathrm i}\\psi})$:\n\\begin{eqnarray} \\fl \nF_{TT} =&& \\hspace*{-12mm}\n\\mbox{$\\frac{1}{6}$} \\left[ 8\\cos(2k) + \\cos k \\cos(lh) \\right] F_{XXXX} \n+ h^2\\cos k \\cos(lh)F_{XXYY}  \\nonumber \\\\ &&\\hspace*{-16mm}\n+ \\mbox{$\\frac{1}{6}$} h^4 \\cos k \\cos(lh)F_{YYYY} \n- \\mbox{$\\frac{2}{3}$} h \\sin k \\sin(lh) F_{XXXY} \n- \\mbox{$\\frac{2}{3}$} h^3 \\sin k \\sin(lh)F_{XYYY} \n\\nonumber\\\\ &&\\hspace*{-16mm}\n+ 6b \\left[ 2\\cos(2k) + \\cos k \\cos(lh) \\right](|F|^2F)_{XX}\n+ 6bh^2\\cos k\\cos(lh)(|F|^2F)_{YY} \\nonumber \\\\ && \\hspace*{-16mm}\n-12b\\sin k \\sin(lh) \\left[  2hFF_X\\overline{F}_Y  + 2hFF_Y \\overline{F}_X\n+ 2h\\overline{F}F_XF_Y + hF^2\\overline{F}_{XY} + 2hF\\overline{F}F_{XY} \\right] \n\\nonumber \\\\  && \\hspace*{-16mm} \n- 3b\\omega^2\\overline{F}^2H_3 - 10\\omega^2 d |F|^4F . \n\\label{hfogen} \\end{eqnarray}\nWe simplify this by restricting attention to stationary breathers.\nAccordingly, only one extra timescale, $T=\\varepsilon^2 t$, is required, and we \nfix the wavenumbers $k$ and $l$ to correspond to one of the wavevectors \n$\\{ \\mathbf{k_1},\\ldots, \\mathbf{k_6} \\}$.  Hence we obtain the equations: \\\\\n\\\\\n$\\mathcal{O}(\\varepsilon^3 {\\mathrm e}^{{\\mathrm i}\\psi})$: \\\\\n\\begin{eqn} \\label{h5nls}\n2{\\mathrm i} \\omega F_T + 3\\nabla^2 F + 3b\\omega^2 |F|^2F = 0,\n\\end{eqn}\n$\\mathcal{O}(\\varepsilon^5 {\\mathrm e}^{{\\mathrm i}\\psi})$: \\\\\n\\begin{eqn} \\label{h550}\nF_{TT} = -\\mbox{$\\frac{3}{4}$} \\nabla^4F - 9b\\nabla^2(|F|^2F) -\n10\\omega^2d|F|^4F , \n\\end{eqn} \nin addition to $\\omega=3$ (from the $\\mathcal{O}(\\varepsilon {\\mathrm e}^{{\\mathrm i}\\psi})$\nequation) and $H_3=0$ (from $\\mathcal{O}(\\varepsilon^3 {\\mathrm e}^{3{\\mathrm i}\\psi})$). \nA consequence of the hexagonal symmetry of the HETL is that all \ndifferentials on the right-hand side of \\eref{h550} are isotropic.  \n\n\nIn order to obtain a generalised NLS equation, we combine the\nhigher-order equation \\eref{h550} with the cubic two-dimensional\nNLS equation \\eref{h5nls}.  The $F_{TT}$ term on the\nleft-hand side of \\eref{h550} is eliminated  by\ndifferentiating \\eref{h5nls} with respect to $T$ and substituting\nfor $F_{TT}$ into \\eref{h550}.  The resulting expression is \n\\begin{eqnarray}\n6 {\\mathrm i} F_T + 3\\nabla^2 F + 27b|F|^2F + \\frac{\\varepsilon^2}{2} \\nabla^4 F\n+ \\frac{9\\varepsilon^2}{4}(40d-27b^2)|F|^4F & \\nonumber \\\\ \\qquad \\qquad \n+ \\frac{27b\\varepsilon^2}{4} \\nabla^2(|F|^2F) - \\frac{9b\\varepsilon^2}{2}|F|^2\\nabla^2F\n- \\frac{9b\\varepsilon^2}{4} F^2\\nabla^2\\overline{F} = 0 , & \\label{hexgennls}\n\\end{eqnarray}\nwhich is isotropic. \n\n\nTo the best of our knowledge, this perturbed form of the NLS equation \nhas not been studied in the literature before; although it is similar to \nand slightly simpler than the corresponding equation derived for the \nsquare lattice \\cite{buts06}.  It is also similar to the perturbed NLS \nequation considered by Davydova \\etal \\citep{dav03}, namely \n\\begin{equation}\n{\\mathrm i} F_T + D \\nabla^2 F + B |F|^2 F + P \\nabla^4 F + K |F|^4 F =0 , \n\\label{davydova} \n\\end{equation} \nwhich is known to have stable soliton solutions.  The anomalous \ndispersion case is $BD>0$, which in our equation (\\ref{hexgennls}) \ncorresponds to $b>0$;   Davydova's criteria for soliton existence is \n$PK>0$ which implies $40d > 27b^2$.  Hence, it is in this parameter \nregime that we seek breathers solutions of the HETL.  The numerical \nresults presented in  \\Sref{hexnumerics} show that long-lived breather \nsolutions are supported by the two-dimensional hexagonal FPU lattice, \nsuggesting that the additional perturbing terms in \\eref{hexgennls}\ndo not destabilise the Townes soliton.\nIn fact, numerical simulations (presented in \\cite{thesis} but not here) of the \ncase $40d<27b^2$ show the breather mode to be long-lived, suggesting \nthat the terms on the second line of (\\ref{hexgennls}) are stabilising. \n\n\nWe have been unable to find a variational formulation of\n\\eref{hexgennls}, and are therefore unable to use the methods of \nDavydova {\\em et al.}\\ \\cite{dav03} and Kuznetsov {\\em et al.}\\ \n\\citep{kuz}.  Alternative possible methods include the modulation \ntheory of Fibich and Papanicolaou \\citep{fib98,fib99}.\n\n\n\\section{Numerical results}  \\label{hexnumerics}\n\n\\subsection{Preliminaries} \\label{hnumpre}\n\n\nUsing a fourth-order Runge-Kutta scheme, we solve the equations \n\\begin{eqnarray}\n\\frac{{\\mathrm d} Q_{m,n}}{{\\mathrm d} t} & = & R_{m,n}, \\nonumber \\\\[1ex]\n\\frac{{\\mathrm d} R_{m,n}}{{\\mathrm d} t} & = & (\\delta^2_I + \\delta^2_J +\n\\delta^2_K)\n\\left[ Q_{m,n} + a Q_{m,n}^2 + b Q_{m,n}^3 + c Q_{m,n}^4 + d\nQ_{m,n}^5 \\right], \\label{heqqfo}\n\\end{eqnarray}\nnumerically.  Introducing the variable $R_{m,n}$ converts the \nsystem of second-order ordinary differential equations \\eref{heqq} \nto an equivalent system of first-order differential equations.  \n\n\nWe present the results of simulations for a range of parameter\nvalues.  From \\eref{velsuv}, the velocity of the envelope $(u,v)$ \ndepends upon the wavevector $\\mathbf{k}=[k,l]^T$, and hence we \nobtain moving breathers by choosing $(k,l) \\in \\mathcal{D}$.  \nIn \\Fref{ellcontnum}, we show the points in  $\\mathcal{D}_1 \\subset  \n\\mathcal{D}$ for which we solve the lattice equations \\eref{heqqfo} \nnumerically.  These points correspond to the wavevectors $\\mathbf{k_1} \n= [\\pi\/3,\\pi\/h]^T$, $\\mathbf{k_a} = [1.4,\\pi\/h]^T$, $\\mathbf{k_b} = \n[0.79,1.7324]^T$ and $\\mathbf{k_c} = [0.8,1.9987]^T$.  \nSelecting $(k,l)$ too near to the boundary of $\\mathcal{D}$ results \nin a sharply elongated breather, which are difficult to simulate as \nthey require a large domain.  Wavevectors close to any of \n$\\{ \\mathbf{k_1},\\ldots, \\mathbf{k_6} \\}$, lead to breather modes with small \nspeeds, a check of breather velocity would then require a long-time \nsimulation.  In practice we have chosen wavenumbers which do not \nlead to a severely elongated breather envelope, and yet have velocities \nwhich result in observable displacements over reasonable times.   We \npresent simulations of stationary breathers in systems with asymmetric \npotentials ($a,c \\neq 0$) in \\Sref{hnra} and, in Sections \n\\ref{hnstsy}--\\ref{hpsi130}, simulations of stationary and moving breathers \nin lattices with symmetric potentials (that is, $a=c=0$ in \\eref{heqqfo}). \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\resizebox{4in}{!}{\\includegraphics{ellcontnum.eps}}\n\\put(-136,114){\\scriptsize $\\mathbf{k_1}$}\n\\put(-89,114){\\scriptsize $\\mathbf{k_a}$}\n\\put(-172,91){\\scriptsize $\\mathbf{k_b}$}\n\\put(-170,140){\\scriptsize $\\mathbf{k_c}$}\n\\put(-87,164){\\footnotesize $\\mathcal{D}_1$}\n\\caption{Wavevectors in $\\mathcal{D}_1 \\subset \\mathcal{D}$ for \nwhich breathers are simulated.}\n\\label{ellcontnum}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Initial data and boundary conditions} \\label{hbcics}\n\n\nWe generate initial data by using the analytic expressions for\nbreather solutions derived in \\Sref{haa}. The formulae for $Q_{m,n}$ \nand $R_{m,n}$ are found in terms of the original discrete variables \n$m$ and $n$, and then shifted horizontally and vertically so that initially \nthe breather lies at the centre of the lattice.   We impose periodic \nboundary conditions for the lattice in both horizontal and vertical \ndirections, converting the two-dimensional arrangement illustrated \nin \\Fref{hexetl} into a two-torus.  In long-time simulations, moving \nbreathers which approach an edge of the lattice reappear from the \nopposite edge.  We select the site $(1,1)$ to lie at the bottom left-hand \ncorner of the arrangement as illustrated in \\Fref{hexbcs}.  Sites along \nthe boundaries and corners are missing between one and four neighbours.  \nWe introduce fictitious sites along the boundaries and corners \nwhere necessary to effect periodic boundary conditions.  \n\n\nAn illustration of a small finite lattice is shown in \\Fref{hexbcs}.  \nThe dots represent capacitors located at lattice sites, and the \nlines represent some of the inter-connecting inductors.  From \nequations \\eref{heqqfo},  the charge $Q_{m,n}$ stored on \neach capacitor depends upon the charge stored on the capacitors \nlocated at its six neighbouring sites,  two in each of the directions \n$\\mathbf{e_i}$, $\\mathbf{e_j}$ and $\\mathbf{e_k}$.  \nFor the sake of clarity, the inductors connecting the capacitors at\nthe center of each hexagon to its six nearest neighbours are not\nshown (see \\Fref{hexetl}).  It is not necessary that the lattice\nshould be ``square,'' meaning that the lattice could comprise $M\n\\times N$ lattice sites, with $M \\neq N$.  However, typically, we\nconsider $M=N$ as in \\Fref{hexbcs}, since the numerical routines\nare simpler to encode when the lattice \\emph{is} square, and in  \nthe examples below, we consider lattices with $N \\leq 50$.  \n\n\n\\addtolength{\\abovecaptionskip}{10pt}\n\\begin{figure}[htbp]\n\\begin{center}\n\\vspace{1.0cm}\n\\resizebox{2.5in}{!}{\\includegraphics{hexbcs.eps}}\n\\put(-204,-10){${Q_{1,1}}$}\n\\put(-10,-10){${Q_{2N-1,1}}$}\n\\put(-173,153){${Q_{2,N}}$}\n\\put(-18,153){${Q_{2N,N}}$}\n\\caption{Periodic boundary conditions for the two-dimensional HETL.}\n\\label{hexbcs}\n\\end{center}\n\\end{figure}\n\\addtolength{\\abovecaptionskip}{-10pt}\n\n\n\\subsection{Numerical computation of breather energy} \\label{hnumen}\n\n\nSince the HETL is lossless the total energy is conserved and \ncan be used as a check of the accuracy of our numerical scheme.   \nWe compute the leading-order energy by expressing the summand \n$e_{m,n}^{(0)}$ \\eref{hentotlo} in terms of the output variables of \nthe numerical routine, namely $Q_{m,n}$ and $R_{m,n}$.  The first \nterm of $e_{m,n}^{(0)}$ is simply dependent on $Q_{m,n}$.  It\nremains to find the currents $I_{m,n}$, $J_{m,n}$ and $K_{m,n}$ in\nterms of $Q_{m,n}$ and $R_{m,n}$.  The details of this calculation \ndepend upon whether the interaction potential is symmetric or asymmetric.\nAs a check of the numerical scheme we verify that the sum is \nconserved and that it agrees with the asymptotic estimate \\eref{henar}.\n\n\n\\subsubsection{Lattices with a symmetric potential. }\n\n\nFirst, we find the current $I_{m,n}$ in terms of $Q_{m,n}$ and\n$R_{m,n}$.  Differentiating the leading-order expression for the breather \ngiven by \\eref{hxs2}, we have (retaining leading-order terms only)\n\\begin{eqn} \\label{hnem}\n\\dot{Q}_{m,n} = R_{m,n} \\sim -2\\varepsilon\\alpha \\omega \\sin \\Phi \\ \\mathrm{sech} (\\beta r),\n\\end{eqn}\nwhere $\\alpha$, $\\beta$ and $r^2$ are as defined in \\Sref{hexsym}.\nComparing the analytic expression \\eref{himov} for the current\n$I_{m,n}$ with the expressions for $Q_{m,n}$ \\eref{hxs2} and\n$R_{m,n}$ \\eref{hnem}, we find\n\\begin{eqn} \\label{hniqr}\nI_{m,n} = \\frac{\\cos(2k)-1}{\\omega^2} R_{m,n} - \\frac{\\sin(2k)}{\\omega}Q_{m,n}.\n\\end{eqn}\nSimilarly, comparing the expressions \\eref{hjmov} and \\eref{hkmov}\nfor the currents $J_{m,n}$ and $K_{m,n}$ respectively with\nequations \\eref{hxs2} and \\eref{hnem}, we find \n\\begin{eqnarray}\nJ_{m,n} & = & \\frac{\\cos(k-lh)-1}{\\omega^2} R_{m,n} -\n\\frac{\\sin(k-lh)}{\\omega}Q_{m,n},\n\\label{hnjqr} \\\\[1ex]\nK_{m,n} & = & \\frac{\\cos(k+lh)-1}{\\omega^2} R_{m,n} -\n\\frac{\\sin(k+lh)}{\\omega}Q_{m,n}. \\label{hnkqr}\n\\end{eqnarray}\nSubstituting these expressions into \\eref{hentotlo}, gives an expression \nfor $E_0$ which does not simplify, so we do not reproduce it here.  \n\n\n\\subsubsection{Lattices with an asymmetric potential. }\n\n\nDifferentiating the leading-order expression \\eref{hxa2} for the\ncharge $Q_{m,n}$ in lattices with an asymmetric potential gives\n\\begin{eqn} \\label{hxa2dot}\n\\dot{Q}_{m,n} = R_{m,n} \\sim \n-2\\varepsilon\\alpha \\omega \\sin \\Theta \\ \\mathrm{sech} (\\beta r) , \n\\end{eqn}\nwhere $\\Theta=\\pi m\/3 +\\pi n + \\Omega t$ with $\\Omega=3$.  Using \\eref{hxa2} \nand \\eref{hxa2dot}, the current $I_{m,n}$  given by \\eref{heni} can be \nexpressed in terms of $Q_{m,n}$ and $R_{m,n}$ as\n\\begin{eqn} \\label{hiqqd}\nI_{m,n} \\sim -\\frac{\\sqrt{3}}{2\\omega} \\left[ \\frac{\\sqrt{3}R_{m,n}}{\\omega} + \nQ_{m,n} \\right]\\!.\n\\end{eqn}\nSimilarly, from  \\eref{henj}--\\eref{henk}, we have\n\\begin{eqn} \\label{hjkqqd}\nJ_{m,n}  = K_{m,n} \\sim -\\frac{\\sqrt{3}}{2\\omega} \n\\left[ \\frac{\\sqrt{3}R_{m,n}}{\\omega} - Q_{m,n} \\right]\\!.\n\\end{eqn}\nThese are equivalent to substituting $k=\\pi\/3$ and $l=\\pi\/h$ into \n(\\ref{hniqr})--(\\ref{hnkqr}).  Inserting the expressions for $I_{m,n}$, \n$J_{m,n}$ and $K_{m,n}$ (\\eref{hiqqd}--\\eref{hjkqqd}) into \\eref{hentotlo} yields \n\\begin{eqn} \\label{hentotqqd}\nE_0 = \\sum_{m,n} \\frac{1}{72 C_0} \\left[ \n45Q_{m,n}^2 + 3R_{m,n}^2 - 2\\sqrt{3}Q_{m,n}R_{m,n} \\right]\\! . \n\\end{eqn}\n\n\n\\subsubsection{Effective breather width. }\n\n\nNumerical computation of the energy as described above allows us to \ncheck that the total lattice energy is conserved, but does not indicate \nwhether a breather changes shape over time.  To remedy this, we define \nbreather widths in the $m$ and $n$ directions, the sum of which we \ndenote $\\mathcal{W}_{\\mathrm{br}}$, where\n\\begin{eqn} \\label{hbrwdth}\n\\mathcal{W}_{\\mathrm{br}}^2 = \\frac{r_{20}}{E_0} + \\frac{r_{02}}{E_0}\n- \\left( \\frac{r_{10}}{E_0} \\right)^2 -  \\left( \\frac{r_{01}}{E_0} \\right)^2,\n\\end{eqn}\nand $r_{10}$, $r_{01}$, $r_{20}$ and $r_{02}$ are defined by\n\\addtolength{\\arraycolsep}{0.5cm}\n\\begin{displaymath}\n\\begin{array}{c c}\nr_{10}=\\sum_{m,n} m e_{m,n},\\quad&r_{20}=\\sum_{m,n}m^2 e_{m,n},\\\\[8pt]\nr_{01}=\\sum_{m,n} hn e_{m,n},&r_{02} = \\sum_{m,n} h^2n^2 e_{m,n}. \n\\end{array}\n\\end{displaymath}\n\\addtolength{\\arraycolsep}{-0.5cm}\nThe variation in $\\mcal{W}_{\\mathrm{br}}$ over time gives a measure of the distortion \nsuffered by a breather.\n\n\n \\subsection{Stationary breather in a lattice with symmetric potential}\n\\label{hnstsy}\n\n\nWe investigate stationary breather solutions in the anomalous dispersion\nregime which corresponds to $b>0$ (see \\eref{hexgennls}).  We set \n$k=\\pi\/3$ and $l=\\pi\/h$, corresponding to $\\mathbf{k_1}$ in \\Fref{ellcontnum}, \nso that the velocities $u$ and $v$ are zero.   Davydova's result \nimplies that we expect to find stable soliton solutions when $PK>0$.  \n{}From \\eref{davydova}, $P=\\varepsilon^2\/2$, hence this inequality implies \n$40d > 27b^2$.  Hence we choose $b=d=1$ as well as $N=30$, $\\varepsilon=0.2$ \nand $\\lambda=1$.  Using the technique outlined in the appendix we \ncalculate $\\alpha=1.0212$, $\\beta=1.8670$.  The breather frequency is \n$\\omega+\\varepsilon^2\\lambda=3.040$, and therefore the period of oscillation is $T=2.0668$. \n\n\nAs is the case for all our simulations, the initial profile of the breather \nis located at the centre of the lattice, as illustrated in  \\Fref{simh1}\\,(a). \nAt $t=0$, the breather energy is $E_0=0.7606$, and $\\mcal{W}_{\\mathrm{br}}=3.74$.\nThe asymptotic estimate for $E_0$ given by \\eref{henar} is 0.7523, \nwhich is only 1\\% different from the numerically obtained value.\nIn \\Fref{simh1}\\,(c) we show the breather after 30 oscillations.  \nPlots of the local energy $e_{m,n}$ at $t=0$\nand $t=30T$ are presented in Figures \\ref{simh1}\\,(b) and\n\\ref{simh1}\\,(d).  After thirty oscillations the breather has shed \na small amount of energy, which is manifested as small amplitude \nradiation throughout the lattice.  Accordingly, the breather \nappears a little distorted in shape compared to its initial profile.  \nIn particular, at $t=30T$, we find $\\mcal{W}_{\\mathrm{br}}=4.21$. The energy of the \nbreather at $t=30T$ is 0.7087, giving $\\Delta E_0\/E_0=-0.0682$, \n\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[Profile at $t=0$.]{\n\\includegraphics[width=.45\\textwidth]{hs1a.eps}}\n\\hspace{.3in}\n\\subfigure[Plot of $e_{m,n}$, $E_0=0.7606$.]{\n\\includegraphics[width=.45\\textwidth]{hs1aen.eps}}\\\\\n\\vspace{.3in}\n\\subfigure[Profile at $t=30T=62.82$.]{\n\\includegraphics[width=.45\\textwidth]{hs1b.eps}}\n\\hspace{.3in}\n\\subfigure[Plot of $e_{m,n}$, $E_0=0.7087$.]{\n\\includegraphics[width=.45\\textwidth]{hs1ben.eps}}\n\\caption{Stationary breather in a lattice with symmetric potential, \nsee Section \\ref{hnstsy} for details.}\n\\label{simh1}\n\\end{figure}\n\n\n\\subsection{Breather moving along a lattice direction ($\\Psi=0\\mbox{$^\\circ$}$)}\n\\label{hpsi0}\n\n\nIn \\Fref{simh2}, we show a simulation of a breather moving along a\nlattice direction parallel to the $m$-axis, that is, $\\Psi=0\\mbox{$^\\circ$}$.  We \nhave chosen $\\mathbf{k} = \\mathbf{k_a} = [1.4,\\pi\/h]^T$ so that $u=0.4445$ \nand $v=0$ and thus $\\Psi=\\tan^{-1}(v\/u) = 0\\mbox{$^\\circ$}$.  The breather \nfrequency is $\\omega=2.9265$, and hence the period $T=2.1470$. Following \nthe calculation outlined in the appendix, the amplitude and width \nparameters are $\\alpha=1.0339$ and $\\beta=1.8440$; the remaining \nparameters being $b=1$, $d=1$, $N=30$, $\\varepsilon=0.1$ and $\\lambda=1$.\n\n\nThe initial profile of the breather is shown in \\Fref{simh2}\\,(a), and at \nthis time, the calculated energy is $E_0=0.5537$, whilst the asymptotic \nestimate \\eref{henar} is $E_0=0.5522$.  It may be observed that the \nbreather is not radially symmetric. In fact, it is slightly elongated in the \ndirection parallel to the $m$-axis, that is, parallel to the direction of \nmotion. This is because the point corresponding to the wavevector \n$\\mathbf{k_a}$ is near the boundary of the region of ellipticity $\\mathcal{D}_1$ \nas illustrated in \\Fref{econt}. \n\n\n\\Fref{simh2}\\,(b) shows the breather at the later time of  $t=21.58T=46.3513$, \nat which time we find $E_0=0.5562$.  By this time, the breather has reached \nthe right-hand edge of the lattice and, owing to periodic boundary conditions, \nit reemerges from the left-hand side as shown in \\Fref{simh2}\\,(c). The \nbreather remains localised, without spreading, though it leaves behind a \nsmall amount of energy in its path (visible in Figures \\ref{simh2}\\,(b) and \n\\ref{simh2}\\,(c)).   The computed energy changes much less than for the \nprevious simulation (here, $\\Delta E_0\/E_0=0.0287$). \n\n\nThe velocity of the breather can be measured from a plot of the energy \n$e_{m,n}$, which is shown in \\Fref{simh2}\\,(d), viewed from directly \nabove the plane of the lattice.  From this plot, we note that  the \nbreather has travelled 42 units at an average speed of 0.42 units per \nsecond, which is 5.5\\% lower than our predicted speed of 0.4445 units \nper second.  This is consistent with a small amount of energy being \nshed as the system transforms from our approximated initial conditions \ninto the precise shape of the breather.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[Profile at $t=0$, $E_0=0.5537$.]{\n\\includegraphics[width=.45\\textwidth]{hs2a.eps}}\n\\hspace{.3in}\n\\subfigure[Profile at $21.58T$, $E_0=0.5562$.]{\n\\includegraphics[width=.45\\textwidth]{hs2b.eps}}\\\\\n\\vspace{.3in}\n\\subfigure[Profile at $t=46.59T$, $E_0=0.5696$.]{\n\\includegraphics[width=.45\\textwidth]{hs2c.eps}}\n\\hspace{.3in}\n\\subfigure[Plot of $e_{m,n}$.]{\n\\includegraphics[width=.45\\textwidth]{hs2cen.eps}}\n\\caption{Breather moving along a lattice direction, $\\Psi=0\\mbox{$^\\circ$}$, \nsee Section \\ref{hpsi0} for details.}\n\\label{simh2}\n\\end{figure}\n\n\n\\subsection{Breather moving at $\\Psi=210\\mbox{$^\\circ$}$} \\label{hpsi210}\n\n\nWe now show that it is possible to simulate breathers moving in\ndirections other than a lattice direction (that is, $\\Psi \\neq 0\\mbox{$^\\circ$}$).  \nWe set $k=0.79$ and $l=1.7324$, which corresponds to $\\mathbf{k_b}$ in\n\\Fref{ellcontnum}.  It may be verified that $u=-0.1999$ and\n$v=-0.1154$ units per second, leading to $\\Psi = 210\\mbox{$^\\circ$}$ as \nrequired.  Although not a lattice vector, this direction is an axis of \nsymmetry of the lattice. For the wavevector $\\mathbf{k_b}$, we have \n$\\omega=2.9675$ and\nhence $T=2.1174$,  the remaining parameters being $b=1$, $d=1$,\n$N=30$, $\\varepsilon=0.1$ and $\\lambda=1$.  The variational parameters\n$\\alpha$ and $\\beta$ are 1.0267 and 1.8568 respectively.  The\nbreather is shown at times $t=25$, 50, 75 and 100 seconds in\n\\Fref{simh6}.  Clearly, the breather does not deform significantly\nas it travels, nor does it radiate much energy.  The initial energy is \ncomputed to be $E_0=0.6306$, and at $t=100$, the energy is 0.6275, \na loss of 0.5\\%.   The asymptotic estimate for the energy is \n$E_0=0.6184$ --- 2\\% different from the numerically computed value.\nThe motion of the breather is charted in \\Tref{tsimh6}, the final \nmeasured values for the velocities $u$ and $v$ give an average \nspeed of 0.2152 units per second, 6.8\\% below the predicted \nspeed of 0.2308 units per second. The angle of travel is \nalmost identical to the expected value of $\\Psi=210\\mbox{$^\\circ$}$.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[Profile at $t=25$, $E_0 = 0.6295$.]{\n\\includegraphics[width=.45\\textwidth]{hs6a.eps}}\n\\hspace{.3in}\n\\subfigure[Profile at $t=50$, $E_0 = 0.6292$.]{\n\\includegraphics[width=.45\\textwidth]{hs6b.eps}}\\\\\n\\vspace{.3in}\n\\subfigure[Profile at $t=75$, $E_0 = 0.6235$.]{\n\\includegraphics[width=.45\\textwidth]{hs6c.eps}}\n\\hspace{.3in}\n\\subfigure[Profile at $t=100$, $E_0 = 0.6275$.]{\n\\includegraphics[width=.45\\textwidth]{hs6d.eps}}\n\\caption{Breather moving at $\\Psi=210\\mbox{$^\\circ$}$, \nsee Section \\ref{hpsi210} for details.}\n\\label{simh6}\n\\end{figure}\n\n\n\\setlength{\\arrayrulewidth}{1pt}\n\\begin{table}\n\\begin{center}\n\\begin{tabular}[htbp]{| c | c | c | c | c | c | c |} \\hline \n{\\scriptsize{Time}} & {\\scriptsize{Horizontal}} & {\\scriptsize{Vertical}} & \n{\\scriptsize{Average horizontal}} & {\\scriptsize{Average vertical}} & {} & {} \\\\[-8pt]  \n{\\scriptsize{(s)}} & {\\scriptsize{displacement}} & {\\scriptsize{displacement}} & \n{\\scriptsize{velocity (units s$^{-1}$)}} & {\\scriptsize{velocity (units s$^{-1}$)}} & \n{\\scriptsize{$\\tan\\Psi$}} & {\\scriptsize{$\\Psi$}} \\\\  \\hline \n{\\scriptsize{$25$}} & {\\scriptsize{$-3.5$}}  & {\\scriptsize{$-2$}}  & {\\scriptsize{$-0.14$}}  & {\\scriptsize{$-0.08$}}  & {\\scriptsize{$0.5714$}} & {\\scriptsize{$209.74\\mbox{$^\\circ$}$}} \\\\ \\hline \n{\\scriptsize{$50$}} & {\\scriptsize{$-8$}}  & {\\scriptsize{$-4.5$}}  & {\\scriptsize{$-0.16$}}  & {\\scriptsize{$-0.09$}}  & {\\scriptsize{$0.5625$}} & {\\scriptsize{$209.36\\mbox{$^\\circ$}$}} \\\\ \\hline \n{\\scriptsize{$75$}} & {\\scriptsize{$-13$}}  & {\\scriptsize{$-7.5$}}  & {\\scriptsize{$-0.1733$}}  & {\\scriptsize{$-0.1$}}  & {\\scriptsize{$0.5770$}} & {\\scriptsize{$209.99\\mbox{$^\\circ$}$}} \\\\ \\hline \n{\\scriptsize{$100$}} & {\\scriptsize{$-18.5$}}  & {\\scriptsize{$-11$}}  & {\\scriptsize{$-0.185$}}  & {\\scriptsize{$-0.11$}}  & {\\scriptsize{$0.5946$}} & {\\scriptsize{$210.74\\mbox{$^\\circ$}$}} \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Summary of breather motion ($\\Psi=210\\mbox{$^\\circ$}$).}\n\\label{tsimh6}\n\\end{table}\n\n\n\\subsection{Breather moving at $\\Psi=130\\mbox{$^\\circ$}$} \\label{hpsi130}\n\n\nWe have presented simulations of breathers which move along axes of \nsymmetry of the lattice.  In \\Sref{hexdell}, from the results of our asymptotic \nanalysis, we found that breather solutions could be constructed for any \ndirection of travel.  In this section, we test the mobility of breathers along \ndirections which do not correspond to axes of symmetry of the lattice.\nWe have successfully propagated breathers in a range of directions, \nand here we show only one such breather, moving at an angle\n$\\Psi=130\\mbox{$^\\circ$}$.  We set $k=0.8$ and $l=1.9987$, from which we \nfind  $u=-0.2160$, $v=0.2575$, $\\Psi = 130\\mbox{$^\\circ$}$, $\\omega=2.9502$ \nand the period is $T=2.1298$.  The remaining parameters are $b=1$, \n$d=1$, $N=30$, $\\varepsilon=0.1$, $\\lambda=1$, $\\alpha=1.0297$ and \n$\\beta=1.8514$.  From Figure \\ref{simh7}, \nEven at 120 seconds, the breather remains localised without suffering\nappreciable degradation nor is much radiation left behind in its wake.   \nInitially the energy $E_0$ is computed as 0.5977, only 0.2\\% different \nfrom the asymptotic estimate of $E_0=0.5965$.   \nAfter 120 seconds, $E_0$ is 0.5897, a change of 1.3\\%.  \nTo find the velocity of the breather, we have recorded its motion\nand presented relevant data in \\Tref{tsimh7}.  From (\\ref{velsuv}), \nthe average speed is predicted to be 0.3254 units per second; \nwhilst our numerical simulation shows an average speed of 0.3361 \nunits per second,  3.2\\% lower than the expected speed.  The \ndirection of motion of the breather is predicted accurately.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[Profile at $t=30$, $E_0 = 0.5961$.]{\n\\includegraphics[width=.45\\textwidth]{hs7a.eps}}\n\\hspace{.3in}\n\\subfigure[Profile at $t=60$, $E_0 = 0.5966$.]{\n\\includegraphics[width=.45\\textwidth]{hs7b.eps}}\\\\\n\\vspace{.3in}\n\\subfigure[Profile at $t=90$, $E_0 = 0.5939$.]{\n\\includegraphics[width=.45\\textwidth]{hs7c.eps}}\n\\hspace{.3in}\n\\subfigure[Profile at $t=120$, $E_0 = 0.5897$.]{\n\\includegraphics[width=.45\\textwidth]{hs7d.eps}}\n\\caption{Breather moving at $\\Psi=130\\mbox{$^\\circ$}$, see Section \n\\ref{hpsi130} for details.}\n\\label{simh7}\n\\end{figure}\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}[htbp]{| c | c | c | c | c | c | c |} \\hline \n{\\scriptsize{Time}} & {\\scriptsize{Horizontal}} & {\\scriptsize{Vertical}} & \n{\\scriptsize{Average horizontal}} & {\\scriptsize{Average vertical}} & {} & {} \\\\[-8pt] \n{\\scriptsize{(s)}} & {\\scriptsize{displacement}} & {\\scriptsize{displacement}} & \n{\\scriptsize{velocity (units s$^{-1}$)}} & {\\scriptsize{velocity (units s$^{-1}$)}} & \n{\\scriptsize{$\\tan\\Psi$}} & {\\scriptsize{$\\Psi$}} \\\\ \\hline \n{\\scriptsize{$30$}} & {\\scriptsize{$-6$}}  & {\\scriptsize{$7$}}  & {\\scriptsize{$-0.2$}}  & {\\scriptsize{$0.2333$}}  & {\\scriptsize{$-1.1667$}} & {\\scriptsize{$130.60\\mbox{$^\\circ$}$}} \\\\ \\hline \n{\\scriptsize{$60$}} & {\\scriptsize{$-11$}}  & {\\scriptsize{$13.5$}}  & {\\scriptsize{$-0.1833$}}  & {\\scriptsize{$0.225$}}  & {\\scriptsize{$-1.2273$}} & {\\scriptsize{$129.17\\mbox{$^\\circ$}$}} \\\\ \\hline \n{\\scriptsize{$90$}} & {\\scriptsize{$-17.5$}}  & {\\scriptsize{$21.5$}}  & {\\scriptsize{$-0.1944$}}  & {\\scriptsize{$0.2389$}}  & {\\scriptsize{$-1.2286$}} & {\\scriptsize{$129.14\\mbox{$^\\circ$}$}} \\\\ \\hline  \n{\\scriptsize{$120$}} & {\\scriptsize{$-25$}}  & {\\scriptsize{$30$}}  & {\\scriptsize{$-0.2083$}}  & {\\scriptsize{$0.25$}}  & {\\scriptsize{$-1.2$}} & {\\scriptsize{$129.81\\mbox{$^\\circ$}$}} \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Summary of breather motion ($\\Psi=130\\mbox{$^\\circ$}$) \nsee Section \\ref{hpsi130} for details.}\n\\label{tsimh7}\n\\end{table}\n\n\n\\subsection{Stationary breather in a lattice with asymmetric potential} \n\\label{hnra}\n\n\nIn Sections \\ref{hnstsy}--\\ref{hpsi130}, we have shown simulations\nof lattices with a symmetric potential, that is, with $a$ and $c$ not \nnecessarily zero in \\eref{heqqfo}.   We now consider the more general \ncase for which the interaction potential is asymmetric ($a\\neq0 \\neq c$).  \nTo illustrate a stationary breather, we generate initial data using \n\\eref{hexqasym},  with the wavevector $\\mathbf{k} = \\mathbf{k_1} = \n[\\pi\/3,\\pi\/h]^T$.  As discussed in \\Sref{hexasymm}, anomalous dispersion \ncorresponds to $b>10a^2\/9$, hence we choose $a=1$, $b=2.5$, $c=0$ \nand $d=1$.  The remaining parameter values are $N=30$, $\\varepsilon=0.1$, \n$\\lambda=1$, hence $\\omega=3$, $T=2.0668$, $\\alpha=0.8665$ and $\\beta=1.8670$. \n\n\nThe breather is shown in \\Fref{simh5} after $10$, $30$ and $40$ full \noscillations.  Initially, we find the breather's width and energy are  \n$\\mcal{W}_{\\mathrm{br}}=7.46$ and $E_0=0.5429$, the latter being only 0.26\\% different \nfrom the asymptotic estimate \\eref{henex} of 0.5416.   The accompanying \nplots of $e_{m,n}$ demonstrate clearly that the breather preserves its \nform and remains localised, even after 80 seconds.  At $t=40T$, we find \nthat $\\mcal{W}_{\\mathrm{br}}=6.83$, a narrowing of the breather by 8\\%.  Also, at $t=40T$, \nthe numerically computed value of the energy $E_0=0.5371$, showing \nthat the energy does not fluctuate significantly.  \n\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[$t=10T=20.69$.]{\n\\includegraphics[width=.45\\textwidth]{hs5b.eps}}\n\\hspace{.3in}\n\\subfigure[Plot of $e_{m,n}$, $E_0=0.5453$.]{\n\\includegraphics[width=.45\\textwidth]{hs5ben.eps}} \\\\\n\\vspace{.3in}\n\\subfigure[$t=30T=41.34$.]{\n\\includegraphics[width=.45\\textwidth]{hs5c.eps}}\n\\hspace{.3in}\n\\subfigure[Plot of $e_{m,n}$, $E_0=0.5425$.]{\n\\includegraphics[width=.45\\textwidth]{hs5cen.eps}} \\\\\n\\vspace{.3in}\n\\subfigure[$t=40T=82.67$.]{\n\\includegraphics[width=.45\\textwidth]{hs5d.eps}}\n\\hspace{.3in}\n\\subfigure[Plot of $e_{m,n}$, $E_0=0.5371$.]{\n\\includegraphics[width=.45\\textwidth]{hs5den.eps}}\n\\caption{Stationary breather in a lattice with an asymmetric potential, \nsee Section \\ref{hnra} for details.}\n\\label{simh5}\n\\end{figure}\n\n\n\\section{Discussion} \\label{hdisc}\n\n\nIn this paper, we have found approximations to discrete breathers\nin a two-dimensional hexagonal FPU lattice with a scalar-valued \nfunction at each node.  We have shown that\nthe lattice equations can be reduced to a cubic NLS equation for\ntwo special cases.  In \\Sref{hexsym}, we obtained moving\nbreathers when the interaction potential is symmetric; in\nthis case an ellipticity criterion for the wavevector is found.\nIn the appendix we summarise our technique for approximating \nsoliton solutions of the two-dimensional NLS equation. \n\n\nIn \\Sref{hexasymm}, we considered lattices with an asymmetric \npotential, in which case a reduction to NLS could only be performed\nfor stationary breathers.   \nThe theoretical methods employed here are similar to those used on \nthe square lattice of \\citep{buts06}; however,  several important \ndifferences emerge in the course of the analysis.  \nWe found that the anomalous dispersion regime corresponds to \n$b>4a^2\/3$ in the square lattice and $b>10a^2\/9$ in the hexagonal.  \nFurthermore we find that stationary breathers involve the \ngeneration of second harmonics in the hexagonal lattice \n($G_2=aF^2\/3$), whereas they are suppressed in the square \nlattice ($G_2=0$).  In both lattices, the quadratic nonlinearity \ngenerates a small amplitude slowly-varying mode ($G_0=-a|F|^2$). \n\n\nFor symmetric interactions we found an associated ellipticity constraint: \nmoving breathers only occur for certain wavevectors, this means that \nall breathers have (i) a relatively high frequency, and (ii) a maximum \nspeed, which depends on the wave vector and hence on the \ndirection of travel, (iii) a threshold energy which also depends on \nwavevector and hence is related to speed and direction of travel.  \nWe find that the threshold energy is lower for breathers at the \nedge of the domain of ellipticity in wavevector space. These breathers \nhave lower frequencies and faster speeds.  The fastest-moving \nbreathers are restricted to moving along lattice directions. \n\n\nWe also presented asymptotic estimates for the breather energy in\nSections \\ref{hexsym} and \\ref{hexasymm}.  As expected, we found a\nminimum energy required to create breathers in the hexagonal\nlattice.  The threshold energy for moving breathers is smaller\nthan that required for stationary breathers and becomes vanishingly\nsmall at the boundary of the domain of ellipticity.\n\n\nIn \\Sref{hex5}, we extended the small amplitude expansion to fifth-order \nand derived a higher-order equation which more correctly describes \nthe shape and stability properties of the breather envelope.  We \nobtained a generalised NLS equation \\eref{hexgennls} with a variety \nof perturbation terms, some of which are known to be stabilising.  This \nequation is slightly simpler than the corresponding equation obtained \nfor the square lattice, namely (3.11) of \\citep{buts06}.  In particular, \nthe higher-order dispersive terms in \\eref{hexgennls} are isotropic, \nreflecting the hexagonal rotational symmetry of the lattice.  For stationary \nbreathers in the case of a symmetric potential we find that the cubic \nnonlinearity does not give rise to third harmonics, that is, $H_3=0$ in \nthe hexagonal lattice, in contrast to $H_3=bF^3\/8$ in the square lattice.\n\n\nIn \\Sref{hexnumerics}, we illustrated  these breather modes, showing \nthat both stationary and moving breathers are long-lived.  The \nbreather profiles change little over time, a small amount of \nenergy is shed due to the initial conditions being only approximate.  \nWe have successfully propagated long-lived breathers moving in \ndirections which are not axes of symmetry of the lattice (for \ninstance, $\\Psi=130\\mbox{$^\\circ$}$),  suggesting that there is no absolute \nrestriction upon the direction of travel; this is in contrast to the \nobservations reported by Marin \\etal \\citep{mar00, mar98} for \nmechanical (two-component) two-dimensional lattices, who could \nonly find breathers which travelled along axes of symmetry of the lattice. \n\n\n\\ack\n\nIAB would like to thank both the UK Engineering and Physical Sciences \nResearch Council for financial assistance, and also Qamran Yaqoob for his \nassistance in the preparation of several diagrams that appear in this paper. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPermutations and multipermutations as information representation formats\nhave a long history, with early applications in communication theory\ndating back to the work of Slepian~\\cite{slepian1965permutationModulation},\nwho proposed using multipermutation codes for transmission in the\npresence of additive white Gaussian noise. More recently, Vinck proposed\nusing permutation codes in the Hamming metric for combatting impulse\nnoise and permanent frequency noise in power grids~\\cite{vinckCodedModulation}.\nPermutation codes have received renewed interest in the past few years\ndue to their promising application in storage systems, such as flash\nmemories~\\cite{bruck2009rank-modulation,barg2012codes,engad2011constantWeightGrayCodes}. \n\nFlash memories are nonvolatile storage units (i.e., storage units\nthat remain operational when unpowered), and are usually used for\narchival or long-term storage. Information is organized in blocks\nof cells, all of which have to be processed jointly during information\nerasure cycles. The gist of the approach underlying permutation coding\nin flash memories, which uses the fact that the memories consist of\nspecially organized cells storing charges, is that information is\nrepresented via the relative order of charge levels of cells rather\nthan their absolute charge levels~\\cite{bruck2009rank-modulation}.\nThis approach, termed \\emph{rank modulation}, alleviates the problems\nof cell over-injection, reduces the need for block erasures, and is\nmore robust to errors caused by charge leakage~\\cite{bruck2009rank-modulation}.\nFor instance, while \\textit{all} absolute values are subject to errors\ncaused by charge leakage, the relative ordering of the quantitative\ndata may remain largely unchanged~\\cite{Bruck}. The modeling assumption\nbehind rank modulation is that only errors swapping adjacently ranked\ncell charges are likely~\\cite{Bruck,barg2012codes}. As a result,\ncode design for flash memories was mainly performed in the domain\nof the Kendall \\textgreek{t} metric, which accounts for small magnitude\nerrors causing swaps of adjacent elements. A thorough treatment of\ncodes in the Kendall metric may be found in~\\cite{barg2012codes}\nand references therein.\n\nIn contrast, a more general error model was proposed by the authors\nin~\\cite{farnoud2013error-correction}, based on the observation\nthat increasing the number of charge levels in order to increase capacity\ndecreases the difference between adjacent charge levels and thus unwanted\nvariations in the charge of a cell may cause its rank to rise above\nor fall below the ranks of several other cells instead of only swapping\ntwo adjacent ranks. In addition, the proposed \\textit{translocation\nerror} model adequately accounts for more general types of error such\nas read-disturb and write-disturb errors. In this context, the distance\nmeasure of interest is the Ulam distance, related to the length of\nthe longest common subsequence of two permutations and consequently,\nthe deletion\/insertion or edit distance~\\cite{levenshtein_perfect}.\nThe Ulam distance has also received independent interest in the bioinformatics\nand the computer science communities for the purpose of measuring\nthe ``sortedness'' of data~\\cite{Gopalan2007Estimatingsortedness}.\nOther metrics used for permutation code construction include the Hamming\ndistance~\\cite{vinckCodedModulation,chu2004constructions} and the\nChebyshev distance (the $\\ell_{\\infty}$ metric)~\\cite{klove2010PermutationArrays,schartz2010correctinglimitedmagnitude}.\n\nMultipermutation codes are a generalization of permutation codes where\neach message is encoded as a permutation of the elements of a multiset.\nMultipermutation codes in the Hamming metric, known as \\emph{constant\ncomposition codes }or \\emph{frequency permutation array}s (FPAs),\nwere studied in several papers including~\\cite{Luo2003CCCZq,Ding2005CombinatorialCCC,Huczynska2006FrequencyPermutationArrays,Chu2006OnCCCs}.\nFor nonvolatile memories, multipermutation coding was proposed by\nEn Gad et al.~\\cite{engad2012trad-offs}, as well as by Shieh and\nTsai~\\cite{Shieh2010Decoding}. These works were motivated by different\nconsiderations -- the former aiming to increase the number of possible\nre-writes between block erasures, and the latter focusing on the advantages\nof multipermutation coding with respect to cell leakage, over-injection\nissues, and charge fluctuations. In addition, multipermutation codes\nwere also recently reported for the Chebyshev distance in~\\cite{Shieh2010Decoding,Shieh2011Computingtheball}\nand for the Kendall \\textgreek{t} distance in~\\cite{buzaglo2013error,sala2013dynamic}.\n\nHere, we continue our study of codes in the Ulam metric for nonvolatile\nmemories by extending it to the level of multipermutation codes. Our\nresults include bounds on the size of the largest multipermutation\ncodes, code constructions using multipermutation codes in the Hamming\nmetric and interleaving as well as permutation codes in the Ulam metric~\\cite{farnoud2013error-correction,farnoud2012translocation}.\nIn the process of analyzing these schemes, we establish new connections\nbetween resolvable balanced incomplete block designs (RBIBDs)~\\cite{rumov1976existence,khare1981simple},\nsemi-Latin squares~\\cite{Preece1983SemiLatin}, and multipermutation\ncodes in the Ulam metric.  As multipermutation codes in the Hamming\nmetric are used in our constructions, we also provide new bounds on\nthe size of these codes, and find their asymptotic capacity. In addition,\nour results include simple decoding schemes for the proposed constructions\nbased on designs and those based on interleaving permutation codes\nin the Ulam metric.\n\nThe paper is organized as follows. In Section~\\ref{sec:Preliminaries-and-Notation},\nwe present the notation used throughout the paper as well as formal\ndefinitions regarding multipermutation codes. In addition, this section\nincludes motivating examples for our work. Section~\\ref{sec:bounds}\nis devoted to bounds on the size of multipermutation codes in the\nUlam and Hamming metrics, as well as to the computation of the asymptotic\ncapacity of these codes. Section~\\ref{sec:Constructions} provides\nconstructions for codes in the Ulam metric. We conclude the paper\nin Section~\\ref{sec:Conclusion} with a summary of our results and\na number of remarks.\n\n\n\\section{Preliminaries and Notation\\label{sec:Preliminaries-and-Notation}}\n\n\\begin{figure*}\n\\captionsetup[subfigure]{labelformat=empty}\\subfloat[$\\mathfrak{m}=(2,1,1,2),$ $\\mathfrak{o}=\\left(\\{2,3\\},\\{1,4\\}\\right)$]{\\includegraphics[width=0.5\\textwidth]{multi-perms_2levels1}\n\n}\\subfloat[$\\mathfrak{m}=(1,2,1,2),$ $\\mathfrak{o}=\\left(\\{1,3\\},\\{2,4\\}\\right)$]{\\includegraphics[width=0.5\\textwidth]{multi-perms_2levels2}\n\n}\n\n\\caption{The two equivalence classes of the code given in~(\\ref{eq:ex-mpcode}).\nThe numbers on the top of the bars indicate cell indices, while the\nheights of the bars represent the charge levels of the indicated cells.\nThe equivalence classes on the left and right correspond to the multipermutations\n$\\mathfrak{m}=(2,1,1,2)$ and $\\mathfrak{m}=(1,2,1,2)$, respectively. Note that each\nmultipermutation can be programmed into the memory as four different\npermutations, each representing a complete ranking of cell charges\nwithout ties.}\n\\label{fig:mp-code-example}\n\\end{figure*}\n\n\n\n\\subsection{Multipermutations, ordered set partitions, and codes}\n\nFor an integer $k$, let $[k]=\\{1,\\dotsc,k\\}$. Furthermore, let $\\mathbb{S}_{n}$\ndenote the symmetric group of order $n!$, i.e., the set of permutations\nof $n$ distinct elements (typically the elements of $[n]$).\n\nA multipermutation is an arrangement of the elements of a multiset.\nFor example, $(2,1,2,3,1,2)$ is a multipermutation of $\\{1,1,2,2,2,3\\}$.\nFor a positive integer $n$ and a multiplicity vector $\\vec{r}=(r_{1},\\dotsc,r_{m})$,\nsuch that $n=\\sum_{i=1}^{m}r_{i}$, we use $\\ms n{\\vec{r}}$ to denote\nthe multiset \n\\[\n\\{\\underbrace{1,\\dotsc,1}_{r_{1}},\\underbrace{2,\\dotsc,2}_{r_{2}},\\dotsc,\\underbrace{m,\\dotsc,m}_{r_{m}}\\}.\n\\]\nA multiset that has $r$ copies of each of its elements is termed\nan \\emph{$r$-regular multiset}. For brevity, we henceforth denote\nthe $r$-regular multiset $\\ms n{(r,\\cdots,r)}$ by $\\ms nr$. An\n\\emph{$r$-regular multipermutation} is a permutation of an $r$-regular\nmultiset. Throughout the paper, we focus on $r$-regular multipermutations.\nMany of the subsequently described results, however, can easily be\nextended to multipermutations of $\\ms n{\\vec{r}}$ for general multiplicity\nvectors $\\vec{r}$.\n\nLet $n$ also denote the number of cells in a block of a flash memory.\nWe assume that $r$ is a positive integer that divides $n$. Consider\na permutation $\\pi\\in\\mathbb{S}_{n}$ that lists the cells in decreasing\norder of charge. For example, $\\pi=(3,2,4,1)$ means that cell $3$\nhas the highest charge, cell 2 has the second-highest charge, and\nso on. The inverse of $\\pi$ is the vector of the ranks of the cells,\n$\\pi^{-1}=(4,2,1,3)$; cell $1$ has rank $4$, cell $2$ has rank\n$2$, and so on. \n\nTo obtain an $r$-regular multipermutation of cell rankings, instead\nof assigning rank $i$ to the element in position $i$ in $\\pi$,\nwe assign rank $i$ to all the elements in positions $\\{(i-1)r+1,\\dotsc,ir\\}$.\nThis multipermutation is denoted by $\\mlp r{\\pi}$, where \n\\[\n\\mlp r{\\pi}(j)=i\\mbox{, iff }(i-1)r+1\\le\\pi^{-1}(j)\\le ir.\n\\]\nFor instance, given $n=4,r=2$, and $\\pi=(3,2,4,1)$, we have $\\mlp 2{\\pi}=(2,1,1,2)$. \n\nObserve that for $\\pi\\in\\mathbb{S}_{n}$, $\\mlp r{\\pi}$ is a multipermutation\nof $\\ms nr$ and that for $r=1$, the multipermutation $\\mlp r{\\pi}$\nreduces to the inverse of $\\pi$, i.e., $\\mlp 1{\\pi}=\\pi^{-1}$. \n\nA multipermutation $\\mathfrak{m}$ of $\\ms nr$ can also be represented as\nan ordered set partition $\\mathfrak{o}$, where the $i$th part of $\\mathfrak{o}$ is\nthe set \n\\[\n\\mathfrak{o}(i)=\\left\\{ j:\\mathfrak{m}(j)=i\\right\\} .\n\\]\nThis definition can naturally be extended to multipermutations of\nother multisets. For $\\pi\\in\\mathbb{S}_{n}$, let $\\orp r{\\pi}$ be\nan ordered set partition where \n\\[\n\\orp r{\\pi}(i)=\\left\\{ j:\\mlp r{\\pi}(j)=i\\right\\} .\n\\]\nFor the aforementioned example $\\pi=(3,2,4,1)$, we have $\\orp 2{\\pi}=(\\{2,3\\},\\{1,4\\})$.\n\nFor $\\pi,\\sigma\\in\\mathbb{S}_{n}$, we write $\\pi\\eqv r\\sigma$ if\n$\\mlp r{\\pi}=\\mlp r{\\sigma}$, and $\\pi\\noteqv r\\sigma$ otherwise.\nIt is easy to show that $\\eqv r$ is an equivalence relation. The\nequivalence class of permutations including $\\pi$ is denoted by $\\eqclass r(\\pi)$,\ni.e., \n\\[\n\\eqclass r(\\pi)=\\{\\sigma:\\sigma\\eqv r\\pi\\}.\n\\]\n\n\nAs an illustration, the equivalence class of $(3,2,4,1)$ under $\\eqv 2$\nequals \n\\begin{multline}\n\\eqclass 2((3,2,4,1))=\\\\\n\\{(3,2,4,1),(2,3,4,1),(3,2,1,4),(2,3,1,4)\\}.\\label{eq:2112}\n\\end{multline}\n In this case, the set $\\eqclass 2((3,2,4,1))$ is isomorphic to the\nsubgroup $\\mathbb{S}_{2}\\times\\mathbb{S}_{2}$ of $\\mathbb{S}_{4}$. \n\nLet $S$ be a set of size $n$. $ $An \\emph{$r$-regular multipermutation\ncode} $\\mpcode{}(n,r)$ over $S$ is a code $C$ whose codewords are\npermutations of $S$ with the property that for any $\\pi\\in C$, $\\eqclass r(\\pi)\\subseteq C$.\nFor example, \n\\begin{multline}\n\\{(2,3,1,4),(3,2,1,4),(2,3,4,1),(3,2,4,1),\\\\\n(1,3,2,4),(3,1,2,4),(1,3,4,2),(3,1,4,2))\\}\\label{eq:ex-mpcode}\n\\end{multline}\nis an $\\mpcode{}(4,2)$ code. We typically assume that $S=[n]$, but\nthe results hold for any set $S$ of size $n$.\n\nEach permutation in $C$ represents an ordering of cell charges. For\nexample $\\pi=(1,3,2,4)$ indicates that the cell 1 has the highest\ncharge, followed by cell 3, and so on. In multipermutation coding,\neach $r$ cells are assigned the same rank and all permutations corresponding\nto the same multipermutation encode the same information. In the previous\nexample, the multipermutation $\\left(2,1,1,2\\right)$ may be represented\nby any of the permutations on the right side of (\\ref{eq:2112}).\n\nAs a result, it is clear that an $\\mpcode{}(n,r)$ code $C$ can be\nrepresented as a set of multipermutations $\\Mlp rC$, where \n\\[\n\\Mlp rC=\\{\\mlp r{\\pi}:\\pi\\in C\\},\n\\]\nor as a set of ordered set partitions $\\Orp rC$, where\n\\[\n\\Orp rC=\\{\\orp r{\\pi}:\\pi\\in C\\}.\n\\]\nFor example, if $C$ is the code given in~(\\ref{eq:ex-mpcode}),\nwe have \n\\begin{align*}\n\\Mlp 2C & =\\{(2,1,1,2),(1,2,1,2))\\},\\\\\n\\Orp 2C & =\\left\\{ \\left(\\{2,3\\},\\{1,4\\}\\right),\\left(\\{1,3\\},\\{2,4\\}\\right)\\right\\} .\n\\end{align*}\nWith slight abuse of notation, for an $\\mpcode{}(n,r)$ $C$, we may\nuse $C$ to mean $\\Mlp rC$ or $\\Orp rC$ if doing so does not lead\nto ambiguity. Similarly, we may consider $C$ to be a set of multipermutations\nor a set of $ $ordered set partitions instead of a set of permutations.\n\nThe \\emph{cardinality} or the \\emph{size} of $C$, denoted by $|C|$,\nequals the number of multipermutations in $\\Mlp rC$, or equivalently,\nthe number of equivalence class of $C$ under the relation $\\eqv r$.\n\nIn what follows, we describe why multipermutation formats are suitable\nfor flash memory coding applications. We start with the readback process.\nTo be able to read the information stored in a flash memory, cells\nwith different ranks must have charge levels that differ by at least\na certain amount $\\Delta$, since if the difference between charge\nlevels of two cells is too small, it cannot be reliably decided which\none had the higher charge level. Hence, in permutation coding, to\nstore a permutation of length $n$, the range of possible charge values\nmust be at least $n\\Delta$ to allow for $n$ different charge levels\ncorresponding to $n$ different ranks. In contrast, an $r$-regular\nmultipermutation of length $n$ has only $n\/r$ ranks and thus it\ncan be stored in a flash memory whose range of possible charge level\nvalues is $n\\Delta\/r$. Specifically, the relative order of charge\nlevels of cells of the same rank of a multipermutation is irrelevant\nas all possibilities correspond to the same multipermutation, i.e.,\nthe same information message.\n\nNote that in order to store information represented by $r$-regular\nmultipermutations, charges are injected to achieve a desired multipermutation\nranking. As it is neither necessary nor possible for cells of the\nsame rank to have precisely the same charge levels, the actual representation\nof such a multipermutation $\\mathfrak{m}$ is some permutation $\\pi$, such\nthat $\\mlp r{\\pi}=\\mathfrak{m}$. The multipermutation is available to the\nuser retrieving information in the form of the cell charge ordering\n$\\pi$. As an illustration, consider Figure~\\ref{fig:mp-code-example}\nfor the code given in~(\\ref{eq:ex-mpcode}). For instance, to store\nthe multipermutation $(2,1,1,2)$, any of the permutations given in~(\\ref{eq:2112})\nmay be programmed into the memory. To retrieve the information, the\nuser reads the permutation, or possibly an erroneous copy of it, and\nperforms error correction to identify the multipermutation corresponding\nto the stored permutation.\n\nNext, we show how multipermutations can achieve a higher information\nrate compared to permutations. Consider a flash memory that can accommodate\n$m$ sufficiently spread charge levels. In a group of $m$ cells of\nsuch a device, one can store a permutation of length $m$. Suppose\n$r$ is a positive integer. It follows that in $mr$ cells, the number\nof possible messages that can be stored is $\\left(m!\\right)^{r}$. \n\nOn the same device, one can store an $r$-regular multipermutation\nof length $mr$ in $mr$ cells. In this case, the number of possible\nmessages equals to the number of possible multipermutations, i.e.,\n$\\frac{\\left(mr\\right)!}{\\left(r!\\right)^{m}}$. It is clear that\nfor $r\\ge2$, we have \n\\[\n\\frac{\\left(mr\\right)!}{\\left(r!\\right)^{m}}>\\left(m!\\right)^{r}.\n\\]\nHence, in this setting, more information messages can be stored if\none uses multipermutations instead of permutations.\n\nAs an illustration, suppose that $m=2$ and $r=10$. Using multipermutations,\nwe can store \n\\[\n\\lg\\frac{\\left(mr\\right)!}{\\left(r!\\right)^{m}}\\approx17.5\\ \\mbox{bits}\n\\]\nwhile using permutations, we can store \n\\[\n\\lg\\left(m!\\right)^{r}=10\\ \\mbox{bits}\n\\]\nin 20 cells. Note that here we considered the uncoded regime.\n\nThe saving in the number of possible charge levels can also be used\nto increase the number of possible re-writes before a block erasure\nbecomes necessary~\\cite{engad2012trad-offs}. As an example, suppose\nthat $5$ charge levels are available. If one uses permutations of\nlength $5$, it is only possible to write once before an erasure becomes\nnecessary, and if one uses permutations of length $3$, it is possible\nto write twice before an erasure. Encoding with $r$-regular multipermutations\nof length $3r$ also provides the ability to write twice before an\nerasure. While both methods, permutation coding and multipermutation\ncoding, allow for writing twice before an erasure, using multipermutations\nleads to a higher information storage rate. For further details on\nmultipermutation re-write codes, we refer the reader to~\\cite{engad2012trad-offs}.\n\n\\begin{figure*}\n\\begin{centering}\n\\captionsetup[subfigure]{labelformat=parens}\\subfloat[$\\mlp 2{\\sigma}=\\mlp 2{\\omega}=(3,1,2,1,2,3).$]{\\includegraphics[width=0.4\\textwidth]{multi-perms_2levels-small}\n\n}\\hspace*{2cm}\\subfloat[$\\mlp 2{\\sigma}=(3,1,2,1,2,3),\\ \\mlp 2{\\omega}=(2,1,2,3,1,3).$]{\\includegraphics[width=0.4\\textwidth]{multi-perms_2levels-large}\n\n}\n\\par\\end{centering}\n\n\\caption{Examples of errors: A small-magnitude charge drop error (a) manifests\nitself as a swap of adjacent ranks in the permutation while a large-magnitude\nerror (b) manifests itself as a translocation in the permutation.\nIn multipermutation coding, charge fluctuations may or may not lead\nto erroneous multipermutations. Assuming $r=2$, in (a) the multipermutation\ndoes not change, while in (b) it does.}\n\\label{fig:leakage-example}\n\\end{figure*}\n\n\nBefore proceeding with an analytical treatment of multipermutation\ncodes in the Ulam metric, we remark that throughout the paper, we\nuse $\\mathbb{Z}^{+}$ to denote the set of positive integers. Whenever\nit is clear from the context, we use the well-known result that $\\ln x!=x\\ln x+O(x)$,\nfor any nonnegative real value $x$. By convention, we adopt $0\\ln0=0$.\n\n\n\\subsection{The Multipermutation Hamming distance}\n\nFor an integer $r$, the \\emph{$r$-regular Hamming distance} (or\nsimply the Hamming distance) $\\hdist r$ between two permutations\n$\\pi,\\sigma\\in\\mathbb{S}_{n}$ is defined as \n\\[\n\\hdist r(\\pi,\\sigma)=\\left|\\left\\{ i:\\mlp r{\\pi}(i)\\neq\\mlp r{\\sigma}(i)\\right\\} \\right|.\n\\]\nIn words, the permutations are first converted into multipermutations,\nwhich are subsequently compared coordinate-wise. The distance $\\hdist r\\left(\\pi,\\sigma\\right)$\nis equivalent to the ordinary Hamming distance between $\\pi$ and\n$\\sigma$. Thus, instead of $\\hdist 1$, we write $\\hdist{}$. \n\nWe observe that \n\\begin{align*}\n\\hdist r(\\pi,\\sigma) & =\\sum_{i=1}^{n\/r}\\left(r-\\left|\\orp r{\\pi}(i)\\cap\\orp r{\\sigma}(i)\\right|\\right)\\\\\n & =\\sum_{i=1}^{n\/r}\\left(\\left|\\orp r{\\pi}(i)\\right|-\\left|\\orp r{\\pi}(i)\\cap\\orp r{\\sigma}(i)\\right|\\right)\\\\\n & =\\sum_{i=1}^{n\/r}\\left|\\orp r{\\pi}(i)\\backslash\\orp r{\\sigma}(i)\\right|.\n\\end{align*}\nFurthermore,\n\\begin{align*}\n\\hdist r(\\pi,\\sigma) & =\\min_{\\pi'\\in R(\\pi)}\\min_{\\sigma'\\in R(\\sigma)}\\hdist{}(\\pi',\\sigma').\n\\end{align*}\n\n\nLet $C$ be an $\\mpcode{}(n,r)$ code. The code $C$ has minimum Hamming\ndistance $d$ if for all $\\pi,\\sigma\\in C$ with $\\pi\\noteqv r\\sigma$,\nwe have $\\hdist r(\\pi,\\sigma)\\ge d$. Equivalently, since for all\n$\\pi\\in C$, $\\eqclass r(\\pi)$ is contained in $C$, the code $C$\nhas minimum Hamming distance $d$ if for all $\\pi,\\sigma\\in C$ with\n$\\pi\\noteqv r\\sigma$, it holds that $\\hdist{}(\\pi,\\sigma)\\ge d$.\nAn $\\mpcode{}(n,r)$ code with minimum Hamming distance $d$ is said\nto be an $\\mpcode H(n,r,d)$ code. The code given in~(\\ref{eq:ex-mpcode})\nis an $\\mpcode H(4,2,2)$ code.\n\n\n\\subsection{Translocation errors and the Ulam distance}\n\nFigure~\\ref{fig:leakage-example} illustrates examples of errors\nin flash memories that may occur due to charge leakage, read-disturb,\nand write-disturb~\\cite{Grupp2009Characterizing}. While errors with\nsmall magnitude represent swaps of adjacent ranks, errors with large\nmagnitude represent \\emph{translocations}~\\cite{farnoud2013error-correction}. \n\nA translocation $\\phi(i,j)$ is a permutation that is obtained from\nthe identity permutation $e$ by moving element $i$ to the position\nof $j$ and shifting all elements between $i$ and $j$, including\n$j$, by one~\\cite{farnoud2013error-correction}. For example, for\n$i<j$,\n\\[\n\\phi(i,j)=(1,\\dotsc,i-1,i+1,i+2,\\dotsc,j,i,j+1,\\dotsc,n).\n\\]\nAs a convention, we assume that $\\phi(i,i)=e$. A \\emph{translocation\nerror} is an error that changes a stored permutation $\\pi$ to $\\pi\\phi(i,j)$,\nwith $i\\neq j$. \n\nA\\emph{ subsequence} of a vector $x=\\left(x\\left(1\\right),x\\left(2\\right),\\dotsc,x\\left(n\\right)\\right)$\nis a sequence $\\left(x\\left(i_{1}\\right),x\\left(i_{2}\\right),\\dotsc,x\\left(i_{k}\\right)\\right)$,\nwhere $i_{1}<i_{2}<\\dotsm<i_{k}$ and $k\\le n$. A \\emph{common subsequence\n}of two vectors $x$ and $y$ is a sequence that is a subsequence\nof both $x$ and $y$. Let the length of the longest common subsequence\nof two permutations $\\pi$ and $\\sigma$ be denoted by $\\lcs{\\pi}{\\sigma}$.\nThe \\emph{Ulam distance $\\tdist{}\\left(\\pi,\\sigma\\right)$} between\ntwo permutations $\\pi$ and $\\sigma$ of length $n$ is defined as\n$ $$n-\\lcs{\\pi}{\\sigma}$. It is straightforward to see that the\nUlam distance between $\\pi$ and $\\sigma$ equals the minimum number\nof translocations required to take $\\pi$ to $\\sigma$~\\cite{farnoud2013error-correction}.\nIt is also well known that the Ulam distance represents the \\textit{edit\ndistance} between two permutations, i.e., the smallest number of insertion\/deletion\npairs needed to transform one permutation into another.\n\nFor $\\pi,\\sigma\\in\\mathbb{S}_{n}$, define the \\emph{($r$-regular)\nUlam distance} $\\tdist r$ on permutations as\n\\[\n\\tdist r(\\pi,\\sigma)=\\min_{\\pi'\\in R(\\pi)}\\min_{\\sigma'\\in R(\\sigma)}\\tdist{}(\\pi',\\sigma').\n\\]\nNote that this distance is a set-distance: it measures the smallest\nUlam distance between two permutations in different equivalence classes.\nFurthermore, the distance $\\tdist r(\\pi,\\sigma)$ equals the minimum\nnumber of translocations required to take a permutation in $\\eqclass r(\\pi)$\nto a permutation in $\\eqclass r(\\sigma)$. \n\nLet $\\closestpair r{\\pi}{\\sigma}$ denote the set \n\\[\n\\{(\\alpha,\\beta):\\alpha\\in\\eqclass r(\\pi),\\beta\\in\\eqclass r(\\sigma),\\tdist r(\\pi,\\sigma)=\\tdist{}(\\alpha,\\beta)\\}.\n\\]\nBy definition of $\\tdist r(\\pi,\\sigma)$, $\\closestpair r{\\pi}{\\sigma}$\nis nonempty.\n\nAn $\\mpcode{}(n,r)$ $C$ has minimum Ulam distance $d$ if for all\n$\\pi,\\sigma\\in C$ with $\\pi\\noteqv r\\sigma$, we have $\\tdist r(\\pi,\\sigma)\\ge d$.\nSuch a code is denoted by $\\mpcode{\\circ}(n,r,d)$. The code given\nin~(\\ref{eq:ex-mpcode}) is an $\\mpcode{\\circ}(4,2,1)$ code, as\n$\\tdist{}\\left((3,2,1,4),(3,1,2,4)\\right)=1.$\n\nUnder minimum distance decoding, an $\\mpcode{\\circ}(n,r,d)$ code\ncan correct $t$ translocation errors iff $d\\ge2t+1$. To see this,\nnote that $d<2t+1$ iff there exists $\\pi,\\sigma\\in C,\\pi\\noteqv r\\sigma,$\nand $\\omega\\in\\mathbb{S}_{n}$ such that $\\tdist{}(\\omega,\\pi)\\le t$\nand $\\tdist{}(\\omega,\\sigma)\\le t$, iff $C$ cannot correct $t$\nerrors.\n\nFor a set $P$ and a permutation $\\pi$, let $\\pi_{P}$ denote the\nprojection of $\\pi$ onto $P$, that is, the sequence obtained by\nonly keeping those elements of $\\pi$ that are in $P$. We find the\nfollowing lemma, proved in our companion paper~\\cite{farnoud2013error-correction},\nuseful in our subsequent derivations. \n\\begin{lem}\n\\label{lem:projection}For sets $P\\subseteq[n]$ and $Q=[n]\\backslash P$,\nand for permutations $\\pi,\\sigma\\in\\mathbb{S}_{n}$, we have \n\\[\n\\tdist{}(\\pi,\\sigma)\\ge\\tdist{}(\\proj{\\pi}P,\\proj{\\sigma}P)+\\tdist{}(\\proj{\\pi}Q,\\proj{\\sigma}Q).\n\\]\n\n\\end{lem}\n\n\\subsection{Relationship between the Ulam and the Hamming metrics}\n\nThe following lemma is an immediate consequence of the definition\nof a translocation.\n\\begin{lem}\n\\label{lem:Atranslocation}A translocation, applied to a permutation,\nchanges at most one element of each rank. That is, for a translocation\n$\\varphi$, a permutation $\\pi$, and $i\\in[n\/r]$, \n\\[\n\\left|\\orp r{\\pi}(i)\\cap\\orp r{\\pi\\varphi}(i)\\right|\\ge r-1.\n\\]\n\n\\end{lem}\nSince there are $n\/r$ ranks, we have\n\\[\n\\hdist r(\\pi,\\pi\\varphi)\\le\\frac{n}{r},\n\\]\nfor a translocation $\\varphi$ and a permutation $\\pi$. Hence, \n\\[\n\\hdist r(\\pi,\\sigma)\\le\\frac{n}{r}\\tdist r(\\pi,\\sigma)\n\\]\nfor $\\pi,\\sigma\\in\\mathbb{S}_{n}$.\n\nWe next upper bound $\\tdist r(\\pi,\\sigma)$ in terms of $\\hdist r(\\pi,\\sigma).$\nThere exist $\\pi'\\in\\eqclass r(\\pi)$, $\\sigma'\\in\\eqclass r(\\sigma)$,\nand a common subsequence of $\\pi'$ and $\\sigma'$ that contains the\nelements of \n\\[\n\\bigcup_{i=1}^{n\/r}\\left(\\orp r{\\pi}(i)\\cap\\orp r{\\sigma}(i)\\right).\n\\]\nHence,\n\\begin{align*}\n\\tdist r(\\pi,\\sigma) & \\le n-\\lcs{\\pi'}{\\sigma'}\\\\\n & \\le n-\\sum_{i=1}^{n\/r}\\left|\\orp r{\\pi}(i)\\cap\\orp r{\\sigma}(i)\\right|\\\\\n & =\\sum_{i=1}^{n\/r}\\left(r-\\left|\\orp r{\\pi}(i)\\cap\\orp r{\\sigma}(i)\\right|\\right)\\\\\n & =\\hdist r(\\pi,\\sigma),\n\\end{align*}\nimplying that $\\tdist r(\\pi,\\sigma)\\le\\hdist r(\\pi,\\sigma).$ \n\\begin{lem}\n\\label{lem:HammingUlam}For $\\pi,\\sigma\\in\\mathbb{S}_{n}$, we have\n\\[\n\\frac{r}{n}\\hdist r(\\pi,\\sigma)\\le\\tdist r(\\pi,\\sigma)\\le\\hdist r(\\pi,\\sigma).\n\\]\n\n\\end{lem}\nThe lemma illustrates the fact that for $r=\\Theta(n)$, the Ulam distance\nis within a constant factor of the Hamming distance, while for $r=o(n)$,\nthe Ulam distance may be much smaller. Consequently, while good codes\nin the Ulam metric allow for substitution error correction, good codes\nin the Hamming metric provide resilience under translocation errors\nonly for a certain limited range of parameters.\n\n\n\\section{Bounds on Size of Multipermutation Codes\\label{sec:bounds}}\n\nIn what follows, we derive bounds on the size of multipermutation\ncodes in the Hamming metric as well as the Ulam metric. For the case\nof the Hamming distance, we find the asymptotic capacity, while for\nthe Ulam distance we provide lower and upper bounds on the capacity.\nWe point out that a number of bounds on multipermutation codes in\nthe Hamming metric were derived in~\\cite{Huczynska2006FrequencyPermutationArrays},\nincluding some simple and some complicated expressions involving Laguerre\npolynomials. Nevertheless, these bounds do not allow for finding a\ncapacity formula for the underlying codes.\n\nLet $A_{H}(n,r,d)$ and $A_{\\circ}(n,r,d)$ denote the maximum cardinalities\nof an $\\mpcode H(n,r,d)$ and an $\\mpcode{\\circ}(n,r,d)$ code, respectively.\nFurthermore, let $\\mathcal{C}_{H}(r,d)$ denote the \\emph{capacity},\ni.e., maximum achievable rate, of multipermutation codes in the Hamming\nmetric, defined as \n\\[\n\\mathcal{C}_{H}(r,d)=\\lim_{n\\to\\infty}\\frac{\\ln A_{H}(n,r,d)}{\\ln n!}\\cdot\n\\]\nThe capacity of multipermutation codes in the Ulam metric, $\\mathcal{C}_{\\circ}(r,d)$,\nis defined similarly.\n\nIn the remainder of the paper, limits are evaluated for $n\\to\\infty$\nunless stated otherwise. We assume that all limits of interest exist\nand we use $\\rho=\\rho(r)=\\lim\\frac{\\ln r}{\\ln n}$, as well as $\\delta=\\delta(d)=\\lim\\frac{d}{n}\\cdot$\n\n\n\\subsection{Multipermutation Codes in the Hamming Metric }\n\nIt was shown by Luo et al.~\\cite{Luo2003CCCZq} that\n\\begin{equation}\nA_{H}(n,r,d)\\le\\frac{d}{r+d-n},\\quad\\mbox{for }r+d>n,\\label{eq:luo-UB}\n\\end{equation}\n and by Huczynska and Mullen~\\cite{Huczynska2006FrequencyPermutationArrays}\nthat\n\\begin{equation}\nA_{H}(n,r,d)\\le\\frac{n!}{r(d-1)!}\\cdot\\label{eq:Huc-UB}\n\\end{equation}\nThe first bound,~(\\ref{eq:luo-UB}), implies that the asymptotic\nrate is zero if $r+d>n$, while the second bound implies that \n\\begin{equation}\n\\mathcal{C}_{H}(r,d)\\le1-\\delta,\\label{eq:simpleUB}\n\\end{equation}\n which also follows from the fact that $\\mathcal{C}_{H}(r,d)\\le\\mathcal{C}_{H}(1,d)$\nand Theorem 11 of our companion paper~\\cite{farnoud2013error-correction},\nstating that $\\mathcal{C}_{H}(1,d)=1-\\delta$. We improve next upon\nthe bound in~(\\ref{eq:simpleUB}) and provide a matching lower bound,\nthereby establishing the capacity of $r$-regular multipermutation\ncodes in the Hamming metric.\n\nLet $S(l,m,r)$ denote the number of sequences of length $l$ over\nthe alphabet $[m]$, with no element appearing more than $r$ times.\nNote that $S(l,m,r)$ equals the number of ordered partitions of a\nset of size $l$ into $m$ sets such that each part has at most $r$\nelements, and where empty subsets are allowed.\n\\begin{lem}\n\\label{lem:Hamming-Singleton}(Singleton bound) For positive integers\n$n,r,d$ such that $r$ divides $n$, we have \n\\[\nA_{H}(n,r,d)\\le\\left(\\frac{n}{r}\\right)^{n-d+1}.\n\\]\n\\end{lem}\n\\begin{IEEEproof}\nConsider an $\\mpcode H(n,r,d)$ code $C$ of size $M$ and let \n\\[\n\\mathfrak{M}=\\Mlp rC=\\{\\mathfrak{m}_{1},\\dotsc,\\mathfrak{m}_{M}\\}\n\\]\ndenote its multipermutation representation. Since the minimum Hamming\ndistance $\\hdist r$ of $C$ is at least $d$, for distinct $i$ and\n$j$, \n\\begin{equation}\n\\sum_{k=1}^{n}\\mathbb{I}\\left(\\mathfrak{m}_{i}(k)\\neq\\mathfrak{m}_{j}(k)\\right)\\ge d,\\label{eq:nExcluded-1-1}\n\\end{equation}\nwhere the indicator function $\\mathbb{I}$ is defined in the standard\nmanner as \n\\[\n\\mathbb{I}(\\mathsf{condition})=\\begin{cases}\n1, & \\quad\\mbox{if }\\mathsf{condition}\\mbox{ is true},\\\\\n0, & \\quad\\mbox{if }\\mathsf{condition}\\mbox{ is false.}\n\\end{cases}\n\\]\nBy removing the last $d-1$ elements of each multipermutation $\\mathfrak{m}_{i},i\\in[M],$\nwe obtain the set $\\mathfrak{M}'=\\left\\{ \\mathfrak{m}'_{1},\\dotsc,\\mathfrak{m}'_{M}\\right\\} $\nof sequences of length $n-d+1$ over $[n\/r]$ where no element appears\nmore than $r$ times.%\n\\footnote{This argument is akin to the approach proposed in~\\cite{barg2012codes}\nfor permutation codes in the Kendall metric.%\n}\n\nSince $d-1$ elements are removed,~(\\ref{eq:nExcluded-1-1}) implies\nthat \n\\[\n\\sum_{k=1}^{n-d+1}\\mathbb{I}\\left(\\mathfrak{m}'_{i}(k)\\neq\\mathfrak{m}'_{j}(k)\\right)=\\sum_{k=1}^{n-d+1}\\mathbb{I}\\left(\\mathfrak{m}{}_{i}(k)\\neq\\mathfrak{m}{}_{j}(k)\\right)\\ge1.\n\\]\nand thus for distinct $i,j\\in[M]$, $\\mathfrak{m}'_{i}$ and $\\mathfrak{m}'_{j}$ are\ndistinct. Hence, we have \n\\begin{align}\nA_{H}(n,r,d) & \\le S\\left(n-d+1,\\frac{n}{r},r\\right).\\label{eq:AH_bound_S}\n\\end{align}\nFurthermore, since $S(n-d+1,n\/r,r)\\le S(n-d+1,n\/r,\\infty),$ we find\n\\[\nA_{H}(n,r,d)\\le S\\left(n-d+1,\\frac{n}{r},\\infty\\right)=\\left(\\frac{n}{r}\\right)^{n-d+1}.\n\\]\n\n\\end{IEEEproof}\nAs shown in the sequel, the bound given in Lemma~\\ref{lem:Hamming-Singleton}\nis sufficiently tight for capacity derivations. Nevertheless, it may\nbe useful to bound $S(l,m,r)$ more tightly. \n\n\\begin{table*}\n\\centering{}\\caption{Bounds on the size $3$-regular multipermutation codes in the Hamming\nmetric of length $9$.}\n\\label{tab:HammingUBs}%\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline \nUpper bound on $A_{H}(9,3,d)$ & $d=1$ & $d=2$ & $d=3$ & $d=4$ & $d=5$ & $d=6$ & $d=7$ & $d=8$ & $d=9$\\tabularnewline\n\\hline \n\\hline \n(\\ref{eq:luo-UB}) \\cite{Luo2003CCCZq} & - & - & - & - & - & - & 7 & 4 & 3\\tabularnewline\n\\hline \n(\\ref{eq:Huc-UB}) \\cite{Huczynska2006FrequencyPermutationArrays} & 120960 & 120960 & 60480 & 20160 & 5040 & 1008 & 168 & 24 & 3\\tabularnewline\n\\hline \nLemma \\ref{lem:Hamming-Singleton} & 19683 & 6561 & 2187 & 729 & 243 & 81 & 27 & 9 & 3\\tabularnewline\n\\hline \n(\\ref{eq:CLT-bonud}) (approximate bound) & 12077 & 4560 & 1700 & 624 & 224 & 79 & 27 & 9 & 3\\tabularnewline\n\\hline \n(\\ref{eq:EGF}) & 1680 & 1680 & 1050 & 510 & 210 & 78 & 27 & 9 & 3\\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table*}\n\n\nIt is easy to see that \n\\begin{equation}\nS(l,m,r)=\\sum_{\\begin{array}{c}\nx_{1}+\\cdots+x_{m}=l,\\\\\n0\\le x_{i}\\le r\n\\end{array}}\\frac{l!}{\\prod_{i=1}^{m}x_{i}!}\\cdot\\label{eq:mulinom}\n\\end{equation}\nwhere the $x_{i}$'s are integers. The exponential generating function\n(EGF) of $S(l,m,r)$ is\n\\[\n\\sum_{l=0}^{\\infty}S(l,m,r)\\frac{z^{l}}{l!}=\\left(\\sum_{i=0}^{r}\\frac{z^{i}}{i!}\\right)^{m}\n\\]\nand thus one can write \n\\begin{equation}\nA_{H}(n,r,d)\\le(n-d+1)!\\left[z^{n-d+1}\\right]\\left(\\sum_{i=0}^{r}\\frac{z^{i}}{i!}\\right)^{n\/r}\\cdot\\label{eq:EGF}\n\\end{equation}\nThe bound given in~(\\ref{eq:EGF}) can be used to find numerical\nupper bounds on the code size, such as those provided in Table~\\ref{tab:HammingUBs}.\nIn addition, it can be used to obtain simple asymptotic bounds using\nmethods described in the classical text~\\cite[Ch.~8]{odlyzko_asymptotic_1995}.\nAs a final note, we point out that the related problems of restricted\nmultisets and restricted integer partitions are far better studied\ncombinatorial entities than the one we addressed above~\\cite[Ch.~21, Sec.~8]{Graham1995Handbook},\nalthough no simple direct connection between these problems and the\nproblem discussed here exist.\n\nAnother approach, which is conceptually much simpler and which applies\nto many other coding-theoretic scenarios is using the Poisson approximation\ntheorem for multinomial variables and the Chernoff bound~\\cite{Deheuvels1988Poissonapproximations,Arenbaev1976AsymptoticBehavioroftheMultinomialDstribution},\nor alternatively, the Central limit theorem~\\cite{hofri1995analysis}.\nAs shown in~\\cite{milenkovic2004probUrmModels,hofri1995analysis},\nthe number of terms in the multinomial summation formula, $m$, may\nrepresent the number of labeled urns into which $l$ labeled balls\nare thrown randomly. The occupancy variables $X_{i}$, $i=1,\\ldots,m$,\nare dependent, since $X_{1}+\\ldots+X_{m}=l$. But in the asymptotic\ncentral domain regime, with $l\/m$ constant, the variables $X_{i}$,\n$i=1,\\ldots,m$, may be viewed as \\emph{independent} Poisson variables\nwith mean $\\lambda=l\/m$. Any result of computations involving independent\nPoisson variables that satisfies the inversion conditions dictated\nby Tauberian theorems described in~\\cite{milenkovic2004probUrmModels}\nmay be \\emph{asymptotically converted} into the correct result by\nsimply replacing $\\lambda$ with $l\/m$. Furthermore, the same approach\nmay be used when dealing with urns and balls that satisfy additional\nconstraints~\\cite{milenkovic2004probUrmModels}.\n\nTo understand the principles behind the Poisson transform method,\nwe follow the analysis in~\\cite{hofri1995analysis} based on\\ \\cite{gonnet1984analysis}.\nThe key observation is that the Poisson distributions satisfy the\nadditivity (infinite divisibility) property, i.e., the property that\nthe sum of independent Poisson random variables is another Poisson\nrandom variable with mean parameter equal to the sum of the parameters\nof the individual variables in the sum. Then, it is straightforward\nto show that for two different ball placement processes, the urn occupancy\nvariables have the same distribution: 1) in the first case, each urn\nreceives balls according to a Poisson distribution with parameter\n$\\lambda$ independently of all other urns; 2) in the second case,\nballs arrive with a Poisson distribution with parameter $\\lambda m$\nand are routed with uniform probability $1\/m$ to one of the urns. \n\nAssume next that $g(m,\\lambda)$ is a quantity of interest where the\ninput to each urn is generated according to model 1). The same quantity\nunder the original urns and balls model with a fixed number $l$ of\nballs is denoted by $f(m,l)$. Using the equivalence between the two\nformulations 1) and 2), one can show that\n\\[\ng(m,\\lambda)=\\sum_{l}f(m,l)\\: P\\left(X_{1}+\\ldots+X_{m}=l\\right),\n\\]\nwhere $X_{1},\\ldots,X_{m}$ are i.i.d Poisson random variables with\nparameter $\\lambda$. As a result, it is straightforward to see that\n$f(m,l)=\\frac{l!}{m^{l}}\\,\\,[\\lambda^{l}]\\left\\{ e^{\\lambda\\, m}g(m,\\lambda)\\right\\} .$ \n\nIn words, $e^{\\lambda\\, m}g(m,\\lambda)$ represents the exponential\ngenerating function over the number of balls $l$ of $f(m,l)$ evaluated\nat $\\lambda\\, m$. Evaluating the coefficient in a generating function\nin the asymptotic domain may be accomplished with the aid of Tauberian\ntheorems (see~\\cite{milenkovic2004probUrmModels}) or classical asymptotic\nanalysis. In the case of the Poisson transform, provided that some\nminor technical conditions are met, it can be shown that $f(m,l)\\simeq\\, g(m,l\/m),$\nwhere $a(x)\\simeq b(x)$ stands for $\\lim_{x\\to\\,\\infty}a(x)\/b(x)=1$.\nIntuitively, the aforementioned result implies that when the dependencies\namong a large number of random variables are weak -- for example,\nonly in terms of a constraint on the total sum of their values --\nthen the variables are asymptotically independent, provided a proper\nchoice of the distribution ensures consistence with the finite-valued\nparameters.\n\nIn the case of interest, we need to find the probability $P\\{X_{i}\\leq r,\\, i=1,\\ldots,m\\}$.\nFor $m,l\\to\\infty$, such that $l\/m$ is a constant, and for $r$\nfixed, this leads to\n\\begin{equation}\nS(l,m,r)\\simeq m^{l}\\,\\left(\\sum_{i=0}^{r}\\,\\exp(-\\lambda)\\,\\frac{\\lambda^{i}}{i!}\\right)^{m},\\notag\n\\end{equation}\nwhere $\\lambda=l\/m$. The asymptotic formula for $S(l,m,r)$ depends\non the relationship between the parameters $r,l,m.$ For $r\\leq l\/m$,\nthe Chernoff bound reads as\n\\[\n\\sum_{i=0}^{r}\\,\\exp(-\\lambda)\\,\\frac{\\lambda^{i}}{i!}\\leq\\frac{\\exp(-\\lambda)(e\\,\\lambda)^{r}}{r^{r}},\n\\]\nso that\n\\begin{equation}\nS(l,m,r)\\lesssim m^{l}\\,\\frac{\\exp(-l\/m)(e\\, l\/m)^{r}}{r^{r}}\\cdot\n\\end{equation}\n\n\nFor the case of interest in our derivation, $r>l\/m$. Whenever $r>10$,\none may use the straightforward Central Limit Theorem approximation\n\\[\n\\sum_{i=0}^{r}\\,\\exp(-\\lambda)\\,\\frac{\\lambda^{i}}{i!}\\simeq\\Phi\\left(\\frac{r+0.5-l\/m}{\\sqrt{l\/m}}\\right),\n\\]\nso that\n\\[\nS(l,m,r)\\simeq m^{l}\\;\\Phi^{m}\\left(\\frac{r+0.5-l\/m}{\\sqrt{l\/m}}\\right),\n\\]\nwhere the function $\\Phi(\\cdot)$ stands for the cumulative distribution\nfunction (CDF) of a standard Gaussian random variable. Since $r>\\frac{n-d+1}{n\/r}$,\nthe preceding relation and (\\ref{eq:AH_bound_S}) imply\n\\begin{equation}\nA_{H}(n,r,d)\\lesssim\\left(\\frac{n}{r}\\right)^{n-d+1}\\;\\Phi^{n\/r}\\left(\\frac{n+2(d-1)r}{2\\sqrt{n(n-d+1)r}}\\right)\\label{eq:CLT-bonud}\n\\end{equation}\n provided that $\\frac{n-d+1}{n\/r}$ is a constant larger than 10.\n\nAs an example, upper bounds on the size of $\\mpcode H(9,3,d)$ are\ngiven in Table~\\ref{tab:HammingUBs}. Note that the bounds of~(\\ref{eq:CLT-bonud}),\nLemma~\\ref{lem:Hamming-Singleton}, and~(\\ref{eq:EGF}) are very\nclose for small values of $d$. Indeed, the right side of~(\\ref{eq:CLT-bonud})\nis bounded above by $(n\/r)^{n-d+1}$ and below by \n\\[\n\\left(\\frac{n}{r}\\right)^{n-d+1}\\left(\\frac{1}{2}\\right)^{n\/r}\n\\]\nand we have \n\\[\n\\lim\\frac{\\ln\\left(\\left(\\frac{n}{r}\\right)^{n-d+1}\\left(\\frac{1}{2}\\right)^{n\/r}\\right)}{\\ln\\left(\\frac{n}{r}\\right)^{n-d+1}}=1\n\\]\nprovided that $\\delta<1$ and $r<n$. Therefore, the bounds of~(\\ref{eq:CLT-bonud})\nand Lemma \\ref{lem:Hamming-Singleton} have the same asymptotic exponent.\n\nThe next lemma provides a lower bound on $A_{H}(n,r,d)$.\n\\begin{lem}\n\\label{lem:Hamming-Gilbert-Varshamov}(Gilbert-Varshamov Bound) We\nhave \n\\[\nA_{H}(n,r,d)\\ge\\frac{n!}{(r!)^{n\/r}\\binom{n}{d-1}\\left(\\frac{n}{r}\\right)^{d-1}}\\cdot\n\\]\n\\end{lem}\n\\begin{IEEEproof}\nThere are $\\frac{n!}{(r!)^{n\/r}}$ multipermutations of $\\ms nr$.\nThe size of a ball of radius $d-1$ in the space of multipermutations\nof $\\ms nr$ endowed with the Hamming distance is bounded above by\n$\\binom{n}{d-1}\\left(\\frac{n}{r}\\right)^{d-1}$ (an exact and complicated\nexpression for the size of the ball may be found in~\\cite{Huczynska2006FrequencyPermutationArrays}).\nThe Lemma follows by a standard application of Gilbert's argument.\\end{IEEEproof}\n\\begin{thm}\n\\label{thm:Hamming-cap}We have\n\\[\n\\mathcal{C}_{H}(r,d)=(1-\\rho)(1-\\delta).\n\\]\n\\end{thm}\n\\begin{IEEEproof}\nFirst, recall that $\\lim$ expressions with no subscripts stand for\n$\\lim_{n\\to\\infty}$. \n\nOn the one hand, from Lemma~\\ref{lem:Hamming-Singleton}, we have\n\\begin{align*}\n\\mathcal{C}_{H}(r,d) & \\le\\lim\\frac{(n-d+1)(\\ln n-\\ln r)}{\\ln n!}\\\\\n & =\\lim\\frac{n\\ln n-n\\ln r-d\\ln n+d\\ln r}{n\\ln n+O(n)}\\\\\n & =1-\\rho-\\delta+\\rho\\delta.\n\\end{align*}\n\n\nOn the other hand, from Lemma~\\ref{lem:Hamming-Gilbert-Varshamov},\nwe easily see that\n\\begin{align*}\n\\mathcal{C}_{H}(r,d) & \\ge\\lim\\frac{\\ln\\left(n!(r!)^{-n\/r}\\binom{n}{d-1}^{-1}\\left(\\frac{n}{r}\\right)^{-d+1}\\right)}{\\ln n!}\\\\\n & =1-\\lim\\frac{(n\/r)\\ln r!+(d-1)\\ln(n\/r)}{\\ln n!}\\\\\n & =1-\\lim\\frac{n\\ln r+d\\ln n-d\\ln r+O(n)}{n\\ln n+O(n)}\\\\\n & =1-\\rho-\\delta+\\rho\\delta,\n\\end{align*}\nwhere we have used the fact that $\\lim\\frac{\\ln\\binom{n}{d-1}}{\\ln n!}=0$.\nThis establishes the claimed result for the asymptotic capacity of\nmultipermutation codes in the Hamming metric.\n\\end{IEEEproof}\n\n\\subsection{Multipermutation Codes in the Ulam Metric}\n\n\n\nUsing Lemma~\\ref{lem:HammingUlam} which implies that $A_{\\circ}(n,r,d)\\le A_{H}(n,r,d)$,\nwe find the following upper bound on $A_{\\circ}(n,r,d)$:\n\\begin{equation}\nA_{\\circ}(n,r,d)\\le A_{H}(n,r,d)\\le S(n-d+1,\\frac{n}{r},r)\\le\\left(\\frac{n}{r}\\right)^{n-d+1}.\\label{eq:Ulam-maxsize-UB}\n\\end{equation}\nThe next lemma provides a lower bound on $A_{\\circ}(n,r,d)$.\n\\begin{lem}\n\\label{lem:GVB-Ulam}(Gilbert-Varshamov Bound) For positive integers\n$n,r,d$ such that $n$ is a multiple of $r$, we have \n\\[\nA_{\\circ}(n,r,d)\\ge\\frac{(n-d+1)!}{\\binom{n}{d-1}(r!)^{2n\/r}}\\cdot\n\\]\n\\end{lem}\n\\begin{IEEEproof}\nLet $\\ball r{\\circ}(u)$ denote the size of a ball of radius $u$\nin $\\mathbb{S}_{n}$ endowed by $\\tdist r$ (note that due to symmetry,\ni.e., left invariance of the Ulam metric, the volume of the ball is\nindependent on the choice of the center). Equivalently, let $\\ball r{\\circ}(u)=\\left\\{ \\pi\\in\\mathbb{S}_{n}:\\tdist r(\\pi,e)\\le u\\right\\} $.\nThe Gilbert bound states that \n\\[\nA_{\\circ}(n,r,d)\\ge\\frac{n!}{\\ball r{\\circ}(d-1)}\\cdot\n\\]\nWe show that $\\ball r{\\circ}(d-1)\\le(r!)^{2n\/r}\\ball 1{\\circ}(d-1)$.\nThe lemma then follows from a result pertaining to the Ulam metric\nwe derived in~\\cite{farnoud2013error-correction}, namely: \n\\[\n\\ball 1{\\circ}(d-1)\\le\\binom{n}{d-1}\\frac{n!}{(n-d+1)!}\\cdot\n\\]\n\n\nThe set $\\left\\{ \\pi\\in\\mathbb{S}_{n}:\\tdist r(\\pi,e)\\le u\\right\\} $\nequals \n\\[\n\\bigcup_{\\sigma\\in\\eqclass r(e)}\\ \\bigcup_{\\pi\\in\\mathbb{S}_{n}:\\tdist{}(\\pi,\\sigma)\\le u}\\eqclass r(\\pi).\n\\]\n Hence,\n\\begin{align*}\n\\ball r{\\circ}(u) & =\\left|\\cup_{\\sigma\\in\\eqclass r(e)}\\cup_{\\pi\\in\\mathbb{S}_{n}:\\tdist{}(\\pi,\\sigma)\\le u}\\eqclass r(\\pi)\\right|\\\\\n & \\le(r!)^{n\/r}\\left|\\cup_{\\pi\\in\\mathbb{S}_{n}:\\tdist{}(\\pi,e)\\le u}\\eqclass r(\\pi)\\right|\\\\\n & \\le(r!)^{n\/r}\\ball 1{\\circ}(u)(r!)^{n\/r}\\\\\n & =(r!)^{2n\/r}\\ball 1{\\circ}(u),\n\\end{align*}\nwhich shows that $ $$\\ball r{\\circ}(d-1)\\le(r!)^{2n\/r}\\ball 1{\\circ}(d-1)$.\n\\end{IEEEproof}\nNext, we improve upon Lemma~\\ref{lem:GVB-Ulam} by finding a sharper\nbound for $\\ball r{\\circ}(u),u\\in\\mathbb{Z}^{+}$. Consider the ball\naround the identity permutation $e$. For a permutation $\\pi$ that\nsatisfies $\\tdist r(\\pi,e)\\le u$, there exists a $\\pi'\\in\\eqclass r(\\pi)$\nthat has a common subsequence $s$ of length $l=n-u$ with some $e'\\in\\eqclass r(e)$.\nThere are \n\\[\nA=\\sum_{\\begin{array}{c}\nx_{1}+\\cdots+x_{n\/r}=l,\\\\\nx_{i}\\in[0,r],\\forall i\n\\end{array}}\\prod_{i=1}^{n\/r}\\binom{r}{x_{i}}x_{i}!\n\\]\n ways of choosing a sequence $s$ of length $l$ such that it is a\nsubsequence of some $e'\\in\\eqclass r(e)$, with $\\prod_{i=1}^{n\/r}\\binom{r}{x_{i}}x_{i}!$\ncounting the number of ways one can choose a subsequence of length\n$l$ with $x_{i}$ elements from rank $i$, \n\\[\n\\orp r{e'}(i)=\\orp re(i)=\\{(i-1)r+1,\\dotsc,ir\\}.\n\\]\n\n\nThe number of ordered partitions $\\mathfrak{o}$ with parts of size equal to\n$r$, such that there exists a $\\pi'$ that satisfies $\\mathfrak{o}=\\orp r{\\pi'}$\nand contains $s$ as a subsequence equals\n\\[\nB=\\sum_{\\begin{array}{c}\nx_{1}+\\cdots+x_{n\/r}=l,\\\\\nx_{i}\\in[0,r],\\forall i\n\\end{array}}\\binom{n-l}{r-x_{1},\\dotsc,r-x_{n\/r}}.\n\\]\nHere, the multinomial $\\binom{n-l}{r-x_{1},\\dotsc,r-x_{n\/r}}$ accounts\nfor the number of ways of choosing the ordered partition $\\mathfrak{o}$ such\nthat the first $x_{1}$ elements of $s$ are in the first part, the\nnext $x_{2}$ elements are in the second part, and so on.\n\nIn addition, there are $(r!)^{n\/r}$ permutations $\\pi$ such that\n$\\pi\\in\\eqclass r(\\pi')$. Hence, \n\\begin{equation}\n\\ball r{\\circ}(u)\\le AB(r!)^{n\/r}.\\label{eq:sizeofBall-1}\n\\end{equation}\n\n\nWith regards to bounding the combinatorial sum $A$, we observe that\n\\begin{align*}\nA & \\le\\binom{l+n\/r-1}{n\/r-1}\\max_{\\begin{array}{c}\nx_{1}+\\cdots+x_{n\/r}=l,\\\\\nx_{i}\\in[0,r],\\forall i\n\\end{array}}\\prod_{i=1}^{n\/r}\\binom{r}{x_{i}}x_{i}!\\\\\n & \\le\\binom{l+n\/r-1}{n\/r-1}\\Bigl(\\frac{r!}{(r-\\frac{l}{n\/r})!}\\Bigr)^{n\/r}\\\\\n & \\le\\binom{2n}{n}\\Bigl(\\frac{r!}{\\frac{r(n-l)}{n}!}\\Bigr)^{n\/r}.\n\\end{align*}\nUsing a similar approach for $B$, we find\n\\[\nB\\le\\binom{2n}{n}\\frac{(n-l)!}{\\left(\\frac{r(n-l)}{n}!\\right)^{n\/r}}\\cdot\n\\]\n Hence, \n\\[\n\\ball r{\\circ}(u)\\le\\binom{2n}{n}^{2}\\Bigl(\\frac{r!}{\\frac{ru}{n}!}\\Bigr)^{2n\/r}u!,\n\\]\n and so, for $d>1$,\n\\begin{align*}\n\\ln\\ball r{\\circ}(d-1) & \\le\\frac{2n}{r}\\ln r!-\\frac{2n}{r}\\ln\\frac{r(d-1)}{n}!\\\\\n & \\qquad+\\ln(d-1)!+O(n)\\\\\n & =2n\\ln r-2(d-1)\\ln\\frac{r(d-1)}{n}\\\\\n & \\qquad+(d-1)\\ln(d-1)+O(n)\\\\\n & =2n\\ln r-2d\\ln r+2d\\ln n-d\\ln d+O(n).\n\\end{align*}\nThis implies that\n\\begin{align}\n\\mathcal{C}_{\\circ}(r,d) & \\ge1-\\lim\\frac{2n\\ln r-2d\\ln r+2d\\ln n-d\\ln d+O(n)}{n\\ln n+O(n)}\\nonumber \\\\\n & =1-2\\rho+2\\delta\\rho-\\delta\\nonumber \\\\\n & =(1-\\delta)(1-2\\rho).\\label{eq:Ulam-cap-LB}\n\\end{align}\n\n\\begin{thm}\n\\label{thm:capacity-Ulam}The capacity of multipermutation codes in\nthe Ulam metric is bounded according to\n\\[\n(1-\\delta)(1-2\\rho)\\le\\mathcal{C}_{\\circ}(r,d)\\le(1-\\delta)(1-\\rho).\n\\]\n\\end{thm}\n\\begin{IEEEproof}\nThe lower bound is given in~(\\ref{eq:Ulam-cap-LB}) while the upper\nbound is a result of~(\\ref{eq:Ulam-maxsize-UB}) and Theorem~\\ref{thm:Hamming-cap}.\n\\end{IEEEproof}\n\n\\section{Constructions\\label{sec:Constructions}}\n\nIn the next subsections, we present several constructions for multipermutation\ncodes in the Ulam metric. One of the key ingredients of our constructions\nis permutation interleaving, which we proposed for Ulam metric code\ndesign in~\\cite{farnoud2013error-correction}. The related idea of\nrestricting certain positions in the codewords to certain values was\nfirst described in~\\cite{klove2010PermutationArrays,schartz2010correctinglimitedmagnitude},\nwhile interleaving in the Chebyshev metric was discussed in\\cite{Shieh2010Decoding}. \n\nFor sequences $\\pi_{1},\\dotsc,\\pi_{k}$, let $\\pi_{1}\\circ_{r}\\pi_{2}\\circ_{r}\\dotsm\\circ_{r}\\pi_{k}$\ndenote the sequence obtained by sequentially interleaving blocks of\n$r$ elements of $\\pi_{i},i\\in[k]$. \n\nFor example, $(1,3,4,2)\\circ_{2}(6,7,8,5)\\circ_{2}(12,10,9,11)=(1,3,6,7,12,10,4,2,8,5,9,11)$. \n\nThis form of interleaving will henceforth be called block interleaving.\nWhenever $r=1$, we simply write $\\circ$ instead of $\\circ_{1}$. \n\n\n\\subsection{Constructions based on almost disjoint sets}\n\nTwo sets $A$ and $B$ are said to be \\emph{at most k-intersecting,\n}if for a given positive integer $k$, one has\n\\[\n|A\\cap B|\\le k.\n\\]\nWhen $k$ is smaller than the size of the sets $A,B$, and the aforementioned\nbound is true, we say that the sets are \\textit{almost disjoint}.\nThe next lemma shows how sets of set partitions with almost disjoint\nparts can be used for constructing multipermutation codes in the Ulam\nmetric.\n\\begin{lem}\n\\label{lem:construct-1}Let $C$ be an $\\mpcode{}(n,r)$ code, and\nsuppose that $t$ is a positive integer such that $2t<r$. If for\nall $\\pi,\\sigma\\in C$ and $i\\in[n\/r]$, we either have $\\orp r{\\pi}(i)=\\orp r{\\sigma}(i)$\nor \n\\begin{equation}\n\\left|\\orp r{\\pi}(i)\\cap\\orp r{\\sigma}(i)\\right|<r-2t,\\label{eq:construct-1}\n\\end{equation}\nthen the code $C$ can correct $t$ translocation errors, that is,\n$C$ is an $\\mpcode{\\circ}(n,r,2t+1)$ code.\\end{lem}\n\\begin{IEEEproof}\nSuppose $\\pi\\in C$ is the (unknown) stored codeword and $\\omega$\nis the retrieved permutation. The Ulam distance between $\\pi$ and\n$\\omega$ is at most $t$ since the codeword $\\pi$ is affected by\nat most $t$ translocation errors. We show that given $\\omega$, $\\orp r{\\pi}$\ncan be uniquely identified. Fix $i\\in[n\/r]$. Since there are at most\n$t$ translocation errors, by Lemma~\\ref{lem:Atranslocation}, we\nhave \n\\begin{equation}\n\\left|\\orp r{\\pi}(i)\\cap\\orp r{\\omega}(i)\\right|\\ge r-t.\\label{eq:t-errors}\n\\end{equation}\nTo identify $\\orp r{\\pi}(i)$ uniquely, it suffices to have $\\left|\\orp r{\\sigma}(i)\\cap\\orp r{\\omega}(i)\\right|<r-t$\nfor all $\\sigma\\in C$ such that $\\mathfrak{o}_{\\pi}^{r}(i)\\neq\\mathfrak{o}_{\\sigma}^{r}(i)$. \n\nSuppose that $\\sigma\\in C$ and $\\mathfrak{o}_{\\pi}^{r}(i)\\neq\\mathfrak{o}_{\\sigma}^{r}(i)$.\nWe use~(\\ref{eq:construct-1}) and~(\\ref{eq:t-errors}) to show\nthat $\\left|\\orp r{\\sigma}(i)\\cap\\orp r{\\omega}(i)\\right|<r-t$. For\nsimplicity, let $ $$B_{\\pi}=\\orp r{\\pi}(i)$, $B_{\\sigma}=\\orp r{\\sigma}(i)$,\nand $B_{\\omega}=\\orp r{\\omega}(i)$. We then have \n\\begin{align*}\n\\left|B_{\\sigma}\\cap B_{\\omega}\\right| & =\\left|B_{\\sigma}\\cap B_{\\omega}\\cap B_{\\pi}^{c}\\right|+\\left|B_{\\sigma}\\cap B_{\\omega}\\cap B_{\\pi}\\right|\\\\\n & \\le\\left|B_{\\omega}\\cap B_{\\pi}^{c}\\right|+\\left|B_{\\sigma}\\cap B_{\\pi}\\right|\\\\\n & \\stackrel{\\mathsf{(a)}}{<}\\left(r-\\left|B_{\\omega}\\cap B_{\\pi}\\right|\\right)+\\left(r-2t\\right)\\\\\n & \\stackrel{(\\mathsf{b})}{\\le}t+r-2t=r-t,\n\\end{align*}\nwhere $B_{\\pi}^{c}$ denotes the complement of $B_{\\pi}$. Inequality\n$\\mathsf{(a)}$ follows from the fact that $\\left|B_{\\omega}\\right|=r$\nand~(\\ref{eq:construct-1}); and inequality $\\mathsf{(b)}$ follows\nfrom~(\\ref{eq:t-errors}). This completes the proof.\\end{IEEEproof}\n\\begin{rem}\n\\label{rem:gen-errors}A code satisfying the condition of Lemma~\\ref{lem:construct-1}\ncan in fact correct a class of errors that is more general than translocation\nerrors. More precisely, the code can correct errors that lead to the\ndisplacement of at most $t$ elements of each rank. In particular,\nthe code can correct $t$ transposition errors, $t$ Hamming errors,\nor any $t$ errors where each error displaces at most one element\nfrom each rank. As an example of the latter type of error, consider\n\\begin{multline*}\n\\left(\\{3,\\mathbf{4}\\},\\{2,6\\},\\{\\mathbf{7},8\\},\\{\\mathbf{1},5\\}\\right)\\xrightarrow{\\mbox{ error }}\\\\\n\\left(\\{3,\\mathbf{1}\\},\\{2,6\\},\\{\\mathbf{4},8\\},\\{\\mathbf{7},5\\}\\right),\n\\end{multline*}\nwhere each rank corresponds to one set in the set partition. We note\nthat each part except for the one listed second has one displaced\n(moved) element.\n\\end{rem}\nA code that satisfies the conditions of Lemma~\\ref{lem:construct-1}\ncan be decoded in time $O(Mnr)$, where $M$ is the size of the code.\nAs before, suppose that $\\pi\\in C$ is the unknown stored codeword\nand $\\omega$ is the retrieved permutation. For each $i\\in[n\/r]$,\nwe must identify a unique set $A(i)\\in\\{\\orp r{\\sigma}(i):\\sigma\\in C\\}$\nsuch that \n\\begin{equation}\n\\left|A(i)\\cap\\orp r{\\omega}(i)\\right|\\ge r-t.\\label{eq:decoding-intersection}\n\\end{equation}\nThe ordered partition representation of $\\pi$ is then $\\orp r{\\pi}=\\left(A(1),\\dotsc,A(n\/r)\\right).$\n\nThe intersection of $\\orp r{\\omega}(i)$ and each of the sets in $\\{\\orp r{\\sigma}(i):\\sigma\\in C\\}$\ncan be trivially found with time complexity $O(r^{2})$. Since there\nare $M$ sets in $\\{\\orp r{\\sigma}(i):\\sigma\\in C\\}$, finding $A(i)$\nfor each $i\\in[n\/r]$ takes $O(Mr^{2})$ steps. Thus $\\orp r{\\pi}$\ncan be identified with complexity $O(Mr^{2}n\/r)=O(Mnr)$. \n\nSince for some code parameters $M$ can be exponential in $n$, the\ntime needed for exhaustive search decoding may be exponential as well.\nHowever, if more information about the structure of the code is available,\ndecoding may be performed much faster, as in the cases of constructions\nbased on grouping elements and Steiner systems discussed in Subsections~\\ref{sub:grouping}\nand \\ref{sub:designs}.\n\nWe pause to briefly comment on the relationship between almost disjoint\nsets of set partitions and intersecting families, in the context of\nthe celebrated Erd\\H{o}s-Ko-Rado (EKR) theorem (see~\\cite{Deza1983EKR22years,Ellis2012Setwise}\nand references therein). A family of subsets of a set is said to be\nintersecting if each pair of subsets have a non-empty intersection.\nThe EKR theorem establishes upper bounds on the size of the largest\nintersecting family. This theorem is also extended to the space of\npermutations where a set of permutations is said to be intersecting\nif each pair of permutations agree in some coordinate~\\cite{frankl-deza,cameron2003intersecting,Godsil2009EKR}.\nIn our formulation, we require the intersections to be small, unlike\nfor intersecting families where the intersection size may be arbitrary\nlarge as long as it is non-zero. Furthermore, we require our subsets\nto be organized into ordered partitions, with the intersection property\nholding only for parts at the same location. Although the code-anticode\ntheorem by Delsarte~\\cite{delsarte1973algebraic,Schwartz2011Optimalpermutationanticodes}\nmay help in establishing bounds on families of subsets intersecting\nin a few elements only, it cannot be used for the specialized ordered\nset partition setting in a simple manner. To the best of our knowledge,\nthe almost disjoint set partition family problem has not been previously\nstudied in the extremal combinatorics literature.\n\nNext, we describe two methods for constructing codes that satisfy\nthe conditions of Lemma~\\ref{lem:construct-1}.\n\n\n\\subsubsection{A Construction based on grouping elements\\label{sub:grouping}}\n\nIf $r$ is a multiple of $2t+1$, the following simple construction\nsatisfies the conditions of Lemma~\\ref{lem:construct-1}. Partition\nthe set $\\left[n\\right]$ in an arbitrary fashion into $n\/(2t+1)$\nparts $E_{1},\\cdots,E_{n\/(2t+1)}$, each of size $2t+1$. Consider\nall ordered partitions $\\mathfrak{o}$ of $[n]$ into $n\/r$ parts of size\n$r$ that place all elements of each $E_{j},j\\in[n\/(2t+1)],$ in the\nsame part. Let $C$ be a code such that its corresponding set of ordered\nset partitions $\\mathfrak{O}_{r}(C)$ consists of the set of aforementioned\npartitions $\\mathfrak{o}$.\n\nAs an illustration, suppose $t=1$, $r=6$, and $n=12$, and let $\\{1,\\dotsc,12\\}$\nbe partitioned as $\\{E_{1},E_{2},E_{3},E_{4}\\}$, with \n\\begin{align*}\nE_{1} & =\\{1,2,3\\}, & E_{2} & =\\{4,5,6\\},\\\\\nE_{3} & =\\{7,8,9\\}, & E_{4} & =\\{10,11,12\\}.\n\\end{align*}\nNext, consider ordered partitions of $\\{1,\\dotsc,12\\}$ that place\nall elements of each $E_{i}$ in the same part, namely,\n\\begin{align*}\n\\mathfrak{o}_{1} & =\\left(\\{\\underline{1,2,3},\\underline{4,5,6}\\},\\{\\underline{7,8,9},\\underline{10,11,12}\\}\\right),\\\\\n\\mathfrak{o}_{2} & =\\left(\\{\\underline{1,2,3},\\underline{7,8,9}\\},\\{\\underline{4,5,6},\\underline{10,11,12}\\}\\right),\\\\\n\\mathfrak{o}_{3} & =\\left(\\{\\underline{1,2,3},\\underline{10,11,12}\\},\\{\\underline{4,5,6},\\underline{7,8,9}\\}\\right),\\\\\n\\mathfrak{o}_{4} & =\\left(\\{\\underline{4,5,6},\\underline{7,8,9}\\},\\{\\underline{1,2,3},\\underline{10,11,12}\\}\\right),\\\\\n\\mathfrak{o}_{5} & =\\left(\\{\\underline{4,5,6},\\underline{10,11,12}\\},\\{\\underline{1,2,3},\\underline{7,8,9}\\}\\right),\\\\\n\\mathfrak{o}_{6} & =\\left(\\{\\underline{7,8,9},\\underline{10,11,12}\\},\\{\\underline{1,2,3},\\underline{4,5,6}\\}\\right).\n\\end{align*}\nThen, the code corresponding to the set of ordered partitions $\\mathfrak{O}=\\{\\mathfrak{o}_{1},\\dotsc,\\mathfrak{o}_{6}\\}$\ncan correct one translocation error.\n\nTo see that $C$ satisfies the conditions of Lemma~\\ref{lem:construct-1},\nconsider $\\mathfrak{o},\\mathfrak{o}'\\in\\Orp rC$ and $i\\in[n\/r]$. Suppose that $\\mathfrak{o}(i)\\neq\\mathfrak{o}'(i)$.\nThere exists $E_{j}$ such that $E_{j}\\subseteq\\mathfrak{o}(i)$ but $E_{j}\\cap\\mathfrak{o}'(i)=\\emptyset$.\nSince $|E_{j}|=2t+1$, we have $\\left|\\mathfrak{o}\\cap\\mathfrak{o}'(i)\\right|<r-2t$.\n\nThe simplicity of this construction allows for fast decoding. Without\nloss of generality, assume that \n\\[\nE_{j}=\\{(j-1)(2t+1)+1,\\dots,j(2t+1)\\}.\n\\]\nSuppose that $\\pi$ is the stored codeword and $\\omega$ is the retrieved\npermutation. For each $i\\in[n\/r]$, we have $E_{j}\\subseteq\\orp r{\\pi}(i)$\nif $|E_{j}\\cap\\orp r{\\omega}(i)|\\ge t+1$. To compute $|E_{j}\\cap\\orp r{\\omega}(i)|$,\n$j\\in\\left[n\/\\left(2t+1\\right)\\right]$, we compare each element of\n$\\orp r{\\omega}(i)$ with $j\\left(2t+1\\right)$, $j\\in\\left[n\/\\left(2t+1\\right)\\right]$.\nThis can be performed in $O\\left(\\frac{rn}{2t+1}\\right)$ steps. Hence,\ndecoding can be performed in time $O(\\frac{n}{r}\\frac{rn}{2t+1})=O(n^{2})$.\n\nLet $d=2t+1.$ The cardinality of the code $C$ equals \n\\[\n\\frac{(n\/d)!}{\\left((r\/d)!\\right)^{n\/r}},\n\\]\nand thus the asymptotic rate is \n\\begin{align*}\n\\lim\\frac{\\frac{n}{d}\\ln\\frac{n}{d}-\\frac{n}{d}\\ln\\frac{r}{d}+O(n)}{n\\ln n+O(n)} & =\\lim\\frac{1}{d}\\frac{\\ln n-\\ln r+O(1)}{\\ln n+O(1)}\\\\\n & =(1-\\rho)\\lim\\frac{1}{d}.\n\\end{align*}\nHence, the asymptotic rate is nonzero iff $d$ is bounded (constant).\nWhile the rate of the code does not approach capacity, it should be\nnoted that, per Remark~\\ref{rem:gen-errors}, the code can correct\nmore general errors than translocation errors.\n\n\n\\subsubsection{Constructions based on combinatorial designs\\label{sub:designs}}\n\nSeveral well-known -- and a number of significantly lesser known --\nfamilies of combinatorial objects are closely related to the notion\nof almost disjoint ordered set partition families. These include block\ndesigns and Latin squares. From the first category, we use \\emph{Steiner\nsystems }and \\emph{resolvable balanced incomplete block designs} and,\nfrom the latter category, we mention \\emph{semi-Latin squares}, representing\na generalization of the well-known family of Latin squares \\cite{stinson2004combinatorial}.\nThe constructions are straightforward consequences of the definition\nof almost disjoint sets, but they provide for a rather limited set\nof code parameters. A more general method, based on interleaving arguments,\nwill be presented in the next subsection.\n\nA Latin square of order $n$ is an $n\\times n$ array such that each\nelement of $[n]$ appears exactly once in each row and exactly once\nin each column. A semi-Latin square with parameters $n$ and $r$\nis an $\\frac{n}{r}\\times\\frac{n}{r}$, array where each cell is an\n$r$-subset of $[n]$ such that each element in $[n]$ appears exactly\nonce in each column and exactly once in each row~\\cite{Preece1983SemiLatin}.\nAn example of a semi-Latin square is shown below, with $n=6,\\, r=2$:\n\n\\begin{table}[H]\n\\noindent \\centering{}%\n\\begin{tabular}{|c|c|c|}\n\\hline \n\\{1,4\\}  & \\{2,5\\}  & \\{3,6\\} \\tabularnewline\n\\hline \n\\{3,5\\}  & \\{1,6\\}  & \\{2,4\\} \\tabularnewline\n\\hline \n\\{2,6\\}  & \\{3,4\\}  & \\{1,5\\} \\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\n\nNote that the definition of a semi-Latin square implies that each\nrow and each column of the square represent a partition of $[n]$.\nHence, we arrive at the following result.\n\\begin{lem}\n\\label{lem:semi-latin-code}The rows of a semi-Latin square with parameters\n$n$ and $r$, viewed as ordered set partitions of $[n]$, form the\nordered set partitions of an $\\mpcode{\\circ}(n,r,r)$ code of cardinality\n$\\frac{n}{r}$.\n\\end{lem}\nThe result is a direct consequence of Lemma~\\ref{lem:construct-1}\nand the fact that no element is repeated in a column of a semi-Latin\nsquare. Unfortunately, the size of a code based on semi-Latin squares\nis small, since the row-column restrictions are too strong for the\npurpose of designing almost disjoint ordered set partition families. \n\nAs stated before, a code that satisfies the conditions of Lemma~\\ref{lem:construct-1}\ncan be decoded in time $O(Mnr)$. This implies that the code of Lemma~\\ref{lem:semi-latin-code}\nis decodable in time $O(n^{2})$.\n\nAnother family of combinatorial objects that allow for constructing\nalmost disjoint ordered set partitions are special types of designs,\nnamely \\emph{resolvable balanced incomplete block designs} and \\emph{resolvable\nSteiner systems}.\n\nA $k$-$(n,r,\\lambda)$\\emph{-design} is a family of $r$-subsets\nof a set $X$ of size $n$, each called a \\emph{block}, such that\nevery $k$-subset of $X$ appears in exactly $\\lambda$ blocks. Such\na design is \\emph{resolvable }if its blocks can be grouped into $m$\nclasses, such that each class forms a partition of $X$. It is known\nthat~\\cite[p.~202]{stinson2004combinatorial}\n\\[\nm=\\lambda\\frac{\\binom{n-1}{k-1}}{\\binom{r-1}{k-1}}\\cdot\n\\]\n\n\nA \\emph{Steiner system} $S(k,r,n)$ is a $k$-$(n,r,1)$-design and\na \\emph{balanced incomplete block design} (BIBD) with parameters $(n,r,\\lambda)$\nis a $2$-$(n,r,\\lambda)$-design. For the purpose of code construction,\nresolvable Steiner systems and Resolvable BIBDs (RBIBDs) are of special\ninterest.\n\nThe following lemma shows that resolvable Steiner systems can be used\nto construct multipermutation codes in the Ulam metric. Resolvable\ndesigns may also be used to construct multipermutation codes in the\nHamming metric, as described by Chu et al.~\\cite{Chu2006OnCCCs}.\nThe aforementioned construction nevertheless does not cater to the\nspecialized requirements posed by the Ulam metric.\n\\begin{lem}\n\\label{lem:k-designs}If a resolvable Steiner system $S(k,r,n)$ exists,\nthen there exists an $\\mpcode{\\circ}(n,r,d)$, where $d$ is an odd\nnumber satisfying $d\\le r-k+1$, of size \n\\[\n\\frac{\\binom{n-1}{k-1}}{\\binom{r-1}{k-1}}\\left(\\frac{n}{r}\\right)!.\n\\]\n\\end{lem}\n\\begin{IEEEproof}\nWe use a Steiner system $S(k,r,n)$ to construct a family of ordered\nset partitions satisfying the conditions of Lemma~\\ref{lem:construct-1}.\n\nLet $m$ denote the number of classes of the Steiner system. The blocks\nof each of the $m$ classes of the Steiner system form an unordered\nset partition. Each unordered set partition gives rise to $\\left(\\frac{n}{r}\\right)!$\nordered set partitions. Hence, in total, we have $m(\\frac{n}{r})!$\nordered set partitions. Let $C$ be a code such that its corresponding\nset of ordered set partitions $\\Orp rC$ is the aforementioned set\nof $m(\\frac{n}{r})!$ partitions. \n\nLet $t=(d-1)\/2$. We have $k\\le r-2t$. Each two blocks in the Steiner\nsystem have less than $k$ elements in common, and consequently, have\nless than $r-2t$ elements in common. It follows that the conditions\nof Lemma~\\ref{lem:construct-1} are satisfied. Hence, $C$ is an\n$\\mpcode{\\circ}(n,r,d)$ of the stated size.\n\\end{IEEEproof}\nThe code described in the preceding lemma can be decoded in time $O(n^{k}r)$\nas follows. Suppose that $\\pi$ is the stored codeword and $\\omega$\nis the retrieved permutation. For each $i\\in[n\/r]$, to find $\\orp r{\\pi}(i)$,\none needs to compute the size of the intersection of $\\orp r{\\omega}(i)$\nwith the blocks of the Steiner system. Computing each intersection\ntakes $O(r^{2})$. Hence, decoding can be performed in time \n\\[\nO\\left(\\frac{n}{r}\\frac{\\binom{n-1}{k-1}}{\\binom{r-1}{k-1}}r^{2}\\right)=O\\left(nr\\binom{n-1}{k-1}\\right)=O\\left(n^{k}r\\right).\n\\]\n\n\nAn RBIBD with parameters $(n,r,\\lambda=1)$ is a resolvable Steiner\nsystem $S(2,r,n)$, and thus can be used for code construction. For\n$\\lambda=1$, the case of interest in all our subsequent derivations,\nthe condition \n\\[\nn=r\\;\\mod r(r-1)\n\\]\nis necessary for the existence of an RBIBD, and it is also known to\nbe asymptotically sufficient for $r\\geq5$~\\cite{rumov1976existence}.\n\nTwo of the most commonly used approaches to constructing RBIBDs are\nbased on finite fields~\\cite{rumov1976existence} and on a simple\ncombinatorial construction~\\cite{khare1981simple}. Using the former\nconstruction, one can derive RBIBDs with parameters $\\lambda=1,$\n$n=p^{\\alpha v}$, and $r=p^{\\alpha}$, with $p$ a prime and $\\alpha$\nand $v$ positive integers. \n\nThe combinatorial construction of~\\cite{khare1981simple} is based\non the following straightforward procedure. Assume that $r$ is prime\nand arrange the $n=r^{2}$ elements of the $n$-set into an $r\\times r$\narray in order. Each row corresponds to one block of size $r$, and\neach array represents a class that partitions the $n$-set. The first\nclass, denoted by $C_{1}$, is shown below for $r=3$:\n\\begin{table}[H]\n\\centering{}%\n\\begin{tabular}{|c|c|c|}\n\\hline \n1  & 2  & 3 \\tabularnewline\n\\hline \n4  & 5  & 6 \\tabularnewline\n\\hline \n7  & 8  & 9 \\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\n\nClass $C_{2}$ is constructed from class $C_{1}$ by taking the transpose.\nEach subsequent class $C_{i}$, for $i\\geq3$, is constructed from\nthe previous class $C_{i-1}$ in the following manner: the cyclically\ncontinued diagonals of $C_{i-1}$ are arranged row-wise, starting\nfrom the main diagonal, and then moving to the left sub-diagonals.\nFor the example with $r=3$, the additional three classes constructed\naccording to the above procedure take the form:\n\\begin{table}[H]\n\\centering{}%\n\\begin{tabular}{|c|c|c|}\n\\hline \n1  & 4  & 7 \\tabularnewline\n\\hline \n2  & 5  & 8 \\tabularnewline\n\\hline \n3  & 6  & 9 \\tabularnewline\n\\hline \n\\end{tabular}~~~~%\n\\begin{tabular}{|c|c|c|}\n\\hline \n1  & 5  & 9 \\tabularnewline\n\\hline \n2  & 6  & 7 \\tabularnewline\n\\hline \n3  & 4  & 8 \\tabularnewline\n\\hline \n\\end{tabular}~~~~%\n\\begin{tabular}{|c|c|c|}\n\\hline \n1  & 6  & 8 \\tabularnewline\n\\hline \n2  & 4  & 9 \\tabularnewline\n\\hline \n3  & 5  & 7 \\tabularnewline\n\\hline \n\\end{tabular} \n\\end{table}\n\n\nNote that the procedure terminates after $r+1$ steps, resulting in\na repetition of class $C_{2}$. The total number of blocks in the\nRBIBD equals $r(r+1)=r^{2}+r$. \n\nThe construction involving cyclic diagonal shifts can be extended\nfor resolvable, \\emph{unbalanced} IBDs with parameters $n=p^{\\alpha}r$,\n$\\alpha\\geq1$, $p$ prime, and block size $r$ which may be an arbitrary\ninteger $\\geq2$. The only difference between a balanced and unbalanced\ndesign is the requirement that any pair of elements appear in \\emph{at\nmost} $\\lambda$ blocks\\ \\cite{khare1981simple}. For the case $\\lambda=1,$\ni.e., any pair of elements appearing zero or one time, the designs\nare known as zero-one concurrence designs; they may be constructed\nby a combination of variety cutting and the diagonalization procedure\ndescribed above. The interested reader is referred to \\cite{khare1981simple}\nfor an in-depth treatment of this construction. \n\nThe aforementioned procedures show that an RBIBD with parameters $(r^{2},r,1)$\nexists, provided that $r$ is am odd prime. Hence, using Lemma \\ref{lem:k-designs},\nwe can obtain the following lemma, which concludes this subsection.\n\\begin{lem}\nSuppose that $r$ is an odd prime. Then, $A_{\\circ}(r^{2},r,r-2)\\ge(r+1)r!$.\n\\end{lem}\n\n\n\n\\subsection{Construction based on  codes with $r$ components \\label{sub:Ulam-rcomps}}\n\nIn this subsection, we present a construction for multipermutation\ncodes in the Ulam metric based on $r$ permutation codes of length\n$n\/r$, interleaved to ensure translocation error protection.\n\nAssume first that $d\\le n\/r$. Consider a partition $\\{P_{1},\\dotsc,P_{r}\\}$\nof $[n]$ into sets of equal size, and the set of codes $\\{C_{1},\\dotsc,C_{r}\\}$,\nwith each $C_{i},i\\in[r],$ being a permutation code of minimum Ulam\ndistance $d$ over $P_{i}$. We form a new code $C$ as follows:\n\\begin{equation}\nC=\\bigcup_{c_{i}\\in C_{i},\\,\\forall\\, i}\\eqclass r(c_{1}\\circ\\dotsm\\circ c_{r}).\\label{eq:interleaved-code}\n\\end{equation}\n\n\\begin{prop}\n\\label{prop:rcomps}The code $C$ given in~(\\ref{eq:interleaved-code})\nis an $\\mpcode{\\circ}(n,r,d)$ code.\\end{prop}\n\\begin{IEEEproof}\nConsider $\\pi',\\sigma'\\in C$ such that $\\pi'\\noteqv r\\sigma'$. By\nconstruction, there exists $\\pi\\in\\eqclass r(\\pi')$ and $\\sigma\\in\\eqclass r(\\sigma')$\nsuch that\n\\begin{align}\n\\sigma & =\\sigma_{1}\\circ\\dotsm\\circ\\sigma_{r},\\quad\\sigma_{i}\\in C_{i},\\nonumber \\\\\n\\pi & =\\pi_{1}\\circ\\dotsm\\circ\\pi_{r},\\quad\\pi_{i}\\in C_{i}.\\label{eq:rcomps1}\n\\end{align}\nSince $\\pi\\noteqv r\\sigma$, there exists an element $j\\in[r]$ such\nthat $\\pi_{j}\\neq\\sigma_{j}$. \n\nWe show that for an arbitrary choice of $\\alpha\\in\\eqclass r(\\pi)$\nand $\\beta\\in\\eqclass r(\\sigma)$, we have $\\tdist{}(\\alpha,\\beta)\\ge d$.\nSince $\\alpha$ and $\\beta$ are chosen arbitrarily, we find that\n\\[\n\\tdist r(\\pi',\\sigma')=\\tdist r(\\pi,\\sigma)=\\min_{\\alpha\\in\\eqclass r(\\pi)}\\min_{\\beta\\in\\eqclass r(\\sigma)}\\tdist{}(\\alpha,\\beta)\\ge d,\n\\]\nwhich completes the proof. \n\nFor $\\alpha\\in\\eqclass r(\\pi)$ and $\\beta\\in\\eqclass r(\\sigma)$\nand each $i\\in[r]$, the order of the elements of $P_{i}$ is the\nsame in $\\pi$ and in $\\alpha$, i.e., $\\proj{\\pi}{P_{i}}=\\proj{\\alpha}{P_{i}}.$\nFurthermore, since $\\pi_{i}\\in C_{i}$, and $C_{i}$ is a code over\n$P_{i}$, we have $\\pi_{i}=\\proj{\\pi}{P_{i}}$. Hence, $\\proj{\\alpha}{P_{i}}=\\pi_{i}$.\nA similar argument holds for $\\sigma$ and $\\beta$, implying that\n$\\proj{\\beta}{P_{i}}=\\sigma_{i}$ for each $i\\in[r]$. So, by Lemma~\\ref{lem:projection},\none can show that\n\\begin{align*}\n\\tdist{}(\\alpha,\\beta) & \\ge\\sum_{i=1}^{r}\\tdist{}\\left(\\proj{\\alpha}{P_{i}},\\proj{\\beta}{P_{i}}\\right)=\\sum_{i=1}^{r}\\tdist{}\\left(\\pi_{i},\\sigma_{i}\\right)\\\\\n & \\ge\\tdist{}\\left(\\pi_{j},\\sigma_{j}\\right)\\ge d,\n\\end{align*}\nwhere the last inequality follows from $\\pi_{j}\\neq\\sigma_{j}$.\n\\end{IEEEproof}\nAs an example, for $n=6$, $r=2$, and $d=2$, consider\n\\begin{align*}\nP_{1} & =\\left\\{ 1,2,3\\right\\} ,\\\\\nP_{2} & =\\left\\{ 4,5,6\\right\\} ,\\\\\nC_{1} & =\\left\\{ \\left(1,2,3\\right),\\left(3,2,1\\right))\\right\\} ,\\\\\nC_{2} & =\\left\\{ \\left(4,5,6\\right),\\left(6,5,4\\right)\\right\\} .\n\\end{align*}\nNote that $C_{1}$ and $C_{2}$ both have Ulam distance equal to 2.\nThe code $C$, constructed according to~(\\ref{eq:interleaved-code})\ncontains\n\\[\n\\begin{array}{cc}\n\\left(1,4,2,5,3,6\\right), & \\left(1,6,2,5,3,4\\right),\\\\\n\\left(3,4,2,5,1,6\\right), & \\left(3,6,2,5,1,4\\right),\n\\end{array}\n\\]\nand their equivalency classes under $\\eqv 2$. For instance, let $\\pi=\\left(1,4,2,5,3,6\\right)$\nand $\\sigma=\\left(3,4,2,5,1,6\\right)$ and consider $\\alpha=\\left(4,1,5,2,3,6\\right)\\in\\eqclass 2\\left(\\pi\\right)$\nand $\\beta=\\left(4,3,5,2,6,1\\right)\\in\\eqclass 2\\left(\\sigma\\right)$.\nIt can be observed that $\\alpha_{P_{1}}=\\pi_{P_{1}}=\\left(1,2,3\\right)$\nand $\\alpha_{P_{2}}=\\pi_{P_{2}}=\\left(4,5,6\\right)$. Similar statements\nhold for $\\beta$ and $\\sigma$. It can also be verified that \n\\[\n\\tdist{}\\left(\\alpha,\\beta\\right)=2.\n\\]\n\n\nFor several constructions of permutation codes in the Ulam metric,\nwe refer the reader to~\\cite{farnoud2013error-correction}. \n\nThe components of the constructed code can be decoded independently.\nAs before, suppose that $\\pi$ is the stored codeword and $\\omega$\nis the retrieved permutation. Since there are at most $t=\\left\\lfloor \\frac{d-1}{2}\\right\\rfloor $\nerrors, we have $\\tdist{}(\\pi,\\omega)\\le t$. By Lemma~\\ref{lem:projection},\nthis implies that $\\tdist{}(\\pi_{P},\\omega_{P})\\le t$ for all $P\\in\\{P_{1},\\dotsc,P_{r}\\}$.\nHence, one can use a decoder for permutation codes in the Ulam metric\nthat can correct $t$ errors. Consequently, $\\orp r{\\pi}$ can be\nidentified from $\\omega_{P},P\\in\\{P_{1},\\dotsc,P_{r}\\},$ through\na parallel decoding process. Note that a simple decoding architecture\nfor a class of codes in the Ulam metric was proposed in our companion\npaper~\\cite{farnoud2013error-correction}, based on Hamming distance\ndecoding of de-interleaved component codes.\n\nAssuming that the cardinality of the codes $C_{i}$ equals $A_{\\circ}(n\/r,1,d)$,\nthe cardinality of $C$ equals $A_{\\circ}(n\/r,1,d)^{r}.$ Recall that\nwe define the cardinality of a multipermutation code as the number\nof its equivalency classes and not the number of its elements. It\nwas proved in~\\cite{farnoud2013error-correction} that \n\\[\nA_{\\circ}(m,1,d)\\ge\\frac{(m-d+1)!}{\\binom{m}{d-1}}\\cdot\n\\]\nHence,\n\\[\nA_{\\circ}(n,r,d)\\ge\\left(\\frac{(n\/r-d+1)!}{\\binom{n\/r}{d-1}}\\right)^{r}.\n\\]\nFurthermore, from the fact that $\\mathcal{C}_{\\circ}(1,d)=1-\\delta$~\\cite{farnoud2013error-correction},\nwe find that \n\\begin{align*}\n\\mathcal{C}_{\\circ}(r,d) & =\\lim\\frac{\\ln A_{\\circ}(n,r,d)}{\\ln n!}\\\\\n & \\ge\\lim\\frac{r\\ln A_{\\circ}(n\/r,1,d)}{\\ln n!}\\\\\n & =\\lim\\frac{\\ln A_{\\circ}(n\/r,1,d)}{\\ln(n\/r)!}\\lim\\frac{r\\ln(n\/r)!}{\\ln n!}\\\\\n & =(1-\\lim\\frac{rd}{n})(1-\\rho).\n\\end{align*}\nIn particular, if $\\lim\\frac{rd}{n}=0$, then $\\mathcal{C}_{\\circ}(r,d)=(1-\\rho).$\n\n\n\\subsection{Construction based on codes in the Hamming metric}\n\nRecall that $\\hdist r(\\pi,\\sigma)\\ge\\tdist r(\\pi,\\sigma).$ Thus,\nif $C$ is an $\\mpcode{\\circ}(n,r,d)$ code, then it is also an $\\mpcode H(n,r,d)$\ncode. We now show that an $\\mpcode{\\circ}(n,r,d)$ code can be obtained\nusing multipermutation Hamming codes of shorter lengths. We refer\nthe reader to~\\cite{Luo2003CCCZq,Ding2005CombinatorialCCC,Huczynska2006FrequencyPermutationArrays,Chu2006OnCCCs}\nfor constructions of multipermutation codes in the Hamming metric.\n\\begin{prop}\n\\label{prop:one-stage}Suppose that $n\/r$ is even and that $d\\le r$.\nLet $P=\\left[\\frac{n}{2}\\right]$, and $Q=[n]\\backslash P$. Additionally,\nlet $C'_{1}$ be an $\\mpcode{\\circ}(\\frac{n}{2},r,d)$ code over $P$\nand $C_{1}$ be an $\\mpcode H(\\frac{n}{2},r,d)$ code over $Q$. The\ncode $C=C'_{1}\\circ_{r}C_{1}$ is an $\\mpcode{\\circ}(n,r,d)$ code.\\end{prop}\n\\begin{IEEEproof}\nLet $\\pi,\\sigma\\in C$ with $\\pi\\noteqv r\\sigma$. Assume that \n\\begin{align*}\n\\pi & =\\pi'_{1}\\circ_{r}\\pi_{1}, & \\sigma & =\\sigma'_{1}\\circ_{r}\\sigma_{1},\n\\end{align*}\nwhere $\\pi_{1}',\\sigma_{1}'\\in C_{1}'$ and $\\pi{}_{1},\\sigma_{1}\\in C_{1}$. \n\nFirst, suppose that $\\pi_{1}'\\noteqv r\\sigma_{1}'$. Then,\n\\begin{align*}\n\\tdist r(\\pi,\\sigma) & =\\min_{\\alpha\\in\\eqclass r(\\pi)}\\min_{\\beta\\in\\eqclass r(\\sigma)}\\tdist{}(\\alpha,\\beta)\\\\\n & \\ge\\min_{\\alpha\\in\\eqclass r(\\pi)}\\min_{\\beta\\in\\eqclass r(\\sigma)}\\tdist{}(\\alpha_{P},\\beta_{P})\\\\\n & \\ge d,\n\\end{align*}\nwhere the first inequality follows from Lemma~\\ref{lem:projection},\nand the second inequality follows from the facts that $\\alpha_{P}\\in\\eqclass r(\\pi_{1}')\\subseteq C_{1}'$,\n$\\beta_{P}\\in\\eqclass r(\\sigma_{1}')\\subseteq C_{1}'$, and that $C_{1}'$\nis an $\\mpcode{\\circ}(n\/2,r,d)$ code.\n\nNext, suppose that $\\pi_{1}'\\eqv r\\sigma_{1}'$. Since $\\pi\\noteqv r\\sigma$,\nwe have $\\pi_{1}\\noteqv r\\sigma_{1}$. Let \n\\[\nD=\\left\\{ x\\in Q:x\\in\\orp r{\\pi_{1}}(i),x\\in\\orp r{\\sigma_{1}}(j),i\\neq j\\right\\} \n\\]\nbe the set of elements of $Q$ that are of different ranks in $\\pi_{1}$\nand $\\sigma_{1}$. Note that $|D|=\\hdist r(\\pi_{1},\\sigma_{1})$.\n\nConsider $\\alpha\\in\\eqclass r(\\pi)$ and $\\beta\\in\\eqclass r(\\sigma)$.\nFor odd values of $i$, we have $\\orp r{\\alpha}(i)=\\orp r{\\beta}(i)$,\nas $\\pi_{1}'\\eqv r\\sigma_{1}'$. \n\nOn the one hand, for any common subsequence of $\\alpha$ and $\\beta$\nthat contains an element of $D$, there exists some odd $i$ such\nthat $\\orp r{\\alpha}(i)=\\orp r{\\beta}(i)$ is not in that subsequence.\nThis implies that the length of the given common subsequence is at\nmost $n-r$. On the other hand, for any common subsequence of $\\alpha$\nand $\\beta$ that does not contain any element of $D$, the length\nof that subsequence is at most \n\\begin{align*}\nn-|D| & =n-\\hdist r(\\pi_{1},\\sigma_{1})\\le n-d.\n\\end{align*}\nHence, the length of any common subsequence of $\\alpha$ and $\\beta$\nis at most \n\\[\n\\max\\{n-d,n-r\\}=n-d\n\\]\nand thus $\\tdist{}(\\alpha,\\beta)\\ge d.$ Since $\\alpha$ and $\\beta$\nare arbitrary elements of $\\eqclass r(\\pi)$ and $\\eqclass r(\\sigma)$,\nrespectively, we find that $\\tdist r(\\pi,\\sigma)\\ge d,$ which completes\nthe proof.\n\\end{IEEEproof}\nOne particularly simple choice for $C_{1}'$ is \n\\begin{equation}\nC_{1}'=\\eqclass r\\left((1,\\dotsc,n\/2)\\right),\\label{eq:one-codeword}\n\\end{equation}\nwhich is a code with cardinality 1. \n\nAs an example, let $n=8$, $r=2$, $d=4$, and \n\\begin{align*}\nC_{1}' & =\\eqclass 2\\left(\\left(1,2,3,4\\right)\\right)\\\\\n & =\\left\\{ \\left(1,2,3,4\\right),\\left(2,1,3,4\\right),\\left(1,2,4,3\\right),\\left(2,1,4,3\\right)\\right\\} ,\\\\\nC_{1} & =\\eqclass 2\\left(\\left(5,6,7,8\\right)\\right)\\cup\\eqclass 2\\left(\\left(7,8,5,6\\right)\\right),\n\\end{align*}\nwhich leads to\n\\[\nC=\\eqclass 2\\left(\\left(1,2,5,6,3,4,7,8\\right)\\right)\\cup\\eqclass 2\\left(\\left(1,2,7,8,3,4,5,6\\right)\\right),\n\\]\nan $\\mpcode{\\circ}\\left(8,2,4\\right)$ code. \n\nFor the case of~(\\ref{eq:one-codeword}), the cardinality of $C$\nequals the cardinality of $C_{1}$, which may be as large as $A_{H}(n\/2,r,d).$\nHence,\n\\[\nA_{\\circ}(n,r,d)\\ge A_{H}(n\/2,r,d)\n\\]\n if $n\/r$ is even and $d\\le r$. With similar arguments, one can\nshow that, if $n\/r$ is odd and $d\\le r$, then \n\\[\nA_{\\circ}(n,r,d)\\ge A_{H}\\left((n+r)\/2,r,d\\right).\n\\]\nFor $d\\le r$ and $\\rho<1$, we have $\\delta=0$. Hence, for $d\\le r$\nand $\\rho<1$, \n\\[\n\\mathcal{C}_{\\circ}(r,d)\\ge\\frac{1}{2}(1-\\rho)(1-2\\delta)=\\frac{1}{2}(1-\\rho).\n\\]\n\n\nTo construct larger codebooks, one may recursively use the construction\nof Prop.~\\ref{prop:one-stage} to design $C_{1}'$. For simplicity,\nsuppose that $n$ and $r$ are both powers of 2. Let \n\\[\nC=\\left(\\left(C'_{k}\\circ_{r}C_{k}\\right)\\circ_{r}C_{k-1}\\circ_{r}\\dotsm\\right)\\circ_{r}C_{1},\n\\]\nwhere each $C_{i},i\\in[k],$ is an $\\mpcode H\\left(n\/2^{i},r,d\\right)$\ncode, $C_{k}'=\\eqclass r\\left((1,\\dotsc,n\/2^{k})\\right)$, and $k$\nis a positive integer satisfying $k\\le\\lg(n\/r)$. The condition $k\\le\\lg(n\/r)$\nis required since we need $n\/2^{k}\\ge r$. Note that this condition\nalso implies that $n\/2^{k}\\ge d$, since $d\\le r$. The code $C$\nis an $\\mpcode{\\circ}(n,r,d)$ code. The cardinality of $C_{i}$ can\nbe as large as $A_{H}\\left(n\/2^{i},r,d\\right)$. Hence, if $n$ and\n$r$ are powers of 2 and $d\\le r$, it holds that\n\\[\nA_{\\circ}(n,r,d)\\ge\\prod_{i=1}^{k}A_{H}\\left(n\/2^{i},r,d\\right).\n\\]\nLet $n=2^{j}$, $r=2^{\\rho j}$, $d\\le r$, where $\\rho$ is a constant\nless than 1, and suppose that $k$ is a constant such that $k\\le\\lg(n\/r)=j(1-\\rho)$.\nFor this regime, we have\n\\begin{align*}\n\\lim_{j\\to\\infty}\\frac{\\ln A_{\\circ}(2^{j},2^{\\rho j},d)}{\\ln n!} & \\ge\\lim_{j\\to\\infty}\\frac{\\lg A_{\\circ}(2^{j},2^{\\rho j},2^{\\rho j})}{\\lg2^{j}!}\\\\\n\\ge & \\sum_{i=1}^{k}\\lim_{j\\to\\infty}\\frac{\\lg A_{H}(2^{j-i},2^{\\rho j},2^{\\rho j})}{\\lg2^{j-i}!}\\frac{\\lg2^{j-i}!}{\\lg2^{j}!}\\\\\n= & \\sum_{i=1}^{k}\\left(1-\\lim_{j\\to\\infty}\\frac{\\rho j}{j-i}\\right)2^{-i}\\\\\n= & (1-\\rho)(1-2^{-k}).\n\\end{align*}\nSince $\\rho<1$, $k$ can be chosen arbitrarily large. Hence, the\nasymptotic rate can be made arbitrary close to $(1-\\rho)$.\n\n\n\\section{Conclusion\\label{sec:Conclusion}}\n\nWe studied a novel rank modulation scheme based on multipermutation\ncodes in the Ulam metric. We also highlighted the close connection\nbetween multipermutation codes in the Hamming metric, also known as\nconstant composition codes and frequency permutation arrays, and codes\nin the Ulam metric.\n\nThe presented results included bounds on the size of multipermutation\ncodes in both the Ulam metric and the Hamming metric; for the case\nof the Hamming metric, these bounds led to the capacity of the codes,\nwhile for the Ulam metric, the bounds led to upper bounds and lower\nbounds for the capacity, with a gap equal to $\\rho(1-\\delta)$. We\nalso presented several construction methods for codes in the Ulam\nmetric using permutation interleaving, semi-Latin squares, resolvable\nSteiner systems, and resolvable balanced incomplete block designs,\namong other techniques.\n\n\n\\paragraph*{Acknowledgment}\n\nThe work was supported by NSF grants CCF 0809895, CCF 1218764, and\nthe Emerging Frontiers for Science of Information, CCF 0939370. The\nauthors would like to thank anonymous reviewers for their insightful\ncomments and Eyal En Gad for several useful discussions.\n\n\\bibliographystyle{ieeetr}\n\\addcontentsline{toc}{section}{\\refname}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intro}\n\nAccording to the hypothesis by Kolmogorov-Obukhov-Corrsin \\citep[KOC;][]{corrsin51,kolmo41,obukhov49} within the inertial-range, the second-order structure function ($S_{2\\theta}^{KOC}$) of potential temperature ($\\theta$) in the vertical direction ($z$) should behave as follows:\n\\begin{equation}\n\\label{KOC1}\nS_{2\\theta}^{KOC} = \\langle \\left(\\Delta \\theta\\right)^2\\rangle  \\sim  \\left(\\overline{\\varepsilon}\\right)^{-1\/3} \\overline{\\chi}_{\\theta} \\left(\\Delta z\\right)^{2\/3}. \n\\end{equation}\nHere $\\Delta z$ is a separation distance that varies within the inertial range $l_0 \\ll \\Delta z \\ll \\Lambda_0$. The inner and outer scales of turbulence are denoted by $l_0$ and $\\Lambda_0$, respectively. The angular bracket and overlines denote ensemble averaging. The dissipation rates of turbulent kinetic energy (TKE; $\\overline{e}$) and variance of temperature ($\\sigma_\\theta^2$) are denoted by $\\overline{\\varepsilon}$ and $\\overline{\\chi}_{\\theta}$, respectively. \n\nIn the literature, Eq.~(\\ref{KOC1}) is commonly re-written as follows: \n\\begin{subequations}\n\\begin{equation}\nS_{2\\theta}^{KOC} = C_T^2 \\left(\\Delta z\\right)^{2\/3},  \n\\label{KOC2}\n\\end{equation}\nwhere $C_T^2$ is the so-called temperature structure parameter which is commonly expressed as \\cite{wyngaard71}: \n\\begin{equation}\nC_T^2 = \\left(\\frac{c}{2} \\right)~\\left(\\overline{\\varepsilon}\\right)^{-1\/3} \\overline{\\chi}_{\\theta} = c \\left(\\overline{\\varepsilon}\\right)^{-1\/3} \\overline{N}_{\\theta}.\n\\label{CT2a}\n\\end{equation}\n\\end{subequations}\nThe variable $\\overline{N}_{\\theta}$ is simply one half of $\\overline{\\chi}_{\\theta}$ and has been used instead of $\\overline{\\chi}_{\\theta}$ in a number of publications (including by Tatarskii~\\cite{tatarskii71}). The proportionality constant $c$ is typically assumed to be in the range of 2.8--3.2. \n\nDirect measurement of $\\overline{\\varepsilon}$, $\\overline{\\chi}_{\\theta}$ (or, $\\overline{N}_{\\theta}$), and $C_T^2$ is difficult in the atmosphere as it involves computations of fine-scale velocity and temperature gradients with spatial resolutions of a few mm (i.e., on the order of $l_0$). As a viable alternative, fifty years ago, Tatarskii~\\cite{tatarskii71} formulated the following equation: \n\\begin{equation}\n    C_T^2 = \\left(\\frac{c}{Pr_t}\\right) L_0^{4\/3} \\Gamma^2, \n    \\label{CT2b}\n\\end{equation}\nwhere $\\Gamma$ is the gradient of mean potential temperature (i.e., $\\partial\\overline{\\theta}\/\\partial z$). $L_0$ is a characteristic length scale; its relationship with $\\Lambda_0$ will be discussed later. $Pr_t$ is known as the turbulent Prandtl number. For simplicity, several studies \\cite[e.g.,][]{beland88,vanzandt78,walters81} in the past have (incorrectly) assumed $Pr_t$ to be equal to unity.\n\nAccording to Eq.~(\\ref{CT2b}), if $L_0$ and $\\Gamma$ can be reliably measured, estimated or prescribed, then $C_T^2$ values can be computed directly. Since it is relatively straightforward to measure or estimate $\\Gamma$ in the atmosphere, over the past decades, several studies focused on the estimation of $L_0$. Some of these studies  \\cite[e.g.,][]{coulman88,dewan93,vanzandt78} proposed empirical regression equations. Others \\cite{basu15,wu20} recommended usage of certain well-known length scales (e.g., Thorpe scale, Ellison scale) as surrogates of $L_0$.   \n\nThe purpose of the present study is two-fold. First, we revisit the derivation of Eq.~(\\ref{CT2b}) following the footsteps of Tatarskii~\\cite{tatarskii71}. Unfortunately, his original derivation was somewhat cryptic and included various implicit assumptions. We have attempted to elucidate on the derivation and the assumptions to the best of our abilities. In addition, we have made a sincere effort to point out some fundamental limitations of Tatarskii's assumptions in the context of atmospheric flows. Next, by relaxing some of these assumptions, we have derived an alternative to Eq.~(\\ref{CT2b}) from fundamental budget equations of stably stratified flows. In our formulation, $L_0$ is directly related to common meteorological variables (e.g., variance of potential temperature) and does not require any ad-hoc prescription. \n\nThe structure of the paper is as follows. In Section~\\ref{Tata}, we delve into the original derivation of Eq.~(\\ref{CT2b}). Its association with turbulence scaling laws in the buoyancy-range is explored in Section~\\ref{Buoyancy}. Our proposed formulation is documented in Section~\\ref{BH21}. The limitations of both Tatarskii's and our proposed formulations for convective mixed layer are discussed in Section~\\ref{CBL}. The findings of this study are summarized in Section~\\ref{Conc}. Finally, in Appendix~1, we derive formal relationships among certain length scales. \n\n\\section{Derivation of Tatarskii's $C_T^2$ Formulation}\n\\label{Tata}\n\nTatarskii utilized both heuristic arguments and turbulence variance budget equations for the derivation of Eq.~(\\ref{CT2b}). They are documented in sections 14 and 15 of \\cite{tatarskii71}. At the outset, it is important to note that these derivations do not account for the buoyancy effects (discussed below). However, in section 17 of \\cite{tatarskii71}, buoyancy effects are mentioned in a different context; there, Tatarskii derived a diagnostic relationship between $L_0$ and $\\Lambda_0$ following a formalism by Bolgiano~\\cite{bolgiano59,bolgiano62} and Obukhov~\\cite{obukhov59}. In this paper, we elaborate on all these derivations albeit with additional clarifications. Furthermore, we have slightly changed some of the notations used by Tatarskii to be consistent with the contemporary literature and also with the rest of our manuscript. For better logical reasoning, we have also changed the sequential ordering of equations used by Tatarskii.\n\n\\subsection{Heuristic Arguments}\n\n The following text has been taken verbatim from Section 14 (page 73) of \\cite{tatarskii71}:  \n\n\\begin{quotation}\n``A gradient of mean temperatures will result in a systematic temperature difference between any two points at different heights. This temperature difference $\\Delta \\theta$ is approximately given by $\\Delta \\theta \\approx \\Gamma \\Delta z$, and its square is $\\left(\\Delta \\theta\\right)^2 \\approx \\Gamma^2 \\left(\\Delta z\\right)^2$. There is also a random temperature difference between these points, whose mean square value is $C_T^2 \\left(\\Delta z \\right)^{2\/3}$. For small $\\Delta z$, $C_T^2 \\left(\\Delta z \\right)^{2\/3}$ is much greater than the gradient term $\\Gamma^2 \\left(\\Delta z\\right)^2$ (i.e., the random temperature differences are much greater than the systematic ones). There is however a certain $\\Delta z_0$ when the two factors become comparable, and for $\\Delta z > \\Delta z_0$ the mean temperature difference is greater than the random difference. This $\\Delta z_0$ is interpreted as the vertical mixing scale. Clearly the ``2\/3 law'' is applicable only over distances $\\Delta z$ not greater than this mixing scale. Therefore $\\Delta z_0$ may be taken equal to the outer scale of turbulence.''\n\\end{quotation}\nEssentially, Tatarskii suggested equating the inertial-range term (`random') with a larger-scale term (`systematic') as follows:\n\\begin{subequations}\n\\begin{equation}\n    C_T^2 \\left(\\Delta z_0 \\right)^{2\/3} = \\Gamma^2 \\left(\\Delta z_0\\right)^2,\n    \\label{Heur1}\n\\end{equation}\n\\begin{equation}\n    \\mbox{or, } C_T^2 =  \\left(\\Delta z_0\\right)^{4\/3} \\Gamma^2.\n    \\label{Heur2}\n\\end{equation}\n\\end{subequations}\nIt is needless to say that Eq.~(\\ref{Heur2}) is virtually identical to Eq.~(\\ref{CT2b}) as long as $\\Delta z_0$ is proportional to $L_0$. In a latter section of this paper, we will show that an equation similar to Eq.~(\\ref{Heur2}) can also be derived by making use of the vertical scaling characteristics of temperature and wind speed in the buoyancy-range ($\\Delta z \\gg \\Lambda_0 $).\n\nEven though the heuristic arguments can in general provide valuable insights, in turbulence research, more rigorous results can be obtained via variance and flux budget equations. Tatarskii \\cite{tatarskii71} made great strides in this direction as discussed next. \n\n\\subsection{Turbulence Variance Budget Equations}\n\nTatarskii's derivation for Eq.~(\\ref{CT2b}) is for a specific type of turbulent flow: uniform shear flow  with an imposed temperature gradient. He further (implicitly) considered temperature to be a passive scalar. This assumption implies that the dynamical evolution of temperature is driven by the velocity field; however, temperature does not modulate the velocity field in any manner. A direct consequence of this assumption is that the buoyancy terms are neglected in the Navier-Stokes momentum and TKE equations. \n\nIn line with non-buoyant, shear flow turbulence literature, Tatarskii assumed the inverse of velocity shear to be a characteristic time-scale; where, velocity shear ($S$) is: \n\\begin{equation}\n    S = \\left(\\frac{\\partial \\overline{u}}{\\partial z}\\right).\n\\end{equation}\nHere, $\\overline{u}$ is the velocity component along the mean wind direction. If $L_0$ is the characteristic length scale, then the velocity scale is simply: $L_0 S$. Then, the eddy viscosity coefficient ($K_M$), commonly parameterized as a product of characteristic length and velocity scales, can be expressed as: \n\\begin{equation}\n    K_M = L_0 \\left(L_0 S\\right) = L_0^2 S. \n    \\label{KM}\n\\end{equation}\nThis is Eq.~(14.23) of \\cite{tatarskii71}.\n\nUnder the assumption of steady-state and horizontal homogeneity, the simplified budget equation for TKE ($\\overline{e}$) can be written as:\n\\begin{equation}\n    \\overline{\\varepsilon} = - \\left( \\overline{u' w'} \\right) S. \n    \\label{EDR1}\n\\end{equation}\nHere, the left hand and right hand sides of this equation represent (molecular) dissipation and shear production of turbulence, respectively. The along-wind component of the momentum flux is $\\overline{u' w'}$. As mentioned earlier, owing to the passive scalar assumption, the buoyancy term is not included in Eq.~(\\ref{EDR1}). Furthermore, the terms with secondary importance (e.g., turbulent transport) are neglected.  \n\nAccording to the K-theory, based on the celebrated hypothesis of Boussinesq in 1877, the momentum flux can be approximated as follows: \n\\begin{equation}\n    \\overline{u' w'} = - K_M S. \n    \\label{uw}\n\\end{equation}\nBy combining Eqs.~(\\ref{EDR1}) and (\\ref{uw}), we get: \n\\begin{equation}\n    \\overline{\\varepsilon} = K_M S^2. \n    \\label{EDR2}\n\\end{equation}\nThis is Eq.~(14.21) of \\cite{tatarskii71}.\n\nSimilar to Eq.~(\\ref{EDR1}), the simplified budget equation of potential temperature variance ($\\sigma_\\theta^2$) can be written as follows: \n\\begin{equation}\n    \\overline{N}_{\\theta} = - \\left( \\overline{w' \\theta'} \\right) \\Gamma, \n    \\label{CHI1}\n\\end{equation}\nwhere $\\overline{w' \\theta'}$ denotes the sensible heat flux. Once again, by using the K-theory, we can write: \n\\begin{equation}\n    \\overline{w' \\theta'} = - K_H \\Gamma. \n    \\label{wT}\n\\end{equation}\nHere $K_H$ is the eddy-diffusivity coefficient and the ratio of $K_M$ and $K_H$ is called the turbulent Prandtl number ($Pr_t$). By combining Eqs.~(\\ref{CHI1}) and (\\ref{wT}), we get:\n\\begin{equation}\n    \\overline{N}_{\\theta} = K_H \\Gamma^2 = \\left(\\frac{K_M}{Pr_t}\\right) \\Gamma^2.\n    \\label{CHI2}\n\\end{equation}\nThis is Eq.~(14.20) of \\cite{tatarskii71}.\n\nNow, via Eqs.~(\\ref{CT2a}), (\\ref{KM}), (\\ref{EDR2}), and (\\ref{CHI2}), we can deduce Eq.~(\\ref{CT2b}) as follows: \n\\begin{widetext}\n\\begin{equation}\nC_T^2 = c \\left( K_M S^2 \\right)^{-1\/3} \\left(\\frac{K_M}{Pr_t}\\right) \\Gamma^2 = \\left(\\frac{c}{Pr_t} \\right) \\left( \\frac{K_M}{S} \\right)^{2\/3} \\Gamma^2 = \\left(\\frac{c}{Pr_t} \\right) L_0^{4\/3} \\Gamma^2. \n\\end{equation}\n\\end{widetext}\nThese equations are documented as Eqs.~(14.22) and (14.24) in \\cite{tatarskii71}.\n\nIt is well accepted in turbulence literature that both shear and buoyancy deform the larger eddies more compared to the smaller ones \\cite[e.g.,][]{smyth00,basu21a}. Corrsin \\cite{corrsin58} postulated that the eddies smaller than $L_C = \\left( \\frac{\\overline{\\varepsilon}}{S^3} \\right)^{1\/2}$ are not affected by shear.  \nIt is important to recognize that Tatarskii's length scale $L_0$ is actually Corrsin's length scale $L_C$. Based on Eqs.~(\\ref{KM}) and (\\ref{EDR2}), we can show that:\n\\begin{equation}\n    L_0 = \\left( \\frac{K_M}{S} \\right)^{1\/2} = \\left( \\frac{\\overline{\\varepsilon}}{S^3} \\right)^{1\/2} = L_C. \n    \\label{LC}\n\\end{equation}\nIn shear flow turbulence, the eddies can be assumed to be isotropic if they are smaller than $L_C$ \\cite{corrsin58,basu21a}. However, for buoyancy-dominated flows, $L_C$ (and $L_0$) may not be a relevant length scale. Tatarskii realized this possibility and advocated for a specific buoyancy length scale in Section 17 of \\cite{tatarskii71}. In the following section, we discuss Tatarskii's strategy and point out its limitations.   \n\n\\section{Buoyancy-Range Scaling}\n\\label{Buoyancy}\n\n\\subsection{Hypotheses of Bolgiano and Obukhov}\n\nOver the years, various scaling-based hypotheses have been proposed for the buoyancy-range (scales larger than the inertial-range) of turbulence. One popular hypothesis was independently formulated by Bolgiano~\\cite{bolgiano59,bolgiano62} and Obukhov~\\cite{obukhov59}. They (henceforth Bolgiano-Obukhov or BO) postulated that in the buoyancy range (i.e., $\\Delta z \\gg \\Lambda_0$), the relevant variables are $\\overline{N}_\\theta$, and $\\beta$; where the buoyancy parameter ($\\beta$) is defined as follows: \n\\begin{equation}\n\\beta = \\left(\\frac{g}{\\Theta_0}\\right).\n\\end{equation}\nHere the gravitational acceleration is denoted by $g$ and $\\Theta_0$ is a reference potential temperature. Via simple dimensional analysis, BO derived the following relationship for second-order structure function of potential temperature in the buoyancy range: \n\\begin{equation}\nS_{2\\theta}^{BO} \\sim \\left(\\overline{N}_\\theta\\right)^{4\/5} \\left( \\beta \\right)^{-2\/5} \\left( \\Delta z \\right)^{2\/5}.\n\\label{BO1}\n\\end{equation}\nA schematic of this relationship is shown in the left panel of Fig.~\\ref{schematic}. In the right panel of this figure, the BO formulation for potential temperature spectra is depicted.   \n\n\\begin{figure*}[ht]\n\\includegraphics[height=1.9 in]{SFSchematic.eps}\n\\includegraphics[height=1.9 in]{SpectraSchematic.eps}\n\\caption{Scaling regimes of atmospheric turbulence. The left and right panels represent second-order structure function and spectra of potential temperature, respectively. The scaling hypothesis by Kolmogorov-Obukhov-Corrsin (KOC) holds in the inertial-range. However, various competing hypotheses exist for the buoyancy range. The scaling relationships by Bolgiano-Obukhov (BO) and Shur-Lumley (SL) are compared in these schematics.}\n\\label{schematic}\n\\end{figure*}   \n\nFrom the schematics in Fig.~\\ref{schematic}, it is evident that at the cross-over point between the inertial-range and the buoyancy-range (i.e., $\\Delta z = \\Lambda_0^{BO}$), both Eqs.~(\\ref{KOC1}) and (\\ref{BO1}) should hold. As a result, we have: \n\\begin{subequations}\n\\begin{equation}\n\\left(\\overline{\\varepsilon}\\right)^{-1\/3} \\overline{N}_{\\theta} \\left(\\Lambda_0^{BO}\\right)^{2\/3} \\sim \\left(\\overline{N}_\\theta\\right)^{4\/5} \\left( \\beta \\right)^{-2\/5} \\left( \\Lambda_0^{BO} \\right)^{2\/5}.    \n\\label{BO2}\n\\end{equation}\nBy algebraic manipulation, we get a length scale: \n\\begin{equation}\n  \\Lambda_0^{BO} \\equiv \\left(\\overline{\\varepsilon}\\right)^{5\/4} \\left(\\overline{N}_\\theta\\right)^{-3\/4} \\left( \\beta \\right)^{-3\/2}.\n  \\label{BO3}\n\\end{equation}\nIn the literature, this specific length scale is known as the Bolgiano-Obukhov length scale. Tatarskii referred to this (buoyancy) length scale in Eq.~(17.1) of \\cite{tatarskii71}. \n\\end{subequations}\n\nTatarskii \\cite{tatarskii71} recognized that $L_0$ is essentially a free parameter in his formulation [i.e., Eq.~(\\ref{CT2b})] and should be prescribed or estimated separately. In order to circumvent this limitation, Tatarskii derived a formal relationship between $L_0$ and a more physically-based length scale $\\Lambda_0^{BO}$ as discussed below. Since Eq.~(\\ref{EDR1}) does not include the buoyancy term, as a first step, Tatarskii used a revised budget equation: \n\\begin{subequations}\n\\begin{equation}\n    \\overline{\\varepsilon} = - \\left( \\overline{u' w'} \\right) S + \\left(\\overline{w'\\theta'}\\right) \\beta. \n    \\label{EDR3}\n\\end{equation}\nThe second term on the right hand side signifies production (unstable condition) or dissipation (stably stratified condition) of TKE due to the buoyancy effects. Using the K-theory [refer to Eqs.~(\\ref{uw}) and (\\ref{wT})] and the definition of turbulent Prandtl number ($Pr_t$), this equation can be rewritten as follows: \n\\begin{equation}\n    \\overline{\\varepsilon} = K_M \\left(S^2 - \\frac{\\Gamma \\beta}{Pr_t} \\right) = K_M S^2 \\left(1 - \\frac{Ri_g}{Pr_t} \\right),\n    \\label{EDR4}\n\\end{equation}\nwhere $Ri_g$ $\\left(= \\beta \\Gamma\/S^2 \\right)$ is the gradient Richardson number. It is a non-dimensional variable which quantifies the strength of atmospheric stability. Non-buoyant conditions are characterized by $Ri_g = 0$.\n\\end{subequations}\n\nNow, by utilizing Eqs.~(\\ref{CHI2}), (\\ref{LC}), (\\ref{BO3}), and (\\ref{EDR4}), the length scale $\\Lambda_0^{BO}$ can be re-written as: \n\\begin{widetext}\n\\begin{subequations}\n\\begin{equation}\n    \\Lambda_0^{BO} = \\left( \\frac{K_M}{S} \\right)^{1\/2} \\left[ \\frac{Pr_t^{3\/4} \\left(1 - Ri_g\/Pr_t \\right)^{5\/4}}{Ri_g^{3\/2}} \\right] = L_0 \\left[ \\frac{Pr_t^{3\/4} \\left(1 - Ri_g\/Pr_t \\right)^{5\/4}}{Ri_g^{3\/2}} \\right],\n    \\label{BO4}\n\\end{equation}\nor,\n\\begin{equation}\nL_0 = \\Lambda_0^{BO} \\left[ \\frac{Ri_g^{3\/2}}{Pr_t^{3\/4} \\left(1 - Ri_g\/Pr_t \\right)^{5\/4}} \\right]. \n\\label{BO5}\n\\end{equation}\n\\end{subequations}\n\\end{widetext}\nThis is essentially Eq.~(17.10) of \\cite{tatarskii71}. \n\nIn principle, if one can measure $\\Lambda_0^{BO}$, then $L_0$ can be estimated via Eq.~(\\ref{BO5}). However, direct estimation of $\\Lambda_0^{BO}$, in our atmosphere, is challenging (if not impossible) since it requires reliable estimates of both $\\overline{\\varepsilon}$ and $\\overline{N}_\\theta$. Apart from this practical limitation, there are more fundamental issues. \n\nIn the literature, only a handful of laboratory and numerical studies of idealized flows \\cite[e.g.,][]{boffetta12,kumar14,niemela00,verma18} have ever documented the existence of the BO scaling. However,  to the best of our knowledge, the observational support for this scaling hypothesis in our atmosphere is non-existent. Thus, the usage of $\\Lambda_0^{BO}$ as an outer (cross-over) scale is questionable.  \n\nAnother critical issue is the incompatibility of $\\Lambda_0^{BO}$ with Eq.~(\\ref{CT2b}). Using Eq.~(\\ref{CT2a}), it is straightforward to manipulate Eq.~(\\ref{BO3}) to arrive at: \n\\begin{equation}\n    C_T^2 = \\left( \\overline{N}_\\theta \\right)^{4\/5} \\left(\\beta \\right)^{-2\/5} \\left( \\Lambda_0^{BO}\\right)^{-4\/15}.\n\\end{equation}\nIt is needless to say that this equation is distinctively different from Eq.~(\\ref{CT2b}). In other words, Tatarskii's strategy to couple the BO hypothesis with his own $C_T^2$ formulation is not justified. However, we can make use of a different scaling formalism to resolve this inconsistency as elaborated below. \n\n\\subsection{Hypotheses of Shur, Lumley, Monin, and Weinstock}\n\nIn the past, several studies \\cite[e.g.,][]{cot89,hovde11,nastrom97,tsuda91} have performed 1-D spectral and structure function analyses of vertical (potential) temperature  and wind speed profiles measured by radiosondes and dropsondes in the atmosphere. They often reported a quasi-universal $\\kappa_z^{-3}$ spectral scaling behavior over extended vertical ranges; where $\\kappa_z$ is the vertical wavenumber. Similarly, in the physical space, the second-order structure function was found to scale as $\\left(\\Delta z\\right)^2$. Please refer to the schematics in Fig.~\\ref{schematic} for a summary of various scaling regimes and associated hypotheses. \n\nTo explain the observed vertical scaling behaviors of (potential) temperature and wind speed profiles, several competing hypotheses have been proposed in the literature \\cite[e.g.,][]{dewan97,lumley64,monin65,shur62,weinstock85}. In this paper, we discuss the hypothesis of Shur~\\cite{shur62} and Lumley~\\cite{lumley64} (henceforth the SL hypothesis) because of its simplicity. Monin~\\cite{monin65} further simplified the SL hypothesis by using dimensional analysis and we follow his approach. He assumed that, in the buoyancy-range of turbulence, $\\beta$ and $\\Gamma$ are the only relevant variables. Then, via dimensional analysis, it is trivial to show that the second-order structure functions should behave as follows: \n\\begin{subequations}\n\\begin{equation}\n    S_{2u}^{SL} \\sim \\beta \\Gamma \\left(\\Delta z\\right)^2,\n    \\label{SLu}\n\\end{equation}\n\\begin{equation}\n    S_{2\\theta}^{SL} \\sim \\beta^0 \\Gamma^2 \\left(\\Delta z\\right)^2 = \\Gamma^2 \\left(\\Delta z\\right)^2.\n    \\label{SLth}\n\\end{equation}\n\\end{subequations}\nAccording to Kolmogorov's hypothesis \\cite{kolmo41}, the following relationship holds in the inertial-range: \n\\begin{equation}\n    S_{2u}^{K41} \\sim \\varepsilon^{2\/3} \\left(\\Delta z\\right)^{2\/3}.\n    \\label{K41u}\n\\end{equation}\n\nBy matching Eq.~(\\ref{SLu}) with Eq.~(\\ref{K41u}) at the cross-over point (i.e., $\\Delta z = \\Lambda_0^{SL}$), we get:\n\\begin{subequations}\n\\begin{align}\n    \\beta \\Gamma \\left(\\Lambda_0^{SL} \\right)^2 & \\equiv \\varepsilon^{2\/3} \\left(\\Lambda_0^{SL}\\right)^{2\/3},\\\\\n\\mbox{or, \\hspace{0.1in}}  \\left(\\Lambda_0^{SL} \\right)^{4\/3} & \\equiv \\left(\\frac{\\varepsilon^{2\/3}}{\\beta \\Gamma} \\right) = \\left(\\frac{\\varepsilon^{2\/3}}{N_{BV}^2} \\right),\\\\\n\\mbox{Therefore, \\hspace{0.1in}} \\Lambda_0^{SL} & \\equiv \\left( \\frac{\\varepsilon}{N_{BV}^3}\\right)^{1\/2} = L_{OZ}.\n\\label{LOZ}\n\\end{align}\n\\end{subequations}\nHere $N_{BV}$ is the so-called Brunt-V\\\"ais\\\"al\\\"a frequency. The length scale $L_{OZ}$ was first proposed by Ozmidov~\\cite{ozmidov1965turbulent}. \n\nIn a similar manner, by matching Eq.~(\\ref{SLth}) against Eq.~(\\ref{KOC2}) at $\\Lambda_0^{SL}$, one can deduce: \n\\begin{subequations}\n\\begin{equation}\n    C_T^2 \\left(\\Lambda_0^{SL} \\right)^{2\/3} \\equiv \\Gamma^2 \\left(\\Lambda_0^{SL}\\right)^2.\n\\end{equation}\nThus, \n\\begin{equation}\n C_T^2 \\equiv \\left(\\Lambda_0^{SL}\\right)^{4\/3} \\Gamma^2 = \\left(L_{OZ} \\right)^{4\/3} \\Gamma^2.\n\\end{equation}\n\\end{subequations}\nThis scaling-based equation is almost identical to Eq.~(\\ref{CT2b}) if $L_0$ is proportional to $L_{OZ}$ and $Pr_t$ is on the order of one. \n\nIn summary, the Bolgiano-Obukhov scaling is not in agreement with Tatarskii's $C_T^2$ formulation. Instead, in a non-rigorous manner, Tatarskii's formulation can be deduced from the well-known vertical scaling characteristics of (potential) temperature  and wind speed profiles. In the following section, we will take a completely different route. Based on the rigorous turbulence variance and flux budget equations, we will derive a revised $C_T^2$ formulation along with a physically-based length scale formulation. We want to emphasize that in contrast to Tatarskii's approach, our formulation does not require an ad-hoc prescription of an outer length scale; the length scale is an integral part of our analytical derivations.   \n\n\\vspace{0.2in}\n\\section{Revised $C_T^2$ Formulation}\n\\label{BH21}\n\nIn a recent study \\cite{basu21c}, we have derived closed-form solutions for a characteristic length scale (denoted as $L_X$) and turbulent Prandtl number ($Pr_t$) based on the conventional budget equations. For brevity, we only report the key findings from \\cite{basu21c} which are directly relevant for the present paper. In \\cite{basu21c}, we used Eqs.~(\\ref{CHI1}), (\\ref{EDR4}), and an additional budget equation for sensible heat flux to deduce:\n\\begin{align}\n    L_X = \\left(\\frac{\\sqrt{Pr_{t0} Pr_t}}{c_\\theta} \\right)\\left(\\frac{\\sigma_\\theta}{\\Gamma} \\right),\n    \\label{LX}\n\\end{align}\nwhere $L_X$ is the characteristic length scale for stably stratified flows. The standard deviation of potential temperature is $\\sigma_\\theta$. The turbulent Prandtl number for non-buoyant flows is denoted by $Pr_{t0}$; it is typically assumed to be equal to 0.85. The coefficient $c_\\theta$ is approximately equal to 2.\n\nIn \\cite{basu21c}, the dissipation rates of TKE ($\\overline{e}$) and variance of temperature ($\\sigma_\\theta^2$) are derived to be respectively as follows:\n\\begin{subequations}\n\\begin{equation}\n    \\overline{\\varepsilon} = \\left(\\frac{\\sigma_w^3}{c_w^3 L_X}\\right),\n    \\label{EDR5}\n\\end{equation}\nand\n\\begin{equation}\n    \\overline{\\chi}_\\theta = \\left(\\frac{2 Pr_{t0}}{c_w c_\\theta^2} \\right) \\left(\\frac{\\sigma_w \\sigma_\\theta^2}{L_X}\\right).\n    \\label{CHI3}\n\\end{equation}\n\\end{subequations}\nHere the standard deviation of vertical velocity is $\\sigma_w$. The coefficient $c_w$ is approximately equal to 1.25. From Eqs.~(\\ref{CT2a}), (\\ref{EDR5}), and (\\ref{CHI3}), we get: \n\\begin{widetext}\n\\begin{equation}\n    C_T^2 = \\left(\\frac{c}{2} \\right) \\left( \\frac{c_w L_X^{1\/3}}{\\sigma_w} \\right) \\left(\\frac{2 Pr_{t0}}{c_w c_\\theta^2} \\right) \\left(\\frac{\\sigma_w \\sigma_\\theta^2}{L_X}\\right) = \\left(\\frac{c Pr_{t0}}{c_\\theta^2} \\right)\\left(\\frac{\\sigma_\\theta^2}{L_X^{2\/3}} \\right).\n    \\label{CT2c}\n\\end{equation}\n\\end{widetext}\nBy substituting $\\sigma_\\theta$ from Eq.~(\\ref{LX}) to Eq.~(\\ref{CT2c}), we find: \n\\begin{equation}\n    C_T^2 = \\left(\\frac{c Pr_{t0}}{c_\\theta^2} \\right) L_X^{4\/3} \\Gamma^2 = \\left(\\frac{c}{Pr_t}\\right) L_X^{4\/3} \\Gamma^2. \n    \\label{CT2d}\n\\end{equation}\nThis equation is identical to Eq.~(\\ref{CT2b}) with one important difference. Here the length scale $L_X$ is defined by Eq.~(\\ref{LX}), and is not a free parameter. Several well-known length scales are explicitly related to $L_X$; please refer to Appendix~1 and \\cite{basu21c} for further details.\n\nEarlier, we mentioned that in the atmospheric optics literature the value of $Pr_t$ is frequently assumed to be equal to unity. However, there is ample evidence in the fluid dynamics and atmospheric turbulence literature that $Pr_t$ monotonically increases with increasing values of $Ri_g$. In this context, a physically-based formulation was derived by \\cite{basu21c}. \n\n\\section{$C_T^2$ in Convective Mixed Layer}\n\\label{CBL}\n\nConvective (unstable) condition typically occur over land during daytime conditions. At offshore locations, such conditions prevail when air temperature is colder than the underlying sea-surface temperature. Under convective conditions, due to intense turbulent diffusion, the so-called 'mixed' layer (ML) develops a few tens of meters above the land or sea-surface. Due to turbulent mixing, (potential) temperature, moisture, and other meteorological variables become uniformly distributed in the ML. In other words,  in the ML, $\\Gamma  = \\frac{\\partial \\overline{\\theta}}{\\partial z} \\approx 0$. As a result, Tatarskii's equation [i.e., Eq.~(\\ref{CT2b})] and our proposed formulation [i.e., Eq.~(\\ref{CT2d})] incorrectly predict $C_T^2 \\approx 0$ in the ML. The reasons behind this unphysical prediction are discussed below.\n\nIn the ML, large coherent structures (called thermals, plumes) are primarily responsible for turbulent mixing. These structures originate near the surface and can reach the top of the ML (on the order of a km or so) in a matter of 10 to 15 minutes. During their ascent, these structures cause `non-local' mixing. As a result, turbulent fluxes are no longer proportional to the local mean gradients. In other words, the application of K-theory is not tenable in the ML. Hence, Eqs.~(\\ref{uw}) and (\\ref{wT}) are not valid in the ML \\cite{holtslag91}. Furthermore, the simplified variance and flux budget equations [e.g., Eq.~(\\ref{EDR4})] are not suitable for the ML. Additional non-local transport terms should be included in them. Unfortunately, such modifications make these equations analytically intractable. \n\nWe recommend the readers to use Eq.~(\\ref{CT2b}) and\/or Eq.~(\\ref{CT2d}) only for stably stratified conditions. Such conditions are omni-present in the free atmosphere (i.e., the layer above the atmospheric boundary layer). Also, nocturnal boundary layers over land are commonly stably stratified. For convective ML, a similarity-based formulation by Kaimal et al.~\\cite{kaimal76} or its variants could be used. On this topic, an interesting study was recently published by \\cite{luce20}. However, more research is highly desired in this arena. \n\n\\section{Concluding Remarks}\n\\label{Conc}\n\nFifty years ago, Tatarskii developed a simple $C_T^2$ formulation which has found wide usage in a range of scientific and engineering disciplines, from astronomy to free-space optical communication. In this paper, we revisit this formulation and point out its limitations. We then propose a revised $C_T^2$ formulation based on turbulence variance and flux budget equations. In contrast to Tatarskii's equation, our formulation includes a novel length scale which is physically-based. Since this length scale is simply dependent on certain variances and mean gradients, it can be estimated from observational data. Such applications will be reported in a separate publication.  \n\n\n\n\n\\begin{acknowledgements}\nWe are truly grateful to Hubert Luce and Lakshmi Kantha for thought-provoking communications and for providing constructive feedback on our work. \n\\end{acknowledgements}\n\n\\section*{Appendix 1: Inter-relationships of Length Scales}\n\\label{Appendix}\n\nIn addition to Eq.~(\\ref{LX}), two alternative definitions of $L_X$ were proposed by \\cite{basu21c}:\n\\begin{subequations}\n\\begin{align}\n    L_X & = c_H \\left( \\frac{\\sigma_w}{S} \\right) \\left( \\frac{1}{\\sqrt{1\/G - Ri_g\/Pr_t}} \\right), \\label{LX1}\\\\\n        & = c_H \\left( \\frac{\\sigma_w}{N_{BV}} \\right) \\left( \\frac{\\sqrt{Ri_g}}{\\sqrt{1\/G - Ri_g\/Pr_t}} \\right).\n    \\label{LX2}\n\\end{align}\n\\end{subequations}\nWhere the coefficient $c_H$ is approximately equal to 0.8. The term $G$, called a `growth factor' \\cite{schumann95,basu21c}, represents the ratio of the production and dissipation terms of the TKE equation. For weakly\/moderately stable condition, $G$ equals to one. Under strongly stratified condition $G$ is expected to be smaller than one. Basu and Holtslag~\\cite{basu21c} proposed the following heuristic parameterization for $G$:\n\\begin{equation}\n    G = \\min \\left(1, Ri_g^{-1} \\right).\n\\end{equation}\n\nAn unique relationship between $L_X$ and the Corrsin's length scale ($L_C$) can be established by utilizing Eq.~(\\ref{LC}), Eq.~(\\ref{EDR5}) and Eq.~(\\ref{LX1}):\n\\begin{align}\n    \\frac{L_X}{L_C} & = \\left( \\frac{1}{\\sqrt{1\/G - Ri_g\/Pr_t}} \\right)^{3\/2}. \n\\end{align}\nWhen $Ri_g \\to 0$, $G = 1$ and $L_X \\to L_C$. \n\nA different relationship can be deduced between the Ozmidov length scale ($L_{OZ}$) and $L_X$ via Eq.~(\\ref{LOZ}), Eq.~(\\ref{EDR5}) and Eq.~(\\ref{LX2}):\n\\begin{align}\n    \\frac{L_X}{L_{OZ}} & = \\left( \\frac{\\sqrt{Ri_g}}{\\sqrt{1\/G - Ri_g\/Pr_t}} \\right)^{3\/2}. \n\\end{align}\nFor $Ri_g \\gg 1$, if $G$ is proportional to $Ri_g^{-1}$, then $L_X \\to L_{OZ}$. \n\nIn summary, the length scale $L_X$ acts as a smooth interpolator between two limiting length scales: $L_C$ and $L_{OZ}$. For this reason, it has been called a hybrid length scale. \n\n\\bibliographystyle{spbasic_updated}     \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n Let $(\\Sigma,g)$ be a compact Riemannian surface without boundary,\n $W^{1,2}(\\Sigma,\\mathbb{R})$ be the usual Sobolev space, namely the completion\n of $C^\\infty(\\Sigma,\\mathbb{R})$ under the norm\n $$\\|u\\|_{W^{1,2}(\\Sigma,\\mathbb{R})}=\\le(\\int_{\\Sigma}\\le(|\\nabla_gu|^2+u^2\\ri)dv_g\\ri)^{1\/2},$$\n where $\\nabla_gu$ denotes the gradient of $u$ and $dv_g$ denotes the volume element with respect to the Riemannian metric $g$.\n  Let $f_k:\\Sigma\\times[0,\\infty)\\ra\n \\mathbb{R}$ be a sequence of functions satisfying the\n following hypotheses:\\\\\n\n \\noindent(H1)\\,\\,$f_k(x,0)=0$, and $f_k(x,t)>0$ for all\n $k$, all $x\\in \\Sigma$, and all $t>0$;\\\\\n (H2)\\,\\,$f_k\\in C^2(\\Sigma\\times[0,+\\infty))$ for each $k$ and $f_k\\ra f_\\infty$ in\n$C^1_{\\rm\n loc}(\\Sigma\\times[0,+\\infty))$ as $k\\ra\\infty$;\\\\\n (H3)\\,\\,for any $\\nu>0$, there exists a constant\n $C_\\nu>0$ such that for all $k$, all $x\\in\\Sigma$, and all $t>0$,\n $$F_k(x,t)\\leq \\nu tf_k(x,t)+C_\\nu,$$\n where\n $$F_k(x,t)=\\int_0^tf_k(x,s)ds$$\n is the primitive of $f_k(x,t)$;\\\\\n (H4)\\,\\,$f_k^\\prime(x,t)\/(tf_k(x,t))\\ra 2$ as $t\\ra+\\infty$\n uniformly in $k\\in\\mathbb{N}$ and in $x\\in\\Sigma$, where $f_k^\\prime$ is the\n derivative of $f_k$ with respect to $t$, moreover there exists a constant $C$ such that\n $|\\nabla_gf_k(x,t)|\\leq C(1+f_k(x,t))$ for all\n $(x,t)\\in\\Sigma\\times\\mathbb{R}$;\\\\\n (H5)\\,\\,there exist $\\psi$, a continuous function with $\\psi(0)=0$, $t_0>0$,\n and $k_0>0$, such that\n  $$|f_k(x,t)\/f_k(y,t)-1|\\leq\\psi({d}_g(x,y))$$\n  for all $t\\geq t_0$, all $k>k_0$, and all $x,y\\in\\Sigma$, where $d_g(\\cdot,\\cdot)$ denotes the\n  geodesic distance between two points of $\\Sigma$.\\\\\n\n  By (H4) we have $f_k(x,t)=f_k(x,t_0)e^{(1+o(1))(t^2-t_0^2)}$ for any given $t_0>0$, where $o(1)\\ra 0$\n  as $t\\ra\\infty$ uniformly in $x\\in\\Sigma$.\n  In view of the Trudinger-Moser embedding\n \\cite{Fontana,Moser,Pohozaev,Trudinger}, we say that $f_k(x,t)$ is of critical exponential\n growth with respect to $t$. A typical example satisfying (H1)-(H5) is\n \\be\\label{sequence}f_k(x,t)=\\lambda_k te^{t^2},\\ee\n where $\\lambda_k$ is a sequence of positive real numbers such that\n $\\lambda_k\\ra\\lambda_\\infty$\n as $k\\ra\\infty$.\n Suppose that for each $k\\in\\mathbb{N}$ we have a smooth function $u_k\\geq\n 0$ satisfying the equation\n \\be\\label{e1}\\Delta_gu_k+\\tau_ku_k=f_k(x,u_k)\\quad{\\rm in}\\quad\\Sigma,\\ee\n where $\\Delta_g$ is the Laplace-Beltrami operator, $\\tau_k$ is a sequence of smooth\n functions such that\n \\be\\label{tauk}\\tau_k\\ra\\tau_\\infty\\,\\,\\,{\\rm in}\\,\\,\\,\n C^0(\\Sigma,\\mathbb{R}),\\,\\,\\,\n \\tau_\\infty(x)>0\\,\\,\\,{\\rm for\\,\\,\\,all}\\,\\,\\, x\\in \\Sigma.\\ee\n Clearly $u_k$ is a critical point of the functional\n \\be\\label{functional}J_k(u)=\\f{1}{2}\\int_\\Sigma\\le(|\\nabla_gu|^2+\\tau_ku^2\\ri)dv_g-\\int_\\Sigma F_k(x,u)dv_g\\ee\n on the Sobolev space $W^{1,2}(\\Sigma,\\mathbb{R})$. The existence of\n nonnegative solutions to equation (\\ref{e1}) in case that $\\tau_k$ is a positive real number\n was studied by  Zhao and the author\n \\cite{Yang-Zhao} by using variational methods. More explicitly, assuming that\n $\\lambda_\\tau=\\lambda_\\tau(\\Sigma)$ is the first eigenvalue of\n the operator $\\Delta_g+\\tau$, where $\\tau>0$ is a constant, we proved that\n the equation $\\Delta_gu+\\tau u=\\lambda ue^{u^2}$ has a nonnegative solution if\n $\\lambda<\\lambda_{\\tau}$. The aim of this\n paper is to study the quantization problem for equation (\\ref{e1}).\n Precisely we shall prove the following result.\\\\\n\n\\noindent{\\bf Theorem 1.1} {\\it Let $(\\Sigma,g)$ be a compact\nRiemannian surface without boundary. Suppose that $u_k\\geq 0$ is a\nsequence of smooth solutions to equation (\\ref{e1}),\n where $\\tau_k$ is a sequence of smooth functions satisfying\n (\\ref{tauk}),\n and $f_k$ is a sequence of functions\n satisfying {\\rm (H1)-(H5)}. Let $J_k$ be as in (\\ref{functional}).\n If $J_k(u_k)\\ra \\beta$ as $k\\ra\\infty$ for some\n $\\beta\\in\\mathbb{R}$, then there exists a nonnegative solution $u_\\infty\\in\n C^1(\\Sigma,\\mathbb{R})$ of the equation\n \\be\\label{3'}\\Delta_gu_\\infty+\\tau_\\infty u_\\infty=f_\\infty(x,u_\\infty(x))\\quad{\\rm in}\\quad\n \\Sigma,\\ee\n and there exists $N\\in \\mathbb{N}$ such that\n $J_k(u_k)=J_\\infty(u_\\infty)+2\\pi N+o(1)$, where $o(1)\\ra 0$ as $k\\ra\\infty$. Here $J_\\infty$ is also as in\n (\\ref{functional}), where $\\tau_k$, $F_k$ are replaced by $\\tau_\\infty$\n and $F_\\infty$ respectively.\n If $N=0$, $u_k\\ra u_\\infty$ strongly in $W^{1,2}(\\Sigma,\\mathbb{R})$ and in\n fact in $C^1(\\Sigma,\\mathbb{R})$.}\\\\\n\n Several works were devoted to prove analogues of Theorem 1.1. In\n \\cite{Adi-Stru}, Adimurthi and Struwe considered a sequence of\n solutions $u_k$ to the equation\n \\be\\label{eucl}\\le\\{\\begin{array}{lll}-\\Delta_{\\mathbb{R}^2} u_k=f_k(x,u_k)\\,\\,\\,{\\rm in}\\,\\,\\,\n \\Omega\\subset\\mathbb{R}^2\\\\[1.5ex]u_k>0\\,\\,\\,{\\rm in}\\,\\,\\,\\Omega,\\,\\,\\,u_k=0\\,\\,\\,{\\rm on}\\,\\,\\,\\p\\Omega,\\end{array}\\ri.\\ee\n where $f_k(x,t)=te^{\\varphi_k(t)}$, $0\\leq\\varphi_k^{\\prime\\prime}(t)\\leq\n 2$ for $t\\geq t_0$ and $\\varphi_k^\\prime(t)\/t\\ra 2$ as $t\\ra\\infty$\n uniformly in $k$. Such a sequence of functions $f_k$ satisfies (H1)-(H5) in case that the Riemannian surface $(\\Sigma,g)$\n is replaced by a smooth bounded domain of $\\mathbb{R}^2$. Assuming that\n $$J_k(u_k)=\\f{1}{2}\\int_\\Omega |\\nabla_{\\mathbb{R}^2} u_k|^2dx-\n \\int_\\Omega F_k(x,u_k)dx\\ra \\beta$$ for $0\\leq\\beta<4\\pi$ and that\n the limit equation does not admit any positive solution with energy\n less than $2\\pi$, they proved that either $u_k\\ra u_\\infty$ strongly in $W_0^{1,2}(\\Omega)$ and\n $u_\\infty$ has energy $\\beta$, or $u_k\\rightharpoonup 0$\n weakly in $W_0^{1,2}(\\Omega)$ and $u_k$ develops one blow-up\n point carrying the energy $2\\pi$. This quantization result was surprisingly refined by Druet\n \\cite{Druet} to the case of all $\\beta\\in\\mathbb{R}$ and general\n nonlinearities of uniform critical growth, analogous to that of the current paper.\n (Blow-up analysis for equation (\\ref{eucl}) with similar nonlinearity was\n also considered by Adimuthi and Druet \\cite{A-D}.) The key point in \\cite{Druet} is\n the gradient estimate (\\cite{Druet}, Proposition 2), through which Druet studied the energy of ${\\varphi}_k$,\n the spherical average of $u_k$ with respect to blow-up\n points,\n instead of $u_k$ itself. Thus he transformed the quantization problem for $u_k$ to the quantization problem\n for ${\\varphi}_k$, which depends only on analysis on certain ordinary differential\n equation and is comparatively easy to be handled.\n Shortly after, using similar idea, Struwe\n \\cite{Struwe} succeeded to get a quantization result for a forth order elliptic\n equation\n $$\\le\\{\\begin{array}{lll}-\\Delta_{\\mathbb{R}^4}^2 u_k=\\lambda_ku_ke^{2u_k^2}\\,\\,\\,{\\rm in}\\,\\,\\,\n \\Omega\\subset\\mathbb{R}^4\\\\[1.5ex]u_k>0\\,\\,\\,{\\rm in}\\,\\,\\,\\Omega,\\,\\,\\,u_k=\\Delta_{\\mathbb{R}^4}u_k=0\n \\,\\,\\,{\\rm on}\\,\\,\\,\\p\\Omega,\\end{array}\\ri.$$\n where $0<\\lambda_k\\ra 0$ as $k\\ra\\infty$, and $u_k\\rightharpoonup\n 0$ weakly in $W^{2,2}(\\Omega)$. Also Lamm, Robert and Struwe \\cite{L-R-S} proved\n a quantization result for the evolution of equation\n (\\ref{eucl}), where $f_k$ is as in (\\ref{sequence}). A recent inspiring work of\n Martinazzi and Struwe \\cite{Mar-Stru} states the following: Let\n $\\Omega\\subset\\mathbb{R}^{2m}$ be a smooth bounded domain, $u_k$\n be a sequence of positive solutions to the equation $-\\Delta_{\\mathbb{R}^2}u_k=\\lambda_k u_k e^{mu_k^2}$\n subject to Dirichlet boundary conditions, where $0<\\lambda_k\\ra 0$\n and $u_k\\rightharpoonup 0$ weakly in $W^{m,2}(\\Omega)$. Assuming $\\Lambda=\\lim_{k\\ra\\infty}\\int_\\Omega\n u_k(-\\Delta_{\\mathbb{R}^{2m}})^mu_kdx<\\infty$, they proved that\n $\\Lambda$ is an integer multiple of $\\Lambda_1=(2m-1)!{\\rm\n vol}(\\mathbb{S}^{2m})$, the total $Q$-curvature of the standard\n $2m$-dimensional sphere. In view of the Trudinger-Moser embedding\n for the space $W_0^{1,n}(\\Omega)$, where $n\\geq 3$ and $\\Omega\\subset\\mathbb{R}^n$\n is a smooth bounded domain, one may ask how about the equation\n \\be\\label{nD}\\le\\{\\begin{array}{lll}-\\Delta_n u_k=\\lambda_ku_k^{\\f{1}{n-1}}e^{u_k^{\\f{n}{n-1}}}\\,\\,\\,{\\rm in}\\,\\,\\,\n \\Omega\\\\[1.5ex]u_k\\geq 0\\,\\,\\,{\\rm in}\\,\\,\\,\\Omega,\\,\\,\\,u_k=0\n \\,\\,\\,{\\rm on}\\,\\,\\,\\p\\Omega.\\end{array}\\ri.\\ee\n Up to now only an energy inequality has been obtained by Adimurthi and the\n author \\cite{Adi-Yang}. Concerning the quantization for\n equation (\\ref{nD}), we have a long way to go. For other works related to\n this kind of\n quantization problems we refer the reader to\n \\cite{Mar-Stru,Struwe} and the references therein.\\\\\n\n For the proof of Theorem 1.1, we follow the lines of \\cite{Druet,L-R-S,Mar-Stru,Struwe}.\n Firstly we use a pointwise estimate on $u_k$ to find all separate blow-up\n points. Specifically we need to deal  carefully with the term $\\tau_k u_k$, which does not appear in the\n Euclidean case. Secondly we establish a gradient estimate for $u_k$.\n This permits us to compare $u_k$ with its spherical\n average with respect to blow-up points. Finally\n we get the quantization result, where we should deal with the extra term $\\tau_k u_k$ again.\n For calculations near blow-up points we prefer to choose\n isothermal coordinates instead of normal coordinates. The advantage of such coordinates is\n that both the Laplace-Beltrami operator $\\Delta_g$ and the gradient\n operator $\\nabla_g$ have simple expressions.\\\\\n\n The remaining part of this paper is organized as follows. In the\n next section we prove a simple property of the weak convergence of $u_k$.\n In Section 3, we locate the blow-up points of $u_k$ and\n describe the asymptotic behavior of $u_k$ near those points. In\n Section 4 we derive a gradient estimate on $u_k$.\n We shall prove quantization results for $u_k$ near the blow-up points in Section\n 5, and complete the proof of Theorem 1.1\n in Section 6.\\\\\n\n Throughout this paper we often denote various constants independent of $k$ by the same\n $C$. In addition, we do not distinguish between sequence and subsequence or points and sequence sometimes.\n The\n reader can easily recognize it from the context.\n\n \\section{Weak convergence}\n\n In this section, we let $u_k\\geq 0$ be a sequence of solutions to\n equation (\\ref{e1}) verifying that\n \\be\\label{beata}J_k(u_k)\\ra\\beta\\,\\,\\, {\\rm as}\\,\\,\\, k\\ra\\infty\\,\\,\\,\n {\\rm for\\,\\, some}\\,\\,\\, \\beta\\in\\mathbb{R},\\ee\n where $J_k$ is defined in (\\ref{functional}).\n Testing equation (\\ref{e1}) by\n$u_k$, we have\n\\be\\label{0.7}\n\\int_\\Sigma\\le(|\\nabla_gu_k|^2+\\tau_ku_k^2\\ri)dv_g=\\int_\\Sigma\nu_kf_k(x,u_k)dv_g. \\ee It follows from (\\ref{beata}) that\n$$\\int_\\Sigma\\le(|\\nabla_gu_k|^2+\\tau_ku_k^2\\ri)dv_g=2\\beta+2\\int_\\Sigma F_k(x,u_k)dv_g+o(1).$$\nHence\n$$\\int_\\Sigma u_kf_k(x,u_k)dv_g=2\\beta+2\\int_\\Sigma F_k(x,u_k)dv_g+o(1).$$\n If $f_k$ satisfies the hypotheses (H1)-(H4), then we have\n\\be\\label{bdd} \\int_\\Sigma u_kf_k(x,u_k)dv_g\\leq C\\ee for some\nconstant $C$. In view of (\\ref{tauk}), it follows from (\\ref{0.7})\nand (\\ref{bdd}) that $u_k$ is bounded in\n$W^{1,2}(\\Sigma,\\mathbb{R})$. Hence there exists some $u_\\infty\\in\nW^{1,2}(\\Sigma,\\mathbb{R})$ such that up to a subsequence,\n$u_k\\rightharpoonup u_\\infty$ weakly in\n$W^{1,2}(\\Sigma,\\mathbb{R})$, $u_k\\ra u_\\infty$ strongly in\n$L^2(\\Sigma,\\mathbb{R})$, and $u_k\\ra u_\\infty$ a.e. in $\\Sigma$.\nSimilarly to \\cite{Druet}, we then get that\n\\be\\label{F}\\lim_{k\\ra\\infty}\\int_\\Sigma F_k(x,u_k)dv_g=\\int_\\Sigma\nF_\\infty(x,u_\\infty)dv_g\\ee\n that  $u_\\infty$ is a weak\nsolution of (\\ref{3'}), and that  $u_\\infty\\in\nC^1(\\Sigma,\\mathbb{R})$. In conclusion we obtained an analogue of (\\cite{Druet}, Lemma1),\nnamely\\\\\n\n\\noindent{\\bf Lemma 2.1} {\\it Let $f_k$ be a sequence of functions\nsatisfying (H1)-(H4). Let $u_k\\geq 0$ be a sequence of\nsolutions to (\\ref{e1}), where $\\tau_k$ is\nas defined in (\\ref{tauk}). If (\\ref{beata}) holds, then $u_k$ is\nbounded in $W^{1,2}(\\Sigma,\\mathbb{R})$, and thus, up to a subsequence,\n$u_k\\rightharpoonup u_\\infty$ weakly in\n$W^{1,2}(\\Sigma,\\mathbb{R})$, where $u_\\infty\\in\nC^1(\\Sigma,\\mathbb{R})$ is a solution to (\\ref{3'}).\nAlso, there holds\n\\be\\label{0.6}\\lim_{k\\ra\\infty}\\int_\\Sigma\\le(|\\nabla_gu_k|^2+\\tau_ku_k^2\\ri)dv_g=\n2\\beta+2\\int_\\Sigma F_\\infty(x,u_\\infty)dv_g.\\ee }\n\n \\section{Multibubble analysis}\n\n In this section we shall use point wise estimate to find blow-up\n points of a sequence of solutions to the equation\n (\\ref{e1}). This technique was first used  by Druet \\cite{Druet} to deal with blow-up analysis for\n solutions to the equation (\\ref{eucl}). Assume $u_k\\geq 0$ is a sequence of solutions to the equation (\\ref{e1})\n and (\\ref{beata}) holds. From (\\ref{0.7}) and (\\ref{bdd}) we can\n find some constant $C$ such that\n\\be\\label{energy-bounded}\\int_\\Sigma\n\\le(|\\nabla_gu_k|^2+\\tau_ku_k^2\\ri)dv_g\\leq C.\\ee\n Then the Sobolev embedding theorem implies that for any $p>1$ there is\n some constant $C$ such that\n \\be\\label{Lp-bdd}\\int_\\Sigma u_k^pdv_g\\leq C.\\ee\n These two properties are very important during the process of exhausting blow-up points.\n Precisely we have the following proposition which is\n analogous to (\\cite{Druet}, Proposition 1), (\\cite{L-R-S}, Theorem 4.2), (\\cite{Mart}, Theorem 1 in the case\n $m=1$) and (\\cite{Adi-Yang}, Proposition 3.1).\n  \\\\\n\n \\noindent{\\bf Proposition 3.1} {\\it Let $(\\Sigma,g)$ be a compact Riemannian surface without boundary,\n $(f_k)$ be a sequence of functions satisfying the hypotheses {\\rm (H1)-(H5)}, and $(u_k)$ be a sequence of smooth\n nonnegative solutions to (\\ref{e1}) such that (\\ref{beata}) holds.\n Assume that $\\max_\\Sigma u_k\\ra+\\infty$\n as $k\\ra\\infty$. Then there exists $N\\in\\mathbb{N}\\setminus\\{0\\}$, and up to a subsequence, there exist\n $N$ sequences of points $x_{i,k}\\ra x_i^\\ast\\in\\Sigma$ and of positive real numbers $r_{i,k}\\ra 0$ as $k\\ra\\infty$, where\n $r_{i,k}$ is defined by\n \\be\\label{scal}r_{i,k}^{-2}=u_k(x_{i,k})f_k(x_{i,k},u_k(x_{i,k})),\\ee\n such that the following hold:\\\\\n\n \\noindent$(i)$ For any $i=1,2,\\cdots,N$, take an isothermal coordinate system\n $(U_i,\\phi_i;\\{x^1,x^2\\})$ near $x_i^\\ast$, where $U_i\\subset\\Sigma$ is a\n neighborhood of $x_i^\\ast$, $\\phi_i: U_i\\ra \\Omega_i\\subset\\mathbb{R}^2$ is a diffeomorphism\n and $\\phi_i(x_i^\\ast)=(0,0)$. If we define\n \\be\\label{blow-up-funct}\\eta_{i,k}(x)=u_k(x_{i,k})(\\widetilde{u}_k(\\widetilde{x}_{i,k}+r_{i,k}x)-u_k(x_{i,k}))\\ee\n for all $x\\in\\Omega_{i,k}=\\{x\\in\\mathbb{R}^2: \\widetilde{x}_{i,k}+r_{i,k}\\in\\Omega_i\\}$,\n where\n $\\widetilde{x}_{i,k}=\\phi_i(x_{i,k})$ and $\\widetilde{u}_k=u_k\\circ\\phi_i^{-1}$,\n then there holds\n $$\\eta_{i,k}(x)\\ra\\eta_\\infty(x)=\\log\\f{1}{1+|x|^2\/4}\\quad{\\rm in}\\quad C^1_{\\rm loc}(\\mathbb{R}^2);$$\n $(ii)$ For any $1\\leq i\\not=j\\leq N$, there holds\n $$\\f{d_g(x_{i,k},x_{j,k})}{r_{i,k}}\\ra+\\infty,\n \\quad{\\rm as}\\quad k\\ra\\infty,$$ where $d_g(\\cdot,\\cdot)$\n denotes the geodesic distance between two points of $\\Sigma$;\\\\\n $(iii)$ Define $R_{N,k}(x)=\\min_{1\\leq i\\leq N}d_g(x,x_{i,k})$ for $x\\in\\Sigma$, then there exists a constant $C>0$ such that\n $$R^2_{N,k}(x)u_k(x)f_k(x,u_k(x))\\leq C$$\n uniformly in $x\\in \\Sigma$ and $k\\in \\mathbb{N}$.\\\\\n\n\\noindent Moreover, given any sequence of points $(x_{N+1,k})$, it\nis impossible to extract a new subsequence from the previous\n one such that $(i)-(iii)$ hold with the sequences $(x_{i,k})$,\n $i=1,\\cdots,N+1$.\\\\\n\nFinally, we have $u_k\\ra u_\\infty$ in $C^1_{\\rm\nloc}(\\Sigma\\setminus\\mathcal{S})$ as $k\\ra\\infty$, where\n$\\mathcal{S}=\\{x_1^\\ast,\\cdots,x_N^\\ast\\}$, and\n $u_\\infty$ is given in Lemma 2.1.}\\\\\n\n {\\it Proof.} Similarly to \\cite{Druet,L-R-S,Mart,Adi-Yang}, we prove the proposition\n  by several steps as follows.\\\\\n\n {\\it Step $1$. The first bubble.}\\\\\n\n Assume $u_k(x_k)=\\max_\\Sigma u_k$. If $u_k(x_k)$ is\n bounded, applying elliptic estimates to equation (\\ref{e1}), we then have\n $u_k\\ra u_\\infty$ in $C^1(\\Sigma,\\mathbb{R})$, where\n $u_\\infty$ is given by Lemma 2.1. Hereafter we assume\n $u_k(x_k)\\ra+\\infty$. Set\n \\be\\label{rk}r_k^{-2}=u_k(x_k)f_k(x_k,u_k(x_k)).\\ee\n It is clear that $r_k\\ra 0$ as $k\\ra\\infty$.\n\n Assume $x_k\\ra x^\\ast$ as $k\\ra\\infty$. Take an isothermal coordinate system $(U,\\phi;\\{x^1,x^2\\})$\n near $x^\\ast$, where $U\\subset\\Sigma$ is a\n neighborhood of $x^\\ast$, $\\phi: U\\ra \\Omega\\subset\\mathbb{R}^2$ is a diffeomorphism\n and $\\phi(x^\\ast)=(0,0)$.\n In such a coordinate system, the metric $g$ can be represented by\n $$g=e^{\\psi}(d{x^1}^2+d{x^2}^2)$$\n for some smooth function $\\psi:\\Omega\\ra\\mathbb{R}$ with\n $\\psi(0,0)=0$. It follows that\n \\be\\label{laplace}\\nabla_g=e^{-\\psi}\\nabla_{\\mathbb{R}^2},\\quad \\Delta_g=-e^{-\\psi}\n \\Delta_{\\mathbb{R}^2},\\ee\n where $\\nabla_{\\mathbb{R}^2}$ and $\\Delta_{\\mathbb{R}^2}$ denote the usual gradient operator\n and the Laplace operator of $\\mathbb{R}^2$ respectively. The\n existence of isothermal coordinate system on Riemannian surface is a well-known fact in Riemannian geometry,\n see for example \\cite{Wu}.\n Define\n \\be\\label{vk}v_k(x)=\\f{\\widetilde{u}_k(\\widetilde{x}_k+r_kx)}{u_k(x_k)}\\ee\n for $x\\in\\Omega_k=\\{x\\in\\mathbb{R}^2:\\widetilde{x}_k+r_kx\\in\\Omega\\}$,\n where $\\widetilde{u}_k=u_k\\circ\\phi^{-1}$,\n $\\widetilde{x}_k=\\phi(x_k)$.\n It follows from (\\ref{e1}), (\\ref{rk}) and (\\ref{laplace}) that $v_k$ satisfies the following equation\n \\be\\label{11}\n -\\Delta_{\\mathbb{R}^2}v_k(x)=e^{\\psi(\\widetilde{x}_k+r_kx)}\\f{\\widetilde{f}_k(\\widetilde{x}_k+r_kx,\n \\widetilde{u}_k(\\widetilde{x}_k+r_kx))}{u_k^2(x_k)f_k(x_k,u_k(x_k))}\n -e^{\\psi(\\widetilde{x}_k+r_kx)}r_k^2\\widetilde{\\tau}_k(\\widetilde{x}_k+r_kx) v_k(x)\\ee\n on $\\Omega_k$,\n where\n $\\widetilde{f}_k(\\widetilde{x}_k+r_kx,t)={f}_k(\\phi^{-1}(\\widetilde{x}_k+r_kx),t)$.\n Note that $u_k(x_k)=\\max_\\Sigma u_k$ and $\\Omega_k\\ra\\mathbb{R}^2$ as $k\\ra\\infty$. It follows from (\\ref{vk})\n that $v_k$ is uniformly bounded in $\\mathbb{B}_R(0)$ for any fixed\n $R>0$. Since $\\psi$ is smooth, $\\psi(0,0)=0$, $\\widetilde{x}_k\\ra(0,0)$\n and $r_k\\ra 0$ as $k\\ra\\infty$, $e^{\\psi(\\widetilde{x}_k+r_kx)}$\n is also uniformly bounded in $\\mathbb{B}_R(0)$ for any fixed\n $R>0$. Furthermore $e^{\\psi(\\widetilde{x}_k+r_kx)}\\ra 1$ locally uniformly in $\\mathbb{R}^2$\n as $k\\ra\\infty$. By (H4) and (H5), we have for all\n $x\\in \\Omega_k$ and all $k$\n \\be\\label{f-bdd}\\f{\\widetilde{f}_k(\\widetilde{x}_k+r_kx,\n \\widetilde{u}_k(\\widetilde{x}_k+r_kx))}{f_k(x_k,u_k(x_k))}\\leq\n C.\\ee\n All these estimates together with (\\ref{tauk}) lead to\n $$\\|-\\Delta_{\\mathbb{R}^2}v_k\\|_{L^\\infty(\\mathbb{B}_R(0))}\\ra 0\\,\\,\\,{\\rm\n as}\\,\\,\\,\n k\\ra\\infty,\\,\\,\\forall R>0.$$\n  Applying elliptic estimates to (\\ref{11}), one gets $v_k\\ra\n v_\\infty$ in $C^1_{\\rm loc}(\\mathbb{R}^2)$, where $v_\\infty$\n satisfies\n $$\\le\\{\\begin{array}{lll}\n -\\Delta_{\\mathbb{R}^2}v_\\infty=0\\,\\,\\,{\\rm in}\\,\\,\\,\\mathbb{R}^2\\\\[1.5ex]\n v_\\infty(0)=1=\\max_{\\mathbb{R}^2}v_\\infty.\n \\end{array}\\ri.$$\n The Liouville theorem for harmonic functions then leads to $v_\\infty\\equiv\n 1$. Therefore\n \\be\\label{C-V}v_k\\ra 1\\quad{\\rm in}\\quad C^1_{\\rm loc}(\\mathbb{R}^2).\\ee\n Now we set\n $$\\eta_k(x)=u_k(x_k)(\\widetilde{u}_k(\\widetilde{x}_k+r_kx)-u_k(x_k)).$$\n In view of (\\ref{e1}), $\\eta_k$ satisfies\n \\bea{\\nonumber}-\\Delta_{\\mathbb{R}^2}\\eta_k(x)&=&e^{\\psi(\\widetilde{x}_k+r_kx)}\\f{\\widetilde{f}_k(\\widetilde{x}_k+r_kx,\n \\widetilde{u}_k(\\widetilde{x}_k+r_kx))}{f_k(x_k,u_k(x_k))}\\\\[1.5ex]&&\n -e^{\\psi(\\widetilde{x}_k+r_kx)}\\widetilde{\\tau}_k(\\widetilde{x}_k+r_kx) r_k^2u_k^2(x_k)v_k(x),\\,\\,\\, x\\in\n \\Omega_k.\\label{eta}\\eea\n We claim that\n \\be\\label{rk-0}r_ku_k^p(x_k)\\ra 0 \\quad{\\rm as}\\quad k\\ra\\infty,\\quad \\forall\n p>1.\\ee Actually,\n it is clear that there exists some constant $c>0$ depending only on\n the diffeomorphism $\\phi$ such that for any fixed $R>0$ and all\n large $k$\n \\be\\label{ball}B_{c^{-1}Rr_k}(x_k)\\subset \\phi^{-1}\\le(\\mathbb{B}_{Rr_k}(\\widetilde{x}_k)\\ri)\\subset\n B_{cRr_k}(x_k).\\ee\n Here and throughout this paper we denote the geodesic ball centered at $x\\in\\Sigma$ with radius $r$ by\n $B_r(x)$, while the Euclidean ball centered at $x\\in\\mathbb{R}^2$ with radius $r$ by $\\mathbb{B}_r(x)$.\n This together with (\\ref{C-V}), the mean value theorem for integral and the H\\\"older\n inequality leads to\n \\bea{\\nonumber}\n r_ku_k^p(x_k)&=&\\f{r_k}{\\pi}\\int_{\\mathbb{B}_1(0)}u_k^p(x_k)dx\\\\{\\nonumber}\n &=&(1+o(1))\\f{r_k}{\\pi}\\int_{\\mathbb{B}_1(0)}\\widetilde{u}_k^p(\\widetilde{x}_k+r_kx)dx\\\\{\\nonumber}\n &\\leq&(1+o(1))\\f{r_k}{\\pi^{1\/3}}\\le(\\int_{\\mathbb{B}_1(0)}\\widetilde{u}_k^{3p}(\\widetilde{x}_k+r_kx)dx\\ri)^{1\/3}\\\\\n &\\leq&(1+o(1))\\f{r_k^{1\/3}}{\\pi^{1\/3}}\\le(\\int_{{B}_{cr_k}(x_k)}{u}_k^{3p}dv_g\\ri)^{1\/3},{\\label{vrg}}\n \\eea\n where $o(1)\\ra 0$ as $k\\ra\\infty$ for any fixed $p>1$. In view of (\\ref{Lp-bdd}),\n our claim (\\ref{rk-0}) follows from (\\ref{vrg}) immediately.\n\n For any fixed $R>0$ we let $\\eta_k^{(1)}$ be a solution to the equation\n \\be\\label{312}\\le\\{\\begin{array}{lll}\n -\\Delta_{\\mathbb{R}^2}\\eta_k^{(1)}=-\\Delta_{\\mathbb{R}^2}\\eta_k\\,\\,\\,{\\rm in}\\,\\,\\,\\mathbb{B}_R(0)\\\\[1.5ex]\n \\eta_k^{(1)}=0\\quad {\\rm on}\\quad{\\p\\mathbb{B}_R(0)}.\n \\end{array}\\ri.\\ee\n In view of (\\ref{eta}), we have by (\\ref{f-bdd}) and (\\ref{rk-0})\n that $\\Delta_{\\mathbb{R}^2}\\eta_k$ is bounded in $L^\\infty_{\\rm\n loc}(\\mathbb{R}^2)$. Applying elliptic estimates to (\\ref{312}), we have\n \\be\\label{bou}\\eta_k^{(1)}\\ra \\eta_\\infty^{(1)}\\quad{\\rm in}\\quad\n C^1(\\mathbb{B}_{R}(0)).\\ee\n  Let $\\eta_k^{(2)}=\\eta_k-\\eta_k^{(1)}$. Then $\\eta_k^{(2)}$\n  satisfies\n \\be\\label{hm}-\\Delta_{\\mathbb{R}^2}\\eta_k^{(2)}=0\\quad{\\rm in}\\quad\n \\mathbb{B}_R(0).\\ee\n It follows from (\\ref{bou}) and $\\eta_k\\leq 0$ that there exists some constant $C$  such that\n $\\eta_k^{(2)}(x)\\leq C$ for all $k$ and all $x\\in \\mathbb{B}_R(0)$.\n Applying the Harnack inequality to (\\ref{hm}), we conclude that\n $\\eta_k^{(2)}$ is uniformly bounded on $\\mathbb{B}_{R\/2}(0)$. Hence\n $\\eta_k$ is also uniformly bounded in $\\mathbb{B}_{R\/2}(0)$. Applying\n elliptic estimates to (\\ref{eta}), we obtain\n $$\\eta_k\\ra \\eta_\\infty\\quad {\\rm in}\\quad\n C^1(\\mathbb{B}_{R\/4}(0)).$$\n This together with (H4), (H5) and (\\ref{C-V}) gives\n \\be\\label{b-c}\\f{\\widetilde{f}_k(\\widetilde{x}_k+r_kx,\n \\widetilde{u}_k(\\widetilde{x}_k+r_kx))}{f_k(x_k,u_k(x_k))}=(1+o(1))e^{(2+o(1))\\eta_\\infty}\n \\ee\n for all $x\\in\\mathbb{B}_{R\/4}(0)$,\n where $o(1)\\ra 0$ as $k\\ra\\infty$ uniformly in $x\\in\n \\mathbb{B}_{R\/4}(0)$. Inserting (\\ref{rk-0}) and (\\ref{b-c}) into\n (\\ref{eta}) and noting that $R>0$ is arbitrary  we obtain\n \\be\\label{bubble}\\le\\{\\begin{array}{lll}\n -\\Delta_{\\mathbb{R}^2}\\eta_\\infty=e^{2\\eta_\\infty}\\,\\,\\,{\\rm in}\\,\\,\\,\\mathbb{R}^2\\\\[1.5ex]\n \\eta_\\infty(0)=0=\\max_{\\mathbb{R}^2}\\eta_\\infty.\n \\end{array}\\ri.\\ee\n Moreover, using (\\ref{bdd}), (\\ref{rk}), (\\ref{C-V}), (\\ref{ball}) and (\\ref{b-c}), we estimate for any fixed $R>0$\n \\bna\n {\\nonumber}\\int_{\\mathbb{B}_R}e^{2\\eta_\\infty}dx&=&\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_{R}(0)}\n \\f{\\widetilde{u}_k(\\widetilde{x}_k+r_k x)\\widetilde{f}_k(\\widetilde{x}_k+r_k x,\\widetilde{u}_k(\\widetilde{x}_k+r_k x))}\n {u_k(x_k)f_k(x_k,u_k(x_k))}dx\\\\\n {\\nonumber}&=&\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_{Rr_k}(\\widetilde{x}_k)}\\widetilde{u}_k(x)\\widetilde{f}_k(x,\\widetilde{u}_k(x))dx\n \\\\\n {\\label{finite}}&\\leq&\\limsup_{k\\ra\\infty}\\int_{B_{cRr_k}(x_k)}u_kf_k(x,u_k)dv_g\\leq C.\n \\ena\n It follows that\n $$\\int_{\\mathbb{R}^2}e^{2\\eta_\\infty(x)}dx<\\infty.$$\n A result of Chen-Li \\cite{C-L} implies that\n \\be\\label{bub}\\eta_\\infty(x)=-\\log(1+|x|^2\/4),\\quad\n x\\in\\mathbb{R}^2.\\ee\n It follows from (\\ref{ball}) that\n $$\\int_{\\mathbb{B}_{c^{-1}Rr_k}(\\widetilde{x}_k)}\\widetilde{u}_k\\widetilde{f}_k\n (x,\\widetilde{u}_k)e^{\\psi(x)}dx\\leq\\int_{B_{Rr_k}(x_k)}u_kf_k(x,u_k)dv_g\\leq\n \\int_{\\mathbb{B}_{cRr_k}(\\widetilde{x}_k)}\\widetilde{u}_k\\widetilde{f}_k\n (x,\\widetilde{u}_k)e^{\\psi(x)}dx.$$\n In view of (\\ref{C-V}) and (\\ref{b-c}), we have\n \\bna\n \\lim_{R\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_{cRr_k}(\\widetilde{x}_k)}\\widetilde{u}_k\\widetilde{f}_k\n (x,\\widetilde{u}_k)e^{\\psi(x)}dx&=&\\lim_{R\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_{c^{-1}Rr_k}(\\widetilde{x}_k)}\\widetilde{u}_k\\widetilde{f}_k\n (x,\\widetilde{u}_k)e^{\\psi(x)}dx\\\\\n &=&\\lim_{R\\ra\\infty}\\int_{\\mathbb{B}_{c^{-1}R}(0)}e^{2\\eta_\\infty}dx=\\int_{\\mathbb{R}^2}e^{2\\eta_\\infty}dx.\n \\ena\n Therefore we obtain by (\\ref{bub})\n  \\be\\label{eng}\n   \\lim_{R\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{B_{Rr_k}(x_k)}u_kf_k(x,u_k)dv_g\n   =\\int_{\\mathbb{R}^2}e^{2\\eta_\\infty}(x)dx=4\\pi.\n  \\ee\n\n {\\it Step 2.} {\\it Multi-bubble analysis.}\\\\\n\n In this step, we shall prove that there exists some positive integer\n $\\ell$ such that the properties $(\\mathcal{B}_\\ell)$ and\n $(\\mathcal{G}_\\ell)$ hold. Namely, there exist $\\ell$ sequences of points $(x_{i,k})\\subset \\Sigma$ such that\n $x_{i,k}\\ra\n x_i^\\ast$ as $k\\ra\\infty$, $1\\leq i\\leq \\ell$, and the following are satisfied:\\\\\n\n \\noindent\n$(\\mathcal{B}_\\ell^1)$ For every $i:1\\leq i\\leq \\ell$, letting\n$r_{i,k}>0$ be given by (\\ref{scal}), $(U_i,\\phi_i;\\{x^1,x^2\\})$ be\nan isothermal coordinate system near $x_i^\\ast$, where $U_i\\subset\\Sigma$ is a neighborhood of $x_i^\\ast$,\n$\\phi_i:U_i\\ra \\Omega_i\\subset\\mathbb{R}^2$ is a diffeomorphism\nwith $\\phi_i(x_i^\\ast)=(0,0)$, and letting $\\eta_{i,k}$ be given by\n(\\ref{blow-up-funct}), we have that $r_{i,k}\\ra 0$ as $k\\ra\\infty$\nand\n$$\\eta_{i,k}(x)\\ra \\eta_\\infty(x)=-\\log(1+|x|^2\/4)\\quad{\\rm in}\\quad\nC^1_{\\rm loc}(\\mathbb{R}^2)\\quad{\\rm as}\\quad k\\ra\\infty;$$\n$(\\mathcal{B}_\\ell^2)$ For all $1\\leq i\\not=j\\leq \\ell$,\n$$\\f{d_g(x_{i,k},x_{j,k})}{r_{i,k}}\\ra\\infty\\quad{\\rm as}\\quad k\\ra\\infty;$$\n $(\\mathcal{B}_\\ell^3)$ The following energy identity holds\n $$\\lim_{R\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\cup_{i=1}^\\ell{B}_{Rr_{i,k}}(x_{i,k})}\n u_kf_k(x,u_k)dv_g= 4\\pi \\ell;$$\n$(\\mathcal{G}_\\ell)$ There exists a constant $C>0$  such that\n$$R_{\\ell,k}^2(x)u_k(x)f_k(x,u_k(x))\\leq C$$\nfor all $x\\in \\Sigma$ and all $k\\in\\mathbb{N}$. Here\n\\be\\label{R}R_{\\ell,k}(x)=\\min_{1\\leq i\\leq \\ell}d_g(x,x_{i,k}).\\ee\n\nFrom Step 1, we know that $(\\mathcal{B}_1)$ holds. Suppose for some\n$\\ell\\geq 1$, $(\\mathcal{B}_\\ell)$ holds but $(\\mathcal{G}_\\ell)$\ndoes not hold. Choose $x_{\\ell+1,k}\\in \\Sigma$ satisfying\n \\bea{\\nonumber}\n R^2_{\\ell,k}(x_{\\ell+1,k})u_k(x_{\\ell+1,k})f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))&=&\\max_{x\\in\\Sigma}\n R_{\\ell,k}^2(x)u_k(x)f_k(x,u_k(x))\\\\&\\ra&+\\infty\\quad{\\rm as}\\quad\n k\\ra\\infty.\\label{R-0}\n \\eea  Let $r_{\\ell+1,k}>0$ be as defined in (\\ref{scal}). It follows from\n (\\ref{scal}), (\\ref{R}), and (\\ref{R-0}) that $r_{\\ell+1,k}\\ra 0$ as\n $k\\ra\\infty$ and\n \\be\\label{rl}\\lim_{k\\ra\\infty}\\f{d_g(x_{\\ell+1,k},x_{i,k})}{r_{\\ell+1,k}}=+\\infty,\n \\quad\\forall 1\\leq i\\leq \\ell.\\ee\n Also we claim that\n \\be\\label{ik}\\lim_{k\\ra\\infty}\\f{d_g(x_{\\ell+1,k},x_{i,k})}{r_{i,k}}=+\\infty,\n \\quad\\forall 1\\leq i\\leq \\ell.\\ee\n Suppose not. There exists some constant $C$ such that for some $1\\leq i\\leq\n \\ell$, there holds\n $${d_g(x_{\\ell+1,k},x_{i,k})}\\leq C{r_{i,k}}\\quad\n {\\rm for\\,\\,\\, all}\\quad k.$$ Hence we have\n \\be\\label{rrr}\n R_{\\ell,k}^2(x_{\\ell+1,k})u_k(x_{\\ell+1,k})f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))\\leq\n Cr_{i,k}^2\n u_k(x_{\\ell+1,k})f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))\n \\ee\n By $(\\mathcal{B}_\\ell^1)$, we estimate\n \\bna\n r_{i,k}^2u_k(x_{\\ell+1,k})f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))&=&\\f{1+o(1)}{\\pi}\n \\int_{\\mathbb{B}_{r_{i,k}}(\\widetilde{x}_{i,k})}\\widetilde{u}_k(x)\\widetilde{f}_k(x,\\widetilde{u}_k(x))\n e^{\\psi_i(x)}dx\\\\\n &\\leq&\\f{1+o(1)}{\\pi}\\int_\\Sigma u_k(x)f_k(x,u_k(x))dv_g.\n \\ena\n This together with (\\ref{bdd}) implies that $r_{i,k}^2u_k(x_{\\ell+1,k})f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))$\n is a bounded sequence, and whence  (\\ref{rrr}) implies that\n $R_{\\ell,k}^2(x_{\\ell+1,k})u_k(x_{\\ell+1,k})f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))$\n is bounded. This contradicts (\\ref{R-0}). Hence our claim\n (\\ref{ik}) holds, and thus $(\\mathcal{B}_{\\ell+1}^2)$ holds.\n\n Assume $x_{\\ell+1,k}\\ra x_{\\ell+1}^\\ast$ as $k\\ra\\infty$. Take an isothermal\n coordinate system $(U_{\\ell+1},\\phi_{\\ell+1};\\{x^1,x^2\\})$ near $x_{\\ell+1}^\\ast$,\n where $U_{\\ell+1}$ is a neighborhood of $x_{\\ell+1}^\\ast$,\n $\\phi_{\\ell+1}:U_{\\ell+1}\\ra\\Omega_{\\ell+1}\\subset\\mathbb{R}^2$ is a diffeomorphism\n with $\\phi_{\\ell+1}(x_{\\ell+1}^\\ast)=(0,0)$. In\n this coordinate system, the metric $g$ can be represented by\n $$g=e^{\\psi_{\\ell+1}}(d{x^1}^2+d{x^2}^2)$$\n for some smooth function $\\psi_{\\ell+1}:\\Omega_{\\ell+1}\\ra\\mathbb{R}$ with\n $\\psi_{\\ell+1}(0,0)=0$. Also we have\n $\\nabla_g=e^{-\\psi_{\\ell+1}}\\nabla_{\\mathbb{R}^2}$ and\n $\\Delta_g=-e^{-\\psi_{\\ell+1}}\\Delta_{\\mathbb{R}^2}$.\n\n Define\n $$v_{\\ell+1,k}(x)=\\f{\\widetilde{u}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x)}{u_k(x_{\\ell+1,k})}$$\n for $x\\in\\Omega_{\\ell+1,k}=\\{x\\in\\mathbb{R}^2:\n \\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x\\in\\Omega_{\\ell+1}\\}$,\n where $\\widetilde{x}_{\\ell+1,k}=\\phi_{\\ell+1}(x_{\\ell+1,k})$, $\\widetilde{u}_k=u_k\\circ\n \\phi_{\\ell+1}^{-1}$.\n Now we prove that\n \\be\\label{vl1}v_{\\ell+1,k}\\ra 1\\quad{\\rm in}\\quad C^1_{\\rm loc}(\\mathbb{R}^2)\\quad {\\rm as}\\quad\n k\\ra\\infty.\\ee\n In view of (\\ref{e1}), $v_{\\ell+1,k}$ satisfies the equation\n \\bea\\label{11k}{\\nonumber}\n -\\Delta_{\\mathbb{R}^2}v_{\\ell+1,k}(x)&=&e^{\\psi_{\\ell+1}(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x)}\\f{\\widetilde{f}_k(\n \\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x,\n \\widetilde{u}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x))}{u_k^2(x_{\\ell+1,k})f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))}\\\\[1.5ex]\n &&-e^{\\psi_{\\ell+1}(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x)}r_{\\ell+1,k}^2\\widetilde{\\tau}_k\n (\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x) v_{\\ell+1,k}(x)\\eea on $\\Omega_{\\ell+1,k}$,\n where $\\widetilde{f}_k(x,t)=f_k(\\phi_{\\ell+1}^{-1}(x),t)$.\n By (\\ref{R-0}), we have\n \\bea\\nonumber\n &&\\widetilde{R}_{\\ell,k}^2(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x)\\widetilde{u}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x)\n \\widetilde{f}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x,\\widetilde{u}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x))\\\\\n \\label{Rleq}&&\\leq\n R_{\\ell,k}^2(x_{\\ell+1,k})u_k(x_{\\ell+1,k})f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k})),\n \\eea\n where $\\widetilde{R}_{\\ell,k}={R}_{\\ell,k}\\circ \\phi_{\\ell+1}^{-1}$.\n Fix any $i$, $1\\leq i\\leq \\ell$. If $x_{\\ell+1}^\\ast\\not=x_i^\\ast$,\n noting that $d_g(\\phi_{\\ell+1}^{-1}(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x),x_{i,k})\\ra d_g(x_{\\ell+1}^\\ast,x_i^\\ast)$\n and $d_g(x_{\\ell+1,k},x_{i,k})\\ra d_g(x_{\\ell+1}^\\ast,x_i^\\ast)$ as\n $k\\ra\\infty$, we then have \\be\\label{ooo}d_g(\\phi_{\\ell+1}^{-1}(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x),x_{i,k})=(1+o(1))\n d_g(x_{\\ell+1,k},x_{i,k}),\\ee where $o(1)\\ra 0$ as $k\\ra\\infty$ uniformly in $x\\in\\mathbb{B}_R(0)$.\n If $x_{\\ell+1}^\\ast=x_i^\\ast$, since the Riemannian distance and the Euclidean distance\n are equivalent in the same local coordinate system, we then have $|\\phi_{\\ell+1}(x_{\\ell+1,k})-\\phi_{\\ell+1}(x_{i,k})|\n =(1+o(1))d_g(x_{\\ell+1,k},x_{i,k})$. Recalling (\\ref{rl}), we\n obtain for all $x\\in\\mathbb{B}_R(0)$\n \\bna\n d_g(\\phi_{\\ell+1}^{-1}(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x),x_{i,k})&=&\n (1+o(1))|\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x-\\phi_{\\ell+1}(x_{i,k})|\\\\\n &=&(1+o(1))d_g(x_{\\ell+1,k},x_{i,k}).\n \\ena\n Hence we have (\\ref{ooo}) in any case. Combining (\\ref{Rleq}) and (\\ref{ooo}), we obtain for  $x\\in\n \\mathbb{B}_R(0)$\n \\bea\\label{ineq}\n &&{v}_{\\ell+1,k}(x)\\f{\\widetilde{f}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x,\\widetilde{u}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x))}\n {f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))}\\nonumber\\\\\n &&\\leq\\f{\\inf_{1\\leq i\\leq \\ell}d_g(x_{\\ell+1,k},x_{i,k})^2}{\\inf_{1\\leq i\\leq\n \\ell}d_g(\\phi_{\\ell+1}^{-1}(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x),x_{i,k})^2}=1+o(1),\n \\eea\n where $o(1)\\ra 0$ uniformly in $x\\in\\mathbb{B}_R(0)$. From (H4), we know that there exists $t_0>0$ such that\n \\be\\label{f-geq}\\f{f_k(x,t_2)}{f_k(x,t_1)}\\geq e^{t_2^2-t_1^2}\\,\\, {\\rm for\\,\\,all}\\,\\, t_1, t_2\\geq t_0,\n \\,\\,{\\rm and\\,\\,all}\\,\\, x\\in\\Sigma.\\ee\n If there exist some $R_0>0$ and a sequence of points $(z_k)\\subset\\mathbb{B}_{R_0}(0)$ such that\n  $v_{\\ell+1,k}(z_k)\\ra \\alpha>1$ as $k\\ra\\infty$, then we conclude by\n (\\ref{f-geq}) and (H5) that\n $${v}_{\\ell+1,k}(z_k)\\f{\\widetilde{f}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}z_k,\n \\widetilde{u}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}z_k))}{f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))}\n \\geq \\f{\\alpha+1}{2}>1$$ for sufficiently large $k$,\n which contradicts (\\ref{ineq}). Therefore we obtain\n $$\\limsup_{k\\ra\\infty}\\|v_{\\ell+1,k}\\|_{L^\\infty(\\mathbb{B}_R(0))}\\leq 1,\\quad\\forall R>0.$$\n When $v_{\\ell+1,k}(x)>1$, we\n have by (\\ref{11k}) and (\\ref{ineq}),\n $\\Delta_{\\mathbb{R}^2}v_{\\ell+1,k}(x)=o(1)$,\n where $o(1)$ is the same meaning as that of (\\ref{ineq}).\n When $v_{\\ell+1,k}(x)\\leq 1$,\n using (H4) and (H5), we also have\n $\\Delta v_{\\ell+1,k}(x)=o(1)$,\n where $o(1)\\ra 0$ as $k\\ra\\infty$ uniformly in all $x$ satisfying $v_{\\ell+1,k}(x)\\leq\n 1$ for sufficiently large $k$. Now applying elliptic estimates to\n equation (\\ref{11k}), we obtain\n $$v_{\\ell+1,k}\\ra v_{\\ell+1,\\infty}\\quad{\\rm in}\\quad C^1_{\\rm loc}(\\mathbb{R}^2),$$\n where $v_{\\ell+1,\\infty}$ is a solution to\n $$\\le\\{\\begin{array}{lll}\n -\\Delta_{\\mathbb{R}^2}v_{\\ell+1,\\infty}=0\\quad{\\rm in}\\quad\\mathbb{R}^2\\\\[1.5ex]\n 0\\leq v_{\\ell+1,\\infty}\\leq 1.\n \\end{array}\\ri.$$\n Note that $v_{\\ell+1,\\infty}(0)=1$. The Liouville theorem for harmonic functions leads to $v_{\\ell+1,\\infty}\\equiv 1$.\n Whence (\\ref{vl1}) holds.\n\n Define another sequence of blow-up functions by\n \\be\\label{blow-up-l}\\eta_{\\ell+1,k}(x)=u_k(x_{\\ell+1,k})(\\widetilde{u}_k(\\widetilde{x}_{\\ell+1,k}\n +r_{\\ell+1,k}x)-u_k(x_{\\ell+1,k})),\\quad x\\in\\Omega_{\\ell+1,k}.\\ee\n In the following, we will prove that $(\\mathcal{B}_{\\ell+1}^1)$ and\n $(\\mathcal{B}_{\\ell+1}^3)$ hold. By (\\ref{e1}), $\\eta_{\\ell+1,k}$ satisfies the\n equation\n \\bea{\\nonumber}-\\Delta_{\\mathbb{R}^2}\\eta_{\\ell+1,k}(x)&=&e^{\\psi_{\\ell+1}(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x)}\n \\f{\\widetilde{f}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x,\n \\widetilde{u}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x))}{f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))}\\\\[1.5ex]&&\n -e^{\\psi_{\\ell+1}(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x)}\\widetilde{\\tau}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}x)\n r_{\\ell+1,k}^2u_k^2(x_{\\ell+1,k})v_{\\ell+1,k}(x)\\label{eta-l1}\\eea\n on $\\Omega_{\\ell+1,k}$.\n We claim that for any fixed $R>0$,\n \\be\\label{upper}\\limsup_{k\\ra\\infty}\\eta_{\\ell+1,k}(x)\\leq 0\n \\,\\,{\\rm uniformly\\,\\,in}\\,\\,x\\in\\mathbb{B}_R(0).\\ee\n For otherwise, we may take a sequence of points\n $(y_k)\\subset\\mathbb{B}_R(0)$ such that $\\eta_{\\ell+1,k}(y_k)\\geq\n \\beta>0$ for all sufficiently large $k$. By (H4), (H5) and\n (\\ref{vl1}), we obtain\n \\bna\n \\f{\\widetilde{f}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}y_k,\n \\widetilde{u}_k(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}y_k))}{f_k(x_{\\ell+1,k},u_k(x_{\\ell+1,k}))}&=&\n (1+o(1))e^{\\widetilde{u}_k^2(\\widetilde{x}_{\\ell+1,k}+r_{\\ell+1,k}y_k)-u_k^2(x_{\\ell+1,k})}\\\\\n &=&(1+o(1))e^{(2+o(1))\\eta_{\\ell+1,k}(y_k)}\\\\\n &\\geq& 1+2\\beta+o(1).\n \\ena\n This together with (\\ref{ineq}) leads to\n $$1+2\\beta+o(1)\\leq 1+o(1),$$\n which is impossible when $k$ is sufficiently large. Hence our claim\n (\\ref{upper}) holds. By (\\ref{vl1}), using the same method of deriving\n (\\ref{rk-0}), we conclude\n \\be\\label{rl10}r_{\\ell+1,k}^2u_{k}^2(x_{\\ell+1,k})\\ra 0\\quad{\\rm as}\\quad k\\ra\\infty.\\ee\n  Combining (\\ref{vl1}) and (\\ref{blow-up-l})-(\\ref{rl10}),\n  similarly as we did in Step 1, we arrive at\n  $$\\eta_{\\ell+1,k}(x)\\ra \\eta_\\infty(x)\\,\\, {\\rm in}\\,\\,\n  C^1_{\\rm loc}(\\mathbb{R}^2)\\,\\,{\\rm as}\\,\\, k\\ra\\infty,$$\n  where $\\eta_\\infty(x)=-\\log(1+|x|^2\/4)$ is the unique solution to\n  (\\ref{bubble}). Hence $(\\mathcal{B}_{\\ell+1}^1)$ holds.\n\n  Moreover, using the same method for proving (\\ref{eng}), we arrive\n  at\n  \\bna\n  \\lim_{R\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{B_{Rr_{\\ell+1,k}}(x_{\\ell+1,k})}u_kf_k(x,u_k)dv_g\n  =\\int_{\\mathbb{R}^2}e^{2\\eta_\\infty(x)}dx=4\\pi.\n  \\ena\n  Thus $(\\mathcal{B}_{\\ell+1}^3)$ holds.\n\n  Actually, we have proved that if $(\\mathcal{B}_\\ell)$ holds but $(\\mathcal{G}_\\ell)$ does not hold,\n  then $(\\mathcal{B}_{\\ell+1})$ holds. Note that\n  \\be\\label{stops}\\int_{\\Sigma}u_kf_k(x,u_k)dv_g\\geq \\sum_{i=1}^{\\ell+1}\n  \\int_{B_{Rr_{i,k}}(x_{i,k})}u_kf_k(x,u_k)dv_g=4(\\ell+1)\\pi.\\ee\n  In view of (\\ref{bdd}), the process must be terminate after finite steps. This ends the proof of Step 2.\\\\\n\n  {\\it Step 3.} {\\it Exhaustion of blow-up points.}\\\\\n\n  It follows from Step 2 that there exists some\n  $\\ell\\in\\mathbb{N}\\setminus\\{0\\}$ and $\\ell$ sequences of points $(x_{i,k})$, $i=1,\\cdots,\\ell$,\n  such that $(\\mathcal{B}_\\ell)$ and $(\\mathcal{G}_\\ell)$ hold. If there\n  exists a sequence of points $(x_{\\ell+1,k})$ of $\\Sigma$ such that\n  after extracting a new subsequence from the previous one, $(\\mathcal{B}_{\\ell+1})$ and\n  $(\\mathcal{G}_{\\ell+1})$ hold, we add this sequence of points, and so\n  on. The process necessarily terminates because of (\\ref{bdd})\n  and (\\ref{stops}).\n  Therefore there exists some $N\\in\\mathbb{N}\\setminus\\{0\\}$ and $N$\n  sequences of points $(x_{i,k})$, $i=1,\\cdots,N$, such that\n  $(\\mathcal{B}_N)$ and $(\\mathcal{G}_N)$ hold and such that, given\n  any sequence of points $(x_{N+1,k})$, it is impossible to extract\n  a new subsequence from the previous one such that $(\\mathcal{B}_{N+1})$\n  and $(\\mathcal{G}_{N+1})$ hold with sequences $(x_{i,k})$,\n  $i=1,\\cdots,N+1$.\\\\\n\n  {\\it Step 4.} {\\it Convergence away from blow-up points.}\\\\\n\n  Set $\\mathcal{S}=\\{x_1^\\ast,\\cdots,x_N^\\ast\\}$. We will prove that\n  $u_k\\ra u_\\infty$ in $C^1_{\\rm loc}(\\Sigma\\setminus\\mathcal{S})$.\n  In view of $(\\mathcal{G}_N)$, given any compact set $K\\subset\\Sigma\\setminus\\mathcal{S}$,\n  there exists a constant $C$ such that\n  $$u_k(x)f_k(x,u_k(x))\\leq C\\,\\,{\\rm for\\,\\,all}\\,\\, x\\in K\n  \\,\\,{\\rm and\\,\\,all}\\,\\, k.$$\n  If $u_k(x)>1$ for some $x\\in K$, then $f_k(x,u_k(x))\\leq C_K$. If\n  $u_k(x)\\leq 1$ for some $x\\in K$, then (H2) implies that\n  $f_k(x,u_k(x))$ is  bounded uniformly in $x$ with $u_k(x)\\leq 1$. Thus, for all $x\\in\n  K$, $f_k(x,u_k(x))$ is bounded in $L^\\infty(K)$. In view of (\\ref{tauk}) and (\\ref{Lp-bdd}), applying elliptic\n  estimates to the equation\n  $$\\Delta_gu_k(x)+\\tau_k(x)u_k(x)=f_k(x,u_k(x)),\\quad {x\\in K},$$\n  we obtain the convergence $u_k\\ra u_0$ in $C^1_{\\rm\n  loc}(\\Sigma\\setminus\\mathcal{S})$.\\\\\n\n  Combining the above four steps, we complete the proof of\n  Proposition 3.1. $\\hfill\\Box$\n\n \\section{Gradient estimate}\n\n Let $u_k\\geq 0$ be a sequence of solutions to (\\ref{e1}). In this section we shall\n establish a gradient estimate on $u_k$, which can be viewed as a version on manifolds of\n (\\cite{Druet}, Proposition 2). Precisely we have the following result.\\\\\n\n \\noindent{\\bf Proposition 4.1} {\\it Let $(\\Sigma,g)$ be a compact Riemannian surface without boundary,\n$f_k$ be a sequence of functions satisfying {\\rm (H1)-(H5)}, and\n$u_k\\geq 0$ be a sequence of smooth solutions to equation\n(\\ref{e1}) such that (\\ref{beata}) holds. Assume that\n$\\max_\\Sigma u_k\\ra+\\infty$ as $k\\ra\\infty$. Let\n$N\\in\\mathbb{N}\\setminus\\{0\\}$ and the sequences $x_{i,k}$,\n$i=1,\\cdots,N$, be given by Proposition 3.1. Then there exists a\nuniform constant $C$ such that\n$$R_{N,k}(x)u_k(x)|\\nabla_gu_k(x)|\\leq C$$\nfor all $x\\in\\Sigma$ and all $k$, where $R_{N,k}(x)$ is defined as in (\\ref{R}).}\\\\\n\n\\noindent{\\it Proof.} Choose $y_k\\in\\Sigma$ such that\n\\be\\label{41}R_{N,k}(y_k)u_k(y_k)|\\nabla_gu_k(y_k)|=\\max_{x\\in\\Sigma}R_{N,k}(x)\nu_k(x)|\\nabla_gu_k(x)|.\\ee Suppose by contradiction that\n\\be\\label{contrad}R_{N,k}(y_k)u_k(y_k)|\\nabla_gu_k(y_k)|\\ra+\\infty\\quad{\\rm\nas}\\quad k\\ra\\infty.\\ee Set \\be\\label{sk}s_k=R_{N,k}(y_k).\\ee By\nProposition 3.1, we have $u_k\\ra u_\\infty$ in $C^1_{\\rm\nloc}(\\Sigma\\setminus\\{x_1^\\ast,\\cdots,x_N^\\ast\\})$, which together\nwith (\\ref{contrad}) implies that $s_k\\ra 0$ as $k\\ra\\infty$.\nWithout loss of generality, we may assume that $y_k\\ra x_1^\\ast$ as\n$k\\ra\\infty$, $x_1^\\ast=\\cdots=x_\\ell^\\ast$ for some $1\\leq \\ell\\leq\nN$, and $x_j^\\ast\\not=x_1^\\ast$ for any $j\\in\\{\\ell+1,\\cdots,N\\}$.\nTake an isothermal coordinate system $(U,\\phi;\\{x^1,x^2\\})$ near\n$x_1^\\ast$, where $U$ is a neighborhood of $x_1^\\ast\\in\\Sigma$,\n$\\phi: U\\ra\\Omega\\subset\\mathbb{R}^2$ is a diffeomorphism with\n$\\phi(x_1^\\ast)=(0,0)$. In this coordinate system the\nmetric $g$ can be represented by $g=e^{\\psi}(d{x^1}^2+d{x^2}^2)$,\nwhere $\\psi:\\Omega\\ra\\mathbb{R}$ is a smooth function with\n$\\psi(0,0)=0$. Denote $\\widetilde{y}_k=\\phi(y_k)$,\n$\\widetilde{u}_k=u_k\\circ\\phi^{-1}$. We\nset\n$$v_k(y)=\\widetilde{u}_k(\\widetilde{y}_k+s_ky)$$\nfor $y\\in\\Omega_k=\\{y\\in\\mathbb{R}^2:\n\\widetilde{y}_k+s_ky\\in\\Omega\\}$. Define\n$$y_{i,k}=\\f{\\widetilde{x}_{i,k}-\\widetilde{y}_k}{s_k}\\in \\Omega_k,\\quad i=1,\\cdots,\\ell,$$\nand\n$$\\widetilde{S}_k=\\{y_{1,k},\\cdots, y_{\\ell,k}\\}.$$\nSince $s_k\\ra 0$, we have $\\Omega_k\\ra\\mathbb{R}^2$ as $k\\ra\\infty$.\nDenote\n$$\\widetilde{S}=\\lim_{k\\ra\\infty}\\widetilde{S}_k.$$\nBy (\\ref{sk}) and the fact $\\psi(0,0)=0$, we have\n \\bna\n d_{\\mathbb{R}^2}(0,\\widetilde{S}_k)&=&\\inf_{1\\leq i\\leq\n \\ell}|y_{i,k}|=\\inf_{1\\leq i\\leq\n \\ell}\\f{|\\widetilde{x}_{i,k}-\\widetilde{y}_k|}{s_k}\\\\\n &=&\\inf_{1\\leq i\\leq\n \\ell}\\f{(1+o(1))d_g(x_{i,k},y_k)}{s_k}\\\\\n &=&1+o(1),\n \\ena\nand thus\n\\be\\label{4.3}{d}_{\\mathbb{R}^2}(0,\\widetilde{S})=1,\\ee\nwhere ${ d}_{\\mathbb{R}^2}(\\cdot,\\cdot)$ denotes the Euclidean\ndistance of $\\mathbb{R}^2$. Clearly, $v_k(y)$ satisfies\n \\be\\label{vvv}\n -\\Delta_{\\mathbb{R}^2}v_k(y)=e^{\\psi(\\widetilde{y}_k+s_ky)}s_k^2\\le(\\widetilde{f}_k(\\widetilde{y}_k+s_ky,\n \\widetilde{u}_k(\\widetilde{y}_k+s_ky))-\n \\widetilde{\\tau}_k(\\widetilde{y}_k+s_ky) v_k(y)\\ri)\\ee\n for $y\\in \\Omega_k$.\n By $(iii)$ of Proposition 3.1, we have\n \\be\\label{point}\\widetilde{R}_{N,k}(\\widetilde{y}_k+s_ky)^2v_k(y)\n \\widetilde{f}_k(\\widetilde{y}_k+s_ky,v_k(y))\\leq C\\ee\n for some constant $C$ independent of $k$. Note that\n \\bea\\nonumber\\widetilde{R}_{N,k}(\\widetilde{y}_k+s_ky)&=&R_{N,k}(\\phi^{-1}(\\widetilde{y}_k+s_ky))\\\\\n \\nonumber&=&\\inf_{1\\leq\n i\\leq\\ell}d_g(\\phi^{-1}(\\widetilde{y}_k+s_ky),x_{i,k})\\\\\n \\nonumber&=&(1+o(1))\\inf_{1\\leq i\\leq\\ell}d_{\\mathbb{R}^2}(\\widetilde{y}_k+s_ky,\\widetilde{x}_{i,k})\\\\\n &=&(1+o(1))s_k d_{\\mathbb{R}^2}(y,\\widetilde{S}_k).\\label{RNK}\n \\eea\n Combining (\\ref{point}) and (\\ref{RNK}), we have\n \\be\\label{vdelta}s_k^2v_k(y)\\widetilde{f}_k(\\widetilde{y}_k+s_ky,v_k(y))\n \\leq \\f{C}{d_{\\mathbb{R}^2}(y,\\widetilde{S}_k)^2},\\ee\n which together with (H1) and (H2) leads to\n \\be\\label{fk}0\\leq s_k^2\\widetilde{f}_k(\\widetilde{y}_k+s_ky,v_k(y))\\leq\n \\f{C}{d_{\\mathbb{R}^2}(y,\\widetilde{S}_k)^2}.\\ee\n  In view of (\\ref{Lp-bdd}), we estimate for any $p>1$ and any $R>0$,\n \\bea\n {\\nonumber}\\int_{\\mathbb{B}_R(0)}(s_k^2v_k(y))^pdy&=&\n s_k^{2p}\\int_{\\mathbb{B}_R(0)}\\widetilde{u}_k(\\widetilde{y}_k+s_ky)^pdy\\\\\n {\\nonumber}&\\leq&\n Cs_k^{2p-2}\\int_{\\Sigma}u_k^pdv_g\\\\{\\label{skvk}}\n &\\ra& 0\\quad {\\rm as}\\quad k\\ra\\infty.\n \\eea\n Denote for any $R>0$\n $$A_{R}=\\mathbb{B}_R(0)\\setminus\\cup_{y\\in \\mathcal{S}}{\\mathbb{B}_{1\/R}(y)}.$$\n Clearly there exists some $R_0>0$ such that $A_{R\/4}$ is necessarily smooth\n bounded domain provided that $R\\geq R_0$. Now we take $R\\geq R_0$.\n In view of (\\ref{tauk}), (\\ref{vvv}), (\\ref{fk}), and (\\ref{skvk}), we arrive at\n $$\\lim_{k\\ra\\infty}\\|\\Delta_{\\mathbb{R}^2}v_k\\|_{L^p(A_R)}= 0,\\quad\\forall R\\geq R_0,\\quad \\forall p>1.$$\n Let $w_k$ satisfy\n $$\\le\\{\\begin{array}{lll}\n -\\Delta_{\\mathbb{R}^2}w_k=-\\Delta_{\\mathbb{R}^2}v_k\\quad{\\rm in}\\quad A_R\\\\[1.5ex]\n w_k=0\\quad {\\rm on}\\quad \\p A_R.\n \\end{array}\\ri.$$\n It follows from (\\ref{skvk}) and elliptic estimates that there exists some function $w$\n such that\n $$w_k\\ra w\\quad{\\rm in}\\quad C^1(\\overline{A_R}).$$\n In particular, $w_k$ is uniformly bounded in $A_R$.\n While $v_k-w_k$ satisfies\n \\be\\label{v-wk}\\le\\{\\begin{array}{lll}\n -\\Delta_{\\mathbb{R}^2}(v_k-w_k)=0\\quad{\\rm in}\\quad A_R\\\\[1.5ex]\n v_k-w_k=v_k\\quad {\\rm on}\\quad \\p A_R.\n \\end{array}\\ri.\\ee\n We claim that\n \\be\\label{vk-0}v_k(0)\\ra+\\infty\\quad{\\rm as}\\quad k\\ra\\infty.\\ee\n For otherwise, $(v_k(0)-w_k(0))$ would be a bounded sequence. Noting that\n $v_k-w_k$ has a lower bound in $A_{R}$, applying Harnack's\n inequality to (\\ref{v-wk}), we obtain\n $$\\|v_k-w_k\\|_{L^\\infty(A_{R\/2})}\\leq C$$\n for some constant $C$ depending only on $R$, and whence $v_k$ is bounded\n in $C^1(A_{R\/4})$. In view of (\\ref{4.3}), this leads to\n  $$v_k(0)|\\nabla_{\\mathbb{R}^2}v_k(0)|\\leq C.$$\n  While (\\ref{41}) and (\\ref{contrad}) implies\n  \\be\\label{vk0-infty}v_k(0)|\\nabla_{\\mathbb{R}^2}v_k(0)|\\ra+\\infty\\quad{\\rm as}\\quad k\\ra\\infty.\\ee\n  This is a contradiction. Hence our claim (\\ref{vk-0}) follows.\n\n  Replacing $v_k$ by $v_k\/v_k(0)$ in the above estimates, we obtain\n  \\be\\label{con-1}\\f{v_k}{v_k(0)}\\ra 1\\quad{\\rm in}\\quad\n  C^1_{\\rm loc}(\\mathbb{R}^2\\setminus\\mathcal{S})\\ee\n  as $k\\ra\\infty$. For $y\\in \\Omega_k$, we set\n  $$\\widetilde{v}_k(y)=\\f{v_k(y)-v_k(0)}{|\\nabla_{\\mathbb{R}^2}v_k(0)|}.$$\n  It follows from (\\ref{41}) and (\\ref{RNK}) that\n  $$v_k(y)|\\nabla_{\\mathbb{R}^2}v_k(y)|\\leq (1+o(1))\\f{v_k(0)|\\nabla_{\\mathbb{R}^2}v_k(0)|}\n  {d_{\\mathbb{R}^2}(y,\\widetilde{S}_k)},\\quad y\\in\\Omega_k\\setminus \\widetilde{S}_k.$$\n  This together with (\\ref{con-1}) gives\n  \\be\\label{grad}|\\nabla_{\\mathbb{R}^2}\\widetilde{v}_k(y)|\\leq \\f{1+o(1)}\n  {d_{\\mathbb{R}^2}(y,\\mathcal{S})},\\ee\n  where $o(1)\\ra 0$ as $k\\ra\\infty$ locally uniformly in\n  $y\\in\\mathbb{R}^2\\setminus\\mathcal{S}$. Since\n  $\\widetilde{v}_k(0)=0$, it follows from (\\ref{grad}) that\n  $\\widetilde{v}_k$ is uniformly bounded in $C^1(A_R)$ for any\n  $R>0$. In view of (\\ref{vvv}) and (\\ref{con-1}), we have\n  \\bea\n  {\\nonumber}-\\Delta_{\\mathbb{R}^2}\\widetilde{v}_k(y)&=&-(1+o(1))\\f{v_k(y)\\Delta_{\\mathbb{R}^2}v_k(y)}{v_k(0)|\\nabla_{\\mathbb{R}^2}v_k(0)|}\\\\\n  &=&\\f{1+o(1)}{v_k(0)|\\nabla_{\\mathbb{R}^2}v_k(0)|}e^{\\psi(\\widetilde{y}_k+s_ky)}s_k^2v_k(y)\\le\\{\n  \\widetilde{f}_k(\\widetilde{y}_k+s_ky,\n v_k(y))-\\widetilde{\\tau}_k(\\widetilde{y}_k+s_ky) v_k(y)\\ri\\}\\quad\\quad{\\label{delt-v}}\n  \\eea\n  for $y\\in \\Omega_k$.\n  Similarly to (\\ref{skvk}), $s_k^2v_k^2$ is bounded in $L^p_{\\rm loc}(\\mathbb{R}^2)$ for\n  any $p>1$. In view of (\\ref{vdelta}) and (\\ref{vk0-infty}), applying elliptic estimates to\n  the equation (\\ref{delt-v}), we have\n  \\be\\label{vk-con}\\widetilde{v}_k\\ra \\widetilde{v}\\quad{\\rm in}\\quad C^1_{\\rm loc}(\\mathbb{R}^2\\setminus \\mathcal{S})\n  \\quad{\\rm as}\\quad k\\ra\\infty,\\ee\n  where $\\widetilde{v}$ satisfies\n  \\be\\label{420} \\Delta_{\\mathbb{R}^2}\\widetilde{v}=0\\quad{\\rm in}\\quad \\mathbb{R}^2\\setminus\\mathcal{S},\\quad\n  \\widetilde{v}(0)=0,\\quad |\\nabla_{\\mathbb{R}^2}\\widetilde{v}(0)|=1,\\ee\n  and\n  \\be\\label{gr-vt}|\\nabla_{\\mathbb{R}^2}\\widetilde{v}(y)|\\leq \\f{1}{d_{\\mathbb{R}^2}(y,\\mathcal{S})},\n  \\quad y\\in\\mathbb{R}^2\\setminus\\mathcal{S}.\\ee\n  Let $\\hat{y}\\in\\mathcal{S}$. For any\n  $0<r<d_{\\mathbb{R}^2}(\\hat{y},\\mathcal{S}\\setminus\\{\\hat{y}\\})\/2$, since\n  \\bna\n  \\int_{\\mathbb{B}_r(\\hat{y})}v_k\\Delta_{\\mathbb{R}^2}v_kdy&=&\\int_{\\mathbb{B}_r(\\hat{y})}\n  \\widetilde{u}_k(\\widetilde{y}_k+s_ky)s_k^2\\Delta_{\\mathbb{R}^2}\\widetilde{u}_k(\\widetilde{y}_k+s_ky)dy\\\\\n  &=&\\int_{\\mathbb{B}_{s_kr}(\\widetilde{y}_k+s_k\\hat{y})}\\widetilde{u}_k(x)\\Delta_{\\mathbb{R}^2}\n  \\widetilde{u}_k(x)dx\\\\\n  &=&-\\int_{\\phi^{-1}(\\mathbb{B}_{s_kr}(\\widetilde{y}_k+s_k\\hat{y}))}u_k\\Delta_g\n  u_kdv_g,\n  \\ena\n  we get by (\\ref{tauk}), (\\ref{bdd}) and (\\ref{Lp-bdd})\n  $$\\le|\\int_{\\mathbb{B}_{r}(\\hat{y})}v_k\\Delta_{\\mathbb{R}^2}v_kdy\\ri|\\leq \\int_\\Sigma\n  \\le(u_kf_k(x,u_k)+\\tau_k u_k^2\\ri)dv_g\\leq C.$$\n  Similarly we have by (\\ref{energy-bounded}) $$\\int_{\\mathbb{B}_r(\\hat{y})}\n  |\\nabla_{\\mathbb{R}^2}v_k|^2dy\\leq\n  \\int_\\Sigma|\\nabla_gu_k|^2dv_g\\leq C.$$\n  It then follows that\n  $$\\int_{\\p\\mathbb{B}_r(\\hat{y})}v_k\\p_\\nu v_k d\\sigma=\\int_{\\mathbb{B}_r(\\hat{y})}\n  |\\nabla_{\\mathbb{R}^2}v_k|^2dy-\\int_{\\mathbb{B}_r(\\hat{y})}v_k\\Delta_{\\mathbb{R}^2}v_kdy=O(1).$$\n  While (\\ref{con-1}) and (\\ref{vk-con}) lead to\n  $$\\int_{\\p\\mathbb{B}_r(\\hat{y})}v_k\\p_\\nu v_k d\\sigma=v_k(0)|\\nabla_{\\mathbb{R}^2}v_k(0)|\n  \\le(\\int_{\\p\\mathbb{B}_r(\\hat{y})}\\p_{\\nu}\\widetilde{v}d\\sigma+o(1)\\ri).$$\n  This together with (\\ref{vk0-infty}) gives for any $0<r<d_{\\mathbb{R}^2}(\\hat{y},\\mathcal{S}\\setminus\\{\\hat{y}\\})\/2$\n  $$\\int_{\\p\\mathbb{B}_r(\\hat{y})}\\p_\\nu \\widetilde{v}d\\sigma=0,$$\n  which leads to\n  $$\n  \\f{d}{dr}\\le(\\f{1}{2\\pi\n  r}\\int_{\\p\\mathbb{B}_r(\\hat{y})}\\widetilde{v}d\\sigma\\ri)=\n  \\f{1}{2\\pi\n  r}\\int_{\\p\\mathbb{B}_r(\\hat{y})}\\p_\\nu\\widetilde{v}d\\sigma=0.\n  $$\n  Hence there exists some constant $\\alpha$ depending only on\n  $\\hat{y}$ such that\n  \\be\\label{mea-const}\\f{1}{2\\pi\n  r}\\int_{\\p\\mathbb{B}_{r}(\\hat{y})}\\widetilde{v}d\\sigma=\\alpha,\\quad\\forall\n  0<r<d_{\\mathbb{R}^2}(\\hat{y},\\mathcal{S}\\setminus\\{\\hat{y}\\})\/2.\\ee\n  Given any $y\\in\\p \\mathbb{B}_{r}(\\hat{y})$. (\\ref{mea-const})\n  permits us to take $y^\\ast\\in \\p \\mathbb{B}_{r}(\\hat{y})$ such\n  that $\\widetilde{v}(y^\\ast)=\\alpha$. It then follows from\n  (\\ref{gr-vt}) that $|\\widetilde{v}(y)-\\alpha|\\leq \\pi$. This\n  indicates that $\\widetilde{v}$ is bounded near $\\hat{y}$. Since\n  this is true for all $\\hat{y}\\in\\mathcal{S}$, we conclude that\n  $\\widetilde{v}$ is a smooth harmonic function in $\\mathbb{R}^2$.\n  By the mean value equality,\n  $$\\int_{\\p\\mathbb{B}_R(0)}\\widetilde{v}d\\sigma=0,\\quad \\forall R>0.$$\n  This together with (\\ref{gr-vt}) implies that $\\widetilde{v}$ is\n  bounded in $L^\\infty(\\mathbb{R}^2)$. Actually we can take $z\\in\\p\\mathbb{B}_R(0)$\n  such that $\\widetilde{v}_k(z)=0$, in view of (\\ref{gr-vt}), we then have\n  for all $y\\in\\p\\mathbb{B}_R(0)$\n  \\bna\n  |\\widetilde{v}_k(y)|=|\\widetilde{v}_k(y)-\\widetilde{v}_k(z)|\\leq\n  \\pi R\\sup_{\\p\\mathbb{B}_R(0)}|\\nabla_{\\mathbb{R}^2}\n  \\widetilde{v}|\\leq 2\\pi,\n  \\ena\n  provided that $R>2\\sup_{\\hat{y}\\in\\mathcal{S}}|\\hat{y}|$.\n   Note again that\n  $\\widetilde{v}(0)=0$. Applying the Liouville theorem to (\\ref{420}), we have $\\widetilde{v}\\equiv\n  0$, which contradicts the fact that\n  $|\\nabla_{\\mathbb{R}^2}\\widetilde{v}(0)|=1$. This completes the proof of the\n  proposition. $\\hfill\\Box$\n\n\n\n  \\section{Quantization}\n\n  In this section we prove quantization results for equation\n  (\\ref{e1}). Let $x_1^\\ast,\\cdots,x_N^\\ast$ be as in Proposition 3.1.\n  For some $1\\leq i\\leq N$, $x_i^\\ast$ is called a {\\it simple} blow-up point\n  if $N=1$ or $x_j\\not=x_i$ for all $j\\in\\{1,\\cdots,N\\}\\setminus\\{i\\}$; Otherwise\n  we call $x_i^\\ast$ a {\\it non-simple} blow-up point. In the following, we distinguish between\n  these two types of points to proceed.\n\n\n\n\n  \\subsection{Quantization for simple blow-up points\\\\}\n\n  Let $x_i^\\ast$ be a simple blow-up point. Take an isothermal coordinate system $(U_i,\\phi_i;\\{x^1,x^2\\})$ near $x_i^\\ast$,\n  where $U_i\\subset\\Sigma$ is a neighborhood  of $x_i^\\ast$ such that $x_j^\\ast\\not\\in \\overline{U}_i$,\n  the closure of $U_i$, for all $j\\in\\{1,\\cdots,N\\}\\setminus\\{i\\}$. As before $\\phi_i:U_i\\ra\\Omega\\subset\\mathbb{R}^2$ is a diffeomorphism\n  with $\\phi_i(x_i^\\ast)=(0,0)$. Particularly we can find some $\\delta>0$ such that $\\mathbb{B}_{2\\delta}(0)\\subset\n  \\Omega$. In this coordinate system, the metric $g$ writes as $g=e^{\\psi_i}(d{x^1}^2+d{x_2}^2)$\n  for some smooth function $\\psi_i:\\Omega\\ra\\mathbb{R}$\n  with $\\psi_i(0,0)=0$. In this subsection we prove the following quantization result. \\\\\n\n  \\noindent{\\bf Proposition 5.1} {\\it Let $u_k$, $u_\\infty$, $\\tau_k$, $\\tau_\\infty$, $x_{i,k}$ and $x_i^\\ast$ be as in Proposition 3.1.\n  Suppose that $x_i^\\ast$\n  is a simple blow-up point. Then up to a subsequence, there exists some positive integer $I^{(i)}$\n  such that\n  \\be\\label{identity}\n  \\lim_{k\\ra\\infty}\\int_{U_i}\n  (|\\nabla_gu_k|^2+\\tau_ku_k^2)dv_g=\n  \\int_{U_i}(|\\nabla_gu_\\infty|^2+\\tau_\\infty\n  u_\\infty^2)dv_g+4\\pi I^{(i)},\n  \\ee\n  where $U_i$ is a neighborhood of $x_i^\\ast$ as above.}\\\\\n\n  In the coordinate system $(U_i,\\phi_i;\\{x^1,x^2\\})$, we write\n  $\\widetilde{x}_{i,k}=\\phi_i^{-1}(x_{i,k})$,\n  $\\widetilde{u}_k(x)=u_k(\\phi_i^{-1}(x))$,\n  $\\widetilde{\\tau}_k(x)=\\tau_k(\\phi_i^{-1}(x))$ and $\\widetilde{f}_k(x,\\widetilde{u}_k(x))=f_k(\\phi_i^{-1}(x),u_k(\\phi_i^{-1}(x)))$\n  for any $x\\in\\Omega$. Moreover for $0<s<t<\\delta$ we define the spherical mean of $\\widetilde{u}_k$, the total energy and\n  the neck energy of $\\widetilde{u}_k$ around $\\widetilde{x}_{i,k}$ by\n  \\be\\label{mean-val}\\varphi_k(t)=\\varphi_k^{(i)}(t)=\\f{1}{2\\pi t}\\int_{\\p\\mathbb{B}_t(\\widetilde{x}_{i,k})}\\widetilde{u}_kd\\sigma,\\ee\n  \\be\\label{energy}\\Lambda_{k}(t)=\\Lambda_k^{(i)}(t)=\\int_{\\mathbb{B}_t(\\widetilde{x}_{i,k})}\n  \\widetilde{u}_k\\widetilde{f}_k(x,\\widetilde{u}_k)dx,\\ee\n  and\n  \\be\\label{neck}N_{k}(s,t)=N_k^{(i)}(s,t)=\\int_{\\mathbb{B}_t(\\widetilde{x}_{i,k})\\setminus \\mathbb{B}_s(\\widetilde{x}_{i,k})}\n  \\widetilde{u}_k\\widetilde{f}_k(x,\\widetilde{u}_k)dx\\ee\n  respectively.\n  We say that the property $(\\mathcal{H}_\\ell)$ holds if there\n  exist sequences\n  $$s_k^{(0)}=0<r_k^{(1)}<s_k^{(1)}<\\cdots<r_k^{(\\ell)}<s_k^{(\\ell)}=o(1)$$\n  such that the following hypotheses are satisfied:\\\\\n\n  \\noindent$(\\mathcal{H}_{\\ell,1})$\\,\\,\\, $\\lim\\limits_{k\\ra\\infty}{r_k^{(j)}}\/{s_k^{(j)}}=\\lim\\limits_{k\\ra\\infty}\n  {s_k^{(j-1)}}\/{r_k^{(j)}}=0$ for all $1\\leq j\\leq\\ell$;\\\\\n  $(\\mathcal{H}_{\\ell,2})$\\,\\,\\,\n  $\\lim\\limits_{k\\ra\\infty}\\varphi_{k}(s_k^{(j)})\/\\varphi_{k}(Lr_k^{(j)})=0$\n  for all $1\\leq j\\leq\\ell$ and all $L>0$;\\\\\n  $(\\mathcal{H}_{\\ell,3})$\\,\\,\\, $\\lim\\limits_{k\\ra\\infty}\\Lambda_k(s_k^{(j)})=4\\pi\n  j$ for all $1\\leq j\\leq\\ell$;\\\\\n  $(\\mathcal{H}_{\\ell,4})$\\,\\,\\,$\\lim\\limits_{L\\ra\\infty}\\lim\\limits_{k\\ra\\infty}\n  \\le(N_k(s_k^{(j-1)},r_k^{(j)}\/L)+N_k(Lr_k^{(j)},s_k^{(j)})\\ri)=0$ for all $1\\leq\n  j\\leq\\ell$.\\\\\n\n\n  To prove Proposition 5.1, we follow the lines of\n  \\cite{L-R-S,Mar-Stru,Struwe}. Precisely we use induction\n  as follows: $(\\mathcal{H}_1)$ holds; if $(\\mathcal{H}_\\ell)$ holds,\n  then either $(\\mathcal{H}_{\\ell+1})$ holds, or\n  \\be\\label{0-}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}N_k(s_k^{(\\ell)},\\delta\/L)=0.\\ee\n  In view of (\\ref{energy}), we have\n  \\bna\n  \\Lambda_k(s_k^{(\\ell)})&=&\\int_{\\mathbb{B}_{s_k^{(\\ell)}}(\\widetilde{x}_{i,k})}\\widetilde{u}_k(x)\n  \\widetilde{f}_k(x,\\widetilde{u}_k(x))dx\\\\\n  &=&(1+o(1))\\int_{\\mathbb{B}_{s_k^{(\\ell)}}(\\widetilde{x}_{i,k})}\\widetilde{u}_k(x)\n  \\widetilde{f}_k(x,\\widetilde{u}_k(x))e^{\\psi_i(x)}dx\\\\\n  &=&(1+o(1))\\int_{\\phi_i^{-1}(\\mathbb{B}_{s_k^{(\\ell)}}(\\widetilde{x}_{i,k}))}\n  u_kf_k(x,u_k)dv_g\\\\\n  &\\leq& (1+o(1))\\int_\\Sigma u_kf_k(x,u_k)dv_g.\n  \\ena\n  This together with (\\ref{bdd}) and $(\\mathcal{H}_{\\ell,3})$ implies that the induction\n  terminates after finitely-many  steps. Letting $\\ell_0$ be the largest\n  integer such that $(\\mathcal{H}_{\\ell_0})$ holds. Since $\\widetilde{x}_{i,k}\\ra 0$ as\n  $k\\ra\\infty$,\n  in view of the last assertion of Proposition 3.1,\n  for any fixed $L>2\/\\delta$,\n  \\be\\label{away}\\lim_{k\\ra\\infty}\\le\\|\\widetilde{u}_k-\\widetilde{u}_\\infty\\ri\\|\n  _{C^1(\\Omega\\setminus\\mathbb{B}_{\\delta\/L}(\\widetilde{x}_{i,k}))}=0.\\ee\n     Moreover it follows from\n  $(\\mathcal{H}_{\\ell_0,3})$ and (\\ref{0-}) (with $\\ell$ replaced by $\\ell_0$) that\n  \\be\\label{inner}\n  \\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\phi_i^{-1}(\\mathbb{B}_{\\delta\/L}(\\widetilde{x}_{i,k}))}u_kf_k(x,u_k)dv_g=\n  \\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_{\\delta\/L}(\\widetilde{x}_{i,k})}\\widetilde{u}_k\\widetilde{f}_k(x,\\widetilde{u}_k)dx\n  =4\\pi\\ell_0.\n  \\ee\n  Recalling equation (\\ref{e1}), we obtain (\\ref{identity}) by combining\n  (\\ref{away}) and (\\ref{inner}) with $I^{(i)}=\\ell_0$, and thus complete the proof of Proposition 5.1.\\\\\n\n  The proof of the above induction process will be divided into the two steps below.\\\\\n\n\n  {\\it Step 1. The property $(\\mathcal{H}_1)$ holds.}\\\\\n\n  For any function $h:\\Omega=\\phi_i(U_i)\\ra\\mathbb{R}$, denote the\n  spherical average of $h$ around $\\widetilde{x}_{i,k}$ by\n  $$\\overline{h}(r)=\\f{1}{2\\pi r}\\int_{\\p\\mathbb{B}_r(\\widetilde{x}_{i,k})}hd\\sigma,\\quad\\forall 0<r<\\delta.$$\n  Let $w_k$  be the unscaled function with respect to the blow-up sequence $\\eta_{i,k}$ as in\n  (\\ref{blow-up-funct}), namely\n  $$w_k(x)=u_k(x_{i,k})(\\widetilde{u}_k(x)-u_k(x_{i,k})),\\quad x\\in\\Omega.$$\n  The decay estimate on $\\overline{w}_k$ near the point $\\widetilde{x}_{i,k}$ is\n  crucial\n  for the property $(\\mathcal{H}_1)$. Precisely we have the following result.\\\\\n\n  \\noindent{\\bf Lemma 5.2} {\\it Given $0<\\epsilon<1$. Let $T_k$\n  be the smallest number such that\n  $\\varphi_{k}(T_k)=\\epsilon u_k(x_{i,k})$. Then\n  ${r_{i,k}}\/{T_k}\\ra 0$ as $k\\ra\\infty$, where $r_{i,k}$ is as in (\\ref{scal}).\n  Moreover, for any $b<2$, there exist some integer $k_0$ and a constant $C$\n  such that when $k\\geq k_0$, we have\n  \\be\\label{w}\\overline{w}_k(r)\\leq b\\log\\f{r_{i,k}}{r}+C\\ee\n  for all $0\\leq r\\leq T_k$ and\n  \\be\\label{n}\\lim_{k\\ra\\infty}\\Lambda_k(T_k)=4\\pi.\\ee}\n  {\\it Proof.} It follows from Proposition 3.1 and the definition of $T_k$\n  that ${r_{i,k}}=o({T_k})$ as $k\\ra\\infty$. In view of (\\ref{e1}), $\\widetilde{u}_k$\n  satisfies the equation\n  \\be\\label{u-tild}-\\Delta_{\\mathbb{R}^2}\\widetilde{u}_k=e^{\\psi_i}(\\widetilde{f}_k(x,\\widetilde{u}_k)-\\widetilde{\\tau}_k\n  \\widetilde{u}_k)\\quad{\\rm in}\\quad \\Omega.\\ee\n  Let $(v_k)$ be a sequence of\n  solutions\n  to\n  \\be\\label{nh}\\le\\{\\begin{array}{lll}-\\Delta_{\\mathbb{R}^2}v_k=e^{\\psi_i} \\widetilde{f}_k(x,\\widetilde{u}_k)\n  &{\\rm on} &\\mathbb{B}_{T_k}(\\widetilde{x}_{i,k})\\\\\n  [1.5ex] v_k=\\widetilde{u}_k&{\\rm\n  on}&\\p\\mathbb{B}_{T_k}(\\widetilde{x}_{i,k}).\\end{array}\\ri.\\ee\n  Then we have by (\\ref{u-tild})\n  \\be\\label{nhh}\\le\\{\\begin{array}{lll}-\\Delta_{\\mathbb{R}^2}(v_k-\\widetilde{u}_k)=e^{\\psi_i}\n  \\widetilde{\\tau}_k\\widetilde{u}_k\n  &{\\rm on} &\\mathbb{B}_{T_k}(\\widetilde{x}_{i,k})\\\\\n  [1.5ex] v_k-\\widetilde{u}_k=0&{\\rm\n  on}&\\p\\mathbb{B}_{T_k}(\\widetilde{x}_{i,k}).\\end{array}\\ri.\\ee\n  Applying elliptic estimates to (\\ref{nhh}), we can find some\n  constant $C$ independent of $k$ such\n  that\n  $$|v_k(x)-\\widetilde{u}_k(x)|\\leq C\\,\\,\\,{\\rm for\\,\\,\\,all}\\,\\,\\,\n  x\\in \\mathbb{B}_{T_k}(\\widetilde{x}_{i,k}).$$\n  Moreover, it follows from Proposition 4.1 that\n  \\be\\label{inf}\\inf_{\\p\\mathbb{B}_{T_k}(\\widetilde{x}_{i,k})}\\widetilde{u}_k\\geq\n  \\varphi_{k}(T_k)-C\\ee\n  for some constant $C$ depending only on the Riemannian metric $g$. Applying the\n  maximum principle to (\\ref{nh}), we have by (\\ref{inf})\n  \\be\\label{lower}\\widetilde{u}_k(x)\\geq \\varphi_{k}(T_k)-C\\,\\,\\,{\\rm for\\,\\,\\,all}\\,\\,\\,\n  x\\in \\mathbb{B}_{T_k}(\\widetilde{x}_{i,k}).\\ee\n  Note that $\\varphi_{k}(T_k)=\\epsilon u_k(x_{i,k})$. For any $0\\leq t\\leq\n  T_k$, we have by (\\ref{lower}) and the fact that $u_k\\ra u_\\infty$ strongly in\n  $L^2(\\Sigma)$\n  \\be\\label{err} u_k(x_{i,k})\\int_{\\mathbb{B}_t(\\widetilde{x}_{i,k})} e^{\\psi_i}\\widetilde{\\tau}_k\\widetilde{u}_kdx\n  \\leq\n  \\f{\\|\\tau_k\\|_{L^\\infty(\\Sigma)}}{\\epsilon}\\int_{\\mathbb{B}_t(\\widetilde{x}_{i,k})}e^{\\psi_i}\n  \\le(\\widetilde{u}_k^2+C\\widetilde{u}_k\\ri)dx=o(1).\\ee\n  For any $Lr_{i,k}\\leq t\\leq T_k$, we obtain by Proposition 3.1\n  \\bea\\label{err1}{\\nonumber}\n  -u_k(x_{i,k})\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}e^{\\psi_i}\\widetilde{f}_k(x,\\widetilde{u}_k)dx\n  &\\leq&-u_k(x_{i,k})\\int_{\\mathbb{B}_{Lr_{i,k}}(\\widetilde{x}_{i,k})}e^{\\psi_i}\\widetilde{f}_k(x,\\widetilde{u}_k)dx\\\\\n  &=&-r_{i,k}^{-2}\\int_{\\mathbb{B}_{Lr_{i,k}}(\\widetilde{x}_{i,k})}\n  e^{\\psi_i}\\f{\\widetilde{f}_k(x,\\widetilde{u}_k)}{f_k(x_{i,k},u_k(x_{i,k}))}dx{\\nonumber}\\\\{\\nonumber}\n  &=&-(1+o(1))\\int_{\\mathbb{B}_L(0)}e^{(2+o(1))\\eta_\\infty}dx\\\\\n  &=&-4\\pi+o(1),\\eea\n  where $o(1)\\ra 0$ as $k\\ra\\infty$ first, and then $L\\ra\\infty$.\n  In view of (\\ref{u-tild}), $\\overline{w}_k$ satisfies\n  $$-\\Delta_{\\mathbb{R}^2}\\overline{w}_k= u_k(x_{i,k})\\overline{e^{\\psi_i}\\widetilde{f}_k(x,\\widetilde{u}_k)}\n  -u_k(x_{i,k}) \\overline{e^{\\psi_i} \\widetilde{\\tau}_k\\widetilde{u}_k}.$$\n  Then we have for any $Lr_{i,k}\\leq t\\leq T_k$\n  \\bna\n  2\\pi\n  t\\overline{w}_k^\\prime(t)&=&\\int_{\\p\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\p_\\nu\\overline{w}_k\n  d\\sigma=\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\Delta_{\\mathbb{R}^2}\\overline{w}_k\n  dx\\\\\n  &=&-u_k(x_{i,k})\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\overline{e^{\\psi_i}\\widetilde{f}_k(x,\\widetilde{u}_k)}dx\n  +u_k(x_{i,k})\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\overline{e^{\\psi_i}\\widetilde{\\tau}_k\n  \\widetilde{u}_k}dx\\\\\n  &=&-u_k(x_{i,k})\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}{e^{\\psi_i}\\widetilde{f}_k(x,\\widetilde{u}_k)}dx\n  +u_k(x_{i,k})\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}{e^{\\psi_i}\\widetilde{\\tau}_k\n  \\widetilde{u}_k}dx\\\\\n  &\\leq& -4\\pi+o(1).\n  \\ena\n  Here we used (\\ref{err}) and (\\ref{err1}) in the last inequality. Thus for any\n  $b<2$, there exists some integer $k_0$ such that\n  $$\\overline{w}_k^\\prime(t)\\leq -\\f{b}{t}\\quad{\\rm for\\,\\,all}\\quad k\\geq k_0.$$\n  This together with Proposition 3.1 leads to\n  \\bna\n  \\overline{w}_k(t)&\\leq&\\overline{w}_k(Lr_{i,k})-b\\log\\f{t}{Lr_{i,k}}{\\nonumber}\\\\\n  &\\leq&\\log\\f{1}{1+L^2}-b\\log\\f{t}{Lr_{i,k}}+o(1){\\nonumber}\\\\\n  &\\leq& b\\log\\f{r_{i,k}}{t}+C\n  \\ena\n  for some constant $C$, all $k\\geq k_0$, and all $Lr_{i,j}\\leq t\\leq\n  T_k$. It follows from Proposition 3.1 again that the above inequality\n  also holds for $0\\leq t\\leq Lr_{i,k}$. Hence (\\ref{w}) holds.\n\n  By (\\ref{w}) and (\\ref{lower}) we have\n  \\bna\n  (\\epsilon-1)u_k^2(x_{i,k})-Cu_k(x_{i,k})\\leq\\overline{w}_k(r)\n  \\leq C,\\quad \\forall r\\in\n  [Lr_{i,k}, T_k].\n  \\ena\n  Hence there holds for $Lr_{i,k}\\leq r\\leq T_k$\n  \\bea\\nonumber\n  \\varphi_{k}^2(r)-u_k^2(x_{i,k})&=&\\le(1+\\f{\\varphi_{k}(r)}{u_k(x_{i,k})}\\ri)\\overline{w}_k(r)\\\\\n  \\nonumber&=&\\le(2+\\f{\\overline{w}_k}{u_k^2(x_{i,k})}\\ri)\\overline{w}_k(r)\\\\\n  \\nonumber&\\leq&(1+\\epsilon+o(1))\\overline{w}_k(r)+(1-\\epsilon+o(1))C\\\\\n  \\label{phi-r}&\\leq& (1+2\\epsilon\/3)b\\log\\f{r_{i,k}}{r}+C,\n  \\eea\n  provided that $k$ is sufficiently large.\n  For $0<r<\\delta$ we denote\n  \\be\\label{theta}\\theta_k(r)=\\theta_k^{(i)}(r)=\\f{1}{2\\pi r}\\int_{\\p\\mathbb{B}_r(\\widetilde{x}_{i,k})}\n  \\widetilde{f}_k(x,\\varphi_k(r))d\\sigma.\\ee\n  Taking $b$ such that\n  $(1+2\\epsilon\/3)b=2+\\epsilon$ in (\\ref{phi-r}) and recalling (H4) and (H5), we\n  can find some constant $C$ such that\n  for $Lr_{i,k}\\leq r\\leq T_k$\n  \\bea\\nonumber\n  \\f{\\theta_{k}(r)}{f_k(x_{i,k},u_k(x_{i,k}))}&=&\\f{\\theta_{k}(r)}\n  {\\widetilde{f}_k(\\widetilde{x}_{i,k},\\varphi_{k}(r))}\n  \\f{\\widetilde{f}_k(\\widetilde{x}_{i,k},\\varphi_{k}(r))}{f_k(x_{i,k},u_k(x_{i,k}))}\\\\\n  \\nonumber&=&(1+o(1))\\f{{f}_k({x}_{i,k},\\varphi_{k}(r))}{f_k(x_{i,k},u_k(x_{i,k}))}\\\\\n  \\nonumber&=&(1+o(1))e^{(1+o(1))(\\varphi_{k}^2(r)-u_k^2(x_{i,k}))}\\\\\n  \\label{thk}&\\leq&C\\le(\\f{r_{i,k}}{r}\\ri)^{2+\\epsilon}\n  \\eea\n  for sufficiently large $k$. For $0<s<t<\\delta$, we define next a function analogous to  (\\ref{neck}) as below.\n  \\be\\label{neck-symm}\\overline{N}_k(s,t)=\\overline{N}_k^{(i)}(s,t)=2\\pi\\int_s^tr\\varphi_k(r)\\theta_k(r)dr.\\ee\n   In view of (\\ref{w}) and (\\ref{thk}), we estimate\n  \\bna\n  \\overline{N}_k(Lr_{i,k},T_k)&=&2\\pi\\int_{Lr_{i,k}}^{T_k}r\\varphi_k(r)\\theta_k(r)dr\\\\\n  &=&2\\pi r_{i,k}^{-2}\\int_{Lr_{i,k}}^{T_k}r\\f{\\varphi_k(r)}{u_k(x_{i,k})}\\f{\\theta_k(r)}\n  {f_k(x_{i,k},u_k(x_{i,k}))}dr\\\\\n  &\\leq&2\\pi(1+o(1))Cr_{i,k}^{\\epsilon}\\int_{Lr_{i,k}}^{T_k}\\f{1}{r^{1+\\epsilon}}dr\\\\\n  &\\leq&2\\pi(1+o(1))C\\epsilon^{-1}L^{-\\epsilon}.\n  \\ena\n  This leads to\n  \\be\\label{nn}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\overline{N}_k(Lr_{i,k},T_k)=0.\\ee\n  Since Proposition 4.1 implies that\n  $$u_k^2(x)-\\varphi_{k}^2(r)\\leq C\\quad{\\rm for \\,\\,all}\\quad x\\in\\p\\mathbb{B}_{r}(\\widetilde{x}_{i,k}),$$\n  there holds\n  $$N_k(Lr_{i,k},T_k)\\leq C\\overline{N}_k(Lr_{i,k},T_k)+o(1).$$\n  This together with (\\ref{nn}) leads to\n  \\be\\label{m1}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}{N}_k(Lr_{i,k},T_k)=0.\\ee\n  By Proposition 3.1,\n  \\bna\n  \\Lambda_k(Lr_{i,k})=\\int_{\\mathbb{B}_{Lr_{i,k}}(\\widetilde{x}_{i,k})}\n  \\widetilde{u}_k\\widetilde{f}_k(x,\\widetilde{u}_k)dx\n  =(1+o(1))\\int_{\\mathbb{B}_L(0)}e^{2\\eta_\\infty}dx.\n  \\ena\n  Hence\n  \\be\\label{m2}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\Lambda_k(Lr_{i,k})=4\\pi.\\ee\n  Thus (\\ref{n}) follows  immediately from (\\ref{m1}) and (\\ref{m2}).\n  $\\hfill\\Box$\\\\\n\n  By Lemma 5.2 we may choose a subsequence $u_k$, numbers $\\epsilon_k\\searrow\n  0$ as $k\\ra\\infty$ and $s_k=T_k(\\epsilon_k)$ with $r_{i,k}\/s_k\\ra\n  0$, $\\varphi_{k}(s_k)\\ra\\infty$ as $k\\ra\\infty$ and such that\n  $$\\lim_{k\\ra\\infty}\\Lambda_k(s_k)=4\\pi,\\quad \\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}{N}_k(Lr_{i,k},s_k)=0,$$\n  while in addition\n  $$\\lim_{k\\ra\\infty}\\f{\\varphi_{k}(s_k)}{\\varphi_{k}(Lr_{i,k})}=0,\\quad\\forall\\, L>0.$$\n  Let $r_k^{(1)}=r_{i,k}$, $s_k^{(1)}=s_k$. Then $(\\mathcal{H}_1)$\n  holds and Step 1 is finished. \\\\\n\n  \\noindent{\\it Step 2. Suppose that $(\\mathcal{H}_\\ell)$ already holds for some integer $\\ell\\geq 1$,\n  namely there exist sequences $s_k^{(0)}=0<r_k^{(1)}<s_k^{(1)}<\\cdots<r_k^{(\\ell)}<s_k^{(\\ell)}=o(1)$\n  such that $(\\mathcal{H}_{\\ell,1})$ up to\n  $(\\mathcal{H}_{\\ell,4})$ hold. Then we shall prove that either\n  $\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}N_k(s_k^{(\\ell)},\\delta\/L)=0$\n  or $(\\mathcal{H}_{\\ell+1})$ holds.}\\\\\n\n  Setting\n  \\be\\label{Pt}P_k(t)=P_k^{(i)}(t)=t\\int_{\\p\\mathbb{B}_t}\\widetilde{u}_k\\widetilde{f}_k(x,\\widetilde{u}_k)d\\sigma,\n  \\quad \\overline{P}_k(t)=\\overline{P}_k^{(i)}(t)=2\\pi t^2\\varphi_k(t)\\theta_k(t)\\ee\n  and assuming $(\\mathcal{H}_\\ell)$ holds,\n  we have the following result.\\\\\n\n  \\noindent{\\bf Lemma 5.3} {\\it There exists a constant $C_0$ depending only on\n  the upper bound of the total energy (\\ref{bdd}) and the Riemannian metric $g$ such that for $s_k^{(\\ell)}\\leq t_k=o(1)$, there holds\n  \\be\\label{lemm}\\overline{N}_{k}(s_k^{(\\ell)},t_k)\\leq \\overline{P}_{k}(t_k)+C_0\n  \\overline{N}_{k}^2(s_k^{(\\ell)},t_k)+o(1),\\ee\n  where $o(1)\\ra 0$ as $k\\ra\\infty$, $\\overline{N}_{k}$ and $\\overline{P}_{k}$ are defined as in\n  (\\ref{neck-symm}) and (\\ref{Pt}) respectively.}\\\\\n\n  \\noindent{\\it Proof.}  We first claim that\n  there exists a constant $C$ depending only on $\\delta$ and the Riemannian metric $g$\n  such that\n  \\be\\label{cla}\\varphi_{k}(s)\\leq \\sup_{\\p\\mathbb{B}_s(\\widetilde{x}_{i,k})}\\widetilde{u}_k\n  \\leq \\inf_{\\p\\mathbb{B}_r(\\widetilde{x}_{i,k})}\\widetilde{u}_k+C\\leq \\varphi_{k}(r)+C\n  \\,\\,\\,{\\rm for\\,\\,all}\\,\\,\\, 0<r<s\\leq \\delta.\\ee\n  To see the last inequality, we set $v_k$ be a positive solution of\n  \\be\\label{nh2}\\le\\{\\begin{array}{lll}-\\Delta_{\\mathbb{R}^2}v_k=e^{\\psi_i} \\widetilde{f}_k(x,\\widetilde{u}_k)\n  &{\\rm in}&\\mathbb{B}_{\\delta}(\\widetilde{x}_{i,k})\\\\\n  [1.5ex] v_k=\\widetilde{u}_k&{\\rm\n  on}&\\p\\mathbb{B}_{\\delta}(\\widetilde{x}_{i,k}).\\end{array}\\ri.\\ee\n  Thus we have by (\\ref{e1})\n  \\be\\label{nhh2}\\le\\{\\begin{array}{lll}-\\Delta_{\\mathbb{R}^2}(v_k-\\widetilde{u}_k)=e^{\\psi_i}\n  \\widetilde{\\tau}_k\\widetilde{u}_k\n  &{\\rm in} &\\mathbb{B}_{\\delta}(\\widetilde{x}_{i,k})\\\\\n  [1.5ex] v_k-\\widetilde{u}_k=0&{\\rm\n  on}&\\p\\mathbb{B}_{\\delta}(\\widetilde{x}_{i,k}).\\end{array}\\ri.\\ee\n  Noting that $\\|e^{\\psi_i}\n  \\widetilde{\\tau}_k\\widetilde{u}_k\\|_{L^p(\\mathbb{B}_{\\delta}(\\widetilde{x}_{i,k}))}$ is\n  bounded for any $p>1$ and\n  applying elliptic regularity estimates to (\\ref{nhh2}), we then find some\n  constant $C=C(\\delta)$ such that\n  \\be\\label{uk}\n  v_k(x)-C\\leq \\widetilde{u}_k(x)\\leq v_k(x)+C\\quad{\\rm\n  for\\,\\,all}\\quad x\\in\\mathbb{B}_{\\delta}(\\widetilde{x}_{i,k}).\n  \\ee\n   By (\\ref{nh2}), we have for $0<r<\\delta$\n    $$-(r\\overline{v}_k^\\prime(r))^\\prime=r\\,\\overline{e^{\\psi_i} \\widetilde{f}_k(x,\\widetilde{u}_k)}.$$\n  Integration from $0$ to $r$ gives\n  $$-r\\overline{v}_k^\\prime(r)=\\int_0^r r\\,\\overline{e^{\\psi_i} \\widetilde{f}_k(x,\\widetilde{u}_k)}dr.$$\n  Hence\n  \\be\\label{mono}\\overline{v}_k^\\prime(r)\\leq 0\\quad {\\rm for\\,\\,all}\\quad 0<r<\\delta.\\ee\n  Now fix $0<r<s\\leq\\delta$. There exist two points\n  $\\xi\\in\\p\\mathbb{B}_{r}(\\widetilde{x}_{i,k})$ and\n  $\\zeta\\in\\p\\mathbb{B}_{s}(\\widetilde{x}_{i,k})$ such that\n  $$v_k(\\xi)=\\overline{v}_k(r),\\quad v_k(\\zeta)=\\overline{v}_k(s).$$\n  This together with the gradient estimate (Proposition 4.1), (\\ref{uk}),\n  and (\\ref{mono}) leads to\n  \\bna\n  \\sup_{\\p\\mathbb{B}_s(\\widetilde{x}_{i,k})}\\widetilde{u}_k&\\leq& \\widetilde{u}_k(\\zeta)+C\n  \\leq v_k(\\zeta)+C\\\\&\\leq& v_k(\\xi)+C\n  \\leq\\inf_{\\p\\mathbb{B}_r(\\widetilde{x}_{i,k})}\\widetilde{u}_k+C.\n  \\ena\n  This confirms our claim (\\ref{cla}).\n\n  Next we calculate\n  \\bna\\theta_{k}^\\prime(r)&=&\\f{d}{dr}\\le(\\f{1}{2\\pi}\\int_0^{2\\pi}\\widetilde{f}_k\n  \\le(\\widetilde{x}_{i,k}^1+r\\cos\\theta,\\widetilde{x}_{i,k}^2+r\\sin\\theta,\\varphi_{k}(r)\\ri)d\\theta\\ri)\\\\\n  &=&\\f{1}{2\\pi}\\int_0^{2\\pi}\\nabla_x\\widetilde{f}_k\n  \\le(\\widetilde{x}_{i,k}^1+r\\cos\\theta,\\widetilde{x}_{i,k}^2+r\\sin\\theta,\\varphi_{k}(r)\\ri)\\cdot\n  (\\cos\\theta,\\sin\\theta)d\\theta\\\\\n  &&+\\f{1}{2\\pi}\\int_0^{2\\pi}\\widetilde{f}_k^\\prime\n  \\le(\\widetilde{x}_{i,k}^1+r\\cos\\theta,\\widetilde{x}_{i,k}^2+r\\sin\\theta,\\varphi_{k}(r)\\ri)\n  \\varphi_k^\\prime(r)d\\theta,\\ena\n  where we write\n  $\\widetilde{x}_{i,k}=(\\widetilde{x}_{i,k}^1,\\widetilde{x}_{i,k}^2)$.\n  In view of (H4), we obtain\n  \\be\\label{der}|\\theta_{k}^\\prime(r)|\\leq C\\le(1+\\theta_{k}(r)+\\varphi_{k}(r)\n  |\\varphi_{k}^\\prime(r)|\\theta_{k}(r)\\ri).\\ee\n  For $s=s_k^{(\\ell)}\\leq t\\leq t_k$, we have by  equation (\\ref{e1}).\n  \\bna\n  -2\\pi\n  t\\varphi_{k}^\\prime(t)&=&-\\int_{\\p\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\p_\\nu\\varphi_{k}\n  d\\sigma=-\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\Delta_{\\mathbb{R}^2}\\varphi_{k}dx\\\\\n  &=&\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\overline{e^{\\psi_i}\\widetilde{f}_k(x,\\widetilde{u}_k)}dx\n  -\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\overline{e^{\\psi_i}\\widetilde{\\tau}_k\\widetilde{u}_k}dx\\\\\n  &=&\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}e^{\\psi_i}\\widetilde{f}_k(x,\\widetilde{u}_k)dx\n  -\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}e^{\\psi_i}\\widetilde{\\tau}_k\\widetilde{u}_kdx.\n  \\ena\n  It follows from (H4), (H5) and Proposition 4.1 that\n  $\\widetilde{f}_k(x,\\widetilde{u}_k)\\leq\n  C(1+\\widetilde{f}_k(x,\\varphi_k(r)))\\leq C(1+\\theta_k(r))$,\n  where $r=|x-\\widetilde{x}_{i,k}|$. Combining (H4), (H5) and\n  (\\ref{cla}), we have\n  $$\\int_{\\mathbb{B}_s(\\widetilde{x}_{k})}\\varphi_k(s)\\theta_k(r)dx\\leq C(1+\\Lambda_k(s)),$$\n  where we used $r=|x-\\widetilde{x}_{i,k}|$.\n  Note that $\\varphi_k(s)\\ra\\infty$ as $k\\ra\\infty$. We then obtain\n  \\bna\n  -2\\pi\n  t\\varphi_{k}^\\prime(t)&\\leq&\n  C\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\le(1+\\theta_{k}(r)\\ri)dx\\\\\n  &\\leq&C\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})\\setminus\\mathbb{B}_{s}\n  (\\widetilde{x}_{i,k})}\\theta_{k}(r)dx+\\f{C}{\\varphi_{k}(s)}\n  \\int_{\\mathbb{B}_{s}(\\widetilde{x}_{i,k})}\\varphi_{k}(s)\\theta_{k}(r)dx+o(1)\\\\\n    &\\leq&C\\overline{N}_k(s,t)+o(1).\n  \\ena\n  This immediately leads to\n  \\be\\label{dr1}\n  -\\pi\\int_s^tr^2\\varphi_{k}^\\prime(r)\\theta_{k}(r)dr\\leq\n  C\\overline{N}_k^2(s,t)+o(1).\n  \\ee\n  Similarly we have\n  \\bea{\\nonumber}\n  -2\\pi\n  t\\varphi_{k}(t)\\varphi_{k}^\\prime(t)&=&-\\int_{\\p\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\varphi_{k}(t)\\p_\\nu\\varphi_{k}\n  d\\sigma=-\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\varphi_{k}(t)\\Delta_{\\mathbb{R}^2}\\varphi_{k}dx\\\\\n  {\\nonumber}&=&\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\varphi_{k}(t)\\overline{e^{\\psi_i}\\widetilde{f}_k(x,\\widetilde{u}_k)}dx\n  -\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\varphi_{k}(t)\\overline{e^{\\psi_i}\\widetilde{\\tau}_k\\widetilde{u}_k}dx\\\\\n  \\label{tph}&=&\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\varphi_{k}(t)\\overline{e^{\\psi_i}\\widetilde{f}_k(x,\\widetilde{u}_k)}dx+o(1),\n  \\eea\n  where the last equality follows from (\\ref{cla}) and $u_k\\ra u_\\infty$ strongly in\n  $L^2(\\Sigma)$. Repeatedly using (\\ref{cla}), we obtain\n  \\bna\n  \\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\varphi_{k}(t)\\overline{e^{\\psi_i}\\widetilde{f}_k(x,\\widetilde{u}_k)}dx\n  &\\leq&C\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})}\\varphi_{k}(t)\\le(1+\\theta_{k}(r)\\ri)dx\\\\\n  &\\leq&C\\int_{\\mathbb{B}_{t}(\\widetilde{x}_{i,k})\\setminus\\mathbb{B}_{s}(\\widetilde{x}_{i,k})}\\le(1+\\varphi_{k}(r)\\ri)\n  \\le(1+\\theta_{k}(r)\\ri)dx\\\\\n  &&+C\\int_{\\mathbb{B}_{s}(\\widetilde{x}_{i,k})\\setminus\\mathbb{B}_{Lr_k^{(\\ell)}}(\\widetilde{x}_{i,k})}\\le(1+\\varphi_{k}(r)\\ri)\n  \\le(1+\\theta_{k}(r)\\ri)dx\\\\\n  &&+C\\int_{\\mathbb{B}_{Lr_k^{(\\ell)}}(\\widetilde{x}_{i,k})}\\le(1+\\varphi_{k}(s)\\ri)\n  \\le(1+\\theta_{k}(r)\\ri)dx\\\\\n  &\\leq&C\\le(\\overline{N}_k(s,t)+\\overline{N}_k(Lr_k^{(\\ell)},s)+\\f{\\varphi_{k}(s)}{\\varphi_{k}(Lr_k^{(\\ell)})}\n  \\le({\\Lambda}_k(Lr_k^{(\\ell)})+o(1)\\ri)\\ri).\n  \\ena\n  This together with\n  (\\ref{tph}), $(\\mathcal{H}_{\\ell,2})$ and $(\\mathcal{H}_{\\ell,4})$ implies\n  \\be\\label{t-p}2\\pi t\\varphi_{k}(t)|\\varphi_{k}^\\prime(t)|\\leq\n  C\\overline{N}_k(s,t)+o(1).\\ee\n  Obviously\n  $$\\int_s^tr^2\\varphi_{k}(r)dr=o(1),\\quad \\int_s^tr^2\\varphi_{k}(r)\\theta_{k}(r)dr=o(1).$$\n  It then follows from (\\ref{der}) and (\\ref{t-p}) that\n  \\bea\\nonumber\n  -\\pi\\int_s^tr^2\\varphi_{k}(r)\\theta_{k}^\\prime(r)dr&\\leq&\\pi C\\int_s^tr^2\\varphi_{k}^2(r)\n  |\\varphi_{k}^\\prime(r)|\\theta_{k}(r)dr+o(1)\\\\\n  &\\leq&C\\overline{N}_k^2(s,t)+o(1).\\label{i-2}\n  \\eea\n  Integration by parts gives\n  \\bna\n  \\overline{N}_k(s,t)&=&\\int_s^t2\\pi\n  r\\varphi_{k}(r)\\theta_{k}(r)dr\\\\\n  &\\leq&\\pi t^2\\varphi_{k}(t)\\theta_{k}(t)-\\pi\\int_s^tr^2\\varphi_{k}^\\prime(r)\\theta_{k}(r)dr\n  -\\pi\\int_s^tr^2\\varphi_{k}(r)\\theta_{k}^\\prime(r)dr.\n  \\ena\n  This together with (\\ref{dr1}) and (\\ref{i-2}) implies (\\ref{lemm}). $\\hfill\\Box$\\\\\n\n  \\noindent{\\bf Lemma 5.4} {\\it Let $C_0$ be the constant as in Lemma 5.3. Let $t_k$ be such that for a subsequence\n  $$s_k^{(\\ell)}<t_k=o(1),\\quad 0<\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},t_k)=\\alpha<\\f{1}{2C_0}.$$\n  Then $s_k^{(\\ell)}=o(t_k)$ as $k\\ra\\infty$, $\\liminf\\limits_{k\\ra\\infty}\\overline{P}_k(t_k)\\geq \\alpha\/2$, and\n  \\be\\label{3-}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},t_k\/L)=0,\\ee\n  where $\\overline{N}_k$ and $\\overline{P}_k$ are as defined in (\\ref{neck-symm}) and (\\ref{Pt})\n  respectively.}\\\\\n\n  \\noindent{\\it Proof.} We first claim that\n  \\be\\label{1-}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},Ls_k^{(\\ell)})=0.\\ee\n  Actually, in view of (\\ref{cla}), we have for $0<t\\leq t_k$\n  \\be\\label{P-N}\\overline{P}_k(t)\\leq C\\overline{N}_k(t\/2,t)+o(1)\\leq C\\overline{P}_k(t\/2)+o(1),\\ee\n  and\n  $$\\overline{N}_k(t,2t)\\leq C\\overline{N}_k(t\/2,t)+o(1).$$\n  In particular, for any $j\\in\\mathbb{N}$ there holds\n  \\bna\n  \\lim_{k\\ra\\infty}\\overline{N}_k(2^{j-1}s_k^{(\\ell)},2^{j}s_k^{(\\ell)})\n  &\\leq&C\\lim_{k\\ra\\infty}\\overline{N}_k(2^{j-2}s_k^{(\\ell)},2^{j-1}s_k^{(\\ell)})\\\\\n  &\\leq&C^{j}\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)}\/2,s_k^{(\\ell)})=0.\n  \\ena\n  If $L\\leq 2^j$, we obtain\n  $$\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},Ls_k^{(\\ell)})\\leq \\lim_{k\\ra\\infty}\\sum_{m=1}^j\n  \\overline{N}_k(2^{m-1}s_k^{(\\ell)},2^ms_k^{(\\ell)})=0.$$\n  Thus our claim (\\ref{1-}) follows immediately. One can see from (\\ref{1-}) that $s_k^{(\\ell)}\/t_k\\ra\n  0$ as $k\\ra\\infty$. By Lemma 5.3,\n  \\be\\label{2-}\n  \\liminf_{k\\ra\\infty}\\overline{P}_k(t_k)\\geq\n  \\f{1}{2}\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},t_k)=\\f{\\alpha}{2}.\n  \\ee\n  Now we show (\\ref{3-}).\n  Assuming the contrary, there holds\n  $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},t_k\/L)=\\beta>0.$$\n  Then we have for any fixed $L\\geq 1$ and all sufficiently large $k$\n  $$\\f{\\beta}{2}\\leq \\overline{N}_k(s_k^{(\\ell)},t_k\/L)\\leq\\overline{N}_k(s_k^{(\\ell)},t_k)<\\f{1}{2C_0}.$$\n  Applying (\\ref{lemm}) with $t_k\/L$ instead of $t_k$, we get\n  $$\\lim_{k\\ra\\infty}\\overline{P}_k(t_k\/L)\\geq \\f{\\beta}{4},$$\n  and then by (\\ref{P-N})\n  $$C\\lim_{k\\ra\\infty}\\overline{N}_k\\le(t_k\/(2L),t_k\/L\\ri)\\geq\\lim_{k\\ra\\infty}\\overline{P}_k(t_k\/L)\\geq \\f{\\beta}{4}.$$\n  Choosing $L=2^m$, $m=0,1,\\cdots,j-1$, we have\n  \\bna\n  \\f{j\\beta}{4}\\leq C\\lim_{k\\ra\\infty}\\overline{N}_k(2^{-j}t_k,t_k)\n  \\leq C(1+\\limsup_{k\\ra\\infty}{\\Lambda}_k(t_k))\\leq C.\n  \\ena\n  We get a contradiction by letting $j\\ra\\infty$ and obtain\n  (\\ref{3-}). $\\hfill\\Box$\\\\\n\n  \\noindent{\\bf Lemma 5.5} {\\it Suppose that\n  \\be\\label{caseA}\\lim_{k\\ra\\infty}\\sup_{s_k^{(\\ell)}<t<t_k}\\overline{P}_k(t)=0\n  \\quad{\\rm for\\,\\,any\\,\\,sequence}\\,\\, t_k\\ra 0\\,\\,{\\rm as}\\,\\, k\\ra \\infty.\\ee\n  Then we have\n  $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},\\delta\/L)=0.$$\n  }\n  {\\it Proof.} In view of Lemma 5.4, it suffices to prove\n  \\be\\label{cd}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\sup_{s_k^{(\\ell)}<t<\\delta\/L}\\overline{P}_k(t)=0.\\ee\n  Indeed, if we take some number $t_{k,L}\\in (s_k^{(\\ell)},\\delta\/L)$ such\n  that\n  $$\\overline{P}_k(t_{k,L})=\\sup_{s_k^{(\\ell)}<t<\\delta\/L}\\overline{P}_k(t),$$\n  then either\n  \\be\\label{ze}\\lim_{k\\ra\\infty}t_{k,L}=0,\\ee\n  or\n  \\be\\label{ze1}\\lim_{k\\ra\\infty}t_{k,L}=t_L^\\ast>0.\\ee\n  In case of (\\ref{ze}), we already have (\\ref{cd}) because of\n  (\\ref{caseA}). While in case of (\\ref{ze1}), we have by using (\\ref{cla})\n  \\bea\\label{eli}\n  \\overline{P}_k(t_{k,L})\n  \\leq Ct_{k,L}^2\\le(1+\\varphi_{k}(t_{L}^\\ast\/2)\\theta_{k}(t_{L}^\\ast\/2)\\ri)\n   \\eea\n    for sufficiently large $k$.\n  Note that\n  $\\p\\mathbb{B}_{t_L^\\ast\/2}(\\widetilde{x}_{i,k})\\subset\\mathbb{B}_{t_L^\\ast}(\\widetilde{x}_{i}^\\ast)\n  \\setminus\\mathbb{B}_{t_L^\\ast\/3}(\\widetilde{x}_{i}^\\ast)$\n  for sufficiently large $k$, and that $t_{k,L}\\leq \\delta\/L\\ra 0$ as $k\\ra\\infty$ first and then $L\\ra\\infty$.\n  Moreover, by Proposition 3.1, we have $u_k\\ra u_\\infty$ in\n  $C^1_{\\rm loc}(\\Sigma\\setminus \\cup_{j=1}^N\\{x_j^\\ast\\},\\mathbb{R})$\n  and $u_\\infty\\in C^1(\\Sigma,\\mathbb{R})$, In particular, $u_\\infty$ is bounded on $\\mathbb{B}_\\delta(x_i^\\ast)$.\n  It then follows from (\\ref{eli})\n  that\n  $$\n  \\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\overline{P}_k(t_{k,L})=0.\n  $$\n  Thus (\\ref{cd}) holds again.  $\\hfill\\Box$\\\\\n\n  If the assumption (\\ref{caseA}) is not satisfied, then (\\ref{P-N})\n  implies that there exists a sequence $t_k\\ra 0$ as $k\\ra\\infty$\n  such that\n  \\be\\label{(5)}\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},t_k)>0.\\ee\n  We shall show that the property $(\\mathcal{H}_{\\ell+1})$ holds.\n  Take $r_k^{(\\ell+1)}\\in(s_k^{(\\ell)},t_k)$ such that up to a\n  subsequence, there holds\n  $$0<\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},r_k^{(\\ell+1)})<\\f{1}{2C_0},$$\n  where $C_0$ is as in Lemma 5.3. It then follows from Lemma 5.4\n  that\n  \\bea\n  \\label{(6)}&&\\lim_{k\\ra\\infty}\\f{s_k^{(\\ell)}}{r_k^{(\\ell+1)}}=0,\\quad\n  \\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},r_k^{(\\ell+1)})>0,\\\\\n  \\label{(7)}&&\\liminf_{k\\ra\\infty}\\overline{P}_k(r_k^{(\\ell+1)})>0,\\quad\n  \\lim_{k\\ra\\infty}\\varphi_{k}(r_k^{(\\ell+1)})=\\infty,\n  \\eea\n  and that\n  \\be\\label{(8)}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\overline{N}_k(s_k^{(\\ell)},r_k^{(\\ell+1)}\/L)=0.\\ee\n  Moreover, we have the following result.\\\\\n\n  \\noindent{\\bf Lemma 5.6} {\\it Up to a subsequence there holds\n  $$\\eta_k^{(\\ell+1)}(x):=\\varphi_{k}(r_k^{(\\ell+1)})\n  \\le(\\widetilde{u}_k(\\widetilde{x}_{i,k}+r_k^{(\\ell+1)} x)-\\varphi_{k}(r_k^{(\\ell+1)})\\ri)\n  \\ra\\eta^{(\\ell+1)}(x)$$ in $C^1_{\\rm loc}(\\mathbb{R}^2\\setminus\\{0\\})$ as $k\\ra\\infty$, where\n  $$\\eta^{(\\ell+1)}(x)=\\log\\f{2}{\\sqrt{\\alpha_0}(1+|x|^2)}$$ and\n  $$\\int_{\\mathbb{R}^2}e^{2\\eta^{(\\ell+1)}}dx=\\f\n{4\\pi}{\\alpha_0}$$\nfor some constant $\\alpha_0>0$.}\\\\\n\n  \\noindent{\\it Proof.} To simplify the notations we write $r_k=r_k^{(\\ell+1)}$, $\\eta_k=\\eta_k^{(\\ell+1)}$,\n  and $\\eta=\\eta^{(\\ell+1)}$.  For any fixed $L>0$, we set\n  \\be\\label{(9)}v_k(x)=\\widetilde{u}_k(\\widetilde{x}_{i,k}+r_k x),\n  \\quad x\\in\\mathbb{B}_L(0)\\setminus\\mathbb{B}_{1\/L}(0).\\ee\n  In view of Proposition 4.1, there exists some constant $C=C(L)$\n  such that\n  $$|\\widetilde{u}_k^2(\\widetilde{x}_{i,k}+r_k x)-\\varphi_{k}^2(r_k)|\\leq C,$$\n  and thus\n  \\be\\label{(11)}|\\varphi_{k}(r_k)\\le(\\widetilde{u}_k(\\widetilde{x}_{i,k}+r_k x)-\\varphi_{k}(r_k)\\ri)|\\leq C.\\ee\n  Hence\n  \\be\\label{(10)}\\eta_k\\quad{\\rm is\\,\\,bounded\\,\\,in}\\quad L^\\infty_{\\rm loc}(\\mathbb{R}^2\\setminus\\{0\\}).\\ee\n  Combining (\\ref{(7)}) and (\\ref{(11)}), we have\n  $$v_k-\\varphi_{k}(r_k)\\ra 0\\quad{\\rm in}\\quad L^\\infty_{\\rm loc}(\\mathbb{R}^2\\setminus\\{0\\})\n  \\quad{\\rm as}\\quad k\\ra\\infty,$$\n  in particular\n  \\be\\label{(13)}\\f{v_k}{\\varphi_{k}(r_k)}\\ra 1\\quad{\\rm in}\\quad L^\\infty_{\\rm loc}(\\mathbb{R}^2\\setminus\\{0\\})\n  \\quad{\\rm as}\\quad k\\ra\\infty.\\ee\n  By the equation (\\ref{e1}), we write for $x\\in\\Omega_k=\\{x\\in\\mathbb{R}^2:\\widetilde{x}_{i,k}+r_k x\n  \\in \\mathbb{B}_\\delta(0)\\}$\n  \\be\\label{(14)}-\\Delta_{\\mathbb{R}^2}\\eta_k(x)=e^{\\psi_i(\\widetilde{x}_{i,k}+r_k x)}\\varphi_{k}(r_k)r_k^2\n  \\widetilde{f}(\\widetilde{x}_{i,k}+r_k x,v_k(x))-e^{\\psi_i(\\widetilde{x}_{i,k}+r_k\n  x)}\\varphi_{k}(r_k)r_k^2\\widetilde{\\tau}_k(\\widetilde{x}_{i,k}+r_kx)\n  v_k(x).\\ee\n  Since $u_k\\ra u_\\infty$ strongly in $L^2(\\Sigma)$, we have by\n  using (\\ref{(13)})\n  \\bea\\nonumber\n  r_k^2\\varphi_{k}^2(r_k)&=&\\f{r_k^2}{3\\pi}\\int_{\\mathbb{B}_2(0)\\setminus\\mathbb{B}_{1}(0)}\\varphi_{k}^2(r_k)dx\\\\\n  \\nonumber&=&(1+o(1))\\f{r_k^2}{3\\pi}\\int_{\\mathbb{B}_2(0)\\setminus\\mathbb{B}_{1}(0)}v_k^2(x)dx\\\\\n  \\nonumber&=&\\f{1+o(1)}{3\\pi}\\int_{\\mathbb{B}_{2r_k}(\\widetilde{x}_{i,k})\\setminus\\mathbb{B}_{r_k}\n  (\\widetilde{x}_{i,k})}\\widetilde{u}_k^2(y)dy\\\\\n  \\label{(15)}&\\ra& 0\\quad{\\rm as}\\quad k\\ra\\infty.\n  \\eea\n  By (\\ref{(7)}) we may assume\n  \\be\\label{(16)}r_k^2\\varphi_{k}^2(r_k)\\theta_{k}(r_k)\\ra \\alpha_0>0.\\ee\n  Moreover, by (H4) and (H5) we have\n  \\bea\n  \\f{\\widetilde{f}_k(\\widetilde{x}_{i,k}+r_k\n  x,v_k(x))}{\\theta_{k}(r_k)}&=&(1+o(1))\\f{\\widetilde{f}_k(\\widetilde{x}_{i,k}+r_k\n  x,v_k(x))}{\\widetilde{f}_k(\\widetilde{x}_{i,k}+r_k\n  x,\\varphi_{k}(r_k))}\\nonumber\\\\\n  \\nonumber&=&(1+o(1))e^{(1+o(1))(v_k^2(x)-\\varphi_{i,k}^2(r_k))}\\\\\\label{(17)}\n  &=&(1+o(1))e^{(2+o(1))\\eta_k(x)}.\n  \\eea\n  Applying elliptic estimates to (\\ref{(14)}), we conclude from\n  (\\ref{(10)}), (\\ref{(15)})-(\\ref{(17)}) that\n  \\be\\label{(18)}\\eta_k\\ra\\eta\\quad{\\rm in}\\quad C^1_{\\rm loc}(\\mathbb{R}^2\\setminus\\{0\\})\n  \\quad {\\rm as}\\quad k\\ra\\infty,\\ee\n  where $\\eta$ satisfies\n  \\be\\label{(19)}-\\Delta_{\\mathbb{R}^2}\\eta=\\alpha_0 e^{2\\eta}\\quad{\\rm in}\\quad \\mathbb{R}^2\\setminus\\{0\\}.\\ee\n  For any $L>0$, (\\ref{(18)}) together with (\\ref{bdd}), (\\ref{(13)}) and (\\ref{(16)})\n  leads to\n  \\bna\n  \\int_{\\mathbb{B}_L(0)\\setminus\\mathbb{B}_{1\/L}(0)}e^{2\\eta}dx&=&\\lim_{k\\ra\\infty}\n  \\int_{\\mathbb{B}_L(0)\\setminus\\mathbb{B}_{1\/L}(0)}e^{2\\eta_k}dx\\\\\n  &=&\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_L(0)\\setminus\\mathbb{B}_{1\/L}(0)}\n  \\f{v_k(x)\\widetilde{f}_k(\\widetilde{x}_{i,k}+r_kx,v_k(x))}{\\varphi_{k}(r_k)\\theta_{k}(r_k)}dx\\\\\n  &=&\\f{1}{\\alpha_0}\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_{Lr_k}(\\widetilde{x}_{i,k})\\setminus\n  \\mathbb{B}_{r_k\/L}(\\widetilde{x}_{i,k})}\\widetilde{u}_k(y)\\widetilde{f}_k(y,\\widetilde{u}_k(y))dy\\\\\n  &\\leq&\\f{C}{\\alpha_0}.\n  \\ena\n  Letting $L\\ra\\infty$, we have\n  $$\\int_{\\mathbb{R}^2}e^{2\\eta}dx<\\infty.$$\n  It follows from (\\ref{cla}), $(\\mathcal{H}_{\\ell,2})$ and $(\\mathcal{H}_{\\ell,4})$ that\n  \\bna\n  \\int_{\\mathbb{B}_{s_k^{(\\ell)}}(\\widetilde{x}_{i,k})}\\varphi_{k}(r_k)\\widetilde{f}_k\n  (y,\\widetilde{u}_k(y))dy&\\leq&\\int_{\\mathbb{B}_{s_k^{(\\ell)}}(\\widetilde{x}_{i,k})\\setminus\n  \\mathbb{B}_{Lr_k^{(\\ell)}}(\\widetilde{x}_{i,k})}\\varphi_{k}(r)\\widetilde{f}_k(y,\\widetilde{u}_k(y))dy\\\\\n  &&+\\f{\\varphi_{k}(s_k^{(\\ell)})}{\\varphi_{k}(Lr_k^{(\\ell)})}\\int_{\\mathbb{B}_{Lr_k^{(\\ell)}}(\\widetilde{x}_{i,k})}\n  \\varphi_{k}(r)\\widetilde{f}_k(y,\\widetilde{u}_k(y))dy+o(1)\\\\\n  &\\leq&\n  N_k(Lr_k^{(\\ell)},s_k^{(\\ell)})+\\f{\\varphi_{k}(s_k^{(\\ell)})}{\\varphi_{k}(Lr_k^{(\\ell)})}\\Lambda_k(Lr_k^{(\\ell)})+o(1)\\\\\n  &\\ra& 0\n  \\ena\n   as $k\\ra\\infty$ first then $L\\ra\\infty$,  that\n  \\bna\n  \\int_{\\mathbb{B}_{r_k\/L}(\\widetilde{x}_{i,k})}\\varphi_{k}(r_k)\\widetilde{f}_k(y,\\widetilde{u}_k(y))dy\n  \\leq N_k(s_k^{(\\ell)},r_k\/L)+o(1)\\ra 0\n  \\ena\n  as $k\\ra \\infty$ first, then $L\\ra\\infty$, and that\n  $$\\int_{\\mathbb{B}_{r_k\/L}(\\widetilde{x}_{i,k})}\\varphi_{k}(r_k)\\widetilde{u}_k(y)dy\\leq\n  \\int_{\\mathbb{B}_{r_k\/L}(\\widetilde{x}_{i,k})}\\widetilde{u}_k^2(y)dy+o(1)\\ra 0$$\n  as $k\\ra\\infty$. Therefore we conclude\n  \\bea\\nonumber\n  \\lim_{L\\ra\\infty}\\limsup_{k\\ra\\infty}\\le|\\int_{\\mathbb{B}_{1\/L}(0)}-\\Delta\\eta_kdx\\ri|&\\leq&\n  \\lim_{L\\ra\\infty}\\limsup_{k\\ra\\infty}\\int_{\\mathbb{B}_{r_k\/L}(\\widetilde{x}_{i,k}))}\n  \\varphi_{k}(r_k)\\widetilde{f}_k(y,\\widetilde{u}_k(y))dy\\\\\\nonumber\n  &&+\\lim_{L\\ra\\infty}\\limsup_{k\\ra\\infty}\\int_{\\mathbb{B}_{r_k\/L}(\\widetilde{x}_{i,k}))}\n  \\varphi_{k}(r_k)\\widetilde{\\tau}_k(y)\\widetilde{u}_k(y)dy\\\\\\label{(21)}\n  &=&0.\n  \\eea\n  Let $\\zeta_k$ be a sequence of solution to the equation\n  \\be\\label{(22)}\\le\\{\\begin{array}{lll}\n  -\\Delta_{\\mathbb{R}^2}\\zeta_k(x)=e^{\\psi_i(\\widetilde{x}_{i,k}+r_kx)}\\varphi_{k}(r_k)r_k^2\n  \\widetilde{f}_k(\\widetilde{x}_{i,k}+r_kx,v_k(x))\\quad{\\rm in}\\quad\n  \\mathbb{B}_1(0)\\\\[1.5ex]\\zeta_k=\\eta_k\\quad{\\rm on}\\quad\n  \\p\\mathbb{B}_1(0).\n  \\end{array}\\ri.\\ee\n  Then in view of (\\ref{(14)}), $\\eta_k-\\zeta_k$ satisfies\n  \\be\\label{(23)}\\le\\{\\begin{array}{lll}\n  -\\Delta_{\\mathbb{R}^2}(\\eta_k-\\zeta_k)(x)=-e^{\\psi_i(\\widetilde{x}_{i,k}+r_k\n  x)}\\varphi_{k}(r_k)r_k^2\\widetilde{\\tau}_k(\\widetilde{x}_{i,k}+r_kx)\n  v_k(x)\\quad{\\rm in}\\quad\n  \\mathbb{B}_1(0)\\\\[1.5ex]\\eta_k-\\zeta_k=0\\quad{\\rm on}\\quad\n  \\p\\mathbb{B}_1(0).\n  \\end{array}\\ri.\\ee\n  Since $u_k$ is bounded in $L^p(\\Sigma)$ for any $p>1$, applying elliptic estimates to\n  (\\ref{(23)}), we get\n  $$\\|\\eta_k-\\zeta_k\\|_{L^\\infty(\\mathbb{B}_1(0))}\\leq C$$\n  for some constant $C$. By (\\ref{(18)}), $\\eta_k$ is uniformly\n  bounded on $\\p\\mathbb{B}_1(0)$. In view of (\\ref{(22)}), the\n  maximum principle implies that there exists some constant $C$\n  such that\n  $$\\zeta_k(x)\\geq -C\\quad{\\rm for\\,\\,all}\\quad x\\in \\mathbb{B}_1(0).$$\n  Hence\n  \\be\\label{(24)}\\eta_k(x)\\geq -C\\quad{\\rm for\\,\\,all}\\quad x\\in \\mathbb{B}_1(0).\\ee\n  By (\\ref{cla}), $\\varphi_{k}(r_k)\\leq v_k(x)+C$ for all $x\\in\\mathbb{B}_{1\/L}(0)$ and $L>1$.  Note that\n  \\bea\\nonumber\n  \\varphi_{k}(r_k)r_k^2\\widetilde{f}_k(\\widetilde{x}_{i,k}+r_kx,v_k(x))&=&\n  \\varphi_{k}(r_k)r_k^2\\theta_{k}(r_k)\\f{\\widetilde{f}_k(\\widetilde{x}_{i,k}+r_kx,v_k(x))}\n  {\\theta_{k}(r_k)}\\\\\\label{60'}\n  &=&(\\alpha_0+o(1))e^{(1+o(1))(v_k^2(x)-\\varphi_{k}^2(r_k))}.\n  \\eea\n  Using the inequality $a^2-b^2\\geq 2b(a-b)$, $a,b\\geq 0$, we get\n  $v_k^2(x)-\\varphi_{i,k}^2(r_k)\\geq 2\\eta_k(x)$ for all $x\\in\\mathbb{B}_1(0)$.\n  Then (\\ref{60'}) leads to\n  \\be\\label{(25)}\\int_{\\mathbb{B}_{1\/L}(0)}e^{\\eta_k}dx\\leq \\f{2}{\\alpha_0}\\int_{\\mathbb{B}_{1\/L}(0)}\n  \\varphi_{k}(r_k)r_k^2\\widetilde{f}_k(\\widetilde{x}_{i,k}+r_kx,v_k(x))dx\\ee\n  for sufficiently large $k$.\n  Combining (\\ref{(14)}), (\\ref{(21)}), (\\ref{(24)}) and (\\ref{(25)}), we obtain\n  \\be\\label{(26)}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_{1\/L}(0)}\\eta_kdx=0.\\ee\n  For any $\\varphi\\in C_0^\\infty(\\mathbb{R}^2)$, integration by\n  parts gives\n  \\bea\\nonumber\n  \\int_{\\mathbb{R}^2}\\eta\\Delta\\varphi dx&=&\\lim_{L\\ra\\infty}\\int_{\\mathbb{R}^2\\setminus\n  \\mathbb{B}_{1\/L}(0)}\\eta\\Delta\\varphi dx\\\\\\label{(27)}\n  &=&\\lim_{L\\ra\\infty}\\le(-\\int_{\\p\\mathbb{B}_{1\/L}(0)}\\eta\\p_\\nu\\varphi d\\sigma\n  +\\int_{\\p\\mathbb{B}_{1\/L}(0)}\\varphi\\p_\\nu\\eta d\\sigma+\n  \\int_{\\mathbb{R}^2\\setminus\n  \\mathbb{B}_{1\/L}(0)}\\varphi\\Delta\\eta dx\\ri).\n  \\eea\n  It is clear that\n  \\bea\\nonumber\n  \\int_{\\p\\mathbb{B}_{1\/L}(0)}\\eta\\p_\\nu\\varphi d\\sigma&=&\\lim_{k\\ra\\infty}\n  \\int_{\\p\\mathbb{B}_{1\/L}(0)}\\eta_k\\p_\\nu\\varphi d\\sigma\\\\\\nonumber\n  &=&\\lim_{k\\ra\\infty}\\le(\\int_{\\mathbb{B}_{1\/L}(0)}\\eta_k\\Delta\\varphi dx+\n   \\int_{\\mathbb{B}_{1\/L}(0)}\\nabla\\eta_k\\nabla\\varphi\n   dx\\ri)\\\\\\label{(28)}\n   &=&\\lim_{k\\ra\\infty}\\le(\\int_{\\mathbb{B}_{1\/L}(0)}\\eta_k\\Delta\\varphi dx+\n   \\int_{\\p\\mathbb{B}_{1\/L}(0)}\\varphi\\p_\\nu\\eta_k d\\sigma-\n   \\int_{\\mathbb{B}_{1\/L}(0)}\\varphi\\Delta\\eta_k dx\\ri).\n  \\eea\n  Moreover, by Proposition 4.1 and (\\ref{cla}), there exists some\n  constant $C$ such that\n  $$|\\nabla\\eta_k(x)|=\\varphi_k(r_k)r_k|\\nabla u_k(\\widetilde{x}_{i,k}+r_k x)|\\leq {C}\/{|x|}$$\n  for all $x\\in\\mathbb{B}_{1\/L}(0)$. This together with (\\ref{(21)})\n  leads to\n  \\bna\n  \\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\p\\mathbb{B}_{1\/L}(0)}\\varphi\\p_\\nu\\eta_k\n  d\\sigma&=&\\varphi(0)\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\p\\mathbb{B}_{1\/L}(0)}\\p_\\nu\\eta_k\n  d\\sigma\\\\&=&\\varphi(0)\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_{1\/L}(0)}\\Delta\\eta_kdx\\\\\n  &=&0.\n  \\ena\n  As a consequence\n  \\be\\label{(29)}\\lim_{L\\ra\\infty}\\int_{\\p\\mathbb{B}_{1\/L}(0)}\\varphi\\p_\\nu\\eta\n  d\\sigma=\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\p\\mathbb{B}_{1\/L}(0)}\\varphi\\p_\\nu\\eta_k\n  d\\sigma=0.\\ee\n  Inserting (\\ref{(19)}), (\\ref{(21)}), (\\ref{(26)}), (\\ref{(28)}) and (\\ref{(29)}) into\n  (\\ref{(27)}), we obtain\n  $$-\\int_{\\mathbb{R}^2}\\eta\\Delta\\varphi dx=\\lim_{L\\ra\\infty}\\int_{\\mathbb{R}^2\\setminus\\mathbb{B}_{1\/L}(0)}\n \\alpha_0e^{2\\eta}\\varphi dx= \\int_{\\mathbb{R}^2}\\alpha_0e^{2\\eta}\\varphi dx.$$\n Therefore $\\eta$ is a distributional solution to the equation\n $$-\\Delta_{\\mathbb{R}^2}\\eta=\\alpha_0 e^{2\\eta}\\quad{\\rm in}\\quad \\mathbb{R}^2.$$\n By the regularity theory for elliptic equations, see for example (\\cite{Aubin}, Chapter 2),\n $\\eta\\in\n C^\\infty(\\mathbb{R}^2)$. By a result of Chen-Li \\cite{C-L},\n $$\\eta(x)=\\log\\f{2}{1+|x|^2}-\\log\\sqrt{\\alpha_0},$$\n and thus\n $$\\int_{\\mathbb{R}^2}e^{2\\eta}dx=\\f\n{4\\pi}{\\alpha_0}.$$ This completes the proof of the lemma.\n$\\hfill\\Box$\\\\\n\nIt follows from Lemma 5.6 that\n$$\n \\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}N_k(r_k^{(\\ell+1)}\/L,Lr_k^{(\\ell+1)})\n =\\alpha_0\\int_{\\mathbb{R}^2}e^{2\\eta^{(\\ell+1)}}dx=4\\pi.\n$$\n This together with (\\ref{(8)}) gives\n $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}N_k(s_k^{(\\ell)},Lr_k^{(\\ell+1)})=4\\pi.$$\n By the inductive hypothesis $(\\mathcal{H}_{\\ell,3})$,\n \\bna\n \\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\Lambda_k(Lr_k^{(\\ell+1)})&=&\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\n \\le(\\Lambda_k(s_k^{(\\ell)})+N_k(s_k^{(\\ell)},Lr_k^{(\\ell+1)})\\ri)\\\\\n &=&4\\pi(\\ell+1).\n \\ena\n\n Now we set\n $w_k^{(\\ell+1)}(x)=\\varphi_k(r_k^{(\\ell+1)})(\\widetilde{u}_k(x)-\\varphi_k(r_k^{(\\ell+1)}))$.\n Similar to Lemma 5.2, we have\\\\\n\n \\noindent{\\bf Lemma 5.7} {\\it For any $\\epsilon>0$, let $T_k^{(\\ell+1)}=T_k^{(\\ell+1)}(\\epsilon)>r_k^{(\\ell+1)}$ be the minimal\n number such that\n $\\varphi_{k}(T_k^{(\\ell+1)})=\\epsilon\\varphi_{k}(r_k^{(\\ell+1)})$.\n Then  $r_k^{(\\ell+1)}\/T_k^{(\\ell+1)}\\ra 0$ as $k\\ra\\infty$.\n Moreover, for any $b<2$ and sufficiently large $k$, $L$, there holds\n $$\\overline{w_k^{(\\ell+1)}}(r)\\leq b\\log\\f{r_k^{(\\ell+1)}}{r}+C\\quad {\\rm for\\,\\,all}\\quad Lr_k^{(\\ell+1)}\\leq\n r\\leq T_k^{(\\ell+1)},$$\n where $C$ is a constant depending only on $\\alpha_0$ and\n $(\\Sigma,g)$, and we have\n $$\\lim_{k\\ra\\infty}N_k(s_k^{(\\ell)},T_k^{(\\ell+1)})=4\\pi.$$}\n {\\it Proof.} Since the proof is completely analogous to that of Lemma\n 4.2, except that instead of Proposition 3.1 we shall use Lemma\n 5.6, the details are omitted here. $\\hfill\\Box$\\\\\n\n For suitable $s_k^{(\\ell+1)}=T_k^{(\\ell+1)}(\\epsilon_k)$, where $\\epsilon_k\\searrow\n 0$ is chosen such that $u_k(s_k^{(\\ell+1)})\\ra\\infty$ as\n $k\\ra\\infty$ and $r_k^{(\\ell+1)}\/s_k^{(\\ell+1)}\\ra 0$ as\n $k\\ra\\infty$. Moreover\n $$\\lim_{k\\ra\\infty}\\Lambda_k(s_k^{(\\ell+1)})=4\\pi(\\ell+1),$$\n and\n $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}N_k(Lr_k^{(\\ell+1)},s_k^{(\\ell+1)})=0.$$\n By the definition of $s_k^{(\\ell+1)}$,\n $$\\lim_{k\\ra\\infty}\\f{\\varphi_{k}(s_k^{(\\ell+1)})}{\\varphi_{k}(Lr_k^{(\\ell+1)})}=0\n \\quad{\\rm for\\,\\,any}\\quad L>0.$$\n Hence\n $(\\mathcal{H}_{\\ell+1})$ holds. This completes Step 2, and thus the proof of Proposition 5.1.\n\n\n\n \\subsection{Quantization for non-simple blow-up points\\\\}\n\n In this subsection, we shall prove a quantization result for non-simple blow-up points.\n We assume that $x_i^\\ast$ is a non-simple blow-up point of order $m$,\n namely there exists a subset $\\{i_1,\\cdots,i_m\\}\\subset\\{1,\\cdots,N\\}$ such that $d_g(x_i^\\ast,x_\\ell^\\ast)=0$\n for all $\\ell\\in\\{i_1,\\cdots,i_m\\}$ and $d_g(x_j^\\ast,x_i^\\ast)>0$\n for all $j\\in \\{1,\\cdots,N\\}\\setminus\\{i_1,\\cdots,i_m\\}$.\n In particular, $i\\in\\{i_1,\\cdots,i_m\\}$.\n  Take an isothermal coordinate system $(U,\\phi;\\{x^1,x^2\\})$ near $x_i^\\ast$,\n  where $U\\subset\\Sigma$ is a neighborhood  of $x_i^\\ast$ such that $x_j^\\ast\\not\\in \\overline{U}$,\n  the closure of $U$ for all $j\\in\\{1,\\cdots,N\\}\\setminus\\{i_1,\\cdots,i_m\\}$, $\\phi:U\\ra\\Omega\\subset\\mathbb{R}^2$ is a diffeomorphism\n  with $\\phi(x_i^\\ast)=(0,0)$. We can find some $\\delta>0$ such that $\\mathbb{B}_{2\\delta}(0)\\subset\n  \\Omega$. In this coordinate system, the metric $g=e^{\\psi}(d{x^1}^2+d{x_2}^2)$ for some smooth function $\\psi:\\Omega\\ra\\mathbb{R}$\n  with $\\psi(0,0)=0$. We shall prove the following result. \\\\\n\n  \\noindent{\\bf Proposition 5.8} {\\it Let $u_k$, $u_\\infty$, $\\tau_k$, $\\tau_\\infty$, $x_{i,k}$ and $x_i^\\ast$ be as in Proposition 3.1.\n  Suppose that $x_i^\\ast$ is a\n  non-simple blow-up point of order $m$ as above. Then up to a subsequence, there exists some positive integer $I$\n  such that\n  \\be\\label{identity2}\n  \\lim_{k\\ra\\infty}\\int_{U}\n  (|\\nabla_gu_k|^2+\\tau_ku_k^2)dv_g=\n  \\int_{U}(|\\nabla_gu_\\infty|^2+\\tau_\\infty\n  u_\\infty^2)dv_g+4\\pi I,\n  \\ee\n  where $U$ is a neighborhood of $x_i^\\ast$ chosen as above.}\\\\\n\n\n   Similarly as\n  before we denote $\\widetilde{x}_{j,k}=\\phi(x_{j,k})$ for\n  $j\\in\\{i_1,\\cdots,i_m\\}$, $\\widetilde{u}_k=u_k\\circ\\phi^{-1}$,\n  $\\widetilde{\\tau}_k=\\tau_k\\circ\\phi^{-1}$, and\n  $\\widetilde{f}_k(x,\\widetilde{u}_k(x))=f(\\phi^{-1}(x),u_k(\\phi^{-1}(x)))$.\n  Let $\\varphi_k=\\varphi_k^{(i)}$, $\\Lambda_k=\\Lambda_k^{(i)}$ and $N_k=N_k^{(i)}$ be as defined in\n  (\\ref{mean-val}), (\\ref{energy}) and (\\ref{neck}) respectively.\n  The proof of Proposition 5.8 will be  divided into several steps below.\\\\\n\n  {\\it Step 1. Blow-up analysis at the scale $o(\\rho_k)$, where\n  $$\\rho_k=\\rho_k^{(i)}=\\f{1}{2}\\inf_{j\\in\\{i_1,\\cdots,i_m\\}\\setminus\\{i\\}}|\\widetilde{x}_{j,k}-\\widetilde{x}_{i,k}|.$$ }\n\n  By Proposition 3.1 we have\n  $\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\Lambda_k(Lr_{i,k})=4\\pi$. Let\n  $r_k^{(1)}=r_{i,k}$.\n   We distinguish the following two\n cases to proceed.\\\\\n\n \\noindent{\\bf Case 1} {\\it there exists some $0<\\epsilon_0<1$ such that for all $t\\in[r_k^{(1)},\\rho_k]$ there holds\n $\\varphi_k(t)\\geq \\epsilon_0\\varphi_k(r_k^{(1)})$};\\\\[1.2ex]\n {\\bf Case 2} {\\it for any $\\epsilon>0$ there exists a minimal $T_k=T_k(\\epsilon)\\in[r_k^{(1)},\\rho_k]$\n such that $\\varphi_k(T_k)=\\epsilon\\varphi_k(r_k^{(1)})$.}\\\\\n\n\n In Case 1, the decay estimate that we established in Lemma 5.2\n remains valid on $[r_k^{(1)},\\rho_k]$. Moreover\n $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\Lambda_k(s_k)=4\\pi$$\n for any sequence $s_k$ satisfying $s_k\/\\rho_k\\ra 0$ and $s_k\/r_k^{(1)}\\ra\\infty$\n as $k\\ra\\infty$. The concentration analysis at scales up to\n $o(\\rho_k)$ is complete.\n\n In Case 2, as before we can find numbers\n $s_k^{(1)}<\\rho_k$ with $\\varphi_k(s_k^{(1)})\\ra\\infty$ as\n $k\\ra\\infty$,\n $\\Lambda_k(s_k^{(1)})\\ra 4\\pi$ as $k\\ra\\infty$,\n and\n $\\varphi_k(s_k^{(1)})\/\\varphi_k(Lr_k^{(1)})\\ra 0$ for any $L\\geq 1$ as $k\\ra\\infty$.\n We proceed by iteration up to some maximal index $\\ell_0\\geq 1$\n where either Case 1 or (\\ref{caseA}) holds with final radii\n $r_k^{(\\ell_0)},s_k^{(\\ell_0)}$, respectively. Hence\n \\be\\label{(72)}\\lim_{k\\ra\\infty}\\Lambda_k(s_k^{(\\ell_0)})=4\\pi\\ell_0,\n \\quad \\lim_{k\\ra\\infty}{\\varphi_k(s_k^{(\\ell_0)})}\/{\\varphi_k(Lr_k^{(\\ell_0)})}=0,\\,\\,\\forall L\\geq 1\\ee\n and\n \\be\\label{73}\\lim_{k\\ra\\infty}N_k(s_k^{(\\ell_0)},t_k)=0\\,\\,\\,{\\rm\n for\\,\\,any\\,\\,sequence}\\,\\,\\,\n t_k=o(\\rho_k).\\ee\n This leads to\n \\be\\label{74}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}N_k(s_k^{(\\ell_0)},\\rho_k\/L)=0.\\ee\n For otherwise, we can find some $\\mu_0>0$ such that up to a\n subsequence\n $$\\lim_{k\\ra\\infty}N_k(s_k^{(\\ell_0)},\\rho_k)\\geq\\lim_{k\\ra\\infty}N_k(s_k^{(\\ell_0)},\\rho_k\/L)\\geq\\mu_0$$\n for all $L\\geq 1$. Take $t_k^\\prime\\in(s_k^{(\\ell_0)},\\rho_k)$ such that\n \\be\\label{75}0<\\lim_{k\\ra\\infty}N_k(s_k^{(\\ell_0)},t_k^\\prime)<\\f{1}{2C_0},\\ee\n where $C_0$ is a constant as in Lemma 5.3. Then by Lemma 5.4 we\n have\n $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}N_k(s_k^{(\\ell_0)},t_k^\\prime\/L)=0.$$\n In view of (\\ref{73}) and (\\ref{75}), there exists some $\\nu_0>0$\n such that up to a subsequence, $t_k^\\prime\\geq \\nu_0\\rho_k$ for all $k$. This immediately implies\n (\\ref{74}) and completes Step\n 1.\\\\\n\nTo proceed, we introduce several terminologies concerning the\nclassification of blow-up points near $x_i^\\ast$.\nDefine a set\n$$\\mathcal{X}=\\mathcal{X}^{(i)}=\\{x_{i_1,k},\\cdots,x_{i_m,k}\\},$$\nwhere each $x_{j,k}$, $j\\in\\{i_1,\\cdots,i_m\\}$, denotes a sequence $(x_{j,k})$. In the sequel we do not distinguish sequences $(x_{j,k})$ and\npoints $x_{j,k}$.\nLet $t_k>0$ be a bounded sequence. For any $j\\in\\{i_1,\\cdots,i_m\\}$, we define a $t_k$-equivalent class associated to the\nsequence $x_{j,k}$ by\n$$[x_{j,k}]_{t_k}:=\\le\\{x_{\\ell,k}:\nd_g(x_{\\ell,k},x_{j,k})=o(t_k),\\,\\,\\ell\\in \\{i_1,\\cdots,i_m\\}\\ri\\}.$$\nThe total number of sequences in $[x_{j,k}]_{t_k}$ is called the order of $[x_{j,k}]_{t_k}$.\nIn particular, the order of $[x_{j,k}]_{\\rho_k^{(j)}}$ is exactly one, while the order of $[x_{j,k}]_{\\delta}$ is $m$. Actually\nwe have $[x_{j,k}]_{\\delta}=\\mathcal{X}$.\n Moreover,\nif $x_{\\ell,k}\\in [x_{j,k}]_{t_k}$, then\n$x_{j,k}\\in[x_{\\ell,k}]_{t_k}$. Also, if\n$[x_{j,k}]_{t_k}\\cap[x_{\\ell,k}]_{t_k}\\not=\\varnothing$, then\n$[x_{j,k}]_{t_k}=[x_{\\ell,k}]_{t_k}$. Hence every subset of $\\mathcal{X}$\ncan be divided into several $t_k$-equivalent classes, any two of\nwhich have no intersection.\n\nFor any $1\\leq \\ell<m$, we say that the property\n$(\\mathcal{A}_\\ell)$ holds for some $t_k$-equivalent class\n$[x_{j,k}]_{t_k}$ of order $\\ell$, if\n  either $(a)$\n  there exist $r_k>0$ and integer $I^{(j)}$ such that for some\n  $\\epsilon_0>0$ and\n all $t\\in[r_k,t_k]$ there holds $\\varphi_k^{(j)}(t)\\geq\n \\epsilon_0\\varphi_k^{(j)}(r_k)$, $\\Lambda_k^{(j)}(Lr_k)\\ra 4\\pi\n I^{(j)}$ and $N_k^{(j)}(Lr_k,t_k\/L)\\ra 0$ as $k\\ra\\infty$ first, and then $L\\ra\\infty$;\n   or\n $(b)$ there exist sequences $r_k<s_k<t_k$ and an integer $I^{(j)}$ such that\n $\\varphi_k^{(j)}(s_k)\/\\varphi_k^{(j)}(Lr_k)\\ra 0$  as $k\\ra\\infty$ for any $L\\geq\n 1$, $\\Lambda_k^{(j)}(t_k\/L)\\ra 4\\pi I^{(j)}$ and $N_k^{(j)}(s_k,t_k\/L)\\ra 0$ as\n $k\\ra\\infty$ first, and then $L\\ra\\infty$.\n While we say that the property $(\\mathcal{A}_m)$ holds, if there exits some $j\\in\\{i_1,\\cdots,i_m\\}$ and\n integer $I^{(j)}$ such that $\\Lambda_k^{(j)}(\\delta\/L)\\ra 4\\pi I^{(j)}$ as $k\\ra\\infty$ first, and then $L\\ra\\infty$.\n\n\n According to Proposition 5.1, when $m=1$, $(\\mathcal{A}_1)$\n holds.\n When $m>1$, we let $\\rho_{k,0}=\\rho_k$ and $\\rho_{k,j}$ $(1\\leq j\\leq m-1)$ be defined as in (\\ref{5.97}) and (\\ref{5.100}) below.\n It follows from Step 1 that $(\\mathcal{A}_1)$ holds for any $t_k$-equivalent class of order one, where\n \\be\\label{tk}t_k\\in\\{\\rho_{k,0},\\cdots,\\rho_{k,m-1}\\}.\\ee\n We now we make an induction procedure on both orders of $t_k$-equivalent class and $m$. Suppose that for some integer $\\nu\\geq 1$, when\n $m=\\nu$, the property $(\\mathcal{A}_\\nu)$ holds; while when $m>\\nu$, the property $(\\mathcal{A}_\\ell)$\n holds for any  $t_k$-equivalent class of order $1\\leq\\ell\\leq \\nu$, where $t_k$ is as in (\\ref{tk}). We shall prove the following:\n When $m=\\nu+1$, the property $(\\mathcal{A}_{\\nu+1})$ holds; When $m>\\nu+1$, the property $(\\mathcal{A}_\\ell)$ holds for any\n $t_k$-equivalent class of order $1\\leq\\ell\\leq \\nu+1$, where $t_k$ is as in (\\ref{tk}).\n Assuming this induction argument is complete, we conclude that $(\\mathcal{A}_m)$ holds for any integer $m$.\n It is easy to see that (\\ref{identity2})\n follows immediately from  $(\\mathcal{A}_m)$ and the fact that\n $u_k\\ra u_\\infty$ in $C^1_{\\rm loc}(\\Sigma\\setminus\\{x_1^\\ast,\\cdots,x_i^\\ast\\})$.\\\\\n\n\n In the next two steps, we shall prove that $(\\mathcal{A}_m)$ holds for $m=\\nu+1$. In Step 4, we shall prove that\n $(\\mathcal{A}_\\ell)$ holds for any\n $t_k$-equivalent class of order $1\\leq\\ell\\leq \\nu+1$, where $t_k$ is as in (\\ref{tk}).\n \\\\\n\n\n\n {\\it Step 2. Blow-up analysis at the scale $\\rho_k$.}\\\\\n\n Let $m=\\nu+1$. Now we turn to carry out blow-up analysis at the scale $\\rho_k$ near\n $\\widetilde{x}_{i,k}$. We first\n assume that for some $L\\geq 1$ there exists some sequence $(x_k)$\n such that $\\rho_k\/L\\leq R_k(x_k)\\leq|x_k-\\widetilde{x}_{i,k}|\\leq L\\rho_k$ and\n \\be\\label{76}|x_k-\\widetilde{x}_{i,k}|^2\\widetilde{u}_k(x_k)\\widetilde{f}_k(x_k,\\widetilde{u}_k(x_k))\\geq \\nu_0>0.\\ee\n By Proposition 4.1 we may assume that $|x_k-\\widetilde{x}_{i,k}|=\\rho_k$. The\n following estimate is important for our subsequent analysis.\\\\\n\n \\noindent{\\bf Lemma 5.9} {\\it Assuming (\\ref{76}), we have $\\varphi_k(\\rho_k)\/\\varphi_k(r_k^{(\\ell_0)})\\ra\n 0$\n as $k\\ra \\infty$.}\\\\\n\n \\noindent{\\it Proof.} If we suppose that there exists some\n $\\epsilon_0>0$ such that $\\varphi_k(\\rho_k)\\geq\n \\epsilon_0\\varphi_k(r_k^{(\\ell_0)})$, then we set\n $$w_k(x)=\\varphi_k(r_k^{(\\ell_0)})(\\widetilde{u}_k(x)-\\varphi_k(r_k^{(\\ell_0)})),\\quad x\\in\\Omega.$$\n Similar to Lemma 5.2, there holds for any $b<2$\n \\be\\label{wdecay}\\overline{w}_k(r)\\leq\n b\\log\\f{r_k^{(\\ell_0)}}{r}+C\\ee\n for all $r\\in [r_k^{(\\ell_0)},\\rho_k]$. Let $\\theta_k$ be as defined in (\\ref{theta}). By (H5) and $(iii)$\n of Proposition 3.1, we find some uniform constant\n $C$ such that\n \\be\\label{rphi}r_k^{(\\ell_0)}\\varphi_k(r_k^{(\\ell_0)})\\theta_k(r_k^{(\\ell_0)})\\leq C.\\ee\n   Hence we obtain\n \\bea\\nonumber\n |x_k-\\widetilde{x}_{i,k}|^2u_k(x_k)\\widetilde{f}_k(x_k,\\widetilde{u}_k(x_k))&\\leq&\n C\\rho_k^2\\varphi_k(\\rho_k)\\theta_k(\\rho_k)\\\\\n \\nonumber&=&C(r_k^{(\\ell_0)})^2\\varphi_k(r_k^{(\\ell_0)})\\theta_k(r_k^{(\\ell_0)})\\le(\\f{\\rho_k}{r_k^{(\\ell_0)}}\\ri)^2\n \\f{\\varphi_k(\\rho_k)}{\\varphi_k(r_k^{(\\ell_0)})}\n \\f{\\theta_k(\\rho_k)}{\\theta_k(r_k^{(\\ell_0)})}\\\\\n \\nonumber&\\leq&\n C\\le({\\rho_k}\/{r_k^{(\\ell_0)}}\\ri)^2e^{(1+o(1))(\\varphi_k^2(\\rho_k)-\\varphi_k^2(r_k^{(\\ell_0)}))}\\\\\n \\nonumber&\\leq&C\\le({\\rho_k}\/{r_k^{(\\ell_0)}}\\ri)^2e^{(1+o(1))(1+\\epsilon_0)\\overline{w}_k(\\rho_k)}\\\\\n &\\leq&C\\le(\\rho_k\/r_k^{(\\ell_0)}\\ri)^{2-(1+o(1))(1+\\epsilon_0)b}\\ra\n 0 \\label{c-t}\n \\eea\n as $k\\ra\\infty$, if we choose $b<2$ such that $(1+\\epsilon_0)b>2$.\n Here the first inequality follows from Proposition 4.1, the second\n one follows from (H4), (H5) and (\\ref{rphi}), while the\n third one is a consequence of our assumption $\\varphi_k(\\rho_k)\\geq\n \\epsilon_0\\varphi_k(r_k^{(\\ell_0)})$, and the last one is implied by (\\ref{wdecay}).\n The contradiction between (\\ref{c-t})\n and (\\ref{76}) ends the proof of the lemma. $\\hfill\\Box$\\\\\n\n Lemma 5.9 implies that for any $\\epsilon>0$ there exists $T_k\\in\n [r_k^{(\\ell_0)},\\rho_k]$ such that\n $\\varphi_k(T_k)=\\epsilon\\varphi_k(r_k^{(\\ell_0)})$. Hence at scales\n up to order $o(\\rho_k)$ we end up with (\\ref{caseA}), where $\\ell$ is replaced by $\\ell_0$. The desired quantization\n result at the scale $\\rho_k$ then is a consequence of the following\n result.\\\\\n\n \\noindent{\\bf Lemma 5.10} {\\it Assuming (\\ref{76}), then up to a\n subsequence we can find some $\\alpha_0\\geq \\nu_0$ such that\n \\be\\label{761}\\lim_{k\\ra\\infty}|x_k-\\widetilde{x}_{i,k}|^2\\widetilde{u}_k(x_k)\\widetilde{f}_k(x_k,\\widetilde{u}_k(x_k))=\\alpha_0.\\ee\n Moreover there exist a finite set $\\mathcal{S}_\\infty\\subset\\mathbb{R}^2$ such that\n $$\\eta_k(x)=\\widetilde{u}_k(x_k)(\\widetilde{u}_k(\\widetilde{x}_{i,k}+\\rho_kx)-\\widetilde{u}_k(x_k))\\ra\\eta(x)=\\log\\f{2}{\\sqrt{\\alpha_0}(1+|x|^2)}$$\n in $C^1_{\\rm loc}(\\mathbb{R}^2\\setminus\\mathcal{S}_\\infty)$ as\n $k\\ra\\infty$.}\\\\\n\n \\noindent{\\it Proof.} It is obvious that (\\ref{761}) holds for some $\\alpha_0\\geq \\nu_0>0$.  Define\n $$v_k(y)=\\widetilde{u}_k(\\widetilde{x}_{i,k}+\\rho_ky)$$\n for $y\\in\\Omega_k=\\{y\\in\\mathbb{R}^2:\\widetilde{x}_{i,k}+\\rho_ky\\in\n \\Omega\\}$. Let\n $$y_{j,k}=\\f{\\widetilde{x}_{j,k}-\\widetilde{x}_{i,k}}{\\rho_k}$$\n and\n $$\\mathcal{S}_k=\\mathcal{S}^{(i)}_{k}=\\le\\{y_{j,k}: j=i_1,\\cdots,i_{\\nu+1}\\ri\\}.$$\n Without loss of generality we assume either $|y_{j,k}|\\ra\\infty$ or\n $y_{j,k}\\ra y_j$, $j=i_1,\\cdots,i_{\\nu+1}$, and we let\n $\\mathcal{S}_\\infty=\\mathcal{S}_{\\infty}^{(i)}$ be the set of accumulation\n points of $\\mathcal{S}_k$. Also we let\n $$y_{0,k}=\\f{x_k-\\widetilde{x}_{i,k}}{\\rho_k}$$\n be the scaled points of $x_k$ for which (\\ref{76}) holds and which\n satisfy $|y_{0,k}|=1$. Moreover we can assume $y_{0,k}\\ra y_0$ as\n $k\\ra\\infty$.\n\n Since $\\widetilde{u}_k(x_k)\\ra\\infty$ by (\\ref{76}) and\n $\\mathcal{S}_\\infty$ is a finite set, we have by using Proposition 4.1 and a standard covering argument that\n \\be\\label{77}v_k-\\widetilde{u}_k(x_k)\\ra 0\\,\\,{\\rm locally\\,\\, uniformly\\,\\,on}\\,\\,\\mathbb{R}^2\\setminus\\mathcal{S}_\\infty\\ee\n as $k\\ra\\infty$. Using the same argument as in the proof of\n Lemma 5.6, we obtain\n $$\\eta_k\\ra\\eta\\quad{\\rm in}\\quad C^1_{\\rm\n loc}(\\mathbb{R}^2\\setminus\\mathcal{S}_\\infty),$$\n where $\\eta\\in C^\\infty(\\mathbb{R}^2\\setminus\\mathcal{S}_\\infty)$\n satisfies the equation\n $$-\\Delta_{\\mathbb{R}^2}\\eta=\\alpha_0 e^{2\\eta}\\quad{\\rm in}\\quad\n \\mathbb{R}^2\\setminus\\mathcal{S}_\\infty.$$\n It follows from (\\ref{77}) that $v_k\/\\widetilde{u}_k(x_k)\\ra 1$ locally uniformly on $\\mathbb{R}^2\\setminus\n \\mathcal{S}_\\infty$. For any $L\\geq 1$ we write\n $$K_L=\\mathbb{B}_L(0)\\setminus(\\cup_{y_j\\in\\mathcal{S}_\\infty}\\mathbb{B}_{\\delta\/L}(y_j)).$$\n Combining (H4), (H5), (\\ref{bdd}) and (\\ref{77}), we can estimate\n \\bna\n \\int_{\\mathbb{R}^2}e^{2\\eta}dx&\\leq&\n \\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{K_L}\\f{v_k(x)}{\\widetilde{u}_k(x_k)}e^{(1+o(1))\\eta_k(1+\\f{v_k(x)}{\\widetilde{u}_k(x_k)})}dx\\\\\n &=&\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{K_L}\\f{\\widetilde{u}_k(\\widetilde{x}_{i,k}+\\rho_kx)\n \\widetilde{f}_k(\\widetilde{x}_{i,k}+\\rho_kx,\\widetilde{u}_k(\\widetilde{x}_{i,k}+\\rho_kx))}\n {\\widetilde{u}_k(x_k)\\widetilde{f}_k(x_k,\\widetilde{u}_k(x_k))}dx\\\\\n &\\leq&\\f{C}{\\nu_0}\\limsup_{k\\ra\\infty}\\int_\\Sigma\n u_kf_k(x,u_k)dv_g\\leq \\f{C}{\\nu_0}.\n \\ena\n\n Since $y_{j,k}\\ra y_j$ as $k\\ra\\infty$, we can take sufficiently large $L$ and $k$ such that\n  $\\mathbb{B}_{1\/L}(y_j)\\subset\\mathbb{B}_{2\/L}(y_{j,k})$ and\n  $\\mathbb{B}_{2\/L}(y_{j,k})\\cap \\mathbb{B}_{2\/L}(y_{\\alpha,k})=\\varnothing$ for any\n  $\\alpha\\not=j$. Moreover let $\\ell$ be the order of the $\\rho_k$-equivalent class $[x_{j,k}]_{\\rho_k}$. Clearly\n  $\\ell\\leq \\nu$.\n  By our inductive assumption,\n  $(\\mathcal{A}_\\ell)$ holds for $[x_{j,k}]_{\\rho_k}$. Noting that Lemma 5.9 excludes the possibility of Case 1\n  with $r_k^{(1)}$ replaced by $r_k^{(\\ell_0)}$, we can find\n  sequences $r_k^{(j)}<s_k^{(j)}$ such that\n  \\be\\label{bz}\\lim_{k\\ra\\infty}\\varphi_k(s_k^{(j)})\/\\varphi_k(Lr_k^{(j)})=0,\\quad\\forall L\\geq 1.\\ee\n  and\n  \\be\\label{nz}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}N_k^{(j)}(s_k^{(j)},\\rho_k\/L)=0,\\ee\n   Note again that $y_{j,k}\\ra y_j$ as $k\\ra\\infty$. There exists some constant $C$,\n    which may depends on $|y_j|$ but\n   not on $k$, such that $|\\widetilde{x}_{j,k}-\\widetilde{x}_{i,k}|\\leq C\\rho_k$.\n   For any\n   ${x}_{\\alpha,k}\\not\\in[{x}_{j,k}]_{\\rho_k}$,\n   we can take some large $L_0$ such that\n   $|\\widetilde{x}_{j,k}-\\widetilde{x}_{\\alpha,k}|\\geq \\rho_k\/(2L_0)$ for all sufficiently large $k$. Recalling that\n   $|x_k-\\widetilde{x}_{i,k}|=\\rho_k$ and applying Proposition\n   4.1, we obtain\n   $$\\widetilde{u}_k(x_k)\\leq \\inf_{\\p\\mathbb{B}_{2\\rho_k\/L_0}(\\widetilde{x}_{j,k})}\\widetilde{u}_k+C$$\n   for some uniform constant $C$. As in the proof of Lemma 5.2, we can\n   find another uniform constant $C$ such that for all $x\\in \\mathbb{B}_{2\\rho_k\/L_0}(\\widetilde{x}_{j,k})$\n   $$\\widetilde{u}_k(x)\\geq \\inf_{\\p\\mathbb{B}_{2\\rho_k\/L_0}(\\widetilde{x}_{j,k})}\\widetilde{u}_k-C.$$\n   These two estimates immediately imply the existence of some uniform constant $C$ such that\n   \\be\\label{ukleq}\\widetilde{u}_k(x_k)\\leq \\widetilde{u}_k(x)+C\\,\\,{\\rm for\\,\\,all}\\,\\,x\\in\\mathbb{B}_{2\\rho_k\/L}\n   (\\widetilde{x}_{j,k}),\n   \\ee provided that $L\\geq L_0$. Note that $g=e^\\psi(d{x^1}^2+d{x^2}^2)$ for some smooth function $\\psi$ with\n    $\\psi(0,0)=0$. By the equation (\\ref{e1}), we have for large $L$\n   \\bna\n   \\int_{\\mathbb{B}_{1\/L}(y_j)}|\\Delta_{\\mathbb{R}^2}\\eta_k|dx&\\leq&\n   \\int_{\\mathbb{B}_{1\/L}(y_j)}\\rho_k^2\\widetilde{u}_k(x_k)\\widetilde{f}_k\\le(\\widetilde{x}_{i,k}+\\rho_kx,v_k(x)\\ri)\n   e^{\\psi(\\widetilde{x}_{i,k}+\\rho_kx)}dx\\\\\n   &&+\\int_{\\mathbb{B}_{1\/L}(y_j)}\\rho_k^2\\widetilde{u}_k(x_k)\\widetilde{\\tau}_k(\\widetilde{x}_{i,k}+\\rho_kx)\n   v_k(x)e^{\\psi(\\widetilde{x}_{i,k}+\\rho_kx)}dx\\\\\n   &\\leq&\\int_{\\mathbb{B}_{2\/L}(y_{j,k})}\\rho_k^2\\widetilde{u}_k(x_k)\\widetilde{f}_k\n   \\le(\\widetilde{x}_{i,k}+\\rho_kx,v_k(x)\\ri)e^{\\psi(\\widetilde{x}_{i,k}+\\rho_kx)}dx\\\\\n   &&+\\int_{\\mathbb{B}_{2\/L}(y_{j,k})}\\rho_k^2\\widetilde{u}_k(x_k)\\widetilde{\\tau}_k(\\widetilde{x}_{i,k}+\\rho_kx)\n   v_k(x)e^{\\psi(\\widetilde{x}_{i,k}+\\rho_kx)}dx\\\\\n   &=&\\int_{\\mathbb{B}_{2\\rho_k\/L}(\\widetilde{x}_{j,k})}\\widetilde{u}_k(x_k)\\le(\\widetilde{f}_k(y,\\widetilde{u}_k(y))\n   +\\widetilde{\\tau}_k(y) \\widetilde{u}_k(y)\\ri)e^{\\psi(y)}dy.\n   \\ena\n   With the help of (\\ref{bz})-(\\ref{ukleq}) and an obvious analogy to (\\ref{cla}), we obtain\n   $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_{1\/L}(y_j)}|\\Delta_{\\mathbb{R}^2}\\eta_k|dx=0,$$\n   analogous to (\\ref{(21)}).\n   In the same way of proving (\\ref{(26)}) we get\n   $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\mathbb{B}_{1\/L}(y_j)}\\eta_kdx=0.$$\n\n   In view of (\\ref{ukleq}), we can find some uniform constant $C$ such that for all $y\\in\\p\\mathbb{B}_{1\/L}(y_j)$\n   $$\\widetilde{u}_k(x_k)\/\\widetilde{u}_k(\\widetilde{x}_{j,k}+\\rho_k y)\\leq C,$$\n   which together with Proposition 4.1 leads to\n   $$\n   |y-y_{j,k}||\\nabla_{\\mathbb{R}^2}\\eta_k(y)|=|\\widetilde{x}_{i,k}+\\rho_k\n   x-\\widetilde{x}_{j,k}|\\widetilde{u}_k(x_k)|\\nabla_{\\mathbb{R}^2} \\widetilde{u}_k(\\widetilde{x}_{i,k}+\\rho_kx)|\\leq C.\n   $$\n   This gives\n   $$|\\nabla_{\\mathbb{R}^2}\\eta_k(y)|\\leq \\f{C}{|y-y_j|}$$\n   for all $y\\in\\p\\mathbb{B}_{1\/L}(y_j)$, provided that $k$ is\n   sufficiently large. Then we obtain an analogy to (\\ref{(29)}),\n   namely, for any $\\varphi\\in C_0^\\infty(\\mathbb{R}^2)$\n   $$\\lim_{L\\ra\\infty}\\int_{\\p\\mathbb{B}_{1\/L}(y_j)}\\varphi\\p_\\nu\\eta\n  d\\sigma=\\lim_{L\\ra\\infty}\\int_{\\p\\mathbb{B}_{1\/L}(y_j)}\\eta\\p_\\nu\\varphi\n  d\\sigma=0.$$\n  This excludes $y_j$ as a singular point of $\\eta$ as in Lemma 5.6. Since $y_j$ is any point of $\\mathcal{S}_\\infty$, we\n  conclude that $\\eta$ is a smooth solution to the equation\n  $$-\\Delta_{\\mathbb{R}^2}\\eta=\\alpha_0 e^{2\\eta}\\quad{\\rm in}\\quad\\mathbb{R}^2.$$\n  The remaining part of the conclusions of the lemma follows from a\n  result of Chen-Li \\cite{C-L}. $\\hfill\\Box$\\\\\n\n  Define a set\n  \\be\\label{ALk}A_{L,k}=\\le\\{x\\in \\Omega: \\rho_k\/L\\leq R_k(x)\\leq|x-\\widetilde{x}_{i,k}|\\leq\n  L\\rho_k\\ri\\}.\\ee\n  It follows from Proposition 4.1 that ${u_k(x)}\/{\\widetilde{u}_k(x_k)}\\ra 1$\n  uniformly in $A_{L,k}$ as $k\\ra\\infty$. Thus by Lemma 5.10, in case of\n  (\\ref{76}) there holds\n  \\bea\n  \\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{A_{L,k}}\\widetilde{u}_k(x)\\widetilde{f}_k(x,\\widetilde{u}_k(x))dx&=&\\alpha_0\n  \\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{A_{L,k}}\\f{\\widetilde{u}_k(x)\\widetilde{f}_k(x,\\widetilde{u}_k(x))}\n  {\\widetilde{u}_k(x_k)\\widetilde{f}_k(x_k,\\widetilde{u}_k(x_k))}dx{\\nonumber}\\\\\n  &=&\\alpha_0\\int_{\\mathbb{R}^2}e^{2\\eta(x)}dx{\\nonumber}\\\\\\label{80}\n  &=&\\alpha_0\\f{4\\pi}{\\alpha_0}=4\\pi.\n  \\eea\n  Let\n  \\be\\label{Xk1}X_{k,1}=X_{k,1}^{(i)}=\\{\\widetilde{x}_{j,k}:\\exists C>0 \\,\\,{\\rm such\\,\\, that}\\,\\,\n  |\\widetilde{x}_{j,k}-\\widetilde{x}_{i,k}|\\leq C\\rho_k\n  \\,\\,{\\rm for\\,\\,all}\\,\\, k\\}.\\ee\n  We can divide $X_{k,1}$ into several $\\rho_k$-equivalent classes with\n  their orders no more than $\\nu$.\n  Recalling our inductive assumption $(\\mathcal{A}_\\ell)$ with $1\\leq\\ell\\leq\\nu$ and using (\\ref{80}),\n  we can find some integer $I$ such that\n  $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\Lambda_k(L\\rho_k)=4\\pi(1+I).$$\n\n  On the other hand, if (\\ref{76}) does not hold, we have\n  \\be\\label{varn}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{A_{L,k}}\\widetilde{u}_k\\widetilde{f}_k(x,\\widetilde{u}_k)dx=0.\\ee\n  The energy estimate at the scale $\\rho_k$ again is\n  finished. \\\\\n\n  {\\it Step 3. Blow-up analysis at scales exceeding\n  $\\rho_k$.}\\\\\n\n  Now we deal with blow-up analysis at scales\n  exceeding $\\rho_k$ near $\\widetilde{x}_{i,k}$. Write\n $$X_{k,0}=\\{\\widetilde{x}_{i_1,k},\\cdots,\\widetilde{x}_{i_m,k}\\}.$$\n Recalling (\\ref{Xk1}), we let\n  \\be\\label{5.97}\\rho_{k,1}=\\rho_{k,1}^{(i)}=\\le\\{\\begin{array}{lll}\\inf\\limits\n  _{\\widetilde{x}_{j,k}\\in X_{k,0}\\setminus X_{k,1}}\\f{|\\widetilde{x}_{j,k}-\\widetilde{x}_{i,k}|}{2} &{\\rm\n  if}\n  &X_{k,0}\\setminus X_{k,1}\\not=\\varnothing\\\\[1.5ex]\n  \\delta, &{\\rm if}&X_{k,0}\\setminus X_{k,1}=\\varnothing.\\end{array}\\ri.\\ee\n  From this definition it follows that $\\rho_{k,1}\/\\rho_k\\ra \\infty$ as $k\\ra\\infty$. Then,\n  using the obvious analogy of Lemma 5.4, either we have\n  $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}N_k(L\\rho_k,\\rho_{k,1}\/L)=0,$$\n  and we iterate to the next scale; or there exist a sequence $t_k$\n  such that $t_k\/\\rho_k\\ra\\infty$, $t_k\/\\rho_{k,1}\\ra 0$ as\n  $k\\ra\\infty$ and up to a subsequence such that\n  \\be\\label{85}P_k(t_k)\\geq \\nu_0>0\\,\\,{\\rm for\\,\\,all\\,\\,large}\\,\\, k.\\ee\n  The argument then depends on whether (\\ref{76}) or (\\ref{varn})\n  holds. In case of (\\ref{76}), as in Lemma 5.9, the bound (\\ref{85})\n  and Lemma 5.10 imply that $\\varphi_k(t_k)\/\\varphi_k(\\rho_k)\\ra 0$ as\n  $k\\ra \\infty$. Then we can argue as in (\\ref{caseA}) for\n  $r\\in[L\\rho_k,\\rho_{k,1}\/L]$ for sufficiently large $L$, and we\n  can continue as before to resolve concentrations in this range of\n  scales.\n\n  In case of (\\ref{varn}) we further need to distinguish whether\n  (\\ref{caseA}) or Case 1 holds at the final stage of our analysis at\n  scales\n  $o(\\rho_k)$. Recalling that in case of (\\ref{caseA}) we have (\\ref{(72)}) and\n  (\\ref{74}), in view of (\\ref{varn}) for a suitable sequence of\n  numbers $s_{k,1}^{(0)}$ such that $s_{k,1}^{(0)}\/\\rho_k\\ra\\infty$,\n  $t_k\/s_{k,1}^{(0)}\\ra\\infty$ as $k\\ra\\infty$ we obtain\n  \\be\\label{sum}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\le(\\Lambda_k(s_{k,1}^{(0)})-\n  \\sum_{\\widetilde{x}_{j,k}\\in\\widetilde{X}_{k,1}}\\Lambda_k^{(j)}(Lr_k^{(\\ell_0^{(j)})})\\ri)=0,\\ee\n  where $\\Lambda_k^{(j)}(r)$ and $r_k^{(\\ell_0^{(j)})}$ are computed\n  as above with respect to the blow-up point $x_{j,k}$ and\n  $\\widetilde{X}_{k,1}$ is the modular set containing all $t_k$-equivalent classes\n  of ${X}_{k,1}$, whence the distance between any two points of\n  $\\widetilde{X}_{k,1}$ is greater than $\\widetilde{\\nu} \\rho_k$ for some\n  constant $\\widetilde{\\nu}>0$. In particular, with such a choice of\n  $s_{k,1}^{(0)}$ we find the immediate quantization result\n  $$\\lim_{k\\ra\\infty}\\Lambda_k(s_{k,1}^{(0)})=4\\pi I$$\n  for some positive integer $I$. Here again we use the inductive assumption that $(\\mathcal{A}_l)$ holds for all\n  $\\rho_k$-equivalent classes of order $\\ell$ with $1\\leq\\ell\\leq \\nu$.\n  While in Case 1 if we assume there is some  $\\epsilon_0>0$ such that\n  \\be\\label{102}\\varphi_k(s_{k,1}^{(0)})\\geq \\epsilon_0\n  \\varphi_k(Lr_k^{(\\ell_0^{(j)})})\\ee\n  for all $r\\in[Lr_{k}^{\\ell_0^{(j)}},s_{k,1}^{(0)}]$, then\n  as  before we have\n  $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}N_k^{(j)}(Lr_k^{(\\ell_0^{(j)})},s_{k,1}^{(0)})=0.$$\n  This contradicts (\\ref{sum}) since $s_{k,1}^{(0)}\/\\rho_k\\ra\\infty$ as $k\\ra\\infty$ and\n  the modular set $\\widetilde{X}_{k,1}$ has at\n  least two elements. This implies that (\\ref{102}) does not hold and\n  up to a subsequence there holds for any $L\\geq 1$\n  $$\\lim_{k\\ra\\infty}\\f{\\varphi_k(s_{k,1}^{(0)})}{\\varphi_k(Lr_k^{(\\ell_0^{(j)})})}=0$$\n  for all $x_{j,k}\\in \\widetilde{X}_{k,1}$ where Case 1 holds. Then\n  we can continue to resolve concentrations on the range\n  $[s_{k,1}^{(0)},\\rho_{k,1}\/L]$ as before.\n\n  We then proceed by iteration. For $\\ell\\geq 2$ we\n  inductively define the sets\n  $$X_{k,\\ell}=X_{k,\\ell}^{(i)}=\n  \\le\\{\\widetilde{x}_{j,k}: \\exists C>0\\,\\,{\\rm such\\,\\, that}\\,\\, |\\widetilde{x}_{j,k}-\\widetilde{x}_{i,k}|\n  \\leq C\\rho_{k,\\ell-1}\n  \\,\\,{\\rm for\\,\\,all}\\,\\, k\\ri\\}$$\n  and let\n  \\be\\label{5.100}\\rho_{k,\\ell}=\\rho_{k,\\ell}^{(i)}=\\le\\{\\begin{array}{lll}\\inf\\limits\n  _{\\widetilde{x}_{j,k}\\in X_{k,0}\\setminus X_{k,\\ell}}\\f{|\\widetilde{x}_{j,k}-\\widetilde{x}_{i,k}|}{2} &{\\rm\n  if}\n  &X_{k,0}\\setminus X_{k,\\ell}\\not=\\varnothing\\\\[1.5ex]\n  \\delta, &{\\rm if}&X_{k,0}\\setminus X_{k,\\ell}=\\varnothing.\\end{array}\\ri.\\ee\n  Iteratively carrying out the above analysis at all scales\n  $\\rho_{k,\\ell}$, exhausting all blow-up points $x_{j,k}$, up\n  to a subsequence we obtain quantization result for\n  $X_{k,0}$. Then Step 3 is finished.\\\\\n\n  It follows from Step 2 and Step 3 that there exists some integer\n  $I$ such that\n  \\be\\label{ast}\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\Lambda_k(\\delta\/L)=4\\pi\n  I,\\ee\n  different analogous to Lemma 5.5. Here and in the sequel, $I$ may denote different integer.\n  Hence the property $(\\mathcal{A}_m)$ holds when $m=\\nu+1$.\\\\\n\n\n  {\\it Step 4. $(\\mathcal{A}_\\ell)$ holds for $1\\leq\\ell\\leq\\nu+1$ when $m>\\nu+1$.}\\\\\n\n  When $m>\\nu+1$, by our inductive assumption, $(\\mathcal{A}_\\ell)$ holds for all $1\\leq\\ell\\leq \\nu$,\n  it suffices to prove that\n  $(\\mathcal{A}_{\\nu+1})$ holds for any $t_k$-equivalent class\n  $[x_{j,k}]_{t_k}$ of order $\\nu+1$, where $j\\in\\{i_1,\\cdots,i_m\\}$ and $t_k$ is as in (\\ref{tk}).\n    This is completely analogous to that $(\\mathcal{A}_m)$ holds in the\n  case of $m=\\nu+1$, which we proved above, except that (\\ref{ast}) is replaced by\n  $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\Lambda_k^{(j)}(t_k\/L)=4\\pi\n  I$$\n  for some integer $I$. We omit the details here. This ends Step 4. $\\hfill\\Box$\\\\\n\n  Proposition 5.8 follows from the property $(\\mathcal{A}_m)$ and the last assertion of Proposition 3.1.\n\n \\section{Completion of the proof of Theorem 1.1}\n In this section, we complete the proof of Theorem 1.1.\n Let $x_{i,k}\\ra x_i^\\ast$ as $k\\ra\\infty$, $1\\leq i\\leq N$, be as in Proposition 3.1.\n In view of possible\n non-simple blow-up points, without loss of generality, we\n may assume for some $q\\leq N$, $x_1^\\ast,\\cdots,x_q^\\ast$ are different from each other and\n $x_\\ell^\\ast\\in\\{x_1^\\ast,\\cdots,x_q^\\ast\\}$ for any $q+1\\leq \\ell\\leq\n N$.\n For any $1\\leq i\\leq q$, we take an isothermal coordinate system $(U_i,\\phi_i;\\{x^1,x^2\\})$\n near $x_i^\\ast$ such that $\\phi_i(x_i^\\ast)=(0,0)$ and $U_i=\\phi_i^{-1}(\\mathbb{B}_\\delta(0))$, where $\\delta$ is chosen\n sufficiently small such that $\\overline{U}_i$ does not contain any $x_j^\\ast$ with $j\\in\\{1,\\cdots,q\\}\\setminus\\{i\\}$.\n It follows from Propositions 5.1 and 5.8 that for some integer\n $I^{(i)}$ there holds\n $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\phi_i^{-1}(\\mathbb{B}_{\\delta\\,\/L}(0))}u_kf_k(x,u_k)dv_g=4\\pi I^{(i)}.$$\n By Proposition 3.1, $u_k\\ra u_\\infty$ in $C^1_{\\rm\n loc}(\\Sigma\\setminus\\{x_1^\\ast,\\cdots,x_q^\\ast\\})$ as $k\\ra\\infty$.\n hence\n $$\\lim_{L\\ra\\infty}\\lim_{k\\ra\\infty}\\int_{\\Sigma\\setminus \\cup_{i=1}^q\\phi_i^{-1}(\\mathbb{B}_{\\delta\\,\/L}(0))}\n u_kf_k(x,u_k)dv_g=\\int_\\Sigma u_\\infty f_k(x,u_\\infty)dv_g.$$\n Combining these two estimates, we obtain\n $$\\lim_{k\\ra\\infty}\\int_{\\Sigma}\n u_kf_k(x,u_k)dv_g=\\int_\\Sigma u_\\infty f_k(x,u_\\infty)dv_g+4\\pi\\sum_{i=1}^qI^{(i)}.$$\n This together with (\\ref{e1}) leads to\n $$\\lim_{k\\ra\\infty}\\int_{\\Sigma}(|\\nabla_gu_k|^2+\\tau_k u_k^2)dv_g\n =\\int_{\\Sigma}(|\\nabla_gu_\\infty|^2+\\tau_\\infty u_\\infty^2)dv_g\n +4\\pi\\sum_{i=1}^qI^{(i)}.$$\n In view of (\\ref{0.6}), or particularly (\\ref{F}), we then have\n $$\\lim_{k\\ra\\infty}J_k(u_k)=J_\\infty(u_\\infty)+4\\pi\\sum_{i=1}^qI^{(i)}.$$\n This completes the proof of Theorem 1.1. $\\hfill\\Box$\\\\\n\n\n\n\n {\\bf Acknowledgements.} This work is supported by the NSFC\n 11171347. The author thanks the referee for his careful reading and\n  valuable suggestions on the first version of this paper.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The Double wire model}\nIn this section we first present the detailed derivations of Eqs.~(\\ref{eq:KitaevG}-\\ref{eq:dbwireH}) in the main text, and then we discuss some alternative derivations of the parent Hamiltonian.  \n\\subsection{Diagonalization of Kitaev's Hamiltonian in momentum space with open boundary}\nTo obtain the Bogoliubov operators in Eq.~(\\ref{eq:alpha_k}) and the form of ground states in Eq.~(\\ref{eq:KitaevG}) in our main text, here we present the momentum space diagonalization of Kitaev's Hamiltonian with $\\Delta=t$\n\\begin{equation}\nH_{\\mathrm{Kitaev}}=\\sum_{j} t(-c^\\dagger_{j}c_{j+1}+ c_{j}c_{j+1}+\\mathrm{H.c.})-\\mu \\hat{N}.\n\\end{equation}\nTo this end we search for Bogoliubov eigenmodes defined as\n\\begin{equation}\n\\alpha_k=\\sum^L_{j=1}(u^k_{j} c_j+v^k_{j} c^\\dagger_j).\n\\end{equation}\nBeing the eigenmodes of $H_{\\mathrm{Kitaev}}$ with energy $E_k>0$, they satisfy $[\\alpha_k,H_{\\mathrm{Kitaev}}]= E_k\\alpha_k$, which gives difference equations on $u^k_{j},v^k_{j}$\n\\begin{eqnarray}\\label{eq:recursive}\n(E_k+\\mu)u^k_j&=&-t(u^k_{j-1}+u^k_{j+1})+t(v^k_{j-1}-v^k_{j+1}),\\nonumber\\\\\n(E_k-\\mu)v^k_j&=&t(v^k_{j-1}+v^k_{j+1})+t(u^k_{j+1}-u^k_{j-1}),\n\\end{eqnarray}\nwith boundary conditions \n\\begin{eqnarray}\\label{eq:obc}\nu^k_0=v^k_0,~u^k_{L+1}=-v^k_{L+1}.\n\\end{eqnarray}\nTo solve these equations, we notice that the ansatz solutions\n\\begin{eqnarray}\nu^k_j=\\lambda(k) e^{i(kj-\\theta_k)}-\\lambda(-k) e^{-i(kj-\\theta_k)},~~v^k_j=e^{i(kj-\\theta_k)}-e^{-i(kj-\\theta_k)}\n\\end{eqnarray}\nwith $\\lambda(k)=-i\\frac{2\\Delta\\sin k}{E_k+\\mu+2t\\cos k}$ and $E_k=\\sqrt{\\mu^2+4t\\mu\\cos k+4t^2}$ satisfy Eq.~(\\ref{eq:recursive}). The boundary conditions in Eq.~(\\ref{eq:obc}) give constraints on the quasi-momentum $k$ and the real parameter $\\theta_k$\n\\begin{eqnarray}\\label{eq:sol1}\nk(L+1)=2\\theta_k+m\\pi,~~m\\in \\mathbf{Z},~~~\\tan\\theta_k=\\frac{2\\Delta\\sin k}{E_k+\\mu+2t\\cos k},\n\\end{eqnarray}\nwhere $0< k< \\pi$ and $0<\\theta_k< \\pi\/2$. The Bogoliubov operators are~(up to normalization)\n\\begin{equation}\\label{eq:alpha_kS}\n\\alpha_k=e^{i\\theta_k}(i\\cos\\theta_k c^\\dagger_{-k}-\\sin\\theta_k c_k)-(k\\to -k)\n\\end{equation}\nAs an aside, we mention that if we replace Eq.~(\\ref{eq:alpha_k}) in our main text by Eq.~(\\ref{eq:alpha_kS}) and use the same matrix in Eq.~(\\ref{eq:dbwireHpp}), then, still following our general construction, we can get a bigger family of number-conserving, short-range interacting Hamiltonians with ground states $|G^{e,o}\\rangle$ depending on $\\mu\/t$, and this method can be generalized to construct parent Hamiltonians for Kitaev's ground states at arbitrary points $(t,\\Delta,\\mu)$~(even including points in the topologically trivial phase).  \n\nAt the $\\mu=0$ point we get especially simple expressions\n\\begin{equation}\nE_k=2t,~~\\theta_k=\\frac{k}{2},~~k=\\frac{m\\pi}{L},~m=1\\ldots(L-1),\n\\end{equation}\nwhich leads to the single wire version of Bogoliubov operators in Eq.~(\\ref{eq:alpha_k}) after normalization. To verify that the expressions given in Eq.~(\\ref{eq:KitaevG}) are indeed the ground states of $H_{\\mathrm{Kitaev}}$, we show that $|G^e\\rangle$ and $|G^o\\rangle$ are annihilated by all $\\alpha_k$. We have\n\\begin{equation}\nc_k|G^e\\rangle=\\frac{1}{\\sqrt{L}}\\sum^L_{j=1}e^{-ikj}c_j~\\exp\\{\\sum_{i<j'}c^\\dagger_{i}c^\\dagger_{j'}\\}|0\\rangle=\\frac{1}{\\sqrt{L}}\\sum_{j,j'}e^{-ikj}\\mathrm{sgn}(j'-j)c^\\dagger_{j'}|G^e\\rangle=[i\\cot\\frac{k}{2}c^\\dagger_{-k}-\\frac{1+(-1)^m}{1-e^{ik}}c^\\dagger_{k=0}]|G^e\\rangle,\n\\end{equation}\nwhere $\\mathrm{sgn}(x)=x\/|x|$ for $x\\neq 0$ and $\\mathrm{sgn}(0)=0$. It follows that\n\\begin{equation}\n[e^{i\\frac{k}{2}}\\sin\\frac{k}{2}c_k-(k\\to-k)]|G^e\\rangle=[e^{i\\frac{k}{2}}i\\cos\\frac{k}{2}c^\\dagger_{-k}-(k\\to-k)]|G^e\\rangle,\n\\end{equation}\nleading to $\\alpha_k|G^e\\rangle=0$. Furthermore, it can be easily checked that $\\{c^\\dagger_{k=0},\\alpha_k\\}=0$ for all $k=m\\pi\/L,~1\\leq m\\leq L-1$, thus $\\alpha_k|G^o\\rangle=\\alpha_kc^\\dagger_{k=0}|G^e\\rangle=-c^\\dagger_{k=0}\\alpha_k|G^e\\rangle=0$. We conclude that Eq.~(\\ref{eq:KitaevG}) indeed gives us the ground states of $H_{\\mathrm{Kitaev}}$ at the point $t=\\Delta,~\\mu=0$.\n\n\\subsection{Detailed derivation of Eq.~(\\ref{eq:dbwireH})}\nTo verify that the combination of Eqs.~(\\ref{eq:Hparen}) and (\\ref{eq:dbwireHpp}) indeed give the local form of Eq.~(\\ref{eq:dbwireH}), we first notice that \n\\begin{eqnarray}\n\\alpha_{k\\sigma}&\\equiv& C_{k\\sigma}-S^\\dagger_{k\\sigma}=\\frac{e^{i\\frac{k}{2}}}{\\sqrt{2}}\\left(i\\cos\\frac{k}{2}c^\\dagger_{-k,\\sigma}-\\sin\\frac{k}{2}c_{k\\sigma}\\right)-(k\\to-k)\\nonumber\\\\\n&=&\\frac{1}{\\sqrt{2L}}\\sum^{L-1}_{j=1}\\sin kj (c_{j+1,\\sigma}+c^\\dagger_{j+1,\\sigma}-c_{j\\sigma}+c^\\dagger_{j\\sigma}),~~k=\\frac{m\\pi}{L},~m=1\\ldots(L-1).\n\\end{eqnarray}\nUsing the completeness and orthonormality of $\\sin kj$,\n\\begin{equation}\n\\sum_{k}\\sin kj \\sin kj'=\\frac{L}{2}\\delta_{jj'},\n\\end{equation}\nwe have \n\\begin{eqnarray}\\label{eq:S_j}\n\\frac{4}{\\sqrt{2L}}\\sum_k (C_{k\\sigma}-S^\\dagger_{k\\sigma})\\sin kj =c_{j+1,\\sigma}+c^\\dagger_{j+1,\\sigma}-c_{j\\sigma}+c^\\dagger_{j\\sigma}\\equiv C_{j\\sigma}-S_{j\\sigma}^\\dagger,~1\\leq j\\leq L-1,\n\\end{eqnarray}\nwhere $C_{j\\sigma}=c_{j+1,\\sigma}-c_{j\\sigma},~S_{j\\sigma}^\\dagger=-c^\\dagger_{j+1,\\sigma}-c^\\dagger_{j\\sigma}$. The parent Hamiltonian given in Eqs.~(4) and (6) can then be expanded in position space~(we use the shorthand $\\sum_{\\boldsymbol k}=\\sum_{k_1,k_2,k_3, k_4}$ and $\\sum_{\\boldsymbol \\sigma}=\\sum_{\\sigma_1,\\sigma_2,\\sigma_3, \\sigma_4}$)\n\\allowdisplaybreaks\n\\begin{eqnarray}\\label{eq:dbwireHpp_S}\nH&=&\\sum_{p_1p_2p_3p_4}H_{p_1 p_2;p_3 p_4}\\hat{A}^\\dagger_{p_1 p_2} \\hat{A}_{p_3 p_4}=\\frac{16}{L^2}\\sum_{\\boldsymbol{k},\\boldsymbol{\\sigma}}\\sum^{L}_{j=1}\\sin k_1j\\sin k_2j\\sin k_3j\\sin k_4j\\nonumber\\\\\n&&{}\\times\\left[p\\delta_{\\sigma_1\\sigma_2\\sigma_3\\sigma_4}+q\\delta_{\\sigma_1\\sigma_2}\\delta_{\\sigma_3\\sigma_4}-r(\\delta_{\\sigma_1\\sigma_3}\\delta_{\\sigma_2\\sigma_4}+\\delta_{\\sigma_1\\sigma_4}\\delta_{\\sigma_2\\sigma_3})\\right] (C^\\dagger_{k_1\\sigma_1}S_{k_2 \\sigma_2}+C^\\dagger_{k_2 \\sigma_2}S_{k_1 \\sigma_1})(S^\\dagger_{k_3\\sigma_3}C_{k_4 \\sigma_4}+S^\\dagger_{k_4 \\sigma_4}C_{k_3 \\sigma_3})\\nonumber\\\\\n&=&\\frac{64}{L^2}\\sum_{\\boldsymbol{k},\\boldsymbol{\\sigma}}\\sum^{L}_{j=1}(C^\\dagger_{k_1\\sigma_1}\\sin k_1j) (S_{k_2 \\sigma_2}\\sin k_2j) (S^\\dagger_{k_3\\sigma_3}\\sin k_3j) (C_{k_4 \\sigma_4}\\sin k_4j) \n\\nonumber \\\\ &&{}\\times \n\\left[p\\delta_{\\sigma_1\\sigma_2\\sigma_3\\sigma_4}+q\\delta_{\\sigma_1\\sigma_2}\\delta_{\\sigma_3\\sigma_4}-r(\\delta_{\\sigma_1\\sigma_3}\\delta_{\\sigma_2\\sigma_4}+\\delta_{\\sigma_1\\sigma_4}\\delta_{\\sigma_2\\sigma_3})\\right]\\nonumber\\\\ \n&=&\\sum_{\\boldsymbol{\\sigma}}\\sum^{L}_{j=1}C^\\dagger_{j\\sigma_1}S_{j\\sigma_2} S^\\dagger_{j\\sigma_3}C_{j \\sigma_4} \n\\left[p\\delta_{\\sigma_1\\sigma_2\\sigma_3\\sigma_4}+q\\delta_{\\sigma_1\\sigma_2}\\delta_{\\sigma_3\\sigma_4}-r(\\delta_{\\sigma_1\\sigma_3}\\delta_{\\sigma_2\\sigma_4}+\\delta_{\\sigma_1\\sigma_4}\\delta_{\\sigma_2\\sigma_3})\\right]\\nonumber\\\\\n&=&(2t-\\beta-\\gamma)\\sum^{L}_{j=1}[C^\\dagger_{aj}S_{aj} S^\\dagger_{aj}C_{aj} +(a\\to b)]+(\\gamma-\\beta)\\sum^{L}_{j=1}[C^\\dagger_{aj}S_{aj} S^\\dagger_{bj}C_{bj}+(a\\leftrightarrow b)]\\nonumber\\\\\n&&+(\\beta+\\gamma)\\sum^{L}_{j=1}(C^\\dagger_{aj}S_{bj}+C^\\dagger_{bj}S_{aj})(S^\\dagger_{aj}C_{bj}+S^\\dagger_{bj}C_{aj}),\n\\end{eqnarray}\nwhere $\\beta=-(q+r)\/2,~\\gamma=(q-r)\/2$ and $t=(p+q-3r)\/2$. By expanding the last line of Eq.~(\\ref{eq:dbwireHpp_S}) we get the form of Eq.~(\\ref{eq:dbwireH}) in the main text~(with $\\alpha=r=-\\beta-\\gamma$).\n\\subsection{Positive region of $\\hat{H}$}\nWe now prove that the matrix $H_{p_1 p_2;p_3 p_4}$ given in Eq.~(\\ref{eq:dbwireHpp}) is positive semi-definite in the triangle region shown in Fig.~\\ref{fig:triangle} in the main text. Notice that $H_{p_1 p_2;p_3 p_4}=H_{k_1 k_2;k_3 k_4}\\cdot \\lambda_{\\sigma_1\\sigma_2;\\sigma_3\\sigma_4}$ with orbital part $H_{k_1 k_2;k_3 k_4}=\\frac{16}{L^2}\\sum^{L}_{j=1}\\sin k_1j\\sin k_2j\\sin k_3j\\sin k_4j$ and spin part $\\lambda_{\\sigma_1\\sigma_2;\\sigma_3\\sigma_4}=p\\delta_{\\sigma_1\\sigma_2\\sigma_3\\sigma_4}+q\\delta_{\\sigma_1\\sigma_2}\\delta_{\\sigma_3\\sigma_4}-r(\\delta_{\\sigma_1\\sigma_3}\\delta_{\\sigma_2\\sigma_4}+\\delta_{\\sigma_1\\sigma_4}\\delta_{\\sigma_2\\sigma_3})$. The orbital part is always positive, since for any vector $f_{k k'}$ we have \n\\begin{equation}\n\\sum_{k_1k_2k_3k_4}f^*_{k_1 k_2}H_{k_1k_2;k_3k_4}f_{k_3k_4}=\\frac{16}{L^2}\\sum_{k_1k_2k_3k_4}\\sum^{L}_{j=1}f^*_{k_1 k_2}\\sin k_1j\\sin k_2j\\sin k_3j\\sin k_4jf_{k_3k_4}=\\frac{16}{L^2}\\sum^{L}_{j=1}f^*_{j}f_{j}\\geq 0,\n\\end{equation}\nwhere $f_j=\\sum_{kk'}f_{kk'}\\sin kj\\sin k'j $. For the spin part $\\lambda_{\\sigma_1\\sigma_2;\\sigma_3\\sigma_4}$, we write it in the matrix form~(assume the order $aa,ab,ba,bb$)\n\n\\begin{eqnarray}\n\\lambda=\n\\begin{bmatrix}\np+q-2r & 0 & 0 & q\\\\\n0 & -r & -r & 0\\\\\n0 & -r & -r & 0\\\\\nq & 0 & 0 & p+q-2r\n\\end{bmatrix}=\n\\begin{bmatrix}\n2t-\\gamma-\\beta & 0 & 0 & \\gamma-\\beta\\\\\n0 & -\\alpha & -\\alpha & 0\\\\\n0 & -\\alpha & -\\alpha & 0\\\\\n\\gamma-\\beta & 0 & 0 & 2t-\\gamma-\\beta\n\\end{bmatrix}=\\lambda_1\\oplus\\lambda_2,\n\\end{eqnarray}\nwith $\\lambda_1=\\begin{bmatrix}-\\alpha & -\\alpha \\\\ -\\alpha & -\\alpha \\end{bmatrix}$ acting on ($ab,ba$) and $\\lambda_2=\\begin{bmatrix}2t-\\gamma-\\beta  & \\gamma-\\beta \\\\ -\\gamma-\\beta & 2t-\\gamma-\\beta  \\end{bmatrix}$ acting on $(aa,bb)$. Thus $\\lambda$ is positive-definite if and only if both $\\lambda_1$ and $\\lambda_2$ are positive-definite. The condition that $\\lambda_1$ is positive-definite gives $-\\alpha>0$, while $\\lambda_2$ is positive-definite gives $|\\gamma-\\beta|<2t-\\gamma-\\beta$, which simplifies to $\\alpha<0,~\\beta<t,~\\gamma<t$, leading to the triangle region in the main text. \n\n\\subsection{Alternative derivations of the parent Hamiltonian}\nThe derivation of $\\hat{H}$ presented above enables us to see how the double wire model follows from our general construction and can be generalized to arbitrary points of Kitaev's model. However, for the double wire parent Hamiltonian constructed in our main text, simpler derivations exist. Actually, Eq.~(\\ref{eq:S_j}) gives us annihilators of the double wire ground state $|G_A\\rangle\\otimes|G_B\\rangle$ in position space. Thus we can directly build the parent Hamiltonian $\\hat{H}$ using the real space version of Eqs.~(\\ref{eq:Hparen}) and (\\ref{eq:A_pp2}) with $C_{j\\sigma}=c_{j+1,\\sigma}-c_{j\\sigma},~S_{j\\sigma}^\\dagger=-c^\\dagger_{j+1,\\sigma}-c^\\dagger_{j\\sigma}$~(or equivalently, directly go to the last line of Eq.~(\\ref{eq:dbwireHpp_S}) without working in momentum space at all), leading to the same Hamiltonian Eq.~(\\ref{eq:dbwireH}) in our main text. This derivation is a direct generalization of the one given in Ref.~\\cite{SDiehl}.\n\n\nAnother simple derivation is based on using an alternative basis of single wire ground states~(at $\\Delta=t,\\mu=0$)~\\cite{AoP2014}\n\\begin{eqnarray}\n|G^\\eta\\rangle&=&(1+\\eta c^\\dagger_1)(1+\\eta c^\\dagger_2)\\cdots(1+\\eta c^\\dagger_L)|0\\rangle,~~\\eta=\\pm 1,\n\\end{eqnarray}\nand observing the following properties\n\\begin{eqnarray}\nc^\\dagger_ic^\\dagger_{i+1}|G^\\eta\\rangle&=&n_i n_{i+1}|G^\\eta\\rangle,~~\nc_{i+1}c_i|G^\\eta\\rangle=\\bar{n}_i \\bar{n}_{i+1}|G^\\eta\\rangle,\\nonumber\\\\\nc^\\dagger_ic_{i+1}|G^\\eta\\rangle&=&n_i \\bar{n}_{i+1}|G^\\eta\\rangle,~~\nc^\\dagger_{i+1}c_i|G^\\eta\\rangle=\\bar{n}_i n_{i+1}|G^\\eta\\rangle,\n\\end{eqnarray} \nwhere $\\bar{n}_i\\equiv 1-n_i$. With this, it is easy to check that the number-conserving single wire operator $[(c^\\dagger_jc_{j+1}+\\mathrm{H.c.})+2(n_j-\\frac{1}{2})(n_{j+1}-\\frac{1}{2})-\\frac{1}{2}]$ annihilates $|G^\\eta\\rangle$. To include interwire couplings, we notice that\n\\begin{eqnarray}\nJ^\\dagger_{||,j}J_{||,j}|G^{\\eta_A,\\eta_B}\\rangle=J^\\dagger_{=,j} J_{=,j}|G^{\\eta_A,\\eta_B}\\rangle= J^\\dagger_{\\times,j} J_{\\times,j}|G^{\\eta_A,\\eta_B}\\rangle=(U^\\square_{j}-U^{(3p)}_{j})|G^{\\eta_A,\\eta_B}\\rangle,\n\\end{eqnarray}\nwhere $\\hat{U}^{\\square}_{j}=n^a_j n^a_{j+1}n^b_jn^b_{j+1},~~\\hat{U}^{(3p)}_j=[n^a_j n^a_{j+1}(n^b_j+n^b_{j+1})+(a\\leftrightarrow b)]$, and $|G^{\\eta_A,\\eta_B}\\rangle=|G^{\\eta_A}_A\\rangle\\otimes|G^{\\eta_B}_B\\rangle$ is the double wire ground state constructed by direct product of single wire ground states. Therefore\n\\begin{equation}\n(\\alpha J^\\dagger_{||,j}J_{||,j}+\\beta J^\\dagger_{=,j} J_{=,j}+\\gamma J^\\dagger_{\\times,j} J_{\\times,j})|G^{\\eta_A,\\eta_B}\\rangle=0,\n\\end{equation}\nfor $\\alpha+\\beta+\\gamma=0$. It then follows that the Hamiltonian constructed in Eq.~(\\ref{eq:dbwireH}) in the main text satisfies $\\hat{H}|G^{\\eta_A,\\eta_B}\\rangle=0$, i.e. $|G^{\\eta_A,\\eta_B}\\rangle$ is an eigenstate of $\\hat{H}$ with zero energy. This method can only tell us that $|G^{\\eta_A,\\eta_B}\\rangle$ is an eigenstate of $\\hat{H}$. To find the positive region of $\\hat{H}$~(where $|G^{\\eta_A,\\eta_B}\\rangle$ become its ground states), we still have to turn to other means. \n\nIn Fig.~\\ref{fig:diag} we draw a pictorial representation of the interaction terms of the parent Hamiltonian Eq.~(\\ref{eq:dbwireH}), including two types of nonlinear terms: interactions and correlated pair tunnelings. \n\\begin{figure}\n\\includegraphics[width=0.5\\linewidth]{diag.png}\n\\caption{\\label{fig:diag} Diagrammatic representation of interaction terms in the double wire parent Hamiltonian in Eq.~(\\ref{eq:dbwireH}). Double arrows represent pair hopping while wavy lines represent interactions. }\n\\end{figure}\n\n\\iffalse\n\\section{The 2D $p_x+ip_y$ mean-field model}\n\\subsection{The ground state $|G_{12,34}\\rangle$}\nIn this section we prove that the state $|G_{12,34}\\rangle$ constructed in Eqs.~(\\ref{eq:Gg}) and (\\ref{eq:g_1234}) in our main text minimizes the boundary term $\\hat{K}_{\\mathrm{bound}}$ up to small corrections, that is, $|G_{12,34}\\rangle$ is indeed a ground state of both the mean-field Hamiltonian $\\hat{K}$ and the number-conserving Hamiltonina $\\hat{H}$ with four vortices.\n\nIn the geometry shown in Fig.~\\ref{fig:holes} in our main text, the boundary term in Eq.~(\\ref{eq:K_bb}) could be expressed as\n\\begin{equation}\\label{eq:K_bvort}\n\\hat{K}_{\\mathrm{bound}}=i\\sum^{2M}_{j=1}\\oint_{C_j}(\\psi^\\dagger_z\\partial_z\\psi_z dz+\\psi^\\dagger_z\\partial_{\\bar{z}}\\psi_z d\\bar{z})=-\\sum^{2M}_{j=1}\\sum_{m\\in \\mathbb{Z}+1\/2}m\\psi^\\dagger_{jm}\\psi_{jm},\n\\end{equation}\nwhere $C_j$ is an anti-clockwise contour surrounding the boundary of the $j$-th vortex, and $\\psi_{jm}=\\frac{1}{\\sqrt{2\\pi}}\\int \\psi_{z_j+(r_0,\\theta)}e^{-im\\theta}d\\theta$ (where $(r_0,\\theta)$ denotes $r_0e^{i\\theta}$) is a boundary operator with angular momentum $m$. \n\\fi\n\n\\iffalse\n\\subsection{The role of the boundary term $\\hat{K}_{\\mathrm{bound}}$}\nIn constructing the parent Hamiltonian in Eqs.~(\\ref{eq:Hparen_pos}) and (\\ref{eq:H_pip}) we mentioned that the boundary term $\\hat{K}_{\\mathrm{bound}}$ must be added along with $H_{\\mathrm{bulk}}$ to stabilize the topological phase. The reason is that the bulk Hamiltonian $\\hat{K}_{\\mathrm{bulk}}$~(and therefore $\\hat{H}_{\\mathrm{bulk}})$ alone contains additional zero energy edge modes~(other than Majoranas), leading to unwanted degeneracy. \n\nTo be specific, consider a general zero energy mode defined as $b=\\int_S [u(z,\\bar{z})\\psi_{z}+v(z,\\bar{z})\\psi^\\dagger_{z}]d^2z$. The zero energy condition $[b,\\hat{K}_{\\mathrm{bulk}}]=0$ simplifies to $[b,2\\partial_{\\bar{z}} \\psi_z-\\psi^\\dagger_z]=[b,2\\partial_z \\psi^\\dagger_z-\\psi_z]=0$, which gives a differential equation on $u(z,\\bar{z})$ and $v(z,\\bar{z})$\n\\begin{equation}\\label{eq:diff_zuv}\n2\\partial_{\\bar{z}}v=u,~~2\\partial_{z}u=v.\n\\end{equation}\nIn the double wall geometry shown in Fig.~\\ref{fig:dbwall} in our main text, the two independent solutions to Eq.~(\\ref{eq:diff_zuv}) are $u(x,y)=u_{\\sigma,k_y}(x)e^{ik_y y},~v(x,y)=v_{\\sigma,k_y}(x)e^{ik_y y}$ with $\\sigma=L,R$ and\n\\begin{eqnarray}\\label{eq:edge}\n\\begin{pmatrix}\nu_{L,k_y}(x)\\\\\nv_{L,k_y}(x)\n\\end{pmatrix}=\n\\begin{pmatrix}\ni\\mu \\\\\n-i\\nu \n\\end{pmatrix}e^{-\\lambda x},~\n\\begin{pmatrix}\nu_{R,k_y}(x)\\\\\nv_{R,k_y}(x)\n\\end{pmatrix}=\n\\begin{pmatrix}\n\\nu\\\\\n\\mu \n\\end{pmatrix}e^{-\\lambda (W-x)},\n\\end{eqnarray}\nwhere $\\lambda=\\sqrt{k_y^2+1^2},~\\mu\/\\nu=\\lambda+k_y$. Thus even the chiral edge modes with non-zero $y-$momentum has exactly zero energy, in contrast to the linear dispersion $E_{k_y}\\propto k_y$ found in literature~\\cite{NRead}. In the multi-vortices geometry shown in Fig.~\\ref{fig:vort}, we can notice that for any solution $(u,v)$ of Eq.~(\\ref{eq:diff_zuv}), $(\\partial_z u,\\partial_zv)$ and $(\\partial_{\\bar{z}} u,\\partial_{\\bar{z}}v)$ are also its solutions. It follows that all solutions to Eq.~(\\ref{eq:diff_zuv}) can be constructed by\n\\begin{equation}\n(v_n,u_n)=(2\\partial_z)^n(v_0,u_0),~(v_{-n},u_{-n})=(2\\partial_{\\bar{z}})^n(v_0,u_0).\n\\end{equation}\nThe corresponding zero energy modes $b_n$ are localized edge modes with nonzero angular momentum $v_n(r,\\theta)\\propto e^{-i(n+1\/2)\\theta},~u_n\\propto e^{-i(n-1\/2)\\theta}$. In a word, all chiral edge modes of $\\hat{K}_{\\mathrm{bulk}}$ have exactly zero energy, leading to infinitely many degenerate ground states. Because by our construction $\\hat{H}_{\\mathrm{bulk}}$ has exactly the same ground states with $\\hat{K}_{\\mathrm{bulk}}$, $\\hat{H}_{\\mathrm{bulk}}$ has exactly the same problem.\n\nIt is expected that adding a boundary term $\\hat{K}_{\\mathrm{bound}}$ into our Hamiltonian completely solves the problem mentioned above, since in the aouble wall geometry, the expression in Eq.~(\\ref{eq:K_bwall}) clearly suggests that $\\hat{K}_{\\mathrm{bound}}$ gives an effective boundary Hamiltonian $\\sum_{k_y} k_y (b_{L,-k_y}b_{L,k_y}+b_{R,k_y}b_{R,-k_y})$ to the chiral edge modes $b_{\\sigma,k_y}$, leading to the linear dispersion relation found in literature. Thus adding $\\hat{K}_{\\mathrm{bound}}$ eliminates the unwanted degeneracy both in the mean-field Hamiltonian $\\hat{K}$ and the number-conserving Hamiltonian $\\hat{H}$.\n\nWe now check that the ground state $|G^e\\rangle$ constructed in Eq.~(\\ref{eq:g_dbwall}) satisfies $\\psi_{0,-k_y}|G^e\\rangle=\\psi_{W,k_y}|G^e\\rangle=0$ for all $k_y>0$ meaning that it minimizes the positive part of $\\hat{K}_{\\mathrm{bound}}$. We have\n\\begin{eqnarray}\n\\psi_{W,k_y}|G_e\\rangle&=&\\frac{1}{\\sqrt{L}}\\int^\\infty_{-\\infty} dy e^{-ik_y y}\\psi_{W,y}|G_e\\rangle\\nonumber\\\\%=\\frac{1}{\\sqrt{L}}\\int^\\infty_{-\\infty} dy e^{-ik_y y}\\partial_{\\psi^\\dagger_{0,y}}|G_e\\rangle\n&=&\\frac{1}{\\sqrt{L}}\\int^\\infty_{-\\infty} dy e^{-ik_y y}\\frac{1}{2\\pi}\\int_S\\frac{1}{W+iy-z'}\\psi^\\dagger_{z'}d^2z'|G_e\\rangle.\n\\end{eqnarray}\nThe integration on $y$ can be done using a contour integral, which gives the result\n\\begin{eqnarray}\n\\int^\\infty_{-\\infty}\\frac{e^{-ik_y y}dy}{W+iy-(x'+iy')}\n=\\begin{cases}\n      0, & k_y>0, \\\\\n      \\pi , & k_y=0,\\\\\n      2\\pi e^{k_y(W-x')-ik_y y'} & k_y<0.\n    \\end{cases}\n\\end{eqnarray}\nIt follows that $\\psi_{W,k_y}|G^e\\rangle=0$ for $k_y>0$. Similarly we can prove that $\\psi_{0,-k_y}|G^e\\rangle=0$ for $k_y>0$. This verifies that $|G^e\\rangle$ minimizes the positive part of $\\hat{K}_{\\mathrm{bound}}$. By using similar methods to the state $|G_{12,34}\\rangle$ constructed in Eq.~(\\ref{eq:g_1234}), we can prove that $|G_{12,34}\\rangle$ minimizes the positive part of $\\hat{K}_{\\mathrm{bound}}$ in multi-vortices geometry.\n\\fi\n\n\\iffalse\n\\subsection{Dimension of ground state subspace with $2M=4$ vortices}\nIn our main text we claimed that the space spanned by $|G_{12,34}\\rangle$, $|G_{13,24}\\rangle$ and $|G_{14,23}\\rangle$ is two dimensional. This can be checked as follow: since \n\\begin{equation}\\label{eq:G_1324}\n|G_{13,24}\\rangle=\\exp\\left[\\frac{1}{2}\\int_S g_{13,24}(z,z') \\psi_z^\\dagger \\psi_{z'}^\\dagger d^2z d^2 z'\\right]|0\\rangle=\\exp\\left[\\frac{1}{2}\\int_S [g_{13,24}(z,z')-g_{12,34}(z,z')] \\psi_z^\\dagger \\psi_{z'}^\\dagger d^2z d^2 z'\\right]|G_{12,34}\\rangle, \n\\end{equation}\nand \n\\begin{eqnarray}\ng_{13,24}(z,z')-g_{12,34}(z,z')=-\\frac{1}{4\\pi}\\frac{(z-z')(z_1-z_4)(z_2-z_3)}{\\sqrt{(z-z_{1234})(z'-z_{1234})}},\n\\end{eqnarray}\nwhere we use the shorthand $(z-z_{1234})=(z-z_1)(z-z_2)(z-z_3)(z-z_4)$ and similarly for $(z'-z_{1234})$. It follows that\n\\begin{eqnarray}\n\\frac{1}{2}\\int_S [g_{13,24}(z,z')-g_{12,34}(z,z')] \\psi_z^\\dagger \\psi_{z'}^\\dagger d^2z d^2 z'=\\alpha_{14,23}\\phi^\\dagger_0\\phi^\\dagger_1,\n\\end{eqnarray}\nwhere $\\alpha_{14,23}=(z_1-z_4)(z_2-z_3)\/(4\\pi)$ and\n\\begin{equation}\n\\phi_0^\\dagger=\\int_S\\frac{\\psi^\\dagger_z}{\\sqrt{z-z_{1234}}}d^2z,~~~~\\phi_1^\\dagger=\\int_S\\frac{z\\psi^\\dagger_z}{\\sqrt{z-z_{1234}}}d^2z.\n\\end{equation}\nTherefore, \n\\begin{equation}\n|G_{13,24}\\rangle=\\exp(\\alpha_{14,23}\\phi^\\dagger_0\\phi^\\dagger_1)|G_{12,34}\\rangle=(1+\\alpha_{14,23}\\phi^\\dagger_0\\phi^\\dagger_1)|G_{12,34}\\rangle,\n\\end{equation}\nand by permuting the indices $3$ and 4, we get\n\\begin{equation}\n|G_{14,23}\\rangle=(1+\\alpha_{13,24}\\phi^\\dagger_0\\phi^\\dagger_1)|G_{12,34}\\rangle.\n\\end{equation}\nIt therefore follows that all the three states $|G_{12,34}\\rangle$, $|G_{13,24}\\rangle$ and $|G_{14,23}\\rangle$ can be expressed as a linear superposition of the two basis states $|G_{12,34}\\rangle$ and $\\phi^\\dagger_0\\phi^\\dagger_1|G_{12,34}\\rangle$, meaning that the space spanned by them is two dimensional.\n\\fi\n\n\\iffalse\n\\subsection{About the normalizability of ground states}\nThe ground states constructed in Eqs.~(\\ref{eq:Gg}) and (\\ref{eq:g_1234}) have the problem of being non-normalizable since the function $g(z,z')$ has a pole at $z=z'$. The physical meaning of non-normalizable ground states can be made clear by considering our model as the limit of a series of well-defined models with normalizable ground states. As an example, we consider the family of mean-field Hamiltonians $\\hat{K}^{(a)}=\\hat{K}^{(a)}_{\\mathrm{bulk}}+\\hat{K}_{\\mathrm{bound}}$ where\n\\begin{eqnarray}\\label{eq:K_bb}\n\\hat{K}^{(a)}_{\\mathrm{bulk}}=\\int_S\\left[2\\partial_z\\psi^\\dagger_z-\\int_S\\delta_a^2(z-z')\\psi_{z'}d^2z'\\right]\\left[2\\partial_{\\bar{z}} \\psi_z-\\int_S\\delta_a^2(z-z'')\\psi^\\dagger_{z''}d^2z''\\right] d^2 z\n\\end{eqnarray}\nand \n\\begin{equation}\n\\delta^2_a(z-z')=\\frac{1}{2\\pi a^2}\\exp(-\\frac{|z-z'|}{a}),\n\\end{equation}\nwith $a>0$ being a real constant. Since $\\mathrm{lim}_{a\\to 0}\\delta^2_a(z-z')=\\delta^2(z-z')$, the mean-field Hamiltonian $\\hat{K}$ in Eq.~(\\ref{eq:K_2Dpp}) can be considered as the $a\\to 0$ limit of $\\hat{K}^{(a)}$. It is easy to check that the ground states of $K^{(a)}$ could still be constructed by the exponential form of Eq.~(\\ref{eq:Gg}) in the main text with $g(z,z')$ satisfying\n\\begin{equation}\\label{eq:gzz_a}\n2\\partial_{\\bar{z}} g(z,z')=\\delta^2_a(z-z').\n\\end{equation}\nWith the sharp delta function in Eq.~(\\ref{eq:gzz}) replaced by a well-defined distribution $\\delta^2_a(z-z')$, the pole of $g(z,z')$ is removed. For example, in regions far from edges and vortices, Eq.~(\\ref{eq:gzz_a}) has solution\n\\begin{equation}\ng_a(z,z')=\\frac{1}{2\\pi(z-z')}\\left[1-\\left(1+\\frac{|z-z'|}{a}\\right)\\exp\\left(-\\frac{|z-z'|}{a}\\right)\\right].\n\\end{equation}\nIn the limit $a\\to 0$, $g_a(z,z')$ gives back Eq.~(\\ref{eq:g_dbwall}) in the main text, but for any nonzero $a$, $g_a(z,z')$ is free of poles and in particular we have $g_a(z,z)=0$. This justifies our use of non-normalizable ground states.\n\\fi\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMany biological processes, such as morphogenesis~\\cite{martin2009pulsed, etournay2015interplay, streichan2018global, maniou2021hindbrain}, wound healing~\\cite{poujade2007collective, brugues2014forces, tetley2019tissue, ajeti2019wound}, and cancer metastasis~\\cite{friedl2009collective, arwert2012epithelial}, require  coordinated motion and shape changes of many cells. An important open question in biology is how the large scale mechanics of biological tissue emerges from the properties of individual cells, which are in turn governed by force-generating proteins within the cytoskeleton and adhesion molecules between cells~\\cite{salbreux2012actin, murrell2015forcing, ladoux2017mechanobiology}. Many theoretical models have been proposed to describe dense epithelia, single layers of very tightly packed cells~\\cite{graner1992simulation, marchetti2013hydrodynamics, banerjee2019continuum, alert2020physical, alt2017vertex}. Among these, vertex models, originally developed from models of soap films~\\cite{nagai2001dynamic}, have proven a powerful starting point for capturing the mechanical properties of epithelia. The vertex model describes the apical surface of a confluent tissue as a polygonal tiling of the plane (Fig.~\\ref{fig:1}a)~\\cite{alt2017vertex, nagai2001dynamic, fletcher2014vertex, farhadifar2007influence, staple2010mechanics, bi2016motility}. Each polygon represents a cell, each edge a cell-cell junction, and each vertex a multicellular junction. Each cell's mechanics are controlled by multiple bio-mechanical processes that were proposed to be effectively described by a mechanical energy determined by deviations of their area and perimeter from preferred values. These preferred values encode bio-mechanical properties such as cadherin molecules concentration, apical ring-contractiliy and more.  Force balance via energy minimization then determines the position of the vertices and thus the shape of cells in the tissue. Topological rearrangements resulting in cell intercalation, cell division and motility have also been incorporated, and the model  has been highly successful in capturing a range of biological processes, such as tissue growth~\\cite{hufnagel2007mechanism}, wound healing~\\cite{tetley2019tissue}, and tissue organization~\\cite{farhadifar2007influence}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{fig1.pdf}\n    \\caption{The vertex model for epithelia. (a)  \n    (a) The apical surface of an epithelium is modeled by a polygonal tiling, with each polygon representing a cell. (b) Vertex model phase diagram in the $p_0$, $r$ plane. Within the blue region cells are unstable and collapse. Within the green region the tissue is in an incompatible state, with neither preferred perimeter nor area achieved, and the ground state is a regular hexagonal lattice. The tissue acts like a solid in response to shear. Within the yellow region both preferred perimeter and area are achieved and cells have a degenerate ground state and the tissue has zero shear modulus. The cell shapes show example energy minima, where a cell may elongate, increase its pointiness using the angle $\\phi$, or increasing its shear tilt angle $\\theta$ as described in Ref.~\\cite{hernandez2022anomalous}, in order to increase its perimeter while maintaining unit area.\n    }\n    \\label{fig:1}\n\\end{figure}\n\n\nIt has been shown that  vertex models exhibit a transition between a fluid-like state and a solid-like state where cells are jammed and unable to rearrange (Fig.~\\ref{fig:1}b). This rigidity transition occurs at constant cell density and is driven by both active processes, such as  fluctuations in cell-edge tension and cell motility~\\cite{bi2015density, bi2016motility, sussman2018anomalous,  tong2021linear, duclut2021nonlinear, duclut2022active,farhadifar2007influence, staple2010mechanics}, as well as by geometric constraints~\\cite{moshe2018geometric,hernandez2022anomalous}. Recent work by us and others has shown that even in the absence of fluctuations and topological rearrangements, vertex models exhibit a rigidity transition associated with geometrical frustration~\\cite{moshe2018geometric,merkel2019minimal,hernandez2022anomalous}.  In the rigid or incompatible state cells are unable to achieve the target values of area and perimeter and the system is under finite tension, with a unique gapped ground state. In the soft or compatible state, cells achieve both target area and perimeter and the ground state has zero energy. Due to the underconstrained nature of the vertex model, however, the liquid ground state is degenerate as for a given $n$-sided polygon there are many shapes that preserve area and perimeter. This allows the system to accommodate small shear deformations by finding  a new zero energy shape, resulting in vanishing shear modulus.\n\nThe linear elastic response of an ordered hexagonal vertex model to external deformations has been examined through calculations of the shear and bulk moduli~\\cite{staple2010mechanics, murisic2015discrete,hernandez2022anomalous}. Staple \\emph{et al.}~\\cite{staple2010mechanics,hernandez2022anomalous} evaluated the elastic moduli and first demonstrated the vanishing of the shear modulus in the compatible state. More recently, we showed that the vanishing of both shear and Young moduli in the soft regime stems from the degeneracy of the compatible ground states, which allows the deformed tissue to spontaneously shear to a new compatible ground state to accommodate the external deformation~\\cite{hernandez2022anomalous}. We additionally discovered that the response  is highly singular at the critical point, with breakdown of linear elasticity and anomalous coupling between compression and shear, as quantified by the development of a new elastic constant~\\cite{hernandez2022anomalous}. \n\nThe above studies  only allow for affine deformations of the cells. This approximation can be viewed as appropriate for determining the short-time response of the vertex model to strain. The vertex model has, however, additional degrees of freedom and can relax stress by moving vertices in a non-affine way. Murisic \\emph{et al.}~\\cite{murisic2015discrete} incorporated these effects by considering the hexagonal lattice as the union of two sub-lattices with a microscopic shift between them and found  that the shear modulus is $2\/3$ softer than previously reported. Tong \\emph{et al.}~\\cite{tong2021linear} used simulations to measure the shear storage modulus and viscosity in both ordered and disordered model tissues.\n\n\nIn this paper, we expand upon previous work  by incorporating simple \\textit{non-affine} deformations. Using a mean field model for a hexagonal lattice, we derive analytic expressions for all the linear elastic moduli of the tissue, and verify these results using simulations. We show that, away from the critical point, the elastic constants of a regular VM satisfy the standard relations of two-dimensional elasticity of isotropic solids. Despite this Hookean relationship, the mechanical linear response exhibits  robust non-affine contributions that can significantly reduce the elastic constants. For instance, the bulk modulus can be softer in the rigid state than in the soft fluid-like state and jumps discontinuously  across  the solid to fluid transition. We highlight several novel behaviors of vertex model elasticity, such as negative Poisson's ratio and a softening of the tissue as the ratio of area to perimeter stiffness increases. We verify our analytical results using numerical vertex model simulations of a regular tissue. \n\nThe remainder of the paper is organized as follows. In Section \\ref{sec:modelandsimulation} we state the vertex model simulation and deformation protocol to extract various elastic constants. In Section \\ref{sec:MFT} we introduce the VM and its mean-field implementation used in the present work, and present a new derivation of the ground states that allows us to quantify the degeneracy of the compatible regime.  In Section \\ref{sec:mechanics}, after  highlighting the distinction between the affine and non-affine deformations allowed in our model, we present results for all the elastic constants. We conclude in Section \\ref{sec:conclusions} with a brief discussion\n\\section{Vertex model: simulation and deformation protocol}\n\\label{sec:modelandsimulation}\n\\subsection{The vertex model of epithelia}\nThe vertex model describes cells in a confluent tissue as polygons of area $A_\\alpha$ and perimeter $P_\\alpha$ (Fig.~\\ref{fig:1}a). The tissue  energy is written as\n\\begin{equation}\n    E_{\\text{tissue}} = \\frac{1}{2}\\sum_\\alpha K(A_\\alpha - A_{\\alpha 0})^2 + \\frac{1}{2} \\sum_\\alpha \\Gamma P_\\alpha^2 + \\sum_{\\langle ij\\rangle} \\Lambda_{ij} L_{ij},\n\\end{equation}\nwhere $\\alpha$ labels individual cells and $\\langle ij\\rangle$ indexes edges connecting vertices $i$ and $j$. The first term  embodies the energy cost of  cell area deformations, with $K$  the area elasticity and $A_{\\alpha 0}$ the preferred or target area. The second term represents active contractility and elasticity of the cytoskeleton, with $\\Gamma$ the contractility. The third term represents interfacial energy between neighboring cells, with $L_{ij}$ the length of edge $ij$ and  $\\Lambda_{ij}$ the associated tension controlled by the interplay of cell-cell adhesion and cortex contractility. The tension can become negative when adhesion overcomes contractile surface forces.\n\n\nThe mechanical force on vertex $i$ with position $\\mathbf{x}_i$ is given by $\\mathbf{F}_i = -\\frac{\\partial E_\\text{tissue}}{\\partial \\mathbf{x}_i}$. The tissue rearranges vertices to locally minimize the energy. This can be described quasi-statically by requiring force balance at each time-step, or dynamically by assuming that vertices relax according to overdamped dynamics where viscous drag balances the mechanical forces: $\\gamma \\frac{\\partial \\mathbf{x}_i}{\\partial t} = \\mathbf{F}_i$, with $\\gamma$ a friction coefficient.\n\nAs the network relaxes, edges may shorten and cells may shrink, resulting in topological rearrangements that reconfigure the network. In T1 transitions, also known as cell-cell intercalations, a junction between two cells shrinks to a point and a new edge is formed, causing two originally neighboring cells to lose contact and two previously unconnected cells to form a new interface. T1 transitions allow the tissue to relax shear stresses through cell rearrangements rather than cell elongation. A T2 transition, also known as cell extrusion, occurs as a cell shrinks to zero area and is replaced by a single vertex. The mechanical state of the tissue is controlled by both topological rearrangements driven by active processes and geometric frustration. Both types of processes can drive transitions between rigid and fluid states. Here we neglect topological rearrangements to focus on the role of geometry.\n\n\nWe further simplify the model by assuming that all cells have the same preferred area $A_{\\alpha 0} = A_0$ and all edges have the same tension $\\Lambda_{ij} = \\Lambda$. The interfacial energy can then be written in terms of the cell perimeter,  $\\sum_{\\langle ij\\rangle} \\Lambda L_{ij} = \\frac{1}{2} \\sum_\\alpha \\Lambda P_\\alpha$, where the factor of $\\frac{1}{2}$ arises because the interfacial energy of each edge is shared by two cells. The tissue energy can then be recast in the form\n\\begin{equation}\n    E_{\\text{tissue}} = \\frac{1}{2}\\sum_\\alpha K(A_\\alpha - A_0)^2 + \\frac{1}{2} \\sum_\\alpha \\Gamma (P_\\alpha - P_0)^2 + E_0\\;,\n\\end{equation}\nwhere $P_0 = - \\frac{\\Lambda}{2 \\Gamma}$ is the preferred perimeter, and $E_0$ is a constant term obtained from completing the square. Since we care about the gradient of energy and not the absolute value, we discard $E_0$ in the following. \n\n\\subsection{Deformation Protocol}\n\nTo numerically obtain the elastic moduli, we simulate the mechanical response of the vertex model under different deformations using a tissue of 4 hexagonal cells in a periodic box of lengths $L_x(0)$ and $L_y(0)$, and area $A(0) = L_x(0) L_y(0)$ determined by energy minimisation, and implemented in the Surface Evolver software~\\cite{brakke1992surface}. First, we use an intermediate rigidity ratio of $r = 0.1$, and test the response across a range of preferred values of the shape index, from $p_0 = 0$ to $p_0 = 4.2$, covering both the compatible and incompatible regimes.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{fig2.pdf}\n    \\caption{ Strain protocols for measuring elastic moduli of the vertex model. (a - c) From the ground state, the periodic box lengths and vertex positions are transformed and constrained according to an affine transformation, shown by the arrows. From the constrained state, the system is relaxed according to tissue-scale or box constrained. (a) The shear modulus is calculated by applying a shear transformation to the box. In the constrained state, every edge has the same tension, producing a net force on the vertices, hence this is not a force-balanced state. After relaxation, forces are balanced through a non-affine transformation on the vertices. During relaxation the box size is fixed. (b) The bulk modulus is calculated by applying an isotropic expansion to the box and vertices. During relaxation the box size is fixed. (c) The Young's modulus and Poisson's ratio are calculated by applying a uniaxial strain to the box and vertices. During relaxation the height of the box may change and vertices may move.}\n    \\label{fig:2}\n\\end{figure}\n\nTo calculate the shear modulus, we deform the ground state (Fig.~\\ref{fig:2}a, left) by applying an initially affine deformation to vertices and the boundaries: $x_i(\\epsilon) = (1 + \\epsilon \/ 2) x_i(0) $, $y_i(\\epsilon) = (1 + \\epsilon \/ 2)^{-1} y_i(0) $, and $L_x(\\epsilon) = (1 + \\epsilon \/ 2) L_x(0)$ and $L_y(\\epsilon) = (1 + \\epsilon \/ 2)^{-1} L_y(0)$, where $\\epsilon = 0.001$ (Fig.~\\ref{fig:2}a, middle). We then allow the vertex positions to relax to an energy minima (Fig.~\\ref{fig:2}a, right), and record the change in tissue energy $\\delta E$ before and after the deformation. The shear modulus is then numerically estimated by $G = \\frac{1}{A(0)}\\frac{2 \\delta E}{\\epsilon^2}$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{fig4.pdf}\n    \\caption{\n    Non-affine deformations allow for a softer mechanical response. (a) Shear modulus $G$, (b) bulk modulus $K$, (c) Young's modulus $Y$, and (d) Poisson's ratio $\\nu$ against target shape index $p_0$ for various rigidity ratios $r$. The constrained values represent elastic moduli where vertices are constrained by the given deformation. The relaxed values lines represent the moduli allowing for non-affine deformations, where vertices may relax, subject to the boundary conditions. Dots represent simulated values. Lines represent analytic values.}\n    \\label{fig:4}\n\\end{figure}\n\nIn the ground state of the incompatible regime, cell edges are under tension and meet at 120$\\degree$ angles. After the initial affine deformation, the angles change and the tissue is no longer in a force-balanced configuration (Fig.~\\ref{fig:2}a, middle). As we allow the tissue to relax, it responds with a non-affine deformation; vertices which are of the same y-coordinate alternate between moving left and moving right during relaxation, returning the angles between edges to a stable 120$\\degree$ configuration (Fig.~\\ref{fig:2}a, right). Such a deformation cannot be described by a single affine transformation, but by two affine transformations applied to different subsets of vertices~\\cite{murisic2015discrete}.\n\nTo demonstrate the importance of this relaxation step, we report the response to two types of deformation protocols: (i) ``constrained'' deformations which are obtained where after deformation of the bounding box the cell vertices are not allowed to move to minimise the energy of the tissue, and (ii) ``relaxed'' deformations where the vertices are allowed to adjust their position to achieve force balance and  the global tissue shape remains controlled by the geometry of the deformed box. Note that in the compatible regime the relaxed state can also be achieved by allowing the tissue to change its shape~\\cite{hernandez2022anomalous}, and the resulting linear elastic constants are the same.\n\nFor an intermediate rigidity ratio $r = 0.1$, we find that the shear modulus decreases as $p_0$ increases and becomes zero at the transition to the compatible regime. In particular, the relaxation step allows cells to decrease their perimeter, and thus system energy after strain, resulting in a relaxed shear modulus that is softer than in the constrained case (Fig.\\ref{fig:3}a). In the compatible regime, the tissue is initially under no tension since the preferred perimeter is achieved. Upon straining the tissue, the perimeter increases and during the subsequent relaxation step the vertices move to reduce the perimeter until the preferred perimeter is achieved again, allowing for the net energy to remain constant, leading to a zero shear modulus.\n\nTo calculate the bulk modulus, we apply the isotropic transformation $x_i(\\epsilon) = (1 + \\epsilon)^\\frac{1}{2} x_i(0) $, $y_i(\\epsilon) = (1 + \\epsilon)^\\frac{1}{2} y_i(0) $, and $L_x(\\epsilon) = (1 + \\epsilon)^\\frac{1}{2} L_x(0)$ and $L_y(\\epsilon) = (1 + \\epsilon)^\\frac{1}{2} L_y(0)$, where $\\epsilon = 0.001$, such that  $A(\\epsilon) = (1 + \\epsilon)A(0)$. During the relaxation step, we allow both the vertices and box lengths to change, subject to the constraint $L_x(\\epsilon) L_y(\\epsilon) = A(\\epsilon)$ (Fig.~\\ref{fig:2}b). The bulk modulus is then given by $K = \\frac{1}{A(0)}\\frac{2 \\delta E}{\\epsilon^2}$.\n\nIn the incompatible regime, force balance requires a constant 120$\\degree$ angle between edges, thus the tissue expands isotropically. We find that the bulk modulus increases as the target shape index $p_0$ increases, and is equal between the relaxed and constrained cases (Fig.\\ref{fig:3}b).\n\nIn the compatible regime, the deformation initially increases the perimeter. During the relaxation step, the tissue responds in a non-affine way to restore its perimeter to its preferred value and so energy change only arises from the area term and we have a bulk modulus $K = 1$. Interestingly, this is lower than the bulk modulus in the incompatible regime just before the transition and thus, there is a discontinuity in the bulk modulus as $p_0$ changes. In contrast, the constrained case is unable to relax the cells perimeters and so has a higher bulk modulus and does not exhibit the discontinuity (Fig.\\ref{fig:3}b).\n\nNext, we apply a uniaxial deformation to calculate the Young's modulus and Poisson's ratio: $x_i(\\epsilon) = (1 + \\epsilon)x_i(0)$ and $L_x(\\epsilon) = (1 + \\epsilon) L_x(0)$ (Fig.~\\ref{fig:1}c). We then allow the vertex positions and box height $L_y(\\epsilon)$ to relax to minimise energy. The Young's modulus is given by $Y = \\frac{1}{A(0)} \\frac{2 \\delta E}{\\epsilon^2}$ and the Poisson's ratio by $\\nu = -\\frac{(L_y(\\epsilon) - L_y(0)) \/ L_y(0)}{(L_x(\\epsilon) - L_x(0)) \/ L_x(0)}$. Note that this definition of the Poison's ratio is equivalent to that in 2D elasticity and therefore its values are limited between $-1 < \\nu < 1$. The extreme case $\\nu=1$ corresponds to incompressible solid, analogous to the case of $\\nu_{3d} = 0.5$ for incompressible 3D solids.\n\nAgain, the tissue undergoes a similar non-affine relaxation as under shear strain, reducing the shear modulus compared to the constrained case (Fig.~\\ref{fig:3}c). In this case, though, we find that the Young's modulus is non-monotonic. For $p_0$ close to zero, the Young's modulus increases as $p_0$ increases. For higher $p_0$, increasing $p_0$ further decreases the Young's modulus towards zero at the transition point, after which the Young's modulus is zero. Interestingly, the Poisson's ratio begins negative for small $p_0$ and increases towards a value of 1 as $p_0$ increases, before remaining 1 in the compatible regime (Fig.~\\ref{fig:3}c). In the constrained case, the Poisson's ratio is actually lower than in the relaxed case for small $p_0$.\n\nThis phenomena highlights the counter-intuitive nature of VM mechanics. In classical elasticity $\\nu = 1$ corresponds to incompressible solids, commonly considered as very stiff. Here we find that the tissue approaches $\\nu = 1$ for higher values of $p_0$ corresponding to compatible tissue with floppy response. This seeming contradiction is resolved by noting that in this limit cells can accommodate rest area and perimeter simultaneously and therefore  upon deformation their area remains intact, just as in incompressible solids.\n\nThe simulations highlight the complex mechanical behaviour of the vertex model to applied tissue-level strains, both in its elastic moduli and the vertex-level non-affine deformations while relaxing the energy. The non-affine relaxation step allows the tissue to reduce its shear modulus and Young's modulus in the incompatible regime, and the bulk modulus in the compatible regime. However, the simulations do not give an intuitive understanding for why the bulk modulus is discontinuous, or why we can get a negative Poisson's ratio. Thus, in the remainder of the paper, we develop a mean-field theory of the vertex model that can account for non-affine relaxation of the tissue under strain to derive analytic expressions for the elastic moduli and understand the source of the complex phenomena mentioned above.\n\n\n\n\\section{Vertex model: mean-field theory and ground states}\n\\label{sec:MFT}\n\n\n\\subsection{Mean-field theory of vertex model}\nTo understand the numerical results, we construct a mean-field theory by assuming that all cells responds equally. In this case the tissue energy is just $E_{\\text{tissue}}=NE$ and one can simply consider the energy $E$ of a single cell. We work in dimensionless units by normalizing the energy with $K A_0^2$ and lengths with $\\sqrt{A_0}$. The  dimensionless single-cell energy is then given by\n\\begin{equation}\n   E = \\frac{1}{2}(a - 1)^2 + \\frac{1}{2}r(p - p_0)^2,\n\\end{equation}\nwhere $a = A \/ A_0$, $p=P\/\\sqrt{A_0}$, $r = \\Gamma \/ K A_0$ is the rigidity ratio, and $p_0 = P_0 \/ \\sqrt{A_0}$ is the target shape index of the cell. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{fig3.pdf}\n    \\caption{Shape parameterization of the vertex model and ground states. (a) Schematic of the vertex model and cell shape parametrization. Cells are defined by the lattice height $h$, width $w$, and angle between edges $\\phi$. (b) The ground state in the solid state is a regular hexagonal lattice, with $\\phi = 2 \\pi \/ 3$. (c) The ground state used in the soft state with $\\phi > 2 \\pi \/ 3$. (d - f) Cell area (d),  cell perimeter (e), and  edge tension (f) vs target shape index $p_0$ for various values of the rigidity ratio $r$. Dots represent simulated values, lines are the  analytical results.}\n    \\label{fig:3}\n\\end{figure}\n\nEach cell consists of horizontal edges of length $l_1$ and diagonal edges of length $l_2$, with  $\\phi$ the angle between horizontal and diagonal edges (Fig.~\\ref{fig:3}a). This parameterization captures the behavior of the tissue observed in our numerical simulations, where it is the angle between edges that changes during relaxation. Although cells have additional degrees of freedom, the description in terms of these three degrees of freedom is sufficient to capture the ground states of the tissue VM, and the response of the tissue under shear and bulk deformations in simulations (Fig.~\\ref{fig:2}). To both examine the ground states and the response to deformation, it is convenient to parametrize each cell in terms of the height $h$ and width $w$, as shown in Fig.~\\ref{fig:3}a, given by\n\\begin{equation}\n    h = 2 l_2 \\sin \\phi\\;,\\hspace{0.2in}\n w = l_1 - l_2 \\cos \\phi\\;.\n   \\label{eq:h-w}\n\\end{equation}\nEach cell then contributes  an area $a = w h$ to the tissue.  We stress that the angle $\\phi$ is distinct from the shear tilt angle $\\theta$ introduced previously in Ref.~\\cite{hernandez2022anomalous}, where cells may tilt or untilt in order to change their perimeter. While the shapes obtained from an initial regular hexagon by varying the shear tilt angle $\\theta$ correspond to affine deformations of the hexagon, those parametrized by $\\phi$ generally correspond to non-affine deformations of the regular hexagon.\nInverting Eqs.~(\\ref{eq:h-w}), we obtain\n\\begin{align}\n& l_1 = w + \\frac{1}{2} h \\cot \\phi\\;,\\\\\n& l_2 = \\frac{1}{2} h \\csc \\phi\\;.\n\\end{align}\nCell area and perimeter can then be written as\n\\begin{align}\n   & a = hw\\;,\\\\\n   & p = 2w + h f(\\phi)\\;,\n\\end{align}\nwhere \n\\begin{equation}\n  f(\\phi)=\\frac{ 2\n   + \\cos \\phi}{ \\sin\\phi}\\;,\n  \\label{eq:fphi}\n\\end{equation} \nresulting in an energy\n\\begin{equation}\n    E = \\frac{1}{2}(hw - 1)^2 + \\frac{1}{2}r(2w +  h f( \\phi) - p_0)^2\\;.\n    \\label{eq:Ephi}\n\\end{equation}\nThis form makes it evident that the VM energy is underconstrained as area and perimeter do not uniquely determine cell shape.\n\n\\subsection{Ground states}\n\nThe ground state configurations are obtained by minimizing the energy with respect to the cell width $w$, height $h$, and angle $\\phi$ and are solutions of the three coupled equations\n\\begin{eqnarray}\n\\label{eq:dEdphi}\n    \\frac{\\partial E}{\\partial \\phi} &=& h r f'(\\phi)(2 w + h f(\\phi) - p_0) = 0\\;,\\\\\n\\label{eq:dEdh}\n    \\frac{\\partial E}{\\partial h} &=& w(hw - 1) + rf(\\phi)(2 w + h f(\\phi) - p_0) = 0\\;,\\\\\n\\label{eq:dEdw}\n    \\frac{\\partial E}{\\partial w} &=& h(hw - 1) + 2r(2 w + h f(\\phi) - p_0) = 0\\;.\n\\end{eqnarray}\nAs shown in previous work, we find a transition at $p_0=p_*$ between two distinct states. For a regular lattice of $n$-sided polygons $p_*$ is given by the isoperimetric value $ p_*(n) = \\sqrt{4 n \\tan(\\pi \/ n)}$, with $p_*(6) = \\sqrt{8 \\sqrt{3}} \\approx 3.72$. The isoperimetric inequality $p\\geq p_*(n)$ provides a lower bound on the perimeter of a regular $n$-sided polygon for given area~\\cite{osserman1978isoperimetric}. For $p_0 > p_*(n)$ the cell is in a geometrically compatible regime, where both preferred area and perimeter may be achieved, and the tissue has zero shear modulus~\\cite{staple2010mechanics, murisic2015discrete, moshe2018geometric, huang2021shear} (Fig.~\\ref{fig:1}b). For $p_0 < p_*(n)$ the cell is in an incompatible regime, where both preferred area and perimeter cannot be simultaneously satisfied, and the tissue behaves like a solid by resisting shear deformation~\\cite{farhadifar2007influence, moshe2018geometric, merkel2019minimal, hernandez2022anomalous}. The corresponding ground state of the tissue is a lattice of identical hexagonal cells (Fig.~\\ref{fig:1}b). As $p_0$ is further lowered the cell may become unstable and collapse to zero area and perimeter (Fig.~\\ref{fig:1}b). Additionally, in a small range of parameters near the collapsing region more exotic ground states exist, with mixed lattices of square and octagonal, or dodecahedral and triangular cells providing lower energy than  hexagonal cells~\\cite{staple2010mechanics}.\n\n\\subsubsection{Compatible State, $p_0 > p_*(6)$}\n\nFor $p_0 > p_*(6)$ Eqs.~(\\ref{eq:dEdphi}) are identically solved by $hw = 1$ and $p=2w + hf(\\phi) = p_0$, and the zero ground state energy vanishes (Fig.~\\ref{fig:3}c-e). We refer to this situation as the compatible state. The ground state configuration is a family of 6-sided polygons parametrized by the angle $\\phi$, with\n\\begin{equation}\n    h = \\frac{p_0 \\pm \\sqrt{p_0^2 - 8 f(\\phi)}}{2f(\\phi)}\\;,\n    \\label{eq:h_comp}\n\\end{equation}\n\\begin{equation}\n    w = \\frac{p_0 \\mp \\sqrt{p_0^2 - 8 f(\\phi)}}{4}\\;,\n    \\label{eq:w_comp}\n\\end{equation}\nwhere both roots are acceptable solutions for a given value of $\\phi$, corresponding to either tall and thin or short and wide cells. It is evident from Eq.~\\eqref{eq:h_comp} and Eq.~\\eqref{eq:w_comp} that \nsuch a solution exists provided $p_0^2 \\ge 8 f(\\phi)$. The function $f(\\phi)$ has a minimum  at $\\phi = \\frac{2 \\pi}{3}$, with $f(\\frac{2 \\pi}{3})=\\sqrt{3}$ corresponding to $p_0 = 2^\\frac{3}{2} 3^\\frac{1}{4} = p_*(6)$. At this value of $p_0$ there is a single zero energy solution that corresponds to a hexagon of unit area. For $p_0>p_*$ there is degenerate continuum of zero energy solutions corresponding to deformed hexagons of unit area, perimeter $p_0$ and  $\\phi\\in\\left[\\frac{2 \\pi}{3},\\phi_m(p_0)\\right]$, with $\\phi_m$ determined by \n$p_0^2 = 8 f(\\phi_m)$ (Fig.~\\ref{fig:3}c). There exist many other parameterizations that can give ground state shapes in the compatible regime, for example, cells becoming tall and thin, cells decreasing the angle $\\phi$ to increase their perimeter, or cells tilting as in Ref.~\\cite{hernandez2022anomalous} (Fig.~\\ref{fig:3}b).\n\n\\subsubsection{Incompatible State, $p_0 < p_*(6)$}\n\nFor $p_0 < p_*$  the cell cannot simultaneously realize the target area and perimeter. We refer to this situation as the incompatible state. Eq.~\\ref{eq:dEdphi} requires $f'(\\phi) = 0$, with solution $\\phi = \\frac{2 \\pi}{3}$, such that $f(\\phi) = \\sqrt{3}$. An intuitive explanation for this fixed angle is that all edges are under identical tension, and so by force balance a junction of three edges must have equally spaced angles. Eqs.~\\eqref{eq:dEdh} and~\\eqref{eq:dEdw} then imply that $\\sqrt{3} h = 2 w$. This gives a perimeter $p = 4w$ and area $a = \\frac{2}{\\sqrt{3}} w^2= p^2 \/ 8 \\sqrt{3}$, which means that the cells are regular hexagons in the incompatible state (Fig.~\\ref{fig:3}b).\n\nWe can combine Eqs.~\\eqref{eq:dEdh} and~\\eqref{eq:dEdw} to obtain a cubic equation for the perimeter\n\\begin{equation}\n    p^3 + \\left(\\frac{r p_*^4 - 2 p_*^2}{2} \\right) p - \\frac{r p_*^4 }{2}p_0 = 0\\;,\n    \\label{eq:p_min}\n\\end{equation}\nwith $p_*^2=8\\sqrt{3}$ and $a=p^2\/p_*^2$. The cubic equation can be solved perturbatively in the limit of low and high rigidity ratio $r$.\n\nAt low rigidity ratio, i.e., $r \\ll 1 \/ p_*^2$, we find\n\\begin{align}\n    p &= p_* - p_*^2 \\left(\\frac{p_* - p_0}{4}\\right) r + \\mathcal{O}(r^2)\\;,\\\\\n    a &= 1 -p_* \\left(\\frac{p_* - p_0}{2}\\right)r + \\mathcal{O}(r^2)\\;.\n\\end{align}\nThe cell remains close in shape to a hexagon of unit area, with a reduction of the perimeter relative to the value $p_*$ (Fig.~\\ref{fig:3}d-e). The tension, given by $2 r (p - p_0) = 2 r \\left(p_* - p_0 \\right) + \\mathcal{O}(r^2)$, decreases monotonically as $p_0$ increases and vanishes at $p_0 = p_*$, where the cell reaches the compatible state and is under no tension  (Fig.~\\ref{fig:3}f). If $p_0$ becomes too small, the stable configuration collapses to a point with zero area and perimeter (Fig.~\\ref{fig:1}b).\n\nFor high rigidity ratio, i.e., $r \\gg 1 \/ p_*^2$, the perimeter and area can be expanded in inverse powers of $r$, with the result\n\\begin{align}\n    p &= p_0 + \\frac{2 p_0}{p_*^2} \\left(1 - \\frac{p_0^2}{p_*^2} \\right)\\frac{1}{r} + \\mathcal{O}\\left(\\frac{1}{r^2}\\right)\\;,\\\\\n    a &= \\frac{p_0^2}{p_*^2} + \\frac{4 p_0^2}{p_*^4} \\left(1 - \\frac{p_0^2}{p_*^2}\\right) \\frac{1}{r} + \\mathcal{O}\\left(\\frac{1}{r^2}\\right)\\;.\n\\end{align}\nIn this limit the cell shape index is close to the target shape index (Fig.~\\ref{fig:3}d-e). The tension $2 r (p - p_0) = \\frac{4 p_0}{p_*^2} \\left(1 - \\frac{p_0^2}{p_*^2} \\right) + \\mathcal{O}\\left(\\frac{1}{r}\\right)$ is nonmonotonic in $p_0$, increasing from almost no tension at $p_0 = 0$ before decreasing as $p_0$ approaches $p_*$ (Fig.~\\ref{fig:3}f).\nAs the rigidity ratio increases the tension saturates to the value  $\\frac{4 p_0}{p_*^2} \\left(1 - \\frac{p_0^2}{p_*^2} \\right)$, which has a maximum value of $2^\\frac{3}{2} 3^{-\\frac{7}{4}} \\approx 0.414$ at $p_0 = 8 \/ \\sqrt{3} \\approx 2.149$ (Fig.~\\ref{fig:3}d). For $p_0 \\leq 0$, the cell collapses to a point with zero area and perimeter (Fig.~\\ref{fig:3}b).\n\nFinally, we note that when topological transitions are allowed, tissues may also unjam and undergo a solid-to-liquid phase transition when cell rearrangements cost zero energy. In disordered realizations of the VM, the unjamming transition occurs at $p_0 \\approx 3.81$, a value close to, but slightly larger than $p_*(5)$~\\cite{bi2015density}. Intuitively, for the tissue to rearrange in a T1 transition with zero energy barrier, two hexagonal cells must momentarily lose an edge and become pentagons while still maintaining their preferred perimeter and area. Cell motility can further promote fluidity and lower the transition point ~\\cite{bi2016motility}.\n\n\\section{Mechanical response of the vertex model}\n\\label{sec:mechanics}\n\n\\subsection{Deformations protocol}\n\nIt is evident from Eq.~(\\ref{eq:Ephi}) that area and perimeter (or equivalently height and width of the box shown in Fig.~\\ref{fig:3}a) do not uniquely specify a polygonal shape. In the compatible regime there is a family of zero energy shapes corresponding to either tilted polygonal shapes obtained by affine deformations or non-affinely deformed polygons  parametrized by the angle $\\phi$. In the incompatible regime, if only affine deformations are allowed, both the ground state and each deformed state are unique for fixed area and perimeter. Allowing non-affine deformations introduces, however, additional degrees of freedom that can lower the energy for a given set of parameters.\n\nIn the following we examine the response of a tissue initially in a ground state to an externally imposed  strain. The deformation is imposed globally on the tissue by changing the shape of the bounding box. Such a deformation uniformly changes the shape of the cells and generally results in a state where individual vertices are no longer force balanced (Fig.~\\ref{fig:2}a, middle). We will refer to this state as the ``constrained'' deformed state. Due to the presence of hidden degrees of freedom the system can, however, lower its energy and relax a state of local force balance. In the compatible regime this relaxation can occur via motion of the vertices that corresponds to non-affine deformations (Fig.~\\ref{fig:1}a, right). In the compatible regime the relaxation can occur either via non-affine deformations with fixed box shape or through a global tilting of the tissue, which entails affine cell deformations, as in Hernandez et al.~\\cite{hernandez2022anomalous}. The elastic constants measured in the ``relaxed'' state of the compatible regime are the same for the two relaxation protocols.\n\n\nOperationally, constrained deformations are achieved by first fixing either the cell height, width, or both, and then transforming the vertices according to the given deformation, as done in Staple \\emph{et al.}~\\cite{staple2010mechanics}. We prevent spontaneous tilting of the tissue, which can be used to soften the mechanical response using only affine deformations in the compatible regime~\\cite{hernandez2022anomalous}.\n\nWe next evaluate the various elastic moduli of the vertex model. As we will see below, a new result of our work is that in the incompatible regime cells can find new deformed states by relaxing through non-affine deformations (Fig.~\\ref{fig:2}), resulting in a softer response than obtained in previous studies~\\cite{staple2010mechanics} (Fig.~\\ref{fig:4}).\n\n\\subsection{Shear Modulus}\n\nTo calculate the shear modulus of the tissue, we apply an area-preserving pure shear deformation, corresponding to $w \\rightarrow w (1 + \\epsilon \/ 2)$ and $h \\rightarrow h (1 + \\epsilon \/ 2)^{-1}$, with $\\epsilon$ the strain. We allow for a non-affine deformation to relax the tissue by minimising energy with respect to the angle $\\phi$ for each value of strain (Fig.~\\ref{fig:3}). The shear modulus is defined as\n\\begin{equation}\n    G = \\frac{1}{a} \\left. \\frac{\\partial^2 }{\\partial \\epsilon^2}\\left( \\min_\\phi E\\right) \\right |_{\\epsilon=0}\\;,\n    \\label{eq:shear}\n\\end{equation}\nwhere the factor of $4$ accounts for the fact that we are considering pure shear instead of simple shear deformations. As area is preserved under pure shear, we  only need to consider the energy cost due to changes in perimeter.\n\n\\subsubsection{Compatible State, $p_0 > p_*$}\n\nIn the compatible case, cells can accommodate shear and maintain their area and perimeter at the target values $a=1$ and $p=p_0$ by changing shape, i.e., by adjusting the angle $\\phi$ to a value other than $2\\pi\/3$. The perimeter of the deformed call is given by $p(\\epsilon,\\phi)=2w(1+\\epsilon \/ 2)+hf(\\phi)\/(1+\\epsilon \/ 2)$. The cell can maintain $p=p_0$ by deforming to a new compatible ground state corresponding to an angle $\\phi^*$ given by the solution of  $p(\\epsilon,\\phi^*)=p_0$. Clearly the energy remains zero, demonstrating that the shear deformation cost no energy and \n\\begin{equation}\n    G = 0\n\\end{equation}\nfor all rigidity ratios and $p_0 > p_*$ (Fig.~\\ref{fig:5}a). This of course only holds up to a maximum value of strain determined by the angle $\\phi_m(p_0)$. Beyond this value the fluid-like compatible tissue stiffens and acquires a finite shear modulus, as mentioned by \\cite{farhadifar2009dynamics}, and recently by~\\cite{huang2021shear}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{fig5.pdf}\n    \\caption{Elastic moduli of the vertex model. (a) Shear modulus, (b) bulk modulus, (c) Young's modulus, and (d) Poisson's ratio against target shape index $p_0$ and rigidity ratio $r$.}\n    \\label{fig:5}\n\\end{figure}\n\n\\subsubsection{Incompatible Case, $p_0 < p_*$}\n\nIn the incompatible case cell edges are under uniform tension. By force balance, this implies that the angle between edges remains $\\phi = \\frac{2 \\pi}{3}$ even under small deformations at the tissue scale. The ground state configuration is a regular hexagon  with perimeter $p = 2w + \\sqrt{3}h$. Using Eq.~\\eqref{eq:shear} and the relations for height and width in terms of the perimeter, $h = \\frac{1}{2\\sqrt{3}}p$ and $w = \\frac{1}{4}p$, we obtain\n\\begin{equation}\n    G =\\frac{r(p - p_0)p}{4a}\\;.\n\\end{equation}\nIn the rigid, incompatible state the shear modulus is a monotonically increasing function of $r$ and vanishes at the transition $p_0 = p_*$ (Fig.~\\ref{fig:4}a, Fig.~\\ref{fig:5}a), in agreement with earlier results~\\cite{murisic2015discrete}. The non-affine deformations of the relaxed tissue allow for a softer response of the tissue, with the shear modulus being a factor of $3\/2$ stiffer when only considering vertices constrained by the affine shear strain~\\cite{farhadifar2009dynamics, staple2010mechanics}, as confirmed by simulations (Fig.~\\ref{fig:4}a).\n\nFor small rigidity ratio ($r \\ll 1 \/ p_*^2$), we can expand $G$ in powers of $r$, with the result\n\\begin{equation}\nG =\\frac{1}{4}p_{*}(p_{*}-p_0) r + \\mathcal{O}(r^2)\\;.\n\\end{equation}\nThe opposite limit of large rigidity ratio ($r \\gg 1 \/ p_*^2$)  yields\n\\begin{equation}\nG=\\frac{1}{2}\\left(1-\\frac{p_0^2}{p_*^2}\\right)-\\frac{1}{p_*^6}\\left(p_*^2-p_0^2\\right)^2\\frac{1}{r} + \\mathcal{O}\\left(\\frac{1}{r^2}\\right)\\;.\n\\end{equation}\n\n\\subsection{Bulk Modulus}\n\nTo calculate the bulk modulus, we change the area $a \\rightarrow a(1 + \\epsilon)$ by rescaling the height $h \\rightarrow h(1 + \\epsilon)^\\frac{1}{2}$ and width $w \\rightarrow w(1 + \\epsilon)^\\frac{1}{2}$, and allow the angle $\\phi$ to vary to minimize the deformation energy. The bulk modulus is then given by\n\\begin{equation}\n    K = \\frac{1}{a}\\left.\\frac{\\partial^2 }{\\partial \\epsilon^2}\\left( \\min_{\\phi} E\\right)\\right|_{\\epsilon=0}\\;,\n\\end{equation}\nwith\n\\begin{equation}\n\\frac{\\partial^2 E}{\\partial \\epsilon^2} = a^2 + r(p - p_0)\\frac{\\partial^2 p}{\\partial \\epsilon^2} + r\\left(\\frac{\\partial p}{\\partial \\epsilon}\\right)^2\\;.\n\\end{equation}\nTo evaluate this expression we need to consider separately the compatible and incompatible states.\n\n\\subsubsection{Compatible state, $p_0>p_*$}\nWe have previously shown that in the compatible case the angle $\\phi$ can adjust to maintain a fixed cell perimeter under small deformations. Thus $\\left. \\frac{\\partial p}{\\partial \\epsilon} \\right|_{\\epsilon=0} = 0$ and $\\left. \\frac{\\partial^2 p}{\\partial \\epsilon^2} \\right|_{\\epsilon=0} = 0$, and the bulk modulus is simply \n\\begin{equation}\n    K = 1\n\\end{equation}\nfor all $r$ and $p_0 > p_*$ (Fig.~\\ref{fig:4}b, Fig.~\\ref{fig:4}b).\n\nBy contrast, if we allow for only affine deformations then the perimeter expands isotropically $p(\\epsilon) = (1 + \\epsilon)^\\frac{1}{2} p(0)$. In this case $\\left. \\frac{\\partial p}{\\partial \\epsilon} \\right|_{\\epsilon=0} = \\frac{1}{2}p(0)$ and $\\left. \\frac{\\partial^2 p}{\\partial \\epsilon^2} \\right|_{\\epsilon=0} = -\\frac{1}{4}p(0)$, giving a bulk modulus equal to\n\\begin{equation}\n    K_{\\text{affine}} = 1 + \\frac{1}{4} r p_0^2,\n\\end{equation}\nwhich can be significantly higher than the non-affine result for high $r$, emphasising the need to consider non-affine displacements (Fig.~\\ref{fig:2}b).\n\n\\subsubsection{Incompatible State, $p_0<p_*$}\n\nIn the incompatible state for $p_0 < p_*$  the angle that minimizes energy remains $\\phi = \\frac{2 \\pi}{3}$ for small perturbations to cell height and width to ensure tension balance  at the cell vertices. The cell then expands isotropically, such that $p(\\epsilon) = (1 + \\epsilon)^\\frac{1}{2} p(0)$, resulting in a bulk modulus\n\n\\begin{equation}\n    K = a + \\frac{1}{4a}r p p_0\\:\n\\end{equation}\nshown in Fig.~\\ref{fig:4}b and Fig.~\\ref{fig:5}b. We note that as $p_0$ approaches the critical value $p_*(6)$ from below,  the bulk modulus has the value $\\rm{lim}_{p_0\\rightarrow p_*^-}K = 1 + \\frac{1}{4}r p_*^2$. On the other hand, in the compatible regime $K = 1$. Thus the bulk modulus exhibits a jump discontinuity at the critical point separating compatible and incompatible states. In contrast, if vertex positions are fixed by uniform dilation without relaxation, the bulk modulus is continuous and higher in the compatible region (Fig.~\\ref{fig:4}b).\n\nIn the limit of low rigidity ratio  ($r \\ll 1 \/ p_*^2$), we find\n\\begin{equation}\n    K = 1 + \\frac{1}{4}p_* (3 p_0 - 2 p_*) r + \\mathcal{O}(r^2)\\;.\n\\end{equation}\nThe bulk modulus increases with $p_0$ up to the critical value, at which point it discontinuously jumps to $1$ for all $p_0 > p_*$, independent of $r$. Interestingly, for $p_0 < \\frac{2}{3} p_*$ the bulk modulus of the incompatible solid is lower than that of the compatible fluid, suggesting that contractility can actually reduce the bulk stiffness of the tissue. Additionally, increasing the rigidity ratio further reduces the bulk modulus for low $p_0$.\n\nIn the limit of high rigidity ratio ($r \\gg 1 \/ p_*^2$), we find\n\\begin{equation}\n    K = \\frac{1}{4} p_*^2 r + \\left(\\frac{3}{2} \\frac{p_0^2}{p_*^2} - \\frac{1}{2} \\right) + \\mathcal{O}\\left(\\frac{1}{r}\\right)\\;.\n\\end{equation}\nThus the bulk modulus increases with the rigidity ratio.\n\nWe find that for low $p_0$ the bulk modulus can be significantly lower in the rigid than in the fluid state, which can affect the rate of spreading monolayers~\\cite{staddon2022interplay}. This is due to the fact that at small  $p_0$ cells have a smaller area while the energy required to deform the cell is proportional to the square of the area change. Therefore it costs more energy to strain a single cell than to strain two cells with half the area, similar to the reduction in effective stiffness obtained when springs are placed in series.\n\nThe bulk modulus is also a non-monotonic function of  the rigidity ratio. For small $r$ the bulk modulus decreases with $r$ as the size of the cell decreases, reaching a minimum near $r = 2 \/ p_*^2 \\approx 0.144$ before increasing linearly in $r$ for high $r$, due to the growing contribution from the perimeter elasticity.\n\n\\subsection{Young's Modulus and Poisson's Ratio}\n\nNext we calculate the Young's modulus and Poisson's ratio (Fig.~\\ref{fig:2}c) by stretching the width of the cell $w \\rightarrow w ( 1 + \\epsilon)$ while allowing the cell height $h$ and angle $\\phi$ free to minimize the energy. The Young's modulus is defined as \n\\begin{equation}\n    Y = \\frac{1}{a} \\left. \\frac{\\partial^2 }{\\partial \\epsilon^2}\\left(\\min_{h,\\phi}  E\\right) \\right|_{\\epsilon = 0}\n\\end{equation}\nand the Poisson's ratio as\n\\begin{equation}\n    \\nu = -\\frac{1}{h} \\frac{\\partial h}{\\partial \\epsilon}= -\\frac{w}{h} \\frac{\\partial h}{\\partial w}.\n\\end{equation}\nTo evaluate $Y$ and $\\nu$ we use the relationship between the linear elastic constants, $Y = \\frac{4KG}{K + G}$ and $\\nu = \\frac{K - G}{K + G}$, which have been shown to hold away from the critical point.\n\n\\subsubsection{Compatible state, $p_0 > p_*$}\n\nIn the compatible state, the ground state degeneracy allows the cell to achieve the target shape index and area for small strain by reducing cell height and finding values of the angle $\\phi$ different from $2\\pi\/3$, i.e., by changing its shape, with no energetic cost. As a result for $p_0 > p_*$ we find\n\\begin{equation}\n    Y = 0\\;,~~~~~\n    \\nu = 1\\;.\n\\end{equation}\nfor all $r$ (Fig.~\\ref{fig:4}c-d, Fig.~\\ref{fig:5}c-d).\n\n\\subsubsection{Incompatible case, $p_0 < p_*$}\n\nIt has been shown that away from the critical point the elastic constants of the VM satisfy the familiar relation of linear elasticity of isotropic solids~\\cite{hernandez2022anomalous}. We can therefore use the relations $Y = \\frac{4KG}{K + G}$ and $\\nu = \\frac{K - G}{K + G}$ to evaluate $Y$ and $\\nu$ in the incompatible regime, with the result\n\\begin{align}\n    Y &=(p-p_0) \\frac{rp\\left(4a^2 + rp p_0 \\right)}{a(4a^2 + rp^2)}\n    \\;,\\\\\n    \\nu &= 1-\\frac{2rp(p-p_0)}{4a^2+rp^2}\n    \\;.\n\\end{align}\nWe find that the Young's modulus is a non-monotonic function of both target shape index and rigidity ratio (Fig.~\\ref{fig:4}c, Fig.~\\ref{fig:5}c). The nonmonotonicity with $p_0$ is most pronounced at intermediate values of $r$, where at small $p_0$ the Young's modulus increases, rather than decrease, with increasing $p_0$.\n\nAt both high and low rigidity ratio, the Poisson's ratio remains close to $1$ for all $p_0$, indicating that the cell preserves its area under deformations (Fig.~\\ref{fig:4}d, Fig.~\\ref{fig:5}d). At intermediate values of the rigidity ratio and small $p_0$, the nonmonotonicity of the Young's modulus results in a negative Poisson's ratio, which indicates that  a tissue stretched in the $x$-direction, also expands in the $y$-direction.\n\nIn comparison to the constrained response, we find that the relaxed response gives a softer Young's modulus for all rigity ratio and target shape index values, since the shear modulus is also softer by a factor of $\\frac{2}{3}$ (Fig.~\\ref{fig:4}c). Similarly, the Poisson's ratio is higher in the comaptible state for all $p_0 < p_*$ (Fig.~\\ref{fig:4}d).\n\nIn the limit of low rigidity ratio ($r \\ll 1 \/ p_*^2$), the Young's modulus and Poisson's ratio are given by\n\\begin{align}\n    Y &= p_* (p_* - p_0) r + \\mathcal{O}(r^2)\\;,\\\\\n    \\nu &= 1 + \\frac{1}{2} p_* (p_0 - p_*) r + \\mathcal{O}(r^2)\\;,\n\\end{align}\nshowing that when  $p_0$ is increased towards the critical point from the solid side ($p_0\\rightarrow p_*^-$) the Young's modulus vanishes and the Poisson's ratio increases towards $1$.\n\nIn the limit of high rigidity ratio ($r \\ll 1 \/ p_*^2$), we obtain approximate expression by expanding in $1\/r$ as\n\\begin{align}\n    Y &= 2 \\left(\\frac{p_*^2 - p_0^2}{p_*^2}\\right) + \\mathcal{O}\\left(\\frac{1}{r}\\right)\\;,\\\\\n    \\nu &= 1 -\\frac{4 (p_*^2 - p_0^2)}{p_*^4}\\frac{1}{r} + \\mathcal{O}\\left(\\frac{1}{r^2}\\right)\\;.\n\\end{align}\nThe Young's modulus also has a maximum value of $2$ at $p_0=0$.\n\n\\subsection{Origin of negative Poisson's ratio}\n\nWe can understand why certain cell parameters give a positive or negative Poisson's ratio by looking at how the energy gradient with respect to cell height changes as we change the cell width. The gradient $\\frac{\\partial E}{\\partial h}$ can be thought of as the effective force acting on the height of the cell, given by\n\\begin{equation}\n    \\frac{\\partial E}{\\partial h} = w(hw - 1) + \\sqrt{3}r(2w + \\sqrt{3} h - p_0).\n\\end{equation}\nIn the ground state, this will be zero. Then, if we vary the cell width but keep the height fixed we can measure the change in force as\n\\begin{equation}\n    \\frac{\\partial^2 E}{\\partial w \\partial h} = 2 h w - 1 + 2 \\sqrt{3} r.\n\\end{equation}\nWhen this value is positive, then as cell width is increased, the effective force acting on cell height increases and so the cell height will decrease as it relaxes to the energy minimum. Consequently, the sign of this value is the same sign as the Poisson's ratio.\n\nThe second term $2\\sqrt{3}r$ comes from the perimeter contribution in the VM and accounts for energy changes due to perimeter elasticity. Since $p \\geq p_0$, the cell is under tension and the perimeter term aims to shrink the cell. Increasing the width further increases the perimeter and so tension increases, providing more force to shrink the cell. Thus, the perimeter elasticity always acts to shrink the cell and contributes to a positive Poisson's ratio.\n\nThe first term, $2hw - 1 = 2a - 1$, represents the energy change due to area elasticity. We can write this as $2w (h - 1 \/ 2w)$, which we can think of as an spring like force with stiffness $2w$ and target height $1\/2w$. As width increases, the height becomes closer to the target height, reducing the strain. At the same time, the effective stiffness $2w$ increases, increasing the pressure. Thus there is a trade off between less restoring force on the cell area versus increased effectiveness of changes in cell height. The net effect on whether this increases or decreases the perimeter depends on the size of the cell area: when cell area $a < \\frac{1}{2}$ the area term acts to increase cell height when width is increased, and for $a > \\frac{1}{2}$ the area term reduces the cell height.\n\nFor high rigidity ratio, the perimeter term dominates and so an increase in cell width leads to a reduction in cell height. For low rigidity ratio, the area term dominates and the cell area is close to $1$ (Fig.~\\ref{fig:3}d), thus an increase in cell width reduces the area pressure and cell height decreases. However, in the intermediate regime for low $p_0$ cell area is small, meaning the area term acts to increase cell width, and the area contribution and perimeter contributions are of comparable size, resulting in negative Poisson's ratio.\n\nWe can calculate the transition to a negative Poisson's ratio exactly at  $p_0 = 0$, which corresponds to the situation where the contribution to cell edge tension from cortical contractility and cell-cell adhesion precisely balance. In this limit the equation for the ground state perimeter, Eq.~\\ref{eq:p_min}, becomes\n\\begin{equation}\n    p \\left(p^2 + \\frac{r p_*^4 - 2 p_*^2}{2} \\right) = 0\n\\end{equation}\nwith  solution \n\\begin{align}\np &= p_* \\sqrt{1 - \\frac{1}{2} r p_*^2}\\;.\\\\\n    a &= \\left(1 - \\frac{1}{2}r p_*^2 \\right)\n\\end{align}\nfor $r < 2 \/ p_*^2 \\approx 0.144$. For $r > 2 \/ p_*^2$ the cell is unstable and collapses to zero area. We can calculate whether cell height increases or decreases when width is increased by calculating how the effective force on the height, $\\frac{\\partial E}{\\partial h}$, changes with width. Substituting our formula for area we find\n\\begin{equation}\n    \\frac{\\partial^2 E}{\\partial h \\partial w} = 1 - \\frac{3}{4} r p_*^2= 1 - 6\\sqrt{3} r\n\\end{equation}\nwhere we have used $p_*^2 = 8 \\sqrt{3}$. Thus for $r > \\frac{4}{3}p_*^2 \\approx 0.096$ and $p_0 = 0$ the tissue has a negative Poisson's ratio. \n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\nIn this paper we studied the linear response of the 2D vertex model by calculating the shear modulus, bulk modulus, Young's modulus and Poisson's ratio, using a mean-field approach that allows for a class of non-affine deformations, which agree well with numerical simulations. We also provide approximate expressions in the limit of high and low rigidity ratio. Our calculations match previous results showing a rigidity transition controlled by purely geometric effects and tuned by the target shape index $p_0$.\n\nFor $p_0 < p_* = 3.772$, the ground state configuration of the tissue is a regular hexagonal lattice of cells with area and perimeter different from their target value. In this regime, referred to as the incompatible, the tissue is rigid, with a gapped or prestressed ground state and finite shear modulus.\n\nFor $p_0 > p_*$, the cells can attain their target area and shape index and there is a continuum manifold of degenerate zero energy states corresponding to deformed hexagons and parametrized by the angle $\\phi$ between adjacent sides of the $6$-sided polygon (Fig.~\\ref{fig:3}a), with $\\phi\\in[2\\pi\/3,\\phi_m(p_0)]$. The tissue is soft because it can accommodate shear deformations with no energy cost by simply finding a new (deformed) zero energy state, i.e., by changing its shape. For $\\phi>\\phi_m(p_0)$ the tissue stiffens and acquires a finite shear modulus. The angle $\\phi_m$ can therefore be interpreted as the threshold of shear-induced stiffening of the liquid state, as first mentioned by \\cite{farhadifar2009dynamics}, and recently shown by \\cite{huang2021shear}. \n\nThe linear response of the tissue to a uniaxial strain reveals a subtle behavior. In the compatible regime the tissue has vanishing Young's modulus and Poisson's ratio is $\\nu = 1$, indicating that the tissue area is conserved. It is important to note, however, that the linear elastic response depends on the protocol of the deformation: the Young's modulus indeed vanishes if the tissue is allowed to change its shape in response to a uniaxial strain, but if the tissue is constrained to retain its shape when stretched uniaxially, then one would measure a finite Young's modulus.\n\nIn the incompatible regime, the Poisson's ratio can become negative at small $p_0$ and intermediate values of the rigidity ratio. This arises because for small cell area upon stretching in one direction the cell has to increase its height in the other direction to maintain its area fixed to the target value.\n \nThe bulk modulus of the tissue has a jump discontinuity at the transition point $p_0=p_*$, where it changes from a larger value in the solid state to a smaller value in the soft state. This discontinuity arises again because cells in the compatible regime can change shape to to maintain their perimeter under small changes in area. In contrast, in the incompatible regime cells must increase their perimeter away from the preferred value, resulting in an increase in energy. We also find that in the incompatible regime, the bulk modulus can decrease below $1$ for small rigidity ratio. This indicates that cell contractility can reduce the stiffness of the tissue, resulting in a larger bulk modulus in the ``soft'' phase than in the ``solid'' phase.\n\nOur results highlight the complex linear elastic behaviour that can arise from the simplest version of the vertex model due to its underconstrained nature. It would be interesting to extend our calculations to the case of disordered cell networks. Additionally, we highlight the importance of allowing for unconstrained degrees of freedom, in this case non-affine deformations, to relax the system and give a softer mechanical response to strain.\n\n\nIn conclusion, our work demonstrates that the vertex model, thought of as a collection of geometric constraints rather than a reference ground state structure, can engender interesting linear mechanical responses. The linear response exhibits a strong non-affine contribution under uniaxial compression and shear, as well as a negative Poisson's ratio. Typically, these two phenomena in crystalline solids require special lattice constructions, whereas in the vertex model exotic mechanical response can be achieved by tuning the relative competition between area and perimeter constraints via $r$ and geometric compatibility via $p_0$.\n\n\\section{Acknowledgements}\nM.C.M. and A.H. were supported by the National Science Foundation Grant No.~DMR-2041459. M.M. was supported by the Israel Science Foundation grant No. 1441\/19. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec1}\n\\IEEEPARstart{G}{enerative} adversarial network (GAN)~\\cite{goodfellow2014generative}, is an algorithmic framework that synthesizes a target data distribution from a given prior distribution. GAN has attracted great attention by achieving great success for various applications including text-to-image translation~\\cite{reed2016generative, hong2018inferring}, image-to-image translation~\\cite{isola2017image, choi2018stargan, zhu2017unpaired}, and image inpainting~\\cite{yu2018free, sagong2019pepsi, shin2020pepsi++}. In general, GAN is composed of two different networks called generator and discriminator. The goal of the generator is to produce the real data distribution to fool the discriminator, whereas the discriminator is trained to classify the real sample from the fake one which is synthesized by the generator. Since the goals of the generator and discriminator are opposed to each other, GAN is difficult to train stably compared with the supervised learning-based convolutional neural networks (CNN). \n\nThe recent studies observed that the sharp gradient space of the discriminator is the major reason for the unstable problem in the GAN training~\\cite{wu2021gradient}. To resolve this problem, some papers~\\cite{miyato2018spectral, gulrajani2017improved, kodali2017convergence, zhang2019consistency} introduced regularization or normalization techniques that inhibit the discriminator from making the sharp gradient space. As the regularization method, the conventional methods~\\cite{gulrajani2017improved, wu2019generalization, wei2018improving, kodali2017convergence, roth2017stabilizing} usually added the regularization term into the adversarial loss function. Among the various regularization approaches, gradient penalty-based regularization methods~\\cite{gulrajani2017improved, wu2019generalization, wei2018improving}, which add the gradient norm as the penalty term, are the most popular techniques and have been widely used. The normalization methods~~\\cite{miyato2018spectral, arjovsky2017wasserstein, kurach2019large} also have been widely studied. Spectral normalization (SN)~\\cite{miyato2018spectral} is the most popular one that controls the Lipschitz constraint on the discriminator by constraining the spectral norm of each layer. Since these regularization and normalization methods can lead the stable GAN training, the recent studies usually employ these techniques when training their networks. \n\nSome researchers studied architectural modules to improve the GAN performance. Miyato~\\textit{et al.}\\@ ~\\cite{miyato2018cgans} introduced a conditional projection technique that provides conditional information to the discriminator by projecting the conditional weight vector to the feature vector. This method significantly boosts the performance of the conditional GAN (cGAN) scheme. Zhang~\\textit{et al.}\\@ ~\\cite{zhang2019self} introduced a self-attention module which guides the generator and discriminator where to attend. Yeo~\\textit{et al.}\\@ ~\\cite{yeo2021simple} introduced a simple yet effective technique, called a cascading rejection module, which produces dynamic features in an iterative manner at the last layer of the discriminator. Park~\\textit{et al.}\\@ ~\\cite{park2021GRB} introduced a generative residual block that not only emphasizes the essential feature but also suppresses the unnecessary one in the side-residual path. On the other hand, some papers investigated new forms of convolution layer to improve the GAN performance. Park~\\textit{et al.}\\@ ~\\cite{park2021generative} proposed a perturbed convolution that prevents the discriminator from falling into the overfitting problem. However, since PConv is developed for the discriminator, it is hard to apply to the generator. Sagong~\\textit{et al.}\\@ ~\\cite{sagong2019cgans} introduced a conditional convolution layer for the generator, which incorporates the conditional information into the convolution operation. Although this method shows the fine performance on cGAN, it has a limitation: this method only can be used for cGAN scheme since it is designed for replacing the conditional batch normalization. Recently, Park~\\textit{et al.}\\@ ~\\cite{park2021generative_conv} proposed a novel convolution layer, called a generative convolution, which controls the convolution weight of the generator according to the given latent vector. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\linewidth]{fig1.pdf}\n\\caption{The overall generator architectures and residual blocks utilized in SNGAN~\\cite{miyato2018spectral, miyato2018cgans} and BigGAN~\\cite{brock2018large}. (a) Generator architecture for SNGAN, (b) Generator architecture for BigGAN, (c) Residual block for SNGAN, (d) Residual block for BigGAN.}\n\\label{fig:fig1}\n\\vspace{-0cm}\n\\end{figure}\n\nThere have been several attempts to newly design the generator and discriminator architectures for boosting the GAN performance. Specifically, in the early stage of the GAN study, Radford~\\textit{et al.}\\@ ~\\cite{radford2015unsupervised} designed the generator and discriminator architectures by stacking the transposed convolution layer and standard convolution layer in series, respectively. These network architectures are simple to build, but generate the low-quality image. Karras~\\textit{et al.}\\@ ~\\cite{karras2017progressive} introduced the GAN with novel training strategy, called a progressive growing skill, which effectively leads the generator and discriminator to produce high-resolution images. Recently, Miyato~\\textit{et al.}\\@ ~\\cite{miyato2018cgans, miyato2018spectral} pberoposed a spectrally normalized GAN~(SNGAN) that build generator and discriminator by stacking multiple residual blocks~\\cite{he2016deep}. To further improve the SNGAN performance, Brock~\\textit{et al.}\\@ ~\\cite{brock2018large} proposed a new generative model, called BigGAN, which has SNGAN-like structure in which noise is additionally input to the residual block. Since SNGAN and BigGAN are easy to implement and train stably, most recent studies employ these networks as their baseline networks. However, even though SNGAN and BigGAN exhibit fine GAN performance, we believe that designing a novel generator and discriminator architectures for improving the GAN performance still remains an open question. Therefore, this brief mainly focuses on developing the novel generator architecture. \n\nA common characteristic of generators in SNGAN and BigGAN is that they are built by stacking multiple residual blocks in series. These generators produce images in a coarse-to-fine manner, where the earlier blocks provide a rough template and the latter ones clarify the details. However, as pointed out in~\\cite{huang2017densely}, even  the residual block is used, the information flow in the generator may be impeded. That means it may be difficult to impart coarse information in the earlier blocks to the latter ones. To alleviate this issue, this brief proposes a novel generator architecture that embraces the strength point of the conventional methods as well as further improves the GAN performance. The proposed method consists of two different branches: the main and auxiliary branches. The main branch learns features by passing through the multiple residual blocks, whereas the auxiliary branch propagates the coarse information in the earlier residual block to the later one. In other words, the goal of the auxiliary branch is to assist the information flow in the generator. In order to combine the features in both branches effectively, we introduce a gated feature fusion module that controls the information flow in those branches. To demonstrate the superiority of the proposed method, this brief presents extensive experimental results using various standard datasets such as CIFAR-10~\\cite{krizhevsky2009learning}, CIFAR-100~\\cite{krizhevsky2009learning}, LSUN~\\cite{yu15lsun}, CelebA-HQ~\\cite{karras2017progressive}, AFHQ~\\cite{choi2020stargan},  and tiny-ImageNet~\\cite{23deng2009imagenet, yao2015tiny}. Furthermore, we provided various ablation studies to demonstrate the generalization ability of the proposed method. Quantitative evaluations show that the proposed method significantly improves GAN performance in terms of Inception score (IS) and Frechet inception distance (FID).  \n\n\\begin{table}[t]\n\\caption{Comparison of the dense block with residual one on the CIFAR-10 and CIFAR-100 datasets in terms of IS and FID. The best results are in bold.}\n\\begin{center}\n\\begin{tabular}{c | c | c | c | c | c | c | c}\n\\hline\n\\multicolumn{2}{c|}{Block name} & \\multicolumn{2}{c|}{Residual} & \\multicolumn{2}{c|}{Residual*} & \\multicolumn{2}{c}{Dense} \\\\\n\\hline\n\\multicolumn{2}{c|}{Parameters} & \\multicolumn{2}{c|}{3.78M} & \\multicolumn{2}{c|}{4.74M} & \\multicolumn{2}{c}{4.74M}\\\\ \n\\hline\nDataset & & IS$\\uparrow$ & FID$\\downarrow$ & IS$\\uparrow$ & FID$\\downarrow$ & IS$\\uparrow$ & FID$\\downarrow$ \\\\\n\\hline\n\\multirow{5}*{CIFAR-10} & trial 1 & 7.70 & 13.98 & 7.74 & 12.93 & 7.96 & 12.41 \\\\\n & trial 2 & 7.80 & 13.69 & 7.68 & 13.34 & 7.95 & 12.20 \\\\\n & trial 3 & 7.87 & 13.09 & 7.79 & 12.93 & 7.92 & 12.04 \\\\\n\\cline{2-8}\n & Mean & 7.79 & 13.58 & 7.74 & 13.07 & \\textbf{7.94} & \\textbf{12.22} \\\\\n\\cline{2-8} \n & Std & 0.08 & 0.45 & 0.05 & 0.24 & \\textbf{0.02} & \\textbf{0.18} \\\\\n\\hline \\hline\n\\multirow{5}*{CIFAR-100} & trial 1 & 7.99 & 17.45 & 7.99 & 17.22 & 8.22 & 15.35 \\\\\n & trial 2 & 7.94 & 17.72 & 7.97 & 16.78 & 8.07 & 16.74 \\\\\n & trial 3 & 7.94 & 17.57 & 8.02 & 16.86 & 8.15 & 15.73 \\\\\n\\cline{2-8}\n & Mean & 7.96 & 17.58 & 8.00 & 16.95 & \\textbf{8.15} & \\textbf{15.94} \\\\\n\\cline{2-8} \n & Std & 0.03 & 0.13 & 0.03 & 0.24 & \\textbf{0.08} & \\textbf{0.72} \\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{table1}\n\\end{table}\nKey contributions of our brief are as follows: First, we propose a novel generator architecture that contains the auxiliary branch and gated feature fusion module, which support the information flow in the generator. Second, with slight additional hardware costs, the proposed method significantly improves the GAN and conditional GAN performances on various datasets in terms of FID and IS scores. For instance, the proposed method improves FID and IS scores on the tiny-ImageNet dataset from 35.13 to 25.00 and 20.23 to 25.57, respectively.  \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.99\\linewidth]{fig2.pdf}\n\\caption{The overall generator architecture of the proposed method. The proposed method contains two different branches: main branch and auxiliary one. }\n\\label{fig:fig2}\n\\vspace{-0cm}\n\\end{figure*}\n\n\\section{Background}\n\\label{sec2}\n\\subsection{Generative Adversarial Network}\n\\label{sec2.1}\nGAN~\\cite{goodfellow2014generative} is a min-max game between two different networks called generator $G$ and discriminator $D$. $G$ is trained to generate visually pleasing images, whereas $D$ is optimized to classify the generated images from real ones. This training process can be formulated as follows:\n\\begin{eqnarray}\n\\label{eq1:ganD}\n       \\lefteqn{L_D = - E_{x\\sim P_\\textrm{data}(x)}[\\log D(x)]}\\nonumber\\\\\n    & {\\qquad \\qquad \\qquad} - E_{z\\sim P_{z(z)}}[\\log(1-D(G(z)))],\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\label{eq1:ganG}\n    L_G = -E_{z\\sim P_{z(z)}}[\\log(D(G(z)))],\n\\end{eqnarray}\nwhere $L_D$ and $L_G$ indicate the objective functions for the discriminator and generator, respectively, whereas $z$ and $x$ mean the noise vector sampled from the Gaussian normal distribution $P_z(z)$ and a real image sampled from data distribution $P_{data}(x)$, respectively. In order to improve the training stability as well as GAN performance, some studies~\\cite{mao2017least, arjovsky2017wasserstein, lim2017geometric, gulrajani2017improved} have been conducted to reformulate the objective function. For instance, Mao~\\textit{et al.}\\@ ~\\cite{mao2017least} introduced the objective function using the least square errors, whereas Arjovsky~\\textit{et al.}\\@ ~\\cite{arjovsky2017wasserstein} proposed Wasserstein GAN~(WGAN) that computes the Wasserstein distance between the real and generated images. Another notable objective function is a hinge-adversarial loss~\\cite{lim2017geometric}, which is widely used in practice~\\cite{yeo2021simple, miyato2018cgans, miyato2018spectral, brock2018large, park2021generative, sagong2019pepsi, shin2020pepsi++, chen2019self, park2021generative_conv, park2021GRB}. On the other hand, cGAN, which produces the class-conditional images, has also been actively studied~\\cite{mirza2014conditional, odena2017conditional, miyato2018cgans}. In general, cGAN utilizes additional conditional information such as class labels or text conditions, in order to produce the condition-specific feature. By optimizing the objective functions for cGAN~\\cite{mirza2014conditional, miyato2018cgans}, the generator can select the image class to be generated. \n\n\\subsection{Residual Block-based Generator}\n\\label{sec2.2}\nIn the field of GAN-based image generation, SNGAN~\\cite{miyato2018cgans, miyato2018spectral} and BigGAN~\\cite{brock2018large} are the most popular base networks since they are simple and exhibit fine performance. The overall generator architectures of those methods are depicted in Fig.~\\ref{fig:fig1}. More specifically, in SNGAN paper, they built the generator using the multiple standard residual blocks. To further improve the GAN performance, Brock~\\textit{et al.}\\@ ~\\cite{brock2018large} introduced a residual block in which the noise vector is additionally input to infer the scaling and shifting parameters of the batch normalization layer. This approach slightly modifies the residual block but still has a similar generator architecture with SNGAN. Indeed, as observed in the previous studies~\\cite{karras2019style, yang2021semantic, wang2021improving}, the generator produces image in a coarse-to-fine manner, where the earlier layers provide a rough template and the latter layers refine the details. That means it needs to convey the information in the earlier layers to the latter ones well. However, since the generators in SNGAN and BigGAN only contain the multiple residual blocks, it may impede the information flow in the network~\\cite{huang2017densely}. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.85\\linewidth]{fig3.pdf}\n\\caption{Framework of the proposed GFFM. BN and Conv indicate the batch normalization~\\cite{ioffe2015batch} and convolution operation, respectively. (a) architecture of GFFM, (b) architecture of gating unit.}\n\\label{fig:fig3}\n\\vspace{-0.5cm}\n\\end{figure*}\n\nTo show the validity of our hypothesis, we conducted simple experiments: we replaced the residual block with the dense block~\\cite{huang2017densely} which propagates the previous information to the latter much better. In our experiments, we used BigGAN as the baseline model and evaluated the GAN performance in terms of IS and FID on the CIFAR-10~\\cite{krizhevsky2009learning} and CIFAR-100~\\cite{krizhevsky2009learning} datasets. The detailed descriptions of the network architectures and evaluation metrics are will be explained in Section~\\ref{subsec:3.2} and~\\ref{subsec:4.1}. Table~\\ref{table1} summarizes the experimental results. As shown in Table~\\ref{table1}, when replacing the residual block with the dense block, the GAN performances are improved on both datasets. In addition, to match the number of network parameters, we conducted ablation studies that increase the channel dimensions of the residual block (Residual* in Table~\\ref{table1}). Although Residual* shows better performance than the residual block, it is still weak compared with the dense block. That means even with the same number of network parameters, the dense block is more effective to achieve the fine GAN performance than the residual block. Based on these results, we expect that the GAN performance could be improved by propagating the information in the earlier layers to the latter ones well. \n\n\\section{Proposed Method}\n\\label{sec3}\n\\subsection{Generator with auxiliary branch}\n\\label{subsec3.1}\nThis brief introduces the novel generator architecture that is composed of main and auxiliary branches. Fig.~\\ref{fig:fig2} illustrates the layout of the proposed method schematically. The goal of the auxiliary branch is to convey the coarse information in the earlier layers to the latter ones. Indeed, an intuitive way to flow information well is to directly sum or concatenate features in the earlier layers to the later ones. However, we should notice that the feature levels in earlier and later layers are different. More specifically, the earlier layers produce high-dimensional features that contain coarse information, whereas the latter ones learn low-dimensional features for image details. Therefore, it may not be the best choice to directly sum or concatenate those features. In addition, the latter layers do not need all information in the earlier layers; they may only need some parts of the information in the earlier layers. That means some procedures are needed to select the appropriate features and combine them successfully. \n\nTo this end, we propose the gated feature fusion module (GFFM) that integrates features in both branches effectively. The architecture of GFFM is depicted in Fig.~\\ref{fig:fig3}, where $f_m$ and $f_a$ indicate the features in the main and auxiliary branches, respectively. In the proposed method, since $f_m$ and $f_a$ have different level of the features, we first utilize the batch normalization (BN)~\\cite{ioffe2015batch} to balance the scales of the features. Then, we blend the normalized features using the gating unit that controls the information flow through a gating mechanism. Specifically, the gating unit needs an input feature $f_i\\in\\mathbb{R}^{h\\times w \\times c_i}$ which will be refined and a side feature $f_s\\in\\mathbb{R}^{h\\times w \\times c_s}$ that contains the information to be used for $f_i$ refinement. Here, $h$ and $w$ indicate the height and width of feature maps, respectively, whereas $c_\\textrm{x}$ means the channel numbers of $f_\\textrm{x}$. By using $f_i$ and $f_s$, the gate feature $f_g \\in\\mathbb{R}^{h\\times w \\times c_g}$, which controls the information passed on in the hierarchy is computed as follows:\n\\begin{equation}\n    f_g = \\sigma \\big[\\textrm{W}_g*(f_i \\odot f_s)\\big],\n\\label{eq3}\n\\end{equation}\nwhere $*$ and $\\odot$ indicate the convolution and channel-concatenation operations, respectively, whereas $\\textrm{W}_g$ is a trainable weight in the convolution layer. $\\sigma$ is a sigmoid function that produces output in the range [0, 1] to control how much to forget or keep the information. In addition, we generate the refinement feature $f_r \\in\\mathbb{R}^{h\\times w \\times c_r}$, which will be newly added to $f_i$, in a similar way. The $f_r$ is defined as \n\\begin{equation}\n    f_r = \\big[\\textrm{W}_f*(f_i \\odot f_s)\\big],\n\\label{eq4}\n\\end{equation}\nwhere $\\textrm{W}_f$ is a trainable weight in the convolution layer. By using these features, \\textit{i.e.}\\@  $f_g$ and $f_r$, we produce the output feature $f_o \\in \\mathbb{R}^{h\\times w \\times c_o}$ as\n\\begin{equation}\n    f_o = \\textrm{W}_o*\\big[f_g \\otimes f_i + (1 - f_g) \\otimes f_r\\big],\n\\label{eq5}\n\\end{equation}\nwhere $\\otimes$ indicates element-wise multiplication and $\\textrm{W}_o$ is a weight in the output convolution layer. Here, we could interpret the Eq.~\\ref{eq5} as follows: the first term $f_g \\otimes f_i$ means that the gating unit decides what it is relevant to keep in the $f_i$ and what information is unnecessary. In contrast, the second term $(1 - f_g) \\otimes f_r$ indicates that $f_i$ is combined with new information coming from $f_s$. That means, the network can effectively control information flow by adding necessary information derived from the earlier layer.\n\nBy using the proposed gating unit, the GFMM produces two different outputs $f^*_m$ and $f^*_a$ which are the refined main and side features, respectively. It is worth noting that the proposed method refines not only $f_m$ but also $f_a$ to store the updated features of the network; $f_a$ acts like memory cell in the gated recurrent unit~\\cite{cho2014learning}. It is worth noting that when producing $f^*_m$, $f_m$ and $f_a$ are used for $f_i$ and $f_s$ of the gating unit, respectively. In contrast, when making $f^*_a$, $f_m$ and $f_a$ are used for $f_s$ and $f_a$ of the gating unit, respectively. Consequently, by adding the GFFM between the residual blocks, the generator produces the image while successfully considering the necessary features in the earlier layers. \n\n\\begin{table*}[t]\n\\caption{Network architectures of the discriminator (top) and generator (bottom) for each image resolution. The RB means the residual block presented in Fig.~\\ref{fig:fig1}(d).}\n\\begin{center}\n\\begin{tabular}{c | c | c | c}\n\\hline\n$32\\times32$ resolution & $128\\times128$ resolution & $256\\times256$ resolution & $512\\times512$ resolution \\\\\n\\hline\nRGB image & RGB image & RGB image & RGB image \\\\\nRB, down, 128 & RB, down, 64 & RB, down, 32 & RB, down, 16\\\\\nRB, down, 128 & RB, down, 128 & RB, down, 64 & RB, down, 32\\\\\nRB, 128 & RB, down, 256 & RB, down, 128 & RB, down, 64\\\\\nRB, 128 & RB, down, 512 & RB, down, 256 & RB, down, 128\\\\\nReLU & RB, down, 512 & RB, down, 512 & RB, down, 256\\\\\nGlobal sum pooling & RB, 512 & RB, down, 512 & RB, down, 512\\\\\nDense, 1 & ReLU & RB, 512 & RB, down, 512\\\\\n & Global sum pooling & ReLU & RB, 512\\\\\n & Dense, 1 & Global sum pooling & ReLU\\\\\n & & Dense, 1 & Global sum pooling\\\\\n & & & Dense, 1 \\\\\n \n\\hline\\hline\n\n$32\\times32$ resolution & $128\\times128$ resolution & $256\\times256$ resolution & $512\\times512$ resolution \\\\\n\\hline\nFC, $4 \\times 4 \\times 256$ & FC, $4 \\times 4 \\times 512$ & FC, $4 \\times 4 \\times 512$ & FC, $4 \\times 4 \\times 512$ \\\\\nRB, up, 256 \/ GFFM, 256 & RB, up, 512 \/ GFFM, 512 & RB, up, 512 \/ GFFM, 512 & RB, up, 512 \/ GFFM, 512 \\\\\nRB, up, 256 \/ GFFM, 256 & RB, up, 512 \/ GFFM, 512 & RB, up, 512 \/ GFFM, 512 & RB, up, 512 \/ GFFM, 512\\\\\nRB, up, 256 \/ GFFM, 256 & RB, up, 256 \/ GFFM, 256 & RB, up, 256 \/ GFFM, 256 & RB, up, 256 \/ GFFM, 256\\\\\nBN, ReLU & RB, up, 128 & RB, up, 128 \/ GFFM, 128  & RB, up, 128 \/ GFFM, 128\\\\\n$3\\times3$ conv, Tanh & RB, up, 64  \/ GFFM, 64 & RB, up, 64 \/ GFFM, 64 & RB, up, 64 \/ GFFM, 64\\\\\n & BN, ReLU & RB, up, 32 \/ GFFM, 32 & RB, up, 32 \/ GFFM, 32\\\\\n& $3\\times3$ conv, Tanh & BN, ReLU  & RB, up, 16 \/ GFFM, 16\\\\\n& & $3\\times3$ conv, Tanh  & BN, ReLU \\\\\n& & & $3\\times3$ conv, Tanh   \\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tableGD}\n\\end{table*}\n\n\\subsection{Implementation Details}\n\\label{subsec:3.2}\nTo reveal the superiority of the proposed method, we conducted extensive experiments using the various standard datasets that are widely used for measuring the GAN performance: CIFAR-10~\\cite{krizhevsky2009learning}, CIFAR-100~\\cite{krizhevsky2009learning}, LSUN~\\cite{yu15lsun}, CelebA-HQ~\\cite{karras2017progressive}, AFHQ~\\cite{choi2020stargan}, and tiny-ImageNet~\\cite{23deng2009imagenet, yao2015tiny} which is a subset of ImageNet~\\cite{23deng2009imagenet}, consisting of the 200 selected classes. Specifically, among the various classes in the LSUN dataset, we employed the church and tower images in our experiments. The resolutions of the CIFAR-10 and CIFAR-100 datasets are $32\\times 32$, whereas we resized the images in LSUN and tiny-ImageNet datasets to $128\\times 128$ pixels. In addition, the CelebA-HQ and AFHQ datasets are used for training the networks that produce the $256\\times 256$ and $512\\times 512$ images. To train the network, we used the hinge-adversarial loss~\\cite{lim2017geometric}. Since all network parameters in the discriminator and generator including the GFFM can be differentiated, we used the Adam optimizer~\\cite{kingma2014adam} and set the user parameters, \\textit{i.e.}\\@  $\\beta _1$ and $\\beta _2$, to 0 and 0.9, respectively.\n\nFor training the CIFAR-10 and CIFAR-100 datasets, we set the learning rate as 0.0002 for both discriminator and generator. The discriminator was updated 5 times using different mini-batches while the generator is updated once. We set a batch size of the discriminator as 64 and trained the generator for 50k iterations. Following the previous papers~\\cite{miyato2018cgans, miyato2018spectral, park2021GRB}, the generator was trained with a batch size twice as large as when training the discriminator. In other words, the generator and discriminator were trained with 128 and 64 batch size, respectively. On the other hand, for training the LSUN and tiny-ImageNet datasets, we employed a two-time scale update rule, called TTUR~\\cite{heusel2017gans}, which set the learning rates of the generator and discriminator to 0.0001 and 0.0004, respectively. Since the TTUR method updates the discriminator a single time when the generator is updated once, we set both batch sizes of the discriminator and the generator to 32. The generator is updated to 300k iterations on the LSUN dataset and 1M iterations on the tiny-ImageNet dataset. In addition, for training the CelebA-HQ and AFHQ datasets, we employed TTUR technique and set both batch sizes of the discriminator and the generator to 16. The generator is updated to 100k iterations. For all datasets, we decreased the learning rate linearly in the last 50,000 iterations. In addition, For training the CIFAR-10 and CIFAR-100 datasets the spectral normalization (SN)~\\cite{miyato2018spectral} was only used for the discriminator, whereas the SN was applied to both generator and discriminator when training the LSUN, CelebA-HQ, AFHQ, and tiny-ImageNet datasets. \n\nIn our experiments, we built generator and discriminator architectures following a strong baseline in~\\cite{brock2018large}, called BigGAN. The detailed discriminator and generator architectures are described in Table~\\ref{tableGD}. In the discriminator, the feature maps were down-sampled by utilizing the average-pooling after the second convolution layer in the residual block. In the generator, the up-sampling (a nearest-neighbor interpolation) operation was located before the first convolution. In the GFFM, we set $c_g = c_r = c_o$ and $c_o$ number for each module is described in Table~\\ref{tableGD}. In addition, we matched the spatial size of $f_m$ and $f_a$ using the nearest neighbor interpolation technique after the BN of $f_a$. When $c_a$ and $c_o$ in the GFFM were different, we used $1 \\times 1$ convolution having $c_o$ output channels before the BN of $f_a$ to match the number of channels. In the last GFFM, we only used $f^*_m$ to produce the image. On the other hand, to train the network in the cGAN framework, we replaced the BN in the generator with the conditional BN~\\cite{dumoulin2017learned, brock2018large} and added the conditional projection layer in the discriminator~\\cite{miyato2018cgans}. The more detailed explanations are presented in~\\cite{miyato2018cgans}.\n\n\n\\begin{table*}[t]\n\\caption{Comparison of the proposed method with conventional ones on the CIFAR-10 and CIFAR-100 datasets in terms of IS and FID. The best results are in bold.}\n\\begin{center}\n\\begin{tabular}{c | c | c | c | c | c | c | c || c | c | c | c | c | c }\n\\hline\n\\multicolumn{2}{c|}{} & \\multicolumn{6}{c||}{GAN} & \\multicolumn{6}{c}{cGAN} \\\\ \n\\hline\n\\multicolumn{2}{c|}{Network} & \\multicolumn{2}{c|}{SNGAN~\\cite{miyato2018cgans}} & \\multicolumn{2}{c|}{BigGAN~\\cite{brock2018large}} & \\multicolumn{2}{c||}{Proposed} & \\multicolumn{2}{c|}{SNGAN~\\cite{miyato2018cgans}} & \\multicolumn{2}{c|}{BigGAN~\\cite{brock2018large}} & \\multicolumn{2}{c}{Proposed} \\\\\n\\hline\nDataset & & IS$\\uparrow$ & FID$\\downarrow$ & IS$\\uparrow$ & FID$\\downarrow$ & IS$\\uparrow$ & FID$\\downarrow$ & IS$\\uparrow$ & FID$\\downarrow$ & IS$\\uparrow$ & FID$\\downarrow$ & IS$\\uparrow$ & FID$\\downarrow$ \\\\\n\\hline\n\\multirow{5}*{CIFAR-10} & trial 1 & 7.79 & 13.81 & 7.70 & 13.98 & 8.06 & 10.92 & 8.06 & 10.26 & 8.09 & 9.36 & 8.27 & 7.96 \\\\\n & trial 2 & 7.76 & 13.29 & 7.80 & 13.69 & 8.11 & 10.96 & 8.02 & 10.37 & 7.97 & 9.62 & 8.20 & 8.20 \\\\\n & trial 3 & 7.76 & 13.29 & 7.87 & 13.09 & 8.12 & 11.12 & 7.99 & 10.01 & 8.02 & 9.38 & 8.32 & 7.83 \\\\\n\\cline{2-14}\n & Mean & 7.77 & 13.46 & 7.79 & 13.58 & \\textbf{8.10} & \\textbf{11.00}  & 8.03 & 10.21 & 8.03 & 9.45 & \\textbf{8.26} & \\textbf{8.00} \\\\\n\\cline{2-14} \n & Std & 0.02 & 0.30 & 0.08 & 0.45 & \\textbf{0.03} & \\textbf{0.11} & 0.03 & 0.18 & 0.06 & 0.15 & \\textbf{0.06} & \\textbf{0.19} \\\\\n\n\\hline \\hline\n\\multirow{5}*{CIFAR-100} & trial 1 & 8.08 & 17.76 & 7.99 & 17.45 & 8.30 & 14.56 & 8.62 & 14.62 & 8.88 & 13.17 & 9.34 & 10.39 \\\\\n & trial 2 & 8.08 & 17.93 & 7.94 & 17.72 & 8.20 & 15.04 & 8.62 & 14.39 & 8.89 & 13.20 & 9.37 & 10.17 \\\\\n & trial 3 & 8.05 & 17.55 & 7.94 & 17.57 & 8.35 & 14.87 & 8.80 & 14.85 & 8.87 & 13.01 & 9.28 & 10.25\\\\\n\\cline{2-14}\n & Mean & 8.07 & 17.75 & 7.96 & 17.58 & \\textbf{8.29} & \\textbf{14.82} & 8.68 & 14.62 & 8.88 & 13.13 & \\textbf{9.33} & \\textbf{ 10.27} \\\\\n\\cline{2-14} \n & Std & 0.02 & 0.19 & 0.03 & 0.13 & \\textbf{0.08} & \\textbf{0.24} & 0.10 & 0.23 & 0.01 & 0.10 & \\textbf{0.05} & \\textbf{0.11}\\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{table2}\n\\end{table*}\n\n\\section{Experimental Results}\n\\label{sec4}\n\\subsection{Evaluation Metric}\n\\label{subsec:4.1}\nFID~\\cite{heusel2017gans} and IS~\\cite{salimans2016improved} are most widely used assessments that measure how well the generator synthesizes the image. More specifically, FID, which computes the Wasserstein distance between the distributions of the real and generated samples in the feature space of the Inception model, is defined as follows: \n\\begin{equation}\n    \\textrm{F}(p,q) = \\| \\mu_p - \\mu_q \\|_2^2 + \\mathrm{trace}(C_p +C_q - 2(C_p C_q)^{\\frac{1}{2}}),\n\\end{equation}\nwhere $ \\{\\mu_p,C_p \\}$ and $\\{\\mu_q,C_q \\}$ are the mean and covariance of the samples with distributions of real and generated images, respectively. Lower FID score indicates that the generated images have better quality. Whileas, Salimans~\\textit{et al.}\\@ ~\\cite{salimans2016improved} demonstrated that IS is strongly correlated with the subjective human judgment of image quality. IS is expressed as \n\\begin{equation}\nI = \\mathrm{exp}(E[D_{KL}(p(l|X)||p(l))]),\n\\end{equation}\nwhere \\textit{l} is the class label predicted by the Inception model~\\cite{26szegedy2016rethinking} trained by using the ImageNet dataset~\\cite{23deng2009imagenet}, and $p(l|X)$ and $p(l)$ represent the conditional class distributions and marginal class distributions, respectively. In contrast with the FID, the better the quality of the generated image, the higher the IS score. In our experiments, we randomly generate 50,000 samples and compute the FID and IS scores using the same number of the real images.\n\n\\begin{table}[t]\n\\caption{Comparison of the proposed method with conventional ones on the CIFAR-10 and CIFAR-100 datasets in terms of IS and FID, when the number of network parameters is same. BigGAN$^\\dagger$ is a BigGAN network built with the dense block. The best results are in bold.}\n\\begin{center}\n\\begin{tabular}{c | c | c | c | c | c | c | c}\n\\hline\n\\multicolumn{2}{c|}{Network} & \\multicolumn{2}{c|}{BigGAN} & \\multicolumn{2}{c|}{BigGAN$^\\dagger$} & \\multicolumn{2}{c}{Proposed} \\\\\n\\hline\n\\multicolumn{2}{c|}{Parameters} & \\multicolumn{2}{c|}{5.68M} & \\multicolumn{2}{c|}{5.68M} & \\multicolumn{2}{c}{5.68M}\\\\ \n\\hline\nDataset & & IS$\\uparrow$ & FID$\\downarrow$ & IS$\\uparrow$ & FID$\\downarrow$ & IS$\\uparrow$ & FID$\\downarrow$ \\\\\n\\hline\n\\multirow{5}*{CIFAR-10} & trial 1 & 7.86 & 12.31 & 7.89 & 11.40 & 8.06 & 10.92 \\\\\n & trial 2 & 7.81 & 12.07 & 7.93 & 11.80 & 8.11 & 10.96 \\\\\n & trial 3 & 7.80 & 12.04 & 7.88 & 11.80 & 8.12 & 11.12 \\\\\n\\cline{2-8}\n & Mean & 7.82 & 12.14 & 7.90 & 11.66 & \\textbf{8.10} & \\textbf{11.00} \\\\\n\\cline{2-8} \n & Std & 0.03 & 0.15 & 0.02 & 0.23 & \\textbf{0.03} & \\textbf{0.11} \\\\\n\\hline \\hline\n\\multirow{5}*{CIFAR-100} & trial 1 & 8.21 & 16.17 & 8.18 & 15.89 & 8.30 & 14.56 \\\\\n & trial 2 & 8.14 & 15.99 & 8.16 & 15.78 & 8.20 & 15.04 \\\\\n & trial 3 & 8.23 & 16.31 & 8.15 & 16.03 & 8.35 & 14.87 \\\\\n\\cline{2-8}\n & Mean & 8.19 & 16.15 & 8.16 & 15.90 & \\textbf{8.29} & \\textbf{14.82} \\\\\n\\cline{2-8} \n & Std & 0.04 & 0.16 & 0.02 & 0.12 & \\textbf{0.08} & \\textbf{0.24} \\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{table3}\n\\end{table}\n\n\n\\subsection{Experimental Results}\n\\label{subsec:4.2}\nTo reveal the effectiveness of the proposed method, we conducted the preliminary experiments of the image generation task on the CIFAR-10 and CIFAR-100 datasets. In our experiments, we trained the network three times from scratch to show that the performance improvement is not due to the lucky weight initialization. Table~\\ref{table2} summarizes the comprehensive results that compare the GAN performance between the proposed and conventional methods. As shown in Table~\\ref{table2}, on both CIFAR-10 and CIFAR-100 datasets, the proposed method consistently outperformed the current state-of-the-art methods, \\textit{i.e.}\\@  SNGAN and BigGAN, in terms of FID and IS scores. These results indicate that the proposed method is more effective than the existing residual block-based generators. In addition, in the cGAN framework, the proposed method showed remarkable performance on both datasets, and exhibited better performance than its counterpart by a large margin. For instance, on the CIFAR-100 dataset, the proposed method achieved the FID of 10.27, which was about 29.75\\% and 21.78\\% better than SNGAN and BigGAN, respectively. Based on these results, we believe that the proposed method can achieve the outstanding performance in both GAN and GAN schemes.\n\n\n\\begin{table}[t]\n\\caption{Comparison of the proposed method with conventional ones on the LSUN-church and LSUN-tower datasets in terms of FID. The best results are in bold.}\n\\begin{center}\n\\begin{tabular}{c | c | c | c | c}\n\\hline\n\\multicolumn{2}{c|}{Network} & SNGAN & BigGAN & Proposed \\\\\n\\hline\nDataset & & FID$\\downarrow$ & FID$\\downarrow$ & FID$\\downarrow$ \\\\\n\\hline\n & trial 1 & 7.83 & 8.18 & 5.59 \\\\\n & trial 2 & 8.16 & 7.88 & 5.56 \\\\\nLSUN- & trial 3 & 8.22 &  8.11 & 5.40 \\\\\n\\cline{2-5}\nchurch & Mean & 8.07 & 8.06 & \\textbf{5.51} \\\\\n\\cline{2-5} \n & Std & 0.21 & 0.15 & \\textbf{0.10} \\\\\n\\hline \\hline\n & trial 1 & 12.42 & 13.89 & 8.77 \\\\\n & trial 2 & 12.29 & 12.38 & 9.90 \\\\\nLSUN- & trial 3 & 12.54 & 11.98 & 8.85 \\\\\n\\cline{2-5}\ntower & Mean & 12.42 & 12.78 & \\textbf{9.17} \\\\\n\\cline{2-5} \n & Std & 0.12 & 0.99 & \\textbf{0.63} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{table4}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{fid_graph.pdf}\n\\caption{Comparison of FID scores over training iterations. Blue, red, and yellow lines indicate the FID scores of proposed method, BigGAN, and SNGAN, respectively.}\n\\label{fig:fig_fid}\n\\vspace{-0.5cm}\n\\end{figure}\n\nOn the other hand, one might perceive that the performance improvement is due to the increased number of network parameters. We agree that the proposed method requires additional network parameters used in the GFFM. To alleviate this concern, we conducted ablation studies that match the network parameters of the conventional methods with that of the proposed one; we increased the channel of the residual block to match the number of network parameters. Since the proposed method follows BigGAN as the baseline model, we compared the performance of the proposed method with that of BigGAN. As presented in Table~\\ref{table3}, even using the same number of network parameters, the proposed method exhibited better performance than the conventional methods. Moreover, since we have already shown that the dense block is more effective than the residual block in Section~\\ref{sec2.2}, we also measured the performance of BigGAN built with the dense block (BigGAN$^\\dagger$ in Table~\\ref{table3}). In our experiments, we adjusted the channel of the dense block to match the number of network parameters. As described in Table~\\ref{table3}, the dense block-based BigGAN exhibited better performance than the residual block-based BigGAN but still showed weaker performance than the proposed method. These results contain two important meanings: First, the performance improvement is not only due to the increased number of network parameters. Second, compared with the residual and dense blocks, the proposed method is more effective to improve the GAN performance by propagating the information in the earlier layers to the later ones successfully. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{is_graph.pdf}\n\\caption{Comparison of IS scores over training iterations. Blue, red, and yellow lines indicate the IS scores of proposed method, BigGAN, and SNGAN, respectively.}\n\\label{fig:fig_is}\n\\vspace{-0cm}\n\\end{figure}\n\n\\begin{table}[t]\n\\caption{Comparison of the final FID and IS scores on the tiny-ImageNet dataset. The best results are in bold.}\n\\begin{center}\n\\begin{tabular}{c | c | c | c | c }\n\\hline\nDataset & Metric & SNGAN & BigGAN & Proposed \\\\\n\\hline\ntiny- & IS$\\uparrow$ & 20.52 & 20.23 & \\textbf{25.57}\\\\\n\\cline{2-5}\nImageNet & FID$\\downarrow$ & 35.42 & 35.13 & \\textbf{25.00} \\\\\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{table:table5}\n\\vspace{-0.3cm}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{fig4.pdf}\n\\caption{Samples of the generated images. (a) Generated images on the LSUN-church dataset, (b) Generated images on the LSUN-tower dataset, (c) Generated images on the tiny-ImageNet dataset.}\n\\label{fig_gen}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{CA-HQ.pdf}\n\\caption{Generated images with $256\\times256$ and $512\\times512$ resolutions on the Celeb-HQ and AFHQ datasets.}\n\\label{fig:CA-HQ}\n\\vspace{-0cm}\n\\end{figure*}\n\nIn order to show the effectiveness of the proposed method when synthesizing images on challenging datasets, we trained the network using the LSUN-church and LSUN-tower datasets. Specifically, on the LSUN-church and -tower datasets, the network is trained by following the experimental setting for GAN. As shown in Table~\\ref{table4}, the proposed method exhibited the notable performance compared to the conventional methods by a large margin. That means the proposed method is also effective to generate the image with the complex scene. Moreover, we trained the network using the tiny-ImageNet dataset, following the experimental setting for cGAN. Since we already proved that the performance improvement is not due to the lucky weight initialization in Tables~\\ref{table2},~\\ref{table3} and~\\ref{table4}, we trained the network a single time from scratch on the tiny-ImageNet dataset. Instead, to show the superiority of the proposed method more reliably, we drew a graph representing the FID and IS scores over training iterations. As shown in Figs.~\\ref{fig:fig_fid} and~\\ref{fig:fig_is}, the proposed method consistently outperformed the conventional methods, \\textit{i.e.}\\@  SNGAN and BigGAN, during the training procedure. The final FID and IS scores are summarized in Table~\\ref{table:table5}. Fig.~\\ref{fig_gen} shows the samples of the generated images. As shown in Fig.~\\ref{fig_gen}, the proposed method is effective to synthesize the images with complex scenes. Based on these results, we concluded that the proposed method can generate visually pleasing images on the challenging datasets. \n\n\\begin{table}[t]\n\\caption{Comparison of the proposed method with conventional ones on the CelebA-HQ and AFHQ datasets in terms of FID. The best results are in bold.}\n\\begin{center}\n\\begin{tabular}{c | c | c | c}\n\\hline\nDataset & resolution & BigGAN & Proposed \\\\\n\\hline\n\\multirow{2}*{CelebA-HQ} & 256$\\times$256 & 16.27 & \\textbf{11.25} \\\\\n\\cline{2-4}\n & 512$\\times$512 & 27.40 & \\textbf{23.43} \\\\\n\\hline \\hline\n \\multirow{2}*{AFHQ} & 256$\\times$256 & 19.02 & \\textbf{16.35} \\\\\n \\cline{2-4}\n & 512$\\times$512 & 22.47 & \\textbf{17.40} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{table:table_CAHQ}\n\\vspace{-0.5cm}\n\\end{table}\n\nFurthermore, to show the ability of the proposed method for generating high-resolution images, we performed additional experiments using the CelebA-HQ~\\cite{karras2017progressive} and AFHQ~\\cite{choi2020stargan} datasets. In this brief, we trained the networks to produce $256\\times256$ and $512\\times512$ images. However, since the CelebA-HQ and AFHQ datasets are composed of a small number of images, it often suffers from the overfitting problem of the discriminator. To alleviate this problem, we employed PConv to the discriminator~\\cite{park2021generative}. The experimental results are presented in Table~\\ref{table:table_CAHQ} and Fig.~\\ref{fig:CA-HQ}. It is worth noting that the proposed method achieved much lower FID scores compared with the conventional methods, producing the visually plausible images. Based on these results, we expected that the proposed method also can be utilized to generate high-resolution images. Indeed, this study does not intend to design an optimal generator and discriminator architectures specialized for the auxiliary branch and GFFM. There might be another network architecture that improves the GAN performance and produces more high-quality images. Instead, this brief focuses on verifying whether it is possible to achieve the better GAN performance by simply adding the auxiliary branch and GFFM. \n\n\\section{Conclusion}\n\\label{sec5}\nThis brief has introduced a novel generator architecture with the auxiliary branch and has shown the remarkable performance of the proposed method on various experiments. In addition, we have introduced the GFFM that effectively blends the features in the main and auxiliary branches. The main advantage of the proposed method is that it significantly boosts the GAN performance with a slight number of additional network parameters. Furthermore, this brief shows the generalization ability of the proposed method in various aspects through high-resolution image generation and extensive experiments. Therefore, we expect that the proposed method is applicable to various GAN-based image generation techniques. However, this brief aims at introducing a novel generator architecture specialized to the GAN. We agree that the proposed method works well for the GAN-based image generation technique, but might cause some issues in other networks used for different applications such as image-to-image translation. There might be another network architecture that shows more generalize ability than the proposed method. Although this brief only covers the effectiveness of the proposed method in the field of GAN-based image generation technique, we have shown that the proposed method boosts the GAN performance significantly. As our future work, we plan to further investigate a novel generator architecture that could cover various applications. \n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:Intro}\n\nExtensions of the Standard Model (SM) with additional gauge symmetries are of particular interest in view of current direct and indirect\nsearches for physics beyond the SM. A feature of such theories is the existence of new gauge bosons.\nThese could provide clear deviations from the SM predictions and have been studied as possible early discoveries\nat the LHC.\n\nConstructions with a single additional $U(1)$ factor correspond to an extra neutral gauge boson $Z'$ \\cite{Langacker:ZpReview, SVZ:Zp, SSVZ:Zp,ABFKS:Zp} and have received a lot of attention, since they are naturally present in a large variety of models, such as Grand Unified Theories\n(GUTs), higher-dimensional models, little or composite Higgs models, superstring constructions.\nHence, the predicted $Z'$ mass spans a wide range of values, from the high GUT scale down to\nthe TeV scale, having different phenomenological consequences. On the other hand,\nit is difficult to determine model independent bounds on the $Z'$ mass, since its relation to\nobservables depends strongly on the mixing of the $Z'$ to the SM $Z$, the strength of the new gauge-coupling constant, the charges of the SM fermions and Higgs doublets, the presence of additional exotic\nfermions necessary to cancel the anomalies, etc..\n\nStill, there exist bounds on the $Z'$ mass and its mixing with the SM $Z$ from precision electroweak data, direct and\nindirect searches at the Tevatron, and interference effects at LEP2 in specific $Z'$ models\n(see~\\cite{Langacker:ZpReview} and references therein). Assuming an electroweak strength for the new gauge coupling constant and family-universal SM fermion charges, most of the canonical $Z'$ models have predictions in agreement with the experimental\nconstraints, for $Z'$ masses not too much below $1$ TeV. However,\nconsidering leptophobic $Z'$ models or models with family non-universal SM fermion charges, the $Z'$\nmass is allowed to be even smaller.\n\nBeyond the simple $U(1)$ group, extended gauge symmetries have also attracted a lot of attention, especially to explain the observed pattern of fermion masses and mixings (some of the earliest papers on this subject are~\\cite{Mohapatra:GaugedSym,BZ:GaugedSym, Ong:GaugeSym, WZ:GaugeSym, Chakrabarti:GaugeSym, MY:GaugeSym, DKW:GaugeSym, CPM:GaugeSym, BC:GaugeSym, Berezhiani:GaugeSym, BK:GaugeSym}). A common feature of these models is the appearance of dangerous flavour-changing-neutral-current (FCNC) contributions, mediated by the new flavour gauge bosons. Hence, a high new physics (NP) scale, usually larger than $1000$ TeV, must be adopted to provide the necessary suppressions. This prevents any NP direct observation at the present and future colliders.\n\nAn alternative approach to provide an explanation of the pattern of fermion masses is considering global flavour symmetries; in particular discrete symmetries have been widely studied in the last years (for a review see \\cite{AF:ReviewDiscreteSyms}). It is interesting to notice that the FCNC contributions are usually less constraining in these models~\\cite{IKOT:LFVA41, FHLM:LFVinA4, HIST:LFVA42, IKOST:LFVDelta54, FHM:Vacuum, FHLM:LFVinSUSYA4, HMP:LFVinSUSYA4SS, FP:RareDecaysA4, MRZ:FVTp, ABMP:Constraining1, ABMP:Constraining2} compared to gauge flavour models, opening the possibility for a lower NP scale and correspondingly for new particles beyond SM that can be detected in present and future colliders.\n\nRecently, a series of papers \\cite{GRV:SU3gauged, Feldmann:SU5gauged, GMS:SU3gaugedLR, AFM:EffectiveGaugeSym} appeared suggesting the possibility of explaining fermion masses and mixings with a NP scale of a few TeV, but with extra gauge symmetries. In these models, the gauge flavour symmetry is the product of non-Abelian $U(3)$ factors. For each gauge-group\ngenerator there is a new gauge boson, with no color or electric charge. In addition, new exotic fermions have to be introduced\nto cancel the anomalies of the new gauge sector.\n\nThe analyses in \\cite{AFM:EffectiveGaugeSym,GRV:SU3gauged, Feldmann:SU5gauged, GMS:SU3gaugedLR} and most papers to date studied almost exclusively the impact of flavour changing heavy neutral gauge bosons on $\\Delta F=2$ transitions and rare $B$ and $K$ decays, leaving their impact on the radiative decay $\\ov{B}\\to X_s\\gamma$ aside. As the latter provides generally a very strong constraint on extensions of the SM, it is important to investigate whether the presence of neutral gauge bosons with flavour violating couplings and new exotic heavy fermions would put the models in question into difficulties and whether generally the role of such new gauge bosons and new fermions in the $\\ov{B}\\to X_s\\gamma$ decay is significant. \n\nHaving this in mind, we point out in this paper two new contributions to the $\\ov{B}\\to X_s\\gamma$ decay, that to our knowledge\nhave not been considered in the literature so far. The first one originates from one-loop diagrams mediated by  \n{\\it heavy neutral} gauge bosons and {\\it exotic} quarks with electric charge $-1\/3$, when {\\it flavour-violating left-handed and right-handed}\ncouplings of these gauge bosons to SM and exotic quarks are present. Then, this new physics contribution to the Wilson\ncoefficient of the dominant operator  is enhanced by $m'_F\/m_b$ with $m_b$ being the mass of the $b$-quark and $m'_F$ the mass of the exotic fermion, in general significantly heavier than SM ones. \n\nAnalogous enhancement is known from the study of left-right symmetric models \\cite{CCFKM:BdecaysRLsym,AI:BSGdecayLRsym,CM:BSGdecayLRsym}, but the absence of new heavy fermions and the smallness of $W_L-W_R$ mixing makes this contribution,  mediated dominantly by {\\it charged} SM gauge bosons with right-handed couplings, sufficiently small to agree with data. \n\nWithin the littlest Higgs model with T-parity, similar contributions to the $\\ov{B}\\to X_s\\gamma$ decay, mediated by heavy neutral gauge bosons and heavy mirror fermions, have already been considered  \\cite{BBPTU:LittlestHiggsT}. However, in this context only left-handed currents are present and in addition the diagrams in question are strongly suppressed by the mixing angles in the T-odd sector and by the GIM mechanism, so that these contributions turn out to be very small.\n\nThe second new contribution stems solely from the heavy neutral gauge bosons, independently of the presence of exotic fermions in the theory. These gauge bosons are integrated out at the scale of the order of their masses, in general much higher than $\\mu_b\\approx m_b$. The resulting {\\it neutral current-current} dimension-six operators affect the $\\ov{B}\\to X_s\\gamma$ rate through QCD mixing with the magnetic dipole operator responsible for the $\\ov{B}\\to X_s\\gamma$ decay. This contribution is present when \nthe new neutral gauge bosons have direct couplings to the $b$ and $s$ quarks.\n\nAs we shall demonstrate explicitly, these new contributions imply constraints on the parameters of models,\ncontaining neutral flavour gauge bosons with couplings described above and with masses within the reach of the LHC. We shall also identify the conditions under which the first of these contributions turns out to be negligible, such as very small mixings among SM and exotic quarks, naturally following from a fermion see-saw mechanism.\n\nWe point out that similar contributions could be considered in $f_j\\to f_i \\gamma$, like $\\mu\\to e\\gamma$\nand $t\\to c\\gamma$, where $f$ is any SM fermion, mediated by exotic fermions with the same electric charge as $f$. The same comments apply to flavour conserving observables like  $(g-2)_\\mu$\nand EDMs, in which dipole operators play the dominant role. Moreover, our QCD analysis of the mixing among neutral current-current and QCD-penguin operators is also applicable to other processes, such as non-leptonic two-body $B$ decays, and other observables, like $\\epsilon'\/\\epsilon$.\n\nWe describe the general framework and fix the notation that will be used throughout the rest of the paper in Sect.~\\ref{sec:Context}. In Sect.~\\ref{sec:analytic} we present the analytical results for the exotic-fermion contribution\nto the $\\ov{B}\\to X_s\\gamma$ decay in the context of only one neutral flavour gauge boson and\none exotic quark with electric charge $-1\/3$. Subsequently, we generalise our results to an arbitrary number of\nsuch gauge bosons and quarks. \n\nIn Sect.~\\ref{sec:QCD} we discuss the second new contribution to the $\\ov{B}\\to X_s\\gamma$ decay. We extend\nthe SM QCD renormalisation group (RG) analysis by taking into account all the new neutral current-current operators,\ngenerated through the exchange of a single neutral gauge boson $A_H$. To this end we calculate the relevant anomalous dimension matrices at the leading order (LO).\n\nIn Sect.~\\ref{sec:pheno} we present our model-independent results in a general form, that can be easily applied to all models with flavour-violating neutral gauge bosons, like $Z'$-models or models with non-Abelian gauge flavour symmetries. We identify three major classes of such models; for each we cast our results into simple formulae, that depend only on the masses and couplings of the new particles. Finally, we illustrate how these formulae can be translated into constraints on the parameter space of representative toy-models. In Sect.~\\ref{sec:concl} we summarise and conclude.\n\n\n\n\\section{Notation and fermion see-saw mechanism}\n\\label{sec:Context}\n\nWe focus on models in which the new physics contributions to\n$\\ov{B}\\to X_s\\gamma$ discussed in the previous section are present. In what follows, we describe a general framework and we fix the notation.\n\nWe consider the general case in which a product of $k\\geq1$ gauge flavour groups,\n\\mbox{$\\cG_f=\\cG_1\\times\\ldots \\times\\cG_k$}, is added to the SM gauge group. Depending\non the dimension $d_i$ of each symmetry factor, $d_1+\\ldots+d_k$ new flavour\ngauge bosons enrich the particle spectrum. In general, the SM fermions $\\psi$ transform\nunder $\\cG_f$ and consequently anomalies from gauge or mixed gauge-gravitational triangles\ncan arise. In this case, it is possible to introduce a new set of exotic fermions $\\Psi$,\nsuitably transforming under the SM and gauge flavour groups to obtain an anomaly-free theory. \n\nOn the other hand, the presence of these new exotic fermions is a quite general feature of\nflavour models, when considering renormalisable ultraviolet completions. This holds not\nonly for gauge flavour symmetries, that we are discussing in this paper, but also for global\nflavour symmetries. Indeed, in these models the new exotic fermions and their mixing with\nthe SM ones account for the mass spectrum of the SM fermions (see \\cite{BGPZ:HeavyFermions} and references therein).\nThis typically happens through a fermion see-saw mechanism where the small masses of\nthe SM fermions are explained by the large masses of their corresponding exotic fermions.\nWithout loss of generality, we consider only the case of $SU(2)_L$-singlet exotic\nfermions. Hence the see-saw Lagrangian can be schematically written as\n\\begin{equation}\n  {\\cal L}} \\def\\cM{{\\cal M}_{\\text{see-saw}}= \\ov{\\Psi}_L\\, M\\, \\Psi_R + \\ov{\\psi}_L\\, Y^D_1\\, \\Psi_R\\, \\phi + \\ov{\\Psi}_L\\, M^D_2\\, \\psi_R+\\text{h.c.}\\,,\n\\label{SeesawLag}\n\\end{equation}\nwhere $\\psi_{L,R}$ ($\\Psi_{R,L}$) comprise all left- and right-handed SM\n(exotic) fermion fields; their dimensions also define the dimensions\nof the matrices $M$, $Y^D_1$ and $M^D_2$. In the same expression,\n$\\phi$ represents the SM Higgs field.\n\nThis see-saw Lagrangian can be made invariant under $\\cG_f$ if both SM and exotic\nfermions belong to the same representation of $\\cG_f$ and $M$, $Y^D_1$, and $M^D_2$ are proportional\nto the identity matrix in the flavour space. This case, however, is phenomenologically not interesting,\nbecause then the flavour symmetry is exactly preserved and it is not possible to reproduce\nthe observed pattern of SM fermion masses and mixings. \n\nOn the other hand, a more realistic situation  appears when we impose the invariance under\n$\\cG_f$ at high scales, by introducing additional degrees of freedom, and then break the flavour symmetries at lower scales to provide the correct SM fermion masses and mixings. In the see-saw Lagrangian, the invariance under $\\cG_f$ and its breaking mechanism can be achieved in\nvarious ways: we can promote $M$, $Y_1^D$, and $M_2^D$, or only some of them, either to spurion fields \n\\`a la MFV \\cite{DGIS:MFV,CGIW:MLFV,DP:MLFV,AIMMN:VMLFV}, or to dynamical scalar fields with non-vanishing\nvacuum expectation values (VEVs) \\cite{FN}, or to fermion condensates \\cite{CG:MFV}. In what follows, we\nsimply treat the terms with $M$, $Y_1^D$, and $M_2^D$, or only some of them, as flavour violating\nterms originating from some non-specified dynamics at the high scale \\cite{AGMR:ScalarPotentialMFV}.\n\nIn the simplest phenomenologically interesting scenario, we discuss, the fields $\\psi_{L}$ ($\\psi_{R}$) and $\\Psi_{R}$ ($\\Psi_{L}$) transform\nunder the same representation of a single flavour group $\\cG_f$. The see-saw Lagrangian is the\nsame as in Eq.~(\\ref{SeesawLag}),  with the only flavour symmetry breaking parameter being\n$M$, while $Y^D_1$ and $M^D_2$ are family universal parameters. In general, the square matrix $M$ has\n non-diagonal hierarchical entries in order to provide the\ncorrect description of the SM fermion masses and mixings. Before finding\nthe mass eigenvalues of the SM and exotic fermions, it is useful\nto diagonalise $M$. In general, $M$ is diagonalised by a bi-unitary transformation\n\\begin{equation}\n\\hat{M}=V^\\dag_L\\, M\\, V_R\\,,\n\\end{equation}\nwhere $V_{L,R}$ are unitary matrices and $\\hat{M}$ is a diagonal matrix. After the absorption of $V_L$ ($V_R$) in $\\Psi_L$ ($\\Psi_R$) and $\\psi_R$ ($\\psi_L$) by a field redefinition and the breaking of the electroweak symmetry, the resulting mass matrix is as follows:\n\\begin{equation}\n\\left(\\ov{\\psi}_L,\\, \\ov{\\Psi}_L\\right)\n\\left(\n        \\begin{array}{cc}\n             0\t\t\t\t\t& Y_1^D\\mean{\\phi} \\times\\mathbb{1}\\\\[2mm]\n\t    M_2^D\\times\\mathbb{1} \t\t& \\hat{M}\\\\\n        \\end{array}\n\\right)\n\\left(\n        \\begin{array}{c}\n             \\psi_R\\\\\n\t    \\Psi_R \\\\\n        \\end{array}\n\\right)\\,.\n\\end{equation}\n\nReabsorbing possible unphysical phases, we can now define the mass eigenstates $f_{L,R}$ and $F_{L,R}$ by performing orthogonal rotations\non the flavour eigenstates $\\psi_{L,R}$ and $\\Psi_{L,R}$. For each generation we may write\n\\begin{equation}\n\\left(\n         \\begin{array}{c}\n             f_{L,R}  \\\\\n             F_{L,R}   \\\\\n         \\end{array}\n\\right)=\n\\left(\n         \\begin{array}{cr}\n            \\cos\\theta_{L,R}  & -\\sin\\theta_{L,R}  \\\\\n            \\sin\\theta_{L,R}  & \\cos\\theta_{L,R} \\\\\n         \\end{array}\n\\right)\\left(\n         \\begin{array}{c}\n              \\psi_{L,R}  \\\\\n              \\Psi_{L,R}  \\\\\n         \\end{array}\n\\right)\\,.\n\\end{equation} \nDenoting with $m_{f_k}$ the mass of the light fermion $f_k$ and with $m'_{F_k}$ that of\nthe heavy exotic fermion $F_k$ and using for brevity the notation $M_1^D\\equiv Y_1^D \\mean{\\phi}$,\nwhat follows is a direct inverse proportionality between $m_{f_k}$ and $m'_{F_k}$, as in the usual see-saw mechanism:\n\\begin{equation}\nm_{f_k}m'_{F_k}=M_1^D\\,M_2^D\\,.\n\\end{equation}\nWe can express such masses in terms of the flavour symmetry breaking terms:\n\\begin{equation}\nm_{f_k}=\\dfrac{\\sin\\theta_{R_k}\\,\\sin\\theta_{L_k}}{\\cos^2\\theta_{R_k}-\\sin^2\\theta_{L_k}}\\hat{M}_k\\,,\\qquad\\qquad\nm'_{F_k}=\\dfrac{\\cos\\theta_{R_k}\\,\\cos\\theta_{L_k}}{\\cos^2\\theta_{R_k}-\\sin^2\\theta_{L_k}}\\hat{M}_k\\,,\n\\end{equation}\nwhere a straightforward calculation gives\n\\begin{equation}\n\\sin\\theta_{L_k}=\\sqrt{\\dfrac{m_{f_k}}{M_2^D}\\dfrac{M_1^D\\,m'_{F_k}-M_2^D m_{f_k}}{m^{\\prime\\,2}_{F_k}-m_{f_k}^2}}\\,,\\qquad\\quad\n\\sin\\theta_{R_k}=\\sqrt{\\dfrac{m_{f_k}}{M_1^D}\\dfrac{M_2^D\\,m'_{F_k}-M_1^D m_{f_k}}{m^{\\prime\\,2}_{F_k}-m_{f_k}^2}}\\,.\n\\label{FormulaSin1}\n\\end{equation}\n\nThese results are exact and valid for all the fermion generations. However, taking\nthe limit in which $m'_{F_k}\\gg m_{f_k}$ and $M_1^D\\approx M_2^D\\equiv M^D$,\nwe find simple formulae that transparently expose the behaviour of the previous expressions.\nIn this limit we find\n\\begin{equation}\nm_{f_k}\\approx \\dfrac{(M^D)^2}{\\hat{M}_k}\\,,\\qquad\\qquad\nm'_{F_k}\\approx \\hat{M}_k\\,,\\qquad\\qquad\n\\sin\\theta_{L_k}\\approx \\sin\\theta_{R_k} \\approx\\sqrt{\\dfrac{m_{f_k}}{m'_{F_k}}}\\,,\n\\label{FormulaSin2}\n\\end{equation}\nas in the usual see-saw scheme in the limit $\\hat{M}_k\\gg M_{1,2}^D$. Notice that these\nsimplified relations are valid for all the fermions, apart from the top quark for which the condition $\\hat{M}_3\\gg M^D_{1,2}$ is not satisfied and large corrections  to Eq.~(\\ref{FormulaSin2}) are expected.\n\nWhen the flavour symmetry is broken, the flavour gauge bosons develop masses proportional to the flavour breaking parameters. It is not possible to enter into details without specifying a particular model and in the following we simply assume that the neutral gauge boson masses are controlled by the parameters $\\hat{M}_k$, as for the exotic fermions.\n\nThe part of the Lagrangian containing the kinetic terms and the usual couplings among\nfermion and gauge bosons is given by \n\\begin{equation}\n{\\cal L}} \\def\\cM{{\\cal M}_{kin}=i\\,\\left(\\ov{\\psi}\\, D\\hspace{-2.5mm}\\big\/\\, \\psi +\\ov{\\Psi}\\, D\\hspace{-2.5mm}\\big\/\\,\\Psi\\right)\\,.\n\\label{StandardKineticTerms}\n\\end{equation}\nSince we are only interested in discussing the couplings between fermions and the new gauge bosons, in writing the relevant covariant derivative we do not consider the coupling terms with the SM gauge bosons but only the following term\n\\begin{equation}\nD^\\mu\\supset i\\, \\dfrac{g_H}{2} \\sum_{a}A_{H_a}^\\mu\\, T_{\\cal R}} \\def\\cV{{\\cal V}^a\\,.\n\\end{equation}\nIn this expression, $g_H$ is the flavour-gauge coupling, $T_{\\cal R}} \\def\\cV{{\\cal V}^a$ are the generators of the flavour\ngroup $\\cG_f$ in the representation ${\\cal R}} \\def\\cV{{\\cal V}$, and $A_{H_a}$ the corresponding\nflavour gauge bosons. The resulting Feynman rules for a single gauge boson $A_H$ in the fermion-mass eigenbasis, ${\\cal F}} \\def\\cG{{\\cal G}\\equiv\\{F,\\,f\\}$, are given by the following relations:\n\\begin{equation}\n\\includegraphics{.\/vertex_general-crop.pdf}\n\\label{vertexGeneral}\n\\end{equation}\nwith\n\\begin{equation}\n\\label{Couplingff}\nA_H\\,\\ov{f}_i\\, f_j:\\begin{cases}\nC_L=\\cos\\theta_{L_i}\\left(V_R^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_R\\right)_{ij}\\cos\\theta_{L_j}+\\sin\\theta_{L_i}\\left(V_L^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_L\\right)_{ij}\\sin\\theta_{L_j}\\\\\nC_R=\\cos\\theta_{R_i}\\left(V_L^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_L\\right)_{ij}\\cos\\theta_{R_j}+\\sin\\theta_{R_i}\\left(V_R^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_R\\right)_{ij}\\sin\\theta_{R_j}\n\\end{cases}\n\\end{equation}\n\\begin{equation}\n\\label{CouplingFF}\nA_H\\,\\ov{F}_i\\, F_j:\\begin{cases}\nC_L=\\sin\\theta_{L_i}\\left(V_R^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_R\\right)_{ij}\\sin\\theta_{L_j}+\\cos\\theta_{L_i}\\left(V_L^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_L\\right)_{ij}\\cos\\theta_{L_j}\\\\\nC_R=\\sin\\theta_{R_i}\\left(V_L^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_L\\right)_{ij}\\sin\\theta_{R_j}+\\cos\\theta_{R_i}\\left(V_R^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_R\\right)_{ij}\\cos\\theta_{R_j}\n\\end{cases}\n\\end{equation}\n\\begin{equation}\n\\label{CouplingfF}\nA_H\\,\\ov{f}_i\\, F_j:\\begin{cases}\nC_L=\\cos\\theta_{L_i}\\left(V_R^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_R\\right)_{ij}\\sin\\theta_{L_j}-\\sin\\theta_{L_i}\\left(V_L^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_L\\right)_{ij}\\cos\\theta_{L_j}\\\\\nC_R=\\cos\\theta_{R_i}\\left(V_L^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_L\\right)_{ij}\\sin\\theta_{R_j}-\\sin\\theta_{R_i}\\left(V_R^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_R\\right)_{ij}\\cos\\theta_{R_j}\n\\end{cases}\n\\end{equation}\n\\begin{equation}\n\\label{CouplingFf}\nA_H\\,\\ov{F}_i\\, f_j:\\begin{cases}\nC_L=\\sin\\theta_{L_i}\\left(V_R^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_R\\right)_{ij}\\cos\\theta_{L_j}-\\cos\\theta_{L_i}\\left(V_L^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_L\\right)_{ij}\\sin\\theta_{L_j}\\\\\nC_R=\\sin\\theta_{R_i}\\left(V_L^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_L\\right)_{ij}\\cos\\theta_{R_j}-\\cos\\theta_{R_i}\\left(V_R^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_R\\right)_{ij}\\sin\\theta_{R_j}\n\\end{cases}\n\\end{equation}\n\nA few comments are in order:\n\\begin{description}\n\\item 1)\\quad These results are valid for both Abelian and non-Abelian flavour gauge\n  symmetries. In particular, barring accidental cancellations,\n  the couplings described in Eqs.~(\\ref{CouplingfF}) and (\\ref{CouplingFf}) are always non-vanishing. The only exception consists in all those $Z'$ models in which the SM fermions have universal charges. In these cases the flavour gauge symmetry $\\cG_f$ is a simple $U(1)$ and the charge universality assures that $T_{\\cal R}} \\def\\cV{{\\cal V}^a\\propto\\mathbb{1}$. Hence, the couplings in Eqs.~(\\ref{CouplingfF}) and (\\ref{CouplingFf}) identically vanish and the couplings in Eqs.~(\\ref{Couplingff}) and (\\ref{CouplingFF}) turn out to be flavour conserving.\n\n\\item 2)\\quad When the flavour symmetry is a product of different groups, \\mbox{$\\cG_f=\\cG_1\\times\\ldots \\times \\cG_k$}, and fermions transform non-trivially under different factors of $\\cG_f$, Eqs.~(\\ref{Couplingff})--(\\ref{CouplingFf})\n  change. In particular, if $\\psi_L$ ($\\Psi_L$) and $\\psi_R$ ($\\Psi_R$) transform under\n  two distinct groups, then only one term in the right-hand side of the previous\n  equations is present (for example in Eq.~(\\ref{Couplingff}), $C_L$ could have either the term \\mbox{$\\cos\\theta_{L_i}\\left(V_R^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_R\\right)_{ij}\\cos\\theta_{L_j}$} or \\mbox{$\\sin\\theta_{L_i}\\left(V_L^\\dag T_{\\cal R}} \\def\\cV{{\\cal V}^a V_L\\right)_{ij}\\sin\\theta_{L_j}$}, but not both terms simultaneously). As a result, even for the simple case $\\cG_f=U(1)^2$ with\n  universal fermion charges, the couplings described in Eqs.~(\\ref{CouplingfF}) and\n  (\\ref{CouplingFf}) are non-vanishing. Notice that in this particular case all the couplings in Eqs.~(\\ref{Couplingff})--(\\ref{CouplingFf}) turn out to be flavour conserving.\n  \n\\item 3)\\quad When the SM fermion masses originate from the see-saw mechanism with heavy exotic fermions, $\\sin\\theta_{L_k}$ and $\\sin\\theta_{R_k}$ are small (cf. Eqs.~(\\ref{FormulaSin1}) and (\\ref{FormulaSin2})). Hence, the couplings described in Eqs.~(\\ref{CouplingfF})  and (\\ref{CouplingFf}) are all suppressed. In the particular case of $m'_{F_k}\\gg m_{f_k}$, this suppression is roughly given by $\\sqrt{m_{f_k}\/m'_{F_k}}$.\n\n\\item 4)\\quad When a theory is anomaly free without the necessity of introducing extra exotic fermions, only the expressions in Eq.~(\\ref{Couplingff}) apply. Still the couplings can be flavour violating. This is the case of $U(1)$ models in which either the charge assignment of the SM fermions is such that no anomalies arise or the Green-Schwarz mechanism \\cite{GS:GSmechanism} is implemented into the theory to compensate for the anomalies. In the specific case of $Z'$ models with universal fermion $U(1)$-charges, the couplings are flavour conserving, as already discussed in 1).\n\\end{description}\n\nAs we will show in Sects.~\\ref{sec:analytic} and \\ref{sec:QCD}, in virtually any gauge model, in which SM fermion masses are explained through a see-saw mechanism with heavy exotic fermions, both new contributions we introduced in Sect.~\\ref{sec:Intro} are present and have potentially interesting effects. Their impact depends on the details of the specific model, according to the above discussion.\n\nOn the other hand, for completeness we will also consider models with gauge symmetries beyond those of the SM that provide either an explanation of the fermion mass patterns through a different mechanism than the see-saw or no explanation at all. In these cases, heavy exotic fermions might not be present in the particle spectrum, as in 4), and the corresponding contributions are absent. However, when present, the couplings listed in Eqs.~(\\ref{Couplingff})--(\\ref{CouplingFf}) have in general a completely different structure. In particular there might be no suppressions of the exotic fermion contributions due to $\\sin\\theta_{L_k}$ and $\\sin\\theta_{R_k}$.\n\nThe presence of exotic fermions can modify the couplings\nof the SM W and Z bosons to fermions with respect to their couplings within the\nSM. As a consequence, non-unitary quark and lepton mixing matrices and\nflavour violating Z couplings\nare usual results of such changes, which are specific to the considered gauge\nflavour model. Also these modifications affect only the Wilson coefficients and do not generate further operators. For this reasons we shall not perform\nsuch a study in our model-independent analysis. A study in a model-dependent\ncontext considering both SM and heavy gauge bosons contributing to the\n$\\ov{B}\\to X_s\\gamma$ branching ratio is in progress \\cite{BCMS:progress}.\n\n\n\n\n\\section{New contributions at the high scale}\n\\label{sec:analytic}\n\\mathversion{bold}\n\\subsection{Effective Hamiltonian for $b\\to s\\gamma$}\n\\mathversion{normal}\n\\label{subsec:ContributionGeneralStructure}\n\nAdopting the overall normalisation of the SM effective Hamiltonian and considering \nas a first step only the dipole operators and the contribution of new\nphysics to their Wilson coefficients, we write the effective Hamiltonian\nrelevant for $b\\to s\\gamma$ at the high matching scale $\\mu_H\\gg M_W$ as: \n\\begin{equation}\n\\label{Heff_at_mu}\n{\\cal H}_{\\rm eff}^{(b\\to s\\gamma)} = - \\dfrac{4 G_{\\rm F}}{\\sqrt{2}} V_{ts}^* V_{tb} \\Big[\\Delta C_{7\\gamma}(\\mu_H) Q_{7\\gamma} + \\Delta  C_{8G}(\\mu_H) Q_{8G} +\\Delta  C'_{7\\gamma}(\\mu_H) Q'_{7\\gamma} +\\Delta  C'_{8G}(\\mu_H) Q'_{8G} \\Big]\\,.\n\\end{equation}\nAt this scale, the $W$ bosons are still dynamical and the usual SM contribution\nto the Wilson coefficients is absent. We will include it after integrating out the $W$\nboson at the lower scale $\\mu_W\\approx \\cO(M_W)$. In our conventions the dipole operators are given by:\n\\begin{equation}\n\\label{O6B}\n\\begin{aligned}\nQ_{7\\gamma}  &=  \\dfrac{e}{16\\pi^2}\\, m_b\\, \\bar{s}_\\alpha\\, \\sigma^{\\mu\\nu}\\, P_R\\, b_\\alpha\\, F_{\\mu\\nu}\\,,\\\\[2mm]          \nQ_{8G}     &=  \\dfrac{g_s}{16\\pi^2}\\, m_b\\, \\bar{s}_\\alpha\\, \\sigma^{\\mu\\nu}\\, P_R\\, T^a_{\\alpha\\beta}\\, b_\\beta\\, G^a_{\\mu\\nu}  \n\\end{aligned}\n\\end{equation}\nand the primed operators are obtained from them after replacing the right-handed projector,\n$P_{R}$, with the left-handed one, $P_L$. In the SM the contributions of the primed\noperators are suppressed by $m_s\/m_b$ relative to those coming from $Q_{7\\gamma}$ and\n$Q_{8G}$. We decompose the Wilson coefficients at the scale $\\mu_W$\nas the sum of the SM contribution and the new one from the exchange of neutral\nflavour gauge bosons, after evolving the new physics contribution using the\nRG running of the operators $Q^{(\\prime)}_{7\\gamma}$ and\n$Q^{(\\prime)}_{8G}$, namely\n\\begin{equation}\nC_i(\\mu_W)=C_i^{SM}(\\mu_W)+\\Delta C_i(\\mu_W)\n\\end{equation}\nand similarly for the primed coefficients. The SM Wilson coefficients\nat LO are \\cite{IL:Loops}\n\\begin{align}\n\\label{c7}\nC^{SM}_{7\\gamma} (\\mu_W) &= \\frac{3 x_t^3-2 x_t^2}{4(x_t-1)^4}\\ln x_t - \\frac{8 x_t^3 + 5 x_t^2 - 7 x_t}{24(x_t-1)^3}\\,,\\\\[2mm]\n\\label{c8}\nC^{SM}_{8G}(\\mu_W) &= \\frac{-3 x_t^2}{4(x_t-1)^4}\\ln x_t - \\frac{x_t^3 - 5 x_t^2 - 2 x_t}{8(x_t-1)^3}\\,,\n\\end{align}\nwhere $x_t\\equiv m_t^2\/M_W^2$. The corresponding primed coefficients are also given by\nthe previous expressions, but with an extra suppression factor of\n$m_s\/m_b$.\n\nThe new physics contributions at $\\mu_W$, $\\Delta C_i(\\mu_W)$, follow from the RG evolution of the Wilson coefficients at $\\mu_H$, $\\Delta C_i(\\mu_H)$, down to $\\mu_W$; Their relations are discussed in Sect.~\\ref{sec:QCD}. We can further decompose the new physics contributions at the high scale as the sum of two pieces, one coming from the exchanges of heavy exotic quarks and the second from the exchanges of light SM down-quarks: respectively,\n\\begin{equation}\n\\begin{aligned}\n\\Delta C_{7\\gamma}(\\mu_H)&=\\Delta C^\\text{heavy}_{7\\gamma}(\\mu_H) + \\Delta C^\\text{light}_{7\\gamma}(\\mu_H)\\,,\\\\[2mm]\n\\Delta C_{8G}(\\mu_H)     &=\\Delta C^\\text{heavy}_{8G}(\\mu_H) + \\Delta C^\\text{light}_{8G}(\\mu_H)\\,.\n\\end{aligned}\n\\label{eq:wilsonatmhTot}\n\\end{equation}\nIn the following we analyse these two contributions separately.\n\n\n\n\n\n\\subsection{Contributions from heavy exotic quark exchanges}\n\nWe begin with the new contributions to $b\\to s \\gamma$ arising from the\nexchange of heavy neutral flavour gauge bosons and heavy exotic quarks. In what follows,\nwe restrict the discussion to the contribution from a single virtual gauge\nboson $A_H$ and a single virtual heavy quark of electric charge $-1\/3$ and mass\n$m'_F$. The results will then be generalised to an arbitrary number of such particles. \nFrom the general Feynman rules listed in Eqs.~(\\ref{CouplingfF}) and (\\ref{CouplingFf}),\nwe read off the flavour changing couplings of $A_H$ to the bottom and strange quarks and\nthe exotic quark $F$ relevant for $b\\to s \\gamma$:\n\\begin{equation}\n\\includegraphics{.\/vertex_heavy-crop.pdf}\n\\label{vertex}\n\\end{equation}\nThe only diagram \\footnote{The corrections on the external legs, necessary for a canonical\nkinetic term, are understood to be contained in finite off-diagonal field renormalisations.}\nwith a virtual $F$ and $A_H$ exchange contributing to the $b\\to s\\gamma$ transition in the\nunitary gauge is shown in Fig.~\\ref{fig:bsgamma}. In the same figure, we also show the\nanalogous diagram contributing to $b\\to s\\, g$. The latter will also contribute to the \n$\\ov{B}\\to X_s\\gamma$ rate via QCD mixing of $Q_{8G}$ in $Q_{7\\gamma}$. In both diagrams,\nthere is no triple gauge boson vertex as the heavy gauge boson $A_H$ considered does\nnot carry electric or colour charge.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics{.\/bsgamma-crop}\n\\caption{\\it Magnetic penguin diagram contributing to $b\\to s\\gamma$ and \n$b\\to s g$ with internal $A_H$ and $F$ exchanges.\n\\label{fig:bsgamma}}\n\\end{figure}\n\nWe further decompose the Wilson coefficients $\\Delta C^\\text{heavy}_i$ at the high\nscale as the sum of the SM-like LL contribution and a new LR one, where $L$ ($R$) stands for the $P_L$ ($P_R$) projector in the vertex of Eq.~(\\ref{vertex}) involving the $s$($b$)-quark:\n\\begin{equation}\n\\begin{aligned}\n\\Delta C^\\text{heavy}_{7\\gamma}(\\mu_H)&=\\Delta^{LL} C^\\text{heavy}_{7\\gamma}(\\mu_H) +\\Delta^{LR}C^\\text{heavy}_{7\\gamma}(\\mu_H)\\,,\\\\[2mm]\n\\Delta C^\\text{heavy}_{8G}(\\mu_H)     &=\\Delta^{LL} C^\\text{heavy}_{8G}(\\mu_H) +\\Delta^{LR}C^\\text{heavy}_{8G}(\\mu_H)\\,.\n\\end{aligned}\n\\label{eq:wilsonatmh}\n\\end{equation}\nThe corresponding primed coefficients to Eq.~(\\ref{eq:wilsonatmh}) are obtained by interchanging $L\\leftrightarrow R$ and including the suppression factor $m_s\/m_b$ in the primed Wilson coefficients.\n\nFollowing \\cite{BBPTU:LittlestHiggsT}, the SM-like contributions LL and RR are\nobtained by noting that their topological structure is the same as the SM {\\it gluon}\nmagnetic penguin. Consequently, the known SM results for the Wilson coefficient of the\ngluon dipole operator, $C^{SM}_{8G}$, can be used to obtain both LL and RR contributions\nafter a suitable redefinition of couplings and masses. Notice that in the neutral\nexchange there is generally no analogue of a GIM mechanism, therefore mass independent terms\nin the SM Wilson coefficients must be kept. We extract them from \\cite{BMU:BSG1,BMU:BSG2}.\nOur results are as follows:\n\\begin{equation}\n\\begin{aligned}\n\\Delta^{LL}C^\\text{heavy}_{7\\gamma}(\\mu_H) &=-\\dfrac{1}{6}\\,\\dfrac{g_H^2}{g_2^2}\\,\\dfrac{M_W^2}{M_{A_H}^2}\\,\\dfrac{C_L^{sF*}\\,C_L^{bF}}{V_{ts}^*\\,V_{tb}}\\,\\left(C_{8G}^{SM}(x)+\\dfrac{1}{3}\\right),\\\\[2mm]\n\\Delta^{LL}C^\\text{heavy}_{8 G}(\\mu_H) &= - 3 \\Delta^{LL}C^\\text{heavy}_{7\\gamma}(\\mu_H)\\,,\n\\end{aligned}\n\\label{LLnew}\n\\end{equation}\nwith $x=m_F^2\/M_{A_H}^2$. The $RR$ contributions are obtained by the\ninterchange $L\\leftrightarrow R$. $C_{8G}^{SM}(x)$ is the SM function\non the right-hand side of Eq.~(\\ref{c8}).\\\\\n\nWe now consider the LR and RL contributions, which have no SM equivalent. To obtain the\nWilson coefficients, we adapt to our case known calculations of $b\\to sg$ in the context\nof $b\\to s\\gamma$ in the left-right symmetric models \\cite{CM:BSGdecayLRsym,BMU:BSG1},\nwhere, however, the process is mediated by charged gauge bosons. In this way,\nwe obtain the following LR Wilson coefficients:\n\\begin{equation}\n\\begin{aligned}\n\\Delta^{LR}C^\\text{heavy}_{7\\gamma}(\\mu_H)&=-\\dfrac{1}{6}\\,\\dfrac{g_H^2}{g_2^2}\\,\\dfrac{M_W^2}{ M_{A_H}^2}\\,\\dfrac{m'_F}{m_b}\\,\\dfrac{C_L^{sF*}\\,C_R^{bF}}{V_{ts}^*\\,V_{tb}}\\,C^{LR}_{8G}(x)\\,,\\\\[2mm]\n\\Delta^{LR}C^\\text{heavy}_{8G}(\\mu_H)&= -3\\Delta^{LR}C^\\text{heavy}_{7\\gamma}(\\mu_H)\\,,\n\\end{aligned}\n\\label{LRnew}\n\\end{equation}\nwith \n\\begin{equation}\nC^{LR}_{8G}(x)=\\dfrac{-3x}{2(1-x)^3}\\ln{x}+ \\dfrac{3 x (x-3)}{4(x-1)^2} -1\\,.\n\\label{CLR8Gcoeff}\n\\end{equation}\nThe analogous RL contributions are obtained by interchanging $L\\leftrightarrow R$. \nIn this case $C^{RL}_{8G}(x)=C^{LR}_{8G}(x)$.\n\nThe following properties should be noted:\n\\begin{description} \n\\item 1)\\quad the LR (RL) contribution in Eq.~(\\ref{LRnew}) dominates over the LL (RR) one of Eq.~(\\ref{LLnew}) in a large part of the parameter space, due to the factor $m'_F\/m_b$; \n\\item 2)\\quad the factor $m'_F\/m_b$ is replaced by $m_t\/m_b$ in the usual left-right symmetric models, therefore in these models the enhancement is less pronounced; \n\\item 3)\\quad $C^{LR}_{8G}(x)$ is a non-vanishing monotonic function of $x$ and takes values in the range $[-1,\\,-1\/4]$ for $x$ from $0$ to $\\infty$; \n\\item 4)\\quad in the decoupling limit these contributions turn out to vanish. Indeed when large masses for the exotic fermions $m'_{F_k}$ are considered, the masses of the neutral gauge bosons $M_{A_H}$ are also large, as we have already commented in Sect.~\\ref{sec:Context}. Therefore the LL (RR) contributions approach zero with $1\/M_{A_H}^2$ and the LR (RL) ones are strongly suppressed by $m'_F\/M_{A_H}^2$.\n\\end{description}\n\n\n\n\n\\subsection{Contributions from light SM quark exchanges}\n\nWe now discuss $\\Delta C^{\\text{light}}_{7\\gamma}$ and $\\Delta C^{\\text{light}}_{8G}$ from\nEq.~\\eqref{eq:wilsonatmhTot}, namely the contributions arising from  the exchange of light down-type quarks in \nloop diagrams analogous to the one in Fig~\\ref{fig:bsgamma}, where the flavour changing couplings of $A_H$ to the SM quarks read\n\\begin{equation}\n\\includegraphics{.\/vertex_light-crop.pdf}\n\\label{vertexLight}\n\\end{equation}\narising from the general Feynman rules listed in Eq.~(\\ref{Couplingff}). In the SM, the light-quark contributions cancel each other at scales well above their masses due to the GIM mechanism. This is in contrast to the present case, where such a mechanism is absent.\n\nFollowing the procedure of the previous section, we decompose the Wilson coefficients at the high scale as the sum of a $LL$ part and a $LR$ part:\n\\begin{equation}\n\\begin{aligned}\n\\Delta C^\\text{light}_{7\\gamma}(\\mu_H)&=\\Delta^{LL} C^\\text{light}_{7\\gamma}(\\mu_H) +\\Delta^{LR}C^\\text{light}_{7\\gamma}(\\mu_H)\\,,\\\\[2mm]\n\\Delta C^\\text{light}_{8G}(\\mu_H)     &=\\Delta^{LL} C^\\text{light}_{8G}(\\mu_H) +\\Delta^{LR}C^\\text{light}_{8G}(\\mu_H)\\,.\n\\end{aligned}\n\\label{eq:wilsonatmhLight}\n\\end{equation}\nIn the framework of effective field theories the light degrees of freedom are treated as massless at the high matching-scale $\\mu_H$.\nTherefore, we only need to account for the first term in the expansion of the\nlight masses and only mass-independent terms in\nEq.~(\\ref{LLnew}) and (\\ref{LRnew}) contribute. For the LL SM-like contributions we find\n\\begin{equation}\n\\begin{aligned}\n  \\Delta^{LL}C^\\text{light}_{7\\gamma}(\\mu_H) &=-\\dfrac{1}{6}\\,\\dfrac{g_H^2}{g_2^2}\\,\n  \t\t\t\t\t\t\\dfrac{M_W^2}{M_{A_H}^2}\\,\\,\\,\n\t\t\t\t\t\t\\sum_{f}\\dfrac{C_L^{sf*}\\,C_L^{bf}}{V_{ts}^*\\,V_{tb}}\\,\n\t\t\t\t\t\t\\left(\\dfrac{1}{3}\\right),\\\\[2mm]\n  \\Delta^{LL}C^\\text{light}_{8 G}(\\mu_H) &= - 3 \\Delta^{LL}C^\\text{light}_{7\\gamma}(\\mu_H)\\,,\n\\end{aligned}\n\\label{eq:LLnewLight}\n\\end{equation}\nand for the LR contribution we have from Eqs.~(\\ref{LRnew}) and (\\ref{CLR8Gcoeff})\n\\begin{equation}\n\\begin{aligned}\n\\Delta^{LR}C^\\text{light}_{7\\gamma}(\\mu_H)&=-\\dfrac{1}{6}\\,\n\t\t\t\t\t\t\\dfrac{g_H^2}{g_2^2}\\,\n\t\t\t\t\t\t\\dfrac{M_W^2}{ M_{A_H}^2}\\,\\,\\,\n\t\t\t\t\t\t\\sum_f\\dfrac{m_f}{m_b}\n\t\t\t\t\t\t\\dfrac{C_L^{sf*}\\,C_R^{bf}}{V_{ts}^*\\,V_{tb}}\\,\n\t\t\t\t\t\t\\left( -1\\right)\\,,\\\\[2mm]\n\\Delta^{LR}C^\\text{light}_{8G}(\\mu_H)&= -3\\Delta^{LR}C^\\text{light}_{7\\gamma}(\\mu_H)\\,,\n\\end{aligned}\n\\label{eq:LRnewLight}\n\\end{equation}\nwhere $f$ stands for the SM down-type quarks. Analogously, we obtain also the results for the primed Wilson coefficients. The\nLL and RR SM-like contributions have already been considered in \\cite{LP:ZpNonUniversal}, while\nthe LR and RL are new. Notice that the latter come with a factor\nof $m_f\/m_b$ and hence, the effects from $d$- and $s$-quarks are suppressed with\nrespect to the $b$-quark contribution.\n\n\n\n\\subsection{Generalisations and the GIM-like mechanism}\n\nIt is easy to extend the previous results to the case of an arbitrary number of neutral\nflavour gauge bosons, $A_{H_i}$, and heavy fermions, $F_k$, by performing the\nfollowing substitution in Eqs.~(\\ref{LLnew}) and (\\ref{LRnew}):\n\\begin{equation}\n\\begin{aligned}\n&\\dfrac{g^2_H}{ M_{A_H}^2}\\,C_{L,R}^{sF*}\\,C_{L,R}^{bF}\\,\\left(C_{8G}^{SM}(x)+\\dfrac{1}{3}\\right)\n\\longrightarrow \n\\sum_{i,k}\\dfrac{g^2_{H_i}}{M_{A_{H_i}}^2}\\,C_{L,R}^{ski*}\\,C_{L,R}^{bki}\\,\\left(C_{8G}^{SM}(x_{ki})+\\dfrac{1}{3}\\right)\\,,\\\\[2mm]\n&g^2_H\\,\\dfrac{{m'_F}}{ M_{A_H}^2}\\,C_{L,R}^{sF*}\\,C_{R,L}^{bF}\\,C^{LR}_{8G}(x)\n\\longrightarrow\n\\sum_{i,k}g^2_{H_i}\\,\\dfrac{{m'_{F_k}}}{M_{A_{H_i}}^2}\\,C_{L,R}^{ski*}\\,C_{R,L}^{bki}\\,C^{LR}_{8G}(x_{ki})\\,,\n\\end{aligned}\n\\label{SumRule}\n\\end{equation}\nwhere $x_{ki}\\equiv {m^{\\prime\\,2}}_{F_k}\/M^2_{A_{H_i}}$ and $C_{L,R}^{(s,b)ki}$ are the\ncouplings among the light quarks $(s,b)$ with $A_{H_i}$ and $F_k$. In\nthis substitution we take $A_{H_i}$ to be the mass eigenstates of the neutral gauge\nbosons. Similarly, we may also generalise Eq.~\\eqref{eq:LLnewLight} and\n\\eqref{eq:LRnewLight}.\n\nNotice that in the case that the masses of the heavy particles span a wide range of values, the generalisation above is only a first approximation: a more precise result follows from a detailed analysis using the results of Sect. \\ref{sec:QCD} considering several threshold scales. This further study goes beyond the scope of the present paper and in the following we only consider a single high matching scale.\n\nThe factor $m'_{F_k}\/M^2_{A_{H_i}}$ can represent a dangerous enhancement of the LR (RL) contribution in Eq.~(\\ref{SumRule}), which is unlikely to vanish, barring accidental cancellations. We identify a series of conditions under which this contribution exactly vanishes:\n\\begin{description}\n\\item 1)\\quad $g_{H_i}=g_{H}$, i.e. there is only one gauge symmetry or all the new gauge coupling constants have the same strength;\n\\item 2)\\quad $m'_{F_k}=m'_{F}$, $M_{A_{H_i}}=M_{A_H}$, i.e. all the exotic fermions and the neutral gauge bosons are degenerate in mass, as in the case of unbroken flavour symmetries; \n\\item 3)\\quad $\\sum_{i}C_{L,R}^{ski*}\\,C_{L,R}^{bki}=\\sum_{i}C_{L,R}^{ski*}\\,C_{R,L}^{bki}=0$, i.e. the couplings are unitary matrices satisfying $C_L=C_R$.\n\\end{description}\nSimilar conditions apply also to the contributions of SM down-type quarks. \n\nThese conditions assure the exact cancellation of the NP contributions. However, similarly to the GIM mechanism, if they are only partially satisfied, the terms in Eq.~(\\ref{SumRule}) do not cancel each other completely and the NP contributions could still be dangerous.\n\nFurthermore, we remark that condition 2) would correspond to degenerate SM fermions and thus these conditions cannot be satisfied in models which successfully explain the SM fermion mass pattern through the see-saw mechanism illustrated in Sect.~\\ref{sec:Context}. In these models, however, the mixings between SM and exotic fermions are small, cf. Eqs.~\\eqref{CouplingfF} and \\eqref{CouplingFf}. This provides a sufficiently strong suppression to safely neglect these contributions. We further discuss this point in Sect.~\\ref{sec:3classes}.\n\n\n\n\n\\section{QCD corrections}\n\\label{sec:QCD}\n\\subsection{General structure}\n\\label{subsec:QCDgeneralstructure}\n\nIn order to complete the analysis of the $b\\to s\\gamma$ decay we have to include QCD\ncorrections, which play a very important role within the SM, enhancing\nthe rate by a factor of $2-3$~\\cite{MABCC:BSGalpha2}. It originates dominantly from the mixing of {\\it charged} current-current operators into\nthe dipole operators and to a smaller extent from the mixing with QCD-penguin operators.\n\nWhen the flavour gauge bosons $A_{H_i}$ and the exotic fermions $F_k$ are integrated\nout at the high scale $\\mu_H$, in addition to the dipole operators discussed at length\nin the previous section, also {\\it neutral} current-current operators corresponding to\na tree-level $A_{H_i}$ exchange are generated together with ordinary QCD-\nand QED-penguin operators.\n\nThe contributions of QCD- and QED-penguin operators, arising from diagrams\nwith $A_{H_i}$ and $F_k$ exchanges, are much less important than within the SM. Indeed,\nin the SM context, QCD-penguins do not have to compete with tree-level FCNC \ndiagrams, as the latter are absent due to the GIM mechanism. However, \nin the present case there are neutral current-current operators originating from\nthe tree-level $A_{H_i}$ exchanges that are not suppressed by $\\alpha_s(\\mu_H)$ and\nloop effects, contrary to QCD-penguins. Thus, for all practical purposes the contributions of\nQCD-penguins from $A_{H_i}$ and $F_k$ exchanges at the high scale $\\mu_H$ can be neglected. Even less important\nare effects from QED-penguins. Note that this is in contrast to dipole operators\nwhich cannot be generated at tree-level, but nevertheless mediate the $b\\to s\\gamma$ decay.\n\nIn what follows we will extend the SM RG analysis by considering\nthe QCD effects of neutral current-current operators at scales $\\mu_b\\leq \\mu\\leq\\mu_H$\ngenerated by diagrams with the exchange of a single neutral gauge boson $A_H$. The\nextension to the case of an arbitrary number of such gauge bosons is straightforward. This\nanalysis does take into account the mixing under QCD renormalisation of these neutral current-current operators 1) into dipole operators, 2) among themselves, 3) into QCD-penguin operators.\nThe latter mixing generates contributions to QCD-penguin operators at scales lower than $\\mu_H$, even if the initial conditions at $\\mu_H$ of the Wilson coefficients of these operators are neglected. Even if this effect is numerically\nnegligible, we include it for completeness as the mixing of QCD-penguin operators and\ndipole operators is taken into account in the SM part.\n\nBefore going into details, let us note that the neutral current-current operators do\nnot mix with the charged current-current operators as their flavour structures\ndiffer from each other. On the other hand, similarly to charged current-current\noperators, neutral current-current operators have an impact on dipole operators\nand QCD-penguins through RG effects without being themselves affected\nby the presence of these two types of operators.\n\nWe would like to emphasise that the QCD analysis presented below is also relevant for other processes, such as non-leptonic two-body $B$ decays, and other observables, like $\\epsilon'\/\\epsilon$.\n\nDenoting symbolically charged current-current, QCD-penguin,\ndipole, and neutral current-current operators and the corresponding primed operators respectively by\n\\begin{equation}\nQ^{cc}\\,,\\qquad\\quad\nQ_P\\,,\\qquad\\quad\nQ_D\\,,\\qquad\\quad\nQ^{nn}\\,,\\qquad\\quad\nQ'_P\\,,\\qquad\\quad\nQ'_D\\,,\\qquad\\quad\nQ^{nn\\,\\prime}\\,,\n\\end{equation}\nthe structure of the one-loop anomalous dimension matrix looks as follows:\n\\begin{equation}\n\\begin{tabular}{l|c|c|c|c|c|c|c|l}\n\\multicolumn{1}{r}{}& \\multicolumn{1}{c}{$Q^{cc}$} & \\multicolumn{1}{c}{$Q_P$}  & \\multicolumn{1}{c}{$Q_D $}\t& \\multicolumn{1}{c}{$Q^{nn}$}\n                                                   & \\multicolumn{1}{c}{} \t& \\multicolumn{1}{c}{}\t\t& \\multicolumn{1}{c}{} \\\\\n\\cline{2-5}\n$Q^{cc}$ \t& $X_1$ \t& $X_2$\t\t& $X_3$ \t& 0 \t&\\multicolumn{1}{c}{}\t&\\multicolumn{1}{c}{}\t&\\multicolumn{1}{c}{}\\\\\n\\cline{2-5}\n$Q_P$ \t\t& 0 \t\t& $X_4$ \t& $X_5$ \t& 0 \t&\\multicolumn{1}{c}{}\t&\\multicolumn{1}{c}{}\t&\\multicolumn{1}{c}{}\\\\\n\\cline{2-5}\n$Q_D$ \t\t& 0 \t\t& 0 \t\t& $X_6$ \t& 0 \t&\\multicolumn{1}{c}{}\t&\\multicolumn{1}{c}{}\t&\\multicolumn{1}{c}{}\\\\\n\\cline{2-5}\n$Q^{nn}$ \t& 0 \t\t& $Y_1$ \t& $Y_2$ \t& $Y_3$ &\\multicolumn{1}{c}{}\t&\\multicolumn{1}{c}{}\t&\\multicolumn{1}{c}{}\\\\\n\\cline{2-8}\n\\multicolumn{1}{c}{} \t   \t&\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{}&\t& $X_4$ & $X_5$ & 0 \t& \\multicolumn{1}{l}{$Q_P'$}\\\\\n\\cline{6-8}\n\\multicolumn{1}{c}{}  \t    \t&\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{}&\t& 0     & $X_6$ & 0 \t& \\multicolumn{1}{l}{$Q_D'$}\\\\\n\\cline{6-8}\n\\multicolumn{1}{c}{}  \t\t&\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{}&\\multicolumn{1}{c}{}&\t& $Y_1$\t& $Y_2$ & $Y_3$ & \\multicolumn{1}{l}{$Q^{nn\\,\\prime}$}\\\\\n\\cline{6-8}\n\\multicolumn{1}{r}{}\t\t&\\multicolumn{1}{r}{}&\\multicolumn{1}{r}{}&\\multicolumn{1}{r}{}&\\multicolumn{1}{r}{}& \\multicolumn{1}{c}{$Q_P'$}  & \\multicolumn{1}{c}{$Q_D' $} & \\multicolumn{1}{c}{$Q^{nn\\,\\prime}$}\n\\end{tabular}\n\\label{AnomalousDimension}\n\\end{equation}\nThe non-vanishing entries denoted by $X_i$ are known from the SM analysis \\cite{CFRS:BSG,BBL:WeakNLO}. The new\nresults in the present paper are the entries denoted by $Y_i$, the initial\nconditions for $Q_D$, given already in Sect. \\ref{sec:analytic}, and the\ncorresponding conditions for $Q^{nn}$, given at the end of this section. \n\nThe RG analysis of the NP contributions can be performed\nindependent from the SM one: that is \n\\begin{equation}\n\\begin{aligned}\n&C_i^{SM}(\\mu_b)=U_{ij}(\\mu_b,\\, \\mu_W)\\,C_j^{SM}(\\mu_W)\\,,\\\\[2mm]\n&C_i^{NP}(\\mu_b)=W_{ij}(\\mu_b,\\,\\mu_H)\\, C_j^{NP}(\\mu_H)\\,,\\\\[2mm]\n&C_i^\\text{total}(\\mu_b)=C_i^{SM}(\\mu_b)+C_i^{NP}(\\mu_b)\\,,\n\\end{aligned}\n\\end{equation}\nwhere $U_{ij}$ and $W_{ij}$ are the elements of the RG evolution\nmatrix. The Wilson coefficients for the primed operators at the high scale are\nin general different from the expression above, but the evolution matrices $U$ and $W$ do not change.\nThe general structure of these matrices in terms of the anomalous dimension\nmatrices and the QCD-$\\beta$-functions is well known \\cite{Buras:LH-lectures}\nand will not be repeated here. Therefore, we present in the next sections our\nresults for the matrices $Y_i$ and the initial conditions for $Q^{nn}$ and\n$Q^{nn\\,\\prime}$ operators.\n\n\n\n\n\\subsection{Operator basis for neutral current-current operators}\n\nThere are $48$ neutral current-current operators generated by the exchange\nof $A_H$, all containing the neutral currents $(\\ov{s}\\,\\gamma_\\mu\\,P_{L,R}\\,b)$\nand the flavour conserving currents $(\\ov{f}\\,\\gamma_\\mu\\,P_{L,R}\\,f)$ with\n$f=u,c,t,d,s,b$. At this scale also the exotic fermions have been already integrated out at $\\mu_H$.\n\nThe general notation for the $48$ $Q^{nn}$ and $Q^{nn\\,\\prime}$ operators in\nquestion will be \n\\begin{equation}\nQ_{1,2}^f(A,B)\\,,\\qquad \\qquad \\text{for} \\quad A,B=\\{L,R\\}\\,.\n\\label{QnnGeneral}\n\\end{equation}\nFor instance\n\\begin{equation}\n\\begin{aligned}\n&Q_1^u(L,R)=\\left(\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_L\\,b_\\beta\\right)\\left(\\ov{u}_\\beta\\,\\gamma^\\mu\\,P_R\\,u_\\alpha\\right)\\,,\\\\[2mm]\n&Q_2^u(L,R)=\\left(\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_L\\,b_\\alpha)(\\ov{u}_\\beta\\,\\gamma^\\mu\\,P_R\\,u_\\beta\\right)\\,,\n\\end{aligned}\n\\end{equation}\nwhere $\\alpha,\\beta=1,2,3$ are colour indices.\n\nMoreover there are 8 additional neutral current-current operators\nthat have to be included, as they have a different structure, when colour\nand flavour structures are considered simultaneously:\n\\begin{equation}\n\\begin{aligned}\n&\\hat{Q}_1^d(A,B)=\\left(\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_A\\,d_\\beta\\right)\\left(\\ov{d}_\\beta\\,\\gamma^\\mu\\,P_B\\,b_\\alpha\\right)\\,,\\\\[2mm]\n&\\hat{Q}_2^d(A,B)=\\left(\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_A\\,d_\\alpha)(\\ov{d}_\\beta\\,\\gamma^\\mu\\,P_B\\,b_\\beta\\right)\\,.\n\\end{aligned}\n\\label{QnnhatGeneral}\n\\end{equation}\nIn this classification, we refer to $Q^{nn}$ [$Q^{nn\\,\\prime}$] as those operators\n$Q_{1,2}(A,B)$ with $A=L$ [$A=R$] and $B=L,R$ [$B=R,L$].\n\nWe point out that the operator basis in Eqs.~(\\ref{QnnGeneral}) and\n(\\ref{QnnhatGeneral}) can be reduced using Fierz transformations. Moreover\nsome relations can be found between the usual charged current-current\nand QCD-penguin operators\n\\begin{equation}\n\\begin{aligned}\n&Q_1=\\left(\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_L\\,c_\\beta\\right)\\left(\\ov{c}_\\beta\\,\\gamma^\\mu\\,P_L\\,b_\\alpha\\right)\\,,\\\\[2mm]\n&Q_2=\\left(\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_L\\,c_\\alpha\\right)\\left(\\ov{c}_\\beta\\,\\gamma^\\mu\\,P_L\\,b_\\beta\\right)\\,,\\\\[2mm]\n&Q_3=\\left(\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_L\\,b_\\alpha\\right)\\sum_{q=u,d,s,c,b}\\left(\\ov{q}_\\beta\\,\\gamma^\\mu\\,P_L\\,q_\\beta\\right)\\,,\\\\[2mm]\n&Q_4=\\left(\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_L\\,b_\\beta\\right)\\sum_{q=u,d,s,c,b}\\left(\\ov{q}_\\beta\\,\\gamma^\\mu\\,P_L\\,q_\\alpha\\right)\\,,\\\\[2mm]\n&Q_5=\\left(\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_L\\,b_\\alpha\\right)\\sum_{q=u,d,s,c,b}\\left(\\ov{q}_\\beta\\,\\gamma^\\mu\\,P_R\\,q_\\beta\\right)\\,,\\\\[2mm]\n&Q_6=\\left(\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_L\\,b_\\beta\\right)\\sum_{q=u,d,s,c,b}\\left(\\ov{q}_\\beta\\,\\gamma^\\mu\\,P_R\\,q_\\alpha\\right)\\,.\n\\end{aligned}\n\\end{equation}\n\nYet, similarly to \\cite{BJL:AnatomyEpsilon,BBH:AnatomyEpsilon} we have decided\nto work with all operators and not use such relations when performing the RG analysis in order to keep the anomalous dimensions in a\ntransparent form. As discussed in \\cite{BJL:AnatomyEpsilon,BBH:AnatomyEpsilon}\none can  work with linearly dependent operators in the process of RG\nanalysis without any problems and use such\nrelations only at the end of the evolution if necessary. However, in the case\nof $C_{7\\gamma}(\\mu_b)$ at LO this is not required. \n\nIn this context we should warn the reader that at NLO level a use of Fierz\nrelations in the RG analysis could lead  to wrong results within\nthe NDR scheme unless Fierz evanescent operators are introduced \n\\cite{JP:QCDinDeltaS,BMU:2loopQCDGamma}.\n\n\n\\subsection{Anomalous dimension matrices}\n\nWe define the LO anomalous dimension matrix through \n\\begin{equation}\n\\hat{\\gamma}(\\alpha_s)=\\dfrac{\\alpha_s}{4\\pi}\\hat{\\gamma}^{(0)}\\,,\n\\end{equation}\nand give below the results for $\\hat{\\gamma}^{(0)}$ (dropping the index $(0)$ to simplify the notation). \n\nThe inspection of the one-loop diagrams contributing to the anomalous\ndimension matrices shows that only 16 operators have to be considered\nin order to find the full matrix. These are\n\\begin{equation}\n\\label{ListOperators}\n\\begin{array}{llll}\nQ_{1,2}^u(L,L)\\,,&\\qquad Q_{1,2}^d(L,L)\\,,&\\qquad Q_{1,2}^s(L,L)\\,,&\\qquad \\hat{Q}_{1,2}^d(L,L)\\,,\\\\[2mm]\nQ_{1,2}^u(L,R)\\,,&\\qquad Q_{1,2}^d(L,R)\\,,&\\qquad Q_{1,2}^s(L,R)\\,,&\\qquad \\hat{Q}_{1,2}^d(L,R)\\,.\n\\end{array}\n\\end{equation}\n\nThe anomalous dimensions for $u$ replaced by $c$ and $t$ are equal to\nthe one of $Q_{1,2}^u$. The ones of $Q_{1,2}^b$ are equal to the ones\nof $Q_{1,2}^s$. The anomalous dimensions of the remaining 28 operators,\nnamely the primed operators obtained by $L\\leftrightarrow R$, are\nthe same as those of the corresponding unprimed operators \n(see Eq.~(\\ref{AnomalousDimension})).\n\nThe mixing between the $Q^{nn}$ operators and $Q_D$, that is the matrix\n$Y_2$, can be extracted from \\cite{CFMRS:BSG,CFRS:BSG} by inspecting the mixing\nbetween QCD-penguin and dipole operators. For the {\\it transposed}\nmatrices we get\n\n\\begin{equation}\n\\hat\\gamma_D^T(L,L)=\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|}\n\\hline\n&&&&&&&&\\\\[-4mm]\n& $Q_1^u$ & $Q_2^u$ & $Q_1^d$ & $Q_2^d$ & $Q_1^s$ & $Q_2^s$ & $\\hat{Q}_1^d$ & $\\hat{Q}_2^d$ \\\\[2mm]\n\\hline\n\\hline\n&&&&&&&&\\\\[-4mm]\n$Q_{7\\gamma}$ & $\\dfrac{416}{81}$ & $0$ & $-\\dfrac{232}{81}$ & 0 & $-\\dfrac{232}{81}$ & $-\\dfrac{232}{81}$ & 0 & $-\\dfrac{232}{81}$ \\\\[2mm]\n\\hline\n&&&&&&&&\\\\[-4mm]\n$Q_{8G}$ & $\\dfrac{70}{27}$ & $3$ & $\\dfrac{70}{27}$ & $3$ & $\\dfrac{151}{27}$ & $\\dfrac{151}{27}$ & $3$ & $\\dfrac{70}{27}$\\\\[2mm]\n\\hline\n\\end{tabular}\n\\end{equation}\n\\begin{equation}\n\\hat\\gamma_D^T(L,R)=\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|}\n\\hline\n&&&&&&&&\\\\[-4mm]\n& $Q_1^u$ & $Q_2^u$ & $Q_1^d$ & $Q_2^d$ & $Q_1^s$ & $Q_2^s$ & $\\hat{Q}_1^d$ & $\\hat{Q}_2^d$ \\\\[2mm]\n\\hline\n\\hline\n&&&&&&&&\\\\[-4mm]\n$Q_{7\\gamma}$ & $-\\dfrac{448}{81}$ & $0$ & $\\dfrac{200}{81}$ & $0$ & $\\dfrac{200}{81}$ & $\\dfrac{16}{9}$ & $-\\dfrac{80}{3}$ & $-\\dfrac{32}{9}$ \\\\[2mm]\n\\hline\n&&&&&&&&\\\\[-4mm]\n$Q_{8G}$ & $-\\dfrac{119}{27}$ & $-3$ & $-\\dfrac{119}{27}$ & $-3$ & $-\\dfrac{173}{27}$ & $-\\dfrac{16}{3}$ & $-4$ & $\\dfrac{8}{3}$ \\\\[2mm]\n\\hline\n\\end{tabular}\n\\end{equation}\n\nThe mixing between neutral current-current operators, that is the\nmatrix $Y_3$, is universally given by two $2\\times 2$ matrices:\n\\begin{equation}\n\\hat\\gamma^{nn}(L,L)=\n\\begin{tabular}{|c||c|c|}\n\\hline\n&&\\\\[-4mm]\n& $Q_1$ & $Q_2$ \\\\[2mm]\n\\hline\n\\hline\n&&\\\\[-4mm]\n$Q_1$ & $-2$ & $6$ \\\\[2mm]\n\\hline\n&&\\\\[-4mm]\n$Q_2$ & $6$ & $-2$ \\\\[2mm]\n\\hline\n\\end{tabular}\\qquad\\qquad\n\\hat\\gamma^{nn}(L,R)=\n\\begin{tabular}{|c||c|c|}\n\\hline\n&&\\\\[-4mm]\n& $Q_1$ & $Q_2$ \\\\[2mm]\n\\hline\n\\hline\n&&\\\\[-4mm]\n$Q_1$ & $-16$ & $0$ \\\\[2mm]\n\\hline\n&&\\\\[-4mm]\n$Q_2$ & $-6$ & $2$ \\\\[2mm]\n\\hline\n\\end{tabular}\n\\end{equation}\nOnly operators with the same flavour-conserving structure $(\\ov{f}\\,\\gamma_\\mu\\,P_{L,R}\\,f)$ mix in this way.\n\nFinally, in the case of $Y_1$ there is a universal mixing of $Q^{nn}$\noperators into QCD-penguin operators:\n\\begin{equation}\n\\hat\\gamma_P=\n\\begin{tabular}{|c||c|c|c|c|}\n\\hline\n&&&&\\\\[-4mm]\n& $Q_3$ & $Q_4$ & $Q_5$ & $Q_6$ \\\\[2mm]\n\\hline\n\\hline\n&&&&\\\\[-4mm]\n$Q^{nn}$ & $-\\dfrac{2}{9}$ & $\\dfrac{2}{3}$ & $-\\dfrac{2}{9}$ & $\\dfrac{2}{3}$ \\\\[2mm]\n\\hline\n\\end{tabular}\n\\end{equation}\nfor\n\\begin{equation}\nQ^{nn}=\\left\\{Q_1^{u,c,t,d,s,b}(L,L),\\, Q_2^{s,b}(L,L),\\, \\hat{Q}_2^d(L,L),\\, Q_1^{u,c,t,d,s,b}(L,R)\\right\\}\n\\end{equation}\nwith no mixing for the remaining operators in Eq.~(\\ref{ListOperators}).\n\n\n\n\n\\subsection{Initial conditions}\n\nThe initial conditions at $\\mu_H$ for the dipole operators have already been presented in Sect.~\\ref{sec:analytic}: see Eqs.~(\\ref{c7}), (\\ref{c8}), (\\ref{LLnew}), (\\ref{LRnew}), (\\ref{eq:LLnewLight}) and (\\ref{eq:LRnewLight}). Here we give the corresponding initial conditions for the neutral current-current operators.\n\nThe initial conditions are determined by integrating out all the\nheavy degrees of freedom at $\\mu_H$ and matching the result to the effective theory. At LO this matching is trivial. In the following we give the result\nfor the simplified case in which only one neutral flavour gauge boson $A_H$\ncontributes to the initial conditions. The generalisation to several\ngauge bosons is obvious.\n\nDenoting the vertex $\\ov{f}_i\\,A_H\\,f_j$ as in Eqs.~(\\ref{vertexGeneral}) and (\\ref{Couplingff}), the general expression for the Wilson\ncoefficient of the neutral current-current operator\n\\begin{equation}\nQ^f_2(A,B)\\equiv (\\ov{s}_\\alpha\\,\\gamma_\\mu\\,P_A\\,b_\\alpha)(\\ov{f}_\\beta\\,\\gamma^\\mu\\,P_B\\,f_\\beta)\\,,\n\\end{equation}\nin the normalisation of Eq.~(\\ref{Heff_at_mu}), is given by\n\\begin{equation}\n\\Delta^{AB}C^f_2(\\mu_H)=-\\dfrac{1}{2}\\dfrac{g_H^2}{g_2^2}\\dfrac{M_W^2}{M_{A_H}^2}\\dfrac{C_A^{sb*}\\,C_B^{ff}}{V_{ts}^*\\,V_{tb}}\\,,\n\\label{InitialConditionQnn}\n\\end{equation}\nwhile all the coefficients $\\Delta^{AB} C_1$ are zero. Similarly, the Wilson coefficient for the $\\hat{Q}^d_{2}(A,B)$ operator is \n\\begin{equation}\n\\Delta^{AB}\\hat{C}_2^d(\\mu_H)=-\\dfrac{1}{2}\\dfrac{g_H^2}{g_2^2}\\dfrac{M_W^2}{M_{A_H}^2}\\dfrac{C_A^{sd*}\\,C_B^{bd}}{V_{ts}^*\\,V_{tb}}\\,,\n\\label{InitialConditionQnnHat}\n\\end{equation}\nwhile $\\Delta^{AB}\\hat{C}_1^d(\\mu_H)$ is again zero.\n\n\\section{Phenomenological analysis}\n\\label{sec:pheno}\n\\mathversion{bold}\n\\subsection{The $\\ov{B}\\to X_s \\gamma$  branching ratio}\n\\mathversion{normal}\n\\label{sec:Br}\n\nThe SM prediction for the $\\ov{B}\\to X_s\\gamma$ branching ratio at\nNNLO \\cite{MABCC:BSGalpha2,GM:BSGdefinition,MS:BSGdefinition} reads,\n\\begin{equation}\nBr(\\ov{B}\\to X_s \\gamma)=(3.15\\pm0.23)\\times 10^{-4}\\,,\n\\label{BRSMNNLO}\n\\end{equation}\nand has been calculated for a photon-energy cut-off $E_\\gamma>1.6$ GeV in the $\\ov{B}$-meson rest frame. This is to be compared\nwith the current experimental value \\cite{PDG2010},\n\\begin{equation}\nBr(\\ov{B}\\to X_s \\gamma)=(3.55\\pm 0.24 \\pm 0.09)\\times 10^{-4}\\,,\n\\label{BRexp}\n\\end{equation}\nfor the same energy cut-off $E_\\gamma$. In the presence of NP the expression\nfor the branching ratio is given as follows:\n\\begin{equation}\nBr(\\ov{B}\\to X_s \\gamma) = R \\left(|C_{7\\gamma}(\\mu_b)|^2+|C'_{7\\gamma}(\\mu_b)|^2+N(E_\\gamma)\\right)\\,,\n\\label{BRtotal}\n\\end{equation}\nwhere $R=2.47\\times10^{-3}$ and $N(E_\\gamma)=(3.6\\pm0.6)\\times10^{-3}$. The parameter $R$ is simply an overall factor, whose determination is discussed in \\cite{GM:BSGdefinition,MS:BSGdefinition}, while $N(E_\\gamma)$ is a non-perturbative contribution. Calculating the NP contributions in the LO approximation, but including the\nSM contribution at the NNLO level we find for $\\mu_b=2.5$ GeV\n\\begin{equation}\nC_{7\\gamma}(\\mu_b)=C^{SM}_{7\\gamma}(\\mu_b)+\\Delta C_{7\\gamma}(\\mu_b)\n\\label{eq:C7efftotal}\n\\end{equation}\nwhere the central value of $C^{SM}_{7\\gamma}(\\mu_b)$ at the NNLO, corresponding to the value in Eq.~(\\ref{BRSMNNLO}), is\ngiven by \\cite{MABCC:BSGalpha2,GM:BSGdefinition,MS:BSGdefinition}\n\\begin{equation}\nC^{SM}_{7\\gamma}(\\mu_b)=-0.3523\n\\label{eq:C7effSM}\n\\end{equation}\nand the NP one, calculated by us at the LO, by\n\\begin{equation}  \n\\begin{split}\n\\Delta C_{7\\gamma}(\\mu_b)=&\\quad\\kappa_7~\\Delta C_{7\\gamma}(\\mu_H) +\\kappa_8~\\Delta C_{8G}(\\mu_H)+\\\\[2mm]\n\t\t\t       &+\\!\\!\\!\\!\\!\\sum_{\\substack{A=L,R\\\\f=u,c,t,d,s,b}}\\!\\!\\!\\!\\! \\kappa^{f}_{LA}~\\Delta ^{LA} C_2^f(\\mu_H)\n\t\t\t        +\\!\\!\\!\\sum_{A=L,R}\\!\\!\\!\\!\\hat{\\kappa}^{d}_{LA}~\\Delta ^{LA} \\hat{C}_2^d(\\mu_H).\n\\end{split}\n\\label{eq:DeltaC7eff}\n\\end{equation}\nHere $\\kappa$'s are the NP magic numbers listed in Tab.~\\ref{tab:c7magicnumbers}, calculated taking $\\alpha_s(M_Z=91.1876\\,\\text{GeV})=0.118$. The primed coefficient $C'_{7\\gamma}(\\mu_b)$ can be obtained from\n(\\ref{eq:C7efftotal})--(\\ref{eq:DeltaC7eff}), by interchanging $L\\leftrightarrow R$\nand taking the initial conditions of the primed Wilson coefficients. In particular, the NP magic numbers listed in Tab.~\\ref{tab:c7magicnumbers} are also valid\nfor the primed case, as QCD is blind to the fermion chirality\\footnote{In\nthe computation of $C'_{7\\gamma}(\\mu_b)$ we are neglecting the SM contributions\n$C_{7\\gamma}^{SM\\,\\prime}(\\mu_b)$ and $C_{8G}^{SM\\,\\prime}(\\mu_b)$, and the NP\ncontributions $\\Delta C'_{7\\gamma}(\\mu_b)$ and $\\Delta C'_{8G}(\\mu_b)$, since all of\nthem are suppressed by $m_s\/m_b$ with respect the unprimed ones. Our numerical\nanalyses confirm that this approximation is consistent with errors below the percent\nlevel.}.\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c||r|r|r|r||r|}\n  \\hline\n  &&&&&\\\\[-4mm]\n  $\\mu_H$\t \t & 200 GeV \t& 1 TeV\t\t& 5 TeV\t\t& 10 TeV & $M_Z$\\\\[1mm]\n  \\hline\\hline\n  $\\kappa_7$\t\t & 0.524\t& 0.457\t\t& 0.408\t\t& 0.390\t &\t 0.566 \\\\\t\t\t\n  $\\kappa_8$\t\t & 0.118\t& 0.125\t\t& 0.129\t\t& 0.130\t & 0.111\t\\\\[1mm]\t\t\n  \\hline\n  &&&&&\\\\[-4mm]\n  $\\kappa_{LL}^{u,c}$\t & 0.039\t& 0.057\t\t& 0.076\t\t& 0.084\t & 0.030\t\\\\\t\t\t\n  $\\kappa_{LL}^{t}$\t &-0.002\t&-0.003\t\t&-0.002\t\t&-0.001\t & --\t\\\\\t\t\t\n  $\\kappa_{LL}^{d}$\t &-0.040\t&-0.057\t\t&-0.072\t\t&-0.079\t & -0.032\t\\\\\t\t\t\n  $\\kappa_{LL}^{s,b}$\t & 0.087\t& 0.090\t\t& 0.090\t\t& 0.090\t & 0.084\t\\\\\t\t\t\n  $\\hat{\\kappa}_{LL}^{d}$& 0.128\t& 0.147\t\t& 0.163\t\t& 0.168\t & 0.116\t\\\\[1mm]\n  \\hline             \n  &&&&&\\\\[-4mm]                                                                            \t\n  $\\kappa_{LR}^{u,c}$\t & 0.085\t& 0.128\t \t& 0.173\t\t& 0.193\t & 0.065\t\\\\\t\t\t\n  $\\kappa_{LR}^{t}$\t & 0.004\t& 0.012\t \t& 0.023\t\t& 0.028\t & --\t\\\\\t\t\t\n  $\\kappa_{LR}^{d}$\t &-0.015\t&-0.025\t\t&-0.036\t\t&-0.041\t & -0.011\t\\\\\t\t\t\n  $\\kappa_{LR}^{s,b}$\t &-0.078\t&-0.092\t \t&-0.106\t\t&-0.111\t & -0.070\t\\\\\t\t\t\n  $\\hat{\\kappa}_{LR}^{d}$& 0.473\t& 0.665\t\t& 0.865\t\t& 0.953\t & 0.383\t\\\\[1mm]\t\n  \\hline                                                                                         \t\n\\end{tabular}\n\\caption{The NP magic numbers for $\\Delta C_{7\\gamma}(\\mu_b)$ defined\nin Eq.~(\\ref{eq:DeltaC7eff}). For completeness, we include in the last column the case of a flavour-violating $Z$. The analogous table for \n$\\Delta C_{8G}(\\mu_b)$ is provided in Appendix \\ref{AppA}.}\n\\label{tab:c7magicnumbers}\n\\end{center}\n\\end{table}\n\nThe SM contribution containing NLO and NNLO QCD corrections exhibits a negligible\n$\\mu_b$-dependence. This is not the case for the NP contribution at LO.\nHowever, we have checked numerically that when the NP contribution enhances the SM value\nby $20\\%$, the $\\mu_b$-dependence in the total branching ratio amounts\nto a $3\\%$ uncertainty for $\\mu_b\\in[2.5,\\,5]$ GeV. For smaller deviations from the SM prediction \nthe uncertainty further decreases. This renders this uncertainty sufficiently small\nfor our purposes.\\\\\n\nBefore concluding a few observations are in order:\n\\begin{description}\n\\item 1)\\quad Similarly to the SM, the magic numbers $\\kappa_7$ and $\\kappa_8$ suppress the initial values $\\Delta C_{7\\gamma}(\\mu_H)$ and $\\Delta C_{8G}(\\mu_H)$. This is due to the QCD RG evolution running down to $\\mu_b$. Furthermore, the suppression of $\\Delta C_{7\\gamma}(\\mu_H)$  increases with $\\mu_H$.\n \n\\item 2)\\quad As in the SM, provided $\\Delta^{AB}C_2^f(\\mu_H)$ and $\\Delta^{AB}\\hat{C}_2^d(\\mu_H)$\n  are sufficiently larger than $\\Delta C_{7\\gamma}(\\mu_H)$, the additive QCD corrections stemming from the mixing of the neutral current-current operators\n  into the dipole operators are dominant. Furthermore the QCD factors $\\kappa_i$ increase in\n  most cases with $\\mu_H$.\n  The most prominent is the coefficient $\\hat{\\kappa}_{LR}^d$ which\n  could even be of $\\cO(1)$, but also $\\kappa_{LR}^{u,c}$, $\\hat{\\kappa}^d_{LL}$ and\n  $\\kappa^{s,b}_{LL,LR}$ are sizable. However, values of the corresponding initial conditions could compensate these coefficients, as in the case of small couplings among SM fermions.\n  \n\\item 3)\\quad The SM and NP primed dipole Wilson coefficients, $ C'_i(\\mu_W)$ and  $\\Delta C'_i(\\mu_H)$, are suppressed by $m_s\/m_b$ and turn out to be numerically negligible. On the other hand, this suppression is absent in the contributions of the primed neutral current-current operators $Q^{nn\\prime}$ and therefore they should be considered in the determination of the branching ratio.\n\\end{description}\n\n\n\n\n\\subsection{Three classes of models}\n\\label{sec:3classes}\n\nThe results for $\\kappa_i$ shown in Tab.~\\ref{tab:c7magicnumbers} are\nmodel-independent and hold for all models in which the neutral gauge\nbosons have flavour violating couplings to fermions as\ndiscussed in Sect.~\\ref{sec:Context}.\nOn the other hand, the initial conditions of the Wilson coefficients are\nmodel-dependent. However, in spite of the large variety of models it is possible to distinguish three main classes, even if\nhybrid situations are also possible:\n\\begin{description}\n  \\item 1)\\quad Models without exotic fermions. This is the case\n  of models that do not aim at explaining the SM fermion masses and mixings\n  or models that do provide such an explanation but without exotic fermions. For instance, $Z'$ constructions in which the theory is \n  anomaly free without extending the fermion spectrum\n  of the SM fall into this class. This is possible when the Green--Schwarz mechanism is implemented in the theory or when the\n  generator of the additional $U(1)$ is\n  a linear combination of the hypercharge $Y$ and $B-L$, where $B$ is the\n  baryon number and $L$ is the lepton number.\n\n  For all models of this class, the new diagrams with  virtual\n  exotic fermions do not contribute. On the other hand, the contributions from the exchange\n  of the light down-type quarks, the neutral current-current operators\n  and the corresponding QCD evolution are still present. As illustrated\n  in our numerical analysis of Sect.~\\ref{sec:ToyModel}, the light-quark\n  contributions turn out to be negligible in the whole parameter space.\n\n\n\\item 2)\\quad Models in which the SM fermion-mass patterns are governed\n  by exotic fermions through a see-saw mechanism as illustrated in\n  Sect.~\\ref{sec:Context}. In this case, the couplings of\n  the neutral gauge bosons to SM and exotic fermions, described by\n  Eqs.~(\\ref{Couplingff})--(\\ref{CouplingFf}), are suppressed by\n  $\\sin\\theta_{L_k,R_k}$ given in Eqs.~(\\ref{FormulaSin1}) and (\\ref{FormulaSin2}).\n  In the specific limit of $m'_{F_k}\\gg m_{f_k}$ and $M_1^D\\approx M_2^D\\equiv M^D$\n  the suppression in $\\sin\\theta_{L_k,R_k}$ is approximately\n  of order $\\cO\\left(\\sqrt{m_{f_k}\/m'_{F_k}}\\right)$ .\n\n  In order to illustrate the size of the contributions from the new diagrams\n  of Sect.~\\ref{sec:analytic}, we redefine the couplings $C_{L,R}^{ski}$\n  and $C_{L,R}^{bki}$ to exhibit the dependence on the suppressing factor. Without\n  loss of generality, we can approximately write\n  \\begin{equation}\n  C_{L,R}^{ski}\\simeq \\sqrt{\\dfrac{m_s}{m'_{F_k}}} \\tilde{C}_{L,R}^{ski}\\,,\\qquad\\qquad\n  C_{L,R}^{bki}\\simeq \\sqrt{\\dfrac{m_b}{m'_{F_k}}} \\tilde{C}_{L,R}^{bki}\\,.\n  \\end{equation}\n  As a result, for the case of arbitrary $A_{H_i}$ and $F_k$, the expressions\n  on the right-hand side of Eqs.~(\\ref{LLnew}) and (\\ref{LRnew}) are given by\n  \\begin{equation}\n  \\begin{aligned}\n    \\Delta^{LL}C^{\\text{heavy}}_{7\\gamma}(\\mu_H) &\\simeq-\\dfrac{1}{6}\\,\\sum_{i,k}\\dfrac{g_{H_i}^2}{g_2^2}\\,\\dfrac{M_W^2}{M_{A_{H_i}}^2}\\,\\dfrac{\\sqrt{m_s\\,m_b}}{m'_{F_k}}\\,\\dfrac{\\tilde{C}_L^{ski*}\\,\\tilde{C}_L^{bki}}{V_{ts}^*\\,V_{tb}}\\,\\left(C_{8G}^{SM}(x_{ki})+\\dfrac{1}{3}\\right),\\\\[2mm]\n    \\Delta^{LR}C^{\\text{heavy}}_{7\\gamma}(\\mu_H)&\\simeq-\\dfrac{1}{6}\\,\\sum_{i,k}\\dfrac{g_{H_i}^2}{g_2^2}\\,\\dfrac{M_W^2}{ M_{A_{H_i}}^2}\\,\\sqrt{\\dfrac{m_s}{m_b}}\\,\\dfrac{\\tilde{C}_L^{ski*}\\,\\tilde{C}_R^{bki}}{V_{ts}^*\\,V_{tb}}\\,C^{LR}_{8G}(x_{ki})\\,,\n  \\end{aligned}\n  \\label{LRnewnew}\n  \\end{equation}\n  and $\\Delta^{(LL,LR)}C^{\\text{heavy}}_{8G}(\\mu_H)=-3\\Delta^{(LL,LR)}C^{\\text{heavy}}_{7\\gamma}(\\mu_H)$.\n  Similar expressions can be written for the primed contributions. Notice\n  that the $m'_{F_k}\/m_b$ enhancement is completely removed from the LR (RL)\n  Wilson coefficient and is replaced by the suppressing factor $\\sqrt{m_s\/m_b}$.\n  The only dependence on $m'_{F_k}$ is in the loop factor $C^{LR}_{8G}(x_{ik})$.\n  On the other hand, an extra inverse power of $m'_{F_k}$ is now present in the LL (RR) contribution, which further suppresses this term. \n\n\nAs we shall explicitly demonstrate in Sect.~\\ref{sec:ToyModel}\n  these contributions turn out\n  to be negligible with respect to those from the neutral current-current operators.\n  The same holds for the light-quark contributions. Hence, when\n  dealing with this class of models, we simply neglect the contributions of\n  exotic fermions and therefore, as in 1), only neutral current-current operators\n  and their mixing with dipole operators are relevant.\n\n\\item 3)\\quad Models with exotic fermions, in which the definition of their couplings to gauge bosons and SM fermions is independent of\n  the mechanism of the SM fermion mass generation. For example, this is the case of models in which\n  these couplings do not arise from the standard kinetic terms of\n  Eq.~(\\ref{StandardKineticTerms}). As a result, $\\sin\\theta_{L_k,R_k}$ can in general be much larger than in\n  the previous classes of models.\n\n  In this case, the expressions in Eqs.~(\\ref{LLnew}) and (\\ref{LRnew}) do not suffer\n  from the additional suppressions of Eq.~(\\ref{LRnewnew}) and in particular the LR (RL)\n  contribution is strongly enhanced by the factor $m'_{F_k}\/m_b$. For the\n  models of this class, the contribution of exotic fermions is the dominant one\n  and all the other contributions can be safely neglected.\n\\end{description}\n\nWe summarise the relevant features of the models in these three classes\nin Tab.~\\ref{tab:Categories}.\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|l||c|c|c|c|}\n  \\hline\n  &&&\\\\[-4mm]\n  Classes of models\t\t\t\t\t\t& $Q_D^\\text{heavy}$ \t& $Q_D^\\text{light}$ \t& $Q^{nn}$,  $Q^{nn\\prime}$ \\\\[1mm]\n  \\hline\\hline\n  &&&\\\\[-4mm]\t\n  1) without exotic fermions  \t\t\t\t& Absent\t\t\t\t& Negligible\t\t& Dominant \\\\\n  2) with exotic fermions and see-saw\t\t\t& Negligible\t\t\t& Negligible\t\t& Dominant \\\\\n  3) with exotic fermions but without see-saw \t& Dominant\t\t\t& Negligible\t\t& Negligible \\\\[1mm]\n  \\hline                                                                                         \t\n\\end{tabular}\n\\caption{\\it Summary of the different classes of models and the\ncorresponding NP contributions to $b\\to s\\gamma$: contributions from\nthe exchange of heavy exotic quarks $Q_D^\\text{heavy}$, from the exchange\nof SM down-type quarks $Q_D^\\text{light}$, and from neutral current-current\noperators $Q^{nn}$ and $Q^{nn\\prime}$.\n}\n\\label{tab:Categories}\n\\end{center}\n\\end{table}\n\n\n\n\\subsection{Model-independent constraints}\n\nIt has been pointed out in \\cite{AGM:SignC7gamma,CMW:SignC7gamma,HW:SignC7gamma,\nGOS:SignC7gamma,GHM:SignC7gamma,AABGS:SO10SusyGutFCNC} that considering the\nmeasurements of $\\ov{B}\\to X_s \\ell^+\\ell^-$ \\cite{BaBar:Bsll,Belle:Bsll} the\nsign of $C_{7\\gamma}(\\mu_b)$ in the presence of NP is likely to be\nthe same as in the SM.\nThis provides a first model-independent constraint for NP contributions: since\n$C^{SM}_{7\\gamma}(\\mu_b)$ is negative, $\\Delta C_{7\\gamma}(\\mu_b)$ should\nalso be negative in order to soften the tension between the central values\nof the SM prediction and the experimental determination of \n$Br(\\overline{B}\\rightarrow X_s\\gamma)$. This holds not only for contributions from unprimed operators, but also when considering the primed\nones due to the relative suppression of $C'_{7\\gamma}(\\mu_b)$ with respect to \n$C_{7\\gamma}(\\mu_b)$, as explicitly confirmed by our numerical analysis.\n\nThe sign of the NP contributions is determined by the product of the initial conditions and the QCD magic numbers in\nTab.~\\ref{tab:c7magicnumbers}. The signs of the $\\kappa_i$ factors are fixed solely by the QCD running, while the sign of the initial conditions \n$\\Delta C_i(\\mu_H)$ depends on the couplings of the new neutral gauge bosons to the fermions.\n\nIn particular, for a model in the first two classes, this constraint translates into the requirement\nthat the second line on the right-hand side of Eq.~(\\ref{eq:DeltaC7eff})\nshould be negative, namely that:\n\\begin{equation}\n\\Delta C_{7\\gamma}(\\mu_b)\\simeq\\sum_{\\substack{A=L,R\\\\f=u,c,t,d,s,b}}\\kappa^{f}_{LA}~\\Delta ^{LA} C_2^f(\\mu_H)+\n\\sum_{A=L,R}\\!\\!\\!\\!\\hat{\\kappa}^{d}_{LA}~\\Delta ^{LA} \\hat{C}_2^d(\\mu_H)<0\\,,\n\\label{Condition1_12}\n\\end{equation}\n where only the couplings listed in Eq.~(\\ref{Couplingff}) are involved. \n \n On the other\n hand, when a model belongs to the third class, the first line on the right-hand\n side of Eq.~(\\ref{eq:DeltaC7eff}) must be negative,\n \\begin{equation}\n \\Delta C_{7\\gamma}(\\mu_b)\\simeq\\kappa_7~\\Delta C_{7\\gamma}(\\mu_H) +\\kappa_8~\\Delta C_{8G}(\\mu_H)<0\\,.\n \\label{Condition1_3}\n \\end{equation} \n  This puts a constraint on the couplings of Eqs.~(\\ref{CouplingfF}) and (\\ref{CouplingFf}).\n\n  Once the coupling constants are chosen such that the NP contributions have\n  the same sign as the SM one, it is possible to further constrain the parameter space\n  by requiring that the predicted branching ratio should not exceed the experimental\n  bound. From Eq.~(\\ref{BRtotal}) we\n  find the constraint\n  \\begin{equation}\n  -\\Delta C_{7\\gamma}(\\mu_b)+1.4 \\left(\\left|\\Delta C_{7\\gamma}(\\mu_b)|^2+|\\Delta C'_{7\\gamma}(\\mu_b)\\right|^2\\right)\\lesssim4.2 (6.1) \\times 10^{-2},\n  \\label{Condition2}\n  \\end{equation}\n  corresponding to the $1\\,\\sigma$ ($2\\,\\sigma$) departure from the experimental value, Eq.~\\eqref{BRexp}.\n  \n  It is straightforward to apply these constraints to all models belonging\n  to one of the three classes discussed above. It is not\n  possible to obtain more insight without specifying a particular model (an analysis will follow in \\cite{BCMS:progress}).\n  However, in the next section, we provide a simplified representative for each class of models\n and show how the constraints discussed so far apply to each of them.\n\n\n\\subsection{Toy-model examples}\n\\label{sec:ToyModel}\n\nFor each class discussed above, we consider a toy-model in order to justify\nthe approximations made in the previous sections and to exemplify the application\nof the model-independent constraints.\n\n\\begin{description}\n  \\item {\\it Classes 1) and 2):}\\quad Here, the relevant initial conditions are those\n  presented in Eqs.~(\\ref{InitialConditionQnn}) and (\\ref{InitialConditionQnnHat}).\n  The coupling constants entering these expressions are $C_{L,R}^{sb}$, $C_{L,R}^{ff}$,\n  $C_{L,R}^{sd}$ and $C_{L,R}^{bd}$, with $f=u,c,t,d,s,b$. To simplify the\n  analysis, we assume that all flavour-violating couplings with a\n  strange flavour are equal to $C_{FV}^{s}$, the two bottom couplings $C_{L,R}^{bd}$ equal to $C_{FV}^{b}$ and all the flavour conserving ones\n  equal to $C_{FC}$. Furthermore, we take\n  $V_{ts}=-0.04047$, $V_{tb}=0.999146$ \\cite{CKMfitter:2005}, assume that $g_H=g_2$, and consider only one heavy neutral gauge boson. In this way\n  we have defined a toy model with only four free parameters: three coupling constants and the mass of the neutral gauge boson.\n  \n  In Fig.~\\ref{fig:C7comparison1} on the left we show the breakdown of \n  $C_{7\\gamma}(\\mu_b)$ in its different contributions\n  as a function of the coupling constants $C_{FV}\\equiv C_{FV}^{s}= C_{FV}^{b}=C_{FC}$, for $M_{A_H}=1$ TeV. For completeness, we also show the exotic-quark contributions for $m'_F=10$ TeV. As expected from the discussion\n  in Sec.~\\ref{sec:3classes}, all the NP contributions apart from the neutral \n  current-current are negligible, justifying our approximations in the previous section.\n  We remark that $C^\\prime_{7\\gamma}(\\mu_b)$ almost coincides with the neutral current-current contribution (red line).\n  In Fig.~\\ref{fig:C7comparison1} on the right we show the value of the coupling constants $C_{FV}$, as\n  a function of $M_{A_H}$, for which the bound in Eq.~(\\ref{Condition2}) is saturated at the \n  $1 \\sigma$ and $2\\sigma$ level. As we can see, for small values of $M_{A_H}$ the\n  couplings are constrained to small values.\n  \n    \\begin{figure}[h!]\n\\centering\n\\includegraphics[]{.\/constraintWRcoup}\n\\hspace*{-10pt}\n\\includegraphics[]{.\/constraintWRscale}\n\\caption{\\it On the left, the different contributions to $C_{7\\gamma}(\\mu_b)$\n(solid line) are plotted as functions of $C_{FV}$: in blue the SM contribution, in red\nthe neutral current-current contribution, in green the overlapping contributions from exotic and SM down-type quarks. The shadowed region is excluded\nimposing the bound in Eq.~(\\ref{Condition2}) at $1\\sigma$ (lighter) and $2\\sigma$ (darker). On the right, we show the value of\n$C_{FV}$ as a function of $M_{A_H}$ for which the bound in  Eq.~(\\ref{Condition2})\nis saturated. Again the shadowed region represents the excluded values.\n\\label{fig:C7comparison1}}\n\\end{figure}\n\nIn Fig.~\\ref{fig:Constraint12} we separately present the implementation of the two model-independent\nconstraints, for $M_{A_H}=1$ TeV. On the left, the constraint on the sign of $\\Delta C_{7\\gamma}(\\mu_b)$ \nmostly reduces the parameter space to cases in which the coupling constants \n$C_{FV}^{s}$ and $C_{FV}^{b}$\nhave opposite signs. In Fig.~\\ref{fig:Constraint12}\non the right, the bound from the branching ratio also applies, further constraining the parameter space.\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[]{.\/constraintWRsign}\n\\hspace*{-10pt}\n\\includegraphics[]{.\/constraintWRbr}\n\\caption{\\it On the left (right), the points satisfying $\\Delta C_{7\\gamma}(\\mu_b)<0$\n(Eq.~(\\ref{Condition2})) in the plane $C_{FV}^{s}$ vs. $C_{FV}^{b}(\\equiv C_{FC})$.\nThe shadowed regions now represent the values for which the bounds are passed: lighter (darker) intensity refers to $-4.2 (6.1)\\times10^{-4}$. \n\\label{fig:Constraint12}}\n\\end{figure}\n\n\\item {\\it Class 3):}\\quad We consider here the case with only one heavy neutral\n  gauge boson and one exotic fermion. The relevant initial conditions are those in\n  Eqs.~(\\ref{LLnew}) and (\\ref{LRnew}). The coupling constants which enter these\n  expressions are $C_{L,R}^{sF}$ and $C_{L,R}^{bF}$. Fixing $g_H=g_2$, $M_{A_H}=1$ TeV\n  and $m'_F=10$ TeV, and identifying the coupling constants $C_{FV}\\equiv C_{L,R}^{sF}=C_{L,R}^{bF}$, we illustrate in Fig.~\\ref{fig:C7comparison2} on the left that the dominant NP contributions stems solely from exotic quarks. For a complete comparison, we also plot the contributions from light quarks and neutral current-current operators, adopting the same conventions as for Fig.~\\ref{fig:C7comparison1}.\n\n  In Fig.~\\ref{fig:C7comparison2} on the right we show the value for the coupling\n  constants $C_{FV}$ as function of $M_{A_H}$ for which the $1\\sigma$ and $2\\sigma$ bounds in Eq.~(\\ref{Condition2})\n  are saturated. As we can see the constraint on the couplings is very strong and only\n  a small part of the shown parameter space survives even for large values of $M_{A_H}$.\n  \n  \\begin{figure}[h!]\n\\centering\n\\includegraphics[]{.\/constraintNRcoup}\n\\hspace*{-10pt}\n\\includegraphics[]{.\/constraintNRscale}\n\\caption{\\it A similar description to Fig.~\\ref{fig:C7comparison1} applies. Here, however, the green (purple) line refers to the exotic-quark (SM-quark) contributions.\n\\label{fig:C7comparison2}}\n\\end{figure}\n\n  In Fig.~\\ref{fig:Constraint3} on the left (right) we show how the couplings are\n  constrained by the expression in Eq.~(\\ref{Condition1_3}) (Eq.~(\\ref{Condition2})):\n  a negative value for $\\Delta C_{7\\gamma}(\\mu_b)$ is only recovered when both\n  $C_L^{sF}$ and $C_{L,R}^{bF}$ have the same sign, while the lower bound\n  in Eq.~(\\ref{Condition2}) provides a very strong constraint on the parameters of the model.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[]{.\/constraintNRsign}\n\\hspace*{-10pt}\n\\includegraphics[]{.\/constraintNRbr}\n\\caption{\\it On the left (right), the points satisfying $\\Delta C_{7\\gamma}(\\mu_b)<0$\n(Eq.~(\\ref{Condition2})) in the plane $C_L^{sF}$ vs. $C_R^{bF}$, for $C_L^{bF}\\in[-1,1]$. The shadowed regions are defined in Fig.~\\ref{fig:Constraint12}. \n\\label{fig:Constraint3}}\n\\end{figure}\n\\end{description}\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nExtensions of the SM in which the gauge group is enlarged by additional symmetries are attractive, because they provide an explanation of the flavour puzzle and predictions testable at colliders. In these models heavy exotic fermions are usually introduced in order to cancel possible anomalies and to justify the observed SM fermion spectrum through the see-saw mechanism. On the other hand, the presence of new heavy neutral gauge bosons and exotic fermions in principle translates into NP contributions to FCNC processes. \n\nIn this paper, we have pointed out two new contributions to the $\\ov{B}\\to X_s\\gamma$ decay that arise in such models. The first is generated through one-loop diagrams mediated by neutral gauge bosons and exotic down-type quarks. The relevance of this contribution depends on the strength of the left-handed and right-handed flavour violating couplings of the neutral gauge bosons to the SM and the exotic quarks. Analogous effects can be present in other processes like $\\mu\\to e\\gamma$ and $t\\to c\\gamma$ as well as flavour conserving observables like $(g-2)_\\mu$ and EDMs, in which dipole operators play the dominant role.\n\nThe second contribution is due to the presence of neutral current-current operators, mediated by neutral gauge bosons, and arises only if the neutral gauge bosons have flavour violating couplings to the SM quarks. Through the QCD mixing with the magnetic dipole operator $Q_{7\\gamma}$, these new neutral current-current operators contribute to $\\ov{B}\\to X_s\\gamma$. To our knowledge these QCD effects have been calculated here for the first time. Furthermore, our QCD analysis of the mixing among neutral current-current, QCD penguin and gluonic dipole operators could also be relevant for other processes, such as non-leptonic two-body $B$ decays, and other observables, like $\\epsilon'\/\\epsilon$.\n\nBeside these new contributions, we have also considered the contributions arising from one-loop diagrams with the exchange of neutral gauge bosons and SM down-type quarks, that have been only partially analysed in the literature.\n\nWe have studied the impact of all these contributions in a model-independent approach and summarised the resulting constraints in Eqs.~(\\ref{Condition1_12})--(\\ref{Condition2}). We have presented these expressions in such a manner that in order to test a specific model, it is sufficient to specify only the values of the couplings of the neutral gauge bosons to SM and exotic quarks and their masses. In particular, it is not necessary to repeat the QCD analysis.\n\nA detailed application of this study on a concrete NP scenario is in progress. Here, without entering into details of a particular model, we have described three representative classes of models and discussed the relevance of these contributions. For models in the first class, the SM spectrum is enriched only by the neutral gauge bosons, but no exotic quarks are present. In this case the contributions from the exotic quarks are absent, but those from the neutral current-current operators turn out to have a potentially observable effect on the branching ratio of $\\ov{B}\\to X_s\\gamma$. The value of the masses of the neutral gauge bosons and the strength of their flavour violating couplings to SM fermions determine the relevance of this effect.\n\nThe second class accounts for models in which the SM fermion masses $m_f$ are explained through the see-saw mechanism with heavy exotic fermions of masses $m'_F$. In this case, the couplings of neutral gauge bosons to SM and exotic fermions are suppressed by terms $m_f\/m'_F$. Therefore, the contributions to $Br(\\ov{B}\\to X_s\\gamma)$ from the exchange of exotic quarks turn out to be negligible with respect to those from the QCD mixing of the neutral current-current and magnetic dipole operators.\n\nThe models in the third class are characterised by the presence of heavy exotic fermions, which either provide the SM fermion masses through a different mechanism than the see-saw or do not participate at all in the explanation of the SM flavour spectrum. In this case, no suppression occurs in the couplings of neutral gauge bosons to SM and exotic fermions and the contributions to $Br(\\ov{B}\\to X_s\\gamma)$ from exotic quarks are enhanced by terms $m'_F\/m_b$, where $m_b$ is the mass of the bottom quark. These contributions dominate over all the others.\n\nFor all the models in the three classes, the contributions from SM down-type quarks turn out to be subdominant in the whole parameter space.\n\nOur analysis shows once more how FCNC processes can put constraints on beyond-SM constructions even before the discovery of new particles in high-energy processes.\n\n\n\n\\section*{Acknowledgments}\nWe thank Joachim Brod and Robert Ziegler for useful comments on the preliminary version of the paper and Mikolaj Misiak, Gerhard Buchalla, and Paride Paradisi for interesting discussions. This work was supported by ERC Advanced Grant ``FLAVOUR'' (267104).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA typical machine learning task involves the minimization of an empirical loss function evaluated over instances within a specific training dataset. When the representation model is built on deep neural networks, millions of parameters are to be optimized; it is a common practice that one invokes a gradient-based optimization method to solve for the minimum of interest \\cite{lecun2015deep}. On the other hand, the datasets for training have been expanding to a very large scale, where the algorithms equipped with stochastic gradients are de facto dominating the practical use of gradient-based optimizers. Therefore, stochastic gradient optimizers have been of significant interest in the community of machine learning research in the era of deep learning, which facilitates various applications of deep learning to real-world problems. \n\n\nOriginated from the stochastic approximation \\cite{robbins1951stochastic,kushner2003stochastic}, various algorithms have been proposed and gained credits for the triumph of deep learning in a broad range of applications, including but not limited to finance \\cite{kim2020can,zhang2018benchmarking}, gaming AI \\cite{silver2016mastering,peng2017multiagent,yang2020overview}, autonomous driving \\cite{grigorescu2020survey,zhou2020smarts} and protein structure discovery \\cite{jumper2021highly}. The momentum methods were conceived along with the advent of back-propagation \\cite{rumelhart1986learning}, which has been proved to have a fundamental physical interpretation \\cite{qian1999momentum} and is very important for fast convergence \\cite{sutskever2013importance}. Adaptive methods, e.g. AdaGrad \\cite{duchi2011adaptive}, RMSProp \\cite{tieleman2012lecture} and Adam \\cite{kingma2014adam}, have demonstrated very fast convergence and robust performance, and now dominating the practical applications. Albeit greatly successful, it may still need more insights on the training dynamics guided by the gradient-based methods in a systematic perspective when one trains deep neural networks for real-world tasks.\n\n\nInvestigations devoted to this topic typically presume the stochastic gradients converge as discretizations of continuous-time dynamical systems, where the dynamical properties of the continuous-time counterpart are studied. The specification of the dynamical systems then lies in identifying the gradient noise. A classical assumption regarding the gradient noise is that it converges asymptotically towards Gaussian random variables due to the classical Central Limit Theorem (CLT)~\\cite{feller:1971,chung2001course}, behind which the variance of stochastic gradients are thought to be finite, and the size of mini-batch is reasonable enough for a good Gaussian approximation. Such assumption applies to typical variational analysis regarding SGD and its variants~\\cite{mandt2017stochastic}. Based on Gaussian noise assumption, some works approximate the dynamic of SGD by Langevin dynamics and proved its escaping efficiency from local minima~\\cite{li2017stochastic,he2019control,zhou2019toward,hu2019diffusion,luo2018thermostat,luo2020replica}. These works assume that the noise covariance of SGD is constant or upper bounded by some constant.\n\nRecently, the structure of the gradient noise has attracted more attention from the community. Many works study SGD from the perspective of its anisotropic nature of noise~\\cite{zhu2019anisotropic,wu2020noisy,xie2020diffusion}. Empirical evidence~\\cite{simsekli2019tail} has also been discovered that the gradient variance may be infinite, and it is the infinite variance that violates the conditions of the classical CLT with the Gaussian approximation. They introduced the generalized CLT which showed that the law of the sum of these i.i.d. variables with infinite variance still converges, but to a different family of heavy-tailed distributions.\n\n \nWe attribute such heavy tails phenomenon as the result of insufficient batch size for the noise (\\emph{i.e.} a sum of limited \\emph{i.i.d.} random variables) to converge to Gaussian rather than the hypothesis that the noise is heavy-tailed by its nature \\cite{simsekli2019tail,simsekli2020fractional}. \nThe investigation is taken through a different lens by evaluating the Gaussianity of noise in terms of the percentage of Gaussian elements for learnable weights in layers during the training procedure with different batch sizes: the gradient noise still exhibits good Gaussianity when the batch size increases. That is, the larger batch size we train, the higher Gaussianity for gradient noise we observe. Notably, there are some similar results showing in~\\cite{panigrahi2019non} where they observe Gaussian behavior for gradient noise distribution at large batch size (over 500). To gain a better understanding and a more quantitative measurement, we involve the Berry-Esseen theorem which characterizes the rate of convergence for CLT~\\cite{chung2001course} and then establish a link between the rate of convergence and the magnitude of the square root of kurtosis of the noise. With such evidence of Gaussian noise, we analyze the dynamics of SGD and unify the discretization scheme w.r.t. the general Langevin equations: the vanilla SGD corresponds to the high-friction limit of the continuous-time dynamics. The learning rate essentially corresponds to the effective temperature of the associated physical system. We provide a simple local approximation of the distribution of learning parameters when the dynamics enter the vicinity of minima.\n\n\n\n\n\nThe main contribution of this paper is twofold.\n1. We demonstrate that the noise within stochastic gradients possesses finite variance during the training procedure of deep neural networks, and therefore the noise is asymptotically Gaussian as is generally conceived due to the application of the classical Central Limit Theorem. We established a link of the rate of convergence to the kurtosis of the noise which can be readily computed.\n2. We analyze the dynamics of stochastic gradient descent and the distribution of parameters, clarifying that the distribution deviates from the Gibbs distribution in the traditional settings. We provide a local approximation to the distribution when the parameters are close to optimal, which resembles an isotropic quadratic potential.\n\n\n\n\n\n\n\n\n\\section{Analysis of Gaussian Behavior}\n\n\\label{analysis}\nIn this section, we show that the gradient noise would not admit infinite variance during neural network training with the analysis on forward and backward propagation under reasonable assumptions.\nAnd we further explain the non-Gaussianity phenomenon from the perspective of the Berry-Esseen theorem.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Preliminaries}\nThe optimization problems for neural networks usually involve minimizing an empirical loss with training data $\\left\\{x_{s}, y_s\\right\\}_{s=1}^{S}$ over the network parameter $\\theta \\in \\mathbb{R}^{d}$.\nTo be more specific, we look for an optimal $\\theta^\\star$, st.\n\\begin{align}\n    \\theta^{\\star}=\\underset{\\theta \\in \\mathbb{R}^{d}}{\\arg \\min }\\left\\{\\mathscr{L}(\\theta) \\triangleq \\frac{1}{|\\mathscr{S}|} \\sum_{x \\in \\mathscr{S}} \\ell(x;\\theta)\\right\\},\n\\end{align}\n\nwhere $\\mathscr{L}: \\mathbb{R}^{d} \\rightarrow \\mathbb{R}$ denotes the empirical loss function in $\\theta$, each $\\ell(x;\\theta)$ denotes the loss over one sample. \nStochastic gradient descent (SGD) is one of the most popular approaches for attacking this problem in practice. \nIt performs parameter update $\\theta_{t+1}=\\theta_{t}-\\alpha \\nabla\\mathscr{L}(\\theta_t)$, using the stochastic gradient $\\nabla\\mathscr{L}(\\theta)$ estimated from the mini-batch $\\mathscr{S} = \\{x_s\\}$ and the learning rate $\\alpha$, where\n\\begin{align}\n\\nabla\\mathscr{L}(\\theta) = \\frac[1pt]{1}{|\\mathscr{S}|} \\sum_{x \\in \\mathscr{S}}\\frac{\\partial\\ell(x;\\theta)}{\\partial{\\theta}}.\n\\end{align}\n\nThe stochastic gradient noise, due to the fact that the gradients can be evaluated on different mini-batches, can be interpreted from additive and multiplicative viewpoints~\\cite{wu2019multiplicative}. \nIn this paper, we adopt the additive formulation and use $\\epsilon$ to represent the additive noise, \n\\begin{align}\n\\epsilon \\triangleq \\nabla\\mathscr{L}(\\theta) - \\nabla\\mathbb{E}\\ell(x;\\theta).\n\\end{align}\n\nWhile $\\theta$ assembles all the parameters in the feed forward neural network, we use $W^{(k)}$ and $b^{(k)}$ as weight matrix and bias vector in the $k$-th layer. Feed forward procedure is\n$u^{(k)} = W^{(k)}a^{(k-1)} + b^{(k)}$, where $a^{(k)}=f_k\\big(u^{(k)}\\big)$ denotes the input of the $k$-th layer (post-activation of the ($k$-1)-th layer), $u^{(k)}$ denotes the pre-activation of the $k$-th layer and $f_k$ denotes the activation function entrywisely applied on the pre-activation. \nThe updating rule for an entry $W_{ij}^{(k)}$ (at the i-th row and the j-th column) in the weight matrix $W^{(k)}$ is \n\\begin{align}\n W_{ij}^{(k)} = W_{ij}^{(k)} - \\alpha \\frac{\\partial\\mathscr{L}}{\\partial W_{ij}^{(k)}}.\n\\end{align}\n\n\n\n\n\\subsection{Central Limit Theorem}\nThe distinction between the requirements of CLT and Generalized CLT is whether the variances of summed variables are finite.\nIn this section, we analyze the variance of gradient noise during neural network training and thus show that the CLT still holds under mild assumptions. Firstly we provide mathematical induction to show that the variance of pre-\/post-activations will not be infinite through the forward propagation. Next, we analytically show that the partial derivative of loss function w.r.t. the output of the neural network and the derivatives of loss function w.r.t. weights will not admit infinite variance.\n\nNote that different steps in the proof have different minimum requirements on the data distribution for the conclusions to hold true.\nThe strongest assumption on the data in our discussion is that the data distribution has finite fourth order moments.\nWhile this is automatically true for empirical distributions, which have been adopted in the literature ~\\cite{jacot2018neural}, it is also true for discrete or continuous distributions with bounded support, e.g., image data with range $[0, 255]$. Preceding the full proof, we show the following lemma for Lipschitz functions which will be used for the variance analysis of activation functions.\\\\\n\n\\noindent\\textbf{Zero mean for SGD noise}\\\\\nDeterministic, full-batch gradient:\n\\begin{align}\n\\phi(\\theta) = \\nabla\\mathbb{E}\\ell(x;\\theta) = \\int_x \\frac{\\partial\\ell(x;\\theta)}{\\partial{\\theta}} \\diff{\\mathscr{D}(x)}.\n\\end{align}\nStochastic, mini-batch gradient:\n\\begin{align}\n\\nabla\\mathscr{L}(\\theta) = \\frac[1pt]{1}{|\\mathscr{S}|} \\sum_{x \\in \\mathscr{S}}\\frac{\\partial\\ell(x;\\theta)}{\\partial{\\theta}} \\approx \\int_x \\frac{\\partial\\ell(x;\\theta)}{\\partial{\\theta}} \\diff{\\mathscr{D}(x)},\n\\end{align}\n\\begin{align}\n\\mathbb{E}[\\nabla\\mathscr{L}(\\theta)] &=\n\\mathbb{E}\\bigg[\\frac[1pt]{1}{|\\mathscr{S}|} \\sum_{x \\in \\mathscr{S}} \\frac{\\partial\\ell(x;\\theta)}{\\partial{\\theta}}\\bigg] \\nonumber\\\\\n&=\\frac[1pt]{1}{|\\mathscr{S}|} \\sum_{x \\in \\mathscr{S}} \\nabla\\mathbb{E}\\ell(x;\\theta) = \\phi(\\theta).\n\\end{align}\n\nSo we can obtain the expectation of stochastic gradient noise\n$\\mathbb{E}[\\epsilon] = \\mathbb{E}[\\nabla\\mathscr{L} - \\mathbb{E}[\\nabla\\mathscr{L}(\\theta)]] = \\mathbb{E}[\\nabla\\mathscr{L} - \\phi] = 0$.\\\\\n\n\n\\noindent\\textbf{Finite variance for SGD noise} \n\\begin{lemma}\\label{lm:lip}\nAssume that a random vector $x\\in\\mathbb{R}^d$ has finite variance on each entry. $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ is a Lipschitz function with some constant $c>0$ and entry-wisely applied on $x$.\nThen the random vector $f(x)\\in\\mathbb{R}^d$ has finite variance on each entry.\n\\end{lemma}\nThe feed forward process could be regarded as recurrent applications of activations and linear transformations.\nWith the finite fourth moments assumption, we know the second order moments of the input data $a^{(0)} :=x$ are finite.\nIf the input to the $k$-th layer, i.e., $a^{(k-1)}$, has finite variance, we show that $a^{(k)}$ has finite variance.\nDenote the mean and co-variance of $a^{(k-1)}$ respectively by $\\bar{a}^{(k-1)}$ and $\\Sigma^{(k-1)}$.\nRecall $u^{(k)}=W^{(k)}a^{(k-1)}+b^{(k)}$, the mean and covariance of $u^{(k)}$ would be\n\\begin{align}\n\\mathbb{E}[u^{(k)}] &= W^{(k)}\\bar{a}^{(k-1)}+b^{(k)},\\\\ \\mathbf{cov}[u^{(k)}] &= W^{(k)}\\Sigma (W^{(k)})^\\top\n\\end{align}\nClearly, the variance of $u^{(k)}$, as part of the covariance $\\mathbf{cov}[u^{(k)}]$, is finite.\nWe apply non-linearity $f_k(\\cdot)$ to $u^{(k)}$ element-wisely and get the activation $a^{(k)}$, i.e. $a^{(k)}=f_k(u^{(k)})$.\nFor commonly adopted activations, i.e., Sigmoid, ReLU, LeakyRelu, Tanh, SoftPlus, etc, they are all Lipschitz functions.\nWith Lemma \\ref{lm:lip}, the variance of $a^{(k)}$ is also finite.\nApplying this logic recursively, we would know for the whole neural network, variances of all pre-\/post-activations are finite. The variance of the final output $a^{(K)}$ is finite as well.\n\nTo use induction for back propagation, we show the variance of the derivative of loss function w.r.t. $a^{(K)}$ is finite.\nHere we investigate two variants of loss functions, i.e., mean square error loss function for regression problems and cross-entropy loss function for classification problems.\nThe mean square error loss is defined as $\\mathscr{L}_{MSE} = ||y-a^{(K)}||^2$.\nSince $\\partial \\mathscr{L}_{MSE}\/\\partial a^{(k)}=-2(y-a^{(k)})$, it is obvious that it has finite variance with our assumption on data and above induction.\nTo complete the discussion, we show the results for cross-entropy loss in the following lemma.\n\\begin{lemma}\nFor cross entropy loss function $\\mathscr{L}_{CE} = -\\sum_{j=1}^M y_j\\log\\big[\\exp{a^{(K)}_j}\/\\sum_{k}\\exp{a^{(K)}_k}\\big]$, the gradient w.r.t. the network output $a^{(K)}$, i.e., $\\partial \\mathscr{L}_{CE}\/\\partial a^{(K)}$, as a function on random variables $a^{(K)}$ and label $y$, has finite variance.\n\\end{lemma}\nThe derivatives of the loss function w.r.t. each entry of the weight matrix admit the following forms by the chain rule,\n\\begin{align}\\label{eq:grad_w}\n\\frac[1pt]{\\partial\\mathscr{L}}{\\partial{W^{(k)}_{ij}}} = \\frac[1pt]{\\partial\\mathscr{L}}{\\partial{u^{(k)}_{i}}}\\frac[1pt]{\\partial{u^{(k)}_{i}}}{\\partial{W^{(k)}_{ij}}}.\n\\end{align}\nNote that the partial gradient $\\partial\\mathscr{L}\/\\partial W_{ij}^{(k)}$ consists of both global information, i.e., $\\partial\\mathscr{L}\/\\partial u^{(k)}_i$, and local information, i.e.,  $\\partial u^{(k)}_i\/\\partial W_{ij}^{(k)}$. For the global information, there holds the following recurrent relationship\n\\begin{align}\\label{eq:grad_recurrent}\n\\frac[1pt]{\\partial\\mathscr{L}}{\\partial{u^{(k-1)}}} = \\bigg( \\big[W^{(k)}\\big]^\\top \\frac[1pt]{\\partial\\mathscr{L}}{\\partial{u^{(k)}}} \\bigg) \\odot f'_{k-1}\\big(u^{(k-1)}\\big),\n\\end{align}\nwhere $\\odot$ denotes the entrywise Hadamard product. Since $\\partial\\mathscr{L}\/\\partial a^{(K)}$ has finite variance and Lipschitz continuous activation functions have bounded (sub-)gradients, $\\partial\\mathscr{L}\/\\partial u^{(K)}$ also has finite variance.\nFrom Equation~\\ref{eq:grad_recurrent} we know $\\partial\\mathscr{L}\/\\partial u^{(k-1)}$ has finite variance.\nBy mathematical induction we conclude that $\\partial\\mathscr{L}\/\\partial u^{(k)}$ has finite variance for all $k$. For the local information, \n\\begin{align}\n\\label{eq:u_tau}\n\\frac[1pt]{\\partial u_{\\tau}^{(k)}}{\\partial W^{(k)}_{ij}}=\n\\begin{cases}\n\\,a_{j}^{(k-1)}& \\text{$\\tau      =      i$}\\\\\n\\,0& \\text{$\\tau      \\neq      i$}\n\\end{cases}\n\\end{align}\n$\\partial u^{(k)}\/\\partial W_{ij}^{(k)}$ is a vector with the $i$-th entry being $a_j^{(k-1)}$ and other entries being $0$'s. \nWith the finiteness assumption on fourth order moments, we can employ Cauchy\u2013Schwarz inequality to bound the variance of $\\partial\\mathscr{L}\/\\partial W^{(k)}_{ij}$, c.f. the supplemental material. \nSince gradient noise is the centered version of the partial gradient $\\partial\\mathscr{L}\/\\partial W^{(k)}_{ij}$\n, it will also have finite variance.\n\n\\subsection{CLT convergence rate}\n\nIn the previous subsection, we showed that the variance of gradient noise is finite under reasonable assumptions on the data and support that CLT will hold.\nThen the theorem about convergence rate of CLT, i.e. Berry\u2013Esseen theorem \\cite{feller:1971,chung2001course} applies.\n\\begin{theorem}[Berry\u2013Esseen theorem]\nThere exists a universal constant $A_0$, s.t. for any i.i.d. random variables $\\{\\chi_s\\}_{s=1}^S$ with $\\mathbb{E}[\\chi_s]=0$, $\\mathbf{var}[\\chi_s]=\\sigma^2$ and $\\beta := \\mathbb{E}[|\\chi_s|^3]<\\infty$, the following inequality holds\n\\begin{align}\\label{eq:betheorem}\n    \\sup_\\chi |F_{|\\mathscr{S}|}(\\chi)-\\Phi(\\chi)| \\le \\frac{\\beta}{\\sigma^3}\\frac{A_0}{\\sqrt{|\\mathscr{S}|}},\n\\end{align}\nwhere $\\Phi$ is the c.d.f. of standard Gaussian distribution and $F_{|\\mathscr{S}|}$ is the c.d.f. of random variable $\\sum_{s=1}^S\\chi_s\/\\sqrt{|\\mathscr{S}|}\\sigma$.\n\\end{theorem}\n\n\n\n\n\n\nMore specifically, assume that neural network is a deterministic function, $x$ is an input datum, $\\ell(x;\\theta)$ is a loss function per datum, $u^{(k)}(x)$ is a pre-activation at $k$-th layer. By Equation~\\ref{eq:grad_recurrent}, we can have\n\\begin{align}\n\\frac[1pt]{\\partial\\mathscr{L}}{\\partial{u^{(k)}}}(x) = \\bigg( \\big[W_{ij}^{(k+1)}\\big]^\\top \\frac[1pt]{\\partial\\mathscr{L}}{\\partial{u^{(k+1)}}}(x) \\bigg) \\odot f'_{k}\\big(u^{(k)}(x)\\big),\n\\end{align}\nwhich is a function of $x$. The mini-batch gradient ($x_s \\in \\mathscr{S}$)\n\\begin{align}\n\\nabla\\mathscr{L}(\\theta) = \\frac[1pt]{1}{|\\mathscr{S}|} \\sum_{s}g_s = \\frac[1pt]{1}{|\\mathscr{S}|} \\sum_{s}\\frac[1pt]{\\partial\\ell(x_s)}{\\partial{W_{ij}^{(k)}}}.\n\\end{align}\nRecall Equation~\\eqref{eq:grad_w} \\& \\eqref{eq:u_tau}, we have \n\\begin{align}\ng_s = \\frac[1pt]{\\partial\\ell(x_s)}{\\partial{W_{ij}^{(k)}}} = \\sum_{\\tau}\\frac[1pt]{\\partial\\ell(x_s)}{\\partial{u^{(k)}_{\\tau}}}\\frac[1pt]{\\partial{u^{(k)}_{\\tau}}}{\\partial{W^{(k)}_{ij}}} = \\frac[1pt]{\\partial\\ell(x_s)}{\\partial{u^{(k)}_{i}}}a_{j}^{(k-1)}\n\\end{align}\nFor each data point, the gradient noise $\\triangle g = g - \\mathbb{E}[g]$ is zero mean. From Inequality Equation~\\eqref{eq:betheorem}, we know that the convergence of CLT is bounded by three quantities. Since $A_0$ is the universal constant and $|\\mathscr{S}|$ is the batch size, we need to estimate the standardized third absolute moment $\\beta\/\\sigma^3$. We define\n\\begin{align}\n\\beta &= \\mathbb{E}[|\\triangle g|^3],\\\\\n\\sigma^2= \\mathbf{var}[\\triangle g] &= \\mathbf{var}[g] = \\mathbf{var}\\bigg[\\frac[1pt]{\\partial\\ell(x_s)}{\\partial{u^{(k)}_{i}}}a_{j}^{(k-1)}\\bigg].\n\\end{align}\nSince the third absolute moment is hard to compute, with $\\sigma^2= \\mathbb{E}[(\\triangle g)^2]$, we provide a ready upper bound using Cauchy-Schwarz Inequality,\n\\begin{align}\n\\frac[1pt]{\\beta}{\\sigma^3} &= \n\\frac[1pt]{\\mathbb{E}[|\\triangle g(\\triangle g)^2|]}{\\sigma^3}\\leq\n\\frac[1pt]{\\sqrt{\\mathbb{E}[(\\triangle g)^2]}\\sqrt{\\mathbb{E}[(\\triangle g)^4]}}{\\sigma^3}\\nonumber\\\\&=\n\\frac[1pt]{\\sigma \\sqrt{\\mathbb{E}[(\\triangle g)^4]}}{\\sigma^3}=\n\\sqrt{\\mathbb{E}\\bigg[\\bigg(\\frac[1pt]{\\triangle g}{\\sigma}\\bigg)^4\\bigg]}\n\\end{align}\nWe denote standard score $z = \\triangle g\/ \\sigma$ which is a random variable with zero mean and unit variance. By the interpretation of kurtosis from~\\cite{moors1986meaning},\n\\begin{align}\n\\mathbb{E}\\left[z^{4}\\right]&=\\mathbf{var}\\left[z^{2}\\right]+\\left[\\mathbb{E}\\left[z^{2}\\right]\\right]^{2}\\nonumber\\\\\n&=\\mathbf{var}\\left[z^{2}\\right]+[\\mathbf{var}[z]]^{2}=\\mathbf{var}\\left[z^{2}\\right]+1\n\\end{align}\nFinally we have the upper bound\n\\begin{align}\n\\frac[1pt]{\\beta}{\\sigma^3} \\leq  \\sqrt{\\mathbf{var}\\left[z^{2}\\right]+1}\n\\end{align}\n\n\nStanding on this ground, we assert that the non-Gaussianity of gradient noise observed in some neural network training is due to the fact that the asymptotic regime has not been reached with that settings.\nIn the following section, we also support our statements with some numerical experiments.\n\n\\section{Dynamics of SGD}\n\nGiven gradient noise is asymptotically Gaussian, the dynamics of gradient descent resembles the motion of a particle within a potential field. The topic can be related to the classical Kramers Escape Problem~\\cite{kramers1940brownian}.\nConsider the procedure of gradient descent optimizers, the parameter vector $\\theta \\in \\mathbb{R}^d$ is updated according to the gradient of loss, which resembles the evolution of position of a particle within the external position-dependent potential $\\phi(\\theta) = \\mathbb{E}\\ell(\\theta)$. With gradient being noisy, the latter is typically described by the Langevin equation and the stochastic gradient noise is effectively the random kicks acting on the particle\n\\begin{align}\n\\label{eq:langevin}\nm\\frac[1pt]{\\diff^2{\\theta}}{\\diff{t}^2} + \\gamma\\frac[1pt]{\\diff{\\theta}}{\\diff{t}} = -\\nabla\\phi(\\theta) + \\eta(\\theta, t)\n\\end{align}\nwhere $\\mathbb{E}\\ell(\\theta)$ is the average of gradients evaluated within a mini-batch also expressed as the underlying potential $\\phi$, and $\\eta(\\theta, t) \\in \\mathbb{R}^d$ is $\\theta$-dependent $\\delta$-correlated random fluctuation. $\\mathbb{E}[\\eta(\\theta, t)\\eta(\\theta, t')^\\top] = \\mathsf{D}(\\theta)\\delta(t - t')$ with $\\mathsf{D}(\\theta)$ denoting the corresponding $\\theta$-dependent diffusion matrix. Discretization w.r.t. time translates the random kicks as\n\\begin{align}\n\\eta(\\theta, t) \\to \\epsilon_n(\\theta) \\sim \\mathscr{N}\\big(0, \\mathsf{C}(\\theta)\\big),\n\\end{align}\nwhere we can express the covariance of SGN $\\epsilon$ as $\\mathsf{C}(\\theta) = \\mathsf{D}(\\theta)\/\\Delta{t}$. The discretization of the Langevin equation \\eqref{eq:langevin} w.r.t. time reads\n{\\small\n\\begin{align}\n\\label{eq:discret_langevin}\nm\\frac{\\Delta^2\\theta_{n}}{\\Delta{t}^2} + \\gamma\\frac{\\Delta\\theta_{n}}{\\Delta{t}} = -\\nabla\\mathbb{E}\\ell(\\theta) + \\epsilon_{n-1}(\\theta) = -\\nabla\\mathscr{L}(\\theta_{n-1}).\n\\end{align}\n}\nLet us define $v_n = \\Delta\\theta_n = \\theta_n - \\theta_{n-1} = \\theta(n\\Delta{t}) - \\theta((n-1)\\Delta{t})$, substituting $v_n$ into Equation~\\eqref{eq:discret_langevin} we can obtain\n\\begin{align}\nm\\frac{v_n - v_{n-1}}{\\Delta{t}^2} + \\gamma\\frac[1pt]{v_n}{\\Delta{t}} = -\\nabla\\mathscr{L}(\\theta_{n-1})\n\\end{align}\nBy re-arranging terms, \n\\begin{align}\nv_n = \\underset{\\rho}{\\underbrace{\\frac{1}{1 + \\gamma\\Delta{t} \/ m}}} v_{n-1} - \\underset{\\alpha}{\\underbrace{\\frac{\\Delta{t}}{\\gamma + m\/\\Delta{t}}}} \\nabla\\mathscr{L}(\\theta_{n-1}).\n\\end{align}\nDefining momentum term as $\\rho$ and learning rate as $\\alpha$,\nwe finally have\n\\begin{align}\n\\label{eq:sgd}\n\\begin{cases}\n\\, v_n = \\rho v_{n-1} - \\alpha \\nabla\\mathscr{L}(\\theta_{n-1})\\\\\n\\, \\theta_n = \\theta_{n-1} + v_n\n\\end{cases}\n\\end{align}\nwhich resembles the formulation of the classical momentum method~\\cite{sutskever2013importance} or method of ``heavy-sphere''~\\cite{polyak1964some}. For high-friction limit, i.e. $\\gamma \\gg m\/\\Delta{t}$, $\\rho \\to 0^+$, $\\alpha \\to \\Delta{t} \/ \\gamma$, Equation~\\eqref{eq:sgd} can be simplified as the update rule for vanilla SGD\n\\begin{align}\n\\theta_n = \\theta_{n-1} - \\alpha \\nabla\\mathscr{L}(\\theta_{n-1}).\n\\end{align}\nThis can be obtained alternatively by leveraging the \\emph{overdamped} Langevin equation, where we omit the second derivative in Equation~\\eqref{eq:langevin} as in high-friction limit the inertial force is negligible comparing to the friction.\n\n\nNow we focus on the Langevin equation which is the analogue of momentum method in continuous time. Associated with the Langevin equation~\\eqref{eq:langevin}, the equation describing the evolution of the distribution $\\pi(\\theta, \\varv, t)$ is the well-known \\emph{Kramers equation} with coordinates $\\theta$ and velocity $\\varv = \\diff{\\theta} \/ \\diff{t} \\in \\mathbb{R}^d$. We write the equation in the form of \\emph{continuity equation} \\cite{risken1996fokker,van1992stochastic} as follows, \n\\begin{align}\n\\label{eq:kramers}\n\\frac[1pt]{\\partial{\\pi}}{\\partial{t}} -\\,\\mathbf{div}\\,\\mathcal{J} = 0\n\\end{align}\nwhere we suppose the unit mass $m = 1$ and unity temperature, $\\mathcal{J}$ is the probability current\n\\begin{align}\n\\label{eq:j}\n\\mathcal{J} =\n\\begin{Bmatrix}\n\\begin{bmatrix} \n-\\varv\\\\[\\jot]\n\\gamma\\varv + \\nabla\\phi(\\theta)\n\\end{bmatrix} + \\frac[1pt]{1}{2}\n\\begin{bmatrix}\n0\\\\\n\\mathsf{D}(\\theta) \\cdot \\nabla_{\\varv}\n\\end{bmatrix}\n\\end{Bmatrix}\\pi(\\theta, \\varv, t)\n\\end{align}\nHowever, one may not always find such a solution for the general Fokker-Planck equation~\\cite{van1992stochastic}. For the steady-state distribution to exists, some conditions have to be imposed on the drift vector as well as the diffusion matrix; it can be shown that under mild conditions the existence of the steady-state distribution for \\eqref{eq:kramers} can be found~\\cite{soize1994fokker}.\n\n\\begin{proposition}\n\\label{p:detailed_balance}\nThe dynamics \\eqref{eq:kramers} in general does not satisfy detailed balance. The steady-state distribution associated with \\eqref{eq:kramers}, if exists, deviates from the classical Gibbs distribution in thermal equilibrium.\n\\end{proposition}\n\nThe deviation has been discovered recently in training neural networks~\\cite{kunin2021rethinking}; Similar phenomena is observed in physics, which is often referred to as \\emph{broken detailed balance}~\\cite{battle2016broken,gnesotto2018broken}.\n\n\\begin{assumption}\n\\label{asm:c2}\nThe potential $\\phi \\in \\mathcal{C}^2$ is second-order differentiable in the vicinity of mimimum $\\theta_*$.\n\\end{assumption}\n\\begin{align}\n\\label{eq:hessian}\n\\mathsf{C}(\\theta) = \\mathbf{cov}[\\nabla\\mathscr{L}(\\theta)] &= \\nabla^2\\phi(\\theta) - \\nabla\\phi(\\theta)\\nabla^\\top\\phi(\\theta) \\notag\\\\\n& \\approx \\nabla^2\\phi(\\theta_*) = \\mathsf{H}(\\theta_*) = \\mathsf{H}_*\n\\end{align}\nThe first approximation is due to the fact that gradient noise variance dominates the gradient mean near critical points~\\cite{zhu2019anisotropic,xie2020diffusion}. When evaluating at the true parameter $\\theta_*$, there is the exact equivalence between the Hessian $\\mathsf{H}$ and Fisher information matrix $\\mathsf{F}=\\nabla^2\\phi(\\theta_*)$ at $\\theta_*$, referring to Chapter 8 of~\\cite{pawitan2001all}.\nAccording to~\\cite{zhang2019algorithmic,xie2020diffusion}, we can approximate the potential in the vicinity of $\\theta_*$ by its second-order Taylor series as\n\\begin{align}\n\\phi(\\theta) \\approx \\phi(\\theta_*) &+ \\nabla^\\top\\phi(\\theta_*) (\\theta - \\theta_*) \\nonumber\\\\&+ \\frac[1pt]{1}{2} (\\theta - \\theta_*)^\\top \\mathsf{H}_* (\\theta - \\theta_*)\n\\end{align}\n\n\nNote that the diffusion matrix $\\mathsf{D}(\\theta) = \\Delta{t}\\mathsf{C}(\\theta)$ is essentially small variables~\\cite{xie2020diffusion,meng2020dynamic}. With $\\Delta{t} \\ll 1$, one may approximate the diffusion by a constant matrix with no dependency w.r.t. $\\theta$ \\cite{luo2017thermostat,luo2020replica}.\n\n\\begin{proposition}\n\\label{p:ss}\nThe steady-state distribution can be approximated when $\\theta$ is in the vicinity of local minimum $\\theta_*$, given Assumption \\ref{asm:c2} applies (Proof in Appendix),\n{\\small\n\\begin{align}\n\\label{eq:steady-state}\n\\pi^{\\mathrm{ss}} \\propto \\exp\\Big[ -\\frac[1pt]{\\gamma}{\\Delta{t}} (\\theta - \\theta_*)^\\top(\\theta - \\theta_*) \\Big] \\exp\\Big[ -\\frac[1pt]{\\gamma}{\\Delta{t}}\\varv^\\top \\mathsf{H}_*^{-1}\\varv \\Big].\n\\end{align}\n}\n\\end{proposition}\nIn the scenario that we assume constant covariance in a small noise regime, the marginal distribution w.r.t. $\\theta$ has the form of Boltzmann distribution with an efficient potential. If $\\Delta{t} \\ll 1$, Equation~\\eqref{eq:steady-state} indicates that the probability peaks at the minima~\\cite{hanggi1990reaction,matkowsky1977exit}. With the covariance as a variable, the distribution exists but may not have a closed form, but we believe the property of Proposition~\\ref{p:ss} provides us with a good approximation.\n\n\n\n\n\n\n\n\n\\section{Experimental Results}\n\nIn this experiment, we aim to verify the Gaussianity of stochastic gradient noise and find what kind of factors can affect the Gaussian behavior. We check the Gaussianity of stochastic gradient noise on different neural network architectures and conduct experiments on different models, namely AlexNet~\\cite{krizhevsky2012imagenet} and Residual Neural Network (ResNet)~\\cite{he2016deep} for classification problems. During the training, we test the Gaussianity of gradient noise for each layer. The data used for both networks is CIFAR-10\\footnote{https:\/\/www.cs.toronto.edu\/~kriz\/cifar.html} dataset which consists of 60,000 colour images ($32 \\times 32$ pixels) in 10 classes. There are 50,000 training images and 10,000 test images, with 6,000 images per class. \n\nIn statistics, Gaussianity testing is a well-studied topic. The Shapiro and Wilk's $W$-test~\\cite{shapiro1965analysis} is found to be the top test in a comparison study~\\cite{razali2011power}. It assumes the null hypothesis that the given set of examples come from a Gaussian distribution. A small p-value provides strong evidence to reject the null hypothesis. The extension of the Shapiro-Wilk test for large samples up to $2000$ \\cite{royston1982extension} (from the original sample size of 50). We use a revised algorithm called \\textbf{AS R94} for fast calculation of the Shapiro-Wilk test \\cite{royston1995remark} in our experiment. More specifically, in each layer we sample points from all the batches of stochastic gradient noise and test whether they are belonging to Gaussian distribution or not, then we calculate the percentage of Gaussianity for all weight dimensions to verify the Gaussianity assumption of gradient noise. During the experiment, we find some extreme cases where the gradient noise samples in one batch follow point mass behavior due to the ReLU activation function. We refer to these distributions as the Dirac delta function~\\cite{dirac1981principles} which is the limit (in the sense of distributions) of the sequence of zero-centered Gaussian distributions. In the experimental results, the Shapiro-Wilk test also recognizes point mass data as Gaussian behavior.\nIt is worth noting that the image data have bounded support, and thus the gradient noise will not have infinite tails. As a result, it is not necessary to use tail index measuring to exam if the components rejected by the Gaussianity test have infinite variance or not -- the variances are finite by nature. See supplementary materials for more discussions related to the tail index.\n\n\\subsection{Gaussian behavior of AlexNet}\n\\label{Alexnet}\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[Conv-layers in AlexNet]{\n\\label{fig:epoch_alexnet_1}\n\\includegraphics[width=0.475\\columnwidth,height=3.3cm]{Plots\/gaussian_epoch1_300.png}}\n\\subfigure[FC-layers in AlexNet]{\n\\label{fig:epoch_alexnet_2}\n\\includegraphics[width=0.475\\columnwidth,height=3.3cm]{Plots\/gaussian_epoch2_300.png}}\n\\vskip -5pt%\n\n\\subfigure[Conv-layers in AlexNet-BN]{\n\\label{fig:epoch_alexnet_bn_1}\n\\includegraphics[width=0.475\\columnwidth,height=3.3cm]{Plots\/gaussian_epoch_bn1.png}}\n\\subfigure[FC-layers in AlexNet-BN]{\n\\label{fig:epoch_alexnet_bn_2}\n\\includegraphics[width=0.475\\columnwidth,height=3.3cm]{Plots\/gaussian_epoch_bn2.png}}\n\\vskip -5pt%\n\\caption{Gaussianity of each layer during training epochs.}\n\\vskip -10pt%\n\\label{fig:epoch_alexnet}\n\\end{figure}\n\nAlexNet~\\cite{krizhevsky2012imagenet} is one of the most successful convolutional neural architectures. In this experiment, we investigate the behavior of stochastic gradients during training. Instead of applying the original AlexNet tailored for ImageNet classification, modifications are necessary before applying AlexNet on datasets like Cifar-10 \\cite{DBLP:conf\/iclr\/ZhangBHRV17}. We adopt some modifications to the architecture such that our network works with different formats of images. The modified AlexNet for Cifar-10 contains 8 learnable layers: the first 5 layers are 2$d$ convolutional layers (Conv-layers), which corresponds to the features extractor, the other 3 layers are fully-connected layers (FC-layers), comprising the classifier for prediction. We remove the \\emph{local response} normalization in the feature extractor, the \\emph{dropout} operations in the classifier, and the average-pooling layer between these two modules. We use non-overlapped max-pooling instead of the overlapped counterparts. The first convolutional layers are rescaled for adaptation on images of Cifar-10: kernel $5\\times 5$, stride $2$, padding $2$. After this refinement of AlexNet for Cifar-10, we further composed a variant with batch normalization (BN), where we insert a batch-norm layer before each ReLU activation after the signal is transformed by either convolutional layers or fully-connected ones. \n\nIn order to understand the change of Gaussian behavior of gradient noise during whole training epochs, we test classic AlexNet and AlexNet with batch-norm layers, show the Gaaussianity results of convolutional layers in Figure~\\ref{fig:epoch_alexnet_1} \\& \\ref{fig:epoch_alexnet_bn_1} and results of fully connection layers in Figure~\\ref{fig:epoch_alexnet_2} \\& \\ref{fig:epoch_alexnet_bn_2}. During total 300 epochs training, we set learning rate is 0.1 before 150 epochs, 0.01 within 150 epochs and 225 epochs, 0.001 from 225 epochs to 300 epochs. The mini-batch size $n=128$, momentum is 0.9 and weight decay is $5*10^{-4}$. From Figure~\\ref{fig:epoch_alexnet}, we can find that in the classic AlexNet, the 3-$rd$ and 4-$nd$ convolutional layers' gaussianity have a decreasing trend with training. But batch normalization helps each convolutional layer to remain strong Gaussian behavior.\n\n\nTo find more properties of gaussianity in the classic AlexNet, we did two experimental cases for two different batch numbers: 200 batches and 600 batches. In each case, we test several different batch size $n$ to see how it affects the Gaussianity of gradient noise. Figure~\\ref{fig:200alexnet} and \\ref{fig:600alexnet} show the same trend of Gaussianity of stochastic gradient noise. The percentage of Gaussianity has a steady increase with the larger batch size. The more details have been displayed in Appendix (Table~\\ref{alexnet_table_200} and \\ref{alexnet_table_600}). We show five convolutional layers' features weight sizes and the final layer's classifier weight size in the last column of the table. For each experiment, we have repeated five times to obtain the more accurate results, which can also be verified by the very small standard derivation values on the table. \n\n\nBased on experimental results, we can find that there is a huge rise on the 4-$th$ convolutional layer, around zero percent Gaussianity with batch size $n=64$ goes up to $56.29\\%$ (on 600 batches) and $73.73\\%$ (on 200 batches) with batch size $n=2048$. They indicate the characteristics of gradient noise can be changed from non-Gaussian to Gaussian behavior if enough batch size is provided. The gradient noise on the first convolutional layer shows the best Gaussian behavior which covers all the batch size settings. When batch size $n=2048$, the gradient noise of the whole neural network method is above $50\\%$ (on 600 batches) and $60\\%$ (on 200 batches). These Gaussian behavior results satisfy our expectations and are consistent with the initial Gaussianity assumption. \n\n\nFrom Figure~\\ref{fig:200alexnet} we have known that the 1-$st$ layer's gaussianity percentage of AlexNet (mini-batch 128 and 200 batches) is 94.12\\%. Shapiro-Wilk test still tells us that about five percent gradient noise does not have Gaussian behavior. Now we utilize tail index estimation~\\footnote{https:\/\/github.com\/josemiotto\/pylevy} and fit the rest non-Gaussian behavior gradient noise to Levy alpha-stable distribution. In Appendix, we show the tail index performance on Figure~\\ref{fig:tail_index_total} and show the example of gradient noise on Figure~\\ref{fig:tail_index_non}. Figure~\\ref{fig:tail_index_first} show the example of gradient noise of the 4-$th$ layer in AlexNet compared with different mini-batch settings.\n\n\\begin{figure}[!]\n\\centering\n\\subfigure[Alexnet (200 batches)]{\n\\label{fig:200alexnet}\n\\includegraphics[width=0.475\\columnwidth,height=3.1cm]{Plots\/200alexnetplot.png}}\n\\subfigure[Alexnet (600 batches)]{\n\\label{fig:600alexnet}\n\\includegraphics[width=0.475\\columnwidth,height=3.1cm]{Plots\/600alexnetplot.png}}\n\\vskip -5pt%\n\\caption{Gaussianity of each layer in Alexnet}\n\\vskip -10pt%\n\\end{figure}\n\n\\subsection{Gaussian behavior of ResNet}\n\\label{Resnet}\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[ResNet (200 batches)]{\n\\label{fig:200resnet}\n\\includegraphics[width=0.475\\columnwidth,height=3.1cm]{Plots\/200resnetplot.png}}\n\\subfigure[ResNet (600 batches)]{\n\\label{fig:600resnet}\n\\includegraphics[width=0.475\\columnwidth,height=3.1cm]{Plots\/600resnetplot.png}}\n\\vskip -5pt%\n\\caption{Gaussianity of each layer in ResNet-56}\n\\vskip -10pt%\n\\end{figure}\n\nA Residual Neural Network (ResNet)~\\cite{he2016deep} is also a widely-used artificial neural network, its structure utilizes skip connections, or shortcuts to jump over some layers. For implementation, we choose official ResNet-20 and ResNet-56 packages from Pytorch~\\footnote{https:\/\/github.com\/pytorch\/vision}. The structure of ResNet is much more complicated, which contains the first convolutional layer, three block layers, and the final linear layer. Each block layer contains 6 convolutional layers with batch normalization for ResNet-20 or 18 convolutional layers with batch normalization for ResNet-56. \n\nWe train ResNet-20 on Cifar10 datasets with 300 epochs; the gaussianity performance of each block layer's first convolutional layer and sixth convolutional layer during whole training epochs is shown in Appendix (Figure~\\ref{fig:epoch_resnet}). ResNet has a very steady gaussianity for gradient noise, it does not change with training epochs. Furthermore, we do two experimental cases with ResNet-56 for two different batch numbers: 200 batches and 600 batches. Figure~\\ref{fig:200resnet} and \\ref{fig:600resnet} show ResNet-56 has a very high percentage of Gaussianity for gradient noise in each layer, which satisfies our Gaussianity assumption very well. Even if the batch size is small as $64$, they still show strong Gaussian behavior with above 85 percent Gaussianity. Besides, we can find similar results from AlexNet, which is the percentage of Gaussianity becomes higher with the larger batch size. The four light gray dashed lines divide three block layers on the plots. We also show these three block layers in Table~\\ref{resnet_table_200} and \\ref{resnet_table_600} in Appendix. In each block, for layout reasons, we only show layer 1, layer 9, and layer 18 in the table. We show the first convolutional layers' features weight dimension sizes, block layers' convolutional layers' features weight dimension sizes, and the final layer's classifier weight dimension size in the last column of the table. We repeat the experiment five times, the results are quite accurate with very small standard derivation errors. \n\n\\subsection{Analysis of Bound}\n\nAccording to Figure~\\ref{fig:200alexnet} and \\ref{fig:600alexnet}, we find that AlexNet's 4-$th$ and 5-$th$ convolutional layer show less Gaussian behavior in stochastic gradient noise compared with other layers. The rate of convergence may be the cause of such a phenomenon. We have introduced the Berry-Esseen theorem which states that the convergence of CLT is bounded by the standardized third absolute moment $\\beta\/\\sigma^3$ and the summation size (batch size) $|\\mathscr{S}|$. We keep as constant the batch size $|\\mathscr{S}|$, and evaluate numerically the values of standardized third absolute moments $\\beta\/\\sigma^3$ to measure the bound of convergence of CLT. We show in Appendix Figure~\\ref{fig:alexnet_distribution} that the empirical distribution of the standardized third absolute moments of all weight dimensions within each layer of AlexNet. Compared with Figure~\\ref{fig:200alexnet} and \\ref{fig:600alexnet}, it is noticeable that 4-$th$ and 5-$th$ convolutional layers have longer right tails and generally get larger values of the standardized third absolute moment, which means their bound is much larger than other layers, so that slow convergence to Gaussian occurs. \n\nLooking into more details in Appendix for 4-$th$ convolutional layer from Figure~\\ref{fig:alexnet_distribution}, we also estimate the quantile values of the third absolute moments distribution corresponding to batch size (from $n=64$ to $n=2048$ in Table~\\ref{fig:200alexnet}) in Appendix Figure~\\ref{fig:linevalues4}. For each batch size in this test, we calculate a quantile value w.r.t. the percentage of Gaussian dimensions. The results are demonstrated as vertical spines in Appendix Figure~\\ref{fig:linevalues4}, the area on the left hand side corresponds roughly to those dimensions that converge towards Gaussian; the spines are shifting right as the batch size grows larger, indicates more dimensions exhibits Gaussian behavior.\n\nFigure~\\ref{fig:600resnet} shows that the percentage of Gaussianity in ResNet-56 Block-3 drops from $94.44\\%$ to $91.58 \\%$ when batch size $n=256$. We select convolutional layer 1 in Block-3  ($94.44\\%$), layer 2 ($94.29\\%$), layer 11 ($90.82\\%$), layer 12 ($90.93\\%$), layer 15 ($93.40\\%$) and layer 16 ($92.60\\%$) to show the empirical distribution of the standardized third absolute moments of weights in Appendix Figure~\\ref{fig:resnet_distribution}; similar results has been shown as AlexNet. The layer with a higher Gaussianity of gradient noise often has a smaller value of bound. The CLT converges faster to Gaussian if the shape of the distribution of the standardized third absolute moments of weights is close to zero. Based on the comparison of Figure~\\ref{fig:alexnet_distribution} and Figure~\\ref{fig:resnet_distribution} in Appendix, we also notice the overall shape of distribution from ResNet is close to zero, only covers very small range around 0.01 to 0.02. It implies that ResNet has a strong Gaussian behavior for stochastic gradient noise. The overall shape of distribution from AlexNet is more flexible, the range reaches 0.2 in some cases. This explains why AlexNet has significant non-Gaussian behavior in some cases when batch size is not large enough.\n\n\n\n\n\n\\section{Conclusion}\n\nIn this paper, we investigated the characteristics of gradient noise during the training of deep neural networks using stochastic optimization methods. We showed that the gradient noise of stochastic methods exhibits finite variance during training.\nThe noise is therefore asymptotically Gaussian, instead of the latest proposal of $\\alpha$-stable framework~\\cite{simsekli2019tail}. To validate our claim that the non-Gaussianity is caused by the non-convergence of CLT, we conducted an empirical analysis on the percentage of Gaussian elements for each trainable layer and established quantitatively a connection of the Gaussianity with the standardized third absolute moment exploited in Berry-Esseen theorem. Our experiments show that the convergence rate of SGD noise is tied with the architecture of neural networks. The experiments show that the rate of convergence has a deep connection with the architecture of the neural networks: for each layer of AlexNet, the distributions of the third absolute moments calculated for all weight dimensions generally have a much wider spectrum, leading towards much worse Gaussianity in certain layers. For ResNet, the layer-wise moments' distributions have a narrower spectrum and smaller values, which validates the observations that the Gaussianity percentages of layers in ResNet are generally beyond $85\\%$. When increasing the batch sizes, the percentage of Gaussianity for both AlexNet and ResNet improves. Based on the evidence of Gaussian noise, we analyze the dynamics of SGD and unify the discretization scheme w.r.t. the general Langevin equations: the vanilla SGD corresponds to the high-friction limit of the continuous-time dynamics. We then provide a simple local approximation of the distribution of learning parameters when the dynamics enter the vicinity of minima. \n\n\n\n\\clearpage\n\n\n\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{Problem Setting}\n\\label{subsec:ps}\nGiven a sequence of triplet constraints $S=\\lbrace(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^{-}})\\rbrace_{t=1}^{T}$ that arrive sequentially, where $\\lbrace\\bm{x_t},\\bm{x_t^{+}},\\bm{x_t^{-}}\\rbrace \\in \\mathcal{R}^d$, and $\\bm{x_t}$ (anchor) is similar to $\\bm{x_t^{+}}$ (positive) but is dissimilar to $\\bm{x_t^{-}}$ (negative), the goal of online adaptive metric learning is to learn a model $\\bm{F}: \\mathcal{R}^d \\mapsto \\mathcal{R}^{d'}$ such that $||\\bm{F}(\\bm{x_t})-\\bm{F}(\\bm{x_t^{+}})||_2\\ll||\\bm{F}(\\bm{x_t})-\\bm{F}(\\bm{x_t^{-}})||_2$. \n\nThe overall aim of the task is to learn a metric model with adaptive complexity while maintaining a high constraint utilization rate. Here, the complexity of $\\bm{F}$ needs to be adaptive so that its hypothesis space is automatically enlarged or shrinked as necessary.\n\nIn practical applications, the input triplet stream can be generated as follows. First, some seed triplets can be constructed from user clicks. Then, more triplets are formed by taking transitive closures\\footnote{If $(\\bm{x_1}, \\bm{x_2})$ and $(\\bm{x_1}, \\bm{x_3})$ are two similar pairs, then $(\\bm{x_2}, \\bm{x_3})$ is a similar pair. If $(\\bm{x_1}, \\bm{x_2})$ and $(\\bm{x_2}, \\bm{x_3})$ are two similar pairs, then $(\\bm{x_1}, \\bm{x_3})$ is a similar pair. If $(\\bm{x_1}, \\bm{x_2})$ is a similar pair and $(\\bm{x_1}, \\bm{x_3})$ is a dissimilar pair,  then $(\\bm{x_2}, \\bm{x_3})$ is a dissimilar pair. If $(\\bm{x_1}, \\bm{x_2})$ is a similar pair and $(\\bm{x_2}, \\bm{x_3})$ is a dissimilar pair, then $(\\bm{x_1}, \\bm{x_3})$ is a dissimilar pair.} over these seed triplets. The generated triplets are inserted into the stream in chronological order of creation.\nIn this paper, we assume the existence of such triplet stream and focus on addressing the challenges of online metric learning in such scenarios.\n\n\\subsection{Overview}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{fig1.jpg}\n\\caption{Overview of the OAHU{}. The chromatic dashed arrows represent update contributions and directions. Each $L_i \\in \\lbrace L_1, L_2, \\ldots \\rbrace$ represents a linear transformation layer followed by a ReLU activation. $\\lbrace E_0, E_1, \\dots \\rbrace$ are the embedding layers connected to corresponding input or hidden layers. Note that $E_0$ here represents a linear metric model, i.e., a linear transformation from the input feature space to the embedding space.}\n\\vspace{-2mm}\n\\label{fig:overview}\n\\end{figure*}\n\nTo learn a metric model with adaptive complexity, we need to address the following question: \\textit{when} and \\textit{how} to change the ``complexity'' of metric model in an \\textit{online} setting? In this section, we will discuss the details of the proposed framework OAHU{} that addresses the question in a data-driven manner.\n\nFigure~\\ref{fig:overview} illustrates the structure of our OAHU{} framework. Inspired by recent works~\\cite{SrinivasB16,HuangSLSW16}, we use an over-complete network and automatically adapt its effective depth to learn a metric function with an appropriate complexity based on input constraints. Consider a neural network with $L$ hidden layers:\nwe connect an independent embedding layer to the network input layer and each of these hidden layers. Every embedding layer represents a space where similar instances are closer to each other and vice-versa. Therefore, unlike conventional online metric learning solutions that usually learns a linear model, OAHU{} is an ensemble of models with varying complexities that share low-level knowledge with each other. For convenience, let $E_l \\in \\lbrace E_0, E_1, E_2, \\ldots, E_L\\rbrace$ denote the $l^{th}$ metric model in OAHU{}, i.e., the network branch starting from input layer to the $l^{th}$ metric embedding layer, as illustrated in Figure~\\ref{fig:overview}. The simplest model in OAHU{} is $E_0$, which represents a linear transformation from the input feature space to the metric embedding space. A weight $\\alpha^{(l)}\\in [0,1]$ is assigned to $E_l$, measuring its importance in OAHU{}.\n\nFor a triplet constraint $(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^{-}})$ that arrives at time $t$, its metric embedding $f^{(l)}(\\bm{x_t^*})$ generated by $E_l$ is\n\n\\small\n\\begin{equation}\n\\label{eq:metic_embedding}\n    f^{(l)}(\\bm{x_t^*}) = h^{(l)}\\Theta^{(l)}\n\\end{equation}\n\\normalsize\nwhere $h^{(l)} = \\sigma(W^{(l)}h^{(l-1)}),$ with $l\\ge 1, l\\in \\mathbb{N}$, and $h^{(0)} = \\bm{x_t^*}$.\n\nHere $\\bm{x_t^*}$ denotes any anchor ($\\bm{x_t}$), positive ($\\bm{x_t^{+}}$) or negative ($\\bm{x_t^{-}}$) instance based on the definition in Section~\\ref{subsec:ps}, and $h^{(l)}$ represents the activation of $l^{\\text{th}}$ hidden layer. \nNote that we also explicitly limit the learned metric embedding $f^{(l)}(\\bm{x_t^*})$ to reside on a unit sphere, i.e., $||f^{(l)}(\\bm{x_t^*})||_2=1$, to reduce the potential model search space and accelerate the  training of OAHU{}. \n\nDuring the training phase, for every arriving triplet $(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^{-}})$, OAHU{} first retrieves the metric embedding $f^{(l)}(\\bm{x_t^*})$ from $l^{th}$ metric model according to Eq.~\\ref{eq:metic_embedding}, and then produces a local loss $\\mathcal{L}^{(l)}$ for $E_l$ by evaluating the similarity and dissimilarity errors based on $f^{(l)}(\\bm{x_t^*})$. Thus, the overall loss introduced by this triplet is given by\n\\small\n\\begin{equation}\n    \\mathcal{L}_{overall}(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^{-}})=\\sum\\limits_{l=0}^L \\alpha^{(l)} \\cdot \\mathcal{L}^{(l)}(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^{-}})\n\\end{equation}\n\\normalsize\n\nSince the new architecture of OAHU{} introduces two sets of new parameters, $\\Theta^{(l)}$ (parameters of $f^{(l)}$) and $\\alpha^{(l)}$, we need to learn $\\Theta^{(l)}$, $\\alpha^{(l)}$ and $W^{(l)}$ during the online learning phase. Therefore, the final optimization problem to solve in OAHU{} at time $t$ is given by\n\\small\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\Theta^{(l)},W^{(l)}, \\alpha^{(l)}}{\\text{minimize}}\n& & \\mathcal{L}_{overall} \\\\\n& \\text{subject to}\n& & ||f^{(l)}(\\bm{x_t^*})||_2 = 1, \\forall l = 0, \\ldots, L.\n\\end{aligned}\n\\end{equation}\n\\normalsize\n\nTo solve the above optimization problem, we propose a novel Adaptive-Bound Triplet Loss (ABTL) in Section~\\ref{subsec:abel} to estimate $\\mathcal{L}^{(l)}$ \nand introduce a novel Adaptive Hedge Update (AHU) method in Section~\\ref{subsec:ahu} for online updating $\\Theta^{(l)}$, $W^{(l)}$ and $\\alpha^{(l)}$.\n\n\\subsection{Adaptive-Bound Triplet Loss}\n\\label{subsec:abel}\n\n\\subsubsection{Limitations of Existing Pairwise\/Triplet Loss} \nExisting online metric learning solutions typically optimize a pairwise or triplet loss in order to learn a similarity metric. A widely-accepted pairwise loss~\\cite{Shalev-ShwartzSN04} and a common triplet loss~\\cite{LiGWZHS18} are shown in Equation~\\ref{eq:pairloss} and ~\\ref{eq:tripletloss} respectively.  \n\\small\n\\begin{equation}\n\\label{eq:pairloss}\n    L(x_t, x_t') = \\max\\lbrace0, y_{t}(d^2(x_{t}, x_{t}')-b)+1\\rbrace\n\\end{equation}\n\\begin{equation}\n\\label{eq:tripletloss}\n    L(x_t, x_t^+, x_t^-) = \\max\\lbrace0, b+d(x_t, x_t^+)-d(x_t, x_t^-)\\rbrace\n\\end{equation}\n\\normalsize\nHere $y_t\\in\\lbrace+1, -1\\rbrace$ denotes whether $x_t$ is similar ($+1$) or dissimilar ($-1$) to $x_t'$ and $b \\in \\mathcal{R}$ is a user-specified fixed margin. Compared to the triplet loss that simultaneously evaluates both similarity and dissimilarity relations, the pairwise loss can only focus on one of the relations at a time, which leads to a poor metric quality. On the other hand, without a proper margin, \nthe loose restriction of triplet loss can result in many failure cases. \nHere, a failure case is a triplet that produces a zero loss value, where dissimilar instances are closer than similar instances.\nAs the example shown in Figure~\\ref{fig:failuremode}, for an input triplet $(x_t, x_t^+, x_t^-)$, although $x_t^+$ is \ncloser to $x_t^-$ and further from $x_t$, models optimizing triplet loss would incorrectly ignore this constraint due to its zero loss value. In addition, selecting an appropriate margin is highly data-dependent, and requires extensive domain knowledge. Furthermore, as the model improves performance over time, a larger margin is required to avoid failure cases. \n\nNext, we describe an adaptive-bound triplet loss that attempts to solve these drawbacks.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{fig2.jpg}\n\\caption{A failure case of triplet loss.}\n\\vspace{-4mm}\n\\label{fig:failuremode}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{fig3.jpg}\n\\caption{Schematic illustration of ABTL.}\n\\vspace{-4mm}\n\\label{fig:abel}\n\\end{figure}\n\n\\subsubsection{Adaptive-Bound Triplet Loss (ABTL)} \nWe define that instances of the same classes are mutually attractive to each other, while instances of different classes are mutually repulsive to each other. Therefore, for any input triplet constraint $(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^{-}})$, there is an attractive loss $\\mathcal{L}_{attr}\\in[0,1]$ between $\\bm{x_t}$ and $\\bm{x_t^{+}}$, and a repulsive loss $\\mathcal{L}_{rep}\\in[0,1]$ between $\\bm{x_t}$ and $\\bm{x_t^{-}}$. In the $l^{th}$ metric model $E_l \\in \\lbrace E_0, \\ldots, E_L \\rbrace$, we focus on a distance measure $D^{(l)}(\\bm{x_i}, \\bm{x_j})$ defined as follows:\n\\small\n\\begin{equation}\n\\label{eq:dist}\n\\begin{split}\nD^{(l)}(\\bm{x_i}, \\bm{x_j}) =& ||f^{(l)}(\\bm{x_i})-f^{(l)}(\\bm{x_j})||_2, \\quad \\forall l=0,1,2,\\ldots,L\\\\\n\\end{split}\n\\end{equation}\n\\normalsize\nwhere $f^{(l)}(\\bm{x})$ denotes the embedding generated by $E_l$. Since $||f^{(l)}(\\bm{x})||_2$ is restricted to be $1$, we have $D^{(l)}(\\bm{x_i}, \\bm{x_j})\\in [0,2]$.\nIn each iteration, let $D^{(l)}_{orig}(\\bm{x_1},\\bm{x_2})$ denote the distance between $\\bm{x_1}$ and $\\bm{x_2}$ before applying the update to $f^{(l)}$ (i.e., updating $\\Theta^{(l)}$ and $\\lbrace W^{(j)}\\rbrace_{j=0}^l$) and $D^{(l)}_{update}(\\bm{x_1},\\bm{x_2})$ denote the distance after applying the update. \n\nFigure~\\ref{fig:abel} illustrates the main idea of ABTL. The objective is to have the distance $D^{(l)}_{update}(\\bm{x_t}, \\bm{x_t^{+}})$ of two similar instances $\\bm{x_t}$ and $\\bm{x_t^{+}}$ to be less than or equal to a similarity threshold $d^{(l)}_{sim}(\\bm{x_t}, \\bm{x_t^{+}})$ so that the attractive loss $\\mathcal{L}^{(l)}_{attr}(\\bm{x_t}, \\bm{x_t^{+}})$ drops to zero; on the other hand, for two dissimilar instances $\\bm{x_t}$ and $\\bm{x_t^{-}}$, we desire their distance $D^{(l)}_{update}(\\bm{x_t}, \\bm{x_t^{-}})$ to be greater than or equal to a dissimilarity threshold $d^{(l)}_{dis}(\\bm{x_t}, \\bm{x_t^{-}})$, thereby reducing the repulsive loss $\\mathcal{L}^{(l)}_{rep}(\\bm{x_t}, \\bm{x_t^{-}})$ to zero. Equation~\\ref{eq:constraint} presents these constraints mathematically.\n\\small\n\\begin{equation}\n\\label{eq:constraint}\n    \\left\\{\\begin{matrix}\n    D^{(l)}_{update}(\\bm{x_t}, \\bm{x_t^{+}}) \\le d^{(l)}_{sim}(\\bm{x_t}, \\bm{x_t^{+}})\\\\ \n    \\\\\n    D^{(l)}_{update}(\\bm{x_t}, \\bm{x_t^{-}}) \\ge d^{(l)}_{dis}(\\bm{x_t}, \\bm{x_t^-})\n    \\end{matrix}\\right.\n\\end{equation}\n\\normalsize\n\nUnfortunately, directly finding an appropriate $d^{(l)}_{sim}(\\bm{x_t}, \\bm{x_t^{+}})$ or $d^{(l)}_{dis}(\\bm{x_t}, \\bm{x_t^-})$ is difficult, since it varies with different input constraints. Therefore, we introduce a user-specified hyper-parameter $\\tau>0$ to explicitly restrict the range of both thresholds.\nLet $\\mathcal{T}^{(l)}_{sim}$ denote the upper bound of $d^{(l)}_{sim}(\\bm{x_t}, \\bm{x_t^{+}})$, and $\\mathcal{T}^{(l)}_{dis}$ denote the lower bound of $d^{(l)}_{dis}(\\bm{x_t}, \\bm{x_t^-})$, \n\n\\small\n\\begin{equation}\n    \\left\\{\\begin{matrix}\n    \\mathcal{T}^{(l)}_{sim} = \\tau \\ge d^{(l)}_{sim}(\\bm{x_t}, \\bm{x_t^{+}})\\\\\n    \\\\\n    \\mathcal{T}^{(l)}_{dis} = 2-\\tau \\leq d^{(l)}_{dis}(\\bm{x_t}, \\bm{x_t^-})\n    \\end{matrix}\\right.\n\\end{equation}\n\\normalsize\n\nWithout loss of generality, we propose\n\n\\small\n\\begin{equation}\n    \\left\\{\\begin{matrix}\n    d^{(l)}_{sim}(\\bm{x_t}, \\bm{x_t^{+}}) = a_1{\\rm e}^{D^{(l)}_{orig}(\\bm{x_t}, \\bm{x_t^{+}})}+b_1\\\\ \n    \\\\\n    d^{(l)}_{dis}(\\bm{x_t}, \\bm{x_t^-}) = -a_2{\\rm e}^{-D^{(l)}_{orig}(\\bm{x_t}, \\bm{x_t^{-}})}+b_2\n    \\end{matrix}\\right.\n\\end{equation}\n\\normalsize\nwhere $a_1$, $b_1$, $a_2$ and $b_2$ are variables need to be determined.\nNote that $d^{(l)}_{sim}(\\bm{x_t}, \\bm{x_t^{+}})$ and $d^{(l)}_{dis}(\\bm{x_t}, \\bm{x_t^-})$ are monotonically increasing and decreasing, respectively. Therefore, $d^{(l)}_{sim}$ preserves the information of original relative similarity, i.e., $d^{(l)}_{sim}(\\bm{x_1}, \\bm{x_2}) < d^{(l)}_{sim}(\\bm{x_1}, \\bm{x_3})$ if $D^{(l)}_{orig}(\\bm{x_1}, \\bm{x_2}) < D^{(l)}_{orig}(\\bm{x_1}, \\bm{x_3})$. This property is critical to many real-world applications that require a fine-grained similarity comparison, such as image retrieval.\n\nThe values of $a_1$, $b_1$, $a_2$ and $b_2$ are then determined by the following boundary conditions:\n\\small\n\\begin{equation}\n\\left\\{\\begin{matrix}\nd^{(l)}_{sim}|_{D^{(l)}_{orig}(\\bm{x_t},\\bm{x_t^{+}})=2}\\leq\\mathcal{T}^{(l)}_{sim}=\\tau &\nd^{(l)}_{sim}|_{D^{(l)}_{orig}(\\bm{x_t},\\bm{x_t^{+}})=0}=0\\\\\nd^{(l)}_{dis}|_{D^{(l)}_{orig}(\\bm{x_t},\\bm{x_t^-})=0}\\ge \\mathcal{T}^{(l)}_{dis}=2-\\tau &\nd^{(l)}_{dis}|_{D^{(l)}_{orig}(\\bm{x_t},\\bm{x_t^-})=2}=2\n\\end{matrix}\\right.\n\\end{equation}\n\\normalsize\n\nThus we have\n\\small\n\\begin{equation}\n    \\left\\{\\begin{matrix}\n    d^{(l)}_{sim}\\big(\\bm{x_t}, \\bm{x_t^{+}}\\big)=\\frac{\\tau}{{\\rm e}^2-1}\\big ({\\rm e}^{D^{(l)}_{orig}(\\bm{x_t}, \\bm{x_t^{+}})}-1\\big ) \\\\\n    \\\\\n    d^{(l)}_{dis}\\big(\\bm{x_t}, \\bm{x_t^{-}}\\big)= -\\frac{2-(2-\\tau)}{1-{\\rm e}^{-2}}\\big({\\rm e}^{-D^{(l)}_{orig}(\\bm{x_t}, \\bm{x_t^{-}})}-1\\big) +(2-\\tau)\n\\end{matrix}\\right.\n\\end{equation}\n\\normalsize\n\nProvided with $d^{(l)}_{sim}$ and $d^{(l)}_{dis}$, the attractive loss $\\mathcal{L}_{attr}$ and repulsive loss $\\mathcal{L}_{rep}$ are determined by\n\n\\small\n\\begin{equation}\n\\label{eq:attr_rep}\n    \\left\\{\\begin{matrix}\n    \\mathcal{L}_{attr}^{(l)}(\\bm{x_t}, \\bm{x_t^{+}}) = \\max \\Big \\lbrace 0,\\frac{1}{2-c_1}D^{(l)}_{orig}(\\bm{x_t}, \\bm{x_t^{+}})-\\frac{c_1}{2-c_1}\\Big \\rbrace\\\\ \n    \\\\\n    \\mathcal{L}_{rep}^{(l)}(\\bm{x_t}, \\bm{x_t^{-}}) = \\max \\Big \\lbrace 0, \\frac{-1}{c_2}D^{(l)}_{orig}(\\bm{x_t}, \\bm{x_t^-})+1 \\Big \\rbrace\n    \\end{matrix}\\right.\n\\end{equation}\n\\normalsize\nwhere $c_1=d^{(l)}_{sim}(\\bm{x_t}, \\bm{x_t^{+}})$ and $c_2=d^{(l)}_{dis}(\\bm{x_t}, \\bm{x_t^{-}})$.\n\nTherefore, the local loss $\\mathcal{L}^{(l)}(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^-})$, which measures the similarity and dissimilarity errors of the input triplet $(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^-})$ in $E_l$, is the average of $\\mathcal{L}^{(l)}_{attr}$ and $\\mathcal{L}^{(l)}_{rep}$:\n\\small\n\\begin{equation}\n\\label{eq:local_loss}\n\\begin{split}\n    \\mathcal{L}^{(l)}(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^-})&=\\frac{1}{2}\\Big(\\mathcal{L}_{attr}^{(l)}(\\bm{x_t}, \\bm{x_t^{+}})+\\mathcal{L}_{rep}^{(l)}(\\bm{x_t}, \\bm{x_t^{-}})\\Big)\\\\\n\\end{split}\n\\end{equation}\n\\normalsize\n\nNext, we answer an important question: What is the best value of $\\tau$? Here, we demonstrate that the optimal range of $\\tau$ is $(0, \\frac{2}{3})$. The best value of $\\tau$ is thus empirically selected within this range via cross-validation.\n\n\n\n\\begin{theorem}\n\\label{theo:1}\nWith $\\tau \\in (0, \\frac{2}{3}) $, by optimizing the proposed adaptive-bound triplet loss, different classes are separated in the metric embedding space.\n\\end{theorem}\n\\begin{proof}\nLet $D(c_1,c_2)$ denotes the minimal distance between classes $c_1$ and $c_2$, i.e., the distance between two closest instances from $c_1$ and $c_2$ respectively. Consider an arbitrary quadrupole $(x_1, x_2, x_3, x_4) \\in \\mathcal{Q}$ where $\\lbrace x_1, x_2\\rbrace \\in c_1$, $\\lbrace x_3,x_4 \\rbrace \\in c_2$, and $\\mathcal{Q}$ is the set of all possible quadrupoles generated from class $c_1$ and $c_2$. Suppose  $(x_2,x_3)$ is the closest dissimilar pair among all possible dissimilar pairs that can be extracted from $(x_1, x_2, x_3, x_4)$ \\footnote{it can always be achieved by re-arranging symbols of $x_1$,$x_2$,$x_3$ and $x_4$}. We first prove that the lower bound of $D(c_1,c_2)$ is given by $\\min\\limits_{(x_1, x_2, x_3, x_4)\\in \\mathcal{Q}} D^{(l)}(x_1, x_4)-D^{(l)}(x_1, x_2)-D^{(l)}(x_3, x_4)$.\nDue to the triangle inequality property of p-norm distance, we have \n\\small\n\\begin{equation}\n\\begin{split}\n    D^{(l)}(x_1, x_4) &\\leq D^{(l)}(x_1, x_2) + D^{(l)}(x_2, x_4)\\\\\n    & \\leq D^{(l)}(x_1, x_2) + D^{(l)}(x_2, x_3) + D^{(l)}(x_3, x_4)\n\\end{split}\n\\end{equation}\n\\normalsize\nTherefore,\n\\small\n\\begin{equation}\n\\begin{split}\n    D(c_1, c_2) &= \\min_{(x_1, x_2, x_3, x_4)\\in \\mathcal{Q}} D^{(l)}(x_2, x_3) \\\\\n    &\\ge \\min_{(x_1, x_2, x_3, x_4)\\in \\mathcal{Q}} D^{(l)}(x_1, x_4)-D^{(l)}(x_1, x_2)-D^{(l)}(x_3, x_4)\n\\end{split}\n\\end{equation}\n\\normalsize\n\nBy optimizing the adaptive-bound triplet loss, the following constraints are satisfied \n\n\\small\n\\begin{equation}\n    \\left\\{\\begin{matrix}\n    D^{(l)}(x_1, x_2) \\le d^{(l)}_{sim}(x_1, x_2) \\le \\mathcal{T}^{(l)}_{sim}\\\\\n    \\\\\n    D^{(l)}(x_3, x_4) \\le d^{(l)}_{sim}(x_3, x_4) \\le \\mathcal{T}^{(l)}_{sim}\\\\\n    \\\\\n    D^{(l)}(x_1, x_4) \\ge d^{(l)}_{dis}(x_1, x_4) \\ge \\mathcal{T}^{(l)}_{dis}\n    \\end{matrix}\\right.\n\\end{equation}\n\\normalsize\n\nThus\n\\small\n\\begin{equation}\n\\begin{split}\n    D(c_1, c_2) \n    &\\ge \\min_{(x_1, x_2, x_3, x_4)\\in \\mathcal{Q}} D^{(l)}(x_1, x_4)-D^{(l)}(x_1, x_2)-D^{(l)}(x_3, x_4)\\\\\n    &\\ge \\mathcal{T}^{(l)}_{dis} - 2\\mathcal{T}^{(l)}_{sim}\\\\\n    &=2-\\tau-2\\tau\\\\\n    &=2-3\\tau\n\\end{split}\n\\end{equation}\n\\normalsize\nif $\\tau \\in (0, \\frac{2}{3})$, we have $3\\tau < 2$. Therefore,\n\\small\n\\begin{equation}\n\\label{eq:final_proof}\n    D(c_1, c_2) \\ge 2-3\\tau > 0\n\\end{equation}\n\\normalsize\nEquation~\\ref{eq:final_proof} indicates that the minimal distance between class $c_1$ and $c_2$ is always positive so that these two classes are separated. \n\n\\end{proof}\n\n\\subsubsection{Discussion}\nCompared to pairwise and triplet loss, the proposed ABTL has several advantages:\n\\begin{itemize}\n    \\item Similar to triplet loss, ABTL simultaneously evaluates both similarity and dissimilarity errors to obtain a high-quality metric.\n    \\item The similarity threshold $d^{(l)}_{sim}(\\bm{x_t}, \\bm{x_t^{+}})$ is always less than or equal to $D^{(l)}_{orig}(\\bm{x_t},\\bm{x_t^{+}})$ and the dissimilarity threshold $d^{(l)}_{dis}(\\bm{x_t}, \\bm{x_t^{-}})$ is always greater than or equal to $D^{(l)}_{orig}(\\bm{x_t},\\bm{x_t^{-}})$. Thus, every input constraint contributes to model update, which eliminates the failure cases of triplet loss and leads to a higher constraint utilization rate.\n    \\item The optimal range of $\\tau$ is provided with a theoretical proof.\n\\end{itemize}\n\n\\subsection{Adaptive Hedge Update (AHU)}\n\\label{subsec:ahu}\nWith adaptive-bound triplet loss, the overall loss in OAHU{} becomes\n\\small\n\\begin{equation}\n\\begin{split}\n    \\mathcal{L}_{overall}(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^{-}})&=\\sum\\limits_{l=0}^L \\alpha^{(l)}\\cdot\\mathcal{L}^{(l)}(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^{-}})\\\\\n    &=\\sum\\limits_{l=0}^L \\frac{\\alpha^{(l)}}{2}\\Big(\\mathcal{L}^{(l)}_{attr}(\\bm{x_t}, \\bm{x_t^{+}})+\\mathcal{L}^{(l)}_{rep}(\\bm{x_t}, \\bm{x_t^{-}})\\Big)\n\\end{split}\n\\end{equation}\n\\normalsize\n\nWe propose to learn $\\alpha^{(l)}$ using the Hedge Algorithm~\\cite{FreundS97}. Initially, all weights are uniformly distributed, i.e., $\\alpha^{(l)}=\\frac{1}{L+1}, l=0,1,2,\\ldots, L$. At every iteration, for each metric model $E_l$, OAHU{} transforms the input triplet constraint into $E_l$'s metric embedding space, where the constraint similarity and dissimilarity errors are evaluated. Thus $\\alpha^{(l)}$ is updated based on the local loss suffered by $E_l$ as:\n\\small\n\\begin{equation}\n\\label{eq:alpha_update}\n    \\alpha^{(l)}_{t+1} \\leftarrow\n    \\begin{cases}\n    \\alpha^{(l)}_t\\beta^{\\mathcal{L}^{(l)}} & \\beta^{\\min\\limits_{l}\\mathcal{L}^{(l)}}\\log\\mathcal{L}^{(l)} > \\beta-1\\\\\n    \\alpha^{(l)}_t[1-(1-\\beta)\\mathcal{L}^{(l)}]& otherwise\n    \\end{cases}\n\\end{equation}\n\\normalsize\nwhere $\\beta \\in (0,1)$ is the discount factor. We choose to update $\\alpha^{(l)}$ based on Equation~\\ref{eq:alpha_update}, to maximize the update of $\\alpha^{(l)}$ at each step. In other words, if a metric model produces a low\/high $\\mathcal{L}^{(l)}$ at time $t$, its associated weight $\\alpha^{(l)}$ will gain as much increment\/decrement as possible. Therefore, in OAHU{}, metric models with good performance are highlighted while models with poor performance are eliminated. \nMoreover, following Sahoo et.al~\\cite{SahooPLH18}, to avoid the model bias issue~\\cite{ChenGS15,GulcehreMVB16}, i.e., OAHU{} may unfairly prefer shallow models due to their faster convergence rate and excessively reduces weights of deeper models, we introduce a smooth factor $s$ used to set a minimal weight for each metric model. Thus, after updating weights according to Equation~\\ref{eq:alpha_update}, the weights are set as: $\\alpha^{(l)}_{t+1}=\\max (\\alpha^{(l)}_{t+1}, \\frac{s}{L+1})$. \nFinally, at the end of every round, the weights $\\alpha$ are normalized such that $\\sum\\limits_{l=0}^L \\alpha^{(l)}_{t+1}=1$.\n\nLearning the parameters $\\Theta^{(l)}$ and $W^{(l)}$ is tricky but can be achieved via gradient descent. By applying Online Gradient Descent (OGD) algorithm, the update rules for $\\Theta^{(l)}$ and $W^{(l)}$ are given by:\n\\small\n\\begin{equation}\n\\label{eq:theta_update}\n\\begin{split}\n    \\Theta^{(l)}_{t+1} &\\leftarrow \\Theta^{(l)}_{t} - \\eta \\nabla_{\\Theta^{(l)}_t} \\mathcal{L}_{overall}(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^{-}})\\\\\n    &= \\Theta^{(l)}_{t} - \\eta \\frac{\\alpha^{(l)}}{2} \\Big( \\nabla_{\\Theta^{(l)}_t}\\mathcal{L}^{(l)}_{attr}(\\bm{x_t}, \\bm{x_t^{+}})+\\nabla_{\\Theta^{(l)}_t} \\mathcal{L}^{(l)}_{rep}(\\bm{x_t}, \\bm{x_t^{-}})\\Big)\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\label{eq:w_update}\n    \\begin{split}\n    W^{(l)}_{t+1} &\\leftarrow W^{(l)}_{t} - \\eta \\nabla_{W^{(l)}_t} \\mathcal{L}_{overall}(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^{-}})\\\\\n    &= W^{(l)}_{t} - \\eta \\sum\\limits_{j=l}^L \\frac{\\alpha^{(j)}}{2} \\Big( \\nabla_{W^{(l)}_t}\\mathcal{L}^{(j)}_{attr}(\\bm{x_t}, \\bm{x_t^{+}})+\\nabla_{W^{(l)}_t} \\mathcal{L}^{(j)}_{rep}(\\bm{x_t}, \\bm{x_t^{-}})\\Big)\n\\end{split}\n\\end{equation}\n\\normalsize\n\nUpdating $\\Theta^{(l)}$ (Equation~\\ref{eq:theta_update}) is simple since $\\Theta^{(l)}$ is specific to $E_l$, and is not shared across different metric models. In contrast, because deeper models share low-level knowledge with shallower models in OAHU{}, all models deeper than $l$ layers contribute to the update of $W^{(l)}$, as expressed in Equation~\\ref{eq:w_update}. Algorithm~\\ref{alg:OAHU{}} outlines the training process of OAHU{}.\n\n\\subsection{Regret Bound}\nOverall, hedge algorithm enjoys a regret bound of $R_T\\leq \\sqrt{TlnN}$~\\cite{FreundS97} where $N$ is the number of experts and $T$ is the number of trials. In our case, $N$ is the number of metric models (i.e., $L+1$) and $T$ is the number of input triplet constraints. The average worst-case convergence rate of OAHU{} is then given by $O(\\sqrt{ln(N)\/T})$. Therefore, OAHU{} is guaranteed to converge after learning from sufficiently many triplets.\n\n\n\\begin{algorithm}[t]\n\\caption{\\textbf{OAHU{}: Online Metric Learning with Adaptive Hedge Update }}\n\\label{alg:OAHU{}}\n\\begin{algorithmic}[1]\n\\REQUIRE Discount Factor $\\beta\\in(0,1)$; Smooth Factor $s$; Control Parameter $\\tau \\in (0, \\frac{2}{3})$; Learning rate $\\eta$; A randomly initialized ANN with $L$ hidden layers that is parameterized by $\\Theta^{(l)}$,$W^{(l)}$ and $\\alpha^{(l)}$.\n\\ENSURE $\\alpha^{(l)}$, $\\Theta^{(l)}$ and $W^{(l)}$\\\\\n\\STATE Initialize $\\alpha^{(l)}=\\frac{1}{L+1}, \\forall l=0,1,\\ldots, L$ \n\\FOR{$t=1,2,\\ldots,T$}\n\\STATE Receive a triplet constraint $(\\bm{x_t}, \\bm{x_t^{+}}, \\bm{x_t^-})$\\\\\n\\STATE Transform and retrieve the metric embedding $f^{(l)}(\\bm{x_t})$, $f^{(l)}(\\bm{x_t^{+}})$ and $f^{(l)}(\\bm{x_t^-})$ from each model $E_l$.\\\\\n\\STATE Evaluate $\\mathcal{L}^{(l)}_{attr}(\\bm{x_t}, \\bm{x_t^{+}})$ and $\\mathcal{L}^{(l)}_{rep}(\\bm{x_t}, \\bm{x_t^{-}})$ (Eq.~\\ref{eq:attr_rep}), $\\forall l=0,1,\\ldots,L$.\n\\STATE Compute $L^{(l)}(\\bm{x_t}, \\bm{x_t^{+}},\\bm{x_t^{-}}), \\forall l=0, 1, \\ldots, L$ as per Eq.~\\ref{eq:local_loss}.\n\\STATE Update $\\Theta^{(l)}_{t+1},\\forall l=0,1,2,\\ldots,L$ as per Eq.~\\ref{eq:theta_update}.\\\\\n\\STATE Update $W^{(l)}_{t+1},\\forall l=0,1,2,\\ldots,L$ as per Eq.~\\ref{eq:w_update}.\\\\\n\\STATE Update $\\alpha^{(l)}_{t+1},\\forall l=0,1,2,\\ldots,L$ as per Eq.~\\ref{eq:alpha_update}.\\\\\n\\STATE $\\alpha^{(l)}_{t+1}=\\max (\\alpha^{(l)}_{t+1}, \\frac{s}{L+1}), \\forall l=0,1,2,\\ldots,L$.\\\\\n\\STATE Normalize $\\alpha^{(l)}_{t+1}$, i.e., $\\alpha^{(l)}_{t+1}=\\frac{\\alpha^{(l)}_{t+1}}{\\sum\\limits_{j=0}^L\\alpha^{(j)}_{t+1}}, \\forall l=0,1,2,\\ldots,L$.\n\\ENDFOR\n\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Application of OAHU{}}\n\\label{subsec:deployment}\n\nIn this part, we discuss the deployment of OAHU{} in different practical applications. Although OAHU{} can be applied in many applications, we mainly focus on the three most common tasks: classification, similarity comparison, and analogue retrieval. Note that the following discussion describes only one possible way of deployment for each task, and other deployment methods can exist.\n\n\\subsubsection{Classification}\n\\label{subsubsec:classification}\nGiven a training dataset $\\mathcal{D}=\\lbrace (\\bm{x_i}, y_i)\\rbrace_{i=1}^n$ where $\\bm{x_i}\\in \\mathcal{R}^d$ is an training instance and $y_i\\in \\lbrace1,2,\\ldots, c \\rbrace$ is its associated label, we determine the label $y\\in \\lbrace1,2,\\ldots,c\\rbrace$ of a test instance $\\bm{x}\\in \\mathcal{R}^d$ as follows:\n\\begin{itemize}\n    \\item For every metric model $E_l$ in OAHU{}, we find $k$ nearest neighbors of $\\bm{x}$ measured by Eq.~\\ref{eq:dist}, each of which is a candidate similar to $\\bm{x}$. Hence, $k(L+1)$ candidates in total are found in OAHU{}. Let $\\mathcal{N} =\\lbrace \\lbrace(\\bm{x_j^{(l)}}, y^{(l)}_j)\\rbrace_{j=1}^k \\rbrace_{l=0}^L=\\lbrace \\mathcal{N}_l \\rbrace_{l=0}^L$ denotes these candidates where $\\bm{x_j^{(l)}}$ is the $j^{th}$ neighbor of $\\bm{x}$ found in $E_l$ and $y^{(l)}_j$ is its associated label. The similarity score between $\\bm{x}$ and $\\bm{x_j^{(l)}}$ is given by \n    \\small\n    \\begin{equation}\n        S(\\bm{x}, \\bm{x_j^{(l)}})={\\rm e}^{-\\frac{D^{(l)}(\\bm{x_j^{(l)}}, \\bm{x})-d_{min}}{d_{max}-d_{min}}}\\cdot \\alpha^{(l)}\n    \\end{equation}\n    \\normalsize\n    where\n    $d_{min}=\\min\\limits_{ \\mathcal{N}_l}D^{(l)}(\\bm{x_j^{(l)}}, \\bm{x})$ and\n    $d_{max}=\\max\\limits_{ \\mathcal{N}_l}D^{(l)}(\\bm{x_j^{(l)}}, \\bm{x})$.\n    \\item Then, the class association score of each class $c_i$ is calculated by:\n    \\small\n    \\begin{equation}\n        S(c_i) = \\sum\\limits_{y_j^{(l)}=c_i} S(\\bm{x}, \\bm{x_j^{(l)}})\n    \\end{equation}\n    \\normalsize\n    \\item The final predicted label $y$ of $\\bm{x}$ in OAHU{} is determined as:\n    \\small\n    \\begin{equation}\n        y = \\argmax\\limits_{c_i} S(c_i)\n    \\end{equation}\n    \\normalsize\n\\end{itemize}\n\\subsubsection{Similarity Comparison}\nThe task of similarity comparison is to determine whether a given pair $(\\bm{x_1}, \\bm{x_2})$ is similar or not. To do this, a user-specified threshold $\\mathcal{T}\\in (0,1)$ is needed. The steps below are followed to perform the similarity comparison in OAHU{}.\n\\begin{itemize}\n    \\item First, for every metric model $E_l$, we compute the distance $D^{(l)}(\\bm{x_1}, \\bm{x_2})$ between $\\bm{x_1}$ and $\\bm{x_2}$. Since $D^{(l)}(\\bm{x_1}, \\bm{x_2}) \\in [0,2]$, we divide $D^{(l)}(\\bm{x_1}, \\bm{x_2})$ by $2$ for normalization.\n    \\item The similarity probability of $(\\bm{x_1}, \\bm{x_2})$ is then determined by\n    \\small\n    \\begin{equation}\n    \\label{eq:sim_com}\n        P(\\bm{x_{1}},\\bm{x_{2}}) = \\sum\\limits_{l=0}^L \\alpha^{(l)}\\cdot p_{l}\n    \\end{equation}\n    \\normalsize\n    where $p_{l}=1$ if $D^{(l)}(\\bm{x_1}, \\bm{x_2})\/2<\\mathcal{T}$; Otherwise, $p_l=0$.\n    \\item $(\\bm{x_1},\\bm{x_2})$ is similar if $ P(\\bm{x_1}, \\bm{x_2})\\ge 0.5$ and is dissimilar if $ P(\\bm{x_1}, \\bm{x_2})< 0.5$.\n\\end{itemize}\n\\subsubsection{Analogue Retrieval}\nGiven a query item $\\bm{x}$, the task of analogue retrieval is to find $k$ items that are most similar to $\\bm{x}$ in a database $\\mathcal{D}$. Note that analogue retrieval is a simplified ``classification'' task where we only find similar items and ignore the labeling issue.\n\\begin{itemize}\n\\item We first retrieve the $k(L+1)$ candidates that are similar to $\\bm{x}$ following the same process in classification task. Let $\\mathcal{N} =\\lbrace \\lbrace \\bm{x_j^{(l)}}\\rbrace_{j=1}^k \\rbrace_{l=0}^L=\\lbrace \\mathcal{N}_l \\rbrace_{l=0}^L$ denote these candidates. The similarity score between $\\bm{x_j^{(l)}}$ and $\\bm{x}$ is:\n\\small\n\\begin{equation}\n\\label{eq:retrieval}\n    Sim(\\bm{x_j^{(l)}}, \\bm{x}) = {\\rm e}^{-\\frac{D^{(l)}(\\bm{x_j^{(l)}}, \\bm{x})-d_{min}}{d_{max}-d_{min}}}\\cdot \\alpha^{(l)}\n\\end{equation}\n\\normalsize\nwhere $d_{min}=\\min\\limits_{ \\mathcal{N}_l}D^{(l)}(\\bm{x_j^{(l)}}, \\bm{x})$ and $d_{max}=\\max\\limits_{ \\mathcal{N}_l}D^{(l)}(\\bm{x_j^{(l)}}, \\bm{x})$.\n\\item Then we sort these items in descending order based on their similarity scores and keep only the top $k$ distinct items as the final retrieval result.\n\\end{itemize}\n\n\\subsection{Time and Space Complexity}\nOverall, the execution overhead of OAHU{} mainly arises from the training of the ANN-based metric model, especially from the gradient computation. Assume that the maximum time complexity for calculating the gradient of one layer is a constant $C$. The overall time complexity of OAHU{} is simply $O(nL^2C)$, where $n$ is the number of input constraints and $L$ is the number of hidden layers. Therefore, with pre-specified $L$ and ANN architecture, OAHU{} is an efficient online metric learning algorithm that runs in linear time.  Let $d$, $S_{hidden}$ and $S_{emb}$ denote the dimensionality of input data, number of units in each hidden layer and metric embedding size respectively. The space complexity of OAHU{} is given by $O\\big(d\\cdot S_{emb} + L\\cdot S_{hidden}(d+S_{emb})+\\frac{L(L-1)}{2}S_{hidden}^2\\big)$.\n\n\\subsection{Extendability}\nAs discussed in Section~\\ref{subsec:deployment}, OAHU{} is a fundamental module that can perform classification, similarity comparison, or analogue retrieval tasks independently. Therefore, it can be easily plugged into any existing deep models (e.g., a CNN or RNN) by replacing the classifier, i.e., fully-connected layers, with OAHU{}. The parameters of those feature-extraction layers (e.g., a convolutional layer) can be fine-tuned by optimizing $\\mathcal{L}_{overall}$ directly. However, in this paper, to prove the validity of the proposed approach, we only focus on OAHU{} itself and leave the deep model extension for future work.\n\n\n\n\\subsection{Baselines}\n    To examine the quality of metrics learned in OAHU{}, we compare it with several state-of-the-art online metric learning algorithms. (1) LEGO~\\cite{JainKDG08} (pairwise constraint): an online metric learning algorithm that learns a Mahalanobis metric based on LogDet regularization and gradient descent. (2) RDML~\\cite{JinWZ09} (pairwise constraint): an online algorithm for regularized distance metric learning that learns a Mahalanobis metric with a regret bound. (3) OASIS~\\cite{ChechikSSB10} (triplet constraint): an online algorithm for scalable image similarity learning that learns a bilinear similarity measure over sparse representations. Finally, (4) OPML~\\cite{LiGWZHS18} (triplet constraint): a one-pass closed-form solution for online metric learning that learns a Mahalanobis metric with a low computational cost. \n\n\\subsection{Implementation}\nWe have implemented OAHU{} using Python 3.6.2 and Pytorch 0.4.0 library. All baseline methods were based on code released by corresponding authors, except LEGO and RDML. Due to the unavailability of a fully functional code of LEGO and RDML, we use our own implementation based on the authors' description~\\cite{JainKDG08,JinWZ09}. Hyper-parameter setting of these approaches, including OAHU{}, varies with different tasks, and will be discussed later for each specific task.\n\n\n\\subsection{Image Classification}\n\\subsubsection{Dataset and Classifier} \nTo demonstrate the validacy of OAHU{} on complex practical applications, we use three publicly available real-world benchmark image datasets, including \\textbf{FASHION-MNIST}\\footnote{https:\/\/github.com\/zalandoresearch\/fashion-mnist}~\\cite{online}, \\textbf{EMNIST}\\footnote{https:\/\/www.nist.gov\/itl\/iad\/image-group\/emnist-dataset}~\\cite{CohenATS17} and \\textbf{CIFAR-10}\\footnote{https:\/\/www.cs.toronto.edu\/~kriz\/cifar.html}~\\cite{Krizhevsky09learningmultiple} for evaluation. To be consistent with other image datasets, we convert images of CIFAR-10 into grayscale through the OpenCV API~\\cite{opencv_library}, resulting in 1024 features. \nThe \\textit{k-NN} classifier is employed here, since it is widely used for classification tasks with only one parameter, and is also consistent with the deployment of OAHU{} discussed in Section~\\ref{subsubsec:classification}. \nDetails of these datasets are listed in ``Image Classification'' part of Table~\\ref{tab:dataset}.\n\n\\begin{table}[t]\n\\centering\n\\resizebox{0.8\\columnwidth}{!}{%\n\\begin{tabular}{|c|c|r|r|r|}\n\\hline\n\\textbf{Task} & \\textbf{Dataset} & \\multicolumn{1}{c|}{\\textbf{\\# features}} & \\multicolumn{1}{c|}{\\textbf{\\# classes}} & \\multicolumn{1}{c|}{\\textbf{\\# instances}} \\\\ \\hline\n\\multirow{3}{*}{\\textbf{Image Classification}} & FASHION-MNIST & 784 & 10 & 70,000 \\\\ \\cline{2-5} \n & EMNIST & 784 & 47 & 131,600 \\\\ \\cline{2-5} \n & CIFAR-10 & 1024 & 10 & 60,000 \\\\ \\hline\n\\textbf{Face Verification} & LFW & 73 & 5749 & 13,233 \\\\ \\hline\n\\multirow{2}{*}{\\textbf{Image Retrieval}} & CARS-196 & 4096 & 196 & 16,185 \\\\ \\cline{2-5} \n & CIFAR-100 & 4096 & 100 & 60,000 \\\\ \\hline\n\\end{tabular}%\n}\n\\caption{Description of Datasets}\n\\vspace{-4mm}\n\\label{tab:dataset}\n\\end{table}\n\n\\begin{table*}[t]\n\\centering\n\\resizebox{0.8\\textwidth}{!}{%\n\\begin{tabular}{|c|l|l|l|l|l|l|l|l|l|c|}\n\\hline\n\\multirow{2}{*}{\\textbf{Methods}} & \\multicolumn{3}{c|}{\\textbf{FASHION-MNIST}} & \\multicolumn{3}{c|}{\\textbf{EMNIST}} & \\multicolumn{3}{c|}{\\textbf{CIFAR-10}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{win\/tie\/loss}}} \\\\ \\cline{2-10}\n & \\multicolumn{1}{c|}{\\textit{Error Rate}} & \\multicolumn{1}{c|}{$F_1$} & \\multicolumn{1}{c|}{$\\mathcal{U}$} & \\multicolumn{1}{c|}{\\textit{Error Rate}} & \\multicolumn{1}{c|}{$F_1$} & \\multicolumn{1}{c|}{$\\mathcal{U}$}  & \\multicolumn{1}{c|}{\\textit{Error Rate}} & \\multicolumn{1}{c|}{$F_1$} &  \\multicolumn{1}{c|}{$\\mathcal{U}$} & \\multicolumn{1}{c|}{} \\\\ \\hline\n\\textbf{LEGO} & 0.35$\\pm$0.01 $\\bullet$ & 0.64$\\pm$0.00 & 0.58$\\pm$0.00 & 0.84$\\pm$0.00 $\\bullet$ & 0.14$\\pm$0.02 & 0.66$\\pm$0.01 & 0.88$\\pm$0.00 $\\bullet$ & 0.05$\\pm$0.02 & 0.59$\\pm$0.00 & \\textbf{3\/0\/0} \\\\ \\hline\n\\textbf{RDML} & 0.22$\\pm$0.01 $\\bullet$ & 0.78$\\pm$0.00 & 0.47$\\pm$0.01 & 0.64$\\pm$0.03 $\\bullet$ & 0.32$\\pm$0.05 & 0.50$\\pm$0.00 & 0.76$\\pm$0.01 $\\bullet$ & 0.24$\\pm$0.01 & 0.50$\\pm$0.01 & \\textbf{3\/0\/0} \\\\ \\hline\n\\textbf{OASIS} & 0.21$\\pm$0.02 $\\bullet$ & 0.79$\\pm$0.01 & 0.14$\\pm$0.00 & 0.41$\\pm$0.01 $\\bullet$ & 0.59$\\pm$0.00 & 0.24$\\pm$0.00 & 0.79$\\pm$0.02 $\\bullet$ & 0.20$\\pm$0.03 & 0.47$\\pm$0.00 & \\textbf{3\/0\/0} \\\\ \\hline\n\\textbf{OPML} & 0.23$\\pm$0.00 $\\bullet$ & 0.78$\\pm$0.00 & 0.30$\\pm$0.01 & 0.43$\\pm$0.01 $\\bullet$ & 0.51$\\pm$0.02 & 0.45$\\pm$0.00 & 0.83$\\pm$0.01 $\\bullet$ & 0.16$\\pm$0.00 & 0.48$\\pm$0.04 & \\textbf{3\/0\/0} \\\\ \\hline\n\\textbf{OAHU{}} & \\textbf{0.18$\\pm$0.00} & \\textbf{0.81$\\pm$0.00} &\n\\textbf{1.00$\\pm$0.00} &\n\\textbf{0.35$\\pm$0.01} & \\textbf{0.64$\\pm$0.01} &\n\\textbf{1.00$\\pm$0.00} & \n\\textbf{0.68$\\pm$0.00} & \\textbf{0.31$\\pm$0.00} &\n\\textbf{1.00$\\pm$0.00} &\n - \\\\ \\hline\n\\end{tabular}%\n}\n\\caption{Comparison of classification performance on competing methods over benchmark image datasets. $\\bullet\/\\circ$ indicates \\textbf{OAHU{}} performs statistically better\/worse (0.05 significance level) than the respective method according to the $p$-values. The statistics of win\/tie\/loss is also included. Both mean and standard deviation of error rates, constraint utilization $\\mathcal{U}$ and macro $F_1$ scores are reported. $0.00$ denotes a value less than $0.005$.}\n\\vspace{-6mm}\n\\label{tab:classification}\n\\end{table*}\n\n\\begin{figure*}[t]\n\\centering\n\\subfloat[FASHION-MNIST\\label{fig:fashionmnist_triplet}]{\\includegraphics[width =0.3 \\textwidth]{fig4a.jpg}}\\hfill\n\\subfloat[CIFAR-10\\label{fig:cifar10_triplet}]{\\includegraphics[width = 0.3\\textwidth]{fig4b.jpg}}\n\\hfill\n\\subfloat[\\label{fig:weight_triplet}]{\\includegraphics[width = 0.3\\textwidth]{fig4c.jpg}}\n\\caption{(a) \/ (b) Error rates of competing methods with increasing number of constraints on FASHION-MNIST \/ CIFAR-10 dataset. (c) The evolvement of metric weight distribution in OAHU{} with increasing number of constraints on FASHION-MNIST dataset. }\n\\vspace{-4mm}\n\\label{fig:error_rate_vary_constraints}\n\\end{figure*}\n\n\\subsubsection{Experiment Setup and Constraint Generation}\n\\label{subsubsec: escg}\nTo perform the experiment, we first randomly shuffle each dataset and then divide it into development and test sets with the split ratio of $1:1$. Hyper-parameters of these baseline approaches were set based on values reported by the authors and fine-tuned via 10-fold cross-validation on the development set. Because OPML applies a one-pass triplet construction process to generate the constraints, we simply provide the training data in the development set as the input for OPML.\nFor other approaches including OAHU{}, we first randomly sample $5,000$ seeds (i.e., pairwise or triplet constraints) from the training data and then construct $5,000$ more constraints by taking transitive closure over these seeds.\nNote that the same number of constraints were adopted by the authors of LEGO~\\cite{JainKDG08} and OPML~\\cite{LiGWZHS18} when comparing their approaches with baselines. Thus our comparison is fair. \nIn OAHU{}, we set $\\beta=0.99$, $L=5$, $S_{hidden}=100$, $s=0.1$, $\\eta=0.3$ and $S_{emb}=50$ as default. The value of $\\tau$ was found by 10-fold cross-validation on the development set.\nTo reduce any bias that may result from random partition, all the classification results are averaged over $10$ individual runs with independent random shuffle processes.\n\n\\subsubsection{Evaluation Metric}\nSimilar to most of the baselines, e.g., LEGO~\\cite{JainKDG08} and OPML~\\cite{LiGWZHS18}, we adopt the error rate and macro $F_1$ score\\footnote{http:\/\/scikit-learn.org\/stable\/modules\/generated\/sklearn.metrics.f1\\_score.html} as the evaluation criterion. Moreover, the $p$-values of student's t-test were calculated to check statistical significance.  In addition, the statistics of win\/tie\/loss are reported according to the obtained p-values. The constraint utilization $\\mathcal{U}$, i.e., the fraction of input constraints that actually contribute to model update, is also reported for each method.\n\n\\subsubsection{Analysis}\nTable~\\ref{tab:classification} compares the classification performance of OAHU{} with baseline methods on benchmark datasets. We observe that LEGO and RDML perform poorly on complex image datasets, e.g., EMNIST ($47$ classes), since they learn a Mahalanobis metric from pairwise constraints, and are biased to either the similarity or dissimilarity relation when updating model parameters. OASIS performs better than OPML because the bilinear metric eliminates the positive and symmetric requirement of the covariance matrix, i.e., $A$ in $d(\\bm{x_1}, \\bm{x_2})=\\bm{x_1}^TA\\bm{x_2}$ which results in a larger hypothesis space~\\cite{ChechikSSB10}. However, OAHU{} outperforms all the baseline approaches by\nproviding not only significantly lower error rate, but also higher $F_1$ score, indicating the best overall classification performance among all competing methods.\n\n\nTo illustrate the effect of different numbers of constraints on the learning of a metric, we vary the  input constraint numbers\\footnote{It is done by first generating $10,000$ constraints as discussed in Section~\\ref{subsubsec: escg} and then uniformly sample the desired number of constraints from them.} for all competing methods except OPML. Here we do not compare with OPML because it internally determines the number of triplets to use via a one-pass triplet construction process. Therefore, explicitly varying the input triplet number for OPML is impossible. Figure~\\ref{fig:fashionmnist_triplet} and Figure~\\ref{fig:cifar10_triplet} show the comparison of error rates on FASHION-MNIST and CIFAR-10 dataset respectively. \nThe large variance of error rates observed in both OASIS and RDML (i.e., error bars in the figures) indicates the strong correlation of their classification performance with the quality of input constraints.\nAll baseline approaches present an unstable, and sometimes, even worse performance with an increasing number of constraints. This is because these baselines are biased to those \"hard\" constraints leading to positive loss values and incorrectly ignore the emerging failure cases (see Figure~\\ref{fig:failuremode}). In contrast, OAHU{} provides the best and most stable classification performance with minimal variance, indicating its robustness to the quality of input constraints and better capability of capturing the semantic similarity and dissimilarity relations in data. \n\nOAHU{} is better mainly because it not only learns from every input constraint by optimizing the adaptive-bound triplet loss (see constraint utilization $\\mathcal{U}$ in Table~\\ref{tab:classification}), but also automatically adapts the model complexity to achieve better performance (see Figure~\\ref{fig:weight_triplet}).\n\n\n\\subsection{Face Verification}\n\\subsubsection{Dataset and Experiment Setup}\nFor face verification, we evaluate our method on Labeled Faces in the Wild Database (LFW)~\\cite{LFWTechUpdate}. \nThis dataset has two views: View 1 is used for development purpose (contains a training and a test set), and View 2 is taken as a benchmark for comparison (i.e., a 10-fold cross validation set containing pairwise images). \nThe goal of face verification in LFW is to determine whether a pair of face images belongs to the same person. \nSince OASIS, OPML, and OAHU{} are triplet-based methods, following the setting in OPML~\\cite{LiGWZHS18}, we adopt the image unrestricted configuration to conduct experiments for a fair comparison. \nIn the image unrestricted configuration, the label information (i.e. actual names) in training data can be used and as many pairs or triplets as one desires can be formulated. We use View 1 for hyper-parameter tuning and evaluate the performance of all competing algorithms on each fold (i.e., 300 intra-person and 300 inter-person pairs) in View 2. Note that the default parameter values in OAHU{} and baselines were set using the same values as discussed in Section~\\ref{subsubsec: escg}. Attribute features of LFW are used in the experiment so that the result can be easily reproduced and the comparison is fair. Attribute features\\footnote{http:\/\/www.cs.columbia.edu\/CAVE\/databases\/pubfig\/download\/lfw\\_attributes.txt} ($73$ dimension) provided by Kumar et al.~\\cite{KumarBBN09} are ``high-level'' features describing nameable attributes such as race, age etc., of a face image. Details of LFW are described in ``Face Verification'' part in Table~\\ref{tab:dataset}.\n\n\\subsubsection{Constraint Generation and Evaluation Metric}\nWe follow the same process described in Section~\\ref{subsubsec: escg} to generate constraints for training purpose. However, we add an additional step here: if a generated constraint contains any pair of instances in the validation set (i.e., all pairs in View 1) or the test set (i.e., all folds in View 2), we simply discard it and re-sample a new constraint. This step is repeated until the new constraint does not contain any pair in both sets. In the experiment, $10,000$ pair or triplet constraints in total are generated for training. \n\nNote, due to the one-pass triplet generation strategy of OPML, we let itself determine the training triplets.\nFor each test image pair, we first compute the distance between them measured by the learned metric, and then normalize the distance into $[0,1]$. The widely accepted Receiver Operating Characteristic (ROC) curve and Area under Curve (AUC) are adopted as the evaluation metric. \n\n\\subsubsection{Analysis}\n\n\\begin{figure}[t]\n\\centering\n\\subfloat[\\label{fig:roc}]{\\includegraphics[width =0.5 \\columnwidth]{fig5a.jpg}}\\hfill\n\\subfloat[\\label{fig:roc_weight}]{\\includegraphics[width = 0.5\\columnwidth]{fig5b.jpg}}\n\\caption{(a) ROC Curves of OAHU{} and contrastive methods on LFW dataset. AUC value of each method is given in bracket. (b) Metric weight distribution of OAHU{}.}\n\\vspace{-4mm}\n\\label{fig:roc_combine}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\subfloat[Failure Cases]{\\includegraphics[width = 0.49\\columnwidth]{fig6a.jpg}}\\hfill\n\\subfloat[Successful Cases]{\\includegraphics[width = 0.49\\columnwidth]{fig6b.jpg}}\n\\caption{Face verification examples in the test set (View 2) ($\\mathcal{T}=0.55$). For images belonging to same person, $P(\\bm{x_1}, \\bm{x_2})$ should be close to $1$ and for images belonging to different people, $P(\\bm{x_1}, \\bm{x_2})$ should drop to $0$.}\n\\vspace{-4mm}\n\\label{fig:fv_cases}\n\\end{figure}\n\nThe ROC curves of all competing methods are given in Figure~\\ref{fig:roc} with corresponding AUC values calculated. It can be observed that the proposed approach OAHU{} outperforms all baseline approaches by providing significantly higher AUC value. As demonstrated in Figure~\\ref{fig:roc_weight}, OAHU{} is better mainly because \nit automatically adapts the model complexity to incorporate non-linear models, which help to separate instances that are difficult to distinguish in simple linear models. Figure~\\ref{fig:fv_cases} presents some example comparisons in the test set for both successful and failure cases. Here, the comparison probability $P(\\bm{x_1}, \\bm{x_2})$ computed based on Eq.~\\ref{eq:sim_com} is expected to approach $1$ if both images are from the same person and should be close to $0$ if they belong to different people. Apparently, although OAHU{} performs much better than baselines, it may still fail in some cases where severe concept drift, e.g. aging and shave, are present.\n\n\n\n\\subsection{Image Retrieval}\n\\subsubsection{Dataset and Experiment Setup}\nTo demonstrate the performance of OAHU{} on image retrieval task, we show experiments on two benchmark datasets, which are CARS-196\\footnote{https:\/\/ai.stanford.edu\/~jkrause\/cars\/car\\_dataset.html}~\\cite{KrauseStarkDengFei-Fei} and CIFAR-100\\footnote{https:\/\/www.cs.toronto.edu\/~kriz\/cifar.html}~\\cite{Krizhevsky09learningmultiple}. \nThe CARS-196 dataset is divided into $8,144$ training images and $8,401$ test images, where each class has been split roughly in a 50-50 ratio. \nSimilarly, we randomly split CIFAR-100 into $30,000$ training images and $30,000$ test images by equally dividing images of each class. \nFor each image in CARS-196 and CIFAR-100, its deep feature, i.e., the output of the last convolutional layer, is extracted from a VGG-19 model pretrained on  the ImageNet~\\cite{ILSVRC15} dataset. \nDetails of these 2 datasets are listed in the ``Image Retrieval'' part of Table~\\ref{tab:dataset}. \nWe form the development set of each dataset by including all its training images.\nFor each dataset, hyper-parameters of baseline approaches were set based on values reported by the authors, and fine-tuned via 10-fold cross-validation on the development set. In OAHU{}, we set $\\beta=0.99$, $L=5$, $S_{hidden}=100$, $s=0.1$, $\\eta=0.3$ and $S_{emb}=50$ as default. The value of $\\tau$ in OAHU{} was found via 10-fold cross-validation on the development set. \n\n\\subsubsection{Constraint Generation and Evaluation Metric}\nWith the training images in the development set, we generate the training constraints in the same way as Section~\\ref{subsubsec: escg}. However, in this experiment, we sample $50,000$ constraints ($25,000$ seeds) for each dataset due to the vast amount of classes included in them. The Recall@K metric~\\cite{JegouDS11} is applied for performance evaluation. Each test image (query) first retrieves K most similar images from the test set and receives a \nscore of 1 if an image of the same class is retrieved among these K images and 0 otherwise. Recall@K averages this score over all the images.\n\n\\subsubsection{Analysis}\n\n\\begin{figure}[t]\n\\centering\n\\subfloat[\\label{fig:recall_cars}]{\\includegraphics[width =0.5 \\columnwidth]{fig7a.jpg}}\\hfill\n\\subfloat[\\label{fig:weight_cars}]{\\includegraphics[width = 0.5\\columnwidth]{fig7b.jpg}}\n\\caption{(a) Recall@K score on the test split of CARS-196. (b) Metric weight distribution of OAHU{} on CARS-196.}\n\\vspace{-4mm}\n\\label{fig:img_retr_cars}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\subfloat[\\label{fig:recall_cifar100}]{\\includegraphics[width =0.5 \\columnwidth]{fig8a.jpg}}\\hfill\n\\subfloat[\\label{fig:weight_cifar100}]{\\includegraphics[width = 0.5\\columnwidth]{fig8b.jpg}}\n\\caption{(a) Recall@K score on the test split of CIFAR-100. (b) Metric weight distribution of OAHU{} on CIFAR-100.}\n\\vspace{-4mm}\n\\label{fig:img_retr_cifar100}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\subfloat[CARS-196]{\\includegraphics[width = 0.49\\columnwidth]{fig9a.jpg}}\\hfill\n\\subfloat[CIFAR-100]{\\includegraphics[width = 0.49\\columnwidth]{fig9b.jpg}}\n\\caption{Examples of successful and failure queries on the CARS-196 and CIFAR-100 datasets using our embedding. Images in the first column are query images and the rest are three most similar images. Best viewed on a monitor zoomed in.}\n\\vspace{-4mm}\n\\label{fig:retrieval_cases}\n\\end{figure}\n\nThe Recall@K scores on the test set of CARS-196 and CIFAR-100 are shown in Figure~\\ref{fig:recall_cars} and Figure~\\ref{fig:recall_cifar100}, respectively. We observe that OAHU{} outperforms all baseline approaches by providing significantly higher Recall@K scores. In addition, the Recall@K score of OAHU{} grows more rapidly than that provided by most baselines. Figure~\\ref{fig:weight_cars} and Figure~\\ref{fig:weight_cifar100} illustrate the metric weight distribution of OAHU{} on CARS-196 and CIFAR-100 respectively. Compared to the baselines that are only linear models, OAHU{} takes full advantage of non-linear metrics to improve the model expressiveness. Figure~\\ref{fig:retrieval_cases} shows some examples of successful and failure queries on both datasets.\nDespite the huge change in the viewpoint, configuration, and illumination, our method can successfully retrieve examples from the same class and most retrieval failures come from subtle visual difference among images of different classes.\n\n\\subsection{Sensitivity of Parameters}\n\n\nThe three main parameters in OAHU{} is the control parameter $\\tau$. \nWe vary $\\tau$ from 0.1 to 0.9 and $S_{emb}$ from 10 to 100 to study their effect on the classification performance. \nWe observe that both classification accuracy (1-error rate) and $F_1$ score significantly drop if $\\tau$ is greater than 0.6 but remain unchanged with various $\\tau$ values if $\\tau\\leq 0.6$. This observation verifies Theorem~\\ref{theo:1} which states that classes are separated if $\\tau \\in (0,\\frac{2}{3})$. On the other hand, the classification accuracy of OAHU{} peaks at embedding size $S_{emb}=50$ where all semantic information are maintained. If we further increase the embedding size, noisy information is going to be introduced which degrades the classification performance. Therefore, we suggest to choose $\\tau=0.1$ and $S_{emb}=50$ as default.\n\n\n\\section{Introduction}\n\\label{Sec:introduction}\n\\input{introduction.tex}\n\n\\section{Related Work}\n\\label{Sec:background}\n\\input{background.tex}\n\n\\section{OAHU}\n\\label{Sec:approach}\n\\input{approach.tex}\n\n\\section{Empirical Evaluation}\n\\label{Sec:evaluation} \n\\input{experiment.tex}\n\n\\section{Limitations and Conclusions}\n\\input{conclusion}\n\\label{Sec:conclusion}\n\n\\section{ACKNOWLEDGMENTS}\nThis material is based upon work supported by NSF award numbers: DMS-1737978,  DGE 17236021. OAC-1828467; ARO award number: W911-NF-18-1-0249; IBM faculty award (Research); and NSA awards.\n\n\\bibliographystyle{ACM-Reference-Format}\n\\balance\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper, we propose to study totally geodesic submanifolds\ninside locally symmetric spaces. Let us start by fixing some\nnotation: $(X^n,Y^m)$ will always refer to a pair of compact locally\nsymmetric spaces of noncompact type, with $Y^m\\subset X^n$ a\ntotally geodesic submanifold.  The spaces $X^n,Y^m$ will be locally\nmodelled on $G\/K$, $G^\\prime\/K^\\prime$ respectively, where $G,G^\\prime$ are a\npair of semisimple Lie groups, and $K,K^\\prime$ are a pair of maximal\ncompact subgroups in the respective $G,G^\\prime$. Note that, since\n$Y^m\\subset X^n$ is totally geodesic, one can view $G^\\prime$ as a\nsubgroup of $G$, and hence one can take $K^\\prime = K\\cap G^\\prime$.\nWe will denote by $X_u=G_u\/K$, $Y_u=G_u^\\prime\/K^\\prime$ the nonnegatively\ncurved dual symmetric spaces to the nonpositively curved spaces $G\/K$,\n$G^\\prime\/K^\\prime$.\n\nNote that for a pair $(X^n,Y^m)$, the submanifold $Y^m$ is always\n{\\it homotopically\\\/} nontrivial.  Indeed, the inclusion induces a\nmonomorphism on the level of fundamental groups.  A more subtle\nquestion is whether the submanifold $Y^m$ is {\\it homologically\\\/}\nnontrivial, ie whether $[Y^m]\\neq 0 \\in H_m(X^n ;\\mathbb R)$ (or in\n$H_m(X^n ;\\mathbb Z)$).  Our first result provides a criterion for\ndetecting when a totally geodesic submanifold $Y^m$ is homologically\nnontrivial (over $\\mathbb R$) in $X^n$.\n\n\n\\begin{Thm}\\label{Theorem1.1}\nLet $Y^m\\hookrightarrow X^n$ be a compact totally geodesic\nsubmanifold of noncompact type inside a compact locally symmetric\nspace of noncompact type, and denote by $\\rho$ the map on cohomology\n$H^m(X_u;\\mathbb R)\\rightarrow H^m(Y_u;\\mathbb R)\\simeq \\mathbb R$ induced by the\nembedding $Y_u\\hookrightarrow X_u$.  Then we have the following:\n\\begin{itemize}\n\\item If $[Y^m]=0 \\in H_m(X^n; \\mathbb R)$ then the map $\\rho$ is identically zero.\n\\item If $\\rho$ is identically zero, and $m\\leq m(\\mathfrak g)$, where $m(\\mathfrak g)$ is the Matsushima\nconstant corresponding to the Lie algebra $\\mathfrak g$ of the Lie group $G$, then\nwe have that $[Y^m]=0 \\in H_m(X^n; \\mathbb R)$.\n\\end{itemize}\n\\end{Thm}\n\nOur proof of this first result is an adaptation of an argument of\nMatsushima \\cite{Mat} and relies on the existence of certain\ncompatible maps (the Matsushima maps) from the real cohomology of\nthe pair of nonnegatively curved duals $(X_u,Y_u)$ to the real\ncohomology of the nonpositively curved pair $(X^n,Y^m)$.  It is\nreasonable to ask whether this map can be realized {\\it\ngeometrically}.  Our second result, extending work of Okun\n\\cite{O1}, shows that this can sometimes be achieved {\\it\nrationally}:\n\n\\begin{Thm}\\label{Theorem1.2}\nAssume that $Y^m\\hookrightarrow X^n$ is a totally geodesic embedding\nof compact, locally symmetric spaces of noncompact type.\nFurthermore, assume that the map $G_u^\\prime\\hookrightarrow G_u$ induced\nby the inclusion $Y\\hookrightarrow X$ is a $\\pi_i$--isomorphism,\nfor $i< m$, and a surjection on $\\pi_m$. Then there exists a finite\ncover $\\bar X$ of $X^n$, and a connected lift $\\bar Y \\subset \\bar\nX$ of $Y^m$, with the property that there exists a {\\it tangential\nmap} of pairs $(\\bar X, \\bar Y) \\rightarrow (X_u, Y_u)$.  If in\naddition we have $\\rk(G_u)=\\rk(K)$ and $\\rk(G_u^\\prime)=\\rk(K^\\prime)$, then the\nrespective tangential maps induce the Matsushima maps on cohomology.\n\\end{Thm}\n\nSince the tangent bundle of the submanifold $Y^m$ Whitney sum with\nthe normal bundle of $Y^m$ in $X^n$ yields the restriction of the\ntangent bundle of $X^n$ to the submanifold $Y^m$, this gives the\nimmediate:\n\n\\begin{Cor}\\label{Corollary1.3}\nUnder the hypotheses of \\fullref{Theorem1.2}, we have that the\npullback of the normal bundle of $Y_u$ in $X_u$ is stably equivalent\nto the normal bundle of $\\bar Y^m$ in $\\bar X^n$.\n\\end{Cor}\n\nIn the previous corollary, we note that if $2m+1\\leq n$, then these\ntwo bundles are in fact isomorphic (see for instance Husemoller \\cite[Chapter 8,\nTheorem 1.5]{H}).\n\nAn example where the hypotheses of the \\fullref{Theorem1.2} are satisfied\narises in the situation where $Y^m$, $X^n$ are real hyperbolic\nmanifolds. Specializing \\fullref{Corollary1.3} to this situation, we\nobtain:\n\n\\begin{Cor}\\label{Corollary1.4}\nLet $Y^m\\hookrightarrow X^n$ be a totally geodesic embedding, where\n$X^n$, $Y^m$ are compact hyperbolic manifolds, and assume that\n$2m+1\\leq n$. Then there exists a finite cover $\\bar X$ of $X^n$,\nand a connected lift $\\bar Y$ of $Y^m$, with the property that the\nnormal bundle of $\\bar Y$ in $\\bar X$ is trivial.\n\\end{Cor}\n\nWhile the hypotheses of \\fullref{Theorem1.2} are fairly technical, we\npoint out that there exist several examples of inclusions\n$Y^m\\hookrightarrow X^n$ satisfying the hypotheses of the theorem.\nThe proof of \\fullref{Corollary1.4}, as well as a discussion of some\nfurther examples will be included at the end of Section 4.  Finally,\nwe will conclude the paper with various remarks and open questions\nin Section 5.\n\n\\vspace{-4pt}\n\\paragraph{Acknowledgements}  \nThis research was partially conducted during the period when B\\,Schmidt was employed by the Clay Mathematics Institute as a Liftoff Fellow. The research of J\\,-F\\,Lafont was partly supported by the National Science Foundation under grant DMS - 0606002.  The authors would like to thank the anonymous referee\nfor pointing out a simplification in our original proof of \\fullref{Proposition5.2}.\n\\vspace{-4pt}\n\n\n\n\n\\section{Background}\n\\vspace{-4pt}\n\nIn this section, we provide some discussion of the statements of our theorems.  We also\nintroduce some of the ingredients that will be used in the proofs of our results.\n\\vspace{-4pt}\n\n\n\\subsection{Dual symmetric spaces}\n\\vspace{-4pt}\n\nLet us start by recalling the definition of dual symmetric spaces:\n\\vspace{-4pt}\n\n\\begin{Def}\nGiven a symmetric space $G\/K$ of noncompact type, we define the\n{\\it dual symmetric space\\\/} in the following manner. Let $G_\\mathbb C$\ndenote the complexification of the semisimple Lie group $G$, and\nlet $G_u$ denote the maximal compact subgroup in $G_\\mathbb C$.  Since $K$\nis compact, under the natural inclusions $K\\subset G\\subset G_\\mathbb C$,\nwe can assume that $K\\subset G_u$ (up to conjugation). The symmetric\nspace dual to $G\/K$ is defined to be the symmetric space $G_u\/K$.\nBy abuse of language, if $X=\\Gamma \\backslash G\/K$ is a locally\nsymmetric space modelled on the symmetric space $G\/K$, we will say\nthat $X$ and $G_u\/K$ are dual spaces.\n\\end{Def}\n\\vspace{-4pt}\n\nNow assume that $Y^m\\hookrightarrow X^n$ is a totally geodesic\nsubmanifold, where both $Y^m$, $X^n$ are locally symmetric spaces of\nnoncompact type.  Fixing a lift of $Y$, we have a totally geodesic\nembedding of the universal covers:\n$$G^\\prime\/K^\\prime =\\tilde Y\\hookrightarrow \\tilde X=G\/K$$\nCorresponding to this totally geodesic embedding, we get a natural commutative diagram:\n$$\\disablesubscriptcorrection\\xysavmatrix{G^\\prime \\ar[r] & G\\\\\nK^\\prime \\ar[r] \\ar[u] & K \\ar[u]}$$\nwhich, after passing to the complexification, and descending to the maximal compacts,\nyields a commutative diagram:\n$$\\disablesubscriptcorrection\\xysavmatrix{G^\\prime_u \\ar[r] & G_u\\\\\nK^\\prime \\ar[r] \\ar[u] & K \\ar[u]}$$\nIn particular, corresponding to the totally geodesic embedding $Y\\hookrightarrow X$,\nwe see that there is a totally geodesic embedding of the dual symmetric spaces $G_u^\\prime\/K^\\prime\n\\hookrightarrow G\/K$.\n\n\n\n\\subsection{Classifying spaces}\n\nFor $G$ a continuous group let $EG$ denote a contractible space\nwhich supports a free $G$--action. The quotient space, denoted $BG$,\nis called a {\\it classifying space\\\/} for principal $G$--bundles. This\nterminology is justified by the fact that, for any topological space\n$X$, there is a bijective correspondence between (1) isomorphism\nclasses of principal $G$--bundles over $X$, and (2) homotopy classes\nof maps from $X$ to $BG$.  Note that the spaces $EG$ are only\ndefined up to $G$--equivariant homotopies, and likewise the spaces\n$BG$ are only defined up to homotopy.  Milnor \\cite{Mil} gave a\nspecific construction, for a Lie group $G$, of a space homotopy\nequivalent to $BG$.  The basic fact we will require concerning\nclassifying spaces is the following:\n\n\\begin{Thm} If $H$ is a closed subgroup of the Lie group $G$, then there exists a natural map\n$BH\\rightarrow BG$ between the models constructed by Milnor; furthermore this map is a fiber\nbundle with fiber the homogenous space $G\/H$.\n\\end{Thm}\n\n\\subsection{Okun's construction}\n\nOkun established \\cite[Theorem 5.1]{O1} the following nice result:\n\n\\begin{Thm}\\label{Theorem2.2} Let $X=\\Gamma \\backslash G\/K$ and $X_u=G_u\/K$ be dual symmetric spaces.\nThen there exists a finite sheeted cover $\\bar X$ of $X$ (ie a subgroup $\\bar \\Gamma$ of\nfinite index in $\\Gamma$, $\\bar X=\\bar \\Gamma \\backslash G\/K$), and a tangential map $k\\co\\bar X\n\\rightarrow X_u$.\n\\end{Thm}\n\nThis was subsequently used by Okun to exhibit exotic smooth\nstructures on certain compact locally symmetric spaces of\nnoncompact type \\cite{O2}, and by Aravinda--Farrell in their\nconstruction of exotic smooth structures on certain quaternionic\nhyperbolic manifolds supporting metrics of strict negative curvature\n\\cite{AF}.  More recently, this was used by Lafont--Roy \\cite{LaR} to\ngive an elementary proof of the Hirzebruch proportionality principle\nfor Pontrjagin numbers, as well as (non)vanishing results for\nPontrjagin numbers of the Gromov--Thurston examples of manifolds with\nnegative sectional curvature.\n\nSince it will be relevant to our proof of the main theorem, we\nbriefly recall the construction of the finite cover that appears in\nOkun's argument for \\fullref{Theorem2.2}.  Starting from the canonical\nprinciple fiber bundle\n$$\\Gamma \\backslash G\\rightarrow \\Gamma \\backslash G\/K=X$$\nwith structure group $K$ over the base space $X$, we can extend the\nstructure group to the group $G$, yielding the flat principle\nbundle:\n$$\\Gamma \\backslash G\\times_K G=G\/K \\times_\\Gamma G \\longrightarrow \\Gamma \\backslash G\/K=X$$\nFurther extending the structure group to $G_\\mathbb C$ yields a flat\nbundle with a complex linear algebraic structure group.  A result of\nDeligne and Sullivan \\cite{DS} implies that there is a finite cover\n$\\bar X$ of $X$ where the pullback bundle is trivial; since $G_u$ is\nthe maximal compact in $G_\\mathbb C$, the bundle obtained by extending the\nstructure group from $K$ to $G_u$ is trivial as well. In terms of\nthe classifying spaces, this yields the commutative diagram:\n$$\\disablesubscriptcorrection\\xysavmatrix{ & G_u\/K \\ar[d]\\\\\n\\bar X \\ar[r] \\ar@{.>}[ur] \\ar[dr]_{\\simeq 0}& BK \\ar[d]\\\\\n & BG_u}$$\nUpon homotoping the bottom diagonal map to a point, one obtains that\nthe image of the horizontal map lies in the fiber above a point,\nie inside $G_u\/K$, yielding the dotted diagonal map in the above\ndiagram.  Okun then proceeds to show that the map to the fiber is\nthe desired tangential map (since the pair of maps to $BK$ classify\nthe respective canonical $K$--bundles on $\\bar X$ and $G_u\/K$, and\nthe canonical $K$--bundles determine the respective tangent bundles).\n\n\n\\subsection{Matsushima's map}\n\nMatsushima \\cite{Mat} constructed a map on cohomology\n$j^*\\co H^*(G_u\/K; \\mathbb R)\\rightarrow H^*(X; \\mathbb R)$ whenever $X$ is a\ncompact locally symmetric space modelled on $G\/K$.  We will require the\nfollowing fact concerning the Matsushima map:\n\n\\begin{Thm}[Matsushima \\cite{Mat}] The map $j^*$ is always injective.\nFurthermore, there exists a constant $m(\\mathfrak g)$ (called the Matsushima\nconstant) depending solely on the Lie algebra $\\mathfrak g$ of the Lie group\n$G$, with the property that the Matushima map $j^*$ is a surjection\nin cohomology up to the dimension $m(\\mathfrak g)$.\n\\end{Thm}\n\nThe specific value of the Matsushima constant for\nthe locally symmetric spaces that are K\\\"ahler can be found in \\cite{Mat}.\nWe also point out the following result of Okun \\cite[Theorem\n6.4]{O1}:\n\n\\begin{Thm}\\label{Theorem2.4} Let $X=\\Gamma \\backslash G\/K$ be a compact locally symmetric space,\nand $\\bar X$, $t\\co \\bar X \\rightarrow G_u\/K$ the finite cover and\ntangential map constructed in \\fullref{Theorem2.2}.  If the groups $G_u$ and\n$K$ have equal rank, then the induced map $t^*$ on cohomology\ncoincides with Matsushima's map $j^*$.\n\\end{Thm}\n\n\n\n\n\n\\section{Detecting homologically essential submanifolds}\n\nIn this section, we provide a proof of \\fullref{Theorem1.1}, which gives a criterion for establishing\nwhen a totally geodesic submanifold $Y\\subset X$ in a locally symmetric space of noncompact\ntype, is homologically nontrivial.\n\n\\begin{proof}[Proof of \\fullref{Theorem1.1}]\nIn order to establish the theorem, we make use of differential\nforms.  If a group $H$ acts on a smooth manifold $M$, we let $\\Omega\n^H(M)$ denote the complex of $H$--invariant differential forms on\n$M$. Let $X=\\Gamma \\backslash G\/K$, $Y=\\Lambda \\backslash G^\\prime\/K^\\prime$ be the pair of\ncompact locally symmetric spaces, and $X_u=G_u\/K$, $Y_u=G_u^\\prime\/K^\\prime$\nbe the corresponding dual spaces.  We now consider the following\nfour complexes of differential forms: (1) $\\Omega ^{G}(G\/K)$, (2) $\\Omega\n^{G^\\prime}(G^\\prime\/K^\\prime)$, (3) $\\Omega ^\\Gamma(G\/K)$ and (4) $\\Omega\n^\\Lambda(G^\\prime\/K^\\prime)$.\n\nWe now observe that the cohomology of the first two complexes can be identified with the\ncohomology of\n$X_u$, $Y_u$ respectively.  Indeed, we have the sequence of natural identifications:\n$$\\Omega ^{G}(G\/K)=H^*(\\mathfrak g,\\mathfrak t)=H^*(\\mathfrak g_u, \\mathfrak t)=\\Omega ^{G_u}(G_u\/K)$$\nThe first and third equalities come from the identification of the complex of harmonic\nforms with the relative Lie algebra cohomology.  The second equality comes via the dual Cartan\ndecompositions: $\\mathfrak g=\\mathfrak t \\oplus \\mathfrak p$ and $\\mathfrak g_u=\\mathfrak t \\oplus i\\mathfrak p$.  Since $X_u=G_u\/K$ is a compact closed\nmanifold, and $\\Omega ^{G_u}(G_u\/K)$ is the\ncomplex of harmonic forms on $X_u$, Hodge theory tells us that the cohomology of the\ncomplex $\\Omega ^{G_u}(G_u\/K)$ is just the cohomology of $X_u$.\nThe corresponding analysis holds for $\\Omega ^{G^\\prime}(G^\\prime\/K^\\prime)$.\n\nNext we note that the cohomology of the last two complexes can be\nidentified with the cohomology of $X$, $Y$ respectively.  This just\ncomes from the fact that the projection $G\/K\\rightarrow \\Gamma \\backslash\nG\/K=X$ induces the isomorphism of complexes $\\Omega ^\\Gamma(G\/K)=\\Omega(X)$,\nand similarly for $Y$.\n\nNow observe that the four complexes fit into a commutative diagram\nof chain complexes:\n$$\\disablesubscriptcorrection\\xysavmatrix{\\Omega ^\\Lambda(G^\\prime\/K^\\prime) & \\Omega ^\\Gamma(G\/K)\\ar[l]_{~\\phi}\\\\\n\\Omega ^{G^\\prime}(G^\\prime\/K^\\prime) \\ar[u]^{j_Y} & \\Omega ^{G}(G\/K)\\ar[l]_{~\\psi}\n\\ar[u]_{j_X}}$$ Let us briefly comment on the maps in the diagram.\nThe vertical maps are obtained from the fact that $\\Gamma\\leq G$, so\nthat any $G$--invariant form can be viewed as a $\\Gamma$--invariant form,\nand similarly for $\\Lambda \\leq G^\\prime$.\n\nFor the horizontal maps, one observes that $G^\\prime\/K^\\prime\n\\hookrightarrow G\/K$ is an embedding, hence any form on $G\/K$\nrestricts to a form on $G^\\prime\/K^\\prime$.  We also have the inclusion\n$\\Lambda\\leq \\Gamma$, and hence the restriction of a $\\Gamma$--invariant form on\n$G\/K$ yields a $\\Lambda$--invariant form on $G^\\prime\/K^\\prime$.  This is the\nhorizontal map in the top row.  One obtains the horizontal map in\nthe bottom row similarly.\n\nNow passing to the homology of the chain complexes, and using the\nidentifications discussed above, we obtain a commutative diagram in\ndimension $m=\\dim(Y)=\\dim(G_u^\\prime\/K^\\prime)$:\n$$\\disablesubscriptcorrection\\xysavmatrix{\\mathbb R \\simeq H^m(Y; \\mathbb R) & & H^m(X;\\mathbb R) \\ar[ll]_{\\phi^*}\\\\\n\\mathbb R \\simeq H^m(G_u^\\prime\/K^\\prime;\\mathbb R) \\ar[u]^{j_Y^*} & & H^m(G_u\/K;\\mathbb R)\n\\ar[ll]_{\\psi^*} \\ar[u]_{j_X^*}}$$ Note that the two vertical maps defined\nhere are precisely the Matsushima maps for the respective locally\nsymmetric spaces.  Since Matsushima's map is always injective, and\nthe cohomology of $H^m(G_u^\\prime\/K^\\prime;\\mathbb R)$ and $H^m(Y; \\mathbb R)$ are both\none-dimensional, we obtain that $j_Y^*$ is an isomorphism.  Likewise\n$j_X^*$ is always injective, and if $m\\leq m(\\mathfrak g)$ then $j_X^*$ is also\nsurjective (and hence $j_X^*$ is an isomorphism as well).  This implies the \nfollowing two facts:\n\\begin{itemize}\n\\item If $\\phi^*$ is identically zero, then $\\psi^*$ is identically zero.\n\\item If furthermore $m\\leq m(\\mathfrak g)$, then both vertical maps are isomorphisms,\nand we have that $\\psi^*$ is identically zero if and only if $\\phi^*$ is identically zero.\n\\end{itemize}\nNow observe that\nboth of the horizontal maps coincide with the maps induced on\ncohomology by the respective inclusions $Y\\hookrightarrow X$ and\n$G_u^\\prime\/K^\\prime \\hookrightarrow G_u\/K$; indeed the maps are obtained by\nrestricting the forms defined on the ambient manifold to the\nappropriate submanifold.  In particular, the map $\\psi^*$ coincides with\nthe map $\\rho$ that appears in the statement of our theorem.  On the other hand, from \nthe Kronecker pairing, the map $\\phi^*$ is nonzero precisely when\n$[Y^m]\\neq 0\\in H_m(X;\\mathbb R)$.  Combining these observations with the\ntwo facts in the previous paragraph completes the proof of \\fullref{Theorem1.1}.\n\\end{proof}\n\n\\begin{Rmk}\n(1)\\qua The Matsushima map is only defined on the real cohomology\n(since it passes through differential forms), and as a result, {\\it\ncannot} be used to obtain any information on torsion elements in\n$H^k(X^n;\\mathbb Z)$.\n\n(2)\\qua We remark that the proof of \\fullref{Theorem1.1} applies equally well to \nlower-dimen\\-sional cohomology (using the fact that Matsushima's map is injective\nin all dimensions), and gives the following lower-dimensional criterion.  Assume\nthat\nthe map $H^k(X^n_u; \\mathbb R) \\rightarrow H^k(Y^m_u;\\mathbb R)$ has image containing a\nnonzero class $\\alpha$, and let $i(\\alpha)\\in H^k(Y^m; \\mathbb R)$ be the nonzero image class\nunder the Matsushima map.  Then the homology class $\\beta\\in H_k(Y^m;\\mathbb R)$ dual\n(under the Kronecker pairing) to $i(\\alpha)$ has nonzero image in $H_k(X^n;\\mathbb R)$ under\nthe map induced by the inclusion $Y^m\\hookrightarrow X^n$.\n\\end{Rmk}\n\n\\section{Pairs of tangential maps}\nIn this section, we proceed to give a proof of \\fullref{Theorem1.2},\nestablishing the existence of pairs of tangential maps from the pair\n$(\\bar X,\\bar Y)$ to the pair $(X_u,Y_u)$.\n\n\\begin{proof}[Proof of \\fullref{Theorem1.2}]\nWe start out by applying\n\\fullref{Theorem2.2}, which gives us a finite cover $\\bar X$ of $X$ with the\nproperty that the natural composite map $\\bar X\\rightarrow\nBK\\rightarrow BG_u$ is homotopic to a point. Note that this map\nclassifies the principle $G_u$ bundle over $\\bar X$.\n\nNow let $\\bar Y\\hookrightarrow \\bar X$ be a connected lift of the\ntotally geodesic subspace $Y\\hookrightarrow X$.  Observe that, by\nnaturality, we have a commutative diagram:\n$$\\disablesubscriptcorrection\\xysavmatrix@R=10pt{ & G_u^\\prime\/K^\\prime \\ar[rr] \\ar[dd] & & G_u\/K \\ar[dd]\\\\\n\\bar Y \\ar[rr] \\ar[dr] \\ar[ddr] & & \\bar X \\ar[dr] \\ar[ddr] & \\\\\n& BK^\\prime \\ar[rr] \\ar[d] & & BK \\ar[d] \\\\\nG_u\/G_u^\\prime \\ar[r] & BG_u^\\prime \\ar[rr] & & BG_u}$$ \nBy Okun's result,\nthe composite map $\\bar X\\rightarrow BG_u$ is homotopic to a point\nvia a homotopy $H\\co\\bar X \\times I\\rightarrow BG_u$.  We would like\nto establish the existence of a homotopy $F\\co\\bar Y\\times I\n\\rightarrow BG_u^\\prime$ with the property that the following diagram\ncommutes:\n$$\\disablesubscriptcorrection\\xysavmatrix{\\bar Y\\times I \\ar[r]^{i\\times \\mathrm{Id}} \\ar[d]_F & \\bar X\\times I \\ar[d]^H\\\\\nBG_u^\\prime \\ar[r] & BG_u}$$\nIndeed, if we had the existence of such a compatible pair of\nhomotopies, then one can easily complete the argument: since each of\nthe vertical columns in the diagram are fiber bundles, we see that\nafter applying the pair of compatible homotopies, the images of\n$(\\bar X,\\bar Y)$ lies in the pair of fibers $(G_u\/K, G_u^\\prime\/K^\\prime)$.\nThis yields a pair of compatible lifts, yielding a commutative\ndiagram of the form:\n$$\\disablesubscriptcorrection\\xysavmatrix{ & G_u^\\prime\/K^\\prime \\ar[rr] \\ar[dd] & & G_u\/K \\ar[dd]\\\\\n\\bar Y \\ar[ru] \\ar[rr] \\ar[dr]  & & \\bar X \\ar[dr] \\ar[ru] & \\\\\n & BK^\\prime \\ar[rr] & & BK } $$\nSince the pair of maps to $BK^\\prime$ (respectively $BK$) classify the\ncanonical $K^\\prime$--bundle structures on $\\bar Y$, $G_u^\\prime\/K^\\prime$\n(respectively the canonical $K$--bundle structure on $\\bar X$,\n$G_u\/K$), and since these bundles canonically determine the tangent\nbundles of these spaces \\cite[Lemma 2.3]{O1}, commutativity of the\ndiagram immediately\ngives us tangentiality of the maps $\\bar Y \\rightarrow G_u^\\prime\/K^\\prime$\n(respectively, of the map $\\bar X\\rightarrow G_u\/K$).\n\n\nIn order to show the existence of the compatible homotopy $F\\co\\bar\nY\\times I \\rightarrow BG_u^\\prime$, we start by observing that the\nbottom row of the commutative diagram is in fact a fibration\n$$ G_u\/G_u^\\prime \\rightarrow BG_u^\\prime \\rightarrow BG_u.$$\nSince $\\bar Y$ is embedded in $\\bar X$, we see that the homotopy $H$\ninduces by restriction a homotopy $H\\co\\bar Y \\times I \\rightarrow\nBG_u$.  Since the bottom row is a fibration, we may lift this\nhomotopy to a homotopy $\\tilde H\\co \\bar Y\\times I\\rightarrow\nBG_u^\\prime$, with the property that $\\tilde H_0$ coincides with the map\n$\\bar Y \\rightarrow BG_u^\\prime$ which classifies the canonical\nprinciple $G_u^\\prime$ bundle over $\\bar Y$.  Unfortunately, we do not\nknow, a priori, that the time one map $\\tilde H_1$ maps $\\bar Y$ to\na point in $BG_u^\\prime$.  Indeed, we merely know that $\\tilde H_1(\\bar\nY)$ lies in the preimage of a point in $BG_u$, ie in the fiber\n$G_u\/G_u^\\prime$.  Our next goal is to establish that the map $\\tilde\nH_1\\co \\bar Y \\rightarrow G_u\/G_u^\\prime$ is nullhomotopic.  If this were\nthe case, we could concatenate the homotopy $\\tilde H$ taking $\\bar\nY$ into the fiber $G_u\/G_u^\\prime$ with a homotopy contracting $\\tilde\nH_1\\co\\bar Y \\rightarrow G_u\/G_u^\\prime$ to a point {\\it within the\nfiber}.  This would yield the desired homotopy $F$.\n\\medskip\n\nIn order to establish that $\\tilde H_1\\co \\bar Y \\rightarrow\nG_u\/G_u^\\prime$ is nullhomotopic, we merely note that we have the\nfibration\n$$G_u^\\prime\\rightarrow G_u\\rightarrow G_u\/G_u^\\prime.$$\nFrom the corresponding long exact sequence in homotopy groups, and\nusing the fact that the inclusion $G_u^\\prime\\hookrightarrow G_u$\ninduces a $\\pi_i$--isomorphism for $i< m$ and a surjection on\n$\\pi_m$, we immediately obtain that $\\pi_i(G_u\/G_u^\\prime)\\cong 0$ for\n$i\\leq m$. Since the dimension of the manifold $\\bar Y$ is $m$, we\ncan now conclude that the map $\\tilde H_1$ is nullhomotopic.\nIndeed, taking a cellular decomposition of $\\bar Y$ with a single\n$0$--cell, one can recursively contract the image of the $i$--skeleton\nto the image of the $0$--cell: the obstruction to doing so lies in\n$\\pi_i(G_u\/G_u^\\prime)$, which we know vanishes.  This yields that\n$\\tilde H_1$ is nullhomotopic, which by our earlier discussion,\nimplies the existence of a tangential map of pairs $(\\bar X,\\bar Y)\n\\rightarrow (X_u,Y_u)$. Finally, to conclude we merely point out the\nOkun has shown (see \\fullref{Theorem2.4}) that in the case where the rank of\n$G_u$ equals the rank of $K$, the tangential map he constructed\ninduces the Matsushima map on cohomology.  Our construction\nrestricts to Okun's construction on both $X$ and $Y$, and from the\nhypothesis on the ranks, so we conclude that the tangential map of\npairs induces the Matsushima map on the cohomology of each of the\ntwo spaces. This concludes the proof of \\fullref{Theorem1.2}.\n\\end{proof}\n\\medskip\n\n\\begin{Rmk}\nWe observe that the argument given above, for the case of a pair $(X^n, Y^m)$,\ncan readily be adapted to deal with any descending chain of totally geodesic\nsubmanifolds.  More precisely, assume that we have a series of totally\ngeodesic embeddings $X^n=Y_k \\supset \\cdots \\supset Y_2\\supset Y_1$, with\nthe property that each $Y_j$ is a closed locally symmetric space of noncompact type.\nFurther assume that, if $(Y_j)_u = (G_j)_u\/K_j$ denotes the compact duals,\nthe maps $(G_j)_u \\hookrightarrow (G_{j+1})_u$\ninduced by the inclusions $Y_j\\hookrightarrow Y_{j+1}$ are $\\pi_i$ isomorphisms\nfor $i< \\dim(Y_j)$ and a surjection on $\\pi_i$ ($i=\\dim(Y_j)$).  Then there exists\na finite cover $\\bar X^n=\\bar Y_k$ of $X^n$, and connected lifts $\\bar Y_j$ of $Y_j$,\nhaving the property that:\n\\begin{itemize}\n\\item we have containments $\\bar Y_j\\subset \\bar Y_{j+1}$, and\n\\item there exists a map $(\\bar Y_k, \\ldots, \\bar Y_1)\\rightarrow \\big((Y_k)_u,\n\\ldots ,(Y_1)_u\\big)$ which restricts to a tangential map from each $\\bar Y_j$ to the\ncorresponding $(Y_j)_u$.\n\\end{itemize}\nThis is shown by induction on the length of a descending chain.  We leave the details\nto the interested reader.\n\\end{Rmk}\n\nWe now proceed to show \\fullref{Corollary1.4}, that is to say, that in the\ncase where $X^n$ is real hyperbolic, and $Y^m\\hookrightarrow X^n$ is\ntotally geodesic, there exists a finite cover $\\bar X$ of $X^n$ and\na connected lift $\\bar Y$ of $Y^m$, with the property that the\nnormal bundle of $\\bar Y$ in $\\bar X$ is trivial.\n\n\\begin{proof}[Proof of \\fullref{Corollary1.4}] We first observe that, provided one could verify the\nhypotheses of \\fullref{Theorem1.2} for the pair $(X^n,Y^m)$, the corollary\nwould immediately follow.  Note that in this case, the dual spaces\n$X_u$ and $Y_u$ are spheres of dimension $n$ and $m$ respectively.\nThis implies that the totally geodesic embedding $Y_u\\hookrightarrow\nX_u$ is in fact a totally geodesic embedding $S^m\\hookrightarrow\nS^n$, forcing the normal bundle to $Y_u$ in $X_u$ to be trivial. But\nnow \\fullref{Corollary1.3} to the \\fullref{Theorem1.2} immediately yields \\fullref{Corollary1.4}.\n\nSo we are left with establishing the hypotheses of \\fullref{Theorem1.2} for\nthe pair $(X^n,Y^m)$.  We observe that in this situation we have\nthe groups $G_u \\cong SO(n+1)$, and $G_u^\\prime \\cong SO(m+1)$.\nFurthermore, there is essentially a unique totally geodesic\nembedding $S^m\\hookrightarrow S^n$, hence we may assume that the\nembedding $G_u^\\prime\\hookrightarrow G_u$ is the canonical one.  But now\nwe have the classical facts that (1) the embeddings\n$SO(m+1)\\hookrightarrow SO(n+1)$ induce isomorphisms on $\\pi_i$ for\n$i< m$ and (2) that the embedding induces a surjection\n$\\pi_m(SO(m+1))\\rightarrow \\pi_m(SO(n+1))$.  Indeed, this is\nprecisely the range of dimensions where the homotopy groups\nstabilize \\cite{Mil2}.  This completes the verification of the\nhypotheses, and hence the proof of \\fullref{Corollary1.4}.\n\\end{proof}\n\nWe now proceed to give an example of an inclusion\n$Y^m\\hookrightarrow X^n$ satisfying the hypotheses of our theorem.\nOur spaces will be modelled on complex hyperbolic spaces, namely we\nhave:\n\\begin{align*}Y^{2m}&= \\Lambda\\backslash \\mathbb C\\mathbb H ^m = \\Lambda\\backslash \\mathit{SU}(m,1) \/ S(U(m)\\times U(1))\\\\\nX^{2n}& = \\Gamma\\backslash \\mathbb C\\mathbb H ^n = \\Gamma\\backslash \\mathit{SU}(n,1) \/ S(U(n)\\times U(1))\\end{align*}\nTo construct such pairs, one starts with the standard inclusion of $\\mathit{SU}(m,1)\\hookrightarrow \\mathit{SU}(n,1)$,\nwhich induces a totally geodesic embedding $\\mathbb C\\mathbb H ^m\\hookrightarrow \\mathbb C\\mathbb H ^n$.  One can\nnow construct explicitly (by arguments similar to those in Gromov and Piatetski-Shapiro \\cite{GP}) an arithmetic uniform lattice\n$\\Lambda \\leq \\mathit{SU}(m,1)$ having an extension\nto an arithmetic uniform lattice $\\Gamma \\leq \\mathit{SU}(n,1)$.  Quotienting out by these lattices gives the\ndesired pair.\n\nLet us now consider these examples in view of our \\fullref{Theorem1.2}.  First of all, we have that the\nrespective complexifications are $G^\\prime_\\mathbb C=\\mathit{SL}(m+1, \\mathbb C)$ and $G_\\mathbb C=\\mathit{SL}(n+1,\\mathbb C)$, with\nthe natural embedding\n$$G^\\prime_\\mathbb C =\\mathit{SL}(m+1,\\mathbb C) \\hookrightarrow \\mathit{SL}(n+1,\\mathbb C)=G_\\mathbb C.$$\nLooking at the respective maximal compacts, we see that $G^\\prime_u=\\mathit{SU}(m+1)$, $G_u=\\mathit{SU}(n+1)$, and the inclusion is\nagain the natural embeddings\n$$G^\\prime_u = \\mathit{SU}(m+1) \\hookrightarrow \\mathit{SU}(n+1) = G_u.$$\nHence the homotopy condition in our theorem boils down to asking\nwhether the natural embedding $\\mathit{SU}(m+1)\\hookrightarrow \\mathit{SU}(n+1)$\ninduces isomorphisms on the homotopy groups $\\pi_i$, where $i\\leq\n\\dim(Y^{2m})=2m$.  But it is a classical fact that the natural\nembedding induces isomorphisms in all dimensions $i<2(m+1)=2m+2$,\nsince this falls within the stable range for the homotopy groups\n(and indeed, one could use complex Bott periodicity to compute the\nexact value of these homotopy groups \\cite{Mil2}).  Finally, we observe\nthat in this context, the dual spaces are complex projective spaces, and the\nembedding of dual spaces is the standard embedding $\\mathbb C`P^m\\hookrightarrow \\mathbb C`P^n$.\nIt is well known that for the standard embedding, we have that the induced map on\ncohomology $H^*(\\mathbb C`P^n) \\rightarrow H^*(\\mathbb C`P^m)$ is surjective on cohomology.\nOur \\fullref{Theorem1.1} now tells us that $Y^{2m}\\hookrightarrow X^{2n}$ is homologically\nnontrivial.  Furthermore, we note that for these manifolds, $\\rk(G_u)=\\rk(K)$ and\n$\\rk(G_u^\\prime)=\\rk(K^\\prime)$, and hence \\fullref{Theorem1.2} tells us that\nthe cohomological map from the proof of \\fullref{Theorem1.1}\ncan be (rationally) realized via a tangential map of pairs.\n\n\\section{Concluding remarks}\n\nWe conclude this paper with a few comments and questions.  First of all, in\nview of our \\fullref{Theorem1.1}, it is reasonable to ask for the converse:\n\n\\medskip\n{\\bf Question}\\qua  Given an element $\\alpha \\in H_m(X^n\n;\\mathbb R)$, is there an $m$--dimensional totally geodesic submanifold\n$Y^m$ with $[Y^m]=\\alpha$?\n\\medskip\n\nA cautionary example for the previous question is provided by the following:\n\n\\begin{Prop}\nLet $X$ be a compact hyperbolic $3$--manifold that fibers over $S^1$, with\nfiber a surface $F$ of genus $\\geq 2$.  Then the homology class\nrepresented by $[F]\\in H_2(X; \\mathbb Z)$ cannot be represented by a\ntotally geodesic submanifold.\n\\end{Prop}\n\n\\begin{proof}\nAssume that there were such a totally geodesic submanifold $Y\n\\subset X$, and observe that since $Y$ is totally geodesic, we have\nan embedding $\\pi_1(Y)\\hookrightarrow \\pi_1(X)$.  Furthermore, since\n$X$ fibers over $S^1$ with fiber $F$, we also have a short exact\nsequence:   \n$$ 0\\rightarrow \\pi_1(F)\\rightarrow \\pi_1(X)\\rightarrow \\mathbb Z \\rightarrow 0 $$\nOur goal is to show that $\\pi_1(Y)\\subset \\pi_1(F)$.  Indeed, if we\ncould establish this containment, one could then argue as follows:\nsince $Y$ is a compact surface, covering space theory implies that\n$\\pi_1(Y)\\subset \\pi_1(F)$ is a finite index subgroup.  Now pick a\npoint $x$ in the universal cover $\\tilde X\\cong \\mathbb H^3$, and consider\nthe subset $\\Lambda_Y \\subset \\partial ^\\infty \\mathbb H^3=S^2$ obtained\nby taking the closure of the $\\pi_1(Y)$--orbit of $x$.  Since $Y$ is\nassumed to be totally geodesic, the subset $\\Lambda\\subset S^2$ is a\ntamely embedded $S^1$ (identified with the boundary at infinity of a\nsuitably chosen totally geodesic lift $\\tilde Y \\cong \\mathbb H^2$ of\n$Y$). On the other hand, since $\\pi_1(Y)$ has finite index in\n$\\pi_1(F)$, we have that $\\Lambda_Y$ must coincide with $\\Lambda_F$,\nthe closure of the $\\pi_1(F)$--orbit of $x$.  But the latter, by a\nwell-known result of Cannon--Thurston is known to be the entire\nboundary at infinity (see for instance Mitra \\cite{Mit}).\n\nSo we are left with establishing that $\\pi_1(Y)\\subset \\pi_1(F)$. In\norder to see this, let us consider the cohomology class $\\alpha_F\\in\nH^1(X; \\mathbb Z)$ which is Poincar\\'e dual to the class $[F]\\in H_2(X;\n\\mathbb Z)$. Now recall that the evaluation of the cohomology class\n$\\alpha_F$ on an element $[\\gamma]\\in H_1(X;\\mathbb Z)$ can be interpreted\ngeometrically as the intersection number of the representing curve\n$\\gamma$ with the surface $F$.  Furthermore, we have that the group\n$H_1(X;\\mathbb Z)$ is generated by the image of $H_1(F;\\mathbb Z)$, under the\ninclusion $F\\hookrightarrow X$, along with an element $[\\eta]\\in\nH_1(X; \\mathbb Z)$ mapping to $[S^1]\\in H_1(S^1;\\mathbb Z)$.  Here $\\eta$ is\nchosen to be a closed loop in $M$ with the property that $\\eta$ maps\nhomeomorphically to the base $S^1$ (preserving orientations) under\nthe projection map.  This gives us the following two facts:\n\\begin{itemize}\n\\item The class $\\alpha_F$ evaluates to $1$ on the element $[\\eta]$, since\n$F\\cap \\eta$ is a single transverse point.\n\\item The class $\\alpha_F$ evaluates to zero on the image of $H_1(F;\\mathbb Z)$ in\nthe group $H_1(X;\\mathbb Z)$.  This follows from the fact that\nthe surface $F$ has trivial normal bundle in $X$, allowing any curve\nin $F$ representing (the image of) an element in $H_1(F; \\mathbb Z)$ to be homotoped to a\ncurve disjoint from $F$.\n\\end{itemize}\nFurthermore, since we are assuming that $[Y]=[F]\\in H_2(X;\\mathbb Z)$, we know that\nwe have an identification of Poincar\\'e duals $\\alpha_Y=\\alpha_F$.\n\nNow let us assume that $\\pi_1(Y)$ is {\\it not\\\/} contained in $\\pi_1(F)$, and\nobserve that this implies that there exists a closed loop $\\gamma\\subset Y$ having\nthe property that under the composition $Y\\hookrightarrow X\\rightarrow S^1$,\nthe class $[\\gamma]\\in H_1(Y;\\mathbb Z)$ maps to $k\\cdot [S^1]\\in H_1(S^1; \\mathbb Z)$ (and\n$k\\neq 0$).  We now proceed\nto compute, in two different ways, the evaluation of the cohomology classes $\\alpha_Y\n=\\alpha _F$ on a suitable multiple of the homology class $[\\gamma]$.\n\nFirstly, from the comments above, we can write $[\\gamma]$ as the sum of\n$k\\cdot [\\eta]$, along with an element $\\beta$, where $\\beta$ lies in the image\nof  $H_1(F; \\mathbb Z)$.  By linearity of\nthe Kronecker pairing, along with the two facts from the previous paragraph, we\nobtain:\n$$\\alpha _F([\\gamma])=\\alpha_F(\\beta) + k\\alpha_F([\\eta]) = k \\neq 0$$\nSecondly, observe that $Y$ is assumed to be embedded in $X$, and\nrepresents the nonzero homology class $[Y]=[F]\\in H_2(X;Z)$. This\nimplies that $Y$ must be orientable, and hence has trivial normal\nbundle in $X$.  In particular, the curve $\\gamma \\subset Y$ can be\nhomotoped (within $X$) to have image disjoint from $Y$.  Since the\ninteger $\\alpha _Y([\\gamma])$ can be computed geometrically as the\nintersection number of the curve $\\gamma$ with $Y$, we conclude that\n$\\alpha_Y([\\gamma])=0$.\n\nCombining the two observations above, we see that\n$0=\\alpha_Y([\\gamma])= \\alpha_F([\\gamma])\\neq 0$, giving us the\ndesired contradiction. This completes the proof of the proposition.\n\\end{proof}\n\nWe observe that Thom \\cite{T} has shown that in dimensions $0\\leq\nk\\leq 6$ and $n-2\\leq k\\leq n$, every integral homology class can be\nrepresented by an immersed submanifold.  In general however, there\ncan exist homology classes which are {\\it not\\\/} representable by\nsubmanifolds (see for instance Bohr, Hanke and\nKotschick \\cite{BHK}).  The question above asks for a more stringent\ncondition, namely that the immersed submanifold in question be\ntotally geodesic.  We believe that the weaker question is also of\nsome interest, namely:\n\n\\medskip\n{\\bf Question}\\qua  Find an example $X^n$ of a compact locally\nsymmetric space of noncompact type, and a homology class in some\n$H_k(X^n; \\mathbb Z)$ which {\\it cannot\\\/} be represented by an immersed\nsubmanifold.\n\\medskip\n\nNow our original motivation for looking at totally geodesic submanifolds\ninside locally symmetric spaces was the desire to exhibit lower-dimensional\nbounded cohomology classes.  In \\cite{LaS}, the authors showed that for\nthe fundamental group $\\Gamma$ of\na compact locally symmetric space of noncompact type $X^n$, the comparison\nmap from bounded cohomology to ordinary cohomology:\n$$H^*_b(\\Gamma)\\rightarrow H^*(\\Gamma)$$\nis {\\it surjective\\\/} in dimension $n$.  The proof actually passed\nthrough the dual formulation, and showed that the $L^1$\n(pseudo)-norm of the fundamental class $[X^n]\\in H_n(X^n; \\mathbb R)$ is\nnonzero.  Now given a totally geodesic embedding $Y\\hookrightarrow\nX$ of the type considered in this paper, it is tempting to guess\nthat the homology class $[Y]$ also has nonzero $L^1$ (pseudo)-norm.\nOf course, this naive guess fails, since one can find examples where\n$[Y]=0\\in H_m(X^n;\\mathbb R)$.  The problem is that despite the fact that\nthe {\\it intrinsic\\\/} $L^1$ (pseudo)-norm of $[Y]$ is nonzero, the\n{\\it extrinsic\\\/} $L^1$ (pseudo)-norm of $[Y]$ is zero.  In other\nwords, one can represent the fundamental class of $Y$ {\\it more\nefficiently} by using simplices that actually {\\it do not lie in\n$Y$} (despite the fact that $Y$ is totally geodesic).  The authors were unable to answer the following:\n\n\\medskip\n{\\bf Question}\\qua Assume that $Y$ and $X$ are compact locally\nsymmetric spaces of noncompact type, that $Y\\subset X$ is a totally\ngeodesic embedding, and that $Y$ is orientable with  $[Y]\\neq 0 \\in\nH_m(X^n; \\mathbb R)$.  Does it follow that the dual cohomology class\n$\\beta\\in H^m(X^n; \\mathbb R)$ (via the Kronecker pairing) has a bounded\nrepresentative?\n\\medskip\n\nNow one situation in which nonvanishing of the $L^1$ (pseudo)-norm would\nbe preserved is the case where $Y\\hookrightarrow X$ is actually a retract of $X$.\nHence one can ask the following:\n\n\\medskip\n{\\bf Question}\\qua If $Y\\subset X$ is a compact totally\ngeodesic proper submanifold inside a locally symmetric space of\nnoncompact type, when is $Y$ a retract of $X$?\n\n\\begin{Rmk}\n(1)\\qua  In the case where $X$ is a (nonproduct) higher rank locally\nsymmetric space of noncompact type, one cannot find a proper\ntotally geodesic submanifold $Y\\subset X$ which is a retract of $X$.\nIndeed, if there were such a submanifold, then the morphism\n$\\rho\\co\\pi_1(X)\\rightarrow \\pi_1(Y)$ induced by the retraction would have\nto be surjective.  By Margulis' normal subgroup theorem \\cite{Mar},\nthis implies that either (1) $\\ker(\\rho)$ is finite, or (2)\nthe image $\\pi_1(Y)$ is finite. Since $Y$ is locally symmetric\nof noncompact type, $\\pi_1(Y)$ cannot be finite, and hence we must\nhave finite $\\ker(\\rho)$. But $\\ker(\\rho)$ is a subgroup of the torsion-free\ngroup $\\pi_1(X)$, hence must be trivial.  This forces $\\pi_1(X)\\cong \\pi_1(Y)$,\nwhich contradicts the fact that the cohomological dimension of $\\pi_1(X)$\nis $\\dim (X)$, while the cohomological dimension of $\\pi_1(Y)$ is $\\dim (Y)<\n\\dim (X)$.  This implies  that no such morphism exists, and hence no such\nsubmanifold exists.  The authors thank C\\,Leininger for pointing out this simple\nargument.\n\n(2)\\qua In the case where $X$ has rank one, the question is more delicate.  Some\nexamples of such retracts can be found in a paper by Long and Reid \\cite{LoR}.\nWe remark that in this case, the application to bounded cohomology is not too\ninteresting, as Mineyev \\cite{Min} has already shown that the comparison map\nin this situation is surjective in all dimensions $\\geq 2$.\n\\end{Rmk}\n\nFinally, we conclude this paper by pointing out that the Okun maps, \nwhile easy to define, are geometrically\nvery complicated.  We illustrate this with a brief comment on the \n{\\it singularities\\\/} of the tangential maps\nbetween locally symmetric spaces of noncompact type and their\nnonnegatively curved dual spaces.  More precisely, for a smooth map\n$f\\co X\\rightarrow X_u$, consider the subset $\\Sing(f)\\subset\nX_u$ consisting of points $p\\in X_u$ having the property that there\nexists a point $q\\in X$ satisfying $f(q)=p$, and $\\ker(df(q))\\neq 0$\n(where $df\\co T_qX\\rightarrow T_pX_u$ is the differential of $f$ at the\npoint $q$).  We can now ask how complicated the set $\\Sing(h)\\subset X_u$ \ngets for $h$ a smooth map within the homotopy class of the Okun maps.\n\n\\begin{Prop}\\label{Proposition5.2}\nLet $X$ be a closed, locally symmetric space of noncompact type, \n$X_u$ the nonnegatively curved dual space, $t\\co\\bar\nX\\rightarrow \\smash{X_u}$ the Okun map from a suitable finite\ncover $\\bar X$ of $X$, and $h$ an arbitrary smooth map in the\nhomotopy class of $t$.  Consider an arbitrary embedded compact\nsubmanifold $N^k\\subset X$, having the property that \\textnormal{(1)} $[N^k]\\neq 0\\in\nH^k(X_u; \\mathbb Z)$, and \\textnormal{(2)} $N^k$ is simply connected.\nThen we have that $\\Sing(h)\\cap N^k\\neq\n\\emptyset$.\n\\end{Prop}\n\n\\begin{proof}\nLet $h$ be a smooth map homotopic to $t$, and assume that\n$\\Sing(h)\\cap N^k=\\emptyset$.  This implies that $dh$ has full\nrank at every preimage point of $N^k\\subset X_u$.  Choose a\nconnected component $S\\subset \\bar X$ of the set $h^{-1}(N^k)$,\nand observe that the restriction of $h$ to $S$ provides a local\ndiffeomorphism (and hence a covering map) to $N^k$.  \nSince $N^k$ is simply connected, $S$ is diffeomorphic to\n$N^k$, and $h$ restricts to a diffeomorphism from $S\\subset \\bar X$\nto $N^k\\subset X_u$.\n\nNext we observe that the homology class represented by $[S]\\in\nH_k(\\bar X;\\mathbb Z)$ is nonzero, since this class has image, under $h$,\nthe homology class $[N_k]\\neq 0\\in H_k(X_u; \\mathbb Z)$.  But \nobserve that $S\\subset \\bar\nX$, being simply connected, supports a cellular decomposition with a \nsingle $0$--cell and {\\it no $1$--cells\\\/}.  In particular,\nthe submanifold $S$ is homotopic to a point in the aspherical space $\\bar X$, \nsince one can recursively contract all the cells of dimension $\\geq 2$ down to \nthe $0$--cell.  This forces $[S]=0\\in H_k(\\bar X;\\mathbb Z)$, giving us the desired \ncontradiction.\n\\end{proof}\n\nWe point out a simple example illustrating this last proposition.  If $X^{2n}$ is \ncomplex hyperbolic, then $X_u=\\mathbb C`P^n$, and one can take for $N^2=\\mathbb C`P^1$\nthe canonically embedded complex projective line.  Topologically, $N^2$ is \ndiffeomorphic to $S^2$, hence is simply connected, while homologically we\nhave that $[N^2]$ is the generator for the cohomology group $H^2(\\mathbb C`P^n; \n\\mathbb Z) \\cong \\mathbb Z$.  \\fullref{Proposition5.2} now applies, and gives us that\n{\\it any\\\/} smooth map $h$ in the homotopy class of the Okun map must have \nsingular set intersecting the canonical $\\mathbb C`P^1$.  A similar example appears\nin the case of quaternionic hyperbolic manifolds, with the singular sets being\nforced to intersect the canonical $\\mathbb O`P^1$ (diffeomorphic to $S^4$) \ninside the dual space $\\mathbb O`P^n$.\n\n\\bibliographystyle{gtart}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\nWe thank the action editor and anonymous reviewers for their\nconstructive feedback. The authors also thank Nikita Moghe, Seraphina\nGoldfarb-Tarrant, Ondrej Bohdal and Heather Lent for their insightful\ncomments on earlier versions of this paper. We gratefully acknowledge\nthe support of the UK Engineering and Physical Sciences Research\nCouncil (grants EP\/L016427\/1 (Sherborne) and EP\/W002876\/1 (Lapata))\nand the European Research Council (award 681760, Lapata).\n\n\n\\section{Methodology}\n\\label{sec:methodology}\n\nWe combine two meta-learning techniques for\ncross-lingual semantic parsing. The first is the {Reptile} algorithm\noutlined in Section~\\ref{sec:rw}. {Reptile} optimizes for dense likelihood\nregions within the parameters (i.e., an \\emph{optimal manifold}) through\npromoting inter-batch generalization \\citep{DBLP:journals\/corr\/abs-1803-02999-REPTILE}.\nStandard Reptile iteratively optimizes the manifold for an improved\ninitialization across objectives. Rapid fine-tuning yields the final task-specific model. The second technique is the first-order approximation of DG-MAML\n\\citep{li-hospedales-metadg,wang-etal-2021-meta-DGMAML}. This\nsingle-stage process optimizes for domain generalization by simulating\n``source'' and ``target'' batches from different domains to\nexplicitly optimize for \\emph{cross-batch} generalization. Our\nalgorithm, \\mbox{\\sc XG-Reptile}, combines these paradigms to optimize\na target loss with the overall learning ``direction'' derived as\nthe \\textit{optimal manifold} learned via {Reptile}. This trains \nan accurate parser demonstrating sample-efficient cross-lingual transfer\nwithin an efficient \\emph{single-stage} learning process.\n\n\n\\subsection{The \\mbox{\\sc XG-Reptile} Algorithm}\n\nEach learning episode of \\mbox{\\sc XG-Reptile} comprises\ntwo component steps: \\mbox{\\emph{intra-task}} learning and\n\\emph{inter-language} generalization to jointly learn parsing and\ncross-lingual transfer. Alternating these processes trains a\ncompetitive parser from multiple languages with low computational\noverhead beyond existing gradient-descent training. Our approach\ncombines the typical two stages of meta-learning to produce a single\nmodel without a fine-tuning requirement.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{imgs\/xgr6.pdf}\n    \\vspace{-2.5em}\n    \\caption{\\small One iteration of \\mbox{\\sc XG-Reptile}. (1) Run\n      $K$ iterations of gradient descent over $K$ support batches to\n      learn $\\phi_{K}$, (2) compute $\\nabla_{macro}$, the difference\n      between $\\phi_{K}$ and $\\phi_{1}$, (3) find the loss on the\n      target batch using $\\phi_{K}$ and (4) compute the final gradient\n      update from $\\nabla_{macro}$ and the target loss.}\n    \\label{fig:xgr_diagram}\n    \\vspace{-1em}\n\\end{figure}\n\n\\paragraph{Task Learning Step}\n\nWe first sample from the high-resource language (i.e.,\n$\\mathcal{S}_{\\rm EN}$) $K$ ``support'' batches of examples,\n$\\mathcal{B}^{S}=\\lbrace\\left(Q_{\\rm\n    EN},~P,~\\mathcal{D}\\right)\\rbrace$. For each of $K$ batches: we\ncompute predictions, compute losses, calculate gradients and adjust\nparameters using some optimizer (see illustration in\nFigure~\\ref{fig:xgr_diagram}). After $K$ successive optimization steps\nthe initial weights in this episode,~$\\phi_{1}$, have been optimized\nto~$\\phi_{K}$. The difference between final and initial weights is\ncalculated as:\n\n\\vspace{-\\baselineskip}\n\\begin{align}\n    \\nabla_{macro} = \\phi_{K} - \\phi_{1} \\label{eq:reptile_step}\n\\end{align}\n\nThis ``macro-gradient'' step is equivalent to a {Reptile} step\n\\citep{DBLP:journals\/corr\/abs-1803-02999-REPTILE}, representing\nlearning an optimal \\textit{manifold} as an approximation\nof overall learning trajectory.\n\n\\paragraph{Cross-Lingual Step}\n\nThe second step samples one ``target'' batch,\n$\\mathcal{B}^{T}=\\left(Q_{l},~P,~\\mathcal{D}\\right)$, from a\nsampled target language (i.e., $\\mathcal{S}_{\n  l}\\subset\\mathcal{S}_{\\rm L}$). We compute the cross-entropy\nloss and gradients from the prediction of the model at $\\phi_{K}$ on\n$\\mathcal{B}^{T}$:\n\n\\vspace{-\\baselineskip}\n\\begin{align}\n    \\mathcal{L}_{T} = \\textrm{Loss}\\left(p_{\\phi_{K}}\\left(Q_{l},~\\mathcal{D}\\right),~P\\right) \\label{eq:cross_lingual_step}\n\\end{align}\n\nWe evaluate the parser at $\\phi_{K}$ on a target\nlanguage we desire to generalize to. We show below that the\ngradient of $\\mathcal{L}_{T}$ comprises the loss at $\\phi_{K}$ and\nadditional terms maximizing the inner product between the high-likelihood\nmanifold and the target loss. The total gradient encourages\nintra-task and cross-lingual learning (see Figure~\\ref{fig:xgr_diagram}).\n\n\n\\begin{algorithm}[!t]\n\\caption{{\\sc XG-Reptile}\\label{xgr-algo}}\n{\\small\n\\begin{algorithmic}[1]\n    \\REQUIRE{Support data, $\\mathcal{S}_{\\rm EN}$, target data, $\\mathcal{S}_{L}$}\n    \\REQUIRE{Inner learning rate, $\\alpha$, outer learning rate, $\\beta$}\n    \\STATE{Initialise $\\theta_{1}$, the vector of initial parameters}\n    \\FOR{$t \\leftarrow 1$~\\textbf{to}~$T$}\n        \\STATE{Copy $\\phi_{1} \\leftarrow \\theta_{t-1}$}\n        \\STATE{Sample $K$ support batches $\\lbrace\\mathcal{B}^{S}\\rbrace_{k=1}^{K}$ from $\\mathcal{S}_{\\rm EN}$}\n        \\STATE{Sample target language $l$ from $L$ languages}\n        \\STATE{Sample target batch~$\\mathcal{B}^{T}$ from $\\mathcal{S}_{l}$}\n        \\FOR{$k \\leftarrow 1$~{\\textbf{to}}~$K$ [Inner Loop]}\n            \\STATE{$\\mathcal{L}^{S}_{k} \\leftarrow {\\rm Forward}\\left(\\mathcal{B}^{S}_{k},~\\phi_{k-1}\\right)$}\n            \\STATE{$\\phi_{k}\\leftarrow{\\rm Adam}\\left(\\phi_{k-1},~\\nabla\\mathcal{L}^{S}_{k},~\\alpha\\right)$\\label{algo:line_grad}} \n        \\ENDFOR\n        \\STATE{Macro grad: ~$\\nabla_{macro}\\leftarrow\\phi_{K}-\\phi_{1}$}\n        \\STATE{Target Step: ~$\\mathcal{L}_{T}\\leftarrow {\\rm Forward}\\left(\\mathcal{B}^{T},~\\phi_{K}\\right)$}\n        \\STATE{Total gradient: $\\nabla_{\\Sigma}=\\nabla_{macro} + \\nabla_{\\phi_K}\\mathcal{L}_{T}$}\n        \\STATE{Update $\\theta_{t} \\leftarrow {\\rm SGD}\\left(\\theta_{t-1},~\\nabla_{\\Sigma},~\\beta\\right)$}\n    \\ENDFOR\n\\end{algorithmic}\n}\n\\end{algorithm}\n\nAlgorithm \\ref{xgr-algo} outlines the \\mbox{\\sc XG-Reptile} process (loss\ncalculation and batch processing are simplified for brevity). We repeat\nthis process over $T$ episodes to train model $p_{\\theta}$ to\nconvergence. If we optimized for target data to align with individual\nsupport batches (i.e., $K=1$) then we may observe batch-level noise in cross-lingual\ngeneralization. Our intuition is that aligning the target gradient with an\napproximation of the task manifold, i.e., $\\nabla_{macro}$, will overcome this\nnoise and align new languages to a more mutually beneficial direction during\ntraining. We observe this intuitive behavior during learning in Section~\\ref{sec:results}.\n\nWe efficiently generalize to low-resource languages by\nexploiting the asymmetric data requirements between steps: one\nbatch of the target language is required for $K$ batches of the source\nlanguage. For example, if $K=10$ then using this $\\frac{1}{K}$\nproportionality requires 10\\% of target-language data relative to\nsupport. We demonstrate in Section~\\ref{sec:results} that we\ncan use a smaller $<\\frac{1}{K}$ quantity per target language to \nincrease sample efficiency. \n\\vspace{-0.25em}\n\n\\paragraph{Gradient Analysis}\n\nFollowing \\citet{DBLP:journals\/corr\/abs-1803-02999-REPTILE}, we\nexpress,~$g_{k}=\\nabla\\mathcal{L}^{S}_{k}$, the gradient in a single\nstep of the inner loop (Line~\\ref{algo:line_grad}) as:\n\n\\vspace{-\\baselineskip}\n\\begin{align}\n\\centering\ng_{k} &= \\bar{g}_i + \\bar{H}_{k}\\left( \\phi_{k}-\\phi_{1}\\right) +\nO\\left(\\alpha^2 \\right) \\label{eq:gi_gradient_taylor_2} \n\\end{align}\nWe use a Taylor series expansion to approximate~$g_k$\nby~$\\bar{g}_{k}$, the gradient at the original point~$\\phi_{1}$, the\nHessian matrix of the gradient at the initial point, $\\bar{H}_{k}$,\nthe step difference between position~$\\phi_{k}$ and the initial\nposition and some scalar terms with marginal influence,\n$O\\left(\\alpha^{2}\\right)$.\n\nBy evaluating Equation~\\eqref{eq:gi_gradient_taylor_2} at $i=1$ and\nrewriting the difference as a sum of gradient steps (e.g.,~Equations\n\\eqref{eq:initial_pt_trick} and \\eqref{eq:diff_trick}), we arrive at\nan expression for $g_{k}$ shown in Equation~\\eqref{eq:gi_gradient_taylor_3} \nexpressing the gradient as an initial component, $\\hat{g}_k$,\nand the product of the Hessian at~$k$, with all prior gradient steps. We\nrefer to \\citet{DBLP:journals\/corr\/abs-1803-02999-REPTILE} for further\nvalidation that the gradient of this product maximizes the\ncross-batch expectation -- therefore promoting intra-task\ngeneralization and learning the optimal manifold. The final gradient (Equation \\eqref{eq:reptile_gradient_taylor})~\nis the accumulation over $g_{k}$ steps and is equivalent to Equation \\eqref{eq:reptile_step}. $\\nabla_{macro}$ comprises both gradients of $K$ steps and\nadditional terms maximizing the inner-product of inter-batch gradients.\n\n\\vspace{-\\baselineskip}\n\\begin{align}\n{\\rm Use}~~& g_{j}=\\bar{g}_{j} + O\\left(\\alpha\\right) \\label{eq:initial_pt_trick} \\\\  \n &~\\phi_{k}-\\phi_{1}=-\\alpha\\sum_{j=1}^{k-1}g_{j} \\label{eq:diff_trick}\\\\\ng_{k} &= \\bar{g}_i  - \\alpha\\bar{H}_{i}\\sum_{j=1}^{k-1}\\bar{g}_{j} + O\\left(\\alpha^2 \\right) \\label{eq:gi_gradient_taylor_3} \\\\\n&\\nabla_{macro} = \\sum_{k=1}^{K}g_{k} \\label{eq:reptile_gradient_taylor}  \n\\end{align}\n\nWe can similarly express the gradient of the target batch as Equation\n\\eqref{eq:target_gradient_taylor} where the term,\n$\\bar{H}_{T}\\nabla_{macro}$, is the cross-lingual generalization\nproduct similar to the intra-task generalization seen above.\n\\begin{align}\ng_{T} &= \\bar{g}_{T} -\\alpha\\bar{H}_{T}\\nabla_{macro} + O\\left(\\alpha^2 \\right) \\label{eq:target_gradient_taylor} \n\\end{align}\n\nEquation~\\eqref{eq:sigma_gradient_taylor} shows an example final \\tabularnewline\ngradient when $K=2$. Within the parentheses are\nthe intra-task and cross-lingual gradient products as components\npromoting fast learning across multiple axes of generalization.\n\n\\vspace{-\\baselineskip}\n\\begin{equation}\n    \\begin{split}\n\\nabla_{\\Sigma} &= g_{1} + g_{2} +  g_{T} \\\\ \n    &=  \\bar{g}_1 +\\bar{g}_2 + \\bar{g}_T \\\\ & -\\alpha\\left(\\bar{H}_{2}\\bar{g}_{1}+\\bar{H}_{T}\\lbrack\\bar{g}_1 +\\bar{g}_2\\rbrack\\right) + O\\left(\\alpha^2 \\right)  \\label{eq:sigma_gradient_taylor} \\\\\n\\end{split}\n\\end{equation}\n\nThe key hyperparameter in \\mbox{\\sc XG-Reptile} is the number of\ninner-loop steps~$K$ representing a trade-off between a manifold\napproximation and target step frequency. At small $K$, the \nmanifold approximation may be poor, leading to sub-optimal learning. At\nlarge $K$, then improved manifold approximation incurs fewer\ntarget batch steps per epoch, leading to weakened cross-lingual\ntransfer. In practice, $K$ is set empirically, and Section~\\ref{sec:results}\nidentifies an optimal region for our task.\n\n\\mbox{\\sc XG-Reptile} can be viewed as generalizing two existing\nalgorithms. Without the $\\mathcal{L}_{T}$ loss, our approach is\nequivalent to {Reptile} and lacks cross-lingual alignment. If $K=1$, then \\mbox{\\sc XG-Reptile} is equivalent to\n\\mbox{\\sc DG-FMAML} \\citep{wang-etal-2021-meta-DGMAML} but lacks\nintra-task generalization across support batches. Our unification of\nthese algorithms represent the best of both approaches and\noutperforms both techniques within semantic parsing. Another perspective is that \\mbox{\\sc XG-Reptile} learns\na \\textit{regularized optimal manifold}, with immediate cross-lingual capability\nas opposed to standard {Reptile} which requires fine-tuning to transfer across tasks.\nWe identify how this contrast in approaches influences cross-lingual transfer in Section \\ref{sec:results}.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe propose \\mbox{\\sc XG-Reptile}, a meta-learning\nalgorithm for few-shot cross-lingual generalization in semantic\nparsing. \\mbox{\\sc XG-Reptile} is able to better utilize fewer samples\nto learn an economical multilingual semantic parser with minimal cost\nand improved sample efficiency. Compared to adjacent training algorithms\nand zero-shot approaches, we obtain more accurate and\nconsistent logical forms across languages similar and dissimilar to\nEnglish. Results on ATIS show clear benefit across many languages and\nresults on Spider demonstrate that \\mbox{\\sc XG-Reptile} is effective\nin a challenging cross-lingual and cross-database scenario. We focus\nour study on semantic parsing, however, this algorithm could be\nbeneficial in other low-resource cross-lingual tasks. In future work\nwe plan to examine how to better align entities in low-resource\nlanguages to further improve parsing accuracy.\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nA semantic parser maps natural language (NL) utterances to logical\nforms (LF) or executable programs in some machine-readable language\n(e.g.,~SQL). Recent improvement in the capability of semantic parsers has\nfocused on domain transfer within English\n\\citep{su-yan-2017-xdomain-paraphrasing,suhr-etal-2020-exploring},\ncompositional generalization\n\\citep{yin-neubig-2017-syntactic-gcd-code-generation,\n  herzig-berant-2021-span,scholak-etal-2021-picard}, and more recently\ncross-lingual methods\n\\citep{multilingsp-and-code-switching-Duong2017,neural-hybrid-trees-susanto2017,polyglotapi-Richardson2018}.\n\nWithin cross-lingual semantic parsing, there has been an effort to\nbootstrap parsers with minimal data to avoid the cost and labor\nrequired to support new languages. Recent proposals include using\nmachine translation to approximate training data for supervised\nlearning\n\\citep{moradshahi-etal-2020-localizing,sherborne-etal-2020-bootstrapping,nicosia-etal-2021-translate-fill-zero-shot-semparse}\nand zero-shot models, which engineer cross-lingual similarity with\nauxiliary losses\n\\citep{van-der-goot-etal-2021-masked,yang-etal-2021-frustratingly-simple-DRS-parsing,sherborne-lapata-2022-zero}. These\nshortcuts bypass costly data annotation but present limitations such\nas ``translationese'' artifacts from machine translation\n\\citep{koppel-ordan-2011-translationese} or undesirable domain\nshift \\citep{sherborne-lapata-2022-zero}. However, annotating a minimally\nsized data sample can potentially overcome these limitations while incurring\nsignificantly reduced costs compared to full dataset translation \\citep{garrette-baldridge-2013-learning}.\n\nWe argue that a few-shot approach is more realistic for an engineer\nmotivated to support additional languages for a database -- as one can\nrapidly retrieve a high-quality sample of translations and\ncombine these with existing supported languages (i.e., English).\nBeyond semantic parsing, cross-lingual few-shot approaches have also\nsucceeded at leveraging a small number of annotations within a variety\nof tasks \\citep[\\emph{inter alia}]{zhao-etal-2021-closer} including\nnatural language inference, paraphrase identification,\npart-of-speech-tagging, and named-entity recognition. Recently, the application\nof meta-learning to domain generalization has further demonstrated capability\nfor models to adapt to new domains with small samples\n\\citep{gu-etal-2018-meta-nmt,li-hospedales-metadg,wang-etal-fewshot-survey}.\n\nIn this work, we synthesize these directions into a meta-learning\nalgorithm for cross-lingual semantic parsing. Our approach \nexplicitly optimizes cross-lingual generalization using fewer training samples per new language without performance degradation. We also require minimal computational overhead beyond standard gradient-descent training and no external dependencies beyond\nin-task data and a pre-trained encoder.  Our algorithm,\n\\textbf{Cross}-Lingual \\textbf{G}eneralization \\textbf{Reptile}\n(\\mbox{\\sc XG-Reptile}) unifies two-stage meta-learning into a single\nprocess and outperforms prior and constituent methods on all\nlanguages, given identical data constraints. The\nproposed algorithm is still model-agnostic and applicable to more tasks \nrequiring sample-efficient cross-lingual transfer.\n\nOur innovation is the combination of both \\mbox{intra-task} and\n\\mbox{inter-language} steps to jointly learn the parsing task and optimal\ncross-lingual transfer. Specifically, we interleave learning \nthe overall task from a high-resource language and learning cross-lingual\ntransfer from a minimal sample of a lower-resource language. Results\non ATIS \\citep{hemphill-etal-1990-atis} in six languages (English,\nFrench, Portuguese, Spanish, German, Chinese) and Spider\n\\citep{yu-etal-2018-spider} in two languages (English, Chinese)\ndemonstrate our proposal works in both single- and cross-domain\nenvironments. Our contributions are as follows: \n\n\\vspace{-0.5em}\n\\begin{itemize}\n\\item We introduce \\mbox{\\sc XG-Reptile}, a\n    first-order meta-learning algorithm for cross-lingual generalization.\n    \\mbox{\\sc XG-Reptile} approximates an \\textit{optimal manifold} using support\n    languages with \\textit{cross-lingual regularization} using target\n    languages to train for explicit cross-lingual similarity.\n    \n\\item We showcase sample-efficient cross-lingual transfer within \n    two challenging semantic parsing datasets across multiple languages.\n    Our approach yields more accurate parsing in a few-shot scenario\n    and demands 10$\\times$ fewer samples than prior methods. \n    \n\\item We establish a cross-domain and cross-lingual \n    parser obtaining promising results for both Spider in English \n    \\citep{yu-etal-2018-spider} and CSpider in Chinese \\citep{Min2019-CSPIDER}.\n\\end{itemize}\n\n\n\n\n\\section{Problem Definition}\n\\label{sec:problem}\n\n\n\\paragraph{Semantic Parsing}\n\nWe wish to learn a parameterized parsing function, $p_{\\theta}$, which\nmaps from a natural language utterance and a relational database\ncontext to an executable program expressed in a logical form (LF) language:\n\\begin{align}\nP = p_{\\theta}\\left(Q,~D\\right) \\label{eq:parser_fn}\n\\end{align}\n\nAs formalized in Equation~\\eqref{eq:parser_fn}, we learn\nparameters,~$\\theta$, using paired data\n$\\left(Q,~P,~\\mathcal{D}\\right)$ where~$P$ is the logical form\nequivalent of natural language question~$Q$. In this work, our LFs are\nall executable SQL queries and therefore grounded in a\ndatabase~$\\mathcal{D}$. A single-domain dataset references only\none~$\\mathcal{D}$ database for all $\\left(Q,~P\\right)$, whereas a\nmulti-domain dataset demands reasoning about unseen databases to\ngeneralize to new queries. This is expressed as a `zero-shot'\nproblem if the databases at test time, $\\mathcal{D}_{\\rm test}$, were\nunseen during training. This challenge demands a parser\ncapable of \\textit{domain generalization} beyond observed\ndatabases. This is in addition to the \\textit{structured prediction}\nchallenge of semantic parsing.\n\n\\paragraph{Cross-Lingual Generalization}\n\nPrototypical semantic parsing datasets express the question, ${Q}$, in\nEnglish only. As discussed in Section~\\ref{sec:intro}, our parser\nshould be capable of mapping from \\emph{additional} languages to\nwell-formed, executable programs. However, prohibitive expense limits\nus from reproducing a monolingual model for each additional language\nand previous work demonstrates accuracy improvement by training\n\\textit{multilingual} models\n\\citep{multilingual-sp-hierch-tree-Jie2014}. In addition to the\nchallenges of structured prediction and domain generalization, we\njointly consider \\textit{cross-lingual generalization}. Training\nprimarily relies on existing English data (i.e., ${Q}_{\\rm EN}$\nsamples) and we show that our meta-learning algorithm in\nSection~\\ref{sec:methodology} leverages a small sample of training data in new\nlanguages for accurate parsing. We  express this sample, $\\mathcal{S}_{l}$, for some language, $l$,\nas: \n\\begin{align}\n\t\\mathcal{S}_{l}&=\\left({Q}_{l},~{P},~\\mathcal{D}\\right)_{i=0}^{N_{l}} \\label{eq:sample_l}\n\\end{align} \nwhere $N_{l}$ is the sample size from~$l$, assumed to be smaller than the\noriginal English dataset (i.e.,~\\mbox{${N}_{l} \\ll N_{\\rm\n    EN}$}). Where available, we extend this paradigm to develop models\nfor $L$ different languages simultaneously in a multilingual setup by\ncombining samples as:\n\\begin{align}\n\t\\mathcal{S}_{\\rm L} &= \\lbrace\\mathcal{S}_{ l_1},\\mathcal{S}_{ l_2},\\ldots,\\mathcal{S}_{l_N}\\rbrace \\label{eq:multilingual_s_outer}\n\\end{align}\nWe can express cross-lingual generalization as:\n\\begin{align}\n\t p_{\\theta}\\left(P~|~Q_{l},~\\mathcal{D}\\right) \\rightarrow p_{\\theta}\\left(P~|~Q_{\\rm EN},~\\mathcal{D}\\right) \\label{eq:xlingual_distributions_approach}\n\\end{align}\nwhere~$p_{\\theta}\\left(P~|~Q_{\\rm EN},~\\mathcal{D}\\right)$ is the\npredicted distribution over all possible output SQL sequences\nconditioned on an English question, $Q_{\\rm EN}$, and a\ndatabase~$\\mathcal{D}$. Our goal is for the prediction from a new language, $Q_{l}$, to converge\ntowards this existing distribution using the same parameters $\\theta$, constrained to fewer\nsamples in~$l$ than English.\n\n\nWe aim to maximize the accuracy of predicting programs on\nunseen test data from each non-English language~$l$. The key challenge\nis learning a performant distribution over each new language with\nminimal available samples. This includes learning to incorporate\neach $l$ into the parsing task and modeling the language-specific\nsurface form of questions. Our setup is akin to few-shot learning; however, the number of examples needed for satisfactory performance is\nan empirical question. We are searching for both minimal sample\nsizes and maximal sampling efficiency. We discuss our sampling\nstrategy in Section~\\ref{sec:sampling_for_xgen} with results at\nmultiple sizes of~$\\mathcal{S}_{L}$ in Section~\\ref{sec:results}.\n\n\n\\section{Results and Analysis}\n\\label{sec:results}\n\nWe contrast \\mbox{\\sc XG-Reptile} to baselines for ATIS in\nTable~\\ref{tab:xg_reptile_atis_1} and present further analysis within\nFigure~\\ref{fig:k_exp_1_2_3}. Results for the multi-domain Spider are\nshown in Table~\\ref{tab:xgr_spider_1}. Our findings\nsupport our hypothesis that \\mbox{\\sc XG-Reptile} is a superior\nalgorithm for jointly training a semantic parser and encouraging\ncross-lingual generalization with improved sample efficiency. Given the same\ndata, \\mbox{\\sc XG-Reptile} produces more mutually beneficial\nparameters for both model requirements with only modifications to the\ntraining loop.\n\n\\vspace{-\\baselineskip}\n\\paragraph{Comparison across Generalization Strategies}\n\nWe compare \\mbox{\\sc XG-Reptile} to established learning algorithms in\nTable \\ref{tab:xg_reptile_atis_1}. Across baselines, we find that\nsingle-stage training, i.e., \\emph{Train-EN$\\cup$All} or\nmachine-translation based models, perform below two-stage\napproaches. The strongest competitor is the\n\\emph{Reptile-EN$\\rightarrow$FT-All} model, highlighting the\neffectiveness of Reptile for single-task generalization\n\\citep{kedia-etal-2021-beyond-reptile}. However, \\mbox{\\sc XG-Reptile}\nperforms above all baselines across sample rates. Practically, 1\\%,\n5\\%, 10\\% correspond to 45, 225, and 450 example pairs, respectively. We\nidentify significant improvements ($p<0.01$; relative to the closest model using an independent t-test) in cross-lingual transfer through jointly learning to parse and multi-language generalization while maintaining single-stage training efficiency.\n\nCompared to the upper bounds, \\mbox{\\sc XG-Reptile} performs above\n\\emph{Monolingual Training} at $\\ge 1\\%$ sampling, which further\nsupports the prior benefit of multilingual modeling\n\\citep{arch-for-neural-multisp-Susanto2017}.\n\\emph{Multilingual Training} is only marginally stronger than \\mbox{\\sc\n  XG-Reptile} at 1\\% and 5\\% sampling despite requiring many more examples. \\mbox{\\sc\n  XG-Reptile}@10\\% improves on this model by an average~$+$1.3\\%. Considering that our upper bound uses $10\\times$ the data of\n\\mbox{\\sc XG-Reptile}@10\\%, this accuracy gain highlights the benefit\nof explicit cross-lingual generalization. This is\nconsistent at higher sample sizes (see Figure\n\\ref{fig:k_exp_1_2_3}(c) for German).\n\nAt the smallest sample size, \\mbox{\\sc XG-Reptile}@1\\%, demonstrates a\n$+$12.4\\% and $+$13.2\\% improvement relative to \\emph{Translate-Train} and\n\\emph{Translate-Test}. Machine translation is often viable for cross-lingual transfer \\citep{conneau2018xnli}. However, we find that mistranslation of named entities\nincurs an exaggerated parsing penalty -- leading to inaccurate logical forms\n\\citep{sherborne-etal-2020-bootstrapping}. This suggests that\nsample quality has an exaggerated influence on semantic parsing\nperformance. When training \\mbox{\\sc XG-Reptile} with MT\ndata, we also observe a lower Target-language average of 66.9\\%. \nThis contrast further supports\nthe importance of sample quality in our context.\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=\\textwidth]{imgs\/k_exp_subfig_v3_9_9_22.png}\n    \\caption{Ablation Experiments on ATIS (a) accuracy against inner loop size $K$ across languages, (b) accuracy against $K$ for German when varying batch size, and (c) accuracy against dataset sample size relative to support dataset from 1\\% to 50\\% for German. For~(b), the $K=1$ case is equivalent to DG-FMAML \\citep{wang-etal-2021-meta-DGMAML}.}\n    \\label{fig:k_exp_1_2_3}\n    \\vspace{-1em}\n\\end{figure*}\n\n \\mbox{\\sc XG-Reptile} improves cross-lingual generalization across all languages at\n equivalent and lower sample sizes. At 1\\%, it\n improves by an average $+$15.7\\% over the closest model,\n \\emph{Reptile-EN$\\rightarrow$FT-All}. Similarly, at 5\\%, we find\n $+$9.8\\% gain, and at 10\\%, we find $+$8.9\\% relative to the closest\n competitor. Contrasting across sample sizes --- our best approach is\n @10\\%, however, this is $+$3.5\\%~above @1\\%, suggesting that smaller\n samples could be sufficient if 10\\% sampling is unattainable. This\n relative stability is an improvement compared to the 17.7\\%, 11.2\\% or\n 10.3\\% difference between @1\\% and @10\\% for other models. This\n implies that \\mbox{\\sc XG-Reptile} better utilizes smaller samples\n than adjacent methods.\n\nAcross languages at 1\\%, \\mbox{\\sc XG-Reptile} improves primarily for\nlanguages dissimilar to English \\citep{ahmad-etal-2019-difficulties}\nto better minimize the cross-lingual transfer gap. For Chinese\n(ZH), we see that \\mbox{\\sc XG-Reptile}@1\\% is $+$26.4\\% above the\nclosest baseline. This contrasts with the smallest gain, $+$8.5\\%~for\nGerman, with greater similarity to English. Our improvement also\nyields less variability across target languages --- the\nstandard deviation across languages for \\mbox{\\sc XG-Reptile}@1\\% is\n1.1, compared to 2.8 for \\emph{Train-EN$\\cup$All} or 7.4 for \n\\emph{Reptile-EN$\\rightarrow$FT-All}.\n\nWe can also compare to {\\sc ZX-Parse}, the method of\n\\citet{sherborne-lapata-2022-zero} which engineers cross-lingual latent\nalignment for zero-shot semantic parsing without data in target\nlanguages. With 45 samples per target language, \\mbox{\\sc\n  XG-Reptile}@1\\% improves by an average of $+$4.9\\%.  \\mbox{\\sc\n  XG-Reptile} is more beneficial for distant languages --\ncross-lingual transfer penalty between English and Chinese is $-$12.3\\%\nfor {\\sc ZX-Parse} compared to $-$5.7\\% in our case. While these\nsystems are not truly comparable, given different data requirements,\nthis contrast is practically useful for comparison between zero- and\nfew-shot localization.\n\n\n\\paragraph{Influence of~$K$ on Performance}\nIn Figure~\\ref{fig:k_exp_1_2_3}(a) we study how variation in the key\nhyperparameter $K$, the size of the inner-loop in\nAlgorithm~\\ref{xgr-algo} or the number of batches used to approximate\nthe \\emph{optimal task manifold} influences model performance across\nlanguages (single run at 5\\% sampling).  When \\mbox{$K=1$}, the model learns\ngeneralization from batch-wise similarity which is equivalent to\n\\emph{DG-FMAML} \\citep{wang-etal-2021-meta-DGMAML}. We empirically\nfind that increasing~$K$ beyond one benefits performance by\nencouraging cross-lingual generalization with the \\emph{task} over a\nsingle \\emph{batch}, and it is, therefore, beneficial to align an\nout-of-domain example with the overall \\textit{direction} of training.\nHowever, as theorized in Section~\\ref{sec:methodology}, increasing $K$\nalso decreases the frequency of the outer step within an epoch leading\nto poor cross-lingual transfer at high~$K$. This trade-off yields an\noptimal operating regime for this hyper-parameter. We use $K=10$ in\nour experiments as the center of this region. Given this setting\nof~$K$, the target sample size must be 10\\% of the support sample size\nfor training in a single epoch. However,\nTable~\\ref{tab:xg_reptile_atis_1} identifies \\mbox{\\sc XG-Reptile} as\nthe most capable algorithm for ``over-sampling'' smaller target\nsamples for resource-constrained generalization.\n\n\\paragraph{Influence of Batch size on performance}\nWe consider two further case studies to analyze \\mbox{\\sc\n  XG-Reptile} performance. For clarity, we focus on German; however,\nthese trends are consistent across all target languages.\nFigure~\\ref{fig:k_exp_1_2_3}(b) examines if the effects of\ncross-lingual transfer within \\mbox{\\sc XG-Reptile} are sensitive to\nbatch size during training  (single run at 5\\% sampling). A dependence between $K$ and batch size\ncould imply that the desired inter-task and cross-lingual\ngeneralization outlined in Equation~\\eqref{eq:sigma_gradient_taylor}\nis an unrealistic, edge-case phenomenon. This is not the\ncase, and a trend of optimal $K$ setting is consistent across many\nbatch sizes. This suggests that $K$ is an independent hyper-parameter\nrequiring tuning alongside existing experimental settings.\n\n\\paragraph{Performance across Larger Sample Sizes} We consider a wider\nrange of target data sample sizes between 1\\% to 50\\% in\nFigure~\\ref{fig:k_exp_1_2_3}(c). We observe that baseline approaches\nconverge to between 69.3\\% and 73.9\\% at 50\\% target sample\nsize. Surprisingly, the improvement of \\mbox{\\sc XG-Reptile} is\nretained at higher sample sizes with an accuracy of 76.5\\%. The benefit of \\mbox{\\sc\n  XG-Reptile} is still greatest at low sample sizes with $+$5.4\\%\nimprovement at 1\\%; however, we maintain a $+$2.6\\% gain over the\nclosest system at 50\\%. While low sampling is the most\neconomical, the consistent benefit of \\mbox{\\sc XG-Reptile}\nsuggests a promising strategy for other cross-lingual tasks.\n\n\n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{@{}llcc|cc@{}}\n\\toprule\n\\multicolumn{2}{l}{} & \\multicolumn{2}{c}{EN} & \\multicolumn{2}{c}{ZH} \\\\ \\midrule\n\\multicolumn{2}{l}{} & Dev & Test & Dev & Test \\\\ \\midrule\n\\multicolumn{6}{l}{\\textit{Monolingual}} \\\\ \\midrule\n\\multicolumn{2}{l}{\\makecell[l]{DG-MAML }} & \\textbf{68.9} &  \\textbf{65.2} & \\textbf{50.4} &  \\textbf{46.9} \\\\\n\\multicolumn{2}{l}{\\makecell[l]{DG-FMAML }} & 56.8 &  --- & 32.5 &  --- \\\\\n\\multicolumn{2}{l}{\\mbox{\\sc XG-Reptile}} & 63.5 &  --- & 48.9 &  --- \\\\ \\midrule\n\\multicolumn{6}{l}{\\textit{Multilingual}} \\\\ \\midrule\n\\mbox{\\sc XG-Reptile} \n & @1\\% & 56.8 &  56.5 & 47.0 &  45.6 \\\\\n & @5\\% & \\textbf{59.6} &  58.1 & 47.3 &  45.6 \\\\\n & @10\\% & 59.2 &  \\textbf{59.7} & \\textbf{48.0} &  \\textbf{46.0} \\\\\n \\bottomrule\n\\end{tabular}\n\\caption{Exact set match accuracy for RAT-SQL trained on Spider (English) and CSpider (Chinese) comparing \\mbox{\\sc XG-Reptile} to DG-MAML and DG-FMAML \\citep{wang-etal-2021-meta-DGMAML}. We experiment with sampling between 1\\% to 10\\% of Chinese examples relative to English. Monolingual and multilingual best results are bolded.\n}\n\\label{tab:xgr_spider_1}\n\\vspace{-1.5em}\n\\end{table}\n\n\\paragraph{Learning Spider and CSpider}\n\nOur results on Spider and CSpider are shown in\nTable~\\ref{tab:xgr_spider_1}. We compare \\mbox{\\sc XG-Reptile}\nto monolingual approaches from  \\citet{wang-etal-2021-meta-DGMAML}\nand discuss cross-lingual results when sampling between 1\\% to 10\\% of CSpider target\nduring training.\n\nIn the \\emph{monolingual setting}, \\mbox{\\sc XG-Reptile} shows\nsignificant improvement ($p<0.01$; using an independent samples t-test) compared to DG-FMAML with $+$6.7\\% for English and $+$16.4\\% for Chinese dev accuracy. This further supports\nour claim that generalizing with a \\textit{task manifold} is superior\nto batch-level generalization. \n\nOur results are closer to DG-MAML \\citep{wang-etal-2021-meta-DGMAML}, a higher-order meta-learning method requiring computational resources and training times exceeding $4\\times$ the requirements for {\\sc XG-Reptile}. \\mbox{\\sc XG-Reptile} yields accuracies $-5.4\\%$ and $-1.5\\%$ below DG-MAML for English and Chinese, where DG-FMAML performs much lower at $-12.1\\%$ (EN) and $-17.9\\%$ (ZH). \nOur results suggest that \\mbox{\\sc XG-Reptile} is a superior first-order meta-learning\nalgorithm rivaling prior work with greater computational demands.\\footnote{We compare against DG-MAML as the\nbest \\emph{public} available model on the \\href{https:\/\/taolusi.github.io\/CSpider-explorer\/}{CSpider leaderboard}\nat the time of writing.}\n\nIn the multilingual setting, we observe that\n\\mbox{\\sc XG-Reptile} performs competitively using as little as 1\\% of\nChinese examples. While training sampling 1\\% and 5\\% perform similarly -- the best model sees 10\\% of CSpider samples during\ntraining to yield accuracy only $-$0.9\\%~(test) below the monolingual DG-MAML model.\nWhile performance does not match monolingual models, the multilingual\napproach has additional utility in serving more users. As a zero-shot\nsetup, predicting SQL from CSpider inputs through the model trained\nfor English, yields 7.9\\%~validation accuracy, underscoring that\ncross-lingual transfer for this dataset is \\mbox{non-trivial}.\n\nVarying the target sample size demonstrates more variable effects for\nSpider compared to ATIS. Notably, increasing the sample size yields\npoorer English performance beyond the optimal \\mbox{\\sc\n  XG-Reptile}@5\\% setting for English. This may be a consequence of\nthe cross-database challenge in Spider --- information shared across\nlanguages may be less beneficial than for single-domain ATIS. The\nleast performant model for both languages is \\mbox{\\sc\n  XG-Reptile}@1\\%. Low performance here for Chinese can be\nexpected, but the performance for English is surprising. We suggest that\nthis result is a consequence of ``over-sampling'' of the target\ndata disrupting the overall training process. That is, for 1\\%\nsampling and optimal $K=4$, the target data is ``over-used''\n$25\\times$ for each epoch of support data. We further observe\ndiminishing benefits for English with additional Chinese\nsamples. While we trained a competitive parser with minimal\nChinese data, this effect could be a consequence of how\nRAT-SQL cannot exploit certain English-oriented learning features (e.g., lexical similarity scores). Future\nwork could explore cross-lingual strategies to unify entity modeling for\nimproved feature sharing.\n\n\\paragraph{Visualizing the Manifold}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=\\textwidth]{imgs\/pca_grid_metrics3.png}\n    \\caption{PCA Visualizations of sentence-averaged encodings for English (EN), French (FR) and Chinese (ZH) from the ATIS test set (@1\\% sampling from Table \\ref{tab:xg_reptile_atis_1}). We identify the regularized weight manifold which improves cross-lingual transfer using \\mbox{\\sc XG-Reptile}. We also improve in two similarity metrics averaged across languages.}\n    \\label{fig:pca}\n    \\vspace{-1em}\n\\end{figure*}\n \nAnalysis of \\mbox{\\sc XG-Reptile} in Section \\ref{sec:methodology} relies on a theoretical basis that first-order meta-learning creates a dense high-likelihood sub-region in the parameters (i.e. \\textit{optimal manifold}). \nUnder these conditions, representations of new domains should cluster within the manifold to allow for rapid adaptation with minimal samples. This contrasts with methods without meta-learning, which provide no guarantees of representation density. However, metrics in Table \\ref{tab:xg_reptile_atis_1} and \\ref{tab:xgr_spider_1} do not directly explain if this expected effect arises. To this end, we visualize ATIS test set encoder outputs using PCA \\citep{doi:10.1137\/090771806} in Figure \\ref{fig:pca}. We contrast English (support) and French and Chinese as the most and least similar target languages. Using PCA allows for direct interpretation of low-dimensional distances across approaches. Cross-lingual similarity is a proxy for manifold alignment -- as our goal is accurate cross-lingual transfer from closely aligned representations from source and target languages \\citep{xia-etal-2021-metaxl, sherborne-lapata-2022-zero}.\n\nAnalyzing Figure \\ref{fig:pca}, we observe meta-learning methods (\\emph{Reptile-EN$\\rightarrow$FT-All},~\\mbox{\\sc XG-Reptile}) to fit target languages closer to the support (English, yellow circle). In contrast, methods not utilizing meta-learning (\\emph{Train-EN$\\cup$All},~\\emph{Train-EN$\\rightarrow$FT-All}) appear less ordered and with weaker representation overlap. Encodings from \\mbox{\\sc XG-Reptile} are less separable across languages and densely clustered, suggesting the regularized manifold hypothesized in Section \\ref{sec:methodology} ultimately yields improved cross-lingual transfer. Visualizing encodings from English in the \\emph{Reptile-EN} model \\textit{before} fine-tuning produces a similar cluster (not shown), however, required fine-tuning results in ``spreading'' leading to less cross-lingual similarity. \n\nWe also quantitatively examine the average encoding change in Figure \\ref{fig:pca} using Cosine similarity and Hausdorff distance \\citep{patra-etal-2019-bilingual} between English and each target language. Cosine similarity is measured pair-wise across parallel inputs in each language to gauge similarity from representations with equivalent SQL outputs. As a measure of mutual proximity between sets, Hausdorff distance denotes a worst-case distance between languages to measure more general ``closeness''. Under both metrics, \\mbox{\\sc XG-Reptile} yields the best performance with the most substantial pair-wise similarity and Hausdorff similarity. These indicators for cross-lingual similarity further support the observation that our expected behavior is legitimately occurring during training. \n\nOur findings better explain \\textit{why} our \\mbox{\\sc XG-Reptile} performs above other training algorithms. Specifically, our results suggest that \\mbox{\\sc XG-Reptile} learns a \\textit{regularized manifold} which produces stronger cross-lingual similarity and improved parsing compared to Reptile \\textit{fine-tuning a manifold}. This contrast will inform future work for cross-lingual meta-learning where \\mbox{\\sc XG-Reptile} can be applied. \n\n\\paragraph{Error Analysis}\n\n\\begin{figure}[t]\n\\centering\n\\begin{tabulary}{\\columnwidth}{@{}LL@{}}\n\\toprule\nEN & {\\footnotesize Show me all flights from San Jose to Phoenix} \\\\ \\midrule\nFR & {\\footnotesize Me montrer tous les vols de San Jos\\`e \\'a Phoenix } \\\\ \\midrule\n$\\times$ & {\\footnotesize SELECT DISTINCT flight\\_1.flight\\_id FROM flight flight\\_1, airport\\_service airport\\_service\\_1, city city\\_1, airport\\_service airport\\_service\\_2, city city\\_2 WHERE flight\\_1.from\\_airport = airport\\_service\\_1.airport\\_code AND airport\\_service\\_1.city\\_code = city\\_1.city\\_code AND city\\_1.city\\_name = \\colorbox{red}{'SAN FRANCISCO'} AND flight\\_1.to\\_airport = airport\\_service\\_2.airport\\_code AND airport\\_service\\_2.city\\_code = city\\_2.city\\_code AND city\\_2.city\\_name =  \\colorbox{red}{'PHILADELPHIA'};} \\\\ \\midrule\n\\checked & {\\footnotesize SELECT DISTINCT flight\\_1.flight\\_id FROM flight flight\\_1, airport\\_service airport\\_service\\_1, city city\\_1, airport\\_service airport\\_service\\_2, city city\\_2 WHERE flight\\_1.from\\_airport = airport\\_service\\_1.airport\\_code AND airport\\_service\\_1.city\\_code = city\\_1.city\\_code AND city\\_1.city\\_name =  \\colorbox{ForestGreen}{'SAN JOSE'} AND flight\\_1.to\\_airport = airport\\_service\\_2.airport\\_code AND airport\\_service\\_2.city\\_code = city\\_2.city\\_code AND city\\_2.city\\_name =  \\colorbox{ForestGreen}{'PHOENIX'};} \\\\ \\bottomrule\n\\end{tabulary}\n\\caption{Contrast between SQL from a French input from ATIS for  \\emph{Train-EN$\\cup$All} and \\mbox{\\sc XG-Reptile}. The entities ``San Jos\\'{e}'' and ``Phoenix'' are not observed in the 1\\% sample of French data but are mentioned in the English support data. The  \\emph{Train-EN$\\cup$All} approach fails to connect attributes seen in English when generating SQL from French inputs ($\\times$). Training with \\mbox{\\sc XG-Reptile} better leverages support data to generate accurate SQL from other languages (\\checked).\n}\n\\label{tab:entity-error}\n\\vspace{-1em}\n\\end{figure}\n\n\nWe can also examine \\textit{where} the improved cross-lingual transfer influences parsing performance. Similar to Figure \\ref{fig:pca}, we consider the results of models using 1\\% sampling as the worst-case performance and examine where \\mbox{\\sc XG-Reptile} improves on other methods on the test set (448 examples) over five languages.\n\nAccurate semantic parsing requires sophisticated entity handling to translate mentioned proper nouns from utterance to logical form. In our few-shot sampling scenario, \\textit{most} entities will appear in the English support data (e.g. ``Denver'' or ``American Airlines''), and \\textit{some} will be mentioned within the target language sample (e.g. ``Mine\\'apolis'' or ``Nueva York'' in Spanish). These samples cannot include all possible entities -- effective cross-lingual learning must ``connect'' these entities from the support data to the target language -- such that these names can be parsed when predicting SQL from the target language. As shown in Figure \\ref{tab:entity-error}, the failure to recognize entities from support data, for inference on target languages, is a critical failing of all models besides \\mbox{\\sc XG-Reptile}. \n\nThe improvement in cross-lingual similarity using \\mbox{\\sc XG-Reptile} expresses a specific improvement in entity recognition. Compared to the worst performing model, \\emph{Train-EN$\\cup$All}, 55\\% of improvement accounts for handling entities absent from the 1\\% target sample but present in the 99\\% English support data. While \\mbox{\\sc XG-Reptile} can generate accurate SQL, other models are limited in expressivity to fall back on using seen entities from the 1\\% sample. This notably accounts for 60\\% of improvement in parsing Chinese, with minimal orthographic overlap to English, indicating that \\mbox{\\sc XG-Reptile} better leverages support data without reliance on token similarity. In 48\\% of improved parses, entity mishandling is the \\textit{sole error} -- highlighting how limiting poor cross-lingual transfer is for our task. \n\nOur model also improves handling of novel \\textit{modifiers} (e.g. ``on a weekday'', ``round-trip'') absent from target language samples. Modifiers are often realized as additional sub-queries and filtering logic in SQL outputs. Comparing \\mbox{\\sc XG-Reptile} to  \\emph{Train-EN$\\cup$All}, 33\\% of improvement is related to modifier handling. Less capable systems fall back on modifiers observed from the target sample or ignore them entirely to generate inaccurate SQL. \n\nWhile \\mbox{\\sc XG-Reptile} better links parsing knowledge from English to target languages -- the problem is not solved. Outstanding errors in all languages primarily relate to query complexity, and the cross-lingual transfer gap is not closed. Furthermore, our error analysis suggests a future direction for optimal sample selection to minimize the error from interpreting unseen phenomena.\n\n\n\\section{Experimental Design}\n\\label{sec:exp_design}\n\nWe evaluate \\mbox{\\sc XG-Reptile} against several comparison systems\nacross multiple languages. Where possible, we re-implement existing\nmodels and use identical data splits to isolate the contribution of\nour training algorithm.\n\n\\subsection{Data}\n\nWe report results on two semantic parsing datasets. First on\n\\textbf{ATIS} \\citep{hemphill-etal-1990-atis}, using the multilingual\nversion from \\citet{sherborne-lapata-2022-zero} pairing utterances in six\nlanguages (English, French, Portuguese, Spanish, German, Chinese) to\nSQL queries. ATIS is split into 4,473 training pairs with 493 and 448\nexamples for validation and testing, respectively. \nWe report performance as execution accuracy to test if\npredicted SQL queries can retrieve accurate database results.\n\nWe also evaluate on \\textbf{Spider} \\citep{yu-etal-2018-spider}, combining\nEnglish and Chinese \\citep[CSpider]{Min2019-CSPIDER} versions as a\ncross-lingual task. The latter translates all questions to Chinese but\nretains the English database. Spider is significantly more\nchallenging; it contains~10,181 questions and 5,693 unique SQL queries\nfor 200 multi-table databases over 138 domains. We use the same split\nas \\citet{wang-etal-2021-meta-DGMAML} to measure generalization to\nunseen databases\/table-schema during testing. This split uses 8,659\nexamples from 146 databases for training and 1,034 examples from 20\ndatabases for validation. The test set contains 2,147 examples from 40\nheld-out databases and is held privately by the authors. To our knowledge, we report the first multilingual approach for Spider by training one model for English and Chinese. Our\nchallenge is now multi-dimensional, requiring cross-lingual and\ncross-domain generalization. Following \\citet{yu-etal-2018-spider}, we\nreport exact set match accuracy for evaluation.\n\n\\subsection{Sampling for Generalization}\n\\label{sec:sampling_for_xgen}\n\nTraining for cross-lingual generalization often uses parallel\nsamples across languages. We illustrate this in\nEquation~\\eqref{eq:sample1} where~$y_1$ is the equivalent\noutput for inputs, $x_{1}$, in each language:\n\\begin{align}\n  {\\rm EN:}&\\left( x_1, y_1 \\right)~{\\rm DE:}\\left( x_1, y_1\n        \\right)~{\\rm ZH:}\\left( x_1, y_1 \\right) \\label{eq:sample1}\n\\end{align}\nHowever, high sample overlap risks trivializing the task because\nmodels are not learning from new pairs, but instead matching only new\n\\textit{inputs} to known outputs. A preferable evaluation will test\ncomposition of novel outputs from unseen inputs:\n\\begin{align}\n  {\\rm EN:}&\\left( x_1, y_1 \\right)~{\\rm DE:}\\left( x_2, y_2 \\right)~{\\rm ZH:}\\left( x_2, y_2 \\right) \\label{eq:sample2}\n\\end{align}\nEquation~\\eqref{eq:sample2} samples exclusive, disjoint datasets for\nEnglish and target languages during training. In other words, this\nprocess is \\textit{subtractive} e.g.,~a 5\\%~sample of German (or Chinese) target\ndata leaves 95\\% of data as the English support. This is similar to\n\\mbox{K-fold} cross-validation used to evaluate across many data\nsplits. We sample data for our experiments with\nEquation~\\eqref{eq:sample2}. It is also possible to use\nEquation~\\eqref{eq:sample3}, where target samples are also disjoint,\nbut we find this setup results in too few English examples for\neffective\nlearning.\n\\begin{align}\n  {\\rm EN:}&\\left( x_1, y_1 \\right)~{\\rm DE:}\\left( x_2, y_2 \\right)~{\\rm ZH:}\\left( x_3, y_3 \\right) \\label{eq:sample3}\n\\end{align}\n\n\\subsection{Semantic Parsing Models}\n\nWe use a Transformer encoder-decoder model\nsimilar to \\citet{sherborne-lapata-2022-zero} for our ATIS experiments. We use the same mBART50\nencoder \\citep{tang2020multilingua-mbart50} and train a Transformer\ndecoder from scratch to generate SQL.\n\nFor Spider, we use the RAT-SQL model \\citep{wang-etal-2020-rat} which\nhas formed the basis of many performant submissions to the\n\\href{https:\/\/yale-lily.github.io\/spider}{Spider leaderboard}. RAT-SQL\ncan successfully reason about unseen databases and table schema using\na novel schema-linking approach within the encoder. We use the version from\n\\citet{wang-etal-2021-meta-DGMAML} with mBERT\n\\citep{devlin-etal-2019-BERT} input embeddings for a unified model between\nEnglish and Chinese inputs. Notably, RAT-SQL can be over-reliant on lexical similarity features between\ninput questions and tables \\cite{wang-etal-2020-rat}. This raises the\nchallenge of generalizing to Chinese where such overlap is null.\nFor fair comparison, we implement identical models as prior work on\neach dataset and only evaluate the change in training algorithm. \nThis is why we use an mBART50 encoder component for ATIS experiments\nand different mBERT input embeddings for Spider experiments.\n\n\\subsection{Comparison Systems}\n\nWe compare our algorithm against several strong baselines and\nadjacent training methods including:\n\\begin{itemize}\n\\item[\\textbf{Monolingual Training}] A monolingual Transformer\nis trained on gold-standard professionally translated data for\n  each new language. This is a monolingual upper bound without\n  few-shot constraints.\\vspace{-0.5em}\n\\item[\\textbf{Multilingual Training}] A multilingual Transformer \n  is trained on the union of all data from the ``Monolingual Training''\n  method. This ideal upper bound uses all data in all languages\n  without few-shot constraints.\\vspace{-0.5em}\n\\item[\\textbf{Translate-Test}] A monolingual Transformer  is trained\n  on source English data ($\\mathcal{S}_{\\rm EN}$). Machine\n  translation is used to translate test data from additional languages into\n  English. Logical forms are predicted from translated data using the English\n  model.\\vspace{-0.5em}\n\\item[\\textbf{Translate-Train}] Machine translation is used to translate\n  English training data into each target language. A monolingual\n  Transformer is trained on translated training data and  logical forms\nare predicted   using this model. \\vspace{-0.5em}\n\\item[\\textbf{Train-EN$\\cup$All}] A Transformer is trained on English\n  data and samples from \\emph{all} target languages together in a single stage i.e.,~$\\mathcal{S}_{\\rm EN}\\cup\\mathcal{S}_{L}$. This is superior to\n  training without English (e.g., on $\\mathcal{S}_{L}$ only), we\n  contrast to this approach for more competitive\n  comparison. \\vspace{-0.5em}\n\\item[\\textbf{TrainEN$\\rightarrow$FT-All}] We first train on English\n  support data, $\\mathcal{S}_{\\rm EN}$, and then fine-tune on target\n  samples, $\\mathcal{S}_{L}$. \\vspace{-0.5em}\n\\item[\\textbf{Reptile-EN$\\rightarrow$FT-All}] Initial training uses {Reptile}\n  \\citep{DBLP:journals\/corr\/abs-1803-02999-REPTILE} on English support\n  data,~$\\mathcal{S}_{\\rm EN}$, followed by fine-tuning on target\n  samples, $\\mathcal{S}_{L}$. This is a typical usage of\n  {Reptile} for training a low-resource multi-domain parser\n  \\citep{chen-etal-2020-low-resource-domain-adaptation-task-oriented-parsing}.\n\\end{itemize}\n\nWe also compare to DG-FMAML \\citep{wang-etal-2021-meta-DGMAML} as a\nspecial case of \\mbox{\\sc XG-Reptile} when~\\mbox{$K=1$}. Additionally,\nwe omit pairwise versions of \\mbox{\\sc XG-Reptile} (e.g., separate models\ngeneralizing from English to individual languages). \nThese approaches demand more computation and demonstrated no significant\nimprovement over a multi-language approach. All Machine Translation uses Google Translate\n\\citep{gtranslate}.\n\n\n\\subsection{Training Configuration}\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{@{}lcc@{}}\n\\toprule\n & ATIS & Spider \\\\ \\midrule\nBatch Size & 10 & 16 \\\\\nInner Optimizer & \\multicolumn{2}{c}{SGD} \\\\\nInner LR & \\multicolumn{2}{c}{$1\\times 10^{-4}$}   \\\\\nOuter Optimizer & \\multicolumn{2}{c}{Adam { \\citep{DBLP:journals\/corr\/KingmaB14}}} \\\\\nOuter LR & $1\\times 10^{-3}$ & $5\\times 10^{-4}$ \\\\\nOptimum $K$ & 10 & 3 \\\\\nMax Train Steps & \\multicolumn{2}{c}{20,000} \\\\\nTraining Time & 12 hours & 2.5 days \\\\ \\bottomrule\n\\end{tabular}%\n\\caption{Experimental Hyperparameters for \\mbox{\\sc XG-Reptile} on ATIS and Spider set primarily by replicating prior work.}\n\\label{tab:parameter_table}\n\\vspace{-1em}\n\\end{table}\n\nExperiments focus on the expansion from English to additional\nlanguages, where we use English as the ``support'' language and\nadditional languages as ``target''. Key hyperparameters are outlined\nin Table \\ref{tab:parameter_table}. We train each model using the\ngiven optimizers with early stopping where model selection is through\nminimal validation loss for combined support and target\nlanguages. Input utterances are tokenized using SentencePiece\n\\citep{kudo-richardson-2018-sentencepiece} and Stanza\n\\citep{qi-etal-2020-stanza} for ATIS and Spider, respectively. All\nexperiments are implemented in Pytorch on a single V100 GPU. We report\nkey results for ATIS averaged over three seeds and five random data splits. For Spider, we submit the best singular model from five random splits to the leaderboard.\n\n\n\n\\section{Related Work}\n\\label{sec:rw}\n\n\\paragraph{Meta-Learning for Generalization}\n\nMeta-Learning\\footnote{We refer the interested reader to\n\\citet{wang-etal-fewshot-survey},\n\\citet{hospedales-meta-learning-survey}, and\n\\citet{Wang2021GeneralizingTU-survey} for more extensive surveys on\nmeta-learning.}  has recently emerged as a promising technique for\ngeneralization, delivering high performance on unseen domains by\n\\emph{learning to learn}, i.e., improving learning over multiple\nepisodes\n\\citep{hospedales-meta-learning-survey,Wang2021GeneralizingTU-survey}. A popular approach is Model-Agnostic Meta-Learning\n\\citep[MAML]{pmlr-v70-finn17a}, wherein the goal is to train a model\non a variety of learning tasks, such that it can solve \\emph{new}\ntasks using a small number of training samples. In effect, MAML\nfacilitates task-specific fine-tuning using few samples in\na two-stage process. MAML requires computing higher-order gradients\n(i.e.,~``gradient through a gradient'') which can often be\nprohibitively expensive for complex models. This limitation has motivated\n\\textit{first-order} approaches to MAML which offer similar\nperformance with improved computational efficiency.\n\nIn this vein, the {Reptile} algorithm\n\\citep{DBLP:journals\/corr\/abs-1803-02999-REPTILE} transforms the\nhigher-order gradient approach into $K$ successive\nfirst-order steps. \n{Reptile}-based training learns an \\textit{optimal manifold} across tasks (i.e., a high-density parameter sub-region biased for strong cross-task likelihood), then\nsimilarly followed by rapid fine-tuning. By learning an optimal \ninitialization, meta-learning proves useful for low-resource\nadaptation by minimizing the data required for out-of-domain tuning on new tasks. \\citet{kedia-etal-2021-beyond-reptile} also demonstrate the utility of Reptile to improve \\emph{single-task} performance. We build on this to examine single-task cross-lingual transfer using the \\textit{optimal manifold} learned with Reptile.\n\n\\paragraph{Meta-Learning for Semantic Parsing}\nA variety of NLP applications have adopted meta-learning in zero- and\nfew-shot learning scenarios as a method of explicitly training for\ngeneralization \\citep{lee-etal-2021-meta-tutorial,\n  hedderich-etal-2021-survey-metanlp}. Within semantic parsing, there\nhas been increasing interest in \\textit{cross-database generalization}, motivated by datasets such as Spider\n\\citep{yu-etal-2018-spider} requiring navigation of unseen\ndatabases\n\\citep{sp-over-many-kbs-Herzig2017, suhr-etal-2020-exploring}.\n\nApproaches to generalization have included simulating source and\ntarget domains \\citep{givoli-reichart-2019-zero-shot-semantic-parsing}\nand synthesizing new training data based on unseen databases\n\\citep{zhong-etal-2020-grounded-adaptation-gazp,\n  xu-etal-2020-autoqa}. Meta-learning has demonstrated fast adaptation\nto new data within a monolingual low-resource setting\n\\citep{huang-etal-2018-natural-structured-query-meta-learning-parsing,guo-etal-2019-coupling-context-meta-learning-parsing,DBLP:journals\/corr\/abs-1905-11499,Sun_Tang_Duan_Gong_Feng_Qin_Jiang_2020-meta-learning-parsing}. Similarly,\n\\citet{chen-etal-2020-low-resource-domain-adaptation-task-oriented-parsing}\nutilize Reptile to improve generalization of a model, trained\non source domains, to fine-tune on new domains. Our\nwork builds on \\citet{wang-etal-2021-meta-DGMAML}, who explicitly\npromote monolingual cross-domain generalization by ``meta-generalizing'' across\ndisjoint domain-specific batches during training.\n\n\\paragraph{Cross-lingual Semantic Parsing}\n\nA surge of interest in cross-lingual NLU has seen the creation of many\nbenchmarks across a breadth of languages\n\\citep{conneau2018xnli,hu2020xtreme,liang-etal-2020-xglue}, thereby\nmotivating significant exploration of cross-lingual transfer\n\\citep[\\emph{inter alia}]{nooralahzadeh-etal-2020-zero-xmaml,xia-etal-2021-metaxl,\n  xu-etal-2021-soft-layer-selection-meta-simpler-xmaml,zhao-etal-2021-closer}.\nPrevious approaches to cross-lingual semantic parsing assume\nparallel multilingual training data\n\\citep{multilingual-sp-hierch-tree-Jie2014} and exploit multi-language\ninputs for training without resource constraints\n\\citep{arch-for-neural-multisp-Susanto2017,\n  neural-hybrid-trees-susanto2017}.\n\nThere has been recent interest in evaluating if machine translation is\nan economic proxy for creating training data in new languages\n\\citep{sherborne-etal-2020-bootstrapping,moradshahi-etal-2020-localizing}.\nZero-shot approaches to cross-lingual parsing have also been explored using\nauxiliary training objectives\n\\cite{yang-etal-2021-frustratingly-simple-DRS-parsing,sherborne-lapata-2022-zero}.\nCross-lingual learning has also been gaining traction in the adjacent\nfield of spoken-language understanding (SLU).\nFor datasets such as MultiATIS \\citep{Upadhyay2018-multiatis},\nMultiATIS++ \\citep{xu-etal-2020-end-multiatis}, and MTOP\n\\citep{li-etal-2021-mtop}, zero-shot cross-lingual transfer has been\nstudied through specialized decoding methods\n\\citep{zhu-etal-2020-dont-parse-insert}, machine translation\n\\citep{nicosia-etal-2021-translate-fill-zero-shot-semparse}, and\nauxiliary objectives \\citep{van-der-goot-etal-2021-masked}.\n\nCross-lingual semantic parsing has mostly remained orthogonal to the\ncross-database generalization challenges raised by datasets such as\nSpider \\citep{yu-etal-2018-spider}. While we primarily present\nfindings for multilingual ATIS into SQL \n\\citep{hemphill-etal-1990-atis}, we also train a parser on both Spider\nand its Chinese version \\citep{Min2019-CSPIDER}. To the best of our\nknowledge, we are the first to explore a multilingual approach to this\ncross-database benchmark. We use {Reptile} to learn the overall task\nand leverage domain generalization techniques\n\\citep{li-hospedales-metadg, wang-etal-2021-meta-DGMAML} for\nsample-efficient cross-lingual transfer.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nNonlinear wave phenomena occur in various systems   in physics,\nengineering, biology and geosciences \\cite{Ba,CV,Ha,Te,Majda,\nPedlosky}. At the macroscopic level, wave phenomena   may be\nmodeled by hyperbolic wave type partial differential equations. We\nconsider the following non-autonomous wave equations with\nnonlinear damping, on a bounded domain $\\Omega$ in $\\mathbb{R}^3$,\nwith smooth boundary $\\partial \\Omega$:\n\\begin{equation}\\label{1.1}\nu_{tt}+h(u_t)-\\Delta u+f(u,t)=g(x,\\,t)\\quad x\\in\\Omega\n\\end{equation}\nsubject to the boundary condition\n\\begin{equation}\\label{1.2}\nu |_{\\partial \\Omega}=0,\n\\end{equation}\nand the initial conditions\n\\begin{equation}\\label{1.3}\nu(x,0)=u_{0}(x),\\quad u_t(x,0)=v_0(x).\n\\end{equation}\nHere $h$ is the nonlinear damping function, $ f$ is the\nnonlinearity, $ g$ is a given external time-dependent forcing, and\n$\\Delta = \\partial_{x_1x_1}+\\partial_{x_2x_2} + \\partial_{x_3x_3}   $ is the Laplace\noperator.\n\nEquation \\eqref{1.1} arises as an evolutionary mathematical model in\nvarious systems. For example, (i) modeling a continuous Josephson\njunction with specific $h, g$ and $ f $ \\cite{LSC}; (ii) modeling a\nhybrid system of nonlinear waves and nerve conduct; and (iii) when\n$h(u_t)=ku_t$ and $f(u)=|u|^ru$, the equation \\eqref{1.1}  models a\nphenomenon in quantum mechanics \\cite{Cha,CY,GM,Te}.\n\nFor the autonomous case of \\eqref{1.1}, i.e., when $f$ and $g$ do\nnot depend on time $t$ explicitly, the asymptotic behaviors of the\nsolutions have been studied extensively in the framework of global\nattractors; see, for example, \\cite{ACH,BV,Ba,CV,Ha} for the\nlinear damping case, and \\cite{CL1,CL2,CL3,Fe,SYZ1} for the\nnonlinear damping case.\n\nIn this paper, we consider the non-autonomous case, especially\nwhen the damping $h$ is   nonlinear  and when the nonlinearity $f$\nhas critical exponent (see below). For a non-autonomous dynamical\nsystem  like \\eqref{1.1}-\\eqref{1.3}, the solution map does not\ndefine a semigroup and instead, it defines a two-parameter\n\\emph{process}, or \\emph{cocycle}. Pullback attractors are\nappropriate geometric objects for describing asymptotic dynamics\nfor cocycles. We will briefly introduce basic concepts for\nnon-autonomous dynamical systems in \\S 3.\nWe will discuss the asymptotic dynamics of \\eqref{1.1}-\\eqref{1.3}\nvia   pullback attractors of the corresponding cocycle. This\ndynamical framework allows us to handle more general\nnon-autonomous time-dependency; for example, the external force\n$g$ needs to be neither almost periodic nor translation compact in\ntime.\n\n\nOur basic assumptions about nonlinear damping $h$, nonlinearity\n$f$ and forcing $g$ are as follows. Let $g(x,t)$ be in the space $\nL_{loc}^2(\\mathbb{R};L^2(\\Omega))$, of locally square-integrable\nfunctions, and assume that the   functions $h$ and $f$ satisfy the\nfollowing conditions:\n\\begin{equation}\\label{1.4}\nh\\in \\mathcal{C}^1(\\mathbb{R}),\\quad h(0)=0,\\quad h~\\text{\\rm\nstrictly increasing},\n\\end{equation}\n\\begin{equation}\\label{1.5}\n\\liminf_{|s|\\to \\infty}h'(s)>0,\n\\end{equation}\n\\begin{equation}\\label{1.6}\n|h(s)|\\leq C_1(1+|s|^p),\n\\end{equation}\nwhere $p \\in [1,\\, 5)$ which will be given precisely later; $f\\in\n\\mathcal{C}^1(\\mathbb{R}\\times \\mathbb{R};\\,\\mathbb{R})$ and\nsatisfies\n\\begin{equation}\\label{1.7}\nF_s(v,s)\\leqslant \\delta^2 F(v,s) +C_{\\delta},\\quad F(v,s)\\geqslant\n-mv^2-C_m,\n\\end{equation}\n\\begin{equation}\\label{1.8}\n|f_v(v,s)|\\leq C_2(1+|v|^{q}),~|f_s(v,s)|\\leq C_3(1+|v|^{q+1}),\n\\end{equation}\n\\begin{equation}\\label{1.9}\nf(v,s)v-C_4F(v,s)+mv^2\\geqslant -C_m,\\quad \\forall ~(v,s)\\in\n\\mathbb{R}\\times \\mathbb{R},\n\\end{equation}\nwhere $0\\leqslant q\\leqslant 2$, $F(v,s)=\\int_0^vf(w,s)dw$ and\n$\\delta$, $m$ are sufficiently small which will be determined in\n$Lemma$ \\ref{l5.3}. The number $q=2$ is called the \\emph{critical\nexponent}, since the nonlinearity $f$ is not compact in this case\n(i.e., for a bounded subset $B\\subset H_0^1(\\Omega)$, in general,\n$f(B)$ is not precompact in $L^2(\\Omega)$). This is an essential\ndifficulty in studying the asymptotic behavior even for the\nautonomous cases \\cite{ACH,BV,Ba,CL1,CL2,CL3,Fe,SYZ1}. The\nassumptions \\eqref{1.4}-{\\eqref{1.6} on $h$ are similar to those in\n\\cite{CL3,Fe,Kh,SYZ1} for the autonomous cases, while the assumption\n$1 \\leq p<5$ is due to the need for estimating\n$\\int_{\\Omega}g(u_t)u$ by $\\int_{\\Omega}g(u_t)u_t$ and\n$\\int_{\\Omega}|\\nabla u|^2$ via Sobolev embedding. Finally, the\nassumptions \\eqref{1.7}-\\eqref{1.9} are similar to the conditions\nused in Chepyzhov \\& Vishik \\cite{CV} for non-autonomous cases\n but with linear damping.\n\n\\medskip\n\nLet us recall some recent relevant research in this area.\n\nThe existence of pullback attractors are established for the\nstrongly dissipative non-autonomous dynamical systems such as those\ngenerated by parabolic type partial differential equations, e.g.,\nthe non-autonomous 2D Navier-Stokes equation and some non-autonomous\nreaction diffusion equations; see\n\\cite{ARS,CLR,CLR1,CR,Ch,CD,CKS,LLR} and the references therein.\nHowever, the situation for the hyperbolic wave type systems  is less\nclear. For the linear damping case $h(v)=kv$ with  a constant $k>0$\nand $q<2$ (subcritical), Chepyzhov \\& Vishik \\cite{CV} have obtained\nthe existence of a uniform absorbing set when $g$ is translation\nbounded in time (i.e., $g\\in L^2_b(\\mathbb{R};L^2(\\Omega))$), and\nthe existence of a uniform attractor when $g$ is translation compact\nin time (i.e., $g\\in L^2_c(\\mathbb{R};L^2(\\Omega))$).\n\nUnder the assumptions that $g$ and $\\partial_t g$ are both in  the space\nof bounded continuous functions\n$\\mathcal{C}_b(\\mathbb{R},L^2(\\Omega ))$, $h$ has bounded positive\nderivative, and furthermore, $f$ is of critical growth (i.e.,\n$q=2$), Zhou \\& Wang \\cite{ZW} have proved the existence of kernel\nsections and obtained uniform bounds of the Hausdorff dimension of\nthe kernel sections. Caraballo et al. \\cite{CKR} have discussed\nthe pullback attractors for the cases of linear damping and\nsubcritical nonlinearity ($q<2$) .\n\n\n As in the autonomous case, some kind of compactness of the\ncocycle is a key ingredient for the existence of pullback\nattractors of cocycles. The corresponding compactness assumption\nin Cheban \\cite{Ch} is that the cocycle   has a compact attracting\nset. Recently, Caraballo et al \\cite{CLR} have established a\ncriterion for the existence of pullback attractors via pullback\nasymptotic compactness, and illustrated their results with the 2D\nNavier-Stokes equation.\n\nFor the autonomous \\emph{linearly} damped wave equations, Ball\n\\cite{Ba} proposed a     method to verify the   asymptotic\ncompactness for the corresponding solution semigroup. This\nso-called energy method has been generalized \\cite{LS, MRW}\n  to some non-autonomous cases. However, for our problem, due\nto the nonlinear damping, it appears   difficult to apply the\nmethod  of Ball \\cite{Ba}. Moreover, a decomposition technique\n\\cite{ACH,CV,Fe,Ha,PZ,SY} has been successfully applied   to\nverify the asymptotic smoothness  of the corresponding solution\nsemigroup for autonomous wave equations.\n\n\nIn this paper, after some preliminaries, we first introduce the\n\\emph{pullback $\\kappa-$contraction} concept, a generalization of\n$\\kappa-$contraction from autonomous systems to non-autonomous\nsystems.  Then we establish a  criterion for the existence of\npullback attractors, in terms of pullback $\\kappa-$contraction or\npullback asymptotic compactness. This criterion is for a class of\n``weakly\" continuous cocycles (i.e., the so-called norm-to-weak\ncontinuous cocycles; see \\S 3 below). Thirdly, we show that the\npullback $\\kappa-$contraction is not equivalent to the pullback\nasymptotic compactness,  unless the cocycle mapping has a nested\nbounded pullback absorbing set (see $Definition$ \\ref{d3.2}\nbelow). This fact   is different from the autonomous semigroup\ncases. Moreover, we propose a   technique  for verifying pullback\nasymptotic compactness. Finally, we apply these results to show\nthe existence of pullback attractors for the non-autonomous\nhyperbolic wave system   \\eqref{1.1}-\\eqref{1.3}.\n\n\n\nDue to the difference between the cases $p=1$ and $1<p<5$ for the\nnonlinear damping exponent $p$, we propose the following two kinds\nof assumptions.\n\\begin{enumerate}\n\\item[] \\hspace{-0.5 cm}{\\bf Assumption  I.}\n\n$h$ satisfies \\eqref{1.4}-\\eqref{1.6} with $p=1$ and there is a\n$C_0$ such that $C_0|u-v|^2\\leqslant (h(u)-h(v))(u-v)$;\n\n$g$ satisfies\n\\begin{equation}\\label{1.10}\n\\int_{-\\infty}^{t}e^{\\beta s}\\int_{\\Omega}|g(x,s)|^2dxds < \\infty\n~~\\text{for each}~~t\\in \\mathbb{R},\n\\end{equation}\nwhere $\\beta(<C_0)$ is constant depending on the coefficients of $h$\nand $f$, which will be determined in the proof of $Lemma$\n\\ref{l5.3};\n\n$f$ satisfies \\eqref{1.7}-\\eqref{1.9}.\n\n\\item[] \\hspace{-0.5 cm}{\\bf Assumption  II.}\n\n$h$ satisfies \\eqref{1.4}-\\eqref{1.6} with $1 \\leqslant p <5$; And\n\\begin{equation}\\label{1.11}\ng \\in L^{\\infty}(\\mathbb{R},\\,L^2(\\Omega));\n\\end{equation}\nIn addition to \\eqref{1.7}-\\eqref{1.9}, $f$ satisfies also\n\\begin{equation}\\label{1.12}\nF_s(v,s) \\leqslant 0 \\quad \\text{for all}~~ (v,s)\\in\n\\mathbb{R}\\times \\mathbb{R}.\n\\end{equation}\n\\end{enumerate}\n\nWe remark that the technical hypotheses \\eqref{1.11} and\n\\eqref{1.12} in {\\it Assumptions $II$} are mainly for the existence\nof pullback absorbing set; see $Lemma$ \\ref{l5.3} below or Haraux\n\\cite{Ha2} for more details. Our method  for verifying the\nasymptotic compactness allows us take some more general assumptions\nthan \\eqref{1.11}-\\eqref{1.12}.\n\nFor convenience, hereafter let $|\\cdot|_p$ be the norm of\n$L^p(\\Omega)~(1\\leqslant p < \\infty)$, and $C$ a general positive\nconstant, which may be different in different estimates.\n\n\\medskip\n\nThis paper is organized as follows. We present some background\nmaterials in \\S 2, then prove a criterion on existence of pullback\nattractors in \\S 3, and a technical method  for verifying pullback\nasymptotic compactness   is presented in \\S 4. Finally, in \\S 5,\nthese abstract results are applied to a non-autonomous wave\nequation with nonlinear damping and critical nonlinearity, to\nobtain the existence of pullback attractors. We conclude the paper\nwith some remarks in \\S 6.\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\n\n\\subsection{Kuratowski measure of non-compactness}\n\nWe briefly review the basic concept about the Kuratowski measure\nof non-compactness and recall its basic properties, which will be\nused to establish a criterion for the existence of pullback\nattractors.\n\\begin{definition}(\\cite{Ha,SY}) \\label{d2.1}\nLet $X$ be a complete metric space and $A$ be a bounded subset of\n$X$. The Kuratowski measure of non-compactness $\\kappa(A)$ of $A$\nis defined as\n\\[\n\\kappa(A)=\\inf\\{\\delta>0|\\text{ A has a finite open cover of sets of\ndiameter} < \\delta\\}.\n\\]\n\\end{definition}\n\nIf $A$ is a nonempty, unbounded set in $X$, then we define\n$\\kappa(A)=\\infty$.\n\nThe properties of $\\kappa(A)$, which we will use in this paper, are\ngiven in the following lemmas:\n\\begin{lemma}(\\cite{Ha,SY}) \\label{l2.2}\nThe Kuratowski measure of non-compactness $\\kappa(A)$ on a\ncomplete metric space $X$ satisfies the following properties:\n\\begin{enumerate}\n\\item[(1)]~$\\kappa(A)=0$ if and only if $\\bar{A}$ is compact,\nwhere $\\bar{A}$ is the closure of $A$;\n\n\\item[(2)]~$\\kappa(\\bar{A})=\\kappa(A)$, $\\kappa(A\\cup B)=\\max\\{\\kappa(A),\\,\\kappa(B)\\}$;\n\n\\item[(3)]~If $A\\subset B$, then $\\kappa(A)\\leqslant \\kappa(B)$;\n\n\\item[(4)]~If $A_t$ is a family of nonempty, closed, bounded sets\ndefined for $t>r$ that satisfy $A_t \\subset A_s$, whenever $s \\leq\nt$, and $\\kappa(A_t)\\to 0$, as $t \\to \\infty$, then\n$\\underset{t>r}{\\cap}A_t$ is a nonempty, compact set in $X$.\n\\end{enumerate}\nIf in addition, $X$ is a Banach space, then the following estimate\nis valid:\n\\begin{enumerate}\n\\item[(5)]~$\\kappa(A+B)\\leqslant \\kappa(A)+\\kappa(B)$\\quad for any\nbounded sets $A,B$ in $ X$.\n\\end{enumerate}\n\\end{lemma}\n\n\\subsection{Some useful properties for nonlinear damping function}\n\nIn the following, we will recall some simple properties of the\nnonlinear damping function $h$, which will be used later.\n\n\\begin{lemma}(\\cite{Fe,Kh})\\label{l2.3}\nLet $h$ satisfy \\eqref{1.4} and \\eqref{1.5}. Then for any $\\delta\n>0$, there exists a constant $C_{\\delta}$   depending on $\\delta$\nsuch that\n\\[\n|u-v|^2\\leqslant \\delta + C_{\\delta}(h(u)-h(v))(u-v)\\quad \\text{for\nall}~u,v\\in \\mathbb{R}.\n\\]\n\\end{lemma}\n\nMoreover, condition \\eqref{1.6} implies that\n\\[\n|h(s)|^{\\frac{1}{p}}\\leqslant C(1+|s|).\n\\]\nTherefore, we have\n\\[\n|h(s)|^{\\frac{p+1}{p}}=|h(s)|^{\\frac{1}{p}}\\cdot |h(s)|\\leqslant\nC(1+|s|)|h(s)|\\leqslant C|h(s)|+Ch(s)\\cdot s.\n\\]\nCombining this estimate with the Young's inequality and\n\\eqref{1.4}, we further obtain that\n\\begin{equation}\\label{2.1}\n|h(s)|^{\\frac{p+1}{p}}\\leqslant C(1+h(s)\\cdot s)\\quad \\text{for\nall}~s\\in \\mathbb{R},\n\\end{equation}\nwhere the constant $C$ is independent of $s$.\n\n\n\n\n\n\n\n\\section{Criterion for the existence of pullback attractors}\n\n\nIn this section, we first recall a few basic concepts  for\nnon-autonomous dynamical systems, including  pullback\n$\\kappa-$contraction, pullback asymptotic compactness and pullback\nattractor. Then we present criteria for existence of pullback\nattractors, in terms of $\\kappa-$contraction or pullback\nasymptotic compactness.\n\n Let $X$ be a complete metric space, which is the state space for\n a non-autonomous dynamical system (NDS).\n As in \\cite{BCD,Ch,CKS}, we define a\nnon-autonomous dynamical system  in terms of a cocycle mapping\n$\\phi$: $\\mathbb{R}^+\\times \\Sigma\\times X \\to X$ which is driven\nby an \\emph{autonomous} dynamical system $\\theta$ acting on a\nparameter space $\\Sigma$. In details, $\\theta=\\{\\theta_t\\}_{t\\in\n\\mathbb{R}}$ is a autonomous dynamical system on $\\Sigma$, i.e., a\ngroup of homeomorphisms under composition on $\\Sigma$ with the\nproperties\n\\begin{enumerate}\n\\item[(i)] $\\theta_0(\\sigma) = \\sigma$ for all $\\sigma \\in \\Sigma$;\n\n\\item[(ii)] $\\theta_{t+\\tau}(\\sigma) = \\theta_t(\\theta_{\\tau}(\\sigma))$ for\nall $t,\\tau\\in \\mathbb{R}$.\n\\end{enumerate}\nThe cocycle mapping $\\phi$ satisfies\n\\begin{enumerate}\n\\item[(i)] $\\phi(0, \\sigma; x) = x$ for all $(\\sigma,\\, x)\\in \\Sigma \\times X$;\n\n\\item[(ii)] $\\phi(s + t,\\, \\sigma; x) = \\phi(s, \\theta_t(\\sigma); \\phi(t, \\sigma;\nx))$ for all $s, t\\in \\mathbb{R}^+$ and all $(\\sigma, x)\\in \\Sigma\n\\times X$.\n\\end{enumerate}\n\nSometimes we say $\\phi$ is a cocycle with respect to (w.r.t.)\n$\\theta$ and denote this by $(\\phi, \\theta)$.\n If,\nin addition, the mapping $\\phi(t, \\sigma; \\cdot):~X\\to X$ is\ncontinuous for each $\\sigma \\in \\Sigma$ and $t\\geqslant 0$, then\nwe call $\\phi$ is a {\\it continuous cocycle}. If the mapping\n$\\phi(t, \\sigma; \\cdot):~X\\to X$ is \\emph{norm-to-weak continuous}\nfor each $\\sigma \\in \\Sigma$ and $t\\geqslant 0$, that is, for each\n$\\sigma \\in \\Sigma$ and $t\\geqslant 0$, norm convergence $x_n \\to\nx$ in $X$ implies weak convergence $\\phi(t, \\sigma;\nx_n)\\rightharpoonup \\phi(t, \\sigma; x)$, then we call $\\phi$ is a\n{\\it norm-to-weak continuous cocycle}. A continuous cocycle is\nobviously also a norm-to-weak continuous cocycle.\n\nFor convenience, hereafter, we will use the following notations:\n\\[\n\\mathcal{B}\\overset{\\vartriangle}{=}\\{B~|~B~\\text{is bounded\nin}~X\\}; \\quad\n\\phi(t,\\,\\sigma;\\,B)\\overset{\\vartriangle}{=}\\{\\phi(t,\\,\\sigma;\\,x_0)~\n|~x_0 \\in B\\}.\n\\]\n\n\\begin{definition}\\label{d3.1} (\\cite{Ch})\nA family of bounded sets $\\mathscr{B} = \\{B_\\sigma\\}_{\\sigma\\in\n\\Sigma}$ of $X$ is called a bounded pullback absorbing set for the\ncocycle $\\phi$ with respect to (w.r.t.) $\\theta$, if for any\n$\\sigma \\in \\Sigma$ and any $B\\in \\mathcal{B}$, there exists\n$T=T(\\sigma, B)\\geqslant 0$ such that\n\\[\n\\phi(t,\\, \\theta_{-t}(\\sigma); B)\\subset B_{\\sigma}\\quad \\text{for\nall}~t\\geqslant T.\n\\]\n\\end{definition}\n\n\n\n\\begin{definition}\\label{d3.4}(\\cite{Ch})\n(\\textbf{Pullback  attractor})\\\\\nA family of nonempty compact sets $\\mathscr{A} =\n\\{\\mathcal{A}_\\sigma\\}_{\\sigma\\in \\Sigma}$ of $X$ is called a\npullback  attractor for the cocycle $\\phi$ w.r.t. $\\theta$, if for\nall $\\sigma \\in \\Sigma$, it satisfies\n\\begin{enumerate}\n\\item[(i)] $\\phi(t, \\sigma; \\mathcal{A}_{\\sigma}) = \\mathcal{A}_{\\theta_t(\\sigma)}$ for all $t \\in\n\\mathbb{R}^+$ \\quad ($\\phi-$invariance);\n\n\\item[(ii)] $\\lim\\limits_{t\\to +\\infty} dist_X(\\phi(t;\n\\theta_{-t}(\\sigma); B),\\,\\mathcal{A}_{\\sigma}) = 0$ for all bounded\nset $B\\subset X$.\n\\end{enumerate}\nOften, $\\mathcal{A}_\\sigma$ is called a fiber at parameter $\\sigma\n\\in \\Sigma$.\n\\end{definition}\n\n\\begin{definition}\\label{d3.5}(\\cite{Ch})\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$, and let $B\\in \\mathcal{B}$. We define the\npullback $\\omega$-limit set $\\omega_{\\sigma}(B)$ as   follows\n\\[\n\\omega_{\\sigma}(B)=\\bigcap_{s\\geqslant\n0}\\overline{\\bigcup_{t\\geqslant s}\\phi(t, \\theta_{-t}(\\sigma);\nB)},  \\;\\;\\;\\;   \\sigma \\in \\Sigma ,\n\\]\nwhere $\\overline{A}$ means the closure of $A$ in $X$.\n\\end{definition}\n\nIf the parameter space $\\Sigma$  contains only one element\n$\\sigma_0$ and $\\theta_t(\\sigma_0)\\equiv\\sigma_0$ for all $t\\in\n\\mathbb{R}$, then $\\varphi$ reduces to a semigroup and all the\nconcepts in $Definitions$ \\ref{d3.1}-\\ref{d3.5} coincide with the\ncorresponding concepts   in autonomous systems. Especially, in the\nautonomous case, the pullback attractor coincides with the global\nattractor; see \\cite{BV,Robinson,SY,Te, Ilyin}. Moreover,\nChepyzhov \\& Vishik \\cite{CV} define the concept of kernel\nsections for non-autonomous dynamical systems, which correspond to\nthe fibers $\\mathcal{A}_{\\sigma}$ in the above Definition\n\\ref{d3.4} of a pullback attractor. Furthermore, similar to the\nautonomous cases, we have also the following equivalent\ncharacterization about the pullback $\\omega$-limit set.\n\n\n\\begin{lemma}\\label{l3.6} (\\cite{Ch})\nFor any $B\\subset \\mathcal{B}$ and any $\\sigma\\in \\Sigma$, $x_0\\in\n\\omega_{\\sigma}(B)$ if and only if there exist $\\{x_n\\}\\subset B$\nand $\\{t_n\\}\\subset \\mathbb{R}^+$ with $t_n \\to +\\infty$ as $n\\to\n\\infty$, such that\n\\[\n\\phi(t_n, \\theta_{-t_n}(\\sigma); x_n) \\to x_0\\quad \\text{as}~n\\to\n\\infty.\n\\]\n\\end{lemma}\n\n\nNow  we define the pullback $\\kappa$-contracting cocycle in terms\n of the Kuratowski non-compactness measure:\n\\begin{definition}\\label{d3.7} (\\textbf{$\\kappa$-contracting cocycle})\\\\\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$. Then $\\phi$ is called pullback\n$\\kappa$-contracting if for any $\\varepsilon>0$, $\\sigma\\in\n\\Sigma$ and any $B\\in \\mathcal{B}$, there is a $T=T(\\varepsilon,\n\\sigma, B)\\geqslant 0$ such that\n\\[\n\\kappa_{_X}\\left(\\phi(t, \\theta_{-t}(\\sigma); B)\\right) \\leqslant\n\\varepsilon \\quad \\text{for all}~t\\geqslant T.\n\\]\n\\end{definition}\n\nFrom the definitions above, we have the following basic fact.\n\\begin{lemma}\\label{c3.8}\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$. If $\\phi$ has a pullback attractor, then $\\phi$\nhas a bounded pullback absorbing set and $\\phi$ is pullback\n$\\kappa$-contracting.\n\\end{lemma}\n\nWe introduce another definition, needed for characterizations of\nexistence of pullback attractors later.\n\\begin{definition}\\label{d3.2}\n(\\textbf{Nested   pullback absorbing set})\\\\\nA family of bounded sets $\\mathscr{B} = \\{B_\\sigma\\}_{\\sigma\\in\n\\Sigma}$ of $X$ is called a nested bounded pullback absorbing set\nfor $\\phi$ w.r.t. $\\theta$ if $\\mathscr{B}$ is a bounded pullback\nabsorbing set, and, moreover, $B_{\\sigma}$ satisfy the nested\nrelation: $B_{\\theta_{-t}(\\sigma)}\\subset B_{\\sigma}$ for any\n$t\\geqslant 0$ and any $\\sigma \\in \\Sigma$.\n\\end{definition}\n\n\\begin{remark}\nThis nested relation appears in some systems arising in physical\napplications.\n For example, the non-autonomous systems considered in \\cite{CKR,Ch,\nCV}  have   nested bounded pullback absorbing sets.\n\n\\end{remark}\n\n\nIn the following, we will present some characterizations for the\npullback $\\kappa$-contracting cocycles.\n\\begin{lemma}\\label{l3.9}\nLet $\\phi$ be a $\\kappa$-contracting cocycle w.r.t. $\\theta$ on\n$\\mathbb{R}^+\\times \\Sigma \\times X$ and have a nested bounded\npullback absorbing set $\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma \\in\n\\Sigma}$. Then for every $\\sigma\\in \\Sigma$,   every bounded\nsequence $\\{x_n\\}_{n=1}^{\\infty}\\subset X$ and every time sequence\n$\\{t_n\\}\\subset \\mathbb{R}^+$ with $t_n\\to +\\infty$ as $n \\to\n\\infty$, we have\n\\begin{enumerate}\n\\item[(i)] $\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,\nx_n)\\}_{n=1}^{\\infty}$ is pre-compact in $X$;\n\n\\item[(ii)] all clusters of $\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,\nx_n)\\}_{n=1}^{\\infty}$ are contained in\n$\\omega_{\\sigma}(B_{\\sigma})$, that is, if\n\\begin{equation*}\n\\phi(t_{n_j},\\,\\theta_{-t_{n_j}}(\\sigma);\\, x_{n_j}) \\to x_0 \\quad\n\\text{as}~j \\to \\infty,\n\\end{equation*}\nthen $x_0\\in \\omega_{\\sigma}(B_{\\sigma})$;\n\n\\item[(iii)] $\\omega_{\\sigma}(B_{\\sigma})$ is nonempty and compact in $X$.\n\\end{enumerate}\n\\end{lemma}\n\n\n{\\bf Proof.} $(i).$ Denote $\\{x_n\\}_{n=1}^{\\infty}$ by $B$. For\nany $\\varepsilon>0$ and for each $\\sigma \\in \\Sigma$, by the\ndefinition of pullback $\\kappa$-contracting cocycle, we know that\nthere exists a $T_0=T_0(\\varepsilon,\\,\\sigma,B_{\\sigma}) >0$ such\nthat\n\\begin{equation}\\label{3.1}\n\\kappa_{_X}\\left(\\phi(t, \\theta_{-t}(\\sigma); B_{\\sigma})\\right)\n\\leqslant \\varepsilon \\quad \\text{for all}~t\\geqslant T_0\n\\end{equation}\nand there exists also a $T_1=T_1(\\varepsilon, \\, \\sigma,B)$ such\nthat\n\\begin{equation}\\label{3.2}\n\\phi(t+T_1,\\,\\theta_{-(t+T_1)}(\\theta_{-T_0}(\\sigma));\\, B) \\subset\nB_{\\theta_{-T_0}(\\sigma)} \\subset B_{\\sigma}\\quad \\text{for all}~\nt\\geqslant 0.\n\\end{equation}\nHence, for any $t\\geqslant 0$, we have\n\\begin{align}\\label{3.3}\n\\phi(t+T_1+T_0,\\,& \\theta_{-(t+T_1+T_0)}(\\sigma);\\, B)\n\\nonumber \\\\\n& = \\phi(T_0,\\,\\theta_{-T_0}(\\sigma);\\,\n\\phi(t+T_1,\\,\\theta_{-(t+T_1)}(\\theta_{-T_0}(\\sigma)); B))\n\\nonumber \\\\\n& \\subset \\phi(T_0,\\,\\theta_{-T_0}(\\sigma);\\,\nB_{\\theta_{-T_0}(\\sigma)}) \\nonumber \\\\\n& \\subset \\phi(T_0,\\,\\theta_{-T_0}(\\sigma);\\, B_{\\sigma}),\n\\end{align}\nand then\n\\begin{equation}\\label{3.4}\n\\bigcup_{t\\geqslant T_0+T_1}\\phi(t, \\theta_{-t}(\\sigma);\\, B)\n\\subset \\phi(T_0,\\,\\theta_{-T_0}(\\sigma);\\, B_{\\sigma}).\n\\end{equation}\n\nTherefore, combining \\eqref{3.1} and \\eqref{3.4}, we have\n\\begin{equation}\\label{3.5}\n\\kappa_{_X}\\left(\\bigcup_{t\\geqslant T_0+T_1}\\phi(t,\n\\theta_{-t}(\\sigma);\\, B)\\right) \\leqslant \\varepsilon.\n\\end{equation}\n\nThen by the properties $(1),(2)$ of $Lemma$ \\ref{l2.2} and\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\, x_n)\\}_{n=n_0}^{\\infty}\n\\subset \\bigcup_{t\\geqslant T_0+T_1}\\phi(t, \\theta_{-t}(\\sigma);\\,\nB)$ for some $n_0$, we know that\n$\\kappa_X(\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,\nx_n)\\}_{n=1}^{\\infty}) \\leqslant \\varepsilon$. Hence by the\narbitrariness of $\\varepsilon$ and property $(1)$ of $Lemma$\n\\ref{l2.2}, we conclude that\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\, x_n)\\}_{n=1}^{\\infty}$ is\npre-compact in $X$.\n\n$(ii).$ Let $x_0$ be a cluster of\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\, x_n)\\}_{n=1}^{\\infty}$, we\nneed to show that $x_0\\in \\omega_{\\sigma}(B_{\\sigma})$. Without\nloss of generality, we assume that\n$\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\, x_n) \\to x_0$ as $n\\to\n\\infty$.\n\nWe claim first that for each sequence $\\{s_m\\}_{m=1}^{\\infty}\n\\subset \\mathbb{R}^+$ satisfying $s_m \\to \\infty$ as $m\\to \\infty$,\nwe can find two sequences $\\{t_{n_m}\\}_{m=1}^{\\infty}\\subset\n\\{t_{n}\\}_{n=1}^{\\infty}$ and $\\{y_m\\}_{m=1}^{\\infty}\\subset\nB_{\\sigma}$ satisfying $t_{n_m} \\to \\infty$ as $m\\to \\infty$, such\nthat\n\\begin{equation}\\label{3.6}\n\\phi(s_m,\\,\\theta_{-s_m}(\\sigma);\\,y_m)=\\phi(t_{n_m},\\,\\theta_{-t_{n_m}}(\\sigma);\\,x_{n_m}).\n\\end{equation}\nIndeed, for each $m\\in \\mathbb{N}$, we can take $n_m$ so large that\n$t_{n_m} \\geqslant s_m$ and\n\\[\ny_m\\overset{\\vartriangle}{=} \\phi(t_{n_m}-s_m,\\,\n\\theta_{-(t_{n_m}-s_m)} (\\theta_{-s_m}(\\sigma)); \\,x_{n_m}) \\in\nB_{\\theta_{-s_m}(\\sigma)} \\subset B_{\\sigma}.\n\\]\nTherefore,\n\\begin{align}\\label{3.7}\n\\phi(t_{n_m},\\,& \\theta_{-t_{n_m}}(\\sigma);\\, x_{n_m}) \\nonumber \\\\\n& = \\phi(s_m+(t_{n_m}-s_m),\\,\n\\theta_{-(s_m+(t_{n_m}-s_m))}(\\sigma);\\, x_{n_m})\n\\nonumber \\\\\n& = \\phi(s_m,\\,\\theta_{-s_m}(\\sigma);\\,\\phi(t_{n_m}-s_m,\\,\n\\theta_{-(t_{n_m}-s_m)}(\\theta_{-s_m}(\\sigma));\\,x_{n_m})) \\nonumber \\\\\n& = \\phi(s_m,\\,\\theta_{-s_m}(\\sigma);\\,y_m).\n\\end{align}\n\nHence,\n\\[\n\\lim\\limits_{m\\to \\infty} \\phi(s_m,\\,\\theta_{-s_m}(\\sigma);\\,y_m) =\n\\lim\\limits_{m\\to \\infty}\\phi(t_{n_m},\\,\\theta_{-t_{n_m}}\n(\\sigma);\\,x_{n_m}) =x_0,\n\\]\nand $y_m\\in B_{\\sigma}$ for each $m\\in \\mathbb{N}$, which implies,\nby the definition of $\\omega_{\\sigma}(B_{\\sigma})$, that $x_0 \\in\n\\omega_{\\sigma}(B_{\\sigma})$.\n\n$(iii).$ The fact that    $\\omega_{\\sigma}(B_{\\sigma})$ is\nnonempty is obvious. Substitute $B$ by $B_{\\sigma}$ in\n\\eqref{3.2}-\\eqref{3.5}, we   obtain that there exists a\n$T_2=T_2(\\varepsilon,B_{\\sigma},\\sigma)$ such that\n\\begin{equation*}\n\\kappa_{_X}\\left(\\overline{\\bigcup_{t\\geqslant T_0+T_2}\\phi(t,\n\\theta_{-t}(\\sigma);\\, B_{\\sigma})}\\right) =\n\\kappa_{_X}\\left(\\bigcup_{t\\geqslant T_0+T_2}\\phi(t,\n\\theta_{-t}(\\sigma);\\, B_{\\sigma})\\right) \\leqslant \\varepsilon.\n\\end{equation*}\nThen by the definition of pullback $\\omega$-limit set and property\n$(4)$ of $Lemma$ \\ref{l2.2}, we know that\n$\\omega_{\\sigma}(B_{\\sigma})$ is compact in $X$. $\\hfill\n\\blacksquare$\n\nA criterion for the existence of pullback attractors is then\nobtained by means of $\\kappa$-contraction.\n\n\\begin{theorem}\\label{t3.10}\n(\\textbf{Sufficient condition for existence of pullback attractors})\\\\\n Let\n$\\phi$ be a continuous cocycle w.r.t. $\\theta$ on\n$\\mathbb{R}^+\\times \\Sigma \\times X$. Then $(\\phi, \\theta)$ has a\npullback attractor provided that\n\\begin{enumerate}\n\\item[(i)]\\ $(\\phi, \\theta)$ has a nested bounded pullback absorbing set $\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma \\in \\Sigma}$;\n\n\\item[(ii)]\\ $(\\phi, \\theta)$ is pullback $\\kappa$-contracting.\n\\end{enumerate}\n\\end{theorem}\n{\\bf Proof.} For any $\\sigma \\in \\Sigma$, we consider a family of\n $\\omega$-limit sets $\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma\\in\n\\Sigma}$:\n\\begin{equation*}\n\\omega_{\\sigma}(B_{\\sigma})=\\bigcap_{s\\geq\n0}\\overline{\\bigcup_{t\\geq  s}\\phi(t, \\theta_{-t}(\\sigma);\nB_{\\sigma})}, \\quad \\sigma \\in \\Sigma.\n\\end{equation*}\n\nBy $Lemma$ \\ref{l3.9} we know that $\\omega_{\\sigma}(B_{\\sigma})$ is\nnonempty and compact in $X$ for each $\\sigma \\in \\Sigma$.\n\nIn the following, we will prove that\n$\\mathscr{A}=\\{\\omega_{\\sigma}(B_{\\sigma})\\}_{\\sigma \\in \\Sigma}$\nis a pullback attractor of $(\\phi,\\,\\theta)$, which will be\naccomplished in two steps.\n\n{\\it Claim 1. For each $\\sigma\\in \\Sigma$ and any $B\\in\n\\mathcal{B}$, we have\n\\begin{equation*}\n\\lim\\limits_{t\\to +\\infty} dist_X(\\phi(t,\\, \\theta_{-t}(\\sigma);\nB),\\,\\omega_{\\sigma}(B_{\\sigma})) = 0.\n\\end{equation*}\n}\n\nIn fact, if {\\it Claim 1} is not true, then there exist\n$\\varepsilon_0>0$, $\\{x_n\\}_{n=1}^{\\infty}\\subset B$ and $\\{t_n\\}$\nwith $t_n\\to +\\infty$ as $n\\to \\infty$, such that\n\\begin{equation}\\label{3.8}\ndist_X(\\phi(t_n,\\, \\theta_{-t_n}(\\sigma);\\,\nx_n),\\,\\omega_{\\sigma}(B_{\\sigma})) \\geqslant \\varepsilon_0 \\quad\n\\text{for}~n=1,2,\\cdots.\n\\end{equation}\n\nHowever, thanks to $Lemma$ \\ref{l3.9}, we know that\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,x_n)\\}_{n=1}^{\\infty}$ is\npre-compact in $X$. Without loss of generality, we assume that\n\\begin{equation}\\label{3.9}\n\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,x_n) \\to x_0  \\quad \\text{as}~n\n\\to \\infty.\n\\end{equation}\nThen $x_0 \\in \\omega_{\\sigma}(B_{\\sigma})$, which is a contraction\nwith \\eqref{3.8}. This complete the proof of {\\it Claim 1}.\n\n{\\it Claim 2. $\\mathscr{A}=\\{\\omega_{\\sigma}(B_{\\sigma})\\}_{\\sigma\n\\in \\Sigma}$ is $\\phi$ invariant, that is,\n\\begin{equation*}\n\\phi(t,\\,\\sigma;\\, \\omega_{\\sigma}(B_{\\sigma}))=\n\\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)})\\quad \\text{for\nall}~t\\geqslant 0,~\\sigma\\in \\Sigma.\n\\end{equation*}\n}\n\nWe first take $x \\in \\phi(t,\\,\\sigma;\\,\n\\omega_{\\sigma}(B_{\\sigma}))$.\n\nThen there is a $y\\in \\omega_{\\sigma}(B_{\\sigma})$ such that\n$x=\\phi(t,\\,\\sigma;\\,y)$, and by the definition of $y$, there exist\n$\\{y_n\\}\\subset B_{\\sigma} \\subset B_{\\theta_t(\\sigma)}$ and $t_n$\nwith $t_n \\to \\infty$ as $n \\to \\infty$ such that\n$y=\\lim\\limits_{n\\to\n\\infty}\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,y_n)$.\n\nTherefore, by the continuity of $\\phi$, as $n \\to \\infty$,\n\\begin{equation}\\label{3.10}\n\\phi(t_n+t,\\,\\theta_{-(t_n+t)}(\\theta_t(\\sigma));\\,y_n) =\n\\phi(t,\\,\\sigma;\\,\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,y_n)) \\to\n\\phi(t,\\,\\sigma;\\,y)=x.\n\\end{equation}\n\nOn the other hand, from $Lemma$ \\ref{l3.9}, we know that\n$\\{\\phi(t_n+t,\\,\\theta_{-(t_n+t)}(\\theta_t(\\sigma));\\,y_n)\\}_{n=1}^{\\infty}$\nis pre-compact in $X$. Without loss of generality, we assume that\n\\begin{equation*}\n\\phi(t_n+t,\\,\\theta_{-(t_n+t)}(\\theta_t(\\sigma));\\,y_n) \\to x_0\n\\in \\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)}) \\quad\n\\text{as} ~n\\to \\infty.\n\\end{equation*}\nThen by the uniqueness of limitation, we have $x=x_0$, which\nimplies that $x\\in\n\\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)})$, and thus\n\\begin{equation}\\label{3.11}\n\\phi(t,\\,\\sigma;\\,\\omega_{\\sigma}(B_{\\sigma})) \\subset\n\\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)}).\n\\end{equation}\nNow, we only need to prove the converse inclusion relation.\n\nLet $z\\in \\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)})$. Then\nthere exist $\\{z_n\\}\\subset B_{\\theta_t(\\sigma)}$ and $t_n$ with\n$t_n \\to \\infty$ as $n \\to \\infty$ such that $z=\\lim\\limits_{n\\to\n\\infty}\\phi(t_n,\\,\\theta_{-t_n}(\\theta_t(\\sigma));\\,z_n)$.\n\nSince $\\{z_n\\}\\subset B_{\\theta_t(\\sigma)}$ is bounded, from\n$Lemma$ \\ref{l3.9}, we know that\n$\\{\\phi(t_n-t,\\,\\theta_{-(t_n-t)}(\\sigma);\\,z_n)\\}_{n=1}^{\\infty}$\nis pre-compact in $X$. Without loss of generality, we assume that\n$\\phi(t_n-t,\\,\\theta_{-(t_n-t)}(\\sigma);\\,z_n) \\to x_0 \\in\n\\omega_{\\sigma}(B_{\\sigma})$ as $n \\to \\infty$. Then by the\ncontinuity of $\\phi$, we have\n\\begin{align}\\label{3.12}\n\\phi(t,\\,\\sigma;\\,x_0)\\leftarrow\n\\phi(t,&\\,\\sigma;\\,\\phi(t_n-t,\\,\\theta_{-(t_n-t)}(\\sigma);\\,z_n))\n\\nonumber \\\\\n& =\\phi(t_n, \\, \\theta_{-(t_n-t)}(\\sigma);\\,z_n) \\nonumber \\\\\n& = \\phi(t_n, \\, \\theta_{-(t_n)}(\\theta_t(\\sigma));\\,z_n) \\to z.\n\\end{align}\nHence, $z= \\phi(t,\\,\\sigma;\\,x_0)$ with $x_0\\in\n\\omega_{\\sigma}(B_{\\sigma})$, which implies\n\\begin{equation}\\label{3.13}\n\\omega_{\\theta_t(\\sigma)}(B_{\\theta_t(\\sigma)}) \\subset\n\\phi(t,\\,\\sigma;\\,\\omega_{\\sigma}(B_{\\sigma})).\n\\end{equation}\n\nCombining \\eqref{3.11} and \\eqref{3.13} we know that {\\it Claim 2}\nis true.\n\nFrom {\\it Claim 1} and {\\it Claim 2}, we complete the proof of\n$Theorem$ \\ref{t3.10}. $\\hfill \\blacksquare$\n\n\\begin{remark}\\label{r3.11}\nIn the proof of $Theorem$ \\ref{t3.10}, the continuity of the cocycle\n$\\phi(t,\\,\\sigma;\\,\\cdot):\\, X\\to X$ can be replaced by the\n``weaker\" continuity; see \\eqref{3.10} and \\eqref{3.12}. That is,\nthe above proof holds for norm-to-weak continuous cocycles; see\n\\cite{ZYS} for autonomous cases.\n\\end{remark}\n\n\n\nSimilar to the definition in Caraballo et al \\cite{CLR}, we define\nthe following {\\it pullback asymptotic compactness} for NDS.\n\\begin{definition}\\label{d3.12}\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$. Then $\\phi$ is called pullback asymptotically\ncompact, if for each $\\sigma\\in \\Sigma$,     every bounded\nsequence $\\{x_n\\}_{n=1}^{\\infty}$, and every time sequence\n$\\{t_n\\}\\subset \\mathbb{R}^+$ with $t_n\\to +\\infty$ as $n \\to\n\\infty$, $\\{\\phi(t_n,\\, \\theta_{-t_n}(\\sigma); \\,\nx_n)\\}_{n=1}^{\\infty}$ is pre-compact in $X$.\n\\end{definition}\n\nIn the framework of pullback attractors, the pullback asymptotic\ncompactness may not be equivalent to the $\\kappa$-contraction if\nthe cocycle only has a general bounded pullback absorbing set. In\nfact, we need this bounded pullback absorbing set to satisfy an\nadditional nesting condition; see the next theorem.\n\nFrom $Lemma$ \\ref{l3.9} we know that if $\\phi$ has a nested\nbounded pullback absorbing set, then $\\phi$ being pullback\n$\\kappa$-contracting implies that $\\phi$ being pullback\nasymptotically compact; furthermore, in the proof of $Theorem$\n\\ref{t3.10}, we note that we indeed only used the pullback\nasymptotic compactness. This, combining with $Lemma$ \\ref{c3.8},\nimplies the following criterion.\n\n\\begin{theorem} \\label{t3.13}(\\textbf{Criterion for existence of pullback\nattractor})\\\\ Let $\\phi$ be a norm-to-weak continuous cocycle\nw.r.t. $\\theta$ on $\\mathbb{R}^+\\times \\Sigma \\times X$ such that\n$(\\phi, \\theta)$ has a nested bounded pullback absorbing set. Then\n$(\\phi, \\theta)$ has a pullback attractor if and only if $(\\phi,\n\\theta)$ is pullback $\\kappa$-contracting, or equivalently,\n$(\\phi, \\theta)$ is pullback asymptotically compact.\n\\end{theorem}\nThat is, under the assumption that $(\\phi, \\theta)$ has a nested\nbounded pullback absorbing set, pullback $\\kappa$-contraction is\nequivalent to pullback asymptotic compactness.\n\nOn the other hand, the authors in \\cite{CLR} have proven that\n$(\\phi, \\theta)$ has a pullback attractor provided that $(\\phi,\n\\theta)$ is pullback asymptotically compact and has a bounded\npullback absorbing set (see $Theorem$ 7 of \\cite{CLR}). In fact,\nfrom the definition of pullback attractor, $Lemma$ \\ref{c3.8},\nTheorems \\ref{t3.10} and \\ref{t3.13},  we   observe that these\nconditions are also necessary. We summarize this result in the\nfollowing theorem.\n\n\\begin{theorem}\\label{t3.14}\n(\\textbf{Another criterion for existence of pullback attractor})\\\\\nLet $\\phi$ be a norm-to-weak continuous cocycle w.r.t. $\\theta$ on\n$\\mathbb{R}^+\\times \\Sigma \\times X$. Then $(\\phi, \\theta)$ has a\npullback attractor if and only if $(\\phi, \\theta)$ is pullback\nasymptotically compact and has a bounded pullback absorbing set.\n\\end{theorem}\n\n$Theorem$ \\ref{t3.13} and $Theorem$ \\ref{t3.14} show that pullback\nasymptotically compact is stronger than pullback\n$\\kappa-$contracting to some extent, which is different from the\nautonomous cases. Note also that $Theorem$ 3.14 is a slight\nimprovement of $Theorem$ 7 in \\cite{CLR}, from continuous cocycles\nto   ``weakly\" continuous cocycles (i.e., norm-to-weak continuous\ncocycles).\n\nAlthough we only use the pullback asymptotic  compactness in our\nlater applications in \\S 5, we think that the pullback\n$\\kappa-$contraction criterion for existence of pullback\nattractors for ``weakly\" continuous cocycles (i.e.,\n norm-to-weak continuous cocycles) is of independent interest and will be\nuseful for other non-autonomous dynamical systems. Another reason\nto present the $\\kappa-$contraction criterion here is that we like\nto highlight a difference with the \\emph{autonomous} systems: In\nnon-autonomous systems, the pullback asymptotic  compactness\ncriterion  and the pullback $\\kappa-$contraction criterion, for\nexistence of pullback attractors, are not equivalent unless when\nthere exists a \\emph{nested} bounded absorbing set ($Theorem$\n\\ref{t3.13}).\n\n\n\\medskip\n\nWe also remark that     the definitions and results in this\nsection can be expressed in the framework of \\emph{processes},\ninstead of cocycles, as   in   \\cite{CV}.\n\n\n\n\n\n\\section{A technical method for verifying pullback asymptotic compactness}\n\nWe now present a convenient method for verifying the pullback\nasymptotic compactness for the cocycle generated by\n\\emph{non-autonomous} hyperbolic type of equations, in order to\napply $Theorem$ \\ref{t3.13} to obtain existence of pullback\nattractors in the next section. This method is partially motivated\nby the methods in \\cite{CL2,CL3,Kh} in some sense; see also in\n\\cite{SYZ2}. In \\cite{CL3}, the authors present a general abstract\nframework for asymptotic dynamics of \\emph{autonomous} wave\nequations.\n\n\\begin{definition}(\\cite{SYZ2})\n Let $X$ be a Banach space  and $B$ be a bounded subset of $X$.\nWe call a function $\\psi(\\cdot,\\cdot)$, defined on $X\\times X$, a\ncontractive function on $B\\times B$ if for any sequence\n$\\{x_n\\}_{n=1}^{\\infty}\\subset B$, there is a subsequence\n$\\{x_{n_k}\\}_{k=1}^{\\infty} \\subset \\{x_n\\}_{n=1}^{\\infty}$ such\nthat\n\\begin{equation*}\n\\lim_{k\\to \\infty}\\lim_{l\\to \\infty}\\psi(x_{n_k},\\,x_{n_l})=0 .\n\\end{equation*}\nWe denote the set of all   contractive functions on $B\\times B$ by\n$Contr(B)$.\n\\end{definition}\n\n\\begin{theorem}\\label{t4.2}\n\\textbf{(Technique for verifying pullback asymptotic compactness)}\n\\\\\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$ and have a nested bounded pullback absorbing set\n$\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma \\in \\Sigma}$. Moreover, assume\nthat for any $\\varepsilon >0$ and each $\\sigma\\in \\Sigma$, there\nexist $T=T(B_{\\sigma}, \\varepsilon)$ and\n$\\psi_{T,\\,\\sigma}(\\cdot,\\cdot) \\in Contr(B_{\\sigma})$ such that\n\\begin{equation*}\n\\|\\phi(T,\\,\\theta_{-T}(\\sigma);x)-\\phi(T,\\,\\theta_{-T}(\\sigma);y)\\|\\leqslant\n\\varepsilon + \\psi_{T,\\,\\sigma}(x,\\,y) \\quad \\text{for all}~x,y\\in\nB_{\\sigma},\n\\end{equation*}\nwhere  $\\psi_{T,\\,\\sigma}$ depends on $T$ and $\\sigma$. Then $\\phi$\nis pullback asymptotically compact in $X$.\n\\end{theorem}\n\n{\\bf Proof.} Let $\\{y_n\\}_{n=1}^{\\infty}$ be a bounded sequence of\n$X$ and $\\{t_n\\}\\subset \\mathbb{R}^+$ with $t_n \\to \\infty$ as\n$n\\to \\infty$}. We need to show that\n\\begin{align}\\label{4.1}\n\\text{$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,y_n)\\}_{n=1}^{\\infty}$\nis precompact in $X$ for each $\\sigma\\in \\Sigma$}.\n\\end{align}\n\nIn the following, we will prove that\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,y_n)\\}_{n=1}^{\\infty}$ has a\nconvergent subsequence via diagonal methods (e.g., see \\cite{Kh}).\n\nTaking $\\varepsilon_m>0$ with $\\varepsilon_m \\to 0$ as $m\\to\n\\infty$.\n\nAt first, for $\\varepsilon_1$, by the assumptions, there exist\n$T_1=T_1(\\varepsilon_1)$ and $\\psi_{1}(\\cdot,\\cdot) \\in\nContr(B_{\\sigma})$ such that\n\\begin{equation}\\label{4.2}\n\\|\\phi(T_1,\\,\\theta_{-T_1}(\\sigma);\\,x)-\\phi(T_1,\\,\\theta_{-T_1}(\\sigma);\\,y)\\|\\leqslant\n\\varepsilon_1 + \\psi_1(x,\\,y) \\quad \\text{for all}~x,y\\in\nB_{\\sigma},\n\\end{equation}\nwhere $\\psi_1$ depends on $T_1$ and $\\sigma$.\n\nSince $t_n\\to \\infty$, for such fixed $T_1$, without loss of\ngenerality, we assume that $t_n$ is so large that\n\\begin{equation}\\label{4.3}\n\\phi(t_n-T_1,\\,\\theta_{-(t_n-T_1)}(\\theta_{-T_1}(\\sigma));\\,y_n) \\in\nB_{\\theta_{-T_1}(\\sigma)} \\subset B_{\\sigma}\\quad \\text{for\neach}~n=1,2,\\cdots.\n\\end{equation}\nSet\n$x_n=\\phi(t_n-T_1,\\,\\theta_{-(t_n-T_1)}(\\theta_{-T_1}(\\sigma));\\,y_n)$.\nThen from \\eqref{4.2} we have\n\\begin{align}\\label{4.4}\n\\|\\phi(t_n,\\,\\theta_{-t_n}&(\\sigma);\\,y_n)-\\phi(t_m,\\,\\theta_{-t_m}(\\sigma);\\,y_m)\\|\n\\nonumber \\\\\n&= \\|\\phi(T_1,\\theta_{-T_1}(\\sigma); x_n)\n-\\phi(T_1,\\theta_{-T_1}(\\sigma);x_m)\\| \\leqslant \\varepsilon_1 +\n\\psi_1(x_n,\\,x_m).\n\\end{align}\n\nDue to the definition of $Contr(B_{\\sigma})$ and\n$\\psi_1(\\cdot,\\cdot) \\in Contr(B_{\\sigma})$, we know that\n$\\{x_n\\}_{n=1}^{\\infty}$ has a subsequence\n$\\{x^{(1)}_{n_k}\\}_{k=1}^{\\infty}$ such that\n\\begin{equation}\\label{4.5}\n\\lim_{k\\to \\infty}\\lim_{l\\to\n\\infty}\\psi_1(x^{(1)}_{n_k},\\,x^{(1)}_{n_l}) \\leqslant\n\\frac{\\varepsilon_1}{2},\n\\end{equation}\nand similar to  \\cite{Kh}, we have\n\\begin{align*}\n\\lim_{k\\to \\infty}\\sup_{p\\in \\mathbb{N}}&\n\\|\\phi(t^{(1)}_{n_{k+p}},\\,\\theta_{-t^{(1)}_{n_{k+p}}}(\\sigma);\\,y^{(1)}_{n_{k+p}})\n-\\phi(t^{(1)}_{n_{k}},\\,\\theta_{-t^{(1)}_{n_{k}}}(\\sigma);\\,y^{(1)}_{n_{k}})\\|\n\\nonumber \\\\\n&\\leqslant \\lim_{k\\to \\infty}\\sup_{p\\in \\mathbb{N}}\\limsup_{l\\to\n\\infty}\n\\|\\phi(t^{(1)}_{n_{k+p}},\\,\\theta_{-t^{(1)}_{n_{k+p}}}(\\sigma);\\,y^{(1)}_{n_{k+p}})\n-\\phi(t^{(1)}_{n_{l}},\\,\\theta_{-t^{(1)}_{n_{l}}}(\\sigma);\\,y^{(1)}_{n_{l}})\\|\n\\nonumber \\\\\n& \\qquad + \\limsup_{k\\to \\infty}\\limsup_{l\\to \\infty}\n\\|\\phi(t^{(1)}_{n_{k}},\\,\\theta_{-t^{(1)}_{n_{k}}}(\\sigma);\\,y^{(1)}_{n_{k}})-\\phi(t^{(1)}_{n_{l}},\\,\\theta_{-t^{(1)}_{n_{l}}}(\\sigma);\\,y^{(1)}_{n_{l}})\\|\n\\nonumber \\\\\n&\\leqslant \\varepsilon_1+\\lim_{k\\to \\infty}\\sup_{p\\in\n\\mathbb{N}}\\lim_{l\\to\n\\infty}\\psi_1(x^{(1)}_{n_{k+p}},\\,x^{(1)}_{n_l}) +\\varepsilon_1+\n\\lim_{k\\to \\infty}\\lim_{l\\to\n\\infty}\\psi_1(x^{(1)}_{n_k},\\,x^{(1)}_{n_l}),\n\\end{align*}\nwhich, combining with \\eqref{4.4} and \\eqref{4.5}, implies that\n\\begin{align*}\n\\lim_{k\\to \\infty}\\sup_{p\\in \\mathbb{N}}\n\\|\\phi(t^{(1)}_{n_{k+p}},\\,\\theta_{-t^{(1)}_{n_{k+p}}}(\\sigma);\\,y^{(1)}_{n_{k+p}})\n-\\phi(t^{(1)}_{n_{k}},\\,\\theta_{-t^{(1)}_{n_{k}}}(\\sigma);\\,y^{(1)}_{n_{k}})\\|\n\\leqslant 4\\varepsilon_1.\n\\end{align*}\n\nTherefore, there is a $K_1$ such that\n\\begin{align*}\n\\|\\phi(t^{(1)}_{n_{k}},\\,\\theta_{-t^{(1)}_{n_{k}}}(\\sigma);\\,y^{(1)}_{n_{k}})\n-\\phi(t^{(1)}_{n_{l}},\\,\\theta_{-t^{(1)}_{n_{l}}}(\\sigma);\\,y^{(1)}_{n_{l}})\\|\n\\leqslant 5\\varepsilon_1 \\quad \\text{for all}~k,l\\geqslant K_1.\n\\end{align*}\n\nBy induction, we obtain that, for each $m\\geqslant 1$, there is a\nsubsequence \\linebreak\n$\\{\\phi(t^{(m+1)}_{n_{k}},\\,\\theta_{-t^{(m+1)}_{n_{k}}}(\\sigma);\\,y^{(m+1)}_{n_{k}})\\}_{k=1}^{\\infty}$\nof\n$\\{\\phi(t^{(m)}_{n_{k}},\\,\\theta_{-t^{(m)}_{n_{k}}}(\\sigma);\\,y^{(m)}_{n_{k}})\\}_{k=1}^{\\infty}$\nand certain $K_{m+1}$ such that\n\\begin{equation*}\n\\|\\phi(t^{(m+1)}_{n_{k}},\\,\\theta_{-t^{(m+1)}_{n_{k}}}(\\sigma);\\,y^{(m+1)}_{n_{k}})-\\phi(t^{(m+1)}_{n_{l}},\\,\\theta_{-t^{(m+1)}_{n_{l}}}(\\sigma);\\,y^{(m+1)}_{n_{l}})\\|\n\\leqslant 5\\varepsilon_{m+1} \\quad \\text{for all}~k,l\\geqslant\nK_{m+1}.\n\\end{equation*}\n\nNow, we consider the diagonal subsequence\n$\\{\\phi(t^{(k)}_{n_{k}},\\,\\theta_{-t^{(k)}_{n_{k}}}(\\sigma);\\,y^{(k)}_{n_{k}})\\}_{k=1}^{\\infty}$.\nSince for each $m\\in \\mathbb{N}$,\n$\\{\\phi(t^{(k)}_{n_{k}},\\,\\theta_{-t^{(k)}_{n_{k}}}(\\sigma);\\,y^{(k)}_{n_{k}})\\}_{k=m}^{\\infty}$\nis a subsequence of\n$\\{\\phi(t^{(m)}_{n_{k}},\\,\\theta_{-t^{(m)}_{n_{k}}}(\\sigma);\\,y^{(m)}_{n_{k}})\\}_{k=1}^{\\infty}$,\nthen,\n\\begin{equation*}\n\\|\\phi(t^{(k)}_{n_{k}},\\,\\theta_{-t^{(k)}_{n_{k}}}(\\sigma);\\,y^{(k)}_{n_{k}})-\n\\phi(t^{(l)}_{n_{l}},\\,\\theta_{-t^{(l)}_{n_{l}}}(\\sigma);\\,y^{(l)}_{n_{l}})\\|\n\\leqslant 5\\varepsilon_{m} \\quad \\text{for all}~k,l\\geqslant\n\\max\\{m,K_{m}\\},\n\\end{equation*}\nwhich, combining with $\\varepsilon_m \\to 0$ as $m\\to \\infty$,\nimplies that\n$\\{\\phi(t^{(k)}_{n_{k}},\\,\\theta_{-t^{(k)}_{n_{k}}}(\\sigma);\\,y^{(k)}_{n_{k}})\\}_{k=1}^{\\infty}$\nis a Cauchy sequence in $X$. This shows that\n$\\{\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,y_n)\\}_{n=1}^{\\infty}$ is\nprecompact in $X$ for each $\\sigma\\in \\Sigma$. We thus complete\nthe proof.\n $\\hfill \\blacksquare$ \\\\\n\nNote that the nested properties and the contractive properties are\nonly used in \\eqref{4.3} and \\eqref{4.2} respectively. We have a\nsimilar corollary for the cocycle without nested pullback absorbing\nset, the proof is similar to that for $Theorem$ \\ref{t4.2} above.\n\\begin{corollary}\\label{c4.3}\nLet $\\phi$ be a cocycle w.r.t. $\\theta$ on $\\mathbb{R}^+\\times\n\\Sigma \\times X$ and have a bounded pullback absorbing set\n$\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma \\in \\Sigma}$. Moreover, assume\nthat for any $\\varepsilon >0$ and each $\\sigma\\in \\Sigma$, there\nexist $T=T(\\sigma, \\varepsilon)$ and $\\psi_{T,\\,\\sigma}(\\cdot,\\cdot)\n\\in Contr(B_{\\theta_{-T}(\\sigma)})$ such that\n\\begin{equation*}\n\\|\\phi(T,\\,\\theta_{-T}(\\sigma);x)-\\phi(T,\\,\\theta_{-T}(\\sigma);y)\\|\\leqslant\n\\varepsilon + \\psi_{T,\\,\\sigma}(x,\\,y) \\quad \\text{for all}~x,y\\in\nB_{\\theta_{-T}(\\sigma)},\n\\end{equation*}\nwhere $\\psi_{T,\\,\\sigma}$ depends on $T$ and $\\sigma$. Then $\\phi$\nis pullback asymptotically compact in $X$.\n\\end{corollary}\n\n\\section{Pullback attractors for a non-autonomous wave equation}\n\nIn this section, we prove the existence of the pullback attractor\nfor the non-autonomous wave system \\eqref{1.1}-\\eqref{1.3}, by\napplying $Theorems$ \\ref{t3.13} and \\ref{t3.14}. We use the method\n(via contractive functions) in \\S 4 to verify the pullback\nasymptotic compactness. This method appears to be very efficient for\nnon-autonomous wave or hyperbolic equations, while the approach in\n\\cite{CLR}, which is an energy method and is different from ours, is\nvery appropriate for some non-autonomous parabolic equations or wave\nequations with \\emph{linear} damping, e.g., see \\cite{MRW}. In fact,\nthe approach in \\cite{CLR} is an energy method and may be seen as a\nnon-autonomous generalization of  Ball's method \\cite{Ba}.\n\n\n\n\n\\subsection{Mathematic setting}\n\nWe consider the non-autonomous wave system \\eqref{1.1}-\\eqref{1.3}\non the \\emph{state space} $X=H_0^1(\\Omega)\\times L^2(\\Omega)$.\n For each $g_0\\in L_{loc}^2(\\mathbb{R};\\,L^2(\\Omega))$,\nwe denote $\\{g_0(s+t)|t\\in \\mathbb{R}\\}$ by $\\mathcal{H}_1(g_0)$.\nFor $f_0(v,s)$ satisfying \\eqref{1.8}-\\eqref{1.9}, we similarly\ndenote $\\mathcal{H}_2(f_0)=\\{f_0(\\cdot,s+t)|t\\in \\mathbb{R}\\}$.\n\n\nLet $\\Sigma=\\mathcal{H}_2(f_0)\\times \\mathcal{H}_1(g_0)$ be the\nparameter space. We define the driving system  $\\theta_t$:\n$\\Sigma\\to \\Sigma$   by\n\\begin{equation}\\label{5.1}\n\\theta_t(f_0(\\cdot),\\,g_0(\\cdot))=(f_0(t+\\cdot),\\,g_0(t+\\cdot)),\n\\;\\; t\\in \\mathbb{R}.\n\\end{equation}\n\nThen, system \\eqref{1.1}-\\eqref{1.3} is rewritten as the following\nsystem\n\\begin{equation}\\label{5.2}\n\\begin{cases}\nu_{tt}+h(u_t)-\\Delta u+f(u,t+s)=g(x,\\,t+s), (x,t)\\in \\Omega\\times \\mathbb{R}^+,\\\\\nu(x,\\,t)|_{\\partial \\Omega}=0, \\\\\nu(x,0)=u_{0}(x),\\quad u_t(x,0)=v_0(x),\n\\end{cases}\n\\end{equation}\nwhere $s\\in \\mathbb{R}$ means the initial symbol, corresponding to\nsome $\\sigma \\in \\Sigma$.\n\nApplying monotone operator theory or Faedo-Galerkin method, e.g.,\nsee \\cite{CV,Li,Sh}, it is known that conditions\n\\eqref{1.4}-\\eqref{1.9} guarantee the existence and uniqueness of\nstrong solution and generalized solution for\n\\eqref{1.1}-\\eqref{1.3}, and the time-dependent terms make no\nessential complications.\n\n\\begin{lemma} \\label{l5.1} (\\textbf{Well-posedness}) \\\\\nLet $\\Omega$ be a bounded subset of $\\mathbb{R}^3$ with smooth\nboundary, and assume that either Assumption  I or Assumption  II\nholds. Then the non-autonomous system \\eqref{5.2} has a unique\nsolution $(u(t),\\,u_t(t))\\in\n\\mathcal{C}(\\mathbb{R}^+;\\,H_0^1(\\Omega)\\times L^2(\\Omega))$ and\n$\\partial_t^2u(t)\\in L^2_{loc}(\\mathbb{R}^+;\\,H^{-1}(\\Omega))$ for\nany initial data $x_0=(u^{0},\\,u^{1})\\in H_0^1(\\Omega)\\times\nL^2(\\Omega)$ and any initial symbol $\\sigma\\in \\Sigma$.\n\\end{lemma}\n\nBy $Lemma$ \\ref{l5.1}, we can define the cocycle as follows:\n\\begin{equation}\\label{5.3}\n\\begin{cases}\n\\phi:\\quad \\mathbb{R}^+\\times \\Sigma\\times X & \\to X, \\\\\n(t,\\sigma,(u^{0}(x),u^{1}(x))) & \\to\n(u^{\\sigma}(t),u^{\\sigma}_t(t)),\n\\end{cases}\n\\end{equation}\nwhere $(u^{\\sigma}(t),u^{\\sigma}_t(t))$ is the solution of\n\\eqref{1.1} corresponding to initial data $(u^{0}(x),u^{1}(x))$ and\nsymbol $\\sigma=(f_0(s+\\cdot),g_0(s+\\cdot))$; and for each\n$(t,\\,\\sigma)\\in \\mathbb{R}^+\\times \\Sigma$, the mapping\n$\\phi(t,\\,\\sigma;\\,\\cdot): X\\to X$ is continuous.\n\n \\emph{Hereafter, we always denote  by $(\\phi,\\,\\theta)$ the\ncocycle defined in \\eqref{5.1} and \\eqref{5.3}.}\n\nWe now prove the following main result.\n\n\\begin{theorem}\\label{t5.2} (\\textbf{Existence of pullback\nattractor})\\\\\nLet $\\Omega$ be a bounded domain of $\\mathbb{R}^3$ with smooth\nboundary. Then under either Assumption   I or Assumption  II, the\nNDS $(\\phi,\\,\\theta)$ generated by the weak solutions of\n\\eqref{1.1}-\\eqref{1.3} has a pullback attractor\n$\\mathscr{A}=\\{\\omega_{\\sigma}(B_{\\sigma})\\}_{\\sigma \\in \\Sigma}$.\n\\end{theorem}\n\n\nWe need a few lemmas before proving this theorem.\n\n\n\n\n\n\\subsection{Pullback absorbing sets} \\label{pullback}\n\n\nIn the following, we deal only with the strong solutions of\n\\eqref{1.1}. The generalized solution case then follows easily by\na density argument. We begin   with the following existence result\non a bounded pullback absorbing set.\n\\begin{lemma} \\label{l5.3}(\\textbf{Pullback absorbing set})\\\\\nLet $\\Omega$ be a bounded domain of $\\mathbb{R}^3$ with smooth\nboundary. Then under either Assumption  I or Assumption II, the NDS\n$(\\phi,\\,\\theta)$ has a bounded pullback absorbing set\n$\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma\\in \\Sigma}$.\n\\end{lemma}\n{\\bf Proof.} For each $\\sigma \\in \\Sigma$, we know that $\\sigma$\nis corresponding to some $s_0$ satisfying that\n$\\sigma=(f(v,s_0+t),g(x,s_0+t))$, and\n$\\phi(t,\\theta_{-t_0}(\\sigma);\\,x_0)$ is the solution of the\nfollowing equation at time $t$:\n\\begin{equation}\\label{5.4}\n\\begin{cases}\nu_{tt}+h(u_t)-\\Delta u+f(u,t-t_0+s_0)=g(x,\\,t-t_0+s_0),\\quad (x,t)\\in \\Omega\\times \\mathbb{R}^+,\\\\\n(u(0),\\,u_t(0))=x_0,\\\\\nu|_{\\partial \\Omega}=0.\n\\end{cases}\n\\end{equation}\n\nUnder {\\it Assumption $II$}, we can repeat what have done in the\nproof of [$Theorem$ 1, Haraux\\cite{Ha2}], to obtain that there exist\na $\\rho$ (which depends only on\n$\\|g\\|_{_{L^{\\infty}(\\mathbb{R},L^2(\\Omega))}}$ and the coefficients\nin \\eqref{1.4}-\\eqref{1.9}) and a $T$ (which depends only on\n$\\|g\\|_{_{L^{\\infty}(\\mathbb{R},L^2(\\Omega))}}$, the coefficients in\n\\eqref{1.4}-\\eqref{1.9} and the radius of $B$) such that for any\n$\\sigma\\in \\Sigma$,\n\\begin{equation}\\label{5.5}\n\\|\\phi(t,\\,\\theta_{-t_0}(\\sigma);\\,x_0)\\|_{_X} \\leqslant \\rho \\quad\n\\text{for all}~T\\leqslant t \\leqslant t_0~\\text{and}~x_0 \\in B.\n\\end{equation}\nHence, for {\\it Assumption $II$}, we can take $B_{\\sigma}\\equiv\n\\{x\\in X~|~\\|x\\|_{_X} \\leqslant \\rho\\}$ for each $\\sigma$.\n\nUnder {\\it Assumption $I$}, we can use the methods as that in the\nproof of Chepyzhov and Vishik [\\cite{CV}, $Lemma$ 4.1, $Proposition$\n4.2, P 121-123], obtain also that there exist $C_{\\beta}$ and\n$\\beta$ (which depend only on the coefficients in\n\\eqref{1.4}-\\eqref{1.9} and $C_0$) and a $T$ (which depends only on\n$\\int_{-\\infty}^{s_0}e^{\\beta}\\int_{\\Omega}|g(x,s)|^2dxds$, the\ncoefficients in \\eqref{1.4}-\\eqref{1.9} and the radius of $B$) such\nthat\n\\begin{equation}\\label{5.6}\n\\|\\phi(t,\\,\\theta_{-t_0}(\\sigma);\\,x_0)\\|^2_{_X} \\leqslant\n\\rho_{_{\\sigma,\\beta}}=C_{\\beta}(1+e^{-\\beta\ns_0}\\int_{-\\infty}^{s_0}e^{\\beta s}\\int_{\\Omega}|g(x,s)|^2dxds)\\quad\n\\forall~T\\leqslant t \\leqslant t_0,~x_0 \\in B.\n\\end{equation}\nTherefore, under {\\it Assumption $I$}, we can take\n$B_{\\sigma}=B^{\\beta}_{\\sigma}= \\{x\\in X~|~\\|x\\|^2_{_X} \\leqslant\n\\rho_{_{\\sigma,\\beta}}\\}$. $\\hfill \\blacksquare$\n\n\\begin{remark}\\label{r5.4}\nFrom \\eqref{5.5} we know that under {\\it Assumption $II$}, the NDS\n$(\\phi,\\,\\theta)$ has a nested bounded pullback absorbing set\n$\\mathscr{B}=\\{B_{\\sigma}\\}_{\\sigma\\in \\Sigma}$.\n\\end{remark}\n\n\n\n\n\\subsection{Pullback asymptotic compactness}\n\nWe new prove  the pullback asymptotic compactness.\n\n\\begin{lemma} \\label{l5.5} (\\textbf{Pullback asymptotic compactness}) \\\\\nUnder either Assumption  $I$ or $II$,   for any bounded sequence\n$\\{x_n\\}_{n=1}^{\\infty}\\in \\mathcal{B}$ and $\\sigma\\in \\Sigma$,\nthe sequence\n$\\phi(t_n,\\,\\theta_{-t_n}(\\sigma);\\,x_n\\}_{n=1}^{\\infty}$ is\nprecompact in $X$.\n\\end{lemma}\nThe idea for the proof is similar to that in Chueshov \\& Lasiecka\n\\cite{CL1,CL2,CL3} and Khanmamedov \\cite{Kh}; see also in\n\\cite{SYZ2} for linear damping and autonomous cases.\n\n\n\n\nIn order to prove this lemma on pullback asymptotic compactness, we\nneed to derive  a few energy inequalities; see\n\\eqref{5.19}-\\eqref{5.22} below.\n\\medskip\n\nWe   first present some preliminaries and notations.\n\nFor each $\\sigma \\in \\Sigma$, we know that $\\sigma$ is corresponding\nto some $s_0$ such that $\\sigma=(f(v,s_0+t),g(x,s_0+t))$. For any\n$x_0^i=(u_0^i,\\,v_0^i)\\in X$ ($i=1,2$), let\n$(u_i(t),u_{i_t}(t))=\\phi(t,\\theta_{-t_0}(\\sigma);\\,x_0^i)$ be the\ncorresponding solution of the following equation at time $t$:\n\\begin{equation}\\label{5.7}\n\\begin{cases}\nu_{tt}+h(u_t)-\\Delta u+f(u,t-t_0+s_0)=g(x,\\,t-t_0+s_0),\\quad (x,t)\\in \\Omega\\times \\mathbb{R}^+,\\\\\n(u(0),\\,u_t(0))=x_0^i,\\\\\nu|_{\\partial \\Omega}=0.\n\\end{cases}\n\\end{equation}\nFor convenience, we introduce notations\n\\begin{equation*}\nf_i(t)=f(u_i(t),\\,t-t_0+s_0),\\quad h_i(t)=h(u_{i_t}(t)),\\quad\nt\\geqslant 0, ~~ i=1,2,\n\\end{equation*}\nand\n\\begin{equation*}\nw(t)=u_1(t)-u_2(t).\n\\end{equation*}\n\nThen  $w(t)$ satisfies\n\\begin{equation}\\label{5.8}\n\\begin{cases}\nw_{tt}+h_1(t)-h_2(t)-\\Delta w+ f_1(t)-f_2(t)=0,\\\\\nw|_{\\partial \\Omega}=0,\\\\\n(w(0),\\,w_t(0))=(u_0^1,\\,v_0^1)-(u_0^2,\\,v_0^2).\n\\end{cases}\n\\end{equation}\n\nWe also define an energy functional\n\\begin{equation}\\label{5.9}\nE_w(t)=\\frac 12\\int_{\\Omega}|w(t)|^2+\\frac 12 \\int_{\\Omega}|\\nabla\nw(t)|^2.\n\\end{equation}\n\nSince the pullback attractors obtained in $Lemma$ \\ref{l5.3} are\ndifferent for \\emph{Assumption I} and $II$, in the following we will\ndeduce different estimations.\n\nWe first deal with the case corresponding to \\emph{Assumption II}:\n\n\\emph{Step 1} Multiplying \\eqref{5.8} by $w_t(t)$, and integrating\nover $[s,\\,T]\\times \\Omega$, we obtain\n\\begin{align}\\label{5.10}\nE_w(T) +\\int_s^T\\int_{\\Omega}(h_1(\\tau)-h_2(\\tau))w_t(\\tau)dxd\\tau\n+\\int_s^T\\int_{\\Omega}(f_1(\\tau)-f_2(\\tau))w_t(\\tau)dxd\\tau =\nE_w(s),\n\\end{align}\nwhere $0\\leqslant s \\leqslant T \\leqslant t_0$. Then\n\\begin{align}\\label{5.11}\n\\int_s^T  \\int_{\\Omega}(h_1(\\tau)-h_2(\\tau))w_t(\\tau)dxd\\tau\n\\leqslant E_w(s)-\n\\int_s^T\\int_{\\Omega}(f_1(\\tau)-f_2(\\tau))w_t(\\tau)dxd\\tau.\n\\end{align}\nCombining with $Lemma$ \\ref{l2.3}, we get that  for any $\\delta\n>0$,\n\\begin{align}\\label{5.12}\n\\int_s^T\\int_{\\Omega}|w_t(\\tau)|^2dxd\\tau \\leqslant |T-s|\\delta\n\\cdot mes(\\Omega) +C_{\\delta}E_w(s) -C_{\\delta}\\int_s^T\n\\int_{\\Omega}(f_1-f_2)w_txd\\tau.\n\\end{align}\n\n\\emph{Step 2} Multiplying \\eqref{5.8} by $w(t)$, and integrating\nover $[0,\\,T]\\times \\Omega$, we get that\n\\begin{align}\\label{5.13}\n\\int_0^T & \\int_{\\Omega}|\\nabla w(s)|^2dxds + \\int_{\\Omega}w_t(T)\\cdot w(T) \\nonumber \\\\\n& = \\int_0^T\\int_{\\Omega}|w_t(s)|^2dxds -\n\\int_0^T\\int_{\\Omega}(h_1-h_2)w + \\int_{\\Omega}w_t(0)\\cdot w(0)  -\n\\int_0^T\\int_{\\Omega}(f_1-f_2)w.\n\\end{align}\n\nTherefore, from \\eqref{5.12} and \\eqref{5.13}, we have\n\\begin{align}\\label{5.14}\n2\\int_0^T&E_w(s)ds \\nonumber \\\\\n& \\leqslant 2\\delta T mes(\\Omega)+2 C_{\\delta}E_w(0) - 2 C_{\\delta}\\int_0^T\\int_{\\Omega}(f_1-f_2)w(t) dxds \\nonumber \\\\\n& \\quad - \\int_{\\Omega}w_t(T)w(T) + \\int_{\\Omega}w_t(0)w(0)-\n\\int_0^T\\int_{\\Omega}(h_1-h_2)w-\\int_0^T\\int_{\\Omega}(f_1-f_2)w.\n\\end{align}\n\nIntegrating \\eqref{5.10} over $[0,\\,T]$ with respect to $s$, we have\nthat\n\\begin{align}\\label{5.15}\nTE_w(T)&+\\int_0^T\\int_s^T\\int_{\\Omega}(h_1(\\tau)-h_2(\\tau))w_t(\\tau) dxd\\tau ds \\nonumber \\\\\n&=-\\int_0^T\\int_s^T\\int_{\\Omega}(f_1-f_2)w_tdxd\\tau ds + \\int_0^TE_w(s) ds \\nonumber \\\\\n& \\leqslant - \\int_0^T\\int_s^T\\int_{\\Omega}(f_1-f_2)w_tdxd\\tau ds+ \\delta T mes(\\Omega)+C_{\\delta}E_w(0)  \\nonumber \\\\\n& \\quad -C_{\\delta}\\int_0^T\\int_{\\Omega}(f_1-f_2)w_tdxds -\\frac 12 \\int_{\\Omega}w_t(T)w(T) +\\frac 12\\int_{\\Omega}w_t(0)w(0) \\nonumber \\\\\n& \\quad -\\frac 12 \\int_0^T\\int_{\\Omega}(h_1-h_2)w-\\frac12 \\int_0^T\n\\int_{\\Omega}(f_1-f_2)w.\n\\end{align}\n\n\\emph{Step 3} We will deal with $\\int_0^T\\int_{\\Omega}(h_1-h_2)w$.\nMultiplying \\eqref{5.7} by $u_{i_t}(t)$,   we obtain\n\\begin{align*}\n\\frac 12\\frac{d}{dt}\\int_{\\Omega}(|u_{i_t}|^2+|\\nabla\nu_i|^2)+\\int_{\\Omega}h(u_{i_t})u_{i_t}+\\int_{\\Omega}f(u_i,t+s_i)u_{i_t}=\\int_{\\Omega}g_iu_{i_t},\n\\end{align*}\nwhich, combining with the existence of bounded uniformly absorbing\nset, implies that\n\\begin{align}\\label{5.16}\n\\int_0^T\\int_{\\Omega}h(u_{i_t})u_{i_t} \\leqslant M_T,\n\\end{align}\nwhere the constant $M_T$ depends on $T$ (which is different from the\nautonomous cases). Then, noticing \\eqref{2.1}, we obtain that\n\\begin{equation}\\label{5.17}\n\\int_0^T\\int_{\\Omega}|h(u_{i_t})|^{\\frac{p+1}{p}}dxds \\leqslant M_T.\n\\end{equation}\n\nTherefore, using H$\\ddot{o}$lder inequality, from \\eqref{5.17} we\nhave\n\\begin{align*}\n|\\int_0^T\\int_{\\Omega}h_iw|  \\leqslant\nM_T^{\\frac{p}{p+1}}\\left(\\int_0^T\\int_{\\Omega}|w|^{p+1}\n\\right)^{\\frac{1}{p+1}},\n\\end{align*}\nwhich implies that\n\\begin{align}\\label{5.18}\n|\\int_0^T\\int_{\\Omega}(h_1-h_2)w|  \\leqslant\n2M_T^{\\frac{p}{p+1}}\\left(\\int_0^T\\int_{\\Omega}|w|^{p+1}\n\\right)^{\\frac{1}{p+1}}.\n\\end{align}\n\nHence, combining \\eqref{5.15} and \\eqref{5.18}, we obtain that\n\\begin{align*}\nE_w(T)& \\leqslant \\delta  mes(\\Omega)-\\frac{1}{T} \\int_0^T\\int_s^T\\int_{\\Omega}(f_1(\\tau)-f_2(\\tau))w_t(\\tau)dxd\\tau ds+ \\frac{C_{\\delta}}{T}E_w(0)  \\nonumber \\\\\n& \\quad -\\frac{C_{\\delta}}{T}\\int_0^T\\int_{\\Omega}(f_1(s)-f_2(s))w_t(s)dxds -\\frac{1}{2T} \\int_{\\Omega}w_t(T)w(T) +\\frac{1}{2T}\\int_{\\Omega}w_t(0)w(0) \\nonumber \\\\\n& \\quad +\\frac 1T\nM_T^{\\frac{p}{p+1}}\\left(\\int_0^T\\int_{\\Omega}|w(s)|^{p+1}dxds\n\\right)^{\\frac{1}{p+1}}-\\frac{1}{2T} \\int_0^T\n\\int_{\\Omega}(f_1(s)-f_2(s))w(s)dxds\n\\end{align*}\nfor any $0\\leqslant T\\leqslant t_0$.\n\nWe define\n\\begin{align}\\label{5.19}\n&\\psi_{T,\\,\\delta,\\,\\sigma}(x_0^1,\\,x_0^2) \\nonumber \\\\\n&=-\\frac{1}{T} \\int_0^T\\int_s^T\\int_{\\Omega}(f_1(\\tau)-f_2(\\tau))w_t(\\tau)dxd\\tau ds  \\nonumber \\\\\n& \\quad -\\frac{C_{\\delta}}{T}\\int_0^T\\int_{\\Omega}(f_1(s)-f_2(s))w_t(s)dxds -\\frac{1}{2T} \\int_{\\Omega}w_t(T)w(T)  \\nonumber \\\\\n& \\quad +\\frac 1T\nM_T^{\\frac{p}{p+1}}\\left(\\int_0^T\\int_{\\Omega}|w(s)|^{p+1}dxds\n\\right)^{\\frac{1}{p+1}}-\\frac{1}{2T} \\int_0^T\n\\int_{\\Omega}(f_1(s)-f_2(s))w(s)dxds.\n\\end{align}\nThen we   have\n\\begin{equation}\\label{5.20}\nE_w(T) \\leqslant \\delta mes(\\Omega)\n+\\frac{1}{2T}\\int_{\\Omega}w_t(0)w(0)+\n\\frac{C_{\\delta}}{T}E_w(0)+\\psi_{T,\\,\\delta,\\,\\sigma}(x_0^1,\\,x_0^2)\n\\end{equation}\nfor any $\\delta >0$, $0\\leqslant T\\leqslant t_0$.\n\n\\medskip\n\nFor the case corresponding to \\emph{Assumption I}:\n\nSince under our general assumption \\eqref{1.10}, as shown in\n\\eqref{5.6}, the pullback attractors may not satisfy the nested\nproperties. Inspired partly by the results in \\cite{CLR,LLR} we will\ndeduce different estimations by the same methods; see \\eqref{5.21}\nand \\eqref{5.22} below.\n\nRepeat \\emph{Step 1} and \\emph{Step 2} above, and just replace the\nmultipliers $w_t(t)$ and $w(t)$ by $e^{\\beta t}w_t(t)$ and $e^{\\beta\nt}w(t)$ respectively, and take into account $\\beta<C_0$, we can\nobtain the following similar estimates\n\\begin{equation}\\label{5.21}\nE_w(T) \\leqslant \\frac{\\alpha}{TC_0}e^{-\\beta\nT}E_w(0)+\\psi'_{T,\\,\\sigma}(x_0^1,\\,x_0^2)\n\\end{equation}\nfor any $\\delta >0$, $0\\leqslant T\\leqslant t_0$, where\n$\\alpha=(C_0+\\beta)\/(C_0-\\beta)$ and\n\\begin{align}\\label{5.22}\n&\\psi'_{T,\\,\\sigma}(x_0^1,\\,x_0^2)\n\\nonumber \\\\\n&=-\\frac{e^{-T\\beta}}{T} \\int_0^T\\int_s^T\\int_{\\Omega}e^{\\beta\n\\tau}(f_1(\\tau)-f_2(\\tau))w_t(\\tau)dxd\\tau ds\n+\\frac{\\alpha}{2T}e^{-\\beta\nT}\\int_{\\Omega}w_t(0)w(0)\\nonumber \\\\\n& \\quad -\\frac{\\alpha}{TC_0}e^{-\\beta\nT}\\int_0^T\\int_{\\Omega}e^{\\beta s}(f_1(s)-f_2(s))w_t(s)dxds -\\frac{C}{T} \\int_{\\Omega}w_t(T)w(T)  \\nonumber \\\\\n& \\quad + M_{_{E_w(0), T,\nC_0,\\beta}}\\left(\\int_0^T\\int_{\\Omega}|w(s)|^{2}dxds\n\\right)^{\\frac{1}{2}}.\n\\end{align}\n\nWith the above energy inequalities, we are now\nready to prove pullback asymptotic compactness. \\\\\n\n{\\bf Proof of $Lemma$ \\ref{l5.5}:} We will deal with\n\\emph{Assumption I} and \\emph{Assumption II} separately.\n\n\\emph{Assumption II}:\n\nFor each $\\sigma\\in \\Sigma$, and for any fixed $\\varepsilon>0$, from\n\\eqref{5.19}, we can take $t_0$ large enough such that\n\\begin{equation}\nE_w(t_0) \\leqslant\n\\varepsilon+\\psi_{t_0,\\,\\delta,\\,\\sigma}(x_0^1,\\,x_0^2)~\\text{for\nall}~x_0^1,\\,x_0^2\\in B_{\\sigma}.\n\\end{equation}\nHence, thanks to $Theorem$ \\ref{t4.2} and $Lemma$ \\ref{l5.3}, it is\nsufficiently to prove that the function\n$\\psi_{t_0,\\,\\delta,\\,\\sigma}(\\cdot,\\,\\cdot)$ defined in\n\\eqref{5.19} belongs to $Contr(B_{\\sigma})$ for each fixed $t_0$.\n\nWe observe from equation \\eqref{5.7} (and also see \\cite{Ha1}) that\nfor any $t_0>0$,\n\\begin{equation}\\label{5.24}\n\\bigcup_{t \\in\n[0,\\,t_0]}\\phi(t,\\,\\theta_{-t_0}(\\sigma);B_{\\sigma})~~\\text{is\nbounded in}~ X,\n\\end{equation}\nand the bound depends only on $t_0$ and $\\sigma$.\n\nLet $(u_n,u_{t_n})$ be the corresponding solution of\n$(u^n_0,v_0^n)\\in B_{\\sigma}$ for problem \\eqref{5.7},\n$n=1,2,\\cdots$. From the observation above, without loss of\ngenerality (or by passing to subsequences), we assume that\n\\begin{equation}\\label{5.25}\nu_n \\to u\\quad  \\star-\\text{weakly in} ~ L^{\\infty}(0,t_0;\\,\nH_0^1(\\Omega)),\n\\end{equation}\n\\begin{equation}\\label{5.26}\nu_n \\to u\\quad  \\text{in} ~L^{p+1}(0,t_0;\\, L^{p+1}(\\Omega)),\n\\end{equation}\n\\begin{equation}\\label{5.27}\nu_{n_t} \\to u_t\\quad  \\star-\\text{weakly in} ~ L^{\\infty}(0,t_0;\\,\nL^2(\\Omega)),\n\\end{equation}\n\\begin{equation}\\label{5.28}\nu_n \\to u\\quad  \\text{in} ~ L^2(0,t_0;\\, L^2(\\Omega))\n\\end{equation}\nand\n\\begin{equation}\\label{5.29}\nu_n(0) \\to u(0)~~\\text{and}~~u_n(t_0) \\to u(t_0)\\quad  \\text{in} ~\nL^4(\\Omega).\n\\end{equation}\nHere we have used the compact embeddings $H_0^1 \\hookrightarrow\nL^{4}$ and $H_0^1 \\hookrightarrow L^{p+1}$ (since $1 \\leq p<5$).\n\nNow, we will deal with each term in \\eqref{5.19} one by one.\n\nFirst, from \\eqref{5.24}, \\eqref{5.29} and \\eqref{5.26} we get that\n\\begin{equation}\\label{5.30}\n\\lim_{n\\to \\infty}\\lim_{m\\to \\infty}\\int_{\\Omega}\n(u_{n_t}(t_0)-u_{m_t}(t_0))(u_n(t_0)-u_m(t_0)) dx = 0,\n\\end{equation}\n\\begin{equation}\\label{5.31}\n\\lim_{n\\to \\infty}\\lim_{m\\to\n\\infty}\\int_0^{t_0}\\int_{\\Omega}|u_n(s)-u_m(s)|^{p+1}dxds=0,\n\\end{equation}\nand from \\eqref{1.8} and \\eqref{5.28}, we further have\n\\begin{equation}\\label{5.32}\n\\lim_{n\\to \\infty}\\lim_{m\\to \\infty}\\int_0^{t_0}\\int_{\\Omega}\n(f(u_n(s),s-t_0+s_0)-f(u_m(s),s-t_0+s_0))(u_n(s)-u_m(s))dxds = 0.\n\\end{equation}\n\nSecond, note that\n\\begin{align*}\n\\int_0^{t_0}\\int_{\\Omega}&\n(u_{n_t}(s)-u_{m_t}(s))(f(u_n(s),s-t_0+s_0)-f(u_m(s),s-t_0+s_0))dxds \\nonumber \\\\\n&  = \\int_0^{t_0}\\int_{\\Omega}\nu_{n_t}(s)f(u_n(s),s-t_0+s_0)+\\int_0^{t_0}\\int_{\\Omega}\nu_{m_t}(s)f(u_m(s),s-t_0+s_0)\\nonumber \\\\\n&\\qquad -\\int_0^{t_0}\\int_{\\Omega}  u_{n_t}(s)f(u_m(s),s-t_0+s_0)-\\int_0^{t_0}\\int_{\\Omega} u_{m_t}(s)f(u_n(s),s-t_0+s_0)\\nonumber \\\\\n& =\\int_{\\Omega} F(u_n(t_0),s_0)-\\int_{\\Omega}\nF(u_n(0),-t_0+s_0)-\\int_0^{t_0}\\int_{\\Omega}F_s(u_n(\\tau),\\tau-t_0+s_0)dxd\\tau\n\\nonumber \\\\\n&\\qquad+\\int_{\\Omega} F(u_m(t_0),s_0)-\\int_{\\Omega}\nF(u_m(0),-t_0+s_0) -\\int_0^{t_0}\\int_{\\Omega}F_s(u_m(\\tau),\\tau-t_0+s_0)dxd\\tau \\nonumber \\\\\n&\\qquad -\\int_0^{t_0}\\int_{\\Omega}\nu_{n_t}(s)f(u_m(s),s-t_0+s_0)-\\int_0^{t_0}\\int_{\\Omega}\nu_{m_t}(s)f(u_n(s)s-t_0+s_0).\n\\end{align*}\nBy   \\eqref{5.25}, \\eqref{5.27}, \\eqref{5.29} and \\eqref{1.8},\ntaking first $m\\to \\infty$ and then $n\\to \\infty$, we obtain that\n\\begin{align}\\label{5.33}\n\\lim_{n\\to \\infty}&\\lim_{m\\to \\infty}\\int_0^{t_0}\\int_{\\Omega}\n(u_{n_t}(s)-u_{m_t}(s))(f(u_n(s),s-t_0+s_0)-f(u_m(s),s-t_0+s_0))dxds \\nonumber \\\\\n&=\\int_{\\Omega} F(u(t_0),s_0)-\\int_{\\Omega}\nF(u(0),-t_0+s_0)-\\int_0^{t_0}\\int_{\\Omega}F_s(u(\\tau),\\tau-t_0+s_0)dxd\\tau\n\\nonumber \\\\\n&\\qquad+\\int_{\\Omega} F(u(t_0),s_0)-\\int_{\\Omega}\nF(u(0),-t_0+s_0) -\\int_0^{t_0}\\int_{\\Omega}F_s(u(\\tau),\\tau-t_0+s_0)dxd\\tau \\nonumber \\\\\n&\\qquad -\\int_0^{t_0}\\int_{\\Omega}\nu_tf(u(s),s-t_0+s_0)-\\int_0^{t_0}\\int_{\\Omega} u_{t}f(u(s),s-t_0+s_0) \\nonumber \\\\\n&=0.\n\\end{align}\nSimilarly, we have\n\\begin{align*}\n\\int_s^{t_0}\\int_{\\Omega}&\n(u_{n_t}(\\tau)-u_{m_t}(\\tau))(f(u_n(\\tau),\\tau-t_0+s_0)-f(u_m(\\tau),\\tau-t_0+s_0))dxd\\tau \\nonumber \\\\\n& =\\int_{\\Omega} F(u_n(t_0),s_0)-\\int_{\\Omega}\nF(u_n(s),s-t_0+s_0)-\\int_s^{t_0}\\int_{\\Omega}F_s(u_n(\\tau),\\tau-t_0+s_0)dxd\\tau\n\\nonumber \\\\\n&\\quad+\\int_{\\Omega} F(u_m(t_0),s_0)-\\int_{\\Omega}\nF(u_m(s),s-t_0+s_0) -\\int_s^{t_0}\\int_{\\Omega}F_s(u_m(\\tau),\\tau-t_0+s_0)dxd\\tau \\nonumber \\\\\n&\\quad -\\int_s^{t_0}\\int_{\\Omega}\nu_{n_t}f(u_m(\\tau),\\tau-t_0+s_0)-\\int_s^{t_0}\\int_{\\Omega}\nu_{m_t}f(u_n(\\tau),\\tau-t_0+s_0).\n\\end{align*}\nSince $|\\int_s^{t_0}\\int_{\\Omega}\n(u_{n_t}(\\tau)-u_{m_t}(\\tau))(f(u_n(\\tau),\\tau-t_0+s_0)-f(u_m(\\tau),\\tau-t_0+s_0))dxd\\tau|$\nis bounded for each fixed $t_0$,   by the Lebesgue dominated\nconvergence theorem, we finally have\n\\begin{align}\\label{5.34}\n\\lim_{n\\to \\infty}&\\lim_{m\\to\n\\infty}\\int_0^{t_0}\\int_s^{t_0}\\int_{\\Omega}\n(u_{n_t}(\\tau)-u_{m_t}(\\tau))(f(u_n(\\tau),\\tau-t_0+s_0)-f(u_m(\\tau),\\tau-t_0+s_0))dxd\\tau ds \\nonumber \\\\\n&= \\int_0^{t_0}\\left(\\lim_{m\\to \\infty}\\lim_{n\\to\n\\infty}\\int_s^{t_0}\\int_{\\Omega}\n(u_{n_t}(\\tau)-u_{n_t}(\\tau))\\right.\n\\nonumber \\\\\n&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad\n(f(u_n(\\tau),\\tau-t_0+s_0)-f(u_m(\\tau),\\tau-t_0+s_0))dxd\\tau\n\\Big)ds \\nonumber \\\\\n&=\\int_0^{t_0} 0ds=0.\n\\end{align}\n\nHence, from \\eqref{5.30}-\\eqref{5.34}, we see that\n$\\psi_{t_0,\\delta,\\sigma}(\\cdot,\\,\\cdot)\\in Contr(B_{\\sigma})$.\n\n\\emph{Assumption I}:\n\nFrom the definition of $\\rho_{_{\\sigma,\\beta}}$ (see \\eqref{5.6}),\nwe have the conclusion: {\\it for each $\\sigma$ and for any\n$\\varepsilon\n>0$, we can take $t_0$ large enough such that $e^{-\\beta\nt_0}\\rho_{_{\\theta_{-t_0}(\\sigma),\\beta}}\\leqslant \\varepsilon$}.\n\nHence, from $Corollary$ \\ref{c4.3} and \\eqref{5.21}, we only need to\nverify that the function $\\psi'_{t_0,\\,\\sigma}(\\cdot,\\,\\cdot)$\ndefined in \\eqref{5.22} belongs to\n$Contr(B_{\\theta_{-t_0}(\\sigma)})$. To this end, we notice that\n$e^{\\beta t}$ is bounded in $[0,t_0]$ and $\\bigcup_{t \\in\n[0,\\,t_0]}\\phi(t,\\,\\theta_{-t_0}(\\sigma);B_{\\theta_{-t_0}(\\sigma)})$\nis bounded in $X$. The remainder is just a repeat of that for\n$\\psi_{t_0,\\,\\delta,\\,\\sigma}(\\cdot,\\,\\cdot)$ above.\n\nThis completes the proof of $Lemma$ \\ref{l5.5}.\n $\\hfill \\blacksquare$\n\n\n\n\n\\subsection{Existence of pullback attractors}\n\nNow we complete the proof of the main result.\n\n {\\bf Proof of\n$Theorem$ \\ref{t5.2}} \\ From $Lemma$ \\ref{l5.3} and $Lemma$\n\\ref{l5.5}, we see that the conditions of $Theorems$ \\ref{t3.13} and\n\\ref{t3.14} are all satisfied respectively and thus we imply the\nexistence of the pullback attractor. $\\hfill \\blacksquare$\n\n\\begin{remark} \\label{kappa}\nIn this section, we obtain the pullback asymptotic compactness for\nthe non-autonomous wave system \\eqref{1.1}-\\eqref{1.3} by the\ntechnique  presented in \\S 4.  This technique is different from the\nmethod in \\cite{CLR}. Due to the existence of nested bounded\npullback absorbing set for \\emph{Assumption II}(Lemma \\ref{l5.3}),\nthe pullback $\\kappa-$contraction is equivalent to pullback\nasymptotic compactness (see $Theorem$ \\ref{t3.13}). Thus, in\nprinciple, we could also use the pullback $\\kappa-$contraction\ncriterion to conclude the existence of pullback attractor, using the\ndecomposition method as in \\cite{Fe,SYZ1} (popular for autonomous\nsystems).\n\n\\end{remark}\n\n\n\n\n\n\n\\section{Some remarks}\n\nIn this paper, we discuss the asymptotic behavior of solutions in\nthe framework of pullback attractors. Another interesting question\nis forward attractors; see \\cite{Ch,CKS} for general discussions\nor \\cite{CKR} for practical applications to wave equations with\ndelays. However, for the forward attraction property to hold, one\nusually needs some uniformity  about the time-dependent terms\n(i.e., about   the symbol spaces \\cite{CV}).  As discussed in\ndetails in \\cite{CLR1,LLR},   for   general non-autonomous\ndissipative systems,  how to obtain the forward attraction\nproperties is an open problem if without this uniformity\nassumption.\n\nFor our problem, under the \\textbf{Assumption  II}   in \\S 1, we\nindeed obtain a bounded uniformly absorbing set in the sense of\n\\cite{CV} in $Lemma$ \\ref{l5.3} (or see Haraux\\cite{Ha1}). If we\nassume further that $g$ satisfies some additional conditions, e.g.,\n$g$ is translation compact or $g\\in\nW^{1,\\infty}(\\mathbb{R};\\,L^2(\\Omega))$, then by the same method, we\ncan verify the family of processes (see \\cite{CV} for more details)\ncorresponding to the non-autonomous wave system\n\\eqref{1.1}-\\eqref{1.3} is uniformly asymptotically compact and thus\nhas a uniform (w.r.t. $\\sigma \\in \\Sigma$) attractor in the sense of\n\\cite{CV}. However, for the case of \\textbf{Assumption I}\n  in \\S 1, it appears difficult to discuss the forward\nattraction  for $g$ satisfying only \\eqref{1.10}.\n\nFor the \\emph{autonomous} case  of \\eqref{1.1}-\\eqref{1.3},\nrecently, Chueshov \\& Lasiecka \\cite{CL3} have shown a   general\nresult   for the existence of global attractor, and they allow $p\n=5$, i.e., the so-called critical interior damping. In their\n\\emph{autonomous} case, it is true that for all $0\\leqslant\ns\\leqslant t$,\n\\begin{equation}\\label{6.1}\n\\int_s^t\\int_{\\Omega}h(u_t)u_tdxd\\tau \\leqslant C_R,\n\\end{equation}\nwhere $C_R$  depends only on the norm of initial data, but\nindependent of time instants $s$ and $t$. However, for our\nnon-autonomous case, this constant may depend on time instants $s$\nand $t$ (e.g., see \\eqref{5.16},\\eqref{5.17}), and thus in our\nproofs, we require (at least, technically) that the growth order of\n$h$ to be strictly less than $5$: $p<5$.\n\nMoreover, in the present paper, we use the {\\it pullback asymptotic\ncompactness} to obtain the existence of pullback attractors of\nnon-autonomous hyperbolic systems. This is mainly based on a\ntechnical method for verifying {\\it pullback asymptotic compactness}\nin \\S 4. However, for other non-autonomous systems\nor using other techniques (e.g., the   decomposition method), the\npullback $\\kappa-$contraction criterion may be more appropriate\nfor proving the existence of pullback attractors.\n\nFinally, we point out that all the contents in this paper can be\nexpressed by the framework of \\emph{processes}, instead of\ncocycles, as   in  \\cite{CKR,CR,CLR1,LLR}; see also \\cite{CV}\n  for more results about processes.\n\\\\\n\n\n\\noindent{\\bf \\Large Acknowledgement}\n\nThe authors would like to thank Edriss Titi and  the referees  for\nhelpful comments and suggestions.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Problem Statement}\n\nThe simulation begins with the reactive gas at rest with the initial condition\n$\\rho_0=p_0=Y_0=1$ and $\\mathbf{v_0}=0$.\nThe domain lies in $x \\in [-3,57]$ and $y \\in [-3,12]$ and reflecting slip walls\nare present on all walls except the exit $x=57$.\nThe heat addition is limited to a circle of radius $R=2$ centered at the origin. \nThe simulation uses a heat of reaction $q=15$, specific heat ratio $\\gamma=1.4$,\nactivation energy $E=13.8$, and pre-exponential factor $B=35$.\nHeat is added between $t_a=0.5$ and $t_b=5.25$.\n\n\n\nThe Parallel Adaptive Wavelet-Collocation Method\n(PAWCM) is used to capture the wide range of scales that are present\n\\cite{Kevlahan2005, Vasilyev2000}.\nThe PAWCM combines second generation wavelets with a prescribed\n error threshold parameter $\\epsilon$ to determine which grid points\nare necessary in order to achieve a prescribed level of accuracy.\nThe hyperbolic solver developed for the PAWCM is used to maintain \nnumerical stability and reduce spurious oscillations across jump\ndiscontinuities~\\cite{Regele2009}. \nThe effective grid resolution for the simulation is $15360\\times3072$.\n\n\n\n\n\n\nFigure \\ref{fig:2D_j9} demonstrates the indirect detonation initiation\nprocess by presenting a series of snapshots of temperature contours\ncorresponding to the same events shown in the video. \nHeat is deposited in a circle in the bottom left hand corner\nfrom $0.5\\le t \\le 5.25$. The rapid deposition of heat creates compression waves\nthat propagate away from the initially heated region. \nBefore $t=2$, the reactants inside the deposition region are\nconsumed in a chemical explosion, which adds additional heat to the deposition\nregion. \nIt is difficult to discern from the contour at $t=2$, but the interface \nbetween the burning and reactive gas forms a rippled surface. It is thought that \nthis is a result of the Darrieus-Landau instability at the burning gas interface\nbecause once the reactants are consumed the growth of surface fluctuations ceases.\n\\begin{figure}[tb]\n \\centering\n   \\includegraphics[width=0.35\\textwidth]{figures\/legend3}\n \\begin{tabular}{cc}\n   \\includegraphics[width=0.465\\textwidth]{figures\/T020_j9-crop_t-red} &\n   \\includegraphics[width=0.465\\textwidth]{figures\/T140_j9-crop_t-red} \\\\\n   \\includegraphics[width=0.465\\textwidth]{figures\/T035_j9-crop_t-red} &\n   \\includegraphics[width=0.465\\textwidth]{figures\/T160_j9-crop_t-red} \\\\\n   \\includegraphics[width=0.465\\textwidth]{figures\/T050_j9-crop_t-red} &\n   \\includegraphics[width=0.465\\textwidth]{figures\/T180_j9-crop_t-red} \\\\\n   \\includegraphics[width=0.465\\textwidth]{figures\/T070_j9-crop_t-red} &\n   \\includegraphics[width=0.465\\textwidth]{figures\/T200_j9-crop_t-red} \\\\\n   \\includegraphics[width=0.465\\textwidth]{figures\/T100_j9-crop_t-red} &\n   \\includegraphics[width=0.465\\textwidth]{figures\/T220_j9-crop_t-red} \\\\\n   \\includegraphics[width=0.465\\textwidth]{figures\/T120_j9-crop_t-red} &\n   \\includegraphics[width=0.465\\textwidth]{figures\/T240_j9-crop_t-red} \\\\\n   \\includegraphics[width=0.46\\textwidth]{figures\/axesbottom2} &\n  \\includegraphics[width=0.46\\textwidth]{figures\/axesbottom2}\n \\end{tabular}\n\\caption{Sequence of temperature contours demonstrate the multidimensional indirect \ndetonation formation process for times $2\\le t \\le 24$.}\n\\label{fig:2D_j9}\n\\end{figure}\n\n\nAt some point shortly after $t=2$ when the compression waves first reflect off\nthe left and bottom walls the compression waves become fully discontinuous shock\nwaves. The reflected and transmitted shocks form Mach stems that propagate in\nthe positive $x$- and $y$-directions. The reflected waves impinge on the \nburnt-unburnt gas interface and induce Richtmyer-Meshkov instabilities, which then\nincrease the fluctuation magnitude at the material interface. This can be initially\nobserved in the $t=3.5$ contour.\n\nAt about $t=2.5$, a second explosion occurs in the lower left corner\nwhen the shock waves reflect off the bottom and left walls and \nraise the pressure in that region in a duration short enough\nthat the temperature rises with pressure. The reactive gas explodes once it has\nreached a sufficient temperature. \n\nIn the $t=5$ frame, the original outward propagating shock wave has reflected off the\nleft and bottom walls and the Mach stems are clearly visible in the temperature\ncontour. On the left wall, the leading edge of the shock wave is just about to\nreflect off the upper wall, starting in the upper left corner.\nWhen reflection occurs on the upper boundary, a hot spot appears in the upper\nleft-hand corner of the channel, characterized by substantial local inertial\nconfinement. This hot spot releases heat and generates compression waves that \npropagate away from the hot spot location. \n\nAt $t=7$, Fig.~\\ref{fig:2D_j9} shows the reflected wave re-enters the reacted region and\nis refracted, which induces an additional longitudinal component to the wave direction.\nThe transverse waves compress and heat previously unreacted fuel pockets, which ignite\nand help produce additional longitudinal waves, as well as sustaining the transverse \nwaves the reverberate off the top and bottom walls.\n\nKelvin-Helmholtz roll-up instabilities are clearly visible in frames $t=[10, 12, 14]$\nat the burnt-unburnt gas interface with a fairly high level of detail. The \nexistence of such a detailed interface serves as an indicator that any numerical\ndiffusion present in the algorithm has been minimized to the point that these \nfeatures are possible to capture.\n\nAt about $t=14$ the heat release rate by the preheated gas begins to escalate.\nThis acceleration in\nheat release can be observed in the temperature contour sequence as the rapid \nconsumption of fuel starting at $t=14$ and ending at $t=24$ with the formation of \nthe over-driven detonation wave emerging from the lead shock front. \n\n\n\\section*{Acknowledgements}\nJ.D.R. would like to thank Guillaume Blanquart for the use of his computational\nresources to perform this simulation.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the centers of many late-type galaxies one finds massive stellar clusters, the nuclear star clusters \n\\citep{Phillips_96,Matthews_97,Carollo_98,Boeker_02}. The nuclear clusters are central light overdensities on a scale of about 5~pc \\citep{Boeker_02}. \nAlso the central light concentration \nof the Milky Way \\citep{Becklin_68}, is a nuclear star cluster \\citep{Philipp_99,Launhardt_02}. \nNuclear clusters are comparably dense as globular clusters, but are typically more massive \\citep{Walcher_05}.\nIn some galaxies the clusters coexist with a supermassive black hole (SMBH), see e.g. \\citet{Graham_09}.\nThe formation mechanism of nuclear stars cluster is debated \\citep{Boeker_10}.\nThere are two main scenarios: on the one hand formation of stars in dense star  clusters, which are possibly globular clusters, followed by cluster \ninfall \\citep{Tremaine_75,Andersen_08,Capuzzo_08}. On the other hand in situ star formation from the cosmological gas inflow \n\\citep{Milsavljevic_04,Emsellem_08}.\n\n\nDue to the proximity of the center of the Milky Way the nuclear cluster of the Milky \nWay can\n be observed in much higher detail than any other nuclear cluster \\citep{Genzel_10}. It is useful to shed light on the properties and the origin\nof nuclear clusters in general.  \nThe access is hampered by the high foreground\n extinction of A$_{\\mathrm{Ks}}=$2.42 \\citep{Fritz_11}. \n Therefore the light profile is uncertain: \\citet{Becklin_68,Haller_96,Philipp_99} find a central light excess with a size of at least 400$\\arcsec$\non top of the bulge. In contrast, \\citet{Graham_09,Schoedel_11a} claim that the nuclear \ncluster transits at 150$\\arcsec$ to the bulge. On a larger scale \\citep{Launhardt_02} find\nthat there is another stellar component between nuclear cluster and bulge,\nan edge-on disk of 3$^{\\circ}$ length, the nuclear disk, which is flattened by a factor five. It corresponds roughly to the central molecular zone \\citet{Launhardt_02}. \nThe flattening is qualitatively confirmed in\n \\citet{Catchpole_90} and \\citet{Alard_01}. Further in, the flattening is less well constrained.\n\\citet{Vollmer_03} mention an ellipsoid of 1.4:1 in the range of about 200$\\arcsec$.\nWithin 70$\\arcsec$ the light distribution seems to be circular \\citep{Schoedel_07} although a quantification is \nmissing therein.    \nThe majority of the stars in the central R$\\approx$2.5 pc are older than 5 Gyrs  \n\\citep{Blum_02,Pfuhl_11}. Only in the center (r$\\approx\\,$0.4 pc) the light\n is dominated by 6 Myrs old stars \\citep{Forrest_87,Krabbe_91,Paumard_06,Bartko_09} \nwith a top-heavy IMF \\citep{Bartko_10,Lu_13}. \n\n\n\nThe mass of the SMBH is well-determined to be $ 4.3 \\times10^6$ M$_{\\odot}$ with an error of less than 10\\% \n\\citep{Gillessen_09,Ghez_09}. In contrast the mass of the nuclear cluster is less well constrained. The mass within \nr$\\leq1$pc is $ 10^6$ M$_{\\odot}$ with about 50 \\%  systematic uncertainty \\citep{Genzel_10}. In the central parsec the potential is dominated by the SMBH which makes measurements of the additional stellar mass  \nmore difficult. \nFurther, possibly due to the surprising core in the profile of the stellar distribution \\citep{Buchholz_09,Do_09,Bartko_10}\nalso recent Jeans modeling attempts \\citep{Trippe_08,Schoedel_09a} to recover the right SMBH mass fail. This possibly biases also the stellar mass determination there. \nAs a result also the newer works of \\citet{Trippe_08} and \\citet{Schoedel_09a} have still about \n50 \\% stellar mass uncertainty in the central parsec, similar to \\citet{Haller_96} and \\citet{Genzel_96} who used fewer radial velocities.\nThe mass determination outside the central parsec is mainly based on relatively few radial velocities of late-type stars, either \nmaser stars \\citep{Lindqvist_92b,Deguchi_04}, or stars with CO band-heads \\citep{Rieke_88,McGinn_89}. With the absence of proper motion information outside the center the extent of anisotropy is there also not well constrained.\nAlso radial velocities of gas in the circumnuclear disk  (CND) were used for mass determinations outside the central \nparsec \\citep{Genzel_85,Serabyn_85,Serabyn_86}.\n\n\nThus, although the central cluster of the Milky Way is the closest\nnuclear cluster, its mass and luminosity profiles are still\npoorly constrained.  Here and in \\citet{Chatzopoulos_14} we improve\nthe constraints on these parameters. In this paper, we first present\nimproved observational data: we extend the area for which all\nthree stellar velocity components are measured, to r$\\approx 4$ pc. We also\nconstruct a surface density map of the nuclear cluster out to r$_{\\mathrm{box}}= 1000\\arcsec$. Then we\npresent a first analysis of the new data, using simple isotropic\nspherical Jeans models. With these assumptions, the analysis is\nrelatively fast and we can easily investigate several systematic effects. Because these models do not fully match the nuclear cluster, we employ\nmore detailed axiymmetric models in \\citet{Chatzopoulos_14} which\nfit the data well.\n\n\nIn Section~\\ref{sec:dataset} we present our data, and describe the extraction of velocities in Section~\\ref{sec:deriv_vel}. In Section~\\ref{sec:lum_prop} we derive \nthe surface\ndensity properties of the nuclear cluster\n and fit it with empirical models. In Section~\\ref{sec:an_method} we describe the kinematic properties mostly qualitatively and\nuse Jeans modeling to estimate the mass of the nuclear cluster.\nWe discuss our results in Section~\\ref{sec:discussion} and conclude in Section~\\ref{sec:conclusions}. Where a distance to the GC needs to be \nassumed, we\nadopt $R_0=8.2\\,$kpc \\citep{Reid_93,Genzel_10,Gillessen_13,Reid_14}.\nThroughout this paper we define the projected distance from Sgr~A* as R and\nthe physical distance from Sgr~A* as r \\citep{Binney_08}.\n\n\n\n\n\n\\section{Data Set}\n\\label{sec:dataset}\n\n\nIn this section we describe the observations used for deriving proper \nmotions, radial velocities, and the luminosity properties of the nuclear \ncluster. \n\n\\subsection{High Resolution Imaging}\n\\label{sec:highres_im}\n\n\nFor deriving proper motions and for determining the stellar \ndensity profile in the center (R$_{\\mathrm{box}}\\approx20\\arcsec$)\nwe use adaptive optics images with a resolution of\n$\\approx 0.080\\arcsec$.\nIn the central parsec we use the same NACO\/VLT images \n\\citep{Lenzen_03,Rousset_03} as described in\n\\citet{Trippe_08} and \\citet{Gillessen_09}. We add images obtained in \nfurther epochs since then, in the 13 mas\/pixel scale matching to the\n\\citet{Gillessen_09} data set and in the 27 mas\/pixel scale extending \nthe \\citet{Trippe_08} data set.\nThe images are listed in Appendix~\\ref{sec:ap_prom_mot}.\n\nFor obtaining proper motions outside\nthe central parsec we use adaptive optics images covering a larger field of view. These \nare four epochs of NACO\/VLT images,\none epoch of MAD\/CAMCAO  at the VLT \\citep{Marchetti_04,Amorim_06} and \none epoch of\nHokupa'a+Quirc \\citep{Graves_98,Hodapp_96} Gemini North images, see Appendix~\\ref{sec:ap_prom_mot}.  \nMost images cover the Ks-band, some are obtained with H-band or narrower \nfilters within the K-band.\nThe VLT images are flat-fielded, bad pixel corrected and sky subtracted.\nIn case of the Gemini data we use the publicly available images\\footnote{Based on the Data Set of the Gemini North Galactic Center Demonstration Science.}. \nThese \nimages are combinations of reduced images with nearly the same pointings.\n\n\n \n\\subsection{Wield Field Imaging}\n\\label{sec:wide_im}\n\n\nTo obtain the structural properties of the nuclear cluster\noutside of the central R$_{\\mathrm{box}}=20\\arcsec$ we use two additional\ndata sets. Here, high resolution is less important, but area coverage and extinction correction are the keys.\n\n\\begin{enumerate}\n\\item Closer to the center we use HST WFC3\/IR data\\footnote{\nBased on observations made with the NASA\/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, \nwhich is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are \nassociated with program 11671  (P.I. A. Ghez).\n}.\nThe central R$_{\\mathrm{box}}\\approx68\\arcsec$ around Sgr~A* are covered in the filters M127, M139 and M153. We use the images in M127 and M153 in our analysis.\n We optimize the data reduction compared to the pipeline in order to achieve Nyquist-sampled final pixels. We use MultiDrizzle to combine the different images in the same filter, with a drop size parameter of 0.6 with boxes and a final pixel \nsize of 60 mas. These choices achieve a high resolution and still samples the image homogeneously enough to avoid pixels without flux.\nWe do not subtract the sky background from the images, since it is \ndifficult to find a source-free region from where one could estimate it.\nSince we use only point source fluxes this does not affect our analysis. \nWe change the cosmic removal parameters to  7.5, 7, 2.3 and 1.9 \nto avoid removal of actual sources. Due to the brightness of the GC sources cosmics are only of minor importance.\nThe final images have an effective resolution of 0.15$\\arcsec$.\n\\item On a larger scale we use the public VISTA Variables in the Via\n Lactea Survey (VVV)  data obtained with VIRCAM \\citep{Saito_12}. We use from data release 1 the central tile 333 in H and Ks-band. \nThe data contain flux calibrated, but not \nbackground subtracted images. The resolution is about 1$\\arcsec$. These images cover more than one square degree around the GC. We only\nuse R$_{\\mathrm{box}}\\approx1000\\arcsec$. \nIn the center the crowding is severe and nearly all sources are saturated in the Ks-band. This is not a limitation, since \nthere we can use the higher resolution images from NACO and WFC3\/IR.\n\\end{enumerate}\n\n\n\\subsection{Spectroscopy}\n\nFor obtaining spectra of the stars we use data cubes obtained with the \nintegral field spectrometer SINFONI \\citep{Eisenhauer_etal2003, \nBonnet_03}.\nWe use data with the combined H+K-band (spectral resolution 1500) and \nK-band (spectral resolution 4000) grating.\nThe spatial scale of the data varies between the smallest pixel scale \n($12.5\\,$mas pixel$^{-1}$ $\\times$ $25\\,$mas pixel$^{-1}$) and the \nlargest scale $125\\,$mas pixel$^{-1}$ $\\times$ $250\\,$mas pixel$^{-1}$. \nThe spatial resolution of the data correspondingly is between 70 \nmas and 2$\\arcsec$. It has been matched at the time of the \nobservations to the stellar density that increases steeply toward\nSgr~A*. Thus, we can detect many sources in the center, and can sample \nalso large areas at large radii.\nWe apply the standard data reduction SPRED \\citep{Abuter_06, \nSchreiber_04} for SINFONI data, including detector calibrations (such as \nbad pixel\ncorrection, flat-fielding, and distortion correction) and cube \nreconstruction. The wavelength scale is calibrated with emission line \nlamps and finetuned with atmospheric\nOH lines.\nFinally, we correct the data for the atmosphere by dividing with a late \nB-star spectrum and multiplying with a blackbody of the same temperature.\n\n\n\n\\section{Determination of Velocities}\n\\label{sec:deriv_vel}\n\nThe primary constraint for estimating the mass of the nuclear cluster \n comes from velocity data. \nAll our velocity dispersions are derived from individual velocities. Wrongly estimated errors for the velocities cause a bias in the calculated dispersions. Therefore, realistic error estimates are essential in our analysis. \n\n\n\n\n\n\n\\subsection{Proper Motions}\n\\label{sec:proper_motion}\n\nWe  combine in this work proper motions obtained from four different data sets: central field, \nextended field, large field and outer field, see Figure~\\ref{fig:_vel_map}.  \\citet{Trippe_08} used the extended field.\n In the central field the stellar crowding is high and a different analysis is needed than further out. Also, \n the number of epochs and their similarity decrease from inside out \nwhich requires a more careful and separated error analysis further out.\nWe give here a short overview of the data and methods used.\nThe details are explained in Appendix~\\ref{sec:ap_prom_mot}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99 \\columnwidth,angle=-90]{f1.eps} \n\\caption{ \nDistribution of stars with radial velocities and proper motions. \nFor the proper motions we combine four different data sets: the central field in the central 2$\\arcsec$. The extended field from 2$\\arcsec$ outwards \nto $\\approx$20$\\arcsec$, the large field from there outwards to $\\approx$40$\\arcsec$ and a separate, outer \nfield in the north. \nThe yellow lines define our \ncoordinate system, shifted Galactic coordinates l$^*$\/b$^*$ \\citep{Deguchi_04, Reid_04}\n where the center is shifted to \nSgr~A*. \n} \n\\label{fig:_vel_map}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{itemize}\n\\item In the central (R$\\leq 2\\arcsec$) we use the same method for astrometry as in \\citet{Gillessen_09}, see  Appendix~\\ref{sec:pm_center}. \nWe now track stars out to r$_{\\mathrm{box}}\\approx2\\arcsec$. With this increased \nfield of view, we more than double the number of stars compared to \\citet{Gillessen_09}.\nWe use 79 stars with a median magnitude of m$_\\mathrm{Ks}=15.45$.\nSome of the old, late-type stars have significant accelerations \\citep{Gillessen_09}, but the curvature of their orbits\nis not important compared to their linear motion, and thus the linear motion approximation is sufficient.\nDue to the sample size the Poisson error of the dispersions dominates all other dispersion errors.\n\\item In the radial range between a box radius of 2$\\arcsec$ and 20$\\arcsec$ (extended field) we expand slightly on the data and method used by \n\\citet{Trippe_08}.  \nWe do not change the field of view and the selection of well isolated stars compared to \\citet{Trippe_08}. \nThe number of stars with velocities has increased a bit due to the addition of new images.\nIn total we have dynamics for 5813 stars in this field. The median magnitude is  m$_\\mathrm{Ks}=15.76$.\nWe obtain a proper motion dispersion of $\\sigma_{\\mathrm{1D}}=2.677 \\pm 0.018$ mas\/yr using all\n stars and averaging the two dimensions.  \nThe error includes only Poisson noise, which is likely the dominating error, because  we measure identical dispersion values and errors for the bright and faint half of \nthe sample, see Appendix~\\ref{sec:tri_fie}. \n\\item Outside a box radius of 20$\\arcsec$ (large field) nearly no proper motions were available, with the exception of a small area in \\citet{Schoedel_09a}, \nwhich however the\nauthors did not use for their analysis. From the images with sufficiently good AO correction we obtain velocities for 3826 stars, see \nAppendix~\\ref{sec:big_fie}. The median magnitude is  m$_\\mathrm{Ks}=15.30$.\nFor the proper motion dispersion we obtain $\\sigma_{\\mathrm{1D}}=2.330 \\pm 0.019$ mas\/yr. \nThe comparison of the dispersion for fainter and brighter stars shows that the error on the dispersion (after subtracting the velocity errors in \nquadrature) of the fainter stars is 2$\\,\\sigma$ larger than for the brighter stars. Since this is barely significant the errors again include only Poisson noise.  \n\\item For expanding our coverage of proper motions to 78$\\arcsec$ we use the outer field. These data do not cover the full circle \naround Sgr~A*, but only a selected field. We have two epochs for this field: \nGemini data from 2000 and NACO data from the 29$^{\\mathrm{th}}$ May\n 2011.  \nWe exclude in this field stars with m$_\\mathrm{Ks}>15.6$, because we obtain a higher velocity dispersion for them than for brighter\n stars, see Appendix~\\ref{sec:out_fie}.\nIn that way we select 633 proper motion stars with a median magnitude of  m$_\\mathrm{Ks}=14.79$.\nFrom these stars  we obtain  $\\sigma_{\\mathrm{1D}}=1.918 \\pm 0.038$ mas\/yr using both dimensions together.\nWe assume again that the Poisson error dominates over other error sources for the stars selected.\n\\end{itemize}\n\n\n\n\\subsection{Radial Velocities}\n\\label{sec:rad_vel}\n\nSince we can resolve the stellar population in the GC, we measure velocities of single stars and obtain\n dispersions by binning stars together. \nWe use two different sources for the velocities (see also Appendix~\\ref{sec:app_rv}):\n\n\n\\begin{itemize}\n\\item Within R$<95\\arcsec$ \nwe use our SINFONI data to measure the radial velocities of the late-type giants in the GC, see Appendix~\\ref{sec:app_our}\n for the details. We obtain radial velocities for 2513 stars.\nThe median velocity error of these velocities is about 8 km\/s as \nwe obtain from comparing independent measurements for multiple covered stars. \nThe line-of-sight velocity dispersion of these stars is  $\\sigma_z=102.2 \\pm 1.4$ km\/s.\nThe velocity errors have less than 1$\\,\\sigma$ influence on the measured dispersion. \nWhen we take into account the uneven distribution in $l^*$ (Figure~\\ref{fig:_vel_map}) with binning (to avoid the influence of rotation)\nwe obtain as the total radial motion of the nuclear cluster \n$6.1 \\pm 3.8$ km\/s. By definition it should be 0.\n This indicates that also our velocity calibration is probably correct. \nIn conclusion it seems likely that Poisson errors are dominating the dispersion uncertainty for our radial velocity sample, and we only include those in the final analysis.\n\n\n\\item\nOutside of 110$\\arcsec$ \n we use radial velocities from the literature \\citep{Lindqvist_92a,Deguchi_04} out to $\\approx\\,$3000$\\arcsec$ (Appendix~\\ref{sec:app_maser}).\nBoth of them used maser radial velocities. We match the two samples and use each star only once. \nAs a side product of this matching, we confirm that the typical velocity uncertainty is less than 3 km\/s as stated in \\citet{Lindqvist_92a}\nand \\citet{Deguchi_04}.\nDue to the big position errors in these radio data it is difficult to find the corresponding IR stars.\nWe therefore exclude from the combined list the eleven stars that overlap spatially with the areas in which we obtained spectra, with the aim of \navoiding using stars twice. Overall we use 261 radial velocities outside the central field. \n\\end{itemize}\n\nThe radial velocity stars sample is not identical with the proper motion stars sample. The majority of the faint proper motions stars have\nno radial velocities and many of the outer radial velocity stars have no proper motion coverage. The radial velocity stars have a median magnitude of m$_{\\mathrm{Ks}}=13.66$, while the proper motion stars are in the median 1.85 magnitudes fainter. We thus use the common (e.g. \\citet{Vandeven_06,Trippe_08,Marel_10}) approach of using different stars for proper motions and radial velocities. Otherwise we would have only 1840 stars in both samples, with an inhomogeneous spatial distribution. The difference in magnitude is likely not a problem, because the stars in both samples are giants. Their different luminosities are mainly caused by different evolutionary stages, not by different ages and masses. Thus, mass dependent effects, like mass segregation, affect both samples in same way and we can safely use all 10368 proper motion and all 2774 radial velocity stars.\n\n\n\\subsection{Sample Cleaning}\n\\label{sec:out_rej}\n\n\nIn order to probe the gravitational potential of the GC out to more than 100$\\arcsec$ we \nneed to use a stellar population in dynamical equilibrium extending over a sufficiently large radial range. \nThe late-type population in the GC is suited.\nA small fraction of the stars in our sample does not belong to this population and is therefore excluded  as far as possible from our analysis. We have the following three exclusion criteria (see Appendix~\\ref{sec:app_clean} for the details):\n\\begin{itemize}\n\\item\n\nThe young stars follow different radial profiles \\citep{Bartko_10} and have different dynamics. \nWe thus exclude the young stars from our sample. These are the early-type stars, the WR-, O- and B-stars \\citep{Paumard_06,Bartko_09,Bartko_10}\nand the red supergiant IRS7,\n that has the same age as the WR\/O-stars of around 6 Myrs  \\citep{Blum_02,Pfuhl_11}.\nThese stars are the most important contamination, especially close to Sgr A*. Due to\nmissing spectra we cannot clean our proper motion sample\ncompletely from these stars. However, we choose the selection criteria such (Appendix~\\ref{sec:app_early})\nthat in all radial ranges not more than about 4\\% of the stars are young. \n\\item\nWe also exclude stars, which due to their low extinction \nbelong to the Galactic disk or bulge, similar to what is done in \\citet{Buchholz_09}, see Appendix~\\ref{sec:app_foreg}.\nBecause in some areas images in a second filter are missing we cannot clean our sample\ntotally from foreground stars.\nSince these stars are not clustered this pollution is nowhere important. \nIntegrated we include maybe about 1\\% foreground stars in the GC sample.\n\\item\nExtreme outliers in velocity are visible in some subsamples (Appendix~\\ref{sec:app_fast}). About  1\\% of the proper motion stars outside of the\n extended field\nare outliers probably caused by measurement flukes. \nIn the maser sample about 5\\% are outliers (\\citet{Lindqvist_92a,Deguchi_04}\n and Section~\\ref{sec:fast_stars}). (See Appendix~\\ref{sec:app_fast} for our procedure of outlier identification.) Their high velocities probably indicate that they belong to another population. \nThe other samples are free from obvious outliers.\n\\end{itemize}\n\n\n\n\\section{Luminosity Properties of the Nuclear Cluster}\n\\label{sec:lum_prop}\n\n\nFor obtaining masses from Jeans modeling it is necessary to know the space density distribution of the tracer population\n\\citep{Binney_08}.\nIn principle the dynamics alone constrain the tracer distribution \\citep{Trippe_08}. However, that constraint is so weak that strong priors are necessary.\nBetter constraints are possibly by using the surface density  \\citep{Binney_08,Genzel_96,Schoedel_09a,Das_11}.\n Given the inhomogeneous and spatially incomplete nature of our dynamics sample, it would be very \ncumbersome (if not impossible) to derive the spatial distribution of tracers from that data set. \n\nIt is easier to derive the tracer surface density of our tracers from other more complete data set.\nThere are two possibilities:\n \\begin{itemize}\n\\item One can use the luminosity profile of the cluster. The usual assumption of a constant mass to light ratio will fail, however, because of the \nyoung stars in the center that dominate the luminosity. Hence, spectral information is needed in addition.\n\\item Another method for the GC cluster is to extract the surface (stellar) density profile. \nIt is also necessary to correct for the early-type stars that contribute an important fraction of all stars in the center.\n\\end{itemize}\n\nWith these data we can obtain the density profile also in radial ranges where we have only very few velocities. Due to projection effects these radii are also important.  \nIn Section~\\ref{sec:number_density} we obtain density maps, the radial profile of the nuclear cluster and also the profile in direction of and \nperpendicular to the Galactic plane. \nIn the Jeans modeling (Section~\\ref{sec:jeans_modelling}) we \nwill actually fit our radial density and dynamics data at once, since also the dynamics yield a weak constraint on the \ntracer profile. However, in order to make this a problem with few parameters, we identify beforehand (Section~\\ref{sec:fitting_profile}) a functional form that gives an\nsatisfying description of the radial distribution of the velocity tracers. \nIn Section~\\ref{sec:flattening} we use the density maps to separate nuclear cluster and nuclear disk.\nFinally, we obtain in Section~\\ref{sec:total_luminosity} the total luminosity of the nuclear cluster.\n\n\n\n\n\\subsection{Deriving Density Profiles}\n\\label{sec:number_density}\n\nFor deriving the light properties  we use three different data sources (Section~\\ref{sec:dataset}).\n In case of the star density we use for all areas the dataset with the highest resolution. In case of the flux density\n we omit the WFC3\/IR data because it is obtained in different filters. \n\n\n\nWe use the following steps to derive the stellar distribution:\n\\begin{itemize}\n\\item We exclude very bright foreground stars and GC clusters like the Arches. \n\\item We correct the star counts for completeness if necessary, see Appendix~\\ref{sec:obt_lum}.\n\\item We exclude stars younger than 10 Myrs from our sample.\n\\item We correct for extinction using two NIR filters. The resulting map is still patchy in some areas because of the optical depth being too high for\n correction.\n\\item We create masks to exclude the emission from these areas. \nThe masks are defined such that the maps appear smooth and symmetric\nin $|l^*|$ and $|b^*|$. \nSince only few pixels are masked out, the overall bias is small.\n\n\\item The areas excluded are masked out for two-dimensional fitting. They are replaced with the average of the other areas at the same $|l^*|$ and $|b^*|$ for creation of radial profiles and for visualization.\n\\end{itemize}\nNot all steps are necessary for all data subsets, and the procedures are described in detail in Appendix~\\ref{sec:obt_lum}.\n\nIn Figure~\\ref{fig:_surf_bri2} we present the profiles obtained.\nThe profiles obtained from the integrated flux and from stellar number counts are \nsimilar but not identical within the errors.\nOn the one hand, in the case of the flux profile the assumption of screen extinction is a simplification.\nOn the other hand due to the high source density in the GC, magnitudes of point sources are less reliable than the extended flux in the GC.\nTo be conservative we use \nboth profiles to fit the mass in our Jeans modeling (Section~\\ref{sec:jeans_modelling}) and \nwe include the scatter between the obtained masses\nin the mass error budget.\nSplitting the number \ncounts based profile into a $|l^*|$ and $|b^*|$ component (Appendix~\\ref{app:meas_flat} and Figure~\\ref{fig:_surf_bri5b}) yields profiles similar to the model of \\citet{Launhardt_02}.\n\n \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f2.eps}\n\\caption{\nRadial distribution of stars. We construct the radial stellar\/flux density profile from \nNACO, WFC3\/IR and VISTA images (in order of increasing field of view).\n} \n\\label{fig:_surf_bri2}\n\\end{center}\n\\end{figure}\n \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f3.eps}\n\\caption{  Stellar density in and orthogonal to the Galactic plane. \nDue to the limited number counts, the density, especially close to the center (R$<68\\arcsec$), is not only measured exactly in that planes, see Appendix~\\ref{app:meas_flat}.\nFrom \\citet{Launhardt_02} we show the model presented in their Figure~12, scaled to our data.\nTheir model does not include Galactic Disk and bulge, which are included in our data.  \n} \n\\label{fig:_surf_bri5b}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Spherical Fitting of the Stellar Density Profile}\n\\label{sec:fitting_profile}\n\n\nWe now fit the density profile assuming spherical symmetry. \nWe relax this assumption in Section~\\ref{sec:flattening}. \n For the Jeans \nmodeling we need a space density profile. Nevertheless, here we firstly parametrize the projected density in order to compare our data set with the \nliterature before we fit the space density. Often we use power laws for first comparisons. The power law of the projected density is defined\nin the following way: $\\Sigma(R)=R^{-\\Delta} $, and the power law of the space density as: $\\rho(r)=r^{-\\delta} $. $\\Delta$ and $\\delta$ are our definitions for power law\nslopes. We use $\\delta_\\mathrm{L}$ for flux and star \ncounts and $\\delta_\\mathrm{M}$ for mass.\n\n\\subsubsection{Nuker models}\n\\label{sec:fitting_profile_nuk}\n\nUsually projected surface densities in the GC were fitted with (broken) power laws \\citep{Genzel_03b}. A generalization to two slopes is the Nuker \nprofile \\citep{Lauer_95}: \n\n \n\\begin{equation}\n\\begin {split}\n\\Sigma(R)=\\Sigma(R_b) 2^{(\\beta-\\Gamma)\/\\alpha} \n\\left(\\frac{R}{R_b}\\right)^{-\\Gamma} \\left[1+\\left(\\frac{R}{R_b}\\right)^{\\alpha}\\right]^{(\\Gamma-\\beta)\/\\alpha}\n\\end{split}\n\\label{eq:nuker_prof}\n\\end{equation}\n\n\nTherein, $R_b$ and $\\Sigma_b$ are the break radius and the density at the break radius. The exponent  $\\Gamma$ is the inner (usually flatter) power law \nslope,  \nand $\\beta$ is the outer (usually steeper) power law slope. The parameter $\\alpha$ is the sharpness of the transition; a large value of $\\alpha$ \nyields a very sharp\ntransition, essentially a broken power law. Using $\\alpha=100$ (fixed) our fits can be compared with the literature.\n\nDue to the break at 220$\\arcsec$ (Figure~\\ref{fig:_surf_bri2}), we restrict the Nuker fits to  $r<220\\arcsec$ (Table~\\ref{tab:_surf_fit1}).\nRows 1 and 2 in Table~\\ref{tab:_surf_fit1} \n give our fits for stellar number counts and the flux profile, respectively. Row 1 can be directly \n compared to the \nliterature. \\citet{Buchholz_09} conducted the largest area (r$<20\\arcsec$) study so far. They obtained\n$\\Gamma=-0.17\\pm0.09 $ and $\\beta=0.70\\pm0.09$ for $R_b=6.0\\arcsec$. This fit is broadly consistent with our data, although we obtain  \nfor this radial range no clear sign for a power law break. The break radius (6$\\arcsec$) of \\citet{Buchholz_09} is smaller than ours, although these authors do not \ncite an error. \nThe binned data of \\citet{Buchholz_09} look similar to our data. The same is true for the data in \\citet{Bartko_10} who do not attempt to fit the\n profile. \nThese works find a very weak increase of the density\nwith radius inside of $\\approx5\\arcsec$ and then a somewhat stronger decrease of the density with radius further out. This is also the \ncase in \\citet{Do_09,Do_13}. \n\\citet{Do_13} used data out to 14$\\arcsec$ and find a single power law with a slope of $\\Delta_\\mathrm{L}=0.16\\pm0.07$. \n\n\nOur data seem to be consistent with a break radius for the late-type stars around 20$\\arcsec$. When counting stars regardless of their age, a smaller break radius of about 8$\\arcsec$ is found \\citep{Genzel_03b,Schoedel_07}. \nFor our large radial coverage, a single power law obviously fails to fit our data, even when we restrict our data to the \nrange for which we have spectral classifications ($R<90\\arcsec$). The best fit broken power law has then $\\chi^2$\/d.o.f$=11.43\/20$ \n(Row~3 in Table~\\ref{tab:_surf_fit1}), while a single power law yields $\\chi^2$\/d.o.f$=53.12\/22$. \n\n\nThis indicates that the \ntransition between the two slopes is softer than for a broken power law. \nWe also test whether a very small $\\alpha$ (very soft transition) can be excluded. However, there are nearly no differences in the $\\chi^2$ for small $\\alpha$; all $\\alpha$ have nearly the same probability. Therefore, we give only upper limits for $\\alpha$: they are $1.15$ and $0.65$ for the flux and star count data, respectively.\nFigure~\\ref{fig:_surf_fit1} shows our data together with some fits.\n\n\nThe older literature used single power law profiles to describe their flux density profiles.  Between 20$\\arcsec$ and 220$\\arcsec$ our data can be fit relatively well by a \nsingle power law of $\\Delta_\\mathrm{L}=0.765 \\pm 0.018$ (stellar density) and of $\\Delta_\\mathrm{L}= 0.915 \\pm 0.015$ (flux density). These slopes show again that the two data sets are not\n consistent.\n\\citet{Becklin_68,Allen_83},\n \\citet{Haller_96} obtain a slope of $\\Delta_\\mathrm{L}=0.8$, while \\citet{Philipp_99} obtain a flatter slope of\n$\\Delta_\\mathrm{L}=0.6$. \n\n\nIn contrast to these data the flux profiles from \\citet{Graham_09}\\footnote{ \\citet{Graham_09} used the profile from \\citet{Schoedel_08a} constructed from public \n2MASS images.} and \\citet{Schoedel_11a} \ndo not follow a single power law, they flatten at about 80$\\arcsec$ \nindicating that the \n bulge dominates already there. Likely this bright bulge is an artifact caused by a missing sky subtraction.\nThe floor level in the profiles of \\citet{Graham_09,Schoedel_11a}, much higher than the bulge floor in the COBE\/DIRBE data of \\citet{Launhardt_02}, fits\nto typical K-backgrounds observation on earth.\nLike  \\citet{Vollmer_03} for 2MASS data  we subtract the light level towards dark clouds as sky. \n None of \\citet{Becklin_68,Allen_83,Haller_96,Philipp_99,Graham_09,Schoedel_11a} use two color information to correct for extinction in contrast to our work.\nThe single power law fit of \\citet{Catchpole_90} to extinction corrected\nstar counts with a slope of $\\Delta_\\mathrm{L}=1.1$ in the radial range from 140$\\arcsec$ to 5700$\\arcsec$\n is not well comparable to our data set due to the different radial range. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f4.eps} \n\\caption{Nuker fits (with $\\alpha=1$) to the late-type stars surface density, either their star \ndensity or their flux density.\nWe present here fits to the inner data (r$<220\\arcsec$, filled dots). The outer data outside the break is presented by \nopen dots.\nWe also plot the fits of \\citet{Buchholz_09} and \\citet{Do_13}.\n} \n\\label{fig:_surf_fit1}\n\\end{center}\n\\end{figure}\n\n\n\\begin{deluxetable*}{lllllllll} \n\\tabletypesize{\\scriptsize}\n\\tablecolumns{9}\n\\tablewidth{0pc}\n\\tablecaption{Nuker fits of the surface density profile \\label{tab:_surf_fit1}}\n\\tablehead{ No. &data source & radial range & $\\alpha$ & $\\chi^2$\/d.o.f & R$_b$ [$\\arcsec$] &  $\\Sigma$(R$_b$) & $\\beta$ &  $\\Gamma$ }\n\\startdata\n1&star density & R$<220\\arcsec$ & 100  \t    &26.66\/25\t& 23 $\\pm$ 3  & 0.86 $\\pm$ 0.09\t &  0.771 $\\pm$ 0.018   & 0.277 $\\pm$ 0.054  \\\\\n2&flux density & R$<220\\arcsec$ & 100  \t    &74.11\/50\t& 24 $\\pm$ 2  & 71.2 $\\pm$ 4.6   &  0.934 $\\pm$ 0.016\t & 0.304 $\\pm$ 0.030  \\\\\n3&star density & R$<90\\arcsec$  & 100  \t    &11.43\/20\t& 13 $\\pm$ 3  & 1.11 $\\pm$ 0.15\t &  0.645 $\\pm$ 0.040  & 0.186 $\\pm$ 0.084  \\\\\n4&star density & R$<220\\arcsec$ & 1\t\t    &15.28\/25\t& 21 $\\pm$ 14 & 0.84 $\\pm$ 0.28\t &  0.972 $\\pm$ 0.089\t & 0.059 $\\pm$ 0.146  \\\\\n5&flux density & R$<220\\arcsec$ & 1\t\t    &50.71\/50\t& 10 $\\pm$ 3   & 110 $\\pm$ 20 &  1.048 $\\pm$ 0.040\t & -0.163 $\\pm$ 0.148 \\\\\n\\enddata\n\n\\end{deluxetable*}\n\n\n\n\n\\subsubsection{$\\gamma$-models}\n\\label{sec:fitting_profile_gam}\n\nFor our Jeans-modeling (Section~\\ref{sec:jeans_modelling})  we need a parametrization of the space density profile $\\rho_{\\mathrm{N}}(r)$. This is connected to the observable surface density profile $\\Sigma(R)$\nby the following projection integral:\n\\begin{equation}\n\\begin {split}\n\\Sigma(R)=2 \\int_{R}^{\\infty} \\rho_{\\mathrm{N}}(r) r dr\/\\sqrt{r^2-R^2} \n\\end{split}\n\\label{eq:abel_1}\n\\end{equation}\n\nWe  use the spherical $\\gamma$-model \\citep{Dehnen_93} (This is equivalent to the $\\eta$-model of \\citet{Tremaine_94} under the transformation $\\gamma=3-\\eta$.):\n\n \\begin{equation}\n\\begin {split}\n\\rho_{\\mathrm{N}}(r)= \\frac{3-\\gamma}{4\\,\\pi}\\frac{L}{r^{\\gamma}}\\frac{a}{(r+a)^{4-\\gamma}}\n\\end{split}\n\\label{eq:eta}\n\\end{equation}\n\n\nTherein $L$ is the total flux, respectively the star counts. \n$a$ is the scale of the core of the model. The density slope is  $-\\gamma$  within the core and -4 at $\\infty$.\n The $\\gamma$-model has a known positive distribution function which we employ in \n\\citet{Chatzopoulos_14}.\nMore complex profiles, like for example a three-dimensional Nuker model, can also fit the data. The projected profiles are nearly independent of the parametrization in that case, e.g. for the star counts\nthe $\\Delta\\chi^2$ between the $\\gamma$-model fit and the Nuker fit is only 1.3. However, the Nuker model has more degrees of freedom. Further, complex profiles contain poorly constrained parameters (e.g. $\\alpha$ in Nuker) for\nour data set; they overfit the data.\n\nThe GC light profile has two breaks (Figure~\\ref{fig:_surf_fit1}).\nThe breaks in the profiles around 200$\\arcsec$ are probably a sign of a two component nature of the nuclear\nlight distribution, as suggested by \\citet{Launhardt_02}. \nThey call the inner component the nuclear (stellar) cluster and the outer one the nuclear (stellar) disk,\nin analogy to other galaxies.\nIn contrast, \\citet{Serabyn_96} assumed that the central\n active star forming zone inside the inactive bulge of the Milky Way consists of a single component,\na central stellar cluster of R$=100\\,$pc.\n\nDue to the breaks we cannot fit the full data range with one $\\gamma$-model. We use instead two independent \n$\\gamma$-models. The use of two models is the main reason that we cannot use the Nuker model since many parameters are then ill determined. \nStill the central slope of the outer component is difficult to determine also for $\\gamma$ models\nfrom the data and we fix it to be flat by setting $\\gamma_\\mathrm{outer}=0$. A smaller value would correspond to a central depression, and values $>0.5$ \ncan create profiles in which the outer component dominates again at very small distances. \n We obtain the fits presented in Row 1 and 2 in Table~\\ref{tab:_surf_fit2}.\n \n\n\n\n\n\\begin{deluxetable*}{lllllllll} \n\\tabletypesize{\\scriptsize}\n\\tablecolumns{9}\n\\tablewidth{0pc}\n\\tablecaption{$\\gamma$-model fits of the surface density profile \\label{tab:_surf_fit2}}\n\\tablehead{ No. &data source & radial range &  $\\chi^2\/d.o.f.$ &$L_\\mathrm{inner}$ & $a_\\mathrm{inner}$ & $\\gamma_\\mathrm{inner}$ &  $K_\\mathrm{outer}$ & $a_\\mathrm{outer}$ }\n\\startdata\n1&star density & all R &29.1\/55 & $6.73\\pm0.88 \\cdot10^4$ stars &$194\\pm 33$'' & $0.90\\pm0.11$ &$7.05\\pm1.49 \\cdot10^6$ stars & $3396\\pm 458$'' \\\\\n2&flux density & all R &220.3\/204 & $3.42\\pm0.14 \\cdot10^6$ mJy &  $117\\pm 10$'' & $0.76\\pm0.08$ & $5.48\\pm0.38 \\cdot10^8$ mJy &$3711\\pm 158$''\\\\\n3&star density & R$<220\\arcsec$ & 18.8\/26 & $28.8\\pm3.4 \\cdot10^4$ stars &$626\\pm 93$'' & $1.14\\pm0.06$ &- & - \\\\ \n4&flux density & R$<220\\arcsec$ &96.87\/51 & $14.52\\pm0.83 \\times10^6$ mJy &  $511\\pm 44$'' & $1.21\\pm0.04$ &- & -\\\\\n\\enddata\n\\end{deluxetable*}\n\n \nThe two best fitting profiles for stars and light are similar (Figure~\\ref{fig:_densities_1}) to each other, but again not consistent within their errors\nas already noticed during the Nuker profile fits.\nThe central slope of the cluster is consistently $\\gamma_\\mathrm{inner}=0.83\\pm0.12$.\nHowever, the small error is a consequence of the functional form that we use. Other functions like Nuker(r) yield a range of inner slopes that\nappears to be consistent with the uncertainty reported by \\citet{Do_09}.\n\nThe outer $\\gamma$-model is needed inside of 220$\\arcsec$. Row 3 and 4 in \nTable~\\ref{tab:_surf_fit2} show fits with a single component. \nThe resulting \n $\\gamma_\\mathrm{inner}$ are unrealistically large ($>1$) when comparing with  \\citet{Do_09}.\n To quantify the mutual consistence of the two data set, we fit the two data sets with the best single $\\gamma$ fit of the other allowing only the scaling to change. \nThe star density data gives $\\chi^2\/d.o.f.$=70.0\/28; the flux density data $\\chi^2\/d.o.f.$=194.0\/53.\n\n \n\n \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f5a.eps} \n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f5b.eps} \n\\caption{Space and projected density fits in comparison with the star and flux density data. \nFor illustration purposes the stellar density data and fits are shifted by a factor 105. We fit two $\\gamma$-models. We also show the models of\n\\citet{Genzel_96,Trippe_08,Schoedel_09a},\n and \\citet{Do_13b}.\n} \n\\label{fig:_densities_1}\n\\end{center}\n\\end{figure}\n\n\n\nThe space density models in the literature do not describe our data well, even when we restrict the comparison to the inner 220$\\arcsec$, \nsee Figure~\\ref{fig:_densities_1}. The density model of \\citet{Schoedel_09a} is a bad fit to both of our data sets ($\\chi^2\/d.o.f.=83.37\/28$ and $\\chi^2\/d.o.f.=494.67\/53$, for star and flux density, respectively).\nThe reason is the combination of a large break radius with a relatively small outer slope. It thus overestimates the density further out and underestimates it in the center. \nThe models of \\citet{Trippe_08}  ($\\chi^2\/d.o.f=117.46\/28$ and $\\chi^2\/d.o.f.=142.71\/53$) and especially \\citet{Genzel_96} ($\\chi^2\/d.o.f.=323.04\/28$ and $\\chi^2\/d.o.f.=165.99\/53$) fit especially our flux data better. \nStill their average $\\chi^2\/d.o.f.$ is worse than our average $\\chi^2\/d.o.f.$\nwhen we fit both data sets with the same parameters\nThe reason is that the core radii of \\citet{Genzel_96,Trippe_08} are small. \nThe different profiles of \\citet{Do_13b}, which are very similar to each other, do not fit our data at all, because the outer slope of the cluster is much too steep and the core too large.\nThese discrepancies with our fit are not surprising, since none of these works used such a large radial coverage as we. \\citet{Schoedel_09a} did not fit the density model, it is a fixed input to their modeling. They were guided by recent literature. \\citet{Trippe_08} only fit dynamic data.\n\\citet{Do_13b} fit density data together with dynamic data, but only inside 12$\\arcsec$. That covers essentially only the core.\nThis shows that it is important to use density data over a scale larger than the core of the nuclear cluster since otherwise\nthe properties of the outer profile, which have influence on the core parameters, cannot be determined accurately.\n\n\n\n\n\\subsection{Flattening of the Cluster}\n\\label{sec:flattening}\n\n\n\nOur profiles show that the axis ratio ($q=b\/a$) of the nuclear cluster increases with radius, see Figure~\\ref{fig:_surf_bri5b}.\nWe measure the flattening binwise, see Appendix~\\ref{app:meas_flat}, and Table~\\ref{tab:_flat1}. \nIn the inner most bin,  $|l^*|<68\\arcsec$, $q=0.80\\pm0.04$. There is some indication in the data (Figure~\\ref{fig:_surf_bri5b}), that the flattening is smaller around 40$\\arcsec$ than around 20$\\arcsec$. That dip is consistent with noise. We do\nnot find a relevant a systematic error source, see Appendix~\\ref{app:flat_syst_err}.\nThe flattening increases further outside of our field, as it is visible in large scale IRAC data and in\n \\citet{Launhardt_02}. \nThey model the inner r$_{\\mathrm{box}}=2^{\\circ}$ of the GC and obtain an axis ratio of 0.2 at \naround $l^*=3100\\arcsec$.  \n\n\n\n\\begin{deluxetable}{ll} \n\\tabletypesize{\\scriptsize}\n\\tablecolumns{2}\n\\tablewidth{0pc}\n\\tablecaption{flattening profile \\label{tab:_flat1}}\n\\tablehead{ $|l^*|$ range &  $q=b\/a$ }\n\\startdata\n 0 to 68$\\arcsec$ & $0.80\\pm0.04$\\\\\n 68 to 130$\\arcsec$ & $0.63\\pm0.03$\\\\\n 130 to 248$\\arcsec$ & $0.58\\pm0.05$\\\\\n 248 to 473$\\arcsec$ & $0.45\\pm0.04$\\\\\n 473 to 1030$\\arcsec$ & $0.32\\pm0.03$\\\\\n\\enddata\n\\end{deluxetable}\n\n\n\n\\begin{figure*}\n\\begin{center}\n\n  \\includegraphics[width=1.19 \\columnwidth]{f6_com7.eps}\n\\caption{Map of the stellar density in the inner r$_{\\mathrm{box}}=1000\\arcsec$. All panels use the same\ncolor scale. Upper left: stellar density from VISTA\/WFC3\/NACO star\ncounts, corrected for completeness and extinction. To the fitted surface brightness data we added smooth contours for illustration.\nUpper right: GALFIT fit to the data. The fit consists of two components, which are\nshown in the lower two panels. Lower left: the central component, a n$=1.46$ Sersic profile, with $q=0.80$ slightly flattened;\nlower right: a Nuker profile ($q=0.26$), which is not well constrained since it extends well outside of our field of view.\n} \n\\label{fig:2dim-map}\n\\end{center}\n\\end{figure*}\n\n\nWe can use our two dimensional data to distinguish between nuclear cluster and nuclear disk, where the outer disk component is \nused to estimate the background for the cluster in the following manner.\nIn the central 68$\\arcsec$ we use the \ndensity profiles shown in Figure~\\ref{fig:_surf_bri5b} in the corresponding quadrants.\nFurther out, we use the VISTA data. Figure~\\ref{fig:2dim-map} shows the resulting map.\nFor finding an empirical description we use GALFIT \\citep{Peng_02}.\nIn the first fits we fix the centers to the known location, and enforce alignment of  the\n inner component with the Galactic Plane, but not of the outer component. (Because we measure the density only in two sectors inside of 68$\\arcsec$ alignment with the Galactic Plane is enforced there by construction.) In this fit the outer component is aligned within 1.1$\\pm1.3^{\\circ}$. The uncertainty of that angle is obtained by four trials, in each we mask out the lower\/upper half in l\/b. In contrast to many other parameters that angle is robust. \n Thus, the misalignment is not significant and we fix the angle to be $0^{\\circ}$.\nOur result is consistent with \\citet{Schoedel_14} who obtain both for the nuclear disk, and the nuclear cluster\nalignment with about 2$^{\\circ}$ uncertainty.\n\n \n\nWe fit the data with Sersic \\citep{Sersic_68} and Nuker profiles. Which of them is used is not important, similar sizes and flattening are  obtained with both.\nHowever, it is difficult to disentangle the inner and the outer component. Small changes in $\\chi^2\/d.o.f.$ result in large differences in most parameters.\nMainly because the flattening has a local minimum around $40\\arcsec$, a free fit results in a large Sersic index for the outer components. In that case the outer component contributes relevantly again in the center, which reduces the flattening of the inner component. Then the nuclear cluster has \n$q\\approx0.91$. \nHowever, it is physically unlikely that outer component is more important in the very center than slightly further out. Since in the case of our $\\gamma$-profiles (Section~\\ref{sec:fitting_profile_gam}) the outer component does not contribute relevantly in the center, we assume that the inside 68$\\arcsec$ measured flattening is identical with flattening of the nuclear cluster as a whole.  \nTo obtain this flattening we use for the nuclear disk a Nuker profile with a flat core, $\\alpha=0.6$, $\\beta=3$, $\\Gamma=0$. This set of parameters approximately models a projected $\\gamma$ model\nin its outer and inner slope (Section~\\ref{sec:fitting_profile_gam}). The fit obtained for the disk is q$=0.264$. \nIn the center we use a Sersic model to enable comparison with the recent literature. \nUnder these assumptions we obtain for the inner component $q=0.80\\pm0.05$, n$=1.46\\pm0.05$, R$_e=127 \\pm 10\\arcsec$, and an integrated count uncertainty of 10\\%.   In contrast to many other parameters, the Sersic parameter is well constrained. It is in all cases between 1.4 and 1.6.\n In the very center the outer component contributes 13\\% of the star counts of the inner component.\n The outer component dominates outside about 104$\\arcsec$.\n \nOur half light radius of 5.0 pc is slightly larger than the preferred value of \\citet{Schoedel_14} (R$_e=4.2\\pm0.4$). However, when the outer Sersic is also free, in their two Sersic fits they obtain $6\\pm0.2$ pc. Our axis ratio is somewhat larger then their value of $q=0.71\\pm0.02$ but within the uncertainties. Our Sersic index n is smaller than their of\nn$=2\\pm0.2$. Since they were not able to use the central parsec in their fit, it is likely that our fit is better in that region. The very center is important for the Sersic index, thus our n is probably better. \n Overall from comparing our and their different fits it is probable that both works underestimate the systematic error in component fitting. The main reason for that fact is the existence of two not clearly separated components in the GC and the high extinction towards the region.\n\n\\subsection{Luminosity of the Nuclear Cluster}\n\\label{sec:total_luminosity}\n\n\n\nTo obtain the Ks-luminosity in the GC \nwe integrate the flux of the old stars (Figure~\\ref{fig:_surf_bri2}).\nThe total extinction-corrected flux within R$<100\\arcsec$ is  $1052\\pm200\\,$Jy. The absolute error of 20\\% contains the uncertainty of the extinction law toward the GC (\\citealt{Fritz_11}: 11\\%), the calibration uncertainty (7\\%), and 14\\% for the differences between\n the stellar density and flux density profiles. The latter we obtain from the scatter between the star counts profile and the flux profile scaled to each other. \n \nTo estimate the total luminosity of the nuclear cluster, we need to estimate its size. We use again different methods, to estimate\nthe systematic error. Firstly, we use  two-dimensional decomposition\nof the star counts in nuclear cluster and nuclear disk (Section~\\ref{sec:flattening}).\\footnote{We do not consider other Galactic components, they are very minor in the center. In the model of \\citet{Launhardt_02} (their Figure~2) these contribute 0.35 mJy\/''$^2$ before extinction correction, while we have 2 mJy\/''$^2$ in total at R$=100\\arcsec$. Extinction corrected, their contribution is even smaller.}\nThe fraction of star counts from the inner component is 67\\% and the fraction the nuclear cluster which is within 100$\\arcsec$ is 44\\%.\nThat leads to a total luminosity of 1599 Jy. Using instead the one dimensional $\\gamma$ decomposition of Section~\\ref{sec:fitting_profile_gam} of the counts leads\nto 5058 Jy. The $\\gamma$-model fit of the flux implies 3420 Jy, see Table~\\ref{tab:_surf_fit2}. \nThe main reason for the different fluxes are the different models: a Sersic can, when it is as here close to exponential, decay fast outside its characteristic radius, while a $\\gamma$-model decays only with its fixed power law of -3.\n\n\nWe obtain absolute luminosities using  R$_0=8.2$ kpc and  M$_\\odot(Ks)=3.28$.\nThereby, we use the 3420 Jy as best estimate for the total flux and the other two values for the error range.\nWe obtain \n$7.4^{+3.5}_{-3.9} \\times 10^7\\,$L$_{\\odot}$  \nfor the total nuclear cluster, consistent with the estimate of $ 6\\pm 3 \\times 10^7\\,$L$_{\\odot}$ of\n\\citet{Launhardt_02} and the $4.1\\pm0.4 \\times 10^7\\,$L$_{\\odot}$  at 4.5~$\\micron$ of  \\citet{Schoedel_14}.\n\n\nThe young O(B)-stars in the center, which are not included in our sample, add 25 Jy in the Ks-band. Although they are\nirrelevant for the light in the Ks-band, this is different for bolometric measurement:\nthe bolometric luminosity of the young stars is  about L$_{UV}\\approx10^{7.5}\\,L_{\\odot}$ and $M_{\\mathrm{bol}}\\approx-14.1$ \\citep{Genzel_10,Mezger_96} \nand thus larger than what we obtain for the old stars, $M_{\\mathrm{bol}}\\approx-13.4$ within R$<100\\arcsec$.\nAlso, the young stars are concentrated on a more than $\\approx1000$ times \n smaller volume than the old stars.\n\n\n\n\n\\section{Kinematic Analysis}\n\\label{sec:an_method}\n\n\nWe now use our kinematic data to characterize the properties of the data and to obtain a rough mass estimate from it.\nIn Section~\\ref{sec:_anisotropy},~\\ref{sec:fast_stars} and \\ref{sec:rotation} we discuss anisotropy, \nfast stars, and rotation, respectively. In  Section~\\ref{sec:binning} we explain and justify the binning\nused in our Jeans modeling and in \\citet{Chatzopoulos_14}.\nIn this work we use isotropic spherical symmetric Jeans modeling \\citep{Binney_08} \nas illustrative models of what can be derived from our data (Section~\\ref{sec:jeans_modelling}). \nWith that relatively simple model we can also explore\nmany systematic error sources easily.\nIn \\citet{Chatzopoulos_14} we use two-integral modeling with self-consistent rotation \\citep{Hunter_93},\nwhich allows us to include intrinsic flattening and rotation.\n\n\n\n\\subsection{Velocity Anisotropy}\n\\label{sec:_anisotropy}\n\n \n\nThe cluster could be anisotropic. One type of anisotropy is radial anisotropy, \nwhich would manifest itself as a difference between the dispersions in tangential and radial direction\n \\citep{Leonard_89,Schoedel_09a}. \n\nWe obtain an estimate of the anisotropy from the proper motions from $\\beta'_{\\mathrm{pm}}(R)=1-[\\sigma_\\mathrm{tan\\,PM}(R)\/\\sigma_\\mathrm{rad\\,PM}(R)]^2$, where $\\sigma_\\mathrm{tan}$ and $\\sigma_\\mathrm{rad}$ are the radial and tangential dispersions of the proper motions, respectively.\n$\\beta'_{\\mathrm{pm}}$ has the advantage that it follows directly from measured properties \nwithout modeling\nand does not depend on $R_0$.\nUneven angular sampling of stars together with  the flattening  which causes $\\sigma_{l^*}>\\sigma_{b^*}$ (Figure~\\ref{fig:_std_v}) can mimic anisotropy in the following way:\nconsider a radial bin that is only covered close to the $b^*$-axis. Then, $\\sigma_{\\mathrm{tan}}\\approx \n\\sigma_{l^*}$ and $\\sigma_{\\mathrm{rad}}\\approx \\sigma_{b^*}$. Since $\\sigma_{l^*}>\\sigma_{b^*}$, $\\beta'_{\\mathrm{pm}}<0$ implies tangential anisotropy.\nThe arguments also hold for uneven angular sampling.\n\\citet{Do_13b} had outside the center only covered close to the $b^*$ axis\nthe Galactic Plane. As expected $\\sigma_{\\mathrm{tan}}>\\sigma_{rad}$ in these data. That is the reason that their fit \nprefers $\\beta_\\infty<0$. \n\n\nTo avoid \na spurious influence of the flattening on the anisotropy we firstly restrict the analysis here to \nr$<$40$\\arcsec$ for having full (not necessarily even) angular coverage.\nSecondly, to even out density fluctuations, we obtain the dispersions by taking the average of the dispersions in two angular bins: one within $\\phi<45^{\\circ}$ to the Galactic Plane and the other with $\\phi> 45^{\\circ}$. To even out coverage fluctuations, we give the two bins equal weight.\n We obtain  $\\beta'_{\\mathrm{pm}}=-0.040 \\pm 0.022$ (Figure~\\ref{fig:_disp_aniso}). \nThe scatter between the bins is consistent with the Poisson errors. Similar to what was found by \\citet{Schoedel_09a} we see that \n $\\sigma_{\\mathrm{tan}}$ is somewhat larger in the center. \nHowever, since $\\chi^2\/d.o.f.=33.71\/39$\n shows that this is not significant. \n\\citet{Schoedel_09a} suggested that the increase in the center could be due to pollution with early-type stars, \nwhich are in average on more tangential orbits \\citep{Genzel_00,Bartko_09,Bartko_10}. We test this hypothesis by using only stars with late-type \nspectra. Integrated over the full field this yields $\\beta'_{\\mathrm{pm}}=0.045 \\pm 0.049$. \nThat is smaller than for all stars, but not significantly different. \n\nBetween 2'' and 5''  $\\beta'_{\\mathrm{pm}}=-0.463 \\pm 0.170$\nwhen using all stars and  $\\beta'_{\\mathrm{pm}}=-0.134 \\pm 0.207$ when using only late-type stars. Since these two values\nare again consistent, pollution is probably not important in this radial range. \n\nThe anisotropy parameter $\\beta$ is defined in 3D coordinates (r) \\citep{Binney_08}. $\\beta$ can be only estimated in full modeling which also needs to account for projection effects. \nIn such model, $R_0$ would also need to be fit and it would also be necessary \\citep{Chatzopoulos_14} to use a flattening model to account for the different effect of the flattening on v$_z$, v$_{l^*}$, and v$_{b^*}$. That is beyond the scope of this paper. \nThe deprojected anisotropy parameter $\\beta$ is more different from 0 than $\\beta'_{\\mathrm{pm}}$ \\citep{Marel_10}. Due to the core-like density profile the difference between  $\\beta'$ and $\\beta_{\\mathrm{pm}}$ can be large especially in the center. Because of projection effects, full modeling is required to constrain the\nradial dependency of the anisotropy \\citep{Marel_10}.\nWhile the overall radial anisotropy is small, other deviations from prefect isotropy exist; see Section~\\ref{sec:rotation}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f7.eps} \n\\caption{\nRadial and tangential dispersions as function of the radius. The points are slightly offset in R from each other for better visibility.\n} \n\\label{fig:_disp_aniso}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Rotation and dynamic main axis}\n\\label{sec:rotation}\n\n\nIn Section~\\ref{sec:flattening} we determined the major and minor axes of the\ndistribution of stars in the nuclear cluster. Here we use the new kinematic\ndata to find the rotation axis of the system. First, we use the\nproper motion data.\n\n\\citet{Trippe_08} interpreted the difference between the velocity\ndispersions $\\sigma_l$ and $\\sigma_b$ (Figure~\\ref{fig:_std_v}) as a sign of rotation.\nHowever, $\\sigma_l>\\sigma_b$ is globally required for any axisymmetric star\ncluster flattened parallel to the Galactic plane. This is independent\nof whether the extra kinetic energy along the Galactic plane is due to\nnet rotation or to higher in-plane velocity dispersions; reversing Lz\nfor any orbit does not change $\\sigma_l$.  Furthermore, spherical clusters\nwith rotation \\citep{Lynden-Bell_60} do not show $\\sigma_l>\\sigma_b$.  Therefore\nthe difference between $\\sigma_l$ and $\\sigma_b$ is ultimately due to the\nflattening, even though most of the flattening of the nuclear cluster is\nin fact generated by additional rotational kinetic energy\n\\citep{Chatzopoulos_14}.\n\nHowever, in any case we can follow the approach of \\citet{Trippe_08}\nto find the kinematic major axis of the nuclear cluster. To this end, we bin the motions in angle (their Figure~7). The pattern is sinusoidal but with more variation close to the peaks. Thus, we use for fitting following function: \\begin{equation} \nf(\\theta)=a+b\\times(|\\theta-\\phi|)^c\\end{equation} \nTherein, theta is the angle relative to the line to the east;\n$\\phi$ the position of the maximum; $a$ is the constant floor, b the peak parameter; a big $b$ implies that the\nmaximum has small width than the minimum.\nWe obtain $b=3.97\\pm0.49$ and $\\phi=58.8\\pm1.2^{\\circ}$\nconsistent with the Galactic plane ($\\phi=58.6^{\\circ}$).\n\n\nLastly, we use the radial velocities, whose\n gradient in the mean radial velocity as a function of $|l^*|$ \\citep{Trippe_08} can only be explained \n by rotation. For fitting the Galactic plane we aim to find the angle for which the radial velocity is constant along the coordinate $x''$, which we obtain by the rotation. We assume cylindrical rotation, since we ignore $y''$. The angle is the rotation axis, rotated by 90$^\\circ$. \n  The advantage of that angle compared to the rotation axis is that it is not necessary to fit at the same time for the possible complex rotation curve, because vertical to it the velocity is identical everywhere. We obtain $53.7\\pm4.0^{\\circ}$ broadly consistent with the Galactic plane.\n \nDue to the finding of \\citet{Feldmeier_14} that radial velocity field is not only a function of l$^*$ and b$^*$ as expected, we divided our velocities in radial bins (Figure~\\ref{fig:_rot_axis}). \nWe can confirm their findings mostly. At medium radii (between 24 and 90$\\arcsec$) the radial velocity plane follows an angle of $45.1\\pm4.2^{\\circ}$. The value is 3.2~$\\sigma$ different from the Galactic Plane. That agrees well with the measurement of \\citet{Feldmeier_14}. Further out and further in the rotation axis aligns with the Galactic plane within the partly large error. \nWe do not find deviations of $\\sigma_z$ in these bins.\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f8.eps} \n\\caption{Orientation of the major (flattening\/rotation) axis.\n} \n\\label{fig:_rot_axis}\n\\end{center}\n\\end{figure}  \n\n\n\n\n\nAlthough it is likely that the rotation is not a function of l$^*$ only, we assume it here to ease comparison with \nmost of the literature.\nFor illustration we bin the radial velocity data (Section~\\ref{sec:rad_vel}) in a less crowded way (Figure~\\ref{fig:_v_rad_fit1}).\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f9.eps}\n\\caption{Average radial velocities from our data and the literature.\n We assume symmetry of the rotation pattern and reverse the sign of the radial velocities\nFor the maser data we use the velocities from \\citet{Lindqvist_92a} and \\citet{Deguchi_04}. \n The data from \\citet{Trippe_08} overlaps within $|l^*|<27\\arcsec$ largely with our data. Further out \\citet{Trippe_08} utilized \na subsets of the dataset of \\citet{McGinn_89}. We also plot the data of \\citet{Rieke_88}.\nWe fit the good (reddish) data without binning by a polynomial for illustration.\n} \n\\label{fig:_v_rad_fit1}\n\\end{center}\n\\end{figure}  \n\n\nInside of  27$\\arcsec$ our velocities are consistent with the velocities of \\citet{Trippe_08}, since the data set is largely identical.\nOutside of 27$\\arcsec$ our new SINFONI radial velocities are on average smaller than \nthe velocities of \\citet{Trippe_08}, who used only a subset of the velocities in \\citet{McGinn_89}, a problem already pointed out\nby \\citet{Schoedel_09a}. Surprisingly, our new high resolution data do not yield the average of the velocities reported by \\citet{McGinn_89},\nbut agree roughly with the lower end of values found. These lower end of values agrees with the \n velocities of single bright stars by \\citet{Rieke_88}.\n\n\n The maser data of \\citet{Lindqvist_92a} and \\citet{Deguchi_04} agree with the lower velocity data of \\citet{McGinn_89} and our CO band head\nvelocities. \\citet{Schoedel_09a} suggested that the differences in the radial velocities in the literature could be a sign of two populations in the GC.\nHowever, in our data we find no sign for any population dependence of the rotation pattern.\nPossibly, the difference in radial velocities in \\citet{McGinn_89} and \\citet{Rieke_88} is an indication that the velocity calibration\nof these old CO band head measurements was more difficult than assumed back then. \nOverall we are confident that our smaller rotation of the cluster compared to \\citet{Trippe_08} is correct and is not population dependent.\n\n\\subsection{Binning}\n\\label{sec:binning}\n\n\nFor simplicity, we choose to bin our data. The loss of information  \\citep{Merritt_94,Feigelson_12,Scott_92}  is small, since we use\na  large amount of data and the variations between the bins are smaller than the errors.\nWe assume symmetry relative to the Galactic plane, as supported by most observations, see Section~\\ref{sec:rotation}.\nThere might be little deviations in the radial velocities. However, since we fit second moments and not velocities and dispersions the impact is very small. \nAn edge-on flattened system has different symmetry properties in proper motion and in radial velocity. That is another reason for our different binnings which is explained in \n Appendix~\\ref{app_binning}. The same bins are also used by \\citet{Chatzopoulos_14}.\nWe have tried also different binnings, which bin only in r and bin proper motions and radial velocities together. \nOur test includes bins of nearly equal size in r, log(r) and nearly equally populated bins.\nThe Jeans masses varies usually by less than the formal fitting errors, sometimes slightly more.\nSince systematic errors are much larger than the fitting error (thus also larger than the binning error) it is therefore not necessary to include binning terms in the errors.\nWe show in Figure~\\ref{fig:_std_v} the binned dispersion data in all three dimensions (l$^*$, b$^*$, z).\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f10a.eps} \n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f10b.eps} \n\\caption{Binned velocity dispersion used for Jeans modeling. The upper panel presents the proper motion data, \nthe lower one the radial velocity data. \n} \n\\label{fig:_std_v}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\subsection{Jeans Modeling}\n\\label{sec:jeans_modelling}\n\n\nIt is obvious from the dispersion difference (Figure~\\ref{fig:_std_v}) in the two proper motion axes\nthat the nuclear cluster is not a spherical, isotropic system, which would have the dispersions in all directions. \nAs shown in \\citet{Chatzopoulos_14}, this difference is caused by\nthe flattening of the NSC; their Figure~10 shows how $\\sigma_l>\\sigma_b>\\sigma_z$\nover the whole range of radii for their favored axisymmetric models.\nWhile we cannot include anisotropy totally, it is very likely minor given our constrains of Section~\\ref{sec:_anisotropy} and the fact that an isotropic rotator fits the data well \\citep{Chatzopoulos_14}.\nThus, anisotropic spherical modeling is at most only a small improvement compared to isotropic spherical modeling. \nTherefore, we use the most simple kind of Jeans modeling \\citep{Binney_08} assuming isotropy and spherical symmetry for mass and light.\nThat simple model is mainly used for illustration of what can be derived from the data. In a simple model,\ntests for other effects are also easier and faster, in contrast to more complex models.\n\n\n Two variants of isotropic Jeans modeling were used in the past for the GC:\n\\begin{itemize}\n\\item \\citet{Genzel_96} and \\citet{Trippe_08}  parametrized the deprojected dispersion.\n\\item \\citet{Schoedel_09a} used a direct mass parametrization.\n\\end{itemize}\nWe follow here \\citet{Schoedel_09a}. \nSince the existence of the SMBH is shown with orbits \\citep{Schoedel_02}, is\nadvantageous when it can be parametrized directly.\n\n\n\n\\subsubsection{Relation of dispersion and mass}\n\nTo relate the measured dispersion to the mass we use the following equation derived from the Jeans equation \\citep{Schoedel_09a,Binney_08}:\n\n\\begin{equation}\n\\begin {split}\n\\sigma_P^2(R)\/G=\\frac{ \\int^{\\infty}_R dr\\, r^{-2} (r^2-R^2)^{1\/2} n(r) M(r)}   {\n\\int^{\\infty}_R dr\\, r (r^2-R^2)^{-1\/2} n(r) \\,\\, ,\n}\n\\end{split}\n\\label{eq:jeans_par_is}\n\\end{equation}\n\nand one needs to choose a parametrization for $M(r)$. The assumption which has\nthe fewest degrees of freedom is a constant mass to light ratio, \n or a theoretically predicted profile of that \nratio \\citep{Luetzgendorf_12}. \nThe GC data sets have been rich enough to \nleave the shape of the extended mass\ndistribution as a free parameter \\citep{Schoedel_09a}.  \nGiven that the presence of the central SMBH is well established in the GC, one\ncan add the SMBH mass explicitly ($M_{\\bullet}$) to $M(r)$, as done in \\citet{Schoedel_09a}.\n\n\n\\subsubsection{Projection}\n\nIt is necessary to deproject the observed radii $R$ into\ntrue 3D radii $r$, for which one needs to know the space tracer density distribution $n(r)$, see Section~\\ref{sec:fitting_profile}. \nThis is done by two projection integrals, namely equation~(\\ref{eq:abel_1}) and the following integral:\n\n\\begin{equation}\n\\begin {split}\n\\Sigma(R) \\sigma_P(R)^2=2 \\int_{R}^{\\infty} n(r) \\sigma(r)^2 r dr\/\\sqrt{r^2-R^2} \n\\end{split}\n\\label{eq:abel_2}\n\\end{equation}\n\n\n\\subsubsection{Averaging the 3D-dispersions}\n\n\n\nIn the Jeans equation a one-dimensional dispersion $\\sigma$ is used. \nWe have data in all three dimensions and thus need to average them. We use proper motions and radial velocities separately\nsince they cover different areas and have different errors. \n\\begin{itemize}\n\n\\item \nFor the radial velocities, there is significant rotation at large radii (Figures~\\ref{fig:_v_rad_fit1}). \nTo roughly take care of rotation we use instead  of  $\\sigma_z$,  $\\langle v^2_{z}\\rangle^{1\/2}$ (see e.g. \\citet{Tremaine_02,Kormendy_13}) in Equation~\\ref{eq:jeans_par_is}. \nOur approach is an approximation to axisymmetric modeling, which we use in \\citet{Chatzopoulos_14}.\n\n\\item By definition of our coordinates, there\nis no rotation term in the proper motions. Still a non-zero mean velocity can appear locally. \nThus we also use $\\langle v^2\\rangle^{1\/2}$.\nThe impact of using this measurement instead of $\\sigma$ is very small.\nWe combine both dimensions in each bin to \n$\\langle v^2_{\\mathrm{pm}}\\rangle=1\/2\\times(\\langle v^2_{l}\\rangle+\\langle v^2_{b}\\rangle)$  \nto reduce the impact of the flattening. \n\\end{itemize}\n\n\n\n\\subsubsection{Errors of the Dispersions}\n\nCalculating the error of the dispersion with $\\delta\\sigma=\\sigma\\times 1\/\\sqrt{2N}$ is problematic when there\nare only a few stars in a bin (especially as in our maser radial velocity data), since the weight of the individual bins will scale inversely with the \ndispersion value. Thus, low dispersion values have too large weights. \nTo obtain errors which do not bias the result, we determine the errors with an iterative procedure: we fit the binned profile $\\sigma(r)$ with an empirical function \n(a fourth order log-log-polynomial). In the first iteration we weight the points according to the observed $\\sigma\\times 1\/\\sqrt{2N}$.\nIn the further iterations we use the value of the polynomial fit ($\\sigma_\\mathrm{fit}$) to obtain the errors: $\\delta\\sigma=\\sigma_\\mathrm{fit}\\times 1\/\\sqrt{2N}$. \n With these errors we repeat the fit and get refined error \nestimates. After four iterations the procedure converges.  \nWe use the same method also for the proper motions. Due to the higher star numbers the effect of this correction\nis smaller there.\n\nAs result our errors are slightly correlated between the different bins. The effect is small.\nAn estimate of its size can be obtained by comparing our masses when using different binning.\nThese differences are smaller than the formal fitting errors, and thus negligible compared to the\nsystematic errors.\n\n\n\n\\subsubsection{Tracer Distribution Profiles}\n\nWe fit our data using three different types of profiles for the three-dimensional tracer distribution:\n\\begin{itemize}\n\\item For most causes we fit a double $\\gamma$-profile (Section~\\ref{sec:fitting_profile}).\n\\item For checking the robustness of these, we also use a single $\\gamma$-profile which we fit only to our inner density data. \n\\item For comparison with the literature, we also use the single component tracer models of\n\\citet{Genzel_96,Trippe_08};\n and \\citet{Schoedel_09a}. \n\\end{itemize}\n\n\n\\subsubsection{Mass Parametrization}\n\n\nOur mass model contains the point mass SMBH at the center and an extended component made of stars. It is justified to ignore gas clouds, since even the most\n massive structure,\nthe circumnuclear disk, has a mass of only a few times $10^4$ M$_{\\odot}$ \\citep{Mezger_96,Launhardt_02,Requena_12}.\nWe use two different ways of \nparametrization for the extended\nmass:\\begin{itemize}\n\\item We use a power law, similar to \\citet{Schoedel_09a}: \n\\begin{equation}\n\\begin{split}\nM(r)=M_{\\mathrm{100}\\arcsec}\\times(r\/100\\arcsec)^{\\delta_\\mathrm{M}}\n\\end{split}\n\\label{eq:mass_param1}\n\\end{equation}\nThe fact that the integrated mass is not finite for $r\\rightarrow\n\\infty$ is not a problem, since our tracer profile $n(r)$ falls more rapidly than $M(r)$. \n\\item We use a constant mass to light ratio for the extended mass:\n\\begin{equation}\n\\begin{split}\nM(r)=M\/L\\times L(r)\n\\end{split}\n\\label{eq:mass_param2}\n\\end{equation}\nThe light is either flux or the star counts (Section~\\ref{sec:lum_prop}).\n\\end{itemize}\n\nFor normalization of each \nwe choose 100$\\arcsec$, since that mass is well determined from our data. \n\n\n\\subsubsection{Fitting}\n\nWe fit simultaneously the surface  density and the projected dispersion data. \nTo the surface density we fit the density model, the double $\\gamma$ model (Section~\\ref{sec:fitting_profile_gam}). \nThe dispersion data depends on both the mass model and the density model, which in that case is the tracer model. Since we fit both at once both model components are constrained by both data sets.\nWhen we use for the mass the power law model, the fit values of the density model are consistent\nwith the fits which only use the surface density (Table~\\ref{tab:_surf_fit2}).\nThis confirms that our fits converge well. For constant M\/L the parameters for the best fitting density model\ndiffer often  by more than 1$\\,\\sigma$ from the ones when we fit only the surface density.  The reason is, that in the second case the density model depends more on the dispersions, because due to constant M\/L the influence of the dispersion on the density model is grater than in the other case.\n Usually, the density model is less concentrated in the full fit than in the density only fit. That is an expression of the low dispersion problem in the center\n(Section~\\ref{sec:mass_smbh}).\nHowever, the density parameter differences are not very big and do not influence the obtained masses relevantly. \nIn one of our fits the central slope ($\\gamma_{\\mathrm{inner}}$) of the inner component of the tracer density is slightly (by only 0.02)\nsmaller than 0.5. This is problematic, because a slope smaller than 0.5 is not possible in the spherical isotropic case\nwhen a central point mass is dominating \\citep{Schoedel_09a}. \nTo retain self-consistency in our simple isotropic models we fix the smaller slope to 0.5 in that case. The fix has no influence on the obtained masses. Also, the inner slope of the outer component is not well constrained and\nwe fix it usually to 0. \nBoth restrictions have no relevant influence on the masses obtained.\nIn the fitting we optimize $\\chi^2$ \\citep{Press_86}: \n\\begin{equation} \n\\chi^2=\\sum\\limits_{i=1}^n (\\frac{x_i-\\mu_i}{\\sigma_i})^2\n\\end{equation}\nTherein $\\mu_i$ is the function which we optimize; it consists of two function, one for density and one for the dispersion. The density function consists of two terms of Equation~\\ref{eq:eta}  projected with Equation~\\ref{eq:abel_1}, while the dispersion function consists of Equation~\\ref{eq:abel_2} using one of our two types of extended mass parameterizations (power law or constant M\/L). In both cases, the dispersion function is projected with Equation~\\ref{eq:jeans_par_is}.\n$x_i$ and $\\sigma_i$\nare the values and errors of the observables which are obtained in bins. \nWe assume Gaussian errors.\n\n\nWe present the fits corresponding to the various choices of how to set up the Jeans model of the nuclear cluster in Table~\\ref{tab:_mass_iso1}. We do not show the cluster properties since they do not differ by much. The\n robustness of the results can be assessed by comparing the different fits. For the fitting itself we first use routine mpfit \\citep{Markwardt_09}. \nSecondly, we also fit the data by using Markoff-Chain Monte Carlo (MCMC) simulations following the method of \\citet{Tegmark_04,Gillessen_09}. Our modification is that we start in the previously found minimum. We never find a better minimum\n than the starting point. \n The errors are usually rather similar with both methods. There is usually some asymmetry, i.e. the errors\n are larger on the positive side. For mass properties that asymmetry is small. It is larger for some\n of the less well defined tracer density parameters. Such asymmetry cannot be found by mpfit.\n In Table~\\ref{tab:_mass_iso1} we give as best value the median value of the accepted MCMC values. As error we give half of the difference between the 84.1\\% and the 15.9\\% quantile. Such errors encompass the central 1~$\\sigma$ range. \n \nInitially, we fit with a free SMBH mass (Rows 1 and 2 in Table~\\ref{tab:_mass_iso1}). However, especially for the power law mass model \n which has one parameter more, the cluster shape, the resulting mass of the SMBH is smaller than the direct mass measurements \nby means of stellar orbits \\citep{Ghez_09,Gillessen_09}. \n We discuss the reason for the small SMBH mass in Section~\\ref{sec:mass_smbh}. The direct cause \nfor the too small SMBH mass is that models with a realistic black hole mass (M$_\\bullet=4.17 \\times 10^6M_{\\odot}$) predict a larger dispersion  within $r<10\\arcsec$ than we measure. That implies that our model is in some aspect incomplete near the black hole.\nIn the following we fix the central mass to M$_\\bullet=4.17 \\times 10^6M_{\\odot}$, corresponding to our fixed distance of $R_0=8.2\\,$kpc, and we neglect the small distance independent uncertainty of 1.5\\% \\citep{Gillessen_09}.\n\n\n\n\\begin{deluxetable*}{llllllllll} \n\\tabletypesize{\\scriptsize}\n\\tablecolumns{10}\n\\tablewidth{0pc}\n\\tablecaption{Jeans model fits \\label{tab:_mass_iso1}}\n\\tablehead{No.& mass & tracer  & tracer &tracer  & dynamics   & M$_\\bullet$ &  M$_{100\\arcsec}$& $\\delta_\\mathrm{M}$&$\\chi^2$\/d.o.f \\\\\n                          &  model &  model & source &range &  range & [$10^6 M_{\\odot}$] &  [$10^6 M_{\\odot}$] & &}\n\n\\startdata\n1  & power law & double $\\gamma$ & stars&all  & all R &  $ 2.26\\pm 0.26$ &  $9.28 \\pm 0.48 $ & $ 0.92 \\pm 0.04 $&185.43\/182 \\\\ \n2  &   M\/L$=$const & double $\\gamma$ & stars&all  & all R &   $ 4.37\\pm 0.13$ &  $5.17 \\pm 0.24 $ & & 276.16\/184 \\\\ \n\\hline\n\\bf 3 &\\bf power law &\\bf  double $\\bf\\gamma$ &\\bf stars&\\bf all &\\bf 10$\\arcsec<$R$<$100$\\arcsec$ &\\bf  4.17  &\\bf  5.81 $\\pm$ \\bf 0.26 &\\bf 1.21 $\\pm$ \\bf 0.05 &\\bf145.83\/137 \\\\ \n\\bf 4  &\\bf power law &\\bf double $\\gamma$ &\\bf  flux &\\bf all  &\\bf  10$\\arcsec<$R$<$100$\\arcsec$ &  \\bf 4.17  & \\bf 6.17 $\\pm$ \\bf 0.23 & \\bf 1.25 $\\pm$ \\bf 0.04  &\\bf 349.51\/286\\\\\n\\bf 5 &\\bf   M\/L$=$const &\\bf double $\\gamma$ &\\bf stars &\\bf all  &\\bf  10$\\arcsec<$R$<$100$\\arcsec$ & \\bf 4.17 &\\bf  5.62 $\\pm$ 0.17  &&\\bf 156.97\/138   \\\\ \n\\bf 6 & \\bf M\/L$=$const &\\bf  double $\\gamma$ &\\bf flux &\\bf all &\\bf  10$\\arcsec<$R$<$100$\\arcsec$ &\\bf  4.17  &\\bf  6.81 $\\pm$ 0.16  &&\\bf 342.58\/287   \\\\\n\\hline\n7  & power law & double $\\gamma$ &stars& all  & R $<100\\arcsec$ &  $4.17 $ &  $4.98 \\pm 0.29 $ & $1.33 \\pm 0.05 $&199.69\/163 \\\\ \n8  &  power law &double $\\gamma$ &  flux &all  & R $<100\\arcsec$ &  $4.17 $ &  $5.35 \\pm 0.23 $ & $1.37 \\pm 0.04 $ &435.79\/312\\\\\n\\hline \n9&  M\/L$=$const &  double $\\gamma$ &stars& all & R $<100\\arcsec$ &  $4.17 $ &  $5.48 \\pm 0.16$ &&195.82\/164    \\\\ \n10  &  M\/L$=$const &  double $\\gamma$ &  flux &all & R $<100\\arcsec$ &  $4.17 $ &  $6.70 \\pm 0.18$ &&414.04\/313  \\\\ \n\n11  & power law & double $\\gamma$ &stars& all  &  R$>$10$\\arcsec$ &  $4.17 $ &  $6.14 \\pm 0.18 $ & $1.12 \\pm 0.03 $&182.36\/157 \\\\ \n12 & power law & double $\\gamma$ &  flux &all  &  R$>$10$\\arcsec$ &  $4.17 $ &  $6.57 \\pm 0.17 $ & $1.12 \\pm 0.03 $&399.37\/306 \\\\ \n13  &  M\/L$=$const &  double $\\gamma$ &stars& all  & R$>$10$\\arcsec$ & $4.17 $ &  $5.58 \\pm 0.15 $ &&240.18\/158  \\\\\\ \n14  &  M\/L$=$const &  double $\\gamma$ &  flux &all & R$>$10$\\arcsec$ &$4.17 $ &  $6.65 \\pm 0.16 $ &&433.36\/307  \\\\ \n15  & power law & double $\\gamma$ &stars& all  & all R & $4.17 $ &  $5.59 \\pm 0.17 $ & $1.19 \\pm 0.03 $&247.59\/183 \\\\ \n16  & power law & double $\\gamma$ &  flux &all & all R &  $4.17 $ &  $6.03 \\pm 0.17 $ & $1.19 \\pm 0.03 $&507.08\/332 \\\\ \n17  &  M\/L$=$const &  double $\\gamma$ &stars& all  & all R & $4.17 $ &  $5.47 \\pm 0.14 $ &&277.14\/185 \\\\ \n18 &  M\/L$=$const &  double $\\gamma$ &  flux &all  & all R &  $4.17 $ &  $6.56 \\pm 0.15 $ &&500.64\/333   \\\\ \n\n\\hline\n19 &power law & single $\\gamma$ & stars&R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$ &   $4.17 $ &  $5.91\\pm 0.21$ & $1.32 \\pm 0.08 $&126.27\/108  \\\\ \n20 &power law & single $\\gamma$ & flux &R$<220\\arcsec$ &  10$\\arcsec<$R$<100\\arcsec$  & $4.17 $ &  $6.27\\pm 0.18$ & $1.36 \\pm 0.07 $ &207.12\/132 \\\\\n21 & M\/L$=$const &  single $\\gamma$ & stars&R$<220\\arcsec$ &  10$\\arcsec<$R$<100\\arcsec$  &  $4.17 $ &  $6.23\\pm 0.16$ &&130.58\/108  \\\\ \n22  &  M\/L$=$const & single $\\gamma$ & flux & R$<220\\arcsec$ &  10$\\arcsec<$R$<100\\arcsec$ &   $4.17 $ &  $6.92\\pm 0.15$  &&205.30\/134  \\\\\n\\hline\n23  & power law & Sch\\\"odel+ 2009 & stars& R$<220\\arcsec$ &  10$\\arcsec<$R$<100\\arcsec$ &   $4.17 $ &  $5.79\\pm 0.25$ & $1.16 \\pm 0.05 $&195.97\/110\\\\\n24  & power law & Trippe+ 2008 & stars& R$<220\\arcsec$ &  10$\\arcsec<$R$<100\\arcsec$ &    $4.17 $ &  $7.22\\pm 0.17$ & $1.20 \\pm 0.04 $&453.40\/110\\\\\n25  &power law & Genzel+ 1996 & stars& R$<220\\arcsec$ & 10$\\arcsec<$R$<100\\arcsec$ &    $4.17 $ &  $6.75\\pm 0.18$ & $1.12 \\pm 0.04 $&237.05\/110\\\\\n26 &  M\/L$=$const & Sch\\\"odel+ 2009 & stars&R$<220\\arcsec$ &   10$\\arcsec<$R$<100\\arcsec$&   $4.17 $ &  $4.66\\pm 0.09$& &218.72\/111  \\\\\n27 & M\/L$=$const & Trippe+ 2008 & stars&R$<220\\arcsec$ &  10$\\arcsec<$R$<100\\arcsec$ &   $4.17 $ &  $7.56\\pm 0.15$& &444.73\/111  \\\\ \n28  &  M\/L$=$const & Genzel+ 1996 & stars& R$<220\\arcsec$&  10$\\arcsec<$R$<100\\arcsec$  &   $4.17 $ &  $6.19\\pm 0.12$ &&243.69\/111 \\\\\n\\hline\n29&  M\/L$=$const &  double $\\gamma$ &stars& all & R $<27\\arcsec$ &  $3.37\\pm0.16 $ &  $7.31 \\pm 0.42$ && 91.93\/120 \n\\enddata\n\\tablecomments{\nJeans model fitting of our dynamics and density data, assuming different mass and tracer models, and different selections for the data. For the fitted surface density (tracer) we use two data sources and restrict us sometimes to\na subset of the available data range (Column 3 and 4). The dynamics data consists of $\\langle v^2\\rangle$ in all three dimensions. We restrict it partly radially.\nThe mass model includes in all cases a central point mass.  M$_{100\\arcsec}$ is the nuclear cluster mass within 100$\\arcsec$. \nIf no error is given for a parameter it is fixed. \nThe literature tracer models are from\n\\citet{Trippe_08,Schoedel_09a},\n and \\citet{Genzel_96}.\n}\n\\end{deluxetable*}\n\n\\subsubsection{Results}\n\n\n The fits which meet our assumption best (Table~\\ref{tab:_mass_iso1}) are in the rows 3, 4, 5, and 6. They use\na range of 10$\\arcsec<R<$100$\\arcsec$ for the dynamics data. The surprisingly low dispersion in the center impacts the cluster mass in case of\nthe fits in rows 7 and 8. These cases\nuse $R<100\\arcsec$ for the dynamics data (i.e. including the central region) and allow the cluster shape to vary with  the power law.\nAll other fits are less influenced by the central dispersion and yield masses $M_{\\mathrm{100}\\arcsec}$ between 5.5 and $6.8 \\times 10^6\\,$M$_{\\odot}$. The fact that the mass does not strongly depend ($\\delta M<15$\\%) on whether we include the central 10$\\arcsec$ shows\nthat the influence of the too small SMBH mass on the cluster mass is much smaller than the effect of the neglected flattening, see Section~5.4.8.\n\nFigure~\\ref{fig:_fits_iso3} illustrates the fits of rows 1, 3, and 5. \nIn this figure it is visible that our models with the right SMBH mass have higher dispersions than the measurements at $R<10\\arcsec$. Probably the models are not correct there. \nIn Figure \\ref{fig:corr_plot1} we show all correlations of the fit in row 5. Some (outer) tracer density parameters are strongly correlated with each other. The cluster mass itself shows only weak correlations with other parameters. That explains the small fit errors for the cluster mass.\n\n Outside of $100\\arcsec$ the fits are also not very good. The reason for that is probably:\n\\begin{itemize}\n\\item  We don't have density data outside of 1000$\\arcsec$, and hence cannot constrain the tracer profile there.\nThe outer slope of the tracer profile is thus not well-constrained.\n\\item The fixed outer slope in the $\\gamma$-model does not allow for variability in the outer slope.\n\\item The assumption of spherical symmetry is a poor approximation outside of $\\approx 300\\arcsec$.\n\\end{itemize}\nTherefore, the mass outside the central $\\approx100\\arcsec$ is less reliable.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f11.eps}\n\\caption{Data and fits of the Jeans modeling. The data within 10$\\arcsec$ and outside 100$\\arcsec$ (gray dots) are somewhat less consistent with \nour simple model; the other data is plotted as black dots. The curves show the fits from rows 1 (green), 3 (red), and 5 (blue) of \nTable~\\ref{tab:_mass_iso1}. \n}\n\\label{fig:_fits_iso3}\n\\end{center}\n\\end{figure}\n\nApart from the fits in rows 7 and 8, the selection of the range for the dynamics data is less important\nthan the choice of the tracer source for the obtained mass. The latter is particularly true when choosing a \nconstant mass-to-light ratio as the mass model. Choosing the stellar number counts or the \nflux as the tracer results in a difference of up to 20~\\% in $M_{100\\arcsec}$ (compare rows 5 and 6 in table). \nThe smaller value occurs for the star density profile. The main reason is that in this profile the inner component has a larger core radius than in case of the flux profile. With a smaller core a bigger part of the mass which causes the dispersion need to be close to the black hole.\nIn Figure~\\ref{fig:corr_plot2} we show the cluster mass probability distribution of row 3 to 6, i.e. the ones which all use the same, likely best, dynamic data. It is also visible that there is a clear anticorrelation between cluster mass and cluster mass slope. \nStill, the best values of both are well defined, in contrast to \\citet{Schoedel_09a}, thanks to our larger radial coverage.\n\nLooking at the $\\chi^2$-values in the table seems to indicate that the fits using the flux density as a tracer profile are worse than \nthe fits using the stellar number counts. \nHowever, the difference is due to the density data, where because of our error calculation \n(Appendix~\\ref{sec:obt_lum}) the $\\chi^2$ of the flux density data is worse (Section~\\ref{sec:fitting_profile}). \nSince we identify no reason to prefer one of the two data sets (Section~\\ref{sec:number_density}), we give of both of them equal weight in the following.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=1.999 \\columnwidth,angle=-90]{f12.eps} \n\\caption{Parameter correlations for model 5 (Table~\\ref{tab:_mass_iso1}). The parameters are the `luminosity' (in star counts), the inner slope parameter and the 'core' radius for the two $\\gamma$ components and the extended mass at 100$\\arcsec$. The black hole mass\nis fixed here. The dark gray corresponds to the 1 $\\sigma$ area, the light gray to the 2 $\\sigma$ area. \nThe parameter correlations are obtained from a Markoff-Chain Monte Carlo simulation.\n} \n\\label{fig:corr_plot1}\n\\end{center}\n\\end{figure*}\n\n\nFrom the sample we exclude rows 7 and 8, because the obtained slope $\\delta_\\mathrm{M}$ are significantly bigger than the slope of stars counts\nand flux profiles. That is unrealistic and is caused by the dispersion problem in the center. Thus, we consider a sample, that contains the 14 fits in the rows 3 - 6 and 9 - 18 \nwhich we all give equal weight. The average mass is\n\\begin{equation}\nM_{100\\arcsec}=(6.09 \\pm 0.53) \\times \\,10^6\\,M_{\\odot} \\,\\, .  \n\\end{equation}\nAs \\citet{Gillessen_09} we calculate the systematic error from the scatter between the different fits.\nThat errors dominates over the smaller fit errors. That way of error calculation has the advantage that\nit uses realistic uncertainties for the extended mass and star distribution. In the case of the mass, it explores the presence and absence of a dark cusp. In the case of the star distribution, it enables a realistic assessment of the uncertainty in the profile, which a single profile cannot provide. \nUsing only rows 3 - 6 yields a very similar result. \n According to Table~\\ref{tab:_mass_iso1}, the main uncertainty is whether the star counts or the flux is used. They have different density profiles which results in different masses. By comparison, the other assumptions (like the mass profile) play a minor role. \n \n\nThe cluster mass slopes are between 1.12 and 1.25 for the cases used. The\nmean is $\\delta_\\mathrm{M}=1.18 \\pm0.06$, where the error again is dominated by the scatter between the different fits. This number is in \nreasonable agreement with the tracer profile, which has a slope of 1.18 (stellar number count based) or 1.06 (flux based) in the \nrange 50$\\arcsec< R <$ 200$\\arcsec$.\n\nIn order to check how important our two-component model is we also fit single $\\gamma$ models to our data (Rows 19 - 22). To ensure, that a single component model is a reasonable fit to the data, we restrict the\nrange of the tracer density data inside the break to the nuclear disk, i.e. $R < 220\\arcsec$.  \nWith this restriction, the masses are slightly larger than before. The main reason is that the core of the inner $\\gamma$ component, which is now the only $\\gamma$,\nis larger in this case.\nUsing the literature profiles of\n\\citet{Genzel_96,Trippe_08}; and \n\\citet{Schoedel_09a} the mass range is larger, from \n$4.66\\times 10^6$ to $7.56\\times 10^6\\,M_{\\odot}$. The main reason for the mass trend between them is the\neffective outer slope. When it is steep as in \\citet{Trippe_08} the expected dispersion at large radii gets smaller.\n\n\nOur data set would allow us to determine the distance to the GC, $R_0$, by means of a statistical parallax \\citep{Genzel_00,Eisenhauer_03,Trippe_08}.\nTo be accurate in $R_0$ it is necessary to have a model which accounts for differences between $\\sigma_l$, $\\sigma_b$ and $\\sigma_z$. Our spherical model cannot provide that, and we\ndefer this to the work of \\citet{Chatzopoulos_14}, where we present self-consistent flattened models.\n\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.74 \\columnwidth,angle=-90]{f13a.eps} \n\\includegraphics[width=0.74 \\columnwidth,angle=-90]{f13b.eps} \n\\includegraphics[width=0.73 \\columnwidth,angle=-90]{f13c.eps} \n\\includegraphics[width=0.73 \\columnwidth,angle=-90]{f13d.eps} \n\\includegraphics[width=0.73 \\columnwidth,angle=-90]{f13e.eps} \n\\includegraphics[width=0.73 \\columnwidth,angle=-90]{f13f.eps} \n\\caption{Cluster mass correlations and probability distributions for model 3, 4, 5 and 6 (Table~\\ref{tab:_mass_iso1}).\n These four cases use the same dynamic data range ($10\\arcsec<R<100\\arcsec$) but use different tracer distributions and extended different mass models. The ones in the left column use the star density as tracer, the ones in the right the flux density. For the two top rows a power law is fitted to nuclear cluster. In the bottom row a constant mass to light ratio is assumed.\nFor all the SMBH is fixed to M$_{\\mathrm{SMBH}}=4.17\\times10^6\\, M_{\\odot}$. The dark gray marks the 1$\\,\\sigma$ area from MCMC,  the light gray the 1$\\,\\sigma$ area.\n} \n\\label{fig:corr_plot2}\n\\end{center}\n\\end{figure*}\n\n\nWe now obtain the mass uncertainty due to the distance uncertainty. \nWe use the latest results for the orbit of the S-stars of $R_0=8.2\\pm 0.34$ kpc \\citep{Gillessen_13}. Due to the small distance-independent SMBH mass error of 1.5\\% \\citep{Gillessen_09}, which is vanishingly small compared to our other uncertainties, it is sufficient\nto derive the mass dependency on distance.\nTo include the distance error in the error budget we repeat the Jeans modeling \nfor $R_0$ of 7.86 and 8.54 kpc. Thus we assume that the distance error is Gaussian, which seems to be true approximately, see Figure~15 in \\citet{Gillessen_09}. We perform full $\\chi^2$ adding in \\citet{Chatzopoulos_14}. We determine the SMBH mass \nfrom the M$_\\bullet-R_0$ relation of\n\\citet{Gillessen_09}: $M_\\bullet \\propto R^{2.19}$. \n We find following mass distance relation: \n\\begin{equation}\nM_{100\\arcsec}=6.09 \\cdot 10^6\n\\times \\left(\\frac{R_0} M_{\\odot} {8.2 \n\\mathrm{kpc}}\\right)^{3.83}  \n\\end{equation}\nThe distance error of \\citet{Gillessen_09} leads to following\nexpression\n\\begin{equation}\nM_{100\\arcsec}=(6.09 \\pm 0.53|_{\\mathrm{fix} R_0}\\pm  \n0.97|_{R_0} ) \\times 10^6  M_{\\odot}\\,\\, . \n\\end{equation}\nThus, the distance-induced error is larger than the other considered errors. \nHowever, we do not consider all error sources in our work. Our model does not include anisotropy and flattening. The inclusion of both can introduce important mass changes, see e.g. \\citet{Marel_10,Dsouza_13}. \nSince we detect signs for flattening but not for anisotropy, flattening is more important. We use a flattened model in \\citet{Chatzopoulos_14}. Comparing with that work it can be clearly seen that the cluster mass is 47\\% bigger when flattening is included. Thus, it is important to include the flattening in nuclear cluster models. Still, our relative errors are useful as an estimate of some systematic uncertainties. While the distance dependent error may improve with a better distance from e.g. the statistical parallax \\citep{Chatzopoulos_14}, it is more difficult to reduce contributions of the light and mass profile uncertainty. They cause an error of about 8.5\\% in our modeling which would be rather similar also in more complex models. That error is larger than the statistical error of 3.5\\% of \\citet{Chatzopoulos_14}, which includes also the distance uncertainty. It is however smaller than their systematical error of 10.1\\%. Their systematic error does not include the tracer profile uncertainty, which is dominant in our modeling, but is dominated by the flattening uncertainty. Overall, our tests show that the uncertainty which is caused by the tracer profile is not the dominant error contribution.\n\n\nFinally, we obtain an estimate for the total mass of the nuclear cluster by using the inner components of the fits with constant M\/L. \nFrom the average and scatter we obtain \\begin{equation}\nM_\\mathrm{NC}=(4.22 \\pm 0.50|_{\\mathrm{fix} R_0}\\pm 0.67|_{R_0}) \\times 10^7  M_{\\odot} \\,\\, .\n\\end{equation}\nSince the outer cluster slope is more uncertain than the mass at 100$\\arcsec$ the mass uncertainty is now larger.\nThe full uncertainty is larger still. It is not present here because the outer slope\nis fixed in $\\gamma$-models. The true total mass uncertainty is probably as in the case of the luminosity \n(Section~\\ref{sec:total_luminosity}) around $\\pm$50\\% of the mass. In contrast, the extended mass within 100$\\arcsec$ is not affected by that uncertainty, because within $\\approx100\\arcsec$ the data are better.\n\n\n\n\\subsection{Velocity Distribution and Fast Stars}\n\\label{sec:fast_stars}\n\n\nThe Jeans modeling only used the first two moments of the velocity distributions, but they contain information beyond that. In particular, stars with\nhigh velocities in the wings of the distributions are interesting. The unbiased space (three-dimensional) velocity for each star is \\citep{Trippe_08}\n\\begin{equation}\n\\begin {split}\nv_{\\mathrm{3D}}=\\sqrt{v_x^2+v_y^2+v_z^2-\\delta v_x^2-\\delta v_y^2-\\delta v_z^2} \\,\\, .\n\\end{split}\n\\label{eq:vel_3d}\n\\end{equation}\n\nSince the three-dimensional velocity is, like the dispersion positive by definition we subtract the errors to get unbiased velocities.\nThe errors of $v_{\\mathrm{3D}}$ do not depend on the value, and in particular there is no indication (especially inside 20'') that the higher velocity stars are caused by measurement problems.\nWe divide the sample into radial bins of 32 stars each, and determine in each the maximum and the median 3D-velocity \n(Figure~\\ref{fig:_max_vel}). The maximum velocity cannot be described by a single power law. Inside of 7.5$\\arcsec$\nit follows a power law slope of $-0.47 \\pm 0.04$, \nclose to a Keplerian slope of -0.5. Other high quantiles like the second fastest star follow similar slopes in the center.  Outside of 7.5$\\arcsec$ these slopes \nappear to be consistent with\nthe single slope of the median velocity, of around $-0.19$.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f14.eps}\n\\caption{\nBinwise median and maximum three-dimensional velocity. We describe the data with (broken) power laws. \nThe error indicates the velocity error of the median velocity stars and the velocity \nerror of the fastest stars, respectively. Fast stars already discussed by \\citet{Reid_07} or \\citet{Schoedel_09a} are marked with open blue circles. \n} \n\\label{fig:_max_vel}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsubsection{Minimum Binding Mass}\n\n\nWe now calculate for all stars the minimum binding mass, i.e. we assume that each star is in the plane of the sky and on a parabolic orbit\naround a point mass. \n In each radial bin we thus can determine the highest mass (Figure~\\ref{fig:_esc_mass}). Inside of 8.3$\\arcsec$ the escape mass \nis close to the SMBH mass. The average escape mass there is $3.67\\times10^6\\,$M$_{\\odot}$.\nThe most significant mass larger than that of the SMBH occurs for the star S111 with \n$4.33 \\pm 0.05 \\times10^6\\,$M$_{\\odot}$, as was noted already by \\citet{Trippe_08} and \\citet{Gillessen_09}.\nHigh escape masses indicate that the stars are close to the SMBH not only in projection, but also in 3D. \nThese fast stars can be used as additional constraints on the late-type density profile close to the SMBH in advanced \ndynamic modeling.\n\\subsubsection{A Distance Estimate}\n\n\nThe number of stars that appear unbound depends on the assumed distance $R_0$. Stars whose velocity is dominated by the\nradial velocity are bound for a large $R_0$, while stars with a large proper motion are bound for small $R_0$. \nIn the following we restrict the used data to r$\\leq8.45\\arcsec$, the area dominated by the SMBH. \nOur sample contains nine stars, which are unbound for some distances between 7.2 and 9.2 kpc. \nFrom these stars we calculate a $\\chi^2$ as a function of $R_0$ summing up in squares the significances of the differences from\nthe minimum binding mass minus the mass inside that radius. We only consider stars when they appear unbound, however not when the minimum binding is smaller than the mass inside that radius. The mass is dominated by the SMBH, for which we use \nM$_{\\bullet}=(3.95 \\pm 0.06 )\\times 10^6 \\,(R_0\/(8.0 \\mathrm{kpc})^{2.19}\\,$M$_{\\odot}$ \\citep{Gillessen_09}. \nThat means we use from that paper only the distance independent mass error, not the distance dependent component.\nFor the extended mass we use our two preferred models, a power-law and a constant M\/L case, see Section~\\ref{sec:cum_mass}.\nThe mass of the SMBH dominates, the extended mass adds at most 10$^5\\,M_{\\odot}$.\nThe error on the SMBH mass we add by exploring the Gaussian 1.5\\% error. In case of the extended mass we give the two cases equal weight and get the error from the difference.\nMinimizing the $\\chi^2$ as a function of $R_0$ yields  $R_0=8.53^{+0.21}_{-0.15}\\,$kpc. \nThe main constraint  comes from 2 stars: S111 and a star with high proper motion, Id 569\n(Figure~\\ref{fig:_dist_esc}). \n\nThis estimate relies on some assumptions: \n\\begin{itemize}\n\\item We assume that all stars are bound. This is justified, since the chance to see an escaping star from a Hills-event \\citep{Hills_88} \nis very low \\citep{Yu_03,Perets_07}. Also the Hills mechanism, which causes stars faster than the local escape velocity at $\\gtrsim 10\\arcsec$ is unlikely to play \na role, as suggested\nby the break around 8$\\arcsec$ visible in Figure~\\ref{fig:_max_vel}).\n\\item The method relies on the mass-distance scaling of the SMBH mass from stellar orbits.  Using the relation from \n\\citet{Ghez_09} we obtain R$_0=8.38^{+0.23}_{-0.25}\\,$kpc.\n\\item We assume that we have correctly debiased the velocities. Since the total velocity error of the two most \nimportant stars is less than 1.5$\\,$\\% of their velocity\nthis assumptions seems to be uncritical.\n\\end{itemize}\nA lower limit on the SMBH mass independent of the M$_{\\bullet}-R_0$ can be obtained by using the smallest recent R$_0$ measurement of 7.2 kpc \\citep{Genzel_10,Bica_06}.\nFor that distance S111 gives $3.6\\times 10^6 M_{\\odot}$, \nconsistent with the orbit-based estimates, and higher than most Jeans-model estimates.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.70 \\columnwidth,angle=-90]{f15.eps} \n\\caption{\nMaximum of the minimum binding mass in bins. The mass errors \nfollow from the 1$\\,\\sigma$ velocity errors We compare these masses with the SMBH mass of \n\\citet{Gillessen_09} and two Jeans-models, model A and C from Table~\\ref{tab:cum_mass}. \nThe fast stars from \\citet{Reid_07,Trippe_08}\nand \\citet{Schoedel_09a} are indicated with open blue circles. \n} \n\\label{fig:_esc_mass}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f16.eps}\n\\caption{Distance constraints from stars, that are close to escaping. The figure shows the mass distance relations for the SMBH from \n\\citet{Gillessen_09}.\nThe mass distance relations for the two stars which are closest to escape is derived from the minimum binding mass. \nRed and black lines mark the value; orange and gray lines the respective 1$\\,\\sigma$ error range.\n} \n\\label{fig:_dist_esc}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\subsubsection{Fast Stars at $R>10\\arcsec$}\n\nOutside of 10$\\arcsec$ most fast stars are unbound to SMBH and nuclear cluster mass (Figure~\\ref{fig:_esc_mass}). These stars are detected in all three velocity dimensions: we have 5, 6, 5, with $|v|>275$ km\/s in either\/or z, $l^*$ and $b^*$, respectively among the stars with all velocity components measured outside of 7$\\arcsec$ in the central and extended sample.\nSome of these stars were already discussed in \n\\citet{Reid_07} and \\citet{Genzel_10}. Our improved mass estimate for the nuclear cluster reinforces the statement that these stars\nare not bound to the GC. Note that placing a star out of the plane of the sky enlarges the discrepancy \\citep{Reid_07}, \ne.g. for a star  at 40$\\arcsec$ the escape velocity decreases out to $z=200\\arcsec$ and then increases  only very slowly to about 165 km\/s at 3600$\\arcsec$, \nwhich is still less than in the plane of the sky. The fact that the maximum velocity decreases for $R>8\\arcsec$ in the same\nway as the median velocity (Figure~\\ref{fig:_max_vel}) excludes that the fast stars are foreground objects, for which \nthe velocity would not depend on radius. The extinction appears to be normal for the fast stars.\n\n\n\\citet{Reid_07} also discussed binaries as a solution for the high velocities. This is excluded by our data set since for some stars the high velocity \nis dominated by long-term proper motions measurements, which cover much more time than what one orbital period would need to be.\n\nFor testing whether the Hills-mechanism is important, the directions of motion of the fast stars can help. We use therefore  the two sided K-S~test on the distribution of proper motion vectors in polar coordinates comparing fast stars with all stars. Fast stars are again selected in radial bins to avoid influence of radial trends.\nWe obtain for different selections borders within the fastest 1\\% that the two distributions are identical, probability usually 0.5, in one case 0.07.  Thus the fast stars are not preferentially moving away from the black hole.\nThat makes it unlikely that the Hills mechanism is the main mechanism.\nThe comparably large number of fast stars makes that also unlikely.\nProbably as already advocated by \\citet{Reid_07}, the best solution is that these stars are on very eccentric orbits in the large scale potential.\nTherein they can obtain higher velocities than the escape velocity determined from the local mass distribution.\n\n\n\nSince the database of  \\citet{Genzel_10} contained many bright, unbound stars, these authors suggested that preferentially young, \nrelaxed, bright TP-AGB stars are on these unbound orbits. This finding might be affected by low number statistics \nand a bias. Bright stars have smaller velocity\nerrors and are therefore easier to identify as significantly unbound. In our sample and excluding the database of \\citet{Genzel_10}, \nwe see no evidence that bright stars   \nare dynamically distinct from the other stars. This also holds for the subsample of medium old TP-AGB stars from \\citet{Blum_02}. Also \n\\citet{Pfuhl_11} found that their two samples of younger and older giants show consistent dynamics.\n\n\\subsubsection{Velocity histograms}\n\nIn Figure~\\ref{fig:_velocity_his} we show the velocity histograms for the three dimensions. In case of the proper motion samples we show\n$R>7\\arcsec$. For the radial velocities, we show two bins,  7$\\arcsec<R<100\\arcsec$ and $R>100\\arcsec$. In the latter\nbin (consisting of the maser stars) we have subtracted off the rotation.  \n\\begin{figure}\n\\begin{center}\n\n \\includegraphics[width=0.70 \\columnwidth,angle=-90]{f17.eps}\n\\caption{Velocity histograms of our dynamics sample.} \n\\label{fig:_velocity_his}\n\\end{center}\n\\end{figure}\n\n\n\n We see deviations from Gaussian distributions in many aspects:\n\n\\begin{itemize}\n\\item The central part of distributions in $l^*$ and $b^*$ are not as peaked as Gaussians (Figure~\\ref{fig:_velocity_his}).\nIn $l^*$ the distribution has a flatter peak than a Gaussian \\citep{Trippe_08,Schoedel_09a}.\nIn $b^*$ the distribution has a slightly, but significantly ($>5\\,\\sigma$), steeper peak than a Gaussian.\nThe flatter peak in $l^*$   is actually a signature of the flattening of the cluster, as shown in \\citet{Chatzopoulos_14}.\n\\item The varying ratio of maximal and median velocity (Figure~\\ref{fig:_max_vel}) is indicative of non-Gaussian wings\nwithin 7$\\arcsec$. \n\\citet{Trippe_08} did not see that\nbecause they did not radially subdivide their sample. Furthermore our sample contains\ntwelve more stars in the central 2$\\arcsec$ with v$_{\\mathrm{3D}}>460$ km\/s than the one from \\citet{Trippe_08}. \nIn that work the only such fast star was S111. \nThe high-velocity wings in all three dimensions are mainly caused by the presence of the SMBH.\n Therefore, because of the higher number of fast stars S111 is less special than discussed in \\citet{Trippe_08}. \n\\item From the 274 stars in the $R>100\\arcsec$-bin, roughly 13 are in the high (radial) velocity wings of the distribution  (Figure~\\ref{fig:_velocity_his}). \nMost of these outliers were already noted by \\citet{Lindqvist_92b} and \\citet{Deguchi_04}.\n\\end{itemize}\n\n\n\\begin{deluxetable*}{lllllllll}\n\\tabletypesize{\\scriptsize}\n\\tablecolumns{9}\n\\tablewidth{0pc}\n\\tablecaption{Fast stars \\label{tab:_fast star}}\n\\tablehead{ID & R.A. [$\\arcsec$] & Dec. [$\\arcsec$]  & v$_{\\mathrm{R.A.}}$ [mas\/yr]&v$_{\\mathrm{Dec.}}$ [mas\/yr]  & v$_{\\mathrm{z}}$ [km\/s]  &v$_{\\mathrm{3D}}$ [km\/s] & v$_{\\mathrm{esc}}$ [km\/s]  & Comment}\n\\startdata\n770 & -1.11 & -0.91 & $-2.78 \\pm 0.02$ & $-7.72 \\pm 0.01$ & $-741 \\pm 5$ &$807 \\pm 5$ & 792 & S111\\\\\n569 & 0.13 & 3.08 & $13.22 \\pm 0.04$ & $-3.6 \\pm 0.05$ & $-92 \\pm 6$ &$540 \\pm 7$ & 541 & \\\\\n4 & 5.68 & -6.33 & $2.96 \\pm 0.09$ & $2.58 \\pm 0.08$ & $-312 \\pm 14$ &$347 \\pm 14$ & 326 & IRS 9\\\\\n4258 & 2.66 & 13.61 & $-8.48 \\pm 0.15$ & $-1.67 \\pm 0.09$ & $-315 \\pm 45$& $458 \\pm 46$ & 255 & fastest v$_{\\mathrm{3D}}$, R$>7\\arcsec$ \\\\\n899 & -8.37 & -12.21 & $-1.98 \\pm 0.15$ & $11.03 \\pm 0.24$ &             &$435 \\pm 11$ & 247 & fastest v$_{\\mathrm{2D}}$, R$>7\\arcsec$\\\\  \n903 & 14.11 & 7.45 & $-1.58 \\pm 0.04$ & $-0.98 \\pm 0.13$ & $379 \\pm 24$ &$385 \\pm 24$ & 238 & only high v$_{\\mathrm{z}}$\\\\\n787 & -6.8 & 18.35 & $-5.96 \\pm 0.13$ & $7.36 \\pm 0.2$ & $-91 \\pm 11$ &$379 \\pm 15$ & 215 & runaway candidate\\\\\n & &  &  &  & & &  &  \\citep{Schoedel_09a}\\\\\n\\enddata\n\\tablecomments{Extraordinarily fast stars sorted by their projected distance from Sgr~A*. The errors of\nthe positions are smaller than 5 mas. v$_{\\mathrm{esc}}$  assumes $R_0=8.2\\,$kpc.\n}\n\\end{deluxetable*}\n\nTable~\\ref{tab:_fast star} gives an overview of the unusually fast stars (R$\\leq20\\arcsec$, further out extremely fast stars are not reliably measured) in our sample. ID 787 is the star which \n\\citet{Schoedel_09a} called a runaway candidate.\nWe find a proper motion that is 3.8$\\,\\sigma$ smaller, and derive a rather normal radial velocity from its late-type spectrum, \nmaking this star considerably less noteworthy. \nThe difference in proper motion is probably due to our better distortion correction at the edge of the field of view\n(Section~\\ref{sec:mass_smbh}). \nThe star with the highest 3D velocity is the giant ID 4258, but it is (projected) close to another bright late-type star, such that\nthe radial velocity error might well be larger than indicated.\nThe outlier fraction appears to be smaller than in the maser sample (Figure~\\ref{fig:_velocity_his}), but at smaller radii the dispersion is higher and thus\nthe identification of outliers is more difficult there. \n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\n\nWe discuss our results in the context of previous results for the nuclear cluster of the Milky Way and of other galaxies.\n\n\n\\subsection{Central Low Dispersion Problem and SMBH Mass}\n\\label{sec:mass_smbh}\n\n\nRow~1 in Table~\\ref{tab:_mass_iso1} shows that fitting for the SMBH mass with a power law profile yields a mass which\nis much smaller than the estimates from stellar orbits \\citep{Gillessen_09,Ghez_09}. \nWe now look closer at that problem.\nIn order to reduce rotation or flattening influence in the result, \n we use here only the data in the central 27$\\arcsec$, where the assumption of \nspherical symmetry is fulfilled approximately. \nThe same effect also occurred in \nearlier works \\citep{Genzel_96,Trippe_08}. However, recently \\citet{Schoedel_09a} and \\citet{Do_13b} obtained \nhigher masses from Jeans modeling of old stars.\n\n\\citet{Do_12} obtained from the three-dimensional motions\nof 248 late-type stars in the central 12$\\arcsec$  masses  between M$_\\bullet=3.77^{+0.62}_{-0.52}\\times 10^6 M_{\\odot}$ and  M$_\\bullet=5.76^{+1.76}_{-1.26}\\times 10^6 M_{\\odot}$, consistent with\nthe orbit-based estimates considering the R$_0$ of the fits. They obtained higher values with anisotropic spherically symmetric Jeans modeling than with isotropic spherically symmetric Jeans modeling. These fits obtain isotropy in the center, but tangential anisotropy  further out. The latter is probably spurious, see Section~\\ref{sec:_anisotropy}.\nThe lowest value is obtained in isotropic modeling.\n\nThe break radius of \\citet{Do_13b} is large and inconsistent with our data, see Figure~\\ref{fig:_densities_1}.\nWe test its influence on the obtained mass.\nWe therefore select from our data the bins inside of 12$\\arcsec$ and fit the SMBH while fixing all other parameters to fit number 1 of \\citet{Do_13b}. We obtain  M$_\\bullet=3.77\\pm0.09\\times 10^6 M_{\\odot}$ fully consistent with \\citet{Do_13b}. (Our error is smaller because we had to fix most parameters.) Thus, our motions are consistent with the motions of \\citet{Do_13b}. Also the binning has no relevant influence. \nThe SMBH mass increases to about $4.5\\times10^6 M_{\\odot}$ when fitting our profiles within 220$\\arcsec$ with a single Nuker profile. In this and the previous fit the cluster mass is zero. When the cluster is included with constant M\/L we obtain  M$_\\bullet=3.33\\pm 0.00|_{\\mathrm{tracer\\,profile}} \\pm0.34|_{\\mathrm{dispersion}}\\times 10^6 M_{\\odot}$ with the Do profile and \n M$_\\bullet=2.86\\pm0.10|_{\\mathrm{tracer\\,profile}}\\pm0.23|_{\\mathrm{dispersion}}\\times 10^6 M_{\\odot}$ \n with our density data. Note that for checking whether the tracer profile is important, only the error which is caused by the tracer profile is relevant. Thus, the reason for the high mass\n of \\citet{Do_13b} is twofold: the tracer profile and the extended mass. \n In conclusion, not only high quality dynamic data is important but also good data of the surface density profile. Further, all components, also the extended mass need to be fit, also in the inner 12$\\arcsec$ due to projection effects.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.74 \\columnwidth,angle=-90]{f18a.eps} \n\\includegraphics[width=0.74 \\columnwidth,angle=-90]{f18b.eps} \n\\includegraphics[width=0.74 \\columnwidth,angle=-90]{f18c.eps} \n\\includegraphics[width=0.74 \\columnwidth,angle=-90]{f18d.eps} \n\\caption{M$_{\\mathrm{SMBH}}$ correlation for model 2 and 29 (Table~\\ref{tab:_mass_iso1}).\nThe top row shows the correlation with cluster mass, the bottom the correlation with  the inner slope $\\gamma_\\mathrm{in}$. \nBoth models use the star counts as tracer density and assume a constant mass to light ratio. The left (model 2) use the full radial range\nof the dynamics. The right (model 29) use only the data inside of 27$\\arcsec$.\n The dark gray shows the 1$\\,\\sigma$ area from MCMC,  the light gray the 1$\\,\\sigma$ area.\n} \n\\label{fig:corr_plot3}\n\\end{center}\n\\end{figure*}\n\n\n\\citet{Schoedel_09a} obtained from isotropic Jeans modeling $M_\\bullet=3.55 \\times 10^6 \\,M_\\odot$, larger than \nour estimate and the earlier works. For their assumed distance of $R_0=8\\,$kpc the SMBH mass of \\citet{Gillessen_09} \nis within the 90\\% probability interval of \\citet{Schoedel_09a}, but in their anisotropic modeling it is excluded by 99\\%. It is interesting to clarify the differences between our analysis and \\citet{Schoedel_09a}, using their publicly available data set.  \n\\begin{itemize}\n\\item Using the \\citet{Schoedel_09a} data, their tracer profile and our two mass parameterizations we get results consistent with what these authors found. For example with the power law and R$_0=8\\,$ kpc we obtain M$_\\bullet=3.33\\pm0.56\\times 10^6 M_{\\odot}$. With our normal R$_=8.2$ kpc the mass increases to M$_\\bullet=3.59\\pm0.60\\times 10^6 M_{\\odot}$.\nHence, the details of the modeling only play a minor role.  \n\\item \nUsing our dynamical data within 27$\\arcsec$ and the tracer profile of \\citet{Schoedel_09a} we retrieve $M_\\bullet=2.88\\pm0.48\\times10^6 M_{\\odot}$ repeating the previous fit with R$_0=8.2$ kpc. That decrease in mass is not because we add also radial velocities, in the opposite: \nwhen we use only our proper motion the mass decreases slightly to $M_\\bullet=2.63\\pm0.60\\times10^6 M_{\\odot}$. When we fit our density data inside the break with a single Nuker profile, in which $\\gamma=0.5$ but the rest is free, we get $M_\\bullet=2.34\\pm0.60\\times10^6 M_{\\odot}$. \nThus, the use of the \\citet{Schoedel_09a} tracer profile increases that SMBH mass by about $0.5 \\times10^6$ $M_{\\odot}$.\nThe use of different proper motion data sets changes the mass by about $0.7 \\times10^6$ $M_{\\odot}$.\nTherefore, differences in the tracer profiles are besides the dispersion data\nthe main reason for the differences in the mass obtained between our work and \\citealt{Schoedel_09a}.\n\\end{itemize}\n\nWe now investigate the most important reason, namely the higher dispersions in the data of  \\citealt{Schoedel_09a}.\n \n\\begin{enumerate}\n\\item The largest relative differences occur within $R<2.5\\arcsec$. There, \\citet{Schoedel_09a} still have a few early-type stars\ncontaminating their sample, since they used the spectroscopy-based, but slightly outdated star list in \\citet{Paumard_06} and \nthe photometric identifications of \\citet{Buchholz_09}. Four certainly early-types are contained in their sample.\nFor example, the well-known fast  early-type\nstar S13 \\citep{Eisenhauer_05}, for which an orbit is known, is part of their sample. In contrast, we use in the central 2.5$\\arcsec$  \nonly stars which we positively identify spectroscopically as late-type stars. The issue is critical in the center,\nsince the early-type stars are more concentrated toward the SMBH than the late-type stars and thus\nshow a higher dispersion in the center. We obtain inside of 1.2$\\arcsec$ $\\sigma=9.44\\pm1$ mas\/yr using the data of \\citet{Schoedel_09a} and $\\sigma=6.92\\pm0.69$ mas\/yr from our proper motions.\nRemoving the early-type stars from the \\citet{Schoedel_09a} sample\nyields a central dispersion consistent with our value.\n\\item At R$>$15$\\arcsec$ differences occur again in the proper motion data. This could be caused by differences in the distortion correction which is more important at large radii.\nAn imperfect distortion correction enlarges dispersions artificially, and thus a smaller measured value is more likely to be correct. \n\\end{enumerate}\nOverall, we believe that our dispersions are more reliable  than the ones of \\citet{Schoedel_09a}. This is also supported by the fact that  we get an extended mass of $M_{100\\arcsec}=5.65\\pm 2.0 \\times10^6 M_\\odot$, consistent with our best value, \nwhen we restrict our dynamics data to the range R$<$27$\\arcsec$ (for fixed SMBH mass with the count profile and constant M\/L) in contrast\nto \\citealt{Schoedel_09a} (Section~\\ref{sec:oth_GC}).\n\n Why do our and other attempts of Jeans-modeling fail to recover the right SMBH mass? \nThe direct cause is that our measured dispersion within 10$\\arcsec$ is smaller than the dispersion in our (isotropic) models\nwhich obtain the right black hole, see Figure~\\ref{fig:_fits_iso3}. \nWhat is the reason for the dispersion difference and thus of the low mass?\n\n\n\\begin{itemize}\n\\item Neglecting anisotropy is probably not the reason. While \\citet{Do_13b} obtain higher SMBH masses in that case, that is mainly caused by the R$_0$-M$_{\\bullet}$ relation. \\citet{Do_13b} obtain R$_0=8.92$ kpc in their free fit, a distance larger than all recent measurements \\citep{Genzel_10}. While the data of \\citet{Schoedel_09a} is problematic in some radial ranges like the very center, we agree about anisotropy: we both get that the cluster is in the inner part slightly (not significant)  tangential anisotropy. Central tangential anisotropy decreases mass estimates, see e.g. \\citet{Genzel_00}. Therefore,  \\citet{Schoedel_09a} obtain a slightly smaller SMBH mass with anisotropic modeling. We therefore assume that the inclusion of anisotropy likely would decrease the obtained SMBH mass slightly.\n\n\\item The binning and the dispersion errors are probably not the reason. Our different binnings obtain all masses between $3.25\\times10^6$ and  $3.38\\times10^6$ M$_{\\odot}$ for the SMBH using the count profile, a constant mass to light ratio and dynamics within 27$\\arcsec$.\nThat makes it also less likely that we underestimated the dispersion errors, since in each of the binnings we use a slightly different\nbias correction, but the results are consistent. The errors are easy to calculate, especially in the center, which is most relevant for the SMBH mass, since measurement errors are significantly smaller than\n Poisson errors.\n\n\n\\item For checking whether we can recover the correct SMBH mass for another $R_0$, we use, as previously, the SMBH mass-$R_0$ relation of \\citet{Gillessen_09} for another $R_0$. Even for an unrealistically large value of\n$R_0=9\\,$kpc the SMBH mass-$R_0$ relation is not recovered. Thus, the SMBH mass underestimation is largely independent of the distance.\n\n\\item  The power law  extended mass parametrization \n yields a smaller mass than using a constant M\/L: \n$M_\\bullet=(2.68\\pm 0.51) \\times 10^6~M_\\odot$  for a free power law slope and $M_\\bullet=(2.98 \\pm 0.17) \\times 10^{6}~M_\\odot$ for a Bahcall-Wulf cusp of $\\delta_M=1.25$.\nWhile a constant mass to light yields  $M_\\bullet=(3.37 \\pm 0.16) \\times 10^{6}~M_\\odot$. Hence, a core in the stellar mass\ninstead of a Bahcall-Wulf cusp may be part of the solution.\n\n\\item \\citet{Trippe_08} and \\citet{Genzel_10} argued that a central core-like structure introduces a bias toward low SMBH masses.\nThis, however is only applicable when a tracer profile without core is used for cored data. If the correct profile contains a core and\nis modeled as such, the central core only increases the error on the central mass. In our models such an uncertainty is included via the free \ndensity profile, yet the SMBH mass falls outside of the error band. In Figure~\\ref{fig:corr_plot3}\nit is visible that the correlation of $\\gamma_{\\mathrm{in}}$ and the SMBH mass is weak. It is not possible to obtain the right mass by changing $\\gamma$ within our model.\nStill, the fact that our flux and star count profiles are not consistent indicates that their errors are probably underestimated. \nNeither produces the right mass, but maybe the true profile is none of them.  A profile with a large enough core radius might\nproduce the right mass. Another possibility is that our functional form is inadequate. Perhaps a profile works with a large radial transition region, in which the profile has a constant slope, \nsomewhat steeper than in the center.\n\n\\item   \nIntroducing a flat true core is unlikely to solve the issue. It is not only inconsistent with the apparent isotropy, \nit also under predicts the number of late-type stars with orbits. \n\n\\item Also the flattening influences the SMBH mass, both by the tracer profile, and by reducing the dispersion in most dimensions,\nsee  \\citet{Chatzopoulos_14}. Also well inside of 27$\\arcsec$ the dispersion is different in l and b, indicating that \nflattening is important also there.\nWe test that case by fitting the data within 27$\\arcsec$ with the flattened 2-integral model of  \\citet{Chatzopoulos_14}.\nWe obtain $M_\\bullet=(3.54\\pm 0.18) \\times 10^6~M_\\odot$. This error also includes the distance uncertainty.\n\\end{itemize}\n\nSummarizing, we have tested several potential solutions to the mass bias. For some we have shown quantitatively that they alone \ncannot correct the bias. Others would requires modeling which goes beyond this work. \nLikely, several together are necessary\nfor a solution.\nThe absence of a dark cusp and another more complex tracer profile\nare maybe the most likely solution.\n Perhaps maximum likelihood modeling\nis needed for our discrete data set \\citep{Dsouza_13}, in particular for properly incorporating the  information the fast stars carry.\n\n\n\n\n\\subsection{Comparison with the M$_\\mathrm{cluster}$ Literature}\n\\label{sec:oth_GC}\n\nOverall, most M$_\\mathrm{cluster}$ values from the literature are similar to our values. \nThe comparison with other works needs some care, since different values for $R_0$ have been used, and hence\nwe scale to $R_0=8.2\\,$kpc.\n We correct the masses in the literature to our distance using a mass scaling with exponent 1 for purely radial velocity based masses, and \nexponent 3 for purely proper motion based ones. \nWe extrapolate from our best estimate $M_{100\\arcsec}$ in two ways: \n on the one hand we use a broken power law with break radius 100$\\arcsec$. The inner slope is $\\delta_\\mathrm{M}=1.232$, the \nouter is $\\delta_\\mathrm{M}=1.126$. (That implies we use inside of 100$\\arcsec$ the average $\\delta_\\mathrm{M}$ of row~3 and 4 of Table~\\ref{tab:_mass_iso1} and outside the average $\\delta_\\mathrm{M}$ of row~11 and 12. The first uses dynamics between 10 and 100$\\arcsec$, the second dynamics outside of 10 $\\arcsec$.)\n On the other hand, we use one of the preferred fits with constant M\/L (Row~5 in Table~\\ref{tab:_mass_iso1})\nMost other mass profiles are similar to one of these cases. \nWe quantify the quality of the comparisons only when errors are given in literature and there is not more than one value in a source without clear preference for one.\n\n\n\\subsubsection{\\citet{Schoedel_09a}}\n\nThe possible mass range of \\citet{Schoedel_09a} within $1\\,$pc is $(0.5-2.2)\\times10^6M_{\\odot}$, similar to our result. \nHowever, when assuming a constant M\/L and isotropy they obtain a mass of $1.6 \\times 10^6 M_{\\odot}$,\nlarger than our value of $0.6\\times 10^6\\,$M$_{\\odot}$ for the same assumptions. At 100$\\arcsec$ their profile yields a\nrather large mass of $16 \\times 10^6\\,$M$_{\\odot}$. \nThe main reason for the difference is their larger dispersion outside of 15$\\arcsec$ (Section~\\ref{sec:mass_smbh}).\n\n\n\\subsubsection{\\citet{Trippe_08}}\nAt 100$\\arcsec$ \\citet{Trippe_08} found a mass that is roughly a factor three larger than ours. \nThe difference is due to the high rotation at large radii as a result of their selective use of data from \\citet{McGinn_89}.\nAt $1\\,$pc where rotation is negligible they found a value of $1.2 \\times 10^6 M_\\odot$ (their Figure~14, gray dashed curve),\nsimilar to our mass in the power law case.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=1.50 \\columnwidth,angle=-90]{f19.eps}\n\\caption{Cumulative mass profile of the GC. The main measurement of this work is the red pentagon at 4 pc, through\nwhich the profiles for the power law or constant M\/L case pass. The latter is slightly preferred. The stellar orbits based value from \\citet{Gillessen_09} is at about 0.002 pc. \n\\citet{Beloborodov_06} used an orbit roulette technique to obtain an enclosed mass.\nIn the work of \\citealt{Schoedel_09a} (gray diamond) no formal error is given, we show the largest range mentioned.\nFor the masses from \\citealt{Trippe_08} (green open circles) no errors are given. The value from \\citet{Serabyn_85} is the violet square.\nFor \\citealt{Serabyn_86} the thick black line is the value and the light gray area gives the error range.\nFrom \\citealt{Lindqvist_92b}  (light violet triangles) and \\citealt{McGinn_89} (green stars) we use the Jeans \nmodeling values. In case of \\citealt{Deguchi_04} (light green line) we show their extended mass model fit using the Boltzmann equation, \nadapting to our assumed SMBH mass.\n We plot Jeans-modeling based values from \\citealt{Genzel_96} (pink dots).\nWe also plot the two-integral fit of  by \\citet{Chatzopoulos_14} (pink curve).\nWe omit the values from \\citet{Genzel_96} and \\citet{McGinn_89}  inside of 0.55 pc, since more accurate values are available there.\n} \n\\label{fig:_mass_lit1}\n\\end{center}\n\\end{figure*}\n\n\n\\subsubsection{Inner Circumnuclear Disk}\n\n\nThe gas in the in circumnuclear disk (CND) can be used to obtain a mass estimate.\n\\citet{Serabyn_85} used the emission of [NeII] 12.8 $\\mu$m from gas streamers \n at the inner edge (including the Western arc) of the CND to obtain its rotation velocity. They found   \n  $ (3.9 \\pm 0.8) \\times 10^6 M_{\\odot}$ at 1.4 pc. \n  That is 1.7$\\,\\sigma$ and  2.4$\\,\\sigma$ less than our constant M\/L and  power law estimates. \nTheir mass is smaller than the mass of the SMBH.\nThe projected rotation velocity of the inner CND of $100\\,$km\/s is well-determined \\citep{Genzel_85,Serabyn_85,Guesten_87,Jackson_93,Christopher_05},\nand hence the reason for the discrepancy has to be in the model assumed.\n Conceptually, the mass derivation of \\citet{Serabyn_85} was simple: they assumed circular motion of all the gas in one ring\n with an inclination of $60-70^{\\circ}$. \n\nIn the HCN J-0 data of the inner CND in \\citet{Guesten_87} the \nvelocities in the southern and western parts follow the model of \\citet{Serabyn_85}, but not in the northern and \neastern parts. The latter can be described by a less inclined ring ($\\approx45^{\\circ}$) with an intrinsic rotation velocity of \n$137\\pm8 $ km\/s. This velocity results in a total mass $(6.1\\pm 0.7)\\times10^6 M_{\\odot}$ somewhat larger than our estimates. Using an average inclination \nbetween \\citet{Serabyn_85} and \\citet{Guesten_87} would hence yield a mass estimate very similar to ours.\n\nHowever, \n\\citet{Zhao_09} find an inclination consistent with \\citet{Serabyn_85}. \nFurther \\citet{Zhao_09} show that a single \norbit, even slightly elliptical, cannot fit all gas streamers in the Western arc, and hence the inner CND is likely more complicated than \nassumed by \\citet{Serabyn_85}. \n\n\nAnother possibility to solve the discrepancy between our result and the result\nof \\citet{Serabyn_85} is that the assumption of \\citet{Serabyn_85} that the line width is due to nongravitional processes\nmay be wrong. If the line width is due to asymmetric drift \\citep{Binney_08} the true circular velocity $v_{\\mathrm{c},\\mathrm{true}}$ would be higher\nthan the measured velocity $v_{\\mathrm{c},\\mathrm{obs}}$ \\citep{Kormendy_13}:  \\begin{equation}\nv^2_{\\mathrm{c},\\mathrm{true}}=v^2_{\\mathrm{c},\\mathrm{obs}}+x \\times \\sigma^2 \n\\end{equation}\nThe factor $x$ depends on the disk profile and other not well known parameters of the \ninner CND. It is between 1 and 3. \nFrom $\\sigma=35\\pm5\\,$km\/s \\citep{Serabyn_85} follows for $x=$3\nan increase in the mass by 30\\% to $ (5.1 \\pm 1.04) \\times 10^6 M_{\\odot}$. This would be consistent with our constant M\/L estimate of $5.27 \\times 10^6 M_{\\odot}$.\n\n \\subsubsection{Outer Circumnuclear Disk}\n\n\\citet{Serabyn_86} used CO 1-0 in the outer parts of the CND for \nestimating the mass using the same method as \\citet{Serabyn_85}.\nThis measurement is consistent with our mass. The agreement argues\nthat other forces besides of gravity are not relevant for at least the outer gas dynamics.\n\n \\subsubsection{\\citet{Rieke_88} }\n\nAt radii larger than $7\\,$pc the gas velocities get unreliable \\citep{Serabyn_86,Guesten_87}, but  stellar velocities are available.\n\\citet{Rieke_88} used a few bright stars and found a dispersion of $75\\,$km\/s between 6$\\arcsec$ and 160$\\arcsec$, independent of radius. With our\ndata we can reject the hypothesis of \\citet{McGinn_89}, proposing that a magnitude effect causes the surprisingly flat\ndispersion curve. Possibly, \\citet{Rieke_88} were limited by the low-number statistics.\n\n\\subsubsection{\\citet{McGinn_89} }\n\nThese authors covered a similar area in size to ours, using integrated velocities and dispersions in large beams.\nBroadly their dispersion data is consistent with our data, and they measure a somewhat stronger radial dispersion trend.\nThey obtain masses similar to our estimates and also find a too small SMBH mass, like most works that apply free Jeans modeling.\nAt the outer edge the masses of \\citet{McGinn_89} are somewhat larger than ours, due to the high rotation velocity in this work\n(Section~\\ref{sec:rotation}).\n\n\n\\subsubsection{\\citet{Lindqvist_92b}}\n\nUsing maser velocities and Jeans modeling out to much larger radii \\citet{Lindqvist_92b} obtained \nmasses consistent with our model. This is not surprising, since we include their velocities in our data set.\nThe main discrepancy occurs around 17 pc, where their estimate is only half of our extrapolation. \nOn the one hand, our modeling has not much flexibility at these large radii. On other hand, their mass profile had possibly too much freedom since the light profile shows no dip there.\n In both our and the analysis of \\citet{Lindqvist_92b} the deviation from spherical symmetry is not considered,\nwhich might lead to significant changes at large radii.\n\n\\subsubsection{\\citet{Deguchi_04} }\n\n\\citet{Deguchi_04} used also maser velocities, but in a somewhat smaller area than \\citet{Lindqvist_92b}. \nThey prefer a result based on a new method developed from the Boltzmann equation. This method includes rotation, but assumes\nspherical symmetry and a fixed extended mass slope of $\\delta_\\mathrm{M}=$1.25. \nTo compare their extended mass result with\nour work we need to adapt their result to our SMBH mass, which is  within their 1.5$\\,\\sigma$ range.\nIn the inner parsec their mass of $(0.90\\pm 0.07)\\times 10^6 M_{\\odot}$ is consistent with our range, but\nat 100$\\arcsec$ their estimate of $(5.06 \\pm 0.39) \\times 10^6 M_{\\odot}$ deviates by about 1.5$\\,\\sigma$ from our estimate. \nDue to their fixed slope their mass deviates also at larger radii from the results in \\citet{Lindqvist_92b}.\nA fixed slope does not describe the stars density in the GC well out to 80 pc.\n\n\\subsubsection{\\citet{Genzel_96}}\n\n\nThis work obtained the mass distribution out to 20 pc from Jeans modeling of radial velocities,\nusing  literature radial velocities in combination with newly measured velocities in the center. Also these authors\nunder predict the SMBH mass. Beyond that, their results are consistent with ours.\n\n\\subsubsection{\\citet{Chatzopoulos_14}}\n\nAs discussed also in Section~\\ref{sec:jeans_modelling}, the modeling of \\citet{Chatzopoulos_14} obtain a larger mass than we do due to the flattening. That mass is also larger than most other estimates between 2 and 50 pc. The reason is usually the same - the other models are not flattened. For example, that is the reason why \\citet{Deguchi_04} obtain a smaller mass.\n\n\\subsubsection{Works Using the Light Distribution}\n\nIt is possible to obtain mass estimates of the extended mass by using the flux and its distribution of the GC.\nIt is however difficult due to  \nfollowing systematic uncertainties in M\/L:  the star formation history, the extinction toward the GC and the IMF of the stars. Thus, in the past \\citet{McGinn_89,Lindqvist_92b,Launhardt_02} used a M\/L obtained at a selected radius with dynamics\ntogether with the flux profile to estimate the mass of the nuclear cluster. Thus the broad agreement of them with our work shows that the mass of \\citet{McGinn_89,Lindqvist_92b,Genzel_96,Genzel_97} agrees with our mass.\nWith the recent agreeing measurements of extinction \\citep{Nishiyama_06b,Fritz_11,Schoedel_09b}\n and star formation history  \\citep{Blum_02,Pfuhl_11} two uncertainties in a purely light derived mass are now reduced. \nStill the IMF essentially cannot be constrained using light information alone \\citep{Pfuhl_11}.\n Since the IMF of the GC is an interesting subject,  we reverse the argument, \nand use the mass to constrain the IMF (Section~\\ref{sec:mass_to_light}).\n\nRecently, \\citet{Schoedel_14} used their light decomposition to infer the total mass of the nuclear cluster\nof $2.1\\pm0.4\\times10^7\\, M_\\odot$. \nThat is only half of our estimate but within the error for the total mass.  The $M\/L=0.5\\pm0.1$ of \\citet{Schoedel_14}\nis nearly identical with our dynamic value (Section~\\ref{sec:mass_to_light}). \nWhile \\citet{Schoedel_14}\nused IRAC2 instead of Ks the impact on M\/L is rather small (at most 38\\%) for the GC star formation history (Section~\\ref{sec:mass_to_light}).\n\n\\subsection{Mass Cusp or Core?}\n\\label{sec:mass_cusp_core}\n\n\n\nThe distribution of old stars does not show the expected central cusp \\citep{Bahcall_76} with the inner $\\delta_\\mathrm{L}=1.25$ slope\n \\citep{Buchholz_09,Do_09,Bartko_10}.\nThe stars follow a shallower slope in the center, \nwhich can either be a true core ($\\delta_\\mathrm{L}=3$) or a shallow cusp ($\\delta_\\mathrm{L}>2$, \\citealt{Do_09}).\nAlso the second case we call core here, since it has much less light in the center than a cusp.\nBy comparing \nour power law mass models (with a central cusp) and the constant M\/L models (with a core)\nwe can constrain the central mass distribution.\n\n\n\\begin{itemize}  \n\\item Given that a core can reduce the low dispersion bias, the Jeans modeling of our data \n favors a core over a cusp. The same is found by \\citet{Schoedel_09a},\nwho used a unrealistically large, fixed break radii for light and especially mass. \nOur results show that the preference of a smaller slope in the center is \nnot only due to the large break radius in \\citet{Schoedel_09a}.\n\\item The presence of a warped disk of young stars  \\citep{Loeckmann_09a,Bartko_09}\nyields an additional constraint.\n \\citet{Ayse_13} found that with\nan isothermal cusp of \nM($r<0.5$pc)$=10^6\\,$M$_{\\odot}$ and a flattening of q$=0.9$ an initial warped disk is too quickly destroyed \n and would not be observable today. For constant M\/L we obtain a smaller mass of M($r<0.5\\,$pc)$\\approx0.15 \\times 10^6\\,$M$_{\\odot}$, which reduces the torque by a factor five, allowing the disk to survive longer. This conclusion\nis not firm, since the warping need not be primordial, and a reduced central flattening might also yield less\nwarping \\citep{Kocsis_11}. \n\\item The most reliable mass measurement around 1.4 pc apart from \nour analysis, that is\n\\citet{Serabyn_85}, is more consistent with a core than with a cusp.\n\\end{itemize}\n\nOverall, it appears likely that the mass profile shows a central core, but better modeling including a solution of the mass bias \nis necessary. \nAnother route for constraining the mass profile worth follow up is using direct accelerations of stars at radii between 1$\\arcsec$ and \n7$\\arcsec$ with\nGRAVITY \\citep{Eisenhauer_08}.\n\n\nOur preference for a core in the mass profile is interesting for theories which aim at explaining the missing light cusp.\nIn some theories the resolvable stars (giants) form a core while dark components (remnants or main sequence stars) \nform a mass cusp. Thus, our result means that mass segregation is less likely to solve the riddle of missing light cusp. \n\\citet{Keshet_09} derive a similar conclusion from different arguments. Another mechanism that destroys a light cusp, but not a \nmass cusp, is the destruction of giants by collisions \\citep{Dale_09}, and hence our results also disfavor that model.\nOne model that could work is presented in \\citet{Merritt_10} who propose that a \nrelatively recent binary black hole (merger) ejected stars. This process would yield a core both in the\nlight and the mass profile. Still, one would need to fine-tune the timing of such a model, since the early-type stars\nare concentrated toward the center, but the giants and even the younger red (super-) giants with ages down to 20 \nMyrs \\citep{Blum_02,Pfuhl_11} are not.\n\n\n\n\\subsection{Cumulative Mass Profile}\n\\label{sec:cum_mass}\n\nAnother way of expressing our results is the cumulative mass profile. In Figure~\\ref{fig:_nc2} we show three\ncases:\n\\begin{itemize}\n\\item We use a constant M\/L model with the best fitting inner slope of -0.81 (Row 5 in Table~\\ref{tab:_mass_iso1}, model A).\n\\item We use a constant M\/L model with the shallowest possible inner slope of -0.5 (and otherwise row 5 in Table~\\ref{tab:_mass_iso1}, model B).\n\\item We use a power law model with a slope of $\\delta_\\mathrm{M}=1.232$ (model C), the average of row 3 and 4 in Table~\\ref{tab:_mass_iso1}\\footnote{Outside of 100$\\arcsec$, a range not relevant here, model C uses\n $\\delta_\\mathrm{M}=1.126$, the average  of rows 11 and 12 in Table~\\ref{tab:_mass_iso1}.}. \n\\end{itemize}\nWe normalize these three models to have the overall best-fitting value of $M_{100\\arcsec} =6.09 \\times 10^6 M_\\odot$. Tabulated values of the profiles can\nbe found in Appendix~\\ref{sec_val_cum_mass}. The largest mass difference between the M\/L$=$const and the power law model is reached\n at about 35$\\arcsec$ with $0.59\\times 10^6 M _{\\odot}$. \nSince this number exceeds our mass error, it is possible to detect dynamically the cusp, provided the low dispersion bias can be solved.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f20.eps}\n\\caption{Cumulative mass profiles, for a power law model or assuming M\/L$_{\\mathrm{Ks}}=$const. The normalization\nis done for $M_{100\\arcsec} = 6.09 \\times 10^6 M_\\odot$ (black square).\nOther masses shown are the ZAMS cumulative mass distribution of the O(B)-star population \\citep{Bartko_10}\nand with a filled disk the CND from\n \\citet{Etxaluze_11,Genzel_85, Mezger_89},  \nand\n \\citet{Requena_12}. The open circle is the CND mass estimate from \\citet{Christopher_05}.\n}\n\\label{fig:_nc2}\n\\end{center}\n\\end{figure}\n\n\nExtrapolating the constant M\/L models down to the regime of the S-stars S2 and S55\/S0-102 \\citep{Meyer_12} \nyields a mass there, which is smaller than that of the individual stars. In this regime, the S-stars dominate, and\ntaking into account their Salpeter-like IMF \\citep{Bartko_10}\nor a potential mass-segregated cusp of stellar remnants \n\\citep{Freitag_06,Hopman_06} \nwould make the dominance even stronger. \nFor power law models it is not clear whether the old stars dominate the mass in the central \narcsecond. \n\nOutside the central arcsecond, the mass of the disks of early-type stars \n\\citep{Bartko_09,Lu_09} may be important. We derive the spatial distribution from Figure~2 in \n\\citet{Bartko_10} and globally deproject to space distances with a factor 1.2 the given projected distances.\nWe estimate the total ZAMS mass of that population to $1.5\\times 10^4\\,M_\\odot$ \\citep{Bartko_10}, \ndominated by the O-stars given the \n top-heavy shape of the IMF. Across the literature \\citep{Paumard_06,Bartko_10,Lu_13} the disk mass is uncertain by a factor 2.5. \nAt $r=2\\arcsec$ the O-stars might be\ncomparable in mass to the old stars, if the constant M\/L model is correct.\n\n\n\nFurther out, the only other component is the circumnuclear disk. Most \npublications \n\\citep{Genzel_85,Etxaluze_11,Mezger_89,Requena_12} agree that its\n mass is a few $10^4\\,M _{\\odot}$ and thus  irrelevant compared to the old \nstars. However, the mass of $\\approx 10^6 M_{\\odot}$ found by \\citet{Christopher_05}\nwould be about a third of our mass estimate at that radius. \n\n\nOur results also yield a new estimate for the sphere of influence of the SMBH \\citep{Alexander_05}, the radial range within which\nthe extended mass is equal to the mass of the SMBH.\nIt is completely consistent with previous values \\citep{Alexander_05,Genzel_10}. Using model A and B, \nwe get r$_{\\mathrm{infl}}=76.3\\pm5.5\\arcsec=3.03\\pm0.25\\,$pc. \n\n\n\n\\subsection{Mass to Light Ratio}\n\\label{sec:mass_to_light}\n\nWe now obtain the mass to light ratio from the fits to flux and dynamics with constant M\/L.\nThe mass to light ratio is directly calculated from the output of these models. \nWe obtain $M\/L=0.51 \\pm 0.12 \\, M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ using $M_{\\mathrm{Ks}\\,\\odot}=3.28$ \\citep{Binney_98}.\nThe error consists of 19~\\% for the light, 10~\\% for the mass, and 8~\\% for the distance uncertainty.\nThe latter number \nfollows from the distance uncertainty of 4.1\\% \\citep{Gillessen_13} multiplied by the exponent 1.83, which is the scaling M\/L with\ndistance given that M scales like 3.83 with distance (Section~\\ref{sec:jeans_modelling}).\nOur M\/L is consistent with the values in \\citet{Pfuhl_11} and \\citet{Launhardt_02}.\nOur error, however, is smaller thanks to the smaller mass error. With the improved mass to light ratio we can constrain the IMF of the old stars \nfurther. \n\n\n\nFirstly, we use the M\/L to determine the IMF slope $\\alpha$, we assume the respective star formation histories for the old stars (older than 10 Myrs) from \\citet{Pfuhl_11}. Their results \nappear to hold also at the larger radii of interest here \\citep{Blum_02}.\nAlready with the accuracy of M\/L of \\citet{Pfuhl_11} it was possible to discard extreme top-heavy IMFs ($\\alpha>-0.6$)\nlike measured for the O-stars in the GC \\citep{Bartko_10}. With our accuracy also the IMF slope\nof $\\alpha=-0.85$ \\citep{Paumard_06} is excluded.\n\n\nSecondly, we concentrate on variants of the usual -2.3 slope below one solar mass, these are the IMFs of \\citet{Salpeter_55,Kroupa_01,Chabrier_03}. \nTo check the result on dependence of the used stellar population model we use the models of \\citet{Bruzual_03} and \\citet{Maraston_05}.\nIn case of the \\citet{Maraston_05} models we obtain \n $M\/L=0.75  M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ and  $M\/L=1.28  M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ for Kroupa and Salpeter, respectively. \nFor the \\citet{Bruzual_03} models we get $M\/L=1.04  M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$\nand $M\/L=0.76  M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ for  Salpeter and Chabrier, respectively. \nThe differences in stellar population models cause the differences of up to 20\\%. (The stellar population model dependences increase with wavelength: at 4.5 $\\mu$m (IRAC2), used by \\citet{Schoedel_14} and \\citet{Feldmeier_14}, the model of \\citet{Bruzual_03} obtains $M\/L=0.71  M_{\\odot}\/L_{\\odot,\\mathrm{IRAC2}}$  for Chabrier and the model of \\citet{Bruzual_03} $M\/L=1.01 M_{\\odot}\/L_{\\odot,\\mathrm{IRAC2}}$ for Kroupa.) \nUsing the star formation history of \\citet{Blum_02} instead of the of \\citet{Pfuhl_11}\ncauses 18\\% smaller M\/L in Ks-band.\n\n\n\nRecently, deviations from the Chabrier IMF  were found for different stellar systems:\non the one hand elliptical galaxies seem \nto have  a Salpeter or more bottom heavy IMF \nas \\citet{Vandokkum_10,Conroy_12} concluded using  population modeling of integrated spectra and as \\citet{Cappellari_06,Cappellari_12}\nconcluded  from advanced dynamic modeling.\nAlready a Salpeter IMF is in strong tension with our ratio. A more bottom heavy IMF  of e.g. a slope of -2.8 has an even larger M\/L (A factor 1.57 more in M\/L compared to Salpeter for the star formation history assumed in \\citet{Cappellari_12,Conroy_12}\nand is thus firmly excluded.\n\n\nIt seems that (metal rich) globular clusters have M\/L smaller than Kroupa \\citep{Kruijssen_09a,Bastian_10,Sollima_12}. This was found in the case of M31 by \\citet{Conroy_12}\n using  population modeling of integrated spectra. In addition, \\citet{Strader_11} found the same by using mass and light measurements.\nPrecisely, \\citet{Strader_11} measured for solar metallicity ($\\mathrm{[Z]}=0\\pm0.3$) $M\/L=0.39  M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$. \nThis is already lower than our measurement, although the GC contains more medium old stars than these globulars. \nIn an attempt to include these younger stars we reduce in our Kroupa and Chabrier models the mass of the stars $>5$ Gyrs to achieve \n$M\/L=0.39  M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$\nfor these ages. We do not change the populations for the younger stars.\nThis model has the motivation that it describes a possible formation scenario of the nuclear cluster in which the majority\nthe old stars originate in big metal rich globular clusters \\citep{Capuzzo_08} and later stars formed locally are added to it.\nThis model results in $M\/L=0.31  M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ deviating by 2$\\,\\sigma$ from our measurement.  \nThis argument against a globular cluster origin of the nuclear cluster is weakened by the fact that\nwhen globulars arrive in the GC, they likely have a different M\/L due to mass segregation. In case of M31 it is argued\nthat the low M\/L is mostly due to a different IMF and not only due evolutionary effects \\citep{Strader_11}. \nHowever, the preferred loss of low mass objects due to internal mass segregation and the outer tidal field\nexists certainly \\citep{Bastian_10,Sollima_12} and should be further enhanced close to the GC. \nThe problem is that, while mass segregation first reduces the M\/L, M\/L is enhanced in late stages during the late remnant dominated state of\nold populations \\citep{Sollima_12,Marks_12}. Thus, without detailed modeling of mass segregation \nthe expected M\/L of a globular cluster dominated nuclear cluster is uncertain.\n\n\nThe GC shows a normal ratio of diffuse light to the total light \\citep{Pfuhl_11}.  \nThis ratio measures the ratio of main sequence stars to giants and decreases monotonically from a bottom-heavy\nto a top-heavy IMF.  Thus it excludes best a cluster dominated by remnants and giants - a possible state of globular\nclusters inspiraling to the GC.\n\nThe Chabrier IMF  common in Galactic disk and bulge \\citep{Bastian_10,Zoccali_00} is close to the M\/L of the medium old and \nold stellar populations of the GC. However, the deviation from it is\n1.7 $\\sigma$ when using the most recent star formation history determination \\citep{Pfuhl_11}. \nThat deviation vanishes when the bias to small masses due to our spherical modeling is corrected, see \\citet{Chatzopoulos_14}. In that case the ratio is  $M\/L=0.76 \\pm 0.18 \\, M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$ matching well the Kroupa\/Chabrier IMF.\nTherefore a globular cluster origin of the nuclear cluster \\citep{Antonini_12,Gnedin_14} is somewhat disfavored by its M\/L and \nL$_{\\mathrm{diffuse}}$\/L$_{\\mathrm{total}}$. \n\n\n\n\\subsection{The Nuclear Cluster of the Milky Way in Comparison}\n\\label{sec:mass_nucl}\n\n\nWe now compare the nuclear cluster of the Milky Way with nuclear clusters in other galaxies.\nThe literature about the mass of other nuclear clusters \\citep{Walcher_05,Barth_09} is sparse and biased toward brighter nuclear clusters. \nWe therefore use mainly the light for comparison, for which ample data from HST imaging are available \\citep{Carollo_02,Boeker_04}.\n\\citet{Boeker_04} studied late-type spiral (Scd to Sm)\nwhile \\citet{Carollo_02} concentrated on early-type spirals (Sa to Sbc).\n For color-correction we use $\\mathrm{'H'}-\\mathrm{Ks}=0.15$ and $\\mathrm{I}-\\mathrm{Ks}=1.1$ for \\citet{Carollo_02} and \\citet{Boeker_04}, respectively. \n We compare the obtained magnitudes \nwith our GALFIT decomposition for the nuclear cluster (Section~\\ref{sec:flattening}), of the double $\\gamma$ model fitting (Section~\\ref{sec:fitting_profile_gam}).\nIn addition we use the cumulative flux as function of radius. The latter is not affected by decomposition uncertainties,  (Figure~\\ref{fig:_comp_nc}).\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.70 \\columnwidth,angle=-90]{f21.eps} \n\\caption{\nSizes and luminosities of nuclear clusters. We show the half light radii for clusters in late-type \\citep{Boeker_04} and early-type\nspiral galaxies \\citep{Carollo_02}.\nFor the Milky Way we present the inner component of our GALFIT decomposition (green box), of the $\\gamma$ model fitting (green triangle) and the total cumulative flux as function of the radius.\n} \n\\label{fig:_comp_nc}\n\\end{center}\n\\end{figure}\n\n\nThe size of the nuclear cluster of the Milky Way is typical. \nIt is smaller than many clusters in \\citet{Carollo_02} and larger than most clusters in \\citet{Boeker_04}. \nThis simply might be a consequence from the fact that the galaxy type of the Milky Way is broadly\nbetween the two samples.\n\n\n\nIt is visible in Figure~\\ref{fig:_comp_nc} that the nuclear cluster of the Milky Way has an unusually high surface brightness.\nNot only its characteristic brightness is high, but \nalso the cumulative brightness profile lies above nearly all other clusters. \nWhat is the reason for that?\n\\begin{itemize}\n\\item\nThe young ($\\approx 6$ Myrs) and medium old ($\\approx 200$ Myrs) stars in GC are likely not the reason for this offset. They are subdominant \ncompared to the old stars in the Ks-band \\citep{Blum_02,Pfuhl_11} and Section~\\ref{sec:total_luminosity}.\nFurther many nuclear clusters \\citep{Rossa_06,Walcher_06} contain significant fractions of young or medium old stars.\n Thus, the M\/L of the GC is probably typical.\n\\item\nExtinction is probably not an issue although we correct for it. \nThe spirals in the sample of \\citet{Carollo_02} and \\citet{Boeker_04} are seen more or less face-on. \nFurther, the mean IR color in \\citet{Carollo_02} yields a \nsmall extinction of A$_{160W}\\approx0.2$. Also the spectroscopic study of \\citet{Rossa_06} obtained small extinctions for the\nsame samples.\n\\item\nThe higher resolution of our data is also not the reason, since we compare the characteristic brightness within 4 pc, \na size which can be resolved in most of the galaxies of \n\\citet{Carollo_02} and \\citet{Boeker_04}.  \n\\item\nThe nuclear cluster of the Milky Way has a projected mass density of $\\log{\\Sigma_e}=5.38 \\pm 0.17$ [M$_{\\odot}$\/pc$^2$]. This is larger than for all 11 nuclear\nclusters in the sample of \\citet{Walcher_05}. \n\\end{itemize}\nWe conclude that the main reason for high surface light density is the high stellar mass density in the nuclear cluster, \ncompared to other galaxies. \n\n\nFor some near-by galaxies a more detailed comparison is possible.\nFor example M33 shows a central star density which is possibly larger than in the nuclear cluster of the Milky Way  \\citep{Lauer_98}.\nM33 does not contain a SMBH \\citep{Merritt_01,Gebhardt_01}, and the old stars - if present at all - are outshined by young stars.\nSome nuclear clusters show flattening and rotation, in case of metallicity the flattening is 16\\% \\citep{Lauer_98} combined with \nsome rotation \\citep{Kormendy_93}.  The large nuclear cluster in NGC4244  (R$_e\\approx10\\,$pc) is visually flattened and has \n$v_{\\mathrm{rot}}\/\\sigma\\approx1$ \\citep{Seth_08}. NGC404 is a similar case \\citep{Seth_10}.\nThus the flattening of the Milky Way nuclear cluster \n is in the large range of possible shapes. \n\n \\citet{Carollo_99} finds that nuclear cluster with M$_V\\lesssim-12$\nare typically associated with signs of circumnuclear star formation, like dust lanes and an HII spectrum.\nIn this respect the cluster of the Milky Way with M$_V\\lesssim-13$ seems typical for its nuclear disk.\nPreferentially, bright nuclear clusters are observed together with  nuclear disks, \n which might be explained \nby a physical coupling of the two systems.   \nSince the GC is dominated by old stars this connection is likely not only valid for  the recent star formation event but also for the older\npopulation. \n\n\\section{Conclusions and Summary}\n\\label{sec:conclusions}\n\n\n\nThe nuclear cluster in the Milky Way is by far the closest and thus can be studied in more detail\nthan other nuclear clusters.\nIn this paper we investigate not only its center, but its full size to compare\nit with such clusters in other galaxies.  \nTo that aim, we obtain its light profile out to 1000$\\arcsec$ and measure motions in all three dimensions out to 100$\\arcsec$,\nexpanding on the works of \\citet{Trippe_08} and \\citet{Schoedel_09a}.\\\\\n\n\\begin{itemize}\n \\item \nWe construct a stellar density map with sufficient resolution for two dimensional structural analysis out to $r=1000\\arcsec$.\nThis map shows in the central 68$\\arcsec$ an axis ratio of\n $q=0.80\\pm0.04$. \n Further out the flattening increases. \nWe fit this map by two components, the inner is a Sersic, the outer a Nuker profile.\nThe more spherical and smaller component is the nuclear cluster, while the more extended component corresponds to the nuclear disk (stellar analog of the central molecular zone).\nThe decomposition is uncertain because the outer component is relatively bright.\nWith our preferred decomposition the nuclear cluster has a half light radius \nof 127$\\arcsec\\approx5.0\\,$pc, a flattening of 1\/0.80, a Sersic index of n$=1.5\\pm0.1$, \nand a total luminosity of M$_{\\mathrm{Ks}}=-15.5$. The outer component has a flattening of about 1\/0.264 and contributes about 13\\% to the projected central flux.\nThe size and luminosity of the nuclear cluster depend on the functional form of the assumed outer slope, which\nis not well constrained by our data. A flatter profile like a $\\gamma$ model yields r$_e\\approx9\\,$pc and \na total luminosity of M$_{\\mathrm{Ks}}\\approx-16.5$.\n\\item Our dynamical analysis shows that\nthe proper motions are radially \nisotropic out to at least 40$\\arcsec$. In the center the velocity distribution shows strong wings\nas expected due to the presence of the SMBH. \nAssuming that the fastest stars in the center are bound, we set a lower bound to the SMBH mass of \n$3.6\\times 10^6\\,$M$_{\\odot}$.\nAssuming in addition that the black hole mass is known to 1.5\\% \\citep{Gillessen_09} for a given distance, we can estimate\n $R_0=8.53^{+0.21}_{-0.15}$ kpc. Our new radial velocities show that the projected rotation velocity increases only weakly outside of 30$\\arcsec$,\nand thus the nuclear cluster rotates less than assumed in \\citet{Trippe_08}.\n\n\n\\item  We use the motions of more than 10000 stars for isotropic, spherically symmetric Jeans modeling. As a tracer profile we use either\n stellar number counts or the light profile, and the mass model is either a power law model or assumes constant M\/L.\nForcing the mass of the SMBH to its known value\n we measure for the extended mass\na power law slope of $\\delta_\\mathrm{M}=1.18\\pm 0.06$, consistent with the light profile. Our best mass estimate\nis obtained at $r_{\\mathrm{3D}}=100\\arcsec$ as an average over the power-law and constant M\/L models. We find\n$M_{100\\arcsec} = (6.09 \\pm 0.53|_{\\mathrm{fix} R_0}\\pm 0.97|_{R_0} ) \\times 10^6 M_{\\odot}$ \nThe error contains\ncontributions from the uncertain surface density data and from the uncertainty in $R_0$. \nDeviations from isotropy and spherical symmetry are not included in our error calculation. The most important\ndeviation is the observed flattening, a model which includes it increases the mass by about 47\\%, see \\citet{Chatzopoulos_14}.\nOur $\\gamma$-modeling yields a total cluster mass of\nM$_\\mathrm{NC}=(4.22 \\pm 0.50|_{\\mathrm{fix} R_0}\\pm 0.67|_{R_0}) \\times 10^7$  M$_{\\odot} $. As in case of the light, the total mass is model dependent. A model with faster outer decay like an exponential, has a smaller total mass than our fit.\n\n \\item The preference for a too small SMBH mass in the Jeans modeling can be interpreted as an argument for a central, core-like structure, unlike\n to the expected cusp \\citep{Bahcall_76}. Hence, the deficit in the center would be present not only in the light, but also in the mass profile.\nThe missing mass cusp makes other explanations for the missing cusp in the light less likely, such as\n mass segregation and giant destruction \n\\citep{Dale_09} less likely.\n However, we think that it is premature to draw this conclusion firmly. One would need to develop a model \nthat fits the surface density data and achieves the right SMBH mass. That needs probably at least one of the \nfollowing elements:  (i) a relatively large core-like structure in the tracer distribution,\nwhich possibly can be found by non-parametric fitting of the density profile, \nor (ii) the inclusion of an outer background which contributes a significant number of stars \nalso in the center. \n\n \\item  We obtain a mass to light ratio of $M\/L=0.51 \\pm 0.12 \\, M_{\\odot}\/L_{\\odot,\\mathrm{Ks}}$. This\nis 1.7$\\,\\sigma$ smaller than the value for a Chabrier IMF. However, since the mass error does not include all contributions, e.g. the contribution from the flattening is not included, it is consistent with a Chabrier IMF.\n\n \\item The obtained half light radius of the nuclear cluster of 4 to 9 pc is typical compared to extragalactic nuclear clusters. However, it is brighter \nand has a higher light and mass density than most other nuclear clusters. \nPossibly, this large density is connected to the nuclear disk further out whose\nsurface brightness also seems high. \n\n \\item The abundance of young stars and molecular clouds in the nuclear disk and the nuclear cluster supports\n the idea that at least the young and medium old stars in the GC formed in-situ. \n Both the nuclear cluster and the nuclear disk are flattened, although the strength of the flattening varies. The different amount of flattening can be interpreted as an argument for a different origin of the two components, a local origin for the disk and a globular cluster origin for the cluster.  The ratio of diffuse light to total light\nappears to be normal, which contradicts a globular cluster origin, since they are likely dominated by stellar remnants when they arrive in the GC. \nFurther, the close association of the bright nuclear cluster with the bright nuclear disk is suggestive of a common origin. Finally, the metallicity of most nuclear cluster stars is close to solar \\citep{Cunha_07,Ryde_15,Do_15} agreeing with an in-situ origin.  Only 5 of the 83 stars of \\citet{Do_15} are compatible with the metallicity [Fe\/H]$\\approx-0.75$ (\\citet{Harris_96}, private communication with Christian Johnson) of bulge globular clusters. Because metallicities are an important discriminator, it would be good to confirm the medium resolution metallicities of \\citet{Do_15} with higher resolution spectroscopy.\nIn conclusion, it seems likely that the majority of the nuclear cluster stars originated not in globular clusters but more locally.\n\n\\end{itemize}\n\nTo improve further the constraints on mass, shape, and origin of the nuclear cluster we  use the data presented here\nfor axisymmetric modeling in \\citet{Chatzopoulos_14}. This lifts the assumptions of spherical symmetry and add rotation.\n\\newline\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\noindent  Tsallis' thermostatistics is today a new paradigm for\nstatistics, with applications\n to several scientific disciplines \\cite{gellmann,1988,PP1997,FMP,TMP}.\n Notwithstanding  its\n manifold applications, some details of the basic thermostatistical formalism\n  remain unexplored. This is why {\\it analytical} results are to be\n  welcome, specially if they are, as the ones to be here\n  investigated, of a very general nature.\n  We will provide this type of results for discrete\n  probability distributions  of fixed variance that maximize Tsallis' entropy.\n\n\\noindent Given a discrete probability distribution (DPD)  $p= \\{\np_{k} \\}$, its Tsallis' information measure (or entropic form) is  defined as\n\\begin{equation}\nH_{q} (  p )  =\\frac{1}{q-1} (  1-\\sum_{k=-\\infty}^{+\\infty}\np_{k}^{q} )  . \\label{dino} \\end{equation} It is a classical result that\nas $q \\rightarrow1,$ Tsallis entropy reduces to\nShannon entropy\n\\[\nH_{1} (  p )  =-\\sum_{k=-\\infty}^{+\\infty}p_{k}\\log p_{k}.\n\\]\nWithout loss of generality, we will here consider only centered\nrandom variables of fixed variance.\n\n\\vskip 2mm \\noindent The aim of this paper is to provide accurate\nestimates of i) the parameters of the DPD's and ii) their behavior.\n  The maximizers of Tsallis' information measure under variance\nconstraint in the {\\it continuous, multivariate case} have been\ndiscussed in \\cite{next2003}. \\vskip 3mm \\noindent  Consider a quantum\nsystem whose eigenstates are characterized by a set of quantum\nnumbers\n that we collectively denote with an integer, running index $k$ (Cf. Eq.\n (\\ref{dino})), that will of course also label the\neigensolutions ($\\vert \\psi_k \\rangle,\\,\\,\\,\\,\\,\\epsilon_k$) of\nthe  pertinent time-independent Schr\\oe dinger equation\n\n\\begin{equation} \\label{0q1} \\mathcal{H}\\, \\vert \\psi_k \\rangle =\n\\epsilon_k\\,\\vert \\psi_k \\rangle,\n \\end{equation} with $\\mathcal{H}$ the Hamiltonian. Let\n $p_k \\equiv \\vert \\psi_k \\vert^2$ be the probability of  finding\n our system in the state $\\vert \\psi_k \\rangle $. The mixed state\n\n \\begin{equation} \\label{0q2} \\rho = \\sum_k\\,p_k\\,  \\vert \\psi_k \\rangle \\langle \\psi_k \\vert, \\end{equation}\ncommutes with the Hamiltonian by construction and represents  thus\na bona fide possible stationary state of the system. \\noindent If we now\nfind a physical quantity $Z$ whose mean value is proportional to\nthe variance, we can interpret $\\rho$ as the mixed state that\nmaximizes Tsallis measure subject to the a-priori known\nexpectation value of such a physical quantity. We will show below\nthat, in these circumstances,  \\fbox{ {\\it universal expressions}\ncan be given for the $p_k$'s, and thus for $\\rho$}.\n\n\n\\noindent We discuss possible applications in the forthcoming section.\nAfterwards, after having introduced some definitions and\nnotations, we characterize the discrete Tsallis maximizers for\nfixed variance in both the $q<1$ and the $q>1$ cases, and discuss\nthermal stability questions. We pass then to\n analyze extensions to the multivariate cases.\n For the sake of completeness,  some of the proofs of\nassertions  referenced to in what follows are given in  Annex.\n\n\\section{Possible physical applications}\n\n\n\n\\noindent Several physical models can be adapted to the scenario\ndescribed above (Cf. Eqs. (\\ref{0q1}) and (\\ref{0q2}))  The\nweights $p_k$ in (\\ref{0q2}))  will be of the Tsallis-power law\nform. With these weights, $\\rho$ maximizes Tsallis' entropy\nsubject to the constraint of a constant variance, which introduces\na Lagrange multiplier that we will call $\\beta$.\n\n\\noindent Given  a physical quantity $Z$ whose mean value is\nproportional to the variance,  $\\rho$ is that mixed state which\nmaximizes Tsallis measure subject to the a-priori known $\\langle Z\n\\rangle-$value. In the examples below $Z$ is the system's energy\n$E$, but many other possibilities can be imagined. Thus, $\\rho$\nwill be the state maximizing Tsallis' $H_q$ for a fixed value of\nthe expectation value $U =\\langle E\\rangle=\nTr\\,[\\mathcal{H}\\,\\rho]$ of the Hamiltonian. As a consequence, the\nmultiplier $\\beta$ can be thought of as an ``inverse temperature\"\n$T$, that is, set $\\beta=1\/(k_BT)$, with $k_B$ the Boltzmann\nconstant. This is so because we are at liberty of imagining that\n$U$ is kept constant because it is in contact with a heat\nreservoir \\cite{reif}. Of course, this is not necessarily the\ncase. $\\rho$ exists by itself and is a legitimate stationary mixed\nstate of our system. But we can think of $\\beta$ as either a\n``real\" or an ``equivalent\" inverse temperature.\n\n \\noindent Consider, for example,   a system for which the energy spectrum\nconsists of a denumerable set   of $N$ ($N$ possibly infinite)\n energy levels labeled by a {\\it quantum} number $k$ with\n $p_{-k}=p_{k},$\nso that all levels exhibit a degeneracy\n\n\\begin{equation} g_k=2\\,\\,\\, for \\,\\,\\,k \\ne 0;\\,\\,\\,k > 0;\\,\\,\\,and\n\\,\\,\\,g_0=1, \\end{equation} i.e., the sums in (\\ref{dino}) run from $0$ up to\n$\\infty$ and each summand is multiplied by $g_k$. Within the\npresent framework, $U$ \\footnote{Mean values linear in the PD are\nemployed here. They are quite legitimate and were used by Tsallis\nin his seminal 1988 paper \\cite{1988}. The Legendre structure of\nthermodynamics is  definitely respected using them in conjunction\nwith Tsallis' entropy \\cite{PP1997}. Recently, Ferri, Martinez and\nPlastino have shown \\cite{FMP} that PD's obtained in this manner\ncan be easily translated into PD's constructed via MaxEnt using\n$q-$expectation values evaluated \\`a la Tsallis-Mendes-Plastino\n\\cite{TMP}, showing thereby that there is a one-to-one\ncorrespondence between the two types of PD.}\n\n\\begin{equation} U= \\langle E \\rangle = \\sum_{k=0}^{+\\infty} g_k p_{k} E_k,\\end{equation}\n becomes numerically equal to the DPD's variance, which by definition is fixed and assumedly known a priori.\n As just stated, we may think,\n if we wish, that our system is in contact with a heat reservoir,\n which fixes the mean energy, and that\n the associated Lagrange multiplier $\\beta$\n can be assimilated to an inverse temperature $T$.\n Among many examples of such a scenario we  mention here:\n\n\n\n\n\n\\begin{itemize}\n\\item\n the planar rotor \\cite{dictio}, where $k$ is the magnetic quantum number\ncorresponding to the azimuthal angle usually denoted by $\\phi$.\nThe level-energies $E_k$ are proportional to  $k^2$ and \\begin{equation} \\label{1} E_k = C_E\nk^2;\\,\\,\\,C_E\\,\\,\\, {\\rm has \\,\\,\\,dimension\\,\\,\\, of\\,\\,\\,\nenergy.}\\end{equation} We have $C_E=\\hbar^2\/2M_I$, with $M_I$ the system's moment of inertia \\cite{Wyllie}.\n For simplicity's sake we take here $C_E=1$, but\nretain, of course, its energy-units.\n\\item the three-dimensional  rigid rotator \\cite{dictio}, although $k$ now means the quantum number $L$ associated\nto orbital angular momentum (for large $k$, the spectrum of (\\ref{1})\nlooks like that of the 3D rigid rotor) and, for all $k \\ge 0$, $g_k=1$.\n\\item  a vibrating string of length $2l$ with fixed ends\n\\cite{mandl}, whose energy eigenvalues are $E_k \\propto\n[k\\pi\/2l]^2$. \\item  the case of a particle (of mass $m$) in a box subject to periodic boundary conditions\n(at $(-L,\\,\\,+L)$) whose energy values are $\\epsilon_k=[2\\pi^2\\hbar^2 k^2\/(mL^2)]$ \\cite{reif}.\n\\end{itemize}\n\n\n\n\n\n\n\n\\section{Definitions and Notations}\n\n\\noindent In what follows, $m$ is a positive real number and $n$  a positive\ninteger. Often, $m$ will take the special form of an odd integer\n$m=2n+1.$\nThe solutions to the problem of maximization of Tsallis\nentropy  ``under variance\nconstraint\" (here equivalent to ``for fixed mean energy\") will be called discrete Tsallis-probability laws (DTPL's) in\nthis manuscript. Since (see preceding Section) fixed mean energy can be thought of as implying contact with a heat reservoir at the fixed temperature $T=1\/(k_B \\beta)$ we will call the pertinent Lagrange multipliers $\\beta$ inverse temperatures in what follows.  DTPL's  are given by the following two theorems.\n\n\\begin{theorem}\n\\noindent If $\\frac{1}{3}<q<1$, for $k_BT=1\/\\beta$, with $k_B$ the Boltzmann\nconstant, the discrete probability law $p$ with zero mean and mean\nenergy $U$ (equivalently, variance $\\sigma^{2}$) that maximizes\nTsallis' entropy is defined as\\footnote{the dependence of $p$ on\n$\\beta= 1\/k_BT$  is omitted for notational\nsimplicity}\n\\begin{equation}\np_{k}=\\Pr \\{  X=k \\}  =f_{q}^{-1} (T  )  (\n1+\\frac{k^{2}}{k_BT} ) ^{\\frac{1}{q-1}}\\;\\;\n\\forall\nk\\in\\mathbb{Z^+},\n\\label{discreteq>1}\n\\end{equation}\nwhere $f_{q} (  T )  $ is the partition function, given by\n\\begin{equation}\nf_{q} ( T )  =\\sum_{k=0}^{+\\infty}g_k\\, (  1+\\frac{k^{2}}\n{k_BT} )  ^{\\frac{1}{q-1}}, \\label{faqle1}\n\\end{equation}\nand $\\beta=1\/k_BT$ is a real positive Lagrangian multiplier such\nthat\n\\[\nU \\equiv \\sigma^{2}=\\sum_{k=0}^{+\\infty}g_k k^{2}p_{k}.\n\\]\nIn the limit $T  \\rightarrow \\infty$, our probability distribution\nconverges to the\n classical discrete Gaussian distribution \\cite{Zb}.\n\\end{theorem}\n\n\\noindent Notice  the existence of a lower  bound $q=1\/3$. Smaller $q$\nvalues are unphysical because  for them the mean energy of the\nmodel (for the PD  (\\ref{discreteq>1})) becomes infinite.\nAdditionally (see below) the system turns out to be thermodynamically\nunstable because the specific heat becomes negative. In this case where $q<1$, we define\nthe real positive parameter $m$ that we need below by\n\\begin{equation}\nm=\\frac{1+q}{1-q}.\n\\label{mforq<1}\n\\end{equation}\n\n\n\n\n\n\\begin{theorem}\n\\noindent If $q>1,$ the discrete probability distribution $p$ with zero mean\nand mean energy $U$ (equivalently, variance\n$\\sigma^{2}$) that maximizes Tsallis entropy is defined as\n\\begin{eqnarray} p_{k}&=&\\Pr \\{  X=k \\}  = \\Bigg\\{\n\\begin{array}\n[c]{ll}\nf_{q}^{-1} ( T )\n ( 1-\\frac{k^{2}}{k_BT} ) ^{\\frac{1}{q-1}} & \\forall\nk\\in [0, \\lfloor k_BT \\rfloor -1 ] \\\\\n0 & \\text{otherwise}\n\\end{array}\n\\label{discreteq<1}\n\\end{eqnarray}\nwhere $f_q  (  T )  $ is the partition function\n\\begin{equation}\nf_{q} ( T )  =\\sum_{k=0}^{ \\lfloor \\beta \\rfloor\n-1}g_k ( 1-\\frac{k^{2}}{k_BT}  ) ^{\\frac{1}{q-1}},\n\\label{faqge1}\n\\end{equation}\n$ \\lfloor k_BT \\rfloor $ denotes the integer part of\n$k_BT$ whose inverse, namely, $\\beta$, is a real positive\nLagrangian multiplier such that\n\\[\nU=\\sigma^{2}=\\sum_{k=0}^{+\\infty}g_k\\, k^{2}p_{k}.\n\\]\n\\end{theorem}\nIn the present case where $q>1$, we define   $m$ as a real positive parameter related to $q$ by\n\\begin{equation}\nm=\\frac{q+1}{q-1},\n\\label{mforq>1}\n\\end{equation}\nto be of use below. Note that this definition differs from definition (\\ref{mforq<1}) in the case $q<1$.\n\n\\noindent We do not give the proofs of theorems (1) and (2) here. That\nlaws defined by (\\ref{faqle1}) and (\\ref{faqge1}) are the\nsolutions of the Tsallis entropy maximization problem can be\nobtained by extending the results of the continuous case as\npresented in \\cite{next2003}.\n {\\it The cases i) $q<1$ and ii) $q>1$ differ essentially by the\nfact that the latter has a finite support.} As far as we know,\nthis  is a novel way, in the $q-$literature, of dealing with the celebrated Tsallis-cut-off \\cite{1988}. \\vskip 3mm\n\n\\noindent We remark on the fact that the forms (\\ref{discreteq>1}) and\n(\\ref{discreteq<1})  coincide with the distribution-form given by\nTsallis in his pioneer 1988 paper \\cite{1988} (see also\n\\cite{TMP}), namely, \\begin{equation} \\label{tsallis} p_k= Z_q^{-1} [1-(q-1)\n\\beta^* k^2]^{1\/(q-1)},\\end{equation} that, for $q  \\rightarrow 1$, tends to\nthe discrete Gaussian  \\cite{1988} \\begin{equation} Z_1^{-1}\n\\,\\exp{[-\\beta^*k^2]}. \\label{gauss}\\end{equation} In the former case (Eq.\n(\\ref{discreteq>1})) we have $q<1$, while in the later (Eq.\n(\\ref{discreteq<1})) $q>1$. Thus, in Eq. (\\ref{discreteq>1}) we\nhave $\\beta= (1-q) \\beta^*$, while for (\\ref{discreteq<1}) $\\beta=\n(q-1) \\beta^*$. Of course, according to (\\ref{discreteq>1}),\n(\\ref{discreteq<1}), and (\\ref{gauss}), for $q \\rightarrow 1$ we\nhave $\\beta^* \\rightarrow \\beta$.\n\\section{Approximate treatments for $q<1$}\n\n\\subsection{The two regimes}\n\n\\noindent Let us recall that, for $q<1$, $m =(1+q)\/(1-q)$. Thus, $m$ is large if\n$q \\rightarrow 1$ (Boltzmann limit (BL)), while a small positive value of $m$ entails\n$q  \\rightarrow -1$. Careful inspection of the partition\nfunction $f_{q} ( T ) $ in the cases $m=3\n  \\leftrightarrow q=1\/2$, $m=5 \\leftrightarrow q=2\/3$,\n and $m=7  \\leftrightarrow q=3\/4$ indicates that two regimes should be distinguished:\n\n\n\\begin{itemize}\n\\item the first regime, called \\textquotedblright power law\\textquotedblright\n\\ regime, corresponds to the case $T \\ll1$ (low temperatures)\n\n\\item the second regime called the \\textquotedblright\nStudent-t\\textquotedblright\\, regime, corresponds to the case $T \\gg1$\n(high temperatures)\n\\end{itemize}\n\n\\noindent Anticipating the results\nprovided by the following theorems, we note that the cases\n\\textquotedblright small $T$\\textquotedblright\/\\textquotedblright\nlarge $T$\\textquotedblright\\ can be translated as joint range\nvalues of parameters $U$ and $m \\equiv  (1+q)\/(1-q):$\ntypically, the large $T$ case corresponds to jointly large values\nof $U$ and $m,$ as expected,  while the small case corresponds to\njointly small values of these parameters. This is illustrated on\nthe curve below (Fig. 1), where the large $T$ property is\ntranslated as $T\\geq100,$ and the small $T$ property as\n$T\\leq0.01.$ Note that the left bound $n=1\/2$ corresponds to\n$m=2n+1=2,$ i.e., $q=1\/3$, and thus to an (unphysical) infinite\n$U$. For economy's sake we set herefrom\n\n\\begin{equation} \\label{defa} a=\\sqrt{k_BT}. \\end{equation}\n\n\\subsection{The power law regime}\n\nIn the power-law regime ($T\\ll1$), a detailed characterization of\nthe distribution can be obtained for all real values of $m$, as\nshown in the following theorem.\n\n\\begin{theorem}\n$\\label{powerlawregime}$\\label{power_law} Assuming that\n$T\\ll1\\,\\,\\,(a \\ll 1)$ and with $m$ defined by (\\ref{mforq<1}), the following approximations hold:\n\n\\begin{enumerate}\n\\item for the partition function\n\\[\nf_{q} (  T )  \\simeq1+2a^{m+1}\\zeta ( m+1 ),\n\\]\n\n\n\\item for the probability law\n\\begin{equation}\np_{k}\\simeq \\bigg\\{\n\\begin{array}\n[c]{ll}\n\\frac{a^{m+1}}{1+2a^{m+1}\\zeta (  m+1 )  }k^{-m-1} & \\text{if }\nk\\neq0\\\\\n\\frac{1}{1+2a^{m+1}\\zeta (  m+1 )  } & \\text{if }k=0\n\\end{array}\n,  \\label{pk_smalla}\n\\end{equation}\nand corresponds thus to a discrete power-law. It tends to a Gaussian for $q  \\rightarrow 1$.\n\n\\item for the mean energy\n\\begin{equation} \\label{mono2} U\\simeq2a^{m+1}\\zeta (  m-1 ). \\end{equation}\n\n\\item If, moreover, $m=2n+1$ with $n\\in\\mathbb{N}$, $n\\geq1$, the characteristic\nfunction can be approximated as\n\\[\n\\phi_{m} (  u )  \\simeq\\frac{1}{ (\n1+2a^{2n+2}\\zeta ( 2n+2 )   )  } (\n1+2a^{2n+2}\\frac{ (  -1 )  ^{n} ( 2\\pi )\n^{2n+2}}{ (  2n+2 )  !}B_{2n+2} ( \\frac{u}{2\\pi\n} )   ),\n\\]\nwhere $B_{n} (  u )  $ denotes the Bernoulli polynomial of order $n.$\n\\end{enumerate}\n\n\\begin{proof}\nthe proof is given in  Annex 1.\n\\end{proof}\n\\end{theorem}\nThe specific heat $C=dU\/dT$ is proportional to $(m+1)$. This is\n{\\it positive}, and thus the system {\\it stable}, for all\n$\\frac{1}{3}<q<1$. \\vskip 3mm \\noindent Notice that, according to\n(\\ref{pk_smalla}), only the ground state is populated at $T=0$, as\nit should. \\vskip 3mm\n\n\\noindent The probability $p_{k}$ decreases exponentially with $k,$\nbeing thus concentrated around its mean value $EX=0.$ As an\nexample, if $a=0.1,$ the following table shows, for several values\nof $q>\\frac{1}{3}$, the\nfirst values of $p_{k}.$\n\\begin{gather*}\n\\begin{tabular}\n[c]{|c|c|c|c|c|}\\hline\n& $k=0$ & $k=\\pm1$ & $k=\\pm2$ & $k=\\pm3$\\\\\\hline\n$q=\\frac{1}{2}$ & $0.999\\,78$ & $9.\\,\\allowbreak997\\,8\\times10^{-5}$ &\n$\\allowbreak6.\\,\\allowbreak248\\,6\\times10^{-6}$ & $\\allowbreak1.\\,\\allowbreak\n234\\,3\\times10^{-6}$\\\\\\hline\n$q=\\frac{3}{5}$ & $0.999\\,98$ & $9.\\,\\allowbreak999\\,8\\times10^{-6}$ &\n$\\allowbreak3.\\,\\allowbreak124\\,9\\times10^{-7}$ & $\\allowbreak4.\\,\\allowbreak\n115\\,1\\times10^{-8}$\\\\\\hline\n$q=\\frac{2}{3}$ & $\\simeq1.\\,\\allowbreak000\\,00$ & $1.\\,\\allowbreak\n000\\,00\\times10^{-6}$ & $\\allowbreak1.\\,\\allowbreak562\\,5\\times10^{-8}$ &\n$\\allowbreak1.\\,\\allowbreak371\\,7\\times10^{-9}$\\\\\\hline\n$q=\\frac{5}{7}$ & $\\simeq\\allowbreak1.\\,\\allowbreak000\\,00$ & $1.\\,\\allowbreak\n000\\,00\\times10^{-7}$ & $\\allowbreak7.\\,\\allowbreak812\\,5\\times10^{-10}$ &\n$4.\\,\\allowbreak572\\,5\\times10^{-11}$\\\\\\hline\n\\end{tabular}\n\\\\\n\\text{Table.1: first values of }p_{k}\\text{ for several values of }q  \\text{ and } a=0.1\n\\end{gather*}\n\n\n\\subsection{The Student-t regime}\n\nIn the case of the Student-t regime (high temperatures), our main\nresult writes as follows:\n\n\\begin{theorem}\n$\\label{student-t}\\forall\\varepsilon>0,$ $\\exists a_{0}>0$ and\n$\\exists\nm_{0}>0$ such that if $a \\geq a_{0}$ and $2<m\\leq m_{0}$ then\n\\begin{equation}\n \\label{inequality}\n \\left \\vert \\sum_{k=0}^{+\\infty}g_k\\, (\n1+\\frac{k^{2}}{a^{2}} )\n^{-\\frac{m+1}{2}}-a\\sqrt{\\pi}\\frac{\\Gamma (  \\frac{m}{2} )  }\n{\\Gamma (  \\frac{m+1}{2} )  } \\right \\vert \\leq\\varepsilon.\n\\end{equation}\n\n\\end{theorem}\n\n\\noindent The proof of this result is given in  Annex 2. The\nvalues of $a_0$ and $m_{0}$ such that (\\ref{inequality}) holds are given in\nFig. 2 for $\\epsilon=10^{-9}$. As a direct consequence of this result, we have the\nfollowing explicit approximations, valid if, simultaneously, i)\n$a\\gg1\\,\\,\\,(T\\gg1)$ and ii) $q  \\rightarrow 1$.\n\n\\begin{proposition}\nIf $a\\gg1\\ $, $m\\gg1$, and $\\frac{a^{2}}{m}\\gg1$, then the\nfollowing approximations hold with high accuracy:\n\n\\begin{itemize}\n\\item for the partition function\n\\[\nf_{q} (  a )  \\simeq a\\sqrt{\\pi}\\frac{\\Gamma (  \\frac{m}\n{2} )  }{\\Gamma (  \\frac{m+1}{2} )  },\n\\]\n\n\n\\item for the probability law (remember that $U=\\sigma^2$)\n\\begin{equation}\np_{k}\\simeq\\frac{\\Gamma (  \\frac{m+1}{2} ) }{\\Gamma (\n\\frac {m}{2} )  \\Gamma ( \\frac{1}{2} )\n\\sigma\\sqrt{m-2}} ( 1+\\frac{k^{2}}{\\sigma^{2} (\nm-2 )  } ) ^{-\\frac{m+1}{2}},\n\\label{pk_largea}\n\\end{equation}\n\n\n\\item while for the mean energy\n\\begin{equation} \\label{mono1} U\\simeq\\frac{a^{2}}{m-2}. \\end{equation}\n\\end{itemize}\n\\noindent {\\rm Note that here}\n\\begin{equation} \\label{r1} (m-2)\\,U= k_B \\,T = \\beta^{-1}, \\end{equation} {\\rm a result that will be needed below.}\n\n\\begin{proof}\nthe expression of the partition function results from theorem (\\ref{student-t}\n). Using a by-product of proof of theorem (\\ref{powerlawregime}),\nthe mean energy $U$ reads\nwrites\n\\[\n U=a^{2} (  \\frac{f_{m-2}}{f_{m}}-1 ) \\simeq\na^{2} (\n\\frac{\\frac{\\Gamma (  \\frac{m-2}{2} )  }{\\Gamma (  \\frac{m-1}\n{2} )  }}{\\frac{\\Gamma (  \\frac{m}{2} )\n}{\\Gamma ( \\frac{m+1}{2} )  }}-1 )\n=\\frac{a^{2}}{m-2}.\n\\]\nAs a consequence, $a\\simeq\\sqrt{U(m-2)}$ and the law writes\n\\[\np_{k}\\simeq\n\\frac{\\Gamma (  \\frac{m+1}{2} )}{\\Gamma ( \\frac {m}{2} ) \\Gamma (\\frac{1}{2}) \\sqrt{U(m-2)}}\n( 1+\\frac{k^2 }{U(m-2)})\n^{-\\frac{m+1}{2}}\n\\]\n\\end{proof}\n\\end{proposition}\nNotice that in this case the specific heat is proportional to\n$(m-2)^{-1}$. This is negative, and thus the system unstable,\nunless $q > 1\/3$. In other words, the system is stable in the\ninterval $]1\/3,1].$ We remark also that, as a consequence of\n(\\ref{mono2}) and (\\ref{mono1}), the mean energy $U$ is an\nincreasing function of $a$ (and thus of $T$), as it\nshould.\n\\begin{remark}\nA remarkable result here is that expression (\\ref{pk_largea}) is\nexactly the sampled version - with the same partition function - of the continuous maximizer of the\nTsallis entropy whose expression is recalled here \\cite{TsallisStudent}:\n\n\\[\nf(x)=\n\\frac{\\Gamma (  \\frac{m+1}{2} )}{\\Gamma ( \\frac {m}{2} ) \\Gamma (\\frac{1}{2}) \\sqrt{U(m-2)}}\n( 1+\\frac{x^2 }{U(m-2)})\n^{-\\frac{m+1}{2}} \\,\\, \\forall x \\in \\mathbb{R}\n\\]\n\nThis entails that, at very high $T$, one can approximate sums over\ndiscrete energy levels by an integral. In order to understand the\nsituation, remember that the results holds for $m\\gg1$, which\nentails $q \\sim 1$, i.e., the Boltzmann limit. In this limit,\nreplacing sums by integrals is a commonplace text-book procedure.\n\\end{remark}\n\n\n\\section{The  $q>1$ instance}\n\n\\subsection{A general result and its consequences}\n\nRecall that in the case $q>1$, $m=(q+1)\/(q-1)$. The equivalent of theorem (\\ref{student-t}) writes as\nfollows:\n\n\\begin{theorem}\n\\label{Student-r}$\\forall\\varepsilon>0,$ $\\exists a_{0}>0$ and\n$\\exists\nm_{0}>0$ such that if $a\\geq a_{0}$ and $m\\geq m_{0}$ then\n\\begin{equation}\n\\left \\vert \\sum_{k=- (  a-1 )  }^{ a-1} (  1-\\frac{k^{2}}{ a^{2}\n} )  ^{\\frac{m-1}{2}}-\\frac{ a\\sqrt{\\pi}\\Gamma (  \\frac{m+1}\n{2} )  }{\\Gamma (  \\frac{m}{2}+1 )  } \\right \\vert \\leq\n\\varepsilon. \\label{approx}\n\\end{equation}\n\n\n\\begin{proof}\nThe proof of this result is given in Annex 3. It essentially\nfollows the steps of the proof of theorem (\\ref{student-t}).\n\\end{proof}\n\\end{theorem}\n\n\\noindent  We depict in Fig. 3 the area (north-east), delimited in the plane $ ( a,m )$, that contains\n values of $a$ and $m$ for which approximation (\\ref{approx})\nholds, with $\\varepsilon=10^{-10}$.\n\n\\begin{corollary}\nFor $ a\\gg1\\,\\,\\,(T\\gg 1)$ the mean energy $U$ verifies\n\\begin{equation}\nU \\simeq\\frac{ a^{2}}{m+2} \\label{varapprox}\n\\end{equation}\n\n\n\\begin{proof}\nDenoting $n=\\frac{m-1}{2}$ and $f_{n}(a)$ for $f_{q}(a)$, the normalization constant verifies\nthe following\nrecurrence\n\\begin{align*}\nf_{n+1} (  a )   &  =\\sum_{k=-a}^{+a} (  1-\\frac{k^{2}}{a^{2}\n} )  ^{n+1}=\\sum_{k=-a}^{+a} (\n1-\\frac{k^{2}}{a^{2}} )\n^{n}-\\frac{1}{a^{2}}\\sum_{k=-a}^{+a}k^{2} (\n1-\\frac{k^{2}}{a^{2}} )\n^{n}\\\\\n&  =f_{n} (  a )  -\\frac{U}{a^{2}}f_{n+1} ( a )\n\\end{align*}\nso that\n\\[\nU =a^{2} (\n1-\\frac{f_{n+1} (  a ) }{f_{n} (  a ) } ) .\n\\]\nMoreover, using the result of theorem (\\ref{Student-r}), we have\n\\[\nf_{n} (  a )  \\simeq\\frac{\\Gamma (\n\\frac{m}{2}+1 )\n}{a\\sqrt{\\pi}\\Gamma (  \\frac{m+1}{2} )  }\n\\]\nwe obtain after some algebra\n\\begin{equation} \\label{ener} U\n\\simeq\\frac{a^{2}}{m+2}. \\end{equation}\n\n\\end{proof}\n\\end{corollary}\nWe see from the form of (\\ref{ener}) that the specific heat $C$ is\nproportional to $1\/(m+2)$. Thus, the is system stable for all $q\n>1$. Also, since $a^2=k_BT$, we see that the mean energy $U$ is an\nincreasing function of $T$, as it should.\n\n\\subsection{Convergence to a sampled Student-r law}\n\nAs a consequence of theorem (\\ref{Student-r}), the Tsallis\nmaximizers can be approximated as follows, for $a$ large enough\n($T$ large enough).\n\n\\begin{theorem}\nIf $a\\gg1$ then the following approximation holds\n\\begin{equation}\np_{k}\\simeq\\frac{\\Gamma (  \\frac{m}{2}+1 )\n}{\\sqrt{\\pi}\\Gamma ( \\frac{m+1}{2} ) \\sqrt{U}\n\\sqrt{m+2}} (  1-\\frac{k^{2}}{U (  m+2 )  } )  ^{\\frac{m-1}{2}}. \\label{sampled}\n\\end{equation}\n\n\n\\begin{proof}\n>From (\\ref{varapprox}), we have\n\\[\na^{2}\\simeq  (  m+2 )U.\n\\]\nand the result follows from Theorem (\\ref{Student-r}).\n\\end{proof}\n\\end{theorem}\n\n\\noindent {\\rm Note that here}\n\\begin{equation} \\label{r2} (m+2)\\, U= k_B \\,T = \\beta^{-1}, \\end{equation} {\\rm a result that will be needed below.}\n\\noindent We remark the similarity to the corresponding $q<1$ probability\nlaw.\n\n\\noindent Eq. (\\ref{sampled}) corresponds to the sampled version of a\ncontinuous Student-r distribution -which maximizes Tsallis entropy\nfor fixed mean energy \\cite{TsallisStudent} - with the same partition function.\n\n\\subsection{Detailed results}\n\nIn the case where $m$ is an odd integer, more detailed results\ncan be obtained concerning the behavior of this probability law.\n\n\\begin{theorem}\n\\label{integercase}For $a\\gg1,$ if $m=2n+1$ is an odd integer with\n$n\\geq2$,\nthen\n\\begin{equation}\nf_{m} (  a )  =\\frac{  2^{2n+1} ( n ! )  ^{2}}{(2n+1)!}a+o (  a^{-2} )  \\label{fapprox}\n\\end{equation}\n\n\nand $\\forall a\\geq1,$ and if $m=3$\n\\[\nf_{3} (  a )  =a\\frac{4}{3}-\\frac{a^{-1}}{3}.\n\\]\nMoreover, the mean  energy $U$ verifies\n\\begin{equation}\nU  =\\frac{a^{2}}{m+2}+o ( a^{-1} ).\n\\label{sigmaapprox}\n\\end{equation}\n\n\n\\begin{proof}\nthe proof is given in Annex 4.\n\\end{proof}\n\\end{theorem}\nThis quasi linearity of $f_m (a )  $ in $a$ for high\ntemperatures is illustrated in Fig. 4, while the quadratic\nbehavior of $U$ vs.  $a$ is depicted in Fig. 5.\n\n\n\n\\section{Convergence to the discrete normal distribution}\n\nWe show now that, in the case where parameter $m$ grows to infinity, both discrete Tsallis distributions corresponding to either $q<1$ or $q>1$ converge to the discrete normal distribution.\n\n\\begin{proposition}\nIf $a\\gg1$ and if the mean energy $U$ is fixed to a constant\nvalue, then, as  $m$ grows to infinity (i.e.\n$q \\rightarrow 1^{\\pm}$), both discrete Tsallis probability distributions\n(\\ref{discreteq>1}) and (\\ref{discreteq<1}) converge to the maximizer of Shannon's discrete entropy,\ni.e., the discrete normal distribution.\n\n\\begin{proof}\nin the case $q<1,$ the result follows by taking $m \\rightarrow +\\infty$ in\n(\\ref{pk_largea}) with\n\\[\n\\lim_{m \\rightarrow +\\infty}\\frac{\\Gamma (  \\frac{m+1}{2} )  }\n{\\Gamma (  \\frac{m}{2} )  \\Gamma (  \\frac{1}{2} )\n\\sqrt{U(m-2})}=\\frac{1}{\\sqrt{2U\\pi}}\n\\]\nand (Cf. (\\ref{r1}))\n\\[\n\\lim_{m \\rightarrow +\\infty} (  1+\\frac{k^{2}}{U (\nm-2 ) } )  ^{-\\frac{m+1}{2}}=\\exp (\n-\\frac{k^{2}}{2U} )  .\n\\]\nIn the case $q>1,$ the result follows similarly by taking $m \\rightarrow\n+\\infty$ in (\\ref{sampled}) with\n\\[\n\\lim_{m \\rightarrow +\\infty}\\frac{\\Gamma (  \\frac{m}{2}+1 )  }\n{\\sqrt{\\pi}\\Gamma (  \\frac{m+1}{2} )  \\sqrt{m+2}}=\\frac{1}{\\sqrt\n{2\\pi}}\n\\]\nand (Cf. (\\ref{r2}))\n\\[\n\\lim_{m \\rightarrow +\\infty} (  1-\\frac{k^{2}}{U ( m+2 ) } )\n^{\\frac{m-1}{2}}=\\exp ( -\\frac{k^{2}}{2U} ).\n\\]\n\n\\end{proof}\n\\end{proposition}\n\\noindent\n\nThis result shows that, for high enough temperatures, and if $m \\rightarrow \\infty$ (or equivalently $q \\rightarrow \\pm 1$) the distribution that maximizes Tsallis'\nentropy  converges to the Boltzmann one, both for $q<1$ and $q>1$.\n Of course, for fixed $T \\gg 1$ and $q \\rightarrow +\\infty$, we obtain the uniform distribution, whereas for $T \\ll 1$ and $q \\rightarrow +\\infty$ we obtain the deterministic case $p_{k} = \\delta_{k}$.\n\n\\section{Extension to the multivariate case}\n\nIn the last part of this communication\n we give a heuristic discussion of\n  the multivariate discrete Tsallis laws with $q<1.$\n\n\\begin{theorem}\n\\label{multivariate}\nif $\\frac{d}{d+2}<q<1$ then the probability\n $ p_{k_{1},\\dots,k_{d}}=Pr\\{X_{1}=k_{1},\\dots,X_{d}=k_{d}\\} $ defined by\n\n\\[\np_{k_{1},\\dots,k_{d}} =f_{m,d}^{-1} (a_{1},\\dots,a_{d} )   (  1+\\sum_{i=1}\n^{d}\\frac{k_{i}^{2}}{a_{i}^{2}} )  ^{-\\frac{m+d}{2}};\\,\\,\n\\,\\,k_{i} \\in \\mathbb{Z}^{+},\n\\]\nmaximizes Tsallis $d$-variate entropy\n\n\\[\nH_{q}(p)=\\frac{1}{q-1} (  1-\\sum_{k_{1},\\dots,k_{d}}p_{k_{1},\\dots,k_{d}}\n^{q} )\n\\]\nfor fixed mean energy\n\\[\n\\sum_{k_{i}}\\,g_{k_{i}}k_{i}^{2}\\,p_{k_{i}}=u_{i}^{2}.\n\\]\nThe partition function $f_{m,d}(a_{1},\\dots,a_{d} )$ is defined as follows:\n\\[\nf_{m,d} (a_{1},\\dots,a_{d} )  =\\sum_{k_{1},\\dots,k_{d}} (  1+\\sum_{i=1}\n^{d}\\frac{k_{i}^{2}}{a_{i}^{2}} )  ^{-\\frac{m+d}{2}}\n\\]\nIn the case where $\\forall i,$ $a_{i}\\gg1,$ this function can be approximated as\n\\[\nf_{m,d} (a_{1},\\dots,a_{d} )\\simeq\\frac{\\pi^{\\frac{d}{2}}\\Gamma (\n\\frac {m}{2} ) }{\\Gamma (  \\frac{m+d}{2} )\n}\\prod_{i=1}^{d}a_{i}.\n\\]\n\n\n\\begin{proof}\nThe proof  can be derived from that given for the continuous case\nin \\cite{next2003}. It is detailed in Annex 5.\n\\end{proof}\n\\end{theorem}\n\nThe family of multivariate discrete Tsallis maximizers verify approximately, as in the continuous case, the property of invariance by marginalization, as described in the following theorem.\n\n\\begin{theorem}\n\\label{marginal}\n\nThe $d^{\\prime}-$variate marginal law ($d'\\,\\leq d-1$) writes\n\\begin{equation}\np_{k_{1},\\dots,k_{d^{\\prime}}}=f_{m,d^{\\prime}} ( a_{1},\\dots,a_{d^{\\prime}} )\n ( 1+\\sum_{i=1}^{d^{\\prime}}\\frac{k_{i}^{2}}{a_{i}^{2}} )\n^{-\\frac {m+d^{\\prime}}{2}}.\n\\label{renormalizability}\n\\end{equation}\n\n\n\\begin{proof}\nthe proof is given in annex 5.\n\\end{proof}\n\\end{theorem}\nWe remark here that property (\\ref{renormalizability}) is nothing but\nrenormalizability: any subsystem of a system distributed\naccording to a discrete Tsallis law is itself distributed\naccording to a discrete Tsallis law. We note, too, that renormalization does not change the value of parameter $m$.\n\n\\section{Conclusions}\n\nIn this report, we have derived {\\it in analytic fashion} some elementary properties of the discrete univariate and multivariate Tsallis laws (probability distributions $p_k$\nrunning over a discrete index $k$) for both $q<1$ and $q>1.$\nMost of the ensuing results are given by approximations, but\nnumerical simulations show that they can be regarded as very\naccurate ones.\n\n\\noindent The discrete probability distributions $p_k$ define, for any Hamiltonian $\\mathcal{H}$ whose eigenvalue equation reads $\\mathcal{H}\\vert \\psi_k\\rangle =\\epsilon_k \\vert \\psi_k\\rangle$ mixed states of the form \\begin{equation} \\label{qqq2} \\rho = \\sum_k\\,p_k\\,  \\vert \\psi_k \\rangle \\langle \\psi_k \\vert, \\end{equation}\nthat commute with the Hamiltonian by construction and represent  thus  bona fide  stationary states of the system defined by $\\mathcal{H}$. These weights $p_k$ are of the Tsallis-power law form. With these weights, $\\rho$ maximizes Tsallis' entropy subject to variance constraint, which introduces a Lagrange multiplier $\\beta$. In this work we have considered situations for which the system's mean energy is proportional to the variance.\n\n\n\\noindent Our present consideration allows one to nitidly appreciate the\neffects of non-extensivity, as, varying $q$, one passes from\nunstable to stable systems and even to unphysical situations of\ninfinite energy.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMagnetic properties of transition metals (TM) are generally determined by the $3d$ valence electrons. Resonant soft X-ray scattering at $L_{2,3}$ absorption edges of TM involves $2p$-to-$3d$ transitions, thus being an element-selective probe with possibility to distinguish magnetic signal from different elements in multicomponent magnets \\cite{fink2013resonant}. Moreover, the spatial coherence of the polarized soft X-ray beams provided by modern synchrotron radiation sources and free-electron lasers give vast opportunities for the lensless imaging using coherent diffraction imaging \\cite{turner2011x,flewett2012method,ukleev2018coherent}, ptychography \\cite{tripathi2011dichroic,shi2016soft}, and holographic techniques \\cite{eisebitt2004lensless,duckworth2011magnetic}. Both coherent resonant soft X-ray scattering (RSXS) imaging and holography allow to perform real-space imaging of the magnetization distribution in thin samples with various environments, such as in high magnetic fields or at low temperatures. Flexibility of the environment and enhanced robustness of these methods against the specimen displacements are significant advantage of the lensless techniques compared to scanning transmission magnetic X-ray microscopy (STXM) \\cite{fischer2015x}, although the possibility of cryogenic STXM imaging also has been recently demonstrated \\cite{simmendinger2018transmission}. Coherent diffraction allows the solution of classical crystallographic inverse problem of phase retrieval by using the iterative reconstruction algorithms applied to the resonant diffraction intensities \\cite{miao1999extending}. X-ray magnetic holography is based on the utilization of the interference between magnetic scattering from the object under investigation and reference wave generated by charge scattering from the prepared source. Imaging experiments using holographic approaches can be realized in a few different ways: Fourier transform holography (FTH) is based on a reference wave scattered from one or multiple small ($30-150$\\,nm) pinholes placed near the sample aperture \\cite{eisebitt2004lensless}. Alternatively, holography with extended reference by autocorrelation linear differential operation (HERALDO) \\cite{guizar2007holography} can be performed. In contrast to FTH, HERALDO technique implies the scattering from extended reference object, such as a narrow slit or a sharp corner, which allows to improve the contrast of the real-space image without compromising the resolution \\cite{zhu2010high}. Moreover, fabrication of the extended reference is less challenging than the array of the reference pinholes \\cite{zhu2010high,buttner2013automatable}. Previously, FTH and HERALDO techniques were successfully applied for imaging of the element-specific magnetic domain patterns in thin films and multilayers with perpendicular magnetic anisotropy \\cite{stickler2010soft,camarero2011exploring,weder2017multi}. Both FTH and HERALDO with references milled at an oblique angle into the masks also allow the imaging at a tilted angle, which is relevant for the spintronic devices with an in-plane magnetic anisotropy, such as spin-valves \\cite{duckworth2013holographic}, and magnetic nanoelements \\cite{tieg2010imaging,parra2016holographic,bukin2016time}. Thus the coherent soft X-ray scattering and imaging are powerful tools to study the spin ordering in multicomponent magnetic compounds with element selectivity.\n \nRecently, several bulk materials that exhibit non-trivial topological spin textures and contain two or more magnetic elements have been discovered: doped B20-type alloys \\cite{adams2010skyrmion, shibata2013towards}, Co-Zn-Mn compounds with $\\beta$-Mn structure \\cite{tokunaga2015new}, molybdenum nitrides \\cite{li2016emergence} and Heusler alloys \\cite{meshcheriakova2014large,phatak2016nanoscale,nayak2017magnetic}. Competition between the magnetic interactions in non-centrosymmetric compounds results in the complex phase diagram. The interplay between exchange interaction, antisymmetric Dzyaloshinskii-Moriya interaction (DMI), and magnetocrystalline anisotropy may cause incommensurate spin phases such as helical, conical and Bloch-type skyrmion lattice states \\cite{bak1980theory,rossler2006spontaneous}.\nThe typical size of a magnetic skyrmion varies in a range from a few to a few hundreds nm which makes them promising candidates for future spintronic applications such as skyrmion racetrack memory and logic devices \\cite{fert2017magnetic}. The skyrmions can be manipulated by current pulses with ultra-low current densities \\cite{iwasaki2013universal}, electric \\cite{white2014electric,okamura2016transition} and microwave fields \\cite{onose2012observation,okamura2013microwave,wang2015driving}, and temperature gradients \\cite{everschor2012rotating,kong2013dynamics,mochizuki2014thermally}. In the past decade skyrmion textures have been extensively studied by means of small-angle neutron scattering (SANS) \\cite{adams2011long,seki2012formation,moskvin2013complex} and Lorentz transmission electron microscopy (LTEM) \\cite{yu2010real,yu2011near}. Also, several groups reported on the resonant X-ray diffraction \\cite{langner2014coupled,zhang2016multidomain}, small-angle scattering \\cite{yamasaki2015dynamical,okamura2017directional,okamura2017emergence,zhang2017room} studies of Bloch-type skyrmions in the chiral magnets and imaging of N\\'eel-type skyrmions stabilized by interfacial DMI \\cite{buttner2015dynamics,blanco2015nanoscale,Woo2016,Woo2017}. Since X-ray magnetic circular dichroism (XMCD) is sensitive to the component of the magnetization parallel to the incident X-ray beam, transmission soft X-ray imaging is a complementary method to LTEM, which is sensitive to the in-plane magnetic flux inside the sample \\cite{chapman1984investigation}. Hence for the N\\'eel-type skyrmions, where the curl of magnetization lies in the sample plane and produce no contrast for LTEM without tilting the sample \\cite{jiang2016mobile,pollard2017observation}, soft X-ray imaging, has been successfully employed for the room-temperature N\\'eel-type skyrmion-hosting thin films \\cite{buttner2015dynamics,blanco2015nanoscale,Woo2016,Woo2017}, but not yet for the thin plates of polar magnets, that order magnetically at cryogenic temperatures \\cite{kezsmarki2015neel,kurumaji2017neel}.\n\nRoom-temperature magnetic ordering of the chiral $\\beta$-Mn-type Co-Zn-Mn alloy \\cite{tokunaga2015new} makes these materials promising for applications. The $\\beta$-Mn-type compound Co$_8$Zn$_8$Mn$_4$ exhibits a transition from the paramagnetic state to a helical or Bloch-type skyrmion lattice state at $T_c \\approx 300$\\,K with a magnetic modulation period of 125\\,nm \\cite{karube2016robust}, and undergoes a spin glass transition\nat $T_g\\approx8$\\,K, probably due to freezing of Mn spins \\cite{karube2018sciadv}. Frustration at the Mn site ultimately results in increment of a spin-glass transition temperature ($T_g\\approx30$\\,K) in the compound with higher Mn concentration Co$_7$Zn$_7$Mn$_6$. Moreover, a low-temperature frustration-induced equilibrium skyrmion phase has been recently found in the latter \\cite{karube2018sciadv}. In the present work we utilized the polarization-dependent soft X-ray magnetic spectroscopy, coherent RSXS, and HERALDO techniques to perform an element-selective study of the magnetic interactions and long-range ordering in Co$_8$Zn$_8$Mn$_4$ compound. The coherence of the synchrotron radiation allowed us to successfully combine small-angle scattering in transmission geometry with coherent diffraction imaging and employ the small-angle scattering patterns for the real-space reconstruction of the local magnetization distribution via iterative phase retrieval algorithm. The coherent diffraction imaging results were compared to the real-space reconstruction results provided by HERALDO. \n\n\\section{Experimental}\n\nX-ray spectroscopy, scattering and imaging experiments were performed at the variable-polarization soft X-ray beamline BL-16A of the Photon Factory (KEK, Japan) \\cite{amemiya2010k} and BL29 BOREAS of the ALBA synchrotron radiation laboratory (Cerdanyola del Vall\\'es, Spain) \\cite{barla2016design}.\n\nExperimental geometry of soft X-ray absorption (XAS) and XMCD experiments are shown in Fig.\\ref{Fig1}a. A bulk polycrystalline Co$_8$Zn$_8$Mn$_4$ specimen was obtained from an ingot grown by Bridgman method as described in Ref.\\cite{karube2016robust}. The sample was polished to remove the potentially oxidized surface layer prior to the experiment. Sample was placed to the vacuum chamber with pressure of $10^{-9}$\\,Torr equipped with a 5\\,T superconducting magnet. XAS and XMCD signals were measured with energy resolution of 0.1\\,eV using the surface-sensitive total electron yield (TEY) method near Co and Mn $L_{2,3}$ absorption edges with right and left circularly polarized (RCP and LCP) X-rays. \n\n\\begin{figure*}\n\\includegraphics[width=16cm]{Fig1.png}\n\\caption{(Color online) Sketch of the (a) XMCD, (b) RSXS and (c) HERALDO experiments. (d) SEM image of the RSXS sample aperture. (e) Thin plate of the Co$_8$Zn$_8$Mn$_4$ fixed onto the membrane. (f) SEM image of the sample aperture and the reference slit drilled in the 1-$\\mu$m-thick gold plate for HERALDO experiment. (g) Co$_8$Zn$_8$Mn$_4$ thin plate fixed onto gold plate by tungsten contact.  The strong white\/black contrast in the top\/bottom parts of the panels (d--g) corresponds to the Au wires sputtered onto the membrane \\cite{okamura2017directional}, that is irrelevant for this study.}\n\\label{Fig1}\n\\end{figure*}\n\nRSXS and HERALDO experiments were performed in the transmission geometry (Figs. \\ref{Fig1}b,c). Commercial silicon nitride membranes from Silson Ltd (Southam, UK) were processed for soft X-ray experiments. The front side of each membrane was coated with $\\approx 4$\\,$\\mu$m-thick layer of gold to absorb the incoming X-ray beam. Further treatment of the membranes and thin plate fabrication were carried out by using focused ion beam (FIB) setup Hitachi NB5000 equipped with scanning electron microscope (SEM). Since the attenuation length for soft X-rays in Co$_8$Zn$_8$Mn$_4$ alloy is of about 100\\,nm, the scattering and imaging experiments were carried out on thin plates of Co$_8$Zn$_8$Mn$_4$ with thickness of 200\\,nm and 150\\,nm, respectively.  Two FIB-thinned plates of a Co$_8$Zn$_8$Mn$_4$ containing (001) plane were cut from the bulk single crystal. For the RSXS experiment a thin plate was attached directly to the Si$_3$N$_4$ membrane (Figs.\\,\\ref{Fig1}d,e). In case of the RSXS sample the aperture with a diameter of 4.5\\,$\\mu$m and asymmetric shape, which provides a better convergence of the phase retrieval algorithm \\cite{fienup1986phase} was drilled in the gold coating (Fig. \\ref{Fig1}d). \n\nIn the case of the HERALDO sample we fabricated the sample aperture by a different fabrication approach: a large hole with a diameter of 6\\,$\\mu m$ was drilled in the gold-coated membrane, and covered by 1\\,$\\mu$m-thick gold plate fabricated by FIB from a bulk specimen. Then, a circular sample aperture with a diameter of 700\\,nm and a reference slit with a length of 1\\,$\\mu$m and width 40\\,nm were milled in the Au plate (Fig. \\ref{Fig1}f). The slit length and distance from the aperture were chosen according to the separation conditions preventing the overlapping of sample autocorrelation and sample-reference cross-correlation at the reconstruction \\cite{guizar2010differentially}. \n\nTo prevent the specimen damage by Ga$^+$ ions in the aperture and reference milling process the thin plate was fixed to the membrane by means of a tungsten contact after the mask treatment (Fig. \\ref{Fig1}e). For both RSXS and HERALDO the thin plates were attached to the corresponding membranes by the single tungsten contacts (Fig. \\ref{Fig1}g) to avoid the possible strain \\cite{shibata2015large,okamura2017emergence}.\n\nThe RSXS setup at Photon Factory, Japan was equipped with a high-vacuum chamber with a background pressure of $10^{-8}$\\,Torr \\cite{yamasaki2015dynamical}. The scattered intensity was collected by an in-vacuum charge-coupled device (CCD) area X-ray detector of $2048\\,\\times\\,2048$ pixels (Princeton Instruments, Trenton, New Jersey, USA). The RSXS endstation MARES was used at ALBA synchrotron \\cite{barla2016design}. Resonant diffraction intensity was collected by a custom-designed CCD detector of $2148\\,\\times\\,2052$ pixels (XCAM Co, ltd, UK).\nSince small-angle scattering intensity is distributed near the transmitted direct beam, a tungsten beamstop was introduced to protect the detector for RSXS experiment, while in case of HERALDO the smaller aperture size allowed us to measure the holograms without using any beamstop.\nThe magnetic field was applied parallel to the incident X-ray beam and perpendicular to the thin plate.  A He-flow-type cryostat was used to control the sample temperature in range from $\\sim 15$\\,K to $\\sim 320$\\,K, as measured by the Si diode thermometers attached next to the sample holder and cryostat head \\cite{yamasaki2013diffractometer}. A radiation shield was wrapped around the sample holder to reduce the heating of the specimen from the warm environment.\n\n\\section{Results and discussions}\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{Fig2.png}\n\\caption{(Color online) (a) XAS spectra of the bulk Co$_8$Zn$_8$Mn$_4$ near Co and Mn $L_{2,3}$ absorption edges: measured points are shown as the solid lines; calculated spectra are shown as the dashed lines. (b) XMCD signal. The measurements were performed at temperature $T=130$\\,K at applied magnetic field $B=0.5$\\,T.}\n\\label{Fig2}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{Fig3.png}\n\\caption{(Color online) (a) Schematic phase diagram of Co$_8$Zn$_8$Mn$_4$ and the procedure of the resonant soft x-ray scattering (RSXS) measurements. The sample is field-cooled (FC) in the applied magnetic field $B = 70$\\,mT from room temperature down to 25\\,K (indicated by blue arrow) and then warmed back to 300\\,K in the same field (indicated by red arrow). Coherent RSXS speckle patterns measured for Co$_8$Zn$_8$Mn$_4$ sample at $E=779$\\,eV corresponding to $L_3$ absorption edge of Co at different temperatures (b) $T=25$\\,K, (c) $T=120$\\,K, (d) $T=150$\\,K and applied field $B=70$\\,mT. White scale bar corresponds to 0.05\\,nm$^{-1}$.}\n\\label{Fig3}\n\\end{figure*}\n\nThe XAS signals averaged between RCP and LCP spectra near the $L_{2,3}$ edges of Mn and Co at $T=130$\\,K are shown in Fig. \\ref{Fig2}a. Surprisingly, despite the metallic nature of the Co$_8$Zn$_8$Mn$_4$ alloy, the Mn absorption shows a multiplet structure at the $L_3$ and $L_2$ edges. Well-resolved peaks at 1.3 and 3.5\\,eV above the absorption maxima at $E=641.0$\\,eV and a doublet structure at the $L_2$ edge, which splits into two maximums at $E=651.7$\\,eV and $E=653.2$\\,eV are clearly observable. Meanwhile, the Co $L_3$ and $L_2$ peak shapes at $E=780.0$\\,eV and $E=794.1$\\,eV are similar to the broad spectrum of metallic Co \\cite{van1991strong,schwickert1998x,dhesi2001anisotropic}. This suggests that the fine structures of Mn $2p \\rightarrow 3d$ transition should result from the localization of Mn $3d$ electrons rather than the oxidation of surface \\cite{miyamoto2003soft,grabis2005element,gilbert2003multiple}: otherwise, the multiplet structure of Co $L_{2,3}$ edges should be either observed \\cite{regan2001chemical,magnuson2002electronic}. To qualitatively illustrate the features of the measured XAS, simulations were performed using the small cluster approach in the Xclaim software tool \\cite{fernandez2015xclaim}. Mn $3d^5$ was assumed as an initial state and Mn $2p^5$ $3d^6$ was the final configuration. The simulated XAS lines were broadened with a Lorentz function with the full width $\\Gamma$ at half maximum (FWHM) of 1\\,eV. The spectrum for Co was calculated using initial $3d^8$ and final $2p^5$ $3d^9$ configurations, which reproduces the measured data despite some broadening of the latter. The observed fine structure of the XAS spectrum of Mn also nicely corresponds to the one calculated from the multiplet effects (Fig. \\ref{Fig2}a).\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{Fig4.png}\n\\caption{(Color online) Temperature dependence of the modulation vector of skyrmion lattice $q_{Sk}$. Inset shows magnetic field dependence of $q_{Sk}$ measured at $T=25$\\,K and $T=50$\\,K.}\n\\label{Fig4}\n\\end{figure}\n\nXMCD signal measured in a magnetic field of $B=0.5$\\,T well above the saturation is shown in Fig. \\ref{Fig2}b for both elements. Despite the magnitude of the XMCD measured at $L_{2,3}$ edges of Mn is about \\sfrac{1}{5} times smaller than XMCD measured at Co $L_{2,3}$ edges, it is clear that the signs of the dichroic signals are the same. XMCD signal at Mn $L_2$ edge is notably suppressed, indicating quenching of the orbital moment.\nThe sum rule analysis \\cite{o1994orbital} allows to estimate the orbital to spin moment ratio for Mn and Co as $\\mu_l$(Mn)\/$\\mu_s$(Mn)=0.03 and $\\mu_l$(Co)\/$\\mu_s$(Co)=0.0025.\nMagnetic field dependence of the element-selective XMCD signals measured at 135\\,K can be found in the Supplementary materials \\cite{supplementary}. The signs and magnitudes of the XMCD signals indicate a ferromagnetic coupling of Co and Mn moments and a partial cancellation of Mn magnetization. This is in a good agreement with the magnetization measurements, which have shown the reduction of magnetization and critical temperature with increment of Mn concentration in $\\beta$-Mn-type Co-Zn-Mn compounds \\cite{tokunaga2015new}. One can assume that while the Co-Co and Co-Mn couplings are ferromagnetic, the Mn-Mn interaction should be antiferromagnetic. This scenario is also suggested by recent neutron diffraction study of the Co-Zn-Mn alloys in a wide composition range accompanied by the density functional theory (DFT) calculations \\cite{bocarsly2019deciphering}.  This is in contrast to the case of Co-Mn alloys where Mn moments tend to align antiparralel to the host Co magnetization \\cite{nakai1978magnetic,menshikov1985magnetic,wildes1992polarized}. On the other hand, the parent $\\beta$-Mn compound shows strongly antiferromagnetic nearest-neighbor correlations in the $12d$ Mn sublattice, while strong ferromagnetic correlations between next-nearest-neighbors were found \\cite{paddison2013emergent}.  Further, recent DFT calculations of the $\\beta-$Mn-type Co-Zn-Mn suggested larger localization of the Mn atoms at $12d$ site than at $8c$, being consistent with our observations \\cite{bocarsly2019deciphering}. For further quantitative discussions on the XAS and XMCD features, such as origin of multiplet spectrum of Mn, and determination of spin and orbital contributions, additional measurements of different Co-Zn-Mn concentrations and spectra calculations from \\textit{ab initio} theory are highly desired.\n\nComplex scattering factor $f$ for resonant magnetic X-ray scattering can be described as\n\n\\begin{equation}\nf=(\\bm{s}\\cdot \\bm{s}') f_c + i (\\bm{s}\\times \\bm{s}')\\cdot \\bm{M}f_{m}^{1} + (\\bm{s}\\cdot\\bm{M})(\\bm{s}'\\cdot\\bm{M})f_{m}^{2},\n\\label{eq1}\n\\end{equation}\nwhere $\\bm{s}$ and $\\bm{s}'$ are polarizations of the incident and scattered X-rays, respectively; $f_c$ is the charge scattering factor; $\\bm{M}$ is the local magnetization; $f_{m}^1$ and $f_{m}^2$ are factors attributed to the magnetic scattering maximized at the resonant condition. The last term in Eq. (\\ref{eq1}) containing scattering factor $f_{m}^2$ is quadratic in $\\bm{M}$ and generally smaller than the other two terms \\cite{hannon1988x}. Therefore the scattering patterns $(|f|^2)$ measured at resonant conditions mainly consist of the squares of charge and magnetic scattering factors $|f_c|^2$ and $|f_{m}^1|^2$, and their interference ($|f_c^* f_{m}^1|$). For linear polarization the measured intensities are dominated by the pure charge and magnetic scattering terms, while those measured using circular polarization contain charge-magnetic cross term \\cite{hill1996resonant,eisebitt2003polarization}. Consequently, the intensity of magnetic scattering can be simply distinguished from the charge scattering by subtracting the diffraction pattern measured 1) at off-resonant condition and appropriately normalized or 2) in the field-induced polarized state of the sample or 3) above the critical temperature.\n\nRSXS experiments were carried out at Mn ($E=640.5$\\,eV) and Co ($E=779$\\,eV) $L_{3}$ edges, where the magnetic scattering intensity was maximized. Far-field RSXS patterns were acquired with an exposition time of 1\\,second (excluding readout time $\\sim 0.4$\\,s) and total 200\\,expositions were averaged. In case of the off-resonant conditions, the absorption was reduced and the measurement time was consequently reduced by a factor of two. RSXS was measured in the temperature range between $T=25$\\,K and $T=270$\\,K during a field \n-warming-after-field-cooling (FWAFC) in $B=70$\\,mT. The sample position was realigned after each temperature change to compensate the effect of the thermal contraction (expansion) of the sample manipulator with a buffer time between the measurements (in average $\\approx 40$\\,minutes) for stabilization. In the present experiment the linear polarization was used to minimize the influence of the charge-magnetic scattering interference term. In order to isolate the charge scattering contribution we have measured the scattering intensity at off-resonance ($E=645$\\,eV and $E=785$\\,eV) conditions.  \n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=15cm]{Fig5.png}\n\\caption{(Color online) Differential scattering patterns  between LCP and RCP X-rays of energies (a) $E=779$\\,eV and (b) $E=640.5$\\,eV the taken at $T=20$\\,K. The colorbar is given for both patterns in arbitrary units. Note that the strong linear patterns inclined upward right should be due to the incomplete subtraction of the charge scattering from the reference slit. (c) Fourier transform of (a) after applying linear differential filter and rotation of the image by 39$^\\circ$. (d), (e) Magnetic texture recorded at Co and Mn $L_3$ edges, respectively. The colorbar is given for both images in arbitrary units. The region outside the field of view is filled with the black background manually for clarity. (f) Fourier ring correlation analysis of the reconstructed images for Co and Mn. Orange line represents calculated half-bit threshold and the dashed line indicates irreversible cross-point between the FRC function and half-bit threshold.}\n\\label{Fig5}\n\\end{center}\n\\end{figure*}\n\nTypical coherent RSXS speckle patterns measured at $E=779$\\,eV at temperatures $T=25$\\,K, $T=120$\\,K, and $T=150$\\,K are shown in Fig.\\ref{Fig3}. The missing area in the left part of each panel is due to the beamstop shadow. Tuning the energy to the Mn $L_3$ edge results in the scaling of the scattering pattern due to the difference in the photons wavelength. Magnetic scattering intensity is weaker at Mn edge, which can be explained by the lower Mn concentration in this compound, stronger absorption, and magnetization reduction, as expected from XMCD experiment. At room temperature and $T=150$\\,K the small-angle scattering patterns demonstrate six-fold symmetry indicating single-domain triangular skyrmion lattice. As it has been already shown by SANS and LTEM experiments, the skyrmion lattice phase can be supercooled by a field cooling process down to the low temperatures \\cite{karube2016robust,nakajima2017skyrmion,morikawa2017deformation}. According to the previous LTEM observations on a Co$_8$Zn$_8$Mn$_4$ thin plate, hexagonal skyrmion lattice to amorphous state transition is reversible and accompanied by the elongation of individual skyrmions along one of the principal crystallographic axes, while the skyrmion density is conserved \\cite{morikawa2017deformation}.\n\nUpon the field cooling (or FWAFC), the RSXS transforms to a homogeneous ring-like pattern corresponding to the intermediate \"amorphous\" phase at the temperature $T\\approx100$\\,K. Below $T\\approx100$\\,K four wide peaks appear at the coherent diffraction speckle pattern. Moreover, below the transition temperature, the skyrmion lattice parameter gradually decreases from $a_{Sk}=112$\\,nm ($T=100$\\,K) to $a_{Sk}=76$\\,nm ($T=25$\\,K). Temperature dependence of the magnetic modulation $q_{Sk}$ is shown in Fig. \\ref{Fig4}. $q_{Sk}$-vector is also dependent on magnetic field similar to the bulk case \\cite{karube2016robust} (inset in Fig. \\ref{Fig4}). Elongation of the $q_{Sk}$ vector is, presumably, caused by increasing antiferromagnetic interaction between Mn ions superimposed on the helical order. Upon cooling from $T=100$\\,K to $T=25$\\,K a coherent RSXS speckle pattern with four-fold symmetry can be observed. According to the previous results, the hexagonal skyrmion lattice \\cite{karube2016robust} is recovered by applying stronger magnetic field of 300-500\\,mT at low temperature. Indeed, the magnitude of the $q_{Sk}$ vector tends to shrink upon increment of the magnetic field (inset in Fig. \\ref{Fig4}). However, in present experiment the magnetic scattering intensity arising from isolated skyrmions is hardly distinguishable in presence of the background even when the charge scattering is subtracted. Corresponding RSXS patterns can be found in the Supplementary materials \\cite{supplementary}.\n\nThe integrated speckle pattern intensities are demonstrating similar features for Co and Mn except the difference in signal-to-noise ratio, which is better at $E=779$\\,eV. Radially integrated azimuthal profiles of the scattering patterns measured using soft X-rays with energies $E=779$\\,eV and $E=640.5$\\,eV can be found in the Supplementary materials \\cite{supplementary}. \n\nFourier transform holography with extended reference allows to reconstruct the real-space image by single Fourier transform of the measured pattern multiplied by the linear differential operator in direction parallel to the reference slit. Circularly polarized soft X-rays with energies matching to Co and Mn $L_3$ edges were used for HERALDO experiment. Far-field diffraction patterns were collected with RCP and LCP to produce the interference pattern and reconstruct the magnetic texture. In this case information about the magnetic texture was simply encoded in the interference term $f_c^* f_{m}^1$ between charge scattering arising from the reference slit and magnetic scattering from the Co$_8$Zn$_8$Mn$_4$ sample. The difference between the holograms taken with the two opposite X-ray helicities provided the isolated interference term which could be inverted to the real-space image via single Fourier transform and linear differential operator \\cite{guizar2007holography}. HERALDO patterns were acquired with an exposition time of 6\\,s (excluding readout time $\\sim 0.4$\\,s), and total 200 expositions were averaged, resulting in a total acquisition time of 20 minutes for each hologram. The sample was cooled down to the temperature $T=20$\\,K while the magnetic field of $B=70$\\,mT was applied along $[001]$ direction parallel to incident soft X-ray beam propagation vector.\n\n\\begin{figure*}\n\\includegraphics[width=16cm]{Fig6.png}\n\\caption{(Color online) Real-space magnetic textures imaged at Co $L_3$ edge ($E=779$\\,eV) at temperatures (a) $T=20$\\,K, (b) $T=100$\\,K and (c) $T=120$\\,K and applied magnetic field $B=70$\\,mT.}\n\\label{Fig6}\n\\end{figure*}\n\nDifferences between the diffraction patterns taken with the two opposite X-ray polarizations at photon energies $E=779$\\,eV and $E=640.5$\\,eV are shown in Figs. \\ref{Fig5}a and b, respectively. The highest harmonics of the interference pattern can be found at $q_{max}=0.2$~nm$^{-1}$ which corresponds to the real-space resolution of 32\\,nm. The difference between Fourier transform images taken with RCP and LCP at Co $L_3$ edge after applying the differential filter in the slit direction is shown in Fig. \\ref{Fig5}c. Reconstructed real-space image shows the sample autocorrelation and two object-reference cross-correlations delivered by both ends of the slits, as well as their complex conjugates. Magnetic contrast is inverted between the reconstructed object and conjugate. By taking into account the widths of magnetic domains where the local magnetization points antiparallel (parallel) to the applied field $B=70$\\,mT at different temperatures (Fig. \\ref{Fig6}), we conclude that the regions with the negative (positive) values in Figs. \\ref{Fig5}d and e correspond to magnetization pointing antiparallel (parallel) to the field.\n\nFigures \\ref{Fig5}d and e show magnification of the sample-reference cross-correlations taken from reconstructions of holograms taken for Co and Mn, respectively. Due to the difference of the wavelengths, the real-space image corresponding to magnetic texture of Mn ions was scaled by factor $E_1\/E_2=779\/640.5\\approx1.22$. The magnetic structures exhibited by magnetic elements Mn and Co are similar to each other within the resolution limit. Signal-to-noise ratio is worse in case of the measurement at Mn $L_3$ edge due to the higher absorption of soft X-rays at $E=640.5$\\,eV and smaller concentration of Mn atoms compared to Co. Different magnitude of the magnetic moment between Co and Mn is also a reason for this. Therefore, the real-space image reconstructed for Mn atoms is slightly blurred (Fig. \\ref{Fig5}e). Additionally the reconstruction resolution is estimated from the Fourier ring correlation (FRC) \\cite{banterle2013fourier}. By using FRC the spatial frequency dependence of the cross-correlation of two real-space magnetic textures obtained from the reconstructions of Co and Mn HERALDO patterns (Fig. \\ref{Fig5}f) is analyzed. The resolution was calculated at the point where the FRC curves irreversibly cross the half-bit threshold \\cite{van2005fourier}. The real-space resolution that was found according to this criteria is similar to the resolution of 32\\,nm determined from the highest interference harmonics observed at the Fourier-space pattern.\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{Fig7.png}\n\\caption{(Color online) Reconstruction of coherent RSXS patterns at (a) $T=25$\\,K, (b) $T=55$\\,K and (c) $T=120$\\,K collected at Co L$_3$ edge. The colorbar for intensity ($z$-scale) is given in arbitrary units. (d) Magnification of the region highlighted by the blue square in (a) is shown for the the different temperatures.}\n\\label{Fig7}\n\\end{figure*}\n\nElongated skyrmions can be observed at the holograms taken both at $E=779$\\,eV and $E=640.5$\\,eV indicating similar magnetic texture of Co and Mn sub-lattices, which coincides with the XMCD and RSXS data. Magnetic holograms were recorded during a FWAFC procedure up to 120\\,K at which the magnetic scattering is highly reduced due to the thermal shrinkage of the magnetic moments. Corresponding evolution of the magnetic texture measured at Co absorption edge is shown in Fig. \\ref{Fig6}. The transformation of the elongated skyrmions to more conventional shape with corresponding expansion takes place at $T=120$\\,K (Fig. \\ref{Fig6}c). Unfortunately, due to a thermal shrinkage of magnetic moments the real-space reconstruction of magnetic texture at higher temperatures can be hardly distinguished from the background fluctuations.\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{Fig8.png}\n\\caption{(Color online) Micromagnetic simulations of Co$_8$Zn$_8$Mn$_4$ $10\\times10$\\,$\\mu$m$^2$ thin plate: $z-$projection of the magnetic texture is shown for various exchange from stiffness parameters $A_{ex}=9.2\\cdot10^{-12}$\\,J\/m (a), $5.8\\cdot10^{-12}$\\,J\/m (b), $3.2\\cdot10^{-12}$\\,J\/m (c). Bottom panels show magnification of the rectangular regions highlighted by the corresponding red boxes.}\n\\label{Fig8}\n\\end{figure*}\n\nSince the sample area probed by HERALDO was limited by the aperture size, we additionally performed a real-space inversion off the measured RSXS patterns by the iterative phase retrieval. Square root of the measured magnetic scattering intensity $\\sqrt{I_{m}}$ shown in Fig. \\ref{Fig3} isolated from the charge scattering as described in Ref. \\cite{ukleev2018coherent}, was used as real part of the Fourier-space constraint. Fixed sample aperture size, shape and orientation were used as real-space support. Pixels, that were missing at the center of diffraction pattern due to the beamstop shadow and subtracted charge scattering, were substituted by the Fourier transform of the support. This approach is similar to the substitution used in the guided hybrid input-output (HIO) algorithm \\cite{chen2007application}. We used the implementation of HIO algorithm \\cite{fienup1982phase} with a feedback parameter $\\beta=0.9$ with assuming positivity and reality constraints for the real-space pattern. Phase-retrieval algorithm tended to stagnate to the local minima after 500 iterations. For each coherent scattering pattern the final real-space image was calculated from average of 500 algorithm trials with random initial phases. Reliability of the reconstructed real-space images was qualitatively and quantitatively examined by comparing the reconstructions performed for the data measured at Co and Mn $L_3$ edges, respectively. Indeed, based on the results of the XMCD and HERALDO experiments the same orientation of the local magnetic moments of Co and Mn atoms was expected.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{Fig9.png}\n\\caption{(Color online) Measured dependence of $q_{Sk}$ over temperature (symbols) and calculated dependence of $q_{Sk}$ over the exchange stiffness ($A_{ex}$) parameter.}\n\\label{Fig9}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=17cm]{Fig10.png}\n\\caption{(Color online) Magnetic field evolution of the $z-$projection of magnetization derived from the micromagnetic simulation of Co$_8$Zn$_8$Mn$_4$ thin plate in magnetic field $B=150$\\,mT (a), $B=250$\\,mT (b), $B=300$\\,mT (c), $B=350$\\,mT (d), $B=400$\\,mT (e), $B=450$\\,mT (f) showing the hexagonal SkX restoration.}\n\\label{Fig10}\n\\end{figure*}\n\nTo quantitatively estimate the reliability of the reconstructions performed for Co and Mn we have introduced the two-dimensional function \n$$R(i,j)=1-\\frac{|\\psi_{E_1}^{*(i,j)} - \\psi_{E_2}^{*(i,j)}|} {|\\psi_{E_1}^{*(i,j)}|},$$\nwhere $\\psi_{E_1}^{*(i,j)}$ and $\\psi_{E_2}^{*(i,j)}$ are values of $(i,j)$ pixels of the reconstructed real-space patterns normalized by maximum value for corresponding image at Co($E_1$) and Mn($E_2$) $L_3$ edges. This normalization is needed to compensate the difference in total magnetic contrast for Mn and Co patterns and transmission coefficient for the photons with different energies. Figure \\ref{Fig7} shows the real-space magnetic patterns reconstructed from data taken at Co absorption edge at different temperatures. At $T=25$\\,K and $T=55$\\,K (Figs. \\ref{Fig7}a,b) an array of magnetic skyrmions elongated along one of the crystallographic axes can be observed. As the temperature increases, some of the elongated skyrmions transform to the conventional circular-shaped (Fig. \\ref{Fig7}c), denoted as amorphous state. Unfortunately, similarly to the HERALDO experiment, the magnetic scattering intensity gradually decreases upon warming due to the thermal shrinkage of the magnetic moments. As a result, the signal to noise ratio is insufficient to reliably reconstruct the real-space image of the single-domain triangular skyrmion lattice at $T>120$\\,K. In the Supplementary information \\cite{supplementary} we show a few examples of the reconstructions performed for the skyrmion lattice and multidomain helical phases at higher temperatures. However, the real-space images obtained for these conditions correspond to the local minima in the solution space and vary sufficiently for the different starting phases. Two-dimensional reliability maps $R(i,j)$ for successful reconstructions can be found in the Supplementary information \\cite{supplementary}. Here we just note that the averaged reliability value $R=\\frac{1}{N}\\sum R(i,j)$, where $N$ is the sample area in pixels, is not less than 89\\% for the datasets measured at 25\\,K.\n\nFrom this aspect, the results of lensless imaging by iterative phase retrieval and HERALDO are consistent with each other. Both techniques provide a spatial resolution of 30\\,nm and sensitive to the signal-to-noise ratio. Data analysis in case of coherent diffraction imaging is more sophisticated compared to HERALDO, but the latter requires advanced sample fabrication. Further attempts to improve the nanofabrication routine are required to obtain better real-space resolution. \n\nThe experimental results for the temperature- and field-dependent transformation of the skyrmion lattice obtained by soft X-ray scattering and imaging agree well with the results of Landau-Lifshitz-Gilbert (LLG) simulations calculated with realistic parameters for Co$_8$Zn$_8$Mn$_4$ thin plate by using mumax$^3$ package \\cite{vansteenkiste2014design}. As hinted by present XMCD experiment and previous magnetization measurements \\cite{tokunaga2015new}, the effective ferromagnetic exchange interaction in Co$_8$Zn$_8$Mn$_4$ system may decrease with temperature due to antiferromagnetic correlations of Mn sub-lattice. Moreover, manifestation of the cubic anisotropy is considerable at low temperatures. For the simulation we used exchange stiffness $A_{ex} = 9.2$\\,pJ\/m and DMI constant $D=0.00053$\\,J\/m measured experimentally by microwave spin-wave spectroscopy \\cite{takagi2017spin}, and cubic anisotropy in the form of an effective field \\cite{vansteenkiste2014design}:\n\\begin{equation*}\n\\begin{aligned}\n\\mathbf{B}_{anis} &= - 2 K_c \/ M_{s} \\big(((\\mathbf{c}_{1} \\cdot \\mathbf{m})\\mathbf{c}_{1})((\\mathbf{c}_{2} \\cdot \\mathbf{m})^2 + (\\mathbf{c}_{3} \\cdot \\mathbf{m})^2)+ \\\\\n&((\\mathbf{c}_{2} \\cdot \\mathbf{m})\\mathbf{c}_{2})((\\mathbf{c}_{1} \\cdot \\mathbf{m})^2 + (\\mathbf{c}_{3} \\cdot \\mathbf{m})^2)+ \\\\\n&((\\mathbf{c}_{3} \\cdot \\mathbf{m})\\mathbf{c}_{3})((\\mathbf{c}_{1} \\cdot \\mathbf{m})^2 + (\\mathbf{c}_{2} \\cdot \\mathbf{m})^2)\\big),\n\\end{aligned}\n\\end{equation*}\nwhere $K_c = 5000$\\,J\/m$^3$ is the 1$^{st}$-order cubic anisotropy constant determined from electron spin resonance (ESR) experiment \\cite{kezmarki2018}; $\\mathbf{c}_{1}$, $\\mathbf{c}_{2}$ and $\\mathbf{c}_{3}$ is a set of mutually perpendicular unit vectors indicating the anisotropy directions (cubic axes), and $M_s=350$\\,kA\/m is the saturation magnetization \\cite{takagi2017spin}. A two-dimensional 200\\,nm-thick plate with area of $10\\times10$\\,$\\mu$m$^2$ with open boundary conditions and elementary cell size $5\\times5\\times200$\\,nm$^3$ was simulated. To follow the experimental field-cooling protocol, we started the simulation with random initial spin configuration in the applied field along $[001]$ direction. The field needed for robust SkX formation was set to $B=150$\\,mT. The discrepancy between the observed and calculated field values is, presumably, caused by demagnetization effects. The characteristic helical pitch $\\lambda=4\\pi A_{ex}\/D$ and skyrmion size are determined by the ratio of the exchange stiffness $A_{ex}$ and the Dzyaloshinskii constant $D$ \\cite{bak1980theory}. Therefore it is reasonable to assume two possible scenarios: 1) the temperature-dependent $q$-vector variation is caused by changing exchange integral due to the antiferromagnetic correlations in Mn sub-lattice towards lower temperatures; 2) the emergence of the $q$-vector elongation can be also caused by enhancement of antisymmetric DMI due to the local non-complanar structure of frustrated Mn. At the moment, the experimental data does not allow to unambiguously distinguish between variations of $A_{ex}$ and $D$. To mimic the first scenario, we reduced the exchange stiffness parameter $A_{ex}$ from the measured value $9.2$\\,pJ\/m to $3.2$\\,pJ\/m in a linear fashion, while the magnetic field $B=150$\\,mT remained constant. Relaxation time of 10 ns was introduced between each step for magnetic texture stabilization. Upon gradual decrease of the exchange interaction in the system, the skyrmion lattice exhibits deformation similar to the LTEM and soft X-ray experiments (Figs. \\ref{Fig7}a--c). Elongation of the skyrmions is manifested due to the overall reduction of the exchange interaction, while its directionality along the cubic axes is dictated by the cubic anisotropy. Notably, this deformation takes place via directional expansion of each topological vortex, but not by merging of the neighboring skyrmions -- in the latter case the total topological charge of the system would decrease, which is opposite to the topological charge conservation scenario reported earlier \\cite{morikawa2017deformation}. Radial averages of the two-dimensional fast Fourier transformation (FFT) patterns of out-of-plane projection of magnetization distribution was used to calculate the $q$-vector magnitude dependence for the resultant magnetic textures. The simulated variation in $q_{Sk}$ induced by change in the exchange stiffness parameter $A_{ex}$ is in the very good qualitative agreement with the temperature dependence (Fig. \\ref{Fig9}). The shoulder in Fig. \\ref{Fig9} at $A_{ex}$ between 5 -- 6\\,pJ\/m corresponds to the smooth transition from the disk-shaped to elongated skyrmions. Therefore we assume that the linear decrease of the exchange interaction in Co$_8$Zn$_8$Mn$_4$ takes place with decreasing temperature. In general, this consideration is consistent with the previous studies of the $\\beta-$Mn alloys, that have shown presence of antiferromagnetic correlation of moments in the $12d$ Mn sublattice \\cite{shiga1994polarized,nakamura1997strong,stewart2009magnetic,paddison2013emergent,bocarsly2019deciphering}. We plan to directly address to this question in our future studies. For example, recently developed spin-wave sensitive SANS technique allows to directly probe $A_{ex}$ of a bulk specimen as a function of temperature \\cite{grigoriev2015spin}.\n\nLLG simulation was either used to probe the magnetic field evolution of the elongated skyrmion texture. The \"field-cooled\" magnetic texture shown in Fig. \\ref{Fig10}c with $A_{ex}=3.2$\\,pJ\/m  was used. Except the discrepancy between the field  magnitude values, the result is consistent with present X-ray and previous LTEM experiments -- restoration of the hexagonal lattice of circular skyrmions from the deformed state by ramping the magnetic field was successfully reproduced (Figs. \\ref{Fig10}a--f).\n\n\\section{Conclusion}\n\nIn conclusion, by means of element-selective soft X-ray circular magnetic dichroism we have revealed the ferromagnetic arrangement of Co and Mn ions in a room-temperature skyrmion-hosting compound Co$_8$Zn$_8$Mn$_4$. Moreover, by using the coherent resonant small-angle soft X-ray scattering and holography with extended reference we can conclude that the topological magnetic texture is the same for both type of atoms in whole temperature range above $T_g$ that is reliable for real-space reconstruction. Our results are consistent with each other and with the previous neutron scattering and Lorentz microscopy experiments and shows the transition from hexagonal skyrmion crystal to elongated skyrmion state that is accompanied by deformation of the individual vortices. Micromagnetic simulation suggests that such transition is driven by decreasing exchange interaction in the system and effect of the cubic anisotropy. This effective decrease of the ratio of symmetric exchange interaction to antisymmetric Dzyaloshinskii-Moriya mimics low-temperature antiferromagnetic frustration of Mn sub-lattice. At lower temperature, antiferromagnetic correlations of Mn atoms is superimposed onto the long-range helical (skyrmion) modulation, resulting in shortening of the helical pitch and deformation of skyrmions. However, this effect is reversible and hexagonal skyrmion lattice from elongated skyrmion state can be restored by increasing magnetic field even when the exchange stiffness is reduced, as learned from the micromagnetic simulation and previous experiments \\cite{karube2016robust,morikawa2017deformation}.\n\nWe have demonstrated first to our knowledge lensless soft X-ray imaging of the magnetic texture at cryogenic temperatures and applied magnetic fields. HERALDO imaging was used with the circularly polarized soft X-rays, while coherent diffraction imaging was performed with a linearly polarized beam. Both methods did not require focusing X-ray optics to perform magnetic imaging with resolution of few tens of nanometers. Practically, the signal to noise ratio sufficient for the successful reconstruction was achieved only in the temperature range from $T=20$\\,K to $T=120$\\,K due to the overall decay of the intensity of the magnetic scattering and charge-magnetic interference with increasing temperature.\n\nSoft X-ray imaging methods allow to simultaneously obtain element-selective real-space information and will be useful for further investigations of non-trivial magnetic textures in thin plates of polar magnets, since N\\'eel-type skyrmions produce no contrast in Lorentz transmission electron microscopy.\n\n\\section*{Acknowledgments}\nThe authors wish to acknowledge P. Gargiani and BOREAS beamline staff for the technical assistance. We also thank T. Honda for providing the membranes. Soft X-ray scattering experiments were performed as a part of the proposals no.: 2015S2-007 (Photon Factory) and 2016081774 (ALBA Synchrotron Light Laboratory).\nThis research was supported in part by PRESTO Grant Number JPMJPR177A from Japan Science and Technology Agency (JST), \"Materials research by Information Integration\" Initiative (MI$^2$I) project of the Support Program for Starting Up Innovation Hub from JST, the Japan Society for the Promotion of Science through the Funding Program for World-Leading Innovative R\\&D on Science and Technology (FIRST Program), and JSPS KAKENHI Grant Number 16H05990. V.U. acknowledges funding from the SNF Sinergia CDSII5-171003 NanoSkyrmionics. M. V. acknowledges additional funding to the MARES endstation by grants MICINN ICTS-2009-02, FIS2013-45469-C4-3-R and  FIS2016- 78591-C3-2-R (AEI\/FEDER, UE).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe study of quantum chaotic systems has gained extensive attention in the last few years. In classical mechanics, the chaos of a system is described as the sensitivity of the particle trajectory to the initial conditions. Or, in other words, the sensitivity of the position of a system in the phase space to the initial conditions. For chaotic systems, the trajectory can deviate exponentially due to small changes in the initial conditions, which is called the butterfly effect. However, quantum chaos is still difficult to be well understood. One important reason is that the concept of trajectory becomes ill-defined after considering quantum effects. One has difficulty in finding a physical observable in quantum systems to measure the extent of chaos. Despite the difficulties in measuring quantum chaos, great progress has been made in the last few years in understanding quantum chaos. To date, many methods have been developed to probe quantum chaos.\n\nIt is worth noting that the study of holography has greatly advanced the understanding of quantum chaotic systems. One of the most exciting and promising examples is the study of operator growth. In a quantum chaotic system, a simple operator becomes extraordinarily complicated after the Heisenberg time evolution. Measuring how an operator becomes more and more \"complex\" from a \"simple\" can be an important indicator in chaotic systems. Similar to the classical case, we expect the measurement to be sensitive enough to the initial conditions, and the deviation caused by different initial conditions (or rather, the complexity of the operator) grows exponentially with time.\n\nThe out-of-time-order correlator (OTOC) method has gained a lot of attention in the study of operator growth \\cite{Rozenbaum:2016mmv,Hashimoto:2017oit,Nahum:2017yvy,Shen:2017kez,Fan:2016ean,Khemani:2017nda,Rozenbaum:2018sns,Pilatowsky-Cameo:2019qxt,Xu:2019lhc,Rozenbaum:2019nwn,Hashimoto:2020xfr}. One reason is that OTOC can reproduce the exponential growth in classical chaotic models. Using the OTOC method, we can generalize the Lyapunov exponent to quantum systems, i.e., the quantum Lyapunov exponent. Another important reason is that, in the context of AdS\/CFT correspondence or quantum gravity, it can be an indicator of possible gravity dual\\cite{Shenker_2014_1,Shenker_2014_2,Maldacena_2016}. Gedanken experiment about shock waves in the black hole background can lead to a maximum bound of Lyapunov exponent\\cite{Maldacena_2016}. This Lyapunov bound is saturated in the SachdevYe-Kitaev (SYK) model \\cite{Sachdev:1992fk}, which implies that the gravity duality of the SYK model may describe a quantum black hole. In addition, the operator size has been argued to correspond to the radial momentum of the particle in the gravity side\\cite{https:\/\/doi.org\/10.48550\/arxiv.1802.01198}. The radial momentum is defined by the maximal volume of the Cauchy surface in the bulk. We will see that this has a very close relation with another important concept in quantum chaotic systems, circuit complexity.\n\nIn parallel developments, circuit complexity (or computational complexity) is another physical observable that has been given significant importance in the study of AdS\/CFT and quantum chaos\\cite{https:\/\/doi.org\/10.48550\/arxiv.1403.5695,https:\/\/doi.org\/10.48550\/arxiv.0804.3401,Stanford_2014,https:\/\/doi.org\/10.48550\/arxiv.1411.0690,Brown_2016,PhysRevLett.116.191301}. In 2014, Susskind and Stanford proposed that the circuit complexity in the boundary state is dual to the maximal volume of the Cauchy surface in bulk. Since then several proposals for the holographic complexity have been proposed, including the CA correspondence \\cite{Brown_2016,PhysRevLett.116.191301} and CV2.0 correspondence \\cite{Couch_2017}. To be more specific, in the study of the eternal AdS-Schwarzchild black hole, it was found that the volume of the wormhole connecting two boundaries grows linearly with time. And an important feature of computational complexity is it grows linearly in chaotic systems until the Heisenberg time $t \\sim e^S$. The conjecture is the growth of the wormhole in bulk is dual to the growth of the complexity on the boundary.\n\nAlthough both the studies of operator growth and circuit complexity have made exciting progress in their respective directions. However, the relation between these two similar but parallel directions has been ambiguous. An emerging concept in recent years, Krylov complexity (or K-complexity in short) \\cite{Parker_2019,Barb_n_2019,Rabinovici_2021,Jian:2020qpp}, seems to bridge the gap between these two fields. It is not only a well-defined physical measure of operator growth but also has the same growth pattern as circuit complexity. In other words, there are possible geometric duals for K-complexity. We will focus on this concept in this paper. \n\nKrylov complexity was first put forward in \\cite{Parker_2019} to measure the universal features of operator growth. Precisely, it measures the extent how an operator spread in the Krylov basis, which is constructed from the Baker-Campbell-Hausdorff formula for the Heisenberg time evolution of chosen operator. The process of constructing the Krylov basis is known as the Lanczos algorithm. When implementing the Lanczos algorithm, we obtain the so-called Lanczos coefficients $b_n$, which compactly contain the information in the Krylov basis. In the chaotic system, Lanczos coefficients are expected to show a linear growth pattern at the beginning and then go into saturation. Also, Lanczos coefficients are argued to have the fastest rate of growth in chaotic systems. In addition, the slope of the linear growth $b_n \\sim a n$ of Lanczos coefficients is conjectured to give the upper bound $\\lambda < 2a$ of the quantum Lyapunov exponent. This upper bound is saturated in the SYK model. The hypothesis itself is even valid beyond the semiclassical system where the Lyapunov exponent is ill-defined. From this point of view, Krylov complexity is important for finding a well-defined physical quantity to measure operator growth in quantum systems.\n\nK-complexity has a growth pattern similar to that of circuit complexity, i.e., exponential growth at the beginning, followed by linear growth, and finally saturation. In the AdS\/CFT correspondence, some possible gravitational duals of K-complexity have been investigated, e.g., the SYK model and JT gravity \\cite{Jian:2020qpp}. Besides, one of the most important drawbacks of circuit complexity is that it is extremely hard to calculate circuit complexity in strongly interacting quantum field theory. While K-complexity seems to have a greater potential to be computed in quantum field theory\\cite{Caputa:2021ori,Caputa:2021sib,Dymarsky:2021bjq}. It is for these reasons that calculating K-complexity in strongly interacting QFT becomes an interesting problem. In this paper, we will show the calculation of K-complexity in the SU(2) Yang-Mills theory. \n\nSU(2) Yang-Mills theory was shown to be chaotic in \\cite{matinyan1981stochasticity}. It is expected to have a gravity dual in the large N and strong coupling limit. So, this model can be a starting point to investigate the relation between K-complexity and AdS\/CFT correspondence. From another point of view, it can be recognized as a rather simple version of the Quantum chromodynamics (QCD). this model has attracted a lot of attention over the years. For these reasons, the SU(2) Yang-Mills theory and related models have attracted attention for many years.\n\nThis paper is organized as follows. In section \\ref{section2}, we review how the SU(2) Yang-Mills theory can be reduced to a coupled harmonic oscillator(CHO) model by dimensional reduction, and analyze the chaotic nature of this model from both classical and quantum perspectives. In section \\ref{section3}, we review the concepts of Krylov complexity and Lanczos coefficients, then summarized our approach to numerically implement the Lanczos algorithm. In section \\ref{section4}, we numerically compute the Krylov complexity and Lanczos coefficients in the CHO model for different coupling constants. We found that Krylov complexity in the CHO model shows a quadratic growth in the early time stage, then enters a linear growth. By analyzing the Lanczos coefficients, we find an upper bound on the quantum Lyapunov exponent. In section \\ref{section5}, we discuss the effect of different energy modes on K-complexity, and how different temperatures and energy affect Lanczos coefficients. We conclude this paper with a brief discussion in section \\ref{section6}.\n\n\\section{Classical and quantum description of coupled harmonic oscillators}\\label{section2}\nIn this section, we review how SU(2) Yang-Mills theory can be reduced to a coupled harmonic model (CHO) by dimensional reduction, following the work of \\cite{Akutagawa_2020,matinyan1981stochasticity}. And then in section \\ref{2.2}, we analyze the CHO model classically. In section \\ref{2.3}, we review the OTOC method and quantum Lyapunov exponent in the CHO model.\n\n\\subsection{Reduction of SU(2) Yang-Mills theory}\\label{2.1}\nThe four-dimensional SU(2) Yang-Mills theory has the following Lagrangian\n\\begin{equation}\n    \\mathcal{L} = (D_\\mu \\phi)^\\dagger (D^\\mu \\phi) - V(\\phi) - \\frac{1}{4} F^a_{\\mu \\nu}F^{a \\mu \\nu}\n\\end{equation}\nwhere the field strength $F^a_{\\mu \\nu} = \\partial_\\mu A^a_\\nu -\\partial_nu A^a_{\\mu} + g \\epsilon^{abc} A_{\\mu}^b A_{\\nu}^{c}$, and $D_\\mu = \\partial_\\mu -i g A^a_\\mu T^a$, $T^a = \\sigma^a\/2$ ($\\sigma^a$ is the Pauli matrix ). Here, the index $a= 1,2,3$ and $\\mu, \\nu = {0,1,2,3}$. The scalar potential $V(\\phi)$ is given by\n\\begin{equation}\n    V(\\phi) = \\mu^2 |\\phi|^2 +\\lambda |\\phi|^4\n\\end{equation}\nThe gauge condition we chose is $A^a_0 = 0$. Furthermore, we assume the fields satisfy the spatial homogeneousness condition.\n\\begin{equation}\n    \\partial_i A^a_j =0, \\ \\ \\ \\partial_i \\phi =0\n\\end{equation}\nThen the remaining degrees of freedom are $A^1_1, A^2_2$. Let $A^1_1=\\sqrt{2} x(t), A^2_2= \\sqrt{2}y(t)$, then the Hamiltonian becomes \n\\begin{equation}\n    H = \\dot{x}^2+\\dot{y}^2 + \\frac{\\omega^2}{4}(x^2+y^2) + g^2 x^2 y^2\n\\end{equation}\nwhere we have introduced the frequency $\\omega = g \\langle \\phi \\rangle_{vac}$.\n\nThis Hamiltonian can also be recognized as a toy model of the Quantum\nchromodynamics (QCD) with the colors $N_C = 2$. Then the coupling constant $g$ controls the non-linear interaction between the colors \\cite{Matinyan:1981dj}, and $\\omega$ denotes the mass, which is finite in the presence of the Higgs mechanism.\n\nIn classical systems, Poincar\u00e9 sections and Lyapunov exponents are often used to measure the chaotic nature of a system. While in quantum mechanics, OTOC is the physical quantity that is most often used to measure the properties of chaotic systems. We will review these methods in the following sections.\n\n\\subsection{Classical analysis of coupled harmonic oscillators} \\label{2.2}\nN-dimensional Coupled harmonic oscillators (CHO) is a quantum mechanical model whose Hamiltonian is the following.\n\\begin{equation}\n    H = \\sum_{i=1}^{N}\\left( p_i^2 + \\frac{\\omega^2}{4} q_i^2 \\right) +\\frac{1}{2}\\sum_{i,j=1}^N g_{ij} q_i^2 q_j^2 \n\\end{equation}\nwhere $\\omega$ is the frequency of the harmonic oscillators. And $g_{ij}$ are the coupling constants, which determine the strength of the nonlinear interaction between different harmonic oscillators. \n\nWhen $g_{ij}$ is zero, the whole system is simplified to N decoupled harmonic oscillators, which is an integrable system. As $g_{ij}$ slowly increases, the non-linear term $g_{ij} q_i^2 q_j^2$ starts to dominate and the model becomes chaotic.\n\nThe Poincar\u00e9 section is an intuitive way to qualitatively measure the chaotic nature of a classical system. We plotted the Poincar\u00e9 sections with energy $E = 1$ for different coupling constants in Figure \\ref{fig120}. It can be seen that when the coupling constant $g_0 = 0.1$, the corresponding Poincar\u00e9 sections show clear orbits, which indicates the system is in a regular phase. The chaotic property of the system starts to appear when $g_0 = 0.3$. Randomly scattered points around the fixed points gradually replace the continuous trajectories. When $g_0 = 0.7$, only a small part of the Poincar\u00e9 section has continuous trajectories, while most of the figure is covered by scattered points. And when the coupling constant $g_0 =1$, the Poincar\u00e9 section is almost completely covered by scattered points, so the system is in the chaos phase. \n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[$g_0 =0.1$]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.1,E=1.png}\n}\n\\quad\n\\subfigure[$g_0 =0.3$]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.3,E=1.png}\n}\n\\quad\n\\subfigure[$g_0 =0.7$]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.7,E=1.png}\n}\n\\quad\n\\subfigure[$g_0 =1$]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=1,E=1.png}\n}\n\\caption{Poincar\u00e9 sections of the CHO model for four different coupling constants $g_0 = 0.1,0.3,0.7,1$. As the coupling constant rises, the clear orbits with strong periodicity gradually become randomly scattered points.}\n\\label{fig120}\n\\end{figure}\n\nOn the other hand, energy is also an important factor that affects the chaos in the system. We show in figure \\ref{fig147} the Poincar\u00e9 section at different energy $E = 1, 3, 5, 15$. We can see the transition from an integrable system to a chaotic system. At $E = 1$, the system is in regular phase. When the energy becomes $E = 3, 5$, the system transforms into a mixed phase. Finally, when $E$ rises to $15$, the system enters the chaos phase.\n\n\\begin{figure}[tbp]\n\\centering\n\\subfigure[E=3]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.1,E=3.png}\n}\n\\quad\n\\subfigure[E=5]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.1,E=5.png}\n}\n\\quad\n\\subfigure[E=7]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.1,E=7.png}\n}\n\\quad\n\\subfigure[E=15]{\n\\includegraphics[width=5.5cm]{Poincare section\/g0=0.1,E=15.png}\n}\n\\caption{Poincar\u00e9 sections in the CHO Hamiltonian system for four different energy $E=3, E=5, E=7, E=15$. When the energy is low, the Poincar\u00e9 section shows the characteristics of integrable systems, i.e., clear periodic orbits. As the energy increases, the system gradually enters the chaotic phase.}\n\\label{fig147}\n\\end{figure}\n\nAnother parameter that affects the chaotic property of the system is time. If the coupling constant $g_0<1$, according to the perturbative theory, when $t$ is small, the nonlinear term in the Hamiltonian is suppressed. So, the system shows the characteristics of an integrable system in the early time stage. After that, chaotic characteristics start to emerge. We analyze in section 4 the time scale of this transition.\n\nThrough the discussion in this section, we find, classically, the chaotic property of the CHO model is governed by three parameters coupling constant $g_0$, energy scale $E$, and the evolution time t. We will further discuss their effects on quantum chaotic systems in sections \\ref{section4} and \\ref{section5}.\n\n\\subsection{OTOC method for coupled harmonic oscillators}\\label{2.3}\nIn classical chaotic systems, the Lyapunov exponent $\\lambda_{cl}$ is defined by the exponential growth of the orbit deflection\n\\begin{equation}\n    \\Delta x(t) \\simeq \\Delta x(0)e^{\\lambda_{cl} t}\n\\end{equation}\nwhere $\\Delta x(0)$ means the small differences in initial conditions, which can cause huge deviations for the orbit afterward. The corresponding exponent is defined the be the classical Lyapunov exponent $\\lambda_{cl}$. This concept can be generalized to quantum systems. The quantum Lyapunov exponent is related to the out-of-time-order correlator (OTOC). In a thermal system, the OTOC is defined by \n\\begin{equation}\n    C_\\beta(t) = -\\langle [\\hat{x}(t),\\hat{p}]^2 \\rangle_{\\beta}\n\\end{equation}\nwhere $\\beta$ is the inverse temperature, $\\langle \\rangle_\\beta$ denotes the thermal expectation. Here we have chosen $\\hat{p}$ as the reference operator. The OTOC is expected to grow exponentially in chaotic system $C_\\beta(t) \\simeq C_\\beta(0)e^{\\lambda t}$. And the quantum Lyapunov exponent is defined to be the exponent $\\lambda$.\n\nQuantum Lyapunov exponent is an important observable to measure the chaotic structure of a system. It was conjectured that the linear growth of Lanczos coefficients gives a maximum bound on quantum Lyapunov exponent \\cite{Parker_2019}. We will analyze how the Lanczos coefficients in the CHO model bound the quantum Lyapunov exponent in section \\ref{section5}.\n\n\\section{Krylov complexity in coupled harmonic oscillators}\\label{section3}\n\\subsection{Brief review of Krylov complexity}\\label{3.1}\n\nIn this subsection, we briefly review the concept of operator growth and Krylov complexity. In a quantum system, the time evolution of an operator is given by\n\\begin{equation}\n    \\mathcal{O}(t) = e^{iHt}\\mathcal{O}e^{-iHt}\n\\end{equation}\nwhere $\\mathcal{O}$ is the operator when $t=0$. We can expand it via the Baker-Campbell-Hausdorff formula\n\\begin{equation}\n    \\mathcal{O}(t) = \\sum_{n=0}^{\\infty} \\frac{(i t)^n}{n!} \\bar{\\mathcal{O}}_n\n\\end{equation}\nwhere $\\bar{\\mathcal{O}}_n$ denotes the n-th commutator of $\\mathcal{O}$ with Hamiltonian\n\\begin{equation}\n    \\bar{\\mathcal{O}}_0 = \\mathcal{O}, \\ \\ \\ \\ \\bar{\\mathcal{O}}_1 = [H,\\mathcal{O}], \\ \\ \\ \\ \\bar{\\mathcal{O}}_2 = [H,[H,\\mathcal{O}]],...\n\\end{equation}\nAlternatively, we can write in another formalism, using the well-known Liouvillian super operator. The Liouvillian super operator acts on the operator $\\mathcal{O}$ by\n\\begin{equation}\n    \\mathcal{L} \\mathcal{O} = [H, \\mathcal{O}]\n\\end{equation}\nIn this way, we can write the time evolution of the operator as\n\\begin{equation}\n    \\mathcal{O}(t) = e^{i\\mathcal{L} t}\\mathcal{O}\n\\end{equation}\n\nInspired by this, we can define a set of orthonormal bases in the operator space, which is called Krylov basis. Krylov basis is constructed by imposing an iterative Gram-Schmidt orthogonalization for $\\mathcal{O}_n$. This process is known as the Lanczos algorithm. However, before introducing the Lanczos algorithm, we need to determine the inner product used in the operator space. following the work of \\cite{Jian:2020qpp}, we take the Wightman inner product\n\\begin{equation}\\label{Wightman}\n    (A|B) = \\langle e^{H \\beta\/2} A^{\\dagger} e^{-H \\beta\/2} B \\rangle_\\beta\n\\end{equation}\nwhere the notation $\\langle ... \\rangle_\\beta$ represents the thermal expectation of operators\n\\begin{equation}\\label{191}\n    \\langle A \\rangle_\\beta =\\frac{ \\Tr (e^{-\\beta H} A)}{Z},\\ \\ \\ \\  Z = \\Tr(e^{-\\beta H})\n\\end{equation}\nWith the definition of inner product, we can start building the orthonormal basis. The Lanczos algorithm process is as follows\n\n\\begin{enumerate}\n\\item Setting the starting conditions:$| \\mathcal{O}_0 ) =  |\\bar{\\mathcal{O}}_0 )\/ \\Vert \\bar{\\mathcal{O}}_0 \\Vert $, \\ \\ \\ \n$| \\mathcal{O}_1 ) =  b_1^{-1}\\mathcal{L}|\\mathcal{O}_0 )$,\\ \\ \\ $b_1 = (\\mathcal{O}_0\\mathcal{L} |\\mathcal{L}\\mathcal{O}_0 )$\n\\item For $n \\geq 2$, operator $A_n$ are constructed iteratively by:\n\\begin{equation}\\label{151}\n|A_n) = \\mathcal{L} |\\mathcal{O}_{n-1} ) - b_{n-1} |\\mathcal{O}_{n-2} )\n\\end{equation}\nThe normalization constant $b_n$ is given by $b_n = (A_n|A_n)^{\\frac{1}{2}}$. Then the basis element is constructed as follows:\n\\begin{equation}\\label{155}\n|\\mathcal{O}_n) = b_n^{-1}|A_n)\n\\end{equation}\n\\item If $b_n = 0$, stop the algorithm.\n\\end{enumerate}\n\nThe normalization constant $b_n$ is known as the Lanczos coefficients. If we consider a system with finite-dimensional Hilbert space. Then the dimension $K$ of the space spanned by the Krylov basis is the value of n when $b_n = 0$. It was proven that Krylov space dimension satisfies $K \\leq D^2-D+1$ \\cite{Rabinovici_2021}, where $D$ is the dimension of Hilbert space. We will see later that this bound of the Krylov space is the reason for the saturation of Krylov complexity. However, if we consider a physical system with an infinite dimensional Hilbert space, where the Krylov space is infinite-dimensional, the condition $b_n = 0$ will never be satisfied. Accordingly, the Krylov complexity will keep growing forever. The CHO model we studied in this paper belongs to the latter one.\n\nOnce we obtained the Krylov basis, we can use it to expand the time-dependent operator $\\mathcal{O}(t)$\n\\begin{equation}\\label{159}\n    |\\mathcal{O}(t)) = \\sum_n i^n \\varphi_n(t) |\\mathcal{O}_n)\n\\end{equation}\nThe Krylov amplitude $\\varphi_n(t)$ is a real number and it indicates the extent to which the operator spreads on the Krylov chain. If we multiply both sides of eq(\\ref{159}) by $(\\mathcal{O}_n|$, we get an explicit expression for $\\varphi_n(t)$.\n\\begin{equation}\\label{amplitude}\n    \\varphi_n(t) = i^{-n} (\\mathcal{O}_n|\\mathcal{O}(t))\n\\end{equation}\nThe sum of the Krylov amplitude $\\varphi_n(t)$ is conserved over time and satisfies the normalization condition. \n\\begin{equation}\\label{171}\n    \\sum_n |\\varphi_n(t)|^2 = 1\n\\end{equation}\n\nSince the CHO model has an infinite dimensional Hilbert space, we must make a truncation $n \\leq N_{cut}$ on the Krylov basis $|\\mathcal{O}_n )$ when implementing the numerical calculation. To ensure the accuracy of the numerical result, we will always keep the difference between eq(\\ref{171}) and 1 no greater than $10^{-5}$. \n\nIf we plug eq(\\ref{159}) in the Heisenberg equation, $\\partial_t \\mathcal{O} = i[H,\\mathcal{O}]$, we can obtain the following recursion relation for $\\varphi_n(t)$\n\\begin{equation}\n    \\partial_t \\varphi_n(t) = b_n \\varphi_{n-1}(t) - b_{n+1} \\varphi_{n+1}(t)\n\\end{equation}\nFrom this, we can see that the time evolution of the Krylov amplitude $\\varphi_n(t)$ is completely determined by the Lanczos coefficient $b_n$. This can also be seen from the matrix representation of the Liouvillian operator $(\\mathcal{O}_n|\\mathcal{L}|\\mathcal{O}_m) = L_{nm}$\n\\begin{equation}\nL = \n    \\begin{pmatrix}\n    0 & b_1 & 0 & ... & 0 \\\\\n    b_1 & 0 & b_2 & ... & 0 \\\\\n    0 & b_2 & 0 & ... & 0 \\\\\n    \\vdots &  & \\vdots & \\vdots & \\vdots\\\\\n     & ... & & ... & b_{K-1}\\\\\n    0 & 0 & ... & b_{K-1} & 0\n    \\end{pmatrix}\n\\end{equation}\n\nWith the previous knowledge laid out, we can next introduce the definition of the Krylov complexity\n\\begin{equation}\\label{K-complexity}\n    C_K = \\sum_n n |\\varphi_n(t)|^2\n\\end{equation}\nwhich can be recognized as the average expectation of the first-order moment of $\\varphi_n(t)$ over the Krylov chain.\n\nKrylov complexity is considered to be an important indicator of chaotic systems. For general integrable systems, it is observed that the Lanczos coefficient shows the following sub-linear growth pattern at $n\\ll \\ln{S}$ (S is the entropy of the system)\n\\begin{equation}\n    b_n \\sim a n^\\delta\n\\end{equation}\nwhere $a$ is a constant coefficient and $\\delta$ satisfies $0<\\delta<1$. Correspondingly, Krylov complexity satisfies polynomial growth in the early time\n\\begin{equation}\\label{241}\n    C_K(t) \\sim (a t)^{\\frac{1}{1-\\delta}}\n\\end{equation}\nFor chaotic systems, Lanczos coefficients satisfy linear growth when n is small.\n\\begin{equation}\n    b_n \\sim a n\n\\end{equation}\nAfter ending the linear growth, Lanczos coefficients enter the saturation plateau. Correspondingly, K-complexity satisfies the exponential growth in the early time stage.\n\\begin{equation}\\label{250}\n   C_K(t) \\sim e^{2 a t}\n\\end{equation}\nAt this point, the coefficient $2 a$ on the exponent is considered to provide an above bound for the quantum Lyapunov exponent $\\lambda$ \\cite{Parker_2019}.\n\\begin{equation}\\label{bound}\n    \\lambda \\leq 2 a\n\\end{equation}\n\nThe exponential growth of the Krylov complexity continues until $t \\sim \\ln(S)$. From then on, the Krylov complexity becomes to grows linearly $C_K(t) \\propto t$. \n\nThe behavior of K-complexity can be traced back to the saturation of Lanczos coefficients $b_n$. For chaotic systems, the Lanczos coefficients $b_n$ reach saturation after $n \\sim \\ln(S)$, which leads to the linear growth of K-complexity.\n\nFor systems with finite-dimensional Hilbert space, the growth of K-complexity will continue until the Heisenberg time $t \\sim e^{S}$, after which it enters the saturation plateau. The Lanczos coefficient $b_n$ starts to decrease rapidly after $n \\sim e^{S}$ until it reaches zero. This operator growth behavior is conjectured to apply to all local operators in chaotic systems \\cite{Parker_2019}. In the case we consider, the K-complexity will never saturate since the Hilbert space is infinite-dimensional. \n\n\\subsection{Numerical computation of Krylov complexity}\nIn this subsection, we describe our approach to numerically calculate the K-complexity in CHO model. The CHO model has the following Schr\u00f6dinger equation.\n\\begin{equation} \\label{Schr\u00f6dinger equation}\n    -\\left(\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}\\right)\\psi_n(x,y)+\\left[ \\frac{\\omega^2}{4}(x^2+y^2) + g_0 x^2y^2 \\right]\\psi_n(x,y) = E_n \\psi_n(x,y)\n\\end{equation}\n\nBy numerical calculation we can obtain a series of eigenvalues $\\{E_n\\}$ and eigenfunctions $\\{\\psi_n(x,y)\\}$ of the Schr\u00f6dinger equation (\\ref{Schr\u00f6dinger equation}). So, to implement the numerical calculation, all we have to do is to represent the Wightman inner product as a function of $ \\{ E_n \\}$ and $\\{\\psi_n(x,y)\\}$.\n\nFollowing the work of \\cite{Akutagawa_2020}. We use $\\hat{x}(t)$ as our local operator, i.e., $\\mathcal{O}(t) = \\hat{x}(t)$. Notice that the CHO model has the following commutation relation\n\\begin{equation}\n  \\begin{aligned}\n    [H,\\hat{x}] = -2ip_x, \\ \\ \\ \\ [H,\\hat{y}] = -2ip_y\n  \\end{aligned}    \n\\end{equation}\n\nTherefore, all local operators in the CHO model can be represented as functions of $\\hat{x}$ and $H$. It is worth noting that the commutation relation is independent of how many oscillators are in the model. So, this computational method can be used in an arbitrary dimensional CHO model. However, in this paper, we will only focus on the two-dimensional CHO model.\n\nIn the Lanczos algorithm, we will only do the inner product among the Krylov basis $\\{\\mathcal{O}_n\\}$. So we just need to express the inner product of the Krylov basis $\\{\\mathcal{O}_n\\}$ as a function of $ \\{ E_n \\}$ and $\\{\\psi_n(x,y)\\}$. According to (\\ref{151}) and (\\ref{155}), $\\mathcal{O}_n$ can be expressed as a polynomial of $\\hat{x}$ and $H$, where the order of $\\hat{x}$ in each term is 1 and the order of $H$ is at most $n$.\n\nFrom (\\ref{Wightman}), the Wightman inner product can be expressed as\n\\begin{equation}\n    (A|B) = \\frac{1}{Z} \\Tr \\left( e^{-H \\beta\/2} A^\\dagger e^{-H \\beta\/2} B \\right)\n\\end{equation}\nAfter inserting the completeness condition $\\sum_l |l\\rangle \\langle l|=1$, we get\n\\begin{equation}\n    (A|B) = \\frac{1}{Z} \\sum_{m,l} e^{-\\frac{\\beta}{2}(E_m+E_l)}A^\\dagger_{ml} B_{lm}\n\\end{equation}\nwhere we introduced $A_{ml} = \\langle m | A | l \\rangle$. Since we only need to consider the Wightman inner product among $\\{|\\mathcal{O}_n)\\}$, and $|\\mathcal{O}_n)$ can be expressed as a polynomial of $\\hat{x}$ and $H$. Therefore, we have\n\\begin{equation}\\label{234}\n\\begin{aligned}\n    (\\mathcal{O}_n)_{lm} &= \\sum_{k=1}^n \\sum_{a+b=k} C_{ab} E_l^a E_m^b \\langle l | \\hat{x} | m \\rangle\\\\\n    (\\mathcal{O}_n)^{\\dagger}_{ml} &= \\sum_{k=1}^n \\sum_{a+b=k} C^{\\prime}_{ab} E_m^a E_l^a \\langle m | \\hat{x} | l \\rangle\n\\end{aligned}\n\\end{equation}\nwhere $C_{ab}$ and $C^{\\prime}_{ab}$ are constant coefficients. According to eq(\\ref{234}), we only need to numerically integrate $\\langle m| x | l \\rangle$ to get the value of the Wightman inner product. Through this method, we can numerically compute the K-complexity and Lanczos coefficients.\n\nOur numerical approach can be summarized as follows\n\n\\begin{enumerate}\n    \\item Numerically solve the Schr\u00f6dinger equation and obtain the corresponding energy eigenvalues $\\{E_n\\}$ and eigenfunctions $\\{\\psi_n(x,y)\\}$.\n    \\item For different $m$ and $l$, compute $\\langle m | \\hat{x} | l \\rangle = \\int dx dy \\ \\psi_n^{\\dagger}(x,y) x \\psi_n(x,y)$ numerically.\n    \\item Implement the Lanczos algorithm by writing the Wightman inner product involved in each step as a function of $\\langle m | \\hat{x} | l \\rangle$, then plug in $\\langle n | \\hat{x} | m \\rangle$ obtained by step 2.\n    \\item Through the Lanczos algorithm, obtain Krylov basis $|O_n)$ and Lancaos coefficient $|b_n)$.\n    \\item Calculate the Krylov amplitude $\\varphi_n$ by (\\ref{amplitude}). The corresponding K-complexity is obtained by plugging $\\varphi_n$ into (\\ref{K-complexity}).\n\\end{enumerate}\n\n\\subsection{Energy spectrum and spectral form factor}\nBefore moving to the calculation of the K-complexity, we show in this section the energy spectrum as well as the spectral form factor of the CHO model. Figure \\ref{energyspectrum} shows the energy spectrum of the CHO model for different coupling constants from the numerical calculation of the Schr\u00f6dinger equation (\\ref{Schr\u00f6dinger equation}).\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=7cm]{energyspectrum.png}\n\\caption{}\n\\label{energyspectrum}\n\\end{figure}\n\nAnother important observable in chaotic systems is the spectral form factor $g(\\beta,t)$. Its behavior can show many characteristics of chaotic systems. And in recent developments it has been related to the information loss problem and AdS\/CFT correspondence \\cite{Dyer:2016pou,Cotler:2016fpe,Cotler:2017jue}. The spectral form factor $g(\\beta,t)$ is defined by\n\\begin{equation}\n    g(\\beta,t) \\equiv \\frac{|Z(\\beta + it)|^2}{Z^2(\\beta)}\n\\end{equation}\nwhere we have analytically continued partition function $Z(\\beta)$ in eq(\\ref{191}) into the complexity plane. In figure \\ref{spectralformfactor} we show the spectral form factor evolution with $t$ for coupling constant $g=0.1$ under different temperatures $T = 1, T=2, T=10$.\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]{\n\\includegraphics[width=5.5cm]{spectralformfactor.png}\n}\n\\quad\n\\subfigure[]{\n\\includegraphics[width=5.5cm]{spectralformfactor2.png}\n}\n\\caption{Spectral form factor $g(\\beta, t)$ as a function of time, for $g_0 = 0.1$ and three different temperatures $T=1,T=2,t=10$. In the early stage, the spectral form factor holds 1. After that, it drops to a dip and then bounces back to a plateau with erratic fluctuations. The fluctuation in the spectral form factor reflects the discrete eigenvalues of the system.}\n\\label{spectralformfactor}\n\\end{figure}\n\nAt low temperatures, $g(\\beta,t)$ seems to first decrease to the dip, and then bounce back to a certain value with erratic fluctuations. At high temperatures, $g(\\beta,t)$ first drops and then rises to a plateau with a relatively small value. We find the spectral form factor is very much reminiscent of that from the random matrix theory \\cite{Dyer:2016pou}, which implies that there may be a random matrix description for the SU(2) Yang-Mills theory.\n\n\n\\section{Krylov Complexity in chaotic transition}\\label{section4}\nIn this section, we numerically compute the behavior of Krylov Complexity and Lanczos coefficient in the CHO model using the method we presented in the previous section. In section \\ref{4.1}, we analyze the time scale for the system's chaotic property to start to emerge as well as the time scale for the K-complexity to grow linearly. In section \\ref{4.2}, we show the growth of Lanczos coefficients with different coupling constants. We found that the Lanczos coefficient grows linearly at the beginning and then goes into saturation. In section \\ref{4.3}, we present the result for the time evolution of Krylov complexity. In this section, we will fix the inverse temperature $\\beta =1$.\n\n\\subsection{Time scale for chaotic transition}\\label{4.1}\nIn the CHO model, K-complexity has two different growth patterns in the early and late time evolution. This is because, according to the BCH formula, at a very short time t, we can treat the nonlinear interaction $g_0 x^2 y^2$ in the Hamiltonian as a perturbative term. And the corresponding operator evolution can be approximated as\n\\begin{equation}\ne^{i H t} \\hat{x} e^{i H t} \\sim (1+ i g_0 \\hat{x}^2 \\hat{y}^2 t) e^{i H_0 t} \\hat{x} e^{i H_0 t} (1- i g_0 \\hat{x}^2 \\hat{y}^2 t) \n\\end{equation}\nwhere $H_0$ is the non-perturbative Hamiltonian. We want to find a time scale when the contribution from the non-linear term becomes important, i.e., the chaotic property of the system begins to emerge.\n\nWe assume that if the correction of the non-linear term for energy exceeds $10\\%$, the effect of the coupling term becomes non-negligible. In other words, when the expectation value of $g_0 \\hat{x}^2 \\hat{y}^2 t$ under the ground state $| \\Omega \\rangle = \\mathcal{N} e^{- \\beta E_n} |n \\rangle$ reaches $10\\% E$ (E is the energy of the system and $\\mathcal{N}$ is the normalization constant), the system starts to become chaotic.\n\nBy calculation, we can find that the expectation of the non-linear term $g_0 \\hat{x}^2 \\hat{y}^2 t$ is roughly\n\n\\begin{equation}\n    \\langle \\Omega | g_0 \\hat{x}^2 \\hat{y}^2 t |\\Omega \\rangle \\sim g_0 t\n\\end{equation}\n\nIf we choose $g_0 = 0.1$, then our chaotic transition time is roughly $t_0=1$. As we will show in the next sections, our estimate here matches our numerical results.\n\nAnother different time scale is the scrambling time $t_*$ at which the K-complexity starts to grow linearly. As stated in early sections, $t_* \\sim \\ln(S)$. The Shannon entropy $S_H$ of our system can be computed by\n\\begin{equation}\n    S_H = -\\sum_n \\mathcal{N}^2 e^{-2 E_n} \\ln(\\mathcal{N}^2 e^{-2 E_n})\n\\end{equation}\nThrough numerical calculation, we find the scrambling time $t_*$ is roughly around $t\\sim1$. We find that this is very close to the chaotic transition time $t_0$ we calculated. We will see in the following sections because the non-linear term of the system is suppressed in the early time, K-complexity in early growth satisfies not exponential growth but power-law growth. And, around the chaotic transition time $t_0$, K-complexity shifts directly from power-law growth to linear growth.\n\n\\subsection{Lanczos coefficients and upper bound for quantum Lyapunov exponent}\\label{4.2}\nLanczos coefficients are an important indicator in chaotic systems. First, the growth of Krylov Complexity is captured by the Lanczos coefficients. The linear growth of Lanczos coefficients corresponds to the exponential growth of K-complexity in the early time. Generally, in chaotic systems, the Lanczos coefficients grow linearly at the beginning, which is called Lanczos ascent. After the linear growth, it enters the saturation plateau. The K-complexity keeps growing linearly while the Lanczos coefficients are on the plateau. Besides, the linear growth of the Lanczos coefficients $b_n \\sim a n $ is expected to give a maximum bound $ \\lambda < 2 a$ for the quantum Lyapunov exponent.\n\nWe show in figure \\ref{fig335} the Lanczos coefficients $b_n$ for different coupling constants. From the figure, we can see that the Lanczos coefficients rise approximately linearly when n is small. Around $n \\sim 12$, the Lanczos coefficients go into saturation, which ensures an asymptotic linear growth of K-complexity. The system we consider has a Hilbert space with infinite dimension, so the Lanczos coefficient stays on the plateau and will not drop to zero. The effect of different coupling constants on the Lanczos coefficient is mainly to change the saturation value of Lanczos coefficients. The greater the coupling constant, the larger the saturation value of Lanczos coefficients. It can be translated to the behavior of K-complexity, i.e., the larger coupling constant will lead to a steeper linear growth for K-complexity.\n\n\\begin{figure}[htbp] \n\\centering\n\\includegraphics[width=0.7\\textwidth]{Lanczo.png} \n\\caption{Lanczos coefficients for different coupling constants $g_0 = 0.1, 0.3, 0.7$ at temperature $\\beta =1$. In the beginning, the Lanczos coefficients $b_n$ increase linearly with $n$. And after about n = 12, Lanczos coefficients enter the saturation plateau and fluctuate around a constant value.} \n\\label{fig335} \n\\end{figure}\n\nNotice that there is an oscillation phenomenon in the Lanczos coefficients. One possible explanation for this oscillation is that the Schr\u00f6dinger equation of Kyrlov amplitude allows the following solution\\cite{Yates:2020lin,Yates:2020hhj,Bhattacharjee:2022vlt}\n\\begin{equation}\n    b_n = f(n)+ (-1)^{n} \\Tilde{f}(n)\n\\end{equation}\nThe asymptotic behavior of the oscillatory function $\\Tilde{f}(n)$ can be derived from the auto-correlation function $C(t) = \\varphi_0(t)$. In general, if the auto-correlation function is power-law decaying $C(t) \\sim t^{-b}$, then the oscillation of the corresponding Lanczos coefficient satisfies a logarithmically decay $\\Tilde{f}(n) \\sim \\ln (n)^{-b}$\\cite{Bhattacharjee:2022vlt}.\n\nAccording to (\\ref{bound}), by computing the slope of the linear growth of Lanczos coefficients $b_n$, we can obtain the above bound of the quantum Lyapunov exponent. \n\n\\begin{equation}\n    \\lambda_{0.1} \\leq 1.57, \\ \\ \\\n    \\lambda_{0.3} \\leq 1.79, \\ \\ \\ \n    \\lambda_{0.7} \\leq 2.09\n\\end{equation}\nThe subscript of $\\lambda$ is the corresponding coupling constant.\n\n\\subsection{K-complexity growth in the chaotic transition}\\label{4.3}\nThrough the perturbative analysis in section \\ref{4.1}, we know that the non-linear term of the Hamiltonian can be neglected when $t$ is small enough. Therefore, the system can be considered as two decoupled harmonic oscillators, which is a system with the Heisenberg-Weyl symmetry. According to \\cite {Caputa:2021sib}, a system with the Heisenberg-Weyl symmetry has a quadratic growing K-complexity. \n\n\\begin{equation}\n    K \\sim t^2\n\\end{equation}\n\nThus, the K-complexity is expected to grow quadratically at the beginning. We show the early growth of K-complexity under different coupling constants in figure \\ref{K-complexity growth}. As expected, in the early time stage, the K-complexity shows a quadratic growth behavior. The dash lines in the figure are fitting functions $C_K(t) = \\gamma t^{2}$. The coefficients $\\gamma$ are respectively $\\gamma_{0.1} = 1.39$, $\\gamma_{0.3} = 1.76$, $\\gamma_{0.7} = 2.23$. Then in the period $0.1<t<1$, K-complexity starts to deviate from quadratic growth and turn into linear growth. The linear growth can be numerically fitted by $C_K(t) = \\eta t + \\xi$. The slop $\\eta$ for different coupling constants are $\\eta_{0.1} = 1.21$, $\\eta_{0.3} = 1.51$, $\\eta_{0.7} = 1.85$. From the analysis of Lanczos coefficients in section \\ref{4.2}, this linear growth will last forever. \n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]{\n\\includegraphics[width=5.5cm]{K-complexity_growth1.pdf}\n}\n\\quad\n\\subfigure[]{\n\\includegraphics[width=5.5cm]{K-complexity_growth2.pdf}\n}\n\\quad\n\\subfigure[]{\n\\includegraphics[width=5.5cm]{K-complexity_growth3.pdf}\n}\n\\caption{K-complexity evolution in different time stages. When t<0.1, the K-complexity shows a quadratic growth. During $0.1<t<1$, K-complexity deviates from the quadratic growth and then grows linearly with time.}\n\\label{K-complexity growth}\n\\end{figure}\n\nAs stated in section \\ref{3.1}, the power-law growth of K-complexity corresponds to integrable systems, while the linear growth corresponds to chaotic systems. Therefore, it can be seen clearly that in the CHO model, there is a transformation from integrability to chaos. The transition time $0.1 < t_{0} < 1$. The exponential growth of K-complexity (\\ref{250}) is also suppressed by this effect.\n\n\\section{Microcanonical K-complexity}\\label{section5}\nAs we saw in the classical analysis, the energy of the system will affect its chaotic nature. In the thermal state, high-modes show more chaotic properties, while low-modes show more integrable properties. In this section, we will analyze the effect of different energies as well as different temperatures on the K-complexity and Lanczos coefficients.\n\\subsection{Contribution from different energy}\nTo investigate the contribution from different energy modes to the K-complexity. we change the Wightman inner product as follows\n\\begin{equation}\\label{379}\n    (A|B)_E \\equiv \\frac{1}{Z} \\sum_{E_n = E} \\langle n | e^{H \\beta \/2} A^\\dagger e^{-H \\beta \/2} B | n \\rangle\n\\end{equation}\nwhere the summation means summing over all the states with degeneracy $E_n = E$. In this way, the original Wightman inner product can be expressed as\n\\begin{equation}\n    (A|B) = \\int dE \\ \\rho(E) e^{-\\beta E} (A|B)_E\n\\end{equation}\nwhere we defined $\\rho(E) = \\sum_{n}(E_n - E) $. Keeping the Krylov basis fixed, according to eq(\\ref{amplitude}), our Krylov amplitude becomes\n\\begin{equation}\n    \\varphi_n(t,E) = i^{-n} (\\mathcal{O}_n|\\mathcal{O}(t))_E\n\\end{equation}\nThis is related to the original Krylov amplitude by\n\\begin{equation}\n    \\varphi_n(t) =\\int dE \\ \\rho(E) e^{-\\beta E} \\varphi_n(t,E)\n\\end{equation}\nWith this definition, we can extract the contribution from different energy sectors to the Krylov amplitude $\\varphi_n(t)$.\n\nWe show in figure \\ref{fig421} the evolution of $|\\varphi_n(t,E)|$ for different energies $E$ in the CHO model with $g_0=0.1$ and $n=10$. It can be seen that after lifting the suppression of the Boltzmann factor $e^{-\\beta E}$, the high-energy modes have a significantly larger contribution to $\\varphi_n(t)$. Notice that when E is small, $\\varphi_n(t,E)$ has a distinct periodicity, which is a characteristic of the integrable system. In other words, the effect of the non-linear term in the CHO model is more on the higher energy modes. Therefore, when the energy of the system is elevated, it deviates from the integrable system and shows stronger chaotic features.\n\n\\begin{figure}[htbp] \n\\centering \n\\includegraphics[width=0.6\\textwidth]{energysector.pdf} \n\\caption{The norm of energy-dependent Krylov amplitude $\\varphi_n(t,E)$ vs evolution time $t$. Here we choose $n=10$ and five different values of energy $E=3.44,E=5.00,E=9.81,E=12.90,E=19.12$. At low energy sectors, $|\\varphi_n(t,E)|$ has a strong periodicity. While for high-energy, the periodicity of $|\\varphi_n(t,E)|$ is broken and the oscillations become random and erratic. on the other hand, high-modes of $|\\varphi_n(t,E)|$ have larger values than low-modes, which means it can contribute more to the K-complexity.} \n\\label{fig421} \n\\end{figure}\n\n\\subsection{Energy and temperature dependence of Lanczos coefficients}\n\nIn the classical limit, the Lanczos coefficients growth rate $a$ can be calculated by the singularity of the analytically continued auto-correlation function $C(t)$. Consider a classical orbit $E = H(x,y,\\frac{\\partial x}{\\partial t},\\frac{\\partial y}{\\partial t})$. Assume the equations of motion have the solution $x(t)=F_E(t)$, and inversely, $t(x)=F_E^{-1}(x)$. $x$ is analytic for all $t\\in \\mathbb{R}$. So the singularity in $x(t)$ has the imagination $\\sigma_* = -i F_E^{-1}(\\infty)$. So, $x(t)$ is analytic in the strip $\\{t:\\Im(t) \\leq \\sigma_* \\}$. Thus, the analytic region of auto-correlation function $C(t) = \\langle x(t) x(0) \\rangle$ can be determined by the singularity in $x(t)$. If we replace the ensemble average by the time average $C(t) = \\frac{1}{T}\\int x(s+t\/2)x(s-t\/2)ds$, then $C(t)$ is analytic in the strip $\\{t:\\Im(t) \\leq  2 \\sigma_* \\}$. According to \\cite{Avdoshkin:2019trj}, $2\\sigma_*$ is related to the Lanczos coefficients growth rate $a$ by $a = \\frac{\\pi}{4 \\sigma_*}$. \n\nIn the high energy limit, the harmonic potential $\\frac{\\omega^2}{4}(x^2+y^2)$ can be ignored. So, the remaining energy is $E = p_x^2+p_y^2+g_0 x^2 y^2$, which allows the scaling transformation, $x\\to \\Lambda x, y\\to \\Lambda y, t\\to \\Lambda^{-1} t, E\\to \\Lambda^4 E$. Plugging the scaling relation to $\\sigma_*$, it turns out that $\\sigma_*$ scales as $\\sigma_* \\propto E^{-1\/4}$. Therefore, the growth rate $a$ scales as $a= \\frac{\\pi}{4 \\sigma_*} \\propto E^{1\/4}$. In the low energy limit, the harmonic potential dominates, so $\\sigma_*$ doesn't scale with the energy. \n\nIn the thermal K-complexity, the contribution from different energy regions is suppressed by the Boltzmann factor $e^{-\\beta E_n}$. This means that increasing temperature will enhance the contribution from higher energy modes. We show in figure \\ref{fig431} the upper bound of quantum Lyapunov exponent $\\lambda_{max} = 2a$ for different temperatures when the coupling constant $g_0=0.1$.\n\n\\begin{figure}[htbp]\n\\centering \n\\includegraphics[width=0.6\\textwidth]{LanczosTemperature.pdf} \n\\caption{Numerical result for the maximum bound of quantum Lyapunov exponent $\\lambda_{max} <2a$ in different temperature. The horizontal coordinate denotes the inverse temperature $\\beta$.} \n\\label{fig431} \n\\end{figure}\n\n\\section{Conclusion and discussion}\\label{section6}\n\nIn this paper, we investigated the Krylov complexity and associated Lanczos coefficients growth in the SU(2) Yang-Mills theory, which can be reduced to a nonlinearly coupled harmonic oscillators model. We developed a method to numerically calculate Krylov complexity in this chaotic model. We computed the growth patterns of K-complexity and Lanczos coefficients for different coupling constants. We observed a transition from integrability to chaos in the CHO model. The K-complexity grows quadratically in the early stage, which is characteristic of integrable systems. After the transition time $t_0$, K-complexity grows linearly, which is characteristic of chaotic systems.\n\nWhile for Lanczos coefficients, we found that it linearly increases with $n$ when $n$ is small and then enters the saturation plateau, which is consistent with the universal hypothesis in \\cite{Parker_2019}. It was conjectured that the quantum Lyapunov exponent $\\lambda$ is upper bounded by the slope of the linear growth of the Lanczos coefficients $a$, i.e.,  $\\lambda < 2a$. Using this relation, we found the upper bound of the quantum Lyapunov exponent in the CHO model at different temperatures and coupling constants.\n\nBy analyzing the microcanonical K-complexity, we extracted the contributions from different energy sectors to the K-complexity and Lanczos coefficients. We found that higher energy modes have a greater contribution to the growth of K-complexity, which means that the higher the energy, the more the system shows the nature of quantum chaos. This is consistent with our classical analysis. Classically, by analyzing the singularity of the auto-correlation function $C(t)$, we obtained that the bound for Lyapunov exponent $\\lambda_{max}$ is proportional to $E^{1\/4}$ in the high-energy limit. In the thermal state, the energy of the system is closely related to the temperature. We calculated the effect of different temperatures on Lanczos coefficients. In addition, we also numerically computed the spectral form factor in the CHO model. We found a behavior similar to that of the random matrix theory, which indicates there might be a random matrix description of SU(2) Yang-Mills theory.\n\nIn conclusion, in SU(2) Yang-Mills theory, Krylov complexity is an interesting observable which encodes many features of quantum chaos. This may have implications for a good definition of quantum chaos. Also, the application of Krylov complexity to strongly interacting field theories may advance the search for its gravity dual. Although our study focuses on the two-dimensional CHO model, our approach can be generalized to higher dimensions. For future research, it will be interesting to see whether Krylov complexity can be computed in more general quantum chaotic systems and quantum field theories.\n\n\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStars form in dense cores of dust and gas within molecular clouds, but remain cocooned in the material of their natal envelopes. Though young stars and protostars can be extremely luminous, the enveloping dust absorbs nearly all of the ultraviolet and optical starlight and reradiates it at longer wavelengths. In the submillimetre regime, the dust emission is generally optically thin. Submillimetre observations thus sample all the dust emission within the cloud along the line of sight, and can be used to determine the column density and mass of material present \\citep{hild83}. Interpretation of the observations, however, depends on the assumed dust grain temperature, composition and size distribution. The effects of composition and grain size on the dust emission are usually combined into a dust opacity, $\\kappa_\\nu$, which can be reasonably fit empirically with a power law dependence on frequency in the submillimetre.\n\n\\begin{table*}\n\\centering\n\\begin{minipage}{106mm}\n\\caption{Positions and peak brightnesses for observed objects}\n\\label{positions}\n\\begin{tabular}{cccccc}\n\\hline\nSource & $\\alpha_{J2000}$ & $\\delta_{J2000}$ & $I_{450}$ & $I_{850}$ & Distance\\\\\n--- & \\fh\\,\\fm\\,\\fs & \\degr\\,\\,\\,\\arcmin\\,\\,\\arcsec & Jy\/8.5\\arcsec~beam & Jy\/15\\arcsec~beam & kpc\\\\\n\\hline\nG10.47 & 18 08 38.2 & $-$19 51 50 & 260 $\\pm$ 65 & 39.0 $\\pm$ 5.9 & 5.8\\\\\nG31.41 & 18 47 34.5 & $-$01 12 43 & 130 $\\pm$ 32 & 24.2 $\\pm$ 3.6 & 7.9\\\\\nG12.21 & 18 12 39.7 & $-$18 24 20 & 121 $\\pm$ 30 & 13.0 $\\pm$ 2.0 & 13.5\\\\\n\\hline\n\\end{tabular}\n\n\\medskip\nThe positions, peak brightnesses at 450\\,\\micron~and 850\\,\\micron, and distances to each of the hot molecular cores observed in this study, ordered by $I_{850}$. Positions and distances are taken from \\citet{hatchell00}, while peak brightnesses were calculated from the archived SCUBA data.\n\\end{minipage}\n\\end{table*}\n\nAt submillimetre wavelengths, dust is generally assumed to radiate as a blackbody at some average temperature $T_d$ modified by the optical depth, $\\tau_\\nu \\propto \\kappa_\\nu$. For optically thin emission, we then find $S_{\\nu} \\propto \\kappa_\\nu \\, B_{\\nu}(T_d)$. Assuming a power law dependence on frequency, the dust opacity $\\kappa_\\nu$ can be parameterized for a given opacity $\\kappa_0$ at frequency $\\nu_0$ as $\\kappa_\\nu=\\kappa_0 (\\nu \/ \\nu_0)^\\beta$, where $\\beta$ is the spectral index of the dust emission. For optically thin emission, the dust continuum emission then follows the power law:\n\\begin{equation}\nS_{\\nu} \\propto \\kappa_{0}(\\frac{\\nu}{\\nu_0})^{\\beta}B_{\\nu}(T_d)\n\\label{eqn:dust_law}\n\\end{equation}\nThrough mathematical modelling, $\\kappa_\\nu$ and $\\beta$ have been calculated for various dust grain compositions and size distributions \\citep{draine84,ossen94,poll94}, which can be compared with astronomical observations. Often, however, various combinations of the dust parameters can reproduce existing observational data with similar accuracy, while dust temperatures and densities can vary dramatically within star forming environments. Calculations often assume $\\beta\\simeq2$ at these wavelengths, based on work by \\citet{hild83}. In practice, however, theoretical dust grain models find spectral indices anywhere between zero and three (see the summary by Goldsmith, Bergin \\& Lis (1997))\\nocite{gold97}, depending on the physical dust properties of the model, while observations have found a similar range of values in different states of the interstellar medium.\n\nIn high mass star forming regions and the cores containing warm protostars, \\citet{will04} find the average frequency dependence of dust emission from more than $60$ high mass protostellar objects to be characterized by $\\beta=0.9\\pm0.4$. This is comparable to values observed in less evolved, colder objects \\citep{gold97,visser98,hoger00,beuther04}, but is systematically lower than spectral indices determined through submillimetre observations of hot molecular cores (HMC) and ultra-compact H\\,{\\small II} (UC\\,H\\,{\\small II}) regions, which are likely the next phases in the evolution of high mass protostars. In these more evolved cores, $\\beta$ values have been determined to be near to \\citep{osorio99} or slightly higher than \\citep{hunter98} $\\sim2$. \n\nThe changing values of $\\beta$ in these studies are indicative of the evolution of the dust itself in the changing physical environments associated with the process of star formation. The growth of dust grains, the formation of icy mantles, and changes in grain composition are likely all significant factors in the variation of the frequency dependence of the dust emission in these regions. In order to understand the properties and evolution of dust in star forming regions, observational methods are required that can discern between temperature and density effects and actual changes in the dust composition and opacity. \n\nTo determine $\\kappa_0$, observations must be calibrated against known dust column densities, which generally requires comparison with observations at shorter wavelengths. In contrast, the spectral index $\\beta$ of the dust opacity can be calculated solely in the submillimetre by comparing observations at two widely separated submillimetre wavelengths. This method provides great leverage on $\\beta$, but is complicated by calibration issues from observing in the different passbands, and has the disadvantage that the value determined for $\\beta$ is dependent on the assumed dust temperature. The Planck function at 450\\,\\micron~is far from the Rayleigh-Jeans limit for dust temperatures less than $\\sim40$\\,K, and even at the higher temperatures expected in high mass star forming regions, the change in emission between passbands could be due to either small changes in temperature or large changes in the dust emissivity. The uncertainties in the determination of $\\beta$ are thus large enough that the subsequently derived physical properties of dusty systems, such as the mass estimates of the material in dense cores in molecular clouds, may range over factors of a few. For this reason, it is beneficial to observe in one passband only.\n\nOne instrument that is well-suited to this problem is a Fourier Transform Spectrometer (FTS). FTS have several advantages and disadvantages compared with other spectrometer designs. The highest spectral resolution of an FTS is determined by the maximum optical path difference between the two beams of the interferometer, which is set by the length of the translation stage. In practice, space limitations require that most submillimetre FTS provide low to intermediate spectral resolution (R\\,$\\sim200$ - R\\,$\\sim2000-3000$ at 345-375\\,GHz), excellent for observations of sources with wide, bright spectral lines and continuum emission. For these studies FTS have a distinct advantage over high spectral resolution but narrow bandwidth heterodyne receivers. The instantaneous bandwidth of an FTS is inherently broad, and in the submillimetre is limited only by the atmospheric transmission windows, enabling a wider spectral coverage within these windows than is possible with the heterodyne receivers currently in use. In the past decade, two groups have successfully used FTS to study the submillimetre emission from several astronomical sources, such as the Orion molecular cloud \\citep{serabyn}, the Sun \\citep{naylor00}, and the atmospheres of planets \\citep{naylor94,davis97}.\n\nA submillimetre FTS operated at low resolution can sample the continuum dust emission from a source with a large enough bandwidth that the change in dust emissivity with frequency, and thus $\\beta$, can be determined solely within one passband. At very low temperatures, it is still difficult to separate the temperature component of the emission from the dust power-law emissivity function, but single bandpass observations are able to place limits on the combined dust temperature and $\\beta$. \n\nIn this study, we present low resolution 350\\,GHz continuum observations of four hot molecular cores performed with a submillimetre FTS of the Mach-Zehnder design (MZ FTS) developed for use at the JCMT by the FTS group at the University of Lethbridge, AB, Canada (described in detail by \\citet{naylor03}). \n\nWe describe in the following the observational procedure and analysis of the data. We compare the dust spectral index values determined for our sources with those calculated from previous submillimetre measurements, and discuss the detrimental effects of the variable submillimetre atmosphere on our sensitivity. Finally, we comment on the continuum measurement capabilities of an imaging FTS currently in preparation for use at the JCMT with SCUBA-2.\n\n\\section{Observations}\n\n\\subsection{Source selection}\n\nThe sources for this pilot study were chosen from a sample of HMC from work by \\citet{hatchell00} for their large submillimetre fluxes and lack of extended structure as shown by \\citet{hatchell00} and \\citet{walsh03}. The sources, G10.47, G12.21 and G31.41, have been previously studied in the submillimetre regime with the Submillimetre Common User Bolometer Array (SCUBA) at the JCMT \\citep{hatchell00, walsh03} and in the radio continuum (\\citet{wc89}, among others). All three have also been detected in numerous molecular lines and maser emission (see the detailed source descriptions in \\citet{hatchell00}). Table \\ref{positions} gives source positions and distances from \\citet{hatchell00}, and 450\\,\\micron~(660\\,GHz) and 850\\,\\micron~(350\\,GHz) peak brightnesses calculated using the archived SCUBA data\\footnote{Guest User, Canadian Astronomy Data Centre, which is operated by the Dominion Astrophysical Observatory for the National Research Council of Canada's Herzberg Institute of Astrophysics}. Each source is associated with one or more UC\\,H\\,{\\small II}~regions near the peak of the submillimetre emission. The hot core in G10.47 contains three UC\\,H\\,{\\small II}~regions, G10.47-0.03A, B and C, while the other sources are coincident with one UC\\,H\\,{\\small II}~region each, G12.21-0.10 and G31.41+0.31. These HMC are very peaked in the 850\\,\\micron~emission, although G31.41 does contain some more extended emission, and the hot cores themselves of these sources are much smaller than the JCMT beam with diameters of only 1-2\\arcsec. Core mass estimates are on the order of a few thousand M$_\\odot$ \\citep{hatchell00}. A fourth HMC, G43.89, was observed by us to only S\/N\\,$\\sim1.5$, too low to analyse with any degree of certainty, and is therefore omitted from further discussion. Observations of Mars were also performed for flux calibration.\n\n\\subsection{Observational method}\nThe observations were performed during 2003 April 21 - 28, using the MZ FTS mounted on the right Nasmyth platform of the JCMT on Mauna Kea, Hawaii. The April 2003 weather was good to excellent with the zenith opacity measured at 225\\,GHz~(1.3\\,mm), $\\tau_{225}$, $\\sim0.07$ for most of the observations, giving precipitable water vapour (pwv) levels averaging $\\sim1.1$\\,mm. At 350\\,GHz, the FTS bandwidth is $\\sim30$\\,GHz. The FTS was operated at low spectral resolution (1.5\\,GHz), resulting in $\\sim\\,20$ resolution elements across the band. The full width half maximum (FWHM) of the 350\\,GHz~beam at the JCMT is $\\sim15\\arcsec$, which was matched by a cold (4\\,K) aperture located at the field stop of the detector system \\citep{naylor99}. The atmospheric emission and transmission window at  350\\,GHz is shown in Figure \\ref{fig:ULTRAM}. \n\nSpectra were obtained in groups, with five spectra taken of a molecular core immediately followed by five spectra of the background sky, +1425\\arcsec~in right ascension from the source. The offset in right ascension was calculated to compensate for the change in airmass of the source due to the Earth's rotation during the time taken to obtain five spectra, allowing the sky measurements to be taken at the same airmass as the original source observations. This is crucial for accurate background sky subtraction, as the brightness and variability of the atmosphere at 350\\,GHz is easily the largest source of uncertainty in our final results. Five scans were obtained approximately every 90 seconds. Each target source was observed for between three and four hours, with approximately half of that time spent on source and half spent on background observations. Due to its brightness, Mars was observed for a half hour. A pointing check performed approximately every 45 minutes resulted in small ($\\sim1-2\\arcsec$) corrections to the pointing. \n\n\\subsection{Data reduction}\n\n\\begin{figure}\n\\includegraphics[width=84mm]{Fig1.ps}\n\\caption{The atmospheric radiance (solid line) in units of Jy\/15\\arcsec~beam and transmission (dash-dotted line) calculated for precipitable water vapour levels of 1.0mm.}\n\\label{fig:ULTRAM}\n\\end{figure}\n\nPreliminary data reduction was performed using processing pipeline software written for the FTS \\citep{naylor03}. The data were first manually screened for transients, such as cosmic rays, which occur approximately once in every ten interferograms and produce large intensity spikes at one or two data points in the interferograms. These spikes are easily detected and can be removed within the pipeline. Some spikes in the interferograms were spread over multiple data points; these interferograms were omitted from the final dataset. Only a handful of interferograms were rejected for this reason. Since the FTS optics and detector have negligible dispersion over the range of interest, a linear phase correction was applied to each interferogram. The individual interferograms were then Fourier transformed. The spectra were then coadded into the on- and off-source groups of five, and the coadded off-source background scans were subtracted from the coadded on-source scans. The coadded, sky-subtracted spectra of G10.47, G12.21, G31.41 and Mars are shown in Figure \\ref{fig:mean_spc}. \n\n\\begin{figure*}\n\\begin{minipage}{170mm}\n\\begin{center}\n\\includegraphics[width=84mm]{Fig2_1.ps}\n\\hfill\n\\includegraphics[width=84mm]{Fig2_2.ps}\\\\\n\\includegraphics[width=84mm]{Fig2_3.ps}\n\\hfill\n\\includegraphics[width=84mm]{Fig2_4.ps}\n\\caption{The raw, coadded, sky-subtracted spectra of G10.47, G12.21, G31.41 and Mars before calibration or atmospheric transmission correction. Error bars are at the 1\\,$\\sigma$ level. Note that the error bars for Mars lie within the line thickness.}\n\\label{fig:mean_spc}\n\\end{center}\n\\end{minipage}\n\\end{figure*}\n\n\nEach background-subtracted spectrum was then corrected for atmospheric transmittance within the band using the technique described by \\citet{davis97}. Updated atmospheric opacities were calculated using the University of Lethbridge Transmission and Radiance Atmospheric Model (ULTRAM; \\citet{chapman}). ULTRAM is a radiative transfer atmospheric model which calculates the radiance and transmission of the atmosphere through a line-by-line, layer-by-layer analysis, and was written with the specific goal of accurately modelling the atmosphere above Mauna Kea. The model parameterizes the atmospheric transmittance in terms of the pwv and airmass, and includes H$_2$O, O$_3$ and O$_2$ as atmospheric absorbers. The airmass of each scan was recorded with each interferogram. The opacity at 225\\,GHz, $\\tau_{225}$, was recorded by a dipping radiometer operated by the Caltech Submillimeter Observatory (CSO). The use of this opacity value in the data analysis introduces some uncertainty into the final results, as the CSO radiometer operates at a fixed azimuth which rarely corresponds with the azimuth of the JCMT observations, and only updates approximately every 10-15 minutes, while the pwv levels in the atmosphere may vary significantly on shorter timescales. This can be seen in Figure \\ref{fig:pwv_vs_cso}, where $\\tau_{225}$ from the CSO is plotted against $\\tau_{225}$ determined from measurements of the pwv recorded by the JCMT's water vapour monitor (which is pointed slightly off the main JCMT beam) during the observations of G10.47. The CSO $\\tau_{225}$ value is often offset from that recorded by the water vapour monitor, and does not reflect the short timescale of the opacity variability. The water vapour monitor, while operating during our observations, was new to the telescope and thus was not incorporated into the data reduction pipeline. In future, the use of the water vapour monitor will provide more accurate atmospheric pwv measurements.\n\n\\begin{figure}\n\\includegraphics[width=84mm]{Fig3.ps}\n\\caption{The opacity at 225\\,GHz measured from the CSO and determined through measurements of the precipitable water vapour by the JCMT's water vapour monitor (WVM) during observations of G10.47. The solid line is the CSO-measured opacity, corrected to match the JCMT observational airmass, which operates at fixed azimuth and updates approximately every 10-15 minutes. The WVM is pointed slightly off the main JCMT beam, and updates every 6 seconds.}\n\\label{fig:pwv_vs_cso}\n\\end{figure}\n\n\nMars observations were then used to calibrate the final, atmospheric transmission corrected, coadded spectra.\n\n\\section{Analysis}\n\n\\subsection{Source signal analysis}\n\nWe first discuss sky subtraction and instrumental contributions to the FTS signal. We then detail calibration issues, which include atmospheric variation and spectral line contamination of the continuum emission. \n\nThe measured signal $S$ as a function of frequency $\\nu$ from an astronomical source through the FTS can be expressed as\n\\begin{eqnarray}\nS(\\nu) &=& R(\\nu)\\,G\\,[\\eta_a\\,J_s(\\nu)\\,\\mbox{e}^{-\\tau(\\nu)} + \\eta_a\\,J_{sky}(\\nu) (1-\\mbox{e}^{-\\tau(\\nu)}) \\nonumber \\\\\n&+& (1-\\eta_a)\\,J_{amb}(\\nu)-J^\\prime(\\nu)] \n\\end{eqnarray}\nwhere the contribution from any additional background has been neglected. Here, $\\tau(\\nu)$ is the atmospheric optical depth as a function of frequency $\\nu$, the telescope aperture efficiency is given by $\\eta_a$,  $R(\\nu)$ is the responsivity or response function of the FTS, and $G$ is the detector gain when observing an astronomical source. For all observations, the gain was set to $G=10^4$. $J_s(\\nu)$ is the true spectrum of the source, $J_{sky}(\\nu)$ is the true spectrum of the atmosphere, $J_{amb}(\\nu)$ is the power received from ambient temperature surfaces of the telescope, and $J^\\prime(\\nu)$ is the power received through the $2^{nd}$ port of the FTS, which is differenced in the interferometric measurement \\citep{naylor00}. The first term therefore represents the power received from the astronomical source itself through the atmosphere and instrument, the second term represents the power received from the atmosphere and the third term represents the power received from the surfaces of the telescope, which are at ambient temperature.\n\nObservations at the background position produce solely the intensity received from the sky, $S_{sky}(\\nu)$, and ambient surfaces:\n\\begin{eqnarray}\nS_{sky}(\\nu) &=& G\\,R(\\nu)\\,[\\eta_a\\,J_{sky}(\\nu)(1-\\mbox{e}^{-\\tau(\\nu)}) \\nonumber \\\\\n&+& (1-\\eta_a)J_{amb}(\\nu)-J^\\prime(\\nu)]\n\\end{eqnarray}\nIf the atmospheric temperature and opacity are assumed to remain unchanged during each pair of source and background scans, the difference of the on- and off-source scans gives the observed power of the source itself, $S_s(\\nu)$:\n\\begin{eqnarray}\nS_s(\\nu)&=& S(\\nu) - S_{sky}(\\nu) = R(\\nu)\\,G\\,\\eta_a\\,J_s(\\nu)\\,\\mbox{e}^{-\\tau_s(\\nu)}\n\\end{eqnarray}\nSky brightnesses at 345\\,GHz~for the typical pwv levels during our observations of $\\sim1.1$\\,mm are $\\sim800$\\,Jy\/beam, an order of magnitude larger than our brightest source, making accurate background subtraction essential. During the observations, measurements of the precipitable water vapour, from the CSO radiometer, and observational airmass were recorded. With this information, the atmospheric model described in Section 2 was used to determine the transmission of the atmosphere in the direction of the source, and thus the contribution of the term $\\mbox{e}^{-\\tau(\\nu)}$ to the observed source intensity $S_s(\\nu)$. Typical values of $\\tau_{225}$ during the observations were $\\sim0.07$, leading to atmospheric transmission levels at 345\\,GHz~of $\\sim75$\\%.\n\nObservations of Mars and a brightness temperature model were used for calibration of the spectra obtained by the instrument. At the time of our observations, Mars had an angular diameter of 8.97\\arcsec. The Mars observations were done in the same on- and off-source pattern as the HMC observations, and the intensity $S_m(\\nu)$ received from Mars after sky subtraction is thus\n\\begin{eqnarray}\nS_m(\\nu)&=& S(\\nu) - S_{sky}(\\nu) = R(\\nu)\\,G\\,\\eta_a\\,J_m(\\nu)\\,\\mbox{e}^{-\\tau_m(\\nu)}\n\\end{eqnarray}\n\nThe brightness temperature of Mars, $T_{B,m}$, on the date of our observations was calculated using the STARLINK FLUXES program \\citep{starlink}. The Mars flux, $J_m(\\nu)$, at the telescope can then be calculated:\n\\begin{equation}\nJ_m(\\nu) = \\frac{2\\nu^2k}{c^2T_{B,m}}\n\\label{eqn:flux}\n\\end{equation}\n\nThe true spectrum of the hot core sources, $J_s(\\nu)$, can then be determined: \n\\begin{equation}\nJ_s(\\nu)=\\frac{S_s(\\nu)\\,\\mbox{e}^{-\\tau_m(\\nu)}}{S_m(\\nu)\\,\\mbox{e}^{-\\tau_s(\\nu)}} J_m(\\nu)\n\\end{equation}\n\nThe final 350\\,GHz spectrum of each source can be seen in Figure \\ref{fig:slopes} in units of Jy\\,\/beam. The derived intensity near the edges of the band has larger error due to the strong water vapour transitions seen in Figure \\ref{fig:ULTRAM}, which define the 350\\,GHz spectral window. Variations in the column abundance of water vapour between source and background measurements result in greater noise at the steepest part of the water vapour line profile.\n\n\\begin{figure}\n\\includegraphics[width=84mm]{Fig4_1.ps}\\\\\n\\includegraphics[width=84mm]{Fig4_2.ps}\\\\\n\\includegraphics[width=84mm]{Fig4_3.ps}\n\\caption{Final calibrated spectra of the hot core sources (top to bottom: G10.47, G31.41, G12.21) with 1$\\sigma$ error bars at 1.5\\,GHz~resolution. The dash-dotted line is the best fit linear slope. The dashed lines indicate the fitting window. Arrows indicate the positions of five molecular emission lines, listed in the text, expected to be strong in HMC. With the exception of the CO(3-2) line at 345.6\\,GHz, no spectral features are coincident with the expected bright line positions.} \\ \\ \\ \\ \\\n\\label{fig:slopes}\n\\end{figure}\n\n\\subsection{Signal-to-noise and atmospheric effects}\n\nThe instrumental signal-to-noise (S\/N) ratio was calculated from the MZ FTS characteristics in \\citet{naylor03} for the spectral resolution and specific observing times for each source, and is solely due to the detector noise of the FTS. The S\/N ratios of the final spectra are compared with the instrumental S\/N ratios in Table \\ref{SN}, with the average S\/N achieved across the band a factor of order $\\sim10$ lower than the instrumental value for all sources. With all detector effects accounted for in the instrumental S\/N ratio, the sensitivity degradation is solely due to the atmospheric variance during our observations. \n\n\\begin{table}\n\\caption{Instrumental and observed S\/N values}\n\\label{SN}\n\\begin{tabular}{cccc}\n\\hline\nObject & Time on source & S\/N$_{Ins}$ & S\/N$_{Obs}$ \\\\\n--- & (hours) & --- & --- \\\\\n\\hline\nG10.47 & $3.5$ & $147$ & $8.5$\\\\\nG31.41 & $4.0$ & $99$ & $8.0$ \\\\\nG12.21 & $4.0$ & $73$ & $7.4$ \\\\\n\\hline\n\\end{tabular}\n\n\\medskip\nThe instrumental and observed signal-to-noise ratio for each source.\n\\end{table}\n\nAt submillimetre wavelengths, the brightness and variability of the sky is an overwhelming obstacle to accurate flux calibration, and observations generally require some form of chopping between source and background measurements in order to remove the sky emission. Ideally, the time between source and background measurements is as short as possible to limit the level of sky variance between observations \\citep{arch02}. While our observational method was designed to minimize the effects of the ever-changing atmosphere on the data by switching between source and background observations every 90\\,s, the atmosphere still contributed substantially to our uncertainties for several reasons. First, the opacity varies on timescales shorter than 90\\,s, as shown in Figure \\ref{fig:pwv_vs_cso}. Second, delays in slewing the telescope to the background position resulted in slight differences between the airmass of the background and source scans, introducing a small systematic effect by effectively offsetting the opacity in the background scan relative to the on-source scan. \n\nAs a quick test, we can think of a small increase in opacity as a small increase in the precipitable water vapour (with the caveat that there are other minor contributions to the opacity). Using atmospheric emissivities calculated for various pwv levels with ULTRAM, we have calculated that a modest jump in pwv of 0.05\\,mm, very reasonable when combining small deviations between source and background airmass with actual changes in the atmospheric pwv levels, increases the atmospheric emission by $\\sim35$\\,Jy\/beam at 350\\,GHz. The pwv values calculated from the CSO $\\tau_{225}$ for the observations contain jumps of this size and a few even larger. For this reason, stable weather is more crucial to this study than particularly good weather. A change in atmospheric emission of 35\\,Jy is comparable to the flux of our brightest source, G10.47, and brighter than the others (see Table \\ref{positions}). As a consequence, some of the subtractions of the source and background spectra produced unphysical, negative flux measurements, while others resulted in impossibly high source brightnesses. When the spectra are coadded, however, on the whole these overly positive and negative scans will average out unless there is a large systematic offset between source and background observations or the pwv levels in the atmosphere are systematically increasing or decreasing. \n\nWhile there was a small systematic offset in the airmass between the on- and off-source scans, it was not large enough to produce significant changes in the sky emission except in the case of G31.41, which was observed while setting. For this source, the offset may have decreased the observed flux by at most 1-2\\,Jy\/beam. During observations of G10.47, the recorded atmospheric opacity steadily increased, although it remained low. After atmospheric transmission corrections, the recorded intensity of this source near the start of the observations was greater by $\\sim10$\\% on average than that recorded later. Again, this may have decreased the observed flux by a few Jy\/beam at most, and is the largest contributor to the uncertainty per data point in the spectrum. There do not appear to be any systematic trends in the atmospheric opacity for the G12.21 observations.\n\n\\begin{figure}\n\\includegraphics[width=84mm]{Fig5_1.ps}\\\\\n\\includegraphics[width=84mm]{Fig5_2.ps}\n\\caption{Sky-subtracted spectra, divided into two groups corresponding to observations made in the first and second halves of each night, then coadded and ratioed for Mars and G10.47. The dash-dotted line indicates the best-fit slope of the ratioed data for G10.47, showing that while the variable atmosphere may introduce offsets into our final observed intensity of the sources, it has not significantly affected the slope of the continuum emission.}\n\\label{fig:source_vs_source}\n\\end{figure}\n\nTo estimate the effects of the varying atmosphere on our spectra, and to determine the level of any systematic effects which may have been introduced into the data, we divided the sky-subtracted spectra into two groups for each source corresponding to observations made in the first and second halves of each night. We then coadded the spectra in these groups, and ratioed the coadded spectrum of the first half with that of the second. Figure \\ref{fig:source_vs_source} shows this ratio for Mars, our calibrator, and G10.47. It is clear that while there is a small shift in overall flux between the early and later observations, within the band there is little overall change in the continuum slope. The fainter sources showed similar results, and we thus expect that while the variable atmosphere may introduce offsets into our final observed intensity of the sources, it has not significantly affected the slope of the continuum emission. \n\n\\subsection{Signal strength}\n\nThe final brightness values recorded by the FTS for each source are listed in Table \\ref{fluxes} with calculated uncertainties. The uncertainty in the final flux value comes from the uncertainty in the observed flux of the source, $S_s(\\nu)$, the observed flux of Mars, $S_m(\\nu)$, the uncertainty in the Mars brightness temperature model, and the uncertainty in the atmospheric transmission, $\\mbox{e}^{-\\tau(\\nu)}$. From the S\/N of the observations, the uncertainty in $S_s(\\nu)$ is $\\sim11-12$\\% for each source. We take an uncertainty in the atmospheric transmission of 10\\%, while the estimated uncertainty for the Mars model is of order 5\\%. The uncertainty in $S_m(\\nu)$ is negligible compared to the other factors due to Mars' brightness. Added in quadrature, this gives an uncertainty in the calibrated source flux of 16\\%. Additionally, the error lobe of the 850\\,\\micron~(350\\,GHz) JCMT beam may contain up to $\\sim5$\\% of the source flux, making the final uncertainties of order 21\\%. \n\n\\begin{table}\n\\caption{Comparison of source brightnesses from SCUBA and the FTS}\n\\label{fluxes}\n\\begin{tabular}{ccc}\n\\hline\nObject & $I_\\nu$ (SCUBA) & $I_\\nu$ (FTS) \\\\\n--- & Jy\/15\\arcsec~beam & Jy\/15\\arcsec~beam \\\\\n\\hline\nG10.47 & $39.0\\pm5.9$ & $44.0\\pm8.8$ \\\\\nG31.41 & $24.2\\pm3.6$ & $24.6\\pm4.9$ \\\\\nG12.21 & $13.0\\pm2.0$ & $27.2\\pm5.4$ \\\\\n\\hline\n\\end{tabular}\n\n\\medskip\nA comparison of the 850\\,\\micron~source brightnesses determined using the FTS with the peak brightnesses recorded by the Submillimetre Common User Bolometer Array. The uncertainties in the SCUBA fluxes are 15\\%, while the uncertainties in the FTS fluxes are 21\\%.\n\\end{table}\n\nThrough calibration with Mars, the observed fluxes of G10.47 and G31.41 agree very well with those recorded by SCUBA; however, G12.21 is much brighter in our observations than the submillimetre continuum measurements indicate, and the values are not within uncertainties. The peak fluxes from the archived SCUBA data for G12.21, which are from two separate observations, agree within uncertainties, suggesting that a variation of the source brightness is unlikely. G12.21 is the faintest source by the SCUBA measurements, and is thus more susceptible to atmospheric subtraction problems than G10.47 and G31.41. Since the largest effect of the variable atmosphere on our observations is to introduce an offset into the overall continuum, and because this offset is cumulative, it is possible that the final flux for this source has been inflated by the bright, variable atmosphere.\n\n\\subsection{Molecular line contamination}\n\nThe sources observed in this study are hot molecular cores. The moderate temperatures of these objects ($T>100$\\,K) cause molecules and elements frozen onto the surface of dust grains during the colder collapse phase to evaporate, releasing these species into the gas phase. The resulting emission lines are often strong in the 350\\,GHz spectral window for HMC, and can therefore contaminate measurements of the submillimetre continuum. The sources in this study have been detected in many molecular lines \\citep{hatchell98} from multiple species present in the 350\\,GHz atmospheric window.\n\nAt our low spectral resolution only very bright, relatively broad lines would contribute noticeably to the dust continuum spectrum. Our spectra show little evidence of bright line contamination, as shown in Figure \\ref{fig:slopes}, where we have indicated the expected positions of $^{13}$CO (3-2), CS (7-6), CO (3-2), HCN (4-3), and HCO$^+$ (4-3) emission lines. These are the strongest, broadest lines observed in the 350\\,GHz window of the hot core G5.89-0.39 \\citep{thompson}, which we expect to be similar to the cores studied here. There is some evidence of small peaks in all the spectra, one of which appears to be at the CO(3-2) transition frequency (345.6\\,GHz); however the data do not allow a conclusive identification. None of the other observed peaks in the spectra are coincident with expected bright line positions. Overall the peaks are small and do not significantly influence the continuum slope, and the contamination of the continuum spectrum by single bright emission lines is therefore not a large problem for this analysis. The concern for this study is then the contamination of the continuum spectrum by a significant number of low-level emission lines that may be present and are not immediately visible, but which may influence the slope of the continuum.\n\nThe frequency coverage of molecular line studies for these sources is much less than that covered by SCUBA and the FTS, so no absolute estimates of the level of contamination have been made \\citep{hatchell00}. Observing at 350\\,GHz with the JCMT, the conversion factor from brightness temperature $T_b$ in Kelvin to flux $S$ in Janskys is approximately $S\\,\\mbox{(Jy)} \\simeq 13 T_b\\,\\mbox{(K)}$ \\citep{hatchell98}. A brightness temperature of 40\\,K thus corresponds to $\\sim 500$\\,Jy.  The line width of strong molecular features, however, is seldom greater than 15\\,km\\,s$^{-1}$ (at least for the bright core of the line) while the spectral resolution of the FTS for this study is $\\sim 1500$\\,km\\,s$^{-1}$, so that individual lines are diluted by at least a factor of 100. Strong molecular lines, with peak brightness $\\sim 40$\\,K, will thus add approximately 5\\,Jy to the measured flux in a given spectral bin. Therefore, we do not expect contamination of the spectra by individual lines by more than approximately 10-20\\%.\n\nThe contamination of the submillimetre continuum of hot core sources by molecular lines can range between 10\\% and (in extreme cases) 60\\% of the total integrated flux \\citep{groes94}. \\citet{johns03} find that the 350\\,GHz continuum emission from their observed protostellar sources, however, is never dominated by molecules other than CO (although line emission from HCN, HNC, CN and CH$_3$OH can contribute a substantial fraction of the contamination for more energetic sources) and line contamination in general is typically less than $\\sim10\\%$ at 350\\,GHz, even for photon-dominated regions. The hot core regions of the sources observed in this study amount to a small fraction of the 350\\,GHz beam (1-2\\arcsec~in diameter compared with 15\\arcsec~FWHM), whereas the more extended emission is from cooler material in which fewer molecules will be excited. Overall, we argue that molecular line contamination of our low resolution 350\\,GHz continuum is minimal, and does not significantly affect the analysis. We recognize, however, that it is important to consider the influence of these low-level molecular lines when making any determination of the slope of the dust continuum emission using low spectral resolution submillimetre observations. \n\n\\section{Spectral index results}\n\n\\subsection{Calculation of the spectral index}\n\nFrom Equation \\ref{eqn:dust_law}, the dust emission $S(\\nu) \\propto \\nu^\\gamma$, where $\\gamma=\\beta+\\alpha$ is the sum of the dust spectral index $\\beta$ and the frequency dependence $\\alpha$ from the Planck function. For small ranges in frequency $\\Delta\\,\\nu$ around a central frequency $\\nu_0$, we can expand this relationship:\n\\begin{eqnarray} \nS(\\nu_0+\\Delta\\,\\nu) &\\propto& (\\nu_0+\\Delta\\,\\nu)^\\gamma \\nonumber \\\\*\n&\\propto& \\nu_0^\\gamma \\,\\biggl(1+\\gamma\\frac{\\Delta\\,\\nu}{\\nu_0}\\biggr)\n\\end{eqnarray}\nThus, for small $\\Delta\\,\\nu\\,\/\\nu_0$, the dust emission is expected to increase linearly with frequency. The FTS passband at 350\\,GHz is $\\simeq30$\\,GHz~wide, or $\\simeq10$\\% of the observing frequency, making the approximation valid. In practice, this window was made slightly smaller due to a decrease in S\/N caused by increased levels of contaminating atmospheric flux near the edges of the band, leaving $\\Delta\\,\\nu\/\\nu\\simeq0.08$ on average. A $\\chi^2$ minimizing linear fit routine was used to determine the slope of the continuum emission inside the passband. All results quoted here have been determined from the unapodized data to retain the highest spectral resolution. The results varied slightly depending on the fraction of the band used (due to increasing error bars at the band edges when using more of the band, and conversely due to a smaller lever arm when using less of the band); however any variations were within uncertainties. \n\nColumn 2 of Table \\ref{tab:betas} shows the best fit continuum emission slope $\\gamma$ for each source with uncertainties determined through the $\\chi^2$ fitting routine, and Figure \\ref{fig:slopes} shows the final spectrum for each source overlaid with the best linear fit. The average reduced $\\chi^2$ value for the fits was $\\sim0.6$, indicating a good fit to the data within the band. We expect, for a good linear fit, that the reduced $\\chi^2$ should be close to unity for a sufficiently large number of data points. The small values of the reduced $\\chi^2$ in our fits suggest that the uncertainty per data point has been overestimated by $\\sim20$\\% for each source. Since the uncertainty in the calculated slope varies as the uncertainty per data point, this implies that the uncertainties in our slopes may be overestimated by $\\sim20$\\%. \n\n\\begin{table}\n\\caption{Spectral indices}\n\\label{tab:betas}\n\\begin{tabular}{cccc}\n\\hline\nSource & $\\gamma_{FTS}$ & $\\beta$ & $\\gamma_{SCUBA}$ \\\\\n\\hline\nG10.47 & $3.6\\pm1.2$  & $1.6\\pm1.2$  & $3.4\\pm0.6$  \\\\\nG31.41 & $4.1\\pm1.2$  & $2.1\\pm1.2$  & $3.3\\pm0.6$  \\\\\nG12.21 & $3.3\\pm1.1$  & $1.3\\pm1.1$  & $4.1\\pm0.6$  \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nFor emission in the Rayleigh-Jeans limit, $\\alpha=2$ and thus the dust spectral index $\\beta=\\gamma-2$. In order for the R-J approximation to apply at 350\\,GHz, $h\\nu\/kT_d\\ll 1$, so $T_d\\gg h\\nu\/k\\sim17$\\,K. Although HMC are warm, they are surrounded by a shell of cold dust. The observed emission is integrated over the line of sight and the beam, which at 15\\arcsec~FWHM is much larger than the hot cores. There is therefore substantial contribution to the observed emission from the colder outer regions along the line of sight. Models of the emission from these sources which take the source temperature and density profiles into account, however, appear to produce spectra which are in the R-J limit at 350\\,GHz \\citep{hatchell00,hatchell03}. When considering only the peak flux, a greater proportion of the emission from the warm inner regions is contained within the single beam, and the R-J approximation still applies. Column 3 of Table \\ref{tab:betas} lists the dust spectral index $\\beta$ calculated for each source assuming the 350\\,GHz emission is in the R-J limit. We find $\\beta$ in the range of $1.3 - 2.1$ with a mean of 1.6.\n\n\n\\subsection{Comparison with SCUBA data}\n\nAs discussed in Section 1, the spectral index of dust emission can also be calculated using observations at two widely separated wavelengths. From Equation \\ref{eqn:dust_law}, we see that we can solve for $\\gamma=\\beta+\\alpha$ with a ratio of the flux at two different wavelengths:\n\\begin{equation}\n\\gamma = \\frac{\\ln(S_{2}\\,\/\\,S_{1})}{\\ln(\\nu_2\\,\/\\,\\nu_1)} = \\beta + \\alpha\n\\label{eqn:scuba_calc}\n\\end{equation}\nFor our sources, archived SCUBA data at 450\\,$\\mu$m (660\\,GHz) and 850\\,$\\mu$m (350\\,GHz) are publicly available. In order to properly compare the flux at the two wavelengths, the 8.5\\arcsec~resolution 450\\,\\micron~data were first convolved to the 15\\arcsec~resolution of the 850\\,\\micron~data. The ratio of the peak brightnesses was then used to calculate $\\gamma$ from Equation \\ref{eqn:scuba_calc}. Column 4 of Table \\ref{tab:betas} gives the $\\gamma$ value, $\\gamma_{SCUBA}$, calculated from the ratio of the 450\\,\\micron~and 850\\,\\micron~fluxes for each source. Within uncertainties, $\\gamma_{SCUBA}$ is consistent with $\\gamma_{FTS}$. The uncertainties in this calculation are dominated by the uncertainty in the source flux at both wavelengths.\n\nIf the R-J approximation applies at both frequencies $\\nu_1$ and $\\nu_2$, then $\\alpha=2$ and $\\beta=\\gamma_{SCUBA}-2$. As shown previously, the emission at 350\\,GHz is likely in the R-J limit, but for the R-J approximation to apply at 660\\,GHz, $T_d\\gg h\\nu\/k\\sim32$\\,K.  For the average dust temperatures likely associated with these hot molecular core sources, the 660\\,GHz emission cannot be assumed to be in the R-J limit. In this case, less power is contributed to the dust emission from the Planck function, and $\\alpha<2$. Using this information, we can use the 350\\,GHz (850\\,\\micron) slope determined using the FTS and $\\gamma_{SCUBA}$ to estimate the temperatures of the observed HMC. \n\nThe spectral indices calculated for G10.47 agree well with $\\alpha\\simeq2$, suggesting that the average dust temperature for this source is high enough ($T_d>60$\\,K) for the emission at both wavelengths to be in the R-J limit. G12.21 is also likely warm, while the results for G31.41 indicate a lower average dust temperature. The uncertainties in these temperature estimates are very large due to the large uncertainties in the evaluated slopes; it is clear, however, that with smaller uncertainties in the value of $\\beta$ determined from FTS observations, it will be possible to independently determine both $T_d$ and $\\beta$ of the dust opacity in star forming regions. \n\n\\subsection{Interpretation}\n\nOur results are consistent with the values of $\\beta$ found in embedded UC\\,H\\,{\\small II}~regions and other HMC. Fitting models of HMC to observational data have found $\\beta=1.6-2$ \\citep{osorio99,church90}. In a study of 17 UC\\,H\\,{\\small II}~regions, \\citet{hunter98} finds an average spectral index of $\\beta=2.0\\pm0.25$ through greybody modelling of IRAS, submillimetre and millimetre observations. Observations and modelling of six compact HII regions, one of which was G10.47, by \\citet{hoare91} find an average $\\beta\\simeq1.5$. They note, however, that the frequency dependence is somewhat affected by the assumed density distribution of the model. The dense ridge in Orion has $\\beta\\simeq1.9$ near embedded infrared sources \\citep{gold97}. Overall, the spectral indices of these warmer, likely more evolved star forming regions tend to be higher than those found in cooler, less evolved dense cores, indicating that dust properties have changed through some mechanism which appears to correlate with protostellar age. \n\nSeveral dust models predict a dust emissivity frequency dependence similar to that found here, but our observations are not sensitive enough to determine a most likely dust composition. Higher values of $\\beta$ have been associated with dust grains covered in thick ice mantles \\citep{aanestad,lismenten}, while many dust grain models with combinations of silicate and graphite compositions find a spectral index of $\\beta\\sim1.5-2$. \\citet{ossen94} tabulate the dust opacity properties for nine different dust distributions, including bare grains versus grains with ice mantles (both thick and thin), and various degrees of dust coagulation. The emissivity results presented in this paper are unable to definitively rule out any of the models. The best fit, however, is for bare grains in relatively dense environments, where coagulation is more pronounced (column 2 of Table 1 in \\citet{ossen94}).\n\nWhile we had hoped to determine $\\gamma$ to an accuracy of $\\pm\\,0.1$, the variable atmosphere at 350\\,GHz degraded the achieved signal-to-noise ratio by a factor $\\ga10$ compared with the instrumental S\/N, leading to uncertainties in our calculation of $\\beta$ of approximately $\\pm1.2$. Even with the atmospheric complications, however, the uncertainties determined here are of the same order as those found when calculating the dust spectral index through multiple wavelength observations of single sources. These results show that with better sky subtraction techniques, it should be possible to determine the spectral index of bright astronomical sources with high accuracy using an FTS. Future instruments with greater sensitivity and precise sky subtraction will enable the measurement of spectral indices of much fainter sources. These results show that careful sky subtraction is \\textit{essential} to future dust continuum emission studies with an FTS.\n\n\\subsection{An imaging FTS for SCUBA-2}\n\n\\begin{table*}\n\\begin{minipage}{135mm}\n\\caption{Source brightnesses required to determine $\\gamma$ in 12 hours of observing}\n\\label{tab:ifts}\n\\begin{tabular}{cccccc}\n\\hline\nWavelength & Frequency & Resolution & 12 hrs 1$\\sigma$ $\\Delta$T & Source brightness & Source brightness \\\\\n\\micron & \\,GHz & \\,GHz & mK & $\\gamma \\pm 0.1$ (Jy\/beam) & $\\gamma \\pm 0.3$ (Jy\/beam) \\\\\n\\hline\n450 & 660 & 3.0 & 1.44 & 4.7 & 1.6 \\\\\n850 & 350 & 3.0 & 0.2 & 0.40 & 0.14 \\\\\n\\hline\n\\end{tabular}\n\n\\medskip\nSource brightnesses required to determine the dust spectral index $\\gamma$ in 12 hours of observing using an IFTS on SCUBA-2.\n\\end{minipage}\n\\end{table*}\n\nSCUBA-2 is the next generation submillimetre instrument currently in development to replace SCUBA in 2006 \\citep{holl03}. An imaging FTS (IFTS) is being developed as an ancillary instrument to SCUBA-2 \\citep{naylor04}. Based on detector noise calculations alone, the IFTS will be at least 10 times more sensitive per pixel than the single bolometer FTS used in these observations \\citep{gom04}. The extremely accurate sky subtraction possible with the IFTS will provide an even more substantial increase in the observational precision. When also considering the variable atmosphere noise, the per-pixel sensitivity of the IFTS increases to at least 100 times greater than in the observations presented in this paper. Using the detector sensitivities in \\citet{gom04}, we have calculated that it will be possible to determine in 12 hours of observing with the IFTS the spectral index $\\gamma=\\beta+\\alpha$ of the submillimetre dust emission at 850\\,\\micron~ to $\\pm0.1$ for sources of only 400\\,mJy\/beam in brightness, and to $\\pm0.3$ for sources only $\\sim140$\\,mJy\/beam in brightness. Similar calculations for observations at 450\\,\\micron~are listed with those at 850\\,\\micron~in Table \\ref{tab:ifts}.\n\n\\section{Summary}\n\nWe have determined the spectral index of the dust emission of three hot molecular cores solely within the 350\\,GHz (850\\,\\micron) passband using an FTS on the JCMT. We find an average dust spectral index $\\beta\\simeq1.6$, in agreement with the spectral indices calculated using archived SCUBA data at 450\\,\\micron~and 850\\,\\micron. These results for $\\beta$ are consistent with measurements of the spectral index in other HMC. The uncertainties in $\\beta$ from these observations are dominated by the difficulties in subtracting the very bright and variable submillimetre atmosphere from the data. With better sky subtraction techniques, as will be possible with an imaging FTS on SCUBA-2, these uncertainties will be greatly reduced. The per-pixel sensitivity of the imaging FTS planned for SCUBA-2 will be greater by a factor of $\\sim100$ than that of the single pixel detector used in this study, and will allow the determination of the spectral index of dust emission of sources with brightnesses of only a few hundred mJy\/JCMT beam. \n\n\\section*{Acknowledgments}\nWe thank the referee, Jenny Hatchell, for her suggestions which improved this paper. We also thank all TSS involved in FTS runs for their invaluable assistance, as well as G. J. Tompkins, B. Gom and I. Chapman for much FTS-related support. Additional thanks to G. Fuller for providing the source list. RKF also thanks B. Hesman for helpful advice and discussion. This work was funded in part by the University of Victoria (RKF) and by grants from NSERC Canada (DJ, DAN and GRD). The JCMT is operated by the Joint Astronomy Centre in Hilo, Hawaii on behalf of the parent organizations Particle Physics and Astronomy Research Council in the United Kingdom, the National Research Council of Canada and The Netherlands Organization for Scientific Research. SCUBA-2 is a jointly funded project through the JCMT Development Fund with substantial additional contributions from the UK Office of Science and Technology and the Canada Foundation for Innovation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe X complex in the Tholen taxonomy comprises the E, M and P classes which\nhave very different inferred mineralogies but which are spectrally similar \nto each other and featureless in the visible wavelengths (Tholen, 1984).  \nBy convention X-types with measured geometric albedos are classified into Tholen E-type\n(high albedo, $p_{v} > 0.3$), M-type (medium albedo, $0.1 < p_{v} < 0.3$), or P-type\n(low albedo, $ p{v} < 0.1$) asteroids (Tholen\\&Barucci, 1989). The X-type\nasteroid indicates an \"EMP\" asteroid for which we do not have albedo\ninformation. \n\n\nThe X-type asteroids are distributed throughout the main belt (Moth\\'e-Diniz et al. 2003) but\ntend to be concentrated around 3.0 AU, and in the Hungaria region (Warner et al. 2009). The X complex includes bodies with very different mineralogies. \nM asteroids may be composed of metals such as iron and \nnickel and may be the progenitors of differentiated iron-nickel\nmeteorites (Gaffey, 1976; Cloutis et al, 1990,\nGaffey et al. 1993).  Enstatite chondrites have also been proposed\nas M-type asteroid analogs (Gaffey 1976; Vernazza et al. 2009; Fornasier et al. 2010; Ockert-Bell et al. 2010). \nA metallic iron interpretation requires that M asteroids are\nfrom differentiated parent bodies that were heated to at least 2000 $^{\\circ} C$ to\nproduce iron meteorites (Taylor, 1992). \n\nE-type asteroids have high albedo values and their surface  compositions \nare consistent with iron-free or iron-poor  silicates  such as enstatite,\nforsterite or feldspar. E-types are thought to be the parent bodies of the\naubrite (enstatite achondrite) meteorites (Gaffey et al. 1989, Gaffey et al.,\n1992, Zellner et al., 1977, Clark et al. 2004a, Fornasier et al. 2008).  \nP-type asteroids have low albedo values, and featureless red spectra. \nP-types are located mainly in the outer region of the main belt (Dahlgren \\& Lagerkvist 1995, \t\nDahlgren et al. 1997, Gil-Hutton \\& Licandro 2010) and in the Jupiter Trojan\nclouds (Fornasier et al. 2004, 2007a). They are probably primitive bodies, however\nthey have no clear meteorite analogues.  P-type asteroids\nare presumed to be similar to carbonaceous chondrites, but with a higher organic content to explain the strong red spectral slopes (Gaffey et al. 1989, Vilas et al. 1994).\n\nIn the last 15 years, several near-infrared spectral surveys have been devoted to the study of the X group and in particular the M and E-type asteroids, revealing that these bodies \nshow great diversity in the infrared, even within the same class. In addition, polarimetric measurements (Lupishko \\&\nBelskaya 1989) and radar observations (Ostro et al., 1991, 2000; Margot and Brown, 2003; Magri et al.,\n2007; Shepard et al., 2008, 2010)  of selected M-type asteroids have revealed surface\nproperties that are in some cases inconsistent with metallic iron. \nMoreover, spectra of M-types are not uniformly featureless as \ninitially believed; \nseveral spectral absorption bands have been detected in the visible and near\ninfrared ranges. Faint absorption bands near 0.9 and 1.9\n$\\mu$m have been identified on the surfaces of some M-types (Hardersen \net al., 2005; Clark et al., 2004b; Ockert-Bell et al., 2008; Fornasier et al., 2010), and of some E-type asteroids (Clark et al. 2004a; Fornasier et al. 2008) and\nattributed to orthopyroxene. A very peculiar absorption band, centered around\n0.49-0.50 $\\mu$m, extending from about 0.41 to 0.55 $\\mu$m, was found on some E-type spectra (Burbine et al, 1998; Fornasier \\& Lazzarin, 2001; Clark et al. 2004a,b;\nFornasier et al., 2007b, 2008), and attributed to the presence of sulfides such as\ntroilite or oldhamite (Burbine et al., 1998, 2002a). Busarev (1998) detected weak\nfeatures tentatively attributed to pyroxenes (at 0.51 $\\mu$m) and oxidized or\naqueously altered mafic silicates (at 0.62 and 0.7 $\\mu$m) in the spectra of 2\nM-type asteroids. \\\\\n\nOf note is the evidence for hydrated minerals on the surfaces of some  E and M-type asteroids (Jones et al, 1990; Rivkin et al, 1995,\n2000, 2002), inferred from  the identification of an absorption feature around 3\n$\\mu$m.  This feature is typically attributed \nto the first overtone of H$_{2}$O and to OH vibrational\nfundamentals in hydrated silicates. If the origin of the 3 $\\mu$m band on E and \nM-type asteroids is actually due to hydrated minerals, then these objects \nare not all igneous as previously believed and the thermal scenario\nin the inner main belt would need revision.\nGaffey et al. (2002) propose alternative explanations for the 3 $\\mu$m band: materials normally considered to be anhydrous containing structural OH; troilite, an anhydrous mineral that shows a feature at 3 $\\mu$m, \nor xenolithic hydrous meteorite components on asteroid surfaces from impacts \nand solar wind implanted H. Rivkin et al. (2002) provide arguments\nrefuting the Gaffey suggestions, and provide evidence in support of the\nhydrated mineral interpretation of the observations.\\\\\n\nThe X complex is consistently identified in the two taxonomies \nrecently developed based on CCD spectra acquired in the visible (Bus \\& Binzel 2002) or in the visible and near infrared range (DeMeo et al., 2009), even though there is not a direct one-to-one correspondence between the Bus and Tholen classes. For instance 92 Undina (Tholen M-type), and 65 Cybele (Tholen P-type) are both  classified as Xc-type in the Bus taxonomy. A description of the X complex classes (X, Xk, Xe, Xc) in the Bus \\& Binzel (2002) classification is summarized in Clark et al. (2004b), while in section 6 we describe the X complex classes in the Bus-DeMeo taxonomy.\n\nAiming to constrain and understand the composition of the\nasteroids belonging to the mysterious X group, we have carried out a\nspectroscopic survey of these bodies in the visible and near infrared range at\nthe 3.5m Telescopio Nazionale Galileo (TNG), in La Palma, Spain, at the 3.5m\nNew Technology Telescope (NTT) of the European Southern \nObservatory, in La Silla,\nChile, and at the Mauna Kea Observatory  3.0 m NASA Infrared Telescope Facility\n(IRTF) in Hawaii, USA. \nResults on asteroids classified as E and M-type following the Tholen \ntaxonomy are presented elsewhere (Fornasier et al. 2008, 2010).\nIn this paper we present new VIS-NIR spectra of 24 asteroids belonging to\nthe X type as defined by Tholen \\& Barucci (1989), that is an \"E--M--P\" type\nasteroid for which albedo information was not available at the time of their\nclassification. To constrain their compositions, we conduct a search for\nmeteorite analogues using the RELAB database, and we model the asteroid\nsurface composition with geographical mixtures of selected minerals when a\nmeteorite match is not satisfactory. \nIn addition, we present an analysis of X complex spectral \nslope values and class distributions in the asteroid main belt, \nwhere we include previously published observations of M and E-type\nasteroids obtained during the same survey (Fornasier et al. 2008, 2010). \n\n\n\\section{Observations and Data Reduction}  \n\n\n[HERE TABLE 1 ]\n\nThe data presented in this work were mainly obtained  during 2 runs \n(February and November 2004) at the Italian Telescopio Nazionale Galileo \nTNG of the European Northern Observatory (ENO) in La Palma, Spain, and 2 runs (August 2005, and January 2007) at the New Technology Telescope (NTT)\nof the European Southern Observatory (ESO), in Chile. Two asteroids (92 Undina, 275 Sapientia),\ninvestigated in the visible range at the NTT and TNG telescopes, were separately\nobserved in the near infrared at the Mauna Kea Observatory  3.0 m NASA Infrared\nTelescope Facility (IRTF) in Hawaii during 2 runs in July 2004 and September 2005 (Table~\\ref{tab1}).\nWe had a total of 11 observing nights. Here\nwe present the results of our observations of 24 X-type asteroids: for 16\nobjects we obtained new visible and near-infrared wavelength spectra (VIS-NIR),\nfor one object we obtained only a near-infrared spectrum (NIR), \nand for 7 objects we obtained only visible range spectra (VIS).\n\nThe instrument setups are the same as used \nin Fornasier et al. (2008, 2010). We refer the reader to those papers\nfor complete\ndescriptions of observations and data reduction techniques.\nThe spectra of the observed asteroids, all normalized at 0.55 $\\mu$m, are shown\nin Figs.~\\ref{fig1}-- \\ref{fig4}, and the observational conditions are reported in Table~\\ref{tab1}.\n\n[Here Figs 1,2,3, and 4]\n\nTo analyze the data, spectral slope values were calculated with linear\nfits to different wavelength regions: $S_{cont}$ is the spectral slope in the whole\nrange observed for each asteroid, $S_{UV}$  in the 0.49-0.55 $\\mu$m range, $S_{VIS}$ is the slope in the 0.55-0.80 $\\mu$m\nrange, S$_{NIR1}$ is the slope in the 1.1-1.6 $\\mu$m range, and S$_{NIR2}$ is\nthe slope in the 1.7-2.4 $\\mu$m range. Values are reported in\nTable~\\ref{slope}. \nBand centers and depths were calculated for each asteroid showing an \nabsorption feature, following the Gaffey et al. (1993) method. \nFirst, a linear continuum was fitted at the edges of the band, \nthat is at the points on the spectrum outside the absorption \nband being characterized. Then the asteroid spectrum was divided \nby the linear continuum and the region of the band was fitted with a \npolynomial of order 2 or more. The band center was then calculated \nas the position where the minimum of the spectral reflectance \ncurve occurs, and the band depth as the minimum in the ratio of the \nspectral reflectance curve to the fitted continuum (see Table~\\ref{band}).\n\n\n[HERE TABLE 2]\n\n\n[HERE TABLE 3]\n\n\\section{Spectral Analysis and Absorption Features}\n\nThe observed asteroids show very different spectral behaviors\n(Figs~\\ref{fig1}-- \\ref{fig4}). Eleven asteroids show at least one absorption\nfeature (Table~\\ref{band}). \n\n50 Virginia (see Fig. 1 and Table 3) shows spectral features centered at 0.43, 0.69, and 0.87 $\\mu$m (Table~\\ref{band}) associated with aqueous alteration products (Vilas et al. 1994, Fornasier et al. 1999), and seems to be a typical 'hydrated' C-type asteroid. \n\nFive asteroids (50 Virginia, 283 Emma, 337 Devosa, 517 Edith, and 1355 Magoeba) show a \nweak band at 0.43 $\\mu$m. For 50 Virginia, 283 Emma, and 517 Edith, \nthe band might be associated with an Fe$^{3+}$  spin-forbidden \ntransition in the iron sulfate jarosite, as suggested by  \nVilas et al. (1993) for low-albedo asteroids. For 337 Devosa and 1355 Magoeba, \nthe band might be associated with chlorites and Mg-rich serpentines,  \nas suggested by King \\& Clark (1989) for enstatite chondrites, \nor to clinopyroxenes such us pigeonite or augite as\nsuggested by Busarev (1998) for M-asteroids. \n\nThe 1355 Magoeba spectrum also shows a band \ncentered at $\\sim$ 0.49 $\\mu$m that\nresembles the absorption seen on some E-type (subclass EII) asteroids (Fornasier \\& Lazzarin 2001; Fornasier et al. 2007b; 2008, Clark et al. 2004a), where it is attributed to sulfides such\nas oldhamite and\/or troilite (Burbine et al. 1998, 2002a, 2002b). Nevertheless, the estimated albedo of 1355 Magoeba has a moderate value (0.267$\\pm$0.095, Gil-Hutton et al. 2007) more consistent with a Tholen M-type classification.\n\n92 Undina has a very weak band centered at $\\sim$ 0.51 $\\mu$m that is similar to the Fe$^{2+}$\nspin-forbidden crystal field transitions in terrestrial and lunar pyroxenes (Burns et\nal. 1973, Adams 1975, Hazen et al. 1978). This band has been previously detected\nby Busarev (1998) in the spectra of two M asteroids, 75 Eurydike and 201\nPenelope. \n\nAn absorption feature has been identified  in the 0.9 $\\mu$m\nregion in the spectra of 4 X-type asteroids that\nhave albedos placing them in the Tholen M-class (92 Undina,\n337 Devosa, 417 Suevia, and 1124 Stroobantia). \nThe band center ranges from 0.86 $\\mu$m to 0.92 $\\mu$m with a\nband depth (as compared to the continuum) of 3--6 \\%.\nAlso 522 Helga, having a low albedo (4\\%)\nconsistent with the Tholen P-type classification, exhibits a faint band centered at 0.94 $\\mu$m.  \nThis band was previously \nreported by several authors in the spectra of Tholen M, E and X-type\nasteroids (Hardersen et al. 2005; Clark et al. 2004a, 2004b, Fornasier\net al. 2008, 2010) and is attributed to low-Fe, low-Ca orthopyroxene\nminerals. \n\nA peculiar feature centered at 1.6 $\\mu$m was found on the spectrum of 758 Mancunia. This band resembles one seen on 755 Quintilia (Fornasier et al., 2010, Feiber-Beyer et al. 2006) whose interpretation is still unknown. No bands due to silicates in the 0.9 and 1.9 $\\mu$m regions have been identified in our spectra, except for a change in slope at around 0.77 $\\mu$m.\n\nFinally, 1122 Nieth shows a strong 0.96 $\\mu$m absorption band (depth of 24\\%) and a steep slope in the NIR region, typical of olivine-rich bodies. \\\\\n\nSeveral asteroids of this survey were observed previously by other authors.\nFive X-type asteroids (50 Virginia, 283 Emma, 337 Devosa, 517 Edith, and 909 Ulla) of the Clark et al. (2004b) survey of the X complex were \nalso studied in this work. The spectral behavior for these objects looks quite similar,\nexcept for 283 Emma (our spectrum is flat in the NIR, while the Clark et\nal. spectrum has a concave shape with a higher spectral gradient),  and 909 Ulla\n(our spectrum has a higher spectral gradient \nas compared to the Birlan et al. (2007) and the Clark et al. (2004b) spectra).\nOckert-Bell et al. (2008, 2010) presented near-infrared spectra of 77 Frigga and 758 Mancunia,\nthat look slightly different compared to ours. In particular, they detect faint spectral bands in the 0.9 and 1.9 $\\mu$m region for 758 Mancunia, and in the 0.9 $\\mu$m region for 77 Frigga, bands that are not detected in our spectra.\n1355 Magoeba and 3447 Burckhalter were observed in the visible\nrange by Carvano et al. (2001), and their data look very similar to ours.\\\\ \nIn sum, comparing our spectra with those in the existing literature, we suggest\nthat asteroids 77 Frigga, 283 Emma, 909 Ulla, and 758 Mancunia may display surface variability as they show different spectral behaviors throughout independent observations. For 758 Mancunia this variability is confirmed also by radar measurements that show the radar cross-section and the polarization ratio to\nvary considerably with rotation phase (Shepard et al. 2008). For the other 3 asteroids, we cannot exclude that some spectral differences may be linked with unresolved observational differences (such as\nbackground stars, or undetected troubles in removing the atmosphere). \n\n \n\n\\section{Tholen Taxonomic Classification}\n\nOur targets were classified as belonging to the Tholen X class on the basis of\ntheir spectra from 0.4 to 1 $\\mu$m and because their albedos were not known\nat the time of classification.\nSince the original X-type classification in the Tholen taxonomy, \nthe albedo value has become available for most of the asteroids \nwe observed (Tedesco et al., 2002). Taking into account this important \ninformation, together with the visible and near infrared (when available) \nspectral behavior,  we suggest a re-classification of the X-type \nasteroids in our sample following the Tholen-classification scheme. \nThe suggested new classifications are reported in Table ~\\ref{slope}.\n\n1122 Nieth (Fig.~\\ref{fig4}), initially classified as X-type in the Tholen taxonomy,\nhas atypical spectral properties that diverge from all other asteroids.\nThe near infrared spectrum of Nieth\nclearly shows a strong 0.96 $\\mu$m absorption band \nand a steep slope in the NIR\nregion. We infer that this asteroid belongs \nto the olivine-rich A-types (used in\nthe Tholen, Bus, and Bus-DeMeo taxonomic systems).\n\n1328 Devota has a very steep featureless red spectrum. Considering its low albedo value (4\\%), \nDevota falls within the D class in the Tholen taxonomy (Fig.~\\ref{fig4}).\n\nWe suggest that the ten asteroids (77 Frigga, 92 Undina, \n184 Dejopeja, 337 Devosa, 417 Suevia, 741 Botolphia, \n758 Mancunia, 1124 Stroobantia, 1146 Biarmia, and 1355 Magoeba) with \nmoderate albedo values (0.1--0.3) are M-type asteroids while those with albedos lower than 10\\% fall in the C\/P classes. \nTo discriminate between C and P-type asteroids, we \nanalyzed spectral slope values in the visible \nrange ($S_{VIS}$ from 0.55 to 0.80 $\\mu$m) and searched for \nthe UV drop-off below $\\sim 0.5 ~\\mu$m that is\ntypical of C-type asteroids. Comparing the $S_{VIS}$ slopes \nwith the $S_{UV}$ slopes (calculated from 0.49 to 0.55 $\\mu$m) \nof the low albedo asteroids, we find two distinct distributions \ncorrelated with the C and P-classes. The 6 asteroids \n(50, 220, 223, 283, 536, and 517) having the UV absorption and\n$S_{vis} < 2 \\%\/10^{3}$\\AA\\ belong to the C-type, while the \n4 asteroids (275, 463, 522, and 909) having \n$3 <S_{vis} < 6 \\%\/10^{3}$\\AA\\ belong to the P-type. \n1902 Shaposhnikov, observed only in the near infrared \nrange, is probably also a P-type, while 3447 Burckhalter, for which the albedo value is 0.336$\\pm$0.164 \n (Gil-Hutton et al. 2007) could possibly be an E-type.\n\n\n\n\\section{Meteorite Spectral Matches and Mixing Models}\n\n\\subsection{Meteorite Analogues}\n\nTo constrain the possible mineralogies of our asteroids, we conducted  \na search for meteorite and\/or mineral spectral matches.\nWe sought matches only for objects observed across the entire \nVIS-NIR wavelength range.  A complete description of \nthe search methodology can be found in Clark et al. (2010) and\nFornasier et al. (2010).  We used the  \npublicly available RELAB spectrum library (Pieters 1983), which  \nconsisted of nearly 15,000 spectra in November of 2008.  RELAB spectra were \nnormalized to 1.0 at 0.55 $\\mu$m, and then a Chi-squared value \nwas calculated for each RELAB spectrum relative to  \nthe normalized input asteroid spectrum. The RELAB spectra were sorted according  \nto Chi-squared values, and then visually examined for dynamic weighting of  \nspectral features by the spectroscopist.  \nGiven similar Chi-squared  \nvalues, a match that mimicked spectral features was preferred over a  \nmatch that did not.  We visually examined the top ~200 Chi-squared  \nmatches for each asteroid and we constrain the search for \nanalogues to those laboratory spectra with\nbrightness (reflectance at 0.55 $\\mu$m) roughly comparable to the albedo of\nthe asteroid.  \nIt must be noted that the asteroid data are disk integrated, \nwhich could mask spectral variations due to differences in \nparticle size, mineralogy, or abundance. However the meteorite \nand mineral spectra are well characterized powders measured under \nwell defined laboratory conditions. It should also be noted that\n an asteroid's albedo \nis defined at phase angle zero,\nwhereas meteorites' reflectance is taken at phase angle  $> 5^{o}$.  \nFor the asteroid albedo value, we used the IRAS  \nalbedo published by Tedesco et al. (2002).  For the darker asteroids  \n(below 10\\% albedo), this is a fairly well-constrained value.  The  \nuncertainty for brighter asteroid albedos (above 10\\% albedo) is  \nrelatively larger. To account for this, we filtered out lab spectra if\nbrightness differed by more than $\\pm$ 3\\% in absolute albedo for the darker  \nasteroids (albedos less than 10\\%), and we filtered out lab spectra if  \nbrightness differed by more than $\\pm$ 5\\% in absolute albedo for the  \nbrighter asteroids  (albedos greater than 10\\%).  For example, if the asteroid' s albedo was 10\\%, we searched over all lab spectra with  \n0.55 $\\mu$m reflectance between 5\\% and 15\\%.  Once the brightness filter was  \napplied, all materials were normalized before comparison by least- \nsquares. \n\nOur search techniques effectively emphasized the  \nspectral characteristics of brightness and shape, and de-emphasized  \nminor absorption bands and other parametric characteristics.  As such,  \nwe suggest that our methods are complementary to band parameter  \nstudies.\n\n[HERE TABLE 4]\n\n[HERE FIGURES 5 and 6]\n\n\nThe best matches between the observed X-type asteroids and meteorites\nfrom the RELAB database are reported\nin Table~\\ref{tab}. We matched only the 15 asteroids \nobserved both in the visible and near infrared range that\nhave published albedo values. Some meteorite matches are quite good, while for some asteroids the best meteorite \nmatch we found does not satisfactorily reproduce both the visual and near infrared\nspectral behavior. In particular our attempt to find a meteorite or mineral match failed for the 3 asteroids 1122 Nieth (A-type), 1328 Devota (D-type), and 1355 Magoeba (M-type).\n\nThe best matches we found \nfor 7 moderate albedo M-type asteroids are presented in Figure~\\ref{fig5met}.  Two asteroids, \n741 Botolphia and 1146 Biarmia were not compared with RELAB spectra\ndue to the lack of near-infrared spectral data. \nWe find that iron or pallasite meteorites are the best matches both for the featureless M-type asteroids and for those having minor absorption bands. This result supports the link between M-type asteroids and iron or pallasite meteorites suggested in the literature by several authors. In this small sample, no enstatite chondrites, which have also been suggested to be linked with M-type asteroids, have been found as the best meteorite analogue.\n\nThe strongest analogy \nwe found is between 758 Mancunia and a sample of the iron meteorite Landes which has a large number of silicate inclusions in a nickel-iron matrix.\nIt must be noted that no specific grain-size sampling\nfilter was applied, and of concern is the fact that Mancunia's best match is with a spectrum obtained of the cut\nslab surface of the Landes iron meteorite. \n\n758 Mancunia has a radar albedo of 0.55$\\pm$0.14 \n(Shepard et al. 2008) one of the highest values measured for the \nasteroids, suggesting a very high metal content. Shepard et al. (2008) also found variations in the radar cross-section and polarization ratio with Mancunia's rotational phase, variations that are not related to shape but to the regolith depth, porosity, or near-surface roughness. The Ockert-Bell et al. (2010) Mancunia spectrum shows absorption bands in the 0.9 and 1.9 $\\mu$m region, and is found to be similar to the ordinary chondrite Paragould. Our spectrum does not show these features, but a change of slope at 0.77 $\\mu$m and a faint (1.3\\% depth) feature centered at 1.6 $\\mu$m, similar to that observed on 755 Quintilia (Fornasier et al. 2010, Fieber-Beyer et al. 2006), whose origin is not yet understood.  Considering both radar and spectral observations, 758 Mancunia could present an heterogeneous surface composition, with a high metal content but also the presence of some silicates that are not uniformly distributed on the surface.\n\nThe Landes iron meteorite sample was found to be the best spectral match also for 337 Devosa, even if the meteorite does not fully reproduced the 0.88 $\\mu$m absorption band of the asteroid. A similar problem occurred for 417 Suevia and 1124 Stroobantia, where different grain sized samples of the iron meteorite DRP78007 are found to be the best spectral matches, but they did not reproduce the asteroids absorption bands in the 0.9 $\\mu$m region. \nWe note that grain size effects can play an important role in the spectral behavior of iron-nickel meteorites. In fact Britt and Pieters (1988) found that\nthe spectra of M-type asteroids show good agreement with those of iron\nmeteorites with surface features in the range of 10 $\\mu$m to 1 mm, that is\nlarger that the wavelength of incident light.  Meteorites with these roughness\nvalues are diffuse reflectors and show the classic red slope continuum of iron,\nwith practically no geometric dependence on reflection. On the other hand, a  \ndecreasing of the meteorite surface roughness changes the reflectance\ncharacteristics:  complex scattering behavior is seen for roughness in the\n0.7-10 $\\mu$m range, while for roughness values $<$ 0.7 $\\mu$m the reflectance\nis characterized by two distinct components, the specular one which is bright and\nred sloped, and the nonspecular one which is dark and flat.  \\\\\n\nFor 77 Frigga we were not able to find any meteoritic\/mineral \nanalog within the medium albedo range that could\nreproduce Frigga's red visible slope and flat near-infrared slope. \nThe best spectral match is shown in Fig.~\\ref{fig5met}.\nThe iron meteorite Chulafinnee\nmodels the spectral behavior below 0.55 $\\mu$m very well,\nand the albedo values are similar, \nhowever the change in spectral slope around 0.9 $\\mu$m is not\nreproduced.\n\nFor 92 Undina we propose two different meteorites: \nthe best spectral match is with a metal-rich powder sample of the pallasite meteorite \nEsquel, whose albedo of 0.14 is lower than the 0.25 value\nof the asteroid. The iron meteorite Babb's Mill\nwas a better numerical fit (minimum chi squared fit)\nand had a more comparable albedo, but was not favored because \nBabb's Mill has a near  infrared spectrum that is \nmuch flatter than that of Undina. \n\n\n\nIn Figure~\\ref{fig6met} we present the best matches \nproposed for six low albedo asteroids. \nIt has long been noticed that reflectance spectra of carbonaceous\nchondrites are similar to those of the low-albedo asteroids\n(Gaffey \\& McCord 1978; Hiroi et al. 1996). In particular, CM meteorites \nshow features due to aqueous alteration processes and are linked\nto low albedo aqueous altered asteroids like the C and G-types (see Vilas et al. 1994, Burbine 1998, Fornasier et al. 1999).   \nIndeed, all the observed low albedo asteroids are best fit with CM meteorites, either unaltered (50 Virginia and 283 Emma) \nor altered by heating episodes (275 Sapientia and 517 Edith) \nor laser irradiation (522 Helga and 909 Ulla) (see Fig.~\\ref{fig6met}). \nThe CM2 MET00639 meteorite matches very well the visible range spectrum of 50 Virginia and in particular the 0.7 $\\mu$m band attributed to $Fe^{2+}\\rightarrow Fe^{3+}$ charge transfer absorptions in phyllosilicate minerals. Nevertheless, the meteorite does not well reproduce the asteroid's near-infrared spectrum. The same meteorite was found as a best match for the near infrared region of 283 Emma, but it does not well reproduce the spectrum in the visible range, where the asteroid is featureless. The absence of the 0.7 $\\mu$m band in the asteroid spectrum led us to discard this as a spectral match.\n\nThe P-type asteroids 275 Sapientia and the C-type 517 Edith have as best match two different samples of the Murchison carbonaceous chondrite heated at 600$^{o}$C (grain size $<$ 63 $\\mu$m) and 700$^{o}$C (63 $\\mu$m $<$  grain size $<$ 125 $\\mu$m), respectively. It must be noted that at these temperatures the phyllosilicates are dehydrated and transformed into olivine and pyroxene and that the 3 $\\mu$m hydratation band vanishes (Hiroi et al., 1996).  \nOur spectral match may indicate that Sapientia and Edith have no hydrated silicates on their surfaces. If hydrated silicates were originally present, the asteroids may have experienced important thermal episodes that dehydrated them.\n\nThe P-type asteroids 522 Helga and 909 Ulla are reproduced by a sample of the CM meteorite Migei after laser irradiation. This meteorite is composed of a black matrix with olivine-rich chondrules, olivine aggregates and individual grains, carbonates, and sulfides (Moroz et al. 2004). Its unaltered spectrum is dark and shows absorption features related to hydrated silicates such as the 2.7-3 $\\mu$m band, the UV-falloff and a 0.75 $\\mu$m band. Once irradiated, the meteorite is dehydrated, the absorption bands are significantly weakened (Moroz et al. 2004), and the material remains dark -- most probably due to abundant submicron inclusions of Fe-rich phases finely dispersed in the glassy mesostasis (Shingareva et al. 2004). \nBecause laser irradiation is a laboratory technique for simulating micrometeorite bombardment, the spectral match between Helga and Ulla and a laser irradiated sample of CM Migei may indicate that the surfaces of these outer belt asteroids, if composed of CM carbonaceous-like materials, may be spectrally reddish due to micrometeoritic bombardment.\n\n\n\\subsection{Mixing Models}\n[HERE FIGURE 7]\n\n[HERE TABLE 5]\n\nTo constrain the surface compositions of the investigated \nX-type asteroids and the materials needed to reproduce the weak 0.9 $\\mu$m band seen on some spectra\nwe considered geographical (spatially segregated) mixtures of \nseveral terrestrial and meteoritic materials in different grain \nsizes: in particular we took into account all the samples \nincluded in the US Geological Digital Spectral Library \n($http:\/\/speclab.cr.usgs.gov\/spectral-lib.html$), in \nthe RELAB database, and in the ASTER spectral library ($http:\/\/speclib.jpl.nasa.gov$),\ntogether with \norganics solids (e.g. kerogens by Clark et al. 1993 and Khare et al. 1991;\nTitan tholins from Khare et al. 1984; and\nTriton tholins from McDonald et al. 1994) and \namorphous carbon (by Zubko et al. 1996).\nThe synthetic spectra  were compared with \nthe asteroid spectra, using the known IRAS albedo, and \nthe VIS-NIR spectral behavior and continuum slopes. \nWe present these matches as non-unique examples of how\ngeographical mixtures can be used to explain some of the\nvariety found in X-type asteroid spectra.  We do not intend\nto indicate that these are unique derivations of the composition\nof these asteroids.  Such an inversion requires much more\ninformation than we currently possess (grain sizes, optical\nconstants, endmembers present, etc.).\nNevertheless, the models we present can\nprovide a first order understanding of\n possible surface composition.\n\n\nIn Figure~\\ref{modelli} and Table~\\ref{models} we present \nmixing models \nfor the 4 X-type asteroids showing the 0.9 $\\mu$m band (this band is\nattributed to low-Fe, \nlow-Ca orthopyroxene and is not reproduced by the meteorite analogues proposed \nin Table~\\ref{tab}). We also present\nthe low albedo asteroids 517 Edith (for which \nthe spectral match with a heated CM \nmeteorite is poor in the 1--2 $\\mu$m region), and 1328 Devota (for which we \ncannot find any satisfactory meteorite analogue).\n\nAs shown in Fig.~\\ref{fig5met} and discussed in Section 5.2 we did not find \na meteorite with a similar albedo value\nthat could match the whole VIS-NIR spectral behavior of \n92 Undina.\nHowever, we were able to produce a model of\nthe surface of this M-type asteroid\nwith a geographical mixture of 99\\% of pallasite Esquel and 1\\% \northopyroxene.\nThis mixture model reproduces the spectral behavior of Undina,\nhowever its albedo is much lower (0.14) than that of the asteroid\n(0.25).\n\nFor 337 Devosa the spectral behavior has been reproduced \nby two different mixtures: one composed of 99\\% pallasite\nEsquel and 1\\% orthopyroxene (for an albedo of 0.14) \n(red line in Fig.~\\ref{modelli}), and one \ncomposed of 98\\% pallasite Esquel and 2\\% goethite (for an albedo of 0.14).\n337 Devosa has spectral properties in the VIS-NIR \nspectral range that are very similar\nto the M-type asteroid 22 Kalliope (see Fornasier et al. 2010). \nFor these asteroids, \nthe 0.9 $\\mu$m band is consistent with a small\namount of anhydrous silicate (such as orthopyroxene), or with \na small amount of goethite (an aqueous alteration product).\n\nFor 417 Suevia the proposed meteoritic analogue (the iron meteorite \nDRP78007, Fig.~\\ref{fig5met}) does not fit the observed 0.9 $\\mu$m band. \nWe found that a geographical mixture of 97\\% IM DRP78007 (RELAB file cdmb47) \nand 3\\% orthopyroxene  models the asteroid spectral behaviour. \nThe albedo of the mixture is 0.16, a bit lower than the value estimated for \nSuevia (0.20).\nFor 1124 Stroobantia the proposed meteorite analogue\n(metallic meteorite DRP78007) also does not fit the faint 0.9\n$\\mu$m band.  For this asteroid, we propose two mixing models. One is a \ngeographical mixture of 98\\% IM MET101A Odessa and 2\\% orthopyroxene, with \nan albedo of 0.13 (blu line in Fig.~\\ref{modelli}). \nThis model reproduces the asteroid spectrum below 1.4 $\\mu$m \nbut not the longer wavelengths.\nAlternatively, a mixture of 96\\% iron meteorite DRP78007 and 4\\% \nolivine (albedo 0.16, red line in Fig.~\\ref{modelli}) fits \nStroobantia \nbeyond 0.6 $\\mu$m, but does not match the spectral region below 0.6 $\\mu$m.  \nAlthough none of the modeled mixtures reproduces all spectral features (albedo, \nspectral slopes, band depth, and band center) of the asteroid, our best inferrence from the modeling is to suggest that the surface composition of Stroobantia is probably consistent with metallic meteorites enriched in silicates.\n\nThese results show that a few percent (less than 3\\%) \nof orthopyroxenes or goethite added to iron or pallasite meteorites \ncan reproduce the weak spectral features around 0.9 $\\mu$m seen on some asteroids \nbelonging to the X-complex. \n\n517 Edith shows a flat and featureless spectrum. The observed spectral behavior appears \nsimilar to the spectrum of pure  amorphous carbon (from Zubko et al., 1996), although we \ncannot exclude the possibility that a small amount \nof silicates may be present and masked by the dark and opaque \nmaterials.\n \n Considering the low albedo value and the featureless and reddish spectral behavior, we classify  1328 Devota as a D-type. Since the Tagish Lake meteorite is usually \nconsidered the best meteorite analog for \nD-type asteroids (Hiroi et al. 2001), we\ncompared the spectrum of Devota with laboratory spectra of\nseveral samples of this meteorite. \nThe spectrum of Devota appears to be redder than that of Tagish Lake. \nThis could be due to the different ages of the asteroids and meteorites.\nThe surface of Devota could be older than\nthe meteorite, and the reddening of the asteroid spectrum could be the result\nof space weathering processes. \nAlternatively, the spectral behavior of Devota can be reproduced with \na geographical mixture of 94\\% Tagish Lake meteorite (RELAB file c1mt11)\nand 6\\% \nTriton tholins (from McDonald et al. 1994), resulting in an albedo of 0.02.\nOur observations of Devota do not cover the wavelengths necessary for detection of the  tholin spectral feature at about 3 $\\mu$m, so\nalthough we cannot constrain the presence of \ntholins with the data in hand, \nwe suggest that some tholin or tholin-like\ncomponent could be the reddening agent producing the observed \nspectral slope between 0.5 and 2.5 $\\mu$m.\n \n\\section{The X complex: overview and discussion}\n\n[HERE TABLE 6 and 7]\n\n[HERE FIGURES 8 and 9]\n\n\nOur survey devoted to the X-complex asteroids as defined by the \nTholen (1984) taxonomy is composed of 78 objects, including the \nE and M-type asteroids already published in Fornasier et al. (2008, 2010).\nOn the basis of their spectral behavior and albedo values,\nwe classified 22 E-types and distributed them \ninto 3 subclasses I, II, and III, \n(see Fornasier et al. 2008 and references therein). \nOne of the objects (3447) was originally classified as X-type and \nwas tentatively attributed to the E(I) class in this work, due to its moderately high albedo value but with a high uncertainty. \nOur survey includes 38 M-types, ten of which (77, 92, 184, 337, 417, 741, 758, 1124, \n1146) were originally classified as X-types, 7 C-types (498 Tokio was \noriginally classified as an M-type, and all the others as X-type),\n1 D (originally classified as X-type), 5 P-type (originally classified as X-type), 3 A-types (2577, 7579 and 1122 which were originally classified as E-types,\nand an  X-type, respectively), and 2 S-types (5806 and 516, which were originally classified as an\nE and M-type, respectively).\nIn Fig.~\\ref{slopea} we show the spectral slope value $S_{VIS}$ calculated \nin the visible wavelength range versus the semimajor axis for all  the E, \nM and C\/P-types observed. As expected, high albedo E-type asteroids populate \nmainly the Hungaria region and the inner part of the main belt, while low \nalbedo C and P-types are located mainly in the outer part of the main belt \nor beyond it, and M-type are located between 2.4 and 3.2 AU. \nThe mean visible spectral slope for the different E, M and P classes are \nvery similar, as expected (Table~\\ref{slopes}). In the NIR range, M-type asteroids having a band in the 0.9 $\\mu$m region have mean spectral slope S$_{NIR1}$ and S$_{NIR2}$ values similar to P-type asteroids. These values are higher than those of the  M-type asteroids without the 0.9 $\\mu$m band. For the E-type asteroids, observations in the infrared range are available only for a very few objects. We give the NIR  mean slope values only for the subtype III (4 asteroids observed in the NIR range: 44 Nysa, 214 Aschera, 317 Roxane, and 437 Rhodia), and the subtype II (64 Angelina, 2867 Steins, and 4660 Nereus), that show the lower S$_{NIR1}$ mean value within the X complex (no data are available in the NIR range for the subtype I).\n\nOne M-type asteroid, 77 Frigga, show a very peculiar spectrum, with a near infrared spectrum similar to that of other M-type bodies, but with a very high $S_{VIS}$ value comparable with that of the D-type Devosa. Nevertheless its moderate albedo value allows us to exclude any possible link with dark P or D-types. Also A or S type asteroids have $S_{VIS}$ value similar to that of Frigga, but the absence of any absorption band due to olivine and pyroxene lets us exclude any link with these asteroid classes. As discussed in section 3, Frigga may show surface variability, and the 3 $\\mu$m band, usually associated with hydrated silicates, has been detected on its surface (Rivkin et al., 2000). Polarimetric data reveals that Frigga has a large inversion angle (Gil-Hutton, 2007), implying that the surface is composed of small particles (comparable to the wavelength), and\/or of a mixture of particles with high contrast in albedo, like refractory inclusions seen on some carbonaceous chondrites (see Belskaya et al. 2010, and references therein).  Our attempt to find a meteorite analogue was unsuccessful as the best match found (the iron meteorite Chulafinnee) does not satisfactory reproduce the asteroid spectrum. No radar observations are available for this body. Additional observations of 77 Frigga will be very important to confirm its surface heterogeneities and possibly to constrain its surface composition. \\\\\nIn our sample 43 objects were observed in the complete \nV+NIR spectral range.\nAmong them 37 have spectral features and albedo values compatible with\nthe X-complex (5 E-types, 29 M-types, and 3 P-types).\nFor all objects we investigated the spectral matches  with\nmeteorites\/minerals or geographical mixture models. \nFor the 5 E-type asteroids (44, 64, 214, 317 and 437) \nwe found good spectral matches with the enstatite\nachondrite meteorites, in several cases enriched with troilite, oldhamite or\northopyroxenes (Fornasier et al. 2008).\nThe 3 P-type asteroids presented in this paper (275, 522, and 909) \nexhibit spectral behaviors and albedo values compatible with carbonaceous chondrite meteorites. \nMost of the M-type asteroids have spectral features and albedo values well represented \nby iron meteorites, pallasites, and enstatite chondrites -- in several cases \nenriched with orthopyroxenes, olivines, or goethite.\n\nOur new spectral observations enhance the available physical information for the observed \nasteroids and allow us to apply the  Bus-DeMeo classification recently published\n(DeMeo et al., 2009). The Bus-DeMeo system is based on the\nasteroids' spectral characteristics over the wavelength range 0.45 to 2.45 $\\mu$m\nwithout taking the albedo into consideration. \nIn the Bus-DeMeo taxonomy, the X complex comprises \n4 types (X, Xk, Xe, Xc) as in the  Bus \\& Binzel (2002) system: \nan asteroid belongs to the Xe-type if the 0.49 $\\mu$m feature is present, \nto the  Xk-type if a feature is present in the 0.8--1.0 $\\mu$m range, \nto the Xc-type if the spectrum is red and featureless with slight \nconcave-down curvature, and to the X-type if the spectrum is straight and\nfeatureless.\nThe Tholen X-type asteroids we observed show different spectral behaviors in the near infrared range. \nWe therefore re-classified 16 of the asteroids presented in this paper and observed both in the VIS and NIR range according to the Bus-DeMeo taxonomy. The corresponding classes are summarized in Table\n~\\ref{slope}. All 5 asteroids showing the faint 0.9 $\\mu$m band \nfall in the Xk-type (337 Devosa also has the 0.43 $\\mu$m band), \nwhile 1122 Nieth, which shows a broad and deep 0.96 $\\mu$m band \nand a steep infrared spectrum, is classified as an A-type.\nThe 5 asteroids having the 0.43 $\\mu$m absorption band (note that this region is outside the wavelength limits where the Bus-DeMeo taxonomy is defined) belong \nto 5 different classes: 337 Devosa is classified as Xk; \n50 Virginia, showing bands due to hydrated materials, is a Ch; \n283 Emma is a C-type; 517 Edith is an Xc; and 1355 Magoeba, \nhaving also the peculiar band at 0.49 $\\mu$m, is classified \nas Xe-type in the Bus-DeMeo taxonomy. \nThe featureless objects belong to the X-type (77, 758, and 909), \nD-type (1328, and 1902), and to the C-type (275).\n\nIn Table~\\ref{busDeMeotx} we report Bus-DeMeo taxonomic classifications\ntogether with the Tholen classifications (existing or proposed by us) \nfor all the asteroids observed during our survey devoted to the X-complex asteroids. \nIn Fig~\\ref{busalbedo} we show the albedo value versus the semimajor axis \nfor the observed asteroids classified with the Bus-DeMeo taxonomy. We aim \nto investigate if possible correlations between the classes of this taxonomy \nand the albedo may exist, although the albedo is not a parameter taken into \naccount in the Bus-DeMeo classification system.  \nWe dispose of the albedo value for 7 out of the 10 Xe asteroids, that show the 0.49$\\mu$m band typical of this class and attributed to sulfides. Most of them \ncorrespond to the high albedo subgroup II of the Tholen E-class, with only two asteroids (132 \nAertha and 1355 Magoeba) being classified as M-type in the Tholen taxonomy. So this peculiar band, attributed to the presence of sulfides like oldhamite, seems not to be exclusively associated with high albedo E-type asteroids. \\\\\nThe Bus-DeMeo Xk-class asteroids show two distinct albedo distributions: \na high albedo group (albedo $>$ 0.4), corresponding to the subgroup III of \nthe Tholen E-class, and a medium albedo group (0.1 $<$ albedo $<$ 0.3) which \nincludes all the Tholen M-types showing a faint feature in the 0.9 $\\mu$m \nregion. The Xk-class includes also a low albedo object, 522 Helga, that we classified here as a P-type in the Tholen taxonomy. \nIron bearing pyroxenes such as orthopyroxene are suggested to cause the feature around 0.9 $\\mu$m, characteristic of the Xk class, and seem to be present on asteroids with very different albedo values. \\\\\nMost of the Bus-DeMeo X-class asteroids have 0.1 $<$ albedo $<$ 0.2, with \nthe exception of 504 Cora (albedo = 0.34, therefore classified as E[I] \nasteroid) and 909 Ulla (albedo=0.034, therefore classified as Tholen P-type).\nAll the C-types have low albedo values, but the Xc and D-types span low \nand moderate albedo values. In particular the asteroid 849 Ara has a steep spectral slope and falls in the D class according to the Bus-DeMeo taxonomy (Fornasier et al. 2010), but its albedo is very high (0.27), excluding a surface composition of organic-rich silicates, carbon, and anhydrous silicates as commonly expected on low-albedo Tholen D-type asteroids.\n\nFrom this comparison, it is evident that the Bus-DeMeo taxonomy is very helpful in constraining the asteroids' surface compostion (for example Xk-type means presence of orthopyroxene), in particular when some absorption bands are present on the spectra. But it is clear that the asteroid albedo is also a very important parameter for constraining the surface properties and also their evolution in time (space weathering effects). We strongly encourage the development of a next-stage Bus-DeMeo taxonomy that includes the important albedo parameter for asteroid composition\/classification.\n\n\\section{Summary}\n\nWe present new visible and near infrared spectra of 24 asteroids belonging to\nthe X-type as defined by Tholen \\& Barucci (1989), that is an \"E--M--P\" type\nasteroid for which albedo information was not available at the time of their\nclassification. The X complex in the Tholen taxonomy is comprised of\nthe E, M and P classes which\nhave very different mineralogies but which are spectrally similar with\nfeatureless spectra in broadband visible wavelengths. \nOur observations reveal a large variety of spectral behaviors within the \nX class, and we identify  \nweak absorption bands on 11 asteroids. We combine our spectra with the albedo \nvalues available since 2002 for the observed bodies to \nsuggest new Tholen-like classifications.  We find:  1 A-type (1122), 1 D-type \n(1328), 1 E-type (possibly, 3447 Burckhalter), 10 M-types (77, 92, 184, 337, 417, \n741, 758, 1124, 1146, and 1355), 5 P-types (275, 463, 522, 909, 1902), and 6 C-types (50, 220, 223, 283, 517, and 536).\nFour new M-type asteroids (92 Undina, 337 Devosa, 417 Suevia, and 1124 \nStroobantia) show a faint band in the 0.9 $\\mu$m region, attributed to low \ncalcium, low iron orthopyroxene. Indeed, several works based on spectral and \nradar observations show that not all the M-type asteroids have a pure metallic \ncomposition (Fornasier et al. 2010 and reference therein).\\\\\nThree low albedo asteroids (50 Virginia, 283 Emma, and 517 Edith) show a weak \nband centered at 0.43 $\\mu$m  that we interpreted as due to Fe$^{3+}$  \nspin-forbidden transition in hydrated minerals (hematite, goethite). \nAlso the medium albedo bodies \n337 Devosa and 1355 Magoeba have the same absorption.  In this case the band \nmay be associated with chlorites and Mg-rich serpentines or pyroxene minerals \nsuch us pigeonite or augite. 50 Virginia shows also \ntwo absorptions centered at $\\sim$ 0.69 and 0.87 $\\mu$m which are typical of \nhydrated silicates. \\\\\nWe performed a search for meteorite and\/or\nmineral spectral matches between the asteroids observed in the visible and \nnear infrared range (with published albedo values) and the RELAB \ndatabase. \\\\\nThe best matches found for all the M-types of our sample are iron or pallasite meteorites as suggested in the literature, however we note that these meteorites often do not reproduce the faint absorption band features (such as the possible orthopyroxene absorption bands at 0.9 $\\mu$m) detected on the asteroids.\nWe tried to constraint the asteroids' surface compositions using geographical mixing models for new M-types having the 0.9 $\\mu$m feature. We found \ngood spectral matches by enriching the iron or pallasite meteorites with small \namounts ($<3\\%$) of orthopyroxene or goethite.   \nFor the low albedo asteroids we found as the best match CM carbonaceous \nchondrites, either unaltered or altered \n(submitted to heating episodes or laser irradiation).\nOur sample includes also two objects whose spectra diverge completely from \nother X complex asteroids: 1328 Devota, a low albedo body with \na red featureless spectrum, suggested to belong to the D-type, and 1122 Nieth, \na medium albedo asteroid with a broad and deep 0.9 $\\mu$m band and a steep \ninfrared spectrum, suggested to belong to the A-type. A \nsynthetic model made with the Tagish \nLake meteorite, considered as the best meteorite analogue of D-type asteroids \n(Hiroi et al 2001), and a reddening agent (Triton tholin) reproduces the \nspectral behavior of 1328 Devota.\n\nThe whole sample of asteroids included in our work is 72 X-type\nobjects (we exclude the A, D and S\/Sq-type asteroids), partly published here and partly  \nalready published in Fornasier et al. (2008) and Fornasier et al. (2010). The analysis of this complete sample \nhas clearly shown that, although\nthe mean visible spectral slopes of M-, E- and P-type asteroids\nare very similar to each other, the differences in albedo indicate major differences in mineralogy and composition.\n\n\\bigskip\n\n{\\bf Acknowledgment} \\\\\nThe authors thank Dr. M. Ockert-Bell, Dr. M. Shepard, and \nDr. A. Migliorini for their help with observations, and A. W. 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Simulation of clustering \nby a modified continuous distribution of ellipsoids. MNRAS 282, 1321--1329.\n\n\n\\newpage\n\n\n{\\bf Tables}\n{\\scriptsize\n       \\begin{center}\n     \\begin{longtable} {|l|l|c|c|c|c|c|c|l|}  \n\\caption[]{Observational circumstances for the observed X type asteroids. Solar analog stars named \"hip\" come from the Hipparcos catalogue, \"la\" from the\nLandolt photometric standard stars catalogue, and \"HD\" from the Henry Draper\ncatalogue}. \n        \\label{tab1} \\\\\n\\hline \\multicolumn{1}{|c|} {\\textbf{Object    }} & \\multicolumn{1}{c|}\n{\\textbf{Night}} & \\multicolumn{1}{c|} {\\textbf{UT$_{start}$}} &\n\\multicolumn{1}{c|} {\\textbf{T$_{exp}$}} &     \\multicolumn{1}{c|}\n{\\textbf{Tel.}} & \\multicolumn{1}{c|} {\\textbf{Instr.}} & \\multicolumn{1}{c|}\n{\\textbf{Grism}} & \\multicolumn{1}{c|} {\\textbf{airm.}} & \\multicolumn{1}{c|}\n{\\textbf{Solar Analog (airm.)}} \\\\  \\hline \n  \n\\endfirsthead\n\\multicolumn{9}{c}%\n{{\\bfseries \\tablename\\ \\thetable{} -- continued from previous page}} \\\\ \\hline \n\\endfoot\n\\hline \\multicolumn{1}{|c|} {\\textbf{Object    }} & \\multicolumn{1}{c|}\n{\\textbf{Night}} & \\multicolumn{1}{c|} {\\textbf{UT$_{start}$}} &\n\\multicolumn{1}{c|} {\\textbf{T$_{exp}$}} &     \\multicolumn{1}{c|}\n{\\textbf{Tel.}} & \\multicolumn{1}{c|} {\\textbf{Instr.}} & \\multicolumn{1}{c|}\n{\\textbf{Grism}} & \\multicolumn{1}{c|} {\\textbf{airm.}} & \\multicolumn{1}{c|}\n{\\textbf{Solar Analog (airm.)}} \\\\  \\hline \n\\endhead\n\\hline \\multicolumn{9}{r}{{Continued on next page}} \\\\ \n\\endfoot\n\\hline \\hline\n\\endlastfoot\nObject  &  Night &  UT$_{start}$  & T$_{exp}$ & Tel. & Instr. & Grism & airm. & Solar Analog (airm.) \\\\\n & & (hh:mm) & (s) & & &   &   \\\\ \\hline\n50 Virginia & 18 Nov. 04 & 04:45 & 120 &  TNG & Dolores & LR-R & 1.09 & hyades64 (1.4) \\\\ \n50 Virginia & 18 Nov. 04 & 04:49 & 180 &  TNG & Dolores & MR-B & 1.10 & hyades64 (1.4) \\\\ \n50 Virginia & 19 Nov. 04 & 04:23 & 360 & TNG & NICS & Amici   & 1.07  & la98-978 (1.17) \\\\\n77 Frigga  & 29 Feb. 04 & 05:12 & 90 & TNG & Dolores & LR-R    & 1.33 & la107684 (1.20)   \\\\\n77 Frigga  & 29 Feb. 04 & 05:15 & 90 & TNG & Dolores & LR-B    & 1.34 & la107684 (1.20)   \\\\\n77 Frigga  & 1 Mar. 04 & 05:27 & 120 & TNG & NICS & Amici   & 1.38  & la102-1081 (1.46) \\\\\n92 Undina     &  20 Jan 07 & 04:29 & 40 & NTT & EMMI &\tGR1 & 1.69 & la98-978 (1.35) \\\\\n92 Undina     & 17 Sep. 05& 11:37 & 2560  & IRTF & SPEX &\tPrism &\t1.15 &\nhyades64 (1.00),la115-270(1.10), \\\\\n& & & & & & & & la93-101(1.10), la112-1333(1.10) \\\\\n184 Dejopeja  &13 Aug 05&06:13&   300& NTT & EMMI &\tGR1 &\t1.07   & HD1835 (1.07) \\\\        \t\n184 Dejopeja  &12 Aug 05 & 06:59 & 360 & NTT & SOFI & GB  &1.11  & hip103579  (1.06) \\\\\n184 Dejopeja  &12 Aug 05 & 07:04 & 360 & NTT & SOFI & GR  &1.12  & hip103579  (1.06) \\\\\n220 Stefania & 21 Nov. 04 & 00:13 & 180 & TNG & Dolores & LR-R  & 1.12 & hyades64 (1.05)  \\\\\n220 Stefania & 21 Nov. 04 & 00:17 & 180 & TNG & Dolores & MR-B  & 1.12 & hyades64 (1.05)  \\\\\n223 Rosa & 20 Nov. 04 & 20:58 & 360 & TNG & Dolores & LR-R  & 1.38 & la115-271 (1.16)      \\\\\n223 Rosa & 20 Nov. 04 & 21:05 & 420 & TNG & Dolores & MR-B  & 1.40 & la115-271 (1.16)   \\\\\n275 Sapientia &13 Aug 05&06:44\t&480& NTT & EMMI &\tGR1 &\t1.09  & HD1835 (1.07) \\\\    \n275 Sapientia & 12 Jul. 04 & 08:50 & 1650 &\tIRTF &\tSPEX &\tPrism &\t  1.30 &\n16CyB(1.20), la107-684(1.10)\\\\\n & & & & & & & &  la110-361(1.10) \\\\\n283 Emma & 16 Nov. 04 & 04:08 & 180 & TNG & Dolores & LR-R & 1.08 & hyades64 (1.03) \\\\\n283 Emma & 16 Nov. 04 & 04:13 &  180 & TNG & Dolores & MR-B & 1.09 & hyades64 (1.03) \\\\\n283 Emma  & 19 Nov. 04 & 01:45 & 240 & TNG & NICS & Amici   & 1.01  & Hyades64 (1.01) \\\\\n337 Devosa    &13 Aug 05&01:31\t&300& NTT & EMMI &\tGR1 &\t1.08 & HD144585 (1.22) \\\\         \t\n337 Devosa    &12 Aug 05 & 03:16 & 280 & NTT & SOFI & GB &1.38  & hip083805  (1.23) \\\\ \n417 Suevia    &13 Aug 05 &08:12\t&600& NTT & EMMI &\tGR1 &\t1.31  & HD1835 (1.07) \\\\       \t\n417 Suevia    &14 Aug 05 &06:04 &480& NTT & SOFI &      GB &   1.15  & la115-271 (1.15) \\\\\n417 Suevia    &14 Aug 05 &06:20 &720& NTT & SOFI &      GR &   1.16  & la115-271 (1.15) \\\\\t \t\n463 Lola & 20 Nov. 04 & 23:48 & 180 & TNG & Dolores & LR-R  & 1.10 & hyades64 (1.05)      \\\\\n463 Lola & 20 Nov. 04 & 23:52 & 240 & TNG & Dolores & LR-R  & 1.11 & hyades64 (1.05)      \\\\\n517 Edith & 16 Nov. 04 & 02:42 & 180 &  TNG & Dolores & LR-R & 1.01 & hyades64 (1.03) \\\\\n517 Edith & 16 Nov. 04 & 02:46 & 180 &  TNG & Dolores & MR-B & 1.01 & hyades64 (1.03) \\\\\n517 Edith &  19 Nov. 04 & 03:50 & 360 & TNG & NICS & Amici   & 1.02  &  Hyades64 (1.01) \\\\\n522 Helga     &14 Aug 05 &07:20 &600& NTT & SOFI &      GB &   1.13  & la115-271 (1.15) \\\\\n522 Helga     &14 Aug 05 &07:38 &900& NTT & SOFI &      GR &   1.12  & la115-271 (1.15) \\\\\n522 Helga     &  20 Jan 07 & 02:49 & 600 & NTT & EMMI &\tGR1 & 1.67 & Hyades64 (1.45) \\\\\n\n536 Merapi & 20 Nov. 04 & 22:23 & 150 & TNG & Dolores & LR-R  & 1.34 & la115-271 (1.28)      \\\\\n536 Merapi & 20 Nov. 04 & 22:27 & 150 & TNG & Dolores & MR-B  & 1.34 & la115-271 (1.28)      \\\\\n741 Botolpia  & 20 Jan 07 & 04:44 & 300 & NTT & EMMI &\tGR1 & 1.70 & la98-978 (1.35) \\\\\n758 Mancunia  &13 Aug 05 &06:05 &180 & NTT & EMMI &\tGR1 &\t1.03  & HD1835 (1.07) \\\\       \t\n758 Mancunia  &12 Aug 05 &06:40 &320 & NTT & SOFI &     GB &   1.05  & hip103572 (1.06) \\\\\n758 Mancunia  &12 Aug 05 &06:46 &320 & NTT & SOFI &     GR &   1.05  & hip103572 (1.06) \\\\\n909 Ulla       & 20 Nov. 04 & 04:41 & 180 & TNG & Dolores & LR-R  & 1.11 &  hyades64 (1.10)  \\\\\n909 Ulla       & 20 Nov. 04 & 04:45 & 240 & TNG & Dolores & MR-B  & 1.11 &  hyades64 (1.10)  \\\\\n909 Ulla       & 21 Nov. 04 & 05:31 & 300 & TNG & NICS & Amici   & 1.17  & la98-978 (1.17)  \\\\\n1122 Neith & 20 Nov. 04 & 03:28 & 120 & TNG & Dolores & LR-R  & 1.14 &  hyades64 (1.10)  \\\\\n1122 Neith  & 20 Nov. 04 & 03:31 & 120 & TNG & Dolores & MR-B  & 1.15 &  hyades64 (1.10)  \\\\\n1122 Neith  & 21 Nov. 04 & 04:19 & 160 & TNG & NICS & Amici   & 1.32  & Hyades64 (1.23) \\\\\n1124 Stroobantia  & 20 Nov. 04 & 03:50 & 300 & TNG & Dolores & LR-R  & 1.02 &  hyades64 (1.10)  \\\\\n1124 Stroobantia  & 20 Nov. 04 & 03:57 & 360 & TNG & Dolores & MR-B  & 1.02 &  hyades64 (1.10)  \\\\\n1124 Stroobantia  & 21 Nov. 04 & 04:40  & 480 & TNG & NICS & Amici   & 1.19  & la98-978 (1.19) \\\\ \n1146 Biarmia & 20 Nov. 04 & 20:10 & 300 & TNG & Dolores & LR-R  & 1.31 &  la115-271 (1.16)  \\\\\n1146 Biarmia & 20 Nov. 04 & 20:19 & 300 & TNG & Dolores & MR-B  & 1.34 &  la115-271 (1.16)  \\\\\n1328 Devota  & 20 Nov. 04 & 02:58 & 300 & TNG & Dolores & LR-R  & 1.01 &  hyades64 (1.02)  \\\\\n1328 Devota  & 20 Nov. 04 & 03:04 & 300 & TNG & Dolores & MR-B  & 1.02 &  hyades64 (1.02)  \\\\\n1328 Devota  & 21 Nov. 04 & 03:45 & 480 & TNG & NICS & Amici   & 1.03  & la98-978 (1.17) \\\\\n1355 Magoeba & 16 Nov. 04 & 04:28 & 300 & TNG & Dolores & LR-R & 1.30 & hyades64 (1.03) \\\\\n1355 Magoeba & 16 Nov. 04 & 04:34 &  360 & TNG & Dolores & MR-B & 1.31 & hyades64 (1.03) \\\\\n1355 Magoeba &  19 Nov. 04 & 03:31 & 480 & TNG & NICS & Amici   & 1.17  & Hyades64 (1.21) \\\\\n1902 Shaposhnikov & 14 Aug 05 & 08:03 & 480 & NTT & SOFI &    GB &   1.20  & hip113948 (1.17) \\\\ \n1902 Shaposhnikov &14 Aug 05 & 08:12 & 480 & NTT & SOFI &    GR &   1.25  & hip113948 (1.17) \\\\ \n3447  Burckhalter      & 20 Nov. 04 & 02:22 & 300 & TNG & Dolores & LR-R  & 1.42 &  hyades64 (1.02)  \\\\\n3447  Burckhalter      & 20 Nov. 04 & 02:29 & 300 & TNG & Dolores & MR-B  & 1.46 &  hyades64 (1.02)  \\\\ \n\\hline\n\\hline\n\\end{longtable}\n\\end{center}\n}\n\\begin{list}{}{}\n\\item \n\\end{list}\n\n\n\n       \\begin{sidewaystable}\n       \\caption{\nPhysical and orbital parameters of the  X-type asteroids\nobserved and selected on the basis of the Tholen taxonomy (Tholen, 1984). The Bus and Bus-DeMeo \nclassifications are also reported, together with the new Tholen-like classification proposed here, \nconsidering the available albedo values (Tedesco et al. 2002; the albedos marked by $^{a}$ come from Gil-Hutton et al. 2007) and the spectral behaviors. The spectral slopes values\nare also given (S$_{UV}$ calculated in the 0.49--0.55 $\\mu$m wavelength range,\nS$_{VIS}$ in the 0.55-0.8 $\\mu$m range, S$_{NIR1}$ in the 1.1-1.6 $\\mu$m range, S$_{NIR2}$ in the 1.7-2.4 $\\mu$m\nrange, S$_{cont}$ in the 0.43-2.40 $\\mu$m for asteroids with VIS and NIR spectra, \nand 0.43-0.92 $\\mu$m for those observed only in the visible range (they are marked with an $^{*}$).  }\n        \\label{slope}\n        \\scriptsize{\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|} \\hline\n\\hline\nAsteroid & Bus &  Bus-DeMeo & NEW TX & albedo &  D (km)  & a\n(AU) & e &    i$^{o}$ & S$_{UV}$ & S$_{Vis}$ & S$_{NIR1}$ & S$_{NIR2}$ & $S_{cont}$\\\\  \\hline\n50 Virginia\t  & Ch &Ch  & C &0.04  &99.82   & 2.6518 & 0.2838 &2.8  & 1.57$\\pm$0.74  & -0.40$\\pm$0.54&  1.39$\\pm$0.84& -0.16$\\pm$0.77 &  1.05$\\pm$0.71 \\\\\n77 Frigga\t  & X  &X   & M &0.14  &69.25   & 2.6674 & 0.1328 &2.4  & 18.23$\\pm$1.32 &  10.16$\\pm$0.57&  2.85$\\pm$0.87&  1.08$\\pm$0.83 &  5.37$\\pm$0.78 \\\\\n92 Undina\t  & Xc &Xk  & M &0.25  &126.42  & 3.1895 & 0.1001 &9.9  &  6.56$\\pm$0.31 & 3.15$\\pm$0.54&  2.30$\\pm$0.76&  1.63$\\pm$0.73 &  2.46$\\pm$0.71 \\\\\n184 Dejopeja\t  & X  &X   & M &0.19  &66.47   & 3.1804 & 0.0763 &1.1  & 5.05$\\pm$1.02  & 3.98$\\pm$0.55&  2.72$\\pm$0.75&  1.38$\\pm$0.75 &  2.57$\\pm$0.71 \\\\\n220 Stephania\t  & C  &$-$ & C &0.07  &31.12   & 2.3482 & 0.2585 &7.5  & 2.14$\\pm$0.32  & 0.36$\\pm$0.55&  --&  -- &  0.74$\\pm$0.72$^{*}$ \\\\\n223 Rosa\t  & C  &$-$ & C &0.03  &87.61   & 3.0920 & 0.1238 &1.9  & 3.99$\\pm$0.61  & 1.76$\\pm$0.57&  --&  -- &  1.86$\\pm$0.73$^{*}$ \\\\\n275 Sapientia\t  &  C &C   & P &0.04   &103.00  & 2.7721 & 0.1615 &4.7  & 7.49$\\pm$0.69  & 4.11$\\pm$0.57&  2.21$\\pm$0.75&  2.13$\\pm$0.73 &  2.38$\\pm$0.72 \\\\\n283 Emma\t  & $-$ &C  & C &0.03  &148.06  & 3.0447 & 0.1499 &7.9  & 2.89$\\pm$0.77  & 1.87$\\pm$0.54&  0.38$\\pm$0.81&  0.24$\\pm$0.79 &  0.73$\\pm$0.72 \\\\\n337 Devosa\t  & X  &Xk  & M &0.16  &59.11   & 2.3835 & 0.1379 &7.8  & 5.57$\\pm$1.28  & 3.95$\\pm$0.57&  3.64$\\pm$0.77&  -- &  3.58$\\pm$0.02 \\\\\n417 Suevia\t  & Xk &Xk  & M &0.20  &40.69   & 2.8006 & 0.1331 &6.6  & 5.88$\\pm$2.33  & 3.31$\\pm$0.76&  5.94$\\pm$0.76&  2.89$\\pm$0.81 &  4.06$\\pm$0.72 \\\\\n463 Lola\t  & $-$ &$-$& P &0.08  &19.97   & 2.3988 & 0.2202 &13.4 & 4.50$\\pm$0.96  & 4.11$\\pm$0.55&  --&  -- &  4.08$\\pm$0.73$^{*}$ \\\\\n517 Edith\t  & C  &Xc  & C &0.04  &91.12   & 3.1558 & 0.1819 &3.1  & 2.61$\\pm$0.78  & 1.88$\\pm$0.54&  1.52$\\pm$0.81&  0.89$\\pm$0.81 &  1.59$\\pm$0.71 \\\\\n522 Helga\t  & X  &Xk  & P &0.04  &101.22  & 3.6287 & 0.0755 &4.4  & 5.64$\\pm$1.09  & 3.79$\\pm$0.55&  3.82$\\pm$0.78&  3.38$\\pm$0.76 &  3.59$\\pm$0.71 \\\\\n536 Merapi\t  & C  &$-$ & C &0.04  &151.42  & 3.5016 & 0.0819 &19.4 & 1.72$\\pm$0.85  & 1.16$\\pm$0.55&  --&  -- &  1.13$\\pm$0.72$^{*}$ \\\\\n741 Botolphia\t  & X  &$-$ & M &0.14  &29.64   & 2.7198 & 0.0678 &8.4  & 6.37$\\pm$0.88  & 3.42$\\pm$0.54&  --&  -- &  3.22$\\pm$0.74$^{*}$ \\\\\n758 Mancunia\t  & $-$ &X  & M &0.13  &85.48   & 3.1861 & 0.1518 &5.6  & 6.79$\\pm$1.05  & 4.06$\\pm$0.56&  2.53$\\pm$0.75&  1.36$\\pm$0.73 &  2.19$\\pm$0.71 \\\\\n909 Ulla\t  & $-$ &X  & P  &0.03  &116.44 & 3.5550 & 0.0959&18.8 &  5.12$\\pm$1.54  & 3.36$\\pm$0.61&  3.20$\\pm$0.89&  1.36$\\pm$0.85 &  3.64$\\pm$0.72 \\\\\n1122 Nieth\t  & $-$ &A  & A  & 0.45 & 12.01 & 2.6068 & 0.2562 &4.7  & 15.42$\\pm$0.58  & 8.03$\\pm$0.58& 15.31$\\pm$1.10&  6.18$\\pm$0.92 &  8.57$\\pm$0.76 \\\\\n1124 Stroobantia  & $-$ &Xk & M  &0.16  &24.65  & 2.9253 & 0.0344 &7.7 & 2.19$\\pm$1.23   & 2.01$\\pm$0.63&  4.73$\\pm$0.88&  3.29$\\pm$0.87 &  3.49$\\pm$0.73 \\\\\n1146 Biarmia\t  & X  &$-$ & M &0.22  &31.14   & 3.0511 & 0.2508 &17.0 & 6.08$\\pm$0.49  & 1.92$\\pm$0.56&  --&  -- &  1.78$\\pm$0.73$^{*}$ \\\\\n1328 Devota\t  & $-$ &D  & D &0.04  &57.11   & 3.5093 & 0.1407 &5.7   & 10.67$\\pm$0.96  & 11.90$\\pm$0.55& 11.70$\\pm$1.10&  7.06$\\pm$1.01 & 11.01$\\pm$0.72 \\\\\n1355 Magoeba\t  & Xe &Xe  & M & 0.27$^{a}$  &12.90   & 1.8535 & 0.0450 &22.8  & 13.42$\\pm$0.98  &  5.63$\\pm$0.55&  2.23$\\pm$0.81& -0.80$\\pm$0.88 &  3.61$\\pm$0.74 \\\\\n1902 Shaposhnikov  & $-$ &D  & P &0.03  &96.86  & 3.9727 & 0.2242 &12.5 & --  & --&  4.08$\\pm$0.77&  2.29$\\pm$0.77 &  -- \\\\\n3447 Burckhalter   & Xc &$-$ & E ? & 0.34$^{a}$  &15.60  & 1.9907 & 0.0285 &20.7 & 16.02$\\pm$1.32  & 4.66$\\pm$0.60&  --&  -- &  6.17$\\pm$0.75$^{*}$ \\\\ \\hline\n\\end{tabular}\n}\n\\end{sidewaystable}\n\n\n\n\n\n       \\begin{table}\n       \\begin{center}\n       \\caption{\nBand center, depth and width for the features detected in the\nasteroid spectra.}\n        \\label{band}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \\hline\n\\hline\nAsteroid & Band Center ($\\mu$m) & Depth (\\%)  & Width ($\\mu$m) \\\\ \\hline\n50 Virginia\t  &0.6920$\\pm$0.0070  &2.2  &0.540--0.839 \\\\\n50 Virginia\t  &0.4310$\\pm$0.0040  &1.1  &0.416--0.445 \\\\\n50 Virginia\t  &0.8660$\\pm$0.0080  &1.2  &0.823--0.911 \\\\\n92 Undina\t  &0.9050$\\pm$0.0080  &2.9  &0.742--1.027 \\\\\n92 Undina\t  &0.5103$\\pm$0.0050  &0.7  &0.487--0.531 \\\\\n283 Emma\t  &0.4310$\\pm$0.0040  &1.4  &0.417--0.443 \\\\\n337 Devosa\t  &0.8820$\\pm$0.0080  &2.8  &0.763--1.017\\\\\n337 Devosa\t  &0.4280$\\pm$0.0040  &3.0  &0.410--0.450 \\\\\n417 Suevia\t  &0.8593$\\pm$0.0100  &5.2  &0.730--1.007 \\\\\n517 Edith\t  &0.4300$\\pm$0.0040  &1.5  &0.416--0.444 \\\\\n522 Helga\t  &0.9400$\\pm$0.0070  &2.3  &0.874--1.004\\\\\n758 Mancunia      &1.5960$\\pm$0.0060  &1.3  &1.520-1.705 \\\\ \n1122 Nieth\t  &0.9650$\\pm$0.0080  &24.3 &0.756--1.564 \\\\\n1124 Stroobantia  &0.9170$\\pm$0.0100  &5.9  &0.807--1.100\\\\\n1355 Magoeba\t  &0.4910$\\pm$0.0070  &3.0  &0.443--0.543\\\\\n1355 Magoeba\t  &0.4300$\\pm$0.0040  &1.9  &0.417--0.445\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n       \\begin{center}\n       \\caption{RELAB Matches (Exp means experimental).}\n        \\label{tab}\n         \\scriptsize{\n\\begin{tabular}{|l|c|c|c|c|l|l|} \\hline\nASTEROID & ALBEDO & BEST FIT & MET    & MET  & NAME & GRAIN SIZE \\\\\n          &       &          &  CLASS & REFL  &  & \\\\ \\hline\n50 Virginia & 0.04 & PH-D2M-032\t  & CM & 0.03 & MET00639    & $<$75$\\mu$m \\\\\n77 Frigga   & 0.14 & MR-MJG-082   & IM & 0.18 & Chulafinnee &  \\\\\n92 Undina   & 0.25 & MR-MJG-083   & IM & 0.23 & Babb's Mill & \\\\\n92 Undina   & 0.25 & MB-TXH-043-H & Pall&0.14 & Esquel      & $<$63 $\\mu$m \\\\\n184 Dejopeja& 0.19 & MB-TXH-043-H & Pall&0.14 & Esquel      & $<$63 $\\mu$m \\\\\n275 Sapientia& 0.04 & MB-TXH-064-C   & CM Exp. & 0.03 & Murchison heated to 600$^{\\circ}$C   & $<$63 $\\mu$m\\\\\n283 Emma    & 0.03 & PH-D2M-032   & CM & 0.03 & MET00639    & $<$75$\\mu$m\\\\\n337 Devosa  & 0.16 & MB-TXH-046   & IM & 0.16 & Landes      & slab \\\\\n417 Suevia  & 0.20 & MB-TXH-047-D & IM & 0.15 & DRP78007    & 75-125$\\mu$m \\\\\n517 Edith   & 0.04 & MB-TXH-064-D & CM-Exp& 0.03 & CM Murchison  heated to 700$^{\\circ}$C & 63--125$\\mu$m \\\\ \n522 Helga   & 0.04 & MA-ATB-068   & CM Exp& 0.05 & CM Migei Laser Irrad. & $<$45$\\mu$m \\\\ \n758 Mancunia& 0.13 & MB-TXH-046   & IM & 0.16 & Landes      & slab \\\\\n909 Ulla    & 0.03 & MA-ATB-068   & CM Exp& 0.05 & CM Migei Laser Irrad. & $<$45$\\mu$m \\\\ \n1124 Stroobantia& 0.16 & MB-TXH-047-A & IM & 0.12 & DRP78007 &  $<$25 \\\\\n\\hline \n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}\n       \\begin{center}\n       \\caption{Geographical mixing models. $^1$ blue line in Fig.~\\ref{modelli},\n$^2$ red line in Fig.~\\ref{modelli}}\n        \\label{models}\n         \\scriptsize{\n\\begin{tabular}{|l|c|l|c|c|} \\hline\nASTEROID & ALBEDO & GEOGRAPHICAL MODEL & FILE SOURCE & MODEL ALBEDO \\\\ \\hline\n92 Undina   & 0.25 &  99\\% pallasite Esquel & RELAB ckmb43 & 0.14 \\\\\n            &      &   1\\% orthopyroxene  & RELAB cbpe40 & \\\\ \n337 Devosa$^1$  & 0.16 &  98\\% pallasite Esquel & RELAB ckmb43 & 0.14 \\\\\n            &      &   2\\% goethite  & ASTER & \\\\\n337 Devosa$^2$  & 0.16 &  98\\% pallasite Esquel & RELAB ckmb43 & 0.14 \\\\\n             &      &   1\\% orthopyroxene  & RELAB cbpe40 & \\\\ \n417 Suevia  & 0.20 &   97\\% iron met. DRP78007 & RELAB cdmb47 & 0.16 \\\\\n            &      &   3\\% orthopyroxene  & RELAB cbpe40 & \\\\ \n517 Edith   & 0.04 & 100\\% amorphous carbon &  Zubko et al., 1996 & \\\\\n1124 Stroobantia$^1$ & 0.16 & 98\\% iron met. MET101A & RELAB c1sc99 & 0.13 \\\\\n            &     & 2\\% orthopyroxene & RELAB  cbpe40 & \\\\\n1124 Stroobantia$^2$ & 0.16 & 96\\% iron met. DRP78007 & RELAB cdmb47  & 0.16 \\\\\n            &     & 4\\% olivine & USGS & \\\\            \n1328 Devota & 0.04 & 94\\% carb. chond. Tagish Lake & RELAB c8mt11 & 0.03 \\\\\n            &      & 6\\% Trit. Tholin & McDonald et al. 1994  & \\\\ \\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n  \\begin{table}\n       \\begin{center}\n       \\caption{Mean spectral slopes, for the different classes of asteroids with the associated standard deviation value. We discard from the following analysis a few asteroids initially classified as\nE\/M\/X in the Tholen taxonomy but whose spectral properties diverge completely\nfrom these classes and who have been since  reclassified as A, S or D-type\n(A: 1122, 2577, and 7579; S\/Sq: 516, and 5806; D: 1328).}\n        \\label{slopes}\n\\begin{tabular}{|l|l|l|l|} \\hline\n\\hline\nAsteroid Class & Mean S$_{VIS}$ & Mean S$_{NIR1}$& Mean S$_{NIR2}$ \\\\ \n               & (\\%\/($10^{3}$ \\AA) & (\\%\/($10^{3}$ \\AA) & (\\%\/($10^{3}$ \\AA) \\\\ \\hline\nE (all)                       & 3.92$\\pm$1.73  & -- & --\\\\\nE[I]                          & 4.60$\\pm$1.18  & -- & -- \\\\\nE[II]                         & 5.15$\\pm$1.31  & 0.85$\\pm$0.30 &  0.32$\\pm$0.15 \\\\\nE[III]                        & 2.43$\\pm$1.12  & 1.07$\\pm$0.81 &  1.09$\\pm$0.87    \\\\\nM with the 0.9 $\\mu$m band    & 3.54$\\pm$0.90  & 3.14$\\pm$1.51 &  1.82$\\pm$1.42   \\\\\nM without the 0.9 $\\mu$m band & 4.00$\\pm$1.97  & 1.73$\\pm$2.07 &  0.81$\\pm$1.21 \\\\\nP                             & 3.34$\\pm$1.69  & 2.66$\\pm$1.65  & 1.83$\\pm$1.25 \\\\\nC                             & 1.77$\\pm$1.94  & 0.48$\\pm$0.68  & 0.17$\\pm$0.35     \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n  \\begin{table}\n       \\begin{center}\n       \\caption{The Tholen (existing or proposed by this work) and the Bus--DeMeo classification for all the asteroids observed withing our spectroscopical survey of the X complex. 1 = this work, 2 = Fornasier et al. 2010, 3 = Fornasier et al. 2008 and reference therein.}\n        \\label{busDeMeotx}\n        \\scriptsize{\n\\begin{tabular}{|l|l|l|l|l|l|l|l|} \\hline\nAsteroid & Tholen & Bus-DeMeo & reference & Asteroid & Tholen & Bus-DeMeo & reference\\\\\n16   & M  & Xk & 2       &       498  & C & Xc & 2          \\\\\n22   & M  & Xk & 2       &   \t 504  & E (I) & -- & 3      \\\\\n44   & E (III) & Xk & 3  &\t 516  & S & Sq & 2          \\\\\n50   & C & Ch & 1        & \t 517  & C & Xc & 1          \\\\\n55   & M  & Xk & 2       &\t 522  & P  & Xk & 1         \\\\\n64   & E (II) & Xe & 3   &\t 536  & C  & C & 1         \\\\\n69   & M  & Xk & 2       &\t 558  & M  & Xk & 2        \\\\\n77   & M  & X  & 1       &\t 620  & E (III) & Xk & 3   \\\\\n92   & M  & Xk & 1       &\t 741  & M  & X  & 1        \\\\\n97   & M  & Xc & 2       &\t 755  & M  & Xk & 2        \\\\\n110  & M  & Xk & 2       &\t 758  & M  & X  & 1        \\\\\n125  & M  & -- & 2\t &\t 785  & M  & X  & 2        \\\\\n129  & M  & Xc & 2\t &\t 849  & M &  D  & 2        \\\\\n132 & E (II) & Xe & 2 \t &\t 860  & M  & X  & 2        \\\\\n135  & M  & Xk & 2\t &\t 872  & M  & Xk & 2        \\\\\n161  & M & Xc & 2\t &\t 909  & P  & X  & 1          \\\\\n184  & M  & X  & 1\t &\t 1025 & E (I) & -- & 3       \\\\\n201  & M  & X  & 2\t &\t 1103 & E (III) & Xk & 3     \\\\\n214  & E (III) & Xk & 3\t &\t 1122 & A & A & 1            \\\\\n216  & M  & Xk & 2\t &\t 1124 & M  & Xk & 1          \\\\\n220  & C  & C & 1        &\t 1146 & M  & X  & 1         \\\\\n223  & C  & C & 1\t &\t 1251 & E (III) & Xk & 3    \\\\\n224  & M  & Xc & 2\t &\t 1328 & D & D & 1           \\\\\n250  & M  & Xk & 2\t &\t 1355 & M  & Xe & 1     \\\\\n275  & P  & C  & 1\t &\t 1902 & P      & D & 1      \\\\\n283  & C  & C  & 1\t &\t 2035 & E (II) & Xe & 3     \\\\\n317  & E (III) & Xk & 3\t &\t 2048 & E (II) & Xe & 3     \\\\\n325  & M  & X  & 2\t &\t 2449 & E (I) & -- & 3      \\\\\n337  & M  & Xk & 1\t &\t 2577 &  A & A &3           \\\\\n338  & M  & Xk & 2\t &\t 2867 & E (II) & Xe & 3     \\\\\n347  & M  & Xk & 2       &\t 3050 & E (III) & Xk & 3     \\\\\n369  & M  & Xk & 2\t &\t 3103 & E (II) & Xe & 3      \\\\\n382  & M  & -- & 2\t &\t 3447 & E (I) ?  & Xc & 1      \\\\\n417  & M  & Xk & 1\t &\t 4660 & E (II) & Xe & 3      \\\\\n418  & M  & X  & 2\t &\t 5806 & S      & S & 3       \\\\\n434  & E (II) & Xe & 3\t &\t 6435 & E (I) & -- & 3      \\\\\n437  & E (III) & Xk & 3\t &\t 6911 & E (II) & Xe & 3     \\\\\n441  & M  & X  & 2\t &\t 7579 & A      & A & 3      \\\\\n463  & P  & -- & 1\t &\t 144898 & E (I) & -- & 3    \\\\ \\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\\newpage\n\n{\\bf Figure captions}\n\n\\vspace{1cm}\n\nFig. 1 - Visible and near infrared spectra  of X-type asteroids.\n\nFig. 2 - Visible and near infrared spectra of X-type asteroids.\n\nFig. 3 - Visible and near infrared spectra of X-type asteroids.\n\nFig. 4 - Visible and near infrared spectra of X-type asteroids.\n\nFig. 5 - Best spectral matches between the observed medium albedo \nasteroids and meteorites from the RELAB database \n(see Table~\\ref{tab} for details of the meteorite samples). \nAll these asteroids are re-classified as M-type \nfollowing the Tholen classification scheme. For 92 Undina \n2 meteorites are proposed: in red the IM Babb's Mill \nand in blue the IM Esquel.\n\nFig. 6 - Best spectral matches between the observed low-albedo asteroids and meteorites from the RELAB database (see Table~\\ref{tab} for details of the meteorite samples). \n\nFig. 7 - Geographical mixing model for the asteroids 92 Undina, 337 Devosa, 417 Suevia, 517 Edith, 1124 Stroobantia, and 1328 Devota (see details of each model in Table~\\ref{models}). \n\nFig. 8 - Spectral slope value ($S_{VIS}$) versus the semimajor axis for the different asteroids observed, classified following the Tholen taxonomy. The size of\nthe symbols is proportional to the asteroids' diameter.\n\nFig. 9 - Albedo versus semimajor axis for the different asteroids observed, classified following the Bus-DeMeo taxonomy. The size of the symbols is proportional to the asteroids' diameter.\n\n\n\\newpage\n\n\n{\\bf Figures}\n\n\\begin{figure*}[b]\n\\includegraphics[width=14cm]{I11726_f1.eps}\n\\caption{}\n\\label{fig1}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f2.eps}\n\\caption{}\n\\label{fig2}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f3.eps}\n\\caption{}\n\\label{fig3}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f4.eps}\n\\caption{}\n\\label{fig4}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f5.eps}\n \\caption{}\n   \\label{fig5met}\n \\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f6.eps}\n \\caption{}\n   \\label{fig6met}\n \\end{figure*}\n\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f7.eps}\n \\caption{}\n   \\label{modelli}\n \\end{figure*}\n\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f8.ps}\n\\caption{}\n\\label{slopea}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{I11726_f9.ps}\n\\caption{}\n\\label{busalbedo}\n\\end{figure*}\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nKitaev quantum spin liquids have an exactly solvable ground state in which the excitations are Majorana fermions \\cite{Kitaev2006,Jackeli2009}, and the $J_{\\textrm{eff}}$ = $\\frac{1}{2}$ layered honeycomb magnet $\\alpha$-RuCl$_3$ has emerged as one of the prominent material candidates\\cite{Plumb2014,Trebst2017,Takagi2019}. In the absence of an applied magnetic field, the ground state has zig-zag antiferromagnetic order ($T_\\textrm{N}$ $\\approx$ 7 K), but upon applying an in-plane magnetic field the ordered state is suppressed ($H_{\\text{C2}} \\approx$ 7 T) and a quantum paramagnetic regime is accessed \\cite{Johnson2015,Kubota2015,Sears2015}. In this field-induced state, a thermal Hall conductivity close to the half-quantized value was reported and discussed as a response of a topological edge state of Majorana fermions \\cite{Kasahara2018a,Nasu2017}. Although that result and interpretation are the subject of ongoing debate \\cite{Kasahara2018a,Yamashita2020,Yokoi2021,Yamashita2020,Czajka2021a,Lefrancois2021,Bruin2022,Czajka2022}, the field-induced state clearly exhibits exotic behavior.\n\nAnother unusual property of the field-induced state was reported recently by Czajka \\textit{et al.}\\cite{Czajka2021a}, who observed a sequence of minima in the magnetic field dependence of thermal conductivity $\\kappa$ and interpreted them as quantum oscillations originating from a Fermi surface of neutral quasiparticles. The amplitudes of these oscillations were shown to grow upon cooling down to $\\sim$1 K, in line with the expected Lifshitz-Kosevich behavior. The present authors, however, have pointed out that two of the reported oscillation minima coincide with known magnetic phase transitions, the sequential collapse of zig-zag order at  $H_{\\text{C1}}$ and $H_{\\text{C2}}$, and that amplitudes of these oscillations collapse upon cooling below 1 K\\cite{Bruin2022}. These observations contradict expectation for conventional quantum oscillations. The origin of the oscillatory features remains to be examined carefully by more comprehensive experiments over a wide range of temperatures and fields. We note that, while the measurements by Czajka \\textit{et al.} were conducted using single crystals grown by a chemical vapor transport (CVT) method, the measurements by the present authors were done on a Bridgman-grown sample (the same batch of single crystals as those used in the original report of half-quantization \\cite{Kasahara2018a}). The possibility of a sample dependence must also be considered when addressing the controversial issue of oscillations.\n\nIn this study we focus on the question of the origin of the oscillatory structures in the magnetic field dependence of thermal conductivity $\\kappa(H)$ of $\\alpha$-RuCl$_3$.  We present detailed measurements of magnetothermal conductivity along the $a$-axis and magnetization with the in-plane field oriented along either the crystal $a$ or $b$ axes, down to the low temperature of 100 mK for single crystals grown by both the Bridgman and the CVT methods. All the experimental results point to magnetic phase transitions as the origin of the oscillatory structures, independent of the crystal growth method.\n\n\\section{Methods}\n\nTwo single crystal pieces grown by a CVT- and a Bridgman technique were measured. The CVT-grown crystal was sourced from the same batch as reported in Ref.~\\onlinecite{Suzuki2021}. The Bridgman-grown crystal was the same as reported in Ref.~\\onlinecite{Bruin2022}. They were both plate-like and similar in size, with approximate dimensions 2 mm x 1 mm x 0.02 mm (length x width x thickness). The crystal structures and their orientations were determined by single-crystal x-ray diffraction at room temperature using a SMART-APEX-II CCD X-ray diffractometer (Bruker AXS, Karlsruhe, Germany) with graphite monochromated MoK$\\alpha$ radiation. The Bridgman sample was best fitted to a C2\/m monoclinic unit cell with lattice parameters a = 6.041(4) \\AA, b = 10.416(8) \\AA, c = 6.088(4) \\AA, $\\beta$ = 108.54(2)$^{\\circ}$, and the CVT sample was determined to be trigonal P3$_1$12 with lattice parameters a = b = 6.012(3) \\AA, c = 17.27(1) \\AA. Lattice parameters were refined with the SAINT subprogram in the Bruker Suite software package\\cite{Bruker2019}, using 448 and 776 reflections, respectively. Despite the differences in the room temperature structures of these two samples, we will see that their low temperature conductivities and magnetic properties are similar to each other.\n\nThermal conductivity was measured using a one heater- two thermometer technique on a dilution refrigerator. RuO$_2$ chip thermometers, calibrated in-situ against a field-calibrated RuO$_2$ reference, were used to determine the temperature gradients. To measure the magnetic field dependence of thermal conductivity, the sample was allowed to thermalize fully at each field before every measurement so that true steady state conditions were achieved. Complementary continuous field sweeps were performed in some cases to examine the detailed field dependence, and these data were cross-checked against steady-state field sweeps to verify consistency. All measurements of $\\kappa$ were performed with heat applied along the crystal $a$-axis. The Bridgman sample $\\kappa$ data with $H \\parallel a$ plotted in Figs. \\ref{fig1}b,c and \\ref{fig2}b,d are reproduced from Ref.~\\onlinecite{Bruin2022}. For the CVT crystal, magnetization data before and after mechanical bending are presented, and all thermal conductivity data for that crystal were measured before it was bent.\n\nThe crystal $a$ and $b$ axes are defined as being perpendicular and parallel to the Ru-Ru bond directions, respectively. In the case of the monoclinic C2\/m structure of the Bridgman crystal, the $a$-axis is uniquely determined as the direction of the layer-to-layer Ru honeycomb shift. In a monoclinic unit cell, the honeycomb axes perpendicular and parallel to Ru-Ru bonds other than the $a$ and $b$ axes are distinct, and we denote these as $a$*,$a$** and $b$*,$b$**, respectively (Fig.~\\ref{fig4}a). In the case of the trigonal CVT crystal, $a$ was defined as the direction of the longest sample dimension. Magnetization measurements were performed on both samples using a Physical Properties Measurement System vibrating sample magnetometer option (Quantum Design, USA) along all 6 named crystal axes.\n\n\\section{Results and discussion}\n\nThe magnetothermal conductivity displays rich structures in both the Bridgman and the CVT crystals, for magnetic field applied along both $a$ and $b$ axes as shown in Figs.~\\ref{fig1}a, \\ref{fig1}b and \\ref{fig1}d. In all the studied crystals and field configurations, the observed minima or kinks in $\\kappa(H)$ coincide with the magnetic fields where the field derivative of magnetization d$M$\/d$H$ has a peak or shoulder, including at the magnetic phase transitions associated with the collapse of zig-zag order at $H_{\\text{C1}}$ and $H_{\\text{C2}}$, which already suggests a link between magnetic transitions and the structures in $\\kappa(H)$. In Fig.~\\ref{fig1}c, the magnetothermal conductivity with $H \\parallel a$ at 2 K for the Bridgman and the CVT crystals are compared together with the data from Ref.~\\onlinecite{Czajka2021a} at 1.75 K. The absolute magnitude of the conductivity of our CVT-grown sample, as well as the sample in Ref.~\\onlinecite{Czajka2021a}, is lower than that of the Bridgman-grown sample, which suggests more disorder in the former. However, the structures are more prominent for the CVT-grown samples, so the degree of disorder correlates with the magnitude of oscillatory features, which goes against expectation for canonical quantum oscillations in metals; the cleaner the sample is, the larger the amplitude of oscillations. The positions of minima occur at nearly the same magnetic fields for all three crystals: at $H_{\\text{1}}\\sim$1.5 T, $H_{\\text{2}}\\sim$4.2 T, $H_{\\text{3}}\\sim$4.8 T, $H_{\\text{C1}}\\sim$5.8 T, $H_{\\text{C2}}\\sim$7 T, $H_{\\text{4}}\\sim$8.3 T, $H_{\\text{5}}\\sim$9 T and $H_{\\text{6}}\\sim$10.5 T (Figs.~\\ref{fig1}a,b), except that the distinction between $H_{\\text{4}}$ and $H_{\\text{5}}$ is unresolved for the Bridgman sample. For the Bridgman sample, the features are clearly less prominent in both $\\kappa(H)$ and d$M$\/d$H$ than in the CVT sample, in particular $H_{\\text{2}}$ and $H_{\\text{3}}$ are very weak in $\\kappa(H)$ and not resolved in d$M$\/d$H$. For magnetic field applied along the $b$-axis we resolve fewer features, and their positions are different than for $H \\parallel a$ (Fig.~\\ref{fig1}d), but the coincidence between structures in $\\kappa(H)$ and maxima or shoulders in d$M$\/d$H$ persists.\n\n\\begin{figure*}\n\\includegraphics[width=0.8\\textwidth, angle=0]{Figure1.jpg}\n\\caption{\\label{fig1} (a) Isotherms of thermal conductivity $\\kappa$ (red) and d$M$\/d$H$ (black) with $H \\parallel a$ at $T$=2 K for the same CVT-grown sample. Open and closed black markers denote d$M$\/d$H$ before and after the sample was lightly bent with tweezers, respectively. Several prominent peaks are enhanced by bending. Dotted lines indicate the locations of minima or kinks in $\\kappa(H)$ and arrows indicate the coincident peaks or shoulder features in d$M$\/d$H$. (b) Isotherms of thermal conductivity $\\kappa$ (blue) and d$M$\/d$H$ (black) with $H \\parallel a$ at $T$=2 K for the same Bridgman-grown sample. Dotted lines indicate the same fields as in panel a. Underlying $\\kappa(H)$ data were reproduced from Ref.~\\onlinecite{Bruin2022} (c) Isotherms of thermal conductivity around 2 K with $H \\parallel a$, comparing the Bridgman (blue) and CVT-grown (red) samples, as well as the result from Ref.~\\onlinecite{Czajka2021a} (green). All principal features are reproduced, and the similarity of the two CVT-grown samples is especially striking. (d) Isotherms of thermal conductivity $\\kappa$ (blue) and d$M$\/d$H$ (black) with $H \\parallel b$ for the Bridgman sample. Dotted lines indicate the locations of minima or kinks in $\\kappa(H)$ and arrows indicate the coincident peaks or shoulder features in d$M$\/d$H$.}\n\\end{figure*}\n\nFor both the Bridgman and the CVT crystals, the amplitudes of the oscillatory structures initially grow upon cooling, but then turn down sharply at lowest temperatures below 1 K. We demonstrate this by first subtracting a background (non-oscillatory) magnetic field dependence $\\kappa_{\\text{bg}}$ at each temperature by fitting a spline through the locations of greatest field derivative in $\\kappa$. By plotting the isotherms of the normalized oscillatory content $(\\kappa-\\kappa_{\\text{bg}})\/\\kappa_{\\text{bg}}$ for $H \\parallel a$ in Figs.~\\ref{fig2}a and \\ref{fig2}b, it is apparent that the amplitudes do not increase monotonically upon cooling, but instead decrease for lowest temperature, especially those of the highest field features. The temperature dependences of the amplitudes at the minima fields (Figs.~\\ref{fig2}c and \\ref{fig2}d) show this turnover between 500 mK - 1 K, which is observed in both samples. We note that the present data are consistent with those reported by Czajka \\textit{et al.} \\cite{Czajka2021a} which were taken above $\\sim$1 K, and we only observe the decrease by further lowering temperature. Our observation speaks against Lifshitz-Kosevich behavior and a quantum oscillation interpretation.\n\n\\begin{figure*}\n\\includegraphics[width=0.8\\textwidth, angle=0]{Figure2.jpg}\n\\caption{\\label{fig2} High-field oscillatory component of $\\kappa(H)$ expressed as $(\\kappa-\\kappa_{\\text{bg}})\/\\kappa_{\\text{bg}}$. (a,b) Field dependences for the CVT and Bridgman-grown samples for temperatures ranging from 0.1 K (purple) to 2.5 K (red). Vertical arrows indicate the well-defined minima at $H_{\\text{C1}}$ = 5.79 T (green) and $H_{\\text{C2}}$ = 7.05 T (orange) as well as two higher field minima arising from transitions at $H_{\\text{4\/5}}$ $\\approx$ 8.52 T and $H_{\\text{6}}$ = 10.47 T (red, purple), which all coincide with peaks or shoulders in d$M$\/d$H$. Whilst the structure at $H_{\\text{4\/5}}$ appears to consist of two minima in the CVT sample, only one broad minimum can be identified in the Bridgman sample. Underlying $\\kappa(H)$ data for the Bridgman sample were reproduced from Ref.~\\onlinecite{Bruin2022}. (c,d) Temperature dependences of the amplitudes of the minima in $(\\kappa-\\kappa_{\\text{bg}})\/\\kappa_{\\text{bg}}$ for the CVT and Bridgman samples for all fields where minima can be identified. Upon cooling from 2.5 K to $\\sim$1 K, the amplitudes grow, but they rapidly collapse upon further cooling below 1 K.}\n\\end{figure*}\n\nWhat is then the origin of the structures in the magnetothermal conductivity? We previously argued\\cite{Bruin2022} that the high-field features observed in the Bridgman sample may be due to magnetic phase transitions or crossovers, as two minima in $\\kappa(H)$ coincide with the magnetic phase transitions at $H_{\\text{C1}}$ and $H_{\\text{C2}}$. The minima would then be caused by soft magnetic scattering of phonons, which dominate the thermal conductivity in $\\alpha$-RuCl$_3$ \\cite{Hentrich2020}. We argue that the high-field transitions at $H_{\\text{4}}$ and $H_{\\text{5}}$ are connected to weak magnetic transitions which have higher transition temperatures than the 7 K main phase. In zero field, signatures of transitions around 13 K \\cite{Johnson2015,He2018,Mi2021} and 10 K \\cite{Sears2015,Kubota2015,He2018,Mi2021} were reported previously in the magnetic susceptibility, the specific heat and the dielectric constant, which were discussed to originate from secondary phases associated with stacking faults. As shown in Figs.~\\ref{fig4}a and \\ref{fig4}b, the 7 K, 10 K and 13 K transitions can be identified as sequential kink-like structures in the temperature dependence of magnetization at a low magnetic field of 1 T. \n\n\\begin{figure*}\n\\includegraphics[width=1\\textwidth, angle=0]{Figure3.jpg}\n\\caption{\\label{fig3} Tracking phase transitions in features of the temperature and field derivatives of magnetization of both the CVT and Bridgman samples. (a-d) d$M$\/d$T$ at constant magnetic fields between 0.1 T and 10 T for $H \\parallel a$ and $H \\parallel b$. Colored triangles indicate the features tracking the 7 K (blue), 10 K (red) and 13 K (brown) transitions. Peaks of unknown origin are marked by black triangles. Dashed lines (Bridgman sample) show additional data taken at half-tesla intervals. (e-h) d$M$\/d$H$ isotherms between 2 K and 10 K for $H \\parallel a$ and $H \\parallel b$ (shifted vertically for clarity). Blue triangles indicate the features tracking the main phase transitions at $H_{\\text{C1}}$ and $H_{\\text{C2}}$, additional colored triangles point at secondary features.}\n\\end{figure*}\n\nThe magnetic transitions in $M(T)$ can be identified more clearly as peaks in the temperature derivative d$M$\/d$T$ as shown in Figs.~\\ref{fig3}a-d, from which we can trace the magnetic field dependence of the phase boundaries. For the 7 K transition (dark blue triangles), the peak in d$M$\/d$T$ becomes hard to define for fields above 6 T due to the rapid decrease of the transition temperature. Instead, well-defined peaks in d$M$\/d$H$ appear at low temperature. They can be traced up to about 7 K (Figs.~\\ref{fig3}e-h) and overlap with the transition temperature determined by the anomalies in d$M$\/d$T$. Taken together, the peaks in d$M$\/d$T$ and d$M$\/d$H$ can be used to construct the entire phase diagram for a given direction of the in-plane magnetic field, including for features independent of the 7 K transition, as is shown in Figs.~\\ref{fig4}c-f. We find that for both $H \\parallel a$ and $H \\parallel b$ the 10 K transition connects to $H_{\\text{4}}$ for the CVT sample and the 13 K transition connects to $H_{\\text{4\/5}}$ for the Bridgman sample. The structures at the 13 K transition for the CVT sample and the 10 K transition for the Bridgman sample are difficult to track below $\\sim$10 K but appear to connect to the $H_{\\text{4\/5}}$ features, although detailed behavior is complex and not fully traceable.\n\n\\begin{figure*}\n\\includegraphics[width=1\\textwidth, angle=0]{Figure4.jpg}\n\\caption{\\label{fig4} (a,b) Temperature dependence of $M\/H$ at $B=1\\textrm{T}$ for the CVT and Bridgman crystals, respectively. The six different magnetic field orientations, separated by $30^{\\circ}$ rotations in the honeycomb plane are sketched in the inset of (a). Kink-like features at the three phase transitions at 7 K, 10 K and 13 K are indicated with arrows. (c-f) Phase diagrams for the CVT and Bridgman crystals with magnetic field along the $a$ and $b$-axes, constructed from the extracted maxima in d$M$\/d$H$ (horizontal error bars) and d$M$\/d$T$ (vertical error bars) (see: Fig.~\\ref{fig3}). The sizes of the error bars represent the widths of the measured features. Solid black lines at low temperature trace the measured locations of minima in $\\kappa(H)$ (see: Fig.~\\ref{fig1}). Dashed lines are guides to the eye tracing possible phase transition lines. (g,h) $M\/H$ at $T=2\\textrm{K}$ as a function of in-plane-field angle $\\phi$ at different magnetic fields, showing the gradual suppression of two-fold modulation upon increasing field.}\n\\end{figure*}\n\nWe conclude that the dip structures in $\\kappa(H)$ at $H_{\\text{4}}$ and $H_{\\text{5}}$ indicated by the black lines at low temperatures in Figs.~\\ref{fig4}c-f coincide reasonably with the phase boundaries connecting to the 10 K and 13 K transitions in zero field, just like the connection between $H_{\\text{C1}}$ and $H_{\\text{C2}}$ and the main 7 K transition, and therefore originate from the high temperature magnetic transitions. The 13 K and 10 K transitions have been discussed to originate from secondary phases closely related to stacking faults sensitive to mechanical deformation \\cite{Cao2016,He2018,Mi2021,Banerjee2016}. The crystals used in this study have a soft, foil-like morphology and are easily deformed. Mechanical bending of the CVT crystal indeed enhances the structure at $H_{\\text{4}}$ and $H_{\\text{5}}$ in d$M$\/d$H$ as shown in Fig.~\\ref{fig1}a, which can be ascribed to the increase of volume of the secondary 10 K and 13 K phases by bending and support these phases as the origin of structure in $\\kappa(H)$ and $M(H)$.\n\n\\begin{figure*}\n\\includegraphics[width=0.8\\textwidth, angle=0]{Figure5.jpg}\n\\caption{\\label{fig5} Locations of minima in $\\kappa(H)$ of the CVT sample at $H_{\\text{C1}}$, $H_{\\text{C2}}$ and $H_{\\text{6}}$ on the $H$-$T$ plane. Background colors indicate regions of negative d$\\kappa(H)$\/d$H$ (pink) and positive d$\\kappa(H)$\/d$H$ (blue) which define the locations of the minima in $\\kappa(H)$ at their boundary. Field scales are expanded to emphasize the convex nature and inverse melting displayed by the transition lines of $H_{\\text{C2}}$ and $H_{\\text{6}}$.}\n\\end{figure*}\n\nIn Fig.~\\ref{fig1}a, we see a bending-induced enhancement of the structures at $H_{\\text{3}}$ and $H_{\\text{6}}$ similar to those at $H_{\\text{4}}$ and $H_{\\text{5}}$. It strongly suggests that the structures at $H_{\\text{3}}$ and $H_{\\text{6}}$ are also linked to the phase transitions of bending-induced secondary phases. As the features in d$M$\/d$H$ at $H_{\\text{3}}$ and $H_{\\text{6}}$ are weakly resolved at 2 K and rapidly disappear upon heating above 4 K (Figs.~\\ref{fig3}e,f), we cannot trace them to corresponding phase transitions at low fields. Instead, we find that additional evidence for the phase transition origin of all the high-field features above $H_{\\text{C1}}$ comes from the small but appreciable temperature dependence of positions of high-field dips in the magnetothermal conductivity. These are not expected for canonical (single frequency) quantum oscillations and support magnetic phase transitions as the origin of the structures in $\\kappa(H)$. The phase transition lines of $H_{\\text{C1}}$, $H_{\\text{C2}}$ and $H_{\\text{6}}$ are traced Fig.~\\ref{fig5}. We also observed temperature dependences for $H_{\\text{4}}$ and $H_{\\text{5}}$ (see: Figs.~\\ref{fig2}a and \\ref{fig2}b) but due to the flatness of the associated minimum in $\\kappa(H)$ the assignment of exact fields to those two features is less certain. The transition line of $H_{\\text{C2}}$ (7 K main phase) as defined by minima in $\\kappa(H)$ has a convex shape below 2.5 K with a field width of $\\sim$70 mT, which agrees very well with the shape of that phase transition determined by a recent measurement of the Gr\\\"uneisen parameter, where it was discussed as an \\textit{inverse melting} due to the higher entropy of the low-field ordered phase at very low temperatures \\cite{Bachus2021}. The similarity in behavior of the features at $H_{\\text{C2}}$ and $H_{\\text{6}}$ confirms that the latter is also a thermodynamic phase transition with an appreciable entropy effect even below 1 K. Suetsugu \\textit{et al.}\\cite{Suetsugu2022} recently reported a rapid increase in $\\kappa(H)$ around 11 T, where we observe the $H_{\\text{6}}$ anomaly, and discussed that it represents a first-order topological transition from a quantum spin liquid with a half-quantized thermal Hall effect to a topologically trivial phase. Although the scenario proposed by Suetsugu \\textit{et al.} is attractive and cannot be excluded by the present data, we note that the similarity between the features at $H_{\\text{6}}$ and $H_{\\text{C2}}$ in $\\kappa(H)$, d$M$\/d$H$ and in their inverse melting behavior means that the origin of $H_{\\text{6}}$ is likely a magnetic phase transition in a secondary phase related to stacking faults. \n\nFinally, we suggest that the feature at $H_{\\text{1}}$, and possibly also that at $H_{\\text{2}}$, might originate from pseudo-spin reorientations within the zig-zag ordered phase, which is supported by the evolution of the in-plane field angle ($\\phi$) dependence of magnetization in this field range. At 1 T (below $H_{\\text{1}}$) $M\/H$ in Figs.~\\ref{fig4}a and \\ref{fig4}b shows a clear two-fold anisotropy implying an easy-axis fixed along $a$, consistent with the reported $ac$-plane orientation of pseudospins in the zig-zag phase \\cite{Cao2016}. The observed easy-axis anisotropy along $a$ is consistent with the monoclinic structure of the Bridgman sample, but in the trigonal CVT sample the origin of the anisotropy is unclear, perhaps a weak easy-axis anisotropy could be induced by strain, or alternatively the low temperature crystal structure may have a lowered symmetry due to the reported structural phase transition around 150 K \\cite{He2018,Mi2021,Park2016}. The antiferromagnetic easy axis will rotate to perpendicular to the field direction when the field is strong enough to overcome the lattice pinning, resulting in a spin-flop transition. Such behavior was previously reported in studies of the magneto-optical dichroism \\cite{Wagner2022} and terahertz spectroscopy \\cite{Wu2018} at $\\sim$1.5 T, where we observe the anomaly at $H_{\\text{1}}$. Increasing the in-plane field from 1 T to 5 T suppresses the easy-axis anisotropy fully in the CVT crystal and substantially in the Bridgman crystal, as can be seen in Figs.~\\ref{fig4}g and \\ref{fig4}h. It is therefore likely that spin-flop or reorientation transitions have happened in between.\n\n\\section{Conclusion}\nIn summary, we show that by measuring the low temperature features in the thermal conductivity and magnetization of two $\\alpha$-RuCl$_3$ single crystals grown with different sample growth techniques, a consistent picture emerges to explain high-field features. Minima in $\\kappa(H)$ coincide with peaks in d$M$\/d$H$, tracing out the magnetic phase transitions of both the 7 K main phase and other likely coexistent secondary phases, including those which have transition temperatures of 10 K and 13 K. These phases are enhanced by sample manipulation and exist in both types of samples. It is interesting to consider that whilst we treat them as essentially independent of the 7 K main phase and coexistent with it, it is clear that they strongly affect the bulk thermal transport, and their existence was shown to suppress the thermal Hall effect rapidly at low temperatures \\cite{Bruin2022}. It would be highly desirable to isolate samples with only one of the 7 K, 10 K or 13 K phases present, but whether this is feasible in bulk crystals is still an open question.\n\n\\begin{acknowledgments}\nWe thank Y. Kasahara, Y. Matsuda, S. Suetsugu and H. Suzuki for insightful discussions and M. Dueller and K. Pflaum for technical assistance. The work done in Germany has been supported in part by the Alexander von Humboldt foundation. H.TAN and N.K. have been supported by JSPS KAKENHI Grant Number JP17H01142 and JP19K03711, respectively. H.TAK has been supported in part by JSPS KAKENHI, Grant Numbers JP22H01180 and JP17H01140. S.L. thanks the Science and Engineering Research Board (SERB), Government of India, for the award of a Ramanujan Fellowship (RJF\/2021\/000050).\n\\end{acknowledgments}\n\n\\section*{Author Declarations}\n\\subsection*{Conflict of Interest}\nThe authors have no conflicts to disclose.\n\n\\section*{Data Availability}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nOver the past several decades, astrophysicists have pushed the ``high-redshift frontier,\" where the most distant known galaxies and quasars reside, farther and farther back -- now encompassing star-forming galaxies at $z \\sim 7$ \\cite{bouwens08} and billion-solar mass black holes shining as quasars at $z \\sim 6.5$ \\cite{fan06}.  Two strategies can extend these efforts even farther over the next decade. The first, direct observations with large ground- and space-based near-infrared telescopes, will teach us about processes internal to these objects.  But a second method promises a beautiful complement to these direct probes:  low-frequency radio arrays using the 21 cm hyperfine line of neutral hydrogen to explore this era through the galaxies' indirect effects on the intergalactic medium (IGM).  Here we will describe the key astrophysical questions that this unique tool can address:  {\\bf What were the properties of high$-z$ galaxies? How did they affect the Universe around them?}\n\n\\section{Scientific Context} \\label{basic}\n\n\\begin{figure}\n\\includegraphics[width=6in]{steve.eps}\n\\caption{Maps ($\\approx 0.6^{\\circ}$ or $94$ comoving Mpc across) of ionized (white) and neutral (black) gas during reionization, showing the features visible in 21 cm maps.  From left to right, the panels assume that increasingly massive galaxies drive reionization (all have $x_{\\rm HI} \\approx 0.5$ and take $z \\approx 7.7$).  From \\cite{mcquinn07}.}\n\\label{fig:pics}\n\\end{figure}\n\nThe 21 cm brightness temperature of an IGM gas parcel at a redshift $z$, relative to the cosmic microwave background (CMB), is \\cite{madau97, furl06-review}\n\\begin{equation}\n\\delta T_b \\approx 25\\;x_{\\rm HI}(1+\\delta) \\, \\left( { 1+z \\over 10} \\right)^{1\/2}\\, \\left[1-\\frac{T_{\\rm CMB}(z)}{T_S}\\right] \\, \\left[ \\frac{H(z)\/(1+z)}{{\\rm d} v_\\parallel\/{\\rm d} r_\\parallel} \\right] \\mbox{ mK},\n\\label{eq:dtb}\n\\end{equation}\nwhere $x_{\\rm HI}$ is the neutral fraction, $\\delta = \\rho\/\\VEV{\\rho}-1$ is the fractional IGM overdensity in units of the mean, $T_{\\rm CMB}$ is the CMB temperature, $T_S$ is the spin (or excitation) temperature of this transition, $H(z)$ is the Hubble constant, and ${\\rm d} v_\\parallel\/{\\rm d} r_\\parallel$ is the line-of-sight velocity gradient.\n\nAll four of these factors contain unique astrophysical information.  The dependence on $\\delta$ traces the development of the cosmic web \\cite{scott90}, while the velocity factor sources line-of-sight ``redshift-space distortions\" that separate aspects of the cosmological and astrophysical signals \\cite{barkana05-vel}.  The other two factors depend strongly on the ambient radiation fields in the early Universe:  the ionizing background for $x_{\\rm HI}$ and a combination of the ultraviolet background (which mixes the 21 cm level populations through the Wouthuysen-Field effect \\cite{wouthuysen52, field58}) and the X-ray background (which heats the gas \\cite{chen04}) for $T_S$.  Fig.~\\ref{fig:pics} shows how $x_{\\rm HI}$ influences the 21 cm signal:  ionized regions appear as ``holes\" in the $\\sim 20 \\mbox{ mK}$ background of neutral gas.\n\n\\begin{SCfigure}\n\\centering\n\\includegraphics[scale=0.35]{f1.eps}\n\\caption{Fiducial histories of the sky-averaged 21 cm brightness temperature, $\\bar{\\delta T}_b$.  The solid blue curve uses a typical Population II star formation history, while the dashed red curve uses only very massive Population III stars.  Both fix reionization to end at $z_r \\approx 7$. From \\cite{furl06-glob}.}\n\\label{fig:global}\n\\end{SCfigure}\n\nFig.~\\ref{fig:global} shows two example scenarios for the sky-averaged 21 cm brightness temperature (see also \\cite{gnedin04}).  In the solid curve (a very typical model), we assume that high-$z$ star formation has similar properties to that at lower redshifts.  When the first galaxies form (here at $z \\sim 25$), the Wouthuysen-Field effect triggers strong 21 cm absorption \\cite{madau97}.  Somewhat later, X-rays -- produced by the first supernovae or black holes -- heat the entire IGM above $T_{\\rm CMB}$, turning this absorption into emission -- which fades and eventually vanishes as ionizing photons from the maturing galaxies destroy the intergalactic HI.\n\nThe red dashed curve shows a scenario in which very massive Population III stars dominate.  Their reduced efficiency for Wouthuysen-Field coupling (relative to their ionizing emissivity) decreases the strength of the absorption era \\cite{furl06-glob}, leading to a markedly different history.  The 21 cm line is clearly sensitive to the basic parameters of high-$z$ star formation.\n\nWe have so far illustrated the 21 cm signal through its all-sky background, but of course it actually fluctuates strongly as individual IGM regions grow through gravitational instability and are heated and ionized by luminous sources.  These fluctuations can ideally be imaged, but over the short term statistical measurements are likely to be more powerful  (see \\S \\ref{mile} below).  Fortunately, as we discuss next (also see Fig.~\\ref{fig:pk}), these statistical fluctuations contain an enormous amount of information about this era.\n\n\\section{Key Questions} \\label{reion}\n\nThe rich physics of the 21 cm line allows us to study the over-arching questions of {\\bf how the first galaxies evolved and affected the Universe around them}.  In particular, we can approach these questions from the following specific directions.  (Note that related cosmological questions, and more detailed IGM physics, are addressed in the companion white paper ``Cosmology from the Highly-Redshifted 21 cm Transition.\")\n\n\\underline{\\it When did reionization occur?}  One of the most dramatic events in the IGM's history was the reionization of intergalactic hydrogen, most likely via ultraviolet photons from star-forming galaxies.  This event marked the point when the small fraction of matter inside galaxies completely changed the landscape of the diffuse IGM gas (and hence future generations of galaxies), rendering it transparent to high-energy photons and heating it substantially.  The 21 cm background provides the \\emph{ideal} probe of reionization.  Its weak oscillator strength (in comparison to Ly$\\alpha$) allows us to penetrate even extremely high redshifts.  We can also image it across the entire sky -- instead of only rare, isolated Ly$\\alpha$  forest lines of sight.  Moreover, unlike the CMB, it is a spectral line measurement, and we can distinguish different redshift slices and study the full history of the ``dark ages\" -- extremely difficult even with a ``perfect\" CMB measurement \\cite{zald08-cmbpol}.  Finally, it directly samples the $ 95\\%$ (or more) of the baryons that reside in the IGM, requiring no difficult inferences about this material from the properties of the rare luminous galaxies. \n\nObservations have provided tantalizing hints about reionization, but even more unanswered questions \\citep{fan06-review}.  For example, CMB observations imply that reionization completed by $z \\sim 10$ \\citep{dunkley08}, but quasar absorption spectra suggest that it may have continued until $z \\sim 6$ \\cite{fan06}, albeit both with substantial uncertainties.  Both the sky-averaged $\\bar{\\delta T}_b$ and the 21 cm power spectrum yield much more precise measures of $\\bar{x}_{\\rm HI}(z)$ (see Fig.~\\ref{fig:global} and the left panel of Fig.~\\ref{fig:pk}).  The fluctuations briefly fade as galaxies ionize their dense surroundings in the first stage of reionization, then increase as large ionized bubbles form, finally fading again as the gas is ionized.  The principal goal of first-generation experiments (now under construction) is to constrain this time evolution \\cite{lidz08-constraint}.\n\n\\underline{\\it What sources were responsible for reionization?}  The 21 cm sky contains much more information about reionization, because the properties of the ionizing sources strongly affect its topology and have quantifiable impacts on the power spectrum.  Most fundamentally, stars produce well-defined ionized regions, while ``miniquasars\" (small accreting black holes) produce much more diffuse features \\cite{zaroubi05}. But the 21 cm background can even distinguish between different stellar reionization scenarios \\cite{lidz08-constraint}.  For example, more massive galaxies produce larger ionized regions \\cite{furl04-bub, mcquinn07}, as shown by the four panels in Fig.~\\ref{fig:pics}.  Photoheating (accompanied by a suppression of galaxy formation due to the increased IGM pressure) also slows the growth of bubbles \\cite{iliev07-selfreg}.  We can thus indirectly constrain the sources of ionization, including even extremely faint galaxies (which probably dominate the stellar mass budget), and gauge the importance of exotic, very massive Population III stars.  \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.35]{f2a.eps}\n\\includegraphics[scale=0.35]{f2b.eps}\n\\caption{\\emph{Left panel:}  Evolution of the spherically-averaged 21 cm power spectrum during reionization, from a numerical simulation of that process.  Here the fluctuations are presented in units of $\\bar{\\delta T}_b^2$ for a fully neutral universe (see eq.~\\ref{eq:dtb}); $\\VEV{x_i}$ is the ionized fraction during reionization.  From \\cite{lidz08-constraint}.  \\emph{Right panels:}   Evolution of the spherically-averaged power spectrum \\emph{before} reionization begins in earnest, including X-ray heating and Wouthuysen-Field coupling fluctuations.  From \\cite{pritchard07}.  In all panels, we use models with parameters similar to the solid curve in Fig.~\\ref{fig:global}.  }\n\\label{fig:pk}\n\\end{center}\n\\end{figure}\n\nOf course, exploring this era is a primary goal of many other forthcoming experiments.  Comparison with, e.g., galaxy surveys will provide a clear and complete picture connecting the small-scale physics of galaxies with landmark events encompassing larger scales and illuminate the relation between galaxies and the IGM, if appropriately wide and deep near-IR surveys can be done \\cite{lidz09}.  Targeted near-IR follow-up of individual HII regions identified in 21 cm will also identify how reionization proceeds on smaller scales.\n\n\\underline{\\it How did the cosmic web evolve?}  Once they leave their sources, ionizing photons must propagate through the IGM, where they can be absorbed by dense pockets of neutral gas around galaxies.  These affect the topology of the ionized bubbles during the tail end of reionization, when the pattern of well-defined ionized bubbles disintegrates into the ``cosmic web\" that we see in the Ly$\\alpha$ forest \\cite{furl05-rec, choudhury08}.  This process also imprints distinctive features on the 21 cm signal and allows us to measure directly the evolution of the IGM's structure, as well as the interactions of ionizing sources with the IGM.\n\n\\underline{\\it Where did the first generations of quasars form, and what were their properties?}  Bright quasars form enormous HII regions in the IGM ($> 40 \\mbox{ Mpc}$ across), visible in the 21 cm sky even after the quasar becomes dormant, because the recombination time is relatively long \\cite{furl08-fossil}.  Studying these regions in detail will constrain the quasar emission mechanism and their lifetimes, luminosity function, and redshift evolution \\cite{wyithe05-qso}.\n\n\\underline{\\it When did the first galaxies form, and what were their properties?}  As mysterious as they are, the galaxies responsible for reionizing the Universe were probably relatively mature.  The 21 cm transition also probes the very first structures to appear in the Universe -- an era most likely inaccessible even to \\emph{JWST}.  The right panel of Fig.~\\ref{fig:pk} shows the 21 cm power spectrum during this early era in one typical model.  The first stars produced a ``soft\" UV background (below the HI ionization edge) that triggered an absorption feature in the 21 cm background (at $z \\approx 20$ in Fig.~\\ref{fig:global}).  The background is strongest near these galaxies, so the absorption fluctuations trace their locations, masses, UV spectra, luminosity function and redshift evolution \\cite{barkana05-ts}.  The $z=19$ curve in Fig.~\\ref{fig:pk} illustrates this era; the fluctuation amplitude can be \\emph{stronger} than that during reionization.\n\n\\underline{\\it When did the first black holes form, and what were their properties?}  Somewhat later, X-rays from the first black holes (and supernovae) heated the Universe (at $z \\approx 15$ in Fig.~\\ref{fig:global}).  Again, the heating was concentrated around the galaxies hosting these sources, so the observable fluctuations trace their redshift evolution, abundance, masses, and spectra \\cite{pritchard07}.  The $z=17$ and 15 curves in Fig.~\\ref{fig:pk} show that the sharp contrast between hot and cold regions produces distinctive troughs in the power spectrum, as well as especially large fluctuation amplitudes.  Thus we can securely identify the era in which the first X-ray sources heated the IGM and then follow their importance throughout the reionization era \\cite{oh01, ricotti05}.\n\n\\underline{\\it How does radiative feedback affect high-z galaxy formation?}  Both the UV and X-ray backgrounds are crucial feedback mechanisms in early galaxy formation.  For example, they strongly affect H$_2$, the primary coolant in primordial gas, and therefore regulate both the possible locations of star formation and its end products \\cite{haiman00, ahn08}.  Moreover, X-ray heating also affects structure formation and clumping in the IGM \\cite{oh03}.  The 21 cm background is the only method to measure  these mechanisms \\emph{directly}.  \n\n\\section{Milestones} \\label{mile}\n\nThe ultimate goal of studying the 21 cm background is to make detailed maps of the IGM throughout the ``dark ages\" and reionization, as in Fig.~\\ref{fig:pics}.  The top axis of Fig.~\\ref{fig:global} shows the observed frequency range for these measurements: well within the low-frequency radio regime.  Unfortunately, this is an extremely challenging band, because of terrestrial interference, ionospheric refraction, and (especially) other astrophysical sources (see \\cite{furl06-review}).  In particular, the polarized Galactic synchrotron foreground has $T_{\\rm sky} \\sim 180 (\\nu\/180 \\mbox{ MHz})^{-2.6} \\mbox{ K}$, at least four orders of magnitude larger than the signal.  For an interferometer, the noise per resolution element (with an angular diameter $\\Delta \\theta$ and spanning a bandwidth $\\Delta \\nu$) is then \\cite{furl06-review}\n\\begin{equation}\n\\Delta T_{\\rm noise} \\sim 20  \\mbox{ mK} \\ \\left( \\frac{10^4 \\mbox{ m$^2$}}{A_{\\rm eff}} \\right) \\, \\left( \\frac{10'}{\\Delta \\theta} \\right)^2 \\, \\left( \\frac{1+z}{10} \\right)^{4.6} \\,  \\left( \\frac{{\\rm MHz}}{\\Delta \\nu} \\, \\frac{100 \\mbox{ hr}}{t_{\\rm int}} \\right)^{1\/2},\n\\label{eq:if-sens}\n\\end{equation}\nwhere $A_{\\rm eff}$ is the effective collecting area and $t_{\\rm int}$ is the integration time.  These angular and frequency scales correspond to $\\sim 30 \\mbox{ Mpc}$.  Here we outline the steps required to explore this era in detail, given the challenges implicit in this huge noise.\n\n{\\bf The all-sky signal:}  The global background illustrated in Fig.~\\ref{fig:global} contains an extraordinary amount of information about the gross properties of galaxies.  These measurements can easily beat down the foreground noise with only a single dipole, so they may provide our first constraints at very high redshifts.  The challenge lies in calibration that is precise enough to extract the signal from instrumental artifacts and the bright foregrounds.  The EDGES experiment \\cite{bowman08} has already set upper limits and hopes to measure this signal to $z \\sim 20$ over the next decade.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=0.35]{f3a.eps}\n\\includegraphics[scale=0.35]{f3b.eps}\n\\caption{\\emph{Left panels:}  Sensitivity of some fiducial arrays to the spherically-averaged power spectrum at $z=10$ and $z=15$, (conservatively) assuming a fully neutral emitting IGM.  The solid and dashed curves take $A_{\\rm eff}=0.1$ and 1 km$^2$, respectively; in both cases we take  4000 hr of observing split over four fields, a total bandwidth $B=8 \\mbox{ MHz}$, $T_{\\rm sky} = (471,\\,1247) \\mbox{ K}$ at $z=(10,15)$, and $N=5000$ stations centered on a filled core and with an envelope out to $R_{\\rm max}=3$~km.  The black squares show the location of the independent $k$ bins in each measurement.  The dotted curve shows the projected MWA sensitivity at $z=10$.  The vertical dotted line corresponds to the bandwidth; modes with smaller wavenumbers are compromised by foreground removal.   \\emph{Right panels:} Sensitivity of the same arrays to angular components of the power spectrum at $z=10$.}\n\\label{fig:sensitivity}\n\\end{center}\n\\end{figure}\n\n{\\bf First-Generation Arrays:}  The arrays now operating or under construction, including the Murchison Widefield Array (MWA), the Precision Array to Probe the Epoch of Reionization (PAPER), LOFAR, the 21 Centimeter Array, and the Giant Metrewave Radio Telescope, have $A_{\\rm eff} \\sim 10^4 \\mbox{ m$^2$}$ and so are limited to imaging only the most extreme ionized regions (such as those surrounding bright quasars).  Nevertheless,  these arrays have sufficiently large fields of view ($> 400^{\\circ2}$) to make reasonably good statistical measurements \\cite{mcquinn06-param, bowman07}.  Fig.~\\ref{fig:sensitivity} shows the projected errors for the MWA at $z=10$.  It will be able to detect fluctuations over a limited spatial dynamic range (thanks to foreground removal and thermal noise) and only at $z < 12$, constraining the timing of reionization and some of the source physics.\n\n{\\bf Second-Generation Arrays:}  Fig.~\\ref{fig:sensitivity} shows that larger telescopes, with $A_{\\rm eff} \\sim 10^5 \\mbox{ m$^2$}$ (and large fields of view), will clearly be needed for precise measurements, and especially to identify the distinctive features in the power spectrum (such as the flattening at $k > 1 \\mbox{ Mpc$^{-1}$}$ during reionization and the troughs during X-ray heating; see Fig.~\\ref{fig:pk}).  Instruments in this class will also be able to measure some more advanced statistics.  This is illustrated in the right panels in Fig.~\\ref{fig:sensitivity}, which consider the components of the redshift-space distortions induced by velocity fluctuations.  These are extremely useful for breaking degeneracies in the signal \\cite{barkana05-vel} (for more information, see the companion white paper ``Cosmology from the Highly-Redshifted 21 cm Line\") but lie beyond the reach of first-generation experiments.\n\n{\\bf Imaging Arrays:}  At $A_{\\rm eff} \\sim 10^6 \\mbox{ m$^2$}$, imaging on moderate scales becomes possible, and statistical constraints become exquisite even at high redshifts  (provided that the large field of view, not strictly necessary for imaging, is maintained; see Fig.~\\ref{fig:sensitivity}).  \n\nPlans for these later generations will evolve as we learn more about ``dark age\" physics  and the experimental challenges ahead; for example, the Long Wavelength Array and other lower-frequency instruments will study the ionospheric calibration required to explore the high-$z$ regime ($z > 12$, or $\\nu < 110 \\mbox{ MHz}$) and help determine the relative utility of a terrestrial Square Kilometer Array or a far-side Lunar Radio Array.  At the same time, we must explore whether innovative new telescope designs more closely aligned with the observables, such as the FFT Telescope \\cite{tegmark09}, can provide cost-effective improvements.\n\nThis roadmap, with accompanying efforts to improve theoretical modeling of the first galaxies and data analysis techniques (such as specialized statistical measures) will ideally position the community to explore the major science questions of high-$z$ galaxy formation.\n\n\\newpage\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\t\tLet $\\mathbb{D}$ be the open unit disc, and let $\\mathbb{T} \\colonequals  \\partial \\mathbb{D}$  be the unit circle. The Hardy space $H^2$ consists  of analytic functions $f$ on $\\mathbb{D}$ such that \n\t\t\\[\n\t\t\\| f\\| _{H^2} \\colonequals\\left(\\sup_{0<r<1}\\int_0^{2\\pi} |f(re^{i\\theta})|^2  \\dfrac{d\\theta}{2\\pi}\\right)^{1\/2}<\\infty.\n\t\t\\]\n\t\tFor an analytic function $g$ on $\\mathbb{D}$, we define the linear transformation \n\t\n\t\t\\[\n\t\tI_gf(z) =\\displaystyle \\int _0^z f(\\zeta)g'(\\zeta)d\\zeta,\\quad f\\in H^2.\n\t\t\\]\n\t\tThese operators have been introduced by Pommerenke \\cite{Pom} who proved that $I_g$ is bounded on $H^2$ if and only if $g$ is of bounded mean oscillation, that is, $g \\in BMOA$. The compactness and membership to Schatten classes have been studied by Aleman and Siskakis in \\cite{AS1}. They proved that $I_g$ is compact if and only if $g$ is in the space of analytic functions of vanishing mean oscillation  $VMOA$, and that $I_g $ belongs to the Schatten class $\\mathcal{S}_p(H^2)$, for $p>1$, if and only if $g$ is in the Besov space $B_p$ given by\n\t\t\\[\n\t\tB_p=\\left  \\{ g \\in H^2:\\ \\displaystyle \\int _\\mathbb{D}\\left ( ( 1-|z|^2)|g'(z)| \\right )^p\\frac{dm(z)}{(1-|z|^2)^2} \\right \\},\n\t\t\\]\n\t\twhere $m$ denotes the normalized Lebesgue measure on $\\mathbb{D}$. They also proved that if ${I_g} \\in \\mathcal{S}_1$ then $g'=0$.\\\\\n\t\t\n\t\t\n\t\tThe weighted Bergman space ${\\mathcal{A}}^2_\\alpha$, with  $\\alpha>-1$, is the space of analytic functions $f$ on $\\mathbb{D}$ for which\n\t\t\\[\n\t\t\\|f\\|_\\alpha \\colonequals \\left((\\alpha+1)\\displaystyle \\int_\\mathbb{D} |f(z)|^2(1-|z|^2)^\\alpha dm(z)\\right)^{1\/2} <\\infty.\n\t\t\\]\n\t\tIn \\cite{AS2}, Aleman and  Siskakis  considered  the operator $I_g$ acting on weighted Bergman spaces. They proved that $I_g$  is bounded on $\\mathcal{A}^2_\\alpha$ (respectively compact) if and only if $g$ is in the Bloch space $\\mathcal{B}$ (respectively in the little Bloch space $\\mathcal{B}_0$), and that  $I_g$ is in the class $\\mathcal{S}_p(\\mathcal{A}^2_\\alpha)$, for $p > 1$, if and only if $g$ is in the space $B_p$. For further information about  \n\t\tthe spaces used in this paper, we refer the reader to the monograph  \\cite{Zhu}.\\\\\n\t\t\n\t\tThis paper is devoted to the study of the asymptotic behavior of the singular values of compact integration operators acting on  $H^2$ or on  ${\\mathcal{A}}^2_\\alpha$.  \n\t\n\t\tBefore stating our main results, we give some basic notations.\n\t\t\\\\\n\t\t\\\\\n\t\tThe Carleson window (or box) associated with an arc $I\\subset \\mathbb{T}$ is defined by\n\t\t\\[\n\t\tW(I)=\\left\\lbrace z\\in\\mathbb{D}:1-|z|\\leqslant | I | \/2\\pi,\\ z\/|z|\\in I\\right\\rbrace,\n\t\t\\]\n\t\twhere $|I|$ denotes the length of $I$. \n\t\tLet $R(I)$ denote the  inner half of $W(I)$ which is given by \n\t\t\\[\n\t\tR(I)= \\left\\{ z\\in \\mathbb{D}:\\ |I|\/4\\pi <1-|z|\\leqslant  | I | \/2\\pi ,z\/|z| \\in I \\right\\}.\n\t\t\\]\n\t\tThe family  $\\left\\{ R(I_{n,k})\\right\\} _{n,k}$ constitutes  a pairwise disjoint covering of $\\mathbb{D}$, where\n\t\t\\[\n\t\tI_{n,k} \\colonequals \\lbrace e^{i\\theta}:2\\pi k\/2^n\\leqslant \\theta <2\\pi(k+1)\/2^n\\rbrace,\\quad n\\geqslant 1\\ \\mbox{and}\\ 1\\leqslant k\\leqslant 2^n,\n\t\t\\]\n\t\tare the dyadic arcs.\\\\ \\\\\n\t\tThe {Bloch} space $\\mathcal{B}$ (respectively the little {Bloch} space $\\mathcal{B}_0$) consists of analytic functions $g$ on $\\mathbb{D}$ such that\n\t\n\t\t\\[\n\t\t\\displaystyle \\sup _{z\\in \\mathbb{D}}(1-|z|^2)|g'(z)| <\\infty \\quad (\\text{respectively } \\displaystyle \\lim _{|z| \\to 1}(1-|z|^2)|g'(z)|=0).\n\t\t\\]\n\t\n\t\n\t\n\t\n\t\n\t\t{Let $dm_g(z) = |g'(z)|^2 (1-|z|^2)^2dm(z)$}. It is established in \\cite{AS2} that $g$ is in the space $\\mathcal{B}$ if and only if\n\t\t\\[ \n\t\t\\frac{m_g(R(I_{n,k}))}{{m(R(I_{n,k}))}}=O(1),\n\t\t\\]\n\t\tand that $g$ is in the space $\\mathcal{B}_0$ if and only if \n\t\t\\begin{equation}\\label{00}\n\t\t\\displaystyle \\lim_{n\\to \\infty} \\sup _{1\\leqslant k \\leqslant 2^n}\\frac{m_g(R(I_{n,k}))}{{m(R(I_{n,k}))}}=0.\n\t\t\\end{equation}\n\t\tHence, $I_g$ is compact on ${\\mathcal{A}}^2_\\alpha$ if and only if  $g$ satisfies assumption (\\ref{00}). In this case, {let $\\left(I_{n}\\right)_{n\\geqslant 1}$ be an enumeration of $\\left(I_{n,k}\\right)_{n,k}$ so that the sequence \n\t\t\t\\[a_n(m_g) \\colonequals \\frac{m_g(R(I_{n}))}{{m(R(I_{n}))}}\n\t\t\t\\] \n\t\t\tis  nonincreasing.}\n\t\t\\\\\n\t\t\\\\\n\t\tThroughout this paper, the notation $A\\lesssim B$ stands for $A\\leqslant cB$ with $c$ is a certain positive constant, and the notation $A\\asymp B$ is used if both $A\\lesssim B$ and $B\\lesssim A$ hold.\n\t\t\\\\\n\t\t\n\t\tFirst, we state the main result of this paper in the case of  Bergman spaces.\n\t\n\t\t\\begin{theo}\\label{Bergman}\n\t\t\tLet $g\\in \\mathcal{B}_0$. There exists an absolute integer $p$ such that\n\t\t\t\\begin{eqnarray}\n\t\t\t\\sqrt{a_{pn}(m_g)} \\lesssim s_{n}(I_g)\\lesssim \\sqrt{a_{[\\frac{n}{p}]}(m_g)},\\quad n\\geqslant 1,\n\t\t\t\\end{eqnarray}\n\t\t\twhere  $[\\frac{n}{p}]$ is the integer part of $\\frac{n}{p}$.\n\t\t\\end{theo}\n\t\t\n\t\tAn immediate consequence of Theorem \\ref{Bergman} is given as follows.\n\t\n\t\t\\begin{cor}\\label{Cor_Bergman}\n\t\t\tLet $g\\in \\mathcal{B}_0$. If $\\eta=(\\eta(n))_n$ is a nonincreasing sequence of positive numbers such that  $\\eta(2n)\\asymp \\eta(n)$, then\n\t\t\t\\[\n\t\t\ts_n^2(I_g) \\asymp \\eta(n) \\Longleftrightarrow a_n(m_g) \\asymp \\eta(n).\n\t\t\t\\]\n\t\t\\end{cor}\n\t\t\n\t\tThe situation in the Hardy space is more subtle. Indeed, the boundedness of $I_g$ on $H^2$ is equivalent to the membership of the symbol $g$ to $ BMOA$, that is,\n\t\t\\[\n\t\t\\| g\\|_{BMOA} \\colonequals |g(0)| + \\sup \\{\\| g\\circ\\varphi _w\\|_{H^2}:\\ w\\in \\mathbb{D}\\}<\\infty,\n\t\t\\]\n\t\twhere $\\varphi _w(z) {\\colonequals} \\frac{w-z}{1-\\bar{w}z}$ is the M\u00f6bius automorphism  of the unit  disc. Recall also that $I_g$ is compact on $H^2$ if and only if $g\\in VMOA$, that is,\n\t\t\\[\n\t\t\\displaystyle \\lim_{|w| \\to 1^-}\\| g\\circ\\varphi _w\\|_{H^2}=0.\n\t\t\\]\n\t\tThe membership of $g$ to $BMOA$ or to $VMOA$ can be expressed using Carleson windows and the measure $d\\nu_g(z) \\colonequals |g'(z)|^2 (1-|z|^2)dm(z)$. Namely, it is proved in \\cite{AS1} that\n\t\t\\[\n\t\tg \\in BMOA\\quad \\iff \\frac{\\nu_g(W(I_{n,k}))}{ |I_{n,k} |}=O(1)\n\t\t\\]\n\t\tand \n\t\t\\[\n\t\tg \\in VMOA \\quad  \\iff \\displaystyle \\lim_{n\\to \\infty} \\sup _{1\\leqslant k \\leqslant 2^n}\\frac{\\nu_g(W(I_{n,k}))}{| I_{n,k}|}=0.\n\t\t\\]\n\t\t\n\t\n\t\n\t\t\\noindent\tFor $g \\in VMOA$, denote by \n\t\t{\\[\n\t\t\t(\\tau _n(\\nu_g))_{n\\geqslant 1}\\colonequals\\left (\\frac{\\nu_g(W(I_{n}))}{{| I_n| }}  \\right )_{n\\geqslant 1}\n\t\t\t\\]\n\t\t\tthe nonincreasing rearrangement of  the sequence $\\left (\\frac{\\nu_g(W(I_{n,k}))}{{| I_{n,k}| }}  \\right )_{n,k}$.}\n\t\n\t\t\\\\\n\t\t\n\t\tThe main result in the case of the Hardy space is the following theorem.\t\n\t\t\n\t\n\t\t\n\t\t\\begin{theo}\\label{Hardy}\n\t\t\tLet $g\\in VMOA$. Then\n\t\t\t\n\t\t\t\\begin{enumerate}[label=$(\\roman*)$,leftmargin=* ,parsep=0cm,itemsep=0cm,topsep=0cm]\n\t\t\t\t\\item\\label{i} There exists an absolute integer $p\\geqslant 1$ such that \n\t\t\t\t\\[\n\t\t\t\ts_{p n}(I_g) \\lesssim  \\sqrt{\\tau_n (\\nu_g)}, \\quad  n\\geqslant 1.\n\t\t\t\t\\]\n\t\t\t\t\\item \\label{ii} There exists  an absolute constant $B>0$ such that\n\t\t\t\t\\[\n\t\t\t\t\\frac{1}{B} \\displaystyle \\sum _{j=1}^n \n\t\t\t\t{\\tau_j(\\nu_g) } \\leqslant  \\displaystyle \\sum _{j=1}^n s_j^2(I_g),\\quad n\\geqslant 1.\n\t\t\t\t\\]\n\t\t\t\\end{enumerate}\n\t\t\\end{theo}\n\t\t\n\t\tUnder some regularity conditions, we obtain a description of the asymptotic behavior of the singular values of $I_g$.\n\t\n\t\t\\\\ \\\\\n\t\t\\noindent Let $\\gamma >0$, a positive nonincreasing sequence $(u_n)_n$ is said to belong to ${\\mathcal{R}}_\\gamma$ if there exists a nonincreasing sequence $(x_n)_n$ such that $n^\\gamma x_n$ increases to infinity and $u_n \\asymp x_n$. If, in addition, there exists $\\alpha \\in (0,\\gamma)$ such that $n^\\alpha x_n$ decreases, $(u_n)_n$ is said to belong to ${\\mathcal{R}} _{\\gamma, \\alpha}$.\n\t\n\t\t\n\t\t\\begin{theo}\\label{B}\n\t\t\tLet $g\\in VMOA$. The following statements hold. \\\\\n\t\t\t\\begin{enumerate}[label=$(\\roman*)$,leftmargin=* ,parsep=0cm,itemsep=0cm,topsep=0cm]\n\t\t\t\t\\item \\label{2i}  Let $\\gamma \\in (0,1\/2)$. If  {$(s_n(I_g))_n \\in {\\mathcal{R}}_\\gamma$ or $(\\tau _n(\\nu_g))_n \\in {\\mathcal{R}}_{2\\gamma}$}. Then\n\t\t\t\t\\[\n\t\t\t\ts_n(I_g) \\asymp\\sqrt{ \\tau _n(\\nu_g)}.\n\t\t\t\t\\]\n\t\t\t\t\\item\\label{3i} Let $\\gamma \\in (0,1)$ and let $\\alpha \\in (0,\\gamma)$. If  {$(s_n(I_g))_n \\in {\\mathcal{R}}_{\\gamma,\\alpha}$ or $(\\tau _n(\\nu_g))_n \\in {\\mathcal{R}}_{2\\gamma,2\\alpha}$}. Then\n\t\t\t\t\\[\n\t\t\t\ts_n(I_g) \\asymp\\sqrt{ \\tau _n(\\nu_g)}.\n\t\t\t\t\\]\n\t\t\t\\end{enumerate}\n\t\t\\end{theo}\n\t\t\\noindent The following corollary is a direct consequence of Theorem \\ref{B}.\n\t\t\\begin{cor}  Let $g \\in VMOA$. Let $\\alpha \\in [0,1[$ and  $\\beta \\geqslant 0$ such that $ \\alpha >0$ or $\\beta >0$ . We have\n\t\t\t\\[\n\t\t\ts_n(I_g) \\asymp \\frac{1}{n^\\alpha \\log^\\beta (n+1)} \\quad \\iff \\tau_n(\\nu_g) \\asymp \\frac{1}{n^{2\\alpha} \\log^{2\\beta} (n+1)}.\n\t\t\t\\]\n\t\t\\end{cor}\n\t\t\n\t\n\t\t\n\t\n\t\t\n\t\n\t\tThe paper is organized as follows. In Section \\ref{sec2}, we prove Theorem \\ref{Bergman}. Section \\ref{sec3} is devoted to the proofs of the main  results in the Hardy space.\n\t\n\t\t\\section{Bergman spaces}\n\t\t\\label{sec2}\n\t\tWe start this section by recalling the definitions of sampling and interpolating sets for Bergman spaces which play an important role in this paper.\\\\\n\t\t\n\t\tLet $Z = (z_n)_{n\\geqslant 1}$ be a sequence of points in $\\mathbb{D}$.\\\\\n\t\t\\begin{enumerate}[label=$\\bullet$]\n\t\t\t\\item  $Z$  is said to be separated if $\\displaystyle \\inf_{i\\neq j}  \\rho(z_i,z_j) > 0$, where $\\rho(z,w) \\colonequals   \\left|\\frac{z-w}{1-\\bar{z}w}\\right|$\n\t\t\tis the pseudohyperbolic metric on $\\mathbb{D}$.\n\t\t\t\n\t\t\t\\item  $Z$ is called a sampling set for ${\\mathcal{A}}^2_\\alpha$ if \n\t\t\t\\[\n\t\t\t\\|f\\|^2_{{\\mathcal{A}}^2_\\alpha} \\asymp \\sum_{n\\geqslant 1} |f(z_n)|^2(1-|z_n|^2)^{2+\\alpha},\\quad f \\in {\\mathcal{A}}^2_\\alpha.\n\t\t\t\\]\n\t\t\t\\item  $Z$  is called an interpolating set for ${\\mathcal{A}}^2_\\alpha$ if for every sequence $(\\lambda_n)_{n \\geqslant 1}$ satisfying \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum_{n\\geqslant 1} |\\lambda_n|^2(1-|z_n|^2)^{2+\\alpha}<\\infty,\n\t\t\t\\]\n\t\t\tthere exists $f$ in ${\\mathcal{A}}^2_\\alpha$ such that $f(z_n)=\\lambda_n$ for all $n\\geqslant 1.$ \n\t\t\\end{enumerate}\n\t\tThe closed graph theorem shows that if $Z$ is an\n\t\tinterpolating set for ${\\mathcal{A}}^2_\\alpha$, then there exists a constant $C$ such that the interpolation problem can be\n\t\tsolved by a function satisfying\n\t\t\\[\n\t\t\\|f\\|_{{\\mathcal{A}}^2_\\alpha} \\leqslant C \\left(\\sum_{n\\geqslant 1} |\\lambda_n|^2(1-|z_n|^2)^{2+\\alpha}\\right)^{1\/2}.\n\t\t\\]\n\t\tThe smallest such constant $C$  is called the constant of interpolation and we denoted it by $M_\\alpha(Z)$. We draw attention to the fact that every interpolating set for ${\\mathcal{A}}^2_\\alpha$ is separated. Interpolating and sampling sets  for Bergman spaces have been studied by many authors, we refer the reader to \\cite{Seip1,Seip2} for more details.\\\\\n\t\t\n\t\n\t\tLet $\\mu$ be a positive Borel measure on $\\mathbb{D}$. Define the operator $T_\\mu$ by\n\t\t\\[ \n\t\tT_\\mu f(z)=\\int_\\mathbb{D} f(w)K^\\alpha(z,w)(1-|w|^2)^\\alpha d\\mu(w),\\  f\\in {\\mathcal{A}}^2_\\alpha,\n\t\t\\]\n\t\twhere $K^\\alpha(z,w)=(1-z\\bar{w})^{-2-\\alpha}$ is the reproducing kernel of ${\\mathcal{A}}^2_\\alpha.$ A simple computation yields\n\t\t\\[\n\t\t\\langle T_\\mu f , f\\rangle_{{\\mathcal{A}}^2_\\alpha} = \\displaystyle \\int _\\mathbb{D}|f(z)|^2 (1-|z|^2)^\\alpha d\\mu (z).\n\t\t\\]\n\t\t\n\t\t\\noindent These operators are called Toeplitz operators on Bergman spaces. Their boundedness was considered by Hasting \\cite{Has}. Later, Luecking obtained necessary and sufficient condition on $\\mu$ for which  $T_\\mu$ belongs to a Schatten class $\\mathcal{S}_p({\\mathcal{A}}^2_\\alpha)$, with $p>0$ (see \\cite{Lue1}). Recently, the asymptotic behavior of the singular values of such operators has been studied on a large class of weighted Bergman spaces (see \\cite{APP, EE}).\\\\ \\\\\n\t\n\t\tIn the specific case of discrete measures, one can obtain explicit lower and upper bounds for the singular values of the associated Toeplitz operators. We have the following two lemmas.\n\t\t\\begin{lem}\\label{separation}\n\t\t\tLet $Z=(z_n)_{n\\geqslant 1}$ be a separated sequence and let $(c_n)_{n\\geqslant 1}$ be a sequence of positive numbers such that  $\\left(c_n(1-|z_n|^2)^{-2}\\right)_{n\\geqslant 1}$  decreases to zero. For $\\mu= \\displaystyle \\sum_{n\\geqslant 1} c_n\\delta_{z_n}$, the operator $T_\\mu$ is compact on ${\\mathcal{A}}^2_\\alpha$ and \n\t\t\t\\[\n\t\t\ts_n(T_\\mu) \\lesssim \\dfrac{c_n}{(1-|z_n|^2)^2}.\n\t\t\t\\]\n\t\t\\end{lem}\n\t\t\n\t\t\\begin{proof}\n\t\t\n\t\t\tFirst, recall \\cite{GGK} that  for a compact operator $T$ on a Hilbert space $H$,\n\t\t\t\\[\n\t\t\ts_n(T)  =  \\inf\\left\\{\\|T-R\\| : R \\colon H\\to H\\ \\text{is of rank}\\ <n \\right\\}.\n\t\t\t\\]\n\t\t\tFor  $\\mu_n = \\displaystyle \\sum_{j=1}^{n-1} c_j\\delta_{z_j}$, the associated operator $T_{\\mu_n}$ is of rank $n-1$. Thus, it suffices to show that \n\t\t\t$$\\|T_\\mu-T_{\\mu_n}\\| \\lesssim \\dfrac{c_n}{(1-|z_n|^2)^2}.$$\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\tLet $f \\in \\mathcal{A}^2_\\alpha$. We have\n\t\t\t\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\langle \\left(T_\\mu-T_{\\mu_n}\\right)f,f\\rangle_{{\\mathcal{A}}^2_\\alpha} & = & \\displaystyle \\sum_{j\\geqslant n} c_j (1-|z_j|^2)^\\alpha|f(z_j)|^2\\\\ \n\t\t\t\t& \\leqslant &  \\frac{c_n}{(1-|z_n|^2)^2} \\displaystyle \\sum_{j\\geqslant n} (1-|z_j|^2)^{2+\\alpha} |f(z_j)|^2.\n\t\t\t\\end{eqnarray*}\n\t\t\tMoreover, since the sequence $Z$ is separated, there exists a small constant $\\delta>0$ such that the discs ${\\mathcal{D}}_\\delta(z_j)=\\{z\\in \\mathbb{D} : |z-z_j|<\\delta(1-|z_j|)\\}$ are pairwise disjoint. \n\t\t\tBy subharmonicity, we get\n\t\t\t\\[\n\t\t\t|f(z_j)|^2(1-|z_j|^2)^{2+\\alpha} \\lesssim \\int_{{\\mathcal{D}}_\\delta(z_j)} |f(z)|^2(1-|z|^2)^\\alpha\\ dm(z).\n\t\t\t\\]\n\t\t\tTherefore\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\langle \\left(T_\\mu-T_{\\mu_n}\\right)f,f\\rangle_{{\\mathcal{A}}^2_\\alpha} & \\lesssim & \\frac{c_n}{(1-|z_n|^2)^2} \\sum_{j\\geqslant n} \\displaystyle\\int_{{\\mathcal{D}}_\\delta(z_j)} |f(z)|^2(1-|z|^2)^\\alpha\\ dm(z)\\\\\n\t\t\t\t& \\lesssim & \\frac{c_n}{(1-|z_n|^2)^2}\\int_{\\mathbb{D}} |f(z)|^2(1-|z|^2)^\\alpha\\ dm(z)\\\\\n\t\t\t\t& = &  \\frac{c_n}{(1-|z_n|^2)^2}\\|f\\| ^2_\\alpha.\n\t\t\t\\end{eqnarray*}\n\t\t\tSince the operator $T_\\mu-T_{\\mu_n}$ is positive, we obtain \n\t\t\t\\[\n\t\t\t\\| T_\\mu -T_{\\mu _n}\\| \\lesssim  \\frac{c_n}{(1-|z_n|^2)^2}.\n\t\t\t\\]\n\t\t\n\t\t\\end{proof}\n\t\t\n\t\t\\begin{lem}\\label{interpolation}\n\t\t\tLet $Z=(z_n)_{n\\geqslant 1}$ be an interpolating set for ${\\mathcal{A}}^2_\\alpha$ and let $(c_n)_{n\\geqslant 1}$ be a sequence of positive numbers such that  $\\left(c_n(1-|z_n|^2)^{-2}\\right)_{n\\geqslant 1}$  decreases to zero. For $\\mu= \\displaystyle \\sum_{n\\geqslant 1} c_n\\delta_{z_n}$, the operator $T_\\mu$ is compact on ${\\mathcal{A}}^2_\\alpha$ and  \n\t\t\t\\[\n\t\t\ts_n(T_\\mu) \\asymp \\frac{c_n}{(1-|z_n|^2)^2}.\n\t\t\t\\]\n\t\t\\end{lem}\n\t\t\n\t\t\\begin{proof}\n\t\t\tThe upper bound result follows from Lemma \\ref{separation}.  To obtain the lower bound, we use an alternative characterization {\\cite[p. 49]{GGK}} of the singular values:\n\t\t\n\t\t\t\\[\n\t\t\ts_n(T_\\mu) =  \\underset{E,\\dim E=n}{\\sup}\\ \\left(\\underset{f\\in E\\setminus\\{0\\}}{\\inf}\\ \\frac{\\|T_\\mu f\\|_{{\\mathcal{A}}^2_\\alpha}}{\\|f\\|_{{\\mathcal{A}}^2_\\alpha}} \\right).\n\t\t\t\\] \n\t\t\tSet\n\t\t\t${\\bf \\mathcal{K}}_Z = span\\{k_{z_j}^\\alpha : j\\geqslant 1\\}$, where $k_{z_j}^\\alpha$ denotes the normalized reproducing kernel of ${\\mathcal{A}}^2_\\alpha$. We choose as subspace $E$ the orthogonal complement in ${\\bf \\mathcal{K}}_Z$ of the linear span of  $\\{k_{z_j}^\\alpha : j > n\\}$. Since $Z$ is an interpolating set for ${\\mathcal{A}}^2_\\alpha$, the family $\\{k_{z_j}^\\alpha : j\\geqslant 1\\}$ is a Riesz basis of ${\\bf \\mathcal{K}}_Z$ and then the vector space $E$ is of  dimension $n$.  \n\t\t\n\t\t\tLet $f\\in E $, we have\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\|f\\|_{{\\mathcal{A}}^2_\\alpha}^2\\ \\leqslant\\ [M_\\alpha(Z)]^{2}\\sum_{j\\geqslant 1} |\\langle f,k_{z_j}^\\alpha\\rangle_{{\\mathcal{A}}^2_\\alpha}|^2\\ =\\ [M_\\alpha(Z)]^{2}\\sum_{j=1}^n |\\langle f,k_{z_j}^\\alpha\\rangle_{{\\mathcal{A}}^2_\\alpha} |^2.\n\t\t\t\n\t\t\t\\end{eqnarray*}\n\t\t\tFurthermore\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\langle T_\\mu f,f \\rangle_{{\\mathcal{A}}^2_\\alpha} & = & \\sum_{j\\geqslant 1} c_j (1-|z_j|^2)^\\alpha \\left| f(z_j) \\right|^2 \\\\\n\t\t\t\t& = & \\sum_{j=1}^n c_j (1-|z_j|^2)^\\alpha \\left| f(z_j) \\right|^2 \\\\\n\t\t\t\t& \\geqslant & \\frac{c_n}{(1-|z_n|^2)^2} \\sum_{j=1}^n |\\langle f,{k_{z_j}^\\alpha}\\rangle_{{\\mathcal{A}}^2_\\alpha}|^2,\n\t\t\t\\end{eqnarray*}\n\t\t\twhich finally leads to\n\t\t\t\\[\n\t\t\ts_n(T_\\mu)\\gtrsim \\left[M_\\alpha(Z)\\right]^{-2} \\dfrac{c_n}{(1-|z_n|^2)^2}.\n\t\t\t\\]\n\t\t\\end{proof} \n\t\n\t\n\t\n\t\n\t\t\n\t\n\t\t\\begin{proof}[\\textbf{Proof of Theorem \\ref{Bergman}}]\n\t\t\tWe start with the the upper bound of the singular values of $I_g$.  Let $Z=(z_n)_{n\\geqslant 1}$ be a  sampling set  for ${\\mathcal{A}}^2_{\\alpha+2}$ such that each $R(I_n)$ contains $N$ points of $Z$, where $N$ is a large integer (see \\cite{Lue2}). We denote these points by $(z_{n,i})_{n\\geqslant 1}$, where $i=1,\\ldots,N$. \n\t\t\n\t\t\tUsing the fact that $Z$ is a sampling set for ${\\mathcal{A}}^2_{\\alpha +2}$ with the subharmoncity of $|g'|^2$, we get\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\int_{\\mathbb{D}}|f(z)g'(z)|^2(1-|z|^2)^{2+\\alpha}dm(z)& \\asymp & \\sum_{n\\geqslant 1}|f(z_{n})g'(z_{n})|^2(1-|z _{n}|^2)^{4+\\alpha}\\\\\n\t\t\t\t& \\lesssim & \\sum_n  |f(z_{n})|^2(1-|z _{n}|^2)^\\alpha \\int _{R(I_n)}|g'(z)|^2(1-|z|^2)^2dm(z)\\\\\n\t\t\t\t& = & \\sum_{n,i}  |f(z_{n,i})|^2(1-|z _{n,i}|^2)^\\alpha \\int _{R(I_n)}|g'(z)|^2(1-|z|^2)^2dm(z)\t\\\\\n\t\t\t\t& = & \\langle \\displaystyle \\sum_{i=1}^N T_{\\mu_i} f,f\\rangle_{{\\mathcal{A}}^2_\\alpha},\n\t\t\t\\end{eqnarray*}\n\t\t\twhere $\\mu_i = \\displaystyle \\sum _{n\\geqslant 1}m_g({R(I_n)})\\delta _{z_{n,i}}.$ This implies\n\t\t\t\\[\n\t\t\tI_g^*I_g  \\lesssim \\displaystyle   \\sum_{i=1}^N  T_{\\mu_i}.\n\t\t\t\\] \n\t\t\n\t\t\t{We use a known inequality of the singular values  from \\cite[p. 30]{GKG}. If $A$ and $B$ are two compact operators, then\n\t\t\t\t\\begin{equation}\\label{eq(2.1)}\n\t\t\t\ts_{m+n-1}(A+B)\\leqslant s_m(A)+s_n(B), \\quad m,n=1,2,\\ldots.\n\t\t\t\t\\end{equation}\n\t\t\t\tThus we obtain\t\n\t\t\t}\t \n\t\t\t\\begin{eqnarray*}\n\t\t\t\ts_{Nn}(I_g^*I_g)  \\lesssim  \\displaystyle \\sum_{i=1}^N s_n(T_{\\mu_i}) \\leqslant  N  \\max_{1\\leqslant i\\leqslant N} s_n(T_{\\mu_i}).\n\t\t\t\\end{eqnarray*}\n\t\t\tThis, together with Lemma \\ref{separation}, gives\n\t\t\t\\[\n\t\t\ts_{Nn}^2(I_g)\\lesssim N   \\max_{1\\leqslant i\\leqslant N} s_n(T_{\\mu_i}) \\lesssim \\frac{m_g({R(I_n)})}{(1-|z_n|^2)^2}.\n\t\t\t\\]\n\t\t\n\t\t\n\t\t\n\t\t\tNow, we turn to prove the lower bound result. Choose a point $\\xi_n$ in $R(I_n)$ so that $|g'(z)| \\leqslant 2|g'(\\xi_n)|$ for all $z \\in R(I_n)$.  Then\n\t\t\t\\begin{eqnarray}\\label{2}\n\t\t\tm_g(R(I_n))= \\int _{R(I_n)}|g'(z)|^2 (1-|z|^2)^2dm(z) \\leqslant |g'(\\xi_n)|^2 (1-|\\xi_n|^2)^4.\n\t\t\t\\end{eqnarray}\n\t\t\tNote that  $(\\xi_n)_{n\\geqslant 1}$ can be written as a finite union of separated sets. Since every separated set itself can be expressed as a finite union of interpolating sets (see \\cite{Seip2}), there exist  $\\Lambda_1, \\Lambda_2,\\cdots ,\\Lambda_M$ such that $\\{\\xi_n, \\ n\\in \\Lambda _i\\}$ is an interpolating sequence for  ${\\mathcal{A}}^2_{\\alpha}$. Therefore \n\t\t\t\n\t\t\n\t\t\t\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\int_\\mathbb{D} |f(z)g'(z)|^2(1-|z|^2)^{\\alpha+2}dm(z)\n\t\t\t\t& \\gtrsim &  \\sum_{i=1}^M  \\sum_{n\\in \\Lambda _i} |f(\\xi_{n})|^2|g'(\\xi_{n})|^2(1-|\\xi_n|^2)^{\\alpha+4}\\\\\n\t\t\t\t& \\geqslant &  \\sum_{i=1}^M  \\sum_{n\\in \\Lambda _i}|f(\\xi_{n})|^2(1-|\\xi_{n}|^2)^\\alpha m_g(R(I_n))\\\\\n\t\t\t\t& = & \\langle \\sum_{i=1}^M T_{\\mu_i} f,f\\rangle_{{\\mathcal{A}}^2_{\\alpha}},\n\t\t\t\\end{eqnarray*}\t\n\t\t\twhere $\\mu_i=\\displaystyle \\sum_{n\\in \\Lambda _i} m_g({R(I_n)})\\delta_{\\xi_{n}}$.\n\t\t\tThe first inequality follows from the fact that $(\\xi_n)_{n\\geqslant 1}$ is a finite union of separated sets. Therefore\n\t\t\t\\[\n\t\t\tI_g^*I_g\\gtrsim \\sum_{i=1}^M T_{\\mu_i} \\geqslant  T_{\\mu_k},\\quad k\\in \\{1,..,M\\}.\n\t\t\n\t\t\t\\] \n\t\t\tThis implies that \n\t\t\t\\[\n\t\t\ts_n(T_{\\mu_k}) \\lesssim s^2_n(I_g),\\quad k\\in \\{1,..,M\\}.\n\t\t\t\\]\n\t\t\tFinally, using the fact that each {set $\\{\\xi_n, \\ n\\in \\Lambda _i\\}$}, for $i=1,\\ldots,M$, is an interpolating set for ${{\\mathcal{A}}^2_{\\alpha}}$ along with Lemma \\ref{interpolation}, we get\n\t\t\t\\[\n\t\t\ts_n(T_{\\mu_k}) \\gtrsim a_n(\\mu_k),\n\t\t\t\\]\n\t\t\twhere $a_n(\\mu _k)$ is the nonincreasing rearrangement of the sequence of the sequence $(\\frac{\\mu _k({R(I_n)})}{m({R(I_n)})})_n$.\n\t\t\tConsequently, there exists an integer $p$ such that $s_{[\\frac{n}{p}]}^2(I_g)\\gtrsim \\frac{m_g({R(I_n)})}{(1-|z_n|^2)^2}.$ \n\t\t\\end{proof}\n\t\n\t\n\t\n\t\t\\section{The Hardy space $H^2$}\\label {hardy}\n\t\t\\label{sec3}\n\t\n\t\tThis section is devoted to prove Theorems \\ref{Hardy} and \\ref{B}. We start by recalling the connection between integration operators acting on the Hardy space $H^2$ and some standard operators.\\\\\\\\\n\t\tLet $\\mu$ be a positive Borel measure on $\\mathbb{D}$, and let $J_\\mu$ denote  the embedding operator from $H^2$ into $L^2(\\mu)$. Carleson proved in \\cite{Car} that $J_\\mu$  is bounded if and only if $\\mu(W(I))\/|I |$ is uniformly bounded for all arcs $I\\subset\\mathbb{T}$. This result can be expressed in terms of the family of the dyadic arcs. \n\t\tIndeed, one can see that $J_\\mu$ is bounded if and only if there exists $C>0$ such that \n\t\t\\[\n\t\t\\mu ( W(I_{n,k}) ) \\leqslant C | I_{n,k}|, \\quad \\quad  n\\geqslant 1\\ \\mbox{and}\\ 1\\leqslant k \\leqslant 2^n.\n\t\t\\]\n\t\tNote also that $J_\\mu$ is compact if and only if \n\t\t\\begin{eqnarray}\\label{compact}\n\t\t\\displaystyle \\lim _{n\\to \\infty} \\displaystyle \\sup _{1\\leqslant k\\leqslant 2^n} \\frac{\\mu ( W(I_{n,k}) )}{| I_{n,k} |}= 0.\n\t\t\\end{eqnarray}\n\t\tWe introduce the operator \n\t\t\\[\n\t\tS_{\\mu}f (z) = \\displaystyle \\int _\\mathbb{D}f(w)K(z,w)d\\mu(w),\\quad f\\in H^2,\n\t\t\\]\n\t\twhere $K(z,w)= (1-z\\bar{w})^{-1}$ is the reproducing kernel of $H^2$. \n\t\n\t\tA straightforward computation shows that\n\t\t\\[\n\t\t\\langle S_\\mu f , f\\rangle_{H^2} = \\displaystyle \\int _\\mathbb{D}|f(z)|^2d\\mu (z)= \\| J_\\mu f\\|^2_{L^2(\\mu)}.\n\t\t\\]\n\t\tThis implies that $S_\\mu$ is bounded, compact, or belongs to a Schatten  class ${\\mathcal{S}}_{p}(H^2)$ if and only if $J_\\mu$ is. In \\cite{Lue1}, Luecking proved that $S_\\mu \\in {\\mathcal{S}}_p(H^2)$, for $p>0$, if and only if \n\t\t\\[\n\t\t\\displaystyle \\sum _{n,k} \\left (\\frac{\\mu (R(I_{n,k}))}{| I_{n,k} |}\\right )^{p}<\\infty.\n\t\t\\]\n\t\tIt is worth noting that Luecking's characterization can also be expressed  in terms of Carleson  windows $W(I_{n,k})$ (see \\cite{LLQRJFA}). However, this is no longer the case for Carleson boundedness criterion (see \\cite{PR}).\\\\\n\t\t\\\\\n\t\tNow, we fix and recall some notations. Let $\\mu$  be a positive Borel measure on $\\mathbb{D}$ which satisfies condition (\\ref {compact}), and let $(I_n(\\mu))_{n\\geqslant 1}$ be an enumeration of  $(I_{n,k})_{n,k}$ such that the sequence $\\left(\\dfrac{\\mu(W(I_n(\\mu)))}{| I_n(\\mu)|}\\right)_{n\\geqslant 1}$ is nonincreasing. According to the notation used in the introduction, we define\n\t\t\\[\n\t\t\\tau _n(\\mu) \\colonequals \\frac{\\mu(W(I_n(\\mu))}{| I_n(\\mu)|}.\n\t\t\\]\n\t\n\t\t\n\t\t\n\t\t\\noindent The following lemma is the key to prove the first assertion of Theorem \\ref{Hardy}.\n\t\t\\begin{lem}\\label{KL}\n\t\t\tLet $\\mu$ be a positive Borel measure on $\\mathbb{D}$ such that $S_\\mu$ is compact on $H^2$. Let $I_n = I_n(\\mu)$, and let $(z_n)_{n\\geqslant 1} $ be a sequence such that $z _n \\in R(I_n)$. Fix an index $n\\geqslant 1$, and set $\\mu_n =\\displaystyle \\sum _{k\\geqslant n}\\mu (R(I_k))\\delta _{z _k}$. We have  \n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\\label{3}  \n\t\t\t\t$\n\t\t\t\t\\displaystyle \\sup _{j\\geqslant 1}\\frac{\\mu_n(W(I_j))}{| I_j|} \\leqslant \\tau_n(\\mu).\n\t\t\t\t$\\\\\n\t\t\t\t\\item\\label{4} $s_n(S_{\\mu _1})\\ \\lesssim \\ \\tau_n(\\mu).$\\\\\n\t\t\t\\end{enumerate}\n\t\t\\end{lem}\t\n\t\t\n\t\t\\begin{proof}\n\t\t\tWe begin with part \\eqref{3}. First, note that if $n=1$, then $\\tau _j(\\mu_1)= \\tau _j(\\mu)$ and the result is obvious.  Now, suppose that  $n\\geqslant 2$. \\\\ \\\\\n\t\t\t{For $j\\geqslant 1$, let $\\Gamma_j=\\lbrace k\\geqslant n \\colon R(I_k) \\subset W(I_j) \\rbrace$. Then\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\t\\mu_n(W(I_j)) &=& \\sum_{k \\in \\Gamma_j} \\mu_n(R(I_k)) = \\sum_{k \\in \\Gamma_j} \\mu(R(I_k))  \\leqslant \\mu(W(I_j)).\n\t\t\t\t\\end{eqnarray*}\t\n\t\t\t\tThus for all $j\\geqslant n$, we have\n\t\t\t}\n\t\t\t\\[\n\t\t\t\\dfrac{\\mu_n(W(I_j))}{| I_j |} \\leqslant \\dfrac{\\mu(W(I_j))}{| I_j |} \\leqslant \\dfrac{\\mu(W(I_n))}{| I_n |} .\n\t\t\t\\]\n\t\t\tLet  $ j\\in \\lbrace 1,\\ldots,n-1\\rbrace$.  If we assume that\n\t\t\t\\begin{eqnarray}\\label{EL1}\n\t\t\t\\dfrac{\\mu_n(W(I_j))}{| I_j |} > \\dfrac{\\mu(W(I_n))}{| I_n |},\n\t\t\t\\end{eqnarray}\n\t\t\twe will end up with a contradiction. Indeed, we will prove that there exists $j_1\\in  \\lbrace 1,\\ldots,n-1\\rbrace$ such that $W(I_{j_1})\\varsubsetneq W(I_j)$ and $\\dfrac{\\mu (W(I_n))}{| I_n |} < \\dfrac{\\mu_n(W(I_{j_1}))}{| I_{j_1}|}$, which is absurd since one can repeat this operation indefinitely.\\\\\n\t\t\t\n\t\t\tSuppose that assumption \\eqref{EL1} is satisfied, and write $W(I_j)= R(I_j)\\cup W(I_{j_1})\\cup W(I_{j'_1})$, where\n\t\t\t\\[\n\t\t\t| I_{j_1} |= | I_{j'_1}|= | I_{j}|\/2. \n\t\t\t\\]\n\t\t\tWithout loss of generality, we assume that $\\mu_n(W(I_{j_1}))\\geqslant \\mu_n(W(I_{j'_1}))$. Then, since $\\mu_n(R(I_j))=0$, we have\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\dfrac{\\mu_n(W(I_j))}{| I_j|} &=& \\dfrac{\\mu_n(W(I_{j_1}))+\\mu_n(W(I_{j'_1}))}{| I_{j_1}|+| I_{j'_1}|}\\leqslant \\dfrac{\\mu_n(W(I_{j_1}))}{| I_{j_1}|}.\n\t\t\t\\end{eqnarray*}\n\t\t\tThis implies  $\\dfrac{\\mu (W(I_n))}{| I_n|} < \\dfrac{\\mu_n(W(I_{j_1}))}{| I_{j_1}|}$ and then $j_1\\in  \\lbrace 1,\\ldots,n-1\\rbrace$. The proof of \\eqref{3} is complete.\\\\\n\t\t\t\n\t\t\tTo prove \\eqref{4}, recall that $s_n(S_{\\mu_1}) =  \\inf\\left\\{\\left\\| S_{\\mu_1}- R\\right\\|\\ :\\ \\text{rank}\\  R<n \\right\\} $. For $R = S_{\\mu_1 -\\mu_n}$, we have \n\t\t\t\\[\n\t\t\ts_n(S_{\\mu_1}) \\leqslant \\| S_{\\mu_1} - S_{\\mu_1-\\mu_n}\\| = \\|S_{\\mu _n}\\| \\lesssim  \\displaystyle \\sup _{j\\geqslant 1}\\frac{\\mu_n(W(I_j))}{| I_j|}.\n\t\t\t\\]\n\t\t\tThe second inequality comes from the embedding Carleson's Theorem. Using the first part of the lemma, we obtain $  s_n(S_{\\mu_1}) \\lesssim \\tau _n(\\mu)$.\n\t\t\\end{proof}\n\t\tThe proof of the second assertion of Theorem \\ref{Hardy} requires the following lemma. We include the proof for completeness.\n\t\t\\begin{lem}\\label{0}\n\t\t\tLet $H$ and $K$ be two Hilbert spaces and  $T: H\\to K$  a compact operator. Suppose that there exist $(u_n)_{n\\geqslant 1}\\subset H$ and $(v_n)_{n\\geqslant 1}\\subset K$ such that \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _n |\\langle u_n, f \\rangle_H |^2\\leqslant C\\|f\\|_H^2 \\ \\text{and} \\ \\displaystyle \\sum _n |\\langle v_n, g\\rangle_K |^2\\leqslant C\\|g\\|_K^2,\\quad f\\in H,\\  g\\in K,\n\t\t\t\\]\n\t\t\tfor some constant $C>0$. Then  we have \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j=1}^n \\left|\\langle Tu_j, v_j\\rangle_{K} \\right|\\leqslant C^2 \\displaystyle \\sum _{j=1}^n s_j(T).\n\t\t\t\\]\n\t\t\\end{lem}\n\t\t\n\t\n\t\t\\begin{proof}\n\t\t\tLet $(e_n)_n$ and $(f_n)_n$ be two orthonormal bases of $H$ and $K$, respectively. Let $A$ and $B$ be the operators given by\n\t\t\t\\[\n\t\t\tAf=\\displaystyle\\sum_n\\langle u_n,f\\rangle_H e_n \\quad \\text{and}\\quad   Bg=\\displaystyle\\sum_n\\langle v_n,g\\rangle_K f_n.\n\t\t\t\\]\n\t\t\tBy assumptions, $A$ and $B$ are bounded and $\\|A\\|, \\|B\\| \\leqslant C$. Note that \n\t\t\tthe adjoint operators $A^*$ and $B^*$ satisfy $A^*e_n=u_n$ and $B^*f_n=v_n$. By Theorem 3.5 of \\cite{GGK}, we have \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum_{j=1}^{n}\\left|\\langle BTA^* e_j,f_j\\rangle \\right|  \\leqslant \\displaystyle \\sum_{j=1}^{n} s_j(BTA^*).\n\t\t\t\\]\n\t\t\tThus\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\displaystyle \\sum_{j=1}^{n}\\left|\\langle Tu_j,v_j \\rangle \\right| &=& \\displaystyle \\sum_{j=1}^{n}\\left|\\langle TA^* e_j,B^*f_j\\rangle \\right |\\\\\n\t\t\t\n\t\t\t\t& = & \\displaystyle \\sum_{j=1}^{n}\\left|\\langle BTA^* e_j,f_j\\rangle \\right|\\\\\n\t\t\t\n\t\t\t\t&\\leqslant& \\displaystyle \\sum_{j=1}^{n} s_j(BTA^*)\\\\\n\t\t\t\t&\\leqslant& \\|B\\|\\|A^*\\|\\displaystyle \\sum_{j=1}^{n}s_j(T)\\leqslant C^2 \\displaystyle \\sum_{j=1}^{n}s_j(T).\n\t\t\t\\end{eqnarray*} \n\t\t\t\n\t\t\\end{proof}\n\t\t\n\t\t\\noindent The classical Littlewood-Paley identity \\cite{Gar} states that\n\t\n\t\t\\[\n\t\t\\| f -f(0)\\|^2 _{H^2}=  \\displaystyle \\int_{\\mathbb{D}}|f'(z)|^2\\log (1\/|z|^2)dm(z).\n\t\t\\]\n\t\n\t\tThis  gives rise to the following formula which relates the operators $I_g$ and $S_{\\nu _g}$. We have\n\t\t\\begin{eqnarray}\\label{H-L}\n\t\t\\langle I^*_gI_g f, f \\rangle_{H^2}  = \\displaystyle \\int _{\\mathbb{D}}|f(z)|^2|g'(z)|^2\\log (1\/|z|^2)dm(z)\n\t\t\\asymp \\langle S_{\\nu_g}f,f \\rangle_{H^2}.\n\t\t\\end{eqnarray}\n\t\n\t\t\n\t\n\t\n\t\t\\begin{proof}[\\textbf{Proof of Theorem \\ref{Hardy}}]\n\t\t\tIt is immediate, from formula (\\ref{H-L}), that if $f\\in H^2$ then $fg'\\in {\\mathcal{A}}^2_1$. We have seen in the proof of the upper bound result from Theorem \\ref{Bergman} that\n\t\t\tthere exist a sampling sequence $Z=(z_n)_{n\\geqslant 1}$ for ${\\mathcal{A}}^2_1$ and an integer $N$ such that each $R(I_n)$ contains  $N$ points of $Z$ denoted by $(z_{n,i})_{n\\geqslant 1}$, where $i=1,\\ldots,N$. Then\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\displaystyle \\int _{\\mathbb{D}}|f(z)g'(z)|^2(1-|z|^2)dm(z)& \\asymp& \\displaystyle \\sum _{n\\geqslant 1} |f(z_{n})g'(z _{n})|^2(1-|z _{n}|^2)^3\\\\\n\t\t\t\t& \\lesssim &  \\displaystyle \\sum _{n\\geqslant 1} |f(z_{n})|^2 \\displaystyle \\int _{R(I_n)}|g'(z)|^2(1-|z|^2)dm(z)\\\\\n\t\t\t\t& = &  \\displaystyle \\sum _{n\\geqslant 1} \\displaystyle \\sum _{i=1}^N|f(z_{n,i})|^2 \\displaystyle \\int _{R(I_n)}|g'(z)|^2(1-|z|^2)dm(z)\\\\\n\t\t\t\t& = & \\langle \\displaystyle \\sum _{i=1}^N S_{\\nu_i} f,f\\rangle_{H^2},\n\t\t\t\\end{eqnarray*}\n\t\t\twhere $\\nu_i = \\displaystyle \\sum _{n\\geqslant 1}\\nu_g(R(I_n))\\delta _{z_{n,i}}.$ {Using inequality \\eqref{eq(2.1)}, we obtain}\n\t\t\t\\[\n\t\t\ts_{Nn}^2(I_g)\\asymp s_{Nn}(S_{\\nu_g}) \\lesssim  \\max_{1\\leqslant i \\leqslant N} s_n(S_{\\nu_i}).\n\t\t\t\\]\n\t\t\t{By} Lemma \\ref{KL}, we deduce  $s_{Nn}^2(I_g) \\lesssim \\tau _n(\\nu_g)$ as desired and the proof of assertion \\ref{i} is complete.\\\\\n\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\t\\noindent To prove assertion \\ref{ii}, we use Lemma \\ref{0}. Let $\\xi_n$ be the center of  $R(I_n)$ and set\n\t\t\t\\[\n\t\t\tu_n(z)=v_n(z)=\\frac{(1-|\\xi_n|^2)^{3\/2}}{(1-\\bar{\\xi}_n z)^2}.\n\t\t\t\\]\n\t\t\tFor $f \\in H^2$, we have\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t|\\langle u _n, f\\rangle _{H^2}|^2 &  \\lesssim &  | u _n (0)f(0)|^2+\\left| \\displaystyle \\int _{\\mathbb{D}}\\frac{(1-|\\xi_n|^2)^{3\/2}}{(1-\\bar{\\xi}_n z)^3} \\overline{f'(z)} (1-|z|^2)dm (z)\\right|^2\\\\\n\t\t\t\t& = & (1-|\\xi_n|^2)^3 |f(0)|^2+|\\langle k_{\\xi_n}^1,f' \\rangle_{\\mathcal{A}^2_1} |^2.\n\t\t\t\\end{eqnarray*}\n\t\t\tThus \n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\displaystyle \\sum _n |\\langle u _n, f\\rangle _{H^2}|^2 &  \\lesssim & \\displaystyle \\sum _n (1-|\\xi_n|^2)^3 |f(0)|^2+ \\displaystyle \\sum _n |\\langle k_{\\xi_n}^1,f' \\rangle_{\\mathcal{A}^2_1} |^2\\\\\n\t\t\t\t& \\lesssim & \\displaystyle \\sum _n (1-|\\xi_n|^2)^3 |f(0)|^2+\\|f'\\|^2_{{\\mathcal{A}}^2_1}\\\\\n\t\t\t\t& \\lesssim & \\|f\\|^2_{H^2}.\n\t\t\t\\end{eqnarray*}\n\t\t\tThe second inequality follows from the fact that $(\\xi _n)_n$ is a separated sequence.\n\t\t\n\t\t\tThen, by Lemma \\ref{0} applied to $S_{\\nu _g}$, there exists an absolute constant $c>0$ such that\n\t\t\t\\[\n\t\t\t\\displaystyle c \\sum _{j=1}^n |\\langle S_{\\nu_g} u_j, u_j\\rangle _{H^2}| \\leqslant \\displaystyle \\sum _{j=1}^n s_j(S_{\\nu_g}).\n\t\t\t\\]\n\t\t\tFurthermore\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\langle S_{\\nu_g} u_n,u_n\\rangle_{H^2} & = & \\displaystyle \\int_\\mathbb{D} \\dfrac{(1-|a_n|^2)^3}{|1-\\bar{a}_n z|^4}|g'(z)|^2(1-|z|^2) dm(z) \\\\\n\t\t\t\t& \\geqslant & \\displaystyle \\int_{W(I_n)}\\dfrac{(1-|a_n|^2)^3}{|1-\\bar{a}_n z|^4}|g'(z)|^2(1-|z|^2) dm(z)\\\\\n\t\t\t\t& \\gtrsim & \\dfrac{1}{|I_n|} \\displaystyle \\int_{W(I_n)} |g'(z)|^2(1-|z|^2) dm(z).\n\t\t\t\\end{eqnarray*}\n\t\t\tCombining these two facts with formula (\\ref{H-L}), we get \n\t\t\t\\[\n\t\t\t\\displaystyle  \\sum _{j=1}^n  \\frac{\\nu_g(W(I_j))}{|I_j|} \\lesssim  \\displaystyle  \\sum _{j=1}^n s_j(S_{\\nu_g}) \\lesssim   \\displaystyle  \\sum _{j=1}^n s^2_j({I_{g}}).\n\t\t\t\\]\n\t\t\\end{proof}\n\t\t\n\t\n\t\t\n\t\n\t\n\t\tWe turn now to prove Theorem \\ref{B}, we start with the proof of the first statement which is completely based on Theorem \\ref{Hardy}.\n\t\t\n\t\t\\begin{proof}[\\textbf{Proof of assertion \\ref{2i} of Theorem \\ref{B}} ]\n\t\t\tSuppose that $(s_n(I_g))_n \\in {\\mathcal{R}} _\\gamma$ for some $\\gamma \\in (0,1\/2)$. Then  there exists $(x_n)_n$ a nonincreasing sequence  of positive numbers such that ${n^{\\gamma} x_n}$ increases to infinity and  $s_n(I_g) \\asymp x_n$.  {For an integer $p\\geqslant 1$, we have  \n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\tp^{-\\gamma}x_n \\leqslant x_{pn} \\leqslant x_n. \n\t\t\t\t\\end{eqnarray*}\n\t\t\t}\n\t\t\tBy assertion \\ref{i} of Theorem \\ref{Hardy}, we have \n\t\t\t\\begin{eqnarray}\\label{UB}\n\t\t\t{x^2_n  \\asymp s_{np}^2(I_g)} \\lesssim \\tau_n(\\nu_g).\n\t\t\t\\end{eqnarray}\t\n\t\t\tMoreover, Theorem \\ref{Hardy} implies\n\t\t\t\\begin{eqnarray*}\\label{LB}\n\t\t\t\tn\\tau_n(\\nu_g) \\leqslant \\displaystyle \\sum_{j=1}^n \\tau_j(\\nu_g)  & \\lesssim & \\displaystyle  \\sum_{j=1}^n s_j^2(I_g)\\\\\n\t\t\t\t&\\lesssim & {\\displaystyle  \\sum_{j=1}^n x^2_j}  \\\\\n\t\t\t\t& \\leqslant & n^{2\\gamma} x^2_n \\displaystyle  \\sum_{j=1}^n \\dfrac{1}{j^{2\\gamma}}. \n\t\t\t\n\t\t\t\n\t\t\t\\end{eqnarray*}\t\n\t\t\t{The last inequality follows from the fact that $(j^{2\\gamma}x_j)_j $ is increasing. Since $2\\gamma <1$, we have\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\t\\sum_{j=1}^n \\dfrac{1}{j^{2\\gamma}} &\\asymp& n^{1-2\\gamma}.\t\n\t\t\t\t\\end{eqnarray*}\t\n\t\t\t\tThus $n\\tau_n(\\nu_g)\t\\lesssim n x^2_n.$}\n\t\t\tThis along with inequality (\\ref{UB}) leads to {$ x_n \\asymp \\tau_n(\\nu_g)$.}\\\\ \\\\\n\t\t\n\t\t\tNow, suppose that  $(\\tau_n(\\nu_g))_n \\in {\\mathcal{R}} _{2\\gamma}$ and let $(x_n)_n$ be a nonincreasing sequence  of positive numbers such that $n^{2\\gamma} x_n$ increases to infinity and  $\\tau_n(\\nu_g) \\asymp x_n$. It follows at once from assertion \\ref{i} of Theorem \\ref{Hardy} that \n\t\t\t{\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\ts_n^2(I_g) \\lesssim x_{[\\frac{n}{p}]} \\asymp x_n. \n\t\t\t\t\\end{eqnarray*}\n\t\t\t}\n\t\t\n\t\t\tLet  $\\kappa$ be a large constant. Assertion \\ref{ii} of Theorem \\ref{Hardy} yields\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\displaystyle \\sum_{j=1}^{\\kappa n} x_j & \\lesssim & \\displaystyle \\sum_{j=1}^n s_j^2(I_g) +  \\displaystyle \\sum_{j=n+1}^{\\kappa n} s_j^2(I_g) \\\\\n\t\t\t\t& \\lesssim & \\displaystyle \\sum_{j=1}^n x_j + \\kappa n s_n^2(I_g).\n\t\t\t\\end{eqnarray*}\n\t\t\t{Using the same line of reasoning as in the previous part, we obtain $\\displaystyle\\sum_{j=1}^n x_j \\lesssim n x_n.$ Therefore\t\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\t\\displaystyle \\sum_{j=1}^{\\kappa n} x_j &\\lesssim & n x_n + \\kappa n s_n^2(I_g).\t\n\t\t\t\t\\end{eqnarray*}\n\t\t\t\tIn contrast\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\t\\displaystyle \\sum_{j=1}^{\\kappa n} x_j  & \\geqslant  & \\kappa n x_{\\kappa n} \\\\ \n\t\t\t\t\t&\\geqslant& \\kappa^{1-2\\gamma} n x_n.\n\t\t\t\t\\end{eqnarray*}\n\t\t\t\tThe first inequality follows from the fact  $(x_j)_j $ is nonincreasing and the second one uses the fact that $(j^{2\\gamma}x_j)_j $ is increasing. \n\t\t\t\tThus\n\t\t\t\n\t\t\t\t\\[\n\t\t\t\t\\kappa^{1-2\\gamma} x_n \\lesssim x_n + \\kappa s_n^2(I_g).\n\t\t\t\t\\]\n\t\t\t}\n\t\t\tFor  $\\kappa$ large enough, we obtain the result.\n\t\t\\end{proof}\n\t\t\n\t\tThe proof of the second statement of Theorem \\ref{B} requires some preparatory results. The following technical lemma is a generalization of Proposition 3.3 in \\cite{LLQRJFA}.\n\t\t\\begin{lem}\\label{lemLLQP}\n\t\t\tLet $h: [0,\\infty[\\to [0,\\infty[$ be an increasing function such that $h(0)=0$. Suppose that there exist $\\varepsilon \\in (0,1)$ and $p\\geqslant 1$ for which $t\\to h(t^\\varepsilon)$ is concave and $t\\to h(t^p)$ is convex. Let $\\mu $ be a positive Borel measure on $\\mathbb{D}$. There exists $B>0$, which depends on $\\varepsilon$ and $p$, such that  \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h\\left (\\frac{\\mu(W{(I_j)})}{| I_j |}\\right ) \\leqslant \\displaystyle \\sum _{j}h\\left (B\\frac{\\mu(R{(I_j)})}{| I_j|}\\right ).\n\t\t\t\\]\n\t\t\\end{lem}\n\t\t\\begin{proof}\n\t\t\tWe use the argument given in \\cite{LLQRJFA}. Let $\\alpha = 1\/\\varepsilon$ and let $\\beta =1\/(1-\\varepsilon)$ be the conjugate of $\\alpha$. Write\n\t\t\t\\[\n\t\t\t{W(I_{n,j})}=\\bigcup_{l\\geqslant n}\\bigcup_{k\\in H_{l,n,j}}  {R(I_{l,k})},\\quad n\\geqslant 1\\ \\mbox{and}\\ 1\\leqslant j\\leqslant 2^n,\n\t\t\t\\]\n\t\t\twhere $H_{l,n,j}=\\left\\lbrace k\\in\\lbrace 1,\\ldots,2^l \\rbrace; \\dfrac{j}{2^n}\\leqslant\\dfrac{k}{2^l}<\\dfrac{j+1}{2^n}\\right\\rbrace.$ Consider $a \\in (1,2)$ such that $2< a^\\beta$. It is proved in \\cite{LLQRJFA} that \n\t\t\t\\[\n\t\t\t\\mu({W(I_{n,j})})^\\alpha \\lesssim \\displaystyle \\sum _{l\\geqslant n; \\ k\\in H_{l,n,k}}a^{(l-n)\\alpha }\\mu({R(I_{l,k})})^\\alpha.\n\t\t\t\\]\n\t\t\tRecall that the function $h_{1\/\\alpha}(t) \\colonequals h(t^{1\/\\alpha})$ is increasing and concave. Consequently,\n\t\t\t\\begin{eqnarray*}\n\t\t\t\th(2^n\\mu({W(I_{n,j})}) )& =& h_{1\/\\alpha} \\left ( (2^n\\mu({W(I_{n,j})}))^\\alpha \\right )\\\\\n\t\t\t\t& \\lesssim & \\displaystyle \\sum _{l\\geqslant n; \\ k\\in H_{l,n,k}}  h_{1\/\\alpha} \\left (a^{(l-n)\\alpha}2^{n\\alpha}\\mu({R(I_{l,k})})^\\alpha \\right )\\\\\n\t\t\t\t& = & \\displaystyle \\sum _{l\\geqslant n; \\ k\\in H_{l,n,k}}  h\\left (\\left (\\frac{a}{2}\\right )^{l-n}2^l\\mu({R(I_{l,k})}) \\right )\\\\\n\t\t\t\t& \\leqslant & \\displaystyle \\sum _{l\\geqslant n; \\ k\\in H_{l,n,k}}  \\left (\\frac{a}{2}\\right ) ^{(l-n)\/p}h(2^l\\mu({R(I_{l,k})}) ).\\\\\n\t\t\t\\end{eqnarray*}\n\t\t\tThe last inequality follows from the fact $h(t^p)$ is convex. Then we obtain \n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\sum_{n= 1}^\\infty\\sum_{j=0}^{2^n-1}h\\left(2^n\\mu({W(I_{n,j})})\\right)&\\lesssim &\\sum_{l=1}^\\infty\\sum_{k=0}^{2^l-1}\\left(\\sum_{(n,j):l\\geqslant n,k\\in H_{l,n,j}}\\left (\\frac{a}{2}\\right ) ^{(l-n)\/p}\\right)h\\left(2^l\\mu({R(I_{l,k})})\\right)\\\\\n\t\t\t\t&\\leqslant&C{(p,\\varepsilon)}\\sum_{l=1}^\\infty\\sum_{k=0}^{2^l-1}h\\left(2^l\\mu({R(I_{l,k})})\\right).\n\t\t\t\\end{eqnarray*}\n\t\t\\end{proof}\n\t\t\n\t\n\t\t\\begin{theo}\\label {trace}\n\t\t\tLet $\\mu$ be a positive Borel measure on $\\mathbb{D}$ such that $J_\\mu$ is compact. Let $h: [0,\\infty[\\to [0,\\infty[$ be a convex  increasing function with $h(0)=0$ and such that $t\\to h(t^\\varepsilon)$ is concave for some $\\varepsilon \\in (0,1)$. Then there exists $B>0$ which depends only on $\\varepsilon $ such that \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h\\left (\\frac{1}{B}\\sqrt{\\tau_j(\\mu)}\\right ) \\leqslant \\displaystyle \\sum _{j}h(s_j(J_\\mu)).\n\t\t\t\\]\n\t\t\\end{theo}\n\t\t\n\t\t\\begin{proof} The proof is based on Lemma \\ref{0}. Let $\\xi_n$ be the center of  ${R(I_n)}$, and set\n\t\t\t\\[\n\t\t\tu_n(z)=\\frac{(1-|\\xi_n|^2)^{3\/2}}{(1-\\bar{\\xi}_n z)^2}\\ \\mbox{and}\\ v_n= c_n \\theta_n \\chi_{{R(I_n)}},\n\t\t\t\\]\n\t\t\twhere $\\theta _n = \\frac{u_n}{|u_n|}$ and $c_n= \\frac{1}{\\sqrt{\\mu ({R(I_n)})}}$ if $\\mu ({R(I_n)}) \\neq 0$.\\\\\n\t\t\tWe proved before \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _n |\\langle u _n, f\\rangle _{H^2}|^2  \\lesssim \\|f\\|^2_{H^2}, \\quad f \\in H^2.\n\t\t\t\\]\n\t\t\tBy Cauchy-Schwarz inequality, we have \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _n |\\langle v_n, v\\rangle _{L^2(\\mu)}|^2 \\leqslant \\displaystyle \\sum _n c^2_n\\mu ({R(I_n)})\\displaystyle \\int_{{R(I_n)}}|v|^2d\\mu =  \\| v \\| ^2_{L^2(\\mu)},\\quad v \\in L^2(\\mu).\n\t\t\t\\]\n\t\t\tNote also that \n\t\t\t\\[\n\t\t\t\\langle J_\\mu u_n, v_n\\rangle _{L^2(\\mu)} = c_n \\displaystyle \\int _{{R(I_n)}}|u_n|d\\mu \\asymp \\sqrt{\\frac{\\mu({R(I_n)})}{| I_n |}}.\n\t\t\t\\]\n\t\t\tApplying Lemma \\ref{0}, we get\n\t\t\t\\[\n\t\t\t\\displaystyle  \\sum _{j=1}^n  \\frac{1}{C}\\sqrt{\\frac{\\mu(R(I_j))}{| I_j |}} \\lesssim  \\displaystyle  \\sum _{j=1}^n s_j(J_\\mu),\n\t\t\t\\]\n\t\t\twhere $C$ is a positive constant. \n\t\t\tThen by Corollary 3.3 of \\cite{GGK}, we obtain\n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h\\left (\\frac{1}{C}\\sqrt{\\frac{\\mu(R{(I_j)})}{| I_j|}}\\right ) \\leqslant \\displaystyle \\sum _{j}h(s_j(J_\\mu)).\n\t\t\t\\]\n\t\t\tThe desired result follows from Lemma \\ref{lemLLQP}.\n\t\t\\end{proof}\n\t\t\\begin{cor}\\label {Ctrace}\n\t\t\tLet $g\\in VMOA$. Let $h: [0,\\infty[\\to [0,\\infty[$ be a convex  increasing function with $h(0)=0$ and such that $t\\to h(t^\\varepsilon)$ is concave for some $\\varepsilon \\in (0,1)$. Then there exists $B>0$ which depends only on $\\varepsilon $ such that \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h\\left (\\frac{1}{B}\\sqrt{\\tau_j(\\nu_g)}\\right ) \\leqslant \\displaystyle \\sum _{j}h(s_j(I_g))).\n\t\t\t\\]\n\t\t\\end{cor}\n\t\t\\begin{proof}\n\t\t\tIt suffices to apply Theorem \\ref{trace} and to remark that $s_n(I_g)= s_n(J_{\\nu_g})$.\n\t\t\\end{proof}\n\t\t\n\t\tThe following lemma is proved in \\cite{EE}.\n\t\t\\begin{lem}\\label{lemEE}\n\t\t\tLet $(a_n)_n$ and $(b_n)_n$ be two positive {nonincreasing} sequences. Suppose  that there exist $\\beta _1 >1$ and $\\beta _2>1 $ such that $(n^{\\beta _1} b_n)_n$ is {nonincreasing} and $(n^{\\beta _2}b_n)_n$ is increasing. Let $\\beta >\\beta _2$  and let $B>0$. If for each positive increasing concave function $h$ such that $h(t^\\beta)$ is convex we have\n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{n}h (\\frac{1}{B}a_n) \\leqslant \\displaystyle \\sum _{n}h (b_n) \\leqslant \\displaystyle \\sum _{n}h(B a_n),\n\t\t\t\\]\n\t\t\tthen $a_n \\asymp b_n$.\n\t\t\\end{lem}\n\t\t\\begin{proof}[\\textbf{Proof of assertion \\ref{3i}  of Theorem \\ref{B}}] \n\t\t\tSuppose that $(s_n(I_g))_n \\in {\\mathcal{R}}_{\\gamma, \\alpha}$. Then there exists a {nonincreasing} sequence $(x_n)_n$ such that $(n^\\gamma x_n)_n$ is increasing, $(n^\\alpha x_n)_n$ is {nonincreasing}, and $x_n \\asymp s_n(I_g)$. We have to prove that $x_n \\asymp \\sqrt{\\tau _n(\\nu_g)}$. \n\t\t\tLet $\\beta>0$ be such that $\\beta \\alpha >1$. Put $\\beta _1= \\beta \\alpha$ and $\\beta _2= \\beta \\gamma$. Let $h$ be a positive increasing concave function such that $h(0)= 0$ and {$h_\\beta \\colonequals h(t^\\beta) $} is convex. The function $h_\\beta$ satisfies the assumptions of Corollary \\ref{Ctrace}, with $\\varepsilon = 1\/\\beta$. Then there exists $B_1$ such that\n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h_\\beta \\left (\\frac{1}{B_1}\\sqrt{\\tau _j(\\nu_g)}\\right ) \\leqslant \\displaystyle \\sum _{j}h_\\beta (x_j).\n\t\t\t\\]\n\t\t\tBy the first assertion of Theorem \\ref{Hardy}, we have\n\t\t\t\\[\n\t\t\tx_n \\asymp s_{p n}(I_g) \\lesssim  \\sqrt{\\tau_n (\\nu_g)}, \\quad  n\\geqslant 1.\n\t\t\t\\]\n\t\t\tThen there exists $B>0$ such that \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h_\\beta\\left (\\frac{1}{B}\\sqrt{\\tau _j(\\nu_g)}\\right ) \\leqslant \\displaystyle \\sum _{j}h_\\beta (x_j)\\leqslant \\displaystyle \\sum _{j}h_\\beta\\left (B\\sqrt{\\tau _j(\\nu_g)}\\right ).\n\t\t\t\\]\n\t\t\tLemma \\ref{lemEE} yields $x_n^{\\beta} \\asymp \\tau ^{\\beta\/2}_n (\\nu_g)$, that is, $x_n \\asymp \\sqrt{\\tau _n(\\nu_g)}$.\\\\\n\t\t\tThe result in the case $(\\tau _n (\\nu_g))_n \\in {\\mathcal{R}}_{2\\gamma, 2\\alpha}$ can be proved in the same way.\n\t\t\\end{proof}\n\t\t\n\t\tUsing Theorems \\ref{Hardy} and \\ref{Ctrace} with  the same line of reasoning as in the proofs of Theorem 1.2 and Proposition 5.2 of \\cite{BEMN}, one can obtain the following result. We omit the proof here.\n\t\n\t\t\\begin{cor}\n\t\t\tLet $g\\in VMOA$. We have\\\\\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item  $s_n(I_g)= o\\left ( \\frac{1}{n}\\right ) \\quad  \\Longrightarrow \\quad I_g =0.$\n\t\t\t\t\\item $s_n(I_g)= O\\left ( \\frac{1}{n}\\right ) \\quad \\iff \\quad g' \\in H^1.$\n\t\t\t\\end{enumerate}\n\t\t\\end{cor}\n\t\n\t\n\t\t\n\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\bibliographystyle{plain}\n\t\n\t\t\\bigskip\n\t\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nAn aspect of great importance in nuclear structure calculations at any\nexcitation energy concerns with the role of the effective interaction.\nAt low energies this problem has generated a considerable body of work\nin the last twenty years. On the contrary, this is a question not \nstudied yet in deep in the literature for higher energies.\n\nGiant resonances show an intricate mixture of multipolarities and the\nstudy of how the interaction affects it is a difficult\ntask. In the quasi--elastic peak region, the problem of the\nlongitudinal and transverse separation has occupied most of the\ninvestigations carried out till now and the discussion of the effects\ndue to changes in the effective interaction have not been considered\nin detail.\n\nAs an example, we mention the considerable number of Random--Phase \nApproximation (RPA) type calculations performed in this energy region,\nmuch of them using residual interactions which include basically a \nzero--range term plus meson--exchange potentials corresponding to\n$\\pi$, $\\rho$ and, eventually, other mesons~\\cite{al84}--\\cite{gi97}. \nAn important point concerning the interaction refers to the values\nchosen for the parameters entering in the zero--range piece.\nHowever, and to the best of our knowledge, only in\nRef.~\\cite{gi97} a certain discussion relative to the effects of\nvarying these parameters can be found. In fact, the common\npractice is to pick an interaction from the literature, which\nusually corresponds to a parameterization fixed for low energy\ncalculations, and afterwards use it to evaluate quasi--free\nobservables sometimes without taking care of the effective theory in \nwhich the interaction was adjusted. It is obvious that, to a certain\nlevel, doubtful results are possible because of the known link\nbetween effective theory and interaction.\n\nIn this work we want to address this question and investigate if \ndifferent parametrizations of the interaction can produce noticeable \ndifferences in the results and the extent to which the use of an \ninteraction fixed for a given effective theory affects the results \nobtained within a different one. In Sec.~II we give the details about \nthe effective theories and interactions used to perform the\ncalculations. In Sec.~III we show and discuss the results we have\nobtained. Finally, we present our conclusions in Sec.~IV.\n\n\\section{EFFECTIVE THEORIES AND INTERACTIONS}\n\nThe first interaction we consider in this work is the so--called \nJ\\\"ulich--Stony Brook interaction~\\cite{sp80} which is an effective \nforce widely used for calculations in the quasi--elastic peak. It is \ngiven as follows: \n\\begin{equation}\n\\label{int-I}\n\\displaystyle\nV^{\\rm I}_{\\rm res} \\, = \\, V_{\\rm LM} \\, + \\, V_\\pi \\, + \\,\n\\tilde{V}_\\rho \\, . \n\\end{equation}\nHere $V_{\\rm LM}$ is a zero--range force of Landau--Migdal type,\nwhich takes care of the short--range piece of the NN\ninteraction: %\n\\begin{equation}\nV_{\\rm LM} \\, = \\,C_0 \\,[ g_0\\, \n{\\mbox{\\boldmath $\\sigma$}}^1\\cdot {\\mbox{\\boldmath $\\sigma$}}^2 \n+\\, g_0^\\prime\\,\n{\\mbox{\\boldmath $\\sigma$}}^1\\cdot {\\mbox{\\boldmath $\\sigma$}}^2\n{\\mbox{\\boldmath $\\tau$}}^1  \\cdot {\\mbox{\\boldmath $\\tau$}}^2]\n\\, .\n\\end{equation}\nOn the other hand, a finite--range component generated by the\n($\\pi$+ $\\rho$)--meson exchange potentials is also included. The\ntilde in $\\tilde{V}_\\rho$ means that the bare $\\rho$--exchange\npotential is slightly modified in order to take into account the\neffect of the exchange of more massive mesons. In particular, a \nfactor $r=0.4$ is multiplying the finite--range non--tensor piece\nof the potential (see Ref.~\\cite{sp80} for details). This force\nwas fitted to reproduce low energy magnetic properties in the\nlead region (specifically, magnetic resonances in $^{208}$Pb and\nmagnetic moments and transition probabilities in the\nneighboring nuclei). The calculations were performed in the\nframework of the RPA and Woods--Saxon single--particle wave\nfunctions were used in the configuration space. The values \n$g_0=0.57$ and $g_0^\\prime=0.717$ (with $C_0=386.04$~MeV~fm$^3$) \nwere found to be adequate to describe the properties \nstudied.\n\nAs previously stated, this interaction has been considered in \ndifferent calculations in the quasi--elastic peak (see\ne.g.~\\cite{gi97}). The problem arise because some of them have been\ndone within the Fermi gas (FG) formalism, with local density\napproximation to describe finite nuclei, in the ring approximation \n(RA), where the exchange terms are not taken into account, and with \nthe full unmodified $\\rho-$exchange potential. Under these \ncircumstances, the possible effects in the nuclear \nresponses due to the inconsistency between the model and the effective\ninteraction could be non--negligible. This is precisely the first aspect\nwe want to investigate. To do that we compare the responses obtained\nwith the J\\\"ulich--Stony Brook interaction with those calculated with\na second effective force of the form: \n\\begin{equation}\n\\label{int-II}\n\\displaystyle\nV^{\\rm II}_{\\rm res} \\, = \\, V_{\\rm LM} \\, + \\, V_\\pi \\, + \\,\nV_\\rho \\, ,\n\\end{equation}\nby considering the same values for the zero--range parameters in both\ncases. The force in Eq.~(\\ref{int-II}) only differs from \n$V^{\\rm I}_{\\rm res}$ in the $\\rho$-potential which, in this case,\ndoes not include any reduction factor. Both RPA and RA effective\ntheories are used to analyze the results.\n\nA second question of interest to us is to determine how the change of\nthe zero--range parameters affects the responses calculated within a\ngiven theory. This will inform us about the necessity of considering\nor not in detail the role of these parameters. This aspect is analyzed\nby considering $V^{\\rm II}_{\\rm res}$ with parameters $g_0$ and \n$g_0^\\prime$ fixed, as in the case of the J\\\"ulich--Stony Brook \ninteraction, to reproduce some low energy properties in the lead\nregion (see details in the next section). It is worth to point out\nthat $V^{\\rm II}_{\\rm res}$ is precisely the interaction used in \npractice in much of the calculations mentioned above and that is why we \nwant to use it for this analysis.\n\nOur analysis focuses on the transverse response functions in the\nquasi--elastic peak. We will not consider the longitudinal ones\nbecause they are strongly influenced by the spin independent pieces of\nthe interaction (in particular, the $f_0$ and $f_0^\\prime$ channels)\nand these are difficult to fix at low energy because of the role\nplayed by the scalar mesons not usually taken into account. \n\n\\section{RESULTS OF THE CALCULATIONS}\n\nThe investigation of the various questions we are interested in has been \ncarried out by comparing different calculations of the transverse (e,e')\nresponses in $^{40}$Ca for three different momentum transfer ($q=300$,\n410 and 550~MeV\/$c$).\n\n\\newpage\n\n\\vspace*{1cm}\n\n\\begin{center}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(100,180)(35,0)\n\\put(0,0){\\ewxy{figu1.ps}{180mm}}\n\\end{picture} \n\\end{center} \n\n\\vspace*{-1.2cm}\n\\noindent\n{\\small {\\sc Fig.~1}. Transverse nuclear responses for $^{40}$Ca, \ncalculated for the three momentum transfers we have considered in this\nwork. Dotted lines correspond to an RPA calculation with \n$V^{\\rm I}_{\\rm res}$, while solid curves represent the RA results for\n$V^{\\rm II}_{\\rm res}$. In both cases the values $g_0=0.57$ and \n$g_0^\\prime=0.717$ (with $C_0=386.04$~MeV~fm$^3$) have been used. \nDashed curves give the free FG responses. In all the calculations a \nvalue of $k_{\\rm F}=235$~MeV\/$c$ has been used.}\n\\vspace{1cm}\n\n\\newpage\n\nFirst we have study the effects produced when an effective\ninteraction, which has been determined in a given effective theory \n(e.g. RPA), is used to calculate (e,e') transverse responses in a \ndifferent framework (e.g. RA).\n\nBy considering the parameterization of Ref.~\\cite{sp80} (that is \n$g_0=0.57$, $g_0^\\prime=0.717$ and $C_0=386.04$~MeV~fm$^3$), we have \ncarried out two different calculations, the results of which are shown\nFig.~1. Therein, solid curves correspond to the calculations performed\nin the FG approach within the RA and with the interaction \n$V^{\\rm II}_{\\rm res}$ in Eq.~(\\ref{int-II}). On the other hand, \ndotted lines have been obtained within the RPA, also for the FG. The \nmodel used in this case is the one developed in Ref.~\\cite{ba96}, \nwhich, contrary to what happens for the RA approach, includes \nexplicitly the exchange terms in the RPA expansion. In this case we \nhave used the interaction $V^{\\rm I}_{\\rm res}$ in Eq.~(\\ref{int-I}) \nand we have adopted the factor $r=0.4$, which is consistent with the \nparameterization used. Also in Fig.~1, we have plotted the free FG \nresponses for comparison (dashed curves). \n\nThe first comment one can draw from these results is that the use of \nthe interaction, as it was fixed at low energy, leads to transverse \nresponses which are quite different from those obtained in the RA\n(with $V^{\\rm II}_{\\rm res}$) calculation, though the differences \nreduce with increasing momentum transfer. As we can see, the results \nobtained in the RPA are peaked at lower energies and this is a clear \nevidence of a more attractive residual interaction. It is \nstraightforward to check this point because the central piece of the \n$V^{\\rm I}_{\\rm res}$ is attractive, while the contrary happens for\n$V^{\\rm II}_{\\rm res}$, at least for $q\\leq 2k_{\\rm F}$. On the other \nhand it is interesting to note how the RA results are more similar to \nthe free response as long as $q$ increases, while the same does not \noccurs for the RPA responses. \n\nObviously, the reason for the discrepancies between both calculations \ncan be ascribed to the two basic ingredients of the effective theories\nused in each case: the exchange terms, which are included in the RPA \ncalculations but not in the RA ones, and the reduction factor $r$ \nmodifying the $\\rho-$exchange potential.\n\nBefore going deeper in this question, it is worth to comment on the\nnuclear wave functions used in the calculations discussed above. As in\nany FG type calculation, plane--waves have been considered here to \ndescribe the single--particle states. The fact that the interaction was \nfixed in a framework which considered microscopic RPA wave functions,\nbased on Woods--Saxon single--particle states, is an obvious \ninconsistency. Despite that, it has been shown~\\cite{am94,am96}\nthat, in this energy region, the details concerning the nuclear wave \nfunctions are not extremely important and, at least to some extent, the \nshell--model response can be reasonably described with the FG model, \nprovided an adequate value of the Fermi momentum, $k_{\\rm F}$, is\nused. In the present work, where we study the response in $^{40}$Ca,\nwe have taken $k_{\\rm F}=235$~MeV\/$c$ which gives a good agreement \nbetween FG and finite nuclei calculations~\\cite{am94}.\n\nWe come back to investigate the reasons for the large discrepancy\nbetween the RPA and RA calculations presented above. To do that we have \ndone two new calculations: RA with $V^{\\rm I}_{\\rm res}$ and RPA with \n$V^{\\rm II}_{\\rm res}$. These calculations have been compared with the\ntwo previous ones by means of the two following quantities:\n\\begin{equation}\n\\label{relexc}\n\\displaystyle\n\\gamma_{\\rm exc}^r(q,\\omega) \\, = \\,\n\\displaystyle\n\\frac{R_T^{{\\rm RPA}(r)}(q,\\omega) \\, - \\, \n      R_T^{{\\rm RA}(r)}(q,\\omega) } \n{R_T^{{\\rm RA}(r)}(q,\\omega)}\n\\end{equation}\nand\n\\begin{equation}\n\\label{relfac}\n\\displaystyle\n\\gamma_r^{\\rm mod}(q,\\omega) \\, = \\,\n\\displaystyle\n\\frac{R_T^{{\\rm mod}(r=1.0)}(q,\\omega) \\, - \\, \n      R_T^{{\\rm mod}(r=0.4)}(q,\\omega) } \n{R_T^{{\\rm mod}(r=0.4)}(q,\\omega) } \\, . \n\\end{equation}\n\nThe first one gives us information about the effect of the consideration\nof the exchange terms in the calculation. The corresponding results have\nbeen plotted in Fig.~2 (left panels). The first aspect to be noted is \nthat the exchange terms produce effects considerably larger for \n$V^{\\rm I}_{\\rm res}$ (solid lines) than for $V^{\\rm II}_{\\rm res}$ \n(dashed curves). These effects reduce with increasing momentum\ntransfer and they are rather small for $V^{\\rm II}_{\\rm res}$ above\n$q=410$~MeV\/$c$. \n\nOn the other hand, the effect of the reduction factor $r$ in the \n$\\rho$--exchange potential is measured with the parameter $\\gamma_r$. \nThe values of this parameter for the two effective models considered, \nthese are RPA and RA, are shown in Fig.~2 (right panels), with solid\nand dashed curves respectively. It is apparent that the effects of \nconsidering the $r$ factor are much larger than those due to the \nexchange. In general they are more important for the RA calculations \nthan for the RPA ones, and reduce the higher $q$ is. \n\n\\newpage\n\n\\vspace*{1cm}\n\n\\begin{center}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(100,180)(30,0)\n\\put(0,0){\\ewxy{figu2.ps}{180mm}}\n\\end{picture} \n\\end{center} \n\n\\vspace*{-1.2cm}\n\\noindent\n{\\small {\\sc Fig.~2}. Left panels: $\\gamma_{\\rm exc}$, in percentage, \nas defined in Eq.~(\\ref{relexc}). Dashed (solid) curves give the results \nobtained for $r=1.0 (0.4)$. Right panels: $\\gamma_r$, in \\%, calculated \nas in Eq.~(\\ref{relfac}), for RPA (solid curves) and RA (dashed curves).}\n\\vspace{1cm}\n\n\\newpage\n\nThe first conclusion to be noted is that when using a given\ninteraction is mandatory to take care of the effective theory where \nits parameterization was fixed. The change of the framework produces \nresults which could not be under control.\n\nThe open question in this respect is how different becomes the \nresponses calculated within different effective theories but with an\ninteraction fixed consistently with the theory. This is the second\naspect we investigate. To do that we have considered the \n$V^{\\rm II}_{\\rm res}$ and have determined the parameters $g_0$ and \n$g_0^\\prime$ of the Landau--Migdal piece in such a way that the \nenergies and B-values of the two $1^+$ states in $^{208}$Pb at 5.85 \nand 7.30~MeV are reproduced. This has been done both in RPA and RA. \nThe reason for choosing these two states lies in their respective \nisoscalar and isovector character, what makes them particularly \nadequate to permit the determination of both parameters almost \nindependently. The values obtained in this procedure are shown in \nTable~I. It is remarkable the small value of $g_0$ needed for the RA\ncalculation. A similar result is found when a pure zero--range \nLandau--Migdal interaction is adjusted, with the same criterion, in \nRPA type calculations (see Refs.~\\cite{co90,hi92}). This points out\nthe importance of the exchange, at least at low energy.\n\n\\vspace{.5cm}\n\\noindent\n{\\small {\\sc TABLE I.} Values of the Landau--Migdal parameters $g_0$ and \n$g_0^\\prime$ obtained in the procedure of fixing the effective interaction \n$V^{\\rm II}_{\\rm res}$ (see text). The values quoted ``RPA'' (``RA'') \ncorrespond to calculations performed with (without) the consideration\nof the exchange terms.}\n\\begin{center}\n\\begin{tabular}{cccc}\n\\hline\\hline\nEffective theory && $g_0$ & $g_0^\\prime$ \\\\ \\hline\nRPA && $~0.470$ & $~0.760$ \\\\\n RA && $~0.038$ & $~0.717$ \\\\\\hline\\hline\n\\end{tabular}\n\\end{center}\n\n\\vspace{1cm}\n\nWith the interaction fixed in this way we have evaluated the\ntransverse responses for the three momentum transfer we are\nconsidering throughout this work. The results are shown in Fig.~3\nwhere dotted (solid) curves correspond to the RPA (RA) calculations.\nDashed lines represent the free FG responses. As we can see, the\ndifferences between the results obtained with the two effective\ntheories are now much smaller than in Fig.~1.\n\n\\newpage\n\n\\vspace*{1cm}\n\n\\begin{center}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(100,180)(35,0)\n\\put(0,0){\\ewxy{figu3.ps}{180mm}}\n\\end{picture} \n\\end{center} \n\n\\vspace*{-1.2cm}\n\\noindent\n{\\small {\\sc Fig.~3}. Transverse nuclear responses for $^{40}$Ca,\ncalculated with the $V^{\\rm II}_{\\rm res}$ interaction. Dotted lines \ncorrespond to an RPA calculation while solid curves represent the RA \nresults. The values of $g_0$ and $g_0^\\prime$ in Table~I have been \nused. Dashed curves give the free FG responses. In all the \ncalculations a value of $k_{\\rm F}=235$~MeV\/$c$ has been used.}\n\n\\vspace{1cm}\n\n\\newpage\n\nTwo points deserve a comment. First, it is clear that the large \ndifferences observed between the RPA calculation here discussed and\nthat shown in Fig.~1 are mainly due to the presence of the reduction \nfactor $r=0.4$ in the $V^{\\rm I}_{\\rm res}$ interaction. Second,\nthe similitude of the results obtained with the two calculations done\nnow with $V^{\\rm II}_{\\rm res}$, shows up the relevance of the link \nbetween effective theories and interactions.\n\nThe last aspect we want to analyze is how the responses calculated in a\ngiven approach change when the zero--range parameters are modified. In\nother words, we want to determine what is the role of these\nparameters. How $g_0^\\prime$ affects the responses is a point which\nhas been investigated with a certain detail in different previous\nworks (see e.g., Ref.~\\cite{ba96}) and then we focus here in $g_0$.\nIts influence can be seen in Fig.~4, where we compare the responses\nplotted in Fig.~3 (solid curves), with those obtained by changing the \n$g_0$ parameter in order to use values considered by different\nauthors. Dashed--dotted curves correspond to $g_0=0$. Dashed lines \nrepresent the responses obtained with $g_0=0.70$ (0.57) for the RPA \n(RA) calculation. The values of $g_0^\\prime$ have not been changed. \nThe first point to be noted is the insensibility of the RA responses \nto the changes in $g_0$. As we can see, strong changes in $g_0$ \nproduce almost no effect on the RA result. This can be easily \nunderstood because in the ring series the $g_0$ contribution is \nweighted with the magnetic moment $\\mu_s^2$ while the $g_0^\\prime$ \npiece appears with $\\mu_v^2$. That means, the $g_0$ contribution is\n$\\mu_s^2\/\\mu_v^2 \\, \\approx \\, 1\/28$ of the $g_0^\\prime$\ncontribution. The situation is different in the RPA case, where the \n$g_0$ contribution is as important as the $g_0^\\prime$ one because \nof the presence of the exchange terms (see Ref.~\\cite{ba96}). This \nmakes that some of the RA calculations performed by other authors can \nbe considered as ``consistent'' in practice, of course despite the \nfact that these parametrizations are unable to reproduce low energy \nproperties. For example, in Ref.~\\cite{gi97}, the parameterization of \nthe J\\\"ulich--Stony Brook interaction was considered and this \ncoincides with one of those used here ($g_0=0.57$ and \n$g_0^\\prime=0.717$).\n\nThe results obtained in this work open a series of questions which we\nconsider worth for nuclear calculations in this energy region. In the\nfollowing we enumerate and comment them:\n\n\\newpage\n\n\\vspace*{.5cm}\n\\begin{center}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(100,170)(24,0)\n\\put(0,0){\\ewxy{figu4.ps}{170mm}}\n\\end{picture} \n\\end{center} \n\n\\vspace*{-1.5cm}\n\\noindent\n{\\small {\\sc Fig.~4}. $R_T$ responses calculated in the RPA (left\npanels) and RA (right panels). Solid curves correspond to\nthe parametrizations of Table~I. Dashed--dotted curves have been \nobtained with $g_0=0$, while the dashed ones correspond to \n$g_0=0.70$ (0.57) for the RPA (RA) calculation,\nwith the same values of $g_0^\\prime$ as for the solid curves.}\n\n\\newpage\n\n\n\\begin{enumerate}\n\n\\item It has been shown that the strength of the tensor piece of \n$V^{\\rm I}_{\\rm res}$ is too strong to describe low energy properties \n(see, e.g., Ref.~\\cite{co90}) and different mechanisms have been \nproposed to cure this problem (core--polarization effects~\\cite{na85},\ntwo--particles two--holes excitations~\\cite{dr86}, {\\it in--medium} \nscaling law~\\cite{br91}, etc.) The role of the tensor part of the \ninteraction in the quasi-elastic peak should be investigated in order \nto establish the effective force to be used.\n\n\\item The presence of the exchange terms increase the sensitivity of \nthe responses to the details of the interaction. How important can be\nthe interference between these terms and other physical mechanisms\nbasic in this energy region (such as, e.g., meson--exchange currents,\nfinal state interactions, short--range correlations, etc.) is a matter\nof relevance in order to fully understand the nuclear response. The\nanalysis of the possible differences between RA and RPA with respect \nto these effects is of special interest in view of the fact that RA \ncalculations are the most usual in the quasi--elastic peak.\n\n\\item The procedure of fixing the interaction is basic in order to deal\nwith the possibility of having an unique framework to calculate the \nnuclear response at any momentum transfer and excitation energy. The \nproblem of developing such ``unified'' model is still unsolved, but\nthe cross analysis of low energy nuclear properties and quasi--elastic\npeak responses could give valuable hints.\n\n\\end{enumerate} \n\n\\section{CONCLUSIONS}\n\nIn this work we have analyzed the role of the effective interaction in\nthe quasi--elastic peak region by comparing the results obtained with\ndifferent effective theories and forces previously fixed in order to \ngive a reasonable description of several low energy nuclear properties.\n\nSome conclusions can be drawn after our analysis. First, it has been \nfound that the interaction plays a role that, similarly to what\nhappens at low excitation energy, cannot be neglected. The particular \npoint to be noted is the necessity of using effective interactions \nwhich have been fixed within an effective theory.\n\nSecond, the procedure we have followed to perform the calculations, \nthat is to determine the interaction at low energy before calculating\nat the quasi--elastic peak, seems to be adequate to look for an\n``unique'' framework to calculate the nuclear response in different\nenergy and momentum regimes. \n\nThe role of the tensor piece of the interaction must be\ninvestigated. At low energy is a basic ingredient of the nuclear\nstructure calculations. Thus it is important to disentangle its\ncontribution in other excitation energy regions. Additionally, it \nseems encouraging to analyze the problem by including other physical \nmechanisms (meson--exchange currents, short--range correlations, \nfinal state interactions, etc.) which are known to be important in \nthe description of the nuclear response and which depend on the \ninteraction. \n\n\\acknowledgements\n\nDiscussions with G. Co' are kindly acknowledged.\nThis work has been supported in part by the DGES (Spain) under \ncontract PB95-1204 and by the Junta de Andaluc\\'{\\i}a (Spain). \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe local mass density of massive black holes residing in the nuclei\nof galaxies (Magorrian et al 1998; Richstone et al 1998) is in good\nagreement with that expected from the intensity of the X-ray\nbackground at 30~keV (Fabian \\& Iwasawa 1999; Salucci et al 1999). The\nX-ray background is assumed to be due to radiatively-efficient\naccretion onto black holes at redshifts $z\\sim 1-2$, with accretion\naccounting for the bulk of their mass. An important requirement is the\npresence of large column densities of absorbing matter, $N_{\\rm H}\\sim\n10^{24}\\hbox{$\\cm^{-2}\\,$}$, around many of the X-ray sources. Intrinsically they\nare assumed to radiate the broad-band spectrum of a typical quasar\n(Elvis et al 1994), with a large UV bump, and a power-law X-ray\ncontinuum of photon index, $\\Gamma\\approx2$, up to a few $100{\\rm\\thinspace keV}$.\nPhotoelectric absorption in this matter then hardens the observed\nspectrum considerably below 30~keV so that the cumulative spectrum\nfrom a population of sources with a range of column densities and\nredshifts resembles that of the observed X-ray Background (Setti \\&\nWoltjer 1989; Madau et al 1994; Celotti et al 1995; Matt \\& Fabian 1994;\nComastri et al 1995; Wilman \\& Fabian 1999). Provided that the sources\nare not too Thomson-thick, it is only above about 30~keV that the\nunabsorbed radiation from growing black holes is observed.\n\nTwo major issues for the model are discussed here; a) the typical mass\nand luminosity of the obscured objects and b) the source of the\nobscuring matter. The first a) is an issue because few luminous\nobscured quasars $L(2-10{\\rm\\thinspace keV})>10^{44}\\hbox{$\\erg\\s^{-1}\\,$}$ have yet been found by\nX-ray (or other waveband) surveys or studies (Boyle et al 1998;\nBrandt et al 1997; Halpern et al 1999), so it might seem that highly\nobscured objects are only of low luminosity. This could be a serious\nproblem for the growth of black holes of mass $10^9\\hbox{$\\rm\\thinspace M_{\\odot}$}$ or more,\nwhich might have luminosities considerably more than that. The second\nb) requires large column densities to obscure most of the Sky as seen\nfrom the sources, in order that most accretion power in the Universe\nis absorbed (Fabian \\& Iwasawa 1999 estimate that about 85 per cent is\nabsorbed). We have previously argued that a circumnuclear starburst is\nresponsible for the obscuration in nearby objects (Fabian et al 1998),\nbut in its simple form could be difficult to sustain if the central\nblack hole takes a Gyr or more to grow to its present mass.\n\nA model is presented here in which a black hole grows with its\nsurrounding stellar spheroid by the accretion of hot gas. The gas is\nalso cooling and forming a distributed gaseous cold component within\nwhich stars slowly form. In the central regions the cold component\nprovides the required column density for absorption of the quasar\nradiation from the accreting black hole. The quasar is assumed to make\na slow wind, which eventually becomes powerful enough to blow away the\nabsorbing gas, probably due to a wind, and an unobscured quasar is\nseen (see Silk \\& Rees 1998 and Blandford 1999 for a similar wind\nmodel). Because the fuel supply for the black hole has also been\nejected however, the quasar dies when its accretion disc is exhausted.\nThe high obscuration phase occurs only during the growing phase and\nthus at redshifts greater than unity which have not been explored yet\nin the hard X-ray band. Reprocessing of the radiation by the absorbing\ngas into far infrared emission may make these objects detectable in\nthe sub-millimetre band.\n\n\\section{Growth of the black hole}\n\nMagorrian et al (1998) find from the demographics of nearby massive\nblack holes that the mass of the hole $M_{\\rm BH}$ is proportional to\nthe mass of the surrounding spheroid $M_{\\rm sph}$ or bulge; $M_{\\rm\nBH}\\approx 0.005M_{\\rm sph}$. There is considerable scatter about this\nrelation of order $\\pm1$~dex. This can be combined with the mass\nfunction of galactic bulges, where most of the mass resides. Salucci et\nal (1999) use the Schechter function luminosity function of bulges to\nobtain the local mass function of black holes. Most of the mass lies\nin black holes of individual mass around the break in the function,\ni.e. $3\\times 10^8\\hbox{$\\rm\\thinspace M_{\\odot}$}$.\n\nTaking this as a typical mass, and assuming growth by accretion with a\nradiative efficiency of $\\eta=0.1 \\eta_{-1}$, the bolometric\nluminosity of the final object $L_{\\rm B}=3\\times 10^{46}-3\\times\n10^{44}\\hbox{$\\erg\\s^{-1}\\,$}$ if it is at one and 0.01 times the Eddington limit\n$L_{\\rm Edd}$, respectively. The Salpeter time for the growth of the\nblack hole, i.e. the mass doubling time, $t_{\\rm s}=3\\times 10^7\n(L_{\\rm Edd}\/L_{\\rm B})\\eta_{0.1}{\\rm\\thinspace yr}.$ If we now assume that the\ninitial mass of all massive black holes is less than that of the\ncentral black hole in our Galaxy, say $\\sim 10^6\\hbox{$\\rm\\thinspace M_{\\odot}$}$, and has grown\nto $3\\times 10^6\\hbox{$\\rm\\thinspace M_{\\odot}$}$ within 3~Gyr, which corresponds to $z\\sim2$,\nthen $8t_{\\rm s}<3\\times 10^9{\\rm\\thinspace yr}$ and the black hole must have grown\nat a rate of at about 10 per cent of the Eddington rate or more. Thus\nover the final $3\\times 10^8{\\rm\\thinspace yr},$ $L_{\\rm B}>3\\times 10^{45}\\hbox{$\\erg\\s^{-1}\\,$}$.\nFrom the work of Elvis et al (1994) the observed 2--10~keV luminosity\nis about 3 per cent of the bolometric one for a quasar so we can\nconclude that $L(2-10{\\rm\\thinspace keV})>10^{44}\\hbox{$\\erg\\s^{-1}\\,$}$ during the growth of a\ntypical black hole and we are dealing with powerful, obscured quasars.\n\nIt is clear that the objects making up the X-ray background, and the\nradiative growth phase of most of the mass in nearby black holes are\nnot represented by any object observed so far. \n\n\\section{The obscuration}\n\nThe spectrum of the X-ray Background requires that most accretion\npower is absorbed. This means that some absorption (say $N_{\\rm\nH}>10^{22}\\hbox{$\\cm^{-2}\\,$}$) occurs in 90 per cent of objects and heavy\nabsorption ($N_{\\rm H}>10^{24}\\hbox{$\\cm^{-2}\\,$}$) in 30--50 per cent of them. 10\nper cent are unabsorbed and are the quasars identified in blue optical\nsurveys or are bright in the soft X-ray band of ROSAT. So far these\ncould be distinct different populations of quasars or different phases\nin the growth of all quasars.\n\nSuch high absorbed fractions mean that the covering fraction of the\nsky by high column density material seen from a growing black hole\nmust approach $4\\pi {\\rm\\thinspace sr}$. This cannot be provided by any thin disc or\nby the standard torus of unified models. The absorbing matter is\nprobably concentrated within the innermost few 100~kpc, or its total\nmass becomes high. It must presumably be cold and fairly neutral, or\nit will not absorb the X-rays. At first sight this is at variance with\nit being space covering, especially if it must be so for a Gyr or\nmore. Such matter should collide and dissipate into a disclike\nstructure.\n\nFollowing our earlier work on a circumnuclear starburst (Fabian et al\n1998), it is plausible that collisions do take place leading to\ndissipation but that some massive star formation occurs in the shocked\nand cooled dense gas. The winds and supernovae from those stars then\nsupply the energy to keep the rest of the cold matter in a chaotic and\nspace covering state.\n\nA more detailed model can be developed by assuming that the black hole\nis growing at the same time as the galaxy does. The stellar spheroid\ncontinues to grow by cooling of the gas heated by gravitational\ncollapse of the protogalaxy. It is likely that the hot phase density\nwhile the galaxy continues to grow is the maximum possible, which\nmeans that the radiative cooling time of the gas equals the\ngravitational infall time. This condition has been studied in the\ncontext of quasar fuelling and growth by Nulsen \\& Fabian (1999). The\ngas accretes in a Bondi flow, probably forming a disc well within the\nBondi radius. The accretion rate is such that it can typically be 10\nper cent of the Eddington value.\n\nThe situation as envisaged here is essentially a maximal cooling flow\nand we can use the properties of observed cooling flows (Fabian 1994\nand references therein) to indicate how the cooled gas is distributed\nand how it may lead to absorption. X-ray observations of cluster\ncooling flows show that the mass deposited by cooling is distributed\nwith $\\dot M(<r)\\propto r$, where $r$ is the radius. The density\ndistribution of the cooled gas is therefore $\\rho\\propto r^{-2}$. If\nthe gravitational potential of the galaxy is isothermal it remains so.\nX-ray absorption with column densities of the order of $10^{21}\\hbox{$\\cm^{-2}\\,$}$\nis also observed in many cluster cooling flows. Although not observed\nin other wavebands, it could represent very cold, dusty gas (Fabian et\nal 1994). \n\nIt is therefore proposed that the cooled gas in protogalaxies does not\nall rapidly form stars but that much of it forms long-lived cold dusty\nclouds. As discussed above, energy from the massive stars helps to\nprevent the clouds rapidly dissipating into a large disc. The inner\nregions of protogalaxies are therefore highly obscured. \n\nThe column density through the cold clouds is obtained from\nintegration of the cold gas density distribution, assumed to be\n$$n=n_0r_0^2r^{-2}.\\eqno(1)$$ If the cold clouds are a fraction $f$ of the\ntotal mass within radius $r, M(<r)=2v^2r\/G,$ then\n$$n_0r_0^2=f{v^2\\over{2\\pi G m_{\\rm p}}}\\eqno(2)$$ and the column density in\nto radius $r_{\\rm in}$ $$N_{\\rm H}={v^2\\over{2\\pi G m_{\\rm p}}}{f\\over\nr_{\\rm in}} =10^{24}f{v_{2.5}^2\\over r_2}\n\\hbox{$\\cm^{-2}\\,$}.\\eqno(3)$$ Here the (line of sight) velocity dispersion of the (isothermal) \nspheroid is\n$300v_{2.5}\\hbox{$\\km\\s^{-1}\\,$}$ and $r_{\\rm in}=100r_2{\\rm\\thinspace pc}.$ It is therefore\nreasonable to assume that column densities at the level required by\nX-ray Background models can be produced in this manner, provided that\n$f$ is fairly high.\n\nIt is interesting that the mass within the radius where the column\ndensity becomes Thomson thick ($N_{\\rm T}=\\sigma_T^{-1}\\sim 2\\times\n10^{24}\\hbox{$\\cm^{-2}\\,$},$ where $\\sigma_{\\rm T}$ is the Thomson electron\nscattering cross section, is given by $$r_{\\rm T}={v^2f\\over{2 \\pi G\nm_{\\rm p} N_{\\rm T}}},\\eqno(4)$$ for which the enclosed mass $$M_{\\rm\nT}={v^4f\\over{2 \\pi G^2 m_{\\rm p} N_{\\rm T}}}.\\eqno(5)$$ $M_{\\rm T}$\nis thus approximately proportional to the mass of the whole spheroid,\n$M_{\\rm sph}$, since approximately $M_{\\rm sph}\\propto v^4$, as\nsuggested by the Faber-Jackson relation amd by galaxy modelling\n(Salucci \\& Persic 1997).  The last authors note that $r_{\\rm\ne}\\propto L_B^{0.7}$ and $M_{\\rm sph}\\propto L_B^{1.35},$ which with\n$v\\propto (M\/r)^{0.5}$ yields a result very close to $M_{\\rm\nsph}\\propto v^4$.\n\nAs the stellar content of the galaxy and the central black hole grow,\nthe accretion radius of the hole also grows. When $M_{\\rm BH}\\sim\nM_{\\rm T}$ the accretion radius is at $r_{\\rm T}$ and the Thomson\ndepth of the obscuring matter around the quasar has dropped to unity. \n\nThe growth of the quasar continues until it runs out of gas. This\ncould be because all the gas has cooled and formed stars. More likely\nit is connected with the central quasar (Silk \\& Rees 1998; Blandford\n1999). Consider the possibility that as well as radiation the quasar\nproduces a wind at velocity $v_{\\rm W}\\ll c$ and with a kinetic power\n$L_{\\rm w}$ which scales with the radiated power $L_{\\rm rad}$ as\n$L_{\\rm w}\/L_{\\rm rad}=L_{\\rm Bol}\/L_{\\rm Edd}$. Thus when the\nbolometric power is 10 per cent of the Eddington value the power of\nthe wind is 10 per cent of the radiated power. Note that winds and\noutflows are observed from many classes of accreting objects (see e.g.\nLivio 1997).\n\nThe ratio of wind to (optically-thin) radiation pressure is then\n$${P_{\\rm w}\\over P_{\\rm rad}}={1\\over 2}{L_{\\rm w}\\over L_{\\rm\nrad}}{c\\over v_{\\rm w}}={1\\over 2}{L_{\\rm Bol}\\over L_{\\rm\nEdd}}{c\\over v_{\\rm w}}.\\eqno(6)$$ For an optically-thin medium of Thomson\ndepth $\\tau_{\\rm T}$, the effective Eddington limit $$L^{\\rm w}_{\\rm\nEdd}=L_{\\rm Edd}{\\tau\\over 2}{v_{\\rm w}\\over c}.\\eqno(7)$$ This means that if\n$v_{\\rm w}\\sim {\\rm 15,000}\\hbox{$\\km\\s^{-1}\\,$}$ and $L_{\\rm Bol}\/L_{\\rm Edd}\\sim 0.1$ then\nthe wind pressure balances gravity. A wind force slightly in excess of\nthis can therefore eject the gas. Since $\\tau \\propto r^{-1}$ and\n$M\\propto r$, this condition applies throughout the galaxy and {\\it\nall} the gas is ejected. (The assumption here that the mass is in a\nthin shell is applicable to the gas as it is swept up.)\n\nThis result can be placed on a surer footing by noting that the force\nbalance between gravity acting on a column of matter at radius $r$ of\ntotal mass $N_{\\rm H} 4\\pi r^2 m_{\\rm p}$ and the outward force due to\na wind $2 L_{\\rm w}\/v$ yields the limiting luminosity $$L^{\\rm w}_{\\rm\nEdd}=2\\pi GMm_{\\rm p}N_{\\rm H} v_{\\rm w}.\\eqno(8)$$ Substituting for $M$\nand $N_{\\rm H}$ in the model galaxy we obtain $$L^{\\rm w}_{\\rm\nEdd}={{{2v^4 f v_{\\rm w}}\\over{G}}}.\\eqno(9)$$ The gas is ejected by the wind\nwhen the wind power exceeds this limit, i.e. when $$L_{\\rm w}>L^{\\rm\nw}_{\\rm Edd}.\\eqno(10)$$ If $L_{\\rm w}=a L_{\\rm Edd}$ then the limit occurs\nwhen $$a {{4 \\pi G M m_{\\rm p} c}\\over \\sigma_{\\rm T}}= {{v^4 f v_{\\rm\nw}}\\over{2G}},\\eqno(11)$$ or at the critical mass $$M_{\\rm c}={v^4\\over{2\\pi\nG^2 m_{\\rm p}N_{\\rm T}}}{v_{\\rm w}\\over c}{f\\over a}={M_{\\rm T}\\over\n2a}{v_{\\rm w}\\over c}.\\eqno(12)$$ Thus if $v_{\\rm w}\\sim 0.1 c$, and $a\\sim\n0.1$ then $M_{\\rm c}\\sim M_{\\rm T}\/2.$ Assuming that most of the mass\nwithin $r_{\\rm T}$ lies in the central black hole again means that\n$M_{\\rm BH}\\sim M_{\\rm T}\\propto M_{\\rm sph}.$ Most of the power\nduring the main growth phase of a massive black hole is then radiated\ninto gas with a Thomson depth of about unity, as required for\nmodelling the X-ray Background spectrum.\n\nClearly $f$ cannot equal unity if a galaxy is to be formed. In\npractice it will be a function of time. The important issue here is\nthat it cannot be small, i.e. it probably lies in the range of 0.1 --\n0.5.\n\nThe black hole mass is therefore proportional to the spheroid mass, as\nobserved (Magorrian et al 1998). The quasar is now unobscured and is\nobservable as an ordinary blue excess object for as long as it has\nfuel. A reasonable estimate would be about a million yr for the disc to\nempty. The ejection phase is tentatively identified with broad\nabsorption line quasars (BAL quasars; see e.g. Weymann 1997).\n\nThe normalization for the relation between $M_{\\rm BH}$  and $M_{\\rm\nsph}$ is obtained by using the Faber-Jackson relation given by Binney\n\\& Tremaine (1992), where $v=220(L\/L_\\star)^{0.25}\\hbox{$\\km\\s^{-1}\\,$}$ and\n$L_\\star=4\\times 10^{10}\\hbox{$\\rm\\thinspace L_{\\odot}$}$ in the V-band, and a mass-to-light\nratio in that band of 6 (both for a Hubble constant of $50\\hbox{$\\kmps\\Mpc^{-1}$}$).\nThe result is $${M_{\\rm T}\\over M_\\star} =0.01\\eqno(13) $$ so\n$${M_{\\rm BH}\\over M_{\\rm sph}}\\approx 0.005,\\eqno(14)$$ for the\nvalues of $a$ and $v_{\\rm w}$ above. This is in good agreement with\nthe relation found by Magorrian et al (1998). In detail, the weak\nvariations in $M\/L$ such as summarized by Salucci \\& Persic (1997)\ncould shift the observed relation from being strictly linear. \n\nNote that the ordinary quasar phase markes the end of the main growth\nphase of both the black hole and its host spheroid. The central black\nhole in galaxies therefore has a profound effect on the whole galaxy,\nin providing a limit to the stellar component. In this context, Silk\n\\& Rees (1998) have shown that a powerful quasar wind could end the\nformation of a galaxy and Blandford (1999) has noted that it could\nprevent the formation of a galactic disc.\n\n\\section{Discussion}\n\nIt has been shown here that a black hole, growing by accretion in the\ncentre of a young, isothermal, spheroidal galaxy which has a large\nmass component of cold gas, will appear highly obscured ($\\tau_{\\rm\nT}>1$). Provided that a significant fraction of the accretion power,\nassumed to be close to the Eddington limit for the black hole, is\nreleased as a sub-relativistic wind, then, as suggested by Silk \\&\nRees (1998) the cold, and accompanying hot, gas components are ejected\nfrom the galaxy when the Thomson depth to the outside $\\tau_{\\rm T}$\ndrops to about unity. The central accretion source then appears as an\nunobscured quasar which lasts until its disc empties. The growth of\nboth the central black hole and the stellar body of the galaxy then\nterminate, unless fresh gas and stars are brought in from outside, for\nexample by a merger. Assuming that most of the mass within the region\nwhere $\\tau_{\\rm T}\\sim 1$ has been accreted into the black hole, it\nis found that its mass $M_{\\rm BH}\\propto M_{\\rm sph}$, the mass of\nthe spheroid of the galaxy.\n\nThe rough explanation for the proportionality is that a more massive\ngalaxy has more cold gas and so requires a more powerful wind to eject\nthe gas. A stronger wind requires a more massive black hole. The\nnormalization results by relating the wind power to the Eddington\nlimit.\n\nThe model accounts for the bulk of the X-ray Background which requires\nthat most accretion power, resulting from the growth of massive black\nholes, is obscured. Also, such obscured accretion leads to agreement\nwith the local mass density of black holes (Fabian \\& Iwasawa 1999).\nMost of the absorbed radiation will be reradiated in the far\ninfrared\/sub-mm bands and contribute to the source counts and\nbackgrounds there. It can plausibly account for some of the sub-mm\nsources recently discovered by SCUBA (Barger et al 1998; Hughes et al\n1998, Blain et al 1999). Indeed it predicts a population of distant\nUltra-Luminous Infrared Galaxies (ULIRGs; Sanders \\& Mirabel 1996)\nassociated with the main growth phase of massive black holes and\ndistinct from the nearby population, which is probably due to mergers\nbriefly fuelling central starbursts and fully grown black holes.\n(Whether radiation from the black hole can then dominate the ULIRG\nemission depends on the Eddington limit, i.e. the mass of the black\nhole, relative to the strength of the starburst; see e.g. Wilman et al\n1999.) The bolometric power of a typical growing black hole must be\nhigh, $L_{\\rm Bol}>10^{46}\\hbox{$\\erg\\s^{-1}\\,$}$; which means an observed 2--10~keV\nluminosity when $N_{\\rm H}\\sim 2\\times 10^{24}\\hbox{$\\cm^{-2}\\,$}$ of about\n$3\\times 10^{-15}\\hbox{$\\erg\\cm^{-2}\\s^{-1}\\,$}$. Current ASCA (e.g. Ueda et al 1998) and\nBeppoSAX (Fiore et al 1999) hard X-ray surveys have not probed deep\nenough to reveal these objects, although they should be obvious in the\ndeeper surveys planned for Chandra and XMM. These surveys will enable\nthe major growth phase of black holes and, with optical\/infrared\nidentifications, their redshift distribution to be studied.\n\nNote that the obscured growth phase of a massive black hole represents\na distinctly different phase from the briefer unobscured phase\npredicted after the gas is ejected and the quasar dies or any later\nphase, obscured or unobscured, when the quasar is revived by a merger\nor other transient fuelling event. These last phases are the ones\nwhich have been observed so far; the major growth phase has not. It is\nunlikely therefore that the properties of the major growth phase are\nobtainable by any simple extrapolation from observations of any\ntransient recent phases, i.e. from studies of the properties of active\ngalaxies at low redshift ($z<0.5$).\n\nIt is important that the cold gas in the young galaxy, which provides\nthe X-ray (and other waveband) obscuration, be metal enriched, and\nprobably dusty. This is likely to be a consequence of continued star\nformation, particularly of massive stars, throughout the galaxy. The\nenergy of stellar winds and supernovae help to keep the cold gas space\ncovering. Note that the injected metals and stellar mass loss will be\ndistributed both as assumed for the stars and the cold gas. Indeed the\nmetallicity of the obscuring gas closest to the black hole may be\nhigher than the solar values, which leads to better fits to the X-ray\nBackground spectrum (Wilman \\& Fabian 1999). This also accounts for\nthe high metallicity inferred in the broad-line region gas for many\nquasars (Hamann \\& Ferland 1999).\n\nThe column density distribution of the gas obscuring the radiation\nfrom the growing black hole will be such that most power is emitted\njust before the gas is ejected, which happens around $\\tau_{\\rm T}\\sim\n1$. The fraction of the power radiated at optical depths greater than\n$\\tau_{\\rm T}$ scales roughly as $\\tau_{\\rm T}^{-1}$. Angular momentum\nand other factors may cause the gas in the young spheroidal galaxy,\nand the wind, to not be completely spherical. Thus when the wind\nejects the gas it may do so most along one axis and only later eject\ngas in other directions (if at all). This means that the column\ndensity distribution for $\\tau_{\\rm T}<1$ is complicated to predict\nand is best found from the shape of the X-ray Background spectrum.\n\nAn important consequence for the stellar radiation in young growing\ngalaxies is that most of it too is obscured. This agrees with recent\nevidence for the star formation history of the Universe from sub-mm and\nother observations. \n\nThe ejection of the metal-rich cold gas in the young galaxy should\nterminate its stellar growth, as well as growth of the black hole. The\nfinal appearance of a galaxy is thus significantly affected by its\ncentral black hole. How far the gas is ejected depends on how long the\nunobscured quasar phase lasts and what the surrounding gas mass and\ndensity is; whether for example the galaxy is in a group or cluster.\nThe most massive black holes will be in the most massive galaxies and\nmay last longest in the unobscured quasar phase. They might also be\nsurrounded by a hot intragroup medium which could prevent much of the\nhotter space-filling phase from being ejected. If a surrounding hot\nphase is a necessary ingredient for a radio source then such objects\nmight be more likely to be radio galaxies.\n\nDuring the ejection phase the quasar might be classed as a BAL and\nlater it might be seen to be surrounded by extended metal-rich\nfilaments, depending on the velocity of ejection of the cold gas. The\nmetal-rich gas, if mixed with surrounding hot intracluster gas, will\nenhance the local metallicity, providing one source for the extensive\nmetallicity gradients found by X-ray spectroscopy around many cD\ngalaxies in clusters (Fukazawa et al 1994). \n\nFinally, it is noted that the model requires a significant power\noutput in the form of a wind associated with the growth of black\nholes. This wind power is dissipated as heat in the surrounding\nmedium. It may have a marked effect on surrounding intracluster gas\n(Ensslin et al 1998; Wu, Fabian \\& Nulsen 1999), possibly contributing\nto the heating required to change the X-ray luminosity--temperature\nrelation, $L_{\\rm x}\\propto T_{\\rm x}^\\alpha$, from the predicted one\nwith $\\alpha\\sim 2$ to the observed one with $\\alpha\\sim 3.$ The\nestimates of Wu et al (1999) indicate that it will also heat the\ngeneral intergalactic medium to a temperature of $\\sim 10^7{\\rm\\thinspace K}$ at\n$z\\sim 1-2$.\n\nIn summary, the growth of both massive black holes and galactic bulges\nis a highly obscured, and related, process, best observed directly in\nthe hard X-ray band and indirectly, through radiation of the absorbed\nenergy, in the sub-mm band.\n\n\n\n\n\\section{Acknowledgements}\n\nI thank the referee for comments and The Royal Society for support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince its inception in 1959 by Erd\\\"os and R\\'enyi \\cite{ER}, the theory of random graphs has developed into \na rapidly growing and widely applicable branch of discrete mathematics, bringing together ideas from graph theory, combinatorics, and probability theory. In one model, a random graph is a subgraph $\\Gamma$ of a complete graph on $n$ vertices such that every edge of the complete graph \nis included in $\\Gamma$ with probability $p$, independently of the other edges.  One is interested in probabilistic features of $\\Gamma$ and their dependence on $p$ when $n$ is large. Here $0<p<1$ is a probability parameter which in general may depend on $n$. The theory of random graphs \n\\cite{AS, B, JLR} offers many spectacular results and predictions, which  play an essential role in various engineering and computer science applications. Random graphs also serve within mathematics as accessible models for other, more complex random \nstructures.  \n\nHigher dimensional analogs of the aforementioned \nErd\\H{o}s--R\\'enyi model were recently suggested and studied by Linial--Meshulam in~\\cite{LM}, and \nMeshulam--Wallach in~\\cite{MW}.\nIn these models, one generates a random $d$-dimensional simplicial complex $Y$ by considering the full $d$-dimensional skeleton of the simplex \n$\\Delta_n$ on vertices $\\{1, \\dots, n\\}$ and retaining $d$-dimen\\-sional faces independently with probability $p$.  Note that in this construction $Y$ contains the $(d-1)$-dimensional \nskeleton of $\\Delta_n$. \nThe work of Linial--Meshulam and\nMeshulam--Wallach provides threshold functions for the vanishing of the\n$(d-1)$-st homology groups of random complexes with coefficients in a~finite\nabelian group. Threshold functions for the vanishing of the $d$-th homology groups were subsequently studied by Kozlov \\cite{Ko}.\n\nIn this paper, we focus on 2-dimensional random complexes. The corresponding probability space $G(\\Delta_n^{(2)}, p)$ of the Linial--Meshulam model is defined as follows. Let $\\Delta_n$ denote the $(n-1)$-dimensional simplex \nwith vertices $\\{1, 2, \\dots, n\\}$. Then $G(\\Delta_n^{(2)}, p)$ denotes the set of all 2-dimensional subcomplexes \n$$\\Delta_n^{(1)}\\subset Y\\subset \\Delta_n^{(2)},$$ containing the one-dimensional skeleton $\\Delta_n^{(1)}$. The probability function \n$\\P: G(\\Delta_n^{(2)},p)\\to {\\mathbf R}$ is given by the formula\n$$\\P(Y) = p^{f(Y)}(1-p)^{{n\\choose 3}-f(Y)}, \\quad Y\\in G(\\Delta_n^{(2)}, p),$$\nwhere $f(Y)$ denotes the number of faces in $Y$.  In other words, each of the 2-dimensional simplexes of $\\Delta_n^{(2)}$ is included in a random 2-complex $Y$ with probability $p$, independently of the other 2-simplexes. As in the case of random graphs, $0<p<1$ is a probability parameter which may depend on $n$.  \nWhen $n$ grows, the model $G(\\Delta_n^{(2)}, p)$ includes all finite $2$-dimensional complexes containing the full 1-skeleton $\\Delta_n^{(1)}$; however, the likelihood of various topological phenomena is dependent on the value of $p$. The theory of deterministic 2-complexes itself is a rich and active field of current research with many challenging open questions, see \\cite{Hog}.\n\nThe fundamental group of a random 2-complex $Y\\in G(\\Delta_n^{(2)}, p)$ was investigated by Babson, Hoffman, and Kahle \\cite{BHK}.  They showed that for \n$p\\gg n^{-1\/2}\\cdot (3\\log n)^{1\/2}$, the group $\\pi_1(Y)$ vanishes asymptotically almost surely (i.e., the probability that $\\pi_1(Y)$ is trivial tends to $1$ as $n\\to\\infty$). \nFor $p\\ll n^{-1\/2-\\epsilon}$, these authors use notions of negative curvature due to Gromov to study the nontriviality and hyperbolicity of $\\pi_1(Y)$.\n\nIn this paper, we show that for $p\\ll n^{-1-\\epsilon}$ a random 2-complex $Y$ is homotopically 1-dimensional, a.a.s.\\footnote{We use the abbreviation a.a.s.~for the phrase \\lq\\lq asymptotically almost surely\\rq\\rq.}\nMore precisely, we show that $Y$ can be collapsed to a graph in finitely many steps. This implies that $Y$ has a free fundamental group and vanishing 2-dimensional homology. Note that the vanishing of 2-dimensional homology in this range of $p$ also follows from a result of Kozlov \\cite{Ko}. In \\cite{CFK}, it is shown that for $p>3\/n$, the homology group $H_2(Y;{\\mathbf Z})$ is nontrivial with probability tending to $1$; see also \\cite{Ko}. Thus, \nfor $p>3\/n$, the random 2-complex $Y$ is homotopically two-dimensional a.a.s. \n\nOur main result  is as follows: \n\n\n\\begin{theorem}\\label{main} \n  (a) If for some $k\\ge 1$ the probability parameter $p$ satisfies\\footnote{Recall that the symbol $a_n\\ll b_n$ means that $a_n>0$ and $a_n\/b_n \\to 0$ as $n \\to \\infty$.}\n\\[\n\\label{ass} p\\ll n^{-1-\\frac{2}{k+1}},\n\\] then a random 2-complex \n$Y\\in G(\\Delta_n^{(2)},p)$ is collapsible to a graph in at most $k$ steps, asymptotically almost surely (a.a.s). \n(b) If for some $k\\ge 1$ the probability parameter $p$ satisfies \n\\[\np\\gg n^{-1-\\frac{1}{3\\cdot 2^{k-1}-1}},\n\\]\nthen $Y$ is not collapsible to a graph in $k$ or fewer steps, a.a.s.\n\\end{theorem}\n\nLoosely speaking, Theorem \\ref{main} combines with previously known results to suggest that {\\it a random 2-complex with vanishing 2-dimensional homology is homotopically one-dimensional. }\n\nTheorem \\ref{main} implies:\n\n\n\\begin{corollary}\nIf for some $k\\ge 1$ the probability parameter $p$ satisfies  $$p\\ll n^{-1-\\frac{2}{k+1}}$$\nthen the fundamental group $\\pi_1(Y)$ of a random 2-complex \n$Y\\in G(\\Delta_n^{(2)},p)$ is free and $H_2(Y;{\\mathbf Z})=0$, a.a.s.\n\\end{corollary}\n\nThe proof of Theorem \\ref{main} is given at the very end of the paper. A key role is played by Theorem \\ref{thm1}, \nwhich states that there exists a finite list of forbidden 2-complexes $\\mathcal L_{k,r}$ with $k\\ge 0$ and $r\\ge 2$, \nsuch that an arbitrary 2-complex of degree at most $r$ (see below) is collapsible to a graph in $k$ steps if and only if it does not contain any of the 2-complexes from $\\mathcal L_{k,r}$. This allows us to reduce the collapsiblity problem to the containment problem for random complexes which was studied in \\cite{CFK}. \n\n\n\\subsection*{Acknowledgments}\nThis research was implemented during visits of M. Farber to FIM ETH Z\\\"urich and Louisiana State University, and a visit of D. Cohen to the University of Z\\\"urich.  Portions of this work were carried out during the Spring of 2010, when the first two authors participated in the Mathematisches Forschungsinstitut Oberwolfach Research in Pairs program. \n  We thank the FIM ETH, LSU, the University of Z\\\"urich, and the MFO for their support and hospitality, and for providing productive mathematical environments.\n\n\\section{Collapsibility of a 2-complex to a graph}  \n\n\\subsection{Basic definitions}\n\nLet $Y$ be a finite 2-dimensional simplicial complex. An edge of $Y$ is called {\\it free} if it is included in exactly one 2-simplex.\n\nThe {\\it boundary} $\\partial Y$ is defined as the union of free edges.  We say that a 2-complex $Y$ is {\\it closed} if $\\partial Y=\\emptyset$. \n\nA $2$-complex \n$Y$ is called {\\it pure} if every maximal simplex is 2-dimensional. By the {\\it pure part} of a 2-complex we mean the maximal pure subcomplex, i.e. the union of all 2-simplexes. \n\n\n\nLet $Y$ be a simplicial 2-complex and let $\\sigma$ and $\\tau$ be two 2-simplexes of $Y$. \nWe say that $\\sigma$ and $\\tau$ are adjacent if they intersect in an edge.\nThe {\\it distance} between $\\sigma$ and $\\tau$, $d_Y(\\sigma, \\tau)$, is the minimal integer $k$ such that there exists a sequence of 2-simplexes $\\sigma=\\sigma_0, \\sigma_1, \\dots, \\sigma_k=\\tau$ with the property that $\\sigma_i$ is adjacent to $\\sigma_{i+1}$ for every $0\\le i<k$. (If no such sequence exists then $d_Y(\\sigma, \\tau)=\\infty$.) The {\\it diameter }\n${\\rm {diam}}(Y)$ is defined as the maximal value of $d_Y(\\sigma, \\tau)$ taken over pairs of 2-simplexes of $Y$. \n\nA simplicial 2-complex is {\\it strongly connected} if it has a finite diameter. \n\nA simplicial 2-complex  has {\\it degree $\\le r$} if every edge is incident to at most $r$ 2-simplexes.\n\nA {\\it pseudo-surface} is a finite, pure, strongly connected 2-dimensional simplicial complex of degree at most $2$ (i.e., every edge is included in at most two 2-simplexes). \n\nMore generally, for an integer $r>0$, an {\\it $r$-pseudo-surface} is a finite, pure, strongly connected 2-dimensional simplicial complex of degree at most $r$.\n\n \n\n\\subsection{Simplicial collapse} Let $Y$ be a 2-complex. A 2-simplex of $Y$ is called {\\it free} if at least one of its edges is free. \nLet $\\sigma_1, \\dots, \\sigma_k$ be all free 2-simplexes in $Y$, and let \n$e_1, \\dots, e_k$ be free edges with $e_i\\subset \\sigma_i$. We say that the complex  \n$$Y'={Y-\\cup_{i=1}^k {\\rm {int}}(\\sigma_i) - \\cup_{i=1}^k {\\rm {int}}(e_i)}$$ \nis obtained from $Y$ by collapsing all free 2-simplexes. Clearly $Y'\\subset Y$ is a deformation retract. The operation $Y\\searrow Y'$ is called a {\\it simplicial collapse}. \nNote that $Y'$ is not uniquely determined if one of the free simplexes of $Y$ has two free edges; however the pure part of $Y'$ (i.e. the union of 2-simplexes of $Y'$) is uniquely determined. \n\nThis process can be iterated $Y'\\searrow Y''$, $Y''\\searrow Y'''$, etc. We denote $Y=Y^{(0)}$, $Y'=Y^{(1)}$, $Y''=Y^{(2)}$ etc. \nThe sequence of subcomplexes $Y^{(0)}\\supset Y^{(1)}\\supset Y^{(2)}\\supset \\dots$ is decreasing and there are two possibilities: either (a)  for some $k$, the complex $Y^{(k)}$ is one-dimensional (a graph), or (b) for some $k$, the complex $Y^{(k)}$ is 2-dimensional and closed, i.e., $\\partial Y^{(k)}=\\emptyset$. \n\n\\begin{definition} We say that $Y$ is collapsible to a graph in at most $k$ steps if $Y^{(k)}$ is a graph. \nWe say that $Y$ is collapsible to a graph in $k$ steps if $Y^{(k)}$ is a graph and $\\dim Y^{(k-1)}=2$. \n\\end{definition}\n\n\nObserve that {\\it if $Y$ is collapsible to a graph in at most $k$ steps then any simplicial subcomplex $S\\subset Y$ is also collapsible to a graph in at most $k$ steps}. \nAt each step one removes the free triangles in $Y^{(i)}$ which belong to $S$. \n\nLet $Y$ be a 2-complex, and consider the sequence of collapses\n\\[\nY^{(0)}\\searrow Y^{(1)}\\searrow Y^{(2)}\\searrow\\dots \\searrow Y^{(k)} \\searrow \\dots.\n\\] \nFor a 2-simplex $\\sigma\\in Y$ define $$ D_Y(\\sigma) = \\sup \\{i; \\, \\sigma\\subset Y^{(i)}\\}\\, \\, \\in \\{0, 1, \\dots, \\infty\\}.$$ \nA 2-simplex $\\sigma$ is free if and only if $D_Y(\\sigma)=0$. \n\nA 2-complex $Y$ is collapsible to a graph in at most $k+1$ steps if and only if $D_Y(\\sigma)\\le  k$ for any $2$-simplex $\\sigma$. \nIf after performing several collapses $Y^{(0)}\\searrow Y^{(1)}\\searrow Y^{(2)}\\searrow\\dots $ \nwe obtain a subcomplex $Y^{(r)}\\subset Y$ with empty boundary $\\partial Y^{(r)}=\\emptyset$, \nthen $Y^{(r)}=Y^{(r+1)}=Y^{(r+2)}=\\dots$ and $D_Y(\\sigma)=\\infty$ for any simplex $\\sigma$ in $Y^{(r)}$.\n\n\\begin{lemma} Let $\\sigma$ be a 2-simplex with $D_Y(\\sigma)=k$ where $0<k<\\infty$. Then one of the edges $e$ of $\\sigma$ has the following property: \nfor any 2-simplex $\\sigma'$ of $Y$ which is incident to $e$ and distinct from $\\sigma$ one has $D_Y(\\sigma')<k$ and there exists a 2-simplex $\\sigma'$ incident to $e$ and distinct from \n$\\sigma$ such that $D_Y(\\sigma')=k-1$. \n\\end{lemma}\n\\begin{proof}\nSince $D_Y(\\sigma)=k$, we know that after $k$ collapses an edge $e$ of $\\sigma$ becomes free. All other simplexes $\\sigma'$ of $Y$ incident to $e$ must have been eliminated in previous steps, i.e., they satisfy $D_Y(\\sigma')<k$. At least one of these simplexes $\\sigma'$ must have been eliminated in step $k-1$ since otherwise $\\sigma$ would have become free earlier. \n\\end{proof}\n\n\\begin{lemma}\\label{ge}\nIf $Z\\subset Y$ is a subcomplex\nand $\\sigma\\subset Z$ is a $2$-simplex, then \n$$D_Z(\\sigma) \\le D_Y(\\sigma).$$\n\\end{lemma}\n\\begin{proof}\nIf a 2-simplex belongs to $Z$ and is not free in $Z$ then it is not free in $Y$. This implies that $Z'\\subset Y'$ and therefore $Z^{(i)}\\subset Y^{(i)}$ for any $i\\ge 1$. \nThus, the maximal $i$ such that $\\sigma$ is contained in $Z^{(i)}$ is less than or equal to the maximal $i$ such that $\\sigma$ is contained in $Y$, which implies the statement of the Lemma.\n\\end{proof}\n\n\n\\subsection{$\\sigma$-accessible boundary}\n\n\\begin{definition}\nLet $Y$ be a 2-complex and let $\\sigma, \\tau$ be two 2-simplexes of $Y$ with $D_Y(\\tau)=0$ and $D_Y(\\sigma)=k\\ge 1$.\n A {\\it collapsing path} from $\\tau$ to $\\sigma$ is a sequence of 2-simplexes $\\tau=\\sigma_0, \\sigma_1, \\dots, \\sigma_{k-1}, \\sigma_k=\\sigma$ \nsuch that $D_Y(\\sigma_i)=i$ and each pair $\\sigma_i$ and $\\sigma_{i+1}$ has a common edge, where $i=0, \\dots, k-1$.\n\\end{definition}\n\nIn a collapsing path, the initial simplex $\\sigma_0=\\tau$ is a free simplex, and hence at least one of its edges belongs to the boundary $\\partial Y$. \n\n\\begin{definition}\\label{defaccess} Given a 2-simplex $\\sigma$, we denote by $A_Y(\\sigma)\\subset \\partial Y$ the union of the edges in $\\sigma_0\\cap \\partial Y$ \nwhich can appear in a collapsing path  $\\sigma_0, \\sigma_1, \\dots, \\sigma_k$  ending at $\\sigma$. We call $A_Y(\\sigma)$ the $\\sigma$-accessible part of the boundary.\n\\end{definition}\nIn Definition \\ref{defaccess}, clearly $k=D_Y(\\sigma)$.\nNote that $A_Y(\\sigma) \\not=\\emptyset$ if and only if $D_Y(\\sigma)<\\infty$. \n\n\\begin{definition} Let $\\sigma$ be a 2-simplex of $Y$ with $D_Y(\\sigma) \\ge 1$. For an edge $e$ of $\\sigma$ define \n$$A_Y(\\sigma, e)\\subset A_Y(\\sigma)$$ as the set of all edges $e'$ of the boundary \n$\\partial Y$ with the property that there exists a collapsing path $\\sigma_0, \\sigma_1, \\dots, \\sigma_k=\\sigma$ such that $e'$ is an edge of $\\sigma_0$ and $e=\\sigma_{k-1}\\cap \\sigma_k$.\n\\end{definition}\n\n\nIf $e_1, e_2, e_3$ are the edges of $\\sigma$ then $A_Y(\\sigma)= \\cup_{i=1}^3 A_Y(\\sigma, e_i)$ and the sets $A_Y(\\sigma,e_i)$ need not be mutually disjoint. \n\n\n\n\\begin{lemma} \\label{dadj} Let $\\sigma$ and $\\sigma'$ be adjacent 2-simplexes of $Z$ with $$D_Z(\\sigma)= D_Z(\\sigma')+1.$$ \nAssume that any collapsing path in $Z$ ending at $\\sigma$ passes through the edge\n$e=\\sigma\\cap \\sigma'$. \nIf $Z$ is embedded as a subcomplex $Z\\subset Y$ and $$D_Z(\\sigma')<D_Y(\\sigma'),$$ then \n$$D_Z(\\sigma)< D_Y(\\sigma).$$\n\\end{lemma}\n\n\\begin{proof}  \nLet $k=D_Z(\\sigma') =  D_Z(\\sigma) - 1$. We must show that $D_Y(\\sigma) \\ge k+2.$ First we claim that the edge $e$ may become free only after at least $k+2$ collapses in $Y$. Assume it is free in $Y$ after $k+1$ collapses. By assumption, $D_Y(\\sigma') \\ge k+1.$ Hence the edge $e$ can only be free after $k+1$ collapses in $Y$ if $\\sigma$ has been removed already before, i.e., $D_Y(\\sigma) \\le k.$ On the other hand, by Lemma \\ref{ge}, \n$D_Y(\\sigma) \\ge D_Z(\\sigma) = k + 1$ which leads to a contradiction. \n\nBy assumption, the two edges of $\\sigma$ different from $e$ are not free in $Z^{(k+1)}$ and hence they are not free in $Y^{(k+1)}$. Thus $D_Y(\\sigma) \\ge k + 2$ as claimed.\n\\end{proof}\n\n\n\nNote that the assumption of Lemma \\ref{dadj} that any collapsing path in $Z$ ending at $\\sigma$ passes through the edge $e$ is equivalent to $A_Z(\\sigma, e')=\\emptyset$ for the two remaining edges $e'\\not=e$ of $\\sigma$.\n\n\n\n\n\n\\begin{lemma}\\label{lm8} Let $Z\\subset Y$ be a subcomplex. If $D_Z(\\sigma) = D_Y(\\sigma)$ for a 2-simplex $\\sigma$ of $Z$ then there is an edge $e$ of $\\sigma$ such that \n$$\\emptyset \\not= A_Z({\\sigma, e}) \\subset A_Y({\\sigma, e})\\subset \\partial Y.$$\n\\end{lemma} \n\n\n\\begin{proof}\nWithout loss of generality, we may assume that $Y$ is obtained from $Z$ by attaching a single $2$-simplex.\n\nThe proof is by induction on $k=D_Y(\\sigma)=D_Z(\\sigma)$.\n\nIn the case $k=0$, there is an edge $e$ of $\\sigma$ that is free in both $Z$ and $Y$.  In particular, $e\\subset\\partial Y$.\n\nWe include the case $k=1$.  Recall that $Z'=Z^{(1)}$ denotes the result of the first collapse of $Z$, $Z\\searrow Z'$.  \nSince $D_Z(\\sigma)=D_Y(\\sigma)=1$, there is an edge $e$ of $\\sigma$ that is free in $Y'$ and hence in  $Z'$.  \nThen every collapsing path $\\tau,\\sigma$ in $Z$ with $e=\\tau\\cap\\sigma$ is also a collapsing path in $Y$. Hence \n$A_Z({\\sigma, e}) \\subset A_Y({\\sigma, e})$.\n\n\nFor the general case, assume that $D_Y(\\sigma)=D_Z(\\sigma)=k$.  After $k$ collapses\n\\[\nZ\\searrow Z^{(1)}\\searrow \\dots \\searrow Z^{(k)},\\quad\nY\\searrow Y^{(1)}\\searrow \\dots \\searrow Y^{(k)},\n\\]\nthe $2$-simplex $\\sigma$ is exposed in both $Z^{(k)}$ and $Y^{(k)}$.  Thus, $\\sigma$ has a free edge $e$\n in $Y^{(k)}$ (and hence in $Z^{(k)}$ as well).  Writing $Z'=Z^{(1)}$ and $Y'=Y^{(1)}$, by induction, we have\n $\\emptyset\\not=A_{Z'}({\\sigma,e}) \\subset A_{Y'}({\\sigma,e})$ so that any collapsing path $\\sigma_1,\\dots,\\sigma_k$ from  $\\sigma_1=\\sigma' \\subset A_{Z'}({\\sigma,e})$ to \n$\\sigma_k=\\sigma$ in $Z'$ is also a collapsing path in $Y'$.  \n Note in particular that every edge of $\\sigma'$ that is free in $Z'$ is also free in $Y'$.  Consequently, for every free triangle $\\tau$ in $Z$ which meets $\\sigma'$ in \nan edge free in $Z'$, the collapsing path\n $\\tau=\\sigma_0,\\sigma_1,\\dots,\\sigma_k$ in $Z$ is a collapsing path in $Y$.  The result follows.\n\\end{proof}\n\n\\begin{corollary}\\label{corgood}\nLet $Z\\subset Y$ be 2-complexes such that for a 2-simplex $\\sigma$ of $Z$ none of the edges $e\\in A_Z(\\sigma)\\subset \\partial Z$ is free in $Y$. Then\n$$ D_Z(\\sigma)+1\\le D_Y(\\sigma).$$\n\\end{corollary}\n\\begin{proof} For a contradiction, assume that $D_Y(\\sigma) \\le D_Z(\\sigma)$.  Then $D_Y(\\sigma) =D_Z(\\sigma)$ by Lemma \\ref{ge}. \nWe may now apply Lemma \\ref{lm8} which claims that there is an edge $e$ of \n$\\sigma$ for which $\\emptyset\\not=A_Z(\\sigma, e)\\subset A_Y(\\sigma, e)\\subset \\partial Y$.\nThis contradicts our assumption that no edge in $A_Z(\\sigma)$ lies on the boundary \n$\\partial Y$. \n\\end{proof}\n\n\\subsection{The list of forbidden $r$-pseudo-surfaces $\\mathcal L_{k,r}$} \n\nFor a pair of integers $k=0, 1, \\dots,$  and $r=2, 3, \\dots$ we denote by $\\mathcal L_{k,r}$ the set of all isomorphism types of \n$r$-pseudo-surfaces $S$ with the following properties:\n\\begin{enumerate}\n  \\item[(a)] Each $S\\in \\mathcal L_{k,r}$ has a specified 2-simplex $\\sigma_\\ast$ (called {\\it the center}). \n  \\item[(b)] If $\\partial S\\not=\\emptyset$ then $D_S(\\sigma_\\ast) =k$. \n \n  \\item[(c)] $d_S(\\sigma_\\ast, \\sigma)\\le k$ for any 2-simplex $\\sigma$. \n  \\end{enumerate}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\resizebox{11cm}{4cm}{\\includegraphics[20,412][548,595]{lone.eps}}\n\\end{center}\n\\caption{Surfaces $\\mathcal L_{1,2}$.}\\label{lone}\n\\end{figure}\n\n\\noindent Note that $\\mathcal L_{0,r}=\\{S\\}$ consists of a single complex $S=\\sigma_\\ast$ (the triangle). \n\nThe set $\\mathcal L_{1,2}$ consists of the three surfaces shown in Figure \\ref{lone}. Each of the surfaces \na, b, c is a union of 4 triangles. The surface c is a tetrahedron, b is a tetrahedron with one face open, and a is a fully flattened tetrahedron. \n\nIt is clear that $\\mathcal L_{k,r}$ is finite and $\\mathcal L_{k,r} \\subset \\mathcal L_{k, r+1}$. \n\n\\begin{example}\\label{ex1}{\\rm Consider the following important family of surfaces $S_k\\in \\mathcal L_{k, 2}$ where $k=0,1,2,\\dots$. The first surface $S_0$ is defined as a single triangle $S_0=\\sigma_\\ast$. The next surface $S_1$ is the\nshown in Figure \\ref{lone} a. Surfaces $S_2$ and $S_3$ are shown in Figure \\ref{sk}. In general, the surface $S_k$ is obtained from $S_{k-1}$ by adding a triangle to every edge of the boundary \n$\\partial S_{k-1}$. It is clear that for the central triangle $\\sigma_\\ast$ of $S_k$, one has $D_{S_k}(\\sigma_\\ast)=k$. Thus $S_k$ is not collapsible to a graph in $k$ steps, but is collapsible in $k+1$ steps.}\n\\end{example}\n\\begin{figure}[h]\n\\begin{center}\n\\resizebox{10cm}{4.5cm}{\\includegraphics[73,382][501,597]{sk.eps}}\n\\end{center}\n\\caption{Surfaces $S_k\\in \\mathcal L_{k,2}$.}\\label{sk}\n\\end{figure}\n\n\n\nThe following Theorem plays a key role in this paper:\n\n\\begin{theorem}\\label{thm1}\nA  2-complex $Y$ of degree at most $r\\ge 2$ is not collapsible to a graph  in $k$ steps, where $k=0, 1, 2, \\dots$, if and only if there is a surface $S\\in \\mathcal L_{k,r}$\nwhich admits a simplicial embedding $S\\to Y$. \n\\end{theorem}\n\nIn the proof, we will use the following statement: \n\n\\begin{lemma}\\label{lm10}\nLet $Y$ be a finite 2-dimensional simplicial complex of degree at most $r$ and let $\\sigma$ be a 2-simplex in $Y$ with $D_Y(\\sigma)=k$, where $k=0, 1, 2, \\dots$.\nThen there exists a surface $S\\in \\mathcal L_{k,r}$ and a simplicial embedding $S\\to Y$ such that the central simplex $\\sigma_\\ast$ of $S$ is mapped onto $\\sigma$. \n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma \\ref{lm10}] We will use induction on $k=D_Y(\\sigma)$. \nFor $k=0$, the statement is obvious. Assume that it is true for all cases with $D_Y(\\sigma)<k$, and consider the situation when $D_Y(\\sigma)=k>0$. If $Y\\searrow Y'$ is the first collapse, then \n$\\sigma\\subset Y'$ and clearl\n$$D_{Y'}(\\sigma)=k-1$$\nand $Y'$ has degree at most $r$. \nBy the inductive hypothesis, there exists $S'\\in \\mathcal L_{k-1,r}$ and a simplicial embedding $S'\\to Y'$, mapping the central simplex of $S'$ onto $\\sigma$. \n\nFor each edge \n$e$ lying in $A_{S'}(\\sigma)$ choose a 2-simplex $\\sigma_e\\subset Y$ as follows. If $e\\subset \\partial Y'$, let $\\sigma_e$ be any free triangle  in $Y$ containing $e$. \nIf $e\\not\\subset \\partial Y'$, let $\\sigma_e$ be any triangle in $Y'$ containing $e$ which is not in $S'$; such $\\sigma_e$ exists since $e\\not\\subset \\partial Y'$. \n\nNext we define a subcomplex $S\\subset Y$ as the union\n$$S=S'\\cup \\bigcup_{e}\n\\sigma_e \\, \\subset Y,$$\nwhere $e$ runs over the edges in $A_{S'}(\\sigma)$. Note that $S$ is finite, pure, and strongly connected since $S'$ is an $r$-pseudo-surface.  \nMoreover, the degree of $S$ is at most $r$ since it is a subcomplex of $Y$. One has \n$D_S(\\sigma)\\ge k$\nby Corollary \\ref{corgood}. More precisely, we obtain that $D_S(\\sigma)=k$\nby Lemma \\ref{ge}. Finally we observe that obviously $d_S(\\sigma, \\sigma')\\le k$ for any 2-simplex $\\sigma'$ of $S$.\nThus, $S\\in \\mathcal L_{k,r}$. \\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm1}] Consider the sequence of successive collapses $Y\\searrow Y^{(1)}\\searrow Y^{(2)} \\searrow Y^{(3)}\\searrow \\dots$. \nWe assume that $Y$ is not collapsible to a graph in $k$ steps, which implies that there are two possibilities: either (a) $Y^{(i)}\\not= Y^{(i+1)}$ for any $i<k$;  or (b) for some $i<k$, \none has $\\partial Y^{(i)}=\\emptyset$. \n\nIn case (a), the complex $Y$ contains a 2-simplex with $D_Y(\\sigma)=k$ and Lemma \\ref{lm10} gives us an embedding of an $r$-pseudo-surface $S\\in \\mathcal L_{k,r}$ into $Y$. \n\nIn case (b), we have\n$\\partial Y^{(i)}=\\emptyset$\nfor some $i<k$. Fix a 2-simplex $\\sigma_\\ast\\in Y^{(i)}$ and consider distances $d_{Y^{(i)}}(\\sigma_\\ast, \\sigma)$ to various 2-simplexes $\\sigma$ of $Y^{(i)}$. If all these distances are less than or equal to $k$, then \n$Y^{(i)}$ belongs to $\\mathcal L_{k,r}$ and we are done. If there are simplexes $\\sigma$ such that $d_{Y^{(i)}}(\\sigma_\\ast, \\sigma)>k$, then consider the subcomplex \n$Z\\subset Y^{(i)}$ defined as the union of all $\\sigma$ with $d_{Y^{(i)}}(\\sigma_\\ast, \\sigma)\\le k$. \n\nClearly $Z$ is not collapsible to a graph in $k$ steps. Therefore, in the sequence of collapses $Z\\searrow Z^{(1)}\\searrow Z^{(2)} \\searrow Z^{(3)}\\searrow \\dots$, we again \nhave either case (a) or (b) as above. In case (a), we apply Lemma \\ref{lm10}; and in case (b), we obtain a subcomplex $S\\subset Z$ with $\\partial S=\\emptyset$ such that\n$d(\\sigma_\\ast, \\sigma)\\le k$ for any $\\sigma\\subset S$. We have $S\\in \\mathcal L_{k,r}$ in either case, completing the proof. \n\\end{proof}\n\n\\section{Collapsibility of a random 2-complex}\n\n\\subsection{The degree sequence}\nRecall that the degree of an edge $e$ in a 2-complex is defined as the number of 2-simplexes which contain $e$. The degree of an edge in a random 2-complex \n$Y\\in G(\\Delta_n^{(2)},p)$ is an integer in the set $\\{0, 1, \\dots, n-2\\}$.\n\n\nLet $X_k: G(\\Delta_n^{(2)},p)\\to {\\mathbf Z}$ be the random variable counting the number of edges of degree $k$ in a random $2$-complex, where $k=0,1, 2, \\dots, n-2$.\nA straightforward calculation reveals that \n\\[\n{\\mathbb E}(X_k) = {n \\choose 2} {{n-2}\\choose k}p^k(1-p)^{n-2-k}.\\]\nThe expectation of the number of edges of degree at least $r$ in a random $2$-complex is \n\\begin{equation} \\label{edegree}\n\\sum_{k=r}^{n-2} {\\mathbb E}(X_k) \\le n^2\\sum_{k=r}^{n-2}(pn)^k \\le \\frac{n^2(pn)^r}{1-pn}.\n\\end{equation}\n\n\n\n\n\\begin{corollary}\\label{cor12} The probability that a random 2-complex $Y\\in G(\\Delta_n^{(2)}, p)$ has an edge of degree at least $r$ is less than or equal to  \n$$\\frac{n^{2+r}p^r}{1-pn}.$$ \nThus, if $$p\\ll n^{-1-\\frac{2}{r}},$$\nthen a random 2-complex $Y\\in G(\\Delta_n^{(2)}, p)$ has no edges of degree $r$ or greater, a.a.s.\n\\end{corollary} \n\\begin{proof} This follows from inequality \\eqref{edegree} by applying the first moment method, see, for instance, \\cite{JLR}. \n\\end{proof}\n\n\n\n\\subsection{The invariant $\\tilde \\mu(S)$.} \nFollowing \\cite{BHK} and \\cite{CFK}, for a $2$-complex $S$ with $v=v(S)$ vertices and $f=f(S)>0$ faces one defines\n\\[\n\\mu(S) = \\frac{v}{f} \\in \\mathbb Q,\n\\]\nand \n\\[\n\\tilde\\mu(S)=\\min_{S'\\subset S} \\mu(S'),\n\\]\nwhere $S'$ runs over all subcomplexes of $S$ or, equivalently, over all pure subcomplexes $S'\\subset S$. Note the following {\\it monotonicity property} of $\\tilde \\mu$:\n\\begin{eqnarray}\\label{mon}\n\\mbox{if} \\quad S\\subset T, \\quad \\mbox{then}\\quad \\tilde \\mu(S) \\ge \\tilde \\mu(T). \n\\end{eqnarray}\n\nThe invariant $\\tilde \\mu$ controls embeddability of finite 2-complexes into random 2-complexes as illustrated by the following result.\n\n\\begin{theorem}[\\cite{CFK}]\\label{tilda} Let $S$ be a finite simplicial complex.  \n\\begin{enumerate}\n  \\item[(a)] If $p\\ll n^{-\\tilde \\mu (S)}$, the probability that $S$ admits a  simplicial embedding into a random 2-complex $Y\\subset G(\\Delta_n^{(2)}, p)$ tends to zero as $n\\to \\infty$;\n  \\item[(b)] If $p\\gg n^{-\\tilde \\mu (S)}$, the probability that $S$ admits a simplicial embedding into a random 2-complex $Y\\subset G(\\Delta_n^{(2)}, p)$ tends to one as $n\\to \\infty$. \n\\end{enumerate}\n\\end{theorem}\n\n\n\\begin{definition} \nA 2-complex $S$ is called balanced if $\\tilde \\mu(S) = \\mu(S)$, or, equivalently, $\\mu(S')\\ge \\mu(S)$ for any subcomplex $S'\\subset S$. \n\\end{definition}\n\nAny triangulated surface is balanced, see \\cite{CFK}.\n\n\\begin{example}{\\rm \nSuppose that a 2-complex $S$ has a free triangle with two free edges, and that the result $S'$ of removing this triangle satisfies $\\mu(S')<1$. Then $\\mu(S)>\\mu(S')$ and $S$ is unbalanced. \nIndeed, if $\\mu(S')=v\/f$, where $v=v(S')$ and $f=f(S')$, then $v<f$ and we have $\\mu(S) = (v+1)\/(f+1)>v\/f$. \nIn this way one produces many unbalance 2-complexes, including 2-disks. }\n\\end{example}\n\nNext, we examine the $\\tilde \\mu$ invariants of 2-complexes $S\\in \\mathcal L_{k, r}$.\n\n\\begin{lemma}\\label{closed} Let $S$ be a closed 2-complex, i.e., $\\partial S=\\emptyset$. Then $\\tilde \\mu(S)\\le 1$. \n\\end{lemma}\n\\begin{proof} Without loss of generality, we may assume that $S$ is connected, since otherwise we can apply the following arguments to a connected component of $S$ and use the monotonicity property (\\ref{mon}). Moreover, we may assume that $S$ is pure, since otherwise we may deal with the maximal pure subcomplex of $S$ instead of $S$. \n\nSuppose first that $H_2(S;{\\mathbf Z}_2)=0$. Then by the Euler--Poincar\\'e theorem, $\\chi(S)\\le 1$, and we have\n$$v-e+f=\\chi(S)\\le 1,\\quad \\mbox{and} \\quad 3f\\ge 2e,$$\nwhere $v, e, f$ denote the numbers of vertices, edges and faces in $S$. In the latter inequality we used the assumptions that $S$ is pure and closed. These inequalities imply\n$$v-f\/2 \\le \\chi(S) \\le 1, \\quad \\mbox{and}\\quad \\mu(S) \\le 1\/2 + 1\/f.$$\nSince $f\\ge 4$ we obtain that $\\tilde \\mu(S) \\le \\mu(S)\\le 3\/4 <1.$\n\nAssume now that $H_2(S;{\\mathbf Z}_2)\\not=0$. We will show that there is a subcomplex $S'\\subset S$ which is also closed, $\\partial S'=\\emptyset$, and satisfies $H_2(S';{\\mathbf Z}_2)={\\mathbf Z}_2$. \nIndeed, consider a nonzero two-dimensional cycle $c=\\sum_{i\\in I} \\sigma_i$ with ${\\mathbf Z}_2$ coefficients, where the $\\sigma_i$ are distinct 2-simplexes of $S$. Let $I' \\subseteq I$ be the minimal subset of the indexing set $I$ for which $c'=\\sum_{i\\in I'} \\sigma_i$ is still a cycle, and let $S'=\\bigcup_{i\\in I'} \\sigma_i$ be the corresponding subcomplex of $S$.  Then clearly $H_2(S';{\\mathbf Z}_2)={\\mathbf Z}_2$ and $S'$ is closed and pure. \n\nBy the Euler--Poincar\\'e theorem, $\\chi(S')\\le 2$, and we have \n$$v'-e'+f'=\\chi(S') \\le 2, \\quad \\mbox{and}\\quad 3f'\\ge 2e',$$\nwhere $v',e ', f'$ denote the numbers of vertices, edges and faces in $S'$. This gives\n$$v'-f'\/2\\le \\chi(S') \\le 2,$$\nand\n\\begin{eqnarray}\\label{two}\n\\mu(S') \\le \\frac{1}{2} + \\frac{2}{f'}. \n\\end{eqnarray}\nSince $f'\\ge 4$, the last inequality gives $\\mu(S')\\le 1$. Finally, we have $\\tilde \\mu(S) \\le \\mu(S') \\le 1$. \n\\end{proof}\n\n\\begin{lemma}\\label{nonclosed} If $S\\in \\mathcal L_{k,r}$ for some $k\\ge 0$, \\, $r\\ge 2$ then one has \n\\begin{eqnarray}\\label{finally}\n\\tilde \\mu(S) \\le 1+\\frac{2}{k+1}.\\end{eqnarray}\n\\end{lemma}\n\\begin{proof} If $S$ is closed the result follows from Lemma \\ref{closed}. \nAssume now that $\\partial S\\not=\\emptyset$. \nLet $\\sigma_\\ast$ be the central simplex of $S$ and let $\\sigma_0, \\sigma_1, \\dots, \\sigma_k=\\sigma_\\ast$ be a collapsing path leading to $\\sigma_\\ast$. \nHere $D_S(\\sigma_i) = i$ and $\\sigma_i\\cap \\sigma_{i+1}$ is an edge, see Definition \\ref{defaccess}. \nThen the union \n$S'=\\cup_{i=0}^k\\sigma_i$ is a subcomplex having exactly $k+1$ faces and at most $k+3$ vertices.\nThus, \n$$\\mu(S') \\le \\frac{k+3}{k+1}=1+\\frac{2}{k+1},$$ \nestablishing (\\ref{finally}). \n\\end{proof}\n\n\n\n\n\\subsection{The threshold for $k$-collapsibility.}\n\\begin{definition}\nLet $\\tilde\\mu_{k,r}$ denote the largest possible value of the invariant $\\tilde\\mu(S)$ for $S$ a forbidden $r$-pseudo-surface, \n\\[\n\\tilde \\mu_{k,r} \\, = \\, \\max_{S\\in \\mathcal L_{k,r}}\\tilde \\mu(S) \\, \\in \\mathbb Q.\n\\]\n\\end{definition}\n\nFor instance, examining the surfaces shown in Figure \\ref{lone} reveals that $\\tilde \\mu_{1,2} =3\/2$.\n\n\\begin{theorem}\\label{thm21} Consider a random 2-complex $Y\\in G(\\Delta_n^{(2)},p)$. \n\\begin{enumerate}\n\\item[(a)] If for some $r\\ge 2$ and $k\\ge 1$, one has $$p\\ll n^{-1-\\frac{2}{r+1}} \\quad \\mbox{and}\\quad p\\ll n^{-\\tilde \\mu_{k,r}},$$ then $Y$ is collapsible to a graph in at most $k$ steps, a.a.s.\n\\item[(b)] If for some $r\\ge 2$ and $k\\ge 1$, one has $p\\gg n^{-\\tilde \\mu_{k,r}}$, then $Y$ is not collapsible to a graph in $k$ or fewer steps, a.a.s.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof} By Corollary \\ref{cor12}, if $p\\ll n^{-1-\\frac{2}{r+1}}$, then a random 2-complex $Y\\in G(\\Delta_n^{(2)},p)$ has degree at most $r$, a.a.s. \nNext, we apply Theorem \\ref{thm1} and examine the embeddability of complexes $S\\in \\mathcal L_{k,r}$ into $Y$. \nBy Theorem \\ref{tilda} (a), if $p\\ll n^{-\\tilde \\mu(S)}$, then \n$S$ does not embed \ninto $Y$, a.a.s. Since $\\tilde \\mu_{k,r}\\ge \\tilde \\mu(S)$, we see that the assumption $p\\ll n^{-\\tilde \\mu_{k,r}}$ implies that no $S\\in \\mathcal L_{k,r}$ can be embedded into $Y$, a.a.s.\nThus, by Theorem \\ref{thm1}, we see that $Y$ is collapsible to a graph in $ k$ or fewer steps. This proves part (a). \n\nTo prove part (b), we apply Theorem \\ref{tilda} (b) to conclude that if $p\\gg n^{-\\tilde \\mu_{k,r}}$, then there exists $S\\in \\mathcal L_{k,r}$ which is embeddable into $Y$, a.a.s. \nThis implies that \n$Y$ is not collapsible to a graph in at most $k$ steps, a.a.s.\n\\end{proof}\n\n\n\n\\begin{example} {\\rm  Consider the surface $S_k\\in \\mathcal L_{k,2}$ introduced in Example \\ref{ex1}. Note that $S_k\\in \\mathcal L_{k,r}$ for any $r\\ge 2$. The numbers of vertices $v_k$ and faces $f_k$ of $S_k$ satisfy\nthe recurrence relations\n\\begin{eqnarray}\\label{musk}v_k=2\\cdot v_{k-1}\\quad \\mbox{and}\\quad f_k = v_{k-1}+f_{k-1}.\\end{eqnarray}\nIndeed, viewing $S_{k-1}$ as a subcomplex of $S_k$, we see that all vertices of $S_{k-1}$ lie on the boundary, and each edge of the boundary of $S_{k-1}$ adds a vertex to $S_k$.  This explains the first equation. For the second, note that the number of new triangles\nin $S_k$ is equal to the number of edges on $\\partial S_{k-1}$. \n\nSince $v_0=3$ and $f_0=1$, solving the recurrence relations \\eqref{musk} yields \n$$v_k = 3\\cdot 2^{k} \\quad \\mbox{and}\\quad f_k = 3\\cdot 2^{k}-2. $$\nConsequently, \n$$\\mu(S_k)= 1+ \\frac{1}{3\\cdot 2^{k-1} -1}.$$\n}\n\\end{example}\n\n\\begin{lemma}\\label{star}\nThe surface $S_k$ is balanced, and hence $$\\tilde \\mu(S_k) = \\mu(S_k)= 1+ \\frac{1}{3\\cdot 2^{k-1} -1}.$$\n\\end{lemma} \n\\begin{proof} \nLet $S$ be a pure subcomplex of $S_k$ with $v=v(S)$ vertices and $f=f(S)$ faces.  Write $v=v_k-m$ and $f=f_k-n$, where \n$v_k$ and $f_k$ are as above and\n$m$ and $n$ are the number of vertices and faces which are in $S_k$, but not in $S$.  We claim that $m=v_k-v \\le f_k-f =n$.  This assertion is established by induction.\n\nThe case $k=0$ is trivial.  So assume inductively that for any $i<k$ and $S'\n\\subset S_i$  a pure subcomplex, we have $v(S_i)-v(S') \\le f(S_i)-f(S')$.\n\nFor a pure subcomplex $S \\subset S_k$ as above, let $S'$ be the pure part of $S\\cap S_{k-1}$. \nThen, $m=m'+m''$ and $n=n'+n''$, where $v(S')=v_{k-1}-m'$, $f(S')=f_{k-1}-n'$, $m''$ is the number of vertices in $S_k\\smallsetminus S_{k-1}$ which are not in $S$, and $n''$ is the number of faces in $S_k\\smallsetminus S_{k-1}$ which are not in $S$.\n\nWe have $m'\\le n'$ by induction.  Observe that the vertices of $S_k\\smallsetminus S_{k-1}$ are in one-to-one correspondence with the faces of $S_k\\smallsetminus S_{k-1}$.  If such a vertex is not in $S$, then the corresponding face cannot be in $S$ either.  Consequently, $m''=n''$, and $m=m'+m''\\le n'+n''=n$, completing the proof of the claim.\n\nIt follows immediately that $\\mu(S)\\ge\\mu(S_k)=\\mu_k$.  Indeed, \\[ \\frac{v}{f}-\\frac{v_k}{f_k}=\\frac{v_k-m}{f_k-n}-\\frac{v_k}{f_k}=\\frac{nv_k-mf_k}{f_k(f_k-n)}=\n\\frac{\\mu_k\nn - m}{f_k-n} \\ge \\frac{n-m}{f_k-n}\\ge 0.\n\\]\nThus, $S_k$ is balanced.\n\\end{proof}\n\n{F}rom Lemmas\n\\ref{nonclosed} and \\ref{star} we obtain:\n\n\\begin{corollary}\\label{cor23} For any $r\\ge 2$ and $k\\ge 0$, one has the following inequalities:\n\\[\n1+ \\frac{1}{3\\cdot 2^{k-1} -1}\\, \\le \\, \\tilde \\mu_{k,r} \\, \\le\\,  1+ \\frac{2}{k+1}.\n\\]\n\\end{corollary}\nNote that the obtained upper and lower bounds for $\\tilde \\mu_{k,r}$ are independent of $r$. \n\nWe believe that $\\tilde\\mu_{k,r}=1+  1\/(3\\cdot 2^{k-1} -1)$.\n\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{main}] \nThe main theorem is now an immediate consequence of Theorem \\ref{thm21} and Corollary \\ref{cor23}:\n\n(a) Assume that $p \\ll n^{-1-2\/(k+1)}$ for some $k \\ge 1$. According to Corollary \\ref{cor23},  \n$\\tilde \\mu_{k,r} \\le 1 + 2\/(k+1).$\nChoosing $r= \\mbox{max} (2,k)$, it then follows from Theorem \\ref{thm21} (a)    \nthat $Y\\in G(\\Delta_n^{(2)}, p)$ is collapsible to a graph in at most $k$ steps, a.a.s. \n\n(b) Assume that $p \\gg  n^{-1-1\/(3\\cdot 2^{k-1} - 1)}$ for some $k \\ge 1$. Then \nby Theorem \\ref{tilda} and Lemma \\ref{star} the surface $S_k$ (see Example \\ref{ex1}) embeds into $Y$, a.a.s.\nSince $S_k$ cannot be collapsed to a graph in $k$ or fewer steps we obtain that $Y$ is not collapsible to a graph in $k$ or fewer steps. \n\\end{proof}\n\n\n\n\n\\newcommand{\\arxiv}[1]{{\\texttt{\\href{http:\/\/arxiv.org\/abs\/#1}{{arXiv:#1}}}}}\n\n\\newcommand{\\MRh}[1]{\\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{MR#1}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn daily conversations, humans create many arguments consisting of implicit, unstated reasoning. \nConsider the following example consisting of two argumentative components: \\textit{Claim} (i.e., declarative statement) and its \\textit{Premise} (i.e., supporting statement):\n\n{\\enumsentence{\\textbf{Claim}: We should introduce compulsory voting.\\\\\n\\textbf{Premise}: It increases voter turnout.} \n\\label{ex:1}}\n\nIn Example~\\ref{ex:1}, it is easy for humans to understand the reason why the premise supports the claim. For example, one might assume that it implies that high voter turnout, resulting from introducing compulsory voting, is good for a fair representation of society.\nHowever, for machines, explicating such implicit reasoning, henceforth \\textit{warrants}~\\cite{toulmin1958use}, is a challenging, but exciting task.\n\nRe-constructing warrants in arguments can play an important role in many argument-related downstream applications, such as argumentative writing and comprehension support~\\cite{qin2010analysis, von2019improve}, improving students critical thinking~\\cite{hillocks2010ej}, and debate systems~\\cite{weber2008learning}.\nFor some of these applications, assessing identification of warrants is an important step in developing a more complex pipeline for overall argument analysis~\\cite{becker2020explaining}.\nFurthermore, warrants can be leveraged to automatically give constructive feedback to users to assist them in the aforementioned argumentative applications. \n\n\nWhile the identification of main argumentative components such as claim and premise have been well explored~\\cite{habernal2017argumentation, ein2020corpus}, research focusing on warrant explication has been lacking due to its extreme complexity. \n\\newcite{habernal-etal-2018-semeval} and~\\newcite{boltuzic-snajder-2016-fill} both demonstrated that simple warrant annotations can be done through crowdsourcing, though the authors acknowledge the high complexity due to subjectivity for reconstructing warrants. \nThey attribute the difficulty to the variety of reasoning patterns possible for framing warrant and multiplicity of warrants between argumentative components.\nHowever, to the best of our knowledge, no work has yet to solve the aforementioned challenges\n\n\n\n\n\nTo tackle the challenge encountered in collecting warrants, in this paper we address the following research question: \\textit{1. Can we devise a methodology that makes it easier to construct high quality warrants?}.\nOur main hypothesis is that since there are multiple reasoning patterns in natural language~\\cite{hitchcock2003toulmin, kock2006multiple}, it must be possible to narrow down the warrants into specific keyword-based patterns, where keywords are utilized from the original argument (i.e., claim and premise), which follows the formal definition of warrants in that they serve as a linking chain between the contents of claim and its premise~\\cite{freeman1992relevance, verheij2005evaluating}.\n\nIn this work, we conduct an extensive annotation study with trained experts and crowd workers to compare three different methodologies for collecting warrants, namely: \\textit{Natural language warrants, User-defined Keyword-based warrants} and \\textit{Pre-defined Keyword-based warrants}.\nFirstly, through each methodology, we collect multiple warrants on top of argumentative texts from IBM-Rank-30k dataset~\\cite{gretz2019large}, a dataset of topic and claim pairs already annotated with quality scores. \nWe analyze the collected warrants and work closely with experts to adapt the annotation task, iterating over how best to approach the novel domains and simplify the annotation guidelines for crowdsourcing to be suitable for both expert and non-expert annotators. \nSecondly, based on our extensive qualitative analysis results on the methodologies, we collect 6,000 warrants annotated for 600 arguments from over 6 diverse topics via \\textit{User-defined Keyword-based warrants}. \nWe publicly release our guidelines and preliminary dataset to facilitate research in automatic warrant generation.\\footnote{\\url{https:\/\/github.com\/keshav2995\/6000_warrants}}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\nImplicit reasonings, commonly referred to as \\textit{warrants}~\\cite{toulmin1958use} or \\textit{enthymemes}~\\cite{walton2008argumentation}, have long been studied to understand the grounds on which a premise lends support to the topic~\\cite{freeman1992relevance}.\nIn other words, a warrant, when made explicit, clearly shows the principle upon which the argument rests~\\cite{pineau2013abuses}.\nIdentification of such warrants has been shown to aid in the argument comprehension process~\\cite{hitchcock2006arguing, becker2017enriching} and in educational research for helping students to make better arguments and improve their critical thinking skills~\\cite{erduran2004tapping,von2019improve}.\n\nNumerous computational approaches have been made towards solving the task of warrant explication in arguments.\nIn an initial attempt,~\\newcite{feng-hirst-2011-classifying} identified argumentation schemes~\\cite{walton2008argumentation} as a means for warrant reconstruction, but they do not approach the task due to absence of training datasets. \n\\newcite{boltuzic-snajder-2016-fill} made a preliminary attempt at crowdsourcing warrants to fill the reasoning gap between a claim and premise. \nThey let each worker write multiple warrants to fill the reasoning gap between a claim and premise pair without any specific guidelines. \nHowever, they concluded that the collected warrants were highly variable in wordings and amount needed to fill the gap.\nIn contrast, our crowdsourcing methodologies provide specific guidelines with numerous examples and restrict workers to write only the most relevant warrant.\n\n\\newcite{becker2017enriching} hire expert annotators to fill missing knowledge pieces between different argument units (counter-argument, topic, premise, rebuttal, etc) in the argmicrotext corpus~\\cite{peldszus2015annotated}.\nHowever, their approach relies on the availability of experts which is an expensive and time-consuming process for collecting warrants on a larger dataset.\nInstead, we perform crowdsourcing with non-experts and show that the quality and cost of warrants collected can be high and inexpensive. \nRecently,~\\newcite{habernal-etal-2018-semeval} proposed a step by step methodology to crowdsource warrants from non-experts, but their error analysis pointed at inherent difficulty in distinguishing patterns between incorrect warrants. \nOn the contrary, in our work, we create various methodologies which can make it easier to analyse good and bad warrants for a given argument.\n\n\\section{Dataset Desiderata}\nTowards creating a preliminary warrant annotated dataset which can be potentially useful in downstream argumentative applications, we require it to fulfill at least the following criteria: i) cover multiple topics, ii), span over diverse premises, and iii) consists of high-quality warrants.\nIn addition, the dataset creation methodology must be cost effective without compromising data quality and easier for experts as well as non-experts to understandably write quality warrants.\nAs mentioned a priori, a dataset with multiple topics is desirable; however, collecting warrants across multiple topics can be both difficult and time consuming.\nThus, we start with small number of topics and define a simple metric to filter a handful of topics (specifically, 3) found in a large, well-known argumentation dataset of diverse premises (\\S~\\ref{sec:sourceargs}).\n\n\\subsection{Source Data (IBM-Rank-30K)}\n\\input{table\/slope.tex}\n\\label{sec:sourceargs}\nWe utilize \\emph{IBM-Rank-30K} dataset~\\cite{gretz2019large}, which contains supporting and opposing stance arguments on 71 common controversial topics.\nSeveral factors motivated our choice of utilizing this dataset:\n(1) Arguments were collected actively from crowdworkers with strict quality control measures as opposed to being extracted from targeted audiences such as debate portals.\nThis represents a vast majority of all the possible arguments that can be made for a given topic.\n(2) Argument structure is not tree-like (i.e., a premise supporting another premise, etc.) but rather flat-list like (i.e., a premise offers direct support to the topic). \nWe believe this will be useful in the pedagogical context, where students may make a declarative statement and generally provide few premises.\n(3) The arguments have already been annotated with point-wise quality for both supporting and opposing stance. \nOur extension on a small subset of it could serve the needs of similar qualitative research in future.\\footnote{We only utilize arguments in  \\emph{IBM-Rank-30K dataset} with supporting stance (i.e., the premise supports the topic) since warrants only exist for arguments which lend support to topic \\cite{toulmin1958use}.}\n\nTo select the most promising topics for testing our crowdsourcing methodologies, we define a simple metric. \nSpecifically, for each topic$(t)$, we estimate the Growth Rate (GR) i.e. slope of diversity of premises$(dp_t)$ as a function of its vocabulary size (V):\n\n\\begin{equation}\n    dp_t = \\frac{(|V(p_t)| - |V(p_t^{50\\%})|)}{(|p_t| - |p_t^{50\\%}|)},\n\\end{equation}\n\nwhere $p_t$ is a set of premises associated with $t$, $p_t^{50\\%}$ is a 50\\% random sample of $p_t$, and $V(p)$ is a set of unique words (i.e. vocabulary size) of $p$ obtained after tokenization and lemmatization.\nThe final growth rate of premises ($dp_t$) for each topic is calculated after averaging over multiple random resampling runs.\n\n\nAs shown in Table~\\ref{table:diversity}, the variety of premises for topics with a lower $dp_t$ depicts higher number of overlapping premises while topics with a higher $dp_t$ might comprise of more variety of debatable premises. \nIt also indicates that the growth rate of diversity strongly depends on the topic i.e. certain topics contain more diverse keywords\/knowledge pieces as compared to others.\nTo proceed with our preliminary warrant crowdsourcing and methodology analysis, we choose 3 topics that fall in the lower band of $dp_t$ value. \nOur choice for the above topics is based on the assumption that if high quality warrants can be collected for relatively low diverse premises then it will be a strong evidence to focus on more diverse premise in future work.\n\n\n\\section{Annotation Study}\nIn this section, we detail how we developed and designed our annotation task to allow for an efficient, reliable collection of warrants with crowdsourcing. \nIn particular, we extensively analyze 3 different warrant crowdsourcing methodologies and highlight the benefits of utilizing the best one through a comparative study.\n\nThe aim of our annotation is to reveal warrants that connect premises in argumentative texts to its claim. \nExplicating warrants, i.e., making the implicit explicit, has been inherently difficult due to varying structural or lexical phrasing chosen by annotators, difference in intuition, and a variety of reasoning patterns in which warrants can be framed.\nTo overcome these challenges, we attempt to conduct a deeper analysis on warrants from a theoretical perspective of reasoning and explore different ways to collect these warrants. \n\n\\subsection{Preliminary Measures}\nWe employ several preliminary measures to ensure the quality of collected annotations for all of our crowdsourcing tasks. \nWe choose Amazon Mechanical Turk (AMT)\\footnote{\\url{www.mturk.com}} as our crowdsourcing platform due to its success in previous argumentation mining tasks~\\cite{habernal-etal-2018-semeval}. As an initial step, we only allowed workers who had $\\geq$ 99\\% acceptance rate, $\\geq$ 5,000 approved Human Intelligence Tasks (HITs) and 100\\% score on our custom Reasoning Qualification Task (RQT) to work on the tasks.\n\nAdditionally, to address any ethical issues~\\cite{adda2011crowdsourcing} raised by our task, we actively monitored multiple pilot tests to ensure workers were satisfied with our task. Simultaneously, we corresponded directly to workers that had questions\/comments on our task.\nCrowdworkers are paid in accordance with the minimum wage calculated by conducting many trials and based on average work-time. \n\n\\subsection{Warrant Collection Methodologies}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{fig\/keyword.png}\n    \\caption{An example of a warrant collected for a claim and premise pair during our User-defined Keyword-based warrant collection trial. We note that claim and premise were shown as 'stance' and 'supporting argument' to avoid using complex terminology during our experiments.}\n    \\label{fig:my_label}\n\\end{figure}\n\nOur choice for the warrant collection methods is based on the formal definition of warrants i.e. warrants are the inference link that fills the reasoning gap between claim and premise~\\cite{toulmin1958use}. \nHence, we consider 3 methodologies: \\textit{natural language warrants}, \\textit{pre-defined keyword-based warrants}, and \\textit{user-defined keyword-based warrants}.\nBonus is paid to workers based on general consensus rather than sole judgement which may help remove bias towards and between workers.\n\n\\paragraph{Natural Language Warrants}\nFor collecting warrants in free-form natural sentences, we follow guidelines similar to~\\newcite{habernal-etal-2018-semeval}.    \nPrior  to  the  main  annotation  task, we  filtered crowdworkers via our custom Reasoning Qualification Test (RQT).\nSelected annotators were presented with a topic and single premise and asked to come up with a warrant that indicates why the premise supports the claim. \n\n\\paragraph{User-defined Keyword-based Warrants}\n\nAs an alternative to natural language sentences which can be variable in structure due to myriad amount of reasoning patterns~\\cite{keith2008toulmin}, we follow a simpler way to restrict the crowdworkers to a specific reasoning pattern. \nTo do this, we follow \\newcite{reisert-etal-2018-feasible}'s argumentation slot-filling templates, where each template encodes an argumentative relation between components. Such templates enable annotators to focus on important keywords in components which are relevant to the missing\/implicit information linking a claim and a premise.\n\nFollowing \\newcite{reisert-etal-2018-feasible}'s annotation study in which expert annotators freely chose slot-fillers, we let the crowdworker freely choose the slot-fillers from both the claim and the premise and additionally fill in the missing\/implicit reasoning between them.\nFor example, Fig.~\\ref{fig:my_label} shows a case in which the warrant connects the ontological elements of both claim and premise, and the claim is a consequence resulting from the premise (i.e., argument from consequence~\\cite{walton2008argumentation}).\n\n\\paragraph{Pre-defined Keyword-based Warrants}\n\\input{table\/method_result.tex}\nTo follow-up on \\newcite{reisert-etal-2018-feasible}'s annotation method, we also would like to determine the quality of warrants when the slot-fillers are pre-defined. Such a method allows for annotators to only write the implicit information between pre-defined keywords from the claim and premise.\nFor choosing keywords, we employ spaCy~\\cite{spacy} and automatically choose the longest verb\/noun phrases from the claim and premise.\n\n\\section{Methodology Comparison and Results}\n\\input{table\/guidelines.tex}\n\nTowards determining which methodology results in the highest quality warrants, we first collect warrants for each methodology using crowdsourcing. For each methodology, we annotate 40 arguments covering 3 different topics with 5 crowdworkers per argument. First, crowdworkers decided whether a warrant can be constructed between the given claim and premise. If so, they were then asked to write the warrant in the correct format.\n\n\\subsection{Filtering}\nFor each methodology, we can collect, at most, 200 warrants, if a worker first identifies that hidden reasoning is necessary to link the claim and its premise\nHowever, after initial filtering we discovered that workers wrote 65, 86 and 80 warrants each respectively out of a possible 200 warrants for the arguments given to them for \\textit{Natural language warrant, User-defined keyword-based} and \\textit{Pre-defined keyword-based warrant} methodologies. \nWe utilize these 231 collected warrants for further analysis.\n\n\\subsection{Results}\nTo analyze the quality of the collected warrants, two expert annotators were asked to judge the quality of the warrants on a scale of 0-2 according to the guidelines as shown in Table~\\ref{tab:quality_guidelines}. \nBoth annotators judged 50 warrants randomly chosen from the pool of collected warrants from each methodology. \nThe Inter-Annotator Agreement (IAA) score between the experts was measured as Krippendorff's alpha ($\\alpha$)~\\cite{krippendorff2011computing}. We also measure the combined average score given to the warrants for each methodology to measure the quality of warrants as shown in Table~\\ref{tab:method_results}.\n\n\\paragraph{Natural language} Expert annotators achieved an overall agreement score of $\\alpha$ = 0.57, indicating a good agreement. The average score given to natural language warrants by both annotators was 1.46. \nOverall, the experts have the highest agreement for warrants collected via this method, but the average score for warrant quality was lowest in this method.\n\n\\paragraph{User-defined} Expert annotators achieved an overall agreement score of $\\alpha$ = 0.56. The average score given to natural language warrants by both annotators was 1.59 which is the highest overall. \nThis suggests that warrants collected via this method might result in higher overall quality of warrants. \n\n\\paragraph{Pre-defined} Expert annotators achieved an overall agreement score of $\\alpha$ = 0.53. The average score given to natural language warrants by both annotators was 1.55. However, since the difference between pre-defined and user-defined warrant collection results is not significant in terms of the Average scores, it may require additional analysis to compare the two methods.\n\nThe overall results are summarized above in Table~\\ref{tab:method_results} and suggest that the difficulty of the task is highly dependent on the method of warrant collection and also on the topic to some extent. \nBetween the 3 methodologies, the average quality scores were highest with user-defined methodology and lowest with natural language.\n\n\n\\subsection{Qualitative Analysis}\n\nTo further analyze the quality of warrants and the quality of the entire crowdsourcing process, we further analyze a sample of the warrants collected via each methodology and annotated by experts. \n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{fig\/example_analysis.png}\n    \\caption{Sample from the warrants collected for our methodology comparison.\n    These warrants were scored the same from both expert annotators.\n    The underlined text denotes keywords from \\textbf{Premise} and \\textbf{Claim} used in pre-defined keyword-based warrant methods. \n    }\n    \\label{fig:example_warrants}\n\\end{figure*}\n\nAs shown in Fig~\\ref{fig:example_warrants}, the warrants shown each have some kind of keyword overlap with the claim\/premise. \nHowever, the difference in scoring warrants might be attributed to different complexities of the way in which warrants are framed.\nSpecifically, even though the warrants encode claim-premise information, the quality of the warrant can essentially be bad.\nThis shows that identifying the correct warrant can still be challenging with automated techniques which extract lexical and word-level features.\n\nWhile keyword-based methods restrict most warrants to a single sentence, the natural language warrants often consist of shorter, multiple sentences, even with character restrictions.\nOverall, we found 23\\% of natural language warrants were compound sentences.\nAlso, such warrants were often found to be paraphrased from the information in the premise or workers simply used a previous premise in place of the warrant. \nAlthough this was mostly seen in natural language warrants, we found a few instances of such type in keyword-based methods also.\nThis might suggest that, although keyword-based methods reduce bad quality warrants in-terms of paraphrasing and incomprehensible warrants, there is still room for improvement.\n\n\\section{Preliminary large-scale corpus}\nBased on our finding that user-defined keyword-based warrant method comparatively results in good warrants, we follow this method to collect a total of 6,000 warrants across 3 topics, annotated for 600 claim-premise pairs. All warrants were limited to have a length between 60 and 200 characters. \nSince this is an ongoing work, we plan to make further analysis on our preliminary dataset in future.\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nIn this work, we tackle the difficult task of explicating warrants in arguments. Due to the complexity of collecting warrants using natural language approaches, we conduct an extensive analysis and annotation studies for determining an appropriate methodology for collecting warrants and determine a user-defined keyword approach results in the highest quality warrants. Based on our finding, we construct a preliminary corpus of 6,000 warrants and make the warrants and guidelines publicly available.\\footnote{\\url{https:\/\/github.com\/keshav2995\/6000_warrants}} \n\nIn our future work, we will annotate our corpus with quality dimensions shown from previous argumentative works~\\cite{wachsmuth2017computational}.\nSimultaneously, we will collect warrants for premises attacking the topic, as our main focus of this work was on premises supporting the original topic. We will also test the usefulness of our model as constructive feedback in a pedagogical setting, such as deploying our system in schools in which students debate and\/or write essays on controversial topics.\nFinally, we will also focus on argumentation in tree-like hierarchical structures, such as a premise supporting\/attacking another premise, which can be discovered in everyday, real-world arguments.\n\n\n\n\\section*{Acknowledgements}\nThis work was partially supported by JST CREST Grant Number JPMJCR20D2 and a project, J200001946, subsidized by the New Energy and Industrial Technology Development Organization (NEDO).\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intro}\n\nClassical Cepheid variable stars are key objects in astronomy\nbecause owing to their radial pulsation and its consequences--mainly \nthe famous period-luminosity ($P$-$L$) relationship--they \nrank among standard candles in establishing the cosmic distance \nscale and serve as test objects of stellar evolution of\nintermediate-mass stars. \n\nCompanions to Cepheids, however, complicate the situation.\nOn the one hand, the contribution of the secondary star to the\nobserved brightness has to be taken into account when involving any\nparticular Cepheid in the calibration of the $P$-$L$ relationship\nand the evolution of binary stars may be quite different from\nthe single star evolution (depending on the separation of the\ncomponents). On the other hand, frequency of binaries (and\nmultiple stars) among classical Cepheid variables is considerable:\nit exceeds 50 per cent for brightest Cepheids \\citep{Sz03a}, \nwhile among the fainter Cepheids an observational selection \neffect encumbers revealing binarity. In a broader aspect, however,\nthis frequent occurrence of binaries among Cepheids is not\nsurprising, it agrees with the general frequency of binary stars\nin the solar neighbourhood: two-third of solar type stars of \nF3-G2 spectral type stars have stellar companions \\citep{AL76}.\n\nDepending on the brightness and temperature differences between\nthe Cepheid and its (either optical or physical) companion, the\nobservable brightness and colour of the unresolved binary\nsystem can differ from the respective value intrinsic to the\nCepheid. If uncorrected for its contribution, the companion\ncan falsify the luminosity and radius of the Cepheid derived by\nusing the Baade-Wesselink method (except its infrared surface \nbrightness method implementation). \n\nThe orbital motion of a Cepheid \nin a binary system can even lead to a wrong trigonometric parallax\nif no allowance is made for binarity. An illustrative example\nfor this adverse effect was shown by \\citet{Szetal11}: all \nnegative Hipparcos values for Cepheids within 2 kpc were solely \nderived in the case of Cepheids with known spectroscopic companions.\n\nCepheids belonging to open clusters are often used for \ncalibrating the $P$-$L$ relationship based on the independently \nderived cluster distances \\citep[e.g.,][]{TB02,T10}. \nFor these calibrating Cepheids it is especially important to correct \nfor the luminosity contribution from the companion.\n\nTherefore, it is essential to study Cepheids individually \nfrom the point of view of binarity before involving them in any\ncalibration procedure (of e.g., $P$-$L$ or period-radius\nrelationship).\n\nHot companions to Cepheids can be effectively discovered by\nultraviolet {\\em spectroscopy}: the {\\it IUE\\\/} satellite was instrumental\nin revealing early type companions \\citep{E92}. Radial velocity\ntime series data (normally in the optical region) obtained during \nat least two widely differing epochs can lead to discovery of\nspectroscopic binaries \\citep[see e.g.,][]{Sz96}.\nRevealing binarity by means of {\\em astrometry} will be available from\nthe data to be obtained during the ESA {\\it Gaia\\\/} space mission\nfrom 2013 on.\nOwing to the regular behaviour of Cepheid pulsation, various \n{\\em photometric} criteria have also been devised for pointing out\nthe presence of Cepheid companions \\citep{Sz03b,KSz09}.\n\nIn this paper we reveal {\\em spectroscopic binarity} of two bright\nCepheids, X~Puppis and XX~Sagittarii, by carefully analysing \nthe radial velocity data published on these variable stars.\nSome new radial velocity data obtained for XX~Sgr have also been \nincluded in the analysis.\n\nIn the case of pulsating variables, like Cepheids, spectroscopic\nbinarity manifests itself in a periodic variation of the\n$\\gamma$-velocity (i.e., the radial velocity of the mass centre\nof the Cepheid). In practice, the orbital radial velocity variation\nof the Cepheid component is superimposed on the radial velocity\nvariations of pulsational origin. To separate the orbital and \npulsational effects, knowledge of the accurate pulsation period\nis essential, especially when comparing radial velocity\ndata obtained at widely differing epochs. Therefore, the pulsation\nperiod and its variations have been determined with the method of\nthe $O-C$ diagram \\citep{S05} for both target Cepheids. Use of the\naccurate pulsation period obtained from the photometric data is a \nguarantee for the correct phase matching of the (usually less\nprecise) radial velocity data.\n\n\\section{X~Puppis}\n\\label{xpup}\n\n\\subsection{Accurate value of the pulsation period}\n\\label{xpup-period}\n\nThe brightness variability of X~Pup (HD\\,60266, $\\langle V \\rangle\n= 8.46$\\,mag.) was revealed by \\citet{K90}.\nDuring the time interval elapsed since the discovery (spanning \nmore than 120 years) the photometric variability was followed \nfirst visually, then photographically, from the 1950s\nphotoelectrically, and in the last decades by CCD photometry. \nAll published observations of this Cepheid radially pulsating \nin the fundamental mode were re-analysed in a homogeneous \nmanner to determine seasonal moments of the normal maxima.\nThese data are collected in Table~\\ref{tab-xpup-oc} whose\ncolumn contain the following pieces of information:\\\\\nColumn~1: Heliocentric normal maxima;\\\\\nCol.~2: Epoch number, $E$, as calculated from Eq.~1:\n\\vspace{-1mm}\n\\begin{equation}\nC = 2\\,454\\,845.0497 + 25.970\\,068{\\times}E \n\\label{xpup-ephemeris}\n\\end{equation}\n\\vspace{-3mm}\n$\\phantom{mmmmm}\\pm0.0926\\phantom{\\,}\\pm 0.000\\,406$\n\n\\noindent (this ephemeris has been obtained by the weighted least squares\nfit to the tabulated $O-C$ residuals);\\\\\n\\noindent Col.~3: the corresponding $O-C$ residual as calculated \nfrom Eq.~\\ref{xpup-ephemeris};\\\\\nCol.~4: weight assigned to the $O-C$ residual (1, 2, or 3 \ndepending on the quality of the light curve leading to \nthe given residual);\\\\\nCol.~5: reference to the origin of data, preceded by the name \nof the observer if different from the author(s) cited.\\\\\n\nThe $O-C$ residuals have been plotted in Fig.~\\ref{fig-xpup-oc}\ntogether with the least squares fitted parabola. This\nparabolic trend corresponds to a continuous period increase\nof $5.375{\\times}10^{-6}$ d\/cycle, i.e., $\\Delta P = 0.007559$\nd\/century. This tiny but non-negligible effect has been caused\nby stellar evolution: while the Cepheid crosses the instability\nregion towards lower temperatures in the Hertzsprung-Russell \ndiagram, its pulsation period is increasing. Superimposed on\nthis parabolic trend, random fluctuations of the period value\nare also seen on a time scale of several years -- see the\nquasi continuously covered part of the deviations from the\nparabolic fit to the $O-C$ residuals after JD\\,2\\,448\\,000 \nin Fig.~\\ref{fig-xpup-deltaoc}. The amplitude of these\nfluctuations is several tenths of a day, an order of magnitude \nlarger than the periodic light-time effect expected in\na binary system (of suitably placed orbital plane) with a \nCepheid component.\n\n\\begin{figure}\n\\includegraphics[height=70mm, angle=0]{szabados1fig1.eps}\n\\caption{$O-C$ diagram of X~Pup based on the residuals\nlisted in Table~\\ref{tab-xpup-oc}. The pulsation period\nof X~Pup is continuously increasing}\n\\label{fig-xpup-oc}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[height=57mm, angle=0]{szabados1fig2.eps}\n\\caption{Deviations of the $O-C$ residuals of X~Pup from \nthe fitted parabola}\n\\label{fig-xpup-deltaoc}\n\\end{figure}\n\n\\begin{table}\n\\caption{$O-C$ residuals of X~Puppis (description of the columns\nis given in Sect.~\\ref{xpup-period})}\n\\begin{tabular}{l@{\\hskip2mm}r@{\\hskip2mm}r@{\\hskip2mm}c@{\\hskip2mm}l}\n\\hline\n\\noalign{\\vskip 0.3mm}\nJD$_{\\odot}$ & $E\\ $ & $O-C$ & $W$ & Data source\\\\\n2\\,400\\,000 + &&&\\\\\n\\noalign{\\vskip 0.3mm}\n\\hline\n\\noalign{\\vskip 0.3mm}\n08872      & $-$1771 & 20     & 1 & Sch\\\"onfeld \\citep{P56}\\\\\n11100.62   & $-$1685 & 15.13  & 1 & \\citet{K90}\\\\\n14136.7    & $-$1568 & 12.7   & 1 & Perry \\citep{P56}\\\\\n14343.0631 & $-$1560 & 11.3195& 1 & \\citet{Z32}\\\\\n14577.1534 & $-$1551 & 11.6792& 1 & \\citet{P97,P99,P03}\\\\\n14890.4    & $-$1539 & 13.3   & 1 & Hartwig \\citep{P56}\\\\\n14890.8    & $-$1539 & 13.7   & 1 & Innes \\citep{P56}\\\\\n18706.3    & $-$1392 & 11.6   & 1 & Worsell \\citep{P56}\\\\\n19040.869  & $-$1379 & 8.543  & 1 & Robinson \\citep{P56}\\\\\n19638.28   & $-$1356 & 8.64   & 1 & \\citet{H28}\\\\\n20001.52   & $-$1342 & 8.30   & 1 & Payne \\& Gaposchkin\\\\\n&&&& \\citep{P56}\\\\\n24545.1    & $-$1167 & 7.1    & 1 & \\citet{B33}\\\\\n26725.6293 & $-$1083 & 6.1632 & 2 & \\citet{O'C34}\\\\\n27115.3559 & $-$1068 & 6.3388 & 1 & \\citet{FK53}\\\\\n32515.29    & $-$860 & 4.50   & 1 & Berdnikov, Mattei \\& \\\\\n            &        &        &   & Beck (2003)\\\\\n34877.0978  & $-$769 & 3.0304 & 2 & Walraven, Muller \\&\\\\\n            &        &        &   & Oosterhoff (1958)\\\\\n35188.9673  & $-$757 & 3.2591 & 1 & \\citet{I61}\\\\\n36149.68    & $-$720 & 3.08   & 1 & \\citet{Betal03}\\\\\n37369.7691  & $-$673 & 2.2572 & 2 & \\citet{Metal64}\\\\\n37369.78    & $-$673 & 2.59   & 1 & \\citet{Betal03}\\\\\n39368.8181  & $-$596 & 1.9289 & 1 & \\citet{T69}\\\\\n40875.2311  & $-$538 & 2.0780 & 3 & \\citet{P76}\\\\\n41653.7755  & $-$508 & 1.5203 & 2 & \\citet{M75}\\\\\n42017.38    & $-$494 & 1.54   & 1 & \\citet{Betal03}\\\\\n42848.4488  & $-$462 & 1.5705 & 1 & \\citet{D77}\\\\\n43082.33    & $-$453 & 1.72   & 1 & \\citet{Betal03}\\\\\n43601.31    & $-$433 & 1.30   & 1 & \\citet{Betal03}\\\\\n44120.65    & $-$413 & 1.24   & 1 & \\citet{Betal03}\\\\\n44457.6939  & $-$400 & 0.6714 & 1 & \\citet{E83}\\\\\n44561.8016  & $-$396 & 0.8988 & 1 & \\citet{B08}\\\\\n44744.01    & $-$389 & 1.32   & 1 & \\citet{Betal03}\\\\\n44898.9264  & $-$383 & 0.4127 & 1 & \\citet{MB84}\\\\\n44977.5009  & $-$380 & 1.0770 & 1 & \\citet{B08}\\\\\n45704.52    & $-$352 & 0.9342 & 1 & \\citet{Betal03}\\\\\n45808.46    & $-$348 & 0.9940 & 1 & \\citet{Betal03}\\\\\n47833.64    & $-$270 & 0.5087 & 1 & \\citet{Betal03}\\\\\n48145.0626  & $-$258 & 0.2904 & 2 & Hipparcos \\citep{ESA97}\\\\\n48716.3278  & $-$236 & 0.2141 & 2 & Hipparcos \\citep{ESA97}\\\\\n49806.8990  & $-$194 & 0.0425 & 3 & \\citet{B08}\\\\\n50118.4982  & $-$182 & 0.0009 & 2 & \\citet{B02}\\\\\n50482.0566  & $-$168 & $-$0.0217 & 2 & \\citet{B02}\\\\\n50560.0343  & $-$165 & 0.0457 & 2 & \\citet{B08}\\\\\n50819.6483  & $-$155 & $-$0.0409 & 2 & \\citet{B08}\\\\\n50897.3144  & $-$152 & $-$0.2850 & 2 & \\citet{B08}\\\\\n51261.0546  & $-$138 & $-$0.1257 & 3 & \\citet{B08}\\\\\n51650.7613  & $-$123 & 0.0300 & 3 & \\citet{B08}\\\\\n51962.2826  & $-$111 & $-$0.0896 & 2 & \\citet{B08}\\\\\n52351.9893   & $-$96 & 0.0661 & 3 & \\citet{B08}\\\\\n52637.6479   & $-$85 & 0.0540 & 3 & \\citet{B08}\\\\\n52689.5032   & $-$83 & $-$0.0309 & 3 & ASAS \\citep{P02}\\\\\n52923.1507   & $-$74 & $-$0.1140 & 3 & ASAS \\citep{P02}\\\\\n53001.1284   & $-$71 & $-$0.0465 & 3 & ASAS \\citep{P02}\\\\\n53001.1284   & $-$71 & $-$0.0465 & 3 & \\citet{B08}\\\\\n53416.5940   & $-$55 & $-$0.1020 & 3 & ASAS \\citep{P02}\\\\\n53754.2896   & $-$42 & $-$0.0172 & 3 & ASAS \\citep{P02}\\\\\n54143.9963   & $-$27 & 0.1384 & 3 & ASAS \\citep{P02}\\\\\n54481.6660   & $-$14 & 0.1973 & 3 & ASAS \\citep{P02}\\\\\n54845.3282   &  0    & 0.2785 & 3 & ASAS \\citep{P02}\\\\\n\\noalign{\\vskip 0.3mm}\n\\hline\n\\end{tabular}\n\\label{tab-xpup-oc}\n\\end{table}\n\n\\subsection{Radial velocity data of X~Pup}\n\\label{xpup-radvel}\n\n\n\\begin{table}\n\\caption{$\\gamma$-velocities of X~Puppis}\n\\begin{tabular}{lccl}\n\\hline\n\\noalign{\\vskip 0.3mm}\nMid-JD & $v_{\\gamma}$ & $\\sigma$ & Data source \\\\\n2\\,400\\,000+ & (km\/s)& (km\/s) & \\\\\n\\noalign{\\vskip 0.3mm}\n\\hline\n\\noalign{\\vskip 0.3mm}\n25590 & 65.0 & 2.0 & \\cite{J37}\\\\\n44666 & 67.4 & 0.5 & \\cite{Cetal01} \\\\\n45065 & 66.5 & 1.5 & \\cite{Betal88} \\\\\n50523 & 70.0 & 0.4 & \\cite{B02} \\\\\n51111 & 71.5 & 0.3 & \\cite{Petal05} \\\\\n54922 & 71.8 & 0.2 & \\cite{Setal11}\\\\\n\\noalign{\\vskip 0.3mm}\n\\hline\n\\end{tabular}\n\\label{tab-xpup-vgamma}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[height=55mm, angle=0]{szabados1fig3.eps}\n\\caption{Temporal drift in the $\\gamma$-velocity\nof X~Puppis}\n\\label{fig-xpup-vrad}\n\\end{figure}\n\nThe available radial velocity data include those obtained by\n\\cite{J37}, \\cite{Cetal01}, Barnes, Moffett \\& Slovak (1988), \n\\cite{B02}, \\cite{Petal05}, and \\cite{Setal11}. The phase curve was\nconstructed from each data set using the actual value of the\npulsation period taking into account the continuous period\nvariation implied by Fig.~\\ref{fig-xpup-oc}.\n (see Sect.~\\ref{xpup-period}). Then the mean\nvalue of the radial velocity (the $\\gamma$-velocity)\nwas determined for each data set. These $\\gamma$-velocities\n(together with their uncertainties are listed in \nTable~\\ref{tab-xpup-vgamma} and also plotted in \nFig.~\\ref{fig-xpup-vrad}. The pattern of the data points\nimplies a monotonically changing $\\gamma$-velocity which is a \nsign of the orbital motion in a spectroscopic binary system.\nThe orbital period can be as long as several decades.\n\nSpectroscopic binarity of X~Pup has to be confirmed by further\nobservations because the earliest data (obtained by Joy about\n80 years ago) are of low quality. Nevertheless, all radial\nvelocity data have been obtained based on the observed wavelength\nof metallic lines, therefore one does not expect a noticeable\nsystematic difference between the various data sets.\n\n\\section{XX~Sagittarii}\n\\label{xxsgr}\n\n\\subsection{Accurate value of the pulsation period}\n\\label{xxsgr-period}\n\nThe brightness variability of XX~Sagittarii (HD\\,169315,\n$\\langle V \\rangle = 8.65$\\,mag.) was discovered by Annie Cannon\n\\citep{P08}. In the first half of the 20th century XX~Sgr\nwas occasionally observed mainly visually, regular \nphotometric data on this Cepheid are available from the\nlast three decades. All photometric data of this single\nperiodic Cepheid pulsating in the radial fundamental mode\nhave been subjected to an $O-C$ analysis whose results are \nsummarized in Table~\\ref{tab-xxsgr-oc}. The $O-C$ residuals in \nthis table have been obtained by using the following final\nephemeris:\n\n\\vspace{-1mm}\n\\begin{equation}\nC = 2\\,452\\,814.4629 + 6.424\\,267{\\times}E \n\\label{xxsgr-ephemeris}\n\\end{equation}\n\\vspace{-3mm}\n$\\phantom{mmmmm}\\pm0.0059\\phantom{\\,}\\pm 0.000\\,003$\n\n\\noindent determined from a weighted least-squares fit to \nall normal maxima listed in Table~\\ref{tab-xxsgr-oc}.\n\nThe $O-C$ diagram of XX~Sgr is plotted in Fig.~\\ref{fig-xxsgr-oc}. \nThe pattern of data can be approximated by a constant period\nbut in the last decades a wave-like pattern is seemingly\nsuperimposed on the $O-C =0$ constant line. However, this \nfeature, can hardly be attributed to a light-time effect\noccurring in a binary system due to the orbital motion\nbecause the amplitude of the $O-C$ variations is too large.\n\n\n\\begin{table}\n\\caption{$O-C$ residuals of XX~Sagittarii}\n\\begin{tabular}{l@{\\hskip2mm}r@{\\hskip2mm}r@{\\hskip2mm}c@{\\hskip2mm}l}\n\\hline\n\\noalign{\\vskip 0.3mm}\nJD$_{\\odot}$ & $E\\ $ & $O-C$ & $W$ & Data source\\\\\n2\\,400\\,000 + &&&\\\\\n\\noalign{\\vskip 0.3mm}\n\\hline\n\\noalign{\\vskip 0.3mm}\n19710.2958  & $-$5153  & 0.0808 & 1 & Zinner \\citep{V32}\\\\\n25100.13    & $-$4314  & $-$0.05   & 1 & \\citet{Z29}\\\\\n25485.6503  & $-$4254  & 0.0192 & 2 & \\citet{V32}\\\\\n25614.0548  & $-$4234  & $-$0.0616 & 1 & \\citet{Z54}\\\\\n26185.8675  & $-$4145  & $-$0.0087 & 1 & \\citet{P38}\\\\\n27117.4641  & $-$4000  & 0.0692 & 1 & \\citet{FK53}\\\\\n29648.5184  & $-$3606  & $-$0.0377 & 1 & \\citet{S48}\\\\\n31646.3938  & $-$3295  & $-$0.1093 & 1 & \\citet{S48}\\\\\n33117.6109  & $-$3066  & $-$0.0494 & 2 & \\citet{Eetal57}\\\\\n37267.7531  & $-$2420  & 0.0163 & 1 & \\citet{Metal64}\\\\\n39265.7249  & $-$2109  & 0.0411 & 2 & \\citet{T69}\\\\\n44019.5564  & $-$1369  & $-$0.0850 & 2 & \\citet{B08}\\\\ \n44771.3477  & $-$1252  & 0.0671 & 2 & \\citet{MB84}\\\\\n44944.8381  & $-$1225  & 0.1023 & 3 & \\citet{B08}\\\\\n48368.8340   & $-$692  & $-$0.0361 & 3 & {\\it Hipparcos\\\/} \\citep{ESA97}\\\\\n48966.3048   & $-$599  & $-$0.0222 & 3 & \\citet{Aetal98}\\\\\n50353.9447   & $-$383  & $-$0.0239 & 3 & \\citet{B08}\\\\\n50893.5982   & $-$299  & $-$0.0089 & 3 & \\citet{B08}\\\\\n51272.6481   & $-$240  & 0.0093 & 3 & \\citet{B08}\\\\\n51651.6659   & $-$181  & $-$0.0047 & 3 & \\citet{B08}\\\\\n52082.1090   & $-$114  & 0.0125 & 3 & ASAS \\citep{P02}\\\\\n52371.1953    & $-$69  & 0.0068 & 3 & \\citet{B08}\\\\\n52499.6717    & $-$49  & 0.0021 & 3 & ASAS \\citep{P02}\\\\\n52814.4674   &   0     & 0.0045 & 3 & ASAS \\citep{P02}\\\\\n52923.6906   &  17     & 0.0152 & 3 & {\\it INTEGRAL\\\/} OMC\\\\\n53161.3645   &  54     & $-$0.0088 & 3 & ASAS \\citep{P02}\\\\\n53546.8451   & 114     & 0.0158 & 3 & ASAS \\citep{P02}\\\\\n53649.5992   & 130     & $-$0.0184 & 3 & {\\it INTEGRAL\\\/} OMC\\\\\n53861.6215   & 163     & 0.0031 & 3 & ASAS \\citep{P02}\\\\\n54272.7729   & 227     & 0.0014 & 3 & ASAS \\citep{P02}\\\\\n54645.3666   & 285     & $-$0.0124 & 3 & ASAS \\citep{P02}\\\\\n54979.4283   & 337     & $-$0.0126 & 3 & ASAS \\citep{P02}\\\\\n\\noalign{\\vskip 0.3mm}\n\\hline\n\\end{tabular}\n\\label{tab-xxsgr-oc}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[height=45mm, angle=0]{szabados1fig4.eps}\n\\caption{$O-C$ diagram of XX~Sgr. The plot can be\napproximated by a constant period}\n\\label{fig-xxsgr-oc}\n\\end{figure}\n\n\\subsection{Radial velocity data of XX~Sgr}\n\\label{xxsgr-radvel}\n\nOver half century the values published by \\citet{J37}\nhad been the only existing radial velocity data on XX~Sgr.\nThe radial velocity survey of \\citet{Betal88} also included\nthis Cepheid, and their two data implied a $\\gamma$-velocity\nvalue different from the one determined Joy's data. \nSuspecting the spectroscopic binary nature we initiated\nnew radial velocity observations (discussed in Sect.~\\ref{feros}).\nIn the meantime, however, some more radial velocity data\nhave been published by \\citet{Betal10} and \\cite{Setal11}.\nThis latter series of observations resulted in a well-covered\nand very accurate radial velocity phase curve which confirms\nthe spectroscopic binarity of XX~Sgr. However, \\cite{Setal11}\ndid not compare their own measurements with the previous data,\nthus they missed to reveal the binarity of this Cepheid.\n\n\\subsection{FEROS observations}\n\\label{feros}\n\n\\begin{table}\n\\caption{Log of the FEROS observations of XX~Sgr}\n\\begin{tabular}{c@{\\hskip3mm}c@{\\hskip3mm}c@{\\hskip3mm}c@{\\hskip3mm}r}\n\\hline\n\\noalign{\\vskip 0.3mm}\nNight       & Obs. ID. & Mid-exposure & Exposure & $v_{\\rm rad}$\\\\\n & & JD 2\\,400\\,000+ & time (s) & (km\/s)\\\\\n\\noalign{\\vskip 0.3mm}\n\\hline\n\\noalign{\\vskip 0.3mm}\n15-16 Apr. &  F1487  & 55\\,667.8105 & 180 & $-$1.98\\\\\n16-17 Apr. &  F1645  & 55\\,668.8165 & 180 &    6.04\\\\\n17-18 Apr. &  F1723  & 55\\,669.8192 & 180 &   15.00\\\\\n18-19 Apr. &  F1805  & 55\\,670.8533 & 180 &   22.48\\\\\n\\noalign{\\vskip 0.3mm}\n\\hline\n\\noalign{\\vskip 0.3mm}\n\\end{tabular}\n\\label{tab-ferosobs}\n\\end{table}\n\nXX~Sgr was observed on four consecutive nights in April, 2011, \nusing the FEROS spectrograph on the MPG\/ESO 2.2\\,m telescope \nin La Silla Observatory, Chile (see Table~\\ref{tab-ferosobs} \nfor details).\nThe Fiber-fed Extended Range Optical Spectrograph (FEROS)\n(Kaufer et al.\\,1999, 2000) has a total wavelength coverage\nof 356-920\\,nm with a resolving power of $R=48\\,000$.\nTwo fibres, with entrance aperture of 2\\farcs7, simultaneously\nrecorded star light and sky background. The detector is\na back-illuminated CCD with 2948$\\times$4096 pixels of 15\\,$\\mu$m\nsize. Basic data reduction was performed using a pipeline package \nfor reductions (DRS), in {\\sc midas} environment. The pipeline performs \nthe subtraction of bias and scattered light in the CCD, orders \nextraction, flatfielding and wavelength calibration with a ThAr \ncalibration frame (the calibration measurements were performed \nat the beginning of each night, using the ThAr lamp).\n\nAfter the continuum normalization of the spectra using \n{\\sc iraf}\\footnote{{\\sc iraf} is distributed by the National Optical \nAstronomy Observatories, which are operated by the Association of \nUniversities for Research in Astronomy, Inc., under cooperative \nagreement with the National Science Foundation.}\nwe determined the radial velocities with the task {\\it fxcor\\\/}, \napplying the  cross-correlation method using a well-matching \ntheoretical template spectrum from the extensive spectral library \nof \\citet{M05}. The velocities were determined in the region \n500-600~nm where a number of metallic lines are present and lack of \nhydrogen lines. We made barycentric corrections to each radial \nvelocity value with the task {\\it rvcorrect\\\/}. The estimated \nuncertainty of the radial velocities is 0.05~km\\,s$^{-1}$.\n\n\n\\subsection{Binarity of XX~Sgr}\n\\label{xxsgr-bin}\n\n\\begin{figure}\n\\includegraphics[height=55mm, angle=0]{szabados1fig5.eps}\n\\caption{Merged radial velocity phase curve of XX~Sgr.\nThere is a striking difference between the \n$\\gamma$-velocities valid for the epoch of Joy's \n(\\citeyear{J37}) data (denoted as small squares) and more \nrecent data (other symbols, see the detailed list in the\ntext)}\n\\label{fig-xxsgr-vrad}\n\\end{figure}\n\nAll radial velocity data have been folded on the accurate\npulsation period taken from the ephemeris given in \nEq.~\\ref{xxsgr-ephemeris},\nso the different data series have been correctly phased with\nrespect to each other. The merged radial velocity phase curve \nis plotted in Fig.~\\ref{fig-xxsgr-vrad}.\n\nThe individual data series are denoted with different symbols:\nsmall squares - \\citet{J37} corresponding to mid-JD\\,2\\,426\\,283; \nopen circles - \\citet{Betal88} with mid-JD\\,2\\,444\\,254; \nfilled circles - \\citet{Setal11} and \\citet{Betal10},\nwith mid-JD\\,2\\,454\\,261 and mid-JD\\,2\\,454\\,730, respectively;\ntriangles: our FEROS data listed in Table~\\ref{tab-ferosobs}, \nwith mid-JD\\,2\\,455\\,669.\n\nThe earliest radial velocity data by \\citet{J37} imply a\nsignificantly different $\\gamma$-velocity than all more recent\nones in spite of the uncertainty of his individual data as\nlarge as 4 km\\,s$^{-1}$. Because the zero point of Joy's system \nis reliable, as discussed by \\citet{Sz96}, there is no systematic \ndifference of instrumental or data treatment origin between Joy's \nand the more recent observational series. The only plausible \nexplanation for the shift in the $\\gamma$-velocity is the orbital \nmotion in a binary system superimposed on the pulsational radial \nvelocity changes.\n\nSpectroscopic binarity of XX~Sgr can hardly be suspected from \nthe radial velocity data obtained in the last 25 years, so the\norbital period is certainly much longer.\n\n\\section{Conclusions}\n\\label{concl}\n\nWe pointed out that two bright Galactic Cepheids,\nX~Puppis and XX~Sagittarii have a variable $\\gamma$-velocity\nwhich implies their membership in spectroscopic binary\nsystems. The available radial velocity data are\ninsufficient to determine the orbital period and\nother elements of the orbit. It is obvious, however,\nthat the orbital period is as long as several decades\nin both cases.\nSuch long orbital periods are not unprecedented among\nclassical Cepheids, cf. the cases of T~Mon and AW~Per \n\\citep[see the on-line data base \nhttp:\/\/www.konkoly.hu\/CEP\/orbit.html\nand its description in][]{Sz03a}.\n\nNeither X~Pup, nor XX~Sgr shows any photometric\nevidence of duplicity based on the criteria discussed\nby \\citet{Sz03b} and \\citet{KSz09} indicating that the\ncompanion cannot be a star much hotter than the \nCepheid component in either case. Further spectroscopic \nobservations are necessary to characterise these binary systems.\n\nRegular monitoring of the radial velocities of a large\nnumber of Cepheids will be instrumental in finding \nmore long-period spectroscopic binaries among Cepheids. \nQuite recently \\citet{Eetal12} reported on their \non-going survey for pointing out binarity of Cepheids \nfrom the existing radial velocity data covering sufficiently \nlong time interval.\n\nWhen determining the physical properties (luminosity, \ntemperature, radius, etc.) of individual Cepheids,\nthe effects of the companion on the observed parameters\n(apparent brightness, colour indices, etc.) have to be\ncorrected for. This type of analysis, however, should be\npreceded by revealing the binarity of the given Cepheid.\n\n\n\\section*{Acknowledgments}\nFinancial support from the ESA PECS Project C98090, ESTEC\nContract No.4000106398\/12\/NL\/KML, the Hungarian OTKA\nGrants K76816 and K83790, the MB08C 81013 mobility grant,\nand the ``Lend\\\"ulet-2009'' Young Researchers\nProgram of the Hungarian Academy of Sciences are gratefully acknowledged.\nA. D. and Cs. K. acknowledges the support of the J\\'anos Bolyai Research\nFellowship of the Hungarian Academy of Sciences.  A. D. was supported by \nthe Hungarian E\\\"otv\\\"os fellowship. We are grateful to the staff at the \n2.2\\,m telescope at La Silla for their kind support during the FEROS\nobservations. The {\\it INTEGRAL\\\/} photometric data, \npre-processed by ISDC, have been retrieved from the OMC Archive at CAB\n\\linebreak(INTA-CSIC). \nCritical remarks by Dr. M. Kun and the referee led to a considerable \nimprovement in the presentation of the results.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe notion of entanglement from quantum physics is now one of the\nmost important resources in current quantum information and quantum\ncomputation theory. Recall that a state is called (fully) separable\nif it is a convex sum of pure product states, and entangled if it is\nnot separable. In multi-partite systems, the notion of entanglement\ndepends on the partitions of subsystems to get various kinds of\npartial entanglement. In the tri-partite system with subsystems $A$,\n$B$ and $C$, we have three kinds of bi-separability, that is,\n$A$-$BC$, $B$-$CA$ and $C$-$AB$ separability according to the\nbi-partitions of the subsystems. We call those {\\sl basic\nbi-separability}. We will denote by $\\aaa$, $\\bbb$ and $\\ccc$ the\nconvex cones consisting of all $A$-$BC$, $B$-$CA$ and $C$-$AB$\nseparable un-normalized states, respectively. In the three qubit\ncases, the convex cones $\\aaa,\\bbb$ and $\\ccc$ are sitting in the\nreal vector space of all three qubit self-adjoint matrices, which is a $64$ dimensional real vector space.\n\nMany authors have considered the intersections and convex hulls for basic bi-separable states, which will be denoted by\n$\\meet$ and $\\join$, respectively. Note that the convex hull of two convex cones coincides with the nonnegative sum.\nAfter it was shown in \\cite{bdmsst} that a three qubit state in $\\aaa\\meet\\bbb\\meet\\ccc$\nneed not to be fully separable as a tri-partite state, several authors have considered intersections and convex sums of the convex cones\n$\\aaa$, $\\bbb$ and $\\ccc$. See \\cite{dur-cirac,dur-cirac-tarrach} for intersections\nof two of them, and \\cite{seevinck-uffink,acin} for convex hull of them.\nSee also \\cite{sz2011,sz2012,sz2015,sz2018,han_kye_bisep_exam,han_kye_pe,Szalay-2019} for further development\nin more general contexts.\nWe recall that a tri-partite state is called genuinely entangled if it does not belong to\n$\\aaa\\join\\bbb\\join\\ccc$.\n\nVery recently, the authors and Szalay \\cite{han_kye_szalay} considered the lattice, dented by ${\\mathcal L}$, generated by\nthree convex cones $\\aaa,\\bbb$ and $\\ccc$ with respect to the two operations of convex hull and\nintersection, and showed that this lattice violates the distributive rules. More precisely,\nit was shown that the following inequalities\n\\begin{align}\n(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)&\\le \\aaa\\meet(\\bbb\\join\\ccc),\\label{dist-ineq}\\\\\n\\aaa\\join(\\bbb\\meet\\ccc)&\\le (\\aaa\\join\\bbb)\\meet(\\aaa\\join\\ccc)\\label{dist-ineq-2}\n\\end{align}\nare strict.\nWe refer to \\cite{birkhoff,{freese}} for general theory of lattices.\n\nFor a convex cone $C$ in a real vector space $V$ with a bi-linear pairing $\\lan\\, \\cdot\\, ,\\, \\cdot \\, \\ran$, the dual cone $C^\\circ$ is defined by\nthe convex cone consisting of all $y\\in V$ satisfying $\\lan x,y\\ran\\ge 0$ for every $x\\in C$.\nFor given two self-adjoint matrices $x=[x_{ij}]$ and $y=[y_{ij}]$, we use the bi-linear pairing\n$$\n\\lan x,y\\ran=\\tr(xy^\\ttt)=\\sum_{ij}x_{ij}y_{ij},\n$$\nwhere $y^\\ttt$ denotes the transpose of $y$.\nWe recall that matrices in the dual cones play the roles of {\\sl witnesses}.\nFor example, we have $\\varrho\\notin (\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)$ if and only if\nthere exists\n$$\nW\\in [(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)]^\\circ\n=(\\aaa^\\circ\\join\\bbb^\\circ)\\meet(\\aaa^\\circ\\join\\ccc^\\circ)\n$$\nsuch that $\\lan W,\\varrho\\ran<0$. See \\cite{han_kye_pe} for the details.\nWe note that the following\ninequalities\n\\begin{align}\n\\aaa^\\circ\\join(\\bbb^\\circ\\meet\\ccc^\\circ)&\\le (\\aaa^\\circ\\join\\bbb^\\circ)\\meet(\\aaa^\\circ\\join\\ccc^\\circ),\\label{dist-ineq-2-dual}\\\\\n(\\aaa^\\circ\\meet\\bbb^\\circ)\\join(\\aaa^\\circ\\meet\\ccc^\\circ)&\\le \\aaa^\\circ\\meet(\\bbb^\\circ\\join\\ccc^\\circ)\\label{dist-ineq-dual}\n\\end{align}\nare also strict, by duality.\n\n\nThe main purpose of this paper is to give criteria for the convex cones arising in the above inequalities\n(\\ref{dist-ineq}), (\\ref{dist-ineq-2}), (\\ref{dist-ineq-2-dual}) and (\\ref{dist-ineq-dual}) in the three qubit cases.\nCriteria will be given in terms of diagonal and anti-diagonal entries.\nCriteria of such kinds have been considered for the convex cone $\\aaa\\join\\bbb\\join\\ccc$ in\n\\cite{{gao},{guhne10},{Rafsanjani}} to get sufficient conditions for genuine entanglement.\nSuch criteria for $\\aaa,\\bbb,\\ccc$ and $\\aaa^\\circ,\\bbb^\\circ,\\ccc^\\circ$ also can be found in\n\\cite[Proposition 5.2]{han_kye_optimal} and \\cite[Theorem 6.2]{han_kye_tri}, respectively,\n(see also Propositions 3.1 and 3.3 of \\cite{han_kye_optimal}) from which\nwe also get criteria for intersections like $\\aaa\\meet\\bbb$ and $\\aaa^\\circ\\meet\\bbb^\\circ$.\nConvex sums like $\\aaa\\join\\bbb$ and $\\aaa^\\circ\\join\\bbb^\\circ$ have been considered in \\cite{han_kye_pe}.\nFinally, we found criteria for the convex cones of the type $\\aaa\\join(\\bbb\\meet\\ccc)$\nin \\cite{han_kye_szalay} in the context of distributive rules. Therefore,\nwe will concentrate on the convex cones of the following types\n\\begin{equation}\\label{list}\n(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc),\\qquad\n\\aaa^\\circ\\join(\\bbb^\\circ\\meet\\ccc^\\circ),\\qquad\n(\\aaa^\\circ\\meet\\bbb^\\circ)\\join(\\aaa^\\circ\\meet\\ccc^\\circ).\n\\end{equation}\nThe whole convex cones of partially separable states we are considering can be drawn in the following diagram with the inclusion relations.\n\\begin{equation}\\label{diag_con}\n\\xymatrix{\n& \\alpha \\join \\beta \\join \\gamma && \\\\\n\\alpha \\join \\beta \\ar[ur] & \\gamma \\join \\alpha \\ar[u] & \\beta \\join \\gamma \\ar[ul] & \\\\\n(\\alpha \\join \\beta) \\meet (\\gamma \\join \\alpha) \\ar[u] \\ar[ur]\n    & (\\alpha \\join \\beta) \\meet (\\beta \\join \\gamma) \\ar[ul] \\ar[ur]\n    & (\\gamma \\join \\alpha) \\meet (\\beta \\join \\gamma) \\ar[ul] \\ar[u] & \\\\\n\\alpha \\join (\\beta \\meet \\gamma) \\ar[u] & \\beta \\join (\\gamma \\meet \\alpha) \\ar[u] & \\gamma \\join (\\alpha \\meet \\beta) \\ar[u]\n&\n\\\\\n\\alpha \\ar[u] & \\beta \\ar[u] & \\gamma \\ar[u] &  \\\\\n\\alpha \\meet (\\beta \\join \\gamma) \\ar[u] \\ar@{>}[uuurr] \\ar@{>}[uuur]\n     &\\beta \\meet (\\gamma \\join \\alpha) \\ar[u] \\ar@{>}[uuul] \\ar@{>}[uuur]\n     & \\gamma \\meet (\\alpha \\join \\beta) \\ar[u] \\ar@{>}[uuull] \\ar@{>}[uuul]\n     &\n\\\\\n(\\alpha \\meet \\beta) \\join (\\gamma \\meet \\alpha) \\ar[u] \\ar@{>}[uuur] \\ar@{>}[uuurr]\n    & (\\alpha \\meet \\beta) \\join (\\beta \\meet \\gamma) \\ar[u] \\ar@{>}[uuul] \\ar@{>}[uuur]\n    & (\\gamma \\meet \\alpha) \\join (\\beta \\meet \\gamma) \\ar[u] \\ar@{>}[uuull] \\ar@{>}[uuul]\n    & \\\\\n\\alpha \\meet \\beta \\ar[u] \\ar[ur] & \\gamma \\meet \\alpha \\ar[ul] \\ar[ur] & \\beta \\meet \\gamma \\ar[ul] \\ar[u] & \\\\\n& \\alpha \\meet \\beta \\meet \\gamma \\ar[ul] \\ar[u] \\ar[ur] &\n}\n\\end{equation}\nWe also have the similar diagram for the dual cones consisting of witnesses.\n\nWe recall that a matrix is called $\\xx$-{\\sl shaped} \\cite{yu07} if all the entries are zero except for diagonal and anti-diagonal entries.\n$\\xx$-shaped states will be called $\\xx$-states. Many important states like GHZ diagonal states belong to this class.\nIf a three qubit state is (partially) separable, then its $\\xx$-part is also (partially) separable,\nand so any separability criteria for \\xx-states\nwill give rise to a necessary criteria for general three qubit states. In this\npaper, we will give necessary conditions for three qubit states\n(respectively, witnesses) to belong to convex cones\nlisted in (\\ref{list}) (respectively, the dual of (\\ref{list})) in\nterms of diagonal and anti-diagonal entries. Such conditions are\nalso sufficient when states\/witnesses are $\\xx$-shaped. Especially,\nwe will find lattice identities for the first and third convex cones\nin (\\ref{list}) when states\/witnesses are $\\xx$-shaped. See\nCorollaries \\ref{identity_P} and \\ref{identity_Q}.\n\nAfter we collect known results for criteria in the next section, we\nwill give criteria for the convex cones of the type\n$(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)$ in Section 3, where we will\nalso show that the lattice ${\\mathcal L}$ is not complemented.\nCriteria for the types $\\aaa^\\circ\\join(\\bbb^\\circ\\meet\\ccc^\\circ)$\nand $(\\aaa^\\circ\\meet\\bbb^\\circ)\\join(\\aaa^\\circ\\meet\\ccc^\\circ)$\nwill be given in Sections 4 and 5, respectively.\nIn Section 6, we restrict our attention on GHZ diagonal states,\nto exhibit all GHZ diagonal states belonging to convex cones considered in the paper.\n\nThe authors are grateful to the referee for bringing their attention to GHZ diagonal states.\n\n\\section{summary of the known criteria}\n\n\nStates and witnesses in the tensor product $M_2\\otimes M_2\\otimes M_2$ of $2\\times 2$ matrices may be written\nas an $8\\times 8$ matrices with respect to the lexicographic order of of indices for subsystems. Then $\\xx$-shaped states\/witnesses\nare of the form\n$$\n\\xx(a,b,z)= \\left(\n\\begin{matrix}\na_1 &&&&&&& z_1\\\\\n& a_2 &&&&& z_2 & \\\\\n&& a_3 &&& z_3 &&\\\\\n&&& a_4&z_4 &&&\\\\\n&&& \\bar z_4& b_4&&&\\\\\n&& \\bar z_3 &&& b_3 &&\\\\\n& \\bar z_2 &&&&& b_2 &\\\\\n\\bar z_1 &&&&&&& b_1\n\\end{matrix}\n\\right),\n$$\nfor $a,b\\in\\mathbb R^4$ and $z\\in\\mathbb C^4$. It is well-known that\nevery GHZ diagonal state \\cite{GHZ} is in this form, and an $\\xx$-state  $\\varrho=\\xx(a,b,z)$\nis GHZ diagonal if and only if $a=b$ and $z\\in\\mathbb R^4$.\nSee \\cite{han_kye_GHZ}.\n\nWe denote by $\\varrho_\\xx$ the $\\xx$-part of a state $\\varrho$, in the obvious sense. If we denote by $\\mathcal L$\nthe lattice generated by $\\aaa$, $\\bbb$ and $\\ccc$ in the three qubit case, then we have\n\\begin{equation}\\label{x-part}\n\\varrho\\in\\sigma \\ \\Longrightarrow \\varrho_\\xx\\in\\sigma,\n\\end{equation}\nfor every $\\sigma\\in\\mathcal L$. In fact, it is easily seen that if (\\ref{x-part}) holds for $\\sigma$ and $\\tau$ then\nit also holds for $\\sigma\\meet\\tau$ and $\\sigma\\join\\tau$. We have already seen that\nthe generators $\\aaa,\\bbb$ and $\\ccc$ of ${\\mathcal L}$ satisfy (\\ref{x-part})\nin \\cite[Proposition 2.2]{han_kye_pe}, and so it follows that (\\ref{x-part}) holds for every $\\sigma\\in\\mathcal L$.\nIf we denote by ${\\mathcal L}^\\circ$ the lattice generated by $\\aaa^\\circ$, $\\bbb^\\circ$ and $\\ccc^\\circ$, then we also have\n\\begin{equation}\\label{x-part-w}\nW\\in\\sigma \\ \\Longrightarrow W_\\xx\\in\\sigma,\n\\end{equation}\nfor every $\\sigma\\in{\\mathcal L}^\\circ$, by the identity $\\lan\\varrho_\\xx,W\\ran = \\lan\\varrho, W_\\xx\\ran$.\n\nBy a {\\sl pair} $\\{i,j\\}$, we always mean throughout this paper an unordered set\nwith {\\sl two} elements among $1,2,3,4$, that is, we assume that $i\\neq j$.\nFor a given three qubit $\\xx$-shaped state $\\varrho=\\xx(a,b,z)$, we\nconsider the inequalities\n\\begin{center}\n\\framebox{\n\\parbox[t][2.5cm]{14.80cm}{\n\\addvspace{0.1cm} \\centering\n$$\n\\begin{array}{ll}\nS_1[i,j]:&\\quad \\min\\{\\sqrt{a_ib_i},\\sqrt{a_jb_j}\\}\\ge \\max\\{|z_i|,|z_j|\\},\\\\\nS_2[i,j]:&\\quad\n\\min\\left\\{\\sqrt{a_ib_i}+\\sqrt{a_jb_j},\\sqrt{a_k b_k}+\\sqrt{a_\\ell b_\\ell}\\right\\}\n   \\ge\\max\\left\\{|z_i|+|z_j|,|z_k|+|z_\\ell|\\right\\},\\\\\nS_3 :&\\quad \\textstyle{\\sum_{j\\neq i}\\sqrt{a_jb_j}\\ge |z_i|},\\quad i=1,2,3,4,\\\\\nS_4[i,j|k,\\ell]:&\\quad\n\\min\\left\\{\\sqrt{a_ib_i}+\\sqrt{a_jb_j},\\sqrt{a_k b_k}+\\sqrt{a_\\ell b_\\ell}\\right\\}\n   \\ge\\max\\left\\{|z_i|+|z_j|,|z_k|+|z_\\ell|\\right\\}.\n\\end{array}\n$$\n}}\n\\end{center}\\medskip\nThe inequalities $S_1$ and $S_2$ are defined for a pair $\\{i,j\\}$, where the pair\n$\\{k,\\ell\\}$ appearing in the inequality $S_2[i,j]$ are chosen so that $\\{i,j,k,\\ell\\}=\\{1,2,3,4\\}$.\nOn the other hand, the inequality $S_4$ is defined for arbitrary two pairs $\\{i,j\\}$ and $\\{k,\\ell\\}$.\nWe note that the inequality $S_4[i,j|k,\\ell]$ holds\nautomatically for any {\\sf X}-states $\\varrho=\\xx(a,b,z)$ when $\\{i,j\\}=\\{k,\\ell\\}$.\nIf $\\{i,j\\}\\cap\\{k,\\ell\\}=\\emptyset$, we note that\nthe three inequalities $S_2[i,j]$, $S_2[k,\\ell]$ and $S_4[i,j|k,\\ell]$ are same.\n\nWe summarize the results for three qubit $\\xx$-shaped states $\\varrho=\\xx(a,b,z)$ as follows:\nFirst of all, it was shown in \\cite[Proposition 5.2]{han_kye_optimal} and\n\\cite[Proposition 3.1]{han_kye_pe} that\n\\begin{itemize}\n\\item\n$\\varrho\\in\\aaa$ if and only if $S_1[1,4]$ and $S_1[2,3]$ hold;\n\\item\n$\\varrho\\in\\bbb$ if and only if $S_1[1,3]$ and $S_1[2,4]$ hold;\n\\item\n$\\varrho\\in\\ccc$ if and only if $S_1[1,2]$ and $S_1[3,4]$ hold.\n\\end{itemize}\nAnalogous results for multi-qubit states are also known in \\cite{han_kye_optimal}.\nAs for the convex hulls of them, the authors showed in\n\\cite[Theorem 5.5]{han_kye_pe} the following:\n\\begin{itemize}\n\\item\n$\\varrho\\in\\bbb\\join\\ccc$ if and only if $S_2[1,4]$ (equivalently $S_2[2,3]$) holds;\n\\item\n$\\varrho\\in\\ccc\\join\\aaa$ if and only if $S_2[1,3]$ (equivalently $S_2[2,4]$) holds;\n\\item\n$\\varrho\\in\\aaa\\join\\bbb$ if and only if $S_2[1,2]$ (equivalently $S_2[3,4]$) holds.\n\\end{itemize}\nOn the other hand, it has been known earlier \\cite{{guhne10},{gao},{Rafsanjani},{han_kye_optimal}}\nthat\n\\begin{itemize}\n\\item\n$\\varrho\\in\\aaa\\join\\bbb\\join\\ccc$\nif and only if $S_3$ holds.\n\\end{itemize}\nSee also \\cite[Proposition 4.5]{han_kye_pe}.\nFinally, the authors and Szalay showed\nin \\cite[Theorem 2.1]{han_kye_szalay} that the following\n\\begin{itemize}\n\\item\n$\\varrho\\in\\aaa\\join(\\bbb\\meet\\ccc)$ if and only if $S_4[i,j|k,\\ell]$ holds\nwhenever $\\{i,j\\}$, $\\{k,\\ell\\}$ are two of $\\{1,2\\}$, $\\{1,3\\}$, $\\{2,4\\}$, $\\{3,4\\}$;\n\\item\n$\\varrho\\in\\bbb\\join(\\ccc\\meet\\aaa)$ if and only if $S_4[i,j|k,\\ell]$ holds\nwhenever $\\{i,j\\}$, $\\{k,\\ell\\}$ are two of $\\{1,2\\}$, $\\{1,4\\}$, $\\{2,3\\}$, $\\{3,4\\}$;\n\\item\n$\\varrho\\in\\ccc\\join(\\aaa\\meet\\bbb)$ if and only if $S_4[i,j|k,\\ell]$ holds\nwhenever $\\{i,j\\}$, $\\{k,\\ell\\}$ are two of $\\{1,3\\}$, $\\{1,4\\}$, $\\{2,3\\}$, $\\{2,4\\}$\n\\end{itemize}\nhold for $\\xx$-states $\\varrho=\\xx(a,b,z)$ in the contexts of distributive rules.\n\n\n\nFor an {\\sf X}-shaped\nself-adjoint matrix $W=\\xx(s,t,u)$ with $s_i,t_i\\ge 0$ and\n$u\\in\\mathbb C^4$, we also consider the following inequalities:\n\\begin{center}\n\\framebox{\n\\parbox[t][1.9cm]{10.00cm}{\n\\addvspace{0.1cm} \\centering\n$$\n\\begin{array}{ll}\nW_1[i,j]: &\\quad \\sqrt{s_it_i}+\\sqrt{s_jt_j}\\ge |u_i|+|u_j|,\\\\\nW_2[i,j]: &\\quad \\sum_{k\\neq j}\\sqrt{s_kt_k}\\ge |u_i|,\\quad \\sum_{k\\neq i}\\sqrt{s_kt_k}\\ge |u_j|,\\qquad \\qquad \\qquad \\qquad\\qquad\\\\\nW_3 :&\\quad \\sum_{i=1}^4\\sqrt{s_it_i}\\ge\\sum_{i=1}^4|u_i|\n\\end{array}\n$$\n}}\n\\end{center}\\medskip\nfor a pair $\\{i,j\\}$.\nThree qubit $\\xx$-shaped witnesses $W=\\xx(s,t,u)$ which are dual of the basic bi-separability\nhave been considered in \\cite[Theorem 6.2]{han_kye_tri} and\n\\cite[Proposition 3.3]{han_kye_pe} as follows:\n\\begin{itemize}\n\\item\n$W\\in\\aaa^\\circ$ if and only if $W_1[1,4]$ and $W_1[2,3]$ hold;\n\\item\n$W\\in\\bbb^\\circ$ if and only if $W_1[1,3]$ and $W_1[2,4]$ hold;\n\\item\n$W\\in\\ccc^\\circ$ if and only if $W_1[1,2]$ and $W_1[3,4]$ hold.\n\\end{itemize}\nOn the other hand, the joins of them have been characterized in\n\\cite[Theorem 5.2]{han_kye_pe}:\n\\begin{itemize}\n\\item\n$W\\in\\bbb^\\circ\\join\\ccc^\\circ$ if and only if $W_2[1,4]$, $W_2[2,3]$ and $W_3$ hold;\n\\item\n$W\\in\\ccc^\\circ\\join\\aaa^\\circ$ if and only if $W_2[1,3]$, $W_2[2,4]$ and $W_3$ hold;\n\\item\n$W\\in\\aaa^\\circ\\join\\bbb^\\circ$ if and only if $W_2[1,2]$, $W_2[3,4]$ and $W_3$ hold.\n\\end{itemize}\nFinally, it was shown in\n\\cite[Theorem 5.5]{han_kye_optimal} and\n\\cite[Proposition 4.2]{han_kye_pe} that\n\\begin{itemize}\n\\item\n$W\\in\\aaa^\\circ\\join\\bbb^\\circ\\join\\ccc^\\circ$  if and only if $W_3$ holds.\n\\end{itemize}\n\n\n\n\\section{Criteria for $(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)$}\n\n\nIn this section, we look for criteria for the convex cones of the type $(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)$. To do this,\nwe first consider the following inequalities;\n\\begin{equation}\\label{dist1}\n\\begin{aligned}\n(x\\meet y)\\join (x\\meet z)\n&\\le x\\meet (x\\join (y\\meet z))\\meet(y\\join (z\\meet x))\\meet (z\\join (y\\meet x))\\\\\n&= x\\meet (y\\join (z\\meet x))\\meet (z\\join (y\\meet x))\\\\\n&\\le x\\meet (y\\join (z\\meet x))\\\\\n&\\le x\\meet (y\\join z),\n\\end{aligned}\n\\end{equation}\nwhich hold in general in an arbitrary lattice.\nFrom the first inequality of (\\ref{dist1}),\nwe have natural necessary conditions. We show that they are sufficient\nfor $\\xx$-states.\n\n\n\\begin{theorem}\\label{dist111}\nFor a three qubit state $\\varrho$ with the $\\xx$-part $\\xx(a,b,z)$, we have the following:\n\\begin{enumerate}\n\\item[(i)]\nif $\\varrho\\in (\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)$, then the\ninequalities $S_1[1,4]$, $S_1[2,3]$ and $S_4[i,j|k,\\ell]$ hold for\nany different pairs $\\{i,j\\}$ and $\\{k,\\ell\\}$;\n\\item[(ii)]\nif $\\varrho\\in (\\bbb\\meet\\ccc)\\join(\\bbb\\meet\\aaa)$, then the\ninequalities $S_1[1,3]$, $S_1[2,4]$ and $S_4[i,j|k,\\ell]$ hold for\nany different pairs $\\{i,j\\}$ and $\\{k,\\ell\\}$;\n\\item[(iii)]\nif $\\varrho\\in (\\ccc\\meet\\aaa)\\join(\\ccc\\meet\\beta)$, then the\ninequalities $S_1[1,2]$, $S_1[3,4]$ and $S_4[i,j|k,\\ell]$ hold for\nany different pairs $\\{i,j\\}$ and $\\{k,\\ell\\}$.\n\\end{enumerate}\nIf $\\varrho=\\xx(a,b,z)$ then the converses also hold.\n\\end{theorem}\n\n\n\\begin{proof}\nWe prove the last one. The necessity follows from (\\ref{dist1}), more precisely from the inequality\n$$\n(\\ccc\\meet\\aaa)\\join(\\ccc\\meet\\beta)\\le\n\\ccc\\meet (\\aaa\\join(\\bbb\\meet\\ccc))\\meet(\\bbb\\join(\\ccc\\meet\\aaa))\\meet (\\ccc\\join(\\aaa\\meet\\bbb)),\n$$\nand the criteria \\cite[Theorem 2.1]{han_kye_szalay} for convex cones in the right side.\nWe prove the converse\nwhen $\\varrho=\\xx(a,b,z)$. To do this, we suppose that $\\varrho$ satisfies $S_1[1,2]$, $S_1[3,4]$\nand $S_4[i,j|k,\\ell]$ for different pairs $\\{i,j\\}$ and $\\{k,\\ell\\}$. In order to show that $\\varrho\\in\n(\\ccc\\meet\\aaa)\\join(\\ccc\\meet\\bbb)$, we suppose\nthat $W=\\xx(s,t,u)\\in (\\ccc^\\circ\\join\\aaa^\\circ)\\meet(\\ccc^\\circ\\join\\bbb^\\circ)$ and\nwill prove $\\lan W,\\varrho\\ran\\ge 0$. So, we assume that $W$ satisfies $W_2[1,3]$,\n$W_2[2,4]$ $W_2[1,4]$, $W_2[2,3]$, and $W_3$.\n\nIf $\\varrho\\in\\aaa$ or $\\varrho\\in\\bbb$, then there is nothing to prove since $\\varrho\\in\\ccc$ by\n$S_1[1,2]$ and $S_1[3,4]$. Therefore, we may assume that\n$\\varrho\\notin\\aaa$ and $\\varrho\\notin\\bbb$. By $\\varrho\\notin\\aaa$, we may assume $|z_4|>\\sqrt{a_1b_1}$ without loss of generality.\nBy $\\varrho\\notin\\bbb$, we have one of the following:\n$$\n|z_3|>\\sqrt{a_1b_1} ,\\qquad |z_1|>\\sqrt{a_3b_3}, \\qquad\n|z_2|>\\sqrt{a_4b_4}, \\qquad  |z_4|>\\sqrt{a_2b_2}.\n$$\nThe second implies $|z_1|+|z_4|>\\sqrt{a_1b_1}+\\sqrt{a_3b_3}$, which violates $S_4[1,4|1,3]$.\nThe third also violates $S_4[2,4|1,4]$. Therefore, we have two cases:\n Either $|z_4|, |z_3| \\ge \\sqrt{a_1b_1}$ or $|z_4| \\ge \\sqrt{a_1b_1}, \\sqrt{a_2b_2}$.\n\nWe consider the case: $|z_4|, |z_3| \\ge \\sqrt{a_1b_1}$.\nBy $S_4[1,i|3,4]$ with $i=2,3,4$, we have\n$$\n\\min\\{ \\sqrt{a_2b_2}, \\sqrt{a_3b_3}, \\sqrt{a_4b_4} \\} \\ge |z_4| + (|z_3|-\\sqrt{a_1b_1}).\n$$\nTherefore, we have\n$$\n\\begin{aligned}\n&\\sqrt{s_2t_2}\\sqrt{a_2b_2} + \\sqrt{s_3t_3}\\sqrt{a_3b_3} + \\sqrt{s_4t_4}\\sqrt{a_4b_4} -|u_4||z_4| \\\\\n&\\phantom{XX} \\ge (\\sqrt{s_2t_2} + \\sqrt{s_3t_3} + \\sqrt{s_4t_4}) \\min \\{ \\sqrt{a_2b_2}, \\sqrt{a_3b_3}, \\sqrt{a_4b_4}) -|u_4||z_4|\\\\\n&\\phantom{XX} \\ge  \\left( \\sqrt{s_2t_2} + \\sqrt{s_3t_3} + \\sqrt{s_4t_4} -|u_4| \\right) |z_4| +\n     (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4})(|z_3| -\\sqrt{a_1b_1}) \\\\\n&\\phantom{XX} \\ge  \\left( \\sqrt{s_2t_2} + \\sqrt{s_3t_3} + \\sqrt{s_4t_4} -|u_4| \\right) \\sqrt{a_1b_1}\n     + (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4})(|z_3| -\\sqrt{a_1b_1}) \\\\\n&\\phantom{XX}=-|u_4|\\sqrt{a_1b_1}+(\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4})|z_3|,\n\\end{aligned}\n$$\nwhere the last inequality follows from $W_2[4,1]$.\nOn the other hand, we also have\n$$\n\\sqrt{s_1t_1}\\sqrt{a_1b_1}-|u_1||z_1|-|u_2||z_2|-|u_3||z_3|\n\\ge\n\\sqrt{s_1t_1}\\sqrt{a_1b_1}-|u_1|\\sqrt{a_1b_1}-|u_2|\\sqrt{a_1b_1}-|u_3||z_3|,\n$$\nby $S_1[1,2]$. Summing up the above two inequalities, we have\n$$\n\\begin{aligned}\n{\\tfrac 12}\\lan W, \\varrho \\ran\n& = {1 \\over 2}\\textstyle\\sum_{i=1}^4 [s_i a_i + t_i b_i + 2 {\\rm Re} (u_i z_i)]\\\\\n& \\ge \\textstyle\\sum_{i=1}^4 (\\sqrt{s_i t_i} \\sqrt{a_i b_i} - |u_i| |z_i|)\\\\\n& \\ge(\\sqrt{s_1t_1}-|u_1|-|u_2|-|u_4|)\\sqrt{a_1b_1}\n  +(\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_3|)|z_3|\\\\\n&\\ge (\\sqrt{s_1t_1}-|u_1|-|u_2|-|u_4|)\\sqrt{a_1b_1}\n  +(\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_3|)\\sqrt{a_1b_1}\\\\\n&=(\\textstyle{\\sum_{i=1}^4\\sqrt{s_it_i}-\\sum_{j=1}^4|u_j|})\\sqrt{a_1b_1},\n\\end{aligned}\n$$\nby $W_2[3,1]$. This is nonnegative by $W_3$.\n\nIt remains to consider the case: $|z_4| \\ge \\sqrt{a_1b_1}, \\sqrt{a_2b_2}$.\nWe use $S_4[1,2|i,4]$ with $i=1,2,3$, to get\n$\\sqrt{a_1b_1} + \\sqrt{a_2b_2} - |z_4| \\ge  \\max\\{ |z_1|, |z_2|, |z_3| \\}$, and so\n$$\n\\begin{aligned}\n(\\textstyle\\sum_{i=1}^4\\sqrt{s_it_i}-|u_4|)(\\sqrt{a_1b_1} + \\sqrt{a_2b_2} - |z_4| )\n&\\ge  (\\textstyle\\sum_{i=1}^3|u_i|) \\max\\{ |z_1|, |z_2|, |z_3| \\}\\\\\n&\\ge \\textstyle\\sum_{i=1}^3|u_i||z_i|,\n\\end{aligned}\n$$\nby $W_3$. Therefore, we have\n$$\n\\begin{aligned}\n \\textstyle\\sum_{i=1}^4 &(\\sqrt{s_i t_i} \\sqrt{a_i b_i} - |u_i| |z_i|) \\\\\n& = \\textstyle\\sum_{i=1}^2\\sqrt{s_it_i}\\sqrt{a_ib_i}\n    + \\left(\\sqrt{s_3t_3}\\sqrt{a_3b_3} + \\sqrt{s_4t_4}\\sqrt{a_4b_4} -|u_4||z_4|\\right) - \\textstyle\\sum_{i=1}^3 |u_i||z_i| \\\\\n& \\ge  \\textstyle\\sum_{i=1}^2\\sqrt{s_it_i}\\sqrt{a_ib_i}+ (\\sqrt{s_3t_3} + \\sqrt{s_4t_4} -|u_4|)|z_4| \\\\\n& \\phantom{XXXXXXXXXXX} - (\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - |u_4| ) (\\sqrt{a_1b_1} + \\sqrt{a_2b_2} - |z_4|),\n\\end{aligned}\n$$\nby $S_1[3,4]$. We continue as follows:\n$$\n\\begin{aligned}\n& =  ( - \\textstyle\\sum_{i \\ne 1} \\sqrt{s_it_i} + |u_4| ) \\sqrt{a_1b_1} + ( - \\textstyle\\sum_{i \\ne 2} \\sqrt{s_it_i} + |u_4| ) \\sqrt{a_2b_2}\\\\\n& \\phantom{XXXXXXXXXXX} + ( \\sqrt{s_1t_1} + \\sqrt{s_2t_2} + 2\\sqrt{s_3t_3} + 2\\sqrt{s_4t_4} - 2|u_4| )|z_4| \\\\\n& \\ge  ( - \\textstyle\\sum_{i \\ne 1} \\sqrt{s_it_i} + |u_4| ) |z_4|\n+ ( - \\textstyle\\sum_{i \\ne 2} \\sqrt{s_it_i} + |u_4| ) |z_4|\\\\\n& \\phantom{XXXXXXXXXXX} + ( \\sqrt{s_1t_1} + \\sqrt{s_2t_2} + 2\\sqrt{s_3t_3} + 2\\sqrt{s_4t_4} - 2|u_4| )|z_4|=0, \\\\\n\\end{aligned}\n$$\nby the inequalities $W_2[4,1]$ and $W_2[4,2]$.\n\\end{proof}\n\nIn order to summarize the role of inequality $S_4[i,j|k,\\ell]$, we consider the following six convex cones\n$$\n\\begin{aligned}\n\\sigma_1&=\\aaa\\join\\bbb,\\\\\n\\sigma_2&=(\\aaa\\join\\bbb)\\meet(\\aaa\\join\\ccc),\\\\\n\\sigma_3&=\\aaa\\join(\\bbb\\meet\\ccc),\\\\\n\\sigma_4&=\\aaa,\\\\\n\\sigma_5&=\\aaa\\meet(\\bbb\\join\\ccc),\\\\\n\\sigma_6&=(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc),\n\\end{aligned}\n$$\nwhich make the chain $\\sigma_1\\supset\\sigma_2\\supset\\sigma_3\\supset\\sigma_4\\supset\\sigma_5\\supset\\sigma_6$\nof inclusions. We provide Table 1 to see which inequalities $S_4[i,j|k,\\ell]$ we need to determine $\\sigma_i$ for $i=1,2,3,4,5,6$.\nWe recall again that both $S_2[i,j]$ and $S_2[k,\\ell]$\ncoincide with $S_4[i,j|k,\\ell]$ when $\\{i,j\\}\\cap\\{k,\\ell\\}=\\emptyset$.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{c||c|c|c|c|c|c|}\n&$\\{1,2\\}$ &$\\{1,3\\}$ &$\\{1,4\\}$ &$\\{2,3\\}$ &$\\{2,4\\}$ &$\\{3,4\\}$\\\\\n\\hline\\hline\n$\\{1,2\\}$ && $\\sigma_3$ &$\\sigma_6$ &$\\sigma_6$  &$\\sigma_3$ &$\\sigma_1$\\\\\n\\hline\n$\\{1,3\\}$ &  &&$\\sigma_6$ &$\\sigma_6$  &$\\sigma_2$ &$\\sigma_3$\\\\\n\\hline\n$\\{1,4\\}$ && &&$\\sigma_5$  &$\\sigma_6$ &$\\sigma_6$\\\\\n\\hline\n$\\{2,3\\}$ &&& &&$\\sigma_6$ &$\\sigma_6$\\\\\n\\hline\n$\\{2,4\\}$ &&&& &&$\\sigma_3$\\\\\n\\hline\n$\\{3,4\\}$ &&&&& &\\\\\n\\hline\n\\end{tabular}\n\\caption{\\small This table shows that $\\sigma_1=\\aaa\\join\\bbb$ is determined by $S_4[1,2|3,4]$.\nThe convex cone $\\sigma_2=(\\aaa\\join\\bbb)\\meet(\\aaa\\join\\ccc)$\nis determined by an extra inequality $S_4[1,3|2,4]$, and $\\sigma_3=\\aaa\\join(\\bbb\\meet\\ccc)$ is determined by the six inequalities\nlabeled by $\\sigma_1$, $\\sigma_2$ and $\\sigma_3$.\nThe convex cone $\\sigma_4=\\aaa$ is determined by another inequalities $S_1[1,4]$ and $S_1[2,3]$ which imply\nthese six inequalities. One more inequality $S_4[1,4|2,3]$ is required in order to determine $\\sigma_5=\\aaa\\meet(\\bbb\\join\\ccc)$,\nand all the inequalities are required to determine $\\sigma_6=(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)$ as well as $S_1[1,4]$ and $S_1[2,3]$.\nWe note that $S_1[i,j]$ may be recovered if we allow $S_4[i,i|j,j]$.\n}\n\\end{center}\n\\end{table}\n\nWe denote by $\\xx$ the real vector space of all three qubit $\\xx$-shaped self-adjoint matrices, and define\n$$\n\\sigma_X:=\\sigma\\cap\\xx=\\{\\varrho_\\xx:\\varrho\\in\\sigma\\},\n$$\nfor each $\\sigma\\in{\\mathcal L}$, where the last identity follows from (\\ref{x-part}).\n Then, it is easily seen that\n\\begin{equation}\\label{homox}\n(\\sigma\\meet\\tau)_\\xx=\\sigma_\\xx\\meet\\tau_\\xx,\\qquad\n(\\sigma\\join\\tau)_\\xx=\\sigma_\\xx\\join\\tau_\\xx,\n\\end{equation}\nand so, we see that\n$$\n{\\mathcal L}_\\xx=\\{ \\sigma_\\xx:\\sigma\\in{\\mathcal L} \\}\n$$\nis the lattice generated by $\\aaa_\\xx$, $\\bbb_\\xx$ and $\\ccc_\\xx$, which are\n$16$ affine dimensional convex bodies sitting in the real vector space $\\xx$.\nWe write\n$$\nP:=[\\aaa\\join(\\bbb\\meet\\ccc)]\\meet[\\bbb\\join(\\ccc\\meet\\aaa)]\\meet[\\ccc\\join(\\aaa\\meet\\bbb)]\n$$\nin the lattice ${\\mathcal L}$. Then we have\n$$\nP_\\xx =[\\aaa_\\xx\\join(\\bbb_\\xx\\meet\\ccc_\\xx)]\\meet[\\bbb_\\xx\\join(\\ccc_\\xx\\meet\\aaa_\\xx)]\\meet[\\ccc_\\xx\\join(\\aaa_\\xx\\meet\\bbb_\\xx)]\n$$\nin ${\\mathcal L}_\\xx$.\nTheorem \\ref{dist111} shows that the following lattice theoretic identities hold among generators\n$\\aaa_\\xx$, $\\bbb_\\xx$ and $\\ccc_\\xx$ of the lattice ${\\mathcal L}_\\xx$.\n\n\n\n\\begin{corollary}\\label{identity_P}\nWe have the following identities\n$$\n\\begin{aligned}\n(\\aaa_\\xx\\meet\\bbb_\\xx)\\join(\\aaa_\\xx\\meet\\ccc_\\xx)&=\\aaa_\\xx\\meet P_\\xx,\\\\\n(\\bbb_\\xx\\meet\\ccc_\\xx)\\join(\\bbb_\\xx\\meet\\aaa_\\xx)&=\\bbb_\\xx\\meet P_\\xx,\\\\\n(\\ccc_\\xx\\meet\\aaa_\\xx)\\join(\\ccc_\\xx\\meet\\bbb_\\xx)&=\\ccc_\\xx\\meet P_\\xx.\n\\end{aligned}\n$$\n\\end{corollary}\n\nWe also write\n$$\nQ:=[\\aaa\\meet(\\bbb\\join\\ccc)]\\join[\\bbb\\meet(\\ccc\\join\\aaa)]\\join[\\ccc\\meet(\\aaa\\join\\bbb)]\n$$\nin the lattice ${\\mathcal L}$. We will also have later the dual identities\nfor $\\aaa_\\xx\\join Q_\\xx$, $\\bbb_\\xx\\join Q_\\xx$ and $\\ccc_\\xx\\join Q_\\xx$\nin the lattice ${\\mathcal L}_\\xx$.\n\n\nIt was asked in \\cite{han_kye_szalay} whether the lattice ${\\mathcal L}$ is complemented or not.\nRecall that a lattice $L$ is called complemented if every $x\\in L$ has a {\\sl complement} $y\\in L$\nsatisfying $x\\meet y=0$ and $x\\join y=1$, where $0$ and $1$ are the least and greatest elements of $L$,\nrespectively. Note that $\\aaa\\meet\\bbb\\meet\\ccc$ and $\\aaa\\join\\bbb\\join\\ccc$\nare the least and the greatest elements of the lattice ${\\mathcal L}$.\nWe will show that $\\aaa$ has no complement in the lattice ${\\mathcal L}$. To do this, we recall\nthe results in \\cite{han_kye_pe}. We denote by $\\Delta$ the collection of eight diagonal states\n$\\xx(E_i,0,0)$ and $\\xx(0,E_i,0)$ with the usual orthonormal basis $\\{E_1,E_2,E_3,E_4\\}$,\nwhich generate extreme rays of every convex cones in ${\\mathcal L}$\nby \\cite[Theorem 4.3]{han_kye_pe}.\nWe also denote by $\\ext (C)$\nthe set of points of a convex cone $C$ which generate extreme rays, and put\n$$\n\\E_\\aaa=\\ext(\\aaa_X)\\setminus \\Delta,\\qquad\n\\E_\\bbb=\\ext(\\bbb_X)\\setminus \\Delta,\\qquad\n\\E_\\ccc=\\ext(\\ccc_X)\\setminus \\Delta.\n$$\nAll the states in $\\E_\\aaa$, $\\E_\\bbb$, $\\E_\\ccc$ and $\\ext(\\aaa_\\xx\\join\\bbb_\\xx\\join\\ccc_\\xx)$\nhave been found in Theorem 3.5 and Theorem 4.6 of \\cite{han_kye_pe}. Especially, we have the following:\n\\begin{itemize}\n\\item\n$\\E_\\aaa$, $\\E_\\bbb$ and $\\E_\\ccc$ are mutually disjoint;\n\\item\n$\\ext(\\aaa_\\xx\\join\\bbb_\\xx\\join\\ccc_\\xx)$ coincides with the disjoint union $\\E_\\aaa\\sqcup \\E_\\bbb\\sqcup\\E_\\ccc\\sqcup\\Delta$.\n\\end{itemize}\nNow, we assume that $\\aaa$ has a complement $\\sigma$ in the lattice ${\\mathcal L}$. Then we have\n$$\n\\aaa_\\xx\\join\\bbb_\\xx\\join\\ccc_\\xx =\\aaa_\\xx\\join\\sigma_\\xx,\\qquad\n\\aaa_\\xx\\meet\\bbb_\\xx\\meet\\ccc_\\xx =\\aaa_\\xx\\meet\\sigma_\\xx,\n$$\nby (\\ref{homox}). Then we have\n$$\n\\E_\\bbb\\cup\\E_\\ccc\n\\subset\\ext(\\aaa_\\xx\\join\\bbb_\\xx\\join\\ccc_\\xx)\n=\\ext(\\aaa_\\xx\\join\\sigma_\\xx)\n\\subset\\ext(\\aaa_\\xx)\\cup\\ext(\\sigma_\\xx),\n$$\nwhich implies $\\E_\\bbb\\cup\\E_\\ccc \\subset\\ext(\\sigma_\\xx)$ by the mutual disjointness of\n$\\E_\\aaa$, $\\E_\\bbb$, $\\E_\\ccc$ and $\\Delta$. We also have\n$$\n\\Delta\\subset\\aaa_\\xx\\meet\\bbb_\\xx\\meet\\ccc_\\xx =\\aaa_\\xx\\meet\\sigma_\\xx\\subset\\sigma_\\xx,\n$$\nand so we have $\\bbb_\\xx\\join\\ccc_\\xx\\subset\\sigma_\\xx$. Therefore, we have\n$$\n\\aaa_\\xx\\meet\\bbb_\\xx\\meet\\ccc_\\xx\\subsetneqq\n\\aaa_\\xx\\meet(\\bbb_\\xx\\join\\ccc_\\xx)\\subset \\aaa_\\xx\\meet\\sigma_\\xx=\\aaa_\\xx\\meet\\bbb_\\xx\\meet\\ccc_\\xx\n$$\nby the criteria in Section 2. This contradiction shows that the lattice ${\\mathcal L}$ is not complemented.\n\n\nReturning to the inequalities in (\\ref{dist1}), we recall that a lattice $L$ is called distributive if all the inequalities in (\\ref{dist1})\nare identities for every $x,y,z\\in L$, and modular if $(x\\meet y)\\join (x\\meet z)$ coincides with\n$x\\meet (y\\join (z\\meet x))$ for every $x,y,z\\in L$. We have considered in \\cite{han_kye_szalay} the $\\xx$-state given by\n$$\n\\varrho_1=\\xx\\left((2,1,1,2)(2,1,1,2),(2,0,1,0)\\right)\\in\\aaa\\meet(\\bbb\\join(\\ccc\\meet\\aaa))\n$$\nin order to show that the lattice ${\\mathcal L}$ is not modular. We also consider\n$$\n\\varrho_2=\\xx\\left( (2,1,1,2),(2,1,1,2), (2,1,0,0)\\right)\\in\\aaa\\meet(\\ccc\\join(\\bbb\\meet\\aaa))\n$$\nto see that the strict inequalities\n$$\n\\begin{aligned}\n(\\aaa_\\xx\\meet\\bbb_\\xx)\\join(\\aaa_\\xx\\meet\\ccc_\\xx)&\\lneqq \\aaa_\\xx\\meet(\\bbb_\\xx\\join(\\ccc_\\xx\\meet\\aaa_\\xx)),\\\\\n(\\aaa_\\xx\\meet\\bbb_\\xx)\\join(\\aaa_\\xx\\meet\\ccc_\\xx)&\\lneqq \\aaa_\\xx\\meet(\\ccc_\\xx\\join(\\bbb_\\xx\\meet\\aaa_\\xx))\n\\end{aligned}\n$$\nhold. If we take meet of these two formulae, we see that the identity holds\nwhen we plug $\\aaa_\\xx$, $\\bbb_\\xx$ and $\\ccc_\\xx$ into $x,y$ and $z$\nin following general inequality\n\\begin{equation}\\label{aaaaa}\n(x\\meet y)\\join (x\\meet z) \\le x\\meet (y\\join (z\\meet x))\\meet (z\\join (y\\meet x)).\n\\end{equation}\nSee also Corollary \\ref{identity_Q}\nfor the dual identity.\nThese identities are very special from the view point of general lattice theory or convex geometry.\nTo see this, we consider the lattice of all convex sets on the plain with respect to the convex hull and intersection.\nIn this lattice, we take a closed disc with a diameter $\\overline{AB}$ and two line segments\n$\\overline{AC}$ and $\\overline{BD}$ so that these line segments touch the disc\nat single points $A$ and $B$, respectively. We plug the disc, the line segments $\\overline{AC}$ and $\\overline{BD}$\ninto $x,y$ and $z$ in (\\ref{aaaaa}). Then the left side is the just line segment $\\overline{AB}$, but the right\nside is the intersection of the two triangles $\\triangle ABC$ and $\\triangle ABD$ inside of the disc,\nwhich is much bigger that the line segment $\\overline{AB}$ in general.\n\n\\section{Criteria for $\\aaa^\\circ\\join(\\bbb^\\circ\\meet\\ccc^\\circ)$}\n\n\nIn this section, we give criteria for the convex cones of the type  $\\aaa^\\circ\\join(\\bbb^\\circ\\meet\\ccc^\\circ)$.\nFor a self-adjoint $W=\\xx(s,t,u)$, we consider the inequality $W_4[i,j]$ which combines the following two inequalities:\n\\begin{center}\n\\framebox{\n\\parbox[t][1.4cm]{10.00cm}{\n\\addvspace{0.1cm} \\centering\n$$\n\\begin{array}{ll}\nW_{4{\\rm a}}[i,j]: &\\quad \\sqrt{s_it_i}+\\sqrt{s_j t_j}+2\\min\\{\\sqrt{s_k t_k},\\sqrt{s_\\ell t_\\ell}\\} \\ge |u_i|+|u_j|,\\\\\nW_{4{\\rm b}}[i,j]: &\\quad \\sqrt{s_it_i}+\\sqrt{s_jt_j}+2(\\sqrt{s_kt_k}+\\sqrt{s_\\ell t_\\ell})\\ge |u_i|+|u_j|+2\\max\\{|u_k|,|u_\\ell|\\}\\\\\n\\end{array}\n$$\n}}\n\\end{center}\\medskip\nfor a pair $\\{i,j\\}$, where $\\{k,\\ell\\}$ is chosen so that $\\{i,j,k,\\ell\\}=\\{1,2,3,4\\}$. We have the following:\n\n\\begin{theorem}\\label{join-statestep4}\nIf a self-adjoint three qubit matrix $W$ with the $\\xx$-part $\\xx(s,t,u)$ belongs to the convex cone\n$$\n\\aaa^\\circ\\join(\\bbb^\\circ\\meet\\ccc^\\circ),\\qquad\n{\\rm(}{\\text respectively}\\\n\\bbb^\\circ\\join(\\ccc^\\circ\\meet\\aaa^\\circ)\\\n{\\text and}\\\n\\ccc^\\circ\\join(\\aaa^\\circ\\meet\\bbb^\\circ)\n{\\rm )}\n$$\nthen $W$ satisfies $W_3$ together with the following:\n\\begin{enumerate}\n\\item[(i)]\n$W_2[i,j]$ whenever $\\{i,j\\}$ is one of $\\{1,2\\}$, $\\{1,3\\}$, $\\{2,4\\}$, $\\{3,4\\}$\n{\\rm (}respectively\n$\\{1,2\\}$, $\\{1,4\\}$, $\\{2,3\\}$, $\\{3,4\\}$ and $\\{1,3\\}$, $\\{1,4\\}$, $\\{2,3\\}$, $\\{2,4\\}${\\rm )};\n\\item[(ii)]\n$W_{4}[i,j]$ whenever\n$\\{i,j\\}$ is one of $\\{1,4\\}$, $\\{2,3\\}$ {\\rm (}respectively\n$\\{1,3\\}$, $\\{2,4\\}$ and $\\{1,2\\}$, $\\{3,4\\}${\\rm )}.\n\\end{enumerate}\nIf $W$ is $\\xx$-shaped, then the converse holds.\n\\end{theorem}\n\n\\begin{proof}\nWe will prove for the convex cone $\\ccc^\\circ\\join(\\aaa^\\circ\\meet\\bbb^\\circ)$.\nSuppose that $W\\in \\ccc^\\circ\\join(\\aaa^\\circ\\meet\\bbb^\\circ)$. The required inequalities $W_2[i,j]$ and $W_3$\nfollow from\n$$\n\\ccc^\\circ\\join(\\aaa^\\circ\\meet\\bbb^\\circ)\\le (\\ccc^\\circ\\join\\aaa^\\circ)\\meet (\\ccc^\\circ\\join\\bbb^\\circ).\n$$\nTo get the inequalities $W_{4{\\rm a}}$ and $W_{4{\\rm b}}$, we\nmay assume that $s_i,t_i>0$ as in the proof of \\cite[Theorem 2.1]{han_kye_szalay}.\nWe consider\n$$\n\\begin{aligned}\n\\varrho_{i,j,k}:=&\\xx(\\sqrt{\\tfrac{t_i}{s_i}}E_i + \\sqrt{\\tfrac{t_j}{s_j}}E_j + 2\\sqrt{\\tfrac{t_k}{s_k}}E_k,\\\\\n                  &\\qquad\\qquad\\qquad \\sqrt{\\tfrac{s_i}{t_i}}E_i + \\sqrt{\\tfrac{s_j}{t_j}}E_j + 2\\sqrt{\\tfrac{s_k}{t_k}}E_k,\\\\\n                 &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad -e^{-{\\rm i}\\theta_i}E_i -e^{-{\\rm i}\\theta_j}E_j), \\\\\n\\varrho_{i,j,k,\\ell}':=&\\xx(\\sqrt{\\tfrac{t_i}{s_i}}E_i + \\sqrt{\\tfrac{t_j}{s_j}}E_j + 2\\sqrt{\\tfrac{t_k}{s_k}}E_k\n                           + 2\\sqrt{\\tfrac{t_\\ell}{s_\\ell}}E_\\ell,\\\\\n                  &\\qquad\\qquad\\qquad \\sqrt{\\tfrac{s_i}{t_i}}E_i + \\sqrt{\\tfrac{s_j}{t_j}}E_j + 2\\sqrt{\\tfrac{s_k}{t_k}}E_k\n                             + 2\\sqrt{\\tfrac{s_\\ell}{t_\\ell}}E_\\ell, \\\\\n                   &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad -e^{-{\\rm i}\\theta_i}E_i - e^{-{\\rm i}\\theta_j}E_j - 2e^{-{\\rm i}\\theta_k}E_k),\n\\end{aligned}\n$$\nwhere $\\theta_m = \\arg z_m$.\nWhen $\\{\\{i,j\\},\\{k,\\ell\\}\\}=\\{\\{1,2\\},\\{3,4\\}\\}$, both of them satisfy $S_1[1,2]$, $S_1[3,4]$, $S_2[1,2]$, and so,\nbelong to $\\ccc\\meet(\\aaa\\join\\bbb)$.\nWe expand $\\langle W, \\varrho_{i,j,k} \\rangle \\ge 0$ and $\\langle W, \\varrho_{i,j,k,\\ell}' \\rangle \\ge 0$\nto get the required inequalities $W_{\\rm 4a}[i,j]$ and $W_{\\rm 4b}[i,j]$.\n\nFor the converse, it suffices to show the\ninequality $\\lan W,\\varrho\\ran\\ge 0$ under the following assumptions:\n\\begin{itemize}\n\\item\n$W=\\xx(s,t,u)$ satisfies $W_3$;\n\\item\n$W=\\xx(s,t,u)$ satisfies\n$W_2[i,j]$ whenever $\\{i,j\\}$ is one of $\\{1,3\\}$, $\\{1,4\\}$, $\\{2,3\\}$, $\\{2,4\\}$;\n\\item\n$W=\\xx(s,t,u)$ satisfies\nboth $W_{4{\\rm a}}[i,j]$ and $W_{4{\\rm b}}[i,j]$\nwhenever\n$\\{i,j\\}$ is one of $\\{1,2\\}$, $\\{3,4\\}$;\n\\item\n$\\varrho=\\xx(a,b,z)\\in\\ccc\\meet(\\aaa\\join\\bbb)$, or equivalently\n$\\varrho$ satisfies\n$S_1[1,2]$, $S_1[3,4]$ and $S_2[1,2]$.\n\\end{itemize}\nBy the inequalities $W_3$, $W_2[1,4]$ and $W_2[2,3]$, we have $W\\in\\bbb^\\circ\\join\\ccc^\\circ$.\nSimilarly, we also have $W\\in \\ccc^\\circ\\join\\aaa^\\circ$ by $W_3$, $W_2[1,3]$ and $W_2[2,4]$.\nIf $\\varrho\\in (\\ccc\\meet\\aaa)\\join(\\ccc\\meet\\bbb)$, then we have $\\lan W,\\varrho\\ran\\ge 0$ by the duality.\nIf $\\varrho\\notin (\\ccc\\meet\\aaa)\\join(\\ccc\\meet\\bbb)$ then both $\\varrho\\notin\\aaa$ and $\\varrho\\notin\\bbb$ hold, since $\\varrho\\in\\ccc$.\nBy $\\varrho\\notin\\aaa$, we may assume that\n$$\n|z_4|>\\sqrt{a_1b_1}\n$$\nwithout loss of generality. As for $\\varrho\\notin\\bbb$, we have the following four possibilities:\n$$\n|z_3|>\\sqrt{a_1b_1}, \\qquad  |z_1|>\\sqrt{a_3b_3}, \\qquad |z_2|>\\sqrt{a_4b_4}, \\qquad  |z_4|>\\sqrt{a_2b_2}.\n$$\nThe second implies $|z_4|>\\sqrt{a_1b_1} \\ge |z_1| > \\sqrt{a_3b_3}\\ge |z_4|$ by $S_1[3,4]$, which is a contradiction.\nBecause the third also implies $|z_2|>\\sqrt{a_4b_4} \\ge |z_4| > \\sqrt{a_1b_1}\\ge |z_2|$ by $S_1[1,2]$, we have\ntwo possibilities, the first and the fourth. We consider the following four cases:\n\\begin{enumerate}\n\\item[(I)]\n$|z_4| \\ge |z_3| \\ge \\sqrt{a_1b_1}$,\n\\item[(II)]\n$|z_3| \\ge |z_4| \\ge \\sqrt{a_1b_1}$,\n\\item[(III)]\n$|z_4| \\ge \\sqrt{a_1b_1} \\ge \\sqrt{a_2b_2}$,\n\\item[(IV)]\n$|z_4| \\ge \\sqrt{a_2b_2} \\ge \\sqrt{a_1b_1}$.\n\\end{enumerate}\n\nFor the case (I), we use the inequality $S_2[1,2]$ to see\n$\\sqrt{a_2b_2} \\ge |z_3|+|z_4|-\\sqrt{a_1b_1}$. Therefore, we have\n\\begin{align*}\n\\tfrac 12\\lan W,\\varrho\\ran\n&\\ge \\textstyle\\sum_{i=1}^4 \\sqrt{s_i t_i} \\sqrt{a_i b_i} - |u_i| |z_i|\\\\\n& = ( \\sqrt{s_2t_2}\\sqrt{a_2b_2} + \\sqrt{s_3t_3}\\sqrt{a_3b_3} + \\sqrt{s_4t_4}\\sqrt{a_4b_4} -|u_4||z_4|) \\\\\n& \\qquad\\qquad\\qquad +( \\sqrt{s_1t_1}\\sqrt{a_1b_1} -|u_1||z_1| -|u_2||z_2| -|u_3||z_3|) \\\\\n& \\ge \\sqrt{s_2t_2}(|z_3|+|z_4|-\\sqrt{a_1b_1}) + \\sqrt{s_3t_3}|z_4| + \\sqrt{s_4t_4}|z_4| - |u_4||z_4| \\\\\n& \\qquad\\qquad\\qquad  +( \\sqrt{s_1t_1}\\sqrt{a_1b_1} -|u_1|\\sqrt{a_1b_1} -|u_2|\\sqrt{a_1b_1} -|u_3||z_3|),\n\\end{align*}\nby $S_1[3,4]$ and $S_1[1,2]$. We continue as follows:\n\\begin{align*}\n & =(\\sqrt{s_2t_2} + \\sqrt{s_3t_3} + \\sqrt{s_4t_4} -|u_4|) |z_4| + (\\sqrt{s_2t_2}-|u_3|)|z_3|\\\\\n& \\qquad\\qquad\\qquad +(\\sqrt{s_1t_1} - \\sqrt{s_2t_2} -|u_1| -|u_2|)\\sqrt{a_1b_1} \\\\\n& \\ge (\\sqrt{s_2t_2} + \\sqrt{s_3t_3} + \\sqrt{s_4t_4} -|u_4|) |z_3| + (\\sqrt{s_2t_2}-|u_3|)|z_3| \\\\\n& \\qquad\\qquad\\qquad +(\\sqrt{s_1t_1} - \\sqrt{s_2t_2} -|u_1| -|u_2|)\\sqrt{a_1b_1}\n\\end{align*}\nby $W_2[4,1]$. This is equal to\n\\begin{align*}\n& = (2\\sqrt{s_2t_2} + \\sqrt{s_3t_3} + \\sqrt{s_4t_4} - |u_3| - |u_4|) |z_3|\\\\\n& \\qquad\\qquad\\qquad+(\\sqrt{s_1t_1} - \\sqrt{s_2t_2} -|u_1| -|u_2|)\\sqrt{a_1b_1} \\\\\n& \\ge (2\\sqrt{s_2t_2} + \\sqrt{s_3t_3} + \\sqrt{s_4t_4} - |u_3| - |u_4|) \\sqrt{a_1b_1}\\\\\n& \\qquad\\qquad\\qquad +(\\sqrt{s_1t_1} - \\sqrt{s_2t_2} -|u_1| -|u_2|)\\sqrt{a_1b_1} \\\\\n& = \\left( \\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - \\textstyle\\sum_{i=1}^4 |u_i| \\right) \\sqrt{a_1b_1}\n\\end{align*}\nby $W_{4{\\rm a}}[3,4]$, which is nonnegative by $W_3$.\nFor the case (II), note that the conclusion and all the conditions on $W$ and $\\varrho$ are invariant\nunder switching the first and the second subsystems, except $|z_3| \\ge |z_4| \\ge \\sqrt{a_1b_1}$.\nIt changes $|z_3| \\ge |z_4| \\ge \\sqrt{a_1b_1}$\ninto $|\\bar z_4| \\ge |\\bar z_3| \\ge \\sqrt{a_1b_1}$, which is exactly the case (I).\n\nFor the case (III), we first note the following inequality\n$$\n|u_3||z_3|\\le\n\\left(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i}-|u_1|-|u_2|- |u_4| \\right) (\\sqrt{a_1b_1}+\\sqrt{a_2b_2}-|z_4|)\n$$\nby $W_3$ and $S_2[1,2]$. Therefore, we have\n\\begin{align*}\n\\textstyle\\sum_{i=1}^4 \\sqrt{s_i t_i} \\sqrt{a_i b_i}& - |u_i| |z_i|\n\\ge ( \\sqrt{s_1t_1}\\sqrt{a_1b_1} + \\sqrt{s_2t_2}\\sqrt{a_2b_2} + \\sqrt{s_3t_3}|z_4| + \\sqrt{s_4t_4}|z_4|) \\\\\n& \\qquad +(  -|u_1|\\sqrt{a_2b_2} -|u_2|\\sqrt{a_2b_2} -|u_4||z_4|) \\\\\n& \\qquad -(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i}-|u_1|-|u_2|- |u_4| ) (\\sqrt{a_1b_1}+\\sqrt{a_2b_2}-|z_4|)\n\\end{align*}\nby\n$S_1[3,4]$ and $S_1[1,2]$.\nThis is equal to the following:\n\\begin{align*}\n& = (\\sqrt{s_1t_1} + \\sqrt{s_2t_2} + 2\\sqrt{s_3t_3} + 2\\sqrt{s_4t_4} -|u_1| -|u_2| -2|u_4|) |z_4| \\\\\n& \\qquad + (-\\sqrt{s_2t_2}-\\sqrt{s_3t_3}-\\sqrt{s_4t_4}+|u_1|+|u_2|+|u_4|)\\sqrt{a_1b_1} \\\\\n& \\qquad +(-\\sqrt{s_1t_1}-\\sqrt{s_3t_3}-\\sqrt{s_4t_4}+|u_4|)\\sqrt{a_2b_2}.\n\\end{align*}\nUsing $W_{4{\\rm b}}[1,2]$, we may replace $|z_4|$ by $\\sqrt{a_1b_1}$ to the smaller quanity\n$$\n(\\sqrt{s_1t_1}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_4|)(\\sqrt{a_1b_1}-\\sqrt{a_2b_2}).\n$$\nThis is nonnegative by $W_2[4,2]$, and so we completed the proof for the case (III).\n\nIt remains to prove the case (IV).\nThe conclusion and all the conditions on $W$ and $\\varrho$ except $|z_4| \\ge \\sqrt{a_2b_2} \\ge \\sqrt{a_1b_1}$\nare invariant under switching the first and the second subsystems and\nthe local unitary operation by $I \\otimes I \\otimes \\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}$. They change\n$|z_4| \\ge \\sqrt{a_2b_2} \\ge \\sqrt{a_1b_1}$ into\n$|z_3| = |\\bar z_3| \\ge \\sqrt{a_2b_2} \\ge \\sqrt{a_1b_1}$, and again into\n$|z_4| \\ge \\sqrt{a_1b_1} \\ge \\sqrt{a_2b_2}$.\nThis is the case (III).\n\\end{proof}\n\n\n\n\\section{Criteria for $(\\aaa^\\circ\\meet\\bbb^\\circ)\\join(\\aaa^\\circ\\meet\\ccc^\\circ)$}\n\nAs for the convex cones of the type $(\\aaa^\\circ\\meet\\bbb^\\circ)\\join(\\aaa^\\circ\\meet\\ccc^\\circ)$,\nwe have the following criteria:\n\n\\begin{theorem}\\label{dist11122}\nFor a three qubit self-adjoint matrix $W$ with the $\\xx$-part $\\xx(s,t,u)$, we have the following:\n\\begin{enumerate}\n\\item[(i)]\nif $W\\in (\\aaa^\\circ\\meet\\bbb^\\circ)\\join(\\aaa^\\circ\\meet\\ccc^\\circ)$, then inequalities\n$W_1[1,4]$, $W_1[2,3]$, $W_3$, $W_2[i,j]$, $W_4[i,j]$ hold for every pair $\\{i,j\\}$;\n\\item[(ii)]\nif $W\\in (\\bbb^\\circ\\meet\\ccc^\\circ)\\join(\\bbb^\\circ\\meet\\aaa^\\circ)$, then inequalities\n$W_1[1,3]$, $W_1[2,4]$, $W_3$, $W_2[i,j]$, $W_4[i,j]$ hold for every pair $\\{i,j\\}$;\n\\item[(iii)]\nif $W\\in (\\ccc^\\circ\\meet\\aaa^\\circ)\\join(\\ccc^\\circ\\meet\\beta^\\circ)$, then inequalities\n$W_1[1,2]$, $W_1[3,4]$, $W_3$, $W_2[i,j]$, $W_4[i,j]$ hold for every pair $\\{i,j\\}$.\n\\end{enumerate}\nIf $W$ is $\\xx$-shaped, then the converses also hold.\n\\end{theorem}\n\n\\begin{proof}\nWe will prove (i) and its converse for $W=\\xx(s,t,u)$. The necessity follows from the inclusion\n$$\n(\\aaa^\\circ\\meet\\bbb^\\circ)\\join(\\aaa^\\circ\\meet\\ccc^\\circ)\n\\le\n\\aaa^\\circ\\meet (\\aaa^\\circ\\join(\\bbb^\\circ\\meet\\ccc^\\circ))\\meet\n(\\bbb^\\circ\\join(\\aaa^\\circ\\meet\\ccc^\\circ))\\meet(\\ccc^\\circ\\join(\\aaa^\\circ\\meet\\bbb^\\circ)),\n$$\nby \\cite[Proposition 3.3]{han_kye_pe} and Theorem \\ref{join-statestep4}.\n\nWe prove the converse when $W=\\xx(s,t,u)$. To do this, we suppose the following:\n\\begin{itemize}\n\\item\n$W=\\xx(s,t,u)$ satisfies\n$W_1[1,4]$, $W_1[2,3]$, $W_3$, $W_2[i,j]$, $W_4[i,j]$ hold for every pair $\\{i,j\\}$;\n\\item\n$\\varrho=\\xx(a,b,z) \\in (\\aaa \\join \\bbb) \\meet (\\aaa \\join \\ccc)$, that is, satisfies $S_2[1,2]$, $S_2[1,3]$,\n\\end{itemize}\nand prove the inequality\n\\begin{equation}\\label{ineqxxxx}\n\\lan W,\\varrho\\ran\\ge 0.\n\\end{equation}\nIf $\\varrho\\in\\aaa$ then we have (\\ref{ineqxxxx}) since $W\\in\\aaa^\\circ$ by $W_1[1,4]$ and $W_1[2,3]$.\nIf $\\varrho\\in\\bbb$ then we have $\\varrho\\in\\bbb\\meet(\\aaa\\join\\ccc)$, and so\nthe inequality (\\ref{ineqxxxx}) follows since $W\\in\\bbb^\\circ\\join(\\ccc^\\circ\\meet\\aaa^\\circ)$.\nWe also have (\\ref{ineqxxxx}) when $\\varrho\\in\\ccc$ by the same reasoning.\n\n\nTherefore, we may assume that $\\varrho\\notin\\aaa$, $\\varrho\\notin\\bbb$ and $\\varrho\\notin\\ccc$.\nBy $\\varrho \\notin \\alpha$, we may assume\n$$\n(A 4)~ |z_4|>\\sqrt{a_1b_1}.\n$$\nBy the assumption $\\varrho\\notin\\beta$, there are four possibilities\n$$\n(B1)~ |z_1|>\\sqrt{a_3b_3}, \\quad (B2)~ |z_2|>\\sqrt{a_4b_4},\n\\quad  (B3)~ |z_3|>\\sqrt{a_1b_1}, \\quad  (B4)~ |z_4|>\\sqrt{a_2b_2}.\n$$\nWe also have the following four cases\n$$\n(C1)~ |z_1|>\\sqrt{a_2b_2}, \\quad  (C2)~ |z_2|>\\sqrt{a_1b_1},\n \\quad (C3)~ |z_3|>\\sqrt{a_4b_4},  \\quad  (C4)~ |z_4|>\\sqrt{a_3b_3},\n$$\nby the assumption $\\varrho\\notin\\ccc$.\nUnder the assumption (A4), we have the implications\n$(B1) \\Rightarrow (C4)$, $(B2) \\Rightarrow (C2)$, $(C1) \\Rightarrow (B4)$ and $(C3) \\Rightarrow (B3)$.\nConversely, the condition (A4) can be implied as\n$(B3), (C4) \\Rightarrow (A4)$ and $(B4), (C2) \\Rightarrow (A4)$.\nHence, it suffices to consider the following eight cases:\n$$\n\\begin{aligned}\n&(A4), (B1); \\quad &(A4), (B2); \\quad &(A4), (C1); \\quad &(A4), (B3), (C2); \\\\\n&(A4), (C3); \\quad &(B3), (C4); \\quad &(B4), (C2); \\quad &(A4), (B4), (C4). \\\\\n\\end{aligned}\n$$\nSwitching the second and third subsystems interchanges\n$$\n(A4), (B1) \\leftrightarrow (A4), (C1), \\quad (A4), (B2) \\leftrightarrow (A4), (C3), \\quad (B3), (C4) \\leftrightarrow (C2), (B4).\n$$\nTherefore, it suffices to consider the following five cases:\n$$\n(B3), (C4) ; \\quad (A4), (B3), (C2); \\quad (A4), (B4), (C4) ; \\quad (A4), (B1); \\quad (A4), (B2).\n$$\n\n[Case I: (B3) and (C4)]. We have\n$|z_4| \\ge \\sqrt{a_3b_3} \\ge |z_3| \\ge \\sqrt{a_1b_1}$, and\n\\begin{align*}\n{\\tfrac 12}\\lan W, \\varrho \\ran\n& \\ge \\textstyle\\sum_{i=1}^4 \\sqrt{s_i t_i} \\sqrt{a_i b_i} - |u_i| |z_i| \\\\\n& \\ge  \\sqrt{s_1t_1}\\sqrt{a_1b_1} + \\sqrt{s_2t_2}(|z_3|+|z_4|-\\sqrt{a_1b_1}) + \\sqrt{s_3t_3}\\sqrt{a_3b_3} + \\sqrt{s_4t_4}|z_4| \\\\\n& \\qquad - |u_1|\\sqrt{a_1b_1} - (\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i}-\\textstyle\\sum_{i \\ne 2} |u_i| )\n                     (\\sqrt{a_1b_1}+\\sqrt{a_3b_3}-|z_4|)\\\\\n& \\qquad - |u_3||z_3|- |u_4||z_4|\n\\end{align*}\nby $S_2[1,2]$, $W_3$ and $S_2[1,3]$. This is equal to\n\\begin{align*}\n& = (\\sqrt{s_1t_1} + 2\\sqrt{s_2t_2} + \\sqrt{s_3t_3} +2\\sqrt{s_4t_4} -|u_1|-|u_3|-2|u_4|) |z_4| \\\\\n& \\qquad +(-\\textstyle\\sum_{i \\ne 3} \\sqrt{s_it_i} + \\sum_{i \\ne 2} |u_i|) \\sqrt{a_3b_3} +(\\sqrt{s_2t_2}- |u_3| ) |z_3| \\\\\n& \\qquad +(-2\\sqrt{s_2t_2} - \\sqrt{s_3t_3} - \\sqrt{s_4t_4} +|u_3| +|u_4|) \\sqrt{a_1b_1}.\n\\end{align*}\nTherefore, applying $W_{\\rm 4b}[1,3]$, we have\n\\begin{align*}\n{\\tfrac 12}\\lan W, \\varrho \\ran\n& \\ge (\\sqrt{s_1t_1} + 2\\sqrt{s_2t_2} + \\sqrt{s_3t_3} +2\\sqrt{s_4t_4} -|u_1|-|u_3|-2|u_4|) \\sqrt{a_3b_3} \\\\\n& \\qquad +(-\\textstyle\\sum_{i \\ne 3} \\sqrt{s_it_i} + \\sum_{i \\ne 2} |u_i| ) \\sqrt{a_3b_3}  +(\\sqrt{s_2t_2}- |u_3| ) |z_3| \\\\\n& \\qquad +(-2\\sqrt{s_2t_2} - \\sqrt{s_3t_3} - \\sqrt{s_4t_4} +|u_3| +|u_4|) \\sqrt{a_1b_1}\\\\\n& = (\\sqrt{s_2t_2} + \\sqrt{s_3t_3} + \\sqrt{s_4t_4} -|u_4|) \\sqrt{a_3b_3}\n    +(\\sqrt{s_2t_2}- |u_3| ) |z_3| \\\\\n& \\qquad +(-2\\sqrt{s_2t_2} - \\sqrt{s_3t_3} - \\sqrt{s_4t_4} +|u_3| +|u_4|) \\sqrt{a_1b_1}.\n\\end{align*}\nBy $W_{2}[4,1]$, $W_{\\rm 4a}[3,4]$ and $|z_3|\\ge \\sqrt{a_1b_1}$, this is greater than or equal to\n\\begin{align*}\n& \\ge (\\sqrt{s_2t_2} + \\sqrt{s_3t_3} + \\sqrt{s_4t_4} -|u_4|) |z_3| +(\\sqrt{s_2t_2}- |u_3| ) |z_3| \\\\\n& \\qquad\\qquad +(-2\\sqrt{s_2t_2} - \\sqrt{s_3t_3} - \\sqrt{s_4t_4} +|u_3| +|u_4|) |z_3|\n= 0.\n\\end{align*}\n\n[Case II: (A4), (B3) and (C2)]. We have $|z_4|, |z_3|, |z_2| \\ge \\sqrt{a_1b_1}$. In this case,\nwe may assume that $|z_2| \\ge |z_3|$ by switching the second and the third subsystems.\nPut\n$$\n\\lambda_2:=|z_2|-\\sqrt{a_1b_1}, \\quad \\lambda_3:=|z_3|-\\sqrt{a_1b_1}, \\quad \\lambda_4:=|z_4|-\\sqrt{a_1b_1},\n$$\nwhich are nonnegative by the assumption.\nWe proceed by considering two subcases.\n\n[Subcase II-1: $\\lambda_2 \\le \\lambda_3 + \\lambda_4$]. We have\n\\begin{align*}\n\\tfrac 12\\lan W, \\varrho \\ran\n& \\ge \\textstyle\\sum_{i=1}^4 \\sqrt{s_i t_i} \\sqrt{a_i b_i} - |u_i| |z_i| \\\\\n& \\ge \\sqrt{s_1t_1}\\sqrt{a_1b_1}  + \\sqrt{s_2t_2}(|z_3|+|z_4|-\\sqrt{a_1b_1})\\\\\n& \\qquad + \\sqrt{s_3t_3}(|z_2|+|z_4|-\\sqrt{a_1b_1}) + \\sqrt{s_4t_4}|z_4|  -|u_1|\\sqrt{a_1b_1} - \\textstyle\\sum_{i=2}^4 |u_i||z_i|,\n\\end{align*}\nby $S_2[1,2]$ and $S_2[1,3]$. By a direct calculation, this is equal to\n\\begin{align*}\n& =\\left(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - \\textstyle\\sum_{i=1}^4 |u_i|\\right) \\sqrt{a_1b_1} \\\\\n& \\qquad + (\\sqrt{s_3t_3}-|u_2|)\\lambda_2\n+ (\\sqrt{s_2t_2}-|u_3|)\\lambda_3\n+ (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_4|)\\lambda_4,\n\\end{align*}\nwhich is greater than or equal to the following\n\\begin{equation}\\label{mmmm}\n(\\sqrt{s_3t_3}-|u_2|)\\lambda_2\n+ (\\sqrt{s_2t_2}-|u_3|)\\lambda_3\n+ (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_4|)\\lambda_4,\n\\end{equation}\nby $W_3$.\nWe note that the sum of the following two terms\n$$\n\\sqrt{s_3t_3}-|u_2|, \\qquad \\sqrt{s_2t_2}-|u_3|\n$$\nare nonnegative by $W_1[2,3]$, and so at most one of them is negative possibly. If both of them are nonnegative, then\nthe proof is complete by $W_2[4,1]$.\nIf $\\sqrt{s_2t_2}-|u_3|<0$, then\nwe replace $\\lambda_3$ in (\\ref{mmmm}) by $\\lambda_2$ which satisfies $\\lambda_2\\ge\\lambda_3$ by $|z_2| \\ge |z_3|$, to get\n$$\n\\begin{aligned}\n\\tfrac 12\\lan W, \\varrho \\ran\n\\ge & (\\sqrt{s_3t_3}-|u_2|)\\lambda_2\n+ (\\sqrt{s_2t_2}-|u_3|)\\lambda_2\n+ (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_4|)\\lambda_4 \\\\\n= & (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}-|u_2|-|u_3|)\\lambda_2\n+ (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_4|)\\lambda_4,\n\\end{aligned}\n$$\nwhich is nonnegative by $W_1[2,3]$ and $W_2[4,1]$ again.\nIf $\\sqrt{s_3t_3}-|u_2|<0$, then we replace $\\lambda_2$ in (\\ref{mmmm})\nby $\\lambda_3+\\lambda_4$ to get\n$$\n\\begin{aligned}\n\\tfrac 12\\lan W, \\varrho \\ran\n\\ge &(\\sqrt{s_3t_3}-|u_2|)(\\lambda_3+\\lambda_4)\n+ (\\sqrt{s_2t_2}-|u_3|)\\lambda_3\n+ (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_4|)\\lambda_4 \\\\\n= & (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}-|u_2|-|u_3|)\\lambda_3\n+ (\\sqrt{s_2t_2}+2\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_2|-|u_4|)\\lambda_4,\n\\end{aligned}\n$$\nwhich is also nonnegative by $W_1[2,3]$ and $W_{\\rm 4a}[2,4]$.\n\n\n[Subcase II-2: $\\lambda_2 \\ge \\lambda_3 + \\lambda_4$]: In this case, we have\n\\begin{align*}\n\\tfrac 12\\lan W, \\varrho \\ran\n& \\ge \\textstyle\\sum_{i=1}^4 \\sqrt{s_i t_i} \\sqrt{a_i b_i} - |u_i| |z_i| \\\\\n& \\ge \\sqrt{s_1t_1}\\sqrt{a_1b_1}\n + \\sqrt{s_2t_2}|z_2| + \\sqrt{s_3t_3}(|z_2|+|z_4|-\\sqrt{a_1b_1}) + \\sqrt{s_4t_4}|z_4| \\\\\n& \\qquad -|u_1|\\sqrt{a_1b_1} - \\textstyle\\sum_{i=2}^4 |u_i||z_i|,\n\\end{align*}\nby $S_2[1,3]$. This is equal to the following:\n\\begin{align*}\n& = \\left(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - \\textstyle\\sum_{i=1}^4 |u_i|\\right) \\sqrt{a_1b_1} \\\\\n& \\qquad + \\left((\\sqrt{s_2t_2}+\\sqrt{s_3t_3}-|u_2|-|u_3|)+|u_3|\\right)\\lambda_2\n-|u_3|\\lambda_3\n+ (\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_4|)\\lambda_4,\n\\end{align*}\nwhich is, by $W_3$ and $W_1[2,3]$, greater than or equal to\n\\begin{align*}\n\\ge &\\left((\\sqrt{s_2t_2}+\\sqrt{s_3t_3}-|u_2|-|u_3|)+|u_3|\\right)(\\lambda_3+\\lambda_4)\n-|u_3|\\lambda_3\n+ (\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_4|)\\lambda_4 \\\\\n= &\\left(\\sqrt{s_2t_2}+\\sqrt{s_3t_3}-|u_2|-|u_3|\\right)\\lambda_3\n+ \\left(\\sqrt{s_2t_2}+2\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_2|-|u_4|\\right)\\lambda_4.\n\\end{align*}\nThis is nonnegative by $W_1[2,3]$ and $W_{\\rm 4a}[2,4]$.\n\n[Case III: (A4), (B4), (C4)]. We have $|z_4| \\ge \\sqrt{a_1b_1}, \\sqrt{a_2b_2}, \\sqrt{a_3b_3}$.\nWe may assume that $\\sqrt{a_2b_2} \\ge \\sqrt{a_3b_3}$ by switching the second and the third subsystems.\nPut\n$$\n\\mu_1:=|z_4|-\\sqrt{a_1b_1}, \\quad \\mu_2:=|z_4|-\\sqrt{a_2b_2}, \\quad \\mu_3:=|z_4|-\\sqrt{a_3b_3}.\n$$\nNote that $\\mu_2\\le\\mu_3$ by assumption. We proceed by considering two subcases.\n\n[Subcase III-1: $\\mu_3 \\le \\mu_1 + \\mu_2$]. In this case, we have\n\\begin{align*}\n\\tfrac 12\\lan W, \\varrho \\ran\n& \\ge \\textstyle\\sum_{i=1}^4 \\sqrt{s_i t_i} \\sqrt{a_i b_i} - |u_i| |z_i| \\\\\n& \\ge \\textstyle\\sum_{i=1}^3 \\sqrt{s_it_i}\\sqrt{a_ib_i} + \\sqrt{s_4t_4}|z_4|  - |u_1|\\sqrt{a_1b_1} - |u_2|(\\sqrt{a_1b_1}+\\sqrt{a_3b_3}-|z_4|)\\\\\n&\\qquad - |u_3|(\\sqrt{a_1b_1}+\\sqrt{a_2b_2}-|z_4|) - |u_4||z_4|,\n\\end{align*}\nby $S_2[1,3]$ and $S_2[1,2]$. By a direct computation, this becomes\n\\begin{equation}\\label{estimate}\n\\begin{aligned}\n& = \\left(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - \\textstyle\\sum_{i=1}^4 |u_i|\\right) \\sqrt{a_1b_1} \\\\\n& \\qquad + (\\textstyle\\sum_{i \\ne 1}\\sqrt{s_it_i}-|u_4|)\\mu_1 + (-\\sqrt{s_2t_2}+|u_3|)\\mu_2 + (-\\sqrt{s_3t_3}+|u_2|)\\mu_3.\n\\end{aligned}\n\\end{equation}\nIf $-\\sqrt{s_2t_2}+|u_3| < 0$, then this is, by $W_2[4,1]$ and $\\mu_2 \\le \\mu_3$, greater\nthan or equal to the following\n$$\n\\begin{aligned}\n&\\ge\n\\left(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - \\textstyle\\sum_{i=1}^4 |u_i|\\right) \\sqrt{a_1b_1}\n     + (-\\sqrt{s_2t_2}+|u_3|-\\sqrt{s_3t_3}+|u_2|)\\mu_3\\\\\n&\\ge\n\\left(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - \\textstyle\\sum_{i=1}^4 |u_i|\\right) \\sqrt{a_1b_1}\n     + (-\\sqrt{s_2t_2}+|u_3|-\\sqrt{s_3t_3}+|u_2|)\\sqrt{a_1b_1}\\\\\n&=(\\sqrt{s_1t_1}+\\sqrt{s_4t_4}-|u_1|-|u_4|)\\sqrt{a_1b_1}\n\\end{aligned}\n$$\nby $W_1[2,3]$ and $\\sqrt{a_1b_1}\\ge \\mu_3$ using $S_2[1,3]$. This is nonnegative by\n$W_1[1,4]$.\nIn the case of $-\\sqrt{s_2t_2}+|u_3| \\ge 0$,\nthe term (\\ref{estimate}) is equal to\n\\begin{align*}\n&= \\left(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - \\textstyle\\sum_{i=1}^4 |u_i|\\right) \\sqrt{a_1b_1}\n        +  (\\textstyle\\sum_{i \\ne 1}\\sqrt{s_it_i}-|u_4|)\\mu_1 \\\\\n& \\qquad +  (-\\sqrt{s_2t_2}+|u_3|)\\mu_2 + (-\\sqrt{s_2t_2}-\\sqrt{s_3t_3}+|u_2|+|u_3|)\\mu_3 + (\\sqrt{s_2t_2}-|u_3|)\\mu_3,\n\\end{align*}\nwhich is greater than or equal to the following\n\\begin{align*}\n& \\ge \\left(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - \\textstyle\\sum_{i=1}^4 |u_i|\\right) \\sqrt{a_1b_1}\n         +  (\\textstyle\\sum_{i \\ne 1}\\sqrt{s_it_i}-|u_4|)\\mu_1 \\\\\n& \\qquad +  (-\\sqrt{s_2t_2}+|u_3|)\\mu_2 + (-\\sqrt{s_2t_2}-\\sqrt{s_3t_3}+|u_2|+|u_3|)\\sqrt{a_1b_1} + (\\sqrt{s_2t_2}-|u_3|)(\\mu_1+\\mu_2)\n\\end{align*}\nby $W_1[2,3]$ and $\\sqrt{a_1b_1}\\ge \\mu_3$. This becomes\n\\begin{align*}\n& = (\\sqrt{s_1t_1}+\\sqrt{s_4t_4}-|u_1|-|u_4|)\\sqrt{a_1b_1} + (2\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_3|-|u_4|)\\mu_1,\n\\end{align*}\nwhich is nonnegative by $W_1[1,4]$ and $W_{\\rm 4a}[3,4]$.\n\n[Subcase III-2: $\\mu_3 \\ge \\mu_1 + \\mu_2$]. We use $S_2[1,3]$ to get the inequality\n\\begin{align*}\n\\tfrac 12\\lan W, \\varrho \\ran\n& \\ge \\textstyle\\sum_{i=1}^4 \\sqrt{s_i t_i} \\sqrt{a_i b_i} - |u_i| |z_i| \\\\\n& \\ge \\textstyle\\sum_{i=1}^3 \\sqrt{s_it_i}\\sqrt{a_ib_i} + \\sqrt{s_4t_4}|z_4| \\\\\n& \\qquad - |u_1|\\sqrt{a_1b_1} - |u_2|(\\sqrt{a_1b_1}+\\sqrt{a_3b_3}-|z_4|)\n- |u_3|\\sqrt{a_3b_3} - |u_4||z_4|,\n\\end{align*}\nwhich is equal to\n\\begin{align*}\n& = \\left(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - \\textstyle\\sum_{i=1}^4 |u_i|\\right) \\sqrt{a_1b_1}\n               + (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_3|-|u_4|)\\mu_1 \\\\\n& \\qquad  - \\sqrt{s_2t_2}\\mu_2 + \\sqrt{s_2t_2}\\mu_3 + (-\\sqrt{s_2t_2}-\\sqrt{s_3t_3}+|u_2|+|u_3|)\\mu_3,\n\\end{align*}\nby a direct calculation. Using $W_1[2,3]$ and $\\mu_3\\le\n\\sqrt{a_1b_1}$, we continue\n\\begin{align*}\n& \\ge \\left(\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i} - \\textstyle\\sum_{i=1}^4 |u_i|\\right) \\sqrt{a_1b_1}\n             + (\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_3|-|u_4|)\\mu_1 \\\\\n& \\qquad  - \\sqrt{s_2t_2}\\mu_2 + \\sqrt{s_2t_2}(\\mu_1+\\mu_2) + (-\\sqrt{s_2t_2}-\\sqrt{s_3t_3}+|u_2|+|u_3|)\\sqrt{a_1b_1} \\\\\n& = (\\sqrt{s_1t_1}+\\sqrt{s_4t_4}-|u_1|-|u_4|)\\sqrt{a_1b_1} + (2\\sqrt{s_2t_2}+\\sqrt{s_3t_3}+\\sqrt{s_4t_4}-|u_3|-|u_4|)\\mu_1,\n\\end{align*}\nwhich is nonnegative by $W_1[1,4]$ and $W_{\\rm 4a}[3,4]$.\n\n[Case IV: (A4), (B1)]. We have $|z_4| \\ge \\sqrt{a_1b_1} \\ge |z_1|\n\\ge \\sqrt{a_3b_3}$. By Case III, we may suppose that\n$\\sqrt{a_2b_2} \\ge |z_4|$. By $W_3$ and $S_2[1,3]$, we have\n\\begin{align*}\n\\tfrac 12\\lan W, \\varrho \\ran\n& \\ge \\textstyle\\sum_{i=1}^4 \\sqrt{s_i t_i} \\sqrt{a_i b_i} - |u_i| |z_i| \\\\\n& \\ge  \\sqrt{s_1t_1}\\sqrt{a_1b_1} + \\sqrt{s_2t_2}|z_4| + \\sqrt{s_3t_3}\\sqrt{a_3b_3} + \\sqrt{s_4t_4}|z_4| - |u_1||z_1|\\\\\n& \\quad  - (\\textstyle\\sum_{i=1}^4 \\sqrt{s_it_i}-\\textstyle\\sum_{i \\ne 2} |u_i| )\n                (\\sqrt{a_1b_1}+\\sqrt{a_3b_3}-|z_4|)- |u_3|\\sqrt{a_3b_3}- |u_4||z_4|\n\\end{align*}\nwhich becomes\n\\begin{align*}\n& = (\\sqrt{s_1t_1} + 2\\sqrt{s_2t_2} + \\sqrt{s_3t_3} +2\\sqrt{s_4t_4} -|u_1|-|u_3|-2|u_4|) |z_4|\\\\\n& \\qquad +(-\\textstyle\\sum_{i \\ne 1} \\sqrt{s_it_i} + \\textstyle\\sum_{i \\ne 2} |u_i| ) \\sqrt{a_1b_1} \\\\\n& \\qquad - |u_1| |z_1|\n +\\left((-\\sqrt{s_1t_1}-\\sqrt{s_4t_4} + |u_1|+|u_4|)-\\sqrt{s_2t_2} \\right) \\sqrt{a_3b_3}.\n\\end{align*}\nUsing $W_{\\rm 4b}[1,3]$ and $W_1[1,4]$, we continue\n\\begin{align*}\n& \\ge (\\sqrt{s_1t_1} + 2\\sqrt{s_2t_2} + \\sqrt{s_3t_3} +2\\sqrt{s_4t_4} -|u_1|-|u_3|-2|u_4|) \\sqrt{a_1b_1} \\\\\n& \\qquad +(-\\textstyle\\sum_{i \\ne 1} \\sqrt{s_it_i} + \\textstyle\\sum_{i \\ne 2} |u_i| ) \\sqrt{a_1b_1} \\\\\n& \\qquad - |u_1| |z_1|  +\\left((-\\sqrt{s_1t_1}-\\sqrt{s_4t_4} + |u_1|+|u_4|)-\\sqrt{s_2t_2} \\right) |z_1|,\n\\end{align*}\nwhich is equal to\n\\begin{align*}\n =(\\sqrt{s_1t_1}  + \\sqrt{s_2t_2}+\\sqrt{s_4t_4} -|u_4|) (\\sqrt{a_1b_1}-|z_1|).\n\\end{align*}\nThis is nonnegative by $W_2[4,3]$.\n\n[Case V: (A4), (B2)]. In this case, we have $|z_2| \\ge \\sqrt{a_4b_4} \\ge |z_4| \\ge \\sqrt{a_1b_1}$].\nBy Case II, we may suppose that $\\sqrt{a_1b_1} \\ge |z_3|$. Using $S_2[1,3]$, we proceed\n\\begin{align*}\n\\tfrac 12\\lan W, \\varrho \\ran\n& \\ge \\textstyle\\sum_{i=1}^4 \\sqrt{s_i t_i} \\sqrt{a_i b_i} - |u_i| |z_i| \\\\\n& \\ge \\sqrt{s_1t_1}\\sqrt{a_1b_1} + \\sqrt{s_2t_2}|z_2| + \\sqrt{s_3t_3}(|z_2|+|z_4|-\\sqrt{a_1b_1}) + \\sqrt{s_4t_4}\\sqrt{a_4b_4} \\\\\n& \\qquad - |u_1|\\sqrt{a_1b_1} - |u_2| |z_2| - |u_3|\\sqrt{a_1b_1}- |u_4||z_4| \\\\\n& = \\left((\\sqrt{s_2t_2} + \\sqrt{s_3t_3} -|u_2|-|u_3|)+|u_3|\\right) |z_2| \\\\\n& \\qquad +\\sqrt{s_4t_4} \\sqrt{a_4b_4}  +(\\sqrt{s_3t_3}-|u_4|) |z_4| +(\\sqrt{s_1t_1}-\\sqrt{s_3t_3} - |u_1|-|u_3|) \\sqrt{a_1b_1}.\n\\end{align*}\nBy $W_1[2,3]$ and $|z_2|\\ge |z_4|$, we have\n\\begin{align*}\n\\tfrac 12\\lan W, \\varrho \\ran\n& \\ge \\left((\\sqrt{s_2t_2} + \\sqrt{s_3t_3} -|u_2|-|u_3|)+|u_3|\\right) |z_4| \\\\\n& \\qquad +\\sqrt{s_4t_4} |z_4|  +(\\sqrt{s_3t_3}-|u_4|) |z_4|  +(\\sqrt{s_1t_1}-\\sqrt{s_3t_3} - |u_1|-|u_3|) \\sqrt{a_1b_1}\n\\end{align*}\nThis is equal to\n$$\n=(\\sqrt{s_2t_2} + 2\\sqrt{s_3t_3} + \\sqrt{s_4t_4} -|u_2|-|u_4|) |z_4|\n+(\\sqrt{s_1t_1}-\\sqrt{s_3t_3} - |u_1|-|u_3|) \\sqrt{a_1b_1}.\n$$\nUsing $W_{\\rm 4a}[2,4]$, we may replace $|z_4|$ by $\\sqrt{a_1b_1}$, to get\n\\begin{align*}\n\\tfrac 12\\lan W, \\varrho \\ran\n& \\ge (\\sqrt{s_2t_2} + 2\\sqrt{s_3t_3} + \\sqrt{s_4t_4} -|u_2|-|u_4|) \\sqrt{a_1b_1} \\\\\n& \\qquad +(\\sqrt{s_1t_1}-\\sqrt{s_3t_3} - |u_1|-|u_3|) \\sqrt{a_1b_1},\n\\end{align*}\nwhich is equal to $\\left(\\sum_{i=1}^4\\sqrt{s_it_i}-|u_i|\\right) \\sqrt{a_1b_1}\\ge 0$ by $W_3$.\nThis completes the proof.\n\\end{proof}\n\nWe recall the definition of $Q$ in Section 3:\n$$\nQ:=[\\aaa\\meet(\\bbb\\join\\ccc)]\\join[\\bbb\\meet(\\ccc\\join\\aaa)]\\join[\\ccc\\meet(\\aaa\\join\\bbb)].\n$$\nThen we have\n$$\nQ^\\circ=[\\aaa^\\circ\\join(\\bbb^\\circ\\meet\\ccc^\\circ)]\\meet[\\bbb^\\circ\\join(\\ccc^\\circ\\meet\\aaa^\\circ)]\n   \\meet[\\ccc^\\circ\\join(\\aaa^\\circ\\meet\\bbb^\\circ)].\n$$\nFor a given $\\sigma\\in{\\mathcal L}$, we define\n$$\n\\begin{aligned}\n\\sigma^\\circ_\\xx := \\sigma^\\circ\\cap \\xx\n&=\\{W\\in\\xx : \\lan W,\\varrho\\ran\\ge 0\\ {\\text{\\rm for every}}\\ \\varrho\\in\\sigma\\}\\\\\n&=\\{W\\in\\xx : \\lan W,\\varrho\\ran\\ge 0\\ {\\text{\\rm for every}}\\ \\varrho\\in\\sigma_\\xx\\}.\n\\end{aligned}\n$$\nThe last identity follows from (\\ref{x-part-w}), and we see that the {\\xx}-part $(\\sigma^\\circ)_\\xx$ of $\\sigma^\\circ$\ncoincides with the dual of $(\\sigma_\\xx)^\\circ$ of $\\sigma_\\xx$ in the space $\\xx$.\nBy Theorem \\ref{dist11122}, we have the\nfollowing:\n\n\\begin{corollary}\nWe have the following identities\n$$\n\\begin{aligned}\n(\\aaa^\\circ_\\xx\\meet\\bbb^\\circ_\\xx)\\join(\\aaa^\\circ_\\xx\\meet\\ccc^\\circ_\\xx)&=\\aaa^\\circ_\\xx\\meet Q^\\circ_\\xx,\\\\\n(\\bbb^\\circ_\\xx\\meet\\ccc^\\circ_\\xx)\\join(\\bbb^\\circ_\\xx\\meet\\aaa^\\circ_\\xx)&=\\bbb^\\circ_\\xx\\meet Q^\\circ_\\xx,\\\\\n(\\ccc^\\circ_\\xx\\meet\\aaa^\\circ_\\xx)\\join(\\ccc^\\circ_\\xx\\meet\\bbb^\\circ_\\xx)&=\\ccc^\\circ_\\xx\\meet Q^\\circ_\\xx.\n\\end{aligned}\n$$\n\\end{corollary}\n\nTaking the dual cones in the vector space $\\xx$, we get the following identities among the\ngenerators $\\aaa_\\xx$, $\\bbb_\\xx$ and $\\ccc_\\xx$ of the lattice\n${\\mathcal L}_\\xx$, which is the lattice theoretic dual identities of those in Corollary \\ref{identity_P}.\n\n\n\\begin{corollary}\\label{identity_Q}\nWe have the following identities\n$$\n\\begin{aligned}\n(\\aaa_\\xx\\join\\bbb_\\xx)\\meet(\\aaa_\\xx\\join\\ccc_\\xx)&=\\aaa_\\xx\\join Q_\\xx,\\\\\n(\\bbb_\\xx\\join\\ccc_\\xx)\\meet(\\bbb_\\xx\\join\\aaa_\\xx)&=\\bbb_\\xx\\join Q_\\xx,\\\\\n(\\ccc_\\xx\\join\\aaa_\\xx)\\meet(\\ccc_\\xx\\join\\bbb_\\xx)&=\\ccc_\\xx\\join Q_\\xx.\n\\end{aligned}\n$$\n\\end{corollary}\n\n\n\\section{Greenberger-Horne-Zeilinger diagonal states}\n\n\nWe recall that an $\\xx$-state $\\xx(a,b,z)$ is GHZ diagonal if and\nonly if $a=b$ and $z\\in\\mathbb R^4$. More generally, we will say\nthat an \\xx-shaped matrix $\\xx(s,t,u)$ is GHZ diagonal if and only\nif $s=t$ and $u\\in\\mathbb R^4$. In this section, we exhibit all the\nGHZ diagonal states which belong to the convex cones considered in\nthis paper. For this purpose, we search for extreme rays of the\ncorresponding convex cones. We consider the following local\noperation\n$$\nU=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix} \\otimes \\begin{pmatrix}0&1\\\\1&0\\end{pmatrix} \\otimes \\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\n$$\nwhich interchanges $|0\\ran$ and $|1\\ran$ in each subsystem $M_2 \\otimes M_2 \\otimes M_2$.\nFor an $\\xx$-shaped self-adjoint three qubit matrix $W=\\xx(s,t,u)$, we define\n$$\n\\widetilde W:=UWU^*=\\xx(t,s,\\bar u).\n$$\nWe also define\n$$\nW_\\GHZ:={W + \\widetilde W \\over 2} = X\\left({s+t \\over 2}, {s+t \\over 2}, {\\rm Re} ~u\\right).\n$$\nThen an \\xx-shaped matrix $W$ is GHZ diagonal if and only if $W=W_\\GHZ$, and the identity\n\\begin{equation}\\label{kjhftyd}\n\\lan W_\\GHZ,\\varrho\\ran=\\lan W,\\varrho_\\GHZ\\ran\n\\end{equation}\nholds for every $\\xx$-shaped $W$ and $\\varrho$, as in the proof of\n\\cite[Theorem 3.2]{han_kye_GHZ}.\nWe also define $W_\\GHZ=(W_\\xx)_\\GHZ$ for a general three qubit self-adjoint matrix $W$.\nThen, $W$ is GHZ diagonal if and only if $W=W_\\GHZ$. Further, the identity (\\ref{kjhftyd}) also holds\nfor every self-adjoint $\\sigma$ and $W$.\nWe denote by $V$ the eight dimensional real vector space consisting of GHZ diagonal matrices.\nIf we take a convex cone $\\sigma\\in{\\mathcal L}$, then\nthe dual cone $(\\sigma\\cap V)^\\circ$ of $\\sigma\\cap V$ in the vector space $V$ coincides\nwith $\\sigma^\\circ\\cap V$, because\n$$\n\\lan W, \\varrho \\ran = \\lan W_\\GHZ, \\varrho \\ran = \\lan W, \\varrho_\\GHZ \\ran \\ge 0,\n$$\nfor $W \\in (\\sigma\\cap V)^\\circ$ and $\\varrho \\in \\sigma$.\nFor GHZ diagonal states, we will use the notation\n\\begin{equation}\\label{GHZ_sym}\n\\xx(a,a,z)=\\xx{a_1~a_2~a_3~a_4\\choose z_1~z_2~z_3~z_4},\n\\end{equation}\nwith real variables $a_i$ and $z_i$ for $i=1,2,3,4$.\n\nWe also note that all the conditions $S_1$, $S_2$, $S_3$ and $S_4$ for GHZ diagonal states are determined by\nfinitely many linear inequalities with respect to eight real variables. For example,\nthe inequality $S_4[i,j|k,\\ell]$ is actually the combination of the following eight linear inequalities:\n$$\na_i+a_j\\ge \\pm z_k \\pm z_\\ell,\\quad\na_k+a_\\ell\\ge \\pm z_i \\pm z_j\n$$\nfor a GHZ diagonal state $\\xx{a_1~a_2~a_3~a_4\\choose z_1~z_2~z_3~z_4}$.\nFor a given convex cone $\\sigma\\in{\\mathcal L}$ considered in this paper,\nthe convex cone $\\sigma\\cap V$ is determined by finitely many hyperplanes which cover all\nmaximal faces of $\\sigma\\cap V$. The same is true for $\\sigma^\\circ\\cap V$.\nIf $\\varrho$ is an extreme ray of $\\sigma\\cap V$,\nthen its dual face\n$$\n\\varrho^\\prime:=\\{W\\in (\\sigma\\cap V)^\\circ: \\lan W,\\varrho\\ran=0\\}\n$$\nof the dual cone $\\sigma^\\circ\\cap V$ is a maximal face. See\n\\cite[Theorem 5.3]{han_kye_pe}. Therefore, we conclude that\n$\\sigma\\cap V$ has only finitely many extreme rays, which must be\northogonal to a hyperplane determining a maximal face\nof the dual cone $\\sigma^\\circ\\cap V$. Considering the\ncoefficients of the characteristic linear inequalities of\n$\\sigma^\\circ\\cap V$, it is straightforward to find all the\ncandidates  of extreme rays of the convex cone $\\sigma\\cap V$. We\ndenote by $\\CE_\\sigma$ the set of candidates for extreme rays of\n$\\sigma\\cap V$ arising in this way. We take a chain of convex cones\nfrom the diagram (\\ref{diag_con}), and list up in {\\sc Table 2} all\nthe candidates in $\\CE_\\sigma$. Note that the diagonal states\n$$\n\\Delta=\\{\\textstyle\n\\xx{1\\,0\\,0\\,0\\choose 0\\,0\\,0\\,0},\\\n\\xx{0\\,1\\,0\\,0\\choose 0\\,0\\,0\\,0},\\\n\\xx{0\\,0\\,1\\,0\\choose 0\\,0\\,0\\,0},\\\n\\xx{0\\,0\\,0\\,1\\choose 0\\,0\\,0\\,0}\n\\}\n$$\nare extreme rays of all the convex cones.\n\nAt this moment, it must  be pointed out that several\ninequalities of Theorem \\ref{dist11122} are redundant. Let us\nconsider the criteria of\n$(\\aaa^\\circ_\\xx\\meet\\bbb^\\circ_\\xx)\\join(\\aaa^\\circ_\\xx\\meet\\ccc^\\circ_\\xx)$\nin Theorem \\ref{dist11122}. One may easily see that $W_1[1,4]$ and\n$W_1[2,3]$ imply the inequalities $W_3$, $W_2[1,2]$, $W_2[1,3]$,\n$W_2[2,4]$, $W_2[3,4]$ and $W_4[1,4]$, $W_4[2,3]$. We may also add one of\n$W_1[1,4]$, $W_1[2,3]$ and one of $W_{4{\\rm a}}[1,2]$, $W_{4{\\rm\na}}[1,3]$, $W_{4{\\rm a}}[2,4]$, $W_{4{\\rm a}}[3,4]$, to get the\ninequalities $W_{4{\\rm b}}[1,2]$, $W_{4{\\rm b}}[1,3]$, $W_{4{\\rm\nb}}[2,4]$ and $W_{4{\\rm b}}[3,4]$.\nFor an example, we get a part of $W_{4{\\rm b}}[1,2]$ from\n$W_1[2,3]$ and $W_{4{\\rm a}}[1,3]$.\nIn {\\sc Table 2}, the states determined by these redundant inequalities are excluded.\n\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|l|c|}\n\\hline $\\sigma$ &\n\\qquad\\qquad\\qquad\\qquad\\qquad$\\CE_\\sigma$\n   & number of $\\CE_\\sigma$\\\\\n\\hline $\\aaa\\meet\\bbb\\meet\\ccc$\n  &$\\Delta$,\\ $\\xx{~1~~1~~1~~1\\choose \\mathsmaller{\\pm}\\!1~\\mathsmaller{\\pm}\\!1~\\mathsmaller{\\pm}\\!1~\\mathsmaller{\\pm}\\!1}$\n  &$4+16=20$\\\\\n\\hline $\\aaa\\meet\\bbb$\n  &add $\\xx{~1\\,0\\,1\\,1\\choose \\mathsmaller{\\pm}\\!1\\,0\\,0\\,0}$\\ \\\n   $\\xx{0\\,~1\\,1\\,1\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,0\\,0}$\\ \\\n   $\\xx{1\\,1\\,~1\\,0\\choose 0\\,0\\,\\mathsmaller{\\pm}\\!1\\,0}$\\ \\\n   $\\xx{1\\,1\\,0\\,~1\\choose 0\\,0\\,0\\,\\mathsmaller{\\pm}\\!1}$\n  &$20+8=28$\\\\\n\\hline $(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)$\n  &add $\\xx{~1\\,1\\,0\\,1\\choose \\mathsmaller{\\pm}\\!1\\,0\\,0\\,0}$\\ \\\n   $\\xx{0\\,1\\,~1\\,1\\choose 0\\,0\\,\\mathsmaller{\\pm}\\!1\\,0}$\\ \\\n   $\\xx{1\\,~1\\,1\\,0\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,0\\,0}$\\ \\\n   $\\xx{1\\,0\\,1\\,~1\\choose 0\\,0\\,0\\,\\mathsmaller{\\pm}\\!1}$\n   &$28+8=36$\\\\\n\\hline $\\aaa\\meet(\\bbb\\join\\ccc)$\n  &add $\\xx{~1\\,2\\,0\\,~1\\choose \\mathsmaller{\\pm}\\!1\\,0\\,0\\mathsmaller{\\pm}\\!1}$\\ \\ \\\n   $\\xx{~1\\,0\\,2\\,~1\\choose \\mathsmaller{\\pm}\\!1\\,0\\,0\\,\\mathsmaller{\\pm}\\!1}$\\ \\ \\\n   $\\xx{2\\,~1\\,~1\\,0\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1\\,0}$\\ \\ \\\n   $\\xx{0\\,~1\\,~1\\,2\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1\\,0}$\n  &$36+48=84$\\\\\n& \\phantom{add} $\\xx{~1\\,~2\\,2\\,~1\\choose \\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!2\\,0\\,\\mathsmaller{\\pm}\\!1}$\\ \\\n   $\\xx{~1\\,2\\,~2\\,~1\\choose \\mathsmaller{\\pm}\\!1\\,0\\,\\mathsmaller{\\pm}\\!2\\,\\mathsmaller{\\pm}\\!1}$\\ \\\n   $\\xx{~2\\,~1\\,~1\\,2\\choose \\mathsmaller{\\pm}\\!2\\,\\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1\\,0}$\\ \\\n   $\\xx{2\\,~1\\,~1\\,~2\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!2}$\n  &\\\\\n\\hline\\hline $\\aaa$\n  &$\\Delta$,\\ $\\xx{~1\\,0\\,0\\,~1\\choose \\mathsmaller{\\pm}\\!1\\,0\\,0\\,\\mathsmaller{\\pm}\\!1}$\\ \\\n   $\\xx{0\\,~1\\,~1\\,0\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1\\,0}$\n   &$4+8=12$\\\\\n\\hline $\\aaa\\join(\\bbb\\meet\\ccc)$\n  &add $\\xx{~1\\,1\\,1\\,0\\choose \\mathsmaller{\\pm}\\!1\\,0\\,0\\,0}$, \\\n   $\\xx{0\\,1\\,1\\,~1\\choose 0\\,0\\,0\\,\\mathsmaller{\\pm}\\!1}$\\ \\\n   $\\xx{1\\,~1\\,0\\,1\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,0\\,0}$\\ \\\n   $\\xx{1\\,0\\,~1\\,1\\choose 0\\,0\\,\\mathsmaller{\\pm}\\!1\\,0}$\n  &$12+8=20$\\\\\n\\hline $(\\aaa\\join\\bbb)\\meet(\\aaa\\join\\ccc)$\n  &add $\\xx{~1\\,~1\\,2\\,0\\choose \\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1\\,0\\,0}$\\ \\\n   $\\xx{~1\\,2\\,~1\\,0\\choose \\mathsmaller{\\pm}\\!1\\,0\\,\\mathsmaller{\\pm}\\!1\\,0}$\\ \\\n   $\\xx{2\\,~1\\,0\\,~1\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,0\\,\\mathsmaller{\\pm}\\!1}$\\ \\\n   $\\xx{2\\,0\\,~1\\,~1\\choose 0\\,0\\,\\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1}$\n  &$20+32=52$\\\\\n &\\phantom{add} $\\xx{~1\\,~1\\,0\\,2\\choose \\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1\\,0\\,0}$\\ \\\n   $\\xx{~1\\,0\\,~1\\,2\\choose \\mathsmaller{\\pm}\\!1\\,0\\,\\mathsmaller{\\pm}\\!1\\,0}$\\ \\\n   $\\xx{0\\,~1\\,2\\,~1\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,0\\,\\mathsmaller{\\pm}\\!1}$\\ \\\n   $\\xx{0\\,2\\,~1\\,~1\\choose 0\\,0\\,\\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1}$\n &\\\\\n\\hline\\hline $\\aaa\\join\\bbb$\n  &$\\Delta$,\\ $\\xx{~1\\,0\\,0\\,~1\\choose \\mathsmaller{\\pm}\\!1\\,0\\,0\\,\\mathsmaller{\\pm}\\!1}$\\ \\\n   $\\xx{0\\,~1\\,~1\\,0\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1\\,0}$\\ \\\n   $\\xx{~1\\,0\\,~1\\,0\\choose \\mathsmaller{\\pm}\\!1\\,0\\,\\mathsmaller{\\pm}\\!1\\,0}$\\ \\\n   $\\xx{0\\,~1\\,0\\,~1\\choose 0\\,\\mathsmaller{\\pm}\\!1\\,0\\,\\mathsmaller{\\pm}\\!1}$\n  &$4+16=20$\\\\\n\\hline $\\aaa\\join\\bbb\\join\\ccc$\n  &add $\\xx{~1\\,~1\\,0\\,0\\choose \\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1\\,0\\,0}$\\ \\\n   $\\xx{0\\,0\\,~1\\,~1\\choose 0\\,0\\,\\mathsmaller{\\pm}\\!1\\,\\mathsmaller{\\pm}\\!1}$\n  &$20+8=28$\\\\\n\\hline\n\\end{tabular}\n\\caption{The convex cones in the first column are increasing\ndownward with respect to inclusion. For each convex cone $\\sigma$ in\nthe first column, we list up all the states in $\\CE_\\sigma$ which\narise from characteristic inequalities for the dual cone\n$\\sigma^\\circ$. The collection $\\CE_\\sigma$ contain all extreme rays\nof the convex cone $\\sigma\\cap V$ consisting of GHZ diagonal states.\nWe prove that all the states in $\\CE_\\sigma$ are indeed extreme.}\n\\end{center}\n \\end{table}\n\n\nIn order to prove that all the candidates are actually extreme rays,\nwe use again duality in the vector space $V$. Suppose that $\\sigma$\nis a convex cone in {\\sc Table 2} with the set $\\CE_\\sigma$\ncontaining all the candidates of extreme rays of $\\sigma\\cap V$. We\nfirst recall that the bi-dual face\n$$\n(\\varrho^\\prime)^\\prime=\\{\\psi\\in\\sigma\\cap V:\n\\lan W,\\psi\\ran=0\\ {\\text{\\rm for every}}\\ W\\in  \\varrho^\\prime\\}\n$$\nis the smallest exposed face of $\\sigma\\cap V$ containing $\\rho$.\nThus, a state $\\varrho\\in\\sigma\\cap V$ is extreme if its bi-dual\nface $\\varrho^{\\prime\\prime}$ consists of nonnegative scalar\nmultiples of $\\varrho$. Because $\\CE_\\sigma$ contains all extreme\nrays of $\\sigma\\cap V$, we see that this is the case if for each\n$\\psi\\in\\CE_\\sigma\\setminus \\{\\varrho\\}$, there exists a witness\n$W_\\psi\\in V$ such that $\\lan W_\\psi,\\varrho\\ran=0$ and $\\lan\nW_\\psi,\\psi\\ran >0$. By taking $W=\\sum_\\psi W_\\psi$, this is\nequivalent to the existence of a witness $W\\in V$ satisfying\n\\begin{equation}\\label{condi}\n\\lan W,\\varrho\\ran=0,\\qquad \\lan W,\\psi\\ran >0\\ {\\text{\\rm for\neach}}\\ \\psi\\in\\CE_\\sigma\\setminus \\{\\varrho\\},\n\\end{equation}\nwhich is a seemingly stronger condition. Note that (\\ref{condi}) actually tells us that\n$W$ belongs to $\\sigma^\\circ\\cap V=(\\sigma\\cap V)^\\circ$, since every state in $\\sigma\\cap V$ is a convex combination of states in $\\CE_\\sigma$.\nGeometrically, it is clear that the condition (\\ref{condi})\nis equivalent to the claim that the dual face $\\varrho^\\prime$ is a maximal face of the dual cone $\\sigma^\\circ\\cap V$.\n\nTo prove that all the states in $\\CE_\\sigma$ are extreme, we begin\nwith the convex cone $\\sigma=\\aaa\\meet(\\bbb\\join\\ccc)$. For\n$\\varrho=\\xx{~1~0~0~0\\choose ~0~0~0~0}$, one can check the witness\n{$W=\\xx{~0~1~1~1\\choose ~0~0~0~0}$} works. For other states like\n$$\n\\textstyle\n\\xx{~1~ 1~ 1~ 1\\choose ~1~ 1~ 1 ~1},\\quad\n\\xx{~1~ 0~ 1~ 1\\choose ~1~ 0~ 0~ 0},\\quad\n\\xx{~1~ 1~ 0~ 1\\choose ~1~ 0~ 0~ 0},\\quad\n\\xx{~1~ 2~ 0~ 1\\choose ~1~ 0~ 0~ 1},\\quad\n\\xx{~1~ 2~ 2~ 1\\choose ~1~ 2~ 0~ 1},\n$$\nthe following witnesses\n$$\n\\textstyle\n\\xx{~1\\,~1\\,~1\\,~1\\choose \\shortminus1\\, \\shortminus1\\, \\shortminus1\\, \\shortminus1},\\quad\n\\xx{\\,1~3~1~1\\choose\n\\shortminus3~0~0~0},\\quad\n\\xx{\\,1~1~3~1\\choose\n\\shortminus3~0~0~0},\\quad\n\\xx{\\,1~1~2~\\,1\\choose\n\\shortminus2~0~0~\\shortminus2},\\quad\n\\xx{~2~\\,1~\\,1~~2\\choose\n\\shortminus1~\\shortminus3~0~\\shortminus1}\n$$\nsatisfy the condition (\\ref{condi}).\nWhen an anti-diagonal of a candidate has minus sign, we take plus sign on the corresponding entry of the witness.\nThe others are similar by symmetry.\n\n\nWe also recall that if $C_1$ and $C_2$ are convex cones and\n$x\\in C_1\\subset C_2$ is extreme in $C_2$ then it is also extreme in $C_1$.\nTherefore, all the candidates in {\\sc Table 2} are really extreme rays for the convex cones\n$\\aaa\\meet\\bbb\\meet\\ccc$, $\\aaa\\meet\\bbb$ and $(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)$.\n\nIt was actually shown in \\cite{han_kye_pe} that all the candidates in {\\sc Table 2} are extreme for\n$\\aaa\\meet\\bbb\\meet\\ccc$, $\\aaa\\meet\\bbb$, $\\aaa$, $\\aaa\\join \\bbb$ and $\\aaa\\join\\bbb\\join\\ccc$.\nTherefore, it remains to consider the convex cone $(\\aaa\\join\\bbb)\\meet(\\aaa\\join\\ccc)$.\nAgain, for the states\n$$\n\\textstyle\n\\xx{1~0~0~1\\choose 1~0~0~1},\\qquad\n\\xx{1~1~1~0\\choose 1~0~0~0},\\qquad\n\\xx{1~1~2~0\\choose 1~1~0~0},\n$$\nthe witnesses\n$$\n\\textstyle\n\\xx{\\,1~1~1~\\,1\\choose\n\\shortminus1~0~0~\\shortminus1}, \\qquad\n\\xx{~1~1~1~3\\choose\n\\shortminus3~0~0~0}, \\qquad\n\\xx{\\,1~~5~1~3\\choose\n\\shortminus3~ \\shortminus5 ~0~0}\n$$\nsatisfy (\\ref{condi}), respectively. This shows that all the candidates in {\\sc Table 2} are really\nextreme rays of the corresponding convex cones.\n\n\n\n\n\\section{Conclusion}\n\nIn this paper, we gave criteria for the convex cones\nlisted in (\\ref{list}). In the plain terminologies, a state\n$\\varrho$ belongs to the convex cone\n$(\\aaa\\meet\\bbb)\\join(\\aaa\\meet\\ccc)$ if and only if it is a mixture of a simultaneously $A$-$BC$ and $B$-$CA$ bi-separable state\nand a simultaneously $A$-$BC$ and $C$-$BA$ bi-separable state.\nWe gave a necessary condition for a three qubit state $\\varrho$\nto have this property in terms of diagonal and anti-diagonal entries of $\\varrho$, and showed\nthat this condition is also sufficient when $\\varrho$ is $\\xx$-shaped.\nWe also found all the GHZ diagonal states which distinguish kinds of partial separability.\nFor example, we can read out from {\\sc Table 2} all the extremal GHZ diagonal states which violate distributive rules.\n\n\n\n\nIt was asked in \\cite{han_kye_szalay} whether the lattice ${\\mathcal L}$ is free or not.\nThe identities in Corollary \\ref{identity_P} actually shows that the lattice ${\\mathcal L}_\\xx$ is not free, because\nwe have exhibited a lattice generated by three elements which give rise to a strict inequality in (\\ref{aaaaa}).\nIt is natural to ask whether the identity holds in (\\ref{aaaaa}) when $(x,y,z)=(\\aaa,\\bbb,\\ccc)$.\nThis is to equivalent to ask whether the following properties\n\\begin{itemize}\n\\item\n$\\varrho$ is $A$-$BC$ separable;\n\\item\n$\\varrho$ is a mixture of $B$-$CA$ separable state and a simultaneously $C$-$AB$ and $A$-$BC$ separable state;\n\\item\n$\\varrho$ is a mixture of $C$-$BA$ separable state and a simultaneously $B$-$AC$ and $A$-$BC$ separable state,\n\\end{itemize}\nfor a three qubit state $\\varrho$ implies that $\\varrho$ is\na mixture of a simultaneously $A$-$BC$ and $B$-$CA$ bi-separable state\nand a simultaneously $A$-$BC$ and $C$-$BA$ bi-separable state. We have seen in this paper that this is the case when\n$\\varrho$ is $\\xx$-shaped. This question must be related with the question\nwhether the lattice ${\\mathcal L}$ is free or not.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Notation}\n\n\n\\begin{NN}\n\n{\\rm Let $A$ be a unital $C^*$-algebra. Denote by $T(A)$ the tracial state space of $A.$ Denote by\n${\\rm Aff}(T(A))$ the space of all real affine continuous functions on $T(A).$ If $\\tau\\in T(A),$ we will also use $\\tau$ for the tracial state $\\tau\\otimes {\\rm Tr}$ on $M_k(A)$ for all integer $k\\ge 1,$ where ${\\rm Tr}$ is the standard trace on $M_k.$\nIf $a\\in A_{s.a.},$ denote by $\\check{a}$ a real affine function in ${\\rm Aff}(T(A))$ defined by $\\check{a}(\\tau)=\\tau(a)$\nfor all $\\tau\\in T(A).$\n\nLet $C$ be another unital $C^*$-algebra. Suppose that $\\gamma: {\\rm Aff}(T(C))\\to {\\rm Aff}(T(A))$ is  a positive linear map.\nWe say it is unital if $\\gamma(1_C)(\\tau)=1.$ We say it is strictly positive, if $a\\in {\\rm Aff}(T(A))_+\\setminus \\{0\\},$ then\n$\\gamma(a)(\\tau)>0$ for all $\\tau\\in T(A).$\n\nSuppose that $\\phi: C\\to A$ is a unital homomorphism. Denote by $h_T: T(A)\\to T(C)$ the affine continuous map induced by $h,$ i.e.,\n$$\nh_T(\\tau)(c)=\\tau\\circ h(c)\\,\\,\\,{\\rm for\\,\\,\\,all}\\,\\,\\, c\\in C.\n$$\nIt also induces a positive linear map\n$h_{\\sharp}: {\\rm Aff}(T(C))\\to {\\rm Aff}(T(A))$  defined by\n$$\nh_{\\sharp}(\\check{a})(\\tau)=\\tau\\circ h(a)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, a\\in C_{s.a}\\,\\,\\,{\\rm and}\\,\\,\\, \\tau\\in T(A),\n$$\nwhere $\\check{a}(\\tau)=\\tau(a)$ for $\\tau\\in T(A).$\n\nIf $\\lambda: T(A)\\to T(C)$ is an affine continuous map, then it\ngives a unital positive linear map $\\lambda_{\\sharp}: {\\rm\nAff}(T(C))\\to {\\rm Aff}(T(A))$ by\n$$\n\\lambda_{\\sharp}(f)(\\tau)=f(\\lambda(\\tau))\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in {\\rm\nAff}(T(C))\\,\\,\\,{\\rm and}\\,\\,\\, \\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A).\n$$\nConversely, a  unital positive linear map $\\gamma: {\\rm\nAff}(T(C))\\to {\\rm Aff}(T(A))$  gives an affine continuous map\n$\\gamma_T: T(A)\\to T(C)$ by\n$$\nf(\\gamma_T(\\tau))=\\gamma(f)(\\tau)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in {\\rm\nAff}(T(C))\\,\\,\\,{\\rm and}\\,\\,\\, \\tau\\in T(C).\n$$\n\nSuppose that $A$ is a unital simple $C^*$-algebra. Then $\\gamma$ is strictly\npositive if and only if $\\gamma_T$ maps $T(A)$ into\n$T_{\\mathtt{f}}(C).$\n\n\nDenote by $\\rho_A: K_0(A)\\to {\\rm Aff}(T(A))$ the positive homomorphism\\, induced by\n$\\rho_A([p])(\\tau)=\\tau(p)$ for all projections\n$p\\in M_{\\infty}(A)$ and $\\tau\\in T(A).$\n\nLet $A$ and $C$ be two unital $C^*$-algebra s and let $\\kappa_0: K_0(C)\\to\nK_0(A)$ be a unital positive homomorphism\\, ($\\kappa_0([1_C])=[1_A]$).\nSuppose that $\\lambda: T(A)\\to T(C)$ is a continuous affine map.\nWe say that $\\lambda$ is compatible with $\\kappa_0,$ if\n$\\tau(\\kappa([p]))=\\lambda(\\tau)(p)$ for all projections $p$ in\n$M_{\\infty}(A).$  Similarly, a unital positive linear map $\n\\gamma: {\\rm Aff}(T(C))\\to {\\rm Aff}(T(A))$ is said to be\ncompatible with $\\kappa_0,$ if\n$\\gamma(\\check{p})(\\tau)=\\tau(\\kappa([p])$ for all projections $p$\nin $M_{\\infty}(C).$ $\\gamma$ is compatible with $\\kappa_0$ if and\nonly if $\\gamma_T$ is so.\n\n\nTwo projections in $A$ are equivalent if there exists a partial isometry $w\\in A$ such that\n$w^*w=p$ and $ww^*=q.$ }\n\n\n\\end{NN}\n\n\\begin{NN}\\label{HomK}\n{\\rm Let $A$ be a unital $C^*$-algebra\\, and let $C$ be a  separable $C^*$-algebra\\,\nwhich satisfies the universal coefficient theorem. By a result of\nDadarlat and Loring (\\cite{DL1}),\n\\begin{eqnarray}\\label{N2-1}\nKL(C,A)=Hom_{\\Lambda}(\\underline{K}(C), \\underline{K}(A)),\n\\end{eqnarray}\nwhere, for any $C^*$-algebra\\, $B,$\n$$\n\\underline{K}(B)=\\oplus_{i=0,1}K_i(B) \\bigoplus_{n=2}^{\\infty}\\oplus_{i=0,1} K_i(B,\\mathbb{Z}\/n\\mathbb{Z}).\n$$\nWe will identify two objects in (\\ref{N2-1}).\nDenote by\n$$\n\\underline{K}_{F,k}(C)=\\bigoplus_{i=0,1}K_i(B) \\bigoplus_{n|k}\\oplus_{i=0,1} K_i(B,\\mathbb{Z}\/n\\mathbb{Z}).\n$$\nIf $K_i(C)$ is finitely generated ($i=0,1$), then there is $k_0\\ge 1$ such that\n$$\nHom_{\\Lambda}(\\underline{K}(C), \\underline{K}(A))\\cong \nHom_{\\Lambda}(F_{k_0}\\underline{K}(C), F_{k_0}\\underline{K}(A))\n$$\n(see \\cite{DL1}). \n}\n\n\\end{NN}\n\n\\begin{df}\\label{KL+}\n\n{\\rm\nDenote by $KL(C,A)^{++}$ the set of those $\\kappa\\in Hom_{\\Lambda}(\\underline{K}(C), \\underline{K}(A))$ such that\n$$\n\\kappa(K_0(C)_+\\setminus\\{0\\})\\subset K_0(A)\\setminus \\{0\\}.\n$$\nDenote by $KL_e(C,A)^{++}$ the set of those $\\kappa\\in\nKL(C,A)^{++}$ such that $\\kappa([1_C])=[1_A].$\n\n\n}\n\\end{df}\n\n\\begin{df}\\label{Dcomp}\n\n{\\rm Let $\\kappa\\in KL_e(C,A)^{++}$  and let  $\\lambda: T(A)\\to\nT(C)$ be a continuous affine map. We say that $\\lambda$ is\ncompatible with $\\kappa$ if $\\lambda$ is compatible with\n$\\kappa|_{K_0(C)}.$ Let $\\gamma: {\\rm Aff}(T(C))\\to {\\rm\nAff}(T(A))$ be a positive linear map. We say $\\gamma$ is\ncompatible with $\\kappa$ if  $\\gamma$ is compatible with\n$\\kappa|_{K_0(C)},$ i.e., $\\tau\\circ\n\\kappa([p])=\\gamma(\\check{p})(\\tau)$ for all projections $p\\in\nM_{\\infty}(C).$ }\n\\end{df}\n\n\n\\begin{NN}\\label{Meas}\n{\\rm Let $C=C(X)$ for some compact metric space $X.$ \nOne has the following short exact sequence:\n$$\n0\\to {\\rm ker}\\rho_C\\to K_0(C)\\to C(X, \\mathbb{Z})\\to 0.\n$$\nIt is then easy to see that, for every projection $p\\in M_{\\infty}(C),$ there is a projection $q\\in C$ and an integer $n$ such that\n$\n\\rho_A([p])=n\\rho_A([q]).\n$\nIt follows that if $C$ is a unital AH-algebra, then for every projection $p\\in M_{\\infty}(C),$ there is a projection\n$q\\in C$ and an integer $n\\ge 1$ such that\n$$\n\\rho_A([p])=n\\rho_A([q]).\n$$\nNote also that in this case  ${\\rm Aff}(T(C))=C_{s.a}.$\nTherefore, in this note, instead of considering a unital positive linear maps $\\gamma: {\\rm Aff}(T(C))\\to {\\rm Aff}(T(A)),$ we may  consider unital positive linear maps $\\gamma: C_{s.a}\\to {\\rm Aff}(T(A)).$ Moreover, $\\gamma$ is compatible with some $\\kappa\\in KL(C,A)^{++},$ if\n$\\gamma(p)(\\tau)=\\tau(\\kappa([p]))$ for all projections $p\\in C$ and $\\tau\\in T(A).$\n\n\n}\n\\end{NN}\n\n\n\\begin{NN}\n{\\rm Let $\\phi, \\psi: C\\to A$ be two maps between $C^*$-algebra s. Let $\\epsilon>0$ and ${\\cal F}\\subset C$ be a subset.\nWe write\n$$\n\\phi\\approx_{\\epsilon} \\psi\\,\\,\\,{\\rm on}\\,\\,\\,{\\cal F},\n$$\nif\n$$\n\\|\\phi(c)-\\psi(c)\\|<\\epsilon\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, c\\in {\\cal F}.\n$$\n}\n\\end{NN}\n\n\\begin{NN}\n{\\rm Let $L: C\\to A$ be a linear map. Let $\\delta>0$ and ${\\cal G}\\subset C$ be a (finite) subset. We say $L$ is $\\delta$-${\\cal G}$-multiplicative if\n$$\n\\|L(ab)-L(a)L(b)\\|<\\delta\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, a, b\\in {\\cal G}.\n$$\n}\n\\end{NN}\n\n\\begin{df}\n{\\rm  Let $A$ be a unital $C^*$-algebra. Denote by $U(A)$ the unitary group of $A.$ Let $B\\subset A$ be another $C^*$-algebra\\, and\n$\\phi: B\\to A$ be a map. We write $\\phi={\\rm ad}\\, u$ for some $u\\in U(A)$ if\n$\\phi(b)=u^*bu$ for all $b\\in B.$\n\nLet $\\phi, \\psi: C\\to A$ be two maps.  We say that $\\phi$ and $\\psi$ are approximately unitarily equivalent if there exists a sequence of unitaries $\\{u_n\\}\\subset A$ such that\n$$\n\\lim_{n\\to\\infty}{\\rm ad}\\, u_n\\circ \\phi(c)= \\psi(c)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, c\\in C.\n$$\n\n}\n\\end{df}\n\n\n\n\\section{Approximate unitary equivalence}\n\nWe begin with the following theorem\n\n\\begin{thm}{\\rm (Theorem 3.6 of \\cite{Lnemb2} and see also Theorem 3.4 of \\cite{Lncd}) } \\label{Om}\nLet $C$ be a unital AH-algebra and let $A$ be a unital simple $C^*$-algebra\\, with tracial rank zero.\nSuppose that $\\phi, \\psi: C\\to A$ are two unital monomorphisms.\nThen there exists a sequence of unitaries $\\{u_n\\}\\subset A$ such that\n$$\n\\lim_{n\\to\\infty}{\\rm ad}\\, u_n\\circ \\psi(c)=\\phi(c)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, c\\in C,\n$$\n if and only if\n$$\n[\\phi]=[\\psi]\\,\\,\\,{\\rm in}\\,\\,\\,KL(C,A)\\,\\,\\,\\text{and}\\,\\,\\, \\tau\\circ\n\\phi=\\tau\\circ \\psi \\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A).\n$$\n\\end{thm}\n\nWe need the following variation of results in \\cite{Lncd}.\n\n\\begin{thm}\\label{1Tcd}\nLet $C$ be a unital $AH$-algebra, let $A$ be a unital simple $C^*$-algebra\\, with $TR(A)=0$ and let  $\\gamma: C_{s.a.}\\to\n{\\rm Aff}(T(A))$ be a unital strictly positive linear map.\n\nFor any $\\epsilon>0$ and any finite subset ${\\cal F}\\subset C,$ there exist $\\eta>0,$  $\\delta>0,$ a finite subset\n${\\cal G}\\subset C,$ a finite subset ${\\cal H}\\subset C_{s.a.}$ and a finite subset ${\\cal P}\\subset \\underline{K}(C)$ satisfying the following:\n\nSuppose that  $L_1, L_2: C\\to A$  are two  unital completely positive linear maps which are $\\delta$-${\\cal G}$-multiplicative  such that\n \\begin{eqnarray}\\label{1Tcd-1}\n[L_1]|_{\\cal P}&=&[L_2]|_{\\cal P},\\\\\n|\\tau\\circ L_i(g)-\\gamma(g)(\\tau)|&<&\\eta\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in {\\cal H},\\,\\,\\,i=1,2.\n\\end{eqnarray}\nThen there is a unitary $u\\in A$ such that\n\\begin{eqnarray}\\label{1Tcd-2}\n{\\rm ad}\\, u\\circ L_2\\approx_{\\epsilon} L_2\\,\\,\\,{\\rm on}\\,\\,\\,{\\cal F}.\n\\end{eqnarray}\n\\end{thm}\n\n\\begin{proof}\nWrite $C={\\overline{\\cup_{n=1}^{\\infty}C_n}},$ where $C_n=P_nM_{r(n)}(C(X_n))P_n,$ where $X$ is a compact \nsubset\nof a finite CW complex and where $P_n\\in M_{r(n)}(C(X_n))$ is a projection.\nLet $\\epsilon>0$ and a finite subset ${\\cal F}\\subset C$ be fixed.  Without loss of generality, we may assume\nthat ${\\cal F}\\subset C_1.$ Let $\\eta_0>0$ such that\n$$\n|f(x)-f(x')|<\\epsilon\/8\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in {\\cal F},\n$$\nif ${\\rm dist}(x,x')<\\eta_0.$\nLet $\\{x_1,x_2,...,x_m\\}\\subset X$ be $\\eta_0\/2$-dense in $X.$ Suppose that\n$O_i\\cap O_j=\\emptyset$ if $i\\not=j,$ where\n$$\nO_j=\\{x\\in X: {\\rm dist}(x,x_j)<\\eta_0\/2s\\},\\,\\,\\,j=1,2,...,m\n$$\nfor some integer $s\\ge 1.$\n\nChoose non-zero element $g_j\\in (C_1)_{s.a}$ such that $0\\le g_j\\le 1$ whose support lies in $O_j,$\n$j=1,2,...,m.$ Note such $g_j$ exists (by taking those in the center for example).\nChoose\n$$\n\\sigma_0=\\min\\{\\inf\\{\\gamma(g_j)(\\tau): \\tau\\in T(A)\\}: 1\\le j\\le m\\}.\n$$\nSince $\\gamma$ is strictly positive, $\\sigma_0>0.$  Set $\\sigma=\\min\\{\\sigma_0\/2, 1\/2s\\}.$\nThen, by Corollary 4.8 of \\cite{Lncd}, such $\\delta>0,$ $\\eta>0,$ ${\\cal G},$ ${\\cal H}$ and ${\\cal P}$ exists.\n\n\n\n\\end{proof}\n\n\\begin{lem}\\label{Lcd}\nLet $X$ be a compact metric space, let $A$ be a unital simple\n$C^*$-algebra\\, with $TR(A)=0$  and let $\\gamma:C(X)_{s.a.}\\to {\\rm Aff}(T(A))$ be\na unital strictly positive linear map.\n\nThen, for any $\\epsilon>0$ and any ${\\cal F}\\subset C(X),$ there exists\n$\\delta>0,$ a finite subset ${\\cal G}\\subset C(X)_{s.a.},$ a set\n $S_1, S_2,...,S_n$ of mutually disjoint clopen subsets\n with $\\cup_{i=1}^n S_i=X,$ satisfying the following:\n\n For any two unital homomorphism s $\\phi_1, \\phi_2: C(X)\\to pAp$ with\n finite dimensional range for some projection $p\\in A$  with $\\tau(1-p)<\\delta$\n such that\n\\begin{eqnarray}\\label{lcd-1}\n[\\phi_1(\\chi_{S_i})]&=&[\\phi_2(\\chi_{S_i})]\\,\\,\\,{\\rm in}\\,\\,\\,\nK_0(A),\\,\\,\\,i=1,2,...,n,\\\\\n|\\tau\\circ \\phi_1(g)-\\gamma(g)(\\tau)|&<&\\delta\\,\\,\\,\\text{and}\\,\\,\\,\\\\\n|\\tau\\circ \\phi_2(g)-\\gamma(g)(\\tau)|&<&\\delta\n\\end{eqnarray}\nfor all $g\\in {\\cal G}$  and for all $\\tau\\in T(A),$ there exist a\nunitary $u\\in U(pAp)$ such that\n\\begin{eqnarray}\\label{lcd-2}\n{\\rm ad}\\, u\\circ \\phi_1\\approx_{\\epsilon} \\phi_2\\,\\,\\,{\\rm on}\\,\\,\\,\n{\\cal F}.\n\\end{eqnarray}\n\n\\end{lem}\n\n\\begin{proof}\nThis follows from \\ref{1Tcd} immediately. There is a sequence of\nfinite CW complex $X_n$ such that\n$C(X)=\\lim_{n\\to\\infty}(C(X_n),h_n),$ where each $h_n$ is a unital\nhomomorphism. Fix $\\epsilon>0$ and a finite subset ${\\cal F}\\subset C(X).$\nWithout loss of generality, we may assume that ${\\cal F}\\subset\nh_K({\\cal F}_K)$ for some integer $K\\ge 1$ and a finite subset\n${\\cal F}_K.$\n\nGiven any finite subset ${\\cal P}\\subset \\underline{K}(C(X)),$ one\nobtains a finite subset ${\\cal Q}_k\\subset \\underline{K}(C(X_k))$\nsuch that $[h_k]({\\cal Q}_k)={\\cal P}$ for some $k\\ge 1.$ Let\n$p_1,p_2,...,p_n$ be mutually orthogonal projections corresponding\nto the connected components of $X_k.$    To simplify notation,\nwithout loss of generality, we may  assume that $k=K.$\n\n\nThere are mutually disjoint clopen sets $S_1, S_2,...,S_n$ of $X$\nwith $\\cup_{i=1}^n S_i=X$ such that $h_k(p_i)=\\chi_{S_i},$\n$i=1,2,...,n.$ Since $\\phi$ and $\\psi$ are homomorphism s with finite\ndimensional range, if\n$$\n[\\phi(\\chi_{S_i})]=[\\psi(\\chi_{S_i})]\\,\\,\\,{\\rm in}\\,\\,\\, K_0(A),\n$$\nthen\n$$\n[\\phi\\circ h_k]=[\\psi\\circ h_k]\\,\\,\\,{\\rm in}\\,\\,\\, KL(C(X_k),A).\n$$\nThis, in particular, implies that\n$$\n[\\phi]|_{\\cal P}=[\\psi]|_{\\cal P}.\n$$\n\nThis above argument shows that the lemma follows from \\ref{1Tcd}.\n\n\\end{proof}\n\n\n\n\n\\begin{df}\\label{D1}\nLet $X$ be a compact metric space which is a compact subset of\nsome finite CW complex $Y.$ Then there exists a decreasing\nsequence of finite CW complexes $X_n\\subset Y$ such that\n$$\nX\\subset X_n\\,\\,\\,{\\rm and}\\,\\,\\, \\lim_{n\\to\\infty}\\text{dist}(X_n, X)=0.\n$$\nDenote by $s_{m,n}: C(X_m)\\to C(X_n)$ (for $n>m$) and $s_n:\nC(X_n)\\to C(X)$ be the surjective homomorphism s induced by the inclusion\n$X_{n+1}\\subset X_n$ and $X\\subset X_n,$ respectively.\n\\end{df}\n\n\n\\begin{lem}\\label{KLL1}\nLet $Y$ be a finite CW complex and $X\\subset Y$ be a compact\nsubset. For any $\\epsilon>0,$ any finite subset ${\\cal F}\\subset C(X),$\nthere exists a finite subset ${\\cal P}\\subset\n\\underline{K}(C(X)),$  an integer $k\\ge 1$ and an integer $N\\ge 1$\nsatisfying the following:\n\nFor any unital homomorphisms $\\phi, \\psi: C(X_m)\\to A$ ($m\\ge k$)\nfor any unital simple $C^*$-algebra\\, with $TR(A)=0$  for which\n$$\n[\\phi]|_{\\cal Q}=[\\psi]|_{\\cal Q}\\,\\,\\,{\\rm in}\\,\\,\\, KL(C(X_m),\nA),\n$$\nwhere $Q\\subset \\underline{K}(C(X_m))$ is a finite subset such\nthat $[s_m]({\\cal Q})={\\cal P},$ then there exists a unitary $U\\in\nM_{N+1}(A)$ such that\n$$\n\\text{ad}\\, U\\circ (\\phi\\oplus \\Phi\\circ s_m)\\approx_{\\epsilon}\n(\\psi\\oplus \\Phi\\circ s_m)\\,\\,\\,\\text{on}\\,\\,\\, s_m^{-1}({\\cal F}),\n$$\n where $\\Phi: C(X)\\to M_N(A)$ is\ndefined by\n\\begin{eqnarray}\\label{KLL1-0}\n\\Phi(f)={\\rm diag}(f(x_1), f(x_2),...,f(x_N))\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in\nC(X_1),\n\\end{eqnarray}\nwhere $\\{x_1, x_2,...,x_N\\}$ is a finite subset of $X.$\n\\end{lem}\n\n\\begin{proof}\nAssume that the lemma were false. Then  there would be  a positive\nnumber $\\epsilon_0>0,$ a finite subset ${\\cal F}_0\\subset C(X),$\nan increasing sequence of finite subsets $\\{{\\cal P}_n\\}\\subset\n\\underline{K}(C(X))$ with $\\cup_n{\\cal P}_n=\\underline{K}(C(X)),$\na sequence of unital $C^*$-algebra s, two subsequences $\\{R(n)\\},\\,\\{k(n)\\}$\nof $\\mathbb{N}$ and two sequences monomorphisms $\\phi_n, \\psi_n:\nC(X_{k(n)})\\to A_n$ such that\n\\begin{eqnarray}\\label{KLL1-1}\n&&[\\phi_n]|_{{\\cal Q}_n}=[\\psi_n]|_{{\\cal Q}_n}\\,\\,\\,\n\\text{in}\\,\\,\\, KK(C(X_{k(n)}), A_n)\\,\\,\\,{\\rm and}\\,\\,\\,\\\\\\label{N-0}\n&&\\hspace{-0.8in}\\limsup_n\\{\\inf\\{\\max\\{\\|u_n^*( \\phi_n\\oplus\n\\Phi_n\\circ s_n)(f))u_n-(\\phi\\oplus \\Phi_n\\circ s_n)(f)\\|: f\\in s_m^{-1}({\\cal F})\\}\\}\\}\\ge\n\\epsilon_0,\n\\end{eqnarray}\nwhere infimum is taken among all possible $\\Phi_n:C(X)\\to\nM_{R(n)}(A_n)$ with the form described above and among all\npossible unitaries $\\{u_n\\}\\subset U(M_{R(n)+1}(A)),$ and where \n${\\cal Q}_n\\subset \\underline{K}(C(X_{k(n)}))$ is a finite subset such that\n$[s_{k(n)}]({\\cal Q}_n)={\\cal P}_n.$\nSince $K_i(C(X_n)$ is finitely generated, by passing to a subsequence, if necessary, \nwithout loss of generality, we may assume (see also the end of \\ref{HomK}) that \n\\begin{eqnarray}\\label{N-00}\n[\\phi_{n+1}\\circ s_{k(n), k(n+1)}]=[\\psi_{n+1}\\circ s_{k(n), k(n+1)}]\\,\\,\\,{\\rm in}\\,\\,\\, KL(C(X_{k(n)}), A),\\,\\,\\,n=1,2,....\n\\end{eqnarray}\nLet $\\phi_n^{(m)}=\\phi_m,$ if $n\\le m,$ $\\phi_n^{(m)}=\\phi_n\\circ\ns_{m,n},$ $\\psi_n^{(m)}=\\psi_m,$ if $n\\le m$ and\n$\\psi_n^{(m)}=\\psi_n\\circ s_{m,n},$ $n=1,2,....$ Denote by\n$H_1^{(m)}, H_2^{(m)} : C(X_{k(m)})\\to \\prod_{n}A_n$ by\n$H_1^{(m)}(f)=\\{\\phi_n^{(m)}\\}$ and\n$H_2^{(m)}(f)=\\{\\psi_n^{(m)}\\}.$ Let $\\pi: \\prod_nA_n\\to\n\\prod_nA_n\/\\bigoplus_nA_n$ be the quotient map. Then\n $\\pi\\circ H_1^{(m)}$ and $ \\pi\\circ H_2^{(m)}$ both have\nspectrum $X.$  Moreover, for each $i,$ all $\\pi\\circ H_i^{(m)}$ gives the same \nhomomorphism\\, $F_i: C(X)\\to \n\\prod_n A_n\/\\bigoplus_n A_n,$ $i=1,2.$\n\nSince $TR(A_n)=0,$ $A_n$ has real rank zero, stable rank one, weakly unperforated $K_0(A_n),$\nby Corollary 2.1 of \\cite{GL2} and (\\ref{N-00})\n$$\n[H_1^{(m+1)}\\circ s_{k(m), k(m+1)}]=[H_2^{(m+1)}\\circ s_{k(m), k(m+1)}]\\,\\,\\,{\\rm in}\\,\\,\\, KL(C(X_{k(m)}), \\prod_nA_n)\n$$\n\nIt follows from Corollary 2.1 of \\cite{GL2} again  that\n$$\n[F_1]=[F_2]\\,\\,\\,{\\rm in}\\,\\,\\, KL(C,\\prod_nA_n\/\\bigoplus_nA_n).\n$$\nIt then follows from  Theorem 1.1 and the Remark 1.1 of \\cite{GL2} that there is an integer $N\\ge 1$ and a\nunitary $W\\in U(M_{N+1}(\\prod_n A_n\/\\bigoplus_n A_n))$ such that\n\\begin{eqnarray}\\label{N-2}\n{\\rm ad}\\,W\\circ (F_2\\oplus H_0)\\approx_{\\epsilon_0\/2} (F_1\\oplus\nH_0)\\,\\,\\,{\\rm on}\\,\\,\\, {\\cal F}_0,\n\\end{eqnarray}\nwhere $H_0: C(X)\\to M_N(\\prod_n A_n\/\\bigoplus_n A_n)$ is defined by\n$H_0(f)=\\sum_{i=1}^N f(x_i)E_i$ for all $f\\in C(X),$ $x_i\\in X$\nand $E_i={\\rm diag}(\\overbrace{0,...,0}^{i-1}, 1, 0,...,0),$\n$i=1,2,...,N.$\n\nThere is a unitary $\\{W_n\\}\\in U(\\prod_n A_n)$ such that\n$\\pi(\\{W_n\\})=W.$ Then, for some sufficiently large $n,$\n\\begin{eqnarray}\\label{N-3}\nW_n^*{\\rm diag}(\\phi_n(f), f(x_1), f(x_2),...,f(x_N))W_n\n\\approx_{\\epsilon_0}(\\psi_n(f), f(x_1), f(x_2),...,f(x_N))\n\\end{eqnarray}\non ${\\cal F}_0.$ This contradicts (\\ref{N-0}).\n\n\n\n\\end{proof}\n\n\\begin{rem}\\label{RM1}\n{\\rm\nThere exists a positive number $\\eta>0$ and integer $N_1>0$ which depend only on $\\epsilon$ and ${\\cal F}$ such that\n$\\{x_1,x_2,...,x_N\\}$ and an integer $N$ can be replaced by any $\\eta$-dense finite subset $\\{\\xi_1,\\xi_2,...,\\xi_{N_1}\\}$\nand integer $N_1.$\n\n\nFrom the proof, we also know that the assumption that $A$ has\ntracial rank zero can be replaced by much weaker conditions (see\nCorollary 2.1 of \\cite{GL2}). The main difference of \\ref{KLL1}\nand results in \\cite{GL2} is that homomorphism s $\\phi$ and $\\psi$ are not\nassumed to be from $C(X)$ to $A.$\n\n}\n\n\\end{rem}\n\n\\section{Monomorphisms from $C(X)$}\n\n\\begin{lem}\\label{KLL2}\nLet $X$ be a finite CW complex and let $A$ be a unital simple\n$C^*$-algebra\\, with real rank zero, stable rank one and weakly unperforated\n$K_0(A).$  Let $e_1,e_2,...,e_m\\in C(X)$ be mutually orthogonal\nprojections corresponding to connected components of $X.$\n\nSuppose that $\\kappa\\in KK(C(X), A)^{++}$ with\n$\\kappa([1_{C(X)}])=[1_A].$ Then, for any projection $p\\in A$ and\nany unital homomorphism\\, $\\phi_0: C(X)\\to (1-p)A(1-p)$ with finite\ndimensional range such that $\\phi_0([e_i])< \\kappa([e_i]),$\n$i=1,2,...,m.$  Then there exists a unital monomorphism $\\phi_1:\nC(X)\\to pAp$ such that\n\\begin{eqnarray}\\label{KLL2-1}\n[\\phi_1+\\phi_0]=\\kappa\\,\\,\\,\\text{in}\\,\\,\\, KK(C(X), A).\n\\end{eqnarray}\n\\end{lem}\n\n\\begin{proof}\nSince $\\sum_{i=1}^m\\kappa([e_i])=[1_A]$ and $A$ has stable rank\none, there are mutually orthogonal projections $p_1,p_2,...,p_m\\in\nA$ such that\n\\begin{eqnarray}\\label{L2-2}\n\\sum_{i=1}^m p_i=1_A\\,\\,\\,{\\rm and}\\,\\,\\, [p_i]=\\kappa([e_i]),\\,\\,\\,i=1,2,...,m\n\\end{eqnarray}\nFrom this it is clear that we may reduce the general case to the\ncase that $X$ is connected.\n\nSo now we assume that $X$ is connected. Then it is easy to see\nthat\n$$\n\\kappa-[\\phi_0]\\in KK(C(X), A)^{++}\n$$\nand $(\\kappa-[\\phi_0])([1_{C(X)}])=p.$ It follows from Theorem 4.7\nof \\cite{LnKT} that there is a monomorphism $\\phi_1: C(X)\\to pAp$\nsuch that\n$$\n[\\phi]=\\kappa-[\\phi_0].\n$$\n\n\n\\end{proof}\n\n\n\\begin{lem}\\label{ML0}\nLet $X$ a compact metric space and let $A$ be a unital simple\n$C^*$-algebra\\, with tracial rank zero. Suppose that $\\gamma: C(X)_{s,a} \\to\n{\\rm Aff}(T(A))$ is a unital strictly positive linear map. Let\n$S_1,S_2,...,S_n$ be a set of mutually disjoint clopen\nsubsets of $X$ with $\\cup_{i=1}^n S_i=X.$  Then for any $\\delta>0$\nand any finite subset ${\\cal G}\\subset C(X)_{s.a},$ there exists a\nprojection $p\\in A$ with $p\\not=1_A$ and a unital homomorphism\\, $h:\nC(X)\\to pAp$ with finite dimensional range such that\n\\begin{eqnarray}\\label{M0-1}\n|\\tau\\circ h(g)-\\gamma(g)(\\tau)|&<&\\delta\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in {\\cal G}\\,\\,\\,\\text{and}\\,\\,\\,\n\\tau\\in T(A), \\,\\,\\,\\text{and}\\,\\,\\,\\\\\\label{M0-1+} \\tau\\circ\nh(\\chi_{S_i})&<&\\gamma(\\xi_{S_i})(\\tau)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A),\n\\end{eqnarray}\n$i=1,2,...,n.$\n\\end{lem}\n\n\\begin{proof}\nPut\n$$\nd=\\min\\{\\delta, \\min\\{\\inf\\{\\gamma(\\chi_{S_i})(\\tau):\\tau\\in\nT(A)\\}:1\\le i\\le n\\}\\}.\n$$\nSince $\\gamma$ is strictly positive, $d>0.$\n\n Let ${\\cal G}_0={\\cal G}\\cup \\{\\chi_{S_1}, \\chi_{S_2},...,\\chi_{S_n}\\}.$\n  It follows from 4.3  of \\cite{Lnhomp}\nthat there is a unital homomorphism\\, $h_0: C(X)\\to A$ with finite dimensional range such that\n\\begin{eqnarray}\\label{M0-2}\n|\\tau\\circ h(g)-\\gamma(g)(\\tau)|<d\/8n \\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in {\\cal G}_0\n\\end{eqnarray}\nand for all $\\tau\\in T(A).$ In particular,\n\\begin{eqnarray}\\label{M0-3}\n|\\tau\\circ h(\\chi_{S_i})-\\gamma(\\chi_{S_i})(\\tau)|<d\/8n\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\,\n\\tau\\in T(A)\n\\end{eqnarray}\n$i=1,2,...,n.$\n\n\nSince $\\rho_A(K_0(A))$ is dense in ${\\rm Aff}(T(A)),$ there\nexists a projection $p_0\\in A$ such that\n\\begin{eqnarray}\\label{M0-4}\nd\/2n<\\tau(p_0)<d\/n \\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A).\n\\end{eqnarray}\n\nNote that $\\tau(p_0)<\\gamma(\\chi_{S_i})(\\tau)$ for all $\\tau\\in\nT(A),$ $i=1,2,...,n.$ Moreover, by (\\ref{M0-3}), \n\\begin{eqnarray}\\label{M0-4+}\n\\tau\\circ h(\\chi_{S_i})> \\gamma(\\xi_{S_i})(\\tau)-d\/8n\\ge\nd-d\/8n>\\tau(p_0).\n\\end{eqnarray}\nfor all $\\tau\\in T(A).$\n\n\nWrite $h_0(f)=\\sum_{k=1}^m f(x_k)e_k$ for all $f\\in C(X),$ where\n$x_k\\in X$ and $\\{e_1,e_2,...,e_k\\}$ is a set of mutually\northogonal projections with $\\sum_{k=1}^m e_k=1_A.$\n\nNote that\n$$\nh_0(\\xi_{S_j})=\\sum_{x_k\\in S_j}e_k.\n$$\nTherefore (by (\\ref{M0-4+}))\n\\begin{eqnarray}\\label{M0-5}\n[p_0]\\le [\\sum_{x_k\\in S_j}e_k].\n\\end{eqnarray}\nBy Zhang's Riesz interpolation property (see \\cite{Z1} ), there are\nprojections $e_k'\\le e_k$ such that\n$$\n[p_0]=[\\sum_{k\\in S_j}e_k'].\n$$\nBy Zhang's half projection theorem (see Theorem 1.1 of \\cite{Z2}),\nfor each $k,$ there is a projection $e_k''\\le e_k'$ such that\n\\begin{eqnarray}\\label{M0-6}\n[e_k'']+[e_k'']\\ge [e_k'].\n\\end{eqnarray}\nThus\n\\begin{eqnarray}\\label{M0-7}\n2[\\sum_{\\chi_k\\in S_i}e_k'']\\ge [p_0],\\,\\,\\,i=1,2,...,n.\n\\end{eqnarray}\nTherefore (by (\\ref{M0-4}) and (\\ref{M0-3}))\n\\begin{eqnarray}\\label{M0-8}\n\\tau(\\sum_{x_k\\in S_i}(e_k-e_k''))&<&\\tau\\circ\nh_0(\\chi_{S_i})-(1\/2)\\tau(p_0)\\\\\n&<& \\tau\\circ h(\\chi_{S_i})-d\/4n\\\\\\label{M0-8+}\n&<& \\gamma(\\chi_{S_i})(\\tau)-d\/8n\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A).\n\\end{eqnarray}\n\n\nLet $p=\\sum_{k=1}^m (e_k-e_k'').$ Then clearly that $p\\not=1.$\nMoreover,\n$$\n\\tau(1-p)<d\/4\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A).\n$$\n\nDefine $h(f)=\\sum_{k=1}^m f(x_k) (e_k-e_k'')$ for all $f\\in C(X).$\nThen\n\\begin{eqnarray}\\label{M0-9}\n|\\tau\\circ h(f)-\\tau\\circ h_0(f)|<\\tau(\\sum_{k=1}^m\ne_k'')=\\tau(1-p)<d\/4<\\delta\n\\end{eqnarray}\nfor all $\\tau\\in T(A).$\n\nThen, by (\\ref{M0-8+}),\n\\begin{eqnarray}\\label{M0-10}\n\\tau\\circ h(\\chi_{S_i})<\\gamma(\\chi_{S_i})(\\tau)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in\nT(A).\n\\end{eqnarray}\n\n\n\n\\end{proof}\n\n\\begin{lem}\\label{ML1}\nLet $X$ a compact metric space  and let $A$ be a unital simple\n$C^*$-algebra\\, with tracial rank zero. Suppose that $\\gamma: C(X)_{s,a} \\to\n{\\rm Aff}(T(A))$ is a unital strictly positive linear map. Let\n$S_1,S_2,...,S_n$ be a set of mutually disjoint clopen\nsubsets of $X$ with $\\cup_{i=1}^n S_i=X.$  Then for any $\\delta>0,$\n$\\eta>0,$  for any integer $N$ and any $\\eta$-dense subset\n$\\{x_1,x_2,...,x_N\\}$ of $X$ and any finite subset ${\\cal\nG}\\subset C(X)_{s.a},$ there exists a projection $p\\in A$ with\n$p\\not=1_A$ and a unital homomorphism\\, $h: C(X)\\to pAp$ with finite\ndimensional range such that\n\\begin{eqnarray}\\label{M1-1}\n|\\tau\\circ h(g)-\\gamma(g)(\\tau)|&<&\\delta\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in {\\cal G}\\,\\,\\,\\text{and}\\,\\,\\,\n\\tau\\in T(A), \\,\\,\\,\\text{and}\\,\\,\\,\\\\\\label{M1-1+} \\tau\\circ\nh(\\chi_{S_i})&<&\\gamma(\\chi_{S_i})(\\tau)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A),\n\\end{eqnarray}\n$i=1,2,...,n,$\n\\begin{eqnarray}\\label{M1-2}\nh(f)=\\sum_{i=1}^N f(x_i)e_i\\oplus h_1(f)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in C(X),\n\\end{eqnarray}\nwhere $h_1: C(X)\\to (1-\\sum_{i=1}^Ne_i)A(1-\\sum_{i=1}^N e_i)$ is a\nunital homomorphism\\, with finite dimensional range and\n$\\{e_1,e_2,...,e_N\\}$ is a set of mutually orthogonal projections\nsuch that $[e_i]=[e_1]\\ge [1-p],$ $i=1,2,...,N.$\n\n\\end{lem}\n\n\\begin{proof}\n  Let $N\\ge 1$ and let\n$\\eta$-dense subset $\\{x_1, x_2,...,x_N\\}$ of $X$ be  given. Let\n$\\eta_0>0$ such that\n\\begin{eqnarray}\\label{M1-3}\n|f(x)-f(x')|<\\delta\/4\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in {\\cal G},\n\\end{eqnarray}\nprovided that ${\\rm dist}(x, x')<\\eta_0.$\n\nChoose $\\eta_0>\\eta_1>0$ such that $B(x_i, \\eta_1)$ intersects\nwith one and only one $S_i$ among $\\{S_1,S_2,...,S_n\\}.$\n\n\n Choose, for each $i,$ a  non-zero function $f_i\\in C(X)$ with $0\\le f\\le 1$\nwhose support is in $B(x_i, \\eta_1\/2).$ Put\n$$\nd_0=\\min\\{ \\inf\\{\\gamma(f_i)(\\tau): \\tau\\in T(A)\\}: 1\\le i\\le N\\}.\n$$\nSo $d_0>0.$\nPut $\\delta_1=\\min\\{\\delta\/8, \\delta_0\/4\\}$ and put ${\\cal G}_1={\\cal\nG}\\cup\\{1_{C(X)}\\}\\cup \\{f_i: i=1,2,...,N\\}.$\n\nNow applying \\ref{ML0}. We obtain a projection $p\\in A$ and a\nunital homomorphism\\, $h_0: C(X)\\to pAp$ such that\n\\begin{eqnarray}\\label{M1-4}\n|\\tau\\circ h_0(g)-\\gamma(g)(\\tau)|&<&\\delta_1\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in {\\cal\nG}_1\\,\\,\\,{\\rm and}\\,\\,\\,\\\\\\label{M1-5}\n \\tau\\circ\nh_0(\\chi_{S_i})&<&\\gamma(\\chi_{S_i})(\\tau)\n\\end{eqnarray}\nfor all $\\tau\\in T(A),$ $i=1,2,...,n.$ Since $1_{C(X)}\\in {\\cal\nG}_1,$ by (\\ref{M1-4}),\n\\begin{eqnarray}\\label{M1-4+}\n\\tau(1-p)<\\delta_1<\\delta_0\/4\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A).\n\\end{eqnarray}\n\nWrite $h_0(f)=\\sum_{j=1}^Lf(\\xi_j)q_j$ for all $f\\in C(X),$ where\n$\\xi_j\\in X$ and $\\{q_1,q_2,...,q_L\\}$ is a set of mutually\northogonal projections with $\\sum_{j=1}^Lq_j=p.$\n\nDefine\n$$\ne_i'=\\sum_{\\xi_j\\in B(x_i, \\eta_1\/2)}q_j,\\,\\,\\,i=1,2,...,N.\n$$\n\n It follows from (\\ref{M1-4}) that, for each\n$i,$\n\\begin{eqnarray}\\label{Ml-6}\n\\tau(e_i')&\\ge & \\tau\\circ h_0(f_i)\\\\\n&>& \\gamma(f_i)(\\tau)-\\delta_1> 3\\delta_0\/4\\ge \\tau(p)\n\\end{eqnarray}\nfor all $\\tau\\in T(A).$ It follows that\n$$\n[e_i']\\ge [p],\\,\\,\\,i=1,2,...,N.\n$$\nThere are projections $e_i\\le e_i'$ such that\n\\begin{eqnarray}\\label{M1-6+}\n[e_i]=[e_1]\\ge [1-p],\\,\\,\\,i=1,2,...,N.\n\\end{eqnarray}\n\n\nDefine\n\\begin{eqnarray}\\label{M1-7}\nh_1(f)&=&\\sum_{\\xi_j\\not\\in \\cup_{i=1}^N B(x_i, \\eta_1\/2)}\nf(\\xi_j)q_j+\\sum_{i=j}^N f(x_j)(e_i'-e_i)\\,\\,\\,{\\rm and}\\,\\,\\,\\\\\nh(f)&=&\\sum_{i=1}^N f(x_i)e_i\\oplus h_1(f)\n\\end{eqnarray}\nfor all $f\\in C(X).$ Since $B(x_j, \\eta_1\/2)$ lies in one of\n$S_i,$\n$$\n\\tau\\circ h(\\chi_{S_i})=\\tau\\circ h_0(\\chi_{S_i})\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in\nT(A),\n$$\n$i=1,2,...,n.$ It follows from  (\\ref{M1-5}) that (\\ref{M1-1+})\nholds. By the choice of $\\eta_0,$ we also have\n\\begin{eqnarray}\\label{M1-8}\n\\|h_0(g)-h(g)\\|<\\delta\/2\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, h\\in {\\cal G}.\n\\end{eqnarray}\nThus,  by (\\ref{M1-4}),  (\\ref{M1-1}) also holds.\n\n\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\\begin{lem}\\label{ML2}\nLet $X$ be a compact metric space and let $A$ be a unital simple\n$C^*$-algebra\\, with tracial rank zero. Suppose that $\\gamma: C(X)_{s.a}\\to\n{\\rm Aff}(T(A))$ is a unital strictly positive linear map which is\ncompatible with a strictly positive homomorphism\\, $\\kappa_0: K_0(C(X))\\to\nK_0(A).$ Fix $\\delta>0,$ $\\eta>0,$ a finite subset ${\\cal F}\\subset\nC(X)_{s.a.},$ an integer $N\\ge 1,$ an $\\eta$-dense subset\n$\\{x_1,x_2,...,x_N\\}$ of $X,$ a finitely many mutually disjoint\nclopen subset $S_1, S_2, ...,S_n\\subset X$ with $\\cup_{i=1}^n\nS_i=X,$ a finite subset set $\\{a_1,a_2,...,a_n\\}\\subset A$ of\nmutually orthogonal projections  with\n$$\n0<a_i< \\kappa_0([\\chi_{S_i})]),\\,\\,\\,i=1,2,...,n,\n$$\na finitely many mutually disjoint clopen subsets\n$\\{F_1,F_2,...,F_{n_1}\\}$ of $X$ with $\\cup_{i=1}^{n_1} F_i=X,$\nand a projection $p$ with $\\tau(p)=\\tau(\\sum_{i=1}^na_i)$ for all\n$\\tau\\in T(A).$\n\nThere is a projection $q\\in A$ such that $[p]\\le [q]$ and a unital\nhomomorphism\\, $h: C(X)\\to qAq$ with finite dimensional range such that\n\\begin{eqnarray}\\label{M2-1}\n|\\tau\\circ h(g)-\\gamma(g)(\\tau)|&<&\\delta\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in {\\cal F}\\,\\,\\,\\text{and}\\,\\,\\,\n\\tau\\in T(A), \\,\\,\\,\\text{and}\\,\\,\\,\\\\\\label{M2-1+} \\tau\\circ\nh(\\chi_{F_i})&<&\\gamma(\\chi_{F_i})(\\tau)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A),\n\\end{eqnarray}\n$i=1,2,...,n,$\n\\begin{eqnarray}\\label{M2-2}\nh(f)=\\sum_{i=1}^N f(x_i)e_i\\oplus h_1(f)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in C(X),\n\\end{eqnarray}\nwhere $h_1: C(X)\\to (1-\\sum_{i=1}^Ne_i)A(1-\\sum_{i=1}^N e_i)$ is a\nunital homomorphism\\, with finite dimensional range and\n$\\{e_1,e_2,...,e_N\\}$ is a set of mutually orthogonal projections\nsuch that $[e_i]=[e_1]\\ge [1-p],$ $i=1,2,...,N.$\n\nMoreover, there exists a projection $p'\\in q$ such that\n\\begin{eqnarray}\\label{M2-3}\np'h(f)&=&h(f)p'\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in C(X)\\,\\,\\,{\\rm and}\\,\\,\\,\n\\end{eqnarray}\n\\begin{eqnarray}\\label{M2-4}\n[h(\\chi_{S_j})p']&=&[a_j],\\,\\,\\,j=1,2,...,n.\n\\end{eqnarray}\n\\end{lem}\n\n\\begin{proof}\nLet\n$$\nd_0=\\min\\{ \\inf\\{\\tau(\\kappa_0([\\chi_{S_i})])-[a_i]): \\tau\\in\nT(A)\\}: 1\\le i\\le n\\}\n$$\nand let\n$$\nd_1=\\inf\\{ \\tau(1-p): \\tau\\in T(A)\\}.\n$$\nThen $d_0, d_1>0.$\nDefine $\\delta_1=\\min\\{\\delta\/4, d_0\/2,d_1\/2\\}$ and ${\\cal G}_1={\\cal\nF}\\cup \\{1_{C(X)}, \\chi_{S_i}, i=1,2,...,n\\}.$ By applying\n\\ref{ML1}, we obtain a projection $q\\in A$ and a unital homomorphism\\, $h:\nC(X)\\to qAq$ with finite dimensional range satisfying the\nfollowing:\n\\begin{eqnarray}\\label{M2-5}\n|\\tau\\circ h(g)-\\gamma(g)(\\tau)|&<&\\delta_1\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in {\\cal G}_1,\\\\\\label{M2-5+}\n\\tau\\circ h(\\chi_{F_j})&<&\\gamma(\\chi_{F_j})(\\tau)\n\\end{eqnarray}\nfor all $\\tau\\in T(A),$ $j=1,2,...,n_1,$\n\\begin{eqnarray}\\label{M2-6}\nh(f)=\\sum_{k=1}^Nf(x_k)e_i\\oplus h_1(f)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in C(X),\n\\end{eqnarray}\nwhere $\\{e_1,e_2,...,e_N\\}$ is a set of mutually orthogonal and\nmutually equivalent projections such that $[e_1]\\ge [1-q],$ and where $h_1: C(X)\\to (q-\\sum_{k=1}^Ne_k)A(q-\\sum_{k=1}^Ne_k)$ is a unital homomorphism\\, with finite dimensional range.\n\nSince $1_{C(X)}\\in {\\cal G}_1$, by the choice of $\\delta_1,$  we\nconclude that $[p]\\le [q].$\n\nMoreover, by (\\ref{M2-5}),\n\\begin{eqnarray}\\label{M2-7}\n\\tau\\circ h(\\chi_{S_i})>\\tau(a_i)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in\nT(A),\\,\\,\\,i=1,2,...,n.\n\\end{eqnarray}\nWrite\n$$\nh(f)=\\sum_{s=1}^L f(\\xi_s)E_s\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in C(X),\n$$\nwhere $\\xi_s\\in X$ and $\\{E_1, E_2,...,E_L\\}$ is a set of mutually\northogonal projections such that $\\sum_{s=1}^L E_s=q.$ By\n(\\ref{M2-7}), one has\n\\begin{eqnarray}\\label{M2-8}\n\\sum_{\\xi_s\\in S_i} E_s\\ge a_i,\\,\\,\\, i=1,2,...,n.\n\\end{eqnarray}\nFor each $i,$ by the Riesz Interpolation Property (\\cite{Z1}), there is a\nprojection $E_s'\\le E_s$ for which $x_s\\in S_i$ such that\n\\begin{eqnarray}\\label{M2-9}\n[\\sum_{\\xi_s\\in S_i}E_s']=[a_i].\n\\end{eqnarray}\n\nPut $p'=\\sum_{s=1}^L E_s'$ Then\n\\begin{eqnarray}\np'h(f)&=&h(f)p'\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in C(X)\\,\\,\\,{\\rm and}\\,\\,\\,\n\\end{eqnarray}\n\\begin{eqnarray}\n[h(\\chi_{S_i})p']=[a_i],\\,\\,\\,i=1,2,...,n.\n\\end{eqnarray}\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\\begin{thm}\\label{TKL1}\nLet $X$ be a compact subset of a finite CW complex and let $A$ be\na unital simple $C^*$-algebra\\, with $TR(A)=0.$  Suppose that $\\kappa\\in\nKL_e(C(X), A)^{++}$ and suppose that there exists a unital\nstrictly positive linear map $\\gamma: C(X)_{s.a}\\to {\\rm\nAff}(T(A))$ which is compatible with $\\kappa.$\n Then there exists a\nunital monomorphism $\\phi: C(X)\\to A$ such that\n$$\n[\\phi]=\\kappa\\,\\,\\,\\text{in}\\,\\,\\, KL(C, A).\n$$\n\\end{thm}\n\n\\begin{proof}\n\n\n\nSuppose that $X\\subset Y,$ where  $Y$ is a finite CW complex.\nLet $X_n\\subset Y$ be a decreasing sequence of finite CW complexes\nfor which \\ref{D1} holds. Suppose that\n$p_{n,1},p_{n,2},...,p_{n,r(n)}$ are  mutually orthogonal\nprojections of $C(X_n)$ which correspond to the connected\ncomponents of $X_n.$ It is clear that we may assume that each\nconnected component of $X_n$ contains at least one point of $X.$\nThis implies that $[s_n]\\in KK(C(X_n), C(X))^{++}.$ It follows\nthat\n\\begin{eqnarray}\\label{TKL1-2}\n[s_n]\\times \\kappa\\in KK(C(X_n), A)^{++}.\n\\end{eqnarray}\n\n\nLet $\\{{\\cal F}_n\\}$ be an increasing sequence of finite subsets\nof $C(X)$ whose union is dense in $C(X).$ Let $\\{\\eta_n\\}$ be a\ndecreasing sequence of positive numbers with\n$\\lim_{n\\to\\infty}\\eta_n=0,$ $\\{{\\cal P}_n\\}$ be an increasing\nsequence of finite subsets of $\\underline{K}(C)$ whose union is\n$\\underline{K}(C),$ let $\\{k(n)\\},\\, \\{N(n)\\}\\subset \\mathbb{N}$ be two\nsequences of integers such that $k(n), N(n)\\nearrow \\infty,$ and\n$\\{x_{n,1},x_{(n,2},...,x_{n,N(n)}\\}$ be $\\eta_n$-dense\n subsets of $X$ which satisfy the requirements of \\ref{KLL1} and \\ref{RM1}\nfor corresponding $\\epsilon_n=1\/2^{n+2}$ and  ${\\cal F}_n.$\n\nBy passing to a subsequence if necessary, we may assume that there\nis a finite subset ${\\cal F}_n'\\subset C(X_{k(n+1)})$ such that\n$s_{k(n+1)}({\\cal F}_n')={\\cal F}_n$ and a finite subset ${\\cal\nQ}_{k(n)}\\subset \\underline{K}(C(X_{k(n)}))$ such that\n$[s_{k(n)}]({\\cal Q}_{k(n)})={\\cal P}_n,$ $n=1,2,....$ We may\nassume that $1_{C(X_{k(n)})}\\in {\\cal F}_n',$ without loss of\ngenerality.\n\nSet $\\kappa_n=[s_{k(n)}]\\times \\kappa.$  Note that\n$\\kappa_n([1_{C(X_{k(n)})}])=[1_A].$\n\n\nLet $\\delta_n$ (in place of $\\delta$), ${\\cal G}_n'\\subset C(X)_{s.a}$\n( in place of ${\\cal G}$), $S_{1,n}, S_{2,n},...,S_{m(n),n}$ (in\nplace of $\\{S_1, S_2,...,\\}$) be a set of disjoint clopen\nsubsets of $X$  with $\\cup_{i=1}^{m(n)}S_i=X$ required by\n\\ref{Lcd} for $\\epsilon_n$ and ${\\cal F}_n,$ $n=1,2,....$ We may assume\nthat $1_{C(X)}\\in {\\cal G}_n',$ $n=1,2,....$\n\nBy taking a refinement of the clopen partition of $X,$ we may\nassume that $s_n(p_{n,i})$ is a finite sum of functions in\n$\\{\\chi_{S_{j,n}}: 1\\le j\\le m(n)\\},$ $i=1,2,...,r(n).$\n\nLet ${\\cal G}_n\\subset C(X_{k(n)})_{s.a.}$ be a finite subsets for\nwhich $s_{k(n)}({\\cal G}_n)={\\cal G}_n',$ $n=1,2,....$\n\nBy applying \\ref{ML1}, we obtain a projection $P_1\\in A$ and a\nunital homomorphism\\, $\\Phi_1': C(X)\\to P_1AP_1$ such that\n\\begin{eqnarray}\\label{T1-10}\n|\\tau\\circ \\Phi_1'(g)-\\gamma(g)(\\tau)|&<&\\delta_1\/2\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in {\\cal\nG}_1',\\\\\n\\tau\\circ \\Phi_1'(\\chi_{S_{j,1}})&<&\\gamma(\\chi_{S_{j,1}})(\\tau)\n\\end{eqnarray}\nfor all $\\tau\\in T(A),$ $i=1,2,...,m(1),$ and\n\\begin{eqnarray}\\label{T1-11}\n\\Phi_1'(f)=\\sum_{i=1}^{N(1)} f(x_{1,i})e_i^{(1)}\\oplus\n\\Phi_{0,1}'(f)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in C(X),\n\\end{eqnarray}\nwhere $\\{e_1^{(1)},e_2^{(1)},...,e_{N(1)}^{(1)}\\}$ is a set of\nmutually orthogonal and mutually equivalent projections with\n$[e_1]\\ge [(1-P_1)]$ and where $ \\Phi_{0,1}':\nC(X)\\to(P_1-\\sum_{i=1}^{N(1)} e_i^{(1)})A((P_1-\\sum_{i=1}^{N(1)}\ne_i^{(1)})$ is a unital homomorphism\\, with finite dimensional range.  Note\nalso, since $1_{C(X)}\\in {\\cal G}_1',$ $\\tau(1-P_1)<\\delta_1\/2$ for\nall $\\tau\\in T(A).$\n\nIt follows from \\ref{KLL2} that there is a unital monomorphism\n$\\phi_1': C(X_{k(1)})\\to (1-P_1)A(1-P_1)$ such that\n\\begin{eqnarray}\\label{T1-12}\n[\\phi_1']+[\\Phi_1'\\circ s_1]=\\kappa_1\\,\\,\\,{\\rm in}\\,\\,\\,\nKK(C(X_{k(1)}), A).\n\\end{eqnarray}\n\nDefine $\\phi_1=\\phi_1'+\\Phi_1'\\circ s_1.$\n\n\nSuppose that, for $1\\le m\\le n,$ there are unital homomorphism s  $\\phi_m': C(X_{k(m)})\\to (1-P_m)A(1-P_m)$  and\n$\\Phi_m': C(X)\\to P_mAP_m$  and a unital (injective) homomorphism\\, \n$\\phi_m=\\phi_m'+\\Phi_m'\\circ s_{k(m)}$ such that\n\\begin{enumerate}\n\\item there are mutually orthogonal and mutually equivalent\nprojections $e_1^{(m)}, e_2^{(m)},...,e_{N(m)}^{(m)}\\in P_mAP_m$\nfor which $[e_1^{(m)}]\\ge [1-P_m],$ and\n$$\n\\Phi_m'(f)=\\sum_{i=1}^{N(m)}f(x_{m,i})e_i^{(m)}\\oplus\n\\Phi_m^{(0)}(f) \\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in C(X)\n$$\nwhere  $\\Phi_m^{(0)}: C(X)\\to\n(P_m-\\sum_{i=1}^{N(m)}e_i^{(m)})A(P_m-\\sum_{i=1}^{N(m)}e_i^{(m)})$\nis a unital homomorphism\\, with finite dimensional range;\n\n\\item  $\\tau\\circ\n\\Phi'(\\chi_{S_{j,m}})<\\gamma(\\chi_{S_{j,m}})(\\tau)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in\nT(A),$ $j=1,2,...,m(m);$\n\n\\item $|\\tau\\circ \\Phi_m'(g)-\\gamma(g)(\\tau)|<\\delta_m\/2\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in\n{\\cal G}_m'$ and for all $\\tau\\in T(A);$\n\n\\item $[P_{m+1}]\\ge [P_m]$ in $K_0(A)$ and $\\tau(1-P_m)<\\delta_m\/2\n\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A);$\n\n\\item there is a projection $P_{m+1}'\\le P_{m+1}$ such that\n$P_{m+1}'\\Phi_{m+1}=\\Phi_{m+1}'P_{m+1}$ and\n$$[\\Phi_{m+1}'(\\chi_{S_{j,m}})P_{m+1}']=[\\Phi_m'(\\chi_{S_{j,m}})]\\,\\,\\,\n{\\rm in}\\,\\,\\,K_0(A),j=1,2,...,m(m);$$\n\n \\item $\\phi_m'$ is a unital monomorphism;\n\n\\item $[\\phi_m]=[\\phi_m']+[\\Phi_m'\\circ s_{k(m)}]=\\kappa_m;$\n\n \\item there exists a unitary $u_m\\in A$ such that\n$$\n{\\rm ad}\\, u_m\\circ \\phi_{m+1}\\circ\ns_{k(m), k(m+1)}\\approx_{1\/2^{m+1}} \\phi_{m}\\,\\,\\,{\\rm on}\\,\\,\\, s_{k(m)}^{-1}({\\cal F}_m\n),\\,\\,\\,m=1,2,...,n-1.\n$$\n\n\\end{enumerate}\n\nIt follows from \\ref{ML2} that there is a projection $P_{n+1}\\in\nA$ and a unital homomorphism\\, $\\Phi_{n+1}': C(X)\\to P_{n+1}AP_{n+1}$\nsatisfying the following:\n\n\\begin{enumerate}\n\\item there are mutually orthogonal and mutually equivalent\nprojections $e_1^{(n+1)}, e_2^{(n+1)},...,e_{N(n+1)}^{(n+1)}\\in\nP_{n+1}AP_{n+1}$ for which $[e_1^{(n+1)}]\\ge [1-P_{n+1}],$ and\n$$\n\\Phi_{n+1}'(f)=\\sum_{i=1}^{N(n+1)}f(x_{n+1,i})e_i^{(n+1)}\\oplus\n\\Phi_{n+1}^{(0)}(f) \\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in C(X)\n$$\nwhere  $\\Phi_{n+1}^{(0)}: C(X)\\to\n(P_{n+1}-\\sum_{i=1}^{N(n+1)}e_i^{(n+1)})A(P_{n+1}-\\sum_{i=1}^{N(n+1)}e_i^{(n+1)})$\nis a unital homomorphism\\, with finite dimensional range;\n\\item  $\\tau\\circ\n\\Phi_{n+1}'(\\chi_{S_{j,n+1}})<\\gamma(\\chi_{S_{j,n+1}})(\\tau)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\,\n\\tau\\in T(A),$ $j=1,2,...,m(n+1);$\n\n\\item $|\\tau\\circ\n\\Phi_{n+1}'(g)-\\gamma(g)(\\tau)|<\\delta_{n+1}\/2\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in {\\cal\nG}_{n+1}'$ and for all $\\tau\\in T(A);$\n\n\n\\item $[P_{n+1}]\\ge [P_n]$ in $K_0(A)$ and\n$\\tau(1-P_{n+1})<\\delta_{n+1}\/2 \\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A);$\n\n\n\\item there is a projection $P_{n+1}'\\le P_{n+1}$ such that\n$P_{n+1}'\\Phi_{n+1}'=\\Phi_{n+1}' P_{n+1}$ and\n$$[\\Phi'_{n+1}(\\chi_{S_{n, j}})P_{n+1}']=[\\Phi'_n(\\chi_{S_{n,\nj}})]\\,\\,\\,{\\rm in}\\,\\,\\,K_0(A),j=1,2,...,m(n).$$\n\\end{enumerate}\n\n\n\n\n\nIt follows from \\ref{KLL2} that there is a unital\nmonomorphism\\linebreak  $\\phi_{n+1}': C(X_{k(n+1)})\\to\n(1-P_{n+1})A(1-P_{n+1})$ such that\n\\begin{eqnarray}\\label{T1-13}\n[\\phi_{n+1}']=\\kappa_{n+1}-[\\Phi_{n+1}'\\circ s_{k(n+1)}]\\,\\,\\,{\\rm\nin}\\,\\,\\, KK(C(X_{k(n+1)},A)\n\\end{eqnarray}\nDefine $\\phi_{n+1}=\\phi_{n+1}'+\\Phi_{n+1}'\\circ s_{k(n+1)}.$\n\nThus $\\phi_{n+1}',$ $\\phi_{n+1}'$ and $\\phi_{n+1}$ satisfy (1),\n(2), (3), (4), (5), (6) and (7).\n\nTo complete the induction, define  $\\Phi_{n+1}'': C(X)\\to\nP_{n+1}'AP_{n+1}'$ by\n$\\Phi_{n+1}''(f)=P_{n+1}'\\Phi_{n+1}'(f)P_{n+1}'$ for all $f\\in\nC(X).$ By (3) and (4),\n$$\n|\\tau\\circ \\Phi_{n+1}''(g)-\\gamma(g)(\\tau)|<\\delta_{n+1}\/2\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in\n{\\cal G}_n\n$$\nfor all $\\tau\\in T(A).$ Note that, by (5), $[P_{n+1}']=[P_n].$\nThere is a unitary $w_n\\in U(A)$ such that\n$$\nw_n^*P_{n+1}'w_n=P_n.\n$$\nThus, by (5) and (3), and by applying \\ref{Lcd}, there exists a\nunitary $v_n\\in U(P_nAP_n)$ such that\n\\begin{eqnarray}\\label{T1-14}\n{\\rm ad}\\, v_n\\circ {\\rm ad}\\, w_n\\circ\n\\Phi_{n+1}''\\approx_{\\epsilon_n} \\Phi_n'\\,\\,\\,{\\rm\n on}\\,\\,\\, {\\cal F}_n.\n \\end{eqnarray}\n Denote $\\Psi_{n+1}'=P'_{n+1}\\Phi_{n+1}P'_{n+1}$ and $\\Psi_{n+1}={\\rm ad}\\, w_n\\circ \\Psi_{n+1}'.$ \n Let $\\phi_{n+1}''={\\rm ad}\\, w_n\\circ \\phi_{n+1}'\\oplus \\Psi_{n+1}.$ \n Now consider $\\phi_n'$ and  $\\phi_{n+1}''\\circ s_{k(n), k(n+1)}.$  By (7) \nand (\\ref{T1-13}),\nwe have\n$$\n[\\phi_{n+1}''\\circ s_{k(n), k(n+1)}]|_{{\\cal Q}_{k(n)}}=[\\phi_n']|_{{\\cal Q}_{k(n)}}.\n$$\nIt follows from (1) and \n\\ref{KLL1} that there exists a unitary $V_n\\in U(A)$ such that\n\\begin{eqnarray}\\label{T1-15}\n{\\rm ad}\\, V_n\\circ (\\phi_{n+1}''\\circ s_{k(n),\nk(n+1)}\\oplus \\Phi_n' \\circ s_{k(n)})\\approx_{\\epsilon_n} \\phi_n'\\oplus\n\\Phi_n'\\circ s_{k(n)}\\,\\,\\,{\\rm on}\\,\\,\\, s_{k(n)}^{-1}({\\cal\nF}_n).\n\\end{eqnarray}\nDefine $u_n=w_n (v_n+(1-P_n)) V_n.$ Then, by (\\ref{T1-14}) and\n(\\ref{T1-15}),\n\\begin{eqnarray}\\label{T1-16}\n{\\rm ad}\\,u_n\\circ \\phi_{n+1}\\approx_{2\\epsilon_n}\\phi_n\\,\\,\\,{\\rm\non}\\,\\,\\, s_{k(n)}^{-1}({\\cal F}_n).\n\\end{eqnarray}\nNote $2\\epsilon_n=1\/2^{n+1}.$\n\n\n\n\n\nThis concludes the induction.\n\nDefine $\\psi_1=\\phi_1$ and $\\psi_{n+1}={\\rm ad}\\,u_n\\circ\n\\phi_{n+1},$ $n=1,2,....$ Then, by (8) above,\n\\begin{eqnarray}\\label{T1-5}\n\\|\\psi_{n}(c)-\\psi_{n+1}\\circ\ns_{k(n),k(n+1)}(c)\\|<1\/2^{n+2}\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, c\\in s_{k(n)}^{-1}({\\cal\nF}_n),\n\\end{eqnarray}\n$n=1,2,....$\n\n\n\n\nFix $m$ and $f\\in {\\cal F}_m,$ let $g\\in s_{k(m)}^{-1}({\\cal\nF}_m)$ such that $s_{k(m)}(g)=f.$\n\nIt follows that $\\{\\psi_n\\circ s_{m,n}(g)\\}_{n\\ge m}$ is a Cauchy\nsequence by (\\ref{T1-5}).\n\nNote that if $g'\\in s_{k(m)}^{-1}({\\cal F}_m)$ such that\n$s_{k(m)}(g)=s_{k(m)}(g'),$ then, for any $\\epsilon>0,$  there exists\n$n\\ge m$ such that\n$$\n\\|s_{k(m), k(n)}(g)-s_{k(m), k(n)}(g')\\|<\\epsilon.\n$$\n\nThus  $h(f)=\\lim_{n\\to\\infty}\\psi_n\\circ s_{m,n}(g)$ is\nwell-defined. It is then easy to verify  that $h$ defines a unital\nhomomorphism\\, from $C(X)$ into $A.$ Since each $\\phi_n$ is injective, it\nis easy to check that $h$ is also injective.\n\nIf  $x\\in {\\cal Q}_m,$ then by (7) above,\n$$\n [h]\\circ [s_{k(m)}](x)=\\kappa_n\\circ [s_{k(m),\nk(n)}](x)=\\kappa\\circ [s_{k(m)}](x).\n$$\nTherefore\n$$\n[h]=\\kappa\\,\\,\\,{\\rm in}\\,\\,\\, KL(C,A).\n$$\n\nIt is also easy to check from (3) and (4) that\n\\begin{eqnarray}\n\\tau\\circ h(g)=\\gamma(\\check{g})(\\tau)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, g\\in C(X)_{s.a}\n\\end{eqnarray}\nand for all $\\tau\\in T(A).$\n\n\n\n\n\n\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{AH-algebras}\n\n\\begin{lem}\\label{2L}\nLet $X$ be a compact subset of a finite CW complex, let\n$C=PM_k(C(X))P,$ where $P\\in M_k(C(X))$ is a projection, and let\n$A$ be a unital simple $C^*$-algebra\\, with tracial rank zero. Suppose that\n$\\kappa\\in KL_e(C, A)^{++}$ and suppose that $\\gamma: C_{s.a.}\\to\n{\\rm Aff}(T(A))$ is  a unital positive linear map which is\ncompatible with $\\kappa.$ Then, for any $\\epsilon>0$ and finite subset\n ${\\cal F}\\subset  C(X),$ there is a unital monomorphism\n $\\phi: C\\to A$ such that\n \\begin{eqnarray}\\label{KLL4}\n [\\phi]=\\kappa\\,\\,\\,{\\rm and}\\,\\,\\, \\tau\\circ \\phi(f)=\\gamma(f)(\\tau)\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A).\n \\end{eqnarray}\n\n\\end{lem}\n\n\n\\begin{proof}\nIt is clear that the case that $C=M_k(C(X))$ follows from\n\\ref{TKL1} immediately. For the general case, there is an integer\n$d\\ge 1$ and a projection $p\\in M_d(C)$ such that $pM_d(C)p\\cong\nM_m(C(X))$ for some integer $m\\ge 1.$ Thus the general case is\nreduced to the case that $C=M_m(C(X)).$\n\n\n\\end{proof}\n\n\n\\begin{thm}\\label{T2}\nLet $C$ be a unital AH-algebra and let $A$ be a unital simple\n$C^*$-algebra\\, with $TR(A)=0.$ Suppose that $\\kappa\\in \\mathrm{KL}_e(C,\nA)^{++}.$  Suppose also that there is a unital strictly positive\nlinear map $\\gamma: C_{s,a}\\to {\\rm Aff}(T(A))$ which is\ncompatible with $\\kappa.$ Then there is a monomorphism $\\phi: C\\to\nA$ such that\n\\begin{eqnarray}\\label{KL-1}\n[\\phi]&=&\\kappa\\,\\,\\,\\textrm{in}\\,\\,\\,\n\\mathrm{KL}(C,A)\\,\\,\\,{\\rm and}\\,\\,\\,\\\\\n\\tau\\circ \\phi(c)&=&\\gamma(c)(\\tau)\n\\end{eqnarray}\nfor all $c\\in C_{s.a.}$ and $\\tau\\in T(A).$\n\\end{thm}\n\n\\begin{proof}\n\nWe may write $C=\\overline{\\cup_{n=1}^{\\infty}C_n},$ where $C_n\n=P_nM_k(C(X_n))P_n,$ where $X_n$ is a compact subset of a finite\nCW complex and $P_n\\in M_k(C(X_n))$ is a projection. We may also\nassume that $1_{C_n}=1_C$ for all $n.$ Denote by $\\imath_n: C_n\\to\nC$  the embedding, $n=1,2,....$\n\nDefine\n$$\n\\kappa_n=\\kappa\\circ [\\imath_n]\\,\\,\\,{\\rm and}\\,\\,\\, \\gamma_n=\\gamma\\circ\n(\\imath_n)_{\\sharp}\n$$\n$n=1,2,....$ Since $\\imath_n$ is injective $\\kappa_n\\in KL_e(C_n,\nA)^{++}$  and $\\gamma_n$ is unital strictly positive. It is also\nclear that $\\gamma_n$ is compatible with $\\kappa_n,$ since $\\gamma$\nis compatible with $\\kappa.$ It follows from \\ref{2L} that there\nis a sequence of unital monomorphisms $\\phi_n: C_n\\to A$ such that\n\\begin{eqnarray}\\label{T2-3}\n[\\phi_n]=\\kappa_n\\,\\,\\,{\\rm and}\\,\\,\\, \\tau\\circ \\phi_n(c)=\\gamma_n(c)(\\tau)\n\\end{eqnarray}\nfor all $c\\in C_{s.a.}$ and $\\tau\\in T(A).$\n\nLet $\\{{\\cal F}_n\\}$ be an increasing sequence of finite subsets\nof $C$ whose union is dense in $C.$ By passing to a subsequence,\nif necessary, without loss of generality, we may assume that\n${\\cal F}_n\\subset C_n.$\n\nIt follows (from  2.3.13 of \\cite{Lnbk}, for example)  that there\nis, for each $n,$ a unital completely positive linear map $L_n:\nC\\to A$ such that\n\\begin{eqnarray}\\label{T2-4}\nL_n\\approx_{1\/2^{n+1}} \\phi_n\\circ \\imath_n\\,\\,\\,{\\rm\non}\\,\\,\\,{\\cal F}_n.\n\\end{eqnarray}\n\nIt follows from Lemma \\ref{2L}, by passing to a subsequence again and by\napplying (\\ref{T2-3}), there is a sequence of unitaries $u_n$ and a\nsubsequence of $\\{k(n)\\}$ such that\n\\begin{eqnarray}\\label{T2-5}\n{\\rm ad}\\, u_n\\circ L_{k(n+1)}\\approx_{1\/2^n} L_{k(n)}\\,\\,\\,{\\rm\non}\\,\\,\\, {\\cal F}_n,\n\\end{eqnarray}\n$n=1,2,....$ Define $\\psi_1=L_1,$ $\\psi_{n+1}={\\rm ad}\\, u_n\\circ\nL_{n+1},$ $n=1,2,....$  Note that $\\{\\psi_n(c)\\}$ is a Cauchy\nsequence in $A$ for each $c\\in {\\cal F}_m.$ Define\n$h(c)=\\lim_{n\\to\\infty}\\psi_n(c).$ It is easy to see that $h$\ngives a unital homomorphism\\, from $C$ into $A.$ Moreover, for each $x\\in\n\\cup_{n=1}^{\\infty}{\\cal F}_n,$\n\\begin{eqnarray}\\label{T2-6}\nh(x)=\\lim_{n\\to\\infty} {\\rm ad}\\, u_n \\circ \\phi_{k(n)}\\circ\n\\imath_{k(n)}\\circ\\cdots \\circ \\imath_n(x).\n\\end{eqnarray}\nSince each $\\phi_n$ is injective, it follows that $h$ is a\nmonomorphism. From (\\ref{T2-6}) and (\\ref{T2-3}), we have\n$$\n[h]=\\kappa\\,\\,\\,{\\rm as\\,\\,\\, well \\,\\,\\, as}\\,\\,\\, \\tau\\circ\nh(c)=\\gamma(c)(\\tau)\n$$\nfor all $c\\in C_{s.a.}$ and $\\tau\\in T(A).$\n\n\n\n\n\n\n\n\n\n\\end{proof}\n\n\n\\begin{cor}\\label{CCX}\nLet $X$ be a compact metric space and let $A$ be a unital simple\n$C^*$-algebra\\, with tracial rank zero. Suppose that $\\kappa\\in\n\\mathrm{KL}_e(C(X), A)^{++}.$ Suppose also that there is a unital\nstrictly positive linear map $\\gamma: C_{s,a}\\to {\\rm Aff}(T(A))$\nwhich is compatible with $\\kappa.$ Then there is a monomorphism\n$\\alpha: C\\to A$ such that\n\\begin{eqnarray}\\label{CKL-1}\n[\\alpha]&=&\\kappa\\,\\,\\,\\textrm{in}\\,\\,\\,\n\\mathrm{KL}(C(X),A)\\,\\,\\,{\\rm and}\\,\\,\\,\\\\\n\\tau\\circ \\phi(c)&=&\\gamma(c)(\\tau)\n\\end{eqnarray}\nfor all $c\\in C(X)_{s.a.}$ and $\\tau\\in T(A).$\n\\end{cor}\n\n\n\n\\begin{exm}\\label{Ex}\n{\\rm Let $X=\\{{-1\\over{n}}: n\\in \\mathbb{N}\\}\\cup [0,1]\\cup\n\\{1+{1\\over{n}}: n\\in \\mathbb{N}\\}\\subset [-1, 2].$ Put $C=C(X).$ Then\n$$\nK_0(C(X))=C(X, \\mathbb{Z}).\n$$\nTake two sequences of positive rational numbers $\\{a_n\\}$ and\n$\\{b_n\\}$ such that $\\sum_{n=1}^{\\infty} a_n =1-\\sqrt{2}\/2$ and\n$\\sum_{n=1}^{\\infty}=\\sqrt{2}\/2.$\n\nDefine a unital positive linear functional $F: C(X)\\to \\mathbb{R}$ as\nfollows:\n$$\nF(f)=\\sum_{n\\in\\mathbb{N}} a_nf({-1\\over{n}})+\\sum_{n\\in \\mathbb{N}} b_n\nf({1\\over{n}})\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, f\\in C(X).\n$$\nLet $D_0=F(C(X, \\mathbb{Z})).$ Note that, if $S$ is a clopen subset which\ndoes not contain $[0,1],$ then $F(S)\\in \\mathbb{Q}.$ If $S\\supset [0,1], $\nThen\n$$\nF(S)=1-F(S_1)\n$$\nfor some clopen subset $S_1\\subset X$ which does not intersect\nwith $[0,1].$ It follows that $D_0\\subset \\mathbb{Q}.$\n\nThis gives a unital positive linear map $F_*: C(X, \\mathbb{Z})\\to \\mathbb{Q}.$ Let\n$p\\in C(X)$ be a projection whose support $\\Omega$ has a non-empty\nintersection with $[0,1].$ Since $\\Omega$ is clopen,\n$\\Omega\\supset [0,1].$ It follows that there is $N\\ge 1$ such that\n${1\\over{k}}\\in \\Omega$ for $|k|\\ge N.$ It follows that\n$$\nF(p)\\ge \\sum_{|k|\\ge N} {1\\over{2^{|k|+1}}}>0.\n$$\nFrom this one sees that $F_*$ is strictly positive.\n\nLet $A$ be a unital simple AF-algebra with\n$$\n(K_0(A), K_0(A), [1_A])=(\\mathbb{Q}, \\mathbb{Q}_+,1).\n$$\nThere is an element $\\kappa\\in KL(C(X), A)$ such that\n$$\n\\kappa|_{K_0(C(X))}=F_*.\n$$\nThus $\\kappa(K_0(C(X))_+\\setminus \\{0\\})\\subset K_0(A)_+\\setminus\n\\{0\\}.$ In other words,  $\\kappa\\in KL_e(C,A)^{++}.$\n\n\n\n\n\n\n\nSuppose that $\\gamma: C_{s.a}\\to {\\rm Aff}(T(A))=\\mathbb{R}$ is unital and\npositive such that\n$$\n\\gamma(\\check{p})(\\tau)=\\tau(\\kappa([p]))\n$$\nfor all projections $p\\in C$ and $\\tau\\in T(A).$ Consider a\npositive continuous function $f\\in C(X)$ with $0\\le f\\le 1$ whose\nsupport $S$ is an open subset of $(0,1).$ Consider\nprojection $p_n(t)=0$ if $t\\not\\in [-1\/n, 1+1\/n]\\cap X$ and\n$p_n(t)=1$ if $t\\in [-1\/n, 1+1\/n]\\cap X.$ Then\n$$\nf\\le p_n,\\,\\,\\, n=1,2,....\n$$\nIt follows that, for all $\\tau\\in T(A),$\n\\begin{eqnarray}\n\\gamma (\\check{f})(\\tau)&\\le &\\gamma(\\check{p}_n)(\\tau)\\\\\n&<&\\sum_{|k|\\ge n} (a_k+b_k)\\to 0\n\\end{eqnarray}\nas $|n|\\to \\infty.$ It follows that\n$$\n\\gamma(\\check{f})(\\tau)=0\\,\\,\\,\\text{for\\,\\,\\,all}\\,\\,\\, \\tau\\in T(A).\n$$\nThis shows that $\\gamma$ is not strictly positive.\n\nIn particular, there is {\\it no} unital monomorphism $\\phi:\nC(X)\\to A$ such that $[\\phi]=\\kappa.$\n\nHow about homomorphism s?  Suppose that there exists a unital homomorphism\\, $h:\nC(X)\\to A$ such that $[h]=\\kappa.$ Let $f\\in C(X)_+$ be so that\nits support is contained in $[0,1].$ Then, as shown above,\n$\\tau(h(f))=0$ for $\\tau\\in T(A).$  Since $A$ is simple, this\nimplies that $h(f)=0.$ It is then easy to see that\n$$\n{\\rm ker}\\,h=\\{f\\in C(X): f|_{X\\setminus (0,1)}=0\\}.\n$$\nThus $C\/{\\rm ker}\\,h\\cong C(Y),$ where $Y=X\\setminus (0,1).$ Let\n$\\phi: C(Y)\\to A$ be the unital homomorphism\\, induced by $h.$ Then $\\phi$\nis a monomorphism.  Let\n$$\nY_1=\\{1+1\/n: n\\in \\mathbb{N}\\}\\cup\\{1\\}\\,\\,\\,{\\rm and}\\,\\,\\, Y_2=\\{-1\/n: n\\in\n\\mathbb{N}\\}\\cup\\{0\\}.\n$$\nThen $Y_1$ and $Y_2$ are  clopen subsets of $Y.$ Let $p_i$ be the\nprojection corresponding to $Y_i,$ $i=1,2.$ Then\n$$\n\\tau(p_1)\\ge \\sum_{n=1}^{\\infty} b_n=1-\\sqrt{2}\/2\\,\\,\\,{\\rm and}\\,\\,\\,\\\\\n\\tau(p_2)\\ge \\sum_{n=1}^{\\infty} a_n=\\sqrt{2}\/2\n$$\nfor $\\tau\\in T(A).$ Since $\\tau(p_1)+\\tau(p_2)=1,$ it follows that\n$$\n\\tau(p_1)=1-\\sqrt{2}\/2\\,\\,\\,{\\rm and}\\,\\,\\, \\tau(p_2)=\\sqrt{2}\/2.\n$$\nThis is impossible since $K_0(A)=\\mathbb{Q}.$\n\n\n}\n\\end{exm}\n\nFrom this we arrive at the following conclusion:\n\n\\begin{prop}\\label{cex}\nThere are compact metric spaces $X$ with dimension one, unital\nsimple AF-algebras $A$ with unique tracial states and  $\\kappa\\in\nKL_e(C,A)^{++}$ which has no  strictly positive affine map from\n${\\rm Aff}(T(C(X))$ to ${\\rm Aff}(T(A))$ compatible with $\\kappa.$\n\nFurthermore, there is no unital homomorphism\\, $\\phi: C(X)\\to A$ such that\n$[\\phi]=\\kappa$ in $KL(C,A).$\n\n\n\\end{prop}\n\n\n\n\n\n\n\n\n\n\n\\\n\n\n\\begin{df}\\label{DMT}\nLet $C$ be a unital AH-algebra which admits a faithful tracial\nstate and let $A$ be a unital simple $C^*$-algebra\\, with\n$T(A)\\not=\\emptyset.$\n\nDenote by $ KLT(C,A)^{++} $ the set of pairs $(\\kappa, \\lambda)$\nwhere $\\kappa\\in KL(C,A)^{++}$  with $\\kappa([1_C])=[1_A]$ and\n$\\lambda: T(A)\\to T_{\\mathtt{f}}(C)$ which is compatible with\n$\\kappa,$ i.e., $\\lambda(\\tau)(p)=\\tau(\\kappa([p])$ for all\nprojections $p\\in M_{\\infty}(C)$ and for all $\\tau\\in T(A).$\n\nDenote by ${\\rm Mon}_{au}^e(C,A)$ the set of approximately unitary\nequivalent classes of unital monomorphisms from $C$ into $A.$\n\n\\end{df}\n\nTo conclude this note, combing the previous result in \\ref{Lncd} (see \\ref{Om}) and \\ref{T2}, we state the following:\n\n\n\\begin{thm}\\label{MT}\nLet $C$ be a unital AH-algebra which admits a faithful tracial\nstate and let $A$ be a unital separable simple $C^*$-algebra\\, with\n$TR(A)=0.$ Then  map\n$$\n\\Lambda: {\\rm Mon}_{au}^e(C,A)\\to  KLT(C,A)^{++}\n$$\ndefined by $\\phi\\mapsto ([\\phi], \\phi_{T})$ is bijective.\n\n\n\n\\end{thm}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe origins of $C^*$-theory and particularly the theory of universal\n$C^*$-algebras generated by operators that satisfy prescribed\nrelations go back to the work of W. Heisenberg, M. Bohr and P. Jordan\non matrix formulation of quantum mechanics, and among the most\nstimulating examples are algebras generated by anti-commutation\nrelations and canonical commutation relations (in the Weyl form). The\ngreat advantage of relations of CAR and CCR type is \\emph{uniqueness\n  of representation}. Namely, due to the celebrated Slawny's theorem,\nsee e.g. \\cite{EL}, the $C^*$-algebras generated by such relations are\ndefined uniquely up to isomorphisms preserving the generators and\nrelations. This \\emph{uniqueness property} is not only a strong\nmathematical tool but also has a significant physical meaning -- if we\nhad no such uniqueness, \\emph{different representations would yield\n  different physics}.\n\nThe aim of the present note is to advertise a program of developing a\ngeneral approach to investigation of uniqueness property and related\nproblems based on exploring the symmetries of relations.  We focus\nhere, as a first attempt, on circular symmetries and propose a\ntwo-step method of investigation universal $C^*$-algebra $C^*(\\mathcal G,\\mathcal R)$\ngenerated by a set of generators $\\mathcal G$ subject to relations $\\mathcal R$ which\ncould be schematically presented as follows:\n$$ \\xymatrix{\\underset{\\text{relations,   circle\n      action}}{(\\mathcal{G},\\mathcal{R}, \\{\\gamma_\\lambda\\}_{\\lambda\n      \\in \\mathbb T})} \\ar[r]^{\\text{step }1} &\\underset{ \\substack{\\text{Hilbert\n      bimodule}\\\\ \\text{(reversible dynamics)}}}{(\\mathcal B_0,\\mathcal B_1)}\n  \\ar[r]^{\\text{step }2\\,\\,\\,\\,}  &\\underset{ \\text{universal\n    }C^*\\text{-algebra}}{C^*(\\mathcal{G},\\mathcal{R})=\\mathcal B_0\\rtimes_{\\mathcal B_1}\n    \\mathbb Z}\n}\n$$\n-- we fix a circle gauge action $\\gamma=\\{\\gamma\\}_{\\lambda \\in \\mathbb T}$\non $C^*(\\mathcal{G},\\mathcal{R})$ which is induced by a circular\nsymmetry in $(\\mathcal{G},\\mathcal{R})$; in the first step we\nassociate to $\\gamma$ a non-commutative reversible dynamical system\nwhich is realized via a Hilbert bimodule $(\\mathcal B_0,\\mathcal B_1)$, and in the\nsecond  step we use this system to determine the uniqueness property\nfor $C^*(\\mathcal{G},\\mathcal{R})$.\n\n\\section{Uniqueness property, universal $C^*$-algebras and gauge\n  actions}\n\nSuppose we are given an abstract set of generators $\\mathcal G$ and a set of\n$^*$-algebraic relations $\\mathcal R$ that we want to impose on\n$\\mathcal G$. Formally $\\mathcal G$ is a set and $\\mathcal R$ is a set consisting of certain\n$^*$-algebraic relations in a free non-unital $^*$-algebra $\\mathbb F$\ngenerated by $\\mathcal G$.  By a \\emph{representation} of the pair $(\\mathcal G,\\mathcal R)$\nwe mean a set of bounded operators $ \\pi=\\{\\pi(g)\\}_{g\\in \\mathcal G}\\subset\nL(H)$ on a Hilbert space $H$ satisfying the relations $\\mathcal R$, and\ndenote by $C^*(\\pi)$ the $C^*$-algebra generated by $\\pi(g)$, $g\\in\n\\mathcal G$.  At this very beginning one faces the following two fundamental\nproblems:\n\\begin{itemize}\n\\item[ 1.]  (\\emph{non-degeneracy problem}) Do there exists a\n  \\emph{faithful representation} of $(\\mathcal G,\\mathcal R)$, i.e. a representation\n  $\\{\\pi(g)\\}_{g\\in \\mathcal G}$ of $(\\mathcal G,\\mathcal R)$ such that $\\pi(g)\\neq 0$ for\n  all $g\\in \\mathcal G$?\n\\item[2.]  (\\emph{uniqueness problem}) If one has two different\n  faithful representation of $(\\mathcal G,\\mathcal R)$, do they generate isomorphic\n  $C^*$-algebras? More precisely, does for any two faithful\n  representations $\\pi_1$, $\\pi_2$ of $(\\mathcal G,\\mathcal R)$ the mapping\n$$\n\\pi_1(g) \\longmapsto \\pi_2(g), \\qquad g\\in \\mathcal G,\n$$\nextends to the (necessarily unique) isomorphism $C^*(\\pi_1)\\cong\nC^*(\\pi_2)$?\n  \n\\end{itemize}\nThe first problem is important and interesting in its own rights, see\n\\cite{kwa-leb1}, \\cite{kwa-doplicher}, however here  we would like  to focus on the second problem and thus \nthroughout we   assume that all the pairs $(\\mathcal G,\\mathcal R)$ under consideration are\nnon-degenerate. We say that\n$(\\mathcal G,\\mathcal R)$ possess \\emph{uniqueness property} if the answer to\nquestion 2 is positive.\n\nAny representation $\\pi$ of $(\\mathcal G,\\mathcal R)$ extends uniquely to a\n$^*$-homomorphism, also denoted by $\\pi$, from $\\mathbb F$ into $L(H)$.\nThe pair $(\\mathcal G,\\mathcal R)$ is said to be \\emph{admissible} if the function\n$|||\\cdot|||:\\mathbb F\\to [0,\\infty]$ given by\n$$\n|||w|||=\\sup\\{\\|\\pi(w)\\|: \\pi \\text{ is a representation of } (\\mathcal G,\\mathcal R)\\}\n$$\nis finite. Plainly, admissibility is a necessary condition for\nuniqueness property and therefore we make it our another standing\nassumption. Then the function $|||\\cdot|||:\\mathbb F\\to [0,\\infty)$ is a\n$C^*$-seminorm on $\\mathbb F$ and its kernel\n$$\n\\mathbb I:=\\{w\\in \\mathbb F: |||w|||=0\\}\n$$\nis a self-adjoint ideal in $\\mathbb F$ -- it is the smallest self-adjoint\nideal in $\\mathbb F$ such that the relations $\\mathcal R$ become valid in the\nquotient $\\mathbb F\/\\mathbb I$.  We\npu\n$$\n C^*(\\mathcal G,\\mathcal R) := \\overline{\\mathbb F\/\\mathbb I}^{|||\\cdot|||}\n$$ \nand call it a \\emph{universal $C^*$-algebra} generated by $\\mathcal G$ subject\nto relations $\\mathcal R$, cf. \\cite{blackadar}. $C^*$-algebra $C^*(\\mathcal G,\\mathcal R)$\nis characterized by the property that any representation of $(\\mathcal G,\\mathcal R)$\nextends uniquely to a representation of $C^*(\\mathcal G,\\mathcal R)$ and all\nrepresentations of $C^*(\\mathcal G,\\mathcal R)$ arise in that manner.  In particular,\n$(\\mathcal G,\\mathcal R)$ possess uniqueness property if and only if any faithful\nrepresentation of $(\\mathcal G,\\mathcal R)$ extends to a faithful representation of\n$C^*(\\mathcal G,\\mathcal R)$.\n\n\\section{Gauge actions -- exploring the symmetries in the relations}\nWe would like to identify the uniqueness property of $(\\mathcal G,\\mathcal R)$ by\nlooking at the symmetries in $(\\mathcal G,\\mathcal R)$. In order to formalize this we use\na natural torus action $\\{\\gamma_{\\lambda}\\}_{\\lambda\\in\\mathbb T^\\mathcal G}$ on\n$\\mathbb F$ determined by the formula\n$$\n\\gamma_\\lambda(g)=\\lambda_g\\, g,\\qquad \\text{ for }\\,\\, g\\in \\mathcal G\n\\,\\,\\text{ and }\\,\\, \\lambda=\\{\\lambda_h\\}_{h\\in \\mathcal G}\\in \\mathbb T^\\mathcal G\n$$\nwhere $\\mathbb T=\\{z\\in \\mathbb C:|z|=1\\}$ is a unit circle.  A closed subgroup\n$G\\subset \\mathbb T^\\mathcal G$ may be considered as a \\emph{group of symmetries in\n  the pair $(\\mathcal G,\\mathcal R)$} if the restricted action\n$\\gamma=\\{\\gamma_{\\lambda}\\}_{\\lambda\\in G}$ leaves invariant the\nideal $\\mathbb I$. Any such group gives rise to a point-wisely continuous\ngroup action on $C^*(\\mathcal G,\\mathcal R)$ and actions that arise in that manner\nare called \\emph{gauge actions}.\n\\\\\nLet us from now on consider the case when $G\\cong \\mathbb T$, that is we have\na circle gauge action $\\gamma=\\{\\gamma_{\\lambda}\\}_{\\lambda\\in \\mathbb T}$ on\n$C^*(\\mathcal G,\\mathcal R)$. Then for  each $n\\in \\mathbb Z$ the formula\n$$\n\\mathcal E_n(b) :=\\int_{\\mathbb T}\\gamma_\\lambda(b)\\lambda^{-n} \\, d\\lambda  \n$$ \ndefines a projection $\\mathcal E_n:C^*(\\mathcal G,\\mathcal R)\\to C^*(\\mathcal G,\\mathcal R)$, called\n\\emph{$n$-th spectral projection}, onto the subspace\n$$\n\\mathcal B_n:=\\{b\\in C^*(\\mathcal G,\\mathcal R): \\gamma_\\lambda(b)=\\lambda^n b\\} \n$$\ncalled \\emph{$n$-th spectral subspace} for $\\gamma$,\ncf. e.g. \\cite{exel1}. Spectral subspaces specify a $\\mathbb Z$-gradation on\n$C^*(\\mathcal G,\\mathcal R)$. Namely, $\\bigoplus_{n\\in\\mathbb Z}\\mathcal B_n$ is dense in\n$C^*(\\mathcal G,\\mathcal R)$, and\n\\begin{equation}\\label{z-gradation relations}\n\\mathcal B_n\\mathcal B_m\\subset \\mathcal B_{n+m},\\,\\,\\, \\mathcal B_n^*=\\mathcal B_{-n} \\,\\,\\text{ for all }\\,\\, n,m \\in \\mathbb Z.\n\\end{equation} \nIn particular, $\\mathcal B_0$ is a $C^*$-algebra -- the fixed point algebra\nfor $\\gamma$, and $\\mathcal E_0:\\mathcal B\\to \\mathcal B_0$ is a conditional expectation.  A\ncircle action on a $C^*$-algebra $\\mathcal B$ is called \\emph{semi-saturated}\n\\cite{exel1} if $\\mathcal B$ is generated as a $C^*$-algebra by its first and\nzeroth spectral subspaces. We note that  every continuous group endomorphism\nof $\\mathbb T$ is of the form $\\lambda\\mapsto \\lambda^n$, for certain\n$n\\in\\mathbb Z$, and hence  it follows that $\\mathcal G\\subset \\cup_{n\\in \\mathbb Z}\\mathcal B_n$. In\nparticular, we have\n\n\\begin{lem} The circle gauge action\n  $\\gamma=\\{\\gamma_{\\lambda}\\}_{\\lambda\\in \\mathbb T}$ on $C^*(\\mathcal G,\\mathcal R)$ is\n  semi-saturated, that is $C^*(\\mathcal G,\\mathcal R)=C^*(\\mathcal B_0,\\mathcal B_1)$ if and only if\n  $\\mathcal G=\\mathcal G_0\\cup \\mathcal G_1$ for some disjoint sets $\\mathcal G_0$, $\\mathcal G_1$ and $\n  \\gamma_\\lambda(g_0)= g_0$, $\\gamma_\\lambda(g_1)= \\lambda g_1$, for\n  all $g_i\\in \\mathcal G_i$.\n\\end{lem}\nWe introduce an important necessary condition for $(\\mathcal G,\\mathcal R)$ to\npossess uniqueness property.\n\\begin{prop}\\label{gauge uniqueness theorem} The following conditions\n  are equivalent:\n  \\begin{itemize}\n  \\item[i)] each faithful representation of $(\\mathcal G,\\mathcal R)$ give rise to a\n    faithful representation of the fixed-point algebra $\\mathcal B_0$.\n  \\item[ii)] each faithful representation $\\pi$ of $(\\mathcal G,\\mathcal R)$ give\n    rise to a faithful representation of $C^*(\\mathcal G,\\mathcal R)$ if and only if\n    there is a circle action $\\beta$ on $C^*(\\pi)$ such that\n$$\n\\beta_z(\\pi(g))=\\pi(\\gamma_z(g)), \\qquad g\\in \\mathcal G.\n$$\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\n  i) $\\Longrightarrow$ ii). It suffices to apply the gauge invariance\n  uniqueness for circle actions, see e.g.  \\cite[2.9]{exel1} or\n  \\cite[4.2]{dr}. ii) $\\Longrightarrow$ i). Assume that $\\pi$ is a\n  faithful representation of $(\\mathcal G,\\mathcal R)$ such that its extension is not\n  faithful on $\\mathcal B_0$.  The spaces $\\{\\pi(\\mathcal B_n)\\}_{n\\in \\mathbb Z}$ form a\n  $\\mathbb Z$-graded $C^*$-algebra and thus by \\cite[4.2]{dr}, there is a\n  (unique) $C^*$-norm $\\|\\cdot\\|_\\beta$ on $\\bigoplus_{n\\in\n    \\mathbb Z}\\pi(\\mathcal B_n)$ such that the circle action $\\beta$ on\n  $\\bigoplus_{n\\in \\mathbb Z}\\pi(\\mathcal B_n)$ established by gradation extends onto\n  the $C^*$-algebra $\\mathcal B=\\overline{\\bigoplus_{n\\in\n      \\mathbb Z}\\pi(\\mathcal B_n)}^{||\\cdot||_\\beta}$. Composing $\\pi$ with the\n  embedding $\\bigoplus_{n\\in \\mathbb Z}\\pi(\\mathcal B_n)\\subset \\mathcal B$ one gets a\n  faithful representation $\\pi'$ of $(\\mathcal G,\\mathcal R)$ which is\n  gauge-invariant but not faithful on $C^*(\\mathcal G,\\mathcal R)$.\n\\end{proof}\nIn the literature the statements showing that the condition ii) in\nProposition \\ref{gauge uniqueness theorem} holds are often\ncalled \\emph{gauge-invariance uniqueness theorems} and therefore we shall\nsay that the triple $(\\mathcal G,\\mathcal R, \\gamma)$ has the \\emph{gauge-invariance\n  uniqueness property} if each faithful representation of $(\\mathcal G,\\mathcal R)$\ngive rise to a faithful representation of the fixed-point algebra\n$\\mathcal B_0$. In particular, this always holds for triples $(\\mathcal G,\\mathcal R,\n\\gamma)$ such that $C^*(\\mathcal G,\\mathcal R)$ can be\nmodeled as relative Cuntz-Pimsner algebra, see \\cite[Sect. 9]{kwa-doplicher}\nand sources cited there.\n\n\\section{From relations to Hilbert bimodules}\nLet us fix a pair $(\\mathcal G,\\mathcal R)$ with a circle gauge action\n$\\gamma=\\{\\gamma_{\\lambda}\\}_{\\lambda\\in \\mathbb T}$.\nIt follows from \\eqref{z-gradation relations} that $\\mathcal B_1$ can be\nnaturally viewed as a \\emph{Hilbert bimodule} over $\\mathcal B_0$, in the\nsense introduced in \\cite[1.8]{BMS}.  Namely, $\\mathcal B_1$ is a\n$\\mathcal B_0$-bimodule with bimodule operations inherited from $C^*(\\mathcal G,\\mathcal R)$\nand additionally is equipped with two $\\mathcal B_0$-valued inner products\n$$\n \\langle a,b\\rangle_{R}:=a^*b, \\qquad  {_{L}\\langle a,b\\rangle}:=ab^*  \n$$\nthat satisfy the so-called imprimitivtiy condition: $ a\\cdot \\langle\nb,c\\rangle_{R}= {_{L}\\langle a,b\\rangle}\\cdot c = ab^*c, $ for all\n$a,b,c\\in \\mathcal B_1$. %\nThus we can consider \\emph{crossed product} $\\mathcal B_1\\rtimes_{\\mathcal B_0} \\mathbb Z$ of\n$\\mathcal B_0$ by \\emph{the Hilbert bimodule} $\\mathcal B_1$ constructed in\n\\cite{aee}, which could be alternatively defined as the universal\n$C^*$-algebra:\n $$\n \\mathcal B_1\\rtimes_{\\mathcal B_0} \\mathbb Z =C^*(\\mathcal G_\\gamma, \\mathcal R_\\gamma)\n $$\n where $\\mathcal G_\\gamma=\\mathcal B_0\\cup \\mathcal B_1$ and $\\mathcal R_{\\gamma}$ consists of all\n algebraic relations in the Hilbert bimodule $(\\mathcal B_0,\\mathcal B_1)$.\n \\begin{prop}\\label{takie tam prr} We have a natural embedding $\n   \\mathcal B_1\\rtimes_{\\mathcal B_0} \\mathbb Z\\hookrightarrow C^*(\\mathcal G,\\mathcal R) $ which is an\n   isomorphism if and only if $\\gamma$ is semi-saturated.  Moreover,\n   if $\\gamma$ is semi-saturated, then the following conditions are\n   equivalent:\n   \\begin{itemize}\n   \\item[i)] $(\\mathcal G,\\mathcal R)$ possess uniqueness property\n   \\item[ii)] $(\\mathcal G,\\mathcal R,\\gamma)$ has gauge-invariance uniqueness\n     property and $(\\mathcal G_\\gamma,\\mathcal R_\\gamma)$ possess uniqueness property\n   \\end{itemize}\n \\end{prop}\n \\begin{proof}\n   Since the homomorphism $ \\mathcal B_1\\rtimes_{\\mathcal B_0} \\mathbb Z\\mapsto C^*(\\mathcal G,\\mathcal R) $\n   is gauge-invariant and injective on $\\mathcal B_0$ it is injective onto the\n   whole $ \\mathcal B_1\\rtimes_{\\mathcal B_0} \\mathbb Z$ by \\cite[2.9]{exel1}. The rest, in view of\n   Proposition \\ref{gauge uniqueness theorem}, is clear.\n\\end{proof}\nThe Hilbert bimodule $(\\mathcal B_0,\\mathcal B_1)$ is an imprimitivity bimodule\n(called also Morita-Rieffel equivalence bimodule), see \\cite{morita},\nif and only if $\\overline{\\mathcal B_1^*\\mathcal B_1}= \\mathcal B_0$ and\n$\\overline{\\mathcal B_1\\mathcal B_1^*}=\\mathcal B_0$. In general, $\\overline{\\mathcal B_1^*\\mathcal B_1}$ and\n$\\overline{\\mathcal B_1\\mathcal B_1^*}$ are non-trivial ideals in $\\mathcal B_0$ and we may\ntreat $\\mathcal B_1$ as a\n$\\overline{\\mathcal B_1\\mathcal B_1^*}-\\overline{\\mathcal B_1^*\\mathcal B_1}$-imprimitivity\nbimodule. This means, cf. \\cite[Cor. 3.33]{morita}, that the induced\nrepresentation functor\n$$\n \\widehat h= \\mathcal B_1\\dashind\n$$\nis a homeomorphism $\\widehat h:\\overline{\\mathcal B_1^*\\mathcal B_1}\\to \\overline{\\mathcal B_1\\mathcal B_1^*}$\nbetween the spectra of $\\overline{\\mathcal B_1^*\\mathcal B_1}$ and\n$\\overline{\\mathcal B_1\\mathcal B_1^*}$. Treating these spectra %\nas open subsets of the spectrum $\\widehat{\\mathcal B}_0$ of $\\mathcal B_0$ we may\ntreat $\\widehat h$ as a partial homeomorphism of $\\widehat{\\mathcal B}_0$.  We shall\nsay that $(\\widehat{\\mathcal B},\\widehat h)$ is a \\emph{partial dynamical system dual\n  to the bimodule} $(\\mathcal B_0,\\mathcal B_1)$. Partial homeomorphism $\\widehat h$ is said\nto be \\emph{topologically free} if for each $n\\in N$ the set of points\nin $\\widehat{\\mathcal B_0}$ for which the equality $\\widehat h^n(x)=x$ (makes sense\nand) holds has empty interior.\n\\begin{thm}[main result]\\label{main result}\n  Suppose that the partial homeomorphism $\\widehat h=B_1\\dashind$ is\n  topologically free.  Then the pair $(\\mathcal G_\\gamma,\\mathcal R_\\gamma)$ possess\n  uniqueness property and in particular\n\\begin{itemize}\n\\item[i)] if $(\\mathcal G,\\mathcal R,\\gamma)$ possess gauge-invariance uniqueness\n  property, then any faithful representation of $(\\mathcal G,\\mathcal R)$ extends to\n  the faithful representation of $ \\mathcal B_1\\rtimes_{\\mathcal B_0} \\mathbb Z\\subset\n  C^*(\\mathcal G,\\mathcal R)$.\n\\item[ii)] if $\\gamma$ is semi-saturated and $(\\mathcal G,\\mathcal R,\\gamma)$ possess\n  gauge-invariance uniqueness property, then $(\\mathcal G,\\mathcal R)$ possess\n  uniqueness property.\n\\end{itemize}\n\\end{thm}\n\\begin{proof}\n  Apply the main result of \\cite{kwa} and Proposition \\ref{takie tam\n    prr}.\n\\end{proof}\n\n\\section{Applications to crossed products and Cuntz-Krieger algebras}\n\nWe show that our main result is a generalization of the so called\nisomorphisms theorem for crossed products by automorphisms (see, for\ninstance, \\cite[pp. 225, 226]{Anton_Lebed} for a brief survey of such\nresults) by applying it to a  crossed  product by an endomorphism which is\nconsidered to be one of the most successful constructions of this\nsort, see \\cite{Ant-Bakht-Leb} and sources cited there. In particular,\nwe shall use this crossed product to identify the uniqueness property\nfor Cuntz-Krieger algebras.\n\n\\subsection{Crossed products by monomorphisms with hereditary range} %\nLet $\\alpha:\\mathcal A\\to\\mathcal A$ be a monomorphism of a unital $C^*$-algebra $\\mathcal A$.\nLet $\\mathcal G=\\mathcal A\\cup \\{S\\}$ and let $\\mathcal R$ consists of all $^*$-algebraic\nrelations in $\\mathcal A$ plus the covariance relations\n\\begin{equation}\\label{covariance conditions for monomorphism}\n  S a S^*= \\alpha(a), \\qquad S^* S =1, \\qquad  a\\in \\mathcal A.\n\\end{equation}\nThen $C^*(\\mathcal G,\\mathcal R)\\cong \\mathcal A\\rtimes_\\alpha \\mathbb N$ is the crossed product of\n$\\mathcal A$ by $\\alpha$, which is equipped with a semi-saturated circle gauge\naction: $ \\gamma_\\lambda(a) =a$, $\\gamma_\\lambda(S)=\\lambda S$, $a\\in\n\\mathcal A$.  Let us additionally assume that $\\alpha(\\mathcal A)$ is a hereditary\nsubalgebra of $\\mathcal A$. This is equivalent to $\\alpha(\\mathcal A)=\\alpha(1)\\mathcal A\\alpha(1)$.\nThen  we have $S^*\\mathcal A S\\subset \\mathcal A$  since for any $a\\in \\mathcal A$\nthere is $b\\in \\mathcal A$ such that $\\alpha(b)=\\alpha(1)a \\alpha(1)$ and therefore\n$$\n  S^* aS =S^*\\alpha(1)a \\alpha(1) S= S^* \\alpha(b) S=S^*S b S^*S=b\\in \\mathcal A.\n$$ \nHence on one hand $\\mathcal A=\\mathcal B_0$ is the fixed point algebra for $\\gamma$ and $\\mathcal B_1=\n\\mathcal B_0S$ is the first spectral subspace.  On the other hand the spectrum\nof the hereditary subalgebra $\\alpha(\\mathcal A)$ may be naturally identified\nwith an open subset of $\\widehat{\\mathcal A}$, see e.g.\n\\cite[Thm~5.5.5]{Murphy}, and then the dual\n$\\widehat{\\alpha}:\\widehat{\\alpha(\\mathcal A)}\\to \\widehat{\\mathcal A}$ to the isomorphism\n$\\alpha:\\mathcal A\\to \\alpha(\\mathcal A)$ becomes a partial homeomorphism of\n$\\widehat{\\mathcal A}$. Under this identification one gets\n$$\n \\widehat{\\alpha}=\\mathcal B_1\\dashind\n$$\nand hence if the partial system $(\\widehat{\\mathcal A},\\widehat{\\alpha})$ dual to\n$(\\mathcal A,\\alpha)$ is topologically free, then $(\\mathcal G,\\mathcal R)$ possess uniqueness\nproperty.\n\n\\subsection{Cuntz-Krieger algebras} \nLet $\\mathcal G=\\{S_i: i=1,...,n\\}$, where $n\\geq 2$, and let $\\mathcal R$ consists\nof the Cuntz-Krieger relations\n\\begin{equation}\\label{Cuntz-Krieger algebras}\n  S_i^*S_i =\\sum_{j=1}^n A(i,j) S_jS_j^*,  \\qquad S_i^* S_k=\n  \\delta_{i,k} S_i^*S_i, \\qquad i,k=1 ,...,n,\n\\end{equation} \nwhere $\\{A(i,j)\\}$ is a given $n\\times n$ zero-one matrix such that\nevery row and every column of $A$ is non-zero, and $\\delta_{i,j}$ is\nKronecker symbol.  Then $C^*(\\mathcal G,\\mathcal R)$ is the Cuntz-Krieger algebra\n$\\mathcal O_A$ and the celebrated Cuntz-Krieger uniqueness theorem,\ncf. \\cite[Thm. 2.13]{CK}, states that the pair $(\\mathcal G,\\mathcal R)$ possess\nuniqueness property if and only if the so called \\emph{condition (I)}\nholds:\n$$\n\\textbf{(I) } \\,\\,\\,\\,\\, \\begin{array}{l} \\text{the space }\n  X_A:=\\{(x_1,x_2,....)\\in\\{1,...,n\\}^\\mathbb N: A(x_k,x_{k+1})=1\\} \\\\\n  \\text{has no isolated points (considered with the product\n    topology)}\\end{array}\n$$\nWe may recover this result applying our method to the standard circle\ngauge action on $\\mathcal O_A$ determined by equations $ \\gamma_\\lambda(S_i)\n=\\lambda S_i$, $i=1,...,n$. Indeed, the fixed point $C^*$-algebra for\n$\\gamma$ coincides with the so called AF-core\n$$\n \\mathcal F_A=\\clsp\\{S_\\mu S^*_\\nu : \\vert\\mu\\vert=\\vert\\nu\\vert =k, \\, k=\\,1,\\,\\dotsc\\}\n$$\nwhere for a multiindex $\\mu = (i_1,\\dots,i_k)$, with $i_j \\in 1,...,n$,\nwe denote by $\\vert \\mu \\vert$ the length $k$ of $\\mu$ and write\n\\,$S_\\mu = S_{i_1}S_{i_2}\\dotsm S_{i_k}$. Moreover, any faithful\nrepresentation of the Cuntz-Krieger relations \\eqref{Cuntz-Krieger\n  algebras} yields a faithful representation of $\\mathcal F_A$, that is\n$(\\mathcal G,\\mathcal R,\\gamma)$ possess gauge-invariance uniqueness\nproperty. Following \\cite{Ant-Bakht-Leb} we put\n$$\nS:=\\sum_{i,j}\\frac{1}{ \\sqrt{n_j}}\\,\\, S_i P_j\n$$\nwhere $n_j=\\sum_{i=1}^n A(i,j)$ and $P_j=S_jS_j^*$,\n$j=1,...,n$.\nA routine computation shows that $S\\mathcal F_A S^*\\subset \\mathcal F_A$, $S^*\\mathcal F_A\nS\\subset \\mathcal F_A$ and $S^*S=1$ ($S$ is an isometry). Hence the mapping\n$\\mathcal F_A\\ni a \\mapsto \\alpha(a):=SaS^* \\in \\mathcal F_A$ is a monomorphism with a\nhereditary range. It\nis uniquely determined by the formula\n\\begin{equation}\\label{endomorphism action on F_A}\n  \\alpha \\Big(S_{i_2\\mu} S^*_{j_2\\nu}\\Big)= \\frac{1}{\n    \\sqrt{n_{i_2}n_{j_2}}} \\sum_{i,j=1}^n S_{i\\,i_2\\mu}\n  S^*_{j\\,j_2\\nu}.\n\\end{equation}\nFrom the construction any representation of relations\n\\eqref{Cuntz-Krieger algebras} yields a representation of\n$(\\mathcal F_A,\\alpha)$ as introduced in the previous subsection. Conversely, if\n$S$ satisfies \\eqref{covariance conditions for monomorphism} where\n$\\mathcal A=\\mathcal F_A$, then one gets representation of \\eqref{Cuntz-Krieger\n  algebras} by putting $S_i:= \\sum_{j=1}^n A(i,j)\\sqrt{n_j} P_i S\nP_j$. Thus we have a natural isomorphism\n$$\n\\mathcal O_A\\cong \\mathcal F_A \\rtimes_\\alpha \\mathbb N\n$$ \nunder which the introduced gauge actions coincide. Hence we may\nidentify the partial dynamical system dual to the Hilbert bimodule\n$(\\mathcal B_1, \\mathcal B_0)$ where $\\mathcal B_0=\\mathcal F_A$ and $\\mathcal B_1=\\mathcal F_AS$ with the partial\ndynamical system $(\\widehat{\\mathcal F_A},\\widehat \\alpha)$ dual to $(\\mathcal F_A,\\alpha)$, as\nintroduced in the previous subsection.\n\\\\\nIn order to identify the the topological freeness of $\\widehat \\alpha$ we define\n$\\pi_\\mu\\in \\widehat{\\mathcal A}$ for any infinite path $\\mu=(i_1,i_2,\\dots)$,\n$A(i_j,i_{j+1})=1$, $j\\in\\mathbb N$, to be the the GNS-representation\nassociated to the pure state $\\omega_\\mu:\\mathcal F_A\\to \\mathbb C$ determined by\nthe formula\n\\begin{equation}\\label{product states equation}\n  \\omega_\\mu(S_\\nu S_\\eta^*)=\n\\begin{cases}\n  1 & \\nu=\\eta=(\\mu_1,...,\\mu_n) \\\\\n  0 & \\text{otherwise}\n\\end{cases}\\qquad \\text{for}\\quad \\vert\\nu\\vert=\\vert\\eta\\vert =n.\n\\end{equation}\nUsing description of the ideal structure in $\\mathcal F_A$ in terms of\nBratteli diagrams \\cite{Bratteli}, similarly as in \\cite{kwa}, one can\nshow that representations $\\pi_\\mu$ form a dense subset of\n$\\widehat{\\mathcal F_A}$ and\n$$\n\\widehat{\\alpha}(\\pi_{(\\mu_1,\\mu_{2},\\mu_3,...)})=\\pi_{(\\mu_{2},\\mu_3,...)},\\qquad\\text{for\n  any }(\\mu_1,\\mu_{2},\\mu_3,...).\n$$\nIn particular, it follows that \\emph{topological freeness of $\\widehat \\alpha$ is\n  equivalent to condition $(I)$}.  Accordingly\n\\begin{quote}\n  \\emph{our main result, Theorem \\ref{main result}, when applied to\n    Cuntz-Krieger relations is equivalent to the Cuntz-Krieger\n    uniqueness theorem.}\n\\end{quote}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the Test Cover problem defined below we are given a set $V=\\{v_1,\\ldots, v_n\\}$ of element called {\\em vertices} and\na family  $\\mathcal{E}=\\{E_1,\\dots,E_m\\}$ of distinct subsets of $V$ called {\\em tests}. We say that a test\n$E_q$ {\\em separates} a pair $v_i,v_j$ of vertices if $|\\{v_i,v_j\\}\\cap E_q|=1.$\nA subset $\\mathcal{T}$ of $\\mathcal{E}$ is called a {\\em test cover}\nif for every pair of distinct vertices $v_i,v_j$ there exists a test $T\\in\\mathcal{T}$ separating them.\n\n\n\\begin{center}\n\\fbox{~\\begin{minipage}{0.9\\textwidth}\n{\\sc TestCover}($p$)\n\n\\textbf{Input:} A set  $V=\\{v_1,\\ldots, v_n\\}$ of vertices, a family  $\\mathcal{E}=\\{E_1,\\dots,E_m\\}$ of tests and an integer $p$.\n\n\\textbf{Task:} Decide whether there exists a test cover with at most $p$ tests.\n\\end{minipage}~}\n\\end{center}\n\n\nThe (optimization version of) Test Cover problem arises naturally in the following general setting of\nidentification problems: Given a set of items (i.e., vertices) and a set\nof binary attributes that may or may not occur in each item, the aim\nis to find a minimum-size subset of attributes (a minimum test cover) such that each\nitems can be uniquely identified from the information on the subset\nof attributes it contains.\nThe Test Cover problem arises in fault analysis, medical diagnostics, pattern recognition, and\nbiological  identification (see, e.g., \\cite{HalHalRav01,HalMinRav01,MorSha85}).\n\nThe {\\sc TestCover}($p$) problem is NP-hard, as was shown by Garey and Johnson \\cite{GarJoh79}. Moreover,\n{\\sc TestCover}($p$) is APX-hard \\cite{HalHalRav01}. There is an $O(\\log n)$-approximation algorithm for the problem \\cite{MorSha85}\nand there is no $o(\\log n)$-approximation algorithm unless P=NP \\cite{HalHalRav01}.\n\n\nThe generic parameterized version \\tcp\\space of {\\sc TestCover}($p$) is defined as follows.\n\n\\begin{center}\n\\fbox{~\\begin{minipage}{0.9\\textwidth}\n{\\sc TestCover}$(p,k)$\n\n\\textbf{Input:} A set $V=\\{v_1,\\dots,v_n\\}$, a family $\\mathcal{E}=\\{E_1,\\dots,E_m\\}$ of subsets of $V$ and two integers $p$ and $k$.\n\n\\textbf{Parameter:} $k$.\n\n\\textbf{Task:} Decide whether there exists a test cover with at most $p$ tests.\n\\end{minipage}~}\n\\end{center}\n\n\nIn this paper, we study two parameterizations of {\\sc TestCover}: the standard parameterization {\\sc TestCover}$(k,k)$ and its dual\\footnote{See Definition \\ref{def:dual} for a formal notion of parameterized duality.} {\\sc TestCover}$(n-k,k)$.\nBoth parameterizations are known to be fixed-parameter tractable \\cite{CroGutJonSauYeo}.\nFixed-parameter tractability of {\\sc TestCover}$(k,k)$ is easy to see using the fact that there is no test cover of size less than $\\lceil \\log n\\rceil$ \\cite{HalHalRav01}. Indeed, if $k< \\log n$ then the corresponding instance of {\\sc TestCover}$(k,k)$ is a {\\sc No}-instance, and if $k\\ge \\log n$ then $n\\le 2^k$ and so $m\\le 2^{2^k}$ allowing us to solve {\\sc TestCover}$(k,k)$ by a brute force fixed-parameter algorithm.\nThe proof in \\cite{CroGutJonSauYeo} that {\\sc TestCover}$(n-k,k)$ is fixed-parameter tractable is much harder.\n\nWe first prove that {\\sc TestCover}$(k,k)$ does not admit a polynomial kernel unless $NP\\subseteq coNP\/poly$. The latter is deemed unlikely as it is known to imply a collapse of the polynomial hierarchy to its third level \\cite{Yap}. Note that our first result solves an open problem \\cite{Fer,Guo}. Our proof uses the cross-composition method recently introduced by Bodlaender et al. \\cite{BodJanKra}.\n\nWe show that {\\sc TestCover}$(k,k)$ does admit a polynomial kernel if the size of each test is bounded by a constant. This constraint is used in some practical applications of {\\sc TestCover} \\cite{HalHalRav01}.  We prove that {\\sc TestCover}$(n-k,k)$ does not admit a polynomial kernel unless $NP\\subseteq coNP\/poly$. This result follows from our first result and a general result on nonexistence of polynomial kernels in dual parameterized problems also obtained in our paper.\nWe use the notion of dual introduced by Chen et al. \\cite{CheFerKanXia} as well as the cross-composition method.\n\nOur paper is organized as follows. In the rest of this section we will give a couple of simple, yet very useful, definitions on {\\sc TestCover}.\nIn the next section we give basics on parameterized algorithms and kernelization as well as on the cross-composition method. In Section \\ref{sec:res}, we prove all our results. In Section \\ref{sec:op} we pose some open problems.\n\nA {\\em partial test cover} $\\mathcal{T}'$ to an instance of \\tcp\\space is a subset of $\\mathcal{E}$ of cardinality at most $p$.\nWe say that $C\\subseteq V$ is a {\\em class} induced by a partial test cover $\\mathcal{T}'$, if $C$ is a maximal set such that there does not exist\na test in $\\mathcal{T}'$ which separates two vertices of $C$. Notice that the classes induced by every partial test cover form a partition of $V$ and every test cover induces $n$ classes. For a positive integer $p$, let $[p]=\\{1,2,\\ldots ,p\\}.$\n\n\\section{Fixed-Parameter Algorithms, Kernels and Cross-composition Method}\n\nA \\emph{parameterized problem} $P$ is a subset $P\\subseteq \\Sigma^* \\times\n\\mathbb{N}$ over a finite alphabet $\\Sigma$. $P$ is\n\\emph{fixed-parameter tractable} if the membership of an instance\n$(x,k)$ of $\\Sigma^* \\times \\mathbb{N}$ in $P$ can be decided by an algorithm of runtime\n$f(k)|x|^{O(1)}$, where $f$ is a function of the\n{\\em parameter} $k$ only (such an algorithm is a \\emph{fixed-parameter} algorithm)\n\\cite{DowneyFellows99,FlumGrohe06,Niedermeier06}.\nGiven a parameterized problem $P$,\na \\emph{kernelization of $P$} is a polynomial-time\nalgorithm that maps an instance $(x,k)$ to an instance $(x',k')$ (the\n\\emph{kernel}) such that (i)~$(x,k)\\in P$ if and only if\n$(x',k')\\in P$, (ii)~ $k'\\leq h(k)$, and (iii)~$|x'|\\leq g(k)$ for some\nfunctions $h$ and $g$. The function $g(k)$ is called the {\\em size} of the kernel.\nIt is well-known \\cite{DowneyFellows99,FlumGrohe06,Niedermeier06} that a decidable parameterized problem $P$ is fixed-parameter\ntractable if and only if it has a kernel. Polynomial-size kernels are of\nmain interest, due to applications \\cite{DowneyFellows99,FlumGrohe06,Niedermeier06}, but unfortunately many fixed-parameter tractable problems\nhave no such kernels unless  coNP$\\subseteq$NP\/poly, see, e.g., \\cite{BDFH09}.\\\\\n\nThe following two definitions and Theorem \\ref{res} were given by Bodlaender \\textit{et al.} \\cite{BodJanKra}.\n\n\\begin{defin}[Polynomial equivalence relation]\\label{polequ}\nAn equivalence relation $\\mathcal{R}$ on $\\Sigma^\\ast$ is called a {\\em polynomial equivalence relation} if the following two conditions hold:\n\\begin{itemize}\n   \\item{There is an algorithm that given two strings $x,y\\in\\Sigma^\\ast$ decides whether $x$ and $y$ belong to the same equivalence class\nin $(|x|+|y|)^{O(1)}$ time.}\n    \\item{For any finite set $S\\subseteq\\Sigma^\\ast$ the equivalence relation $\\mathcal{R}$ partitions the elements of $S$ into at most\n$(\\max_{x\\in S}|x|)^{O(1)}$ equivalence classes.}\n\\end{itemize}\n\\end{defin}\n\n\\begin{defin}[Cross-composition]\\label{crocom}\nLet $L\\subseteq\\Sigma^\\ast$ be a problem and let $Q\\subseteq\\Sigma^\\ast\\times\\mathbb{N}$ be a parameterized problem. We say that $L$\n{\\em cross-composes} into $Q$ if there is a polynomial equivalence relation $\\mathcal{R}$ and an algorithm which, given $t$ strings $x_1,\\dots,x_t$\nbelonging to the same equivalence class of $\\mathcal{R}$, computes an instance $(x^\\ast,k^\\ast)\\in\\Sigma^\\ast\\times\\mathbb{N}$ in time\npolynomial in $\\sum_{i=1}^t|x_i|$ such that:\n\\begin{itemize}\n   \\item {$(x^\\ast,k^\\ast)\\in Q$ if and only if $x_i\\in L$ for some $1\\leq i\\leq t$.}\n  \\item{$k^\\ast$ is bounded by a polynomial in $\\max_{i=1}^t|x_i|+\\log t$.}\n\\end{itemize}\n\n\\end{defin}\n\n\\begin{theo}\\label{res}\nIf some problem $L$ is $NP$-hard under Karp reductions and $L$ cross-composes into the\nparameterized problem $Q$ then there is no polynomial kernel for $Q$ unless $NP\\subseteq coNP\/poly$.\n\\end{theo}\n\n\\section{Results}\\label{sec:res}\n\n\\begin{theo}\\label{nopolker}\n\\tcpk\\space does not admit a polynomial kernel, unless $NP\\subseteq coNP\/poly$.\n\\end{theo}\n\\begin{proof}\nWe will use Theorem \\ref{res} to show the result, hence we need an $NP$-hard problem $L$ which cross-composes into \\tcpk.\nFor this purpose, it is possible to use \\tc.\nAn instance of this problem is a triple $(V,\\mathcal{E},p)$. We say that two triples $(V_1,\\mathcal{E}_1,p_1)$\nand $(V_2,\\mathcal{E}_2,p_2)$ are equivalent if $|V_1|=|V_2|$ and $p_1=p_2$.\n\nIt is not difficult to see that this defines a polynomial\nequivalence relation on \\tc\\space (see Definition \\ref{polequ}). To see that the second condition of Definition \\ref{polequ} holds, observe that the relation\nthus defined partitions a finite set $S$ into at most $O(n\\cdot m)$ equivalence classes, assuming that for every $(V,\\mathcal{E},p)\\in S$ we have\nthat $|V|\\leq n$ and $|\\mathcal{E}|\\leq m$ (since $p\\leq|\\mathcal{E}|$).\n\nConsider now $t$ instances $Q_1,\\dots,Q_t$ of \\tc, belonging to the same equivalence class. For every $i\\in[t]$,\nlet $V=\\{v_1,\\dots,v_n\\}$ be the set of vertices of $Q_i$, $\\mathcal{E}_i=\\{E_1^i,\\dots,E_{m_i}^i\\}$ its set of tests and\n$p$ be the upper bound on the size of the solution.\nLet $l=2\\lceil(\\log t)\/2\\rceil$.\nWe will construct an instance $Q^\\ast$ of \\tcpk\\space such that $Q^\\ast$ has a solution of size at most $k=2l+p$\nif and only if at least one of $Q_1,\\dots,Q_t$ has a solution of size at most $p$.\nThis will be enough to prove our result (see Definition \\ref{crocom} and Theorem \\ref{res}). \\\\\n\nThe vertex set of $Q^\\ast$ is defined in the following way:\n$$V(Q^\\ast)=\\{v_1,\\dots,v_n\\}\\cup(\\cup_{j=1}^{2l}\\{y_j,x_1^j,\\dots,x_p^j\\})\\cup\\{a_1,\\dots,a_l\\}.\n$$\nThe tests in $Q^\\ast$ are defined as follows. Consider some $i\\in[t]$ and let $(i_1,\\dots,i_l)$ be its binary representation\n(e.g. if $i=11$ and $l=6$, $(i_1,\\dots,i_l)=(1,1,0,1,0,0)$). For every $i$, consider the following family of sets $\\mathcal{S}_i$\n(where subscripts are taken modulo $k$):\n$$\\mathcal{S}_i=\\bigl\\{\\{x_{h}^1,x_{h+i_1}^2,x_{h}^3,x_{h+i_2}^4,x_{h}^5,x_{h+i_3}^6,\\dots,x_{h}^{2l-1},x_{h+i_l}^{2l}\\}:\nh=1,\\dots,p\\bigr\\}.\n$$\nDefine $\\mathcal{E}(Q^\\ast)=\\mathcal{E}^\\ast\\cup\\widetilde{\\mathcal{E}}_1\\cup\\dots\\cup\\widetilde{\\mathcal{E}}_t$, where\n$$\\mathcal{E}^\\ast=\\bigl\\{\\{a_j,y_{2j-1},x_1^{2j-1},x_2^{2j-1},x_3^{2j-1},\\dots,x_p^{2j-1}\\},\\{a_j,y_{2j},x_1^{2j},x_2^{2j},x_3^{2j}\n\\dots,x_p^{2j}\\}:j\\in[l]\\bigr\\}\n$$\nand\n$$\\widetilde{\\mathcal{E}}_i=\\{(S\\cup E_j^i):j\\in[m_i],S\\in\\mathcal{S}_i\\}.\n$$\nThis completes the definition of $Q^\\ast$.\n\nAssume that there is a solution to $Q^\\ast$ using at most $2l+p$ tests. Since $a_i$ and $y_{2i-1}$ need to be separated and\nsimilarly for $a_i$ and $y_{2i}$ for all $i$, all $2l$ tests in $\\mathcal{E}^\\ast$ must be used (because they are the only tests\nwhere $a_i$, $y_{2i-1}$ and $y_{2i}$ appear); notice also that they are enough to separate them.\nMoreover, as every set in $\\widetilde{\\mathcal{E}}_i$ (for all $i$) only intersects $\\mathcal{X}_i=\\{x_1^i,x_2^i,\\dots,x_p^i\\}$\nin one vertex and as all vertices in $\\mathcal{X}_i$ have to be separated from $y_i$, the tests not in $\\mathcal{E}^\\ast$ all have to contain\na distinct vertex from $\\mathcal{X}_i$\n\\footnote{Notice that this ensures that a solution to $Q^\\ast$ has \\textit{at least} $2l+p$ vertices. Therefore we are actually showing that\n$Q^\\ast$ has a solution of size exactly $4\\lceil(\\log t)\/2\\rceil+p$\nif and only if at least one of $Q_1,\\dots,Q_t$ has a solution of size at most $p$.}.\n\nConsider now a test $E$ in $\\widetilde{\\mathcal{E}}=\\bigcup_{j=1}^t\\widetilde{\\mathcal{E}}_j$. Its intersection with $\\mathcal{X}_i\n\\cup\\mathcal{X}_{i+1}$, for every $i\\in\\{1,3,\\dots,2l-1\\}$, contains only vertices which are in $S$ for some $S\\in\\bigcup_{j=1}^t\\mathcal{S}_j$.\n\nSuppose $S\\in\\mathcal{S}_r$ for some $r$ (this ensures that $E\\in\\widetilde{\\mathcal{E}}_r$)\nand suppose that it is equal to\n$$\\{x_{h}^1,x_{h+r_1}^2,\\dots,x_{h}^{2l-1},x_{h+r_l}^{2l}\\},$$\nwhere $(r_1,\\dots,r_l)$ is the binary representation of $r$ and $h\\in[p]$.\n\nThe intersection with $\\mathcal{X}_i\\cup\\mathcal{X}_{i+1}$ is given by the black\nvertices in next figure if $r_{\\frac{i+1}{2}}=0$:\n\n\\begin{equation*}\n\\begin{tikzpicture}\n\n\\node (pun1) at (3,0) {$\\dots$};\n\\node (pun2) at (6,0) {$\\dots$};\n\n\\node (pun3) at (3,-1) {$\\dots$};\n\\node (pun4) at (6,-1) {$\\dots$};\n\n\\tikzstyle{every node}=[draw,circle,fill=white,minimum size=4pt,inner sep=0pt]\n\n\\foreach \\number in {1,2}\n{\n\\node (N1 + \\number) at (\\number,0) [label=above:$x_\\number^i$] {};\n}\n\n\\node[fill=black] (N1quattro) at (4,0) [label=above:$x_{h}^i$] {};\n\\node (N1cinque) at (5,0) [label=above:$x_{h+1}^i$] {};\n\n\\node (N1sette) at (7,0) [label=above:$x_{p-1}^i$] {};\n\\node (N1otto) at (8,0) [label=above:$x_p^i$] {};\n\n\n\\foreach \\number in {1,2}\n{\n\\node (N2 + \\number) at (\\number,-1) [label=below:$x_\\number^{i+1}$] {};\n}\n\n\\node[fill=black] (N2quattro) at (4,-1) [label=below:$x_{h}^{i+1}$] {};\n\\node (N2cinque) at (5,-1) [label=below:$x_{h+1}^{i+1}$] {};\n\n\\node (N2sette) at (7,-1) [label=below:$x_{p-1}^{i+1}$] {};\n\\node (N2otto) at (8,-1) [label=below:$x_p^{i+1}$] {};\n\n\\end{tikzpicture}\n\\end{equation*}\n\nIf, on the contrary, $r_{\\frac{i+1}{2}}=1$, the intersection is given by the black vertices in next figure:\n\n\\begin{equation*}\n\\begin{tikzpicture}\n\n\\node (pun1) at (3,0) {$\\dots$};\n\\node (pun2) at (6,0) {$\\dots$};\n\n\\node (pun3) at (3,-1) {$\\dots$};\n\\node (pun4) at (6,-1) {$\\dots$};\n\n\\tikzstyle{every node}=[draw,circle,fill=white,minimum size=4pt,inner sep=0pt]\n\n\\foreach \\number in {1,2}\n{\n\\node (N1 + \\number) at (\\number,0) [label=above:$x_\\number^i$] {};\n}\n\n\\node[fill=black] (N1quattro) at (4,0) [label=above:$x_{h}^i$] {};\n\\node (N1cinque) at (5,0) [label=above:$x_{h+1}^i$] {};\n\n\\node (N1sette) at (7,0) [label=above:$x_{p-1}^i$] {};\n\\node (N1otto) at (8,0) [label=above:$x_p^i$] {};\n\n\n\\foreach \\number in {1,2}\n{\n\\node (N2 + \\number) at (\\number,-1) [label=below:$x_\\number^{i+1}$] {};\n}\n\n\\node (N2quattro) at (4,-1) [label=below:$x_{h}^{i+1}$] {};\n\\node[fill=black] (N2cinque) at (5,-1) [label=below:$x_{h+1}^{i+1}$] {};\n\n\\node (N2sette) at (7,-1) [label=below:$x_{p-1}^{i+1}$] {};\n\\node (N2otto) at (8,-1) [label=below:$x_p^{i+1}$] {};\n\n\\end{tikzpicture}\n\\end{equation*}\n\nThese are the only possible types of intersection.\n\nNotice that, if $E$ is part of the solution, in order to cover all vertices in $\\mathcal{X}_i\\cup\\mathcal{X}_{i+1}$ (for every $i$)\nit is possible to use only tests in $\\widetilde{\\mathcal{E}}_r$. Otherwise, if a test in $\\widetilde{\\mathcal{E}}_s$\nis used (with $r\\neq s$),\nthere exists $i^\\ast$ such that $r_{i^\\ast} \\neq s_{i^\\ast}$ and it is impossible to cover all vertices\nin $\\mathcal{X}_{i^\\ast}\\cup\\mathcal{X}_{i^\\ast+1}$.\nTherefore, if there is a solution to $Q^\\ast$ using $2l+p$ tests, then $2l$ tests are taken from $\\mathcal{E}^\\ast$ and $p$ tests from\n$\\widetilde{E}_r$ for some $r$. This ensures that $Q_r$ has a solution using at most $p$ tests. \\\\\n\nTo prove the opposite direction, assume that $Q_r$ has a solution of size at most $p$. Using all sets in $\\mathcal{E}^\\ast$ and carefully choosing\n$p$ sets from $\\widetilde{\\mathcal{E}}_r$\n\\footnote{The $p$ tests from $\\widetilde{\\mathcal{E}}_r$ have to contain the tests used in a solution to $Q_r$ and tests $S$ which are chosen\naccordingly to the binary representation of $r$, in order to cover all vertices in $\\bigcup_{j=1}^{2l}{\\mathcal{X}_j}$.},\nit is possible to obtain a solution to $Q^\\ast$ with at most $2l+p$ tests.\nThis completes the proof.\n\\end{proof}\n\n\\begin{defin}\nThe generic parameterized problem \\tcpr\\space is defined as \\tcp, except that for an instance $(V,\\mathcal{E},p,k)$\nwe have that $r\\leq|V|$ and $|E|\\leq r$ for every $E\\in\\mathcal{E}$.\n\\end{defin}\n\nAs the next theorem shows, this additional condition enables us to obtain a kernel which is linear in the number of vertices\nin the case of \\tcprk.\n\n\\begin{theo}\\label{polver}\n\\tcprk\\space admits a kernel with at most $r\\cdot k-(\\lfloor\\log r\\rfloor-1)\\cdot r$ vertices.\n\\end{theo}\n\\begin{proof}\nNotice that $s$ tests can create at most $2^s$ classes: more specifically, adding a test to a partial test cover which induces $t$ classes\nproduces a partial test cover with at most $2t$ classes. This happens because in order to produce a new class, the test must contain a subset\n$C'$ of a class $C$ such that $C'$ is not empty and it does not coincide with $C$ and if this happens for every class, the number of classes doubles.\nAt the same time, since every test contains at most $r$ vertices, this can happen with at most $r$ different classes.\nTherefore, given a partial test cover which induces $t$ classes, adding a new test can create at most $\\min\\{t,r\\}$ new classes.\nUsing $k$ tests it is possible to create at most $2^{\\lfloor\\log r\\rfloor}+(k-\\lfloor\\log r\\rfloor)r$ classes,\nsince the first $\\lfloor\\log r\\rfloor$ tests can create at most $2^{\\lfloor\\log r\\rfloor}$ classes and\nevery other test increases the number of classes by at most $r$.\nIt follows that we have a {\\sc No}-instance unless\n$$n\\leq 2^{\\lfloor\\log r\\rfloor}+(k-\\lfloor\\log r\\rfloor)r\\leq r+(k-\\lfloor\\log r\\rfloor)r=kr-(\\lfloor\\log r\\rfloor -1)r.\n$$\n\\end{proof}\n\n\\begin{cor}\\label{cor1}\n\\tcprk\\space admits a polynomial kernel when $r$ is a constant.\n\\end{cor}\n\\begin{proof}\nTheorem \\ref{polver} ensures the existence of a kernel with at most $r\\cdot k-(\\lfloor\\log r\\rfloor-1)\\cdot r$ vertices. This means that the number\n$m$ of tests is at most $$\\sum_{s=1}^r\\binom{rk}{s}\\leq\\sum_{s=1}^r\\left(\\frac{erk}{s}\\right)^s\\in O(r(erk)^r)=O(k^r).$$\n\\end{proof}\n\n\\begin{cor}\\label{cor2}\nConsider an instance $(V,\\mathcal{E},k)$ of \\tcpk\\space and let $r=\\max_{E\\in\\mathcal{E}}|E|$. If $r\\in O(n^{1-\\varepsilon})$ for some\n$\\varepsilon>0$, then the problem admits a kernel which is polynomial in the number of vertices.\n\\end{cor}\n\\begin{proof}\nBy Theorem \\ref{polver}, $n<kr$. Now, for some positive constant $c$ it holds that $kr<kcn^{1-\\varepsilon}$, which ensures that\n$n^{\\varepsilon}<kc$, that is $n<(kc)^{\\frac{1}{\\varepsilon}}$.\n\\end{proof}\n\nThe proof of Theorem \\ref{nopolker} can be modified to show the non-existence of a polynomial kernel for the parameterized problem \\tcpn.\nHowever, it is possible to obtain this result from a more general theorem. First of all, consider a definition of dual problem equivalent to the one\ngiven by Chen \\textit{et al.} \\cite{CheFerKanXia}:\n\n\\begin{defin}\\label{def:dual}\nLet $P$ be a parameterized problem, i.e., $P\\subseteq \\Sigma^* \\times\n\\mathbb{N}$. A mapping $s:\\Sigma^*\\rightarrow\\mathbb{N}$ is a {\\em size function} for $P$ if\n\\begin{itemize}\n\\item{$0\\leq k\\leq s(x)$ for every $(x,k)\\in P$, and}\n\\item{$s(x)\\leq |x|$ for every $x\\in  \\Sigma^*$.}\n\\end{itemize}\nThe {\\em dual} $P_d$ of a problem $P$ respectively to the size function $s$ is the problem corresponding\nto the language (i.e., the set of {\\sc Yes}-instances)\n$P_d=\\{(x,s(x)-k):\\ (x,k)\\in P\\}$.\n$P_d$ is also called the $s$-{\\em dual} of $P$.\n\\end{defin}\n\nNotice that the $s$-dual of the $s$-dual of a problem is again the original problem.\n\n\\begin{theo}\\label{dualnopolker}\nGiven a parameterized problem $P$, if $P$ admits a cross-composition from a NP-hard problem $L$, let $(x,k)$ be the instance\nwhich is associated to the $t$ strings $x_1,\\dots,x_t$ by the cross-composition algorithm. If there exists a size function\n$s(x)$ which is bounded by a polynomial in\n$\\max_{i=1}^t|x_i|+\\log t$,\nthen the $s$-dual problem $P_d$ does not\nadmit a polynomial kernel, unless $NP\\subseteq coNP\/poly$.\n\\end{theo}\n\\begin{proof}\nWe will show that if the hypothesis hold, $P_d$ admits a cross-composition from the same problem $L$. Hence we conclude using Theorem \\ref{res}.\nBy Definition \\ref{crocom}, we know that there exists $i$ such that $x_i\\in L$ if and only if $(x,k)\\in P$.\nBy definition of $s$-dual problem, this is equivalent to $(x,s(x)-k)\\in P_d$.\nThis provides a cross-composition for $P_d$, as long as the parameter $k'=s(x)-k$ is a polynomial in $\\max_{i=1}^t|x_i|+\\log t$,\nthat is as long as $s(x)$ is a polynomial in $\\max_{i=1}^t|x_i|+\\log t$.\n\\end{proof}\n\nNow, notice that \\tcpn\\space is the $s$-dual problem of \\tcpk, with $s(Q)=n$. Notice also that in Theorem \\ref{nopolker},\n$s(Q^*)=n+2l\\cdot(k+1)+l$, which is a polynomial in $n+l\\leq\\max_{i=1}^t|Q_i|+\\log t$. Hence, the hypothesis of Theorem \\ref{dualnopolker}\nholds and there is no polynomial kernel for \\tcpn. Thus, we have the following:\n\n\\begin{theo}\nThe problem \\tcpn{}\ndoes not admit a polynomial kernel, unless $NP\\subseteq coNP\/poly$.\n\\end{theo}\n\n\\section{Open Problems}\\label{sec:op}\n\nIn Corollaries \\ref{cor1} and \\ref{cor2} we give  sufficient conditions for the existence of kernels of polynomial size and polynomial in the number of vertices, respectively. It would be interesting to obtain extensions of the corollaries.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\\label{intro}\n\n\tIn recent years, photometric redshift galaxy surveys such as the Sloan Digital Sky Survey (SDSS) \\citep{2017arXiv171103234K}, the Dark Energy Survey (DES) \\citep{2016MNRAS.460.1270D}, and the future Large Synoptic Survey Telescope (LSST) \\citep{2008arXiv0805.2366I} and Euclid \\citep{2012SPIE.8442E..0ZA}, have arisen as powerful probes of the Large Scale Structure (LSS) of the universe and of dark energy. The main advantage of these surveys is their ability to retrieve information from a vast number of objects, yielding unprecedented statistics for different observables in the study of LSS. Their biggest drawback is the lack of line-of-sight precision and the systematic effects associated with it. Thus, well constrained systematic effects and robust observables are required in order to maximize the performance of such surveys. In this context, simple observables such as the galaxy number counts serve an important role in proving the robustness of a survey. In particular, the galaxy Counts-in-Cells (CiC), a method that consists of counting the number of galaxies in a given three-dimensional or angular aperture, has been shown to provide valuable information about the LSS~\\citep{Peebles1980,1990MNRAS.247P..10E,Gaztanaga:1993ru,Bernardeau:1993qu,1998ApJ...497...16S} and gives an estimate of how different systematic effects can affect measurements. CiC can provide insights to higher-order statistical moments of the galaxy counts without requiring the computation resources of other methods \\citep{Gil-Marin:2014sta}, such as the three- or four-point correlation functions. \n\n\\noindent Understanding the relation between galaxies and matter (galaxy bias) is essential for the measurements of cosmological parameters \\citep{Gaztanaga:2011yi}. The uncertainties in this relation strongly increase the errors in the dark energy equation of state or gravitational growth index \\citep{Eriksen:2015hqa}. Thus, having a wide variety of complementary methods to determine galaxy biasing can help break degeneracies and improve the overall sensitivity for a given galaxy survey. \n\n\\noindent In this paper we present a method to extract information from the galaxy CiC. Using this method, we measure the projected (angular) galaxy bias (linear and non-linear) in both simulations and observational data from DES, we compare the measured and predicted linear and non-linear bias, and we test for the presence of systematic effects. This dataset is ideal for this study since it has been already used for CiC in \\citet{Clerkin:2016kyr}, where it was found that the galaxy density distribution and the weak lensing convergence ($\\kappa_{WL}$) are well described by a lognormal distribution. The main difference between our study and \\citet{Clerkin:2016kyr} is that our main goal is to provide a measurement of the galaxy bias, whereas \\citet{Clerkin:2016kyr} study convergence maps.\n\n\\noindent \\citet{2017arXiv171005045G} also perform CiC in DES data. Combining gravitational lensing information and CiC, they measure the galaxy density probability distribution function (PDF) and obtain cosmological constraints using the \\texttt{redMaGiC} selected galaxies~\\citep{2016MNRAS.461.1431R} in DES Y1A1 photometric data~\\citep{2018ApJS..235...33D}. In our case we measure the moments of the galaxy density contrast PDF and compare them to the matter density contrast PDF from simulations (with the same redshift distributions) to study different biasing models, in a different galaxy sample (DES-SV).\n\n\\noindent Throughout the paper, we assume a fiducial flat $\\Lambda$CDM+$\\nu$ (one massive neutrino) cosmological model based on Planck 2013 + WMAP polarization + ACT\/SPT + BAO, with parameters \\citep{Ade:2013zuv} $\\omega_b = 0.0222$, $\\omega_c = 0.119$, $\\omega_\\nu = 0.00064$, $h = 0.678$, $\\tau = 0.0952$, $n_s = 0.961$ and $A_s=2.21\\times10^{-9}$ at a pivot scale $\\overline{k}=0.05\\rm{Mpc}^{-1}$ (yielding $\\sigma_8 = 0.829$ at $z = 0$), where $h \\equiv H_0 \/100 \\rm{km\\ s}^{-1}\\rm{Mpc}^{-1}$ and $\\omega_i\\equiv \\Omega_i h^2$ for each species $i$.\n\n\\noindent \nThe paper is organized as follows: in Section \\ref{sec:data_sample}, we present the data sample used for our analysis. First, we present the simulations used to test and validate the method and afterwards, the dataset in which we perform our measurements. In Section \\ref{sec:CiC}, we present the CiC theoretical framework and detail our method to obtain the linear and non-linear bias. Section \\ref{sec:moments} and \\ref{sec:DES-SV} present the CiC moments and bias calculations for the MICE simulation and DES-SV dataset, respectively. In Section \\ref{sec:sys}, we study the systematic uncertainties in our method. Finally, in Section \\ref{sec:conclusions}, we include some concluding remarks about this work.\n\\section{Data sample}\\label{sec:data_sample}\n\n\n\\subsection{Simulations}\\label{sec:sim}\n\nIn order to test and validate the methodology presented in this paper, we use the MICE simulation~\\citep{2008MNRAS.391..435F, 2010MNRAS.403.1353C}. MICE is an N-body simulation with cosmological parameters following a flat $\\Lambda$CDM model with $\\Omega_m=0.25$, $\\Omega_\\Lambda=0.75$, $\\Omega_b=0.044$, $n_s = 0.95$, and $\\sigma_8 = 0.8$. The simulation covers an octant of the sky, with redshift z, between 0 and 1.4 and contains 55 million galaxies in the lightcone. The simulation has a comoving size $L_{box} = 3072h^{-1}$Mpc and more than $8\\cdot10^9$ particles ~\\citep{Crocce:2013vda}. The galaxies in the MICE simulation are selected following the procedure in \\citet{2016MNRAS.455.4301C}, imposing the threshold $i_{evol}<22.5$. The MICE simulation has been extensively studied in the literature~\\citep{2011MNRAS.411..277S,2016MNRAS.455.4301C,Hoffmann:2015mma,Pujol:2015wna,2018MNRAS.476.1071G}, including measurements of the higher-order moments in the dark matter field~\\citep{2008MNRAS.391..435F}, providing an ideal validation sample.\n\n\n\n\\subsection{The DES SV Benchmark Data Sample}\\label{sec:benchmark}\n\nIn this paper we perform measurements of the density contrast distribution and its moments on the DES Science Verification (SV) photometric sample \\footnote{This sample is available at \\url{https:\/\/des.ncsa.illinois.edu\/releases\/sva1}} (Figure \\ref{fig:bench}). The DES Science Verification observations were taken using DECam on the Blanco 4m Telescope near La Serena, Chile, covering over 250 deg$^{2}$ at close to DES nominal depth. From this sample we make selection cuts in order to recover the LSS Benchmark sample~\\citep{2016MNRAS.455.4301C}. By doing this we minimize the possible two-point systematic effects and we ensure completeness. We focus on the SPT-E field, since it is the largest contiguous field and the best analyzed, with $60^{\\circ}<$ RA $<95^{\\circ}$, and $-60^{\\circ} <$ Dec $< -40^{\\circ}$ considering only objects with $ 18 < i < 22.5$ where $i$ is \\texttt{MAG\\_AUTO} as measured by \\texttt{SExtractor}~\\citep{Bertin1996} in the i-band. The star-galaxy separation is performed by selecting objects such that \\texttt{WAVG\\_SPREAD\\_MODEL} $> 0.003$.  The total area considered for our study is then 116.2 deg$^{2}$ with approximately 2.3 million objects and a number density $n_{g} = 5.6$ arcmin$^{-2}$. Several photo-z estimates are available for these data~\\citep{2014MNRAS.445.1482S}. We will focus on the TPZ \\citep{2013MNRAS.432.1483C} and BPZ~\\citep{Benitez:1998br} catalogs. We use the same 5 redshift bins used in \\citet{2016MNRAS.455.4301C}. We use the redshift distributions from \\citet{2014MNRAS.445.1482S}, which are depicted in Figure \\ref{fig:selection_function}. These distributions have been obtained by comparing the DES-SV photometric sample including spectroscopic data from zCOSMOS~\\citep{2007ApJS..172...70L,2009ApJS..184..218L} and VVDS Deep~\\citep{2013A&A...559A..14L} among other datasets. For more details about the photometric redshift measurement and calibration, we refer the reader to~\\citet{2014MNRAS.445.1482S}.\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{benchmark\/sv.png}\n\\caption{Footprint of the DES SV benchmark sample~\\citep{2016MNRAS.455.4301C}. We use approximately 2.3 million objects contained within this area for our studies.}\n\\label{fig:bench}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{benchmark\/phi.pdf}\n\\caption{Redshift distribution of the galaxies in each photometric redshift bin using TPZ (solid line) and BPZ (dashed line) in DES-SV benchmark data from \\citet{2016MNRAS.455.4301C}. These distributions have been obtained by stacking the photometric redshift probability density functions of galaxies in a spectroscopic subsample detailed in ~\\citet{2014MNRAS.445.1482S}. Lighter lines represent higher redshift slices.}\n\\label{fig:selection_function}\n\\end{figure}\n\\noindent\nSeveral measurements of the linear bias have been performed using this field~\\citep{2016MNRAS.455.4301C,2016MNRAS.456.3213G,2018MNRAS.473.1667P}, making it ideal for this study.\n\\section{Theoretical framework and methodology}\\label{sec:CiC}\n\\subsection{Counts-in-cells}\nCounts-in-Cells~\\citep{Peebles1980} is a method used to analyze the LSS based on dividing a galaxy survey into cells of equal volume ($V_{\\rm{pix}}$) and counting the number of galaxies in each cell, ($N_{\\rm{gal}}$). This method has also been extensively used in the literature to characterize the galaxy distribution~\\citep{1990MNRAS.247P..10E,Gaztanaga:1993ru,Bernardeau:1993qu,1998ApJ...497...16S}, and, recently, even the neutral hydrogen in simulations~\\citep{2018arXiv180809968L}. In the case of photometric redshift surveys, the lack of precision in the redshift determination makes angular aperture cells more appealing. Numerous examples of applications of CiC using angular aperture cells can be found in the literature~\\citep{Gaztanaga:1993ru,2002ApJ...570...75S,Ross:2006zr,2011ApJ...729..123Y,2013MNRAS.435....2W}.\n\nIt is particularly useful to work with the density contrast, $\\delta_{i}$ in each cell (or pixel), $i$, defined as:\n\\begin{equation}\\label{eq:delta}\n\\delta_{i}\\equiv\\frac{\\rho_{i}}{\\langle\\rho\\rangle}-1 \n\\end{equation}\nwhere $\\rho_{i}\\equiv\\frac{N_{i,\\rm{gal}}}{A_{i,\\rm{pix}}}$ is galaxy density in the pixel of area $A_{i,\\rm{pix}}$ and $\\langle\\rho\\rangle$ is the mean density. In this work, we are going to use $\\langle \\cdots \\rangle$ to denote statistical averages. Given these definitions, it follows that $\\langle \\delta \\rangle = 0$.\n\nIn order to study the statistical properties of the density contrast distribution, $\\delta$, we are interested in the measurement of the average of the $J$-point correlation functions, $\\overline{w}_J(\\theta)$, in a cell of solid angle $A = 2\\pi(1-\\cos{\\theta})$~\\citep{Gaztanaga:1993ru}:\n\\begin{equation} \\label{eq:Ipcfw}\n\\overline{w}_J(\\theta) = \\frac{1}{A^J}\\int_{A}dA_1...dA_Jw_J(\\theta_1,...,\\theta_J), \\, J \\geq 2\n\\end{equation}\nwith $dA_{i}=\\sin{\\theta_{i}}d\\theta_{i}d\\phi_{i}$ and $w_{J}(\\theta)$ the $J$-point angular correlation function. \n\nTo estimate the angle-averaged $J$-point correlation function, $\\overline{w}_{J}(\\theta)$, we use the \\textbf{corrected connected moments}, $\\langle \\delta^{J} \\rangle_{c}$, taking into account the discrete nature of CiC and assuming Poisson-like shot-noise contributions as introduced by~\\citet{Gaztanaga:1993ru}. In particular, we are interested in terms up to $J=4$:\n\\begin{equation}\n\\begin{gathered}\n\\overline{w}_{2}(\\theta)=\\langle\\delta^2\\rangle_c=\\langle\\delta^2\\rangle-\\frac{1}{\\overline{N}}\\\\\n\\overline{w}_{3}(\\theta)=\\langle\\delta^3\\rangle_c=\\langle\\delta^3\\rangle-\\frac{3}{\\overline{N}}\\langle\\delta^2\\rangle_c-\\frac{1}{\\overline{N}^2}\\\\\n\\overline{w}_{4}(\\theta)=\\langle\\delta^4\\rangle_c=\\langle\\delta^4\\rangle-3\\langle\\delta^2\\rangle^2-\\frac{7}{\\overline{N}^2}\\langle\\delta^2\\rangle_c-\\frac{6}{\\overline{N}}\\langle\\delta^3\\rangle_c-\\frac{1}{\\overline{N}^3}\n\\end{gathered}\n\\end{equation}\n\nwhere $\\overline{N}=\\frac{N_{\\rm{gal}}^{tot}*A_{\\rm{pix}}}{A_{\\rm{tot}}}$, and $N_{\\rm{gal}}^{tot}$ the total number of galaxies, $A_{\\rm{tot}}$ the total area, and $A_{\\rm{pix}}$ the area of the pixel.\n\nFor our study we use the \\textbf{rescaled connected moments} $S_J$ defined as:\n\\begin{equation}\\label{eq:hierarchyw}\nS_J\\equiv\\frac{\\overline{w}_J(\\theta)}{[\\overline{w}_2(\\theta)]^{J-1}}, \\hspace{4pt} J>2\n\\end{equation}\n\\begin{equation}\nS_{2} = \\overline{w}_{2}(\\theta)\n\\end{equation}\nIn most previous studies, the cells considered were spheres with radii of varying apertures (\\citet{Peebles1980}, \\citet{Bernardeau:1993qu}). We perform our measurements of the projected (angular) density contrast by dividing the celestial sphere into \\texttt{HEALpix} pixels \\citep{Gorski2005}. \n\\noindent\nFor our study we vary the \\texttt{HEALPix} parameter $N_{side}$ from 32 to 4096 (\\textit{i.e.} apertures ranging from $1.83^\\circ$ to $0.014^\\circ$). The angular aperture, $\\theta$, is estimated as the square root of the pixel area. According to equation (\\ref{eq:Ipcfw}) there is a dependence on the boundaries of the cell and thus on the shape that we choose for the pixels. \\citet{Gaztanaga:1993ru}, estimates CiC for square cells of side $l$ in a range $l = 0.03^\\circ-20 ^\\circ$ and compares to the average correlation functions $\\overline{w}_2(\\theta)$. The agreement between the two estimates indicates that square cells give very similar results to circular cells when the sizes of the cells are scaled to $\\theta=l\/\\sqrt{\\pi}$. Using data from MICE, we perform several tests to see that the concrete shape of the pixel, when it is close to a regular polygon, does not affect the measured moments despite boundary effects (Appendix \\ref{sec:appendix_b}). Furthermore, when working with the acquired observational data, the geometry of the survey becomes complicated. A discussion of how we deal with this is found in Appendix \\ref{sec:appendix_c}. \n\\noindent\nThe error bars throughout this paper are estimated using the bootstrap method~\\citep{Efron79,2006ASPC..357..271M,astroMLText}. This choice is mainly due to the lack of number of samples for large pixel sizes that might limit the precision of other methods such as jack-knife, given that the latter depends highly on the number of samples as pointed out in \\citep{2009MNRAS.396...19N}. Figure \\ref{fig:errors} shows agreement between the uncertainties computed using the jackknife and bootstrap methods for a randomly chosen redshift bin in the MICE simulation. We use $N_{b}=\\min{\\left(N_{pix},100\\right)}$ bootstrap and jack-knife realizations of the density contrast distribution to estimate our errors, where $N_{pix}$ is the total number of unmasked pixels in our map. \n\n\\begin{figure}\n\\includegraphics[scale=0.45]{micecheck\/moments_0.pdf}\n\\caption{Moments of the density contrast distribution as a function of cell scale in the MICE simulation for the redshift bin $0.2<z<0.4$, with jackknife errors (solid circles) and bootstrap errors (solid triangles). The results for a given scale $\\theta$ have been separated in the figure for visualization purposes, being the blue triangles the ones shown at the nominal measured scale.}\n\\label{fig:errors}\n\\end{figure}\n\n\n\\subsection{Galaxy bias}\\label{sec:bias}\n\nOne of the most important applications of the CiC observable is the determination of the galaxy bias. We observe the galaxy distribution and use it as a proxy to the underlying matter distribution. Both baryons and dark matter structures grow around primordial overdensities via gravitational interaction, so these distributions should be highly correlated. This relationship is called the \\textit{galaxy bias}, which measures how well galaxies trace the dark matter.\n\\noindent\nGalaxy biasing was seen for the first time analyzing the clustering of different populations of galaxies \\citep{1978ApJ...221....1D,1980ApJ...236..351D}. The theoretical relation between galaxy and mass distributions was suggested by \\citet{Kaiser:1984sw} and developed by \\citet{1986ApJ...304...15B}. Since then, many different prescriptions have arisen \\citep{Fry:1992vr,Bernardeau:1995ty,Mo:1995cs,Sheth:1999mn,Manera:2009ak,2011MNRAS.415..383M}. However, there is no generally accepted framework for galaxy biasing. While the galaxy and dark matter distribution are related, the exact relation depends on galaxy formation \\citep{PressSchechter}, galaxy evolution \\citep{Nusser:1993sx,Tegmark:1998wm,2000ApJ...531....1B}, and selection effects. Bias depends strongly on the environment. Using dark matter simulations, \\citet{Pujol:2015wna} show how the halo bias is determined by local density and not by halo mass. Several studies have demonstrated the different behaviors of early-type and late-type galaxies at both small and large scales \n\\citep{Ross:2006zr,Willmer:1999hf,2002ApJ...571..172Z,2002MNRAS.332..827N}. To have a good estimate of the real matter distribution, it is convenient to use a galaxy sample as homogeneous as possible. \nWith the linear bias $b(z)$ approximation, we can relate the matter fluctuations $\\delta_m$ with the fluctuations in the galaxy distribution $\\delta_g$:\n\\begin{equation} \n\\delta_g=b\\delta_m\n\\end{equation}\nIn the linear approximation, up to scalings, all statistical properties are preserved by the biasing and the observed galaxy properties reflect the matter distribution on large scales, as long as we consider only two-point statistics. However, in the general case, it is highly unlikely that the relation is both local and linear. Non-local dependencies might come from some properties such as the local velocity field or derivatives of the local gravitational potential \\citep{Fry:1992vr,Scherrer:1997hp}. Bias also depends on redshift \\citep{Fryevolution,Tegmark:1998wm}.\n\\noindent\nWhen non-Gaussianities are taken into account, linear bias fails to be a good description. If we want to measure higher orders we can assume that the (smoothed) galaxy density can be written as a function of the mass density and expand it as a Taylor series (assuming a local relation) \\citep{friemannGaz,Fry:1992vr}:\n\\begin{equation}\\label{eq:exp}\n\\delta_g=f(\\delta)=\\sum_{k=0}^\\infty\\frac{b_k}{k!}\\delta^k_m\n\\end{equation}\nThe linear term $b_1=b$ is the usual linear bias. Using this expansion we can relate the dark matter and the galaxy density contrast moments using the following relationships~\\citep{Fry:1992vr}:\n\\begin{equation}\nS_{2, mod}=b^{2}S_{2m}\n\\label{eq:S2_S2m}\n\\end{equation}\n\\begin{equation}\nS_{3, mod}=b^{-1}(S_{3m}+3c_2)\n\\label{eq:S3_S3m}\n\\end{equation}\n\\begin{equation}\nS_{4, mod}=b^{-2}(S_{4m}+12c_2S_{3m}+4c_3+12c_2^2)\n\\label{eq:S4_S4m}\n\\end{equation}\nwhere $c_k=b_k\/b$ for $k\\geq2$, the subscript $m$ refers to the underlying matter distribution, and the subscript $mod$ to the galaxy distribution. We will refer to this model as \\textbf{local}.\n\n\\citet{Bel:2015jla} point out that ignoring the contribution from the non-local bias can affect the linear and non-linear bias results. As a consequence, we analyze the case when the non-local contribution is included. To do so, we substitute $c_{2}$ by $c'_{2}=c_{2}-\\frac{2}{3}\\gamma_{2}$, where $\\gamma_{2}$ is the so-called non-local bias parameter \\citep{Bel:2015jla}. We will refer to this model as \\textbf{non-local}.\n\nNote that we omit the terms higher than 3rd order because, as we will show later, we have very limited sensitivity to $b_{3}$, and expect to have no sensitivity to $b_{4}$. \n\n\\subsection{Estimating the projected linear and non-linear bias}\nThe relations in equations (\\ref{eq:S2_S2m}-\\ref{eq:S4_S4m}) refer to the three-dimensional case and connect an observed galaxy distribution with its underlying dark matter distribution, both tracing the same redshift range and cosmological parameters. We assume that this bias model is also valid for the projected moments (we will check the validity of this assumption later). Moreover, given the measurements in a dark matter simulation with the same redshift distribution and angular footprint as our galaxy dataset, we estimate the linear and non-linear bias of these galaxies using equations (\\ref{eq:S2_S2m}-\\ref{eq:S4_S4m}). Note that these relations apply when we are comparing two datasets with the same value for $\\sigma_{8}$ parameter. In the case that $\\sigma_{8} \\neq \\sigma_{8,m}$ we will have to correct the resulting bias so,\n\\begin{equation}\nb_{corr} = b_{uncorr}\\frac{\\sigma_{8,m}}{\\sigma_{8}}.\n\\label{eq:bias_corr}\n\\end{equation}\nWe will use this correction in Section~\\ref{ssec_sys_s8}. We also take advantage of the fact that the skewness and kurtosis depend weakly with the cosmological parameters~\\citep{Bouchet:1992uh}. In particular, a 5\\% variation choosing $\\Omega_{m}=0.25$ translates to a variation of $0.2\\%$ in the measured $S_{3m}$, which is much smaller than the statistical fluctuations that we expect from our samples. In the case of $S_{4m}$ our sensitivity is even lower, making it safe to use a simulation with the same footprint and redshift distribution, as long as the variation in the cosmological parameters is small. However, this is not necessarily true for the case of $S_{2m}$, where the dependency on the cosmological parameters is higher. We check this using equation (\\ref{eq:Ipcfw}) to compute the projected $S_{2m}$ for two different sets of cosmological parameters: our fiducial Planck cosmology~\\citep{Ade:2013zuv} and a model with $\\Omega_{m}=0.2$. We use a Gaussian selection function $\\phi(z)$ with $\\sigma_{z}=0.05(1+z)$ since this is representative of the datasets that we analyze in this work. After this, we check the ratio:\n\\begin{equation}\n\\delta p_{ij}(z,\\theta) = \\frac{S_{2m,i}(z,\\theta)}{S_{2m,j}(z,\\theta)}\\frac{D^{2}_{+,j}(\\bar{z})}{D^{2}_{+,i}(\\bar{z})}\n\\end{equation}\nfor the different redshift slices considered in our analysis, where the subscripts $i$ and $j$ correspond to two different sets of cosmological parameters and $D_{+}(\\bar{z})$ is the linear growth factor \\citep{Peebles1980,Heath1977} evaluated at the mean redshift of the considered slice. This gives us an upper limit to the expected variation in $S_{2m}$ to consider in our analysis. In Figure~\\ref{fig:variation_S2} we can see that the variation is within 12\\% of the linear prediction, thus, we conservatively assign 12\\% systematic error to $S_{2m}$ due to this variation.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Systematics\/ratio_S2.pdf}\n\\caption{Ratio between $S_{2,m}$ for $\\Omega_{m}=0.2$ (the minimum allowed by Planck priors) and $S_{2,m}$ for our fiducial model as a function of the cell aperture angle $\\theta$. The different lines represent different redshift bins. We see that the variation is within 12\\% of the linear prediction.}\n\\label{fig:variation_S2}\n\\end{figure}\nUnder these conditions we perform a simultaneous fit to $b$, $b_{2}$, $b_{3}$ and $\\gamma_{2}$. In order to do so, we consider the likelihood:\n\\begin{multline}\n\\log{\\mathcal{L}} = -\\frac{1}{2}\\sum_{k=2}^{4}\\sum_{i,j}\\left[S_{k,g}(\\theta_{i})-S_{k,mod}(\\theta_{i})\\right]\\\\\nC^{-1}_{k,ij}\\left[S_{k,g}(\\theta_{j})\n-S_{k,mod}(\\theta_{j})\\right] = -\\frac{\\chi^{2}}{2}\n\\label{eq:loglike}\n\\end{multline}\nwhere $S_{k,g}$ are the measured galaxy moments and $S_{k,mod}$ are the models in equations (\\ref{eq:S2_S2m}), (\\ref{eq:S3_S3m}), and (\\ref{eq:S4_S4m}). We checked that the measured $S_{k}$ follow a Gaussian distribution. The covariances $C_{k,ij}$ are computed as follows:\n\\begin{equation}\nC_{k,ij}=\\frac{N_{u,pix}(\\theta_{i})}{N_{u,pix}(\\theta_{j})}2^{2(j-i)}\\sigma_{k}(\\theta_{i})\\sigma_{k}(\\theta_{j})\n\\end{equation}\nwith $N_{u,pix}(\\theta_{i})$ being the number of pixels used in an aperture, $\\theta_{i}$. Note that, since we are using \\texttt{HEALPix}, which imposes a fixed grid, and we are not repeating the measurements in translated\/rotated galaxy fields, we are re-using the same galaxies for different scales, so the factor $\\frac{N_{u,pix}(\\theta_{i})}{N_{u,pix}(\\theta_{j})}2^{2(j-i)}$ accounts for the induced correlation due this reuse. We assume that the errors in the dark matter moments and the errors in the galaxy moments are not correlated and add them in quadrature, so: \\begin{equation}\n\\sigma_{k}(\\theta_{i})=\\sqrt{\\sigma_{k,g}^{2}(\\theta_{i})+\\sigma_{k,m}^{2}(\\theta_{i})}\n\\end{equation}\nwhere $\\sigma_{k,g\/m}(\\theta_{i})$ is the standard deviation of the $k$-th (galaxy or matter) moment in an aperture $\\theta_{i}$ computed using bootstrapping.\n\nWe use the following flat priors:\n\\begin{itemize}\n\\item $0 < b < 10$.\n\\item $-10 < b_{2} < 10$.\n\\item $-10 < b_{3} < 10$.\n\\item $\\gamma_{2}=0$ (or in the case of non-local model $-10 < \\gamma_{2} < 10$).\n\\end{itemize}\nThese priors have been chosen to prevent unphysical results. We evaluate the likelihood and obtain the best fit values and their uncertainties by performing a MCMC using the software package \\texttt{emcee}~\\citep{2013PASP..125..306F}. Summarizing, the method works as follows:\n\\begin{enumerate}\n\\item Measure CiC moments using \\texttt{HEALPix} pixels in the galaxy sample.\n\\item Measure CiC moments using the same pixels and selection function in a dark matter simulation with comparable cosmological parameters.\n\\item Evaluate statistical and systematic uncertainties in the measured moments.\n\\item Obtain best fit $b, b_{2}, b_{3}$, (and $\\gamma_{2}$ in the non-local model) using MCMC with the models from equations (\\ref{eq:S2_S2m}-\\ref{eq:S4_S4m}).\n\\end{enumerate}\n\nIn summary, in the local model we fit 3 free parameters, whereas in the non-local model we fit 4.\n\n\\citet{Hoffmann:2015mma} present a prediction for the non-linear bias as a function of the linear bias in the three-dimensional case: \n\\begin{eqnarray}\n\\centering\nb_{2}&=&b^{2}-2.45 b+1.03\\\\\nb_{3}&=&b^{3}-7.32 b^{2}+10.79\nb-3.90\n\\label{eq:hoffman}\n\\end{eqnarray}\nWe will use these predictions to test the compatibility between the 3D and the measured projected values for the non-linear bias.\n\n\\section{RESULTS IN SIMULATIONS} \\label{sec:moments}\n\nIn order to validate this method, we first compute the CiC moments in the MICE simulation (in both galaxies and DM) using a Gaussian selection function $\\phi(z)$ with $\\sigma_z=0.05(1+z)$. This $\\sigma_z$ is similar to the photometric redshifts found in the data using TPZ ~\\citep{Kind:2013eka} and BPZ ~\\citep{Benitez:1998br}. We split our sample into 5 photometric redshift bins: $z \\in [0.2, 0.4], [0.4, 0.6], [0.6, 0.8], [0.8, 1.0], [1.0, 1.2]$, mirroring the choice in \\citep{2016MNRAS.455.4301C}. Then we do the same with the SV data sample presented in Section \\ref{sec:benchmark} with TPZ photometric redshifts.\n\n\\subsection{Angular moments for MICE}\\label{sec:mice_cic}\n\nFigure \\ref{fig:micemomphotoz} shows the moments of the density contrast distribution as a function of the cell scale for the different photometric redshift bins. We observe that the moments follow the expected trend, that is, lower redshift bins have higher values for the higher-order moments since non-linear gravitational collapse has a larger effect on these. This is true for all measurements except for the last two redshift bins of the variance, $S_{2}$. This can be due to the magnitude cuts, since the galaxy populations are different at different redshifts. We also see that the larger the cell scale, the smaller the variance $S_{2}$, since larger cell scales should be more homogeneous. The skewness and the kurtosis in linear scales ($\\theta>0.1^\\circ$) are constant and of the same order of magnitude as the expected values ($S_{3} \\approx 34\/7$, $S_{4} \\approx 60712\/1323$) ~\\citep{Bernardeau:1993qu}. The behavior at non-linear scales is due to the non-linearities of the MICE simulation.\n\n\\begin{figure}\n\\includegraphics[scale=0.45]{micephotoz\/moments.pdf}\n\\caption{Moments of the density contrast distribution as a function of cell scale in the MICE simulation with Gaussian photometric redshift ($\\Delta z=0.2$ $\\sigma_z=0.05(1+z)$) for different redshift bins. The results for a given scale $\\theta$ have been separated in the figure for visualization purposes.}\n\\label{fig:micemomphotoz}\n\\end{figure}\n\\subsection{Projected galaxy bias in MICE simulation}\\label{sec:mice_nlresults}\n\nWe smear the true redshift with the proper selection function in the MICE dark matter field, obtained from a dilution of the dark matter particles (taking $1\/700$ of the particles). \\citet{2016MNRAS.459.3203C} demonstrate that the dilution of the dark matter field does not impact their statistics and using the measured moments from the previous section we proceed to perform a simultaneous fit for $b$, $b_{2}$, and $b_{3}$ using the local, non-linear bias model from equations (11,12,13). The fit results are summarized in Figure \\ref{fig:nl_bias_summary_mice}. We can see the impact of changing the range of $\\theta$ considered in the fit. In this case we see that including scales smaller than $0.1^{\\circ}$, where non-linear clustering has a large impact, affects the $b_{2}$ results. This, together with the fact that the reduced $\\chi^{2}$ minimum value doubles when including $\\theta = 0.05^{\\circ}$ clearly shows that we should not consider scales smaller than $\\theta = 0.1^{\\circ}$. We can see as well that $b_{3}$ is compatible with zero and that we have a limited sensitivity to it, given the area used. Thus, the choice of ignoring terms of orders higher than $b_{3}$ becomes a good approximation. However, for $b_{2}$ we are able to measure a significant non-zero contribution. We can also see that the predicted values for the 3D non-linear bias parameter $b_{2}$ are not in good agreement at small scales, while there is an indication of better agreement at larger scales. This suggests that the 2D and 3D values for $b_{2}$ might be compatible at larger scales, in agreement with \\citet{2011MNRAS.415..383M} who show that the local bias is consistent for scales larger than $R>30-60$ Mpc\/$h$. They also show that the values of $b_1$ and $b_2$ vary with the scale and converge to a constant value around $R>30-60$ Mpc\/$h$, which means that the values that we measure here have not yet fully converged. The prediction for $b_{3}$ seems to be compatible with the estimated values given the size of the error bars. These results show that we should consider $b_{2}$ as a first order (small) correction to the linear bias model at these scales for projected (angular) measurements.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45,trim={0 2.5cm 0 2cm}, clip]{mcmc\/nl_bias_mice_summary_v2.pdf}\n\\caption{Linear and non-linear bias results as a function of redshift for MICE data with Gaussian photo-z. The different marker shapes represent the best-fit results considering different ranges of the aperture angle $\\theta$. For the solid triangles we consider the range from $0.05^{\\circ}$ to $0.92^{\\circ}$, open circles are our fiducial case with $0.11^{\\circ} < \\theta < 0.92^{\\circ}$, for the solid circles, we take out the smallest scale in our fiducial case and in open triangles we take out the largest scale. The top panel shows the projected linear bias $b$ as a function of redshift, the middle panel shows the best-fit results for the projected $b_{2}$, and $b_{3}$ is shown in the lower panel. The shaded region corresponds to the 3D predicted values using equations (\\ref{eq:hoffman}). The results for a given redshift $z$ have been separated in the figure for visualization purposes.}\n\\label{fig:nl_bias_summary_mice}\n\\end{figure}\nThe individual fits can be seen in Appendix \\ref{app:fits}.\n\\subsection{Verification and biasing model comparison}\nIn order to verify this method and check if the local non-linear model considered induces certain systematic biases on the results, we check that the measured linear bias is compatible with corresponding measurements from the two point correlation function (Figure \\ref{fig:linear_bias_mice}). In particular, we use the best fit parametrization from \\citet{2016MNRAS.455.4301C}:\n\\begin{equation}\nb_{best}(z) = 0.98 + 1.24z - 1.72z^{2} + 1.28z^{3}\n\\end{equation}\nIn Figure~\\ref{fig:linear_bias_mice}, we can see that the local and non-local bias are in agreement, most likely due to the scale range that we are dealing with and the projection effects due to the size of the redshift slices. In this figure, we can also notice that the reduced chi-square for both models is similar, and that they are well below one. Given that the number of degrees of freedom is small, it is still possible that these values are correct, however, it is unlikely that this happens for all redshift bins. This suggests that, in agreement with \\citet{2009MNRAS.396...19N}, bootstrapping uncertainties are overestimated. However, we prefer to use these conservative uncertainties, rather than state uncertainties that are too optimistic since, one of the main goals of this work will be to state the statistical significance on the non-linear $b_{2}$ term. Another interesting feature in Figure~\\ref{fig:linear_bias_mice} is that the uncertainties in $b_{2}$ for the non-local model are considerably larger than in the local model. This is due to the fact that $\\gamma_{2}$ is highly correlated with $b_{2}$, which makes the posterior distribution for $b_{2}$ much wider, increasing the resulting uncertainty.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45,trim={0 1.6cm 0 2cm}, clip]{mcmc\/bias_models_comparo_mice.pdf}\n\\caption{(Top) Comparison between the MICE simulation bias obtained using CiC with different biasing models: non-local (solid triangles), and local (open triangles). We also show the best-fit from \\citet{2016MNRAS.455.4301C} (Figure 17) as reference. The middle panel shows the equivalent results for $b_{2}$. This is done for Gaussian photo-z with $\\sigma_{z}=0.05(1+z)$. (Bottom) total reduced chi-square for each of the models when fitting the moments to obtain the bias.}\n\\label{fig:linear_bias_mice}\n\\end{figure}\n\n\\section{RESULTS IN DES-SV data}\n\\label{sec:DES-SV}\n\\subsection{Angular moments for DES - SV}\\label{sec:benchcic}\nUsing the same footprint, selection cuts, and redshift bins as in \\citet{2016MNRAS.455.4301C}, we compute the moments of the density contrast distribution for the SV data. These results are depicted in Figure \\ref{fig:benchmom} as a function of cell scale for different redshift bins. Here, as in the case of MICE, the variance decreases with the scale. The skewness and the kurtosis are also constant and of the same order of magnitude as the theoretical values within errors. The largest differences when compared with the simulation are in the non-linear regime due to the different way non-linearities are induced in the simulation and in real data. We also compare to the results from CFHTLS found in \\citet{2013MNRAS.435....2W}. We find a similar general behavior, as well as the same order of magnitude in the measured $S_{3}$ and $S_{4}$. However, we do not expect the same exact results since the redshift distributions from CFHTLS do not match exactly the corresponding distributions in the DES-SV data.\n\n\\begin{figure}\n\\includegraphics[scale=0.45]{benchmark\/moments.pdf}\n\\caption{Moments of the density contrast distribution of the DES SV benchmark sample as a function of cell scale, for five different redshift bins and different scales. The results for a given scale $\\theta$ have been separated in the figure for visualization purposes. We compare with the results from \\citet{2013MNRAS.435....2W} for CFHTLS marked with solid lines of different colors for the different redshift bins: navy $(0.2 < z < 0.4)$, cyan $(0.4 < z < 0.6)$, lime $(0.6 < z < 0.8)$, yellow $(0.8 < z < 1.0)$.}\n\\label{fig:benchmom}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45,trim={0 2.5cm 0 2cm}, clip]{mcmc\/nl_bias_tpz_summary_v2.pdf}\n\\caption{Linear and non-linear bias results as a function of redshift for DES-SV data. Systematic uncertainties from Section \\ref{sec:sys} are already included in these results, excluding the uncertainties associated to the modeling. The different marker shapes represent the best-fit results considering different ranges of aperture angle $\\theta$. For the solid triangles we consider the range from $0.05^{\\circ}$ to $0.92^{\\circ}$, open circles symbolize our fiducial case with $0.11^{\\circ} < \\theta < 0.92^{\\circ}$, in solid circles, we take out the smallest scale in our fiducial case and, in open triangles, we take out the largest scale. The shadowed region corresponds to the 3D predicted values using equations (\\ref{eq:hoffman}). The top panel shows the projected linear bias $b$ as a function of redshift, the middle panel shows the best-fit results for the projected $b_{2}$ and $b_{3}$ is shown in the lower panel. The results for a given redshift $z$ have been separated in the figure for visualization purposes.}\n\\label{fig:bnl_bench_2}\n\\end{figure}\n\n\\subsection{Projected galaxy bias in DES - SV}\nRepeating the procedure that we used for the MICE galaxy simulation, we analyze the DES - SV data and the MICE dark matter simulation, and compare their moments. In Figure \\ref{fig:bnl_bench_2} we can see the results of simultaneously fitting for $b$, $b_{2}$ and $b_{3}$. The measurements in this figure include the systematic uncertainties are introduced in Section 6. The resulting $b$ is corrected by the ratio of $\\sigma_{8}$ between MICE and our adopted fiducial cosmology using equation (\\ref{eq:bias_corr}). The fit results can be seen in Figure \\ref{fig:fits_data_gaussian_nl}. In this case, we detect a non-zero value for $b_{2}$. We check the probability of $b_{2}$ being zero by computing:\n\\begin{equation}\n\\chi^{2}_{z}=\\sum_{i,j=1,N_{zbins}}\\hat{b}_{2,i}\\mathcal{C}_{2,ij}^{-1}(z)\\hat{b}_{2,j}\n\\end{equation}\nThe sum runs for all the redshift bins. $\\hat{b}_{2}$ is the weighted average of the fit results with the different fitting ranges and $\\mathcal{C}_{2,ij}(z)$ is the covariance matrix for $b_{2}$. Taking into account the correlations between different redshift bins:\n\\begin{equation}\n\\mathcal{C}_{2,ij}(z)=\\frac{N_{ij}N_{ji}}{N_{ii}N_{jj}}\\Delta\\hat{b}_{2,i}\\Delta\\hat{b}_{2,j}\n\\end{equation}\nwith $N_{ij}$ is the number of galaxies observed in the photo-z bin $i$ from the true-z bin $j$ and $\\Delta\\hat{b}_{2,i}$ is the weighted uncertainty in $\\hat{b}_{2,i}$ for the photo-z bin $i$.\nThe value of $\\chi^{2}_{z} = 64.75$ with 4 degrees of freedom, so the probability is essentially 0, making clear that the overall value of $b_{2}$ is non-zero for the local model. However, we lack the sensitivity necessary to detect a non-zero $b_{3}$. \n\n\\noindent\nWe also check the measurement of linear bias obtained in this work and compare it with previous measurements on the same dataset Figure~\\ref{fig:linear_bias_comparison}. The measurements are generally in good agreement with each other showing the robustness of the method.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.45,trim={0.8cm 0.5cm 0 1.5cm}, clip]{mcmc\/bias_comparo.pdf}\n\\caption{Bias obtained from second order CiC, including systematic uncertainties from Section \\ref{sec:sys}, compared with the 2-point correlation study \\citep{2016MNRAS.455.4301C}, the CMB-galaxy cross-correlations study \\citep{2016MNRAS.456.3213G}, galaxy-galaxy lensing \\citep{2018MNRAS.473.1667P}, and the shear+density analysis~\\citep{2016MNRAS.459.3203C}. The points for the same $z$ have been separated in the horizontal axis for visualization purposes.}\n\\label{fig:linear_bias_comparison}\n\\end{figure}\n\\noindent\nFuture DES data will have a considerably larger area and, as previous MICE measurements show, these measurements will improve. Here we also use the skewness and the kurtosis of dark matter from the MICE dark matter simulation, as those quantities hardly depend on the cosmology~\\citep{Bouchet:1992uh}. We also find that our results are similar to those in \\citet{Ross:2006zr}. We do not expect them to be equal as the samples are different and the bias depends strongly on the population sample.\n\n\\section{Systematic Errors}\\label{sec:sys}\n\nIn this section, we explore the effects that several potential sources of systematic uncertainty have on our moment measurements. Since our main observable is related to the number of galaxy-counts in a given redshift interval, we are interested in observational effects that can affect this number. The main potential sources of systematic uncertainties are changes in airmass, seeing, sky brightness, star-galaxy separation, galactic extinction, and the possible errors in the determination of the photometric redshift. In order to evaluate their effects, we use the maps introduced in \\citet{2016ApJS..226...24L}. To account for the stellar abundance in our field we proceed as in \\citet{2016MNRAS.455.4301C} and use the USNO-B1 catalog \\citep{2003AJ....125..984M}. We also use the SFD dust maps ~\\citep{1998ApJ...500..525S}.\n\\noindent\nWhat follows is a detailed step-by-step guide to our systematic analysis: we select one of the aforementioned maps and locate the pixels where the value of the systematic is below the percentile level $t$. We compute the moments of the density contrast distribution in these pixels and their respective errors using bootstrap. We change the threshold to $t+5$, repeat the process, and evaluate the difference between the moments calculated using this threshold divided by the moments in the original footprint $\\Delta S_{i}(t)$\/$\\langle S_{i} \\rangle$. An example of the results of this procedure can be found in Figure \\ref{fig:sys_example}. Note that the plot showing the variation of the moments with USNOB shows less points in the horizontal axis. Due to the discrete nature of the map of stellar counts, the 50th and 60th percentiles of the $\\delta$ distribution of the stellar counts are the same, in order to avoid these problems, we make less bins in this case.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Systematics\/sysplot_bin0_FWHM_i.pdf}\n\\includegraphics[width=0.9\\textwidth]{Systematics\/sysplot_bin0_FWHM_g.pdf}\n\\includegraphics[width=0.9\\textwidth]{Systematics\/sysplot_bin0_SKYBRITE_i.pdf}\n\\includegraphics[width=0.9\\textwidth]{Systematics\/sysplot_bin0_USNOB.pdf}\n\\caption{Dependence of the moments $S_{i}$ with the variation in the value of potential systematic effects. We show an example for $N_{side}=2048$ in the redshift bin $0.2 < z < 0.4$ using TPZ. The left column shows the behavior for $S_{2}$, the middle column shows $S_{3}$, and the last column shows the results for $S_{4}$. The first row corresponds to the results for the seeing in i-band, the second row shows the results for seeing in g-band, the third shows the sky-brightness in i-band. Finally the last row shows the evolution of the moments with the variation in the number of stars per pixel.}\n\\label{fig:sys_example}\n\\end{figure*}\n\\noindent\nWe consider that a systematic effect is present if the average of $\\Delta S_{i}(t)$\/$\\langle S_{i} \\rangle$ is different from zero at a $2\\sigma$ confidence level or above for the different values of $t$ from the 50th tile to the 100th tile. Then, we assign a systematic uncertainty equal to the value of this average. To be conservative, we consider these effects as independent, so we add them in quadrature. We summarize the main systematic effects observed in each redshift bin of our sample:\n\\begin{itemize}\n\\item Bin $0.2 < z < 0.4$:\n\\begin{itemize}\n\\item Seeing in i-band: we assign a 3\\% systematic uncertainty in $S_{4}$.\n\\item Seeing in z-band: we assign a 2.5\\% systematic uncertainty in $S_{4}$.\n\\item Sky-brightness r-band: we assign a 1\\% systematic uncertainty in $S_{4}$.\n\\item Sky-brightness i-band: we assign a 1\\% systematic uncertainty in $S_{4}$.\n\\item Airmass in g-band: we assign a 1\\% uncertainty in $S_{4}$.\n\\item Airmass in r-band: we assign a 1\\% uncertainty in $S_{4}$.\n\\item Airmass in i-band: we assign a 1\\% uncertainty in $S_{4}$.\n\\item USNO-B stars: We assign a 4\\% uncertainty to $S_{2}$, 7\\% uncertainty to $S_{3}$, and 9\\% to $S_{4}$.\n\\end{itemize}\n\\item Bin $0.4 < z < 0.6$:\n\\begin{itemize}\n\\item Seeing in z-band: We assign a 1.5\\% uncertainty to $S_{4}$.\n\\item USNO-B stars: We assign a 4\\% uncertainty to $S_{2}$, 3\\% uncertainty to $S_{3}$, and 4\\% to $S_{4}$.\n\\end{itemize}\n\\item Bin $0.6 < z < 0.8$:\n\\begin{itemize}\n\\item Seeing in g-band: We assign a 2\\% to $S_{4}$.\n\\item Seeing in r-band: We assign a 2\\% to $S_{4}$.\n\\item Sky-brightness i-band: We assign a 1.5\\% uncertainty to $S_{3}$, and 3\\% systematic uncertainty to $S_{4}$.\n\\item Airmass in g-band: We assign a 2.5\\% uncertainty to $S_{4}$.\n\\item Airmass in r-band: We assign a 2\\% uncertainty to $S_{4}$.\n\\item Airmass in z-band: We assign a 1.5\\% uncertainty to $S_{3}$, and 3\\% uncertainty to $S_{4}$.\n\\item USNO-B stars: We assign a 3\\% uncertainty to $S_{3}$, and 5\\% uncertainty to $S_{4}$.\n\\end{itemize}\n\\item Bin $0.8 < z < 1.0$:\n\\begin{itemize}\n\\item Seeing in g-band: We assign a 2\\% uncertainty to $S_{4}$.\n\\item Sky-brightness in i-band: We assign a 2\\% uncertainty to $S_{3}$, and a 3.5\\% uncertainty to $S_{4}$.\n\\item Airmass in g-band: We assign a 2\\% uncertainty to $S_{4}$.\n\\item Airmass in r-band: We assign a 3\\% uncertainty to $S_{4}$.\n\\item USNO-B stars: We assign a 3\\% uncertainty to $S_{4}$.\n\\end{itemize}\n\\item Bin $1.0 < z < 1.2$:\n\\begin{itemize}\n\\item The measurement of $S_{4}$ in this bin is dominated by systematics.\n\\item Sky-brightness i-band: We assign 2\\% to $S_{3}$.\n\\item Sky-brightness z-band: We assign 3\\% to $S_{3}$.\n\\item USNO-B stars: We assign a 4.5\\% uncertainty to $S_{3}$.\n\\end{itemize}\n\\end{itemize}\n\\noindent\nThe estimated systematic errors for the bias are propagated from the estimation of the systematics in $S_2$, $S_3$, and $S_4$. Their behavior is compatible with the systematics found in \\citet{2016MNRAS.455.4301C}. We use the same data masking, excluding regions with large systematic values to recover $w(\\theta)$. The linear bias is more robust using CiC since the variance, $S_{2}$, is less affected by the small scale power induced by the systematics given that these scales are smoothed out. On the other hand, the non-linear bias is more sensitive to the presence of systematics because they can induce asymmetries in the density contrast distribution. \n\n\\subsection{Photometric redshift}\n\nPhotometric redshift is one of the main potential sources for systematic effects in photometric surveys like DES. We have repeated the analysis in DES-SV data for a second estimate of the photometric redshift using BPZ ~\\citep{Benitez:1998br}. In Figure \\ref{fig:biasbenchbpz} we compare the results for the two photometric redshift codes and we see that they are in good agreement. The linear bias seems to be the most affected by the choice of a photometric redshift estimator but the results do not show any potential systematic biases. For the non-linear bias we get remarkably consistent results, showing the robustness of this method.\n\n\\begin{figure}\n\\includegraphics[scale=0.45]{mcmc\/tpz_bpz_results.pdf}\n\\caption{Bias obtained in the SV data from second order CiC for TPZ (solid blue circles) and BPZ (green crosses). The results for a given redshift $z$ have been separated in the figure for visualization purposes.}\n\\label{fig:biasbenchbpz}\n\\end{figure}\n\n\\subsection{Biasing models}\nApart from the terms that we considered in our model, \\citet{Bel:2015jla} found that non-local bias terms are responsible for the overestimation of the linear bias from the three-point correlation in \\citet{Pollack:2013alj,Hoffmann:2015mma,2011MNRAS.415..383M} but that they should not significantly affect second-order statistics. As we mentioned previously in Section 5, we do not expect these terms to have a significant impact on our estimations because we analyze projected quantities over considerable volumes (note that we integrate in the cell and in the redshift slice). Having said that, we test the local and non-local models and find the results depicted in Figure \\ref{fig:model_comp_sv}. We can see, as in the case of the simulation, that both models are consistent within errors. This means that choosing the local model does not introduce any systematic uncertainties in our linear bias measurements. However, it affects the $b_{2}$ measurements and their uncertainty since the new parameters introduced with these more complicated models are correlated with them. We check the probability of $b_{2}$ being zero for the different models and obtain the results in table \\ref{tab:bias_models}. We find $b_{2}$ to be different from zero at a 3-$\\sigma$ level in the worst case (non-local).\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nBias model & $\\chi^{2}$ & $p$-value & ndof\\\\\n\\hline\nLocal & 64.75 & $3\\times 10^{-13}$ & 4\\\\\nNon-local & 12.63 & 0.013 & 4\\\\\n\\hline\n\\end{tabular}\n\\caption{Comparison on the null hypothesis for $b_{2}$ in DES-SV data for the different bias models considered in this work.}\n\\label{tab:bias_models}\n\\end{table}\nWe also can see that in the first bin, none of the models fit the data well, which is not surprising, given that the range of (comoving) scales is very small ($\\sim 1 - 20$ Mpc $h^{-1}$) and non-linear clustering dominates.\n\nFinally, we are not considering stochastic models and we are assuming a Poisson shot-noise. This means that our measured $b_{2}$ could be entangled with stochasticity~\\citep{1998ApJ...504..601P,2013PhRvD..87l3523S}. We leave the study of stochasticity to future works. \n\\begin{figure}\n\\includegraphics[scale=0.45]{mcmc\/bias_models_comparo_sv}\n\\caption{(Top) Comparison between the linear bias results obtained with CiC for SV using different biasing models: non-local (solid triangles), and local (open triangles) using the TPZ sample. (Middle) Comparison between $b_{2}$ results for the same models as above. (Bottom) total reduced chi square for each of the models.}\n\\label{fig:model_comp_sv}\n\\end{figure}\n\\subsection{Value of $\\sigma_{8}$}\n\\label{ssec_sys_s8}\nAs mentioned in previous sections, our bias estimation depends linearly on the value of $\\sigma_{8}$. Thus, if the actual value of $\\sigma_{8}$ is different from our assumed fiducial value, our results will be biased, and we have to correct for the difference using equation \\ref{eq:bias_corr}. This is why, we introduce a systematic uncertainty of $1.4\\%$ (the uncertainty level in $\\sigma_{8}$ from \\citet{Ade:2013zuv}) that we add in quadrature to the statistical errors in the final estimation of the bias.\n\n\\section{Conclusions}\\label{sec:conclusions} \n\nCiC is a simple but effective method to obtain the linear and non-linear bias. A good measurement of the galaxy bias is essential to maximize the performance of photometric redshift surveys because it can introduce a systematic effect on the determination of cosmological parameters. The galaxy bias is highly degenerate with other cosmological parameters and an independent method to determine it can break these degeneracies and improve the overall sensitivity to the underlying cosmology. In this paper we have developed a method to extract the bias from CiC. We use the MICE simulation to test our method and then perform measurements on the public Science Verification data from the Dark Energy Survey. The strength of this method is that it is based on a simple observable, the galaxy number counts, and is not demanding computationally. \n\nWe check that our linear bias measurement from CiC agrees with the real bias in the MICE simulation. Figure \\ref{fig:linear_bias_mice} shows an agreement between our measurement and the one obtained using the angular two-point correlation function. We then obtain the linear bias in the SV data and find that it is in agreement with previous bias measurements from other DES analyses. In Figure \\ref{fig:linear_bias_comparison}, we see that the CiC values are compatible with the two-point correlation study~\\citep{2016MNRAS.455.4301C}, the CMB-galaxy cross-correlations study~\\citep{2016MNRAS.456.3213G}, and the galaxy-galaxy lensing~\\citep{2018MNRAS.473.1667P}, and we demonstrate that these results are robust to the addition of new parameters in the biasing model, such as the non-local bias. Finally, we compute the non-linear bias parameters up to third order. We detect a significant non-zero $b_{2}$ component. It appears that the 2D and 3D predictions of the non-linear bias are in better agreement at larger scales, as expected. However, given the uncertainties associated with these quantities, it is difficult to draw any conclusions from $b_{3}$ despite its compatibility with the expected 3D prediction. When more data is available, we plan to check if we can improve our constraints on $b_{3}$ and whether the agreement with the 3D prediction improves as well. The systematic errors are in general lower than the statistical errors, in agreement with the systematic study done by~\\citep{2016MNRAS.455.4301C}.\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA problem stemming from physics and soon becoming an important mathematical\nproblem is to recover geometry of a manifold from the knowledge of the\neigenvalues of a natural differential operator. This problem originates in the\ntheory of radiation (how to determine the state density of the electromagnetic\nwave in a given cavity) and is formulated by Kac as \"Can one hear the shape of\na drum?\" \\cite{Kac}.\n\nWhat the original problem asked is that how many eigenvalues are smaller than\na given number $\\lambda$. This problem is formulated as to seek the so-called\ncounting function $N\\left(  \\lambda\\right)  $. For a given spectrum $\\left\\{\n\\lambda_{n}\\right\\}  $, the counting function is directly defined to be%\n\\begin{equation}\nN\\left(  \\lambda\\right)  =\\text{the\\ number of eigenvalues (with multiplicity)\nsmaller than }\\lambda\\text{,}\\label{NLmd}%\n\\end{equation}\ni.e.,%\n\\begin{equation}\nN\\left(  \\lambda\\right)  =\\sum_{\\lambda_{n}<\\lambda}1.\\label{NLsgm1}%\n\\end{equation}\nIn other words, the counting function represents the number of the eigenstates\nwhose eigenvalues are smaller than $\\lambda$. Nevertheless, the counting\nfunction $N\\left(  \\lambda\\right)  $ is very difficult to calculate and there\nis no general method for calculating $N\\left(  \\lambda\\right)  $ in\nmathematics \\cite{Berger}. This is because when calculating $N\\left(\n\\lambda\\right)  $, one often encounters some unsolved problems in number\ntheory. For example, when calculating the counting function for the spectrum\nof the Laplace operator on a tori, one encounters the Gauss circle problem in\nnumber theory. The Hardy-Littlewood-Karamata Tauberian theorem gives the first\nterm of the asymptotic expansion of $N\\left(  \\lambda\\right)  $, but does not\nprovide any information beyond the first-order term \\cite{Berger}.\n\nFor a given spectrum $\\left\\{  \\lambda_{n}\\right\\}  $, one also introduces the\nheat kernel,\n\\begin{equation}\nK\\left(  t\\right)  =\\sum_{n}e^{-\\lambda_{n}t},\\label{Kt}%\n\\end{equation}\nwhich is an another important function describing the relation between the\neigenvalues of the operator and the geometrical property of the manifold. The\nheat kernel is relatively easy to calculate and some methods for calculating\nheat kernels have been developed \\cite{Vas}. In special, Elizalde \\textit{et\nal.} developed a very effective approach for calculating $K\\left(  t\\right)  $\n\\cite{ER,Eli,BEK,BEKL,EBK}.\n\nThere is no doubt that there must exist a relation between the counting\nfunction $N\\left(  \\lambda\\right)  $ and the heat kernel $K\\left(  t\\right)\n$, since they are both determined by the same spectrum $\\left\\{  \\lambda\n_{n}\\right\\}  $. Nevertheless, the relation between $N\\left(  \\lambda\\right)\n$ and $K\\left(  t\\right)  $ is still an open problem in mathematics\n\\cite{Berger}.\n\nThough the definition of counting function $N\\left(  \\lambda\\right)  $, eq.\n(\\ref{NLmd}) or (\\ref{NLsgm1}), is comprehensive and intuitive, other than the\ndefinition of heat kernel $K\\left(  t\\right)  $, eq. (\\ref{Kt}), the\ndefinition of $N\\left(  \\lambda\\right)  $ is not suitable for practical\ncalculations. In fact, if starting from this definition to calculate the\ncounting function, one may always encounter the mathematical difficulty\nmentioned above. In this paper, we will propose an expression for $N\\left(\n\\lambda\\right)  $, which has a similar form with the definition formula of\n$K\\left(  t\\right)  $, eq. (\\ref{Kt}), and is more operable than the\ndefinition (\\ref{NLsgm1}). This expression can be regarded as an alternative\ndefinition formula for $N\\left(  \\lambda\\right)  $.\n\nAs a main result of this paper, we provide a relation\\ between the counting\nfunction $N\\left(  \\lambda\\right)  $ and the heat kernel $K\\left(  t\\right)\n$. This problem interests us from both purely mathematical viewpoint and\npractical viewpoint. This relation allows us to calculate $N\\left(\n\\lambda\\right)  $ from $K\\left(  t\\right)  $ which, as mentioned above, is\noften relatively easier to calculate.\n\nThe relation between the counting function $N\\left(  \\lambda\\right)  $ and the\nheat kernel $K\\left(  t\\right)  $ presented in this paper is an integral\ntransformation. However, in practice the heat kernel is often given in the\nform of a series expansion and in most cases only the first several expansion\ncoefficients can be obtained \\cite{Vas}. This requires us to integrate term by\nterm. Nevertheless, the series expansion of $K\\left(  t\\right)  $ is not\nconvergent uniformly. As a result, the integral of some terms will be\ndivergent. The counting function $N\\left(  \\lambda\\right)  $ is of course\nfinite; the divergences are caused by illegally integrating a nonuniformly\nconvergent series term by term. For dealing with this problem, we provide a\nrenormalization procedure to remove the divergences.\n\nIn fact, the results of this paper provide two approaches for calculating the\ncounting function $N\\left(  \\lambda\\right)  $: (1) by the expression for\n$N\\left(  \\lambda\\right)  $ presented in this paper, one can calculate\n$N\\left(  \\lambda\\right)  $ directly, and (2) by the relation between\n$N\\left(  \\lambda\\right)  $ and $K\\left(  t\\right)  $, one can first calculate\nthe heat kernel $K\\left(  t\\right)  $ and then calculate $N\\left(\n\\lambda\\right)  $ from $K\\left(  t\\right)  $. As a comparison, we calculate\n$N\\left(  \\lambda\\right)  $ for a $D$-dimensional box by both these two\napproaches, respectively.\n\nAs an example of the calculation of counting functions by the relation between\n$N\\left(  \\lambda\\right)  $ and $K\\left(  t\\right)  $, we calculate $N\\left(\n\\lambda\\right)  $ from $K\\left(  t\\right)  $ for three-dimensional balls.\n\nAs another example, we calculate the counting function from the heat kernel\nfor the minus Laplace operator in a two-dimensional region with irregular\nshape and nontrivial topology. For two-dimensional heat kernels, there are two\nknown results: the heat kernel for a simply-connected region \\cite{NPD} and a\nhypothesis made by Kac on the heat kernel for the multiply-connected case\n\\cite{Kac}. In the present paper, we first generalize the heat kernel for a\nsimply-connected region given in \\cite{NPD} to the case of a\nmultiply-connected region which is bounded by a smooth but irregular curve and\nwith some holes. This result proves Kac's conjecture. Then, we calculate the\ncounting function for the two-dimensional multiply-connected region from the\nheat kernel by the relation between $N\\left(  \\lambda\\right)  $ and $K\\left(\nt\\right)  $.\n\nAs applications, we discuss the problem that how to calculate the asymptotic\nexpressions for eigenvalue spectra and state densities from the counting\nfunction $N\\left(  \\lambda\\right)  $. By $N\\left(  \\lambda\\right)  $, we\nconstruct an equation for eigenvalues and then provide an approximate solution\nin which the eigenvalue is expressed as a function of heat kernel\ncoefficients. Moreover, we also provide two expressions for state densities by\nthe relation between $N\\left(  \\lambda\\right)  $ and $K\\left(  t\\right)  $ and\nby the expression of $N\\left(  \\lambda\\right)  $ provided in the present paper.\n\nThe counting function and the heat kernel are interesting in both mathematics\nand physics:\n\nIn mathematics, the relation between the spectrum of the Laplace operator on a\nRiemannian manifold and the geometry of this Riemannian manifold is an\nimportant subject \\cite{Berger,Berard,Milnor,GWW}, and the problem of spectral\nasymptotics is one of the central problems in the theory of partial\ndifferential operators \\cite{Ivrii}. One of the main problems is to seek the\nasymptotic expansions of the counting function $N\\left(  \\lambda\\right)  $ and\nthe heat kernel $K\\left(  t\\right)  $. The general relation between $N\\left(\n\\lambda\\right)  $ and $K\\left(  t\\right)  $ is still unknown. Especially,\nthere is no general method for calculating the asymptotic expansion of the\ncounting function $N\\left(  \\lambda\\right)  $.\n\nIn physics, the spectrum of the Laplace operator on a Riemannian manifold can\nbe directly applied to boundary quantum fields. Moreover, reconstructing the\ngeometrical property of a system from an eigenproblem is an interesting and\nimportant problem. For example, Aurich \\textit{et al.} reconstruct the shape\nof the universe from the result of the eigenproblem \\cite{ALSH}. The counting\nfunction is directly related to the Casimir effect, the spectrum problem, and\nthe state density, etc. Moreover, there are many studies on heat kernels. In\nquantum field theory, it is of crucial importance to evaluate the one-loop\ndivergences, and the control of the ultra-violet divergences can be achieved\nby using heat kernel regularization methods \\cite{Vas,BCP,Avr}. The heat\nkernel expansion becomes a standard tool in the calculations of vacuum\nenergies \\cite{BK}, the Casimir effect \\cite{Mar,EMN}, and quantum anomalies\n\\cite{MV}. Now a large amount of research has been devoted to quantum gravity\nbased on the heat kernel expansion, including semiclassical approaches\n\\cite{Sha}, black hole thermodynamics \\cite{CO}, wormhole physics \\cite{KB},\nheat kernels in curved space \\cite{Sal}, and supergravity theories\n\\cite{BH,Hoo}. In addition, the heat kernel expansion is also used to study\nstring theory \\cite{KKS,BN,MSN} and noncommutative field theory\n\\cite{FGV,ABBP,GIV}. The relation between the counting function and the heat\nkernel allows one to introduce the concept of the counting function into these\nfields. Moreover, there exist some indirect ways to construct the heat kernel\nwithout solving the heat equation, and the heat kernel coefficients for\nvarious manifolds have been discussed \\cite{Vas,BEK,PN,DK}, which makes the\nheat kernel expansion a very convenient and effective tool. By the relation\nprovided in the present paper, such results of the heat kernel can be directly\napplied to the problem of the counting function.\n\nIn section \\ref{definitionofNL}, we provide an expression for the counting\nfunction $N\\left(  \\lambda\\right)  $, which is more operable compared to the\ndefinition (\\ref{NLsgm1}) and can be regarded as an alternative definition\nformula for the counting function. In section \\ref{relation}, we point out a\nrelation between the counting function $N\\left(  \\lambda\\right)  $ and the\nheat kernel $K\\left(  t\\right)  $. In section \\ref{calculating}, we provide a\nrenormalization procedure for removing the divergences appearing in the\ncalculation of $N\\left(  \\lambda\\right)  $ from a nonuniformly convergent\nseries expansion of $K\\left(  t\\right)  $. In section \\ref{D-box}, we compare\nthe two approaches for calculating $N\\left(  \\lambda\\right)  $ using a\n$D$-dimensional box as an example. In section \\ref{3Dballs}, we calculate the\ncounting function of a three-dimensional ball from the heat kernel. In section\n\\ref{2Dmul}, for two-dimensional cases, we first generalize the heat kernel in\na simply-connected region to a multiply-connected region, and then calculate\nthe counting function in a multiply-connected region from the heat kernel. The\nresult of the heat kernel in the multiply-connected region obtained in the\npresent paper proves Kac's hypothesis. In sections \\ref{spectrum} and\n\\ref{statedensity}, as applications of the counting function, we discuss how\nto calculate the asymptotic expressions for spectra and state densities from\n$N\\left(  \\lambda\\right)  $. The conclusions are summarized in section\n\\ref{dc}.\n\n\\section{An expression for the counting function: an alternative definition\nformula\\label{definitionofNL}}\n\nThe definition formula for the counting function $N\\left(  \\lambda\\right)  $,\neq. (\\ref{NLsgm1}), is not suitable for practical calculations. In fact,\nstarting from this definition to calculate $N\\left(  \\lambda\\right)  $, one\nmay encounter many difficulties, e.g., the unsolved problems in number theory\n\\cite{Berger}. Contrarily, the definition formula for the heat kernel, eq.\n(\\ref{Kt}), is more operable. In this section, we present an expression for\n$N\\left(  \\lambda\\right)  $, which has a similar form with that of the heat\nkernel $K\\left(  t\\right)  $, eq. (\\ref{Kt}). Such an expression can be\nregarded as an operable definition formula for $N\\left(  \\lambda\\right)  $.\n\n\\begin{theorem}%\n\\begin{equation}\nN\\left(  \\lambda\\right)  =\\lim_{\\beta\\rightarrow\\infty}\\sum_{n}\\frac\n{1}{e^{\\beta\\left(  \\lambda_{n}-\\lambda\\right)  }+1}.\\label{asymptotic}%\n\\end{equation}\n\n\\end{theorem}\n\n\\Proof{\nObserving that\\[\n\\lim_{\\beta\\rightarrow\\infty}\\frac{1}{e^{\\beta\\left(  \\lambda_{n}-\\lambda\\right)  }+1}=\\left\\{\n\\begin{array}\n[c]{cc}1, & \\text{ when }\\lambda_{n}<\\lambda,\\\\\n0, & \\text{ when }\\lambda_{n}>\\lambda,\n\\end{array}\n\\right.\n\\]\nwe have\\[\n\\lim_{\\beta\\rightarrow\\infty}\\sum_{n}\\frac{1}{e^{\\beta\\left(  \\lambda\n_{n}-\\lambda\\right)  }+1}=\\sum_{\\lambda_{n}<\\lambda}1=N\\left(  \\lambda\\right)\n.\n\\]\n}\n\nComparing with eq. (\\ref{NLsgm1}), eq. (\\ref{asymptotic}) is obviously more\noperable, since eq. (\\ref{asymptotic}) converts the partial sum in eq.\n(\\ref{NLsgm1}), $\\sum_{\\lambda_{n}<\\lambda}$, into a sum over all possible\nvalues, $\\sum_{\\lambda_{n}<\\infty}$. This will, of course, make the\ncalculation easier.\n\n\\section{The relation between counting function and heat kernel\n\\label{relation}}\n\nIn this section, we provide a relation between the counting function $N\\left(\n\\lambda\\right)  $ and the heat kernel $K\\left(  t\\right)  $. This relation is\nan interesting mathematical result and is useful for practical calculations.\nAs mentioned above, the calculation of the counting function $N\\left(\n\\lambda\\right)  $ is more difficult than that of the heat kernel $K\\left(\nt\\right)  $. With the relation between $N\\left(  \\lambda\\right)  $ and\n$K\\left(  t\\right)  $ given in the following, one can first calculate\n$K\\left(  t\\right)  $ and then calculate $N\\left(  \\lambda\\right)  $ from the\nresult of $K\\left(  t\\right)  $.\n\nThe relation between $N\\left(  \\lambda\\right)  $ and $K\\left(  t\\right)  $ is\nas follows.\n\n\\begin{theorem}%\n\\begin{equation}\nK\\left(  t\\right)  =t\\int_{0}^{\\infty}N\\left(  \\lambda\\right)  e^{-\\lambda\nt}d\\lambda.\\label{K-N}%\n\\end{equation}\n\n\\end{theorem}\n\n\\Proof{\nThe generalized Abel partial summation formula reads\\begin{equation}\n\\sum_{u_{1}<\\lambda_{n}\\leq u_{2}}b\\left(  n\\right)  f\\left(  \\lambda\n_{n}\\right)  =B\\left(  u_{2}\\right)  f\\left(  u_{2}\\right)  -B\\left(\nu_{1}\\right)  f\\left(  u_{1}\\right)  -\\int_{u_{1}}^{u_{2}}B\\left(  u\\right)\nf^{\\prime}\\left(  u\\right)  du, \\label{abel}\\end{equation}\nwhere $\\lambda_{i}\\in\\mathbb{R}$,\n$\\lambda_{1}\\leq\\lambda_{2}\\leq\\cdots \\leq\\lambda_{n}\\leq\\cdots$,\nand $\\lim_{n\\rightarrow\\infty}\\lambda_{n}=\\infty$. $f\\left(\nu\\right)  $ is a continuously differentiable function on $\\left[\nu_{1},u_{2}\\right]  $ $\\left(  0\\leq u_{1}<u_{2}\\text{,\n}\\lambda_{1}\\leq u_{2}\\right)  $, $b\\left(  n\\right)  $ $\\left(\nn=1,2,3,\\cdots\\right)  $ are arbitrary complex numbers, and\n$B\\left(  u\\right)  =\\sum_{\\lambda_{n}\\leq u}b\\left(  n\\right)  $.\nWe apply the generalized Abel partial summation formula, eq.\n(\\ref{abel}), with $f\\left(  u\\right)  =e^{-u\\left( s-s_{0}\\right)\n}$ and $b\\left(  n\\right)  =a_{n}e^{-\\lambda_{n}s_{0}}$, where\n$s$, $s_{0}\\in\\mathbb{C}$. Then\\begin{equation}\nA\\left(  u_{2},s\\right)  -A\\left(  u_{1},s\\right)  =A\\left(  u_{2},s_{0}\\right)  e^{-u_{2}\\left(  s-s_{0}\\right)  }-A\\left(  u_{1},s_{0}\\right)\ne^{-u_{1}\\left(  s-s_{0}\\right)  }+\\left(  s-s_{0}\\right)  \\int_{u_{1}}^{u_{2}}A\\left(  u,s_{0}\\right)  e^{-u\\left(  s-s_{0}\\right)  }du,\n\\label{Au12s}\\end{equation}\nwhere\\begin{equation}\nA\\left(  u,s\\right)  =\\sum_{\\lambda_{n}\\leq u}a_{n}e^{-\\lambda_{n}s}.\n\\label{Aus}\\end{equation}\nSetting $a_{n}=1$ in eq. (\\ref{Aus}), we find\\[\nA\\left(  \\lambda,0\\right)  =\\sum_{\\lambda_{n}\\leq\\lambda}1=N\\left(\n\\lambda\\right)  ,\n\\]\nthe counting function, and\\[\nA\\left(  \\infty,t\\right)  =\\sum_{n}e^{-\\lambda_{n}t}=K\\left(  t\\right)  ,\n\\]\nthe heat kernel. By eq. (\\ref{Aus}), we also have $A\\left(\n0,t\\right)  =0$.\nThen, by eq. (\\ref{Au12s}), we have\\begin{equation}\nK\\left(  t\\right)  =A\\left(  \\infty,t\\right)  -A\\left(  0,t\\right)  =t\\int\n_{0}^{\\infty}N\\left(  \\lambda\\right)  e^{-\\lambda t}d\\lambda.\n\\end{equation}\nThis is just eq. (\\ref{K-N}). }\n\n\\begin{theorem}%\n\\begin{equation}\nN\\left(  \\lambda\\right)  =\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty}K\\left(\nt\\right)  \\frac{e^{\\lambda t}}{t}dt,\\text{ \\ \\ }c>\\lim_{n\\rightarrow\\infty\n}\\frac{\\ln n}{\\lambda_{n}}.\\label{N-K}%\n\\end{equation}\n\n\\end{theorem}\n\n\\Proof{\nBy the Perron formula, we have\\begin{equation}\n\\sum_{\\mu_{n}<x}a_{n}=\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty}f\\left(\nt\\right)  \\frac{x^{t}}{t}dt, \\label{Perron}\\end{equation}\nwhere\\begin{equation}\nf\\left(  s\\right)  =\\sum_{n=1}^{\\infty}\\frac{a_{n}}{\\mu_{n}^{s}}, \\label{fs}\\end{equation}\nand $c$ is a constant which is greater than the abscissa of absolute\nconvergence of the Dirichlet series $f\\left(  s\\right)  $. Setting\n\\[\na_{n}=1\\text{ and }\\mu_{n}=e^{\\lambda_{n}}\\]\nin eq. (\\ref{fs}), we obtain the heat kernel,\\[\nf\\left(  t\\right)  =\\sum_{n=1}^{\\infty}e^{-\\lambda_{n}t}=K\\left(  t\\right)  .\n\\]\nThe abscissa of absolute convergence of $f\\left(  t\\right)  $\nequals its abscissa of convergence, equaling\n$\\overline{\\lim}_{n\\rightarrow\\infty}\\ln\nn\/\\lambda_{n}=\\lim_{n\\rightarrow\\infty}\\ln n\/\\lambda_{n}$. Thus,\nby eq.\n(\\ref{Perron}), we have\\[\nN\\left(  \\lambda\\right)  =\\sum_{\\lambda_{n}<\\lambda}1=\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty}K\\left(  t\\right)  \\frac{e^{\\lambda t}}{t}dt,\n\\]\nand $c>\\lim_{n\\rightarrow\\infty}\\ln n\/\\lambda_{n}$. This proves\nthe theorem. }\n\nClearly, the relation between the counting function $N\\left(  \\lambda\\right)\n$ and the heat kernel $K\\left(  t\\right)  $ obtained above is an integral\ntransformation and the corresponding inverse transformation.\n\n\\section{Calculating $N\\left(  \\lambda\\right)  $ from an asymptotics of\n$K\\left(  t\\right)  $\\label{calculating}}\n\nIn principle, we can calculate the counting function $N\\left(  \\lambda\\right)\n$ from the heat kernel $K\\left(  t\\right)  $ or calculate $K\\left(  t\\right)\n$ from $N\\left(  \\lambda\\right)  $ by the relation (\\ref{N-K}) or (\\ref{K-N})\ndirectly. However, the relation given above is an integral transformation.\nTherefore, when the heat kernel $K\\left(  t\\right)  $ is in the form of a\nseries expansion, practically, the above relation is available only when this\nintegral transformation can be applied to each term of the series, or, in\nother words, this series must can be integrated term by term. If the series is\nnot uniformly convergent, even though the integral of the sum function is\nconvergent, the integrals of some of the terms may be divergent. In this case,\nwe need a renormalization procedure to remove the divergences.\n\n\\subsection{An asymptotics for $N\\left(  \\lambda\\right)  $}\n\nConcretely, the expansion of heat kernel $K\\left(  t\\right)  $ can be\nexpressed as \\cite{BEK}%\n\\begin{equation}\nK\\left(  t\\right)  =\\left(  4\\pi t\\right)  ^{-D\/2}\\sum_{k=0,\\frac{1}%\n{2},1,\\cdots}^{\\infty}B_{k}t^{k},\\label{Ktseries}%\n\\end{equation}\nwhere $B_{k}$ is the heat kernel coefficient, $D$ is the dimension of space.\nIn fact, in practice only the first several heat kernel coefficients can be\nobtained, so we have to integrate term by term. Substituting eq.\n(\\ref{Ktseries}) into eq. (\\ref{N-K}) and exchanging the order of integration\nand summation gives%\n\n\\begin{equation}\nN\\left(  \\lambda\\right)  =\\left(  4\\pi\\right)  ^{-D\/2}\\sum_{k=0,\\frac{1}%\n{2},1,\\cdots}^{\\infty}B_{k}I_{D}\\left(  k\\right)  ,\\label{Nbare}%\n\\end{equation}\nwhere%\n\\begin{equation}\nI_{D}\\left(  k\\right)  =\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty}%\nt^{-D\/2+k}\\frac{e^{\\lambda t}}{t}dt.\\label{intDk}%\n\\end{equation}\nNevertheless, for a power series, unless the series is uniformly convergent,\nit is not permissible to integrate term by term, i.e., the order of\nintegration and summation cannot be exchanged. When calculating an integral of\na series that is not uniformly convergent term by term, one may find that the\nintegrals of some terms diverge. In our case, it can be directly seen that\nonly when $k<D\/2+1$, $I_{D}\\left(  k\\right)  $ is convergent. That is to say,\nthe term with $k\\geq D\/2+1$ needs to be renormalized for obtaining a finite result.\n\n\\subsection{Renormalization}\n\nThe integral (\\ref{intDk}) is divergent when $k\\geq D\/2+1$. For obtaining the\ncounting function $N\\left(  \\lambda\\right)  $ from the series (\\ref{Nbare}),\nwe need to make sense of these divergent integrals. In other words, we need a\nrenormalization procedure to remove the divergences.\n\nIt can be directly seen that eq. (\\ref{intDk}) is the inverse Laplace\ntransformation of the function $t^{-D\/2+k-1}$. Thus for $k<D\/2+1$, we have\n\\cite{AS}\n\\begin{equation}\nI_{D}\\left(  k\\right)  =\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty\n}t^{-D\/2+k-1}e^{\\lambda t}dt=\\frac{1}{\\Gamma\\left(  1+D\/2-k\\right)  }%\n\\lambda^{D\/2-k},\\text{ }\\left(  k<D\/2+1\\right)  .\\label{intF}%\n\\end{equation}\nThe definition domain for the gamma function $\\Gamma\\left(  1+D\/2-k\\right)  $\nis $k<D\/2+1$. Now, we deal with the case of $k\\geq D\/2+1$. When $-D\/2+k-1$ is\na non-negative integer, the integral (\\ref{intF}) equals $\\delta^{\\left(\n-D\/2+k-1\\right)  }\\left(  \\lambda\\right)  $ \\cite{Handbook}, where\n$\\delta^{\\left(  n\\right)  }\\left(  \\lambda\\right)  $ is the $n$th derivative\nof $\\delta\\left(  \\lambda\\right)  $. When $-D\/2+k-1$ is a positive half\ninteger, we can redefine the gamma function by analytically continuing the\ngamma function via\\ the recurrence relation $\\Gamma\\left(  x\\right)  =\\frac\n{1}{x}\\Gamma\\left(  x+1\\right)  $ \\cite{Stone}. Concretely, when defining the\ngamma function as $\\Gamma\\left(  x\\right)  =\\frac{1}{x}\\Gamma\\left(\nx+1\\right)  $, the definition domain of the gamma function is changed to\n$k<D\/2+2$. Repeating this procedure will eventually analytically continue the\ngamma function to the whole real axis.\n\nFrom the above result we can see that, like that in the perturbative expansion\nof the scattering amplitude in quantum field theory, the first several terms\nare convergent (like tree diagrams) and the other terms are divergent and\nneeded to be renormalized (like loop diagrams). For clarity, write the\ncounting function $N\\left(  \\lambda\\right)  $ in two parts:%\n\\begin{equation}\nN\\left(  \\lambda\\right)  =N_{T}\\left(  \\lambda\\right)  +N_{L}\\left(\n\\lambda\\right)  ,\n\\end{equation}\nwhere%\n\\begin{equation}\nN_{T}\\left(  \\lambda\\right)  =\\left(  4\\pi\\right)  ^{-D\/2}\\sum_{k=0,\\frac\n{1}{2},1,\\cdots}^{\\frac{D}{2}+\\frac{1}{2}}B_{k}I_{D}\\left(  k\\right)\n\\label{NT}%\n\\end{equation}\nis the convergent part and%\n\\begin{equation}\nN_{L}\\left(  \\lambda\\right)  =\\left(  4\\pi\\right)  ^{-D\/2}\\sum_{k=\\frac{D}%\n{2}+1,\\frac{D}{2}+\\frac{3}{2},\\cdots}^{\\infty}B_{k}I_{D}^{R}\\left(  k\\right)\n\\label{NL}%\n\\end{equation}\nis the renormalized divergent part, where $I_{D}^{R}\\left(  k\\right)  $\ndenotes the renormalized result. $N_{L}\\left(  \\lambda\\right)  $ is only a\nhigher-order contribution to the counting function and is often negligible.\n\nAfter renormalization, eqs. (\\ref{NT}) and (\\ref{NL}) can be written as%\n\\begin{align}\nN\\left(  \\lambda\\right)   & =\\sum_{k=0,\\frac{1}{2},1,\\cdots}^{\\infty}%\nC_{k}\\lambda^{D\/2-k}+\\sum_{l=0,1,2,\\cdots}^{\\infty}\\left(  4\\pi\\right)\n^{-D\/2}B_{1+D\/2+l}\\delta^{\\left(  l\\right)  }\\left(  \\lambda\\right)\n\\nonumber\\\\\n& =\\sum_{k=0,\\frac{1}{2},1,\\cdots}^{\\frac{D}{2}}C_{k}\\lambda^{D\/2-k}%\n+\\sum_{k=\\frac{D}{2}+\\frac{1}{2},\\frac{D}{2}+\\frac{3}{2},\\cdots}^{\\infty}%\nC_{k}\\lambda^{D\/2-k}+\\sum_{k=\\frac{D}{2}+1,\\frac{D}{2}+2,\\cdots}^{\\infty\n}\\left(  4\\pi\\right)  ^{-D\/2}B_{k}\\delta^{\\left(  k-\\left(  \\frac{D}%\n{2}+1\\right)  \\right)  }\\left(  \\lambda\\right) \\label{NLbyCk}%\n\\end{align}\nwith the counting function coefficient%\n\\begin{equation}\nC_{k}=\\left(  4\\pi\\right)  ^{-D\/2}\\frac{B_{k}}{\\Gamma\\left(  1+D\/2-k\\right)\n}.\\label{Ck}%\n\\end{equation}\nIt should be emphasized that the delta-function terms in eq. (\\ref{NLbyCk})\nwill contribute the zero-point energy; this is, such contributions will play\nan essentially important role in the calculation of the Casimir effect.\n\n\\section{Comparison of the two approaches for calculating $N\\left(\n\\lambda\\right)  $: $D$-dimensional boxes \\label{D-box}}\n\nThe results presented in sections \\ref{definitionofNL} and \\ref{relation} show\nthat there are two approaches for calculating the counting function $N\\left(\n\\lambda\\right)  $. The first is to directly calculate $N\\left(  \\lambda\n\\right)  $ by eq. (\\ref{asymptotic}), the operable expression of $N\\left(\n\\lambda\\right)  $ presented in this paper; the second is to first calculate\nthe heat kernel $K\\left(  t\\right)  $ and then to calculate $N\\left(\n\\lambda\\right)  $ from $K\\left(  t\\right)  $ by the relation between $N\\left(\n\\lambda\\right)  $ and $K\\left(  t\\right)  $. In this section, as an example,\nwe calculate $N\\left(  \\lambda\\right)  $ by the above two approaches\nrespectively for a rectangular $D$-dimensional box.\n\nThe spectrum of the minus Laplace operator of a $D$-dimensional rectangle box reads%\n\n\\begin{equation}\n\\lambda_{n_{1},n_{2},\\cdots,n_{D}}=\\pi^{2}\\left(  \\frac{n_{1}^{2}}{L_{1}^{2}%\n}+\\frac{n_{2}^{2}}{L_{2}^{2}}+\\cdots+\\frac{n_{D}^{2}}{L_{D}^{2}}\\right)\n,~~n_{i}=1,2\\cdots,\n\\end{equation}\nwhere $L_{i}$ is the side length.\n\nFirst, we directly calculate the counting function $N\\left(  \\lambda\\right)  $\nfrom eq. (\\ref{asymptotic}):%\n\\begin{equation}\nN\\left(  \\lambda\\right)  =\\lim_{\\beta\\rightarrow\\infty}\\sum_{n_{1}%\n,n_{2},\\cdots,n_{D}}\\frac{1}{e^{\\beta\\left(  \\lambda_{n_{1},n_{2},\\cdots\n,n_{D}}-\\lambda\\right)  }+1}.\n\\end{equation}\nBy the Euler-MacLaurin formula $\\sum\\limits_{n=0}^{\\infty}F(n)=\\int\n_{0}^{\\infty}F(n)dn+\\frac{1}{2}F(0)-\\frac{1}{2!}B_{2}F^{\\prime}(0)-\\frac\n{1}{4!}B_{4}F^{\\prime\\prime\\prime}(0)+\\cdots$, where $B_{2}$, $B_{4}$,\n$\\cdots$ are Bernoulli numbers, we achieve%\n\\begin{align}\nN\\left(  \\lambda\\right)   &  =\\left(  4\\pi\\right)  ^{-D\/2}\\left[  \\frac\n{V}{\\Gamma\\left(  D\/2+1\\right)  }\\lambda^{D\/2}-\\sum_{i=1}^{D}\\frac{V}{L_{i}%\n}\\frac{\\sqrt{\\pi}}{\\Gamma\\left(  D\/2+1\/2\\right)  }\\lambda^{D\/2-1\/2}\\right.\n\\nonumber\\\\\n&  +\\left.  \\sum_{i<j}^{D}\\frac{V}{L_{i}L_{j}}\\frac{\\pi}{\\Gamma\\left(\nD\/2\\right)  }\\lambda^{D\/2-1}+\\cdots+\\left(  -1\\right)  ^{D}\\pi^{D\/2}\\right]\n,\\label{NfromK}%\n\\end{align}\nwhere $V=$ $L_{1}L_{2}\\cdot\\cdots\\cdot L_{D}$ is the volume.\n\nAlternatively, we can first calculate the heat kernel from the definition\nformula of $K\\left(  t\\right)  $,\n\\begin{equation}\nK\\left(  t\\right)  =\\sum_{n_{1},n_{2},\\cdots,n_{D}}e^{-\\lambda\\left(\nn_{1},n_{2},\\cdots,n_{D}\\right)  t}.\n\\end{equation}\nA direct calculation by the Euler-MacLaurin formula gives the heat kernel\ncoefficients:%\n\\begin{align}\nB_{0}  &  =V,\\ B_{1\/2}=-\\sqrt{\\pi}\\sum_{i=1}^{D}\\frac{V}{L_{i}},\\ B_{1}%\n=\\pi\\sum_{i<j}^{D}\\frac{V}{L_{i}L_{j}},\\cdots,\\nonumber\\\\\nB_{\\nu\/2}  &  =\\left(  -1\\right)  ^{\\nu}\\pi^{\\nu\/2}\\sum_{i_{1}<i_{2}%\n<\\cdots<i_{\\nu}}^{D}\\frac{V}{L_{i_{1}}L_{i_{2}}\\cdots L_{i_{\\nu}}%\n}.\\label{Bkbox}%\n\\end{align}\nBy the relation between the counting function coefficient and the heat kernel\ncoefficient, eq. (\\ref{Ck}), we also achieve eq. (\\ref{NfromK}).\n\nFrom the above result, we can see the equivalence between these two approaches.\n\n\\section{Calculating $N\\left(  \\lambda\\right)  $ from $K\\left(  t\\right)  $:\nthree-dimensional balls \\label{3Dballs}}\n\nIn this section, as an example, we calculate the counting function $N\\left(\n\\lambda\\right)  $ from the series expansion of the heat kernel $K\\left(\nt\\right)  $ for a three-dimensional ball. It should be emphasized that for\ncalculating the heat kernel $K\\left(  t\\right)  $, it is not needed to know\nthe eigenvalue spectrum in advance \\cite{BEK}.\n\nFor a three-dimensional ball, from eq.(\\ref{Ck}), the expansion coefficients\nof the counting function $N\\left(  \\lambda\\right)  $ can be obtained as%\n\\begin{equation}\nC_{k}=\\left(  4\\pi\\right)  ^{-3\/2}\\frac{B_{k}}{\\Gamma\\left(  5\/2-k\\right)\n}.\\label{Ck3ball}%\n\\end{equation}\nThe heat kernel coefficients for Dirichlet, Neumann, and Robin boundary\nconditions for three-dimensional balls are calculated explicitly in \\cite{BEK}.\n\nTake the case of Dirichlet boundary condition as an example. Ref. \\cite{BEK}\ngives the first $21$ heat kernel coefficients. From this, we can obtain the\nfirst $21$ counting function coefficients $C_{k}$ by eq. (\\ref{Ck3ball}). Only\ntaking the tree-diagram-like part into account, from eq. (\\ref{NT}), we have\n\\begin{equation}\nN\\left(  \\lambda\\right)  =\\frac{2}{9\\pi}R^{3}\\lambda^{3\/2}-\\frac{1}{4}%\nR^{2}\\lambda+\\frac{2}{3\\pi}R\\lambda^{1\/2}-\\frac{1}{48}-\\frac{2}{315\\pi}%\n\\frac{1}{R\\lambda^{1\/2}}.\\label{NT3Ball}%\n\\end{equation}\nIt can be directly checked that the tree-diagram-like part provides the main\ncontribution to the $N\\left(  \\lambda\\right)  $.\n\n\\section{Multiply-connected cases: two-dimensional $K\\left(  t\\right)  $ and\n$N\\left(  \\lambda\\right)  $\\label{2Dmul}}\n\nAn interesting special case is about two-dimensional counting function and\ntwo-dimensional heat kernel, which is just the original problem formulated by\nKac as \"Can one hear the shape of a drum?\". The first result of this problem\nis proved by Weyl \\cite{Weyl} and then improved by Pleijel \\cite{Pleijel} and\nKac \\cite{Kac}. In \\cite{Kac}, Kac pointed out a special relation between\n$N\\left(  \\lambda\\right)  $ and $K\\left(  t\\right)  $: in two dimensions, the\nleading-order term of $N\\left(  \\lambda\\right)  $ is accidentally equal to the\nleading-order term of $K\\left(  t\\right)  $. However, beyond the leading\norder, there is no further result. Moreover, for the multiply-connected case,\nKac made a hypothesis about the topological contribution on the heat kernel\n$K\\left(  t\\right)  $: the contribution from the nontrivial topology\n(connectivity) is in proportion to the Euler-Poincar\\'{e} characteristic number.\n\nIn this section, (1) with the help of the relation between $N\\left(\n\\lambda\\right)  $ and $K\\left(  t\\right)  $ given above, we give a proof for\nKac's result: the leading-order term of $N\\left(  \\lambda\\right)  $ equals the\nleading-order term of $K\\left(  t\\right)  $; (2) based on the Gauss-Bonnet\ntheorem, we first generalize the simply-connected heat kernel $K\\left(\nt\\right)  $ given in \\cite{NPD} to the multiply-connected case (this result\nproves Kac's conjecture: in two dimensions, the topology contribution is\nproportion to the Euler-Poincar\\'{e} characteristic number), and, then,\ncalculate the two-dimensional\\ $N\\left(  \\lambda\\right)  $ in a\nmultiply-connected region with the help of the relation between $N\\left(\n\\lambda\\right)  $ and $K\\left(  t\\right)  $.\n\n\\subsection{The leading contribution of $N\\left(  \\lambda\\right)  $ and\n$K\\left(  t\\right)  $ in two dimensions}\n\nUsing the relation between $N\\left(  \\lambda\\right)  $ and $K\\left(  t\\right)\n$, we first prove that when the number of eigenstates per unit interval (the\ndensity of eigenstates) $\\rho\\left(  \\lambda\\right)  $ is a constant,\n$N\\left(  \\lambda\\right)  $ equals $K\\left(  t\\right)  $ in the limit\n$\\lambda\\rightarrow\\infty$ or $t\\rightarrow0$, i.e., in the limit\n$\\lambda\\rightarrow\\infty$ or $t\\rightarrow0$,%\n\\begin{equation}\nN\\left(  \\lambda\\right)  =K\\left(  \\frac{1}{\\lambda}\\right)  \\text{\\ or\n\\ }N\\left(  \\frac{1}{t}\\right)  =K\\left(  t\\right)  .\\label{lttendsinf}%\n\\end{equation}\n\n\nThe proof is straightforward. In the limit $\\lambda\\rightarrow\\infty$ or\n$t\\rightarrow0$, the summations can be converted into integrals:%\n\\begin{align}\nN\\left(  \\lambda\\right)   &  =\\sum_{\\lambda_{n}<\\lambda}1=\\int_{0}^{\\lambda\n}\\rho\\left(  \\lambda^{\\prime}\\right)  d\\lambda^{\\prime},\\\\\nK\\left(  t\\right)   &  =\\sum_{n}e^{-\\lambda_{n}t}=\\int_{0}^{\\infty}\\rho\\left(\n\\lambda^{\\prime}\\right)  e^{-\\lambda^{\\prime}t}d\\lambda^{\\prime}.\n\\end{align}\nIf $\\rho\\left(  \\lambda\\right)  =C$, where $C$ is a constant, then%\n\\begin{align}\nN\\left(  \\lambda\\right)   &  =C\\lambda,\\\\\nK\\left(  t\\right)   &  =\\frac{C}{t}.\n\\end{align}\nThis proves eq. (\\ref{lttendsinf}).\n\nIn two dimensions, the state density for the eigenstate of the minus Laplace\noperator is a constant, so the leading contribution of the counting function\nequals the leading contribution of the heat kernel. This is just the case\nappeared in Kac's work \\cite{Kac}.\n\n\\subsection{$K\\left(  t\\right)  $ and $N\\left(  \\lambda\\right)  $ in\nMultiply-connected regions}\n\nFor achieving the counting function in a multiply-connected region by the\nrelation between $K\\left(  t\\right)  $ and $N\\left(  \\lambda\\right)  $, we\nneed to start with the heat kernel in a multiply-connected region. For\ntwo-dimensional heat kernels, there are two known results: in the\nsimply-connected case, ref. \\cite{NPD} provides a series expansion of the heat\nkernel; in the multiply-connected case, Kac makes a hypothesis on the heat\nkernel \\cite{Kac}. In the following, based on the Gauss-Bonnet theorem, we\nfirst generalize the simply-connected result of heat kernel $K\\left(\nt\\right)  $ given in \\cite{NPD} to the multiply-connected case. The result can\nbe viewed as a proof of Kac's hypothesis.\n\nIn the simply-connected case, the heat kernel for a two-dimensional plane\nbounded by a smooth curve $\\Gamma_{s}$ and with the Dirichlet boundary\ncondition on $\\Gamma_{s}$ reads \\cite{NPD}%\n\\begin{equation}\nK\\left(  t\\right)  =\\frac{S}{4\\pi t}-\\frac{L}{8\\sqrt{\\pi t}}+\\frac{1}{12\\pi\n}\\int_{\\Gamma_{s}}k\\left(  s\\right)  ds+\\cdots,\\label{Kt2Dsimply}%\n\\end{equation}\nwhere $S$ is the area of the region, $L$ is the length of $\\Gamma_{s}$,\n$k\\left(  s\\right)  $ is the curvature of the curve $\\Gamma_{s}$ at the point\n$s $.\n\n\\begin{figure}[ptb]\n\\begin{center}\n\\includegraphics[width=5in]{pic.eps} \\label{fig}\n\\end{center}\n\\caption{Converting a multiply-connected region to a simply-connected region.\n}%\n\\end{figure}\n\nTo generalize the result of the simply-connected region, eq. (\\ref{Kt2Dsimply}%\n), to the multiply-connected case, we first convert the multiply-connected\nregion bounded by $\\Gamma_{m}$ (figure 1a) to a simply-connected one bounded\nby a piecewise smooth simple closed curve $\\Gamma_{s}$ (figure 1b). In the\nsimply-connected region illustrated in figure 1b, the first two terms which\nare proportional to the area and the perimeter of the region, respectively,\ncan be calculated by the method in the simply-connected case directly. For the\nthird term, by the Gauss-Bonnet theorem, we can calculate the integral of the\ncurvature along $\\Gamma_{s}$. The Gauss-Bonnet theorem reads \\cite{Chern}\n\\begin{equation}%\n{\\displaystyle\\sum\\limits_{i}}\n\\left(  \\pi-a_{i}\\right)  +\\int_{\\Gamma_{s}}k\\left(  s\\right)  ds+%\n{\\displaystyle\\iint}\nKd\\sigma=2\\pi\\chi,\\label{GB}%\n\\end{equation}\nwhere $a_{i}$ is the interior angle of $\\Gamma_{s}$ at each vertex, $K=0$ in\nthe present case is the Gauss curvature, and $\\chi$ is the Euler-Poincar\\'{e}\ncharacteristic number. From figure 1, since the integrals along $l_{i}^{in}$\nand $l_{i}^{out}$ ($i=1,\\cdots,r$) cancel each other in pairs, we achieve%\n\\begin{equation}\n\\int_{\\Gamma_{m}}k\\left(  s\\right)  ds=\\int_{\\Gamma_{s}}k\\left(  s\\right)\nds,\\label{mtos}%\n\\end{equation}\nwhere $r$ is the number of holes in the region. Moreover, it is also easy to\nsee that%\n\\begin{equation}%\n{\\displaystyle\\sum\\limits_{i=1}^{r}}\n{\\displaystyle\\sum\\limits_{\\alpha=1}^{4}}\n\\left(  \\pi-a_{i}^{\\alpha}\\right)  =r2\\pi.\n\\end{equation}\nThe region bounded by $\\Gamma_{s}$ in figure 1b is simply-connected, so\n$\\chi=1$. Thus, from eq. (\\ref{GB}), we have%\n\\begin{equation}\n\\int_{\\Gamma_{s}}k\\left(  s\\right)  ds=2\\pi\\left(  1-r\\right)  .\\label{intr}%\n\\end{equation}\nBy the approach given by \\cite{MV}, we can calculate the first two terms.\nThen, with eqs. (\\ref{Kt2Dsimply}), (\\ref{mtos}), and (\\ref{intr}), we have%\n\\begin{equation}\nK\\left(  t\\right)  =\\frac{S}{4\\pi t}-\\frac{L}{8\\sqrt{\\pi t}}+\\frac{1-r}%\n{6}.\\label{Ktmul}%\n\\end{equation}\nThis is just the result that Kac hypothesized in \\cite{Kac}.\n\nBy the relation between $N\\left(  \\lambda\\right)  $ and $K\\left(  t\\right)  $,\nwe can directly calculate the counting function in the two-dimensional\nmultiply-connected region:%\n\\begin{equation}\nN\\left(  \\lambda\\right)  =\\frac{S}{4\\pi}\\lambda-\\frac{L}{\\pi}\\sqrt{\\lambda\n}+\\frac{1-r}{6}.\n\\end{equation}\n\n\n\\section{Calculating spectrum from $N\\left(  \\lambda\\right)  $\\label{spectrum}%\n}\n\nFrom the counting function $N\\left(  \\lambda\\right)  $, we can directly obtain\nthe asymptotic expression for the spectrum of the system \\cite{CH}. In view of\nthe fact that one can obtain the heat kernel $K\\left(  t\\right)  $ and,\naccordingly, obtain the counting function $N\\left(  \\lambda\\right)  $ without\nknowing the spectrum in advance \\cite{BEK}, this result can serve as an\napproach for calculating the spectrum.\n\nThe counting function is the number of eigenvalues smaller than $\\lambda$, so\nfor the $n$-th eigenvalue $\\lambda_{n}$, we have%\n\\begin{equation}\nN\\left(  \\lambda_{n}\\right)  =n.\\label{NLbdeqn}%\n\\end{equation}\nTherefore, from eqs. (\\ref{NLbdeqn}) and (\\ref{NLbyCk}), we have%\n\\begin{equation}\n\\left(  4\\pi\\right)  ^{-D\/2}\\sum_{k=0,\\frac{1}{2},1,\\cdots}^{\\infty}%\n\\frac{B_{k}}{\\Gamma\\left(  1+D\/2-k\\right)  }\\lambda_{n}^{D\/2-k}%\n=n.\\label{NLbdR}%\n\\end{equation}\nThe eigenvalue $\\lambda_{n}$ can be solved from eq. (\\ref{NLbdR}).\n\nFor approximately solving eq. (\\ref{NLbdR}), we assume that $\\lambda_{n}$ can\nbe expanded as%\n\\begin{equation}\n\\lambda_{n}=%\n{\\displaystyle\\sum\\limits_{m=0}^{\\infty}}\n\\alpha_{m}n^{\\left(  2-m\\right)  \/D}.\\label{Lbdn}%\n\\end{equation}\nThen we have%\n\\begin{align}\n\\lambda_{n}  &  =4\\pi\\Gamma^{2\/D}\\left(  \\frac{D}{2}+1\\right)  \\frac{1}%\n{B_{0}^{2\/D}}n^{2\/D}-4\\sqrt{\\pi}\\frac{\\Gamma^{1+1\/D}\\left(  \\frac{D}%\n{2}+1\\right)  }{D\\Gamma\\left(  \\frac{D}{2}+\\frac{1}{2}\\right)  }\\frac{B_{1\/2}%\n}{B_{0}^{1+1\/D}}n^{1\/D}\\nonumber\\\\\n&  +\\left[  \\frac{D\\Gamma^{2}\\left(  \\frac{D}{2}\\right)  }{4\\Gamma^{2}\\left(\n\\frac{D}{2}+\\frac{1}{2}\\right)  }\\frac{B_{1\/2}^{2}}{B_{0}^{2}}-\\frac{B_{1}%\n}{B_{0}}\\right]  n^{0}+\\cdots.\n\\end{align}\n\n\nAs examples, we list some spectra obtained by this approach for various\ndimensional balls with Dirichlet, Neumann, and Robin boundary conditions,\nrespectively. The heat kernel coefficients are given in \\cite{BEK}.\n\nFor a three-dimensional ball, the spectrum for the Dirichlet and the Neumann\nor Robin boundary conditions is%\n\n\\begin{equation}\n\\lambda_{n}=\\frac{3}{2}\\left(  6\\pi^{2}\\right)  ^{1\/3}\\frac{1}{R^{2}}%\nn^{2\/3}\\pm\\frac{3}{8}\\left(  6\\pi^{2}\\right)  ^{2\/3}\\frac{1}{R^{2}}%\nn^{1\/3}+\\left(  \\frac{27}{64}\\pi^{2}-2\\right)  \\frac{1}{R^{2}},\n\\end{equation}\nwhere $R$ is the radius. In this equation and following, the upper sign stands\nfor the Dirichlet boundary condition and the lower sign for the Neumann or\nRobin boundary condition.\n\nFor a four-dimensional ball, the spectrum is%\n\n\\begin{equation}\n\\lambda_{n}=\\frac{8}{R^{2}}n^{1\/2}\\pm\\frac{16\\sqrt{2}}{3}\\frac{1}{R^{2}%\n}n^{1\/4}-\\frac{26}{9}\\frac{1}{R^{2}},\n\\end{equation}\nand for a five-dimensional ball, the spectrum is%\n\n\\begin{equation}\n\\lambda_{n}=\\frac{1}{2}\\left(  450\\sqrt{2}\\pi\\right)  ^{2\/5}\\frac{1}{R^{2}%\n}n^{2\/5}\\pm\\frac{15\\pi}{32}\\left(  3600\\pi\\right)  ^{1\/5}\\frac{1}{R^{2}%\n}n^{1\/5}+\\left(  \\frac{1125}{1024}\\pi^{2}-\\frac{20}{3}\\right)  \\frac{1}{R^{2}%\n}.\n\\end{equation}\n\n\n\\section{The state density\\label{statedensity}}\n\nIn this section, we provide two approaches to achieve the state density based\non the above results.\n\n\\subsection{Calculating the state density from $N\\left(  \\lambda\\right)  $ and\n$K\\left(  t\\right)  $}\n\nThe first approach is straightforward. The meaning of the counting function\n$N\\left(  \\lambda\\right)  $ is the number of the states whose eigenvalues are\nsmaller than $\\lambda$, so the state density reads%\n\\begin{equation}\n\\rho\\left(  \\lambda\\right)  =\\frac{dN\\left(  \\lambda\\right)  }{d\\lambda\n}.\\label{density}%\n\\end{equation}\nThus, from eq. (\\ref{NLbyCk}), we have%\n\\begin{align}\n\\rho\\left(  \\lambda\\right)   & =\\left(  4\\pi\\right)  ^{-D\/2}\\sum_{k=0,\\frac\n{1}{2},1,\\cdots}^{\\infty}\\frac{B_{k}}{\\Gamma\\left(  D\/2-k\\right)  }%\n\\lambda^{D\/2-k-1}+\\sum_{l=0,1,2,\\cdots}^{\\infty}\\left(  4\\pi\\right)\n^{-D\/2}B_{1+D\/2+l}\\delta^{\\left(  l+1\\right)  }\\left(  \\lambda\\right)\n.\\nonumber\\\\\n& =\\left(  4\\pi\\right)  ^{-D\/2}\\sum_{k=0,\\frac{1}{2},1,\\cdots}^{\\frac{D}%\n{2}-\\frac{1}{2}}\\frac{B_{k}}{\\Gamma\\left(  D\/2-k\\right)  }\\lambda\n^{D\/2-k-1}+\\left(  4\\pi\\right)  ^{-D\/2}\\sum_{k=\\frac{D}{2}+\\frac{1}{2}%\n,\\frac{D}{2}+\\frac{3}{2},\\cdots}^{\\infty}\\frac{B_{k}}{\\Gamma\\left(\nD\/2-k\\right)  }\\lambda^{D\/2-k-1}\\nonumber\\\\\n& +\\sum_{k=\\frac{D}{2}+1,\\frac{D}{2}+2,\\cdots}^{\\infty}\\left(  4\\pi\\right)\n^{-D\/2}B_{k}\\delta^{\\left(  k-\\frac{D}{2}\\right)  }\\left(  \\lambda\\right)\n\\label{sd}%\n\\end{align}\nFrom this result, one can obtain the state density once he knows the heat\nkernel coefficients.\n\n\\subsection{Calculating the state density from the spectrum}\n\nAn alternative way for obtaining the state density is to start with the\nexpression of the counting function given in section \\ref{definitionofNL}.\n\nDerivating both sides of eq. (\\ref{asymptotic}) by $\\lambda$ gives%\n\\begin{equation}\n\\rho\\left(  \\lambda\\right)  =\\frac{dN\\left(  \\lambda\\right)  }{d\\lambda}%\n=\\lim_{\\beta\\rightarrow\\infty}\\sum_{n}\\frac{\\beta e^{\\beta\\left(  \\lambda\n_{n}-\\lambda\\right)  }}{\\left[  e^{\\beta\\left(  \\lambda_{n}-\\lambda\\right)\n}+1\\right]  ^{2}}.\\label{density2}%\n\\end{equation}\nOr, approximately, we can convert the summation over $n$ to an integral:%\n\\begin{equation}\n\\rho\\left(  \\lambda\\right)  =\\lim_{\\beta\\rightarrow\\infty}%\n{\\displaystyle\\int}\ndn\\frac{\\beta e^{\\beta\\left(  \\lambda_{n}-\\lambda\\right)  }}{\\left[\ne^{\\beta\\left(  \\lambda_{n}-\\lambda\\right)  }+1\\right]  ^{2}}.\n\\end{equation}\nThen, we can directly calculate the state density from the spectrum.\n\n\\section{Conclusions\\label{dc}}\n\nIn this paper, we reveal a relation between the counting function $N\\left(\n\\lambda\\right)  $ and the heat kernel $K\\left(  t\\right)  $ and provide an\noperable expression for $N\\left(  \\lambda\\right)  $. By the relation, one can\ncalculate the counting function from a known heat kernel, and \\textit{vice\nversa}. By the expression of the counting function presented in this paper,\none can achieve the counting function by a direct calculation.\n\nThe relation between $N\\left(  \\lambda\\right)  $ and $K\\left(  t\\right)  $ is\nan integral transformation and its inverse transformation. When calculating\n$N\\left(  \\lambda\\right)  $ from $K\\left(  t\\right)  $ by this relation,\nhowever, one may encounter the problem of divergence. This is because in most\ncases the expression of $K\\left(  t\\right)  $ is in the form of a power series\n(in fact, often only first several heat kernel coefficients can be obtained),\nbut the series is not uniformly convergent. As a result, when applying the\nintegral transformation to the series, one needs to integrate term by term.\nHowever, to integrate term by term is not feasible: the integration of some\nterms will diverge. For removing the divergences, we develop a renormalization procedure.\n\nThe results of this paper provide two approaches for calculating the counting\nfunctions: one is to calculate $N\\left(  \\lambda\\right)  $ directly from the\nexpression of $N\\left(  \\lambda\\right)  $ given in section\n\\ref{definitionofNL} and the other is based on the relation between $N\\left(\n\\lambda\\right)  $ and $K\\left(  t\\right)  $ given in section \\ref{relation}.\nAs a comparison between the two approaches, we calculate the counting function\nfor $D$-dimensional boxes by these two approaches, respectively.\n\nAs applications of the relation between $N\\left(  \\lambda\\right)  $ and\n$K\\left(  t\\right)  $, we also calculate the counting functions for\nthree-dimensional balls and two-dimensional multiply-connected irregular\nregions. For calculating the counting function of a two-dimensional\nmultiply-connected region, we generalize the result of the heat kernel of a\ntwo-dimensional simply-connected region to a multiply-connected one, and the\nresult proves Kac's conjecture on the two-dimensional multiply-connected heat kernel.\n\nMoreover, we also present approaches for calculating eigenvalue spectra and\nstate densities from the counting function.\n\n\\acknowledgments{We are very indebted to Dr. G. Zeitrauman for his\nencouragement. This work is supported in part by NSF of China under\nGrant No. 10605013 and the Hi-Tech Research and Development\nProgramme of China under Grant No. 2006AA03Z407.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\subsection{Segmentation}\n\n\\input{figs\/fig_comparison} \n\\input{figs\/inbreast_pred_viz}\n\\input{figs\/curves}\n\nAs a baseline for comparison we fine-tune state-of-the-art DeepLabV3 \\cite{deeplabv32017} semantic segmentation model.\nThe baseline DeepLabV3 model has $\\sim$40\\textbf{M} trainable parameters (two orders of magnitude more than ours) and a receptive field of roughly $900\\times 900$ pixels ($\\sim\\times 200$ than ours).\nWe were unable to compare to \\cite{nature1,ref18}, since they provide neither code nor data for comparison.\n\nFig.~\\ref{fig:comparison} and ~\\ref{fig:inbreast_pred_vis} show segmentation examples.\nOur model outputs per-pixel ``calcification probability\" in the range of $\\left[0, 1\\right]$ depicted as a heat map from blue (low probability) to red (high probability) in the figures.\nOur method delineates the microcalcifications more accurately than the baseline.\nMore importantly, in clusters of microcalcifications DeepLab's results are ``smeared\" and do not allow to distinguish between the different microcalcifications. \nIn our segmentation, on the other hand, one can distinguish between the different microcalcifications, which is an important feature for diagnosis.\n\n\nWe further used the annotations of the test sets\nfor quantitative evaluation.\nFirst, we checked accuracy on a ``calcification level\"; that is, we partitioned the annotations and the predictions according to their connected components. \nWe treat each such component as a single calcification. \nFig.~\\ref{fig:results_curves}~(a) shows FROC \\cite{froc} measuring the calcification detection rate as a function of the average number of false alarms per mammogram. \nOur model performs better than the baseline detecting more microcalcifications for any given false alarm rate.\n\nMeasuring accuracy at the calcification level is good for assessing performance on isolated microcalcifications where pixel-level accuracy is not required as long as all lesions are being detected.\nHowever, when dealing with dense clusters of microcalcifications accurate \\emph{pixel-level} prediction is crucial, see, e.g. Fig.~\\ref{fig:comparison}.\nTherefore, we further computed ROC and precision-recall curves measuring \\emph{per-pixel} accuracy as shown in Fig.~\\ref{fig:results_curves}~(b)~and~(c).\nDue to the severe imbalance between the classes, the precision-recall curve is more meaningful as it is not affected by the size of the negative class. The table in Fig~\\ref{fig:results_curves}.d\nsummarizes the graphs of Fig.~\\ref{fig:results_curves}~a-c by showing the Area Under Curve (AUC) of the ROC curve and Average Precision Score (APS) of the precision-recall curves \\cite{aps} (in both, higher is better). As can be seen, our method outperforms the baseline on both datasets in all performance measures.\n\n\n\n\n\n\n\\subsection{Tracking Changes over Time}\nOne important aspect of periodic screening is the ability to reliably and efficiently compare two scans of the same patient taken at different times.\nHere we exemplify how our predicted segmentation can significantly assist clinicians in this otherwise tedious task.\nGiven two screening mammograms of the same patient taken in two different times,\nthe clinician marks a crude region around the suspicious cluster in both mammograms.\nOur accurate segmentation facilitates easy comparison and progression assessment between the two scans.\nAs can be seen in Fig ~\\ref{fig:inbreast_pred_vis}, our method segmenting the calcifications accurately while the baseline DeepLabV3 results are ``smeared\" and do not allow to distinguish between the different microcalcifications.\nOur accurate segmentation allows to \\emph{automatically} extract important statistics (e.g., number of calcifications in a cluster, mean size etc.) to assess temporal changes in follow-up screenings as shown in Fig.~\\ref{fig:temporal_comparison2}.\n\n\n\\input{figs\/temporal_comparison_2}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Summary for Program}\n\n\n\n\\section{Introduction}\n\\input{intro}\n\n\\section{Related Work}\n\\input{relatedwork}\n\n\\section{Method}\n\\input{method}\n\n\\section{Experimental Results}\n\\input{exp_results}\n\n\n\\section{Conclusion}\n\\input{conclusions}\n\n\\subsection*{Acknowledgments}\nDr Bagon is a Robin Chemers Neustein Artificial Intelligence Fellow.\nAdditionally, Dr Bagon is supported by a research grant from the Carolito Stiftung. \n\n\n\\subsection*{Model}\nIn contrast to popular deep models designed with large receptive fields and relying on contextual cues (e.g., \\cite{deeplabv32017,fpn,unet}), we specifically design our FCN model to have a small receptive field.\nThe rationale behind this design choice is that calcification residues are local phenomena and therefore contextual information does not provide discriminative information.\nFig.~\\ref{fig:netarch} shows the architecture of our FCN.\nInspired by Inception modules \\cite{inception}, the basic building block of our model has 4 parallel 2D convolutions of varying kernel sizes.\nThis allows for capturing both \\emph{micro} as well as \\emph{macro} calcifications.\nThe output of all four convolutions is then concatenated to form the output of the block.\nAn additional $5\\times 5$ conv layer outputs a scalar prediction score per-pixel.\nThis design is very ``slim\": it has only $\\sim$450K trainable parameters and a receptive field of $45 \\times 45$ pixels allowing it to rapidly spot minute details in high-res mammograms.\n\n\\subsection*{Training}\nThe main difficulty in training a pixel-wise segmentation model of tiny microcalcifications in high-res mammograms is the severe class\/non-class imbalance: there are about 1K negatives for each positive pixel.\nAs a result, the gradients of the binary cross-entropy loss function are overwhelmed by the majority negative class in a way that masks important but weak updates for the positive class.  \nNa\\\"ively training a model under these conditions simply yields a constant ``negative\" prediction, which is mostly accurate, but meaningless.\nWe use online hard negative mining \\cite{hardnegativemining} as an integral part of our training procedure, computing the gradients of the loss function of only few hard negative examples in each iteration.\nWe maintain a ratio of 1:3 between positive and hard negative examples discarding the gradients from all the rest of the negatives.\nThis way gradients resulting from positive examples contribute roughly equally to negative examples allowing the model to converge to non-trivial accurate predictions.\nNote that the imbalance in this case is so severe that simple modifications to the loss function (e.g., \\cite{focalloss}) were not effective.\n\nTo further enrich our training examples we used random data augmentations: \nrotation within the interval $\\left[-35\\degree, 35\\degree\\right]$ and horizontal flipping.\nWe trained our model with Stochastic Gradient Descent (SGD) with momentum 0.9 and a fixed learning rate $\\lambda=0.001$ for 500-1000 epochs.\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\section{Introduction}\n\nThe recent success of neural networks in a range of tasks connected with logical inference in natural language is remarkable. Foremost among such systems are those employing transformer-based language models \\cite{Vaswani2017}  optimised on large corpora in an unsupervised manner \\cite{Devlin2018} and then further fine-tuned on task-specific datasets \\cite{Bowman2015,rajpurkar2016squad}.\n\\ignore{Superficially, the performance of these systems is remarkable.} \nHowever, concerns  persist regarding so-called ``data-set artefacts''~\\cite{gururangan2018annotation,Schlegel2020a,Schlegel2020}. Have netural networks really acquired the principles of reasoning with natural language, or are they merely responding to superficial patterns in the data?\n\n\\ignore{Applications of neural networks to inference in formal languages typically employ different architectures---for example graph neural networks encoding logical syntax such as NeuroSAT~\\cite{nnFrag:slblmd18} or collections of sampled possible worlds as in PossibleWorldNet~\\cite{ nnFrag:esakg18}---or alternatively proceed more obliquely, using neural networks to select inference rules or proof strategies~\\cite{nnFrag:kcs17,nnFragrlbs20,nnFrag:ps20,nnFrag:hrwap21}.}\n\n\\begin{figure}[!t]\n    \\centering\n    \\resizebox{1.02\\columnwidth}{!} {\n    \\input{figure}\n    }\n    \\resizebox{1\\columnwidth}{!}{\n    \\input{figure2}\n    }\n\n    \\caption{Depiction of our approach to evaluate reasoning capabilities in neural networks.}\n    \\label{fig:summary}\n\\end{figure}\n  \n\n\\ignore{\nThere are reasons to be cautious. One recent line of research in natural language inference reveals that\ncrowd-sourced datasets are prone to so-called ``data-set artefacts''~\\cite{gururangan2018annotation,Schlegel2020a}.\nFor instance, {\\em shorter} hypotheses are more likely to be entailed by their respective premises than are longer ones. The fact that models learn to rely on combinations of such clues casts further doubt on whether their seemingly impressive performance is really  evidence of their ability to reason with language.  \nIndeed, data sets used to train systems for performing inference in formal languages may be prone to broadly similar difficulties.\nNotwithstanding the success of the NeuroSAT system in solving the satisfiability problem for propositional clause sets,\neven its authors suggest the detection of \\textit{unsatisfiability} by neural networks appears to depend on the existence of \nsmall, \\textit{unsatisfiable cores} in the training and test sets ({\\em op.~cit.~p.~8}). Put bluntly, \nthe problems the network is given, though large, may nevertheless be, in a well-defined sense, \\textit{easy}. (Similar remarks  apply to\nother applications of neural networks to hard combinatorially problems such as, for instance, graph-colourability~\\cite{nnFrag:lpal19}.)\nSuch considerations suggest that the performance of neural networks on inference problems, both in natural and formal languages, may be based on capacities orthogonal to the grasp of any logical principle or generally applicable rule. }\n\n\\ignore{\nThere are reasons to be cautious. One recent line of research in natural language inference reveals that\ncrowd-sourced datasets are prone to so-called ``data-set artefacts''~\\cite{gururangan2018annotation,Schlegel2020a}.}\n\n\nIn fact, two strands of research may be discerned in recent work on natural language inference (NLI). The first formulates the central task as follows: given a pair of sentences in some natural language, a {\\em premise}, $P$ and a {\\em hypothesis}, $H$, determine whether \n$P$ (i)  \\textit{entails} or (ii) \\textit{contradicts} (i.e.~entails the negation of) $H$; or (iii) $P$ is {\\em neutral} with respect to $H$. \nFor example, the premise\n\\textit{A person rides his bicycle in the sand beside the ocean} is taken to entail the hypothesis \\textit{A person is on a beach}. The primary issue is the possibility of learning such a mapping on the basis of some data set consisting of many such sentence pairs, each  tagged with the a `correct' (i.e.~gold standard) label as determined by human judgement, typically obtained by crowd-sourcing.\nExamples include the RTE dataset~\\cite{nnFrag:BDDGM09} and the (much larger) SNLI and MNLI datasets~\\cite{Bowman2015,Williams2018}. \nNeural-network models achieve impressive accuracy on this task \\cite{Chen2017EnhancedInference,Devlin2018}. \nThe nature of the challenge here is (deliberately)  mixed:\non the one hand, solving the NLI problem requires a grasp of the meaning\nof various closed-class words  ({\\em a}, {\\em not}, \\dots) \ntogether with an appreciation of the semantics of the grammatical constructions involved. On the other hand, almost all the entailments or contradictions encountered rely on common sense knowledge (for example, that racing a bicycle involves riding it), which it is the job of the system to acquire. The supposition is that these pieces of commonsense knowledge mediate the entailment of the hypothesis (or its negation)\nfrom the premise. The logical basis of the inferences, so reconstructed, is typically straightforward,\namounting to simple syllogism-like inferences; but the mediating commonsense knowledge is coded in `soft' form, matching the approximate, probabilistic nature of natural language inference.  The operative notion of inference\nthus eludes a formal \ndefinition~\\cite{Sugawara2018}\nultimately resulting in debatable labels \\cite{Schlegel2020}\n\nTh\nsecond strand of research investigates the ability of neural networks to recognize formally valid entailments {\\em in sensu stricto}, but nevertheless couched in fragments of natural language. Examples include~\\newcite{Salvatore2019,Richardson2019a, nnFrag:rs21} and \\newcite{Talmor2020}. Here the focus is on  learning the (potentially complex) inferential patterns inherent in the logical syntax of the closed-class words and grammatical constructions defining the fragment of language under investigation.\nFor example, the premises\n\\textit{Every doctor who is not a philosopher is a baker}, {\\em John is a doctor} and {\\em John is not a baker}\nentail the hypothesis \\textit{John is a philosopher}.\nNo commonsense knowledge is required here: the given premises either entail the given hypothesis formally, or they do not. Since entailment in a formally defined fragment of language is a matter of mathematical fact,\nnot human judgement, inferential problems may be constructed artificially, using random sampling over a grammar, with the correct answers determined by an automated theorem prover (ATP). The question is not whether neural networks can be trained to mimic human annotators' judgements, but rather, whether they can learn the logical principles governing language in question. Note that, because it is formal validity rather than commonsense plausibility that is at issue here, inference tasks of interest typically involve not a \\textit{single} premise $P$, \nbut rather, a \\textit{collection} of premises. \nIndividual sentences seldom yield any non-trivial formal entailments.\n\n\\ignore{Closer analysis of some widely-used NLI datasets reveals three striking\nfeatures of the challenge posed. \nThus, to perform well on this NLI task, a system has to solve two quite different sorts of problem: (i) understanding basic logical syntax, and (ii) inducing meaning postulates. \n\nThe reliance on implicit meaning postulates explains the second striking feature of the NLI \n\n\nYet, from a logical point of view,  when sentences are short\nthe problem of recognising entailments becomes interesting only when those entailments depend on multiple, interacting premises. Here, of course,\nthe induced meaning postulates supply the additional (implicit) premises that render the entailments informative. Yet,\nmany of the entailments encountered appear to involve little more than syllogism-like inferences.\n}\n\n\nThese two strands of research reflect a difference in motivation. The former aims to understand inferences apparently performed by humans in everyday linguistic settings---inferences which are necessarily messy and approximate in character. \nFrom a formal point of view, many of the alleged entailments or contradictions\nare in fact judgements of \\textit{probability}. Thus, for example the premise {\\em Alice is holding a dog} does not, logically speaking, contradict the hypothesis {\\em Alice is holding a cat}, even assuming that no dog is a cat; yet such labellings abound in NLI datasets, reflecting the judgements made by (and instructions given to) the human annotators who generate them. \n\\ignore{Human annotators have a notoriously weak grasp of the concept of logico-syntactic entailment familiar---for better or for worse---to most logicians, as evidenced by inherent disagreements between different annotators of gold labels \\cite{Pavlick2019}, ultimately resulting in data that can be factually wrong \\cite{Schlegel2020}. \nWhatever the practical value of teaching a neural network to mimic these judgements, its scientific value must be open to question.}\nBy contrast, the second (strictly logical) strand of research is motivated principally by a desire to understand the rule-based character of natural language syntax and semantics, and in particular, by the question of whether a neural network can learn the rules in question. \n\nIn this paper, we report on\na large-scale empirical study to investigate whether transformer-based language models can learn the logical syntax of natural language. Specifically, we consider performance on\nthe problem of determining the satisfiability (logical consistency)\nof sets of English sentences featuring\nthe\ndeterminers\n{\\em every}, {\\em some}, {\\em no}, the negative adverb {\\em not}, \nand relative clauses,\nin the context of a vocabulary of \ncount nouns and\n transitive verbs. \n Importantly, the sentences in question are drawn from various different \\textit{fragments} of English, each characterized by a particular range of the grammatical constructions investigated. \nIn each case,\ndata sets---both for training and evaluation---are artificially generated \nin order to test the system's grasp of the underlying logic. Figure~\\ref{fig:summary} illustrates the approach, giving an example problem instance for the very simplest of the fragments considered.\nWe find that\nstate-of-the-art deep learning-based approaches achieve impressive performance on this task. However, in-depth analysis of their generalisation capabilities reveals a more nuanced pattern, suggesting a tendency to overfit to the parameters that control the data generation, rather than learning the underlying logical principles.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\t\t\n\n\\section{Fragments of Language}\n\nBy a {\\em fragment} of a natural language, we understand a collection of sentences forming a naturally delineated subset of that language, and equipped with a truth-conditional semantics commanding the assent of its native speakers.\nFor example, consider the fragment of natural language corresponding to the {\\em classical syllogistic} given to us by~\\citet[Book A]{nnFrag:aristotlePA}.\nIn its English-language version, it  consists of the sentence forms\n\\begin{center}\n\t\\begin{minipage}{10cm}\n\t\t\\begin{tabbing}\n\t\t\t\\small Every $p$ is a $q$ \\hspace{2cm} \\= \n\t\t\t\\small No $p$ is a $q$ \\\\\n\t\t\t\\small Some $p$ is a $q$               \\> \n\t\t\t\\small Some $p$ is not a $q$,\n\t\t\\end{tabbing}\n\t\\end{minipage}\n\\end{center}\nwith schematic variables $p$ and $q$ substituted by common (count) nouns. \nThis fragment, which we here denote \\sS, can be used to formulate examples such as the one shown in Figure~\\ref{fig:summary}.\n\nFor present purposes, we may consider the truth conditions of an English sentence to be given by translation to a formal language such as first-order logic, in a way which uncontroversially reconstructs the operative notion of logical entailment. Thus, for example, the sentence forms \\sS{}  correspond to the respective logical forms\n$\\forall x(p(x) \\rightarrow \\pm q(x))$ and $\\exists x(p(x) \\wedge \\pm q(x))$, where $\\pm$ indicates either the presence or absence of negation ($\\neg$). An argument in \\sS{} is regarded as valid precisely when the first-order translations of its premises entail---in the familiar logical sense---the first-order translation of its conclusion.\n\nConsider now the argument:\n\\begin{equation}\n\\text{\\begin{minipage}{6cm} \\small\n\\textit{Some artist hates no beekeeper};  \\textit{every beekeeper hates some artist}; \\textit{therefore some artist is not a beekeeper}.\n\\end{minipage}}\n\\label{eq:arg1}\n\\end{equation}\nThis argument is again intuitively valid\n(though this takes a little thought to see). On the other hand, \nit cannot be sensibly cast in the\nsyllogistic, because it\nso obviously\nhinges on intrinsically relational information.\n(Observe the alternation of subject and object of the verb {\\em hates} in the two premises.) Is there, then, a larger fragment of English in which it might be expressed? Take\nthe {\\em relational syllogistic}, denoted \\sR, to be the fragment of English obtained by adding to \\sS{} the sentence-forms:\n\\begin{center}\n\t\\begin{minipage}{10cm}\n\t\t\\begin{tabbing}\n\t\t\t\\small Every $p$ $r$s every\/some $q$ \\hspace{0.25cm} \\= \n\t\t\t\\small Some $p$ $r$s every\/some $q$\\\\\n\t\t\t\\small No $p$ $r$s every\/any $q$ \\> \n\t\t\t\\small Some $p$ does not $r$ every\/any $q$,\n\t\t\\end{tabbing}\n\t\\end{minipage}\n\\end{center}\nwhere $p$ and $q$ are common count nouns and $r$ a transitive verb. \nAs with \\sS{}, so too with \\sR{}: the truth-conditions of sentences\ncan be captured by translation to first-order logic in a way which faithfully reconstructs the intuitive notion of validity. \nFor example, ``Some artist hates no beekeeper'' may be rendered as $\\exists x(\\mbox{artist}(x) \\wedge \\forall y (\\mbox{beekeeper}(y) \n\\rightarrow \\neg \\mbox{hate}(x,y)))$, and so on. \nUnder these semantics, argument \\eqref{eq:arg1} is confirmed as a valid argument in the fragment \\sR.\n\nThere are other ways to extend the fragment \\sS, of course. \nOne (very modest) such extension is actually \nfeatured in the works of~\\citet[Ch.~10]{nnFrag:aristotleCdI}. Let us say that the {\\em extended classical syllogistic}, denoted \\sSd, is the fragment of English which adds to \\sS{} the sentence-forms\n\\begin{center}\n\t\\begin{minipage}{10cm}\n\t\t\\begin{tabbing}\n\t\t\t\\small Every non-$p$ is a $q$ \\hspace{1cm} \\= \n\t\t\t\\small Some non-$p$ is a $q$\\\\\n\t\t\t\\small No non-$p$ is a $q$                 \\> \n\t\t\t\\small Some non-$p$ is not a $q$,\n\t\t\\end{tabbing}\n\t\\end{minipage}\n\\end{center}\ncorresponding to the first-order formulas \n$\\forall x(\\neg p(x) \\rightarrow \\pm q(x))$ and $\\exists x(\\neg p(x) \\wedge \\pm q(x))$. As we might say, \\sSd{} adds `noun-level negation' to \\sS. \nFollowing in the same vein, we can extend \\sR{} with noun-level negation, yielding sentences such as \n\\emph{``No non-carpenter admires any non-artist''}\nand so on. We call this fourth fragment the {\\em extended relational syllogistic}, denoted \\sRd. Again, translation into first-order logic is completely standard.\n\nLet $\\cL$ be a fragment of some natural language. A set of \\cL-sentences is said to be \\textit{satisfiable} if there is a structure making the logical translations of these sentences true. The \\textit{satisfiability problem} for $\\cL$, denoted $\\ensuremath{\\mbox{\\rm Sat}}(\\cL)$, is the problem of determining whether a given finite set of sentences of $\\cL$ is satisfiable. Provided $\\cL$ is equipped with a mechanism for sentence negation (as are all the fragments considered here), any procedure for solving $\\ensuremath{\\mbox{\\rm Sat}}(\\cL)$ immediately yields a procedure for recognizing logical entailments in $\\cL$, since an argument is valid just in case its premises together with the negation of its conclusion is not satisfiable.\nTherefore, we have no use for the familiar classification of\nnatural language inference problems as {\\em entailment}, {\\em contradiction} and {\\em neutral} (and still less the four-way classification of~\\citet{nnFrag:tlcxhj21}): the satisfiability problem is as general as we need. As the fragment $\\cL$ becomes more expressive, the\ncorresponding satisfiability problem $\\ensuremath{\\mbox{\\rm Sat}}(\\cL)$ will, in general, become\nmore increasingly difficult. However, as we shall see, the details of the resulting trade-off between expressiveness and ease of inference are rather intricate. The focus of the present paper is whether neural networks can learn to solve problems such as $\\ensuremath{\\mbox{\\rm Sat}}(\\cL)$ for various fragments $\\cL$\nsuch as \\sS{}, \\sSd{}, \\sR{} and \\sRd{}. \n\nWe remark that the approach taken here is parallel to that taken with respect to the fragments GRL and RCL in~\\cite{nnFrag:rs21}.\nThere are two notable differences, however, Firstly, the fragments considered here are, from a grammatical point of view, more basic, and less clearly a natural language version of propositional logic clauses.\nIn particular, GRL includes the `sentence' \\textit{If carrot and not steak then apples}; and even the more natural RCL is limited to the constructions \\textit{Every $X$ who is (not) a $Y$ is (not) a $Z$}.  \nTo allow comparison between this work and the present study \nwe consider the minimal extension of \\sS{} by means of \\textit{relative clauses}, thus allowing the additional sentence-forms\n\\begin{center}\n\t\\begin{minipage}{10cm}\n\t\t\\begin{tabbing}\n\t\t\\small\n\t\t\tEvery $o$ who is a $p$ is a $q$ \\hspace{.5cm} \\= \\small No $o$ who is a $p$ is a $q$  \\\\\n\t\t\\small\tSome $o$ who is a $p$ is a $q$               \\> \\small Some $o$ who is a $p$ is not a $q$\n\t\t\\end{tabbing}\n\t\\end{minipage}\n\\end{center}\nDenote this fragment by \\sSrel. Similarly, denote by \\sSrelN{} the same fragment additionally allowing negative relative clauses, such as ``Every $o$ who is not a $p$ is a $q$'' and so on. The fragment \n\\sSrelN{} is actually an extension of RCL.\n\nAn important motivation for the rather more general approach taken here is that \nthe satisfiability problem $\\ensuremath{\\mbox{\\rm Sat}}(\\cL)$ for various fragments $\\cL$ of English may be studied from a purely complexity-theoretic point of view. Thus, for example, it is known that the problems $\\ensuremath{\\mbox{\\rm Sat}}(\\sS)$, $\\ensuremath{\\mbox{\\rm Sat}}(\\sSd)$ and $\\ensuremath{\\mbox{\\rm Sat}}(\\sR)$ are \\textsc{NLogSpace}-complete, while $\\ensuremath{\\mbox{\\rm Sat}}(\\sRd)$ is \\textsc{ExpTime}-complete \\cite{nnFrag:p-hm09}. \nSimilarly, the satisfiability \nproblems for GRL and RCL are easily seen to be \\textsc{NPTime}-complete, as is the problem \n$\\ensuremath{\\mbox{\\rm Sat}}(\\sSrelN)$; by contrast, the problem \n$\\ensuremath{\\mbox{\\rm Sat}}(\\sSrel)$\nis \\textsc{PTime}-complete~\\cite[Theorem 7]{nnFrag:ph14}. \nThe question naturally arises as to whether the ability of neural networks to learn to solve these various satisfiability problems correlates with these complexity-theoretic differences. \n\n\\begin{table*}[t]\n    \\centering\n    \\begin{tabularx}{1\\textwidth}{p{.05\\textwidth}XX}\n         \\textbf{Frag.} & \\textbf{Templates} & \\textbf{Example sentence} \\\\\n         \\hline\n         \\multirow{2}{*}{\\sSd} & Every\/No (non-)$p$ is a $q$. & Every artist is a beekeeper. \\\\\n          & Some (non-)$p$ is (not) a $q$. & Some carpenter is not a dentist. \\\\\n         \\hdashline\n         \\multirow{4}{*}{\\sR} & \\emph{all of \\sS{} and:} & \\\\\n          & Every\/Some $p$ $r$s every\/some $q$. & Every artist chases some beekeeper. \\\\\n          & Some $p$ does not $r$ every\/any $q$. & Some beekeeper does not chase any artist. \\\\\n          & No $p$ $r$s every\/any $q$. & No beekeeper bewitches any artist. \\\\\n         \\hdashline\n         \\multirow{4}{*}{\\sRd} & \\emph{all of \\sSd{} and:} & \\\\\n          & Every\/Some (non-)$p$ $r$s every\/some (non-)$q$. & Every non-artist chases some beekeeper. \\\\\n          & Some (non-)$p$ does not $r$ every\/any (non-)$q$. & Some beekeeper does not chase any non-artist. \\\\\n          & No (non-)$p$ $r$s every\/any (non-) $q$. & No non-beekeeper bewitches any non-artist. \\\\\n          \\hdashline\n         \\multirow{2}{*}{\\sSrel} \n          & Every\/Some\/No $o$ who is a $p$ is a $q$. & Every artist who is a dentist is a carpenter. \\\\\n          & Some $o$ who is a $p$ is not a $q$. & Some dentist who is a hunter is not a spy. \\\\\n          \\hdashline\n          \\multirow{2}{*}{\\sSrelN} & \\emph{all of \\sSrel{} and:} & \\\\\n          & Every\/Some\/No $o$ who is not a $p$ is a $q$. & Every artist who is a not dentist is a carpenter. \\\\\n          & Some $o$ who is a not $p$ is not a $q$. & Some dentist who is a not hunter is not a spy. \\\\\n          \\hline\n    \\end{tabularx}\n    \\caption{Templates used to generate the problem instances for all five fragments. Round brackets () denote optionals, forward slashes \/ denote alternatives.}\n    \\label{tab:examples}\n\\end{table*}\n\nTable~\\ref{tab:examples\nshows all templates used to generate the datasets for each of the fragments. Code to generate the datasets, allowing control of the parameters discussed above, is included in the supplementary material available on github\\footnote{\\url{https:\/\/github.com\/schlevik\/nlr}}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Random problems} \n\\label{sec:generation}\nIn this section  we outline the construction of collections of sets of formulas in the fragments \\sSd, \\sR, \\sRd, \\sSrel{} and \\sSrelN, in which each generated set is labelled as \\textit{satisfiable} or \\textit{unsatisfiable}. (The fragment \\sS{} is not interestingly different\nfrom \\sSd{}, and will not be considered in the experiments reported here.\nThese collections are partitioned into training and evaluation sets, enabling us to test the ability of neural networks to learn to recognise satisfiability under a range of conditions. \n\nAny sentence in $\\sSd$ translates to a formula having either of the forms $\\forall x(\\pm p(x) \\rightarrow \\pm q(x))$ or $\\exists x(\\pm p(x) \\wedge \\pm q(x))$. We may thus generate a sentence of $\\sSd$ pseudo-randomly by selecting a universal sentence with probability $p_u$, a negated subject with probability $p_{\\bar s}$ and a negated object with probability $p_{\\bar o}$, and then choosing $p$ and $q$ at random from some collection of $n$ nouns. By\ncarrying out this process $s$ times, we obtain a random set $\\Phi$ of \\sSd-sentences ($|\\Phi| = s$).  For definiteness, we set $p_{\\bar s}=p_{\\bar o}=0.5$ and $p_u = 0.8$. In addition, we remove any inconsistent sentences, such as ``Some $p$ is not a $p$.''  In \nchoosing the various parameters, we fix $n\/s=0.8$, which, as we have empirically established,\nkeeps the proportion of satisfiable instances at roughly 50\\%. Collections\nof such problem instances are created for various values of $s$.  The ratio $n\/s=0.8$ corresponds (for \\sSd) roughly to the critical region of SAT problems studied in~\\cite{nnFrag:rs21}, \nwhere it was shown that learning on data from this region is more effective than learning from uniformly sampled data. However, we need to be wary of assuming that such instances are difficult---an issue addressed in Section 4. \n\nFor \\sRd, we proceed similarly, generating $s$ sentences at random over a fixed collection of $n$ nouns and $v$ transitive verbs. Every sentence in \\sRd{} is either a sentence in \\sSd{} or translates to a formula having one of the forms $\\forall x (\\pm p(x) \\rightarrow \\gamma)$ or \n$\\exists x (\\pm p(x) \\wedge \\gamma)$, where $\\gamma$ is either $\\forall y(\\pm q(y) \\rightarrow \\pm r(x,y))$ or\n$\\exists y(\\pm q(y) \\wedge \\pm r(x,y))$. Call sentences in $\\sRd{} \\setminus  \\sSd{}$ {\\em relational sentences}. We \nmay thus generate a sentence of \\sRd{}  pseudo-randomly by choosing to produce a relational sentence with probability \n$p_r$. If we choose to produce a non-relational sentence (i.e.~a sentence in \\sSd), we proceed as above; otherwise a negated verb is chosen with probability $p_{\\bar v}$, a universally quantified $\\gamma$ with probability \n$p_{uu}$; the other parameters, $n$, $v$, $p_u$, $p_s$ and $p_o$ are interpreted as before. By setting  \n$p_{\\bar s}=p_{\\bar o}=0$, we guarantee that every generated relational sentence has a non-negated subject and a non-negated object, and hence is\na sentence of $\\sR$.  By repeating this process $s$ times, we obtain an set $\\Phi$ of sentences in \\sRd{} (or \\sR). \nWhen generating instances of $\\ensuremath{\\mbox{\\rm Sat}}(\\sR)$, we thus set \n$p_r=0.2$, $p_{\\bar s}=p_{\\bar o}=0$, \n$p_{\\bar v}=0.5$ and $p_u = p_{uu} = 0.8$; in addition, we fix  $n\/s=0.6$ and $v\/s = 0.15$, which, as we have empirically established,\nkeeps the ratio of satisfiable to non-satisfiable instances at roughly 50\\%. \nLikewise, when generating instances of $\\ensuremath{\\mbox{\\rm Sat}}(\\sRd)$, we set $p_{\\bar s}=p_{\\bar v}=0.5$, with the other parameters as for $\\sR$; however, we adjust $n\/s$ to $0.64$ in order to maintain the ratio of satisfiable to non-satisfiable instances. \n\nFinally, for fragments with\nrelative clauses, \\sSrel{} and \\sSrelN, in addition to the parameters introduced for \\sSd, we control the probability of negated relative clauses by the parameter $p_{\\bar r}$. By setting  \n$p_{\\bar r}=0$ we guarantee that every generated sentence has an un-negated relative clause, and hence is\na sentence of \\sSrel{}. We additionally set $p_{\\bar o}=0.5$ and $p_u=0.8$, and for \\sSrelN{}, we set $p_{\\bar r}=0.5$.\nWe empirically establish that setting $(n-0.225)\/s~=~0.59$ yields a probability of satisfiability of approximately 0.5\nfor both fragments.\n\nThe satisfiability of a generated problem instance is determined using the Vampire automated theorem prover (ATP) \\cite{Kovacs2013First-OrderVampire}, which (assuming termination) either reports that the input set is satisfiable or outputs a proof of a contradiction.\nWe record, for each generated problem instance, whether it  is satisfiable, and, if not, the number $l$ of lines in the\ndiscovered proof of a contradiction (\\emph{proof length}) as well as the number of input sentences, $d$, used in that proof. The ATP terminated on all generated problem instances; there is, however, no general guarantee that the proofs found are the shortest possible.\n\nGenerated sentences are created in the relevant fragments of English with templates depicted in Table~\\ref{tab:examples}\nand realised with dictionaries of nouns that describe categories for unary predicates (e.g. ``artist'', ``beekeeper'') and transitive verbs for binary predicates (e.g. ``admires'', ``bewitches''). We use distinct vocabularies for training and evaluation data and use words with non-overlapping semantic fields.\nUnless stated otherwise, to maintain comparability between the different datasets, we generate $60000$ examples for training and $8000$ for evaluating model accuracy. This is achieved by generating $3750$ and $500$ (for training and evaluation sets, respectively) examples for each number of sentences $s$ between $15$ and $30$.\n\n\n\\section{Constructed problems} \nOne attractive feature of the fragments \\sS, \\sSd{} and \\sR{} is that their satisfiable sets of formulas admit of a simple graph-theoretic characterization. This gives us an additional means of creating data sets comprising challenging problem instances.\n\nWe\nillustrate with \\sS{}. Let a set $\\Phi$ of \\sS{}-sentences be given, and let $p_1, \\dots, p_n$ be the common nouns (predicates) \noccurring in $\\Phi$. Now let $V$ be the set of expressions $p_i(x)$ or $\\neg p_i(x)$ ($1 \\leq i \\leq n$). We call the elments of $V$ {\\em literals} and let the variables $\\ell$ and $m$ range over $V$. If $\\ell \\in V$, denote by $\\bar{\\ell}$ the opposite literal obtained by adding or removing the negation sign as appropriate. Now let $E$ be the set of ordered pairs of literals $(\\ell, m)$ such that \n$\\Phi$ contains a sentence formalized as\neither $\\forall x(\\ell \\rightarrow m)$ or as\n$\\forall x(\\bar{m} \\rightarrow \\bar{\\ell})$.\nThus, $G_\\Phi= (V,E)$ is a directed graph. Write\n$\\ell \\Rightarrow_\\Phi m$ if there is a path in $G_\\Phi$ from $\\ell$ to $m$. \nIt can be shown \n\\citep[Sec.~3]{nnFrag:p-hm09} that a set of \\sS-formulas $\\Phi$ is satisfiable if and only if $\\Phi$ contains no sentence $\\exists x (\\ell \\wedge  m)$ such that either: \\textup{(i)} $\\ell \\Rightarrow_\\Phi \\bar{\\ell}$; \\textup{(ii)} $m \\Rightarrow_\\Phi \\bar{m}$ or \\textup{(iii)} $\\ell \\Rightarrow_\\Phi \\bar{m}$.\nThus, determining (un)satisfiability in \\sS{} amounts to detecting certain\n{\\em forbidden configurations} (in this case: paths) in the directed graph $G_\\Phi$. We regard the length of the\npath  (if it exists) as the {\\em size}\nof the forbidden configuration. Satisfiability in \\sSd{} is characterized similarly: we simply have to check that, in addition,\n$V$ contains no pair of opposite literals \n$o$ and $\\bar{o}$ such that $o \\Rightarrow \\bar{o}$ and $\\bar{o} \\Rightarrow o$. Again, if a set of sentences of \\sSd{} is inconsistent, then it contains a forbidden configuration having a well-defined {\\em size}.\n\nThis gives us controlled way to generate hard problem instances. Consider the fragment \\sSd.\nWe begin by simply constructing a forbidden configuration having a given size, $d$. To obtain an \\textit{unsatisfiable} problem instance $\\Phi$ of size $s$, we then add $s - d$ random sentences, checking (using a simple algorithm) that doing so does not create any smaller forbidden configurations. To obtain a \\textit{satisfiable} problem instance of size $s$, we reverse one of the implications in the forbidden configuration, and check that the added $s - d$ random sentences do not cause an unsatisfiability. In effect, $d$ is a guaranteed difficulty level; it yields a lower bound on the proof length required to establish unsatisfiability. \nAt the same time, the satisfiable and unsatisfiable instances thus generated are not easily distinguished by any superficial characteristics. We denote the problem $\\ensuremath{\\mbox{\\rm Sat}}(\\sSd)$ constructed as just described with $d$ in a specified range as $\\ensuremath{\\mbox{\\rm Sat}}(\\sSd_{[\\cdot,\\cdot]})$. Thus, for example, in $\\sSd_{[2,6]}$, unsatisfiable problem instances have $2 \\leq d \\leq 6$.\n\nFor $\\sR$,  the corresponding characterization of unsatisfiability in terms of forbidden configurations involves several cases~\\citep[Sec.~4]{nnFrag:p-hm09}. \nFor simplicity, we generate difficult instances by focusing on just one of these types of forbidden configuration, which we call an $\\forall\\forall$-configuration.\nWe begin by constructing an\n$\\forall\\forall$-configuration of specific size $6d$ (for $d \\geq 1$).\n(Such a collection of sentences is always unsatisfiable.)\nTo obtain an \\textit{unsatisfiable} problem instance of size $s$,\nwe add $s - 6d$ randomly generated sentences, again checking that \nthe resulting unsatisfiability is due entirely to the forbidden configuration.\n\\textit{Satisfiable} instances are then obtained by reversing one of the implications in the $\\forall\\forall$-configuration, and checking that this does not lead to unsatisfiability. Denote the problem $\\ensuremath{\\mbox{\\rm Sat}}(\\sR)$ constructed as just described with $d$ in a specified range as $\\ensuremath{\\mbox{\\rm Sat}}(\\sR_{\\langle\\cdot,\\cdot\\rangle})$. For example, in $\\sR_{\\langle 1,2\\rangle}$, inconsistent problem instances are inconsistent because of $\\forall\\forall$-configurations of size 6 or 12\n \nIt is not possible to characterize $\\ensuremath{\\mbox{\\rm Sat}}(\\sRd)$ in terms of forbidden configurations in this simple way. As a substitute, we use the proof-lengths of the proofs found by the ATP as a rough guide to difficulty. \nTo generate difficult instances of \\sRd, therefore, we first generate random instances as in \nSec.~\\ref{sec:generation};  we then  filter out those unsatisfiable instances with short proof-lengths (as reported by the ATP), and then remove satisfiable instances at random to preserve the proportion of satisfiable instances overall.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experimental Setup}\nIn the following study, we investigate whether neural networks, and in particular state-of-the-art transformer-based language models \\cite{Devlin2018} can learn to perform satisfiability checks on examples representing the selected fragments. First, we investigate whether they can do so in principle, by optimising and evaluating classifiers on training and evaluation data drawn from the same distribution. Second, in an attempt to understand whether they reliably learn the underlying logical principles that govern (un-)satisfiability, we evaluate their \\emph{generalisation capabilities} on out-of-distribution evaluation data.  We do so by altering various parameters of the training and evaluation data generation to control the structure\nof the generated problem instances.\n\n\n\n\n\n\n\nMore specifically, we cast the problem of determining satisfiability as binary text classification and conduct experiments with three transformer based language models, RoBERTa, XLnet and Electra \\cite{Liu2019c,Yang2019a,Clark2020ELECTRA:Generators}.which are further described in the Appendix. We employ pre-trained language models because initial experiments showed that these tend to converge faster compared to training from scratch, despite the fact that their pre-training objectives bears little similarity to the task at hand. Following \\newcite{Devlin2018}, we represent each problem in plain English text prepended by the special \\texttt{[CLS]} token as input to the language model. The text is embedded using the language model, and the embedding of the \\texttt{[CLS]} token, as output by the final layer of the language model, is projected into a two-dimensional space, representing the odds of the problem being satisfiable. During inference, we pick the highest logit as the model's prediction and during training we minimise the cross-entropy loss between the logits and the expected class, thus optimising the parameters of the language model to produce the expected prediction conditioned on the input. We keep the choice of hyper-parameters (detailed in Appendix) consistent across experiments. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n    \n\n\n\\section{Results and Analysis}\n\nWe report and analyse the results of the conducted empirical study.\nFor all results we measure the error as a confidence interval at $\\alpha = 0.05$,  using asymptotic normal approximation and omit reportage for brevity, as all measures are in the range of at most\ntwo percent points. We average results obtained for all three language models.\n\n    \n\\begin{table}[!tb]\n    \\centering\n    \\begin{tabular}{c|ccccc}\n   \n    \\multirow{1}{*}{\\textbf{Problem size}} & \\sSd & \\sR & \\sSrel & \\sSrelN & \\sRd \\\\\n    \n    \\hline\n        $15\\leq s \\leq 30$ &  76 & 93 & 94 & 80 & 74\\\\\n    \\hdashline\n       \n       \n       \n       \n        $30\\leq s \\leq 40$\\footnotemark[1] & 64 & 89 & 94 & 82 & 72 \\\\\n        $40\\leq s \\leq 45$ & 63 & 86 & - &  -  & 71 \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{Accuracy of optimised models trained on random examples consisting of $15$ to $30$ sentences ($s$) and evaluated on longer random problem instances.}\n    \\label{tab:res-longer}\n\\end{table}\n\n\\footnotetext[1]{$37$ for \\sSrel{} and \\sSrelN{} to fit transformers' $512$-token limit}\n\\textbf{\\emph{Transformers perform well on random examples in all fragments.}} \nTo seek evidence for the first question we train and evaluate separate models on randomly generated problems instances of each of the five fragments. The results are reported the first row of Table~\\ref{tab:res-longer} and suggest that the models perform well on randomly generated problems in all fragments, even on the \\textsc{ExpTime}-complete $\\sRd{}$ fragment. Surprisingly, the obtained accuracies do not seem to correspond to the complexity classes of the elicited fragments. Note that the results reported in this table are not necessarily that of the best-performing model; for example, by training the RoBERTa model longer and on a larger dataset representing the \\sRd{} fragment, we obtain accuracy scores of up to $81\\%$ (from original $79\\%$, see Table~\\ref{tab:res-longer-roberta} in the Appendix). However, to maintain comparability and as the difference is marginal, we use the same training budget for all models optimised on different fragments. The task appears non-trivial, as a simple LSTM-based classifier was not able to outperform the majority class baseline even on the simplest \\sSd{} fragment and after explicit hyper-parameter optimisation. \n\nTo investigate whether this performance generalises with the size of the problem instances, we generate evaluation sets with $30 \\leq s \\leq45$ sentences for longer problems. Note that we are constrained to problems of the size of up to $512$ tokens, as a technical constraint of the pre-trained language model, hence we do not investigate generalisation capabilities to problems beyond 45 (37 for \\sSrel{}, \\sSrelN{}) sentences.\nThe remainder of Table~\\ref{tab:res-longer} shows that models generalise consistently to problems larger than seen during training, with the notable exception of the model optimised on \\sSd{}, which exhibits the most significant drop of over $10$ percent points.\n\nSuperficially, Table~\\ref{tab:res-longer} appears to indicate good generalisation performance. However, it is important to realise that simply increasing the number of sentences does not make the reasoning problems harder, as witnessed by the fact that the number of sentences, $d$, required to prove contradiction does not increase: on average, $d=3.50$ for examples with  $15$ to $30$ sentences, and  $d=3.58$  for problems with $30$ to $45$ sentences.\nThus, the forbidden configurations the network is identifying are, for the most part, small, even for large numbers of sentences. A complimentary analysis in the Appendix reveals further, that these forbidden configurations are likely to be common between different fragments.\n\n\n\n\\textbf{\\emph{Transformers are not robust to distribution shifts.}}\nMoving on to the second question, we investigate whether the optimised models truly pick up the reasoning patterns as intended or rather overfit to their training data as an artefact of the configuration of parameters controlling the stochastic generation. Approximating the ``hardness'' of a problem instance by its proof length $l$, we find that for all fragments, the overwhelming majority (ranging from 29\\% in \\sS{} to 86\\% in \\sSrelN{}) of contradictory examples have short proof lengths of $12$ or less, indicating that random problems are, in fact, unsatisfiable for trivial reasons which are easy to show (See also histogram in Appendix). Thus, we collect ``hard'' examples by \\mbox{(over-) generating} a large body of examples and then filtering by proof length and refer to them as \\emph{random hard} problem instances. Naturally, in this way we can only capture the hardness of \\emph{inconsistent} examples, therefore the evaluation focuses on those.\n\nWhen comparing the (out-of-distribution) performance of models on ``easy'' and ``hard'' non-satisfiable problem instances with proof length of at least $42$ (Table~\\ref{tab:res-harder}, second and third rows), on average, there is a gap of $25$ percent points,\nThis suggests that models in fact overfit to simple problems that tend to dominate the\nrandomly generated datasets without picking up the general principles governing reasoning in the corresponding fragments. Models optimised on ``hard'' training examples  (proof length $\\geq 22$, complemented with an equal number of random consistent problems), generalise well to even harder problems (proof length $\\geq 42$, Table~\\ref{tab:res-harder}, last row).  This suggests that the\nmodels can learn to classify hard problem instances when presented explicitly by supplying appropriate training data. \nThis performance does not carry over to simpler problems, however, as models optimised on harder problems exhibit a drop in accuracy when evaluated on simpler problems, as the penultimate row of Table~\\ref{tab:res-harder} shows. In conjunction, these observations suggest that the models tend to overfit to patterns in the generated data arising from the parameterisation of the generation algorithm, rather than learning to perform satisfiability checking in a more general sense. In other words,  just like a bad student of logic, they appear to ``learn the proofs'' rather than the logical principles behind the proofs required for successful systematic generalisation.\n\n\n\n \n    \n\n\\begin{table}[!tb]\n    \\centering\n    \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{cc|ccccc}\n    \n    \\textbf{Train} $l$ & \\textbf{Eval} $l$ & \\sSd & \\sR & \\sSrel & \\sSrelN & \\sRd \\\\\n    \n    \\hline\n        \n        \n         \\multirow{3}{*}{\\begin{tabular}{@{}c@{}}$l \\geq 6$ \\\\ \\emph{(easy)}\\end{tabular}} & satisfiable & 61 & 87 & 90 & 53 & 58 \\\\\n          & $l \\leq 12$ & 98 & 99 & 99 & 70 & 98\\\\\n         & $l \\geq 42$ & 84 & 74 & 70 &  42 & 70 \\\\\n         \\hdashline\n         \\multirow{3}{*}{\\begin{tabular}{@{}c@{}}$l \\geq 22$ \\\\ \\emph{(hard)}\\end{tabular}} &satisfiable & 57 & 89 & 86 & 50 & 52 \\\\\n          & $l \\leq 12$ & 68 & 61 &33 & 61& 72\\\\\n         & $l \\geq 42$ &100 & 94 & 99 &  96 & 73 \\\\\n        \\hline\n    \\end{tabular}\n    }\n    \\caption{Accuracy of models trained on random satisfiable and easy\/hard insatisfiable examples, and evaluated on random satisfiable and easy\/hard insatisfiable examples with proof length $l\\leq12$ and $l \\geq 42$.}\n    \\label{tab:res-harder}\n\\end{table}\n\n\n\\textbf{\\emph{Transformers seem unable to reliably learn the distinct reasoning patterns.}} \nFinally, as determining non-satisfiability for the fragments \\sSd{} and \\sR{} involves detection of one of a handful of forbidden configurations, we generate data that exposes precisely these forbidden configurations.\n\nThe four top rows of Table~\\ref{tab:res-constructed} show that models optimised on \\emph{random} and \\emph{random hard} datasets in \\sSd{} and \\sR{} pick these patterns up to a varying degree: while all models perform better than chance, the evaluation performance drops considerably, when comparing to in-distribution performance.\nIn the case of \\sSd{}, breaking down the performance by chain depth, as shown in Figure~\\ref{fig:breakdown}, reveals that models optimised on \\emph{random} data perform best on examples with chain length two (essentially detecting inconsistencies of the form ``All $p$ are $q$'', ``All $q$ are $s$'', ``Some $p$ are not $s$'') and fail to generalise beyond that, while models trained on \\emph{random hard} data perform best on examples with chain length five, with their performance deteriorating for examples with longer and shorter chains. This reinforces the previous point that these models appear to have learned to identify problems that have proofs with similar structure to those in their training data rather than learning to solve the problem in a more general sense.\n\nWhen we both optimise and evaluate models on the constructed datasets, we find that they are capable of learning these inconsistency patterns well, and generalise to harder problems unseen during training: the RoBERTa model optimised on \\sSd{} with chain lengths between two and six performs just as well on evaluation data with chain lengths of up to 10 (Table~\\ref{tab:res-constructed}, row five). However, other models fail to learn these patterns from constructed \\sSd{} data, suggesting that it is a challenging task. Similarly, all models optimised on \\sR{} generalise to unseen lengths of the $\\forall\\forall$-configurations (Table~\\ref{tab:res-constructed}, row six).\nHowever, even when seemingly learning these patterns, models fail to reliably transfer these capabilities beyond the constructed cases, as evidenced by the poor generalisation performance when trained on the constructed datasets and evaluated on randomly generated data  (Table~\\ref{tab:res-constructed}, bottom). For \\sR{}, the bad generalisation capability is expected, as  the constructed dataset does not contain inconsistency patterns other than the $\\forall\\forall$-configurations (e.g. examples where only the non-relational statements are inconsistent). However, this is inconsistent with the case of \\sSd{}, as all possible inconsistency patterns are covered in the constructed dataset, yet the optimised RoBERTa models fail to transfer to random \\sSd{} data.\n\n    \n\n\n\n\\begin{table}[!tb]\n    \\centering\n    \\begin{tabular}{l l c}\n    \\textbf{Train On} & \\textbf{Evaluate On} & \\textbf{Accuracy} \\\\\n    \\hline\n   \n   \n    \\tikzmarkin{d}(1.01,-0.15)(-0.2,0.3)$\\sSd$ & \\setbordercolor{rred!35}\\tikzmarkin{f}(1.2,-0.15)(-0.2,0.3)$\\sSd_{[2,6]}$ & $65$ \\\\ \n    $\\sSd_{l\\geq22}$ & $\\sSd_{[2,6]}$ & $57$ \\\\\n    $\\sR$ & $\\sR_{\\langle1,2\\rangle}$ & $53$ \\\\\n    $\\sR_{l\\geq22}$ \\tikzmarkend{d} & $\\sR_{\\langle1,2\\rangle}$ & $58$ \\\\\n    \n   \n   \n   \n    \\setbordercolor{rred!35}\\tikzmarkin{g}(0.1,-0.19)(-0.2,0.3)$\\sSd_{[2,6]}$ & $\\sSd_{[2,10]}$ &  $96$\\footnotemark[2] \\\\\n   \n    $\\sR_{\\langle1,2\\rangle}$ & $\\sR_{\\langle1,3\\rangle}$ \\tikzmarkend{f} & $100$ \\\\\n   \n   \n   \n   \n    $\\sSd_{[2,6]}$ &\\tikzmarkin{e}(1.93,-0.19)(-0.2,0.3)$\\sSd$ & $54$\\footnotemark[2] \\\\\n    $\\sR_{\\langle1,2\\rangle}$ & $\\sR$ & $48$ \\\\\n   \n    $\\sSd_{[2,6]}$+$\\sR_{\\langle1,2\\rangle}$\\tikzmarkend{g}&$\\sR$\\tikzmarkend{e}& $88$\\footnotemark[2] \\\\\n    \n    \\hline\n    \\end{tabular}\n    \\caption{Accuracy  of model optimised and evaluated on {\\color{bblue} randomly} generated and {\\color{rred}constructed} datasets in \\sSd and \\sR{}. The datasets $\\sSd_{l\\geq22}$ and $\\sR_{l\\geq22}$ have unsatisfiable examples with at least proof length $22$, while $\\sSd_{[\\cdot,\\cdot]}$, $\\sR_{\\langle\\cdot,\\cdot\\rangle}$ have constructed chains in the bracketed range.}\n    \\label{tab:res-constructed}\n\\end{table}\n\\footnotetext[2]{Results for RoBERTa only, as other models failed to converge on constructed \\sSd{} data.}\n\n\\pgfplotsset{\n\/pgfplots\/my legend\/.style={\nlegend image code\/.code={\n\\draw[thick,ggreen,dotted](-0.05cm,0cm) -- (0.3cm,0cm);%\n   }\n  }\n}\n\n\\begin{figure}[tb]\n    \\centering\n    \\begin{tikzpicture}\n\\begin{axis}[\n    height=15em,\n    width = 1\\columnwidth,\n    ybar=5pt,\n    ymin=40,\n    ymax=100,\n    scaled y ticks = false,\n    extra y ticks = 50,\n    extra y tick labels={},\n   \n    extra y tick style={grid=major,major grid style={thick,draw=ggreen,dotted}},\n    bar width=4.25pt,\n    x tick label style={},\n    major x tick style = transparent,\n    ylabel={Accuracy},\n    xlabel={Chain Length},\n    x tick label as interval,\n    xmajorgrids=true,\n    ymajorgrids=true,\n   \n    legend columns=1,\n        xticklabels={$2$,$3$,$4$,$5$, $6$},\n    xtick={0,1,2,3,4,5},\n    xmin=0, xmax=5,\n    ]\n    \n    \\addplot+[legend entry=\\sSd, bar shift=-2.5pt, color=bblue, fill=bblue] \n    coordinates {\n        (0.5,91.69) (1.5,65.31) (2.5,59.69) (3.5,59.13) (4.5,60.37)\n        };\n    \\addplot+[legend entry=\\sSd$_{l\\leq 22}$, bar shift=+2.5pt, color=rred, fill=rred] \n    coordinates {\n        (0.5,51.8) (1.5,54.25) (2.5,58.38) (3.5,63.94) (4.5,62.31)\n        };\n    \\addlegendimage{my legend}\n    \\addlegendentry{Random Guess}\n \n \n\\end{axis}\n\\end{tikzpicture}\n    \\caption{Accuracy by chain length for RoBERTa models optimised on \\sSd{} and \\sSd$_{l\\leq 22}$, evaluated on \\sSd$_{[2,6]}$.\n    }\n    \\label{fig:breakdown}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work and Conclusion}\n\nThe work reported in this paper represents a continuation of various studies carried out on ``challenge sets'', proposed to investigate the ability of (optimised) neural networks to perform different kinds of reasoning. Such studies range from monotonicity entailments \\cite{nnFrag:GRP20,Yanaka2020} to probing lexical knowledge \\cite{Glockner2018} to logical connectives \\cite{Salvatore2019,Richardson2019a} or verb symmetry \\cite{Mitra2019}. These studies are related to ours, in that they seek to isolate capabilities of interest and perform controlled experiments using synthetic datasets. They have in common that models optimised on crowd-sourced datasets, such as MNLI, \\cite{Williams2018}, perform poorly on the challenge set data exhibiting the elicited phenomenon, but fine-tuning the optimised model on portions of these data improves the performance. \n\nHowever, as \\newcite{Rozen2019} show, good scores after the fine-tuning probably stem from the fact that the investigated model has learned to adapt to the regularities of the challenge set rather than learning a general notion of the investigated phenomenon. Our analysis is similar: transformer-based neural networks perform well on randomly generated data, but this performance is brittle, and the models overfit to the problem space as set out by the dataset generation method. They do not generalise well to examples outside of that space, suggesting that they do not generalise \\emph{systematically} \\cite{Fodor1988ConnectionismAnalysis}, i.e. they struggle to identify a finite set of rules and to apply them repeatedly. Similar to our findings, research on systematic generalisation suggests that neural networks tend to generalise without systematicity in supervised learning scenarios \\cite{Johnson2017CLEVR:Reasoning,Lake2017GeneralizationNetworks,Goodwin2020}, although these studies, unlike ours, did not concern pre-trained models. \n\nOne might argue that it is unfair to expect a statistical model that relies on correlations to learn patterns that are not obviously present in the data. However, it seems that such claims are being made in the literature \\cite{Clark2020a}.\nWhile we observe that models optimised on constructed examples generalise well to  harder problems unseen during training, we also show that this capability appears not to transfer to examples that are only superficially different. Thus, in our experiments, similar to \\newcite{nnFrag:rs21}, we find that the optimised models are not able to reliably disentangle and acquire the different reasoning patterns required to successfully complete the task of determining satisfiability.\n\nOur study highlights one of the issues with empirically postulating neural networks various capabilities by means of good performance on challenge sets. They have only negative predictive power \\cite{Gardner2020}: while low performance indicates the lack of a capability, the converse does not necessarily hold. This can be taken as a motivation to develop formal verification methods for neural networks \\cite{Shi2020RobustnessTransformers}, or investigate worst-case bounds for different phenomena \\cite{Raghunathan2018,Jia2019}.\n\nOur formulated task naturally allows us to expand the scope of the controlled experiments: for example, by increasing the closed-class vocabulary.\nAnother possible avenue is to focus on improving the systematic generalisation of neural approaches, for example by providing the formulas required to prove that a set of sentences is unsatisfiable as additional supervision signal, or by relying on modular or model-based approaches \\cite{Andreas2016NeuralNetworks,Lake2013One-shotProcess}.\n\n\\section*{Limitations}\n\nThe design of the study limits our findings by design - by removing the need of inducing\nmeaning postulates (i.e. commonsense reasoning and world knowledge) we explicitly focus\non logico-syntactic capabilities, analogous to how \"reasoning\" is defined for symbolic\napproaches to AI. Arguably, In \"real world\" application scenarios, commonsense reasoning\nand world knowledge cannot be fully disconnected from the requirement to perform\nreasoning, which allows our inquiry to draw fundamental conclusions about the capabilities\nof transformer-based language models rather than to make recommendations which are of\nimmediate relevance to practitioners.\n\nOur study also suffers from the inductive dilemma. We find that multiple transformer-based\nlanguage models follow the trends reported in this paper, specifically that they fail to robustly\nidentify the reasoning patterns necessary for reliably determining satisfiability in the elicited\nfragments. However, due to the empirical nature of this research, this finding is of course not\na guarantee that some neural architecture (transformer-based or otherwise) could still\nperform well, when tested in our out-of-distribution evaluation settings.\n\n\\section*{Acknowledgements}\nThe authors would like to acknowledge the use of the Computational Shared Facility at The University of Manchester and thank the anonymous reviewers from the ARR December 2021 cycle for their valuable feedback.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe physical states of a bosonic string moving on a torus form a nice Lie algebra called the fake monster algebra (\\cite{B2}). The study of this algebra has led to the definition of vertex algebras, generalized Kac-Moody algebras and the proof of the moonshine conjectures (\\cite{B4}). The fake monster algebra is the simplest example of a generalized Kac-Moody algebra. It is easy to extend the definition of generalized Kac-Moody algebras to superalgebras. Since the physical states of a compactified bosonic string form a generalized Kac-Moody algebra one would expect that the physical states of a compactified superstring give a realization of a generalized Kac-Moody superalgebra. Some evidence for the existence of such an algebra is given in \\cite{NRS} and \\cite{B4}. In this paper we show that this conjecture is true. In a special case we get a particularly nice algebra which we call the fake monster superalgebra.\n\nWe explain the construction in more detail. The superstring has bosonic and fermionic degrees of freedom. The bosonic degrees are represented by a vertex algebra constructed from a lattice. The fermionic degrees can be bosonized and then also be described by such an algebra. One also has to include some ghost coordinates to define a fermion emission vertex. Thus the Fock space of a compactified superstring can be described by the vertex algebra of some suitable lattice. This vertex algebra carries representations of various algebras the most important being the $N=2$ superconformal algebra. The representation of this algebra is used to construct the BRST operator $Q$ satisfying $Q^2=0$. The physical states of the superstring are now represented by the cohomology of this operator. More precisely for nonzero momentum there are infinitely many isomorphic copies of vector spaces called pictures representing the physical states. We restrict to the canonical ghost pictures. The vertex algebra induces a product on the vector space of physical states that defines the structure of a Lie superalgebra on this space. This Lie superalgebra has a number of nice properties. It is graded by the 10 dimensional momentum lattice. The dimensions of the root spaces can be described easily. All roots have zero or negative norm. It has an invariant supersymmetric bilinear form and carries a representation of the $N=1$ supersymmetry algebra. In particular it is a generalized Kac-Moody superalgebra.\n If the momentum lattice of the superstring is the unique unimodular even 10 dimensional Lorentzian lattice then the denominator function is an automorphic form with known product expansion. This allows us to determine the simple roots in this case.\n\nMost of the ideas necessary for the above construction come from physics and are described in the physics literature (cf. \\cite{FMS},\\cite{LT},\\cite{K} and \\cite{LZ2}). References \\cite{B5} and \\cite{R2} are also important for this work.\n\nWe describe the sections of this paper in more detail.\n\nIn the first section we recall some facts about vertex algebras graded by abelian groups. We show that in the lattice construction of these algebras the locality axiom is a simple consequence of Wick's Theorem.\n\nIn the second section we describe some results about generalized Kac-Moody superalgebras and state the characterization theorem that will be applied to deduce the first main result of this paper.\n\nIn the next section we construct the vertex algebra of the 10 dimensional superstring moving on a torus from a rational 18 dimensional lattice. We give the operator product expansions of various fields. They are used to construct an action of the $N=2$ superconformal algebra and to construct the nilpotent BRST operator.\n\nIn the last section we describe the cohomology spaces that represent the physical states and define a Lie bracket on the space of physical states. We derive a number of properties of this Lie superalgebra, e.g. a root space decomposition, the multipicities of the roots, an action of the $N=1$ supersymmetry algebra etc.. These properties imply that this algebra is a generalized Kac-Moody superalgebra. Finally we define the fake monster superalgebra which is a special case of the above construction with the Lorentzian lattice $II_{9,1}$ as momentum lattice and determine the simple roots of this algebra using the product expansion of its denominator function.\n\n\\section{Vertex algebras}\n\nIn this section we recall some results on vertex algebras. A more detailed exposition can be found in \\cite{K2} and \\cite{DL}. We give a simple proof of the locality axiom in the lattice construction.\n\n\\subsection{Definition and some properties} \n\nLet $\\Gamma$ be a finite abelian group and $\\Delta : \\Gamma \\times \\Gamma \\rightarrow \\mathbb{Q}\/\\mathbb Z$ a symmetric map bilinear mod $\\mathbb Z$. This implies that $\\Delta$ takes it values in $\\frac{1}{g}\\mathbb Z$ where $g$ is an exponent of $\\Gamma$. Let $\\eta \\, :\\, \\Gamma \\times \\Gamma \\rightarrow \\mathbb C^*$ be bimultiplicative. Then $\\eta(\\gamma_1,\\gamma_2)$ is a root of unity, i.e. $\\eta(\\gamma_1,\\gamma_2)^g=1$.\n\nSuppose that \n\\begin{equation} V=\\bigoplus_{\\gamma \\in \\Gamma} V_{\\gamma} \\ee \nis a $\\Gamma$-graded \ncomplex vector space and \n\\begin{eqnarray} \nV & \\rightarrow & (\\mbox{End} V)[[z^{\\frac{1}{g}},z^{-\\frac{1}{g}}]] \\\\\na & \\mapsto     & a(z)=\\sum_{n\\in \\frac{1}{g}\\mathbb Z} a_nz^{-n-1}\n\\end{eqnarray}\na parity preserving state-field correspondence, i.e. for $a\\in V_{\\gamma_1}, \nb\\in V_{\\gamma_2}$ \n\\begin{equation} \\label{dpe}\na(z)b = \\sum_{n\\in {\\mathbb Z}+\\Delta(\\gamma_1,\\gamma_2)} a_nb\\, z^{-n-1} \n\\ee\nand\n\\begin{equation}\na_nb \\in  V_{\\gamma_1+\\gamma_2} \n\\ee\nis zero for $n$ sufficiently large. \n\nThe field $a(z)$ is also written $Y(a,z)$ in the literature.\n\n$V$ is a $\\Gamma$-graded vertex algebra if it satisfies the following \nconditions\n\\begin{enumerate}\n\\item There is an element $1\\in V_0$, called vacuum, with \n      \\begin{equation} 1(z)a=a \\qquad \\mbox{and} \\qquad a(z)1\\big|_{z=0}=a. \\ee \n\\item The operator $D$ on $V$ defined by $Da=a_{-2}1$ satisfies \n      \\begin{equation} [D,a(z)]=\\partial a(z). \\ee\n\\item The locality condition \n      \\begin{equation}\n      i_{z,w}(z-w)^n\\, a(z)b(w)-\\eta(\\gamma_1,\\gamma_2)i_{w,z}(z-w)^n \\, b(w)a(z)=0\n      \\label{loc} \\ee\n      holds for $a\\in V_{\\gamma_1}, b\\in V_{\\gamma_2}$ and \n      $n\\in \\mathbb Z+\\Delta(\\gamma_1,\\gamma_2)$ sufficiently large.\n      Here $i_{z,w}(z-w)^n$ is the binomial expansion of $(z-w)^n$ in the \n      domain $|z|>|w|$, i.e. \n      $i_{z,w}(z-w)^n=\\sum_{k\\geq 0}(-1)^k{n \\choose k}z^{n-k}w^k$.\n      In the expresssion $i_{w,z}(z-w)^n$ we must be precise about the roots of      unity. We choose $i_{w,z}(z-w)^n=e^{i\\pi n}i_{w,z}(w-z)^n=\n      e^{i\\pi n}\\sum_{k\\geq 0}(-1)^k{n \\choose k}w^{n-k}z^k$.\n\\end{enumerate} \nNote that $D$ leaves the spaces $V_{\\gamma}$ invariant.\n\nFor the rest of the section assume that $V$ is a $\\Gamma$-graded vertex \nalgebra. \n\nWe now describe some consequences of the definition. \n\nLocality implies $\\eta(\\gamma_1,\\gamma_2)e^{\\Delta(\\gamma_1,\\gamma_2)}=\n                  \\eta(\\gamma_2,\\gamma_1)^{-1}e^{-(\\gamma_2,\\gamma_1)}$.\n\nFor all $a$ in $V$ one has\n\\begin{eqnarray} \na(z)1   &=& e^{zD}a \\\\\n(Da)(z) &=& \\partial \\, a(z) \\label{Dpa}\\\\\ne^{wD} a(z) e^{-wD} &=& a(z+w) \\quad\\mbox{in the domain }|z|>|w|.\n\\end{eqnarray}\n\nFor $a\\in V_{\\gamma_1}$ and $b\\in V_{\\gamma_2}$ the following symmetry formula can be proved as in the superalgebra case.\n\\begin{equation} \na(z)b=\\eta(\\gamma_1,\\gamma_2)e^{zD}b(-z)a\n\\, .\n\\ee\n\nThe most important property of vertex operators is Borcherds' identity.\n\\begin{p1}\nLet $a\\in V_{\\gamma_1},\\, b\\in V_{\\gamma_2},\\, c\\in V_{\\gamma_3}$ and \n$n\\in \\mathbb Z+\\Delta(\\gamma_1,\\gamma_2),\\, k\\in {\\mathbb Z}+\\Delta(\\gamma_1,\\gamma_3)$. Then we have \n\\[ \\sum_{j\\geq 0} {n \\choose j} (-1)^j \n          \\big\\{ a_{n+k-j}b_{m+j}c  \n         -\\eta({\\gamma_1},{\\gamma_2})e^{i\\pi n} b_{m+n-j}a_{k+j}c \\big\\} \\]  \n\\[ = \\sum_{j\\geq 0} {k \\choose j} (a_{n+j}b)_{k+m-j}c \\]\n\\end{p1}\n\nWe will use some special cases of Borcherds' identity very often in the following sections. If $\\Delta(\\gamma_1,\\gamma_3)$ is zero we can put $k=0$ to get the associativity formula \n\\begin{equation} \n(a_{n}b)_{m}c =\n\\sum_{j\\geq 0} {n \\choose j} (-1)^j \n          \\big\\{ a_{n-j}b_{m+j}c  \n         -\\eta({\\gamma_1},{\\gamma_2})e^{i\\pi n} b_{m+n-j}a_{j}c \\big\\}.\n\\ee\nIf $\\Delta(\\gamma_1,\\gamma_2)=0$ the commutator formula \n\\begin{equation}  \na_{k}b_{m}c  \n         -\\eta({\\gamma_1},{\\gamma_2})b_{m}a_{k}c =\n\\sum_{j\\geq 0} {k \\choose j} (a_{j}b)_{k+m-j}c \n\\ee \nholds.\nNow suppose that $a_{n+j}b=0$ for $j\\geq 1$. Then \n\\begin{equation} \\label{bisc}\n\\! \\! (a_{n}b)_{k+m}c = \n\\sum_{j\\geq 0} {n \\choose j} (-1)^j \n          \\big\\{ a_{n+k-j}b_{m+j}c  \n         -\\eta({\\gamma_1},{\\gamma_2})e^{i\\pi n} b_{m+n-j}a_{k+j}c \\big\\}. \n\\ee\nIn contrast to the associativity formula we need not assume here that $\\Delta(\\gamma_1,\\gamma_3)$ is zero.\n\nWe conclude this section with some remarks.\n\\pagebreak[3]\n\nAn element $\\omega\\in V_{0}$ is called Virasoro element of central charge \n $c$ if it satisfies the following 3 conditions\n\\begin{enumerate}\n\\item[] The operators $L_m=\\omega_{m+1}$ give a representation of the \n          Virasoro algebra of central charge $c$, i.e.\n          \\[ [L_m,L_n]=(m-n)L_{m+n}+ \\frac{m^3-m}{12}\\delta_{m+n,0}\\,c\\, . \\] \n\\item[] $L_0$ is diagonizable on $V$.\n\\item[] $D=L_{-1}$.\n\\end{enumerate}\n\nIn conformal field theory the vertex operator of a state $a$ with $L_0 a=h a$  is expanded as \n$a(z)=\\sum  a_{(n)}z^{-n-h}$, i.e. the expansion of a field depends on its conformal weight $h$. The relation between the two modes is given by \n$a_n= a_{(n+1-h)}$ resp. $a_{(n)}=a_{n-1+h}$. \n\n\\subsection{Construction from rational lattices} \\label{lc}\n\nNow we describe how vertex algebras can be constructed from rational lattices.\nThe proof given here only uses Wick's theorem (cf. \\cite{K2}).\n \nLet $L$ be a rational lattice of finite rank and $L_0\\neq 0$ an even sublattice of $L$ with $L\\subset L_0^*$. Then $(L,L_0)\\subset \\mathbb Z\\,$.\nLet $L=L_0\\cup L_1\\cup \\ldots \\cup L_n$ with $L_i=\\delta_i + L_0$ be the coset decomposition of $L$ with respect to $L_0$. Put $\\Gamma=L\/L_0=\\{\\gamma_0,\\gamma_1,\\ldots , \\gamma_n\\}$ where $\\gamma_i$ corresponds to $L_i$ and let $g$ be an exponent of $\\Gamma$. \nDefine $\\Delta\\, :\\, \\Gamma \\times \\Gamma \\rightarrow \\mathbb Q\/\\mathbb Z$ by \n$\\Delta(\\gamma_i,\\gamma_j)=-(\\delta_i,\\delta_j) \\, \\mbox{mod}\\, \\mathbb Z$.\nThis map is well-defined and bilinear mod $\\mathbb Z\\,$. \nLet $\\eta \\, :\\, \\Gamma \\times \\Gamma \\rightarrow \\mathbb C^*$ be a bimultiplicative map with\n\\begin{equation} \\eta(\\gamma_j,\\gamma_j)=e^{i\\pi (\\delta_j,\\delta_j)}\\, . \\label{nc} \\ee\nThis condtion is quite restrictive but necessary for consistency.\n\nLet $\\varepsilon : L\\times L\\rightarrow {\\mathbb C}^*$ be a 2-cocycle satisfying $\\varepsilon(\\alpha,\\beta)=B(\\alpha,\\beta)\\varepsilon(\\beta,\\alpha)$ where\n$B(\\alpha,\\beta)=e^{-i\\pi (\\alpha,\\beta)}\\eta(\\gamma_i,\\gamma_j)$ for $\\alpha \\in L_i$ and $\\beta \\in L_j$ and $\\varepsilon(0,\\alpha)=\\varepsilon(\\alpha,0)=1$. Note that $B$ is a bimultiplicative map from $L\\times L$ to ${\\mathbb C}^*$ and that $B(\\alpha,\\alpha)=1$ by (\\ref{nc}).\n\nIt is easy to construct such a 2-cocycle. Let $\\{\\alpha_1,\\ldots, \\alpha_m\\}$ be a $\\mathbb Z$-basis of $L$. Define \n\\[ \\begin{array}{lclr} \n\\varepsilon(\\alpha_i,\\alpha_j) &=& a_{ij} & i\\leq j \\\\\n\\varepsilon(\\alpha_i,\\alpha_j) &=& B(\\alpha_i,\\alpha_j)\n    \\varepsilon(\\alpha_j,\\alpha_i)   & i>j \n\\end{array} \\]\nwith some $a_{ij}\\in {\\mathbb C}^*$ and extend to $L$ by bimulticativity. This gives us a 2-cocycle $\\varepsilon : L\\times L\\rightarrow {\\mathbb C}^*$ satisfying $\\varepsilon(\\alpha,\\beta)=B(\\alpha,\\beta)\\varepsilon(\\beta,\\alpha)$ and $\\varepsilon(0,\\alpha)=\\varepsilon(\\alpha,0)=1$.\n\nExtend the symmetric form of $L$ to $h=L{\\otimes}_{\\mathbb Z}{\\mathbb C}$. Define the infinite dimensional Heisenberg algebra $\\hat{h}=h\\otimes {\\mathbb C}[t,t^{-1}]\\oplus {\\mathbb C}c$ with products $[h_1(m),h_2(n)]=m\\delta_{m+n,0} \\langle h_1,h_2 \\rangle c$ and $[h_1(m),c]=0$. Then $\\hat{h}^-=h\\otimes t^{-1}{\\mathbb C}[t^{-1}]$ is an abelian subalgebra of $\\hat{h}$ and $S(\\hat{h}^-)$ is the \nsymmetric algebra of polynomials in $\\hat{h}^-$.  \n  \nLet ${\\mathbb C}[L]$ be the group algebra of $L$ with basis $\\{ e^{\\alpha} |\\, \\alpha \\in L \\}$ and products $e^{\\alpha}e^{\\beta}=e^{\\alpha + \\beta}$.\n\nThe vector space \n\\begin{equation} V=S(\\hat{h}^-)\\otimes {\\mathbb C}[L] \\ee\ndecomposes as\n\\begin{equation} V=\\bigoplus_{\\gamma_i \\in \\Gamma} V_{\\gamma_i}  \\ee\nwhere \n\\begin{equation} V_{\\gamma_i}=S(\\hat{h}^-)\\otimes {\\mathbb C}[L_i]\\, . \\ee\n\nThere is a natural action of $\\hat{h}$ on $V$. \n\nWe define the vertex operator of $e^{\\alpha}$ as\n\\begin{equation} \ne^{\\alpha}(z) = e^{\\alpha}(z)^{+}\\, e^{\\alpha}(z)^{-}\n\\ee\nwhere\n\\begin{equation} e^{\\alpha}(z)^{+}=e^{\\alpha}c_{\\alpha}e^{\\sum_{m>0}{\\alpha}(-m)\\frac{z^m}{m}}\n                     =e^{\\alpha}c_{\\alpha}\\sum_{m\\geq 0}S_m(\\alpha)z^m \\ee\nand\n\\begin{equation}\n  e^{\\alpha}(z)^{-}=z^{\\alpha(0)}e^{-\\sum_{m>0}{\\alpha}(m)\\frac{z^{-m}}{m}} \\, .\n\\ee\nThe linear operator $c_{\\alpha}$ acts on ${\\mathbb C}[L]$ as $c_{\\alpha} e^{\\beta}=\\varepsilon(\\alpha,\\beta) e^{\\beta}$. \n\nFor $h(-n-1),\\, n\\geq 0$ put\n\\begin{equation} h(-n-1)(z) = \\partial_z^{(n)} h(z) \\ee\nwith $h(z)=\\sum_{n\\in {\\mathbb Z}}h(n)z^{-n-1}$ and \n$\\partial_z^{(n)}=\\frac{\\partial_z}{n!}$.\n\nThe bosonic normal ordering $: h_1(n_1)\\cdots h_k(n_k) :$ of Heisenberg generators is defined by putting all $h(n)$ with $n<0$ (\"creation operators\") to the left of those with $n\\geq 0$ (\"annihilation operators\").\n\nDefine \n\\begin{eqnarray} \n\\lefteqn{ \\left( h_1(-n_1-1)\\cdots h_k(-n_k-1) e^{\\alpha}\\right)(z)  } \n\\nonumber \\\\\n&= e^{\\alpha}(z)^{+}\n    \\, : h_1(-n_1-1)(z) \\cdots h_k(-n_k-1)(z) : \\, \n   e^{\\alpha}(z)^{-} &\n\\end{eqnarray}\nand extend this definition linearly to $V$ to get a parity preserving state-field correspondence $V \\rightarrow (\\mbox{End} V)[[z^{\\frac{1}{g}},z^{-\\frac{1}{g}}]]$.\n\n\\begin{thp1}\nWith this structure $V$ is a vertex algebra graded by $\\Gamma$. The vacuum is given by $1\\otimes e^0$.\n\\end{thp1}\n{\\it Proof:}\nWe only prove the locality. Using\n\\begin{equation}\ne^{\\alpha}(z)^{-}e^{\\beta}(w)^{+} =\n(z-w)^{(\\alpha,\\beta)}e^{\\beta}(w)^{+}e^{\\alpha}(z)^{-} \n                                   \\quad |z|>|w| \n\\ee\nwe get for $\\alpha\\in L_i$ and $\\beta \\in L_j$ in the domain $|z|>|w|$\n\\[ e^{\\alpha}(z) e^{\\beta}(w) = \n    (z-w)^{(\\alpha,\\beta)}e^{\\alpha}(z)^{+} e^{\\beta}(w)^{+}\n                          e^{\\alpha}(z)^{-} e^{\\beta}(w)^{-} \\, . \\]\nWhile the commutator $[e^{\\alpha}(z)^{-},e^{\\beta}(w)^{-}]$ is zero \n$e^{\\alpha}(z)^{+}$ and $e^{\\beta}(w)^{+}$ do not commute. \n\nLet $n\\in {\\mathbb Z}+{\\Delta}(\\gamma_i,\\gamma_j)$. Then\n\\[ i_{z,w}(z-w)^n e^{\\alpha}(z) e^{\\beta}(w)\n          -\\eta(\\gamma_i,\\gamma_j)i_{w,z}(z-w)^n e^{\\beta}(w) e^{\\alpha}(z)=0 \\]\nis equivalent to \n\\[ e^{\\alpha}c_{\\alpha}e^{\\beta}c_{\\beta}i_{z,w}(z-w)^{n+(\\alpha,\\beta)}\n   -\\eta(\\gamma_i,\\gamma_j)e^{-i\\pi (\\alpha,\\beta)}\n   e^{\\beta}c_{\\beta}e^{\\alpha}c_{\\alpha}i_{w,z}(z-w)^{n+(\\alpha,\\beta)}=0 \\]\n\nNote that $n+(\\alpha,\\beta)$ is an integer by the definition of $\\Delta$. For $n$ sufficiently large this integer is positive. Then \n$i_{z,w}(z-w)^{n+(\\alpha,\\beta)}=i_{w,z}(z-w)^{n+(\\alpha,\\beta)}$. Hence \n$e^{\\alpha}(z)$ and $e^{\\beta}(w)$ are local if \n\\begin{equation} \ne^{\\alpha}c_{\\alpha}e^{\\beta}c_{\\beta}=B(\\alpha,\\beta)e^{\\beta}c_{\\beta}e^{\\alpha}c_{\\alpha} \\, .\n\\ee\nSince $B$ is bimultiplicative this equation is equivalent to the equations\n\\begin{eqnarray}\n\\varepsilon(\\alpha,\\beta) &=& B(\\alpha,\\beta) \\varepsilon(\\beta,\\alpha) \\\\\n\\varepsilon(\\alpha,\\beta)\\varepsilon(\\alpha+\\beta,\\gamma) &=&\n\\varepsilon(\\beta,\\gamma)\\varepsilon(\\alpha,\\beta+\\gamma) \\, . \n\\end{eqnarray}\nBut these are satisfied by construction of $\\varepsilon$ and $B$. It follows that the fields $e^{\\alpha}(z)$ and $e^{\\beta}(w)$ are local.\n\nNow we want to prove the locality of the fields of two arbitrary states in $V$.\nFor that it is sufficient to prove it for two states of the form\n\\begin{eqnarray*}\na &=& h_1(-n_1-1)\\cdots h_i(-n_i-1) e^{\\alpha} \\\\\nb &=& k_1(-m_1-1)\\cdots k_j(-m_j-1) e^{\\beta}\\, . \n\\end{eqnarray*}\nWe do this by induction on $i+j$. The case of $i+j=0$ has been done above.\nWe can assume that $j\\geq 1$ so that we can write \n$b=c\\otimes d$ with $c=k_1(-m_1-1)$.\nThen \n\\[ b(w)=c(w)^{+} d(w) + d(w) c(w)^{-} \\]\nwhere \n\\[ c(w)^{-}=k_1(-m_1-1)(w)^{-}\n=\\sum_{l\\geq 0}(-1)^l {l \\choose m_1}k_1(l-m_1)w^{-l-1} \\]\nonly contains annihilation operators and \n\\[ c(w)^{+}=k_1(-m_1-1)(w)^{+}\n=\\sum_{l< 0}(-1)^l {l \\choose m_1}k_1(l-m_1)w^{-l-1}\\]\nonly creation operators. We get\n\\begin{eqnarray}\n\\lefteqn{ \ni_{z,w}(z-w)^n a(z) b(w) -\\eta(\\gamma_i,\\gamma_j)i_{w,z}(z-w)^nb(w)a(z)} \n\\label{ind} \\\\\n&=& c(w)^{+}\\big\\{i_{z,w}(z-w)^n a(z) d(w)\n     -\\eta i_{w,z}(z-w)^n d(w)a(z) \\big\\} \\nonumber \\\\\n& & +\\big\\{i_{z,w}(z-w)^n a(z) d(w)\n     -\\eta i_{w,z}(z-w)^n d(w)a(z) \\big\\}\n     c(w)^{-} \\nonumber \\\\\n& & +\\big\\{i_{z,w}(z-w)^n [a(z),c(w)^{+}]d(w)\n     -\\eta i_{w,z}(z-w)^n d(w)[c(w)^{-},a(z)]\\big\\} \\, . \\nonumber\n\\end{eqnarray}\nThe first two terms vanish for $n\\in {\\mathbb Z}+{\\Delta}(\\gamma_i,\\gamma_j)$\nsufficiently large by the induction hypothesis. Hence we need to calculate \nthe commutators. To simplify notations we write $a_l=h_l(-n_l-1)$. Then\n$a(z)=e^{\\alpha}(z)^{+}:a_1(z)\\cdots a_i(z): e^{\\alpha}(z)^{-}$ and\n\\begin{eqnarray}\n\\lefteqn{ [a(z),c(w)^{+}] } \\\\\n&=& e^{\\alpha}(z)^{+}:a_1(z)\\cdots a_i(z):[\\, e^{\\alpha}(z)^{-},c(w)^{+}]\n    \\nonumber \\\\\n& & +\\, e^{\\alpha}(z)^{+}[\\,:a_1(z)\\cdots a_i(z):\\, ,c(w)^{+}]\\,  \n    e^{\\alpha}(z)^{-} \\, . \\nonumber\n\\end{eqnarray}\nUsing $[e^{\\alpha}(z)^{-},h(l)]=-z^l(\\alpha,h)e^{\\alpha}(z)^{-}$ for $l<0$ \nwe get\n\\[ [e^{\\alpha}(z)^{-},c(w)^{+}]=-i_{z,w}(z-w)^{-m_1-1}(\\alpha,k_1)e^{\\alpha}(z)^{-}\n       \\, . \\]\nWick's theorem (cf. \\cite{K2}) yields \n\\[ \n[:a_1(z)\\cdots a_i(z):\\, ,c(w)^{+}]=\\sum_{l=1}^{i}[a_l(z)^{-},c(w)]\\, \n\\hat{a}_l(z) \\]\nwhere $\\hat{a}_l(z)$ is $:a_1(z) \\cdots a_i(z):$ with $a_l(z)$ dropped.\nThe commutator relations of the Heisenberg algebra imply\n\\[ [a_l(z)^{-},c(w)]=(-1)^{n_l }{ n_l+m_1+1 \\choose m_1} (n_l+1) (h_l,k_1)\n    i_{z,w}(z-w)^{-n_l-m_1-2} \\]\nso that\n\\begin{eqnarray}\n\\lefteqn{ [a(z),c(w)^{+}] } \\label{com1}\\\\\n&=& -i_{z,w}(z-w)^{-m_1-1}(\\alpha,k_1)a(z) \\nonumber \\\\\n& & +\\sum_{l=1}^{i}(-1)^{n_l }{ n_l+m_1+1 \\choose m_1} (n_l+1) (h_l,k_1)\n    i_{z,w}(z-w)^{-n_l-m_1-2} (\\hat{a}_le^{\\alpha})(z) \\, .\\nonumber\n\\end{eqnarray}\nNote that in the case $i=0$ the second term is zero. \n$[c(w)^{-},a(z)]$ can be calculated similarly. We find\n\\begin{eqnarray}\n\\lefteqn{[c(w)^{-},a(z)] } \\label{com2}\\\\\n&=& -i_{w,z}(z-w)^{-m_1-1}(\\alpha,k_1)a(z) \\nonumber \\\\\n& & +\\sum_{l=1}^{i}(-1)^{n_l }{ n_l+m_1+1 \\choose m_1} (n_l+1) (h_l,k_1)\n    i_{w,z}(z-w)^{-n_l-m_1-2} (\\hat{a}_le^{\\alpha})(z) \\, .\\nonumber \\,\n\\end{eqnarray}\nAgain in the case $i=0$ the second term vanishes.\n\nPutting (\\ref{com1}) and (\\ref{com2}) into the last term of (\\ref{ind})\nwe see that also this term is zero for $n$ sufficiently large by the induction hypothesis. This finishes the proof of the theorem. \\hspace*{\\fill} $\\Box$\\\\\n\nNote that $e^{\\alpha}(z)^{+}=\\sum_{n\\geq 0}D^{(n)}e^{\\alpha}c_{\\alpha}z^n$.\n\nThe following simple formulas are useful\n\\begin{eqnarray*} \nk(-1)_{n} &=& k(n) \\\\ \ne^{\\alpha}_n e^{\\beta} &=& \n    \\varepsilon(\\alpha,\\beta) S_{-n-1-(\\alpha,\\beta)}(\\alpha)e^{\\alpha+\\beta} \\, .\n\\end{eqnarray*}\n\nThe first 3 Schur polynomials are\n\\begin{eqnarray*} \nS_0(\\alpha) &=& 1 \\\\\nS_1(\\alpha) &=& \\alpha(-1) \\\\\nS_2(\\alpha) &=& \\textstyle{\\frac{1}{2}} \\left(\\alpha(-1)^2+\\alpha(-2)\\right) \\, .\n\\end{eqnarray*}\n\n$V$ contains a one-parameter family of Virasoro elements. \n\nIf $L$ is nondegenerate then $V$ is simple.\n\n\\subsection{Invariant bilinear forms} \\label{ibf}\n\nWe will need some results on bilinear forms on vertex superalgebras. They have been studied in detail in \\cite{NRS}. There the even and the odd part of a vertex superalgebra are distinguished by their $L_0$-eigenvalues. Here we will consider a slighty different situation.\n\nLet $V$ be a vertex superalgebra with Virasoro element. Suppose that $L_1$ acts locally nilpotent on $V$ and that the eigenvalues of $L_0$ are all integral. \n\nDefine the adjoint vertex operator of $a\\in V$ with $L_0 a=ha$ by  \n\\begin{equation} \na(z)^*= \\sum_{n\\in \\mathbb Z} a_n^* z^{-n-1}\n\\ee\nwhere\n\\begin{equation}\na_n^* = (-1)^h \\sum_{m \\geq 0} \\left( \\frac{L_1^m}{m!} a \\right)_{2h-n-m-2} \n\\, .\n\\ee\nFor example $L_n^* = \\omega_{n+1}^* = \\omega_{1-n} = L_{-n}$. \nThe adjoint vertex operators satisfy a number of identities. \n\nA bilinear form $(\\, , \\, )$ on $V$ is called invariant if\n\\begin{equation} (a_n b,c) = (-1)^{|a||b|} (b, a_n^*c) \\, . \\ee\nAn invariant bilinear form is supersymmetric, i.e. \n\\begin{equation} (a,b) = (-1)^{|a||b|} (b,a) \\ee\nand states with different $L_0$-eigenvalues are orthogonal.\n\nLet $V_h$ be the subspace of states with $L_0$-eigenvalue $h$.\nWe can define a linear functional $f$ on $V_0$ by putting $f(a)=(1,a)$.\nThis functional vanishes on $L_1V_1$.\nThe correspondence between invariant bilinear foms on $V$ and linear functionals on $V_0\/L_1V_1$ is injective. It is even surjective. Given an element $f$ of the dual of $V_0\/L_1V_1$ then the bilinear form defined by \n$(a,b)=(f \\circ \\pi)(a_{-1}^*b)$, where $\\pi$ is the projection from $V$ to \n$V_0\/L_1V_1$, is invariant and induces $f$. This implies \n\\begin{thp1}\nThe space of invariant bilinear forms on $V$ is naturally\nisomorphic to the dual of $V_0\/L_1V_1$.\n\\end{thp1}\n\n\\section{Generalized Kac-Moody superalgebras}\n\nIn this section we describe some results about generalized Kac-Moody superalgebras (cf. \\cite{R1}, \\cite{R2}, \\cite{GN} and also \\cite{B1},\\ldots,\\cite{B4}).\nUnless otherwise specified the results will hold over the real or complex numbers.\n\n\\subsection{Definitions and properties}\n\nLet $a=(a_{ij})_{i,j \\in I}$ be a real square matrix, where $I$ is some finite or countable index set and $\\tau$ a subset of $I$, with the following properties\n\\[ \\renewcommand{\\arraystretch}{1.3} \\begin{array}{ll}\n a_{ij} =a_{ji} \\\\\n a_{ij} \\leq 0                    & \\mbox{if} \\quad i\\neq j \\\\\n 2a_{ij}\/a_{ii}\\in \\mathbb Z      & \\mbox{if} \\quad a_{ii} >0 \\\\\n  a_{ij}\/a_{ii}\\in \\mathbb Z      & \\mbox{if} \\quad a_{ii} >0 \\; \n                            \\mbox{and} \\; i\\in \\tau \\, .\n\\end{array}\\]\nLet $\\hat{G}$ be the Lie superalgebra with generators $e_i, f_i, h_{ij}$ and relations\n\\[ \\renewcommand{\\arraystretch}{1.3} \\begin{array}{l}\n {[} e_i, f_j ] = h_{ij}  \\\\\n {[} h_{ij}, e_k ] =  \\delta_{ij} a_{ik}e_k  \\\\\n {[} h_{ij}, f_k ] = -\\delta_{ij} a_{ik}f_k  \\\\\n (\\mbox{ad}\\, e_i)^{1-2a_{ij}\/a_{ii}}e_j = \n (\\mbox{ad}\\, f_i)^{1-2a_{ij}\/a_{ii}}f_j = 0 \n \\quad \\mbox{if} \\quad a_{ii}>0 \\\\\n {[} e_i,f_j ] = [f_i,f_j] = 0 \\quad \\mbox{if} \\quad a_{ij} = 0 \\\\\n\\mbox{deg } e_i =\\mbox{deg } f_i = 0 \\quad \\mbox{if} \\quad i\\not\\in\\tau \\\\\n\\mbox{deg } e_i =\\mbox{deg } f_i = 1 \\quad \\mbox{if} \\quad i\\in \\tau \n\\end{array} \\] \nWe list some properties of $\\hat{G}$. The elements $h_{ij}$ are zero unless the $i$-th and $j$-th columns of $a$ are equal. The nonzero $h_{ij}$ are linearly independent and span a commutative subalgebra $\\hat{H}$ of $\\hat{G}$. The elements $h_{ii}$ are usually denoted $h_i$. Every nonzero ideal of $\\hat{G}$ has nonzero intersection with $\\hat{H}$. The center of $\\hat{G}$ is in $\\hat{H}$ and contains all the elements $h_{ij}$ with $i\\neq j$. \n\nA Lie superalgebra $G$ is called a generalized Kac-Moody superalgebra if $G$ is the semidirect product $(\\hat{G}\/C)\\cdot D$ where $C$ is a subspace of the center of $\\hat{G}$ containing the the nonzero $h_{ij}$ and $D$ an Abelian even subalgebra of $G$ such that the elements $e_i$ and $f_i$ are all eigenvectors of $D$. \n\nLet $G$ be a generalized Kac-Moody superalgebra. We keep the above notations for the images in the quotient. \n\nThe root lattice $Q$ of $G$ is the free abelian group generated by the elements $\\alpha_i$,$ \\, i\\in I$, with the bilinear form defined by $(\\alpha_i,\\alpha_j)=a_{ij}$. The elements $\\alpha_i$ are called simple roots. $G$ is graded by the root lattice if we define the degree of $e_i$ as $\\alpha_i$ and the degree of $f_i$ as \n$-\\alpha_i$. We have the usual definitions of roots and root spaces. A root is called real if it has positive norm and imaginary else. \n\n$H=\\hat{H}\\oplus D$ is a commutative even subalgebra of $G$ called Cartan subalgebra. There is a unique invariant supersymmetric bilinear form on $G$ satisfying $(h_i,h_j)=a_{ij}$. We have a natural homomorphism of abelian groups from $Q$ to $H$ sending $\\alpha_i$ to $h_i$. By abuse of terminology the images of roots under this map are also called roots. It is important to note that this map is in general not injective. It is possible that $n>1$ imaginary simple roots have the same image $h$ in $H$. In this case we call $h$ a root of multiplicity $n$. If the matrix $a$ has infinitely many identical columns this multiplicity may even be infinite. The root space of root $\\alpha\\in Q$ is either even or odd. The root space of the corresponding element in $H$ has in general an even and an odd part. \n\nUnder certain conditions\none can also prove a character formula (\\cite{R1},\\cite{GN}).\n\n\\subsection{A characterization theorem}\n\nWhile most generalized Kac-Moody algebras posses an almost positive definite contravariant bilinear form in the super case this does not even hold for the finite dimensional algebras. A more general characterization is the following (cf. \\cite{R2}, Theorem 3.2) \n\\begin{thth1} \\label{ut}\nLet $G$ be a real Lie superalgebra satisfying the following conditions \n\\begin{enumerate}\n\\item\n$G$ has a selfcentralizing even subalgebra $H$, such that $G$ is the direct\nsum of the eigenspaces of $H$ and all the eigenspaces are finite dimensional.\nThe nonzero eigenvalues of $H$ having on $G$ are called roots of $G$.\n\\item\n$G$ has a nonsingular invariant supersymmetric bilinear form.\n\\item\nThere is an element $h\\in H$ such that the centralizer of $h$ in $G$ is $H$ \nand there exist only finitely many roots $\\alpha$ of $G$ with \n$\\left| \\alpha(h) \\right| < M$ for any real number $M$\n(such an element $h$ is called a regular element). \nA root $\\alpha$ is called positive or negative depending on wether its value on $h$ is positive or negative.\n\\item\nAll roots are either of finite type or of infinite type.\n(A root $\\alpha$ is of finite type if for all roots $\\beta$, $\\beta+n\\alpha$ is a root for only finitely many integers $n$. A root $\\alpha$ is of infinite type if, either $\\alpha$ has zero norm and is not of finite type, or for all roots \n$\\beta\\not\\in \\{\\alpha,\\frac{1}{2}\\alpha,2\\alpha\\}$ with $(\\alpha,\\beta)(\\alpha,\\alpha)>0$, $\\beta+n\\alpha$ is a root for all positive integers $n$ unless $\\alpha$ and $\\beta$ are both positive or negative, $\\beta$ is of finite type and norm zero and $\\alpha-\\beta$ is a root.)\n\\item\nLet $\\alpha$ and $\\beta$ be orthogonal positive (resp. negative) roots of\ninfinite type or of norm 0.  If $x\\in G_{\\alpha}$ and $[x, G_{-\\gamma}]=0$ for\nall roots $\\gamma$ for which $0< \\left| \\gamma (h) \\right| <\n\\left| \\alpha(h) \\right|$, then $[x,G_{\\beta}]=0$.\n\\end{enumerate}\nThen $G$ is a direct sum of a generalized Kac-Moody superalgebra, finite\ndimensional simple Lie superalgebras \nand affine Lie superalgebras. \n\\end{thth1}\n\n\\section{The vertex algebra of the compactified superstring}\n\nIn this section we construct a vertex algebra that represents the Fock space of a chiral superstring moving on a 10 dimensional torus and describe its symmetries. \n \n\\subsection{The construction of the vertex algebra}\n\nNow we construct the vertex algebra of the compactified superstring using the lattice construction described in section \\ref{lc}. \n\nThe 10 dimensional superstring has 10 bosonic and 10 fermionic degrees of freedom. We represent the bosonic degrees by a 10 dimensional even lattice and the fermionic degrees in bosonized form by the weightlattice of the Lie algebra $D_5\\,$. The conformal ghosts and superghosts give 3 additional lattice dimensions. One of them carries a negative metric and must be described together with the fermionic lattice (cf. \\cite{LT}). \n\nHence we define\n\\begin{equation} L = L^X \\oplus L^{\\psi,\\phi} \\oplus L^{\\chi,\\sigma} \\ee\nwhere $L^X$ is a 10 dimensional even Lorentzian lattice, \n$L^{\\psi,\\phi}=\\{ (x_1,\\ldots,x_6)\\in {\\mathbb R}^{5,1}\\, | \\, \\mbox{all }\nx_i\\in {\\mathbb Z}\\, \\mbox{ or all } x_i\\in {\\mathbb Z}+\\frac{1}{2}\\}$\nand $L^{\\chi,\\sigma}={\\mathbb Z}^2$\n \nThe lattice $L^{\\psi,\\phi}$ is a 6 dimensional rational lattice. \nThe elements $(x_1,\\ldots,x_6)$ with integral $x_i$ and $\\sum x_i$ even form an even sublattice of $L^{\\psi,\\phi}$. The quotient of $L^{\\psi,\\phi}$ by this lattice is isomorphic to ${\\mathbb Z}_2 \\times {\\mathbb Z}_2$ and will be written $\\Gamma(L^{\\psi,\\phi})=\\{ 0,V,S,C\\}$. We have the coset decomposition $L^{\\psi,\\phi}=L^{\\psi,\\phi}_0 \\cup L^{\\psi,\\phi}_V \\cup L^{\\psi,\\phi}_S \\cup L^{\\psi,\\phi}_C$ with\n\\begin{eqnarray*} \nL^{\\psi,\\phi}_V &=& (0,\\ldots,0,1) + L^{\\psi,\\phi}_0 \\\\\nL^{\\psi,\\phi}_S &=& ({\\textstyle \\frac{1}{2}},\\ldots,\n                     {\\textstyle \\frac{1}{2}},{\\textstyle \\frac{1}{2}})\n                                                  + L^{\\psi,\\phi}_0 \\\\\nL^{\\psi,\\phi}_C &=& ({\\textstyle \\frac{1}{2}},\\ldots,\n                     {\\textstyle \\frac{1}{2}},-{\\textstyle \\frac{1}{2}})\n                                                  + L^{\\psi,\\phi}_0 \n\\end{eqnarray*}\nWe can write the conjugacy classes of $L^{\\psi,\\phi}$ also in terms of the classes of $D_5^*$ and $D_1^*$. The lattice $D_n^*$ has the decomposition \n$0(D_n^*)\\cup V(D_n^*)\\cup  S(D_n^*)\\cup C(D_n^*)$ where \n$0(D_n^*)$ is the even lattice with elements \n$(x_1,\\ldots,x_n)\\in {\\mathbb Z}^n$ \nsuch that $\\sum x_i$ is even, and\n\\begin{eqnarray*}\nV(D_n^*) &=& (0,\\ldots,0,1)+0(D_n^*) \\\\ \nS(D_n^*) &=& ({\\textstyle\\frac{1}{2}},\\ldots,{\\textstyle\\frac{1}{2}},{\\textstyle\\frac{1}{2}})+0(D_n^*) \\\\\nC(D_n^*) &=& ({\\textstyle\\frac{1}{2}},\\ldots,{\\textstyle\\frac{1}{2}},-{\\textstyle\\frac{1}{2}})+0(D_n^*) \\, . \n\\end{eqnarray*}\nThe quotient of $D_n^*$ by $0(D_n^*)$ is ${\\mathbb Z}_2\\times {\\mathbb Z}_2$ for $n$ even and ${\\mathbb Z}_4$ for $n$ odd. \nWith this notation we can write \n\\[ \\renewcommand{\\arraystretch}{1.3} \\begin{array}{c|c}\nL^{\\psi,\\phi} & (D_5^*,D_1^*) \\\\ \\hline \n       0      & (0,0)\\cup (V,V) \\\\\n       V      & (0,V)\\cup (V,0) \\\\\n       S      & (S,S)\\cup (C,C) \\\\\n       C      & (C,S)\\cup (C,S) \n\\end{array} \\]\nAn element of $L^{\\psi,\\phi}$ has even normsquare if it is in $L^{\\psi,\\phi}_0$ and odd normsquare else. \nThe vectors $\\phi^i,\\, i=1,\\ldots,5$ with $1$ in the i-th position and zero else and \n$s^6=({\\textstyle \\frac{1}{2}},\\ldots,{\\textstyle \\frac{1}{2}})$ form a \n$\\mathbb Z$-basis of $L^{\\psi,\\phi}$\nThe sublattice $E_{5,1}=L^{\\psi,\\phi}_0 \\cup L^{\\psi,\\phi}_S$ is an integral unimodular Lorentzian lattice.\n\nThe lattice $L^{\\chi,\\sigma}={\\mathbb Z}^2$ can be decomposed as \n$L^{\\chi,\\sigma}=L^{\\chi,\\sigma}_0 \\cup L^{\\chi,\\sigma}_1$ into the elements of even and odd norm. The quotient $\\Gamma(L^{\\chi,\\sigma})=\\{ 0, 1\\}$ is isomorphic to ${\\mathbb Z}_2$. A $\\mathbb Z$-basis of $L^{\\chi,\\sigma}$ is given by the elements $\\chi=(1,0)$ and $\\sigma=(0,1)$. \n\nNow the sum $L^X \\oplus L^{\\psi,\\phi}_0 \\oplus L^{\\chi,\\sigma}_0$ is an even sublattice of $L$. The quotient $\\Gamma(L)$ of $L$ by this sublattice is $\\Gamma(L^{\\psi,\\phi})\\times \\Gamma(L^{\\chi,\\sigma})$. \n\nTo construct a vertex algebra from $L$ we need a bilinear map $\\eta$ from $\\Gamma(L) \\times \\Gamma(L)$ to ${\\mathbb C}^*$. There is a number of such maps but we need a special one to assure that the modes of the physical bosons and fermions that we will define have the correct commutation and anticommutation properties.\n\nThe following definition of $\\eta$ turns out to be appropriate.\n\n\\[ \\renewcommand{\\arraystretch}{1.3} \\begin{array}{c|cccccccc}\n      & (0,0) & (V,0) & (S,0) & (C,0) & (0,1) & (V,1) & (S,1) & (C,1) \\\\ \\hline\n(0,0) &   1   &   1   &   1   &   1   &   1   &   1   &   1   &   1   \\\\\n(V,0) &   1   &  -1   &   y   &  -y   &   -1  &   1   &   -y  &   y   \\\\\n(S,0) &   1   &  -y   &  -1   &   y   &   -1  &   y   &   1   &   -y  \\\\\n(C,0) &   1   &   y   &  -y   &  -1   &   1   &   y   &   -y  &   -1  \\\\\n(0,1) &   1   &  -1   &  -1   &   1   &   -1  &   1   &   1   &   -1  \\\\\n(V,1) &   1   &   1   &  -y   &  -y   &   1   &   1   &   -y  &   -y  \\\\\n(S,1) &   1   &   y   &   1   &   y   &   1   &   y   &   1   &    y  \\\\  \n(C,1) &   1   &   -y  &   y   &   -1  &   -1  &   y   &   -y  &    1\n\\end{array} \\]\n\n$y$ can be chosen as $+1$ or $-1$. The column denotes the first and the row the second argument of $\\eta$. Note that $\\eta$ is not symmetric, e.g. \n$\\eta ((S,0),(V,0))=y$ and $\\eta ((V,0),(S,0))=-y$. One can check that this map is bilinear. It clearly satisfies condition (\\ref{nc}).\n\nLet $\\{\\alpha^1,\\dots,\\alpha^{10}\\}$ be a basis of $L^X$. Then \n$\\{ \\alpha^1,\\dots,\\alpha^{10},\\phi^1,\\dots,\\phi^5,s^6,\\chi,\\sigma \\}$ is an \nordered basis of $L$. As 2-cocycle we choose the one constructed from this basis as described above with the $a$'s being as follows. Let $i\\leq j$. Then define $a_{ii}=(-1)^{\\frac{1}{2}(\\alpha^i,\\alpha^i)}$ for $i=1,\\ldots,10$ and $a_{ij}=1$\nin the other cases. This definition will give nice formulas.\n\nWe define the vertex algebra $V$ of the compactified superstring as the vertex algebra constructed from the lattice $L$ with $\\eta$ and $\\varepsilon$ given as above. Hence\n\\[\nV =          V_{(0,0)} \\oplus V_{(V,0)} \\oplus V_{(S,0)} \\oplus V_{(C,0)} \n      \\oplus V_{(0,1)} \\oplus V_{(V,1)} \\oplus V_{(S,1)} \\oplus V_{(C,1)}\\, .\n\\]\n$V$ can be decomposed into the Neveu-Schwarz and Ramond sector given by \n\\[ V^{NS}=V_{(0,0)} \\oplus V_{(V,0)} \\oplus V_{(0,1)} \\oplus V_{(V,1)} \\]\nand   \n\\[ V^R=V_{(S,0)} \\oplus V_{(C,0)} \\oplus V_{(S,1)} \\oplus V_{(C,1)} \\, . \\]\n\nThe vector space \n\\begin{equation} V^{GSO} =  \nV_{(0,0)} \\oplus  V_{(S,1)} \\oplus V_{(0,1)} \\oplus V_{(S,0)} \\ee\ndescribes the Fock space of a GSO-projected superstring. $V^{GSO}$ is a vertex superalgebra with even part $V_{(0,0)} \\oplus  V_{(S,1)}$ and odd part  $V_{(0,1)} \\oplus V_{(S,0)}$.\n\n\\subsection{The symmetries of the compactified superstring}\n\nIn this section we show that $V$ carries representations of the $N=1$ and $N=2$ superconformal algebras and use them to construct the BRST operator.  \n\nFirst we will consider the matter sector.\n\nLet $\\{ x^1,\\dots,x^{10} \\}$ be a basis of ${\\mathbb R}\\otimes_{\\mathbb Z} L^X$ with $(x^{\\mu},x^{\\nu})= g^{\\mu \\nu}$ where $g^{\\mu \\nu}$ is the diagonal matrix with entries $g^{1 1}=\\ldots=g^{9 9}=1$ and $g^{10\\, 10}=-1$. The inverse of $g^{\\mu \\nu}$ is denoted by $g_{\\mu \\nu}$. These matrices can be used to raise and lower indices. The bosons $x^{\\mu}(-1)$ satisfy \n$x^{\\mu}(-1)_0x^{\\nu}(-1)=0,\\, x^{\\mu}(-1)_1x^{\\nu}(-1)=g^{\\mu \\nu}$ and $x^{\\mu}(-1)_nx^{\\nu}(-1)=0$ for $n\\geq 2$. By (\\ref{dpe}) this is equivalent to the operator product expansion\n\\begin{equation} x^{\\mu}(-1)(z)\\,x^{\\nu}(-1)= g^{\\mu \\nu} z^{-2} + \\ldots \\ee\nwhere the dots denote nonsingular terms.\n\nNext we define 10 complex fermions \n\\[ \\Psi^{\\pm i} = e^{\\pm \\phi^i} \\quad i=1,\\ldots,5 \\, .  \\]\nSince the complex fermions are elements of $V_{(V,0)}$ their vertex operators have integral expansions when acting on the Neveu-Schwarz sector and half-integral expansions when acting on the Ramond sector. They satisfy\n\\begin{equation} \\Psi^{i}(z)\\Psi^{-j}= \\delta^{ij} z^{-1}+\\ldots \\, . \\ee\nThe 10 real fermions \n\\[ \\renewcommand{\\arraystretch}{1.3} \\begin{array}{lcl c lcl}\n \\psi^1 & = &  \\frac{1}{\\sqrt 2} (\\Psi^1 + \\Psi^{-1}) & \\qquad &\n \\psi^2 & = &  \\frac{i}{\\sqrt 2} (\\Psi^1 - \\Psi^{-1}) \\\\\n        & \\vdots &   & & & \\vdots &                  \\\\ \n \\psi^7 & = &  \\frac{1}{\\sqrt 2} (\\Psi^4 + \\Psi^{-4}) & &\n \\psi^8 & = &  \\frac{i}{\\sqrt 2} (\\Psi^4 - \\Psi^{-4}) \\\\\n\n \\psi^9 & = &  \\frac{1}{\\sqrt 2} (\\Psi^5 + \\Psi^{-5}) & & \n \\psi^{10} & = & \\frac{1}{\\sqrt 2} (\\Psi^5 - \\Psi^{-5})\n\\end{array} \\]\nhave the operator product expansion  \n\\begin{equation}  \\psi^{\\mu}(z)\\psi^{\\nu} = g^{\\mu \\nu} z^{-1} + \\ldots \\, . \\ee\nUsing the commutator formula this implies \n\\begin{equation}  \\{ \\psi^{\\mu}_m,\\psi^{\\nu}_n \\} = g^{\\mu \\nu} \\delta_{m+n+1} \\ee\nwhere $m$ and $n$ are either both integral or both half-integral depending on wether the $\\psi$'s act on the Neveu-Schwarz or Ramond sector.\n\nDefine the supercurrent and the energy momentum tensor of the matter sector by \n\\begin{equation} \\tau^M = g_{\\mu \\nu} x^{\\mu}(-1)_{-1}\\psi^{\\nu} \n           = x_{\\nu}(-1)_{-1}\\psi^{\\nu}              \\ee\nand\n\\begin{equation}\n  \\omega^M = {\\textstyle \\frac{1}{2}} \\tau^M_{\\; 0}\\,\\tau^M \n           = {\\textstyle \\frac{1}{2}} x_{\\nu}(-1)_{-1}x^{\\nu}(-1)\n             +{\\textstyle \\frac{1}{2}}  (D\\psi_{\\nu})_{-1}\\psi^{\\nu} \\, . \n\\ee\nNote that we assume the summation convention in these expressions. \nWe remark that the energy momentum tensor can also be written as\n\\[ \\omega^M={\\textstyle \\frac{1}{2}} x_{\\nu}(-1)_{-1}x^{\\nu}(-1) \n+ {\\textstyle \\frac{1}{2}}\\phi^1(-1)_{-1}\\phi^1(-1) + \\ldots \n+ {\\textstyle \\frac{1}{2}}\\phi^5(-1)_{-1}\\phi^5(-1) \\, .\\]\nIt is easy to calculate the following operator product expansions\n\\begin{eqnarray}\n\\omega^M(z)x^{\\mu}(-1) &=& x^{\\mu}(-1)z^{-2} + Dx^{\\mu}(-1)z^{-1} + \\dots \\\\\n\\omega^M(z)\\psi^{\\mu}  &=& {\\textstyle \\frac{1}{2}}\\psi^{\\mu}z^{-2} \n                           + D\\psi^{\\mu}z^{-1} + \\dots \n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\tau^M(z)x^{\\mu}(-1) &=& \\psi^{\\mu}z^{-2} + D\\psi^{\\mu}z^{-1} + \\dots \\\\\n\\tau^M(z)\\psi^{\\mu}  &=& x^{\\mu}(-1)z^{-1} + \\dots \\, .\n\\end{eqnarray}\n\nTo obtain a representation of the $N=2$ superconformal algebra we define \n\\begin{equation}    j^M = \\phi^1(-1) + \\ldots + \\phi^5(-1) \\, . \\ee\nand decompose the supercurrent \n\\begin{equation}   \\tau^M = \\tau^{M+} + \\tau^{M-} \\ee\nwith\n\\[ \\tau^{M+} = \\sum_{i=1}^5 h^i(-1)_{-1}\\Psi^i \\, ,\\qquad\n   \\tau^{M-} = \\sum_{i=1}^5 h^{-i}(-1)_{-1}\\Psi^{-i} \\]\nwhere $h^1 =  (x^1+ix^2)\/{\\sqrt 2},\\ldots,\n       h^4 =  (x^7+ix^8)\/{\\sqrt 2},\n       h^5 =  (x^9-x^{10})\/{\\sqrt 2}$ \nand $ h^{-1} = (x^1-ix^2)\/{\\sqrt 2},\\ldots,\n      h^{-4} = (x^7-ix^8)\/{\\sqrt 2},\n      h^{-5} = (x^9+x^{10})\/{\\sqrt 2}$. \nThese elements satisfy $(h^i,h^j)=(h^{-i},h^{-j})=0$ and $(h^{i},h^{-j})=\\delta^{ij}$.\n\nUsing the associativity formula one proves the following \n\\begin{p1}\nThe above fields satisfy the operator product expansions\n\\[ \\renewcommand{\\arraystretch}{1.3} \n\\begin{array}{rlccclclcl}\n\\omega^M(z) & \\hspace{-0.3cm}\\omega^M &=& {\\textstyle \\frac{15}{2}} z^{-4} \n                   &+& 2 \\omega^M z^{-2} &+& D\\omega^M z^{-1} &+& \\ldots  \\\\\n\\omega^M(z) & \\hspace{-0.3cm}\\tau^{M\\pm} &=& \n                              {\\textstyle \\frac{3}{2}} \\tau^{M\\pm} z^{-2}\n                               &+& D\\tau^{M\\pm} z^{-1} &+& \\dots & & \\\\\n\\omega^M(z) & \\hspace{-0.3cm}j^M  &=& j^M z^{-2} \n                               &+& Dj^M z^{-1} &+& \\dots & & \\\\\nj^M(z) & \\hspace{-0.3cm}\\tau^{M\\pm} &=& \\pm \\tau^{M\\pm} z^{-1} &+& \\dots & & \n                                                                       & &  \\\\\nj^M(z) & \\hspace{-0.3cm}j^M &=& 5 z^{-2} &+& \\dots & &   & &  \n\\end{array}\\]\n\\[ \\renewcommand{\\arraystretch}{1.3} \n\\begin{array}{rlclclclcl}\n\\tau^{M+}(z) & \\hspace{-0.3cm}\\tau^{M-} &=& 5 z^{-3} &+& j^M z^{-2}  \n                           &+& (\\omega^M+ {\\textstyle \\frac{1}{2}}Dj^M) z^{-1}\n                           &+& \\dots \\\\\n\\tau^{M-}(z) & \\hspace{-0.3cm}\\tau^{M+} &=& 5 z^{-3} &-& j^M z^{-2}  \n                           &+& (\\omega^M - {\\textstyle \\frac{1}{2}}Dj^M) z^{-1}\n                           &+& \\dots \n\\end{array}\\]\nThe expansions $\\tau^{M+}(z)\\tau^{M+}$ and $\\tau^{M-}(z)\\tau^{M-}$ contain only nonsingular terms.\n\\end{p1}\nNow define operators $L^M_{m}=\\omega^M_{m+1}$ and \n$G^{M\\pm}_{m}=\\tau^{M\\pm}_{m+\\frac{1}{2}}$. Then the commutator formula and the above operator product expansions imply \n\\[ \\renewcommand{\\arraystretch}{1.3} \n\\begin{array}{lcl}\n {[} L^M_{m}, L^M_{n} {]} & = & (m-n)L^M_{m+n} + \\frac{1}{12}m(m^2-1)\n                                                  \\delta_{m+n}c^M \\\\\n {[} L^M_{m}, G^{M\\pm}_{n} {]} & = & (\\frac{1}{2}m-n) G^{M\\pm}_{m+n} \\\\\n {[} L^M_{m}, j^M_{n} {]}         & = & -n j^M_{m+n} \\\\\n {[} j^M_m, G^{M\\pm}_{n} {]}   & = & \\pm G^{M\\pm}_{m+n} \\\\\n {[} j^M_m,j^M_n {]}    & = & \\frac{1}{3}m \\delta_{m+n}c^M \\\\\n \\{ G^{M+}_m,G^{M-}_n\\} & = & L^M_{m+n} + \\frac{1}{2}(m-n) j^M_{m+n} \n                        + \\frac{1}{6}(m^2-\\frac{1}{4}) \\delta_{m+n}c^M \\\\\n \\{ G^{M+}_m,G^{M+}_n\\} & = & \\{ G^{M-}_m,G^{M-}_n\\} = 0\n\\end{array} \\]\nwhere the central charge is $c^M=15$. The expansions of the fields $\\tau^{M\\pm}(z)$ and $j^M(z)$ depand on where they act. However the corresponding algebras are all isomorphic to the $N=2$ superconformal algebra. The isomorphism is called spectral flow. \n\nThe expansions for the $\\tau^{M\\pm}$ give \n\\[ \\tau^M(z)\\tau^M = 10 z^{-3} + 2\\omega^{M} z^{-1} + \\ldots \\] \nso that the operators $G^{M}_{m}=\\tau^M_{m+\\frac{1}{2}}=G^{M+}_m+G^{M-}_m$ satisfy \n\\[ \\renewcommand{\\arraystretch}{1.3}\n\\begin{array}{lcl}\n {[} L^M_{m}, L^M_{n} {]} & = & (m-n)L^M_{m+n} + \\frac{1}{12}m(m^2-1)\n                                                  \\delta_{m+n}c^M \\\\\n {[} L^M_{m}, G^M_{n} {]} & = & (\\frac{1}{2}m-n)G^M_{n} \\\\\n  \\{ G^{M}_m,G^{M}_n \\}   & = & 2 L^M_{m+n} \n                       + \\frac{1}{3} (m^2-\\frac{1}{4}) \\delta_{m+n}c^M\n\\end{array} \\]\nwhere $m$ and $n$ are half-integral in the Neveu-Schwarz sector and integral in the Ramond sector. The corresponding 2 superalgebras are called Neveu-Schwarz and Ramond algebra.\n\nNow we show that there is another representation of the $N=2$ superconformal algebra on $V$ coming from the ghost fields.\n\nLet $\\phi=(0,\\ldots,0,1)\\in L^{\\psi,\\phi}$. \nNote that $\\varepsilon (\\phi, \\phi) =-1$ and \n$\\varepsilon (\\phi-\\chi,-\\phi+\\chi)=\\varepsilon (\\sigma, \\sigma)=1$.\nDefine the ghost fields\n\\begin{equation} b=e^{-\\sigma}\\, , \\quad  c=e^{\\sigma} \\ee\nand\n\\begin{equation} \\beta = \\chi(-1)_{-1}e^{-\\phi+\\chi} \\, , \\quad \\gamma = e^{\\phi-\\chi} \\, . \\ee\nThen\n\\[ c(z)b = z^{-1}+\\ldots \\quad \\mbox{and} \\quad \n   \\gamma(z)\\beta = z^{-1}+\\ldots \\: .              \\]\nThe other expansions between these fields contain only non singular terms.\nThis implies that \n\\[ \\{ c_m ,b_n \\} = \\delta_{m+n+1} \\quad \\mbox{and} \\quad\n    [ \\gamma_m,\\beta_n ] = \\delta_{m+n+1} \\]\nare the only nontrivial commutation relations. \n\nNow we define the supercurrent and the energy momentum tensor of the ghost sector by\n\\begin{equation} \\tau^{Gh}=c_{-1}(D\\beta)+{\\textstyle\\frac{3}{2}}(Dc)_{-1}\\beta\n              -2b_{-1}\\gamma \\ee\nand \n\\begin{equation} \\omega^{Gh} = {\\textstyle \\frac{1}{2}} \\tau^{Gh}_{\\; 0}\\,\\tau^{Gh} \n             = 2(Dc)_{-1}b - (Db)_{-1}c  \n               - {\\textstyle \\frac{3}{2}}(D\\gamma)_{-1}\\beta\n               - {\\textstyle \\frac{1}{2}}(D\\beta)_{-1}\\gamma \\, .\\ee\nThe energy momentum tensor can be written as \n\\begin{equation} \n\\omega^{Gh} = \\omega^{\\phi} + \\omega^{\\chi} + \\omega^{\\sigma} \n\\ee\nwhere the elements\n\\begin{eqnarray*}\n\\omega^{\\phi}   &=& -{\\textstyle \\frac{1}{2}}\\phi(-1)_{-1}\\phi(-1)\n                         -\\phi(-2) \\\\  \n\\omega^{\\chi}   &=& {\\textstyle \\frac{1}{2}}\\chi(-1)_{-1}\\chi(-1) \n                         +{\\textstyle \\frac{1}{2}}\\chi(-2)          \\\\\n\\omega^{\\sigma} &=& {\\textstyle \\frac{1}{2}}\\sigma(-1)_{-1}\\sigma(-1)\n                         + {\\textstyle \\frac{3}{2}}\\sigma(-2)\n\\end{eqnarray*} \ngenerate representations of the Virasoro algebra on $V$ of central charges \n$13,-2$ and $-26$.\n\nTo get a representation of the $N=2$ superconformal algebra we define\n\\begin{equation} j^{Gh} = -3\\phi(-1) + 2 \\sigma(-1)\\, . \\ee\nand decompose the supercurrent as\n\\begin{equation} \\tau^{Gh}=\\tau^{Gh+}+\\tau^{Gh-} \\ee\nwith \n\\begin{equation}\n    \\tau^{Gh+}=c_{-1}(D\\beta)+{\\textstyle\\frac{3}{2}}(Dc)_{-1}\\beta \\, , \\quad\n    \\tau^{Gh-}=-2b_{-1}\\gamma \\, .\n\\ee\nAs for the matter fields we get by using the associativity formula \n\\begin{pp1} \nThe above fields satisfy the following operator product expansions\n\\[ \\renewcommand{\\arraystretch}{1.3}\n\\begin{array}{rlccclclcl}\n\\omega^{Gh}(z) & \\hspace{-0.3cm}\\omega^{Gh} \n                        &=& -{\\textstyle \\frac{15}{2}} z^{-4} \n                        &+& 2 \\omega^{Gh} z^{-2} \n                        &+& D\\omega^{Gh} z^{-1} &+& \\ldots  \\\\\n\\omega^{Gh}(z) & \\hspace{-0.3cm}\\tau^{Gh\\pm} \n                        &=& {\\textstyle \\frac{3}{2}} \\tau^{Gh\\pm} z^{-2}\n                        &+& D\\tau^{Gh\\pm} z^{-1} &+& \\dots & & \\\\\n\\omega^{Gh}(z) & \\hspace{-0.3cm}j^{Gh}  \n                        &=& j^M z^{-2} \n                        &+& Dj^{Gh} z^{-1} &+& \\dots & & \\\\\nj^{Gh}(z) & \\hspace{-0.3cm}\\tau^{Gh\\pm} \n                        &=& \\pm \\tau^{Gh\\pm} z^{-1} &+& \\dots & &   & &  \\\\\nj^{Gh}(z) & \\hspace{-0.3cm}j^{Gh} \n                        &=& -5 z^{-2} &+& \\dots & &   & &  \n\\end{array}\\]\n\\[ \\renewcommand{\\arraystretch}{1.3} \n\\begin{array}{rlclclclcl}\n\\tau^{Gh+}(z) & \\hspace{-0.3cm}\\tau^{Gh-} \n                     &=& -5 z^{-3} &+& j^{Gh} z^{-2}  \n                     &+& (\\omega^{Gh}+ {\\textstyle \\frac{1}{2}}Dj^{Gh}) z^{-1}\n                     &+& \\dots \\\\\n\\tau^{{Gh}-}(z) & \\hspace{-0.3cm}\\tau^{Gh+} \n                     &=& -5 z^{-3} &-& j^{Gh} z^{-2}  \n                     &+& (\\omega^{Gh} - {\\textstyle \\frac{1}{2}}Dj^{Gh}) z^{-1}\n                     &+& \\dots \n\\end{array}\\]\nThe expansions $\\tau^{Gh+}(z)\\tau^{Gh+}$ and $\\tau^{Gh-}(z)\\tau^{Gh-}$ contain only nonsingular terms.\n\\end{pp1}\nHence the operators $L^{Gh}_{n}=\\omega^{Gh}_{n+1}, \nG^{Gh\\pm}_{n}=\\tau^{Gh\\pm}_{n+\\frac{1}{2}}$ and $j^{Gh}_n$ give a representation of the $N=2$ superconformal algebra of central charge $c^{Gh}=-15$. \n\nAs before the opertors $L^{Gh}_{n}$ and $G^{Gh}_{n}=G^{Gh+}_{n}+G^{Gh-}_{n}$\ngive a representation of the Neveu-Schwarz resp. Ramond algebra in the Neveu-Schwarz resp. Ramond sector of central charge $c^{Gh}=-15$.\n\nWe give some expansions of the energy momentum tensor and the supercurrent of the ghost sector. They can be used to prove the next proposition.\n\\[ \\renewcommand{\\arraystretch}{1.3} \n\\begin{array}{lcccccl}\n\\omega^{Gh}(z)b &=& 2b z^{-2} &+& Db z^{-1} &+& \\ldots \\\\\n\\omega^{Gh}(z)c &=& -c z^{-2} &+& Dc z^{-1} &+& \\ldots \\\\\n\\omega^{Gh}(z)\\beta &=& \\frac{3}{2}\\beta z^{-2} &+& D\\beta z^{-1} &+&\\ldots \\\\\n\\omega^{Gh}(z)\\gamma &=& -\\frac{1}{2}\\gamma z^{-2} &+& D\\gamma z^{-1} \n                                       &+&\\ldots \\\\\n                     & &   & &   & & \\\\ \n\\tau^{Gh}(z)b &=& -\\frac{3}{2}\\beta z^{-2} &-& \\frac{1}{2}D\\beta z^{-1} \n                                           &+& \\ldots \\\\\n\\tau^{Gh}(z)c      &=& &-& 2\\gamma z^{-1} &+& \\ldots \\\\\n\\tau^{Gh}(z)\\beta  &=& &-& 2b z^{-1}       &+&\\ldots \\\\\n\\tau^{Gh}(z)\\gamma &=& c z^{-2} &-& \\frac{1}{2}Dcz^{-1} &+&\\ldots \n\\end{array}\\]\n\nWe define $\\omega=\\omega^{M}+\\omega^{Gh}$ and $\\tau=\\tau^{M}+\\tau^{Gh}$.\nThen $\\omega$ is a Virasoro element, in particular $D=L_{-1}$. The operators $L_{n}=\\omega_{n+1}$ and $G_{n}=\\tau_{n+\\frac{1}{2}}$ give a representation of the Neveu-Schwarz and the Ramond algebra of central charge zero.\n\nLet \n\\begin{equation} j^{BRST}= c_{-1}(\\omega^M+{\\textstyle \\frac{1}{2}}\\omega^{Gh})\n              +\\gamma_{-1}(\\tau^M+{\\textstyle \\frac{1}{2}}\\tau^{Gh})\\, . \\ee\nThe zero mode of $j^{BRST}$ is called BRST operator and denoted by $Q$.\nUsing the associativity formula and the above expansions we get\n\n\\pagebreak[3]\n\\begin{pp1}\nThe BRST operator transforms the above fields as\n\\begin{eqnarray} \nQb      &=& \\omega \\\\\nQc      &=& -c_{-2}c-\\gamma_{-1}\\gamma \\\\\nQ\\beta  &=& \\tau \\\\\nQ\\gamma &=& \\gamma_{-2}c - {\\textstyle \\frac{1}{2}}c_{-2}\\gamma \\\\\nQ\\omega &=& 0 \\\\\nQ\\tau   &=& 0 \\, .\n\\end{eqnarray}\n\\end{pp1}\nThese formulas imply \n\\begin{eqnarray} \n\\{Q,b_n\\} &=& \\omega_n \\\\\n\\{Q,c_n\\} &=& -(c_{-2}c+\\gamma_{-1}\\gamma)_n \\label{qc} \\\\\n{[}Q,\\beta_n{]}  &=& \\tau_n \\\\\n{[}Q,\\gamma_n{]} &=& (\\gamma_{-2}c - {\\textstyle \\frac{1}{2}}c_{-2}\\gamma)_n \n                                     \\label{qg}\\\\\n{[}Q,\\omega_n{]}   &=& 0 \\\\\n\\{ Q,\\tau_n \\} &=& 0 \\, .\n\\end{eqnarray}\n\nEquations (\\ref{qc}) and (\\ref{qg}) together with the explicit expressions for $Q\\omega^{Gh}$ and $Q\\tau^{Gh}$ show that $Qj^{BRST} = Dv$ with\n$  v= \\frac{1}{4}c_{-1}\\gamma_{-1}\\tau^M \n      + \\frac{1}{4}\\gamma_{-1}\\gamma_{-1}\\gamma_{-1}\\beta$ \n      $- \\frac{1}{2}\\gamma_{-1}\\gamma_{-1}c_{-1}b\n      + \\frac{3}{2}c_{-3}c + \\frac{3}{2}\\gamma_{-2}\\gamma$ .\nHence \\begin{equation} Q^2=0 \\, . \\ee\n\nWe derive a decomposition of $Q$ that will be useful for calculations in the next section. Write $\\omega^{Gh}=\\omega^{b,c} + \\omega^{\\beta,\\gamma}$  where \n\\[ \\renewcommand{\\arraystretch}{1.3}\n\\begin{array}{c}\n\\omega^{b,c}=2(Dc)_{-1}b - (Db)_{-1}c=\\omega^{\\sigma} \\\\\n\\omega^{\\beta,\\gamma}\n =-\\frac{3}{2}(D\\gamma)_{-1}\\beta - \\frac{1}{2}(D\\beta)_{-1}\\gamma \n =\\omega^{\\phi}+\\omega^{\\chi} \\, .\n\\end{array} \\]\nThen\n$j^{BRST}=c_{-1}(\\omega^M+\\omega^{\\beta,\\gamma}\n                                      +\\frac{1}{2}\\omega^{b,c})\n          + \\gamma_{-1}\\tau^M - \\gamma_{-1}\\gamma_{-1}b \n          + \\frac{3}{4}D(\\beta_{-1}\\gamma_{-1}c)$ \nimplies that \n\\begin{equation} Q = Q_0 + Q_1 +Q_2 \\ee\nwith\n\\begin{eqnarray*}\nQ_0 &=& \\left(c_{-1}(\\omega^M+\\omega^{\\beta,\\gamma}\n                     +{\\textstyle \\frac{1}{2}}\\omega^{b,c})\\right)_0 \\\\\nQ_1 &=& \\left(\\gamma_{-1}\\tau^M\\right)_0 \\\\\nQ_2 &=& -\\left(\\gamma_{-1}\\gamma_{-1}b\\right)_0 \\, .\n\\end{eqnarray*} \nThe following lemma is easy to prove.\n\\begin{lp1} \\label{ecl}\nLet $v$ be a state satisfying\n\\begin{eqnarray*}\n (L^M_0+L^{\\beta,\\gamma}_0)v &=& v \\\\\n (L^M_n+L^{\\beta,\\gamma}_n)v &=& 0 \\quad\\mbox{for}\\quad n\\geq 1 \\\\\n  L^{b,c}_n v &=& 0  \\quad\\mbox{for}\\quad n\\geq -1 \\, .\n\\end{eqnarray*}\nThen \\[ Q_0v =D(c_{-1}v) \\, .\\]\n\\end{lp1}\n\nFinally we construct operators that induce gradings on $V$. Let $j=a\\phi(-1)+b\\chi(-1)+c\\sigma(-1)$. The zero mode \n$j_0=a\\phi(0)+b\\chi(0)+c\\sigma(0)$ of $j$ measures the ghost charge of a state.\nThe BRST element $j^{BRST}$ is an eigenvector of $j_0$ iff $a+b+c=0$.\nWe define \n\\[ \\renewcommand{\\arraystretch}{1.3}\n\\begin{array}{lcl}\n j^{N} &=& -\\chi(-1)+\\sigma(-1) \\\\\n j^{P} &=& -\\phi(-1)  +\\chi(-1) \\, .\n\\end{array} \\]\nThe corresponding zero modes are called ghost number and ghost picture \noperator. \nA state has integral ghost picture if it is in the Neveu-Schwarz sector and half-integral ghost picture if it is in the Ramond sector. \nThe elements of $V_{(.., 0)}$ have even and the elements of $V_{(.., 1)}$ odd ghost number.\n\nWe list the $L_0$-eigenvalue and the ghost number and ghost picture of some \nstates.\n\\[ \\renewcommand{\\arraystretch}{1.3}\n\\begin{array}{l|ccc}\n            & L_0                & j^{N}_0 & j^{P}_0 \\\\ \\hline\ne^{n\\phi}   & -\\frac{1}{2}n(n+2) & 0       & n       \\\\\ne^{n\\chi}   & \\frac{1}{2}n(n-1)  & -n      & n       \\\\\ne^{n\\sigma} & \\frac{1}{2}n(n-3)  & n       & 0       \\\\\nj^{BRST}    &  1                 & 1       & 0\n\\end{array} \\]\nThe last row implies \n\\[ [j^{N}_0,Q]=Q \\quad\\mbox{ and }\\quad [j^{P}_0,Q]=0 \\, . \\]\nHence the BRST operator increases the ghost number by one and leaves the ghost picture unchanged.\n\n\\section{The Lie algebra of physical states}\n\nIn this section we construct the Lie superalgebra of physical states of the 10 dimensional superstring moving on a torus and study its properties.\nSection \\ref{mrs} contains the main results of this paper. \n\n\\subsection{The small algebra}\n\nHere we define a subalgebra of the vertex algebra of the compactified superstring which is called small algebra. This is necessary to define the picture changing operator which is essential for the construction of the Lie superalgebra of physical states. We also construct a bilinear form on the small algebra.\n\nDefine the elements $\\xi=e^{\\chi}$ and $\\eta=e^{-\\chi}$. We have \n\\[ \\renewcommand{\\arraystretch}{1.3}\n\\begin{array}{lcrll}\n\\xi(z)\\eta   &=& z^{-1}  & +& \\ldots \\\\\nD \\xi(z)\\eta &=& -z^{-2} & +& \\ldots \\, .\n\\end{array} \\]\nThe expansions $D\\xi(z)D\\xi$ and $\\eta(z)\\eta$ have no singular terms so that\n\\[ \\renewcommand{\\arraystretch}{1.3}\n\\begin{array}{lcl}\n\\{ (D\\xi)_m,\\eta_n \\}   &=& -m\\delta_{m+n} \\\\\n\\{ (D\\xi)_m,(D\\xi)_n \\} &=& 0 \\\\\n\\{ \\eta_m, \\eta_n \\}    &=& 0 \\, .\n\\end{array}\\]\n\nWe define the small algebra $V_S$ as the subalgebra of $V$ generated \nby the states $e^{\\gamma},D\\xi,\\eta,b,c$ with \n$\\gamma\\in L^X \\oplus L^{\\psi,\\phi}$. Note that $\\beta$ and $\\gamma$ are in the small algebra.\n\nLet\n\\[ X= Q\\xi = (Q_0 + Q_1 + Q_2)\\xi \n           = c_{-1}(D\\xi) + \\tau^M_{-1}e^{\\phi} + D(\\eta_{-1}e^{2\\phi}_{-1}b)\n                             + (D\\eta)_{-1}e^{2\\phi}_{-1}b \\, .\n\\]\nThis is an element in $V_{(0,0)}$. The mode $X_{-1}=(Q\\xi)_{-1}=\\{Q,\\xi_{-1}\\}$ is an endomorphism of the small algebra called picture changing operator.\n\n\\begin{p1}\nThe picture changing operator has the following properties\n\\[ \\renewcommand{\\arraystretch}{1.3}\n\\begin{array}{lcl}\n{[} X_{-1}, Q  \\, ] &=& 0 \\\\\n{[} X_{-1}, L_0 \\,] &=& 0 \\\\\n{[} X_{-1}, b_n \\,] &=& -(D\\xi)_{n-1} \\, .\n\\end{array}  \\]\nIn particular $[ X_{-1}, b_1 ] =0$.\n\\end{p1}\n{\\em Proof:} The first relation is clear. The second one follows from $L_0\\xi=0$. $X_k=Q\\xi_k+\\xi_kQ$ and $\\xi_kb=0$ for $k\\geq 0$ imply that for $k\\geq 0$\n\\begin{eqnarray*}\nX_kb &=& \\xi_kQb \\\\\n     &=& \\xi_k\\omega \\\\\n     &=& -[ L_{-2},\\xi_k]1 \\\\\n     &=& -\\sum_{m\\geq 0}{-1 \\choose m} (L_{m-1}\\xi)_{-1+k-m} 1 \\\\\n     &=& -(D\\xi)_{k-1}1 \n\\end{eqnarray*}\nsince $L_{m-1}\\xi=0$ for $m\\geq 1$. By the commutator formula we have\n\\[ [X_{-1},b_n\\,]=\\sum_{k\\geq 0}{-1 \\choose k} (X_kb)_{-1+n-k} =(X_0b)_{n-1}\n               =-(D\\xi)_{n-1} \\, . \\]\n\\hspace*{\\fill} $\\Box$\\\\\n\n$X$ has ghost number $0$ and ghost picture $1$. This implies\n\\[ [j^{N}_0,X_{-1}]=0 \\quad\\mbox{ and }\\quad [j^{P}_0,X_{-1}]=X_{-1} \\, . \\]\n\nIt is easy to see that the eigenvalues of $L_0$ on $V^{GSO}$ are all integral. Moreover $L_1$ acts locally nilpotent on $V^{GSO}$. The same holds for the subalgebra $V^{GSO}_S=V_S\\cap V^{GSO}$ so that we can apply the results of section \\ref{ibf} to construct a bilinear form on this subalgebra.\nSince $e^{3\\sigma-2\\phi}$ is not in the image of $L_1$ we can define an invariant bilinear form $(\\, ,\\, )$ on $V^{GSO}_S$ by setting \n$(e^{3\\sigma-2\\phi},1)=1$. It is easy to prove\n\\begin{pp1}\n$(\\, ,\\, )$ is nondegenerate on $V^{GSO}_S$. \n$(e^{\\gamma+n\\sigma},e^{\\gamma'+n'\\sigma})$ with \n$\\gamma,\\gamma'\\in L^X \\oplus L^{\\psi,\\phi}$ is nonzero if $\\gamma+\\gamma'=-2\\phi$ and $n+n'=3$\nand zero else.\n\\end{pp1}\n\nWe calculate the adjoints of some operators. $b$ and $c$ have $L_0$-eigenvalue $2$ and $-1$ and are annihilated by $L_1$. Hence \n\\begin{equation} b_n^* = b_{2-n} \\quad\\mbox{ and }\\quad c_n^*=-c_{-4-n}\\, . \\ee\nFrom $L_1j^{BRST}=\\frac{3}{2}Dc,\\, \nL_1^2j^{BRST}=-3c$ and $L_1^nj^{BRST}$ $=0$ for $n\\geq 2$ we get\n\\begin{equation} Q^*=-Q \\,. \\ee\nFinally $L_0X=L_1X=0$ gives \n\\begin{equation} X_{-1}^* = X_{-1} \\, . \\ee \n\n\\subsection{The cohomology spaces}\n\nWe now construct the cohomology spaces that represent the physical states of the compactified superstring and determine their dimensions.\n\nThe space \n\\[ B = V^{GSO}_S \\cap \\mbox{Ker } b_1 \\]\nis graded by the lattice $L^X$. We denote the homogeneous subspaces by \n$B(\\alpha)$. The operators $L_0,\\, j_0^N$ and $j_0^P$ are simultaneously diagonizable on $B(\\alpha)$ so that we can write \n\\[ B(\\alpha)=\\bigoplus B(\\alpha)_{p,n}^k \\]\nwhere $k,n$ and $p$ denote the eigenvalues of $L_0,\\, j_0^N$ and $j_0^P$.\nWe define spaces $C(\\alpha)_{p,n}=B(\\alpha)_{p,n}^0,\\, C(\\alpha)=\\bigoplus C(\\alpha)_{p,n}$ and $C=\\bigoplus C(\\alpha)$.\n\nThe following proposition can be proved by analyzing the $L_0$-contributions of the ghost sector. \n\\begin{pp1}\nLet $\\alpha$ be in $L^X$. Then\n\\begin{enumerate}\n\\item[]\nThe space $B(\\alpha)_{p,n}^k$ is finite dimensional.\n\\item[]\nThe space $B(\\alpha)_{-1}^k=\\bigoplus_{n\\in \\mathbb Z} B(\\alpha)_{-1,n}^k$ is\nfinite dimensional.\n\\item[]\nThe space $B(\\alpha)_{-\\frac{1}{2}}^k\n=\\bigoplus_{n\\in \\mathbb Z} B(\\alpha)_{-\\frac{1}{2},n}^k$ is in general not \nfinite dimensional.\nFor example the state\n\\[ e^{\\lambda+(n-\\frac{1}{2})\\phi}_{-1}\\eta_{-n}\\ldots\\eta_{-1}c\\, , \\]\nwhere $\\lambda=(\\pm\\frac{1}{2},\\ldots,\\pm\\frac{1}{2})$ is in $C(D_5^*)$ if $n$ is even and in $S(D_5^*)$ if $n$ is \nodd, is in the kernel of $b_1$, has $L_0$-eigenvalue zero, ghost picture $-\\frac{1}{2}$ and ghost number $n$.\n\\item[]\nThe spaces $C(\\alpha)_{-1,1}$ and $C(\\alpha)_{-\\frac{1}{2},1}$ are zero for \n$(\\alpha,\\alpha)>0$.\n\\end{enumerate}\nNow let $\\alpha$ be of norm zero. Then\n\\begin{enumerate}\n\\item[]\n$C(\\alpha)_{-1,1}$ is 10 dimensional and spanned by the states \n$\\psi^{\\mu}_{-1}e^{-\\phi}_{-1}c_{-1}e^{\\alpha}$. \n\\item[]\n$C(\\alpha)_{-1,0}$ is spanned by $e^{-2\\phi}_{-1}(D\\xi)_{-1}c_{-1}e^{\\alpha}$.\n\\item[]\nThe 16 states \n$e^{\\lambda_{\\dot{\\beta}}-\\frac{1}{2}\\phi}_{-1}c_{-1}e^{\\alpha}$ \nwith \n$\\lambda_{\\dot{\\beta}}=(\\pm\\frac{1}{2},\\ldots,\\pm\\frac{1}{2})\\in C(D_5^*)$ \nform a basis of $C(\\alpha)_{-\\frac{1}{2},1}$.\n\\item[]\nThe space $C(\\alpha)_{-\\frac{1}{2},0}$ is zero.\n\\item[]\nThe 16 states \n$e^{\\lambda_{\\beta}-\\frac{3}{2}\\phi}_{-1}c_{-1}e^{\\alpha}$ \nwith $\\lambda_{\\beta}=(\\pm\\frac{1}{2},\\ldots,\\pm\\frac{1}{2})\\in S(D_5^*)$ \nare a basis of $C(\\alpha)_{-\\frac{3}{2},1}$.\n\\item[]\n$C(\\alpha)_{-\\frac{3}{2},0}$ is spanned by the 16 states\n$e^{\\lambda_{\\dot{\\beta}}-\\frac{5}{2}\\phi}_{-1}(D\\xi)_{-1}c_{-1}e^{\\alpha}$\nwhere \n$\\lambda_{\\dot{\\beta}}=(\\pm\\frac{1}{2},\\ldots,\\pm\\frac{1}{2})$ is in \n$C(D_5^*)$.\n\\end{enumerate}\n\\end{pp1}\n\nSince $b_1^*=b_1$ and $\\{b_1,c_{-2}\\}=1$ the bilinear form on $C$ induced from $(\\, ,\\, )$ is zero. We can construct a nontrivial bilinear form on \n$C$ by defining \n\\[ (u,v)_C=(c_{-2}u,v)\\, . \\]\nThen\n\\begin{pp1} \\label{bf1}\n$(\\, ,\\, )_C$ pairs nondegenerately $C(\\alpha)_{p,m}$ and \n$C(\\beta)_{q,n}$ where $\\alpha+\\beta=0,\\, p+q=-2$ and $m+n=2$. In particular\n$(\\, ,\\, )_C$ is nondegenerate on $C$.\n\nFurthermore $(Qu,v)_C$ is up to a sign equal to $(u,Qv)_C$.\n\\end{pp1}\n{\\em Proof:} $\\{b_1,c_{-2}\\}=1$ implies that $c_{-2}$ is an injective on the kernel of $b_1$. If $u\\in C(\\alpha)_{p,m}$ is nonzero then there is an element $v$ in $V^{GSO}_S$ such that $(c_{-2}u,v)$ is nonzero. Since $[L_0,c_{-2}]=0$ this element has $L_0$-eigenvalue zero. Clearly it has ghost number $2-m$, ghost picture $-2-q$ and momentum $-\\alpha$. The commutator $\\{b_1,c_{-2}\\}=1$ shows that $v$ is in the kernel of $b_1$. \n\nFor $u,v\\in C$ \n\\begin{eqnarray*}\n(Qu,v)_C &=& (c_{-2}Qu,v) \\\\\n                &=& (c_{-2}Qb_1c_{-2}u,v) \\\\\n                &=& -(c_{-2}b_1Qc_{-2}u,v) \\\\\n                &=& -(Qc_{-2}u,v) \\\\\n                &\\sim & (c_{-2}u,Qv) \\\\\n                &=& (u,Qv)_C\n\\end{eqnarray*}\n\\hspace*{\\fill} $\\Box$\\\\\nSince $Q^2=0$ the sequence \n\\[ \\ldots\\stackrel{Q}{\\longrightarrow} C(\\alpha)_{p,n-1} \n         \\stackrel{Q}{\\longrightarrow} C(\\alpha)_{p,n}\n         \\stackrel{Q}{\\longrightarrow} C(\\alpha)_{p,n+1}\n         \\stackrel{Q}{\\longrightarrow} \\ldots \\]\ndefines a complex. We denote the cohomology groups by $H(\\alpha)_{p,n}$. Define  \nthe spaces $H(\\alpha)=\\bigoplus H(\\alpha)_{p,n}$ and $H=\\bigoplus H(\\alpha)$. \nWe investigate the structure of the space $H$.\n\nProposition \\ref{bf1} and the fact that the spaces $C(\\alpha)_{p,n}$ are \nfinite dimensional give \n\\begin{pp1}\n$(\\, ,\\, )_C$ induces a nondegenerate bilinear form \n$(\\, ,\\, )_H$ on $H$.\n\\end{pp1}\n\nAs in \\cite{LZ1} one can prove\n\\begin{thp1}\nLet $\\alpha\\neq 0$. Then \n\\[ H(\\alpha)_{-1,n}=0 \\quad \\mbox{for } n\\neq 1\\,. \\]\n\\end{thp1}\n\nWe use this result and the Euler-Poincar\\'{e} principle to calculate the dimension of $H(\\alpha)_{-1,1}$ for $\\alpha\\neq 0$.\nDefine numbers $c(n)$ by\n\\begin{eqnarray}\n \\sum c(n)q^n &=& 8 \\prod_{m=1}^{\\infty}\\left(\\frac{1+q^m}{1-q^m}\\right)^8 \\\\\n              &=& 8+128q+1152q^2+7680q^3+42112q^4+\\ldots \\, .\n\\end{eqnarray}\nThe asymptotic behaviour of the $c(n)$ is given by \n\\begin{equation} c(n)\\sim {\\textstyle \\frac{1}{2}}\n                         n^{-\\frac{11}{4}}e^{2\\pi\\sqrt{2n}} \\, . \\ee\nFor $n\\geq 10$ the right hand side gives a good approximation for $c(n)$.\n\nWe have\n\\begin{thp1}\nLet $\\alpha\\neq 0$. Then \n\\[ \\mbox{dim}\\, H(\\alpha)_{-1,1}=c({\\textstyle -\\frac{1}{2}}\\alpha^2)\\,. \\]\n\\end{thp1}\n{\\em Proof:} Let $\\alpha\\neq 0$. \nSince the space $C_{-1}(\\alpha)=\\bigoplus_{n\\in \\mathbb Z} C_{-1,n}(\\alpha)$ is finite dimensional we can apply the Euler-Poincar\\'{e} principle to get\n\\[ - \\mbox{dim}\\, H(\\alpha)_{-1,1}=  \\sum(-1)^n \\mbox{dim}\\,C(\\alpha)_{-1,n}\\, . \\]\nThe right hand side is the constant term in\n\\[ \\mbox{tr}\\, (-1)^{j_0^N}q^{L_0} \\]\nwhere the trace has to be evaluated on \n$B(\\alpha)_{-1}=\\bigoplus_{k,n \\in \\mathbb Z}B(\\alpha)_{-1,n}^k$.\nSince the spaces $B(\\alpha)_{-1}^k$ are finite dimensional $\\mbox{tr}\\, (-1)^{j_0^N}q^{L_0}$ is well defined as a formal Laurent series in $q$.\nBy looking at a basis of $B(\\alpha)_{-1}$ we find \n\\begin{enumerate}\n\\item[] \nThe 10 bosonic coordinates $x^{\\mu}$ give a contribution\n\\[    q^{\\frac{1}{2}\\alpha^2} \\varphi(q)^{-10} \\]\nto the trace where $\\varphi(q) = \\prod_{n=1}^{\\infty} (1-q^n)$.\n\\item[] \nThe $(\\psi,\\phi)$-$(D\\xi,\\eta)$ system gives a contribution \n\\[ \\varphi(q)^{-7}\\sum (-1)^m q^{\\frac{1}{2}\\lambda^2}\n      q^{-\\frac{1}{2}p^2+\\frac{1}{2}m(m+1)+\\frac{1}{2}} \\]\nwhere the sum extends over $(\\lambda,-p-1)\\in L_0^{(\\psi,\\phi)}$ and \n$m\\in \\mathbb Z$ with $m\\geq |p|$. \nNote that the sum is a well defined power series in $q$.\n\\item[]\nThe $(b,c)$ system contributes a factor\n\\[ -q^{-1}\\varphi(q)^2 \\]\nto the trace.\n\\end{enumerate}\nHence the trace is given by\n\\[ -q^{\\frac{1}{2}\\alpha^2-1}\\varphi(q)^{-15}\n      \\sum (-1)^m q^{\\frac{1}{2}\\lambda^2}\n           q^{-\\frac{1}{2}p^2+\\frac{1}{2}m(m+1)+\\frac{1}{2}}\\, \\]\nA technical calculation \nshows that this is equal to\n\\[ - 8 q^{\\frac{1}{2}\\alpha^2} \\frac{\\varphi(q^2)^{8}}{\\varphi(q)^{16}}\\,. \\]\nThe constant term in this expression is $-c(-\\frac{1}{2}\\alpha)$.\nThis proves the theorem. \\hspace*{\\fill} $\\Box$\\\\\n\nWe will see that for $\\alpha\\neq 0$ the spaces $H(\\alpha)_{-1,1}$ and \n$H(\\alpha)_{-\\frac{1}{2},1}$ are related by a supersymmetry transformation so that they have the same dimension. \n\nThe picture changing operator commutes with $Q$ and therefore defines a map on $H$. The following proposition (cf. \\cite{BZ}) describes an important property of this map.\n\\begin{pp1} \\label{piso}\nFor $\\alpha\\neq 0$ the map $X_{-1}: H(\\alpha)_{p,n}\\rightarrow H(\\alpha)_{p+1,n}$ is an isomorphism.\n\\end{pp1}\n{\\em Proof:} $\\alpha=\\alpha_{\\nu}x^{\\nu}$ has a nonzero component $\\alpha_{\\mu}$. \nThe element $\\tilde{P}^{\\mu}=\\psi^{\\mu}_{-1}e^{-\\phi}$ has ghost number $0$ and ghost picture $-1$. The zero mode $\\tilde{P}^{\\mu}_0$ is the momentum operator in the $-1$ picture. It satisfies \n$[Q,\\tilde{P}^{\\mu}_0]=(Q\\tilde{P}^{\\mu})_0=(Q_0\\tilde{P}^{\\mu})_0\n=(D(c_{-1}\\tilde{P}^{\\mu}))_0=0$. \nHence $\\tilde{P}^{\\mu}_0$ defines a map from $H(\\alpha)_{p,n}$ to $H(\\alpha)_{p-1,n}$. One can show that $\\tilde{P}^{\\mu}_0$ is a scalar multiple of the inverse of $X_{-1}$ \\linebreak[3] (cf. \\cite{BZ}). \\hspace*{\\fill} $\\Box$\\\\\n \nHence for nonzero momentum the spaces $H(\\alpha)_{p,1}$ with integral or half-integral ghost picture $p$ are all isomorphic. They represent the bosonic resp. fermionic physical states of momentum $\\alpha$. The ghost pictures $-1$ and \n$-\\frac{1}{2}$ are called canonical ghost pictures because they give the easiest description of the physical states. The fermionic physical states also admit a simple description in the $-\\frac{3}{2}$ picture since the contribution of the $\\phi$-charge to the $L_0$-value in this picture is the same as in the $-\\frac{1}{2}$ picture. \n\nBefore we can determine the cohomology spaces in the canonical ghost pictures for $\\alpha^2=0$ explicitly we have to introduce some more notations.\n\nLet \n\\[ S^{\\dot{\\alpha}}=e^{\\lambda_{\\dot{\\alpha}}-\\frac{1}{2}\\phi} \\quad\\mbox{and}\\quad \n   S^{\\alpha}=e^{\\lambda_{\\alpha}-\\frac{3}{2}\\phi}                              \\]\nwhere \n$\\lambda_{\\dot{\\alpha}}=(\\pm\\frac{1}{2},\\ldots,\\pm\\frac{1}{2})$ is a conjugate spinor of $D_5^*$, i.e. has an odd number of $-$ signs, and\n$\\lambda_{\\alpha}=(\\pm\\frac{1}{2},\\ldots,\\pm\\frac{1}{2})$ is a spinor of $D_5^*$ and has an even number of $-$ signs.\n\nWe define $\\Gamma$-matrices $\\Gamma^{\\mu}$ by\n\\begin{eqnarray}\n(\\psi^{\\mu}_{-1}e^{-\\phi})(z)S^{\\dot{\\alpha}} &=& \\frac{1}{\\sqrt{2}}\n{\\Gamma^{\\mu\\, \\dot{\\alpha}}}_{\\beta}S^{\\beta}z^{-1} + \\ldots \\\\\n(\\psi^{\\mu}_{-1}e^{\\phi})(z)S^{\\alpha} &=& \\frac{1}{\\sqrt{2}}\n{\\Gamma^{\\mu\\, \\alpha}}_{\\dot{\\beta}}S^{\\dot{\\beta}}z^{1} + \\ldots \\, .\n\\end{eqnarray}\nThe upper index denotes the row and the lower index the column.\n\\begin{pp1}\nWe have\n\\begin{eqnarray}\n(\\psi^{\\mu}_{-1}e^{\\phi})_{-2}(\\psi^{\\nu}_{-1}e^{-\\phi})_{0}S^{\\dot{\\alpha}} &=& \n\\psi^{\\mu}_{-\\frac{1}{2}}\\psi^{\\nu}_{-\\frac{1}{2}}S^{\\dot{\\alpha}} \\\\\n(\\psi^{\\mu}_{-1}e^{-\\phi})_{0}(\\psi^{\\nu}_{-1}e^{\\phi})_{-2}S^{\\alpha} &=& \n\\psi^{\\mu}_{-\\frac{1}{2}}\\psi^{\\nu}_{-\\frac{1}{2}}S^{\\alpha}\\, .\n\\end{eqnarray}\n\\end{pp1}\n{\\em Proof:}\n\\begin{eqnarray*}\n(\\psi^{\\mu}_{-1}e^{\\phi})_{-2}(\\psi^{\\nu}_{-1}e^{-\\phi})_0S^{\\dot{\\alpha}} &=&\n(\\psi^{\\mu}_{-1}e^{\\phi})_{-2} \n\\psi^{\\nu}_{-\\frac{1}{2}}e^{-\\phi}_{-\\frac{1}{2}}S^{\\dot{\\alpha}} \\\\\n &=& (\\psi^{\\mu}_{-\\frac{3}{2}}e^{\\phi}_{-\\frac{3}{2}}\n       -e^{\\phi}_{-\\frac{5}{2}}\\psi^{\\mu}_{-\\frac{1}{2}})\n      \\psi^{\\nu}_{-\\frac{1}{2}}e^{-\\phi}_{-\\frac{1}{2}}S^{\\dot{\\alpha}} \\\\\n &=& -e^{\\phi}_{-\\frac{5}{2}}\\psi^{\\mu}_{-\\frac{1}{2}}\n        \\psi^{\\nu}_{-\\frac{1}{2}}e^{-\\phi}_{-\\frac{1}{2}}S^{\\dot{\\alpha}}\\\\\n &=& -\\psi^{\\mu}_{-\\frac{1}{2}}\\psi^{\\nu}_{-\\frac{1}{2}}\n     e^{\\phi}_{-\\frac{5}{2}}e^{-\\phi}_{-\\frac{1}{2}}S^{\\dot{\\alpha}}\\\\\n &=& \\psi^{\\mu}_{-\\frac{1}{2}}\\psi^{\\nu}_{-\\frac{1}{2}} S^{\\dot{\\alpha}} \n\\end{eqnarray*}\nWe have used (\\ref{bisc}) repeatedly.\nNote that\n$e^{\\phi}_{-\\frac{5}{2}}e^{-\\phi}_{-\\frac{1}{2}}S^{\\dot{\\alpha}}=-S^{\\dot{\\alpha}}$\nsince \n\\[ \\varepsilon(-\\phi,\\lambda_{\\dot{\\alpha}}-{\\textstyle \\frac{1}{2}}\\phi)\n       \\varepsilon(\\phi,\\lambda_{\\dot{\\alpha}}-{\\textstyle \\frac{3}{2}}\\phi)\n       =\\varepsilon(\\phi,\\lambda_{\\dot{\\alpha}}-{\\textstyle \\frac{1}{2}}\\phi)^{-1}\n        \\varepsilon(\\phi,\\lambda_{\\dot{\\alpha}}-{\\textstyle \\frac{1}{2}}\\phi)\n        \\varepsilon(\\phi,\\phi)^{-1}=-1\\,. \\]\nThe proof of the other equation is analogous.  \\hspace*{\\fill} $\\Box$\\\\\n\nThe commutator $ \\{ \\psi^{\\mu}_{-\\frac{1}{2}},\\psi^{\\nu}_{-\\frac{1}{2}} \\}\n                 = g^{\\mu \\nu}$ implies \n\\begin{equation} \\{ \\Gamma^{\\mu},\\Gamma^{\\nu}\\}=2g^{\\mu \\nu} \\ee\nso that the $\\Gamma$-matrices give a Weyl representation of the Clifford algebra $C(9,1)$. The matrix $\\Gamma^{11}=\\Gamma^{1}\\ldots \\Gamma^{10}$ anticommutes with the $\\Gamma^{\\mu}$ and satisfies $(\\Gamma^{11})^2=1$. More precisely $\\Gamma^{11}$ is diagonal and with diagonal entries $\\pm(1,\\ldots,1,-1,\\ldots,-1)$.\nThe operators $\\frac{1}{2}(1\\pm \\Gamma^{11})$ are projection operators on the spinors and conjugate spinors.\n\nWe define a charge conjugation matrix $C$ by\n\\begin{eqnarray}\nS^{\\alpha}(z)S^{\\dot{\\beta}} &=& C^{\\alpha \\dot{\\beta}}e^{-2\\phi}z^{-2}+\\ldots \\\\\nS^{\\dot{\\alpha}}(z)S^{\\beta} &=& C^{\\dot{\\alpha} \\beta}e^{-2\\phi}z^{-2}+\\ldots  \\, .\n\\end{eqnarray}\nThe definition implies that $C$ is antisymmetric and invertible. We have\n\\begin{pp1} \\label{susyv}\nThe spinor fields have the operator product expansions\n\\begin{eqnarray*}\nS^{\\dot{\\alpha}}(z)S^{\\dot{\\beta}} &=& \\frac{1}{\\sqrt{2}}\n                                  (\\Gamma_{\\mu}C)^{\\dot{\\alpha} \\dot{\\beta}}\n                                  \\psi^{\\mu}_{-1}e^{-\\phi} z^{-1} + \\ldots \\\\\nS^{\\alpha}(z)S^{\\beta} &=&            \\frac{1}{\\sqrt{2}}      \n                                 (\\Gamma_{\\mu}C)^{\\alpha \\beta}\n                                  \\psi^{\\mu}_{-1}e^{-3\\phi} z^{-3} + \\ldots \n\\end{eqnarray*}\n\\end{pp1}\n{\\em Proof:} \nIt is easy to see that the products $S^{\\dot{\\alpha}}_nS^{\\dot{\\beta}}$ vanish for $n\\geq 1$. We make the ansatz $S^{\\dot{\\alpha}}_{\\, 0}S^{\\dot{\\beta}}=\nM_{\\mu}^{\\dot{\\alpha} \\dot{\\beta}}\\psi^{\\mu}_{-1}e^{-\\phi}$. Equation (\\ref{bisc}) with \n$n=0,\\, k=1$ and $m=0$ gives\n\\[ ((\\psi^{\\mu}_{-1}e^{-\\phi})_0S^{\\dot{\\alpha}})_1S^{\\dot{\\beta}}\n= (\\psi^{\\mu}_{-1}e^{-\\phi})_1 S^{\\dot{\\alpha}}_0S^{\\dot{\\beta}} \\, .\\]\nInserting the definitions and using again equation (\\ref{bisc}) we get\n\\[ \\frac{1}{\\sqrt{2}}\n     {\\Gamma^{\\mu\\, \\dot{\\alpha}}}_{\\gamma} C^{\\gamma \\dot{\\beta}}e^{-2\\phi} \n   = M_{\\nu}^{\\dot{\\alpha} \\dot{\\beta}}\n     (\\psi^{\\mu}_{-1}e^{-\\phi})_1(\\psi^{\\nu}_{-1}e^{-\\phi})  \n   = M_{\\nu}^{\\dot{\\alpha} \\dot{\\beta}} g^{\\mu \\nu} e^{-2\\phi}\\, . \\]\nThis implies the first equation.\\\\\nThe second equation is proved in the same way.\n\\hspace*{\\fill} $\\Box$\\\\\nClearly $\\Gamma^{\\mu} C$ is symmetric and \n$C^{-1}\\Gamma^{\\mu}C=-{\\Gamma^{\\mu}}^T$.\n\nNow we describe the cohomology spaces for $\\alpha^2=0$.\n\\pagebreak[3]\n\\begin{pp1}\nLet $\\alpha\\in L^X$ with $\\alpha^2=0$ and $\\alpha\\neq 0$. Then\n\\begin{enumerate}\n\\item[]\n$H(\\alpha)_{-1,1}$ is generated by the massless vectors\n$|\\xi,\\alpha\\rangle = \\xi_{\\mu}\\psi^{\\mu}_{-1}e^{-\\phi}_{-1}c_{-1}e^{\\alpha}$\nsatisfying $(\\xi,\\alpha)=\\xi_{\\mu}\\alpha^{\\mu}=0$. The state \n$\\alpha_{\\mu}\\psi^{\\mu}_{-1}e^{-\\phi}_{-1}c_{-1}e^{\\alpha}$ is trivial in the \ncohomology. $H(\\alpha)_{-1,1}$ has dimension $8$.\n\\item[]\n$H(\\alpha)_{-\\frac{1}{2},1}$ also has dimension $8$ and is spanned by the spinors $|u,-\\frac{1}{2},\\alpha \\rangle = u_{\\dot{\\beta}}S^{\\dot{\\beta}}_{-1}c_{-1}e^{\\alpha}$ satisfying the massless Dirac equation \n$\\alpha_{\\mu}{\\Gamma^{\\mu\\, \\dot{\\beta}}}_{\\gamma} u_{\\dot{\\beta}}=0$.\n\\item[]\n$H(\\alpha)_{-\\frac{3}{2},1}$ is generated by the massless spinors $|u,-\\frac{3}{2},\\alpha \\rangle = u_{\\beta}S^{\\beta}_{-1}c_{-1}e^{\\alpha}$. The states with\n$\\alpha_{\\mu}{\\Gamma^{\\mu \\, \\beta}}_{\\dot{\\delta}}u_{\\beta}=0$ are BRST-trivial.\n$H(\\alpha)_{-\\frac{3}{2},1}$ also has dimension $8$.\n\\end{enumerate}\nLet $\\alpha=0$. Then\n\\begin{enumerate}\n\\item[]\n$H(\\alpha)_{-1,1}$ has dimension $10$. The vectors \n$P^{\\mu}=-\\psi^{\\mu}_{-1}e^{-\\phi}_{-1}c$ form a basis of this space.\n\\item[]\n$H(\\alpha)_{-\\frac{1}{2},1}$ has dimension $16$ and a basis \n$Q^{\\dot{\\alpha}}=S^{\\dot{\\alpha}}_{-1}c$.\n\\item[]\n$H(\\alpha)_{-\\frac{3}{2},1}$ has dimension $16$ and a basis \n$Q^{\\alpha}=S^{\\alpha}_{-1}c$.\n\\end{enumerate}\n\\end{pp1} \n{\\em Proof:} We can write $\\xi_{\\mu}\\psi^{\\mu}_{-1}e^{-\\phi}_{-1}c_{-1}e^{\\alpha}=c_{-1}w$ with $w=\\xi_{\\mu}\\psi^{\\mu}_{-1}e^{-\\phi}_{-1}e^{\\alpha}$. \nThen by Lemma \\ref{ecl}\n\\[ Qc_{-1}w = \\{Q,c_{-1}\\}w-c_{-1}Qw = -c_{-1}Q_1w \\, .\\]\nUsing equation (\\ref{bisc}) with $n=k=-1$ and $m=1$ we get \n\\[ Q_1w =(\\gamma_{-1}\\tau^M)_0 w \n        = \\sum_{j\\geq 0}(\\gamma_{-2-j}\\tau^M_{j+1}+\\tau^M_{-j}\\gamma_{-1+j})w \n        = \\gamma_{-2}\\tau^M_1 w  \\, .\\]\nNow $\\tau^M_1w = (\\xi,\\alpha) e^{-\\phi}_{-1}e^{\\alpha}$. Calculating \n$Qe^{-2\\phi}_{-1}(D\\xi)_{-1}c_{-1}e^{\\alpha}$ with Lemma \\ref{ecl} shows that  \n$\\alpha_{\\mu}\\psi^{\\mu}_{-1}e^{-\\phi}_{-1}c_{-1}e^{\\alpha}$ is trivial in cohomology.\nThis proves the first statement. \\\\\nThe massless spinor \n$u_{\\dot{\\beta}}S^{\\dot{\\beta}}_{-1}c_{-1}e^{\\alpha}$ is in the kernel of $Q$ iff \n$Q_1w=0$ where $w=u_{\\dot{\\beta}}S^{\\dot{\\beta}}_{-1}e^{\\alpha}$. Equation \n(\\ref{bisc}) gives \n\\[ Q_1 w = \\sum_{j\\geq 0}( \\gamma_{-\\frac{3}{2}-j}\\tau^M_{j+\\frac{1}{2}}\n                        +\\tau^M_{-\\frac{1}{2}-j}\\gamma_{-\\frac{1}{2}+j})w \n         = \\gamma_{-\\frac{3}{2}}\\tau^M_{\\frac{1}{2}} w  \\]\nand \n\\begin{eqnarray*} \n\\tau^M_{\\frac{1}{2}} w \n&=& (x_{\\nu}(-1)_{-1}\\psi^{\\nu})_{\\frac{1}{2}} w \\\\\n&=& x_{\\nu}(-1)_0\\psi^{\\nu}_{-\\frac{1}{2}} w \\\\\n&=& \\alpha_{\\nu}u_{\\dot{\\beta}}e^{\\alpha}_{-1}\\psi^{\\nu}_{-\\frac{1}{2}} S^{\\dot{\\beta}}\\\\\n&=& -\\alpha_{\\nu}u_{\\dot{\\beta}}e^{\\alpha}_{-1}\\psi^{\\nu}_{-\\frac{1}{2}} \n    e^{\\phi}_{-\\frac{5}{2}}e^{-\\phi}_{-\\frac{1}{2}}S^{\\dot{\\beta}} \\\\\n&=& \\alpha_{\\nu}u_{\\dot{\\beta}}e^{\\alpha}_{-1}e^{\\phi}_{-\\frac{5}{2}}\n    \\psi^{\\nu}_{-\\frac{1}{2}}e^{-\\phi}_{-\\frac{1}{2}}S^{\\dot{\\beta}} \\\\\n&=& \\frac{1}{\\sqrt{2}}\\alpha_{\\nu}u_{\\dot{\\beta}}{\\Gamma^{\\nu\\, \\dot{\\beta}}}_{\\gamma}\n    e^{\\phi}_{-\\frac{5}{2}}S^{\\gamma}_{-1}e^{\\alpha}\n\\end{eqnarray*}\nwhere we have used \n$S^{\\dot{\\beta}}=-e^{\\phi}_{-\\frac{5}{2}}e^{-\\phi}_{-\\frac{1}{2}}S^{\\dot{\\beta}}$. \nHence $Q_1w=0$ iff \n$\\alpha_{\\nu}{\\Gamma^{\\nu\\,\\dot{\\beta}}}_{\\gamma}u_{\\dot{\\beta}}=0$.\nSince $C(\\alpha)_{-\\frac{1}{2},0}=0$ this proves the second statement.\\\\\nIt is easy to see that the elements in $C(\\alpha)_{-\\frac{3}{2},1}$ are in the kernel of $Q$. Thus we have to calculate the image of $C(\\alpha)_{-\\frac{3}{2},0}$ \nunder $Q$. We have \n\\[ Qu_{\\dot{\\delta}}e^{\\lambda_{\\dot{\\delta}}-\\frac{5}{2}\\phi}_{-1}(D\\xi)_{-1}\n   c_{-1}e^{\\alpha}\n   = Qc_{-1}w = -c_{-1}Q_1w \\]\nwith $w=u_{\\dot{\\delta}}e^{\\lambda_{\\dot{\\delta}}-\\frac{5}{2}\\phi}_{-1}\n      (D\\xi)_{-1}e^{\\alpha}$.\nAs above \n\\[ Q_1w = (\\gamma_{-1}\\tau^M)_0w  \n        = \\sum_{j\\geq 0}( \\gamma_{-\\frac{3}{2}-j}\\tau^M_{j+\\frac{1}{2}})w\n        = \\gamma_{-\\frac{3}{2}}\\tau^M_{\\frac{1}{2}} w \\, .\\] \nNow\n\\begin{eqnarray*}\n\\tau^M_{\\frac{1}{2}} w\n&=& \\alpha_{\\mu}u_{\\dot{\\delta}}(D\\xi)_{-1}e^{\\alpha}_{-1}\n    \\psi^{\\mu}_{-\\frac{1}{2}}e^{\\lambda_{\\dot{\\delta}}-\\frac{5}{2}\\phi} \\\\\n&=& \\alpha_{\\mu}u_{\\dot{\\delta}}(D\\xi)_{-1}e^{\\alpha}_{-1}\n    \\psi^{\\mu}_{-\\frac{1}{2}}\n    (e^{-2\\phi}_0e^{\\lambda_{\\dot{\\delta}}-\\frac{1}{2}\\phi}) \\\\\n&=& \\alpha_{\\mu}u_{\\dot{\\delta}}(D\\xi)_{-1}e^{\\alpha}_{-1}e^{-2\\phi}_0\n    \\psi^{\\mu}_{-\\frac{1}{2}}e^{\\lambda_{\\dot{\\delta}}-\\frac{1}{2}\\phi} \\\\\n&=& \\frac{1}{\\sqrt{2}}\\alpha_{\\mu}u_{\\dot{\\delta}}\n    {\\Gamma^{\\mu \\, \\dot{\\delta}}}_{\\gamma}\n    (D\\xi)_{-1}e^{\\alpha}_{-1}e^{-2\\phi}_0\n    e^{\\phi}_{-\\frac{5}{2}}\n    e^{\\lambda_{\\gamma}-\\frac{3}{2}\\phi},\n\\end{eqnarray*}\nwhere we have used \n$\\varepsilon(-2\\phi,\\lambda_{\\dot{\\delta}}-{\\textstyle\\frac{1}{2}}\\phi) =1$\nand \n\\begin{eqnarray*}\n\\lefteqn{Q_1w} \\\\ \n&=& \\frac{1}{\\sqrt{2}}\\alpha_{\\mu}u_{\\dot{\\delta}}\n    {\\Gamma^{\\mu \\, \\dot{\\delta}}}_{\\gamma}e^{\\alpha}_{-1}\n    \\gamma_{-\\frac{3}{2}}(D\\xi)_{-1}e^{-2\\phi}_0\n    e^{\\phi}_{-\\frac{5}{2}}e^{\\lambda_{\\gamma}-\\frac{3}{2}\\phi}  \\\\\n&=& \\frac{1}{\\sqrt{2}}\\alpha_{\\mu}u_{\\dot{\\delta}}\n    {\\Gamma^{\\mu \\, \\dot{\\delta}}}_{\\gamma}e^{\\alpha}_{-1}\n    (e^{\\phi}_{-1}\\eta)_{-\\frac{3}{2}}(D\\xi)_{-1}e^{-2\\phi}_0\n    e^{\\phi}_{-\\frac{5}{2}}e^{\\lambda_{\\gamma}-\\frac{3}{2}\\phi}  \\\\\n&=& \\frac{1}{\\sqrt{2}}\\alpha_{\\mu}u_{\\dot{\\delta}}\n    {\\Gamma^{\\mu \\, \\dot{\\delta}}}_{\\gamma}e^{\\alpha}_{-1}\n    \\sum_{j\\geq 0}(e^{\\phi}_{-\\frac{9}{2}-j} \\eta_{2+j}\n                   -\\eta_{1-j}e^{\\phi}_{-\\frac{7}{2}+j}) \n    (D\\xi)_{-1}e^{-2\\phi}_0\n    e^{\\phi}_{-\\frac{5}{2}}e^{\\lambda_{\\gamma}-\\frac{3}{2}\\phi} \\\\\n&=& \\frac{1}{\\sqrt{2}}\\alpha_{\\mu}u_{\\dot{\\delta}}\n    {\\Gamma^{\\mu \\, \\dot{\\delta}}}_{\\gamma}e^{\\alpha}_{-1}\n    \\eta_1 (D\\xi)_{-1} e^{\\phi}_{-\\frac{7}{2}}e^{-2\\phi}_0\n    e^{\\phi}_{-\\frac{5}{2}}e^{\\lambda_{\\gamma}-\\frac{3}{2}\\phi} \\\\\n&=& -\\frac{1}{\\sqrt{2}}\\alpha_{\\mu}u_{\\dot{\\delta}}\n    {\\Gamma^{\\mu \\, \\dot{\\delta}}}_{\\gamma}e^{\\alpha}_{-1}\n    e^{\\lambda_{\\gamma}-\\frac{3}{2}\\phi} \n\\end{eqnarray*}\nsince\n\\pagebreak[3] \n\\begin{eqnarray*}\n\\lefteqn{\n   \\varepsilon(\\phi,\\lambda_{\\gamma}-{\\textstyle \\frac{3}{2}}\\phi)\n   \\varepsilon(-2\\phi,\\lambda_{\\gamma}-{\\textstyle \\frac{1}{2}}\\phi)\n   \\varepsilon(\\phi,\\lambda_{\\gamma}-{\\textstyle \\frac{5}{2}}\\phi)} \\\\\n&=& \\varepsilon(\\phi,\\lambda_{\\gamma}-{\\textstyle \\frac{3}{2}}\\phi)\n    \\varepsilon(-2\\phi,\\lambda_{\\gamma}-{\\textstyle \\frac{1}{2}}\\phi) \n    \\varepsilon(\\phi,\\lambda_{\\gamma}-{\\textstyle \\frac{3}{2}}\\phi)\n    \\varepsilon(\\phi,-\\phi) \\\\\n&=& \\varepsilon(2\\phi,\\lambda_{\\gamma}-{\\textstyle \\frac{3}{2}}\\phi)\n    \\varepsilon(-2\\phi,\\lambda_{\\gamma}-{\\textstyle \\frac{1}{2}}\\phi) \n    \\varepsilon(\\phi,\\phi)^{-1} \\\\\n&=& \\varepsilon(2\\phi,-\\phi) \\varepsilon(\\phi,\\phi)^{-1} \\\\\n&=& -1 \\, . \n\\end{eqnarray*}\nHence \n\\[ Q u_{\\dot{\\delta}}e^{\\lambda_{\\dot{\\delta}}-\\frac{5}{2}\\phi}_{-1}\n   (D\\xi)_{-1}c_{-1}e^{\\alpha}\n   = -\\frac{1}{\\sqrt{2}}\\alpha_{\\mu}{\\Gamma^{\\mu \\, \\dot{\\delta}}}_{\\gamma}\n     u_{\\dot{\\delta}} S^{\\gamma}_{-1}c_{-1}e^{\\alpha}\\, . \\]\nThus $v_{\\beta}S^{\\beta}_{-1}c_{-1}e^{\\alpha}$ is BRST-trivial iff \n$v_{\\beta}=\\alpha_{\\mu}{\\Gamma^{\\mu \\, \\dot{\\delta}}}_{\\beta}u_{\\dot{\\delta}}$\nwhich is equivalent to \n$\\alpha_{\\mu}{\\Gamma^{\\mu \\, \\beta}}_{\\dot{\\delta}}v_{\\beta}=0$.\nThis proves the third statement. \\\\\nThe rest of the proposition is now clear.\n\\hspace*{\\fill} $\\Box$\\\\\n\nWe conclude this section with some remarks on the picture changing operator. \nFirst we give two examples. \nThe massless vector is given in the $0$ picture by\n\\begin{equation}  X_{-1}|\\xi,\\alpha \\rangle = \n\\left( \\xi_{\\mu}x^{\\mu}(-1)_{-1}+\\alpha_{\\nu}\\xi_{\\mu}\n  \\psi^{\\nu}_{-1}\\psi^{\\mu}_{-1}\\right) e^{\\alpha}_{-1}c \n  + \\xi_{\\mu}\\psi^{\\mu}_{-1}e^{\\alpha}_{-1}\\gamma \\, . \\ee\nFor $|u,-\\frac{3}{2},\\alpha \\rangle = u_{\\beta}S^{\\beta}_{-1}e^{\\alpha}_{-1}c$ we have\n\\begin{equation} X_{-1} |u,{\\textstyle -\\frac{3}{2}},\\alpha \\rangle \n   = \\frac{1}{\\sqrt 2} \\alpha_{\\mu}{\\Gamma^{\\mu\\,\\beta}}_{\\dot{\\gamma}} \n     u_{\\beta}S^{\\dot{\\gamma}}_{-1}c_{-1}e^{\\alpha} \\, . \\ee\nAlthough $\\mbox{dim }H(\\alpha)_{-\\frac{1}{2},1}=H(\\alpha)_{-\\frac{3}{2},1}=16$ for \n$\\alpha=0$ the last equation implies\n\\begin{pp1} \\label{ncgp}\nLet $\\alpha=0$. Then $X_{-1}H(\\alpha)_{-\\frac{3}{2},1}=0$.\n\\end{pp1}\nThis shows that Proposition \\ref{piso} does not hold for zero momentum.\n\n\\subsection{The Lie algebra and the Lie bracket}\n\nWe define a product on $V^{GSO}_S$ that induces a product on the cohomology space $H$ (cf. \\cite{LZ2}) and construct a Lie bracket on the physical states using the picture changing operator.\n\nFor $u,v$ in $V^{GSO}_S$ define\n\\begin{equation} \\{u,v\\}=(-1)^{|u|}(b_0u)_0v \\ee\nwhere $|u|$ denotes the parity of $u$ with respect to $V^{GSO}_S$. Then\n\\begin{pp1}\\label{qder}\nFor $u,v$ in $V^{GSO}_S$ we have\n\\begin{eqnarray*}\n(-1)^{|u|}\\{ u,v \\} &=& b_1(u_{-1}v)-(b_1u)_{-1}v-(-1)^{|u|} u_{-1}(b_1v) \\\\\nQ\\{ u,v \\} &=& \\{Qu,v\\}+(-1)^{|u|+1}\\{u,Qv\\} \\, .\n\\end{eqnarray*}\n\\end{pp1}\nSince $b_1^2=0$ the first equation implies that the bracket closes on $B$.\nIt is easy to see that it also closes on $C$. The second equation shows that it projects down to the cohomology $H$. Hence $\\{\\, ,\\, \\}$ defines a product\n$H(\\alpha)_{p,m}\\times H(\\beta)_{q,n}\\rightarrow H(\\alpha+\\beta)_{p+q,m+n-1}$.\nThe product satisfies (cf. \\cite{LZ2})\n\\begin{pp1} \\label{bsym}\nLet $u,v$ and $w$ be representatives of elements in $H$. Then\n\\[ \\renewcommand{\\arraystretch}{1.5}\n\\begin{array}{c}\n \\{ u,v \\}+ (-1)^{(|u|+1)(|v|+1)}\\{ v,u \\} = 0 \\\\\n  (-1)^{(|u|+1)(|w|+1)}\\{u,\\{v,w\\}\\} +  (-1)^{(|v|+1)(|u|+1)}\\{v,\\{w,u\\}\\} \\\\\n                  + (-1)^{(|w|+1)(|v|+1)}\\{w,\\{u,v\\}\\} =0 \n\\end{array} \\]\n\\end{pp1}\nNow we show that the bracket invariant under picture changing.\n\\begin{pp1} \\label{pinv}\nLet $u,v\\in H$. Then\n\\[ X_{-1}\\{u,v\\}=\\{X_{-1}u,v\\}=\\{u,X_{-1}v\\} \\]\n\\end{pp1}\n{\\em Proof:}\nFrom Proposition (\\ref{qder}) we get\n\\[ (-1)^{|u|}X_{-1} \\{ u,v \\} - (-1)^{|u|}\\{ u, X_{-1}v\\} \n              = b_1 [X_{-1},u_{-1}]v \\, . \\]\nNow\n\\[ [X_{-1},u_{-1}]=\\sum_{k\\geq 0} {-1 \\choose k} (X_ku)_{-k-2}\n                  =\\sum_{k\\geq 0} {-1 \\choose k} (Q\\xi_ku)_{-k-2} \\]\nand \n\\[ (Q\\xi_ku)_{-k-2}= \\left( j_0^{BRST} (\\xi_ku) \\right)_{-k-2}\n                   = Q(\\xi_ku)_{-k-2}\\pm (\\xi_ku)_{-k-2}Q \\]\nimply\n\\[\n[X_{-1},u_{-1}]v = \\sum_{k\\geq 0} {-1 \\choose k}Q(\\xi_ku)_{-k-2}v \\\\\n                 = \\sum_{k\\geq 0} \\frac{(-1)^{k+1}}{k+1} Q \n                     \\left( (D\\xi)_{k+1}u \\right)_{-k-2}v \\, .\n\\]\nNote that $\\left((D\\xi)_{k+1}u\\right)_{-k-2}v$ is an element of $V^{GSO}_S$\nwith $L_0$-eigenvalue zero. $b_1Q=L_0-Qb_1$ shows then that \n$b_1 [X_{-1},u_{-1}]v$ is in the image of $Q$. This proves \n$X_{-1}\\{u,v\\}=\\{u,X_{-1}v\\}$. The Proposition now follows from the symmetry of the product. \\hspace*{\\fill} $\\Box$\\\\\n\nLian and Zuckerman have defined in \\cite{LZ2} another product $u\\cdot v=u_{-1}v$. By the derivation property of zero modes this product defines a product on the cohomology. It satisfies $u\\cdot v =(-1)^{|u| |v|}v\\cdot u$. Then as above $X_{-1}(u\\cdot v) -u\\cdot (X_{-1}v)=[X_{-1},u_{-1}]v$ is zero in cohomology so that this product is also invariant under $X_{-1}$, i.e. \n$X_{-1}(u\\cdot v)=u\\cdot (X_{-1}v)=(X_{-1}u)\\cdot v$.\n\nDefine ${\\tilde G}(\\alpha)={\\tilde G}(\\alpha)_0\\oplus {\\tilde G}(\\alpha)_1$ where \n${\\tilde G}(\\alpha)_0 = H(\\alpha)_{-1,1}$ and ${\\tilde G}(\\alpha)_1 = H(\\alpha)_{-\\frac{1}{2},1}$.\nLet \\[ {\\tilde G} = \\bigoplus_{\\alpha\\in L^X} {\\tilde G}(\\alpha) \\]\nand define a ${\\mathbb Z}_2$-grading on ${\\tilde G}$ by setting ${\\tilde G}_0=\\bigoplus {\\tilde G}(\\alpha)_0$\nand ${\\tilde G}_1=\\bigoplus {\\tilde G}(\\alpha)_1$.\n\nWe define a product on ${\\tilde G}$ by\n\\[ [u,v] = X_{-1} \\{u,v\\} \\quad \\mbox{ if $u$ or $v$ is in ${\\tilde G}_0$} \\]\nand \n\\[ [u,v] = \\{u,v\\}  \\quad \\mbox{ if $u$ and $v$ are in ${\\tilde G}_1$}\\, .\\]\nNote that the parity of $u$ as an element of ${\\tilde G}$ is equal to $|u|+1$ \nwhere $|u|$ denotes the parity of $u$ as an element in $V^{GSO}_S$. \nBy Propositions \\ref{bsym} and \\ref{pinv} we have\n\\begin{pp1}\nWith this product ${\\tilde G}$ is a Lie superalgebra.\n\\end{pp1}\nDefine $G$ as ${\\tilde G}$ without ${\\tilde G}(0)_1$, i.e. without the supersymmetry charges $Q^{\\dot{\\alpha}}$. Then Proposition \\ref{ncgp} shows that $G$ is a subalgebra of ${\\tilde G}$. We will see that $G$ is in contrast to ${\\tilde G}$ a generalized Kac-Moody superalgebra. Since $G$ represents the physical states of the compactified superstring in the canonical ghost pictures we call $G$ the Lie superalgebra of physical states. The even part of $G$ contains the bosonic and the odd part the fermionic physical states. We have already seen that $G$ has no real roots. This corresponds to the fact that the superstring has no tachyons after GSO-projection. \n\n\\subsection{Properties of the Lie superalgebra of physical states}\n\nIn this section we derive some properties of the Lie superalgebras $G$ and ${\\tilde G}$. In the next section they will be used to show that $G$ is a generalized Kac-Moody superalgebra. \n\nWe define a bilinear form on $G$ by setting\n\\[ \\renewcommand{\\arraystretch}{1.5}\n\\begin{array}{lcrcl} \n\\langle u, v \\rangle &=& (u,v)_H         & &        \\mbox{if } u,v \\in G_0 \\\\\n\\langle u, v \\rangle &=& -(\\tilde{u},v)_H & \\mbox{with } \n                        u=X_{-1}\\tilde{u}\\, & \\mbox{if } u,v \\in G_1\n\\end{array} \\]\nand zero else. Note that by Proposition \\ref{ncgp} this definition does not extend to $\\tilde{G}$. We have\n\\begin{pp1}\nThe bilinear form $\\langle\\, ,\\,\\rangle$ pairs nondegeneratly $G(\\alpha)$ with $G(-\\alpha)$.\nFurthermore $\\langle\\, ,\\,\\rangle$ is supersymmetric and invariant.\n\\end{pp1}\n{\\em Proof:}\nThe first part of the proposition is clear. It is also easy to see that \n$\\langle u, v \\rangle=\\langle v,u \\rangle$ for $u,v\\in G_0$. Let $u,v \\in G_1$. Then\n\\begin{eqnarray*}\n\\langle u,v \\rangle &=&  -(c_{-2}\\tilde{u}, X_{-1}\\tilde{v}) \\\\\n            &=&  -(c_{-2}\\tilde{u}, X_{-1}b_1c_{-2}\\tilde{v}) \\\\\n            &=&  -(b_1X_{-1}c_{-2}\\tilde{v}, c_{-2}\\tilde{u}) \\\\\n            &=&   (X_{-1}c_{-2}\\tilde{v}, b_1c_{-2}\\tilde{u}) \\\\\n            &=&   (X_{-1}c_{-2}\\tilde{v},\\tilde{u}) \\\\\n            &=&   (c_{-2}\\tilde{v},X_{-1}\\tilde{u}) \\\\\n            &=&   (c_{-2}\\tilde{v},u) \\\\\n            &=&  - \\langle v,u \\rangle \\, .\n\\end{eqnarray*} \nHence $\\langle u, v \\rangle$ is supersymmetric.\\\\\nTo prove the invariance it is sufficient to consider two cases.\nFirst let $u,v$ and $w$ be in $G_0$. Then\n\\begin{eqnarray*} \n\\langle [u,v],w \\rangle \n&=& - \\langle [v,u],w \\rangle \\\\\n&=& - (-1)^{|v|} (c_{-2}b_1v_{-1}X_{-1}u,w) \\\\\n&=&   (v_{-1}X_{-1}u , b_1c_{-2}w) \\\\\n&=&   (v_{-1}X_{-1}u , w) \\\\\n&=& - (X_{-1}u, v_{-1}^*w) \\\\\n&=& - (X_{-1}u, v_{-1}w )\n\\end{eqnarray*}\nWe explain the last equality. Recall the definition\n\\[ v_{-1}^* = v_{-1}+\\sum_{n\\geq 1}\\left(\\frac{L_1^n}{n!}v\\right)_{-n-1}\\, . \\]\nFor $n\\geq 1$ we have\n\\[ L_1^nv = \\{Q,b_2\\}L_1^{n-1}v = Qb_2L_1^{n-1}v \\]\nSince $Q$ is a derivation this implies that only $v_{-1}w$ contributes to the product. Now\n\\begin{eqnarray*} \n\\langle u,[v,w] \\rangle &=& (-1)^{|v|}(c_{-2}u,b_1X_{-1}v_{-1}w) \\\\\n                &=& -(b_1X_{-1}v_{-1}w,c_{-2}u) \\\\\n                &=& -(X_{-1}v_{-1}w,u) \\\\\n                &=& -(v_{-1}w,X_{-1}u) \\\\\n                &=& -(X_{-1}u,v_{-1}w)\n\\end{eqnarray*}\nproves the invariance in this case. Now let $u,v\\in G_1$ and $w\\in G_0$.\nThen\n\\begin{eqnarray*} \n\\langle [u,v],w \\rangle &=& \\langle [v,u], w \\rangle \\\\\n                &=& (-1)^{|v|}(c_{-2}b_1 v_{-1}u, w) \\\\\n                &=& (v_{-1}u,b_1c_{-2}w) \\\\\n                &=& (v_{-1}u,w)\\\\\n                &=& (u,v_{-1}^*w) \\\\\n                &=& (u,v_{-1}w)\n\\end{eqnarray*}\nby the same argument as above and \n\\begin{eqnarray*}\n\\langle u,[v,w] \\rangle &=& - \\langle [v,w] ,u \\rangle \\\\\n                &=& (-1)^{|v|}(c_{-2}b_1v_{-1}w,u ) \\\\\n                &=& (v_{-1}w,b_1c_{-2}u) \\\\\n                &=& (v_{-1}w,u) \\\\\n                &=& (u,v_{-1}w) \n\\end{eqnarray*}\nshowing the invariance in this case.\n\\hspace*{\\fill} $\\Box$\\\\\n\nSimple calculation gives the bilinear form on the massless states\n\\begin{pp1}\nLet $\\alpha\\in L^X$ with $\\alpha^2=0$ and $\\alpha\\neq 0$. Then\n\\[ \\renewcommand{\\arraystretch}{1.3}\n\\begin{array}{rcl}\n \\langle \\, \\xi,\\alpha\\,|\\, \\zeta, -\\alpha \\, \\rangle &=& \\xi_{\\mu} \\zeta_{\\nu} g^{\\mu \\nu}\\\\\n \\langle \\, X_{-1}\\,|\\,u,{-\\textstyle \\frac{3}{2}},\\alpha \\rangle , \n                                |\\,v,{-\\textstyle \\frac{1}{2}},-\\alpha \\, \\rangle  \n                        &=&  u_{\\beta}v_{\\dot \\gamma} C^{\\beta \\dot \\gamma} \n\\end{array} \\]\nFor $\\alpha =0$ we have \n\\[  \\langle P^{\\mu} , P^{\\nu} \\rangle = g^{\\mu \\nu}\\, . \\]\n\\end{pp1}\nHence the map $j$ from $H=G(0)_0$ to ${\\mathbb C}\\otimes_{\\mathbb Z}L^X$ defined by $P^{\\mu}\\mapsto x^{\\mu}$ is an isometry. We have\n\\begin{pp1} \nLet $h\\in H, \\, x\\in G(\\alpha)$ and $y\\in G(-\\alpha)$. Then\n\\begin{eqnarray*}\n{[} h,x {]} &=& (h,\\alpha) x \\\\\n{[} x,y {]} &=& \\langle x, y \\rangle j^{-1}(\\alpha)\n\\end{eqnarray*}\n\\end{pp1}\n{\\em Proof:} From $b_0X_{-1}P^{\\mu}=-x^{\\mu}(-1)$ we get \n\\begin{equation} \n[P^{\\mu}, x] = \\{X_{-1}P^{\\mu},x\\}= -(b_0X_{-1}P^{\\mu})_0 =\\alpha^{\\mu}x \\, .\n\\ee\nThis proves the first statement. The second follows from the nondegeneracy of \n$\\langle\\, ,\\, \\rangle$. Let $h\\in H$. Then\n\\[ \\langle\\, [x,y]-\\langle x, y\\rangle j^{-1}(\\alpha), h\\rangle=\n   \\langle x,[y,h]\\, \\rangle - \\langle x, y\\rangle (\\alpha , j(h)) =0 \\, .\\]\n\\hspace*{\\fill} $\\Box$\\\\\nWe give a simple application of this result.\n\\begin{pp1}\n$G$ is simple.\n\\end{pp1}\n{\\em Proof:} Let $I\\neq 0$ an ideal in $G$. Then by Proposition 1.5 in \n\\cite{K1} $I$ is graded by $L^X$. $I$ clearly has an element of degree $\\alpha\\neq 0$ in $L^X$. This implies that $I\\cap H$ is nonempty. Let $h$ be a nonzero element in $I\\cap H$. Since $\\langle \\, , \\, \\rangle$ is nondegenerate the elements $h'$  \nwith $\\langle h,h' \\rangle \\neq 0$ span $H$. $I$ contains the root spaces $G(\\alpha')$ corresponding to these $h'$. This implies that the $h'$ are in $I$. Hence $I\\cap H=H$ and $I=G$. \\hspace*{\\fill} $\\Box$\\\\\n\nThis implies that $G$ is the derived algebra of ${\\tilde G}$.\n\nThe next proposition shows that the zero momentum states $P^{\\mu}$ and \n$Q^{\\dot{\\alpha}}$ generate a representation of the $N=1$ supersymmetry algebra on $G$ and ${\\tilde G}$. \n\\begin{pp1}\nThe zero momentum states satisfy the following algebra.\n\\[ \\renewcommand{\\arraystretch}{1.5}\n\\begin{array}{c}\n [ P^{\\mu} , P^{\\nu} ] = [ P^{\\mu},Q^{\\dot{\\alpha}} ] = 0 \\\\\n \\{ Q^{\\dot{\\alpha}},Q^{\\dot{\\beta}} \\} = \n         \\frac{1}{\\sqrt 2} (\\Gamma_{\\mu}C)^{\\dot{\\alpha}\\dot{\\beta}} P^{\\mu}  \n\\end{array} \\]\n\\end{pp1}\n{\\em Proof:} The first two products are clear. \nWe do the calculations for the third product. \n\\begin{eqnarray*}\n\\{ Q^{\\dot{\\alpha}},Q^{\\dot{\\beta}} \\} \n&=& (b_0 S^{\\dot{\\alpha}}_{-1}c)_0S^{\\dot{\\beta}}_{-1}c \\\\\n&=& - c_{-1} S^{\\dot{\\alpha}}_0 S^{\\dot{\\beta}} \\\\\n&=& - c_{-1}\\frac{1}{\\sqrt 2} (\\Gamma_{\\mu}C)^{\\dot{\\alpha}\\dot{\\beta}} \n             (\\psi^{\\mu}_{-1}e^{-\\phi}) \\\\\n&=& \\frac{1}{\\sqrt 2} (\\Gamma_{\\mu}C)^{\\dot{\\alpha}\\dot{\\beta}} P^{\\mu}\n\\end{eqnarray*} \nby Proposition \\ref{susyv}.\n\\hspace*{\\fill} $\\Box$\\\\\nHence the spaces $G(\\alpha)_0$ and $G(\\alpha)_1$ are related by a supersymmetry transformation for $\\alpha\\neq 0$. This implies \n\\begin{thp1}\nLet $\\alpha\\neq 0$. Then the dimension of the root space $G(\\alpha)_1$ is given by\n\\[ \\mbox{dim }G(\\alpha)_1 = c({\\textstyle -\\frac{1}{2}}\\alpha^2) \\, . \\]  \n\\end{thp1}\n\nThe following technical result is proved by direct calculation.\n\\begin{pp1}\nLet $\\alpha$ and $\\beta$ be proportional zero norm vectors in \n$L^X\\backslash \\{0\\}$. \nThen their root spaces commute.           \n\\end{pp1}\n\nConsider the real vector space spanned by the states \n\\[    x^{\\mu_1}(-1)_{-m_1}\\ldots x^{\\mu_i}(-1)_{-m_i}\n      \\psi^{\\nu_1}_{-n_1}\\ldots \\psi^{\\nu_j}_{-n_j} \n      e^{-\\phi}_{-1}e^{\\alpha}_{-1}c \\]\nwhere $j$ is odd and the $m$'s and $n$'s are positive integers and define the \nsubspace $P(\\alpha)_{-1}$ of states $v$ satisfying  \n$L^M_0 v= \\frac{1}{2}v$ and $L^M_nv=G^M_{n-\\frac{1}{2}}v=0$ for $n\\geq 1$.\nThen $P(\\alpha)_{-1}$ is $Q$-closed. We denote the image in $H(\\alpha)_{-1,1}$ also \nby $P(\\alpha)_{-1}$. Then the complexification of $P(\\alpha)_{-1}$ is \n$H(\\alpha)_{-1,1}$ (cf. e.g. \\cite{P}).\nSimilarly the states \n\\[    x^{\\mu_1}(-1)_{-m_1}\\ldots x^{\\mu_i}(-1)_{-m_i}\n      \\psi^{\\nu_1}_{-n_1-\\frac{1}{2}}\\ldots \n      \\psi^{\\nu_j}_{-n_j-\\frac{1}{2}} \n      S^{\\dot{\\beta}}_{-1}e^{\\alpha}_{-1}c        \\]\nwhere $j$ is even and the $m$'s and $n$'s are again positive integers\ngenerate a real vector space and we define $P(\\alpha)_{-\\frac{1}{2}}$ as the \nsubspace of states $v$ satisfying $L^M_0 v= \\frac{5}{8}v$ and \n$L^M_nv=G^M_{n-1}v=0$ for $n\\geq 1$. $P(\\alpha)_{-\\frac{1}{2}}$ is $Q$-closed and its image in $H(\\alpha)_{-\\frac{1}{2},1}$ generates this space.\n\nNow choose a basis of spinors such that the $\\Gamma$-matrices defined above \nsatisfy the Majorana and the Weyl condition. Denote the sum of the \n$P(\\alpha)_{-1}$ over $\\alpha$ in $L^X$ as $P_{-1}$ and the sum of the \n$P(\\alpha)_{-\\frac{1}{2}}$ with $\\alpha\\neq 0$ as $P_{-\\frac{1}{2}}$.\nThen one can prove\n\\begin{pp1}\n$P_{-1} \\oplus P_{-\\frac{1}{2}}$ is a real form of $G$. The bilinear form on $G$ induces a real bilinear form on this space. \n\\end{pp1}\n\n\\subsection{The main theorems} \\label{mrs}\n\nIn this section we state the main results of this paper.\n\\begin{thp1} \\label{mt1}\nThe Lie superalgebra $G$ has the following properties.\n\\begin{enumerate}\n\\item[]\n$G$ is graded by $L^X$. We define $H$ as the subspace of degree zero, so that\n$G=H\\oplus \\bigoplus_{\\alpha\\neq 0}G(\\alpha)$.\n\\item[]\n$G$ has a invariant supersymmetric bilinear form that pairs nondegenerately\n$G(\\alpha)$ with $G(-\\alpha)$.\n\\item[]\n$H$ is an abelian selfcentralizing even subalgebra of $G$. $H$ is isometric to ${\\mathbb C}\\otimes_{\\mathbb Z}L^X$ and $[h,x]=(h,\\alpha)x$ for $h\\in H$ and \n$x\\in G(\\alpha)$ so that the $G(\\alpha)$ are eigenspaces of $H$. $H$ is called the Cartan subalgebra of $G$. \n\\item[]\nThe root spaces $G(\\alpha)=G(\\alpha)_0+G(\\alpha)_1$ are finite dimensional and\n\\linebreak  \n$\\mbox{dim}\\, G(\\alpha)_0 = \\mbox{dim}\\, G(\\alpha)_1 = c(-\\frac{1}{2}\\alpha^2)$.\nHence the roots of $G$ are the nonzero $\\alpha\\in L^X$ with $\\alpha^2\\leq 0$.  \n\\item[]\n$G$ is simple.\n\\item[]\n$G$ is acted on by the $N=1$ superymmetry algebra.\n\\item[]\nThe root spaces of two proportional norm zero roots commute.\n\\item[]\n$G$ is a generalized Kac-Moody superalgebra.\n\\end{enumerate}\n\\end{thp1}\n\\pagebreak[3]\n{\\em Proof:} Using the root space decomposition and the nondegeneracy of $\\langle \\, ,\\, \\rangle$ it is easy to see that $H$ is selfcentralizing. The last statement follows from Theorem \\ref{ut}. As regular element one can take any nonzero negative norm vector. The roots of $G$ are all of infinite type. Note that in a Lorentzian lattice any two positive imaginary roots have inner product at most zero and zero only if they are both multiples of the same norm zero vector.\n\\hspace*{\\fill} $\\Box$\\\\\n\nNow we consider the case that $L^X$ is the unique unimodular even Lorentzian \nlattice $II_{9,1}$. In this case we call $G$ the fake monster superalgebra.\nThe lattice $II_{9,1}$ can be described as the lattice of all points \n$(x_1,\\ldots,x_9,x_{10})\\in {\\mathbb R}^{9,1}$ with all $x_i\\in {\\mathbb Z}$ or all $x_i\\in {\\mathbb Z}+\\frac{1}{2}$ and which have integer inner product with $(\\frac{1}{2},\\ldots,\\frac{1}{2},\\frac{1}{2})$. In a Lorentzian lattice there are two cones of negative norm vectors. We define one of them as the positive cone. Then we have\n\\begin{thp1}\nThe simple roots of the fake monster superalgebra are the norm zero vectors in the closure of the positive cone of $II_{9,1}$. Their multiplicities as even and odd roots are equal to 8. \n\\end{thp1}\n{\\em Proof:}\nLet $K$ be the generalized Kac-Moody superalgebra with Cartan subalgebra \n${\\mathbb C}\\otimes_{\\mathbb Z}II_{9,1}$ and simple roots as stated in the theorem. The simple roots of $G$ are determined by the root multiplicities because of the denominator formula. Hence it is sufficient to show that $G$ and $K$ have the same root multiplicities so that they are isomorphic. \nThe denominator identity (cf. \\cite{GN}) for $K$ is \n\\begin{equation} \\prod_{\\alpha\\in\\Delta_{+}}\n   \\frac{ (1-e(\\alpha))^{ \\mbox{mult}_0 \\, \\textstyle{\\alpha} } }\n        { (1+e(\\alpha))^{ \\mbox{mult}_1 \\, \\textstyle{\\alpha} } }\n        = e(\\rho) \\sum_{w\\in W} \\mbox{det}(w)w(T)  \\ee\nwhere $T$ is the sum \n\\[ T=e(-\\rho)\\sum \\epsilon(\\mu)e(\\mu) \\]\n with $\\mu$ running over all sums of imaginary simple roots and $\\epsilon(\\mu)=(-1)^n$ if $\\mu$ is the sum of $n$ pairwise perpendicular imaginary simple roots, which are distinct unless they are odd and of norm zero, and $\\epsilon(\\mu)=0$ else. Note that $e(\\alpha)$ and $e(\\mu)$ are elements of the group algebra of $II_{9,1}$. We can calculate the right hand side of the identity, i.e. the denominator function of $K$. Since $K$ has no real simple roots the Weyl group of $K$ is trivial. The Weyl vector is zero. Hence we only have to calculate $\\sum \\epsilon(\\mu)e(\\mu)$. $\\mu$ is the sum of pairwise perpendicular imaginary simple roots iff it is the multiple of a primitive norm zero vector in $II_{9,1}$ since two imaginary simple roots can only be are orthogonal if they are proportional. Taking care of the multiplicities of the simple roots as elements in the Cartan subalgebra we find\n\\[ \\sum \\epsilon(\\mu)e(\\mu) = 1 + \\sum a(\\lambda)e(\\lambda) \\]\nwhere $a(\\lambda)$ is the coefficient of $q^n$ of \n\\[ \\prod_{m\\geq 0} \\left( \\frac{1-q^m}{1+q^m} \\right)^8 =\n   1-16q+112q^2-448q^3+\\ldots \\]\nif $\\lambda$ is $n$ times a primitive norm zero vector in the closure of the positive cone and zero else.   \nBorcherds has shown (cf. \\cite{B5}, Example 13.7) that  \n\\[ 1 + \\sum a(\\lambda)e(\\lambda) = \\prod_{\\alpha\\in\\Delta_{+}}\n   \\frac{ (1-e(\\alpha))^{ c(-\\frac{1}{2}\\alpha^2) }}\n        { (1+e(\\alpha))^{ c(-\\frac{1}{2}\\alpha^2) }} \\]\nThis implies that the multiplicity of a root $\\alpha$ of $K$ is given by $c(-\\frac{1}{2}\\alpha^2)$. \\hspace*{\\fill} $\\Box$\\\\ \n\n\\begin{cp1}\nThe denominator identity of the fake monster superalgebra is \n\\[ \\prod_{\\alpha\\in\\Delta_{+}}\n   \\frac{ (1-e(\\alpha))^{ c(-\\frac{1}{2}\\alpha^2) }}\n        { (1+e(\\alpha))^{ c(-\\frac{1}{2}\\alpha^2) }} =\n    1 + \\sum a(\\lambda)e(\\lambda) \\]\nwhere $a(\\lambda)$ is the coefficient of $q^n$ of \n\\[ \\prod_{m\\geq 0} \\left( \\frac{1-q^m}{1+q^m} \\right)^8 =\n   1-16q+112q^2-448q^3+\\ldots \\]\nif $\\lambda$ is $n$ times a primitive norm zero vector in the closure of the positive cone and zero else. \n\\end{cp1}\n\nThe denominator function of the fake monster superalgebra is the right hand side of the denominator identity. It is an automorphic form of weight $4$ for a subgroup of $O_{10,2}({\\mathbb R})$ (cf. \\cite{B5}, Example 13.7). \n\nFinally we describe the construction of the fake monster superalgebra $G$ by generators and relations. Let $\\{ \\alpha_i \\,|\\, i\\in I\\}$ the set of simple roots of $G$. Taking $8$ even and $8$ odd copies of each $\\alpha_i$ we can calculate the Cartan matrix of $G$. Then we define as described in section 3 the superalgebra $\\hat{G}$ by generators and relations.\nThe subalgebra $\\hat{H}$ of $\\hat{G}$ is the direct sum of a $16^2=256$ dimensional space for each simple root $\\alpha_i$. \nThe center $C$ of $\\hat{G}$ has index $10$ in $\\hat{H}$. The fake monster superalgebra is the quotient of $\\hat{G}$ by $C$.\n\n\n\\section*{Acknowledgments}\n\nI thank R. E. Borcherds for stimulating discussions. Furthermore I want to thank N. Berkovits and the referee for useful comments. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nAn outstanding problem in stellar astrophysics concerns\nthe identification of physical processes responsible for the heating\nof outer stellar atmospheres and the acceleration of stellar winds\n(see reviews by \\citealt{nara90,nara96} and \\citealt{gued07}).\nFor the Sun and other types of stars with surface convection zones,\nacoustic heating has been identified as most likely responsible for\nbalancing the ``basal\" flux emission \\citep[e.g.,][]{buch98,cunt07}.\nOn the other hand, it is well known that most,\nif not all stars also exhibit a large amount of magnetic activity.\nThus, the chromospheres of main-sequence stars, including the Sun, are\nexpected to be significantly shaped by magnetically heated structure\n\\citep[e.g.,][]{saar94,schri96}.\n\nThere is a large body of previous work devoted to the description of\nthe two-component structure of stellar chromospheres.  In these models\nthe magnetic component of the chromosphere is typically heated by\nenergy dissipation of longitudinal flux tube waves.  \\cite{cunt99}\ncomputed two-component theoretical chromosphere\nmodels for K2~V stars with different levels of magnetic activity\nwith the filling factor for the magnetic component determined\nfrom an observational relationship between the measured magnetic\narea coverage and the stellar rotation period.  For stars with\nvery slow rotation, they were able to reproduce the basal flux\nlimit of chromospheric emission previously identified with\nnon-magnetic regions.  Most notably, however, \\cite{cunt99}\ndeduced a relationship between the Ca~II~H+K emission\nand the stellar rotation rate that is consistent with the\nrelationship previously obtained by observations; see also\n\\cite{cunt98} for earlier results.\n\nFurther studies for a large spectral range of stars were given\nby \\cite{fawz02} based on specified values for the\nmagnetic filling factor.  They concluded that heating by\nacoustic and longitudinal flux tube waves is able to explain\nmost of the observed range of chromospheric activity as gauged\nby the Ca~II and Mg~II lines.  On the other hand, indirect\nevidence for non-wave (i.e., reconnective) heating was also\ndeduced needed to explain the structure of the\nhighest layers of stellar chromospheres.\n\nThis type of models, as well as envisioned future models of\nchromospheric heating and emission, partially motivated by\nthe quest of investigating the effects of UV and EUV emission\non planetary atmosphere and (potentially) the evolution of life\n\\citep[e.g.,][]{guin03,lamm03,gued07,cunt10}, require the\ncontinuation of detailed simulations of magnetic wave energy\ngeneration, including studies on longitudinal tube waves in\ndifferent types of stars, particularly main-sequence stars.\nThis latter goal is the focus of the present paper.\n\nPrevious work on the calculation of longitudinal tube waves\nhas been based on progress made by \\cite{musi94} who\ncorrected the Lighthill-Stein theory by incorporating an\nimproved description of the spatial and temporal spectrum\nof the turbulent convection and utilized the corrected theory\nfor calculating revised stellar acoustic wave energy fluxes\n\\citep{ulms96,ulms99}.  This type of work focused\non the generation of acoustic waves; however, it did not consider \nstellar magnetic fields.  Considering the fundamental\nimportance of magnetic heating in most, if not all stars,\na set of papers focused on the study of longitudinal and\ntransverse tube wave generation has been pursued\n\\citep[e.g.][]{musi89,musi95,ulms98}.  \nIn subsequent work, \\cite{ulms01} used the\napproach developed by \\cite{ulms98} to compute\nthe wave energy fluxes carried by longitudinal tube waves\npropagating along thin and vertically oriented magnetic flux\ntubes that are embedded in atmospheres of late-type stars.\nThis numerical approach supplemented previous work by\n\\cite{musi00}, who analytically calculated the\nlongitudinal wave energy fluxes generated in\nstellar convective zones.\n\n\\begin{figure}\n    \\begin{minipage}[t]{0.50 \\textwidth}\n    \\includegraphics[width=0.999 \\textwidth]{FC_fig1.eps}\n    \\end{minipage}\n\\caption{Diagram of a flux tube embedded into a stellar convective zone.\nThe squeezing point of the tube is assumed to be located at optical depth\n$\\tau_{5000} = 1$, coinciding with the ``stellar surface\". Credit: P. Ulmschneider.\n\\label{fig1}}\n\\end{figure}\n\nIn the numerical approach by \\cite{ulms98},\nlongitudinal tube waves are generated as a result of the\nsqueezing of a thin, vertically oriented magnetic \nflux tube by external pressure fluctuations produced \nby the turbulent motions in a stellar photosphere and convection \nzone, which correspond to associated velocity fluctuations.\nHence, to compute the pressure fluctuations imposed \non the tube, it is required to know the external turbulent \nmotions.  The motions are modeled by specifying the rms \nvelocity amplitude and using an extended Kolmogorov turbulent\nenergy spectrum with a modified Gaussian frequency factor\n\\citep{musi94}.\n\nThe main advantage of this approach is\nthat it is not restricted to linear waves and that it\nallows for occasionally large-amplitude waves observed\non the Sun at the photospheric level\n\\citep[e.g.,][]{mull85,komm91,nesi93,mull94} and also seen in detailed\ntime-dependent simulations of solar and stellar convection\n\\citep[e.g.,][]{nord90,nord91,catt91,stef93,nord97}.\nHorizontal flow patterns are a notable candidate process \nfor the initiation of wave modes with respect to flux tubes\n(see Fig. 1); see \\cite{stei09a,stei09b} for recent models of\nconvective flows for the Sun based on up-to-date simulations\nextending toward the scale of supergranules.\n\nThe code used by \\cite{ulms98}\nwas originally developed by \\cite{herb85} who treated \nmagnetic flux tubes in the so-called thin flux tube \napproximation and described them mathematically by using a set \nof one-dimensional, time-dependent and nonlinear MHD equations.\nIt allows to compute the instantaneous and time-averaged \nlongitudinal tube wave energy fluxes and the corresponding wave \nenergy spectra.  It requires specifying the strength of the \nmagnetic field inside the flux tube and the height in the stellar \natmosphere where the squeezing is assumed to take place.\nThe code has previously been used to calculate  \nwave energy fluxes and spectra for longitudinal tube waves \npropagating in the solar atmosphere (see \\citealt{fawz98}\nfor models of different spreading factors), and to investigate the \ndependence of these fluxes on the magnetic field strength, the rms \nvelocity amplitude of turbulent motions, and the location of \nthe squeezing in the atmosphere; for recent models for\nother stars with non-solar metallicities see \\cite{fawz10}.\n\nThe reason for reinvestigating the generation of\nlongitudinal flux tube waves in main-sequence stars\nis three-fold.  First, we would like to use realistic\ncombinations of ($T_{\\rm eff}$, $\\log g$), with $T_{\\rm eff}$\nas stellar effective temperature and $\\log g$ as surface\ngravity, for main-sequence stars guided by up-to-date\nstudies by R.~L. Kurucz and D.~F. Gray.  Note that\n$\\log g$ is typically close to 4.5 (see Table 1).\nPrevious models by \\cite{ulms01} and others\nhave been pursued for either $\\log g = 4$ or $5$, thus\nresulting in unnecessary interpolation errors.\nSecondly, we would like to investigate the amount of\nupward propagating longitudinal wave energy flux\nfor a wider range of stellar convective and magnetic\nparameters, notably the mixing length $\\alpha$ amid\nrecent progress made through models by \\cite{stei09a,stei09b}\nand others.  Thirdly, we would like to deduce a\nfitting formula for the wave energy flux that\nallows insight into the role of the relevant parameters\nconcerning that flux and, furthermore, offers a\nmore universal use.\n\nOur paper is structured as follows:  In Sect. 2, we comment\non the parameters of theoretical main-sequence stars.\nAdditionally, we summarize the method for the computation\nof longitudinal tube waves as well as the construction of\nstellar flux tube models.  Our results are given in Sect. 3.\nFinally, in Sect. 4 we present the summary and conclusions.\n\n\\begin{table}\n\\caption{Theoretical main-sequence stars}\n\\centering\n\\vspace{0.05in}\n\\vspace{0.05in}\n\\begin{tabular}{l c c c c}\n\\hline\n\\hline\n\\noalign{\\vspace{0.03in}}\nSp. Type & $T_{\\rm eff}$ & $R$         & $M$         & $\\log~g$ \\\\\n...      & (K)           & ($R_\\odot$) & ($M_\\odot$) & ...      \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\n F0~V  &  7178 &    1.620  &   1.600  &   4.223 \\\\\n F1~V  &  7042 &    1.541  &   1.560  &   4.255 \\\\\n F2~V  &  6909 &    1.480  &   1.520  &   4.279 \\\\   \n F3~V  &  6780 &    1.453  &   1.480  &   4.283 \\\\   \n F4~V  &  6653 &    1.427  &   1.440  &   4.287 \\\\    \n F5~V  &  6528 &    1.400  &   1.400  &   4.292 \\\\    \n F6~V  &  6403 &    1.333  &   1.330  &   4.312 \\\\    \n F7~V  &  6280 &    1.267  &   1.260  &   4.333 \\\\    \n F8~V  &  6160 &    1.200  &   1.190  &   4.355 \\\\    \n F9~V  &  6047 &    1.155  &   1.120  &   4.362 \\\\    \n G0~V  &  5943 &    1.120  &   1.050  &   4.360 \\\\    \n G1~V  &  5872 &    1.100  &   1.022  &   4.364 \\\\    \n G2~V  &  5811 &    1.080  &   0.994  &   4.368 \\\\   \n G3~V  &  5760 &    1.037  &   0.967  &   4.392 \\\\   \n G4~V  &  5708 &    0.993  &   0.940  &   4.417 \\\\   \n G5~V  &  5657 &    0.950  &   0.914  &   4.443 \\\\   \n G6~V  &  5603 &    0.937  &   0.888  &   4.443 \\\\   \n G7~V  &  5546 &    0.923  &   0.863  &   4.443 \\\\   \n G8~V  &  5486 &    0.910  &   0.838  &   4.443 \\\\   \n G9~V  &  5388 &    0.870  &   0.814  &   4.469 \\\\   \n K0~V  &  5282 &    0.830  &   0.790  &   4.497 \\\\   \n K1~V  &  5169 &    0.790  &   0.766  &   4.527 \\\\   \n K2~V  &  5055 &    0.750  &   0.742  &   4.558 \\\\   \n K3~V  &  4973 &    0.730  &   0.718  &   4.567 \\\\   \n K4~V  &  4730 &    0.685  &   0.694  &   4.608 \\\\   \n K5~V  &  4487 &    0.640  &   0.670  &   4.651 \\\\   \n K6~V  &  4294 &    0.601  &   0.643  &   4.689 \\\\   \n K7~V  &  4133 &    0.565  &   0.614  &   4.722 \\\\   \n K8~V  &  4006 &    0.533  &   0.582  &   4.749 \\\\   \n K9~V  &  3911 &    0.505  &   0.547  &   4.770 \\\\   \n M0~V  &  3850 &    0.480  &   0.510  &   4.783 \\\\    \n\\noalign{\\vspace{0.03in}}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\section{Methods}\n\n\\subsection{Comments on the theoretical main-sequence stars}\n\nStellar parameters for theoretical main-sequence stars have been\ndeduced by \\cite{gray05}; see his Table B.1.  His values,\nnotably $T_{\\rm eff}$ and $\\log g$, serve as basis for the\npresent study.  We also improved the accuracy of the $\\log g$\nvalues if more accurate values for the stellar masses and\nstellar radii were given.  For stellar spectral types with\nno data given, we calculated those data using biparabolic\ninterpolation.  The stellar data are summarized in Table~1.\n\nAnother set of spectral models has been constructed by\nR.~L.~Kurucz and collaborators.  They take into account millions\nor hundred of millions of lines for a large array of atoms and molecules;\nsee, e.g., \\cite{cast04} and \\cite{kuru05} for technical details.\nThese models indicate very similar effective temperatures compared\nto the models by \\cite{gray05} for most types of stars.  However,\nstellar spectral types of K5~V and below, the indicated effective\ntemperatures of R.~L.~Kurucz are consistently lower noting that\nthe difference amounts to nearly 300~K for spectral type M0~V.\nTherefore, we assumed average values between the models by\nD.~F. Gray and R.~L.~Kurucz for stars of spectral spectral type\nK5~V and M0~V in the following.\n\n\\begin{figure}\n    \\begin{minipage}[t]{0.50 \\textwidth}\n    \\includegraphics[width=0.999 \\textwidth]{FC_fig2.eps}\n    \\end{minipage}\n\\caption{Root mean square turbulent velocities at the squeezing point ($\\tau_{5000}=1$)\nfor the set of theoretical main-sequence stars for different values of the\nmixing-length parameter $\\alpha$.\n\\label{fig2}}\n\\end{figure}\n\n\\begin{figure}\n    \\begin{minipage}[t]{0.50 \\textwidth}\n    \\includegraphics[width=0.999 \\textwidth]{FC_fig3.eps}\n    \\end{minipage}\n\\caption{\nPower spectra of the instantaneous wave energy flux in\nflux tubes with $B\/B_{\\rm eq} = 0.85$ for\nF5~V, G5~V, K5~V, and M0~V stars (from top to bottom)\nas a function of circular frequency $\\omega$.  The\nmixing-length parameter is assumed as $\\alpha = 2$.\n\\label{fig3}}\n\\end{figure}\n\n\\begin{table}\n\\caption{Wave energy flux$^a$ for different parameters $\\alpha$ and $\\eta$}\n\\centering\n\\vspace{0.05in}\n\\vspace{0.05in}\n\\begin{tabular}{l c c c c}\n\\hline\n\\hline\n\\noalign{\\vspace{0.03in}}\nSp.Type & $T_{\\rm eff}$ & $F_{\\rm LTW}$ & $F_{\\rm LTW}$ & $F_{\\rm LTW}$ \\\\\n...     & (K)           & ...           & ...           & ...           \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\n... & ... & $\\eta = 0.75$ & $\\eta = 0.85$ & $\\eta = 0.95$ \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\n\\multicolumn{2}{c}{$\\alpha = 1.5$} & & \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\nF5~V\t&  6528  &\t6.03E8  &  4.05E8  &  9.41E7  \\\\\nF8~V  &  6160  &  5.59E8  &  3.10E8  &  9.14E7  \\\\\nG0~V\t&  5943  &\t4.75E8  &  2.94E8  &  7.91E7  \\\\\nG2~V\t&  5811  &\t4.50E8  &  2.62E8  &  7.91E7  \\\\\nG5~V\t&  5657  &\t3.56E8  &  2.20E8  &  6.53E7  \\\\\nG8~V\t&  5486  &  3.27E8  &  1.98E8  &  5.85E7  \\\\\nK0~V  &  5282  &\t2.54E8  &  1.56E8  &  4.77E7  \\\\\nK2~V\t&  5055  &\t1.92E8  &  1.10E8  &  3.42E7  \\\\\nK5~V\t&  4487  &\t1.14E8  &  6.83E7  &  2.12E7  \\\\\nK8~V\t&  4006  &\t4.47E6  &  2.87E6  &  9.82E5  \\\\\nM0~V\t&  3850  &\t2.26E6  &  1.33E6  &  5.36E5  \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\n\\multicolumn{2}{c}{$\\alpha = 1.8$} & & \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\nF5~V\t&  6528  &\t8.99E8  & 5.00E8   &  1.25E8  \\\\\nF8~V  &  6160  &  7.51E8  & 4.62E8   &  1.15E8  \\\\\nG0~V\t&  5943  &\t5.69E8  & 3.40E8   &  9.73E7  \\\\\nG2~V\t&  5811  &\t5.63E8  & 3.02E8   &  9.05E7  \\\\\nG5~V\t&  5657  &\t5.29E8  & 2.78E8   &  7.96E7  \\\\\nG8~V\t&  5486  &  3.98E8  & 2.37E8   &  7.32E7  \\\\\nK0~V  &  5282  &\t2.93E8  & 1.82E8   &  5.60E7  \\\\\nK2~V\t&  5055  &\t2.69E8  & 1.63E8   &  4.72E7  \\\\\nK5~V\t&  4487  &\t1.30E8  & 8.15E7   &  2.50E7  \\\\\nK8~V\t&  4006  &\t5.36E6  & 3.50E6   &  1.24E6  \\\\\nM0~V\t&  3850  &\t2.28E6  & 1.44E6   &  6.49E5  \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\n\\multicolumn{2}{c}{$\\alpha = 2.0$} & & \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\nF5~V\t&  6528  &\t1.12E9  &  6.07E8  &  1.47E8  \\\\\nF8~V  &  6160  &\t8.51E8  &  4.92E8  &  1.34E8  \\\\\nG0~V\t&  5943  &\t6.77E8  &  3.96E8  &  1.11E8  \\\\\nG2~V\t&  5811  &\t6.35E8  &  3.65E8  &  1.06E8  \\\\\nG5~V\t&  5657  &\t5.81E8  &  3.35E8  &  1.01E8  \\\\\nG8~V\t&  5486  &\t4.79E8  &  2.86E8  &  8.05E7  \\\\\nK0~V  &  5282  &\t3.39E8  &  1.95E8  &  6.39E7  \\\\\nK2~V\t&  5055  &\t2.98E8  &  1.86E8  &  5.81E7  \\\\\nK5~V\t&  4487  &\t1.44E8  &  9.19E7  &  2.86E7  \\\\\nK8~V\t&  4006  &\t6.79E6  &  4.37E6  &  1.57E6  \\\\\nM0~V\t&  3850  &\t3.32E6  &  2.11E6  &  7.62E5  \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\end{tabular}\n\\vspace{0.05in}\n\\begin{list}{}{}\n\\item[]$^a$The unit of $F_{\\rm LTW}$ is erg~cm$^{-2}$~s$^{-1}$.\n\\end{list}\n\\end{table}\n\n\\subsection{Convective zone models and turbulent velocities}\n\nThe method for calculating wave energy fluxes carried by \nlongitudinal tube waves adopted in the present paper has been\ndescribed in detail by \\cite{ulms01}.\nThus, it is not necessary to present an intricate discussion in\nthe following.  In the solar application, it is possible to\nselect many model parameters and characteristic values directly\nfrom observations.  However, for stars other than the Sun such\ndata are mostly unavailable.   Therefore, we need to discuss in\nsome detail the physical reasoning behind our choice of relevant\nparameters used in our calculations.\n\nIn the current approach, the magnetic flux tubes are embedded\nin nonmagnetized photospheric convection zones (see Fig.~1).\nConsidering that the\ninteraction between the flux tubes and the convective\nturbulence is the driving mechanism for the generation of\nlongitudinal tube waves, among other waves, models of the\nstellar convection zones are required.  Guided by previous\nstudies, it is assumed that the squeezing of the tube\nis symmetric with respect to the tube axis.  \nThe computed pressure fluctuations are subsequently\ntranslated into gas pressure and magnetic field fluctuations\ninside the tube assuming horizontal pressure balance.\nFinally, the internal velocity perturbation resulting\nfrom the internal pressure fluctuation is calculated. \nThis internal velocity served as a boundary condition \nin the numerical simulation of the generation of the\nlongitudinal tube waves. \n\nBoth numerical simulations of stellar convection and mixing length \nmodels show that the maximum convective velocities occur at optical \ndepths of $\\tau_{5000}\\approx 10$ to 100.  For example, \\cite{stef93}\nfound in his time-dependent solar numerical convection calculations \nthat maximum convective velocities $v_{\\rm CMax} \\simeq 2.8$ km~s$^{-1}$\nare reached at $\\tau_{5000} \\approx 50$ and that these values can be \nreproduced via mixing length theory with a mixing length parameter \nof $\\alpha \\simeq 2$.  The value $\\alpha = 2$ is furthermore\nindicated by time-dependent hydrodynamic simulations of stellar convection\nfor stars other than the Sun \\citep{tram97} as well as by a \ncareful fitting of evolutionary tracks of the Sun with its present \nluminosity, effective temperature and age \\citep{schro96}.  \n\nNevertheless, there is still some debate about the most appropriate\nvalue of $\\alpha$.  \\cite{nord90} originally pursued\ndetailed numerical simulations based on 3-D hydrodynamics coupled with\n3-D non-grey radiative transfer for stars similar to Procyon (F5~IV-V),\n$\\alpha$ Cen~A (G2~V), $\\beta$~Hyi (G2~IV), and $\\alpha$ Cen~B (K1~V).\nThey concluded a mixing-length parameter of $\\alpha = 1.5$ (or\nslightly higher), although the mixing length concept appeared to be\nproblematic at photospheric heights.  A mixing length of 1.5 was also\nused by \\cite{cunt99} in their two-component theoretical chromosphere\nmodels for K2~V stars with different levels of magnetic activity.\nEven though the deduced relationship between the Ca~II~H+K emission\nand the stellar rotation rate was found to be largely consistent with the\nobserved relationship, the agreement could probably be improved if\na somewhat higher longitudinal wave energy flux, corresponding to a\nslightly larger mixing length parameter, was adopted.  Recently,\n\\cite{stei09a,stei09b} pursued updated state-of-the-art simulations\nof solar convection zone extending toward the scale of supergranules\nindicating a mixing length parameter of $\\alpha \\simeq 1.8$.\nFor these reasons, we will calculate a set of models concerning\nwave energy generation of longitudinal tube wave for a set of\n$\\alpha$ values, which are $\\alpha$ = 1.5, 1.8, and 2.0.\n\n\\subsection{Computation of stellar magnetic flux tube models}\n\nOur treatment of stellar convection associated with the facilitation\nof stellar flux tube models is akin to that described by\n\\cite{ulms96}.  In this approach, information is needed\nabout velocities of turbulent motions in the overshooting layer near\nthe stellar surface, where the squeezing of the magnetic flux tube\nis assumed to occur.  \nSteffen's numerical calculations show that the rms velocities decrease \ntoward the solar surface and reach a plateau in the overshooting layer.  \nBetween $\\tau_{5000}=1$ and $10^{-4}$, he finds values of $v_{\\rm rms} = \n1.4$ km~s$^{-1}$, which are essentially independent of height.\n\\cite{ulms98} adopted for the Sun a variety of observed rms velocity \namplitudes $u_t$ in the range $0.9<u_t<1.9$ km~s$^{-1}$, and showed the \ndependence of the computed fluxes on this velocity.  \nFor stars, these velocities cannot be determined from observations; thus,\nwe follow \\cite{ulms01} by assuming that the rms velocity\nfluctuations at the squeezing points are given by $u_t = v_{\\rm CMax}\/2$.\nThe values of $v_{\\rm t}$ and\n$v_{\\rm CMax}$ (see Fig.~2) are evaluated from stellar convection zone\nmodels based on the adopted mixing length parameter.\nThe numerical factor 2 used in our calculations ensures that\nin our approach the considered convective velocities are always\nlower than the local speed of sound.\n\nAfter specifying the rms velocities that are responsible for the \nwave generation, we also determine the height in stellar \natmospheres, where the most efficient squeezing of magnetic flux tubes \ntakes place.  We are guided by earlier studies performed by\n\\cite{ulms98}, who pointed out that shifting the height of the\nexcitation point did not change much the resulting wave \nenergy fluxes for the Sun.  Based on these results, we take the \nsqueezing point to be located at optical depth $\\tau_{5000} = 1$ for \nall considered stars.  This depth is commonly taken as the zero \nheight level in stellar atmosphere computations.\n\n\n\\begin{table*}\n\\caption{Wave energy flux$^a$ for different values of $\\eta$ based on $\\alpha = 2$} \n\\centering\n\\vspace{0.05in}\n\\vspace{0.05in}\n\\begin{tabular}{l c c c c c c c c c c}\n\\hline\n\\hline\n\\noalign{\\vspace{0.03in}}\nSp.Type & $T_{\\rm eff}$ & $F$(tot) & $F$(prop) & $F$(prop+up) & $F$(tot) & $F$(prop) & $F$(prop+up) & $F$(tot) & $F$(prop) & $F$(prop+up) \\\\\n...     & (K)           & ...      & ...       & ...          & ...      & ...       & ...          & ...      & ...       & ...          \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\n... & ... & \\multicolumn{3}{c}{$\\eta = 0.75$} & \\multicolumn{3}{c}{$\\eta = 0.85$} & \\multicolumn{3}{c}{$\\eta = 0.95$} \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\nF2~V\t&  6909  &   ...   &   ...   &   ...   &  5.18E8 &  4.37E8 &\t6.26E8 & 1.17E8 &\t1.01E8 &\t1.41E8 \\\\\nF5~V\t&  6528  &\t8.23E8 &  6.97E8 &  1.12E9 &  5.04E8 &  4.13E8 &\t6.07E8 & 1.27E8 &\t1.08E8 &\t1.47E8 \\\\\nF8~V  &  6160  &\t6.76E8 &  6.02E8 &  8.51E8 &  4.10E8 &  3.59E8 &\t4.92E8 & 1.09E8 &\t9.48E7 &\t1.34E8 \\\\\nG0~V\t&  5943  &\t5.45E8 &  4.71E8 &  6.77E8 &  3.22E8 &  2.76E8 &\t3.96E8 & 9.75E7 &\t8.11E7 &\t1.11E8 \\\\\nG2~V\t&  5811  &\t4.91E8 &  4.26E8 &  6.35E8 &  2.96E8 &  2.53E8 &\t3.65E8 & 8.70E7 &\t7.56E7 &\t1.06E8 \\\\\nG5~V\t&  5657  &\t4.75E8 &  3.93E8 &  5.81E8 &  2.73E8 &  2.32E8 &\t3.35E8 & 8.45E7 &\t7.22E7 &\t1.01E8 \\\\\nG8~V\t&  5486  &\t3.86E8 &  3.29E8 &  4.79E8 &  2.38E8 &  2.02E8 &\t2.86E8 & 6.76E7 &\t5.87E7 &\t8.05E7 \\\\\nK0~V  &  5282  &\t2.95E8 &  2.30E8 &  3.39E8 &  1.66E8 &  1.39E8 &\t1.95E8 & 5.47E7 &\t4.68E7 &\t6.39E7 \\\\\nK2~V\t&  5055  &\t2.52E8 &  2.08E8 &  2.98E8 &  1.55E8 &  1.26E8 &\t1.86E8 & 4.86E7 &\t3.90E7 &\t5.81E7 \\\\\nK5~V\t&  4487  &\t1.29E8 &  8.93E7 &  1.44E8 &  8.08E7 &  5.93E7 &\t9.19E7 & 2.53E7 &\t1.83E7 &\t2.86E7 \\\\\nK8~V\t&  4006  &\t5.55E6 &  3.36E6 &  6.79E6 &  3.74E6 &  2.14E6 &\t4.37E6 & 1.25E6 &\t6.95E5 &\t1.57E6 \\\\\nM0~V\t&  3850  &\t2.29E6 &  1.36E6 &  3.32E6 &  1.57E6 &  8.23E5 &\t2.11E6 & 5.37E5 &\t2.86E5 &\t7.62E5 \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\end{tabular}\n\\vspace{0.05in}\n\\begin{list}{}{}\n\\item[]$^a$The unit of $F$(tot), $F$(prop), and $F$(prop+up) is erg~cm$^{-2}$~s$^{-1}$.  Note that \n$F$(prop+up) $\\equiv$ $F_{\\rm LTW}$.\n\\end{list}\n\\end{table*}\n\nFor the computation of stellar magnetic flux tubes we consider\nthe ``thin flux tube approximation\" \\citep[e.g.,][]{spru81}.\nTwo dimensional modeling of magnetic flux tubes by \\cite{hasa03}\nindicates that this approach renders reasonable results\nif the models do not extend beyond a small number of scale heights above\nthe stellar surface; note that this approximation is fully compatible\nwith our study.  In the following,\nwe augment the commonly used concept of solar magnetic flux tubes\n\\citep[e.g.,][]{sten78,sola93} to stellar flux tubes; see\n\\cite{sola96} for further discussion.  Stellar magnetic flux tubes\nat the stellar surface are assumed to have diameters of roughly equal\nto the local pressure scale height like in case of the Sun.\n\nFor the Sun, the magnetic flux tubes within the solar photosphere\nhave field strengths on the order $B_0 = 1500$~G \\citep[e.g.,][]{sola93}.\nTaking $p_e=1.17\\cdot 10^5\\ \\rm dyn\\ cm^{-2}$ from model C of\n\\cite{vern81} at the height $z=0$ where $\\tau_{5000}=1$ yields an\nequipartition field strength ($B_{\\rm eq}^2\/8\\pi = p_e$) of $B_{\\rm eq}=1716$~G.\nThis corresponds to a ratio $B\/B_{\\rm eq}=0.875$, which may or may not be\ntypical for stellar flux tubes.  Since this ratio is likely\nto vary even on the Sun \\citep[e.g.,][]{schri00}, it is appropriate\nto consider a range of that ratio for the sake of comprehensiveness.\nTherefore, we take $B\/B_{\\rm eq} = 0.75$, 0.85, and 0.95 for\nour stellar flux tube models.  This allows us to deduce an appropriate\nset of energy fluxes for upward propagating longitudinal tube waves for\nspecified values of $\\alpha = 1.5$, 1.8, and 2.0, resulting in\na total of 9 models per theoretical target star.\n\n\\begin{figure}\n    \\begin{minipage}[t]{0.50 \\textwidth}\n    \\includegraphics[width=0.999 \\textwidth]{FC_fig4a.eps}\n    \\includegraphics[width=0.999 \\textwidth]{FC_fig4b.eps}\n    \\includegraphics[width=0.999 \\textwidth]{FC_fig4c.eps}\n    \\end{minipage}\n\\caption{\nDifferent types of energy fluxes (see text) for the set of theoretical main-sequence\nstars for $\\alpha = 2$.  Results are given for $B\/B_{\\rm eq} = 0.75$ (top),\n0.85 (middle), and 0.95 (bottom).\n\\label{fig4}}\n\\end{figure}\n\n\\begin{figure}\n    \\begin{minipage}[t]{0.50 \\textwidth}\n    \\includegraphics[width=0.999 \\textwidth]{FC_fig5.eps}\n    \\end{minipage}\n\\caption{\nUpward propagating energy fluxes, also referred to as $F_{\\rm LTW}$,\nfor the set of theoretical main-sequence stars for $\\alpha = 2$ and\ndifferent values of $\\eta = B\/B_{\\rm eq}$.\n\\label{fig5}}\n\\end{figure}\n\n\\section{Results and discussion}\n\n\\subsection{Computation of wave energy fluxes}\n\nThe instantaneous and time-averaged tube wave energy fluxes are computed\nby employing a modified time-dependent wave code based on an earlier version\nby \\cite{herb85}.  The external turbulent motion is translated into\ninternal pressure and velocity fluctuations.  The time-averaged wave energy\nfluxes at the squeezing point are computed for each combination of\neffective temperature $T_{\\rm eff}$, surface gravity $\\log g$,\n$\\eta = B\/B_{\\rm eq}$ and mixing-length parameter $\\alpha$ for flux tubes\nembedded in the atmospheres of the main-sequence stars as considered.\n\nFor the evaluation of reliable time-averaged wave energy fluxes and due to the\nspiky nature of the instantaneous wave energy fluxes, the wave propagation code\nhas to be run over a time of about $35 P_{D}$ with $P_{D}$ as the wave period\nof the Defouw cut-off frequency \\citep{defo76}.  The generated waves include\nboth propagating and non-propagating wave energy fluxes.  We apply a high pass\nfilter at the Defouw cut-off frequency, $\\omega_{D}$, on the velocity and pressure\nfluctuations inside the flux tubes to separate the propagating waves and \nto compute the time-averaged upward propagating wave components.\nThe position of the filter is indicated by the low frequency cut-off depicted\nin Fig.~3, which features a comparison of the\npower spectra for different types of stars.  Note that the power spectra\nreach their maxima at roughly $2 \\omega_{D}$ and decrease toward\nhigher frequencies.\n\nFigure~4 and 5 as well as Table~2 and 3 show the computed wave energy\nof the magnetic flux tube. Actually, Table 3 for the case of $\\alpha = 2$\nalso distinguishes between the total energy flux, propagating energy flux\nand upward propagating energy flux; see \\cite{ulms01} for detailed definitions.\nThe wave energy flux $F_{\\rm LTW}$ identified as upward propagating energy\nflux shows a characteristic behaviour as function of the model parameters,\nnotably the stellar spectral type (or $T_{\\rm eff}$).  It is found that\nthe value of $F_{\\rm LTW}$ decreases by about a factor of 1.7 between G0~V\nand K0~V stars, and drops by almost two orders of magnitude between K0~V\nand M0~V stars.  This is a direct consequence of the action of stellar\nconvection, noting that in relatively hot main-sequence stars there is an\nenhanced efficiency in the creation of turbulence, reflected in higher\nvalues of the rms velocity $u_t$ (see Fig.~2).\n\nAdditionally, we also found that the wave energy flux $F_{\\rm LTW}$ shows\nsignificant dependence on the magnetic field strengths inside of the\ntubes, given as $\\eta = B\/B_{\\rm eq}$ as well as some minor dependence\non the mixing-length parameter $\\alpha$.  In fact, the wave energy\nfluxes are found to increase with decreasing magnetic field strengths\ninside the flux tube, a phenomenon that can be explained by the decreasing\nstiffness of the magnetic tube; see Fig.~5 for details.  This behaviour\nwas previously also pointed out by \\cite{ulms01}.\n\nMoreover, it is found that the wave energy flux for upward propagating waves\nis also somewhat sensitive to the choice of $\\alpha$ concerning the stellar\nconvective zone.  Note that an increase of the mixing length from $\\alpha$ = 1.5\nto 2.0 amplifies the simulated longitudinal wave energy fluxes by a factor of\napproximately 1.45, which is most notable for relatively hot main-sequence stars.\nThis can be explained by the fact that for higher values of $\\alpha$, the convective\nzones are more efficient concerning the generation of turbulence, thus resulting\nin thus higher values of the rms velocity $u_t$ (see Fig.~2).\n\n\\subsection{Derivation of fitting formulas}\n\nThe amount of data for $F_{\\rm LTW}$ for stars of different spectral types \nand the various choices for $\\eta$ and the mixing-length parameter $\\alpha$\nare a strong motivation for the derivation of a fitting formulas.  Our main focus\nare stars between $T_{\\rm eff} = 6000$~K (i.e., spectral type F9.5~V) and\nstars of spectral-type mid-K.  Stars below mid-K are increasingly dominated\nby flare activity or other processes as, e.g., magnetic reconnection\n\\citep[e.g.,][]{nara96}, which makes the existence of an accurate\nfitting formula less urgent.  Nevertheless, we extended our\nfitting formula toward stars of spectral type M0~V (with $T_{\\rm eff} =\n3850$~K) while reproducing the wave energy flux of M0~V stars with a\nprecision of 10$^{-3}$.\n\nHowever, no attempt has been made to accurately\nreproduce the knee in the wave energy flux function near F8~V as this\nwould have led to an additional complication in our fitting formula without\nobvious merits.  This approach is motivated by the finding that energy\ndissipation by longitudinal tube waves appears to be less important in\ndwarfs stars of spectal type mid-K to M compared to, e.g., flare heating\nas pointed out by \\cite{fawz02} in their comparison between empirical\nand theoretical radiative chromospheric emission losses.\n\nOur formula is given as follows:\n\\begin{equation}\nF_{\\rm LTW} \\ = \\ 2.55 \\cdot 10^8 \\\n       T_{\\rm rel}^\\epsilon \\ e^{{\\mu} T_{\\rm dif}} \\ Z(\\alpha,\\eta)  \\ \\\n       {\\rm erg~cm}^{-2}~{\\rm s}^{-1} \n\\end{equation}\n\\noindent\nwith $T_{\\rm rel} = T_{\\rm eff} \/ T_{{\\rm eff},\\odot}$ noting that\n$T_{{\\rm eff},\\odot} = 5777$~K.  In addition, $Z(\\alpha,\\eta)$ is\ngiven as\n\\begin{equation}\nZ(\\alpha,\\eta) \\ = \\ \\alpha^{1.25} \\cdot \\Bigl({\\eta \\over 0.75}\\Bigr)^\\gamma\n\\end{equation}\nwith $\\eta = B\/B_{\\rm eq}$ and $\\gamma = - 4.5 - 30 \\cdot \\vert \\eta - 0.85 \\vert$.\nFurthermore, $T_{\\rm dif}$ is defined as Min$(0,T_{\\rm eff} - 4500$~K),\nwhich means that $T_{\\rm dif}$ is zero for $T_{\\rm eff} \\ge 4500~{\\rm K}$\nand takes a negative value otherwise.  Note that the parameters\n$\\epsilon$ and $\\mu$ of Eq.~(1) weakly depend on $\\alpha$ and $\\eta$;\nsee Table~4 and 5 for detailed information.  If a reduced level of\naccuracy is permitted, it might be appropriate to use average values for\n$\\alpha$ and $\\mu$, given as $\\alpha = 5.21$ and $\\mu = 4.66 \\cdot 10^{-3}$,\nrespectively.\n\nOur formula for the upward propagating longitudinal wave energy flux has\nbeen subjected to thorough testing for stars of spectral type F9.5~V, G2~V,\nG5~V, G8~V, K0~V, K2~V, and K5~V.  The simulated data for the F9.5~V star\nwere obtained via logarithmic interpolation between the data for the F8~V\nand G0~V stars.  Detailed information on the tests is given in Appendix A.\n\n\\begin{table}\n\\caption{Data for the parameter $\\epsilon$}\n\\centering\n\\vspace{0.05in}\n\\vspace{0.05in}\n\\begin{tabular}{l c c c}\n\\hline\n\\hline\n\\noalign{\\vspace{0.03in}}\n$\\alpha$ & $\\eta = 0.75$ & $\\eta = 0.85$ & $\\eta = 0.95$ \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\n 1.5  & 5.24 & 5.30 & 4.95  \\\\\n 1.8  & 5.51 & 5.11 & 4.87  \\\\\n 2.0  & 5.67 & 5.31 & 4.90  \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Data for the parameter $\\mu$}\n\\centering\n\\vspace{0.05in}\n\\vspace{0.05in}\n\\begin{tabular}{l c c c}\n\\hline\n\\hline\n\\noalign{\\vspace{0.03in}}\n$\\alpha$ & $\\eta = 0.75$ & $\\eta = 0.85$ & $\\eta = 0.95$ \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\noalign{\\vspace{0.03in}}\n 1.5  &  4.78E-3 &  4.69E-3 &  4.45E-3 \\\\\n 1.8  &  4.95E-3 &  5.04E-3 &  4.55E-3 \\\\\n 2.0  &  4.47E-3 &  4.53E-3 &  4.49E-3 \\\\\n\\noalign{\\vspace{0.03in}}\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\section{Summary and conclusions}\n\nWe studied the generation of longitudinal waves in stellar magnetic flux\ntubes of theoretical main-sequence stars.  Our results are commensurate\nwith those obtained from previous studies, especially the work by\n\\cite{ulms01}.  Our investigations show that through\nnonlinear time-dependent responses of stellar magnetic flux tubes to\ncontinuous and impulsive external turbulent pressure fluctuations,\nlongitudinal tube waves were effectively produced via dipole emission. \nFurthermore, the shapes of the computed power spectra were found\nto be similar for stars of different effective temperature.  Moreover,\nthe longitudinal wave energy fluxes are found to increase with\nhigher effective temperature, i.e., stars of earlier spectral types.\n\nAs part of our study, we investigated the role of the magnetic field\nstrength inside of the tube $B$ as well as that of the adopted\nconvective model characterized by the mixing-length parameter $\\alpha$\nconcerning the generated wave energy flux.  We found that the\ncomputed wave energy flux strongly depends on the strength of the\nmagnetic field as already discussed in previous studies\n\\citep[e.g.,][]{ulms98,ulms01}.\nFor a given spectral type, the flux is considerably higher in tubes\nwith a field strength of $B\/B_{\\rm eq} = 0.75$ compared\nto $B\/B_{\\rm eq} = 0.95$, although the difference as a function of\nspectral type is not as large as previously pointed out by\n\\cite{ulms01} owing to the differences in stellar surface gravity\nfor the different types of stars.  Note that the difference concerning\nthe magnetic field strength of $B\/B_{\\rm eq} = 0.75$ and 0.95 is found\nto be a factor of 7.6, 6.1, 5.3, and 4.4 (for $\\alpha = 2$) for stars\nof spectral type F5~V, G0~V, K0~V, and M0~V, respectively.  This\ndifference exhibits a noticeable, albeit little dependence on the\nmixing-length parameter $\\alpha$ for stars hotter than G2~V,\nrooted in the behaviour of the adopted root mean square velocity at\nthe squeezing point of the tube.\n\nAnother aspect of our study was to consider a limited range of\nthe mixing-length parameter $\\alpha$, which are 1.5, 1.8, and 2.0.\nIt was found that an increase of the mixing length from $\\alpha$ = 1.5\nto 2.0 enhances the computed energy fluxes by a factor of about 1.45,\ncorresponding to a proportionality of $\\alpha^{1.25}$.\nThis relationship can be compared with previous findings for\nacoustic energy generation, which show a dependence such as\n$\\alpha^{2.8}$ \\citep{bohn84} or $\\alpha^{3.8}$ in updated models\nby \\cite{musi94}.  The relatively weak influence of\n$\\alpha$ on the amount of upward propagating wave energy flux\n$F_{\\rm LTW}$ is apparently due to the fact that the latter is\nlargely controlled by magnetic processes, as also evidenced by\nthe strong influence of $\\eta$ up to $\\eta^{-7.5}$,\nrather than controlled by convective processes, even though\nthe latter are key for the excitement of the tubes subsequently\nresulting in magnetic wave generation.\n\nNote that the strong dependence of the generated wave energy fluxes\non the stellar and magnetic parameters is in general agreement\nwith the findings from previous studies, although some noticeable\ndifferences are attained.  The wave energy flux $F_{\\rm LTW}$\ndecreases by about a factor of 1.7 between the G0~V and K0~V stars, and\ndrops by almost two orders of magnitude between the K0~V and M0~V stars.\nFor $\\alpha = 2$ at a fixed value of $T_{\\rm eff} = 5000$~K,\n\\cite{ulms01} deduced a fitting formula for the behaviour\nof $F_{\\rm LTW}$ as function of $B\/B_{\\rm eq}$, which is found to be\ncommensurate with the results obtained in our current study.  On the\nother hand, the fitting formula for the wave energy flux given in our\npaper is more general than any of the previously deduced formulas\nas it is applicable to a large range of stellar effective temperatures\nand furthermore allows insight into the role of the governing magnetic\nand convective parameters concerning the amount of generated upward\npropagating wave energy flux.  Therefore, it is of interest to future solar\nand stellar physics studies as it allows flexibility both concerning the\nmixing-length parameter $\\alpha$ and the magnetic parameter\n$\\eta = B\/B_{\\rm eq}$.  Note that even for the Sun, $\\eta$ is expected\nto exhibit considerable spatial and temporal fluctuations across the\nsurface as implied by previous studies of solar physics research; see,\ne.g., \\cite{schri00} for background information.\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStudies of parton saturation and the unitarization of high energy amplitudes began more than 20 years ago with the work of Gribov, Levin and Ryskin \\cite{GLR} and has now evolved into a very active field of research \\cite{reviews}. The simplest modern QCD evolution equation that describes parton saturation in the high energy limit is the Balitsky--Kovchegov (BK) equation~\\cite{Kovchegov1,Balitsky}. The BK equation is a ``mean-field'' approximation which neglects higher order correlators present in the so-called Balitsky \\cite{Balitsky} and JIMWLK equations (see \\cite{reviews} and references therein). Nevertheless, because the corrections neglected in the mean field approximation are likely small at phenomenologically interesting energies \\cite{EGM}, it is expected to be accurate for dipole scattering on large nuclei and at least a phenomenological model for saturation in scattering on protons. The equation has the correct high-energy limit at small parton densities, when it reduces to the BFKL equation \\cite{BFKL}, but it has a non-linear term which reduces the power-like growth with energy.\n\nThe BK equation is derived at leading logarithmic (LL) accuracy and thus reduces to the LL BFKL equation in the linear regime. The next-to-leading logarithmic (NLL) corrections to the BFKL equation were computed in \\cite{NLLBFKL} and were found to be negative and very large. As the BFKL kernel is a central part of the BK equation, it is important to understand the effect of the NLL corrections on the solutions of the BK equation.\n\nIn fact the NLL BFKL kernel is larger than the leading order kernel, and for any reasonable value of $\\alpha_s$ it is negative; the pomeron intercept becomes less than one for $\\alpha_s\\gtrsim 0.15$. The kernel also has two complex conjugate saddle points which lead to oscillating cross sections. The perturbative expansion of the BFKL kernel is therefore highly unstable and far from converging. These problems are discussed in detail in \\cite{NLLproblems}.\n\nA cure was proposed by Ciafaloni, Colferai, and Salam \\cite{CCS}, who realized that the large corrections come from collinear contributions of the DGLAP type. They devised the so-called $\\omega$-resummation, which makes the NLL kernel compatible with renormalization group (RG) requirements through matching to the DGLAP limit and resummation of spurious poles. A somewhat different scheme has been proposed in \\cite{ABF}. Many groups have also studied ways of including some of the most important NLL effects in the LL kernel \\cite{cconstraint,rapveto,Khoze:2004hx}. A common feature of these schemes is that they modify the functional dependence of the Mellin space BFKL kernel $\\chi(\\gamma)$ so that it acquires a dependence also on the $\\omega$ variable. This will be discussed at length below. In a slight abuse of terminology I will henceforth refer to all these schemes as ``RG improved'' kernels.\n\nAnother important part of the NLL corrections is the running of the QCD coupling. It is, however, not clear how to incorporate the effect of the running coupling in the BK and Balitsky--JIMWLK equations, and recent studies have come to different conclusions \\cite{running}. It is also reasonable, but by no means obvious, to assume that the NLL corrections to the BFKL part of the BK equation will be the same as for the BFKL equation. Since the way to include the running coupling is unclear it is therefore useful to study the BK equation with NLL corrections to the BFKL kernel but with a fixed coupling.\n\nThe onset of saturation corrections is governed by the saturation scale $Q_s$---the momentum that separates the regions dominated by linear evolution and saturation. Because of the non-linear term in the BK equation it is difficult to obtain analytical solutions, but at large momenta the amplitude is small enough that one may consider the linear equation only (the BFKL equation) together with appropriate boundary conditions. This has been used in a series of papers \\cite{Mueller:1999wm,IancuItakuraMcLerran,Dionysis1,Dionysis2} to compute the rapidity dependence of $Q_s=Q_s(Y)$. Recently the method of traveling waves has been developed~\\cite{MP1,MP2,MP3}, which allows performing these calculations in a more systematic way.\nAsymptotically the saturation scale increases exponentially as rapidity increases \\cite{Mueller:1999wm,IancuItakuraMcLerran}, but, as we will discuss below, there is also a factor with a sub-asymptotic $Y$-dependence which slows down the increase \\cite{Dionysis1,Dionysis2,MP1}.\n\nSaturation is intimately related to geometric scaling, which was discovered in the HERA data in Ref.\\ \\cite{geometricscaling}. It refers to a scaling of the $\\gamma^* p$ cross section for small Bjorken-$x$, where the cross section $\\sigma^{\\gamma^* p}(x,Q^2)$ is not a function of $x$ and $Q^2$ separately, but only through the combination $\\tau=Q^2\/Q_s^2(x)$.\nAs we shall see, the BK equation predicts geometric scaling of the amplitude and a specific pattern of violation of this scaling.\n\nIn this paper I make an analytical study using the method of traveling wave fronts to compute the higher-order corrections and suggest a method which is simple to implement for any higher order corrected kernel (see also \\cite{RE-DIS2005}). I will do this for the case of a fixed strong coupling $\\bar\\alpha$. The case of an RG-improved kernel with running coupling was very recently done by Peschanski and Sapeta \\cite{PS} and will therefore not be studied here. The present paper should therefore be taken together with \\cite{PS} for a treatment of the RG-improved BK equation.\n\nThe outline of the paper is as follows. Section \\ref{II} contains a brief review of the traveling wave front method for analyzing the BK equation and section \\ref{III} discusses higher order corrections to the BFKL kernel. In Section \\ref{IV},  I then generalize the method described in Section \\ref{II} to deal with RG-resummed kernels. I show how to compute analytic results for the saturation scale and the shape of the solution in momentum space above the saturation scale, restricting to the case of fixed coupling. This is developed for a general form of the resummed kernel and then applied to various schemes proposed in the literature in Section \\ref{V}. In Section \\ref{VI}, finally, I perform a ``diffusive'' differential equation approximation, which illustrates that some important qualitative properties of the kernel change when including RG corrections.\n\n\n\n\\section{The BK equation and traveling waves} \\label{II}\n\nThe BK equation in position space \ndescribes the scattering amplitude ${\\cal N}(\\vec x_{01},Y)$ of a dipole on a target (originally a large nucleus), where the target is specified by the initial condition. For large and homogeneous targets (target radius much larger than the involved dipoles in the cascade) one may neglect the dependence on the impact parameter and Fourier transform to momentum space. The equation can then be written on the form~\\cite{Kovchegov2}\n\\begin{equation}\n\\partial_Y {\\cal N} = \\bar\\alpha \\chi_0(-\\partial_L) {\\cal N} - \\bar\\alpha {\\cal N}^2.\n\\label{LLBK-eq-fixed}\n\\end{equation}\nwhere $\\bar\\alpha=N_c \\alpha_s\/\\pi$, $L=\\ln(k^2\/k_0^2)$, with $k_0$ some low momentum scale, e.g., $\\Lambda_{QCD}$, and $\\partial_Y\\equiv\\partial\/\\partial Y$, $\\partial_L\\equiv\\partial\/\\partial L$. The quantity ${\\cal N}(\\vec k,Y)$ which we will study in the following is related to the unintegrated gluon distribution of the target. The non-linear term in Eq.\\ \\eq{LLBK-eq-fixed} is particularly simple in this representation and is not acted on by the kernel operator.\nLinearizing Eq.\\ \\eq{LLBK-eq-fixed} one arrives at the BFKL equation \\cite{BFKL}.\n\nThe operator $\\chi_0(-\\partial_L)$ is given by the characteristic BFKL function \n\\begin{equation}\n\\chi_0(\\gamma) = 2\\psi(1) - \\psi(\\gamma) - \\psi(1-\\gamma),\n\\label{chi0}\n\\end{equation}\nthe Mellin transform of the LL BFKL kernel in momentum space \\cite{BFKL}. The explicit form of the term $\\chi_0(-\\partial_L) {\\cal N}$ is a convolution $K_\\text{BFKL}\\otimes{\\cal N}$ of the BFKL integral kernel $K_\\text{BFKL}$ and the gluon distribution, but it can be formally represented as an infinite order differential operator through the power series\n\\begin{equation}\n\\chi_0(-\\partial_L) {\\cal N} = \\left[ \\chi_0(\\gamma_0) \n+ (-\\partial_L-\\gamma_0) \\chi_0'(\\gamma_0)\n+ \\frac{1}{2!}(-\\partial_L-\\gamma_0)^2 \\chi_0''(\\gamma_0) +\\dots \\right] {\\cal N}\n\\label{series-kernel}\n\\end{equation}\nwhere $\\gamma_0$ is a constant. \n\nThe approximations involved in deriving the BK equation are (i) large $N_c$, (ii) no correlations between different dipoles,  and (iii) leading logarithmic (LL) resummation. In this paper we will investigate the corrections to the third point. The second point has been studied a lot in the last years, e.g., in \\cite{Iancu:2004es,EGM}.\n\nMunier and Peschanski~\\cite{MP1,MP2,MP3} found an interesting relationship between non-linear evolution equations in QCD (in particular the BK equation) and statistical physics studies of \\emph{traveling wave fronts}. These are wave fronts that separate a stable state from an unstable state, which invades the unstable state by propagating in space with an essentially fixed shape. Such fronts are described by non-linear diffusion equations with a growth term and a reduction term \\cite{Fisher,KPP,travelingwaves,EvS}. \n\nThey observed~\\cite{MP1} that if one truncates the series \\eq{series-kernel} and keeps only the terms up to order $\\partial_L^2$ one obtains the differential equation\n\\begin{equation}\n\\partial_Y {\\cal N} = \\bar\\alpha \\left[ \\chi_0(\\gamma_0) \n+ (-\\partial_L-\\gamma_0) \\chi_0'(\\gamma_0)\n+ \\frac{1}{2!}(-\\partial_L-\\gamma_0)^2 \\chi_0''(\\gamma_0)\\right] {\\cal N} - \\bar\\alpha {\\cal N}^2\n\\label{LLBK-KPP}\n\\end{equation}\nwhich, with a suitable change of variables $Y \\propto t$ and $L = A x + B t$ (with $A,B$ some constants), gives the equation\n\\begin{equation}\n\\partial_t u = \\partial_x^2 u +  u -  u^2.\n\\label{FKPP}\n\\end{equation}\nThis is the celebrated Fisher--Kolmogorov--Petrovsky-Piscunov\n(FKPP) equation~\\cite{Fisher,KPP}, which under certain conditions leads to  traveling wave solutions $u=u(x,t)$.\nThe condition is that the initial conditions are steep enough, i.e.\\ decay like $\\exp(-\\lambda x)$ at large $x$ with $\\lambda$ large enough, see~\\cite{travelingwaves,EvS} for more details. I will not discuss the initial conditions further here, but will assume that they are steep enough, as is the case in QCD~\\cite{MP1}.\n\n\nThe traveling wave property means essentially that the solutions are saturated wave fronts in the space variable $x$, which propagate forward in space as the time $t$ increases, with a fixed (or nearly fixed) shape. The speed of propagation $v(t)$ of this front has been known for a long time~\\cite{Bramson}; it approaches an asymptotic speed $v^*$ like\n\\begin{equation}\nv(t) = v^* - \\frac{3}{2 \\lambda^* t} + {\\cal O}(t^{-3\/2})\n\\label{velocity2}\n\\end{equation}\nwhere for the FKPP equation $v^*=2$ and $\\lambda^*=1$. The result for the saturation scale given in~\\cite{MP1} is obtained from this formula by using the correct change of variables to get the approximate BK equation \\eq{LLBK-KPP} in the form of the FKPP equation, Eq.\\ \\eq{FKPP}. That is, when studying the BK equation in the traveling wave framework, we study the evolution of a front that propagates in the ``space'' variable $L\\sim\\ln k^2$ as the ``time'' $Y$ increases.\n\nThe $O(t^{-3\/2})$ term in the velocity relation for FKPP-like equations has been found by Ebert and van Saarloos~\\cite{EvS}, namely\n\\begin{equation}\nv(t) = v^* - \\frac{3}{2 \\lambda^* t}\\left( 1 - \\sqrt{\\frac{\\pi}{(\\lambda^*)^2 D t}}\\right)  + {\\cal O}(t^{-2})\n\\label{velocity}\n\\end{equation}\nwhere $\\lambda^*$ is related to the wave number and has the dimensions of inverse length, and $D$ is the diffusion constant. (For the FKPP equation \\eq{FKPP}, $D=1$.) It has also been shown that these are all universal terms in the velocity relation, i.e., all other terms depend on the initial conditions. The three constants $v^*$, $\\lambda^*$ and $D$ \\emph{completely determine} the velocity relation of the traveling waves, and thus the saturation scale in the case of the BK equation. These constants are calculated from the dispersion relation for Fourier modes of the solutions of the equation under study.\n\n\nThe most important point of the approach of Munier and Peschanski may be that the velocity that one calculates is \\emph{independent} of the exact form of the non-linearity, i.e., it is enough to analyze the linearized integro-differential equation, i.e., the BFKL equation, which of course is much easier than the full BK equation. This does not mean that one gets saturation from the linear equation---the non-linearity is crucial in selecting the velocity \\eq{velocity}. See Section III of Ref.\\ \\cite{EvS} for a thorough discussion of this point.\n\n\n\\section{Higher order corrections to the BFKL and BK equations} \\label{III}\n\nThe full NLL BFKL kernel is given by $\\chi(\\gamma)=\\chi_0(\\gamma)+\\bar\\alpha \\chi_1(\\gamma)$ where $\\chi_0$ is the LL BFKL kernel and $\\chi_1$ is the second order contribution to the BFKL kernel and is given in Refs.\\ \\cite{NLLBFKL}. The full BK equation has not been derived at NLL  accuracy---only the BFKL equation is known. There will also be corrections to the non-linear term, but it is reasonable to assume that the linearized equation will be the same as the NLL BFKL equation. Since the observables studied here are, as explained above, determined by the linear equation we can study the BK equation with NLL corrections only in the linear term and obtain the correct $Y$-dependence of the saturation scale. The absolute normalization of $Q_s$ is not constrained by the method. The normalization of the wave front, however, is given by the full non-linear evolution of the initial condition and may change if the non-linear term is changed. The same holds for the wave front in the saturation region, but here we only aspire to calculating it in the region above the saturation scale.\n\nUsing the improved kernels mentioned in the Introduction, the BFKL kernel is modified to the form $\\chi(\\gamma,\\omega)$ depending on both $\\gamma$ and $\\omega$. \nThe RG-improved BK equation is then non-local in both $L$ and $Y$ and mixes transverse and longitudinal degrees of freedom (see also Section \\ref{diffsection}). It can be written\n\\begin{equation}\n\\partial_Y {\\cal N} = \\bar\\alpha \\chi(-\\partial_L,\\partial_Y) {\\cal N} - \\bar\\alpha {\\cal N}^2\n\\end{equation}\nregardless of the specific form of the kernel. Here one can take either fixed or running coupling. The evolution equation is now non-local in rapidity, whereas the uncorrected kernel (LL or NLL) only is non-local in momentum. This is where the traveling wave method reveals its power---it can easily be generalized to kernels much more complicated than LL BFKL (and similarly the FKPP equation can be turned into higher order differential equations). \n\nThe first study of higher order corrections to the saturation scale was to my knowledge done by Golec-Biernat et al.\\ \\cite{Golec-Biernat:2001if} in their numerical study of the BK equation, by implementing the kinematical constraint of Ref.\\ \\cite{cconstraint}. The first analytical study was performed by Triantafyllopoulos \\cite{Dionysis2} using BFKL evolution with absorbing boundary conditions \\cite{Dionysis1}. The higher corrections in this case were taken into account through the use of the resummed NLL kernel of Refs. \\cite{CCS} with a running coupling constant. Khoze et al.\\ \\cite{Khoze:2004hx} have made another BFKL study where they compute the asymptotic term in the saturation scale using a resummed LL kernel along the lines of~\\cite{CCS}, i.e., a simpler variant than the one in \\cite{Dionysis2}. Chachamis et al.\\ \\cite{Chachamis:2004ab} used the BFKL kernel with a rapidity veto \\cite{rapveto} in the BK equation and performed a numerical solution of this BK equation. Finally, Gotsman et al.\\ \\cite{Gotsman:2004xb} studied a DGLAP-corrected BFKL kernel similar to the one used in \\cite{Khoze:2004hx}. All these studies naturally find a reduced growth with energy of the saturation momentum.\n\n\n\\section{Method for calculation} \\label{IV}\n\n\\subsection{The leading logarithmic BK equation}\\label{LL-BFKL-fixed}\n\nThe linearized equation corresponding to \\eq{LLBK-eq-fixed} is the LL BFKL equation, which can be solved analytically by Mellin transformation; the solution is given by the double inverse Mellin transform\\footnote{Note that I use a slightly different convention than e.g.~\\protect\\cite{MP1} for the $\\gamma$-dependence of the exponent.}\n\\begin{equation}\n{\\cal N}(k,Y) = \\int_{\\cal C} \\frac{d\\gamma}{2\\pi i} \\int_{{\\cal C}^\\prime} \\frac{d\\omega}{2\\pi i} {\\cal N}_0(\\gamma,\\omega) \n\\frac{e^{-(1-\\gamma) L +\\omega Y}}{\\omega - \\bar\\alpha \\chi_0(\\gamma)}\n\\end{equation}\nwhere ${\\cal N}_0(\\gamma,\\omega)$ encodes the initial conditions. This is analogous to the Fourier--Laplace transform used in \\cite{EvS} to analyze the FKPP equation leading to a dispersion relation between wave number and frequency for the Fourier modes\\footnote{Working with Mellin transforms instead of Fourier--Laplace transforms, the quantities $\\omega$ and $k^*$ are real instead of purely imaginary. Appropriate changes must therefore be made to the formulas of \\protect\\cite{EvS}.}. In the present case we obtain a dispersion relation for Mellin moments by picking up the pole in the denominator to perform the $\\omega$-integral. This gives\n\\begin{equation}\n{\\cal N}(k,Y) = \\int_{\\cal C} \\frac{d\\gamma}{2\\pi i} \\widetilde {\\cal N}_0(\\gamma)\ne^{-(1-\\gamma) L + \\bar\\alpha \\chi_0(\\gamma) Y}.\n\\end{equation}\nWe thus have the dispersion relation $\\omega(\\gamma) = \\bar\\alpha \\chi_0(\\gamma)$ for the solution of the BK equation. The evolution variable is now rapidity $Y$, which corresponds to time $t$ in the FKPP equation.\n\n\nLet us now see how easily the saturation scale is computed using the Ebert--van Saarloos (EvS) method \\cite{EvS}. The calculation consists of finding the three parameters $v^*$, $\\lambda^*=k^*=1-\\gamma_c$, and $D$.\nThe linearized LL-BK equation (i.e., the LL-BFKL equation) is in Mellin $\\gamma$-space written as\n\\begin{equation}\n\\partial_Y \\widetilde\\phi(\\gamma,Y) = \\bar\\alpha \\chi_0(\\gamma) \\widetilde\\phi(\\gamma,Y)\n\\end{equation}\nwhere $\\widetilde\\phi(\\gamma,Y)$ denotes the Mellin transform of ${\\cal N}(k,Y)$ with respect to $L$. \nFollowing the general formalism of EvS Section V-D, we write this as\n\\begin{equation}\n\\left[\\partial_Y + \\widetilde{\\mathsf T}(\\gamma)\\right] \\widetilde\\phi(\\gamma,Y) = 0\n\\label{eqn}\n\\end{equation}\nwhere $-\\widetilde{\\mathsf T} \\widetilde\\phi$ is the Mellin transform with respect to $k^2$ of $K_\\text{BFKL}\\otimes{\\cal N}$, i.e., \n$\\widetilde{\\mathsf T} = -\\bar\\alpha \\chi_0(\\gamma)$.\nDefine also $\\hat{\\mathsf S}(\\gamma,\\omega)$, the Mellin transform with respect to $Y$ of the left hand side of \\eq{eqn}, i.e., it is the double Mellin transform of the BFKL equation.\nIn general, for differential equations, $\\hat{\\mathsf S}$ and $\\widetilde{\\mathsf T}$ are $M \\times M$ matrices where $M$ is the order of the highest time derivative.\n\nThe dispersion relation for Mellin moments is obtained from the requirement $S(\\gamma,\\omega)=0$ where $S$ is the characteristic function $S(\\gamma,\\omega) = \\det \\hat{\\mathsf S}(\\gamma,\\omega)= \\omega -\\bar\\alpha \\chi_0(\\gamma)$. This again yields\n$\\omega(\\gamma) = \\bar\\alpha \\chi_0(\\gamma)$\nas expected. One now finds the asymptotic velocity $v^*$ and the saddle point $k^* = \\lambda^* = 1-\\gamma_c$ by solving the set of equations [EvS Eqs.\\ (5.60) and (5.66)]\n\\begin{equation}\n\\begin{cases}\n\\displaystyle v^* = -\\left. \\frac{d\\omega(\\gamma)}{d\\gamma}\\right|_{\\gamma_c} = -\\bar\\alpha \\chi^{\\prime}_0(\\gamma_c)\\\\\n\\displaystyle v^* = \\frac{\\omega(\\gamma_c)}{1-\\gamma_c} = \\frac{\\bar\\alpha \\chi_0(\\gamma_c)}{1-\\gamma_c}.\n\\end{cases}\n\\label{gammac-eqs}\n\\end{equation}\nThe relevant saddle point for evolution with constant amplitude is therefore not the usual BFKL saddle point $\\gamma_0=1\/2$ but is given by the solution of the equation $\\chi^{\\prime}_0(\\gamma_c)(1-\\gamma_c)+\\chi_0(\\gamma_c)=0$ as was shown already in \\cite{GLR}. A numerical solution of \\eq{gammac-eqs} gives $\\gamma_c=0.373$. \nFinally, the diffusion coefficient $D$ is obtained from\n\\begin{equation}\nD = \\left.\\frac{1}{2!} \\frac{d^2\\omega(\\gamma)}{d\\gamma^2}\\right|_{\\gamma_c}\n= \\frac{\\bar\\alpha}{2} \\chi^{\\prime\\prime}_0(\\gamma_c).\n\\label{D}\n\\end{equation}\nThe speed of the front is now given by Eq.\\ \\eq{velocity}\nand the position is then simply\n\\begin{equation}\nx(Y)= \\ln [Q_s^2(Y)\/k_0^2]\t =  v^* Y - \\frac{3}{2\\lambda^*} \\ln Y\n- \\frac{3}{(\\lambda^*)^2} \\sqrt{\\frac{\\pi}{D Y}} \n\\end{equation}\nwhich, inserting $v^*$, $\\lambda^*$and $D$ gives the result (neglecting $k_0$)\n\\begin{equation}\n\\ln Q^2_s(Y) = \\bar\\alpha \\frac{\\chi_0(\\gamma_c)}{1-\\gamma_c} Y - \\frac{3}{2(1-\\gamma_c)} \\ln Y - \\frac{3}{(1-\\gamma_c)^2} \\sqrt{\\frac{2\\pi}{\\bar\\alpha\\chi^{\\prime\\prime}_0(\\gamma_c)}} \\frac{1}{\\sqrt{Y}}\n+{\\cal O}(1\/Y).\n\\label{Qs-LL-fixed}\n\\end{equation}\nThis expression for the saturation scale of a system whose evolution is described by Eq.\\ \\eq{LLBK-eq-fixed}\nwas first obtained in \\cite{MP3}. These three terms are universal. The first term, which gives the asymptotic exponential rise of the saturation scale used e.g.\\ in the saturation model of~\\cite{GBW}, was obtained already in \\cite{GLR} from the double logarithmic approximation, and from LL BFKL in \\cite{IancuItakuraMcLerran}.\nThe second term was obtained in \\cite{Dionysis1} using BFKL evolution with an absorbing boundary to mimic the non-linearities and was confirmed in \\cite{MP1,MP2} using the traveling wave method.\\footnote{Note that when comparing the results of \\protect\\cite{Dionysis1} and \\protect\\cite{MP1,MP2} there is a difference of a factor of 2 due to the convention for the definition of the BFKL characteristic function $\\chi_0$. Furthermore, there are some terms in the results of \\protect\\cite{Dionysis1} that are non-universal and subleading in the classification used here.} \n\n\nThe shape of the wave front in the so-called leading edge region can also be computed in the  traveling wave framework. Ebert and van~Saarloos~\\cite{EvS} compute the solution at the same level of approximation as the velocity \\eq{velocity}, i.e., including all universal terms. Brunet and Derrida~\\cite{BrunetDerrida} use a somewhat simpler method which allows computing the solution at the level of approximation of \\eq{velocity2}, which gives simpler results. This method was employed in Ref.\\ \\cite{MP2} to compute the solution of the LL BK equation in the leading edge region for both fixed and running coupling. The result is\n\\begin{align}\n{\\cal N}(k,Y) &= \\frac{1}{\\sqrt{D}}\n\\ln\\left(\\frac{k^2}{Q_s^2(Y)}\\right) \n\\left(\\frac{k^2}{Q_s^2(Y)}\\right)^{\\gamma_c-1}\n\\exp\\left(-\\frac{1}{4 D Y}\\ln^2\\left(\\frac{k^2}{Q_s^2(Y)}\\right) \n\\right)\n\\\\\n&= \\sqrt{\\frac{2}{\\bar\\alpha\\chi^{\\prime\\prime}_0(\\gamma_c)}}\n\\ln\\left(\\frac{k^2}{Q_s^2(Y)}\\right) \n\\left(\\frac{k^2}{Q_s^2(Y)}\\right)^{\\gamma_c-1}\n\\exp\\left(-\\frac{1}{2\\bar\\alpha\\chi^{\\prime\\prime}(\\gamma_c)Y}\\ln^2\\left(\\frac{k^2}{Q_s^2(Y)}\\right) \n\\right) \\nonumber\n\\end{align}\nup to a constant factor. This result has been obtained both using an absorbing boundary~\\cite{Dionysis1} and traveling waves~\\cite{MP2}.\n\nThe agreement between these completely different methods both for the saturation scale and the wave front shape is good evidence that the traveling wave method is reliable and general. Furthermore, the numerical study of the BK equation in \\cite{EGM} shows very good agreement at large $Y$ with Eq.\\ \\eq{Qs-LL-fixed} and the corresponding analytical formulas for ${\\cal N}(k,Y)$ in the leading edge region.\n\nThe description in terms of operators $\\hat{\\mathsf S}$ and $\\widetilde{\\mathsf T}$ \nmight seem unnecessarily formal, but a main result of the analysis of EvS~\\cite{EvS} is that the method generalizes to much more general types of equations than differential equations of the FKPP type,\\footnote{They have to be equivalent in the sense that they must be reaction--diffusion equation with second-order diffusion. The running coupling case, for example, leads to a different equivalence class which has $D=0$.} for example higher order differential equations, integral equations with a kernel in either time (memory kernels) or in space (non-local kernels), or even difference equations. \n\nThus, the kernel in momentum or configuration space might have any complicated structure, but if one knows the Mellin (or Fourier) transform of the kernel, one can solve for the dispersion relation and compute $v^*, \\lambda^*$ and $D$ (in fact, in the following treatment the higher order corrections are implemented in Mellin space rather than in momentum space). \nThis dispersion relation can lead to non-algebraic equations, however, unlike for the FKPP equation. \nThis generalization will be exploited below for the RG-resummed case.\n\n\n\n\\subsection{RG-improved BK}\\label{NLL-BK-fixed}\n\nLet me now generalize to RG-improved kernels at fixed coupling. The dispersion relation now reads\n\\begin{equation}\n\\omega(\\gamma,\\bar\\alpha) = \\bar\\alpha \\chi(\\gamma,\\omega(\\gamma,\\bar\\alpha))\n\\label{NLLdisp}\n\\end{equation}\nwhich implicitly defines the function $\\omega$. For each value of $\\gamma$ \nand $\\bar\\alpha$ Eq.\\ \\eq{NLLdisp} gives a transcendental equation, which can in principle be solved numerically for $\\omega(\\gamma)$. This is not what we will do here.\n\nThe asymptotic velocity $v^*$ and saddle point $\\lambda^*=1-\\gamma_c$ are as earlier found from the equations\n\\begin{align}\nv^* = & -\\left. \\frac{d\\omega(\\gamma,\\bar\\alpha)}{d\\gamma}\\right|_{\\gamma_c}\n\\label{v1}\\\\\nv^* = & \\ \\frac{\\omega(\\gamma_c,\\bar\\alpha)}{1-\\gamma_c},\\label{v2}\n\\end{align}\nbut since $\\omega(\\gamma,\\bar\\alpha)$ does not have an explicit form it is not as straightforward to obtain the required quantities. For LL there is a saddle point of the characteristic function $\\chi_0(\\gamma)$ determined from the equation given above, but now the function $\\chi=\\chi(\\gamma,\\omega)$ has an additional variable. This means that we will have to determine two constants $\\gamma_c$ and $\\omega_c$.\nTo keep notation simple, let us introduce the shorthand \n$\\chi_c = \\chi(\\gamma_c,\\omega_c)$, \n$\\chi^{\\prime}_c = \\partial_\\gamma \\chi(\\gamma_c,\\omega_c)$  and \n$\\dot\\chi_c = \\partial_\\omega \\chi(\\gamma_c,\\omega_c)$.\n\nFirst, we require $\\omega_c = \\bar\\alpha \\chi(\\gamma_c,\\omega_c)$. Observing that on one hand, $v^*$ is given by\n\\begin{equation}\nv^* = -\\left. \\frac{d\\omega}{d\\gamma}\\right|_{\\gamma_c} = -\\bar\\alpha  \\left.\\left[\n\\frac{\\partial\\chi(\\gamma,\\omega)}{\\partial\\gamma} + \n\\frac{\\partial\\chi(\\gamma,\\omega)}{\\partial\\omega}\n\\frac{d\\omega}{d\\gamma} \\right] \\right|_{\\gamma_c}, \\label{dwdg}\n\\end{equation}\nso that \n\\begin{equation}\nv^* = -\\left. \\frac{d\\omega(\\gamma)}{d\\gamma}\\right|_{\\gamma_c}\n= -\\frac{\\bar\\alpha \\chi^{\\prime}_c}{1 - \\bar\\alpha \\dot\\chi_c},\n\\end{equation}\nand using Eq.\\ \\eq{v2} we obtain\n$-\\chi^{\\prime}_c (1-\\gamma_c) = \\chi_c \\, [1 - \\bar\\alpha \\dot\\chi_c] = \\chi_c  - \\omega_c \\,\\dot\\chi_c$. We therefore have the system of equations\n\\begin{equation}\n\\begin{cases}\n\\chi^{\\prime}_c (1-\\gamma_c) + \\chi_c  - \\omega_c \\, \\dot\\chi_c=0\\\\\n\\omega_c = \\bar\\alpha \\chi_c\n\\end{cases}\n\\label{gcwc-eq}\n\\end{equation}\nThe first equation reduces to the corresponding LL one if there is no dependence on $\\omega$ in the kernel.\n\nThe advantage of knowing both $\\gamma_c$ and $\\omega_c$ is obvious---there exists a closed form for all the constants that will appear in the expression for the saturation scale, and it is not necessary to use functions that can only be defined numerically and to numerically differentiate these functions. It is much more straightforward to just numerically solve \\eq{gcwc-eq}.\n\nOnce we have determined $\\gamma_c$ and $\\omega_c$, the other constants $v^*$ and $D$ are fixed (remember $\\lambda^*=1-\\gamma_c$). We are then in a position to write down the expression for the velocity of the front,\n\\begin{equation}\nv(Y) = v^* - \\frac{3}{2 (1-\\gamma_c) Y}\\left( 1 - \\sqrt{\\frac{\\pi}{(1-\\gamma_c)^2 D Y}}\\right) \n\\label{velocityY}\n\\end{equation}\nwhere $v^*=\\omega_c\/(1-\\gamma_c)$ and $D$ is obtained by applying the chain rule as in Eq.\\ \\eq{dwdg} twice;\n\\begin{equation}\nD = \\frac{1}{2} \\, \\left. \\frac{d^2\\omega}{d\\gamma^2}\\right|_{\\gamma_c}\n= \\frac{\\omega_c}{2\\chi^{\\prime}_c (\\gamma_c-1)}  \n\\left( \\chi^{\\prime\\prime}_c + 2\\, \\frac{\\omega_c}{1-\\gamma_c}\\, \\dot\\chi^{\\prime}_c \n+ \\left(\\frac{\\omega_c}{1-\\gamma_c}\\right)^2 \\ddot\\chi_c \\right).\n\\label{diffusionD}\n\\end{equation}\nFinally this gives the saturation scale,\n\\begin{equation}\n\\ln Q^2_s(Y) = \\frac{\\omega_c}{1-\\gamma_c} Y - \\frac{3}{2(1-\\gamma_c)} \\ln Y -  \n\\sqrt{\\frac{18\\pi \\chi^{\\prime}_c}{\\omega_c (\\gamma_c-1)^3 Y}}\n\\left( \\chi^{\\prime\\prime}_c + 2\\, \\frac{\\omega_c}{1-\\gamma_c}\\, \\dot\\chi^{\\prime}_c \n+ \\left(\\frac{\\omega_c}{1-\\gamma_c}\\right)^2 \\ddot\\chi_c \\right)^{-1\/2}.\n\\label{Qs-NLL-fixed}\n\\end{equation}\n\nAll the discussion so far is perfectly general in that we have not chosen any particular form for the NLL BFKL kernel, we have only anticipated that it will depend on both $\\gamma$ and $\\omega$.\n\n\\section{Application to RG-improved kernels}\\label{applications}\\label{V}\n\nThe method from the last section can now be easily applied to some of the RG-improved kernels suggested in the literature. I will look at the rapidity veto model of \\cite{rapveto}, the kinematical constraint model of \\cite{cconstraint}, and the resummed model of \\cite{Khoze:2004hx}. I will thus solve Eqs.\\ \\eq{gcwc-eq} for these models numerically using Mathematica, and use the obtained values for $\\gamma_c,\\omega_c$ in the expression \\eq{Qs-NLL-fixed} for the saturation scale and in the generalized expression \\eq{redfront} below for the shape of the wave front.\n\n\\subsection{Rapidity veto}\nThe idea of a rapidity veto in the kernel \\cite{rapveto} is that by forbidding gluon emissions that are very close in rapidity one accounts for a part of the NLL corrections. This was studied numerically in the context of the BK equation in Ref.\\ \\cite{Chachamis:2004ab}.\n\nThe form of the kernel is \n\\begin{equation}\n\\chi(\\gamma,\\omega)=\\chi_0(\\gamma) e^{-\\eta \\omega}\n\\label{chi-rapveto}\n\\end{equation}\nwhere $\\eta$ is the minimum rapidity interval imposed on two subsequent emissions, and is taken as a parameter. A numerical solution for $\\omega(\\gamma)$ is shown in Fig.\\ \\ref{CC-fig}.\n\nThe equations \\eq{gcwc-eq} thus take the form\n\\begin{align}\n\\omega_c = \\ &  \\bar\\alpha \\chi_0(\\gamma_c) e^{-\\eta \\omega_c} \\nonumber\\\\\n\\chi^{\\prime}_0(\\gamma_c) \\gamma_c e^{-\\eta \\omega_c} = \\ &\n\\chi_0(\\gamma_c) e^{-\\eta \\omega_c} \\left( 1 + \\bar\\alpha \\eta \\chi_0(\\gamma_c) e^{-\\eta \\omega_c}\\right)\n\\end{align}\nwhich in fact for this factorized form of the kernel allows eliminating $\\omega_c$ from the result so that one arrives at the following equation for $\\gamma_c$:\n\\begin{equation}\n\\frac{\\chi^{\\prime}_0(\\gamma_c)(1-\\gamma_c)}{\\chi_0(\\gamma_c)}+1 = -\\bar\\alpha \\eta \\chi_0(\\gamma_c) \\exp\\left(1+\\frac{\\chi^{\\prime}_0(\\gamma_c)(1-\\gamma_c)}{\\chi_0(\\gamma_c)}\\right)\n\\label{gc-rapveto}\n\\end{equation}\nwhich, taking into account the difference in convention, $\\gamma \\to 1-\\gamma$, is the result for $\\gamma_c$ found in Ref.\\ \\cite{Chachamis:2004ab}.\nNote that for $\\eta\\to 0$ this reduces to the LL result. \n\nThe authors of \\cite{Chachamis:2004ab} give an analytic result for the saturation scale with only the asymptotic term for the velocity and then solve the BK equation numerically. But using the present method it is easy to find the subleading terms for the saturation scale. To do this we need to find the diffusion coefficient given by Eq.\\ \\eq{diffusionD}, \n\\begin{equation}\nD =  \\frac{\\bar\\alpha \\chi_0(\\gamma_c)}{2\\chi^{\\prime}_0(\\gamma_c)(\\gamma_c-1)} e^{-\\eta \\omega_c} \n\\left( \\chi^{\\prime\\prime}_0(\\gamma_c) - 2\\eta\\,\\chi^{\\prime}_0(\\gamma_c) \\, \\frac{\\bar\\alpha \\chi_0(\\gamma_c)e^{-\\eta \\omega_c}}{1-\\gamma_c}\\, \n+ \\eta^2 \\, \\chi_c(\\gamma_c)\\left(\\frac{\\bar\\alpha \\chi_0(\\gamma_c)e^{-\\eta \\omega_c}}{1-\\gamma_c}\\right)^2  \\right)\n\\label{D-rapveto}\n\\end{equation}\nwith\n\\begin{equation}\ne^{-\\eta \\omega_c}=\\exp \\left(1+\\frac{\\chi^{\\prime}_0(\\gamma_c)(1-\\gamma_c)}{\\chi_0(\\gamma_c) }\\right).\n\\end{equation}\nThus, the saturation scale with all universal terms reads\n\\begin{equation}\n\\ln Q_s^2(Y) = \\bar\\alpha \\frac{\\chi_0(\\gamma_c)}{1-\\gamma_c} Y \\exp \\left(1+\\frac{\\chi^{\\prime}_0(\\gamma_c)(1-\\gamma_c)}{\\chi_0(\\gamma_c) }\\right)\n- \\frac{3}{2(1-\\gamma_c)} \\ln Y\n- \\frac{3}{(1-\\gamma_c)^2} \\sqrt{\\frac{\\pi}{D Y}} \n\\end{equation}\nwith $\\gamma_c$ and $D$ given by Eqs.\\ \\eq{gc-rapveto} and \\eq{D-rapveto}.\n\n\n\n\\subsection{Kinematical constraint}\n\nThe so-called kinematical constraint, or consistency constraint \\cite{cconstraint}, imposes a cut-off on the virtualities of gluons in the real emission part of the BFKL kernel, motivated by the requirement that in the high energy kinematics considered, the virtualities of the exchanged gluons should be dominated by their transverse momenta. It is also a part of the resummed framework of \\cite{CCS}. \n\nThis constraint has been shown to reproduce a large part of the NLL-corrections to the pomeron intercept. It has been used in several phenomenological applications of BFKL dynamics. Because the value of the pomeron intercept is reduced, the results show a reduced growth of cross sections with energy, and generally fit data better than LL BFKL.\n \nThe constraint implies shifting the $\\gamma$ in the kernel by $\\pm\\omega\/2$,\n\\begin{equation}\n\\chi(\\gamma,\\omega)=2\\psi(1)-\\psi\\left(\\gamma+\\frac{\\omega}{2}\\right)-\\psi\\left(1-\\gamma+\\frac{\\omega}{2}\\right),\n\\label{chi-CC}\n\\end{equation}\nsuch that, in contrast to the kernel \\eq{chi-rapveto} with rapidity veto discussed above, the kernel \\eq{chi-CC} does not have poles in $\\gamma=0,1$.\n\n\\begin{figure}[th]\n\\epsfig{file=cc-and-rapveto.eps,width=0.7\\columnwidth}\n\\caption{Characteristic functions $\\omega(\\gamma,\\bar\\alpha=0.2)$. The solid line is LL BFKL, Eq.\\ \\protect\\eq{chi0}, the dashed line is the kinematical constraint, Eq.\\ \\protect\\eq{chi-CC}, and the dotted line is the rapidity veto, Eq.\\ \\protect\\eq{chi-rapveto}, for $\\eta=1.87$, which gives the same value for the pomeron intercept $\\omega(\\frac{1}{2},\\bar\\alpha)$ as the dashed line.\\label{CC-fig}}\n\\end{figure}\n\n\nIn Fig.\\ \\ref{CC-fig} the solutions of the implicit equation for $\\omega(\\gamma,\\bar\\alpha)$ for a fixed $\\bar\\alpha=0.2$ for the kernel \\eq{chi-CC} is compared with the LL kernel $\\chi_0$. I also show the kernel \\eq{chi-rapveto} with the rapidity veto for a choice of $\\eta$ that gives the same pomeron intercept as the kinematical constraint.\n\nThe kernel \\eq{chi-CC} above is symmetric with respect to the interchange $\\gamma\\to 1-\\gamma$, just as the original LL BFKL kernel. It is thus written for a symmetric scale choice $s_0$ in the evolution variable $Y=\\ln (s\/s_0)$, were the scales on both sides of the ladder appear in $s_0$. In the present analysis we would like to make an asymmetric scale choice. In such a case the kinematical constraint should be formulated slightly differently \\cite{CCS}, namely one writes the kernel as\n\\begin{equation}\n\\chi(\\gamma,\\omega)=2\\psi(1)-\\psi\\left(\\gamma\\right)-\\psi\\left(1-\\gamma+\\omega\\right),\n\\label{chi-CC-asy}\n\\end{equation}\nwhich is now asymmetric with respect to $\\gamma\\to 1-\\gamma$. This form will be used below.\n\n\n\n\\subsection{Khoze--Martin--Ryskin--Stirling model}\n\nKhoze, Martin, Ryskin and Stirling (KMRS) \\cite{Khoze:2004hx} proposed a simpler model for the NLL corrected LL kernel, which is based on the approach of \\cite{CCS} but with a fixed coupling. This kernel has also been used to estimate the effect of higher order corrections in diffractive $\\gamma^*\\gamma^*\\to\\rho\\rho$ production \\cite{EPSW} which provides a possibility to test it against the full NLL calculation \\cite{Ivanov:2005gn}.\n\nThe characteristic function is expressed as\n\\begin{equation}\n\\chi(\\gamma,\\omega) = \\chi_0(\\gamma) + \\bar\\alpha \\chi_1(\\gamma,\\omega)\n\\label{durhamchi}\n\\end{equation}\nwhere $\\chi_0(\\gamma)$ is the usual BFKL function \\eq{chi0}. The correction piece $\\chi_1(\\gamma,\\omega)$ is given by\n\\begin{equation}\n\\bar\\alpha \\chi_1(\\gamma,\\omega) =\n\\frac{1+ \\omega A_1(\\omega)}{\\gamma} -\n\\frac{1}{\\gamma} + \n\\frac{1+ \\omega A_1(\\omega)}{1-\\gamma+\\omega} -\n\\frac{1}{1-\\gamma} -\n\\omega \\chi_0^{\\text{ht}}(\\gamma), \n\\end{equation}\nwhere $A_1(\\omega)$ is related to the Mellin transform of the DGLAP splitting function and $\\chi_0^{\\text{ht}}$ is the higher twist part of $\\chi_0$,\n\\begin{equation}\n\\chi_0^{\\text{ht}}(\\gamma) = \\chi_0(\\gamma) - \\frac{1}{\\gamma} - \\frac{1}{1-\\gamma}\n= 2\\psi(1) - \\psi(1+\\gamma) -\\psi(2-\\gamma).\n\\end{equation}\nWe make the approximation that $n_F=0$ in the evolution, which gives $A_1(\\omega)\\simeq -11\/12$. It is possible to include the effects of quarks \\cite{Khoze:2004hx} but they will be neglected here.\n\nThe characteristic kernel \\eq{durhamchi} is shown in Fig.~\\ref{durham-fig1} together with the LL BFKL kernel and the asymmetric consistency constraint. An important feature of the kernel  \\eq{durhamchi} is that it has no pole at $\\gamma=0$. Instead it approximately fulfills energy conservation through $\\gamma(\\omega=1)\\approx 0$ (exact energy conservation implies equality).\n\n\\begin{figure}[t]\n\\epsfig{file=ll-cc-durham.eps,width=0.7\\columnwidth}\n\\caption{Characteristic functions $\\omega(\\gamma,\\bar\\alpha=0.2)$. The solid line is LL BFKL, Eq.\\ \\protect\\eq{chi0}, the dotted line is the asymmetrical kinematical constraint, Eq.\\ \\protect\\eq{chi-CC-asy}, and the dashed line is the KMRS model \\protect\\eq{durhamchi}.\\label{durham-fig1}}\n\\end{figure}\n\n\n\n\\subsection{Comparison of results}\n\nLet us now compare the results obtained from the different suggested kernels. We will focus on two quantities, the derivative of the saturation scale and the reduced front shape.\n\nThe logarithmic derivative \n\\begin{equation}\n\\lambda_s(Y) \\equiv \\frac{\\partial \\ln Q_s^2}{\\partial Y}\n\\label{satscder}\n\\end{equation}\ngives a measure of how fast the fronts propagate toward large momenta and should asymptotically approach the constant value $v^*$. In Fig.\\ \\ref{lambda-FC}, the results for $\\lambda_s(Y)$ obtained from the three different kernels are plotted together with the LL BFKL result. It is clear that the NLL corrections decrease $v^*$, but also that the approach to the asymptotics is somewhat smoothed out. This can be traced to the smaller values of the diffusion constants. For experimentally observable rapidities it seems that the velocity is a decreasing function of $Y$. Remember, however, that for smaller rapidities there can be appreciable effects coming from the non-universal terms that are not included here (see \\cite{EGM} for numerical investigations of this).\n\n\\begin{figure}[th]\n\\epsfig{file=lambda-FC.eps,width=0.7\\columnwidth}\n\\caption{Derivative of the saturation scale, $\\lambda_s(Y)$, as defined in Eq.\\ \\protect\\eq{satscder} for LL BFKL, the kinematical constraint (KC), the rapidity veto, and the KMRS kernel.\\label{lambda-FC}}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\epsfig{file=redfront-FC.eps,width=0.7\\columnwidth}\n\\caption{Reduced front shape as defined in Eq.\\ \\protect\\eq{redfront}, \nobtained from LL BFKL, the kinematical constraint, and the KMRS kernel, and \nplotted for rapidities $Y=2,4,6,8,10$. \n\\label{redfront-FC}}\n\\end{figure}\n\nThe reduced front shape is obtained as the combination \n\\begin{align}\n{\\cal M}(k,Y) &\\equiv {\\cal N}(k\/Q_s(Y),Y) \\times (k^2\/Q_s^2(Y))^{\\gamma_c}\\\\\n&= \\frac{1}{\\sqrt{D}}\n\\ln\\left(\\frac{k^2}{Q_s^2(Y)}\\right) \n\\exp\\left(-\\frac{1}{4 D Y}\\ln^2\\left(\\frac{k^2}{Q_s^2(Y)}\\right) \n\\right)\n\\label{redfront}\n\\end{align}\nwhich removes the exponential part and quantifies the deviation from geometric scaling. For perfect geometric scaling the reduced front should not depend on the rapidity. In Fig.\\ \\ref{redfront-FC} the reduced fronts are plotted for the LL BFKL kernel, the consistency constraint and the KMRS model. The curves have widths which reflect the diffusion constants in the diffusive breaking of geometric scaling, and it is obvious that the RG-improved kernels lead to a slower diffusive breaking of geometric scaling.\n\n\n\n\n\n\\section{Diffusive differential equation approximation}\\label{diffsection}\\label{VI}\n\nIn \\cite{MP1} the BK equation was approximated as a partial differential equation, which could be transformed into the FKPP equation by expanding the BFKL kernel to second order. The RG-improved BK equation may similarly be approximated as a differential equation, which will be a generalization of the FKPP equation. It is instructive to study this approximation since we then obtain closed analytical expressions and the structure of the modifications to the traveling wave equations may be easier appreciated.\n\nThe BFKL function $\\chi$ can be Taylor expanded around the point $(\\gamma_c,\\omega_c)$ to second order,\n\\begin{eqnarray}\n\\chi(\\gamma,\\omega) &\\approx& \n\\chi \n+ \\chi^{\\prime} (\\gamma-\\gamma_c)\n+ \\dot\\chi (\\omega-\\omega_c) \\nonumber\\\\\n&+& \\frac{1}{2} \\left[\n\\chi^{\\prime\\prime} (\\gamma-\\gamma_c)^2\n+ \\ddot\\chi (\\omega-\\omega_c)^2 \n+  2 \\dot\\chi^{\\prime} \n(\\gamma-\\gamma_c)(\\omega-\\omega_c)\n\\right]\n\\end{eqnarray}\nwhere in this Section the subscript $c$ on $\\chi$ is omitted for simplicity of notation, so that\n$\\chi = \\chi(\\gamma_c,\\omega_c)$,\n$\\chi^{\\prime} = \\partial_\\gamma \\chi(\\gamma_c,\\omega_c)$,\n$\\dot\\chi = \\partial_\\omega \\chi(\\gamma_c,\\omega_c)$,\n$\\dot\\chi^{\\prime} = \\partial_\\gamma \\partial_\\omega\\chi(\\gamma_c,\\omega_c)$,\nand so on. The BK equation for ${\\cal N}={\\cal N}(L,Y)$ in the diffusive approximation then takes the form\n\\begin{eqnarray}\n\\partial_Y {\\cal N} &=& \\bar\\alpha\\left[ \n\\frac{\\chi^{\\prime\\prime} }{2} \\, \\partial_L^2\n+ \\frac{\\ddot\\chi}{2}\\, \\partial_Y^2\n+ (\\chi^{\\prime\\prime} \\gamma_c - \\chi^{\\prime} + \\dot\\chi^{\\prime} \\omega_c)\\, \\partial_L \n- (\\ddot\\chi \\omega_c - \\dot\\chi + \\dot\\chi^{\\prime} \\gamma_c)\\, \\partial_Y\n\\right. \\nonumber\\\\\n&&\\left.\n- \\dot\\chi^{\\prime} \\,\\partial_L\\partial_Y + \\left( \\chi \n- \\chi^{\\prime} \\gamma_c - \\dot\\chi \\omega_c \n+ \\frac{\\chi^{\\prime\\prime} {{\\gamma_c }^2}}{2} \n+ \\dot\\chi^{\\prime} \\gamma_c \\omega_c \n+ \\frac{\\ddot\\chi {{\\omega_c }^2}}{2}\n\\right)\n\\right] {\\cal N} - \\bar\\alpha {\\cal N}^2\n\\label{BK-NLL-diffusive}\n\\end{eqnarray}\nwhich at first glance looks more symmetric in $L$ and $Y$ than the LL approximation.\nThe equation contains a cross term derivative and a term proportional to $\\partial_L$, but the variable change\n\\begin{equation}\n\\begin{cases}\nt = A Y + B L \\\\\nx = C Y + D L\n\\label{firstchange}\n\\end{cases}\n\\end{equation}\ngives an equation on the form\n\\begin{equation}\n\\tau\\, \\partial_t^2 u + \\partial_t u\n= \\partial_x^2 u +  u -  u^2. \\label{ext-KPP}\n\\end{equation}\nThe explicit expressions for the constants $\\tau$, $A$, $B$, $C$, and $D$ in terms of $\\chi,\\chi^{\\prime},\\dot\\chi$ etc.\\ are not very illuminating and are given in the Appendix. The function $u$ is defined as\n\\begin{eqnarray}\nu(x,t) &=& \\text{const} \\times  \n{\\cal N}\\left(L=\\frac{A x -C t}{AD-BC},Y=\\frac{D t- B x}{AD-BC}\\right).\n\\end{eqnarray}\nRemember that all the coefficients are just pure numbers once the saddle point values $\\gamma_c$ and $\\omega_c$ have been found.\n\n\nEq.\\ \\eq{ext-KPP} is an extended FKPP equation with a higher time derivative. Note that in LL BFKL the evolution occurs in rapidity, since the time variable $t \\propto Y$, but in the present case the evolution instead occurs in $t = A Y + B L$. The $\\omega$-dependent kernel thus mixes evolution in rapidity and momentum.\n\nWe can now analyze Eq.\\ \\eq{ext-KPP} in the same way as we analyzed the full linearized BK equation in the LL or NLL cases. The result is\n\\begin{eqnarray}\nv^* &=& \\frac{2}{\\sqrt{1+ 4\\tau}}\\\\\n\\lambda^* &=& \\sqrt{1+ 4\\tau}\\\\\nD &=& \\frac{1}{(1+ 4\\tau)^2}.\n\\end{eqnarray}\nThis is to be compared with the results for the first-order in time FKPP equation, \n$v^*=2, \\lambda^*=1, D=1$. Taking into account the transformation back to the variables $L$ and $Y$ we again see that the asymptotic velocity is reduced compared to the LL value. \n\nIt turns out, however,  to be much simpler to accept a ``convection term'' proportional to $\\partial_x$ in the equation. This makes it possible to keep the $t \\propto Y$ property and the change of variables is much simpler since the only constraint is to get rid of the mixed derivative term instead of getting rid of several terms. In fact, such a convection term will only change the velocity by an additive constant.\n\nThus, if we instead in Eq.\\ \\eq{BK-NLL-diffusive} make the variable change\n\\begin{equation}\n\\begin{cases}\nt = A Y  \\\\\nx = C Y + D L,\n\\label{secondchange}\n\\end{cases}\n\\end{equation}\nwe get the transformed equation\n\\begin{equation}\n\\tau\\, \\partial_t^2 u + \\partial_t u\n= \\partial_x^2 u + \\sigma \\, \\partial_x u + u -  u^2. \\label{ext2-KPP}\n\\end{equation}\nwith\n\\begin{eqnarray}\nu(x,t) &=&  \\text{const} \\times  \n{\\cal N}\\left(L=\\frac{x}{D} -\\frac{C t}{A D},Y=\\frac{t}{A}\\right).\n\\end{eqnarray}\nThe coefficients $\\tau$, $\\sigma$, $A$, $C$, and $D$ are again given in the Appendix.\nThis equation has an additional linear derivative in $x$, with a ``convection coefficient'' $\\sigma$. With this change of variables we therefore see the problem in a different way; the evolution in time is now evolution in rapidity  as in LL instead of in rapidity and momentum as with the earlier variable change. There is a higher order time derivative slowing down the spreading, as well as a convection term damping the wave.\nUnfortunately the expressions for $v^*$, $\\lambda^*$ and $D$ are now quite complicated and I choose to not write them here.\n\nThe interesting points here are that when passing to RG-improved kernels, the spreading velocity decreases, and the equation becomes more symmetric in $L$ and $Y$. In terms of linearized partial differential equation approximations, the LL BFKL equation is first order in time and second order in space while the DGLAP equation is first order in space and second order in time. The RG-improved equation which contains higher order corrections is instead second order in both time and space. This duality is explicit in Eqs.\\ \\eq{ext-KPP} and \\eq{ext2-KPP}, reminding one of the approach of Ref.~\\cite{ABF}. \n\n\n\\section{Summary and outlook}\n\nI have proposed a simple way to use the traveling wave framework to compute the energy dependence of the saturation scale for RG-improved BFKL kernels with fixed QCD coupling. This completes the similar computation done for running coupling in \\cite{PS}. Higher order corrections are found to suppress both the increase with energy of the saturation scale and the speed of the diffusive spread of the front that breaks geometric scaling. The results obtained from three of the different kernels proposed in the literature are somewhat different in size. I have also studied the differential equation approximation of the evolution equation and found that the RG-resummation qualitatively changes the obtained equations.\n\nThe study of RG-improvements to the BK equation at fixed or running coupling is the first step in understanding higher order corrections to saturation equations. Recent studies of how to rigorously include a running coupling into the BK and Balitsky--JIMWLK equations are not conclusive \\cite{running}, and the full NLL corrections to the BK equations have not been computed. It is not clear if the linearized version of the NLL BK equation would be the same as the NLL BFKL equation, but it is a reasonable assumption. Therefore, at this time, it is useful to consider the effect of RG-resummed kernels on the solutions of the BK equation. \n\nIt would be interesting to compare the results obtained in this paper to numerical simulations for various kernels. The results could also be used in comparing to experimental data along the lines of e.g.~\\cite{Iancu:2003ge}. Another interesting question is to ask what happens when fluctuations~\\cite{Iancu:2004es} are included in the picture. \n\n\n\n\\begin{acknowledgments}\nI am very grateful to St\\'ephane Munier for support, discussions and helpful suggestions. I also want to thank him and Robi Peschanski for useful comments on the manuscript.\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s:intro}\n\n\nSample regularity\nof random fields and stochastic processes has been first studied by Kolmogorov in the 1930's, as reported in~\\cite{S37}, and extended by Chentsov in~\\cite{C56}. The Kolmogorov--Chentsov theorem states the existence of continuous modifications of stochastic processes and derives bounds on the exponent of sample H\\\"older continuity. This result has been extended in many ways, for example: for random fields on cubes~\\cite{A10}, for random fields on the sphere~\\cite{LS13}, for random fields on metric spaces~\\cite{P09, HJ91}, and for random fields with values in more general spaces~\\cite{PZ07, MS03}. For a review on the literature and the history of the problem we refer the reader to the introduction of~\\cite{P09}. \n\nThe first objective of this paper\nis to extend these results to random fields on domains.\nIn fact, we also show sample differentiability\nunder suitable further assumptions on the random fields.\nThe second objective \nis to extend these results\nto random fields on manifolds.\n\nIn contrast to H\\\"older continuity of stochastic processes and random fields, sample differentiability has hardly been studied so far, partly because the Brownian motion and the related\nstochastic processes are almost surely nowhere differentiable. \nIn recent years, however, the question of smoothness of random fields (beyond H\\\"older continuity) has become more important. In particular, higher spatial smoothness of solutions of stochastic partial differential equations can be exploited to improve the order of convergence of numerical algorithms. \nOther examples\ninclude solutions of random partial differential equations as presented in~\\cite{C12}, or lognormal random fields on the sphere in the modeling of ice crystals (cf.~\\cite{NMF04,LS13}). Results on existence of sample differentiable modifications for stochastic processes are presented in~\\cite{CL67} and~\\cite{L78}. Furthermore, differentiability on~$\\mathbb{R}^n$ of Gaussian random \nfields is discussed in~\\cite{AT07} and first order sample derivatives are obtained in~\\cite{P10} with a differential and integral calculus in quadratic mean. As a first approach to manifolds, existence of higher order derivatives of isotropic Gaussian random fields on the sphere is shown in relation with the decay of the corresponding angular power spectrum in~\\cite{LS13}. To the best of our knowledge, this paper is the first to provide an extension of the Kolmogorov--Chentsov theorem to show sample differentiability of random fields on domains of cone type as well as on manifolds. \n\nThe main technical device of our proof is the Sobolev embedding theorem,\nas was sketched in~\\cite[Proof of Theorem 3.4]{DPZ92}.\nIn general, \nit\nprovides a modification (in space) of the function in question,\nand \none would need to show that the resulting random field\nis a modification of the original one.\nWe circumvent this complication by\nshowing existence of a\nsample continuous modification first.\nOur results are stated in terms of spaces of continuously differentiable functions of fractional order,\nwhich characterize the order of differentiability and the ``remaining'' H\\\"older continuity of the highest order derivatives.\n\nThe paper is organized as follows. In Section~\\ref{s:pre} we introduce the necessary basics on Sobolev spaces, Sobolev embeddings, manifolds, and random fields, as well as our notation. Section~\\ref{s:main} contains our two main results, which state the existence of H\\\"older continuous and differentiable modifications of random fields on domains of cone type, and on sufficiently smooth manifolds. The proofs can be found in the same section.\n\n\\section{Preliminaries}\n\\label{s:pre}\n\nIn this preparatory section\nwe collect the notions required \nto obtain our main results in Section~\\ref{s:main}\nin the order that is needed later on. \nTherefore, we start with the introduction of Sobolev spaces on domains, and recall the appropriate variant of the Sobolev embedding theorem.\nWe move on to define manifolds,\nand spaces of H\\\"older continuous and differentiable functions on them. \nFinally,\nwe introduce random fields on manifolds and associated properties.\n\n\nWe briefly recall the theory of (fractional) Sobolev spaces \nand spaces of H\\\"older continuous and differentiable \nfunctions on domains. \nFor details, \nwe refer the reader to \nthe standard literature\n\\cite{Adams2003, Evans1998, Triebel1978}.\n\nFor any $t > 0$,\nthe integer part~$\\floor{t} \\in \\mathbb{N}_0$ and the fractional part~$\\sawtooth{t} \\in [0, 1)$ of~$t$ \nare uniquely determined by $t = \\floor{t} + \\sawtooth{t}$.\nFor a multi-index $\\alpha \\in \\mathbb{N}_0^n$ we set $|\\alpha| := \\sum_{i=1}^n \\alpha_i$.\n\nA subset $D \\subset \\mathbb{R}^n$ is called a \\emph{domain}\nif it is nonempty, open, and connected.\nIf $D$ is a domain, we define for $t \\in \\mathbb{N}_0$\n  \\begin{equation*}\n   \\|f\\|_{\\bar{C}^t(D)}\n    := \\sum_{|\\alpha|\\le t} \\sup_{x \\in D} |\\partial^\\alpha f(x)|,\n  \\end{equation*}\nwhere $\\partial^\\alpha := \\partial^{|\\alpha|}\/(\\partial x_1^{\\alpha_1} \\cdots \\partial x_n^{\\alpha_n})$\nis the classical partial derivative,\nand for noninteger $t > 0$\n  \\begin{equation*}\n   \\norm{f}_{\\bar{C}^t(D)}\n    := \n    \\norm{f}_{\\bar{C}^{\\floor{t}}(D)}\n\t+ \n\t\\sum_{|\\alpha| \\le \\floor{t}}\n\t\\sup_{\\substack{x,y \\in D \\\\ x \\neq y}}\n\t\\frac{|\\partial^\\alpha f(x) - \\partial^\\alpha f(y)|}{|x-y|^{\\sawtooth{t}}}.\n  \\end{equation*}\n  %\n  %\nFor $t > 0$ we define the \\emph{H\\\"older spaces}\n  \\begin{equation*}\n   \\bar{C}^t(D)\n    := \n    \\left\\{ \n    f : D \\rightarrow \\mathbb{R} ;\n    f\n    \\text{ is $\\floor{t}$ times continuously differentiable and }\n    \\|f\\|_{\\bar{C}^t(D)} < + \\infty\n    \\right\\}\n    .\n  \\end{equation*}\n\nThe \\emph{Lebesgue space~$L^p(D)$}, $p \\in [1, \\infty)$, comprises \nall measurable functions $u:D \\rightarrow \\mathbb{R}$ \nfor which\n$\n\t\\|u\\|_{L^p(D)}^p\n\t:= \\int_D |u(x)|^p \\, dx\n$\nis finite.\nFunctions that are equal for almost every $x \\in D$ are identified.\nFor $p \\in [1, \\infty)$ and $k \\in \\mathbb{N}$,\nthe \\emph{Sobolev space $W^k_p(D)$} is defined by\n  \\begin{equation*}\n   W^k_p(D)\n    := \\{ u \\in L^p(D) : \\partial^\\alpha u \\in L^p(D) \\text{ for all } 0 \\le |\\alpha| \\le k\\},\n  \\end{equation*}\n  where $\\partial^\\alpha$ denotes the distributional partial derivative.\n  %\n  Equipped with the norm $\\|\\cdot\\|_{W^k_p(D)}$ given by\n  \\begin{equation*}\n   \\|u\\|_{W^k_p(D)}^p\n    := \\sum_{0 \\le |\\alpha| \\le k} \\|\\partial^\\alpha u\\|_{L^p(D)}^p,\n  \\end{equation*}\nit becomes a Banach space (see e.g.~\\cite[Thm.~3.3]{Adams2003}).\n\nFinally, we extend the definition of Sobolev spaces to nonintegers for bounded domains $D$ of cone type\nfollowing\n\\cite[Def.~4.2.3]{Triebel1978}.\nAs noted there,\nexamples of bounded domains of cone type \ninclude\nopen cubes\nand\nbounded domains with a smooth (or $C^1$) boundary.\n\n\\begin{definition}\n\tA bounded domain~$D$ is said to be of \\emph{cone type}\n\tif there exist \n\tdomains $U_1, \\ldots, U_m$,\n\tand cones $C_1, \\ldots, C_m$,\n\twhich may be carried over by rotations into the cone of height~$h$\n\t\\[\n\t\tK_h := \\{ x = (x',x_n) \\in \\mathbb{R}^n : 0 < x_n < h, |x'| < a x_n\\}\n\t\\]\n\twith fixed $a>0$ such that \n\t$\\partial D \\subset \\bigcup_{j=1}^m U_j$\n\tand \n\t$(U_j \\cap D) + C_j \\subset D$ for all $j = 1, \\ldots, m$.\n\\end{definition}\n\n\nThe Sobolev spaces of fractional smoothness \nare obtained by setting\n  \\begin{equation*}\n   W^s_p(D) :=\n    \\{\n\t\t\tu \\in W^{\\floor{s}}_p(D)\n\t\t\t:\n\t\t\t\\|u\\|_{W^s_p(D)} < \\infty\n\t\t\\},\n  \\end{equation*}\nwhere\n\\begin{equation*}\n \\|u\\|^p_{W^s_p(D)}\n  := \\|u\\|_{L^p(D)}^p\n    + \\sum_{|\\alpha| = \\floor{s}} \\int_{D \\times D} \\frac{|\\partial^\\alpha u(x) - \\partial^\\alpha u(y)|^p}{|x-y|^{n+\\sawtooth{s}p}} \\, dx \\, dy.\n\\end{equation*}\nThe defined norm~$\\|\\cdot\\|_{W^s_p(D)}$ is equivalent to the norm induced by the real method of interpolation by Remark~4.4.2\/2 in~\\cite{Triebel1978}. Together with \\cite[Theorem~4.6.1(e)]{Triebel1978}, we deduct the following theorem.\n\\begin{theorem} \\label{t:Wsp-Ct}\n\tLet $D \\subset \\mathbb{R}^n$ be a bounded domain of cone type, \n\t$1 < p < \\infty$, and $t \\ge 0$. \n\t%\n\tThen for all $s > t + n\/p$ one has a continuous embedding\n\t$\n\t\tW^s_p(D) \\hookrightarrow \\bar{C}^t(D).\n\t$\n\tThe embedding is still valid for $s = t + n\/p$ if $t \\notin \\mathbb{N}_0$.\n\\end{theorem}\n\n\n\nWe generalize the spaces of H\\\"older continuous and differentiable functions~$\\bar{C}^t$ to manifolds by imposing these properties on charts.\nBefore doing so, \nwe recapitulate the necessary geometric definitions and properties.\nFor more details on manifolds, we refer the reader to \ne.g.~\\cite{Klingenberg1995,Lang1999,Lee2009,W87}.\n\n\nIf $A \\subset \\mathbb{R}^n$ is any subset, $m \\in \\mathbb{N}$, and $k \\in \\mathbb{N} \\cup \\{ 0, \\infty \\}$,\nthen $f : A \\to \\mathbb{R}^m$\nis \\emph{$k$ times continuously differentiable}\nor \\emph{of class $C^k$}\nif for every $x \\in A$\nthere exists an open $\\mathcal{O}_x \\subset \\mathbb{R}^n$ containing $x$\nand\n$g : \\mathcal{O}_x \\to \\mathbb{R}^m$ of class $C^k$\nthat coincides with $f$ on $A$.\nSuch $f$ are collected in $C^k(A; \\mathbb{R}^m)$.\nLooking ahead,\nin order to avoid technicalities\nwe will only consider \nmanifolds \n\\emph{without manifold boundary},\nsuch as the Euclidean space or a sphere therein.\nThe first step \nis the definition of an atlas.\n\n\\begin{definition}\n\tLet $M$ be a set, $r \\in \\mathbb{N} \\cup \\{ 0, \\infty \\}$, and $n \\in \\mathbb{N}$.\n\t%\n\tA \\emph{$C^r$ $n$-atlas~$\\mathcal{A}$ on $M$}\n\tis a collection of \n\t\\emph{charts} $(U_i, \\varphi_i)$, $i \\in I$,\n\tindexed by an arbitrary set $I$,\n\tsatisfying the following:\n\t\\begin{enumerate}\n\t\\item\n\t\t$U_i \\subset M$ and $\\bigcup_{i \\in I} U_i = M$,\n\t\\item\n\t\t$\\varphi_i : U_i \\to \\varphi_i(U_i) \\subset \\mathbb{R}^n$\n\t\tis a bijection\n\t\tand for any $i, j \\in I$,\n\t\t$\\varphi_i(U_i \\cap U_j)$ is open in $\\mathbb{R}^n$,\n\t\\item\n\t\t$\n\t\t\t\\varphi_i \\circ \\varphi_j^{-1} : \n\t\t\t\\varphi_j(U_i \\cap U_j) \\to \\varphi_i(U_i \\cap U_j)\n\t\t$\n\t\tis a $C^r$ diffeomorphism\n\t\tfor any $i, j \\in I$.\n\t\\end{enumerate}\n\\end{definition}\n\nIn the following, \nwe omit the dimension~$n$ \nin reference to an atlas.\nTwo $C^r$ atlases on a set~$M$\nare called equivalent\nif \ntheir union\nis again a $C^r$ atlas on $M$.\nThis indeed defines an equivalence relation\non the $C^r$ atlases on $M$.\nThe union\nof all atlases in\nsuch an equivalence class\nis again an atlas in the equivalence class,\ncalled the \\emph{maximal $C^r$ atlas}.\nThe topology on~$M$ induced by any maximal $C^r$ atlas\nis the empty set together with arbitrary unions of \nits chart domains.\n\n\t\n\\begin{definition}\n\tLet $n \\in \\mathbb{N}$ and $r \\in \\mathbb{N} \\cup \\{ 0, \\infty \\}$.\n\t%\n\tA \\emph{$C^r$ $n$-manifold $M$}\n\tis a set $M \\neq \\emptyset$ \n\ttogether with a maximal $C^r$ atlas $\\mathcal{A}(M)$\n\tsuch that the induced topology\n\tis Hausdorff and paracompact.\n\\end{definition}\n\nRecall that in a Hausdorff topological space\ndistinct points have disjoint open neighborhoods,\nand a topological space\nis called paracompact\nif\nevery open cover admits\na locally finite open cover \n(i.e., for any point, there is an open neighborhood \nwhich intersects only finitely many members of the collection)\nwhich\nrefines the original cover.\nUsually, \nthe maximal atlas is not mentioned explicitly\nand \n$M$ is understood to be equipped with the induced topology.\nWe will say ``a chart on $M$'' to refer to a chart\nin $\\mathcal{A}(M)$.\nFurther, we say that an atlas $\\mathcal{A}$ on $M$\nis an ``atlas for $M$''\nif it is equivalent to $\\mathcal{A}(M)$.\nAny open subset $U \\subset M$ canonically inherits\nthe manifold structure of $M$.\n\n\n\\begin{definition}\n\tLet $M$ be a $C^r$ $n$-manifold\n\tand\n\t$k \\geq 0$ an integer.\n\t%\n\tA function $f : M \\to \\mathbb{R}$\n\tis said to be of class $C^k$,\n\tdenoted by $f \\in C^k(M)$,\n\tif $f \\circ \\varphi^{-1} \\in C^k( \\varphi(U) )$\n\tfor every  chart $(U, \\varphi)$ on~$M$.\n\t%\n\tThe \\emph{support} $\\mathop{\\mathrm{supp}} f$ of $f \\in C^0(M)$ is\n\tthe closure of the set $\\{ x \\in M : f(x) \\neq 0 \\}$.\n\n\tFor any $t \\geq 0$,\n\ta function $f : M \\to \\mathbb{R}$ is said to be \n\t\\emph{continuous}\n\t(\\emph{locally of class $\\bar{C}^t$})\n\tif\n\tfor any $x \\in M$\n\tthere exists an open connected subset $V \\subset M$,\n\t$x \\in V$,\n\tsuch that\n\tfor any chart $(U, \\varphi)$\n\twith\n\t$U \\subset V$,\n\tthe composite function\n\t$f \\circ \\varphi^{-1} : \\varphi(U) \\to \\mathbb{R}$\nis continuous\n(of class $\\bar{C}^t$).\n\n\\end{definition}\n\nA useful technical device\nis the partition of unity\ndefined next.\n\n\\begin{definition}\n\tLet $M$ be a $C^r$ $n$-manifold\n\t%\n\tand let $\\mathcal{U} = \\{ U_i \\}_{i \\in I}$ be an open cover of $M$.\n\t%\n\tA \\emph{$C^r$ partition of unity subordinate to $\\mathcal{U}$}\n\tis \n\ta collection\n\t$\\{ \\psi_i \\}_{i \\in I} \\subset C^r(M)$\n\tsuch that\n\t\\begin{enumerate}\n\t\\item\n\t\t$0 \\leq \\psi_i(x) \\leq 1$ for all $i \\in I$ and $x \\in M$,\n\t\\item\n\t\tthere exists\n\t\ta locally finite open cover\n\t\t$\\{ V_i \\}_{i \\in I}$ of~$M$\n\t\twith\n\t\t$\\mathop{\\mathrm{supp}} \\psi_i \\subset V_i \\cap U_i$,\n\t\t%\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\item\n\t\t$\\sum_{i \\in I} \\psi_i(x) = 1$ for all $x \\in M$\n\t\t(where the sum is finite by the previous assertion).\n\t\\end{enumerate}\n\\end{definition}\n\nThe assumed paracompactness of $M$\nimplies\nthe existence of such partitions of unity,\nsee \n\\cite[Chapter II, Corollary 3.8]{Lang1999}\nor\n\\cite[Theorem 1.73]{Lee2009}, which is stated in the following proposition.\n\n\\begin{proposition}\\label{p:partof1}\n\tLet $M$ be a $C^r$ $n$-manifold.\n\t%\n\tLet $\\mathcal{U} = \\{ U_i \\}_{i \\in I}$ be an open cover of $M$.\n\t%\n\tThen there exists a $C^r$ partition of unity\n\tsubordinate to $\\mathcal{U}$.\n\\end{proposition}\n\n\n\n\n\n\nWe close the preparatory section\nby introducing\nrandom fields on manifolds.\nRandom fields on domains are defined accordingly. \nIn what follows, let $(\\Omega, \\mathcal{F}, P)$ be a probability space.\n\\begin{definition}\nLet $M$ be a $C^r$ $n$-manifold and let $\\mathcal{B}(M)$ denote its Borel $\\sigma$-algebra.\nA mapping $X:\\Omega \\times M \\rightarrow \\mathbb{R}$ \nthat is $(\\mathcal{F} \\otimes \\mathcal{B}(M))$-measurable \nis called a \\emph{(real-valued) random field} on the manifold~$M$.\nA random field $Y$ is \na \\emph{modification} of \na random field $X$ if $P(X(x) = Y(x)) = 1$ for all $x \\in M$.\nFor any $t \\geq 0$,\na random field $X$ on $M$ is said to be \n\\emph{continuous} (\\emph{locally of class $\\bar{C}^t$})\nif\n$X(\\omega)$ is continuous (locally of class $\\bar{C}^t$)\nfor all $\\omega \\in \\Omega$.\n\\end{definition}\nWe note that \nif $M$ is endowed with a metric \n(say, given by a Riemannian metric),\nand the resulting metric space is separable and locally compact,\nthen\nmeasurability of $X(x)$ for all $x \\in M$\nand\ncontinuity in probability of $X$\nimply\n$(\\mathcal{F} \\otimes \\mathcal{B}(M))$-measurability of $X$\n(cf.~\\cite{P09_1}).\n\n\\section{H\\\"older continuity and differentiability of random fields}\\label{s:main}\n\nThis section\ncontains our main results\non \nH\\\"older continuity and differentiability of random fields.\nWe begin by considering random fields on domains of cone type. \nAs indicated in the introduction,\nresults on sample H\\\"older continuity \non different types of domains\nare well-known\n(see, e.g., \\cite{A10,DPZ92,P09,K02}), \nbut\nsample differentiability has not been of main interest so far\n(see, however, \\cite{AT07,P10} for the available results). \nWe prove sample \nH\\\"older continuity and differentiability properties\nin Theorem~\\ref{t:KC-D}\nby revisiting\nthe approach of~\\cite[Proof of Theorem 3.4]{DPZ92}\nvia \nthe Sobolev embedding theorem.\nWe then address\nsample H\\\"older and differentiability properties \nof random fields on manifolds\nin Theorem~\\ref{t:KC-M}.\n\n\n\nWe now state our \nversion of the Kolmogorov--Chentsov\\xspace theorem on domains of cone type.\n\n\\begin{theorem}\n\\label{t:KC-D}\n Let $D \\subset \\mathbb{R}^n$ be a bounded domain of cone type \n and\n let\n $X : \\Omega \\times D \\to \\mathbb{R}$ be a random field on~$D$.\n %\n Assume that there exist $d \\in \\mathbb{N}_0$, $p>1$, $\\epsilon \\in (0, p]$, and $C>0$ such that\n the weak derivatives $\\partial^\\alpha X$ are in $L^p(\\Omega \\times D)$\n and\n  \\begin{equation}\n   \\label{e:t:KC-D:E}\n   \\E( |\\partial^\\alpha X(x) - \\partial^\\alpha X(y)|^p)\n    \\le C \\, |x-y|^{n + \\epsilon}\n  \\end{equation}\n for all $x,y \\in D$ and\n any multi-index $\\alpha \\in \\mathbb{N}_0^n$ with $|\\alpha| \\leq d$.\n Then $X$ has a modification that is locally of class $\\bar{C}^t$ for all $t < d + \\min\\{\\epsilon\/p,1-n\/p\\}$.\n\\end{theorem}\n\n\nWe remark that \n\\eqref{e:t:KC-D:E} with $\\epsilon > p$ for $\\alpha = 0$ would imply \nthat \nalmost every sample of \nthe random field is actually a constant function\n(cf.~\\cite[Proposition~2]{Br02}).\n\nThe proof is given in two steps.\nIn the following lemma\nwe first obtain a continuous modification of~$X$\nbased on~\\cite[Theorem~2.3.1]{K02},\nwhich we again denote by~$X$.\nIn the second step,\nwe prove Theorem~\\ref{t:KC-D}\nby\ninvoking the Sobolev embedding on~$X(\\omega)$\nfor all $\\omega \\in \\Omega$.\nSince $X(\\omega)$ is continuous for all $\\omega \\in \\Omega$,\nthis does not modify the random field,\nand there is no need to prove measurability\nof a modified field and that it is actually a modification.\nAlternatively,\none could use the last step of \\cite[Proof of Theorem 2]{Schilling2000-ExpoMath}. \n\nLet us start by showing the existence of a continuous modification.\n\n\\begin{lemma} \\label{l:K-D}\n\tUnder the assumptions of Theorem~\\ref{t:KC-D},\n\t$X$ admits a continuous modification.\n\\end{lemma}\n\n\\begin{proof\n Observe that \n the\n domain~$D$ \n equipped with the usual Euclidean metric $|\\cdot - \\cdot|$\n is a totally bounded pseudometric space in the sense of~\\cite{K02}; \n indeed, its metric entropy \n $\\mathsf{D}(\\delta) := \\mathsf{D}(\\delta; {D}, |\\cdot - \\cdot|)$ \n is bounded by $\\mathsf{D}(\\delta) \\le \\tilde{C} \\delta^{-n}$ \n for all $\\delta > 0$ and some constant $\\tilde{C}>0$, \n since the domain ${D}$ \n can be embedded into a $n$-dimensional closed cube of finite diameter.\n %\n We set $\\Psi(r) := C r^{n+\\delta}$, \n where the constant~$C$ is provided by~\\eqref{e:t:KC-D:E},\n and $f(r) := r^{\\delta\/p}$ for $r \\ge 0$. \n %\n The integrals\n   $\\int_0^1 r^{-1} f(r) \\, dr\n    = p\/\\delta\n    $\n as well as\n   $\\int_0^1 \\mathsf{D}(r) \\Psi(2r) f(r)^{-p} \\, dr\n    \\le C \\tilde{C} 2^{n+\\delta}\n  $\n are finite.\n %\n Therefore, \\cite[Theorem~2.3.1]{K02} shows the existence of a continuous modification of~$X$.\n\\end{proof}\n\n\nHaving obtained a continuous modification, we are set to continue with the proof of Theorem~\\ref{t:KC-D}.\n\n\\begin{proof}[Proof of Theorem~\\ref{t:KC-D}]\n Assume without loss of generality\n that the random field $X$ is continuous\n (otherwise apply Lemma~\\ref{l:K-D}).\n Consider arbitrary $0<\\nu<\\min\\{(n+\\epsilon)\/p,1\\}$ and $\\alpha \\in \\mathbb{N}_0^n$ with $|\\alpha|=d$.\n Since\n  \\begin{equation*}\n   (\\omega,x,y) \\mapsto\n    \\frac{|\\partial^\\alpha X(\\omega,x) - \\partial^\\alpha X(\\omega,y)|^p}{|x-y|^{n + \\nu p}}\n  \\end{equation*}\n is $(\\mathcal{F} \\otimes \\mathcal{B}(D\\times D))$-measurable, \n we apply Fubini's theorem and hypothesis \\eqref{e:t:KC-D:E} to obtain\n  \\begin{align*}\n\t  \\E \\left(\n\t    \\int_{D \\times D} \\frac{|\\partial^\\alpha X(x)-\\partial^\\alpha X(y)|^p}{|x-y|^{n + \\nu p}} \\, dx \\, dy\n\t  \\right)\n    & = \\int_{D \\times D} \\frac{\\E(|\\partial^\\alpha X(x)-\\partial^\\alpha X(y)|^p)}{|x-y|^{n + \\nu p}} \\, dx \\, dy\n    \\\\\n    & \\le C \\int_{D \\times D} |x-y|^{n + \\epsilon - (n + \\nu p)} \\, dx \\, dy\n    .\n  \\end{align*}\n The last integral is finite due to $\\epsilon - \\nu p > -n$.\n %\n With the $L^p(\\Omega\\times D)$ integrability assumptions\n on $X$ and its derivatives of order $d$,\n this implies \n that\n  \\begin{align*}\n   \\E ( \\|X\\|_{W^{d+\\nu}_p(D)}^p )\n    & = \\E( \\|X\\|_{L^p(D)}^p)\n      + \n    \\sum_{|\\alpha| = d}\n\t  \\E \\left(\n\t    \\int_{D \\times D} \n\t    \\frac\n\t    {|\\partial^\\alpha X(x)-\\partial^\\alpha X(y)|^p}\n\t    {|x-y|^{n + \\nu p}}\n\t    \\, dx \\, dy\n\t  \\right)\n  \\end{align*}\n  is finite,\n and therefore there exists $\\Omega' \\in \\mathcal{F}$ with $P(\\Omega') = 1$ \nsuch that $X(\\omega) \\in W^{d+\\nu}_p(D)$ for all $\\omega \\in \\Omega'$. \nConsider as continuous modification of~$X$ the random field\n$\\tilde{X} := \\mathds{1}_{\\Omega'} X$, where $\\mathds{1}_{\\Omega'}$ is the indicator function of~$\\Omega'$.\nBy the Sobolev embedding theorem~\\ref{t:Wsp-Ct}, \nwe get that $\\tilde{X}(\\omega) \\in \\bar{C}^t(D)$ for all \n$\\omega \\in \\Omega$ and all\n$t < d + \\nu - n\/p$.\nSince $0<\\nu<\\min\\{(n+\\epsilon)\/p,1\\}$\nwas arbitrary, the claim follows.\n\\end{proof}\n\nWe remark that $1-n\/p$ is positive only for $p > n$.\nIn the case that $n \\geq p$, \nwe obtain in Theorem~\\ref{t:KC-D}\nonly lower sample differentiability order $\\floor{t}$\nthan the assumed weak differentiability order $d$.\n\n\\begin{rem}\n\\label{r:weaker_differentiability_X}\n The assumptions in Theorem~\\ref{t:KC-D} can be weakened. If $d \\neq 0$, it is sufficient that~\\eqref{e:t:KC-D:E} holds for $|\\alpha| = d$, and that $X$ has a continuous modification as provided by Theorem~2.3.1 in~\\cite{K02} under the assumption that \\eqref{e:t:KC-D:E} holds for $\\alpha = 0$ and some $\\epsilon > 0$.\n\\end{rem}\n\nWe apply Theorem~\\ref{t:KC-D} in the following example to a Brownian motion on the interval,\nrecovering\nthe classical \nproperty of H\\\"older continuity with exponent $\\gg < 1\/2$.\n\n\\begin{example}\n\tIf $X$ is a Brownian motion on the interval~$[0,T] \\subset \\mathbb{R}^1$, $T< + \\infty$,\n\tthen Assumption~\\eqref{e:t:KC-D:E} is satisfied\n\tfor $\\alpha = 0$,\n\tany $p \\geq 2$, and $\\epsilon = p\/2 - 1$.\n\t%\n\tThus\n\t$X$ admits a \n\tmodification that is locally of class $\\bar{C}^t$\n\tfor any \n\t$0 < t < \\sup_{p \\geq 2} (p\/2 - 1) \/ p = 1\/2$,\n\twhich is the well-known result.\n\\end{example}\n\n\n\nWe are now ready to generalize Theorem~\\ref{t:KC-D}\nto random fields on manifolds.\n\n\\begin{theorem}\n\\label{t:KC-M}\n Let $M$ be a $C^r$ $n$-manifold, $r > 0$,\n and let $X : \\Omega \\times M \\to \\mathbb{R}$ be a random field on $M$.\n %\n Assume that there exist $d \\in \\mathbb{N}_0$, $p>1$, and $\\epsilon \\in (0,p]$ \n such that\n for any chart $(U, \\varphi)$ on $M$\n with bounded $\\varphi(U) \\subset \\mathbb{R}^n$,\n there exists $C_\\varphi > 0$\n such that\n the weak derivatives of\n $X_\\varphi := X \\circ \\varphi^{-1}$ satisfy \n $\\partial^\\alpha X_\\varphi \\in L^p(\\Omega \\times \\varphi(U))$ and\n  \\begin{equation*}\n   \\E( |\\partial^\\alpha X_\\varphi(x) - \\partial^\\alpha X_\\varphi(y)|^p)\n   \\le\n   C_\\varphi \\, |x-y|^{n + \\epsilon}\n  \\end{equation*}\n for all $x,y \\in \\varphi(U)$ and\n any multi-index $\\alpha \\in \\mathbb{N}_0^n$ with $|\\alpha| \\leq d$.\n %\n %\n %\n Then $X$ has a modification that is locally of class $\\bar{C}^t$ for all $t < d + \\min\\{\\epsilon\/p,1-n\/p\\}$ with $t \\le r$.\n\\end{theorem}\n\n\\begin{proof}\n\tTo obtain the continuous modification\n\twe first construct a locally finite atlas with \n\tcoordinate domains that are bounded and of cone type.\n\t%\n\tOn each of these charts, a modification of $X$\n\tis provided by Theorem~\\ref{t:KC-D}.\n\t%\n\tUsing a partition of unity we then \n\tpatch together a modification of~$X$\n\twith the desired properties.\n\t\n\tFor each $x \\in M$, let $(\\tilde{U}_x, \\tilde\\varphi_x)$ be \n\ta chart on $M$ with $x \\in \\tilde{U}_x$.\n\t%\n\tLet $D_x \\subset \\tilde\\varphi_x(\\tilde{U}_x)$ be \n\tan open ball of positive radius centered at $\\tilde\\varphi_x(x)$.\n\t%\n\tDefine $U_x := \\tilde\\varphi_x^{-1}(D_x)$ and $\\varphi_x := \\tilde\\varphi_x|_{U_x}$.\n\t%\n\tLet $\\mathcal{A} := \\{ (U_x, \\varphi_x) : x \\in M \\}$ be the resulting atlas for $M$,\n\twhich we will index\n\tby\n\t$\\Phi := \\{ \\varphi : (U, \\varphi) \\in \\mathcal{A} \\}$.\n\t\n\tNow, for each $(U_\\varphi, \\varphi) \\in \\mathcal{A}$,\n\tthe coordinate domain $\\varphi(U_\\varphi)$\n\tis a bounded domain with smooth boundary,\n\tin particular of cone type.\n\t%\n\tWith our assumptions on $X_\\varphi$,\n\tTheorem~\\ref{t:KC-D}\n\tprovides a modification\n\t$Y^\\varphi$ of the random field\n\t$X_\\varphi : \\Omega \\times \\varphi(U_\\varphi) \\to \\mathbb{R}$\n\ton $\\varphi(U_\\varphi)$,\n\twhich is \n\tlocally of class $\\bar{C}^t$\n\tfor any fixed $t < d + \\min\\{\\epsilon\/p,1-n\/p\\}$,\n\tfor each $\\varphi \\in \\Phi$.\n\t\n\tLet $\\{ \\psi_\\varphi \\}_{ \\varphi \\in \\Phi }$ \n\tbe a $C^r$ partition of unity\n\tsubordinate to $\\{ U_\\varphi \\}_{\\varphi \\in \\Phi}$,\n\twhich exists by Proposition~\\ref{p:partof1}.\n\t%\n\tDefine $Y : \\Omega \\times M \\to \\mathbb{R}$\n\tby\n\t$Y := \\sum_{\\varphi \\in \\Phi} \\psi_\\varphi Y^\\varphi \\circ \\varphi$.\n\t%\n\tSince the covering $\\{ \\mathop{\\mathrm{supp}} \\psi_\\varphi \\}_{ \\varphi \\in \\Phi }$\n\tis locally finite,\n\tthe sum is well-defined on a neighborhood of any $x \\in M$.\n\t%\n\tFurthermore, $Y$ is a random field on~$M$ because all $\\varphi \\in \\Phi$ are $C^r$ diffeomorphisms and therefore at least continuous.\n\t%\n\tMoreover, it is a modification of~$X$\n\tby the properties of the partition of unity.\n\t%\n\tOwing to the fact that $r \\geq t$,\n\tthe random field\n\t$Y$\n\tis locally of class $\\bar{C}^t$. \n\\end{proof}\n\nWe finish this section with two comments.\nFirst,\nsince we only used the assumptions on the random field on the charts to apply Theorem~\\ref{t:KC-D}, it is clear that Remark~\\ref{r:weaker_differentiability_X} carries over to Theorem~\\ref{t:KC-M}.\nSecond,\nfor an example of random fields on manifolds, we refer the reader to \\cite{LS13}.\nTherein,\nisotropic Gaussian random fields\non\nthe unit sphere in~$\\mathbb{R}^3$\nare considered\nand\nsample regularity is obtained by direct calculations.\n\n\\section*{Acknowledgment}\n\n\nThe work was supported in part by ERC AdG no.~247277. \nThe authors thank Sonja Cox, Sebastian Klein, Markus Knopf, J\\\"urgen Potthoff, and Christoph Schwab\nfor fruitful discussions and helpful comments,\nas well as Ren\\'e Schilling for pointing out reference~\\cite{Schilling2000-ExpoMath}. \nThe first author acknowledges the hospitality of \nthe Seminar for Applied Mathematics\nduring summer 2013.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nGalaxy clustering has traditionally been one of the most important ways to extract \ncosmological information. Galaxies are \nnot a faithful tracer of dark matter, as their clustering strength is biased relative to the dark matter. \nHowever, they are expected to follow the same gravitational \npotential as the dark matter and hence have the same velocities. This is not observable through \nangular clustering, which is only sensitive to correlations transverse to the line of sight. \nIt is however detectable in redshift surveys, because the redshift of the galaxy does not \nprovide information only on the radial distance, but also on the radial velocity through the Doppler shift. \nThis induces anisotropies in the clustering, which are generically called redshift space distortions (RSD) \n\\cite{1998ASSL..231..185H}. \nThey provide an opportunity to extract information on the dark matter clustering directly. \nOn large scales clustering of galaxies along the line of sight is enhanced\nrelative to the transverse direction due to peculiar motions and this allows\none to determine the ratio of logarithmic rate of growth $f$ to bias $b$ \\cite{1987MNRAS.227....1K}. Combining \nthe statistics from different lines of sight \none can eliminate the unknown bias and measure \ndirectly the logarithmic rate of growth times the amplitude. \n\nIt has been argued that using RSD information could greatly increase our knowledge of cosmological \nmodels, including tests of dark energy and general relativity \\cite{2009MNRAS.397.1348W,2009JCAP...10..007M,2011arXiv1104.3862B}. \nGalaxy clustering has clear \nadvantages over the alternatives such as weak lensing: it is intrinsically 3-dimensional, thus providing \nbetter statistics, and it has high signal to noise. \nWhile most of the predictions in the literature are model dependent, a generic statement can be made that if systematic\neffects were perfectly understood RSD would be one of the most powerful techniques for such studies. \nThe main problem with RSD is that nonlinear velocity effects extend to rather large scales and \ngive rise to a scale dependent and angular dependent clustering signal. It is easy to see these effects in any \nreal redshift survey: one sees elongated features along the line of sight, \ncalled the fingers-of-god (FoG) effect, \nwhich are caused by random velocities inside virialized objects such as \nclusters, which scatter galaxies \nalong the radial direction in redshift space, even if they have a localized \nspatial position in real space. \nThis is just an extreme example and other related effects, such as nonlinear infall streaming \nmotions, also cause nonlinear corrections. \nThis means that one needs to understand these and separate them from the nonlinear evolution of the dark \nmatter and from the nonlinear relation between the galaxies and the dark matter, both of which also give rise to \na scale dependent bias \\cite{1998MNRAS.301..797H}. \n\nSeveral recent studies have investigated these nonlinear effects \n\\cite{2004PhRvD..70h3007S,2007MNRAS.374..477T,2010PhRvD..82f3522T,2011MNRAS.410.2081J,2011arXiv1103.3614T,2011arXiv1105.4165R,2011arXiv1105.5007S}, \nsome limiting the analysis \nto dark matter only and some also including galaxies or halos. The common denominator of these studies is \nthat they are based on various ansatzes for the scale and angular dependence of RSD, typically \ncombined with some perturbation theory analysis. This has the advantage of having just a few free parameters, so \nthat if the ansatz is accurate one can model the effects accurately. The reverse is also true and the problem is \nthat it is difficult to make general statements regarding the range of validity for any given model. \nThere is another problem connected to perturbation theory, in that the usual perturbation theory makes a single stream\napproximation, which we know breaks down on small scales inside the virialized halos \n(indeed, FoG are a manifestation of multi-streaming on small scales).  \n\nIn this paper we \npresent a different approach to RSD: we use a distribution function approach to \nshow that one can make a series expansion of RSD, which is convergent on \nsufficiently large scales \nand we derive the most general form of RSD correlator allowed by the\nsymmetries. In this paper we present the formal derivations and conceptual implications, \nreserving all the applications to future work. \nThe structure of this paper is as follows: in section 2 we develop the distribution function approach\nto RSD and derive the helicity decomposition. In section 3 we discuss the power spectra, and use rotation symmetries to derive the most general \nform of the RSD correlator. We also discuss the lowest order contributions\nand connect them to physical quantities such as density, momentum, stress energy tensor etc. \nThis is followed by a discussion in section 4. \n\n\n\\section{Redshift-space distortions from the distribution function}\n\\label{sec:theory}\nThe exact evolution of collisionless particles is described by the\nVlasov equation \\cite{1980lssu.book.....P}.  Following the discussion by\n\\cite{2011JCAP...04..032M}, we start from the distribution function of\nparticles $f(\\mathbf{x},\\mathbf{q},\\tau)$ at a phase-space position $(\\mathbf{x},\\mathbf{q})$ and \nat conformal time $\\tau$ in order\nto derive the perturbative redshift-space distortions.  Here $\\mathbf{x}$ is the \ncomoving position and $\\mathbf{q}=\\mathbf{p}\/a=m\\mathbf{u}$ is the comoving momentum, where \n$\\mathbf{u}=d\\mathbf{x}\/d\\tau$. \nIn the following we will omit the time dependence, i.e we will \nwrite $f(\\mathbf{x},\\mathbf{q})$. \nThe density\nfield in real space is obtained by averaging the distribution function\nover momentum:\n\\begin{equation}\n  \\rho\\left(\\mathbf{x}\\right)\\equiv m_p \\int d^3 \\mathbf{q} ~f\\left(\\mathbf{x},\\mathbf{q}\\right),\n\\end{equation}\nwhere $m_p$ is the particle mass\nand $a=1\/(1+z)$ is the scale factor ($z$ is the redshift).\nIn redshift space the position is distorted by peculiar\nvelocities, thus the comoving redshift-space coordinate for a particle is given\nby $\\mathbf{s}=\\mathbf{x}+\\hat{r}~u_\\parallel\/ \\mathcal{H}$, where $\\hat{r}$ is\nthe unit vector pointing along the observer's line of sight, $u_\\parallel$ is \nthe radial velocity, \n$m_pu_\\parallel=q_\\parallel = \\mathbf{q}\\cdot \\hat{r}$, and $\\mathcal{H}=aH$, where $H$ is the Hubble\nparameter.  Then the mass density in redshift space is given by\n\\begin{equation}\n  \\rho_s\\left(\\mathbf{s}\\right)=\n  m_p~ \\int d^3\\mathbf{x}~ d^3\\mathbf{q}~ f\\left(\\mathbf{x},\\mathbf{q}\\right)\n  \\delta^D\\left(\\mathbf{s}-\\mathbf{x}-\\hat{r}\\frac{u_\\parallel}{  \\mathcal{H}}\\right)=\n  m_p~ \\int d^3\\mathbf{q}~ f\\left(\\mathbf{s}-\\hat{r}\n  \\frac{u_\\parallel}{  \\mathcal{H}},\n  \\mathbf{q}\\right)~. \\label{eq:rhos}\n\\end{equation}\nBy Fourier transforming equation \\ref{eq:rhos}, we find\n\\begin{eqnarray}\n  \\rho_s\\left(\\mathbf{k}\\right)&=&\n  m_p~ \\int d^3\\mathbf{x}~ d^3\\mathbf{q}~ f\\left(\\mathbf{x},\\mathbf{q}\\right)\n  e^{i \\mathbf{k}\\cdot \\mathbf{x} + i k_\\parallel u_\\parallel\/  \n  \\mathcal{H}} \\nonumber \\\\ &=&\n  m_p~ \\int d^3\\mathbf{x}~ e^{i \\mathbf{k}\\cdot \\mathbf{x} } \n  \\int d^3\\mathbf{q}~ f\\left(\\mathbf{x},\\mathbf{q}\\right)\n  e^{i k_\\parallel u_\\parallel\/  \\mathcal{H}} ~,  \\label{eq:rhok}\n\\end{eqnarray}\nwhere $\\mathbf{k}$ is the wavevector in redshift space, corresponding to the\nredshift-space coordinate $\\mathbf{s}$.\n\nNow we expand the second integral in equation \\ref{eq:rhok} as a Taylor series \nin $k_\\parallel u_\\parallel\/  \\mathcal{H}$,\n\\begin{eqnarray}\n  m_p~ \\int d^3\\mathbf{q}~ f\\left(\\mathbf{x},\\mathbf{q}\\right)\n  e^{i k_\\parallel u_\\parallel\/  \\mathcal{H}} \n  &=&m_p~ \\int d^3\\mathbf{q}~ f\\left(\\mathbf{x},\\mathbf{q}\\right)\n  \\sum_{L=0}\\frac{1}{L!} \\left(i k_\\parallel u_\\parallel\/  \\mathcal{H}\\right)^L\n  \\nonumber \\\\&=&\\bar{\\rho}\\left[\\sum_{L=0}\\frac{1}{L!}\n    \\left(\\frac{i k_\\parallel}{\\mathcal{H}}\\right)^L T_\\parallel^L(\\mathbf{x})\\right] ~,\n\\end{eqnarray}\nwhere $\\bar{\\rho}$ is the mean mass density and\n\\begin{equation}\n  T_\\parallel^L(\\mathbf{x})={m_p \\over \\bar{\\rho}} \n  ~ \\int d^3\\mathbf{q}~ f\\left(\\mathbf{x},\\mathbf{q}\\right) \n  u_\\parallel^L\n, \\label{eq:q_def}\n\\end{equation}\nwhere in the last expression the integral over phase space assures that the quantity \nis defined in terms of a\nsum over all particles at position $\\mathbf{x}$. \nFor a single stream at $\\mathbf{x}$ this is just \n$(1+\\delta(\\mathbf{x})) u_\\parallel^L(\\mathbf{x})$.\nThese\nare thus  radial components of the moments of the distribution function and\nthe distribution function description allows for inclusion of both bulk velocities and\nmulti-streamed velocities.  \nNote that these quantities are\nmass-weighted, and so well-defined for any system: one just sums \nover all the particles in the system weighting each one by the appropriate \npower of their radial velocities. If the field needs to be defined on a grid \na simple assignment scheme of particles to the grid \nsuffices, and empty grid cells are assigned a value of 0. \nWe note this to contrast it with volume weighted quantities which need to \nbe \ndefined even if there are no particles assigned to a given grid cell, which is \noften impossible \nfor sparse biased tracers, specially in underdense regions. We return to this issue later. \n\n The Fourier component of the\ndensity fluctuation in redshift space is\n\\begin{equation}\n  \\delta_s(\\mathbf{k})= \n  \\sum_{L=0}\\frac{1}{L!}\n  \\left(\\frac{i k_\\parallel}{\\mathcal{H}}\\right)^L T_\\parallel^L(\\mathbf{k}) ~,\n  \\label{eq:deltak}\n\\end{equation} \nwhere $T_\\parallel^L(\\mathbf{k})$ is the Fourier transform of $T_\\parallel^L(\\mathbf{x})$.\n\\begin{equation}\nT_\\parallel^L(\\mathbf{k})=\\int d^3\\mathbf{x}~ T_\\parallel^L(\\mathbf{x})\n  e^{i \\mathbf{k}\\cdot \\mathbf{x}}.\n\\end{equation}\nFor $L=0$ we have\n$T_\\parallel^0(\\mathbf{k})=\\delta(\\mathbf{k})$, \nthe density fluctuation in real space. \n\n\\subsection{Angular decomposition of moments of distribution function}\n\nThe objects $T_\\parallel^L(\\mathbf{x})$ introduced in equation \\ref{eq:q_def}\nare radial components of moments of the\nthe distribution function, which are rank $L$ tensors,\n\\begin{equation}\nT^L_{i_1,i_2,..i_L}={m_p \\over \\bar{\\rho}}\n  ~ \\int d^3\\mathbf{q}~ f\\left(\\mathbf{x},\\mathbf{q}\\right)u_{i_1}u_{i_2}...u_{i_L}.\n\\label{eqTfulldef}\n\\end{equation}\nThe real-space density field corresponds to\n$L=0$, i.e. zeroth moment, \nthe $L=1$ moment corresponds to the momentum density, \n$L=2$ gives the stress energy density tensor etc.  \nThese objects are symmetric under exchange of any two indices and \nhave $(L+1)(L+2)\/2$ independent components. They can be decomposed into \nhelicity eigenstates under rotation around $\\mathbf{k}$, as we do next. \n\nSince translational symmetry guarantees that each Fourier mode is only correlated \nwith itself, we can work with each Fourier mode separately, and add them  appropriately\nin the end when we discuss the power spectra. By symmetry we may take  $\\mathbf{k}$ to be along $z$-axis. \nWe can decompose the distribution function into spherical harmonics, \n\\begin{equation}\nf(\\mathbf{k},q,\\theta,\\phi)=\\sum_{l=0}^{\\infty}\\sum_{m=-l}^{m=l}f_l^m(\\mathbf{k},q)Y_{lm}(\\theta,\\phi),\n\\label{flm}\n\\end{equation}\nwhere $q$ is the amplitude of the momentum (often a term $(-i)^{l}\\sqrt{{4 \\pi \\over 2l+1}}$ is inserted into this \nexpansion, but we will drop all such terms here). \nThe components $f_l^m(\\mathbf{k},q)$ are helicity eigenmodes (i.e., eigenmodes of angular momentum component in $z$-direction \n$L_z=-i\\partial\/\\partial \\phi$) and under rotation by angle $\\psi$ around the $z$-axis \nthey transform as \n\\begin{equation}\nf_l^m(\\mathbf{k},q)'=e^{im\\psi}f_l^m(\\mathbf{k},q).\n\\end{equation}\nThis follows from the transformation properties of spherical harmonics. \nA quantity which transforms under rotation according to this equation is said to \nhave helicity $m$. A quantity with helicity $0$ is called a scalar, that with helicity \n$m=\\pm 1$ is called a vector and that with $m=\\pm 2$ a tensor, but the expansion goes to \narbitrary values of $m$. \n\nMoments of the distribution function are defined in terms of integrals of velocity moments \nover the distribution function. \nWe can define helicity eigenstates of moments of the distribution function as \n\\begin{equation}\nT_l^{L,m}(\\mathbf{k})={4\\pi m_p \\over \\bar{\\rho}}~\\int q^2dq u^L f_l^m(\\mathbf{k},q).\n\\label{tllm}\n\\end{equation}\nNote that each term $T_{l}^{L,m}$ contains $L$ powers of velocity \n$u$ ($u\\equiv\\left|\\mathbf{u}\\right|$, and recall that $\\mathbf{q} = m \\mathbf{u}$). \n\nA general rank $L$ tensor $T^L_{i_1,i_2,..i_L}$ can be decomposed into \n$2L+1$ helicity eigenmodes $T_L^{L,m}$ ($m=-L,..,L$) and \nadditional components  \nformed \nby a product of a scalar $u^2$ with rank $L-2$ tensors. This gives \nadditional $2(L-2)+1$ \nhelicity eigenmodes \n$T_{L-2}^{L,m}$ ($m=-(L-2)..(L-2)$), and additional components formed \nagain by $u^2$ and rank $L-4$ tensor, which gives additional\n$2(L-4)+1$ helicity eigenmodes\n$T_{L-4}^{L,m}$ ($m=-(L-4)..(L-4)$) etc. \n\nFor the lowest terms we have\nrank 1 tensor $T_i^1$, momentum density, which is a 3-vector in the usual geometrical context, \nand can be decomposed into a $m=0$ helicity scalar component $T_1^{1,0}$ \nand two $m=\\pm 1$ helicity vector components $T_1^{1,\\pm 1}$. \nA general symmetric rank 2 tensor $T_{ij}^2$ has 6 independent components. These \ncan be decomposed into an isotropic rank 0 helicity scalar term $T^{2,0}_0=(1+\\delta)u^2$, \nwhich corresponds to the energy density, and 5 $l=2$ components: one helicity scalar part of anisotropic \nstress tensor $T^{2,0}_2$, two helicity vector part of anisotropic stress tensor $T^{2,\\pm 1}_2$\nand two helicity tensor part of anisotropic stress tensor $T^{2,\\pm 2}_2$.\nAt $L=3$ we have 10 independent components, of which 7 are $T^{3,m}_3$, i.e. $l=3$, $m=-3,..,3$, \nand 3 are $T^{3,m}_1$, $l=1$, $m=-1,0,1$ tensors, \nformed by taking isotropic $u^2$ and multiplying it with a 3-vector $u_i$, the latter of which can be \ndecomposed into $m=-1,0,1$ components. \n\nOne can show that $2L+1+2(L-2)+1+2(L-4)+1+...=(L+1)(L+2)\/2$, so this decomposition \ngives the required number of independent components of a general symmetric tensor of rank $L$. \nIn analysis of \ngeneral relativity it is customary to expand the metric and stress energy tensor into scalar ($m=0$), vector ($m=\\pm 1$)\nand tensor ($m=\\pm 2$) helicity modes (SVT decomposition). No higher order helicity modes \nare needed, since only tensors of rank 0, 1 and 2 enter \ninto the description of the metric and energy momentum tensor. \nIn contrast the moments of distribution function contain tensors of arbitrary rank and the \nexpansion in equations \\ref{flm},\\ref{tllm} is the appropriate generalization of the SVT decomposition.\n\nSo far we worked in the basis defined by $\\mathbf{k}$ pointing in $z$ direction. \nIn general we are interested in computing the components of the moments in the radial direction $\\hat{r}$. \nIf $\\hat{r}$ is parallel to $\\mathbf{k}$ then only $m=0$ components contribute, while for a general direction all of them do.\nThe angular dependence of the moments is obtained\nby performing a rotation of the basis from $z||\\mathbf{k}$ to $z'||\\hat{r}$. \nWe can achieve this by rotating by $\\phi$ around $z$ and then by $\\theta$ around the axis perpendicular to $z$, $z'$, \nso in terms of the general rotation by 3 Euler angles we have $T_{l}^{L,m'}=\\sum_{-m}^{m}D^l_{m,m'}(\\phi,\\theta,0)T_{l}^{L,m}$, \nwhere $D^l_{m,m'}(\\phi,\\theta,\\phi')$ is the general rotation matrix of spin $l$ associated with the 3 Euler angles $\\phi$, $\\theta$, $\\phi'$ \n(we do not need to perform the rotation around $z'$ by $\\phi'$). \nSince $u_{\\parallel}$ is invariant under rotation around $z'||\\hat{r}$ only $m'=0$ survive. \nThe rotation matrix is given by the spherical harmonics, $D^l_{0,m}(\\phi,\\theta,0)=\\sqrt{4\\pi\/(2l+1)}Y_{lm}(\\theta,\\phi)$. \nCombining all together we find\n\\begin{equation}\nT_{\\parallel}^L(\\mathbf{k})=\\sum_{(l=L,L-2,..)}\\sum_{m=-l}^{m=l}n_l^LT_l^{L,m}(\\mathbf{k})Y_{lm}(\\theta,\\phi),\n\\label{tpl}\n\\end{equation}\nwhere $N_l^L$ is a constant independent of angle whose numerical value  will not be needed. \n\n\n\\section{Power spectra}\\label{sec:power_th}\nWe will adopt a plane-parallel approximation, where   \nonly the angle between the line of sight and the Fourier mode \nneeds to be specified. \nThe redshift-space power spectrum is defined as $\\langle \\delta_s(\\mathbf{k})\\delta_s^*(\\mathbf{k}')\\rangle=P^{ss}(\\mathbf{k})\\delta_D(\\mathbf{k}-\\mathbf{k}')$.\nEquation \\ref{eq:deltak} gives,\n\\begin{equation}\n  P^{ss}(\\mathbf{k})=\n  \\sum_{L=0}^{\\infty}\\sum_{L'=0}^{\\infty}\\frac{\\left(-1\\right)^{L'}}{L!~L'!}\n  \\left(\\frac{i k_\\parallel}{\\mathcal{H}}\\right)^{L+L'} P_{LL'}(\\mathbf{k}) \\label{eq:p_ss0} ~,\n\\end{equation}\nwhere $P_{LL'}(\\mathbf{k})\\delta(\\mathbf{k}-\\mathbf{k}')=\\langle T^{L}_\\parallel(\\mathbf{k}) (T^{*L'}_\\parallel(\\mathbf{k}') \\rangle$.  \nNote that $P_{LL'}(\\mathbf{k})=P_{L'L}(\\mathbf{k})^*$ so that the\ntotal result is real valued, as expected. Thus we only need to consider the terms $P_{LL'}(\\mathbf{k})$ with \n$L\\le L'$, each of which comes with a factor of 2 if $L \\ne L'$ and 1 if $L=L'$. \nWe can also write $k_{||}\/k=\\cos \\theta=\\mu$, \n\\begin{equation}\n  P^{ss}(\\mathbf{k})=\\sum_{L=0}^{\\infty}\\frac{1}{L!^2}\\left(\\frac{ k\\mu}{\\mathcal{H}}\\right)^{2L} P_{LL}(\\mathbf{k}) +\n  2\\sum_{L=0}^{\\infty}\\sum_{L'>L}\\frac{\\left(-1\\right)^{L'}}{L!~L'!}\n  \\left(\\frac{i k\\mu}{\\mathcal{H}}\\right)^{L+L'} P_{LL'}(\\mathbf{k}) \\label{eq:p_ss} ~.\n\\end{equation}\n\n\nNext we want to insert the helicity decomposition of equation \\ref{tpl} and \nconsider the implications of rotational symmetry on the power spectrum. \nEach term $P_{LL'}(\\mathbf{k})$ contains products of multipole moments \n\\begin{equation}\nT_l^{L,m}Y_{lm}(\\theta,\\phi)[T_{l'}^{L',m'}Y_{l'm'}(\\theta,\\phi)]^* \\propto e^{i(m-m')\\phi}.\n\\end{equation}\nUpon averaging over the azimuthal angle $\\phi$ of Fourier modes all the terms \nwith $m\\ne m'$ vanish. Another way to state this is that upon rotation by angle $\\Psi$ the correlators\npick up a term $e^{i(m-m')\\Psi}$, and in order for the power spectrum to be rotationally invariant we \nrequire $m=m'$. Putting it all together we find\n\\begin{equation}\nP_{LL'}(\\mathbf{k})=\\sum_{(l=L,L-2,..)}\\sum_{(l'=L',L'-2,..;\\; l'\\ge l)}\\sum_{m=0}^{l}P^{L,L',m}_{l,l'}(k)P_l^m(\\mu)P_{l'}^m(\\mu),\n\\label{pll}\n\\end{equation}\nwhere $P_l^m(\\mu=\\cos \\theta)$ are the associated Legendre polynomials, which determine the $\\theta$ angular dependence\nof the spherical harmonics, $Y_{lm}(\\theta,\\phi)=\\sqrt{(2l+1)(l-m)!\/4\\pi(l+m)!}P_l^m(\\cos \\theta) e^{im\\phi}$.\nWe \nabsorbed all of the terms that depend on $l$ and $m$ and various constants into the definition of power spectra $P^{L,L',m}_{l,l'}(k)$, \nreplaced the two helicity states $\\pm m$ by a single one with $m>0$, since their $\\theta$ angular dependencies are the same, \nand we absorbed the factor of 2 into the definition of $P^{L,L',m}_{l,l'}(k)$. We also require $l'\\ge l$ and \nabsorb the factor of 2 into the definition of $P^{L,L',m}_{l,l'}(k)$ since the two terms have the same\nangular structure. \nNote that due to statistical isotropy the spectra $P^{L,L',m}_{l,l'}(k)$ depend only on amplitude of $k$, \ni.e. we have \n\\begin{equation}\nP^{L,L',m}_{l,l'}(k) \\propto \\langle T_l^{L,m}(\\mathbf{k})(T_{l'}^{L',m}(\\mathbf{k}))^* \\rangle .\n\\end{equation}\nAll the angular structure is thus in associate Legendre polynomials \n$P_l^m(\\mu)$. \n\nEquations \\ref{eq:p_ss} and \\ref{pll} are the main result of this paper. \nThey show that there exists a well defined expansion in terms of cross and auto-power spectra of velocity \nmoments. The expansion parameter is roughly defined as $k\\mu u\/\\mathcal{H}$, where \n$u$ is related to a typical gravitational velocity of the system (which should be of the order of hundreds of km\/s, \nbut note that we take higher and higher powers of these velocities in the series). \nThe expansion is convergent if the expansion parameter is less than unity. \n\nIn terms of perturbation theory there is a close, but not one to one, relation between the lowest order of perturbation \ntheory and the order of the moment expansion. \nAssuming $\\delta$ and $ku\/\\mathcal{H}$ are of the same order,  \nthe lowest order of the contribution in terms of powers of power spectrum (i.e., quadratic in $\\delta$) is  \n$(L+L')\/2$ if \n$L+L'$ is even and $L>0$, and $(L+L'+1)\/2$ if odd and $L>0$, while for $L=0$ it is $L'\/2+1$ if $L'$ even \nor $(L'+1)\/2$ if $L'$ odd, but of course all \nhigher order terms also enter. \n\nThese equations also show that there is a close relation between the order of the moments and their \nangular dependence. To understand the angular dependence we first note that \nassociated Legendre polynomial $P_l^m(\\mu)$ contains powers from 1 to $\\mu^{l-m}$ for even $l$, and from\n$\\mu$ to $\\mu^{l-m}$, for odd $l$, \nand is always multiplied with a power of $(1-\\mu^2)^{m\/2}$. \nThus $P_{l,l'}^{L,L',m}(k)$ gets multiplied with powers of $(1-\\mu^2)^{m}$ or $\\mu (1-\\mu^2)^{m}$ to \n$\\mu^{l+l'-2m}(1-\\mu^2)^{m}$, so the highest order is $\\mu^{l+l'}$. In addition we have $\\mu^{L+L'}$ dependence in \nequation \\ref{eq:p_ss}, so the lowest contribution in \npowers of $\\mu$ to \n$P^{ss}(k)$ is $\\mu^{L+L'}$ if $L+L'$ is even or $\\mu^{L+L'+1}$ if $L+L'$ is odd, \nand the highest is $\\mu^{2(L+L')}$. Thus for $P_{00}(\\mathbf{k})$ the only angular term is isotropic, \nfor $P_{01}(\\mathbf{k})$ the only angular term is $\\mu^2$, $P_{11}(\\mathbf{k})$ and $P_{02}(\\mathbf{k})$ contain both $\\mu^2$ and $\\mu^4$ etc. \nNote that only even powers of $\\mu$ enter in the final expression, as required by the symmetry. \nWe now proceed to look in more detail at the lowest order terms. \n\n\\subsection{$P_{00}(\\mathbf{k})$: the isotropic term}\n\nAt the lowest order in the expansion we have correlation of real space density $T_{||}^0=\\delta(\\mathbf{k})$ with itself. \nDensity is a scalar of rank 0, $P_0(\\mu)=1$. The power spectrum is isotropic\n$P_{00}(\\mathbf{k})=P_{00}^{000}(k)$. \nThis term is just the real space power spectrum and of \ncourse does not have any $\\mu$ dependence since it is independent of redshift space distortions. \nFor small values of $\\mu$ this term always dominates, and in the limit $\\mu=0$ the transverse \npower spectrum becomes the real power spectrum $P_{00}(k)$. The real space power spectrum agrees with the\nlinear one on large scales, $P_{00}(k)=P_{\\rm lin}(k)$, slightly dips below the linear one around $k \\sim 0.1h\/Mpc$, \nwhile on even smaller scales the nonlinear corrections cause it to increase over the linear one. \n\n\\subsection{$P_{01}(\\mathbf{k})$}\n\nAt the next order in our expansion (not in perturbation theory PT, see below) we have correlations between the density \n$T_{||}^0(\\mathbf{k}) =\\delta(\\mathbf{k})$ and radial component of momentum density \n$T_{||}^1(\\mathbf{k})=[(1+\\delta)u_{||}](\\mathbf{k})$. \nMomentum density can be decomposed into a scalar ($m=0$) $T_1^{1,0}$\nand two vector ($m=\\pm 1$) components $T_1^{1,\\pm 1}$, but only the scalar part correlates with the density $T_0^{0,0}$, \nwhich is a scalar. \nThus the only contribution comes from $P^{0,1,0}_{0,1}(k) \\propto \\langle T_0^{0,0}(\\mathbf{k})(T_1^{1,0}(\\mathbf{k}))^* \\rangle$, \n\\begin{equation}\nP_{01}(\\mathbf{k})=P^{0,1,0}_{0,1}(k)\\mu,\n\\end{equation}\nwhere we used $P_1^0(\\mu)=\\mu$. \n\nThe scalar mode of momentum can be obtained from the divergence of momentum \nand related to $\\dot{\\delta}$ using the continuity equation,\nwhich in terms of our quantities is\n\\begin{equation}\n\\dot{T}_0^{0,0}+ikT_1^{1,0}=0.\n\\label{ce}\n\\end{equation}\nThis is an exact relation (for conserved quantities), in the sense that the \nvector part of momentum does \nnot contribute to it, since \nit vanishes upon taking the divergence (i.e., vector components are orthogonal to $\\mathbf{k}$ and the dot product is zero). \n\nFrom this we get that\n\\begin{equation}\nP_{01}(\\mathbf{k})=-ik^{-1}\\mu P_{\\delta,\\dot{\\delta}}(k)=-{i\\mu \\over 2k} {dP_{00}(k) \\over dt}.\n\\end{equation}\nThe total contribution from this term to $P^{ss}(k)$ is \n\\begin{equation}\nP^{ss}_{01}(\\mathbf{k})=\\mu^2 \\mathcal{H}^{-1} {dP_{00}(k) \\over d\\tau}=\\mu^2 {d P_{00}(k) \\over d\\ln a}.\n\\end{equation}\nThis is an exact relation for dark matter, valid also in the nonlinear regime. \nIt shows that this term can be obtained directly from the redshift evolution of \nthe dark matter power spectrum $P_{00}(k)$. On large scales it agrees with the linear theory predictions. \nIf we write \n$P_{00}(k)=D(a)^2P_{\\rm lin}(k)$, with $D(a)$ the linear growth rate and $f=d\\ln D\/d\\ln a$, then \nwe find $P^{ss}_{01}(\\mathbf{k})=2f\\mu^2P_{\\rm lin}(k)$. \nWe thus see that this order is of the same order in PT as the first order $P_{00}(k)$, the well known \nKaiser result \\cite{1987MNRAS.227....1K}.\nOn smaller scales we expect the term to deviate from\nthe linear one, just as for $P_{00}(k)$. \n\n\\subsection{$P_{11}(\\mathbf{k})$}\n\nThe next term is the correlation of the momentum density $T_{||}^1(\\mathbf{k})$ with itself.\nIn this case the scalar ($m=0$) $T_1^{1,0}(k)$ \ncorrelates with itself and the vector ($m=\\pm 1$) components $T_1^{1,\\pm 1}(k)$ also \ncorrelate with itself, so both components of \nmomentum contribute, \n\\begin{equation}\nP_{11}(\\mathbf{k})=P^{1,1,0}_{1,1}(k)[P_{1}^{0}(\\mu)]^2+P_{1,1}^{1,1,1}(k)[P_{1}^{1}(\\mu)]^2.\n\\end{equation}\nIn terms of the contribution to the redshift space power spectrum this gives \n\\begin{equation}\nP^{ss}_{11}(\\mathbf{k})=\\mathcal{H}^{-2}k^2\\mu^2[P^{1,1,0}_{1,1}(k)\\mu^2+P^{1,1,1}_{1,1}(k)(1-\\mu^2)]. \n\\label{pss11}\n\\end{equation}\nThe scalar part of the momentum is the one that contributes to the continuity equation \\ref{ce}. \nIn linear perturbation theory only the scalar contribution is non-zero and $P^{1,1,0}_{1,1}(k)=f^2P_{\\rm lin}(k)$. \nThis term is also of linear order and collecting all terms at this order we obtain the usual expression \\cite{1987MNRAS.227....1K}\n\\begin{equation}\nP^{ss}_{\\rm lin}(\\mathbf{k})=(1+f\\mu^2)^2P_{\\rm lin}(k). \n\\end{equation}\n\nHowever, we see from the above that there will be another contribution to both $\\mu^2$ and $\\mu^4$ \nterms from the vector part of momentum correlator $P_{1,1}^{1,1,1}(k) \\propto \\langle |T_1^{11}(k)|^2\\rangle$, \nwhich comes in at the second order in power spectrum. \nThis vector part is often called the vorticity part of the momentum. \nIn general this term is non-zero because vorticity of momentum does not vanish, \neven if vorticity of velocity vanishes for a single streamed fluid  \n\\cite{2002PhR...367....1B}.\nAs seen from equation \\ref{pss11} this term always {\\it adds} power to $\\mu^2$ term and subtracts it in $\\mu^4$ term (but is combined with \na positive contribution from the scalar part in $\\mu^4$ term). \n\n\\subsection{$P_{02}(\\mathbf{k})$}\n\nAt orders higher than $P_{11}(\\mathbf{k})$ we no longer have any linear contributions, hence these terms are usually \nnot of interest for extracting the cosmological information. \nHowever, these terms, including what is sometimes called the Fingers-of-God \n(FoG) effect, are known to be important on \nfairly large scales. \nHere we will limit the discussion to some general statements of their $k$ and \n$\\mu$ dependence, \nleaving their more precise calculations to future work. \n\nThere are two different terms that contribute to this term,\n\\begin{equation}\nP_{02}(\\mathbf{k})=P^{0,2,0}_{0,0}(k)[P_{0}^{0}(\\mu)]^2+P^{0,2,0}_{0,2}(k)P_0^0(\\mu)P_{2}^{0}(\\mu).\n\\end{equation}\nIn terms of the contribution to the redshift space power spectrum this gives\n\\begin{equation}\nP^{ss}_{02}(\\mathbf{k})=-\\left({k \\mu \\over \\mathcal{H}}\\right)^2\\left[P^{0,2,0}_{0,0}(k)+{1 \\over 2}P^{0,2,0}_{0,2}(k)(3\\mu^2-1)\\right]. \n\\end{equation}\nThe first term is the correlation between the isotropic part of the mass weighted square of velocity, i.e. the energy density,\n$T_0^{2,0}=(1+\\delta)u^2$ and the density field $T_0^{0,0}=\\delta$. The second term comes from the scalar part \nof the anisotropic stress $T_2^{2,0}$ correlated with the density $T_0^{0,0}=\\delta$. \n\nOn physical grounds we expect the first term \nto be large in systems with a large rms velocity resulting in a term scaling as \n$P^{0,2,0}_{0,0}(k) \\sim P_{00}(k)\\sigma^2$, where $\\sigma^2$ has units of velocity squared, but is not \nsimply the volume averaged velocity squared (see below). The contribution of this term to $P^{ss}$\ngoes as $-(k\\mu\/\\mathcal{H})^2\\sigma^2 P_{00}(k)$, i.e. it is a damping term suppressing the linear power spectrum, \nwith the effect increasing towards higher $k$ (smaller scales). \nThis is the lowest order FoG term, which we see contributes as $(k\\mu)^2$ dependence and so affects the $\\mu^2$ term.  \nIt is a damping term that is always negative, while the corresponding $\\mu^2$ term from $P_{11}$ always adds power. \nThe scalar anisotropic stress-density correlator $P_{0,2}^{0,2,0}(k)$ \nalso contributes to $\\mu^2$ angular term, as well as to $\\mu^4$ angular term, \nand is formally of the same order in perturbation theory as \n$P^{0,2,0}_{0,0}(k)$, \nbut is likely to be smaller\non physical grounds that velocity dispersion in virialized objects \nis isotropic and hence has a small anisotropic stress. \n\n\\subsection{$P_{03}(\\mathbf{k})$, $P_{12}(\\mathbf{k})$, $P_{04}(\\mathbf{k})$, $P_{13}(\\mathbf{k})$ \nand $P_{22}(\\mathbf{k})$}\n\nSince the lowest order in $\\mu$ is $(L+L')$ or $L+L'+1$, \nthe terms of order higher than $P_{02}(\\mathbf{k})$ do not  contribute to $\\mu^2$ \nterm. \nAt the next order in $\\mu$ we have terms of order $\\mu^4$. At this order there\nare \n7 terms that contribute. $P_{11}(\\mathbf{k})$ and $P_{02}(\\mathbf{k})$ we already discussed: \nwhile $P_{11}(\\mathbf{k})$ has a linear order term and \nis expected to dominate on large scales, $P_{02}(\\mathbf{k})$ is second order in power\nspectrum. They both come with a prefactor of $k^2$. \nAt the next order we have $P_{03}(\\mathbf{k})$ and $P_{12}(\\mathbf{k})$, both second order in power spectrum \nand each multiplied by $k^3$, followed by $P_{13}(\\mathbf{k})$ and $P_{22}(\\mathbf{k})$, also second order in power spectrum, but \neach multiplied by $k^4$, and by $P_{04}(\\mathbf{k})$, third order in power spectrum. \nAll of these terms also\ncontribute to terms of higher order in $\\mu^{2j}$, up to \n$\\mu^6$ or $\\mu^8$ for these terms. \n\nOne can see from this discussion that the angular structure of higher order \nterms is considerably more \ncomplex than that of lower order terms and that all the terms even in powers of\n$\\mu$ are being generated by \nRSD. However, there is a connection between the angular order, powers of $k$ \nand lowest order in perturbation \ntheory, such that only the low powers of $k$ and low lowest order of PT \ncontribute to the lowest orders in $\\mu$. \nThus, at low values of $k\\mu$, the series is convergent. To make these \nstatements more quantitative \na numerical or perturbative analysis is required, which will be presented \nelsewhere \\cite{Okumura2011}.\n\n\\subsection{Shot noise and connections between the correlators}\n\nThe correlators at the same order in powers of velocity, i.e. equal $L+L'$, \ncontain nontrivial cancellations among them. \nTo see this assume velocity is constant over a region of space \n$r \\sim k_0^{-1}$. \nFor example, large scale bulk flows lead to correlated velocities on small \nscales, giving rise to \nnearly equal velocities between nearby particles.  On scales smaller than this \nvelocity coherence scale, $k>k_0$, we  \ncan pull out these constant velocity terms from the correlators, to obtain  \n\\begin{equation}\nP_{LL'}(k)\\delta_D(\\mathbf{k}-\\mathbf{k}')=\\langle [(1+\\delta)u_{\\parallel}^L](\\mathbf{k})[(1+\\delta)\nu_{\\parallel}^L]^*(\\mathbf{k}')\\rangle \\sim P_{00}(k) \\langle u_{\\parallel}^{L+L'} \n\\rangle \\delta_D(\\mathbf{k}-\\mathbf{k}')\n,\n\\end{equation}\nwhere $\\langle u_{\\parallel}^{L+L'} \\rangle$ is just a number corresponding to \nthe spatial average of this term. \nSo these terms are all equal as long as $L+L'$ is the same. \n\nThese terms enter into the sum in equation \\ref{eq:p_ss0} with different \nprefactors and opposite signs, leading to \ncancellations between them. The lowest order example is that of $P_{11}$ and \n$P_{02}$, which enter with equal prefactors but \nopposite signs, canceling any such contributions from each other. \nThis is not surprising: bulk flows lead to ``rigid body'' displacements \nof particles but do not contribute to FoG effects, so their contribution to \n$P_{02}$ must be canceled. \nAs a result, only velocity dispersion type contributions lead to FoG effects. \n\nIn the extreme case this argument can be applied to the shot noise\nfor these correlators, which is the \ncontribution to the power spectrum caused by discreteness\nof tracers. It is well known that the shot noise of a density field sampled by \ntracers of number density $\\bar{n}$ is \ngiven by $P_{00}(k)=\\bar{n}^{-1}$. Analogous calculation for the moments gives \n\\begin{equation}\nP_{LL'}(k)=\\bar{n}^{-1}\\langle u_{\\parallel}^{L+L'}\\rangle.\n\\end{equation}\nThis expression is exact, since by definition a discrete tracer population only has a single value of velocity at any given position, \nso $\\langle u_{\\parallel}^{L+L'} \\rangle$ will be the same for any pair of $L,L'$ such that $L+L'$ is the same. \nThese shot noise terms can be large if the tracer is sparse, i.e. if $\\bar{n}$ is small. \nHowever, the argument above shows that these terms enter with opposiste signs in the final result and so these shot \nnoise contributions cancel in the total sum of \\ref{eq:p_ss0}. This is expected: the only shot noise contribution \nto the total RSD power spectrum $P^{ss}(k)$ should be $\\bar{n}^{-1}$. \nThese examples show that these velocity moments are connected, and \nit is more natural to consider\nthem together, such as $P_{11}(k)-P_{02}(k)$, where the shot noise and the bulk flow terms cancel out. \n\n\\subsection{Relation to Legendre moments}\n\nIn RSD analyses it is customary to integrate $P^{ss}(\\mathbf{k})$ over the lowest order Legendre polynomials to obtain \nmoments $P^{ss}_l(k)$, \n\\begin{equation}\nP^{ss}_{l}(k)=(2l+1)\\int_0^1 P^{ss}(\\mathbf{k})P_l(\\mu)d\\mu,\n\\end{equation}\nwhere $P_l(\\mu)$ are the ordinary Legendre polynomials, $P_0(\\mu)=1$, \n$P_2(\\mu)=(3\\mu^2-1)\/2$ and $P_4(\\mu)=(35\\mu^4-30\\mu^2+3)\/8$.\nOnly the lowest 3 orders contain contributions from linear terms, so the analysis is usually limited to $l=0,2,4$. \nThe 0 moment is just the spherical average of the power spectrum in redshift space. \nThe advantage of this expansion is that in a typical survey the moments are uncorrelated on scales small compared to \nthe size of the survey. \n\nMoments in even $l$ can be viewed as an alternative way to expand in terms of even powers of $\\mu$. However, \nthe expansion given in equation \\ref{pll} is {\\it not} an expansion in Legendre polynomials, since it contains \nproducts of associated Legendre polynomials (including squares of ordinary Legendre polynomials of both even and odd\norders). Hence there is no orthogonality between the moments of distribution function and Legendre moments $P^{ss}_{l}(k)$. So \nif we expand the angular dependence of any given term $P^{ss}_{LL'}(k)$ into Legendre polynomials, we will \ngenerate all orders up to $l=2(L+L')$. This means for example that all terms\nwill contribute to the monopole $l=0$, all except $P^{ss}_{00}(k)$ to quadrupole \n$l=2$ and all but $P^{ss}_{00}(k)$ and $P^{ss}_{10}(k)$ to hexadecupole $l=4$. \nAs a result we always have an infinite number of terms $P_{LL'}(\\mathbf{k})$ contributing to any given\nLegendre series term $P^{ss}_l(k)$, with \nhigher and higher powers of $k$. This expansion is thus considerably more \ncomplex than the expansion in powers of $\\mu^{2j}$, \nwhich has a finite number of terms for any given value of $j$. \n\nThe discussion above suggests it may be more beneficial to fit for powers of $\\mu^{2j}$ rather than \nfor Legendre moments, for example by fitting for $\\mu^0$, $\\mu^2$ and $\\mu^4$ terms, which contain \nlinear order contributions, together with higher order terms $\\mu^6$, $\\mu^8$ etc., which we do not care for and \ncan marginalize over in the end. \nHowever, Legedre moments are uncorrelated while powers of $\\mu^{2j}$ are strongly \ncorrelated, so a marginalization over higher order terms will lead to a large increase in errors  for higher $k$, given that these\nterms become very large at high $k$ compared to lowe order terms. So \nthis can only work if sufficiently strong priors are adopted for higher order terms $\\mu^6$, $\\mu^8$ etc. Such priors could \ncome from simulations extracting individual higher order terms or from a parametrized model. This is pursued further in \\cite{Okumura2011}.\n\n\\subsection{Applications to galaxies and issues of bias} \n\nThe relation to other tracers such as galaxies is a rich subject worth exploring further with this method. \nIn this paper we focus primarily on the dark matter, but all the derivations remain unchanged if the dark matter \nparticles are replaced with some other tracers, such as galaxies or halos. \nIn large scale structure we usually define bias as the ratio of galaxy power spectrum (shot noise subtracted) to \nmatter power spectrum, \n$b^2(k)=P_{00}^{gg}(k)\/P_{00}^{mm}(k)$. \nWe can generalize the concept of bias to \n\\begin{equation}\nb_{LL'}(\\mathbf{k})={P_{LL'}^{gg}(\\mathbf{k}) \\over P_{LL'}^{mm}(\\mathbf{k})}, \n\\label{bll}\n\\end{equation}\nwhere $P_{LL'}^{gg}(\\mathbf{k})$ is galaxy correlator and $P_{LL'}^{mm}(\\mathbf{k})$ is the corresponding dark matter term. \nIn linear theory we have $b_{00}=b_1^2$, $b_{01}=b_1$ and $b_{11}=1$, independent of scale or angle, where \nlinear bias is defined as $\\delta_g=b_1\\delta_m$. \nTwo ways to extract cosmological information from RSD are either by combining $P_{00}$ and $P_{01}$ to eliminate $b_1$, or \nto use $P_{11}$ directly.  \n\nBefore discussing RSD further it is useful to draw a comparison to weak lensing. In case of weak lensing \nwe can measure both projected dark matter density or galaxy density, so we can perform \na joint correlation analysis of galaxy clustering and weak lensing, where the \ngalaxy auto-correlation is proportional to $b^2$ times matter correlations, cross-correlation between galaxies and \nweak lensing signal around them induced by the dark matter is proportional to $b$ (the so called galaxy-galaxy lensing), \nwhile the weak lensing auto-correlation is independent of bias. \nTwo ways to extract the signal are either using just shear-shear correlations tracing matter-matter correlations, \nor combining galaxy auto-correlation with galaxy-galaxy lensing to eliminate bias. \nThis latter has higher signal to noise but is complicated due to the fact that bias is \nscale independent and the scale dependence depends on the \ngalaxy properties \\cite{2010PhRvD..81f3531B}. To understand when this happens it is useful to expand galaxy density perturbation \nto second order in matter density, $\\delta_g=b_1\\delta_m+b_2\\delta_m^2$. \nThe second order terms will become important when they cannot be neglected against the first order terms,\nso the expansion parameter is $(b_2\/b_1)\\delta_m$. Since $\\delta_m^{\\rm rms}$ increases on small scales this scale \ndependent bias increases towards small scales. Typically we have $|b_2\/b_1|<0.4$ \\cite{2010PhRvD..81f3531B} \nand the corrections become \nimportant at $k \\sim 0.1h\/Mpc$, where $\\delta_m^{\\rm rms} \\sim 0.5$.  \n\nReturning back to RSD, our formalism is directly applicable to galaxies, except that \nall the velocity moments are mass weighted for the dark matter, $T_{\\parallel}^{L,m}=\n(1+\\delta_m)u_{\\parallel}^L$, and number density weighted for the galaxies, $T_{\\parallel}^{L,g}= \n(1+\\delta_g)u_{\\parallel}^L$. \nWhat this shows is that if the density distribution of galaxies differs from that of the\ndark matter then all the correlators of velocity moments will differ from each other, even those that appear independent of bias, \nsuch as $P_{11}$. \nIn reality thus the predictions of linear bias model will be modified, \nbecause even if galaxies are faithful tracers of the dark matter velocities at a given position, the weighting of\nthe velocity moments differs: in one case they are weighted by the dark matter mass, in the other by the number of \ngalaxies, and the two \ndiffer in their spatial distribution. This will result in scale dependence of the higher order \nbias terms $b_{LL'}$, just like it does for the $b_{00}$ itself \\cite{2010PhRvD..82d3515H}. \n\nTo quantify this further, for the lowest order momentum density term and for linear bias\nwe must compare correlation of $(1+b_1\\delta_m)u_{\\parallel}$ with itself  to give $P_{11}^{gg}(\\mathbf{k})$ \nor with $b_1\\delta_m$  to give $P_{01}^{gg}(\\mathbf{k})$. The auto-correlation will give the result that agrees with the dark matter \nonly for $b_1=1$, or if $b_1\\delta_m \\ll 1$. In the same limit the cross-correlation will give linear bias $b_1$. \nSo momentum density becomes velocity in the limit $b_1\\delta_m \\ll 1$,\nwhile requiring it to be scale independent relative to dark matter  \nrequires something like $(b_1-1)\\delta_m \\ll 1$, which for typical LRG galaxies ($b_1 \\sim 2$) \nis in fact a more stringent requirement than that of a scale independent bias condition \ndiscussed above \n$(b_2\/b_1$ versus $b_1$). This  suggests \nthat the scale dependence of the momentum density bias terms $b_{01}$ and $b_{11}$ \ndefined in equation \\ref{bll} extends to larger scales\nthan scale dependent bias of density $b_{00}$. \n\nThe conclusion from this discussion is that the scale dependence of bias terms \ninvolving momentum density is a real concern in RSD and likely \nextends to relatively large scales ($k<0.1h\/Mpc$). \nIn terms \nof the angular decomposition in powers of $\\mu$, \nthe discussion of the scale dependent bias of RSD can be divided into a $\\mu^2$ term, which depends \nentirely on $b_{01}$ in $P_{01}$, and the $\\mu^4$ term, for which \nthe scale dependence of $b_{11}$ term above is applicable,\nsince that is the term that does not vanish on large scales in linear theory. \nIn this sense RSD analysis is not the equivalent of a joint galaxy-weak lensing analysis, \nsince weak lensing auto-correlation truly traces the dark matter directly, \nwhile in RSD \nthis limit is achieved only on relatively large scales where \n$\\delta_g^{\\rm rms} \\ll 1$. \n\nThe discussion so far completely ignored FoG effects: for $\\mu^2$ term \nthese are encoded in $P_{02}$ and in vector part of $P_{11}$, which  \nunlike $P_{02}$ adds power rather than removes it, and these terms \nhave their own physical interpretation and scale dependence \nunrelated to the scale dependent bias discussion above. \nWhile they can partially cancel the effects discussed above they are unlikely to \nachieve this exactly. \nIn most of the literature so far only the scale dependence induced by FoG effects was discussed \n(although see \\cite{2004PhRvD..70h3007S,2011arXiv1105.4165R}). \nThe simple linear bias model predicts that FoG effects scale with bias squared: the leading order term scales as $b_1^2$ \nboth in $P_{02}^{gg} \\propto \n\\langle [b_1\\delta v_{\\parallel}^2](\\mathbf{k})b_1\\delta(-\\mathbf{k})$ and in $P_{11}^{gg}\\propto [b_1\\delta v_{\\parallel}]^2$. \nIf we write $P_{02}^{gg}(k)-P_{11}^{gg}(k)=P_{00}^{gg}(k)\\sigma^2$, then $\\sigma$ is independent of bias, \nsince $P_{00}^{gg}(k) \\propto b_1^2P_{00}^{mm}(k)$. \n\n\\section{Discussion}\n\nIn this paper we present a distribution function approach to redshift space distortions. \nWe show that the redshift space density can be expressed in terms of a\nsum over velocity moments and the redshift \nspace power spectrum can be expressed in terms of correlators between the Fourier \ncomponents of these moments. \nThese moments are simple objects to calculate in any system: they are calculated by simply taking appropriate powers \nof radial velocity and summing over all particles. The lowest order moments are density, momentum density, stress \nenergy density etc. \n\nWe have decomposed the moments into helicity \neigenstates based on their transformation properties under rotation around the direction of the Fourier mode, \na generalization of SVT decomposition in cosmological perturbation theory. \nWe use rotational invariance to derive all of the allowed correlator terms, showing that only terms with \nthe same helicity can contribute to the correlators. \nThe moments of distribution function are complicated objects with many terms allowed by symmetries, specially at higher order, \nleading to a complicated angular and scale dependence, \nsuggesting that treatments of RSD cannot be fully successful with simple ansatzes, such as the popular FoG velocity\ndispersion model with one free parameter \\cite{2011MNRAS.410.2081J,2011arXiv1103.3614T}. \n\nDespite the complexity of the general RSD description some general statements can be made. \nThe lower order terms generally only \ncontribute to low orders of expansion in $\\mu^2$, where $\\mu$ is the angle between the Fourier mode and the line of sight. \nAs an example, we have shown that only the scalar part of the momentum density correlates with the density, and \nthis term can be written in terms of a time derivative of the power spectrum. This term only contributes to $\\mu^2$\nangular dependence and contains a linear order term. But \nthere is also the vector part of the momentum density-momentum density correlation, \nthe (scalar) energy density-density correlation, and the scalar part of anisotropic stress density-density correlation, \nall of which also contribute to the $\\mu^2$ term. \nThey are all nonlinear and cannot dominate on very large scales, but likely \ndominate on small scales. \nThe energy density-density correlation term is the term most closely related to the \nFoG velocity dispersion effect and is always negative, suppressing the power, \nbut the other terms are formally of the same order in perturbation theory. \nWe have shown that the vorticity part of momentum always adds to the RSD power of $\\mu^2$ term, and hence acts in the opposite direction to \nthe FoG term. Our analysis cannot address which term has a larger amplitude, but it would be interesting to see \nif there are any systems where the terms that add power dominate over those that suppress it. \nThe next angular term has $\\mu^4$ dependence and we identified 7 terms that contribute to it, of which one, \nscalar part of $P_{11}$, contains a linear contribution that does not vanish on large scales. \n\nThe fact that there are a finite number of velocity moment terms at each order of $\\mu^{2j}$ expansion should be contrasted to\nthe popular Legendre multipoles expansion (monopole, quadrupole and hexadecupole contain cosmological information),\nwhich receive contributions from all orders in moments of distribution function. \nThis suggests that a better behaved analysis may be possible if instead of a multipole analysis the analysis is performed in \nterms of a $\\mu^{2j}$ expansion, with the lowest 3 orders containing cosmological information and the rest treated as nuisance \nparameters. \n\nIt is important to emphasize that these moments are mass weighted quantities, and \nno volume averaged quantities ever enter into our expressions. \nThis relates to one of the long standing issues in the treatment of RSD: many of the past treatments \\cite{2004ApJ...606..702T,2011MNRAS.410.2081J,2011arXiv1103.3614T} have \nassumed that RSD trace correlations between velocities and dark matter and that the FoG effects multiply these \ndensity-velocity and velocity-velocity correlations, where FoG quantities are also\ndefined as volume weighted quantities such as velocity dispersion $\\sigma^2=\\langle u^2 \\rangle$. \nBut these volume weighted quantities \nare not well defined, specially for sparse biased systems such as galaxies or clusters. \nFor a biased tracer with $b>1$ one finds that voids with no tracers in them are enlarged, \nsince, for $\\delta<0$, $1+b\\delta$ is closer to 0 than $1+\\delta$. \nThis has forced some workers to use the dark matter velocity field instead, \nwith unpredictable results \\cite{2011arXiv1103.3614T}. \nOur expansion shows that it is more natural to define RSD in terms of mass or number weighted quantities, such as momentum density or \nenergy density, the former replacing velocity and the latter replacing velocity dispersion. \nMass and number weighted moments such as momentum or energy density\nare well defined even in voids (where they are simply zero). In this paper we show that there is a consistent \nexpansion using mass weighted moments, and that the expansion is convergent on large scales. \n\nThe fact that all RSD quantities are density weighted also suggests that RSD effects will differ \nif the galaxy number density distribution differs from mass density distribution. We have shown that even a linear \nbias model induces scale dependent bias of the momentum density correlators, and that this scale dependence is \nlikely to show up on relatively large scales, $k<0.1{\\rm h\/Mpc}$. \nThe success of RSD in extracting cosmological information depends entirely on our ability to model \nthese various bias terms and relate them to each other. Similarly, \nthe success of the approach presented here in modeling RSD depends on our ability to extract these moments from simulations and \ndata and on our ability to model them with analytic models, such as perturbation theory. \nProviding physical interpretation of the terms, as done here, could enable one to develop more effective modeling, \nor provide a better physical understanding of limitations of RSD in extracting cosmological information.\nFor example, it is relatively straight-forward to include the bias induced scale dependence\neffect at the lowest order of PT and we will present the results elsewhere \\cite{Vlah2011}. \nIn this paper we have focused on theory, conceptual issues and general symmetries, while applications to simulations \nand perturbation theory will be presented in \nupcoming work \\cite{Okumura2011,Vlah2011}. \n\n\\begin{acknowledgments}\nWe thank Teppei Okumura and Zvonimir Vlah for helpful discussions. \nThis work is supported by the DOE, the Swiss National Foundation under contract 200021-116696\/1 and WCU grant R32-10130. \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFor certain problems and domains, collecting real-world data can be difficult and expensive.\nSimulations are used to mimic real-world stimuli and can provide a clear, valid, easily reproducible digital representation of a system, often founded on principles of mathematics and physics.\nHowever, oftentimes simulations may not be able to capture the full complexity of real-world data due to poor assumptions or bias built into the model. This leads to a gap between synthetic and real data distributions, and ultimately a machine learning solution biased towards the larger synthetic dataset.\nThis is a common problem in the medical industry: if we want to study heart disease, collecting electrocardiogram (ECG) heartbeat signals from real subjects can be a time-consuming and tedious process, as medical data needs to be properly classified and anonymized, often requiring a trained professional.\nThere are many existing simulations that mimic a healthy heartbeat \\citep{PhysioBank, simple_ecg_sim, advanced_ecg_sim, NeuroKit2}, but they poorly capture the noise seen in the real data.\nThe complementary unhealthy heartbeat data has yet to be simulated consistently well.\n\nGenerative Adversarial Networks (GANs) \\citep{GAN} have seen great success in generating synthetic data samples of different types such as images, speech, text, etc., by having two neural networks compete against one another.\nFurthermore, researchers have modified GANs to be used in combination with simulators to take advantage of simulated and unsupervised learning, in the SimGAN learning method \\citep{SimGAN}.\nRather than having the GAN start from random noise as its input, as is traditionally done with GANs, a simulator's output is fed to the GAN's input. This change in procedure allows the GAN to take advantage of the simulator's annotations while producing more realistic samples compared to the original simulator.\nHowever, GANs are predominantly used for image generation or 2D data types, and 1D GAN data synthesis has not seen as many real world applications.\nDespite the successes of these models and methods, deep generative models are still difficult to train and evaluate. This difficulty stems from generative models' lack of objective truth. Without truth, there are not always easy quantitative metrics for determining how realistic the models' outputs are.\nCreating a successful network architecture and finding the optimal hyperparameters for it are also challenges, as it often requires a researcher to manually tune the models through trial and error.\nOn top of that, a variety of loss functions \\citep{GAN, WGAN}, network architectures \\citep{DCGAN, styleGAN}, and training techniques \\citep{mbd_feature_matching_loss, WGAN-GP, DRAGAN} have all been proposed, so it is difficult to select which ones would work best for a specific domain.\n \nWhile evolutionary computation (EC) has been utilized before to help optimize and search for successful GAN architectures \\citep{E-GAN} and loss functions \\citep{TaylorGAN}, we could not find an evolutionary system that can perform neural architecture search for a SimGAN, adjust custom loss functions, train and evaluate models across multiple objectives, and optimize hyperparameters simultaneously while being customizable to fit any data problem, including 1D data.\n\nThus, in this paper, we pose the following research questions:\n\\begin{itemize}\n\\item \\textbf{RQ1:} How effective are SimGANs for generating realistic 1D signals from simulated inputs? Are SimGANs able to shift the synthetic data distribution closer to the real data distribution? Are they applicable to real world problems like ECG heartbeat synthesis?\n\\item \\textbf{RQ2:} What are examples of effective quantitative metrics for evaluating SimGAN outputs during and after training? \n\\item \\textbf{RQ3:} How can we utilize evolutionary computation to search for novel SimGAN architectures and optimize existing models? Can we automate this process so that training, evaluation, optimization, and selection of models can be done simultaneously while being flexible enough to be applied to a multitude of problems? \n\\end{itemize}\n\nThe main contributions of this paper are:\n\\begin{enumerate}\n    \\item Improvements to the SimGAN learning approach and architecture for 1D data\n    \\item New feature-based quantitative metrics for evaluating generated outputs\n    \\item An open source software implementation of ezCGP, an evolutionary computation framework for optimizing complex machine learning pipelines\n    \\item Examples of SimGAN outputs mimicking abnormal heart conditions from the MIT-BIH arrhythmia database \\citep{MIT-BIH} and the effects of using the outputs for heartbeat classification\n\\end{enumerate}\n\nThe rest of the paper is organized as follows; Section \\ref{Background} presents a brief overview of important concepts and related work. \nSection \\ref{Dataset} describes the dataset and simulators we used. \nSection \\ref{Methods} describes our SimGAN improvements and ezCGP's evolutionary process. \nThe results and discussion are in Section \\ref{ResultsAndDiscussion}. Finally, conclusions are drawn and future work is outlined in Section \\ref{ConclusionAndFutureWork}.\n\n\\section{Background}\n\\label{Background}\nThis section introduces some basic concepts in GAN training, the SimGAN learning method, our ezCGP framework, and summarizes relevant studies related to this research.\n\\subsection{General GAN training}\nGANs \\citep{GAN} typically train two deep neural networks (DNNs): a generator ($G$), which generates synthetic samples, and a discriminator ($D$), which predicts if the synthetic samples are real or fake.\nThese two networks are placed in an adversarial setup and can be seen as a two-player minimax game for the networks to compete against each other.\nA GAN uses two loss functions: one for generator training and one for discriminator training. \nThe generator loss is calculated by how likely the generator is able to fool the discriminator, while the discriminator loss is calculated by how likely it is able to predict correctly. In other words, $D$ and $G$ try to find parameters to optimize the objective function $V(G, D)$:\n\\begin{equation}\n    \\underset{G}{\\min} \\text{ } \\underset{D}{\\max} \\text{ } V(D, G) = \\mathbb{E}_{x}[\\log D(x)] + \\mathbb{E}_{z}[\\log(1 - D(G(z)))]. \n\\end{equation}\nWhere $\\mathbb{E}_{x}$ is the expected value over all real data instances, $D(x)$ is the discriminator's prediction if real data instance $x$ is real, $G(z)$ is the generator's output when given input $z$, $D(G(z))$ is the discriminator's prediction if a fake instance is real, and $\\mathbb{E}_{z}$ is the expected value over all generated fake instances $G(z)$.\n\n\\subsection{SimGAN Overview}\nWith the explosion of deep learning models in recent years, it has become more important to collect as much data as possible.\nHowever, this is not always possible, and learning from synthetic data may not achieve the desired performance due to a gap between synthetic and real data distributions.\nTo address the limitations in accessibility of real-world data or the poor assumptions and overly simplified simulations, researchers proposed a modification to the GAN framework named SimGAN \\citep{SimGAN} that utilizes simulated images and unsupervised learning, such that synthetic images from a simulator are used as inputs instead of random vectors.\nThe model tries to improve the realism of the simulator's output by trying to learn features from the real data, and then \"refining\" the simulated data to better match what is found in the real-world. Hence, the generative model is named the refiner ($R$).\n\nAs such, SimGANs have two new loss functions.\nThe refiner uses a self-regularization loss to minimize the difference between the simulated input and the resultant refined output, with the underlying idea being that the refined outputs should be a transformation of the original synthetic data and should not deviate too far from the original simulated conditions. The formula is defined below:   \n\\begin{equation}\n    V(\\theta) = - \\underset{i}{\\sum} \\log(1 - D(R_{\\theta}(x_{i}))) + \\lambda || R_{\\theta} (x_{i}) - x_{i}||_{1}.\n\\end{equation}\nThe first half of the equation is the same as the GAN generator loss.\nIn the second half, $\\lambda$ is a regularization weight constant, $R_{\\theta} (x_{i}) - x_{i}$ is the difference between the refiner output and the original input, and $||.||_{1}$ is the L1 norm.\n\nThe other loss is a local adversarial loss used by the discriminator network. \nAny local patch sampled from the refined image should have similar statistics to a real image patch.\nTherefore, rather than defining a discriminator network that was trained on the entire global image, the authors used a discriminator that classifies all local image patches separately then aggregates them to predict whether the sample is real or fake.\n\nAdditionally, they introduced the idea of using a history buffer, where refined images generated by the previous steps of the refiner are sampled during each iteration of discriminator\ntraining, and the discriminator loss function is computed by summing the predictions on the current batch of refined images and the sampled history of images. This is meant to improve the stability of adversarial training, as the discriminator network tends to only focus on the latest refined\nimages and the refiner network\nmay reintroduce previously seen artifacts that the discriminator has\nforgotten about. \n\n\\subsection{Easy Cartesian Genetic Programming (ezCGP)}\nezCGP is an end-to-end evolutionary Cartesian Genetic Programming framework designed to be highly flexible and customizable to any researcher's task.\nThe graph representation of a genome, instead of a traditional tree representation, lends itself better to represent the graph architecture of a neural network. \nThe framework also introduces the novel idea of compartmentalizing the genome into a sequence of 'blocks', each with their own evolutionary rules: a unique set of operators and hyperparameters, mating and mutation strategies, number of genes, and evaluation method. Each block can be thought as a conceptual component of an algorithm; by segmenting, we allow the algorithm to be evolved without mixing components from one concept to another. This allows for full representations of complex machine learning pipelines in evolutionary computation.\nA more in-depth explanation of how it is applied in this work is included in Section \\ref{Methods}. We share our code online for further research and experimentation.\n \\footnote{\\url{https:\/\/github.com\/ezCGP\/ezCGP}}\n\n\\subsection{Related Work}\nMedical data and ECG synthesis is challenging since biological systems are dynamic with how the various parts of the body interact. Studies have attempted to use GANs to generate sufficiently realistic medical data \\citep{TSTR} and one study has shown a successful example of using SimGANs to generate biologically plausible ECG signals\\citep{SimGAN_ECG}.\nThe resultant synthetic data generated was shown to improve ECG classification when using the new synthetic data for training ECG classifiers.\nIn this example, they use the ECGSYN ECG simulator  \\citep{advanced_ecg_sim} and tailor a specific self-regularization loss based on the simulator's system of ordinary differential equations during the training process.\nUnfortunately, the approach only generates examples for single heart beats, which are extremely short in nature and are not always useful when diagnosing heart disease. \n\nIntegrating evolutionary techniques into traditional GAN approaches is not a novel idea.\nEvolutionary GAN (E-GAN) \\citep{E-GAN} evolves a population of generators, where the mutation method selects among different loss functions to train the generators, which then compete against a single discriminator. \nCOEGAN \\cite{COEGAN} utilizes neuroevolution to coevolve discriminator and generator architectures.\nMulti-objective E-GAN (MO-EGAN) \\citep{multi-EGAN} instead treats E-GAN training as a multi-objective optimization problem, and uses Pareto dominance, defined as when some other outcome is weakly preferred by all individuals and strictly preferred by at least one individual, to select the best solutions across the selected objectives that measure diversity and quality.\nOther approaches propose improving GANs by discovering customized loss functions for each of its networks.\nFor example, TaylorGAN \\citep{TaylorGAN} treats the GAN losses as Taylor expansions and optimizes custom definitions through multi-objective evolution. These losses were meant to act as an alternative to traditional GAN losses such as Wasserstein loss \\citep{WGAN} or minimax loss. \nLipizzaner \\cite{Lipizzaner} uses spatial coevolution to evolve and train a grid of GANs simultaneously, and Mustangs \\cite{Mustangs} builds upon E-GAN and Lipizzaner by mutating the loss function across the coevolution grid.\n\n\\section{Dataset}\n\\label{Dataset}\n\n\\subsection{ECG Dataset}\nFor a real-world example of 1D signals that are difficult to collect, we applied our approach to ECG heartbeat data to help generate realistic samples of various arrhythmias. In our approach to the problem, we want it to be generic towards the simulated source of data, i.e. it should work for multiple simulators, and we want our generated signals to be longer which better represent what doctors would actually use to diagnose heart disease.\nWe use ECG recordings taken from the MIT-BIH Arrhythmia Database  \\citep{MIT-BIH} for real heartbeat data, as the database is a public well-established source of data for ECG heartbeat classification tasks.\nThe database contains 48 half-hour ECG records obtained from patients, where each record contains two 30-minute ECG lead signals collected with a sampling rate of 360 samples per second per channel. The database has annotations for heartbeat class information verified by independent experts.\nHowever, a single heartbeat alone is not always useful for true heart disease classification, as such, longer recorded signals often prove more useful.\nFor this purpose, we have segmented the dataset samples into 10 second long intervals, where an entire segment is flagged as \"abnormal\" if any beats are not annotated as normal by the experts. In particular for training our SimGANs we selected 32 samples of uncommon abnormal signals, an example of which is shown in Figure \\ref{abnormal_example}, but the whole dataset of both abnormal and normal heartbeats, roughly about 2000 and 5000 signals respectively, were later used for our ECG classifiers described in Section \\ref{ECG_Classifier}.\n\n\\begin{figure}[!htbp]\n  \\centering\n  \\includegraphics[trim=0 20 0 50, clip, width=\\linewidth]{figures\/abnormal_example.png}\n  \\caption{Example of an ECG heartbeat sample with an abnormal condition from the MIT-BIH Arrhythmia Database. Specifically, this sample contains an instance of a left bundle branch block beat around the 2500 mark}\n  \\Description{Abnormal heartbeat example}\n  \\label{abnormal_example}\n\\end{figure}\n\nFor our simulated dataset, we use the NeuroKit2 \\citep{NeuroKit2} package as our ECG simulator to collect samples.\nNeuroKit2 is an open source Python package meant to provide easy access to advanced bio-signal processing routines based on existing cited works.\nFor our heartbeat ECG use case, NeuroKit2 can simulate healthy heartbeats and uses either a simple simulation based on Daubechies wavelets \\citep{simple_ecg_sim}, which roughly approximate a cardiac cycle, or a more complex simulation based on ECGSYN \\citep{advanced_ecg_sim}. \nWe utilize both simulators and our simulated dataset contains an equal number of samples from both. \nBoth simulators can specify conditions like duration, sampling rate, and heart rate, but ECGSYN can add synthetic noise drawn from a Laplacian distribution and even mimic a heart's random fluctuations between some beats.\nTo match our real ECG data from the MIT-BIH database, we use the same duration and sampling rate of 10 seconds and 360 samples per second respectively. Roughly 2500 samples were generated from both simulators with a range from 50 through 100 beats per minute for heart rate and up to 10 percent noise added for a total of 5000 samples. An example of a signal generated from both simulators is shown in Figure \\ref{ECGSim_examples}. \n\n\\begin{figure}[!htbp]\n  \\centering\n  \\includegraphics[trim=0 20 0 50, clip, width=\\linewidth]{figures\/ecg_sim_examples.png}\n  \\caption{Generated ECG heartbeat samples from NeuroKit2, the top plot is using the Daubechies wavelets simulator and the bottom plot is using ECGSYN}\n  \\Description{Generated samples from NeuroKit2}\n  \\label{ECGSim_examples}\n\\end{figure}\n\nFrom these examples, we can see that the real samples often include noise from the collection sensors, and often are more complex and variable in the sequences of heartbeats.\n\n\n\\section{Research Methods}\n\\label{Methods}\nSimGANs are notoriously difficult to train and evaluate, as during training we must balance the learning process between the refiner and discriminator. There is often no objective quantitative measure of how good a generated output is. Our training process is largely inspired by the original SimGAN framework. However, we propose new additions to the framework based on current research and apply the approaches to 1D data during training.\n\nWe add in specialized network architectures, generalizable feature-based quantitative evaluation metrics, and an evolutionary framework to help search and optimize potential neural architectures and hyperparameters. \n\n\\subsection{Training Configuration}\n\nWe use the minimax loss described earlier as our adversarial loss between the refiner and discriminator, as just a simple binary cross entropy loss. For our self-regularization loss in our 1D space, it is actually simpler than the original, as we measure the absolute difference between two signals' amplitudes at each point or bin. In terms of implementation, this is a simple L1 Loss.\n\nThe original SimGAN authors noticed that the refiner network tends to over-emphasize certain features to fool the discriminator networks, leading to the refiner network producing artifacts, so they proposed using a local adversarial loss. In our 1D use case, we quickly noticed that using just a local adversarial loss would lead to a collapse of the waveform and would not resemble the true wave structure, as signals and other forms of 1D data are often sequential and meaning is derived from ordering. Therefore, to help prevent the formation of artifacts, we propose a Siamese dual discriminator network where we ensemble a global discriminator network and a local discriminator network together to capitalize on both a global and local loss. The local discriminator network is fed segments of the signal, of specified equal length, then the network predicts if each segment is real or fake. The signal segment predictions are then averaged to obtain an aggregated prediction for the whole signal, which is then added to the prediction from the global discriminator network. Together they average their predictions to form the unified loss, which is backpropagated to update both network weights. Similarly, the local discriminator network also utilizes the history buffer and trains on previous iterations of refined waveforms.\n\nRecent GAN approaches try to enforce a soft Lipschitz constraint using gradient penalties \\citep{WGAN-GP}, to help bound the losses to some extent and help training stability, as the hypothesis is that mode collapse, when the refiner learns a single example that fools the discriminator and uses it over and over, is the result of the competing game converging to bad local equilibria. As such, during training, we chose to use DRAGAN \\citep{DRAGAN} as our gradient penalty.\n\n\\subsection{Evaluation Metrics}\n\\label{sec:Evaluation}\nThese evaluation metrics are methods to measure a GAN's performance after it has completed training. Evaluating GANs is an open problem, as there is no objective truth in most cases. How does one classify if a generated signal is \"real\" enough? One way is simple visual qualitative analysis: \"does it look right?\" However, this often requires domain knowledge and is not scalable. The following metrics are a quantitative way to measure model performance. \n\n\\subsubsection{Feature Evaluation}\nOne method for evaluating the refiner outputs is extracting key features from the real data, then comparing it to the features from the refined outputs. For every signal, we can effectively extract a feature vector. The following are a list of features we utilized, that can be generalized to multiple domains: \n\\begin{enumerate}\n    \\item Amplitude\/height difference between peaks \n    \\item Amplitude\/height ratio between peaks \n    \\item Distance between peaks\n    \\item Trough heights\n    \\item Peak to trough amplitude\/height difference\n    \\item Peak to trough distance\n    \\item Area under the signal's curve\n    \\item Number of times the signal changes direction\n    \\item Roughness of signal measured by a difference between a rolling mean of values and the current value\n\\end{enumerate}\nNaturally, if the domain is deeply understood, more specific features can be used. We have ways to visualize the feature distributions so that we can qualitatively determine if a refiner truly transforms the simulated data to better reflect the real data distribution, or if one refiner is better than another based on one feature or another. \n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[trim=0 5 0 25, clip, scale=0.4]{figures\/violin_plot_sample.png}\n  \\caption{Example of a violin plot visualization of feature distributions of the number of times the signal changes direction. Note that we can qualitatively assess if the simulated distribution approaches the real distribution.}\n  \\Description{Violinplot example }\n\\end{figure}\nWe use distance measurements like Kullback\u2013Leibler divergence (KL-Div), Wasserstein Distance, or Kolmogorov\u2013Smirnov tests (KS-Stat) to quantitatively determine how far the refined output features are from our expected real features. We typically utilize KS-Stat as our distance metric, as it can be a nonparametric test of the equality between two discrete distributions. However, these distance metrics expect the distributions to be the same size and effectively \"map\" instances to one another. This is difficult for SimGANs as the simulated data size is expected to be much larger than the real data size, and there is no true mapping between real and simulated data. Therefore, to use distance metrics, we repeatedly sample the distributions until we are confident every signal has been sampled, and then average the distances calculated by our metric.\n\nAnother metric is a statistical t-test for unequal distributions to measure how likely that the two samples are drawn from the same distribution. We use a one-tailed unequal variance Welch's t-test to compare the distributions of the extracted features from the refined and real signals. Unlike the traditional use of t-tests where we look for the p-value to be small to show there is a significant difference between two distributions, we want to show that there is no significant difference between the distribution of features between the real and refined data. Therefore, we want our p-value to be close to $1$. We can use the t-test on each feature individually, then keep track of the number of features where the simulated distribution was significantly different from the real distribution, and average the resultant p-values across all the features.\n\n\\subsubsection{FID Score}\nFrechet Inception Distance \\cite{fidscore} was originally used to measure the quality of generated images.\nFID scores measure the difference between sets of generated and ground truth images via comparing the distributions (specifically the mean and co-variance) of model activations of each set.\nAn oracle network, typically Inception \\cite{inception}, is used for computing the activations.\nUnfortunately there is not enough data to train a custom oracle network for the ECG dataset. Therefore, we convert ECG waveforms to 2D images via stacking the waveform and zero-padding the width to the input shape of Inception. \n\n\\subsubsection{Tournament Evaluation}\nTournament evaluation is a peer to peer evaluation method that is domain agnostic \\citep{tournament}. The underlying idea is that we have our SimGAN population compete with themselves. So every refiner and discriminator will be paired with one another for every possible combination and the refiner is trying to fool the discriminator it is paired against. Each network is assigned an initial rating, based on the Glicko \\citep{glicko} system, and as they win or lose matches, their rating will go up or down. The idea is that good refiners will be able to fool many types of discriminators if their outputs are close to the real signals.\nHowever, as this method is peer-to-peer, it does not provide a direct measure of how well the refiners are performing and changes with the population. In fact, it is entirely possible that the entire population performs poorly. It should be noted that during our experiments, we noticed that the tournament would sometimes reward refiners that produced random noise, as the discriminators were not trained on random noise for very long. So despite a high tournament ranking, the resultant refiner is not guaranteed to generate good data. However, since GANs don't converge easily, a model at an earlier step can easily produce better outputs than ones in later steps. Therefore, we log each SimGAN at different intervals and use tournament evaluation to select the best point in time to represent the individual SimGAN as it is sent through ezCGP.\n\n\\subsection{Initial Seed}\nWith our new training improvements, we define a hand-designed SimGAN architecture that we hypothesized would be successful, and used it as an initial seed for our evolutionary process. Both the refiner and discriminator networks are inspired by DCGAN \\citep{DCGAN} for deep convolutional network GANs where ResNet \\citep{resnet} blocks are used as our convolutions. To further improve our discriminator so that it makes more accurate predictions while providing useful feedback to the refiner, we utilize the Siamese dual discriminator described earlier, a mini-batch discrimination \\citep{mbd_feature_matching_loss} layer to discriminate between whole mini-batches of samples rather than between individual samples, and a feature extractor layer which extracts the features described above. This seed is visualized in Figure \\ref{seed}.\n\n\n\\begin{figure*}[!htbp]\n  \\centering\n  \\includegraphics[width=.9\\linewidth]{figures\/SeededIndividual.pdf}\n  \\caption{Architecture of our seeded individual for ezCGP. Notice the three distinct genome blocks. Note that only the active genes are shown.}\n  \\Description{seeded example }\n  \\label{seed}\n\\end{figure*}\n\n\\subsection{ezCGP Optimization}\nDue to the compartmentalization of a genome in ezCGP, we define our genome as the following sequence of blocks: one block to build the architecture of the refiner network, another block to build the architecture of the discriminator network, and a third block to set the network training hyperparameters. Each block has its own set of operators, genes, and rule set to fit its definition. For our problem, we defined operators to act as various different neural network layer types, to better search for SimGAN improvements. A wide variety of operators and blocks allows for easy evolution and neural architecture search, especially if an initial seed is given. \nExample operators include: convolutional layers, concatenation layers, linear layers, pooling layers, dropout layers, batch normalization layers, ResNet blocks, activation layers, flatten layers, mini-batch discrimination layers, and custom feature extract layers.\n\nEach operator has its own set of hyperparameters that can be tuned, but hyperparameters that guide the individual's learning process can also be tuned, such as optimizer types, learning rates, number of training steps, regularization weights, etc. All together these operators can easily allow us to build and define deep neural networks. Mating is defined to be strictly the exchange of whole blocks between parents; each block is set to mate with 33 percent probability. Mutation is defined as either a change of a gene's position in a block, a change of the operation used at a gene, or the change of the hyperparameter requested by the operation. The two neural network blocks have their genetic material mutated with a probability of 20 percent, and the remaining block mutated with a probability of 10 percent. After training, each individual was scored by the four objectives mentioned earlier: minimize the FID score, minimize the KS statistic, minimize the number of features where the simulated distributions were significantly different from the real distributions, maximize the average p-value of the feature distributions. The initial population size was set to 4, as a successful genome can easily spawn many children and a low initial population is used to prevent potential computational constraints such as GPU memory. We maintained a hall-of-fame with a max size of 40, and the next generation was selected using NSGA-II \\citep{NSGA-II} from a pool of individuals: the previous generation's offspring and hall-of-fame. Algorithm \\ref{ezCGP_algo} shows the pseudo-code for the evolution process. We ran this for 48 hours on a single computer with 1 TeslaV100-PCIE 32GB GPU, 128 GB RAM, and a Xeon-Gold6126 CPU.\n\n\n\\begin{algorithm}\n\\caption{ezCGP Evolution}\n\\begin{algorithmic} \n\\STATE $init\\_population()$\n\\WHILE{not converged}\n\\STATE{$parents=get\\_parent\\_list()$}\n\\FOR{$p1,p2$ in $parents$}\n\\FOR{$i^{th}block$ in $range(num\\_blocks)$}\n\\STATE{$roll=random\\_number()$}\n\\IF{$roll < i^{th}block\\_prob\\_mate$}\n\\STATE{$population += mate(p1,p2)$}\n\\ENDIF\n\\ENDFOR\n\\ENDFOR\n\n\\FOR{$indiv$ in $population$}\n\\FOR{$i^{th}block$ in $range(num\\_blocks)$}\n\\STATE{$roll=random\\_number()$}\n\\IF{$roll < i^{th}block\\_prob\\_mutate$}\n\\STATE{$mutate(indiv)$}\n\\ENDIF\n\\ENDFOR\n\\ENDFOR\n\n\\FOR{$indiv$ in $population$}\n\\STATE{$train(indiv)$}\n\\STATE{$score(indiv)$}\n\\ENDFOR\n\n\\STATE{$population = select\\_population(indiv)$}\n\n\\ENDWHILE\n\n\\end{algorithmic}\n\\label{ezCGP_algo}\n\\end{algorithm}\n\n\\section{Results and Discussion}\n\\label{ResultsAndDiscussion}\n\n\\subsection{Novel Configurations}\nezCGP searches for neural architectures, so over time we should see improved configurations for SimGANs across the population. Figure \\ref{evolved1} and \\ref{evolved2} are examples of evolved networks. In total over 40 individuals were created by the end of generation three. \n\n\\begin{figure*}[!htbp]\n  \\centering\n  \\includegraphics[width=.9\\linewidth]{figures\/Evolved_Individual_No_Feature.pdf}\n  \\caption{Architecture of an evolved individual from ezCGP. Notice the discriminator network opts not to use the feature extractor and adds additional dropout layers, causing a stronger regularization effect to occur. The refiner network uses a different activation function for the first ResNet block.}\n  \\Description{Evolved individual 1 example }\n  \\label{evolved1}\n\\end{figure*}\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[scale=0.5]{figures\/Evolved_Individual_Concat.pdf}\n  \\caption{Architecture of another evolved individual's discriminator network from ezCGP. Notice the network takes advantage of concatenate layers to use features from previous convolutions.}\n  \\Description{Evolved individual 2 example }\n  \\label{evolved2}\n\\end{figure}\n\nThe evolved refiner genomes did not have as many significant changes as the discriminators. We hypothesize this is due to how we implemented a ResNet block as a large single gene-operator with less evolvability, and thus generally shows good performance overall and overpowers the discriminators. \n\n\n\\subsection{Quality of Generated ECG Data}\nFigure \\ref{ECGSim_examples} shows examples where we can see that the real samples for ECG heartbeat signals are often rougher than the simulated examples caused by noise from the collection sensors. Not only that, more complex conditions like blockages are not possible to generate with the given simulators. However, after training our seeded SimGAN model using the real and simulated data, we can qualitatively observe in Figure \\ref{ECG_Gen_0} that the SimGAN's refined waveforms include noise and features present in the real data, though it may not be as principled as the simulation.\n\n\\begin{figure}[!htbp]\n  \\centering\n  \\includegraphics[trim=0 65 0 65, clip, width=\\linewidth]{figures\/seeded_output.png}\n  \\caption{Generated ECG heartbeat sample from our seeded SimGAN individual in generation 0.}\n  \\Description{Generated sample from Gen 0 }\n  \\label{ECG_Gen_0}\n\\end{figure}\n\nTo analyze the effects of evolution, we select a Pareto individual in our last generation from ezCGP and compare how the optimized refiner transforms the same wave. Figure \\ref{ECG_Gen_5} shows improvement as the refined signal contains characteristics beyond learned noise.\n\n\\begin{figure}[!htbp]\n  \\centering\n  \\includegraphics[trim=0 65 0 65, clip, width=\\linewidth]{figures\/evolved_output.png}\n  \\caption{Generated ECG heartbeat sample from our optimized SimGAN individual from generation 9. }\n  \\Description{Generated sample from Gen 9}\n  \\label{ECG_Gen_5}\n\\end{figure}\n\n\n\\subsection{Empirical Analysis}\n\\label{ECG_Classifier}\nSynthetic data is useful when it is sufficiently similar to real data. A TSTR score empirically shows that the data is similar enough to use, where the synthetic data is used to train a model, which is  tested on real data \\citep{TSTR}. We extend TSTR score by training the same type of classifier on a subset of real data, a combination of real and simulated data, and a combination of real and refined data. We test the trained classifiers on the same withheld set of real signals. \n\nAs Table \\ref{Tab:comp} shows, we train four types of binary classifier models on the ECG signals output from our trained SimGAN models to predict whether a signal contains normal or abnormal characteristics. We can see that there is a performance boost in F1 scores after the models are trained on the refined data compared to just the real or real and simulated data. This indicates that the refined training data is likely closer to the real test set than the simulated training data was.  The refiner showcases how we can balance the training data with realistic abnormal heartbeats that were created from simulated healthy heartbeats. Similarly, the data our evolved SimGAN individual produces that the classifiers trained on sometimes performs better than the initial seeded individual, which suggests the evolved refiner's signals are of higher quality. Our refined signals are also model agnostic, as multiple models show similar improvement after being trained on the same data. \n\n\n\n\n\n\\begin{table\n\\centering\n\\caption{F1-Scores of ECG classifiers with dataset variations}\n\\resizebox{\\linewidth}{!}{%\n\\begin{tabular}{lrrrr}\n\\toprule\n& AdaBoost\n& Gradient Boosted Tree \n& Multi-Layer Perceptron\n& 1D-SqueezeNet \\citep{SqueezeNet}\\\\ \n\\midrule\nReal Waveforms \n& 0.205 & 0.256 & 0.293 & 0.875 \\\\ \nReal + Simulated Normal \n& 0.176 & 0.243 & 0.313 & 0.854 \\\\\nReal + Seed-refined Abnormal  \n& \\textbf{0.247} & 0.341 & \\textbf{0.491} & 0.884\\\\\nReal + Evolved-refined Abnormal  \n& 0.227 & \\textbf{0.354} & 0.467 & \\textbf{0.916}\\\\ \\bottomrule\n\\end{tabular}%\n}\n\\label{Tab:comp}\n\\end{table}\n\n\\section{Conclusion and Future Work}\n\\label{ConclusionAndFutureWork}\nIn this paper we show a method for using genetic evolution in conjunction with SimGANs to refine simulated data to better fit the real distribution.\nWe applied this method to an electrocardiogram dataset to generate refined waveforms, and show that using the refined waveforms to train an ECG classifying method will improve the classifier performance over the unrefined waveforms. \n\nWe could potentially adapt this work to other forms of one-dimensional data, such as audio or electroencephalography data. Simulated data will help fill the datasets, but there is still a worry that simulated and real-world data are separable. The method could also be modified to work for unevenly-spaced time-series data.\n\nThere is ample room for improvements in the evolution process. We can perform efficiency updates to encourage more diverse architectures to be found. Other seeds could be tested with various architectures that work well on one-dimensional data (such as transformers). Data preprocessing blocks and methods can be added as gene-operators, allowing ezCGP to specify individuals that both change model architecture\/hyperparameters and the preprocessing for its inputs. Our dual discriminator network could be generalized for ensembled networks for both discriminator and refiner networks, and could lend itself to co-evolution solutions.\n\nGiven the unclear nature of evaluating GANs, multiple objectives are beneficial for evaluating them. However, too many objectives can make the selection process overly complex. While Section \\ref{sec:Evaluation} showed a multitude of objectives for evaluation, we only used four for selection. Future work could explore creating more representative objectives or utilizing a many-objective selection algorithm.  \n\nSince we have described a multitude of different losses and configurations which can be used to train SimGAN models, we can see that it quickly becomes very complex and there is no definitive reason for choosing one loss over another. There were  many other losses and training techniques that have shown success in other studies that we chose not to include, such as Wasserstein Loss \\citep{WGAN} and the original WGAN-GP gradient penalty \\citep{WGAN-GP}. Therefore, we propose to model the loss function as a symbolic regression problem, where all the losses and evaluations will be used, each weighted by an evolved constant, that can be optimized.\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\\subsection{Notation}\nFor the response generation task,\nlet $M$ denote the input word sequence (message) $M=\\{m_1,m_2,...,m_I\\}$. %\n$R$ denotes the word sequence in response to $M$, where $R=\\{r_1,r_2,...,r_J,$ {\\it EOS}\\xspace{}\\} and $J$ is the length of the response (terminated by an {\\it EOS}\\xspace token). $r_t$ denotes a word token that is associated with a $K$ dimensional distinct word embedding $e_t$. $V$ is the vocabulary size. \n\n\n\\subsection{Speaker Model}\nOur first model is the Speaker Model, which \nmodels the respondent alone.\nThis model represents each individual speaker as a vector or embedding, which encodes \nspeaker-specific information (e.g., dialect, register, age, gender, personal information) that influences the content and style of her responses. Note that these attributes are not explicitly annotated, which would be tremendously expensive for our datasets. Instead, our model manages to cluster users along some of these traits (e.g., age, country of residence) based on the responses alone.\n\nFigure \\ref{fig1} gives a brief illustration of the Speaker Model. \nEach speaker $i\\in [1,N]$ is associated with a user-level representation $v_i\\in\\mathbb{R}^{K\\times 1}$. \n As in standard {{\\textsc{Seq2Seq}}}\\xspace models, we first encode message  $S$ into a vector representation $h_S$ using the source LSTM. \nThen for each step in the target side, hidden units are obtained by \ncombining the representation produced by the target LSTM at the previous time step, the word representations at the current time step, and the speaker embedding $v_i$:\n\\begin{equation}\n\\left[\n\\begin{array}{lr}\ni_t\\\\\nf_t\\\\\no_t\\\\\nl_t\\\\\n\\end{array}\n\\right]=\n\\left[\n\\begin{array}{c}\n\\sigma\\\\\n\\sigma\\\\\n\\sigma\\\\\n\\text{tanh}\\\\\n\\end{array}\n\\right]\nW\\cdot\n\\left[\n\\begin{array}{c}\nh_{t-1}\\\\\ne_{t}^s\\\\\nv_i\\\\\n\\end{array}\n\\right]\n\\end{equation}\n\\begin{equation}\nc_t=f_t\\cdot c_{t-1}+i_t\\cdot l_t\n\\end{equation}\n\\begin{equation}\nh_{t}^s=o_t\\cdot \\text{tanh}(c_t)\n\\end{equation}\nwhere $W\\in \\mathbb{R}^{4K\\times 3K}$. \nIn this way, speaker information is encoded and \ninjected into the hidden layer at each time step and thus helps predict personalized responses throughout the generation process.\nThe Speaker embedding $\\{v_i\\}$ is shared across all conversations that involve speaker $i$.  $\\{v_i\\}$ are learned by back propagating word prediction errors to each neural component during training. \n\nAnother useful property of this model is that it \nhelps {\\it infer} answers to questions even if the evidence is not readily present in the training set.\nThis is important as\nthe training data does not contain explicit \ninformation about every\nattribute of each user\n(e.g., gender, age, country of residence).\nThe model learns speaker representations based on conversational content produced by different speakers, and speakers producing similar responses tend to have similar embeddings, occupying nearby positions in the vector space. \nThis way, the training data of speakers nearby in vector space help increase the generalization capability of the\nspeaker model. For example, consider two speakers $i$ and $j$\nwho sound distinctly British, and who are therefore close in speaker \nembedding space. Now, suppose that, in the training data, speaker $i$ was asked {\\it Where do you live?} and responded {\\it in the UK}. Even if speaker $j$ was never asked the same question, this answer can help influence a good response from speaker $j$, and this without explicitly labeled geo-location information.\n\\begin{comment}\nWhen a speaker is asked a questions like {\\it Which company do you work for?} or {\\it Where do you live?}, the evidence of which is not contained in the training set, the model can benefit from speakers that the current speaker resemble, take advantage of information of them and make inference. \n \n  the hometown of the speaker, the company he works for or his\/her food preference for dinner for most of users as long as they don't explicitly mention these aspects.  But the model learns \n\\end{comment}\n  \n\\subsection{Speaker-Addressee Model}\nA natural extension of the Speaker Model is a model that is sensitive to speaker-addressee interaction patterns within the conversation. Indeed,\nspeaking style, register, and content does not vary only with the identity of the speaker, but also with that of the addressee.\nFor example, in scripts for the TV series {\\it Friends} used in some of our experiments, the character Ross often \ntalks differently to his sister Monica than to Rachel,\nwith whom he is engaged in an on-again off-again relationship throughout the series. \n\\begin{comment}\nModeling speaker-to-speaker interacting patterns requires that both of the speakers are involved in a long-term interactions with each other, from which their interacting patterns can be learned. \n\\end{comment}\n\nThe proposed Speaker-Addressee Model operates as follows:\nWe wish to predict how speaker $i$ would respond to a message produced by speaker $j$. Similarly to the Speaker model, we associate each speaker with a $K$ dimensional speaker-level representation, namely $v_i$ for user $i$ and $v_j$ for user $j$. \nWe obtain an interactive representation $V_{i,j}\\in \\mathbb{R}^{K\\times 1}$ by linearly combining user vectors $v_i$ and $v_j$\nin an attempt to model the interactive style of user $i$ towards user $j$,\n\\begin{equation}\nV_{i,j}=\\text{tanh}(W_1\\cdot v_i+W_2\\cdot v_2)\n\\end{equation}\nwhere $W_1, W_2\\in \\mathbb{R}^{K\\times K}$. \n$V_{i,j}$  is then linearly incorporated into LSTM models at each step in the target: \n\\begin{equation}\n\\left[\n\\begin{array}{lr}\ni_t\\\\\nf_t\\\\\no_t\\\\\nl_t\\\\\n\\end{array}\n\\right]=\n\\left[\n\\begin{array}{c}\n\\sigma\\\\\n\\sigma\\\\\n\\sigma\\\\\n\\text{tanh}\\\\\n\\end{array}\n\\right]\nW\\cdot\n\\left[\n\\begin{array}{c}\nh_{t-1}\\\\\ne_{t}^s\\\\\nV_{i,j}\\\\\n\\end{array}\n\\right]\n\\end{equation}\n\\begin{equation}\nc_t=f_t\\cdot c_{t-1}+i_t\\cdot l_t\\\\\n\\end{equation}\n\\begin{equation}\nh_{t}^s=o_t\\cdot \\text{tanh}(c_t)\n\\end{equation}\n$V_{i,j}$ \ndepends on both speaker and addressee and\nthe same speaker will thus respond differently to a message from different interlocutors. \nOne potential issue with Speaker-Addressee modelling is the difficulty involved in collecting a large-scale training dataset in which each speaker \nis involved in conversation with a wide variety of people. Like the Speaker Model, however, the Speaker-Addressee Model derives generalization capabilities from speaker embeddings.\nEven if the two speakers\nat test time ($i$ and $j$) were never involved in the same conversation in the training data, two speakers $i'$ and $j'$ who are respectively close in embeddings may have been, and this can help  modelling how $i$ should respond to $j$. \n\n\\subsection{Decoding and Reranking}\nFor decoding, \nthe N-best lists  are generated using the decoder with beam size \\mbox{$B=200$}.\nWe set a maximum length of 20 for the generated candidates. \nDecoding operates as follows: At each time step, \nwe first examine all \\mbox{$B\\times B$} possible next-word candidates, and add all hypothesis ending with an {\\it EOS}\\xspace token to the N-best list. We then preserve the top-$B$ unfinished hypotheses and move to the next word position. \n\nTo deal with the issue that {{\\textsc{Seq2Seq}}}\\xspace models tend to generate generic and commonplace responses such as {\\it I don't know}, we follow \\newcite{li2015diversity} by reranking the generated N-best list \nusing a scoring function that linearly combines \n a length penalty and the log likelihood of the source given the target:\n\\begin{equation}\n\\log p(R|M,v)+\\lambda\\log p(M|R)+\\gamma |R| \n\\end{equation}\nwhere $p(R|M,v)$ denotes the probability of the generated response given the message $M$ and the  respondent's speaker ID. \n$|R|$ denotes the length of the target and $\\gamma$ denotes the associated penalty weight. We optimize $\\gamma$ and $\\lambda$ on N-best lists of response candidates generated from the development set using MERT \\cite{mert} by optimizing {{\\sc Bleu}}\\xspace.\nTo compute $p(M|R)$, \nwe train an inverse {{\\textsc{Seq2Seq}}}\\xspace model by swapping messages and responses. We trained standard {{\\textsc{Seq2Seq}}}\\xspace models for $p(M|R)$  with no speaker information considered.  \n\n\n\\subsection{Twitter Persona Dataset} \n\\paragraph{Data Collection}\nTraining data for the {Speaker Model} was extracted from the Twitter FireHose for the six-month period beginning January 1, 2012.\nWe limited the sequences to those where the responders had engaged in at least 60 (and at most 300) 3-turn conversational interactions during the period, in other words, users who reasonably frequently engaged in conversation. This yielded a set of 74,003 users who took part in a minimum of 60 and a maximum of 164 conversational turns (average: 92.24, median:  90). \nThe dataset extracted using responses by these ``conversationalists'' contained 24,725,711 3-turn sliding-window (context-message-response) conversational \nsequences.\n\nIn addition, we sampled 12000 3-turn conversations from the same user set from the Twitter FireHose for the three-month period beginning July 1, 2012, and set these aside as development, validation, and test sets (4000 conversational sequences each). Note that development, validation, and test sets for this data are single-reference, which is by design. Multiple reference responses would typically require acquiring responses from different people, which would confound different personas.\n\n\\paragraph{Training Protocols} We trained four-layer {{\\textsc{Seq2Seq}}}\\xspace models on the Twitter corpus following the approach of \\cite{sutskever2014sequence}.\nDetails are as follows: \n\\begin{itemize}\n\\reduceVerticalSpace\n\\item 4 layer LSTM models with 1,000 hidden cells for each layer.\n\\item Batch size is set to 128.\n\\item Learning rate is set to 1.0.\n\\item Parameters are initialized by sampling from the uniform distribution $[-0.1,0.1]$.\n\\item Gradients are clipped to avoid gradient explosion with a threshold of 5.\n\\item Vocabulary size is limited to 50,000.\n\\item Dropout rate is set to 0.2.\n\\end{itemize}\nSource and target LSTMs use different sets of parameters.\nWe ran 14 epochs, and \ntraining took roughly a month to finish on a Tesla K40 GPU machine. \n\nAs only speaker IDs of responses were specified when compiling the Twitter dataset, experiments on this dataset were limited to the {Speaker Model}.  \n\n\\subsection{Twitter Sordoni Dataset} \n\nThe Twitter Persona Dataset was collected for this paper for experiments with speaker ID information. \nTo obtain a point of \ncomparison with prior state-of-the-art work \\cite{Sordoni2015,li2015diversity}, \nwe measure our baseline (non-persona) LSTM model against prior \nwork on the dataset of \\cite{Sordoni2015}, which we call the Twitter Sordoni Dataset. \nWe only use\nits test-set portion, which contains\nresponses for 2114 context and messages. \nIt is important to note that \nthe Sordoni dataset offers up to 10 references per message, while the \nTwitter Persona dataset has only 1 reference per message. Thus {{\\sc Bleu}}\\xspace scores cannot be compared across the two Twitter datasets ({{\\sc Bleu}}\\xspace scores on 10 references \nare generally much higher than with 1 reference).\nDetails of this dataset are in \\cite{Sordoni2015}.\n\n\\subsection{Television Series Transcripts} \n\\paragraph{Data Collection} For the dyadic Speaker-Addressee Model we used scripts from the American television comedies {\\it Friends}\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/Friends}} and {\\it The Big Bang Theory},\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/The_Big_Bang_Theory}} available from Internet Movie Script Database (IMSDb).\\footnote{\\url{http:\/\/www.imsdb.com}}\nWe collected 13 main characters from the two series in a corpus of 69,565 turns. \nWe split the corpus into training\/development\/testing sets, with development and testing sets each of about 2,000 turns. \n\n\\paragraph{Training} \nSince the relatively small size of the dataset does not allow for training an open domain dialog model, we adopted a domain adaption strategy where we first trained a standard {{\\textsc{Seq2Seq}}}\\xspace models using a much larger OpenSubtitles (OSDb) dataset \\cite{tiedemann2009news}, and then adapting the pre-trained model to the TV series dataset. \n\nThe OSDb dataset is a large, noisy, open-domain dataset containing roughly 60M-70M scripted lines spoken by movie characters. \nThis dataset does not specify which character speaks each subtitle line, which prevents us from inferring speaker turns. \nFollowing Vinyals et al. (2015), we make the simplifying assumption that each line of subtitle constitutes a full speaker turn.\\footnote{This introduces a degree of noise as consecutive lines are not necessarily from the same scene or two different speakers.}\nWe trained standard {{\\textsc{Seq2Seq}}}\\xspace models on OSDb dataset, following the protocols already described in Section 5.1. \nWe run 10 iterations over the training set.\n\nWe initialize word embeddings and LSTM parameters in the Speaker Model and the Speaker-Addressee model using parameters learned from OpenSubtitles datasets. \nUser embeddings are randomly initialized  from $[-0.1,0.1]$. \nWe then ran 5 additional epochs until the perplexity on the development set stabilized. \n\n\\subsection{Evaluation}\nFollowing \\cite{Sordoni2015,li2015diversity}\nwe used {{\\sc Bleu}}\\xspace \\cite{Papineni2002BLEU} \nfor parameter tuning and evaluation. \n{{\\sc Bleu}}\\xspace has been shown to correlate well with human judgment on the response generation task, as demonstrated in \\cite{galley2015deltableu}. \n\\begin{comment}\n\\cite{li2015diversity} also adopted  {\\it distinct-1} and {\\it distinct-2} to calculating the number of distinct unigrams and bigrams in generated responses \nto measure the degree of diversity. The value is scaled by total number of generated tokens to avoid favoring long sentences. \nWe also report the degree of diversity following the protocols defined in \\cite{li2015diversity}.  \n\\end{comment}\nBesides {{\\sc Bleu}}\\xspace scores, we also report perplexity as an indicator of model capability.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lc}\nSystem                                    & {{\\sc Bleu}}\\xspace \\\\ \\hline\nMT baseline \\cite{ritter2011data}         & 3.60\\% \\\\ \\hline\nStandard LSTM MMI \\cite{li2015diversity}  & 5.26\\% \\\\\nStandard LSTM MMI (our system)            & 5.82\\% \\\\ \\hline\n{\\it Human}                               & {\\it 6.08\\%}\\\\\n\\end{tabular}\n\\caption{{{\\sc Bleu}}\\xspace on the Twitter Sordoni dataset (10 references). We contrast our baseline against an SMT baseline \\cite{ritter2011data}, and the best result \\cite{li2015diversity} on the established\ndataset of \\cite{Sordoni2015}.\nThe last result is for a human oracle, but it is not directly comparable as the oracle {{\\sc Bleu}}\\xspace is computed in a leave-one-out fashion, having one less reference available. We nevertheless provide\nthis result to give a sense that these {{\\sc Bleu}}\\xspace scores of 5-6\\% are not unreasonable.}\n\\label{twitter-baselines}\n\\end{table}\n\n\\subsection{Baseline}\nSince our main experiments are with a new dataset (the Twitter Persona Dataset), we first show that our LSTM baseline is competitive with the state-of-the-art \\cite{li2015diversity} on an established\ndataset, the Twitter Sordoni Dataset \\cite{Sordoni2015}.\nOur baseline is simply our implementation of the LSTM-MMI of \\cite{li2015diversity}, so results should be relatively close to their reported results.\nTable~\\ref{twitter-baselines} summarizes our results against prior work.\n\\begin{comment}\nThe comparison is particularly important because the\ntest set of \\cite{Sordoni2015,li2015diversity} offers up to 10 references, while ours is single-reference (a multi-reference test set, would involve different respondents and confound different personas, which we want to avoid).\n\\end{comment}\nWe see that our system actually does better than \\cite{li2015diversity}, and\nwe attribute the improvement to a larger training corpus, the use of dropout during training, and possibly to the ``conversationalist'' nature of our corpus.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ccc}\nModel&Standard LSTM&Speaker Model \\\\\\hline\nPerplexity&47.2&42.2 ($-10.6\\%$) \\\\\n\\end{tabular}\n\\caption{Perplexity for standard {{\\textsc{Seq2Seq}}}\\xspace and the Speaker model \non the Twitter Persona development set.}\n\\label{twitter-per}\n\\end{table}\n\\begin{table}\n\\centering\n\\begin{tabular}{lll}\nModel&Objective& {{\\sc Bleu}}\\xspace \\\\\\hline\nStandard LSTM &MLE& 0.92\\% \\\\\nSpeaker Model & MLE&1.12\\% (+21.7$\\%$) \\\\\\hline\nStandard LSTM &MMI& 1.41\\% \\\\\nSpeaker Model & MMI&1.66\\% (+11.7$\\%$) \\\\\n\\end{tabular}\n\\caption{\n{{\\sc Bleu}}\\xspace on the Twitter Persona dataset (1 reference), for the\nstandard {{\\textsc{Seq2Seq}}}\\xspace model and the Speaker model using as objective either maximum likelihood (MLE) or maximum mutual information (MMI).}\n\\label{twitter-bleu}\n\\end{table}\n\n\\begin{table*}\n\\centering\n\\begin{tabular}{cccc}\nModel&Standard LSTM&Speaker Model& Speaker-Addressee Model \\\\\\hline\nPerplexity&27.3&25.4 ($-7.0\\%$)& 25.0 ($-8.4\\%$)  \\\\\n\\end{tabular}\n\\caption{Perplexity for standard {{\\textsc{Seq2Seq}}}\\xspace and persona models on the TV series dataset.}\n\\label{tv-per}\n\\end{table*}\n\\begin{table*}\n\\centering\n\\begin{tabular}{ccccc}\nModel&Standard LSTM&Speaker Model& Speaker-Addressee Model \\\\\\hline\nMLE&1.60\\%& 1.82\\% ($+13.7\\%$)& 1.83\\%  ($+14.3\\%$) \\\\\nMMI&1.70\\%& 1.90\\% ($+10.6\\%$) &1.88\\% ($+10.9\\%$) \\\\\\hline\n\\end{tabular}\n\\caption{\n{{\\sc Bleu}}\\xspace on the TV series dataset (1 reference), for the\nstandard {{\\textsc{Seq2Seq}}}\\xspace and persona models.}\n\\label{tv-bleu}\n\\end{table*}\n\n\\subsection{Results}\nWe first report performance on the Twitter Persona dataset.\nPerplexity is reported in Table \\ref{twitter-per}. We observe about a $10\\%$ decrease in perplexity for the Speaker model over the standard {{\\textsc{Seq2Seq}}}\\xspace model. \nIn terms of {{\\sc Bleu}}\\xspace scores (Table~\\ref{twitter-bleu}), a significant performance boost \nis observed for \n the Speaker model over the standard {{\\textsc{Seq2Seq}}}\\xspace model, yielding an increase of $21\\%$\nin the maximum likelihood (MLE) setting and $11.7\\%$ for mutual information setting (MMI). \nIn line with findings in \\cite{li2015diversity}, we observe a consistent performance boost introduced by \nthe MMI objective function \nover a standard {{\\textsc{Seq2Seq}}}\\xspace model based on the MLE objective function. \nIt is worth noting that our persona models are more beneficial to the MLE models\nthan to the MMI models. This result is intuitive as the persona models help make \nStandard LSTM MLE outputs more informative and less bland, and thus make the use \nof MMI less critical.\n\\begin{comment}\nIt is noting that the difference between MLE persona models and MLE {{\\textsc{Seq2Seq}}}\\xspace models is greater than that between MMI persona model and MMI {{\\textsc{Seq2Seq}}}\\xspace models. \nSuch a result is intuitive since both persona models and {{\\textsc{Seq2Seq}}}\\xspace models use the same $p(M|R)$ for later MMI reranking, which is obtained without user-level information. Such a reranking process would dilute the discrepancy between the persona models and vanilla {{\\textsc{Seq2Seq}}}\\xspace models. \n\\end{comment}\n\n\nFor the TV Series dataset,  perplexity and {{\\sc Bleu}}\\xspace  scores are respectively  reported in Table \\ref{tv-per} and Table \\ref{tv-bleu}.\nAs can be seen, the Speaker and Speaker-Addressee models respectively achieve perplexity values of 25.4 and 25.0 on the TV-series dataset, $7.0\\%$ and $8.4\\%$ percent lower than the correspondent standard {{\\textsc{Seq2Seq}}}\\xspace models. \nIn terms of {{\\sc Bleu}}\\xspace score,  we observe a similar performance  boost as on the Twitter dataset, in which the Speaker model\nand the Speaker-Addressee model \n outperform the standard {{\\textsc{Seq2Seq}}}\\xspace model by $13.7\\%$ and $10.6\\%$.\nBy comparing the Speaker-Addressee model against the Speaker model on the TV Series dataset, we do not observe a significant difference.  \nWe suspect that this is primarily due to the relatively small size of the dataset where the interactive patterns might not be fully captured. \nSmaller values of perplexity are observed for the Television Series dataset than the Twitter dataset, the perplexity of which is over 40, presumably due to the noisier nature of Twitter dialogues. \n\\subsection{Qualitative Analysis}\n\n\\begin{table}\n\\setlength{\\tabcolsep}{4pt}\n\\center\n{\\small\n\\begin{tabularx}{\\linewidth}{rl} \\hline\n{{\\it message}}\\xspace   & Where is your hometown?\\\\\\hline\n{\\it baseline}    & I was born in Canada.\\\\\\hline\n\\User{1}   & I'm from England.\\\\\n\\User{2}   & I'm from Manchester.\\\\\n\\User{3}   & I'm from Liverpool.\\\\\n\\User{4}   & England. You?\\\\\n\\User{5}   & My hometown.\\\\\n\\User{6}   & I'm from Texas.\\\\\n\\User{7}   & I'm from LA.\\\\\n\\User{8}   & England.\\\\\n\\User{9}   & I was born here.\\\\\n\\User{10}  & I was born in the us.\\\\\\hline\\hline\n{{\\it message}}\\xspace   & What company do you work for?\\\\\\hline\n{\\it baseline}    & I work for a company.\\\\\\hline\n\\User{1}   & I don't have a job.\\\\\n\\User{2}   & I'm a manager.\\\\\n\\User{3}   & I'm working. {\\color{red} $\\#$}\\\\\n\\User{4}   & I work for a company. \\\\\n\\User{5}   & I'm working. {\\color{red} $\\#$}\\\\\n\\User{6}   & Customer service.\\\\\n\\User{7}   & I work in retail.\\\\\n\\User{8}   & I work in a cafe.\\\\\n\\User{9}   & I work at a gas station.\\\\\n\\User{10}  & I don't know {\\color{red} $\\#$} \\\\\\hline\\hline\n{{\\it message}}\\xspace   & What did you have for dinner?\\\\\\hline\n{\\it baseline}    & I had fish and chips.\\\\\\hline\n\\User{1}   & I had spag bol.\\\\\n\\User{2}   & Chicken and chips.\\\\\n\\User{3}   & Chicken and rice.\\\\\n\\User{4}   & Fish and chips.\\\\\n\\User{5}   & I had spag bol.\\\\\n\\User{6}   & I had Mexican food.\\\\\n\\User{7}   & Salad...\\\\\n\\User{8}   & I had chicken and chips.\\\\\n\\User{9}   & I had spag bol.\\\\\n\\User{10}  & Pizza.\\\\\\hline\n\\end{tabularx}\n}\n\\caption{Responses generated by the baseline (LSTM-MMI) and the Speaker Model for ten randomly selected users, without cherry picking. {\\color{red} $\\#$} indicates poor-quality responses produced by the system.}\n\\label{example1}\n\\end{table}\n\n\n\\paragraph{Diverse Responses by Different Speakers}\nTable \\ref{example1} represents responses generated by persona models in response to three different input questions. We randomly selected 10 speakers (without cherry-picking) from the original Twitter dataset. We collected their user level representations from a speaker look-up table and integrated them into the decoding models.  The model tends to generate specific responses for different people in response to the factual questions.\\footnote{There appears to be a population bias in the training set that favors British users.}  \n\nTable \\ref{addressees} shows responses generated from the {\\it Speaker-Addressee Model} using the TV-series dataset. Interestingly, we regularly observe \nthat this model is sensitive to the identity of the addressee, generating words specifically targeted at that addressee (e.g., her name).\nFor example, the model produces {\\it Of course, I love you, Emily} in response to an input from {\\it Emily}. Also, the model generates {\\it Of course I love you. ( kisses him)}, where the pronoun ``him'' accurately identifies the gender of the addressee. \n\n\n\\paragraph{Human Evaluation} We conducted a human evaluation of outputs from the Speaker Model, using \na crowdsourcing service. \nSince we cannot expect crowdsourced human judges to know or attempt to learn the ground truth of Twitter users who are not well-known public figures, we designed our experiment to evaluate the consistency of outputs associated with the speaker IDs. To this end, we collected 24 pairs of questions for which we would expect responses to be consistent if the persona model is coherent.  For example, responses to the questions {\\it What country do you live in?} and {\\it What city do you live in?} would be considered consistent if the answers were {\\it England} and {\\it London} respectively, but not if they were {\\it UK} and {\\it Chicago}.  Similarly, the responses to {\\it Are you vegan or vegetarian?} and {\\it Do you eat beef?} are consistent if the answers generated are {\\it vegan} and {\\it absolutely not}, but not if they are {\\it vegan} and {\\it I love beef}.  We collected 20 pairs of outputs for randomly-selected personas provided by the Speaker Model for each question pair (480 response pairs total). We also obtained the corresponding outputs from the baseline MMI-enhanced {{\\textsc{Seq2Seq}}}\\xspace system. \n\n\\begin{table}\n\\setlength{\\tabcolsep}{4pt}\n\\center\n{\\small\n\\begin{tabularx}{\\linewidth}{rl} \\hline\n{{\\it message}}\\xspace & Do you love me?\\\\\\hline\n{\\it baseline} & Of course, I love you.  \\\\\\hline\n& {\\it addressee: {\\it Emily}, speaker: Sheldon}\\\\\n{\\it response} & Of course, I love you, Emily.\\\\\\hline\n& {\\it addressee: {\\it Emily}, speaker: Penny}\\\\\n{\\it response} & Of course, I love you, Emily.\\\\\\hline\n& {\\it addressee: {\\it Leonard}, speaker: Penny}\\\\\n{\\it response} & Of course, I love you, Leonard.\\\\\\hline\n& {\\it addressee: {\\it Monica}, speaker: Chandler}\\\\\n{\\it response} & Of course I love you. I want to marry you.\\\\\\hline\n& {\\it addressee: {\\it  Rachel}, speaker: Chandler}\\\\\n{\\it response} & Of course I love you.\\\\\\hline\n& {\\it addressee: {\\it  Ross}, speaker: Chandler}\\\\\n{\\it response} & Of course I love you.\\\\\\hline\n& {\\it addressee: {\\it  Ross}, speaker: Rachel}\\\\\n{\\it response} & Of course I love you. (kisses him) \\\\\\hline\n& {\\it addressee: {\\it  Emily}, speaker: Rachel}\\\\\n{\\it response} & Of course I love you. \\\\\\hline\n\\end{tabularx}\n}\n\\caption{Responses to {\\it Do you love me?} from the {\\it Speaker-Addressee} model on the TV-series dataset using different addressees and speakers.}\n\\label{addressees}\n\\end{table}\n\n\\begin{table}\n\\setlength{\\tabcolsep}{4pt}\n\\center\n{\\small\n\\begin{tabularx}{\\linewidth}{rl} \\hline\n{\\bf User1} &\\\\\\hline\n{{\\it message}}\\xspace    & Where is your hometown?\\\\\n{{\\it response}}\\xspace   & I'm from England.\\\\\n{{\\it message}}\\xspace    & Where are you from?\\\\\n{{\\it response}}\\xspace   & I'm from England.\\\\\n{{\\it message}}\\xspace    & In which city do you live now?\\\\\n{{\\it response}}\\xspace   & I live in London.\\\\\n{{\\it message}}\\xspace    & In which country do you live now?\\\\\n{{\\it response}}\\xspace   & I live in England.\\\\\\hline\n{{\\it message}}\\xspace    & What is your major?\\\\\n{{\\it response}}\\xspace   & Business. You?\\\\\n{{\\it message}}\\xspace    & What did you study in college?\\\\\n{{\\it response}}\\xspace   & I did business studies.\\\\\\hline\n{{\\it message}}\\xspace    & How old are you?\\\\\n{{\\it response}}\\xspace   & I'm 18.\\\\\n{{\\it message}}\\xspace    & What is your age?\\\\\n{{\\it response}}\\xspace   & I'm 18.\\\\\\hline\\hline\n{\\bf User2} \\\\\\hline\n{{\\it message}}\\xspace    & Where is your hometown? \\\\\n{{\\it response}}\\xspace   & I live in Indonesia.\\\\\n{{\\it message}}\\xspace    & Where are you from?\\\\\n{{\\it response}}\\xspace   & England, you?\\\\\n{{\\it message}}\\xspace    & In which city do you live now?\\\\\n{{\\it response}}\\xspace   & I live in Jakarta.\\\\\n{{\\it message}}\\xspace    & In which country do you live now?\\\\\n{{\\it response}}\\xspace   & I live in Indonesia.\\\\\\hline\n{{\\it message}}\\xspace    & What is your major?\\\\\n{{\\it response}}\\xspace   & Business, you?\\\\\n{{\\it message}}\\xspace    & What did you study in college?\\\\\n{{\\it response}}\\xspace   & Psychology, you?\\\\\\hline\n{{\\it message}}\\xspace    & How old are you?\\\\\n{{\\it response}}\\xspace   & I'm 18.\\\\\n{{\\it message}}\\xspace    & What is your age?\\\\\n{{\\it response}}\\xspace   & I'm 16.\\\\\\hline%\n\\end{tabularx}\n}\n\\caption{Examples of speaker consistency and inconsistency generated by the Speaker Model}\n\\label{example2}\n\\end{table}\n\n\nSince our purpose is to measure the gain in consistency over the baseline system, we presented the pairs of answers system-pairwise, i.e., 4 responses, 2 from each system,  displayed on the screen, and asked judges to decide which of the two systems was more consistent.  The position in which the system pairs were presented on the screen was randomized.  \nThe two systems were judged on 5-point zero-sum scale, assigning a score of 2 (-2) if one system was judged more (less) consistent than the other, and 1 (-1) if one was rated ``somewhat'' more (less) consistent.  Ties were assigned a score of zero. Five judges rated each pair and their scores were averaged and remapped into 5 equal-width bins. After discarding ties, we found the persona model was judged either ``more consistent'' or ``somewhat more consistent'' in 56.7\\% of cases. If we ignore the ``somewhat more consistent'' judgments, the persona model wins in 6.1\\% of cases, compared with only 1.6\\% for the baseline model. \nIt should be emphasized that the baseline model is a strong baseline, \n\\begin{comment}\nThe two systems were judged on a 5-point scale, assigning a score of 2 (respectively -2) if the persona system was judged much more (respectively less) consistent than the baseline, 1 (respectively -1) if ``mostly'' more (respectively less) consistent, and 0 otherwise. \nFive judges rated each pair, and scores were averaged across judges.\\footnote{To turn this average back into a 5-point scale, we mapped average scores into 5 equal-size bins.}\nAfter removing ties\nwe found that\nthe persona model (respectively baseline) was judged either ``more consistent'' or ``somewhat more consistent'' than the baseline (respectively persona model) in 56.7\\% (respectively 43.3\\%) of the cases. Ignoring ``somewhat more consistent\" judgments, the persona model is more consistent in 6.0\\% of the cases, while the baseline only 1.6\\% of the time.\nIt should be stressed that the latter is a strong baseline, \n\\end{comment}\nsince it represents the consensus of all 70K Twitter users in the dataset\\footnote{{\\it I'm not pregnant} is an excellent consensus answer to the question {\\it Are you pregnant?}, while {\\it I'm pregnant} is consistent as a response only in the case of someone who also answers the question {\\it Are you a guy or a girl?} with something in the vein of {\\it I'm a girl}.}.\n\nTable \\ref{example2} illustrates how consistency is an emergent property of two arbitrarily selected users. The model is capable of discovering the relations between different categories of location such as London and the UK, Jakarta and Indonesia. However, the model also makes inconsistent response decisions, generating different answers in the second example in response to questions asking about age or major. \nOur proposed persona models integrate user embeddings into the LSTM, and thus can be viewed as encapsulating a trade-off between a persona-specific generation model and a general conversational model. \n\\begin{comment}\nHere we qualitatively show examples about speaker consistency to identify both the advantages and limitations of the proposed models. We manually make up different sets of questions, and each set consists of 2 or 3 questions. These questions are either paraphrases such as {\\it How old are you?} and {\\it What's your age?} or related questions such as {\\it Where are you from?} and {\\it Where do you live?}. In our experiments for consistency analysis, we set maximum response length to 8, as to make it easier for humans (especially crowd workers) to judge factual consistency. \n\nwe observe a general tendency to generate consistent responses for individual speakers. A  broader analysis across the dataset shows that the model can handle some categories of  \nlocation related questions pretty well, and is also capable of discovering the relations between different categories of location names such as London and the UK, Jakarta and Indonesia as shown in the examples. However, the model also frequently make inconsistent response decisions: generating different answers in the second example in response to questions asking about age or major. \nOur explanation is as follows: the proposed persona models integrate user embeddings into decoding, and can be viewed as a trade-off between personal specific generation model and the general conversational model. \nThe combined model would lean towards the general one, which is a safer choice\nwhen the influence from user-level embedding is significant or in other words, clues presented from the training data is not evident enough. \nFrom the algorithmic perspective, the process of integrating speaker embeddings can be viewed as \nadding bias to the standard {{\\textsc{Seq2Seq}}}\\xspace model, the influence of which can be limited. At the same time, we also wish to maintain a balance between speaker-level evidence and the overall response model to avoid overfitting. However to address these issues remains our long-term goal in dialogue literature. \n\\end{comment}\n\n\n\\section{Introduction} \n\n\n\\label{sec:intro}\n\\input{01-intro-clean.tex}\n\n\\section{Related Work}\n\\label{sec:related}\n\\input{02-relatedwork-clean.tex}\n\n\\section{Sequence-to-Sequence Models}\n\\label{sec:seq2seq}\n\\input{03-seq2seq-clean.tex}\n\n\\section{Personalized Response Generation}\n\\label{sec:models}\n\\input{04-models-clean.tex}\n\n\\section{Datasets}\n\\label{sec:data}\n\\input{05-datasets-new-clean.tex}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\input{06-experiments-clean.tex}\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\\input{07-conclusion-clean.tex}\n\n\\section*{Acknowledgments}\n\nWe with to thank Stephanie Lukin, Pushmeet Kohli, Chris Quirk, Alan Ritter, and Dan Jurafsky for helpful discussions.\n\n\\bibliographystyle{acl2016}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Other Related Work}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminary Results}\n\\label{app:prelim}\n\n\n\\begin{lemma}[Young's inequality]\n\\label{lem:Young}\nGiven two same-dimensional vectors $\\mathbf u, \\mathbf v \\in \\mathbb R^d$, the Euclidean inner product can be bounded as follows:\n$$\\left\\langle \\mathbf u, \\mathbf v \\right\\rangle \\leq \\frac{\\norm{\\mathbf u}^2}{2 \\gamma} + \\frac{\\gamma \\norm{\\mathbf v}^2}{2}$$\nfor every constant $\\gamma > 0$.\n\\end{lemma}\n\n\\begin{lemma}[Strong Concavity]\nA function $g: \\mathcal X \\times \\mathcal Y$ is strongly concave in ${\\mathbf y}$, if there exists a constant $\\mu > 0$, such that for all ${\\mathbf x} \\in \\mathcal X$, and for all ${\\mathbf y}, {\\mathbf y}' \\in \\mathcal Y$, the following inequality holds.\n$$g({\\mathbf x}, {\\mathbf y}) \\leq g({\\mathbf x}, {\\mathbf y}') + \\left\\langle \\nabla_{\\by} g({\\mathbf x}, {\\mathbf y}'), {\\mathbf y}' - {\\mathbf y} \\right\\rangle - \\frac{\\mu}{2} \\norm{{\\mathbf y} - {\\mathbf y}'}^2.$$\n\\end{lemma}\n\n\\begin{lemma}[Jensen's inequality]\n\\label{lem:jensens}\nGiven a convex function $f$ and a random variable $X$, the following holds.\n$$f \\left( \\mathbb E [X] \\right) \\leq \\mathbb E \\left[ f(X) \\right].$$\n\\end{lemma}\n\n\\begin{lemma}[Sum of squares]\n\\label{lem:sum_of_squares}\nFor a positive integer $K$, and a set of vectors $x_1, \\hdots, x_K$, the following holds:\n\\begin{align*}\n    \\norm{\\sum_{k=1}^K x_k}^2 \\leq K \\sum_{k=1}^K \\norm{x_k}^2.\n\\end{align*}\n\\end{lemma}\n\n\\begin{lemma}[Quadratic growth condition \\cite{schmidt16lin_conv_PL_kdd}]\n\\label{lem:quad_growth}\nIf function $g$ satisfies Assumptions \\ref{assum:smoothness}, \\ref{assum:PL_y}, then for all $x$, the following conditions holds\n\\begin{align*}\n    g(x) - \\min_{z} g(z) & \\geq \\frac{\\mu}{2} \\norm{x_p - x}^2, \\\\\n    \\norm{\\nabla g(x)}^2 & \\geq 2 \\mu \\left( g(x) - \\min_z g(z) \\right).\n\\end{align*}\n\\end{lemma}\n\n\\subsection{Local SGD}\n\\label{app:local_SGD}\nLocal SGD is the algorithm which forms the basis of numerous Federated Learning algorithms \\cite{konevcny16federated, fedavg17aistats}.\nEach client running Local SGD (\\cref{alg_local_SGD}), runs a few SGD iterations locally and only then communicates with the server, which in turn computes the average and returns to the clients. \nThis approach saves the limited communication resources of the clients, without sacrificing the convergence guarantees.\n\nThe algorithm has been analyzed for both convex and nonconvex minimization problems.\nWith identical distribution of client data, Local SGD has been analyzed in \\cite{stich18localSGD_iclr, stich20error_fb_jmlr, khaled20localSGD_aistats, spiridonoff21comm_eff_SGD_neurips} for (strongly) convex objectives, and in \\cite{wang21coopSGD_jmlr, zhou18localSGD_ijcai} for nonconvex objectives.\nWith heterogeneous client data Local SGD has been analyzed in \\cite{khaled20localSGD_aistats, koloskova20unified_localSGD_icml} for (strongly) convex objectives, and in \\cite{jiang18linear_neurips, haddadpour19conv_FL_arxiv, koloskova20unified_localSGD_icml} for nonconvex objectives.\n\n\\begin{algorithm}[ht]\n\\caption{Local SGD}\n\\label{alg_local_SGD}\n\\begin{algorithmic}[1]\n\t\\STATE{\\textbf{Input: }{\\small${\\mathbf x}_0^i = {\\mathbf x}_0$}, for all $i \\in [n]$, step-size $\\eta$, $\\tau$, $T$}\n\t\\FOR[At all clients $i=1,\\hdots, n$]{$t=0$ to $T-1$}\n\t    \\STATE{Sample minibatch ${\\xi^i_{t}}$ from local data}\n        \\STATE{${\\mathbf x^i_{t+1}} = {\\mathbf x^i_t} - \\eta \\nabla g_i ({\\mathbf x^i_t}; {\\xi^i_{t}})$}\n       \n        \\IF{$t+1$ mod $\\tau = 0$}\n            \\STATE{Clients send $\\{ {\\mathbf x^i_{t+1}} \\}$ to the server}\n            \\STATE{Server computes averages ${\\mathbf x_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_{t+1}}$, and sends to all the clients}\n            \\STATE{${\\mathbf x^i_{t+1}} = {\\mathbf x_{t+1}}$, for all $i \\in [n]$}\n        \\ENDIF\n\t\\ENDFOR\n\t\\STATE{\\textbf{Return: }${\\bar{\\bx}_T}$ drawn uniformly at random from $\\{ {\\mathbf x_t} \\}$, where ${\\mathbf x_t} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_t}$}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{lemma}[Local SGD for Convex Function Minimization \\cite{khaled20localSGD_aistats}]\n\\label{lem:local_SGD_khaled}\n\nSuppose that the local functions $\\{ g_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, and are all convex.\\footnote{The result actually holds under slightly weaker assumptions on the noise and heterogeneity.}\nSuppose, the step-size $\\eta$ is chosen such that $\\eta \\leq \\min \\left\\{ \\frac{1}{4 L_f}, \\frac{1}{8 L_f (\\tau - 1)} \\right\\}$.\nThen, the iterates generated by Local SGD (\\cref{alg_local_SGD}) algorithm satisfy\n\\begin{align*}\n    \\mathbb E \\left[ g({\\bar{\\bx}_T}) \\right] - g({\\mathbf x}^*) \\leq \\frac{1}{T} \\sumtT \\mathbb E \\left[ g({\\mathbf x_t}) - g({\\mathbf x}^*) \\right] \\leq \\frac{4 \\norm{{\\mathbf x}_0 - {\\mathbf x}^*}^2}{\\eta T} + \\frac{20 \\eta \\sigma^2}{n} + 16 \\eta^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_x^2 \\right),\n\\end{align*}\nwhere ${\\bar{\\bx}_T} \\triangleq \\frac{1}{T} \\sumtT {\\mathbf x_t}$.\n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\section{Nonconvex-PL (NC-PL) Functions: Local SGDA (\\texorpdfstring{\\cref{thm:NC_PL}}{Theorem 1})} \\label{app:ncpl}\nIn this section we prove the convergence of \\cref{alg_local_SGDA} for Nonconvex-PL functions, and provide the complexity and communication guarantees.\n\nWe organize this section as follows. First, in \\cref{sec:NC_PL_int_results} we present some intermediate results, which we use to prove the main theorem. Next, in \\cref{sec:NC_PL_thm_proof}, we present the proof of \\cref{thm:NC_PL}, which is followed by the proofs of the intermediate results in \\cref{sec:NC_PL_int_results_proofs}.\nWe utilize some of the proof techniques of \\cite{mahdavi21localSGDA_aistats}.\nHowever, the algorithm we analyze for NC-PL functions is different. Also, we provide an improved analysis, resulting in better convergence guarantees.\n\nThe problem we solve is\n\\begin{align*}\n    \\min_{{\\mathbf x}} \\max_{{\\mathbf y}} \\left\\{ f({\\mathbf x}, {\\mathbf y}) \\triangleq \\frac{1}{n} \\sum_{i=1}^n f_i({\\mathbf x}, {\\mathbf y}) \\right\\}.\n\\end{align*}\nWe define\n\\begin{align}\n    \\Phi ({\\mathbf x}) \\triangleq \\max_{{\\mathbf y}} f({\\mathbf x}, {\\mathbf y}) \\quad \\text{and} \\quad {\\mathbf y}^* ({\\mathbf x}) \\in \\operatornamewithlimits{arg\\,max}_{{\\mathbf y}} f({\\mathbf x}, {\\mathbf y}).\n   \n\\end{align}\nSince $f({\\mathbf x}, \\cdot)$ is $\\mu$-PL, ${\\mathbf y}^*({\\mathbf x})$ \\textit{need not} be unique.\n\nFor the sake of analysis, we define \\textit{virtual} sequences of average iterates:\n\\begin{align*}\n    & {\\mathbf x_t} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_t}; \\quad {\\mathbf y_t} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf y^i_t}.\n\\end{align*}\nNote that these sequences are constructed only for the sake of analysis. During an actual run of the algorithm, these sequences exist only at the time instants when the clients communicate with the server.\nWe next write the update expressions for these virtual sequences, using the updates in Algorithm \\ref{alg_local_SGDA}.\n\\begin{equation}\n    \\begin{aligned}\n        {\\mathbf x_{t+1}} &= {\\mathbf x_t} - \\eta_x \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}}) \\\\\n        {\\mathbf y_{t+1}} &= {\\mathbf y_t} + \\eta_y \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}})\n    \\end{aligned}\n    \\label{eq:NC_PL_update_avg}\n\\end{equation}\nNext, we present some intermediate results which we use in the proof of \\cref{thm:NC_PL}. To make the proof concise, the proofs of these intermediate results is relegated to \\cref{sec:NC_PL_int_results_proofs}.\n\n\\subsection{Intermediate Lemmas} \\label{sec:NC_PL_int_results}\n\nWe use the following result from \\cite{nouiehed19minimax_neurips19} about the smoothness of $\\Phi(\\cdot)$.\n\n\\begin{lemma}\n\\label{lem:Phi_smooth_nouiehed}\nIf the function $f({\\mathbf x}, \\cdot)$ satisfies Assumptions \\ref{assum:smoothness}, \\ref{assum:PL_y} ($L_f$-smoothness and $\\mu$-PL condition in ${\\mathbf y}$), then $\\Phi ({\\mathbf x})$ is $L_{\\Phi}$-smooth with $L_{\\Phi} = \\kappa L\/2 + L$, where $\\kappa = L\/\\mu$ is the condition number.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lem:NC_PL_Phi_decay_one_iter}\nSuppose the local client loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:PL_y} and the stochastic oracles for the local functions satisfy \\cref{assum:bdd_var}.\nThen the iterates generated by \\cref{alg_local_SGDA} satisfy\n\\begin{equation}\n    \\begin{aligned}\n        \\mathbb E \\left[ \\Phi ({\\mathbf x_{t+1}}) \\right] & \\leq \\mathbb E \\left[ \\Phi ({\\mathbf x_t}) \\right] - \\frac{\\eta_x}{2} \\mathbb E \\norm{\\nabla \\Phi ({\\mathbf x_t})}^2 - \\frac{\\eta_x}{2} \\left( 1 - L_{\\Phi} \\eta_x \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\nonumber \\\\\n        & \\quad + \\frac{2 \\eta_x L_f^2}{\\mu} \\mathbb E \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + 2 \\eta_x L_f^2 \\Delta_{t}^{\\bx,\\by} + \\frac{L_{\\Phi} \\eta_x^2 \\sigma^2}{2 n},\n    \\end{aligned}\n\\end{equation}\nwhere, we define $\\Delta_{t}^{\\bx,\\by} \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left( \\left\\| {\\mathbf x^i_t} - {\\mathbf x_t} \\right\\|^2 + \\left\\| {\\mathbf y^i_t} - {\\mathbf y_t} \\right\\|^2 \\right)$, the synchronization error.\n\\end{lemma}\n\n\n\\begin{lemma}\n\\label{lem:NC_PL_phi_error}\nSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_hetero},\nand the stochastic oracles for the local\nfunctions satisfy \\cref{assum:bdd_var}.\nFurther, in \\cref{alg_local_SGDA}, we choose step-sizes $\\eta_x, \\eta_y$ satisfying $\\eta_y \\leq 1\/\\mu$, $\\frac{\\eta_x}{\\eta_y} \\leq \\frac{1}{8 \\kappa^2}$.\nThen the following inequality holds.\n\\begin{equation}\n    \\begin{aligned}\n        & \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left( \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right) \\\\\n        & \\leq \\frac{2 \\left( \\Phi ({\\mathbf x}_0) - f({\\mathbf x}_0, {\\mathbf y}_0) \\right)}{\\eta_y \\mu T} + \\frac{2 L_f^2}{\\mu \\eta_y} \\left( 2 \\eta_x (1 - \\eta_y \\mu) + \\eta_y \\right) \\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\bx,\\by} + (1 - \\eta_y \\mu) \\frac{\\eta_x}{\\eta_y \\mu} \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\nabla \\Phi({\\mathbf x_t})}^2 \\nonumber \\\\\n        & \\quad + \\left[ (1 - \\eta_y \\mu) \\frac{\\eta_x^2}{2} \\left( L_f + L_{\\Phi} \\right) + \\eta_y L_f^2 \\eta_x^2 \\right] \\frac{2}{\\eta_y \\mu T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\\\\n        & \\quad + \\frac{\\sigma^2}{\\mu n} \\left( \\eta_y L_f + 2 L_f^2 \\eta_x^2 \\right) + \\frac{(1 - \\eta_y \\mu)}{\\mu \\eta_y} \\frac{\\eta_x^2 \\sigma^2}{n} \\left( L_f + L_{\\Phi} \\right).\n    \\end{aligned}\n   \n\\end{equation}\n\\end{lemma}\n\n\\begin{remark}[Comparison with \\cite{mahdavi21localSGDA_aistats}]\nNote that to derive a result similar to \\cref{lem:NC_PL_phi_error}, the analysis in \\cite{mahdavi21localSGDA_aistats} requires the additional assumption of $G_x$-Lipschitz continuity of $f(\\cdot, {\\mathbf y})$.\nAlso, the algorithm we analyze (Local SGDA) is simpler than the algorithm analyzed in \\cite{mahdavi21localSGDA_aistats} for NC-PL functions.\n\\end{remark}\n\n\n\\begin{lemma}\n\\label{lem:NC_PL_consensus_error}\nSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_hetero},\nand the stochastic oracles for the local\nfunctions satisfy \\cref{assum:bdd_var}.\nFurther, in \\cref{alg_local_SGDA}, we choose step-sizes $\\eta_x, \\eta_y \\leq \\frac{1}{8 \\tau L_f}$.\nThen, the iterates $\\{ {\\mathbf x^i_t}, {\\mathbf y^i_t} \\}$ generated by \\cref{alg_local_SGDA} satisfy\n\\begin{equation}\n    \\begin{aligned}\n        \\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\bx,\\by} & \\triangleq \\frac{1}{T} \\sum_{t=0}^{T-1} \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left( \\left\\| {\\mathbf x^i_t} - {\\mathbf x_t} \\right\\|^2 + \\left\\| {\\mathbf y^i_t} - {\\mathbf y_t} \\right\\|^2 \\right) \\nonumber \\\\\n        & \\leq 2 (\\tau-1)^2 \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 6 (\\tau-1)^2 \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right).\n    \\end{aligned}\n    \\label{eq:lem:NC_PL_consensus_error}\n\\end{equation}\n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\texorpdfstring{\\cref{thm:NC_PL}}{Theorem 1}}\n\\label{sec:NC_PL_thm_proof}\nFor the sake of completeness, we first state the full statement of \\cref{thm:NC_PL} here.\n\n\\begin{theorem*}\nSuppose the local loss functions $\\{ f_i \\}_i$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, and the global function $f$ satisfies \\cref{assum:PL_y}.\nSuppose the step-sizes $\\eta_x, \\eta_y$ are chosen\nsuch that $\\eta_y \\leq \\frac{1}{8 L_f \\tau}$, $\\frac{\\eta_x}{\\eta_y} = \\frac{1}{8 \\kappa^2}$, where $\\kappa = \\frac{L_f}{\\mu}$ is the condition number.\nThen for the output ${\\bar{\\bx}_T}$ of \\cref{alg_local_SGDA}, the following holds.\n\\begin{align}\n    \\mathbb E \\norm{\\nabla \\Phi ({\\bar{\\bx}_T})}^2 = & \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\nabla \\Phi ({\\mathbf x_t})}^2 \\nonumber \\\\\n    & \\leq \\underbrace{\\mathcal O \\left( \\kappa^2 \\left[ \\frac{\\Delta_{\\Phi}}{\\eta_y T} + \\frac{L_f \\eta_y \\sigma^2}{n} \\right] \\right)}_{\\text{Error with full synchronization}} + \\underbrace{\\mathcal O \\left( L_f^2 \\kappa^2 (\\tau-1)^2 \\left[ \\eta_y^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) + \\eta_x^2 \\varsigma_x^2 \\right] \\right)}_{\\text{Error due to local updates}},\n    \\label{eq_proof:thm_NC_PL}\n\\end{align}\nwhere $\\Phi({\\mathbf x}) \\triangleq \\max_{\\mathbf y} f({\\mathbf x}, {\\mathbf y})$ is the envelope function, $\\Delta_{\\Phi} \\triangleq \\Phi ({\\mathbf x}_0) - \\min_{\\mathbf x} \\Phi ({\\mathbf x})$.\nUsing $\\eta_y = \\sqrt{\\frac{n}{L_f T}}$ and $\\eta_x = \\frac{1}{8 \\kappa^{2}} \\sqrt{\\frac{n}{L_f T}}$, we get\n\\begin{align}\n    & \\mathbb E \\norm{\\nabla \\Phi ({\\bar{\\bx}_T})}^2 \\leq \\mathcal O \\left( \\frac{\\kappa^2 \\left( \\sigma^2 + \\Delta_{\\Phi} \\right)}{\\sqrt{n T}} + \\kappa^2 (\\tau-1)^2 \\frac{n \\left( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 \\right)}{T} \\right). \\nonumber\n   \n\\end{align}\n\\end{theorem*}\n\n\\begin{proof}\nWe start by summing the expression in \\cref{lem:NC_PL_Phi_decay_one_iter} over $t = 0, \\hdots, T-1$.\n\\begin{align}\n    \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left[ \\Phi ({\\mathbf x_{t+1}}) -  \\Phi ({\\mathbf x_t}) \\right] & \\leq - \\frac{\\eta_x}{2} \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\nabla \\Phi ({\\mathbf x_t})}^2 - \\frac{\\eta_x}{2} \\left( 1 - L_{\\Phi} \\eta_x \\right) \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\nonumber \\\\\n    & \\quad + \\frac{2 \\eta_x L_f^2}{\\mu} \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left[ \\Phi ({\\mathbf x_t}) - F({\\mathbf x_t}, {\\mathbf y_t}) \\right] + 2 \\eta_x L_f^2 \\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\bx,\\by} + \\frac{L_{\\Phi} \\eta_x^2 \\sigma^2}{2 n}. \\label{eq_proof:thm:NC_PL_1}\n\\end{align}\nSubstituting the bound on $\\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\bx,\\by}$ from \\cref{lem:NC_PL_consensus_error}, and the bound on $\\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left[ \\Phi ({\\mathbf x_t}) - F({\\mathbf x_t}, {\\mathbf y_t}) \\right]$ from \\cref{lem:NC_PL_phi_error}, and rearranging the terms in \\eqref{eq_proof:thm:NC_PL_1}, we get\n\\begin{align}\n    & \\frac{\\mathbb E \\Phi ({\\mathbf x}_T) - \\Phi ({\\mathbf x}_0)}{T} \\nonumber \\\\\n    & \\leq - \\underbrace{\\left( \\frac{\\eta_x}{2} - (1 - \\eta_y \\mu) \\frac{2 \\eta_x^2 L_f^2}{\\eta_y \\mu^2} \\right)}_{\\geq \\eta_x\/4} \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\nabla \\Phi ({\\mathbf x_t})}^2 \\nonumber \\\\\n    & \\quad - \\underbrace{\\frac{\\eta_x}{2} \\left( 1 - L_{\\Phi} \\eta_x - \\frac{8 L_f^2}{\\mu^2 \\eta_y} \\left[ (1 - \\eta_y \\mu) \\frac{\\eta_x^2}{2} \\left( L + L_{\\Phi} \\right) + \\eta_y L_f^2 \\eta_x^2 \\right] \\right)}_{\\geq 0} \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\nonumber \\\\\n    & \\quad + \\left[ \\frac{2 \\eta_x L_f^2}{\\mu} \\left( \\frac{2 L_f^2}{\\mu} + \\frac{4 \\eta_x L_f^2 (1 - \\eta_y \\mu)}{\\mu \\eta_y} \\right) + 2 \\eta_x L_f^2 \\right] \\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\quad + \\frac{2 \\eta_x L_f^2}{\\mu} \\left[ \\frac{2 \\left( \\Phi ({\\mathbf x}_0) - f({\\mathbf x}_0, {\\mathbf y}_0) \\right)}{\\eta_y \\mu T} + \\frac{\\sigma^2}{\\mu n} \\left( \\eta_y L_f + 2 L_f^2 \\eta_x^2 \\right) + \\frac{(1 - \\eta_y \\mu)}{\\mu \\eta_y} \\frac{\\eta_x^2 \\sigma^2}{n} \\left( L_f + L_{\\Phi} \\right) \\right] + \\frac{L_{\\Phi} \\eta_x^2 \\sigma^2}{2 n}. \\label{eq_proof:thm:NC_PL_2}\n\\end{align}\nHere, $\\frac{\\eta_x}{2} - \\frac{2 \\eta_x^2 (1-\\mu \\eta_y)L_f^2}{\\mu^2 \\eta_y} \\geq \\frac{\\eta_x}{4}$ holds since $\\frac{\\eta_x}{\\eta_y} \\leq \\frac{1}{8 \\kappa^2}$.\nAlso, $1 - L_{\\Phi} \\eta_x - \\frac{8 L_f^2}{\\mu^2 \\eta_y} \\left[ (1 - \\eta_y \\mu) \\frac{\\eta_x^2}{2} \\left( L + L_{\\Phi} \\right) + \\eta_y L_f^2 \\eta_x^2 \\right] \\geq 0$ follows from the bounds on $\\eta_x, \\eta_y$.\nRearranging the terms in \\eqref{eq_proof:thm:NC_PL_2} and using \\cref{lem:NC_PL_consensus_error}, we get\n\\begin{align}\n    & \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\nabla \\Phi ({\\mathbf x_t})}^2 \\leq \\frac{4 \\left( \\Phi ({\\mathbf x}_0) - \\mathbb E \\Phi ({\\mathbf x}_T) \\right)}{\\eta_x T} \\nonumber \\\\\n    & \\quad + \\frac{4}{\\eta_x} 2 \\eta_x L_f^2 \\left[ 1 + 2 \\kappa^2 + 4 \\kappa^2 \\frac{\\eta_x}{\\eta_y} \\right] 2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) \\right] \\nonumber \\\\\n    & \\quad + \\frac{4}{\\eta_x} \\left[ \\frac{4 \\eta_x \\kappa^2}{\\eta_y} \\frac{\\left( \\Phi ({\\mathbf x}_0) - f({\\mathbf x}_0, {\\mathbf y}_0) \\right)}{T} + \\frac{2 \\eta_x \\kappa^2 \\sigma^2}{n} \\left( \\eta_y L_f + 2 L_f^2 \\eta_x^2 \\right) + \\frac{2 \\eta_x \\kappa^2}{\\eta_y} \\frac{\\eta_x^2 \\sigma^2}{n} \\left( L_f + L_{\\Phi} \\right) \\right] + \\frac{4}{\\eta_x} \\frac{L_{\\Phi} \\eta_x^2 \\sigma^2}{2 n} \\nonumber \\\\\n    & \\overset{(a)}{\\leq} \\frac{4 \\Delta_{\\Phi}}{\\eta_x T} + 8 L_f^2 \\left[ 2 + 2 \\kappa^2 \\right] 2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) \\right] \\nonumber \\\\\n    & \\quad + \\frac{16 \\kappa^2 \\Delta_{\\Phi}}{\\eta_y T} + \\frac{8 \\kappa^2 \\sigma^2}{n} \\left( \\eta_y L_f + 2 L_f^2 \\eta_x^2 \\right) + \\frac{8 \\kappa^2 \\eta_x}{\\eta_y} \\frac{\\eta_x \\sigma^2}{n} \\left( L_f + L_{\\Phi} \\right) + \\frac{2 L_{\\Phi} \\eta_x \\sigma^2}{n} \\nonumber \\\\\n    & \\overset{(b)}{\\leq} \\frac{4 \\Delta_{\\Phi}}{\\eta_x T} + 192 L_f^2 \\kappa^2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 + \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right] + \\frac{16 \\kappa^2 \\Delta_{\\Phi}}{\\eta_y T} + \\frac{8 \\kappa^2 \\sigma^2}{n} \\left( \\eta_y L_f + 2 L_f^2 \\eta_x^2 \\right) + \\frac{4 L_{\\Phi} \\eta_x \\sigma^2}{n} \\nonumber \\\\\n    & = \\mathcal O \\left( \\frac{\\Delta_{\\Phi}}{\\eta_x T} + \\frac{L_{\\Phi} \\eta_x \\sigma^2}{n} + \\kappa^2 \\left[ \\frac{\\Delta_{\\Phi}}{\\eta_y T} + \\frac{L_f \\eta_y \\sigma^2}{n} \\right] + L_f^2 \\kappa^2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 + \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right] \\right). \\nonumber \\\\\n    & = \\underbrace{\\mathcal O \\left( \\kappa^2 \\left[ \\frac{\\Delta_{\\Phi}}{\\eta_y T} + \\frac{L_f \\eta_y \\sigma^2}{n} \\right] \\right)}_{\\text{Error with full synchronization}} + \\underbrace{\\mathcal O \\left( L_f^2 \\kappa^2 (\\tau-1)^2 \\left[ \\eta_y^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) + \\eta_x^2 \\varsigma_x^2 \\right] \\right)}_{\\text{Error due to local updates}}. \\tag{$\\because \\kappa \\geq 1$}\n   \n\\end{align}\nwhere, we denote $\\Delta_{\\Phi} \\triangleq \\Phi ({\\mathbf x}_0) - \\min_{\\mathbf x} \\Phi ({\\mathbf x})$.\n$(a)$ follows from $\\frac{\\eta_x}{\\eta_y} \\leq \\frac{1}{8 \\kappa^2}$;\n$(b)$ follows since $\\kappa \\geq 1$ and $L_{\\Phi} \\geq L_f$.\nTherefore, $\\frac{8 \\kappa^2 \\eta_x}{\\eta_y} \\frac{\\eta_x \\sigma^2}{n} (L_f + L_{\\Phi}) \\leq \\frac{\\eta_x \\sigma^2}{n} (L_f + L_{\\Phi}) \\leq \\frac{2 L_{\\Phi} \\eta_x \\sigma^2}{n}$, which results in \\eqref{eq_proof:thm_NC_PL}.\n\nUsing $\\eta_y = \\sqrt{\\frac{n}{L_f T}}$ and $\\eta_x = \\frac{1}{8 \\kappa^{2}} \\sqrt{\\frac{n}{L_f T}} \\leq \\frac{\\eta_y}{8 \\kappa^2}$, and since $\\kappa \\geq 1$, we get\n\\begin{align}\n    & \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left\\| \\nabla \\Phi ({\\mathbf x_t}) \\right\\|^2 \\leq \\mathcal O \\left( \\frac{\\kappa^2 \\left( \\sigma^2 + \\Delta_{\\Phi} \\right)}{\\sqrt{n T}} + \\kappa^2 (\\tau-1)^2 \\frac{n}{T} \\left[ \\sigma^2 + \\frac{\\varsigma_x^2}{\\kappa^4} + \\varsigma_y^2 \\right] \\right). \\nonumber\n   \n\\end{align}\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{cor:NC_PL_comm_cost}]\nWe assume $T \\geq n^3$.\nTo reach an $\\epsilon$-accurate point, i.e., ${\\mathbf x}$ such that $\\mathbb E \\left\\| \\nabla \\Phi ({\\mathbf x}) \\right\\| \\leq \\epsilon$, we need\n\\begin{align*}\n    \\mathbb E \\left\\| \\nabla \\Phi ({\\bar{\\bx}_T}) \\right\\| = \\left[ \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left\\| \\nabla \\Phi ({\\mathbf x_t}) \\right\\|^2 \\right]^{1\/2} \\leq \\mathcal O \\left( \\frac{\\kappa \\sqrt{\\sigma^2 + \\Delta_{\\Phi}}}{(nT)^{1\/4}} + \\kappa (\\tau-1) \\sqrt{\\frac{n \\left( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 \\right)}{T}} \\right).\n\\end{align*}\nIf we choose $\\tau = \\mathcal O \\left( \\frac{T^{1\/4}}{n^{3\/4}} \\right)$, we need $T = \\mathcal O \\left( \\kappa^4\/(n \\epsilon^4) \\right)$ iterations, to reach an $\\epsilon$-accurate point.\nThe number of communication rounds is $\\mathcal O \\left( \\frac{T}{\\tau} \\right) = \\mathcal O \\left( (n T)^{3\/4} \\right) = \\mathcal O \\left( \\kappa^3\/\\epsilon^3 \\right)$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proofs of the Intermediate Lemmas}\n\\label{sec:NC_PL_int_results_proofs}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:NC_PL_Phi_decay_one_iter}]\n\\label{proof:lem:NC_PL_Phi_decay_one_iter}\nIn the proof, we use the quadratic growth property of $\\mu$-PL function $f({\\mathbf x}, \\cdot)$ (\\cref{lem:quad_growth}), i.e.,\n\\begin{align}\n    \\frac{\\mu}{2} \\norm{{\\mathbf y} - {\\mathbf y}^*({\\mathbf x})}^2 \\leq \\max_{{\\mathbf y}'} f({\\mathbf x}, {\\mathbf y}') - f({\\mathbf x}, {\\mathbf y}), \\quad \\forall \\ {\\mathbf x},{\\mathbf y} \\label{eq:quad_growth_PL}\n\\end{align}\nwhere ${\\mathbf y}^*({\\mathbf x}) \\in \\operatornamewithlimits{arg\\,max}_{{\\mathbf y}'} f({\\mathbf x}, {\\mathbf y}')$.\nSee \\cite{mahdavi21localSGDA_aistats} for the entire proof.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:NC_PL_consensus_error}]\nWe define the separate synchronization errors for ${\\mathbf x}$ and ${\\mathbf y}$\n\\begin{align*}\n    \\Delta_{t}^{\\bx} \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left\\| {\\mathbf x^i_t} - {\\mathbf x_t} \\right\\|^2, \\qquad \\Delta_{t}^{\\by} \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left\\| {\\mathbf y^i_t} - {\\mathbf y_t} \\right\\|^2,\n\\end{align*}\nsuch that $\\Delta_{t}^{\\bx,\\by} = \\Delta_{t}^{\\bx} + \\Delta_{t}^{\\by}$.\nWe first bound the ${\\mathbf x}$- synchronization error $\\Delta_{t}^{\\bx}$.\nDefine $s = \\lfloor t\/\\tau \\rfloor$, such that $s \\tau + 1 \\leq t \\leq (s+1) \\tau - 1$.\nThen,\n\\begin{align}\n    \\Delta_{t}^{\\bx} & \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left\\| {\\mathbf x^i_t} - {\\mathbf x_t} \\right\\|^2 \\nonumber \\\\\n    &= \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\norm{\\Big( {\\mathbf x}^i_{s \\tau} - \\eta_x \\sum_{k=s\\sync}^{t-1} \\nabla_{\\bx} f_i ({\\mathbf x}_k^i, {\\mathbf y}_k^i; \\xi^i_k) \\Big) - \\Big( {\\mathbf x}_{s \\tau} - \\eta_x \\frac{1}{n} \\sum_{j=1}^n \\sum_{k=s\\sync}^{t-1} \\nabla_{\\bx} f_j ({\\mathbf x}_k^j, {\\mathbf y}_k^j; \\xi_k^j) \\Big)}^2 \\tag{see \\eqref{eq:NC_PL_update_avg}} \\\\\n    &= \\eta_x^2 \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\norm{\\sum_{k=s\\sync}^{t-1} \\nabla_{\\bx} f_i ({\\mathbf x}_k^i, {\\mathbf y}_k^i; \\xi^i_k) - \\frac{1}{n} \\sum_{j=1}^n \\sum_{k=s\\sync}^{t-1} \\nabla_{\\bx} f_j ({\\mathbf x}_k^j, {\\mathbf y}_k^j; \\xi_k^j)}^2 \\tag{$\\because {\\mathbf x}^i_{s \\tau} = {\\mathbf x}_{s \\tau}, \\forall \\ i \\in [n]$} \\\\\n    & \\overset{(a)}{\\leq} \\eta_x^2 \\frac{1}{n} (t-s\\tau) \\sum_{k=s\\sync}^{t-1} \\sum_{i=1}^n \\mathbb E \\Big\\| \\nabla_{\\bx} f_i ({\\mathbf x}_k^i, {\\mathbf y}_k^i; \\xi^i_k) - \\nabla_{\\bx} f_i ({\\mathbf x}_k^i, {\\mathbf y}_k^i) + \\nabla_{\\bx} f_i ({\\mathbf x}_k^i, {\\mathbf y}_k^i) - \\nabla_{\\bx} f_i ({\\mathbf x}_k, {\\mathbf y}_k) + \\nabla_{\\bx} f_i ({\\mathbf x}_k, {\\mathbf y}_k) \\nonumber \\\\\n    & \\qquad - \\nabla_{\\bx} f ({\\mathbf x}_k, {\\mathbf y}_k) - \\frac{1}{n} \\sum_{j=1}^n \\left( \\nabla_{\\bx} f_j ({\\mathbf x}_k^j, {\\mathbf y}_k^j, \\xi_k^j) - \\nabla_{\\bx} f_j ({\\mathbf x}_k^j, {\\mathbf y}_k^j) + \\nabla_{\\bx} f_j ({\\mathbf x}_k^j, {\\mathbf y}_k^j) - \\nabla_{\\bx} f_j ({\\mathbf x}_k, {\\mathbf y}_k) \\right) \\Big\\|^2 \\nonumber \\\\\n    & \\overset{(b)}{=} \\frac{\\eta_x^2 (t-s\\tau)}{n} \\sum_{k=s\\sync}^{t-1} \\sum_{i=1}^n \\mathbb E \\Bigg[ \\norm{\\nabla_{\\bx} f_i ({\\mathbf x}_k^i, {\\mathbf y}_k^i; \\xi^i_k) - \\nabla_{\\bx} f_i ({\\mathbf x}_k^i, {\\mathbf y}_k^i)}^2 + \\Big\\| \\frac{1}{n} \\sum_{j=1}^n \\left( \\nabla_{\\bx} f_j ({\\mathbf x}_k^j, {\\mathbf y}_k^j, \\xi_k^j) - \\nabla_{\\bx} f_j ({\\mathbf x}_k^j, {\\mathbf y}_k^j) \\right) \\Big\\|^2 \\nonumber \\\\\n    & + \\Big\\| \\nabla_{\\bx} f_i ({\\mathbf x}_k^i, {\\mathbf y}_k^i) - \\nabla_{\\bx} f_i ({\\mathbf x}_k, {\\mathbf y}_k) + \\nabla_{\\bx} f_i ({\\mathbf x}_k, {\\mathbf y}_k) - \\nabla_{\\bx} f ({\\mathbf x}_k, {\\mathbf y}_k) - \\frac{1}{n} \\sum_{j=1}^n \\left( \\nabla_{\\bx} f_j ({\\mathbf x}_k^j, {\\mathbf y}_k^j) - \\nabla_{\\bx} f_j ({\\mathbf x}_k, {\\mathbf y}_k) \\right) \\Big\\|^2 \\Bigg] \\nonumber \\\\\n    & \\overset{(c)}{\\leq} \\frac{\\eta_x^2 (\\tau-1)}{n} \\sum_{k=s\\sync}^{t-1} \\sum_{i=1}^n \\mathbb E \\Bigg[ \\sigma^2 + \\frac{\\sigma^2}{n} + 3 \\norm{\\nabla_{\\bx} f_i ({\\mathbf x}_k^i, {\\mathbf y}_k^i) - \\nabla_{\\bx} f_i ({\\mathbf x}_k, {\\mathbf y}_k)}^2 + 3 \\norm{\\nabla_{\\bx} f_i ({\\mathbf x}_k, {\\mathbf y}_k) - \\nabla_{\\bx} f ({\\mathbf x}_k, {\\mathbf y}_k)}^2 \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + 3 \\Big\\| \\frac{1}{n} \\sum_{j=1}^n \\left( \\nabla_{\\bx} f_j ({\\mathbf x}_k^j, {\\mathbf y}_k^j) - \\nabla_{\\bx} f_j ({\\mathbf x}_k, {\\mathbf y}_k) \\right) \\Big\\|^2 \\Bigg] \\nonumber \\\\\n    & \\overset{(d)}{\\leq} \\frac{\\eta_x^2 (\\tau-1)}{n} \\sum_{k=s\\sync}^{t-1} \\sum_{i=1}^n \\mathbb E \\Bigg[ \\sigma^2 + \\frac{\\sigma^2}{n} + 3 L_f^2 \\left[ \\norm{{\\mathbf x}_k^i - {\\mathbf x}_k}^2 + \\norm{{\\mathbf y}_k^i - {\\mathbf y}_k}^2 \\right] + 3 \\varsigma_x^2 \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + \\frac{3}{n} \\sum_{j=1}^n L_f^2 \\left[ \\norm{{\\mathbf x}_k^j - {\\mathbf x}_k}^2 + \\norm{{\\mathbf y}_k^j - {\\mathbf y}_k}^2 \\right] \\Bigg] \\nonumber \\\\\n   \n    &= \\eta_x^2 (\\tau-1) \\sum_{k=s\\sync}^{t-1} \\left[ \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\varsigma_x^2 + 6 L_f^2 \\left( \\Delta_{k}^{\\bx} + \\Delta_{k}^{\\by} \\right) \\right], \\nonumber\n   \n\\end{align}\nwhere $(a)$ follows from \\cref{lem:sum_of_squares};\n$(b)$ follows from \\cref{assum:bdd_var} (unbiasedness of stochastic gradients);\n$(c)$ follows from \\cref{assum:bdd_var} (bounded variance of stochastic gradients);\n$(d)$ follows from \\cref{assum:smoothness}, \\ref{assum:bdd_hetero}, and Jensen's inequality (\\cref{lem:jensens}) for $\\| \\cdot \\|^2$.\n\nFurthermore, $\\Delta_{t}^{\\bx} = 0$ for $t = s \\tau$. Therefore,\n\\begin{align}\n    \\sum_{t=s\\tau}^{(s+1)\\tau-1} \\Delta_{t}^{\\bx} = \\sum_{t=s\\tau+1}^{(s+1)\\tau-1} \\Delta_{t}^{\\bx} & \\leq \\eta_x^2 (\\tau-1) \\sum_{t=s\\tau+1}^{(s+1)\\tau-1} \\sum_{k=s\\sync}^{t-1} \\left[ \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\varsigma_x^2 + 6 L_f^2 \\left( \\Delta_{k}^{\\bx} + \\Delta_{k}^{\\by} \\right) \\right] \\nonumber \\\\\n    & \\leq \\eta_x^2 (\\tau-1)^2 \\sum_{t=s\\tau+1}^{(s+1)\\tau-1} \\left[ \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\varsigma_x^2 + 6 L_f^2 \\Delta_{t}^{\\bx,\\by} \\right]. \\label{eq_proof:lem:NC_PL_x_consensus_error}\n\\end{align}\nThe ${\\mathbf y}$- synchronization error $\\Delta_{t}^{\\by}$ following a similar analysis and we get.\n\\begin{align}\n    \\sum_{t=s\\tau}^{(s+1)\\tau-1} \\Delta_{t}^{\\by} & \\leq \\eta_y^2 (\\tau-1)^2 \\sum_{t=s\\tau+1}^{(s+1)\\tau-1} \\left[ \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\varsigma_y^2 + 6 L_f^2 \\Delta_{t}^{\\bx,\\by} \\right]. \\label{eq_proof:lem:NC_PL_y_consensus_error}\n\\end{align}\nCombining \\eqref{eq_proof:lem:NC_PL_x_consensus_error} and \\eqref{eq_proof:lem:NC_PL_y_consensus_error}, we get\n\\begin{align}\n    \\sum_{t=s\\tau}^{(s+1)\\tau-1} \\Delta_{t}^{\\bx,\\by} & \\leq (\\tau-1)^2 \\left[ \\tau \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\tau \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) + 6 L_f^2 \\left( \\eta_x^2 + \\eta_y^2 \\right)  \\sum_{t=s\\tau+1}^{(s+1)\\tau-1} \\Delta_{t}^{\\bx,\\by} \\right]. \\nonumber\n   \n\\end{align}\nUsing our choice of $\\eta_x, \\eta_y$, we have $6 L_f^2 \\left( \\eta_x^2 + \\eta_y^2 \\right) (\\tau - 1)^2 \\leq 1\/2$, then\n\\begin{align}\n    \\sum_{t=s\\tau}^{(s+1)\\tau-1} \\Delta_{t}^{\\bx,\\by} & \\leq 2 \\tau (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) \\right] \\nonumber \\\\\n    \\Rightarrow \\frac{1}{T} \\sum_{s=0}^{T\/\\tau - 1} \\sum_{t=s\\tau}^{(s+1)\\tau-1} \\Delta_{t}^{\\bx,\\by} = \\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\bx,\\by} & \\leq 2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) \\right]. \\nonumber\n   \n\\end{align}\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:NC_PL_phi_error}]\nUsing $L_f$-smoothness of $f({\\mathbf x}, \\cdot)$,\n\\begin{align}\n    f({\\mathbf x_{t+1}}, {\\mathbf y_t}) &+ \\left\\langle \\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t}), {\\mathbf y_{t+1}} - {\\mathbf y_t} \\right\\rangle - \\frac{L_f}{2} \\norm{{\\mathbf y_{t+1}} - {\\mathbf y_t}}^2 \\leq f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\nonumber \\\\\n    \\Rightarrow f({\\mathbf x_{t+1}}, {\\mathbf y_t}) & \\leq f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\eta_y \\left\\langle \\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t}), \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}}) \\right\\rangle + \\frac{\\eta_y^2 L_f}{2} \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}})}^2 \\tag{using \\eqref{eq:NC_PL_update_avg}} \\\\\n    \\Rightarrow \\mathbb E f({\\mathbf x_{t+1}}, {\\mathbf y_t}) & \\leq \\mathbb E f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\eta_y \\mathbb E \\left\\langle \\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t}), \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}) \\right\\rangle \\nonumber \\\\\n    & \\qquad + \\frac{\\eta_y^2 L_f}{2} \\left[ \\frac{\\sigma^2}{n} + \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\right] \\tag{\\cref{assum:bdd_var}} \\\\\n    &= \\mathbb E f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\frac{\\eta_y}{2} \\mathbb E \\norm{\\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t})}^2 - \\frac{\\eta_y}{2} \\left( 1 - \\eta_y L_f \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\nonumber \\\\\n    & \\qquad + \\frac{\\eta_y}{2} \\mathbb E \\norm{\\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t}) - \\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) + \\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + \\frac{\\eta_y^2 L_f \\sigma^2}{2n} \\nonumber \\\\\n    & \\leq \\mathbb E f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\frac{\\eta_y}{2} \\mathbb E \\norm{\\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t})}^2 - \\frac{\\eta_y}{2} \\left( 1 - \\eta_y L_f \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\nonumber \\\\\n    & \\qquad + \\eta_y L_f^2 \\mathbb E \\norm{{\\mathbf x_{t+1}} - {\\mathbf x_t}}^2 + \\eta_y L_f^2 \\Delta_{t}^{\\bx,\\by} + \\frac{\\eta_y^2 L_f \\sigma^2}{2n}, \\label{eq_proof:lem:NC_PL_phi_error_1}\n\\end{align}\nwhere \\eqref{eq_proof:lem:NC_PL_phi_error_1} follows from Jensen's inequality (\\cref{lem:jensens}) for $\\norm{\\cdot}^2$, \\cref{assum:smoothness} and Young's inequality (\\cref{lem:Young}) for $\\gamma = 1$, $\\left\\langle \\mathbf a, \\bf b \\right\\rangle \\leq \\frac{1}{2} \\norm{\\mathbf a}^2 + \\frac{1}{2} \\norm{\\mathbf b}^2$.\nNext, note that using \\cref{assum:bdd_var}\n\\begin{align}\n    \\mathbb E \\norm{{\\mathbf x_{t+1}} - {\\mathbf x_t}}^2 &= \\eta_x^2 \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}})}^2 \\leq \\eta_x^2 \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + \\frac{\\eta_x^2 \\sigma^2}{n}. \\label{eq_proof:lem:NC_PL_phi_error_2a}\n\\end{align}\nAlso, using \\cref{assum:PL_y},\n\\begin{align}\n    \\norm{\\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t})}^2 \\geq 2 \\mu \\left( \\max_{\\mathbf y} f({\\mathbf x_{t+1}}, {\\mathbf y}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right) = 2 \\mu \\left( \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right). \\label{eq_proof:lem:NC_PL_phi_error_2b}\n\\end{align}\nSubstituting \\eqref{eq_proof:lem:NC_PL_phi_error_2a}, \\eqref{eq_proof:lem:NC_PL_phi_error_2b} in \\eqref{eq_proof:lem:NC_PL_phi_error_1}, and rearranging the terms, we get\n\\begin{align}\n    & \\eta_y \\mu \\mathbb E \\left( \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right) \\nonumber \\\\\n    & \\leq \\mathbb E f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\mathbb E f({\\mathbf x_{t+1}}, {\\mathbf y_t}) - \\frac{\\eta_y}{2} \\left( 1 - \\eta_y L_f \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + \\frac{\\eta_y^2 L_f \\sigma^2}{2n} \\nonumber \\\\\n    & \\quad + \\eta_y L_f^2 \\left[ \\eta_x^2 \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + \\frac{\\eta_x^2 \\sigma^2}{n} \\right] + \\eta_y L_f^2 \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    \\Rightarrow & \\mathbb E \\left( \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\right) \\nonumber \\\\\n    & \\leq (1 - \\eta_y \\mu) \\mathbb E \\left( \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right) - \\frac{\\eta_y}{2} \\left( 1 - \\eta_y L_f \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + \\frac{\\eta_y^2 L_f \\sigma^2}{2n} \\nonumber \\\\\n    & \\quad + \\eta_y L_f^2 \\left[ \\eta_x^2 \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + \\frac{\\eta_x^2 \\sigma^2}{n} \\right] + \\eta_y L_f^2 \\Delta_{t}^{\\bx,\\by}. \\label{eq_proof:lem:NC_PL_phi_error_3}\n\\end{align}\nNext, we bound $\\mathbb E \\left( \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right)$.\n\\begin{align}\n    & \\mathbb E \\left[ \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right] \\nonumber \\\\\n    &= \\underbrace{\\mathbb E \\left[ \\Phi ({\\mathbf x_{t+1}}) - \\Phi ({\\mathbf x_t}) \\right]}_{I_1} + \\mathbb E \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\underbrace{\\mathbb E \\left[ f({\\mathbf x_t}, {\\mathbf y_t}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right]}_{I_2}\n\\end{align}\n$I_1$ is bounded in \\cref{lem:NC_PL_Phi_decay_one_iter}.\nWe next bound $I_2$. Using $L_f$-smoothness of $f(\\cdot, {\\mathbf y_t})$,\n\\begin{align}\n    & f({\\mathbf x_t}, {\\mathbf y_t}) + \\left\\langle \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}), {\\mathbf x_{t+1}} - {\\mathbf x_t} \\right\\rangle - \\frac{L_f}{2} \\norm{{\\mathbf x_{t+1}} - {\\mathbf x_t}}^2 \\leq f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\nonumber \\\\\n    \\Rightarrow I_2 &= \\mathbb E \\left[ f({\\mathbf x_t}, {\\mathbf y_t}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right] \\nonumber \\\\\n    & \\leq \\eta_x \\mathbb E \\left\\langle \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}), \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}}) \\right\\rangle + \\frac{\\eta_x^2 L_f}{2} \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}})}^2 \\nonumber \\\\\n    & \\leq \\eta_x \\mathbb E \\left\\langle \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}), \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}) \\right\\rangle + \\frac{\\eta_x^2 L_f}{2} \\left[ \\frac{\\sigma^2}{n} + \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\right] \\tag{\\cref{assum:bdd_var}} \\\\\n    & \\leq \\frac{\\eta_x}{2} \\mathbb E \\left[ \\norm{\\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t})}^2 + \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\right] + \\frac{\\eta_x^2 L_f}{2} \\left[ \\frac{\\sigma^2}{n} + \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\right] \\nonumber \\\\\n    & \\leq \\eta_x \\mathbb E \\left[ \\norm{\\nabla \\Phi({\\mathbf x_t})}^2 + \\norm{\\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - \\nabla \\Phi({\\mathbf x_t})}^2 \\right] + \\frac{\\eta_x^2 L_f \\sigma^2}{2 n} + \\frac{\\eta_x}{2} \\left( 1 + \\eta_x L_f \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\nonumber \\\\\n    & \\overset{(a)}{\\leq} \\eta_x \\mathbb E \\norm{\\nabla \\Phi({\\mathbf x_t})}^2 + \\eta_x L_f^2 \\mathbb E \\norm{{\\mathbf y_t} - {\\mathbf y}^*({\\mathbf x_t})}^2 + \\frac{\\eta_x^2 L_f \\sigma^2}{2 n} + \\frac{\\eta_x}{2} \\left( 1 + \\eta_x L_f \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\nonumber \\\\\n    & \\leq \\eta_x \\mathbb E \\norm{\\nabla \\Phi({\\mathbf x_t})}^2 + \\frac{2 \\eta_x L_f^2}{\\mu} \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\frac{\\eta_x^2 L_f \\sigma^2}{2 n} + \\frac{\\eta_x}{2} \\left( 1 + \\eta_x L_f \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2. \\label{eq_proof:lem:NC_PL_phi_error_4}\n\\end{align}\nwhere $(a)$ follows from \\cref{assum:smoothness} and \\cref{lem:Phi_smooth_nouiehed}. \nAlso, recall that ${\\mathbf y}^*({\\mathbf x}) \\in \\operatornamewithlimits{arg\\,max}_{{\\mathbf y}'} f({\\mathbf x}, {\\mathbf y}')$.\n\\eqref{eq_proof:lem:NC_PL_phi_error_4} follows from the quadratic growth property of $\\mu$-PL functions (\\cref{lem:quad_growth}).\nSubstituting the bounds on $I_1, I_2$ from \\cref{lem:NC_PL_Phi_decay_one_iter} and \\eqref{eq_proof:lem:NC_PL_phi_error_4} respectively, in \\eqref{eq_proof:lem:NC_PL_phi_error_3}, we get\n\\begin{align}\n    & \\mathbb E \\left( \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\right) \\nonumber \\\\\n    & \\leq (1 - \\eta_y \\mu) \\left( 1 + \\frac{4 \\eta_x L_f^2}{\\mu} \\right) \\mathbb E \\left( \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right) \\nonumber \\\\\n    & \\quad + (1 - \\eta_y \\mu) \\left[ - \\frac{\\eta_x}{2} \\mathbb E \\norm{\\nabla \\Phi ({\\mathbf x_t})}^2 - \\frac{\\eta_x}{2} \\left( 1 - L_{\\Phi} \\eta_x \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + 2 \\eta_x L_f^2 \\Delta_{t}^{\\bx,\\by} + \\frac{L_{\\Phi} \\eta_x^2 \\sigma^2}{2 n} \\right] \\nonumber \\\\\n    & \\quad + (1 - \\eta_y \\mu) \\left[ \\eta_x \\mathbb E \\norm{\\nabla \\Phi({\\mathbf x_t})}^2 + \\frac{\\eta_x^2 L_f \\sigma^2}{2 n} + \\frac{\\eta_x}{2} \\left( 1 + \\eta_x L_f \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\right] \\nonumber \\\\\n    & \\quad - \\frac{\\eta_y}{2} \\left( 1 - \\eta_y L_f \\right) \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + \\frac{\\eta_y^2 L_f \\sigma^2}{2n} \\nonumber \\\\\n    & \\quad + \\eta_y L_f^2 \\left[ \\eta_x^2 \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + \\frac{\\eta_x^2 \\sigma^2}{n} \\right] + \\eta_y L_f^2 \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\leq \\left( 1 - \\frac{\\eta_y \\mu}{2} \\right) \\mathbb E \\left( \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right) + \\frac{\\eta_y^2 L_f \\sigma^2}{2n} + \\frac{\\eta_y L_f^2 \\eta_x^2 \\sigma^2}{n} + \\eta_y L_f^2 \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\quad + \\left[ (1 - \\eta_y \\mu) \\frac{\\eta_x^2}{2} \\left( L_f + L_{\\Phi} \\right) + \\eta_y L_f^2 \\eta_x^2 \\right] \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\nonumber \\\\\n    & \\quad + (1 - \\eta_y \\mu) \\left[ \\frac{\\eta_x}{2} \\mathbb E \\norm{\\nabla \\Phi({\\mathbf x_t})}^2 + \\frac{\\eta_x^2 L_f \\sigma^2}{2 n} + 2 \\eta_x L_f^2 \\Delta_{t}^{\\bx,\\by} + \\frac{L_{\\Phi} \\eta_x^2 \\sigma^2}{2 n} \\right], \\label{eq_proof:lem:NC_PL_phi_error_5}\n\\end{align}\nwhere we choose $\\eta_x$ such that $(1 - \\eta_y \\mu) \\left( 1 + \\frac{4 \\eta_x L_f^2}{\\mu} \\right) \\leq \\left( 1 - \\frac{\\eta_y \\mu}{2} \\right)$. \nThis holds if $\\frac{4 \\eta_x L_f^2}{\\mu} \\leq \\frac{\\eta_y \\mu}{2} \\Rightarrow \\eta_x \\leq \\frac{\\eta_y}{8 \\kappa^2}$.\nSumming \\eqref{eq_proof:lem:NC_PL_phi_error_5} over $t=0, \\hdots, T-1$, and rearranging the terms, we get\n\\begin{align}\n    & \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left( \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\right) \\nonumber \\\\\n    & \\leq \\left( 1 - \\frac{\\eta_y \\mu}{2} \\right) \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left( \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right) + L_f^2 \\left( 2 \\eta_x (1 - \\eta_y \\mu) + \\eta_y \\right) \\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\quad + \\left[ (1 - \\eta_y \\mu) \\frac{\\eta_x^2}{2} \\left( L_f + L_{\\Phi} \\right) + \\eta_y L_f^2 \\eta_x^2 \\right] \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + (1 - \\eta_y \\mu) \\frac{\\eta_x}{2} \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\nabla \\Phi({\\mathbf x_t})}^2 \\nonumber \\\\\n    & \\quad + \\frac{\\eta_y^2 L_f \\sigma^2}{2n} + \\frac{\\eta_y L_f^2 \\eta_x^2 \\sigma^2}{n} + (1 - \\eta_y \\mu) \\left[ \\frac{\\eta_x^2 L_f \\sigma^2}{2 n} + \\frac{L_{\\Phi} \\eta_x^2 \\sigma^2}{2 n} \\right]. \\nonumber\n   \n\\end{align}\nRearranging the terms, we get\n\\begin{align}\n    & \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left( \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right) \\nonumber \\\\\n    & \\leq \\frac{2}{\\eta_y \\mu} \\left[ \\frac{\\Phi ({\\mathbf x}_0) - f({\\mathbf x}_0, {\\mathbf y}_0)}{ T} - \\frac{\\mathbb E \\left( \\Phi ({\\mathbf x}_T) - f({\\mathbf x}_T, {\\mathbf y}_T) \\right)}{ T} \\right] + \\frac{2 L_f^2}{\\mu \\eta_y} \\left( 2 \\eta_x (1 - \\eta_y \\mu) + \\eta_y \\right) \\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\quad + \\left[ (1 - \\eta_y \\mu) \\frac{\\eta_x^2}{2} \\left( L_f + L_{\\Phi} \\right) + \\eta_y L_f^2 \\eta_x^2 \\right] \\frac{2}{\\eta_y \\mu T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + (1 - \\eta_y \\mu) \\frac{\\eta_x}{\\eta_y \\mu T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\nabla \\Phi({\\mathbf x_t})}^2 \\nonumber \\\\\n    & \\quad + \\frac{\\eta_y L_f \\sigma^2}{\\mu n} + \\frac{2 L_f^2 \\eta_x^2 \\sigma^2}{\\mu n} + \\frac{(1 - \\eta_y \\mu)}{\\mu \\eta_y} \\left[ \\frac{\\eta_x^2 L_f \\sigma^2}{n} + \\frac{L_{\\Phi} \\eta_x^2 \\sigma^2}{n} \\right] \\nonumber \\\\\n    & \\leq \\frac{2 \\left( \\Phi ({\\mathbf x}_0) - f({\\mathbf x}_0, {\\mathbf y}_0) \\right)}{\\eta_y \\mu T} + \\frac{2 L_f^2}{\\mu \\eta_y} \\left( 2 \\eta_x (1 - \\eta_y \\mu) + \\eta_y \\right) \\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\bx,\\by} \\tag{$\\because \\Phi ({\\mathbf x}_T) \\triangleq \\operatornamewithlimits{arg\\,max}_{\\mathbf y} f({\\mathbf x}_T, {\\mathbf y})$} \\\\\n    & \\quad + \\left[ (1 - \\eta_y \\mu) \\frac{\\eta_x^2}{2} \\left( L_f + L_{\\Phi} \\right) + \\eta_y L_f^2 \\eta_x^2 \\right] \\frac{2}{\\eta_y \\mu T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + (1 - \\eta_y \\mu) \\frac{\\eta_x}{\\eta_y \\mu T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\nabla \\Phi({\\mathbf x_t})}^2 \\nonumber \\\\\n    & \\quad + \\frac{\\eta_y L_f \\sigma^2}{\\mu n} + \\frac{2 L_f^2 \\eta_x^2 \\sigma^2}{\\mu n} + \\frac{(1 - \\eta_y \\mu)}{\\mu \\eta_y} \\left[ \\frac{\\eta_x^2 L_f \\sigma^2}{n} + \\frac{L_{\\Phi} \\eta_x^2 \\sigma^2}{n} \\right], \\nonumber\n   \n\\end{align}\nwhich concludes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\section{Nonconvex-PL (NC-PL) Functions: Momentum Local SGDA (\\texorpdfstring{\\cref{thm:NC_PL_mom}}{Theorem 2})} \\label{app:NC_PL_mom}\nIn this section we prove the convergence of \\cref{alg_NC_momentum} for Nonconvex-PL functions, and provide the complexity and\ncommunication guarantees.\n\nWe organize this section as follows. First, in \\cref{sec:NC_PL_mom_int_results} we present some intermediate results. \nNext, in \\cref{sec:NC_PL_mom_thm_proof}, we present the proof of \\cref{thm:NC_PL_mom}, which is followed by the proofs of the intermediate results in \\cref{sec:NC_PL_mom_int_results_proofs}.\n\nAgain, the problem we solve is\n\\begin{align*}\n    \\min_{{\\mathbf x}} \\max_{{\\mathbf y}} \\left\\{ f({\\mathbf x}, {\\mathbf y}) \\triangleq \\frac{1}{n} \\sum_{i=1}^n f_i({\\mathbf x}, {\\mathbf y}) \\right\\}.\n\\end{align*}\nWe define\n\\begin{align}\n    \\Phi ({\\mathbf x}) \\triangleq \\max_{{\\mathbf y}} f({\\mathbf x}, {\\mathbf y}) \\quad \\text{and} \\quad {\\mathbf y}^* ({\\mathbf x}) \\in \\operatornamewithlimits{arg\\,max}_{{\\mathbf y}} f({\\mathbf x}, {\\mathbf y}). \\label{eq:Phi_defn}\n\\end{align}\nSince $f({\\mathbf x}, \\cdot)$ is $\\mu$-PL (\\cref{assum:PL_y}), ${\\mathbf y}^*({\\mathbf x})$ is not necessarily unique.\n\nFor the sake of analysis, we define \\textit{virtual} sequences of average iterates and average direction estimates:\n\\begin{align*}\n    & {\\mathbf x_t} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_t}; \\quad {\\mathbf y_t} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf y^i_t}; \\\\\n    & \\Tbx_{t+\\frac{1}{2}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\Tbx^i_{t+\\frac{1}{2}}; \\quad \\Tby_{t+\\frac{1}{2}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\Tby^i_{t+\\frac{1}{2}}; \\\\\n    & {\\mathbf d_{x,t}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf d^i_{x,t}}; \\quad {\\mathbf d_{y,t}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf d^i_{y,t}}.\n\\end{align*}\nNote that these sequences are constructed only for the sake of analysis. During an actual run of the algorithm, these sequences exist only at the time instants when the clients communicate with the server.\nWe next write the update expressions for these virtual sequences, using the updates in Algorithm \\ref{alg_NC_momentum}.\n\\begin{equation}\n    \\begin{aligned}\n        & \\Tbx_{t+\\frac{1}{2}} = {\\mathbf x_t} - \\eta_x {\\mathbf d_{x,t}}, \\qquad {\\mathbf x_{t+1}} = {\\mathbf x_t} + \\alpha_t \\left( \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right) \\\\\n        & \\Tby_{t+\\frac{1}{2}} = {\\mathbf y_t} + \\eta_y{\\mathbf d_{y,t}}, \\qquad {\\mathbf y_{t+1}} = {\\mathbf y_t} + \\alpha_t \\left( \\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t} \\right) \\\\\n        & {\\mathbf d_{x,t+1}} = (1 - \\beta_x \\alpha_t) {\\mathbf d_{x,t}} + \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}) \\\\\n        & {\\mathbf d_{y,t+1}} = (1 - \\beta_y \\alpha_t) {\\mathbf d_{y,t}} + \\beta_y \\alpha_t \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\by} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}).\n    \\end{aligned}\n    \\label{eq:NC_mom_update_avg}\n\\end{equation}\nNext, we present some intermediate results which we use in the proof of \\cref{thm:NC_PL_mom}. To make the proof concise, the proofs of these intermediate results is relegated to \\cref{sec:NC_PL_mom_int_results_proofs}.\n\n\\subsection{Intermediate Lemmas} \\label{sec:NC_PL_mom_int_results}\n\nWe use the following result from \\cite{nouiehed19minimax_neurips19} about the smoothness of $\\Phi(\\cdot)$.\n\n\\begin{lemma}\n\\label{lem:Phi_PL_smooth_nouiehed}\nIf the function $f({\\mathbf x}, \\cdot)$ satisfies Assumptions \\ref{assum:smoothness}, \\ref{assum:PL_y} ($L_f$-smoothness and $\\mu$-PL condition in ${\\mathbf y}$), then $\\Phi ({\\mathbf x})$ is $L_{\\Phi}$-smooth with $L_{\\Phi} = \\kappa L_f\/2 + L_f$, where $\\kappa = L_f\/\\mu$, and \n$$\\nabla \\Phi(\\cdot) = \\nabla_{\\bx} f(\\cdot, {\\mathbf y}^*(\\cdot)),$$\nwhere ${\\mathbf y}^*(\\cdot) \\in \\operatornamewithlimits{arg\\,max}_{\\mathbf y} f(\\cdot, {\\mathbf y})$.\n\\end{lemma}\n\n\n\\begin{lemma}\n\\label{lem:NC_PL_mom_Phi_1_step_decay}\nSuppose the loss function $f$ satisfies Assumptions \\ref{assum:smoothness}, \\ref{assum:PL_y}, and the step-size $\\eta_x$, and $\\alpha_t$ satisfy $0 < \\alpha_t \\eta_x \\leq \\frac{\\mu}{4 L_f^2}$.\nThen the iterates generated by Algorithm \\ref{alg_NC_momentum} satisfy\n\\begin{align}\n    \\Phi ({\\mathbf x_{t+1}}) - \\Phi ({\\mathbf x_t}) & \\leq - \\frac{\\alpha_t}{2 \\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{4 \\eta_x \\alpha_t L_f^2}{\\mu} \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + 2 \\eta_x \\alpha_t \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2, \\nonumber\n   \n\\end{align}\nwhere $\\Phi (\\cdot)$ is defined in \\eqref{eq:Phi_defn}.\n\\end{lemma}\n\n\nNext, we bound the difference $\\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t})$.\n\n\n\\begin{lemma}\n\\label{lem:NC_PL_mom_phi_error}\nSuppose the loss function $f$ satisfies Assumptions \\ref{assum:smoothness}, \\ref{assum:PL_y}, and the step-sizes $\\eta_x, \\eta_y$, and $\\alpha_t$ satisfy $0 < \\alpha_t \\eta_y \\leq \\frac{1}{2 L_f}$, $0 < \\alpha_t \\eta_x \\leq \\frac{\\mu}{8 L_f^2}$, and $\\eta_x \\leq \\frac{\\eta_y}{8 \\kappa^2}$.\nThen the iterates generated by Algorithm \\ref{alg_NC_momentum} satisfy\n\\begin{align}\n    \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) & \\leq \\left( 1 - \\frac{\\alpha_t \\eta_y \\mu}{2} \\right) \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] - \\frac{\\alpha_t}{4 \\eta_y} \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 \\nonumber \\\\\n    & \\quad + \\frac{\\alpha_t}{2 \\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\alpha_t \\eta_y \\norm{\\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2. \\nonumber\n   \n\\end{align}\n\\end{lemma}\n\n\nThe next result bounds the variance in the average direction estimates ${\\mathbf d_{x,t}}, {\\mathbf d_{y,t}}$ \\eqref{eq:NC_mom_update_avg} w.r.t. the partial gradients of the global loss function $\\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}), \\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t})$, respectively.\n\n\\begin{lemma}\n\\label{lem:NC_PL_mom_grad_var_bound}\nSuppose the local loss functions $\\{ f_i \\}$ satisfy \\cref{assum:smoothness}, and the stochastic oracles for the local functions $\\{ f_i \\}$ satisfy \\cref{assum:bdd_var}.\nFurther, in \\cref{alg_NC_momentum}, we choose $\\beta_x = \\beta_y = \\beta$, and $\\alpha_t$ such that $0 < \\alpha_t < 1\/\\beta$.\nThen the following holds.\n\\begin{equation}\n    \\begin{aligned}\n        & \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - {\\mathbf d_{x,t+1}} \\right\\|^2 \\leq \\left( 1 - \\frac{\\beta \\alpha_t}{2} \\right) \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 + \\frac{\\beta^2 \\alpha_t^2 \\sigma^2}{n} \\\\\n        & \\quad + \\frac{2 L_f^2 \\alpha_t}{\\beta} \\mathbb E \\left( \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\left\\| \\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t} \\right\\|^2 \\right) + \\beta \\alpha_t \\frac{1}{n} \\sum_{i=1}^n L_f^2 \\mathbb E \\left( \\left\\| {\\mathbf x^i_{t+1}} - {\\mathbf x_{t+1}} \\right\\|^2 + \\left\\| {\\mathbf y^i_{t+1}} - {\\mathbf y_{t+1}} \\right\\|^2 \\right),\n    \\end{aligned}\n    \\label{eq:lem:NC_PL_mom_grad_var_bound_x}\n\\end{equation}\n\\begin{equation}\n    \\begin{aligned}\n        & \\mathbb E \\left\\| \\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - {\\mathbf d_{y,t+1}} \\right\\|^2 \\leq \\left( 1 - \\frac{\\beta \\alpha_t}{2} \\right) \\mathbb E \\left\\| \\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}} \\right\\|^2 + \\frac{\\beta^2 \\alpha_t^2 \\sigma^2}{n} \\\\\n        & \\quad + \\frac{2 L_f^2 \\alpha_t}{\\beta} \\mathbb E \\left( \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\left\\| \\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t} \\right\\|^2 \\right) + \\beta \\alpha_t \\frac{1}{n} \\sum_{i=1}^n L_f^2 \\mathbb E \\left( \\left\\| {\\mathbf x^i_{t+1}} - {\\mathbf x_{t+1}} \\right\\|^2 + \\left\\| {\\mathbf y^i_{t+1}} - {\\mathbf y_{t+1}} \\right\\|^2 \\right).\n    \\end{aligned}\n    \\label{eq:lem:NC_PL_mom_grad_var_bound_y}\n\\end{equation}\n\\end{lemma}\nNotice that the bound depends on the disagreement of the individual iterates with the \\textit{virtual} global average: $\\mathbb E \\left\\| {\\mathbf x^i_{t+1}} - {\\mathbf x_{t+1}} \\right\\|^2$, $\\mathbb E \\left\\| {\\mathbf y^i_{t+1}} - {\\mathbf y_{t+1}} \\right\\|^2$, which is nonzero since $\\tau > 1$, and the clients carry out multiple local updates between successive rounds of communication with the server.\nNext, we bound these synchronization errors.\nHenceforth, for the sake of brevity, we use the following notations:\n\\begin{align*}\n    \\Delta_{t}^{\\bx,\\by} & \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left( \\left\\| {\\mathbf x^i_t} - {\\mathbf x_t} \\right\\|^2 + \\left\\| {\\mathbf y^i_t} - {\\mathbf y_t} \\right\\|^2 \\right), \\\\\n    \\Delta_{t}^{\\bdx} & \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left\\| {\\mathbf d^i_{x,t}} - {\\mathbf d_{x,t}} \\right\\|^2, \\\\\n    \\Delta_{t}^{\\bdy} & \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left\\| {\\mathbf d^i_{y,t}} - {\\mathbf d_{y,t}} \\right\\|^2.\n\\end{align*}\n\n\n\\begin{lemma}\n\\label{lem:NC_PL_mom_cons_errs_recursion}\nSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness},  \\ref{assum:bdd_hetero}, and the stochastic oracles for the local functions $\\{ f_i \\}$ satisfy \\cref{assum:bdd_var}.\nFurther, in \\cref{alg_NC_momentum}, we choose $\\beta_x = \\beta_y = \\beta$, and $\\alpha_t$ such that $0 < \\alpha_t < 1\/\\beta$.\nThen, the iterates $\\{ {\\mathbf x^i_t}, {\\mathbf y^i_t} \\}$ and direction estimates $\\{ {\\mathbf d^i_{x,t}}, {\\mathbf d^i_{y,t}} \\}$ generated by Algorithm \\ref{alg_NC_momentum} satisfy\n\\begin{align}\n    \\Delta_{t+1}^{\\bx,\\by} & \\leq (1+c_1) \\Delta_{t}^{\\bx,\\by} + \\left( 1 + \\mfrac{1}{c_1} \\right) \\alpha_t^2 \\left( \\eta_x^2 \\Delta_{t}^{\\bdx} + \\eta_y^2 \\Delta_{t}^{\\bdy} \\right), \\qquad \\text{ for any constant } c_1 > 0\n    \\label{eq:lem:NC_PL_mom_xy_cons_errs_recursion} \n    \\\\\n    \\Delta_{t+1}^{\\bdx} & \\leq (1-\\beta \\alpha_t) \\Delta_{t}^{\\bdx} + 6 L_f^2 \\beta \\alpha_t \\Delta_{t+1}^{\\bx,\\by} + \\beta \\alpha_t \\left[ \\sigma^2 \\left( 1 + \\mfrac{1}{n} \\right) + 3 \\varsigma_x^2 \\right],\n    \\label{eq:lem:NC_PL_mom_p_cons_errs_recursion}\n    \\\\\n    \\Delta_{t+1}^{\\bdy} & \\leq (1-\\beta \\alpha_t) \\Delta_{t}^{\\bdy} + 6 L_f^2 \\beta \\alpha_t \\Delta_{t+1}^{\\bx,\\by} + \\beta \\alpha_t \\left[ \\sigma^2 \\left( 1 + \\mfrac{1}{n} \\right) + 3 \\varsigma_y^2 \\right].\n    \\label{eq:lem:NC_PL_mom_q_cons_errs_recursion}\n\\end{align}\n\\end{lemma}\n\n\n\\begin{lemma}\n\\label{lem:NC_PL_mom_induct_bd_cons_error_xy}\nSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness},  \\ref{assum:bdd_hetero}, and the stochastic oracles for the local functions $\\{ f_i \\}$ satisfy \\cref{assum:bdd_var}.\nFurther, in \\cref{alg_NC_momentum}, we choose $\\beta_x = \\beta_y = \\beta$, and step-sizes $\\eta_x, \\eta_y, \\alpha_t$ such that \n$\\alpha_t \\equiv \\alpha \\leq \\min \\left\\{ \\frac{\\beta}{6 L_f^2 (\\eta_y^2 + \\eta_x^2)}, \\frac{1}{16 \\beta \\tau} \\right\\}$ for all $t$, and $L_f^2 (\\eta_y^2 + \\eta_x^2) \\leq \\frac{\\beta^2}{6}$.\nSuppose $s \\tau + 1 \\leq t \\leq (s+1) \\tau -1$ for some positive integer $s$ (i.e., $t$ is between two consecutive synchronizations).\nAlso, let $1 \\leq k < \\tau$ such that $t - k \\geq s \\tau + 1$.\nThen, the consensus error satisfies\n\\begin{align}\n    \\Delta_{t}^{\\bx,\\by} \\leq (1 + 2 k \\theta) \\Delta_{t-k}^{x,y} + 2 k \\mfrac{\\alpha}{\\beta} (1-\\beta \\alpha) \\left( \\eta_x^2 \\Delta_{t-k-1}^{{\\mathbf d_x}} + \\eta_y^2 \\Delta_{t-k-1}^{{\\mathbf d_y}} \\right) + k^2 (1+\\theta) \\Upsilon,\n    \\label{eq:lem:NC_PL_mom_induct_bd_cons_error_xy}\n\\end{align}\nwhere, $\\theta = c_1 + 6 L_f^2 \\alpha^2 (\\eta_y^2 + \\eta_x^2)$, $c_1 = \\frac{\\beta \\alpha}{1 - \\beta \\alpha}$, and $\\Upsilon = \\alpha^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\mfrac{1}{n} \\right) + 3 \\eta_x^2 \\varsigma_x^2 + 3 \\eta_y^2 \\varsigma_y^2 \\right]$.\n\\end{lemma}\n\n\n\\begin{cor}\n\\label{cor:NC_PL_mom_induct_bd_cons_error_xy}\nSince the clients in Algorithm \\ref{alg_NC_momentum} communicate with the server every $\\tau$ iterations, for all $t = 0, \\hdots, T-1$, then under the conditions of \\cref{lem:NC_PL_mom_induct_bd_cons_error_xy}, the iterate consensus error is bounded as follows.\n\\begin{align*}\n    \\Delta_{t}^{\\bx,\\by} \\leq \\Theta \\left( (\\tau - 1)^2 \\alpha^2 \\left( \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 + \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) \\right).\n\\end{align*}\n\\end{cor}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\texorpdfstring{\\cref{thm:NC_PL_mom}}{Theorem 2}}\n\\label{sec:NC_PL_mom_thm_proof}\n\nFor the sake of completeness, we first state the full statement of \\cref{thm:NC_PL_mom}, in a slightly more general form.\n\n\\begin{theorem*}\nSuppose the local loss functions $\\{ f_i \\}_i$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, and the global function $f$ satisfies \\cref{assum:PL_y}.\nSuppose in \\cref{alg_NC_momentum}, \n$\\beta_x = \\beta_y = \\beta = 3$, $\\alpha_t \\equiv \\alpha \\leq \\min \\big\\{ \\frac{\\beta}{6 L_f^2 (\\eta_y^2 + \\eta_x^2)}, \\frac{1}{48 \\tau} \\big\\}$, for all $t$, and the step-sizes $\\eta_x, \\eta_y$ are chosen such that $\\eta_y \\leq \\frac{\\mu}{8 L_f^2}$, and $\\frac{\\eta_x}{\\eta_y} \\leq \\frac{1}{20 \\kappa^2}$, where $\\kappa = L_f\/\\mu$ is the condition number.\nThen the iterates generated by \\cref{alg_NC_momentum} satisfy\n\\begin{equation}\n    \\begin{aligned}\n        & \\frac{1}{T} \\sum_{t=0}^{T-1} \\left[ \\frac{1}{\\eta_x^2} \\mathbb E \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{2 L_f^2}{\\mu} \\mathbb E \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 \\right] \\\\\n        & \\leq \\underbrace{\\mathcal O \\left( \\frac{\\kappa^2}{\\eta_y \\alpha T} + \\frac{\\alpha}{\\mu \\eta_y} \\frac{\\sigma^2}{n} \\right)}_{\\text{Error with full synchronization}} + \\underbrace{\\mathcal O \\Big( (\\tau - 1)^2 \\alpha^2 \\left( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 \\right) \\Big)}_{\\text{Error due to local updates}}.\n    \\end{aligned}\n    \\label{eq_proof:thm_NC_PL_mom_conv_rate}\n\\end{equation}\nRecall that $\\sigma^2$ is the variance of stochastic gradient oracle (\\cref{assum:bdd_var}), and $\\varsigma_x, \\varsigma_y$ quantify the heterogeneity of local functions (\\cref{assum:bdd_hetero}).\nWith $\\alpha = \\sqrt{\\frac{n}{T}}$ in \\eqref{eq_proof:thm_NC_PL_mom_conv_rate}, we get\n\\begin{align}\n    & \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left[ \\frac{1}{\\eta_x^2} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{L_f^2}{\\mu} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 \\right] \\nonumber \\\\\n    & \\qquad \\leq \\mathcal O \\left( \\frac{\\kappa^2 + \\sigma^2}{\\sqrt{nT}} \\right) + \\mathcal O \\left( \\frac{n (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 \\right)}{T} \\right). \\nonumber\n\\end{align}\n\\end{theorem*}\n\n\\begin{remark}[Convergence results in terms of $\\norm{\\Phi (\\cdot)}$]\nThe inequality \\eqref{eq:thm:NC_PL_mom} results from the following reasoning.\n\\begin{align}\n    \\norm{\\nabla \\Phi ({\\mathbf x_t})} &= \\norm{\\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y}^*({\\mathbf x_t}))} \\tag{\\cref{lem:Phi_PL_smooth_nouiehed}} \\\\\n    & \\leq \\norm{\\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y}^*({\\mathbf x_t})) - \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t})} + \\norm{\\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t})} \\tag{Triangle inequality} \\\\\n    & \\leq L_f \\norm{{\\mathbf y}^*({\\mathbf x_t}) - {\\mathbf y_t}} + \\norm{\\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}}} + \\norm{{\\mathbf d_{x,t}}} \\tag{\\cref{assum:smoothness}} \\\\\n    &= L_f \\sqrt{\\frac{2}{\\mu} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right]} + \\norm{\\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}}} + \\frac{1}{\\eta_x} \\norm{\\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t}}. \\tag{quadratic growth of $\\mu$-PL functions (\\cref{lem:quad_growth})}\n    \\label{eq:NC_PL_mom_compare_metrics}\n    \\\\\n    \\Rightarrow \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{\\nabla \\Phi({\\mathbf x_t})}^2 & \\leq \\frac{3}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left( \\frac{1}{\\eta_x^2} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{2 L_f^2}{\\mu} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 \\right). \\nonumber\n\\end{align}\n\\end{remark}\n\n\\begin{proof}[Proof of \\cref{thm:NC_PL_mom}]\n\n\nMultiplying both sides of \\cref{lem:NC_PL_mom_phi_error} by $10 L_f^2 \\eta_x \/ (\\mu^2 \\eta_y)$, we get\n\n\\begin{align}\n    & \\frac{10 L_f^2 \\eta_x}{\\mu^2 \\eta_y} \\Big[ \\left[ \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\right] - \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\Big] \\nonumber \\\\\n    & \\leq - \\frac{5 \\eta_x \\alpha_t L_f^2}{\\mu} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] - \\frac{5 \\kappa^2 \\alpha_t \\eta_x}{2 \\eta_y^2} \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 \\nonumber \\\\\n    & \\quad + \\frac{5 L_f^2 \\alpha_t}{\\mu^2 \\eta_y} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + 10 \\kappa^2 \\eta_x \\alpha_t \\norm{\\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2. \n    \\label{eq_proof:thm:NC_PL_mom_1}\n\\end{align}\n\n\nDefine\n\\begin{align}\n    \\mc E_{t} \\triangleq \\Phi({\\mathbf x_t}) - \\Phi^* + \\frac{10 L_f^2 \\eta_x}{\\mu^2 \\eta_y} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right]. \\nonumber\n   \n\\end{align}\nThen, using \\cref{lem:NC_PL_mom_phi_error} and \\eqref{eq_proof:thm:NC_PL_mom_1}, we get\n\\begin{align}\n    \\mc E_{t+1} - \\mc E_{t} & \\leq - \\left( \\frac{\\alpha_t}{2 \\eta_x} - \\frac{5 L_f^2 \\alpha_t}{\\mu^2 \\eta_y} \\right) \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 - \\frac{\\eta_x \\alpha_t L_f^2}{\\mu} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] - \\frac{5 \\kappa^2 \\alpha_t \\eta_x}{2 \\eta_y^2} \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 \\nonumber \\\\\n    & \\quad + 2 \\eta_x \\alpha_t \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 + 10 \\kappa^2 \\eta_x \\alpha_t \\norm{\\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2 \\nonumber \\\\\n    & \\leq - \\frac{\\alpha_t}{4 \\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 - \\frac{\\eta_x \\alpha_t L_f^2}{\\mu} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] - \\frac{5 \\kappa^2 \\alpha_t \\eta_x}{2 \\eta_y^2} \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 \\nonumber \\\\\n    & \\quad + 2 \\eta_x \\alpha_t \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 + 2 \\alpha_t \\eta_y \\norm{\\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2. \\label{eq_proof:thm:NC_PL_mom_2}\n\\end{align}\nwhere, $- \\frac{\\alpha_t}{2 \\eta_x} + \\frac{5 \\kappa^2 \\alpha_t}{\\eta_y} \\leq - \\frac{\\alpha_t}{4 \\eta_x}$, since $\\eta_x \\leq \\frac{\\eta_y}{20 \\kappa^2}$. \nNext, we choose $\\beta_x = \\beta_y = \\beta = 3$, and define\n\\begin{align}\n    \\mathfrak{E}_{t} \\triangleq \\mc E_{t} + \\frac{2 \\eta_x}{\\mu \\eta_y} \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 + \\frac{2 \\eta_x}{\\mu \\eta_y} \\left\\| \\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}} \\right\\|^2, \\quad t \\geq 0. \\nonumber\n   \n\\end{align}\nThen, using the bounds in \\cref{lem:NC_PL_mom_grad_var_bound} and \\eqref{eq_proof:thm:NC_PL_mom_2}, we get\n\\begin{align}\n    \\mathbb E \\left[ \\mathfrak{E}_{t+1} - \\mathfrak{E}_{t} \\right] & \\leq - \\left( \\frac{\\alpha_t}{2 \\eta_x} - 2 \\frac{2 \\eta_x}{\\mu \\eta_y} \\frac{2 L_f^2 \\alpha_t}{3} \\right) \\mathbb E \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 - \\frac{\\eta_x \\alpha_t L_f^2}{\\mu} \\mathbb E \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\nonumber \\\\\n    & \\quad - \\left( \\frac{2 \\eta_x}{\\mu \\eta_y} \\frac{3 \\alpha_t}{2} - 2 \\alpha_t \\eta_x \\right) \\mathbb E \\left[ \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 + \\left\\| \\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}} \\right\\|^2 \\right] \\nonumber \\\\\n    & \\quad - \\left( \\frac{5 \\alpha_t \\kappa^2 \\eta_x}{2 \\eta_y^2} - 2 \\frac{2 \\eta_x}{\\mu \\eta_y} \\frac{2 L_f^2 \\alpha_t}{3} \\right) \\mathbb E \\left\\| \\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t} \\right\\|^2 + 2 \\frac{2 \\eta_x}{\\mu \\eta_y} 3 \\alpha_t L_f^2 \\Delta_{t+1}^{\\bx,\\by} + 2 \\frac{2 \\eta_x}{\\mu \\eta_y} \\frac{9 \\alpha_t^2 \\sigma^2}{n} \\nonumber \\\\\n    & \\leq - \\frac{\\alpha_t}{4 \\eta_x} \\mathbb E \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 - \\frac{\\eta_x \\alpha_t L_f^2}{\\mu} \\mathbb E \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] -\\frac{\\alpha_t \\kappa^2 \\eta_x}{\\eta_y^2} \\mathbb E \\left\\| \\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t} \\right\\|^2 \\nonumber \\\\\n    & \\quad - \\frac{2 \\alpha_t \\eta_x}{\\mu \\eta_y} \\mathbb E \\left[ \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 + \\left\\| \\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}} \\right\\|^2 \\right] + \\frac{4 \\eta_x}{\\mu \\eta_y} \\left[ 3 \\alpha_t L_f^2 \\Delta_{t+1}^{\\bx,\\by} + \\frac{9 \\alpha_t^2 \\sigma^2}{n} \\right] \\label{eq_proof:thm:NC_PL_mom_3}\n\\end{align}\n\nHere, using $\\eta_y\\leq 1\/(8 L_f) \\leq 1\/(8 \\mu)$ and $\\eta_y\\geq 20 \\eta_x \\kappa^2$, we simplify the coefficients in \\eqref{eq_proof:thm:NC_PL_mom_3} as follows\n\\begin{align*}\n    & - \\frac{\\alpha_t}{2 \\eta_x} \\left( 1 - \\frac{16 \\eta_x^2 L_f^2}{3 \\mu \\eta_y} \\right) = - \\frac{\\alpha_t}{2 \\eta_x} + \\frac{\\alpha_t}{2 \\eta_x} \\frac{16 \\mu \\eta_y \\kappa^2}{3} \\frac{\\eta_x^2}{\\eta_y^2} \\leq - \\frac{\\alpha_t}{2 \\eta_x} + \\frac{\\alpha_t}{2 \\eta_x} \\frac{16}{3} \\frac{1}{8} \\frac{1}{400 \\kappa^2} \\leq -\\frac{\\alpha_t}{4 \\eta_x} \\tag{$\\because \\kappa \\geq 1$} \\\\\n    & -\\left( \\frac{2 \\eta_x}{\\mu \\eta_y} \\frac{3 \\alpha_t}{2} - 2 \\eta_x \\alpha_t \\right) \\leq - \\frac{3 \\eta_x \\alpha_t}{\\mu \\eta_y} + \\frac{2 \\eta_x \\alpha_t}{8 \\mu \\eta_y} \\leq - \\frac{2 \\eta_x \\alpha_t}{\\mu \\eta_y}, \\tag{$\\because 1 \\leq 1\/(8 \\mu \\eta_y)$} \\nonumber \\\\\n    & - \\left( \\frac{5 \\alpha_t \\kappa^2 \\eta_x}{2 \\eta_y^2} - \\frac{4 \\eta_x}{\\mu \\eta_y} \\frac{2 L_f^2 \\alpha_t}{3} \\right) = \\frac{\\alpha_t \\kappa^2 \\eta_x}{\\eta_y^2} \\left( -\\frac{5}{2} + \\frac{8}{3} \\eta_y \\mu \\right) \\leq \\frac{\\alpha_t \\kappa^2 \\eta_x}{\\eta_y^2} \\left( -\\frac{5}{2} + \\frac{1}{3} \\right) \\leq -\\frac{\\alpha_t \\kappa^2 \\eta_x}{\\eta_y^2}. \\tag{$\\because 1 \\leq 1\/(8 \\mu \\eta_y)$}\n\\end{align*}\nSumming \\eqref{eq_proof:thm:NC_PL_mom_3} over $t=0, \\hdots, T-1$ and rearranging the terms, we get\n\\begin{align}\n    & \\frac{1}{T} \\sum_{t=0}^{T-1} \\frac{\\alpha_t \\eta_x}{4} \\left[ \\frac{1}{\\eta_x^2} \\mathbb E \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{4 L_f^2}{\\mu} \\mathbb E \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\frac{8}{\\mu \\eta_y} \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 \\right] \\nonumber \\\\\n    & \\qquad \\leq \\frac{1}{T} \\sum_{t=0}^{T-1} \\frac{4 \\eta_x}{\\mu \\eta_y} \\left[ 9 \\alpha_t^2 \\frac{\\sigma^2}{n} + 3 \\alpha_t L_f^2 \\Delta_{t+1}^{\\bx,\\by} \\right] + \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left[ \\mathfrak{E}_{t} - \\mathfrak{E}_{t+1} \\right]. \\nonumber\n\\end{align}\n\nWe choose $\\alpha_t = \\alpha$ for all $t$. $\\frac{1}{8 \\mu \\eta_y} \\geq 1$. Also, $\\mathfrak{E}_{t} \\geq 0, \\forall \\ t$. Therefore,\n\\begin{align}\n    & \\frac{1}{T} \\sum_{t=0}^{T-1} \\left[ \\frac{1}{\\eta_x^2} \\mathbb E \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{2 L_f^2}{\\mu} \\mathbb E \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 \\right] \\nonumber \\\\\n    & \\qquad \\leq \\frac{4 \\mathfrak{E}_0}{\\eta_x \\alpha T} + \\frac{1}{T} \\sum_{t=0}^{T-1} \\frac{16}{\\mu \\eta_y} \\left[ 9 \\alpha \\frac{\\sigma^2}{n} + 3 L_f^2 \\Delta_{t+1}^{\\bx,\\by} \\right] \\tag{$\\because \\mathfrak{E}_{t} \\geq 0$ for all $t$} \\\\\n   \n    & \\qquad \\leq \\mathcal O \\left( \\frac{\\mathfrak{E}_0}{\\eta_x \\alpha T} + \\frac{\\alpha}{\\mu \\eta_y} \\frac{\\sigma^2}{n} \\right) + \\mathcal O \\left( \\frac{L_f^2}{\\mu \\eta_y} (\\tau - 1)^2 \\alpha^2 \\left( \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 + \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) \\right) \\tag{\\cref{cor:NC_PL_mom_induct_bd_cons_error_xy}} \\\\\n    & \\qquad = \\mathcal O \\left( \\frac{\\kappa^2}{\\eta_y \\alpha T} + \\frac{\\alpha}{\\mu \\eta_y} \\frac{\\sigma^2}{n} \\right) + \\mathcal O \\left( \\kappa^2 \\mu (\\tau - 1)^2 \\alpha^2 \\left( \\eta_y \\left( \\sigma^2 + \\varsigma_y^2 \\right) + \\frac{\\eta_x^2}{\\eta_y} \\left( \\sigma^2 + \\varsigma_x^2 \\right) \\right) \\right) \\nonumber \\\\\n    & \\qquad = \\mathcal O \\left( \\frac{\\kappa^2}{\\eta_y \\alpha T} + \\frac{\\alpha}{\\mu \\eta_y} \\frac{\\sigma^2}{n} \\right) + \\mathcal O \\left( (\\tau - 1)^2 \\alpha^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) + \\mu (\\tau - 1)^2 \\alpha^2 \\left( \\eta_x \\left( \\sigma^2 + \\varsigma_x^2 \\right) \\right) \\right) \\tag{$\\because \\eta_y \\leq \\frac{\\mu}{8 L_f^2}, \\frac{\\eta_x}{\\eta_y} \\leq \\frac{1}{20 \\kappa^2}$} \\\\\n    & \\qquad \\leq \\underbrace{\\mathcal O \\left( \\frac{\\kappa^2}{\\eta_y \\alpha T} + \\frac{\\alpha}{\\mu \\eta_y} \\frac{\\sigma^2}{n} \\right)}_{\\substack{\\text{Single client} \\\\\n    \\text{convergence error}}} + \\underbrace{\\mathcal O \\Big( (\\tau - 1)^2 \\alpha^2 \\left( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 \\right) \\Big)}_{\\text{Error due to local updates}}. \\tag{$\\because \\mu \\eta_x \\leq 1$}\n   \n\\end{align}\nFinally, since $\\mathfrak{E}_0$ is a constant, and using $\\eta_y \\geq 20 \\eta_x \\kappa^2$, we get \\eqref{eq_proof:thm_NC_PL_mom_conv_rate}.\n\nFurther, with $\\alpha = \\sqrt{\\frac{n}{T}}$ in \\eqref{eq_proof:thm_NC_PL_mom_conv_rate}, we get\n\\begin{align}\n    & \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left[ \\frac{1}{\\eta_x^2} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{2 L_f^2}{\\mu} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 \\right] \\nonumber \\\\\n    & \\qquad \\leq \\mathcal O \\left( \\frac{\\kappa^2 + \\sigma^2}{\\sqrt{nT}} \\right) + \\mathcal O \\left( \\frac{n (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 \\right)}{T} \\right). \\nonumber\n\\end{align}\n\\end{proof}\n\n\n\n\n\n\\begin{proof}[Proof of \\cref{cor:NC_PL_mom_comm_cost}]\nWe assume $T \\geq n^3$.\nTo reach an $\\epsilon$-accurate point, we note that using Jensen's inequality\n\\begin{align}\n    & \\min_{t \\in [T-1]} \\mathbb E \\left[ \\frac{1}{\\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\| + L_f \\sqrt{\\frac{2}{\\mu} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right]} + \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\| \\right] \\nonumber \\\\\n    & \\leq \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left[ \\frac{1}{\\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\| + L_f \\sqrt{\\frac{2}{\\mu} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right]} + \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\| \\right] \\nonumber \\\\\n    &  \\leq \\left[ \\frac{3}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left( \\frac{1}{\\eta_x^2} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{2 L_f^2}{\\mu} \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 \\right) \\right]^{1\/2} \\nonumber \\\\\n    & \\leq \\mathcal O \\left( \\frac{\\kappa + \\sigma}{(n T)^{1\/4}} \\right) + \\mathcal O \\left( \\tau \\sqrt{\\frac{n \\left( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 \\right)}{T}} \\right), \\nonumber\n\\end{align}\nwhere we use $\\sqrt{a+b} \\leq \\sqrt{a} + \\sqrt{b}$.\nHence,  we need $T = \\mathcal O \\left( \\kappa^4\/(n \\epsilon^4) \\right)$ iterations, to reach an $\\epsilon$-accurate point.\nWe can choose $\\tau \\leq \\mathcal O \\left( \\frac{T^{1\/4}}{n^{3\/4}} \\right)$ without affecting the convergence rate.\nHence, the number of communication rounds is $\\mathcal O \\left( \\frac{T}{\\tau} \\right) = \\mathcal O \\left( (n T)^{3\/4} \\right) = \\mathcal O \\left( \\kappa^3\/\\epsilon^3 \\right)$. \n\\end{proof}\n\n\n\\subsection{Proofs of the Intermediate Lemmas}\n\\label{sec:NC_PL_mom_int_results_proofs}\n\n\n\\begin{proof}[Proof of \\cref{lem:NC_PL_mom_Phi_1_step_decay}]\nUsing $L_{\\Phi}$-smoothnes of $\\Phi(\\cdot)$ (\\cref{lem:Phi_PL_smooth_nouiehed})\n\\begin{align}\n    & \\Phi ({\\mathbf x_{t+1}}) - \\Phi ({\\mathbf x_t}) \\leq \\langle \\nabla \\Phi({\\mathbf x_t}), {\\mathbf x_{t+1}} - {\\mathbf x_t} \\rangle + \\frac{L_{\\Phi}}{2} \\left\\| {\\mathbf x_{t+1}} - {\\mathbf x_t} \\right\\|^2 \\nonumber\n   \n    \\\\\n    & \\quad = \\alpha_t \\langle \\nabla \\Phi({\\mathbf x_t}), \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\rangle + \\frac{L_{\\Phi} \\alpha_t^2}{2} \\left\\| {\\mathbf x_{t+1}} - {\\mathbf x_t} \\right\\|^2 \\tag{see updates in \\eqref{eq:NC_mom_update_avg}} \\\\\n    & \\quad = \\alpha_t \\langle {\\mathbf d_{x,t}}, \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\rangle + \\alpha_t \\langle \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}}, \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\rangle \\nonumber \\\\\n    & \\qquad + \\alpha_t \\left\\langle \\nabla \\Phi({\\mathbf x_t}) - \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}), \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\rangle + \\frac{L_{\\Phi} \\alpha_t^2}{2} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2. \\label{eq:lem:NC_PL_mom_Phi_1_step_decay_1}\n\\end{align}\nNext, we bound the individual inner product terms in \\eqref{eq:lem:NC_PL_mom_Phi_1_step_decay_1}.\n\\begin{align}\n    \\alpha_t \\langle {\\mathbf d_{x,t}}, \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\rangle &= -\\frac{\\alpha_t}{\\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2, \\label{eq:lem:NC_PL_mom_Phi_1_step_decay_2a} \\\\\n    \\alpha_t \\langle \\nabla \\Phi({\\mathbf x_t}) - \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}), \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\rangle & \\overset{(a)}{\\leq} \\frac{\\alpha_t}{8 \\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\alpha_t 2 \\eta_x \\left\\| \\nabla \\Phi({\\mathbf x_t}) - \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) \\right\\|^2, \\nonumber\n   \n    \\\\\n    & \\overset{(b)}{\\leq} \\frac{\\alpha_t}{8 \\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + 2 \\eta_x \\alpha_t L_f^2 \\left\\| {\\mathbf y}^*({\\mathbf x_t}) - {\\mathbf y_t} \\right\\|^2, \\nonumber \\\\\n    & \\leq \\frac{\\alpha_t}{8 \\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{4 \\eta_x \\alpha_t L_f^2}{\\mu} \\left[ f({\\mathbf x_t}, {\\mathbf y}^*({\\mathbf x_t})) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right], \\nonumber \\\\\n    & = \\frac{\\alpha_t}{8 \\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{4 \\eta_x \\alpha_t L_f^2}{\\mu} \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right], \\label{eq:lem:NC_PL_mom_Phi_1_step_decay_2b} \\\\\n    \\alpha_t \\langle \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}}, \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\rangle & \\leq \\frac{\\alpha_t}{8 \\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + 2 \\eta_x \\alpha_t \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2, \\label{eq:lem:NC_PL_mom_Phi_1_step_decay_2c}\n\\end{align}\nwhere \\eqref{eq:lem:NC_PL_mom_Phi_1_step_decay_2a} follows from the update expression of \\textit{virtual} averages in \\eqref{eq:NC_mom_update_avg}; \n$(a)$ and \\eqref{eq:lem:NC_PL_mom_Phi_1_step_decay_2c} both follow from Young's inequality \\cref{lem:Young}\n(with $\\gamma = 4 \\eta_x$); \n$(b)$ follows from \\cref{lem:Phi_PL_smooth_nouiehed} and $L_f$-smoothness of $f({\\mathbf x_t}, \\cdot)$ (\\cref{assum:smoothness});\nand \\eqref{eq:lem:NC_PL_mom_Phi_1_step_decay_2b} follows from the quadratic growth condition of $\\mu$-PL functions (\\cref{lem:quad_growth}).\nSubstituting \\eqref{eq:lem:NC_PL_mom_Phi_1_step_decay_2a}-\\eqref{eq:lem:NC_PL_mom_Phi_1_step_decay_2c} in \\eqref{eq:lem:NC_PL_mom_Phi_1_step_decay_1}, we get\n\\begin{align*}\n    \\Phi ({\\mathbf x_{t+1}}) - \\Phi ({\\mathbf x_t}) & \\leq - \\left( \\frac{3 \\alpha_t}{4 \\eta_x} - \\frac{L_{\\Phi} \\alpha_t^2}{2} \\right) \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{4 \\eta_x \\alpha_t L_f^2}{\\mu} \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + 2 \\eta_x \\alpha_t \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2.\n\\end{align*}\nNotice that for $\\alpha_t \\leq \\frac{\\mu}{4 \\eta_x L_f^2}$, $\\frac{L_{\\Phi} \\alpha_t^2}{2} \\leq \\kappa L_f \\alpha_t^2 \\leq \\frac{\\alpha_t}{4 \\eta_x}$. Hence the result follows.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:NC_PL_mom_phi_error}]\nUsing $L_f$-smoothness of $f({\\mathbf x}, \\cdot)$ (\\cref{assum:smoothness}),\n\\begin{align}\n    f({\\mathbf x_{t+1}}, {\\mathbf y_t}) &+ \\left\\langle \\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t}), {\\mathbf y_{t+1}} - {\\mathbf y_t} \\right\\rangle - \\frac{L_f}{2} \\norm{{\\mathbf y_{t+1}} - {\\mathbf y_t}}^2 \\leq f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\nonumber \\\\\n    \\Rightarrow f({\\mathbf x_{t+1}}, {\\mathbf y_t}) & \\leq f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\alpha_t \\left\\langle \\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t}), \\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t} \\right\\rangle + \\frac{\\alpha_t^2 L_f}{2} \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2. \\label{eq_proof:lem:NC_PL_mom_phi_error_1}\n\\end{align}\nNext, we bound the inner product in \\eqref{eq_proof:lem:NC_PL_mom_phi_error_1}.\n\\begin{align}\n    & - \\alpha_t \\left\\langle \\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t}), \\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t} \\right\\rangle = - \\alpha_t \\eta_y \\left\\langle \\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t}), {\\mathbf d_{y,t}} \\right\\rangle \\tag{using \\eqref{eq:NC_mom_update_avg}} \\\\\n    &= -\\frac{\\alpha_t \\eta_y}{2} \\left[ \\norm{\\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t})}^2 + \\norm{{\\mathbf d_{y,t}}}^2 - \\norm{\\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t}) - \\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) + \\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2 \\right] \\nonumber \\\\\n    & \\leq -\\alpha_t \\eta_y \\mu \\left[ \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right] - \\frac{\\alpha_t}{2 \\eta_y} \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 + \\alpha_t \\eta_y \\left[ L_f^2 \\norm{{\\mathbf x_{t+1}} - {\\mathbf x_t}}^2 + \\norm{\\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2 \\right] \\label{eq_proof:lem:NC_PL_mom_phi_error_2}\n\\end{align}\nwhere, \\eqref{eq_proof:lem:NC_PL_mom_phi_error_2} follows from the quadratic growth condition of $\\mu$-PL functions (\\cref{lem:quad_growth}),\n\\begin{align}\n    \\norm{\\nabla_{\\by} f({\\mathbf x_{t+1}}, {\\mathbf y_t})}^2 \\geq 2 \\mu \\left( \\max_{\\mathbf y} f({\\mathbf x_{t+1}}, {\\mathbf y}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right) = 2 \\mu \\left( \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right). \\nonumber\n   \n\\end{align}\nSubstituting \\eqref{eq_proof:lem:NC_PL_mom_phi_error_2} in \\eqref{eq_proof:lem:NC_PL_mom_phi_error_1}, we get\n\\begin{align}\n    f({\\mathbf x_{t+1}}, {\\mathbf y_t}) & \\leq f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) -\\alpha_t \\eta_y \\mu \\left[ \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right] - \\frac{\\alpha_t}{2 \\eta_y} \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 + \\frac{\\alpha_t^2 L_f}{2} \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 \\nonumber \\\\\n    & \\quad + \\alpha_t \\eta_y \\left[ L_f^2 \\norm{{\\mathbf x_{t+1}} - {\\mathbf x_t}}^2 + \\norm{\\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2 \\right]. \\nonumber\n\\end{align}\nRearranging the terms we get\n\\begin{align}\n    \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) & \\leq \\left( 1-\\alpha_t \\eta_y \\mu \\right) \\left[ \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\right] - \\frac{\\alpha_t}{2} \\left( \\frac{1}{\\eta_y} - \\alpha_t L_f \\right) \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 \\nonumber \\\\\n    & \\quad + \\alpha_t \\eta_y \\left[ L_f^2 \\norm{{\\mathbf x_{t+1}} - {\\mathbf x_t}}^2 + \\norm{\\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2 \\right]. \\label{eq_proof:lem:NC_PL_mom_phi_error_3}\n\\end{align}\n\nNext, we bound $\\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t})$.\n\\begin{align}\n    & \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) = \\Phi ({\\mathbf x_{t+1}}) - \\Phi ({\\mathbf x_t}) + \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\underbrace{f({\\mathbf x_t}, {\\mathbf y_t}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t})}_{I}. \\label{eq_proof:lem:NC_PL_mom_phi_error_4}\n\\end{align}\nNext, we bound $I$. Using $L_f$-smoothness of $f(\\cdot, {\\mathbf y_t})$,\n\\begin{align}\n    & f({\\mathbf x_t}, {\\mathbf y_t}) + \\left\\langle \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}), {\\mathbf x_{t+1}} - {\\mathbf x_t} \\right\\rangle - \\frac{L_f}{2} \\norm{{\\mathbf x_{t+1}} - {\\mathbf x_t}}^2 \\leq f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\nonumber \\\\\n    \\Rightarrow I &=  f({\\mathbf x_t}, {\\mathbf y_t}) - f({\\mathbf x_{t+1}}, {\\mathbf y_t}) \\nonumber \\\\\n    & \\leq -\\alpha_t \\left\\langle \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}), \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\rangle + \\frac{\\alpha_t^2 L_f}{2} \\norm{\\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t}}^2 \\nonumber \\\\\n    &= -\\alpha_t \\left\\langle \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - \\nabla \\Phi({\\mathbf x_t}), \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\rangle -\\alpha_t \\left\\langle \\nabla \\Phi({\\mathbf x_t}), \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\rangle + \\frac{\\alpha_t^2 L_f}{2} \\norm{\\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t}}^2 \\nonumber \\\\\n    & \\leq \\frac{\\alpha_t}{8 \\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{4 \\eta_x \\alpha_t L_f^2}{\\mu} \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\tag{using \\eqref{eq:lem:NC_PL_mom_Phi_1_step_decay_2b}} \\\\\n    & \\quad + \\Phi ({\\mathbf x_t}) - \\Phi ({\\mathbf x_{t+1}}) + \\frac{\\alpha_t^2 L_{\\Phi}}{2} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\frac{\\alpha_t^2 L_f}{2} \\norm{\\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t}}^2 \\tag{smoothness of $\\Phi$ (\\cref{lem:Phi_PL_smooth_nouiehed})} \\\\\n    & = \\Phi ({\\mathbf x_t}) - \\Phi ({\\mathbf x_{t+1}}) + \\frac{4 \\eta_x \\alpha_t L_f^2}{\\mu} \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\frac{\\alpha_t}{2} \\left( \\frac{1}{4 \\eta_x} + 2 \\alpha_t L_{\\Phi} \\right) \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 \\tag{$\\because L_f \\leq L_{\\Phi}$}.\n\\end{align}\nUsing the bound on $I$ in \\eqref{eq_proof:lem:NC_PL_mom_phi_error_4} and then substituting in \\eqref{eq_proof:lem:NC_PL_mom_phi_error_3}, we get\n\\begin{align}\n    & \\Phi ({\\mathbf x_{t+1}}) - f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\nonumber \\\\\n    & \\leq (1 - \\alpha_t \\eta_y \\mu) \\left[ \\left( 1 + \\frac{4 \\eta_x \\alpha_t L_f^2}{\\mu} \\right) \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\frac{\\alpha_t}{2} \\left( \\frac{1}{4 \\eta_x} + 2 \\alpha_t L_{\\Phi} \\right) \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 \\right] \\nonumber \\\\\n    & \\quad - \\frac{\\alpha_t}{2} \\left( \\frac{1}{\\eta_y} - \\alpha_t L_f \\right) \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 + \\alpha_t \\eta_y \\left[ L_f^2 \\norm{{\\mathbf x_{t+1}} - {\\mathbf x_t}}^2 + \\norm{\\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2 \\right] \\nonumber \\\\\n    & \\overset{(a)}{\\leq} \\left( 1 - \\frac{\\alpha_t \\eta_y \\mu}{2} \\right) \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\frac{\\alpha_t}{2} \\left( \\frac{1}{4 \\eta_x} + 2 \\alpha_t L_{\\Phi} + 2 \\eta_y L_f^2 \\alpha_t^2 \\right) \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 \\nonumber \\\\\n    & \\quad - \\frac{\\alpha_t}{2} \\left( \\frac{1}{\\eta_y} - \\alpha_t L_f \\right) \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 + \\alpha_t \\eta_y \\norm{\\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2. \\nonumber \\\\\n    & \\overset{(b)}{\\leq} \\left( 1 - \\frac{\\alpha_t \\eta_y \\mu}{2} \\right) \\left[ \\Phi ({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + \\frac{\\alpha_t}{2 \\eta_x} \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 - \\frac{\\alpha_t}{4 \\eta_y} \\norm{\\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t}}^2 + \\alpha_t \\eta_y \\norm{\\nabla_{\\by} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{y,t}}}^2. \\nonumber\n   \n\\end{align}\nwhere in $(a)$ we choose $\\eta_x$ such that $(1 - \\alpha_t \\eta_y \\mu) \\left( 1 + \\frac{4 \\eta_x \\alpha_t L_f^2}{\\mu} \\right) \\leq \\left( 1 - \\frac{\\alpha_t \\eta_y \\mu}{2} \\right)$. \nThis holds if $\\frac{4 \\eta_x \\alpha_t L_f^2}{\\mu} \\leq \\frac{\\alpha_t \\eta_y \\mu}{2} \\Rightarrow \\eta_x \\leq \\frac{\\eta_y}{8 \\kappa^2}$, where $\\kappa = L_f\/\\mu \\geq 1$ is the condition number.\nFinally, $(b)$ follows since $\\alpha_t \\eta_y \\leq \\frac{1}{2 L_f}$ and $\\alpha_t \\leq \\frac{\\mu}{8 \\eta_x L_f^2} = \\frac{1}{8 \\eta_x \\kappa L_f}$. Therefore,\n\\begin{align*}\n    & 2 \\alpha_t L_{\\Phi} \\leq 4 \\kappa \\alpha_t L_f \\leq \\frac{1}{2 \\eta_x} \\tag{$L_{\\Phi} \\leq 2 \\kappa L_f$} \\\\\n    & 2 \\eta_y L_f^2 \\alpha_t^2 \\leq 2 \\eta_y \\alpha_t \\frac{\\mu}{8 \\eta_x} \\leq \\frac{\\mu}{8 \\eta_x} \\frac{1}{L_f} \\leq \\frac{1}{8 \\eta_x}.\n\\end{align*}\n\\end{proof}\n\n\n\\begin{proof}[Proof of \\cref{lem:NC_PL_mom_grad_var_bound}]\nWe prove \\eqref{eq:lem:NC_PL_mom_grad_var_bound_x} here. The proof for \\eqref{eq:lem:NC_PL_mom_grad_var_bound_y} is analogous.\n\\begin{align}\n    & \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - {\\mathbf d_{x,t+1}} \\right\\|^2 \\nonumber \\\\\n    &= \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - (1 - \\beta_x \\alpha_t) {\\mathbf d_{x,t}} - \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}) \\right\\|^2 \\tag{see \\eqref{eq:NC_mom_update_avg}} \\\\\n    &= \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - (1 - \\beta_x \\alpha_t) {\\mathbf d_{x,t}} - \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) \\right. \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\left.- \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n \\left( \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}) - \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) \\right) \\right\\|^2 \\nonumber \\\\\n    & \\overset{(a)}{=} \\mathbb E \\left\\| (1 - \\beta_x \\alpha_t) \\left( \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - {\\mathbf d_{x,t}} \\right) + \\beta_x \\alpha_t \\left( \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) \\right) \\right\\|^2 \\nonumber \\\\\n    & \\qquad \\qquad \\qquad + \\beta_x^2 \\alpha_t^2 \\mathbb E \\left\\| \\frac{1}{n} \\sum_{i=1}^n \\left( \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}) - \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) \\right) \\right\\|^2 \\nonumber \\\\\n    & \\leq (1 + a_1) (1 - \\beta_x \\alpha_t)^2 \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - {\\mathbf d_{x,t}} \\right\\|^2 \\nonumber \\\\\n    & \\qquad + \\beta_x^2 \\alpha_t^2 \\left( 1 + \\dfrac{1}{a_1} \\right) \\mathbb E \\left\\| \\frac{1}{n} \\sum_{i=1}^n \\left( \\nabla_{\\bx} f_i ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) \\right) \\right\\|^2 + \\beta_x^2 \\alpha_t^2 \\frac{\\sigma^2}{n}. \\label{eq:proof_lem:NC_PL_mom_grad_var_bound_x_3}\n\\end{align}\nHere, $(a)$ follows from Assumption \\ref{assum:bdd_var} (unbiasedness of stochastic gradients),\n\\begin{align*}\n    & \\mathbb E \\left\\langle (1 - \\beta_x \\alpha_t) \\left( \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - {\\mathbf d_{x,t}} \\right) + \\beta_x \\alpha_t \\left( \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) \\right), \\right. \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\left. \\frac{1}{n} \\sum_{i=1}^n \\left( \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}) - \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) \\right) \\right\\rangle \\nonumber \\\\\n    &= \\mathbb E \\left\\langle (1 - \\beta_x \\alpha_t) \\left( \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - {\\mathbf d_{x,t}} \\right) + \\beta_x \\alpha_t \\left( \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) \\right), \\right. \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\left. \\frac{1}{n} \\sum_{i=1}^n \\left( \\mathbb E \\left[ \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}) \\right] - \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) \\right) \\right\\rangle = 0. \\tag{Law of total expectation}\n\\end{align*}\nAlso, \\eqref{eq:proof_lem:NC_PL_mom_grad_var_bound_x_3} follows from Assumption \\ref{assum:bdd_var} (independence of stochastic gradients across clients), and \\cref{lem:Young}\n(with $\\gamma = a_1$).\nNext, in \\eqref{eq:proof_lem:NC_PL_mom_grad_var_bound_x_3}, we choose $a_1$ such that $\\left( 1 + \\frac{1}{a_1} \\right) \\beta_x \\alpha_t = 1$, i.e., $a_1 = \\frac{\\beta_x \\alpha_t}{1 - \\beta_x \\alpha_t}$. \nTherefore, $(1-\\beta_x \\alpha_t) (1 + a_1) = 1$. Consequently, in \\eqref{eq:proof_lem:NC_PL_mom_grad_var_bound_x_3} we get,\n\\begin{align}\n    & \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - {\\mathbf d_{x,t+1}} \\right\\|^2 \\nonumber \\\\\n    & \\leq (1 - \\beta_x \\alpha_t) \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) + \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 + \\beta_x^2 \\alpha_t^2 \\frac{\\sigma^2}{n} \\nonumber \\\\\n    & \\quad + \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n L_f^2 \\mathbb E \\left[ \\left\\| {\\mathbf x_{t+1}} - {\\mathbf x^i_{t+1}} \\right\\|^2 + \\left\\| {\\mathbf y_{t+1}} - {\\mathbf y^i_{t+1}} \\right\\|^2 \\right] \n    \\tag{Jensen's inequality with $\\norm{\\cdot}^2_2$; \\cref{assum:smoothness}}\n   \n    \\\\\n    & \\leq (1 - \\beta_x \\alpha_t) \\left[ (1 + a_2) \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 + \\left( 1 + \\dfrac{1}{a_2} \\right) \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) \\right\\|^2 \\right] + \\beta_x^2 \\alpha_t^2 \\frac{\\sigma^2}{n} \\nonumber \\\\\n    & \\quad + \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n L_f^2 \\mathbb E \\left[ \\left\\| {\\mathbf x_{t+1}} - {\\mathbf x^i_{t+1}} \\right\\|^2 + \\left\\| {\\mathbf y_{t+1}} - {\\mathbf y^i_{t+1}} \\right\\|^2 \\right], \\label{eq:proof_lem:NC_PL_mom_grad_var_bound_x_5}\n\\end{align}\nIn \\eqref{eq:proof_lem:NC_PL_mom_grad_var_bound_x_5}, we choose $a_2 = \\frac{\\beta_x \\alpha_t}{2}$. Then, $(1-\\beta_x \\alpha_t) \\left( 1 + \\frac{\\beta_x \\alpha_t}{2} \\right) \\leq 1 - \\frac{\\beta_x \\alpha_t}{2}$, and $(1-\\beta_x \\alpha_t) \\left( 1 + \\frac{2}{\\beta_x \\alpha_t} \\right) \\leq \\frac{2}{\\beta_x \\alpha_t}$. Therefore, we get\n\\begin{align}\n    & \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - {\\mathbf d_{x,t+1}} \\right\\|^2 \\nonumber \\\\\n    & \\leq \\left( 1 - \\frac{\\beta_x \\alpha_t}{2} \\right) \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 + \\frac{2}{\\beta_x \\alpha_t} L_f^2 \\mathbb E \\left[ \\left\\| {\\mathbf x_{t+1}} - {\\mathbf x_t} \\right\\|^2 + \\left\\| {\\mathbf y_{t+1}} - {\\mathbf y_t} \\right\\|^2 \\right] \\nonumber \\\\\n    & \\qquad + \\beta_x^2 \\alpha_t^2 \\frac{\\sigma^2}{n} + \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n L_f^2 \\mathbb E \\left[ \\left\\| {\\mathbf x_{t+1}} - {\\mathbf x^i_{t+1}} \\right\\|^2 + \\left\\| {\\mathbf y_{t+1}} - {\\mathbf y^i_{t+1}} \\right\\|^2 \\right] \\nonumber \\\\\n    &= \\left( 1 - \\frac{\\beta_x \\alpha_t}{2} \\right) \\mathbb E \\left\\| \\nabla_{\\bx} f({\\mathbf x_t}, {\\mathbf y_t}) - {\\mathbf d_{x,t}} \\right\\|^2 + \\frac{2 L_f^2 \\alpha_t}{\\beta_x} \\mathbb E \\left[ \\left\\| \\Tbx_{t+\\frac{1}{2}} - {\\mathbf x_t} \\right\\|^2 + \\left\\| \\Tby_{t+\\frac{1}{2}} - {\\mathbf y_t} \\right\\|^2 \\right] \\nonumber \\\\\n    & \\qquad + \\beta_x^2 \\alpha_t^2 \\frac{\\sigma^2}{n} + \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n L_f^2 \\mathbb E \\left[ \\left\\| {\\mathbf x_{t+1}} - {\\mathbf x^i_{t+1}} \\right\\|^2 + \\left\\| {\\mathbf y_{t+1}} - {\\mathbf y^i_{t+1}} \\right\\|^2 \\right],\n   \n\\end{align}\nFinally, we choose $\\beta_x = \\beta$.\nThis concludes the proof.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of \\cref{lem:NC_PL_mom_cons_errs_recursion}]\nFor the sake of clarity, we repeat the following notations:\n$\\Delta_{t}^{\\bx,\\by} \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left( \\left\\| {\\mathbf x^i_t} - {\\mathbf x_t} \\right\\|^2 + \\left\\| {\\mathbf y^i_t} - {\\mathbf y_t} \\right\\|^2 \\right)$, $\\Delta_{t}^{\\bdx} \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left\\| {\\mathbf d^i_{x,t}} - {\\mathbf d_{x,t}} \\right\\|^2$ and $\\Delta_{t}^{\\bdy} \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left\\| {\\mathbf d^i_{y,t}} - {\\mathbf d_{y,t}} \\right\\|^2$.\n\nFirst we prove \\eqref{eq:lem:NC_PL_mom_xy_cons_errs_recursion}.\n\\begin{align}\n    \\Delta_{t+1}^{\\bx,\\by} & \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left( \\left\\| {\\mathbf x^i_{t+1}} - {\\mathbf x_{t+1}} \\right\\|^2 + \\left\\| {\\mathbf y^i_{t+1}} - {\\mathbf y_{t+1}} \\right\\|^2 \\right) \\nonumber \\\\\n    & = \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left( \\left\\| \\left( {\\mathbf x^i_t} - {\\mathbf x_t} \\right) - \\eta_x \\alpha_t \\left( {\\mathbf d^i_{x,t}} - {\\mathbf d_{x,t}} \\right) \\right\\|^2 + \\left\\| \\left( {\\mathbf y^i_t} - {\\mathbf y_t} \\right) + \\eta_y\\alpha_t \\left( {\\mathbf d^i_{y,t}} - {\\mathbf d_{y,t}} \\right) \\right\\|^2 \\right)\n   \n    \\tag{from \\eqref{eq:NC_mom_update_avg}} \\\\\n    & \\leq \\frac{1}{n} \\sum_{i=1}^n \\left[ (1 + c_1) \\mathbb E \\left( \\left\\| {\\mathbf x^i_t} - {\\mathbf x_t} \\right\\|^2 + \\left\\| {\\mathbf y^i_t} - {\\mathbf y_t} \\right\\|^2 \\right) + \\alpha_t^2 \\left( 1 + \\dfrac{1}{c_1} \\right) \\mathbb E \\left( \\eta_x^2 \\left\\| {\\mathbf d^i_{x,t}} - {\\mathbf d_{x,t}} \\right\\|^2 + \\eta_y^2 \\left\\| {\\mathbf d^i_{y,t}} - {\\mathbf d_{y,t}} \\right\\|^2 \\right) \\right]\n   \n    \\tag{from \\cref{lem:Young}, with $\\gamma = c_1$}\n    \\\\\n    &= (1+c_1) \\Delta_{t}^{\\bx,\\by} + \\left( 1 + \\dfrac{1}{c_1} \\right) \\alpha_t^2 \\left( \\eta_x^2 \\Delta_{t}^{\\bdx} + \\eta_y^2 \\Delta_{t}^{\\bdy} \\right). \\nonumber\n\\end{align}\nNext, we prove \\eqref{eq:lem:NC_PL_mom_p_cons_errs_recursion}.\nThe proof of \\eqref{eq:lem:NC_PL_mom_q_cons_errs_recursion} is analogous, so we skip it here.\n\\begin{align}\n    & \\Delta_{t+1}^{\\bdx} \\triangleq \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left\\| {\\mathbf d^i_{x,t+1}} - {\\mathbf d_{x,t+1}} \\right\\|^2 \\nonumber \\\\\n    &= \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left\\| (1 - \\beta_x \\alpha_t) \\left( {\\mathbf d^i_{x,t}} - {\\mathbf d_{x,t}} \\right) + \\beta_x \\alpha_t \\Big( \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}) - \\frac{1}{n} \\sum_{j=1}^n \\nabla_{\\bx} f_j ({\\mathbf x^j_{t+1}}, {\\mathbf y^j_{t+1}}; {\\xi^j_{t+1}}) \\Big) \\right\\|^2 \\tag{from \\eqref{eq:NC_mom_update_avg}} \\\\\n    & \\leq (1 + c_2) (1 - \\beta_x \\alpha_t)^2 \\Delta_{t}^{\\bdx} + \\left( 1 + \\dfrac{1}{c_2} \\right) \\frac{\\beta_x^2 \\alpha_t^2}{n} \\sum_{i=1}^n \\mathbb E \\left\\| \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}) - \\frac{1}{n} \\sum_{j=1}^n \\nabla_{\\bx} f_j ({\\mathbf x^j_{t+1}}, {\\mathbf y^j_{t+1}}; {\\xi^j_{t+1}}) \\right\\|^2 \n   \n    \\tag{\\cref{lem:Young} (with $\\gamma = c_2$)} \\\\\n    & \\overset{(a)}{=} (1 - \\beta_x \\alpha_t) \\Delta_{t}^{\\bdx} \\nonumber \\\\\n    & \\ + \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\Bigg\\| \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}) - \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) + \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) - \\nabla_{\\bx} f_i ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) + \\nabla_{\\bx} f_i ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\nonumber \\\\\n    & \\quad - \\frac{1}{n} \\sum_{j=1}^n \\left( \\nabla_{\\bx} f_j ({\\mathbf x^j_{t+1}}, {\\mathbf y^j_{t+1}}; {\\xi^j_{t+1}}) - \\nabla_{\\bx} f_j ({\\mathbf x^j_{t+1}}, {\\mathbf y^j_{t+1}}) + \\nabla_{\\bx} f_j ({\\mathbf x^j_{t+1}}, {\\mathbf y^j_{t+1}}) - \\nabla_{\\bx} f_j ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) + \\nabla_{\\bx} f_j ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\right) \\Bigg\\|^2 \\nonumber\n   \n    \\\\\n    & \\overset{(b)}{\\leq} (1 - \\beta_x \\alpha_t) \\Delta_{t}^{\\bdx} + \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\Bigg[ \\left\\| \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}}) - \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) \\right\\|^2 \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad + \\left\\| \\frac{1}{n} \\sum_{j=1}^n \\left( \\nabla_{\\bx} f_j ({\\mathbf x^j_{t+1}}, {\\mathbf y^j_{t+1}}; {\\xi^j_{t+1}}) - \\nabla_{\\bx} f_j ({\\mathbf x^j_{t+1}}, {\\mathbf y^j_{t+1}}) \\right) \\right\\|^2 \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad + \\Big\\| \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) - \\nabla_{\\bx} f_i ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) + \\nabla_{\\bx} f_i ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad - \\frac{1}{n} \\sum_{j=1}^n \\left( \\nabla_{\\bx} f_j ({\\mathbf x^j_{t+1}}, {\\mathbf y^j_{t+1}}) - \\nabla_{\\bx} f_j ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\right) - \\nabla_{\\bx} f ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\Big\\|^2 \\Bigg]\n   \n    \\nonumber \\\\\n    & \\overset{(c)}{\\leq} (1 - \\beta_x \\alpha_t) \\Delta_{t}^{\\bdx} + \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n \\Bigg[ \\sigma^2 + \\frac{\\sigma^2}{n} + 3 \\mathbb E \\left\\| \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}) - \\nabla_{\\bx} f_i ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\right\\|^2 \\nonumber \\\\\n    & + 3 \\mathbb E \\left\\| \\nabla_{\\bx} f_i ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) - \\nabla_{\\bx} f ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\right\\|^2 + 3 \\mathbb E \\left\\| \\frac{1}{n} \\sum_{j=1}^n \\left( \\nabla_{\\bx} f_j ({\\mathbf x^j_{t+1}}, {\\mathbf y^j_{t+1}}) - \\nabla_{\\bx} f_j ({\\mathbf x_{t+1}}, {\\mathbf y_{t+1}}) \\right) \\right\\|^2 \\Bigg]\n   \n    \\nonumber \\\\\n    & \\overset{(d)}{\\leq} (1 - \\beta_x \\alpha_t) \\Delta_{t}^{\\bdx} + \\beta_x \\alpha_t \\frac{1}{n} \\sum_{i=1}^n \\Bigg[ \\sigma^2 + \\frac{\\sigma^2}{n} + 3 L_f^2 \\mathbb E \\left( \\left\\| {\\mathbf x^i_{t+1}} - {\\mathbf x_{t+1}} \\right\\|^2 + \\left\\| {\\mathbf y^i_{t+1}} - {\\mathbf y_{t+1}} \\right\\|^2 \\right) + 3 \\varsigma_x^2 \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\qquad + 3 L_f^2 \\frac{1}{n} \\sum_{j=1}^n \\mathbb E \\left( \\left\\| {\\mathbf x^j_{t+1}} - {\\mathbf x_{t+1}} \\right\\|^2 + \\left\\| {\\mathbf y^j_{t+1}} - {\\mathbf y_{t+1}} \\right\\|^2 \\right) \\Bigg]\n   \n    \\nonumber \\\\\n    &= (1 - \\beta_x \\alpha_t) \\Delta_{t}^{\\bdx} + 6 \\beta_x \\alpha_t L_f^2 \\Delta_{t+1}^{\\bx,\\by} + \\beta_x \\alpha_t \\left[ \\sigma^2 \\left( 1 + \\dfrac{1}{n} \\right) + 3 \\varsigma_x^2 \\right]. \\nonumber\n\\end{align}\nIn $(a)$ we choose $c_2$ such that $\\left( 1 + \\frac{1}{c_2} \\right) \\beta_x \\alpha_t = 1$, i.e., $c_2 = \\frac{\\beta_x \\alpha_t}{1 - \\beta_x \\alpha_t}$ and $(1-\\beta_x \\alpha_t) (1 + c_2) = 1$; \n$(b)$ follows from Assumption \\ref{assum:bdd_var} (unbiasedness of stochastic gradients);\n$(c)$ follows from Assumption \\ref{assum:bdd_var} (bounded variance of stochastic gradients, and independence of stochastic gradients across clients), and the generic sum of squares inequality in \\cref{lem:sum_of_squares};\n$(d)$ follows from \\cref{assum:smoothness} ($L_f$-smoothness of $f_i$) \\cref{assum:bdd_hetero} (bounded heterogeneity across clients).\n\nFinally, we choose $\\beta_x = \\beta$.\nThis concludes the proof of \\eqref{eq:lem:NC_PL_mom_p_cons_errs_recursion}.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of \\cref{lem:NC_PL_mom_induct_bd_cons_error_xy}]\nSubstituting \\eqref{eq:lem:NC_PL_mom_p_cons_errs_recursion}, \\eqref{eq:lem:NC_PL_mom_q_cons_errs_recursion} from Lemma \\ref{lem:NC_PL_mom_cons_errs_recursion} in \\eqref{eq:lem:NC_PL_mom_xy_cons_errs_recursion}, we get\n\\begin{equation}\n    \\begin{aligned}\n        \\Delta_{t+1}^{\\bx,\\by} & \\leq \\left\\{ 1+c_1 + \\left( 1 + \\mfrac{1}{c_1} \\right) 6 L_f^2 \\beta \\alpha^3 (\\eta_x^2 + \\eta_y^2) \\right\\} \\Delta_{t}^{\\bx,\\by} + \\left( 1 + \\mfrac{1}{c_1} \\right) \\alpha^2 (1 - \\beta \\alpha) \\left( \\eta_x^2 \\Delta_{t-1}^{\\bdx} + \\eta_y^2 \\Delta_{t-1}^{\\bdy} \\right) \\\\\n        & \\qquad + \\left( 1 + \\mfrac{1}{c_1} \\right) \\beta \\alpha^3 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\mfrac{1}{n} \\right) + 3 \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right].\n    \\end{aligned}\n    \\label{eq_proof:lem:NC_PL_mom_induct_bd_1}\n\\end{equation}\nUsing $c_1 = \\frac{\\beta \\alpha}{1 - \\beta \\alpha}$ in \\eqref{eq_proof:lem:NC_PL_mom_induct_bd_1} gives us\n\\begin{align}\n    \\Delta_{t+1}^{\\bx,\\by} & \\leq \\left\\{ 1+c_1 + 6 L_f^2 \\alpha^2 (\\eta_x^2 + \\eta_y^2) \\right\\} \\Delta_{t}^{\\bx,\\by} + \\frac{\\alpha}{\\beta} (1 - \\beta \\alpha) \\left( \\eta_x^2 \\Delta_{t-1}^{\\bdx} + \\eta_y^2 \\Delta_{t-1}^{\\bdy} \\right) \\nonumber \\\\\n    & \\qquad + \\alpha^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\mfrac{1}{n} \\right) + 3 \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right] \\nonumber \\\\\n    & = \\left( 1 + \\theta \\right) \\Delta_{t}^{\\bx,\\by} + \\frac{\\alpha}{\\beta} (1 - \\beta \\alpha) \\left( \\eta_x^2 \\Delta_{t-1}^{\\bdx} + \\eta_y^2 \\Delta_{t-1}^{\\bdy} \\right) + \\Upsilon, \\label{eq_proof:lem:NC_PL_mom_induct_bd_2}\n\\end{align}\nwhere we define $\\theta \\triangleq c_1 + 6 L_f^2 \\alpha^2 (\\eta_x^2 + \\eta_y^2)$.\n\nNow, we proceed to prove the induction. For $k=1$, it follows from \\eqref{eq_proof:lem:NC_PL_mom_induct_bd_2} that \\eqref{eq:lem:NC_PL_mom_induct_bd_cons_error_xy} holds. Next, we assume the induction hypothesis in \\eqref{eq:lem:NC_PL_mom_induct_bd_cons_error_xy} holds for some $k > 1$ (assuming $t-1-k \\geq s \\tau + 1$). We prove that it also holds for $k+1$.\n\\begin{align}\n    \\Delta_{t}^{\\bx,\\by} & \\leq (1 + 2 k \\theta) \\Delta_{t-k}^{{\\mathbf x},{\\mathbf y}} + 2 k \\mfrac{\\alpha}{\\beta} (1-\\beta \\alpha) \\left( \\eta_x^2 \\Delta_{t-k-1}^{{\\mathbf d_x}} + \\eta_y^2 \\Delta_{t-k-1}^{{\\mathbf d_y}} \\right) + k^2 (1+\\theta) \\Upsilon \\tag{Induction hypothesis} \\\\\n    & \\leq \\left\\{ (1 + 2 k \\theta) (1 + \\theta) + 2 k \\mfrac{\\alpha}{\\beta} (1-\\beta \\alpha) (\\eta_x^2 + \\eta_y^2) 6 L_f^2 \\beta \\alpha \\right\\} \\Delta_{t-k-1}^{x,y} \\tag{\\cref{lem:NC_PL_mom_cons_errs_recursion}, \\eqref{eq_proof:lem:NC_PL_mom_induct_bd_2}} \\\\\n    & \\quad + \\left\\{ (1 + 2 k \\theta) \\frac{\\alpha}{\\beta} (1 - \\beta \\alpha) + 2 k \\mfrac{\\alpha}{\\beta} (1-\\beta \\alpha)^2 \\right\\} \\left( \\eta_x^2 \\Delta_{t-k-2}^{{\\mathbf d_x}} + \\eta_y^2 \\Delta_{t-k-2}^{{\\mathbf d_y}} \\right) \\nonumber \\\\\n    & \\quad + \\left[ 1 + 2k \\theta + k^2 (1 + \\theta) \\right] \\Upsilon + 2 k \\mfrac{\\alpha}{\\beta} (1-\\beta \\alpha) \\beta \\alpha \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\mfrac{1}{n} \\right) + 3 \\eta_x^2 \\varsigma_x^2 + 3 \\eta_y^2 \\varsigma_y^2 \\right] \\nonumber \\\\\n    & \\leq \\left\\{ (1 + 2 k \\theta) (1 + \\theta) + 2 k (1-\\beta \\alpha) (\\theta - c_1) \\right\\} \\Delta_{t-k-1}^{x,y} \\tag{see definition of $\\theta$ in \\cref{lem:NC_PL_mom_induct_bd_cons_error_xy}} \\\\\n    & \\quad + \\left[ 1 + 2 k \\theta + 2 k (1-\\beta \\alpha) \\right] \\frac{\\alpha}{\\beta} (1 - \\beta \\alpha) \\left( \\eta_x^2 \\Delta_{t-k-2}^{{\\mathbf d_x}} + \\eta_y^2 \\Delta_{t-k-2}^{{\\mathbf d_y}} \\right) \\nonumber \\\\\n    & \\quad + \\left[ 1 + 2k \\theta + k^2 (1 + \\theta) + 2 k (1-\\beta \\alpha) \\right] \\Upsilon. \\tag{see definition of $\\Upsilon$ in \\cref{lem:NC_PL_mom_induct_bd_cons_error_xy}}\n\\end{align}\nNext, we see how the parameter choices in \\cref{lem:NC_PL_mom_induct_bd_cons_error_xy} satisfy the induction hypothesis.\nBasically, we need to satisfy the following three conditions:\n\\begin{align}\n    \\begin{aligned}\n        (1 + 2 k \\theta) (1 + \\theta) + 2 k (1-\\beta \\alpha) (\\theta - c_1) & \\leq 1 + 2 (k+1) \\theta, \\\\\n        1 + 2 k \\theta + 2 k (1-\\beta \\alpha) & \\leq 2(k+1), \\\\\n        1 + 2k \\theta + k^2 (1 + \\theta) + 2 k (1-\\beta \\alpha) & \\leq (k+1)^2 (1+\\theta).\n    \\end{aligned}\n    \\label{eq:NC_PL_mom_induct_bd_cons_error_xy:param_condition}\n\\end{align}\n\\begin{enumerate}\n    \\item The first condition in \\eqref{eq:NC_PL_mom_induct_bd_cons_error_xy:param_condition} is equivalent to\n    \\begin{align}\n        \\theta + 2 k \\theta^2 + 2 k (1-\\beta \\alpha) (\\theta - c_1) & \\leq 2 \\theta. \\label{eq:cond_theta_1}\n    \\end{align}\n    Recall that in \\cref{lem:NC_PL_mom_induct_bd_cons_error_xy}, $\\theta - c_1 = 6 L_f^2 \\alpha^2 (\\eta_y^2 + \\eta_x^2)$. \n    If $6 L_f^2 \\alpha^2 (\\eta_y^2 + \\eta_x^2) \\leq \\min \\{ c_1, \\theta^2 \\}$, a \\textit{sufficient} condition for \\eqref{eq:cond_theta_1} is\n    \\begin{align*}\n        4 k \\theta^2 \\leq \\theta \\quad \\Rightarrow \\quad \\theta \\leq 1\/4k.\n    \\end{align*}\n    Since $\\theta \\leq 2 c_1$ and $c_1 \\leq 2 \\beta \\alpha$ (if $\\alpha \\leq 1\/(2 \\beta)$), this is satisfied if $\\alpha \\leq \\frac{1}{16 \\beta k}$.\n    Next, we verify that $6 L_f^2 \\alpha^2 (\\eta_y^2 + \\eta_x^2) \\leq \\min \\{ c_1, \\theta^2 \\}$ holds.\n    \\begin{itemize}\n        \\item $6 L_f^2 \\alpha^2 (\\eta_y^2 + \\eta_x^2) \\leq c_1$ follows from the condition $\\alpha \\leq \\frac{\\beta}{6 L_f^2 (\\eta_y^2 + \\eta_x^2)}$ (since $c_1 \\geq \\beta \\alpha$).\n        \\item $6 L_f^2 \\alpha^2 (\\eta_y^2 + \\eta_x^2) \\leq \\theta^2$ follows from the condition $L_f^2 (\\eta_y^2 + \\eta_x^2) \\leq \\frac{\\beta^2}{6}$ (since $\\theta \\geq c_1 \\geq \\alpha \\beta$).\n    \\end{itemize}\n   \n        \n   \n   \n   \n   \n   \n    \\item The second condition in \\eqref{eq:NC_PL_mom_induct_bd_cons_error_xy:param_condition} is equivalent to\n    \\begin{align*}\n        2k (\\theta - \\beta \\alpha) \\leq 1.\n    \\end{align*}\n    A \\textit{sufficient} condition for this to be satisfied is $\\theta \\leq \\frac{1}{2k}$, which, as seen above, is already satisfied if $\\alpha \\leq \\frac{1}{16 \\beta k}$.\n    \\item The third condition in \\eqref{eq:NC_PL_mom_induct_bd_cons_error_xy:param_condition} is equivalent to\n    \\begin{align*}\n        1 + 2k \\theta + 2 k (1-\\beta \\alpha) & \\leq 2k (1+\\theta) + (1+\\theta) \\\\\n        \\Leftrightarrow - 2k \\beta \\alpha & \\leq \\theta.\n    \\end{align*}\n    which is trivially satisfied.\n\\end{enumerate}\nHence, the parameter choices in \\cref{lem:NC_PL_mom_induct_bd_cons_error_xy} satisfy the induction hypothesis, which completes the proof.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of \\cref{cor:NC_PL_mom_induct_bd_cons_error_xy}]\nFor $k = k_0$ such that $(t-k_0-1) \\mod \\tau = 0$, then by \\cref{alg_NC_momentum} $$\\Delta_{t-k_0-1}^{{\\mathbf x}, {\\mathbf y}} = \\Delta_{t-k_0-1}^{{\\mathbf d_x}} = \\Delta_{t-k_0-1}^{{\\mathbf d_y}} = 0.$$\nFrom \\cref{lem:NC_PL_mom_cons_errs_recursion}, $\\Delta_{t-k_0}^{{\\mathbf x}, {\\mathbf y}} = 0$. \nUsing this information in \\cref{lem:NC_PL_mom_induct_bd_cons_error_xy}, we get\n\\begin{align*}\n    \\Delta_{t}^{\\bx,\\by} & \\leq (1 + 2 k_0 \\theta) \\Delta_{t-k_0}^{x,y} + k_0^2 (1+\\theta) \\Upsilon \\\\\n    & \\leq (\\tau - 1)^2 \\alpha^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\mfrac{1}{n} \\right) + 3 \\eta_x^2 \\varsigma_x^2 + 3 \\eta_y^2 \\varsigma_y^2 \\right]. \\tag{Using $\\Upsilon$ from \\cref{lem:NC_PL_mom_induct_bd_cons_error_xy}}\n\\end{align*}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\section{Nonconvex-Concave Functions: Local SGDA+ (\\texorpdfstring{\\cref{thm:NC_C}}{Theorem 1})} \\label{app:NC_C}\n\n\n\\begin{algorithm}[ht]\n\\caption{Local SGDA+ \\cite{mahdavi21localSGDA_aistats}}\n\\label{alg_local_SGDA_plus}\n\\begin{algorithmic}[1]\n\t\\STATE{\\textbf{Input: }{\\small${\\mathbf x}_0^i = \\widetilde{\\bx}_0 = {\\mathbf x}_0, {\\mathbf y}_0^i = {\\mathbf y}_0$}, for all $i \\in [n]$; step-sizes $\\eta_x, \\eta_y$; $\\tau$, $T$, $S, k=0$}\n\t\\FOR[At all clients $i=1,\\hdots, n$]{$t=0$ to $T-1$}\n\t    \\STATE{Sample minibatch ${\\xi^i_{t}}$ from local data}\n        \\STATE{${\\mathbf x^i_{t+1}} = {\\mathbf x^i_t} - \\eta_x \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}})$}\n        \\STATE{${\\mathbf y^i_{t+1}} = {\\mathbf y^i_t} + \\eta_y \\nabla_{\\by} f_i (\\Tbx_{k}, {\\mathbf y^i_t}; {\\xi^i_{t}})$}\n        \\IF{$t+1$ mod $\\tau = 0$}\n            \\STATE{Clients send $\\{ {\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}} \\}$ to the server}\n            \\STATE{Server computes averages ${\\mathbf x_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_{t+1}}$, \n            ${\\mathbf y_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf y^i_{t+1}}$, and sends to all the clients}\n            \\STATE{${\\mathbf x^i_{t+1}} = {\\mathbf x_{t+1}}$, ${\\mathbf y^i_{t+1}} = {\\mathbf y_{t+1}}$, for all $i \\in [n]$}\n        \\ENDIF\n        \\IF{$t+1$ mod $S = 0$}\n            \\STATE{Clients send $\\{ {\\mathbf x^i_{t+1}} \\}$ to the server}\n            \\STATE{$k \\gets k+1$}\n            \\STATE{Server computes averages $\\Tbx_{k} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_{t+1}}$, and sends to all the clients}\n        \\ENDIF\n\t\\ENDFOR\n\t\\STATE{\\textbf{Return: }${\\bar{\\bx}_T}$ drawn uniformly at random from $\\{ {\\mathbf x_t} \\}$, where ${\\mathbf x_t} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_t}$}\n\\end{algorithmic}\n\\end{algorithm}\n\nWe organize this section as follows. First, in \\cref{sec:NC_C_int_results} we present some intermediate results, which we use in the proof of \\cref{thm:NC_C}. Next, in \\cref{sec:NC_C_thm_proof}, we present the proof of \\cref{thm:NC_C}, which is followed by the proofs of the intermediate results in \\cref{sec:NC_C_int_results_proofs}.\n\n\\subsection{Intermediate Lemmas} \\label{sec:NC_C_int_results}\n\n\\begin{lemma}\n\\label{lem:NC_C_Phi_smooth_decay_one_iter}\nSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, \\ref{assum:concavity}, \\ref{assum:Lips_cont_x}.\nThen, the iterates generated by \\cref{alg_local_SGDA_plus} satisfy\n\\begin{align}\n    \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_{t+1}}) \\right] & \\leq \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_t}) \\right] + \\eta_x^2 L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + 2 \\eta_x L_f^2 \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\quad + 2 \\eta_x L_f \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] - \\frac{\\eta_x}{8} \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2. \\nonumber\n\\end{align}\nwhere $\\Delta_{t}^{\\bx,\\by} = \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left( \\left\\| {\\mathbf x^i_t} - {\\mathbf x_t} \\right\\|^2 + \\left\\| {\\mathbf y^i_t} - {\\mathbf y_t} \\right\\|^2 \\right)$ is the synchronization error at time $t$.\n\\end{lemma}\n\nNext, we bound the difference $\\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right]$.\n\n\\begin{lemma}\n\\label{lem:NC_C_Phi_f_diff}\nSuppose the local functions satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, \\ref{assum:Lips_cont_x}.\nFurther, suppose we choose the step-size $\\eta_y$ such that $\\eta_y \\leq \\frac{1}{8 L_f \\tau}$.\nThen the iterates generated by \\cref{alg_local_SGDA_plus} satisfy\n\\begin{align}\n    \\frac{1}{T} \\sumtT \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] & \\leq 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y S} + \\frac{20 \\eta_y \\sigma^2}{n} + 16 \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right). \\nonumber\n\\end{align}\n\\end{lemma}\n\n\n\\begin{lemma}\n\\label{lem:NC_C_consensus_error}\nSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_hetero},\nand the stochastic oracles for the local\nfunctions satisfy \\cref{assum:bdd_var}.\nFurther, in \\cref{alg_local_SGDA}, we choose step-sizes $\\eta_x, \\eta_y \\leq \\frac{1}{8 \\tau L_f}$.\nThen, the iterates $\\{ {\\mathbf x^i_t}, {\\mathbf y^i_t} \\}$ generated by \\cref{alg_local_SGDA_plus} satisfy\n\\begin{align}\n    \\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\by} & \\triangleq \\frac{1}{T} \\sum_{t=0}^{T-1} \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left( \\left\\| {\\mathbf y^i_t} - {\\mathbf y_t} \\right\\|^2 \\right) \\leq 2 (\\tau-1)^2 \\eta_y^2 \\left[ \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\varsigma_y^2 \\right], \\nonumber \\\\\n    \\frac{1}{T} \\sum_{t=0}^{T-1} \\Delta_{t}^{\\bx} & \\triangleq \\frac{1}{T} \\sum_{t=0}^{T-1} \\frac{1}{n} \\sum_{i=1}^n \\mathbb E \\left( \\left\\| {\\mathbf x^i_t} - {\\mathbf x_t} \\right\\|^2 \\right) \\leq 2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3\\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) \\right]. \\nonumber\n    \\label{eq:lem:NC_C_consensus_error}\n\\end{align}\n\\end{lemma}\n\n\n\n\\subsection{Proof of \\texorpdfstring{\\cref{thm:NC_C}}{Theorem 3}}\n\\label{sec:NC_C_thm_proof}\nFor the sake of completeness, we first state the full statement of \\cref{thm:NC_C} here.\n\n\\begin{theorem*}\nSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, \\ref{assum:concavity}, \\ref{assum:Lips_cont_x}.\nFurther, let $\\norm{{\\mathbf y_t}}^2 \\leq D$ for all $t$.\nSuppose the step-sizes $\\eta_x, \\eta_y$ are chosen such that $\\eta_x, \\eta_y \\leq \\frac{1}{8 L_f \\tau}$.\nThen the iterates generated by \\cref{alg_local_SGDA_plus} satisfy\n\\begin{align}\n    \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 & \\leq \\frac{8 \\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + \\frac{320 \\eta_y L_f \\sigma^2}{n} + 16 L_f \\left[ 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y S} \\right] \\nonumber \\\\\n    & \\quad + 64 L_f^2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) + 4 \\eta_y^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) \\right]. \\nonumber\n   \n\\end{align}\nWith the following parameter values:\n\\begin{align*}\n    \\eta_x = \\Theta \\left( \\frac{n^{1\/4}}{T^{3\/4}} \\right), \\qquad \\eta_y = \\Theta \\left( \\frac{n^{3\/4}}{T^{1\/4}} \\right), \\qquad S = \\Theta \\left( \\sqrt{\\frac{T}{n}} \\right),\n\\end{align*}\nwe can further simplify to\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\mathcal O \\left( \\frac{1}{(nT)^{1\/4}} \\right) + \\mathcal O \\left( \\frac{n^{1\/4}}{T^{3\/4}} \\right) + \\mathcal O \\left( \\frac{n^{3\/2} (\\tau-1)^2}{T^{1\/2}} \\right) + \\mathcal O \\left( (\\tau-1)^2 \\frac{\\sqrt{n}}{T^{3\/2}} \\right). \\nonumber\n   \n\\end{align}\n\\end{theorem*}\n\n\n\\begin{proof}\nWe sum the result in \\cref{lem:NC_C_Phi_smooth_decay_one_iter} over $t = 0$ to $T-1$ and rearrange the terms to get\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\frac{8}{\\eta_x} \\frac{1}{T} \\sumtT \\left( \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_t}) \\right] - \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_{t+1}}) \\right] \\right) + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + 16 L_f \\frac{1}{T} \\sumtT \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + 16 L_f^2 \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\leq \\frac{8}{\\eta_x T} \\left[ \\Phi_{1\/2L_f} ({\\mathbf x}_0) - \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x}_T) \\right] \\right] + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + 16 L_f^2 \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\quad + 16 L_f \\left[ 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y S} + \\frac{20 \\eta_y \\sigma^2}{n} + 16 \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) \\right] \\tag{\\cref{lem:NC_C_Phi_f_diff}} \\\\\n    & \\leq \\frac{8 \\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + \\frac{320 \\eta_y L_f \\sigma^2}{n} + 16 L_f \\left[ 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y S} \\right] \\nonumber \\\\\n    & \\quad + 64 L_f^2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) + 4 \\eta_y^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) \\right], \\tag{\\cref{lem:NC_C_consensus_error}}\n\\end{align}\nwhere $\\widetilde{\\Delta}_{\\Phi} = \\Phi_{1\/2L_f} ({\\mathbf x}_0) - \\min_{\\mathbf x} \\Phi_{1\/2L_f} ({\\mathbf x})$.\n\nIf $D = 0$, we let $S=1$. Else, let $S = \\sqrt{\\frac{2 D}{\\eta_x \\eta_y G_x \\sqrt{G_x^2 + \\sigma^2\/n}}}$. Then we get\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\frac{8 \\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + \\frac{320 \\eta_y L_f \\sigma^2}{n} + 64 L_f \\sqrt{\\frac{2 D \\eta_x G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}}}{\\eta_y}} \\nonumber \\\\\n    & \\quad + 64 L_f^2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) + 4 \\eta_y^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) \\right], \\label{eq_proof:thm_NC_C_1}\n\\end{align}\nFor $\\eta_y \\leq 1$, the terms containing $\\eta_y^2$ are of higher order, and we focus only on the other terms containing $\\eta_y$, i.e., \n\\begin{align*}\n    64 L_f \\left[ \\frac{5 \\eta_y \\sigma^2}{n} + \\sqrt{\\frac{2 D \\eta_x G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}}}{\\eta_y}} \\right].\n\\end{align*}\nTo optimize these, we choose $\\eta_y = \\left( \\frac{n}{10 \\sigma^2} \\right)^{2\/3} \\left( 2 D \\eta_x G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{1\/3}$. Substituting in \\eqref{eq_proof:thm_NC_C_1}, we get\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\frac{8 \\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + 320 L_f \\left( 10 \\frac{\\sigma^2}{n} D \\eta_x G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{1\/3} \\nonumber \\\\\n    & \\quad + 200 L_f^2 (\\tau-1)^2 \\left[ 4 \\eta_x^{2\/3} \\left( \\frac{n}{10 \\sigma^2} \\right)^{4\/3} \\left( 2 D G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{2\/3} \\left( \\sigma^2 + \\varsigma_y^2 \\right) + \\eta_x^2 \\left( \\sigma^2 + \\varsigma_x^2 \\right) \\right],\n    \\label{eq_proof:thm_NC_C_2}\n\\end{align}\nAgain, we ignore the higher order terms of $\\eta_x$, and only focus on\n\\begin{align*}\n    \\frac{8 \\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + 320 L_f \\left( 10 \\frac{\\sigma^2}{n} D \\eta_x G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{1\/3}.\n\\end{align*}\nWith $\\eta_x = \\left( \\frac{3}{40 L_f T} \\right)^{3\/4} \\left( 10 \\frac{\\sigma^2}{n} D G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{-1\/4}$,\nand absorbing numerical constants inside $\\mathcal O (\\cdot)$ we get,\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\mathcal O \\left( \\left( \\sigma^2 D G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{1\/4} \\frac{L_f^{3\/4}}{(nT)^{1\/4}} \\right) \\nonumber \\\\\n    & \\quad + \\mathcal O \\left( \\frac{L_f^{1\/4}}{T^{3\/4}} \\left( \\frac{\\sigma^2}{n} D G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{-1\/4} \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) \\right) \\nonumber \\\\\n    & \\quad + \\mathcal O \\left( \\frac{L_f^{3\/2} (\\tau-1)^2}{T^{1\/2}} \\left( \\frac{n}{\\sigma^2} \\right)^{3\/2} \\left( D G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{1\/2} \\left( \\sigma^2 + \\varsigma_y^2 \\right) \\right), \\nonumber \\\\\n    & \\quad + \\mathcal O \\left( (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_x^2 \\right) \\frac{\\sqrt{L_f}}{T^{3\/2}} \\left( \\frac{\\sigma^2}{n} D G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{-1\/2} \\right), \\label{eq_proof:thm_NC_C_3} \\\\\n    & \\leq \\mathcal O \\left( \\frac{\\sigma^2 + D + G_x^2}{(nT)^{1\/4}} \\right) + \\mathcal O \\left( \\frac{n^{1\/4}}{T^{3\/4}} \\right) + \\mathcal O \\left( \\frac{n^{3\/2} (\\tau-1)^2}{T^{1\/2}} \\right) + \\mathcal O \\left( (\\tau-1)^2 \\frac{\\sqrt{n}}{T^{3\/2}} \\right),\n    \\label{eq_proof:thm_NC_C_4}\n\\end{align}\nwhere in \\eqref{eq_proof:thm_NC_C_4}, we have dropped all the problem-specific parameters, to show dependence only on $\\tau, n, T$.\n\nLastly, we specify the algorithm parameters in terms of $n,T$.\n\\begin{itemize}\n    \\item $\\eta_x = \\left( \\frac{3}{40 L_f T} \\right)^{3\/4} \\left( 10 \\frac{\\sigma^2}{n} D G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{-1\/4} = \\Theta \\left( \\frac{n^{1\/4}}{T^{3\/4}} \\right)$,\n    \\item $\\eta_y = \\left( \\frac{n}{10 \\sigma^2} \\right)^{2\/3} \\left( 2 D \\eta_x G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} \\right)^{1\/3} = \\Theta \\left( \\frac{n^{3\/4}}{T^{1\/4}} \\right)$,\n    \\item $S = \\sqrt{\\frac{2 D}{\\eta_x \\eta_y G_x \\sqrt{G_x^2 + \\sigma^2\/n}}} = \\Theta \\left( \\sqrt{\\frac{T}{n}} \\right)$.\n\\end{itemize}\n\n\\end{proof}\n\n\n\\begin{proof}[Proof of \\cref{cor:NC_C_comm_cost}]\nWe assume $T \\geq n^7$.\nTo reach an $\\epsilon$-accurate point, i.e., ${\\bar{\\bx}_T}$ such that $\\mathbb E \\left\\| \\nabla \\Phi_{1\/2L_f} ({\\bar{\\bx}_T}) \\right\\| \\leq \\epsilon$, we need\n\\begin{align*}\n    \\mathbb E \\left\\| \\nabla \\Phi_{1\/2L_f} ({\\bar{\\bx}_T}) \\right\\| & \\leq \\left[ \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left\\| \\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t}) \\right\\|^2 \\right]^{1\/2} \\nonumber \\\\\n    & \\leq \\mathcal O \\left( \\frac{1}{(nT)^{1\/8}} \\right) + \\mathcal O \\left( \\frac{n^{1\/8}}{T^{3\/8}} \\right) + \\mathcal O \\left( \\frac{n^{3\/4} (\\tau-1)}{T^{1\/4}} \\right) + \\mathcal O \\left( (\\tau-1) \\frac{n^{1\/4}}{T^{3\/4}} \\right).\n\\end{align*}\nWe can choose $\\tau \\leq \\mathcal O \\left( \\frac{T^{1\/8}}{n^{7\/8}} \\right)$ without affecting the convergence rate $\\mathcal O \\left( \\frac{1}{(nT)^{1\/8}} \\right)$.\nIn that case, we need $T = \\mathcal O \\left( \\frac{1}{n \\epsilon^8} \\right)$ iterations to reach an $\\epsilon$-accurate point.\nAnd the minimum number of communication rounds is \n$$\\mathcal O \\left( \\frac{T}{\\tau} \\right) = \\mathcal O \\left( (n T)^{7\/8} \\right) = \\mathcal O \\left( \\frac{1}{\\epsilon^7} \\right).$$\n\\end{proof}\n\n\n\n\n\\subsection{Proofs of the Intermediate Lemmas}\n\\label{sec:NC_C_int_results_proofs}\n\n\\begin{proof}[Proof of \\cref{lem:NC_C_Phi_smooth_decay_one_iter}]\nWe borrow the proof steps from \\cite{lin_GDA_icml20, mahdavi21localSGDA_aistats}. Define $\\widetilde{{\\mathbf x}}_{t} = \\operatornamewithlimits{arg\\,min}_{\\mathbf x} \\Phi ({\\mathbf x}) + L_f \\norm{{\\mathbf x} - {\\mathbf x_t}}^2$, then using the definition of $\\Phi_{1\/2L_f}$, we get\n\\begin{align}\n    \\Phi_{1\/2L_f} ({\\mathbf x_{t+1}}) & \\triangleq \\min_{\\mathbf x} \\Phi ({\\mathbf x}) + L_f \\norm{{\\mathbf x} - {\\mathbf x_{t+1}}}^2 \\nonumber \\\\\n    & \\leq \\Phi (\\widetilde{{\\mathbf x}}_t) + L_f \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_{t+1}}}^2.\n    \\label{eq:lem:NC_C_Phi_smooth_decay_one_iter_1}\n\\end{align}\nUsing the ${\\mathbf x^i_t}$ updates in \\cref{alg_local_SGDA_plus},\n\\begin{align}\n    & \\mathbb E \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_{t+1}}}^2 = \\mathbb E \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t} + \\eta_x \\frac{1}{n} \\sumin \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}})}^2 \\nonumber \\\\\n    &= \\mathbb E \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 + \\eta_x^2 \\mathbb E \\norm{\\frac{1}{n} \\sumin \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}})}^2 + 2 \\eta_x \\mathbb E \\left\\langle \\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}, \\frac{1}{n} \\sumin \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}) \\right\\rangle \\tag{\\cref{assum:bdd_var}} \\\\\n    & \\leq \\mathbb E \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 + \\eta_x^2 \\mathbb E \\norm{\\frac{1}{n} \\sumin \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + \\frac{\\eta_x^2 \\sigma^2}{n} + 2 \\eta_x \\mathbb E \\left\\langle \\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}, \\nabla_{\\bx} f ({\\mathbf x_t}, {\\mathbf y_t}) \\right\\rangle \\nonumber \\\\\n    & \\quad + \\eta_x \\mathbb E \\left[ \\frac{L_f}{2} \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 + \\frac{2}{L_f} \\norm{\\frac{1}{n} \\sumin \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}) - \\nabla_{\\bx} f ({\\mathbf x_t}, {\\mathbf y_t})}^2 \\right] \\tag{\\cref{lem:Young}} \\\\\n    & \\leq \\mathbb E \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 + \\eta_x^2 \\left( \\mathbb E \\norm{\\frac{1}{n} \\sumin \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 + \\frac{\\sigma^2}{n} \\right) + 2 \\eta_x \\mathbb E \\left\\langle \\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}, \\nabla_{\\bx} f ({\\mathbf x_t}, {\\mathbf y_t}) \\right\\rangle \\nonumber \\\\\n    & \\quad + \\frac{\\eta_x L_f}{2} \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 + 2 \\eta_x L_f \\Delta_{t}^{\\bx,\\by}\n    \\label{eq:lem:NC_C_Phi_smooth_decay_one_iter_2}\n\\end{align}\nwhere \\eqref{eq:lem:NC_C_Phi_smooth_decay_one_iter_2} follows from \\cref{assum:smoothness}.\nNext, we bound the inner product in \\eqref{eq:lem:NC_C_Phi_smooth_decay_one_iter_2}.\nUsing $L_f$-smoothness of $f$ (\\cref{assum:smoothness}):\n\\begin{align}\n    \\mathbb E \\left\\langle \\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}, \\nabla_{\\bx} f ({\\mathbf x_t}, {\\mathbf y_t}) \\right\\rangle & \\leq \\mathbb E \\left[ f(\\widetilde{{\\mathbf x}}_t, {\\mathbf y_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) + \\frac{L_f}{2} \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 \\right] \\nonumber \\\\\n    & \\leq \\mathbb E \\left[ \\Phi(\\widetilde{{\\mathbf x}}_t) - f({\\mathbf x_t}, {\\mathbf y_t}) + \\frac{L_f}{2} \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 \\right] \\nonumber \\\\\n    & = \\mathbb E \\left[ \\Phi(\\widetilde{{\\mathbf x}}_t) + L_f \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 \\right] - \\mathbb E f({\\mathbf x_t}, {\\mathbf y_t}) - \\frac{L_f}{2} \\mathbb E \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 \\nonumber \\\\\n    & \\leq \\mathbb E \\left[ \\Phi({\\mathbf x_t}) + L_f \\norm{{\\mathbf x_t} - {\\mathbf x_t}}^2 \\right] - \\mathbb E f({\\mathbf x_t}, {\\mathbf y_t}) - \\frac{L_f}{2} \\mathbb E \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 \\tag{by definition of $\\widetilde{{\\mathbf x}}_t$} \\\\\n    & \\leq \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) - \\frac{L_f}{2} \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 \\right]. \\label{eq:lem:NC_C_Phi_smooth_decay_one_iter_3}\n\\end{align}\nSubstituting the bounds in \\eqref{eq:lem:NC_C_Phi_smooth_decay_one_iter_2} and \\eqref{eq:lem:NC_C_Phi_smooth_decay_one_iter_3} into \\eqref{eq:lem:NC_C_Phi_smooth_decay_one_iter_1}, we get\n\\begin{align}\n    \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_{t+1}}) \\right] &\\leq \\mathbb E \\Phi (\\widetilde{{\\mathbf x}}_t) + L_f \\left[ \\mathbb E \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 + \\eta_x^2 \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) \\right] + \\frac{\\eta_x L_f^2}{2} \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 + 2 \\eta_x L_f^2 \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\qquad + 2 \\eta_x L_f \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) - \\frac{L_f}{2} \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 \\right] \\nonumber \\\\\n    & \\leq \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_t}) \\right] + \\eta_x^2 L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + 2 \\eta_x L_f^2 \\Delta_{t}^{\\bx,\\by} - \\frac{\\eta_x L_f^2}{2} \\mathbb E \\norm{\\widetilde{{\\mathbf x}}_t - {\\mathbf x_t}}^2 \\nonumber \\\\\n    & \\quad + 2 \\eta_x L_f \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\nonumber \\\\\n    & = \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_t}) \\right] + \\eta_x^2 L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + 2 \\eta_x L_f^2 \\Delta_{t}^{\\bx,\\by} - \\frac{\\eta_x}{8} \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\nonumber \\\\\n    & \\quad + 2 \\eta_x L_f \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right]. \\nonumber\n\\end{align}\nwhere we use the result $\\nabla \\Phi_{1\/2L_f} ({\\mathbf x}) = 2 L_f ({\\mathbf x} - \\widetilde{{\\mathbf x}})$ from Lemma~2.2 in \\cite{davis19wc_siam}. This concludes the proof.\n\\end{proof}\n\n\n\\begin{proof}[Proof of \\cref{lem:NC_C_Phi_f_diff}]\nLet $t = kS + 1$ to $(k+1) S$, where $k = \\lfloor T\/S \\rfloor$ is a positive integer.\nLet $\\Tbx_{k}$ is the latest snapshot iterate in \\cref{alg_local_SGDA_plus}. Then\n\\begin{align}\n    & \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\nonumber \\\\\n    &= \\mathbb E \\left[ f({\\mathbf x_t}, {\\mathbf y}^*({\\mathbf x_t})) - f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) \\right] + \\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right] + \\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\nonumber \\\\\n    & \\leq \\mathbb E \\left[ f({\\mathbf x_t}, {\\mathbf y}^*({\\mathbf x_t})) - f(\\Tbx_{k}, {\\mathbf y}^*({\\mathbf x_t})) \\right] + \\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right] + G_x \\mathbb E \\norm{\\Tbx_{k} - {\\mathbf x_t}} \\nonumber \\\\\n    & \\leq 2 G_x \\mathbb E \\norm{\\Tbx_{k} - {\\mathbf x_t}} + \\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right]. \\label{eq_proof:lem:NC_C_Phi_f_diff_1}\n\\end{align}\nwhere, \\eqref{eq_proof:lem:NC_C_Phi_f_diff_1} follows from $G_x$-Lipschitz continuity of $f(\\cdot, {\\mathbf y})$ (\\cref{assum:Lips_cont_x}), and since ${\\mathbf y}^*(\\cdot) \\in \\operatornamewithlimits{arg\\,max}_{\\mathbf y} f(\\cdot, {\\mathbf y})$.\nNext, we see that\n\\begin{align*}\n    & \\mathbb E G_x \\norm{\\Tbx_{k} - {\\mathbf x_t}} \\leq \\eta_x S G_x \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}},\n\\end{align*}\nThis is because ${\\mathbf x^i_t}$ can be updated at most $S$ times between two consecutive updates of $\\widetilde{\\bx}$. Also, at any time $t$,\n\\begin{align*}\n    \\mathbb E \\norm{\\frac{1}{n} \\sumin \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}})}^2 & = \\mathbb E \\norm{\\frac{1}{n} \\sumin \\left[ \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}}) - \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}) \\right]}^2 + \\mathbb E \\norm{\\frac{1}{n} \\sumin \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t})}^2 \\\\\n    & \\leq \\frac{\\sigma}{n} + G_x^2,\n\\end{align*}\nwhere the expectation is conditioned on the past.\nTherefore, from \\eqref{eq_proof:lem:NC_C_Phi_f_diff_1} we get\n\\begin{align}\n    & \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\leq 2 \\eta_x G_x S^2 \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right]. \\label{eq_proof:lem:NC_C_Phi_f_diff_2}\n\\end{align}\nNext, we bound $\\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right]$.\nSince in localSGDA+, during the updates of $\\{ {\\mathbf y^i_t} \\}$, for $t = kS + 1$ to $(k+1) S$, the corresponding ${\\mathbf x}$ remains constant at $\\Tbx_{k}$. Therefore, for $t = kS + 1$ to $(k+1) S$, the ${\\mathbf y}$ updates behave like maximizing a concave function $f(\\Tbx_{k}, \\cdot)$.\nWith $\\{ {\\mathbf y^i_t} \\}$ being averaged every $\\tau$ iterations, these ${\\mathbf y^i_t}$ updates can be interpreted as iterates of a Local Stochastic Gradient Ascent (Local SGA) algorithm.\n\nUsing \\cref{lem:local_SGD_khaled} for Local SGD (\\cref{alg_local_SGD}), and modifying the result for concave function maximization, we get\n\\begin{align*}\n    \\frac{1}{S} \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right] & \\leq \\frac{4 \\norm{{\\mathbf y}_{kS+1} - {\\mathbf y}^*(\\Tbx_{k})}^2}{\\eta_y S} + \\frac{20 \\eta_y \\sigma^2}{n} + 16 \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) \\nonumber \\\\\n    & \\leq \\underbrace{\\frac{4 D}{\\eta_y S} + \\frac{20 \\eta_y \\sigma^2}{n}}_{\\substack{\\text{error with full} \\\\ \\text{synchronization}}} + \\underbrace{16 \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right)}_{\\text{error due to local updates}}. \\nonumber\n\\end{align*}\nSubstituting this bound in \\eqref{eq_proof:lem:NC_C_Phi_f_diff_2}, we get\n\\begin{align*}\n    & \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\leq 2 \\eta_x G_x S^2 \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y} + \\frac{20 \\eta_y \\sigma^2 S}{n} + 16 S \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right).\n   \n\\end{align*}\nSumming over $k = 0$ to $T\/S - 1$, we get\n\\begin{align*}\n    & \\frac{1}{T} \\sum_{k=0}^{T\/S-1} \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\leq 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y S} + \\frac{20 \\eta_y \\sigma^2}{n} + 16 \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right).\n   \n\\end{align*}\n\\end{proof}\n\n\n\\begin{proof}[Proof of \\cref{lem:NC_C_consensus_error}]\nThe proof follows analogously to the proof of \\cref{lem:NC_PL_consensus_error}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\section{Nonconvex-One-Point-Concave Functions: Local SGDA+ (\\texorpdfstring{\\cref{thm:NC_1PC}}{Theorem 4})} \\label{app:NC_1PC}\n\nThe proof of \\cref{thm:NC_1PC} is similar to the proof of \\cref{thm:NC_C}.\nWe organize this section as follows. First, in \\cref{sec:NC_1PC_int_results} we present some intermediate results, which we use in the proof of \\cref{thm:NC_1PC}. Next, in \\cref{sec:NC_1PC_thm_proof}, we present the proof of \\cref{thm:NC_1PC}, which is followed by the proofs of the intermediate results in \\cref{sec:NC_1PC_int_results_proofs}.\nIn \\cref{app:NC_1PC_tau_1}, we prove convergence for the full synchronized Local SGDA+.\n\n\\subsection{Intermediate Lemmas} \\label{sec:NC_1PC_int_results}\n\nThe main difference with the nonconvex-concave problem is the bound on the difference $\\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right]$.\nIn case of concave functions, as we see in \\cref{lem:NC_C_Phi_f_diff}, this difference can be bounded using standard results for Local SGD (\\cref{lem:local_SGD_khaled}), which have a linear speedup with the number of clients $n$ (notice the $\\frac{\\eta_y \\sigma^2}{n}$ term in \\cref{lem:NC_C_Phi_f_diff}).\nThe corresponding result for minimization of smooth one-point-convex function using local SGD is an open problem.\nRecent works on deterministic and stochastic quasar-convex problems (of which one-point-convex functions are a special case) \\cite{gasnikov17acc_quasar_convex_arxiv, hinder20near_opt_star_convex_colt, jin20quasar_convex_arxiv} have achieved identical (within multiplicative constants) convergence rates, as smooth convex functions, for this more general class of functions, using SGD.\nThis leads us to conjecture that local SGD should achieve identical communication savings, along with linear speedup (as in \\cref{lem:local_SGD_khaled}), for one-point-convex problems.\nHowever, proving this claim formally remains an open problem.\n\nIn absence of this desirable result, we bound $\\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right]$ in the next result, but without any linear speedup in $n$.\n\n\\begin{lemma}\n\\label{lem:NC_1PC_Phi_f_diff}\nSuppose the local functions satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, \\ref{assum:Lips_cont_x}, \\ref{assum:1pc_y}.\nFurther, suppose we choose the step-size $\\eta_y$ such that $\\eta_y \\leq \\frac{1}{8 L_f \\tau}$.\nThen the iterates generated by \\cref{alg_local_SGDA_plus} satisfy\n\\begin{align}\n    \\frac{1}{T} \\sumtT \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] & \\leq 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y S} + 20 \\eta_y \\sigma^2 + 16 \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right). \\nonumber\n\\end{align}\n\\end{lemma}\n\n\n\n\\subsection{Proof of \\texorpdfstring{\\cref{thm:NC_1PC}}{Theorem 4}}\n\\label{sec:NC_1PC_thm_proof}\nFor the sake of completeness, we first state the full statement of \\cref{thm:NC_1PC} here.\n\n\\begin{theorem*}\nSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, \\ref{assum:Lips_cont_x}, \\ref{assum:1pc_y}.\nFurther, let $\\norm{{\\mathbf y_t}}^2 \\leq D$ for all $t$.\nSuppose the step-size $\\eta_y$ is chosen such that $\\eta_y \\leq \\frac{1}{8 L_f \\tau}$.\nThen the output ${\\bar{\\bx}_T}$ of \\cref{alg_local_SGDA_plus} satisfies\n\\begin{equation}\n    \\begin{aligned}\n        \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 & \\leq \\mathcal O \\left( \\frac{\\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + \\eta_y L_f \\sigma^2 + L_f \\left[ \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{D}{\\eta_y S} \\right] \\right) \\\\\n        & \\quad + \\mathcal O \\left( L_f^2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) + \\eta_y^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) \\right] \\right),\n       \n    \\end{aligned}\n    \\label{eq:thm:NC_1PC}\n\\end{equation}\nwhere {\\small$\\widetilde{\\Delta}_{\\Phi} \\triangleq \\Phi_{1\/2 L_f} ({\\mathbf x}_0) - \\min_{\\mathbf x} \\Phi_{1\/2 L_f} ({\\mathbf x})$}.\nWith the following parameter values:\n\\begin{align*}\n    \\eta_x = \\Theta \\left( \\frac{1}{T^{3\/4}} \\right), \\qquad \\eta_y = \\Theta \\left( \\frac{1}{T^{1\/4}} \\right), \\qquad S = \\Theta \\left( \\sqrt{T} \\right),\n\\end{align*}\nwe get\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\mathcal O \\left( \\frac{1}{T^{1\/4}} \\right) + \\mathcal O \\left( \\frac{1}{T^{3\/4}} \\right) + \\mathcal O \\left( \\frac{(\\tau-1)^2}{T^{1\/2}} \\right) + \\mathcal O \\left( \\frac{(\\tau-1)^2}{T^{3\/2}} \\right).\n    \\label{eq:thm:NC_1PC_conv_rate}\n\\end{align}\n\\end{theorem*}\n\n\\begin{cor}\n\\label{cor:NC_1PC_comm_cost}\nTo reach an $\\epsilon$-accurate point, i.e., ${\\mathbf x}$ such that $\\mathbb E \\| \\nabla \\Phi_{1\/2L_f} ({\\mathbf x}) \\| \\leq \\epsilon$,\nthe stochastic gradient complexity of \\cref{alg_local_SGDA_plus} is $\\mathcal O (1\/\\epsilon^8)$.\nThe number of communication rounds required for the same is $T\/\\tau = \\mathcal O ( 1\/\\epsilon^{7} )$.\n\\end{cor}\n\n\\begin{remark}\nNote that the only difference between the convergence rates for NC-1PC functions in \\eqref{eq:thm:NC_1PC_conv_rate}, and for NC-C functions in \\eqref{eq:thm:NC_C_conv_rate} is the absence of $n$ from the leading $\\mathcal O (1\/T^{1\/4})$ term.\nThis implies we do not observe a linear speedup in $n$ in this case.\nAs stated earlier, this limitation stems from the fact that even for simple minimization of one-point-convex functions, proving linear speedup in convergence rate in the presence of local updates at the clients is an open problem.\n\\end{remark}\n\n\n\\begin{proof}\nWe sum the result in \\cref{lem:NC_C_Phi_smooth_decay_one_iter} over $t = 0$ to $T-1$ and rearrange the terms to get\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\frac{8}{\\eta_x} \\frac{1}{T} \\sumtT \\left( \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_t}) \\right] - \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_{t+1}}) \\right] \\right) + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + 16 L_f \\frac{1}{T} \\sumtT \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] + 16 L_f^2 \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\leq \\frac{8}{\\eta_x T} \\left[ \\Phi_{1\/2L_f} ({\\mathbf x}_0) - \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x}_T) \\right] \\right] + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + 16 L_f^2 \\Delta_{t}^{\\bx,\\by} \\nonumber \\\\\n    & \\quad + 16 L_f \\left[ 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y S} + 20 \\eta_y \\sigma^2 + 16 \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) \\right] \\tag{\\cref{lem:NC_1PC_Phi_f_diff}} \\\\\n    & \\leq \\frac{8 \\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + 320 \\eta_y L_f \\sigma^2 + 16 L_f \\left[ 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y S} \\right] \\nonumber \\\\\n    & \\quad + 32 L_f^2 (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 \\left( 1 + \\frac{1}{n} \\right) + 3 \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) + 8 \\eta_y^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) \\right], \\tag{\\cref{lem:NC_PL_consensus_error}}\n\\end{align}\nwhere $\\widetilde{\\Delta}_{\\Phi} = \\Phi_{1\/2L_f} ({\\mathbf x}_0) - \\min_{\\mathbf x} \\Phi_{1\/2L_f} ({\\mathbf x})$.\nFollowing similar technique as in the proof of \\cref{thm:NC_C}, using the following parameter values,\n\\begin{align*}\n    S = \\Theta \\left( \\sqrt{T} \\right), \\qquad \\eta_x = \\Theta \\left( \\frac{1}{T^{3\/4}} \\right), \\qquad \\eta_y = \\Theta \\left( \\frac{1}{T^{1\/4}} \\right),\n\\end{align*}\nwe get the following bound.\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\mathcal O \\left( \\frac{\\sigma^2 + D + G_x^2}{T^{1\/4}} \\right) + \\mathcal O \\left( \\frac{1}{T^{3\/4}} \\right) + \\mathcal O \\left( \\frac{(\\tau-1)^2}{T^{1\/2}} \\right) + \\mathcal O \\left( \\frac{(\\tau-1)^2}{T^{3\/2}} \\right),\n    \\label{eq_proof:thm_NC_1PC_4}\n\\end{align}\nwhich completes the proof\n\\end{proof}\n\n\n\\begin{proof}[Proof of \\cref{cor:NC_1PC_comm_cost}]\nTo reach an $\\epsilon$-accurate point, i.e., ${\\mathbf x}$ such that $\\mathbb E \\left\\| \\nabla \\Phi_{1\/2L_f} ({\\mathbf x}) \\right\\| \\leq \\epsilon$, we need\n\\begin{align*}\n    \\mathbb E \\left\\| \\nabla \\Phi_{1\/2L_f} ({\\bar{\\bx}_T}) \\right\\| & \\leq \\left[ \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left\\| \\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t}) \\right\\|^2 \\right]^{1\/2} \\nonumber \\\\\n    & \\leq \\mathcal O \\left( \\frac{1}{T^{1\/8}} \\right) + \\mathcal O \\left( \\frac{1}{T^{3\/8}} \\right) + \\mathcal O \\left( \\frac{\\tau-1}{T^{1\/4}} \\right) + \\mathcal O \\left( \\frac{\\tau-1}{T^{3\/4}} \\right).\n\\end{align*}\nWe can choose $\\tau \\leq \\mathcal O \\left( T^{1\/8} \\right)$ without affecting the convergence rate $\\mathcal O \\left( \\frac{1}{T^{1\/8}} \\right)$.\nIn that case, we need $T = \\mathcal O \\left( \\frac{1}{\\epsilon^8} \\right)$ iterations to reach an $\\epsilon$-accurate point.\nAnd the minimum number of communication rounds is \n$$\\mathcal O \\left( \\frac{T}{\\tau} \\right) = \\mathcal O \\left( T^{7\/8} \\right) = \\mathcal O \\left( \\frac{1}{\\epsilon^7} \\right).$$\n\\end{proof}\n\n\n\n\n\\subsection{Proofs of the Intermediate Lemmas}\n\\label{sec:NC_1PC_int_results_proofs}\n\n\n\\begin{proof}[Proof of \\cref{lem:NC_1PC_Phi_f_diff}]\nThe proof proceeds the same way as for \\cref{lem:NC_C_Phi_f_diff}.\nLet $t = kS + 1$ to $(k+1) S$, where $k = \\lfloor T\/S \\rfloor$ is a positive integer.\nLet $\\Tbx_{k}$ is the latest snapshot iterate in \\cref{alg_local_SGDA_plus}. From \\eqref{eq_proof:lem:NC_C_Phi_f_diff_2}, we get\n\\begin{align}\n    & \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\leq 2 \\eta_x G_x S^2 \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right]. \\label{eq_proof:lem:NC_1PC_Phi_f_diff_2}\n\\end{align}\nNext, we bound $\\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right]$.\nSince in \\cref{alg_local_SGDA_plus}, during the updates of $\\{ {\\mathbf y^i_t} \\}$, for $t = kS + 1$ to $(k+1) S$, the corresponding ${\\mathbf x}$ remains constant at $\\Tbx_{k}$. Therefore, for $t = kS + 1$ to $(k+1) S$, the ${\\mathbf y}$ updates behave like maximizing a concave function $f(\\Tbx_{k}, \\cdot)$.\nWith $\\{ {\\mathbf y^i_t} \\}$ being averaged every $\\tau$ iterations, these ${\\mathbf y^i_t}$ updates can be interpreted as iterates of a Local Stochastic Gradient Ascent (Local SGA) (\\cref{alg_local_SGD}).\n\nHowever, since the function is no longer concave, but one-point-concave, we lose the linear speedup in \\cref{lem:local_SGD_khaled}, and get\n\\begin{align*}\n    \\frac{1}{S} \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right] & \\leq \\frac{4 \\norm{{\\mathbf y}_{kS+1} - {\\mathbf y}^*(\\Tbx_{k})}^2}{\\eta_y S} + 20 \\eta_y \\sigma^2 + 16 \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) \\nonumber \\\\\n    & \\leq \\underbrace{\\frac{4 D}{\\eta_y S} + 20 \\eta_y \\sigma^2}_{\\substack{\\text{error with full} \\\\ \\text{synchronization}}} + \\underbrace{16 \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right)}_{\\text{error due to local updates}}. \\nonumber\n\\end{align*}\nSubstituting this bound in \\eqref{eq_proof:lem:NC_1PC_Phi_f_diff_2}, we get\n\\begin{align*}\n    & \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\leq 2 \\eta_x G_x S^2 \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y} + 20 \\eta_y \\sigma^2 S + 16 S \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right).\n   \n\\end{align*}\nSumming over $k = 0$ to $T\/S - 1$, we get\n\\begin{align*}\n    & \\frac{1}{T} \\sum_{k=0}^{T\/S-1} \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\leq 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{4 D}{\\eta_y S} + 20 \\eta_y \\sigma^2 + 16 \\eta_y^2 L_f (\\tau-1)^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right).\n   \n\\end{align*}\n\\end{proof}\n\n\n\n\\subsection{With full synchronization}\n\\label{app:NC_1PC_tau_1}\nIn this subsection, we discuss the case when the clients perform a single local update between successive communications $\\tau = 1$.\nThe goal of the results in this subsection is to show that at least in this specialized case, linear speedup can be achieved for NC-1PC functions.\n\n\\begin{lemma}\n\\label{lem:NC_1PC_Phi_f_diff_tau_1}\nSuppose the local functions satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, \\ref{assum:Lips_cont_x}, \\ref{assum:1pc_y}.\nFurther, suppose we choose the step-size $\\eta_y$ such that $\\eta_y \\leq \\frac{1}{2 L_f}$.\nThen the iterates generated by \\cref{alg_local_SGDA_plus} satisfy\n\\begin{align}\n    \\frac{1}{T} \\sumtT \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] & \\leq 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{D}{2 \\eta_y S} + \\frac{\\eta_y \\sigma^2}{n}. \\nonumber\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows similar technique as in \\cref{lem:NC_C_Phi_f_diff}. From \\eqref{eq_proof:lem:NC_C_Phi_f_diff_2}, we get\n\\begin{align}\n    & \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\leq 2 \\eta_x G_x S^2 \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right]. \\label{eq_proof:lem:NC_1PC_Phi_f_diff_tau_1}\n\\end{align}\nWe only need to bound the second term in \\eqref{eq_proof:lem:NC_1PC_Phi_f_diff_tau_1}.\nWith $\\tau = 1$, the ${\\mathbf y^i_t}$ updates reduce to minibatch stochastic gradient ascent, with batch-size $\\mathcal O (n)$. Using the result for stochastic minimization of $\\gamma$-quasar convex functions (for one-point-concave functions, $\\gamma = 1$) using SGD (Theorem~3.3 in \\cite{jin20quasar_convex_arxiv}), we get\n\\begin{align*}\n    \\frac{1}{S} \\sum_{t=kS+1}^{(k+1)S} \\mathbb E \\left[ f(\\Tbx_{k}, {\\mathbf y}^*(\\Tbx_{k})) - f(\\Tbx_{k}, {\\mathbf y_t}) \\right] \\leq \\frac{D}{2 \\eta_y S} + \\frac{\\eta_y \\sigma^2}{n},\n\\end{align*}\nwhich completes the proof.\n\\end{proof}\n\n\nNext, we state the convergence result.\n\n\\begin{theorem*}\nSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, \\ref{assum:Lips_cont_x}, \\ref{assum:1pc_y}.\nFurther, let $\\norm{{\\mathbf y_t}}^2 \\leq D$ for all $t$.\nSuppose the step-size $\\eta_y$ is chosen such that $\\eta_y \\leq \\frac{1}{2 L_f}$.\nThen the output ${\\bar{\\bx}_T}$ of \\cref{alg_local_SGDA_plus} satisfies\n\\begin{equation}\n    \\begin{aligned}\n        & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\mathcal O \\left( \\frac{\\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + \\frac{\\eta_y L_f \\sigma^2}{n} + L_f \\left[ \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{D}{\\eta_y S} \\right] \\right),\n       \n    \\end{aligned}\n    \\label{eq:thm:NC_1PC_tau_1}\n\\end{equation}\nwhere {\\small$\\widetilde{\\Delta}_{\\Phi} \\triangleq \\Phi_{1\/2 L_f} ({\\mathbf x}_0) - \\min_{\\mathbf x} \\Phi_{1\/2 L_f} ({\\mathbf x})$}.\nWith the following parameter values:\n\\begin{align*}\n    S = \\Theta \\left( \\sqrt{\\frac{T}{n}} \\right), \\qquad \\eta_x = \\Theta \\left( \\frac{n^{1\/4}}{T^{3\/4}} \\right), \\qquad \\eta_y = \\Theta \\left( \\frac{n^{3\/4}}{T^{1\/4}} \\right),\n\\end{align*}\nwe get\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\mathcal O \\left( \\frac{1}{(n T)^{1\/4}} \\right) + \\mathcal O \\left( \\frac{n^{1\/4}}{T^{3\/4}} \\right). \\nonumber\n\\end{align}\n\\end{theorem*}\n\n\\begin{cor}\n\\label{cor:NC_1PC_comm_cost_tau_1}\nTo reach an $\\epsilon$-accurate point, i.e., ${\\mathbf x}$ such that $\\mathbb E \\| \\nabla \\Phi_{1\/2L_f} ({\\mathbf x}) \\| \\leq \\epsilon$,\nthe stochastic gradient complexity of \\cref{alg_local_SGDA_plus} is $\\mathcal O (1\/n \\epsilon^8)$.\n\\end{cor}\n\n\n\\begin{proof}\nWe sum the result in \\cref{lem:NC_C_Phi_smooth_decay_one_iter} over $t = 0$ to $T-1$. Since $\\tau = 1$, $\\Delta_{t}^{\\bx,\\by} = 0$ for all $t$. Rearranging the terms, we get\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\frac{8}{\\eta_x} \\frac{1}{T} \\sumtT \\left( \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_t}) \\right] - \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x_{t+1}}) \\right] \\right) + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) \\nonumber \\\\\n    & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad + 16 L_f \\frac{1}{T} \\sumtT \\mathbb E \\left[ \\Phi({\\mathbf x_t}) - f({\\mathbf x_t}, {\\mathbf y_t}) \\right] \\nonumber \\\\\n    & \\leq \\frac{8}{\\eta_x T} \\left[ \\Phi_{1\/2L_f} ({\\mathbf x}_0) - \\mathbb E \\left[ \\Phi_{1\/2L_f} ({\\mathbf x}_T) \\right] \\right] + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) \\nonumber \\\\\n    & \\quad + 16 L_f \\left[ 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{D}{2 \\eta_y S} + \\frac{\\eta_y \\sigma^2}{n} \\right] \\tag{\\cref{lem:NC_1PC_Phi_f_diff_tau_1}} \\\\\n    & \\leq \\frac{8 \\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + 8 \\eta_x L_f \\left( G_x^2 + \\frac{\\sigma^2}{n} \\right) + \\frac{16 \\eta_y L_f \\sigma^2}{n} + 16 L_f \\left[ 2 \\eta_x G_x S \\sqrt{G_x^2 + \\frac{\\sigma^2}{n}} + \\frac{D}{2 \\eta_y S} \\right], \\nonumber\n\\end{align}\nwhere $\\widetilde{\\Delta}_{\\Phi} = \\Phi_{1\/2L_f} ({\\mathbf x}_0) - \\min_{\\mathbf x} \\Phi_{1\/2L_f} ({\\mathbf x})$.\nFollowing similar technique as in the proof of \\cref{thm:NC_C}, using the following parameter values,\n\\begin{align*}\n    S = \\Theta \\left( \\sqrt{\\frac{T}{n}} \\right), \\qquad \\eta_x = \\Theta \\left( \\frac{n^{1\/4}}{T^{3\/4}} \\right), \\qquad \\eta_y = \\Theta \\left( \\frac{n^{3\/4}}{T^{1\/4}} \\right),\n\\end{align*}\nwe get the following bound.\n\\begin{align}\n    & \\frac{1}{T} \\sumtT \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t})}^2 \\leq \\mathcal O \\left( \\frac{\\sigma^2 + D + G_x^2}{(n T)^{1\/4}} \\right) + \\mathcal O \\left( \\frac{n^{1\/4}}{T^{3\/4}} \\right). \\nonumber\n\\end{align}\n\n\\end{proof}\n\n\n\\begin{proof}[Proof of \\cref{cor:NC_1PC_comm_cost_tau_1}]\nWe assume $T \\geq n$.\nTo reach an $\\epsilon$-accurate point, i.e., ${\\mathbf x}$ such that $\\mathbb E \\left\\| \\nabla \\Phi_{1\/2L_f} ({\\mathbf x}) \\right\\| \\leq \\epsilon$, since\n\\begin{align*}\n    \\mathbb E \\left\\| \\nabla \\Phi_{1\/2L_f} ({\\bar{\\bx}_T}) \\right\\| \\leq \\left[ \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\left\\| \\nabla \\Phi_{1\/2L_f} ({\\mathbf x_t}) \\right\\|^2 \\right]^{1\/2} \\leq \\mathcal O \\left( \\frac{1}{(n T)^{1\/8}} \\right) + \\mathcal O \\left( \\frac{n^{1\/8}}{T^{3\/8}} \\right),\n\\end{align*}\nwe need $T = \\mathcal O \\left( \\frac{1}{n \\epsilon^8} \\right)$ iterations.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\section{Additional Experiments}\n\\label{app:add_exp}\n\n\\begin{algorithm}[ht]\n\\caption{Local SGDA+ \\cite{mahdavi21localSGDA_aistats}}\n\\label{alg_mom_local_SGDA_plus}\n\\begin{algorithmic}[1]\n\t\\STATE{\\textbf{Input:} {\\small${\\mathbf x}_0^i = \\widetilde{\\bx}_0 = {\\mathbf x}_0, {\\mathbf y}_0^i = {\\mathbf y}_0$, $\\mathbf d_{x,0}^i = \\nabla_{\\bx} f_i ({\\mathbf x}^i_0, {\\mathbf y}^i_0; \\xi^i_0)$, $\\mathbf d_{y,0}^i = \\nabla_{\\by} f_i ({\\mathbf x}^i_0, {\\mathbf y}^i_0; \\xi^i_0)$} for all $i \\in [n]$; step-sizes $\\eta_x, \\eta_y$; synchronization intervals $\\tau, S$; $T, k = 0$}\n\t\\FOR[At all clients $i=1,\\hdots, n$]{$t=0$ to $T-1$}\n\t    \\STATE{{\\small$\\Tbx^i_{t+\\frac{1}{2}} = {\\mathbf x^i_t} - \\eta_x {\\mathbf d^i_{x,t}}$, \n        $\\ {\\mathbf x^i_{t+1}} = {\\mathbf x^i_t} + \\alpha_t ( \\Tbx^i_{t+\\frac{1}{2}} - {\\mathbf x^i_t} )$}}\n        \\STATE{{\\small$\\Tby^i_{t+\\frac{1}{2}} = {\\mathbf y^i_t} + \\eta_y {\\mathbf d^i_{y,t}}$, $\\ {\\mathbf y^i_{t+1}} = {\\mathbf y^i_t} + \\alpha_t ( \\Tby^i_{t+\\frac{1}{2}} - {\\mathbf y^i_t} )$}}\n        \\STATE{Sample minibatch ${\\xi^i_{t+1}}$ from local data}\n        \\STATE{{\\small${\\mathbf d^i_{x,t+1}} = (1 - \\beta_x \\alpha_t) {\\mathbf d^i_{x,t}} + \\beta_x \\alpha_t \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}})$}}\n        \\STATE{{\\small${\\mathbf d^i_{y,t+1}} = (1 - \\beta_y \\alpha_t) {\\mathbf d^i_{y,t}} + \\beta_y \\alpha_t \\nabla_{\\by} f_i (\\Tbx_{k}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}})$}}\n\t  \n   \n   \n        \\IF{$t+1$ mod $\\tau = 0$}\n           \n           \n           \n           \n           \n            \\STATE{Clients send $\\{ {\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}} \\}$ to the server}\n            \\STATE{Server computes averages ${\\mathbf x_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_{t+1}}$, \n            ${\\mathbf y_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf y^i_{t+1}}$, and sends to all the clients}\n            \\STATE{${\\mathbf x^i_{t+1}} = {\\mathbf x_{t+1}}$, ${\\mathbf y^i_{t+1}} = {\\mathbf y_{t+1}}$, for all $i \\in [n]$}\n            \\STATE{${\\mathbf d^i_{x,t+1}} = 0$, ${\\mathbf d^i_{y,t+1}} = 0$, for all $i \\in [n]$}\n        \\ENDIF\n        \\IF{$t+1$ mod $S = 0$}\n            \\STATE{Clients send $\\{ {\\mathbf x^i_{t+1}} \\}$ to the server}\n            \\STATE{$k \\gets k+1$}\n            \\STATE{Server computes averages $\\Tbx_{k} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_{t+1}}$, and sends to all the clients}\n        \\ENDIF\n\t\\ENDFOR\n\t\\STATE{\\textbf{Return: }${\\bar{\\bx}_T}$ drawn uniformly at random from $\\{ {\\mathbf x_t} \\}$, where ${\\mathbf x_t} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_t}$}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Fair Classification}\nBatch-size of $32$ is used. Momentum parameter $0.9$ is used only in Momentum Local SGDA (\\cref{alg_NC_momentum}) and corresponds to $\\alpha \\beta$ in the pseudocode.\n\n\\begin{table}[ht]\n\\begin{center}\n\\caption{Parameter values for experiments in \\cref{sec:exp_fair}}\n\\begin{tabular}{llll}\n\\hline\nParameter &  &  &  \\\\\n\\hline\nLearning Rate $(\\eta_y)$ & $0.02$ & $2 \\times 10^{-3}$ & $2 \\times 10^{-4}$ \\\\\nLearning Rate $(\\eta_x)$ & $0.016$ & $1.6 \\times 10^{-3}$ & $1.6 \\times 10^{-4}$ \\\\\nCommunication rounds & 150 & 75 & 75 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Robust Neural Network Training}\nBatch-size of $32$ is used. Momentum parameter $0.9$ is used only in Momentum Local SGDA+ (\\cref{alg_mom_local_SGDA_plus}) and corresponds to $\\alpha \\beta$ in the pseudocode. $S = \\tau^2$ in both \\cref{alg_local_SGDA_plus} and \\cref{alg_mom_local_SGDA_plus}.\n\n\\begin{table}[ht]\n\\begin{center}\n\\caption{Parameter values for experiments in \\cref{sec:exp_fair}}\n\\begin{tabular}{llll}\n\\hline\nParameter &  &  &  \\\\\n\\hline\nLearning Rate $(\\eta_y)$ & $0.02$ & $2 \\times 10^{-3}$ & $2 \\times 10^{-4}$ \\\\\nLearning Rate $(\\eta_x)$ & $0.016$ & $1.6 \\times 10^{-3}$ & $1.6 \\times 10^{-4}$ \\\\\nCommunication rounds & 150 & 75 & 75 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[ht]\n     \\centering\n     \\includegraphics[width=0.55\\textwidth]{figures\/RobustNN\/CIFAR10_test_loss.pdf}\n     \\caption{Robust test loss for the CIFAR10 experiment shown in \\cref{sec:exp_robustnn}. The test loss in \\cref{eq:exp_robustnn} is computed using some steps of gradient ascent to find an estimate of ${\\mathbf y}^*$.}\n   \n\\end{figure}\n\n\\begin{figure}[ht]\n     \\centering\n     \\begin{subfigure}\n         \\centering\n         \\includegraphics[width=0.45\\textwidth]{figures\/RobustNN\/fashionMNIST_test_loss.pdf}\n       \n         \\label{fig:robustnn_fashionmnist_loss}\n     \\end{subfigure}\n     \\begin{subfigure}\n         \\centering\n         \\includegraphics[width=0.45\\textwidth]{figures\/RobustNN\/fashionMNIST_test_acc.pdf}\n       \n         \\label{fig:robustnn_fashionmnist_acc}\n     \\end{subfigure}\n     \\caption{Comparison of the effects of $\\tau$ on the performance of Local SGDA and Momentum Local SGDA algorithms, for the robust NN training problem on the FashionMNIST dataset, with the VGG11 model. The figures show the robust test loss and robust test accuracy. \\label{fig:robustnn_fashionmnist}}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\ifx\n\\newpage\n\\section{Extra: Nonconvex-Concave (NC-C) case} \nWe organize this section as follows. First, in  we present some intermediate results, which we use to prove the main theorem. Next, in , we present the proof of \\cref{thm:NC_C}, which is followed by the proofs of the intermediate results in \n\n\\subsection{Intermediate Lemmas} \n\n\n\\subsection{Proof of \\cref{thm:NC_C}}\n\n\n\\subsection{Proofs of the Intermediate Lemmas}\n\n\\subsection{Algorithm}\nThe algorithm is modified from the single-client approach proposed in \\cite{rafique18WCC_oms}, and incorporates local updates at the clients and periodic communication with the server.\n\n\n\\begin{equation}\n    \\min_{{\\mathbf x}} \\max_{{\\mathbf y}} \\left\\{ {F_{\\gamma, \\psi}} ({\\mathbf x}, {\\mathbf y}; \\bar{{\\mathbf x}}, \\bar{{\\mathbf y}}) \\triangleq f({\\mathbf x}, {\\mathbf y}) + \\frac{1}{2\\gamma} \\left\\| {\\mathbf x} - \\bar {\\mathbf x} \\right\\|^2 - \\frac{1}{2 \\psi} \\left\\| {\\mathbf y} - \\bar {\\mathbf y} \\right\\|^2 \\right\\}\n\\end{equation}\nwhere $\\gamma < \\frac{1}{L}$.\nFor every $t$, the problem \n\\begin{align*}\n    \\min_{{\\mathbf x}} \\max_{{\\mathbf y}} {F_{\\gamma, {\\psi_t}}} ({\\mathbf x}, {\\mathbf y}; \\bar{{\\mathbf x}}_t, \\bar{{\\mathbf y}}_t)\n\\end{align*}\nis a strongly convex-strongly concave minimax problem, which we solve using the Local SGDA algorithm proposed in \\cite{mahdavi21localSGDA_aistats}. \nThe function ${F_{\\gamma, {\\psi_t}}} ({\\mathbf x}, {\\mathbf y}; \\bar{{\\mathbf x}}_t, \\bar{{\\mathbf y}}_t)$ is $\\mu_x$-strongly convex in ${\\mathbf x}$, and $\\mu_y$-strongly concave in ${\\mathbf y}$, where\n$\\mu_x = \\frac{1}{\\gamma}-L$ and $\\mu_y = \\frac{1}{{\\psi_t}}$.\nAlso, ${F_{\\gamma, {\\psi_t}}}$ is $L_x$-smooth in ${\\mathbf x}$, and $L_y$-smooth in ${\\mathbf y}$, where $L_x = L_f + \\frac{1}{\\gamma}$, and $L_y = L_f + \\frac{1}{{\\psi_t}}$.\n{\\color{blue}We define $\\mu = \\min \\left\\{ \\mu_x, \\mu_y \\right\\}$ and $\\widetilde{L}_f = \\max \\left\\{ L_x, L_y \\right\\}$.}\n\n\\subsection{Analysis}\n\\paragraph*{Update Equations}\n\\begin{equation}\n    \\begin{aligned}\n        {\\mathbf{x}^i_{t,k+1}} &= \\mathcal P_{\\mathcal X} \\left[ {\\mathbf{x}^i_{t,k}} - \\eta_k \\left( \\Tilde{\\nabla}_x f_i ({\\mathbf{x}^i_{t,k}}, {\\mathbf{y}^i_{t,k}}) + \\frac{1}{\\gamma} \\left( {\\mathbf{x}^i_{t,k}} - \\bar{{\\mathbf x}}_t \\right) \\right) \\right] \\\\\n        {\\mathbf{y}^i_{t,k+1}} &= \\mathcal P_{\\mathcal Y} \\left[ {\\mathbf{y}^i_{t,k}} + \\eta_k \\left( \\Tilde{\\nabla}_y f_i ({\\mathbf{x}^i_{t,k}}, {\\mathbf{y}^i_{t,k}}) - \\frac{1}{{\\psi_t}} \\left( {\\mathbf{y}^i_{t,k}} - \\bar{{\\mathbf y}}_t \\right) \\right) \\right]\n    \\end{aligned}\n\\end{equation}\n\n\n\\paragraph*{Convergence Result from \\cite{mahdavi21localSGDA_aistats} for the Strongly-Convex-Strongly-Concave Case}\n\n\\begin{theorem*}\nIf each local function $\\{ f_i \\}$ satisfies Assumptions \\ref{assum:smoothness}, \\ref{assum:concavity}, \\ref{assum:bdd_var}. \nChoose $\\eta_x = \\eta_y = \\eta_k = \\frac{8}{\\mu (k + a)}$, then the iterates generated by \\cref{alg_local_SGDA} satisfy\n\\begin{align}\n    \\mathbb E \\left[ \\norm{{\\mathbf x}_K - {\\bx^{\\star}}}^2 + \\norm{{\\mathbf y}_K - {\\by^{\\star}}}^2 \\right] \\leq \\mathcal O \\left( \\frac{a^3}{K^3} + \\frac{\\kappa^2 \\tau^2 (\\varsigma_x + \\varsigma_y) }{\\mu K^2} + \\frac{\\kappa^2 \\tau^2 \\sigma^2}{\\mu K^2} + \\frac{\\sigma^2}{\\mu^2 n K} \\right).\n\\end{align}\nWith synchronization frequency $\\tau = \\sqrt{K\/n}$, we get\n\\begin{align}\n    \\mathbb E \\left[ \\norm{{\\mathbf x}_K - {\\bx^{\\star}}}^2 + \\norm{{\\mathbf y}_K - {\\by^{\\star}}}^2 \\right] \\leq \\mathcal O \\left( \\frac{a^3}{K^3} + \\frac{\\kappa^2 (\\varsigma_x + \\varsigma_y) }{\\mu n K} + \\frac{\\kappa^2 \\sigma^2}{\\mu n K} + \\frac{\\sigma^2}{\\mu^2 n K} \\right).\n\\end{align}\n\\end{theorem*}\n\n\n\\paragraph*{Extension of Lemma 3.1 in \\cite{mahdavi21localSGDA_aistats}}\n$({\\bx_t^{\\star}}, {\\by_t^{\\star}})$ is the saddle-point of\n\\begin{align*}\n    \\min_{{\\mathbf x} \\in \\mathcal X} \\max_{{\\mathbf y} \\in \\mathcal Y} \\left\\{ {F_{\\gamma, {\\psi_t}}} ({\\mathbf x}, {\\mathbf y}; \\bar{{\\mathbf x}}_t, \\bar{{\\mathbf y}}_t) \\triangleq f({\\mathbf x}, {\\mathbf y}) + \\frac{1}{2 \\gamma} \\norm{{\\mathbf x} - \\bar{{\\mathbf x}}_t}^2 - \\frac{1}{2 {\\psi_t}} \\norm{{\\mathbf y} - \\bar{{\\mathbf y}}_t}^2 \\right\\}.\n\\end{align*}\n\n\n\\begin{theorem}{Convergence Rate}\n\\label{thm:nc_c_conv_rate}\n\\begin{align}\n    \\mathbf E_{t, K_t} &= \\mathbb E \\left[ \\norm{\\mathbf x_{t,K_t} - {\\bx_t^{\\star}}}^2 + \\norm{\\mathbf y_{t,K_t} - {\\bx_t^{\\star}}}^2 \\right] \\nonumber \\\\\n    &\\leq \\mathcal O \\left( \\frac{a_t^3}{K_t^3} \\right) + \\mathcal O \\left( \\frac{\\kappa_t^2 \\tau (\\tau - 1)}{\\mu K_t^2} \\left( \\sigma^2 + \\delta_x + \\delta_y \\right) \\right) + \\mathcal O \\left( \\frac{\\sigma^2}{\\mu^2 n K_t} \\right).\n   \n\\end{align}\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\paragraph*{Global Algorithm:}\n\\begin{align*}\n    {F_{\\gamma, {\\psi_t}}} ({\\mathbf x}, {\\mathbf y}; \\bar{{\\mathbf x}}_t, \\bar{{\\mathbf y}}_t) & \\triangleq f({\\mathbf x}, {\\mathbf y}) + \\frac{1}{2 \\gamma} \\norm{{\\mathbf x} - \\bar{{\\mathbf x}}_t}^2 - \\frac{1}{2 {\\psi_t}} \\norm{{\\mathbf y} - \\bar{{\\mathbf y}}_t}^2, \\\\\n    {F_{\\gamma}} ({\\mathbf x}, {\\mathbf y}; \\bar{{\\mathbf x}}_t) & \\triangleq f({\\mathbf x}, {\\mathbf y}) + \\frac{1}{2 \\gamma} \\norm{{\\mathbf x} - \\bar{{\\mathbf x}}_t}^2.\n\\end{align*}\nNext,\n\\begin{align}\n    & \\max_{{\\mathbf y}} {F_{\\gamma}} ({\\mathbf x}_{t,K_t}, {\\mathbf y}; \\bar{{\\mathbf x}}_t)  - \\min_{{\\mathbf x}} \\max_{{\\mathbf y}} {F_{\\gamma}} ({\\mathbf x}, {\\mathbf y}; \\bar{{\\mathbf x}}_t) \\nonumber \\\\\n    & \\leq \\max_{{\\mathbf y}} {F_{\\gamma, {\\psi_t}}} ({\\mathbf x}_{t,K_t}, {\\mathbf y}; \\bar{{\\mathbf x}}_t, \\bar{{\\mathbf y}}_t)  - \\min_{{\\mathbf x}} \\max_{{\\mathbf y}} {F_{\\gamma, {\\psi_t}}} ({\\mathbf x}, {\\mathbf y}; \\bar{{\\mathbf x}}_t, \\bar{{\\mathbf y}}_t) + \\frac{D_y^2}{2 {\\psi_t}} \\nonumber \\\\\n   \n\\end{align}\nusing $\\left( L + \\frac{1}{\\gamma} - \\frac{1}{{\\psi_t}} \\right)$-smoothness of ${F_{\\gamma, {\\psi_t}}}$.\nDefine $\\Phi ({\\mathbf x}) \\triangleq \\max_{{\\mathbf y}} f({\\mathbf x}, {\\mathbf y})$. \nThen, $\\max_{{\\mathbf y}} {F_{\\gamma}} ({\\mathbf x}, {\\mathbf y}; \\bar{{\\mathbf x}}) = \\Phi({\\mathbf x}) + \\frac{1}{2 \\gamma} \\norm{{\\mathbf x} - \\bar{{\\mathbf x}}}^2$.\nWe denote ${\\bar{\\bx}_{t+1}} = {\\mathbf x}_{t,K_t}$, ${\\bx_t^{\\star}} = \\operatornamewithlimits{arg\\,min}_{{\\mathbf x}} \\max_{{\\mathbf y}} {F_{\\gamma}} ({\\mathbf x}, {\\mathbf y}; \\bar{{\\mathbf x}})$.\nFrom ()\n, we see that for some constant $c_1 > 0$ (following proof of Theorem 4.2 in \\cite{rafique18WCC_oms})\n\\begin{align}\n    & \\mathbb E \\left[ \\Phi({\\bar{\\bx}_{t+1}}) + \\frac{1}{2 \\gamma} \\norm{{\\bar{\\bx}_{t+1}} - {\\bar{\\bx}_t}}^2 - \\Phi({\\bx_t^{\\star}}) - \\frac{1}{2 \\gamma} \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}}^2 \\right] \\leq c_1 \\frac{\\sigma^2 {\\psi_t}^2}{n K_t} + \\frac{D_y^2}{2 {\\psi_t}} \\\\\n    \\Rightarrow & \\mathbb E \\Phi({\\bar{\\bx}_{t+1}}) \\leq \\mathbb E \\left[ \\Phi({\\bx_t^{\\star}}) + \\frac{1}{2 \\gamma} \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}}^2 - \\frac{1}{2 \\gamma} \\norm{{\\bar{\\bx}_{t+1}} - {\\bar{\\bx}_t}}^2 \\right] + c_1 \\frac{\\sigma^2 {\\psi_t}^2}{n K_t} + \\frac{D_y^2}{2 {\\psi_t}} \\\\\n    \\Rightarrow & \\mathbb E \\Phi({\\bar{\\bx}_{t+1}}) \\leq \\mathbb E \\Phi({\\bx_t^{\\star}}) + \\frac{1}{2 \\gamma} \\mathbb E \\left( \\frac{1}{3} \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}}^2 + 4 \\norm{{\\bar{\\bx}_{t+1}} - {\\bx_t^{\\star}}}^2 \\right) + c_1 \\frac{\\sigma^2 {\\psi_t}^2}{n K_t} + \\frac{D_y^2}{2 {\\psi_t}} \\\\\n    \\Rightarrow & \\mathbb E \\Phi({\\bar{\\bx}_{t+1}}) \\leq \\mathbb E \\Phi({\\bx_t^{\\star}}) + \\frac{1}{6 \\gamma} \\mathbb E \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}}^2 + c \\left( c_1 \\frac{\\sigma^2 {\\psi_t}^2}{n K_t} + \\frac{D_y^2}{2 {\\psi_t}} \\right) \\\\\n    \\Rightarrow & \\mathbb E \\Phi({\\bar{\\bx}_{t+1}}) \\leq \\mathbb E \\Phi({\\bar{\\bx}_t}) - \\frac{1}{3 \\gamma} \\mathbb E \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}}^2 + c \\left( c_1 \\frac{\\sigma^2 {\\psi_t}^2}{n K_t} + \\frac{D_y^2}{2 {\\psi_t}} \\right) \\\\\n    \\Rightarrow & \\mathbb E \\Phi({\\bar{\\bx}_{t+1}}) \\leq \\mathbb E \\Phi(\\bar{{\\mathbf x}}_0) - \\frac{1}{3 \\gamma} \\sum_{t=0}^{T-1} \\mathbb E \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}}^2 + c \\sum_{t=0}^{T-1} \\left( c_1 \\frac{\\sigma^2 {\\psi_t}^2}{n K_t} + \\frac{D_y^2}{2 {\\psi_t}} \\right).\n\\end{align}\nwhere\n\\begin{align}\n    \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}}^2 - \\norm{{\\bar{\\bx}_{t+1}} - {\\bar{\\bx}_t}}^2 &= \\left( \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}} + \\norm{{\\bar{\\bx}_{t+1}} - {\\bar{\\bx}_t}} \\right) \\left( \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}} - \\norm{{\\bar{\\bx}_{t+1}} - {\\bar{\\bx}_t}} \\right) \\nonumber \\\\\n    & \\leq \\left( 2 \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}} + \\norm{{\\bar{\\bx}_{t+1}} - {\\bx_t^{\\star}}} \\right) \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_{t+1}}} \\nonumber \\\\\n    & = 2 \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}} \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_{t+1}}} + \\norm{{\\bar{\\bx}_{t+1}} - {\\bx_t^{\\star}}}^2 \\nonumber \\\\\n    & = \\frac{1}{3} \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}}^2 + 4 \\norm{{\\bar{\\bx}_{t+1}} - {\\bx_t^{\\star}}}^2.\n\\end{align}\nRearranging the terms and summing over $t=0,\\hdots, T-1$, we get\n\\begin{align}\n    \\mathbb E \\norm{{\\bx_{\\tau}^{\\star}} - \\bar{{\\mathbf x}}_{\\tau}}^2 \\leq \\frac{1}{T} \\sum_{t=0}^{T-1} \\mathbb E \\norm{{\\bx_t^{\\star}} - {\\bar{\\bx}_t}}^2 \\leq \\frac{3 \\gamma}{T} \\left( \\Phi(\\bar{{\\mathbf x}}^0) - \\Phi^* + c \\sum_{t=0}^{T-1} \\left( c_1 \\frac{\\sigma^2 {\\psi_t}^2}{n K_t} + \\frac{D_y^2}{2 {\\psi_t}} \\right) \\right).\n\\end{align}\nSince $\\mathbb E \\left[ \\text{Dist} \\left( \\mathbf 0, \\partial \\Phi ({\\bx_{\\tau}^{\\star}}) \\right)^2 \\right] \\leq \\frac{1}{\\gamma^2} \\mathbb E \\norm{{\\bx_{\\tau}^{\\star}} - {\\bar{\\bx}_t}}^2$, we can ensure $\\mathbb E \\left[ \\text{Dist} \\left( \\mathbf 0, \\partial \\Phi ({\\bx_{\\tau}^{\\star}}) \\right)^2 \\right] \\leq \\epsilon^2$ by choosing\n\\begin{align*}\n    T = \\frac{3}{\\gamma \\epsilon^2} \\left( \\Phi(\\bar{{\\mathbf x}}^0) - \\Phi^* + c \\sum_{t=0}^{T-1} \\left( c_1 \\frac{\\sigma^2 {\\psi_t}^2}{n K_t} + \\frac{D_y^2}{2 {\\psi_t}} \\right) \\right)\n\\end{align*}\nUsing ${\\psi_t} = \\mathcal O \\left( \\frac{1}{\\epsilon^2} \\right), K_t = \\mathcal O \\left( \\frac{1}{n \\epsilon^6} \\right)$, we get\n\\begin{align*}\n    \\frac{1}{T} \\sum_{t=0}^{T-1} \\left( c_1 \\frac{\\sigma^2 {\\psi_t}^2}{n K_t} + \\frac{D_y^2}{2 {\\psi_t}} \\right) \\leq \\mathcal O \\left( \\epsilon^2 \\right).\n\\end{align*}\nTotal IFO complexity\n\\begin{align}\n    \\sum_{t=0}^{T-1} K_t &= \\mathcal O \\left( \\frac{1}{\\gamma \\epsilon^2} \\frac{1}{n \\epsilon^6} \\right) = \\mathcal O \\left( \\frac{1}{\\gamma n \\epsilon^8} \\right).\n\\end{align}\n\\fi\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nIn the recent years, minimax optimization theory has found relevance in several modern machine learning applications including \nGenerative Adversarial Networks (GANs) \\cite{goodfellow14GANs_neurips, arjovsky17WGANs_icml, gulrajani17improved_WGANs_neurips},\nadversarial training of neural networks \n\\cite{sinha17certifiable_robust_iclr, madry18adversarial_iclr, wang21adversarial_minmax_neurips},\nreinforcement learning \\cite{dai17learning_aistats, dai18sbeed_RL_nonlin_FA_icml},\nand robust optimization \\cite{namkoong16SG_DRO_neurips, namkoong17var_regular_neurips, mohri19agnosticFL_icml}. \nMany of these problems lie outside the domain of classical convex-concave theory \\cite{daskalakis21constr_minmax_sigact, hsieh21limits_minmax_icml}.\n\n\\ificml\n\\begin{table*}[t]\n\t\\begin{center}\n\t\t\\begin{threeparttable}\n\t\t\t\\caption{Comparison of different local-updates-based algorithms proposed to solve \\eqref{eq:problem}, in terms of the number of stochastic gradient computations (per client) and the number of communication rounds needed to reach an $\\epsilon$-stationary solution (see \\cref{defn:stationarity}) of \\eqref{eq:problem}. \n\t\t\t\tHere, $\\kappa = L_f\/\\mu$ is the condition number (see Assumptions \\ref{assum:smoothness}, \\ref{assum:PL_y}).\n\t\t\t}\n\t\t\t\\label{table:comparison}\n\t\t\t\\vskip 0.15in\n\t\t\t\\begin{small}\n\t\t\t\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\t\\hline\n\t\t\t\t\tFunction Class & Work & \\makecell{Number of Communication \\\\ Rounds} & \\makecell{Stochastic Gradient \\\\ Complexity} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multirow{3}{*}{\\begin{tabular}[c]{@{}c@{}} \\underline{N}on\\underline{C}onvex- \\\\ \\underline{S}trongly-\\underline{C}oncave \\\\\n\t\t\t\t\t\t\t(NC-SC) \\end{tabular}} &\n\t\t\t\t\tBaseline ($n=1$) \\cite{lin_GDA_icml20}\n\t\t\t\t\t& - & $\\mathcal O ( \\kappa^3\/\\epsilon^{4} )$ \\\\\n\t\t\t\t\t& \\cite{mahdavi21localSGDA_aistats} & {\\small$\\mathcal O ( \\kappa^8\/(n^{1\/3} \\epsilon^4) )$} & {\\small$\\mathcal O ( \\kappa^{12}\/(n \\epsilon^6) )$} \\\\\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}\\textbf{This Work} (Theorems \\ref{thm:NC_PL}, \\ref{thm:NC_PL_mom}) & \\cellcolor{Gainsboro!60} {\\color{red}$\\mathcal O ( \\kappa^3\/\\epsilon^{3} )$} & \\cellcolor{Gainsboro!60} {\\color{red}$\\mathcal O ( \\kappa^4\/(n \\epsilon^{4}) )$} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multirow{4}{*}{\\begin{tabular}[c]{@{}c@{}} \\underline{N}on\\underline{C}onvex-\\underline{PL} \\\\ (NC-PL) \\end{tabular}} &\n\t\t\t\t\t\\cellcolor{Gainsboro!60} \\begin{tabular}[c]{@{}c@{}}\n\t\t\t\t\t\tBaseline ($n=1$) \\\\ \\textbf{This Work} (Theorems \\ref{thm:NC_PL}, \\ref{thm:NC_PL_mom}), \\cite{yang21NCPL_arxiv}\\tnote{a}\n\t\t\t\t\t\\end{tabular}\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}- & \\cellcolor{Gainsboro!60} {\\color{red}$\\mathcal O ( \\kappa^4\/\\epsilon^{4} )$} \\\\\n\t\t\t\t\t& \\cite{mahdavi21localSGDA_aistats}\\tnote{b} & {\\small$\\mathcal O \\left( \\max \\left\\{ \\frac{\\kappa^2}{\\epsilon^4}, \\frac{\\kappa^4}{n^{2\/3} \\epsilon^4} \\right\\} \\right)$}\n\t\t\t\t\n\t\t\t\t\t& {\\small$\\mathcal O \\left( \\max \\left\\{ \\frac{\\kappa^3}{n \\epsilon^6}, \\frac{\\kappa^6}{n^2 \\epsilon^6} \\right\\} \\right)$} \\\\\n\t\t\t\t\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}\\textbf{This Work} (Theorems \\ref{thm:NC_PL}, \\ref{thm:NC_PL_mom}) & \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O ( \\kappa^3\/\\epsilon^{3} )$} & \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O ( \\kappa^4\/(n \\epsilon^{4}) )$} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multirow{3}{*}{\\begin{tabular}[c]{@{}c@{}} \\underline{N}on\\underline{C}onvex- \\\\ \\underline{C}oncave (NC-C) \\end{tabular}} &\n\t\t\t\t\tBaseline ($n=1$) \\cite{lin_GDA_icml20}\n\t\t\t\t\t& - & $\\mathcal O ( 1\/\\epsilon^{8} )$ \\\\\n\t\t\t\t\t& \\cite{mahdavi20dist_robustfl_neurips}\\tnote{c} \n\t\t\t\t\t& $\\mathcal O (1\/\\epsilon^{12})$ & $\\mathcal O ( 1\/\\epsilon^{16} )$ \\\\\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}\\textbf{This Work} (Theorem \\ref{thm:NC_C})\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}{\\color{red}\n\t\t\t\t\t\t$\\mathcal O (1\/\\epsilon^7)$}\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}{\\color{red}\n\t\t\t\t\t\t$\\mathcal O (1\/(n \\epsilon^8))$} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multirow{5}{*}{\\begin{tabular}[c]{@{}c@{}} \\underline{N}on\\underline{C}onvex- \\\\ \\underline{1}-\\underline{P}oint-\\underline{C}oncave \\\\\n\t\t\t\t\t\t\t(NC-1PC) \\end{tabular}} &\n\t\t\t\t\t\\cellcolor{Gainsboro!60} \n\t\t\t\t\tBaseline ($n=1$) \\textbf{This Work} (\\cref{thm:NC_1PC})\n\t\t\t\t\t& \\cellcolor{Gainsboro!60} - \n\t\t\t\t\t& \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O ( 1\/\\epsilon^{8} )$} \\\\\n\t\t\t\t\t& \\cite{mahdavi21localSGDA_aistats} & $\\mathcal O ( n^{1\/6}\/\\epsilon^{8} )$ & $\\mathcal O ( 1\/\\epsilon^{12} )$ \\\\\n\t\t\t\t\t& \\cite{liu20dec_GANs_neurips} & $\\widetilde{\\mathcal O} ( 1\/\\epsilon^{12} )$\\tnote{d} & $\\mathcal O ( 1\/\\epsilon^{12} )$ \\\\\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}\n\t\t\t\t\t\\textbf{This Work} (\\cref{thm:NC_1PC})\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O ( 1\/\\epsilon^{7} )$} & \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O ( 1\/\\epsilon^{8} )$} \\\\\n\t\t\t\t\t& \\cellcolor{Gainsboro!60} \\textbf{This Work} ($\\tau = 1$) (\\cref{app:NC_1PC_tau_1})\\tnote{e}\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O ( 1\/(n \\epsilon^{8}) )$} & \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O ( 1\/(n \\epsilon^{8}) )$} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\\end{tabular}\n\t\t\t\t\\begin{tablenotes}\n\t\t\t\t\t\\small\n\t\t\t\t\t\\item[a] We came across this work during the preparation of this manuscript.\n\t\t\t\t\t\\item[b] Needs the additional assumption of $G_x$-Lipschitz continuity of $f(x,y)$ in $x$.\n\t\t\t\t\t\\item[c] The loss function is nonconvex in ${\\mathbf x}$ and linear in ${\\mathbf y}$.\n\t\t\t\t\t\\item[d] Decentralized algorithm. Requires $\\mathcal O (\\log(1\/\\epsilon))$ communication rounds with the neighbors after each update step.\n\t\t\t\t\t\\item[e] This is fully synchronized Local SGDA.\n\t\t\t\t\\end{tablenotes}\n\t\t\t\n\t\t\t\\end{small}\n\t\t\t\\vskip -0.1in\n\t\t\\end{threeparttable}\n\t\\end{center}\n\\end{table*}\n\\else\n\\begin{table*}[t]\n\t\\begin{center}\n\t\t\\begin{threeparttable}\n\t\t\t\\caption{Comparison of different local-updates-based algorithms proposed to solve \\eqref{eq:problem}, in terms of the number of stochastic gradient computations (per client) and the number of communication rounds needed to reach an $\\epsilon$-stationary solution (see \\cref{defn:stationarity}) of \\eqref{eq:problem}. \n\t\t\t\tHere, $\\kappa = L_f\/\\mu$ is the condition number (see Assumptions \\ref{assum:smoothness}, \\ref{assum:PL_y}).\n\t\t\t}\n\t\t\t\\label{table:comparison}\n\t\t\t\\vskip 0.15in\n\t\t\t\\begin{small}\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\t\t\\hline\n\t\t\t\t\tFunction Class & Work & \\makecell{Number of Communication \\\\ Rounds} & \\makecell{Stochastic Gradient \\\\ Complexity} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multirow{4}{*}{\\begin{tabular}[c]{@{}c@{}} \\underline{N}on\\underline{C}onvex- \\\\ \\underline{S}trongly-\\underline{C}oncave \\\\\n\t\t\t\t\t\t\t(NC-SC) \\end{tabular}} &\n\t\t\t\t\tBaseline ($n=1$) \\cite{lin_GDA_icml20}\n\t\t\t\t\t& - & $\\mathcal O \\left( \\frac{\\kappa^3}{\\epsilon^{4}} \\right)$ \\\\\n\t\t\t\t\t& \\cite{mahdavi21localSGDA_aistats} & $\\mathcal O \\left( \\frac{\\kappa^8}{n^{1\/3} \\epsilon^4} \\right)$ & $\\mathcal O \\left( \\frac{\\kappa^{12}}{n \\epsilon^6} \\right)$ \\\\\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}\\textbf{This Work} (Theorems \\ref{thm:NC_PL}, \\ref{thm:NC_PL_mom}) & \\cellcolor{Gainsboro!60} {\\color{red}$\\mathcal O \\left( \\frac{\\kappa^3}{\\epsilon^{3}} \\right)$} & \\cellcolor{Gainsboro!60} {\\color{red}$\\mathcal O \\left( \\frac{\\kappa^4}{n \\epsilon^{4}} \\right)$} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multirow{4}{*}{\\begin{tabular}[c]{@{}c@{}} \\underline{N}on\\underline{C}onvex-\\underline{PL} \\\\ (NC-PL) \\end{tabular}} &\n\t\t\t\t\t\\cellcolor{Gainsboro!60} \\begin{tabular}[c]{@{}c@{}}\n\t\t\t\t\t\tBaseline ($n=1$) \\\\ \\textbf{This Work} (Theorems \\ref{thm:NC_PL}, \\ref{thm:NC_PL_mom}), \\cite{yang21NCPL_arxiv}\\tnote{a}\n\t\t\t\t\t\\end{tabular}\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}- & \\cellcolor{Gainsboro!60} {\\color{red}$\\mathcal O \\left( \\frac{\\kappa^4}{\\epsilon^{4}} \\right)$} \\\\\n\t\t\t\t\t& \\cite{mahdavi21localSGDA_aistats}\\tnote{b} & $\\mathcal O \\left( \\max \\left\\{ \\frac{\\kappa^2}{\\epsilon^4}, \\frac{\\kappa^4}{n^{2\/3} \\epsilon^4} \\right\\} \\right)$\n\t\t\t\t\n\t\t\t\t\t& $\\mathcal O \\left( \\max \\left\\{ \\frac{\\kappa^3}{n \\epsilon^6}, \\frac{\\kappa^6}{n^2 \\epsilon^6} \\right\\} \\right)$ \\\\\n\t\t\t\t\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}\\textbf{This Work} (Theorems \\ref{thm:NC_PL}, \\ref{thm:NC_PL_mom}) & \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O \\left( \\frac{\\kappa^3}{\\epsilon^{3}} \\right)$} & \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O \\left( \\frac{\\kappa^4}{n \\epsilon^{4}} \\right)$} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multirow{3}{*}{\\begin{tabular}[c]{@{}c@{}} \\underline{N}on\\underline{C}onvex- \\\\ \\underline{C}oncave (NC-C) \\end{tabular}} &\n\t\t\t\t\tBaseline ($n=1$) \\cite{lin_GDA_icml20}\n\t\t\t\t\t& - & $\\mathcal O ( \\epsilon^{-8} )$ \\\\\n\t\t\t\t\t& \\cite{mahdavi20dist_robustfl_neurips}\\tnote{c} \n\t\t\t\t\t& $\\mathcal O (\\epsilon^{-12})$ & $\\mathcal O ( \\epsilon^{-16} )$ \\\\\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}\\textbf{This Work} (Theorem \\ref{thm:NC_C})\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}{\\color{red}\n\t\t\t\t\t\t$\\mathcal O (\\epsilon^{-7})$}\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}{\\color{red}\n\t\t\t\t\t\t$\\mathcal O \\left( \\frac{1}{n \\epsilon^8} \\right)$} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\multirow{6}{*}{\\begin{tabular}[c]{@{}c@{}} \\underline{N}on\\underline{C}onvex- \\\\ \\underline{1}-\\underline{P}oint-\\underline{C}oncave \\\\\n\t\t\t\t\t\t\t(NC-1PC) \\end{tabular}} &\n\t\t\t\t\t\\cellcolor{Gainsboro!60} \n\t\t\t\t\tBaseline ($n=1$) \\textbf{This Work} (\\cref{thm:NC_1PC})\n\t\t\t\t\t& \\cellcolor{Gainsboro!60} - \n\t\t\t\t\t& \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O ( \\epsilon^{-8} )$} \\\\\n\t\t\t\t\t& \\cite{mahdavi21localSGDA_aistats} & $\\mathcal O \\left( \\frac{n^{1\/6}}{\\epsilon^{8}} \\right)$ & $\\mathcal O ( \\epsilon^{-12} )$ \\\\\n\t\t\t\t\t& \\cite{liu20dec_GANs_neurips} & $\\widetilde{\\mathcal O} ( \\epsilon^{-12} )$\\tnote{d} & $\\mathcal O (\\epsilon^{-12})$ \\\\\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}\n\t\t\t\t\t\\textbf{This Work} (\\cref{thm:NC_1PC})\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O ( \\epsilon^{-7} )$} & \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O ( \\epsilon^{-8} )$} \\\\\n\t\t\t\t\t& \\cellcolor{Gainsboro!60} \\textbf{This Work} ($\\tau = 1$) (\\cref{app:NC_1PC_tau_1})\\tnote{e}\n\t\t\t\t\t& \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O \\left( \\frac{1}{n \\epsilon^{8}} \\right)$} & \\cellcolor{Gainsboro!60}{\\color{red}$\\mathcal O \\left( \\frac{1}{n \\epsilon^{7}} \\right)$} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\\end{tabular}\n\t\t\t\t\\begin{tablenotes}\n\t\t\t\t\t\\small\n\t\t\t\t\t\\item[a] We came across this work during the preparation of this manuscript.\n\t\t\t\t\t\\item[b] Needs the additional assumption of $G_x$-Lipschitz continuity of $f(x,y)$ in $x$.\n\t\t\t\t\t\\item[c] The loss function is nonconvex in ${\\mathbf x}$ and linear in ${\\mathbf y}$.\n\t\t\t\t\t\\item[d] Decentralized algorithm. Requires $\\mathcal O (\\log(1\/\\epsilon))$ communication rounds with the neighbors after each update step.\n\t\t\t\t\t\\item[e] This is fully synchronized Local SGDA.\n\t\t\t\t\\end{tablenotes}\n\t\t\t\\end{small}\n\t\t\n\t\t\\end{threeparttable}\n\t\\end{center}\n\\end{table*}\n\\fi\n\n\nIn this work, we consider the following smooth\nnonconvex minimax distributed optimization problem:\n\\ificml\n{\\small\n\t\\begin{align}\n\t\t\\min_{{\\mathbf x} \\in \\mathbb R^{d_1}} \\max_{{\\mathbf y} \\in \\mathbb R^{d_2}} \\Big\\{ f({\\mathbf x}, {\\mathbf y}) := \\frac{1}{n} \\sum_{i=1}^n f_i({\\mathbf x}, {\\mathbf y}) \\Big\\},\n\t\t\\label{eq:problem}\n\t\\end{align}\n}%\n\\else\n\\begin{align}\n\t\\min_{{\\mathbf x} \\in \\mathbb R^{d_1}} \\max_{{\\mathbf y} \\in \\mathbb R^{d_2}} \\Big\\{ f({\\mathbf x}, {\\mathbf y}) := \\frac{1}{n} \\sum_{i=1}^n f_i({\\mathbf x}, {\\mathbf y}) \\Big\\},\n\t\\label{eq:problem}\n\\end{align}\n\\fi\nwhere $n$ is the number of clients, and $f_i$ represents the local loss function at client $i$, defined as $f_i({\\mathbf x}, {\\mathbf y}) = \\mathbb E_{\\xi_i \\sim \\mathcal D_i} \\left[ L({\\mathbf x}, {\\mathbf y}; \\xi_i) \\right]$.\nHere, $L(\\cdot, \\cdot; \\xi_i)$ denotes the loss for the data point $\\xi_i$, sampled from the local data distribution $\\mathcal D_i$ at client $i$.\nThe functions $\\{ f_i \\}$ are smooth, nonconvex in ${\\mathbf x}$, and concave or nonconcave in ${\\mathbf y}$.\n\nStochastic gradient descent ascent (SGDA) \\cite{heusel17gans_neurips, daskalakis18GANs_iclr}, a simple generalization of SGD \\cite{bottou18optML_siam}, is one of the simplest algorithms used to iteratively solve \\eqref{eq:problem}. \nIt carries out alternate (stochastic) gradient descent\/ascent for the min\/max problem.\nThe exact form of the convergence results depends on the (non)-convexity assumptions which the objective function $f$ in \\eqref{eq:problem} satisfies with respect to $\\mathbf{x}$ and $\\mathbf{y}$. For example,  strongly-convex strongly-concave (in ${\\mathbf x}$ and ${\\mathbf y}$, respectively), non-convex-strongly-concave, non-convex-concave, etc.\n\nMost existing literature on minimax optimization problems is focused on solving the problem at a single client.\nHowever, in big data applications that often rely on multiple sources or \\textit{clients} for data collection \\cite{xing2016strategies}, transferring the entire dataset to a single \\textit{server} is often undesirable. Doing so might be costly in applications with high-dimensional data, or altogether prohibitive due to the privacy concerns of the clients \\cite{leaute13protecting}.\n\nFederated Learning (FL) is a recent paradigm \\cite{konevcny16federated, kairouz19advancesFL_arxiv} proposed to address this problem.\nIn FL, the edge clients are not required to send their data to the server, improving the privacy afforded to the clients. Instead, the central server offloads some of its computational burden to the clients, which run the training algorithm on their local data. The models trained locally at the clients are periodically communicated to the server, which aggregates them and returns the updated model to the clients.\nThis infrequent communication with the server leads to communication savings for the clients. \nLocal Stochastic Gradient Descent (Local SGD or FedAvg) \\cite{fedavg17aistats, stich18localSGD_iclr} is one of the most commonly used algorithms for FL.\nTight convergence rates along with communication savings for Local SGD have been shown for smooth convex \\cite{khaled20localSGD_aistats, spiridonoff21comm_eff_SGD_neurips} and nonconvex \\cite{koloskova20unified_localSGD_icml} minimization problems. \nSee \\cref{app:local_SGD} for more details.\nDespite the promise shown by FL in large-scale applications \\cite{yang18FL_google_arxiv, bonawitz19towardsFL_arxiv}, much of the existing work focuses on solving standard minimization problems of the form $\\min_{\\mathbf{x}} g(\\mathbf{x})$.\nThe goals of distributed\/federated minimax optimization algorithms and their analyses are to show that by using $n$ clients, we can achieve error $\\epsilon$, not only in $n$ times fewer total iterations, but also with fewer rounds of communication with the server. This means that more local updates are performed at the clients while the coordination with the central server is less frequent.\nAlso, this $n$-fold saving in computation at the clients is referred to as \\textit{linear speedup} in the FL literature \\cite{jiang18linear_neurips, yu19icml_momentum, yang21partial_client_iclr}.\nSome recent works have attempted to achieve this goal for convex-concave \\cite{mahdavi20dist_robustfl_neurips, hou21FedSP_arxiv, liao21local_AdaGrad_CC_arxiv}, for nonconvex-concave \\cite{mahdavi20dist_robustfl_neurips}, and for nonconvex-nonconcave problems \\cite{mahdavi21localSGDA_aistats, reisizadeh20robustfl_neurips, guo20DeepAUC_icml, yuan21FedDeepAUC_icml}.\n\n\nHowever, in the context of stochastic smooth nonconvex minimax problems, the convergence guarantees of the existing distributed\/federated approaches are, to the best of our knowledge, either asymptotic \\cite{shen21fedmm_arxiv} or suboptimal \\cite{mahdavi21localSGDA_aistats}.\nIn particular, they do not reduce to the existing baseline results for the centralized minimax problems $(n=1)$. \nSee \\cref{table:comparison}.\n\n\n\\paragraph{Our Contributions.}\nIn this paper, we consider the following four classes of minimax optimization problems and refer to them using the abbreviations given below:\n\\ificml\n\\newline\n1) NC-SC: \\underline{N}on\\underline{C}onvex in ${\\mathbf x}$, \\underline{S}trongly-\\underline{C}oncave in ${\\mathbf y}$,\n2) NC-PL: \\underline{N}on\\underline{C}onvex in ${\\mathbf x}$, \\underline{PL}-condition in ${\\mathbf y}$ (\\cref{assum:PL_y}),\n3) NC-C: \\underline{N}on\\underline{C}onvex in ${\\mathbf x}$, \\underline{C}oncave in ${\\mathbf y}$,\n4) NC-1PC: \\underline{N}on\\underline{C}onvex in ${\\mathbf x}$, \\underline{1}-\\underline{P}oint-\\underline{C}oncave in ${\\mathbf y}$ (\\cref{assum:1pc_y}).\n\\newline\n\\else\n\\begin{enumerate}\n\t\\item NC-SC: \\underline{N}on\\underline{C}onvex in ${\\mathbf x}$, \\underline{S}trongly-\\underline{C}oncave in ${\\mathbf y}$,\n\t\\item NC-PL: \\underline{N}on\\underline{C}onvex in ${\\mathbf x}$, \\underline{PL}-condition in ${\\mathbf y}$ (\\cref{assum:PL_y}),\n\t\\item NC-C: \\underline{N}on\\underline{C}onvex in ${\\mathbf x}$, \\underline{C}oncave in ${\\mathbf y}$,\n\t\\item NC-1PC: \\underline{N}on\\underline{C}onvex in ${\\mathbf x}$, \\underline{1}-\\underline{P}oint-\\underline{C}oncave in ${\\mathbf y}$ (\\cref{assum:1pc_y}).\n\\end{enumerate}\n\\fi\nFor each of these problems, we improve the convergence analysis of existing algorithms or propose a new local-update-based algorithm that gives a better sample complexity. \nA key feature of our results is the linear speedup in the sample complexity with respect to the number of clients, while also providing communication savings. We make the following main contributions, also summarized in \\cref{table:comparison}.\n\n\\begin{itemize}[leftmargin=*]\n\t\\setlength\\itemsep{-0.5em}\n\t\\item For NC-PL functions (\\cref{sec:NC_PL}), we prove that Local SGDA\n\n\thas {\\small$\\mathcal O (\\kappa^4\/(n \\epsilon^{4}))$} gradient complexity, and {\\small$\\mathcal O (\\kappa^3\/\\epsilon^{3})$} communication cost (\\cref{thm:NC_PL}).\n\n\tThe results are optimal in $\\epsilon$.\\footnote{Even for simple nonconvex function minimization, the complexity guarantee cannot be improved beyond {\\small$\\mathcal O (1\/\\epsilon^{4})$} \\cite{arjevani19lower_stoch_NC_arxiv}. Further, our results match the complexity and communication guarantees for simple smooth nonconvex minimization with local SGD \\cite{yu19icml_momentum}.}\n\tTo the best of our knowledge, this complexity guarantee does not exist in the prior literature even for $n=1$.\\footnote{During the preparation of this manuscript, we came across the centralized minimax work \\cite{yang21NCPL_arxiv}, which achieves {\\small$\\mathcal O (\\kappa^4\/ \\epsilon^{4})$} complexity for NC-PL functions. However, our work is more general since we incorporate local updates at the clients.}\n\t\\item Since the PL condition is weaker than strong-concavity, our result also extends to NC-SC functions.\n\tTo the best of our knowledge, ours is the first work to prove optimal (in $\\epsilon$) guarantees for SDGA in the case of NC-SC functions, with $\\mathcal O (1)$ batch-size. This way, we improve the result in \\cite{lin_GDA_icml20} which necessarily requires {\\small$\\mathcal O (1\/\\epsilon^2)$} batch-sizes.\n\tIn the federated setting, ours is the first work to achieve the optimal (in $\\epsilon$) guarantee.\n\t\\item We propose a novel algorithm (Momentum Local SGDA - \\cref{alg_NC_momentum}), which achieves the same theoretical guarantees as Local SGDA for NC-PL functions\n\t(\\cref{thm:NC_PL_mom}), and also outperforms Local SGDA in experiments.\n\t\\item For NC-C functions (\\cref{sec:NC_C}), we utilize Local SGDA+ algorithm proposed in \\cite{mahdavi21localSGDA_aistats}\\footnote{\\cite{mahdavi21localSGDA_aistats} does not analyze NC-C functions.}, and prove {\\small$\\mathcal O (1\/(n \\epsilon^{8}))$} gradient complexity, and {\\small$\\mathcal O (1\/\\epsilon^{7})$} communication cost (\\cref{thm:NC_C}).\n\tThis implies linear speedup over the $n=1$ result \\cite{lin_GDA_icml20}.\n\t\\item For NC-1PC functions (\\cref{sec:NC_1PC}), using an improved analysis for Local SGDA+,\n\twe prove {\\small$\\mathcal O (1\/\\epsilon^{8})$} gradient complexity, and {\\small$\\mathcal O (1\/\\epsilon^{7})$} communication cost (\\cref{thm:NC_1PC}).\n\tTo the best of our knowledge, this result is the first to generalize the existing {\\small$\\mathcal O (1\/\\epsilon^{8})$} complexity guarantee of SGDA (proved for NC-C problems in \\cite{lin_GDA_icml20}), to the more general class of NC-1PC functions. \n\\end{itemize}\n\n\n\\section{Related Work}\n\\label{sec:related_work}\n\n\\subsection{Single client minimax}\n\n\nUntil recently, the minimax optimization literature was focused largely on convex-concave problems \\cite{nemirovski04prox_siam, nedic09subgradient_jota}.\nHowever, since the advent of machine learning applications such as GANs \\cite{goodfellow14GANs_neurips}, and adversarial training of neural networks (NNs) \\cite{madry18adversarial_iclr}, the more challenging problems of nonconvex-concave and nonconvex-nonconcave minimax optimization have attracted increasing attention.\n\n\\paragraph{Nonconvex-Strongly Concave (NC-SC) Problems.}\nFor stochastic NC-SC problems, \\cite{lin_GDA_icml20} proved {\\small$\\mathcal O (\\kappa^3\/\\epsilon^{4})$} stochastic gradient complexity for SGDA.\nHowever, the analysis necessarily requires mini-batches of size {\\small$\\Theta (\\epsilon^{-2})$}.\nUtilizing momentum, \\cite{qiu20single_timescale_ncsc} achieved the same {\\small$\\mathcal O (\\epsilon^{-4})$} convergence rate with {\\small$\\mathcal O (1)$} batch-size.\n\\cite{qiu20single_timescale_ncsc, luo20SREDA_ncsc_neurips} utilize variance-reduction to further improve the complexity to {\\small$\\mathcal O (\\kappa^3\/\\epsilon^{3})$}.\nHowever, whether these guarantees can be achieved in the federated setting, with multiple local updates at the clients, is an open question.\nIn this paper, we answer this question in the affirmative.\n\n\\paragraph{Nonconvex-Concave (NC-C) Problems.}\nThe initial algorithms \\cite{nouiehed19minimax_neurips19, thekumparampil19NC_C_neurips, rafique18WCC_oms} for deterministic NC-C problems all have a nested-loop structure. For each ${\\mathbf x}$-update, the inner maximization with respect to ${\\mathbf y}$ is approximately solved. Single-loop algorithms have been proposed in subsequent works by \\cite{tomluo_1_loop_ncc_neurips20, lan_unified_ncc_arxiv20}.\nHowever, for stochastic problems, to the best of our knowledge, \\cite{lin_GDA_icml20} is the only work to have analyzed a single-loop algorithm (SGDA), which achieves {\\small$\\mathcal O (1\/\\epsilon^{8})$} complexity.\n\n\\paragraph{Nonconvex-Nonconcave (NC-NC) Problems.}\nRecent years have seen extensive research on NC-NC problems \\cite{mertikopoulos18optMD_SP_iclr, diakonikolas21NC_NC_aistats, daskalakis21constr_minmax_sigact}.\nHowever, of immediate interest to us are two special classes of functions.\n\\newline\n1) Polyak-{\\L}ojasiewicz (PL) condition \\cite{polyak63PL} is weaker than strong concavity, and does not even require the objective to be concave.\nRecently, PL-condition has been shown to hold in overparameterized neural networks \\cite{charles18generalization_icml, liu22overparameter_NN_elsevier}.\nDeterministic NC-PL problems have been analyzed in \\cite{nouiehed19minimax_neurips19, yang20NCNC_VR_neurips, fiez21NC_PL_SC_neurips}.\nDuring the preparation of this manuscript, we came across \\cite{yang21NCPL_arxiv} which solves stochastic NC-PL minimax problems. \nStochastic alternating gradient descent ascent (Stoc-AGDA) is proposed, which achieves {\\small$\\mathcal O (\\kappa^4\/\\epsilon^{4})$} iteration complexity. \nFurther, another single-loop algorithm, \\textit{smoothed GDA} is proposed, which improves dependence on $\\kappa$ to {\\small$\\mathcal O (\\kappa^2\/\\epsilon^{4})$}.\n\\newline\n2) One-Point-Concavity\/convexity (1PC) has been observed in the dynamics of SGD for optimizing neural networks \\cite{li17relu_neurips, kleinberg18icml}.\nDeterministic and stochastic optimization guarantees for 1PC functions have been proved in \\cite{gasnikov17acc_quasar_convex_arxiv, hinder20near_opt_star_convex_colt, jin20quasar_convex_arxiv}.\nNC1PC minimax problems have been considered in \\cite{mertikopoulos18optMD_SP_iclr} with asymptotic convergence results, and in  \\cite{liu20dec_GANs_neurips}, with $\\mathcal O (1\/\\epsilon^{12})$ gradient complexity.\nAs we show in \\cref{sec:NC_1PC}, this complexity result can be significantly improved.\n\n\n\\subsection{Distributed\/Federated Minimax}\n\nRecent years have seen a spur of interest in distributed minimax problems, driven by the need to train neural networks over multiple clients \\cite{liu20dec_GANs_neurips, chen20dist_GAN_quantize_arxiv}.\nSaddle-point problems and more generally variational inequalities have been studied extensively in the context of decentralized optimization by \\cite{beznosikov20dist_SP_arxiv, gasnikov21dec_stoch_EG_VI_arxiv, beznosikov21dist_sp_neurips, rogozin21dec_local_global_var_cc_arxiv, xian21dec_ncsc_storm_neurips}.\n\nLocal updates-based algorithms for convex-concave problems have been analyzed in \\cite{mahdavi20dist_robustfl_neurips, hou21FedSP_arxiv, liao21local_AdaGrad_CC_arxiv}.\n\\cite{reisizadeh20robustfl_neurips} considers PL-PL and NC-PL minimax problems in the federated setting.\nHowever, the clients only communicate min variables to the server. \nThe limited client availability problem of FL is considered for NC-PL problems in \\cite{xie21NC_PL_FL_arxiv}. \nHowever, the server is responsible for additional computations, to compute the global gradient estimates.\nIn our work, we consider a more general setting, where both the min and max variables need to be communicated to the server periodically.\nThe server is more limited in functionality, and only computes and returns the averages to the clients.\n\\cite{mahdavi20dist_robustfl_neurips} shows a suboptimal convergence rate for nonconvex-linear minimax problems (see \\cref{table:comparison}). \nWe consider more general NC-C problems, improve the convergence rate, and show linear speedup in $n$.\n\n\n\\paragraph{Comparison with \\cite{mahdavi21localSGDA_aistats}.}\nThe work most closely related to ours is \\cite{mahdavi21localSGDA_aistats}. \nThe authors consider three classes of smooth nonconvex minimax functions: NC-SC, NC-PL, and NC-1PC. \nHowever, the gradient complexity and communication cost results achieved are suboptimal.\nFor all three classes of functions, we provide tighter analyses, resulting in improved gradient complexity with improved communication savings.\nSee \\cref{table:comparison} for a comprehensive comparison of results.\n\n\n\\section{Preliminaries}\n\\label{sec:prelim}\n\n\\ificml\n\\paragraph{Notations.} Throughout the paper, we let $\\norm{\\cdot}$ denote the Euclidean norm $\\norm{\\cdot}_2$.\nGiven a positive integer $m$, the set of numbers $\\{ 1, 2, \\hdots, m \\}$ is denoted by $[m]$. Vectors at client $i$ are denoted with superscript $i$, for e.g., ${\\mathbf x}^i$.\nVectors at time $t$ are denoted with subscript $t$, for e.g., ${\\mathbf y}_t$.\nAverage across clients appear without a superscript, for e.g., {\\small${\\mathbf x_t} = \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_t}$}.\nWe define the gradient vector as {\\small$\\nabla f_i({\\mathbf x}, {\\mathbf y}) = \\left[ \\nabla_{\\bx} f_i({\\mathbf x}, {\\mathbf y})^{\\top}, \\nabla_{\\by} f_i({\\mathbf x}, {\\mathbf y})^{\\top} \\right]^{\\top}$}.\nFor a generic function {\\small$g({\\mathbf x}, {\\mathbf y})$}, we denote its stochastic gradient vector as {\\small$\\nabla g({\\mathbf x}, {\\mathbf y}; \\xi^i) = \\left[ \\nabla_{\\bx} g({\\mathbf x}, {\\mathbf y}; \\xi^i)^{\\top}, \\nabla_{\\by} g({\\mathbf x}, {\\mathbf y}; \\xi^i)^{\\top} \\right]^{\\top}$}, where $\\xi^i$ denotes the randomness.\n\\else\n\\paragraph{Notations.} Throughout the paper, we let $\\norm{\\cdot}$ denote the Euclidean norm $\\norm{\\cdot}_2$.\nGiven a positive integer $m$, the set of numbers $\\{ 1, 2, \\hdots, m \\}$ is denoted by $[m]$. Vectors at client $i$ are denoted with superscript $i$, for e.g., ${\\mathbf x}^i$.\nVectors at time $t$ are denoted with subscript $t$, for e.g., ${\\mathbf y}_t$.\nAverage across clients appear without a superscript, for e.g., ${\\mathbf x_t} = \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_t}$.\nWe define the gradient vector as $\\nabla f_i({\\mathbf x}, {\\mathbf y}) = \\left[ \\nabla_{\\bx} f_i({\\mathbf x}, {\\mathbf y})^{\\top}, \\nabla_{\\by} f_i({\\mathbf x}, {\\mathbf y})^{\\top} \\right]^{\\top}$.\nFor a generic function $g({\\mathbf x}, {\\mathbf y})$, we denote its stochastic gradient vector as $\\nabla g({\\mathbf x}, {\\mathbf y}; \\xi^i) = \\left[ \\nabla_{\\bx} g({\\mathbf x}, {\\mathbf y}; \\xi^i)^{\\top}, \\nabla_{\\by} g({\\mathbf x}, {\\mathbf y}; \\xi^i)^{\\top} \\right]^{\\top}$, where $\\xi^i$ denotes the randomness.\n\\fi\n\n\\paragraph{Convergence Metrics.} Since the loss function $f$ is nonconvex, we cannot prove convergence to a global saddle point. \nWe instead prove convergence to an \\textit{approximate} stationary point, which is defined next.\n\n\\begin{definition}[$\\epsilon$-Stationarity]\n\t\\label{defn:stationarity}\n\tA point $\\widetilde{\\bx}$ is an $\\epsilon$-stationary point of a differentiable function $g$ if $\\norm{\\nabla g (\\widetilde{\\bx})} \\leq \\epsilon$.\n\\end{definition}\n\n\\begin{definition}\n\tStochastic Gradient (SG) complexity is the total number of gradients computed by a single client during the course of the algorithm.\n\\end{definition}\n\nSince all the algorithms analyzed in this paper are single-loop and use a $\\mathcal O (1)$ batchsize, if the algorithm runs for $T$ iterations, then the SG complexity is $\\mathcal O (T)$.\n\nDuring a communication round, the clients send their local vectors to the server, where the aggregate is computed, and communicated back to the clients.\nConsequently, we define the number of communication rounds as follows.\n\n\\begin{definition}[Communication Rounds]\n\tThe number of communication rounds in an algorithm is the number of times clients communicate their local models to the server.\n\\end{definition}\nIf the clients perform $\\tau$ local updates between successive communication rounds, the total number of communication rounds is $\\lceil T\/\\tau \\rceil$.\nNext, we discuss the assumptions that will be used throughout the rest of the paper.\n\n\\begin{assump}[Smoothness]\n\t\\label{assum:smoothness}\n\tEach local function $f_i$ is differentiable and has Lipschitz continuous gradients.\n\tThat is, there exists a constant $L_f > 0$ such that at each client $i \\in [n]$, for all ${\\mathbf x}, {\\mathbf x}' \\in \\mathbb R^{d_1}$ and ${\\mathbf y}, {\\mathbf y}' \\in \\mathbb R^{d_2}$,\n\t\\ificml\n\t\\newline\n\t$\\left\\| \\nabla f_i({\\mathbf x}, {\\mathbf y}) - \\nabla f_i({\\mathbf x}', {\\mathbf y}') \\right\\| \\leq L_f \\left\\| ({\\mathbf x}, {\\mathbf y}) - ({\\mathbf x}', {\\mathbf y}') \\right\\|$.\n\t\\else\n\t\\begin{align*}\n\t\t\\left\\| \\nabla f_i({\\mathbf x}, {\\mathbf y}) - \\nabla f_i({\\mathbf x}', {\\mathbf y}') \\right\\| \\leq L_f \\left\\| ({\\mathbf x}, {\\mathbf y}) - ({\\mathbf x}', {\\mathbf y}') \\right\\|.\n\t\\end{align*}\n\t\\fi\n\\end{assump}\n\n\\begin{assump}[Bounded Variance]\n\t\\label{assum:bdd_var}\n\tThe stochastic gradient oracle at each client is unbiased with bounded variance, i.e., there exists a constant $\\sigma > 0$ such that at each client $i \\in [n]$, for all ${\\mathbf x}, {\\mathbf y}$,\n\t\\ificml\n\t$\\mathbb E_{\\xi_i} [ \\nabla f_i({\\mathbf x}, {\\mathbf y}; \\xi^i) ] = \\nabla f_i({\\mathbf x}, {\\mathbf y})$, and $\\mathbb E_{\\xi_i} \\| \\nabla f_i({\\mathbf x}, {\\mathbf y}; \\xi^i) - \\nabla f_i({\\mathbf x}, {\\mathbf y}) \\|^2 \\leq \\sigma^2.$\n\t\\else\n\t\\begin{align*}\n\t\t\\mathbb E_{\\xi_i} [ \\nabla f_i({\\mathbf x}, {\\mathbf y}; \\xi^i) ] &= \\nabla f_i({\\mathbf x}, {\\mathbf y}), \\\\\n\t\t\\mathbb E_{\\xi_i} \\| \\nabla f_i({\\mathbf x}, {\\mathbf y}; \\xi^i) - \\nabla f_i({\\mathbf x}, {\\mathbf y}) \\|^2 & \\leq \\sigma^2.\n\t\\end{align*}\n\t\\fi\n\t\n\\end{assump}\n\n\\begin{assump}[Bounded Heterogeneity]\n\t\\label{assum:bdd_hetero}\n\tTo measure the heterogeneity of the local functions $\\{ f_i({\\mathbf x}, {\\mathbf y}) \\}$ across the clients, we define\n\t\\ificml\n\t\\newline\n\t{\\small$\\varsigma_x^2 = \\sup_{{\\mathbf x} \\in \\mathbb R^{d_1}, {\\mathbf y} \\in \\mathbb R^{d_2}} \\frac{1}{n} \\textstyle \\sum_{i=1}^n \\left\\| \\nabla_{\\bx} f_i({\\mathbf x}, {\\mathbf y}) - \\nabla_{\\bx} f({\\mathbf x}, {\\mathbf y}) \\right\\|^2,$}\n\t\\newline\n\t{\\small$\\varsigma_y^2 = \\sup_{{\\mathbf x} \\in \\mathbb R^{d_1}, {\\mathbf y} \\in \\mathbb R^{d_2}} \\frac{1}{n} \\textstyle \\sum_{i=1}^n \\left\\| \\nabla_{\\by} f_i({\\mathbf x}, {\\mathbf y}) - \\nabla_{\\by} f({\\mathbf x}, {\\mathbf y}) \\right\\|^2.$}\n\tWe assume that $\\varsigma_x$ and $\\varsigma_y$ are bounded.\n\t\\else\n\t\\newline\n\t\\begin{align*}\n\t\t\\varsigma_x^2 &= \\sup_{{\\mathbf x} \\in \\mathbb R^{d_1}, {\\mathbf y} \\in \\mathbb R^{d_2}} \\frac{1}{n} \\textstyle \\sum_{i=1}^n \\left\\| \\nabla_{\\bx} f_i({\\mathbf x}, {\\mathbf y}) - \\nabla_{\\bx} f({\\mathbf x}, {\\mathbf y}) \\right\\|^2, \\\\\n\t\t\\varsigma_y^2 &= \\sup_{{\\mathbf x} \\in \\mathbb R^{d_1}, {\\mathbf y} \\in \\mathbb R^{d_2}} \\frac{1}{n} \\textstyle \\sum_{i=1}^n \\left\\| \\nabla_{\\by} f_i({\\mathbf x}, {\\mathbf y}) - \\nabla_{\\by} f({\\mathbf x}, {\\mathbf y}) \\right\\|^2.\n\t\\end{align*}\n\tWe assume that $\\varsigma_x$ and $\\varsigma_y$ are bounded.\n\t\\fi\n\\end{assump}\n\n\n\\section{Algorithms and their Convergence Analyses}\n\\label{sec:algo_theory}\n\nIn this section, we discuss local updates-based algorithms to solve nonconvex-concave and nonconvex-nonconcave minimax problems.\nEach client runs multiple update steps on its local models using local stochastic gradients.\nPeriodically, the clients communicate their local models to the server, which returns the average model.\nIn this section, we demonstrate that this leads to communication savings at the clients, without sacrificing the convergence guarantees.\n\n\nIn the subsequent subsections, for each class of functions considered (NC-PL, NC-C, NC-1PC), we first discuss an algorithm.\nNext, we present the convergence result, followed by a discussion of the gradient complexity and the communication cost needed to reach an $\\epsilon$ stationary point.\nSee \\cref{table:comparison} for a summary of our results, along with comparisons with the existing literature.\n\n\n\\subsection{Nonconvex-PL (NC-PL) Problems} \\label{sec:NC_PL}\n\nIn this subsection, we consider smooth nonconvex functions which satisfy the following assumption.\n\n\\begin{assump}[Polyak {\\L}ojasiewicz (PL) Condition in ${\\mathbf y}$]\n\t\\label{assum:PL_y}\n\tThe function $f$ satisfies $\\mu$-PL condition in ${\\mathbf y}$ ($\\mu > 0$), if for any fixed ${\\mathbf x}$: 1) $\\max_{{\\mathbf y}'} f({\\mathbf x}, {\\mathbf y}')$ has a nonempty solution set; \n\t2) {\\small$\\norm{\\nabla_{\\by} f({\\mathbf x}, {\\mathbf y})}^2 \\geq 2 \\mu ( \\max_{{\\mathbf y}'} f({\\mathbf x}, {\\mathbf y}') - f({\\mathbf x}, {\\mathbf y}) )$}, for all ${\\mathbf y}$.\n\\end{assump}\n\nFirst, we present an improved convergence result for Local SGDA (\\cref{alg_local_SGDA}), proposed in \\cite{mahdavi21localSGDA_aistats}. Then we propose a novel momentum-based algorithm (\\cref{alg_NC_momentum}), which achieves the same convergence guarantee, and has improved empirical performance (see \\cref{sec:exp}).\n\n\n\\paragraph{Improved Convergence of Local SGDA.}\nLocal Stochastic Gradient Descent Ascent (SGDA) (\\cref{alg_local_SGDA}) proposed in \\cite{mahdavi21localSGDA_aistats}, is a simple extension of the centralized algorithm SGDA \\cite{lin_GDA_icml20}, to incorporate local updates at the clients. At each time $t$, clients updates their local models $\\{ {\\mathbf x^i_t}, {\\mathbf y^i_t} \\}$ using local stochastic gradients $\\{ \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}}), \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}}) \\}$.\nOnce every $\\tau$ iterations, the clients communicate $\\{ {\\mathbf x^i_t}, {\\mathbf y^i_t} \\}$ to the server, which computes the average models $\\{ {\\mathbf x_t}, {\\mathbf y_t} \\}$, and returns these to the clients.\nNext, we discuss the finite-time convergence of \\cref{alg_local_SGDA}.\nWe prove convergence to an approximate stationary point of the envelope function $\\Phi({\\mathbf x}) = \\max_{\\mathbf y} f({\\mathbf x}, {\\mathbf y})$.\\footnote{Under Assumptions \\ref{assum:smoothness}, \\ref{assum:PL_y}, $\\Phi$ is smooth \\cite{nouiehed19minimax_neurips19}.}\n\n\\begin{algorithm}[ht]\n\t\\caption{Local SGDA \\cite{mahdavi21localSGDA_aistats}}\n\t\\label{alg_local_SGDA}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE{\\textbf{Input: }{\\small${\\mathbf x}_0^i = {\\mathbf x}_0, {\\mathbf y}_0^i = {\\mathbf y}_0$}, for all $i \\in [n]$; step-sizes $\\eta_x, \\eta_y$; $\\tau$, $T$}\n\t\t\\FOR[At all clients $i=1,\\hdots, n$]{$t=0$ to $T-1$}\n\t\t\\STATE{Sample minibatch ${\\xi^i_{t}}$ from local data}\n\t\t\\STATE{${\\mathbf x^i_{t+1}} = {\\mathbf x^i_t} - \\eta_x \\nabla_{\\bx} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}})$}\n\t\t\\STATE{${\\mathbf y^i_{t+1}} = {\\mathbf y^i_t} + \\eta_y \\nabla_{\\by} f_i ({\\mathbf x^i_t}, {\\mathbf y^i_t}; {\\xi^i_{t}})$}\n\t\t\\IF{$t+1$ mod $\\tau = 0$}\n\t\t\\STATE{Clients send $\\{ {\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}} \\}$ to the server}\n\t\t\\STATE{Server computes averages ${\\mathbf x_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_{t+1}}$, \n\t\t\t${\\mathbf y_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf y^i_{t+1}}$, and sends to all the clients}\n\t\t\\STATE{${\\mathbf x^i_{t+1}} = {\\mathbf x_{t+1}}$, ${\\mathbf y^i_{t+1}} = {\\mathbf y_{t+1}}$, for all $i \\in [n]$}\n\t\t\\ENDIF\n\t\t\\ENDFOR\n\t\t\\STATE{\\textbf{Return: }${\\bar{\\bx}_T}$ drawn uniformly at random from $\\{ {\\mathbf x_t} \\}_{t=1}^T$, where ${\\mathbf x_t} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_t}$}\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\begin{theorem}\n\t\\label{thm:NC_PL}\n\tSuppose the local loss functions $\\{ f_i \\}_i$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, and the global function $f$ satisfies \\cref{assum:PL_y}.\n\tSuppose the step-sizes $\\eta_x, \\eta_y$ are chosen such that {\\small$\\eta_y \\leq \\frac{1}{8 L_f \\tau}$, $\\frac{\\eta_x}{\\eta_y} \\leq \\frac{1}{8 \\kappa^2}$}, where {\\small$\\kappa = L_f\/\\mu$} is the condition number.\n\tThen, for the output ${\\bar{\\bx}_T}$ of \\cref{alg_local_SGDA}, the following holds.\n\t\\ificml\n\t\\vspace{-3mm}\n\t{\\small\n\t\t\\begin{equation}\n\t\t\t\\begin{aligned}\n\t\t\t\t& \\mathbb E \\norm{\\nabla \\Phi ({\\bar{\\bx}_T})}^2 \\leq \\underbrace{\\mathcal O \\left( \\kappa^2 \\left[ \\frac{\\Delta_{\\Phi}}{\\eta_y T} + \\frac{\\eta_y \\sigma^2}{n} \\right] \\right)}_{\\text{Error with full synchronization}} \\\\\n\t\t\t\t& \\qquad + \\underbrace{\\mathcal O \\left( \\kappa^2 (\\tau-1)^2 \\left[ \\eta_y^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) + \\eta_x^2 \\varsigma_x^2 \\right] \\right)}_{\\text{Error due to local updates}},\n\t\t\t\\end{aligned}\n\t\t\t\\label{eq:thm:NC_PL}\n\t\t\\end{equation}\n\t}%\n\t\\else\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t& \\mathbb E \\norm{\\nabla \\Phi ({\\bar{\\bx}_T})}^2 \\leq \\underbrace{\\mathcal O \\left( \\kappa^2 \\left[ \\frac{\\Delta_{\\Phi}}{\\eta_y T} + \\frac{\\eta_y \\sigma^2}{n} \\right] \\right)}_{\\text{Error with full synchronization}} + \\underbrace{\\mathcal O \\left( \\kappa^2 (\\tau-1)^2 \\left[ \\eta_y^2 \\left( \\sigma^2 + \\varsigma_y^2 \\right) + \\eta_x^2 \\varsigma_x^2 \\right] \\right)}_{\\text{Error due to local updates}},\n\t\t\\end{aligned}\n\t\t\\label{eq:thm:NC_PL}\n\t\\end{equation}\n\t\\fi\n\twhere {\\small$\\Phi(\\cdot) \\triangleq \\max_{\\mathbf y} f(\\cdot, {\\mathbf y})$} is the envelope function, {\\small$\\Delta_{\\Phi} \\triangleq \\Phi ({\\mathbf x}_0) - \\min_{\\mathbf x} \\Phi ({\\mathbf x})$}.\n\tUsing {\\small$\\eta_x = \\mathcal O ( \\frac{1}{\\kappa^2} \\sqrt{\\frac{n}{T}} )$, $\\eta_y = \\mathcal O ( \\sqrt{n\/T} )$}, we can bound {\\small$\\mathbb E \\norm{\\nabla \\Phi ({\\bar{\\bx}_T})}^2$} as\n\t\\ificml\n\t{\\small\n\t\t\\begin{align}\n\t\t\t& \\mathcal O \\Big( \\frac{\\kappa^2 ( \\sigma^2 + \\Delta_{\\Phi} )}{\\sqrt{n T}} + \\kappa^2 (\\tau-1)^2 \\frac{n ( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 )}{T} \\Big).\n\t\t\t\\label{eq:thm:NC_PL_conv_rate}\n\t\t\\end{align}\n\t}\n\t\\else\n\t\\begin{align}\n\t\t& \\mathcal O \\Big( \\frac{\\kappa^2 ( \\sigma^2 + \\Delta_{\\Phi} )}{\\sqrt{n T}} + \\kappa^2 (\\tau-1)^2 \\frac{n ( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 )}{T} \\Big).\n\t\t\\label{eq:thm:NC_PL_conv_rate}\n\t\\end{align}\n\t\\fi\n\\end{theorem}\n\n\\begin{proof}\n\tSee \\cref{app:ncpl}.\n\\end{proof}\n\n\\begin{remark}\n\t\\label{rem:NC_PL_local_SGDA_1}\n\tThe first term of the error decomposition in \\eqref{eq:thm:NC_PL} represents the optimization error for a fully synchronous algorithm ($\\tau = 1$), in which the local models are averaged after every update.\n\tThe second term arises due to the clients carrying out multiple $(\\tau > 1)$ local updates between successive communication rounds.\n\tThis term is impacted by the data heterogeneity across clients $\\varsigma_x, \\varsigma_y$.\n\tSince the dependence on step-sizes $\\eta_x, \\eta_y$ is quadratic, as seen in \\eqref{eq:thm:NC_PL_conv_rate}, for small enough $\\eta_x, \\eta_y$, and carefully chosen $\\tau$, having multiple local updates does not impact the asymptotic convergence rate $\\mathcal O (1\/\\sqrt{nT})$.\n\\end{remark}\n\n\\begin{cor}\n\t\\label{cor:NC_PL_comm_cost}\n\tTo reach an $\\epsilon$-accurate point ${\\bar{\\bx}_T}$, assuming $T \\geq \\Theta (n^3)$, the stochastic gradient complexity of \\cref{alg_local_SGDA} is $\\mathcal O (\\kappa^4\/(n \\epsilon^4))$.\n\tThe number of communication rounds required for the same is $T\/\\tau = \\mathcal O ( \\kappa^3\/\\epsilon^{3} )$.\n\\end{cor}\n\n\\begin{remark}\n\t\\label{rem:NC_PL_local_SGDA_2}\n\tOur analysis improves the existing complexity results for Local SGDA \\cite{mahdavi21localSGDA_aistats}.\n\tThe analysis in \\cite{mahdavi21localSGDA_aistats} also requires the additional assumption of $G_x$-Lipschitz continuity of $f(\\cdot, {\\mathbf y})$, which we do not need.\n\tThe complexity result is optimal in $\\epsilon$.\\footnote{In terms of dependence on $\\epsilon$, our complexity and communication results match the corresponding results for the simple smooth nonconvex minimization with local SGD \\cite{yu19icml_momentum}.}\n\tTo the best of our knowledge, this complexity guarantee does not exist in the prior literature even for $n=1$.\\footnote{During the preparation of this manuscript, we came across the centralized minimax work \\cite{yang21NCPL_arxiv}, which achieves {\\small$\\mathcal O (\\kappa^4\/ \\epsilon^{4})$}, using stochastic alternating GDA.}\n\tFurther, we also provide communication savings, requiring model averaging only once every $\\mathcal O ( \\kappa\/(n \\epsilon) )$ iterations.\n\\end{remark}\n\n\\begin{remark}[Nonconvex-Strongly-Concave (NC-SC) Problems]\n\t\\label{rem:NC_PL_local_SGDA_3}\n\tSince the PL condition is more general than strong concavity, we also achieve the above result for NC-SC minimax problems.\n\tMoreover, unlike the analysis in \\cite{lin_GDA_icml20} which necessarily requires $\\mathcal O (1\/\\epsilon^{2})$ batch-sizes, to the best of our knowledge, ours is the first result to achieve $\\mathcal O (1\/\\epsilon^{4})$ rate for SGDA with $\\mathcal O (1)$ batch-size.\n\\end{remark}\n\n\n\\paragraph{Momentum-based Local SGDA.}\n\nNext, we propose a novel momentum-based local updates algorithm (\\cref{alg_NC_momentum}) for NC-PL minimax problems.\nThe motivation behind using momentum in local updates is to control the effect of stochastic gradient noise,\nvia historic averaging of stochastic gradients.\nSince momentum is widely used in practice for training deep neural networks, it is a natural question to ask, whether the same theoretical guarantees as Local SGDA can be proved for a momentum-based algorithm.\nA similar question has been considered in \\cite{yu19icml_momentum} in the context of smooth minimization problems.\n\\cref{alg_NC_momentum} is a local updates-based extension of the approach proposed in \\cite{qiu20single_timescale_ncsc} for centralized problems.\nAt each step, each client uses momentum-based gradient estimators {\\small$\\{ {\\mathbf d^i_{x,t}}, {\\mathbf d^i_{y,t}} \\}$} to arrive at intermediate iterates {\\small$\\{ \\Tbx^i_{t+\\frac{1}{2}}, \\Tby^i_{t+\\frac{1}{2}} \\}$}.\nThe local updated model is a convex combination of the intermediate iterate and the current model.\nOnce every $\\tau$ iterations, the clients communicate {\\small$\\{ {\\mathbf x^i_t}, {\\mathbf y^i_t}, {\\mathbf d^i_{x,t}}, {\\mathbf d^i_{y,t}} \\}$} to the server, which computes the averages {\\small$\\{ {\\mathbf x_t}, {\\mathbf y_t}, {\\mathbf d_{x,t}}, {\\mathbf d_{y,t}} \\}$}, and returns these to the clients.\\footnote{The direction estimates {\\small$\\{ {\\mathbf d^i_{x,t}}, {\\mathbf d^i_{y,t}} \\}$} only need to be communicated for the sake of analysis. In our experiments in \\cref{sec:exp}, as in Local SGDA, only the models are communicated.}\n\n\\ificml\n\\begin{algorithm}[ht]\n\t\\caption{Momentum Local SGDA}\n\t\\label{alg_NC_momentum}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE{\\textbf{Input:} {\\small${\\mathbf x}_0^i = {\\mathbf x}_0, {\\mathbf y}_0^i = {\\mathbf y}_0$, $\\mathbf d_{x,0}^i = \\nabla_{\\bx} f_i ({\\mathbf x}^i_0, {\\mathbf y}^i_0; \\xi^i_0)$, $\\mathbf d_{y,0}^i = \\nabla_{\\by} f_i ({\\mathbf x}^i_0, {\\mathbf y}^i_0; \\xi^i_0)$} for all $i \\in [n]; \\eta_x, \\eta_y, \\tau, T$}\n\t\t\\FOR[At all clients $i=1,\\hdots, n$]{$t=0$ to $T-1$}\n\t\t\\STATE{{\\small$\\Tbx^i_{t+\\frac{1}{2}} = {\\mathbf x^i_t} - \\eta_x {\\mathbf d^i_{x,t}}$, \n\t\t\t\t$\\ {\\mathbf x^i_{t+1}} = {\\mathbf x^i_t} + \\alpha_t ( \\Tbx^i_{t+\\frac{1}{2}} - {\\mathbf x^i_t} )$}}\n\t\t\\STATE{{\\small$\\Tby^i_{t+\\frac{1}{2}} = {\\mathbf y^i_t} + \\eta_y {\\mathbf d^i_{y,t}}$, $\\ {\\mathbf y^i_{t+1}} = {\\mathbf y^i_t} + \\alpha_t ( \\Tby^i_{t+\\frac{1}{2}} - {\\mathbf y^i_t} )$}}\n\t\t\\STATE{Sample minibatch ${\\xi^i_{t+1}}$ from local data}\n\t\t\\STATE{{\\small${\\mathbf d^i_{x,t+1}} = (1 - \\beta_x \\alpha_t) {\\mathbf d^i_{x,t}} + \\beta_x \\alpha_t \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}})$}}\n\t\t\\STATE{{\\small${\\mathbf d^i_{y,t+1}} = (1 - \\beta_y \\alpha_t) {\\mathbf d^i_{y,t}} + \\beta_y \\alpha_t \\nabla_{\\by} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}})$}}\n\t\t\\IF{$t+1$ mod $\\tau = 0$}\n\t\t\\STATE{Clients send $\\{ {\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}, {\\mathbf d^i_{x,t+1}}, {\\mathbf d^i_{y,t+1}} \\}$ to the server}\n\t\t\\STATE{Server computes averages {\\small${\\mathbf x_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_{t+1}}$}, \n\t\t\t{\\small${\\mathbf y_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf y^i_{t+1}}$}, {\\small${\\mathbf d_{x,t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf d^i_{x,t+1}}$}, \n\t\t\t{\\small${\\mathbf d_{y,t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf d^i_{y,t+1}}$}, and sends to the clients}\n\t\t\\STATE{${\\mathbf x^i_{t+1}} = {\\mathbf x_{t+1}}$, ${\\mathbf y^i_{t+1}} = {\\mathbf y_{t+1}}$, ${\\mathbf d^i_{x,t+1}} = {\\mathbf d_{x,t+1}}$, ${\\mathbf d^i_{y,t+1}} = {\\mathbf d_{y,t+1}}$, for all $i \\in [n]$}\n\t\t\\ENDIF\n\t\t\\ENDFOR\n\t\t\\STATE{\\textbf{Return: }${\\bar{\\bx}_T}$ drawn uniformly at random from $\\{ {\\mathbf x_t} \\}$, where ${\\mathbf x_t} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_t}$}\n\t\\end{algorithmic}\n\\end{algorithm}\n\\else\n\\begin{algorithm}[ht]\n\t\\caption{Momentum Local SGDA}\n\t\\label{alg_NC_momentum}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE{\\textbf{Input:} ${\\mathbf x}_0^i = {\\mathbf x}_0, {\\mathbf y}_0^i = {\\mathbf y}_0$, $\\mathbf d_{x,0}^i = \\nabla_{\\bx} f_i ({\\mathbf x}^i_0, {\\mathbf y}^i_0; \\xi^i_0)$, $\\mathbf d_{y,0}^i = \\nabla_{\\by} f_i ({\\mathbf x}^i_0, {\\mathbf y}^i_0; \\xi^i_0)$} for all $i \\in [n]; \\eta_x, \\eta_y, \\tau, T$\n\t\t\\FOR[At all clients $i=1,\\hdots, n$]{$t=0$ to $T-1$}\n\t\t\\STATE{$\\Tbx^i_{t+\\frac{1}{2}} = {\\mathbf x^i_t} - \\eta_x {\\mathbf d^i_{x,t}}$, \n\t\t\t$\\ {\\mathbf x^i_{t+1}} = {\\mathbf x^i_t} + \\alpha_t ( \\Tbx^i_{t+\\frac{1}{2}} - {\\mathbf x^i_t} )$}\n\t\t\\STATE{$\\Tby^i_{t+\\frac{1}{2}} = {\\mathbf y^i_t} + \\eta_y {\\mathbf d^i_{y,t}}$, $\\ {\\mathbf y^i_{t+1}} = {\\mathbf y^i_t} + \\alpha_t ( \\Tby^i_{t+\\frac{1}{2}} - {\\mathbf y^i_t} )$}\n\t\t\\STATE{Sample minibatch ${\\xi^i_{t+1}}$ from local data}\n\t\t\\STATE{${\\mathbf d^i_{x,t+1}} = (1 - \\beta_x \\alpha_t) {\\mathbf d^i_{x,t}} + \\beta_x \\alpha_t \\nabla_{\\bx} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}})$}\n\t\t\\STATE{${\\mathbf d^i_{y,t+1}} = (1 - \\beta_y \\alpha_t) {\\mathbf d^i_{y,t}} + \\beta_y \\alpha_t \\nabla_{\\by} f_i ({\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}; {\\xi^i_{t+1}})$}\n\t\t\\IF{$t+1$ mod $\\tau = 0$}\n\t\t\\STATE{Clients send $\\{ {\\mathbf x^i_{t+1}}, {\\mathbf y^i_{t+1}}, {\\mathbf d^i_{x,t+1}}, {\\mathbf d^i_{y,t+1}} \\}$ to the server}\n\t\t\\STATE{Server computes averages \n\t\t\t\\begin{align*}\n\t\t\t\t{\\mathbf x_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_{t+1}}, \\quad {\\mathbf y_{t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf y^i_{t+1}}, \\quad {\\mathbf d_{x,t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf d^i_{x,t+1}}, \\quad {\\mathbf d_{y,t+1}} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf d^i_{y,t+1}}\n\t\t\t\\end{align*}\n\t\t\t\\hspace{7mm} and sends to the clients}\n\t\t\\STATE{${\\mathbf x^i_{t+1}} = {\\mathbf x_{t+1}}$, ${\\mathbf y^i_{t+1}} = {\\mathbf y_{t+1}}$, ${\\mathbf d^i_{x,t+1}} = {\\mathbf d_{x,t+1}}$, ${\\mathbf d^i_{y,t+1}} = {\\mathbf d_{y,t+1}}$, for all $i \\in [n]$}\n\t\t\\ENDIF\n\t\t\\ENDFOR\n\t\t\\STATE{\\textbf{Return: }${\\bar{\\bx}_T}$ drawn uniformly at random from $\\{ {\\mathbf x_t} \\}$, where ${\\mathbf x_t} \\triangleq \\frac{1}{n} \\sum_{i=1}^n {\\mathbf x^i_t}$}\n\t\\end{algorithmic}\n\\end{algorithm}\n\\fi\n\nNext, we discuss the finite-time convergence of \\cref{alg_NC_momentum}.\n\n\\begin{theorem}\n\t\\label{thm:NC_PL_mom}\n\tSuppose the local loss functions $\\{ f_i \\}_i$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, and the global function $f$ satisfies \\cref{assum:PL_y}.\n\tSuppose in \\cref{alg_NC_momentum}, \n\t$\\beta_x = \\beta_y = \\beta = 3$, {\\small$\\alpha_t \\equiv \\alpha \\leq \\min \\big\\{ \\frac{\\beta}{6 L_f^2 (\\eta_y^2 + \\eta_x^2)}, \\frac{1}{48 \\tau} \\big\\}$}, for all $t$, and the step-sizes $\\eta_x, \\eta_y$ are chosen such that $\\eta_y \\leq \\frac{\\mu}{8 L_f^2}$, and $\\frac{\\eta_x}{\\eta_y} \\leq \\frac{1}{20 \\kappa^2}$, where {\\small$\\kappa = L_f\/\\mu$} is the condition number.\n\tThen, for the output ${\\bar{\\bx}_T}$ of \\cref{alg_NC_momentum}, the following holds.\n\t\\ificml\n\t{\\small\n\t\t\\begin{equation}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\mathbb E \\norm{\\nabla \\Phi ({\\bar{\\bx}_T})}^2 & \\leq \\underbrace{\\mathcal O \\Big( \\frac{\\kappa^2}{\\eta_y \\alpha T} + \\frac{\\alpha}{\\mu \\eta_y} \\frac{\\sigma^2}{n} \\Big)}_{\\text{Error with full synchronization}} \\\\\n\t\t\t\t& + \\underbrace{\\mathcal O \\big( (\\tau - 1)^2 \\alpha^2 ( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 ) \\big)}_{\\text{Error due to local updates}},\n\t\t\t\\end{aligned}\n\t\t\t\\label{eq:thm:NC_PL_mom}\n\t\t\\end{equation}\n\t}%\n\t\\else\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t\\mathbb E \\norm{\\nabla \\Phi ({\\bar{\\bx}_T})}^2 & \\leq \\underbrace{\\mathcal O \\Big( \\frac{\\kappa^2}{\\eta_y \\alpha T} + \\frac{\\alpha}{\\mu \\eta_y} \\frac{\\sigma^2}{n} \\Big)}_{\\text{Error with full synchronization}} + \\underbrace{\\mathcal O \\big( (\\tau - 1)^2 \\alpha^2 ( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 ) \\big)}_{\\text{Error due to local updates}},\n\t\t\\end{aligned}\n\t\t\\label{eq:thm:NC_PL_mom}\n\t\\end{equation}\n\t\\fi\n\twhere $\\Phi(\\cdot) \\triangleq \\max_{\\mathbf y} f(\\cdot, {\\mathbf y})$ is the envelope function.\n\tWith {\\small$\\alpha = \\sqrt{n\/T}$}, the bound in \\eqref{eq:thm:NC_PL_mom} simplifies to\n\t\\ificml\n\t{\\small\n\t\t\\begin{align}\n\t\t\t\\mathcal O \\Big( \\frac{\\kappa^2 + \\sigma^2}{\\sqrt{n T}} + (\\tau-1)^2 \\frac{n ( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 )}{T} \\Big).\n\t\t\t\\label{eq:thm:NC_PL_mom_conv_rate}\n\t\t\\end{align}\n\t}\n\t\\else\n\t\\begin{align}\n\t\t\\mathcal O \\Big( \\frac{\\kappa^2 + \\sigma^2}{\\sqrt{n T}} + (\\tau-1)^2 \\frac{n ( \\sigma^2 + \\varsigma_x^2 + \\varsigma_y^2 )}{T} \\Big).\n\t\t\\label{eq:thm:NC_PL_mom_conv_rate}\n\t\\end{align}\n\t\\fi\n\\end{theorem}\n\n\\begin{proof}\n\tSee \\cref{app:NC_PL_mom}.\n\\end{proof}\n\n\\begin{remark}\n\tAs in the case of \\cref{thm:NC_PL}, the second term in \\eqref{eq:thm:NC_PL_mom} arises due to the clients carrying out multiple ($\\tau > 1$) local updates between successive communication rounds. \n\tHowever, the dependence of this term on $\\alpha$ is quadratic. Therefore, as seen in \\eqref{eq:thm:NC_PL_mom_conv_rate}, for small enough $\\alpha$ and carefully chosen $\\tau$, having multiple local updates does not affect the asymptotic convergence rate $\\mathcal O (1\/\\sqrt{nT})$.\n\\end{remark}\n\n\n\n\\begin{cor}\n\t\\label{cor:NC_PL_mom_comm_cost}\n\tTo reach an $\\epsilon$-accurate point ${\\bar{\\bx}_T}$, assuming $T \\geq \\Theta (n^3)$, the stochastic gradient complexity of \\cref{alg_NC_momentum} is $\\mathcal O (\\kappa^4\/(n \\epsilon^4))$.\n\tThe number of communication rounds required for the same is $T\/\\tau = \\mathcal O ( \\kappa^3\/\\epsilon^{3} )$.\n\\end{cor}\n\nThe stochastic gradient complexity and the number of communication rounds required are identical (up to multiplicative constants) for both \\cref{alg_local_SGDA} and \\cref{alg_NC_momentum}.\nTherefore, the discussion following \\cref{thm:NC_PL} (Remarks \\ref{rem:NC_PL_local_SGDA_2}, \\ref{rem:NC_PL_local_SGDA_3}) applies to \\cref{thm:NC_PL_mom} as well.\nWe demonstrate the practical benefits of Momentum Local SGDA in \\cref{sec:exp}.\n\n\n\\subsection{Nonconvex-Concave (NC-C) Problems} \\label{sec:NC_C}\n\nIn this subsection, we consider smooth nonconvex functions which satisfy the following assumptions.\n\n\\begin{assump}[Concavity]\n\t\\label{assum:concavity}\n\tThe function $f$ is concave in ${\\mathbf y}$ if for a fixed ${\\mathbf x} \\in \\mathbb R^{d_1}$, for all  ${\\mathbf y}, {\\mathbf y}' \\in \\mathbb R^{d_2}$,\n\t\\ificml\n\t\\newline\n\t$f({\\mathbf x}, {\\mathbf y}) \\leq f({\\mathbf x}, {\\mathbf y}') + \\left\\langle \\nabla_{\\by} f({\\mathbf x}, {\\mathbf y}'), {\\mathbf y} - {\\mathbf y}' \\right\\rangle$.\n\t\\else\n\t\\begin{align*}\n\t\tf({\\mathbf x}, {\\mathbf y}) \\leq f({\\mathbf x}, {\\mathbf y}') + \\left\\langle \\nabla_{\\by} f({\\mathbf x}, {\\mathbf y}'), {\\mathbf y} - {\\mathbf y}' \\right\\rangle.\n\t\\end{align*}\n\t\\fi\n\\end{assump}\n\n\\begin{assump}[Lipschitz continuity in ${\\mathbf x}$]\n\t\\label{assum:Lips_cont_x}\n\tFor the function $f$, there exists a constant $G_x$, such that for each ${\\mathbf y} \\in \\mathbb R^{d_2}$, and all ${\\mathbf x}, {\\mathbf x}' \\in \\mathbb R^{d_1}$,\n\t\\ificml\n\t$\\norm{f({\\mathbf x}, {\\mathbf y}) - f({\\mathbf x}', {\\mathbf y})} \\leq G_x \\norm{{\\mathbf x} - {\\mathbf x}'}$.\n\t\\else\n\t\\begin{align*}\n\t\t\\norm{f({\\mathbf x}, {\\mathbf y}) - f({\\mathbf x}', {\\mathbf y})} \\leq G_x \\norm{{\\mathbf x} - {\\mathbf x}'}.\n\t\\end{align*}\n\t\\fi\n\\end{assump}\n\nIn the absence of strong-concavity or PL condition on ${\\mathbf y}$, the envelope function $\\Phi({\\mathbf x}) = \\max_{\\mathbf y} f({\\mathbf x}, {\\mathbf y})$ defined earlier need not be smooth.\nInstead, we use the alternate definition of stationarity, proposed in \\cite{davis19wc_siam}, utilizing the Moreau envelope of $\\Phi$, which is defined next.\n\n\\begin{definition}[Moreau Envelope]\n\tA function $\\Phi_{\\lambda}$ is the $\\lambda$-Moreau envelope of $\\Phi$, for $\\lambda > 0$, if for all ${\\mathbf x} \\in \\mathbb R^{d_1}$,\n\t\\ificml\n\t\\newline\n\t$\\Phi_\\lambda({\\mathbf x}) = \\min_{{\\mathbf x}'} \\Phi ({\\mathbf x}') + \\frac{1}{2 \\lambda} \\norm{{\\mathbf x}' - {\\mathbf x}}^2$.\n\t\\else\n\t\\begin{align*}\n\t\t\\Phi_\\lambda({\\mathbf x}) = \\min_{{\\mathbf x}'} \\Phi ({\\mathbf x}') + \\frac{1}{2 \\lambda} \\norm{{\\mathbf x}' - {\\mathbf x}}^2.   \n\t\\end{align*}\n\t\\fi\n\\end{definition}\n\nA small value of $\\norm{\\nabla \\Phi_\\lambda({\\mathbf x})}$ implies that ${\\mathbf x}$ is near some point $\\widetilde{\\bx}$ that is \\textit{nearly stationary} for $\\Phi$ \\cite{drusvyatskiy19wc_mathprog}.\nHence, we focus on minimizing $\\norm{\\nabla \\Phi_\\lambda({\\mathbf x})}$.\n\n\\paragraph{Improved Convergence Analysis for NC-C Problems.}\n\nFor centralized NC-C problems, \\cite{lin_GDA_icml20} analyze the convergence of SGDA.\nHowever, this analysis does not seem amenable to local-updates-based modification.\nAnother alternative is a double-loop algorithm, which approximately solves the inner maximization problem $\\max f({\\mathbf x}, \\cdot)$ after each ${\\mathbf x}$-update step.\nHowever, double-loop algorithms are complicated to implement.\n\\cite{mahdavi21localSGDA_aistats} propose Local SGDA+ (see \\cref{alg_local_SGDA_plus} in \\cref{app:NC_C}), a modified version of SGDA \\cite{lin_GDA_icml20}, to resolve this impasse.\nCompared to Local SGDA, the ${\\mathbf x}$-updates are identical.\nHowever, for the ${\\mathbf y}$-updates, stochastic gradients $\\nabla_{\\by} f_i (\\widetilde{\\bx}, {\\mathbf y^i_t}; {\\xi^i_{t}})$ are evaluated with the $x$-component fixed at $\\widetilde{\\bx}$, which is updated every $S$ iterations.\n\nIn \\cite{mahdavi21localSGDA_aistats}, Local SGDA+ is used for solving nonconvex-one-point-concave (NC-1PC) problems (see \\cref{sec:NC_1PC}).\nHowever, the guarantees provided are far from optimal (see \\cref{table:comparison}).\nIn this and the following subsection, we present improved convergence results for Local SGDA+, for NC-C and NC-1PC minimax problems.\n\n\\begin{theorem}\n\t\\label{thm:NC_C}\n\tSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, \\ref{assum:concavity}, \\ref{assum:Lips_cont_x}.\n\tFurther, let $\\norm{{\\mathbf y_t}}^2 \\leq D$ for all $t$.\n\tSuppose the step-sizes $\\eta_x, \\eta_y$ are chosen such that $\\eta_x, \\eta_y \\leq \\frac{1}{8 L_f \\tau}$.\n\tThen, for the output ${\\bar{\\bx}_T}$ of \\cref{alg_local_SGDA_plus},\n\t\\ificml\n\t{\\small\n\t\t\\begin{equation}\n\t\t\t\\begin{aligned}\n\t\t\t\t& \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\bar{\\bx}_T})}^2 \\leq \\underbrace{\\mathcal O \\Big( \\frac{\\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + \\eta_x \\Big( G_x^2 + \\frac{\\sigma^2}{n} \\Big) \\Big)}_{\\text{Error with full synchronization I}} \\\\\n\t\t\t\t& \\qquad + \\underbrace{\\mathcal O \\Big( \\frac{\\eta_y \\sigma^2}{n} + \\Big[ \\eta_x G_x S \\sqrt{G_x^2 + \\sigma^2\/n} + \\frac{D}{\\eta_y S} \\Big] \\Big)}_{\\text{Error with full synchronization II}} \\\\\n\t\t\t\t& \\qquad + \\underbrace{\\mathcal O \\Big( (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 + \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) \\right] \\Big)}_{\\text{Error due to local updates}},\n\t\t\t\\end{aligned}\n\t\t\t\\label{eq:thm:NC_C}\n\t\t\\end{equation}\n\t}%\n\t\\else\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t\\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\bar{\\bx}_T})}^2 & \\leq \\underbrace{\\mathcal O \\left( \\frac{\\widetilde{\\Delta}_{\\Phi}}{\\eta_x T} + \\eta_x \\Big( G_x^2 + \\frac{\\sigma^2}{n} \\Big) \\right) + \\mathcal O \\left( \\frac{\\eta_y \\sigma^2}{n} + \\Big[ \\eta_x G_x S \\sqrt{G_x^2 + \\sigma^2\/n} + \\frac{D}{\\eta_y S} \\Big] \\right)}_{\\text{Error with full synchronization}} \\\\\n\t\t\t& \\qquad + \\underbrace{\\mathcal O \\Big( (\\tau-1)^2 \\left[ \\left( \\eta_x^2 + \\eta_y^2 \\right) \\sigma^2 + \\left( \\eta_x^2 \\varsigma_x^2 + \\eta_y^2 \\varsigma_y^2 \\right) \\right] \\Big)}_{\\text{Error due to local updates}},\n\t\t\\end{aligned}\n\t\t\\label{eq:thm:NC_C}\n\t\\end{equation}\n\t\\fi\n\twhere {\\small$\\Phi_{1\/2L_f}({\\mathbf x}) \\triangleq \\min_{{\\mathbf x}'} \\Phi ({\\mathbf x}') + L_f \\norm{{\\mathbf x}' - {\\mathbf x}}^2$}, {\\small$\\widetilde{\\Delta}_{\\Phi} \\triangleq \\Phi_{1\/2 L_f} ({\\mathbf x}_0) - \\min_{\\mathbf x} \\Phi_{1\/2 L_f} ({\\mathbf x})$}.\n\n\tUsing {\\small$S = \\Theta ( \\sqrt{T\/n} )$}, {\\small$\\eta_x = \\Theta \\left( \\frac{n^{1\/4}}{T^{3\/4}} \\right)$}, {\\small$\\eta_y = \\Theta \\left( \\frac{n^{3\/4}}{T^{1\/4}} \\right)$}, the bound in \\eqref{eq:thm:NC_C} simplifies to\n\t\\ificml\n\t{\\small\n\t\t\\begin{equation}\n\t\t\t\\begin{aligned}\n\t\t\t\t& \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\bar{\\bx}_T})}^2 \\leq \\underbrace{\\mathcal O \\Big( \\frac{1}{(nT)^{1\/4}} + \\frac{n^{1\/4}}{T^{3\/4}} \\Big)}_{\\text{Error with full synchronization}} \\\\\n\t\t\t\t& \\qquad + \\underbrace{\\mathcal O \\Big( \\frac{n^{3\/2} (\\tau-1)^2}{T^{1\/2}} + (\\tau-1)^2 \\frac{\\sqrt{n}}{T^{3\/2}} \\Big)}_{\\text{Error due to local updates}}.\n\t\t\t\\end{aligned}\n\t\t\t\\label{eq:thm:NC_C_conv_rate}\n\t\t\\end{equation}\n\t}%\n\t\\else\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t& \\mathbb E \\norm{\\nabla \\Phi_{1\/2L_f} ({\\bar{\\bx}_T})}^2 \\leq \\underbrace{\\mathcal O \\left( \\frac{1}{(nT)^{1\/4}} + \\frac{n^{1\/4}}{T^{3\/4}} \\right)}_{\\text{Error with full synchronization}} + \\underbrace{\\mathcal O \\left( \\frac{n^{3\/2} (\\tau-1)^2}{T^{1\/2}} + (\\tau-1)^2 \\frac{\\sqrt{n}}{T^{3\/2}} \\right)}_{\\text{Error due to local updates}}.\n\t\t\\end{aligned}\n\t\t\\label{eq:thm:NC_C_conv_rate}\n\t\\end{equation}\n\t\\fi\n\\end{theorem}\n\n\n\\begin{proof}\n\tSee \\cref{app:NC_C}.\n\\end{proof}\n\n\\ificml\n\\begin{remark}\n\t\\label{rem:NC_C_local_SGDA_plus_1}\n\tThe first two terms in the error decomposition in \\eqref{eq:thm:NC_C}, represent the optimization error for a fully synchronous algorithm.\n\tThis is exactly the error observed in the centralized case \\cite{lin_GDA_icml20}.\n\tThe third term arises due to multiple ($\\tau > 1$) local updates.\n\tAs seen in \\eqref{eq:thm:NC_C_conv_rate}, for small enough $\\eta_y, \\eta_x$, and carefully chosen $S, \\tau$, this does not impact the asymptotic convergence rate {\\small$\\mathcal O (1\/(nT)^{1\/4})$}.\n\\end{remark}\n\\else\n\\begin{remark}\n\t\\label{rem:NC_C_local_SGDA_plus_1}\n\tThe first term in the error decomposition in \\eqref{eq:thm:NC_C}, represents the optimization error for a fully synchronous algorithm.\n\tThis is exactly the error observed in the centralized case \\cite{lin_GDA_icml20}.\n\tThe second term arises due to multiple ($\\tau > 1$) local updates.\n\tAs seen in \\eqref{eq:thm:NC_C_conv_rate}, for small enough $\\eta_y, \\eta_x$, and carefully chosen $S, \\tau$, this does not impact the asymptotic convergence rate $\\mathcal O (1\/(nT)^{1\/4})$.\n\\end{remark}\n\\fi\n\n\\begin{cor}\n\t\\label{cor:NC_C_comm_cost}\n\tTo reach an $\\epsilon$-accurate point, i.e., ${\\mathbf x}$ such that {\\small$\\mathbb E \\| \\nabla \\Phi_{1\/2L_f} ({\\mathbf x}) \\| \\leq \\epsilon$},\n\tassuming {\\small$T \\geq \\Theta (n^7)$},\n\tthe stochastic gradient complexity of \\cref{alg_local_SGDA_plus} is {\\small$\\mathcal O (1\/(n \\epsilon^8))$}.\n\tThe number of communication rounds required is {\\small$T\/\\tau = \\mathcal O ( 1\/\\epsilon^{7} )$}.\n\\end{cor}\n\n\n\\begin{remark}\n\t\\label{rem:NC_C_local_SGDA_plus_2}\n\tOurs is the first work to match the centralized ($n=1$) results in \\cite{lin_GDA_icml20} ({\\small$\\mathcal O ( 1\/\\epsilon^{8} )$} using SGDA), and provide linear speedup for $n>1$ with local updates.\n\tIn addition, we also provide communication savings, requiring model averaging only once every {\\small$\\mathcal O ( 1\/(n \\epsilon) )$} iterations.\n\\end{remark}\n\n\n\n\n\\subsection{Nonconvex-One-Point-Concave (NC-1PC) Problems} \\label{sec:NC_1PC}\n\nIn this subsection, we consider smooth nonconvex functions which also satisfy the following assumption.\n\n\\begin{assump}[One-point-Concavity in ${\\mathbf y}$]\n\t\\label{assum:1pc_y}\n\tThe function $f$ is said to be one-point-concave in ${\\mathbf y}$ if fixing ${\\mathbf x} \\in \\mathbb R^{d_1}$, for all  ${\\mathbf y} \\in \\mathbb R^{d_2}$,\n\t\\ificml\n\t$\\left\\langle \\nabla_{\\by} f({\\mathbf x}, {\\mathbf y}'), {\\mathbf y} - {\\mathbf y}^*({\\mathbf x}) \\right\\rangle \\leq f({\\mathbf x}, {\\mathbf y}) - f({\\mathbf x}, {\\mathbf y}^*({\\mathbf x}))$,\n\t\\else\n\t\\begin{align*}\n\t\t\\left\\langle \\nabla_{\\by} f({\\mathbf x}, {\\mathbf y}'), {\\mathbf y} - {\\mathbf y}^*({\\mathbf x}) \\right\\rangle \\leq f({\\mathbf x}, {\\mathbf y}) - f({\\mathbf x}, {\\mathbf y}^*({\\mathbf x})),   \n\t\\end{align*}\n\t\\fi\n\twhere ${\\mathbf y}^*({\\mathbf x}) \\in \\operatornamewithlimits{arg\\,max}_{\\mathbf y} f({\\mathbf x}, {\\mathbf y})$.\n\\end{assump}\n\nDue to space limitations, we only state the sample and communication complexity results for \\cref{alg_local_SGDA_plus} with NC-1PC functions. The complete result is stated in \\cref{app:NC_1PC}.\n\n\\begin{theorem} \n\t\\label{thm:NC_1PC}\n\tSuppose the local loss functions $\\{ f_i \\}$ satisfy Assumptions \\ref{assum:smoothness}, \\ref{assum:bdd_var}, \\ref{assum:bdd_hetero}, \\ref{assum:Lips_cont_x}, \\ref{assum:1pc_y}.\n\tFurther, let {\\small$\\norm{{\\mathbf y_t}}^2 \\leq D$} for all $t$.\n\tThen, to reach a point ${\\mathbf x}$ such that {\\small$\\mathbb E \\| \\nabla \\Phi_{1\/2L_f} ({\\mathbf x}) \\| \\leq \\epsilon$}, the sample complexity of \\cref{alg_local_SGDA_plus} is {\\small$\\mathcal O (1\/\\epsilon^8)$}, and the number of communication rounds required is {\\small$\\mathcal O ( 1\/\\epsilon^{7} )$}.\n\\end{theorem}\n\n\\begin{remark}\n\t\\label{rem:NC_1PC_local_SGDA_plus_2}\n\tSince one-point-concavity is more general than concavity, for $n=1$, our gradient complexity result $\\mathcal O (1\/\\epsilon^8)$ generalizes the corresponding result for NC-C functions \\cite{lin_GDA_icml20}. To the best of our knowledge, ours is the first work to provide this guarantee for NC-1PC problems. We also reduce the communication cost by requiring model averaging only once every $\\mathcal O ( 1\/\\epsilon )$ iterations.\n\tFurther, our analysis improves the corresponding results in \\cite{mahdavi21localSGDA_aistats} substantially (see \\cref{table:comparison}).\n\\end{remark}\n\n\n\\section{Experiments}\n\\label{sec:exp}\n\nIn this section, we present the empirical performance of the algorithms discussed in the previous sections.\nTo evaluate the performance of Local SGDA and Momentum Local SGDA, we consider the problem of fair classification \\cite{mohri19agnosticFL_icml, nouiehed19minimax_neurips19} using the FashionMNIST dataset \\cite{xiao17fashionMNIST}.\nSimilarly, we evaluate the performance of Local SGDA+ and Momentum Local SGDA+, a momentum-based algorithm (see \\cref{alg_mom_local_SGDA_plus} in \\cref{app:add_exp}), on a robust neural network training problem \\cite{madry18adversarial_iclr, sinha17certifiable_robust_iclr}, using the CIFAR10 dataset.\nWe conducted our experiments on a cluster of 20 machines (clients), each equipped with an NVIDIA TitanX GPU. Ethernet connections communicate the parameters and related information amongst the clients.\nWe implemented our algorithm based on parallel training tools offered by PyTorch 1.0.0 and Python 3.6.3.\nAdditional experimental results, and the details of the experiments, along with the specific parameter values can be found in \\cref{app:add_exp}.\n\n\n\\subsection{Fair Classification}\n\\label{sec:exp_fair}\nWe consider the following NC-SC minimax formulation of the fair classification problem \\cite{nouiehed19minimax_neurips19}.\n\\ificml\n{\\small\n\t\\begin{align}\n\t\t\\min_{\\mathbf x} \\max_{{\\mathbf y} \\in \\mathcal Y} \\sum_{c=1}^C y_c F_c({\\mathbf x}) -\\frac{\\lambda}{2} \\norm{{\\mathbf y}}^2,\n\t\t\\label{eq:exp_fair_2}\n\t\\end{align}\n}%\n\\else\n\\begin{align}\n\t\\min_{\\mathbf x} \\max_{{\\mathbf y} \\in \\mathcal Y} \\sum_{c=1}^C y_c F_c({\\mathbf x}) -\\frac{\\lambda}{2} \\norm{{\\mathbf y}}^2,\n\t\\label{eq:exp_fair_2}\n\\end{align}\n\\fi\nwhere ${\\mathbf x}$ denotes the parameters of the NN, $F_1, F_2, \\hdots, F_C$ denote the individual losses corresponding to the $C(=10)$ classes, and {\\small$\\mathcal Y = \\{ {\\mathbf y} \\in \\mathbb R^C: y_c \\geq 0, \\sum_{c=1}^C y_c = 1 \\}$}.\n\n\\ificml\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{figures\/FairClassifier\/fashionMNIST_test_acc.pdf}\n\t\\vspace{-3mm}\n\t\\caption{Comparison of the effects of increasing $\\tau$ on the performance of Local SGDA and Momentum Local SGDA algorithms, for the fair classification problem on the FashionMNIST dataset, with a VGG11 model. The figure shows the test accuracy for the worst distribution. \\label{fig:fairclass_fashionmnist}}\n\\end{figure}\n\\else\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.55\\textwidth]{figures\/FairClassifier\/fashionMNIST_test_acc.pdf}\n\t\\vspace{-3mm}\n\t\\caption{Comparison of the effects of increasing $\\tau$ on the performance of Local SGDA and Momentum Local SGDA algorithms, for the fair classification problem on the FashionMNIST dataset, with a VGG11 model. The figure shows the test accuracy for the worst distribution. \\label{fig:fairclass_fashionmnist}}\n\\end{figure}\n\\fi\n\nWe ran the experiment with a VGG11 network.\nThe network has $20$ clients.\nThe data is partitioned across the clients using a Dirichlet distribution $\\text{Dir}_{20}(0.1)$ as in \\cite{wang19FL_iclr}, to create a non-iid partitioning of data across clients. We use different values of synchronization frequency $\\tau \\in \\{1, 5, 10\\}$.\nIn accordance with \\eqref{eq:exp_fair_2}, we plot the worst distribution test accuracy in \\cref{fig:fairclass_fashionmnist}.\nWe plot the curves for the number of communications it takes to reach $50\\%$ test accuracy on the worst distribution in each case.\nFrom \\cref{fig:fairclass_fashionmnist}, we see the communication savings which result from using higher values of $\\tau$, since fully synchronized SGDA ($\\tau = 1$) requires significantly more communication rounds to reach the same accuracy.\nWe also note the superior performance of Momentum Local SGDA, compared to Local SGDA.\n\n\\subsection{Robust Neural Network Training}\n\\label{sec:exp_robustnn}\nNext, we consider the problem of robust neural network (NN) training, in the presence of adversarial perturbations \\cite{madry18adversarial_iclr, sinha17certifiable_robust_iclr}.\nWe consider a similar problem as considered in \\cite{mahdavi21localSGDA_aistats}.\n\\ificml\n{\\small\n\t\\begin{align}\n\t\t\\min_{\\mathbf x} \\max_{\\norm{{\\mathbf y}}^2 \\leq 1} \\sum_{j=1}^N \\ell \\left( h_{\\mathbf x} (\\mathbf a_i + {\\mathbf y}), b_i \\right), \\label{eq:exp_robustnn}\n\t\\end{align}\n}%\n\\else\n\\begin{align}\n\t\\min_{\\mathbf x} \\max_{\\norm{{\\mathbf y}}^2 \\leq 1} \\sum_{j=1}^N \\ell \\left( h_{\\mathbf x} (\\mathbf a_i + {\\mathbf y}), b_i \\right), \\label{eq:exp_robustnn}\n\\end{align}\n\\fi\nwhere ${\\mathbf x}$ denotes the parameters of the NN, ${\\mathbf y}$ denotes the perturbation, $(a_i, b_i)$ denotes the $i$-th data sample.\n\\ificml\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{figures\/RobustNN\/CIFAR10_test_acc.pdf}\n\t\\vspace{-3mm}\n\t\\caption{Comparison of the effects of $\\tau$ on the performance of Local SGDA and Momentum Local SGDA algorithms, for the robust NN training problem on the CIFAR10 dataset, with the VGG11 model. The figure shows the robust test accuracy. \\label{fig:robustnn_cifar10}}\n\\end{figure}\n\\else\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.55\\textwidth]{figures\/RobustNN\/CIFAR10_test_acc.pdf}\n\t\\vspace{-3mm}\n\t\\caption{Comparison of the effects of $\\tau$ on the performance of Local SGDA and Momentum Local SGDA algorithms, for the robust NN training problem on the CIFAR10 dataset, with the VGG11 model. The figure shows the robust test accuracy. \\label{fig:robustnn_cifar10}}\n\\end{figure}\n\\fi\n\nWe ran the experiment using a VGG11 network, with the same network and data partitioning as in the previous subsection.\nWe use different values of $\\tau \\in \\{1, 5, 10\\}$.\nFor both Local SGDA+ and Momentum Local SGDA+, we use $S = \\tau^2$.\nIn \\cref{fig:robustnn_cifar10}, we plot the robust test accuracy. \nFrom \\cref{fig:robustnn_cifar10}, we see the communication savings which result from using higher values of $\\tau$, since for both the algorithms, $\\tau = 1$ case requires significantly more communication rounds to reach the same accuracy.\nWe also note the superior performance of Momentum Local SGDA+, compared to Local SGDA+ to reach the same accuracy level.\n\n\n\\section{Concluding Remarks}\n\\label{sec:conclude}\nIn this work, we analyzed existing and newly proposed distributed communication-efficient algorithms for nonconvex minimax optimization problems.\nWe proved \\textit{order-optimal} complexity results, along with communication savings, for several classes of minimax problems. Our results showed linear speedup in the number of clients, which enables scaling up distributed systems.\nOur results for nonconvex-nonconcave functions improve the existing results for centralized minimax problems.\nAn interesting future direction is to analyze these algorithms for more complex systems with partial and erratic client participation \\cite{gu21mifa_neurips, ruan21device_part_FL_aistats}, and with a heterogeneous number of local updates at each client \\cite{joshi20fednova_neurips}.\n\n\n\n\\nocite{lin20near_opt_det_colt, nesterov18book, yoon21acc_ncc_icml, ouyang21lower_cc_bilinear_mathprog, wang20improved_cc_neurips, li21lower_bd_NCSC_neurips, lei21stability_Minimax_icml, zhang21NCSC_uai, kiyavash20catalyst_neurips, lee21NCNC_structured_neurips, lu20HiBSA_NC_C_ieee, tran20hybrid_NCLin_neurips, jin20local_opt_NCNC_icml, liang20proxGDA_KL_iclr, luo21near_opt_FS_cc_arxiv, xie20lower_FS_cc_icml, gasnikov21decen_deter_cc_icoa, ozdaglar19dec_prox_sp_arxiv, richtarik21dist_VI_comm_arxiv, gasnikov21dec_person_FL_arxiv, jacot18NTK_neurips}\n\n\\section*{Acknowledgements}\n\t\n\n\n\t\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn recent years, the ultrathin graphite films, i.e. $N$ ($>1$) stacked\ngraphene layers (NSGL's), are successfully fabricated.\\cite{Novoselov1,\n  Berger} Extensive studies have been devoted to these systems\ndue to their competitive capability for the design of novel\nnano-devices. It's desirable to carry out a through symmetry analysis to figure\nout the corresponding electron and phonon spectra, and to reveal the\nrelevant selection rules and the optical activities. Moreover, to our\nknowledge, although there are some studies on the electronic\nstructure,\\cite{Peeters1} no existing works concern the phonon dispersions\nfor the NSGL's. Recently it is reported that the\nfrequencies of optical C-C stretching mode in the NSGL's decrease with\nincreasing $N$.\\cite{Ferrari, Gupta, Anindya} The amplitude of this\nred shift is about $3 - 5$, $5 - 6$, and 8~cm$^{-1}$ in these three\nexperiments respectively.  Theoretical explanation of this red shift\nis required.\n\nIn this paper, the symmetry analysis is referred for both the AB- and\nAA-stacked lattice structures, while the latter has the point group\n$D_{6h}$ irrespective of the even-oddness of $N$.  We classify the\nphonon normal modes at $\\Gamma$ point and determine their Raman and\nInfra-red (Ir) properties. We\ngeneralize the force constant model of graphene\\cite{Aizawa} into\nNSGL's and it can be applied to calculate the phonon dispersions for the NSGL's in AB- or\nAA-stacking with arbitrary layer number $N$. \n\nFor the intra-layer\noptical C-C stretching mode with frequency around 1600~cm$^{-1}$, it \nis Raman active for all NSGL's and the calculated frequencies exhibit\nlayer-number dependence as that the frequency decreases with $N$\nincreasing. The red shift values for the AB- (AA-) stacked systems are\nabout 2~cm$^{-1}$ (4~cm$^{-1}$) which are in consistent with the\nexperimental measurements. In the medium frequency range around 800~cm$^{-1}$,\nthe out-of-plane optical mode is Ir active for AB-stacked structure but\nneither Raman nor Ir active in AA-stacking. Its frequency behaviors a blue\nshift as layer number increasing. There is an interesting inter-layer optical\nmode in the low frequency region which is Raman active in the NSGL's with $N$ even (ENSGL's) while \nIr active in NSGL's with $N$ odd (OENSGL's). Its\nfrequency value depends on the layer number $N$ more sensitively and\nincreases from 106~cm$^{-1}$ (94.5~cm$^{-1}$) to 149.8~cm$^{-1}$\n(133.6~cm$^{-1}$) for the AB- (AA-) stacked NSGL's, which is of an\norder less than those intra-layer optical modes. Phonon dispersions\nfor the AA-stacked 3-dimensional (3D) graphite are also discussed.\n\nThe present paper is organized as follows. In Sec.~II, the lattice\nconfiguration is illustrated for the NSGL's.  Sec.~III is devoted to\nthe symmetry analysis for the phonon modes. The vibrational potential\nenergy is discussed in Subsec.~IV~A, while the main results and\nrelevant discussions on phonon spectrum calculations are presented in\nSubsec.~IV~B. The paper ends with a brief summary in Sec.~V.\n\n\n\n\\section{lattice configuration}\n\\subsection{AB-stacked}\n\\begin{figure}\n  \\begin{center}\n    \\scalebox{1.2}[1.4]{\\includegraphics[width=7cm]{unitcell.eps}}\n  \\end{center}\n  \\caption{The sketch of the configurations of AB-stacked in (a) and \n      AA-stacked in (b) for multi-layer graphene.}\n  \\label{Fig:unitcell}\n\\end{figure}\n\nIt is known that graphene is a single layer of carbon atoms with the\nhoneycomb lattice configuration which is characterized by the $D_{6h}$\nsymmetry.\\cite{Dresselhaus} The 3D graphite is AB-stacked honeycomb\nlattice, where the B layers are achieved by shifting the A layers\nalong one of its first-nearest carbon-carbon bonds in the horizontal\nplane as shown in Fig.~\\ref{Fig:unitcell}(a). The space group of the\n3D graphite is non-symmorphic group $D_{6h}^{4}$ with non-primitive\ntranslation $ \\vec\\tau=\\frac{1}{2}\\vec{c}$ (primitive translation\n$\\vec t=n_1 \\vec a_1+n_2 \\vec a_2+ n_3\\vec c$).\\cite{Brillson} The\ndistance between two adjacent layers is about $\\frac{c}{2}=3.35\\AA$\nwhich is much larger than the bond length between two nearest-neighbor\natoms in the plane, $b=1.42\\AA$.\n\nThe NGSL's are also constructed by AB-stacked honeycomb lattice, but\nwith limited number of layers. Although the structures of each layer of\n3D graphite and NSGL's are the same, the corresponding symmetry groups\nare different since the displacement symmetry along $\\vec{c}$ axis no\nlonger exists for NSGL's, so does the symmetry associated with\n$\\vec\\tau$. Now the symmetry group becomes a direct product of a 3D\npoint group and a 2D translational group. And the point groups are\ndifferent for ENSGL's and ONSGL's as mentioned in\nRef.~\\onlinecite{Manes}. For the ENSGL's, a center of inversion\n$\\sigma_{i}$ occurs in the middle of atom 4 in the $\\frac {N}{2}$-th\nlayer and 5 in the $\\frac {N}{2}+1$-th layer as shown in\nFig.~\\ref{Fig:unitcell}(a). There are one 3-fold main axis in the\ndirection perpendicular to the layers and three 2-fold axes,\n$C_{2}''$, perpendicular to the main axis and at angles of $\\pi\/3$ to\neach other. All these symmetry operations together constitute the\npoint group $D_{3d}=\\{E, 3C_{3}, 3C_{2}''\\}\\times\\{E, \\sigma_{i}\\}$.\nIn the ONSGL's, instead of a center of inversion there is a reflection\nsymmetry $\\sigma_{h}$ with the middle layer as its reference plane. A\n3-fold main axis exists in the direction of $z$-axis and three 2-fold\naxes, $C_{2}'$, are perpendicular to it.  We notice that these three\n2-fold axes $C_{2}'$ are one to one perpendicular to those of\n$C_{2}''$. Consequently, the symmetry group for the ONSGL's is\n$D_{3h}=\\{E, 3C_{3}, 3C_{2}'\\}\\times\\{E, \\sigma_{h}\\}$.\n\nThe environments for an atom in the graphite and NSGL's are different\nfrom that in 2D graphene. For each carbon atom in the graphene layer,\nthere are three nearest-neighbor carbon atoms and six next-nearest-\nneighbors. There are four carbon atoms 1, 2, 3, 4 in a unit cell of\ngraphite, as represented in Fig.~\\ref{Fig:unitcell}(a). For atom 4 in\nA layer, there are two inter-layer nearest-neighbors in each of the\ntwo adjacent layers, with the distance $c$. Also, in each of the two\nadjacent layers, there are three inter-layer next-nearest-neighbor\natoms around atom 4 with distance $\\sqrt{b^{2}+(\\frac{c}{2})^{2}}$. As illustrated\nin Fig.~\\ref{Fig:unitcell}(a), the adjacent environment for atom 3 is\nquite different from that of atom 4. It has no inter-layer\nfirst-neighbors with the same distance as $\\frac{c}{2}$. However, atom 3 has six\ninter-layer neighbors in each of the two adjacent layers with the same\ndistance as that of the second-nearest-neighbor of atom 4.\n\\subsection{AA-stacked}\nThe AA-stacked NSGL's (AA-NSGL's) are constructed by AA-stacked\nhoneycomb lattice where all layers have the same configuration. In the\nAA-stacked system, the ENSGL's, ONSGL's and the 3D graphite have the\nsame point group $D_{6h}$ which is the symmetry of the graphene.  As\nshown in Fig.~\\ref{Fig:unitcell}(b), the environments for carbon\natoms in the AA-stacked NSGL's are quite different from those in AB-stacked\nsystems. For each atom, there are two inter-layer nearest-neighbors in\neach of the two adjacent layers with the distence $\\frac{c}{2}$. And each atom\nhas six inter-layer second-nearest-neighbors with distance\n$\\sqrt{b^{2}+(\\frac{c}{2})^{2}}$ in its two adjacent layers. We notice here that\nin the AA-stacked 3D graphite, there are only two atoms in the unit\ncell and the primitive translation along $c$-axis is $\\vec{c}\/2$, which\nis half of the correspondence in the AB-stacked 3D graphite.\n\n\n\\section{symmetry analysis for the phonon modes}\nThe dynamical representation\n$\\Gamma^{dyn}=\\Gamma^{v}\\bigotimes\\Gamma^{atom}$ can be decomposed\ninto the irreducible representations of the symmetry group, with the\nlattice displacements as the bases, where $\\Gamma^{v}$ is the vector\nrepresentation and $\\Gamma^{atom}$ is the permutation representation\nof the group. By applying the projection operator technique, we carry\nout the decomposition of the dynamical representation into the\nirreducible representations for the NSGL's with $N$ even and odd\nrespectively. According to Elliott,\\cite{Elliott} the Ir\nactive phonon modes belong to the irreducible representations\ndecomposed from the vector representation $\\Gamma^{v}$, while the\nRaman active modes correspond to the irreducible representations shown\nup in the decomposition of a six-dimensional representation with bases\nas the quadratic forms: $x^{2}+y^{2}$, $z^{2}$, $x^{2}-y^{2}$, $xy$,\n$yz$, and $zx$. The three acoustic modes with zero frequency at the\n$\\Gamma$ point, which correspond to the vector representation\n$\\Gamma^{v}$, are excluded in the consideration of Ir and Raman active\nmodes. For comparison the corresponding results for the graphene and\n3D graphite are also listed in the following. The symbol for the\nirreducible representations we used here is the notation used in\nRef.~\\onlinecite{Eyring} which is the most commonly used in the\ntreatment of molecules.\n\\begin{table*}[t]\n  \\caption{The symmetry analysis for the phonon modes at the\n    $\\Gamma$ point of the NSGL's with AA- or AB-stacking.\n    Phonon modes are classified by the irreducible\n    representations of $\\Gamma^{dyn}$ in the fourth column.\n    The irreducible representations of the Ir and Raman\n    active modes are listed in the fifth and sixth column respectively.}\n  \\label{Tab:SymmetryAnalysis}\n  \\begin{ruledtabular}\n    \\begin{tabular}{|l|c|c|c|c|c|}\n      &&group&$\\Gamma^{dyn}$&$\\Gamma^{Ir}$&$\\Gamma^{R}$\\\\\n      \\hline\n      graphene\\cite{Dresselhaus}&&$D_{6h}$&$A_{2u}\\bigoplus B_{2g}\\bigoplus E_{1u}\\bigoplus E_{2g}$&\/&$E_{2g}$\\\\\n      \\hline &ENSGL's&$D_{3d}$\\cite{Manes}&$N(A_{1g}\\bigoplus A_{2u}\\bigoplus E_{g}\\bigoplus E_{u})$&$(N-1)A_{2u}\\bigoplus (N-1)E_{u}$&$NA_{1g}\\bigoplus NE_{g}$\\\\\n      AB-stacked&ONSGL's&$D_{3h}$\\cite{Manes}&$(N-1)A_{1}'\\bigoplus (N+1)A_{2}''\\bigoplus (N+1)E'$&$NA_{2}''\\bigoplus N E'$&$(N-1)A_{1}'\\bigoplus NE'\\bigoplus (N-1)E''$\\\\\n      &&& $\\bigoplus (N-1)E''$ &&\\\\\n      &3D\\cite{Mani}&$D_{6h}^{4}$&$2(A_{2u}\\bigoplus B_{2g}\\bigoplus\n      E_{1u}\\bigoplus\n      E_{2g})$&$A_{2u}\\bigoplus E_{1u}$&$2E_{2g}$\\\\\n      \\hline\n      &ENSGL's&$D_{6h}$&$\\frac{N}{2}(A_{1g}\\bigoplus A_{2u}\\bigoplus B_{1u}\\bigoplus B_{2g}\\bigoplus E_{1u}$&$(\\frac{N}{2}-1)(A_{2u}\\bigoplus E_{1u})$&$\\frac{N}{2}(A_{1g}\\bigoplus E_{1g}\\bigoplus E_{2g})$\\\\\n      &&& $\\bigoplus E_{1g}\\bigoplus E_{2g}\\bigoplus E_{2u})$ &&\\\\\n      AA-stacked&ONSGL's&$D_{6h}$&$\\frac{N-1}{2}(A_{1g}\\bigoplus B_{1u}\\bigoplus E_{1g}\\bigoplus E_{2u})$&$\\frac{N-1}{2}(A_{2u}\\bigoplus E_{1u})$&$\\frac{N-1}{2}(A_{1g}\\bigoplus E_{1g})\\bigoplus \\frac{N+1}{2}E_{2g}$\\\\\n      &&& $\\bigoplus \\frac{N+1}{2}(A_{2u}\\bigoplus B_{2g}\\bigoplus E_{1u}\\bigoplus E_{2g})$ &&\\\\\n      &3D&$D_{6h}$&$A_{2u}\\bigoplus B_{2g}\\bigoplus E_{1u}\\bigoplus E_{2g}$&\/&$E_{2g}$\\\\\n    \\end{tabular}\n  \\end{ruledtabular}\n\\end{table*}\n\n\nThe symmetry analysis for the phonon modes and the Raman and Ir modes\nare classified in Table.~\\ref{Tab:SymmetryAnalysis}. In the\nAB-stacked NSGL's, since the $\\sigma_{i}$ symmetry and $\\sigma_{h}$\nsymmetry can not coexist in the ENSGL's or ONSGL's, we can see two\nstraightforward consequences from the above classification for the Ir\nand Raman active modes. Firstly, in the ENSGL's, phonon modes can not\nbe Ir and Raman active simultaneously (which is also true for the\ngaphene and graphite). However, in the ONSGL's, the $N$ $E'$ modes are\nboth Ir and Raman active. This is because there is no inversion center\nin the ONSGL's. Secondly, among the optical modes with their\nvibrational displacements perpendicular to the constituent layers,\nthere is an exotic mode oscillating with each layer as a whole but\nalternatively from layer to layer. It belongs to the $A_{1g}$ in the\nENSGL's and $A''_{2}$ in the ONSGL's. Since the $\\sigma_{h}$ operation\nexists only in the ONSGL's, this mode ($\\omega_{1}$ mode) is Raman\nactive in the ENSGL's while Ir active in the ONSGL's.\n\nIn the AA-stacked NSGL's with $N$ even or odd, the symmetry group is\n$D_{6h}$ which includes both $\\sigma_{i}$ and $\\sigma_{h}$. As a\nresult, phonon modes can not be Ir and Raman active simultaneously.\nThe $\\omega_{1}$ mode mentioned above belongs to the $A_{1g}$ in the\nENSGL's and $A_{2u}$ in the ONSGL's. This mode is Raman active in the\nENSGL's while Ir active in the ONSGL's which is the same as the\nAB-stacked NSGL's. Nevertheless, its vibrational mode favors to take\nthe maximum advantage of the inter-layer interactions. It would be\nsensitive and useful experimentally to identify the even-oddness of\nthe NSGL's with a few layers.\n\n\n\\section{calculation for the phonon dispersion}\n\\subsection{Vibrational potential energy}\nThe vibrational potential energy for a graphene sheet can be described\nby five quadratic terms with the rigid rotational symmetry\nimplemented.\\cite{Aizawa, Jiang2} They are the 1st and 2nd\nnearest-neighbor stretching, the in-plane bond angle variations, the\nout-of-surface bond bending and the bond twisting energies. From the\nmodality of atomic movements, we can also classify the inter-layer\nvibrational potential terms into three types: The first one describes\nthe stretching movements between the two atoms located in the adjacent\nlayers. The second describes the relative movement between the two\npairs of atoms with a common one as an apex. That is this type of\nmovements involves three atoms to form one bond in a layer and another\nconnecting the two nearest layers. The third involves more than three\natoms according to the specific bond configurations. As shown in\nFig.~\\ref{Fig:unitcell}, there is only one inter-layer\nnearest-neighbor carbon-carbon bond in each unit cell (the bond\nbetween atoms 1 and 4). So that just the twisting potential on the\ninter-layer bond is encountered here. The whole of these terms is\nactually a modified valence force field model to account some interactions for far away atoms\nin response to the bond charge effect in certain extent. Since the\ninter-layer bonds are much longer than that in the plane, all above three \ntype interactions are one or two orders\nless than their counterparts in layer and they\nthemselves have comparable contributions. In the following, the\ninter-layer terms are written in the AB-stacked system and they can be\nsimilarly generalized to the AA-stacked system.\n\n\\begin{table}[t]\n  \\caption{Comparison of several mode frequencies (in the unit of\n    cm$^{-1}$) for the AB-stacked 3D graphite between the our calculation results and the\n    experimental values.\\cite{Nicklow, Maultzsch}}\n  \\label{Tab:Fit}\n  \\begin{ruledtabular}\n    \\begin{tabular}{cllll}\n      Reps &$A_{1}^{'}$ & $E_{2g}$ & $A_{2u}$ & $E_{2g}$ \\\\\n      \\hline\n      experiments& 30\\cite{Nicklow} & 40\\cite{Nicklow} & 868\\cite{Maultzsch} & 1586\\cite{Maultzsch}\\\\\n      theory &30.2 & 42.7 & 869.9 & 1586.6 \\\\\n    \\end{tabular}\n  \\end{ruledtabular}\n\\end{table}\n\n\n(1). The inter-layer bond stretching energies $V^{(int)}_l\n(V^{(int)}_{sl})$ have the form as:\n\\begin{eqnarray}\n  \\sum_{i,j}\\frac{\\hat{k}_l}{2}[(\\vec{u}_{i}-\\vec{u}_{j})\\cdot\n  \\vec{e}_{ij}^{l}]^{2},\n  \\label{Eq:Potential1}\n\\end{eqnarray}\nwhere $\\vec{u}_{i}$ $(\\vec{u}_{j})$ is the displacement vector of the\natom $i$ $(j)$ and $\\vec{e}_{ij}^{l}$ is the unit vector from atom $i$\nto atom $j$. If the summation is taken over the nearest-neighbored\ninter-layer pair of atoms, the corresponding force constant is denoted\nas $\\hat{k}_l$ while for next nearest-neighbor inter-layer pairs we\nhave the force constant as $\\hat{k}_{sl}$.\n\n\n(2). For the three atoms 1, 4 and $i$, where $i$ is the in-plane\nnearest neighbor of atom 1 (see Fig.~\\ref{Fig:unitcell}), we found\nthat under a specific configuration with atom $i$ rather than atom 1\nas an apex, while the force being along the corresponding bond\ndirection instead of perpendicular direction, a correlation term\n$\\hat{k}_{rr}$ has the most and sensitive contribution to the layer\ndependence of the inta-layer C-C stretching optical modes,\n\\begin{eqnarray}\n  \\frac{\\hat{k}_{rr}}{2}\\sum_{i}[(\\vec{u}_{1}-\\vec{u}_{i})\\cdot\n  \\vec{e}_{i1}^{l}-(\\vec{u}_{4}-\\vec{u}_{i})\\cdot\n  \\vec{e}_{i4}^{l}]^2. \\nonumber\n\\end{eqnarray}\nActually the two square terms in above modality have already been\naccounted in the in-plane and inter-plane stretching terms\nrespectively. Only the across term is left,\n\\begin{eqnarray}\n  V_{rr}=-\\hat{k}_{rr}\\sum_{i}[(\\vec{u}_{1}-\\vec{u}_{i})\n  \\cdot \\vec{e}_{i1}^{l}]\n  [(\\vec{u}_{4}-\\vec{u}_{i})\\cdot \\vec{e}_{i4}^{l}]\\; ,\n  \\label{Eq:Potential4}\n\\end{eqnarray}\nwhich weakens the interaction between two adjacent layers.  The\npositive definite condition for getting real frequencies is\n$\\hat{k}_{sl}\\geq \\hat{k}_{rr}$.\n\n\n\n(3). The twisting potential for an inter-layer bond between atoms 1\nand 4 is coming from the two sets of three nearest-neighbors of atoms\n1 and 4 respectively. It can be described as\n\\begin{eqnarray}\n  V_{tw}=\\frac{\\hat{k}_{tw}}{2}[\\sum_{i}(\\vec{u}_{i}-\\vec{u}_{1})\\cdot \\vec{e}_{i}^{\\theta}\n  -\\sum_{j}(\\vec{u}_{j}-\\vec{u}_{4})\\cdot\n  \\vec{e}_{j}^{\\theta}]^{2},\n  \\label{Eq:Potential3}\n\\end{eqnarray}\nwhere $\\sum_{i}$ and $\\sum_{j}$ represent the summation over the three\nintra-plane first-nearest-neighbors for atoms 1 and 4\nrespectively. $\\vec{e}_{i}^{\\theta}=\\vec{e}_{z}\\times\n\\vec{e}_{1i}^{l}$ is the tangential unit vector in the plane formed by\nthree atoms 1, 4, and $i$. The expression in quadratic form as a whole\nensures a proper definition for the torsion angle. For pure rotations\naround the bond, this expression gives zero torsion consistently. In\ncontrast, the bond is most severely twisted when the three neighbors\naround atom 1 and those of atom 4 rotate reversely.\n\n\nWe stress here that, all of the above four inter-layer vibrational\npotential energy terms satisfy the rigid rotational symmetry\nrequirements\\cite{Popov, Mahan, Jiang2} which guarantees the existence\nof the flexure modes in the low dimensional systems. Although we establish the vibrational\npotential terms based on the analysis to the modality of movements,\nthe bond charge effect especially along the perpendicular direction\nhas been involved by extending the valence force field beyond the nearest\nneighbors. Comparing, for example, the above $\\hat{k}_{rr}$ term with\nthat $V_{b-b}$ in Ref.~\\onlinecite{Mahan2}, which\nis followed from the bond-charge model, they have the same negative\ncross term.\n\n\\subsection{Results and discussion}\n\n\n\\begin{figure}\n  \\begin{center}\n    \\scalebox{1.0}[1.0]{\\includegraphics[width=7.6cm]{3DFitLow.eps}}\n  \\end{center}\n  \\caption{Phonon dispersion for the 3D graphite in the low-frequency\n    region.  Solid dots are the experimental results of\n    Ref.~\\onlinecite{Nicklow}.  Our theoretical calculations are shown\n    in lines.}\n  \\label{Fig:3DFitLow}\n\\end{figure}\n\\begin{figure}\n  \\begin{center}\n    \\scalebox{1.0}[1.0]{\\includegraphics[width=7.6cm]{3DFitHigh.eps}}\n  \\end{center}\n  \\caption{Phonon dispersion for the 3D graphite in the high-frequency\n    region.  Solid dots are the experimental results.\\cite{Maultzsch,\n      Mohr} In Refs.~\\onlinecite{Maultzsch, Mohr}, those phonon wave\n    vectors $\\vec{q}$, which were not exactly along the ~$\\Gamma$-M or\n    ~$\\Gamma$-K-M direction, were projected onto the closest\n    high-symmetry direction.  Lines are our theoretical calculations.}\n  \\label{Fig:3DFitHigh}\n\\end{figure}\n\n\n\n\nThe five intra-layer force constants we used in the following are\ntaken from Ref.~\\onlinecite{Jiang} with a minor modification. We\nadjust the four inter-layer force constants to fit the experimental\nvalues of four modes in 3D graphite as shown in Table~\\ref{Tab:Fit}.\nThe fitting error for phonon modes is kept less than $7\\%$. The\ninter-layer force constants are then fitted as\n$\\hat{k}_{l}=0.77$~Nm$^{-1}$, $\\hat{k}_{sl}=0.95$~Nm$^{-1}$,\n$\\hat{k}_{tw}=0.64$~Nm$^{-1}$, $\\hat{k}_{rr}=0.9$~Nm$^{-1}$.\n\n\n\n\nBase on the above fitted vibrational potential energy with nine terms,\nwe calculate the dispersion curves for the AB-stacked graphite. As\nillustrated in Figs~\\ref{Fig:3DFitLow} and ~\\ref{Fig:3DFitHigh}, our\ntheoretical calculations meet the experimental results not only in the\nlow frequency \\cite{Nicklow} but also in the high frequency regions.\n\\cite{Maultzsch, Mohr} The excellent consistency with the experimental data\nshows that our model and parameters are reasonable and applicable.\n\n\n\n\\begin{table}[t]\n  \\caption{The Raman and Ir mode frequencies (in the unit of cm$^{-1}$)\n    for the AB-stacked 3D graphite, AB-stacked 2-layer and AA-stacked 3D grphite are listed. \n    The irreducible representations\n    are presented in the brackets following the frequency values.}\n  \\label{Tab:Raman}\n  \\begin{ruledtabular}\n    \\begin{tabular}{|l|cc|cc|}\n      &Raman&&Infra-red&\\\\\n      \\hline\n      AB- 3D & 42.7 ($E_{2g}$) & 1586.7 ($E_{2g}$) & 869.9 ($A_{2u}$) & 1588.2 ($E_{1u}$)\\\\\n      \\hline\n      AB- & 30.2 ($E_{g}$) & 106 ($A_{1g}$) & 868.7 ($A_{2u}$) & 1588.1 ($E_{u}$)\\\\\n      2-layer & 867.4 ($A_{1g}$) & 1587.3 ($E_{g}$)\\\\\n      \\hline\n      AA- 3D & 1584.7 ($E_{2g}$) &  & & \\\\\n    \\end{tabular}\n  \\end{ruledtabular}\n\\end{table}\n\n\\begin{figure}\n  \\begin{center}\n    \\scalebox{1.0}[1.0]{\\includegraphics[width=7.6cm]{ModeIntraLayer.eps}}\n  \\end{center}\n  \\caption{The frequency value for the optical C-C stretching mode vs\n    the layer number $N$.  Lines are draw to guide eyes.}\n  \\label{Fig:ModeIntraLayer}\n\\end{figure}\n\\begin{figure}\n  \\begin{center}\n    \\scalebox{1.0}[1.0]{\\includegraphics[width=7.6cm]{ModeOut.eps}}\n  \\end{center}\n  \\caption{The frequency value for the out-of-plane optical mode vs\n    the layer number $N$. This mode is Ir active in the AB stacking while \n    it is neither Ir nor Raman active in the AA stacking. Lines are draw to guide eyes.}\n  \\label{Fig:ModeOut}\n\\end{figure}\n\\begin{figure}\n  \\begin{center}\n    \\scalebox{1.1}[1.2]{\\includegraphics[width=7.6cm]{ModeInterLayer.eps}}\n  \\end{center}\n  \\caption{The frequencies of the inter-layer optical mode vs the layer\n    number $N$. Datas for the AB and AA stacked NSGL's are designated by \n    pentagrams and circles, respectively. The Raman and Infra-red activities \n    for this mode are displayed by the full and empty symbols, respectively.\n    The broken and dashed lines correspond to the frequencies for the AB-stacked \n    and AA-stacked graphite, respectively.}\n  \\label{Fig:ModeInterLayer}\n\\end{figure}\n\\begin{figure}\n  \\begin{center}\n    \\scalebox{1.0}[1.0]{\\includegraphics[width=7.6cm]{2Layer3DLow.eps}}\n  \\end{center}\n  \\caption{In the low frequency region, there is significant\n    difference between 3D graphite and the 2-layer graphene.}\n  \\label{Fig:2Layer3DLow}\n\\end{figure}\n\nWith the above force constants, we can calculate the phonon dispersion\nfor NSGL's of the AA- or AB-stacking with an arbitrary layer\nnumber $N$. In Fig.~\\ref{Fig:ModeIntraLayer}, the calculated frequency\nof the intra-layer optical C-C stretching mode is represented with\ndifferent stacked styles and layer number $N$. The layer dependence of\nthe frequency shows up a red shift behavior which is in agreement with\nthe experimental measurements. The frequency value for this mode is\nabout 1588~cm$^{-1}$ in the single graphene layer and decreases with\nincreasing $N$ and almost saturates at $N=10$. The limit is\n1586.7~cm$^{-1}$ (1584.7~cm$^{-1}$) in the AB- (AA-) stacked system\nrespectively. The amount of red shift value in our calculation\ncorresponds excellently with that measured by experiments within the ranges $3 - 5$, $5 -\n6$, and 8~cm$^{-1}$ in Refs.~\\onlinecite{Ferrari}, \\onlinecite{Gupta}\nand \\onlinecite{Anindya}, respectively.\n\nThe out-of-plane optical mode, belonging to the $A_{2u}$ ($B_{2g}$)\nirreducible representation in the AB (AA) stacking, is Ir active in the AB-stacking\nyet inactive in the AA-stacking irrespective of the even-oddness of the \nlayer number $N$ and is useful in determining whether the NSGL's is of AB or AA stacking. \nAs shown in Fig.~\\ref{Fig:ModeOut}, frequencies for \nthis mode depend on the layer number $N$ and increase from 864.8~cm$^{-1}$\nto 872.6~cm$^{-1}$ in both the AB and AA stacking. In contrast to the C-C \nstretching optical mode, this mode frequency exhibits a blue shift type\nlayer dependence which could be identified with the development of the experimental\ntechnique.\n\nFor the inter-layer optical mode, the layer number dependence of the\nfrequency value is shown in Fig.~\\ref{Fig:ModeInterLayer}. This mode\ntakes the greatest advantage of the inter-layer interation and is\nconsiderably dependent on the layer number $N$ and the stack style AB\nor AA. In case of $N=2$, the $\\omega_{1}$ mode has the frequency values\n106~cm$^{-1}$ and 94.5~cm$^{-1}$ for the AB- and AA-stacked NSGL's\nrespectively. The frequencies of the $\\omega_1$ mode increase with\nincreasing $N$ and almost come to the limit values at $N=10$. The limit\nvalues are 149.8~cm$^{-1}$ and 133.6~cm$^{-1}$ for the AB- and\nAA-stacked NSGL's respectively. The frequency differences as well as the\nRaman versus Ir (see Sec. III) of the $\\omega_{1}$ mode in NSGL's with\ndifferent layers might inspire considerably experimental interest in\nthe $\\omega_1$ mode.\n\n\nWe then calculate the phonon dispersion for the 2-layered AB stacking, in\ncomparison with that of the 3D graphite. The most significant\ndifference between the 2-layer graphene and the 3D graphite lies in\nthe low-frequency region around the $\\Gamma$ point as shown in\nFig.~\\ref{Fig:2Layer3DLow}. The frequencies of the low-frequency\noptical modes in the 2-layer graphene are much smaller than their\ncounterparts in the 3D graphite. The frequencies of the Raman and Ir\nactive modes are shown in the third line of Table~\\ref{Tab:Raman}\namong which the two $A_{1g}$ modes have the frequency value\n$\\omega_{1}$=106~cm$^{-1}$ and $\\omega_{2}$=867.4~cm$^{-1}$. In fact,\n$\\omega_{1}$ and $\\omega_{2}$ modes are the above mentioned $NA_{1g}$\nmodes of the ENSGL's specified to $N=2$.\n\n\n\n\\begin{figure}\n  \\begin{center}\n    \\scalebox{1.0}[1.0]{\\includegraphics[width=7.6cm]{AA3Dgraphite.eps}}\n  \\end{center}\n  \\caption{The phonon dispersion along some high-symmetry directions\n    for the AA-stacked 3D graphite. There are only six branches in the\n    figure, since the unit cell in the AA-stacked 3D graphite contains\n    two atoms.}\n  \\label{Fig:AA3Dgraphite}\n\\end{figure}\n\nWe further calculate the phonon dispersion curve for the AA-stacked 3D\ngraphite as shown in Fig.~\\ref{Fig:AA3Dgraphite}. Since the unit cell\ncontains only two atoms in contrast to that of the AB-stacked\ngraphite, there are six branches of phonon dispersion. Along $\\Gamma A$ in\nthe Brillouin zone, the lowest and highest branches which correspond\nto the in-plane acoustic and optical vibrational modes are doubly\ndegenerate, while the remaining two branches describing the\nout-of-plane vibration are non-degenerate. At the $A$ point in\nthe Brillouin zone, there is a phase factor difference of $\\pi$\nbetween two adjacent layers for the out-of-plane motion, among which\nthe $\\omega_{1}$ mode has the frequency value of 133.6~cm$^{-1}$. In\nthe fourth line of Table~\\ref{Tab:Raman}, the Raman and Ir active\nmodes for the AA-stacked 3D graphite are listed which are to be\nconfirmed in future experiments.\n\n\n\n\n\n\\section{conclusion}\nBased upon a thorough investigation of the lattice symmetry of the\nNSGL's, the Raman and Ir properties, in particular its layer dependence, is\nsystematically studied. With a proposed generalized vibrational\npotential, we further calculate the phonon dispersion of various modes\nof the AB- or AA-stacked NSGL's where the layer dependence is also\nstressed. The calculated frequencies of optical C-C stretching mode \nexhibit a red shift as layer number increasing in both the AB- and \nAA-stacked NSGL's, and the shift value 2~cm$^{-1}$ (4~cm$^{-1}$) for \nAB- (AA-) stacked NSGL's is in good consistent with the experimental measurements. \nThe out-of-plane optical mode with frequency around about 800~cm$^{-1}$ is \nIr active in AB-stacked structure yet neither Raman nor Ir active\nin AA-stacking. Its frequency shows a blue shift layer dependence.\nWe also predict that the frequency of the inter-layer optical mode increases with $N$\nincreasing. Since this mode is more sensitive to the layer number $N$, \nit should be experimentally interesting in\ndetermining the lattice structure properties of the NSGL's.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThermoelectric effect (i.e. the appearance of an electric current upon application of a thermal gradient) in superconductors remains an intriguing topic that attracts a lot of interest over last decades \\cite{NL}.\nWhile usually the magnitude of this effect in generic metals and superconductors remains small (being proportional to the ratio between temperature and the Fermi energy $T\/\\varepsilon_F$), it can increase by orders of magnitude provided electron-hole symmetry is lifted, e.g., due to spin-dependent scattering of electrons. This situation was predicted to occur in a variety of structures, such as superconductors doped with magnetic impurities \\cite{Kalenkov12}, superconductor-ferromagnet hybrids with the density of states spin-split by the exchange and\/or Zeeman fields \\cite{Machon,Ozaeta} or superconductor-normal metal (SN) bilayers with spin-active interfaces \\cite{KZ14,KZ15}. In accordance with theoretical predictions large thermoelectric currents were recently observed in superconductor-ferromagnet tunnel junctions in high magnetic fields \\cite{Beckmann}.\n\nLarge thermoelectric effect was also observed in multiterminal hybrid SNS structures with no magnetic inclusions \\cite{Venkat1,Venkat2,Petrashov03,Venkat3}. Being exposed to a temperature gradient such structures (frequently called Andreev interferometers) were found to develop a thermopower signal which magnitude was not restricted by a small parameter $T\/\\varepsilon_F$ and, furthermore, turned out to be a periodic function of the superconducting phase difference $\\chi$ across the corresponding SNS junction. The latter observation indicates that macroscopic quantum coherence can play an important role and poses a question about the relation between thermoelectric and Josephson effects in the systems under consideration. Subsequent theoretical analysis \\cite{Seviour00,KPV,VH,VP} indeed demonstrated that the thermoelectric effect in Andreev interferometers can be large and confirmed the periodic dependence of the thermopower on the phase difference $\\chi$. At the same time, there appear to be no general consensus in the literature concerning the basic physical origin of this effect. While the authors \\cite{KPV} emphasized an important role of electron-hole imbalance, Virtanen and Heikkil\\\"a \\cite{VH}, on the contrary, proposed that in Andreev interferometers a non-vanishing thermopower signal can be generated even provided electron-hole symmetry is maintained and that the dominant part of this signal can be directly related to the difference between the {\\it equilibrium} values for the Josephson current at temperatures $T_1$ and $T_2$. Subsequently, Volkov and Pavlovskii \\cite{VP} argued that in general no such simple relation between the thermopower and the Josephson current can be established and that under certain conditions the former can still remain large even though the latter gets strongly suppressed by temperature effects.\n\nThe existing theory \\cite{Seviour00,KPV,VH,VP} merely deals with the experimentally relevant diffusive limit in which case the analysis may become rather cumbersome forcing the authors either to employ numerics or to resort to various approximations. On the other hand, in order to clarify the physical origin of a large thermoelectric effect in Andreev interferometers one could take a different route focusing the analysis on the ballistic limit. A clear advantage of this approach is the possibility to employ the notion of semiclassical electron trajectories and in this way to treat the problem exactly. Below we will follow this route.\n\nThe structure of the paper is as follows. In Sec. \\ref{model} we define our\nmodel and describe the quasiclassical formalism which will be employed in our\nfurther analysis. In Sec. \\ref{riccati} we demonstrate that the quasiclassical\nEilenberger equations supplemented by Zaitsev boundary conditions can be conveniently resolved with the aid\nof the so-called Riccati parameterization of the Green functions. Specific\nquasiparticle trajectories and their contributions to electric currents are\nidentified in Sec. \\ref{quas} where we also discuss the conditions for\nelectron-hole symmetry violation in our system. In Sec. \\ref{thermo} we\nevaluate the thermoelectric voltages induced by applying a temperature gradient\nand demonstrate that these voltages can be large because of the presence of\nelectron-hole asymmetry in Andreev interferometers. Sec. \\ref{josephson} is\ndevoted to further analysis of the relation between thermoelectric and Josephson\neffects. Some technical details of our calculation are displayed in Appendix.\n\n\\section{The model and quasiclassical formalism}\n\\label{model}\nLet us consider an NSNSN structure shown in Fig. \\ref{nsnsn3-fig}. Two\nsuperconductors with phase difference $\\chi = \\chi_1 - \\chi_2$ are connected to\na normal wire. The two ends of this normal wire are maintained at temperatures $T_1$\nand $T_2$ and voltages $V_1$ and $V_2$ respectively. Within our model we deliberately choose to disregard electron scattering on impurities and boundary imperfections, in which case electron motion is ballistic and\ncan be conveniently described in terms of quasiclassical trajectories. In\naddition, we will disregard inelastic electron relaxation inside our system by\nassuming the inelastic relaxation length to exceed the system size. In this case\ninelastic electron relaxation can only occur deep inside normal terminals\n$\\mathrm{N_1}$ and $\\mathrm{N_2}$.\n\n\\begin{figure}\n\\centerline{ \\includegraphics[width=80mm]{nsnsn3-fig} }\n\\caption{(Color online) Normal wire connected to normal and superconducting leads. Normal leads are maintained at temperatures $T_1$ and $T_2$ and voltages $V_1$ and $V_2$ respectively. An example of a quasiclassical electron trajectory relevant for the thermoelectric effect under consideration is also illustrated.}\n\\label{nsnsn3-fig}\n\\end{figure}\n\nIn what follows we will employ the quasiclassical theory of superconductivity based on the Eilenberger equations \\cite{Eil,bel}. Under the above assumptions adopted within our model these equations take the form\n\\begin{equation}\n\\left[ \\hat\\Omega , \\hat g^{R,A,K} \\right]\n+\ni\\bm{v}_F \\nabla \\hat g^{R,A,K} (\\bm{p}_F, \\bm{r}, \\varepsilon) =0,\n\\quad\n\\check g^2 =1,\n\\end{equation}\nwhere $\\hat g^{R,A,K}$ are energy-integrated retarded, advanced and Keldysh $2\\times 2$ matrix\nGreen functions. The matrices $\\hat\\Omega$ and $\\check g$ have the following\nstructure\n\\begin{equation}\n\\hat \\Omega=\n\\begin{pmatrix}\n\\varepsilon & \\Delta \\\\\n-\\Delta^* & -\\varepsilon\n\\end{pmatrix},\n\\quad\n\\check g=\n\\begin{pmatrix}\n\\hat g^R & \\hat g^K  \\\\\n0 & \\hat g^A\n\\end{pmatrix},\n\\end{equation}\nwhere $\\Delta$ and $\\varepsilon$ are respectively the superconducting order parameter and the\nquasiparticle energy. Electric current density can be expressed in terms of the\nKeldysh Green function in the standard manner as\n\\begin{equation}\n\\bm{j}(\\bm{r})= -\\dfrac{e N_0}{4} \\int d \\varepsilon\n\\left< \\bm{v}_F \\Sp [\\hat \\tau_3 \\hat g^K(\\bm{p}_F, \\bm{r},\n\\varepsilon) ] \\right>,\n\\label{current}\n\\end{equation}\nwhere $\\bm{p}_F=m\\bm{v}_F$ is the electron Fermi momentum vector, $\\hat\\tau_3$\nis the Pauli matrix in the Nambu space and  $N_0=mp_F\/(2\\pi^2)$ is the normal density of states at\nthe Fermi level. Here the angular brackets $\\left<\\cdots\\right>$ denote\naveraging over\nthe Fermi momentum directions.\n\n\\section{Riccati parameterization and boundary conditions}\n\\label{riccati}\nFor the system under consideration the Eilenberger equations can be solved exactly.\nIn order to proceed we restrict our analysis to quasiparticles propagating in the normal metal with subgap energies $|\\varepsilon| < |\\Delta|$ and employ the so-called Riccati parameterization for the retarded and advanced Green functions \\cite{Schopohl95}\n\\begin{equation}\n\\hat g^{R,A}=\\pm\n    \\hat N^{R,A}\n    \\begin{pmatrix}\n    1+\\gamma^{R,A} \\tilde \\gamma^{R,A} & 2\\gamma^{R,A} \\\\\n    -2 \\tilde \\gamma^{R,A} & -1- \\tilde \\gamma^{R,A}  \\gamma^{R,A} \\\\\n    \\end{pmatrix},\n    \\label{graparam}\n\\end{equation}\nwhere $\\gamma^{R,A}$, $\\tilde \\gamma^{R,A}$ are Riccati amplitudes and\n\\begin{equation}\n\\hat N^{R,A}=\n    \\begin{pmatrix}\n    (1-\\gamma^{R,A} \\tilde \\gamma^{R,A})^{-1} & 0 \\\\\n    0 & (1-\\tilde \\gamma^{R,A}  \\gamma^{R,A} )^{-1} \\\\\n    \\end{pmatrix}.\n    \\label{nrparam}\n\\end{equation}\nParameterization of the Keldysh Green function $\\hat g^K$ also contains the two distribution functions  $x$ and $\\tilde x$. It reads \\cite{Eschrig00}\n\\begin{equation}\n\\hat g^K=\n2\n\\hat N^R\n\\begin{pmatrix}\nx - \\gamma^R  \\tilde x  \\tilde \\gamma^A &\n-\\gamma^R  \\tilde x + x  \\gamma^A \\\\\n-\\tilde \\gamma^R  x + \\tilde x \\tilde \\gamma^A &\n\\tilde x - \\tilde \\gamma^R  x  \\gamma^A \\\\\n\\end{pmatrix}\n\\hat N^A.\n\\label{gkparam}\n\\end{equation}\n\nIn the normal metal (i.e. for $\\Delta \\equiv 0$) Riccati amplitudes $\\gamma^{R,A}$, $\\tilde \\gamma^{R,A}$ and distribution functions  $x$, $\\tilde x$ obey the following simple equations\n\\begin{gather}\ni\\bm{v}_F \\nabla \\gamma^{R,A} = -2 \\varepsilon \\gamma^{R,A},\n\\quad\ni\\bm{v}_F \\nabla \\tilde \\gamma^{R,A} = 2 \\varepsilon \\tilde \\gamma^{R,A},\n\\label{gammaeq}\n\\\\\ni\\bm{v}_F \\nabla x =0, \\quad i\\bm{v}_F \\nabla \\tilde x =0.\n\\label{xeq}\n\\end{gather}\nWithin the quasiclassical approximation adopted here quasiparticles propagate\nalong the straight line trajectories between each two scattering events which\ncan only occur at the boundaries of the normal metal wire. Of interest for us\nhere is to describe quasiparticle scattering at the interfaces between the\nnormal metal and each of the two superconductors. This task is accomplished in\nthe standard manner with the aid of the Zaitsev boundary conditions\n\\cite{Zaitsev84} for the quasiclassical Green functions rewritten in terms of\nthe above Riccati amplitudes and the distribution functions\\cite{Eschrig00}.\n\nConsider, for instance, a quasiparticle propagating from the first normal terminal, being reflected at each of the two NS interfaces and leaving the system through the second normal terminal. The corresponding quasiparticle trajectory is indicated in Fig. \\ref{nsnsn3-fig}. It is important to emphasize that here we only account for quasiparticles with subgap energies which cannot penetrate into superconductors suffering either normal or Andreev reflection at both NS interfaces. It is easy to verify that the distribution functions for such quasiparticles take the form\n\\begin{gather}\nx =\n\\left[1- \\gamma^R \\tilde \\gamma^A\\right] x_1,\n\\quad\n\\tilde x =\n\\left[1- \\tilde \\gamma^R \\gamma^A \\right] \\tilde x_2\n\\end{gather}\nobeying both Eilenberger equations and the corresponding boundary conditions at NS\ninterfaces. Here $\\gamma^{R,A}$ and $\\tilde \\gamma^{R,A}$ are Riccati amplitudes\nalong the trajectory, $x_1$ and $\\tilde x_2$ are asymptotic values of the\ndistribution functions $x$ and $\\tilde x$ respectively at the initial and final trajectory points. Then we obtain\n\\begin{equation}\n\\hat g^K = \\dfrac{x_1}{2}\n(1+\\hat g^R)(1 - \\hat g^A)\n+\n\\dfrac{\\tilde x_2}{2}\n(1 - \\hat g^R)(1 + \\hat g^A).\n\\end{equation}\nMaking use of the asymptotic conditions\n\\begin{gather}\n\\gamma^R_1 = \\tilde \\gamma^A_1 =0, \\quad \\tilde \\gamma^R_2 = \\gamma^A_2 = 0,\n\\label{as1}\n\\end{gather}\nwhich hold respectively in the initial and final\ntrajectory points we recover simple expressions for the Keldysh Green function\ninside the normal leads, i.e.\n\\begin{gather}\n\\hat g^K_1=\n2\n\\begin{pmatrix}\nx_1  &  x_1  \\gamma^A_1 \\\\\n-\\tilde \\gamma^R_1  x_1 & \\tilde x_2 - \\tilde \\gamma^R_1 \\gamma^A_1 (x_1 + \\tilde x_2)\n\\\\\n\\end{pmatrix}\n,\n\\\\\n\\hat g^K_2=\n2\n\\begin{pmatrix}\nx_1 - \\gamma^R_2 \\tilde \\gamma^A_2 ( x_1 + \\tilde x_2) & - \\gamma^R_2  \\tilde x_2 \\\\\n\\tilde x_2 \\tilde \\gamma^A_2 & \\tilde x_2\\\\\n\\end{pmatrix}.\n\\end{gather}\n\nThe expressions for Riccati amplitudes $\\gamma^{R,A}$ and $\\tilde \\gamma^{R,A}$ are derived from Eqs. \\eqref{gammaeq}  supplemented by the boundary conditions at the NS interfaces. The latter can be expressed in the form \\cite{Eschrig00}\n\\begin{gather}\n\\gamma^R_{\\text{out}}=\n\\dfrac{\n(\\mathcal{R}-\\gamma^R_S \\tilde \\gamma^R_S) \\gamma^R_{\\text{in}} + \\mathcal{D} \\gamma^R_S\n}{\n- \\mathcal{D} \\tilde \\gamma^R_S \\gamma^R_{\\text{in}} +(1- \\mathcal{R}\\gamma^R_S \\tilde \\gamma^R_S)\n},\n\\label{bound1}\n\\\\\n\\gamma^A_{\\text{in}}=\n\\dfrac{\n(\\mathcal{R}-\\gamma^A_S \\tilde \\gamma^A_S) \\gamma^A_{\\text{out}} + \\mathcal{D} \\gamma^A_S\n}{\n- \\mathcal{D} \\tilde \\gamma^A_S \\gamma^A_{\\text{out}} +(1- \\mathcal{R}\\gamma^A_S \\tilde \\gamma^A_S)\n},\n\\label{bound2}\n\\\\\n\\tilde \\gamma^R_{\\text{in}}=\n\\dfrac{\n(\\mathcal{R}-\\gamma^R_S \\tilde \\gamma^R_S) \\tilde \\gamma^R_{\\text{out}} + \\mathcal{D} \\tilde\n\\gamma^R_S\n}{\n- \\mathcal{D} \\gamma^R_S \\tilde \\gamma^R_{\\text{out}} +(1- \\mathcal{R} \\gamma^R_S \\tilde\n\\gamma^R_S)\n},\n\\label{bound3}\n\\\\\n\\tilde \\gamma^A_{\\text{out}}=\n\\dfrac{(\\mathcal{R}-\\gamma^A_S \\tilde \\gamma^A_S) \\tilde \\gamma^A_{\\text{in}}\n+\n\\mathcal{D} \\tilde \\gamma^A_S}{\n- \\mathcal{D} \\gamma^A_S \\tilde \\gamma^A_{\\text{in}}\n+(1- \\mathcal{R}\\gamma^A_S \\tilde \\gamma^A_S)\n\\label{bound4}\n},\n\\end{gather}\nwhere the Riccati amplitudes denoted by the subscripts ``in'' and ``out''\nparameterize retarded and advanced Green function for respectively incoming and outgoing\nmomentum directions, $\\mathcal{D}=1-\\mathcal{R}$ denotes the\nnormal transmission of the corresponding NS interface and $\\gamma^{R,A}_S$, $\\tilde \\gamma^{R,A}_S$ are\nthe Riccati amplitudes in the superconductor defined as\n\\begin{gather}\n\\gamma_S^R = \\dfrac{\\varepsilon - \\sqrt{\\varepsilon^2 -\n|\\Delta|^2}}{\\Delta^*},\n\\quad\n\\tilde \\gamma_S^A = [\\gamma_S^R]^*,\n\\\\\n\\tilde \\gamma_S^R = \\dfrac{\\varepsilon - \\sqrt{\\varepsilon^2 -\n|\\Delta|^2}}{\\Delta},\n\\quad\n\\gamma_S^A = [\\tilde \\gamma_S^R]^*,\n\\end{gather}\nwhere the branch of the square root is chosen so that $\\sgn \\Img \\sqrt{z} =\n\\sgn \\Img z$.\n\nFrom Eqs. \\eqref{bound1}-\\eqref{bound4} it is easy to observe that at subgap energies $|\\varepsilon| < |\\Delta|$  the boundary conditions at the NS interface acquire the same form for four different functions $\\gamma^R$, $1\/\\tilde \\gamma^R$, $1\/\\tilde \\gamma^A$, and $\\gamma^A$, i.e.\n\\begin{gather}\nw_{\\text{out}}=\n\\dfrac{\n(\\mathcal{R}-\\gamma^R_S \\tilde \\gamma^R_S) w_{\\text{in}} + \\mathcal{D} \\gamma^R_S\n}{\n- \\mathcal{D} \\tilde \\gamma^R_S w_{\\text{in}} +(1- \\mathcal{R}\\gamma^R_S \\tilde \\gamma^R_S)\n},\n\\\\\nw=\\gamma^R, 1\/\\tilde \\gamma^R, 1\/\\tilde \\gamma^A, \\gamma^A.\n\\label{w}\n\\end{gather}\nHere we employed the identities $\\gamma_S^R \\tilde \\gamma_S^A =1$ and $\\tilde\n\\gamma_S^R \\gamma_S^A =1$ which hold in the relevant energy interval\n$|\\varepsilon| < |\\Delta|$. Furthermore, Eqs. \\eqref{gammaeq} demonstrate that\nthe four functions \\eqref{w} obey the same equation. With this in mind it is\nstraightforward to verify that the following combination of the Riccati\namplitudes\n\\begin{equation}\n\\dfrac{1 - \\gamma^R \\tilde \\gamma^A}{1 - \\gamma^R \\tilde \\gamma^R}\n\\dfrac{1 - \\tilde \\gamma^R \\gamma^A}{1 - \\gamma^A \\tilde \\gamma^A}\n\\end{equation}\nremains constant along the trajectory at subgap energies. Then making use of Eqs. \\eqref{as1} we obtain the relation between Riccati amplitudes in the initial and final points of the quasiclassical trajectory:\n\\begin{equation}\n\\tilde \\gamma^R_1 \\gamma^A_1\n=\n\\gamma^R_2 \\tilde \\gamma^A_2.\n\\end{equation}\n\nAsymptotic behavior of Riccati amplitudes at the beginning and at the end of\nelectron trajectories is\ndirectly related to transmission and reflection probabilities of the corresponding\nprocesses. These probabilities will be explicitly evaluated in the next section.\n\n\\section{Quasiclassical trajectories and electron-hole asymmetry generation}\n\\label{quas}\nAs we already pointed out, quasiparticles with subgap energies can only propagate inside the normal part of our system being unable to penetrate deep into the superconductors S$_1$ and S$_2$. Let us classify such electron trajectories relevant for the thermoelectric effect under consideration.\n\nThere exist electron trajectories starting in one of the terminals\n($\\mathrm{N_1}$ or $\\mathrm{N_2}$) and going back to the same terminal without\nhitting any\nof the two NS interfaces. These trajectories do not contribute to any current flowing in our system and, hence, can be safely ignored in our subsequent\nconsideration. Of more relevance are electron trajectories which propagate from the first to the second terminal (or vice versa).\nProvided these trajectories ``know nothing about superconductivity'' (even \nthough some of them can hit at least one of the NS interfaces)\nthey contribute to the dissipative Ohmic current $(V_1-V_2)\/R_{0}$ flowing\nbetween the terminals $\\mathrm{N_1}$ and $\\mathrm{N_2}$. Here\n\\begin{equation}\n\\dfrac{1}{R_{0}}=\n2 e^2 N_0 \\int \\left< \\bm{v}_F \\Theta_{12}(\\bm{p}_F,\\bm{r}) \\right> d\n\\bm{\\Sigma}_1,\n\\label{sharvin}\n\\end{equation}\nis the inverse Sharvin resistance of the normal wire and the function $\\Theta_{12}(\\bm{p}_F,\\bm{r})$ equals to unity for\nall electron trajectories connecting the first and the second terminals and to zero otherwise.\nHere and below averaging $\\left< ...\\right>$ includes only the directions of $\\bm{v}_F$ corresponding to electron trajectories going\nout of the first terminal and $\\int \\cdots d\\bm{\\Sigma}_{1(2)}$ denotes the integral over the cross-section of the normal lead $N_{1(2)}$.\n\nThe remaining contributions to the currents $I_1$ and $I_2$ in the normal terminals (see Fig. \\ref{nsnsn3-fig})\nare due to electron trajectories which directly involve at least one of the superconductors. In what follows for the sake of simplicity we will restrict our analysis to trajectories which may hit each of the NS interfaces only once assuming that the contribution of more complicated trajectories is negligible. This can easily be achieved by a proper choice of the system geometry (e.g., by assuming the cross sections of both NS interfaces to be sufficiently small). There are electron trajectories which originate in the first terminal, hit one of the\nNS interfaces and go back to the same terminal. Making use of the formalism\ndescribed in the previous sections we evaluate the Keldysh Green function on\nsuch trajectories and then derive the expression for the current with the aid of\nEq. \\eqref{current}. E.g., in the case of the first terminal we obtain\n\\begin{multline}\nI_{1}^{loc}=\neN_0\n\\int\n\\left<\n\\bm{v}_{F}\\Theta_{11}^S(\\bm{p}_F,\\bm{r})\n|\\tilde \\gamma_1^R(\\bm{p}_F,  \\bm{r}, \\varepsilon)|^2\n\\right>\n\\\\\\times\n\\left[\n-\\tilde x_1 (\\varepsilon)\n-\nx_1(\\varepsilon)\n\\right] d \\varepsilon d \\bm{\\Sigma}_1,\n\\label{i1loc}\n\\end{multline}\nwhere the function $\\Theta_{11}^S(\\bm{p}_F,\\bm{r})=1$ for electron trajectories which start inside the\nfirst terminal, hit one of the two NS interfaces and return back to the same\nterminal $\\mathrm{N_1}$ and $\\Theta_{11}^S(\\bm{p}_F,\\bm{r})=0$ otherwise. The\nexpression for $I_{2}^{loc}$ can be obtained from Eq. \\eqref{i1loc} simply by\nreplacing the indices $1 \\leftrightarrow 2$.\nWe also note that the equilibrium distribution functions $x_{1,2}$ and $\\tilde\nx_{1,2}$ in the bulk normal electrodes $\\mathrm{N_1}$ and $\\mathrm{N_2}$ are\ndefined by the standard expressions\n\\begin{gather}\nx_{1,2}=\\tanh\\dfrac{\\varepsilon - eV_{1,2}}{T_{1,2}},\n\\quad\n\\tilde x_{1,2}=-\\tanh\\dfrac{\\varepsilon + eV_{1,2}}{T_{1,2}}.\n\\label{distr}\n\\end{gather}\n\nWhat remains is to account for the trajectories connecting two different\nterminals and touching either only one of the two NS interfaces or both these\ninterfaces one after the other. The latter situation is illustrated in Fig.\n\\ref{nsnsn3-fig}. As before, for these trajectories we set\n$\\Theta_{12}^S(\\bm{p}_F,\\bm{r})=1$, whereas $\\Theta_{12}^S(\\bm{p}_F,\\bm{r})=0$\nfor all other trajectories. Again evaluating the Keldysh component of the Green\nfunction matrix and making use of Eq. \\eqref{current}, we get\n\\begin{multline}\nI^{nl} = eN_0\n\\int\n\\left<\n\\bm{v}_{F}\\Theta_{12}^S(\\bm{p}_F,\\bm{r})\n|\\tilde \\gamma_1(\\bm{p}_F,  \\bm{r}, \\varepsilon)|^2\n\\right>\n\\\\\\times\n\\left[\n-\\tilde x_2 (\\varepsilon)\n-\nx_1(\\varepsilon)\n\\right] d \\varepsilon d \\bm{\\Sigma}_1.\n\\label{nl}\n\\end{multline}\n\nCollecting all the above contributions, we determine the currents $I_1$ and $I_2$ flowing into respectively the first and the second normal terminals:\n\\begin{gather}\nI_1 = (V_1-V_2)\/R_{0}+I_1^{loc}(V_1)+I^{nl}(V_1,V_2,\\chi),\n\\label{i1}\n\\\\\nI_2 =(V_2-V_1)\/R_{0}+ I_2^{loc}(V_2) +I^{nl}(V_1,V_2,\\chi).\n\\label{i2}\n\\end{gather}\n\nIn order to proceed we need to evaluate the Riccati amplitude $\\tilde \\gamma_1^R(\\bm{p}_F,\n\\bm{r}, \\varepsilon)$ in the beginning of the corresponding trajectory. For simplicity let us assume that both temperatures $T_{1,2}$ and voltages $V_{1,2}$ remain well below the superconducting gap, i.e. $T_{1,2}, eV_{1,2} \\ll |\\Delta|$. In this case it follows immediately, e.g., from Eqs. \\eqref{i1loc}-\\eqref{nl} that electron transport in our system is dominated by quasiparticles with energies well in the subgap range\n\\begin{equation}\n|\\varepsilon| \\ll |\\Delta|.\n\\label{subgap}\n\\end{equation}\n\nConsider first the quasiclassical electron trajectory that begins in the\nterminal $\\mathrm{N_1}$,\nhits one of the NS interfaces and returns back to the same terminal. Making use of the analysis\ndeveloped in the previous section, under the condition \\eqref{subgap} one readily finds\n\\begin{equation}\n|\\tilde \\gamma_1^R(\\bm{p}_F,  \\bm{r},\n\\varepsilon)|^2 = \\mathcal{D}^2\/(1+\\mathcal{R})^2,\n\\label{BTK}\n\\end{equation}\nwhere, as before, $\\mathcal{D}=\\mathcal{D}_{1,2}=1-\\mathcal{R}_{1,2}$ is the normal transmission of the corresponding NS interface. Likewise, for the trajectories which start in the first terminal, hit\nthe interfaces NS$_1$ and NS$_2$  and then go towards the terminal\n$\\mathrm{N_2}$ (as illustrated\nin Fig. \\ref{nsnsn3-fig}), in the limit \\eqref{subgap} we obtain\n\\begin{multline}\n|\\tilde \\gamma_1^R|^2 =\n1-\\frac{16\\mathcal{R}_1\\mathcal{R}_2}\n{\\bigl|\n(1+\\mathcal{R}_1)(1+\\mathcal{R}_2)\n+\\mathcal{D}_1\\mathcal{D}_2e^{i(\\chi + 2\\varepsilon d\/v_F)}\n\\bigr|^{2}},\n\\label{as2}\n\\end{multline}\nwhere $d$ is the effective distance covered by a quasiparticle between the two\nscattering events at $\\mathrm{NS_1}$ and $\\mathrm{NS_2}$ interfaces. Here and\nbelow we assume that this distance obeys the condition $d \\gg v_F\/|\\Delta|$.\nCombining the results \\eqref{BTK} and \\eqref{as2} with Eqs.\n\\eqref{i1loc}-\\eqref{nl} we can easily evaluate the currents $I_{1,2}$\n\\eqref{i1}, \\eqref{i2}.\n\nBefore we complete this calculation let us briefly discuss the physical meaning of the above results.\nIt is straightforward to observe that the asymptotic value $\\tilde \\gamma^R_1\n\\gamma^A_1=|\\tilde \\gamma_1^R (\\varepsilon )|^2$ in the beginning of the\ncorresponding trajectory defines the Andreev reflection probability, i.e.\nthe probability for an incoming electron with energy $\\varepsilon$ to be\nreflected back as a hole. This observation is well illustrated, e.g., by Eq.\n\\eqref{BTK} which is nothing but the standard BTK result \\cite{BTK}. Making\nuse of general symmetry relations for the Green functions one can also\ndemonstrate that the probability for an incoming hole to be reflected back as an\nelectron equals\nto $|\\tilde \\gamma_1^R(-\\varepsilon)|^2$. Thus, from Eq. \\eqref{as2} we conclude that scattering on two NS interfaces generates {\\it electron-hole symmetry violation}\n\\begin{equation}\n|\\tilde \\gamma_1^R(\\varepsilon)|^2\\neq |\\tilde \\gamma_1^R(-\\varepsilon)|^2\n\\end{equation}\nfor quasiparticles propagating from $\\mathrm{N_1}$- to $\\mathrm{N_2}$-terminals\nalong the trajectories displayed in Fig. \\ref{nsnsn3-fig} provided the\nsuperconducting phase difference $\\chi$ takes an arbitrary value not equal to\nzero or $\\pi$ and provided normal transmissions of both NS interfaces obey the\ncondition $0<\\mathcal{D}_{1,2}<1$. Below we will demonstrate that this\nelectron-hole asymmetry yields a large thermoelectric effect in the system\nunder consideration.\n\n\n\\section{Thermoelectric voltage}\n\\label{thermo}\nLet us now evaluate the currents $I_1$ \\eqref{i1} and $I_2$ \\eqref{i2}. In order to recover the local BTK terms $I^{loc}_{1,2}$ we explicitly specify the contributions from electron trajectories scattered at the first\nand the second NS interfaces by splitting $\\Theta_{11}^S \\to \\Theta_{11}^{S_1}+\\Theta_{11}^{S_2}$ (and similarly for $\\Theta_{22}^S$). Then combining Eqs. \\eqref{i1loc}, \\eqref{distr} with \\eqref{BTK}, at low voltages and temperatures $eV_{1,2}, T_{1,2} \\ll |\\Delta|$ we get\n\\begin{equation}\nI^{loc}_{1,2} = V_{1,2}\/R_{1,2},\n\\end{equation}\nwhere $R_{1,2}$ define the standard BTK low temperature resistance of the SN interfaces, i.e.\n\\begin{multline}\n\\dfrac{1}{R_1}=\n4e^2N_0\n\\int\n\\Biggr<\n\\bm{v}_{F}\n\\Biggl[\n\\dfrac{\\mathcal{D}_1^2}{(1+\\mathcal{R}_1)^2}\n\\Theta_{11}^{S_1}(\\bm{p}_F,\\bm{r})\n+\\\\+\n\\dfrac{\\mathcal{D}_2^2}{(1+\\mathcal{R}_2)^2}\n\\Theta_{11}^{S_2}(\\bm{p}_F,\\bm{r})\n\\Biggr]\n\\Biggl>\nd \\bm{\\Sigma}_1.\n\\label{BTK2}\n\\end{multline}\nNote that for simplicity in Eq. \\eqref{BTK2} we disregard multiple scattering effects \\cite{FN0} and account for quasiparticle trajectories which hit only one of the two NS interfaces ignoring, e.g., the contribution of trajectories of the type $\\Theta_{11}^{S_1S_2}(\\bm{p}_F,\\bm{r})$ which hit both NS interfaces. If necessary, the latter contribution can easily be recovered, however, it may only yield renormalization of subgap resistances $R_{1,2}$ (cf., e.g., the last term in Eq. \\eqref{rnl} below) and does not play any significant role in our further analysis.\n\nIn contrast, the trajectories of the type $\\Theta_{12}^{S_1S_2}(\\bm{p}_F,\\bm{r})$ displayed in Fig. \\ref{nsnsn3-fig} give an important contribution to the non-local current $I^{nl}$ and they should\nnecessarily be accounted for along with trajectories $\\Theta_{12}^{S_1}$ and $\\Theta_{12}^{S_2}$.\nCollecting all these contributions \\cite{FN}, with the aid of Eqs. \\eqref{distr}, \\eqref{nl} and \\eqref{as2}\nwe obtain\n\\begin{equation}\nI^{nl}= (V_1+V_2)\/R^{nl}_+\n+\n\\tilde I^{nl}(T_{1,2},V_{1,2},\\chi),\n\\end{equation}\nwhere we defined\n\\begin{multline}\n\\dfrac{1}{R^{nl}_{\\pm}}\n=\n2e^2N_0\n\\int\n\\Biggr<\n\\bm{v}_{F}\n\\Biggl[\n\\dfrac{\\mathcal{D}_2^2\\Theta_{12}^{S_2}(\\bm{p}_F,\\bm{r})}{(1+\\mathcal{R}_2)^2}\n\\pm\\dfrac{\\mathcal{D}_1^2\\Theta_{12}^{S_1}(\\bm{p}_F,\\bm{r})}{(1+\\mathcal{R}_1)^2}\n\\\\+\n\\dfrac{\\mathcal{R}_1 \\mathcal{D}_2^2 \\pm \\mathcal{R}_2 \\mathcal{D}_1^2}{\n(1+ \\mathcal{R}_1 \\mathcal{R}_2) (\\mathcal{R}_1 + \\mathcal{R}_2)}\n\\Theta_{12}^{S_1S_2}(\\bm{p}_F,\\bm{r})\n\\Biggr]\n\\Biggl>\nd \\bm{\\Sigma}_1,\n\\label{rnl}\n\\end{multline}\nand $\\tilde I^{nl} (T_{1,2}, V_{1,2}, \\chi)$ represents the term sensitive to electron-hole asymmetry in our system.\nPerforming the corresponding energy integral (see Appendix), we get\n\\begin{multline}\n\\tilde I^{nl}= -eN_0\n\\int\n\\Biggr<\n\\dfrac{32\\pi\\bm{v}_{F}\\Theta_{12}^{S_1S_2}(\\bm{p}_F,\\bm{r})\\mathcal{R}_1\n\\mathcal{R}_2\\beta}{\n(1+ \\mathcal{R}_1 \\mathcal{R}_2) (\\mathcal{R}_1 + \\mathcal{R}_2)\n}\n\\\\\\times\n\\Biggl[\nT_2 W(\\beta, t_2, \\chi - v_2)-T_1 W(\\beta, t_1, \\chi + v_1)\n\\Biggr]\n\\Biggl>\nd \\bm{\\Sigma}_1,\n\\label{tnl}\n\\end{multline}\nwhere\n\\begin{equation}\nW(\\beta,t,\\chi) = \\Img\n\\sum_{n \\geqslant 0}\n\\dfrac{e^{i\\chi} e^{-t(2n+1)}}{1 + \\beta e^{i\\chi} e^{-t(2n+1)}}\n\\label{W}\n\\end{equation}\nand we defined\n\\begin{equation}\n\\beta = \\dfrac{\\mathcal{D}_1 \\mathcal{D}_2}{(1+\\mathcal{R}_1)(1+\\mathcal{R}_2)},\n\\end{equation}\n$t_{1,2} =2\\pi T_{1,2} d\/v_F$ and $v_{1,2}=2eV_{1,2} d\/v_F$. Note that here and below\nthe parameter $d$ depends on the particular electron trajectory and, hence, the function\n$W$ cannot be taken out of the angular brackets indicating averaging over the directions of $\\bm{v}_{F}$.\n\nThe expression for $W$ \\eqref{W} gets simplified in the limits of high and low temperatures,\ni.e. \\begin{equation}\nW(\\beta, t, \\chi)\n=\n\\begin{cases}\ne^{-t}\\sin \\chi , & t \\gg 1,\n\\\\\n\\dfrac{1}{2t\\beta}\n\\arctan\n\\left(\n\\dfrac{\\beta \\sin \\chi}{1 + \\beta \\cos \\chi}\\right), & t \\ll 1.\n\\end{cases}\n\\end{equation}\n\n\nThe current $\\tilde I^{nl}$ is responsible for the large thermoelectric effect in the system under consideration. In order to illustrate this fact let us disconnect both normal terminals from external leads in which case one obviously has\n\\begin{equation}\nI_1=I_2\\equiv 0.\n\\label{discon}\n\\end{equation}\nThen in the absence of a temperature gradient (i.e. for $T_1=T_2$) both voltages vanish identically $V_1=V_2=0$. If, however, temperatures $T_1$ and $T_2$ take different values non-zero {\\it thermoelectric} voltages\n$V_{1,2}=V_{T1,2}$ are induced in our system. Introducing the renormalized subgap and\nSharvin resistances\n\\begin{gather}\n\\dfrac{1}{\\tilde R_{1,2}}\n=\n\\dfrac{1}{R_{1,2}}\n-\n\\dfrac{2}{R^{nl}_+},\n\\quad\n\\dfrac{1}{\\tilde R_{0}}\n=\n\\dfrac{1}{R_{0}}\n-\n\\dfrac{1}{R^{nl}_+},\n\\end{gather}\nrewriting Eqs. \\eqref{i1}, \\eqref{i2} in the form\n\\begin{gather}\nI_1 = \\dfrac{V_{T1} - V_{T2}}{\\tilde R_{0}}+\\dfrac{V_{T1}}{\\tilde R_1}\n+ \\tilde I^{nl},\n\\label{i11}\n\\\\\nI_2 = \\dfrac{V_{T2} - V_{T1}}{\\tilde R_{0}}+\\dfrac{V_{T2}}{\\tilde R_2} +\n\\tilde I^{nl}\n\\label{i22}\n\\end{gather}\nand resolving these equations with respect to $V_{T1}$ and $V_{T2}$ together\nwith Eq. \\eqref{discon}, we arrive at the result\n\\begin{equation}\nV_{T1,2} =\n\\dfrac{\n\\tilde R_{2,1}(\\tilde R_{1,2}+\\tilde R_{0}\/2)}{\n\\tilde R_1+ \\tilde R_2+\\tilde R_{0}}\n\\tilde I^{nl}(T_{1,2},V_{T1,2},\\chi),\n\\label{vt12}\n\\end{equation}\nwhich determines the magnitude of the thermoelectric voltages $V_{T1,2}$ induced by a nonzero temperature gradient $T_2-T_1$.\n\nEq. \\eqref{vt12} -- together with the expression for $\\tilde I^{nl}$ \\eqref{tnl} -- constitutes the central result of this work.\nIt demonstrates that the thermoelectric effect in Andreev interferometers is in general not reduced by the small parameter $T\/\\varepsilon_F$ and\nremains well in the measurable range. The key physical reason for this behavior is the presence of electron-hole\nasymmetry generated for quasiparticles moving between two normal terminals and being scattered at two interfaces NS$_1$ and NS$_2$.\nAccording to Eq. \\eqref{as2} this asymmetry is generated for any value of the phase difference $\\chi$ between superconductors S$_1$ and S$_2$\nexcept for $\\chi =0,\\pi$ and for all values of the interface transmissions $\\mathcal{D}_{1,2}$ except for $\\mathcal{D}_{1,2}=0,1$.\n\nThese observations emphasize a crucial role played by coherent Andreev reflections at both NS interfaces.\nIndeed, the effect trivially vanishes  $\\tilde I^{nl}\\equiv 0$ in the absence of Andreev reflection at any of the interfaces,\ni.e. for $\\mathcal{D}_{1(2)}=0$. Remarkably, electron-hole asymmetry is {\\it not} generated also at full transmissions $\\mathcal{R}_{1(2)}=0$\n(cf. Eq. \\eqref{as2}) and, hence, the current $\\tilde I^{nl}$ \\eqref{tnl} also vanishes provided complete Andreev reflection\n($|\\tilde \\gamma_1^R|^2=1$) is realized at any of the NS interfaces. Moreover, bearing in mind that Andreev reflection does not violate\nquasiparticle momentum conservation one can immediately conclude that, e.g., for  $\\mathcal{R}_{1}=0$ electron scattering at the first\nNS interface can only contribute to the BTK resistance of this interface but not to the non-local current $\\tilde I^{nl}$.\nThis is because an incident electron and a reflected hole propagate along the\nsame (time-reversed) trajectories starting and ending\nin one and the same normal terminal. This observation is specific for ballistic systems and has the same physical origin as,\ne.g., the effect of vanishing crossed Andreev reflection contribution to the average non-local current in NSN structures with ballistic electrodes and\nfully transparent NS interfaces \\cite{KZ07}.\n\nWe would like to emphasize that the validity of our analysis is not restricted to the limit of small temperature\ngradients and, hence, our results apply at any values $T_2-T_1$ provided both temperatures are kept well in the subgap range.\nIn the limit $T_2-T_1 \\gg v_F\/d$ the magnitudes of both thermal voltages $V_{T1,2}$ \\eqref{vt12} and\nthe non-local current $\\tilde I^{nl}$ \\eqref{tnl} depend only on the lower of the two temperatures ($T_1$) and become practically\nindependent of the higher one ($T_2$). Then the maximum values of $\\tilde I^{nl}$ which could possibly be reached at a given temperature\n$T_1$ can roughly be estimated as\n\\begin{equation}\n\\tilde I^{nl}\n\\sim\n\\begin{cases}\ne\\mathcal{N}_{\\mathrm{ch}}T_1 e^{-2\\pi T_1d\/v_F} , & T_1 \\gg v_F\/(2\\pi d),\n\\\\\n e\\mathcal{N}_{\\mathrm{ch}}v_F\/d, &T_1 \\ll v_F\/(2\\pi d),\n\\end{cases}\n\\label{estim}\n\\end{equation}\nwhere $\\mathcal{N}_{\\mathrm{ch}} \\sim p_F^2S$ is the number of conducting channels in a metallic wire with an effective cross section $S$. The parameter\n$d$ here should be understood as an effective distance between the two NS interfaces. The estimate \\eqref{estim} demonstrates that in the optimum case\nthe current $\\tilde I^{nl}$ can be of the same order as the critical Josephson current of ballistic SNS junctions \\cite{Kulik}\nwith similar parameters $\\mathcal{N}_{\\mathrm{ch}}$ and $v_F\/d$. In the low \ntemperature limit $T_{1,2} \\ll v_F\/(2\\pi d)$ and small temperature difference \n$|T_1-T_2| \\ll T_{1,2}$ Eqs. \\eqref{tnl}, \\eqref{W} yield $\\tilde \nI^{nl} \\propto (T_1+T_2)(T_1-T_2)$ in a qualitative agreement with the results \\cite{Seviour00,KPV,VP,JW}.\n\nLet us also note that the thermoelectric signal \\eqref{tnl}, \\eqref{vt12} \nderived within our model is described by an \\textit{odd} periodic function of \nthe phase difference $\\chi$. This result goes in line with a general symmetry \nanalysis \\cite{VH2} both for diffusive and ballistic structures. At the same \ntime, it is worth pointing out that odd as well as even $2\\pi$-periodic phase \ndependencies of the thermopower were observed in experiments \n\\cite{Venkat1,Venkat2,Petrashov03,Venkat3}. Possible explanations of such \nobservations were proposed by a number of authors. E.g., Titov \\cite{Titov} \nargued that an even in $\\chi$ thermopower response can occur for certain \ngeometries of Andreev interferometers as a result of the charge imbalance \nbetween the chemical potential of Cooper pairs in superconductor and the one for \nquasiparticles in the normal metal. Jacquod and Whitney \\cite{JW} analyzed \nthermoelectric effects in Andreev interferometers within the scattering \nformalism and attributed an even in $\\chi$ behavior of the observed thermopower \nto mesoscopic fluctuation effects.\n\n\n\\section{Josephson current and thermoflux}\n\\label{josephson}\nIn order to further investigate possible relation  between $\\tilde I^{nl}$ and the Josephson current between the two S-terminals\nlet us evaluate the current $I_{12}$ flowing in the middle part of the normal wire, see Fig. \\ref{nsnsn3-fig}. Under the condition\n\\eqref{discon} $I_{12}$ determines the total current between two superconductors S$_1$ and S$_2$. The whole calculation is carried out in the same manner as\nthat for the currents $I_1$ and $I_2$. Evaluating the contributions from all quasiparticle trajectories going through the central part\nof the normal wire, we obtain\n\\begin{equation}\nI_{12}=I_T+I_S^{\\rm tot},\n\\end{equation}\nwhere the term $I_T$ represents the thermoelectric current which has the form\n\\begin{equation}\nI_T=\\frac{V_{T1}}{R_{1}^+}\n-\\frac{V_{T2}}{R_{2}^-},\\quad \\frac{1}{R_{1,2}^{\\pm}}=\\frac{1}{R_{1,2}^*}+\\dfrac{1}{R_{0}}\\pm\\dfrac{1}{R^{nl}_{-}},\n\\label{IT}\n\\end{equation}\n$R_{1(2)}^*=R_{1(2)}|_{\\mathcal{D}_{1(2)}=0}$, while the current $I_S^{\\rm tot}$ defines the supercurrent between the two superconductors.\nIt can be split into two parts:\n\\begin{equation}\nI_S^{\\rm tot}=I_{S_1S_2}(\\chi )+ I_S(\\chi, T_{1,2},V_{T1,2}),\n\\end{equation}\nwhere $I_{S_1S_2}(\\chi )$ is the equilibrium Josephson current evaluated for closed quasiparticle trajectories confined between\nthe two S-terminals \\cite{GZ02} (such trajectories, if exist, ``know nothing''\nabout the normal terminals $\\mathrm{N_1}$ and $\\mathrm{N_2}$ and\nfor this reason were not considered above) and\n\\begin{multline}\nI_S(\\chi, T_{1,2},V_{T1,2})=-e N_0\n\\int\n\\Biggr<16 \\pi\\bm{v}_{F}\\Theta_{12}^{S_1S_2}(\\bm{p}_F,\\bm{r})\n\\\\\\times\n \\dfrac{\n\\mathcal{R}_1\\mathcal{R}_2\\beta}{(\\mathcal{R}_1 + \\mathcal{R}_2)(1 + \\mathcal{R}_1\\mathcal{R}_2)}\n\\Biggl[\n\\frac{1+\\mathcal{R}_2^2}{\\mathcal{R}_2} T_1 W(\\beta, t_1, \\chi +v_1)\n\\\\\n+\n\\frac{1+\\mathcal{R}_1^2}{\\mathcal{R}_1} T_2 W(\\beta, t_2, \\chi -v_2)\n\\Biggr]\n\\Biggl>\nd \\bm{\\Sigma}_1\n\\label{J}\n\\end{multline}\nrepresents the contribution of open trajectories of the type $\\Theta_{12}^{S_1S_2}$ connecting\nthe normal terminals $\\mathrm{N_1}$ and $\\mathrm{N_2}$. In Eq. \\eqref{J} we\nagain defined $v_{1,2}=2eV_{T1,2} d\/v_F$.\n\nComparing the above expressions for the thermoelectric current $I_T \\propto \\tilde I^{nl}$ and for the supercurrent $I_S^{\\rm tot}$\nwe conclude that in general there exists no simple relation between these two currents. Only in the tunneling limit  $\\mathcal{R}_{1,2} \\to 1$\none can observe that Eqs. \\eqref{tnl} and \\eqref{J} are defined by almost the same integrals except for different signs\nin front of the two $W$-functions in the square brackets. If, furthermore, we assume that the temperature difference exceeds the parameter\n$v_F\/d$, in the tunneling limit we obtain $\\tilde I^{nl}\\simeq I_S$ for $T_1-T_2 \\gg v_F\/d$ and $\\tilde I^{nl}\\simeq -I_S$ for $T_2-T_1 \\gg v_F\/d$.\n\nNote that in the presence of a temperature gradient the expression \\eqref{J} represents a {\\it non-equilibrium} contribution to the \nJosephson current which explicitly depends on both $T_{1,2}$ and $V_{T1,2}$. In the absence of this gradient, i.e. at $T_1=T_2=T$ and $V_{T1,2}=0$\nthe term \\eqref{J} reduces to its equilibrium form and we can define\n\\begin{equation}\nI_J(\\chi ,T)=-I_S(\\chi, T_{1,2}=T,V_{T1,2}=0).\n\\label{IJ}\n\\end{equation}\nThen in the tunneling limit $\\mathcal{R}_{1,2} \\to 1$ and provided both thermal voltages remain small $2eV_{T1,2} \\ll v_F\/d$ we may write\n\\begin{equation}\nV_{T1,2}\\propto \\tilde I^{nl}\\simeq \\frac12[I_J(\\chi ,T_1)-I_J(\\chi ,T_2)].\n\\label{VJ}\n\\end{equation}\nTo a certain extent this relation resembles the result \\cite{VH} derived in the diffusive limit.\nNote, however, that within our model Eq. \\eqref{VJ} is valid only under quite stringent conditions and, furthermore, the term $I_J(\\chi ,T)$ \\eqref{IJ} can be interpreted as a total equilibrium Josephson current only if we neglect the contribution $I_{S_1S_2}(\\chi )$. Most importantly, it is clear from our analysis that the relation \\eqref{VJ} by no means implies that large thermoelectric voltages $V_{T1,2}$ could possibly occur provided electron-hole symmetry in our system is maintained.\n\nThe effects discussed here can be conveniently measured, e.g., in a setup similar to that employed in recent experiments \\cite{Petrashov16}. One can connect two superconductors in a way to form a superconducting loop.\nInserting an external flux $\\Phi_x$ into this loop one would be able to control the phase difference $\\chi =2\\pi \\Phi_x\/\\Phi_0$, where $\\Phi_0$ is the superconducting flux quantum. By measuring the total flux inside\nthe loop $\\Phi=\\Phi_x+\\mathcal{L}I_{12}$ with and without a temperature gradient one could easily determine\nthe value of the magnetic thermoflux\n\\begin{equation}\n\\Phi_T = \\mathcal{L}[I_{12}(\\chi, T_{1,2},V_{T1,2})-I_{12}(\\chi, T)],\n\\end{equation}\nwhere $\\mathcal{L}$ is an effective inductance of a superconducting loop and $ I_{12}(\\chi, T)=I_{12}(\\chi, T_{1,2}=T,V_{T1,2}=0)$ is the equilibrium Josephson current at temperature $T$.\n\nIn conclusion, we demonstrated that large thermoelectric effect in multiterminal ballistic normal-superconducting hybrid structures is caused by electron-hole asymmetry generated for quasiparticles propagating between two normal terminals (kept at different temperatures) and suffering coherent Andreev reflection at two NS interfaces. At sufficiently high temperature gradients the thermoelectric voltages $V_{T1,2}$ depend only on the lowest of the two temperatures. The $2\\pi$-periodic dependence of $V_{T1,2}$ on the superconducting phase difference $\\chi$ is determined self-consistently and is strongly non-sinusoidal at low enough temperatures. Although the temperature dependence of $V_{T1,2}$ roughly resembles that of the equilibrium Josephson current in SNS junctions there exists no fundamental relation between these two quantities. Further information can be obtained by analyzing the behavior of the magnetic thermoflux induced in Andreev interferometers by applying\na thermal gradient.\n\nThis work was supported in part by RFBR grant No. 15-02-08273.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe observation of baryon number violation (BNV) would address a number of open questions in modern physics. BNV is required to understand the matter-antimatter symmetry of the universe~\\cite{Sakharov:1967dj}. Many models which explain  non-zero neutrino masses also prescribe BNV~\\cite{Mohapatra:1980qe}. Even within the Standard Model (SM) baryon number is subject only to an approximate conservation law. At the perturbative level baryon number conservation arises due to the specific matter content in the SM, and corresponds to a so-called  ``accidental\" symmetry. The SM predicts BNV to occur via rare non-perturbative electroweak instanton processes~\\cite{Adler:1969gk,'tHooft:1976fv} (the quantum number $B-L$ is respected by the SM and not $B$ and $L$ separately).  Furthermore, precision tests of the Equivalence Principle~\\cite{Adelberger:1990xq} offer no evidence for a long range force coupled to baryon number and thus a local gauge symmetry forbidding BNV. Consequently, BNV occurs as a generic feature of many proposed extensions to the SM~\\cite{Barbier:2004ez}.  A promising means of searching for BNV is via the observation of the $\\Delta B=2$ process, neutron-antineutron oscillation~\\cite{Mohapatra:2009wp,Phillips:2014fgb}. In this paper, a proposed new experiment~\\cite{EOInnbar} to look for such oscillations at the European Spallation Source (ESS) is outlined. The experiment would be sensitive to oscillation probabilities up to three orders of magnitude lower than has previously been obtained using free neutrons.\n\n\nThere exists a symbiosis between neutron-antineutron oscillations and neutrino physics via the quantum number $B-L$. A popular model explaining non-zero neutrino mass is the see-saw mechanism~\\cite{see-saw}. In this approach neutrinos possess a Majorana component and lepton number is violated by two units. Evidence for $\\Delta L=2$ processes are sought with, eg,  double neutrinoless beta decay searches~\\cite{Elliott:2012sp}. Since $B-L$ (the true anomaly-free SM symmetry) is also violated by two units it would be natural to expect $\\Delta B=2$ processes. In addition to the complementarity with neutrino physics, neutron-antineutron oscillation features in a number other models of new physics. Examples include $R$-parity violating supersymmetry~\\cite{Barbier:2004ez} and post-sphaleron baryogesesis~\\cite{Babu:2006xc}. Values of the BNV mass scale for which observable oscillations take place exceed those attainable at colliders.  Using a six-fermion BNV operator and dimensional reasoning mass scales of $10-1000$~TeV are obtained while other approaches (also leading to an observable signature) predict scales near the grand unified mass~\\cite{Mohapatra:2009wp}.  A further motivation for searching for oscillations was recently provided by the observation that such processes violate not only baryon number but also $CP$~\\cite{Berezhiani:2015uya}, thereby addressing two of the Sakharov conditions~\\cite{Sakharov:1967dj} for baryogenesis.\n\nSetting aside the substantial theoretical motivation,  a strictly experimentalist consideration of BNV hunting highlights the importance of neutron-antineutron oscillation searches.  In an oscillation experiment only the violation of baryon number is sought, and not that of other hitherto conserved quantities. Single nucleon decay searches (eg, $p\\rightarrow \\pi^0 e^+$) require lepton number violation ; among other reasons this ensures angular momentum conservation. Only searches for free neutron oscillation~\\cite{Fidecaro:1985cm, BaldoCeolin:1994jz} and anomalous nuclear decays, under the neutron oscillation~\\cite{nnbartrapped} or dinucleon decay-hypothesis~\\cite{dinucleon}, offer high precision sensitivity to BNV-only processes. The most competitive limits for the free neutron oscillation time have hitherto been produced at ILL~\\cite{BaldoCeolin:1994jz} ($\\sim 3 \\times 10^8$s) and Super-Kamiokande~\\cite{Abe:2011ky} ($\\sim 1 \\times 10^8$s, after a correction for nuclear effects).\n\nOf the class of experiments which search for BNV-only processes, free neutron oscillation searches possess both the cleanest experimental and theoretical environments in which to perform the search and quantify the results of the search. Owing to improvements in neutronics, particle identification technology and a longer running time, the proposed experiment at the ESS~\\cite{EOInnbar} will have a sensitivity in oscillation probability which is to up to three orders of magnitude greater than at ILL.\n\nThis paper is organised as follows. Descriptions of the ESS and the neutron moderator are given. The plan for the transmission of neutrons to the detector is outlined, followed by a description of the detector. Finally, the collaboration which aims to conduct the experiment, as well a provisional timescale for the work, are briefly described.\n\n\\section{Overview of ESS and the proposed experiment}\\label{sec:ess}\nCurrently under construction, the ESS is a multi-disciplinary research laboratory which will house the world's most powerful neutron source~\\cite{ESS-TDR}. The ESS will comprise a 2.86 ms long proton pulse at 2 GeV energy at a repetition rate of 14 Hz which impacts on a rotating tungsten target. Spallation neutrons emerging from a system of moderators and reflectors are delivered to the beam ports and are then guided with a neutron supermirror to the instrument.  For the experiment described here, neutrons would be transported through in a vacuum through a magnetically shielded beam pipe over $200--300$m to a target with which anti-neutrons could annihilate. Magnetic shielding is necessary to suppress the energy split between neutron and antineutron states ($\\Delta E=\\bar{\\mu} \\cdot \\bar{B}$) which would occur in a $B$-field due do the particles' dipole moments $\\pm \\bar{\\mu}$, and which would inhibit the oscillation process. A detector surrounding the target would record the final states emerging from an annihilation as well as monitoring background processes.\n\nThe quantity of merit for a neutron-antineutron search is $N_n \\cdot t^2$ where $N_n$ is the free neutron flux reaching the target and $t$ is the free flight time of the neutron.\n\nFor a high sensitivity (high $N_n \\cdot t^2$) search the following criteria must be met:\n\\begin{itemize}\n\\item The moderator must deliver a beam of slow, cold neutrons (energy $< 5$meV) at high intensity, maximising $t$ and $N_n$, respectively.  An overall lower neutron spectrum emission also increases the transport efficiency of the supermirror neutron reflector.\n\\item The beam port must correspond to a large opening angle for neutron emission.\n\\item A long beamline to increase $t$.\n\\item Long running time.\n\\end{itemize}\n\n\nA number of factors drive the improvement in sensitivity of the proposed experiment compared to the work at ILL. An important contribution to increased sensitivity is due to the use of a large elliptical focusing supermirror reflector which directs off-axis neutrons to the detector with a single reflection\\footnote{Each reflection effectively ''resets the clock\" prior to a putative neutron antineutron oscillation.}. Furthermore, a larger detector with enhanced particle identification is possible, as is a longer running time. \n\n\\section{Moderator system}\nA conceptual overview of the ESS and moderator system is given Figure~\\ref{fig:wheelandmoderator}. A ''butterfly\" moderator design has been chosen which comprises cold regions of para-$H_2$ at around $20$~K, and water at ambient temperature . The upper (lower) moderator has a height of 3cm (6cm). This represents the most optimal choice for the brightness of the ESS. In addition, it provides a flexibility such that the placing of the instruments can be chosen according to which moderator is most appropriate for the experiment. For the work proposed here, a beam port is available such that neutrons from both moderators can be exploited.\n\n\n\\begin{figure}[tb]\n  \\setlength{\\unitlength}{1mm}\n  \\begin{center}\n\\includegraphics[width=0.80\\linewidth, angle=90]{wheelandmoderator.pdf}\n    \\end{center}\n    \\vspace{-2.25cm}\n  \\caption{Overview of the ESS and the moderator system.\n  }\n  \\label{fig:wheelandmoderator}\n\\end{figure}\n\nFigure~\\ref{fig:neutronics} shows a cross sectional view of the moderator system from the suppermirror. The cold moderators are shown in red.  Here, the $x$ and $y$ coordinates are axes map out the horizontal plane. The figures to the top and left  show the relative guiding efficiency by a truncated focusing ellipsoid mirror centred on the middle point as a function of the point of emission. The transport efficiency for neutrons produced in the cold region is $\\sim 10$\\% of that expected for neutrons produced at the ellipsoid's focal point. This drop in efficiencies near the cold region illustrates the need for a sophisticated mirror configuration which fully takes into account the design of the moderator. At present a ''clover\" assembly of four quarter ellipsoids, each one centred on a moderator, is being considered, as is a simpler design comprising an ellipsoid focused on aa cold moderator.\n\n\n\\begin{figure}[tb]\n  \\setlength{\\unitlength}{1mm}\n  \\begin{center}\n\\includegraphics[width=1.00\\linewidth, angle=-90]{neutronics.pdf}\n    \\end{center}\n    \\vspace{-2.75cm}\n  \\caption{ View of the butterfly moderators from the supermirror showing the cold moderators in red. The top and left figures show the relative guiding efficiency by a mirror centred on the middle point as a function of the point of emission.\n  }\n  \\label{fig:neutronics}\n\\end{figure}\n\n\nAn increase to the nominal angular acceptance for produced neutrons is another area in which sensitivity can be enhanced. As seen in Figure~\\ref{fig:moderatorandreflector} losses will occur due to the presence of the Fe shield and Be reflector system. Figure~\\ref{fig:moderatorandreflector} shows a possible adjustment to the design of the beam port for the proposed work. Here, the parts of the shield and reflector system would be removed to allow a greater conical penetration corresponding to an increase in sensitivity of $\\sim 2$. Inserts could be made such that the full system is restored for experiments after the proposed work. The study of such a scheme including its on the other experiments is underway.\n\n\\begin{figure}[tb]\n  \\setlength{\\unitlength}{1mm}\n  \\begin{center}\n\\includegraphics[width=0.80\\linewidth, angle=0]{reflectormoderator.pdf}\n    \\end{center}\n    \\vspace{-.75cm}\n  \\caption{ Nominal (white region to the left of the neutron source) and enlarged (region enclosed by dashed lines) conical penetration through the Be reflector and Fe shield).\n  }\n  \\label{fig:moderatorandreflector}\n\\end{figure}\n\n\\begin{figure}[tb]\n  \\setlength{\\unitlength}{1mm}\n  \\begin{center}\n\\includegraphics[width=0.80\\linewidth, angle=0]{shielding.pdf}\n    \\end{center}\n    \\vspace{-1.75cm}\n  \\caption{ Schematic overview of the planned shielding.\n  }\n  \\label{fig:shield}\n\\end{figure}\n\n\n\\section{Magnetics}\nAs mentioned in Section~\\ref{sec:ess}, the neutrons must be transported in a magnetically shielded vaccum. For the proposed work this corresponds to a vacuum of $10^{-5}$ mbar and a magnetic field of less than 5nT along the neutron flight path.\n\nThe target vaccum can be achieved with a vacuum chamber comprising highly non-magnetic materials, eg Al, with turbo molecular pumps, mounted outside of the magnetically shielded area. Magnetic fields of  less than 5 nT have been achieved over large volumes (see, for example, Ref.~\\cite{m1}). For the planned experiment, a shielding concept will be used based on an aluminium vacuum chamber, a two layer passive shield made from magnetizable alloy for transverse shielding, and end sections made from passive and active components for longitudinal shielding, as shown in Figure~\\ref{fig:shield}.\n\n\\section{Detector}\nDetector design is guided by the need for high efficiency for antineutron detection and the ability to maintain a low background yield. A typical annihilation signature would be a multi-pion final state. A schematic diagram of the detector is given in Figure~\\ref{fig:detector}. The following components are envisaged and studies on the various technology options underway:\n\n\\begin{itemize}\n\\item A annihilation target. One option is a $^{12}C$ disk of diameter $1$~m and which is 100~$\\mu$m thick. An alternative is a target made of $^{10}Be$ which may have better potential to capture photons from background processes. A two target system is also being considered; an antineutron would annihilate in the first target so a second target could be used for background monitoring.\n\\item A charged particle tracker, necessary for the determination of pion momenta and the vertex position. Different technologies are under consideration, for a straw tube-based drift chamber or a Time Projection Chamber. However, any tracking system will need at least some layers with fast readout (i.e. straws) to allow tracking information to be included in the trigger.\n\\item A calorimeter must accurately measure photon and pion energies in order to reconstruct the final state invariant mass. Depending on the technology choice, precision timing information from the calorimeter or a calorimeter+time-of-flight configuration would be available. This is necessary for establishing the time of an annihilation event, and for rejecting false vertex reconstructions due to cosmic ray showers. The calorimeter will also need to handle high pile-up rates from gamma production at the target.\n\\item A trigger exploiting all read-out channels enabling a highly selective system to collect signal and background candidate events.\n\\item A dedicated cosmic veto system to reject background.\n\\end{itemize}\n\n\\begin{figure}[tb]\n  \\setlength{\\unitlength}{1mm}\n  \\begin{center}\n\\includegraphics[width=0.80\\linewidth, angle=90]{detector.pdf}\n    \\end{center}\n    \\vspace{-2.15cm}\n  \\caption{Schematic overview of the detector.\n  }\n  \\label{fig:detector}\n\\end{figure}\n\n\\section{Collaboration and time-scales}\nA growing collaboration has been formed with the aim of carrying out the proposed work. Working groups corresponding to specific aspects of the experiment have been established and a number of workshops have taken place. An expression of interest (EOI) with signatories from 26 institutes and 8 countries has been sent~\\cite{EOInnbar}.\n\nA provisional time-scale consists of the preparation a Technical Design Report in 2017, construction in 2019, commissioning in 2022, and data-taking ''physics\" runs in 2023-2025.\n\n\n\\section{Summary}\nStrong theoretical motivations addressing open questions in modern physics such as the matter-antimatter asymmetry and the possible majorana nature of the neutrino imply the existence of BNV processes. A promising means of searching for BNV is via the neutron-antineutron oscillation signature. A new high precision search is being planned for neutron-antineutron oscillation processes at the ESS. The sensitivity to the oscillation probability would be around three orders of magnitude compared with the last such experiment. A collaboration to carry has been formed with the aim of performing the experiment.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nNeuro\nNeuro\n\nThe New Brain Sciences and the Management of the Mind\n\nNikolas Rose and Joelle M. Abi-Rached\n\nPRINCETON UNIVERSITY PRESS\n\nPrinceton and Oxford\nCopyright \u00a9 2013 by Princeton University Press\n\nPublished by Princeton University Press, 41 William Street, Princeton, New Jersey 08540\n\nIn the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW\n\npress.princeton.edu\n\nCover image: Joan Mir\u00f3 (1893\u20131983), _Head of a Catalan Peasant_ , 1925. Oil on canvas.\n\nMus\u00e9e National d'Art Moderne, Centre Pompidou, Paris. Peter Willi \/ The Bridgeman Art Library.\n\n\u00a9 2012 Successi\u00f3 Mir\u00f3 \/ Artists Rights Society (ARS), New York \/ ADAGP, Paris.\n\nReproduction, including downloading of Joan Mir\u00f3 works is prohibited by copyright laws and international conventions without the express written permission of Artists Rights Society (ARS), New York.\n\nAll Rights Reserved\n\nLibrary of Congress Cataloging-in-Publication Data\n\nRose, Nikolas S.\n\nNeuro : the new brain sciences and the management of the mind \/ Nikolas Rose and Joelle M. Abi-Rached.\n\np. cm.\n\nIncludes bibliographical references and index.\n\nISBN 978-0-691-14960-8 (hardcover : alk. paper) \u2014 ISBN 978-0-691-14961-5\n\n(pbk. : alk. paper) 1. Neuropsychology. I. Abi-Rached, Joelle M., 1979\u2013 II. Title.\n\nQP360.R655 2013\n\n612.8\u2014dc23 2012023222\n\nBritish Library Cataloging-in-Publication Data is available\n\nThis book has been composed in Minion Pro and Ideal Sans\n\nPrinted on acid-free paper. \u221e\n\nPrinted in the United States of America\n\n10 9 8 7 6 5 4 3 2 1\nFor Diana, as always.  \nNR\n\nFor my parents, May and Maroun.  \nJAR\nContents | Acknowledgments _ix_\n\n---|---  \n|\n\nAbbreviations _xi_\n\n|\n\n**Introduction _1_**  \nBeyond Cartesianism? _3_  \nGoverning through the Brain _6_  \nOur Argument _9_  \nHuman Science? _23_\n\nOne | **The Neuromolecular Brain _25_**  \nHow Should One Do the History of the Neurosciences? _28_  \nInfrastructure _38_  \nA Neuromolecular Style of Thought _41_  \nEnter Plasticity _47_  \nA Neuromolecular and Plastic Brain _51_\n\nTwo | **The Visible Invisible _53_**  \nThe Clinical Gaze _55_  \nInscribed on the Body Itself _56_  \nOpen Up a Few Brains _61_  \nSeeing the Living Brain _65_  \nThe Epidemiology of Visualization _74_  \nThe New Engines of Brain Visualization _80_\n\nThree | **What's Wrong with Their Mice? _82_**  \nArtificiality? _85_  \nModels1, _Models 2, Models3, Models4_ (and _Possibly Models 5_) _92_  \nThe Specificity of the Human _102_  \nTranslation _104_  \nLife as Creation _108_\n\nFour | **All in the Brain? _110_**  \nTo Define True Madness _113_  \nThe Burden of Mental Disorder _125_  \nAll in the Brain? _130_  \nNeuropsychiatry and the Dilemmas of Diagnosis _137_\n\nFive | **The Social Brain _141_**  \nThe \"Social Brain Hypothesis\" _143_  \nPathologies of the Social Brain _148_  \nSocial Neuroscience _151_  \nSocial Neuroscience beyond Neuroscience _156_  \nGoverning Social Brains _160_\n\nSix | **The Antisocial Brain _164_**  \nEmbodied Criminals _167_  \nInside the Living Brain _173_  \nNeurolaw? _177_  \nThe Genetics of Control _180_  \nNipping Budding Psychopaths in the Bud _190_  \nSculpting the Brain in Those Incredible Years _192_  \nGoverning Antisocial Brains _196_\n\nSeven | **Personhood in a Neurobiological Age _199_**  \nThe Challenged Self _202_  \nFrom the Pathological to the Normal _204_  \nThe Self: From Soul to Brain _213_  \nA Mutation in Ethics and Self-Technologies? _219_  \nCaring for the Neurobiological Self _223_\n\nConclusion | **Managing Brains, Minds, and Selves _225_**  \nA Neurobiological Complex _225_  \nBrains In Situ? _227_  \nCoda: The Human Sciences in a Neurobiological Age _232_\n\nAppendix | How We Wrote This Book _235_\n\n|\n\nNotes _237_\n\n|\n\nReferences _277_\n\n|\n\nIndex _325_\nAcknowledgments\n\nThis book arises from research funded by the United Kingdom's Economic and Social Research Council in the form of a three-year professorial research fellowship to Nikolas Rose (grant number RES-051-27-0194), and we are happy to acknowledge this support and the opportunities that it provided. We have also benefited greatly from the work with our colleagues in the European Neuroscience and Society Network (ENSN), funded by the European Science Foundation (ESF), and the numerous events organized by this network that bring researchers from the human sciences together with leading neuroscience researchers in the spirit of critical friendship within which we have written this book: thanks to Trudy Dehue, Giovanni Frazzetto, Cornelius Gross, Ilpo Hel\u00e9n, Kenneth Hugdahl, Ilse Kryspin-Exner, Klaus-Peter Lesch, Linsey McGoey, Cordula Nitsch, Jo\u00e3o Arriscado Nunes, Andreas Roepstorff, Ilina Singh, and Scott Vrecko. We would also like to acknowledge our lively conversations with Francisco Ortega and Fernando Vidal, and to thank as well the many members of the international network of social scientists working in the field of neuroscience for so many productive discussions on its social dimensions. The research for this book was carried out within the wonderfully supportive community of the BIOS Research Centre at the London School of Economics, who also helped with organization of the final \"Brain, Self, and Society\" conference and follow-up workshop, \"Personhood in a Neurobiological Age,\" held in London on September 13\u201314, 2010: special thanks go to Btihaj Ajana, Valentina Amorese, Rachel Bell, Astrid Christoffersen-Deb, Megan Clinch, Caitlin Connors, Des Fitzgerald, Amy Hinterberger, John MacArtney, Sara Tocchetti, and our excellent administrator Victoria Dyas and outstanding manager, Sabrina Fernandez. We also wish to thank our editorial team at Princeton University Press for their support for this book and their careful work on our text.\nAbbreviations\n\nAD: | Alzheimer's disease\n\n---|---\n\nADHD: | attention deficit hyperactivity disorder\n\nAPA: | American Psychiatric Association\n\nAPIRE: | American Psychiatric Institute for Research and Education\n\nBNA: | British Neuroscience Association\n\nBOLD: | blood-oxygen-level-dependent (contrast imaging)\n\nBPD: | bipolar disorder (sometimes termed \"manic depression\")\n\nBRA: | Brain Research Association\n\nCaltech: | California Intitute of Technology\n\nCNRS: | Centre National de la Recherche Scientifique (France)\n\nCNVs: | copy number variations\n\nCT scan: | Computerized tomography (formerly computerized axial tomography or CAT scan)\n\nDALYs: | disability adjusted life years\n\n_DSM_ : | _Diagnostic and Statistical Manual of Mental Disorders_\n\nDSPD: | dangerous and severe personality disorder\n\nEBC: | European Brain Council\n\nECNP: | European College of Neuropsychopharmacology\n\nEEG: | electroencephalography\n\nENSN: | European Neuroscience and Society Network\n\nESF: | European Science Foundation\n\nESRC: | Economic and Social Research Council (U.K.)\n\nFDA: | Food and Drug Administration (U.S.)\n\nfMRI: | functional magnetic resonance imaging\n\nGWAS: | genome-wide association studies\n\nIBRO: | International Brain Research Organization\n\nICD: | International Classification of Diseases\n\nLSD: | lysergic acid diethylamide\n\nMAOA: | monoamine oxidase A\n\nMCI: | mild cognitive impairment\n\nMIT: | Massachusetts Institute of Technology\n\nMRC: | Medical Research Council (U.K.)\n\nMRI: | magnetic resonance imaging\n\nNESARC: | National Epidemiologic Survey on Alcohol and Related Conditions (U.S.)\n\nNGBRI: | not guilty by reason of insanity\n\nNHS: | National Health Service (U.K.)\n\nNIDA: | National Institute on Drug Abuse (U.S.)\n\nNIH: | National Institutes of Health (U.S.)\n\nNIMH: | National Institute of Mental Health (U.S.)\n\nNMR: | nuclear magnetic resonance\n\nNRP: | Neurosciences Research Program (U.S.)\n\n_OED_ : | _Oxford English Dictionary_\n\nPET: | positron emission tomography\n\nRSA: | Royal Society for the Encouragement of Arts, Manufactures and Commerce (U.K.)\n\nSfN: | Society for Neuroscience\n\nSNPs: | single nucleotide polymorphisms (pronounced \"snips\")\n\nSPECT: | single photon emission computed tomography\n\nSSRI: | selective serotonin receptor inhibitor\n\nUC Berkeley: | University of California, Berkeley\n\nUCLA: | University of California, Los Angeles\n\nWHO: | World Health Organization\n\nYLDs: | years lived with disability\nNeuro\n**Introduction**\n\nWhat kind of beings do we think we are? This may seem a philosophical question. In part it is, but it is far from abstract. It is at the core of the philosophies we live by. It goes to the heart of how we bring up our children, run our schools, organize our social policies, manage economic affairs, treat those who commit crimes or whom we deem mentally ill, and perhaps even how we value beauty in art and life. It bears on the ways we understand our own feelings and desires, narrate our biographies, think about our futures, and formulate our ethics. Are we spiritual creatures, inhabited by an immaterial soul? Are we driven by instincts and passions that must be trained and civilized by discipline and the inculcation of habits? Are we unique among the animals, blessed or cursed with minds, language, consciousness, and conscience? Are we psychological persons, inhabited by a deep, interior psyche that is shaped by experience, symbols and signs, meaning and culture? Is our very nature as human beings shaped by the structure and functions of our brains?\n\nOver the past half century, some have come to believe that the last of these answers is the truest\u2014that our brains hold the key to whom we are. They suggest that developments in the sciences of the brain are, at last, beginning to map the processes that make our humanity possible\u2014as individuals, as societies, and as a species. These references to the brain do not efface all the other answers that contemporary culture gives to the question of who we are. But it seems that these other ways of thinking of ourselves\u2014of our psychological lives, our habitual activities, our social relations, our ethical values and commitments, our perceptions of others\u2014are being reshaped. They must now be grounded in one organ of our bodies\u2014that spongy mass of the human brain, encapsulated by the skull, which weighs about three pounds in an adult and makes up about 2% of his or her body weight. This 'materialist' belief has taken a very material form. There has been a rapid growth in investment of money and human effort in neurobiological research, a remarkable increase in the numbers of papers published in neuroscience journals, a spate of books about the brain for lay readers, and many well-publicized claims that key aspects of human affairs can and should be governed in the light of neuroscientific knowledge. A host of neurotechnologies have been invented\u2014drugs, devices, techniques\u2014that seem to open ourselves up to new strategies of intervention through the brain. In the societies of what we used to call the West, our brains are becoming central to understanding who we are as human beings.\n\nFor many in the social and human sciences, these developments are profoundly threatening. Their unspoken premise, for at least the past century, has been that human beings are _freed_ from their biology _by virtue of that biology_ \u2014that we come into the world unfinished and that our individual capacities, mores, values, thoughts, desires, emotions (in short, our mental lives), as well as our group identities, family structures, loyalties to others, and so forth are shaped by upbringing, culture, society, and history. Practitioners of these disciplines can point, with good reason, to the disastrous sociopolitical consequences of the biologization of human beings: from eighteenth-century racial science to twentieth-century eugenics, and more recently to the reductionist simplifications of sociobiology and evolutionary psychology. To give priority to the biological in human affairs, it seems, is not only to ignore what we have learned from more than two centuries of social, historical, and cultural research, but also to cede many of the hard-won disciplinary and institutional achievements of the social and human sciences to others, and to risk their sociopolitical credibility.\n\nWe are sympathetic to these anxieties, and there is much truth in them. One only has to glance at the wild overstatements made by many of the popular writers in this field, the recurring overinterpretation of the findings from animal research in neurobiology, the sometimes willful misrepresentation of the significance of the images generated by brain scanning, not to mention the marketing on the Internet of many dubious products for improving brain power, to realize that there is much scope for critical sociological analysis and for cultural investigation of the contemporary lure of the brain sciences. But in this book, while we certainly seek to develop tools for a critical relation to many of the claims made in both serious and popular presentations of neuroscience, we also seek to trace out some directions for a more affirmative relation to the new sciences of brain and mind.\n\nWe do this for two reasons. On the one hand, there is no reason for those from the social and human sciences to fear reference to the role of the human brain in human affairs, or to regard these new images of the human being, these new ontologies, as fundamentally threatening. These disciplines have managed to live happily with the claims of psychoanalysis and dynamic psychologies, even though these disciplines also see much of human individual mental life and conduct as grounded in processes unavailable to our consciousness. Perhaps, then, we should not be so wary of the reminder that we humans are, after all, animals\u2014very remarkable ones, indeed, but nonetheless not the beneficiaries of some special creation that sets us in principle apart from our forebears. And, on the other hand, the new brain sciences share much with more general shifts within contemporary biological and biomedical sciences: at their most sophisticated, they are struggling toward a way of thinking in which our corporeality is in constant transaction with its milieu, and the biological and the social are not distinct but intertwined. Many of the assumptions and extrapolations that are built into the heterogeneous endeavor of neurobiology are ripe for critique. But at a time when neurobiology, however hesitantly, is opening its explanatory systems to arguments and evidence from the social sciences, perhaps there is a relation beyond commentary and critique that might be more productive. We will return to these issues in our conclusion. For now, though, it is sufficient to say that it is in the spirit of critical friendship between the human sciences and the neurosciences that we have written this book.\n\nBeyond Cartesianism?\n\nMind is what brain does. This little phrase seems to encapsulate the premise of contemporary neurobiology. For many, it now merely states the obvious. But it was not always so. In 1950, the BBC broadcast a series of talks titled _The Physical Basis of Mind_ , introduced by the eminent neurophysiologist Sir Charles Sherrington, with contributions from leading philosophers, psychiatrists, and neurologists (Laslett 1950). In his opening remarks, Sherrington pointed out that half a century earlier he had written, \"We have to regard the relation of mind to brain as still not merely unsolved, but still devoid of a basis for its very beginning\"; in 1950, at the age of ninety-two, he saw no reason to change his view: \"Aristotle, two thousand years ago, was asking how the mind is attached to the body. We are asking that question still.\" Whatever their differences, all the distinguished contributors to this series agreed that this debate over the relation between mind and body, between mind and brain, had lasted many centuries, and that it was unlikely that a consensus would soon be reached as to whether there _was_ a physical basis for the mind in the brain, let alone what that basis was, or where it was, or how such a basis should be conceptualized.\n\nA half century later, Vernon Mountcastle, celebrated for his fundamental discoveries about the structure of the cerebral cortex, contributed the introductory essay on \"Brain Science at the Century's Ebb\" to a special issue of _Daedalus_ (the journal of the American Academy of Arts and Sciences) devoted to the state of brain research. \"The half-century's accumulation of knowledge of brain function,\" he wrote, \"has brought us face to face with the question of what it means to be human. We make no pretention that solutions are at hand, but assert that what makes man human is his brain.... Things mental, indeed minds, are emergent properties of brains\" (Mountcastle 1998, 1). Minds are properties of that organ of the body that we term the brain. And brains makes humans human, because the minds that constitute their humanity emerge from their brains. Mountcastle spoke here for most of those working in the field that had come to call itself neuroscience. In a way that they did not quite understand and yet that they could not doubt, the human mind did indeed have a physical basis in the human brain. And that brain, however remarkable and complex, was an organ like any other organ, in principle open to neuroscientific knowledge: the 'explanatory gap' between the processes of brain and the processes of mind they somehow produce was beginning to narrow and would, in time, be closed.\n\nOf course, many would say, there is nothing much new here. Have we not known that the brain is the seat of consciousness, will, emotion, and cognition since the Greeks? Closer to our present, from the nineteenth century onward, especially in Europe and North America, there was an intense focus on the significance of the brain to human character, to human mental pathologies, and to the management of the moral order of society: while many may now scoff at the attempts of the phrenologists to read intellectual and moral dispositions in the shape and contours of the skull, few would dispute the pioneering work on brain anatomy and function memorialized in the brain areas designated by the names of Wernicke, Broca, Flechsig, and their colleagues (Hagner 1997, 2001; Hagner and Borck 2001). If we wanted further evidence that there was nothing new about the salience long accorded to brain research, we could point to the fact that in the first six decades of the twentieth century more than twenty scientists were honored by the award of a Nobel prize for discoveries concerning the nervous system\u2014from Santiago Ramon y Cajal in 1906 to John Eccles, Alan Hodgkin, and Andrew Huxley in 1963.\n\nNor is there anything particularly novel in the challenge that contemporary neuroscientists mount to dualism. For example, in the early decades of the twentieth century, Charles Sherrington sought to develop an integrated theory of brain and mind, and this was the prelude to a host of neurological, psychological, and philosophical attempts to clarify the mind-body relation; it also led to a host of worries about the implications for the higher human values of morality, autonomy, wholeness, and individuality (R. Smith 2002). Like their contemporaries today, neurologists and brain researchers in the first half of the twentieth century certainly believed\u2014and claimed\u2014that their research had uncovered mechanisms of the brain that would have major social implications. The gradual acceptance of the usefulness of the electroencephalograph in the 1930s, and the image of the electrical brain that it seemed to embody, appeared to some (notably William Grey Walter) to offer the possibility of objective diagnoses of psychiatric conditions, and indeed of normal characteristics; it was thought of \"as a kind of truth machine or electrical confessional\" that would reveal the workings of the human mind and enable public access to private mental life, and also have implications for the management of everything from child rearing to the choice of marriage partners (R. Hayward 2002, 620\u201321ff.). Perhaps, then, when neuroscientist Michael Gazzaniga titled his recent book _Human: The Science behind What Makes Us Unique_ (Gazzaniga 2008a), claiming that advances in research on the brain will reshape our understandings of who we are, he was only the latest in a long tradition.\n\nYet, as the twenty-first century began, there was a pervasive sense, among the neuroscientific researchers themselves, among clinicians, commentators, writers of popular science books, and policymakers that advances in our understanding of the human brain had implications that were nothing short of revolutionary. Much had indeed changed in the fifty years between Sherrington's pessimism and Mountcastle's optimism. By the end of the twentieth century, the term _neuroscience_ slipped easily off the tongue, yet it dates only to 1962. The Society for Neuroscience (SfN) was formed in 1969 and held its first major conference in 1979, which was attended by about 1,300 people; by 1980 it attracted about 5,800 people; by 1990 this number had grown to more than 13,000; and by 2000 it reached more than 24,000. Alongside this annual event there were now dozens of other conferences and workshops organized by more specialist associations of brain researchers, each with its own membership, websites, and newsletters, along with undergraduate and graduate programs in neuroscience, 'boot camps' for those who sought a rapid immersion in the field, and much more. These activities were not confined to the United States but spanned Europe, Japan, China, and many other countries. This was not a unified field: there were many different formulations of the problems, concepts, experimental practices, professional allegiances, and so forth. But nonetheless, by the start of the twenty-first century, a truly global infrastructure for neuroscience research had taken shape.\n\nThese organizational changes were accompanied by a remarkable burgeoning of research and publishing. While in 1958 there were only some 650 papers published in the brain sciences, by 1978 there were more than 6,500. By 1998 this figure had risen to more than 17,000, and in 2008 alone more than 26,500 refereed papers were published on the neurosciences in more than four hundred journals. In the wake of the decade of the 1990s, which U.S. President George Bush designated \"the decade of the brain,\" things seemed to shift into a new phase, with discussions of the crucial role of the brain for individuals and society in the light of advances in neuroscience moving from the specialized literature into a wider domain. In the subsequent ten years, dozens of books were published suggesting that we have witnessed the birth of new sciences of brain and\/or mind, and drawing on research findings to illustrate these claims (Andreasen 2001; Kandel 2006; Restak 2006; Iacoboni 2008; Rizzolatti and Sinigaglia 2008; Begley 2009; Lynch and Laursen 2009). These books, and the regular newspaper articles and television programs about these discoveries and their importance, are now almost always accompanied by vibrant visual illustrations derived from brain imaging of the living brain in action as it thinks, feels, decides, and desires (Beaulieu 2002): the brain has entered popular culture, and mind seems visible in the brain itself.\n\nGoverning through the Brain\n\nBy the turn of the century, it seemed difficult to deny that the neurosciences had, or should have, something to say about the ways we should understand, manage, and treat human beings\u2014for practices of cure, reform, and individual and social improvement. Across the first half of the twentieth century, the prefix _psy_ \\- was attached to a great many fields of investigation of human behavior, seeming to link expertise and authority to a body of objective knowledge about human beings (Rose 1989); now the prefix _neuro_ \\- was being invoked in the same way. Psychiatry was an obvious niche, not least because of the belief, since the 1950s, that new pharmacological treatments had been discovered that were effective because they acted directly on the neurobiology underpinning mental disorder. While the term _neuropsychiatry_ had been used as early as the 1920s (for example, Schaller 1922) and gained popularity in the European literature in the 1950s (for example, Davini 1950; Garrard 1950; Hecaen 1950), by the start of the twenty-first century the term was being used in a very specific sense\u2014to argue that the future of psychiatry lay in the integration of insights from genetics and neurobiology into clinical practice (Healy 1997; J. Martin 2002; Sachdev 2002, 2005; Yudofsky and Hales 2002; Lee, Ng, and Lee 2008).\n\nBut while psychiatry might seem an obvious niche for neuroscience, it was not alone in using the _neuro_ -prefix to designate a novel explanatory framework for investigating phenomena previously understood in social, psychological, philosophical, or even spiritual terms. Thus we now find _neurolaw_ , which, especially in its U.S. version, claims that neuroscientific discoveries will have profound consequences for the legal system, from witness interrogation to ideas about free will and programs of reform and prevention; the first papers proposing this term appeared in the mid-1990s (Taylor, Harp, and Elliott 1991; Taylor 1995, 1996). We encounter _neuroeconomics_ , which argues for the importance of studying the neurobiological underpinnings of economic behavior such as decision making. We read of _neuromarketing_ ( _Lancet_ 2004; Lee, Broderick, and Chamberlain 2007; Renvois\u00e9 and Morin 2007; Senior, Smythe, Cooke, et al. 2007); _neuroaesthetics_ (Zeki 1993, 1999), which concerns the neuronal basis of creativity and of perceptions of beauty; _neuroergonomics_ , which studies brain and behavior at work; _neurophilosophy_ (Churchland 1986, 1995); and _neurotheology_ , or the neuroscience of belief and spirituality (Trimble 2007).\n\nFor some, even the capacity to think ethically, to make moral judgments, is a brain kind of thing (see for example, Greene, Sommerville, Nystrom, et al. 2001; Tancredi 2005; Koenigs, Young, Adolphs, et al. 2007). And, reuniting apparent rivals for a knowledge of the human mind, we find _neuropsychoanalysis_ ; proponents always remind their readers that Freud was a neurologist and hoped for just such an integration in his early _Project for a Scientific Psychology_ (Bilder and LeFever 1998). We see _neuroeducation_ \u2014thus Johns Hopkins University established an initiative bringing together educators and brain science researchers to \"magnify the potential for current findings to enrich educational practice,\" and the University of London launched its own platform on \"educational neuroscience.\" We see _social neuroscience_ , which, in the words of the journal of that name, \"examine[s] how the brain mediates social cognition, interpersonal exchanges, affective\/cognitive group interactions [and] the role of the central nervous system in the development and maintenance of social behaviors.\" Centers, institutes, and laboratories in social neuroscience were established at the Max Planck Institute, New York University, the University of Chicago, UCLA, Columbia University, and elsewhere.\n\nFor others, the new brain sciences had important implications for the reform of social policy and welfare (cf. Blank 1999). In 2009, the United Kingdom's Institute for Government was commissioned by the Cabinet Office to produce a report on the implication of neuroscience for public policy. The following year the nation's Royal Society\u2014the oldest scientific academy in the world\u2014launched a project called Brain Waves to investigate developments in neuroscience and their implications for policy and for society. In 2009, the French government's _Centre d'Analyse Strat\u00e9gique_ launched a new program dedicated to inform public policy based on neuroscientific research. And there is much more of the same. Hence, perhaps inevitably given the contemporary ethicalization of biomedical matters, we have seen the rise of a new professional enterprise for worrying about all this: _neuroethics_ (Marcus 2002; Moreno 2003; Kennedy 2004; Illes 2006; Farah 2007; Levy 2007). It appears that to understand what is going on when people engage in social interactions with one another, when they feel empathy or hostility, when they desire products and buy goods, when they obey rules or violate laws, when they are affected by poverty or child abuse, when they do violence to others or themselves, and indeed when they fall in love or are moved by works of art, we should turn to the brain.\n\nWhat are we to make of all this? How has it come about that in the space of half a century, the neurosciences have become such a repository of hope and anticipation? How have they emerged from the laboratory and the clinic, and have not only entered popular culture, but have become practicable, amenable to being utilized in practices of government? And with what consequences? We know that there are close linkages between the ways in which human beings are understood by authorities, and the ways in which they are governed. The various psychological conceptions of the human being in the twentieth century had a major impact on many practices: on understanding and treatment of distress; on conceptions of normality and abnormality; on techniques of regulation, normalization, reformation, and correction; on child rearing and education; on advertising, marketing, and consumption technologies; and on the management of human behavior in practices from the factory to the military. Psychological languages entered common usage across Europe and North America, in Australasia, in Latin America, and in many other countries. Psychological training affected professionals from child guidance counselors and social workers to human resource managers. In the process, our very ideas of our selves, identity, autonomy, freedom, and self-fulfillment were reshaped in psychological terms (Rose 1979, 1985, 1988, 1989, 1996, 1999; Miller and Rose 1988, 1990, 1994, 1995a, 1995b, 1997; K. Danziger 1990; Hacking 1995).\n\nWill these developments in the neurosciences have as significant a social, political, and personal impact? If we now consign the Cartesian split of mind and body to history and accept that mind is nothing more than what the brain does, does that mean that neuroscientists, after years of toiling in relative obscurity, are poised to become nothing less than \"engineers of the human soul\"? It is undoubtedly too early for a considered judgment; it is far from clear what we would see if we were to look back on these events from the twenty-second century. Despite all the grand promises and expectations generated by neuroentrepreneurs, we cannot know for certain whether any lasting new bodies of expertise will emerge, nor can we foretell the role of neurobiology in the government of conduct across the next decades.\n\nIn this book, we abstain from speculation wherever possible. We also seek to distance ourselves from the overgeneralized critiques of 'neuromania' and other fundamentally defensive reactions from the social and human sciences. For while we raise many technical and conceptual problems with these new ways of thinking and acting, and point to many premature claims and failed promises of translation from laboratory findings to treatments and policies for managing human miseries and ailments, we also find much to appreciate in many of these attempts to render human mental life amenable to explanation and even to intervention.\n\nAnd, unlike many of our disciplinary colleagues, we do not think that the social and human sciences have anything to fear, provided that they maintain an appropriate critical awareness, from our new knowledge of the brain. The _Oxford English Dictionary_ ( _OED_ ) provides a definition of criticism that matches our aims. Rather than fault finding or passing censorious judgment, we are critical here in the sense of \"exercising careful judgement or observation; nice, exact, accurate, precise, punctual.\" It is in that critical spirit that we aim to describe the new ways of thinking about the nature of the human brain and its role in human affairs that are taking shape, to consider the problems around which these have formed and the conceptual and technical conditions that have made it possible to think in these new ways, and to analyze the ways in which these developments have been bound up with the invention of novel technologies for intervening upon human beings\u2014governing conduct though the brain, and in the name of the brain.\n\nOur Argument\n\nIn the course of the intertwined investigations that make up this book, we argue that a number of key mutations\u2014conceptual, technological, economic, and biopolitical\u2014have enabled the neurosciences to leave the enclosed space of the laboratory and gain traction in the world outside. It may be helpful to summarize these here, to help guide readers through the rather detailed analyses contained in the following chapters.\n\nConcepts and Technologies\n\nOver the course of the half century that we focus on in this book, the human brain has come to be anatomized at a molecular level, understood as plastic and mutable across the life-course, exquisitely adapted to human interaction and sociality, and open to investigation at both the molecular and systemic levels in a range of experimental setups, notably those involving animal models and those utilizing visualization technologies. This has generated a sense of human neurobiology as setting the conditions for the mental lives of humans in societies and shaping their conduct in all manner of ways, many of which are not amenable to consciousness. Each of the major conceptual shifts that led to the idea of the neuromolecular, plastic, and visible brain was intrinsically linked to the invention of new ways of intervening on the brain, making possible new ways of governing through, and in the name of, the brain. Yet despite the ontological changes entailed, and the emerging belief that so much of what structures human thoughts, feelings, desires, and actions is shaped by nonconscious neurobiological processes, few of those who work in this area believe that humans are mere puppets of their brains, and the emerging neurobiologically informed strategies for managing human conduct are rarely if ever grounded in such a belief. Neurobiological conceptions of personhood are _not_ effacing other conceptions of who we are as human beings, notably those derived from psychology. On the contrary, they have latched on to them in the many sites and practices that were colonized by psychology across the twentieth century\u2014from child rearing to marketing, and transformed them in significant ways. In this way, and through these processes, our contemporary 'neurobiological complex' has taken shape. Let us say a little more about some of these developments.\n\nThe central conceptual shift that we chart in the chapters that follow is the emergence of a neuromolecular vision of the brain. By this we mean a new scale at which the brain and nervous system was conceptualized, and a new way in which their activities were understood. At this molecular level, the structure and processes of the brain and central nervous system were made understandable as material processes of interaction among molecules in neurons and the synapses between them. These were conceived in terms of the biophysical, chemical, and electrical properties of their constituent parts. At this scale, in a profoundly reductionist approach, despite the recognition that there was much that could not yet be explained, there seemed nothing mysterious about the operations of the nervous system. Mental processes\u2014cognition, emotion, volition\u2014could be explained in entirely material ways, as the outcome of biological processes in the brain, understood as an organ that was, in principle, like any other, even if, in the case of humans and many other animals, it was far more complex than any other organ. While the explanatory gap still remained, and the move from the molecular level to that of mental processes was highly challenging, the dualism that had haunted philosophy and the sciences of mental life increasingly seemed anachronistic.\n\nThe project of neuroscience\u2014for it was indeed an explicit project to create interactions between researchers from the whole range of disciplines that focused on the brain, from mathematics to psychology\u2014had as its aim to revolutionize our knowledge of the brain, and in so doing, radically to transform our capacities to intervene in it. One key transactional point was psychiatric pharmacology\u2014that is to say, the development of pharmaceuticals to treat mental disorder. The emergence of this neuromolecular gaze was intrinsically intertwined with the development of psychopharmacology and the increasing resort to drugs for treating people diagnosed with mental illness, first within, and then outside the asylum walls. Many key findings about molecular mechanisms were made in the course of trying to identify the mode of action of those drugs, almost always using animal models. Indeed, we argue that animal models were epistemologically, ontologically, and technologically crucial to the rise of neuroscience. Research using such models focused on the molecular properties of drugs that appeared to act on mental states and behavior, and hence almost inescapably led to the belief that the anomalies in those mental states could and should be understood in terms of specific disturbances, disruptions, or malfunctions in neuromolecular processes. Since the drugs seemed to affect the components of neurotransmission, this led both to the triumph of the chemical view of neurotransmission over the electrical view that had previously been dominant, and to the belief that malfunctions in neurotransmission underpinned most if not all mental disorders.\n\nThe two founding myths of psychopharmacology\u2014the monoamine hypothesis of depression and the dopamine hypothesis of schizophrenia provided ways of organizing these linkages conceptually and technologically. Both have proved mistaken, perhaps fundamentally wrong. However, this 'psychopharmacological imaginary' enabled the growth of novel transactions between laboratory, clinic, commerce, and everyday life. In particular, it was linked to the growing associations between the pharmaceutical companies, the neurobiological research community, and the profession of psychiatry. It was associated with many inflated statements about the therapeutic potency of the compounds being produced and marketed, with the growing routinization of the use of psychoactive drugs that claimed to be able to manage the travails of everyday life by acting on the brain, and with the reshaping of distress in ways that might best accord with the vicissitudes of an increasingly competitive and profitable market for pharmaceuticals.\n\nThere were, of course, many who were critical of these new relationships. Critics denounced the medicalization of social problems, linked this to an analysis of psychiatry\u2014and in particular of biological psychiatry\u2014as an apparatus of social control, and argued that the profession, its explanatory claims, its diagnostic categories, and its preference for drugs as a first line of intervention, resulted from its capture by the pharmaceutical industry. For many of these critics, aware of the doleful history of eugenics, genetic explanations of mental disorders were particularly distasteful. Despite the certainty of psychiatric geneticists that mental disorders had a genetic basis, critics correctly pointed out that the repeated claims to have discovered 'the gene for' schizophrenia, manic depression, and many other conditions were always followed by failures of replication. However as the twentieth century came to a close, a radical transformation in the styles of thought that characterized genetics made a different approach possible. This focused on variations at a different level\u2014at the level of changes in single bases in the DNA sequences themselves, and the ways in which such small variations in the sequence might affect the nature of the protein synthesized or the activity of the enzyme in question, with consequences for susceptibility to certain diseases or response to particular drugs.\n\nThis molecular vision of genomic complexity thus mapped onto the vision of the neuromolecular brain. It thus became possible to move beyond studies of heritability in lineages and families to seek the specific genomic variants and anomalies that had consequences for susceptibility to certain diseases or pathological conditions such as impulsive behavior. One now looks for the variations that increase or decrease the activity of an enzyme, the operation of an ion channel, or the sensitivity of a receptor site, and which, in all their multiple combinations, underpin all differences in human mental functioning, whether these be deemed normal variations or pathologies. Further, one tries to locate these within the environmental or other conditions that provoke or inhibit the onset of such conditions. As we have moved to such a neurogenomics of susceptibilities and resiliencies, new translational possibilities appear to emerge for neuroscience to engage with strategies of preventive intervention in the real world, whether via early identification and treatment of mental disorder or of neurodegenerative diseases, or in enabling preventive intervention to steer children from a pathway that will lead to antisocial behavior and crime.\n\nAlongside psychopharmacology and psychiatric genomics, there was a third pathway, equally significant in our view, for the transactions between the knowledge of the brain and interventions in human lives\u2014the growing belief that, at least when it comes to the human brain, neither structure nor function is inscribed in the genes or fixed at birth. One term has come to designate this new way of thinking\u2014 _plasticity_. The neural architecture of the brain was now located in the dimension of time\u2014not just the time of development from fertilization to birth and into the early years of life, but also throughout the life-course, through adolescence, into adulthood, and indeed across the decades. While it had long been recognized that plasticity existed at the level of the synapse\u2014that synaptic connections constantly formed or were pruned in response to experience\u2014new ideas of plasticity were taken to mean that a wider 'rewiring' was also possible. Notable, here, were the results of work on rehabilitation after stroke in humans, and related work with animal models, which showed that the primate brain could remap itself after injury and that this process could be accelerated by neurobiologically informed practices of rehabilitation\u2014an argument that was commercialized in the development of a number of therapeutic methods, often patented by the neuroscientific researchers themselves.\n\nAt the other end of life, researchers argued that experience in the very early days and months following birth, perhaps even in utero, shaped the brain in fundamental ways through modifying gene methylation. Epigenetic arguments sought to establish the ways in which experience 'gets under the skin' at the level of the genome itself. In particular, it seemed, early maternal behavior toward offspring might so shape their neural development to affect not only the behavior of offspring over their whole life span, but also their own maternal behavior. There now seemed to be a mechanism to pass these environmentally acquired characteristics of the brain down the generations. And finally, the long-held dogma that no new neurons were produced after the first years of life was itself overturned with the finding that in humans, neurogenesis or the growth of new nerve cells in the brain, was possible throughout adult life and might be stimulated or inhibited by environmental factors from nutrition to cognitive activity. No matter that many doubts remained about the translation of these findings from animals to humans, and the interpretation of these results. The brain now appeared as an organ that was _open_ to environmental inputs at the level of the molecular processes of the genome, shaping its neural architecture and its functional organization, with consequences that might flow down the generations. The implications were clear: those who were concerned about the future of our children, and the conduct and welfare of the adults they would become, needed to recognize, and to govern, these processes of shaping and reshaping our plastic brains.\n\nIf these three imaginaries\u2014of pharmacology, neurogenomics, and neuroplasticity\u2014provided pathways linking the work of brain labs to interventions in the everyday world, so too did a fourth: the visual imaginary, associated in particular with the development of powerful technologies of brain imaging. While the skull initially proved an impenetrable barrier to techniques of medical imaging such as X-rays, there were early attempts\u2014notably by Edgar Adrian\u2014to explore the electrical activity of the living brain using electroencephalography (Adrian and Matthews 1934). However, the fundamental shift in the visibility of the living brain was linked to the development of computerized tomography (CT) scanning in the 1970s and magnetic resonance imaging (MRI) in the 1980s. These produced images of the structure and tissues of the brain that were, to all intents and purposes, equivalent to the images produced of any other bodily tissues. They were simulations, of course, not simple photographs, but they were open to confirmation by physical interventions into the imaged tissues.\n\nTwo further developments, positron emission tomography (PET) and functional magnetic resonance imaging (fMRI), seemed to produce identical images of something with a very different ontological status\u2014not the _structure_ of the brain but its _functioning_ , its activity as its human host engaged in certain tasks or experienced certain emotions. We seemed to be able to see the neural correlates of the activities of mind itself in real time. And once we did, it seemed impossible to doubt that mind is what brain does. As these technologies became more widely available to researchers, thousands of papers were published claiming to identify the neural correlates of every human mental state from love to hate, from responses to literature to political allegiances; by 2011, such publications were appearing at a rate of about six hundred a month. Despite the well-known technical problems, assumptions, and limitations of these technologies, and the fact that they do not speak for themselves and must be interpreted by experts, the images have undoubted powers of persuasion, and their apparent ability to track mental processes objectively, often processes outside the awareness of the individual themselves, have proved persuasive in areas from neuromarketing to policies on child development. The belief that we can see the mind in the living brain, can observe the passions and its desires that seemingly underlie normal and pathological beliefs, emotions, and behaviors, has been a key element in the claim that neuroscience can provide useful information about the government of human beings, the conduct of their conduct in the everyday world.\n\nGoverning the Future\u2014through the Brain\n\nWe should beware of scientific or technological determinism. Truths and technologies make some things possible, but they do not make them inevitable or determine the sites in which they find a niche. Different societies, cultures, and sociopolitical configurations offer different opportunities for the new brain sciences. Nonetheless, there is one feature of contemporary biopolitics that has proved particularly welcoming to the image of the molecular, visible, and plastic brain\u2014that which concerns the future. Contemporary biopolitics is infused with futurity, saturated with anticipations of imaged futures, with hope, expectation, desire, anxiety, even dread. The future seems to place a demand not just on those who govern us but also on all those who would live a responsible life in the present (O'Malley 1996).\n\nNo doubt this widespread sense of obligation to take responsibility for the future is not unique to advanced liberal democracies in the early twenty-first century. Biopolitics, since at least the eighteenth century, has been future-oriented. From earliest politics of the population, governing vitality operates on axis of time and orients to the future, and images of the future are intrinsic to biopolitical thought and strategies from the politics of health in the eighteenth century, to concerns with the degeneracy of the population in the nineteenth century, through the rise of eugenics and the birth of strategies of social insurance in the first half of the twentieth century. Today we are surrounded by multiple experts of the future, utilizing a range of technologies of anticipation\u2014horizon scanning, foresight, scenario planning, cost-benefit analyses, and many more\u2014that imagine those possible futures in different ways, seeking to bring some aspects about and to avoid others. It would not be too much of an exaggeration to say that we have moved from the risk management of almost everything to a general regime of futurity. The future now presents us neither with ignorance nor with fate, but with probabilities, possibilities, a spectrum of uncertainties, and the potential for the unseen and the unexpected and the untoward. In the face of such futures, authorities have now the obligation, not merely to 'govern the present' but to 'govern the future.' Such futurity is central to contemporary problematizations of the brain.\n\nOne feature of these imagined futures is the growing burden of brain disorder. Public funding for research in the new brain sciences, not just in the so-called Decade of the Brain but from at least the 1960s, when initiatives such as the Neurosciences Research Program (NRP) were established, has almost always been linked the belief that conquering this new frontier of the brain will, in an unspecified time line, lead to major advances in tackling that burden. The idea of _burden_ , here, has disturbing resonances for those who know their history (Proctor 1988; Burleigh 1994). Nonetheless, in a very different sociopolitical context than that of eugenics, psychiatrists, lobby groups, and international organizations make dire predictions about the rising numbers of those in the general population who suffer from depression and other mental disorders, not to mention the \"dementia time bomb\": a recent estimate was that, in any one year, more than one-third of the population of the European Union could be diagnosed with such a brain disorder (Wittchen, Jacobi, et al. 2011). In the emerging style of thought that we trace in this book, brain disorders encompass everything from anxiety to Alzheimer's disease, and often include both addictions and obesity\u2014all, it seems, have their origin in the brain. These disorders, the majority of which are undiagnosed, lead to many days lost from work and many demands on medical and other services, costing European economies hundreds of billions of euros. The corollary seems obvious: to reduce the economic burden of mental disorder, one should focus not on cure but on prevention. And prevention means early intervention, for the sake of the brain and of the state.\n\nChildren are the key\u2014children who are at risk of mental health problems as they grow up. Many pathologies\u2014ADHD, autism, schizophrenia, bipolar disorder, dementias\u2014are now reframed as _developmental_ and hence amenable to early detection and ideally to preventive intervention. This logic can then be extended from mental disorders to antisocial conduct, resulting in the attempt to discover biomarkers in the brain or in the genes of young children that might predict future antisocial personality or psychopathy. In this logic, one first identifies susceptibility, and one then intervenes to minimize the chances of that unwanted eventuality coming about, in order to maximize both individual and collective well-being and to reduce the future costs of mental health problems. Earlier is almost always better\u2014as the mantra has it. Earlier usually means during childhood, because the brain of the developing child is more 'plastic,' believed to be at its most open to influences for the good (and for the bad)\u2014and hence leads to intensive interventions in the parenting of those thought to be potentially at risk. This is the rationale of \"screen and intervene\" (Singh and Rose 2009; Rose 2010b). Neuroscientifically based social policy thus aims to identify those at risk\u2014both those liable to show antisocial, delinquent, pathological, or criminal behavior and those at risk of developing a mental health problem\u2014as early as possible and intervene presymptomatially in order to divert them from that undesirable path.\n\nAt the other end of life, many argue for early intervention to forestall the development of Alzheimer's disease and other dementias, which has led to regular announcements of tests claiming to identify those at risk, the rise of the prodromal category of \"mild cognitive impairment,\" the growing number of \"memory clinics\" to diagnose such brain states and prescribe interventions to ameliorate them, and much research, so far largely unsuccessful, to find effective forms of intervention into the dementing brain (discussed in Whitehouse and George 2008).\n\nIn the era of the neuromolecular and plastic brain, those who advocate such strategies think of neurobiology not as destiny but _opportunity_. Many believe that to discover the seeds of problematic conduct in the brain will reduce stigma rather than increase it, despite research showing the reverse (Phelan 2002, 2006). Those seeking biomarkers for psychopathy, even when they believe that there is a clear, genetically based, neurobiological basis for antisocial conduct, argue that neurobiology informs us about susceptibility but not inevitability. Their wish to identify the gene-environment interactions, which provoke vulnerability into frank psychopathy, is linked to a hope for protective strategies, for \"the goal of early identification is successful intervention\" (Caspi, McClay, Moffitt, et al. 2002; Kim-Cohen, Moffitt, Taylor, et al. 2005; Odgers, Moffitt, Poulton, et al. 2008).\n\nInterventions sometimes involve behavior therapy, cognitive therapy, and psychopharmaceuticals. But the preferred route to the problematic child\u2014as so often in the past\u2014is through the parents. In the age of the plastic brain, many undesirable neurobiological traits appear to be malleable by changing the ways parents deal with their vulnerable children (Dadds, Fraser, Frost, et al. 2005; Hawes and Dadds 2005, 2007). Such arguments for early intervention have been strengthened over recent years by the proliferation of brain images seeming to show the consequences of early adverse environments on the developing brain of the child (Perry 2008). On the one hand, these images provide powerful rhetorical support for early intervention into the lives of the most disadvantaged families, in the name of the individual, familial, and social costs of the developing brain, and hence future lives, of their children. On the other hand, in situating the origins of all manner of social problems and undesirable forms of conduct so firmly in neurobiology, even in a neurobiology that is itself shaped by environment, we see a repetition of a strategy that we have seen innumerable times since the nineteenth century\u2014to prevent social ills by acting on the child through the medium of the family: a neurobiological explanation for the persistence of social exclusion in terms of a 'cycle of deprivation' grounded in the inadequate parenting provided by the socially deprived.\n\nEconomies of the Brain\n\nThese arguments about the burden\u2014in this case the economic costs\u2014of brain disorders and the increasing faith in the economic benefits to be gained through strategies of prediction and preventive medicine, have been one important factor for the growth of public investment in neuroscientific research. The National Institutes for Health (NIH) in the United States, their equivalent in the U.K. research councils, the European Commission's Framework Programmes, the European Research Council, and the European Science Foundation have all invested in this work, as have private foundations and charitable bodies, such as the MacArthur Institute in the United States and the Wellcome Trust in the United Kingdom. Given the diversity of these funding sources, the total sums involved are hard to estimate. However, in the United States as of 2006, it was estimated that the combined commitment to neuroscience research of the NIH, the pharmaceutical industry, large biotech firms, and large medical device firms increased from $4.8 billion in 1995 to $14.1 billion in 2005\u2014adjusted for inflation, this meant a doubling of investment, which was more or less in line with that for biomedical research as a whole.\n\nIn the case of neuroscience, over half of this investment came from industry, a proportion that remained around that level over that decade, despite the fact that investment in this area was not matched by an increase in new pharmaceuticals coming on to the market: the U.S. Food and Drug Administration (FDA) approved only \"40 new molecular entities for indications within the neurosciences from 1995 to 2005, with the annual number of approvals remaining relatively stagnant during this period\" (Dorsey, Vitticore, De Roulet, et al. 2006). However, commenting on developments over the decade from 2000 to 2010, and into the near future, leading neuroscientists have expressed their fears that public funding for research in neuroscience in many major industrialized countries\u2014notably the United States, the United Kingdom, and Japan\u2014has not kept up with inflation and has been affected by the recession; they argue that this is producing something of a crisis and does not match the scale of the social costs of what they variously refer to as brain diseases, mental disorders, or brain disorders (Amara, Grillner, Insel, et al. 2011). In the United Kingdom, in early 2011, the British Neuroscience Association (BNA) estimated overall U.K. neuroscience research funding from public sources to be in the region of \u00a3200 million per year. In the United States, despite a one-time supplement to the NIH of $10 billion as a result of the American Recovery and Reinvestment Act of 2009, funding in the second decade of the century was expected to be flat.\n\nCommercial investment in neuroscience is, of course, linked to the belief that there is a growing global market for the drugs, diagnostics, and devices that are being produced. Zack Lynch of NeuroInsights, perhaps the leading organization appraising the neurotechnology market, has estimated that the global value generated by the neurotechnology industry in 2009 was around $140 billion, slightly down from that in previous years (a fall attributed to the drop in cost of neuroimaging devices), but that over the previous decade the average annual growth had been on the order of 9%. As we have already remarked, many critics have viewed the business strategies of pharmaceutical companies as a key factor in reshaping psychiatry toward neuropsychiatry, influencing its diagnostic systems and the use of psychiatric drugs as the main form of therapy. In fact, global neuropharmaceutical revenue actually slowed over 2009, largely as a result of the failure of new drugs to come onto the market and the replacement of patented products by generics, although there is now some evidence that the pharmaceutical companies are withdrawing investment for the development of novel psychiatric compounds that are based on the hypotheses of specificity of etiology and treatment that had informed their strategies since the 1960s. But other key neurotechnology areas showed growth, including neurodiagnostic technologies ranging from brain scanners to biomarkers, and neurodevices ranging from cochlear implants, through brain stimulation technologies, to neuroprosthetics purporting to arrest memory decline and optimize attention.\n\nWe can see immediately that it would be misleading to separate the academic and the industrial components of this neuroeconomy: what Stephen Shapin has termed \"the new scientific life\" entails an entrepreneurial spirit on the part of both researchers and universities, and the search for intellectual property and the rhetoric of knowledge transfer are endemic (Shapin 2008). Thus, for example, David Nutt, president-elect of the BNA, commenting on the effects of the decisions by GlaxoSmithKline and Merck in 2011 to close their U.K. neuroscience research sites, with a loss of about one thousand jobs in neuroscience, remarked that it not only \"removes the only major site of job opportunities for neuroscience graduates and postdoctoral researchers outside academia\" but \"the company pull-out directly impacts research spend[ing] in neuroscience as many PhD students and postdoctoral researchers were funded by, or in partnership with, this industry.\"\n\nIndeed, such partnership funding has been a major element in the strategy of the U.K. Medical Research Council. Many of the start-up companies in the neuroindustry are initiated by research scientists with the assistance of venture capitalists, and in turn, their investments are fueled by expectations about future market growth and market opportunities generated by organizations such as IMS Health and NeuroInsights (\"the neurotech market authority\") whose own business models depend on their effective servicing of these expectations. Running across all actors in this new configuration is a 'translational imperative': an ethos that demands and invests in research that is predicted to generate returns of investment\u2014in therapies and in products. This translational imperative now inflects every serious research proposal or grant application in the United States, the United Kingdom, continental Europe, and many other regions. It has arisen, at least in part, because of failure: the failure of many hopes that an increased knowledge of basic biological processes of body and brain would lead inevitably to better therapies for individuals and valuable products for national bioeconomies. If that return in health and wealth, rather than advances in knowledge for its own sake, was the quid pro quo for public investment in basic research, it seemed that the researchers had not kept their side of the bargain. The response of the funding organizations\u2014the NIH in the United States (Zerhouni 2003, 2005) and the MRC in the United Kingdom (Cooksey 2006; National Institute for Health Research 2008)\u2014was to argue for a radical reshaping of the pathways from research to application, and a specific focus, organizationally and financially, on 'translational medicine.'\n\nOf course, there is no simple path from 'bench to bedside' or vice versa. It is rare for a single piece of research, or even the research program of a single group or lab, to translate on its own. The time frames over which research findings are integrated into novel therapies or products are often of the order of decades; the acceptance and utilization of research findings often depends more on social, political, and institutional factors than on the inherent productiveness of the research itself. Nonetheless, we have seen the formation of a number of 'translational platforms'\u2014sites of diverse material exchanges, from knowledge production to decision making and commercial transactions, from innovation to technical assemblages of material entities, where diverse agents and agencies, practices and styles of thought, discourses and apparatuses, converge in the name of the promissory benefits of translational neuroscientific research. This is, of course, an agonistic space, especially when it comes to the role of pressure groups. Some such groups are strongly opposed to the neuro-biomedicalization of conditions such as autism or depression, while others seek to shape biomedical funding toward their own translational demands as a sign of the new democratization of biomedical research (Heath, Rapp, and Taussig 2004). Many activist pressure groups of patients or their families are funded in part by donations from those very commercial corporations and pharmaceutical companies that stand to benefit from increased public awareness and concern about the disorders in question (Moynihan 2002, 2008; Moynihan and Cassels 2005). These new vital economies are tangled webs and permit of no easy ethical judgments.\n\nIn the analyses that follow, we do not argue that there is something essentially malign in the intertwining of researchers' hopes for academic success, hopes for a cure for one's loved ones, hopes for private financial advantage for individual scientists and for companies, and hopes for public economic benefits in terms of health. Indeed, these intertwinings characterize contemporary biomedicine in what Carlos Novas has termed our contemporary \"political economy of hope\" (Novas 2006). But we do point to zones where such entanglements may be highly contentious, for example, where there are powerful but hidden financial links between researchers and those who stand to benefit from the claims made in their research.\n\nEven where frank corruption is not entailed, we have seen significant changes in the economics of research in the life sciences generally, which certainly extend to neuroscience. Steven Shapin quotes a writer in _Science_ magazine in 1953: \"The American scientist is not properly concerned with hours of work, wages, fame or fortune. For him an adequate salary is one that provides decent living without frills or furbelows.... To boil it down, he is primarily interested in what he can do for science, not in what science can do for him.\" This description certainly seems to fit some of the key figures in the early years of neuroscience, and it undoubtedly fits some of those working in the field today. But it is doubtful if it characterizes those many others who would be scientific entrepreneurs; even for those for whom the research itself is the prize\u2014the much-cited article, recognition by peers, career advancement\u2014the economy of neuroscience as a whole, and hence the comportment of most of those who work in this field, has had to change.\n\nIn the United States, the passage of the Bayh-Dole Act in 1980 gave up federal rights to intellectual property that arose from research supported from government funds, allowing those property rights and the value that might flow from them to be claimed by universities or by individual scientists (Kennedy 2001). The ostensible aim was to avoid important scientific discoveries lying idle, and thus not contributing to national wealth and well-being. But this act, coupled with other legal changes, opened biomedical research to a flood of private investment and venture capital. Trends in the United States were followed across much of the rest of the world, and the new opportunities were embraced enthusiastically by research universities, which set up technology transfer offices, participated in start-up companies, and entered into the murky realm of patenting, licensing, distribution of royalties, and complex commercial relations with the corporate world.\n\nBut many argued that the principles that should govern scientific knowledge were being compromised by the drive for intellectual property and the growing entanglements between universities, researchers, industry, and knowledge claims. For example, in 2002, the Royal Society of London set up a study to ask \"whether the use of laws which encourage the commercial exploitation of scientific research is helping or hindering progress in fields such as genetics.\" The conclusion of its study was that \"[i]ntellectual property rights (IPRs) can stimulate innovation by protecting creative work and investment, and by encouraging the ordered exploitation of scientific discoveries for the good of society.... [But] the fact that they are monopolies can cause a tension between private profit and public good. Not least, they can hinder the free exchange of ideas and information on which science thrives.\"\n\nWhile questions of patenting and intellectual property have been crucial in reshaping the neuroeconomy, we would argue that this intense capitalization of scientific knowledge, coupled with the other pressures on researchers to focus on, and to maximize, the impact of their research, has additional consequences. It exacerbates tendencies to make inflated claims as to the translational potential of research findings, and, where those potentials are to be realized in commercial products, to a rush to the market to ensure that maximum financial returns are achieved during the period of a patent. It produces many perverse incentives (Triggle 2004). These include the possibilities of 'corporate capture,' where universities, departments, or research centers are significantly funded by commercial companies that gain priority rights to patent and commercialize discoveries that are made. They increase the already existing incentives on researchers selectively to report positive findings of research\u2014an issue that is particularly relevant where the studies are funded by commercial companies. And, as we show in the chapters that follow, they can lead researchers (or the press releases issued by their university communications departments) to overclaim the generalizability of studies carried out with small samples and to imply that studies with animals will, very soon, lead to therapeutic developments for humans.\n\nBut from our point of view, above and beyond these specific problems generated by the new forms of scientific life, there is a more general question about truth. For if one has a path-dependent theory of truth\u2014if you believe as we do that it is difficult to make things become true in science, and among the necessities for making things become true today are the funds to enable the research to proceed\u2014then the decisions by public, private, and commercial bodies as to which areas of research to fund, and the often un-acknowledged intertwining of promises, hopes, anticipations, expectations, and speculations that are involved, play a key role in shaping our contemporary regimes of truth about persons and their mental lives.\n\nBrains and Persons\n\nMany critics have suggested that the rise of neurobiology is leading to a kind of reductionism in which mental states are reduced to brain states, human actions are generated by brains rather than conscious individuals, and the key dimensions of our humanness\u2014language, culture, history, society\u2014are ignored (Ehrenberg 2004; Vidal 2005; Vidal 2009; Tallis 2011). For some, this rests on a philosophical error: attributing to brains capacities that can only properly be attributed to people (Bennett and Hacker 2003). For others, it is the apotheosis of contemporary individualism\u2014a turn away from social context to a vision of society as an aggregate of isolated individuals (Martin 2000, 2004). There is an analogy here with the concerns that social scientists expressed about 'geneticization' before and after the Human Genome Project. Dire warnings of genetic reductionism, genetic discrimination, and hereditary fatalism proved wide of the mark. And as with genomics, so with neuroscience. This is partly because of the findings of the research itself. But it is also because of the way such arguments are aligned with existing conceptions of personhood and regimes of self-fashioning in advanced liberal societies (Novas and Rose 2000; Rose and Novas 2004; Rose 2007c).\n\nWe argue that despite their apparent contradictions, neurobiological research emphasizing the role of nonconscious neural processes and habits in our decisions and actions can\u2014and does\u2014happily coexist with long-standing ideas about choice, responsibility, and consciousness that are so crucial to contemporary advanced liberal societies (Rose 1999). Such societies are premised on the belief that adult human beings, whatever the role of biology and biography, are creatures with minds, who have the capacity to choose and to intend on the basis of their mental states. Humans can be held accountable for the outcomes of those decisions and intentions, even when they are shaped by those nonconscious forces, except in specific circumstances (compulsion, mental disorder, brain injury, dementia, automatism, etc.). Indeed, it is hardly radical to suggest that human beings are swayed by forces that come from beyond their consciousness. In most cultures and most human practices, individuals have believed in the importance of fates, passions, instincts and drives, unconscious dynamics, and the like. And, as a result, they have been urged, and taught, to govern these forces in the name of self-control, whether by spiritual exercises, by prayer and mortification, by the inculcation of habits, by learning how to govern one's will through inhibition, by understanding the dynamics of projection and denial, by consciousness raising, or a multitude of other techniques.\n\nSimilarly, the recognition of nonconscious neurobiological factors in our mental lives does not lead policymakers to propose that we should resign ourselves before these neural forces. On the one hand, it leads them to believe that those who govern us should base their strategies on knowledge of these nonconscious neurobiological mechanisms, for example, by creating settings that make it easier for human beings to make the right decisions, with nonconscious cues that steer them in the desired directions (Thaler and Sunstein 2008). Likewise, they urge our authorities to nurture the nonconscious bonds of fraternity and good citizenship and to minimize those that weaken culture and character (Brooks 2011). They suggest that our governors need to recognize the nonconscious roots of valued moral understandings such as empathy and the sense of fairness, and to reinforce them\u2014to build social policies on the neurobiological evidence that our beliefs, actions, and aspirations are shaped more by our evolved emotional judgments and the habits that have become ingrained in our nonconscious neural pathways, than by conscious, rational deliberations. On the other hand, they argue that we as individuals should seek to manage these underlying processes, to become conscious of them, to develop mindfulness, to work responsibly to improve ourselves as persons, to reform our habits through acting on our brains: we have entered, they claim, the age of neurological reflexivity (Rowson 2011).\n\nIn this emerging neuro-ontology, it is not that human beings _are brains_ , but that we _have brains_. And it is in this form\u2014that our selves are shaped by our brains but can also shape those brains\u2014that neuroscientific arguments are affecting conceptions of personhood and practices of self-fashioning. In the final decades of the twentieth century in the West, we saw the rise of a \"somatic ethic\" in which many human beings came to identify and interpret much of their unease in terms of the health, vitality, or morbidity of their bodies, and to judge and act upon their soma in their attempts to make themselves not just physically better but also to make themselves better persons (Rose 2007c). The contemporary cultural salience of the brain does not mark the emergence of a new conception of personhood as \"brainhood\" (as suggested by Vidal 2009), in which persons become somehow conceived of as identical to, or nothing more than, their brains. Rather, we are seeing this somatic ethic gradually extending from the body to the embodied mind\u2014the brain.\n\nThe pedagogies of 'brain awareness' and the rise of practices and devices for working on the brain in the service of self-improvement thus find their locus within a more general array of techniques for working on the somatic self in the name of maximizing our well-being. While it is true that much neuroscience presents a picture of brains as isolated and individualized, an alternative image is also taking shape. In this image, the human brain is evolved for sociality, for the capacity and necessity of living in groups, for the ability to grasp and respond to the mental states of others: human brains are both shaped by, and shape, their sociality. As various aspects of sociality are attributed to human brains, neurobiological self-fashioning can no longer so simply be criticized as individualistic and asocial. It is now for the _social_ good that parents need to understand the ways in which their earliest interactions with their children shape their brains at the time when they are most plastic, to recognize the ways in which they learn to understand other minds, to enhance their capacities for empathy and the inherent emotional ability of their brains to respond positively to fairness and commitment to others, to maximize the mental capital and moral order of society as a whole. And it is in the name of improving the well-being of our societies that each of us is now urged to develop a reflexive understanding of the powers of these nonconscious determinants of our choices, our affections, our commitments: in doing so, we will no longer be passive subjects of those determinants, but learn the techniques to act on them in order to live a responsible life. Once more, now in neural form, we are obliged to take responsibility for our biology, to manage our brains in order to bear the responsibilities of freedom.\n\nHuman Science?\n\nWhat, then, of the relations between neuroscience and the human and social sciences? There is much to be critical of in the rise of these technologies of the neurobiological self. Experimental practices in laboratory settings still fail adequately to address the fact that neither animal nor human brains exist in isolation or can be understood outside their milieu and form of life. Conceptions of sociality in social neuroscience are frequently impoverished, reducing social relations to those of interactions between individuals, and ignore decades of research from the social sciences on the social shaping and distributed character of human cognitive, affective, and volitional capacities. Strategies of intervention, of governing through the brain, are based on many dubious assumptions and often blithely ignorant of the likely social consequences of their endeavors.\n\nArguments claiming that neuroscience offers a radical and challenging reconceptualization of human personhood and selfhood are often confused and are based on the very culturally and historically specific premises that that they claim to explain. Many, if not all, of the claims made for _neuro_ products claiming to improve mental capacities by brain stimulation of various types have no basis in research, although they appeal to long-standing cultural beliefs that mental capacities are like physical capacities and can be improved by exercise, training, and rigorous ascetic control of the body. Practices such as 'mindfulness' have swiftly migrated from being self-managed radical alternatives to other forms of 'governing the soul' to become yet another element in the armory of the psychological, psychiatric, and lifestyle experts trying to persuade their clients to improve themselves by becoming mindful. And we can see how the practices of self-improvement, focusing on enhancing each person's capacity to manage themselves flexibly and adaptably in a world of constantly changing demands, do aim to produce the forms of subjectivity that might be able to survive in the new patterns of work and consumption that have taken shape over the past twenty years.\n\nYet, as we argue throughout this book, new styles of thought are beginning to emerge in neuroscience that recognize the need to move beyond reductionism as an explanatory tool, to address questions of complexity and emergence, and to locate neural processes firmly in the dimensions of time, development, and transactions within a milieu. These offer the possibilities of a more positive role for the social and human sciences, an opportunity to seize on the new openness provided by conceptions of the neuromolecular, plastic, and social brain, and to move beyond critique to find some rapprochement. The human sciences have nothing to fear from the argument that much of what makes us human occurs beneath the level of consciousness; indeed, many have long embraced such a view of our ontology (Ellenberger 1970). If we take seriously the renewed assault on human narcissism from contemporary neurobiology, we may find the basis of a radical way of moving beyond notions of human beings as individualized, discrete, autonomous, coherent subjects, free to choose. Neuroscience may seem an unlikely ally of progressive social thought, but its truth effects could surprise us.\nChapter One\n\nThe Neuromolecular Brain\n\nThose of us who have lived through the period in which molecular biology grew up... realized that here was an enormous opportunity for a new synthesis... [an] approach to understanding the mechanisms and phenomena of the human mind that applies and adapts the revolutionary advances in molecular biology achieved during the postwar period. The breakthrough to precise knowledge in molecular genetics and immunology\u2014\"breaking the molecular code\" resulted from the productive interaction of physical and chemical sciences with the life sciences. It now seems possible to achieve similar revolutionary advances in understanding the human mind.\n\n\u2014Francis O. Schmitt, \"Progress Report on the Neurosciences Research Program,\" 1963\n\nIn the summer of 1961, at the Massachusetts Institute of Technology (MIT), Francis O. Schmitt and a small group of collaborators began to lay the plans for a project that was as simple to state as it was ambitious to imagine: to do for the brain what Watson and Crick had so recently done for the gene when they discovered the structure of DNA and \"cracked the genetic code.\" In the process they were to invent what they would come to term _neuroscience_ (Swazey 1975, 531). \"To this end, new devices are being forged,\" wrote Schmitt in 1967, \"with one of which I am closely identified. This is the Neurosciences Research Program (familiarly known as NRP), being a group of 34 gifted scientists\u2014mathematicians, physicists, chemists, neurobiologists, neurologists, psychologists, and psychiatrists\u2014who meet regularly to study the problem and who sponsor a center and a center staff which arranges work sessions on relevant topics of conceptual importance and which communicates the harvest of the work sessions in the program's Bulletin\" (Schmitt 1967, 562).\n\nThe NRP had grown out of the first Intensive Study Program (ISP) that took place in 1966 at the University of Colorado at Boulder, where more than a hundred participants from diverse professional backgrounds gathered to \"unify the disparate neurosciences\" and shed the light on the emerging trends, issues, hypotheses, and findings that could further advance our knowledge of the human brain (Quarton, Melnechuk, Schmitt et al. 1967). The community brought together by this first study program defined itself by its set of objects of study: the brain, the nervous system, and their relation to their arising \"products\" (for example, learning, memory, sleep, arousal, reflex, etc.). The challenge, as they saw it, was to bring together the disparate fields of neurologists, behavioral scientists, physicists, chemists, and mathematicians into meaningful communication with one another under the rubric of what Schmitt (1970)\u2014in one of the first journal articles to mention _neuroscience_ in its title\u2014was to call \"brain and behavior.\" In other words, the intention was to integrate not just the _neuro_ disciplines\u2014neurophysiology, neuroanatomy, neurochemistry, and neurology\u2014but also the _psy_ disciplines of psychology, psychoanalysis, and psychiatry, as well as some other less obvious fields such as immunology (Schmitt 1990). Their approach was characterized by the attempt to combine analysis at many levels: cellular, molecular, anatomical, physiological, and behavioral (Quarton, Melnechuk, Schmitt, et al. 1967). Twenty years later it seemed easy to state the program's aim: it was to answer the question \"What is neuroscience?\" (Schmitt 1985, 1990).\n\nThe terms _neuroscience_ and _neuroscientist_ now seem the obvious ones for this kind of research on the brain. But neither the name nor its implications were immediately evident to those visionaries at the time. Judith Swazey tells us that Schmitt first used the terms \"mental biophysics\" and \"biophysics of the mind\" to encapsulate his attempt to bring together the \"wet, the moist and the dry\" disciplines that addressed central nervous system functioning (Swazey 1975). He and his colleagues later worked under the title \"the Mens Project\" (adopting the Latin term for mind) and initially applied to the NIH for a grant for a \"Neurophysical Sciences Study Program\"\u2014which was approved in just two weeks. The project only became the \"Neurosciences Research Program\" in a subsequent grant application to NASA and the term _neuroscience_ made its first public appearance (in the plural) in 1963 in the title of the _Neurosciences Research Program Bulletin_ of that year.\n\nSchmitt was a biophysicist who was recruited to MIT in 1941 at the request of MIT's president, Karl Compton, and its then vice-president, Vannevar Bush, author of the classic text on the endless frontier of science published in 1945 (Bush 1945). Compton and Bush were keen on establishing a \"new kind of biology\" or \"biological engineering\" that would combine physics, mathematics, and chemistry (Bloom 1997). This vision, as we have suggested, was explicitly informed by the rapid developments in molecular biology during the 1950s (Schmitt 1990). The new kind of biology that Schmitt was called upon to establish at MIT was a molecular biology heavily rooted in biophysics, biochemistry, and biophysical chemistry (Schmitt 1990). Swazey tells us that when Schmitt was sketching out his plans in 1961, \"he listed nine 'basic disciplines' for mental biophysics: solid state physics, quantum chemistry, chemical physics, biochemistry, ultrastructure (electron microscopy and x-ray diffraction), molecular electronics, computer science, biomathematics, and literature research\" (Swazey 1975, 531). Physical chemistry, bioelectric studies and computer science seemed to be essential to study the \"physical basis\" of memory, learning, consciousness, and cognitive behavior.\n\nIn a later paper the list was expanded to twenty-five areas of study, and in its progress report to the NIH in 1963, the NRP enumerated fourteen disciplines represented in their program: chemistry, biochemistry, neurochemistry, physical chemistry, mathematics, physics, biophysics, molecular biology, electrophysiology, neurobiology, neuroanatomy, psychology, psychiatry, and clinical medicine. This was in danger of becoming an enterprise without limits! By 1967, however, Schmitt created a simpler diagram that divided the neurosciences into \"three intellectual areas\": _molecular neurobiology_ , working at the molecular, subcellular, and cellular level; _neural science_ , dealing with \"net characteristics,\" including neuroanatomy, neurophysiology, and neuropathology; and _behavioral or psychological science_ , or what he called \"molecular psychology\" and which he\u2014like so many of his successors\u2014found the most challenging to integrate (Schmitt 1967, 562).\n\nSwazey argued that the NRP had a \"catalytic role in creating and promoting the growth of a _neuroscience_ community.... [T]he term and concepts of 'neuroscience,' of a new type of integrated multidisciplinary and multilevel approach to the study of the brain and behavior, came into being with the NRP\" (Swazey 1975). While those working in this area would retain their specific disciplinary allegiances\u2014molecular biologists, neuroanatomists, biophysicists, psychologists, and so on\u2014they would also locate themselves within this emerging scientific project with a common goal: they would think of themselves as _neuroscientists_. By 1964, this aspect of the project had achieved some success. The term _neuroscientist_ was first used publicly in an article in the journal _Technological Review_ titled \"A neuroscientist sees 'invisible colleges' as a means of meeting the information problem,\" with specific reference to the \"invisible college\" of the recently created Neuroscience Program \"activated and made 'visible' in Boston.\" To create the neuroscientist, then, required, first and foremost, the creation of a matrix\u2014the invisible college\u2014that could bind those who identified with this term into some kind of community.\n\nFor Schmitt, as for other prominent scientists of the time, it would be through the unification or \"synthesis\" of these diverse and separate disciplines and through the reductionist molecular approach (i.e., a bottom-up approach) that substantial breakthroughs could be delivered (cf. Kay 1993). He based this belief on his perception that the success of molecular biology, genetics, immunology, and biophysics arose from the multidisciplinary dialogue between three traditionally separate disciplines: physics, chemistry, and biology. \"It now seems possible,\" Schmitt wrote in a 1963 progress report, \"to achieve similar revolutionary advances in understanding the human mind.... By making full use of [the approaches of physiology and the behavioral sciences] and by coupling them with the conceptual and technical strengths of physics, chemistry, and molecular biology, great advances are foreseeable\" (quoted in Swazey 1975, 530). This was not to be simply a natural evolution of scientific knowledge; it was an explicit project to anatomize the workings of mind in terms of brain, to be undertaken at a novel, neuromolecular level: \"[M]olecular neuropsychology is becoming ripe for study by the methods and concepts that proved significant in molecular genetics and molecular immunology. It seems inevitable that the problem of the mechanisms of brain function will soon become the primary intellectual salient, offering the greatest promise for the life sciences\u2014indeed, as Schroedinger has suggested, of all science\" (Schmitt 1967, 561\u201362). This, then, was to be the promise, and the mission, of the new molecular vision of the brain that was born in 1962.\n\nHow Should One Do the History of the Neurosciences?\n\nBut surely this claim, that the neurosciences were born in 1962 is absurd, or at the least, purely semantic. As we have already mentioned, human beings have been concerned with the brain, the nerves, let alone the mind, since time immemorial, and rigorous reflections, often based on empirical examples, go back at least to the Greeks. So why focus on 1962? To explain this, it is necessary to reflect for a moment on different ways of writing the history of the neurosciences.\n\nConsider, for example, one of the first articles published in French to use the term _neuroscience_ , written by Andr\u00e9 Holley in 1984: its title was \"Les neurosciences: Unit\u00e9 et diversit\u00e9\" (Holley 1984). Holley argued that there is one object of study, and one objective of these new sciences of the brain, namely, \"the knowledge of the nervous system, its function and the function of the diverse phenomena that emerge from it\" (Holley 1984, 12; our translation). Yet, says Holley, on the other hand there are a plurality of methods and concepts. Thus, for him, the neurosciences were born with the convergence of all these sciences around one object of study and concern: the brain. This account of unity in the face of conceptual and methodological diversity seems to fit the account given in most 'authorized' versions of the history, which usually take the form of chronologies of the great thinkers, establishing a lineage of precursors for the present (Garrison and McHenry 1969; DeJong 1982; Brazier 1984, 1988; Finger 1994). These are often supplemented, for the modern period, by compilations of autobiographical accounts of key figures in the manner of oral history (Healy 1996, 1998, 2000; Squire 1996; Tansey 2006).\n\nThere are also a number of institutional versions of this history: many university departments of neuroscience have created their own accounts or time lines of the milestones\u2014key discoveries, key texts, key persons. Some versions are presented from the perspective of the various disciplinary trajectories\u2014from the perspective of neurology, neurochemistry, or neuroimaging. For example, the Society for Neuroscience (SfN), has its own history and has produced eleven DVDs of interviews with prominent neuroscientists. Along the same lines, accounts of other societies and organizations have been published as well, for example, the International Neuropsychological Symposium, founded in 1951 (Boller 1999); the International Brain Research Organization (IBRO), founded in 1961 (Marshall, Rosenblith, Gloor, et al. 1996); the International Society for Neurochemistry, founded in 1965 (McIlwain 1985); and the European Brain and Behavior Society, founded in 1968 (Marzi and Sagvolden 1997).\n\nIn addition, numerous memoirs have been written by famous neuroscientists of the twentieth century (Ram\u00f3n y Cajal 1937; Schmitt 1990; Hodgkin 1992; Kandel 2006), as well as abundant biographies and articles on the contributions and achievements of particular individuals, either in the form of \"neuro-anniversaries\" (Eling 2001) or as part of the development of a particular concept or a discovery, for example, the discovery of acetylcholine by Henry Dale (Tansey 2006) or the discovery of dopamine by Arvid Carlsson (Benes 2001). Other published papers name this or that historical figure as the true founder of neuroscience (Changeux 1985, 1997; Wills 1999). Alongside these accounts of founding fathers\u2014and they were mostly men\u2014we find accounts of the major discoveries, such as the synapse (Rapport 2005), and of major controversies, such as the famous \"war of the soups and the sparks\" between the proponents of electrical transmission and neurotransmission in the central nervous system (Valenstein 2005). And, starting in the 1990s, the history of neuroscience becomes a kind of subdiscipline in its own right, with the founding of the _Journal of the History of the Neurosciences_ , and then of the International Society for the History of the Neurosciences (Rosner 1999).\n\nWhat are they for, all these histories? What role do they play for the neurosciences themselves? As in other truth-claiming disciplines, a historical sensibility is internal to the science itself. It is performative: it establishes a respectable past for neuroscientific knowledge, thus securing a basis for its future. Knowingly or not, such histories solidify the scientificity of the present state of the science whose tale they tell by presenting it as the culmination of a long trajectory of the progress of knowledge. Georges Canguilhem terms such accounts \"recurrent history\"\u2014the history written and rewritten by almost every science, viewing the past from the perspective of its own regime of truth. These histories establish a division between the sanctioned and the lapsed. The sanctioned: a more or less continuous sequence that has led to the present\u2014a past of genius, of precursors, of influences, of obstacles overcome, crucial experiments, discoveries, and the like. The lapsed: a history of false paths, of errors and illusions, of prejudice and mystification, and all those books, theories, arguments, and explanations associated with the past of a system of thought but incongruous with its present. To point to this use of history within science is not to denounce it but to characterize its constitutive role: using the past to help demarcate a contemporary regime of truth, to police the present, and to try to shape the future.\n\nCentral to such a history is a succession of predecessors. This quest for precursors fabricates \"an artifact, a counterfeit historical object\" and ignores the fact that the precursor is the creature of a certain history of science, and not an agent of scientific progress; Canguilhem quotes Koyr\u00e9: \"It is quite obvious (or should be) that no-one has ever regarded himself as the 'forerunner' of someone else.... [T]o regard anyone in this light is the best way of preventing oneself from understanding him\" (Koyr\u00e9 1973, 77; quoted in Canguilhem 1994, 51). Indeed, as Canguilhem points out (1968, 21): \"The precursor is a thinker who the historian believes he can extract from one cultural frame in order to insert him into another, which amounts to considering concepts, discourses and speculative experimental acts as capable of displacement or replacement in an intellectual space in which reversibility of relations has been obtained by forgetting the historical aspect of the object dealt with.\"\n\nFor such histories, we can reinterpret the arguments of those precursors in terms of our own concepts and problems. The objects of study\u2014the brain, the nerves\u2014remain static, intact, brute entities in a world external to thought, which are merely grasped more or less adequately by a succession of thinkers. But the kind of historical epistemology practiced by Canguilhem recognizes the historicity of the very object of explanation, the set of problems, issues, phenomena that an explanation is attempting to account for. In a way, Andr\u00e9 Holley is right when he says that the neurosciences (in the plural form) are centered on one object of study: \"la connaissance du syst\u00e8me nerveux, de son fonctionnement et des ph\u00e9nom\u00e8nes qui emergent de ce fonctionnement\" (Holley 1984, 12). He insists on this 'unity': \"pluralit\u00e9 des methods, diversit\u00e9 des concepts mais unit\u00e9 de l'objet\" (ibid.).\n\nYet the contemporary brain is not the object that formed the focus of nineteenth-century neurologists, nor that explored by Sherrington and Adrian in the first half of the twentieth century. In this chapter we will suggest that the new style of thought that has taken shape in the form of neuroscience has so modified the object of thought, the brain, that that it appears in a new way, with new properties and new relations and distinctions with other objects. And yet, as pointed out by Hans-J\u00f6rg Rheinberger in another context, the research object\u2014what he terms the \"epistemic thing\"\u2014has a \"characteristic, irreducible vagueness\" (Rheinberger 1997, 28). That is because this object embodies what is not yet known. The object, in our case, the neuromolecular brain, is both there and not there. Rheinberger quotes Latour: \"[T]he new object, at the time of its inception, is still undefined. [At] the time of its emergence, you cannot do better than explain what the new object is by repeating the list of its constitutive actions. [The] proof is that if you add an item to the list you _redefine the object_ , that is, you give it a new shape\" (Latour 1987, 87\u201388, quoted in Rheinberger 1997, 29). That is no doubt true, yet we will suggest in this chapter that while the object of the new brain sciences is elusive, vague, shifting, and while it constantly refers back, and gains some of its power, from that brain that we think we have always known, that mass inside the skull, a new style of thought has taken shape, which has provided the possibility for the neurosciences to achieve their current status, to move out of the laboratory and the clinic, to become a kind of expertise in understanding and intervening in human conduct in many different practices.\n\nIt is, of course, a mistake to think of historical time as continuous: the time of history is not that of a calendar, it is intercalated by discontinuous events (Canguilhem, 1968; Foucault 1972) or \"turbulences\" (Serres 1977). The birth of neuroscience marks something like an event. Yet even a discontinuous genealogy cannot do without dates, symbolic landmarks to alert us to something like an 'event.' It is in that sense that we mark this event with a date\u20141962. This is the point at which a number of distinct lines of thought and practice seem to come together to create a difference. It is beyond our capacity to undertake a genealogy of this event here. So, at the risk of simplification, let us try to orient ourselves by distinguishing, purely for heuristic purposes, three intertwining pathways that seem to intersect in that event\u2014the path through the nerves, the path through the brain, and the path through madness.\n\nThe Path through the Nerves\n\nNeuroscience, neurology, neuroanatomy, neuroembryology, neurophysiology, neurochemistry, and the burgeoning _neuro_ fields, from neuroeducation and neuroeconomics, to neurophilosophy and neuromarketing all have in common the prefix _neuro_ -, which is taken from the ancient Greek _ne\u00fbron_ or the Latin _nervus_ , meaning \"of or relating to nerves or the nervous system.\" The _OED_ dates the first use of _neuro_ -in words of Greek origin to the seventeenth century with the introduction of the term _neurology_ and in the eighteenth and early nineteenth centuries in loanwords from post-classical Latin, as with the terms _neurotomy_ (the anatomical dissection of nerves), _neurography_ (the anatomical description of nerves), and _neuralgia_ (pain in the area served by a nerve). In French, the prefixes _n\u00e9vro_ -and _neuro_ -both occur from the late seventeenth century onward (1690, in _n\u00e9vrologie_ ; 1691, in _neurologie_ ). In German, the prefix makes its first appearance probably in 1856 with the term _Neuroglia._ In Italian, the _OED_ dates the first use of _neurologia_ in 1764 and _nevralgia_ in 1828.\n\nThomas Willis is thought to have coined the term _neurology_ to refer to the \"Doctrine of Nerves\": the first entry in the _OED_ for the term _neurology_ is 1681, the date of publication of Samuel Pordage's translation of Willis's 1664 influential _Cerebri anatome_. The use of that term to describe a treatise on the nervous system dates back to 1704, and the word gradually came to refer to the branch of science or medicine dealing with the nervous system and its diseases and disorders. Wilhelm Griesinger, in Berlin, founded the _Archiv f\u00fcr Psychiatrie und Nervenkrankheiten_ in 1867, and the American Neurological Association was formed in 1875. The _OED_ tells us that term _neuropath_ \u2014for both the doctor who attributes diseases to disorders of the nerves, and to those suffering from or susceptible to nervous diseases\u2014dates from the 1880s.\n\nAll histories of these matters credit Golgi with inventing the silver salt staining method that enabled the visualization of the elements of the nervous system, although he believed that they formed a single continuous network, and Cajal with establishing the doctrine that neurons were discrete cells that were the elementary units of the brain. The _OED_ finds the first English use of the term _neuron_ in a paper by Alex Hill, published in _Brain_ in 1891, summarizing the research of Camillo Golgi, Santiago Ram\u00f3n y Cajal, Albert von K\u00f6lliker and Wilhelm His Sr. But Jean-Pierre Changeux tells us: \"As to the term 'neuron,' we owe it neither to Ram\u00f3n y Cajal nor to His, but to Wilhelm Waldeyer (1891) about whom Ram\u00f3n y Cajal wrote that all he had done was to 'publish in a daily paper a r\u00e9sum\u00e9 of his research and invent the term neuron' \" (Changeux 1997, 28). In 1906, despite their bitter disagreements, and Cajal's attacks on Golgi's reticular or neural net theory, Golgi and Cajal jointly received the Nobel Prize for Physiology or Medicine for their discoveries of the structure of the nervous system.\n\nThe French Soci\u00e9t\u00e9 de Neurologie was formed in 1899, and its German equivalent in 1907. We find _neuroanatomy_ used in 1900, the first reference to _neurobiology_ in 1906, and _neuroembryology_ emerged in the 1930s. The first chair in _neurochemistry_ was established in 1945, the _Journal of Neurochemistry_ started publishing in 1956, the International Society for Neurochemistry was founded in 1965, and the American Society for Neurochemistry was inaugurated in 1969. The mode of action of the neuron was gradually unraveled over this period: in 1926, Edgar Adrian published his recordings of impulses for individual sensory nerve fibers; in 1939, Hodgkin and Huxley published their classic paper reporting the recording of action potentials from inside a nerve fiber; and in 1944, Joseph Erlanger and Herbert Gasser were awarded the Nobel Prize in Physiology or Medicine \"for their discoveries relating to the highly differentiated function of single nerve fibres.\" _Neuropharmacology_ appears in the late 1940s, used in relation to the introduction and discovery of more effective antiepileptics (Toman 1949). In 1951, the International Neuropsychological Symposium was formed under the leadership of cofounder Henry H\u00e9caen, and it met almost every year from 1951 until 1998 for the purpose of promoting knowledge and understanding of brain functions and cognate issues on the borderland of neurology, psychology, and psychiatry. The 1950s saw the famous \"war of soups and sparks\"\u2014the disputes between the believers in electrical neurotransmission and those who believed that neurotransmission in the central nervous system was chemical\u2014and the gradual victory of the chemical view of neurotransmission (cf. Valenstein 2005). And thus we arrive, more or less, at 1962.\n\nThe Path via the Brain\n\nMost historians agree that contemporary research in neuroscience builds on studies of the brain that flourished in Europe in the nineteenth century. It was at this point, it seems, that the brain began to be truly understood as an organ analagous to others, and that specific areas and structures in the brain came to be seen as responsible for certain functions\u2014such as those that are still known by the names of their discoverers\u2014Broca, Wernicke, Flechsig, for example. The journal _Brain_ was first published in England in 1878, calling itself \"A Journal of Neurology\"\u2014thus making it clear that our separation of nerves and brain is purely heuristic! It had four editors. J. C. Bucknill and James Crichton-Browne both had backgrounds as medical superintendents of insane asylums; both were prolific authors and advocates for what was then often termed psychological medicine. David Ferrier and John Hughlings Jackson were neurologists working on the localization of brain functions, work that we discuss in more detail in chapter 2.\n\nIn the half-century that followed, work on brain structure and function continued, notably by Charles Sherrington and his group in the United Kingdom, using technologies of stimulation or ablation of specific of brain areas. Others, notably Sherrington's colleague Edgar Adrian, used technologies of visualization, such as the electroencephalogram. A meeting of electroen-cephalographers in London in 1947 led to the establishment of an International Federation of Electroencephalography and Clinical Neurophysiology, and the proposal for an International Brain Research Organization arose from a meeting of this group in Moscow in 1958 initiated by the French clinical neurophysiologist Henri Gastaut. IBRO was established as a component of the United Nations Educational, Scientific, and Cultural Organization (UNESCO), which was founded in 1951. Initial proposals for UNESCO had included an \"Institut du Cerveau\" (Institute for the Brain), which was not accepted (McIlwain 1985). IBRO was constituted with seven panels\u2014in neuroanatomy, neurochemistry, neuroendocrinology, neuropharmacology, neurophysiology, behavioral sciences and neurocommunications, and biophysics. This multidisciplinary structure of collaboration between approaches and across national borders was retained when it became an independent body: again note the year of its foundation\u20141961.\n\nThe Path via Insanity\n\nWe arrive at roughly the same point in the 1960s if we follow the path of those dealing with those deemed insane or otherwise mentally afflicted. In fact, the journal _Brain_ , not surprisingly, given the interests of two of its founding editors, was much concerned with the ways in which brain damage or disorder could produce symptoms like those of hysteria or mania, as well as loss of speech, hearing, memory, or other functions. Broca, Wernicke, Flechsig, and others studied the brains of those who died in lunatic asylums in their attempts to discover the basis of mental afflictions in the brain. Many also studied those who had suffered brain injuries, for example, those who drew conclusions from emblematic figures like Phineas Gage, who had suffered changes in character after an injury to specific regions of his brain caused by that famous tamping iron (Macmillan 2000). There were those like the French psychiatrist B\u00e9n\u00e9dict Morel, whose understanding of madness as a physical condition was closely linked to their ideas of inheritance and the problem of degeneration (Morel 1852\u201353, 1860). The birth of psychiatric genetics was intrinsically bound up with the eugenic style of thought that was prevalent in Europe and North America in the first half of the twentieth century, although the pathways for the action of the supposed inherited diathesis were unknown.\n\nIt is true that this period was also that of the early development of dynamic psychotherapies, but most of those who worked in this tradition were\u2014like Freud himself\u2014quite open to the belief that the propensities to the disorders they addressed were organic and inherited, although once more the locus of these disorders in the nerves or the brain was a matter of conjecture. The first half of the twentieth century saw the birth of movements to address the 'maladaption' and 'maladjustment' of the instincts that underpinned the troubles of everyday existence at work or in the family, such as the mental hygiene movement. But in the asylums, physical treatments were flourishing: shock therapies using electroshock-or insulin-induced coma, malaria therapies (for neurosyphilis), and psychosurgery (Sargant and Slater 1944; Sargant, Hill, and Slater, 1948). Somehow, it seemed, the brain was the locus of mental pathology, and potentially therefore the target for therapy, but just how and where remained elusive.\n\nIn 1936, Egas Moniz reported the first results of his technique of frontal lobotomy. These were apparently inspired by a report of the calming effects of destruction of the prefrontal brain area of monkeys and chimpanzees; Moniz's version of this procedure involved the destruction of a portion of the frontal lobes, first by injection of alcohol, and later by cutting nerve fibers with a surgical leucotome (Moniz 1948, 1954; Valenstein 1986; Pressman 1998). These methods were taken up by Walter Freeman and James Watts in the United States, who invented a technique to enter the brain through the eye socket using an instrument modeled on the ice pick (Freeman and Watts 1942, 1950). By 1948, lobotomies had been performed on about twenty thousand patients worldwide. In 1949, Moniz was awarded the Nobel Prize in Physiology or Medicine for this work.\n\nJoseph Zubin, in his article on \"Abnormalities of Behavior\" for _Annual Reviews of Psychology_ in 1952, notes: \"The tremendous impact that psychosurgery has made on current investigations is evidenced by the following books and monographs which were published between 1949 [ _sic_ ] and 1951\" (Halstead 1947; Mettler 1949; Freeman and Watts 1950; Partridge 1950; Fulton 1951; Greenblatt, Arnot, and Solomon 1951) and the number of symposia that focused on this topic (Association for Research in Nervous and Mental Disease 1948; Rees, Hill, and Sargeant 1949; Congr\u00e8s International de Psychiatrie 1950; Halstead 1950; U.S. National Institute of Mental Health 1951), as well as the studies of the brain that showed the impact of this research (Hebb 1949; Golla and Richter 1950; Penfield and Rasmussen 1950; Porteus 1950). Psychosurgery, he remarks, acted as a \"shot in the arm\" for psychiatry in the 1940s, and while \"the mechanisms whereby the operation brings about improvement remain unknown.... By the end of the third month, most of the psychological functions have returned to their preoperative level, or have even exceeded it\" (Zubin 1952, 262\u201363). Subsequent experience was to prove him wrong.\n\nThe enthusiasm for physical treatments in the 1940s and 1950s, along with the conviction that they had a scientific basis and major therapeutic implications for psychiatry, led to the formation in 1945 of the Society for Biological Psychiatry, whose mission was to encourage \"the study of the biological causes of and treatments for psychiatric disorders.\" Edward Shorter credits Stanley Cobb, one of the originators of this society, with being the founder of biological psychiatry in the United States, not only building the Department of Psychiatry at Massachusetts General Hospital beginning in 1934 (Shorter 1997, 263) but also promoting psychoanalysis. The opposition between the biological and the psychoanalytic does not emerge until the 1960s. When the first International Congress on Psychiatry was held in Paris in 1950, sessions on Psychotherapy and Psychoanalysis (chaired by Franz Alexander) were presented alongside those on Genetic and Eugenic Psychiatry (chaired by Torsten Sj\u00f6gren), on Shock Therapy (chaired by Josef Handesman), and on Cerebral Anatomy and Physiology in the Light of Lobotomy and Topectomies (chaired by Frederick Golla) (Congr\u00e8s International de Psychiatrie, 1950). Present at the meeting were Denis Hill, who talked on the use of the EEG in psychiatry, and Max Reiss and Derek Richter, both of whom were then working on the psychiatric implications of hormones and endocrinol-ogy. The overall chairman of the Congress was Jean Delay, who was the emblem of the new bridge between psychiatry and neurochemistry via psychiatric drugs that would rapidly come to replace most, if not all, of the more direct physical interventions into the brain (whose promises were only to be enthusiastically explored again half a century later).\n\nThis is not the place to recount the history of psychopharmacology in any detail; a rapid sketch will serve our purposes. The effects of chlorpromazine were first reported by Henri Laborit working at Val-de-Gr\u00e2ce military hospital in Paris in 1951, and in 1952, Jean Delay and Pierre Deniker published the apparently dramatic results of administering this drug to psychotic patients at the Saint Anne Hospital in Paris (Delay 1953). In a series of transactions between France and North America, the drug was widely taken up by those working in psychiatric hospitals in North America during the 1950s (Horwitz 2002). The first results of another drug, reserpine, on psychotic patients were reported in 1954, and by 1955, when Jean Delay organized an international colloquium on psychopharmacology in Paris, it seemed as if at last drugs had been developed that acted on the symptoms of mental disorder by virtue of their specific effects in the brain. Once more, however, this was not framed as an opposition between physical and psychological treatments, for it was argued that one of the main advantages of the drugs was that it made the patients who had been treated more amenable to psychotherapy.\n\nThroughout the 1950s, a number of key papers begin to discuss the effects of other drugs, notably isoniazid, iproniazid, reserpine, and imipramine on disturbed or depressed patients. Gradually the theory took shape that all these novel psychotherapeutic agents produced their effects on mood and conduct by acting on the levels of monamines in the brain (Salzer and Lurie 1953; Kline 1954; Delay and Buissoj 1958; Kuhn 1958). Bernard (\"Steve\") Brodie, who decided to enter psychopharmacology \"after hearing about the sensational clinical actions of the new antipsychotic drugs and the ability of the hallucinogenic LSD to block the effects of serotonin on various peripheral organs\" (Carlsson 1990, 486), developed new analytical methods that enabled him and his colleagues to demonstrate that the effects of reserpine were linked to its ability to deplete serotonin in the brain (Brodie, Pletscher, and Shore 1955; Pletscher, Shore, and Brodie 1955; Shore, Silver, and Brodie 1955). Arvid Carlsson, who had worked with Brodie, generalized this to other catecholamines, notably to dopamine (Carlsson, Lindqvist, Magnusson, et al. 1958; Carlsson 1959; Carlsson and Lindqvist 1963; Dahlstrom and Carlsson 1986), and began to formulate the link between the depletion of dopamine by reserpine and the symptoms of Parkinson's disease.\n\nAlthough by the 1950s it was accepted that neurotransmission in the _peripheral_ nervous system was chemical, as late as 1960, according to Arvid Carlsson, there was still considerable resistance to the argument that chemical neurotransmission happened in the brain and was probably the dominant form of neurotransmission. However, the work of Carlsson and his collaborators over the next few years, involving many technical developments, established the neuronal localization of dopamine, noradrenaline, and serotonin in the central and peripheral nervous system, and mapped the major monoaminergic pathways and the sites of action of the major psychotropic drugs. According to Carlsson, by 1965 a \"paradigm shift\" had taken place, and it was now almost universally accepted that the monoamines played the key role in neurotransmission in both peripheral and central nervous systems and that their presence and actions were clearly linked to psychiatric symptoms (Dahlstrom and Carlsson 1986; Carlsson 1990, 2001).\n\nA kind of informal consensus began to develop in which each of the major classes of neurotransmitters then known was allocated to a specific disorder\u2014catecholamines to mood disorders, serotonin to anxiety, dopamine to psychosis, and acetylcholine to dementia (Healy 1997). In 1963, Carlsson and Lindqvist published what is usually taken as the first formulation of the dopamine hypothesis of schizophrenia, describing the interaction between the antipsychotic drugs chlorpromazine and haloperidol, and the level of the dopamine metabolite, 3-methoxytyramine, in the mouse brain\u2014thus suggesting that high levels of dopamine might be linked to the symptoms and that reduction in dopamine might be the mode of effect of the drug (Carlsson and Lindqvist 1963).\n\nIn 1965 Joseph Schildkraut published his paper on the catecholamine hypothesis of depression: \"what has been designated 'the catecholamine hypothesis of affective disorders,' proposes that some, if not all, depressions are associated with an absolute or relative deficiency of catecholamines, particularly norepinephrine, at functionally important adrenergic receptor sites in the brain. Elation, conversely, may be associated with an excess of such amines\" (Schildkraut 1965, 509). This paper, which hedged its conclusions with many reservations, marked the moment when a new language was assembled together, one that would come to shape the style of thinking and research in psychopharmacology and the forms of clinical practice in psychiatry. Indeed, Schildkraut claimed that he decided to publish first in the _American Journal of Psychiatry_ , a clinical journal, because he saw his work as more than about catecholamines\u2014it was about the \"notion that pharmacology can become a bridge, linking neuroscience and clinical psychiatry\"\u2014though he later published a version in a journal that would be read by the psychopharmacologists, when he discovered that they didn't read the clinical journals (Schildkraut 2000, 124).\n\nThe significance of the hypothesis was not limited either to the catecholamines or to depression. What one had to do, from this point onward, was to understand mental disorders in terms of their characteristic patterns of biogenic amines in the brain and to understand the action of actual or potential psychiatric drugs in terms of their actions on the secretion, breakdown, depletion, and reuptake of these amines in the synapses or the receptors. A whole continent of potential experiments, clinical trials, and drug development was opened for exploration. One might examine the amines in the brain directly, if one could. One might examine them in experimental animals. One might examine the behavior of synapses or receptors by using in vitro cultures. One might examine them or their breakdown products in the blood or urine of psychiatric patients. One might, in a reverse move, seek to refine psychiatric diagnoses in terms of amine malfunctions. This, then, was the path to a future neuropsychiatry (Healy 1997).\n\nThe dopamine hypothesis of schizophrenia and the monoamine hypothesis of depression were the two founding myths of the psychopharmacological imaginary. Both have proved to be wrong, perhaps fundamentally so. However, for the biological psychiatrists, who would gradually come to dominate the field, the biogenic amine system in the brain would increasingly become the obligatory passage point of all accounts of mental disorder. If there were environmental or biographical factors in the etiology of disorder, they would have their effects by acting through this mechanism. If there were genetic factors, they would work through their impact on this mechanism. If there were psychodynamic factors, well, they too would work via this mechanism. Even evidence that seemed to conflict with the argument would have to be explained in analogous neurochemical terms. Once more, note the date when this neuromolecular vision of the mode of action of psychiatric drugs was formulated\u2014the early 1960s.\n\nInfrastructure\n\nDespite so many histories of neuroscience, most social historians working in this area agree that we do not have anything approaching a sociological account of the development of the neurosciences\u2014that is, at a minimum, something that would link these developments to their social, cultural, perhaps even political conditions of emergence. We have not attempted such an account here. Our claim is a limited one: that the event here was a new way of seeing the brain\u2014a 'neuromolecular gaze.' And in the chapters that follow, we will argue that it is this new way of envisaging the brain and nervous system, and the style of thought and intervention to which it is linked, that has established the conditions that allow for the burgeoning of the neurosciences over the past five decades, and hence for the birth of our contemporary neurobiological complex. This is not to say that Schmitt was the founder of the neurosciences or of the new complex of sciences of mind and behavior that takes shape after 1962. Rather, his NRP is emblematic; it exemplifies a moment, dispersed across different sites, cultures, and countries, in which something new was coming into existence\u2014a neuromolecular vision of the brain as an intelligible organ that was open to knowledge.\n\nBut a vision itself is seldom enough. A vision needs some kind of infra-structure. Many different infrastructural forms can be found in scientific research. We have already encountered a number in our rapid overview of pathways to neuroscience. Many involve a kind of discipleship, where a charismatic scientist infuses certain beliefs and commitments into those who are trained at his or her feet. Others involve the founding of learned journals developed to promulgate a particular kind of identity for those who publish, contribute to, and read them. In some, a university has played a key role, for example where a university department employs those who share and transmit the vision of the professor. Another ancient form of infrastructure is the scientific society or the academy. And one more form, that of an 'invisible college,' refers us back to the phrase used by Robert Boyle in the 1640s to designate a network of intellectuals without an institutional form, exchanging ideas, information, and speculation on specific topics (Crane 1972). And a framework has also been provided by scientific institutes\u2014usually located within or attached to universities, with a physical form, an organizational structure, a group of faculty, students, visitors, and so forth.\n\nSchmitt was certainly aware of the significance of infrastructure. While he did sometimes speak of the NRP as an invisible college, he spent much time and energy pondering and organizing the rather material conditions under which his project might best flourish. Beginning with the idea of a bricks-and-mortar research center or institute, Schmitt concluded that this was not the best form to foster the novel mode of scientific communication and interaction that he sought\u2014he wanted to establish a different kind of center \"oriented towards probing and evaluating the state of the art, seeking to map new research and conceptual directions, and providing new modes of education for the communication among a worldwide network of scientists\" (Swazey 1975, 533). He took the idea of a study program from those he already knew from biophysics, and chose a structure in which a core group of individuals from the different disciplines would sponsor a series of such programs\u2014conferences and workshops\u2014with a relatively small central physical base independent of any university for the co-location of staff and for meetings of the associates.\n\nThis was indeed the form the NRP took, with premises located within the American Academy of Arts and Sciences in Brookline, Massachusetts, undertaking a set of activities\u2014study programs, focused conferences, teaching activities to introduce those with no knowledge of neuroscience to the fundamentals of brain structure and function\u2014plus the _NRP Bulletin_ which would disseminate the results of all these endeavors. The NRP thus involved many of the key elements of infrastructure: a charismatic leader, a network of loyal students, a shared language and nomenclature, a shared moral economy among its members, and, perhaps more than anything, a certain way of seeing the object of investigation, rendering it thinkable and actionable via a neuromolecular gaze. But the NRP was no 'origin'; rather, it was exemplary of something happening at several other sites, around other key individuals. For Schmitt was not alone in his endeavor to forge a community among those who were addressing this organ of the brain from diverse disciplinary perspectives.\n\nIn their own 'recurrent history' of neuroscience, Eric Kandel and his colleagues choose David Rioch at the Walter Reed Army Institute of Research as the key early figure, and they single out Stephen Kuffler, who created the first Department of Neurobiology at Harvard Medical School in 1966, as the person who drove the project of a unified neuroscience together, \"surmount[ing] the conventional divisions that had separated the subdisciplines of the neurosciences, and succeed[ing] in establishing a unified discipline that represented the various subfields\" (Cowan, Harter, and Kandel, 2000, 347). They argued that it was Kuffler's work that inspired the formation of similar departments in other universities and led to the formation of the Society for Neuroscience in 1969 \"under the leadership of the psychologist Neil Miller, the biochemist Ralph Gerard, and the neurophysiologist Vernon Mountcastle,\" which in turn brought together researchers who had previously been associated with their own disciplinary societies\u2014as physiologists, anatomists, biochemists, or psychologists (ibid.). What was taking shape here was a rather different form of infrastructure from that developed in the NRP, linked to a different project of neuro-interdisciplinarity. Rioch certainly facilitated interdisciplinarity, as David Hubel later reflected: \"In the neuropsychiatry division [of Walter Reed] David Rioch had assembled a broad and lively group of young neuroscientists, notably M.G.F. Fuortes and Robert Galambos in neurophysiology, Walle Nauta in neuroanatomy, Joseph Brady and Murray Sidman in experimental psychology and John Mason in chemistry.... As in Montreal, the focus was on the entire nervous system, not on a subdivision of biological subject matter based on methods.\"\n\nSo indeed did Kuffler and his group at Johns Hopkins\u2014a group particularly focused on the electrical physiology of the nervous system, and that included Vernon Mountcastle, David Hubel, and Torsten Wiesel. The scope of these collaborations widened when Kuffler moved to Harvard to set up a new Department of Neurobiology in 1966 (Katz 1982). As Timothy Harrison puts it:\n\nIn the 1960's, many scientists in and outside Kuffler's orbit realized that their grasp of biology was too limited. A prime example of this limitation was that electrical recording techniques could give only electrical answers and information, whereas new questions constantly suggested themselves. What are optimal nerve membrane ionic and biochemical requirements for nerve impulse propagation? What are the brain's energy sources? Do they differ from those of peripheral nerves? To answer such questions, an army of scientists developed new areas of neurobiologic research. Molecular and sub-cellular physiologists were interested in regulation of protein synthesis. Other scientists, specifically neurochemists, immunologists, molecular geneticists and several varieties of biophysicists all focused on the biology of the nervous system. Many of these neurobiologists gathered in Kuffler's new department at Harvard.... Kuffler's department inspired many other scientists trying to organize their departments and research in neurobiology. In its way Harvard served as a microcosm of a greater and broader scientific world now available to the handful of interested scientists who started the Society of Neuroscience in 1969. (Harrison 2000, 177)\n\nThe achievement here was both conceptual\u2014the need for some kind of integration of these distinct approached to the nervous system\u2014and topographical\u2014for, as the autobiographies by those associated with these developments show, it was the co-location in a single physical building, often along a single corridor, of researchers working on different questions, using different methods and approaches\u2014that enabled ideas and techniques to travel and to hybridize (Purves 2010).\n\nAs the researchers and postdocs moved from mentor to mentor, from the United States to London, to Australia, to Germany and back again, encountering enthusiasms and resistances to these new integrative approaches in equal measure, the linkages became stronger. By the end of the 1960s, then, neuroscience, as an interdisciplinary project with a neuromolecular gaze, was acquiring a spreading network of trained disciples of such charismatic people as Schmitt and Kuffler, although it took longer for other departments to form under the title of neurobiology. By 1970, Schmitt was able to point to the \"recent appearance of several journals and series of books with the words _brain_ and _neuroscience_ in their titles, and the formation of new multidisciplinary societies concerned with neuroscience. These include the Brain Research Association of the United Kingdom, the Society for Neuroscience of the United States, the European Brain and Behavior Organization and new specialty groups such as the International Neurochemical Society and the American Society for Neurochemistry\" (Schmitt 1970, 1006). And Schmitt would have agreed with Kandel and his colleagues when they write that the existence of the new discipline of neuroscience was institutionalized in a new way when, a \"decade later, in 1978, the first volume of the _Annual Review of Neuroscience_ appeared... a clear recognition that this new synthesis of disciplines had matured and could stand on an equal footing with other fully mature basic science disciplines\" (Cowan, Harter, and Kandel 2000, 348).\n\nA Neuromolecular Style of Thought\n\nIn 1962, the International Brain Research Organization began its world survey of research facilities and manpower in the brain sciences, focusing on three broad areas: first, the basic sciences modified by the prefix _neuro-_ : neuroanatomy, neurochemistry, neuroendocrinology, neuropathology, neuropharmacology, and neurophysiology; second, a group of disciplines consisting of neurocommunications and biophysics, including cybernetics and information theory, mathematical modeling, nerve cell and membrane biophysics, and neuromolecular biology; and third, the behavioral sciences, encompassing animal behavior, behavioral genetics, psycholinguistics, sensation and perception, and certain aspects of anthropology, psychology, psychiatry, and sociology. By 1968, using rather generous criteria, it could identify 880 different research groups and 4,245 investigators identified with the neurosciences in the United States alone (International Brain Research Organization 1968). And in the forty years that followed, it seemed self-evident that neuroscience was indeed a discipline, with all the implications of the term\u2014institutional organization; authorities of truth; incorporation into the academy; textbooks and training; norms of experimentation, evidence, and argument; and much more.\n\nBut nonetheless, it is a discipline whose boundaries seem blurred and whose internal configuration is complex and shifting. Papers are published in hundreds of different specialist journals as well as those, such as _Nature_ , intended for a more general scientific audience. The Neuroscience Citation Index (NSCI), which covers more than 350 journals, divides the neurosciences into eleven fields: Behavioral Neurology, Cardiovascular Disease and Metabolism, Developmental Neuroscience, Electroencephalography, Epilepsy Research, Molecular Brain Research, Neural Networks, Neurogenetics, Neuroimaging, Neurosurgery, and Psychopharmacology. Even in areas that might seem closely related to the outsider, interdisciplinary divisions into distinct organizations are evident, each with its own international conferences, recurrent themes, key reference points, and forms of argument and experimentation.\n\nThus, despite the claims that neuroscience is now a discipline with a unified object of study and concern, what we have here is something different. But this heterogeneity and contestation is not surprising. For, as we have seen, the creation of a neuroscience community was not intended to eradicate the specific disciplinary allegiances of those who formed it, but rather to create a common space within which they could interact. In any event, we agree with those historians and philosophers of science who have pointed out that disunity, rather than unity, characterizes the sciences (Dupr\u00e9 1993; Rosenberg 1994; Galison and Stump 1996; Hacking 1996; Galison 1997). There are a number of research fronts, with different sources of public and private research funding directed to different aspects and research strategies, and the multiplication of specialist institutes on everything from basic research on brain and behavior, to clinical and translational research. Efforts at unification, even when merely in the form of a consolidated textbook or introduction to neuroscience, can thus be seen as strategic, with political, cultural, epistemological, and ethical implications (Galison and Stump 1996).\n\nYet there is another way in which this appearance of diversity is misleading, because the event that we have traced to the 1960s is not really one of disciplinary formation. It is more an event in epistemology and ontology, in the nature of the object of the neurosciences and the forms of knowledge that can render it into thought. Is there any unity in the thought styles that characterize the diverse neuroscientific communities built around specific disciplines and research fronts? Is there something like an overarching \"style of thought\" (Fleck [1935] 1979)? Ian Hacking, drawing on A. C. Crombie more than Fleck, has identified six \"styles of reasoning\" in the sciences: (1) deducing from postulates, (2) experimental exploration and measurement of observational relations, (3) hypothetical construction of analogical models, (4) use of comparison and taxonomy to establish order, (5) statistical analysis of regularities in populations and the calculation of probabilities, and (6) analysis in terms of genesis or historical development (Hacking 1992b). John Forrester added another: (7) \"thinking in cases\" (Forrester 1996). Approached from this perspective, we would not find just single style of reasoning in neuroscience: in the same paper or the work of a single research group, we can often see thinking in terms of models, use of the laws of large numbers, analyses of the genesis and development of neuronal structures, concerns with traits or disorders, questions of taxonomy and classification, and even reasoning in cases.\n\nYet, approached in a different way, closer to the spirit of Ludwik Fleck, it is indeed possible, by 1970, to discern something like a style of thought what we term a \"neuromolecular style of thought.\" At the risk of oversimplification, let us enumerate some of its key structuring principles:\n\n1. The brain is an organ like any other (even if much more complex).\n\n2. Like other organs, many basic neural processes and structures have been conserved by evolution, so that one can posit, and indeed identify, similarities not merely in other primates, but also in other vertebrate animals, and indeed, in some cases, even in other phyla. Thus one can use animal models to understand these features that humans share with other creatures.\n\n3. Neural processes in the brain, and their common features across species, can and should be anatomized at a molecular level\u2014that is to say, each event in the brain can and should in principle be traced to identifiable molecular events\u2014thus investigations should proceed in a reductionist form, that is to say, by exploring these in the simplest possible systems.\n\n4. At that level, the key processes are those of neurotransmission, that is to say, communication along and between neurons. Communication is a combination of chemical transmission (across the synaptic cleft) and electrical transmission (down the neuron itself), and neurons are of different types, in part depending on the neurotransmitters that they use\u2014dopamine, serotonin, and so forth.\n\n5. Neurotransmission also entails the function of multiple other entities: ion channels, transporters, receptors, enzymes that catalyze or metabolize neurotransmitters at different rates, and so forth. Variations in each of these elements have functional significance and can in principle account for processes at higher levels.\n\n6. Different parts of the brain\u2014both different structural areas and different types of neurons using different neurotransmitters\u2014have different evolutionary histories, and have been evolved for different mental functions, and these networks of interconnected neurons can be identified anatomically and visualized by observing levels of activation in different brain regions when the organism is undertaking different tasks or is in different cognitive, emotional, or volitional states.\n\n7. All mental processes reside _in the brain_ (where else could they reside!), and each mental process will reflect, or be mediated by, or have something variously described as a correlate, an underpinning, or a basis, in brain events.\n\n8. Thus any mental state or process (normal or abnormal), and the behavior associated with that state or process (normal or abnormal), will have a relation\u2014exactly what relation is in dispute\u2014with a potentially observable material process in the organic functioning of the neuromolecular processes in the brain.\n\nExplicitly or implicitly, those who belonged to the invisible college of neuroscience shared this neuromolecular vision of the brain. However intellectually challenging it was in practice, in principle the brain was to be understood as an organ like any other organ (Schmitt 1967). The mental functions of perception, cognition, emotion, volition, and so forth could all be accounted for, in principle, by processes operating at the molecular level that was their substrate, and that underpinned or subserved them.\n\nInitially, the focus of this reductionist style of thought was on features that were common to all members of a species and especially those that were conserved across species by evolution. But what of variation? It was, of course, well understood that there were variations in all of the above, from individual to individual and across the life span, as well as across species. The belief that these variations were evolutionarily intelligible, meant, of course, that they were shaped by genetics, as did the fact that they could be affected by mutations, both naturally occurring and artificially induced though mutagenesis. This is not the place to trace the history of neurogenetics or the role played by such key figures as Seymour Benzer in identifying the genetic bases of behavioral characteristics in his favored model animal, the fruit fly _Drosophila melanogaster_. In this style of thought, genes were conceived of as more or less individualized, distinct units of inherited information that determined the synthesis of proteins. They could occur in a small number of distinct forms, or alleles, which could affect the phenotype of the animal concerned\u2014either its physical properties or in its behavior. These could be investigated in the laboratory by artificially inducing mutations in the genes of individual animals\u2014that is to say by mutagenesis\u2014and then by breeding from those animals and observing the effects in their offspring, and if necessary, in further generations. A whole range of technologies were used to study these changes and the way in which they were carried in pedigrees or lineages, and to explore the relations between genes, brains, and behavior, not just in laboratory studies of animals, but also in field studies of the genetics of mental disorders in humans. While few questioned the advances made in behavioral genetics of animals, the use of these explanatory regimes in humans was highly socially and politically controversial, not least because of the historical association with eugenics: genetic accounts of mental illness appeared to naturalize mental distress, to make it a matter of pathological brains not of pathogenic social experiences, and to legitimate physical methods of intervention, notably through the use of drugs.\n\nIn the aftermath of the sequencing of the human genome, however, a new style of thought took shape that may still speak of genes, but that focuses on variations at a different level\u2014at the molecular level of the base pairs themselves that code for proteins or that act to regulate gene expression. Variations between individuals were now thought to be related to variations at this molecular level, where one base might be substituted for another, thus affecting the nature of the protein or the activity of the enzyme that was synthesized. Conceived in this way, molecular genomics came into alignment with molecular neurobiology. Research could now explore the genomic basis of variations in the molecular components of neural activity\u2014of the neurotransmitters or the enzymes that metabolize them, the structures of receptor sites, ion channels, and so forth, and, indeed, of the factors that influence methylation and gene expression, or neurogenesis.\n\nThis molecular vision of genomic complexity thus mapped onto the vision of the neuromolecular brain. In this way of thinking, the properties and variations in the former underpinned all differences in human mental functioning, whether these be deemed normal variations or pathologies. Other molecular-level variations in DNA sequences were discovered, notably copy number variations (CNVs), where a relatively large region of sequence on a chromosome, sometimes within a coding sequence, sometimes involving several coding sequences, may be repeated several times. Taken together with research that was beginning to grasp the mysterious operations of the stretches of DNA previously thought of as 'junk' as well as the complex operations of RNA in gene transcription, by the first decade of the twenty-first century, genomics was a world away from notions of simple genetic determinisms of the 'gene for schizophrenia' variety. As we shall see in later chapters, this shift in thought was linked to a very significant shift in strategies of intervention and prevention.\n\nDespite constant references to the social importance of the project of neuroscience by those who sought funding and recognition for this new discipline, and, indeed, the initial concerns of many of its key figures with problems of mental illness and practices of psychotherapy, until the closing decades of the twentieth century, most of the actual research undertaken in the institutions and laboratories of the new discipline was primarily focused on investigations of what were thought to be basic neural processes at the neuromolecular level. The reductionist approach that started to gain more prominence in neurobiology in the 1950s and 1960s aimed at studying primarily known and simple behaviors, usually in their normal, or physiological, state: \" _normal_ mentation: of perception, motor coordination, feeling, thought, and memory\" (Kandel 1982, 299). It seemed that, in order to account for abnormal mental states, one first needed something like a map of the so-called normal nervous system: one had to proceed \"one cell at a time\" (Kandel 2006, 55). One needed to reduce the complex phenomenon to a simple or basic process. One needed to chart the normal neuronal circuitry involved in simple and normal behaviors; inducing abnormalities was indeed important experimentally, but only in order to assess their implications for the understanding of normal patterns and functions.\n\nNonetheless, the burgeoning neurosciences were full of hopes and also claims about their social relevance. Thus Schmitt, writing on the \"promising trends in neuroscience\" in 1970 suggested that it has \"relevance to social needs\" (Schmitt 1970, 1008): research in animals on hormonal reaction to psychosocial stress may have relevance to ghetto disturbances; basic and clinical research in neuroscience may help address the drug menace to society evidenced by \"extensive use of various psychotomimetic substances by adolescents and young adults and the widespread addiction in suburbia to 'energizing' and tranquillizing drugs\"; work with brain stimulation may help investigate \"the neural, hormonal, biochemical and genetic mechanisms of aggressive and other aberrant behavior,\" and so forth. Such speculations, however, remained gestural; Schmitt concluded, \"Perhaps still more important to man's progress is the possibility of a better understanding of _basic_ mental processes which, in the end, is responsible for the advancement of science and society\" (Schmitt 1970, 1008; emphasis added).\n\nNonetheless, the neuromolecular vision of the brain that was taking shape blurred two historically significant boundaries, and in so doing, paved the way for neuroscience to leave the lab for the world outside. _First_ , what one might term the Cartesian boundary, so crucial to psychiatry since its beginnings in the mid-nineteenth century, between organic and functional disorders\u2014the former being regarded as originating in identifiable lesions in the brain, the latter being considered to be a disturbance of mental functioning in which the symptoms have no known or detectable organic basis, and whose origin is thought to be in life history, biography, stress, or some other experience. In the neuromolecular style of thought, all mental disorders and disturbances must, in principle, and in some potentially identifiable way be related to anomalies within the brain. _Second_ , the distinction between 'states' and 'traits,' which had grounded the division between psychiatry and psychology. States were in-termittent periods of illness\u2014a person had been well, became depressed, and then, as a result of treatment or merely of time, returned to more or less his or her normal state of mind. Traits were pervasive features of mind or character, attributes one was born with, perhaps that one inherited, aspects of one's personality, such as, for example, that one was introverted or of a melancholy disposition. On this distinction between states and traits, a disciplinary stand-off between psychiatry and psychology was erected in the mid-twentieth century. Psychiatry would deal with abnormal states, for after all, these were illnesses, which could be treated and perhaps even cured, and psychiatry was a medical discipline. Psychology, on the other hand, would deal with traits, for these were aspects of character and personality, even if they were problematic or pathological: they could not be treated or cured, although they could be assessed and perhaps even managed better via forms of psychological therapy. But if both states and traits essentially were variations of the same molecular mechanisms, that distinction blurred, and along with it the distinction between personality disorders and psychiatric illnesses\u2014perhaps, even, the disciplinary divide between psychology and psychiatry when it comes to intervention.\n\nAnd indeed, it was in and through practices of intervention, and research aimed at generating interventions, that the conceptual architecture of the neuromolecular brain was established. We have already remarked on the crucial role of psychopharmacological research in identifying processes of neurotransmission and in explicating its elements\u2014as receptor sites, reuptake mechanisms, and the like\u2014all of which were first invoked as hypothetical entities to explain differences in action of various drugs on the brain, and then, through sophisticated experiments in vitro and in vivo, accorded an anatomical reality. The researchers, and the pharmaceutical companies who were funding much of the research, became convinced, not only that this was the pathway for the development of powerful, specific, and effective psychiatric drugs, but also that one could divide up the psychiatric nosology according to the pharmaceuticals that would act on each condition, and then develop and market the drug for each.\n\nIt became increasingly common, and acceptable, to seek to modulate troublesome mental states\u2014from the minor anxieties of family and work to the major disturbances of thoughts and feelings of schizophrenia and depression\u2014by drugs that, according to their manufacturers and marketers, acted precisely on specific sites in the brain (Rose 2003b, 2004). Beyond the walls of any asylum, when troubles arose in the family, the school, the workplace, the prison, the military, and in the management of daily life, a plausible response\u2014often the first response\u2014was to seek a drug that would mitigate the problems by acting on the brain. The management of everyday life through neurobiology was no longer associated with science fiction. This neuromolecular vision of the brain was fundamental to the rise of 'psychopharmacological societies' over the closing decades of the twentieth century.\n\nAs the rising rates of consumption of minor and major tranquilizers, antidepressants, and antipsychotic medications showed, many were happy to accept these designations, or were persuaded or coerced to consume the products of these styles of thought. But at the same time, a vocal movement began to take shape to contest them. The antipsychiatric movements of the 1960s and 1970s disputed the expansion of the categories of mental disorder to ways of thinking or acting that were at the least only unconventional, and at the most, were challenges to the existing social order. And they disputed the conviction that the use of psychiatric drugs should be understood as treatment of a disorder of the brain, rather than normalization of those who were socially challenging or disruptive. We return to these issues in chapter 4. However, whatever the fate of this sociopolitical challenge to the explanatory system and social practice of psychopharmacology, along with its founding mythologies of the neurobiological specificity of psychiatric classifications and the neurobiological specificities of the mode of action of drugs, the neuromolecular vision of the brain to which these were linked was to underpin research in the neurosciences for the next half century.\n\nEnter Plasticity\n\nIt is no surprise to find that the research problems addressed by developmental neurobiologists were increasingly shaped by evidence and arguments from this growing neuromolecular vision of the brain. Those developments are illustrated by the pathway of research from the 1950s onward, which eventually led to the discovery of Nerve Growth Factor by Rita Levi-Montalcini and the identification of its molecular mechanisms in the 1970s (Levi-Montalcini 1982). Toward the end of the account of this journey that she published in 1982, Levi-Montalcini asked a key question: \"Is the formation of neuronal circuits in the central nervous system and the establishment of specific connections between nerve fibers and peripheral end organs rigidly programmed and unmodifiable, or are nerve fibers endowed with sufficient plasticity to allow for deviation from predetermined routes, in response to chemical signals issued during neurogenesis and regeneration from neuronal and non-neuronal cells?\" (ibid., 356). The growing attention to these questions of development, neurogenesis, and plasticity was to establish another key condition for the movement of neuroscience from the laboratory to the world. For it was linked to the growing belief that, when it comes to the _human_ brain at least, neither structure nor function were fixed at birth, or inscribed in the genes. The neural architecture of the brain, it seemed, had to be located in the element of temporality\u2014not just the time of development from fertilization to birth and into the early years of life, but also the time of the life-course, through adolescence, into young adulthood, and indeed across the decades.\n\nBy the close of the twentieth century, the brain had come to be envisaged as mutable across the whole of life, open to environmental influences, damaged by insults, and nourished and even reshaped by stimulation\u2014in a word _plastic_. An ISI Web of Knowledge search reveals a steadily increasing rate of journal articles on this theme throughout the 1990s and 2000s. And in the first decade of the present century, more than a dozen semipopular books on this theme were published, mostly stories of hope invested in the power of the brain to change across a lifetime (for example, Doidge 2008; Begley 2009; Arden 2010). While the details of the history of the idea of neuroplasticity are disputed by specialists, a certain version is now entering common sense. The Wikipedia entry in December 2010 embodies this common sense when it asserts that neuroplasticity is \"the ability of the human brain to change as a result of one's experience,\" that the brain is \"plastic\" and \"malleable,\" and that the \"discovery of this feature of the brain is rather modern; the previous belief amongst scientists was that the brain does not change after the critical period of infancy.\"\n\nThe belief that up until about 1970, the human brain was viewed as immutable after the period of its development in utero and in early childhood has become something of a modern myth, though no less important for that. It is true that most researchers followed Ram\u00f3n y Cajal's dogma from 1928 that: \"[o]nce development has ended, the fonts of growth and regeneration of the axons and dendrites dried up irrevocably. In adult centers, the nerve paths are something fixed and immutable: everything may die, nothing may be regenerated\" (Ram\u00f3n y Cajal 1928, quoted from Rubin 2009). Other researchers demonstrated the importance of 'critical periods' in development of various sensory abilities such as vision, hearing, language, and certain other capacities in humans. However, there was general acceptance that synaptic connections among neurons were constantly being created and pruned across the life of the organism, as a result of neural activity arising from experience\u2014even Cajal, it seems, used the word _plasticity_ in this context (Berlucchi and Buchtel 2009). This premise was most clearly stated in Hebb's famous postulate from 1949 that \"any two cells or systems of cells that are repeatedly active at the same time will tend to become 'associated,' so that activity in one facilitates activity in the other\" (Hebb 1949, 70)\u2014usually glossed as 'what fires together, wires together.'\n\nA series of developments from the 1960s onward established neuroplasticity as a premise for those interested in the implications of neuroscience for therapy and policy\u2014and a premise entirely congruent with the hopeful ethos that would imbue the life sciences over the closing decades of the twentieth century. It had long been known from neurological studies of patients with brain damage that the brain could somehow recover some of the functionality lost as a result of stroke or injury that had caused a lesion in a key area, suggesting that other areas of the adult brain were capable of taking over those lost functions. But Paul Bach-y-Rita's famous work on the rehabilitation of patients with brain injury would firmly establish this as the underpinning of therapy in such cases (Bach-y-Rita 1967). Other research carried out on animals, notably monkeys, strengthened this belief in plasticity.\n\nIn their popular account in _The Mind and the Brain: Neuroplasticity and the Power of Mental Force_ , Jeffrey Schwartz and Sharon Begley suggest that one can trace the discovery of neuroplasticity to the Silver Spring monkey experiments in the 1970s (Schwartz and Begley 2002). In these experiments, carried out by at the Institute of Behavioral Research in Silver Spring, Maryland, Edward Taub cut the afferent nerves that carried sensations from the limbs of the monkeys to their brains and then explored the ways in which they might be trained to use those limbs despite the supposed absence of sensations. An undercover animal rights activist reported Taub's work to police, the monkeys were seized by federal agents, and a series of trials ensued in which the primary investigator, Edward Taub, was accused of animal cruelty and initially convicted, although that verdict was overturned on appeal. Following these trials, the monkeys were eventually killed, and on dissection it appeared that significant cortical remapping had occurred; publications in the 1990s took this as evidence of the plasticity of the somato-sensory cortex in adult primates despite long-term deprivation of sensation (Jones and Pons 1998).\n\nTaub eventually took a position at the University of Alabama and developed his findings into a program called constraint-induced movement therapy, or more simply Taub therapy, that \"empowers people to improve the use of their limbs, no matter how long ago their stroke or traumatic brain injury.\" By the time of the publication of the research into the brains of these monkeys, there was other evidence that showed that the mapping of sensory functions such as vision onto the cortex could be redrawn even into adulthood (Wall, Kaas, Sur, et al. 1986; Merzenich, Recanzone, Jenkins, et al. 1988; Jenkins, Merzenich, Ochs, et al. 1990). Like Taub, Merzenich was also to go on to found a number of companies seeking to apply such findings therapeutically to humans\u2014to turn them into a technology. These include Scientific Learning, which uses his Fast for Word software, and Posit Science which seeks to develop behavioral therapies, including brain training software called Cortex and Insight. Plasticity was to become one of the key dimensions of the matrix that linked the laboratory, the corporation, and the everyday world.\n\nA further dimension was added by the work of Elizabeth Gould and her group, which seemed directly to contradict Cajal's 'dogma.' Early claims to have discovered the formation of new neurons in the brains of birds and some other species had been discounted: leading authorities, notably Pasko Rakic, expressed the view that neurogenesis in the adult brain of primates could not occur, as the generation of new neurons would interfere with the neuronal storage of learning (Rakic 1985). But in the 1990s, using newly developed DNA labeling techniques, Gould and her colleagues were able to identify neurogenesis in adult mammals, first in adult rodents, where it was enhanced by training on learning tasks (Gould, Tanapat, Hastings, et al. 1999), and later in primates, where it was modulated by hormones and by stress (Gould and McEwen 1993; Cameron and Gould 1994; Cameron, McEwen, and Gould 1995; Gould, Reeves, Graziano, et al. 1999).\n\nThere remained many doubts about the functional implications of such newly generated neurons, let alone the extent to which such properties might be harnessed therapeutically. But by 2004, Gould herself was drawing social and policy implications, arguing that early adverse experience inhibits structural plasticity in responses to stress in adulthood, as did other factors such as social isolation (Mirescu, Peters, and Gould 2004; Stranahan, Khalil, and Gould 2006). By the end of the decade, she and her colleagues drew on evidence from neurogenesis to emphasize the need to understand the impacts of parenthood on the developing brain of the child to avoid potentially serious consequences for cognition and mental health (Leuner, Glasper, and Gould 2010). The idea of neurogenesis, and that external inputs might stimulate or inhibit it, was rapidly seized upon in sociopolitical debates. For some progressive thinkers, it supported what they had believed all along\u2014that the environment\u2014by which they seemed to mean that ill-defined social domain generated by political action\u2014was a, perhaps _the_ , crucial determinant of human abilities. For others, for whom the environment was thought of, principally, as the family, it seemed to confirm that pathological consequences flowed from poor family environments, and hence that intensive family intervention was the key to addressing social problems (Allen and Duncan-Smith 2008).\n\nThese arguments for early intervention in the name of the brain drew support from another body of work that framed plasticity in terms of epigenetics. Michael Meaney and his group had been carrying out research in rodents from the 1980s on the effects of early experiences of maternal care on the developing brains of offspring. In the first decade of the twenty-first century, in a number of very widely cited papers, they came to understand these effects in terms of epigenetic programming, arguing that the mother's behavior toward her pup shapes the expression of genes in its brain through altering methylation, and that this shapes neuronal development, which in turn has consequences for that pup's later behavior toward its own offspring\u2014hence providing a mechanism whereby these changes could be passed down the generations (Meaney and Stewart 1979; Meaney, Aitken, Bodnoff, et al. 1985; Champagne, Chretien, Stevenson, et al. 2004; Champagne and Meaney 2006; Szyf, McGowan, and Meaney 2008).\n\nBy 2009, Meaney and his colleagues were suggesting that these findings could be extended to humans, for example, suicide victims with a history of child abuse (McGowan, Sasaki, D'Alessio, et al. 2009; Meaney and Ferguson-Smith 2010). The brain now seemed open to environmental inputs, not just at the level of the synapse, at the level of cortical mapping, or at the level of the neuron and neurogenesis, but at the level of the molecular processes of the genome with consequences that might pass from parents to children and even on to _their_ children. The developing brain of the child now seemed to be a key site through which a range of social problems could be understood; once mapped onto the brain, paradoxically, they became more, not less amenable to intervention\u2014governing through, and in the name of the plastic brain.\n\nA Neuromolecular and Plastic Brain\n\nThe processes and structures of the brain and central nervous system were now understandable as material processes of interaction among molecules in nerve fibers and the synapses between them, in terms of the chemical and electrical properties of their constituent parts. While the explanatory gap still remained, and the move from the molecular level to mental processes of volition, emotion, and cognition was recognized to be highly challenging, the dualism that had haunted philosophy and the sciences of mental life seemed increasingly anachronistic. In this profoundly reductionist approach, despite the recognition that there was much that could not yet be explained, there seemed nothing about the operations of the nervous system that could not potentially be explained in terms of the biophysical properties of its component parts.\n\nBut the growing acceptance of plasticity acted as something of a coun-terweight to such reductionism. Ideas about plasticity and the openness of brains to environmental influences, from initial evidence about nerve development, through the recognition that synaptic plasticity was the very basis of learning and memory, to evidence about the influence of environment on gene expression and the persistence throughout life of the capacity to make new neurons\u2014all this made this neuromolecular brain seem exquisitely open to its milieu, with changes at this molecular level occurring throughout the course of a human life and thus shaping the growth, organization, and regeneration of neurons and neuronal circuits at time scales from the millisecond to the decade. This was an opportunity to explore the myriad ways in which the milieu got 'under the skin,' implying an openness of these molecular processes of the brain to biography, sociality, and culture, and hence perhaps even to history and politics. It could act as an antidote to suggestions of unidirectional causality. For if that openness was a condition of normal development, perhaps abnormal development might be understood in terms of such influences as well, and perhaps, by modulating those influences, one could indeed open the brain\u2014and the person\u2014up to calculated intervention at the neural level.\n\nHence the plastic brain itself became thinkable as open to government by experts. It simultaneously became open to action by each individual themselves. Dozens of Internet sites were developed to instruct us how to harness brain plasticity for our personal growth\u2014informed by books such as that by Norman Doidge\u2014for \"your brain constantly alters its structure according to your thoughts and actions has enormous implications for your personal growth, particularly in the areas of behavior change.\" Correctly informed about plasticity, we could rewire our brains for love, rewire corporate brains, and so much more (Zohar 1997; Begley 2007; Lucas 2012). The plastic brain becomes a site of choice, prudence, and responsibility for each individual.\nChapter Two\n\nThe Visible Invisible\n\nThe biological basis of mental illness is now demonstrable: no one can reasonably watch the frenzied, localized activity in the brain of a person driven by some obsession, or see the dull glow of a depressed brain, and still doubt that these are physical conditions rather than some ineffable sickness of the soul. Similarly, it is now possible to locate and observe the mechanics of rage, violence and misperception, and even to detect the physical signs of complex qualities of mind like kindness, humour, heartlessness, gregariousness, altruism, mother-love and self-awareness.\n\n\u2014Rita Carter, _Mapping the Mind_ , 1998\n\nNeuroscience, and the study of the activity of the brain, is beginning to bring its own illumination to our understanding of how art works, and what it is. I have come to see the delight in making connections\u2014of which metaphor-making is one of the most intense\u2014as perhaps the fundamental reason for art and its pleasures. Philip Davis, at Liverpool University, has been working with scientists on responses to Shakespeare's syntax, and has found that the connecting links between neurons stay \"live\"\u2014lit up for longer\u2014after responding to Shakespeare's words, especially his novel formations of verbs from nouns, than they do in the case of \"ordinary\" sentences.\n\n\u2014A. S. Byatt, \"Novel Thoughts,\" 2007\n\nIn mid-February 2007, a press release generated considerable interest in the popular media. Titled \"The Brain Scan That Can Read People's Intentions,\" it reported the work of an international team led by John-Dylan Haynes at the Max Planck Institute for Human Cognitive and Brain Sciences in Germany. The team had conducted some laboratory experiments imaging the brains of subjects who were confronted with a simple task in which they had to decide between a small number of possible choices. The scans showed activity in particular brain regions _before_ the action was undertaken; it seemed that one could predict the later choice from the earlier pattern of activation. On this basis, Haynes sketched out a mind-reading scenario in which such scans could reveal a person's intentions in advance of actions, and he called for an ethical debate about the implications. Neuroethicists pitched in, expressing their worries to the journalists, happy to take this prospect seriously. Of course, this scenario was a fantasy; the highly artificial and simplistic laboratory experiment told us nothing about the formation of actual complex human intentions to undertake meaningful actions in the messy real world. But it was premised on a belief that is coming to acquire the status of truth: that experiments scanning the brains of individuals in laboratory settings to identify patterns of activation while they undertake simple tasks or games can reveal the workings of the human mind in real time\u2014and, moreover, that they can show exactly which regions of the brain are involved in exactly what specific mental states, not just emotion and cognition but also intentionality, volition, and more. It seems that these new technologies of visualization have finally and objectively revealed the physical basis of human mental life in patterns of activity in the living brain.\n\nIt would be misleading to claim that, by the end of the twentieth century, brain imagers believed that the age-old dilemmas concerning the relations of mind and brain had been resolved. Most neuroscientists are careful to speak of 'the neural correlates of mental processes,' avoiding the language of causes, which suggests that the neural is prior to and determinative of such states, and the language of identity, which suggests that mental states are simply neural states. Their press releases and media accounts are less cautious. And the images produced by brain scans are certainly central to the new ways in which this issue of mind and brain is being posed, not just in the laboratory, or in psychiatry, but in the popular media, in the arts, and in many other areas of human life. It is not simply that many now believe it to be possible to locate each faculty of the human mind\u2014emotion, fear, self-control\u2014in a specific region of the brain; it is also the belief that one might, in principle, use this new knowledge of the brain to target interventions into the mind on these faculties in practices from the therapeutic to the artistic and the commercial.\n\nWe have many good studies of the intertwined social, technological, and conceptual history of medical imaging in general (Kevles 1997; Burri and Dumit 2007; Saunders 2008), and of brain imaging in particular (Dumit 1997, 2003; Beaulieu 2000a, 2000b, 2002, 2004; Roepstorff 2001, 2002, 2007). While we draw upon these studies in this chapter, our focus is quite specific\u2014it is on the diverse attempts to render 'mind' thinkable by means of images. Without pretense to a genealogy of our contemporary forms of visualization, we explore some of the technologies that have been invented and the kinds of information that they seem to provide. We examine the modes of interpretation and arguments that have been developed around those technologies and their products. And we sketch out the new ways of thinking about and articulating the relations of the corporeal, the mental, and the psychopathological that these entail and support. This visual imaginary has been one pathway along which neuroscience has been able to move out of the laboratory and into the territory of everyday life, and to play a role in the management of normal and problematic conduct.\n\nThe Clinical Gaze\n\nWhen we speak of rendering the mind visible, we do not just mean the act of seeing. We mean the whole configuration by means of which a certain way of seeing becomes possible and can be articulated. The ways of seeing that characterize a scientific practice, or a mode of intervention, do not arise from an isolated act of looking, but from a conjunction of different elements and practices. One dimension is _spatial and temporal_. This distribution occurs along a number of planes: across the individual body (localization), across the collective body (social factors in causation and recovery), across time (the life-course and the course of illness), across generations (family pathology, heredity). A second dimension of this conjunction is _technical_ or perhaps _technological_. It consists in the means, the apparatus and devices, that render that which is observed into marks, lines, colors, spaces and edges, patterns and patterning. As many recent studies from the social and cultural sciences have demonstrated, the images produced by the electroencephalographs, the CT scanners, the magnetic resonance imaging machines, and all the other technical devices of contemporary brain mapping are no more 'true to nature' than those produced by the other \"engines of visualization,\" such as photographs, so tellingly analyzed by Patrick Maynard (Maynard 1997). But it is only through such artifacts that we have become able to see that which is not present to our un-augmented eyes.\n\nA third dimension of this rendering visible, perhaps the most significant, consists in the practices within which acts of seeing are enmeshed. They are always bound up with specific ways of _practicing_ , occurring within specific sites and by means of particular techniques of intervention upon those who are the subjects of that act of seeing. When it comes to seeing the brain, seeking to discover within its fleshy volume the traces of the pathological or normal mental processes that the brain might embody, it involves the designation of those who have the authority to see: doctors, neurologists, researchers, psychopharmacologists, geneticists, and now, of course, the imagers. It also involves the subjectification of those who are spoken about\u2014subjectification in the sense that living creatures become subjects of these visualizing technologies only as a consequence of certain technical interventions, and subjectification in another sense, in the case of humans, whose sense of themselves may well be transformed as a result of the images of their brains with which they are presented. The act of seeing, or the practice of seeing, also involves a particular locale for the act of visualization\u2014the designation of a case in the asylum, the 'demonstration' of the patient in the clinic, the dissection of the brain in the laboratory, the scanning of the subject in the imaging suite. A real space is required that is all too often absent when the image produced in the scanner is being interpreted. To render visible, that is to say, requires conditions of possibility within a larger networked, distributed, assembled field of intensities and powers\u2014connecting up such diverse sites as the clinic, the lab, the pharmaceutical company, popular literature, and the mass media. Seeing, rendering visible, is thus part epistemology, part topography, part technology, part objectification and subjectification, part network of forces, even part ethics.\n\nOur argument in this chapter is intended, in part, as a homage to Michel Foucault and to his 1963 book, _Naissance de la clinique_ , translated into English under the title _The Birth of the Clinic: An Archaeology of Medical Perception_ (Foucault 1973). Alan Sheridan, the translator, remarks that the word _gaze_ would have been a better choice in the subtitle than _perception_ \u2014the book is about the formation of a particular clinical gaze\u2014a way of seeing, saying, and doing in relation to illness, the body, life itself. It charts a fundamental shift in the medical gaze, which occurred in Europe in the early nineteenth century. Previously medicine had focused on a two-dimensional space of tissues and symptoms, but from that time forth, the gaze of the doctor would plunge into the interior of the body. To diagnose an illness would be to interpret symptoms in terms of the internal organic malfunctions that were their cause. In life, the gaze of the doctor would now need to be augmented, initially by such devices as the stethoscope. In death, that which had been rendered into thought while life persisted would be confirmed\u2014or disconfirmed\u2014by the dissection of the corpse. This new way of seeing was inscribed into the anatomical textbooks and embodied in a multitude of anatomical models, educating the eye of the doctor by progressively revealing the layers of skin, muscles, blood vessels, nerves, and organs that made up the interiority of the living being. It was the body itself that had become ill. Through a knowledge of how that illness appeared in the sick body, it became possible to understand what the absence of illness might look like\u2014to envisage the functioning of those interior processes of the body to produce the state we took to be health. To know normality, it was necessary to see pathology. But if this was the case for the body, what then for the afflictions of the mind?\n\nInscribed on the Body Itself\n\nSander Gilman is the preeminent historian of ways of \"seeing the insane\" (Gilman 1982). Gilman tells us that, from the sixteenth century onward, to speak of madness was also to visualize it. Images of madness were central to the ways in which the mad person was turned into a proper object for rational knowledge and for the practice of a cure. Madness was represented iconically through the association of the mad person with images such as that of the divided stick or the pinwheel. It was represented in the complexion of the mad person, where the black bile that was at the root of the condition was manifested in the dark hue of complexion, and this also linked madness to the blackness of skin that was consequent upon the soul falling from grace. It embraced the posture, gesture, and movements of the figure. The melancholic was seated, eyes cast down, clothing disheveled, hands clenched or hidden to manifest a turning away from useful labor. The maniac raved in semi-nakedness, arms akimbo, and mouth agape. The epileptic and the ecstatic shared a posture: arms and legs flailing, body in the fixed curve known as the _arc de cercle_ that would become, for Jean-Martin Charcot at the Salp\u00eatri\u00e8re, the emblem of the hysterical crisis.\n\nFrom the mid-eighteenth century onward, from the essay on the passions in Diderot and d'Alembert's _Encyclop\u00e9die_ through Johann Lavater's _Physiognomische Fragmente_ (1774\u201378), madness became associated with a specific physiognomy, initially an emphasis on the fixed aspects of skull shape and facial appearance, which could be derived from particular techniques of inscription of the head on a homogeneous and regular plane of two dimensions. Some disputed the claim that there was a direct relation between psychopathological states and these fixed physical characteristics. They argued that other visible features\u2014expression, gesture, posture\u2014were the real external signs of the inner pathology. But in either case, few doubted that madness was inscribed on the surface of the individual, that, as Arthur Schopenhauer was to put it in his work on physiognomy published in the mid-nineteenth century, \"the outer man is a graphic reproduction of the inner and the face the expression and revelation of his whole nature\" (quoted in Gilman 1982, 164).\n\nEvery student of psychiatry is told the story of Philippe Pinel, acclaimed as the radical physician of the French Revolution who publicly struck the chains from the inmates at Bic\u00eatre in Paris in October 1793, restoring the mad person to the status of an individual citizen with rights, and inaugurating 'moral treatment.' The enlightened asylums of the early nineteenth century also flooded the inmates with a new light, a light that that made them individually visible in new ways. Pinel himself introduced illustrations of the insane in his _Trait\u00e9 m\u00e9dico-philosphique sur l'ali\u00e9nation mentale, ou la manie_ (Pinel 1801). These compared the physiognomy and skulls of \"an idiot\" and \"a maniac\" in a style familiar to Lavater and the craniologists, with a special emphasis on the proportionality of the parts and the relation of head size to body size: these drawings of particular individuals seemed to show that for each case, appearance was the key to diagnosis. In the ensuing years, Pinel's pupil Jean Esquirol documented the newly visible varieties of insanity in his _Dictionnaire des sciences m\u00e9dicales_ (1812\u201322). He commissioned some two hundred drawings of the insane at the Salp\u00eatri\u00e8re, focusing both on facial expression and on bodily comportment, each of which was used to illustrate the varieties of insanity that he described. When he published his collected papers in 1830, he selected twenty-seven of these illustrations and appended them in the form of an _Atlas_ (Esquirol 1838).\n\nThe use of the word _atlas_ in such contexts is conventionally traced to Gerard Mercator, who adopted it \"to honour the Titan, Atlas, King of Mauritania, a learned philosopher, mathematician, and astronomer.\" It appears first to have been used posthumously on the title page of a collection of his maps published by his son in 1595: the illustration of Atlas with the globe of the earth in his hands as the frontispiece was added to later editions (Hall and Brevoort 1878). The atlas format was taken up by William Playfair, \"who, in 1786 published a collection of forty-four diagrams and charts, the content of which was in the nature of political economy, with the title _The Commercial and Political Atlas_ : he \"founded a tradition... in which collections of charts and figures appertaining to disciplines like biology, medicine, physics as well as statistical overviews were also put into the category 'atlas' \" (Krausse 1998, 11\u201312). The lines, curves, and shapes in the diagrams that Playfair produced, Krausse suggests, enable the eye to reckon without calculating\u2014to commit information to memory in seconds that would take days to assimilate if it were in the form of words or numbers.\n\nThis was equally true of the lines and curves of Esquirol's drawings. They certainly recycled some of the old themes in the positioning of hands and body. They also followed the physiognomies in their ways of rendering the shape and proportions of the skull. But these plates did something more. They were part of a fundamental individualization that rendered madness into a possible object for a clinical medicine. The laborious procedures of observation and documentation that accompanied moral treatment visualized each insane person as an individual case, but one whose uniqueness was intelligible because it could be charted and measured in terms of the general norms of types of madness, their characteristics, etiology, and prognosis. Esquirol gave a detailed description of the visual appearance characteristic of each type of madness, accompanied by an account of a specific case\u2014an individual diagnosed with mania, dementia, or lypemania\u2014focusing not just on the course of that person's condition and its symptoms in speech, behavior, posture, and comportment, but also his or her visual appearance\u2014the color of the hair, the shape of the eyebrows, the complexion of the skin, the look of the face, and much more. In this way, the image was fused to the biography in the form of the case, and inscribed at the heart of psychiatric epistemology and diagnostic practice. As Sander Gilman argues, such combinations of description, case study, and picture bridge the theoretical and the observable (Gilman 1982). The patient is rendered into thought in terms of the theory; the means of diagnosis have merged with the object of diagnosis itself. In these images of the mad person, the norms of psychiatric thought seem to have merged with the individual subject of psychiatric practice in the portrayal of each specific case.\n\nEsquirol was not the first to use portrayals of the inmates of asylums to document the varieties of madness through appearance. As Gilman shows us, Charles Bell had done so at the very start of the nineteenth century in his vividly illustrated _Essays on the Anatomy of Expression in Painting_ (Bell 1806) focusing in particular on the fear and terror of the mad, and Alexander Morrison, having visited Esquirol in Paris, published his own series of plates in his _Outline of Lectures on Mental Diseases_ of 1825 (Albin 2002), which rapidly went into two further editions and led to his later _Physiognomy of Mental Diseases_ (Sani, Jobe, Smith, et al. 2007). And the idea that madness could be captured in the image of the mad person was pervasive well beyond the asylum. From the drawings of William Blake and Thomas Rowlinson to those of Goya and Delacroix, whatever their stylistic differences, to see madness in the face, in the posture, in the comportment of the body, was to know it for what it was.\n\nClinical medicine in the nineteenth century mutated fundamentally when the gaze of the doctor focused beneath the skin, into the interior of the patient's body. But across the second half of the nineteenth century, the gaze of the mad doctors remained stubbornly on the surface; the images played the same epistemological role when they moved from drawings to still photographs and then to moving images. Whether in the photographs that Hugh Diamond presented to the Royal Society in 1856, in William Hood's collection of Henry Hering's photographs of the inmates of the Bethlem Hospital of which he was the first superintendent (Gale and Howard 2003; Logan 2008), in Dietrich George Kieser's _Elemente der Psychiatrik_ of 1855 (Kieser 1855), in Max Leidsdorf's _Lehrbuch der Psychischen Krankheiten_ of 1865 (Leidesdorf 1865), in the nosological archives created in the San Clemente Hospital in Venice in 1873, in the photographs taken at the Devon County Lunatic Asylum used by Bucknill and Tuke to illustrate their _Manual of Psychological Medicine_ (Bucknill and Tuke 1874), or in Henri Dagonet's _Nouveau trait\u00e9 \u00e9l\u00e9mentaire et practique des maladies mentales_ (Esman 1999), the photographs seemed to show, apparently without requiring the artifice of the portrait painter, that the invisible states of the human soul were visible in the proportions of the mouth, the hue of the complexion, the expression on the face, the posture of the body and the arrangement of the limbs (most of these examples are discussed in detail in Gilman [1982]). Even Darwin was initially tempted to use similar photographs\u2014supplied by James Crichton Browne, the medical director of the West Riding Asylum at Wakefield\u2014to support his theory of the innate bases for the expression of the emotions. But Darwin recognized the circularity of the procedures by which the images were produced: the unfortunate individuals who appeared in these photographs were clothed and posed in terms of prevailing beliefs as to how emotions were manifested in visual expression\u2014they could hardly serve as evidence for the inherited or universal basis of those expressions (Gilman 1982, 185).\n\nIn this continuous work of visual documentation of asylum inmates, we can learn a lot, not merely about the form of the gaze that the mad doctors directed to their patients, but also of the power of the visual in seeming to render the interior world of the human mind into thought\u2014its passions, affections, thoughts\u2014in the form of images. Such images were simultaneously to satisfy the requirements of knowledge (the image as both evidence and proof), of the clinic (the image as diagnostic), of pedagogy (for these pictures were teaching tools), of museumology (for the picture would enter the archive and the plethora of scientific collections gathered across the nineteenth century), of aesthetics (for there was a certain style of representing the interior world here that would go beyond the world of science), and of the popular imagination. Understanding, here, is played out, almost without the need for language, through the apparently empirical form of the image. Despite its objectivity effect, a whole scientific and cultural imaginary underpins these images of mental states\u2014this remains true today, despite the technological gulf between these images and those generated in the brain scanners of our own times.\n\n\"Behold the truth.... I am nothing more than a photographer; I inscribe what I see.\" With this quote from Jean-Martin Charcot, Georges Didi-Huberman opens the chapter on \"Legends of Photography\" in his provocative psychoanalytic study of _The Invention of Hysteria_ (Didi-Huberman 2003, 29). Charcot, who worked in the Salp\u00eatri\u00e8re from the 1850s to 1893, represented the apotheosis of this tradition of visualization of the troubled or deranged mind through the appearance and comportment of the body and the expressions of the face. In 1877, he founded the _Iconographie photographique de la Salp\u00eatri\u00e8re_. It was published until 1880 and resumed publication under his editorship in 1888, as the _Nouvelle iconographie_. In this form it was to publish twenty-eight volumes until its demise in 1918. It illustrated not only the faces and postures of the varieties of madness, but introduces time into these pictorial representations, for example, providing a series of pictures demonstrating the sequences of epileptic attacks, hysteria, and the like. Vision was the key to grasping the nature and progress of the condition: the troubles of the mind were now intelligible to the extent that, before all else, they were visible. Charcot is, of course, now infamous for the way these demonstrations were carefully staged, and for how they relied on extensive coaching, suggestion, and simulation: yet the photographs became their inscribed and durable counterparts. Charcot's most famous pupil, Freud, referred to him in his obituary as a \"visuel,\" one who sees (Freud [1893] 1962, 12). But Freud, and the psychoanalysis and dynamic psychiatries that followed, turned away from the image and paid scant attention to the visual appearance of the patient\u2014 _the voice_ was the royal road to the unconscious in all of the 'talking cures' that were to follow.\n\nThe chapters on cretinism, dementia praecox, general paresis, manic depression, and idiocy in Kraepelin's _Textbook on Psychiatry_ ([1899] 1902) all contained the carefully posed photographs familiar from earlier works, but these images were no longer accorded a particular diagnostic significance. In his _Lectures on Clinical Psychiatry_ (translated into English in 1904) Kraepelin describes the patient to his audience: the clothes, the demeanor, the facial expression of emotion. But these are important only for what they reveal of the everyday life of the individual, they are not clues to the nature of his or her condition. The significance of the image of the mad person as the outward and visible sign of an internal constitutional defect or malady was beginning to fade (Kraepelin 1904). Kraepelinian diagnosis was based on the course of the condition and on etiology and prognosis as manifested in conduct, speech, emotion, thought. Illustrations still play a role, but this is, precisely, to illustrate\u2014the pictures in Kraepelin's texts are presented neither as pedagogic nor as diagnostic. But nonetheless, in the closing decades of the nineteenth century, different kinds of image were beginning to become important\u2014images of the brain itself.\n\nOpen Up a Few Brains\n\nOf course, even nineteenth-century proponents of moral medicine like Esquirol thought madness was a disease of the brain. But it was in the work of Franz Joseph Gall that we can find the first modern articulation of the idea that the brain is a differentiated organ whose different regions are responsible for different aspects of human mental life. While the physiognomists sought to correlate the form of the face and the proportions of the skull with mental characteristics, Gall's view was that it was the cerebral cortex that was the key (English and Annand 2010; Gall [1822] 1835). The cortex was composed of distinct \"brain organs\"\u2014regions associated with different faculties\u2014whose distribution could be represented visually to show exactly where under the external skull each organ was located. The size of each brain organ was linked both to its power in the individual and to the configuration of the part of the skull above it. Thus it could be detected externally by examining the form of the skull and the shape and size of the various regions, not just in death, but also in life. Condemned for his materialism in Austria, Gall traveled around Europe, giving demonstrations of his skill at brain dissection, and collecting material\u2014brains and skulls\u2014especially from prisons and asylums.\n\nGall's craniological work had many disciples; under the name of phrenology, it was promoted in Europe by his former assistant, Johann Spurzheim, and by George Combe. It was institutionalized in phrenological societies and journals and was enthusiastically taken up by many in America. However, by the end of the 1830s, it had lost most of its credibility as a scientific analysis of mental functions, notably at the hands of Pierre Flourens, whose experimental work creating localized brain lesions in animals and documenting their consequences seemed to disprove the claim that there was a close relation between cerebral locales and specific behavioral capacities (Flourens 1824). Nonetheless, phrenology retained considerable popular appeal, especially through the visual technologies developed by Spurzheim and Combe\u2014the familiar charts, diagrams, and model phrenological heads that showed the different mental faculties distributed across the brain and mapped to specific locations on the external surface of the skull.\n\nGall's fate was to be derided by association with the role of phrenology in fortune telling and showmanship. But the concept of cerebral localization was to entrance European neurologists in the second half of the twentieth century. In 1861, Paul Broca seemed to have reestablished the truth of localization theory, when he described eight patients with loss of speech, each of whom had lesions in the third left frontal convolution (Broca 1861). He developed this thesis further, with arguments based on specific case studies, such as that of Leborgne (the famous \"Tan\"), claiming to have demonstrated the centrality of the frontal lobes in many higher functions, and he also developed the thesis of cerebral dominance, arguing for the distinct functions of the left and right cortex. Neurology seemed to have shown without doubt that the brain was an organ composed of regions with specific structural properties that were the basis of the various human mental functions.\n\nIn 1867, Wilhelm Griesinger, professor of psychiatry in Berlin from 1865 to 1868, founded the _Archive for Psychiatry and Nervous Diseases_ , and this journal provided a focus and a forum for a succession of German psychiatrists and students of brain structure. It also stimulated the development of psychiatric clinics in Germany attached to universities, which helped provide the necessary infrastructure for this organic gaze. In the 1870s and 1880s, Theodore Meynert in Vienna pored over his microscope scrutinizing stained sections of the brains from psychiatric patients in a search for pathological lesions in the cerebral cortex, the frontal lobes, and the nerve fibers themselves. More work on cerebral localization was carried out by Carl Wernicke, also in Vienna, who in 1874 identified the specific part of the brain that, if damaged, led to an incapacity to understand the spoken word or to speak comprehensibly. Wernicke spent the rest of his days trying\u2014and failing\u2014to discover similar links between psychiatric symptoms and brain abnormalities.\n\nPaul Flechsig worked in Leipzig from 1877 to 1922, seeking to localize the basis of many neurological disorders, and mapping the human brain into multiple areas according to the sequences in which they became myelinated. He used this as the basis for his speculations that particular areas in the cortex were most important for the exercise of intellectual functions, for the formation of mental images, the naming of objects, and so forth, and sought to confirm these by postmortem studies of the brains of patients with general paresis and other brain diseases. Flechsig had trained himself in psychiatry, visiting asylums in Germany and spending time with Charcot in Paris (Flechsig's work is usefully discussed in Finger 1994, 308\u201310). Daniel Paul Schreber, whose case was to be made famous by Sigmund Freud, spent time in his clinic, and Flechsig was the target of Schreber's allegations of \"soul murder\" (Schatzman 1973, 30\u201331). However limited his therapeutic achievements, Flechsig's cortical maps were highly influential visualizations of cerebral functions. He believed that what he termed \"the posterior association center\"\u2014rather than the frontal lobes\u2014was the part of the brain most important for intellectual functions, thus, like so many other neuroanatomists, linking the size of regions to intellectual capacity.\n\nWe should not underestimate the risky radicalism of many of the early researchers on cerebral localization: most held explicitly materialist philosophical positions, asserting that human mental capacities could be explained without reference to any metaphysics or religion, and without invoking an immaterial soul. We can see this clearly in the work of the Society for Mutual Autopsy studied by Jennifer Hecht (2003). This organization was established by a group of French anthropologists, psychologists, artists, and intellectuals in the 1870s, and was directly inspired by Paul Broca, who was one of the founding members, along with other neurologists and intellectuals. It was organized around specifically atheist values\u2014notably the belief that there was no soul, only brain. Those who established the society were troubled by the fact that the only brains available for dissection were those of the abnormal, the damaged, the insane\u2014not of the normal or the intellectuals. Membership was based on an agreement to examine each other's brains after death to see to what extent their eminence was inscribed within the brain itself, and what difference their lives may have made to the configuration of the cerebral fissures\u2014a relationship that proved to be problematic when some of the more eminent intellects proved to have small brain volumes, and in some cases, the convolutions of their brains were not fine, as had been hoped, but somewhat coarse.\n\nIf the brain was the seat of mental life, it seemed self-evident that there must be a relation between the intellect and the size and configuration of the brain, if the calculations were done correctly. Given the ways in which racial science had long linked the hierarchy of races to the configuration of face and skull, it is not surprising that race was one focus of these investigations of the brain. Samuel Morton was one of the first systematically to measure brain size by calculating the volume of cranial capacity (initially filling the skull with mustard seed, then with lead shot) and comparing average skull volume by race (Morton 1844). But Broca was probably the most rigorous of all craniometrists (this work is discussed in detail by Stephen Jay Gould [1981, esp. chap. 3]). Gould points to Broca's careful refinement of the methods of craniometry and his undoubted respect for empirical and statistical methods. Nonetheless, Gould argues, Broca systematically distorted his data, perhaps unwittingly, and applied all manner of dubious corrections in order to make his results conform with his preconceptions concerning the differences in relative size of brain regions\u2014for example frontal versus posterior cranial regions\u2014in \"inferior\" versus \"superior\" races, and between men and women. A few neurologists disputed the basic premise of a relation between brain size and intelligence, but similar endeavors occupied many members of the anthropometric societies that flourished in many European countries and in the United States (Chimonas, Frosch, and Rothman 2011).\n\nWhatever the rationale for many of these early explorations of the shape, size, and anatomy of the brain, and despite the repeated failure of attempts to link intellect to brain size, a vision of the brain as an internally differentiated organ was taking shape, in which specific regions were associated with speech, language and comprehension, and other mental faculties. For obvious reasons, experiments to explore these relations between region and function were mostly undertaken on animals and involved creating lesions in specific brain areas and studying the consequences. The names of the researchers are known to all modern students of neuroanatomy. Fritsch and Hitzig's experiments on dogs in the 1870s seemed to identify a localized motor area in the cortex (republished as Fritsch and Hitzig 2009). David Ferrier's subsequent experiments on a range of animals, notably his work with monkeys, seemed to support the existence of distinct cortical areas for sensory and motor functions: the results were presented in the late 1870s and in the 1880s culminated in successive editions of _The Functions of the Brain_ , replete with maps of brain areas associated with different function (Ferrier 1876, 1886). Others followed a similar route.\n\nThe architecture of the brain was also being opened to visualization in another way, thanks to a different technique\u2014that of staining. The nerve cell was first described in the form we know it today by Otto Deiters in 1865, leading to much dispute as to whether such cells formed a continuous network or whether the cells were distinct and contiguous. Camillo Golgi, who invented his method of silver staining in the 1870s, was an ardent supporter of the first hypothesis; Santiago Ram\u00f3n y Cajal used Golgi's method of staining to support the second. Jean-Pierre Changeux quotes the account Cajal gave in 1909 of what he saw when he looked at a section of nervous tissue that had been left around for a few days in M\u00fcller's fluid and then immersed in silver nitrate: \"Sections were made, dehydrated, illuminated, and observed. An unexpected spectacle!... Everything looked simple, clear, unconfused. Nothing remained but to interpret. One merely had to look and note\" (Changeux 1997, 26).\n\nLook and note: vision, once more, seems to reveal the truth without interpretation. Done correctly, everything looked clear. From 1903 to 1908, Korbinian Brodmann, working with Oscar Vogt in the Neurobiological Laboratory of Berlin, used another staining technique, that developed by Franz Nissl, for his work on neuroanatomy. He published a series of papers in the _Journal f\u00fcr Psychologie und Neurologie_ , setting out the results of his comparative studies of the mammalian cortex; these became the basis of the maps of the cerebral cortex of humans, monkeys, and other mammals contained in _Vergleichende Lokalisationslehre der Grosshirnrinde in ihren Prinzipien dargestellt auf Grund_ (Rheinberger 2000). These maps delineated the so-called Brodmann areas of the cortex, which, in the words of Laurence Garey, \"must be among the most commonly reproduced figures in neurobiological publishing\" (in his introduction to J. Smith and Berlin 1999). The brain now appeared as a differentiated organ, with an anatomy that was consistent within species and comparable between species, and which could be localized and classified according to a standardized nomenclature agreed upon across laboratories and researchers.\n\nFerrier's experiments had been carried out at the West Riding Lunatic Asylum, with the support of the asylum director, James Crichton-Browne; in 1878, Ferrier and Crichton-Browne together with John C. Bucknill\u2014who had founded and edited the _Journal of Mental Science_ in 1853\u2014founded _Brain: A Journal of Neurology_. However, despite this close association of neuroanatomy with the asylum, and despite dissecting innumerable brains of deceased asylum inmates, nineteenth-century neurologists were unable to identify any visible abnormalities in the brain that correlated with abnormalities of thought, conduct, mood, or will. Charcot had carried out autopsies of his patients at the Salp\u00eatri\u00e8re, but was unable to find lesions that correlated with mental disturbances, except those that involved physical disabilities such as the scleroses. Freud had himself carried out neurological studies of the brain in the 1880s and was apparently a supporter of the theory of a continuous network of nerves. But while he remained committed in principle to the ultimate neurological basis of the processes that he hypothesized, these attempts to visualize psychiatric disorders in terms of lesions in the brains of dead patients, extracted, sliced, stained, and magnified, were derided by psychoanalysts in the early twentieth century. And it is true that they had little to offer psychiatrists. As Nissl concluded in 1908: \"It was a bad mistake not to realize that the findings of brain anatomy bore no relationship to psychiatric findings, unless the relationships between brain anatomy and brain function were first clarified, and they certainly have not been up to the present\" (quoted in Shorter 1997, 109).\n\nThe mapping of the functional architecture of the brain became ever more developed throughout the twentieth century (Frackowiak 2004). Yet even if anatomical variations between brains did have the functions attributed to them, they seemed to have no therapeutic implications. The study of the potential links between mental disorder and the anomalous brain did not seem able to generate any clinical interventions, and was only possible after the patient had died. How could one make this neuronal anatomy of the organ of the mind amenable to clinical thought and practice, let alone to those other practices concerned with the mental states and conduct of the living human being?\n\nSeeing the Living Brain\n\nCould one see the activity of the living human brain? Could the gaze of the neurologist actually plunge into the interior of that organ in life, rather than observing it dead, extracted, sliced, stained, and preserved, and thus inferring functions from accidentally produced lesions or experiments with animals? Central to the conquest of the living brain for knowledge in the twentieth century was the invention, assembly, refinement, and commercialization of a whole variety of devices that could \"prosthetically augment... the range of the scientific observer's vision, allowing the perception of deep, minute, mobile and generally imperceptible structures\" (L. Cartwright 1995, 83) within the skull of a living person. In the process, our very understanding of what it is to be human being\u2014that is to say, an individual with a brain that makes mental life possible, has been irreversibly transformed.\n\nThe first conquest of the living brain by vision was to reveal its structure. At the turn of the nineteenth and twentieth centuries, X-rays transformed the way in which the interior of the body could be seen, and indeed transformed the very idea of what it was to have such a body. Bettyann Holzmann Kevles quotes an early enthusiast describing the experience as \"the mind walk[ing] in among the tissues themselves\" and from that time on, as she points out, much intellectual effort was spent visualizing what was once invisible: that which was once hidden was now amenable to a new kind of knowledge (Kevles 1997, 2). But when it came to the brain, while a great deal was known from brain dissections postmortem, the opacity of the skull made it impossible to produce X-ray pictures of the living brain. This was a challenge for surgeons seeking to understand symptoms that appeared to originate in damage to the brain from accidents, blood clots, or tumors.\n\nWhat was needed, it seemed, was some kind of contrast agent that would show up the configuration of brain tissue in relation to the surrounding fluid. Some surgeons had already achieved results in a few cases where air had somehow entered the space occupied by the cerebrospinal fluid. These findings were drawn upon in 1918 by an American neurosurgeon, Walter Dandy, who developed a method that involved draining some of the cerebrospinal fluid via a lumbar puncture while simultaneously injecting an equivalent volume of air: the air found its way into the skull, filling the ventriculi, and subsequent X-rays revealed the position and size of tumors or blood clots. This rather painful technique, known as pneumoencephalography, enabled at least the gross features of the cerebral anatomy of living patients to be visualized, and hence surgeons were able to confirm the origins in the brain of symptoms such as extreme headaches or strange bodily sensations (Kevles 1997, 97\u2013103).\n\nEgas Moniz, who had returned to his original career in neurology after a successful spell in Portuguese politics, notably as ambassador to Madrid and minister of foreign affairs, is now famous or infamous for his development of the frontal lobotomy in the 1930s, for which he received the Nobel Prize in Physiology or Medicine in 1949. However, prior to that, he was already well known for inventing the technique of cerebral angiography, in which a contrast agent\u2014initially sodium iodide and later thorium dioxide\u2014enabled the distribution of blood and other fluids in the brain to be captured on film when the head was X-rayed. This technique, developed in 1927, enabled both normal and abnormal blood vessels in and around the brain to be visualized: X-rays of the skull now made visualization of intracranial tumors, vascular abnormalities, and aneurysms possible.\n\nOf course, in some cases it proved necessary to actually expose the brain of an individual to the direct vision of the surgeon during an operation. No one made more of this opportunity than Wilder Penfield. While neurosurgeons had previously operated on the brain to remove tumors and scar tissue thought to be the focus of epileptic attacks, such operations often caused further scarring, with very undesirable consequences. Penfield, working in Montreal from the mid-1920s, developed a novel procedure involving the removal of considerably more tissue; initially controversial, this actually proved more effective (Penfield 1927). The procedure involved shaving the head of the patient, opening a \"trapdoor\" in the skull, and removing the protective layers of tissue to expose the surface of the cortex. The patient remained conscious at all times. The surgeon then stimulated different points on the cortex with a wire carrying a small electrical current, and the patient reported what he or she felt\u2014a tingling in the left hand, or a flashing light, each understood, by Penfield, as indicating the part of the cortex where that organ or function was represented. The aim, in part, was to identify the region where the sensations that precede a fit were elicited. This is the so-called aura, which is different for each patient and sometimes involves a strong smell or flashing lights. Stimulation of different areas of the cortex would help identify the scar tissue or other lesion that was the focus of the seizure\u2014a process aided initially by X-rays or ventriculograms, and later by the use of an EEG method developed by Hebert Jasper. Further, the mapping of vital functions enabled the surgeon to identify which regions should not be damaged in the operation if one was to preserve those functions. In the course of undertaking hundreds of these operations, Penfield was also able to produce his famous visual maps of the sensory and motor cortex, indicating the locations of different functions and sensory zones on its surface (Penfield and Erickson 1941; Penfield and Rasmussen 1950; Penfield, Jasper, and Macnaughton 1954).\n\nIn addition to mapping sensory and motor functions, this method of cortical stimulation produced some unexpected results. When some regions of the temporal lobes were stimulated in some patients, they recalled what seemed to be precise memories of long-forgotten events, sometimes mundane, sometimes frightening. In other cases, stimulation produced intense perceptual illusions or hallucinations, also usually seeming to recreate specific past experiences, such as an orchestra playing\u2014the patient would hum along to the tune. The sensations, memories, and hallucinations would disappear when the probe was removed. For the first time, it seemed, Penfield had achieved the cerebral localization of conscious experiences and memories, although he doubted whether these were actually integrated in the cortex.\n\nBut, starting in the 1950s, other methods would be found to enable the vision of the investigator to penetrate the skull without the need for surgery. The first major development in imaging brain structure was computerized axial tomography\u2014the CAT scan or CT scan. Kevles recounts the long and winding road that finally led to the development of a scanner that would use computer algorithms to assemble the results of repeated X-ray 'slices' across the brain to produce an image of an anatomical slice, showing the internal structure to the extent that it was made up of tissues of different densities. Many relatively independent steps were involved\u2014Bracewell's work in the 1950s developing mathematical techniques for constructing imagelike maps of astronomical bodies from multiple 'strips' of data; Olendorf's work in Los Angeles rigging up a device that sent collimated beams of high-energy particles through a plane in a model head; Cormack's work to develop a computer that could reconstruct images from X-ray scans.\n\nBut it was Godfrey Newbold Hounsfield who integrated all the elements into a functioning scanner that could image the internal structure of the brain in sufficient detail to be used for diagnostic and research purposes. Hounsfield developed and built the device for Electrical and Music Industries Limited (EMI), which financed this work with funds generated from the sale of records by bands such as the Beatles, together with support from the Department of Health and Social Security, which was interested in the possibility of early detection of brain tumors, and from the United Kingdom's Medical Research Council. CAT scanning\u2014now simply CT scanning or CT\u2014enabled three-dimensional images to be produced of any section of the body, including the brain. Indeed, the first CT scanner was designed for the head, and Hounsfield demonstrated it in 1971 at Atkinson Morley's Hospital in London. Cormack and Hounsfield won the 1979 Nobel Prize in Physiology or Medicine for this work, and such scanners\u2014rapidly improving in speed and acuity\u2014and scans\u2014in pictorial rather than numerical form\u2014rapidly became part of the everyday world of physicians treating brain injuries and neurological diseases in the clinic and hospital.\n\nThe development of the CT scanning technology was closely paralleled by another method for visualizing body structure, including brain structure\u2014magnetic resonance imaging, or MRI (Kevles 1997). The principle of what was initially termed nuclear magnetic resonance (NMR) was that the protons in atoms would become magnetized and align themselves when exposed to a strong magnetic field; when that field was altered, the protons would also realign themselves. In 1946, Felix Bloch (Harvard) and Edward Purcell (Stanford) independently published papers that showed that the protons in a spinning nucleus would resonate when placed in an alternating magnetic field at a particular frequency: when the magnetic field was turned on, the protons would align themselves, and when it was turned off, they would 'relax.' The times taken between their original state and their relaxed state varied between substances, and could be detected by a receiver and turned into images.\n\nInitially, the technique was used in chemistry to explore the structure of molecules, but in the 1950s, as the machines became more widespread, researchers began experimenting with slices of tissue\u2014the images picked up variations in the water in tissues by focusing on spin in hydrogen atoms. In 1959, the first NMR scans were conducted to measure rates of blood flow in living mice by Jay Singer at Berkeley, and at the start of the 1970s, in an episode of apparently independent discovery plagued by controversy and competition, Raymond Damadian and Paul Lauterbur worked out the techniques for deriving images by reconstructing them from multiple scans across a body: Damadian was looking for a device that would map malignancies, and Lauterbur hoped to image fluids. In 1973, Lauterbur used NMR to create cross-sectional images in much the same manner as CT and published the results in a paper in _Nature_ (Lauterbur 1973). There was immediate interest not only because the technique was free of ionizing radiation but because it produced better body images than CT because of its sensitivity to soft issues.\n\nMeanwhile, in the United Kingdom, Peter Mansfield was developing a related way to use NMR to image first solids, then liquids, in the human body; working with others, he developed a machine that would produce three-dimensional images, not by reconstructing them from successive slices, but by using complex algorithms to manipulate data. By the early 1980s, facilitated by the success of CT scanning and by the advantage that NMR did not use ionizing radiation, the machines that undertook what became known as MRI scanning entered clinical use. During the 1980s, many technical refinements of MR were made, and a number diagnostic MR applications were undertaken: MRI received FDA approval for clinical use in 1985. Of course problems remained, not least the great expense of the machines, the need to take special precautions around the very high magnetic fields that they produced (in supercooled magnets, cooled with liquid helium), and the disorienting effects on humans of the strong magnetic field itself, when the organ imaged is the brain. Nonetheless, given its capacity to reveal variations in soft tissue, the brain was one of the key targets of MRI scanning, which was used to detect injuries caused to babies by violent shaking, to reveal the demyelination of nerve fibers that was at the root of the previously mysterious symptoms of multiple sclerosis, and to identify the anomalies, lesions, and tumors that underlay many neurological disorders. Mansfield and Lauterbur were awarded the Nobel Prize in Physiology or Medicine in 2003 for their work in developing the technology.\n\nBut how to go from imaging _structure_ to imaging _function_? CT and MRI scanning, however useful for detecting structural abnormalities, did not image what the living brain was _doing_. Of course, conclusions could be drawn from structure, however tentatively, where structural anomalies were thought to be linked to functional anomalies. For example, in the trial of John Hinckley for attempting to assassinate Ronald Reagan, CT scans were introduced by the defense, who argued that they showed abnormalities in the appearance of the brain\u2014in its gyri and sulci\u2014and that autopsies had shown these to be characteristic of people diagnosed with schizophrenia. However, this route was indirect\u2014was there a more direct route to render visible the invisible operations of the mind?\n\nFrom the 1930s onward, some investigators had explored normal and abnormal brain activity using electroencephalography (EEG). Edgar Adrian had demonstrated that the Berger rhythm, discovered by Hans Berger in the 1920s, was genuine and not artifactual, reporting his findings in a famous paper of 1934 that argued that the rhythm was linked to attention (Adrian and Matthews 1934). But it was positron emission tomography, or PET, that first seemed to enable the gaze of the investigator to visualize mind in brain. PET appeared to permit direct imaging of very specific processes in the brain as it undertook certain tasks, by showing how certain radiolabeled molecules, such as glucose or drugs, were taken up into brain tissue over time. As Joseph Dumit puts it in his ethnographic study: \"PET was the first noninvasive technology to permit direct quantitative assessment of regional physiological processes in the brain.... PET allowed scientists to make exacting measurements of blood flow, glucose uptake and dopamine receptor uptake\" (Dumit 2003, 22).\n\nThere is a long and generally accepted history of the precursors of PET\u2014those whose basic work established the various elements that would be brought together in the technology. But there is a short and rather disputed history of rivalry and competition when it comes to the last stages in developing PET itself, in particular between the groups around Michael Phelps, Michael Ter-Pogossian, and Henry Wagner (Dumit [2003] provides an excellent account of these rival versions of the history). PET assembled together three basic elements: the idea that a molecule could be labeled with a small amount of radioactivity and traced as it circulated through the body and was metabolized in particular cells; the idea of a device that could attach radioactive tracers attached to a range of molecules of interest; and idea of a scanner that could identify the location of the radioactively labeled molecules at multiple time points, and compile these data into an image.\n\nAll accounts credit Georg Van Hervsey with discovering the tracer principle in the early decades of the twentieth century, for which he received the Nobel Prize in Chemistry in 1943: he showed that \"radioisotopes of elements participate in biochemical and physiological processes in the same way as the chemicals they have replaced\"; that organisms absorb, circulate, metabolize, and excrete these labeled molecules; that their radioactivity decays in a predictable way; and that this process can be traced over time through the various organs and can be traced externally by a detector such as a Geiger counter (Kevles 1997, 202). Hervsey initially used naturally occurring radioisotopes. However, after the Curies had shown, in 1934, that artificial radioisotopes could be produced by bombarding elements with atomic particles, it became possible to create radioisotopes of a whole range of elements that were taken up in particular organs and tissues and in specific metabolic processes. At around the same time, Ernest Lawrence invented a machine, later termed a cyclotron, that used this process of bombarding elements with high-speed neutrons to produce a whole host of radioisotopes, and these began to be used in medical research, even though, at first, there was some concern that they might produce biological damage from radiation effects. In 1965, the first medical cyclotron was installed at the Hammersmith Hospital in London, followed by installations at Massachusetts General Hospital and Washington University's Mallinckrodt Institute of Radiology, also in 1965.\n\nThe next step was the development of the scanning technology. In the 1950s, David Kuhl worked out methods to create images by detecting and localizing the emissions of the radiolabeled ligands, and in particular of a radiolabeled form of deoxyglucose, termed FDG, which, like blood, would be taken up specifically in areas of cerebral activity. This work led to the development of the first scanner that used the method of imaging multiple slices through the brain and using algorithms to compile them together into a three-dimensional image\u2014the process known as single photon emission computerized tomography, or SPECT. SPECT was first used to map brain function by tracking blood flow by Niels Lassen in 1972 (see Lassen 1978). It is the question of priorities in the research that led from SPECT to PET that is most disputed by the participants: the short version of this period is that Phelps, working in Ter-Pogossian's lab, used versions of the algorithms developed for CT scanning to generate similar images from the positron emissions\u2014first dubbed PETT (Ter-Pogossian, Phelps, Hoffman, et al. 1975).\n\nIn the subsequent years, techniques were improved, leading to the commercialization of PET devices, although these remained costly and complex. This was partly because, in order to use them, a lab or clinic had to have a cyclotron close at hand: the radioisotopes that were required had to be produced in cyclotrons, but they decayed rapidly and had to be used in a short time; a considerable crew of technicians and others were required to deploy the procedures. Nonetheless, PET was a powerful research tool, one which, for example, enabled Henry Wagner, who became a powerful promoter of the virtues of nuclear medicine, to image the dopamine receptor (Wagner, Burns, Dannals, et al. 1983). PET was also used in clinical studies of cancer, heart disease, and many other areas, as well as in imaging studies of motor activity, attention, cognition, and various psychiatric disorders, even though its diagnostic capacities proved limited.\n\nWhat scale was appropriate, and possible, for imaging neuronal activity? Some argued that such imaging should ideally be at the molecular level, and that nuclear medicine should become molecular imaging. The earliest studies to quantify neurotransmission and neurochemistry in humans were published in the 1980s (Wagner, Burns, Dannals, et al. 1983; Holman, Gibson, Hill, et al. 1985). Twenty years later, advocates of this approach argued for \"the visual representation, characterization, and quantification of biological processes at the cellular and subcellular levels within intact living organisms\"; they believed that \"the change in emphasis from a non-specific to a specific approach represents a significant paradigm shift, the impact of which is that imaging can now provide the potential for understanding of integrative biology, earlier detection and characterization of disease, and evaluation of treatment\" (Massoud and Gambhir 2003, 545). Shankar Vallabhajosula, the author of one of a flurry of textbooks on molecular imaging published in the first decade of the twenty-first century, claimed that it would be the future of clinical practice: \"although molecular imaging is not necessarily new, what is new is 'molecular and anatomic correlation' \" (Vallabhajosula 2009). The aim would be \"to integrate patient-specific and disease-specific molecular information with traditional anatomical imaging readouts\" with the eventual hope of achieving \"non-invasive or minimally invasive molecular diagnostic capabilities, better clinical risk stratification, more optimal selection of disease therapy, and improved assessment of treatment efficacy\" (ibid., 4).\n\nFrom this perspective, the goal was to create a molecular and neuromolecular therapeutics. The Society for Molecular Imaging (SMI) was founded in 2000 (Provenzale and Mukundan 2005); the European Society for Molecular Imaging (ESMI) was created in 2006; and the journal _Molecular Imaging and Biology_ was launched in 2005 as the official journal of the Academy of Molecular Imaging (AMI), the SMI, and the ESMI. According to Henry Wagner, molecular imaging can help create a more \"cost-benefit knowledge-based health care system\" by shifting the focus and the gaze to specific molecules that become the \"nexus for understanding disease, developing both preventive measures and effective therapies and monitoring treatment efforts\" (Wagner 2006). By 2009, when Wagner published a book titled _Brain Imaging: The Chemistry of Mental Activity_ , it seemed that the technologies for rendering the mental visible in the material, in the chemistry of tissues, fluids, and membranes, had fused with the neuromolecular conception of the brain (Wagner 2009).\n\nIt was another development, however, that was to underpin the proliferating belief that one could see the functional activity of the living brain\u2014to see the mind in the brain. This was functional MRI or fMRI. MRI, as we have seen, focused on structure. How could one get from structure to function? Speed was one part of the answer to this question. As the MRI technology became faster, and the algorithms became more sophisticated, it became possible to take successive MRI images and link them together to capture up to 110 frames per second. The second requirement was to find a tissue that, on the one hand could be imaged, and on the other could be linked to functional activity in the brain. What could be more closely linked to levels of activity in a body than blood? And blood was one of the tissues that could be imaged by MRI techniques. It was via the medium of blood, and the long-standing observations that correlated increased brain activity with increased blood flow, that the functional activity of the living brain\u2014mental states themselves\u2014would seem to be captured in a scanner.\n\nThe link between brain activity and blood flow goes back to the 1870s, when Angelo Mosso, a prominent Italian physiologist, made the observation in humans that local blood flow to the brain is related to brain function: he published his results in 1880 (Mosso 1880). The actual physiological relation between brain function and blood flow was further explored in 1890 by Charles Roy and Charles Sherrington (Roy and Sherrington 1890). In 1948, Seymour Kety and Carl Schmidt developed the method of measuring cerebral blood flow by using the nitrous oxide method (Kety and Schmidt 1948), and in 1955, Kety and colleagues at the University of Pennsylvania and the NIH used radioisotopes to track brain blood flow in cats (Landau, Freygang, Roland, et al. 1955; Freygang and Sokoloff 1958; Kety 1960). This 1955 paper from Kety's group is often claimed to be the first quantification of blood flow in the brain related to brain function. Blood had another great advantage: its magnetic properties. Faraday had carried out studies of the magnetic properties of hemoglobin in 1845, and almost a century later, Linus Pauling and Charles Coryell found that oxygenated and deoxygenated hemoglobin have different magnetic susceptibilities (Pauling and Coryell 1936).\n\nSeiji Ogawa, working at the AT&T Bell Laboratory in New Jersey, recognized that these findings, taken together, made functional imaging possible. He based his approach on the observation that brain cells, when activated, increase their uptake of energy from oxygenated blood, which then becomes deoxygenated (Kevles 1997, 198). Ogawa realized that the deoxygenation of the blood in the regions of the brain where activity was taking place could be imaged. As Kevles explains: \"[V]enous blood, lacking a bound oxygen molecule, is paramagnetic, and when placed in a magnetic field changes the magnetism all around it. This distortion of the magnetic field, in turn, affects the magnetic resonance of nearby water protons. It amplifies their signal as much as 100,000 times. Ogawa called this effect BOLD (blood-oxygen-level-dependent) contrast imaging, and he published his first paper using this method in 1989 (Kevles 1997, 198). Other research groups were experimenting with the use of contrast agents with MRI to demonstrate changes in brain blood volume produced by physiological manipulations of brain blood flow; after some animal studies in the late 1980s, the first study using this approach with human volunteers to explore task activation brain mapping was published in 1991 (Belliveau, Kennedy, McKinstry, et al. 1991). However, the BOLD response, which required no administration of an external contrast agent, was to become the key to the widespread use of functional MRI (Ogawa, Lee, Kay, et al. 1990).\n\nAs functional brain imaging, whether by PET or by fMRI, came to supplement images of anatomical structures, we now seemed able to observe metabolic activity in living, functioning brains in four dimensions\u2014three of space and one of time. The very interior processes of the living and dynamic brain, its anatomical normalities and pathologies, its activity in delusions and normal perception, its secretion or blockade of neurotransmitters, could be rendered visible and correlated with a phenomenology of mental life. At last, it now seems that we can see the physical basis of mind in the activities of the living brain. To see is to know, and to know is to see. But what is it that is being seen, what is happening when, to quote Andreas Roepstorff, \"knowing becomes seeing\"? (Roepstorff 2007)\n\nThe Epidemiology of Visualization\n\nIn 2008, Nikos Logothetis reported that\n\na recent database (ISI\/Web of Science) query using the keywords \"fMRI,\" or \"functional MRI\" or \"functional magnetic resonance imaging\" returned over 19,000 peer reviewed articles.... About 43% of papers explore functional localization and\/or cognitive anatomy associated with some cognitive task or stimulus\u2014constructing statistical parametric maps from changes in haemodynamic responses from every point in the brain. Another 22% are region of interest studies examining the physiological properties of different brain structures, analogous to single-unit recordings; 8% are on neuropsychology; 5% on the properties of the fMRI signal; and the rest is on a variety of other topics including plasticity, drug action, experimental designs and analysis methods. (Logothetis 2008, 869)\n\nWhen we repeated this exercise in October 2010, the result was even more staggering\u201453,662 papers were identified, starting with one published in 1980 (Holland, Moore, and Hawkes in the _Journal of Computer Assisted Tomography_ ), 146 papers in 1990, 2,139 in 2000, 4,725 in 2005, and almost 7,000 in the one year of 2009 alone. Alongside the categories that Logothetis reports, we find papers relating changes in brain activity to responses to art in general and to the work of specific painters; to responses to music and to specific composers or performers; papers discussing the neurocognitive processes of religious leaders; studies of brain activation related to the use of language, metaphor, or responses to various novelists; papers that monitor brain activity in response to television commercials or to grief after the loss of a child; as well as hundreds of studies purporting to image love, hate, fear, and other emotions, all manner of cognitive processes, and volition and acts of will when individuals in scanners are given simple tasks to undertake. What is one to make of this industry of visualization?\n\nOne cannot fail to be tempted by the results of all this research\u2014has not the activities of the living mind within living brain now come into view? Yet many of the researchers in this area, including many of those actually responsible for generating the statistical maps that make the images possible, feel profoundly uneasy about this translation of their quantitative data into pictorial form and the conclusions, both professional and popular, that are drawn from these studies. As indeed do the social scientists who have explored these developments. For the sake of simplicity, we can group their concerns under four headings: localization, the lab, pixels to pictures, and evidence to interpretation.\n\nAre Brain Functions Localized?\n\nAre brain functions localized, and in what sense? And why should this be important? One of the most vociferous critics of the emphasis on localization in the practices of imaging has been Jerry Fodor who is a philosopher and cognitive scientist. Writing in the popular magazine the _London Review of Books_ , he argued:\n\nI want, to begin with, to distinguish between the question whether mental functions are neurally localized in the brain, and the question where they are neurally localized in the brain. Though I find it hard to care about the second, the first clearly connects with deep issues about how the mind works;... [but] just what is the question about the mind-brain relation in general, or about language in particular, that turns on where the brain's linguistic capacities are? And if, as I suspect, none does, why are we spending so much time and money trying to find them? It isn't, after all, seriously in doubt that talking (or riding a bicycle, or building a bridge) depends on things that go on in the brain somewhere or other. If the mind happens in space at all, it happens somewhere north of the neck. What exactly turns on knowing how far north? (Fodor 1999)\n\nThis argument actually comes rather strangely from someone who is one of the strongest proponents of the mental ontology that seems quite consistent with localization, that is to say, a 'modular' theory of mind (Fodor 1983, 2001). Nonetheless, many others, including many neuroscientists, have also questioned a strict conception of localization, one that is based on the idea of hard-wired connections of neurons, clustered together, along with the belief that there can be a one-to-one mapping of specific mental processes to specific brain regions common to all 'normal' human beings' brains.\n\nConsider this analysis from Russell Poldrack:\n\nOne cannot infer that the activated regions are necessary or sufficient for the engagement of the mental process. Indeed, there are well-known examples of cases in which regions that are activated during a task are not necessary for the task. For example, the hippocampus is activated during delay classical conditioning, but lesions to the hippocampus do not impair this function.... Inferences from lesion studies are limited by the fact that the brain may often have multiple ways to perform a cognitive process.... Cognitive neuroscientists have generally adopted a strongly modular approach to structure\u2013function relationships, perhaps driven by the facile leap to localizationist conclusions from lesion and neuroimaging results. Despite the longstanding appreciation for the importance of functional integration within the neuroimaging literature, the widespread use of functional and effective connectivity analyses has not yet come about. Given that many cognitive processes may be distinguished not by activity in specific regions but by patterns of activity across regions, there is reason for caution regarding many of the inferences that have been driven by highly modular approaches. (Poldrack 2008)\n\nNeuroscientists critical of many of the conclusions drawn from brain imaging further point out the problem of scale. For despite all its increases in acuity, fMRI can only measure mass action. Nikos Logothetis has examined the three hundred top-cited fMRI studies and estimated that a typical voxel size (before filtering of data that is likely to increase this by a factor of two or three) contains 5.5 million neurons, between 2.2 \u00d7 1010 and 5.5 \u00d7 1010 synapses, 22 km of dendrites, and 220 km of axons. Leaving aside all the other complications that will shape the blood-oxygen-level variations within this cube of brain tissue that is somewhere between 9 and 16 mm square and around 7 mm in thickness\u2014notably, the balance in every single neuron between excitation and inhibition\u2014this is, to say the least, a large and heterogeneous population of cells.\n\nIn any event, how are we to know what scale is appropriate for examining mental processes? What proportion, in what configuration of the 1010 neurons and 1014 connections in the human cortex alone should be analyzed in relation to any task? Logothetis quotes neuroanatomist Valentino Braitenberg: \"It makes no sense to read a newspaper with a microscope\" (Logothetis 2008); one might add, as pointed out by Hauke Heekeren, of the Max Planck Institute for Human Development, that it makes no sense to read a book from a photograph of the bookshelf. No wonder that Logothetis concludes that the \"limitations of fMRI are not related to physics or poor engineering, and are unlikely to be resolved by increasing the sophistication and power of the scanners; they are instead due to the circuitry and functional organization of the brain, as well as to inappropriate experimental protocols that ignore this organization\" (Logothetis 2008: 876\u201377).\n\nThe Lab: Place or Non-Place?\n\nTypically, reports of brain imaging experiments pay little attention to the material properties of the laboratory setting. This is unfortunate, for, as ethnographers such as Simon Cohn have pointed out, the scanning facility is not a 'non-place' for those who are being scanned, it is a particular and rather unusual arrangement of space, persons, machinery, sounds, and sights, not to mention the experience of being in the scanner itself. What are the subjects thinking and doing when they are in the scanner? The scanned brain is in a human body, the body of a human being who is being paid to lie in a machine to perform a task that the brain will never be confronted with in the world outside. These features, that cannot enter the modeling of the relation between the stimulus and response contained within the space of the scanner, cannot enter into the analysis that the scanning software uses to generate the images (Cohn 2008).\n\nYet as Cohn also points out, those social relations of the scanning facility, the \"brief but intimate personal relations\" between the researchers and their subjects, are central to the possibility of the scanning experiment. What impact do the social relations of the scanning facility have on the mental life of the scanned subject? How can the researchers set aside, for the purposes of the experiment, the arguments about the nature of 'the social brain' that neuroscientists themselves are generating\u2014that the human brain is particularly evolved for interactions, including adapting itself in all sorts of ways to its attributions of wishes and intentions to others? How might this experience and response to sociality differ between subjects, for example, in the scanning of healthy volunteers and hospitalized patients diagnosed with depression? To what extent can there be generalization across subjects who are experiencing and understanding the scanning situation in different ways, let alone beyond the lab setting?\n\nThese are, indeed, key questions. What is involved here, as we will discuss in the next chapter in relation to animal models, is the creation of a certain kind of \"setup\" (Latour 1987) that brings together, within a particular institutional form and practice, experimenters, assorted technicians from various backgrounds, devices, concepts, theories, and subjects. Only through this setup is it possible to generate the data that are required to generate the images that can then be subject to interpretation at leisure and then deployed in presentations, articles, and books to make arguments. The brain of the individual is thus given the opportunity to produce something that it would not otherwise be able to produce. But that setup, with all its sociality, contingency, specificity, and embedded meaning, disappears in the interpretation of the measurements produced. This is not necessarily a problem in itself: it is an inescapable part of the incredibly difficult process by which data of this sort can be generated. But it does raise some rather important questions about what can be concluded from such data, or rather, from the images that the data are used to generate. What is the salience of the image, when it emerges from the secluded and simplified world of the laboratory, and makes claims for its relevance to the understanding of conduct in the wild and messy world of everyday life?\n\nPixels to Pictures\n\nThe artifactual character of the images produced in the various technologies of brain scanning has attracted much critical attention (Dumit 1997, 1999; Beaulieu 2000b, 2002; Roepstorff 2004). In the analysis and presentation of the scan data, computerized methods are used, involving choosing among a whole range of available algorithms to transform quantitative information into images of domains that can be explored spatially. This is a technically highly complex rendition, generated by mapping quantitative data on the emission of each positron in a scan of a particular subject (carefully chosen, screened, prepared) in four dimensions onto a morphological space usually drawn from a standardized MR atlas of slices of the typical or average brain. The process involves transforming, smoothing, warping, and stretching the data on each individual to fit the standard anatomical space; this is necessary to make the images comparable and cumulable. And the colors, so impressive in the proliferation of such images in the domains of nonspecialists, are similarly artifactual\u2014there are no reasons apart from convention why areas seemingly highly active should be represented in reds and those less active in blues. Yet of course, those colors carry cultural resonances that have great significance for the meanings that are attributed to the images, at least when they are represented beyond the laboratory or the journal article.\n\nJoseph Dumit identifies four key stages in the amalgam of social and technical processes that generates the image (Dumit 2003). First, _devising the experimental design_ : choosing participants, defining them (normal, schizophrenic, drug naive, etc.), controlling prior states, specifying the task unambiguously. Second, _managing the technical process_ of measuring brain activity: conducting the scan, compiling the data, algorithmically reconstructing a three-dimensional map of activity. Third, _making the data comparable_ : normalizing the brainsets, warping them onto a standard anatomical atlas of brain space, then combining brainsets by manipulating the datasets. Fourth, _making the data presentable_ : coloring, contrasting, and production of the images. Much of this is handled by standard software packages, notably statistical parametric mapping (SPM). While such packages are regularly updated and incorporate input from the researchers in the field, like other such programs, many assumptions, some dating back many years, are 'black-boxed' within them\u2014in this case literally, as the software is designed to work 'straight out of the box.'\n\nThe sociologists of scientific knowledge have given us many ways to think of the process of black-boxing in scientific research. But here, in the image itself, as the process disappears into the product, the illusion of transparency is crucial to the effects of truth. To quote another ethnographer of brain imaging, this process does not so much \"image\" mind in brain, but constitutes a new object: \"the mind-in-the-brain\" (Beaulieu 2000b). Like the diagnostic space opened by Bucknill and Tuke's carefully posed pictorial representations of the varieties of madness, then, this new molecular vision opens onto an irreal space of thought and interventions. It represents not by the _simulacrum_ but by the _simulation_ \u2014by generating a model of processes at the level of the living brain, a model that is, precisely, capable of modeling in apparently precise images and flows, the functioning and malfunctioning of those process under particular conditions.\n\nIt is undoubtedly true that such images provide powerful support for particular theories of brain functioning, by purporting to allow researchers to actually observe the mind\/brain seeing, hearing, smelling, thinking, desiring, emoting, willing, or hallucinating. Yet the power of these images is located at their weakest point\u2014the point where simulation claims not merely to be a way of isolating certain elements of a more complex system for analysis (as, for example, in a computer simulation of an earthquake), but to be the thought image of that system itself. Despite their apparent realism, claims to account for the phenomenology of mental states in terms of these simulated functions localized in the living brain require an act of faith in all those elements on which the image is premised. The visual imaginary that is the result is, in this sense, no different from that of the phrenologists of the nineteenth century.\n\nEvidence and Interpretation\n\nOf course, the experiments, and the images they produce, do not demonstrate _nothing_. But _what_ they demonstrate is a matter of dispute. The 'objectivity effect' of the image seems to be clear: studies have shown that the presentation of a brain image can positively influence the reader's judgment about the quality of the reasoning of the article in which that image appears (McCabe and Castel 2008) and that of verbal descriptions of neuroimaging results can make weak arguments more persuasive\u2014even when they are irrelevant to the argument (Weisberg, Keil, Goodstein, et al. 2008). We shall see in a later chapter, however, that this objectivity effect seems to be limited by the characteristics of the practice in which the image is deployed. For the image, despite suggestions to the contrary, and despite all the arguments black-boxed within it, does not speak for itself\u2014it has to be spoken for. And in areas where those who speak in the name of the image are challenged, for example, in the criminal justice system, the image itself may get caught up in the dynamics of controversy.\n\nIn other domains too, the interpretations necessary to give images their salience have come under significant criticism from imagers themselves. Let us take a single example here, a paper titled \"Does Rejection Hurt? An fMRI Study of Social Exclusion\" published in _Science_ in 2003 (Eisenberger, Lieberman, and Williams 2003). Here is how the abstract describes the study:\n\nA neuroimaging study examined the neural correlates of social exclusion and tested the hypothesis that the brain bases of social pain are similar to those of physical pain. Participants were scanned while playing a virtual balltossing game in which they were ultimately excluded. Paralleling results from physical pain studies, the anterior cingulate cortex (ACC) was more active during exclusion than during inclusion and correlated positively with self-reported distress. Right ventral prefrontal cortex (RVPFC) was active during exclusion and correlated negatively with self-reported distress. ACC changes mediated the RVPFC-distress correlation, suggesting that RVPFC regulates the distress of social exclusion by disrupting ACC activity.\n\nThis illustrates the problems of this style of reasoning rather clearly. First, one creates a game that an individual can play in a scanner, that is taken to simulate a 'social' interaction (although the relation between virtual ball tossing and any salient social situation is somewhat tenuous). Second, one adjusts one's scanning parameters until one is able to observe increased brain activity\u2014or rather an increased BOLD response\u2014in a particular region of the brain\u2014or rather, a region of the standard brain space onto which your data has been warped, in this case the anterior cingulate cortex, or ACC. Next, one refers to previous experiments claiming that this brain region shows an increase in activation in another condition: in this case, the ACC has been shown to be activated when the subject experiences physical pain. Then one concludes that, because the same region is activated in both conditions\u2014here in both social exclusion and in physical pain processing\u2014the two are causally linked: hence social exclusion hurts.\n\nAmong the many rather obvious flaws in this chain of reasoning is the fact that activation in the same brain region has been claimed in multiple tasks. For example, changes in amygdala activity have been found to occur in experiments seeking to generate or simulate a whole range of mental states: fear, reward, fairness, moral status, subjective reports of beauty, trustworthiness, and more. So to make such arguments work, either more and more baroque alignments must be made, or the experimenters must, as in this case, be exceptionally selective. The authors of the study cited above conclude: \"This study suggests that social pain is analogous in its neurocognitive function to physical pain, alerting us when we have sustained injury to our social connections, allowing restorative measures to be taken. Understanding the underlying commonalities between physical and social pain unearths new perspectives on issues such as why physical and social pain are affected similarly by both social support and neurochemical interventions... and why it 'hurts' to lose someone we love\" (Eisenberger, Lieberman, and Williams 2003, 292). On the basis of such extrapolations in hundreds of papers using fMRI to measure changes in brain activity when subjects undergo tasks said to simulate features of our form of life in the scanner, an empire of brain imaging research has been built.\n\nThe New Engines of Brain Visualization\n\nWhat, then, are the consequences of this artful construction of the mind? Should we object to artfulness itself? This would be unwise, not least because it would tie us, even if tacitly, to the aspiration for an untenable realism. Knowledge, as a whole tradition of thinkers about science have reminded us, always entails a break away from the immediate and the concrete to the visualization and manipulation of artificial objects moved by hypothetical forces in abstract space. It is not the irreality of this image of the mind that should concern us, but the attribution of false concreteness to that image by researchers, policymakers, and popular interpreters. The implications of these attributions of false concreteness are best discussed in relation to specific practices, and will be examined in our later chapters. But, more generally, can we say that we have traveled up a gradient of objectivity as we have moved from Esquirol's drawings through Diamond's posed photographs, Charcot's movies, Brodmann's maps, and Penfield's homunculi to the luminous images of fMRI investigations that stud the pages of our contemporary scientific journals?\n\nIt is hard to free ourselves from the photographic illusion that might support such a belief in progress toward the real. But it is nonetheless misleading. In saying this, we are not suggesting a fundamental rejection of the truth effects of brain imaging. To attend to the artifice of the image is not to critique it but to recognize the nature of the objectivity, the facticity that it produces. The image should be understood not as a picture to be judged via a criterion of realism, but as a tool, a move in an argument, an instrument in an intervention, to be judged by criteria of rationality, validity, or efficacy. Advances in clinical medicine from the nineteenth century onward went hand in hand with the penetration of the gaze of the doctor into depths of the body itself. There are now many examples of analogous advances linked to the structural imaging of the brain\u2014in the detection of tumors, the identification of lesions, the mapping of the damage caused by injury or stroke. Thanks to such images, the mind of the neuroscientist, the neurologist, and the psychiatrist does now seem able to \"walk among the tissues themselves.\" But however similar the images of brain _function_ are to those of brain _structure_ , they mislead if they seem to allow the mind of the neuroscientist to walk among thoughts, feelings, or desires. Technology alone, even where it appears to measure neural activity, cannot enable the gaze to bridge the gap between molecules and mental states. We should not fall victim to the illusion that visualization itself has\u2014or could\u2014resolve the question of the relations between minds and brains. To recognize this is not to undermine the work of brain scanning, far from it. On the contrary, by enabling us critically to evaluate the different kinds of claims made in behalf of the diverse components of this avalanche of images, an analysis of these new engines of visualization can and should help improve the robustness of the truths they create.\nChapter Three\n\nWhat's Wrong with Their Mice?\n\nWhat a pitiful, what a sorry thing to have said that animals are machines bereft of understanding and feeling, which perform their operations always in the same way, which learn nothing, perfect nothing, etc.!\n\n\u2014Voltaire, _Dictionnaire philosophique portatif_ , 1764\n\nTread softly when approaching a mouse model of a human psychiatric disease.\n\n\u2014Jacqueline Crawley, _What's Wrong with My Mouse? Behavioral Phenotyping of Transgenic and Knockout Mice_ , 2007\n\nWe take the title of this chapter from Jacqueline Crawley's _What's Wrong with My Mouse?_ from which we have learned a great deal about the world of animal modeling of behavioral disorders (Crawley 2007). Given the limitations that have long surrounded experimental intervention into the brains of living humans, animal research has been crucial to the project of neuroscience, from the molecular dissection of neurobiological processes such as neurotransmission, to research on plasticity and neurogenesis, and of course, to the development and testing of psychiatric drugs. But why frame the chapter in such terms\u2014why might there be 'something wrong' with the mice\u2014or for that matter the flies, rats, macaques, and the other creatures\u2014that populate the labs, and the papers, of those who use animal models, and model animals, in the experimental investigation of the neurobiological underpinnings of human behavior? The question of _translation_ poses this issue most clearly, for if there are persistent failures in using the findings in the laboratory to develop effective therapeutic interventions in practices that seek to treat, cure, or reform humans, then it might be sensible to ask whether there is 'something wrong' with the practices of animal modeling and experimentation themselves.\n\nSociologists, anthropologists, historians and philosophers have written extensively on models in science. There have been analyses of scientific modeling as a practice and as a \"style of reasoning\"\u2014sometimes referred to as \"the third dimension of science\" (Morgan and Morrison 1999; Thagard, Magnani, and Nersessian 1999; de Chadarevian and Hopwood 2004). There has been a range of ethnographic and historical studies that have focused on the social relations within which models are shaped and used among communities of modelers (e.g., Star and Griesemer [1989] 1999; L\u00f6wy 1992; Kohler 1994; Gaudilli\u00e8re and L\u00f6wy 1998; Lemov 2005; Franklin 2007; Shostak 2007; Friese 2009). But surprisingly, little has been written from the social and human sciences on animal models of _behavior_. And even less has been written on animal models _in psychiatry_ \u2014where the models are neither conceptual representations (although they embody these) nor three dimensional artifacts (although they are in three dimensions), but living, breathing, developing organisms (adding the dimension of vitality and temporality) that are of interest because of their _behavior_ (adding a further dimension to the model), and of behavior deemed pathological or abnormal, which complicates things even further.\n\nWhen we have discussed with colleagues from the social and human sciences the use of animals to explore issues relating to human cognition, emotion, volition, and their pathologies, the initial response has usually been skeptical. When we have described the methods of assessment used in such laboratory studies, for example, the use of the elevated plus maze in the study of anxiety, or the forced swim test in the study of depression, the reaction is often one of laughter. How, our naive colleagues ask, can anyone believe that there can be any similarity between mouse _behavior_ in this strange apparatus in an artificial laboratory setting and the rich meaningful, culturally embedded, historically shaped, linguistically organized, situationally framed, _experience_ of depression, anxiety, and whatever else in our everyday human world\u2014especially, when we find it difficult enough to describe these human feelings in rigorous ways, to compare them between persons and societies, or to distinguish what is normal from what is not?\n\nThe immediate response of researchers who use such animal models in their work is to point to similarities in the genomes of the two species, in the structure of mouse and human brain, in patterns of brain activation, in neural mechanisms at the cellular and molecular level, in responses to drugs and so forth, perhaps with reference to evolution and the principle of conservation across species when it comes to the most basic aspects of living organisms, including their brains. To which our colleagues in the human sciences might well respond\u2014if they had read Wittgenstein\u2014that this is all very well, but \"[o]nly of a living human being and what resembles (behaves like) a living human being, can one say it has sensations; it sees, is blind; hears, is deaf; is conscious or unconscious\" (Wittgenstein 1958, \u00a7 281)\u2014or, to extend the argument, it is only of a person that one can say that he or she experiences anxiety or feels depressed.\n\nThey might also suggest that the animal modelers were guilty of what Bennett and Hacker call the \"mereological fallacy\" (Bennett and Hacker 2003): that talk of similarities in genes or brains between animals and humans as-signs properties to genes or brains that properly belong only to persons. And so, they might continue, showing similarities in genes or brains between animals and humans does not resolve the problem. For while the genes or the brain may be involved\u2014certainly are involved\u2014in anxiety and depression in humans, while there may be brain activity associated with these complex affects, they are distributed across whole embodied persons in specific social situations whose meaning is given by the whole complex of resources that we can briefly sum up as 'culture.' In short, the way in which humans are in the world, their emotions, actions, and desires, depends on the way in which they give meaning to themselves and to their world, their models of that world, their beliefs about themselves and that world, and the affects associated with those beliefs. While those may have conditions of possibility in brain states and gene states, they do not reflect them. Indeed such an idea is meaningless because, _as a result of their biology_ , humans are the _only_ creatures that live, act, feel, and will in a world of language, meanings, beliefs\u2014sedimented and stabilized in artifacts, buildings, forms of life, human social practices, and much more.\n\nResearchers who use animal models are familiar with such reactions and have several responses. They claim that they are well aware of the problems of anthropomorphism and are taught from the outset not to speak of anxiety or depression in mice but of 'anxiety-like' or 'depression-related' states, because \"[t]here is no way for a human investigator to know whether a mouse is feeling afraid, anxious, or depressed. These are subjective emotional experiences, existing in the mind and body of the individual\" (Crawley 2007, 227). Thoughtful modelers such as Crawley insist that, at most, it is possible to develop a behavioral test for mice that has some validity for a symptom of the human condition, or an assay that is sensitive to therapeutic drug responses in humans (261). Nonetheless, those within the community of animal modelers in psychiatric research and behavioral neuroscience have pointed to at least six sets of problems that afflict such work: (1) the problematic homologies and analogies between human and animal behaviors, especially in relation to psychiatry; (2) the precarious assumption of common mechanisms; (3) the difficulty of the phenotype\u2014of the behavioral classification of mouse models of disorders generated by knockouts or other techniques, as these are seldom open to validation other than by means of pointing to apparent similarities in the observable behavior of the animal and the human; (4) the difficulties of modeling human stressors in animals; (5) the weakness of current tests, such as forced swim test, the elevated plus maze, and so forth for simulating variations in _human_ behavior; and (6) the fact that many trials are not properly blinded or randomized. This is not to mention the further difficulties arising from the great variety between strains, between labs, between particular practices for managing the animals, and so forth, all of which bedevil comparisons between results obtained by different research groups.\n\nThese technical difficulties are of great importance. However, we want to locate them in the context of some more fundamental issues. We can system-atize these into four interconnected themes that will structure this chapter. The first three are rather general and conceptual, the fourth more specific and practical.\n\n\u2022 First, the question of the _artificiality_ of the laboratory situation within which animal experiments are conducted.\n\n\u2022 Second, the idea of a _model_ in behavioral and psychiatric research.\n\n\u2022 Third, the fundamental objection from the social and human sciences\u2014the _specificity_ of the human and the elision of history and human (as opposed to animal) sociality.\n\n\u2022 Finally, the problem of _translation_. For if the criticisms from the social and human sciences are correct\u2014of the artificiality of the laboratory experiment; of the idea of an animal model of human behavior; of the blindness of animal modeling to the realities of the social and cultural lives that shape human mental life and human actions, then the translational problems of behavioral and neuroscientific work on animals would be inevitable. And the question would then be, to what extent are any of these problems remediable.\n\nArtificiality?\n\nFirst Stabilization: The Experimenter\n\nFor those of us who have undergone training in biology, the cry \"what's wrong with my mouse?\"\u2014or fruit fly or pigeon\u2014is usually one of desperation from novitiates whose dratted creatures will not manifest the attributes or behavior that our textbooks, professors, and demonstrators assure us is there if only we knew how to look: how to mix the reagents, to manage the apparatus, to observe the experiment, to note the results, and so forth. This common experience, banal as it is, may serve to highlight one set of considerations when we are thinking about laboratory studies using animals as models of human behavior. The activity of modeling must be seen as a practice, a craft work, and one that involves training and inculcation of routines and habits. This is a process that involves a triple stabilization\u2014of the experimenter, of the setup, and, of course, of the subject (the fly, mouse, rat, pigeon, etc.). No stabilization along these three dimensions\u2014no laboratory, no result, no paper; and, for those who fail the test of stabilization, no more experimenter. So the stabilization of the researcher\u2014or in most cases the research team\u2014is the first task. We will not dwell on this process of creating the experimenter here, though there is much to be said about it, for we all know that this is what scientific apprenticeship is intended to achieve\u2014to embed these ways of seeing and doing (and many other controls over conduct)\u2014into the bodily and mental habits of the experimenter so that all the strange activities invoved in animal work become a kind of second nature.\n\nSecond Stabilization: The Setup\n\nLet us say more, however, about the idea of a setup, of which the experimenter is one element. We have taken the term _setup_ from Bruno Latour (Latour 1987), who uses it to refer to the _dispositif_ that brings together experimenters, colleagues from a range of disciplines, institutions, apparatuses, data, concepts, theories, subjects (in this case the animals themselves) and much more in order to produce what he terms \"inscriptions.\" Inscriptions are the visual records of the experiment, turned into data that can be taken away from the lab, and worked upon long after the experiment itself is concluded and combined with many other inscriptions, in order to produce an argument. In that process, Latour suggests, the complex, heterogeneous, fragile, crafted assemblage of the setup retreats into the background, as its outcomes, in highly processed forms, appear in graphs, diagrams, formulas, and the like.\n\nWhile some regard this account as a criticism, from our own point of view, to characterize the process through which data are produced is not to seek to undermine it. It is just to point out that this is one way of enabling the animal and its behavior to 'appear' in a form that enables one to do scientific work on it. One consequence, of course, is that the animals appear in a particular way. They do not appear as individual creatures, with their own peculiarities, as they are seen in the actual practice of the experiment, and as they are known by those who work in the animal labs\u2014who often have a finely tuned knowledge of the specific 'character' of each animal, and an awareness of its 'emotional state' at the time of its entry to the experiment. These individual differences among individual living organisms, which persist despite all the work of stabilization of the animals that we discuss later, disappear as the animal is transformed into a series of measurements that are aggregated with others and turned into the group data that can work in experimental analysis.\n\nThis is not the only way to work with animals, of course\u2014one could observe them in their natural habitat\u2014but this would require a different setup. Latour, who has done some interesting work in primatology with Shirley Strum, therefore suggests that it is not helpful to analyze this in terms of an opposition between naturalness and artificiality. We devise different setups to make visible, to elicit, features of the lives and potentialities of the animals that were previously invisible or hard to discern. We can make use of something that Latour points out in a debate with primatologists, drawing on the way in which Thelma Rowell speaks of her work on sheep\u2014we \"give the mouse the opportunity to show what it can do,\" we give it a chance to show us something, perhaps even give it a chance to enter into behavior in which it would not otherwise engage (Latour 2002). There are, of course, other ways to give the mouse, or the monkey, the chance to show what it can do\u2014maybe better ways, maybe ways that have different consequences. But without some setup, producing some kinds of inscriptions that can be worked upon, the animal would not have a chance to show what it can do in a way that can enter scientific debate. So it is no criticism of animal models to say they produce their phenomena only in a certain setup. But this does exemplify one of the two translation problems that we are addressing\u2014the translation from those features that are given permission to appear within the setup of the lab to the features that appear in the clinic and the world beyond.\n\nA comment by Georges Canguilhem in his 1965 essay \"The Living and Its Milieu\" is relevant here. Discussing the work of Kurt Goldstein, he writes, \"The situation of a living being commanded from the outside by the milieu is what Goldstein considers the archetype of a catastrophic situation. And that is the situation of the living in a laboratory. The relations between the living and the milieu as they are studied experimentally, objectively, are among all possible relations, those that make the least sense biologically; they are _pathological_ relations\" (Canguilhem [1965] 2008, 113; emphasis added). To live, to be an individual living being, Canguilhem argues, is to _organize_ one's milieu\u2014and to deprive the experimental animal of precisely that opportunity is to vitiate a crucial aspect of that mode of living that one wishes to analyze. And yet, as we shall see, this seems to be precisely the situation of the animal in behavioral neuroscience. Is this, then a catastrophic situation, and if so, with what consequences?\n\nWe need to situate this question in the more general context of the laboratory disciplines, which use a very particular sort of setup to produce truth. They make use of apparatus, instruments, and procedures in order to create phenomena in the laboratory that did not exist before: the truthfulness of their statements is then mapped onto the phenomena produced by that experimental apparatus (Shapin and Schaffer 1985; Hacking 2009, 43). Historians of science tell us that this laboratory form of life emerged across Europe in the seventeenth century (Salomon-Bayet 1978). The _OED_ defines a laboratory as \"a building set apart for conducting practical investigations in natural science, orig. and esp. in chemistry, and for the elaboration or manufacture of chemical, medicinal, and like products\" and finds its first use in the early seventeenth century. Laboratory researchers may bridle at the way in which some of these early usages think of the laboratory as place for alchemy, but undoubtedly, the emergence of laboratories led to a new way of making truths.\n\nThis way of truth-making was controversial at the outset, but is now so ordinary that its artifactual nature has become almost imperceptible. In the laboratory sciences, we no longer derive truth from speculation, nor from the observation of natural phenomena in the wild. From this time forward, scientific truths will have to answer to the phenomena that are produced in a novel, enclosed and most unnatural setting which is neither public nor is it exactly private: the phenomena and procedures are observed and recorded by reliable witnesses and only then communicated to others by means of words spoken or written. So the apparatus, the setting, the witnesses (which include, of course, the apparatus and the inscriptions it produces), the form of writing, the kind of fact, the means of making things true\u2014all of these are specific to, constitutive of, the laboratory as a style of thought, and one that is embedded in the emerging form of scientific life. And while, in the early days of the laboratory, the emphasis was on the trustworthiness and reliability of the human witnesses to the experiment, today the crucial role of trust is retained, but displaced to other mechanisms such as peer review and the rhetoric of replication (Shapin 1994, 2008). So, this is the second stabilization\u2014of the apparatuses and the instruments.\n\nIf we are to understand the role of model animals and animal models in experimental investigations in neuroscience, then, it will not suffice to denounce them for their artificiality. This would require us to denounce most of what has come to be accepted as truth within the laboratory sciences, for those truths too have been created, manufactured, or at least exposed, intensified, given a chance, made visible, stabilized, within the peculiar world of the laboratory. But perhaps another line of criticism is more salient\u2014that of self-vindication. Ian Hacking has argued that, in the case of those kinds of science that \"study phenomena that seldom or never occur in a pure state before people have brought them under surveillance\" (1992a, 34), the setups of the laboratory science are \"self-vindicating\":\n\nAs a laboratory science matures, it develops a body of theory and types of apparatus and types of analysis that are mutually adjusted to each other.... They become... \"a closed system\" that is essentially unrefutable.... The theories of the laboratory sciences... persist because they are true to phenomena produced or even created by apparatus in the laboratory and are measured by instruments that we have engineered.... Our theories are at best true to the phenomena that were elicited by instrumentation in order to get a good mesh with theory. The process of modifying the workings of instruments\u2014both materially (we fix them up) and intellectually (we redescribe what they do)\u2014furnishes the glue that keeps our intellectual and material world together. It is what stabilizes science. (1992a, 30, 58)\n\nAnimal research in behavioral neuroscience certainly studies phenomena that do not occur in nature but are produced in the laboratory setup. So is such research also self-vindicating? If so, the translation problem would inescapably follow.\n\nThis is a question raised within the community of animal modelers themselves. Thus Helene Richter, together with Joseph Garner and Hanno W\u00fcrbel, argues that \"mounting evidence indicates that even subtle differences in laboratory or test conditions can lead to conflicting test outcomes. Because experimental treatments may interact with environmental conditions, experiments conducted under highly standardized conditions may reveal local 'truths' with little external validity\" (Richter, Garner, and W\u00fcrbel 2009). Could one move from such local truths\u2014the truths of a particular laboratory\u2014to the truths of the lifeworld of persons, or even the lifeworld of mice? From our perspective, the key test should not merely be a one-off demonstration of a capacity to translate, as, for example, in the claim that some pharmaceuticals tested on animals seem effective when used to treat analogous conditions in humans. We need an integrated way of escaping self-vindication by establishing a continual relation of transaction between the world of the laboratory and that lifeworld world of humans that it seeks to elucidate. If the laboratory neuroscience of animal models cannot establish those relations, then their problem, perhaps, is not a technical matter, to be rectified by tinkering in various ways, but a conceptual matter about the very nature of the animal models themselves as they function in behavioral neuroscience. We will return to this point later. First, however, let us turn to the third stabilization, that of the subjects, the animals themselves.\n\nThird Stabilization: The Animals\n\nThe stabilization of animals used in modeling has involved two aspects\u2014the selection of species to use in modeling, and the stabilization of the creatures within that species to render them into 'model animals' appropriate for experimentation. How does one choose one's model animal? No doubt some animals are exceptionally convenient for laboratory research on behavior. Curt Richter, in his charmingly titled piece \"Experiences of a Reluctant Rat Catcher,\" extols the Norway rat, which he studied for more than half a century, beginning in the 1920s: \"If someone were to give me the power to create an animal most useful for all types of studies on problems concerned directly or indirectly with human welfare, I could not possibly improve on the Norway rat\" (Richter 1968, 403). He lists the following advantages of his favored creature (377\u201378):\n\n_Norway Rat's Qualification for Behavior Studies_\n\nAt this point I should like to call attention to the advantages that the rat offers for behavior studies as well as for scientific studies in general.\n\n1. Its dietary needs are very nearly the same as man's. This explains how observations made on it can readily be extrapolated to man. It is a very stable and reliable animal.\n\n2. It reproduces readily.\n\n3. It is just the right size for all kinds of anatomical and physiological studies.\n\n4. Its short lifespan of about three years makes it possible to study factors involved in growth, development and aging.\n\n5. It is a very clean animal. Given the opportunity, it will keep itself perfectly clean.\n\n6. It is well equipped with tools for manipulation of its environment: with very sharp and strong teeth, very dexterous paws and feet, and strong hind legs.\n\n7. It is well equipped with sensory organs: for tasting, smelling, seeing, hearing, and tactile orienting (whiskers and tail).\n\n8. It is available in large numbers in both the domesticated and wild forms.\n\nWhen the _Scientist_ produced its supplement on \"Biology's Models\" in 2003, its account of the criteria for the choice of model organism was articulated in pragmatic terms very similar to Richter's. Introducing their collection of articles on the \"motley collection of creatures\" that \"fly, swim, wiggle, scurry, or just blow in the wind\" that \"for the scientific community... has been elevated above all other species,\" the editors argue that \"[r]esearchers selected this weird and wonderful assortment from tens of millions of possibilities because they have common attributes as well as unique characteristics. They're practical: A model must be cheap and plentiful; be inexpensive to house; be straightforward to propagate; have short gestation periods that produce large numbers of offspring; be easy to manipulate in the lab; and boast a fairly small and (relatively) uncomplicated genome. This type of tractability is a feature of all well-used models\" (Bahls, Weitzman, and Gallagher 2003, 5).\n\nBut how should one choose one's model organism? Is there one standard species that one should use to study everything, or are there some species that are particularly suitable for studying specific questions? For Hans Krebs, the answer was the latter: he articulated what he terms \"the August Krogh principle,\" which was that \"[f]or many problems there is an animal on which it can be most conveniently studied\" (Krebs 1975). From this perspective, close attention to diversity is an important feature of biological research on animals, because that enables one to select the particular animal most appropriate for the purpose at hand. Nonetheless, while a range of animals have been used as model organisms\u2014the fly, the worm, the zebra fish, the mouse, the rat, the guinea pig, the macaque\u2014the assumption has almost always been that what one is able to study most clearly in _each_ is an exemplar of principles and processes that have generality across _all_ \u2014or many\u2014species. One studies the fly or the mouse, not because one has a particular interest in the fly or the mouse, but because, one hopes that this will enable one to explore issues, processes, and functions that go beyond those specific animals. The particular species is thus not so much a model as an _example_ of general principles, no doubt based on the idea that such general principles are preserved, or conserved, across species because of the conservatism of evolution: \"Two premises underlie the use of simple organisms in medical research. First, most of the important biologic processes have remained essentially unchanged throughout evolution\u2014Rose. that is, they are conserved in humans and simpler organisms. Second, these processes are easier to unravel in simple organisms than in humans\" (Hariharan and Haber 2003, 2457). But are these assumptions so unproblematic?\n\nCheryl Logan, who has examined the history of test animals in physiology, argues that this presupposition of uniformity across species diversity arises, at least in part, from the pragmatics of the modern research process. The use of a small number of 'standardized' animal species and strains for experimental research began around the start of the twentieth century. Until then, Logan points out, experimenters tended to stress the importance of exploring their hypotheses in a number of different species: \"Experimental physiologists gave many reasons for the choice of test animals, some practical and others truly comparative. But, despite strong philosophical differences in the approaches they represented, the view that it was best to incorporate as many species as possible into research... was widespread.... Authors aimed for generality, but they treated it as a conclusion that would or would not follow from the examination of many species\" (Logan 2002, 329). However, there were numerous criticisms of the variations in results found between different laboratories, and Logan argues that this was partly responsible for an increasing emphasis on standardization, which was also linked to the growth and industrialization of scientific research, and the commercialization of animal breeding for research. This, she suggests, led to a restriction on the number of species used. \"As animals were increasingly viewed as things that were assumed to be fundamentally similar, scientific generality became an _a priori_ assumption rather than an empirical conclusion\" (Logan 2002, 329). The standardized animals were thus regarded as laboratory materials, along with other materials that had to be combined in an experimental setup, and\u2014like those other materials, measures, scales, and instruments\u2014their uniformity was both required and, after a while, assumed. If they were obtained from reputable sources, they were, indeed, stable and stabilized.\n\nThe animals conventionally used as experimental materials have been raised through many generations of interbreeding of closely related individuals in order to create organisms that can be treated as interchangeable in repeated experimental situations. This has required much labor\u2014often a labor of love\u2014over many years and many generations\u2014as the historical studies by Robert Kohler of the fly (Kohler 1993, 1994) and Karen Rader of the mouse (Rader, 1998, 2004) have shown. Thus Evelyn Fox Keller is uncharacteristically misleading when she suggests that \"unlike mechanical and mathematical models (and this may be the crucial point), model organisms are exemplars or natural models\u2014not artefactually constructed but selected from nature's very own workshop\" (Keller 2002, 51). For while those species that are to become models may be selected from nature, the model animals are indeed artifactually constructed, and it takes a lot to sustain them as such\u2014not only to breed, feed, and manage them, but crucially to standardize them and the experiments undertaken with them\u2014it takes animal houses, trained persons, specially adapted feed cages, constant tending and familiarity, and much more. We thus agree with Canguilhem when he writes that \"the study of such biological materials, whose elements are a given, is literally the study of an _artefact_ \" (Canguilhem [1965] 2008, 13). We have here another problem of translation, for even the translation between the stabilized model animal and its wild type contemporaries is assumed rather than demonstrated. This brings us directly to the issue of models themselves.\n\n_Models 1, Models2, Models3, Models4_ (and Possibly _Models 5_)\n\nThere are, of course, many types of models. For our current purposes, we would like to distinguish four: conceptual models ( _models_ _1_ ); mechanical or material models ( _models_ _2_ ); animal models of structures, functions, and diseases ( _models_ _3_ ); and animal models of behavior ( _models_ _4_ ). And perhaps there is even a fifth: animal models of human psychopathology ( _models_ _5_ ). Most epistemological work has been done on the first two of these\u2014conceptual models and mechanical models. A conceptual model\u2014 _models_ _1_ in our little classification\u2014such as those in theoretical physics, is a kind of diagram of theoretical entities and the dynamic relations between them, which can be manipulated in thought\u2014nowadays often in silica on-screen or in virtual reality\u2014by varying values and the like in order to explore their consequences. Nancy Cartwright suggests that, in such cases, \"the model is the theory of the phenomenon\" (Cartwright 1983, 159). Or, in a slightly different vein, that we should view such models as instruments that mediate between experiments and theory, not to be evaluated by their truth but by their ability to perform in particular experimental and practical contexts, \"go-betweens\" between the world of things and the world of theories (Cartwright, 1997; Morgan and Morrison 1999). Others have suggested their value is more practical\u2014as Evelyn Fox Keller puts it, a model \"suggests experiments, it enables one to predict the consequences of particular interventions, and it prompts the posing of new questions\" (Keller 2000, S78).\n\nThe brain sciences have not been without their conceptual models that perform such functions. For example, the catecholamine hypothesis of affective disorders discussed in chapter 1, formulated by Schildkraut in 1965 explicitly as a heuristic device, brought together a range of empirical evidence and conceptualization into a diagram of the mode of action of the monoamines, and this functioned as a model to organize much research and explanation in this area for some four decades (Schildkraut 1965). But, as this example illustrates, a model of this sort, which abstracts and generalizes from the experimental moment and which then structures the path of future experiments and the interpretation of results, runs the risk of becoming self-vindicating. Indeed, the life span of the monoamine hypothesis of depression is a very good case in point. Simplified and abstracted, and represented in icons and simulations (Stahl 1996), this theory structured clinical and experimental thought and practice and underpinned drug development and the interpretation of the effects of the drugs so developed (Kramer 1992).\n\nMost recent work in the history of science and technology has focused on mechanical or material models\u2014which we think of as _models_ _2_. Soraya de Chadarevian and Nick Hopwood have referred to these kinds of models as \"the third dimension of science.\" Their focus is on such things as wooden ships and plastic molecules, displays of stuffed animals, artfully constructed model bridges and other artifacts that \"purported to bring the tiny, the huge, the past, or the future within reach, to make fruitful analogies, to demonstrate theories.... [A]ll are three-dimensional (3-D), and as such, their advocates claimed, displayed relations that could not easily be represented on paper\" (de Chadarevian and Hopwood 2004, 1). Such models combine a certain sense of realism with the advantage of docility and manipulability. One can hold them in one's hands, examine them from different perspectives, and subject them to various stresses and other insults to see how they respond.\n\nBut the history of biology\u2014more or less since the invention of the term at the start of the nineteenth century\u2014has been coextensive with the use of a different kind of model\u2014one that we term _models_ _3_. These use certain animals, or parts of animals, to elucidate what are thought to be the basic features of the structure and function of organs, tissues, systems, cells, and other corporeal elements, which are believed to be generalizable to other living creatures, notably to humans. This has certainly been the case in research on the nervous system. The frog was the animal of choice for the eighteenth-and nineteenth-century practice of neurophysiology. Frogs were used for the study of the concept of reflex and the discovery by Galvani of \"animal electricity,\" thus displacing the long-standing belief in \"animal spirits\" (Meulders 2006; Harr\u00e9 2009), and paving the way for the birth of a new avenue of inquiry: electrophysiology. In the 1950s, after his work with Eccles and Kuffler in Australia in the 1930s, Bernard Katz used the neuromuscular junction of the frog as a model system to explore synaptic transmission.\n\nThe squid was also a model of this sort\u2014or at least its giant axon was. Initially discovered by J. Z. Young in 1936 (Keynes 2005), the giant squid axon was subsequently used by several prominent scientists in the twentieth century, notably Alan Hodgkin and Andrew Huxley, who shared the Nobel Prize in Physiology or Medicine in 1963 for uncovering the ionic mechanisms in nervous cells. One chose the animal on the basis of its suitability for the purpose at hand\u2014for instance, Kuffler, and a number of those who worked with him in the 1960s, chose the medicinal leech because of its relatively small number of neurons, whose size enabled them to be observed using an ordinary microscope and to be recorded by using intracellular microelectrodes. Not that the researcher was interested in the leech\u2014or the crayfish, the lobster, the mudpuppy, or the sea slug, to enumerate some of the creatures popular with neurobiologists\u2014but because each in some way provided a simple system for studying basic neurobiological mechanisms believed to be universal and hence generalizable. And, it was hoped, one would eventually be able to build up from such basic knowledge of isolated specific mechanisms to a knowledge of the whole\u2014this was the promise, and the premise, of neuroreductionism.\n\nThese brief examples serve to demonstrate that model organisms are rather different from the classes of models that we have termed _models_ _1_ and _models_ _2_. This is, in part, because they do not exist in just three dimensions\u2014as in the model of a bridge, for example\u2014but in four. The fourth dimension is time. They have\u2014indeed they exploit\u2014the added feature of vitality\u2014birth, development, change, and death. Model animals are certainly 'simplifications' in one sense: because they are thought to have eliminated much of the genetic variation that is found in the wild type and because of the apparent standardization of their rearing and their living environment. But like other models, they are complex and heterogeneous products of persons, artifacts, theories, concepts, apparatus, and much craft work\u2014to use the term coined by Carrie Friese, they are \"model assemblages\" wherein heterogeneous things with varying historical legacies are brought together in order to pursue certain goals (Friese 2009).\n\nYet despite that labor of standardization, the animals used as _models_ _3_ remain individual living creatures. It may be hard to think this of the fly, the worm, or the zebra fish, but when one reaches rodents, it is hard to ignore the fact that each mouse is a unique life form struggling to survive in its own way. Animals, unlike most other model entities, are also dynamic, homeostatic, self-regenerating, and self-compensating, continually adapting in various ways to internal and external signals. As Canguilhem put it in his essay titled \"Machine and Organism,\" animals are polyvalent: \"An organism has greater latitude of function than a machine.... Life... is experience, that is to say, improvisation, the utilization of occurrences\" (Canguilhem [1965] 2008, 90). Indeed, it is often this internal systemic vitality that is precisely the issue for investigation. Perhaps this is why, to paraphrase Canguilhem again, it is only of living creatures\u2014and not of machines\u2014that one can distinguish the normal and the pathological (ibid., 88\u201390).\n\nThis vitality, this purposiveness, this 'holism,' this teleology of each individual organism has implications for the particular problems of animal models: the practice of treating one organism as a model for another. Every biology student has heard of Fran\u00e7ois Jacob's confident assertion that what is true for _Escherichia coli_ is also true for the elephant (Jacob 1988). Canguilhem takes a different view. In his classic paper \"Experimentation in Animal Biology,\" he emphasizes the specificity of the animal that is used in each experimental demonstration:\n\nIt must be said that, in biology, the specificity of an object of observation or experiment limits unpredictably any logical generalization.... [I]t would often be both prudent and honest to add to the title of a chapter of physiology that it concerns the physiology of a certain animal, so that the laws of phenomena, which almost always bear the name of the person who formulated them, would also bear the name of the animal used for the experiment: the dog for conditioned reflexes, the pigeon for equilibration, the hydra for regeneration, the rat for vitamins and maternal comportment, the frog (that \"Job of biology\") for reflexes, the sea urchin for the fertilization and segmentation of the egg, the drosophila for heredity, the horse for the circulation of blood. (Canguilhem [1965] 2008, 11)\n\nIn the experimental approach of biology, he argues, we need to accept \"that no experimentally acquired fact (whether it deals with structures, functions, or comportments) can be generalized either from one variety to another within a single species, or from one species to another, or from animal to man without express reservations\" (Canguilhem [1965] 2008, 12). When it comes to the action of caffeine on striated muscles, for example, what is true for the green frog is not for the red frog; when it comes to the reflex, the cat differs from the rabbit but resembles the newt, and when it comes to the repair of bone fractures, the normal development of the callus in humans differs from that in dogs. That is to say, there can be no _assumption_ of common mechanisms between different species or strains of animals: even rather basic mechanisms are _not_ conserved by evolution but differ, even between closely related varieties, let alone between animals and humans.\n\nIf Canguilhem may be regarded as a rather obsolete source, more recent studies have similar findings. Thus Andrea Gawrylewski, in her article \"The Trouble with Animal Models,\" says: \"A recent study at the Massachusetts Institute of Technology shows distinct differences between gene regulation in humans and mouse liver, particularly how the master regulatory proteins function. In a comparison of 4,000 genes in humans and mice, the researchers expected to see identical behavior, that is, the binding of transcription factors to the same sites in most pairs of homologous genes. However, they found that transcription factor binding sites differed between the species in 41% to 89% of the cases\" (Gawrylewski 2007, 44).\n\nThis issue becomes more important when one begins directly to approach issues of disease. _Models_ _3_ have long been used to study human disease, either by making use of an apparently analogous pathology that occurs spontaneously in the animal, by breeding the animal to manifest the disease, or otherwise inducing it experimentally. Mice, for example, have served as models for the study of cancer since the elucidation of the nature of leukaemias in the bone marrow of rats in 1933 by the French oncologist and hematologist Jean Bernard (Degos 2006; Irino 2006). However, translation is not as straightforward as this might imply. As Francesco Marincola put it in the opening editorial of his new _Journal of Translational Medicine_ , animal models \"do not represent the basic essence of human diseases.... Prestigious journals, however, appear more fascinated with the modern mythology of transgenic and knock-out mice than the humble reality of human disease\" (Marincola 2003, 1, 3). Or, in the words of Thomas Insel, director of the National Institute of Mental Health (NIMH), in a paper titled \"From Model Animals to Animal Models\":\n\n[M]any biological psychiatrists seem to assume that a mouse is a small rat, a rat is a small monkey, a monkey is a small human, and that all of these are \"models\" for studying abnormal behavior or abnormal brain function in humans. The failure to attend to species differences not only ignores the opportunity for understanding mechanisms of diversity, it will doom anyone who wants to make facile comparisons. The broad use of laboratory mice and rats for clinical drug development has demonstrated repeatedly that not only are these species inconsistent predictors of clinical response, they are poor predictors of toxicity in humans. (Insel 2007, 1337)\n\nInsel provides a convenient link to what we have termed _models_ _4_ : animal models of _behavior_. The use of such models dates at least back to the nineteenth century. David Ferrier's studies in the 1870s, carried out while he was the director of the laboratory of experimental neurology at the West Riding Lunatic Asylum, involved ablating cortical regions in dogs and monkeys in the 1870s to cast light on the issue of the localization of functions in the human brain. Better known are the experiments in Pavlov's laboratory in 1912, notably the work of Yerofeeva on behavioral disturbances in dogs caused by conditioning techniques, which were followed by Pavlov's own experiments from the early 1920s on \"experimental neurosis\" (Broadhurst 1960; Abramson and Seligman 1977). That type of neurosis was very different from the human neuroses as they were conceived by Freud or the other dynamic psychologies of the time\u2014the Pavlovian neurosis was either \"excitatory\" or \"inhibitory\" as opposed to the \"ideal normal type in which both opposing nervous processes exist in equilibrium\" (Pavlov 1941, 73).\n\nRelated work was carried out in the United States in the early 1940s, where a number of research groups created experimental neuroses and other conditioned reflexes primarily in sheep. Models, here, were experimental me-diums through which a behavior, like salivation or other physiological phenomena like respiration and heart rate could be measured and tested. From this point onward, we begin to see the use of animal models for therapeutic investigation. For example, in the 1930s, phenobarbital, a sedative drug, was investigated by Liddell and his group, who tested its effect on the \"excitatory neurosis\" in sheep (Liddell, Anderson, Kotyuka, et al. 1935). Pavlov also used bromides, another sedative drug, in the treatment of the experimental neurosis of his dogs (Pavlov 1941). Indeed, experimental neurosis was investigated in all kinds of species\u2014sheep, pigs, rats, cats, and monkeys\u2014with a view to the psychiatric relevance of the findings: this was an \"experimental approach to psychiatry\" (Gantt 1936, 1942, 1953; Liddell 1938, 1947). This brings us to Harry Harlow, whose early work was on the effects of brain lesions and radiation on learning in monkeys, but who is now best known for his studies of infant macaque monkeys and their wire and cloth mother surrogates, as models for the importance of mother love for normal human development\u2014or, in the words of the title of his presidential address to the sixty-sixth annual convention of the American Psychological Association\u2014on \"the nature of love\" (Harlow 1958).\n\nNeurosis, anxiety, depression, love\u2014how is one to characterize the _human states_ that the animal models model? Let us take one example\u2014aggression. Jacqueline Crawley opens her chapter \"Emotional Behaviors: Animal Models of Psychiatric Diseases\" with the firm injunction \"Step one: _Don't anthropomorphize!_ \" (Crawley 2007, 227). In her discussion of animal models of emotional behavior, for example, in anxiety research, she expresses great caution about the dangers of anthropomorphization; we must always talk not of anxiety in a mouse but of \"anxiety-like behavior.\" But surprisingly, in her discussion of \"social behaviors\" she does not write of \"aggression-like behavior\" but of \"aggressive behavior.\" Perhaps we can learn something from this apparent contradiction.\n\nIt seems that, for Crawley and many others working in this field, aggressive behavior can be identified without reference to internal states. The behavior shown by mice in the lab is self-evidently aggressive and reflects that which is present in aspects of the \"natural mouse repertoire\": namely, attacks that \"serve the functions of defending progeny, mates, and self from attack by a predator, defending the resources of the home territory, gaining access to a sexual partner, and achieving dominant status within the social group\" (Crawley 2007, 209). It is on this basis that strains of mice can be developed that are more or less aggressive, and knockouts can be created, entailing null mutants in many of the genes affecting aspects of the neurotransmitter system\u2014subtypes of serotonin receptors, monoamine oxidase, the serotonin transporter, nitric oxide, adrenergic receptors, adenosine receptors, Substance P, vasopressin, and many more. On the basis of the behavioral definition of aggression in mice, these variations certainly increase aggressive behavior\u2014in mice. But what relevance does this have for the kinds of actions in humans that we think of as aggression? In what sense are these artificially produced, engineered mice, developed in close association with the setups, apparatus, and testing procedures that measure their behavior, a mouse model of human aggression?\n\nAnd, while \"don't anthropomorphize\" might be a maxim ritually invoked, how about the following breathless translation between species when it comes to aggression:\n\nPsychopathological violence in criminals and intense aggression in fruit flies and rodents are studied with novel behavioral, neurobiological, and genetic approaches that characterize the escalation from adaptive aggression to violence.... One goal is to delineate the type of aggressive behavior and its escalation with greater precision; second, the prefrontal cortex (PFC) and brainstem structures emerge as pivotal nodes in the limbic circuitry mediating escalated aggressive behavior.... By manipulating either the fruitless or transformer genes in the brains of male or female flies, patterns of aggression can be switched with males using female patterns and vice versa.... New data from feral rats point to the regulatory influences on mesocortical serotonin circuits in highly aggressive animals via feedback to autoreceptors and via GABAergic and glutamatergic inputs. Imaging data lead to the hypothesis that antisocial, violent, and psychopathic behavior may in part be attributable to impairments in some of the brain structures (dorsal and ventral PFC, amygdala, and angular gyrus) subserving moral cognition and emotion. (Miczek, de Almeida, Kravitz, et al. 2007, 11,803)\n\nWhile few make the leap from flies to humans in quite such a startling manner, there is a long history of ethological studies of aggression, with many arguing for its universality across species and drawing analogies between humans and animals. We see this, for example, in the work of John Paul Scott (Scott 1958), but the most emblematic of authors here is Konrad Lorenz (Lorenz 1964; see also Ardrey 1966; Tinbergen 1968). J. Z. Young notes, in his classic _Introduction to the Study of Man_ , that the approach of these authors was undoubtedly shaped by the doleful history of human conflict across the twentieth century and indeed by the threat of nuclear war in the 1960s (Young 1974). Even among ethologists, arguments that human conflict manifests internal states of aggression, even in animals, were highly contentious and were criticized by eminent ethologists such as Robert Hinde (1967, 1970).\n\nOf course, most of those from the social sciences were skeptical to say the least. And with good reason. Many circumstances commonly thought of as human aggression\u2014fighting in wars, violence between gangs of youth, knife crime, for example\u2014are not necessarily accompanied by an internal mental state of aggression. Soldiers fight for their comrades, rarely because they feel aggressive toward their enemies; indeed, an aggressive soldier is often a liability. Gang members fight for social recognition or esteem, and for many other reasons related more to personal anxieties and social values than to aggression. Knife crime is as much a matter of fear as it is of aggression, and it of course depends on fashion, availability, town planning, alcohol, and much more. Suicide bombers kill as an act of faith or resistance and rarely out of misanthropy. Genocide, especially where groups mobilized by nationalist or other rhetoric brutally murder others with whom they have lived side by side for years, is another example that indicates that even apparently highly aggressive actions by human beings are shaped and often instigated by language and the mobilization of meanings, ideologies, and memories.\n\nAre these actions based on the same evolutionary mechanisms that generated aggressive behavior in mice? Doubtful. Do drugs that reduce aggressive behavior in mice do the same in humans? Alcohol (when one can get a mouse to ingest it) does not seem to increase aggression in mice, while it certainly facilitates aggressive actions in some humans\u2014but only in some circumstances (in other circumstances it facilitates singing, sex, or sleep). Is aggression part of \"the natural human repertoire\"? Given the enormous historical, geographical, cultural, and religious variations in what we consider human aggression, and the varieties of types of behavior that are often grouped under that one name, it seems problematic to place all this under one common heading and to imagine common neural underpinnings, let along to seek correlates of superficially analogous behavior in other species and then to grant those correlates a causal status. And even if such behavior did have neural correlates, at what point and in what contexts would it become pathological? And, more relevant to the present discussion, what are the analogies between the 'natural' repertoires placed under this description in animals and those called aggression in humans? For example, according to Crawley, observations of aggression in mice entail such behaviors as general body sniffs, anogenital sniffs, tail rattling, rump biting, and allogrooming (Crawley 2007, 213\u201317): behaviors hardly thought of as components of aggression in humans, although some would undoubtedly elicit a punch on the (human) nose if displayed on the streets of London or New York on a Friday night.\n\nThis brief overview of some of the more obvious problems in using animal models to explore the genetic or neural correlates of human aggression brings us to _models_ _5_ \u2014animal models of human psychiatric disorders. It is commonly thought that chlorpromazine, the first of the modern generation of psychiatric drugs, was developed by serendipity while the French pharmaceutical company Rh\u00f4ne-Poulenc was seeking potential molecules that might be marketed as antihistamines. However, even at this early point, the properties of novel molecules were first explored through experiments on animals. The first experimental setups that used the elevated plus maze to explore anxiety in animals began in the 1950s, but it was not until the 1980s that this apparatus was validated as a _measure_ of anxiety and then used to assess the efficacy of anxiolytic drugs (Pellow, Chopin, File, et al. 1985; Pellow and File 1986). The role of the third edition of the _Diagnostic and Statistical Manual of Mental Disorders_ ( _DSM-III_ ) published in 1980 by the American Psychiatric Association was crucial here (American Psychiatric Association 1980). As we discuss in detail in chapter 4, _DSM-III_ and its successors provided checklists of apparently objectively observable phenotypic features that could then be reproduced in animals. That is to say, one could try to produce a breeding line of mice or other animals that would exhibit these overt and visible behavioral characteristics, initially by conventional breeding techniques or by the use of drugs to modify animal behavior and later via transgenic and knockout technologies.\n\nThis procedure produces a paradoxical looping effect, in which the testing regime plays a key role. Suppose one is trying to develop a psychiatric drug for a given condition included in the _DSM_. One first uses selective breeding or genetic modification to produce a strain of mice that show analogous behaviors to those listed in the _DSM_ as criteria for diagnosis of the disorder in humans. Whether or not they do have those analogous characteristics is assessed, in large part, by their performance on certain tests. For instance, the elevated plus maze is used to assess anxiety by measuring how much time a mouse or other rodent spends in the exposed arms of the maze, as opposed to the enclosed parts\u2014presumably on the basis of an simple analogy that an 'anxious' mouse will 'prefer' to be in an enclosed space, while a less 'anxious' mouse will be more ready to explore. The forced swim test and the tail suspension test are thought to assess depression, based on the idea that depression in humans is characterized by 'learned helplessness'\u2014that is to say, the tendency for the depressed person to give up rapidly in the face of obstacles. Hence, it is suggested, one can assess the level of 'depression' in a mouse by measuring the length of time that it will continue to swim and make attempts to escape when immersed in a bath of cold water before switching to mere floating, or the length of time that it will struggle when suspended by its tail before merely hanging passively: each is taken as a measure of learned helplessness and hence of depression.\n\nOne then takes individuals from the strain of mice that seem to exhibit high levels of anxiety or depression, administers the drug to them, and sees whether the anxious or depressed behavior is mitigated. Once more, one uses the same tests\u2014elevated plus maze, forced swim test, tail suspension test, and so forth\u2014to assess the behavioral change. If the results are positive, then (subject to all the necessary protocols and phases of drug regulation) one trials the drugs on humans who are selected for the trial on the basis of the _DSM_ criteria. One then interprets the results of these trials as evidence that the drug is indeed treating the condition that one has modeled in the animal. One then explores the gene expression pattern, the patterns of brain activation, or the structural features of the brains of the animals, in an attempt to learn about the molecular mechanisms of the condition in humans. The _DSM_ diagnosis is thus the fulcrum of such a setup. The self-vindicating loop is designed to find the neural correlates of the _DSM_ category that shaped the research design in the first place. And, given that such experiments seek to reduce, eliminate, or ignore all factors other than those modeled in the individual animal\u2014that is to say, those that can be thought of as internal to its genome and brain\u2014the translation to the humans entails an almost inescapable reduction of human action and human affect to expressions of such somatic determinants.\n\nCrawley titles her excellent book on animal models in behavioral neuroscience _What's Wrong with My Mouse?_ because this is the question repeatedly addressed to her by researchers who are trying to work out what they have actually modeled when they have genetically modified an animal (Crawley 2007). Have they produced a mouse (and hence a potential mouse line) that is depressed, anxious, aggressive, schizophrenic, or what? The process of characterizing an animal phenotype of a human disorder created by genetic manipulation is no easy matter, even in the hands of an expert such as Crawley:\n\nEvaluating the behavioral phenotype of a new line of transgenic or knockout mice involves a multi-tiered approach that begins with the systematic assessment of general health, home cage behaviors, neurological reflexes, sensory abilities, and motor functions. Intact performance on these initial measures of procedural abilities allows investigators to proceed with specific complex tests in the behavioral domain of interest (e.g. anxiety-like tests, feeding, learning and memory tasks). If preliminary analyzes indicate deficits in procedural abilities, then investigators may be able to select or modify complex behavioral tests to avoid false-positive results and artifactual interpretations of genotype differences.... In addition, careful attention to the baseline behaviors of the background strain(s), and to the possibility that flanking genes and environmental factors may affect phenotypic effects, will maximise the correct attribution of interesting phenotypic anomalies to the targeted gene mutation. (Bailey, Rustay, and Crawley 2006, 129)\n\nBut this technical sophistication does not remove the fundamental problem. Even if one goes through all these onerous steps, what is one to say of the relation between the phenotype that accompanies the knockout and the human actions and emotions that it is supposed to model? How is that translation accomplished?\n\nThomas Insel has some cautionary words on such an approach. Taking Fragile X syndrome as an example, he argues that the animal model approach\n\nmight have created a cognitive deficit in a mouse or rat to resemble Fragile X... and then hunted for altered molecular and cellular function that could be targets for drug development. This approach to model Fragile X might work if cognitive deficits were simple lesions with a few predictable causes, as, for instance, in a model of ischemic stroke. This approach might work if we knew that the deficit created in the mouse was identical to the cognitive deficit in the human syndrome, as, for instance, in a lesion model of Parkinson's syndrome. This approach might work if we knew much more about the pathophysiology of the human syndrome, as, for instance, in diabetes. The problem for autism, anorexia nervosa, schizophrenia, and mood and anxiety disorders is that we know so little about the biology of these disorders; there is no credible way to validate (or to falsify) an \"animal model\" to learn more about the pathophysiology of the human syndrome. (Insel 2007, 1338)\n\nThe problem, in Insel's view, is that these animal models are analogies rather than homologies\u2014they produce a phenotype that superficially resembles the condition in humans, but the condition in humans is complex, with multiple causes; in the absence of a clear understanding of the pathophysiology, the model animal is merely a simulacrum. Insel himself has some suggestions as to how to overcome such simulation-based experimental practices, and hence, he believes, to have more success in translation from the laboratory to the clinic and from animals to humans. But before turning to these, let us consider the primary objection from the human sciences: the specificity of the human.\n\nThe Specificity of the Human\n\nA belief in the specificity of human beings has been foundational for the social and human sciences. Humans are unique among living beings, it is argued, in having language, culture, and history. Human action differs from animal behavior as a wink differs from a blink: the one has meaning, the other, though it involves the same set of physiological pathways, does not. Even behavior in humans that resembles that in animals bears only a superficial similarity, as human action embodies intention, and intentions are linguistically and culturally shaped. Human affects may seem similar to those of animals, but that is misleading. Although, as Darwin argued in the late nineteenth century, there are similarities in \"the expression of the emotions in animals and man\" (Darwin [1872] 2009), there are also very significant historical and cross-cultural variations in the recognition, experience, naming, and consequences of human emotions (Lutz and White 1986; Reddy 2001; Oatley 2004). Even the expression of pain in humans, is a communicative act, as Erving Goffman points out in his wonderful little essay on response cries. The utterance \"ouch\" on hitting one's thumb with a hammer is a call for attention, a request for sympathy, a complaint about having been required to undertake a task unwillingly (\"now see what you've made me do!\"); it is not an immediate physiological and instinctive response to the sensation of the blow (Goffman 1978).\n\nThe role of historical, social, and cultural processes is even more evident in human psychopathologies, which are shaped, organized, differentiated, and framed in terms of language and meaning. The categories of the diagnostic manuals certainly do not 'cut nature at its joints,' and psychiatric phenotypes from schizophrenia to anxiety are notoriously heterogeneous. The thresholds of disorder\u2014where one might mark the divisions of normality and pathology\u2014are difficult to define and vary across history and culture, as is shown in the difficulties in delimiting attention deficit hyperactivity disorder or antisocial personality disorder, or in differentiating clinical depression from normal sadness. Features often considered to be unique symptoms of psychiatric disorder, for example hearing voices, often turn out to be quite widespread in the population, and only contingent factors lead to them being treated as diagnostic indicators (Romme and Escher 1993). Cognitive psychiatrists, whatever one may think of their general claims, are surely correct in the argument that beliefs are of crucial importance in the development and experience of psychiatric symptoms, and there is some plausibility to their argument that \"disruption or dysfunction in psychological processes is a final common pathway in the development of mental disorder [and that] biological and social factors, together with a person's individual experiences, lead to mental disorder through their conjoint effects on those psychological processes\" (Kinderman 2005, 206).\n\nThus, until about a decade ago, the standard social science position would be to deny, and decry, all endeavors that sought to account for human thoughts, beliefs, actions, and psychopathologies by relating them to supposedly analogous matters in nonhuman animals. Anthropomorphism was a cardinal error, as was its reciprocal, the biologization of the human. Such biologization, or animalization, was the province of those deemed reactionary, part of a lineage linked historically with racism and eugenics, and more recently with the reductionist simplicities of evolutionary psychology and sociobiology, with their beliefs in fixed human capacities and predilections, laid down over millennia, and merely played out in human social life and relations.\n\nBut perhaps things are changing. This is not merely because animal rights activists have repeatedly challenged the privilege that humans accord to themselves over other animals and ecologists have located humans squarely within a natural ecosphere. It is also because a theoretical avant-garde is beginning to question the dichotomy between animals and humans, which they now see as perhaps the final unchallenged legacy of the Enlightenment. Donna Haraway has pointed to the ways in which humans have evolved along with their companion species and has explored the anthropomorphism that seems endemic to the human condition itself\u2014which may have provided some benefits for humans, though probably not for animals (Haraway 2003, 2007). But further, today, the most radical theoretical move of all is to question the ontological divide between humans and animals. For is it not the final rebuff to the narcissism of the human, to recognize that we are, after all, just animals\u2014that the divide between humans and all other beings needs to be overcome, along with all the other humanist dichotomies of self and other, reason and passion, mind and body, society and nature, organism and machine (Wolfe 2009)?\n\nFor present purposes, we do not need to follow that line of argument to its conclusion. We merely need to refuse the simplifications that see human animals as the only living creatures who exhibit the features of vitality that we have pointed to earlier\u2014polyvalence, individuality, improvisation, and constant dynamic transactions with their milieu. What would it mean, then, for those who conduct this work with animals in psychiatry to 'think with the animal' in this sense\u2014to locate their experimental results, and, even more important, their experimental procedures, in the mental life of the animal, and indeed in the form of life of the animal? Perhaps this is impossible. We might recall Wittgenstein once more, and ask whether, if a lion could talk, we would understand him (Wittgenstein 1958, 223). But we should not satisfy ourselves by repeating the nostrum that only humans have language or that animals' communication is radically different, a matter of signals eliciting 'fixed action patterns' rather than of signs embedded in systems of meaning. A host of evidence argues to the contrary, from Gregory Bateson's studies of play among mammals in his essay \"A Theory of Play and Fantasy\" (Bateson 1972) through discussions of linguistic competence in great apes (Maturana and Varela 1987) to studies of the domestication and training of animals, such as Paul Patton's intriguing use of his knowledge of French philosophy to explore his communication with his horse in the practice of dressage (Patton 2003).\n\nAnthropomorphism is almost inescapable among those who work with animals, whether in or outside the lab. But perhaps we can learn a better way of attributing mental life to animals, a better way of thinking with the animal. If we were to couple this kind of attitude in our animal experimentation with a genuine zoomorphism, then, this would challenge the standard social science critique of modeling humans, in whole or in part, in animal experimentation. Not that humans and animals would be rendered equivalent, but that the capacities, pathologies, and behaviors of animals, like those of humans, would have to be located in their form of life and their constant dynamic interchanges with the specificities of their milieu. The price of breaching of this imagined boundary is, of course, to trouble the ethical sensibilities of those who undertake such experiments\u2014though such concern with the welfare of laboratory animals is somewhat hypocritical in a culture where so many living creatures are bred for slaughter to serve human desires. But even if one can get beyond the old oppositions between anthropomorphism and anthropodenial, the assumptions underlying animal models and model animals remain problematic and need to be opened to empirical examination. And that brings us back to the place where that examination of the distinctions between humans and other animals has consequences for medical and psychiatric practice\u2014the question of translation.\n\nTranslation\n\nMany leading biological, biomedical, and neuroscientific researchers have criticized the conception of translational medicine as a pathway \"from bench to bedside\" and suggested that their criticism might in turn challenge the usage of animal models. Thus Sydney Brenner has argued, \"I think that the focus should be the other way around. We should have \"from bedside to bench\" research by bringing more basic science directly into the clinic. We should also stop working exclusively on model organisms like mice and shift the emphasis to humans, viewing each individual patient as an experimental model\" (Brenner 2008, 9).\n\nLeading neuropsychiatric researchers such as Steven Hyman, former director of the NIMH, and Thomas Insel, its current director, share Brenner's critical assessment of much work with animal models, and his view that the failures of results from animal research to translate to effective understanding of the molecular underpinnings of human disorders, let alone to result in the radical improvement of pharmaceutical or other treatments that was promised, should lead us to question some of the fundamental premises on which this work has been based. Thomas Insel, in the paper discussed earlier, suggests that a more effective approach would be what he terms \"reverse translation,\" starting not from the creation of model organisms whose behavior bears a phenotypical resemblance to a human psychopathology, but from the identification of the molecular pathways in humans with mental disorders, and then modeling those pathways in animals: \"The good news is that studies of mental disorders might finally be able to exploit reverse translation, moving from studies in humans with a mental disorder to understand how risk genes or abnormal proteins alter brain development and function.... The next decade should see a shift from creating abnormal animal behaviors that phenotypically resemble aspects of mental disorders (i.e., animal models) to increasingly understanding the mechanisms of disease by developing animals with the molecular and cellular abnormalities found in mental disorders (i.e., model organisms)\" (Insel 2007, 1339).\n\nAt first sight this seems a very significant conceptual shift, embodied in a distinct idea of the model, and a distinct setup. One identifies molecular anomalies in humans diagnosed with a condition such as schizophrenia; one seeks to find their genomic basis, perhaps using genome-wide association studies (GWAS); and one then tries to produce animals with a similar genomic anomaly and investigates to see if they show the same disrupted molecular pathways and also display the same behavioral anomaly\u2014if so, one has grounds for arguing that one has modeled one element of the pathway for the disorder. Once more, however, this runs into the problem of replicating\u2014perhaps self-vindicating\u2014a particular _DSM_ -derived model of a psychiatric disorder as an amalgam of discrete elements, each of which has a specific neurobiological basis. Insel himself seems to hint at that problem:\n\nConditioned fear in a rat is not an anxiety disorder, and social defeat in a mouse is not major depressive disorder. Yes, we have made great progress in \"models\" of individual features of psychiatric syndromes. For example, in common laboratory animals, conditioned fear has been highly informative of human fear. But how well does this paradigm serve as a model of any human syndrome? At best, we can focus on singular features of disorders, such as vulnerability to conditioned fear, assuming that human syndromes are composites of isolated abnormal behavior, cognition, or emotion. This assumption remains unproven. (Insel 2007, 1337)\n\nBut even so, this argument retains the idea of identifying singular features and of isolating, characterizing, and modeling what are taken to be the distinct components of a mental disorder. Perhaps we should remind ourselves of a caution expressed by Claude Bernard over 150 years ago in his _Introduction to the Study of Experimental Medicine_ : \"Physiologists and physicians must never forget that a living being is an organism with its own individuality. We really must learn, then, that if we break up a living organism by isolating its different parts, it is only for the sake of ease in experimental analysis and by no means in order to conceive them separately. Indeed when we wish to ascribe to a physiological quality its value and true significance, we must always refer it to this whole and draw our final conclusion only in relation to its effects in the whole\" (Bernard [1865] 1957, 88\u201389). For Bernard, then, when it comes to a living creature, the whole is not merely the sum of its parts. Even if we had an effective animal homology, based on a specific neurobiological pathway, for one aspect of the particular behavior in question, can we isolate one feature from the individuality of the organism as a whole? How can we avoid the self-vindication of _DSM_ -type neuropsychiatry?\n\nPerhaps this problem is best illustrated by an example. In July 2007, a team from Johns Hopkins University published a paper that was referred to as \"the first mouse model of schizophrenia\" in the press release that accompanied it: \"Johns Hopkins researchers have genetically engineered the first mouse that models both the anatomical and behavioral defects of schizophrenia, a complex and debilitating brain disorder that affects over 2 million Americans. In contrast to current animal studies that rely on drugs that can only mimic the manifestations of schizophrenia, such as delusions, mood changes and paranoia, this new mouse is based on a genetic change relevant to the disease. Thus, this mouse should greatly help with understanding disease progression and developing new therapies.\"\n\nThe work, by Akira Sawa's lab, was published in a paper titled \"Dominant-Negative DISC1 Transgenic Mice Display Schizophrenia-Associated Phenotypes Detected by Measures Translatable to Humans\" (Hikida, Jaaro-Peled, Seshadri, et al. 2007). The researchers generated mice that make an incomplete, shortened form of a protein thought to play an important role in neural development\u2014called DISC1 (short for \"Disrupted-in-Schizophrenia-1\")\u2014in addition to the regular type. The short form of the protein attaches to the full-length one, disrupting its normal duties. As these mice matured, they became more agitated when placed in an open field, had trouble finding hidden food, and did not swim as long as regular mice; such behaviors allegedly parallel the hyperactivity, smell defects, and apathy observed in schizophrenia patients. The mice were also subjected to MRI and showed certain brain defects, notably enlarged lateral ventricles. The press release reports Sawa as noting that the defects in these mice were not as severe as those typically seen in people with schizophrenia, because more than one gene is required to trigger the clinical disease. \"However, this mouse model will help us fill many gaps in schizophrenia research,\" he said. \"We can use them to explore how external factors like stress or viruses may worsen symptoms. The animals can also be bred with other strains of genetically engineered mice to try to pinpoint additional schizophrenia genes.\"\n\nHere we see the idea of reverse translation\u2014and its inescapable link to _DSM_ diagnoses. The belief that there is a relation between DISC1 and schizophrenia is based on a number of family studies, starting in 2000, that show statistical associations between mutations in this coding sequence and a psychiatric diagnosis. However, the original study was in a Scottish family with a very wide range of psychiatric symptoms\u2014including bipolar affective disorder, unipolar affective disorder, and conduct disorder: the authors themselves remark that this may be somewhat \"atypical\" (Millar, Wilson-Annan, Anderson, et al. 2000). The significance of this gene in the original Scottish family is still controversial, as is the specificity of the link to a particular psychiatric diagnosis of schizophrenia, and the mechanisms of action in the brain are also disputed (Ishizuka, Paek, Kamiya, et al. 2006). The functional properties of the protein expressed by the DISC1 gene seem to relate to some very basic pathways and processes in the brain at key points in development, and it appears to interact with a wide variety of proteins linked to neural growth, signal transduction pathways, and proteins linked to neocortical development (Morris, Kandpal, Ma, et al. 2003). Thus one might expect disruption of this gene, and hence of these various pathways, to be nonspecific to this particular diagnosis, and this is indeed the case: \"genetic analysis has implicated DISC1 in schizophrenia, schizoaffective disorder, bipolar affective disorder, and major depression\" (Hennah, Thomson, Peltonen, et al. 2006, 409).\n\nNonetheless, the mouse knockout made on the basis of the debatable claim concerning DISC1 is deemed a mouse model of basic pathways in schizophrenia, and then attempts are made to see if behaviors in the mouse parallel those associated with the diagnosis in humans. This relation is established by a rather simplistic set of analogies of just the sort that troubled Insel in his criticism of earlier animal model research: agitation of the mouse when placed in an open field reminds us of hyperactivity, trouble finding hidden food reminds us of smell defects apparently shown by persons diagnosed with schizophrenia, and poor performance in the forced swim test reminds us of apathy\u2014or even \"anhedonia\" (Hikida, Jaaro-Peled, Seshadri, et al. 2007, 14,503)\u2014observed in those diagnosed with schizophrenia. For those apprenticed in animal models in psychiatry (though not perhaps to others) the effect is, to quote John Medina \"evocative of some of the information processing anomalies observed in patients with mood disorders and psychoses\" (Medina 2008, 26). But would we be surprised\u2014given the basic pathways being disrupted here\u2014that mouse behavior was somewhat abnormal in a whole variety of ways? No. Do we know how widely distributed are disruptions in DISC1 in the general population of those without a psychiatric diagnosis, and in different countries or regions? No. Do we know whether DISC1 has the same developmental pathways and functions in mouse and human? No. In what sense, then, do we have \"a mouse model of schizophrenia\"?\n\nWhat indeed, about schizophrenia itself\u2014a diagnosis given its name a century ago, whose symptomatology and prognosis have varied greatly across that period (changing dramatically, for example, when those living under that diagnosis were no longer subject to long-term confinement in psychiatric institutions) and where recent failures in GWAS confirm what has long been argued by critics\u2014that this is a problematic label for an array of heterogeneous anomalies in human mental life and social skills. If we are not very careful with our thinking, reverse translation from humans into animal models will indeed get caught in the spirals of self-vindication of the diagnosis through these multiple translations between the level of _DSM_ , the level of the genome, and the level of the rodent performance in those setups we have discussed earlier.\n\nLife as Creation\n\nAnimal models have been crucial in the experimental practices of the brain sciences and central to the development of neuroscience over the past fifty years. Yet we have argued in this chapter that they have some quite fundamental flaws when used as the basis of claims concerning human mental life and psychopathology. Many of our criticisms, though perhaps not our conclusions, are shared by the most careful researchers who work in this field and, as we have seen, some are trying to rethink the practice and conceptualization of animal models and model animals.\n\nHowever, it seems we are left with more questions than answers. How can we avoid self-vindication in behavioral neuroscience? Can one resolve some of these problems by a move toward 'ecological validity' in animal experimentation? Can one develop forms of assessment of animal behavior that do not rely on crude analogies with human stressors, such as the elevated plus maze or the forced swim test? Can we avoid self-vindication by moving beyond the checklist approach to mental health problems, with all the presumptions embodied in the _DSM_? Can we overcome the epistemological and ontological divide between animal work on human mental states and all we know of humans\u2014and of animals? Or are model organisms in psychiatry doomed to disappear, as some suggest (Debru 2006)? Will they be displaced by the power of novel biotechnologies to create stem cell lines from tissue derived from those diagnosed with disorders and to use these for the conduct of experiments and the trialing of therapies, or by simulation in computer models in silica, or by virtual experimental situations, or by a return to the human as its own model (Rheinberger 2006; Brenner 2008)?\n\nPerhaps there is something to be learned from our earlier insistence on the animal experiment as craft work\u2014for given that modeling is craft work, is there a way in which we can setup the animals differently? Can we reimag-ine the epistemological status of animal models? And perhaps, just perhaps, a genuine interdisciplinary, transdisciplinary dialogue between animal modelers in neuroscience and those from the social and human sciences can contribute to this. So let us conclude by returning once more to the reflections of Georges Canguilhem in his essay on \"Experimentation in Animal Biology,\" written almost half a century ago: \"The use of concepts and intellectual tools [of biology] is thus at once both inevitable and artificial. We will not conclude from this that experimentation in biology is useless or impossible. Instead, keeping in mind Bernard's formulation that 'life is creation,' we will say that the knowledge of life must take place through unpredictable conversions, as it strives to grasp a becoming whose meaning is never so clearly revealed to our understanding as when it disconcerts it\" (Canguilhem [1965] 2008, 22).\nChapter Four\n\nAll in the Brain?\n\nNothing can be more slightly defined that the line of demarcation between sanity and insanity. Physicians and lawyers have vexed themselves with attempts at definition in a case where definition is impossible. There has never yet been given to the world anything in the shape of a formula upon this subject which may not be torn to shreds in five minutes by any ordinary logician. Make the definition too narrow, it becomes meaningless; make it too wide, the whole human race are involved in the drag-net. In strictness, we are all mad as often we give way to passion, to prejudice, to vice, to vanity; but if all the passionate, prejudiced, vicious, and vain people in this world are to be locked up as lunatics, who is to keep the key of the asylum?\n\n\u2014 _The Times_ (London), July 22, 1854\n\nPsychiatry is a biomedical science. This is not a trivial statement. Nor could psychiatry have been called a science twenty years ago.... [T]oday a remarkable body of new knowledge relates neurobiology to psychiatric disorders.\n\n\u2014Philip Berger and Harlow Brodie, preface to _Biological Psychiatry_ , 1986\n\nHere's how doctors decide which mental or neurological disorder their troubled patients suffer from: they ask questions like, \"Are you hearing voices?\" and \"Do you feel like people are out to get you?\"... Not all that different from how they used to do it about 100 years ago. New techniques are set to radically change that approach... relying more on changes in physiology than in behavior.... Scientists are increasingly turning to biomarkers\u2014such as genes or proteins in tissues, blood and body fluids\u2014to distinguish between symptomatically indistinct illnesses.... Using techniques that have only become available in the past few years, scientists are looking at the brain, serum and spinal fluid and asking which combinations of genes or proteins might be expressed differently in these disorders. The field is still far from ready for the clinic, but holds great promise: markers could allow for earlier treatment and a better prognosis, ultimately enabling scientists to intervene even before the full-blown disease strikes.... The most challenging aspect might yet turn out to be redefining a 100-year-old classification system.\n\n\u2014Amanda Haag, \"Biomarkers Trump Behavior in Mental Illness Diagnosis,\" 2007\n\nIn 1953, Dr. Henry Yellowlees, chief medical officer at the United Kingdom's Department of Health, published a small volume described as \"commonsense psychiatry for lay people\"; its title was _To Define True Madness_. This phrase is, of course, a quote from Shakespeare's _Hamlet_ , published in 1602. Polonius, in act 2, scene 2, having told the King that his noble son is mad, then pauses for a second: \"Mad call I it; for to define true madness, What is't but to be nothing else but mad? But let that go.\" Psychiatry, since its invention in the nineteenth century, has found it less easy to let that question go\u2014the question of what it actually is to be 'mad,' of the boundaries of madness and sanity, and of the differentiation between the various forms that madness\u2014whether termed lunacy, mental illness, or mental disorder\u2014can take. Indeed, it has been haunted by this problem, framing it as the problem of diagnosis, with theorists, practitioners, and critics returning to it again and again (cf. Porter 2002, 1). Some feel that we are, at last, on the verge of putting these troubles behind us, that neurobiology has enabled psychiatry to become a science, and as psychiatry becomes neuropsychiatry it will be able to answer, finally, the question of who is 'a suitable case for treatment.' Will an appeal to the brain, or to specific genetic sequences associated with particular neurobiological anomalies, or to the new visibility apparently conferred by neuroimaging, finally enable clinicians to delineate the boundaries of normality, and to differentiate between disorders? Will neuroscience at last provide psychiatry with the objectivity that it seeks?\n\nWhile the term _neuropsychiatry_ had been in use both in English and French since the early twentieth century and probably since the late nineteenth century, it was not until after the Second World War that it gained popularity in the European and American literature. While it was used in a variety of senses, by the start of the twenty-first century the term was being used to argue that the future of psychiatry lay in the integration of insights from genetics and neurobiology into clinical practice (Healy 1997; J. Martin 2002; Sachdev 2002, 2005; Yudofsky and Hales 2002; Lee, Ng, and Lee 2008). There is a weak version and a strong version of this argument. Some advocate the formation of neuropsychiatry as a particular professional specialty, appropriate for those disorders whose understanding and treatment requires both neurology and psychiatry. This, for example, is the argument of Yudofsky and Hales in an article titled \"Neuropsychiatry and the Future of Psychiatry and Neurology\":\n\n[There are] many problems inherent in separating psychiatry and neurology into discrete medical specialties. First, the separation perpetuates mind-body dualism that is a perennial source of the public's confusion about whether or not mental illnesses have biological bases and whether psychiatry is a credible medical specialty. This confusion is also a source of the stigmatization of the mentally ill and leads to a lack of parity in the reimbursement for psychiatric treatments with those for other medical conditions. Second, this separation of the specialties promotes the condensation of complex brain disorders into simplistic categorizations that are based on single symptoms.... [N]europsychiatry... bridges conventional boundaries imposed between mind and matter as well as between intention and function [and] vaults the guild-based theoretical and clinical schisms between psychiatry and neurology. (2002, 1262)\n\nOthers argue more expansively and either propose, or assume, that all psychiatric disorders are ultimately grounded in anomalies in the brain and nervous system\u2014that is to say, the disorders that psychiatry deals with are neurological disorders. This, for instance, is the U.S. Surgeon General writing in 1999: \"Mental disorders are characterized by abnormalities in cognition, emotion or mood, or the highest integrative aspects of behavior, such as social interactions or planning of future activities. These mental functions are all _mediated_ by the brain. It is, in fact, a core tenet of modern science that behavior and our subjective mental lives _reflect_ the overall workings of the brain. Thus, symptoms related to behavior or our mental lives _clearly reflect_ variations or abnormalities in brain function\" (United States, Center for Mental Health Services 1999, 39; emphases added).\n\nWe will not dwell on the slippery terms used here\u2014 _mediated_ , _reflect_ , _clearly reflect_ \u2014we should not expect the Surgeon General to resolve the age-old problem of mind and brain. However, he is just one of the many who invest their hopes in a fusion of psychiatry and neuroscience. It will, some hope, overcome the legacies of the Cartesian division between mind and body that are still enshrined in the distinction of organic and functional disorders in psychiatry, proving conclusively that neurobiological anomalies underpin mental disorders. It could help challenge stigma, they believe, by demonstrating that psychiatric disorders are genuine conditions, not just 'all in the mind' and hence not problems for which the sufferers themselves should be held responsible. And, as we have said, some expressed the hope that such developments could overcome the multiple problems that have surrounded psychiatric diagnosis, generating new methods that rely on measurable and objective markers (Hyman 2003; Haag 2007; G. Miller 2010). Many made the linked argument that this would also enable early, presymptomatic identification of incipient disorders, allowing early intervention and preventive treatment. In this chapter we focus upon this question of diagnosis and explore some of the hopes for the relations between neuroscience and psychiatry from this perspective. For, as we have already suggested, this question of diagnosis goes to the heart of the key question, not just for neuropsychiatry, but for psychiatry in general\u2014the question of what is it to be mad, what kind of a thing is a 'mental disorder'?\n\nTo Define True Madness\n\nSome of us are old enough to have lived through\u2014indeed participated in\u2014the contentious disputes of the decades of the 1960s and 1970s that troubled psychiatry to its core. At that time madness, as mental illness, came to be regarded as an 'essentially contested category.' These contests were not restricted to academic debate. They entered popular imagination, in films like _One Flew over the Cuckoo's Nest_ (1975) or _Morgan: A Suitable Case for Treatment_ (1966), and penetrated general cultural awareness in the array of critiques that became known as antipsychiatry. David Rosenhan, in a classic and widely known experiment published in 1973, used this instability of diagnosis to ask a key question: \"if sanity and insanity exist, how shall we know them?\" Pseudopatients (students with no existing psychiatric problems) presented themselves at several psychiatric hospitals, claiming to hear voices that corresponded with prevailing existential philosophies\u2014\"thud,\" \"empty,\" \"hollow\"\u2014and each was admitted to a psychiatric ward. Their histories in their case records were reworked by their psychiatrists to fit prevailing conceptions of the etiology of schizophrenia. None were unmasked by psychiatric professionals, and all were eventually discharged with the diagnosis of \"schizophrenia in remission.\" For Rosenhan, the message was stark: normality is hard to diagnose.\n\nThese criticisms were linked with others, familiar from the sociology of deviance then in its heyday\u2014that a psychiatric diagnosis was actually the outcome of a social process of labeling; that mental illness was a residual category, applied to those whose violation of cultural norms could not be explained in any other way; that diagnosis is the outcome of a social process and retrospectively gathers together all manner of previous episodes in a life history and reframes them as evidence of developing mental breakdown; and that the label, once applied, is sticky and hard to dislodge, leading to stigma, exclusion, and isolation. In labeling deviance as mental illness, psychiatry, it was claimed, was actually creating the forms of damaged and disordered selfhood that it claimed to explain. If these concerns had been restricted to critical social scientists, they would have been of minor significance. But, as we shall see, others were also troubled by psychiatric diagnosis, including psychiatrists themselves.\n\nWhat is diagnosis? A layperson might say that a medical diagnosis is intended to identify the nature of the disease or condition that ails an individual, in order to clarify its causes and suggest an appropriate treatment. Its Greek root \u03b4\u03b9\u03b1\u03b3\u03bd\u03c9\u03c3\u03c4\u03b9\u03ba\u03cc\u03c2 ( _diagnostikos_ ) means \"capable of discerning.\" The _OED_ gives a similar definition, dating it to Willis in 1681: \"Determination of the nature of a diseased condition; identification of a disease by careful investigation of its symptoms and history; also, the opinion (formally stated) resulting from such investigation.\"\n\nRobert Kendell, who in 1975 was a professor of psychiatry at the University of Edinburgh, published _The Role of Diagnosis in Psychiatry_ in the throes of these debates over the objective status of psychiatric diagnoses. He opens the book with a quote from Daniel Hack Tuke, the great-grandson of William Tuke, founder of the York Retreat: \"Eighty years ago Hack Tuke observed that 'The wit of man has rarely been more exercised than in the attempt to classify the morbid mental phenomena covered by the term insanity,' and went on to add that the result had been disappointing. The remark remains as apposite now as it was then\" (Kendell 1975, vii). Kendell argues that \"the failure of both psychiatrists and psychologists to develop a satisfactory classification of their subject matter, or even to agree on the principles on which that classification should be based, is a most serious barrier to fruitful research into the aetiology of mental illness and even into the efficacy of treatment regimes\" (ibid.).\n\nOthers took the fact that psychiatrists could not agree on the application of their own categories to indicate that those categories themselves were unstable subjective judgments. The decision by the American Psychiatric Association (APA) in 1973 to remove homosexuality as a category of disorder, the regular ridiculing of psychiatric evidence in legal cases, and the association of psychiatry with confinement for political reasons in the Soviet Union also led many to wonder about the capacity of psychiatrists to legitimately categorize an individual as suffering from a particular mental disorder, or even a mental disorder at all. We shall return to these debates presently. But if psychiatry ran into so much trouble over its diagnoses, one might be forgiven for thinking that this occurred for clinical reasons\u2014that diagnoses were failing to identify an underlying disease from a set of symptoms or to clarify etiology and enable treatment and prognosis. However, a brief glance at the history of psychiatric classification is enough to show us that such categorization has not been a purely clinical matter.\n\nMany have written histories of attempts to classify diseases, have described the work of pioneers in medical nosology, and have examined the debates over the status of such classifications\u2014whether they were, for example, 'natural' or 'artificial' (see, e.g., Roy Porter's chapter in L. Conrad et al. 1995). Critical sociologists, especially in the United States, have focused on the development of the _Diagnostic and Statistical Manuals_ produced by the APA and have tended to regard the whole enterprise as suspect, as part of an entrepreneurial strategy to expand the empire of psychiatry and enhance its power and legitimacy by pathologizing everyday life, hand in glove with a pharmaceutical industry looking to increase the market for its wares (Kutchins and Kirk 1997). Such overly general critiques usually fail to consider the work of classification in psychiatry in other national contexts, and greatly underestimate the level of awareness of those undertaking the exercises about the problematic epistemological and ontological issues involved in classification. It is not only sociologists who understand the performative character of categorization, as the debates among those responsible for creating and using the manuals themselves testify. It is, therefore, worth sketching the development of psychiatric classification and examining some of the debates within psychiatry itself.\n\nAlthough we can find many earlier attempts at classifying forms of madness, notably Robert Burton's _The Anatomy of Melancholy_ , published in 1621, or Thomas Sydenham's _Epistolary Dissertation on the Hysterical Affections_ of 1682, the project of classification in modern psychiatry can be traced to the visibility cast on inmates by the reformed asylum of the nineteenth century. Indeed, the asylum provided key conditions for the birth of psychiatry, because it first assembled together the diverse characters who were to form its object. It was only the asylum that seemed to confer some homogeneity upon those disparate individuals\u2014displaced, desolate, despairing, deranged\u2014who were collected within its walls. What first linked these troubled and troublesome people was not a medical gaze, but a civilizing project: the asylum was the place for all those who offended the norms of self-conduct and self-control that were beginning to grid the emerging urban, liberal social orders in the mid-nineteenth century (Foucault [1961] 2005; Rothman 1971; Castel 1976; Scull 1979; Goldstein 2002).\n\nAs doctors conquered the space of asylum over the second half of the nineteenth century, guarding its points of entry and of exit, and managing its interior world, the assorted figures who were confined within became psychiatry's subjects. The doctors of the mad began to search for a single classificatory schema that would embrace them all, that would grasp them all as sufferers from varieties of a common condition\u2014madness, mental illness, or mental disorder. These schemes of classification drew upon older themes, it is true, but attempted to give them clinical utility, linking each category to a description intended to educate the gaze of the doctor, and often adding some case histories and recommendations as to appropriate treatment (Esquirol 1838, 1845, [1845] 1965; Bucknill and Tuke 1874).\n\nIn the United States, this clinical desire to classify was supplemented by another, a _statistical_ project driven by the national census, and its demand that every person within the national territory should be enumerated (on the history of the U.S. Census, see Alonso and Starr 1987; Anderson 1988). The 1830 U.S. census had counted the deaf, dumb, and blind, and the 1840 U.S. census added a count of the insane and idiots, distinguished by race and by mode of institutional or other support. When the results of this census were published in 1841, the total number of those reported as insane or feebleminded was more than seventeen thousand, of whom nearly three thousand were black. The rate of insanity among free blacks was eleven times higher than that of slaves and six times higher than that of the white population.\n\nFor those who opposed abolition, like U.S. Vice President John C. Calhoun, these census figures proved that blacks were congenitally unfit for freedom: \"Here is proof of the necessity of slavery. The African is incapable of self-care and sinks into lunacy under the burden of freedom. It is a mercy to him to give him the guardianship and protection from mental death\" (Calhoun, 1843; quoted in Deutsch 1944, 473). Abolition, he believed, far from improving the condition of \"the African,\" worsened it: where \"the ancient relations\" between the races had been retained, the condition of the slaves had improved \"in every respect\u2014in number, comfort, intelligence and morals\" (quoted in Gilman 1985, 137). A political dispute about madness and slavery followed, waged in the language of numbers (Deutsch 1944).\n\nBy the time of the 1880 U.S. census, seven categories were used to count the institutionalized population of the mad\u2014mania, melancholia, monomania, paresis, dementia, dipsomania, and epilepsy. It was in order to provide the U.S. census with a better basis for its classification of those confined that the first official U.S. nosology of mental illnesses, the _Statistical Manual for the Use of Institutions for the Insane_ , was prepared in 1918 by the American-Medico Psychological Association (the successor of the Association of Medical Superintendents of American Institutions for the Insane, which was to become the American Psychiatric Association in 1921) and the National Committee for Mental Hygiene (Grob 1991). That manual was the forerunner of the _Diagnostic and Statistical Manuals_ that would follow from the 1950s onward.\n\nThe history of these manuals has often been told (Spitzer, Williams, and Skodol 1980; Bayer and Spitzer 1985; Grob 1991; M. Wilson 1993; Spitzer and Williams 1994; A. Young 1995, 89\u2013117; Spitzer 2001; Cooper 2004; Mayes and Horwitz 2005). The first editions, as we have said, focused on classification of those who were confined in the asylums, with their growing populations of inmates. This remained their main concern until the 1940s. However, by then a different kind of psychiatry was beginning to flourish outside the asylum and was difficult to ignore. From the spas and water treatments of the nineteenth century, through Mesmerism and the work of the nerve doctors like Breuer and Freud, and through the pathologies of children; the problems of the maladjusted worker or soldier; the work of psychiatric social workers in courts, schools, and elsewhere; and in the growth of individual and group psychotherapeutics, a much wider territory for psychiatry had taken shape, not defined by the walls of the asylum. This was the territory of anxieties and minor mental troubles, of problems of the adjustment of the individual to the demands of family and workplace, of delinquency and criminality\u2014in short, the territory of the neuroses and of problems of conduct (Rose 1989).\n\nHence it is not surprising that the tenth edition of the _Statistical Manual_ included classifications for the psychoneuroses (hysteria, compulsive states, neurasthenia, hypochondriasis, reactive depression, anxiety state, anorexia nervosa, and mixed psychoneuroses) and primary behavior disorders (adult maladjustment and primary behavior disorders in children), although it nonetheless conceived of these states, like the other conditions it named, primarily as somatic conditions. It is also not surprising that the term _neuropsychiatry_ was used at the time to refer to the study and management of those _neuroses_ precipitated by industrialization and transformations at the workplace. Thus, for example, in an article published in 1921 under the title \"The Service of Neuropsychiatry to Industrial Medicine,\" Dr. Harold Wright defines neuropsychiatry as a \"synthetic\" specialty and the neuropsychiatrist as \"a synthesist\" who \"considers the personality as a whole with all its physical and psychological faults of adjustment. And if the psychiatrist or general medical man is to do effective work in an industrial plant, he must consider things outside of the plant as well as those influences which are present within it and within the patient's body\" (Wright 1921, 465).\n\nIn the Second World War, in the United States, as in the United Kingdom, psychiatrists were widely deployed in the military apparatus, in branches ranging from psychological warfare and propaganda, through personnel selection and allocation, to the diagnosis and rehabilitation of those invalided out of the forces with mental troubles apparently brought on by the conflict (Rose 1989; Shephard 2003). A whole array of minor neuropathologies were now being diagnosed and treated by psychiatrists, and at the same time, psychiatry began to argue that these mild mental pathologies were at the root of problems in many social institutions and practices; they thus argued that there was a positive role for their expertise in the promotion of mental health. As Grob puts it (1991, 427): \"In the postwar era, the traditional preoccupation with the severely mentally ill in public mental hospitals slowly gave way to a concern with the psychological problems of a far larger and more diverse population as well as social problems generally. Persuaded that there was a continuum from mental health to mental illness, psychiatrists increasingly shifted their activities away from the psychoses toward the other end of the spectrum in the hope that early treatment of functional but troubled individuals would ultimately diminish the incidence of the more serious mental illnesses.\"\n\nThe American psychiatrist William C. Menninger, head of the newly created Neuropsychiatric Division of the U.S. Army in 1943, perhaps captured this approach best. In treating war neurosis, he wrote, one should take into account the \"whole environment,\" that is, \"everything outside ourselves, the thing to which we have to adjust\u2014our mates and our in-laws, the boss and the work, friends and enemies, bacteria and bullets, ease and hardship\" (W. Menninger [1947] 1994, 352); in the postwar years, he was to play an important role in the revision of psychiatric nomenclature (Houts 2000), However, it was his brother, Karl Menninger, who made the most uncompromising statements of this idea that there was an essential unity of all mental troubles: they were failures to adapt to pressures of the environment, and hence conditions should be explained and symptoms interpreted in psychosocial terms (K. Menninger 1965; cf. M. Wilson 1993). For Karl Menninger, \"most people have some degree of mental illness at some time, and many of them have a degree of mental illness most of the time\" (K. Menninger 1965, 33). Hence, as William Menninger put it in 1947: \"Psychiatry should place high priority on its efforts to provide the 'average' person with psychiatric information he can apply to his own problems.... The public wants this education.... Very possibly it may increase the number of patients who seek help from a psychiatrist, just as a campaign about cancer or tuberculosis increases the number of patients who go to doctors about these problems\" (W. Menninger [1947] 1994, 79).\n\nIt was in the context of this radically expanded field that the _Diagnostic and Statistical Manual_ , published in 1952, was prepared by the APA's Committee on Nomenclature and Statistics (American Psychiatric Association 1952). This manual construed mental disorders as reactions of the personality to psychological, social, and biological factors. Following the lesson of wartime, both minor and major mental troubles were included in the classificatory framework\u2014although distinguished as neuroses and psychoses. Psychiatry, it was argued, now had a key role outside the hospital, in addressing all those minor troubles, and in prevention. Accurate diagnosis was the key to competent prognosis and appropriate treatment, especially in the name of preventive intervention. And diagnosis required interpretation: symptoms needed to be understood as meaningful responses to events in the life of the patient.\n\nIt was not until sixteen years later, in 1968, that the first revision of the _DSM_ was published. The revision arose, according to its foreword, from the need for American psychiatrists to collaborate with those who were preparing and using the new revision of the International Classification of Diseases (ICD-8), which had been approved by the WHO and was to come into use in 1968 (American Psychiatric Association 1968, vii). _DSM-II_ , as it was known, was developed by a Committee on Nomenclature and Statistics of the APA, with advice from a team of three consultants\u2014Bernard Glueck, Morton Kramer (who had worked on _DSM-I_ ), and Robert Spitzer (then director of the Evaluation Unit of Biometrics Research of the New York Psychiatric Institute). For Kramer, the earlier versions of ICD were unsuitable \"for use in the United States for compiling statistics on the diagnostic characteristics of patients with mental disorders or for indexing medical records in psychiatric treatment facilities\" (ibid., xi). The APA committee worked in consultation with the WHO team that was revising the ICD, though \"the Committee had to make adjustments\" to conform to U.S. usage. _DSM-II_ was 134 pages long and had 180 disease classifications organized into ten categories. The aim was \"to avoid terms which carry with them _implications_ regarding either the nature of a disorder or its causes\" (ibid., vii). The foreword, by Ernest Gruenberg, who was chairman of the APA's Committee on Nomenclature and Statistics, drew particular attention to the renaming of \"schizophrenic reaction\" to \"schizophrenia\" to eliminate such causal assumptions, adding, somewhat poignantly, \"Even if it had tried, the Committee could not establish agreement about what this disorder is; it could only agree on what to name it\" (ibid., ix).\n\nOn the occasion of the publication of _DSM-II_ , Robert Spitzer, together with Paul Wilson, who had done much of the editing work on the _Manual_ , published an account of the new diagnostic nomenclature and the ways in which it differed both from _DSM-I_ and from ICD-8. They claimed that the new classification system was not a \"return to a Kraeplinian way of thinking, which views mental disorders as fixed disease entities\" but an attempt to avoid terms that carry implications about the causes of a disorder, especially where there is controversy over its nature or cause (Spitzer and Wilson 1968, 1621). While _DSM-II_ reduced the previous emphasis on disorders as reactions to precipitating life experiences, it did not eliminate this suggestion entirely, and it also retained, and indeed extended, the categories of neurosis. It also retained the categorical distinctions between mental retardation, organic brain syndromes, neuroses, and personality disorders (American Psychiatric Association 1968). Spitzer and Wilson concluded their presentation thus: \"With the adoption of _DSM-II_ , American psychiatrists for the first time in history will be using diagnostic categories that are part of an international classification of diseases. While this is an important first step, it is only an agreement to use the same sets of categories for classifying disorders\" (Spitzer and Wilson 1968, 1629).\n\nDespite these high hopes, by the time of its publication in 1968, the expansive project of psychosocial psychiatry that _DSM-II_ sought to underpin was already under attack (cf. M. Wilson 1993). As we have seen, over the 1960s, antipsychiatry, in its various manifestations, cast doubt on the conceptual underpinning and sociopolitical consequences of this broadening of the remit of psychiatry. The debate in the early 1970s over the inclusion or exclusion of homosexuality as a disease category seemed to some to mark a turning point (Bayer 1981). The text of _DSM_ was altered for every printing of _DSM-II_ , and in the seventh printing homosexuality was removed as a disease category, seemingly confirming the view of critics that its categories were just sociopolitical labeling of behavior considered to be socially undesirable by the authorities at any given time.\n\nWithin psychiatry itself, concern began to mount about the lack of standardized criteria for diagnosis and the poor reliability of diagnosis in practice\u2014studies such as that by Rosenhan exacerbated this anxiety. Further, as funding for psychiatry was restricted in the United States in the 1970s, government and private health insurers became concerned about the lack of specificity of classification, the confusions of terminology, and the difficulty of establishing clear boundaries between those who were eligible for treatment\u2014and hence for insurance reimbursement\u2014because they were suffering from an illness and those who were not, even if they were receiving some kind of psychological therapy. The response by many within the profession was to emphasize the classical medical model of disease and to argue that this should form the basis and the limits of psychiatry's vocation, which should not seek an extension of its ministrations to the ills of society as a whole (Klerman 1978).\n\n_DSM-III_ , published in 1980, is often seen as a response to this crisis in legitimacy of psychiatry (American Psychiatric Association 1980; Mayes and Horwitz 2005). It had begun with the perceived imperative to narrow the definition of mental disorder, to raise the threshold for diagnosis of illness, and to set out a classificatory scheme based on observable symptoms. This was no simple \"return to Kraepelin\" (Decker 2007), but it owed a great deal to the so-called Feighner criteria, associated with the Department of Psychiatry at Washington University School of Medicine in St. Louis (Kendler, Munoz, and Murphy 2010). These were most influentially set out in a 1972 article by John Feighner and his colleagues as diagnostic criteria to be used in psychiatric research (Feighner, Robins, Guze, et al. 1972). In contrast to _DSM-II_ , say the authors, \"in which the diagnostic classification is based upon the 'best clinical judgment and experience' of a committee and its consultants, this communication will present a diagnostic classification validated primarily by follow up and family studies\" (ibid., 57). The diagnostic criteria that they presented for fourteen psychiatric illnesses each contained a general clinical description together with a checklist of observable symptoms, with the stipulation that a certain number of these must be present to warrant a diagnosis, and a criterion concerning the time scale of the condition and the implications of preexisting psychiatric conditions. The authors referred to the desirability of \"laboratory studies\"\u2014chemical, physiological, radiological, and anatomical, which they considered to be \"generally more reliable, precise and reproducible than are clinical descriptions\" but remarked that \"[u]nfortunately, consistent and reliable laboratory findings have not yet been demonstrated in the more common psychiatric disorders\" (ibid.). Nonetheless, they concluded that what they presented was \"a synthesis [of existing information] based on data rather than opinion or tradition\" (ibid., 62).\n\nThis paper was receiving some two hundred citations a year by the mid-1970s (Garfield 1989; Kendler, Munoz, and Murphy 2010), and by 1974, two of the authors consolidated this way of thinking into a textbook on psychiatric diagnosis (Woodruff, Goodwin, and Guze 1974). But the view that this was in some way a return to Kraepelin was hard to shake off, despite the protestations of the authors. Thus, writing in 1978, while head of the Alcohol, Drug Abuse and Mental Health Administration, Gerald Klerman referred to them as \"neo-Kraeplinian,\" and portrayed them as engaged in a long struggle against what they felt to be an entrenched Meyerian domination of the field in the NIMH in the United States and the MRC in the United Kingdom (Klerman 1978).\n\nKlerman, like Spitzer, had been working since the early 1970s with psychiatric research groups developing \"research diagnostic criteria\" based on the Feighner approach. He argued that this classificatory ethos does not necessarily imply a commitment to a biological etiology for psychiatric problems, let alone a biological approach to treatment\u2014indeed Klerman himself was a key contributor to the development of interpersonal psychotherapy. However, Spitzer and his team _were_ committed to biological approaches from the outset; they were working on the basis of a commission from the NIMH as part of their Collaborative Program on the Psychobiology of the Depressive Disorders: \"biological and clinical collaborative programs... initiated in the early 1970s [spanned] research on nosology, genetics, neurochemistry, neuroendocrinology, and psychosocial factors... [arising from the] excitement created by the introduction of the new antidepressant drugs and the controversy surrounding some inspired hypotheses about the potential genetic and biochemical causes of the major depressive psychoses\" (Katz, Secunda, Hirschfeld, et al. 1979, 765).\n\nBy 1975, writing with Eli Robins, a biological psychiatrist who was one of the authors of the Feighner paper, Spitzer was commending the Feighner approach to diagnostic classification for the forthcoming _DSM-III_ , in contrast to that taken in _DSM-II_ (Spitzer, Endicott, and Robins 1975, 1187\u201388). Urging the use of what they now termed \"specified diagnostic criteria\" in _DSM-III_ , Spitzer and his colleagues argued that these would not eliminate clinical judgment or displace validity in the quest for reliability, but would improve training of psychiatrists and improve communication; in conclusion they contended that \"[t]he use of such criteria also forces clinicians to share their definitions of clinical terms and concepts, overcoming the tendency to be idiosyncratic.... We believe that the major potential beneficial effect of including specified diagnostic criteria in _DSM-III_ would be to improve the reliability and validity of routine psychiatric diagnosis. This in turn should improve the value of the standard nomenclature for all of its many uses, clinical, research and administrative\" (ibid., 1191).\n\n_DSM-III_ was, indeed, reshaped along these lines. In particular, much to the irritation of psychodynamic psychiatrists (Frances and Cooper 1981), it eliminated the use of the term _neurosis_ , which was thought inescapably tainted by psychodynamic theory, although in a moment of appeasement to pressures from psychoanalytic practitioners, the team preparing the revision agreed to put some reference to this term in parentheses, next to the disorders that had been termed neuroses in _DSM-II_. Thus 300.40 \"Dysthymia\" was followed by \"(or Depressive Neurosis),\" and the section headed \"Anxiety Disorders\" followed this title with \"(or Anxiety and Phobic Neuroses)\": \"the term 'neurotic disorder' is used in DSM-III without any implication of a special [and invisible] etiological process\" (American Psychiatric Association 1980, 9\u201310; quoted from A. Young 1995, 100).\n\nAlthough much criticized over the years of its development, its publication marked a crucial moment in the development of American psychiatry. _DSM-III_ and its subsequent versions were sanctioned by the NIMH, and diagnoses using its classification became required elements in research protocols, in articles submitted for publication in psychiatric journals, in the management of clinical trials and recruitment of subjects, in reimbursement criteria of health management organizations (HMOs), and in training programs for psychiatrists. _DSM-III_ stressed that diagnosis must be based on a pattern of symptoms that was not merely an expectable response to an event, but a manifestation of a dysfunction in the person: \"Neither deviant behavior... nor conflicts that are primarily between the individual and society are mental disorders unless... a symptom of a dysfunction in the person\" (American Psychiatric Association 1980). Its 494 pages classified 265 distinct diagnoses; the revised version of 1987 was 567 pages and classified 292 diagnoses (American Psychiatric Association 1987). Each was to be diagnosed in terms of a set of \"observable\" criteria, and each of these categories was construed as a distinct disorder, with a unique etiology and prognosis, amenable to a specific kind of treatment.\n\nBut despite the fact that the demand for revision of _DSM-II_ grew, in part, out of disquiet with the expanding scope of psychiatry, _DSM-III_ did not restrict the field of action of the experts of mental disorder. Its rejection of the fundamental division of neuroses and psychoses, together with its claims to provide neutral and theory-free empirical descriptions of symptom patterns did quite the reverse. It actually reinforced the claim that psychiatrists could diagnose as mental disorders conditions ranging from troublesome conduct in children to men's problems in getting an erection. All the travails of everyday life now seemed to come within its purview. And, indeed, by the 1980s this expanded territory of psychiatry had become a reality. At the risk of overgeneralization, we could say that up until the 1960s, there was a reasonably clear distinction\u2014professional, clinical, conceptual\u2014between what one might term 'asylum psychiatry,' based in the mental hospitals and often associated with the use of compulsory powers of confinement, and the psychiatry of everyday life, of the neuroses, of minor mental troubles of childhood, of psychotherapy and psychoanalysis. The former were the focus of the disquiet of the critics and were associated with the use of harsh physical treatments, drugs, and electroshock; the latter were associated with talking cures and increasingly with the use of minor tranquilizers such as Valium and the other benzodiazepines. But by the 1980s, in almost all developed countries a re-configuration and reunification of the territory of psychiatry was under way (Castel, Castel, and Lovell 1982). The antipsychiatric critiques of the custodial and segregative project of the asylum were welcomed by many psychiatrists, who were frustrated by the fact that they were confined to the mental hospitals along with their patients, shut off from the world of therapeutic optimism outside the asylum, and from the general hospitals and general medicine where the real progress appeared to be being made.\n\nThe movement for deinstitutionalizing the mental hospitals was much more than a shift of the incarcerated population from one locale to another. A new configuration of sites for the practice of psychiatry was taking shape, together with their populations\u2014halfway houses; residential homes; specialized units for children, alcoholics, anorexics, or drug users; and clinics of many different sorts. Children, delinquents, criminals, alcoholics, vagrants, the work shy, unhappy sexual partners, troubled husbands and wives all became possible objects for this extended practice of psychiatry. The unrivaled dominance of the doctor gave way to teams of multiple professionals with diverse claims to expertise. That monologue of reason about madness, of which Foucault wrote so eloquently, was supplemented by an array of sites where the incitement of the patient or client to speech had become obligatory.\n\nThe voice of the patient, expressed in therapies, group sessions, confessional literature, and the like was initially encouraged by professionals for reasons of therapy: few regarded it as the pathway to a transformation of the relations of power and authority within the psychiatric system. However, it presaged the emergence of psychiatric users into the mainstream of debates about psychiatry and mental health, with narratives of illness and recovery, pressure groups, and support networks playing an increasing role in psychiatric politics. By the 1980s, psychiatry had become much more than a specialty for the management of a small minority of persons unable to live in the world of work, family, and civility\u2014it had become a widespread \"discipline of mental health\" whose rationale was not so much cure as \"coping\"\u2014helping troubled individuals manage themselves in their everyday lives (Rose 1986). And, by the closing decade of the twentieth century, this focus had widened to a positive vocation for the maximization of individual potential, the minimization of sadness and anxiety, the promotion of well-being, even happiness (Rose 1989, 1999).\n\nThe diagnostic manuals of our own age are, in a certain respect, analogous to those that arose from within the nineteenth century asylums. Like those 'atlases' that we discussed in chapter 2, they suggest that there are fundamental connections between all those who come under the gaze of the psychiatrist, a gaze now not confined to the asylum, or the closed world of the hospital, but directed across the open space of the community. Now, however, these connections among the inhabitants of this territory are established through making connections between the 'molar' and the 'molecular.' That is to say, the diversity of problems in managing a life in the everyday world is nonetheless unified\u2014at least in aspiration\u2014at the level of the brain. For it is in the brain, in its patterns of activity, in the structures of its regions and networks, in the functioning of its systems of neurotransmission, in its capacities to adapt to stimuli, and in its molecular and cellular processes that these heterogeneous problems find their substrate. But we are anticipating our argument. Let us return to the _DSM_.\n\nFrom the 1980s onward, the expanded yet differentiated idea of mental disorder that we have described became indelibly inscribed in the style of thought of modern psychiatry. _DSM-IV_ , published in 1994, runs to 886 pages and continues the trend of identifying even more distinct conditions. It classifies some 350 distinct disorders, from acute stress disorder to voyeurism (American Psychiatric Association 1994). It cautions that individuals within any diagnostic group are heterogeneous: its categories are only intended as aids to clinical judgment. But it promotes an idea of specificity in diagnosis that is linked to a conception of specificity in underlying pathology. The broad categories of the start of the twentieth century\u2014depression, schizophrenia, neurosis\u2014no longer appeared adequate: it seems that pathologies of mood, cognition, will, or affect must be dissected at a finer level of discrimination.\n\nIn the course of these developments, key distinctions began to wither away. The distinction between inside and outside the asylum began to blur in the 1950s. The distinction between the neuroses and the psychoses began to blur in the 1960s. By the 1980s, the distinction between mental and physical disorders itself began to blur. As _DSM-IV_ put it: \"although this volume is titled the _Diagnostic and Statistical Manual of Mental Disorders_ , the term _mental disorder_ unfortunately implies a distinction between 'mental' disorders and 'physical' disorders that is a reductionist anachronism of mind\/body dualism\" (1994, xxi). Over the 1980s, another key distinction, which had been the basis of a division of professional labor between psychiatrists and psychologists, also began to blur\u2014the distinction between states and traits. As discussed in chapter 1, this distinction between a relatively enduring feature of personality (a trait) and a particular episode of illness (a state) were the province of distinct disciplines\u2014psychology and psychiatry\u2014distinct explanatory regimes, and distinct modes of intervention, as well as grounding legal distinctions between those who were, or were not 'treatable' for their condition. But by the 1980s, it was being argued that there was no such ontological distinction between states of illness and traits of personality: for example, episodes both of illness, such as depression, and variations in traits, such as shyness or hostility, could be altered by psychiatric drugs (Knutson, Wolkowitz, Cole, et al. 1998). In the new neuromolecular gaze, variations in both states and traits could be analyzed in terms of the same molecular mechanisms, differing in many things, of course, but not in their underlying nature.\n\n_DSM-IV_ included, for the first time, a \"clinical significance\" criterion, which requires that, to be classified as a mental disorder, symptoms must cause \"clinically significant distress or impairment in social, occupational, or other important areas of functioning\" (1994, 7). In the words of Robert Spitzer and Jerome Wakefield, \"In response to concerns that the DSM criteria are overly inclusive, the clinical significance criterion attempts to minimize false positive diagnoses in situations in which the symptom criteria do not necessarily indicate pathology\" (Spitzer and Wakefield 1999, 1856). But Spitzer and Wakefield suggest that this delimitation, though helpful, is not in itself sufficient to eliminate the problem of false positives, because it still does not sufficiently consider questions of social context, and discriminate between normal and abnormal reactions to \"psychosocial stress\u2014it does not restrict the diagnosis of a disorder to situations where there is what they term 'a dysfunction' \" (ibid., 1863). Their cautions do not seem to have influenced the ways in which _DSM\u2013IV_ has been used to estimate the prevalence of psychiatric disorder. It seemed that, according to its criteria, the diagnosable psychiatric population was no longer a small minority; quite the reverse. Psychiatric morbidity was now common, widespread, almost ubiquitous. And, for many, this 'burden of mental disorder' had to be placed firmly on the agenda of policymakers.\n\nThe Burden of Mental Disorder\n\nHow many people suffer from psychiatric illness? This deceptively simple question turns out to be rather difficult to answer. One indicator is the high and increasing rate of use of psychiatric drugs (Rose 2006a, 2006b). But this is not the only one. In 2001, the WHO published a report titled _Mental Health: New Understanding, New Hope_. In the report, based on its own epidemiological work and that of others, it estimated that: \"mental and neurological conditions account for 30.8% of all years lived with disability (YLDs).... Six neuropsychiatric conditions figured in the top twenty causes of disability (YLDs) in the world, these being unipolar depressive disorders, alcohol use disorders, schizophrenia, bipolar affective disorder, Alzheimer's and other dementias, and migraine\" (World Health Organization 2001, 26). And it concluded: \"By the year 2020, if current trends for demographic and epidemiological transition continue, the burden of depression will increase to 5.7% of the total burden of disease, becoming the second leading cause of DALYs (disability adjusted life years) lost. Worldwide it will be second only to ischemic heart disease for DALYs lost for both sexes. In the developed regions, depression will then be the highest ranking cause of burden of disease\" (ibid., 30).\n\nEpidemiological studies of the prevalence of psychiatric disorders in the general population seem to give empirical support to this picture. Both in Europe and in the United States, such research claimed very high rates of prevalence of untreated psychiatric illness, as diagnosed by checklist interviews based on the descriptions of disorders in _DSM-IV_. The classic studies were undertaken by a team led by Ronald Kessler, originally trained as a sociologist and later a professor of health policy at Harvard Medical School. By 2003, the first of these surveys, published in 1994, had received more citations than any other paper published in the field of psychiatry and psychology (Kessler, McGonagle, Zhao, et al. 1994). In an interview at that time, Kessler noted that his survey, carried out by a household interview survey, was fundamentally dependent on the \"operational precision\" of the _DSM_ diagnostic criteria and the structured diagnostic interview developed by Lee Robins.\n\nKessler believed that the first of its findings\u2014that as many as half of all Americans have met criteria for some mental disorder at some time in their lives\u2014was so significant because \"it addressed the issue of stigma that has for so long interfered with rational thinking about mental illness. The mentally ill are not some distinct set of 'them' out there who are distinct from 'us' sane people. Instead, the vast majority of us have been touched by some form of mental illness at some time in our lives either through personal experience or through the illness of a close loved one. In many cases these illnesses are either mild or transient or both, but they certainly should not be considered in any way foreign.\" Kessler and his colleagues repeated the survey in 2005 with rather similar results: in any one year, 26.2% of adult Americans reported having symptoms that would qualify them for a _DSM-IV_ diagnosis of mental disorder, with the anxiety disorders leading at 18.1%, followed by mood disorders, notably depression, at 9.5% and impulse control disorders at 8.9%\u2014\"a total of over 57 million Americans diagnosable with mental disorder in any one year\" (Kessler, Chiu, Demler, et al. 2005). The authors concluded: \"About half of Americans will meet the criteria for a _DSM-IV_ disorder sometime in their life, with first onset usually in childhood or adolescence\" (Kessler, Berglund, Demler, et al. 2005).\n\nWe are not speaking here of the kinds of conditions that might previously been thought of as mild neuroses. Take, for example, personality disorder. While we often think of personality disorder as a relatively rare condition, the 2001\u20132 National Epidemiologic Survey on Alcohol and Related Conditions (NESARC), which administered a standardized diagnostic instrument to 43,093 adults over eighteen years old in the United States, reported that almost 15% of Americans, or 30.8 million adults, met _DSM-IV_ diagnostic criteria for at least one personality disorder (Grant, Hasin, Stinson, et al. 2004). The most common types of this disorder diagnosed were obsessive-compulsive personality disorder (16.4 million people), followed by paranoid personality disorder (9.2 million), and antisocial personality disorder (7.6 million). Other personality disorders affecting substantial numbers of Americans were schizoid (6.5 million), avoidant (4.9 million), histrionic (3.8 million), and dependent (1 million) (Bender 2004). This survey excluded borderline, schizotypal, and narcissistic disorders; however, NIMH estimates that borderline personality disorder alone affects a further 5.5 million Americans (National Institute for Mental Health 2001). All in all, then, these figures suggest that some 20% of Americans have diagnosable personality disorders. Which, of course, does make one wonder about the very idea of a normal personality.\n\nThese estimates have been much criticized, with many suggesting that they are the consequence of the _DSM_ itself (Kutchins and Kirk 1997). But they are not confined to the United States, nor to the use of the _DSM_. In 2005, a task force of the European College of Neuropsychopharmacology published the results of a Europe-wide survey of the size and burden of mental disorders in Europe, which claimed that some 27%, or 82.7 million of the adult EU population (eighteen to sixty-five years of age) \"is or has been affected by at least one mental disorder in the past 12 months.... [T]he most frequent disorders are anxiety disorders, depressive, somatoform and substance dependency disorders.... Only 26% of cases had any consultation with professional health care services, a finding suggesting a considerable degree of unmet need\" (Wittchen and Jacobi 2005, 357). It further estimated that over their lifetimes 50% of the EU population would be affected by at least one diagnosable psychiatric disorder, if lifetime risk was considered, and that mental disorders were associated with costs of more than \u20ac290 billion, the majority of which were not health-care costs (ibid., 355\u201356).\n\nThe matter was put plainly by the European Brain Council (EBC): \"There are an estimated 127 million Europeans currently living with a brain disorder out of a population of 466 million. The total annual cost of brain disorders in Europe was estimated to \u20ac386 billion in 2004\" (European Brain Council 2005, x). The diagnostic and epidemiological data used in these estimates were not solely based on the _DSM_ , but included the diagnostic criteria more often used in Europe, such as the ICD-9 or -10, or other diagnostic assessment scales. For the EBC, not just neurodegenerative disorders and major mental illnesses are considered brain disorders, but so are addiction, depression, anxiety and panic disorder\u2014what else could they be? How can we understand this simultaneous extension and cerebralization of the idea of mental abnormality?\n\nLet us return to the United States and the _DSM_. It is, of course, something of a paradox that while a key objective of those developing _DSM_ had been the _delimitation_ of the territory of psychiatry, the argument now is reversed. Kessler, speaking in 2003, said that the initial skepticism about his findings had been based on the error that \"the term 'mentally ill' is being taken too seriously.... [W]e invest the term 'mentally ill' with excess meaning. A number of common mental illnesses, like adjustment disorders and brief episodes of depression, are usually mild and self-limiting.\"  In his view, not everything classified in the _DSM_ \u2014and hence included in his estimates\u2014required treatment. However, this was a little disingenuous as, in the same year of this interview, he and his colleagues argued that it was important to include mild disorders in the forthcoming _DSM-5_ because mild disorders may progress to major ones, and diagnosis at an early stage provides an opportunity for preventive intervention (Kessler, Merikangas, Berglund, et al. 2003). Indeed, most of those who use these data to urge policymakers to focus funding and energy on mental health argue along the same lines: the burden of untreated mental disorder shown by such studies, they argue, shows that we need a step change in our capacity to understand and to treat these disorders, and in our capacity to screen our populations for them, to pick up presymptomatic susceptibilities and prevent the development of disorder.\n\nThus, Steven Hyman opens his paper \"A Glimmer of Light for Neuropsychiatric Disorders\" by referring explicitly to estimates of the prevalence of psychiatric disorders\u2014or what he terms \"neuropsychiatric disorders\"\u2014and suggesting that neuropsychiatry may soon be able to meet the challenge they pose:\n\nNeuropsychiatric disorders severely compromise the well-being of those affected, with their negative effects on general health and on the ability of children to learn and of adults to work. These disorders have a relatively high prevalence[,]... can have an early onset (for example, autism in childhood and schizophrenia in young adulthood) or a relapsing\u2013remitting course (as in mood and anxiety disorders and obsessive\u2013compulsive disorder), and often have disabling symptoms. For these reasons, they exert, in aggregate, a devastating impact on the families involved and on the human capital of a society.... Despite the disease burden attributable to neuropsychiatric disorders, and despite significant research, their mechanisms of pathogenesis and precise genetic and non-genetic risk factors have remained stubbornly out of reach. Although sustained and clever exploitation of clinical observations has produced a useful pharmacopoeia, almost all recently introduced psychotherapeutic drugs are based on reverse engineering the mechanisms of existing drugs. Arguably, no new drug targets or therapeutic mechanisms of real significance have been identified for more than four decades.... This parlous state of affairs is finally beginning to improve, in part through the application of new genomic technologies coupled to advances in neuroscience. (Hyman 2008, 890)\n\nTwo questions thus seem relevant. First, how much credibility can we give to these estimates? And second, why should we look to neuropsychiatry for the answer?\n\nLet us begin with the first question. The debate about the utility of these epidemiological studies has not abated. Some have suggested that the high prevalence rates reported arose because researchers failed to address such issues as symptom threshold, severity and level of impairment, and duration of symptoms (Regier, Kaelber, Rae, et al. 1998; Regier 2000). Insel and Fenton believe that this was indeed a problem for earlier epidemiological research: \"Many critical issues were not addressed by earlier studies. While the overall 12-month prevalence of any mental illness was reported to be in the range of 30%, significant questions about the disability associated with these syndromes remain. How severe are the disorders reported to be present in 30% of the population? What is the economic and public health impact of these conditions? How long is the delay between onset and diagnosis?\" (Insel and Fenton 2005, 590).\n\nBut their comment is in a commentary on more recent studies by Kessler and his colleagues that replicated the earlier research and sought to refine it; they claimed that the majority of those diagnosed by such methods are rated as having a serious or moderate disorder, with substantial disruption to their ability to carry out their normal activities and, further, that retrospective studies such as theirs actually _underestimate_ prevalence because respondents underreport the symptoms that they have had in the past, perhaps because of poor recall (Kessler, Berglund, Demler, et al. 2005a; Kessler, Chiu, Demler, et al. 2005; Wang, Berglund, Olfson, et al. 2005; Wang, Lane, Olfson, et al. 2005). Thus, in their commentary, Insel and Fenton concur with those who believe that such surveys do clearly demonstrate the high level of unmet need for mental health care. If this was not enough, researchers using a different, 'prospective' approach, based on their work with the Dunedin cohort study in New Zealand, argue that this method confirms the underestimation of prevalence in existing research and suggest that the actual lifetime prevalence rate of _DSM_ -diagnosed mental disorders is about twice that shown in the retrospective studies such as those by Kessler and his team (Moffitt, Caspi, Taylor, et al. 2009).\n\nOthers, however, continue to question the high prevalence rates and the methods used to generate them. Those from the team developing _DSM-5_ suggested technical changes, for example, requiring more evidence of severity (Narrow, Rae, Robins, et al. 2002; Narrow, Kuhl, and Regier 2009). Others warned those developing _DSM-5_ of the pitfalls they might face. Thus Allen Frances, who was chair of the _DSM-IV_ task force, argued that the problems in delimiting the diagnostic criteria in _DSM-IV_ had produced the consequences of contributing to \"the false epidemic of autism and attention-deficit disorder and a forensic disaster that has led to the inappropriate psychiatric commitment of sexually violent offenders\": he warned those responsible for producing _DSM-5_ of the danger of exacerbating these problems if they produce a set of criteria that \"capture\" prodromal and subthreshold conditions (Frances 2010, 2).\n\nPerhaps the most through interrogation of the methods used to create the prevalence data on the burden of brain disorders has been undertaken by Allan Horwitz and Jerome Wakefield (Horwitz and Wakefield 2007). Their principal argument is that psychiatric diagnoses using _DSM_ criteria generate many false positives because they fail adequately to discriminate between 'normal' responses to experiential factors\u2014that is to say, normal anxiety, normal sadness, or normal grief in the face of the difficulties that life presents us with\u2014and pathological conditions requiring specialized medical intervention. The presenting features of these normal responses overlap with those designated as symptoms of disorder in _DSM_ , and because _DSM_ fails to recognize the importance and take adequate account of context, it thus fails to differentiate between normal reactions that have a clear cause (for example, loss of job, ending of a relationship, reaction to a life-threatening medical condition, etc.) and those that, because of their severity, duration, or other features, that is, those without a cause, reveal that something has gone wrong that requires specialized psychiatric intervention.\n\nWhen framed as 'symptoms,' these normal reactions may well appear to psychiatrists, or those conducting community epidemiological surveys, as indicating a high level of undiagnosed psychiatric disorder in the general population. While clinicians may be able to make good and appropriate use of _DSM_ -type diagnostic procedures in clinical settings, they are not appropriate for use in the community. Even the clinical significance criterion introduced into _DSM-IV_ is not very helpful here, they argue, because normal reactions can indeed reach the threshold of significant distress or impairment in functioning in one's allocated social roles without indicating some underlying harmful dysfunction. The apparent epidemic that such community studies seem to reveal, and that mental health policy is urged to address, is, in their view, merely a consequence of the inappropriate decontextualized conceptual framework used in the research\u2014it is an epidemic of false positives.\n\nIt is not surprising, then, that in the light of these continued questions over the validity of diagnostic methods, some argue that there is really only one way to settle this matter\u2014to identify some biological abnormality, lesion, malfunction, or marker that will demonstrate objectively, once and for all, who is, and who is not, suffering from a mental disorder. Will neuroscience enable us, at last, 'to define true madness'?\n\nAll in the Brain?\n\nOur chapter opened with a quote from Amanda Haag, but we do not have to look to science journalists for the hope that neuroscience will resolve the diagnostic problems of psychiatry by demarcating disorders in terms of their underlying neurobiological bases. Let us quote Steven Hyman again. In 2003, writing while director of the NIMH for a popular audience in _Scientific American_ , he put it thus: \"By combining neuroimaging with genetic studies, physicians may eventually be able to move psychiatric diagnoses out of the realm of symptom checklists and into the domain of objective medical tests. Genetic testing of patients could reveal who is at high risk for developing a disorder such as schizophrenia or depression. Doctors could then use neuroimaging on the high-risk patients to determine whether the disorder has actually set in. I do not want to sound too optimistic\u2014the task is daunting. But the current pace of technological development augurs well for progress.\"\n\nThe hope at that time was threefold. First, in pinning psychiatric classification and categorization to a neurobiological basis, using identifiable markers in genetics and in patterns of brain activation, it would be possible not just to make accurate distinctions between normality and pathology, but also to distinguish subtypes of pathology according to their organic and functional pathways. Psychiatric classification would move from the surface to the interior, from symptom checklists to the brain itself. Second, in doing so, one would gain understandings of etiology, which would not only enable accurate diagnosis and prognosis, but also would make it possible to screen presymptomatic individuals thought to be at risk and intervene preventively before the disease developed. Third, on this basis, one might be able to develop effective treatments that target the neurobiological basis of mental disorders.\n\nOne organizing focus for these hopes of reform was, once again, the revision process for the _DSM_. The process of revision of the fourth edition of that manual began in 1999, with the _DSM-5_ scheduled for publication in 2013. In 2002, David Kupfer, Michael First, and Darrel Regier published _A Research Agenda for DSM-V_ , which among other things, argued for greater neurobiological research on nosological issues\u2014that is to say, questions of classification of disorders (Kupfer, First, and Regier 2002). The volume collected five \"white papers\" from different work groups, the first substantive one of which was titled \"Neuroscience Research Agenda to Guide Development of a Pathophysiologically Based Classification System.\" The authors\u2014Dennis Charney, David Barlow, Kelly Botteron, Jonathan Cohen, David Goldman, Raquel Gur, Keh-Ming Lin, Juan L\u00f3pez, James Meador-Woodruff, Steven Moldin, Eric Nestler, Stanley Watson, and Steven Zalcman stated the problem boldly, if pessimistically, early in their discussion: \"At the risk of making an overly broad statement of the status of neurobiological investigations of the major psychiatric disorders noted above, it can be concluded that the field of psychiatry has thus far failed to identify a single neurobiological phenotypic marker or gene that is useful in making a diagnosis of a major psychiatric disorder or for predicting response to psychopharmacologic treatment\" (Charney, Barlow, Botteron, et al. 2002, 33).\n\nThey argued that it would be years\u2014and possibly decades\u2014before a fully explicated etiologically and pathophysiologically based classification system for psychiatry would exist. Nonetheless, at the close of their review of a whole variety of methods\u2014genetic research, brain imaging, postmortem studies, and work with animal models\u2014relevant to elucidating the pathophysiology of mental disorders, they allowed themselves to imagine:\n\nWe speculate that single genes will be discovered that map onto specific cognitive, emotional, and behavioral disturbances but will not correspond neatly to currently defined diagnostic entities. Rather, it will be discovered that specific combinations of genes will relate to constellations of abnormalities in many brain-based functions\u2014including but not limited to the regulation of mood, anxiety, perception, learning, memory, aggression, eating, sleeping, and sexual function\u2014that will coalesce to form disease states heretofore unrecognized. On the other hand, genes that confer resilience and protection will also be identified, and their interaction with disease-related genes will be clarified. The impact of environmental factors on gene expression and phenotype expression will be defined. The ability to discover intermediate phenotypes will be improved with advances in techniques such as neuroimaging. This will all lead to novel therapeutic targets of greater efficacy and specificity to disease states. Prediction of therapeutic response will be possible through genetic analysis and phenotype analysis. Disease prevention will become a realistic goal. (Charney, Barlow, Botteron et al., 2002: 70)\n\nAnd they even suggested a hypothetical radical revision of the _DSM_ axes. Instead of the current multiaxial system (introduced in _DSM-III_ ), which provides information for clinical disorders (Axis I), personality disorders and mental retardation (Axis II), any general medical condition (Axis III), psychosocial and environmental problems (Axis IV), and global functioning (Axis V), they proposed the following: Axis I for _genotype_ , identifying symptom-or disease-related genes, resiliency genes, and genes related to therapeutic responses and side effects to specific psychotropic drugs; Axis II for _neurobiological phenotype_ , such as identifying intermediate phenotypes discovered by neuroimaging, cognitive evaluation, and neurophysiological testing that could aid in selecting targeted pharmacotherapies and psychotherapies and monitoring the neurobiological response; Axis III for _behavioral phenotype_ , severity and frequency of specific cognitive, emotional, and behavioral disturbances; Axis IV for _environmental modifiers or precipitants_ to be evaluated in the context of genotype; and Axis V for _therapeutics._\n\nSeven years later, the prospects for a radical revision of diagnostic criteria along neurobiological lines looked, if anything, even more remote than they did in 2002. When Regier and his colleagues from the team overseeing the revision discussed the conceptual basis of _DSM-5_ , their diagnosis of the problems of _DSM-IV_ was damning, and it seemed that the prospects for a neurobiological solution were remote (Regier, Narrow, Kuhl, et al. 2009). They were particularly concerned by the lack of clear separation between _DSM_ -defined syndromes, as there seemed to be high levels of \"comorbidity\" between mood, somatic, and anxiety syndromes. The question, for them, was how to \"update our classification to recognize the most prominent syndromes that are actually present in nature, rather than in the heuristic and anachronistic pure types of previous scientific eras\" (ibid., 646).\n\nThe hope that psychopharmaceuticals would 'cut nature at its joints'\u2014that one might differentiate disorders by identifying which conditions responded to which drugs\u2014now seemed forlorn. The SSRIs had proved not to be specific for depression but were now prescribed for a range of anxiety, mood, and eating disorders; the 'atypical antipsychotics' were now licensed not only for schizophrenia but also bipolar disorder and for treatment-resistant major depression. More troubling, given so much investment in genomic research, the _DSM-5_ revision team was skeptical about the prospects for large-scale genetic studies using GWAS: they thought it would be \"unlikely to find single gene underpinnings for most mental disorders, which are more likely to have polygenetic vulnerabilities interacting with epigenetic factors (that switch genes on and off) and environmental exposures to produce disorders\" (Regier, Narrow, Kuhl, et al. 2009, 646). But hope does not die easily, and they remained optimistic that, in some way mental disorder syndromes could be redefined so that they were useful diagnostically and related to objective neurobiological features\u2014that one could recognize the \"dimensional\" nature of disorders, yet still set clear thresholds between pathology and normality.\n\nIndeed by the end of the decade, the prevailing attitude among neuropsychiatric researchers seemed to be a combination of disappointment in the present with hope for the future. Take, for example, Chou and Chouard, in their introduction to a _Nature_ \"Insights\" supplement published in October 2008:\n\nDespite decades of... research, the prevalence of neuropsychiatric diseases has not decreased. Our understanding of the biological mechanisms of diseases such as mood disorders, schizophrenia and autism is frustratingly limited. And, although it has long been clear that most such diseases have a strong genetic component, the identities of the genes involved have proved elusive. There is also a lack of reliable biological markers for characterizing these diseases and, perhaps unsurprisingly, treatment options are far from optimal in terms of efficacy and specificity. There is, however, some cause for optimism. Recent advances in genomic technology and large-scale studies are helping to identify genetic variants associated with diseases. In addition, new animal models of disorders such as depression and autism are providing ways to test hypotheses about the underlying neuropathology\u2014at the molecular, neural-circuit and behavioral levels.... The hope is that developments such as these will lead to integrative approaches for designing better therapeutic strategies. (Chou and Chouard 2008, 889)\n\nOthers writing in the same issue were less optimistic and doubted the very premises on which neuropsychiatric hope was based. Take, for example, Steven Hyman, who, as we have seen, was an optimist in 2003 and even quite up-beat when writing for a popular audience five years later. Yet in the same year, writing for his peers, he expressed a rather different opinion: that we have made little progress even in those disorders where conventional measures of heritability based on family studies indicate a strong genetic component, such as autism, schizophrenia, and bipolar disorder:\n\nEfforts to identify risk-conferring alleles... have been largely unrewarding.... The underlying genetics of common neuropsychiatric disorders has proved highly complex.... [T]here is much evidence that similar neuropsychiatric symptoms can result from different combinations of genetic risk factors [and] that the same genetic variant may be associated with multiple DSM-IV diagnoses.... The identification of common risk-conferring variants has... proved extremely challenging in most cases, because of their relatively small contribution to the disease phenotype... because of the diverse genetic, environmental and random factors that lead to these common disease. (Hyman 2008, 891)\n\nHyman, concurring with several generations of sociological critics, now concluded that \"what we think of as a single 'disease' is not in a strict sense a homogeneous entity for which there is a 'Platonic' ideal phenotype. Common diseases are more likely to represent families of diseases that share major pathophysiological and symptomatic features, but can differ in important characteristics such as age of onset, severity of symptoms, rate of progression, and response to treatment\" (2008, 891). He argued that this problem of the phenotype is one of the reasons why GWAS fail in psychiatry, but that they also fail because they do not take account of the fact that genes for psychiatric disorders are pleiotropic in effects (i.e., a single gene can produce many effects), and crucially that the pathways from genomics to phenotype are complex and not understood. And, perhaps most damaging for the hopes that were vested in neurobiology to resolve the diagnostic problems of psychiatry, he questioned the view that it might ever be possible to draw clear boundaries between ill and well\u2014it is, he thinks, better to think of \"dimensions\" without sharp boundaries.\n\nThis pessimistic conclusion on the current state of neuropsychiatry was echoed in the same issue of _Nature_ by Krishnan and Nestler, who discussed the molecular neurobiology of depression (Krishnan and Nestler 2008). Nestler, who is Nash Family Professor of Neuroscience, chairman of the Department of Neuroscience, and director of the Brain Institute at the Mount Sinai Medical Center in New York, was one of the authors of the chapter on neurobiology in the 2002 _Research Agenda for DSM-V._ He and his coauthor were also skeptical of the prospects for GWAS in depression, not least because of the problems of comorbidities. They argued, further, that brain imaging studies were flawed because of simplistic ideas about localization (for instance, that the amygdala is the locus for fear and anxiety; the nucleus accumbens the locus for reward; and so forth). \"Such artificial distinctions are of limited heuristic value,\" they believe, as \"depressive symptoms are probably mediated by dysfunction in a diffuse series of neural networks.\" (ibid., 895). Earlier theories of depression, like the monoamine hypothesis, are false: \"[A]lterations in central monoamine function might contribute marginally to genetic vulnerability.... [T]he cause of depression [is not] a simple deficiency of central monoamines.\" (ibid.). And the effects of drugs that act on serotonin levels, they suggested, are actually due to secondary changes in molecular and cellular plasticity. They concluded: \"Collectively, these studies highlight the weaknesses of attempts to generate a 'unified theory' of depression. Mechanisms that promote depressive symptoms in response to stress differ markedly between different neural circuits and can also be distinct from changes that underlie depression in the absence of external stress ('endogenous depression'). In addition, neuroplastic events that are required for antidepressant efficacy need not function through the reversal of stress-induced plasticity... and might function through separate and parallel circuits\" (Krishnan and Nestler 2008, 898). In short, at the neurobiological level, \"enormous gaps in the knowledge of depression and its treatment persist... [and researchers] must follow a systems approach that acknowledges the powerful bidirectional interactions between peripheral organs and the brain\" (ibid., 901).\n\nIn February 2010, the APA released the proposed draft diagnostic criteria for the fifth edition of the _DSM_. Among the changes proposed were several relevant to our discussion here: the current diagnoses of autistic disorder, Asperger's disorder, childhood disintegrative disorder, and pervasive developmental disorder (not otherwise specified) were to be incorporated into a single diagnostic category of \"autism spectrum disorders\"; there were various changes to the categories relating to substance abuse and dependence, which were to be incorporated into a new category of \"addiction and related disorders\"; a new category of \"behavioral addictions\" was proposed, with gambling as the sole disorder; there was to be consideration of a new \"risk syndromes\" category, with information to help clinicians identify earlier stages of some serious mental disorders, such as neurocognitive disorder (dementia) and psychosis; and there were other changes designed to assist clinicians in better differentiating the disorders of children. Dimensional assessments would be added to diagnostic categories to permit clinicians to evaluate the severity of symptoms. And arabic numerals were to replace the previously used roman numerals\u2014thus the new edition would be, not _DSM-V_ but _DSM-5.0_ , to enable future revisions to be identified as _DSM-5.2_ , etc.\u2014to make _DSM_ \"a living document.\"\n\nMany of those who had high hopes for neuropsychiatry expressed disappointment at the way things were shaping up. One prolific commentator, Dr. Charles Parker, writing in his CorePsychBlog three days after the draft criteria were published, expostulated:\n\n_The Earth Is Flat! DSM 5 Points at the Tips of Icebergs\u2014Only What You See_\n\nThe tip-toe progress with these new superficial labels for office appearances misses altogether the complexity of new brain and body science. Neuroscience evidence is easily available, often paid for by insurance, and remains almost completely ignored by psychiatry. Real facts, not labels, will foretell the changes necessary for psychiatric practice, for treatment strategies to evolve with the rapidly evolving new science.... [T]he new DSM5 conclusions are based almost completely upon 19th century vertical thinking, insufficient feedback with patients in the office, and questions that ignore modern neurophysiology and metabolism. Psychiatrists will remain speculative with dreams and fantasies, while hard evidence from molecular and cellular physiology remains in the closet\u2014frequently derided as quackery. Interestingly, psychiatry is held to a different standard on evidence than the rest of medicine, as SPECT brain imaging and the measurement of neurotransmitter biomarkers is still derided as non-specific, while patently non-specific biomarkers, such as cholesterol screening, are accepted uniformly in general medicine. It's time for diagnostic change\u2014but DSM5 is already old news.\n\nOthers, however, did identify something new emerging in the proposals for the new version of _DSM_ \u2014the inclusion of some subthreshold or prodromal conditions. These were conditions where clear symptoms of disorder were not yet visible, but where there was believed to be high risk of progressing to a frank disorder\u2014a perception intrinsically linked with the belief in the value of screening to identify those at risk at the presymptomatic stage and to intervene early to avert or mitigate the development of the disorder. Two areas were notable here: psychosis risk syndrome and minor neurocognitive disorder. Let us focus on the first.\n\nEarly intervention for those at risk of psychosis had advocates dating back at least to the 1990s, with some arguing that best results would be obtained by intervention at as early a point as possible, even before any outright symptoms had emerged (McGorry et al. 2009). During the first decade of this century, a group of international collaborators, with key investigators at the Institute of Psychiatry in London, developed a program for early diagnosis of psychosis, to be followed by intervention with atypical antipsychotics, perhaps combined with the prescription of some essential fatty acids and cognitive behavioral therapy. It appeared that general practitioners could be educated to make early diagnosis (Power et al. 2007) and that it might be possible to use imaging technologies to assist accurate diagnoses at the prodromal stage (Pantelis et al. 2003; Howes et al. 2007; Meisenzahl et al. 2008; Smieskova et al. 2010), although the efficacy of these early interventions with drugs and other measures in preventing further development of the disorder was not yet clear (Gafoor, Nitsch, and McCrone 2010). Some suggested that cost savings would be generated by such interventions with those deemed to be at high risk (Valmaggia et al. 2009). Others, however, were more skeptical, arguing that prodromal interventions were not clinically justified, generated high levels of false positives, that no significant differences arising from early intervention were found on follow-up, and that there were significant problems from the side effects of antipsychotic medication (De Koning et al. 2009; Bosanac, Patton, and Castle 2010).\n\nMore radically, Nil Kaymaz and Jim van Os, writing in the journal _Psychosis_ , took particular issue with the proposed diagnosis itself, arguing that the category of Psychosis Risk Syndrome \"is based on the notion that labeling people with invalid diagnostic terms has more clinical relevance than simply addressing care needs, is contingent on elusive sampling strategies posing as precise diagnostic criteria, and is associated with a false-positive rate of at least 90% in the year after diagnosis. In the twenty-first century, opinion-based diagnostics continues to pose a threat to the process of diagnostic revision\" (Kaymaz and van Os 2010). Not, however, that these authors were opposed to early intervention in itself; indeed, although they recognized that all current examples of such intervention generate large numbers of false positives, they remained of the view that early intervention \"improves subsequent course and outcome given that psychotic disorders, arising during adolescence and early adulthood, disrupt important maturational tasks and increase the risk of long term exclusion from meaningful societal participation\" (ibid., 100). Despite their assertion that \"the hypothesis of valid nosological entities is very weak,\" they look to resolve this issue not by an appeal to biomarkers, but to the generic diagnosis already in _DSM-IV_ : \"Psychotic Disorder, Not Otherwise Specified\" (Kaymaz and van Os 2010)\n\nSome leading psychiatrists, however, warned of the dangers of these proposals for early intervention. In the month before the publication of the draft criteria for _DSM-5.0_ , Allen Frances, who had been the chair of the _DSM-IV_ task force, had expressed his concern that the subthreshold diagnoses would \"medicalize normality\":\n\nThe most appealing subthreshold conditions (minor depression, mixed anxiety depression, minor cognitive disorder, and prepsychotic disorder) are all characterized by nonspecific symptoms that are present at extremely high frequencies in the general population. These proposed \"disorders\" might well become among the most common diagnoses in the general population\u2014particularly once they are helped along by drug company marketing\u2014resulting in excessive use of medications that often have serious long-term complications associated with weight gain. Early case finding is a wonderful goal, but it requires a happy combination of a specific diagnostic test and a safe intervention. Instead, we would now have the peculiarly unhappy combination of a wildly false-positive set of criteria coupled with potentially dangerous interventions.... Altogether, in my view, the costs and risks of the subthreshold diagnoses far outweigh any possible current gains. (Frances 2010)\n\nHowever, as we shall see in many of the areas discussed in this book, the logic of 'screen and intervene,' the identification of prodromal or subthreshold disorders coupled with interventions in the name or preventing or mitigating the condition, has proved hard to resist.\n\nNeuropsychiatry and the Dilemmas of Diagnosis\n\nCould _DSM 5.0_ avoid those dangers of neurobiology's medicalization of normality by utilizing biomarkers, by grounding diagnostic classification on objective anomalies identified in the brain? The situation did not look very encouraging. In the draft criteria, positive references to the current utility of biomarkers are rare. One does find reference to \"a positive genetic test for dominantly inherited AD, or as the field develops, evidence that certain imaging markers, atrophy of medial temporal lobe structures on MRI, temporoparietal hypometabolism on FDG-PET, amyloid deposition on PET scanning or markers for tau and abeta in the CSF\" as a requirement if one is to make a prediction of Alzheimer's disease in cases of minor neurocognitive disorder, but the working group does not consider that this would be appropriate in cases where the diagnosis is Mild Cognitive Impairment (MCI). But this reference to a biomarker is an exception. Each of the pathways that neuropsychiatry has attempted to trace through the brain seems to run, not into the bright uplands of clarity, but into the murky, damp, misty, and mysterious forests of uncertainty. Where, then, do matters stand in the question of the role of neuroscience in the diagnosis and delimitation of mental disorders?\n\nDespite the penetrating gaze of neuroscience, which has opened up the brain to vision in so many ways, as we have seen in chapter 2, psychiatric classification remains superficial. This neuromolecular vision seems incapable of grounding the clinical work of psychiatry in the way that has become routine in other areas of medicine. Despite the conviction of most practitioners that they deal with conditions that have a corporeal seat in the brain of the afflicted individual, psychiatry has failed to establish the bridge that, from the nineteenth century on, underpinned the epistemology of modern clinical medicine\u2014the capacity to link the troubles of the troubled and troubling individuals who are its subjects with the vital anomalies that underpin them.\n\nWhy, then, has neuropsychiatry not been able to \"self-vindicate\"\u2014to use Ian Hacking's term (Hacking 1992a)? Of course, in part this is because it is not a laboratory science, for while its research may, in part, follow the protocols of an experimental laboratory discipline, the phenomena that it addresses\u2014persons deemed to be ailing in particular ways\u2014are not laboratory subjects. One might have thought that _DSM_ criteria would enable such self-vindication\u2014after all, they are used to select subjects for entry into neuropsychiatric research studies, which then become the basis for hypotheses, trials, developments of drugs, labeling and treatment by relevant personnel, changes in self-perception, and so forth. And as we have seen in chapter 3, the behaviors of animals in animal models of psychopathologies\u2014what we called _models_ _5_ \u2014are indeed assessed according to those very _DSM_ criteria. But although the criteria have summoned up an image of a mental disorder in thought\u2014a specific genetic sequence or set of sequences linked to anomalies in the neurotransmitter system or other aspects of neurobiology causally related to a particular array of cognitive, affective, volitional, or behavioral dysfunctions\u2014this assemblage has proved impossible to summon up in reality, despite the expenditure of millions of dollars and thousands of person-hours. The phenomenon itself, that is to say, has proved resistant. Why should this be?\n\nNo doubt a certain quantum of realism is essential here. Genomics, once welcomed as a potential savior, has not lived up to its early promises. The style of thought that underpins GWAS has proved to be radically oversimple when applied to the field of mental disorder. The response of neuropsychiatrists committed to this way of thinking has been to appeal to _complexity_. They suggest that many genes are likely to be involved in any psychiatric disorder. They argue that any gene sequence that might be involved is likely to be pleiotropic. They contemplate the likelihood that there may be many routes to the final common pathways that generate the phenotypical symptoms. And they try to incorporate in their explanatory systems something that, elsewhere within psychiatry, has always been well known: that the manifestation of a disorder is an outcome of complex and individual variable interactions between genomics, developmental processes, epigenetics, neuroplasticity, life experiences, and hormonal and nutritional states.\n\nMore fundamentally fatal to self-vindication has been the recognition that psychiatric diagnoses, however much they may appear to be stabilized by the circular transactions between the diagnostic manuals and the training of the clinical gaze of the psychiatrist, seem to have a kind of irreducible fuzziness, so that selection of cases and controls on the basis of phenotypes is unable to frame studies in a way that is capable of generating clear results. Of course, social scientists have argued for decades that sociocultural belief systems play a crucial role in shaping not only the recognition of unease by others, but also its shaping into phenotypical symptoms. Hence it is no surprise to them to find that there are no good routes from psychiatric phenotypes to SNPs; that in almost every case, GWAS signals have been weak in psychiatry; and that functional pathways that have been implicated are those that are basic to neurodevelopment, and not specific to psychiatric disorders.\n\nFor the same reason, the drive to discover biomarkers, though stimulated by a whole constellation of forces\u2014commercial interests, public health concerns, employer and health service concerns, and pressure from individuals and families\u2014has yielded no validated results. Even if some markers are found in the future, all available evidence suggests that they will, at best, indicate marginal increases in susceptibility in particular environmental circumstances, and that these will be modulated by other genetic and neurobiological factors that are related to _resilience_ \u2014a term that is itself now becoming the basis of an expanding program of research. Further, despite the fervent beliefs in the virtues of early diagnosis and intervention, there is little evidence that presymptomatic intervention is more effective than intervention at the time when problematic conduct is shown. Premature use of such biomarkers within a strategy of screen and intervene would produce large numbers of false negatives and false positives, generate stigma rather than reduce it, expand interventions and their costs and downsides, and start many unnecessarily on a psychiatric career (Rose 2008; Singh and Rose 2009). Perhaps there is a lesson here about the dangerous lure of the 'translational imperative'\u2014neurobiological work in these areas may have been valuable in exploring brain mechanisms, but it may also be wise for those working in this area to avoid premature claims about the implications of their results for humans and clinical practice.\n\nIt is clear that neuropsychiatry has a long way to go in developing a style of thought adequate to the complexity of its object, when that object is an ailment suffered by an individual human being living in a social world. Few would doubt the enormous advances in our understanding of neural processes that have been made since the 1960s. But neurobiology will not solve psychiatry's current diagnostic dilemmas. Neurobiology cannot bear the weight placed on diagnosis in psychiatry for many reasons. First, allocation to a diagnostic category does not tell one much about the severity of symptoms, their duration, the level of impairment involved, the amount of distress caused to the individual or the family, and the extent of need for care. Further, diagnosis has low predictive validity and tells us little about the course of development of the problem, or the specific social, environmental, or other factors that may shape this course. More fundamentally, neuropsychiatry won't solve the dilemmas of diagnosis, because diagnosis is not solely about the nature or causes of a disorder. As we have seen, diagnoses in psychiatry answer to many different and competing demands: statistical record keeping for institutions and clinicians, entry into research trials and studies, eligibility for treatment, choice of intervention, prognosis, predictions of need for care, eligibility for work, eligibility for compensation via insurance, and epidemiology and public health policies, including predictions of need and distribution of resources. Perhaps no single system could ever bear this weight, but it is certain that one based on the brain alone cannot.\n\nAt root, the neurobiological project in psychiatry finds its limit in the simple and often repeated fact: mental disorders are problems of persons, not of brains. Mental disorders are not problems of brains in labs, but of human beings in time, space, culture, and history. And, indeed, so is diagnosis; however well-trained and stabilized the gaze of the doctor, however regulated by criteria and augmented by neuromolecular indicators, diagnosis is a practice governed by its own rules\u2014rules that are different in epidemiology, in cohort studies, in research, in the clinic, and in life. Has neuropsychiatry brought us any closer to resolving that centuries-old problem of how to define 'true madness'\u2014to find what one recent author referred to as \"a brain based classification of mental illness\" (Miller 2010)? At present, one must answer in the negative.\nChapter Five\n\nThe Social Brain\n\nA primate specialization which has now been amply documented by behavioral scientists, but ignored for the most part by neuroscientists, is the specialization for social cognition. What I mean by the term \"social cognition\" is the following: _Social cognition is the processing of any information which culminates in the accurate perception of the dispositions and intentions of other individuals_. While many non-primates (for example, ants) can interact in highly specific ways with others of their kind, it appears that primates, especially those most closely related to ourselves, have developed a unique capacity to perceive psychological facts (dispositions and intentions) about other individuals. This capacity appears to distinguish primate social behavior from that of other orders.\n\n\u2014Leslie Brothers, \"The Social Brain,\" 1990\n\nAn... hypothesis offered during the late 1980s was that primates' large brains reflect the computational demands of the complex social systems that characterize the order.... There is ample evidence that primate social systems are more complex than those of other species. These systems can be shown to involve processes such as tactical deception and coalition-formation... which are rare or occur only in simpler forms in other taxonomic groups. Because of this, the suggestion was rapidly dubbed the Machiavellian intelligence hypothesis, although there is a growing preference to call it the social brain hypothesis.\n\n\u2014Robin Dunbar, \"The Social Brain Hypothesis,\" 1998\n\nPhilosophers and psychologists have long wondered about \"this thing called empathy.\" Only recently social neuroscience has begun to provide some support in this endeavor. Initial findings are encouraging that we will some day have a better understanding of why, when, and how we experience empathy and whether we can use that knowledge to increase prosocial behavior and an intersubjectivity that is grounded in a better understanding of ourselves and of others.\n\n\u2014Tania Singer and Claus Lamm, \"The Social Neuroscience of Empathy,\" 2009\n\nHuman beings, to state the obvious, are social creatures. We have platitudes by the dozen to assure us that 'no man is an island.' We live in groups\u2014families, communities, societies. We work collaboratively in organizations, fight in bands and armies, take pleasure in events where we gather together to dance, party, watch or play sports. We interact in pairs, and small and large groups, whether in love or in hate, in teams and gangs, and in everyday activities. We care for one another and experience sympathy, empathy, or a sense of obligation to some, though not to others. And so forth. But why? What accounts for this apparently intrinsic and ubiquitous sociality of our species? Is it, as one might say, in our nature, as we primates have evolved to become human? And, if so, does that nature, that biology, that neurobiology, account not only for the fact that we humans are social, but also for the ways that sociality has varied across history, between cultures, among different practices?\n\nFor the human sciences, the constitutive role of human sociality is undoubtedly a premise, but it is one that tends to be framed in a particular way. What _is_ in our nature is the fact that we humans are born incomplete, requiring interaction with others, and enmeshing in language, meaning, and culture, to be completed\u2014to paraphrase Simone de Beauvoir, we are not born, but become social. Absent that, humans remain mere animals; and for some, the repetitive stories of wild children, abandoned, isolated, living without human contact, seem to testify to that truth (Hirst and Woolley 1982). But what if it was the case that our sociality, like that of our primate ancestors, was a consequence of the evolved capacities of our brains, if our group formation was shaped by our neurobiology, if our empathy was not merely social but neural? And what if our neurobiology does not just establish the _conditions_ for sociality but also the _forms_ that it takes? What if our friendships and our loves, our communal ties and hates, our social and cultural lives take the shape that they do as only a consequence of our neuronal architecture? What then for the social and human sciences? Would such a neurobiologically grounded conception of human sociality represent a fundamental threat to the social and human sciences, as suggested by some of the critics discussed in earlier chapters? Or does it represent a new opportunity for collaboration across the 'two cultures,' one area where the 'critical friendship' we have proposed might be possible and valuable? And whether or not the claims made about the neural basis of human sociality are true, what might be the wider implications of such arguments being _considered truthful_ \u2014what might follow for how we are governed or govern ourselves\u2014how we manage our brains?\n\nThe \"Social Brain Hypothesis\"\n\nIn 1985, neuroscientist Michael Gazzaniga published a book called _The Social Brain_ (Gazzaniga 1985b). It was based on the work that he had done, initially with Roger Sperry, on so-called split-brain animals\u2014where the corpus callosum bridging left and right hemispheres had been lesioned\u2014and other evidence from clinical studies of individuals with brain lesions. This work seemed to show that damaging or destroying specific areas of the brain affected very specific aspects of cognition, language, and memory. On this basis he argued that the brain was not an indivisible whole but a confederation of relatively independent modules, each of which was relatively discrete and had specific functions. This modular way of thinking about the brain has been accepted by many.\n\nBut it is not in this sense of a kind of confederation of modules that the phrase _social brain_ has become so popular. Rather, the term has come to stand for the argument that the human brain, and indeed that of some other animals, is specialized for a collective form of life. One part of this argument is evolutionary: that the size and complexity of the brains of primates, including humans, are related to the size and complexity of their characteristic social groups. To put it simply, it is argued that the computational demands of living in large, complex groups selected for brains that had the capacities for interaction required to live in such social groups. _Social group_ is used here in the sense attributed to it by ethologists\u2014the characteristic number of animals such as primates who interact with one another in a relatively stable pattern and who differentiate themselves from other groups of the same species (see, for example, Bateson and Hinde 1976).\n\nAs far as humans are concerned, the evolutionary psychologist and anthropologist Robin Dunbar suggested that the number 150\u2014subsequently known as the Dunbar number\u2014specifies the optimal size of self-managing collectivities of human beings bonded together with ties of reciprocity, trust, and obligation, such as the clan or the village (Dunbar 1993). He and those who followed him suggested that this group size was intrinsically related to the size and structure of the brain as it had evolved to manage the complex tasks required to collaborate with others, to solve problems, maintain cohesion, and deal with others as enemies or collaborators; even today, it is claimed, it remains significant in shaping the optimal size of the functional groupings in the military and other human organizations, and for contemporary self-assembled networks (Dunbar 1996, 1998, 2003a, 2003b; C. Allen 2004; Dunbar and Shultz 2007; Shultz and Dunbar 2007).\n\nBut the social brain hypothesis is more than a general account of the role of brain size: for in this thesis, the capacities for sociality are neurally located in a specific set of brain regions shaped by evolution, notably the amygdala, orbital frontal cortex, and temporal cortex\u2014regions that have the function of facilitating an understanding of what one might call the 'mental life' of others. This faculty became termed _social cognition_. A key paper by Leslie Brothers in 1990 drew on evidence from work with macaques, and from humans with various neurological syndromes, to develop the hypothesis that humans, together with some primates, had a \"specialization for social cognition\": \" _the processing of any information which culminates in the accurate perception of the dispositions and intentions of other individuals_ \" (Brothers 1990, quoted from Brothers 2002, 367; emphasis in original). Brothers suggested that face recognition, with a specific neural basis, was crucial for this capacity and that social cognition formed a separate module subserved by a discrete neural system. Brothers, who herself was trained in psychiatry and later in psychoanalysis, developed this insight into a popular book, _Friday's Footprint_ , which argued that human brains are \"designed to be social\"\u2014that is to say, to attribute mental lives to human bodies (Brothers 1997).\n\nBut she could not have anticipated the ways in which, over the next twenty years, a body of work grew up that turned increasingly toward a neurobiological account of human behavior. Indeed, she later criticized this movement on Wittgensteinian grounds, and attributed it, in part at least, to the dominance of hidden interests, such as those of the pharmaceutical industry (Brothers 2001). But despite such concerns, this way of thinking has underpinned a growing body of work that focused on the identification of these neural bases and emphasized their determining role in social behavior: a body of research that called itself variously the social brain sciences, social cognition, or social cognitive neuroscience (Cacioppo 1994; Adolphs 2003; Heatherton, Macrae, and Kelley 2004). Ralph Adolphs has described this as \"an uneasy marriage of two different approaches to social behavior: sociobiology and evolutionary psychology on the one hand, and social psychology on the other.\" Nonetheless, he argues that \"neuroscience might offer a reconciliation between biological and psychological approaches to social behavior in the realization that its neural regulation reflects both innate, automatic and cognitively impenetrable mechanisms, as well as acquired, contextual and volitional aspects that include self-regulation\" (Adolphs 2003, 165; capitalization as in original), because there seems to be a relation between some biological factors that we share with other species and other psychological factors that are unique to humans.\n\nDoes neurobiology now claim to have resolved the question of how I know what is going on in your mind? This has long been the concern of philosophers, playwrights\u2014and, of course, of psychologists\u2014for my relations to you are shaped by my beliefs about your beliefs, what I think that you are thinking, what I take to be your motives\u2014the phenomenon of ascription. In the late 1970s, Premack and Woodruff coined the phrase that social neuroscience would adopt to describe such ascription\u2014the way humans and some other primates think about others and take account of what others are thinking in their own conduct\u2014such creatures had what they termed a \"theory of mind\" (Premack and Woodruff 1978). It was by means of a theory of mind that we are able \"to attribute mental states to other people,\" in particular to attribute beliefs, and false beliefs, to them (Adolphs 2003, 171).\n\nOf course there was no reason, in principle, to seek to localize such a capacity in specific brain regions, but in the developing style of thought within the experimental neurosciences that we discuss in chapter 2, this was exactly what followed: it appeared that theory of mind entails the activity of particular brain regions, including the amygdala, the medial and orbital prefrontal cortex, and the superior temporal gyrus. Now, of course, the reciprocal of such a claim was the argument that lesions or anomalies in the sites that underpin theory of mind might produce different kinds of deficits in social abilities\u2014such as the inability to detect social faux pas. Many are cautious about such localization, not least because while a specific area might be activated when a particular task is carried out, often a lesion in that area does not result in an impairment in carrying out that task, or else it produces only subtle and very specific changes. Indeed, because of the plastic nature of the brain, discussed in chapter 1, such deficits may take unpredictable turns. Thus Adolphs concludes: \"Although the data that I have reviewed... converge on several key brain structures that mediate social cognition, it cannot be over-emphasized that the causal role of these structures remains unclear.... This probably reflects the considerable redundancy and plasticity of the brain.... Caution should be taken in attempts to predict people's behavior from knowledge about their brains\" (Adolphs 2003, 174).\n\nNonetheless, despite the difficulties entailed by localization, theory of mind has come to be a central concern for social brain theorists and a key element in their explanatory repertoire. One dimension has come to be termed _mentalization_ \u2014a kind of built-in intersubjectivity in human experience, perhaps shared with some other primates. Mentalization refers to the \"largely automatic process by which we 'read' the mental states of others\" and which helps us make predictions about their future actions (Frith 2007b, 671). Much emphasis has been placed on the apparent phenomenon termed _mirroring_ that was reported in macaque monkeys in the early 1990s and in humans soon after: the existence of a so-called mirror system in the brain that was mapped in 1999 (Di Pellegrino, Fadiga, Fogassi, et al. 1992; Gallese, Fadiga, Fogassi, et al. 1996; Rizzolatti, Fadiga, Gallese, et al. 1996; Iacoboni, Woods, Brass, et al. 1999; Hari 2004; Rizzolatti and Craighero 2004).\n\nWhen an individual observes a movement being carried out by another\u2014usually a 'conspecific' but sometimes an individual of another species\u2014a small number of neurons are activated in those areas of the brain that are also activated when the individual carries out that same movement him-or herself. If I watch you reaching for a banana, there is some activation in the same areas that would be activated more intensively if I was reaching for that banana myself\u2014the original observations were made by chance in monkeys and initially termed \"monkey see, monkey do\"\u2014a phrase that did not travel so well (Iacoboni 2003). The mirror system thus seems to embody a neural basis for imitation learning, \"the capacity to learn an action from seeing it done\" (the definition from Thorndyke in 1889 is quoted in Rizzolatti and Craighero 2004, 172).\n\nNow this capacity is apparently present only in humans and some apes, leading some to believe that it has been crucial for human development, not only for the learning of complex motor skills, but for interpersonal communication and learning social skills. V. S. Ramachandran extrapolated further than most, arguing that this mirror system, and the capacity for imitation that it enables, facilitated the propagation of culture and was \"the driving force behind 'the great leap forward' in human evolution\" and \"set the stage for the emergence, in human hominids, of a number of uniquely human abilities, such as proto-language... empathy, 'theory of other minds,' and the ability to 'adopt another's point of view.' \"\n\nBut mirror theorists developed their argument beyond their original claims about simple imitation or simulation (Gallese and Goldman 1998). It was not merely some kind of mechanical imitation of action that was at stake, but the capacity to grasp the intention behind an action\u2014for example, to understand someone's intention to drink when reaching for a glass, but not when reaching in an identical manner for some other object. This means that I don't just understand your intentions through the mirroring mechanism: I also feel your feelings. For mirroring is not restricted to the observation of another _carrying out a movement_ , but is extended to the observation of another _experiencing an emotion_ such as joy or pain, at least when this other is a conspecific.\n\nThus Chris Frith explains that the notion of a \"mirror system in the brain arises from the observation that the same brain areas are activated when we observe another person experiencing an emotion as when we experience the same emotion ourselves\" (Frith 2007b, 673). It appeared that this challenged the conventional, rationalist accounts of ascription\u2014that we understand the intentions of others by creating theories about what lies behind their appearance or their acts, on the basis of our own commonsense psychology about their desires and their beliefs about how to achieve them. It was not through _theorizing_ about others, but through _feeling what they feel_ , that we understand the mental states of others. I really do 'feel your pain.'\n\nBy around 2005, driven largely by the work of the research groups that had formed around the original mirror neuron researchers, mirror neurons had moved on from imitation and language to account for everything from a feeling of disgust when observing others showing disgust (Wicker, Keysers, Plailly, et al. 2003; Rizzolatti 2005) to providing a unified general theory of the neural basis for interpersonal relations (Metzinger and Gallese 2003; Gallese, Keysers, and Rizzolatti 2004; Iacoboni, Molnar-Szakacs, Gallese, et al. 2005). Ramachandran, with characteristic hubris, predicted that \"mirror neurons will do for psychology what DNA did for biology: they will provide a unifying framework and help explain a host of mental abilities that have hitherto remained mysterious and inaccessible to experiments.\"\n\nIt was not long, however, before these claims for mirror neurons were being contested. Critics argued that the claims made were philosophically, conceptually, and empirically problematic and that there was little or no evidence that these mirror neurons actually could achieve the effects claimed for them\u2014of understanding the intentions from observing actions or appearances of others, especially when it came to complex interpretations of social behavior or of the communicative acts of others\u2014and indeed that there was actually rather little evidence for the existence of mirror neurons in humans (Jacob and Jeannerod 2005; Borg 2007; Dinstein 2008; Dinstein, Thomas, Behrmann, et al. 2008; Lingnau, Gesierich, and Caramazza 2009). Skeptical voices suggested that the enthusiasm for these claims derived, not from their scientific bases or importance, but because the neuroscientists concerned were carried away by their desires to make their research seem socially important and relevant outside their own narrow disciplinary domain.\n\nThe controversy continues. But perhaps because of such doubts, many of those working in social neuroscience chose not to base their arguments on mirror neurons, but on the more limited claim that the social brain consists of brain areas specifically attuned to the recognition of different aspects of conspecifics\u2014individuals of the same species\u2014such as faces, motion, and attribution of mental states, some of which seem present at birth, and others of which develop over time. As Sarah-Jayne Blakemore puts it: \"The social brain is defined as the complex network of areas that enable us to recognize others and evaluate their mental states (intentions, desires and beliefs), feelings, enduring dispositions and actions\" (Blakemore 2008, 267). Even without reference to mirroring, this is the basis of the argument that there is a neural basis for many human social phenomena, such as empathy (Decety and Meyer 2008; Singer and Lamm 2009). Such arguments rapidly became central to a way of thinking about the human brain as specialized for sociality through the built-in capacity to understand the beliefs and intentions of others (Kilner and Frith 2008; Rizzolatti and Fabbri-Destro 2008).\n\nFor social brain theorists, then, the human brain, as it has evolved from our primate forebears, actually embodies the capacity for intersubjectivity: when I observe your actions, or the visible signs of your inner states, areas of my own brain activate that enable me to understand the intentions or feelings that lie behind those observable features, and hence to feel a tiny trace of what it would take, what it would mean, for me to act or feel that way myself. Of course, on the one hand this proposal, whether it is neurobiologically accurate or not, is sociologically anodyne. Few would doubt that there are neurobiological conditions for our capacities to impute feelings, thoughts, and intentions to others, or for our capacities to engage in social interaction. For the human sciences, the possibility of such imputations and attributions has long been taken as a premise; what has been crucial has been the analysis of the varying forms of such attribution, and the ways in which they have been shaped by culture, history, language, and by highly variable regimes of meaning and value.\n\nHence, from this perspective, it is not clear what would be gained by attempts to localize such feelings in particular brain regions such as the anterior insula or anterior cingulate cortex, or in suggesting that particular brain regions provided the causal basis for such capacities. The interesting question from the human sciences would concern the ways in which, given that apparently universal basis, we might account for that historical and cultural variability in the ways in which individuals do, in fact, attribute beliefs and intentions to other entities, and the variability in the forms and limits of the empathy or fellow feeling that they exhibit to others. But the human sciences would not be alone in such concerns. When Tania Singer and Claus Lamm reviewed the neuroscience of empathy in 2009, they agreed that this research is beginning \"to reveal the mechanisms that enable a person to feel what another is feeling\" (Singer and Lamm 2009, 81). But they were exceptionally cautious about claims for universality, not only arguing for very careful distinctions to be made between different forms of \"affect sharing\"\u2014empathy, sympathy, compassion, and so forth\u2014but also showing that \"a clearcut empirical demonstration of the link between empathy and prosocial behavior is still missing\" (ibid., 84). Further, they concluded, we know little of variations across a life span, almost nothing about individual differences, have not sufficiently recognized the fact that affect sharing is highly contextually specific and far from universal, and know almost nothing about the malleability of these features and the ways in which such sharing of affects may be influenced by training and experience\u2014the latter, they claim, having potentially \"enormous implications for education and society as a whole\" (ibid., 93).\n\nAs we have seen in previous chapters, such expressions of caution about what we know today coupled with optimism about future possibilities are characteristic of leading researchers in the new brain sciences. But the lure of translation is hard to resist. Whatever their consequences in other areas, a number of researchers saw, in these arguments, a way in which some important problems of mental health in humans could be reframed. Were not many human mental disorders characterized by problems in precisely those capacities explored by social brain theorists? Crucially, if these disorders were recoded as pathologies of the social brain, they appeared to offer the prospect of various forms of neurobiologically based interventions.\n\nPathologies of the Social Brain\n\nCould conditions such as schizophrenia and autism be disorders of the social brain, arising from disruptions in the evolved neural circuits that implement social behavior? As early as the mid-1990s, Chris Frith and his colleagues were suggesting that those diagnosed with schizophrenia had deficits in their theory of mind and problems in \"mentalizing\" (Corcoran, Mercer, and Frith 1995; Frith and Corcoran 1996). Over the next ten years, this idea was developed into a full-scale social brain theory of schizophrenia (Russell, Rubia, Bullmore, et al. 2000; Sharma, Kumari, and Russell 2002), largely based on neuroimaging studies that seemed to show that patients with such a diagnosis had abnormal activation patterns in regions thought to be involved in social cognition. In a characteristic reverse move, it could then be suggested that the social brain networks in the medial prefrontal cortex, the prefrontal cortex, the amygdala, and the inferior parietal lobe were the neural systems underlying disorders in theory of mind, emotion perception, and self-agency in such patients (Brunet-Gouet and Decety 2006).\n\nFor some, mirror neurons, once more, were involved. Mirror neuron theorists suggested that pathologies such as autism or schizophrenia arise because of deficits in that capacity for mind reading found in all presumably normal human beings (Iacoboni 2008). For example, mirror neuron deficits might lie behind the false attribution of auditory hallucinations to external sources found in some patients diagnosed with schizophrenia (Arbib and Mundhenk 2005). Mirror neuron deficits might provide the neural substrate for deficits in mentalization in autism\u2014a view proposed in a popular piece in _Scientific American_ by Ramachandran and Oberman under the title of \"Broken Mirrors\" (Ramachandran and Oberman 2006).\n\nOthers, however, managed without mirror neurons. Well before the advent of the mirroring arguments, Uta Frith and Simon Baron-Cohen had argued that autism and autism spectrum disorders were linked to deficits in the capacity of an individual to mentalize, that is to say, to construct a theory of the minds of others in order to attribute beliefs and intentions to their actions (Baron-Cohen, Leslie, and Frith 1985); it was only later that they looked to neurobiology for reinforcement, although in combination with psychological arguments (Baron-Cohen, Tager-Flusberg, and Cohen 2000; Frith and Hill 2003; Hill and Frith 2003; Baron-Cohen and Belmonte 2005). Baron-Cohen and Belmonte concluded their 2005 review by hoping that it will eventually be possible to identify the way in which \"large numbers of genetic biases and environmental factors converge to affect neural structure and dynamics, and the ways in which such abnormalities at the neural level diverge through activity-dependent development to produce an end state in which empathizing abilities are impaired and systemizing abilities are augmented. Such a unified understanding of the psychobiology of autism will offer targets for intervention on many levels\" (Baron-Cohen and Belmonte 2005, 119): not a single mention of mirror neurons here.\n\nThere were other routes to explanations of mental disorders in terms of the social brain. Thomas Insel, who was appointed director of the NIMH in 2002, had previously adopted a different research strategy to explore the neurobiology of sociality. In his famous studies of prairie voles, he had argued that the neuropeptides oxytocin and vasopressin played crucial roles in the establishment of enduring social bonds among these creatures, and suggested that differences between the reproductive, parenting, and affiliative behaviors of the microtine voles and the asocial behavior of the closely related montane voles were related to differences in the patterns of oxytocin and vasopressin receptors in their brains (Insel and Shapiro 1992; Winslow, Hastings, Carter, et al. 1993; Insel and Young 2000).\n\nIn 2004, he linked these arguments into more general neurobiological accounts of the way in which the brain \"processes social information\"\u2014as he and Russell Fernald phrased it when they were \"searching for the social brain\" (Insel and Fernald 2004). They singled out four dimensions of such processing of social information\u2014epigenesis; sensory specializations for social perception, such as smell; imprinting or early social learning based on the exceptional plasticity of the infant brain; and the presence of specialized neural circuits for social recognition, also often involving the effect of olfaction on specific brain regions. Insel argued that these evolved mechanisms for processing social information or social signals might provide a way of understanding human psychopathology that offered possibilities for translation into the clinic. \"Results from these studies and some recent functional imaging studies in human subjects begin to define the circuitry of a 'social brain,' \" they concluded (Insel and Fernald 2004, 697). Not only does this conception of a brain evolved for sociality help us understand why isolation and social separation are linked to many types of disease, but it also suggests that conditions such as autism and schizophrenia could be viewed as neurodevelopmental disorders characterized by abnormal social cognition and corresponding deficits in social behavior. Thus \"social neuroscience offers an important opportunity for translational research with an impact on public health\" (ibid.).\n\nBy the time that Insel was appointed to the NIMH, others were also suggesting that the idea of the social brain could become an integrating zone for a new kind of social psychiatry. Thus Bakker and colleagues argued that the social brain provided the physiological focus that could be a unifying foundation for psychiatry \"The social brain is defined by its function\u2014namely, the brain is a body organ that mediates social interactions while also serving as the repository of those interactions. The concept focuses on the interface between brain physiology and the individual's environment. The brain is the organ most influenced on the cellular level by social factors across development; in turn, the expression of brain function determines and structures an individual's personal and social experience\" (Bakker, Gardner, Koliatsos, et al. 2002, 219). While not expecting social psychiatrists to resolve the mind-brain problem, we may note, once again, the phrasing\u2014\"the expression of brain function determines and structures... experience\"\u2014that is called into service to bridge the explanatory gap between the neurobiological and the experiential.\n\nThe link between psychopathology and this complex notion of the social brain, in which arguments from evolution and comparative studies of animals, explanations based on mirror neurons, and findings from genomics and neurochemistry are brought together, is perhaps best exemplified in the work of Klaus-Peter Lesch. Lesch's classic paper of 1996 identified a polymorphism in the promoter region of the 5-HTT (serotonin) transporter gene\u2014basically a region of the gene sequence that came in two version, short or long\u2014where possession of one or two copies of the short variants led to a lower expression of serotonin in the brain and was associated with greater levels of anxiety and personality traits often linked to depression and suicidal behavior (Lesch, Bengel, Heils, et al. 1996). In subsequent work, Lesch argued that the 5-HTT transporter was a kind of controller that was responsible for fine-tuning the serotonin system and hence for the regulation of emotion and disorders of emotion regulation, within which he included such conditions as depression, substance abuse, eating disorder, and autism. Further studies in rhesus monkeys, where there was a different but analogous polymorphism, seemed to show that stress in early life reinforced the effect of the low-acting form, suggesting that 5-HTT was a susceptibility gene for depression.\n\nFor Lesch, these studies reveal a set of complex interactions between single nucleotide polymorphism (SNP)-level genetic variations, \"embryonic brain development[,] and synaptic plasticity, particularly in brain areas related to cognitive and emotional processes,\" which \"transcends the boundaries of behavioral genetics to embrace biosocial science\" (Lesch 2007, 524). Life stress interacts with variations in the 5-HTT genotype to influence patterns of brain activation, provoking different cognitive and emotional states. And Lesch turns to Iacoboni's mirror neuron theory (Iacoboni 2005) to argue that the regions affected by allelic variations in 5-HTT genotype are none other than those \"that integrate imitation-related behavior, from which social cognition and behavior have evolved\" (Lesch 2007) and that epigenetic programming of emotionality by maternal behavior demonstrates the ways in which environmental influences can remodel neuronal organization in the course of development. \"We are clearly a long way from fully understanding the evolutionary and neural mechanisms of social cognitive phenomena and social functioning,\" he concludes. \"But the potential impact of 5-HTT variation on social cognition transcends the boundaries of behavioral genetics to embrace biosocial science and create a new social neuroscience of behavior\" (Lesch 2007, 528). Here, in perhaps its most complex formulation to date, seeking to incorporate evidence from molecular genomics, evolutionary theory, brain imaging, and cognitive neuroscience, we have the new image of the social brain.\n\nSocial Neuroscience\n\nThis emergent, though far from homogeneous, discourse on the social brain seems to escape some of the ritual criticisms leveled by social scientists at biological explanations of human conduct\u2014reductionism, essentialism, individualism, and the like. There is certainly a genealogical link back from today's social brain to the ethological studies of human behavior that were popular in the 1960s and early 1970s, but there is little of sociobiology here, either in its scholarly or popular forms. The teleological tropes of evolutionary psychology are, if not absent, then mitigated. There is no appeal to a fixed human nature to account for either virtuous or vicious conduct, and humans are not portrayed as naked apes. If evolution has provided humans with their behavioral attributes through selection for certain neural capacities, these are not those of unmitigated self-interest, but rather of empathy, sociality, and reciprocity, enabled by imitation learning, the ability to form a theory of other minds, and the plasticity of the human brain. All these are in turn shaped by gene expression that is modulated by biographical, social, and environmental inputs in utero, infancy, and across the life-course. The human brain is now portrayed as evolved for sociality and culture\u2014open, mutable, and in constant transactions with its milieu, whether these be at the level of the epigenetic modification of gene expression; the interactions of gene and environment in development; the interaction of evolved visual, olfactory, or other sensory capacities and relations between individuals in a milieu; or the role of mirror neurons in the attribution of beliefs, intentions, and mental states to others (Cacioppo and Berntson 2004). A discipline of _social neuroscience_ has become possible.\n\nMost of those who write the history of social neuroscience agree that the beginnings of the discipline emerge in the 1990s, if by that one means the invention of a name and the framing of a project (Cacioppo, Berntson, and Decety 2011). Certainly some of the psychological ethologists working in the 1960s and 1970s discussed the social functions of human intellect and the evolutionary pressures for intellectual development (e.g., Humphrey 1976). However, the first to use the term _social neuroscience_ in print seem to have been two psychologists, John Cacioppo and Gary Berntson, who published an article in _American Psychologist_ in 1992 titled \"Social Psychological Contributions to the Decade of the Brain\":\n\n[T]he brain does not exist in isolation but rather is a fundamental but interacting component of a developing or aging individual who is a mere actor in the larger theater of life. This theater is undeniably social, beginning with prenatal care, mother-infant attachment, and early childhood experiences and ending with loneliness or social support and with familial or societal decisions about care for the elderly. In addition, mental disorders such as depression, schizophrenia, and phobia are both determined by and are determinants of social processes; and disorders such as substance abuse, juvenile delinquency, child and spousal abuse, prejudice and stigmatization, family discord, worker-dissatisfaction and productivity, and the spread of acquired immunodeficiency syndrome (AIDS) are quintessentially social as well as neurophysiological phenomena. (Cacioppo and Berntson 1992, 1020)\n\nThe term _social_ was used here to index the fact that (a) neurochemical events influence social processes and (b) social processes influence neurochemical events. However, despite references in the paper to economic factors and to social systems, the social processes that the authors sought to integrate into their \"multilevel analysis\" were not placed under the sign of sociology but rather of social psychology. Cacioppo and Berntson used the term _social neuroscience_ only in the concluding section of their paper, and it was followed by a question mark. The question mark expressed a concern about potential for reductionism and the need to recognize different levels of explanation: \"[T]he brain is a single, pivotal component of an undeniably social species,\" but we must recognize that \"the nature of the brain, behavior, and society is... or-derly in its complexity rather than lawful in its simplicity\" (Cacioppo and Berntson 1992, 1027). Two years later, Cacioppo used the term more affirmatively in the title of an article in 1994 discussing the effect of social and psychological factors on the immune system (Cacioppo 1994), and by the time that he and others published their entry under that title in the _International Encyclopedia of the Social and Behavioral Sciences_ in 2001 (Cacioppo, Bernston, Neil, et al. 2001), the question mark had been firmly left behind.\n\nA discipline needs an infrastructure (Star and Ruhleder 1996; Star 1999)\u2014a fact of which the practitioners are usually very well aware. Cacioppo identifies the first infrastructural element in a grant to him from the Behavior and Neuroscience Division of the National Science Foundation in the 1980s to establish a five-year interdisciplinary research program, although it appears from the resulting edited volume that this research was actually focused on the neurophysiological, psychoneuroendocrinological, and psychoneuroimmunological foundations of psychophysiology (Cacioppo and Tassinary 1990). Following that, he identifies other infrastructural elements, ranging from the establishment of the Summer Institute in Cognitive Neuroscience, to the use of EEG technology and the study of \"event-related potentials\" (ERPs) and other measures of brain activation within experimental social psychology, and eventually to fMRI scanning, which made it possible to study neural activity by noninvasive means as humans engaged in various tasks in laboratory settings. Over the 1990s, the number of papers published that used the term _social neuroscience_ gradually increased, from one or two a year at the start of the decade to around twenty per year at the end, but this number rapidly increased after the turn of the century\u2014there were 36 in 2001, 128 in 2006, and 279 in 2009. By the start of the twenty-first century, MIT Press was publishing, largely under Cacioppo's influence, a book series on social neuroscience, about which it quoted a review in _Science_ : \"In a very short time, social neuroscience has come from almost nothing to become one of the most flourishing research topics in neurobiology.\"\n\nThe Center for Cognitive and Social Neuroscience at Chicago, founded in 2004 and headed by Cacioppo, provided a crucial institutional basis for social neuroscience. As it describes itself, the center focuses on the study of the biological systems that \"implement\"\u2014an intriguing term\u2014social processes and behavior. In many different publications since, Cacioppo has asserted his own definition of social neuroscience as \"an interdisciplinary approach devoted to understanding how biological systems implement social processes and behavior and to using biological concepts and methods to inform and refine theories of social processes and behavior...,\" and his work explores the effects of social environment\u2014notably social isolation\u2014on everything from gene expression to hormonal regulation. This is the direction that his own work has taken, both in his general accounts of the discipline (Norman, Cacioppo, and Berntson 2010) and in his specific studies of the physical and mental consequences of a preeminently culturally shaped human experience\u2014loneliness (Hawkley and Cacioppo 2010). Further, Cacioppo and his coworkers made a move that opens a space for the entry of more conventional arguments from the human sciences, stressing that the effects of loneliness were not merely those of actual isolation but also those where people perceive that they live in isolation (Cacioppo and Patrick 2008). For along with the entry of human perception, rather than raw external sensory input, comes culture, language, meaning, history, and all the other dimensions known to the cultural sciences. An interdisciplinary effort, it seemed, might be on the cards. Or was it?\n\nWhen the _Journal of Personality and Social Psychology_ published a special section on \"Social Neuroscience\" in its October 2003 issue, the editors defined the field in terms that owed a lot to Cacioppo: \"Social neuroscience is an integrative field that examines how nervous, endocrine, and immune systems are involved in sociocultural processes. Being nondualist in its view of humans, it recognizes the importance of understanding the neural, hormonal, and immunological processes giving rise to and resulting from social psychological processes and behaviors\" (Harmon-Jones and Devine 2003, 590). However, the bulk of the papers were rather different: they used various imaging and localization technologies to identify the patterns of brain activity associated with tasks performed in the traditional experimental setups of social psychology (e.g., Cunningham, Johnson, Gatenby, et al. 2003) and considered the potential of brain imaging for the discipline (Willingham and Dunn 2003). Indeed, in the same collection, Cacioppo himself expressed some reservations about that direction of travel and the way it framed its research questions (Cacioppo, Berntson, Lorig, et al. 2003). But when, in 2004, a range of journals in psychology, such as _Current Directions in Psychological Science_ , and neurobiology, such as _Current Opinion in Neurobiology_ , published expository and review papers on social neuroscience, they mainly followed the brain localization route (Heatherton 2004; Jackson and Decety 2004; Ochsner 2004).\n\nIt is no surprise, given its focus, that when _NeuroImage_ published its special section on \"Social Cognitive Neuroscience\" in the December 2005 issue under the editorship of Mathew Lieberman, whose own work focused on the use of functional neuroimaging to investigate social cognition, it was indeed the power of imaging that was central, notably its role in revealing the nonconscious impact of situational factors on motives, attitudes, behaviors, perceptions, beliefs, and reflective thought (Lieberman 2005). And the bulk of what came to call itself social and cognitive neuroscience follows this direction, adding various versions of brain imaging to the methodological armory of experimental social psychology. Indeed, the journal _Social Neuroscience_ , which commenced publication in 2006, defines its focus as \"how the brain mediates social cognition, interpersonal exchanges, affective\/cognitive group interactions, and related topics that deal with social\/personality psychology\"; the editors define social neuroscience as \"the exploration of the neurological underpinnings of the processes traditionally examined by, but not limited to, social psychology\" (Decety and Keenan 2006).\n\nIn the same year, the journal _Social Cognitive and Affective Neuroscience_ published its first issue. It asserted that it was devoted to \"publishing research on the social and emotional aspects of the human mind by applying the techniques from the neurosciences and providing a home for the best human and animal research that uses neuroscience techniques to understand the social and emotional aspects of the human mind and human behavior.\" However, the opening editorial, once more written by Mathew Lieberman, was quite clear about the type of work it wished to publish: \"[S]ubmitted research should report new empirical investigations using neuroscience techniques (including, but not limited to, fMRI, MRI, PET, EEG, TMS, single-cell recording, genetic imaging, psychopharmacological perturbations, and neuropsychological lesion techniques) to examine social and affective processes\" (Lieberman 2006, 2). Location and visualization, it seemed, were the paths to understanding the social brain.\n\nTo know is to see; and, as we have seen, many believed that to see the brain meant to localize. By 2010, localization by means of brain imaging had become the method of choice for much social neuroscience, especially within psychology departments. Although scanning was usually an intensely individual experience for any experimental subject, researchers instructed their subjects to carry out tasks in the scanner that they believed simulated social interactions, and then tried to find the regions of the brain that were activated. This style of investigation was not merely an American or European preoccupation. By 2010, the Culture and Cognitive Neuroscience Laboratory based at Peking University was receiving research funding for \"Neuroimaging studies of human aggressive and affiliative behaviors\" (funded by the Ministry of Science and Technology of the People's Republic of China), \"Modulation of empathy by culture and social relationship\u2014A transcultural cognitive neuroscience study\" (funded by the National Natural Science Foundation of China), and a study of the \"Cognitive and neural mechanisms underlying emotional influences on risk behaviors\" (funded by the National Natural Science Foundation of China).\n\nNonetheless, tensions remained within the field itself. When the Society for Social Neuroscience was established in 2010, under the leadership Cacioppo and Decety, and headquartered with Cacioppo in Chicago, it framed its mission in terms that owed much to Cacioppo: the impact that social structures\u2014from dyads, through families, neighborhoods, and groups to cities and civilizations\u2014have on the brain and nervous system of the individual \"through a continuous interplay of neural, neuroendocrine, metabolic and immune factors on brain and body, in which the brain is the central regulatory organ and also a malleable target of these factors.... Social neuroscience is the interdisciplinary academic field devoted to understanding how biological systems implement social processes and behavior, and how these social structures and processes impact the brain and biology. A fundamental assumption underlying social neuroscience is that all social behavior is implemented biologically.\"\n\nEach of the two sessions in its first conference in November 2010 focused on one dimension. The first session examined such matters as the social regulation of brain gene expression and aggression in honey bees; measuring social behavior in the rat and mouse; neuropeptides, bonding, and social cognition in voles to man; and perspectives on stress, brain, and body health. The other featured evidence from neuroimaging in areas ranging from fronto-temporal dementia to reasoning processes in game theory. Cacioppo and Decety commented that despite the agenda for integration set out in the early project for social neuroscience, \"contemporary social neuroscience investigations to elucidate the genetic, hormonal, cellular, and neural mechanisms of social behavior have grown from two largely separate root systems: one based on data from humans grounded in the discipline of psychology and the other based on data from animal models grounded in biology and biomedicine.... With a few notable exceptions in each group, there has been little communication between these two groups of scientists. In our view, this is a gap that must be filled for social neuroscience to reach its full potential\" (Cacioppo and Decety 2011). In many ways, the relation of social neuroscience to the human sciences more generally depends on which of these two paths is followed, for they point to two very different conceptions of sociality.\n\nSocial Neuroscience beyond Neuroscience\n\nBy 2005, social neuroscience was attracting the interest of those funding research on types of human social behavior considered problematic. The NIH, the U.S. National Institute on Drug Abuse (NIDA), the U.S. National Institute of Alcohol Abuse and Alcoholism (NIAAA), the U.S. National Institute on Aging (NIA), and the U.S. Institute of Neurosciences, Mental Health and Addiction (INMHA), and the Canadian Institutes of Health Research (CIHR) were calling for proposals under the heading of social neuroscience to \"act as a catalyst for the emerging area of social neuroscience in order to elucidate fundamental neurobiological mechanisms of social behavior.... Clinical and preclinical research supported by this initiative must take a multidisciplinary, multilevel approach to a social behavioral research question (or set of questions) that is framed at the behavioral level (e.g., social cognition, social development, social interaction, and social aspects of emotion).\"\n\nIn the same year in the United Kingdom, the Economic and Social Research Council carried out a 'scoping' workshop on social neuroscience and issued an enthusiastic media briefing titled \"Social Neuroscience Will Shed New Light on Human Behavior\" that asserted this \"new domain of study is set to spearhead an assault on what has been described as science's final frontier: understanding, even decoding the complex interconnected web that mediates between the brain, and the mind and human behavior. Already it is making us think again about humanity's understanding of itself and what this means for the norms and dynamics of how we behave in society.... The Economic and Social Research Council fully intends that the UK should not be left behind in this exciting and potentially extremely valuable field.\" Together with the Medical Research Council, in 2007 it was calling for cross-disciplinary grants in this area, arguing that \"[e]normous advances are being made in our understanding of the links between the mind, brain, innate traits, society, culture and behavior, whether normal or abnormal.\"\n\nSome of those within the social and human sciences were open to the opportunities presented by these new ways of thinking. At the 2008 meeting of the American Anthropological Association, a group convened a panel on \"the encultured brain: neuroanthropology and interdisciplinary engagement.\" They argued in their proposal that neuroanthropology\n\ncan help to revitalize psychological anthropology, promote links between biological and cultural anthropology, and strengthen work in medical and linguistic anthropology.... Neuroscience has increasingly produced basic research and theoretical models that are surprisingly amenable to anthropology. Rather than \"neuro-reductionist\" or determinist approaches, research has increasingly emphasized the role of environment, body, experience, evolution, and behavior in shaping, even driving organic brain development and function. At the same time, the complexity of the brain makes a mockery of attempts to pry apart \"nature\" from \"nurture,\" or to apportion credit for specific traits. Research on gene expression, endocrine variability, mirror neurons, and neural plasticity all beg for comparative data from across the range of human variation\u2014biological and cultural.... Anthropology has much to offer to and much to learn from engagement with neuroscience.\n\nAt almost the same moment, in November 2008, the first Social Brain Conference convened in Barcelona, focused on \"a neurocognitive approach to fairness.\" It included sessions titled \"From Animosity to Empathy: Neuroimaging Studies on the Building Block of Fairness\" and \"Genetic Markers for Good and Bad Cooperators: A Biological Approach to Fairness in Economic Exchanges\" alongside those on \"Sacred Values: Anthropological Perspectives on Fairness in Social Conflicts.\" Political figures were among the speakers, for instance, Lord John Alderdice, who played an important role in brokering the peace deal in Northern Ireland. The representatives from social neuroscience included psychologists using brain imaging, researchers on economic game theory and social groups, as well as Tania Singer, whose work on moral feelings we have already encountered, and who published the first imaging studies of pain empathy and the relation of neural signatures of empathy with cooperation and competition (Singer, Seymour, O'Doherty, et al. 2004). The organizers were modest about whether this new approach to issues from nationalism, terrorism, and suicide bombers to cooperation, fairness, and empathy would form the basis for \"ruling human communities according to high standards of values and morality,\" as some had hoped (Atran 2009, 2), but undoubtedly the social brain hypothesis was enabling some anthropologists to reframe their research such that it seemed to address pressing social issues and perhaps even inform public policy.\n\nAnthropology has always had a biological branch, so perhaps its openness to the social brain hypothesis is not so surprising. Similarly, we should not be surprised that social neuroscience also seemed attractive to many social psychologists. One of the most articulate spokespersons was Chris Frith, for whom social neuroscience seemed to promise this fragile discipline some objectivity at last (Frith 2007a). Frith placed the social brain firmly within the forms of sociality studied within experimental social psychology\u2014sociality as a kind of interactional nexus: \"what the social brain is for\" is that it \"allows us to interact with other people,\" and specifically, \"the function of the social brain is to enable us to make predictions [which need not be conscious and deliberated] during social interactions\" (Frith 2007b, 671). The themes that have most engaged such researchers\u2014empathy, trust, love, social bonding, maternal-infant relations, deception, ascriptions of beliefs and feelings to others, decision making\u2014all fit easily within a premise that social life is, in essence, a life of interactions. The preconditions for such interactions may lie in some form of socialization or development, but the capacity for enactment of those interactions in particular ways\u2014or the failure to enact them\u2014is, at any one instant, located within the capacities and attributes of the interacting entities. The social brain is construed as the fleshy and embodied material substrate that enables (subserves, implements, underpins) the transactions between self and other\u2014in pairs, families, friendships, and social networks. This interactive conception of sociality is embodied in the experimental setups of those whose preferred technology is that of brain imaging.\n\nImaging technologies, with their focus on the localization of function and the experimental manipulations they require and utilize, impose their own constraints and logics as to what can be thought. As with a long tradition of experimental psychology, they make it seem unproblematic to draw upon evidence that occurs in the laboratory setting where such interactions can be simulated, because the other features of the laboratory\u2014its specific properties as a location full of meaning, expectations, dense with a different kind of sociality for all the participants\u2014may somehow be deleted from the analysis (Cohn 2008). Such premises lead, usually almost imperceptibly, to a theory of sociality that renders it in terms of the capacities required to maintain interactions between individuals in small groups, because that is all the sociality that can enter the experimenters' purview. That explicit or implicit theory is then operationalized in experiments in laboratory situations that appear to simulate such interactions, and, through the use of recording machines of one sort or another, the outcomes are turned into data that will then stand in for the experiment itself. The data produced by this technology are then used to support the theory that has enabled its production. In the process, the tool or technology has, as it were, merged with the theory itself. We can see here a reciprocal relation between concepts and technologies\u2014a slightly more recursive example of what Gerd Gigerenzer once termed \"tools to theories,\" in which physical tools, such as computers and scientific instruments, and intellectual tools, such as statistics or intelligence tests, start off as the means to produce data and end up providing the theories of the data that they themselves have produced. Where human mental life is concerned, scientific tools are not neutral: \"the mind has been recreated in their image\" (Gigerenzer 1991, 19).\n\nCacioppo and those who think like him seek a conception of sociality that goes far beyond interpersonal interactions: \"As a social species, humans create emergent organizations beyond the individual\u2014structures that range from dyads, families, and groups to cities, civilizations, and international alliances. These emergent structures evolved hand in hand with neural, hormonal, cellular, and genetic mechanisms to support them because the consequent social behaviors helped humans survive, reproduce, and care for offspring sufficiently long that they too survived to reproduce.\" The work he and his colleagues have done on social isolation, for example, is placed in the context of a range of other sociological and epidemiological evidence and acknowledges the fact that humans \"are an irrepressibly meaning-making species, as well, and a large literature has developed showing that perceived social isolation (i.e., loneliness) in normal samples is a more important predictor of a variety of adverse health outcomes than is objective social isolation.... [The] social environment, therefore, is fundamentally involved in the sculpting and activation\/inhibition of basic structures and processes in the human brain and biology.\"\n\nBut while many social psychologists have been eager to ally themselves with these new styles of thought and research, the same cannot be said for sociologists. Warren TenHouton ascribes the first use of the term _neurosociology_ to a 1972 paper on the impact of different cultures on the development of the left and right hemispheres of the brain (Bogen, DeZure, TenHouten, et al. 1972). Five years later, in a paper in the _Journal of Social and Evolutionary Systems_ , TenHouton argued: \"The case for neurosociology begins with the observation that no social relationship or social interaction can possibly be carried out in the absence of human mental activity. It follows immediately that, as scientists, sociologists must find ways to describe and measure such mental functions and cognitive structure.... Just as neuropsychology has far reaching implications for general psychology, so also, then, does neurosociology for general sociology\" (TenHouten 1997, 8). The case was carried forward by David Franks, who described neurosociology as \"the nexus between neuroscience and social psychology\" (Franks 2010; cf. Franks and Smith 2001), taking up themes of emergence and neuroplasticity to argue that neuroscience can be brought into alignment with the sociology of embodiment as well as with insights from interactionist sociology and the pragmatism of George Herbert Mead and the Chicago pragmatists. But he is a rather lonely voice: few sociologists seem to believe that the explanatory models of social neuroscience have implications for their often unarticulated conception of human beings as sense making creatures, shaped by webs of signification that are culturally and historically variable and embedded in social institutions that owe nothing substantial to biology.\n\nGoverning Social Brains\n\nGordon Allport opened his historical review of social psychology in 1954 by quoting from Giambattista Vico's phrase of 1725: \"government must conform to the nature of the men governed\" (Vico, quoted in Allport 1954, 1). Despite the commonly held view that governing is an essentially pragmatic affair, attempts to conduct human conduct are always based on certain premises about people, even if these are often tacit and frequently incoherent. That is to say, there are intrinsic links between 'ontological' assumptions (about what human beings are really like), 'epistemological' questions (about the kinds of knowledge practices according to which they are best understood), and 'governmental' questions (about how the conduct of human beings might best be conducted according to their nature). No doubt for much of our history, these assumptions have been derived from a mixture of folk wisdom, theological doctrines, philosophies, and the accumulated experience of those who have exercised authority in different domains. Across the twentieth century, the _psy_ -disciplines came to infuse these assumptions, not just in relation to identifying and managing aberrant individuals, but also in understanding and governing groups and collectivities. Psychology, that is to say, became a 'social' science (Rose 1996). It is thus not surprising that social neuroscience too is becoming an expertise with a social vocation, claiming the capacity to provide an objective knowledge about key features of human social conduct and of human pathology that can and should underpin interventions.\n\nIn his book _Mirroring People: The New Science of How We Connect with Others_ , Iacoboni remarks: \"Unfortunately, Western culture is dominated by an individualistic, solipsistic framework that has taken for granted the assumption of a complete separation between self and other.... Against this dominant view, mirror neurons put the self and the other back together again\" (Iacoboni 2008, 155). And he concludes by asking how, despite the fact that we are \"geared toward empathy and sharing of meaning\" our world is so filled with violence and atrocities. We are at a point, he asserts, \"at which findings from neuroscience can significantly influence and change our society and our understanding of ourselves. It is high time we consider this option seriously\" (ibid., 271). Others share this view that significant implications follow from the recognition that the human brain is profoundly social. Sarah-Jayne Blakemore argues that understanding the social brain should help us adjust educational practices in the light of knowledge about the impact of specific types of environmental input on social cognition (Blakemore 2010). The United Kingdom's Royal Society for the Encouragement of Arts, Manufactures and Commerce (RSA) launched its Social Brain project in 2009, based on the belief stated by its director, Mathew Taylor, that we are entering an age of \"neurological reflexivity.\"\n\nBecoming aware of the social and biological conditions that underpin our actions can, it seems, help each of us develop our competencies and understand and manage our cognitive frailties. But more important, our authorities have much to learn from the lessons of social neuroscience, notably that much of our behavior results from automatic, nonconscious, and habitual reactions to social situations, and that although our brains are inherently disposed to value cooperation and altruism, they are sadly myopic and focus on things close to us temporally, spatially, and emotionally (Grist 2009, 2010). For the Social Brain project, neuroscience has shown us that \"the brain is fundamentally social and largely automatic\" and thus that policies based on \"the notion of a rational, profit maximizing individual who makes decisions consciously, consistently and independently\" was at best a very partial understanding of who we are; hence it was not surprising that social policies based on this idea were hardly successful (Rowson 2011). While recognizing that individual behavior was largely habitual and shaped by nonconscious dynamics did not lead to clear \"policy levers,\" individuals can be aided to become more \"reflexively aware\" of the way their brains shape their lives, and hence move from passivity in the face of such constraints to proactively shaping their lives with an awareness of them. We can anticipate many similar initiatives over the next decade.\n\nElliot Aronson's social psychology textbook _The Social Animal_ was first published in 1972. Much praised and much translated, and now in its tenth edition, the book deals with topics from conformity to interpersonal attrac-tion, and manages its task without any reference to the role of the human brain. But when David Brooks published his book with the same title, _The Social Animal_ , in 2011, he looked first to the brain to understand human sociality, drawing wide implications from neuroscientific arguments about the myriad nonconscious processes occurring each millisecond that make humans exquisitely finely attuned to other human beings throughout their lives. Seeking to dethrone consciousness from its role in shaping human relationships, the book claimed to show \"the relative importance of emotion over reasons, character over IQ, emergent, organic systems over mechanistic ones, and the idea that we have multiple selves over the idea that we have a single self\" (Brooks 2011). The subtitle of the book, _How Success Happens_ , points to its pedagogic ambition\u2014if we understand these profoundly unconscious forces shaping decisions and our affects, whether as individuals, as those in authority, or as politicians, we will not only learn so much more about ourselves, we will learn why so many attempts to improve our societies have failed in the past to combat educational underachievement, achieve social mobility, and tackle welfare dependency and crime. Through social neuroscience and the recognition of the nonrational processes of our social brains, it appears that human beings\u2014at least those in the kinds of societies envisaged by such authors\u2014will know better how to achieve social and political goals.\n\nDespite the radical claims of many of those who write about the implications of understanding the role of nonconscious processes of the social brain in shaping human lives, the policy implications they extrapolate are usually entirely familiar: for example, that the social environment has a role in promoting the individual; stressful and deprived environments are harmful; good parenting and family support is desirable; educational input should take account of critical periods in children's development; and social institutions need to override short-term impulses (now arising in the brain rather than the instincts or the psyche) and foster commitment and social control. Perhaps arguments from neuroscience are merely being invoked to give such proposals a sheen of objectivity\u2014for they are often criticized as arising from hopes rather than facts. Yet the very fact that invoking the brain increases the power of a truth claim shows us something about the way we are coming to understand ourselves today.\n\nOne is reminded of an earlier movement in which a theory of the human mind with the status of truth, this time derived from dynamic psychology, grounded a program for transforming policy. This was of course the movement for 'mental hygiene,' whose many adherents and advocates argued that authorities from politicians to industrialists, and from teachers to parents, needed to reform their practices in the light of the new understanding forged by dynamic psychology of what shaped us as human beings, in order to maintain and maximize the adjustment of the individual to his or her social milieu (Rose 1985). Today, it is no longer a matter of the social shaping of the dynamic forces of instincts inscribed in our biology by evolution producing adjustment or maladjustment with personal, social, and industrial costs. Now, the inescapable nexus where these forces meet is the brain. All agree that social factors are very important. But everything that is considered social now must pass through the brain and have effects via the brain. Parenting, biography, experience, diet, alcohol, drugs, stress, and lifestyle all pass through the brain, shaping and reshaping that brain at the very same time as those capacities and attributes\u2014cognition, emotion, conduct, disorder, resilience, and the like\u2014are shaped by the brain. Hence we have a social brain, in the sense that it now seems that capacities that are crucial to society are a matter of brains. We have a social brain in that the brain has evolved to favor a certain type of sociality manifested in all the interactions between persons and groups that come naturally to humans in our social lives. And we have a social brain in that this organ is now construed as malleable, open to, and shaped by, social interactions\u2014shaping sociality as it is itself reshaped by it.\n\nWe should not diagnose neuroreductionism here. It is not that human beings have become conceived of as mere puppets of their brains, far from it. Human beings are not thought of as identical with their brains, or reduced to their brains or determined by their brains. Mental states have not disappeared; an interior psychological realm of thoughts, intentions, beliefs, and will is still accorded a role. Indeed, as we have seen, in the social neuroscience project, elements drawn from social psychology are frequently invoked. Not only are human beings understood as persons with mental states that are in constant transactions with their neurobiology; they also have the responsibility\u2014presumably via an exercise of will\u2014to nurture their mental capacities in the light of a knowledge of their brains. We, as persons, must adopt the mental states, the habits, the relationships and forms of life appropriate for this work on our brains\u2014we must shape them as they shape us. And we must look after our brains, so that they may continue to look after us. As responsible subjects obliged to manage ourselves in the name of our own health, it seems now we have the added obligation of fulfilling our responsibilities to others by caring for our mutable, flexible, and valuable social brains.\nChapter Six\n\nThe Antisocial Brain\n\nA difference between the brains of psychopaths and ordinary people has been identified.... Research by British scientists using advanced brain-scanning techniques has revealed that a critical connection between two regions of the brain appears to be abnormal in psychopaths. The findings are preliminary and do not show that brain abnormality causes psychopathy but they suggest a plausible biological explanation for the antisocial and amoral behavior that characterizes the condition. If the link to brain wiring could be proved it would raise the prospect of using brain scans to help in diagnosing psychopaths.... Professor Murphy said the findings offered the most compelling evidence yet that altered brain anatomy might be involved in psychopathy.\n\n\u2014\" **Brains of Psychopaths Are Different** , British Researchers Find,\" _The Times_ (London), August 3, 2009\n\nThe exceptionally strong influence of early experience on brain architecture makes the early years of life a period of both great opportunity and great vulnerability for brain development. An early, growth-promoting environment, with adequate nutrients, free of toxins, and filled with social interactions with an attentive caregiver, prepares the architecture of the developing brain to function optimally in a healthy environment. Conversely, an adverse early environment, one that is inadequately supplied with nutrients, contains toxins, or is deprived of appropriate sensory, social, or emotional stimulation, results in faulty brain circuitry. Once established, a weak foundation can have detrimental effects on further brain development, even if a healthy environment is restored at a later age.\n\n\u2014 **J. Cameron** , G. J. Duncan, N. A. Fox, et al., \"The Timing and Quality of Early Experiences Combine to Shape Brain Architecture,\" 2007\n\nThe headline on _Sky News_ on April 26, 2010, read: \"Murdered Gangster's Brain Donated to Science: Scientists Have Been Given the Chance to Get Inside the Mind of One of Australia's Most Notorious Gangsters.\" It was reporting the fact that Roberta Williams had given \"experts\" permission to examine the brain of her husband, Carl Williams, who had been murdered in Melbourne's maximum security Barwon Prison less than three years into a thirty-five year sentence. \"I believe it's to help with research,\" Mrs. Williams is quoted as saying, \"and might explain why guys like Carl do the violent things they do.\"\n\nA month earlier, in March 2010, _Nature_ carried a news feature titled \"Science in Court: Head Case\" (Hughes 2010). Virginia Hughes reported on the brain imaging\u2014using fMRI\u2014of Brian Dugan, who was serving two life sentences in DuPage County Jail for two murders committed in the 1980s, and was facing the prospect of the death penalty for having confessed to an earlier killing. Dugan had been taken to Chicago's Northwestern Memorial Hospital \"to meet one of the few people who might help him to avoid that fate: Kent Kiehl, a neuroscientist at the University of New Mexico,\" Hughes wrote. \"Kiehl has been amassing data on men such as Dugan for 16 years. Their crimes are often impulsive, violent, committed in cold blood and recalled without the slightest twinge of remorse. They are psychopaths, and they are estimated to make up as much as 1% of the adult male population, and 25% of male prisoners.\" And Hughes went on to tell her readers that Kiehl had, by that time, used fMRI to scan more than one thousand inmates, often using a mobile scanner that he took to the prisons. Kiehl's studies, she reported, show that \"the brains of psychopaths tend to show distinct defects in the paralimbic system, a network of brain regions important for memory and regulating emotion\"\u2014work that he believes will \"eliminate the stigma against psychopaths and find them treatments so they can stop committing crimes\" (Hughes 2010, 340).\n\nKiehl is among a number of neuroscientists who believe that advances in genomics and brain imaging are, at last, enabling researchers to detect anomalies in the specific brain regions that are responsible for the pattern of behavior shown by violent and impulsive individuals. For most social scientists, these claims are just the latest episodes in a history of biological criminology dating from at least the early nineteenth century, and indeed are coextensive with the history of modern psychiatry\u2014a history of attempts to find the roots of criminality in the criminal body itself, replete with bold claims, misplaced fears, false hopes, and sometimes tragic failures. It is, however, not just sociologists and historians who are doubtful about claims made by brain researchers for the significance of their work for the criminal justice system. ABC in Australia reported on the Carl Williams case under the headline \"Carl Williams' Brain Unlikely to Reveal Secrets\" and quoted Professor Brian Dean from the Mental Health Research Institute in Melbourne on the \"contentious history\" of examining brains and the lack of good evidence linking violence in men to a change within a particular brain structure, while commenting more optimistically about the role of brain research in depression and schizophrenia.\n\nVirginia Hughes in _Nature_ (Hughes 2010) notes that some researchers, such as Ruben Gur of the Brain Behavior Laboratory at the University of Pennsylvania, who has testified in court on the basis of structural brain scans, believe that, unlike psychiatric \"soft data,\" \"the brain scan doesn't lie\" where structural brain anomalies are concerned. They remain skeptical, however, about functional imaging, which they feel is not reliable enough for legal settings. Similarly, Helen Mayberg, a neurologist from Emory School of Medicine who uses brain imaging in studies of depression, has testified in several court cases _against_ the use of brain scans: \"It is a dangerous distortion of science that sets dangerous precedents for the field.\" Among the reasons for criticisms are the fact that, as discussed in chapter 2, fMRI measures BOLD responses that are difficult to interpret, is based on small samples, usually compares differences between groups and is difficult to extrapolate to single cases, and tells us nothing about the mental state of an individual at the time when the crimes were committed, which may have been many years before the scan was performed.\n\nIn the Dugan case, as in the other cases to which we will refer in this chapter, the court was not convinced by the evidence from neurobiology. But Kiehl is not alone in believing that neuroscientific evidence can and should play an increasingly important role in the criminal justice system. In this chapter we will examine these arguments that claim that human antisocial behavior\u2014notably impulsivity, aggression, and related forms of criminal conduct\u2014have neurobiological roots. We argue that while neurobiological evidence from genomics or functional brain imaging is likely to have limited traction in the criminal courtroom itself, a new diagram is nonetheless emerging in the criminal justice system as it encounters developments in the neurosciences. This does not entail a challenge to doctrines of free will or an exculpatory argument that \"my brain made me do it,\" as some have suggested. Rather it is developing around the themes of susceptibility, prediction, and precaution that have come to infuse many aspects of criminal justice systems as they have come to focus on questions of risk\u2014risk assessment, risk management, and risk reduction (Feeley and Simon 1992, 1994; O'Malley 1996, 1998, 2000; Ericson and Haggerty 1997; Pratt 2000; Baker and Simon 2002).\n\nThis is an element in a wider shift in strategies and policies in many societies toward preemption of unwanted incidents, including criminal offences. In criminal justice systems, especially in the English-speaking world, this has focused particularly on certain types of offending behavior involving violence, impulsivity, repeat offences, danger, sexual predation, and the like. We term the new diagram of control taking shape 'screen and intervene,' and in this chapter we trace out this new configuration and consider some of the consequences. Whatever the virtue of arguments for preventive intervention in relation to health, in the case of neurobiologically based strategies for preventive crime control, we argue that policies of screen and intervene\u2014which have a family resemblance to other strategies for 'governing the future' in the name of security\u2014are likely to contribute to a further widening of the net of the apparatus of control to the 'precriminal' or 'predelinquent,' and to play a part in the new ways in which subjects and subjectivities are governed in the name of freedom in an age of insecurity.\n\nEmbodied Criminals\n\nIn 1890, the same year that he published his classic study _The Criminal_ (Ellis 1890a), extolling the European tradition of scientific analysis of the bodies and brains of criminals to an English audience, Havelock Ellis summarized his analysis in an article published in the _Journal of Mental Science_ (Ellis 1890b). He traced the pathway from the physiognomic epigrams of Homer and Aristotle to Galen and Lavater, via a brief mention of Johann Christian August Grohmann, whose 1820 work he quoted: \" 'I have often been impressed in criminals, and especially in those of defective development, by the prominent ears, the shape of the cranium, the projecting cheek-bones, the large lower jaws, the deeply-placed eyes, the shifty, animal-like gaze' \" (Ellis 1890b, 3). He then arrived at Franz Joseph Gall, that \"unquestionable scientific genius\" who was the founder of criminal anthropology and for whom \"the varying development of the brain was the cause of the divergent mental and moral qualities of the individual.\" For Ellis, it was Gall's \"unceasing study of all the varieties of the brain and of the living head that he could find... throughout Europe, in lunatic asylums and in prisons, as well as among the ordinary population\" that led Gall to foresee \"the extent of the applications of the science that he was opening up to medicine and to law, to morality and to education\" (ibid).\n\nGall had indeed studied the criminal brain, and he provided a full account of the \"carnivorous instinct\" or the \"disposition to murder\"\u2014which was one of the twenty-seven faculties that he hypothesized were linked to different regions of the human brain (Gall [1822] 1835, 4:50\u2013119).  This was based on assessments of the differences in skull shape between carnivorous and non-carnivorous animals, and on observations of the skulls of a parricide and a murderer: \"[T]here was, in each, a prominence strongly swelling out immediately over the external opening of the ear\" (Gall [1822] 1835, 4:50, quoted from R. Young 1970, 39). This was supported by further inspections of the heads of animals, as well as by an examination of the skulls of many famous criminals and the busts and portraits of others. These observations, together with much doleful evidence of cruel, sadistic, murderous, and tyrannical behavior of humans led Gall to the conclusion that this propensity to violence against members of his own species was innate in man and particularly prominent in certain individuals\u2014and that the outward mark of their cruel and bloody character could be seen in this particular prominence on their skulls. \"There can be no question,\" he said, \"of culpability or of justice in the severe sense; the question is of the necessity of society preventing crime. The measure of culpability, and the measure of punishment cannot be determined by a study of the illegal act, but only by a study of the individual committing it\" (Gall [1822] 1835, quoted from Ellis 1901, 31).\n\nOne of Gall's earliest followers in the exploration of the criminal brain was Moriz Benedikt, whose text of 1879, _Anatomische Studien an Verbrecher-gehirnen f\u00fcr Anthropologen, Mediciner, Juristen und Psychologen bearbeitet_ , was translated into English in 1881 as _Anatomical Studies upon Brains of Criminals_. The moral sense, he believed, was located in parts of the occipital lobes that were absent in apes and criminals (Verplaetse 2004). Benedikt, according to Verplaetse, was a freethinker and committed materialist\u2014it was in the brain, not in the soul, that one should look for human moral sensibility. So it was with Paul Broca: the Society for Mutual Autopsy, whose foundation in Paris in 1876 he inspired, was organized around specifically atheist values\u2014notably the belief that there was no soul, only brain (Hecht 2003). As we saw in chapter 2, while Broca may now be remembered for his identification of the brain area that still bears his name, he was also a keen measurer of brain size, in particular seeking the differences between races and between men and women. He was also rather interested in the brains of lunatics and criminals. It might have been assumed that criminals, on the whole considered of low intellect, would have smaller than average brains. The reverse turned out to be the case\u2014the criminal brain was large\u2014something that Broca attributed to the manner of death itself in those whose brains were available to him\u2014sudden death by execution, he reasoned, diminished the original deficit in brain size.\n\nEllis, in the paper we have just discussed (1890b), selects Broca as the founder of the modern science of anthropology and credits him with initiating the special science of criminal anthropology. The next significant advance, in his view, was the conception introduced by the French psychiatrist B\u00e9n\u00e9dict Morel: that the criminal was one of the \"types,\" or forms, taken by degeneration (Morel 1857). This idea was developed by Prosper Despine in his _Psychologie naturelle_ of 1868 and led to Henry Maudsley's comment in _Responsibility in Mental Disease_ (1872) that \"[i]t is a matter of observation that this criminal class [instinctive criminals] constitutes a degenerate or morbid variety of mankind, marked by peculiarly low physical and mental characteristics\" (Maudsley 1872, 30). He considered such criminals morally insane individuals who should be treated in the same way as the intellectually insane. \"Circumstances combined,\" Ellis writes, \"to render possible, for the first time, the complete scientific treatment of the criminal man as a human variety\" (1890b, 9).\n\nIn 1872, inspired by Gall, Benedikt, and Broca, and after early attempts to understand the insane by the use of various instruments of measurement, Cesare Lombroso published the results of some investigations he had made on prisoners at Padua in _Revista delle Discipline Carcerarie_. And by 1876, he published the first volume of _L'uomo delinquente_ , in which he \"perceived the criminal as, anatomically and physiologically, an organic anomaly\" (Ellis 1890b, 9). Ellis acknowledged the many criticisms of this work, notably Lombroso's somewhat indiscriminate collection of facts, but then opined that Lombroso \"was, as he has himself expressed it, the pollen-conveying insect, and the new science which he fecundated has grown with extraordinary rapidity. A continuous stream of studies from books of the most comprehensive character down to investigations into minute points of criminal anatomy or physiology is constantly pouring forth\" (Ellis 1890b, 10). Ellis points to work in criminal anthropology not only in Italy, but in France, Germany, Portugal, Spain, Poland, and Russia, and to the various international congresses on criminal anthropology. \"In Great Britain alone,\" he laments, \"during the last fifteen years there is no scientific work in criminal anthropology to be recorded.... Fifty years ago English men sought to distinguish themselves by the invention of patent improved tread mills and similar now antiquated devices to benefit the criminal. We began zealously with the therapeutics of crime; it is now time to study the criminal's symptomatology, his diagnosis, his pathology, and it is scarcely possible to imagine that in these studies England will long continue to lag so far behind the rest of the civilized world\" (Ellis 1890b, 14\u201315).\n\nFor in Great Britain, as Ellis noted with regret, despite Maudsley's initial commitment to criminal anthropology, the uptake of these ideas was slow. It is true that at the Exeter meeting of the British Association in 1869, Dr. G. Wilson had read a paper on \"The Moral Imbecility of Habitual Criminals as Exemplified by Cranial Measurements\": he measured 464 heads of criminals and found that habitual thieves presented well-marked signs of insufficient cranial development, especially anteriorly. And soon after came the study of more than five thousand prisoners that was published in the _Journal of Mental Science_ in 1870 by J. Bruce Thomson, resident surgeon to the General Prison for Scotland at Perth, pointing to the large numbers who showed morbid appearance at postmortem. This was followed by a series of papers on \"The Morbid Psychology of Criminals\" in the _Journal of Mental Science_ by David Nicholson (Nicolson 1873a, 1873b, 1874, 1875a, 1875b). But it was Italy that had become the home of criminal anthropology and of all the sciences connected with crime and the criminal. We are all too familiar with the charts of degenerate ears, jaws, and noses, along with the measurements of skulls and other physical characteristics that were created by Lombroso and his followers, initially understood in terms of atavism and later recast in the language of degeneracy. It was partly as a result of the proselytizing by Havelock Ellis that the work of the criminal anthropologists\u2014notably of the Italians Lombroso, Ferri, and Garafalo\u2014became much better known in the English-speaking world in the 1890s.\n\nIn the United States, criminal anthropology seemed, at first, to have a hopeful future. At least a dozen books on this topic were published by American authors between 1890 and 1910, repeating Lombroso's catalogue of physical deformities and anomalies of behavior\u2014laziness, frivolity, use of argot, tendency to adorn body and surroundings with hieroglyphics, moral insensibility, and emotional instability (Rafter 1992). Although some expressed misgivings, in America, Britain, and continental Europe, the search for a science of man and a knowledge of the criminal, the delinquent, and the degenerate based on the objectivity of numbers\u2014height, weight, relative proportions of various body parts, and the like\u2014generated an extraordinary flurry of activity. Heights of criminals were compared with those of other sections of society to determine whether, as some thought criminals were inferior in stature: for example, \"Mr. Jacobs, in a paper communicated to the _Journal of the Anthropological Institute_ ,\" determined that the average stature of the London criminal, at 64.3 inches, was some five inches less than that of the well-to-do classes, and very nearly the same as that of the \"East End Jew\" (Morrison 1891, 179). Many focused on the question of impaired intelligence, which encompassed not only intellectual feebleness but moral weakness, and argued that criminals suffered from a defective moral sense\u2014arguments that would feed into a growing concern with the problems of the feebleminded and the need for eugenic strategies to counter their threat to the race and the nation.\n\nBut while many in the English-speaking world were initially impressed with the claims of criminal anthropology, in Great Britain, most medical professionals interested in the specific links between insanity and crime expressed their doubts, as for example, did the influential Scottish psychiatrist Thomas Clouston (Clouston 1894). This is not to say that research into 'pathological' brains was nonexistent. We have already encountered James Crichton-Browne, one of the founding editors of _Brain_ , and medical superintendent of the West Riding County Asylum in Yorkshire, where David Ferrier carried out his experiments on cerebral localization. As Cathy Gere has pointed out (Gere 2003), Crichton-Browne sought to turn his institution into a center for neurological research, appointing W. Lloyd Andriezen as his pathologist in 1872: Andriezen's study of more than one hundred human brains was published in _Brain_ in 1894 (Andriezen 1894). Yet, as David Garland has argued, the psychiatric and medico-legal framework within which early criminological science developed in Great Britain differed from the Lombrosian tradition: it \"was a therapeutically oriented discipline based upon a classification system of psychiatric disorders [within which] there were a variety of conditions which criminals were typically said to exhibit\u2014insanity, moral insanity, degeneracy, feeble-mindedness, etc. But generally speaking, _the_ criminal was not conceived as _a_ psychological type\" (Garland 1988, 3).\n\nFurther, the medico-legal approach to the criminal was institutionally based within the prison establishment and wary of challenging the basic doctrines of legal responsibility as they were elaborated in the courts. Indeed, by the 1890s when Havelock Ellis was proselytizing, most of the leading figures\u2014Needham, Hack Tuke, Nicolson, Maudsley\u2014were explicitly criticizing the excesses of criminal anthropology. Maudsley, having initially been attracted by criminal anthropology, now became one of its most influential critics, arguing \"first, there is no general criminal constitution, predisposing to and, as it were, excusing crime; second,... there are no theories of criminal anthropology so well-grounded and exact as to justify their introduction into a revised criminal law\" (in his \"Remarks on Crime and Criminals,\" quoted in Garland 1988, 4). Or, as David Nicolson put it in his presidential address to the Medico-Psychological Association in 1895:\n\nWriters give us a copious and precise history of the anatomical configuration, the physiological eccentricities, the complexion, the shapes of the ear and nose, the tattoo marks, etc., in certain criminals. We get a striking and elaborated account of their numerous fearful crimes, of their atrocious mental peculiarities and hideous moral obliquities. This analytical and biological process is applied by those who call themselves criminalists to a comparatively small group of criminals; and by implication, and even more directly, it is made applicable to criminals generally. The whole picture is by some writers exaggerated to distortion as regards even the few, and it is in its main features so spurious and unfair as regards the many that it becomes impossible to regard the conclusions or assumptions to be either authentic or authoritative. (Nicolson 1895, 579)\n\nNicolson proposed a complex classification of types of crime and criminals, linked to an assessment of the most appropriate methods of dealing with each, arguing that if the criminal brain was significant, this was only where education and training had prevented the normal development of the brain. In this, his approach was typical of British criminology as it was to develop across the first three decades of the twentieth century: it was to be largely an exercise in individual classification and administrative management of the criminal around the different sites of punishment and reform, using forms of psychiatric and psychological knowledge that undoubtedly made reference to heredity, but moved away from a focus on the exterior of the body and the organic structures of the brain toward an internal mental realm\u2014it was the criminal mind, as formed by domestic circumstances, education, training, and its moral milieu, not the degenerate criminal brain, that was at issue.\n\nIn any event, in the decades at the turn of the century, many of those with skills in neurosurgery and access to brains of criminals cast doubt on the evidence suggesting that there were specific anomalies in the brains of criminals. In Canada, as early as 1882, William Osler rejected Benedikt's argument: he examined the brains of two executed murderers, compared them with thirty-four controls, and found no differences (Osler 1882a, 1882b). In words that may have some resonance today, he wrote that \"as society is at present constituted it cannot afford to have a class of criminal automata and to have every rascal pleading faulty grey matter in extenuation of some crime\" (quoted from Couldwell, Feindel, and Rovit 2004, 711). And in the United States, the anatomist Franklin Paine Mall, writing in 1909, considered that racial comparison of brains spoke volumes about the misleading nature of such claims: \"It certainly would be important if it could be shown that the complexity of the gyri and sulci of the brain varied with the intelligence of the individual, that of genius being the most complex, but the facts do not bear this out, and such statements are only misleading. I may be permitted to add that brains rich in gyri and sulci, of the Gauss type, are by no means rare in the American negro\" (quoted from Finger 1994, 308).\n\nBy the end of the first decade of the twentieth century, these attempts to ground differences in intellect and morality in variations in the gross features of the brain\u2014weight, complexity of gyri and sulci and the like\u2014had more or less run their course, at least in the English-speaking world, as evidence failed to confirm earlier reports, and pointed to some problematic implications. Nonetheless, criminal anthropology continued to flourish into the twentieth century in a number of countries in continental Europe\u2014one of which was Germany. As Michael Hagner has shown, the work of neurologists and psychiatrists on the criminal brain was to become a significant element within what he terms \"the cultivation of the cortex\" in German neuroanatomy (Hagner 2001). From our point of view, two essential elements are consolidated in the work of Meynert, Flechsig, Forel, and Oscar and C\u00e9cile Vogt, so well analyzed by Hagner, and to which arguments about the antisocial brain will return almost a century later. The first is the conception of the brain as a complex organ that can be anatomized into multiple functional areas, and where anomalies in one or more areas have specific and determinate consequences for normal and pathological conduct. The second is the belief that important sociopolitical issues are at stake in an understanding of the structures and functions of the human brain, together with all that contributes to their healthy or unhealthy development. As August Forel put it in 1907: \"As a science of the human in man, neurobiology forms the basis of the object of the highest human knowledge which can be reached in the future.... It will... also have to provide the correct scientific basis for sociology (and for mental hygiene, on which a true sociology must be based)\" (quoted from Hagner 2001, 553).\n\nA true sociology based on the brain\u2014these words have a renewed resonance a century later. Whatever their many differences, Meynert, Flechsig, Forel, and the Vogts mapped out a picture of the brain as an organ differentiated into distinct regions, sometimes thought of along a temporal axis running from the oldest, primitive parts wherein were located the basic drives for survival and reproduction, to the newest, higher regions of the cortex, where impulses from the sense organs were associated with one another. It was in the cortex, it seemed, where complexes developed over time as a result of experience and the progressive myelination of the nerve fibers. It was the cortex that coordinated these into images of the world and into ideas, and in the cortex where the civilizing and rational controls over those lower impulses were located. The anatomists and pathologists waged an interdisciplinary battle with the developing experimental psychologists, namely, Wilhelm Wundt and his followers. Oscar Vogt termed their position one of \"neurobiology\" and linked their arguments with those from what he termed \"neuropsychology.\" At the same time, the Vogts, working with Korbinian Brodmann, undertook \"the most complete structural and functional mapping of the cortex possible, and integrated neurophysiological studies and brain surgery into their research in order to find a physiological correlate for every element of consciousness\" (Hagner 2001, 554).\n\nThe Vogts examined many brains of eminent men in the 1920s and 1930s, including those of Lenin, Forel, and Ferdinand Tonnies, but they also collected the brains of criminals and thought they had observed \"a weak development of the third cortical layer\" in some feebleminded criminals. They linked this analysis to programmatic proposals for both positive eugenics (\"the deliberate marriage of men and women with similar talents\") and the prevention of the \"pre-destined development\" of some individuals into criminals (Hagner 2001, 556\u201357). The Vogts proposed and established the Kaiser Wilhelm Institute for Brain Research in 1914, which increasingly focused on the genetic underpinnings of brain pathology. But in 1933, Oscar Vogt was dismissed from his position as director, and according to Hagner, the Vogts themselves turned away from their ideas of cultivation through breeding and did not advocate forced sterilization or euthanasia. However, those themes were enthusiastically pursued by Oscar Vogt's successor as director of the institute, Hugo Spatz, and his colleague Julius Hallervorden. Spatz and Hallervorden received many brains from the Aktion T4 extermination campaign. However, despite the many measures enacted by the Nazis to target, sterilize, incarcerate, and murder habitual criminals and others thought to be psychopaths or asocial, the focus of these Nazi brain explorers was not directly on the criminal brain, but on the brains of those deemed feebleminded or mentally ill.\n\nInside the Living Brain\n\nIn June 1989, the Max Planck Society announced plans to give a respectful burial to some ten thousand glass slides with sections of brain tissue that had been collected by Hallervorden; the slides were cremated and buried in Munich a year later (Dickman 1989). There was little interest in research on the criminal brain in the period following the end of the Second World War, no doubt, in part as a reaction to such research on those murdered in the Nazi period. Yet four decades later, at the same time as the fate of the Hallervorden collection was being debated in Germany, the relation between the brain and criminality was to be reopened within another style of thought, in a very different sociopolitical climate, with other technical means, and in living subjects. In 1987, the _British Journal of Psychiatry_ published a study by Nora Volkow and Lawrence Tancredi from the University of Texas of four individuals \"with a history of repetitive purposeless violent behavior,\" using three neuroimaging techniques: EEG, CT scan, and PET. They reported the following results: \"Three patients showed spiking activity in left temporal regions, and two showed CT scan abnormalities characterized by generalized cortical atrophy. The PET scans for the four cases showed evidence of blood flow and metabolic abnormalities in the left temporal lobe. Two patients also had derangement in the frontal cortex. The patients showing the largest defects with the PET scans were those whose CT scans were reported as normal\" (Volkow and Tancredi 1987, 668).\n\nA year earlier, when Thomas Grisso, a leading forensic psychiatrist at the University of Massachusetts Medical School, had reviewed the tools available to those conducting assessments for forensic purposes, the repertoire seemed limited to traditional symptom description and diagnosis, together with history-taking about head injuries, brain insults, and epilepsy, and psychological tests of intelligence and personality, including some specifically developed for forensic use (Grisso 1986). In this context, and given the many criticisms of the validity and forensic utility of the psychological tests, it was perhaps not surprising that Volkow and Tancredi's study, probably the first to use such neuroimaging techniques in relation to violent individuals, was hailed by Daniel Martell, an American forensic neuropsychologist, as foreshadowing a larger role for the neurobehavioral sciences in criminal law: \"[T]he particular appeal of neuropsychological evidence in the criminal context [is] the expert's ability to bring quantified, normative data on brain-behavior relationships to bear in support of what have traditionally been professional opinions based on mental status examinations and clinical interview techniques.\" (Martell 1992b, 315).\n\nMartell noted that such imaging studies nonetheless raised the thorny issue of how to relate specific brain abnormalities to individual behavior in particular cases: \"while neuroimaging technologies can provide powerful evidence of the presence, type, location, size, and angle of brain lesions, they provide the trier of fact with little information about the behavioral sequelae of a given lesion\" (Martell 1992b, 324). Hence, he suggested, it is precisely in this area that the neuropsychologist, using a battery of other tests, is best prepared to provide evidence to inform the decision-making process in the court: the \"very best neuropsychological assessments represent the highest level of contemporary objective psychological evaluation [and, with] the highest standards of thoroughness, accuracy, and care in the administration, scoring, interpretation, and reporting of findings... speak adequately to the presence or absence of mental disease or defect, and its relationship to specific criminal-legal referral issues\" (ibid., 318).\n\nThe early 1990s saw a series of papers from Martell and others, arguing for a new neurobiology of violent crime and pointing in particular to the results of research using EEGs and brain scanning of violent offenders (Kahn 1992; Martell 1992a; Volavka, Martell, and Convit 1992; Pincus 1993; Rojas-Burke 1993; Dolan 1994). The direction of the brain scanning research itself, here as elsewhere, was toward localization. Within the style of thought that had taken shape in brain imaging, if there was something anomalous in the brains of violent offenders, it had to be locatable in a specific brain region. But where?\n\nIn 1990, Antonio Damasio and colleagues at the University of Iowa Hospitals and Clinics published a paper drawing on the clinical observation that adults with previously normal personalities who sustain damage to the ventromedial frontal cortex develop defects in decision making and planning revealed in abnormal social conduct (Damasio, Tranel, and Damasio 1990). They suggested that this might be a result of inability to activate the somatic states related to previous experiences of punishment and reward, hence depriving the individual of the automatic devices regulating conduct; they claimed to find confirmation in the abnormal autonomic responses of patients with frontal lobe damage. The belief in a link between regional (frontal) brain abnormalities and problems of impulse control seemed to be firming up.\n\nIn 1991, Joseph Tonkonogy published the results of head CT and MRI scanning studies that had been undertaken since 1987 at the Worcester State Hospital. Reporting on five patients with violent behavior and local lesions in the anterior-inferior temporal lobe, he argued that \"violent behavior in psychiatric patients may manifest as a primary symptom of tissue loss in the areas adjacent to the amygdaloid nuclei or in the amygdala itself, especially when combined with a history of seizure disorders.... To the best of our knowledge, this is the first report of local tissue loss in the anterior temporal lobe in the group of psychiatric patients with violent behavior\"\u2014although the author noted that such patients formed only a relatively small percentage of all violent psychiatric patients in the Worcester State Hospital (Tonkonogy 1991, 189\u201390, 193).\n\nAs this work gathered pace, one of the first to reflect on its more general implications was Adrian Raine, a former airline accountant turned experimental psychologist, who had worked for a spell in high security prisons prior to his academic career, first at Nottingham University and then at the University of Southern California. In 1993, Raine published _The Psychopathology of Crime: Criminal Behavior as a Clinical Disorder_. He reviewed the evolutionary, genetic, and neurobiological evidence and sought to rehabilitate biological investigations of criminal behavior. His aim was to free them from their associations with racism, suggesting that many of the critics were attacking a straw man: he argued that there were good grounds for considering at least some types of offending behavior as a clinical disorder. What was still lacking, he contended, was a concerted effort to integrate biological and social theories with empirical evidence, in particular to resolve currently tangled issues concerning the relations of causality between the biological, psychological, and social factors involved (Raine 1993, 288).\n\nAcross the 1990s, the apparent insights to be gained from using the technology of brain imaging, which had now become relatively easy for researchers to access, encouraged many other investigators to use PET and fMRI to study the brains of individuals with a history of violence or aggression, and indeed to extend the groups studied to those diagnosed with personality disorder (Goyer, Cohen, Andreason, et al. 1994). Raine himself continued his studies of the brains of murderers. He argued that \"seriously violent offenders pleading not guilty by reason of insanity or incompetent to stand trial are characterized by prefrontal dysfunction\" and suggested that \"deficits localized to the prefrontal cortex may be related to violence in a selected group of offenders,\" although he accepted that the group that he studied\u2014those accused of murder or manslaughter undergoing examination in relation to a potential plea of not guilty by reason of insanity (NGRI) or incompetence to stand trial\u2014might not be typical of violent offenders in the community (Raine, Buchsbaum, Stanley, et al. 1994). By 1997, he and his colleagues were arguing that \"[t]he advent of brain imaging research has recently made it possible for the first time to directly assess brain functioning in violent individuals,\" and their research followed up the suggestion that frontal brain regions were implicated in addition to the temporal cortex (Raine, Buchsbaum, and LaCasse 1997, 496). Using a larger sample group of forty-one people (thirty-nine men and two women) accused of murder or manslaughter who were undergoing examination in relation to a potential plea of NGRI or incompetence to stand trial\u2014a group the researchers designated somewhat misleadingly as \"murderers\"\u2014Raine and his colleagues concluded that significant brain abnormalities were present. In their discussion, the researchers conceded that this was both a relatively specific subgroup of offenders, and also one that was itself clinically heterogeneous. They emphasized that the findings did not demonstrate that violence was caused by biology alone, nor that murderers pleading NGRI were not responsible for their actions, nor that PET could be used diagnostically, nor that their findings \"could speak to the issue of the cause (genetic or environmental) of the brain dysfunction,\" nor could they be taken as specific to violence in particular, since a nonviolent criminal control group was not used (Raine, Buchsbaum, and LaCasse 1997, 505). But their abstract stated their findings in less cautious terms: \"Murderers were characterized by reduced glucose metabolism in the prefrontal cortex, superior parietal gyrus, left angular gyrus, and the corpus callosum, while abnormal asymmetries of activity (left hemisphere lower than right) were also found in the amygdala, thalamus, and medialtem-poral lobe. These preliminary findings provide initial indications of a network of abnormal cortical and subcortical brain processes that may predispose to violence in murderers pleading NGRI\" (ibid., 495).\n\nIn 2000, Raine and his team published their much-cited study claiming to find reduced prefrontal gray matter volume and reduced autonomic activity in people diagnosed with antisocial personality disorder (Raine, Lencz, Bihrle, et al. 2000). By this time, they were able to refer to more than a dozen studies that had used different brain imaging techniques to explore anomalies in the brains of alcoholics and other substance abusers, those diagnosed with sociopathic behavior or antisocial personality disorder with convictions for violence, as well as those diagnosed with mood disorders or schizophrenia, and of course their own studies of \"murderers.\" Their conclusion was that prefrontal structural deficits may underlie the low arousal, poor fear conditioning, lack of conscience, and decision-making deficits in antisocial, psychopathic behavior.\n\nSome forensic psychiatrists and neurobiologists expressed caution about the claims that frontal lobe dysfunction was strongly associated with, or predictive of, violent behavior (Brower and Price 2001). But despite such doubts, in the first decade of the twenty-first century, an expanding research program in what Raine termed \"neurocriminology\" took shape in many labs and in many countries, using brain imaging techniques to seek the neurological basis of psychopathy, antisocial behavior, impulsive violence, and aggression (Blair 2004, 2005; Blair, Newman, Mitchell, et al. 2006; Rogers, Viding, Blair, et al. 2006; Crowe and Blair 2008). Biological criminology was no longer, it seemed, linked inseparably with the pseudoscience of criminal anthropology and its focus on the exterior of the bodies of those forming a criminal class. The gaze of the neurocriminologist could now plunge into the interior of the living criminal brain and discover therein the roots of the violence and failures of impulse control in each dangerous individual. But with what consequences? For if the roots of such socially undesirable behavior were neural, was that exculpatory or inculpatory? Was it grounds for mitigating the responsibility and the culpability of offenders because their capacities to constrain their conduct were diminished? Or did it increase the need for their incarceration and exclusion from society on the grounds of their incorrigibility? On these crucial matters, the leading researchers remained ambivalent.\n\nNeurolaw?\n\nWhat, then, would be the impact of this research in neurocriminology within the legal system? When the _neuro_ -prefix came to law in the 1990s, neurolaw was initially focused on the question of traumatic brain injury and the role of neuropsychologists as expert witnesses in the increasing number of cases involving claims for damages (Taylor, Harp, and Elliott 1991; Taylor 1995, 1996). But by 2004, when the Dana Foundation convened a meeting of twenty-six neuroscientists for a conference on law and neuroscience, the focus of the debate was firmly on criminal law and the relations of the brain to the criminal justice system (Garland 2004). The commissioned papers marked out what was to become the territory for neuroethical discussions in this area for the next decade.\n\nWhat, if any, were the implications for the legal doctrines of free will and responsibility? asked Michael Gazzaniga and Megan Steven. Stephen Morse, writing under the title \"New Neuroscience, Old Problems,\" rehearsed his argument that legal systems operated with their own \"folk psychological\" conception of human beings as intentional persons possessing mental states that were the causes of their actions, and hence could be held responsible for them, and that neuroscientific evidence was not likely to pose a radical challenge to this view. And Henry (Hank) Greeley discussed issues of \"Prediction, Litigation, Privacy and Property,\" focusing, presciently, on the likelihood that the real impact of neuroscience would probably not be in the courtroom, where legal reasoning and evidential rules were likely to retain hegemony, but would lie outside the courtroom, in the pretrial and posttrial practices of the criminal justice system. This overall verdict was reiterated a couple of years after that event by the editor of the DANA volume: \"Much discussion has focused on 'ultimate' issues such as free will, determinism (genetic or mechanistic), and their effect on whether the concept of criminal culpability will be undone by new scientific discoveries. This seems unlikely, at least in the near future\" (Garland and Frankel 2006, 103).\n\nThis has not deterred those who continue to speculate about these themes. However, as predicted by most of the contributors to the DANA debate, the courts or juries, at least in adversarial criminal justice systems, have proved resistant to changing their own conceptions of responsibility, or their ways of attributing guilt, in the light of neurobiological evidence (Rose 2000, 2007c, 2008). They do not seem to find evidence from the neurosciences about factors predisposing to a criminal act any more relevant to legal notions of responsibility than arguments or evidence from the other 'positive sciences' of human conduct\u2014psychiatry, psychology, sociology\u2014despite the frequent suggestions by commentators that the rhetoric of objectivity surrounding brain scans and other neurobiological evidence will radically enhance their plausibility. Anglo-American legal systems conceptualize their subjects\u2014with specific exceptions\u2014as individuals with minds, or mental states, who intend the acts they commit, and who foresee their outcome to the extent that any reasonable person could so do (Morse 2004, 2006, 2008; Morse and Hoffman 2007). If they are to mitigate this presupposition, they require compelling evidence that some state or event\u2014an illness, sleepwalking, coercion from a third party\u2014was directly and causally related to the commission of the act in question. Probabilistic arguments, to the effect that persons of type A, or with condition B, are in general more likely to commit act X, or fail to commit act Y, hold little or no sway in the process of determining guilt.\n\nFurther, as we have seen in chapter 2, evidence from brain scans or other neurobiological tests does not 'speak for itself'\u2014it must be spoken for. Images do not convince by means of their own intrinsic plausibility: interlocutors are required, and expert witnesses such as the clinicians who have requested and interpreted the brain scans of the accused must make the claim that the scan shows relevant anomalies. As in other areas of criminal trials in adversarial systems, these claims will usually be contested by other expert witnesses, who argue that it shows no such thing. Judges advise juries to take into account many factors, including their own judgments as to what might be a reasonable and plausible account of the circumstances that led to the commission of the act in question. It is no surprise, therefore, to find that despite the repetitive discussion of this issue by philosophers and ethicists, there is no evidence, to date, of neurobiological testimony concerning _functional_ abnormalities in the brain of the accused affecting the verdict in any recent murder trial in the United States or the United Kingdom (Rose 2007c, chap. 8).\n\nBut this is not to say that neurobiological arguments about the neural basis of violent or antisocial conduct will have no traction in the criminal justice system or in crime control policy. Initially, as might be expected, the effects are likely to appear in relation to those kinds of persons who have long been treated differently by the legal system\u2014notably children and women, but also those who are thought to be incapable of understanding the nature or meaning of their acts, and hence, in the criminal law at least, not capable of bearing responsibility. As far as children are concerned, the age of legal subjectivity for the purpose of attributing criminal responsibility has always been somewhat arbitrary, contested, and variable across jurisdictions. There is some evidence that neurobiological arguments may have an impact here, for example, in the much-discussed judgment of the U.S. Supreme Court in _Roper v. Simmons_ , which barred capital punishment for juvenile offenders under the age of eighteen. The Court was presented with an amicus brief drawn up by a group of New York lawyers at the request of a number of key American professional organizations, including the American Medical Association, the American Psychiatric Association, the American Society for Adolescent Psychiatry, and the National Mental Health Association (Haider 2005). Taking advice from a panel of scientists, the brief argued that juvenile brains are anatomically different and that the frontal lobes, \"the part of the brain responsible for reasoning, impulse control, cost-benefit calculation and good judgment is not fully developed. This means that adolescents are inherently more prone to risk-taking behavior, and less capable of governing impulses, than adults.... These studies were presented to the Court as evidence that adolescents are biologically different\" (Haider 2005, 371\u201372).\n\nWe should, however, be wary of reading too much into this case. The judgment itself did not explicitly refer to neurobiological evidence, or the adolescent brain, although it did refer to some sociological research on adolescence, and of course to commonsense knowledge about the impulsiveness of youth. The brief was submitted in the context of a fairly major mobilization of liberal U.S. academics, lawyers, and medical professionals against the death penalty, seeking any opportunity to mitigate or limit its use. Nonetheless, despite the well-known cultural and historical specificity of the notion of adolescence, and the relative recency of the association of this period in life with risk taking and poor impulse control\u2014which would rule out any underlying evolutionary changes in the rate of brain development\u2014a number of U.S. researchers argue that such characteristics are a consequence of 'the developing brain' and the neural mechanisms responsible for processing emotions and incentives (Casey, Jones, and Hare 2008). It is doubtful whether such arguments will have much influence on questions of _culpability_ in the courtroom, but they are being referred to in debates on the age of criminal responsibility. And in the United States, there are suggestions that they will increasingly impact on sentencing decisions for adolescents, especially those concerning the legitimacy of the death penalty and the sentence of life imprisonment (Maroney 2009, 2011).\n\nWill neurobiological arguments reshape those other categories in which persons are released from the category of legal subject\u2014for example, in relation to the question of unfitness to plead? It is too early to tell, but in any event, such cases are rare. It is more likely that neuroscience, like the other positive sciences since the nineteenth century, will gain its traction outside the agonistic space of the courtroom. We will see its impact in criminal justice policy, notably policies aimed at prevention, in the identification of risky persons, and in the agonized debates over the existence and management of psychopathy. Further, we will see the effects of neuroscience in the investigatory process and in the procedures following arrest and leading up to trial\u2014these are crucial, since only a small percentage of cases actually reach trial, as in many instances the suspect is either released or pleads guilty prior to trial, sometimes to a lesser charge.\n\nAnd we will also see the impact after the verdict has been delivered, in the determination of sentence and the appropriate institutional disposition, in decisions as to release from prison, and of course, in the probation service. Indeed, in the DANA event mentioned earlier, one of the neuroscientists expressed concerns over just these issues: \"[A significant fear], both in the behavioral genetics area and with [neuroscience], is that there are no [rules of evidence] that control the use of these kinds of technologies in the preformal stages of criminal processes. When it gets to the formality of sentencing, the cry will come up, but the ability of judges and prosecutors to make decisions about whether they're going to initiate charges, [whether] they're going to accept diversion [from criminal prosecution] for people, et cetera\u2014using [neuroscience tests] that haven't been validated\u2014is a serious risk that the technology poses\" (quoted in Garland and Frankel 2006, 106).\n\nBut before turning to examine some areas where notions of the antisocial brain are indeed entering these arenas, let us turn from evidence about brain development to examine the challenges posed by research in behavioral genomics.\n\nThe Genetics of Control\n\nAdrian Raine's 1993 book, _The Psychopathology of Crime_ , contained one chapter on the genetics of crime. It was largely concerned with defending those who were undertaking this work at the time against the charges of genetic reductionism, political bias, racism, or eugenics; from allegations that they rule out environmental factors; and from accusations that they believe that genetics is destiny. Nonetheless, he argued: \"Overall, data from genetic studies [he is referring to twin and adoption studies] supports the prediction made by sociobiological theory that heritable influences in part underlie criminal behavior\" (Raine 1993, 79). Brain scanning studies of violence and personality disorder in the 1990s made little reference to genetic research, and, when they did, generally noted that they were unable to determine whether the anomalies that they reported had a genetic base or resulted from environmental insults. But a new genomic approach to antisocial conduct was taking shape that would prove more compatible with the emerging image of the impulsive brain.\n\nThis is not the place to rehearse the depressing history of research on the genetics of criminality. From about 1990, something began to change in the way in which the relations between genetics, neurobiology, and crime were construed. It was no longer a question of the search for a gene for crime, or even of a gene for aggression (even though this was the way that it was sometimes reported in the popular press). Indeed, these studies no longer sought to explain criminality in general, but focused on the impulsive behaviors that were thought to be involved in many criminal or antisocial acts, in particular, acts of aggression. These behaviors seemed to involve something that would once have been understood in terms of the will but that now fell under the rubric of _impaired impulse control_. And those searching for a genetic basis for such impairment now conceptualized the issue in neuromolecular terms\u2014that is to say, they tried to identify variations in SNPs in the coding sequences that affected neuronal circuits in the brain, notably in the neurotransmitter system.\n\nAt the risk of repeating a familiar story, let us begin with the paper by Han Brunner and his team\u2014based in Nijmegen in the Netherlands\u2014published in _Science_ in 1993 under the title \"Abnormal Behavior Associated with a Point Mutation in the Structural Gene for Monoamine Oxidase A\" (Brunner, Nelen, Breakefield, et al. 1993). For if there was one paper that can be said to mark a mutation in the thought style of neurocriminology, this was it. Like much genetic research at the time, Brunner's team was conducting a _kindred study_ \u2014that is to say, the researchers studied a problem that seemed to pass down a lineage. In this case, it was \"a large kindred in which several males are affected by a syndrome of borderline mental retardation and exhibit abnormal behavior, including disturbed regulation of impulsive aggression\" (ibid., 578). It seems that the kindred in question came to light because an unaffected maternal great-uncle, aware that there was a pattern of violent behavior among his male relatives, perhaps going back to the eighteenth-century, had constructed a detailed family history of what he thought was an inherited mental disability; this led a sister of one of the affected males to come to Brunner's genetic clinic seeking a genetic test to see if she was a carrier of this hereditary trait.\n\nBrunner and his team collected samples and histories from fourteen affected males and ten nonaffected individuals in the extended family; their first report, initially submitted to the _American Journal of Human Genetics_ in 1992, focused largely on the question of X-linked mental retardation. It did, however, relate this to previous research on the neurotransmitter anomalies that might underlie disturbed aggression regulation, notably the work by a veteran of earlier struggles over the legitimacy of biological psychiatry, Herman van Praag (van Praag, Asnis, Kahn, et al. 1990; van Praag 1991; van Praag, Brown, Asnis, et al. 1991). The later paper in _Science_ , however, specifically made the connection between human impulsive aggression and low or absent monoamine oxidase A (MAOA); it placed this in the context of a dozen studies of aggressive behavior in animals and humans published over the previous decade that had implicated altered metabolism of neurotransmitters\u2014serotonin, and to a lesser extent dopamine and noradrenaline\u2014and suggested an as yet unknown genetic basis.\n\nBrunner and his colleagues were aware that their kindred was rather special; all the affected males had a single point mutation, in which a C was replaced by a T at nucleotide 296 in the coding sequence, changing a gluta-mine (CAG) codon to a termination (TAG) codon; this resulted in complete deficiency (that is to say, total absence) of MAOA in affected males. MAOA is specifically implicated in the breakdown of serotonin, norepinephrine, and epinephrine, and therefore it seemed reasonable to postulate that a lack or deficiency in this enzyme could be related to an increase in available levels of these neurotransmitters in the central nervous system (and indeed in the other organs where it is present, such as the liver and gastrointestinal tract). Actually, although this was not a concern of Brunner and his group, one might wonder how, in the absence of such a key enzyme for neurotransmitter functioning, these unfortunate individuals survived at all\u2014a testament to the redundancy within human neurotransmitter systems and its homeodynamic mechanisms.\n\nBrunner and his colleagues reported that impulsive, rather than premeditated, aggression in humans had been linked to low concentrations of the breakdown product of serotonin, 5-HIAA,  in cerebrospinal fluid, and that \"this may be caused by absent MAOA activity in these subjects\"; they also noted that \"inhibition of MAOA activity has not been reported to cause aggressive behavior in humans,\" although they speculated that \"deficiencies throughout adult life might have different consequences\" (Brunner, Nelen, Breakefield, et al. 1993, 579). They concluded their paper with the suggestion that would spark a whole research program. For while their kindred was marked by complete absence of MAOA, they pointed to the wide range of MAOA activity in the normal population: \"one could ask whether aggressive behavior is confined to complete MAOA deficiency\" (ibid., 590). Could this, then, be some kind of dimension, where levels of aggression were linked to variations in MAOA activity, rather than the complete absence of the enzyme?\n\nThis paper attracted great interest, not only among researchers but also in the international press, who dubbed the mutation that had been identified \"the aggression gene,\" and in the legal profession\u2014very soon after publication the defense lawyers in the murder trial of Stephen Mobley, in the U.S. state of Georgia, sought unsuccessfully to introduce it as evidence to mitigate their client's responsibility for the crime for which he had been convicted. Brunner himself, clearly troubled by the implications, attempted to cast doubt on extrapolations from this rare kindred to more general genetic explanations of aggression and violence. By 1996, in a CIBA Foundation event on the genetics of crime, he was arguing that his research gave no support for the notion of an aggression gene, despite having been interpreted in this way by the popular press: \"[T]he notion of an 'aggression gene' does not make sense, because it belies the fact that behavior should and does arise at the highest level of cortical organization, where individual genes are only distantly reflected in the anatomical structure, as well as in the various neurophysiological and biochemical functions of the brain.... [A]lthough a multitude of genes must be involved in shaping brain function, none of these genes by itself encodes behavior\" (Brunner 1996, 16).\n\nBrunner soon turned away from this area of work, but this paper sparked a veritable industry of research, both in humans and in animal models, on the relations between aggression and monoamine oxidase, and more generally on neurotransmitter anomalies that might be correlated with aggression. As far as the animal studies are concerned, one of the leading labs was that of Klaus Miczek at Tufts University. Miczek, who we mentioned briefly in chapter 3, had been working on animal models of aggression since the early 1970s, initially examining the effects of various drugs of abuse\u2014cannabis, amphetamines, alcohol\u2014and later focusing on the role of the monoamines. He was also a member of the National Academy of Sciences's somewhat controversial panel on \"Understanding and Preventing Violence\" from 1989 to 1992. Throughout the 1990s, he and his research group at Tufts published a series of papers arguing for the importance of experimental ethological studies of aggression in animals, mostly in rodents, to pinpoint the most appropriate selection of pharmaceutical interventions in clinical situations to modify aggressive behavior in humans. They suggested that such studies would enable one to select precise receptor sites that could be targeted by drugs to reduce situationally specific forms of aggressive, defensive, and submissive behavior (e.g., Miczek, Weerts, Haney, et al. 1994).\n\nReviewing their own and other relevant research almost a decade later, Miczek and his team asserted: \"Aggressive outbursts that result in harm and injury present a major problem for the public health and criminal justice systems, but there are no adequate treatment options\" (Miczek, Fish, DeBold, et al. 2002, 434). It seemed that obstacles \"at the level of social policy, institutional regulation, and scientific strategy in developing animal models continue to impede the development of specific anti-aggressive agents for emergency and long-term treatments\" (ibid.). There were, they conceded, no such things as specifically pathological _mean genes_ , as \"aggressive behaviors serve mostly adaptive functions in all phyla, rather than being behavioral aberrations\"; however, the authors nonetheless argued that \"[b]ehavioral and neurobiological analyzes of aggression in various animal species contribute to understanding human violence and to the development of therapeutic interventions\" (ibid., 435). Their aim was to identify the neural circuits underlying different types of aggression, so as to ascertain the particular serotonin, dopamine, and GABA receptor sites that could be targeted by psychopharmaceuticals in order to modulate human aggression.\n\nThe authors noted that a usual definition of aggression in humans entails an _intent_ to do harm. However, like others working in this field, they were not detained for long by the problems that this invocation of a mental state might entail. They proceeded on the basis that certain behavioral patterns in animals can be used as models for aggression in humans, supporting their argument with reference to studies of higher primates; laboratory studies with humans, for example, those involving simulation of violent or aggressive encounters with fictional opponents; and the effects of various forms of reinforcement or drug treatment. They also pointed to neurobiological evidence concerning the effects of varying levels of neurotransmitter activity in humans, primates, and rodents, which seemed to demonstrate a homology between the neurobiological mechanisms in human and animal aggression. This enabled them to draw upon their rodent models to explore the specific neurotransmitters and receptors that seem to be entailed in either activation or reduction of different types of aggression, with a special interest in individual differences in escalated\u2014that is, non\u2013species typical\u2014aggressive behavior. The premise of the argument was that this neuromolecular dissection of the bases of aggression might lead to the development and use of pharmacological interventions to control escalated aggression in humans.\n\nThis argument for a \"pharmacotherapeutic approach for managing violent and aggressive [human] individuals\" was made more explicit in a paper published two years later, in which the authors lamented the fact that\n\n[t]he development of selective anti-aggressive agents, once referred to as \"serenics,\" [here they refer to the work of Berend Olivier who proposed this term (Olivier, Mos, Raghoebar et al., 1994)] has been sporadic and, at present, appears to be inactive. There are many reasons for this calamitous situation, including a lack of enthusiasm by regulatory agencies to approve medicines for a non-disease and the reluctance of targeting neurobiological sites for an ostensibly social dysfunction. The latter issue derives from a conceptual problem that divorces aggressive experiences from neurobiological mechanisms. (Miczek, Faccidomo, de Almeida, et al. 2004, 344)\n\nDespite this \"calamitous\" situation and the reluctance of authorities to license pharmaceuticals whose main indication for use was behavioral control, the authors conclude: \"It can be anticipated that currently developed tools for targeting the genes that code for specific subtypes of serotonin receptors will offer new therapeutic options for reducing aggressive behavior, and the 5-HT1B receptor appears to be a promising target\" (ibid., 336).\n\nBy 2007, this time writing with Adrian Raine, Miczek noted with regret that the 2002 WHO _World Report on Violence and Health_ completely ignored neuroscience; nonetheless, they argued, neurobiological research on aggressive behavior was emerging from its shameful past of eugenics and lobotomies, and that \"the schism between these approaches [social and biological] promises to be overcome by advancing our knowledge of the molecular events through which social experiences sculpt future aggressive acts. Insights into the gene-environment interactions are critical for the way in which the criminal justice and the public health systems deal with aggression and violence.... [E]merging successes in behavioral and molecular biology of aggression research... may have important implications not only for diagnosis, prevention, and treatment but also for guidance of public and judicial policies\" (Miczek, de Almeida, Kravitz, et al. 2007, 11,803).\n\nYet despite this apparent openness to social and environmental factors in the genesis of aggressive and violent conduct, they summed up their argument in their abstract, cited in chapter 3, suggesting that genetic and imaging research on aggression in fruit flies and feral rats leads to the hypothesis that \"antisocial, violent, and psychopathic behavior may in part be attributable to impairments in some of the brain structures (dorsal and ventral PFC, amygdala, and angular gyrus) subserving moral cognition and emotion\" (ibid.). Despite the fact that this argument would have most human scientists, and many neurobiologists working with animal models, reeling at the bizarre jumps of logic entailed in the connections between flies, rats, and humans, they concluded by arguing for a \" _neuroethical_ discussion\" of the implications among representatives of the criminal justice system, psychiatry, neuroscience, and the social sciences (ibid., 11,805). One might be forgiven for thinking that critical scientific analysis of the conclusions drawn by these researchers might preclude the necessity for discussion of their supposed implications.\n\nDespite the publicity around Han Brunner's study discussed earlier, the specific role of MAOA in aggression was not central to animal studies in the 1990s, with the exception of the work of Isabelle Seif and colleagues based at the CNRS in Paris. Referring directly to Brunner's work, Seif and her group created a line of transgenic mice with a deletion in the gene encoding MAOA, thus providing an animal model of MAOA deficiency. This increased serotonin concentrations in the brains of the mouse pups by up to ninefold, and also increased levels of norepinephrine concentrations in pup and adult brains. It appeared that (perhaps unsurprisingly given such an uplift in neurotransmitter levels at an early stage in brain development), behavioral abnormalities followed: \"Adults manifested a distinct behavioral syndrome, including enhanced aggression in males,\" and the authors concluded that their study \"supports the idea that the particularly aggressive behavior of the few known human males lacking MAOA is not fostered by an unusual genetic background or complex psychosocial stressors but is a more direct consequence of MAOA deficiency\" (Cases, Seif, Grimsby, et al. 1995, 1763, 1766). However, by 1999, Schuback, and a group that included one of the researchers on Brunner's original study, Xandra Breakefield, screened a large sample, including males with mental retardation, those with a history of sexually deviant behavior, and other patient groups, and found no evidence at all for mutations disrupting the MAOA gene; they concluded that such mutations were not common in humans (Schuback, Mulligan, Sims, et al. 1999). One might have thought that this would spell the end, or at least the beginning of the end, for MAOA-linked hypotheses concerning impulsive and violent criminal behavior. But this was not to be the case.\n\nLacking much else to go on, researchers would continue to search for MAOA links whenever they had the chance. And MAOA was to come back into prominence via another route, with the first significant publication from Avshalom Caspi and Terrie Moffitt and their team from their work on the Dunedin Multidisciplinary Health and Development Study. This was a cohort study of 1,037 people born in a coastal region of the South Island of New Zealand between April 1, 1972 and March 31, 1973, and still living in the area three years later, who were assessed every few years up to and beyond the age of twenty-six.  Caspi and Moffitt and their team drew on the earlier studies of the links between deficiencies in MAOA activity and aggression in mice and humans\u2014including of course, that of Brunner's group\u2014and argued that although \"[e]vidence for an association between MAOA and aggressive behavior in human populations remains inconclusive,... [c]ircumstantial evidence suggests that childhood maltreatment predisposes most strongly to adult violence among children whose MAOA is insufficient to constrain maltreatment-induced changes to neurotransmitter systems\" (Caspi, McClay, Moffitt, et al. 2002, 851).\n\nThey supported this claim with reference to animal studies showing that early stress alters levels of norepinephrine, serotonin, and dopamine in ways that can be long-lasting (Francis and Meaney 1999; Bremner and Vermetten 2001) and reports linking variations in norepinephrine and serotonin levels to aggressive behavior in humans, although they did acknowledge that no study had shown that MAOA plays a role in these variations. Nonetheless, they proceeded to test the hypothesis \"that MAOA genotype can moderate the influence of childhood maltreatment on neural systems implicated in antisocial behavior\" by seeking to correlate the number of tandem repeats in the promoter region of the MAOA gene and response to childhood maltreatment. Dividing the sample on the basis of repeat numbers into \"low MAOA activity\" and \"high MAOA activity\"; classifying childhood maltreatment as severe, probable, and none; and assessing outcomes in terms of _DSM-IV_ conduct disorder, convictions for violent crimes, personality assessments of disposition toward violence, and anecdotal reports of antisocial personality disorder, they generated the finding that was to launch a host of studies of 'gene-environment interaction.' Maltreated males with high levels of MAOA activity did not have elevated antisocial scores, while maltreated males with low levels of MAOA activity were more likely to have adolescent conduct disorder, violent convictions, and to score high on measures of antisocial conduct. Hence the conclusion: \"a functional polymorphism in the MAOA gene moderates the impact of early childhood maltreatment on the development of antisocial behavior in males\" (Caspi, McClay, Moffitt, et al. 2002, 853). It appeared that some genetic variants could both _increase_ risk of antisocial behavior in the face of stressful childhood experiences, while other variants might be protective. The team speculated that the findings had practical implications for drug treatment. \"Both attributable risk and predictive sensitivity indicate that these findings could inform the development of future pharmacological treatments\" (ibid.).\n\nIn the subsequent eight years, this study was cited some 1,700 times. Some tried to replicate it and failed; others criticized various aspects of the methodology or the analysis. However, from our perspective, these problems are not the key issue. More to the point is that, as with the Brunner study, this argument seemed to encapsulate a whole style of thought concerning the role of genetic variation in generating behavioral phenotypes, not just violent and antisocial conduct but a whole range of psychiatric conditions and problematic conduct. Caspi and Moffitt made this explicit in a paper published four years later. Rather than proceeding by trying to link genetic variations to disorders, or even seeking to find endophenotypes, they suggested that the \"gene-environment interaction approach assumes that genes moderate the effect of environmental pathogens on disorders\" (Caspi and Moffitt 2006, 584). Given that we know a great deal about the environmental pathogens and risk factors for many mental disorders from depression to antisocial behavior, and heterogeneity of response between individuals characterizes even the most overwhelming of environmental traumas, we can now look to genetics to help us understand why some succumb and others do not: \"The hypothesis of genetic moderation implies that differences between individuals, originating in the DNA sequence, bring about differences between individuals in their resilience or vulnerability to the environmental causes of many pathological conditions of the mind and body.\" (Caspi and Moffitt 2006, 584). And they suggested that we should go beyond genetic epidemiology: \"Neuroscience can complement psychiatric genetic epidemiology by specifying the more proximal role of nervous system reactivity in the gene-environment interaction\" (ibid.).\n\nIn particular, Caspi and Moffitt's group focused on three genes where high and low activity variants could be linked to susceptibility or resilience. The first concerned variants of the MAOA genotypes and their link with susceptibility to childhood maltreatment, which we have already discussed. The second concerned the long and short versions of the promoter region of 5-HTT, the serotonin transporter gene, where individuals carrying one or two copies of the short version were believed to be more susceptible to depression following stressful life events (Caspi, Sugden, Moffitt, et al. 2003). The third was the influence of a polymorphism in the catechol-O-methyltransferase (COMT) gene, in which individuals carrying the valine allele were thought likely to develop severe psychiatric symptoms following cannabis use, while those with two copies of the COMT methionine allele seemed to exhibit no such adverse effects (Caspi, Moffitt, Cannon, et al. 2005).\n\nWe were, it seems, ready for a new relationship between psychiatric genetics and neuroscience, in which they would work together to identify the neurobiological pathways that accounted for the variations in susceptibility or resilience between individuals to environmental insults, in order to clarify \"[h]ow an environmental pathogen, especially one that is psycho-social in nature, get[s] under the skin to alter the nervous system and generate mental disorders\" (Caspi and Moffitt 2006, 589). Despite its neuromolecular form, the basic form of this argument is reminiscent of debates in the mid-nineteenth century on hereditary dispositions, predisposing and exciting causes, and the like in the inheritance of insanity and all manner of socially problematic conduct. So what, today, does this imply for understanding of, and intervention into, problems of antisocial and aggressive conduct?\n\nIn 2008, Joshua Buckholtz and Andreas Meyer-Lindenberg summed up current thinking on the neurobiology of antisocial conduct along the following lines (Buckholtz and Meyer-Lindenberg 2008). Antisocial and violent conduct is a major social problem, which also generates significant economic costs, and is hence an important target for government intervention. Such conduct runs in families, and a small proportion of families in any community account for a large proportion of its violent crime. Hence risk for antisocial aggression is transmitted intergenerationally. While twin studies have shown significant heritability, they have been unable to identify the specific genetic bases of risk. However, they say, Brunner's study began to clarify this, by demonstrating the link between MAOA deficiencies and human aggression. This link has been further demonstrated in rodents, where MAOA knockouts demonstrate increased aggression, which is correlated with elevated levels of serotonin and noradrenaline. MAOA has a key role in regulating release and clearance of serotonin and noradrenaline in monaminergic synapses in brain, suggesting that aggression is also regulated by such levels. Levels of MAOA at or before birth have long-term effects into adulthood. While the complete deficiency of MAOA in the population is rare, there are polymorphisms in the MAOA gene\u2014basically long and short versions of a VNTR (variable-number tandem repeat). This determines whether an individual has a low level of monoamine oxidase activity (MAOA-L) or a high level (MAOA-H). In vitro studies have failed to demonstrate the relevance of these differences for in vivo levels of MAOA-activity in adults; nonetheless, much research has sought to link these differences to aggression and impulsivity in animals and humans.\n\nBuckholtz and Meyer-Lindenberg note that direct association studies have been inconclusive but argue that gene-environment studies have shown a consistent relationship between MAOA-L and the risk of impulsive violence in those males who have experienced early-life adversity. Other studies in animals have also shown a link with variations in other genes that modulate serotonin levels in brain, such as the promoter region of the serotonin transporter gene. Neuroimaging studies are beginning to demonstrate that there are differences between MAOA-L and MAOA-H alleles in cortical circuitry relevant to emotional regulation, notably in the medial prefrontal cortex, which has a role in modulating amygdala function, which is well known to be crucial in fear extinction, emotion regulation, and temperamental variation; the medial prefrontal cortex is also well known to be modulated by serotonin levels. These variations in neural circuitry can be shown empirically to be associated with individual differences in temperamental traits, notably increased reactivity to threat cues; increased tendency to experience anger, frustration, and bitterness; and reduced sensitivity to cues for prosocial behavior. These alterations in corticiolimbic structure, function, and connectivity may underlie MAOA's role in genetic susceptibility for aggression as a result of the crucial role played by the implicated circuitry in social evaluation and decision making. Thus, they conclude, \"although MAOA is not a 'violence gene' per se, susceptibility alleles might bias brain development towards alterations in function and structure which, in combination with other factors, predispose the development of antisocial behavior.\" Yet they also admit that in the absence of such factors, inheritance of the low-activity version of MAOA \"is completely compatible with psychiatric health\" (Buckholtz and Meyer-Lindenberg 2008, 127).\n\nOnce again, we can see the logic of 'bioprediction' that is taking shape here: screen and intervene. This is not a search for the biological roots of crime but for the biological susceptibilities that increase the likelihood of manifesting undesirable behavior involving violent, impulsive, or aggressive conduct, and the apparent weakening of the conventional workings of guilt and remorse. The focus is on risk identification, intervention, precaution, and prevention. The neurobiological basis here is not conceived of as genes for crime or even genes for aggression, but rather as specific variations at the level of SNPs, in the sequences that are believed to code for the various neurobiological components of behavior control and the neurochemical anomalies that correlate with low impulse control or aggression. This genomic analysis is combined with imaging research on localization of functions, seeking to identify the brain regions or neural circuits activated in aggression or impulsivity, carried out using brain scanning techniques to identify characteristic patterns of brain activity. This neurobiological approach allies itself well with its sociopolitical milieu, for it has taken shape in a context in which crimes of this sort\u2014impulsive, aggressive, antisocial\u2014are understood not merely as infractions of the law but as problems of public insecurity and generators of economic burden. Within these control strategies of precaution, prevention, and preemption, the question shifts from that of response to the offense after the act, to programs of prediction and prevention that identify those at risk on the basis of a kind of algorithm that combines genetic and neurobiological factors with those relating to family life, parental behavior, poverty, housing, and other environmental factors. For these violent or impulsive behavior at least, crime, is reframed as a problem of public health.\n\nThe stage is set for the reinvention of strategies for governing childhood and family life in the name of crime prevention. On the one hand, we can find approaches that focus on the neurobiological underpinnings of those at risk of antisocial conduct or psychopathy and advocate screening to identify children at risk, initially using well-known scales, but hopeful of the possibility of identifying biomarkers in brains or genes. On the other, we can see approaches that seek to act upon the developing brain of the child who is at risk by shaping his or her interpersonal milieu; they aim to maximize the potential of each child by targeting those 'incredible years.' But, as we will see, strategies developed in the second approach merge with those proposed from the first\u2014and for both, the key actors are not the child, but the parents.\n\nNipping Budding Psychopaths in the Bud\n\nShould one screen children to identify those at risk of developing antisocial behavior or even psychopathy? In the 1990s, the case for this approach was made by Donald Lynam, first at the University of Kentucky and later at Purdue University (Lynam 1996, 1998; Lynam and Gudonis 2005). But if screeing is desirable, what methods might be appropriate? Population-level screening is controversial in many regions, for example, in the United Kingdom, though some argue for its cost-effectiveness and utility even at the preschool age, given certain conditions (Wilson, Minnis, Puckering, et al. 2009). In some areas of the United States, screening programs for children's behavioral problems using non-neurobiological forms of assessment have already been tried. The most controversial was the Texas Childhood Medication Algorithm Project (TCMAP)\u2014a development of the adult version (TMAP), which was designed to screen adults for major psychiatric conditions. The version for children proved particularly contentious, with claims that it was acting as a powerful marketing tool for pharmaceutical companies: it was suspended in September 2008. Other U.S. screening programs for children have included Teenscreen, employed for the early identification of mental illness in teenagers, notably screening for depression to reduce suicide risk.\n\nWhile there are a multitude of screening programs that seek to predict future violence on the basis of current behavior or symptoms, at the time of writing, there seem to be no screening programs in general use for antisocial conduct that involve biomarkers (genetic profiling, brain imaging, etc.) or endophenotypes, such as hormone levels. Indeed, as we discuss in chapter 4, despite the hopes raised by developments in genomics and in brain imaging, it has proved difficult, if not impossible, to develop validated and reliable biomarkers for _any_ psychiatric condition that can be used to make a diagnosis, even in a clinical context. Thus, those researching the possibilities of screening for future antisocial behavior have tended to rely largely on a behavioral assessment of features, notably those termed \"callous and unemotional traits\" (J. Hill 2003; Moran, Ford, Butler, et al. 2008), derived ultimately from Cleckley's _Mask of Sanity_ \u2014published in 1941 (Cleckley [1941] 1976), operationalized by Robert Hare in _The Hare Psychopathy Check List\u2014Revised_ (Hare 1991; cf. Hare 1999), and then reformatted for children (Frick and Hare 2001). But endeavors to go beyond the behavioral and the symptomatic, to diagnose through the brain itself, are certainly at the research stage.\n\nLet us take one extremely influential example to illustrate this line of thinking: the program of research funded by the United Kingdom's Medical Research Council, the Department of Health, and the Home Office, and carried out at the Institute of Psychiatry, King's College London (Viding, Blair, Moffitt, et al. 2005; Odgers, Moffitt, Poulton, et al. 2008; Viding, Jones, Frick, et al. 2008; Viding, Larsson, and Jones 2008). The research has argued as follows: First, that one particular subtype of antisocial behavior, that with \"callous and unemotional traits\" (AS-CU) is a precursor of adult psychopathy. Second, that the characteristics of childhood AS-CU and adult psychopathy map onto one another: in both children and adults we see lack of empathy, lack of guilt and remorse, and shallow affect and manipulative conduct, and in such children we can also see precursors of adult violence, such as cruelty to animals and enjoyment of aggression. Third, if we look at studies of psychopaths, psychological tests show them to be poor at empathizing with others and bad at recognizing their fear and sadness, and brain scans show them to have a disruption of one particular part of the brain, the amygdala, which is, so it seems, implicated in these affective defects. Fourth, if we take twin pairs aged around seven years, some monozygotic and some dizygotic, who have been assessed for AS-CU and carry out fMRI scans on them, we find the same deficits in the brain's \"affect circuitry\" in the AS-CU subjects as we see in adult psychopaths. Finally, if we use classical twin study methods, taking environmental factors into account, we find a much higher correlation of these brain patterns in the genetically identical twins than in nonidentical twins\u2014that is, the trait is highly heritable and suggests a strong genetic vulnerability. Hence, it is argued, we should search for the precise SNP-level variations that underlie this vulnerability. We should uses SNPs and scans to search for budding psychopaths.\n\nAs is now obvious, what these researchers are searching for is a vulnerability, a _susceptibility_. It is not surprising, then, that they are closely linked to the Caspi and Moffitt research group and hope to identify the gene-environment interactions that provoke vulnerability into frank psychopathy (Caspi, Mc-Clay, Moffitt, et al. 2002; Kim-Cohen, Moffitt, Taylor, et al. 2005; Odgers, Moffitt, Poulton, et al. 2008). Their message, they insist, is _not_ fatalism. This is how Viding and McCrory put it in a presentation to a 2007 conference commissioned by the United Kingdom's Department of Health; the Department for Children, Schools and Families; the Youth Justice Board; and the Cabinet Office: \"the goal of early identification is successful intervention.\" And the means to be used were multiple\u2014perhaps behavior therapy, cognitive therapy, and even psychopharmaceuticals\u2014for the aim is to reshape brain mechanisms in order to nip those budding psychopaths in the bud.\n\nIn fact, the path most emphasized is one that addresses the problem indirectly. The route to the problematic child, as so often in the past, is to be through the parents. For research now claims that these undesirable traits can be ameliorated by such means (Dadds, Fraser, Frost, et al. 2005; Hawes and Dadds 2005, 2007). That is to say, one can modulate the traits, and perhaps even their neural underpinnings, by training and 'empowering' parents\u2014or rather 'caregivers'\u2014in parenting skills. Perhaps, some say, what is required is \"multisystemic therapy\" that will improve parental disciplinary practices, increase family affection, divert children from antisocial peers to prosocial ones, improve schoolwork, engage children in positive recreation, improve family-community relations, and empower the family to solve future difficulties. But in any event, the familiar answer is that one should govern the child at risk by governing the risky family.\n\nSculpting the Brain in Those Incredible Years\n\nWho could question that the child is a most incredible human being? And that the child needs everything that he or she can receive in order to withstand the emotional and social challenges that life will inevitably bring, especially in the teenage years? And who could doubt the virtue of interventions that seek both to ensure and sustain resilience in the child, to prevent a future of delinquency, drug abuse, and violence, with all the accompanying misery to child, family, and victims, not to mention the economic costs? Thus, in the United States more than a decade ago, Mark Cohen calculated the various lifetime costs associated with a typical criminal career, and based on the fact that such careers start young, and in an attempt to provide a basis for the cost-effectiveness of early preventive intervention programs, estimated that the \"monetary value of saving a high-risk youth\" was between $1.7 million and $2.3 million (Cohen 1998, 5). In the United Kingdom, in 2006, Stephen Scott, Martin Knapp, and their colleagues estimated that individuals who show conduct disorder at the age of ten will, by the time they are twenty-eight, have \"cost society\" up to ten times more than controls without this diagnosis (Scott, Knapp, Henderson, et al. 2001; Romeo, Knapp, and Scott 2006). All seem agreed about the implications: surely we need strategies for early identification and intervention to safeguard our children from such miserable futures, and our societies from such a social and economic burden (Harrington and Bailey 2003; Kim-Cohen, Moffitt, Taylor, et al. 2005; Margo and Stevens 2008).\n\nNo wonder, then, that we see a whole array of individuals, pressure groups, and organizations seeking to promote early intervention in the name of prevention\u2014many of which can be found on the pages of the website Prevention Action. These programs are often pioneered by individuals with an almost evangelical zeal and commitment. There is Carolyn Webster-Stratton with her \"Incredible Years\" training series, which has expanded from its start in the United States to England, Wales, Ireland, Norway, Australia, and New Zealand. There is Clay Yeager, a onetime probation officer, turned Pennsylvania oilman, with his consultancy working to introduce evidence-based programs for prevention such as \"Nurse Family Partnerships\" and \"Communities That Care\" into child welfare and youth justice systems in Florida and several other U.S. states. There is Steve Aos in Washington, advising the legislature on issues such as \"the economic costs and benefits of investment choices in juvenile justice.\" Of course, much of this work does not explicitly refer to brains, let alone to neurobiology\u2014Webster-Stratton, for example, relies on theories of social learning\u2014but others, such as Tom Dishion, combine their practical work promoting the importance of early intervention with brain imaging research\u2014the article on his work found at Prevention Action is titled \"Putting Brain Science Back on the Streets of Los Angeles.\"\n\nIn the United Kingdom, this agenda was proposed most passionately by Iain Duncan Smith's Institute of Social Justice. Duncan Smith, a former leader of the Conservative Party, teamed up with Graham Allen, a member of Parliament from the Labour Party, to produce a report in 2008 titled _Early Intervention: Good Parents, Great Kids, Better Citizens_ (Allen and Duncan Smith 2008). After outlining the problems and costs of social dysfunction, violence, drugs, alcohol, and family breakdown, they argued that from birth through age three was the vital period when the right social and emotional inputs must be made to build the human foundations of a healthy, functioning society and that the key agents to provide those inputs were not the state but the parents. Why this age range? How many times have we heard this period emphasized in social policy, most recently in the arguments in the 1970s concerning the cycle of deprivation and the need for policies of intervention in the preschool years, such as the Head Start program in the United States? The report refers to many longitudinal studies to make its case that early neglect and lack of appropriate parental input leads to these later problems, and that youngsters who do not have the right kind of emotional and familial input at this age grow up to be antisocial adults who go on to raise antisocial children\u2014thus perpetuating the cycle.\n\nWhile the conceptual bases of Head Start and analogous schemes for preschool intervention in the United Kingdom were psychological, the rationale is now articulated in terms of the brain\u2014because the brain's capacity for change is believed to be at its greatest in this period: \"the human brain has developed to 85 per cent of its potential at age three (and 90 per cent at age four)\" (Allen and Duncan Smith 2008, 48). Because the human being is born with the brain underdeveloped, and its completion depends on input after birth, \"sculpting the young brain\" and programming the development of the baby from the first moment after birth is of the highest importance. \"The structure of the developing infant brain is a crucial factor in the creation (or not) of violent tendencies, because early patterns are established not only psychologically but at the physiological level of brain formation\" (ibid., 57). The mechanism is the growth and pruning of synapses in response to experience\u2014synapses that become \"hard-wired\" by repeated use.\n\nBut not only that\u2014the brains of deprived children are actually smaller, or so it is claimed: \"The brain of an abused or neglected child is significantly smaller than the norm: the limbic system (which governs emotions) is 20\u201330 per cent smaller and tends to have fewer synapses; the hippocampus (responsible for memory) is also smaller. Both of these stunted developments are due to decreased cell growth, synaptic and dendrite density\u2014all of which are the direct result of much less stimulation (e.g. sight, sound, touch) than is required for normal development of the brain\"; this claim is supported by the reproduction of images from Bruce Perry's work at the Child Trauma Academy (Perry 2002). The role of cortisol, neurotransmitter modulation, sensitive windows, and the whole repertoire of neurobiological accounts of child development is deployed to produce the objectivity effect that in previous periods was generated by reference to dynamic and developmental psychology. Even attachment, that stalwart of developmental psychopathology, is now inscribed in the brain: \"Babies who are healthily attached to their carer can regulate their emotions as they mature because the cortex, which exercises rational thought and control, has developed properly. However, when early conditions result in underdevelopment of the cortex, the child lacks an 'emotional guardian' \" (ibid., 62). It seems obvious that intensive intervention on early parenting, on the parents of today, and on those at risk of being the dysfunctional parents of tomorrow is the path required to break the cycle of antisocial and violent conduct that destroys lives and costs our societies so dearly. Who could disagree?\n\nWell, as it happens, in April 2010, this question of whether there was a link between child deprivation, brain development, and future social problems became rather controversial. Dr. Bruce Perry reproached Iain Duncan Smith\u2014who was about to become a minister in the Conservative-led coalition government that came to power in the United Kingdom in May 2010\u2014for using his brain scan images of children who had suffered extreme neglect to illustrate a more general claim that neglect or family breakdown changes brain development. Perry himself had written extensively on the neurodevelopmental consequences of ill treatment and neglect of children, notably opening one of his more cited pieces with the arresting phrase \"Children are not resilient, children are malleable\" (Perry 1997). In this article, he drew on neurobiological research and studies of neglected children, including the Romanian orphans who were discovered living in appalling conditions in Romania in the years following the fall of the Ceausescu regime in 1989, to argue for the malign consequences of early violence and neglect on different brain areas and functions, and in particular, claiming that the consequences were that those who had suffered in these ways were themselves more prone to violence and aggression in adolescence and adulthood as a result. Perry called for policies that provided parents with the resources and information to optimize their children's developing brains, an argument he developed in a number of other scientific publications (Perry 2002, 2008), and in a popular book arguing for the crucial role of empathy, its roots in the developing brain, and its social and personal consequences\u2014apparently under threat from current cultural and technological changes (Szalavitz and Perry 2010).\n\nThese disputes about whether one could extrapolate from the neurobiological consequences of cases of extreme neglect did not deflect those who argued for early intervention in the name of the brain. In the U.K. debate, Duncan Smith was backed by Camila Batmanghelidjh, charismatic founder of one of the country's leading early intervention charities, Kids Company, arguing that lack of care does affect brain development, and can be com-bated by intensive interactions in preventive interventions. A month later, Kids Company launched \"Peace of Mind,\" a major campaign to raise \u00a35 million to fund research into vulnerable children's brains. Supporters were asked to go online to access a virtual brain with one million neurons; for each \u00a35 they donated, they received a \"virtual neuron,\" and they were asked to invite their social networks to do the same\u2014the virtual neurons they purchased would connect together to form clusters in the virtual brain. Supported by a section of the website summarizing genetic and neurobiological research on child maltreatment and the brain, the conclusion was that \"[s]ustained child abuse and neglect has a devastating impact on a child's brain often creating a lifetime of vulnerability, tragically increasing the risk of suicide, violence, anti-social behavior, depression and drug addiction.\" This was, it seemed, a nonpartisan issue: in January 2011, Graham Allen, still a Labour member of Parliament, was asked by the Conservative British Prime Minister David Cameron to produce a report on early intervention. Allen used the very same brain images on its cover and deployed essentially the same brain-based arguments in his proposals for intensive early intervention for preschool children (Allen 2011).\n\nThese arguments for the benefits of early intervention seem hard to deny. As in earlier programs, such as Head Start, they have generated considerable research seeking to evaluate preventive programs for intensive intervention for those thought to be at risk. As we have noted, currently, children at risk are not identified using biomarkers but because either they, or their parents, have come to the notice of social workers, teachers, doctors, or other authorities. And the intervention programs do not use interventions targeted directly at neural processes. As we have seen, the key is thought to be the training of parents\u2014and teachers\u2014in managing the behavioral difficulties of their children. Consider, for example, the randomized controlled trial by Stephen Scott at the Institute of Psychiatry in London, evaluating an intervention using the Incredible Years program on six-year-old children identified by screening all children of this age at eight schools for antisocial behavior. The study concluded that, at the cost of \u00a32,380 ($3,800) per child, a twenty-eight-week program showed that by targeting multiple risk factors (ineffective parenting, conduct problems, ADHD symptoms, and low reading ability), and by training parents, \"[e]ffective population based intervention to improve the functioning of [children] with antisocial behavior is practically feasible.... Including halving the rate of oppositional defiant disorder... there is the prospect that children's long term mental health will be improved, including better school attainments and less violence\" (Scott, Sylva, Doolan, et al. 2010, 48, 56).\n\nBy way of the brain, then, we reach a conclusion that does not differ greatly from arguments reaching back to the late nineteenth century about the effects of the early years on later propensities to problematic conduct (Platt 1977; Donzelot 1979; Rose 1985). From Mary Carpenter's campaigns for colonies for dangerous and perishing children (Carpenter 1851), through the social problem group and \"the submerged ten percent\" in the early twentieth century, via the mental hygiene movement in the 1930s, and the arguments for setting up child welfare services in the years after the Second World War (Rose 1985), to the contentious concept of the \"cycle of deprivation\" in the 1970s (Joseph 1972; Jordan 1974), and the interventions from the Head Start program to the Sure Start program\u2014we find the repeated arguments that one should minimize a host of social ills, including criminal and antisocial conduct, by governing the child through its family. In each generation, unsurprisingly, these arguments are made on the basis of whatever happens to be the current mode of objectivity about the development of children\u2014habits, the will, instinct theory, psychoanalysis, and today the brain. Social justice, it seems, lies not in tackling the causes of structural inequality, poverty, poor housing, unemployment, and the like, but in managing parents in the name of the formation of good citizens.\n\nGoverning Antisocial Brains\n\nIt may seem odd to have concluded our discussion of antisocial brains not by debating questions of free will and criminal responsibility, but by exploring proposals to train parents in ways of rearing their young children. But if there is one key lesson that should be learned from the genealogy of crime-control strategies, it is that the criminal justice system itself has rarely been considered the most important strategic site for interevention (D. Garland 1985; Rose 1985). The most significant changes occurring today at the intersection of crime-control strategies and neuroscience do not take the form that concerns many neuroethicists. They do not entail a challenge to doctrines of free will in the courtroom\u2014despite the flurry of books claiming and discussing such a shift (Libet, Freeman, and Sutherland 1999; Dennett 2003a; B. Garland 2004; Spence 2009). Nor do they undermine the notion of the autonomous legal subject. Indeed, the reverse is perhaps closer to the truth\u2014strategies seek to create the kinds of persons who _can_ take responsibility for their actions, and they attempt to enhance such self-control by acting on the brain.\n\nDespite the concerns of those who raise neurobiological doubts about autonomy and free will, these kinds of interventions go entirely with the grain of folk psychology, with its long-standing emphases on the importance of the early years\u2014creating the adult through shaping the child. Indeed, they are entirely consonant with programs, strategies, and techniques since the second half of the nineteenth century that have sought (and usually failed) to address the problem of crime by early intervention to shape the family into a mechanism for creating civilized citizens (Donzelot 1979; Rose 1989; Parton 1991). Today, it is _brains at risk_ \u2014at risk of becoming risky brains\u2014that are to be addressed indirectly and preventively. As in earlier strategies, two senses of risk are brought into alignment. The first is the desire to identify _risky individuals_ \u2014that is to say, those who will present a future risk to others\u2014before the actual harm is committed. The second is the hope that one might be able to identify _individuals at risk_ \u2014those whose particular combination of biology and life history makes them themselves susceptible to some future condition\u2014here personality disorder, impulsivity, aggressivity, or whatever, but more generally susceptibility for any psychiatric disorder. And of the many options available once one has identified such children, the strategies that are emerging most forcefully, at least in the United States and the United Kingdom, are strategies of _optimization_ and strategies of hope\u2014strategies that aim, in the words of the Kids Company Peace of Mind campaign, to address both \"[t]he individual cost to the young person in terms of emotional wellbeing, mental and physical health, and education\" and \"[t]he cost to society of antisocial behavior in terms of police and criminal justice resources, public safety, NHS resources, and the continuation of the cycle of abuse and violence.\"\n\nIt is too early to say whether neurogenetics and brain imaging will play an important role in the screening programs to identify those at risk. And it is too early to say whether interventions directly addressing the brain, perhaps by drugs, will come to supplement or supplant those based on parent, teacher, and nurse training. But it is undoubtedly the case that the reference to neurobiological consequences of childhood neglect will continue to provide such programs with crucial support, that research into such consequences will expand, and that the brain will increasingly be seen to lie at the root of anti-social and dangerous conduct, even if its effects are manifested psychologically in poor impulse control and failures of the will. Whatever the fate of the high-profile endeavors to identify anomalies in the brains of adult murderers, or to discover the neural anomalies that might finally resolve disquiets over the category of psychopathy, we can certainly anticipate future strategies to govern antisocial citizens by acting upon their developing brains.\nChapter Seven\n\nPersonhood in a Neurobiological Age\n\nIn the course of centuries the _na\u00efve_ self-love of men has had to submit to two major blows at the hands of science. The first was when they learnt that our earth was not the centre of the universe but only a tiny fragment of a cosmic system of scarcely imaginable vastness. This is associated in our minds with the name of Copernicus, though something similar had already been asserted by Alexandrian science. The second blow fell when biological research destroyed man's supposedly privileged place in creation and proved his descent from the animal kingdom and his ineradicable animal nature. This re-evaluation has been accomplished in our own days by Darwin, Wallace and their predecessors, though not without the most violent contemporary opposition. But human megalomania will have suffered its third and most wounding blow from the research of the present time which seeks to prove to the ego that it is not even master in its own house, but must content itself with scanty information of what is going on unconsciously in its mind. We psycho-analysts were not the first to utter this call to introspection; but it seems to be our fate to give it its most forcible expression and to support it with empirical material which affects every individual.\n\n\u2014 **Sigmund Freud** , _Introductory Lectures on Psychoanalysis_ , 1916\n\nHumans are probably the only animals that have made conjectures about what goes on inside the body.... Today we are engaged in the quest of understanding our brain. When we have finally worked out the details, this remarkable organ, for all of its lofty pretensions\u2014seat of intellect, home of the soul\u2014will very likely join other remarkable pieces of biological machinery. Remarkable, certainly, but not mysterious or possessed of any supernatural qualities.\n\n\u2014 **Mark Bear and Leon Cooper** , \"From Molecules to Mental States,\" 1998\n\nSometime in the twenty-first century, science will confront one of its last great mysteries: the nature of the self. That lump of flesh in your cranial vault not only generates an \"objective\" account of the outside world but also directly experiences an internal world\u2014a rich mental life of sensations, meanings, and feeling. Most mysteriously, your brain also turns its view back on itself to generate your sense of self-awareness.\n\n\u2014 **V. S. Ramachandran** , _The Tell-Tale Brain: Unlocking the Mystery of Human Nature_ , 2011\n\nThe self an illusion created by our brain. This is the message of a host of recent semipopular books published by neuroscientists over the past decade. Not long ago, few would have had the temerity to approach this issue in such a direct manner. But now many neuroscientists are willing to take up the challenge posed by Freud at the start of the previous century: to find the neural bases of the processes that he lodged, in his view only provisionally, in the psyche (Pribram and Gill 1976). Like Freud, today's neuroscientists challenge the view that consciousness is the master in its own house and argue that much that is conventionally ascribed to the realm of the mental, of conscious deliberation and willed intention, actually occurs somewhere else\u2014now not in some immaterial unconscious but in the brain.\n\nIn one of the earlier and most provocative formulations, Francis Crick asserted: \"You, your joys and your sorrows, your memories and your ambitions, your sense of personal identity and free will, are in fact no more than the behavior of a vast assembly of nerve cells and their associated molecules. As Lewis Carroll's Alice might have phrased it: 'You're nothing but a pack of neurons' \" (Crick 1994). In similar vein, Patricia Churchland claimed: \"Bit by experimental bit, neuroscience is morphing our conception of what we are. The weight of evidence now implies that it is the _brain_ , rather than some non-physical stuff, that feels, thinks, and decides. That means there is no soul to fall in love. We do still fall in love, certainly, and passion is as real as it ever was. The difference is that now we understand those important feelings to be events happening in the physical brain\" (Churchland 2002, 1).\n\nRather less boldly, this is how Joseph LeDoux opens his book _The Synaptic Self_ : \"The bottom-line point of this book is 'You are your synapses.' Synapses are the spaces between brain cells, but are much more. They are the channels of communication between brain cells, and the means by which most of what the brain does is accomplished\" (LeDoux 2002, ix). For the psychologist-turned-neuroscientist, Chris Frith, in his book _Making Up the Mind: How the Brain Creates Our Mental World_ , it is the brain that \"creates the world,\" producing not only \"the illusion that we have direct contact with objects in the physical world\" but also that \"our own mental world is isolated and private\": \"through these two illusions we experience ourselves as agents, acting independently upon the world\" (Frith 2007a, 17). For the philosopher Thomas Metzinger, writing under the title _The Ego Tunnel: The Science of the Mind and the Myth of the Self_ , despite the way we think and talk, \"nobody has ever _been_ or _had_ a self\": conscious experience is a creation of our brains, an internal construct in what he terms \"the ego tunnel,\" which also creates \"an internal image of the person-as-a-whole,\" which is \"the phenomenal ego, the 'I' or 'self' as it appears in conscious experience.\" But \"you are constitutionally unable to realize that all this is just the content of a simulation in your brain... not reality itself but an image of reality... a complex property of the global neural correlate of consciousness\" (Metzinger 2009, 1\u201312).\n\nHow should we assess such claims? Many from the social and human sciences regard this neurobiologization of the self as the most challenging feature of contemporary neuroscience. It seems to threaten the very conception of the human being that lies at the heart of their work: the idea that personhood is a matter of internal mental states\u2014consciousness, intention, beliefs, and the like\u2014existing in a uniquely human psychological realm of mind, embodied in a self-conscious subjectivity, and created in a world of meaning, culture, and history. Thus the French sociologist Alain Ehrenberg, writing in 2004, suggested that some neuroscientists were going beyond explorations of the neuronal machinery, and hopes for a new neuropsychiatry, to argue for a philosophical identity between knowledge of the brain and knowledge of the self: the \"sujet parlant\" was giving way to the \"sujet c\u00e9r\u00e9bral\"\u2014a \"cerebral subject\" (Ehrenberg 2004).\n\nThe \"Brainhood Project,\" established by Fernando Vidal and colleagues in 2006, took up this idea: \"Starting in the second half of the 20th century, the quality or condition of being a brain ranked as the defining property of human beings.... From science fiction in writing and film to neurophilosophy and the practices of intensive care and organ transplantation, humans came to be thought of not merely as having a brain, but as being a brain. Of course, many other properties of human persons were recognized. Nevertheless, whole or in part, the brain emerged as the only organ truly indispensable for the existence of a human self and for defining individuality.\"\n\nIn this chapter we take a different approach. The conception of the person that is emerging in neurobiology does not quite match the version attacked by critics from the _Geisteswissenschaften_. Indeed, neurobiological arguments in this area are beset with slippages and confusions, both linguistic and conceptual, between different notions of person and self, of consciousness and its relation to selfhood, of identity and personhood, of the relevance of questions of will and intention to self and consciousness, and much more. Further, the social and human sciences have not been so committed to the psychological ideal of the person or the self as is sometimes suggested. Before turning to neuroscience, then, it is worth beginning by reminding ourselves of some earlier challenges from within the human sciences to the idea of the bounded, conscious, constitutive self.\n\nThe Challenged Self\n\nIn our 'individualist' times, almost all aspects of political, ethical, and cultural thought hinge upon the things we believe about persons: their nature, their capacities, their freedoms and self-actualization, their rights and duties. These ideas about ourselves are central to debates about law, morality, justice, and the rights of minorities; about sexual relations, lifestyles, marriage, parenthood, and bioethical issues related to the beginning and end of life; about consumption, tastes, marketing and economic life; about politics and conflicts over national identities; and much more. They are entangled with other notions such as autonomy, agency, individuality, control, choice, self-realization, self-esteem, free will, and liberty. Human beings who inhabit our contemporary Western cultures understand themselves in terms of a kind of 'inwardness,' as endowed with a personal interiority, to which only they have direct access, in which lies their deepest truths and their truest thoughts, feelings, and wishes. It is this inwardness, inscribed in memories, capacities, and habits, that assures human beings that they _are_ \u2014that they have an enduring identity rather than being an ever-changing matrix of fleeting thoughts, feelings, and desires. This individualized inwardness, with its interiority and its continuity, is both unique to each and common to all (it was not always so, consider slaves, women, 'primitives,' children, idiots, the mad): it grounds the claim of each individual to membership in the human race and to the consideration that such membership implies.\n\nIn the 1970s and 1980s, many social historians, anthropologists, and sociologists turned their attention to the question of 'the person' and 'the self' and their historical and cultural variability. In the last essay he published\u2014delivered in French as the Huxley Memorial Lecture in 1938\u2014Marcel Mauss asserted his belief that \"there has never existed a human being who has not been aware, not only of his body, but at the same time also of his individuality, both spiritual and physical.\" But, he argued, the _notion_ or _concept_ of the person or the self was \"a category of the human mind\" that had evolved over time, and has taken a different form in different societies, according to their \"systems of law, religion, customs, social structure and mentality\": the \"cult of the self\" and the respect for the self was very recent (Mauss [1938] 1985, 3). Almost four decades later, Clifford Geertz argued something similar, writing that \"[t]he Western conception of the person as a bounded, unique, more or less integrated motivational and cognitive universe, a dynamic center of awareness, emotion, judgment and action, organized into a distinctive whole and set contrastively against other such wholes and against a social and natural background is, however incorrigible it may seem to us, a rather peculiar idea within the context of the world's cultures\" (Geertz 1974, 31).\n\nIn the years that followed, in the growing literature on the historical and cultural variability of conceptions of personhood and ideas of the self, most historians, philosophers, anthropologists, and sociologists agreed with Mauss and Geertz, as did many social psychologists engaged in comparative research on 'self-concepts' in different cultures (Dodds 1968; Shweder and LeVine 1984; Marsella, De Vos, and Hsu 1985; Roland 1988; Sampson 1988; Markus and Kitayama 1991). In different ways, they argued that ideas of the self or the person in Japanese society, among the native people of North America, in ancient Greece, in premodern Europe were different: they did not share our sense of personal uniqueness, of individual autonomy, of bodily integrity and biographical continuity. The contemporary Western conception of the self\u2014individualized, bounded, with interior depth and temporal continuity, self-possessed, autonomous, free to choose\u2014was not natural, given, or universal, it was a historical and cultural achievement.\n\nMore radically, some social theorists returned to Sigmund Freud and to the blow that psychoanalysis struck to human narcissism, arguing that the sense of oneself as having full insight into one's own interior psychological world, let along mastery over it, is an illusion. As Freud put it: \"the ego... is not even master in its own house, but must content itself with scanty information of what is going on unconsciously in its mind\" (Freud [1916] 1953\u201374, 285). Jacques Lacan, interpreting Freud in the spirit of French postwar structural linguistics, asserted that the self is radically excentric to itself: \"the philosophical _cogito_ is at the centre of the mirage that renders modern man so sure of being himself,... but yet I am not what I think I am or where I think I am\" (Lacan [1957] 1977, 165, 171). A critical evaluation of a certain narcissistic mirage, image or 'ideology' of the universal, individuated, self-aware, self-conscious person, as both eternal fact and unimpeachable value, thus lay at the heart of much critical social thought in the closing decades of the twentieth century (Rose 1996).\n\nIn this light, the character of the current debate between the human sciences and the neurosciences seems paradoxical. While the human sciences have been happy to accept that our current conception of the 'naturally' autonomous, self-conscious person is an artifact of history, culture, meaning, and language, and that consciousness is far from sovereign in human affairs, most react with hostility when practitioners of the new sciences of the brain announce, usually with an irritating sense of daring, the death of our illusions about this same conscious, self-identical, autonomous person. It would seem prudent, therefore, to abstain from both celebration and critique, in order to explore the extent to which a new image of the self or person is taking shape, where, and with what consequences.\n\nIn his magisterial survey, _Sources of the Self_ , the philosopher Charles Taylor traced the formation of the notion of the self, our modern sense of self, through the canonical works of Western philosophy (Taylor 1989). He argued that while there were deep and distributed supports in contemporary culture for the forms of thought that underpin psychological notions of selfhood\u2014autonomy, depth, the capacity to choose and bear responsibility for those choices\u2014this was nonetheless a function of a historically limited mode of self-interpretation, which had spread from the West to other parts of the globe, but which had a beginning in time and space and may have an end.\n\nFrom a different perspective, that of the genealogy of personhood, we come to a similar conclusion. But, we suggest, we need to look beyond philosophy if we are to discern the signs of any such 'end' of this mode of self-interpretation today. For philosophers do not live in a separate world, at least when it comes to arguments concerning human nature or ideas of self and identity. It is not only that their arguments need to be located 'in context' as the saying goes (Tully 1988, 1993), but that philosophers draw, sometimes explicitly but usually implicitly, upon the empirical knowledge and cultural beliefs of the times and places they inhabit. Their apparently abstract considerations of the nature of humans have been bound up in all kinds of interesting ways with endeavors to shape, train, and govern those human beings, both individually and collectively. Descartes drew upon his experiences witnessing dissections and on contemporary anatomical knowledge of the body and brain to articulate his ideas for the location of the soul and his doctrines of healthy living; Locke, Condorcet, and others drew upon images and stories of wild children in their thought experiments about the existence of innate ideas and the role of the sensations (Rose 1985). Despite their often expressed disdain for 'folk psychology,' philosophers since at least the mid-nineteenth century have, explicitly or implicitly, taken up ideas from the knowledges about human beings articulated in the positive sciences\u2014biology, psychology, neurology, and the brain sciences\u2014even when apparently in dispute with them.\n\nIf we are considering the transformations immanent in our present, then, we would do better to turn our gaze away from the great thinkers of our age and to explore the changes in language, technique, and judgment that arise in more mundane settings, in the face of more pragmatic concerns about the government of conduct. That is to say, we should examine mutations in knowledges with a lower epistemological profile\u2014not so much in philosophy as in the human sciences and indeed in neuroscience itself. As for neuroscience, we will find once more, to paraphrase Canguilhem, that when it comes to matters of our ontology, while the normal seems prior to the abnormal, it is historically secondary. Again and again, in what follows, we will see the extent to which it is through the analysis of pathology\u2014accidental, temporarily induced, or experimentally produced\u2014that neurobiology has made its assault on the question of the mind and the self.\n\nFrom the Pathological to the Normal\n\nEarly on in his genial book _Making Up the Mind: How the Brain Creates Our Mental World_ , Chris Frith wrestles with the question of dualism and materialism. \"There can,\" he writes, \"be changes in the activity in my brain without any changes in my mind. On the other hand I firmly believe that there cannot be changes in my mind without there also being changes in brain activity\"\u2014a sentence that he annotates with a footnote declaring, \"I am not a dualist\" (Frith 2007a, 23). However, he immediately follows this with the following somewhat dualist pronouncement that \"everything that happens in my mind (mental activity) is caused by, or at least depends upon, brain activity\"\u2014a sentence that he annotates with a footnote declaring, \"I am a materialist. But I admit that I sometimes sound like a dualist\" (ibid.). He says that he talks of \"the brain 'not telling me everything it knows' or 'deceiving me'... because this is what the experience is like\" (ibid.). So we can ask questions like \"How does my brain know about the physical world?\"\u2014that is to say, we can ask what he terms \"a mind question\" as if it was a brain question. It is not clear how far this gets us in resolving those complex and slippery ways in which we are coming to shuffle the ways we speak of minds, brains, bodies, selves, persons, conscious intentions, unconscious decisions, and so forth. Frith's book is conducted as an imaginary debate with a mythical professor of English who points out the obvious error that brains can no more 'know' things than an encyclopedia that contains information about the world 'knows' about the world: even if the information is encoded in neurons rather than letters, we still seem to require some entity to 'read' it.\n\nBut nonetheless, his book is devoted to demonstrating that it is indeed the brain, not the mind, nor the conscious self, that creates the mental world in which humans live, including their capacity to act, interpret the actions of others, and inhabit a realm of meaning and culture. The route to this conclusion is an experimental one, making use of evidence from neuropathologies, or of the creation in the laboratory of some kind of artificial pathology, where damage to an area of the brain creates anomalies in what seems to be the normal experience of selfhood. In each case, that is to say, the technology for generating evidence depends upon breaching of the normal ways in which we inhabit our worlds. Breaching here, not in the sociological sense, where breaching is a method to render visible the taken for granted rules and presuppositions that make the micropractices of everyday life possible. But breaching the usual relations that hold between the brain and the world. And in so doing, the accidents and experiments that are drawn upon seem to demonstrate that certain central capacities of normal personhood are, in fact, creations of nonconscious neural mechanisms and that it is these nonconscious mechanisms that create the illusions within each individual's self-awareness\u2014the illusions of coherent, conscious, self-controlling capacities that are attributed to the normal person.\n\n**Lessons from Lesions**\n\n\"By studying patients... who have deficits and disturbances in the unity of self, we can gain deeper insights into what it means to be human\" (Ramachandran 2011, 288). Perhaps it is obvious that the pathway from mind to brain should begin with lesions. What could give a clearer indication of the role of brain in mental life than the lacks, distortions, and other strange consequences of physical damage to particular regions of the brain? As we have seen in previous chapters, contemporary histories of brain localization ritualistically invoke Phineas Gage and his unfortunate accident with that tamping iron (Macmillan 2000), and our current knowledge of functional brain regions still bears the names of neurologists (Broca, Wernicke, Fleschig) who studied lesions to cognitive functions such as speech and language. The 1987 edition of the translation of the Soviet neurologist, A. R. Luria's _The Man with a Shattered World_ (Luria 1973), the account of a man who loses his speech ability following a brain trauma during the Second World War, was introduced by Oliver Sacks, whose own romantic stories of the strange, sometimes wonderful, sometimes poignant worlds of those with damaged brains have become so popular (Sacks 1976, 1984, 1985, 1991). And yet, at least up to the 1990s, these authors did not frame their research, their findings, or their narratives in the way that has become routine today: in terms of the lessons that we could learn from such episodes about the ancient problem of the relations between mind and brain.\n\nWhat, then, are the contemporary neurobiological 'lessons from lesions' about the self\u2014the study of what the neurologist and psychiatrist Todd Feinberg terms \"neuropathologies of the self,\" where \"a brain lesion causes a profound and specific alteration in the patient's personal identity or personal relationships between the self and the world\" (Feinberg 2011, 75)? We can start with the famous split-brain experiments dating back to the early 1960s conducted by Michael Gazzaniga, with Roger Sperry, Joseph Bogen, and P. J. Vogel (the history of these experiments is reviewed in Gazzaniga 2005). These _callostomies_ involved severing the corpus callosum in human patients suffering from intractable epilepsy in an attempt to control the seizures. While it was initially claimed that they had no cognitive effects in humans, Roger Sperry and his colleagues showed that similar operations in animals led to very severe and profound deficits. Much research followed on the consequences in humans, for instance, in relation to the hemispheric localization of different aspects of language, vision, and causality, but only one aspect is relevant here\u2014that bearing on issues of the self.\n\nWhen a patient has his or her corpus callosum severed, Gazzaniga asks, \"does each disconnected half brain have its own sense of self... could it be that each hemisphere has its own point of view, its own self-referential system that is truly separate and different from the other hemisphere?\" He argues that there is indeed a lateralization, although not a complete one, in which \"the left hemisphere is quick to detect a partial self-image, even one that is only slightly reminiscent of the self, whereas the right brain needs an essentially full and complete picture of the self before it recognizes the image as such.... [T]he data indicate that a sense of self arises out of distributed networks in both hemispheres\" (Gazzaniga 2005, 657, 658). But however it may be affected by the operation, Gazzaniga's premise is clear: that it is indeed the brain that, through its patterned networks of neurons, produces the sense of self.\n\nOthers draw their evidence for the neural creation of the illusion of self from disturbances concerning the boundaries and components of the body. One that is regularly cited is the 'rubber hand' phenomenon (see, e.g., Metzinger 2009). Here individuals observing a rubber hand will sometimes attribute sensation to that hand rather than to their own. For example, experimental subjects will make this false attribution if they see the 'alien hand' being stroked by a brush while their own equivalent hand is stroked in the same way but is hidden from sight. After a few minutes, they will 'feel' the stroking on the rubber hand, even though it is separate from their body, rather than in their own, hidden, hand. It seems that the brain has attributed the sensation to a physically distinct object within its field of vision, and in the process, it has somehow incorporated that alien object into the body (Botvinick and Cohen 1998; Tsakiris and Haggard 2005).\n\nAnother lesson along these lines comes from the phenomenon of 'phantom limbs.' It has long been recognized in the literature that patients who have had a limb amputated often continue to experience sensations from that missing limb, sometimes believing that they can still move the absent hand or fingers, or that the amputated limb is present but locked in space, or, worse, feeling excruciating pain arising from that absent part of the body. This evidence shows, suggests Frith, that these are \"phantoms created in the brain\"; a conclusion that is further strengthened by the reverse situation, where individuals who have suffered damage to a particular area of the brain feel that they have had a limb amputated\u2014that they have 'lost' their hand, arm, leg, or foot. This shows, for those who think like Frith, that \"you do not have privileged access to knowledge about your own body\"\u2014that our feeling of effortless conscious control over our physical being in the world is an illusion created by the brain by \"hiding from us all the complex processes that are involved in discovering about the world\" (Frith 2007a, 72, 81).\n\nHence, as Gazzaniga and Cooney put it, neurological disorders illuminate the structure of human consciousness: \"subjective awareness emerges from the interactions of specialized, modular components in a distributed neural network. In addition, the activity of such components is united cognitively by an interpretive process that occurs in the left hemisphere of the human brain.... [T]he specialized nature of the component neural processors suggests a potential explanation for how specific lesions can selectively alter or eliminate the contents of particular aspects of subjective experience, with no resulting experience of deficit for the brain-damaged individual\" (Cooney and Gazzaniga 2003, 164). These, then, are the lessons from lesions.\n\n**Irreal Visions of a Real World**\n\nFrith, provocative as ever, titles chapter 5 of his book \"Our Perception of the World Is a Fantasy That Coincides with Reality.\" \"The remarkable thing about our perception of the physical world in all its beauty and detail is that it seems so easy.... But this very experience that our perception of the physical world is easy and immediate is an illusion created by our brains\" (Frith 2007a, 111). He points to various contributions to this way of understanding the work of the brain\u2014information theory, for example, which shows that messages can be recreated from physical events by eliminating noise and errors, and by the use of redundancy. He also notes the evidence that expectations and prior knowledge shape perception and that some of this prior knowledge is \"hard wired into the brain through millions of years of evolution\"\u2014for example, that light comes from above (the sun), and so our interpretation of patterns of shadows on objects interprets them as convex or concave on that assumption, as many optical illusions show (ibid.).\n\nThen there is Bayesian theory and what it tells us about the ways the brain creates beliefs or models of the world on the basis of successive predictions that are then compared with the patterns of activity detected from the senses, predictions that are linked to our actions in the world and the feedback they provide. What I perceive, then, is not the world, but \"my brain's model of the world,\" which combines the ambiguous cues from all my senses with a picture built up by past experience to generate predictions of what ought to be out there in the world, which I constantly test and refine through my actions (Frith 2007a, 132). This, it seems, is what is demonstrated by all those familiar optical illusions, from the Ames room through the Necker cube to the Rubin vase figure: \"our perceptions are fantasies that coincide with reality\"\u2014or perhaps we should say, that \"usually\" coincide with reality. It is not so much a question, says Frith, of whether the brain's model of the world is true, but whether it works\u2014enables us to take appropriate actions in order to survive. And all this happens effortlessly, outside of any consciousness. Our perception of the physical world is, in short, a fantasy constructed by our brains. And as for the physical world, so for the mental world\u2014we create models of the mental world of others in the same way, recognizing the motion of dancing figures from patterns of spots of light alone, inducing the goals and intentions of others into their movements and the direction of their eyes, imitating others, actually or mentally, and automatically imputing to them the same intentions as we ourselves would have if we were acting in this way (the line of thought extrapolated in the rhetoric of mirror neurons discussed in chapter 5) and much more.\n\nFrith, as we have noted, conducts his book as an imaginary debate with a skeptical professor of English, into whose mouth he places the supposed objections from the human sciences. As one would anticipate, she is distressed by the absence of meaning, language, and culture from these accounts, and the attribution to brains of capacities that should properly only be attributed to persons\u2014thinking, feeling, wanting, knowing, and so forth. But, beyond such objections, how should the human sciences respond to the fundamental proposition that the world we humans live in is not an immediate perception of some external reality? As we have seen, this has, in fact, been the founding presumption of much contemporary social thought, enabling a historical and cultural interrogation of both the form and the function of changing beliefs about persons and selves. If one was of a radical political bent, one could look to the French Marxist philosopher Louis Althusser, to whom the autonomous self is the central element in contemporary ideology, \"a representation of the imaginary relationship of individuals to their real conditions of existence\" (Althusser 1971, 153). Or if one was of a more philosophical bent, one could look to the American philosopher Nelson Goodman's notion of \"irrealism\"\u2014which he describes as a kind of radical relativism with rigorous constraints, in which many world versions\u2014some conflicting and irreconcilable with others\u2014are equally right (Goodman 1978, 1983). But it seems that, in the matter of the creation of worlds, dethroning the self-possessed self and its illusions of reality by culture and language is one thing, and dethroning the perceptual capacities of the self-possessed self by the nonconscious, automatic, evolutionarily shaped functioning of neurons and their hard- and soft-wired interconnections is quite another.\n\n**Libetism**\n\nIn 1983, Benjamin Libet and his colleagues published an article in _Brain_ titled \"Time of Conscious Intention to Act in Relation to Onset of Cerebral Activity (Readiness-Potential): The Unconscious Initiation of a Freely Voluntary Act\"\u2014an article that has become a standard reference for almost every account of the implications of neurobiology for mind-brain relations and the idea of free will. Perhaps it is not too much to describe the way of thinking that was engendered by this article as _Libetism_. What then is Libetism?\n\nLibet's experiment relied on a finding that a certain pattern of brain activity in prefrontal motor areas that is termed the \"readiness-potential\" (RP), which can be measured by electrodes placed (externally) on the scalp, had been observed to occur up to a second or more before an individual is aware of having initiated an act. Libet and his colleagues worked with six college students, divided into two groups of three, each sitting in a \"partially reclining position in a lounge chair\" with recording equipment attached to their heads and a cathode ray oscilloscope in front of them into which they were to stare without blinking. They instructed each subject \"when he felt like doing so, to perform the quick, abrupt flexion of the fingers and\/or the wrist of his right hand.\" In a second experiment, they instructed each subject \" 'to let the urge to act appear on its own at any time without any preplanning or concentration on when to act,' that is, to try to be 'spontaneous' in deciding when to perform each act; this instruction was designed to elicit voluntary acts that were freely capricious in origin\" (Libet, Gleason, Wright, et al. 1983, 625). The time of initiation of the RP was recorded using the monitoring equipment attached to the scalp.\n\nWhy were the subjects asked to stare without blinking at the oscilloscope? Well, this had a revolving beam like the second hand of a clock, with spots marked on the circumference, and the subjects had to remember, and subsequently report, the \"clock position\" on that oscilloscope. In some trials each had to report the position at \"the time of appearance of his conscious _awareness of 'wanting' to perform_ a given self-initiated movement. The experience was also described as an 'urge' or 'intention' or 'decision' to move, though subjects usually settled for the words 'wanting' or 'urge.' \" In other trials they had to report \"the time of subject's _awareness_ that he\/she _'actually moved'_ \" or, in a third series, where the movement was triggered by an external stimulus, the \"time of _awareness of the sensation_ elicited by the near-threshold stimulus pulse to the back of the hand, delivered at randomly irregular times unknown to the subject\" (Libet, Gleason, Wright, et al. 1983, 627). Of course, there were many other technical details and much statistical analysis of the results. And the conclusion?\n\n\"It is clear that neuronal processes that precede a self-initiated voluntary action, as reflected in the readiness-potential, generally begin substantially _before_ the reported appearance of conscious intention to perform that specific act. This temporal difference of several hundreds of milliseconds appeared fairly consistently\" (Libet, Gleason, Wright, et al. 1983, 635). Thus, it seems, \"the brain decides\" to act before awareness of, or conscious initiation of, that act:\n\nSince onset of RP regularly begins at least several hundreds of milliseconds before the appearance of a reportable time for awareness of any subjective intention or wish to act, it would appear that some neuronal activity associated with the eventual performance of the act has started well before any (recallable) conscious initiation or intervention could be possible. Put another way, the brain evidently \"decides\" to initiate or, at the least, prepare to initiate the act at a time before there is any reportable subjective awareness that such a decision has taken place. It is concluded that cerebral initiation even of a spontaneous voluntary act, of the kind studied here, can and usually does begin _unconsciously._ The term \"unconscious\" refers here simply to all processes that are not expressed as a conscious experience; this may include and does not distinguish among preconscious, subconscious or other possible nonreportable unconscious processes.... These considerations would appear to introduce certain constraints on the potential of the individual for exerting conscious initiation and control over his voluntary acts. (Ibid., 640)\n\nSo, if the brain decides to act prior to conscious initiation of an act, where then lies the possibility for free will\u2014a retrospective rationalization of the act, giving the subject the illusion that he or she has actually consciously willed it, whereas in fact it was the brain that decided? Libet and his colleagues allow two ways out\u2014the subject might decide to veto or abort the performance\u2014for both the RP and the awareness of the decision appeared _before_ the actual movement in question. Or \"[i]n those voluntary actions that are not 'spontaneous' and quickly performed, that is, in those in which conscious deliberation (of whether to act or of what alternative choice of action to take) precedes the act, the possibilities for conscious initiation and control would not be excluded by the present evidence\" (Libet, Gleason, Wright, et al. 1983, 641). This is what we have termed _Libetism_.\n\n\"The result,\" observes Frith, \"had such a vast impact outside psychology because it seemed to show that even our simplest voluntary actions are predetermined. We think we are making a choice when, in fact, our brain has already made the choice\" (Frith, 2007a, 67). Of course, to most people who pause to consider this experiment, the idea that this tells us anything about the exercise of human will in any of the naturally occurring situations where individuals believe they have made a conscious choice\u2014to take a holiday, choose a restaurant, apply for a job, vote for a political party, approach a potential girlfriend or boyfriend, write a letter of complaint to some authority\u2014is absurd. Not only is the task involved in Libet-like experiments trivial, and the perspective of the subjects of the experiment on their role ignored, but the results simply meant that those subjects reported their awareness of that decision\u2014the \"urge\"\u2014a few milliseconds after a brain change occurred\u2014a fact that seems to be about the process of reporting rather than the process of deciding. Actually, one has to be a strange kind of dualist, first to distinguish, and then to reconnect these two in a relation of causality. And, of course, even experimental psychologists find flaws in the welter of conclusions about free will that Libet and others built on such findings. For example, Frith points out that, to put it mildly, the freedom of these subjects was somewhat constrained\u2014they had a task to perform and an experimenter to satisfy: \"The participants are not actually making free choices about action. They are playing a complex game with the experimenter\" (ibid., 187).\n\nAlthough current neurobiological understandings of volition are significantly more complex and subtle (see Haggard 2008 for an excellent review), Libetism is alive and well, especially in debates with neuroscientists and philosophers about the idea of free will (Frith 2002; Wegner 2002). When the minds (or brains) of those who think like this turn to the implications for the law and legal notions of responsibility, Libetism is often coupled with some of the other arguments we have reviewed in this section and presented without the qualifications that Libet himself makes. Here, for instance, was the British neuropsychiatrist Sean Spence, whose brain presumably initiated the writing of this in the mid-1990s:\n\nThe purpose of this paper is to demonstrate that a conscious free will (in the sense of consciousness initiating action) is incompatible with the evidence of neuroscience, and the phenomenology described in the literature of normal creativity, psychotic passivity, and the neurological syndrome of the alien limb or hand. In particular the work of Libet and others who have directly stimulated the brain and measured activity from its surface leads to the conclusion that subjective states (be they sensory or intentional) are preceded by predictive neural activity... \"decisions\" to act arise prior to our conscious awareness of them. Thus our \"decision\" or \"freedom\" is illusory (if by these terms we mean conscious phenomena). It occurs after activity has been initiated.... The decision does not make itself. It is a product or correlate of preceding neuronal events. (Spence 1996, 75, 83\u201384)\n\nMichael Gazzaniga is a little more cautious, writing in _Neuron_ in 2008 under the title \"The Law and Neuroscience\":\n\nOne thing we are certain of is that the \"work\" in the brain happens before we are consciously aware of our mental struggles. Researchers have, since as early as 1965, advanced our understanding of the fact that much of the work is done at the subconscious level (Kornhuber and Deecke, 1965; Libet et al., 1979; Soon et al. 2008). A decision, for example, can be predicted several seconds before the subject consciously decides. If it is simply the brain, working up from its unconscious neural elements, that causes a person to act (even before he or she is aware of making a decision), how can we hold any person liable for his or her mental decisions? To hold someone responsible for his or her actions, one must find a \"there\" there. Is a little guy pulling the levers in your head producing a free-floating you? Modern neuroscience, of course, tells us the answer is \"no.\" (Gazzaniga 2008b, 413)\n\nWhere, then, does this leave the question of the self and the person? In the brief epilogue at the end of his book, Frith turns his mind, or his brain, to this issue. With all this talk of the brain, and how much it achieves without any conscious or willed engagement, without thought or awareness, how is one to account for the feeling of \"I\"\u2014that I think, that I intend, that I act? Perhaps there is an area in the brain that corresponds to this \"I\"ness\u2014but surely not a kind of homunculus, for that would merely be to replicate the question that it was intended to solve. No, if the brain does so much that we usually attribute to consciousness, then the perception that one is a conscious \"I\"\u2014and that others are also such \"I\"s\u2014must itself be produced by the brain as a result of evolution: \"This is my brain's final illusion:... [to] create an autonomous self.\" But he goes on to tell us that his is not a book about consciousness, although, as he points out \"[a]fter the age of about 50, many neuroscientists feel they have sufficient wisdom and experience to set about solving the problem of consciousness\" (Frith 2007a, 189). He adds the acerbic footnote: \"Whether or not they have ever done any experimental work on the topic.\"\n\nPerhaps it would be unfair to think of Thomas Metzinger in the context of this remark, for although he is a theoretical philosopher, he works closely with cognitive neuroscientists. Nonetheless, his argument is put provocatively: human beings \"do not have selves.\" He argues that \"we live our conscious lives in our own Ego Tunnel, lacking direct contact with outside reality but possessing an inward, first person perspective.\" This ego tunnel, which has evolved for controlling and predicting our behavior and understanding the behavior of others, creates a robust feeling of being in contact with an outside world, and a sense of immediate contact with our selves\u2014\"the self is not a thing but a process... we are 'selfing' organisms: At the very moment we wake up in the morning, the physical system, that is\u2014ourselves\u2014starts the process of 'selfing' \" (Metzinger 2009, 207\u20138). But do those who have done the experimental work on this topic share this view of the constitutive illusion of the human self?\n\nThe Self: From Soul to Brain\n\n\"An Ape with a Soul\" is the title that V. S. Ramachandran gives to the penultimate chapter of his popular book _The Tell-Tale Brain: A Neuroscientist's Quest for What Makes Us Human_ (2011). It opens it with an account of a patient called Jason, who suffered a serious head injury in a car accident, followed by a long period in a semiconscious state of akinetic mutism. Jason seemed alert, could move his eyes, but could not recognize anyone, talk, or comprehend speech\u2014except in one context. If his father, whom he neither recognized nor spoke to directly, telephoned him from another room, he became alert and talkative on the phone. For Ramachandran, Jason was a person when he was on the phone but not otherwise\u2014a person, he suggests, requires \"the ability to form rich, meaningful metarepresentations, which are essential not only to our uniqueness as a species, but also our uniqueness as individuals and our sense of self\" (Ramachandran 2011, 246).\n\nIt is, it seems, _metarepresentations_ that are the key. Like other neuroscientists, despite some difference of terminology, Ramachandran distinguishes between what he terms first-order sensory representations of external objects\u2014which evolved earlier, and humans share with many other animals\u2014and metarepresentations\u2014representations of representations, created by a part of the brain that evolved later and works \"by processing the information from the first brain into manageable chunks that can be used for a wider repertoire of more sophisticated responses, including language and symbolic thought\" (Ramachandran 2011, 246). He distinguishes seven dimensions, which together create a sense of self \"in the normal human brain,\" each of which may be disrupted or eliminated where a region of the brain is damaged or nonfunctional: _unity_ \u2014a myriad of sensory experiences somehow cohere to give the sense that one is a single individual; _continuity_ \u2014the ability to project oneself back and forward in time with a continuing identity; _embodiment_ \u2014the sense that one's body, and only parts of one's own body such as one's limbs, belong to oneself; _privacy_ \u2014that one's experiences of sensations, qualia, and one's mental life are one's own, alone; _social embedding_ \u2014Rose. that we are part of, and can interact with, a social environment; _free will_ \u2014that one is able consciously to \"choose between alternative courses of action with the full knowledge that you could have chosen otherwise\" (ibid., 252); and _self-awareness\u2014_ an aspect of selfhood that is, for Ramachandran, axiomatic.\n\nIt is hardly necessary to point out that Ramachandran's sense of self consists precisely in those features of selfhood that Charles Taylor suggests arise from a historically limited mode of self-interpretation. For the human sciences, the time of such a history is that of human culture. For Ramachandran, the time is that of evolution: these features are indeed understood as the outcome of evolved features of human neurobiology, creating the sense of unified selfhood arising from the human brain itself\u2014seemingly independent of its historical or cultural moorings. At the root of his analysis, as for many other neurobiologists, is the view that neural organization is not a unitary whole but a multiplicity, an assemblage that has come together over the course of evolution.\n\nAs Francisco Varela puts it: \"brains are not logical machines, but highly co-operative, nonhomogeneous.... [T]he entire system resembles a patchwork of subnetworks assembled by a complicated history of tinkering, rather than an optimized system resulting from some clean unified design... networks whose abilities are restricted to specific, concrete, cognitive activities as they interact with each other\" (Varela 1992, 321). Whether it is Michael Gazzaniga's modular conception of the brain as \"organized into relatively independent functioning units that work in parallel\" apart from our conscious verbal selves, contributing to consciousness but often producing behavior without conscious intervention (Gazzaniga 1985b, 4), or Marvin Minsky's idea of mind as a society of multiple agents with limited and circumscribed abilities (Minsky 1986), the message is clear: mental activity is made up from the interconnections and interactions between many relatively distinct components, each with its own mode of action. And yet, out of this assemblage, for humans at least, unless interrupted by pathology or tricked by experiment, a sense of a punctual, unified, continuous, autonomous, and self-conscious self always emerges.\n\nThe neurobiological self, then, is not to be equated with consciousness. Presenting a paper at an event he co-organized under the auspices of the New York Academy of Sciences titled \"The Self: From Soul to Brain,\" Joseph LeDoux sought to characterize an emerging body of thought within neuroscience that articulated this \"broader view of the self\" as a multiplicity with several distinct aspects, many of which are nonconscious (LeDoux 2003, 296\u201397). He argued that the notion of the self can be thought of along an evolutionary continuum: while only humans have the unique aspects of the self that are made possible by their brains, other animals have the kinds of selves made possible by _their_ brains. As summarized by Henry Moss, in his review of the conference, for LeDoux: \"our 'selves' are our brains, or, more accurately, the synaptic connections and systems that capture our experiences and memories, and express our habits and dispositions\" (Moss 2003, 2). \"Because you are a unique individual, the particular multifaceted aspects of the self that define 'you' are present in your brain alone. And in order for you to remain who you are from minute to minute, day to day, and year to year, your brain must somehow retain the essence of who you are over time.... [T]he self is essentially a memory, or more accurately, a set of memories\" (LeDoux 2003, 298).\n\nBut LeDoux is at pains to stress that this way of understanding our selfhood is not at the expense of all others\u2014the view of the self as psychological, social, or spiritual in nature and not neural: \"My assertion that synapses are the basis [of] your personality _does not assume that your personality is determined by synapses; rather, it's the other way around._ Synapses are simply the brain's way of receiving, storing, and retrieving our personalities, as determined by all the psychological, cultural, and other factors, including genetic ones.... A neural understanding of human nature in fact broadens rather than constricts our sense of who we are\" (LeDoux 2003, 302\u20133; emphasis added). The self _is_ the brain, the self is _based on_ the synapses, the synapses are ways of storing a multitude of inputs that constitute our personality, and the self is a set of memories. One suspects this constant slippage of terminology masks a deeper confusion about what it is that is being written about here. And the matter is made even less clear when LeDoux invokes the preeminently psychological concept of _personality_ , which he does not define, as the locus where nature and nurture interact in the \"wiring of synapses\" that makes possible the self, while still arguing that the source of identity and continuity of the person remains nonconscious\u2014the brain itself.\n\nDespite these diverse, terminologically slippery, and somewhat confused accounts, neuroscientists such as Ramachandran and LeDoux consider that _coherence_ is the very condition of normal selfhood. LeDoux is typical in suggesting that when such coherence fails, mental health deteriorates\u2014indeed mental illness, it seems, centrally involves the failure of that sense of coherent selfhood. But if the self is, or arises from, or is based upon, a multiplicity, how does the _impression_ of unity and coherence necessary for normality emerge? LeDoux is somewhat vague, suggesting that all the different systems in the brain experience the same world, and that all the systems are interconnected and use the same monoamines, and that this, somehow, enables most of us, most of the time, to hold our self together.\n\nIn her contribution to the event, Patricia Churchland, however, addresses the problem of coherence directly. She focuses on what she terms the \"self-concept.\" \"[I]t is useful to start,\" she says, \"with the idea that one's self-concept is a set of organizational tools for 'coherencing' the brain's plans, decisions, and perceptions,\" and it turns out that there are remarkably diverse ways in which individuals refer to themselves; thinking of these in terms of \"the self-representational capacities of the brain thus deflates the temptation to think of the self as a singular entity and encourages the idea that self-representing involves a plurality of functions, each having a range of shades, levels, and degrees\" (Churchland 2003, 32). This suggests that various levels of such coherencing can be found in other species, as a result of the nature of their particular brains. She draws on diverse evidence for her suggestion that there is a physical basis in the brain for the capacities for self-representation\u2014not just evidence of the neural bases for the representation of the functions of the internal milieu of the body, but also split-brain studies, developmental studies of the capacity for self-representation in children and the fading of such capacities in dementia, studies of amnesic subjects with various lesions who seem to have lost all their autobiographical information, confusion about self\/nonself boundaries in schizophrenia, patients with lesions in various regions who show limb-denial, alien hand syndrome, personality change, and the like.\n\nAnd she follows Damasio in arguing that the self\/nonself distinction provides the basis for consciousness, as it builds capacities to distinguish between internal and external representations, creating a metarepresentational model of the relation between inner and outer worlds: \"as wiring modifications enable increasingly sophisticated simulation and deliberation, the self-representational apparatus becomes correspondingly more elaborate, and therewith the self\/not-self apparatus. On this hypothesis, the degrees or levels of conscious awareness are upgraded in tandem with the self-representational upgrades\" (Churchland 2003, 37). \"The self thus turns out to be identifiable not with a nonphysical soul, but rather with a set of representational capacities of the physical brain\" (ibid., 31).\n\nNow this is clearer, and indeed accords with the kinds of arguments made in the human sciences\u2014the self is an imaginary representation of the real relations that any human being has to its internal world and external milieu. Leaving aside the _re_ -in representation\u2014which always misleadingly directs us to asking what is it that is present prior to its re-presentation\u2014we can now ask the interesting questions about how experience, culture, language, and so forth shape these representations. But that is not the direction pursued by most neuroscientists. Instead, some, entranced by the apparent capacities of neuroimaging to localize anything referred to by a 'mind' kind of word in some brain region activated during a task carried out in a scanner, ask where in the brain this representational self can be found (Feinberg and Keenan 2005; Northoff, Qin, and Feinberg 2011). One way to approach this is familiar: study those who appear to have a deficit in some aspect of self-representation (Sinigaglia and Rizzolatti 2011, 71).\n\nThus Baron-Cohen and others committed to the importance of theory of mind have suggested that the \"mindblindness\" or theory of mind deficits they find in those diagnosed with autism spectrum disorders may also affect the ways in which the individual becomes aware of his or her own experience and can use that self-awareness in understanding the experience, or the minds, of others (Carruthers 2009). Some suggest that perhaps this might be located in an anomaly in the right temporo-parietal junction, which may be part of a distributed network of brain regions that is 'recruited' for mentalizing about both self and other (Lombardo, Chakrabarti, Bullmore, et al. 2010; Lombardo and Baron-Cohen 2011). Others have taken different routes to discover the neural correlates of the various components of such metarepresentational capacities, seeking to identify and localize the neural systems of subsystems responsible for each aspect considered crucial for the emergence of the self: first-person perspective in spatial navigation, first-person narrative comprehension, self-face recognition, self-memory, awareness of the boundaries of one's own body, awareness of relation of self to social environment, and much more. These studies are well reviewed by Gillihan and Farah, who conclude that \"many of the claims for the special status of self-related processing are premature given the evidence and that the various self-related research programs do not seem to be illuminating a unitary, common system, despite individuals' subjective experience of a unified self\" (Gillihan and Farah 2005).\n\nThis is not surprising given the fact that most of those carrying out these studies are either confused or imprecise about what they mean by 'the self,' and this imprecision is manifested when they redeploy brain imaging tasks and procedures devised for other purposes to try to localize 'selfhood' in a particular brain region (Zahavi and Roepstorff 2011). Paradoxically, however, the weaknesses of attempts to 'locate' the self in the brain serves to support the general thesis\u2014that the conscious feeling of selfhood, as self-awareness, coherent identity, willful agency, and the like emerges out of the transactions among a whole array of evolutionarily distinct and functionally separate subsystems, whose modes of operation not only exceed that self-awareness and constitute it, but are, in principle, unavailable to it.\n\nThere is, as we have already mentioned, a long tradition of research on cultural variability in self-concepts. There was a revival of interest in this work in the sociology and anthropology of the 1980s: most argued that cultures shaped the self by acting on collective beliefs, practices of socialization, and individual psychologies. Can the brain take on the role that the psyche once occupied\u2014can these brain-based conceptions of the self open themselves to culture? Today, it seems, the answer is yes: for as we have seen in previous chapters, the twenty-first century brain is not an organ whose fate is fixed at the moment of conception or birth, but is plastic and open to external influences throughout development and indeed throughout life.\n\nFor LeDoux, as we have seen, selves are memories, and memories are patterns of synaptic interconnections. How are these patterns of synaptic interconnections established? In a tradition that goes back to Donald Hebb's maxim that \"what fires together, wires together\" (Hebb 1949), it is now 'experience' that shapes the patterns of neural connections that are strengthened\u2014and what is experience, if not cultural? This, for example, is the argument developed by Naomi Quinn in her contribution to _The Self: From Soul to Brain_ , titled \"Cultural Selves\": \"Thus, to LeDoux's picture of how neural processes engage multiple neural systems to endow each person with a self, must be added a description of how a very different set of processes, those that enter the cultural patterning of experience, shape and enhance that self.... The cultural patterning of the world we live in is so far-reaching and so all-encompassing that it cannot help but pattern our synaptic connections\" (Quinn 2003, 146). In this neural version of the social shaping of the self, all culturally variable aspects of selfhood familiar from psychological accounts\u2014habit, comportment, distinctions between good and bad behavior, practices for self-management, \"cultural schema\"\u2014are inscribed into the developing brain itself through the patterns of synaptic connections formed during child rearing and socialization.\n\nYet despite this belated entry of culture, we are left with something of a paradox. For while evolved neural processes and pathways provide the _capacities_ for self-representation, for this metarepresentation of representations that becomes conscious and amenable to awareness, what accounts for the _form_ of such representations? By this we mean, not the 'content' of the self\u2014for this is what is addressed by the claims about the role of cultural schema and the like\u2014but the _sense of self_ itself. Is that form\u2014unitary, punctual, continuing, attributing autonomy and free will to itself\u2014given in the neural capacities and hence universal? Or is it socially and historically variable, as argued by the cultural and historical sciences?\n\nPerhaps this is merely a new version of that double-faced conception of the self that was articulated by Marcel Mauss many decades ago. Mauss, as we have seen, argued that the _category_ of the self or the person is historically and culturally variable\u2014by which he means, he says, the notion or concept that people in different ages have formed of it. Yet, to repeat, he asserts that \"there has never existed a human being who has not been aware, not only of his body, but at the same time also of his individuality, both spiritual and physical.\" In other words, while the _form_ of the self is variable across cultures and epochs, he believes that the _sense of self_ is universal. It is worth quoting the remarks with which he continues: \"The psychology of this awareness has made immense strides over the last century, for almost a hundred years. All neurologists, French, English and German, among them my teacher Ribot, our esteemed colleague Head, and others, have amassed a great deal of knowledge about this subject and the way this particular awareness is formed, functions, deteriorates, deviates and dissolves.... My subject is entirely different and independent of this\" (Mauss [1938] 1985, 2).\n\nShould we be content with this distinction between what we owe to nature and what we owe to culture? Should we delegate the task of understanding the biological conditions of awareness, consciousness, and the sense of self to neurobiology and allocate to an \"entirely different\" enterprise the task of exploring the ways that humans at different times and places come to speak about selves, allocate statuses and capacities to them, mark them culturally, differentiate among them, and so forth? These questions undoubtedly inhabit distinct registers. But the distinction between an esoteric scientific discourse and an everyday lay understanding is itself a fact of history\u2014scientific truths and cultural truths are not always so distinct. What happens, then, when these two dimensions transect and intertwine? What happens when the quotidian conception that human beings have of themselves is itself shaped in part by the truth claims made in an esoteric language of science? This certainly was the case for psychological conceptions of personhood across the twentieth century. Is it also our contemporary condition in relation to neurobiology?\n\nA Mutation in Ethics and Self-Technologies?\n\nDespite their many differences, it is possible to identify a general picture of 'the self' and 'the person' emerging from contemporary developments in the brain sciences. In this picture, consciousness, with its image of the person as unified, purposive, intentional, and self-aware, is by no means the master in its own house. Consciousness is an effect, a metarepresentation, which creates an illusory, but probably evolutionarily advantageous, sense of coherence and self-direction. In fact, however, our conscious selves are not aware of, let alone in control of, the multitude of neural processes that enable us to live our lives: to receive, interpret, and respond to sensory inputs, to create an internal sense of our embodiment, to manage our motor skills and comportment, to remember and sometimes to recall previous experiences, to respond to many features in our environment and to humans and other animate entities, and to select among, and carry out, various options for action. It is not just that these, and many other processes essential to survival happen outside of our consciousness. It is that they are _in principle_ unavailable to consciousness in normal circumstances, although some can become amenable to conscious control under particular conditions or with appropriate training.\n\nThere is not much in this account that is new or startling to those from the _Geisteswissenschaften._ The novelty of contemporary neurobiological arguments lies elsewhere, in the claims about the processes that enable all this to occur. But does this renewed recognition of the limits of our awareness and conscious control over the neural infrastructure that enables us to consider ourselves as selves have any relevance to the way in which we are, to use Foucault's phrase, \"constituted as moral subjects of our own actions\" (Foucault 1984, 49)? Is it now these nonconscious neural processes that are considered as the constitutors of moral action, the origin of morally relevant conduct, the subject of moral evaluation, the target of self-improvement? Has the new prominence of the brain in these truth discourses effaced older forms of selfhood, rendering personhood exclusively, or largely, a matter of brains? The answer to the first set of questions is a qualified yes\u2014at least for some of our authorities, though perhaps not yet for many individuals themselves, those neural processes, now attributed names, locations, and mechanisms, have achieved salience and acquired a seeming practicability. But the answer to the latter is certainly no: personhood has not become 'brainhood.'\n\nThe brain has certainly become a rich register for narratives for self-fashioning. There are now many neurobiological technologies of selfhood, that is to say, practices that seek to mold, shape, reform, or improve aspects of one's person\u2014moods, emotions, cognition, desire\u2014by acting on or through the brain. But with what consequences for personhood? Let us briefly consider perhaps the longest established of these contemporary neurotechnologies\u2014psychopharmaceuticals.\n\nDespite the claims of some that such drugs promise a painless modulation of personality (Kramer 1992), and the reciprocal worries of some who believe the drugs offer a shallow and facile chemical manipulation of one's selfhood (President's Council on Bioethics (U.S.) and Kass 2003), there is little evidence here of a reduction of personhood to the brain. It is not only that the drugs do not work as advertised\u2014they do not have the remarkable therapeutic properties once claimed for them, and their mode of operation is far less specific than initially thought (Moncrieff and Cohen 2005; Moncrieff 2008). It is also that what such drugs promise is not designer moods or designer selves; they do not offer a limitless and painless recrafting of one's personhood through a simple chemical redesign of the brain (Rose 2003, 2007). The image that they offer is different, and it appeals to those who believe that, for whatever reason, they have lost touch with themselves, that they fail to recognize themselves in the person that they have become. The promise they offer is not that you will become someone new, but that you may be able to get your life back, to become yourself again, to feel, once more, like the \"real me\" or \"better than well\" (Kramer 1992, x). And to get your life back, to return to the 'real you' that you were once, that you that your loved ones remember, you must do more than consume a pill: this requires work on the self\u2014introspection, self-monitoring, perhaps cognitive behavior therapy\u2014all implying the existence of persons with brains, not persons who are their brains.\n\nThe two most powerful empirical studies of the way in which individuals engage psychopharmaceuticals in their everyday lives and practices of self-fashioning are David Karp's _Is It Me or My Meds?_ (Karp 2006) and Emily Martin's _Bipolar Expeditions_ (Martin 2007; see also Martin 2006). Each contains compelling firsthand accounts of individuals seeking to manage psychiatric conditions that they consider, at least in part, to be biological, based upon, arising from, or otherwise linked to, anomalies in their neurobiology. But in no case do these individuals consider themselves to be mere fleshly puppets of their brains or their neurotransmitters: they are persons with brains, and persons who believe that they experience their brain states by means of what we might loosely call a psychological sense of their own consciousness, and who need, as a matter of their psychological and subjective responsibilities, to manage their moods through the modulated use of psychoactive medication. It is true that they employ a vocabulary in which their personhood is shaped by their biology, and altered by their medication. And they wonder if their changing moods are caused by the condition itself, by the interaction of the medication upon the condition, or by their own actions and their interactions with family, friends, or professionals.\n\nWhile it has certainly become possible for human beings today to understand their actions, emotions, desires, and moods in a neurobiological language, their sense that they are inhabited by a psychology remains, and remains crucial. For it is through that psychological selfhood that they are required to enact the responsibilities that they have now acquired over the consequences of their troubled and troublesome brains. While one might now have the opportunity of modulating specific aspects of personhood by neurochemical means, it remains the responsibility of the person to manage himself or herself by acting on the brain. Indeed this reminds, us, if we need reminding, that our own time is not unique in having multiple and often inconsistent and contradictory repertoires for describing and managing personhood and selfhood, which are activated and deployed in different contexts and practices, and in different departments of life. If over the past century a psychological sense of the person has come to dominate in our ways of thinking, speaking, and acting in so many practices, and interpreting the actions of others, it is unlikely easily to be effaced by neurobiology.\n\nBut drugs are not the only neurotechnologies. There are now multiple attempts to persuade individuals to adopt or use technologies that claim to act directly to modulate the brain and its development. For example, we have seen the emergence of 'brain education' or 'brain awareness programs.' The Dana Foundation, in partnership with the Society for Neuroscience, runs Brain Awareness Weeks, described as \"the global campaign to increase public awareness about the progress and benefits of brain research. Every March BAW unites the efforts of organizations worldwide in a week-long celebration of the brain. During BAW campaign partners organize activities to educate and excite people of all ages about the brain and brain research. Events are limited only by the organizers' imaginations. Examples include open days at neuroscience laboratories; museum exhibitions about the brain; lectures on brain-related topics; displays at malls, libraries, and community centers; and classroom workshops.\" They involve downloadable coloring sheets for children, based on the Dana Alliance's Mindboggling booklet series; downloadable puzzles for all ages, based on the content of Dana Alliance publications; tests of neuroscience know-how; tools for educating policymakers; and activities organized by many universities and organizations in the United Kingdom and the United States, as well as in many other countries, such as Malaysia.\n\nPlasticity plays an explicit rhetorical role in arguments for such interventions on the brain. Thus the International Brain Education Association supports brain education across the world, basing its philosophy on neuroplasticity:\n\nA few decades back, scientists thought that people could have very limited influence over their brains. It was assumed that by the time people reached adulthood, their brain connections were permanently and indelibly in place.... Recently we have discovered the neuroplasticity of the brain. Right up to the end of your life, you can restructure and adapt your brain according to your needs. Your capacity to change your brain depends on your choice, nothing is predetermined.... The brain is the only organ that exerts influence on the world outside your body as well as inside your body.... IBREA's message is that if we all chose to use our brains to pursue the common good and bring humanity together under a common system of life values, our lives and our experience in this Earth will become much more meaningful and hopeful.\n\nOr consider Power Brain Education, which disseminates programs that \"teach how to use the brain well, through physical conditioning and balance, sensory awareness, emotional regulation, concentration, and imagination.\" There is also the Brain Gym movement founded by Paul Dennison, although its promoters are honest enough to explain: \"The investigation of the neuroscience that underpins Brain Gym is an eagerly awaited project for the future. We can only at the present time hypothesise about why without actually knowing.\" And this is not to mention all the organizations offering techniques for brain fitness, brain health, or brain improvement, such as SharpBrains, whose 2011 summit was titled \"Retooling Brain Health for the 21st Century,\" or Neurobics, \"a unique new system of brain exercises based on the latest scientific research from leading neurobiology labs around the world... the first and only program scientifically based on the brain's ability to produce natural growth factors called neurotrophins that help fight off the effects of mental aging.\" And, closely related, one sees the invention or rebranding of an array of nutritional products or other devices purporting to improve brain fitness\u2014brain foods, brain drinks, brain-boosting games for Nintendo, and so forth.\n\nBut one can also act on one's brain by acting on one's mind. This is most clearly exemplified in the intersections of neuroscience and Buddhism, both in its pure versions in the Mind and Life Institutes and in the somewhat de-based version of 'mindfulness' being taught in sites ranging from psychotherapy to business management. But there is much more. A few minutes on the Internet will lead to dozens of philanthropic organizations, quasi-religious evangelicals for peace, and profit-seeking enterprises organized around this premise\u2014your brain is amazing; it is flexible; it can be trained, developed, improved, optimized: learn to use it well for your own benefit and for that of your society, perhaps even for the world. It is not that you have _become_ your brain, or that you are identical with your brain, but you can act on your brain, even if that brain is not directly available to consciousness, and in so acting, you can improve yourself\u2014not as a brain, but as a person.\n\nCaring for the Neurobiological Self\n\nOur argument in this chapter has been that the emerging neuroscientific understandings of selfhood are unlikely to efface modern human beings' understanding of themselves as persons equipped with a deep interior world of mental states that have a causal relation to our action. Rather, they are likely to add a neurobiological dimension to our self-understanding and our practices of self-management. In this sense, the \"somatic individuality\" (Novas and Rose 2000; Rose 2007c), which was once the province of the _psy_ -sciences, is spreading to the _neuro_ -sciences. Yet _psy_ is not being displaced by _neuro_ : neurobiological conceptions of the self are being construed alongside psychological ones. We extend our hopes and fears over our biomedical bodies to that special organ of the brain; act upon the brain as on the body, to reform, cure, or improve ourselves; and have a new register to understand, speak, and act upon ourselves\u2014and on others\u2014as the kinds of beings whose characteristics are shaped by neurobiology.\n\nThis is not a biology of fate or destiny\u2014although we may sometimes describe ourselves as fated, for example, when diagnosed with a neurodegenerative disorder\u2014but a biology that is open for intervention and improvement, malleable and plastic, and for which we have responsibility to nurture and optimize. If our contemporary ethical practices are increasingly 'somatic,' that is to say, organized around the care of our corporeal selves, aided by rules for living articulated by somatic experts from doctors to dieticians, we should not be surprised if care of the brain becomes one aspect of this 'somatic ethics' (Rose 2007c). But in all these neurotechnologies, the self that is urged to take care of its brain retains the form of selfhood that took shape in the late twentieth century\u2014the commitment to autonomy and self-fulfillment, the sense of personal responsibility for one's future, the desire for self-optimization, the will, often the obligation, to act on our selves to improve ourselves over the whole of our lives. That is to say, we are not seeing personhood replaced by brainhood, but the emergence of a new register or dimension of selfhood, _alongside_ older ones\u2014a dimension in which we can understand and take care of ourselves, in part, by acting on our brains.\n\nAdded those other obligations, we are now acquiring a new one: the obligation to take care of our brain\u2014and of the brains of our families and children\u2014for the good of each and of all. For as we have seen throughout the book, brains are now construed as persistently plastic, molded across the life-course and hence requiring constant care and prudence. There is clearly an \"elective affinity,\" to use Max Weber's term, between this emphasis on plastic, flexible brains and more general sociopolitical changes that prioritize individual flexibility across the life span to accommodate to rapidly changing economic demands, cultural shifts, and technological advances\u2014and that demand a constant labor of self-improvement on the part of today's citizens. Indeed, that link is made explicitly in the marketing of many of the neurotechnologies we have discussed. But it would be too easy to merely map the neural onto the social in this way. Such socioreductionism would be a short-cut, sidestepping the work of analysis required to link these developments in the neurosciences and neurotechnologies with their sociopolitical context. The wish to fashion the self is not a recent phenomenon, nor is the belief that the continuous work of improving the self is a virtuous exercise of freedom. In the liberal societies of the West, from around the 1960s, at least for some of the middle classes and for many young people, such self-fashioning became no longer the privilege of the elite, the philosopher, the dandy, or the aesthete. The radical democratization of self-fashioning over the closing decades of the twentieth century has been taken into new territory with the spectacular diversification of authorities of the self in the age of the Internet. What is novel, then, is not the aspiration to shape, improve, fashion oneself, but the source of authority that underpins it, the technologies that it deploys, and the target or substance upon which it operates\u2014the brain itself.\nConclusion\n\nManaging Brains, Minds, and Selves\n\nAre developments in the neurosciences transforming our conceptions of what it is to be a human being? If so, how have they come to achieve this role, what forms are these transformations taking, and with what consequences? These questions, with which we began this book, lead to a subsidiary one, for if such transformations are occurring, how should the social and human sciences respond, given that individual and collective human beings are their own privileged objects of investigation? Of course, it is far too early to reach any definitive diagnosis. As we have argued, while investigations into the brain and nervous system can be traced back many centuries, _neuroscience_ is barely half a century old. And it would be misleading to search for any single figure of the human that inhabits all practices at any time and place, let alone across an epoch: many different conceptions of personhood can and do coexist, often incompatible and sometimes contradictory. Further, we need to be wary of the tedious tendency to proclaim the novelty of our own times, or to suggest that we are at the end of one age or on the verge of something new. Yet it is hard to ignore the pervasiveness of references to the brain and neuroscience, the growth of research and scientific publishing that we have documented, and the frequency of popular accounts of new discoveries about the brain in the mass media and in popular science books. Even more significant, we are beginning to see neuroscientific arguments deployed in relation to questions of policy and practice in areas ranging from child rearing to behavioral change. How, then, should we assess these developments?\n\nA Neurobiological Complex\n\nIn this book, we have described some of the key mutations\u2014conceptual, technological, economic, and biopolitical\u2014that have enabled the neurosciences to leave the enclosed space of the laboratory and gain such traction in the world. In this process, the human brain has come to be anatomized at a molecular level, understood as plastic, and mutable across the life-course, believed to be exquisitely adapted to human interaction and sociality, and open to investigation at both the molecular and molar scales in a range of experimental setups, notably those involving animal models and those utilizing neuroimaging technologies. However, the hopes that advances in basic neurobiological knowledge would translate into radical improvements in our capacities to understand and treat troubled and troublesome individuals have largely been disappointed. Nor has it yet proved possible to articulate clearly the way in which brain states give rise to mental states, and still less how they enable conscious experience: despite the proliferation of terms to describe this relationship, the explanatory gap remains.\n\nThe avalanche of neuroscientific research and its popular presentations are generating a growing belief, among policymakers and in public culture, that human neurobiology sets the conditions for the lives of humans in societies and shapes human actions in all manner of ways not amenable to consciousness. Yet, contrary to the predictions of many neuroethicists and social critics, neurobiological conceptions of personhood do not portray humans as mere puppets of their brains and have not effaced other conceptions, notably those from psychology, of who we are as human beings. Nor have they presented a monotonous picture of 'neuronal man' as isolated, individualized, and asocial. Instead, they have argued that the human brain, like that of some of our primate forebears, is evolved for a collective form of life, to make the attributions of internal mental states, beliefs, and intentions to our conspecifics that enable interaction among conspecifics, and the formation of groups both small and large to pursue common aims. And, as a corollary, neuroscientists have argued not only that human interactions in dyads, groups, and larger collectivities depend on these features of our neurobiology, but also that some of the difficulties suffered by those diagnosed with psychiatric disorders arise from problems within the neurobiological circuits that underpin and sustain these capacities for interaction with others. In these and many other ways, then, the normal functioning of the human brain has been constituted as crucial for sociality, and all manner of problems of sociality have been linked back to anomalies in neurobiology. This has enabled neurobiological ways of thinking to infuse the analyses of problems of individual and collective human conduct in the many sites and practices that were colonized by the _psy_ -disciplines across the twentieth century.\n\nThe proliferation of the _neuro_ -prefix is, no doubt, often merely a matter of fashion. Yet when 'neuro-drinks' promise to improve our brain power, or when 'neuro-trainers' claim that their courses will enable us to take full control of our lives, we can recognize the growing power of the belief, however banal, that by acting on our brains, we can improve our mental life and enhance our everyday capacities to meet the ceaseless demands for self-development and self-management that characterize our contemporary form of life. When various kinds of neurotherapy offer to make invisible mental processes visible, for example, via neurofeedback, and to give us the power to transform them by specialized techniques, we see something of the new configuration of personhood, in which the human mind can know and act on the brain that gives birth to it, in order to transform itself. When legal debates have to confront arguments from neuroscience, and neuroscientists argue that their research has relevance for issues from the sentencing of offenders to the age of criminal responsibility, we can see the growing salience of this new topography of the human being, even where brain-based claims do not displace psychological accounts and fail to convince judges and juries in the courtroom.\n\nWhen some authorities look to neuroscience to help resolve the intractable difficulties of risk predictions\u2014whether it be risk of developing a mental disorder or risk of committing a violent offence\u2014we can see signs of the rise of new forms of expertise whose apparent access to objective knowledge of the determinants of human conduct can legitimate the exercise of social and political authority. When contemporary proponents of long-standing arguments about the importance of a child's earliest years in determining its character and shaping its future life chances try to influence policy by reframing these in terms of our growing knowledge of the vulnerable developing brain, and supplement their arguments with images from brain scanning and data from neuroscientific experiments, we can see signs of the potential emergence of a new regime of truth about our nature as human beings.\n\nIt is too early to diagnose the emergence of a full-blown 'neurobiological complex,' or a radical shift from _psy_ -to _neuro_ -. But neurobiology is undoubtedly reconfiguring some of the ways in which individual and collective problems are made intelligible and amenable to intervention. The passage of neuroscience from the seclusion of the laboratory to the unruly everyday world, and the new styles of thought concerning the intelligible, visible, mutable, and tractable brain that characterize the new brain sciences are beginning to reshape some of the ways in which human beings, at least in advanced liberal societies, are governed by others. They are also providing those of us who live within their purview with new techniques by which we can hope to improve ourselves, manage our minds, and optimize our life chances by acting on our brains.\n\nBrains In Situ?\n\nHow should the social and human sciences respond to these developments? The most common response has been to argue that neuroscience is reductionist. Of course, there are many sophisticated philosophical discussions of reductionism, but this book is not a philosophical treatise, and our discussion here is not intended to be a contribution to philosophy. Rather, we want to consider some of the different ways in which this allegation has been made from the social and human sciences and consider where we stand today, in the light of our discussion in previous chapters.\n\nA first question of reductionism concerns the relation between the laboratory and the world, between the necessary simplification of the experimental setup in neuroscientific research and the inescapable complexity and messiness of the world in which humans and other animals live their lives. Of course, all experimental setups, even 'naturalistic,' observational ones, require a certain simplification, and there are many virtues in the quest of the laboratory sciences for the elimination of extraneous variables and the rigorous paring down that is required to demonstrate causal relations. But we have suggested that the persistent failure to translate laboratory findings to real-world situations\u2014for example, in animal models of psychiatric conditions or in brain imaging experiments of human volition\u2014may represent problems that are not technical but fundamental.\n\nFor, as so many ethnographic studies of laboratory life have shown, the laboratory is not a 'non-space': it is itself a real configuration of persons, devices, and techniques, which imposes its own characteristics on the data generated, as well as the interpretations made of them, and all the more so when the subjects are individual living creatures, be they humans or other animals. This is not a critique of artifice or artificiality, merely a recognition that the things that are made to appear in these laboratory setups are not those that appear in non-laboratory spaces, and their relation to what occurs in such other places and spaces is far from simple. All too often, as we have pointed out, these issues are ignored or set to one side by laboratory researchers. There are certainly some within the research community of neuroscience who recognize these problems and are seeking ways\u2014both conceptual and methodological\u2014to address them. Sometimes they can be addressed in part by seeking more ecological validity in the lab itself. Sometimes they can be addressed by trying to explore the ways the laboratory situation itself shapes the data that are generated. The first step on this journey, however, is to recognize the issue and to acknowledge it, however slightly, in the reports and interpretation of experimental results.\n\nAmong science policymakers, and among many researchers, there is now a ritual repetition of the theme that 'translation' from laboratory findings to clinical applications is the major challenge for the twenty-first century. We would agree. But we have also argued that the current pressures for neuroscientists to make premature claims of applicability may have perverse effects. Neuroscientists might well be advised to be frank about the conceptual and empirical questions that translation entails, rather than suggesting that the outcome of a series of experiments with fruit flies or feral rats has something to tell us about human violence, or that brain scans of individuals when they are exposed to images of differently colored faces in an fMRI machine has something to tell us about the neurobiological basis of racism.\n\nThe _translational imperative_ under which all are now forced to work in-duces such reckless extrapolations. Perhaps one can have some sympathy for the researchers' dilemma, caught between the necessary simplifications of the experimental design and the promises of rapid applicability demanded by those who fund the research. There is a role for the social sciences here in defending scientists from these demands. For there is a wealth of historical and sociological research on the relations between scientific findings and social innovations that undermines these naive views of linear translational pathways and their associated 'roadmaps.' Social research also suggests that the translational imperative, leading as it does to cycles of promises and disappointments, is likely to fuel just that distrust in science that it is intended to alleviate.\n\nIt is not entirely misleading to suggest that the move from the lab to the social world is a move from the simple to the complex\u2014although the lab itself is complex in its own way. We will address this issue later, for it goes to the heart of the criticism that neuroscience 'reduces' the social and cultural. However, there is another criticism of reductionism that concerns complexity. It addresses the approach that we described in our account of the early work carried out under the neuromolecular style of thought\u2014the idea that one should first observe the basic neuromolecular processes in the simplest possible systems and then, scale up to the level of intracellular transactions, circuits, networks, regions, and finally to the scale of the whole brain. This kind of reductionism has taught us a great deal. However, such a bottom-up approach also presents enormous difficulties in moving up the scales, and in addressing the kinds of phenomena that are created when tens, dozens, thousands, millions of neurons are in communication and transaction with one another and with their milieu.\n\nSome neuroscientific projects are directly addressing this issue, for example, in the work on phenomics that we cited in an earlier chapter, where it is framed in terms of the challenge of relating phenotypes to genotypes, and seeks to go beyond the impoverished explanatory architecture of interaction ritually invoked in the expression _GxE_. But the issues are not merely those concerning the technical complexity of mapping multiplicities of transactions, cascades, circuits, and feedback loops. They are also conceptual. At what scale _should_ different processes or functions of the brain be conceptualized? This raises a rather fundamental issue about the kind of biology that is appropriate here. We have much sympathy with the view of the microbiologist and physicist Carl Woese, when he argued that we need a new biology for a new century: criticizing mechanistic models that have transformed biology into an engineering discipline focused on applications at the expense of understanding, he argued that \"biology is at the point where it must choose between two paths: either continue on its current track, in which case it will become mired in the present, in application, or break free of reductionist hegemony, reintegrate itself, and press forward once more as a fundamental science. The latter course means an emphasis on holistic, 'nonlinear,' emergent biology\u2014with understanding evolution and the nature of biological form as the primary, defining goals of a new biology\" (Woese 2004, 185). Yet despite the great interest in studying the emergent properties of networks within complex systems that one can see in many scientific domains\u2014atomic physics, information science, and of course in neuroscience\u2014these remain exceptionally difficult to specify with precision. It has to be said, however, that the social and human sciences are hardly exemplary here: too often, appeals to complexity are a sign of explanatory failure, not explanatory success. Here, then, is a further challenge for genuine conceptual collaboration across the human sciences and the life sciences.\n\nThere is a further dimension to the charge of reductionism from the social and human sciences. This is the argument that neuroscience, in isolating brains and their components as the focus of investigation and explanation, ignores or denies the fact that, in humans and other animals, brains only have the capacities that they do because they are embodied, organs in a body of organs with all its bloody, fluid, and fleshly characteristics. There is, of course, a long philosophical debate about brains in vats (Putnam 1981), but the problems of solipsism and the like are not our concern here. Rather, like our colleagues from the social and human sciences, we are troubled by the misleading assumption that one can understand neural processes outside those of the bodies that they normally inhabit. To put it simply, as far as vertebrates are concerned at least, brains in situ function within complex pathways of afferent and efferent nerves reaching both the interior and the periphery of the body; brains are infused with blood containing all manner of active chemicals that carry nutrition, hormonal signals, and much else. And all of this and more is shaped by time\u2014at levels from the microsecond to the hour, by the times of circadian rhythms, and by the times of cell growth and cell death. Stomachs, intestines, lungs, kidneys, and livers\u2014let alone limbs\u2014do these play no part, or merely a peripheral one, in the faculties and functions studied by neuroscience\u2014in memory, in thought, in feelings, in desires? Psychiatric conditions shape and are shaped by mobilizations of feelings and functions across the body. Emotions course through the veins, engage the heart and the lungs, the bowels and the genitals, the muscles, the skin and face. As for cognition, do we not think, literally here, with hands and eyes? As Elizabeth Wilson puts it, an exclusive focus on the brain \"narrows the geography of mind from a diverse, interconnected system (a mutuality of moods-objects-neurotransmitters-hormones-cognitions-affects-attachments-tears-glands-images-words-gut) to a landscape within which the brain, as sovereign, presides over psychological events\" (Wilson 2011, 280). This not a plea for some kind of mystical recognition of wholeness: to criticize the isolation of the properties of the brain from those of the organism of which it is a part is to demand that neurobiology addresses more adequately the biological reality of the processes it seeks to explain.\n\nA further problem of reductionism, perhaps the most crucial challenge from the social and human sciences, concerns sociality, culture, and history. This is the argument that the neurosciences, even when they make reference to the environment, and even when they accept a role for social and cultural factors, perhaps even when they ask how experience gets under the skin, reduce this complex extraneuronal milieu\u2014of practices, meanings, symbols, swarming collectivities, multiplicities of historically sedimented human practices\u2014to a small number of discrete 'inputs' into neurobiological systems, ideally able to be represented in the form of quantifiable variables. Hence the repeated, not always playful, challenge to sociologists to identify 'the socio-ome' to be set alongside the genome, the proteome, the metabolome and all the other '-omes' so beloved of the biosciences at the end of the twentieth century.\n\nThe belief that one might be able to specify a stable, objectively given, factorially representable, quantitatively measurable socio-ome, generalizable across time and space, is anathema to most from the social and human sciences. It is, however, possible to identify some ways in which a more limited engagement might be possible. We have already contrasted two ways of thinking about sociality in social neuroscience: On the one hand, we find the impoverished conception of sociality that can be found in most research that utilizes brain imaging in laboratory settings\u2014where 'social' seems to refer to sequential transactions between isolated individuals or simulations of such transactions in scanners. On the other, we encounter the more subtle arguments of those who seek to understand sociality in terms of the multiple networks of collective association that characterize human existence, in families, groups, villages, towns, nations, and beyond, and the neurobiological consequences of isolation for humans evolved to live in such ecological settings. We have argued that the latter, for all their current weaknesses, open a pathway for collaboration beyond critique.\n\nConsider, for example, the way we might explore the mental consequences of urban life. The differences, and the potential linkages, between sociological and neurobiological approaches can be seen if we contrast the recent work of Meyer-Lindenberg's group on urbanicity, stress, and mental disorder with Georg Simmel's classic study of the metropolis and mental life. Meyer-Lindenberg and his group \"show, using functional magnetic resonance imaging in three independent experiments, that urban upbringing and city living have dissociable impacts on social evaluative stress processing in humans. Current city living was associated with increased amygdala activity, whereas urban upbringing affected the perigenual anterior cingulate cortex, a key region for regulation of amygdala activity, negative affect and stress\" (Lederbogen, Kirsch, et al. 2011, 498). Simmel locates the psychological form of metropolitan individuality in the shift of humans from rural to urban living, in the space, texture, and temporality of the urban experience, in the constant encounter with differences in urban interactions, in the novel economic and occupational forms that enmesh those who live in cities, and in the constant tension between the equalization of all and the uniqueness of each that traverses the collective life of the metropolis (Simmel 1903). The neuroscientists argue that their work provides the basis for \"a new empirical approach for integrating social sciences, neurosciences and public policy to respond to the major health challenge of urbanization\" (Lederbogen, Kirsch, et al. 2011, 500). Yet they ignore a century of empirical research and conceptualization in sociology, social history, and cultural geography on these issues, on the ways in which they play out in the mental lives of the inhabitants of these environments, and on the policy difficulties that are entailed in transforming the experiences and consequences of contemporary forms of urban life. The challenge both to neuroscientists and to social scientists is to go beyond mutual critique\u2014with each side accusing the other of gestural reference to the mental or the social\u2014to a kind of conversation in which experimental reductionism does not imply the mindless empiricism so long criticized in the social sciences, and where experimental rigor does not become the enemy of social, historical, and cultural complexity.\n\nCoda: The Human Sciences in a Neurobiological Age\n\nFor the human sciences, we suggest, there is nothing to fear in the rise to prominence of neurobiological attempts to understand and account for human behavior. It is important to point out the many weakness in the experimental setups and procedures, for example, in the uses of animal models and in the interpretations of brain imaging data generated in the highly artificial social situations of the laboratory. It is also important to challenge the attempts by neuroscientists and popular science writers to extrapolate wildly from small experiments to human practices, to attach the _neuro_ -prefix to so many areas, and to overclaim the novelty of their findings and their implications for everything from child rearing to marketing, and from criminal justice to political strategy. However, we should not be surprised to find, in contemporary neurosciences, all the features of inflated expectations, exaggerated claims, hopeful anticipations, and unwise predictions that have been so well analyzed in other areas of contemporary biotechnologies (Brown, Rappert, and Webster 2000; Webster, Brown, and Rappert 2000; Brown 2003; Brown and Michael 2003; Brown and Webster 2004).\n\nHistorical and genealogical analysis can and should identify the conditions of possibility for the current prominence of neuroscientific styles of thought and the practices where these have gained traction. Science and technology studies can and should analyze the setups that have generated the truths of contemporary neuroscience and the impact of the sociopolitical shaping of contemporary science on these expectational and promissory landscapes. We need detailed social research to map out the interconnections between pathways for the production of truth and hopes about the generation of profit that have shaped investment, research priorities, and the interpretation of evidence and its implications. We need to analyze the new practices for governing human conduct that are taking shape, the new powers of authority and expertise that deploy the _neuro_ -prefix to legitimate themselves, the new ethical obligations associated with the valorization of the brain as a key sociopolitical resource, and the new modes of subjectivation that are being born. While we are not witnessing a wholesale revolution, in which the psychological domain brought into existence in the twentieth century is consigned to history by the rise of neuroscience, our conceptions of human personhood, of the topography of the human being, are undoubtedly being reshaped by these processes.\n\nIt is important constantly to remind neuroscientists that neither animal nor human brains exist in individualized isolation. That if evolution has acted on the brain, it has done so to the extent that brains are constitutively embodied in living creatures, dwelling in space and in time, interacting in small and large groups on which they depend for their existence, striving to survive, inhabiting and remaking their milieu across the course of their lives. That when it comes to humans, and perhaps other primates, cognitive capacities, affective flows, and the powers of volition and the will are not individuated, but distributed. That they are enabled by a supra-individual, material, symbolic, and cultural matrix\u2014a matrix that the capacities of the human brain have themselves made possible, and that shapes the developing human brain (and the body within which it is situated) from the moment of conception to the moment of death. Whatever the virtues of reductionism as a research strategy, then, an account of the operation of the living brain in a living animal that does not recognize these transactions, that does not realize that the boundaries of skull and skin are not the boundaries of the processes that shape the irreal interior world in the brain, that does not recognize that human capacities and competencies emerge out of, and are possible only within, this wider milieu made and remade by living creatures, shaped by history, marked by culture in ways ranging from the design of space and material objects to the management of action and interaction and organization of time itself, will be scientifically flawed.\n\nIn the necessity for this criticism, there is also opportunity. There are many opportunities for a more positive role for the social and human sciences that engages directly with these truth claims, that seizes on the new openness provided by conceptions of the neuromolecular, plastic, and social brain to find some rapprochement. Opportunities exist to bring to bear the evidence from more than a century of empirical research on all those simplifications about sociality that currently inform even the most sophisticated accounts of the interactions between neurogenetics, brain development, and milieu, even the most subtle explanations of the affective shaping of human interaction, even most social theories of social neuroscience. The human sciences have nothing to fear from the argument that much of what makes us human occurs beneath the level of consciousness or from the endeavors of the new brain sciences to explore and describe these processes. If we take seriously the combined assault on human narcissism from historical and genealogical investigations, from cultural anthropology, from critical animal studies, and from contemporary neurobiology, we may indeed find the basis of a radical, and perhaps even progressive, way of moving beyond illusory notions of human beings as individualized, discrete, autonomous, coherent subjects who are, or should be, 'free to choose.'\n\nWe will undoubtedly find support for the core contention of the human sciences, that human societies are not formed by aggregations of such isolates, each bounded by the surfaces of its individual body. We will find much evidence that disproves the idea that the nature of humans is to seek to maximize self-interest, and hence to challenge the view that to govern in accord with human nature is to require each individual to bear the responsibilities and culpabilities of his or her selfish choices. Such unpredictable conversations between the social sciences and the neurosciences may, in short, enable us to begin to construct a very different idea of the human person, human societies, and human freedom. We have tried to show, in this book, how neuroscience has become what it is today; let us conclude with a simple hope for the future: that neuroscience should become a genuinely human science.\nAppendix\n\nHow We Wrote This Book\n\nIn this book we have drawn principally on arguments made by 'authorities,' that is to say, by those who claim to speak authoritatively about scientific research on the brains of human beings and other animals, and how humans should, in the light of that, be governed. We have examined the scientific literature as published in book and articles in the peer-reviewed journals, as well as popular books written by neuroscientists. We have looked at legal debates and law cases where neuroscientific evidence has been drawn on or referred to. We have researched the policy literature and policy proposals that make reference to neurobiology. We are aware, of course, that many of these matters have been the topic of public debate in the mass media and on the Internet, but, with some limited exceptions, we have not investigated these wider discussions and responses\u2014while undoubtedly of great significance, they would have required a different kind of study.\n\nWe have, however, drawn extensively on a wider array of informal interactions with neuroscientists, lawyers, and regulators. Many of these have been made possible through the work of the European Neuroscience and Society Network (ENSN) funded by the European Science Foundation (ESF), which has brought neuroscientists, psychiatrists, social scientists, historians, philosophers, regulators, and lawyers together in conferences, workshops, symposia, and neuroschools over the past five years. In addition to the many opportunities to explore the key issues in intense yet informal dialogue, we have also been fortunate to be able to spend time with researchers in a number of laboratories, notably Cornelius Gross's laboratory at the European Molecular Biology Laboratory facility, Monterotondo, outside Rome; the imaging facilities at the University of Vienna; Klaus-Peter Lesch's Laboratory of Translational Neuroscience in the Department of Psychiatry, Psychosomatics and Psychotherapy at the University of Wurzburg; and Kenneth Hugdahl's Bergen fMRI Group, at the Department of Biological and Medical Psychology of the University of Bergen.\n\nOur research on the legal issues raised by the new brain sciences was informed by a special interdisciplinary conference on Law and Neuroscience, also funded by the ESF, which we helped organize and which took place in Italy in October 2009, and by the involvement of one of us in the law and neuroscience module of the U.K. Royal Society's Brain Waves project. Our research on animal models was informed by our involvement in some heated discussions at a major international conference of animal modelers at the European Molecular Biology Laboratory facilities in Heidelberg, and by the involvement of one of us in the U.K. Academy of Medical Sciences working group on research with animals containing human materials, which involved quite extensive discussion of animal modeling in neuroscience. And one of us has also been involved in many discussions of these issues with regulators and lawyers, through membership of the U.K.'s Nuffield Council on Bioethics and other advisory groups. While we do not directly refer to these and other interactions, in Europe, North America, China, and Latin America, they have engaged us within a community of interlocutors, who have helped us form, test, and revise the arguments we make in this book. None of our colleagues in any of these encounters bears any responsibility for the arguments we make here, but their openness and friendship have shaped the tone of our argument, our belief in the possibilities of critical and affirmative dialogue, and our wish to avoid the many simplistic stereotypes that can be found in much critical writing on neuroscience from the side of the social sciences.\nNotes\n\n**Introduction**\n\n. Our focus in this book is on \"advanced liberal\" societies (Rose 1999), but we should note that these developments are by no means confined to Europe, the Americas, and Australasia, and indeed that we have seen many novel attempts at rapprochement, via the brain, with practices of meditation, especially those from the Buddhist tradition (see for example, the activities of the Mind and Life Institute: <http:\/\/www.mindandlife.org>). Further, there is another whole pathway of research that we barely touch on in the account that follows\u2014cognitive science and artificial intelligence. While there are many links between those developments and the ones we analyze in this book, to address them in any detail would overburden an already lengthy account. A good starting point for interested readers would be Elizabeth Wilson's study of Alan Turing (Wilson 2010). See also the discussion of the work of Warren McCulloch by Lily Kay (Kay 2002).\n\n. There is a growing body of work in the human sciences that shares this view, recognizing that human beings are, after all, living creatures, and seeking a more affirmative relation with biology. We can see this in the rise of 'corporeal' feminism, in 'affect studies,' and in projects to create a new materialism. While we do not discuss these endeavors here, we are sympathetic to them, although we are critical of the way in which some authors in this vein borrow very selectively from work in the life sciences to support pre-formed conceptual, political, or ethical commitments. In the main, however, we have not engaged with this literature in the present study.\n\n. Quoted from Lewis (1981, 215). We have drawn extensively on Lewis's account of this debate in this paragraph. Although it is undoubtedly true that the question dates back to Aristotle and was posed again by Descartes, the idea that it was answerable by a positive knowledge, rather than by philosophy, dates back to the start of the nineteenth century.\n\n. The phrase \"the physical basis of mind\" has a long history\u2014for example, it is the title of a nineteenth-century book by George Henry Lewes (Lewes 1877). Contributors to the BBC series included Edgar Adrian, who was Sherrington's pupil and shared the Nobel Prize with him in 1932 for his work on the function of neurons; Elliot Slater, who was a central figure in psychiatric genetics in the postwar years; Gilbert Ryle, who rehearsed his critique from _The Concept of Mind_ written the year before (Ryle 1949) that there was no \"ghost in the machine\"; and Wilder Penfield, pioneer of neurosurgery in epilepsy, who argued that there was a coordinating center in the upper brain stem that, together with areas of the cortex, was the seat of consciousness and that this was \"the physical basis of mind\"\u2014not mind itself, but the place through which \"the mind connected to the brain\" (a view he was to elaborate in subsequent years; cf. Penfield 1975).\n\n. The set of essays was later published as a book simply titled _The Brain_ (Edelman and Changeux 2001).\n\n. As we discuss in chapter 1, Francis O. Schmitt of the Massachusetts Institute of Technology used it in 1962 to designate his Neurosciences Research Program, and the term _neuroscientist_ was first used soon after (Swazey 1975).\n\n. Data from <http:\/\/www.sfn.org\/static\/amstats\/amstatsgraph.html>; consulted December 2010. By the end of the decade, annual attendance at this event was more than thirty thousand scientists, and around four thousand nonscientists, including many staffing industry or pharmaceutical displays.\n\n. Data from LSE's Brain, Self, and Society project, 2007\u201310 (<http:\/\/www.esrc.ac.uk\/my-esrc\/grants\/RES-051-27-0194\/read>).\n\n. The British Library catalog lists 433 books with the terms _mind_ and _brain_ in the title, the earliest, by Spurzheim and Gall, dating from 1815. It contains 62 books with the words _mind_ and _brain_ in their titles published in the thirty-five years between 1945 and 1980. A further 68 were published in the decade between 1981 and 1990, 109 between 1991 and 2000, and 122 between 2001 and 2009. By 1984, John Searle thought he had resolved this problem (Searle 1984, 1992), as did Jean-Pierre Changeux, whose _Neuronal Man_ was published in English a year after Searle's lectures (Changeux 1985), although many, including Paul Ricoeur, were not convinced (Changeux and Ricoeur 2002). Francis Crick, whose reductionist text _The Astonishing Hypothesis_ was published in 1994 argued for a physical basis for consciousness\u2014the \"claustrum\" in his last paper written with Christof Koch (Crick and Koch 2005). We will come across many versions of these debates and many words used to characterize the mind-brain relation in the course of this book, and we will return to this issue in our conclusion.\n\n. There have been earlier uses of the prefix. For example in his biography of Wilder Penfield cited above, Lewis notes that, in the 1940s, whenever a new researcher was recruited to the Montreal Neurology Institute, Penfield \"would add the prefix 'neuro' to his or her speciality\": thus when a chemist was recruited his speciality would become Neurochemistry, photography would become Neurophotography, and there was Neuropsychology, Neurocytology, Neuropathology and Neuroanatomy\u2014the prefix was intended to draw all together towards a common purpose (Lewis 1981, 192).\n\n. In 1996, Peter Shizgal and Kent Conover sought to describe the neurobiological substrate for choice in rats using a normative economic theory, followed in 1999 by Michael Platt and Paul Glimcher's publication in _Nature_ of \"Neural Correlates of Decision Variables in Parietal Cortex\"; by the early years of this century, there was a steady trickle of publications on these issues (Montague 2003; Sanfey, Rilling, Aronson, et al. 2003; Schultz 2003). A Society for Neuroeconomics was established in 2005; neuroeconomics programs and laboratories were set up at a number of U.S. universities (e.g., Caltech, UC Berkeley, UCLA, Stanford, Duke University); several handbooks and textbooks have now been published under titles such as _Decision Making and the Brain_ (Glimcher 2003, 2009); and popular magazines began to report excitedly on the new science of decision making (Adler 2004).\n\n. For Johns Hopkins, see <http:\/\/education.jhu.edu\/nei>; for developments at the University of London on \"educational neuroscience,\" see <http:\/\/www.educationalneuroscience.org.uk>.\n\n. <http:\/\/www.informaworld.com\/smpp\/title~db=all~content=t741771143>; consulted April 2010.\n\n. The report is titled _MINDSPACE: Influencing Behavior through Public Policy_ and can be downloaded at <http:\/\/www.instituteforgovernment.org.uk\/content\/133\/mindspace-influencing-behavior-through-public-policy>; consulted January 2011.\n\n. <http:\/\/royalsociety.org\/policy\/projects\/brain-waves>.\n\n. The program is called \"Neurosciences et politiques publiques\" (<http:\/\/www.strategie.gouv.fr\/article.php3?id_article=1029>; consulted January 2011).\n\n. Although the term was apparently coined by Yuri Olesha, it is conventionally attributed to Josef Stalin, who used it in a speech to Soviet writers at the house of Maxim Gorky in 1932: \"The production of souls is more important than the production of tanks.... And therefore I raise my glass to you, writers, the engineers of the human soul\" (for some details, see <http:\/\/www.newworldencyclopedia.org\/entry\/Yury_Olesha>).\n\n. We have omitted most references here, as each of the points that we make in this overview is discussed in detail in the chapters that follow.\n\n. We use the term _complex_ to bring to mind the interconnected resonances of this term. The _Oxford English Dictionary Online_ includes the following: \"A whole comprehending in its compass a number of parts, _esp._ (in later use) of interconnected parts or involved particulars,... An interweaving, contexture.... A group of emotionally charged ideas or mental factors, unconsciously associated by the individual with a particular subject... hence _colloq._ , in vague use, a fixed mental tendency or obsession.\"\n\n. The critical literature here is large, and we can only cite some representative samples. David Healy has been the most significant chronicler of the history of the links that we outline here, and the most persistent critic of the overclaiming made by the pharmaceutical companies (Healy 1996, 1997, 1998, 2000, 2004). Some of these points were tellingly made by Lauren Mosher, in his letter of res-ignation from the American Psychiatric Association after thirty-five years\u2014he said it should now be known as \"the American Psychopharmacological Association\"; see <http:\/\/www.successfulschizophrenia.org\/stories\/mosher.html>. Recently, Marcia Angell, former editor of the _New England Journal of Medicine_ , has also become an outspoken critic (Relman and Angell 2002; Angell 2004); see also her 2011 review of a number of polemical books on the \"epidemic of mental illness\" (<http:\/\/www.nybooks.com\/articles\/archives\/2011\/jun\/23\/epidemic-mental-illness-why\/?pagination=false>). The most persistent, and perceptive, critic of \"medicalization\" in psychiatry has been Peter Conrad (Conrad 1976, 2005; Conrad and Schneider 1992). Our own view of the limits of the medicalization thesis and our alternative approach has been spelled out in a number of places, and we will not repeat it here (Rose 1994, 2006a, 2007a).\n\n. These transformations in genomics are discussed in much more detail in Rose (2007c). We can note here that in this new vision, it was also at this molecular level that evolutionary pressures operated, and that evolved species' differences in behavior were to be understood. We discuss this further in relation to animal models in chapter 3, and also in relation to the 'evolution of the social brain' in chapter 5.\n\n. The intriguing title of the U.K. House of Commons Public Administration Select Committee's Second Report for the Session 2006\u20137 was indeed \"Governing the Future\"; see <http:\/\/www.publications.parliament.uk\/pa\/cm200607\/cmselect\/cmpubadm\/123\/123i.pdf>; consulted July 1, 2011.\n\n. In the case of the NRP, funding was chiefly drawn from federal agencies\u2014not only from the NIH and later the NIMH, but also NASA, NSF, and even the Office of Naval Research (ONR). This was complemented by funds from the Rockefeller Foundation and a private charity fund related to the NRP: the NRF, or Neurosciences Research Foundation (Schmitt 1990).\n\n. This is the widespread belief in biomedicine that Charles Rosenberg calls \"reductionist means to achieve necessarily holistic ends\" (Rosenberg 2006).\n\n. For example, the World Alzheimer Reports. The one for 2009 contained estimates of future prevalence (<http:\/\/www.alz.co.uk\/research\/files\/WorldAlzheimerReport.pdf>). That for 2010 focused on the economic cost (<http:\/\/www.alz.co.uk\/research\/files\/WorldAlzheimerReport2010ExecutiveSummary.pdf>).\n\n. As the reputable and influential commentator Polly Toynbee put it in the _Guardian_ newspaper, December 4, 2010, \"the brain hardens\" by the age of three.\n\n. The quote is from a presentation by Viding and McCrory to a 2007 conference commissioned by the United Kingdom's Department of Health; the Department for Children, Schools and Families; Youth Justice Board; and the Cabinet Office that followed the publication of the Social Exclusion Action Plan (2006) and the Care Matters white paper (2007). See <http:\/\/www.personalitydisorder.org.uk\/resources\/emerging-pd>; consulted January 4, 2011.\n\n. See, for example, Beddington, Cooper, Field, et al. (2008), framed recently in the form of a \"Grand Challenge of Global Mental Health\" (Collins, Patel, Joestl, et al. 2011).\n\n. According to the _NIH Almanac_ (<http:\/\/www.nih.gov\/about\/almanac>), funding for the NIMH remained more or less constant in the five years from 2005 to 2010, at between $1.4 billion and $1.5 billion, despite inflation at around 4% over this period. However, this was a significant increase from 1995, where the figure was just over $0.5 billion, and indeed from 1965, where it was around $186 million (data from <http:\/\/www.nih.gov\/about\/almanac\/appropriations\/index.htm#one>; consulted July 18, 2011). An indication of the complexity of deriving meaningful data in this area is that this covers all grants from the NIMH, not only for neuroscience, and that, as Dorsey et al. point out \"NIMH separated from NIH in 1967 and was raised to bureau status in PHS, became a component of PHS's Health Services and Mental Health Administration (HSMHA), later became a component of ADAMHA (successor organization of HSMHA), and rejoined the NIH in 1993.\" Another estimate shows significant variations in NIH funding for research in different mental disorders over the period from 2005 to 2009 (<http:\/\/report.nih.gov\/rcdc\/categories>); while NIH funding for some disorders\u2014notably depression and autism\u2014had increased by between 20% and 25% in the five years from 2005 to 2009, that for other conditions such as schizophrenia had decreased by the same amount, and funding for research on Parkinson's, Alzheimer's, and ADHD had actually decreased by between 30% and 40%\u2014<http:\/\/brainposts.blogspot.com\/2009\/08\/nih-trends-in-clinical-neuroscience.html>; consulted July 18, 2011. Of course, these figures represent only a very partial glimpse of public funding for neuroscience and do not take account of inflation, which was between 3% and 4% over the first decade of the twenty-first century.\n\n. These data are from a presentation given by Zack Lynch in 2010 at the fifth annual Neurotech Investing and Partnering Conference; we would like to thanks Zack Lynch for making this presentation available to us and allowing us to quote from it.\n\n. In 2011, the European College of Neuropsychopharmacology (ECNP) expressed concern that pharmaceutical companies were withdrawing their own research funding in this area, despite the growing burden of brain diseases (see http:\/\/www.guardian.co.uk\/science\/2011\/jun\/13\/research-brain-disorders-under-threat; consulted July 1, 2011).\n\n. See <http:\/\/www.bbsrc.ac.uk\/funding\/news\/2011\/110211-bbsrc-neuroscience-funding.aspx>; consulted July 1, 2011.\n\n. For IMS Health (\"revealing the insights within the most comprehensive market intelligence available\"), see <http:\/\/www.imshealth.com\/portal\/site\/imshealth>. For NeuroInsights (\"track market dynamics, develop investment strategies, identify partnering opportunities, and analyze comparables across this $143 billion global industry), see <http:\/\/www.neuroinsights.com>.\n\n. Sir David Cooksey, who reviewed U.K. health research funding at the request of the then government, concluded that the United Kingdom was at risk of failing to reap the full economic, health, and social benefits that its public investment in health research should generate, and argued that there were \"two key gaps in the translation of health research: translating ideas from basic and clinical research into the development of new products and approaches to treatment of disease and illness; and\u2014implementing those new products and approaches into clinical practice\" (Cooksey 2006, 3). It is these translational gaps that led to attempts to reform the whole research funding system to overcome professional, financial, and institutional barriers to translation.\n\n. These issues are discussed in more detail in _Translating Neurobiological Research_ , a 2009 report produced by the LSE's BIOS Centre for the Medical Research Council.\n\n. We use the term _translational platforms_ for zones of knowledge exchange and hybridization of techniques, practices, and styles of thought whose driving rationale is the translation of potential products and artifacts produced in the lab or the clinic into meaningful and useful clinical and social applications of clinical, commercial, or social value. They can be thought of as a special case of what Peter Keating and Albert Cambrosio call \"biomedical platforms\" (Keating and Cambrosio 2003). The idea of a translational platform derived from our mapping of seventy thousand peer-reviewed articles published in 2008 in almost three hundred high-impact journals; see Abi-Rached (2008); Abi-Rached, Rose, and Mogoutov (2010).\n\n. This is especially problematic in the much-criticized links between psychiatrists and the pharmaceutical companies. One particularly notorious example was the promotion of the diagnosis of pediatric bipolar disorder in the United States by Dr. Joseph Biederman, leading to a fortyfold increase in the diagnosis over the decade from 1994 to 2003 and the off-label use of antipsychotics, notably Risperdal, manufactured by Janssen, a division of Johnson and Johnson, who funded the research center that he directed. Some details of this case are given in a press report in the _New York Times_ in November 2008 (http:\/\/www.nytimes.com\/2008\/11\/25\/health\/25psych.html?bl&ex=1227762000&en=ab700f6adb9c70e5&ei=5087%0A; consulted April 4, 2011).\n\n. <http:\/\/www.lrb.co.uk\/v30\/n06\/shap01_.html>; consulted April 4, 2008.\n\n. The following passage derives more or less verbatim from a paper given by NR, \"Commerce versus the Common Good,\" at the LSE Asia Forum in Singapore in April 2008, and subsequently published as Rose (2009). Note that in theory the federal government retains the right to reclaim intellectual property and real-locate it elsewhere\u2014the so-called march-in provision. The march-in provision of the act, 35 U.S.C. \u00a7 203, implemented by 37 C.F.R. \u00a7 401.6, authorizes the government, in certain specified circumstances, to require the funding recipient or its exclusive licensee to license a federally-funded invention to a responsible applicant or applicants on reasonable terms, or to grant such a license itself.\n\n. For example, according to the U.S. Association of University Technology Managers Report in 2006, $45 billion in R&D expenditures were received by U.S. academic centers in that year alone; 697 new products were introduced into the market; and as a result of these relations, a total of 4,350 new products were introduced from Financial Year 98 through Financial Year 06; 12,672 licenses and options were managed, each yielding active income to a university, hospital, or research center; and 5,724 new spinouts were launched from 1980 to 2006 (Association of University Technology Managers, 2007); <http:\/\/www.autm.net\/Surveys.htm>; consulted July 1, 2011.\n\n. Quoted from its press release of August 29, 2002 (<http:\/\/royalsociety.org\/news.asp?id=2507>). The report published in 2003 concluded: \"Intellectual property rights (IPRs) can stimulate innovation by protecting creative work and investment, and by encouraging the ordered exploitation of scientific discoveries for the good of society.... A narrow focus on research most likely to lead directly to IPRs would damage the health of science in the longer term.... Patents can provide valuable, although sometimes expensive, protection for inventions. They therefore encourage invention and exploitation, but usually limit competition. They can make it impracticable for others to pursue scientific research within the areas claimed, and because inventions cannot be patented if they are already public knowledge, they can encourage a climate of secrecy. This is anathema to many scientists who feel that a free flow of ideas and information is vital for productive research. Additionally, research by others may be constrained by patents being granted that are inordinately broad in scope\u2014a particular risk in the early stages of development of a field. This is bad for science and bad for society\" (Royal Society of London 2003, v).\n\n. For example, in the celebrated Berkeley-Novartis deal, a pharmaceutical company, Novartis, gave UC Berkeley $25 million over five years, in exchange for the first rights to license any discovery that was made on the basis of research that was supported by Novartis funds; critics of such arrangements within the university were reported to have suffered various forms of disadvantage, including in relation to tenure processes (Triggle 2004, 144).\n\n. It is notable that bodies such as the NIH have recently begun to address this issue of conflicts of interest\u2014see its 2004 \"Conflict of Interest\" report (<http:\/\/www.nih.gov\/about\/ethics_COI_panelreport.pdf>). The scientific journals are also increasingly preoccupied with this issue.\n\n. While some have argued that neuroscience will challenge ideas of free will and responsibility in the criminal justice system, given evidence on the nonconscious initiation of apparently willed acts (Greene and Cohen 2004), this does not seem to be the case. As we discuss in chapter 6, while neurobiological evidence from CT and MRI scans showing structural damage or anomalies in the brain has begun to play a role, for example, in claims for compensation after injury, this has not fundamentally challenged the logics of personhood within those systems. Indeed, to the extent that individual' responsibility does come to be understood as somehow compromised by their neurobiology, as in the case of psychopathy, the response of the authorities of control, in a sociopolitical context that emphasizes precaution and preemption, is likely to be more, rather than less, severe\u2014as in the rising numbers of individuals who are already detained, without reference to neuroscientific evidence, on the grounds that they pose a significant and continuing risk to others. This issue is discussed further in the \"Neuroscience and Law\" module of the U.K. Royal Society's Brain Waves project, in which one of us was involved (Royal Society 2011).\n\n. On inhibition, see R. Smith (1992).\n\n. We can see similar arguments coming from the Social Brain project in the United Kingdom (Rowson 2011).\n\n. These self-technologies have been the subject of research by Francisco Ortega, who uses the term _neuroascesis_ to describe them (<http:\/\/www.neuroculture.org\/brain_gym.pdf>; consulted July 1, 2011).\n\nChapter One. The Neuromolecular Brain\n\n. Schmitt is quoted from Worden, Swazey, Adelman, et al. (1975, 529\u201330).\n\n. This chapter draws on earlier analyses published as Abi-Rached and Rose (2010). Some of the information is presented in more detail in three working papers for the Brain, Self, and Society project written by JAR (<http:\/\/eprints.lse.ac.uk\/view\/lseauthor\/Abi-Rached,_Joelle_M=2E.xhtml>). Some parts are also derived from a series of talks given by NR at the University of California in February and March 2007. Thanks to Sara Lochlan Jain for the invitation to Stanford, to Charis Thompson for the invitation to UC Berkeley, to Jenny Reardon for the invitation to UC Santa Cruz, to Norton Wise for the invitation to UCLA, and to all those who participated in the discussions. A version of the paper was given by NR as the Bochner Lecture delivered at the Scientia Institute of Rice University in Houston, Texas on March 9, 2009. NR wishes to thank his hosts, especially Susan Macintosh, and those at Rice and at Baylor College of Medicine who commented on the presentation. We have also drawn on an unpublished research report, \"The Age of Serotonin,\" funded by the Wellcome Trust Biomedical Ethics Program, and jointly written by NR, Mariam Fraser, and Angelique Pratt.\n\n. Schmitt uses \"wet, dry, and moist\" to refer to three different kinds of biophysics. \"Wet biophysics\" refers to the biochemical study of macromolecules and other cellular elements in their \"normal aqueous environment.\" \"Dry biophysics\" is concerned with the study of cellular constituents or organisms as \"systems\" in mathematical or natural models. Finally, \"moist biophysics\" is the study of properties of the central nervous system through \"bioelectric studies.\" See Swazey (1975, 532); Schmitt (1990, 199).\n\n. The citation to the first usage is given in the _Oxford English Dictionary Online._\n\n. SfN was founded in 1969 (see <http:\/\/web.sfn.org\/home.aspx>; consulted April 2010). It is the world's largest association for neuroscience, with more than forty thousand members by 2010. For a historical account of the SfN, see Doty (1987). Six autobiographical volumes from the SfN are available at <http:\/\/www.sfn.org\/index.cfm?pagename=HistoryofNeuroscience_autobiographies>; consulted April 2010. Similarly, interviews with prominent British neuroscientists have been conducted in a project at UCL titled \"Today's Neuroscience, Tomorrow's History\" and supported by a Wellcome Trust Public Engagement grant in the History of Medicine (<http:\/\/www.ucl.ac.uk\/histmed\/audio\/neuroscience>; consulted April 2010).\n\n. There is a plethora of books and articles on brain scientists and their particular contributions to the neurosciences. For example, Changeux's _Neuronal Man_ (1985, 1997), Gross's _Brain, Vision, Memory_ (1999), or Finger's _Origins of Neuroscience_ (1994). Some other examples include a recent book by leading French scholars on the contribution of prominent French scientists to the neurosciences (Debru, Barbara, and Cherici 2008); numerous references on the contribution of British neuroscientists (Gardner-Thorpe 1999; Rose 1999; C.U.M Smith 1999, 2001; Parkin and Hunkin 2001; Wade and Bruce 2001; Laporte 2006); a special double issue of the _Journal of the History of the Neurosciences_ 16 (1\u20132) (January\u2013June 2007) on the contribution of Russian neuroscientists; and numerous references on the contribution of this or that Nobel laureate to our understanding of the brain and the nervous system (Sourkes 2006; Langmoen and Apuzzo 2007).\n\n. In this account we have drawn on Rose (1996, 42\u201343).\n\n. Cajal's lecture was titled \"The Structure and Connexions of Neurons\" (<http:\/\/nobelprize.org\/nobel_prizes\/medicine\/laureates\/1906\/cajal-lecture.xhtml>), and Golgi's bore the title \"The Neuron Doctrine\u2014Theory and Facts\" (<http:\/\/nobelprize.org\/nobel_prizes\/medicine\/laureates\/1906\/golgi-lecture.xhtml>).\n\n. For more on the beginnings of neurochemistry, see McIlwain (1985, 1988); Bachelard (1988); Agranoff (2001).\n\n. In the early 1960s, he named his laboratory \"Groupe de Neuropsychologie et de Neurolinguistique\" (Boller 1999).\n\n. An overview of its historical development can be found in Boller (1999).\n\n. Of course, one could mention various other trajectories, for example, those of neural networks, cybernetics, communication theory, computing, and artificial intelligence, the latter of which was, of course, bound up with hopes for its usefulness for military and intelligence applications, which provided funding and support. Warren McCulloch also thought that these developments had significant implications for psychiatry, though his argument seems not to have convinced others. Kay argues that \"the particular configurations of these tools and images bore the unmistakable marks of a new, postindustrial episteme: an emergent military technoculture of communication, control, and simulations. Within its regimes of signification, life and society were recast as relays of signals and as information systems... exemplifying the new episteme of the information age\" (Kay 2002, 610). And we could also consider the borderlands with psychology, as it disciplined itself over the first half of the twentieth century, especially the intersections with psychophysiology. But this is not the place to write this history.\n\n. For more, see Bigl and Singer (1991); Marshall, Rosenblith, Gloor, et al. (1996).\n\n. J. Richter (2000) argues that the IBRO was the successor of the Brain Commission founded in 1903 in London. Although the commission was dissolved at the outbreak of the First World War, it was the first international organization whose aim was the creation of national brain research institutes and the promotion of international cooperation. One notable example is the Russian psychiatrist, Bechterev, who served on the Brain Commission and who pioneered long before neuroanatomist David McKenzie Rioch (mentioned by Cowan, Harter, and Kandel 2000) the physiological underpinnings of psychiatric illness and mental behavior. Bechterev was actually the founder of the first institute that looked at psychiatry from a physiological perspective, the so-called Psycho-Neurological Institute, founded in 1907 in St. Petersburg (Richter 2000).\n\n. In organizational terms, the United Kingdom seems to have lagged behind: the Brain Research Association (BRA) was founded in 1968. In 1996, the association decided to rebrand itself by changing the name to the British Neuroscience Association (Abi-Rached 2012).\n\n. We discuss Moniz's techniques of cerebral angiography using X-rays to visualize intracranial tumors, vascular abnormalities, and aneurysms in a subsequent chapter.\n\n. Moniz shared the Nobel Prize with Walter Hess, who developed an early brain mapping technique focused on the diencephalon (<http:\/\/nobelprize.org\/nobel_prizes\/medicine\/laureates\/1949\/hess-lecture.xhtml>). He was, of course, not the first to intervene on the brain. Those who have written on this history, notably Valenstein (1986) and Swayze (1995), point out that Burckhardt, the superintendent of a Swiss mental hospital, conceived neurosurgical approaches based on early studies in cerebral localization. He reported his results in 1891. However, his procedure was considered reckless, since one of his patients died soon after the surgery and two others developed seizures. There were no further attempts until 1910: Puusepp, a Russian neurosurgeon, resected fibers in the parietal and frontal lobes of patients suffering from manic-depressive psychosis. But results were published after the advent of prefrontal leucotomy, and in any case the procedure had limited success. In 1936, the Geneva neurosurgeon Ody resected the right prefrontal lobe of a patient suffering from \"childhood-onset schizophrenia.\" The result was published before Moniz's first publication and was inspired by Cushing's belief that mental symptoms could be reversed by resection of cerebral (frontal lobe) tumors. In 1935, Moniz attended the Second International Neurologic Congress in London where Charles Jacobsen presented work that he and John Fulton had done in bilaterally ablating the prefrontal cortex of two chimpanzees. Swayze comments: \"Although in his memoirs Moniz downplayed the influence of this report, stating that he had conceived the idea of a neurosurgical approach 2 years before, most observers, including Fulton and Freeman, believed that the animal results were the primary impetus for his taking such a bold step in November 1935\" (Swayze 1995).\n\n. See Cobb's obituary (<http:\/\/www.harvardsquarelibrary.org\/unitarians\/cobb.xhtml>).\n\n. Note that there are more interconnections than our \"three paths\" might imply: Richter was one of the pioneers in neurochemistry and was instrumental in establishing the IBRO under the aegis of UNESCO in 1961. He also helped establish\u2014though the extent of his involvement is disputed\u2014the Brain Research Association (BRA) in 1968, which was renamed in 1997 as the British Neuroscience Association (BNA). For more on the history of the BRA, see Abi-Rached (2012).\n\n. A further related element, which we will not discuss here, is the history of the medical uses of LSD (Dyck 2008).\n\n. See his 2001 lecture on acceptance of the Nobel Prize in Physiology or Medicine, reprinted as Carlsson (2001).\n\n. Some very useful papers on these issues are contained in a special issue of _Science in Context_ 14 (4) (2001), edited by Michael Hagner and Cornelius Borck. There is much more sociological work on the history and anthropology of psychiatry, some of which touches on biological psychiatry, usually in the context of reductionism and eugenics. And there is a growing body of ethnographic work that touches on disputes over biological psychiatry. A few empirical investigations have been carried out by social scientists. Historians of science have charted changing ideas of mind and brain (Harrington 1987, 2008). There are also studies of changing treatment of depression in clinical practice (Healy 1997, 2004; Kramer 1997; Elliott 2003); narratives of users of antidepressants (Stepnisky 2007); ethnographic accounts of transformations in psychiatric training and clinical practice (Lakoff 2005); and investigations of specific disorders: for example, post traumatic stress disorder (A. Young 1995), bipolar affective disorder (E. Martin 2007), and autism spectrum disorder (Silverman 2004). Others have studied the cultural impact of brain imaging (Dumit 1997, 2003; Beaulieu 2000b; Joyce 2008).\n\n. Susan Leigh Star was at the forefront of reflections on infrastructure from the perspective of ethnography and science studies (Star 1999). We share her emphasis on the importance of this, but, for present purposes, not the inventory of issues that were relevant for her ethnographic attention to the minutiae of space and interactions.\n\n. Horace Magoun has pointed to the key role of such research institutes in the twentieth-century history of neurology and brain research\u2014for example, of Ranson's Institute of Neurology at Northwestern University, with its work on the role of the hypothalamus and the lower brainstem in the 1930s\u2014pointing to the way that this can stabilize the lives of young scientists by providing them with space, equipment, mentors, and collaborators and by fostering a particular style of work (Magoun 1962, 1974).\n\n. Of course, those writing these histories tend to give the credit to those with whom they worked: Kandel worked in Kuffler's department, while Rioch is not mentioned in the 622 pages of _The Neurosciences: Paths of Discovery_ , which celebrates Schmitt and the NRP (Worden, Swazey, Adelman, et al. 1975).\n\n. In his autobiographical account on his website (<http:\/\/hubel.med.harvard.edu\/bio.htm>).\n\n. Purves's autobiographical account of his own movement, and his training at the hands of mentors including Eccles, Katz, Kuffler, and many others, provides a telling example of the ways in which these conceptual, methodological, and personal interconnections formed.\n\n. Among many honors, Kuffler was awarded the F. O. Schmitt Prize in Neuroscience in 1979.\n\n. Those approaching the issue from the perspective of investment opportunities and marketing divide the field differently; thus the entrepreneurial market analyst Zack Lynch maps the \"neurotechnology industry\" into three main sectors: neuropharmaceutical, neurodevices, and neurodiagnositic (Lynch and Lynch 2007).\n\n. See, for example, Aaron Panofsky's analysis of the diverse and rivalrous research groups working on the genetics of behavior, which we draw upon below (see note 32). Thanks to Aaron Panofsky for discussing this with NR and for letting him see three chapters of his doctoral thesis, which discuss these issues in great detail, as well as considering some of the issues of scientific fields, which we also consider here (Panofsky 2006).\n\n. See the acclaimed biography of Benzer, _Time, Love, Memory: A Great Biologist and His Quest for the Origins of Behavior_ (Weiner 1999).\n\n. Aaron Panofsky has analyzed differentiation of those who work on the genetics of behavior into a number of separate yet overlapping groups and associations (Panofsky 2006).\n\n. Of course, others had questioned this division before this time, for example, Sir Archibald Garrod in his idea of \"chemical individuality\" (Garrod 1902; Childs 1970).\n\n. This distinction was not merely a semantic or institutional one. For example, in the United Kingdom, this conception of the difference between treatable states of illness and untreatable variations in personality meant that those problematic individuals with \"personality disorder\" could not be legally confined under various Mental Health Acts until 2007\u2014confinement had to enable treatment, and a personality disorder, a \"trait,\" was not treatable.\n\n. For the rising rates of the use of psychiatric drugs, see Rose (2003a, 2004, 2006a, 2006b) and chapter 7 of _The Politics of Life Itself_ (Rose 2007c).\n\n. <http:\/\/en.wikipedia.org\/wiki\/Neuroplasticity>; consulted December 2010.\n\n. Wikipedia provides a good account of the developments that led up to the investigations of the brains of these monkeys in 1990, despite earlier decisions by NIH that no further research should be carried out on them, and that in 1990 indicated significant reorganization of the sensory cortex (<http:\/\/en.wikipedia.org\/wiki\/Silver_Spring_monkeys>; consulted December 2010).\n\n. For Taub's official biography at the University of Alabama, which makes a rather brief reference to the research on monkeys, see <http:\/\/www.uab.edu\/psychology\/primary-faculty\/11-primary-faculty\/27-dr-edward-taub>. For the therapy, see his statement at <http:\/\/www.taubtherapy.com>; consulted December 2010: \"Providing The Most Effective Stroke Therapy In The World. Taub Therapy, widely recognized as the most innovative form of CI therapy, empowers people to improve the use of their limbs, no matter how long ago their stroke or traumatic brain injury (TBI) occurred. The most effective stroke rehabilitation program in the world, Taub Therapy has been proven to be over 95% successful in helping patients in the clinic regain significant movement. Through the one-on-one encouragement of a therapist, patients can relearn to use their affected limb by restricting the use of the unaffected one. By causing neurons to 'rewire' themselves, Taub Therapy not only changes the brain, it changes lives.... Taub Therapy gives patients hope that they can recapture the life they had before suffering a stroke or TBI.\"\n\n. For Scientific Learning (\"Fit Brains Learn Better\"), see <http:\/\/www.scilearn.com\/our-approach\/our-scientists\/merzenich>; for Posit Science (\"Proven in Labs and Lives: The Posit Science Brain Fitness Programs Dramatically Improve Cognitive Performance\"), see <http:\/\/www.positscience.com\/science\/global-science-team\/merzenich>.\n\n. This story has been told many times with different emphases (Gross 2000; Gage 2002; Rubin 2009); for our present purposes, this brief outline is sufficient.\n\n. In the wake of her publications on these issues, Gould was invited to speak at many debates on exactly this topic, for instance at the United Kingdom's Royal Society of Arts in 2010 (<http:\/\/www.thersa.org\/events\/video\/archive\/elizabeth-gould>).\n\n. Some have suggested a link, perhaps even a causal one, between the popularity of these ideas of plasticity and a sociopolitical emphasis in \"neo-liberalism\" on flexibility in employment relations (Rubin 2009). Catherine Malabou has tried to distinguish this sociopolitical emphasis on flexibility from a more progressive idea of plasticity (Malabou 2008). The affinities are intriguing. However, we do not find it useful to invoke some generalized actor such as neo-liberalism in explaining either their origins or their implications. Rather, as we have suggested in this chapter, ideas of plasticity bring the brain sciences into line with the 'hopeful' ethos of the life sciences in contemporary societies\u2014that we can, and should, seek to improve ourselves by using techniques based in biomedical knowledge to act on our biology. This opens multiple routes for authorities, and individuals themselves, to act on human conduct and capacities by means of, and in the name of, the brain.\n\n. <http:\/\/www.best-personal-growth-resources.com\/brain-plasticity.xhtml>.\n\nChapter Two. The Visible Invisible\n\n. Writing in the U.K. _Times Literary Supplement_ on November 28, 2007 (<http:\/\/entertainment.timesonline.co.uk\/tol\/arts_and_entertainment\/the_tls\/article2960112.ece>).\n\n. Of course, much philosophical ink has been spent on debating this issue, and some philosophical critics claim it has merely been misleadingly reframed by attributing to brains and their components matters that can only properly be attributed to organisms, and in particular to humans (see Bennett and Hacker 2003).\n\n. As we shall see throughout this book, a variety of terms are used to express this relationship, all of them somewhat vague, and none of them specifying precisely how to overcome the explanatory gap between the activities of molecules within the human brain and the phenomenology of the experience of an inner world of thoughts, feelings, memories, and desires by a human being. We return to this question in our conclusion.\n\n. The English version of the title of chapter 9 of that book is \"The Visible Invisible.\"\n\n. These were shown in a wonderful exhibition produced at the turn of the century titled \"Spectacular Bodies\" (Kemp and Wallace 2000).\n\n. Foucault's well-known words are worth repeating: \"The gaze plunges into the space that it has given itself the task of traversing.... In the anatomo-clinical experience, the medical eye must see the illness spread before it... as it penetrates into the body, as it advances into its bulk, as it circumvents or lifts its masses, as it descends into its depths. Disease is no longer a bundle of characters disseminated here and there over the surface of the body and linked together by statistically observable concomitances and successions.... [I]t is the body itself that has become ill\" (Foucault [1963] 1973, 136).\n\n. We repeat in this section some of the arguments that can be found in chapter 7 of Rose (2007c).\n\n. Of course, as many careful histories show, one should take this triumphalism with a pinch of salt\u2014see, for example, Goldstein (2002).\n\n. The title page of the 1595 edition shows an image of a man, presumably Mercator, measuring a globe held in his hands; the image of Atlas replaced this in later editions: see, for example, that of 1633. There is a considerable historiographical literature on these issues, with some dispute about timing and rationale for the use of the term _atlas_ , which we do not discuss here. The more familiar image of Atlas with the globe on his shoulders dates back at least to Roman times but is often wrongly attributed to Mercator's collections of maps.\n\n. Diamond was founder and president of the Royal Photographic Society of London, established in 1853, and of the _Photographic Journal_ : he is often credited with taking the first photographs of asylum inmates in the Surrey County Asylum in 1851. On Diamond, see Didi-Huberman, cited below, and Packard (1962).\n\n. These are presented and discussed in chapter 3 of Didi-Huberman (1982) (see also Didi-Huberman and Charcot 2003), who also notes that a session of the French Medico-Psychological Society in Paris in 1867 was organized around the theme of the application of photography to the study of mental illness.\n\n. There is a close relation between photography in the asylum and the use of photography in the form of criminal anthropology by the police forces that were established in many countries at this time: we discuss some of this in a later chapter.\n\n. We have drawn on aspects of the analysis provided by Georges Didi-Huberman, cited above, although not its psychoanalytic interpretations. We have had to sidestep a large literature on the historical transformation of modes of scientific representation and the history of 'truth to nature' in the image, most tellingly analyzed by Lorraine Daston and Peter Galison in their account of how objectivity, as truth to nature, shifts in the nineteenth century from the image of the ideal form hidden within the empirical exemplars, to the image of the example itself, in all its anomalies and specificities (Daston and Galison 1992). We will return to some of these issues later in the chapter.\n\n. Finger credits Emanuel Swedenborg's work as an earlier example, though his treatises on the brain were only 'discovered' in 1868 (Finger 2000, 119).\n\n. The term was not used by Gall, and according to the _OED_ , was probably coined by the American psychiatrist Benjamin Rush in 1811 to designate the science of the mind.\n\n. Supporters of Gall argued that such animal experiments were beside the point, but they did not prevail against their critics.\n\n. Many of these images collected together by John van Wyhe can be found at <http:\/\/www.historyofphrenology.org.uk\/overview.htm>.\n\n. Stephen Jay Gould has discussed these attempts in detail, reanalyzed the available data, and pointed to the many ways in which those engaged in these brain-measuring activities fudged and distorted their data to fit their preconceptions concerning the differences in relative size of brain regions\u2014for example, frontal versus posterior cranial regions\u2014in \"inferior\" versus \"superior\" races, and between men and women (S. Gould 1981, see chap. 3).\n\n. Golgi and Cajal were jointly awarded the Nobel Prize in Physiology or Medicine in 1906 despite their bitter rivalry over this issue.\n\n. We will come across Vogt's work again in our investigation of \"the antisocial brain.\"\n\n. We will not deal directly with brain banking here. Cathy Gere tells us that what happened was a remarkable shift not only in the techniques of preservation of specimens (from chemical fixation to cryopreservation) but perhaps a more remarkable change in the rationale of their collection and the wider web of entanglements they found themselves part of. She argues: \"In the 1960s, brain archiving underwent the greatest social, technological, and organizational transformations in its two hundred year history. Brains began to be archived prospectively, rather than for specific research projects, with tissue samples made available upon application\u2014the advent of brain 'banking' \" (Gere 2003, 408). To fulfill that prospective usage, brains had to be frozen rather than fixed, which also meant that the biochemical processes that were preserved could later be studied at the molecular level. The shift from brain collections to brain banks indicates another change. A bank does not merely refer to a store of things for future use, it also refers to a stock that has some value; in other words, it refers to some sort of capital. Because these vital processes could now be stored, brain banks became part of a broader \"economy of vitality.\"\n\n. The use of thorium was unfortunate, as this substance often leaked into the brain, causing brain tumors.\n\n. We have drawn on the account given by Penfield's biographer (Lewis 1981). Penfield's work inaugurated a continuing technique for exploring the living brain using the opportunity presented by patients awaiting surgery for the removal of the localized brain lesions thought to be the focal points triggering seizures. For a recent example, see Abott (2010).\n\n. Another imaging technology was ultrasound. In 1937, Karl and Friedrich Dussik used ultrasound to obtain rather crude images of the brain. In 1953, Lars Leksell used ultrasound to diagnose a hematoma in an infant's brain\u2014a technique he used in 1956 to identify the midline in the adult brain. However, although ultrasound was used as a diagnostic tool, it did not play a major role in brain visualization.\n\n. The history of MRI and the controversy over the respective contributions of Lauterbur and Damadian are discussed by Amit Prasad (Prasad 2005, 2007).\n\n. We discuss some subsequent uses of brain scans in the courtroom in chapter 6. See also chapter 8 of Rose (2007c).\n\n. The first symposium on the topic of molecular neuroimaging, with the theme \"envisioning the future of neuroscience,\" was held in May 2010 at the National Institutes of Health in Bethesda, Maryland, to provide \"an overview of the potential of molecular neuroimaging in improving our understanding and management of critical CNS pathophysiological processes, such as neurodegeneration, brain tumors, and psychiatric disease.\" The conference program is available at www.molecularimagingcenter.org\/docs\/Brain2010\/MI_Brain_PrelimProgram.pdf; consulted July 2010.\n\n. Many new technologies are currently under development to increase the acuity of imaging and to reduce the need for cumbersome apparatus so that imaging may occur in more naturalistic settings. For example, there are developments in the use of the EEG to investigate connectivity between brain regions, and, more particularly, in the use of near infrared spectroscopy (NIRS).\n\n. In 2009, there were two widely cited indications of this disquiet. First was the widely reported fMRI experiment by Craig Bennett and colleagues that showed apparent variations in levels of brain activation in a dead salmon when exposed to various tasks. The poster by Craig Bennett, Abigail Baird, Michael Miller, and George Walford was titled \"Neural Correlates of Interspecies Perspective Taking in the Post-Mortem Atlantic Salmon\" and pointed out the dangers of false positives and the need for correction for chance (<http:\/\/prefrontal.org\/files\/posters\/Bennett-Salmon-2009.pdf>), but it received much wider attention when reported in _Wired_ magazine (<http:\/\/www.wired.com\/wiredscience\/2009\/09\/fmrisalmon>). Second was the paper by Ed Vul and colleagues, widely circulated under its original title of \"voodoo correlations,\" arguing that mistakes in statistical procedures in many cases had led to the \"Puzzlingly High Correlations in fMRI Studies of Emotion, Personality, and Social Cognition\" (Vul, Harris, et al. 2009).\n\n. Personal communication.\n\n. This, perhaps, suggests some caution in relation to recent attempts to image activity in single neurons and relate this to specific tasks or even specific thoughts.\n\n. Science and technology scholars have addressed similar issues in relation to digital images in other areas, for example, in Lynch's examination of digital image processing in astronomy (M. Lynch 1991). More recently, the manipulation of images in scientific publications, with the ready availability of software packages to manage such processes, has become a matter of controversy: see the discussion of this issue in _Nature_ in 2009 (<http:\/\/www.nature.com\/news\/2009\/091009\/full\/news.2009.991.htm>). Note that there are differences between different kinds of data manipulation. A technology such as fMRI requires choice of parameters such as those of threshold and color; it is good practice to note these explicitly in the publication. This differs from the silent use of techniques to enhance the visibility of an effect in an illustration in a scientific article. It is different again from the deliberate manipulation of data to mislead readers. However, the precise borderlines are sometimes difficult to define.\n\n. SPM is usefully discussed by researchers at the UCL Wellcome Trust Centre for Neuroimaging (<http:\/\/www.fil.ion.ucl.ac.uk\/spm>).\n\n. See our discussion of the antisocial brain in chapter 6.\n\n. Thanks to Hauke Heekeren, now of the Max Planck Institute for Human Development, for his account of these problems at an ENSN workshop, which we have reproduced here.\n\n. Molecules and Minds was the title of a provocative collection of essays by the neurobiologist Steven Rose (1986), whose critique of reductionist explanations has many similarities to our own position.\n\nChapter Three. What's Wrong with Their Mice?\n\n. Voltaire, _The Philosophical Dictionary_ , trans. H. I. Woolf (New York: Knopf, 1924), available at <http:\/\/history.hanover.edu\/texts\/voltaire\/volanima.xhtml>.\n\n. This chapter had its origin in discussions at the ENSN Neuroschool on Behavioral Genetics held in 2008 at the European Molecular Biology Laboratory (EMBL) outside Rome, and we benefited from reading the work in progress by one of those who attended that event. Nicole Nelson's doctoral dissertation is now completed (\"Epistemic Scaffolds in Animal Behavior Genetics,\" PhD diss., Cornell University), and an abstract is available at <http:\/\/nicolenelson.net\/abstract.xhtml>. An earlier version of the chapter was presented as a keynote lecture at the EMBL conference on \"Translating Behavior: Bridging Clinical and Animal Model Research,\" held in Heidelberg in November 2009, and we would like to thank Cornelius Gross, Klaus-Peter Lesch, and Haldor Stefansson for their invitation to give \"the social science lecture\" at that event and the participants in the conference for their critical comments. The preparation of this paper was greatly facilitated by the comments of our colleague Carrie Friese on an early draft, and we also thank her for letting us see some of her own forthcoming work on this topic, especially a draft version of her paper \"Models of Cloning, Models for the Zoo: Rethinking the Sociological Significance of Cloned Animals\" (now published as Friese [2009]).\n\n. Thanks to Carrie Friese for pointing us to this ethnographic and sociological literature on models.\n\n. Although, as we shall discuss later, evolutionary theory has also been used by some to argue against animal experimentation, explicitly contesting the principle of conservation (LaFollette and Shanks 1996).\n\n. Note that Wittgenstein is not so clearly on the side of those who draw a bright line between humans and other animals. See also Vicki Hearne's impassioned argument that some animals also have desires to achieve goals and the capacity for moral understanding (Hearne 1986). Thanks to one of our reviewers for suggesting that we make this point more explicit.\n\n. The mereological fallacy is that of ascribing to the part that which is properly a property of the whole.\n\n. Nicole Nelson's ethnographic research provides us with several examples of these arguments and of the anxiety (if we may put it like that) expressed by the behavioral geneticists about their validity: see her dissertation, cited in note 2.\n\n. See, for example, Hogg (1996); Geyer and Markou (2000); Richter, Garner, and W\u00fcrbel (2009).\n\n. These issues have certainly been addressed in the literature on animal models; for example, Geyer and Markou distinguish a number of different types of validity: predictive, constructive or convergent, discriminant, etiological, and face validity (Geyer and Markou 2000). This characterization problem is precisely the issue addressed by Jacqueline Crawley.\n\n. Of course, as we saw with brain imaging in the previous chapter, this question arises in different ways in all laboratory research in neuroscience.\n\n. Of course, this was a key argument in the work of Ludwik Fleck, who had a particular interest in the training in these ways of seeing (Fleck [1935] 1979).\n\n. Lynch provides us with some compelling ethnographic descriptions of the transformation of individual living creatures into data (M. Lynch 1989).\n\n. The _OED_ gives the following: \"1605 TIMME _Quersit._ III. 191 Wee commonly prouide that they bee prepared in our laboratorie. 1637 B. JONSON _Mercury Vind._ Induction, A Laboratory or Alchymist's workehouse.\"\n\n. We say \"rhetoric of replication\" because, of course, few have an interest in precisely replicating a published study, so replication is almost conducted through a series of displacements.\n\n. Ian Hacking is having fun by emphasizing the stability of the sciences against all the talk in the philosophy of science about scientific revolutions and so forth. So we are more than a little mischievous in using his arguments here.\n\n. Note that the term _model organism_ is a relatively recent addition to the terms used to describe these experimental animals and animal models. As argued by Jean Gayon, the expression emerged in the 1980s, notably in relation to the funding priorities of the U.S. National Institutes of Health (Gayon 2006).\n\n. This quote is also used by Angela Creager, Elizabeth Lunbeck, and Norton Wise in the introduction to their interesting collection of essays on the roles of models, cases, and exemplary narratives in scientific reasoning (Creager, Lunbeck, and Wise 2007).\n\n. Note that Steven Rose, arguing that the \"young chick is a powerful model system in which to study the biochemical and morphological processes underlying memory formation\" titles his article \"God's Organism?\"\u2014albeit with a question mark (Rose 2000).\n\n. Logan quotes Krogh: \"[T]he route by which we can strive toward the ideal [of generality] is by a study of the vital functions in all their aspects _throughout the myriad of organisms_. We may find out, nay, we will find out before very long the essential mechanisms of mammalian kidney function, but the general problem of excretion can be solved only when excretory organs are studied wherever we find them and in _all their essential modifications_ \" (C. Logan 2002, 331; Logan's emphasis).\n\n. Creager et al. also make this point, citing the work of Jessica Bolker to point out that \"some biologists argue that the highly canalised development of _Drosophila_ that makes its development so easy to study in the laboratory also makes it a relatively poor evolutionary representative of its own phylum\" (Creager, Lunbeck, and Wise 2007, 6; referring to Bolker [1995]). It is also relevant to note that the selection of the so-called wild type\u2014against which the models are compared\u2014is also a complex matter, as Rachel Ankeny points out in the same collection (Ankeny 2007).\n\n. There are, of course, many different classifications of types of model. One distinction, that we believe goes back to Clifford Geertz, is the distinction between \"models of,\" which are representations or devices of things that are seemingly inaccessible to vision, and \"models for,\" which can be thought of as designs that might aid in the later construction of an actual artifact (Geertz 1973). Some of these issues are helpfully discussed in McCarty (2003). For other classification schemes and discussions, see, for example, Black (1962); Suppes (1962); Achinstein (1968); Nouvel (2002); Harr\u00e9 (2009).\n\n. There are many different versions of the idea of a conceptual model, many of which are well discussed in Morrison and Morgan (1999).\n\n. As we suggested in earlier chapters, this model is almost certainly misleading, perhaps fundamentally mistaken. In the first decade of this century, researchers increasingly questioned the model proposed by Schildkraut, grounded in levels of neurotransmitter present in the synaptic cleft, and argued that that the principal mode of action of the selective serotonin class of psychopharmaceuticals is not an increase in the availability of neurotransmitter in the synapse, but effects on gene expression and neurogenesis (Santarelli, Saxe, Gross, et al. 2003; Sapolsky 2004). However, this model was the basis of one of the foundational myths of psychopharmacology.\n\n. For them, the other two dimensions are theories and experiments.\n\n. Besides animals (like rodents, frogs, dogs, rabbits, and fruit flies), model organisms also include plants (see Leonelli [2007]) and microorganisms such as _Escherichia coli_ and yeasts\u2014the latter are especially popular in molecular and cellular biology. For more on the evolution of model organisms and their phylogenetic diversity and relatedness see Hedges (2002).\n\n. Katz was awarded the Nobel Prize in Physiology or Medicine in 1970 for his discoveries, together with Ulf von Euler and Julius Axelrod (<http:\/\/nobelprize.org\/nobel_prizes\/medicine\/laureates\/1970\/katz-bio.xhtml>).\n\n. Dale Purves, who provides a very good account of this early work, argues that the entry of molecular biologists such as Sydney Brenner, Seymour Benzer, and others into neurobiology in the 1970s \"rapidly changed the scene with respect to simple nervous systems. With surprising speed, the neurogenetics and behavior of rapidly reproducing invertebrates, such as the roundworm _Cenorhabitis elegans_ and the fruit fly _Drosophila melanogaster_ , supplanted leeches, lobsters, and sea slugs as the invertebrates of choice, with molecular genetics considered the best way to ultimately understand their nervous systems\" (Purves 2010, 32).\n\n. Other models, for example, those in chemistry, can now include a temporal dimension through the use of computer animation and simulation (Francoeur and Segal 2004). Thanks to one of our reviewers for pointing to this work.\n\n. The question of the 'simplification' of the environment is now coming under scrutiny, with critics within the animal modeling community arguing for the replacement of the impoverished environment that has been conventional with an 'enriched' environment thought to confer more 'ecological validity' on the animals' behavior and their neural development: recent research has explored the behavioral and neurobiological changes that occur when animals are placed in such environments and has considered the implications for previous work using animal models of mental disorder (Laviola, Hannan, et al. 2008).\n\n. Friese, whose work (2009) has focused on the cloning of model animals in the preservation of endangered species in zoos, develops this term drawing on the terminology of Deleuze and Guattari ([1980] 1987). She suggests that the cloned animals function as models that validate in practice the theoretical assumptions of somatic cell nuclear transfer, especially interspecies transfer, as well as models for a particular way in which research on reproduction could be conducted in order to improve biomedical technologies and outcomes that might have important commercial applications\u2014see also Sarah Franklin's work on the creation of \"Dolly\" by somatic cell nuclear transfer (Franklin 2007).\n\n. \"Tout ce qui est vrai pour le Colibacille est vrai pour l'\u00e9l\u00e9phant\" (Jacob 1988). Note that Mark Pallen, from whom we have quoted the French original of this remark, has demonstrated that, as he puts it: \"By contrast, in the postgenomic era, our bioinformatics and laboratory-based studies lead us to conclude that what is true of one strain of _E. coli_ is not even true of another strain from within the same species!\"\u2014a finding that rather confirms our own view, and that of Canguilhem (and indeed, according to Pallen, that of Charles Darwin) (Pallen 2006, 116).\n\n. See chapters 5\u20136 in LaFollette and Shanks (1996) and Pallen (2006).\n\n. It is relevant to point out that Insel himself has long made use of voles as a model organism to study the adult human social brain, or at least with the aim of exploring some aspects of human behavior, and he makes frequent references to the probable implications for psychiatric disorders such as autism (e.g. Insel, O'Brien, and Leckman 1999), although he argues that the work with animals and other genomic research should only be the precursors to suggest hypotheses to be followed up in clinical studies in humans. Thanks to Zoltan Sarnyai for forcefully pointing this out to us at the event we cite in note 2 above.\n\n. The director of the West Riding Lunatic Asylum was James Crichton-Browne: as we saw in chapter 1, Ferrier and Crichton-Browne, together with Hughlings Jackson, founded the journal _Brain_ in 1878. We discuss some further aspects of this work in chapter 6.\n\n. Harlow's work has been continued into the genomic era by Stephen Suomi. Suomi, working with Klaus-Peter Lesch, who is credited with discovering the effects of the short or long alleles of the promoter region of the serotonin transporter gene on risk for various psychiatric problems, now correlates the responses of infant monkeys to styles of mothering with single nucleotide polymorphism (SNP) variants affecting the activity of the serotonin transporter gene (Lesch, Bengel, Heils, et al. 1996; Bennett, Lesch, Heils, et al. 2002).\n\n. This injunction is the title of one of the manuscripts by Nicole Nelson referred to earlier, and she also points out, on the basis of ethnographic observation in labs, how difficult this is for those working in the area to achieve.\n\n. See also Scott (1958).\n\n. This from an author who, elsewhere, has given some serious and thoughtful consideration to problems in translation and in animal models (Miczek and de Wit 2008).\n\n. The authors conclude their paper by asserting: \"Neuroethical concerns require discussion in open discourse among representatives of the criminal justice system, psychiatry, neuroscience, and the social sciences.\" (Miczek, de Almeida, Kravitz, et al. 2007, 11,805). Perhaps, however, it would be well for them to consider their own conceptual problems before suggesting that others need to consider the consequences of their tendentious extrapolations. We will come across Miczek and his group again in chapter 6, where we discuss the neurobiology of aggressive and impulsive conduct and its implications for the criminal justice system.\n\n. This despite explicit arguments to the contrary, for example, by John Paul Scott (Scott 1969). Of course, there is a huge literature here that we cannot discuss\u2014for example, the careful argument by Tinbergen (1968). One could also argue that the issue of internal states could be eliminated altogether if it could be established or proved that aggressive behavior in humans (rape, fighting) is correlated with brain states and\/or genomics, and that such similar states could be replicated in animals, but this nonetheless runs the risk of self-vindication. We return to some of these issues in chapter 6.\n\n. Note that Rodgers has pointed to some similar difficulties with the modeling of 'anxiety' in animals as if it were a unitary phenomenon (Rodgers 1997).\n\n. Chlorpromazine was marketed as Thorazine in the United States and as Largactil in Europe. Animal work actually began soon after the compound known as RP-4560 was first synthesized by the chemist Paul Charpentier in December 1950, and before it was made available for clinical use in 1951; however, the results were not published until after the clinical reports. See Lopez-Munoz, Alamo, Cuenca, et al. (2005, 117).\n\n. The elevated plus maze consists of a crosslike structure of four arms elevated above the floor, two of which are open, and two of which are enclosed at the sides (though open at the top), meeting at an open platform at the junction of the arms. The experimenters measure the amount of time the mouse spends in each arm (together with other behavior, such as 'nose poking,' when the mouse peers out from an enclosed arm into the open platform).\n\n. Some suggest an interpretation that is precisely the reverse: that the more alert the mouse, the quicker it will recognize that escape is impossible in these circumstances, and will preserve its energy by ceasing futile struggle.\n\n. Fragile X is a genetic condition in humans that leads to mental impairments of a range of severity: it is commonly believed to arise from a triplet repeat (GCG) on the X chromosome that affects the proteins involved in neural development.\n\n. There is a long history of the creation of simulacra in biology that are then taken to exemplify real features of the phenomenon simulated. Evelyn Fox Keller gives us the example of St\u00e9phane Leduc's attempts in the first decades of the twentieth century to overcome the distinction between life and non-life by creating \"osmotic organisms\" by introducing metallic salts and alkaline silicates into a solution of water glass, thus generating structures showing \"a quite dramatic similarity to the growth and form of ordinary vegetable and marine life\" (Keller 2002, 26).\n\n. These debates were central to the so called _methodenstreit_ at the end of the nineteenth century about the distinct epistemological character of the human and social sciences, supposedly arising from their distinct ontologies, and has continued across the history of sociology and allied disciplines. The argument for a fundamentally distinct epistemology in the social sciences is particularly associated with German thought and the work of Max Weber, but it came to the boil again in the 1970s, for example, in the writings of Peter Winch and Alasdair MacIntyre (MacIntyre 1962; Winch 1990).\n\n. We explore these questions further in chapter 4.\n\n. As we were writing an early draft of this paper, _Le Monde_ held a three-day public event under the title \"Qui sont les animaux?\" as part of the twenty-first annual forum \"Le Monde Le Mans,\" November 13\u201316, 2009 (http:\/\/forumlemans.univ-lemans.fr\/index.php?option=com_content&task=view&id=12&Itemid=27; consulted November 24, 2009).\n\n. Patton was writing in the context of a wave of recent philosophical and conceptual work on animals in the human sciences, notably by Vicki Hearne (1986). Much of this is discussed helpfully by Cary Wolfe (2003).\n\n. On the application of the idea of \"forms of life\" across species, see J. Hunter (1968).\n\n. Animal experiments in medical research, while contentious, are rigorously regulated in most countries, requiring specific ethical approval, licenses, assurances of good practice in the animal facilities, and the general principles of \"the three R's\"\u2014replacement, reduction, and refinement\u2014see, for example, <http:\/\/www.understandinganimalresearch.org.uk\/homepage>.\n\n. A somewhat similar view has been expressed by Francesco Marincola, editor in chief of the _Journal of Translational Medicine_ , in the editorial quoted earlier (Marincola 2003).\n\n. This is, of course, a proposal that has been made many times since Claude Bernard wrote: \"Certains chimistes... raisonnent du laboratoire \u00e0 l'organisme, tandis qu'il faut raisonner de l'organisme au laboratoire\" (Bernard 1858\u201377, 242).\n\n. There is a parallel argument in contemporary genomics, where Caspi and Moffitt have suggested that one should not frame explanations in terms of genetic causes of particular types of behavior modified by environmental effects, but in terms of environmental causes of disorders, and then seek the genomic variants that increase or decrease susceptibility or resilience to those environmental causes (Caspi and Moffitt 2006).\n\n. \"Hopkins Team Develops First Mouse Model of Schizophrenia,\" July 30, 2007 (<http:\/\/www.physorg.com\/news105036711.xhtml>; consulted November 2009).\n\n. To complicate the picture, since the turn of the century, many have begun to argue that the comorbidity between these disorders arises because they are linked neurodevelopmental disorders, whose commonalities are masked by diagnostic and clinical practice (Rutter, Kim-Cohen, et al. 2006; Owen, O'Donovan, et al. 2011). We return to this issue in a later chapter, in relation to arguments for screening and early intervention.\n\n. The research on DISC 1 and neural development has, in the main, been carried out in rodents (for example, in Brandon, Handford, Schurov, et al. [2004], all the primary references cited are to work on the rodent brain).\n\n. Although three papers published in _Nature_ in mid-2009 were optimistic in their claims (International Schizophrenia Consortium, 2009), most commentary on the papers was downbeat; for example, the discussion in the Schizophrenia Research Forum (dated July 3, 2009) pointed out: \"There are no break-out candidate genes, though there is support for previous linkage findings, several new candidates, as well as statistical modeling that supports the notion of genetic overlap between schizophrenia and bipolar disorder. Perhaps surprisingly, none of the studies alone identified any genetic marker with significant association to the disease (by the commonly applied genome wide significance benchmark of p < 5 \u00d7 10\u20138 [i.e., p < 5 \u00d7 10-8])\" (<http:\/\/www.schizophreniaforum.org\/new\/detail.asp?id=1532>; consulted November 20, 2009).\n\n. Some of these problems are alluded to in a discussion by John Medina (2008).\n\n. This is one use suggested for the synthetically engineered cells created by Craig Venter and his team (Gibson, Glass, Lartigue, et al. 2010); other work is currently under way with stem cell lines created by the induced pluripotent stem cell technique (iPS cells) from people diagnosed with neurodegenerative diseases, as discussed at <http:\/\/www.timesonline.co.uk\/tol\/news\/science\/medicine\/article7005401.ece>; consulted June 2011).\n\n. This is one of the promises of the Human Brain Project (<http:\/\/www.humanbrainproject.eu>; consulted June 1, 2011).\n\n. For an example of the use of virtual reality\u2013based experiments, see Baas, Nugent, Lissek, et al. (2004).\n\n. This way of putting things was suggested to us by Carrie Friese.\n\nChapter Four. All in the Brain?\n\n. Thanks to Peter Kinderman, who drew our attention to this quote.\n\n. Vol. 8 of _American Handbook of Psychiatry_ (New York: Basic Books, 1986), xiv.\n\n. Our title for this chapter is an allusion to that well-known aphorism, that this or that experience is nothing to worry about\u2014it is \"all in the mind\"\u2014which is also the title of a program on mental health issues on BBC radio.\n\n. Some will remember this phrase from the title of a 1966 movie directed by Karel Reisz that raises the question of the borderlines of eccentricity and mental disorder.\n\n. For a review of the concept of neuropsychiatry, see Berrios and Markov\u00e1 (2002).\n\n. In the French medical literature, Dr. Lucien Lagriffe used the term _neuro-psychiatrie_ in a 1910 article titled \"A propos du diagnostic de la paralysie g\u00e9n\u00e9rale\" in _Annales_ M\u00e9dico-Psychologique (June 27, 1910: 323): \"... le diagnostic s'impose \u00e0 quiconque a des notions, m\u00eame seulement \u00e9l\u00e9mentaires, de neuro-psychiatrie.\" In the Anglo-American literature, a few articles published in the 1920s mention neuropsychiatry as a 'synthetic' specialty encompassing clinical as well as anatomical and physiological aspects of the nervous system and approaches to mental and psychological conditions (e.g., Schaller 1922). _Neuropsychiatry and the War_ , a literature review on military neuropsychiatry, was published by the War Work Committee of the U.S. National Committee for Mental Hygiene in 1918. A neuropsychiatric branch of the U.S. Army Medical Department was founded in 1942. A year later it became a full-fledged division because of the increasing demand for 'neuropsychiatrists' during the Second World War; the division encompassed psychiatry, neurology, preventive psychiatry, and clinical psychology (Anderson, Bernucci, et al. 1966).\n\n. Thomas Willis, Sedleian Professor of Natural Philosophy at Oxford, was a neuroanatomist and the author of the very influential _Anatomy of the Brain_ , which assigned to the striatum the function of linking brain with mind. Of course, it is possible to find earlier uses of the term. Gaspar Peucer (1525\u20131602), a German physician, mathematician, reformer, and scholar, wrote about doctors' \"signs and diagnosis\" in his popular _Commentarius de praecipuis divinationum generibus_ published in Wittenberg in 1553, which was translated into French by Simon Goulard in 1584 as _Les devins ou commentaries des principales sortes de divinations_ (Anvers 1584). Peucer tells us that the signs are themselves divided into \"healthy, sick or neutral,\" that is, they reflect the bodily \"constitutions.\" They get further divided into \"diagnosis, prognosis and anamnestics.\" Diagnosis, Peucer writes, is a heuristic device, as it \"helps discover the exact disease which is presently in the ailed body.\" This definition of _diagnosis_ is repeated in Blankaart's _A Physical Dictionary_ of 1684, and extended to include the cause of the disease and the affected bodily part besides the disease process itself: \"Diagnosis is the knowledge of present signs or a knowledge whereby we understand the present condition of a Distemper; and it is three-fold, either a right instigation of the _part affected_ , of the _disease itself_ , or of its _cause_.\"\n\n. We have discussed some of these early attempts in Chapter 2, in connection with the question of visualizing mental states.\n\n. Allan Young has argued that those who were responsible for drafting _DSM-III_ \u2014notably Robert Spitzer\u2014\"identified themselves, in a self-conscious way, with the nosological perspective of... Emil Kraepelin\" (A. Young 1995, 95). Spitzer himself denies that he was a neo-Kraeplinian (Horwitz and Wakefield 2007).\n\n. Actually, much of the credit for these goes to Eli Robins, a biological psychiatrist, head of the Department of Psychiatry at Washington University School of Medicine (Kendler 2009): he was awarded the APA Foundations' Fund Prize in 1982 for his contributions.\n\n. While the 'molar' level is the tangible and visible body, the body of organs, of tissues and limbs, the 'molecular' body is, as we have seen in chapter 1, the one envisaged by the molecular gaze at the scale of neurotransmitters, of molecules and ionic pumps, of receptors and signals across neural networks. For more, see the chapter \"Molecular Biopolitics\" in Rose (2007c).\n\n. By this, we do not mean that it was uncontested\u2014of course, the contemporary history of psychiatry is riven by conceptual and factual disputes, and this extends to the criteria for diagnosis and the utility of the _DSM_. But the idea of a single discipline, with an explanatory regime that should embrace all these conditions nonetheless is the premise for the contemporary discipline of psychiatry.\n\n. <http:\/\/www.in-cites.com\/papers\/DrRonaldKessler.xhtml>; consulted June 2010. Lee Robins is the wife of Eli Robins, whose work on the Feigner criteria we discussed earlier.\n\n. There have been numerous other studies in the United States along these lines, such as the Epidemiological Catchment Area Study (Regier and Robins 1991). One of the authors (Darrel Regier) has had a key role in the task force developing _DSM-5_.\n\n. The EBC includes both the direct costs of medical and social care and the indirect costs of lost workdays and productivity: \"Direct medical expenditures alone totalled \u20ac135 billion, comprising inpatient stays (\u20ac78 billion), outpatient visits (\u20ac45 billion) and drug costs (\u20ac13 billion). Attributable indirect costs resulting from lost workdays and productivity loss because of permanent disability caused by brain disorders and mortality were \u20ac179 billion, of which the mental disorders are the most prevalent. Direct non-medical costs (social services, informal care and other direct costs) totalled \u20ac72 billion. Mental disorders amounted to \u20ac240 billion and hence constitute 62% of the total cost (excluding dementia), followed by neurological diseases (excluding dementia) totalling \u20ac84 billion (22%). Neurosurgical diseases made up a smaller fraction of the total cost of brain disorders\" (European Brain Council 2005, x).\n\n. <http:\/\/www.in-cites.com\/papers\/DrRonaldKessler.xhtml>; consulted June 2010.\n\n. Note that some similar issues arose in earlier epidemiological studies, for example, the work of Hollinshead and Redlich (Hollinshead 1999) and the Midtown Manhattan Study (Srole, Langner, et al. 1962).\n\n. There is, of course, a tradition of critique of medicalization in the social sciences, most recently focusing on the role of the pharmaceutical industry in expanding the empire of conditions amenable to treatment with drugs. On \"the new engines of medicalization,\" see Conrad (2005). We have discussed this kind of analysis in detail elsewhere and will not rehearse our views here (Rose 1994, 2006a): our focus in this chapter is specifically on its implications of, and for, neuroscience.\n\n. Note that Kessler and colleagues argue precisely the reverse\u2014that it is important to include mild disorders in _DSM-5_ because mild disorders may progress to major ones, and diagnosis at an early stage provides an opportunity for preventive intervention (Kessler, Merikangas, Berglund, et al. 2003).\n\n. This criterion was introduced into _DSM-IV_ largely as a result of the leadership of Allen Frances.\n\n. Wakefield has argued repeatedly that the so-called harmful dysfunction criterion could enable a completely \"factual\" basis for determining that a mental condition is a medical one: \"According to the HD analysis, a condition is a disorder if it is negatively valued ('harmful') and it is in fact due to a failure of some internal mechanism to perform a function for which it was biologically designed (i.e., naturally selected)\" (Wakefield 2007, 149). This will, he believes, enable medical practitioners to distinguish disorders from negative mental conditions that should not be disorders: he is referring to conditions such as ignorance, lack of skill, lack of talent, low intelligence, illiteracy, criminality, bad manners, foolishness, and moral weakness.\n\n. Indeed, this route seems to be implied by Wakefield's rather different appeal to biology as a means to protect contemporary _DSM_ -based psychiatry against a repeat of the earlier backlash from antipsychiatry on the grounds that it is again, pathologizing and medicalizing difference.\n\n. According to the American Psychiatric Association's _DSM-5_ website in March 2010, the process for revising the _DSM_ began with a brief discussion between Steven Hyman, MD (then-director of the NIMH), Steven M. Mirin, MD (then-medical director of the APA), and David J. Kupfer, MD (then-chair of the American Psychiatric Association Committee on Psychiatric Diagnosis and Assessment) at the NIMH in 1999. The planning process included experts in family and twin studies, molecular genetics, basic and clinical neuroscience, cognitive and behavioral science, development throughout the life span, and disability. Planning work groups were created, including groups covering developmental issues, gaps in the current system, disability and impairment, neuroscience, nomenclature, and cross-cultural issues. Darrel A. Regier was recruited from the NIMH to serve as the research director for the APA and to coordinate the development of _DSM-5_. A series of five white papers were published together in a volume titled _A Research Agenda for DSM-5_ , published in 2002; other papers were published; research was undertaken by the American Psychiatric Institute for Research and Education (APIRE), with Regier as the principal investigator; and thirteen conferences were held from 2004 to 2008 to help develop the research base for the _DSM-5_ task force and work groups and for WHO as it develops revisions of the International Classification of Diseases. In 2006, Kupfer was appointed as chair and Regier as vice-chair of the task force to oversee the development of _DSM-5_. The work groups have developed draft _DSM-5_ diagnostic criteria, with the release of the final, approved _DSM-5_ planned for May 2013. See <http:\/\/www.dsm5.org\/about\/Pages\/DSMVOverview.aspx>.\n\n. They added: \"Ethnicity and culture represent important factors that should be included in all of these research endeavors,\" but little was said to indicate that they believed this suggestion had serious epistemological implications, rather than being merely a nod to political correctness.\n\n. Indeed, it has been argued by several critics that this very claim of specificity owed more to marketing than to either neurobiology or to the results of clinical trials.\n\n. Perhaps the most concerted attempt to recognize the complexity of the interactions that generate \"phenotypes\" of mental disorder is being undertaken by the Consortium for Neuropsychiatric Phenomics at UCLA (<http:\/\/www.phenomics.ucla.edu>).\n\n. <http:\/\/www.dsm5.org\/Newsroom\/Pages\/PressReleases.aspx>.\n\n. <http:\/\/www.corepsychblog.com\/2010\/02\/psychiatric-diagnosis>; consulted June 2010.\n\n. Van Os, a Dutch psychiatrist and epidemiologist, had earlier written an editorial for the _British Journal of Psychiatry_ , based on epidemiological and related research, arguing that the diagnosis of schizophrenia should be abolished, in favor of \"salience syndrome,\" on the basis that psychosis was best understood as aberrant salience regulation (van Os 2009).\n\n. Note that David Kingdon, Lars Hansen, and Douglas Turkington, writing in the same issue of _Psychosis_ and also aware of the problem that the diagnosis \"extends the breadth of conditions included as mental disorders and could lead to stigmatization and inappropriate treatment with associated side-effects for individuals, most of whom will never develop psychosis,\" think that nonetheless it \"would focus attention on provision, and improve evaluation of, early intervention services. It might also lead to earlier availability of evidence based interventions, stimulate research and provide an agreed definition of psychosis risk\" (Kingdon, Hansen, and Turkington 2010).\n\n. This despite the fact that in the article they reference, van Os and Kapur point to \"key developments in biology, epidemiology, and pharmacology of schizophrenia\" such as recent molecular genetic findings, demonstrable alterations in brain structure and changes in dopamine neurotransmission, and the effectiveness of pharmacological treatments that block the dopamine system on delusions and hallucinations. They argue that a \"clear genetic susceptibility exists in schizophrenia; however, what one inherits is not the illness, but altered brain development, shared partly with developmental disorders, such as autism, and partly with affective disorders such as bipolar disorder.\" But the susceptibility only leads to the abnormal dopamine release that causes frank psychotic symptoms when this vulnerability is combined with environmental insults. Nevertheless, \"100 years after being so named, research is beginning to understand the biological mechanisms underlying the symptoms of schizophrenia and the psychosocial factors that moderate their expression\" (van Os and Kapur 2009, 641).\n\n. There is a proposal to rename this category \"Attenuated Psychotic Symptoms Syndrome\" (Cornblatt and Correll 2010).\n\n. AD is Alzheimer's disease; FDG-PET is fluorodeoxyglucose positron emission tomography; and CSF is cerebrospinal fluid.\n\n. The very diagnosis of Mild Cognitive Impairment is somewhat controversial (see Macher 2004; Rose 2010a). \"Some claim that the diagnosis can be made with relative certainty, others say that the phenotype is too blurry to be of much clinical or predictive use. Estimates of prevalence in those aged over 65 range from 5.2% to 16.8% (Golomb, Kluger, Ferris, et al. 2004, 354). Some claim MCI is an early stage of Alzheimer's and will progress to full Alzheimer's; others dispute this inevitable progression. Some claim that recent advances in brain imaging enable the visualization of plaques and tangles at an early stage, and hence allow prediction of which among those who meet the behavioral criteria of MCI will progress to Alzheimer's, although the evidence from the Nun Study and elsewhere suggests we should be wary. Some say that the diagnosis enables early treatment, which will slow progression; others doubt that such treatment is available. There is much to be said, here, given the blurry character of Alzheimer's itself, of the apparent capacity of a diagnosis of MCI to shift an individual onto a social and experiential pathway to Alzheimer's. Nonetheless, since the naming of this phenomena in 1988 re-organised a complex and competing field of categories and definitions, citations have increased exponentially\u2014according to the Institute for Scientific Infor-mation's Web of Science, there was one article on this topic in 1990, 158 in 2000, and 943 in 2007\" (Rose 2010a, 77). A further 2,017 papers were published on this topic from 2008 to July 2011. For helpful discussions of the utility of this diagnosis, see Brayne (2007); Whitehouse and George (2008).\n\n. We discuss self-vindication in chapter 3.\n\n. This is also true of many common complex, nonpsychiatric conditions; see Rose (2007c).\n\n. As we discuss in a subsequent chapter, the logics of screen and intervene have recently come into question in relation both to breast cancer (G\u00f8tzsche and Nielsen 2009) and to prostate cancer\u2014in the latter case, Richard Ablin, the inventor of the prostate specific antigen test, widely used in the United States as a screening test, referred to the screening program as a \"public health disaster\" (<http:\/\/www.nytimes.com\/2010\/03\/10\/opinion\/10Ablin.xhtml>).\n\nChapter Five. The Social Brain\n\n. A first version of this chapter was given by Nikolas Rose at the \"Global Minds\" conference held at Aarhus University, Denmark, in November 2008. Thanks to Andreas Roepstorff and Nils Buband for organizing the event, and to those participants who gave helpful comments. In developing it for this chapter, we would like to thank John Cacioppo for responding to an e-mail out of the blue, and for sending us some very helpful papers prior to publication.\n\n. Book 2 of _The Second Sex_ opens with, \"One is not born, but rather becomes, a woman\" (Beauvoir 1952).\n\n. Of course, wild children have historically been enrolled in many different arguments for different purposes, for example, by Locke, Condillac, and the sensationalist philosophers in relation to the issue of innate ideas, and more recently, in debates over \"critical periods\" in the development of capacities such as vision and language. Some of these issues are discussed in Rose (1985).\n\n. We discuss these split-brain experiments in chapter 7.\n\n. A rather different version of the modularity thesis was asserted a little earlier by Jerry Fodor, although his modules were delineated in terms of their functional properties, not their localizability in the brain (Fodor 1983). Of course, to some extent, modularity is both premise and outcome of the work on visualization discussed in chapter 2, and indeed, as we saw, Fodor\u2014for whom modularity is a conceptual and philosophical issue rather than an anatomical one\u2014is one of those who have cast doubt on the significance of that project of localization.\n\n. In 2009, Dunbar was the recipient of an award from the United Kingdom's British Academy for a project titled \"From Lucy to Language: The Archaeology of the Social Brain,\" which \"aimed to bring together archaeologists, evolutionary psychologists, social anthropologists, sociologists and linguists to reconstruct our ancestors' social lives and behavior from the archaeological evidence of bones and tools. New models developed for understanding primate behavior can now be applied to the hard evidence of our ancestors to help us understand how our brains have enlarged three-fold since early hominid Lucy, four million years ago\" (<http:\/\/www.britac.ac.uk\/arp>; consulted April 2011).\n\n. Dunbar gives a convenient account of this thesis in a lecture given at Gustavus Adolphus College in September 2009 (<http:\/\/www.youtube.com\/watch?v=i98XpBFWPrI>). Dunbar also makes a great deal of the role of brain size for pair bonding, and the anthropomorphic and somewhat teleological style of thought here is characteristic.\n\n. \"Mirror Neurons and the Brain in a Vat\" (<http:\/\/www.edge.org\/3rd_culture\/ramachandran06\/ramachandran06_index.xhtml>). See also Ramachandran's earlier essay, \"Mirror Neurons and Imitation Learning as the Driving Force behind the 'Great Leap Forward' in Human Evolution\" (<http:\/\/www.edge.org\/3rd_culture\/ramachandran\/ramachandran_p1.xhtml>; both consulted on November 12, 2008).\n\n. While the original observations of mirroring of actions seem to have occurred when monkeys observed human actions, there seems to be some ambiguity in the literature as to whether the mirroring that supposedly underpins empathy for the emotions of another is limited to cases where that other is of the same species, and whether it is more pronounced when that other has a prior emotional relationship with the subject.\n\n. <http:\/\/www.edge.org\/3rd_culture\/ramachandran\/ramachandran_p1.xhtml>. We would like to thank Daniel Margulies for sharing the research that he conducted in 2009 in the LSE's BIOS Centre on the rise and fall of mirror neuron theory.\n\n. See, for example, Olivier Morin's blog written in November 2008, <http:\/\/www.cognitionandculture.net\/Olivier-s-blog\/do-we-have-mirror-neurons-at-all.xhtml>; consulted April 1, 2010).\n\n. See, for example, the dispute over a 2010 paper from Marco Iacoboni's group claiming for the first time to have made recordings from human mirror neurons (Mukamel, Ekstrom, et al. 2010), also at <http:\/\/neurocritic.blogspot.com\/2010\/04\/mirror-neurons-join-marilyn-monroe.xhtml>.\n\n. Tania Singer, now at University College London, was a key early proponent of this line of work (e.g., Singer, Seymour, O'Doherty, et al. 2006). Another key figure is Ernst Fehr, who is based at the University of Zurich (<http:\/\/www.econ.uzh.ch\/faculty\/fehr.xhtml>). They are joined by a growing number of behavioral economists whose research usually entails various game playing scenarios.\n\n. Some anthropologists, notably Eve Danziger, have argued that capacities such as the ability to read the intentions of others are by no means universal, but are culturally specific (Danziger 2006, 2010).\n\n. This suggestion has troubled some, who wonder why, if the social brain arises in response to evolutionary pressures, such pathologies have persisted. Some suggest that they are the price that humans pay for other valuable attributes, such as language (Crow 2004); others hypothesize that they confer evolutionary advantages in certain settings (Bosman, Brunetti, and Aboitiz 2004); others believe that they persist because they share a common genetic basis with the evolving circuitry of the social brain (Burns 2004, 2006). This is not the place critically to evaluate these forms of argument.\n\n. They drew on their own earlier work (Oberman, Hubbard, McCleery, et al. 2005; Oberman and Ramachandran 2007) as well as the claims by Rizzolatti and Gallese cited above. See also Dapretto, Davies, Pfeifer, et al. (2005).\n\n. According to Essential Science Indicators, this paper had been cited 1,160 times by May 2007.\n\n. See the very helpful interview with Lesch (<http:\/\/www.in-cites.com\/papers\/KPLesch.xhtml>; consulted November 13, 2008).\n\n. See also Canli and Lesch (2007); Lesch (2008).\n\n. Thanks to John Cacioppo for letting us see a version of this history prior to publication.\n\n. Data derived from Web of Science citation index (November 2010) for articles with the topic \"social neuroscience.\"\n\n. http:\/\/mitpress.mit.edu\/catalog\/item\/default.asp?ttype=2&tid=10683; consulted November 18, 2010).\n\n. <http:\/\/ccsn.uchicago.edu>. The Social Neuroscience Lab at NYU, directed by a psychologist, David Amodio, was established at around the same time (<http:\/\/www.psych.nyu.edu\/amodiolab>), and many other similar centers and labs followed, from Queensland, Australia, to Beijing, China. It is beyond the scope of this chapter to map out the networks of collaboration among a relatively small number of enthusiasts that established the basis for this new discipline.\n\n. \"The social environment, therefore, is fundamentally involved in the sculpting and activation\/inhibition of basic structures and processes in the human brain and biology.... [S]ocial isolation or perceived social isolation (loneliness) gets under the skin to affect social cognition and emotions, personality processes, brain, biology, and health.\" Both this quotation and the one in the text are from his website at the University of Chicago (<http:\/\/psychology.uchicago.edu\/people\/faculty\/cacioppo\/index.shtml>; consulted November 1, 2010).\n\n. Lieberman had hosted an early conference on social cognition at UCLA in 2001. On Lieberman, see <http:\/\/www.scn.ucla.edu\/people\/pdf\/APA07.pdf>. Lieberman has published extensively with his wife, Naomi Eisenberger, on the neural basis of social rejection, notably \"Why Rejection Hurts\" (Eisenberger, Lieberman, and Williams 2003), which we discuss later, and which is seen by many neuroimagers as a textbook example of the flawed reasoning known as \"reverse inference.\"\n\n. <http:\/\/www.psy.pku.edu.cn\/LABS\/CSCN_lab\/research.xhtml>; consulted November 26, 2010.\n\n. <http:\/\/s4sn.org\/drupal>; consulted November 26, 2010.\n\n. <http:\/\/s4sn.org\/drupal\/?q=node\/3>; consulted November 30, 2010.\n\n. They refer here to two histories: Cacioppo, Berntson, and Decety (2011); Matusall, Kaufmann, and Christen (2011).\n\n. <http:\/\/grants.nih.gov\/grants\/guide\/rfa-files\/RFA-DA-06-004.xhtml>; consulted November 15, 2008.\n\n. http:\/\/www.esrc.ac.uk\/ESRCInfoCentre\/about\/CI\/CP\/research_publications\/index27.aspx?ComponentId=6545&SourcePageId=6557; consulted November 15, 2008.\n\n. It called for applications focused on \"[u]nderstanding normal and abnormal cognition, social and economic behavior of individuals and groups, and the neuropsychological basis of such behavior (including, associated disorders), particularly utilising innovative tools and methods\" and on \"[s]ocial, economic and occupational factors affecting mental health, mental wellbeing and mental illness\" (ESRC website; consulted November 15, 2008; page no longer active).\n\n. Posted on <http:\/\/neuroanthropology.net\/2008\/03\/04\/neuroanthropology-session-at-the-aaa-conference>; consulted November 15, 2008.\n\n. <http:\/\/www.nyas.org\/valconf>; consulted November 15, 2008.\n\n. This is at the heart of Alain Ehrenberg's critique of the social brain\u2014that the social is reduced to the interaction between two naturalized human individuals, whereas humans as social actors operate in a shared world of cultures, institutions, meanings, and values, and human affects such as empathy arise only in certain kinds of societies and in relation to those persons to whom those societies deem it appropriate to empathize\u2014they are not natural but social (Ehrenberg 2008). However, presumably, social neuroscientists would retort that features of the brain provide the _capacity_ for empathy, but that different societies and cultures shape it in particular historically contingent ways.\n\n. <http:\/\/psychology.uchicago.edu\/people\/faculty\/cacioppo\/index.shtml>; consulted November 2010.\n\n. Ibid.\n\n. Note that they refer to this as a neurosociologic theory, not a neurosociology.\n\n. Franks also wrote the entry on neurosociology under this name for the online _Blackwell Encyclopedia of Sociology_ and, in an interesting case of conceptual entrepreneurship, he also seems to have been responsible for the Wikipedia entry\u2014comments by those editing that entry argue that the term is now obsolete and is synonymous with _social neuroscience_ : see <http:\/\/en.wikipedia.org\/wiki\/Neurosociology>.\n\n. One important social science discipline that has shown an openness to these new ways of thinking is economics. A number of economists have used versions of social neuroscience to explore decision making in consumption, trading, and other areas of economic activity, contesting images of rational choice, most often using the game-playing methodologies that have become habitual, and which are conveniently adaptable for use in brain scanners. While these endeavors have, to date, remained largely in the world of academia, another version, claiming to be based on an understanding of the brain, has achieved much popular exposure and has been picked up by governments in the United Kingdom and elsewhere. So-called 'nudge' techniques seek to make it easier for people to make behavioral change in desired directions, not by instructing them but by changing the incentive structures in their environment and the largely nonconscious influences on decisions (Thaler and Sunstein 2008). The claim of such arguments to rest on an understanding of the brain derived from neuroscience is hollow, however, and the suggestion that these indirect modes of behavior shaping are in some way novel is historically ignorant\u2014governments and other authorities have utilized such techniques since at least the mid-nineteenth century (Rose and Miller 1992; Miller and Rose 2008).\n\n. <http:\/\/www.thersa.org\/events\/vision\/vision-videos\/matthew-taylor>.\n\n. Grist was the lead researcher on the Social Brain project, a role later taken over by Jonathan Rowson.\n\n. This phrase was also the title of an anthology of papers on ethology published in 1969 (Parkin 1969) and an introductory sociology book favoring an approach loosely based on social evolution (Runciman 1999).\n\n. Emily Martin, whose excellent work in this area we have already discussed, has consistently warned of the danger of neuroreductionism, suggesting that brain-based views of personhood were intrinsically reductionist, but that they would only become widely deployed in a culture in which a \"brain-centered\" view of the person made cultural sense (Martin 2000); however, we argue that a different path has been followed by many, in which brain-based views of personhood come more closely into alignment with what Martin refers to as \"our everyday mental concepts\"\u2014that is to say, an ontology informed by a century of cultural assimilation of psychological conceptions of human beings.\n\nChapter Six. The Antisocial Brain\n\n. Significantly, this passage was quoted by Graham Allen in his _Early Intervention: The Next Steps; An Independent Report to Her Majesty's Government_ (2011).\n\n. <http:\/\/news.sky.com\/home\/world-news\/article\/15620258>; consulted December 1, 2010.\n\n. Another individual who has made something of a media career out of these claims is James (Jim) Fallon of the University of California, Irvine\u2014see for instance, his recent appearance on U.S. National Public Radio (NPR) (<http:\/\/www.npr.org\/templates\/story\/story.php?storyId=127888976>).\n\n. <http:\/\/www.abc.net.au\/news\/2010-04-27\/carl-williams-brain-unlikely-to-reveal-secrets\/412892>; consulted December 1, 2010.\n\n. <http:\/\/www.nature.com\/news\/2010\/100317\/full\/464340a.xhtml>; consulted December 1, 2010.\n\n. This shift is discussed in more detail in chapter 8 of Rose (2007c).\n\n. Our account here depends on that given by Robert M. Young (1970, 39). Young points to the crucial difference between Gall's work and that of later localization researchers such as Ferrier\u2014Gall's faculties were hypothesized, then localized, whereas the work of such researchers as Ferrier started from evidence derived from accidental or artificial brain lesions in humans or in animals (Gall was opposed to animal experiments on the grounds of their cruelty).\n\n. Note that Benedikt's argument was criticized at the time by the celebrated psychiatrist and neurologist Theodor Meynert, who in 1876 wrote: \"[A]ttempts to localize complex phenomena of cerebral life, which only find expression in contact with social life, are not worthy of discussion. The scientific research towards the localization of cerebral powers has to restrict itself to simple functional energies, such as perception, motion, association, judgement or moods\" (quoted in Verplaetse 2004, 314).\n\n. We discuss the Society for Mutual Autopsy in chapter 2. Note that this was just one of a number of similar societies that, unhappy with the fact that most brains for study were those obtained from asylums, prisons, or almshouses, sought to collect, study, and preserve the brains of eminent persons, and to examine them in an attempt to identify the physical basis of their prowess in the size of the brain or the convolutions of the cortex (Gere 2003, 403\u20136). We will not repeat here the now familiar criticisms of craniometry in general, and Broca in particular, for their attempts to identify the peculiarities of the brains of different and inferior races, or to find, in the morphology of the brain, the basis for the differences between men and women (Gould 1981).\n\n. In 1880, the year of Broca's own death, a large study showed an average increase of 11 grams in the brains of 119 assassins, murderers, and thieves when compared with brains from non-criminals.\n\n. We can also note the work of Lorenzo Tenchini, who founded the Museum of Criminal Anthropology in Parma, and, alongside his other work in medicine and psychiatry, engaged in a lifelong study of the brains of criminals and the relationship between brain anatomy and criminal behavior (Lorusso, Cristini, and Porro 2007).\n\n. We have drawn on David Garland's helpful discussion here (Garland 1988).\n\n. Note that, in his contribution to the discussion that followed this address, Thomas Clouston, himself a former president of the association, disagreed strongly: \"No doubt most of us who have looked through the books of Lombroso and Havelock Ellis and others are inclined to admit that it is a little overdone by some of our continental brethren, but to say that the mass of criminals in this country are merely criminals by want of opportunity of doing good, by want of education, and not by their organization, is absolutely contrary to the results of psychological investigation for the last fifty years. I once had occasion to carefully examine the inmates of the Edinburgh prison, and if there was one thing that impressed itself upon me it was that I had to do with a degenerate aggregation of human beings\"(Clouston, in discussion following Nicolson 1895, 589). Other contributors to the discussion supported Clouston in this debate.\n\n. We have drawn on Hagner's excellent account in what follows.\n\n. Vogt, who had links with the Moscow Brain Institute, was called as a neurologist to consult on Lenin's illness; after his death he received Lenin's brain for histological study: the story is told by Paul Gregory (2008). Vogt had been Forel's pupil and then his collaborator\u2014they cofounded the _Journal f\u00fcr Psychologie und Neurologie_ , later renamed the _Journal f\u00fcr Hinforschung_ (Journal of Brain Research). As for Tonnies, he came from Vogt's hometown and had been his mentor and friend (see \"Vogt, Oskar Georg Dieckmann,\" _Complete Dictionary of Scientific Biography_ (<http:\/\/www.encyclopedia.com\/doc\/1G2-2830906176.xhtml>; consulted July 29, 2011).\n\n. Volkow, later to become director of NIDA, is Leon Trotsky's great-granddaughter; she grew up in the house in Mexico City where Trotsky was killed by a blow from an ice axe to his brain, though not from a repetitively violent offender, but from an undercover NKVD agent.\n\n. Then at the Forensic Neuropsychology Laboratory, Kirby Forensic Psychiatric Center, and Nathan S. Kline Institute for Psychiatric Research, New York University School of Medicine.\n\n. See, for example, presentations from Kent Kiehl and Adrian Raine at the March 2011 symposium on neuroscience and the law organized by the U.K. Royal Society's Brain Waves project and the U.S. Sackler Foundation. Transcripts of their talks (and talks from some of the other key figures in this debate such as Hank Greely and Stephen Morse) are available at <http:\/\/sites.nationalacademies.org\/PGA\/stl\/PGA_062477>; consulted July 10, 2011.\n\n. We will not discuss developments in other legal areas, for example, in civil law and in claims for damages relating to brain injury (where the term _neurolaw_ was originally coined), where things are very different; see our earlier note on this.\n\n. Jurisdictions vary in how they make such exceptions in relation to age, gender, medical conditions, and the like, some of which we discuss below. Of course, as noted above, there are differences between different domains of law\u2014tax law, shipping law, the law relating to specific types of activity such as driving a motorized vehicle, and so forth. Our discussion here focuses on the criminal justice system in jurisdictions with common law and adversarial systems, where the court acts as a supposedly impartial overseer of a combat between prosecution and defense, rather than, as in inquisitorial systems such as those in most countries in continental Europe, where the court is actively involved in the investigation of the offense. It is a regrettable fact that most published debate on the matters under discussion in this chapter also relates to such systems, and there is little published information on how these issues are being worked out in other jurisdictions.\n\n. Probabilities do, of course, figure in deliberations in civil cases, where the standard of proof is not 'beyond reasonable doubt' but 'the balance of probabilities.' As one reviewer of an earlier draft pointed out, there have been discussions of the use of Bayesian statistics in legal proceedings, but so far with little evidence of success (Fienberg and Kadane 1983; Taroni, Bozza, et al. 2010).\n\n. This issue is tellingly illustrated in the attempts by Dr. Richard J. Konkol to explain what was shown on his SPECT scans, when he gave evidence at the 1999 trial of Kip Kinkel, who was accused of the Oregon school shootings. In this case, the State chose not to cross-examine Konkol. The transcript is available at <http:\/\/www.pbs.org\/wgbh\/pages\/frontline\/shows\/kinkel\/trial\/konkol.xhtml>; consulted June 1, 2010.\n\n. There are many cases where brain scan evidence of structural abnormalities has been introduced, where this is relevant to questions of brain damage, as in cases where compensation is at stake. At the time of writing, December 2011, these are currently under research by Susan Wolf in the United States (<http:\/\/www.law.umn.edu\/facultyprofiles\/wolfs.xhtml#I4thmUSlx6Uir73faZ-K_w>) and by Lisa Claydon in Europe (<http:\/\/law.uwe.ac.uk\/staff-directory\/dol-staff\/lisa-claydon.aspx>); results are not yet published. In criminal trials in the United States, there was, of course, the famous case of John Hinckley Jr., found not guilty by reason of insanity in his 1982 trial after shooting Ronald Reagan: a CAT scan was introduced as part of the psychiatric evidence for the defense, although there is no indication that this was decisive. After this verdict, many U.S. states changed their legal codes to abolish the verdict of not guilty by reason of insanity, which mandates a psychiatric disposal, and replaced it with the verdict of guilty but insane, which allows the courts to choose among the full range of options for sentence.\n\n. We confine ourselves here to questions of violent or antisocial conduct in the criminal justice system and crime control policy. We will not discuss the many other areas where neurobiological evidence may be considered relevant, for example, in witness credibility, capacity to stand trial, etc.\n\n. There is no evidence that we know of, as yet, of neurobiological arguments being brought to bear in cases of infanticide\u2014one key area where the English law, since the 1920s, has established a partial defense in murder cases, as juries were reluctant to convict mothers who killed their newborn children if this meant a mandatory death sentence. The Law Commission of England and Wales reviewed this legislation in 2006, but it remains in force at the time of writing.\n\n. Jay Aronson has provided some helpful discussions of these issues (J. Aronson 2007, 2009).\n\n. For example, in 2002, the Supreme Court ruled by a majority of six to three, in the case of _Atkins v. Virginia_ , that executing individuals who are mentally retarded or mentally handicapped violated the Eighth Amendment ban on cruel and unusual punishment; the threshold for such a ban was left to individual states, but is usually deemed to come into effect with an IQ below 70. This case engendered much discussion. Wikipedia provides an accessible account of the case (<http:\/\/en.wikipedia.org\/wiki\/Atkins_v._Virginia>). Despite this Supreme Court ruling, and the appeals to it by his lawyers, in 2009 Bobby Wayne Woods, with an IQ below the threshold, was put to death by lethal injection for the rape and murder of the eleven-year-old daughter of his girlfriend (<http:\/\/www.guardian.co.uk\/world\/2009\/dec\/04\/texas-execution-mental-disability-iq>; both websites were consulted in July 2011).\n\n. Note that, in England, this issue is under investigation by the Law Commission as we write (<http:\/\/www.justice.gov.uk\/lawcommission\/areas\/unfitness-to-plead.htm>; consulted July 10, 2011). The commission published a consultation paper in October 2010, which does discuss issues of brain damage in relation to unfitness to plead, but apart from recommending the continued involvement of psychiatrists in the assessment, makes no reference to evidence from neuroscience. Some of these issues are discussed in Peay (2010).\n\n. Again note that our focus here is on the antisocial brain; we will therefore not discuss the use of neuroscientific arguments and neurotechnologies in other legal areas, for example, in relation to 'deception'\u2014as in the invention of neural lie detector technology. These issues are under ongoing investigation by Andrew Balmer of the University of Sheffield and have been discussed in a number of helpful conference papers (<http:\/\/sheffield.academia.edu\/AndrewBalmer\/CurriculumVitae>). See also Moriarty (2009). However, there is one emerging possibility here that is relevant\u2014the potential use of neural lie detection technologies in risk assessment of certain groups of offenders, for example, sexual offenders.\n\n. For more detail see Rose (2007c).\n\n. The online edition of the MIT newspaper, the _Tech_ , of October 22, 1993, reported it thus: \"More than three decades ago, a Dutch schoolteacher, troubled by a pattern of violence among his male relations, traced the pattern's origin to a couple who married in 1780. He concluded his kin must be suffering from an inherited mental disability. Pretending to be a dispassionate outsider, he then wrote up his notes under the title 'A Curious Case.' The teacher has long since died. But Friday, his 'curious case' earns a page in the annals of science, as a team of researchers from the Netherlands and the United States report that some men in his family harbor a mutant gene that predisposes them to aggressive behavior\" (<http:\/\/tech.mit.edu\/V113\/N51\/agression.51w.xhtml>; consulted June 10, 2010).\n\n. Van Praag was one of the group of researchers in the 1960s, influenced by the emergence of the first antidepressants and antipsychotic drugs, as well as the evidence from the use of LSD, who argued that psychiatric disorders had a biological basis. Initially working on the biological basis of endogenous depression and its links with serotonin, he later argued that genetically based variations in serotonin levels were linked to aspects of personality such as impulsivity and irritability. According to David Healy and others, his lectures on biological psychiatry in the Netherlands were picketed by students and he received death threats from antipsychiatry activists (Healy 1997). Further details of his work can be found in an interview conducted with him when he was elected to become an honorary member of the European College of Neuropsychopharmacology in 2007: (<http:\/\/matters.ecnp.nl\/number13\/interview3.shtml>; consulted December 1, 2010).\n\n. 5-hydroxyindoleacetic acid (5-HIAA), the main metabolite of serotonin in the human body, is used in urine samples to determine the body's levels of serotonin.\n\n. For further details, including the Mobley trial, see Rose (2007c).\n\n. As of June 2010, Brunner was professor of medical genetics and head of the department of human genetics at the Radboud University Nijmegen Medical Centre, focusing his work on the genetics of disease (<http:\/\/www.ncmls.eu\/NCMLS\/MenuStructures\/PI\/theme3\/HanBrunner.asp>; consulted June 10, 2010).\n\n. For Miczek's biography, see <http:\/\/ase.tufts.edu\/psychology\/peopleMiczek.htm>; consulted December 1, 2010.\n\n. The work of this panel is discussed in chapter 8 of Rose (2007c).\n\n. The members of this cohort were assessed on a range of measures at age three, and then at ages five, seven, nine, eleven, thirteen, fifteen, eighteen, twenty-one, and twenty-six, with future assessments scheduled for ages thirty-two, thirty-eight, forty-four, and fifty. Retention rates at age twenty-six were over 95%.\n\n. We discuss this discovery by Klaus Peter Lesch in chapter 5.\n\n. Much of the text in this section is derived verbatim from Rose (2010a).\n\n. Interestingly, Lynam's early research was carried out with Terrie Moffitt.\n\n. <http:\/\/www.dshs.state.tx.us\/mhprograms\/tmapover.shtm>; consulted February 24, 2009.\n\n. <http:\/\/www.pharmalot.com\/2008\/08\/texas-suspends-psych-drug-program-for-kids>; consulted December 1, 2010.\n\n. <http:\/\/www.teenscreen.org\/about>. For the controversy, see, for example, <http:\/\/www.teenscreentruth.com>; both websites consulted January 4, 2011. Note that many of the critics are widely believed to be organized by the Citizens Commission on Human Rights (CCHR), which is funded by Scientology.\n\n. The problems surrounding the use of biomarkers as predictors in psychiatry are discussed in detail in Singh and Rose (2009).\n\n. Those who work in this area distinguish between a relatively large group of children with 'conduct disorder' (around 7%), a smaller group within this with 'anti-social personality' (around 3%), and an even smaller group with 'psychopathy' (1%), and argue that each has its own characteristic pattern of neural anomalies. Some suggest that research is needed to establish which particular interventions are most appropriate for which group. For our purposes, these distinctions are not relevant.\n\n. This work is based on the Twins Early Development Study (TEDS) (<http:\/\/www.iop.kcl.ac.uk\/departments\/?locator=336>): \"a large-scale longitudinal study of twins from early childhood through adolescence. The twins were assessed longitudinally at 2, 3, 4, 7, 9, 10, 12, 14 and currently 16 years of age in order to investigate genetic and environmental contributions to change and continuity in language, cognitive and academic abilities and behavior problems from multivariate quantitative and molecular genetic perspectives. The twins were identified from birth records of twins born in the UK in 1994\u201396. More than 15,000 pairs of twins have been enrolled in TEDS and the participating families are representative of the UK population. TEDS data indicate that both genetic and environmental influences are important in nearly all areas of behavioral development.... The TEDS dataset is proving valuable in genome-wide association research that tries to identify some of the many genes responsible for the ubiquitous heritability of behavior.\" See also <http:\/\/www.teds.ac.uk>; consulted January 4, 2011.\n\n. The conference followed the publication of the Social Exclusion Action Plan (2006) and the \"Care Matters\" white paper (2007). See <http:\/\/www.personalitydisorder.org.uk\/resources\/emerging-pd>; consulted January 4, 2011.\n\n. These are the aspects emphasized in the presentation on \"Multi-Systemic Therapy\" by Brigitte Squire at the 2007 conference cited above (<http:\/\/www.personalitydisorder.org.uk\/news\/wp-content\/uploads\/brigitte-squire.pdf>; consulted January 4, 2011).\n\n. The phrase _incredible years_ derives from the well-known training series, described as follows on the organization's website: \"The Incredible Years, our award-winning parent training, teacher training, and child social skills training approaches have been selected by the U.S. Office of Juvenile Justice and Delinquency Prevention as an 'exemplary' best practice program and as a 'Blueprints' program. The program was selected as a 'Model' program by the Center for Substance Abuse Prevention (CSAP).... The Incredible Years Parents, Teachers, and Children Training Series has two long-range goals. The first goal is to develop comprehensive treatment programs for young children with early onset conduct problems. The second goal is the development of cost-effective, community-based, universal prevention programs that all families and teachers of young children can use to promote social competence and to prevent children from developing conduct problems in the first place. The purpose of the series is to prevent delinquency, drug abuse, and violence\" (<http:\/\/incredibleyears.com>; consulted January 4, 2011).\n\n. The details for the United States that follow in this paragraph are derived from the Prevention Action website (<http:\/\/www.preventionaction.org>; consulted January 4, 2011).\n\n. <http:\/\/www.preventionaction.org\/putting-brain-science-back-streets-los-angeles>.\n\n. <http:\/\/www.ChildTrauma.org>; consulted January 4, 2011.\n\n. According to the _Guardian_ on April 9, 2010, in a speech recorded by a member of the audience, Duncan Smith said: \"We keep going back, and as you track back you begin to realise that actually, for far too many people in society crime began before they were born. And that is a really strongly held belief of mine now, and more and more work was done in both social science, but even particularly now backed up by neuroscience demonstrates that the damage that we start children with, is damage that they keep, and that damage becomes more and more difficult as they go through.... We now know that we can pretty much figure out where an 18-year-old will be at the time that they are two and a half or three years old.\" He added the inability of a child to have \"imbibed the concept of empathy\" from its parent could have profound impacts on its later life. Perry said that these remarks were \"an oversimplification\" that \"greatly misrepresents the way we would explain the impact of neglect or trauma on the developing brain.\" He added: \"to oversimplify this way is, essentially, to distort.\" \"I do believe that overstating and misunderstanding the neurobiology can lead to confusion, anger, distortion and potentially to bad policy,\" he said, adding that the claims appeared to be \"a terrible distraction from the important issues related to the need to create family friendly, and developmentally informed policy that is aware and informed about the importance of early childhood and brain development\" (<http:\/\/www.guardian.co.uk\/politics\/2010\/apr\/09\/iain-duncan-smith-childrens-brains>; consulted January 4, 2010).\n\n. There were a number of studies of the developmental consequences of the extreme neglect suffered by Romanian orphans (Rutter, Beckett, Castle, et al. 2007; Rutter, Sonuga Barke, and Castle 2010). Brain images from the Romanian orphans, and stories of their neglect and recovery, gained considerable popular attention. For an example of the public reporting, see <http:\/\/americanradioworks.publicradio.org\/features\/romania\/b1.xhtml>; consulted 4 January 4, 2011.\n\n. <http:\/\/www.kidspeaceofmind.org>; consulted January 4, 2011.\n\n. For other evaluations of parenting programs in the United Kingdom, including those using the Incredible Years approach, see Edwards, C\u00e9illeachair, Bywater, et al. (2007); Hutchings, Bywater, Daley, et al. (2007); Jones, Daley, Hutchings, et al. (2008).\n\n. <http:\/\/www.kidspeaceofmind.org>; consulted January 4, 2011.\n\nChapter Seven. Personhood in a Neurobiological Age\n\n. Earlier versions of this chapter were presented at various places, including the University of California Los Angeles, the Centre for Research on Socio-Cultural Change, and the Universities of Durham and Exeter. Thanks to all for comments. We also return here to some themes discussed in a number of papers dating back to the 1980s, some of which were brought together under the title _Inventing Ourselves: Psychology, Power, and Personhood_ (Rose 1996).\n\n. For a very small selection of neuroscience books on these issues written for a non-specialist audience, in addition to the work of Francis Crick, Joseph LeDoux, Chris Frith, Thomas Metzinger, Julian Baggini, and the Churchlands cited below, see Gazzaniga (2008a); Greenfield (2008); Damasio (2010); Baggini (2011).\n\n. We should note that one of the present authors suggested some time ago that we had witnessed the birth of what he termed \"the neurochemical self\" (Rose 2003a, 2003b, 2004)\u2014a term that was meant to imply, not a wholesale mutation in personhood, but the availability of a neurochemical register within which individuals could describe, judge, and seek to modulate their mental states and ailments.\n\n. <http:\/\/www.brainhood.net>; consulted June 2009.\n\n. We use this compendious German term for the sciences of the spirit, incorporating the humanities, the social sciences and many other disciplines, as opposed to the natural and mathematical sciences, the _Naturwissenschaften_.\n\n. Of course, discussion of what Mauss really meant depends a great deal on questions of translation. We return to this question later in this chapter. See also note 18, below.\n\n. By the phrase _positive science_ we do not mean to imply approval of their epistemological bases, let alone of their normative claims\u2014merely that these endeavors justify their truth claims by reference to empirical evidence in a characteristic manner.\n\n. We should note that each author finds his or her own way of making sense of these; for example, Feinberg chooses a quasi-Freudian language when he asserts: \"The neuropathologies of the self are a group of highly complex and multi-determined syndromes in which brain dysfunction causes a profound and specific disturbances [ _sic_ ] of ego boundaries and ego functions (ego disequilibrium). This disturbance facilitates the emergence of primitive ego functions and primitive psychological defenses that are essential for full creation of the NPS. The neuropathology in the majority of these cases involves the right hemisphere with a predilection for frontal involvement especially within right medial heteromodal association cortices. This latter region is hypothesized to play a critical role in the establishment of ego boundaries and to mediate the relationship between self and world. According to this hypothesis, the preservation and activation of the largely verbal defenses, such as verbal denial, projection, splitting, and fantasy may be the result of the remaining, and presumably relatively intact left verbal hemisphere. The evidence suggests that given the fact these immature defenses and fantasies are preserved and even activated in the presence of right hemisphere damage, the emergence of the mature defenses in childhood, and their functioning in adulthood, critically depends up [ _sic_ ] right hemisphere functions\" (Feinberg 2011, 79).\n\n. We have discussed some aspects of these experiments, and of Gazzaniga's conclusions from them, in previous chapters.\n\n. The references that Gazzaniga draws on to reach this conclusion include Turk, Heatherton, Kelley, et al. (2002); Cooney and Gazzaniga (2003); Turk, Heatherton, Macrae, et al. (2003).\n\n. Oliver Sacks recounts a particularly striking case of this phenomenon, one he terms \"internal amputation\" in _A Leg to Stand On_. However, his experience comes, not from brain damage but from the immobilization of his badly broken leg in a plaster cast for several months\u2014suggesting that the relation between the limb and the brain can equally work in the reverse direction (Sacks 1984).\n\n. The philosopher Daniel Dennett has been a persistent critic of Libet's work and his conclusions (Dennett 2003a, 2003b).\n\n. The title of this section is taken from a conference held by the New York Academy of Sciences on September 26\u201328, 2002, and a subsequent volume containing essays by many of the participants (LeDoux, Debiec, and Moss 2003).\n\n. This has been an argument particularly developed by Jaak Panksepp, who has suggested that humans share with many other animals what he terms a \"core self\" that is capable of integrating information about the animals' (or humans') own internal body states and their goal orientations, generating the affects or emotions that promote the behaviors required for survival; see, for example, Panksepp (2005); Northoff and Panksepp (2008).\n\n. Theory of mind is discussed in chapter 5.\n\n. Others follow a similar route in suggesting that it is humans' access to their own experiences of self that underpins insight into the mental states of others considered as similar selves. Thus those committed to the significance of mirror neurons, discussed in chapter 5, also draw on evidence from those diagnosed with autism spectrum disorder, claiming this is linked to a disruption in the capability of the mirror neuron system to represent the action goals of others, and to link those to the self's own action goals\u2014for it is this that \"critically contributes to shaping our experience of ourselves and of other selves, providing us with a multilayered motor representation both of our own and of others' action possibilities [and] paves the way for the higher-level forms of self-and other-awareness generally thought to be at the core of our full-fledged sense of self and sense of others\" (Sinigaglia and Rizzolatti 2011, 72).\n\n. Of course, not all of those who contribute to the emerging discipline of cultural neuroscience frame their arguments in terms of neuroplasticity; others prefer to think in terms of gene-environment interactions and consequential cultural variation in patterns of gene expression in neural development and pathways (Chiao 2009). This is not the place to review the very extensive neurobiologically inflected literature on culture and cognition.\n\n. Many have sought to clarify Mauss's argument. When he titled his talk \"Une cat\u00e9gorie de l'esprit humanine: La notion de personne, celle de 'moi',\" what exactly did he mean by these terms? (Carrithers, Collins, and Lukes 1985). As we have mentioned earlier, the issue is somewhat bedeviled by translation. The original text, in French, reads: \"Pas plus que de linguistique, je ne vous parlerai de psychologie. Je laisserai de c\u00f4t\u00e9 tout ce qui concerne le 'moi,' la personnalit\u00e9 consciente comme telle. Je dirai seulement: il est \u00e9vident, surtout pour nous, qu'il n'y a jamais eu d'\u00eatre humain qui n'ait eu le sens, non seulement de son corps, mais aussi de son individualit\u00e9 spirituelle et corporelle \u00e0 la fois.... Mon sujet est tout autre, et est ind\u00e9pendant. C'est un sujet d'histoire sociale. Comment, au cours des si\u00e8cles, \u00e0 travers de nombreuses soci\u00e9t\u00e9s, s'est lentement \u00e9labor\u00e9, non pas le sens du 'moi,' mais la notion, le concept que les hommes des divers temps s'en sont cr\u00e9\u00e9s? Ce que je veux vous montrer, c'est la s\u00e9rie des formes que ce concept a rev\u00eatues dans la vie des hommes des soci\u00e9t\u00e9s, d'apr\u00e8s leurs droits, leurs religions, leurs coutumes, leurs structures sociales et leurs mentalit\u00e9s\" (<http:\/\/classiques.uqac.ca\/classiques\/mauss_marcel\/socio_et_anthropo\/5_Une_categorie\/Une_categorie.xhtml>; consulted July 1, 2011). It is often argued that Mauss was trying to 'sociologize' Kant's universal categories, but as Steven Lukes points out, for Kant \"the self, or thinking subject ( _das Ich_ ), is _not_ a category, but 'the vehicle of all concepts' and itself transcendental but not knowable,\" presupposed by all experience but not amenable to knowledge (Carrithers, Collins, and Lukes 1985, 283). And when Mauss spoke of ideas, was he concerned with specialized, articulated knowledge, with more or less formalized discourses, or was he concerned with the implicit presuppositions of lay beliefs or indigenous psychologies? These and other issues are debated in the volume cited above.\n\n. As Mauss refers to Ribot, we might note here that Th\u00e9odule Ribot was one of many in the late nineteenth century (including Sigmund Freud) who argued that consciousness is but a small part of a much larger domain of mental life\u2014for example, asserting in his _Maladies de la personalit\u00e9_ of 1885 that \"what emerges of the _moi_ into full consciousness is only a small part of what remains buried although active. The conscious personality is never more than a small part of physical personality.... [I]t is by studying how the _moi_ unravels [ _se defait_ ] that we can understand how it was originally constructed\" (quoted in Nicolas and Murray 1999, 290).\n\n. The idea of \"self-fashioning\" was most articulately developed by Stephen Greenblatt in his study _Renaissance Self-Fashioning: From More to Shakespeare_ (1980). Self-fashioning, for Greenblatt, is the self-conscious process of fashioning character or identity, which almost always occurs in relation to one or more authorities, often in competition, and usually in order to exclude or minimize some other form of the self that is perceived as alien, strange, or in some way threatening. We differ in our usage here, placing less emphasis on fashioning that concentrates on the outward person, and in our recognition that self-fashioning is not a process primarily undertaken in the element of language.\n\n. We know a great deal about the history of the use of these drugs and the rise in consumption of the different classes of psychiatric drugs in different countries and geographical regions over the last two decades of the twentieth century, especially of those newer drugs claiming to be based on neurobiological research and to target precise neural pathways that underpin disorders such as depression, anxiety, or attention deficit hyperactivity disorder. See chapter 7 of Rose (2007c) and also Rose (2006b).\n\n. There have been many fictional versions of such a fantasy, and its costs, most recently in Alan Glynn's _Limitless_ (2011).\n\n. <http:\/\/www.dana.org\/brainweek>; consulted April 1, 2011.\n\n. <http:\/\/ibreaus.org\/site\/why-brain-education\/>; consulted April 1, 2011. The IBREA also enunciated the \"Brain Declaration\": \"World-renowned scholars, thinkers, and social activists gathered in Seoul, South Korea on June 15, 2001, to attend the New Millennium World Peace Humanity Conference. It was the first Humanity Conference held to highlight the values of Humanity, the Earth, and the Brain.... The adoption of the Declaration of Humanity reflected a fundamental shift in consciousness, capable of resolving the problems facing the human race. The Declaration became the philosophical foundation of Brain Education and was later adapted and renamed the Brain Declaration. Essentially, the Brain Declaration is a set of statements that affirm the power of the human brain: I declare that I am the master of my brain; I declare that my brain has infinite possibilities and creative potential; I declare that my brain has the right to accept or refuse any information or knowledge that it is offered; I declare that my brain loves humanity and the earth; I declare that my brain desires peace\" (<http:\/\/www.ibreaus.org\/newdesign2011\/?page_id=13>).\n\n. <http:\/\/www.powerbrainedu.com>; consulted April 1, 2011.\n\n. <http:\/\/www.braingym.org.uk\/about\/about.htm>; consulted April 1, 2011.\n\n. See, for example, SharpBrains (<http:\/\/www.sharpbrains.com>; consulted April 1, 2011).\n\n. <http:\/\/www.neurobics.com\/explain.xhtml>; consulted April 1, 2011\u2014one of many such programs that appears to be a commercial spin-off developed by a research neurobiologist.\n\n. For the Mind and Life Institute, see <http:\/\/www.mindandlife.org>; consulted April 1, 2011. The work of Francisco Varela was one important element in this relationship, and he is remembered in the institute's annual Varela Awards. Varela was trained in medicine and biology and did his doctorate on information processing in the retina with Torsten Wiesel. He trained as a Buddhist monk in Tibet in the 1970s (Hayward and Varela 1992; Varela, Thompson, and Rosch 1993).\n\n. For example, Ilchi Lee (<http:\/\/www.ilchi.com>; consulted April 1, 2011): \"Ilchi Lee is a pioneering Brain Philosopher and Educator. He has developed brain training programs that are widely used in many organizations around the world. In his homeland, South Korea, his programs have been adopted as mind and body training methods in the Ministry of Education, Samsung Corporation, and the Korean Army.... Ilchi Lee has spent several decades investigating ways to develop the potential of the human brain. Through his life-long pursuit of brain-centered training methods and programs, hundreds of thousands of people around the world have achieved the benefits of healthier bodies, improved learning, business success, and personal empowerment.\"\n\nConclusion. Managing Brains, Minds, and Selves\n\n. To aid the readability of our conclusion, we have chosen, in the main, not to include bibliographic references where we are summarizing arguments made in previous chapters.\n\n. Neuronal Man is the title of a much-reprinted book by the eminent French neuroscientist Jean-Pierre Changeux (1985, 1997).\n\n. GxE is the conventional notation used to indicate that there are \"interactions\" between genes\u2014or a specific genomic sequence\u2014and environmental factors.\n\n. Woese distinguishes what he terms \"empirical reductionism\"\u2014the methodological analysis of constituent parts and their relations\u2014from \"fundamentalist reductionism\"\u2014by which he means a metaphysical belief that living systems can be completely understood in terms of the specific properties of their constituent parts\u2014a view that came to prominence with the rise of molecular genetics, and that, in his view, denies what classically trained biologists took for granted: the existence of emergent properties.\n\n. This issue is perceptively discussed in two papers by Cathy Gere (2004a, 2004b).\n\n. The theorist who has pursued this line of thought most rigorously is Elizabeth Wilson, and there are many affinities between her project of \"gut feminism\" and the issues that we raise here. For some early products of this work, see Wilson (2004a, 2004b).\nReferences\n\nAbi-Rached, J. M. 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See also antisocial behavior\n\nAlexander, Franz,\n\nAllen, Graham, ,\n\nAllport, Gordon,\n\nAlthusser, Louis,\n\nAlzheimer's disease, , , , , 261n34\n\nAmerican Society for Neurochemistry, ,\n\nAndriezen, W. Lloyd,\n\nanimal models: anthropomorphism and, , 97\u201398, 103\u20134; and artificiality of laboratory as research setting, 85\u201392; behavioral studies, , , , 96\u201399; cerebral localization and, ; computer simulations as replacement for, 108\u20139; and conceptual models, ; and DSM categories, 105\u20138; ethics of animal-based research, , 247n37; generalization and, 94\u201395; and human psychopathologies, , 99\u2013102, ; and human sociality, , 150\u201351; and human specificity, 102\u20134; vs. mechanical or material models, 92\u201393; mereological fallacy and, 83\u201384; reverse translation, 106\u20137; self-vindication and laboratory research, 88\u201389, 92\u201393, , 105\u20139; \"setup\" and, , 85\u201389, , , ; species selection for, 89\u201390; specificity of individual organisms, 94\u201395; and stabilization of the experimenter, 85\u201386; stabilization (standardization) of animals used, 89\u201391; structures, functions, and disease models (models3), 92\u201395; \"style of reasoning\" and, 82\u201383; translation of findings, , , 95\u201396, 104\u20138\n\nantisocial behavior: \"bioprediction\" of, ; brain imaging and, , 165\u201366; and \"callous and unemotional traits\" (AS-CU), ; costs of, , 192\u201393, , ; diagnosis of antisocial personality disorders, 102\u20133, , 176\u201377; and environmental influences on brain development, , , 187\u201390; gene-environment interaction and, ; genomics and, , , 180\u201390, 181\u201389, 261n31, 269n31; linked to physical abnormalities in the brain, , , , , ; as neurobiological in origin, ; parental input and, ; preventive interventions, , 15\u201316, 188\u201396; \"screen and intervene\" control strategy, , , 166\u201367, 189\u201391, . See also criminality; criminal justice system\n\nAos, Steve,\n\nAronson, Elliot,\n\nasylums: \"asylum psychiatry\" and treatment of mental disorders, ; and clinical gaze, 55\u201359, , , 249n10, 249n12; as setting for research, , ; as setting for treatment of mental disorders, , , ,  Atkins v. Virginia, 268n27 autism spectrum disorders: and abnormal social cognition in, 148\u201350; as developmental, , , ; diagnosis of, , , 240n29; mentalizing deficits and, 148\u201350, 216\u201317, 273n16; mirror neurons and, 273n16\n\nautopsies, 61\u201365, 165\u201366, , 266n9\n\nBach-y-Rita, Paul,\n\nBakker, C.,\n\nBarlow, David,\n\nBaron-Cohen, Simon, ,\n\nBateson, Gregory,\n\nBatmanghelidjh, Camila,\n\nBayh-Dole Act (1980), 19\u201320\n\nBear, Mark,\n\nBegley, Sharon,\n\nbehavior: animal models and, 96\u201399; nonconscious processes and, 21\u201323, 161\u201362, , 265n40. See also antisocial behavior\n\nBelmonte, M.,\n\nBenedikt, Moriz,\n\nBennett, M. P., 83\u201384\n\nBenzer, Seymour,\n\nBerger, Hans,\n\nBerger, Philip,\n\nBernard, Claude, ,\n\nBernard, Jean,\n\nBerntson, Gary, 152\u201353\n\nBeyesian theory,\n\nbiopolitics, , 13\u201316,\n\nbipolar disorder, , , , , , 241n37, 257n59\n\nBlakemore, Sarah-Jayne, ,\n\nBloch, Felix,  blood flow, brain function imaging and, , 71\u201373\n\nBotteron, Kelly,\n\nBrady, Joseph,\n\nbrain: anatomical structures in, , ,  (see also localization of brain functions); autopsy and investigation of the, 61\u201365, 165\u201366, , 266n9; conceptualization of, 216, 225\u201326,  (see also social brain); evolution and structures in, , 243\u201344; as focus of research, 32\u201333; function as observable,  (see also function, visualization of brain); interconnected synaptic network of neurons in, , , , ; mental processes as brain events, ; mereological fallacy and, 83\u201384; mind\/brain relationship, 3\u20134, , , , 204\u20135, ; the molecular brain, 9\u201310, , 225\u201326; as object, , , ; as organ, , 32\u201333, 43\u201344, , , ; as part of interconnected system of the body, ; physical measurement of, , 168\u201369, ; self and, , , 205\u20137, 212\u201317; \"social brain\" hypothesis (see social brain)\n\nbrain education, , 221\u201322\n\nBraitenberg, Valentino,\n\nBreakefield, Xandra,\n\nBrenner, Sydney,\n\nBroca, Paul, , , 62\u201363,\n\nBrodie, Bernard \"Steve,\"\n\nBrodmann, Korbinian, , ,\n\nBrooks, David, 161\u201362\n\nBrothers, Leslie, ,\n\nBrowne, James Crichton,\n\nBrunner, Han, 181\u201383, 185\u201386,\n\nBuckholtz, Joshua, 188\u201389\n\nBucknill, John C., , , ,\n\nBurton, Robert,\n\nBush, George H. W.,\n\nBush, Vannevar, 26\u201327\n\nByatt, A. S.,\n\nCacioppo, John, 152\u201356,\n\nCalhoun, John C.,\n\nCameron, David,\n\nCameron, J.,  Canguilhem, Georges, 29\u201331, , , 94\u201395, ,\n\nCarlsson, Arvid, , 36\u201337\n\nCarter, Rita,\n\nCartwright, Nancy,\n\nCaspi, Avshalom, 186\u201388, 191\u201392, 256n55\n\nCAT scans (computerized axial tomography), 68\u201370\n\nChadarevian, Soraya de,\n\nChangeux, Jean-Pierre, ,\n\nCharcot, Jean-Martin, , , ,\n\nCharney, Dennis,\n\nchemical neurotransmission, 29\u201332,\n\nchildren: attribution of criminal responsibility to, ; brain development of abused or neglected, , 186\u201390, 193\u201396, 271n54, 271n55; childrearing and neuroscience, 8\u20139; diagnosis of mental disorders in, , , 241n37; and governing through the brain, ; preventive intervention during childhood, 15\u201316, , 190\u201398; psychopharmaceuticals and, , 241n37; and social neuroscience,\n\nChou, I. H.,\n\nChouard, T.,\n\nChurchland, Patricia, , 215\u201316\n\nclinical gaze: asylums as site of, 55\u201359, , , 249n10, 249n12; individualization and, , 57\u201361, , , 86\u201387, 103\u20136; and neuromolecular research, ; object\/subject relationship and, ; as practice of seeing, 55\u201356; spatial and temporal dimensions of, ; technology and (see imaging technologies)\n\nClouston, Thomas,\n\nCobb, Stanley,\n\nCohen, Jonathan,\n\nCohen, Mark,\n\nCohn, Simon, 76\u201377\n\nCombe, George, 61\u201362\n\nCooney, J. W.,\n\nCooper, Leon,\n\nCormack, A. M.,\n\ncortical stimulation,\n\nCoryell, Charles,\n\nCrawley, Jacqueline, , , , 99\u2013101\n\nCrichton-Browne, James, , 59\u201360, ,\n\ncriminality: anatomy and, 167\u201377; criminal anthropology as field of study, 167\u201372, , 249n12; diagnostic criteria and, ; free will and criminal responsibility, , , 177\u201378, 196\u201397, 242n44; as insanity, ; and lack of empathy, , , 271n54. See also criminal justice system\n\ncriminal justice system: attribution of criminal responsibility, ; brain images as evidence in criminal court, , ; deceit and lie detection, 269n29; insanity as exculpatory (NGRI), ; intent and criminal responsibility, ; mental capacity and criminal responsibility, , 268n27\n\nCT scans (computerized axial tomography), 68\u201370\n\nDagonet, Henri,\n\nDale, Henry,\n\nDamadian, Raymond,\n\nDANA foundation, , 177\u201378,\n\nDandy, Walter,\n\nDarwin, Charles, ,\n\nDean, Brian, 165\u201366\n\nde Beauvoir, Simone,\n\nDecety, Jean,\n\ndecision making: Libetism and awareness of intent, 209\u201313; nonconscious processes and, , 21\u201323, 161\u201362, , , 209\u201315\n\nDeiters, Otto,\n\nDelay, Jean, 35\u201336\n\ndementia, 14\u201315, , , ,\n\nDeniker, Pierre,\n\nDennison, Paul,\n\ndepression, ; costs of, , ; diagnosis of, , , 125\u201327, , 240n29; monoamine hypothesis and, , , 92\u201393, 134\u201335; and pharmaceutical intervention, , , ; screening and suicide prevention, ; serotonin and, 150\u201351, , 269n32\n\nDespine, Prosper,\n\ndiagnosis: antipsychiatry and critiques of, ; and classification or categorization of disorders, , , , 114\u201316, 118\u201320, 122\u201324, , , , , , ; and definition of insanity, 110\u201312, ; errors in (including false positives), , , , 129\u201330, 250n29; estimated prevalence of mental disorders, , 127\u201329, ; Feighner criteria, 120\u201321; genetic testing and, 130\u201332; manuals for, ,  (see also Diagnostic and Statistical Manual of Mental Disorders (DSM)); neuroimaging technology and, , , 250n29; objectivity and, 4, 111\u201316, 130\u201333, 137\u201338, ; population demographics and, 115\u201316, , ; of prodromal or mild disorders, , , , , , 136\u201337, 259n19; of Psychosis Risk Syndrome, 136\u201337; self-vindication and, 105\u20138, 138\u201339; and threshold of disorder or abnormality, , , ,  (see also of prodromal or mild disorders under this heading); and visualization, 60\u201361\n\nDiagnostic and Statistical Manual of Mental Disorders (DSM), 105\u20138, 131\u201337; causal assumptions avoided in, 118\u201319; DSM II revision, 118\u201320; DSM III revision, 99\u2013100, 114\u201315, 120\u201322; DSM IV revision, 123\u201325; DSM 5 revision, , , 131\u201337, 259n19, 260n23; first formulation of, ; self-vindication and, 105\u20138\n\nDiamond, Hugh, ,\n\nDidi-Huberman, Georges,\n\nDishion, Tom,\n\nDomasio, Antonio,\n\ndopamine hypothesis, , 36\u201337\n\ndrugs: substance abuse as mental disorder, , , , 176\u201377. See also pharmaceuticals\n\ndualism, 204\u20135\n\nDumit, Joseph, ,\n\nDunbar, Robin, ,\n\nDuncan, G. J.,\n\nEccles, John, ,\n\neconomics: brain education products and, 222\u201323; costs of antisocial behavior, , 192\u201393, , ; costs of intervention, 136\u201337, , , , ; costs of mental disorders, , 16\u201317, , , 259n15; intellectual property rights and research, 19\u201320; market for pharmaceuticals, ; neuroeconomics as discipline, , 265n40; profit as research motive, , ; of research, 15\u201319, 232\u201333\n\nelectrical neurotransmission, 29\u201332,\n\nelectroencephalography (EEG), ,\n\nEllis, Havelock, 167\u201369\n\nembodiment: and brain as part of interconnected system, ; of mental disorders, 56\u201361; neural process as embodied, ; \"rubber hand\" and phantom limb phenomena, ; and selfhood,\n\nempathy, , 21\u201322, 141\u201342, 146\u201349, , 157\u201358, , ; as biological in origin, 143\u201344; criminal behavior and lack of, , , 271n54; mirroring and, 263n9; social cognition and, 143\u201344; as social rather than biological in origin, 264n35\n\nenvironment: and antisocial behavior, , , 187\u201390; gene-environment interactions, , , , 191\u201392, 256n55, 261n31,\n\n270n47, 273n17, 275n3; influences on brain development, , , , , , ; intervention and, 15\u201316; laboratory setting as, , , ; milieu, transaction or interaction with, 23\u201324, , , , 103\u20134, , , , , , , ; plasticity and response to, ; resilience and, , , 187\u201388, ; social brain and, 150\u201351, 159\u201362, 264n24; social interaction as, 230\u201332,\n\nepigenetics, , , , , ,\n\nErlanger, Joseph,\n\nEsquirol, Jean, 57\u201359,\n\neugenics, , , , , , , , , 180\u201381, 246n22\n\nEuropean Brain and Behavior Society,\n\nEuropean Neuroscience and Society Network,\n\nEuropean Society for Molecular Imaging (ESMI),\n\nevolution: of brain and neurons, , 243\u201344;\n\nand conservation across species, , , 90\u201391, ; social behavior and brain, 143\u201352, , , ; and use of animal models, , 90\u201395; variation and, . See also evolutionary psychology\n\nevolutionary psychology, , , 143\u201344,\n\nFaraday, Michael,\n\nFarah, M. J.,\n\nFeighner, John,\n\nFeighner criteria, 120\u201321\n\nFenton, W. S., 128\u201329\n\nFernald, Russell,\n\nFerrier, David, , 64\u201365, ,\n\nFirst, Michael,\n\n5-HTT (serotonin) transporter gene, , 187\u2013188\n\nFlechsig, Paul, , , 62\u201363,\n\nFleck, Ludwick, 42\u201343\n\nFlourens, Pierre,\n\nFodor, Jerry, , 262n5\n\nFood and Drug Administration (US FDA),\n\nForel, August, 172\u201373\n\nFoucault, Michel, , , , , , 248n6. See also clinical gaze\n\nFox, N. A.,\n\nFragile X syndrome, 101\u20132\n\nFrances, Allen,\n\nFranks, David,\n\nFreeman, Walter, , 245n17\n\nfree will, ; and criminal responsibility, , , 177\u201378, , 196\u201397, 242n44; impulse control and, ; Libetism and, , -Rose. 14; nonconscious processes and, 21\u201323, 209\u201319, 242n44; self-awareness and, , , 209\u201319\n\nFreud, Sigmund, 6\u20137, , , , , , 199\u2013200,\n\nFriese, Carrie,\n\nFrith, Chris, , , , 200\u2013201, 204\u20135, 207\u20139, 211\u201312\n\nFrith, Uta,\n\nFritsch, G.,\n\nfunction, visualization of brain, 74\u201380; blood flow and, , 71\u201373; functional magnetic resonance imaging (fMRI), , , , , , 165\u201366; and localization, 75\u201376, ; vs. structural, , 69\u201370\n\nfunctional MRI (fMRI), , , , , , 165\u201366\n\nfunding for research: activist groups and, 18\u201319; brain disorders and, ; public vs. private industrial research, 16\u201318; in social neuroscience, 156\u201357\n\nFuortes, M.G.F.,\n\nfuturity, 13\u201314\n\nGage, Phineas, , 205\u20136\n\nGalambos, Robert,\n\nGall, Franz Joseph, 61\u201362, 167\u201368, 266n7\n\nGarner, Joseph, 88\u201389\n\nGasser, Herbert,\n\nGastaut, Henri,\n\nGawrylewski, Andrea,\n\ngaze. See clinical gaze\n\nGazzaniga, Michael, 4\u20135, , 177\u201378, 206\u20137, ,\n\nGeertz, Clifford,\n\ngenetics, ; and animal models, ; antisocial behavior and, , , 180\u201390, 181\u201389, 261n31, 269n31; diagnosis and genetic testing, 130\u201332; epigenetic factors, , , , , , ; 5-HTT (serotonin) transporter gene, , 187\u201388; gene-environment interactions, , , , 191\u201392, 256n55, 261n31, 270n47, 273n17, 275n3; genomics and therapeutic advances, 131\u201332, ; mental disorders as inheritable, 33\u201334, 133\u201334, 257n59, 261n13; molecular genomics, 44\u201345; neurogenetics and mutation, ; neurogenomics, 11\u201312, , , ; phenomics and genomics, ; psychiatric genomics, , 131\u201332, 180\u201390; sequencing of human genome, 44\u201345; and social neuroscience, 151\u201352\n\nGere, Cathy,\n\nGigerenzer, Gerd,\n\nGillihan, S. J.,\n\nGilman, Sander, 56\u201359\n\nGlueck, Bernard,\n\nGoffman, Erving,\n\nGoldman, David,\n\nGoldstein, Kurt,\n\nGolgi, Camillo, 31\u201332,\n\nGolla, Frederick,\n\nGould, Elizabeth,\n\nGould, Stephen Jay, , 249n18\n\ngoverning through the brain, , 13\u201316; environment and brain development, ; intervention and preemptive crime control, 166\u201367, 197\u201398; intervention to influence brain development, 195\u201398; neuroscience as influence on policy, ; and nonconscious processes, 21\u201322; personal intention free choice and accountability, ; plasticity and, , ; \"screen and intervene\" policies and, , 166\u201367, 189\u201391, ; social brain theory and, , 160\u201363\n\nGreeley, Henry \"Hank,\"\n\nGriesinger, Wilhelm, ,\n\nGrisso, Thomas,\n\nGrohmann, Johann Christian August,\n\nGruenberg, Ernest, 118\u201319\n\nGur, Raquel,\n\nGur, Ruben,\n\nHaag, Amanda, 110\u201311\n\nHacker, P.M.S., 83\u201384\n\nHales, R. e., 111\u201312\n\nHandesman, Josef,\n\nHaraway, Donna,\n\nHarlow, Harry, 96\u201397\n\nHaynes, John-Dylan, 53\u201354\n\nHearne, Vicki, 252n5\n\nHebb, Donald,\n\nH\u00e9caen, Henry,\n\nHecht, Jennifer,\n\nHering, Henry,\n\nHervsey, Georg Van,\n\nHess, Walter, 245n17\n\nHill, Alex,\n\nHill, Denis,\n\nHinde, Robert,\n\nHis, Wilhelm, Sr.,\n\nHitzig, E.,\n\nHodgkin, Alan, ,\n\nHolley, Andr\u00e9, ,\n\nHood, William,\n\nHopwood, Nick,\n\nHorwitz, Allan,\n\nHounsfield, Godfrey N.,\n\nHubel, David,\n\nHughes, Virginia, 165\u201366\n\nHuxley, Andrew, , ,\n\nHyman, Steven, , 127\u201328, , 133\u201334\n\nhysteria, , , ,\n\nIacoboni, M.: mirror neuron theory, 145\u201346, , , 160\u201361\n\nimaging technologies: computerized tomography (CT), 12\u201313; diagnosis and, ; electroencephalogram, ; electroencephalography (EEG), , ; as evidence in criminal cases, , ; fMRI and the functioning brain, , , , , , 165\u201366; and images as artifacts or simulations, 77\u201379, ; interpretation of, ; magnetic resonance imaging (MRI), , 68\u201369, 72\u201373, , ; and objectivity, , 158\u201359, ; photography, 59\u201361, 249n12; positron emission tomography (PET), 70\u201371, ; and processes in the living brain, 53\u201354, 65\u201374 (see also fMRI and the functioning brain under this heading); single photon emission computerized tomography (SPECT), ; and social brain, ; staining and, ; structural vs. functional, , 69\u201370; X-rays, , 66\u201368\n\nindividualization, ; and the clinical gaze, , 57\u201361, , , 86\u201387, 103\u20136; and objectification, ; and organisms used in animal research, 90\u201391, 94\u201395; reductionism and, 20\u201321, ; social neuroscience and the individual, 151\u201352, 160\u201361; and visualization of the brain, 20\u201322\n\ninfrastructures for neuroscience, 38\u201341, 151\u201356\n\nInsel, Thomas, , 101\u20132, 105\u20137, 128\u201329, 149\u201350, 254n33\n\nintellectual property rights, 19\u201320\n\ninterdisciplinary nature of neuroscientific research, 25\u201328, 39\u201342, , 155\u201356, 229\u201330\n\nInternational Brain Research Organization\n\n(IBRO), ,\n\nInternational Neuropsychological Symposium, ,\n\nInternational Society for Neurochemistry,\n\nintervention: during childhood brain development, 15\u201316, , 190\u201398; costs of, 136\u201337, , , , ; focus on the developing brain, ; genetic predisposition and, ; parental involvement and, 15\u201316; prodromal diagnosis and, , 136\u201337, 259n19; risk and, 117\u201318, , ; \"screen and intervene\" logic of, , , , 166\u201367, 189\u201390, 197\u201398, 262n37; and susceptibility, 191\u201392\n\n\"invisible college\" and neuroscience research, , ,\n\nJackson, John Hughlings,\n\nJasper, Hebert,\n\nKandel, Eric,\n\nKarp, David,\n\nKatz, Bernard, , 254n26\n\nKaymaz, Nil, 136\u201337\n\nKeller, Evelyn Fox, 91\u201392, 256n36\n\nKendell, Robert,\n\nKessler, Ronald, 125\u201329, 259n19\n\nKety, Seymour,\n\nKevles, Bettyann Holzmann, , ,\n\nKiehl, Kent,\n\nKieser, Dietrich George,\n\nKlerman, Gerald, 120\u201321\n\nKnapp, Martin,\n\nKoyr\u00e9, A.,\n\nKraepelin, Emil, 60\u201361\n\nKramer, Morton,\n\nKrausse, J.,\n\nKrishnan, V.,\n\nKuffler, Stephen, 39\u201341,\n\nKuhl, David,\n\nKupfer, David,\n\nlaboratory as setting for research, 76\u201377; artificiality of, 85\u201392, ; and diagnostic criteria, ; as impoverished environment, 254n29; reductionism and, 228\u201329; self-vindication and, ; \"setup\" and, 76\u201378, 85\u201389, , ; sociality and, 158\u201359; social world as alternative to, 228\u201332; \"truth-making\" and, 87\u201388\n\nLaborit, Henri,\n\nLamm, Claus, 141\u201342,\n\nLatour, Bruno, , ,\n\nLauterbur, Paul,\n\nLawrence, Ernest,\n\nLeDoux, Joseph, 214\u201315\n\nlegal system. See criminal justice system\n\nLeidsdorf, Max,\n\nLesch, Klaus-Peter, 150\u201351, 255n35\n\nLevi-Montalcini, Rita,\n\nLiddell, H. S.,\n\nLin, Keh-Ming,\n\nlobotomies, 34\u201335, ,\n\nlocalization of brain functions: and complex or social function, 31\u201333, 266n8; imaging technologies and, , 75\u201379, , 154\u201355, 158\u201359, , 175\u201376; lesions and evidence of, , 61\u201362, , 205\u20136, 266n7; modularity thesis and, , 262n5; physical experimentation (surgery, direct stimulation, ablation) and, , 61\u201365, , , 245n17, 250n23; and selfhood, 216\u201317; social brain and, ; \"theory of mind\" or mentalizing and, , 144\u201348, 216\u201317\n\nLogan, Cheryl,\n\nLogothetis, Nikos, ,\n\nLombroso, Cesare, 168\u201369\n\nL\u00f3pez, Juan,\n\nLorenz, Konrad,\n\nLynch, Zack,\n\nmagnetic resonance imaging (MRI), , 68\u201369, 72\u201373, ,\n\nMansfield, Peter,\n\nMarincola, Francesco,\n\nMartell, Daniel,\n\nMartin, Emily,\n\nMason, John,  Massachusetts Institute of Technology (MIT), 25\u201326\n\nmaterialism, , , , , 204\u20135, 237n2\n\nMaudsley, Henry, 168\u201371\n\nMauss, Marcel, , , 273n18\n\nMayberg, Helen,\n\nMaynard, Patrick,\n\nMead, George Herbert,\n\nMeador-Woodruff, James,\n\nMeaney, Michael,\n\nmedia and popular culture, 53\u201354, 225\u201326\n\nMedical Research Council (UK),\n\nmemory, , , , , , ,\n\nMenninger, William C., 117\u201318\n\nmental disorders: costs of, , 16\u201317, , 259n15; defined as deviance or abnormality, ; deinstitutionalizing, 122\u201323; embodiment of, 56\u201361; environment and psychosocial concept of, 117\u201318; estimated prevalence of, 125\u201329; minor mental problems treated as, , 116\u201317, , 122\u201325, 136\u201337; organic vs. functional, ; in popular culture, ; psychopathology and the social brain, 148\u201351; risk as motive for intervention or treatment, 117\u201318, , ; \"social brain\" and, 148\u201351; stereotyped or posed images of, 60\u201361; visualization of madness or insanity, 56\u201361. See also specific disorders\n\nMerzenich, M.,\n\nMetzinger, Thomas, 212\u201313\n\nMeyer-Lindenberg, Andreas, 188\u201389,\n\nMeynert, Theodore,\n\nMiczek, Klaus, 97\u201398, 183\u201385, 255n39\n\nMild Cognitive Impairment, , , 261n34\n\nmind, physical basis of, 3\u20134, , , 73\u201374, , 237n4, 238n9; mind\/brain relationship, 3\u20134, , , , , 204\u20135, , . See also brain\n\nmind, theory of, 144\u201345; localization and, , ; \"mindblindness\" and autism spectrum disorders, 216\u201317\n\nmindfulness, , ,\n\nMinsky, Marvin,\n\nmirror neurons, 145\u201352, , 160\u201361, , 263n8, 263n9, 273n16\n\nMobley, Stephen,\n\nmodularity thesis, 75\u201376, , , 262n5\n\nMoffitt, Terrie, 186\u201388, 191\u201392, 256n55\n\nMoldin, Steven,\n\nmolecular brain, 9\u201311, 30\u201331, 38\u201339, , 46\u201347, 225\u201326\n\nmolecular imaging, 71\u201372,\n\nMoniz, Egas, , 66\u201367, 245n17\n\nmonoamine hypothesis, , , , 92\u201393, 134\u201335\n\nMorel, B\u00e9n\u00e9dict, 33\u201334,\n\nMorse, Stephen, 177\u201378\n\nMorton, Samuel,\n\nMoss, Henry, 214\u201315\n\nMosso, Angelo, 72\u201373\n\nMountcastle, Vernon, 3\u20134, 39\u201340\n\nNational Institutes for Health (NIH),\n\nNauta, Walle,\n\nNerve Growth Factor (neurotrophins), 48\u201349,\n\nnerves: brain as part of nervous system, 32\u201333, , , ; \"Doctrine of Nerves,\" 31\u201332; neurogenesis and plasticity, ; neurotransmission, , ; as single, continuous network, 31\u201332, 64\u201365; staining and visualization of, 31\u201332, ; and study of neurology, 31\u201332\n\nNestler, Eric, ,\n\nneuro: use of term, , , 226\u201327, 232\u201333\n\nneuroaesthetics,\n\nneuroanthropology,\n\nneuroeducation,\n\nneuroembryology,\n\nneuroergonomics,\n\nneuroethics, , , ; and neurolaw, 177\u201380, , , 255n39; and personhood,\n\nneurogenesis,\n\nneurogenomics,\n\nneurolaw, , 177\u201380, , , 255n39\n\nneuromarketing,\n\nneuromolecular research: emergence of molecular biology and, 26\u201328; genomics and, 11\u201312, , ; molecular imaging, 71\u201372, ; pharmaceuticals and, , 46\u201347; plasticity of the brain, ; as style of thought, 41\u201351; and vision of the neuromolecular brain, 9\u201311, 30\u201331, 38\u201339, 46\u201347\n\nneurophilosophy,\n\nneuropsychiatry, , 127\u201328; and the identification of insanity, 110\u201312; use of term,\n\nneuroscience: economic research and, 265n40; growth of field, ; as interdisciplinary, 25\u201328, 39\u201342, , 155\u201356, 229\u201330; \"invisible college\" and, , , ; profit motive and, 232\u201333; social sciences and, 228\u201334; as style of thought, 41\u201351, 232\u201333; use of term, ,\n\nNeurosciences Research Program (NRP), , , , 38\u201339\n\nneurosociology. See social neuroscience neurotransmitters, , 253n23; electrical vs. chemical transmission debate, 29\u201332, ; functional brain imaging and, 73\u201374; genetics and, , , , ; pyscho-pharmaceuticals and, , 46\u201347, 184\u201385, 253n23. See also dopamine hypothesis; monoamine hypothesis\n\nNissl, Franz, 64\u201365\n\nnonconscious processes, , 21\u201323, 161\u201362, ; and self-awareness, , , 214\u201315\n\nnormality\/abnormality, 7\u20138; criminality linked to physical abnormality, 167\u201368; diagnostic thresholds for disorders, , , , ,  (see also prodromal diagnoses); mental disorders linked to brain abnormalities, , 110\u201312, 167\u201368 (see also specific disorders); minor mental problems treated as abnormal, 136\u201337; normal mentation, studies of, 45\u201346; normal reactions as \"symptoms,\" 129\u201330; and observation of process in the brain, ; and pervasive diagnosis of mental disorders, 125\u201329; visualization of pathology,\n\nNovas, Carlos,\n\nnuclear magnetic resonance (NMR), 68\u201369\n\nNutt, David,\n\nOberman, L. M.,\n\nobjectivity: and the brain as self, ; child development and, , , ; diagnosis of mental disorders, 111\u201314, 130\u201333, 137\u201338; experimental \"setup\" and, 76\u201378; imaging technology and, , 79\u201381, ; interpretation and visualization of the brain, 79\u201380; laboratory experimentation and, ; objective measures and psychological evaluation, , ; \"objectivity effect,\" , 79\u201381, ; and scientific tools as neutral, ; social neuroscience and, , 160\u201362; and stabilization of the experimenter, 85\u201386; and stabilization of the \"setup,\" 85\u201389; and \"truth to nature,\" 249n13. See also truth\n\nOgawa, Seiji,\n\nParker, Charles, 135\u201336\n\npatents, , , 19\u201320, 242n41\n\nPatton, Paul,\n\nPauling, Linus,\n\nPavlov, Ivan,\n\nPenfield, Wilder,\n\nPerry, Bruce, 194\u201395, 271n54\n\npersonhood, 20\u201323, 226\u201327; and animal research, 83\u201384; of the mental patient, . See also selfhood\n\nphantom limb phenomenon,\n\npharmaceuticals: animal models and drug development, 99\u2013100; antipsychiatric movements and, ; children and psychopharmaceuticals, ; genomics and psychopharmaceuticals, 131\u201332; industrial investment in brain research, ; intervention and use of, 46\u201347, , , 183\u201385; and management of everyday life, , ; and neuromolecular brain, , 46\u201347; and self-fashioning or modification of personality, 220\u201321\n\nPhelps, Michael,\n\nphenomics,\n\nphotography, 59\u201361, 249n12\n\nphrenology, , 61\u201362\n\nThe Physical Basis of Mind (BBC radio series), 3\n\nphysiognomy, ,\n\nPinel, Philippe,\n\nplasticity, , 11\u201312, 225\u201326; animal research and evidence for, 48\u201351; and brain education technologies, , 221\u201322; discovery of nerve growth factor and, 48\u201349; and intervention, ; neuroplasticity defined, ; selfhood and social or cultural influence, ; and synaptic networks in the brain, ,\n\nPlayfair, William,\n\nPoldrack, Russell, 75\u201376\n\npopular culture, 225\u201326\n\nPordage, Samuel,\n\npositron emission tomography (PET), 70\u201371,\n\nPremack, D., 144\u201345\n\npresentation of data, decisions regarding,\n\nprodromal diagnoses, , , , , , 136\u201337\n\npsychiatric genetics\/genomics, ; antisocial behavior and, 180\u201390; eugenics and, ; intervention and genetic predisposition, ; and psychopharmaceuticals, , 131\u201332\n\npsychiatry, ; neuroscience and, 6\u20137; as preventive, ; shift away from asylum, 122\u201323\n\npsychoanalysis,\n\npsychology,\n\npsychopharmacology, 10\u201311, 35\u201337, 46\u201347, 132\u201333, 220\u201321, 239n20, 253n23\n\nPsychosis Risk Syndrome, 136\u201337\n\npsychosurgery, 34\u201335, 245n17; animals and experimental, ; lobotomies, 34\u201335, ,\n\nQuinn, Naomi, 217\u201318\n\nracism, , 115\u201316\n\nRaine, Adrian, 175\u201377, 180\u201381,\n\nRakic, Pasko,  Ramachandran, V. S., , , , , 213\u201314\n\nRam\u00f3n y Cajal, Santiago, , 31\u201332, 48\u201349,\n\n\"recurrent history,\" 29\u201331,\n\nreductionism, 20\u201321, 43\u201344, 51\u201352, 228\u201331,\n\nRegier, Darrel,\n\nReiss, Max,\n\nresilience, , , 187\u201388, ,\n\nRheinberger, Hans-J\u00f6rg,\n\nRichter, Curt, 89\u201390\n\nRichter, Derek,\n\nRichter, Helene, 88\u201389\n\nRioch, David, 39\u201340\n\nrisk: neuroscience and risk prediction, ; preventive intervention and, 117\u201318, ,\n\nRoepstorff, Andreas,\n\nRoper v. Simmons,\n\nRosenhan, David, , ,\n\nRowell, Thelma, 86\u201387\n\nRoy, Charles,\n\n\"rubber hand\" phenomenon,\n\nSawa, Akira,\n\nSchildkraut, Joseph, , , 253n23\n\nschizophrenia, 36\u201337; abnormality of the brain and, 69\u201370, 176\u201377, , 245n17; animal models and, 106\u20138; costs of, ; as developmental, , 261n13; diagnosis of, , , , 118\u201319, , 240n29, 260n29; dopamine hypothesis and, , 36\u201337; genetics and heritability of, 133\u201334, 257n59, 261n13; incidence of, ; mentalizing deficits and abnormal social cognition in, 148\u201350; pharmaceutical treatment of, ,\n\nSchmidt, Carl,\n\nSchmitt, Francis O., 25\u201328, 38\u201341, , 45\u201346\n\nSchopenhauer, Arthur,\n\nSchreber, Daniel Paul, 62\u201363\n\nSchwartz, Jeffrey,\n\nScott, John Paul,\n\nScott, Stephen,\n\nSeif, Isabelle, 185\u201386\n\nself-fashioning, 21\u201322, 220\u201321, , 274n20\n\nselfhood: brain as generator of, , , 205\u20137, 212\u201315; \"brainhood\" as dimension of, , , , ; brain injury and neuropathologies of the self, 205\u20137, ; coherence and, 215\u201316; consciousness and, 214\u201316, 274n19; continuity or coherence and, 202\u20133, 213\u201316; as cultural construct, 202\u20134, 217\u201318; as embodied in the brain, 214\u201315, ; free will and, 209\u201315; interiority and, 202\u20133, ; and mentalizing or theory of the mind, 216\u201317; mereological fallacy and, 83\u201384; and metarepresentation, 213\u201318; personality and, , ; responsibility for the neurobiological self, 223\u201324; self-fashioning, 21\u201322, 220\u201321, , 274n20; subjective awareness, , , , , , , 217\u201318\n\nself-vindication, 88\u201389, 92\u201393, , 105\u20139, 138\u201339\n\n\"setups,\" experimental, 76\u201377; and animal-model research, , 85\u201389, , , ; and brain imaging, ; and laboratory as setting for research, , ; and objectivity, 76\u201378, 85\u201389; stabilization of, 85\u201389\n\nShapin, Steven, ,\n\nSheridan, Alan,\n\nSherrington, Charles, , , , ,\n\nshock therapies, ,\n\nShorter, Edward,\n\nSidman, Murray,\n\nSimmel, Georg,\n\nSinger, Jay,\n\nSinger, Tania, 141\u201342, ,\n\nsingle photon emission computerized tomography (SPECT),\n\nSj\u00f6gren, Torsten,\n\nskull: as barrier to imaging via X-ray, , 66\u201367; physiognomy or phrenology, , 61\u201362, 167\u201368\n\nSmith, Iain Duncan, 193\u201395, 271n54\n\nsocial brain: autism spectrum disorders and, , 216\u201317; described and defined, ; evolution and, 143\u201352, , , ; funding for research on, 156\u201357; imaging technologies and, , 158\u201359; intersubjectivity and, ; localization of function and, ; mirror neuron theory and, 145\u201352, , , , 263n8, 263n9, 273n16; pychopathology and, 148\u201351; \"social brain\" hypothesis, 143\u201348; and social world as research setting, 228\u201332; visualization of,\n\nsociality: brain evolution for, 143\u201352, , , ; and laboratory as setting for research, 158\u201359; neurobiology and emergence of, ; social cognition, 143\u201344; social group, defined, ; and social world as research setting, 227\u201334. See also social brain\n\nsocial neuroscience: anthropology and, 158\u201359; and biological explanations for human behavior, 151\u201354, , ; as emerging discipline, 151\u201355; funding for, 156\u201357; genetics and, 151\u201352, ; and the individual, 151\u201352, 160\u201361; objectivity and, , 160\u201361; social brain theory and, 144\u201345, , ; sociality and, , , , ; social psychology and, 159\u201362, ; sociology and, ; \"theory of mind\" and, 144\u201345\n\nSociety for Biological Psychiatry,\n\nSociety for Molecular Imaging (SMI),\n\nSociety for Mutual Autopsy, , , 266n9\n\nSociety for Neuroscience (SfN), , 28\u201329\n\nSociety for Social Neuroscience, 155\u201356\n\nsociology, 159\u201360\n\nsoul, , , , 199\u2013200, , ; madness and notions of, , ; self concept and, , 213\u201318\n\nSpatz, Hugo,\n\nSpence, Sean, 211\u201312\n\nSperry, Roger, ,\n\nSpitzer, Robert, , , , ,\n\nSpurzheim, Johann, 61\u201362\n\nstaining,\n\nstates vs. traits,\n\nSteven, Megan, 177\u201378\n\n\"style of thought\": and neuroscience, 41\u201351, 232\u201333; and use of animal models, 82\u201383\n\nsubjectivity: neuroscience and subjectivation, . See also objectivity\n\nSuomi, Stephen, 255n35\n\nSwazey, Judith, 26\u201328\n\nSydenham, Thomas,\n\nTancredi, Lawrence, 173\u201374\n\nTaub, Edward, 49\u201350\n\nTaylor, Charles, ,\n\nTenHouton, Warren, 159\u201360\n\nTer-Pogossian, Michael, 70\u201371\n\n\"theory of mind\": and autism disorders, 216\u201317; localization and modular, , 144\u201345; mentalizing and, 145\u201348, 216\u201317; \"mindblindness\" and autism disorders, 216\u201317; social brain hypothesis and, 143\u201345\n\nThomson, J. Bruce,\n\nTonkonogy, Joseph,\n\ntranslation of findings: and improved treatments, ; translational imperative as context for research, 228\u201329; unjustified extrapolation and,\n\ntruth, ; histories and, ; images and objective, , , 78\u201381, , 249n13; laboratory science and, , 87\u201389, ; neuroscience and, , , ; research and truth-making, 232\u201333; science and truth claims, 172n7, , 232\u201333\n\nTuke, Daniel Hack, , , , 170\u201371\n\nVallabhajosula, Shankar,\n\nvan Os, Jim, 136\u201337\n\nVarela, Francisco, , 275n29\n\nvisualization of the brain, , 225\u201326; artifactual nature of images, 78\u201381; the clinical gaze and, 55\u201356; interpretation and, 79\u201380; and the mind or the interior, 59\u201361. See also imaging technologies\n\nVogt, C\u00e9cile, 172\u201373\n\nVogt, Oscar, , 172\u201373\n\nVolkow, Nora, 173\u201374\n\nVoltaire,\n\nvon K\u00f6lliker, Albert,\n\nWagner, Henry, 70\u201372\n\nWakefield, Jerome,\n\nWaldeyer, Wilhelm,\n\nWalter, William Grey,\n\nWatson, Stanley,\n\nWatts, James,\n\nWeber, Max,\n\nWebster-Stratton, Carolyn,\n\nWernicke, Carl, , , ,\n\nWiesel, Torsten,\n\nWilliams, Carl, 165\u201366\n\nWillis, Thomas, , , 258n7\n\nWilson, Elizabeth,\n\nWittgenstein, L., , , 252n5\n\nWoese, Carl,\n\nWoodruff, G., 144\u201345\n\nWright, Harold,\n\nW\u00fcrbel, Hanno, 88\u201389\n\nX-rays, , 66\u201368\n\nYeager, Clay,\n\nYellowlees, Henry,\n\nYoung, J. Z., ,\n\nYoung, R.,\n\nYudofsky, S. C., 111\u201312\n\nZalcman, Steven,\n\nZubin, Joseph, 34\u201335\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nImmanuel Kant (1724\u20131804).\nKANT\n\n_Roger Scruton_\n\nSTERLING and the distinctive Sterling logo are registered trademarks of   \nSterling Publishing Co., Inc.\n\n**Library of Congress Cataloging-in-Publication Data**\n\nScruton, Roger.   \nKant : a brief insight \/ Roger Scruton. -- Illustrated ed.   \np. cm.   \nISBN 978-1-4027-7901-5   \n1. Kant, Immanuel, 1724-1804. I. Title   \nB2798.S37 2010   \n193--dc22\n\n2010013457\n\n10 9 8 7 6 5 4 3 2 1\n\nPublished by Sterling Publishing Co., Inc.   \n387 Park Avenue South, New York, NY 10016\n\nPublished by arrangement with Oxford University Press, Inc.\n\n\u00a9 1982, 2001 by Roger Scruton   \nIllustrated edition published in 2010 by Sterling Publishing Co., Inc.   \nAdditional text \u00a9 2010 Sterling Publishing Co., Inc.\n\nDistributed in Canada by Sterling Publishing   \n  Canadian Manda Group, 165 Dufferin Street   \nToronto, Ontario, Canada M6K 3H6\n\nBook design: The DesignWorks Group\n\nPlease see picture credits for image copyright information.\n\nPrinted in China   \nAll rights reserved\n\nSterling ISBN 978-1-4027-7901-5\n\nFor information about custom editions, special sales, premium and   \ncorporate purchases, please contact Sterling Special Sales   \nDepartment at 800-805-5489 or specialsales@sterlingpublishing.com.\nPREFACE TO THE   \nREVISED EDITION\n\nIN REVISING THIS INTRODUCTION I have tried to take account of recent scholarship, and to offer a more reliable guide to both primary and secondary sources. I have also amended the text where it seemed to me to be inaccurate or misleading, and added a chapter on Kant's political philosophy\u2014a subject that has attracted increasing attention in the twenty years since this book first appeared. The interpretation of Kant that I offer is by no means the only one, and I have had to take sides in controversies that are of ever-increasing intricacy. In the guide to Further Reading I indicate books that will help to rectify the inevitable one-sidedness of an introduction.\n\n_Malmesbury_  \nJANUARY 2001 \nPREFACE TO THE   \nFIRST EDITION\n\nI HAVE TRIED TO PRESENT Kant's thought in a modern idiom, while presupposing the least possible knowledge of philosophy. Since Kant is one of the most difficult of modern philosophers, I cannot hope that I have made every aspect of this thought intelligible to the general reader. It is not clear that every aspect of his thought has been intelligible to anyone, even to Kant. The depth and complexity of Kant's philosophy are such that it is only after complete immersion that the importance of its questions, and the imaginative power of its answers, can be understood. Kant hoped to draw the limits of the human understanding; he found himself compelled to transcend them. Readers should therefore not be surprised if they have to read this introduction more than once in order to appreciate Kant's vision. To share that vision is to see the world transformed; to acquire it cannot be the labor of a single day.\n\nThe first draft of this book was written in Prague. I am grateful to Dr. Ladislav Hejd\u00e1nek, not only for the invitation to speak to his seminar on the topic of the categorical imperative, but also for the example he has set in obeying it. I have benefited from Ruby Meager, Mark Platts, and Dorothy Edgington, who commented on a later draft, and from the students of London University who have, over the last decade, made the teaching of Kant's philosophy so rewarding. I have also benefited, in more ways than I can express, from the kindness of Lenka  , to whom this book is dedicated.\n\n_London_  \nMAY 1981 \nCONTENTS\n\n**ONE**| Life, Works, and Character  \n---|---  \n**TWO**| The Background of Kant's Thought  \n**THREE**| The Transcendental Deduction  \n**FOUR**| The Logic of Illusion   \n**FIVE**| The Categorical Imperative   \n**SIX**| Beauty and Design   \n**SEVEN**| Enlightenment and Law  \n**EIGHT**| Transcendental Philosophy  \n|   \n_Further Reading_  \n _Index_  \n _Picture Credits_\nABBREVIATIONS\n\nQUOTATIONS ARE MOSTLY FROM the English translations referred to in the Further Reading. However, where these translations have seemed to me to be misleading or inelegant, I have used my own.\n\nA. _Critique of Pure Reason_ , first edition\n\nB. _Critique of Pure Reason_ , second edition\n\n_P. Critique of Practical Reason_\n\n_J. Critique of Aesthetic Judgement_\n\n_T. Critique of Teleological Judgement_\n\n_G. Groundwork of the Metaphysic of Morals_\n\n_F. Prolegomena to any Future Metaphysics_\n\n_M. The Metaphysics of Morals_\n\nPage references to those works are to the standard German edition, or else, in the case of the _Critique of Pure Reason_ , to the original pagination of the first and second editions. For English translations, see Further Reading.\n\nI have also occasionally referred to the following:\n\nI. \"Inaugural Dissertation,\" contained in G. B. Kerferd, D. K. E. Walford, and P. G. Lucas (eds.), _Kant: Selected_ _Pre-Critical Writings_ (Manchester, 1968)\n\n_K. The Kant\u2013Eberhard Controversy_ , by H. E. Allison (Baltimore and London, 1973)\n\n_L. Lectures on Ethics_ , tr. L. Infield, new edition (New York, 1963)\n\n_C. Kant's Philosophical Correspondence: 1759\u201399_ , ed. and tr. Arnulf Zweig (Chicago, 1967)\n\nR. Hans Reiss (ed.), _Kant: Political Writings_ , tr. H. B. Nisbet (Cambridge, 1970). This contains three works that I refer to: _An Answer to the Question: \"What Is Enlightenment?\"_ ( _What Is Enlightenment?_ , for short); _On the Common_ _Saying: \"This may be true in theory, but it does not apply in_ _practice\"_ ( _Common Saying_ , for short), and _Perpetual Peace:_ _A Philosophical Sketch_ ( _PP_ for short).\n\nAll italics that appear in quotations from Kant are his own.\n\n**Kant's early life and education,** as well as his later moral thinking, were shaped by the German pietism movement, which stressed work, duty, and prayer. \"The Broad and Narrow Road,\" an illustration depicting the principles of the pietist movement, draws on Matthew 7:13\u201314: \"Enter through the narrow gate. For wide is the gate and broad is the road that leads to destruction, and many enter through it. But small is the gate and narrow the road that leads to life, and only a few find it.\"\n[ONE\n\nLife, Works, and Character ](Roge_9781402782107_epub_c7_r1.html#d6e1)\n\nTHE GREATEST MODERN PHI LOSOPHER was moved by nothing more than by duty. His life, in consequence, was unremarkable. For Kant, the virtuous man is so much the master of his passions as scarcely to be prompted by them, and so far indifferent to power and reputation as to regard their significance as nothing beside that of duty itself. Having confined his life so that he could act without strain according to this ideal, Kant devoted himself to scholarship, entirely governed by congenial routines. The little professor of K\u00f6nigsberg has thus become the type of the modern philosopher: bounded in a nutshell, and counting himself king of infinite space.\n\nImmanuel Kant was born in K\u00f6nigsberg in 1724, the fourth of the nine children of a poor harness maker. Kant's parents were simple people and devout pietists. At that time, pietism, a reformist movement within the Lutheran Church, held powerful sway among the lower and middle classes in Germany, consoling hardship with the idea of the sacredness of work, duty, and prayer; its vision of the sovereignty of conscience was to exert a lasting influence on Kant's moral thinking. Although in some respects anti-intellectual, it was also one of the major forces behind the spread of education in late-seventeenth-century Germany, and a pietist school had been established in K\u00f6nigsberg. To this school Kant, whose talents had been recognized by a wise and benevolent pastor, was sent at the age of eight. It is fortunate for posterity that such an education should have been offered to one of Kant's lowly origins; it was perhaps less fortunate for the young Kant himself, whose gratitude toward his tutors was so mingled with distaste for their oppressive zeal that in later years he forbore all mention of his early schooling. Some impression of its nature can be gathered from a remark in the little treatise on education, edited from Kant's lecture notes late in his life:\n\nMany people imagine that the years of their youth are the pleasantest and best of their lives; but it is not really so. They are the most troublesome; for we are then under strict discipline, can seldom choose our friends, and still more seldom have our freedom.\n\nIn a letter sent to Kant by a former school friend, at a time when both had become famous, the philologist David Ruhnken remarked: \"Thirty years have passed since the two of us groaned beneath the pedantically gloomy, but not entirely worthless, discipline of those fanatics.\" And it is undeniable that Kant emerged from his schooling with a considerable weight of gloom, together with a remarkable self-discipline. His early manhood was partly devoted to using the second to overcome the first. In this he was almost completely successful. Despite straitened circumstances, a deformed and diminutive body, and the loss both of a father whom he respected and of a mother whom he deeply loved, Kant soon became one of the most popular citizens of K\u00f6nigsberg, welcomed everywhere for his grace, wit, and ready conversation.\n\nKant entered the university of his native city at the age of sixteen and graduated from it six years later. Being unable to secure an academic position, he took work as a private tutor in various households. It was not until the age of thirty-one that he obtained a post at the university, as private docent, an unsalaried employment that conferred the privilege of giving public lectures, and the chance of securing a meagre reward through private tuition. By then Kant had already published works on dynamics and mathematics. He had also acquired, through the connections that his position as private tutor made available, the social ease that was to earn him the title of _der sch\u00f6ne Magister_.\n\n**Kant's home** in K\u00f6nigsberg, Germany.\n\nK\u00f6nigsberg was then a city of some dignity, with fifty thousand inhabitants and an important garrison. As a seaport serving the trading interests of Eastern Prussia, it contained a bustling and variegated population, including Dutch, English, Poles, and Russians. The university, founded in 1544 as the Collegium Albertinum, was a cultural center of some importance, although so much sunk in provincial obscurity by the mid-eighteenth century that Frederick the Great, visiting the city as crown prince in 1739, described it as \"better suited to the training of bears than to becoming a theatre of the sciences.\" Frederick ascended the throne the following year, and did his best to spread into this corner of his kingdom the high culture and intellectual toleration that characterized his reign. Kant, who had already determined to value truth and duty above all things, was therefore fortunate to find that his university offered no major impediments to the pursuit of either. It was perhaps this, as much as his passionate attachment to his birthplace, that prompted him to wait so long for his first appointment, and to continue waiting thereafter, for another fifteen years, before being granted the professorship that he desired. During this period Kant several times refused offers from other German universities, and continued conscientiously to deliver, in the house where he lodged, the lectures that established his reputation. His intellectual labors were devoted mainly to mathematics and physics, and at the age of thirty-one he published a treatise on the origin of the universe that contained the first formulation of the nebular hypothesis. His duties required him, however, to lecture on a wide variety of subjects, including physical geography, about which he became, perhaps because of his reluctance to travel, an acknowledged authority and, in the opinion of one Count Purgstall (who admired the philosopher greatly), a conversational bore.\n\nIt is to some extent by chance that Kant's professorship was in metaphysics and logic, rather than in mathematics or natural science. From this point in his career, however, Kant devoted his energies entirely to philosophy, rehearsing in his lectures the thoughts that he was to publish, ten years later, in works that earned him a reputation as the greatest luminary in Germany. The philosopher J. G. Hamann records that it was necessary to arrive in Kant's lecture room at six in the morning, one hour before the professor was due to appear, in order to obtain a place, and Kant's pupil Jachmann has this to say of the performance:\n\nKant had a peculiarly skilful method of asserting and defining metaphysical concepts, which consisted, to all appearances, in carrying out his inquiries in front of his audience; as though he himself had just begun to consider the question, gradually adding fresh determining concepts, improving bit by bit on previously established explanations, and finally arriving at a definitive conclusion of his treatment of the subject, which he had thoroughly examined from every angle, having given the completely attentive listener not only a knowledge of the subject, but also an object lesson in methodical thought . . .\n\nAnd in a letter to a friend, the same writer speaks of Kant's lectures on ethics:\n\nIn these he ceased to be merely a speculative philosopher and became, at the same time, a spirited orator, sweeping the heart and emotions along with him, as well as satisfying the intelligence. Indeed, it was a heavenly delight to hear his sublimely pure ethical doctrine delivered with such powerful philosophic eloquence from the lips of its very creator. How often he moved us to tears, how often he stirred our hearts to their depths, how often he lifted up our minds and emotions from the shackles of self-seeking egoism to the exalted self-awareness of pure free-will, to absolute obedience to the laws of reason and to the exalted sense of our duty to others!\n\nJachmann is more fulsome than false, and Kant's fame as a speaker, both in private and in public, earned him wide recognition, long before the publication of his greatest works.\n\nKant's private life is often parodied as one of clockwork routine, fastidious, donnish, and self-centered. It is said (because Heine said it) that the housewives of K\u00f6nigsberg would set their clocks by his time of passing; it is said (because Kant once said it) that his constant concern for his bodily condition displayed a morbid hypochondria; it is also said that the bareness of his house and furnishings displayed an indifference to beauty, and that the punctuality of his routine disguised a cold and even frozen heart.\n\nIt is true that Kant's life was, if not mechanical, at least highly disciplined. His manservant had instructions to wake him each morning at five and to tolerate no malingering. He would work until seven at his desk, dressed in nightcap and robe, changing back into these garments at once when he had returned from his morning lectures. He remained in his study until one, when he took his single meal of the day, following it, irrespective of the weather, by a walk. He took this exercise alone, from the eccentric conviction that conversation, since it causes a person to breathe through the mouth, should not take place in the open air. He was averse to noise, twice changing lodging in order to avoid the sound of other people, and once writing indignantly to the director of police, commanding him to prevent the inmates of a nearby prison from consoling themselves with the singing of hymns. His aversion to music other than military marches was indeed notorious, as was his total indifference to the visual arts\u2014he possessed only one engraving, a portrait of Rousseau, given to him by a friend.\n\n**An engraved portrait** of Swiss-born French philosopher Jean-Jacques Rousseau (1712\u201378) was the only picture in Kant's possession.\n\nKant was aware of the accusations that the intellect draws to itself. And it was to the subject of this engraving that he turned in self-justification, saying that he should have regarded himself as much more worthless than the common laborer had not Rousseau convinced him that the intellect could play its part in restoring the rights of man. Like all people given to the life of the mind, Kant was in need of the discipline that he imposed on himself. Far from crippling his moral nature, his routine enabled him to flower in the ways best suited to his genius. His love of solitude was balanced by an equal love of company. He would invariably have guests at his midday meal, inviting them on the same morning lest they should be embarrassed by the need to refuse some other invitation, and providing for each a pint of claret, and, if possible, some favorite dish. He conversed, to the delight and instruction of his companions, until three, endeavoring to end the meal in laughter (as much, however, from a conviction that laughter promotes digestion, as from any natural inclination toward it). Kant's writings contain many flashes of satire, and satire, indeed, was his favorite reading. His indifference to music and painting must be set against his love of poetry; even his concern for his health was little more than a consequence of the Kantian philosophy of duty. He neither admired nor enjoyed the sedentary life, but regarded it, nevertheless, as indispensable to the exercise of his intellect. Herder, one of the greatest and most passionate writers of the Romantic movement, attended Kant's lectures and afterward vigorously opposed their influence. He nevertheless thought highly of Kant himself, and summarized his character in these words:\n\nI had the good fortune to make the acquaintance of a philosopher, who was my teacher. Though in the prime of life, he still had the joyful high spirits of a young man, which he kept, I believe, into extreme old age. His open brow, built for thought, was the seat of indestructible serenity and gladness. A wealth of ideas issued from his lips, jest and wit and good humour were at his bidding, and his instructional lecture was also the most fascinating entertainment.\n\n**Kant and his table partners** by Emil Doerstling, ca. 1900.\n\nWith the same spirit with which he examined Leibniz, Wolff, Baumgarten, Crusius and Hume, and analysed the laws of nature expounded by the physicists Kepler and Newton, he appraised the currently appearing writings of Rousseau, his _Emile_ and his _H\u00e9loise_ , as he did every fresh discovery in natural science which came to his notice, estimated their value and returned, as always, to an unbiased knowledge of nature and of the moral worth of man.\n\nThe history of mankind, of nations and of nature, natural science, mathematics and his own experience were the well-springs which animated his lectures and his everyday life. He was never indifferent to anything worth knowing. No intrigue, no sectional interests, no advantage, no desire for fame ever possessed the slightest power to counteract his extension and illumination of truth. He encouraged and gently compelled people to think for themselves: despotism was alien to his nature. This man, whom I name with the deepest gratitude and reverence, is Immanuel Kant; I recall his image with pleasure.\n\nKant's duties as a university teacher required him to lecture on all aspects of philosophy, and for many years he devoted the major part of his intellectual efforts to teaching, and to publishing short and undeveloped books and papers. His greatest achievement\u2014the _Critique of Pure_ _Reason\u2014_ was also his first major publication, appearing in 1781, when Kant was fifty-seven. Of this work he wrote to Moses Mendelssohn: \"Although the book is the product of twelve years of reflection, I completed it hastily, in perhaps four or five months, with the greatest attentiveness to its content but less care about its style and ease of comprehension\" ( _C._ 105\u20136). In an attempt to alleviate the difficulties presented by the _Critique_ , he published a short _Prolegomena to Any Future Metaphysic Which Shall Lay_ _Claim to Being a Science_ (1783), in which brilliant polemic is combined with an obscure condensation of the _Critique_ 's most offending passages. For the second edition of the _Critique_ , in 1787, Kant rewrote the most forbidding sections; since the result is equally difficult, commentators have come to agree that the opacity of Kant's work stems not so much from the style as from the thought itself. Despite its difficulty, however, the work rapidly became so famous that the \"critical philosophy\" was being advocated, taught, opposed, and sometimes even censored and persecuted, throughout the German-speaking world. Kant's self-confidence increased, and he was able to write in 1787 to K. L. Reinhold (who did much to popularize Kant's ideas): \"I can assure you that the longer I continue in my path the less worried I become that any contradiction . . . will ever significantly damage my system\" ( _C._ 127). The influence of Kant's first _Critique_ is justly summarized by Mme. de Sta\u00ebl, when, thirty years after the first edition, she wrote that \"when at length the treasures of thought which it contains were discovered, it produced such a sensation in Germany, that almost all which has been accomplished since, in literature as well as in philosophy, has flowed from the impulse given by this performance.\"\n\nDuring the twelve years of reflection to which Kant refers in his letter to Mendelssohn he published almost nothing, and his earlier (\"precritical\") writings are of peripheral interest to the student of his mature philosophy. However, once the critical philosophy had achieved expression, Kant continued, with increasing confidence, to explore its ramifications. The _Critique of Pure Reason_ dealt in a systematic way with metaphysics and the theory of knowledge; it was followed by the _Critique of Practical_ _Reason_ (1788), concerned with ethics, and the _Critique of Judgement_ (1790), concerned largely with aesthetics. Many other works were added to those, and Kant's collected writings in the so-called _Berliner Ausgabe_ now fill thirty-two volumes. Of these other works, three will particularly concern us: the _Prolegomena_ , already mentioned, the _Groundwork of the_ _Metaphysic of Morals_ , which appeared in 1785, before the second _Critique_ , and contains a compelling expression of Kant's moral theory, and _The Metaphysics of Morals_ , a late work published in 1797, which contains Kant's ideas on politics and law.\n\nDuring the reign of Frederick the Great, K\u00f6nigsberg breathed the air of enlightenment, and Kant enjoyed the esteem of Frederick's ministers, in particular of von Zedlitz, the minister of education, to whom the _Critique of Pure Reason_ is dedicated. A marked change occurred when Frederick William II ascended the throne. His minister, Woellner, exerted great influence, and, becoming responsible for religion in 1788, attempted to bring religious toleration to an end. Kant's _Religion Within the Limits of Reason Alone_ was published in 1793, under the imprint of the K\u00f6nigsberg philosophy faculty, thus escaping censorship on a point of law. Woellner, mightily displeased, wrote in the king's name to Kant, charging him to give an account of himself. Kant replied with a solemn promise to his sovereign not to engage in public discussion of religion, either through lectures or through writings. Kant regarded himself as absolved from this promise by the monarch's death. Nevertheless, this conflict with authority caused him much pain and bitterness. Kant prided himself on being a loyal subject, despite republican sympathies, which he once expressed in the presence of an Englishman with such vivacity as to call forth a challenge to a duel, and with such eloquence as to overcome both the challenge and the opinions of the man who had offered it. (The man in question, Joseph Green, was a merchant in K\u00f6nigsberg, and subsequently became Kant's closest friend.)\n\nKant enjoyed the company of women (provided that they did not pretend to understand the _Critique of Pure Reason_ ) and twice contemplated marriage. On each occasion, however, he hesitated long enough to ensure that he remained unwed. One day, his disreputable and drunken manservant appeared at table in a yellow coat. Kant indignantly ordered him to take it off and sell it, promising to make good the financial loss. He then learned with amazement that the servant had been married, was a widower, and was now to marry again, the yellow coat having been purchased for the occasion. Kant was appalled at these revelations, and never again looked on his servant with favor. His attitude to marriage was curiously disenchanted, and, although he defended the institution in _The_ _Metaphysics of Morals_ , he also described the married state as an agreement between two people for the \"reciprocal use of each other's sexual organs\" ( _C._ 235, _M._ 277). However, in his early _Observations on the Sentiment of_ _the Beautiful and the Sublime_ (1764), Kant had written eloquently on the distinction between the sexes. He was radically opposed to the view that men and women partake of a common nature, which alone suffices to determine the character of their relations; instead, he assigned to women a charm, beauty, and capacity to melt the heart that are foreign to the more \"sublime,\" \"principled,\" and \"practical\" sex to which he belonged. This description of women accords with Kant's description of natural beauty. It was nature, above all, that stirred his emotions, and it was to scenes of natural beauty that his mother had taken him as a child, so as to awaken his feelings toward the things she loved. It is possible to discern in his evocations, both of feminine charm and of natural beauty, the residue of erotic feelings which, had they been more actively expressed, might well have broken the routine to which our intellectual history is so heavily indebted.\n\n**K\u00f6nigsberg university's** \"old building,\" where Kant taught.\n\nKant gave his last official lecture in 1796. By that time his faculties had begun to decline and a somber melancholy had replaced his former gaiety. Fichte describes him as seeming to lecture in his sleep, waking with a start to his half-forgotten subject matter. Soon he lost his clarity of mind, his ability to recognize old friends, even his ability to complete simple sentences. He faded into insensibility, and passed from his blameless life on February 12, 1804, unaccompanied by his former intellectual powers. He was attended to his grave by people from all over Germany, and by the whole of K\u00f6nigsberg, being acknowledged even in his senility as the greatest glory of that town. His grave crumbled away and was restored in 1881. His remains were moved in 1924, to a solemn neoclassical portico attached to the cathedral. In 1950 unknown vandals broke open the sarcophagus, and left it empty. By that time K\u00f6nigsberg had ceased to be a center of learning, had been absorbed, following its brutal destruction by the Red Army, into the Soviet Union, and had been renamed in honor of one of the few of Stalin's henchmen to die of natural causes. A bronze tablet remains fixed to the wall of the castle, overlooking the dead and wasted city, bearing these words from the concluding section of the _Critique of_ _Practical Reason_ :\n\n**This copy of the memorial tablet** for Kant that appears in the castle wall is on the base of a bust of Kant at the Universidade Federal de Minas Gerais in Brazil.\n\nTwo things fill the heart with ever renewed and increasing awe and reverence, the more often and the more steadily we meditate upon them: _the starry firmament above and the moral law within._\n\nIt is fortunate for the inhabitants of Kaliningrad that they are daily reminded of two things that they may still admire.\n\n**Title page of the first edition** of Kant's _Critique of Pure Reason_.\n[TWO\n\nThe Background of Kant's Thought ](Roge_9781402782107_epub_c7_r1.html#d6e2)\n\nTHE _CRITIQUE OF PURE REASON_ is the most important work of philosophy to have been written in modern times; it is also one of the most difficult. It poses questions so novel and comprehensive that Kant judged it necessary to invent technical terms with which to discuss them. These terms have a strange beauty and compulsion, and it is impossible to acquire a full appreciation of Kant's work without experiencing the order and connectedness that his vocabulary imposes upon the traditional problems of philosophy. Nevertheless, the gist of Kant's thought can be expressed in a lowlier idiom, and in what follows I shall try to eliminate as many of his technicalities as I can. The task is not easy, since there is no accepted interpretation of their meaning. While a \"picture\" of the Kantian system is common to all who have commented on it, there is no agreement whatsoever as to the strength, or even as to the content, of his arguments. A commentator who presents clear premises and clear conclusions will invariably be accused of missing Kant's argument, and the only way to escape academic censure is to fall into the verbal mannerisms of the original. It has become, in recent years, slightly easier to risk this censure. Contemporary Kantian studies\u2014in particular in Britain and America\u2014have tended to the view that the obscurity of Kantian scholarship is often a product of its confusion. In order not to attribute the confusion to Kant, scholars have labored hard to elicit from, or at least to impose upon, the first _Critique_ an interpretation that renders it intelligible. I shall try to do the same, and shall be more influenced by these contemporary studies than I am able to acknowledge.\n\nThe first problem posed by the interpretation of the _Critique of Pure_ _Reason_ is this: what are the questions that it hopes to answer? Kant wrote, in the preface to the first edition:\n\nIn this enquiry I have made completeness my aim, and I venture to assert that there is not a single metaphysical problem which has not been solved, or for the solution of which the key at least has not been supplied. (A., p. xiii)\n\nWhile this represents the ambition, if not the achievement, of the first _Critique_ , Kant was in fact motivated by more specific interests. If we turn to the historical antecedents of Kant's argument, we can extract from the philosophical controversies that influenced him certain major subjects of dispute. The most important we find to be the problem of objective knowledge, as this had been posed by Descartes. I can know much about myself, and this knowledge often has a character of certainty. In particular it is senseless, according to Descartes, to doubt that I exist. Here, doubt only confirms what is doubted. _Cogito ergo sum_. In this case, at least, I have objective knowledge. The fact that I exist is an objective fact; it is a fact about the world and not just about someone's perception. Whatever the world contains, it contains the thinking being who I am. Kant's contemporary Lichtenberg pointed out that Descartes ought not to have drawn this conclusion. The \"cogito\" shows that there is a thought, but not that there is an \"I\" who thinks it. Kant, similarly dissatisfied with Descartes's argument, and with the doctrine of the soul that flowed from it, felt that the certainty of self-knowledge had been wrongly described. It is true that, however skeptical I may be about the world, I cannot extend my skepticism into the subjective sphere (the sphere of consciousness): so I can be immediately certain of my present mental states. But I cannot be immediately certain of what I am, or of whether, indeed, there is an \"I\" to whom these states belong. These further propositions must be established by argument, and that argument had yet to be found.\n\n**French philosopher** Ren\u00e9 Descartes (1596\u20131650), who coined the famous phrase \" _Cogito ergo sum_ \" (\"I think, therefore I am\"), is shown here in a portrait (1649) by Dutch painter Frans Hals.\n\nWhat is the character of this immediate and certain knowledge? The distinguishing feature of my present mental states is that they are as they seem to me and seem as they are. In the subjective sphere, being and seeming collapse into each other. In the objective sphere they diverge. The world is objective because it can be other than it seems to me. So the true question of objective knowledge is: how can I know the world as it is? I can have knowledge of the world as it _seems_ , since that is merely knowledge of my present perceptions, memories, thoughts, and feelings. But can I have knowledge of the world that is _not_ just knowledge of how it seems? To put the question in slightly more general form: can I have knowledge of the world that is not just knowledge of my own point of view? Science, common sense, theology, and personal life all suppose the possibility of objective knowledge. If this supposition is unwarranted, then so are almost all the beliefs that we commonly entertain.\n\nAmong Kant's immediate predecessors, two in particular had provided answers to the question of objectivity that were sufficiently decisive to command the attention of the intellectual world. These were G. W. Leibniz (1646\u20131716) and David Hume (1711\u201376); the first claimed that we could have objective knowledge of the world uncontaminated by the point of view of any observer; the second claimed (or at least seemed to his contemporaries to claim) that we could have objective knowledge of nothing.\n\nLeibniz was the founding father of Prussian academic philosophy. His thought, left to the world in succinct and unpublished fragments, had been built into a system by Christian von Wolff (1679\u20131754), and applied and extended by Wolff's pupil, the former pietist A. G. Baumgarten (1714\u201362). The Leibnizian system had met with official censure during Kant's youth, since it made such claims for reason as to threaten those of faith; for a time Wolff was forbidden to teach. But the system was restored to favor under Frederick the Great, and became the orthodox metaphysics of the German Enlightenment. Kant respected this orthodoxy, and to the end of his days would use Baumgarten's works as texts for his lectures. But Hume's skepticism made a deep impression on him, and introduced new problems that he felt could be answered only by overthrowing the Leibnizian system. These problems, concerning causality and a priori knowledge (that is, knowledge not based in experience), were combined with the question of objectivity to form the peculiar subject matter of the first _Critique_.\n\nLeibniz belonged to the school of thought now generally labeled \"rationalist,\" and Hume to the school of \"empiricism,\" which is commonly contrasted with it. This convenient, though contentious, division of his predecessors into rationalists and empiricists is in fact due to Kant. Believing that both philosophies were wrong in their conclusions, he attempted to give an account of philosophical method that incorporated the truths, and avoided the errors, of both. Rationalism derives all claims to knowledge from the exercise of reason, and purports to give an absolute description of the world, uncontaminated by the experience of any observer. It is an attempt to give a God's-eye view of reality. Empiricism argues that knowledge comes through experience alone; there is, therefore, no possibility of separating knowledge from the subjective condition of the knower. Kant wished to give an answer to the question of objective knowledge that was neither as absolute as Leibniz's nor as subjective as Hume's. The best way to make his unique position intelligible is to begin by summarizing the two views that he strove to reject.\n\nLeibniz believed that the understanding contains within itself certain innate principles, which it knows intuitively to be true, and which form the axioms from which a complete description of the world can be derived. These principles are necessarily true, and do not depend upon experience for their confirmation. Hence they lead to a description of the world as it is, not as it appears in experience or to a circumscribed \"point of view.\" At the same time, the \"points of view\" that are characteristic of individuals can be fitted into the rational picture of the world. Leibniz recognized a division in thought between subject and predicate (in the sentence \"John thinks,\" the subject-term is \"John\" and the predicate-term is \"thinks\"). He believed that this division corresponds to a distinction in reality between substance and property. The fundamental objects in the world are substances. These, he thought, must be self-dependent, unlike the properties that inhere in them: for example, a substance can exist without thinking, but no thinking can exist without a substance. Being self-dependent, substances are also indestructible, except by a miracle. He called them \"monads,\" and his model for the monad was the individual soul, the thinking substance, as this had been described by Descartes. From this idea, he derived his \"perspectiveless\" picture of the world, relying on two fundamental laws of reason: the Principle of Contradiction (a proposition and its negation cannot both be true), and the Principle of Sufficient Reason (nothing is true that has no sufficient explanation). By means of ingenious and subtle arguments and making the fewest possible assumptions, he arrived at the following conclusions.\n\n**German philosopher** Gottfried Wilhelm Leibniz (1696\u20131716) is shown here in a portrait by Johann Friedrich Wentzel from about 1700.\n\nThe world consists of infinitely many individual monads, which exist neither in space nor in time, but eternally. Each monad is different in some respect from every other (the famous \"identity of indiscernibles\"). Without that assumption objects cannot be individuated in terms of their intrinsic properties; a point of view then becomes necessary from which to tell things apart, and it is Leibniz's contention that the real nature of the world can be given from no point of view. The point of view of each monad is simply a way of representing its internal constitution; it does not represent the world as it is in itself. Each monad mirrors the universe from its own point of view, but no monad can enter into real relation, causal or otherwise, with any other. Even space and time are intellectual constructs, through which we make our experience intelligible, but which do not belong to the world as such. However, by the principle of \"pre-established harmony,\" the successive properties of every monad correspond to the successive properties of every other. So we can describe our successive states of mind as \"perceptions,\" and the world will \"appear\" to each monad in a way that corresponds to its appearance to every other. There is a system among appearances, and within this system it makes sense to speak of spatial, temporal, and causal relations; of destructible individuals and dynamic principles; of perception, activity, and influence. These ideas, and the physical laws that we derive from them, depend for their validity on the underlying harmony among points of view that they describe. They do not yield knowledge of the real world of monads except indirectly, on account of our assurance that the way things appear bears the metaphysical imprint of the way things are. When two watches keep exact time together I might be tempted to think that the one causes the other to move. This is an example of a merely apparent relation. In some such way Leibniz argued that the whole world of commonsense belief and perception is no more than an appearance or \"phenomenon.\" But it is, Leibniz said, a \"well-founded phenomenon.\" It is no illusion, but a necessary and systematic offshoot of the operation of those rational principles that determine how things really are. The real substances, because they are described and identified from no point of view, are without phenomenal characteristics. Reality itself is accessible to reason alone, since only reason can rise above the individual point of view and participate in the vision of ultimate necessities, which is also God's. Hence reason must operate through \"innate\" ideas. These are ideas that have been acquired through no experience and that belong to all thinking beings. They owe their content not to experience but to the intuitive capacities of reason. Among these ideas is that of substance, from which all Leibniz's principles ultimately flow.\n\nHume's vision is in some measure the opposite of Leibniz's. He denies the possibility of knowledge through reason, since reason cannot operate without ideas, and ideas are acquired only through the senses. The content of every thought must be given, in the last analysis, in terms of the experiences that warrant it, and no belief can be established as true except by reference to the sensory \"impressions\" that provide its guarantee. (This is the general assumption of empiricism.) But the only experience that can confirm anything for me is my experience. The testimony of others, or of records, the formulation of laws or hypotheses, the appeal to memory and induction\u2014all these depend for their authority on the experiences that guarantee them. My experiences are as they seem, and seem as they are, for here \"seeming\" is all that there is. Hence there is no problem as to how I can know them. But in basing all knowledge on experience, Hume reduces my knowledge of the world to knowledge of my point of view. All claims to objectivity become spurious and illusory. When I claim to have knowledge of objects existing externally to my perceptions, all I can really mean is that those perceptions exhibit a kind of constancy and coherence that generates the (illusory) idea of independence. When I refer to causal necessities, all I am entitled to mean is the regular succession among experiences, together with the subjective sense of anticipation that arises from that. As for reason, this can tell us of the \"relations of ideas\": for example, it can tell us that the idea of space is included in that of shape, or that the idea of a bachelor is identical with that of an unmarried man. But it can neither generate ideas of its own, nor decide whether an idea has application. It is the source only of trivial knowledge derived from the meanings of words; it can never lead to knowledge of matters of fact. Hume took his skepticism so far as to cast doubt upon the existence of the self (the entity that had provided the model for Leibniz's monad), saying that neither is there a perceivable object that goes by this name, nor is there any experience that would give rise to the idea of it.\n\nSuch skepticism, reaching back into that very point of view from which skepticism begins, is intolerable, and it is not surprising that Kant was, as he put it, aroused by Hume from \"dogmatic slumbers\" ( _F._ 9). The parts of Hume's philosophy that most disturbed him concerned the concept of causality. Hume had argued that there is no foundation for the belief in necessities in nature: necessity belongs to thought alone, and merely reflects the \"relations of ideas.\" It was this that led Kant to perceive that natural science rests on the belief that there are real necessities, so that Hume's skepticism, far from being an academic exercise, threatened to undermine the foundations of scientific thought. Kant did indeed have a lasting quarrel with Leibniz, and with the Leibnizian system. But it was the sense that the problems of objectivity and of causal necessity are ultimately connected that led him toward the outlook of the _Critique of Pure Reason_. It was only then that he perceived what was really wrong with Leibniz, through his attempt to show what was really wrong with Hume. He came to think as follows.\n\n**English philosopher** David Hume (1711\u201376) is shown here in an engraving after an original portrait by G. Phillips.\n\nNeither experience nor reason is alone able to provide knowledge. The first provides content without form, the second form without content. Only in their synthesis is knowledge possible; hence, there is no knowledge that does not bear the marks of reason and of experience together. Such knowledge is, however, genuine and objective. It transcends the point of view of the person who possesses it, and makes legitimate claims about an independent world. Nevertheless, it is impossible to know the world \"as it is in itself,\" independent of all perspective. Such an absolute conception of the object of knowledge is senseless, Kant argues, since it can be given only by employing concepts from which every element of meaning has been refined away. While I can know the world independently of my point of view on it, what I know (the world of \"appearance\") bears the indelible marks of that point of view. Objects do not depend for their existence upon my perceiving them; but their nature is determined by the fact that they _can_ be perceived. Objects are not Leibnizian monads, knowable only to the perspectiveless stance of \"pure reason\"; nor are they Humean \"impressions,\" features of my own experience. They are objective, but their character is given by the point of view through which they can be known. This is the point of view of \"possible experience.\" Kant tries to show that, properly understood, the idea of \"experience\" already carries the objective reference that Hume denied. Experience contains _within_ itself the features of space, time, and causality. Hence, in describing my experience I am referring to an ordered perspective on an independent world.\n\nIn order to introduce this novel conception of objectivity (to which he gave the name \"transcendental idealism\") Kant began from an exploration of a priori knowledge. Among true propositions, some are true independently of experience, and remain true however experience varies: these are the a priori truths. Others owe their truth to experience, and might have been false had experience been different: these are the a posteriori truths. (The terminology here was not invented by Kant, although it owes its popularity to Kant's frequent use of it.) Kant argued that a priori truths are of two kinds, which he called \"analytic\" and \"synthetic\" (A. 6\u201310). An analytic truth is one like \"All bachelors are unmarried\" whose truth is guaranteed by the meaning, and discovered through the analysis, of the terms used to express it. A synthetic truth is one whose truth is not so derived but that, as Kant puts it, affirms something in the predicate that is not already contained in the subject. It is a truth like \"All bachelors are unfulfilled,\" which (supposing it to be true) says something substantial about bachelors and does not merely reiterate the definition of the term used to refer to them. The distinction between the analytic and the synthetic involved novel terminology, although similar distinctions can be found in earlier philosophers. Aquinas, inspired by Boethius, defines a \"self-evident\" proposition as one in which the \"predicate is contained in the notion of the subject,\" and a similar idea is to be found in Leibniz. What is original, however, is Kant's insistence that the two distinctions (between the a priori and the a posteriori, and between the analytic and the synthetic), are of a wholly different nature. It is mere dogmatism on the part of empiricists to think that they must coincide. And yet for the empiricist view to be true, there cannot be synthetic a priori knowledge: synthetic truths can be known only through experience.\n\nThe empiricist position has been taken in recent times by the logical positivists of the \"Vienna Circle,\" who argued that all a priori truths are analytic, and drew the conclusion that any metaphysical proposition must be meaningless, since it could be neither analytic nor a posteriori. It was already apparent to Kant that empiricism denies the possibility of metaphysics. And yet metaphysics is necessary if foundations are to be provided for objective knowledge: without it, there is no conceivable barrier against the skepticism of Hume. So the first question of all philosophy becomes \"How is synthetic a priori knowledge possible?\" Or, to put it another way, \"How can I come to know the world through pure reflection, without recourse to experience?\" Kant felt that there could be no explanation of a priori knowledge that divorces the object known from the perspective of the knower. Hence he was skeptical of all attempts to claim that we can have a priori knowledge of some timeless, spaceless world of the \"thing-in-itself \" (that is, any object defined without reference to the \"possible experience\" of an observer). I can have a priori knowledge only of the world that I experience. A priori knowledge provides support for, but it also derives its content from, empirical discovery. Kant's _Critique_ is directed in part against the assumption that \"pure reason\" can give content to knowledge without making reference to experience.\n\n**German philosopher** Moritz Schlick (1882\u20131936), leader of the Vienna Circle, is shown here in a photograph taken ca. 1935.\n\nAll a priori truths are both necessary and absolutely universal: these are the two signs whereby we can discern, among our claims to knowledge, those items that, if they are true at all, are true a priori. For it is obvious that experience could never confer necessity or absolute universality on anything; any experience might have been otherwise, and experience is necessarily finite and particular, so that a universal law (which has indefinitely many instances) could never be truly confirmed by it. No one should really doubt that there is synthetic a priori knowledge: Kant gave as the most conspicuous example mathematics, which we know by pure reasoning, but not by analyzing the meanings of mathematical terms. There ought to be a philosophical explanation of the a priori nature of mathematics, and Kant attempted to provide it in the opening sections of the _Critique_. But he also drew attention to other examples, of a more puzzling kind. For instance, the following propositions seem to be true a priori: \"Every event has a cause\"; \"The world consists of enduring objects which exist independently of me\"; \"All discoverable objects are in space and time.\" These propositions cannot be established through experience, since their truth is presupposed in the interpretation of experience. Moreover, each claims to be true, not just on this or that occasion, but universally and necessarily. Finally, it is just such truths as these that are required for the proof of objectivity. Hence the problem of objectivity and the problem of synthetic a priori knowledge are ultimately connected. Moreover, the vital role played by the truths given above in all scientific explanation persuaded Kant that a theory of objectivity would also provide an explanation of natural necessity. Such a theory would then give a complete answer to the skepticism of Hume.\n\nWhat, then, are Kant's aims in the first _Critique_? First, in opposition to Hume, to show that synthetic a priori knowledge is possible, and to offer examples of it. Second, in opposition to Leibniz, to demonstrate that \"pure reason\" alone, operating outside the constraints placed on it by experience, leads only to illusion, so that there is no a priori knowledge of \"things-in-themselves.\" It is normal to divide the _Critique_ into two parts, in accordance with this division of the subject, and to describe the first part as the \"Analytic,\" the second as the \"Dialectic.\" While this division does not correspond exactly to Kant's division of chapters (which is exceedingly complex and bristles with technicalities), it is sufficiently close not to be misleading. The terms \"analytic\" and \"dialectic\" are Kant's: and so is the bifurcation of the argument. In the first part Kant's defense of objectivity is expounded, and it is with the argument of the \"Analytic\" that I shall begin, for, until it is grasped, it will be impossible to understand the nature either of Kant's metaphysics, or of the moral, aesthetic, and political theories that he later derived from it.\n\n**The subjective deduction,** in trying to show what is involved in making a judgment, concentrates on the nature of the mind, particularly on the nature of belief, sensation, and experience. This image of the human head in profile, labeled with brain functions, is by German philosopher and theologian Albertus Magnus (ca. 1200\u20131280) and appeared in his _Philosophia pauperum, sive_ _Philosophia naturalis_.\n[THREE\n\nThe Transcendental Deduction ](Roge_9781402782107_epub_c7_r1.html#d6e3)\n\nKANT'S ANSWER TO THE FUNDAMENTAL question of metaphysics\u2014 \"How is synthetic a priori knowledge possible?\"\u2014contains two parts. I shall call these, following Kant's own terminology (A., p. xvi), the \"subjective\" and the \"objective\" deductions. (Kant uses the term _deduction_ in the legal sense, as in the deduction of a title to land\u2014that is, a proof that we have the right to something, in this case, the right to apply certain concepts or \"categories.\") The subjective deduction had been partly adumbrated in Kant's inaugural dissertation of 1770 (I. 54 ff.), and consists in a theory of cognition. It tries to show what is involved in making a judgment: in holding something to be true or false. It concentrates on the nature of the mind, in particular on the nature of belief, sensation, and experience. Its conclusions are presented as part of a general theory of the \"understanding\" (the faculty of judgment). Kant repeatedly emphasizes that the theory is not to be construed as empirical psychology. It is not, nor does it purport to be, a theory of the workings of the _human_ , as opposed to some other, intelligence. It is a theory of the understanding as such, telling us what it is, and how it must function if there are to be judgments at all. In all philosophical discussion of these matters, Kant argues, we are talking \"not about the origin of experience, but about what lies in it\" ( _F._ 63). And he compares such purely philosophical questions to that \"analysis of concepts\" that has since become so fashionable. Kant wishes to draw the limits of the understanding. If there are things that cannot be grasped by the understanding, then all assertions about them are meaningless.\n\nThe objective deduction consists in a positive attempt to establish the content of a priori knowledge. The argument here proceeds, not by an analysis of the faculties of knowledge, but by an exploration of its grounds. What are the presuppositions of experience? What has to be true if we are to have even that bare point of view that the skeptics ascribe to us? If we can identify these presuppositions, then they will be established as true a priori. For their truth follows, not from the fact that we have this or that experience, but from the fact that we have experience _at_ _all_. Hence they depend upon no particular experience for their verification, and can be established by reasoning alone. They will be true in every world where the skeptical question can be asked (in every comprehensible world). And this is tantamount to their being necessary. I cannot conceive of their falsehood, since I cannot conceive of myself as part of a world that refutes them.\n\nKant calls this argument the \"transcendental deduction,\" and the resulting theory \"transcendental idealism.\" The word _transcendental_ needs some explanation. An argument is transcendental if it \"transcends\" the limits of empirical enquiry, so as to establish the a priori conditions of experience. We must distinguish transcendental from empirical argument (B. 81); the former, unlike the latter, leads to \"knowledge which is occupied not so much with objects as with the mode of our knowledge of objects _in so far as this mode of knowledge is possible a priori_ \" (B. 25). The word _transcendental_ is also used by Kant in another sense, to refer to \"transcendental objects.\" These are objects that transcend experience\u2014 that is, objects which are not disclosed to empirical investigation, being neither observable themselves, nor causally related to what is observable. These \"transcendental objects\" present a problem of interpretation to which I shall return in Chapter 4.\n\nI shall also return to the interpretation of \"transcendental idealism.\" Briefly, this theory implies that the laws of the understanding, laid down in the subjective deduction, are the same as the a priori truths established in the objective deduction. It implies, in other words, a very special kind of harmony between the capacities of the knower and the nature of the known. It is because of this harmony that a priori knowledge is possible.\n\nIt follows from this theory that the \"forms of thought\" that govern the understanding, and the a priori nature of reality, are in exact correspondence. The world is as we think it, and we think it as it is. Almost all the major difficulties in the interpretation of Kant depend upon which of those two propositions is emphasized. Is it our thought that determines the a priori nature of the world? Or is it the world that determines how we must think of it? The answer, I believe, is \"neither, and both.\" But only at the very end of this book will that answer be clear.\n\nSelf-Consciousness\n\nI have referred, adopting Kant's usage, to \"our\" understanding and experience. Who are \"we\"? Kant's use of the first person plural is a device of a very special character. He is not, as I said, engaging in a psychological study of \"creatures like us.\" Nor is he speaking with some abstract authorial voice for which no true subject can be found. He means the term \"we\" to denote indifferently any being who can use the term \"I\": anyone who can identify himself as the subject of experience. The starting point of all Kant's philosophy is the single premise of self-consciousness, and the first two of his three _Critiques_ concern themselves respectively with the questions: \"What must a self-conscious being think?\" and \"What must he do?\" Self-consciousness is a deep phenomenon, with many layers and aspects. It is not every being that can know his own experience (for whom the \"I think\" can accompany all his perceptions, as Kant expresses his crucial emendation of Descartes [B. 131\u201332]). But it is only such a being who can pose the skeptical question: \"Are things as they seem to me (as my experience represents them)?\" The argument explores, therefore, the presuppositions of this self-consciousness. Kant's conclusion can be summarized thus: the conditions that make skepticism possible also show it to be false.\n\nThe Transcendental Synthesis\n\nI shall deal first with the subjective deduction\u2014the theory of the \"subjective conditions\" of judgment. There are two sources from which our knowledge is drawn: sensibility and understanding. The first is a faculty of intuitions ( _Anschauungen_ ): it includes all the sensory states and modifications that empiricists think to be the sole basis of knowledge. The second is a faculty of concepts. Since concepts have to be _applied_ in judgments, this faculty, unlike sensibility, is active. It is a mistake of empiricism, Kant argued, not to have understood this crucial point, and to have construed all concepts of the understanding on the model of sensations. (Thus for Hume a concept is simply a faded relict of the \"impression\" from which it derives.) The corresponding mistake of rationalism is to think of sensation as a kind of confused aspiration toward conceptual thought. Thus Kant summarized the famous dispute between Leibniz and Locke in the following way: \"Leibniz _intellectualised_ appearances, just as Locke . . . _sensualised_ the concepts of the understanding.\" In fact, however, there are two faculties here, irreducible the one to the other; they \"can supply objectively valid judgements of things only in _conjunction_ with each other\" (A. 271, B. 327).\n\nJudgment requires, then, the joint operation of sensibility and understanding. A mind without concepts would have no capacity to think; equally, a mind armed with concepts, but with no sensory data to which they could be applied, would have nothing to think about. \"Without sensibility no object would be given to us, without understanding no object would be thought. Thoughts without content are empty; intuitions without concepts are blind\" (A. 51, B. 75). Judgment requires what Kant calls a \"synthesis\" of concept and intuition, and only in this synthesis is true experience (as opposed to mere \"intuition\") generated. This synthesis is somewhat confusingly described by Kant: sometimes it seems to be a \"process\" whereby experience is generated, at other times a kind of \"structure\" that experience contains. In any event, it seems to have two stages: the \"pure\" synthesis, whereby intuitions are grouped together into a totality, and then the act of judgment, in which the totality is given form through a concept (A. 79, B. 104). This synthesis is not meant to be a psychological fact; it is a \"transcendental\" as opposed to an \"empirical\" synthesis. In other words, it is presupposed in (self-conscious) experience, and not derived from it. I do not lay hold of my experience and then subject it to synthesis. For the very act of \"laying hold\" presupposes that this synthesis has occurred. Suppose I attempt to describe how things seem to me, as I sit writing at this desk. I am at once engaged in the activity of subsuming my sensory awareness under concepts (such concepts as those of desk and writing). I can represent my experience to myself only by describing \"how things seem\": and that is to use the concepts of the understanding. Conversely, none of my concepts would be intelligible without the experiences that exhibit their application.\n\nA Priori Concepts\n\nIt is an assumption of empiricism that all concepts are derived from, or in some manner reducible to, the sensory intuitions that warrant their application. There can be no concept without the corresponding sensory stimulus, and it is in terms of such stimulus that the meaning of a concept must be given. Kant argued that this assumption is absurd. The empiricists confuse experience with sensation. Experience can provide the grounds for the application of a concept, because it already contains a concept, in accordance with the \"synthesis\" just described. Sensation, or intuition, contains no concept, and provides grounds for no judgment. Until transformed by mental activity, all sensation is without intellectual structure, and therefore provides grounds for no belief. If we understand experiences, then it is because they already contain within themselves the concepts that we supposedly derive from them. Whence came these concepts? Not from the senses. There must therefore be some repertoire of concepts contained within the understanding itself, and which defines the forms of its activity.\n\nIt follows that the Leibnizian theory of \"innate ideas\" is substantially correct. There are concepts that cannot be given through experience because they are presupposed in experience. They are involved in every apprehension of the world that I can represent as mine; not to possess them is to have, not experience, but mere intuition, from which no knowledge can be derived. These \"a priori concepts\" of the understanding prescribe the basic \"forms\" of judgment. All other concepts can be seen as \"determinations\" of them\u2014that is, as special cases, more or less adulterated by the reference to observation and experiment.\n\nKant called these fundamental concepts \"categories,\" borrowing a term that had been put to similar (but less systematic) use by Aristotle. The categories are our forms of thought. One such category is the concept that lies at the origin of the Leibnizian system: the category of substance. A substance is that which is able to exist independently, and which supports the properties that depend upon it. The concept \"chair\" is a special, empirical determination of the general concept of substance. It can be acquired only by someone who already grasps that general concept, for only such a person would be able to interpret his experience in the requisite way. Another category is the one that had been subjected to such skeptical attack by Hume: the category of cause. It is not surprising that much of the argument of the Analytic concerns the ideas of substance and causality, as Kant wished us to understand them. However, he gave a list of twelve categories in all, and found to his satisfaction that they corresponded to all the disputes of traditional metaphysics.\n\nThe Subjective Deduction\n\nIt seems to follow from the above account that, if we are to have knowledge at all, our intuitions must permit application of the categories. To speak more directly: it must seem to us that we are confronted by substances, causes, and the rest. So we can know a priori that every comprehensible world (every world that could contain self-consciousness) must also have the appearance dictated by the categories. It can have that appearance only if it appears to obey certain \"principles.\" A principle specifies the conditions under which a category gains application. The sum of all principles defines the extent of our claims to a priori knowledge.\n\nWhat does this \"subjective deduction\" of the categories establish? The answer seems to be this: we must think in terms of the categories, and must therefore accept as true the principles that govern their application. So the world must ordinarily appear to us in such a way that we _can_ accept these principles. Self-consciousness requires that the world must appear to conform to the categories. This assertion contains the essence of what Kant called his \"Copernican Revolution\" in philosophy. Previous philosophers had taken nature as primary, and asked how our cognitive capacities could lay hold of it. Kant takes those capacities as primary, and then deduces the a priori limits of nature. This is the first important step in his answer to Hume. It is well for Hume to assert that our knowledge has its foundation in experience. But experience is not the simple concept that Hume supposed it to be. Experience contains intellectual structure. It is already organized in accordance with the ideas of space, time, substance, and causality. Hence there is no knowledge of experience that does not point toward a world of nature. Our point of view is intrinsically a point of view _on_ an objective world.\n\n**This portrait of Immanuel Kant** at the age of forty-four was created by Johann Gottlieb Becker in 1768.\n\nBut does that answer the skeptic? Surely, he will say, even if Kant were right, and the world must appear to us in this way, does it also have to _be_ as it _appears_? Even if we are compelled to think that the categories apply, it does not follow that they really _do_ apply. We have yet to pass from the description of our point of view to a description of the world. The problem therefore remains, how \"subjective conditions of thought can have objective validity\" (A. 89, B. 122). As Kant argues (B. 141), there can be no judgment without objectivity. Clearly then, an \"objective deduction\" of the categories is required: an argument that will show that the world, and not just our experience of the world, is in conformity with the a priori principles of the understanding.\n\nForms of Thought and Forms of Intuition\n\nBefore proceeding to that objective deduction, we must return to the earlier sections of the _Critique of Pure Reason_ , in which Kant writes in general terms about the nature of sensibility. Kant believed that he had arrived at his list of categories by a process of abstraction. Suppose I describe what I now see: a pen writing. The concept \"pen\" is a special \"determination\" of the wider concept \"artifact,\" itself a determination of \"material object,\" and so on. The limit of this train of abstraction is the a priori concept that each stage exemplifies: the concept of substance. Beyond that point we cannot abstract further, without ceasing to think. Likewise \"writing\" is a determination of \"action,\" which is a determination of \"force,\" and so on: the category here being that of cause or explanation, beyond which the understanding cannot proceed. By these, and similar, thought experiments, Kant supposed that he had isolated, through his list of the twelve categories, all the forms of judgment, and so given an exposition of the concept of objective truth. Hence the proof of the objectivity of our claims to knowledge involves the demonstration of the \"objective validity\" of the categories, and of the principles presupposed in their application.\n\nThere are two ideas, however, that, despite their importance to science and to the objective view of the world, are not included in Kant's list of categories. These are the ideas of space and time, which he described, not as concepts, but as forms of intuition. Space and time are discussed in the opening section of the first _Critique_ , the \"Transcendental Aesthetic.\" The word _aesthetic_ here derives from the Greek for sensation, and indicates that the subject matter of this section is the faculty of sensibility, considered independently of the understanding. Kant argued that space and time, far from being concepts applicable to intuitions, are basic _forms_ of intuition, meaning that every sensation must bear the imprint of temporal, and sometimes of spatial, organization.\n\nTime is the form of \"inner sense,\" that is, of all states of mind, whether or not they are referred to an objective reality. There could not be a mental state that is not in time, and time is made real to us through this organization in our experience. Space is the form of \"outer sense\"\u2014that is, of those \"intuitions\" that we refer to an independent world and that we therefore regard as \"appearances\" of objective things. Nothing can appear to me as independent of myself without also being experienced as \"outside,\" and therefore as spatially related to, myself. Space, like time, forms part of the organization of my sensibility. My very sense-impressions bear the form of space, as is evidenced in the phenomenon of the \"visual field.\"\n\nBut why deny that space and time are a priori concepts? They are a priori, but they cannot be concepts, since concepts are general, admitting of a plurality of instances. Kant argued that there is of necessity only one space, and only one time. All spaces form parts of a single space, and all times parts of a single time. Kant sometimes expresses this point by saying that space and time are not concepts but \"a priori intuitions.\"\n\nThere is, in Kant's philosophy, a rage for order that leads him to attempt to solve as many philosophical questions as possible through each distinct part of his system. One motive for treating space and time separately from the categories of the understanding is in order to suggest an explanation for the fact that there are two kinds of synthetic a priori truth: mathematics and metaphysics. It seems that there ought to be a different explanation for each, since mathematics is self-evident, and obvious to all thinking beings, whereas metaphysics is essentially disputed, a matter over which people argue interminably; \"infinitely removed from being as evident as the proposition that twice two makes four\" (A. 733, B. 761). Mathematics possesses, indeed, all the immediacy and indubitability of intuition itself, whereas the metaphysical principles, which derive from thought alone, are necessarily contentious. By construing mathematics as an a priori science of intuition, Kant thought that he could show why this is so. In mathematics we are dealing with \"a priori intuitions\"; this automatically gives a content to our thought that is absent from the abstract employment of the categories. The conclusions of mathematics are arrived at both \"a priori and immediately\" (A. 732, B. 760), whereas those of metaphysics must be derived by laborious argument.\n\nWhether or not we accept Kant's explanation, it is a distinctive feature of his philosophy that he took mathematical truths to be synthetic a priori. At the same time he refused to countenance Plato's explanation of this synthetic a priori status, as deriving from the peculiar nature of mathematical objects: the abstract, immutable numbers and Forms. Kant also disagreed with Hume and Leibniz, both of whom had argued that mathematics is analytic. The question is, therefore, how can mathematics be synthetic a priori, and yet not provide knowledge of some mysterious and unobservable realm? In the _Inaugural Dissertation_ Kant argued that the a priori nature of geometry derives from the subject, rather than from the object, of mathematical thought (I. 70\u201372). It is this that led to his mature view that there can be a priori knowledge of space only if space enters into the nature of perception. Hence the theory of space as a \"form\" of intuition.\n\nObjective knowledge has, then, a double origin: sensibility and understanding. And, just as the first must \"conform to\" the second, so must the second \"conform to\" the first; otherwise, the transcendental synthesis of the two would be impossible. What does it mean to say that the understanding must \"conform to\" the sensibility? Since time is the general form of all sensibility, the claim amounts to this: the categories must find their primary application in time, and be \"determined,\" or limited, accordingly. Thus the concept of substance must have, as its primary instance, not the monads of Leibniz, nor the abstract objects of the Platonic supersphere, but ordinary temporal things, which endure through time, and are subject to change. If such things are objective, then, by the theory that space is the form of outer sense, they must also be spatial. So to prove the \"objective validity\" of the concept of substance is not to prove that the world consists of monads. The world consists rather of ordinary spatio-temporal objects. The philosophical proof of objectivity establishes the existence, not of an abstract, perspectiveless, world, but of the commonsense world of science and everyday perception: the very world that both Humean skepticism and Leibnizian metaphysics had thrown in doubt. It was therefore important to Kant to show, in his proof of objectivity, that the beliefs that he justified corresponded exactly to the laws that Newtonian science had laid down for all perceivable things.\n\nThe Transcendental Unity of Apperception\n\nThe \"objective\" deduction of the categories begins from the premise of self-consciousness, described, in characteristic language, as the \"transcendental unity of apperception.\" It is important to understand this phrase, which contains in embryo much of Kant's philosophy. \"Apperception\" is a term taken from Leibnizian metaphysics; it refers to any experience of which the subject is able to say \"this is mine.\" In other words, \"apperception\" means \"self-conscious experience.\" The unity of apperception consists in the \"'I think' which can accompany all my perceptions\" (B. 131\u201332), to borrow again Kant's version of Descartes. It consists of my immediate awareness that simultaneous experiences belong to me. I know immediately that this thought, and this perception, are equally mine, in the sense of belonging to the unity of consciousness that defines my point of view. Doubt is here impossible: I could never be in the position that Dickens in _Hard Times_ attributes to Mrs. Gradgrind on her deathbed, knowing that there is a pain in the room somewhere, but not knowing that it is mine. This apprehension of unity is called \"transcendental\" because I could never derive it from experience. I could not argue that, because this pain has such a quality, and this thought such another, they must belong to a single consciousness. If I did that, I could make a mistake; I could be in the absurd position of ascribing to myself some pain, thought, or perception that belonged, not to me, but to someone else. So the unity that I apprehend in my point of view is not a conclusion from experience, but a presupposition of experience. Its basis \"transcends\" anything that experience could establish. As Kant sometimes puts it, the unity of consciousness \"precedes\" all the data of intuition (A. 107).\n\nThe transcendental unity of apperception provides the minimal description of our point of view. I can know at least one thing: that there is a unity of consciousness. To doubt this is to cease to be self-conscious, and so to cease to find significance in doubt. The task is to show that this point of view is possible only in an objective world.\n\nThe Transcendental Deduction\n\nThe objective deduction has the same \"transcendental\" character as its premise. It is concerned to show that the truth of its conclusion is not a deduction from experience but presupposed in the _existence_ of experience. The transcendental unity of apperception is possible only if the subject inhabits the kind of world that the categories describe: an objective world, in which things may be other than they seem. To put the argument in a nutshell: the unity of apperception describes the condition of subjectivity, in which everything is as it seems and seems as it is. But no one can have the point of view of subjectivity who does not have knowledge of objective truths. He must therefore belong to a world of things that can be other than they seem and that exist independently of his own perspective.\n\nAn argument to that effect is hard to find, and it is significant that Kant was so dissatisfied with \"The Transcendental Deduction\" that he rewrote it entirely for the second edition of the _Critique_ , changing its emphasis from the subjective conclusion given above toward the objective deduction presently under discussion. Even so, the result is very obscure and a further passage\u2014\"The Refutation of Idealism\"\u2014was added in order to make the emphasis on objectivity more persuasive. The point of this \"refutation\" was to show \"that we have experience, and not merely _imagination_ , of outer things\" by demonstrating that \"even our inner experience [ _Erfahrung_ ], which for Descartes is indubitable, is possible only on the assumption of outer experience [i.e., experience of an objective world]\" (B. 275). The flavor of the argument is more apparent than its substance. At least the following three thoughts seem to be involved in it:\n\n(i) _Identifying experience_. It is a common empiricist assumption that I can know my experience simply by observing it. But this is not so. I do not observe my experience, but only its object. Any knowledge of experience must therefore involve knowledge of its object. But I can have knowledge of the object only if I identify it as continuous. Nothing can have temporal continuity without also having the capacity to exist when unobserved. Its existence is therefore independent of my perception.\n\n(ii) _Identity through time_. I can identify experience as mine only if I locate it in time. I must therefore ascribe it to a subject who exists in time and endures through time. My unity requires my continuity. But to endure is to be substantial, and nothing can be substantial unless it also enters into causal relations. I endure only if my past explains my future. Otherwise there is no difference between genuine duration and an infinite sequence of momentary selves. If I can be conscious of my experience at all, I can therefore conclude that I belong to a world to which such categories as substance and cause are correctly applied, since they are correctly applied to me. A condition of self-consciousness is, therefore, the existence of just that objective order which my experience suggests to me.\n\n(iii) _Ordering in time_. I have privileged knowledge of present experience. To know my experience is therefore to know it as present. This is possible only if I distinguish in experience between now and then. Hence there is a \"temporal order\" inherent in perception, and I must know this order if I am to know myself. But I can know it only if I can make reference to independent objects, and the regularities that govern them. Only then will I be able to describe time as a dimension in which experience occurs, rather than as a series of unrelated instants. The reality of time is presupposed in experience. And the reality of time presupposes the reality of an objective sequence. It is only by reference to that sequence, and to the enduring objects that structure it, that I can identify my own perception.\n\nNone of those thoughts is clearer in the original than in my brief r\u00e9sum\u00e9. It is fair to say that the transcendental deduction has never been considered to provide a satisfactory argument. In all its versions it involves a transition from the unity of consciousness to the identity of the subject through time. Hume pointed out that the slide from unity to identity is involved in all our claims to objective knowledge; he also thought that it could never be justified. Kant did not find the terms with which to answer Hume. Nevertheless his enterprise has appealed to many subsequent philosophers, and arguments similar to those adumbrated in the transcendental deduction have been revived in recent years, most notably by Ludwig Wittgenstein (1889\u20131951). In the famous \"private-language\" argument in his _Philosophical Investigations_ Wittgenstein argues that there can be no knowledge of experience that does not presuppose reference to a public world. I can know my own experience immediately and incorrigibly, but only because I apply to it concepts that gain their sense from public usage. And public usage describes a reality observable to others besides myself. The publicity of my language guarantees the objectivity of its reference. Wittgenstein's argument\u2014which has seemed persuasive to many\u2014shares the premise and the conclusion of the transcendental deduction. It relies, however, not on metaphysical doctrines about time, but on theories of reference and meaning. It does not argue directly that subjects depend upon objects. Instead it shows that subjects depend upon _communities_ of subjects, and therefore on the publicly observable world that establishes their shared frame of reference.\n\nIf valid, the transcendental deduction achieves a result of immense significance. It establishes the objectivity of my world while assuming no more than my point of view on it. Descartes, in the proof of the external world offered in his _Meditations_ , tried to step outside the point of view of the subject by establishing the existence of an omniscient God: the world is then validated as an object of God's (perspectiveless) awareness. The essence of Kant's \"transcendental\" method lies in its egocentricity. All the questions that I can ask I must ask from the standpoint that is mine; therefore, they must bear the marks of my perspective, which is the perspective of \"possible experience.\" The answer to them is not to be found in the attempt to rise to the standpoint of the reasoning being who knows without experience, but to find, within my experience itself, the response to skeptical enquiry. The transcendental method finds the answer to every philosophical question in the presuppositions of the perspective from which it must be asked.\n\nPrinciples\n\nKant argues that every category corresponds to a principle, whose truth is presupposed in its application. Principles are \"rules for the objective employment\" of the categories (A. 161, B. 200). These principles are a priori truths, and they have two aspects. They say how we must think if we are to think at all; and, in their objective aspect, they say how the world must be if it is to be intelligible. Through these principles, the categories \"prescribe laws a priori to appearances, and therefore to nature\" (B. 163). That is to say, they lay down synthetic a priori truths concerning the world of everyday and scientific observation. They are not a priori truths about \"things-in-themselves.\" They do not give knowledge of a world described without reference to our perception. Synthetic a priori knowledge is only of \"things which may be the objects of possible experience\" (B. 148). In this area, the principles state objective and necessary truths, since the categories \"relate of necessity and a priori to the objects of experience, for the reason that only by means of them can any object of experience be thought\" (A. 110). The limitation to the \"objects of possible experience\" is vital to Kant's philosophy, and he repeatedly emphasizes that \"outside the field of possible experience there can be no synthetic a priori principles\" (A. 248, B. 305). We must always bear in mind that Kant wishes to admit a priori knowledge, but also to deny the possibility of the perspectiveless metaphysics of Leibniz.\n\nI have chosen to concentrate on the categories of substance and cause. Kant associated these with a third, that of community, or reciprocal interaction. These concepts had been the principal objects of Hume's skeptical attack. They were also crucial to Leibniz's metaphysics, substance being the idea from which Leibniz began, and the \"Principle of Sufficient Reason,\" associated by Kant with causality (A. 200\u2013201, B. 246), being the principle that governed his argument. Kant's discussion of these concepts is of great and lasting interest. He placed them at the heart of the question of objectivity, and expressed, in their analysis, what he took to be the metaphysical foundations of physical science. He derives their associated principles in a section called \"Analogies of Experience,\" and it is here that Kant makes some of his most revolutionary suggestions concerning the nature of objective knowledge.\n\nKant's division of all philosophical thought into elements of four and three is obsessive. But there is a special reason, when talking of substance and causality, for his insistence that there should be a third basic concept associated with these; namely, that Kant wished his results to correspond to the three Newtonian laws of motion. Scientific explanation depends upon principles of method: being presupposed in scientific enquiry, these principles cannot be proved through it. Kant believed that such principles would be reflected in basic scientific laws; and it is one of the tasks of metaphysics to provide grounds for their acceptance.\n\nThe physical science of Kant's day seemed to assume a priori the existence of universal causation, and of reciprocal interaction. It assumed that it must explain, not the existence of matter, but the changes undergone by it. It assumed the need for a law of conservation, according to which, in all changes, some fundamental quantity remains unaltered. It is just such assumptions, Kant thought, that had guided Newton in the formulation of his laws of motion. Kant therefore attempts, in deriving his principles, to establish the \"validity of universal laws of nature as laws of the understanding\" ( _F._ 74), arguing further that all the fundamental laws of the new astronomy can be seen, on reflection, to rest on principles that are valid a priori ( _F._ 83).\n\nThe attempt to uphold the Newtonian mechanics is mixed with an attack on Hume's skepticism about causality. Kant tries to show that causal relations are necessary, both in the sense that it is necessary that objects enter into them (there is no event without a cause), and also in the sense that they are themselves a species of necessary connection.\n\n**Newton's** ** _Principia_** **:** the mathematics of motion.\nThe \"Analogies\" contain too many arguments to be treated in detail here. Like other crucial passages of the Analytic, they were substantially rewritten for the second edition, the actual statement of the \"principles\" undergoing significant revisions. But two theses deserve singling out on account of their subsequent importance. First, Kant argues that all explanation of change requires the postulation of an unchanging substance, and uses this as proof of the validity of a fundamental \"law of conservation\" in science. His arguments provide one of the most important insights into the nature of scientific method since Descartes, and a clear move in the direction of the modern idea of the \"unity of science.\" According to one version of this idea, there is a single law of conservation involved in the explanation of every change, and hence a single stuff (for example, energy), whose laws of transformation govern the whole of nature.\n\nSecond, Kant defends, in the second Analogy, the important metaphysical thesis that \"the relation of cause to effect is the condition of the objective validity of our empirical judgements\" (A. 202, B. 247). It is only because we can find causal connections in our world that we can postulate its objectivity. This is a consequence of the connection between objectivity and duration: it is because things endure that I can distinguish their appearance from their reality. But they can endure only if there is a thread of causal connection that unites their temporal parts. This table is as it is _because_ it was as it was. The dependence of objectivity on causality is matched by a similar dependence of causality on substance: it is only on the assumption of enduring things that our causal laws gain application. \"Causality leads to the concept of action, this in turn to the concept of force, and thereby to the concept of substance\" (A. 204). We could put the point briefly by saying that we discover how things really are only by finding causes for how they seem, and we find causes only by postulating a realm of enduring things.\n\nSo the thought of an independent object involves thoughts about causality, and causality, Kant argues, is a species of necessity. To know any truth about the world is therefore to have knowledge of necessities: of what must be and of what might have been. This thesis radically contradicts the empiricist view that there are no necessities in nature, and also those modern theories that believe that our basic thoughts about the actual involve mental operations that are simpler than those involved in grasping the necessary and the possible.\n\nAll these striking theses emerge from an attempt to give what one might call a fully \"enriched\" account of the objectivity of the physical world. Any philosopher who takes Kant's arguments seriously will see how difficult it is to \"reduce\" the idea of an objective order to a mere synopsis of experience. Hence he will recognize the implausibility, not only of the skepticism that Kant attacked, but also of the empiricist theory of knowledge from which it began.\n\nConclusion\n\nThe principles are valid a priori, but only in the case of the \"objects of possible experience.\" The empiricists were wrong in rejecting the possibility of such a priori principles, but right in their assumption that our own perspective on the world is in some measure a constituent of our knowledge. We can know the world a priori only insofar as it is possible for it to present an appearance to our point of view. It is vain to attempt to rise above that point of view and know the world \"as it is in itself,\" independently of any possible perception. Thus all attempts to prove Leibniz's Principle of Sufficient Reason \"have, by the universal admission of those concerned, been fruitless\" (A. 783, B. 811). But the \"transcendental\" equivalent of the principle, Kant argues, is provable: the principle of sufficient reason then becomes the law of causality, the law that every event in the empirical world is bound by causal connections. This is an a priori truth, but only of the \"world of appearance,\" and therefore only of events in time. Despite Kant's swaggering self-confidence in this matter, many Leibnizians felt that they _had_ offered a proof of the Principle of Sufficient Reason\u2014Baumgarten, for one. And one of Baumgarten's disciples, Eberhard, attempted to show that the critical philosophy was no more than a restatement of the Leibnizian system, the Principle of Sufficient Reason being both necessary to Kant's enterprise and provable a priori. This was one of the few criticisms of the _Critique_ to which Kant felt obliged to reply (in _On a Discovery_ , published in 1790). Kant demonstrates the impossibility of proving the Principle \"without relation to sensible intuition\" ( _K._ 113), and emphasizes the helplessness of pure reason to advance beyond concepts to any substantial truth about the world. The Principle of Sufficient Reason can be established, he argues, only by his \"transcendental\" method, which relates the objects of knowledge to the capacities of the knower, and proves a priori only those laws that determine the conditions of experience.\n\nThe vain attempt of the Leibnizians to found knowledge in \"pure reason\" alone is the topic of the Dialectic. In following the outline of Kant's criticisms we shall be able to see exactly what he meant by \"transcendental idealism,\" and why he thought it could provide the middle course between empiricism and rationalism.\n\n**Kant divided metaphysics into three parts:** rational psychology, cosmology\u2014represented in this engraving by the armillary sphere at Kant's side\u2014and theology.\n[FOUR\n\nThe Logic of Illusion ](Roge_9781402782107_epub_c7_r1.html#d6e4)\n\nWHILE THE UNDERSTANDING, properly employed, yields genuine, objective, knowledge, it also contains a temptation to illusion. It is this temptation that Kant attempts to diagnose and criticize in his examination of \"pure reason,\" and once again his argument has a subjective and an objective side. He describes a particular faculty, reason, in its illegitimate employment; he also demolishes all the claims to knowledge that this faculty tempts us to make. His subject matter in the Dialectic is rationalist metaphysics, and he divides metaphysics into three parts\u2014rational psychology, concerning the nature of the soul; cosmology, concerning the nature of the universe and our status within it; and theology, concerning the existence of God. He then goes on to argue that each proceeds in accordance with its own kind of illusory argument, not toward truth, but toward fallacy. The diagnosis of these errors follows a common pattern. Each attempt by pure reason to establish the metaphysical doctrines toward which it is impelled transgresses the limits of experience, applying concepts in a manner that is \"unconditioned\" by the faculty of intuition. Since it is only through the synthesis of concept and intuition that judgment is formed, this attempt cannot produce knowledge. On the contrary, concepts, divorced from their \"empirical conditions,\" are empty. \"The pure concepts of the understanding can _never_ admit of _transcendental_ but _always_ only of _empirical_ employment\" (A. 246, B. 303). At the same time, since concepts contain within themselves this tendency toward \"unconditioned\" application, it is an inevitable disease of the understanding that \"pure reason\" should usurp its functions, and the categories be transformed from instruments of knowledge into those instruments of illusion that Kant calls \"ideas.\"\n\nHere we need to take another look at transcendental idealism, in order to see exactly what it is that Kant regards himself as having established in the Analytic. We find a crucial ambiguity in Kant's doctrines, which persists through the second and third _Critiques_.\n\nAppearance and Reality\n\nKant often describes transcendental idealism as the doctrine that we have a priori knowledge only of \"appearances\" and not of \"things as they are in themselves\" (or \"things-in-themselves\"). His followers and critics disputed heatedly over the \"thing-in-itself.\" Moses Mendelssohn thought of it as a distinct entity, so that an appearance is one thing, the thing-in-itself another. Others, under the influence of Kant's pupil J. S. Beck, took the phrase \"thing-in-itself\" to refer to a way of describing the very same object that we also know as an appearance. Kant lends support to this second interpretation in his correspondence with Beck, in many passages of the _Critique of Practical Reason_ , and in a letter to one of its principal advocates: \"All objects that can be given to us can be conceptualised in two ways: on the one hand, as appearances; on the other hand, as things in themselves\" ( _C._ 103n). But there is no doubt that his mind was not made up about the matter, and this led to a crucial ambiguity in the critical philosophy.\n\nKant also says that the categories can be applied to \"phenomena,\" but not to \"noumena.\" A phenomenon is an \"object of possible experience,\" whereas a noumenon is an object knowable to thought alone, and which it does not make sense to describe as an object of experience. It is natural to connect the two distinctions, and to assume that Kant believes appearances, or phenomena, to be knowable through experience, and \"things-in-themselves\" to be mere noumena, not knowable at all since nothing is knowable in thought alone. Kant says, for example, that the concept of a \"noumenon\" can be used only negatively, to designate the limit of our knowledge, and not positively, to designate things as they are in themselves. So that \"the division of objects into phenomena and noumena, and the world into a world of the senses and a world of the understanding, is . . . quite inadmissible in the positive sense\" (A. 255, B. 311). In which case, the \"thing-in-itself \" is not an entity, but a term standing proxy for the unrealizable ideal of perspectiveless knowledge.\n\nThe ambiguity in Kant's account is illustrated by the term \"appearance,\" which is taken sometimes in a transitive, sometimes in an intransitive, sense. Kant sometimes writes as though appearances are \"appearances _of_ \" something, whose reality is hidden from us. At other times, he writes as though appearances were independent entities, which derive their name from the fact that we observe and discover their nature. In this second sense, the word _appearance_ corresponds to our idea of a physical object. An appearance can be observed; it is situated in space; it enters into causal relations with other appearances and with the being who observes them. It is governed by scientific laws, and is or can be an object of discovery. It may be as it seems, and also other than it seems. It can have both secondary qualities (qualities that an object possesses only in relation to a particular sensory experience) and primary qualities (qualities that belong to its inner constitution) (A. 28\u201329). In short, appearances possess all the characteristics of physical objects. To say that an appearance, so understood, is also an appearance _of_ something is surely incoherent. For, by Kant's own theory, the \"something\" that supposedly underlies appearance could only be a \"noumenon,\" and about noumena nothing positive can be said. In particular, it makes no sense (or, at least, is devoid of content) to say that a noumenon causes, or stands in any other relation to, an appearance. (Kant's critics\u2014such as Mendelssohn and G. E. Schulze\u2014were quick to seize on this point, and put it forward as the principal weakness of the transcendental philosophy.) Sometimes indeed Kant writes as though every object is an appearance of some \"thing-in-itself,\" arguing that appearances must be \"grounded\" in a \"reality\" that is itself not merely appearance. But, since this is inconsistent with his theory of knowledge (which allows him neither to know nor even to mean what he purports to say), we must, for the moment, consider the thing-in-itself to be a nonentity. As we shall see, there are countervailing reasons that caused Kant to deny his own official theory. But it is impossible to proceed if we allow these reasons, and the rival doctrines that stem from them, to gain precedence at this juncture.\n\nPhenomena and Noumena\n\nThe first edition of the _Critique_ contained a lengthy exposition of the theory of transcendental idealism. This was deleted by Kant from the second edition, perhaps because it gave too much encouragement to the ambiguity described above. Kant also added the section called \"The Refutation of Idealism\" (see above), which attempts to provide a positive proof of objectivity, and a disproof of the \"empirical idealism\" attributed to Berkeley. Empirical idealism is the view that \"empirical\" objects are nothing but perceptions, and that the world of science has no reality beyond the experience of the observer. All empirical objects become \"ideal\" entities, with no reality outside our conception of them. By contrast, Kant argues, transcendental idealism is a form of empirical realism: it implies that empirical objects are real.\n\nKant's assertion that transcendental idealism entails empirical realism is difficult to interpret. He argues, for example, that space and time are empirically real and also transcendentally ideal (A. 28, B. 44). This could mean that, if we take an empirical perspective, so to speak, then we acknowledge the reality of space and time; while, from the transcendental point of view, these things are \"nothing at all\" (A. 28, B. 44). However, the idea of a transcendental point of view is, as Kant recognizes, highly contentious. It is not a point of view that is available to us, and therefore not something of which we can have a positive conception. A simpler reading of Kant's theory is the following: empirical objects are real, whereas transcendental objects are ideal. A transcendental object is not perceivable, and does not belong to the world of space, time, and causality. The \"monad\" of Leibniz is such an object, and must therefore always remain a mere idea in the mind that conceives it, with no independent reality. But what are empirical objects? The answer that suggests itself is \"whatever objects are discovered or postulated through experience.\"\n\nThe above states a metaphysical view that exactly corresponds to Kant's theory of knowledge. In the last analysis, it does not matter whether we describe the distinction between the empirical and the transcendental object as a distinction between that which exists and that which does not, or as a distinction between that which is knowable and that which is not. For, to borrow a remark of Wittgenstein's, \"a nothing would do as well as a something about which nothing can be said.\" It seems then that the distinction between empirical and transcendental objects corresponds again to that between phenomena and noumena, the first being knowable, the second unknowable, since the concept of a noumenon can be employed only negatively, in order to mark out the limits of experience. All noumena are transcendental objects, and all transcendental objects, being merely \"intelligible\" (that is, not knowable through experience), are noumena. So it would seem that the three distinctions coincide: phenomenon\/noumenon; empirical object\/transcendental object; appearance\/thing-in-itself. In a lengthy discussion of Leibniz, which forms the bridging passage between the Analytic and the Dialectic, this is, in effect, what Kant says (especially A. 288\u201389, B. 344\u201346). Many scholars do not accept this interpretation; but it seems to me that, if we do not accept it, we attribute to Kant more inconsistency than his dexterity can sustain.\n\nIt is not surprising, therefore, to find that the ambiguity that surrounds Kant's conception of \"appearance\" attaches also to that of the \"phenomenon.\" This term is interpreted, sometimes as denoting a real perceivable object that exists independently of the observer, sometimes as denoting a mental \"representation\" (or, in modern parlance, \"intentional object\"). Under the latter interpretation the phenomenon becomes subjective (it is as it seems, and seems as it is). To establish its existence is to provide no guarantee of objectivity. No significant distinction would then remain between Kant's position (that we can have knowledge of phenomena) and the empirical idealism that he claims to reject. Even Leibniz admitted more objectivity into our \"point of view\" than the empirical idealist, since he introduced the notion of the \"well-founded phenomenon\" (see above), according to which the stability of appearances is sufficient to warrant the distinction between being and seeming. Leibniz's approach to the \"phenomenal world\" certainly influenced Kant, more perhaps than he cared to admit. But he wanted to go further in the direction of establishing the objectivity of the world of appearance, not less far.\n\nIt is clear from all this that the only acceptable interpretation of the terms _phenomenon_ , _appearance_ , and _empirical object_ is as denoting items in the physical world. Appearances include tables, chairs, and other such visible things; they also include entities that are observable only through their effects, such as atoms (A. 442, B. 470) and the most distant stars (A. 496, B. 524). These \"theoretical\" entities also have the reality that comes from existence in space and time, and from the order imposed by the categories. They too are perceivable, in the sense of entering into specific causal relations with the mind that knows them. In other words, any object of scientific investigation is a \"phenomenon,\" and all phenomena are knowable in principle. But nothing else is knowable. \"Nothing is really given us save perception and the empirical advance [i.e., scientific inference] from this to other possible perceptions\" (A. 493, B. 521). As for the idea of a noumenon, this is \"not the concept of an object\" but a \"problem unavoidably bound up with the limitation of our sensibility\" (A. 287, B. 344).\n\nThe Unconditioned\n\nKant's intention in the Dialectic is to show that we cannot know the \"world as it is,\" meaning the world conceived apart from the perspective of the knower. We must not, as Kant puts it, aspire to \"unconditioned\" knowledge. At the same time, it seems inevitable that we should do so. Every time we establish something by argument, we assume the truth of the premise. The premise therefore describes the \"condition\" under which the conclusion is true. But what about the truth of this condition? That too must be established by argument, and will turn out to possess its truth only \"conditionally.\" Hence reason (in its guise as inference) inevitably leads us to search for the \"unconditioned,\" the ultimate premise whose truth is derived from no other source. This \"idea\" of reason contains the source of all metaphysical illusion. For all knowledge that we can legitimately claim is subject to the \"conditions\" of possible experience. To aspire to knowledge of the unconditioned is to aspire beyond the conditions that make knowledge possible.\n\nThe effort of transcendence is, Kant argues, inevitable. Not only do we seek to transcend the conditions contained in the possibility of experience. We also seek to know the world \"as it is,\" free from the conditions to which it may be subject by such categories as substance and cause. These enterprises are in fact one and the same; for, as the Analytic shows, the two sets of conditions are identical. In each case the advance of reason toward the \"unconditioned\" is the pursuit of knowledge untainted by perspective. Reason always aims to view the world, as Leibniz had viewed it, from no point of view.\n\n\"Pure Reason\"\n\nFrom the standpoint of Kant's theory of knowledge, reason must be seen as the highest of cognitive faculties, under which all that is distinctive of self-knowledge is subsumed. Apart from its use in understanding (forming judgments), reason can be employed legitimately in two further ways: practically, and in carrying out inferences. Practical reason cannot be regarded as a branch of the understanding, since it does not issue in judgments (it makes no claims about the true and the false). Nevertheless its use is legitimate: I can reason what to do, and my action can be a legitimate outcome of this process. Inference, which is the practice of deriving the logical consequences of a judgment, is also legitimate. But, unlike understanding, it employs no concepts of its own (for an inference that, as it were, \"adds a concept\" to the premise is for that reason invalid).\n\nIt is only when _pure_ reason enters our thoughts that the \"logic of illusion\" begins to beguile us. Pure reason is distinguished by the fact that it tries to make judgments of its own, using, not concepts, but \"ideas\" from which all empirical conditions have been removed. The logic of illusion is \"dialectical\": it must inevitably end in fallacy and contradiction. This tendency toward fallacy is not accidental, but intrinsic. There is no way in which reason can set out to know the world through \"ideas\" and avoid the errors that lie in wait for it. These errors have already been committed, just as soon as we leave the knowable realm of experience and embark on the journey toward the \"unconditioned\" world beyond. At the same time there is no way in which we can avoid the temptation toward this vain journey into the transcendental. Our very possession of a point of view on the world creates the \"idea\" of a world seen from no point of view. Thus we strive always \"to find for the conditioned knowledge of the understanding, the unconditioned, whereby its unity might be brought to completion\" (A. 307, B. 364).\n\nPure Reason and Metaphysics\n\nI have already drawn attention to Kant's important threefold division of the subject matter of speculative metaphysics. In \"transcendental psychology,\" reason generates its doctrines of the soul; in rational cosmology, it attempts to describe the world in its \"unconditioned totality\"; in theology, it creates the idea of the perfect being who presides over a transcendent world. By a contrivance, Kant reconciles this division with the traditional view, that \"metaphysics has as the proper object of its enquiries three ideas only: God, freedom and immortality\" (B. 395n). The idea of freedom is assigned to cosmology, on the grounds that all the metaphysical problems created by this idea stem from the belief that there is something\u2014the moral agent\u2014that is both in the world of nature and also outside it. This is one of the most important of Kant's theories. But I shall postpone discussion of it until the chapter that follows.\n\nCosmology\n\nThe illusions of cosmology are called \"antinomies.\" An \"antinomy\" is the peculiar fallacy which enables us to derive both a proposition and its negation from the same premise. According to Kant, antinomies are not genuine contradictions, since both of the propositions that constitute them are false, being based on a false assumption. He entitles this \"kind of opposition _dialectical_ , and that of contradictories _analytical_ \" (A. 504, B. 532). He offers various tortuous explanations of how a proposition and its negation can both be false, and it is perhaps unnecessary to be detained by this puzzling piece of logic. Kant's point is that, in deriving each side of an antinomy, the same false assumption must be made. The purpose of his \"critique\" is to root out this false assumption and show it to stem from the application of one of reason's \"ideas.\" The assumption involved in cosmology is that we can think of the world in its \"unconditioned totality.\" This would be possible only by transcending the perspective of \"possible experience\" and trying to see all nature as a whole, from a point of view outside of it. The illusory nature of this idea is displayed by the fact that, from the premise of such a transcendent perspective, a \"dialectical\" contradiction follows.\n\nSuppose, for example, that I allow myself to entertain the idea of the whole world of nature, as it is situated in space and time. If I now attempt to apply the idea in judgment, I must pass beyond my empirical vision in order to grasp the totality of all objects of experience. I must attempt to envisage the whole of nature, independently of my particular perspective within it. If I could do that, I could ask myself: Is this totality limited or unlimited in space and time? Does it, or does it not, have boundaries? I find myself equally able to prove both conclusions. I can prove, for example, that the world must have a beginning in time (otherwise an infinite sequence of events must have already elapsed, and, Kant argues, a \"completed infinity\" is an absurd idea). I can equally prove that it must have no beginning in time (for if it had a beginning, then there must be a reason why it began when it did, which is to suppose, absurdly, that a particular time has a causal property, or a capacity to \"bring into being,\" independently of the events that occupy it).\n\nA similar contradiction can be derived from the assumption that the world as a whole has an explanation for its existence. From that assumption, I can prove that the world is causally self-dependent, consisting of an infinite chain of causes linking each moment to its predecessor. For the idea of a beginning to this series\u2014the idea of a \"first cause\"\u2014is absurd, leading automatically to the question \"What caused the beginning?,\" to which there is no coherent answer. I can equally prove that the world is causally dependent, deriving its existence from some being that is \"cause of itself,\" or _causa sui_. For if there were no such being, then none of the causes in the sequence of nature would explain the existence of its effect. In which case, nothing in nature would have an explanation, and it would be impossible to say why anything should exist at all.\n\nSuch antinomies result from the attempt to reach beyond the perspective of experience to the absolute vantage point from which the totality of things (and hence the world \"as it is in itself\") can be surveyed. If we suppose nature to be a thing-in-itself\u2014that is, if we remove from the concept of nature the reference to any possible experience through which it is observed\u2014then the proofs of the antinomies are well grounded. \"The conflict which results from the propositions thus obtained shows, however, that there is a fallacy in this assumption, and so leads us to the discovery of the true constitution of things, as objects of the senses\" (A. 507, B. 537). The idea of \"absolute totality\" holds only of \"things-in-themselves\" (A. 506, B. 534), which is to say, of nothing knowable. For example, the concept of cause, which we can apply within the realm of empirical objects in order to designate their relations, becomes empty when lifted out of that realm and applied to the world as a whole. It is then applied beyond the empirical conditions that justify its application, and so leads inevitably to contradiction.\n\nNevertheless, Kant argues, these antinomies are not to be lightly dismissed as errors no sooner perceived than avoided. The assumption of totality that generates them is both the cause and the effect of all that is most serious in science. Suppose we were to accept the \"big bang\" hypothesis concerning the origin of the universe. Only a shortsighted person would think that we have then answered the question of how the world began. For what caused the bang? Any answer will suppose that something already existed. So the hypothesis cannot explain the origin of things. The quest for an origin leads us forever backward into the past. But either it is unsatisfiable\u2014in which case, how does cosmology explain the _existence_ of the world?\u2014or it comes to rest in the postulation of a _causa sui_ \u2014in which case, we have left the scientific question unanswered and taken refuge in theology. Science itself pushes us toward the antinomy, by forcing us always to the limits of nature. But how can I know those limits if I cannot also transcend them?\n\nKant engages in an elaborate diagnosis of the antinomies, arguing that the two sides always correspond to rationalism and empiricism, respectively, and he takes the opportunity to explore again the various errors of those philosophies. The resulting discussion is exceedingly complex, and has produced as much philosophical commentary as anything in the first _Critique_. Kant's love of system leads him to combine arguments of very different import and quality. But behind the profusion of reasoning lies one of the acutest discussions of scientific method to have issued from the pen of a philosopher. A reading of Kant's discussion inspired thinkers as diverse as Hegel and Einstein, and few can fail to be puzzled by the problems that Kant unearths. How _can_ I view the world in its totality? And yet, how can there be explanation if I cannot? How can I explain the existence of anything, if I cannot explain the existence of everything? If I am confined forever within my own point of view, how can I penetrate the mystery of nature?\n\nTheology\n\nI have discussed the first and fourth of Kant's antinomies. The latter brings us to theology, and to Kant's subsequent suggestion that the idea of a _causa sui_ is itself empty, and can be applied in judgment only so as to generate a contradiction. In the chapter entitled \"The Ideal of Reason,\" Kant reviews the traditional arguments for the existence of God, and imposes upon them a now famous classification. There are, he says, only three kinds of argument for God's existence, the \"cosmological,\" the \"ontological,\" and the \"physico-theological.\" The first kind comprises all arguments which, proceeding from some contingent fact about the world, and from the question \" _Why_ is it so?,\" postulate the existence of a necessary being. A version of this is the \"first cause\" argument discussed above: only if the series of causes begins in a _causa sui_ can any contingent fact be finally explained. Of the second kind are all arguments which, in order to free themselves from contingent premises (which, because they might be false, might also be doubted), attempt to prove the existence of God from the concept of God. Of the third kind are all arguments from \"design,\" which begin from the premise of some good in nature, and argue by analogy to the perfection of its cause.\n\nKant says of the argument from design that it \"always deserves to be mentioned with respect. It is the oldest, the clearest, and the most consonant with human reason. It enlivens the study of nature, just as it itself derives its existence and gains ever new strength from that source\" (A. 623, B. 651). In the third _Critique_ he explains in more detail why this argument appealed to him. Kant was no atheist, and was perturbed by Hume's posthumous _Dialogues on Natural Religion_ , the German version of which appeared as the _Critique_ was being prepared for the press. There was time for Kant to wonder about Hume's motives, but not to consider his arguments (A. 745, B. 773). Hume foreshadows Kant's own critique of rational theology. But he is particularly dismissive of the argument from design, arguing that it proves either nothing at all (since there is no genuine analogy between the perfections of nature and the perfections of art) or, at best, the existence of a being no more perfect than the world of his creation. Kant too was unsatisfied by the argument from design. But, although he thought of it as invalid, at the same time he felt it to be the expression of a true presentiment. Hence he tried to provide a wholly novel elucidation of it, from the point of view not of the first but of the third _Critique._\n\n**Kant claimed that there are only three kinds** of argument about the existence of God: cosmological, ontological, and physico-theological. This manuscript illumination from a thirteenth-century French Bible posits God in the cosmological sense, as the great architect of the universe.\n\nConsidered as an intellectual proof, the argument from design can never be more cogent than the cosmological proof, upon which it depends (A. 630, B. 658). For design can be imputed to the world only on the assumption that the world has an explanation. That assumption carries no weight unless the cosmological proof is valid. We must show that we can step outside nature, in order to postulate the existence of a transcendent, necessary being\u2014a being upon whom the entire order of things depends. Such a being must be self-caused; moreover, his existence cannot be a merely contingent fact, since, if that were so, he would be part of the chain of natural events, rather than an explanation of them. Something can exist necessarily only if its existence is implied by its concept. The existence of God must _follow_ from the concept of God, since it is only by logical relations that a concept can explain anything. So, in the end, all three arguments reduce to the ontological argument, which tells us that God must exist, since existence belongs to the very concept of God.\n\nIn its traditional forms the ontological argument proves not only the existence, but also the perfection, of God. Neither of the other arguments seems able to prove God's absolute perfection: once again, they must rely on the ontological argument to give intellectual grounds for religious sentiment. The ontological argument proceeds as follows. God is an all-perfect being. Among perfections we must count moral goodness, power, and freedom. But we must also count existence. For the concept of an existent _x_ is the concept of something more perfect than an _x_ ; to take away existence is to take away perfection. So existence is a perfection. Hence it follows from the idea of God, as an all-perfect being, that God exists.\n\nKant's rebuttal of this argument is famous for its anticipation of a doctrine of modern logic\u2014the doctrine that existence is not a predicate. (When I say that John exists, is bald, and eats oysters, I attribute to John not three properties, but two.) Leibniz, in his discussion of contingency, had already recognized that existence is quite different from ordinary predicates. Nevertheless, he accepted the ontological argument, and did not see the logical consequence for his philosophy. To say that an _x_ exists is to add nothing to its concept: it is to say that the concept has an instance. Indeed, there is already a fallacy (Kant even says a contradiction) involved in introducing existence into the concept of a thing (A. 597, B. 625). For it then becomes empty to assert that the thing exists. The assertion makes no advance from concept to reality. And so it affirms the existence of nothing. The ontological argument says that existence is a perfection. But it cannot be a perfection, since it is not a property. If the argument were valid, Kant thinks, it would follow that the judgment that God exists expresses an analytic truth. The theory that existence is not a predicate implies, however, that all existential propositions are synthetic (A. 598, B. 626).\n\nThe Regulative Employment of the Ideas of Reason\n\nHaving completed his denunciation of the \"logic of illusion,\" Kant goes on to argue, at considerable length, and in a relaxed, expansive style appropriate to a writer whose intellectual labor is ended, that there is, after all, a legitimate use for the ideas of reason. Such ideas as that of unconditioned totality, and of the perfect creator who exists of necessity, generate illusions when considered in their \"constitutive\" role: that is, when considered as descriptions of reality. The correct way to see them, however, is as \"regulative principles\" (A. 644, B. 672). If we act as if these ideas were true of reality, then we are led to formulate true hypotheses. The ideas of order and totality, for example, lead us to propose ever wider and simpler laws, in terms of which the empirical world becomes ever more intelligible. This \"regulative\" employment of the ideas is an employment from within the standpoint of experience. The constitutive employment attempts to transcend that standpoint, into the illusory realm of reason. The contradictions do not stem from the ideas themselves (which are not contradictory but merely empty); they stem from their wrong application. By way of emphasizing their regulative function, Kant sometimes refers to the ideas of reason as _ideals_.\n\nThus \"the ideal of a supreme being is nothing but a _regulative principle_ of reason, which directs us to look upon all connection in the world as if it originated from an all-sufficient and necessary cause\" (A. 619, B. 647). Considered thus it is the source, not of illusion, but of knowledge. The knowledge that it leads to remains circumscribed by the conditions of possible experience: in other words, it conforms to the categories, and does not reach beyond their legitimate territory into a transcendent realm. The idea \"does not show us how an object is constituted, but how, under its guidance, we should _seek_ to determine the constitution and connection of the objects of experience\" (A. 671, B. 699). Thus reason is led back from its vain speculations to the empirical world, trading the illusions of metaphysics for the realities of empirical science.\n\nThe Soul\n\nKant's discussion of the soul, and of the concept of the \"self\" from which it gains its initial description, is among the most subtle parts of his philosophy. The account is given in two complex arguments: the first at the beginning of the Dialectic, in which he attacks the rationalist doctrines of the soul; the second in the third antinomy, and in the _Critique of Practical_ _Reason_ , in which he describes the nature of morality.\n\nThe \"Paralogisms of Pure Reason,\" which deal with the rationalist doctrines of the soul, were substantially revised for the second edition, perhaps because this aspect of the critical philosophy forms part of the multifaceted argument of the transcendental deduction, with which Kant had remained so dissatisfied. Kant's argument in the Analytic had begun by recognizing the peculiar reality of self-consciousness. I have a privileged awareness of my states of mind, and this is an \"original\" or \"transcendental\" act of understanding. The rationalists had sought to deduce from this privileged knowledge a specific theory of its object. They had thought that, because of the immediacy of self-awareness, the self must be a genuine object of consciousness. In the act of self-awareness I am presented with the \"I\" that is aware. I can maintain this self-awareness even while doubting every other thing. Moreover, I am necessarily aware of my unity. Finally, I have an intuitive sense of my continuity through time: this cannot be derived from the observation of my body, or from any other external source. It therefore seems natural to conclude that I know myself to be substantial, indivisible, enduring, perhaps even immortal, on the basis of self-awareness alone. Such, Kant thought, was Descartes's argument. Nor is the conclusion an eccentricity of the \"Cartesian\" view of consciousness. Like all the illusions of reason, it is one into which we are tempted just as soon as we begin to reflect on the datum before us. Every rational being must be tempted to think that the peculiar immediacy and inviolability of self-awareness guarantee its content. In the midst of every doubt, I may yet know this thing that is me, and, reason assures me, this intimate acquaintance with my own nature gives grounds for the belief in the immateriality of the soul. Furthermore, I cannot conceive of my own nonexistence, since every attempt to do so postulates the \"I\" as conceiver.\n\nThe reasoning is erroneous, since it moves from the purely _formal_ unity of apperception to the substantial unity affirmed in the doctrine of the soul. \"The unity of consciousness which underlies the categories is . . . mistaken for an intuition of the subject as object, and the category of substance is then applied to it\" (B. 421). Although the transcendental unity of apperception assures me that there is a unity in my present consciousness, it tells me nothing else about the kind of thing that bears it. It does not tell me that I am a substance (that is, an independently existing object) as opposed to an \"accident\" or property. (For example, it does not refute the view that the mind is a complex property of the body.) It is \"quite impossible, by means of . . . simple self-consciousness, to determine the manner in which I exist, whether it be as substance or accident\" (B. 420). If I cannot deduce that I am a substance, so much the less can I deduce that I am indivisible, indestructible, or immortal. The unity of consciousness does not even assure me that there is something in the empirical world to which the term _I_ applies. For the peculiar features of self-consciousness summarized under the idea of a transcendental unity of apperception are simply features of a \"point of view\" on the world. The \"I\" as thereby described is not part of the world but a perspective upon it (a way things seem). \"For the I is not a concept but only a designation of the object of inner sense in so far as we know it through no further predicate\" ( _F._ 98). To study the peculiarities of our self-awareness is, then, to study no item _in_ the world. It is rather to explore that limiting point of empirical knowledge. \"The subject of the categories cannot by thinking the categories acquire a concept of itself as an object of the categories\" (B. 422). It is no more possible for me to make the \"I\" into the object of consciousness than it is to observe the limits of my own visual field. \"I\" is the expression of my perspective, but denotes no item within it. To suppose otherwise is to pass illegitimately from subject to object; it is to suppose that the subject of consciousness can also, _as subject_ , be the object of its own awareness.\n\nThe conclusion that Kant draws is this. There is a gap between the premise of \"transcendental psychology\"\u2014the transcendental unity of apperception\u2014and its conclusion\u2014the substantiality of the soul. Because the first describes a point of view on the world, and the second an item in the world, it is impossible that reason should take us validly from the one to the other. Whether correct or not, Kant's suggestion has provided the cornerstone of many subsequent philosophies of the self, from that of Schopenhauer to those of Husserl, Heidegger, and Wittgenstein.\n\nSometimes Kant implies that the \"I\" of self-awareness refers to a transcendental object. For it might seem that, having proved that the \"I\" is not part of the empirical world, Kant has given us reason to refer it to the world of the thing-in-itself that lies beyond experience. This is not a legitimate conclusion from his argument, but, on the contrary, another, more subtle reiteration of the fallacy that it was designed to expose. Nevertheless, it is a conclusion that Kant was inclined to countenance, since, without it, he thought morality would not be possible. Kant sought for a positive theory of the soul, not through pure reason, but through practical reason (B. 430\u201331). In order to understand this theory we must therefore explore his description of the moral life of the rational agent.\n\n**Detail from a game board** for the Mansion of Happiness, \"an instructive, moral, and entertaining amusement.\"\n[FIVE\n\nThe Categorical Imperative ](Roge_9781402782107_epub_c7_r1.html#d6e5)\n\nKANT'S _CRITIQUE OF PRACTICAL REASON_ was preceded by a brilliant r\u00e9sum\u00e9 of his moral viewpoint, the _Groundwork of the Metaphysic of_ _Morals_. These works treat of \"practical reason\": in using this expression Kant was consciously reviving the ancient contrast between theoretical and practical knowledge. All rational beings recognize the distinction between knowing the truth and knowing what to do about it. Judgments and decisions may each be based on, and amended through, reason, but only the first can be true or false. Hence there must be an employment of our rational faculties that does not have truth, but something else, as its aim. What is this something else? Aristotle said happiness; Kant says duty. It is in the analysis of the idea of duty that Kant's distinctive moral vision is expressed.\n\nSuppose we establish the objectivity of judgment, and provide the necessary metaphysical grounding to those scientific principles that underlie the process of discovery. There remains another problem of objectivity, raised by practical, rather than theoretical, knowledge. Can we know what to do objectively, or must we simply rely on our subjective inclination to guide us? It is to this problem that Kant addressed himself, producing the most metaphysical and abstract basis that has ever been given for the common intuitions of morality.\n\nThe Antinomy of Freedom\n\nThe starting point of Kant's ethics is the concept of freedom. According to his famous maxim that \"ought implies can,\" the right action must always be possible: which is to say, I must always be free to perform it. The moral agent \"judges that he can do a certain thing because he is conscious that he ought, and he recognises that he is free, a fact which, but for the moral law, he would never have known\" ( _P._ 30). In other words, the practice of morality forces the idea of freedom upon us. But, Kant argues, this idea, viewed theoretically, contains a contradiction, and he displays this contradiction in the third antinomy of the first _Critique._\n\nEvery change that occurs in the order of nature has a cause: this is an \"established principle\" of the Analytic, and \"allows of no exception\" (A. 536, B. 564). If this is so, then every event in nature is bound in chains of ineluctable necessity. At the same time I think of myself as the originator of my actions, giving rise to them spontaneously, under the influence of no external constraint. If my action is part of nature, this seems to contradict the view that every event in nature is bound by causal necessity. If it is not part of nature, then it falls outside the realm of causal connection, and my will is the originator of nothing in the natural world.\n\nThere is only a contradiction here if I really am free. Kant is sometimes content to argue merely that I must think of myself as free. It is a presupposition of all action in the world\u2014and hence of reasoned decisions\u2014that the agent is the originator of what he does. And, Kant suggests, I cannot forsake this idea without losing the sense of myself as agent. The very perspective of reason that sees the world as bound in chains of necessity also sees it as containing freedom. Occasionally Kant goes further, and argues for the \"primacy\" of practical reason ( _P._ 120\u201321), meaning that all thought is an exercise of freedom, so that, if practical reason were impossible, we could not think coherently. In which case, the certainty of my freedom is as great as the certainty of anything. (This argument occurs, in more rhetorical form, in the writings of Sartre, whose existentialist doctrine of the moral life owes much to Kant.) If this is true, then of course the antinomy of freedom becomes acute: we are compelled by practical reason to accept that we are free, and by understanding to deny it.\n\nKant felt that there must be a solution to this antinomy, since in the practical sphere the employment of reason is legitimate. It is indeed practical reason that tells me what I am. The illusory progress of pure reason toward self-contradiction ought not to forbid practical reason, through which the antinomy must therefore be resolved. Pure reason leaves, as it were, a \"vacant place\" in its account of the world, where the moral agent should be. \"This vacant place is filled by pure practical reason with a definite law of causality in an intelligible world . . . namely, the moral law\" ( _P._ 49). This new \"law of causality\" is called \"transcendental freedom,\" and it defines the condition of the moral agent. The law of cause and effect operates only in the realm of nature (the empirical realm). Freedom, however, belongs, not to nature, but precisely to that \"intelligible\" or transcendental realm to which categories like causality do not apply. I exist in the world of nature, as one \"appearance\" among others. But I also exist as a \"thing-in-itself,\" bound, not by causality, but by the laws of practical reason. It is not that I am two things, but rather one thing, conceived under two contrasting aspects. Thus, \"there is not the smallest contradiction in saying that a thing in appearance (belonging to the world of sense) is subject to certain laws, from which the very same as a thing in itself is independent.\" Moreover, the moral agent must always \"conceive and think of himself in this two-fold way\" ( _G._ 453). Freedom, then, is a transcendental \"idea,\" without application in the empirical world. And, in knowing ourselves to be free, we know ourselves at the same time as part of nature and as members of a transcendental world.\n\n**Title page from** the first edition of Kant's _Grundlegung zur Metaphysik_ _der Sitten_ ( _Groundwork of the_ _Metaphysic of Morals_ ).\n\nThe Transcendental Self\n\nThe doctrine of transcendental freedom is both puzzling and appealing. Its appeal lies in its promise of access to the transcendental; its puzzling quality comes from Kant's previous proof that such access is impossible. By Kant's own argument, there is nothing to be known, and nothing meaningfully to be said, about the transcendental world. Kant recognizes the difficulty, and admits that the \"demand to regard oneself _qua_ subject of freedom as a noumenon, and at the same time from the point of view of physical nature as a phenomenon in one's own empirical consciousness\" is \"paradoxical\" ( _P._ 6). And he even goes so far as to say that, while we do not comprehend the fact of moral freedom, \"we yet comprehend its _incomprehensibility_ , and this is all that can fairly be demanded of a philosophy which strives to carry its principles to the very limit of human reason\" ( _G._ 463).\n\nWe can go some way toward explaining Kant's doctrine if we relate the \"transcendental freedom\" that underlies practical reason to the \"transcendental unity of apperception\" that underlies our knowledge of nature. Our perspective on the world contains two distinct aspects; and neither the unity of consciousness, nor transcendental freedom, can be deduced from our knowledge of the empirical world. But they are each guaranteed a priori as preconditions of the knowledge that we have. The first is the starting point for all our knowledge of truths; the second the starting point for all deliberation. They are transcendental, not in the positive sense of involving knowledge of a transcendental object, but in the negative sense of being presupposed by the legitimate employment of our rational powers. Hence they lie at the limit of what can be known. Freedom, being a perspective _on_ the empirical world, cannot also be part of it. The knowledge of our own freedom is therefore a part of the \"apperception\" that defines our perspective. (Authority for this interpretation can be found in the first _Critique_ , notably A. 546\u201347, B. 574\u201375.)\n\nPure reason attempts to know the transcendental world through concepts. In other words, it attempts to form a positive conception of noumena. This attempt is doomed to failure. Practical reason, however, not being concerned in the discovery of truths, imposes no concepts on its objects. It will never, therefore, lead us into the error of forming a positive conception of the transcendental self. We know this self only practically, through the exercise of freedom. While we cannot translate this knowledge into judgments about our nature, we can translate it into some other thing. This other thing is given by the laws of practical reason, which are the synthetic a priori principles of action. Just as there are a priori laws of nature that can be derived from the unity of consciousness, so too are there a priori laws of reason that can be derived from the perspective of transcendental freedom. These will not be laws about the true and the false: they will have no part to play in description, prediction, and explanation. They will be practical laws, concerning what to do. The free agent will be bound by them in all his practical reasoning, since acceptance of them is a presupposition of the freedom without which practical reason is impossible.\n\nIt is true that Kant wavers between the view that the transcendental self is a kind of perspective, and the view that it is a distinct noumenal thing. He even attempts to resuscitate through practical reason all those theoretical conclusions concerning God, the soul, and immortality that had been dismissed as illusions of pure reason. I shall not, for the present, follow Kant into these mysterious regions. But the reader should bear in mind that I have only postponed the deep metaphysical problem that Kant's ethical doctrine creates.\n\nThe Problem of Practical Reason\n\nKant's idea of freedom becomes clearer when seen in the context of the problem that it was supposed to solve. Rational beings exist not only as self-conscious centers of knowledge, but also as agents. Their reason is not detached from their agency, but forms a constitutive part of it; which is to say that, for a rational being, there is not only action, but also the _question_ of action (the question \"What shall I do?\"), and this question demands a reasoned answer. My rationality is expressed in the fact that some of my actions are intentional (they issue, to use Kant's term, from my \"will\"). Of all such actions the question can be asked: Why do that? This question asks, not for a cause or explanation, but for a reason. Suppose someone asks me why I struck an old man in the street. The answer: \"Because electrical impulses from my brain precipitated muscular contractions, and this resulted in my hand making contact with his head\" would be absurd and impertinent, however accurate as a causal explanation. The answer: \"Because he annoyed me\" may be inadequate, in that it gives no good reason, but it is certainly not absurd. Reasons are designed to justify action, and not primarily to explain it. They refer to the grounds of an action, the premises from which an agent may conclude what to do.\n\nPractical reasons concern either ends or means. If I have an end in view, then I may deliberate over the means to achieve it. All philosophers agree that such reasoning exists; but many of them think that it shows no special \"practical\" employment of the rational faculties. It is merely theoretical reason, put to use. Kant himself is of this view, arguing that \"precepts of skill\" (how to find the means to one's ends) are merely theoretical principles ( _P._ 25\u201326). Skeptical philosophers go further, and argue, with Hume, that there is no other use for reason in practical matters. All reasoning concerns means. Reason can neither generate, nor justify, the ends of our activity, since, in Hume's words, \"reason is, and ought only to be, the slave of the passions.\" It is from the \"passions\" that our ends are drawn, since it is passion, and passion only, that provides the ultimate motive to act. Reason can persuade us to act only when we are already motivated to obey it. If that is the case, then, Kant held, there can be no objective practical knowledge. For there can be no way in which reason will settle the question of what to do.\n\n**David Hume's claim** that \"reason is, and ought only to be, the slave of the passions\" appeared in _A Treatise of Human Nature_ (1739), the first-edition title page of which is shown here.\n\nKant held that practical reason is possible. He affirmed the (commonsense) belief that reason may constrain and justify, not only the means, but also the ends. In which case there may be an objective exercise of practical reason. It would be objective because based in reason alone, recommending ends of action to all rational beings, irrespective of their passions, interests, and desires. For this to be possible, however, reason must, as Hume argued, not only justify but also motivate our actions. If reason did not also _prompt_ me to act, then reason would play no part in the process of decision making. So it would not be practical. Now, if reason generates only judgments about the world and inferences therefrom, it is hard to see how it _can_ be a motive to act. That such and such a judgment is true or false may prompt me to all kinds of action, depending upon my ends. If these ends stem from the \"passions,\" then reason has no part in determining them. The only way in which reason can become practical, therefore, is by issuing not in judgments but in imperatives. An imperative does not describe the world; it addresses itself to an agent, and, if he accepts it, determines what he does. If, therefore, there are imperatives that arise simply from the exercise of reason, then reason alone can move us to action. \"In a practical law, reason determines the will immediately\" ( _P._ 25).\n\nAutonomy of the Will\n\nKant's moral philosophy emerges from the amalgamation of the idea of transcendental freedom with that of an imperative of reason. He believes that reasoning about ends must always suppose just the kind of transcendental freedom that his metaphysics claims to be possible. Freedom is the power to will an end of action for myself. Any derivation of my ends from an external source is at the same time a subjection of myself to that source. And any natural process that governs my action confers on me the unfreedom of its cause. I then become the passive channel through which natural forces find their enactment. If my action is called unfree, it is because there is a sense in which it is not truly _mine._\n\nAn action that originates in me can be attributed only to me, and is therefore in a real sense mine. In respect of such an action I am free. I act freely whenever I act, and unfreely whenever some other agency acts through me. This raises the question what am I? The obvious answer is \"a transcendental self,\" since that explains my freedom from the causality of nature. But Kant now supplements this answer with a theory of the will. An action originates in me whenever I decide on the action, simply on the basis of considering it. I do not consult my desires, interests, or any other \"empirical conditions,\" since that is to subject myself to the causality of nature. I simply reflect on the action, and choose it for its own sake, as an end in itself. This is the paradigm of a free action: one that is brought into being by reason alone. Such an action can be attributed, Kant thought, to no \"natural\" force, no chain of \"empirical\" causality. It arises spontaneously out of the rational processes that constitute my will.\n\nFreedom, then, is the ability to be governed by reason. The imperatives of reason discussed in the last section are \"laws of freedom\": principles whereby reason determines action. Hence there is a \"causality of freedom\" in addition to the \"causality of nature,\" and freedom is nothing more than obedience to the first, perhaps in defiance of the second. This ability to be motivated by reason alone Kant called the autonomy of the will, and he contrasted it with the \"heteronomy\" of the agent whose will is subject to external causes. Kant designates as external any cause that belongs to the \"causality of nature\"\u2014that is, any cause that is not founded in reason alone. An action that springs from desire, emotion, or interest is therefore \"heteronomous.\"\n\nKant now develops his conception of the autonomous agent. This is an agent who is able to overcome the promptings of all heteronomous counsels, such as those of self-interest and desire, should they be in conflict with reason. Such a being postulates himself as a \"transcendental being\" in that he defies the causality of nature and refers the grounds of his actions always to the \"causality of freedom.\" Only an autonomous being has genuine ends of action (as opposed to mere objects of desire), and only such a being deserves our esteem, as the embodiment of rational choice. The autonomy of the will, Kant goes on to argue, \"is the sole principle of all moral laws, and of all duties which conform to them; on the other hand, heteronomy of the will not only cannot be the basis of any obligation, but is, on the contrary, opposed to the principle thereof, and to the morality of will\" ( _P._ 43). Because autonomy is manifest only in the obedience to reason, and because reason must guide action always through imperatives, autonomy is described as \"that property of the will whereby it is a law to itself\" ( _G._ 440). It is also \"the ground of the dignity of human nature and of every rational nature\" ( _G._ 436).\n\nMetaphysical Difficulties\n\nWe must now return to the metaphysical problem of transcendental freedom. Two difficulties in particular stand out, since they correspond to difficulties that Kant himself had discovered in the rationalist metaphysics of Leibniz. First, how is the transcendental self to be individuated? What makes _this_ self _me_? If the essential feature is reason and the agency that springs from it, then, since the laws of reason are universal, how am I to be distinguished from any other being who is subject to them? If the essential feature is my \"point of view\" on the world, how is Kant to avoid the Leibnizian view of the self as a monad, defined by its point of view, but existing outside the world that it \"represents\" thereby, incapable of entering into real relation with anything contained in it?\n\nSecond (or rather to continue the objection), how is the transcendental self related to the empirical world? In particular, how is it related to its own action, which is either an event in the empirical world or else totally ineffective? I must exist, as Kant acknowledges, both as an \"empirical self,\" within the realm of nature, and as a transcendental self, outside it. But, since the category of cause applies only to nature, the transcendental self remains forever ineffective. In which case, why is its freedom so valuable? Kant relies on the view that the category of cause denotes a relation in time (of before and after); whereas the relation of a reason to that which springs from it is not temporal at all ( _P._ 94\u201395). His tortuous discussion of this in the first _Critique_ (A. 538\u201341, B. 566\u201369) fails to make clear how a reason offered to the transcendental self can motivate (and so explain) an event in the empirical world.\n\nKant's preferred stance toward these difficulties is summarized in the assertion that the idea of ourselves as members of a purely \"intelligible\" realm to which categories do not apply \"remains always a useful and legitimate idea for the purpose of rational belief, although all knowledge stops at its threshold\" ( _G._ 462). At the same time, Kant continued to regard the paradox of human freedom as unavoidable: we could never solve it through theoretical reason, while practical reason assured us only that it _has_ a solution. However, we must acknowledge \"the right of pure reason in its practical use to an extension which is not possible in its speculative use\" ( _P._ 50). Hence we can accept the verdict of practical reason on trust. Of course, we can always raise the question of freedom anew: it then becomes \"How is practical reason possible?\" We know that it _is_ possible, for without it our perspective on the world would vanish. But \"how pure reason can be practical\u2014 to explain this is beyond the power of human reason\" ( _G._ 461).\n\nKant is able to derive, with compelling logic, an entire system of commonsense morality from the premise of transcendental freedom. Since the paradox of freedom remains unsolved at the end of this derivation, Kant may not be wholly wrong in his suggestion that we shall never be able to comprehend it.\n\nHypothetical and Categorical Imperatives\n\nThere is a division in practical thought between hypothetical and categorical imperatives. The first typically begin with an \"If . . . ,\" as in \"If you want to stay, be polite.\" Here the end is hypothetical, and the imperative states the means to it. The validity of all such imperatives can be established in accordance with the \"supreme principle\" that \"whoever wills the end, wills the means.\" This principle, Kant argues, is analytic (it derives its truth from concepts alone). But, although hypothetical imperatives may be valid, they can never be objective, since they are always _conditional_. They give a reason only to the person who has the end mentioned in the antecedent (in the example given above, the person who wants to stay), and are binding on no one else. This is true even of the \"counsels of prudence,\" which tell us what to do in order to be happy. For the concept of happiness refers to what we unavoidably desire, by virtue of our human nature; the hypothetical imperatives that refer us to our happiness apply universally, since human nature is the common condition of all rational agents; but they do not apply _necessarily_. \"What everyone already wants unavoidably, of his own accord, does not belong under the concept of _duty_ ; for duty is a _necessitation_ to an unwillingly adopted end\" ( _M._ 385). It seems then that all hypothetical imperatives remain subjective, conditional on the individual's desires, and none of them corresponds to a true \"command of reason.\"\n\nCategorical imperatives do not typically contain an \"if.\" They tell you what to do _unconditionally_. They may nevertheless be defended by reasons. If I say \"Shut the door!,\" then my command is arbitrary unless I can answer the question \"Why?\" If I answer it to your satisfaction, then the imperative binds you. If the answer refers to some independent interest of yours, however, the imperative also ceases to be categorical, as in \"If you want to avoid punishment, shut the door.\" Only if the answer represents the action _itself_ as an end do we have a nonarbitrary categorical imperative. The signal of this is the presence of an \"ought\": \"You ought to shut the door.\" In the categorical \"ought\" we have the true imperative of reason.\n\nIt is not implausible to follow Kant in aligning this familiar distinction between hypothetical and categorical imperatives with the distinction between reasoning over means and reasoning over ends. The problem of practical reason then becomes \"How are categorical imperatives possible?\" Furthermore, Kant argues, morality can be expressed only in categorical imperatives. \"If duty is a conception that is to have any import and real legislative authority for our actions, it can only be expressed in categorical imperatives and not at all in hypothetical imperatives\" ( _G._ 425). Obedience to a hypothetical imperative is always obedience to the condition expressed in its antecedent. It therefore always involves heteronomy of the will. Obedience to a categorical imperative, however, since it springs from reason alone, must always be autonomous. Thus, Kant also aligns the distinction between the two kinds of imperatives with that between heteronomy and autonomy, and thereby associates the problem of categorical imperatives with that of transcendental freedom: \"what makes categorical imperatives possible is this, that the idea of freedom makes me a member of an intelligible world\" ( _G._ 454).\n\nBut, if categorical imperatives are to be possible, they too require a supreme principle, which will show how reason discovers them. Hence, practical reason is faced with a problem as general as the problem of theoretical reason, and requiring the same kind of answer. We must show how synthetic a priori _practical_ knowledge is possible. The supreme principle of hypothetical imperatives is, as we have seen, analytic. Hence, hypothetical imperatives make no real demands on the agent, but simply relate his ends to the means needed to secure them. Categorical imperatives, however, make real and unconditional demands: they are, in that sense, synthetic. Moreover, since their foundation lies in reason alone, they must be based a priori. The very form of a categorical imperative prevents it from deriving authority from any other source\u2014for example, from a desire, need, or interest, or any other \"empirical condition\" of the agent. In particular, there is no way in which categorical imperatives can be deduced from \"the particular attributes of human nature\" ( _G._ 425). Thus, Kant rejects all the usual systems of ethics of his day, arguing that they are unable to explain the \"unconditional necessity\" that attaches to the moral law. This necessity is explained only by a theory with an a priori basis. Hence, all reference to empirical conditions\u2014even to the most established facts of human nature\u2014must be excluded from the grounds of morality ( _G._ 427).\n\nThe Categorical Imperative\n\nThe supreme principle of categorical imperatives is called _the_ categorical imperative, on the assumption that there is, or ought to be, only one such principle (perhaps to avoid the possibility of conflicting duties [ _L._ 20]). In fact, the principle is restated in five variant forms, two of which involve new conceptions. It is therefore normal to consider the categorical imperative as a composite law of reason, with at least three separate parts. The law is derived, in its first and most famous formulation, as follows.\n\nIf we are to find an imperative that recommends itself on the basis of reason alone, then we must abstract from all the distinctions between rational agents, discounting their interests, desires, and ambitions, and all the \"empirical conditions\" that circumscribe their actions. Only then will we base our law in practical reason alone, since we will have abstracted from any other ground. By this process of abstraction I arrive at the \"point of view of a member of an intelligible world\" ( _G._ 454). This is a point of view outside my own experience, which could therefore be adopted by any rational being, whatever his circumstances. The law that I formulate will then be an imperative that applies universally, to all rational beings. When deciding on my action as an end, I will be constrained by reason to \"act only on that maxim which I can at the same time will as a universal law\" (my paraphrase from _G._ 421). (The term _maxim_ means both \"principle\" and \"motive\": a categorical imperative always commands, and always lays down a law.) The principle is in one sense \"formal\": that is, it commands nothing specific. At the same time it is synthetic, since it legislates among all the possible ends of action, allowing some, and forbidding others. For example, it forbids the breaking of promises, for to will the universal breaking of promises is to will the abolition of promising, hence to will the abolition of the advantage that accrues to breaking promises, and so to will the abolition of my motive. Such forbidden ends of action are shown, by their confrontation with the supreme moral law, to involve the agent in a contradiction.\n\nKant regarded this first formulation of the categorical imperative as the philosophical basis of the famous golden rule, that we should do as we would be done by. It is a priori in that it is based in what can recommend itself to reason alone. This explains its right to a \"universal\" form, and to the kind of necessity embodied in the categorical \"ought.\"\n\nThe categorical imperative finds the ends of action by abstracting from everything but the fact of rational agency. Rational agency must therefore provide its own ends. The autonomous being is both the agent and the repository of all value, and exists, as Kant puts it, \"as an end in himself.\" If we are to have values at all, we must value (respect) the existence and endeavors of rational beings. In this way autonomy prescribes its own limit. The constraint on our freedom is that we must respect the freedom of all: how else can our freedom issue in _universal_ laws? It follows that we must never use another without regard to his autonomy; we must never treat him as a means. This brings us to the second major formulation of the categorical imperative, as the law that I must \"so act as to treat humanity, whether in my own self or in that of another, always as an end, and never as a means only\" (my paraphrase from _G._ 429). \"Humanity\" here covers all rational beings, and the distinction between those beings that can and those that cannot be regarded as mere means is what we are referring to in distinguishing _things_ from _persons_. This distinction is the foundation of the concept of a \"right.\" If someone asks whether an animal, a child, or an inanimate object has rights, he is asking whether the categorical imperative applies to it, in this second, most powerful, form.\n\nWe cannot, then, treat rational agents merely as the means through which external forces operate: to do so is to deny their autonomy and therefore to violate their claim to our respect. In abstracting toward the moral law, I am always deferring to the sovereignty of reason. So, while the form of my law is universal (the first imperative), its content must derive from its application to rational beings as ends in themselves (the second imperative). Hence I must always think of the moral law as a piece of universal legislation, which binds rational beings equally. I am thereby led to the idea of \"the will of every rational being as a universally legislative will\" ( _G._ 449). This conception in turn leads to another, that of a \"kingdom of ends,\" in which the universal legislation to which we all willingly bow when acting autonomously has become a law of nature. So \"every rational being must so act as if he were by his maxims in every case a legislating member of the universal kingdom of ends\" (my paraphrase from _G._ 433\u201335). This third imperative suggests that all speculation about ends is also the postulation of an ideal world, in which things are as they ought to be and ought to be as they are. In this kingdom nothing conflicts with reason, and the rational being is both subject and sovereign of the law that there obtains.\n\nMoral Intuitions\n\nKant believes that the various formulations of the categorical imperative can be derived by reflection on the single idea of autonomy, and that this alone is sufficient to recommend them to every rational being. He also thinks that they lend support to ordinary moral thought, just as the synthetic a priori principles of the understanding uphold our common scientific knowledge. It is a singular merit of Kant's moral system that it imposes order on an intuitive vision of morality. This vision is not the property of one person only, but (as Kant and many others have thought) of all people everywhere. Kant had been much influenced by the third Lord Shaftesbury (1671\u20131713) and his followers\u2014the so-called British Moralists\u2014who had argued that certain fundamental moral principles were not matters of individual preference, but rather, when reduced to their true basis in the human soul, universally acceptable, recording the unspoken agreement of rational beings everywhere. Kant accepted that view. But he attempted to free it from the study of \"the niggardly provision of our stepmother, nature\" ( _G._ 394). It was of the greatest importance to him, therefore, that his theory should generate the axioms of an intuitive morality. \"We have simply showed by the development of the universally received notion of morality that an autonomy of the will is inevitably connected with it\" ( _G._ 449). It is instructive to list some of the common intuitions that Kant's theory explains.\n\n(i) _The content of morality_. Common morality enjoins respect for others and for oneself; it forbids exceptions in one's own favor; it regards all people as equal before the moral law. These are immediate consequences of the categorical imperative. In its second formulation, moreover, the imperative lends support to quite specific, and universally accepted, laws. It forbids murder, rape, theft, fraud, and dishonesty, along with all forms of arbitrary compulsion. It imposes a universal duty to respect the rights and interests of others, and a rational requirement to abstract from personal involvement toward the viewpoint of the impartial judge. Thus it encapsulates fundamental intuitions about justice, together with a specific and intuitively acceptable moral code.\n\n**Kant was influenced by English philosopher** and politician Anthony Ashley-Cooper, third Earl of Shaftesbury, who, with the other British Moralists, argued that certain moral principles were universally acceptable, not matters of personal preference.\n\n**Justice** \"giving to each that which is owed to him.\" This allegorical engraving of Justice was made in the late nineteenth century for the U.S. Bureau of Engraving and Printing.\n\n(ii) _The force of morality_. In Kant's view, the motive of morality is quite different from that of interest or desire. It rules us absolutely and necessarily; we feel its power even when we are most defying it. It is not one consideration to be balanced against others, but rather a compelling dictate that can be ignored but never refuted. This accords, he thinks, with common intuition. If someone is told that he can satisfy the greatest of his lusts, only on condition that he will afterward be hanged, then he is sure to refuse the offer. But if he is told that he must betray his friend, bear false witness, kill an innocent, or else be hanged, then his interest in his own life counts for nothing in determining what he ought to do. He may bow to the threat; but only with a consciousness of doing wrong; and the moral law itself, unlike any motive of desire, propels him onward to destruction.\n\n(iii) _The good will_. In the moral judgment of action we refer the consequences produced to the agent who produced them. Unlike the intentional or the negligent, the unforeseeable and unintended are never blamed. Moral judgment is directed, not to the effects of an action, but to the good or bad intention that it shows. Hence, in Kant's famous words, \"nothing can possibly be conceived in the world, or even out of it, which can be called good without qualification, except a good will\" ( _G._ 393). Kant's theory accords exactly with this common intuition. All virtue is contained in autonomy, all vice in its absence, and all morality is summarized in the imperatives that guide the will.\n\n(iv) _The moral agent_. Underlying intuitive morality is a view of the moral agent. Moral agents are differently motivated and differently constituted from agencies that occur in nature. Their actions have not only causes, but also reasons. They make decisions for the future, and so distinguish their intentions from their desires. They do not suffer their desires always to overcome them, but sometimes resist and subdue them. In everything they are both active and passive, and stand as legislators among their own emotions. The moral agent is an object not only of affection and love (which we may extend to all of nature) but also of esteem, and commands our esteem to the extent that the moral law is manifest in him. In all these intuitive distinctions\u2014between reason and cause, intention and desire, action and passion, esteem and affection\u2014 we find aspects of the vital distinction that underlies them, that between person and thing. Only a person has rights, duties, and obligations; only a person acts for reasons in addition to causes; only a person merits our esteem. The philosophy of the categorical imperative explains this distinction and all those that reflect it. It also explains why the \"respect for persons\" lies embedded in every moral code.\n\n(v) _The role of law_. Someone may act as the good person acts, but this fact cast no credit on him, since his motive is self-interest. The person who helps another in distress only when he sees some advantage to himself in doing so is not acting out of duty, even if he is doing what duty commands. Hence we must distinguish action _according to_ duty from action _from_ duty, and confine our praise to the second. \"The former ( _legality_ ) is possible even if inclinations alone have been the determining principles of the will; but the latter ( _morality_ ) has moral value, which can be placed only in this: that the action is done out of duty\" ( _P._ 81). This is a clear and intuitively acceptable consequence of Kant's theory. So too is the more theoretical proposition, that the fundamental concept in moral thought is not goodness, but obligation ( _P._ 60\u201365).\n\n(vi) _Reason and the passions_. We recognize in all our moral efforts that there may be a conflict between duty and desire. There thus arises, in every moral being, the idea of conscience as an independent motive, able to legislate among desires and so to forbid or permit them. Kant distinguishes the \"good will\" of the moral agent from the \"holy will\" that acts always without resistance from desire. The \"holy will\" needs no imperative ( _P._ 32), since it bends automatically in the direction of duty, whereas the ordinary agent stands always in need of principles, since his inclination is to thwart them. This sense of the conflict between reason and the passions is a widespread intuition. Kant takes it, however, to somewhat counterintuitive extremes. He believes that the motive of benevolence, so dear to empiricist morality, is a species of mere inclination, and therefore morally neutral. \"It is a very beautiful thing to do good to men from love to them and from sympathetic benevolence, or to be just from love of order, but this is not yet the true moral maxim\" ( _P._ 82). Kant seems to have more praise for the misanthropy that does good against every inclination, than for the expressions of cheerful benevolence. It is in his ability to _resist_ inclination that the worth of the moral agent resides. When, for example, someone who desires death goes on living out of a duty of self-preservation, only then, and for the first time, does self-preservation, in ceasing to be an instinct, become a sign of moral worth.\n\n(vii) _Intention and desire_. Kant's moral philosophy also helps us to understand our peculiar predicament in another way, by highlighting the fact that we, as moral agents, can intend what we do not desire, and desire what we do not intend. This distinction between intention and desire has no place in animal behavior: we explain what animals do in terms of what they want, with no remainder. But human motivation lies across a metaphysical divide\u2014the very same divide between freedom and nature, reason and cause, subject and object, person and thing, which Kant tried to expound, explain, and justify in terms of his transcendental philosophy. Through reasoning we make up our mind to act; and having done so we _intend_ what we have also decided, regardless of whether we want it. This extraordinary feature of the human condition invites an explanation: and Kant is one of the few philosophers to have provided one.\n\nMorality and the Self\n\nThose intuitions already lead us, Kant thinks, toward the idea of a transcendental freedom. If that idea is nonsense, then so is all our moral and practical thought. For, if we examine the intuitions systematically, we see that they distinguish the moral agent from his desires, and the free, reason-governed, autonomous nature of the person, from the unfree, passive (or, in Kant's term, \"pathological\") nature of the animal. Surely, then, we must face up to the paradox of freedom. For we do think of ourselves both as empirical beings, bound by the laws of causality, and as transcendental beings, obedient to imperatives of reason alone.\n\nWe must remind ourselves again of the methodological character of the \"transcendental object.\" This phrase designates, not an object of knowledge, but a limit to knowledge, defined by the perspective of practical reason. I am constrained by reason to view the world as a \"field of action,\" and hence to postulate the freedom of my will. Only from that postulate can I deliberate at all. From this perspective of practical reason I arrive inexorably at the categorical imperatives that compel my action. In one interpretation, the theory of the self and its agency is not a theory of how things _are_ in a transcendental world, but of how things _seem_ in the empirical world. \"The conception of an intelligible world is then only a _point of view_ which reason finds itself compelled to take outside appearances, _in order to conceive itself as_ _practical_ \" ( _G._ 458). The rational agent requires, then, a special perspective on the world: he sees his actions under the aspect of freedom, and, while what he sees is the same as what he sees when studying the world scientifically, his practical knowledge cannot be expressed in scientific terms. He seeks not causes but reasons, not mechanisms but rational ends, not descriptive laws but imperatives. Thus Kant answers in the affirmative the question posed in the first _Critique_ : \"is it possible to regard one and the same event as being in one aspect merely an effect of nature and in another aspect due to freedom?\" (A. 543, B. 571). But the answer, referring as it does to a transcendental perspective, is such that we can comprehend only its incomprehensibility ( _G._ 463).\n\nThe Objectivity of Morals\n\nIf my freedom and the laws that guide it are constituted by a particular way things seem, how can I consider the commands of morality to be objectively valid? Reason alone, Kant argues, compels me to accept the categorical imperative; hence it has \"objective necessity.\" It does not matter that this compulsion issues from a point of view. For the same is true of the compulsion to accept the a priori laws of science. If reason is both a motive to action, and also impelled by its nature toward definite precepts, what more can be demanded by way of objectivity? When someone asks \"Why should I be moral?,\" he is asking for a reason. An answer that refers to his interests binds him only \"conditionally\": if his interests change, then the answer loses its force. But the perspective of practical reason is able to rise above all \"conditions,\" and this is part of what is implied in its \"transcendental\" nature. It generates imperatives that bind unconditionally. In which case it answers the question \"Why be moral?\" for all rational agents, irrespective of their desires. The person with evil desires can no more escape the force of reason's answer than the person who desires what is right.\n\nSo conceived, the task of proving the objectivity of morality is less great than that of proving the objectivity of science, despite popular prejudice to the contrary. For the faculty of the understanding requires two \"deductions,\" one to show what we must believe, the other to show what is true. Practical reason, which makes no claims to truth, does not stand in need of this second, \"objective,\" deduction. It is enough that reason compels us to think according to the categorical imperative. There is nothing further to be proved about an independent world. If sometimes we speak of moral truth, and moral reality, this is but another way of referring to the constraints that reason places on our conduct.\n\nThe Moral Life\n\nThe moral nature of rational beings resides in their ability to impregnate all their judgments, motives, and affections with the universal demands of practical reason. Even in our most private and intimate encounters, reason covertly abstracts from the immediate circumstances and reminds us of the moral law. Those philosophers\u2014such as Shaftesbury, Hutcheson, and Hume\u2014who had stressed the importance to morality of the emotions were not wrong, except in their faulty concept of emotion. The moral life involves the exercise of anger, remorse, indignation, pride, esteem, and respect. And these are all emotions, since they are not subject to the will. But at the heart of such emotions is a respect for the moral law, and an abstraction from present and immediate conditions. It is this that justifies their place in the moral life of rational beings. \" _Respect_ is a _tribute_ which we cannot refuse to merit, whether we will or not; we may indeed outwardly withhold it, but we cannot help feeling it inwardly\" ( _P._ 77). Kant's discussion of pride and self-respect ( _L._ 120\u201327) is remarkably acute. By emphasizing the element of reason that lies embedded in human emotion, he is able to rebut the charge that \"moral feeling\" plays no part in the motivation of the rational person ( _L._ 139). But any theory that pays attention to the complexity of moral feeling must emphasize the imperative that lies at its heart.\n\nThe process of abstraction leads us, Kant thought, in a metaphysical direction. The moral life imposes an intimation of transcendental reality; we feel compelled toward the belief in God, in immortality, and in a divine ordinance in nature. These \"postulates of practical reason\" are as ineluctable a product of moral thought as the imperative that guides us. Pure reason falls over itself in the attempt to prove the existence of God and the immortality of the soul. But, \"from the practical point of view the _possibility_ [of these doctrines] must be assumed, although we cannot theoretically know and understand it\" ( _P._ 4). This obscure concession to the claims of theology is hard to accept. But some light is cast on Kant's intention by the third of his _Critiques_ , to the examination of which we now must turn.\n\n**Kant suggested that only in the** aesthetic experience of nature could people grasp the relation of their faculties to the world\u2014insight granted through natural beauty. _Chalk Cliffs on R\u00fcgen_ (1818), by German Romantic landscape painter Caspar David Friedrich (1774\u20131840), was inspired by Kant.\n[SIX\n\nBeauty and Design ](Roge_9781402782107_epub_c7_r1.html#d6e6)\n\nIT IS NOT GOD'S COMMAND that binds us to morality, but morality that points to the possibility of a \"holy will.\" Kant warns against the \"fanaticism, indeed the impiety, of abandoning the guidance of a morally legislative reason in the right conduct of our lives, in order to derive guidance directly from the idea of the Supreme Being\" (A. 819, B. 847). Kant's writings on religion exhibit one of the first attempts at the systematic demystification of theology. He criticizes all forms of anthropomorphism, and expounds, in his _Religion Within the Limits_ _of Reason Alone_ (1793), a \"hermeneutical rule\" of \"moral interpretation.\" All scripture and religious doctrine that conflict with reason must be interpreted allegorically, so as to express moral insights that gain vivacity, rather than validity, from their religious expression. The attempt to make the idea of God intelligible through images, and so to subsume God under the categories of the empirical world, is self-contradictory. If God is a transcendental being, then there is nothing to be said of him from our point of view except that he transcends it. If he is not a transcendental being, then he no more deserves our respect than any other work of nature. Under the first interpretation we can respect him only because we respect the moral law that points toward his existence. In the second interpretation, we could respect him only as a subject of the moral law that governs his activity.\n\nKant's demythologized religion was not uncommon among his contemporaries. He differed, however, in appropriating the images of traditional religion for the veneration of morality. The worship due to God becomes reverence and devotion for the moral law. The faith that transcends belief becomes the certainty of practical reason that surpasses understanding. The object of esteem is not the Supreme Being, but the supreme attribute of rationality. The moral world is described as the \"realm of grace\" (A. 815, B. 844), the actual community of rational beings as the \"mystical body\" in the world of nature (A. 808, B. 836), and the Kingdom of God to which mortals aspire is transmuted into the Kingdom of Ends that they make real through their self-legislation. It is not surprising to learn from one of Jachmann's letters that \"many evangelists went forth [from Kant's lectures on theology] and preached the gospel of the Kingdom of Reason.\"\n\nNevertheless, Kant accepted the traditional claims of theology, and even tried to resuscitate them under the obscure doctrine of the \"postulates of practical reason.\" Moreover, he felt that one of the traditional arguments for God's existence, the argument from design, contains a vital clue to the nature of creation. It is in the third _Critique_ , at the end of an account of aesthetic experience, that Kant attempts to reveal his meaning.\n\nThe Third Critique\n\nThe _Critique of Judgment_ is a disorganized and repetitious work, which gains little from Kant's struggle to impose on its somewhat diffuse subject matter the structure of the transcendental philosophy. A contemporary who attended Kant's lectures on aesthetics recorded that \"the principal thoughts of his _Critique of Judgment_ [were] given as easily, clearly, and entertainingly as can be imagined.\" Kant was seventy-one when he came to write the work, however, and there seems little doubt that his mastery of argument and of the written word was beginning to desert him. Nevertheless, the third _Critique_ is one of the most important works of aesthetics to have been composed in modern times; indeed, it could fairly be said that, were it not for this work, aesthetics would not exist in its modern form. Kant's most feeble arguments were here used to present some of his most original conclusions.\n\nKant felt the need to explore in the _Critique of Judgment_ certain questions left over from the first two _Critiques_. Moreover, he wished to provide for aesthetics its own \"faculty,\" corresponding to understanding and practical reason. The faculty of judgment \"mediates\" between the other two. It enables us to see the empirical world as conforming to the ends of practical reason, and practical reason as adapted to our knowledge of the empirical world. Kant believed that \"judgment\" has both a subjective and an objective aspect, and divided his _Critique_ accordingly. The first part, concerned with the subjective experience of \"purposiveness\" or \"finality,\" is devoted to aesthetic judgment. The second, concerned with the objective \"finality\" of nature, is devoted to the natural manifestation of design. I shall concentrate on the first, which ties up the loose ends of the critical system.\n\nThe eighteenth century saw the birth of modern aesthetics. Shaftesbury and his followers made penetrating observations on the experience of beauty; Burke presented his famous distinction between the beautiful and the sublime; Batteux in France and Lessing and Winckelmann in Germany attempted to provide universal principles for the classification and judgment of works of art. The Leibnizians also made their contribution, and the modern use of the term _aesthetic_ is due to Kant's mentor A. G. Baumgarten. Nevertheless, no philosopher since Plato had given to aesthetic experience the central role in philosophy that Kant was to give to it. Nor had Kant's predecessors perceived, as he perceived, that both metaphysics and ethics must remain incomplete without a theory of the aesthetic. Only a rational being can experience beauty; and, without the experience of beauty, the exercise of reason is incomplete. It is only in the aesthetic experience of nature, Kant suggests, that we grasp the relation of our faculties to the world, and so understand both our own limitations, and the possibility of transcending them. Aesthetic experience intimates to us that our point of view is, after all, only _our_ point of view, and that we are no more creators of nature than we are creators of the point of view from which we observe and act on it. Momentarily we stand outside that point of view, not so as to have knowledge of a transcendent world, but so as to perceive the harmony that exists between our faculties and the objects in relation to which they are employed. At the same time we sense the divine order that makes this harmony possible.\n\nThe Problem of Beauty\n\nKant's aesthetics is based on a fundamental problem, which he expresses in many different forms, eventually giving to it the structure of an \"antinomy.\" According to the \"antinomy of taste,\" aesthetic judgment seems to be in conflict with itself: it cannot be at the same time aesthetic (an expression of subjective experience) and also a judgment (claiming universal assent). And yet all rational beings, simply in virtue of their rationality, seem disposed to make these judgments. On the one hand, they feel pleasure in an object, and this pleasure is immediate, not based in any conceptualization of the object, or in any inquiry into cause, purpose, or constitution. On the other hand, they express their pleasure in the form of a judgment, speaking \"as if beauty were a quality [ _Beschaffenheit_ ] of the object\" ( _J._ 211), thus representing their pleasure as objectively valid. But how can this be so? The pleasure is immediate, based in no reasoning or analysis; so what permits this demand for universal agreement?\n\nHowever we approach the idea of beauty, we find this paradox emerging. Our attitudes, feelings, and judgments are called aesthetic precisely because of their direct relation to experience. Hence no one can judge the beauty of an object that he has never heard or seen. Scientific judgments, like practical principles, can be received \"at second hand.\" I can take you as my authority for the truths of physics, or for the utility of trains. But I cannot take you as my authority for the merits of Leonardo, or for the beauties of Mozart, if I have seen no work by the one or heard none by the other. It would seem to follow from this that there can be no rules or principles of aesthetic judgment. \"A principle of taste would mean a fundamental premise under the condition of which one might subsume the concept of an object, and then, by a syllogism, draw the inference that it is beautiful. That, however, is absolutely impossible. For I must feel the pleasure immediately in the perception of the object, and I cannot be talked into it by any grounds of proof\" ( _J._ 285). It seems that it is always experience, and never conceptual thought, that gives the right to aesthetic judgment, so that anything that alters the experience of an object alters its aesthetic significance (which is why poetry cannot be translated). As Kant puts it, aesthetic judgment is \"free from concepts,\" and beauty itself is not a concept. Hence we arrive at the first proposition of the antinomy of taste: \"The judgement of taste is not based on concepts; for, if it were, it would be open to dispute (decision by means of proofs)\" ( _J._ 338).\n\nHowever, such a conclusion seems to be inconsistent with the fact that aesthetic judgment is a form of _judgment_. When I describe something as beautiful, I do not mean merely that it pleases _me_ : I am speaking about _it_ , not about myself, and, if challenged, I try to find reasons for my view. I do not _explain_ my feeling, but give _grounds_ for it, by pointing to features of its object. And any search for reasons has the universal character of rationality. I am in effect saying that others, insofar as they are rational, ought to feel just the same delight as I feel. This points to the second proposition of Kant's antinomy: \"the judgement of taste is based on concepts; for otherwise . . . there could be no room even for contention in the matter, or for the claim to the necessary agreement of others\" ( _J._ 338\u201339).\n\nThe Synthetic a Priori Grounds of Taste\n\nKant says that the judgment of beauty is grounded not in concepts but in a feeling of pleasure; at the same time this pleasure is postulated as universally valid, and even \"necessary.\" The aesthetic judgment contains an \"ought\": others ought to feel as I do, and, to the extent that they do not, either they or I am wrong. It is this that leads us to seek reasons for our judgments. The terms \"universality\" and \"necessity\" refer us to the defining properties of the a priori. It is clear that the postulate that others ought to feel as I do is not derived from experience: it is, on the contrary, a presupposition of aesthetic pleasure. Nor is it analytic. Hence its status must be synthetic a priori.\n\nThe argument is very slippery. The \"necessity\" of the judgment of taste has little to do with the necessity of the a priori laws of the understanding, nor does its universality issue in a definite principle. Kant sometimes recognizes this, and speaks of aesthetic _pleasure_ rather than aesthetic judgment as universally valid, and so a priori ( _J._ 289). Nevertheless, he was convinced that aesthetics raises precisely the same problem as all philosophy. \"The problem of the critique of judgement . . . is part of the general problem of transcendental philosophy: How are synthetic a priori judgements possible?\" ( _J._ 289).\n\nKant offers a \"transcendental deduction\" in answer. It is only fifteen lines long, and wholly inadequate. He lamely says: \"what makes this deduction so easy is that it is spared the necessity of having to justify the objective reality [= application] of a concept\" ( _J._ 290). In fact, however, he argues independently for an a priori component in the judgment of taste, and for the legitimacy of its \"universal\" postulate.\n\nObjectivity and Contemplation\n\nKant's concern is, as always, with objectivity. Aesthetic judgments claim validity. In what way can this claim be upheld? While the objectivity of theoretical judgments required a proof that the world is as the understanding represents it to be, no such proof was necessary for practical reason. It was enough to show that reason constrained each agent toward a set of basic principles. In aesthetic judgment the requirement is weaker still. We are not asked to establish principles that will compel the agreement of every rational being. It is sufficient to show how the thought of universal validity is possible. In aesthetic judgment we are only \"suitors for agreement\" ( _J._ 237). It is not that there are valid rules of taste, but rather that we must _think_ of our pleasure as made valid by its object. People may doubt that aesthetic judgment contains even a _claim_ to objectivity. But that is usually because they fail to consider the aesthetic judgments that really matter to them. When a beloved landscape is vandalized or a beautiful old town laid waste, people feel affronted, wounded, and indignant. They agitate for laws that will prevent such things, form committees to protect what they love, and campaign with all their energies to deter the spoilers. If this does not suggest a claim to objective validity, it is hard to know what does.\n\nKant distinguishes sensory from contemplative pleasures. The pleasure in the beautiful, although it is \"immediate\" (arising from no conceptual thought), nevertheless involves a reflective contemplation of its object. The pure judgment of taste \"combines delight or aversion immediately with the bare contemplation of the object . . .\" ( _J._ 242). Aesthetic pleasure must therefore be distinguished from the purely sensuous pleasures of food and drink. It can be obtained only through those senses that also permit contemplation (which is to say, through sight and hearing).\n\nThis act of contemplation involves attending to the object not as an instance of a universal (or concept), but as the particular thing that it is. The individual object is isolated in aesthetic judgment and considered \"for its own sake.\" But contemplation does not rest with this act of isolation. It embarks on a process of abstraction that exactly parallels the process whereby practical reason arrives at the categorical imperative. Aesthetic judgment abstracts from every \"interest\" of the observer, who does not regard the object as a means to his ends, but as an end in itself (although not a moral end). The observer's desires, aims, and ambitions are held in abeyance in the act of contemplation, and the object regarded \"apart from any interest\" ( _J._ 211). This act of abstraction is conducted while focusing on the individual object in a \"singular [ _einzelne_ ] judgement\" ( _J._ 215). Hence, unlike the abstraction that generates the categorical imperative, it leads to no universal rule. Nevertheless, it underlies the \"universality\" of the subsequent judgment. It is this that enables me to \"play the part of judge in matters of taste\" ( _J._ 205). Having abstracted from all my interests and desires, I have, in effect, removed from my judgment all reference to the \"empirical conditions\" that distinguish me, and referred my experience to reason alone, just as I refer the ends of action when acting morally. \"Since the delight is not based on any inclination of the subject (or on any other deliberate interest) . . . he can find as reason for his delight, no personal conditions to which his own subjective self might alone be party\" ( _J._ 212). In which case, it seems, the subject of aesthetic judgment must feel compelled, and also entitled, to legislate his pleasure for all rational beings.\n\nDisinterest is the sign of an \"interest of reason,\" and occurs whenever rational agents set aside their own desires and strive to look on the world as God might look on it, with a view to judgment. This we do when deciding what is right, when presiding in a court of law, when assessing a proof, and\u2014strange though it may seem\u2014when contemplating the world of appearances. Disinterested contemplation is a recognition that the object matters\u2014matters so much that our interests have no bearing on our judgment. If you find this thought both strange and persuasive, then you will also recognize the genius of Kant, in making it the central premise of his aesthetics.\n\nImagination and Freedom\n\nWhat aspect of rationality is involved in aesthetic contemplation? In the \"subjective deduction\" of the first _Critique_ (see above, p. 39), Kant had argued for the central role of imagination in the \"synthesis\" of concept and intuition. Imagination transforms intuition into datum; we exercise imagination whenever we attribute to our experience a \"content\" that represents the world. When I see the man outside my window, the concept \"man\" is present in my perception. This work of impregnating experience with concepts is the work of imagination.\n\nKant thought that imagination could also be \"freed from\" concepts (that is, from the rules of the understanding). It is this \"free play\" of the imagination that characterizes aesthetic judgment. In the free play of imagination, concepts are either wholly indeterminate, or if determinate not applied. An example of the first is the imaginative \"synthesis\" involved in seeing a set of marks as a pattern. Here there is no determinate concept. There is nothing to a pattern except an experienced order, and no concept applied in the experience apart from that indeterminate idea. An example of the second is the \"synthesis\" involved in seeing a face in a picture. Here the concept \"face\" enters the imaginative synthesis, but it is not applied to the object. I do not judge that this, before me, is a face, but only that I have imaginative permission, as it were, so to see it. The second kind of \"free play\" is at the root of our understanding of artistic representation. Kant was more interested in the first kind, and this led him to a formalistic conception of the beautiful in art.\n\nThe free play of the imagination enables me to bring concepts to bear on an experience that is, in itself, \"free from concepts.\" Hence, even though there are no rules of taste, I can still give grounds for my aesthetic judgment. I can give reasons for my pleasure, while focusing on the \"singularity\" that is its cause.\n\nHarmony and Common Sense\n\nKant valued art less than nature, and music least among the arts, \"since it plays merely with sensations\" ( _J._ 329). Nevertheless, the example of music provides a good illustration of Kant's theory. When I hear music, I hear a certain organization. Something begins, develops, and maintains a unity among its parts. This unity is not indeed _there_ in the notes before me. It is a product of my perception. I hear it only because my imagination, in its \"free play,\" brings my perception under the indeterminate idea of unity. Only beings with imagination (a faculty of reason) can hear musical unity, since only they can carry out this indeterminate synthesis. So the unity is a perception of mine. But this perception is not arbitrary, since it is compelled by my rational nature. I perceive the organization in my experience as objective. The experience of unity brings pleasure, and this too belongs to the exercise of reason. I suppose the pleasure, like the melody, to be the property of all who are constituted like me. So I represent my pleasure in the music as due to the workings of a \"common sense\" ( _J._ 295), which is to say, a disposition that is at once based in experience, and common to all rational beings.\n\nBut how is it that the experience of unity is mixed with pleasure? When I hear the formal unity of music, the ground of my experience consists in a kind of compatibility between what I hear and the faculty of imagination through which it is organized. Although the unity has its origin in me, it is attributed to an independent object. In experiencing the unity I also sense a harmony between my rational faculties and the object (the sounds) to which they are applied. This sense of harmony between myself and the world is both the origin of my pleasure and also the ground of its universality.\n\nOne who feels pleasure in simple reflection on the form of an object, without having any concept in mind, rightly lays claim to the agreement of everyone, although this judgement is empirical and a singular judgement. For the ground of this pleasure is found in the universal, though subjective, condition of reflective judgements, namely the final harmony of an object . . . with the mutual relation of the faculties of cognition (imagination and understanding), which are requisite for every empirical cognition. ( _J._ 191)\n\nForm and Purposiveness\n\nIt seems, then, that our pleasure in beauty has its origin in a capacity, due to the free play of imagination, first to experience the harmonious working of our own rational faculties, and second to project that harmony outward on to the empirical world. We see in objects the formal unity that we discover in ourselves. This is the origin of our pleasure, and the basis of our \"common sense\" of beauty. And it is \"only under the presupposition . . . of such a common sense that we are able to lay down a judgement of taste\" ( _J._ 238).\n\nKant distinguishes \"free\" from \"dependent\" beauty, the first being perceived wholly without the aid of conceptual thought, the second requiring prior conceptualization of the object. When I perceive a representational picture, or a building, I can have no impression of beauty until I have first brought the object under concepts, referring in one case to the content expressed, in the other to the function performed ( _J._ 230). The judgment of such \"dependent\" beauty is less pure than the judgment of \"free\" beauty, and would become pure only for the person who had no conception of the meaning or function of what he saw ( _J._ 231). The purest examples of beauty are therefore \"free.\" Only in the contemplation of such examples are our faculties able to relax entirely from the burdens of common scientific and practical thought, and enter into the free play that is the ground of aesthetic pleasure. Examples of this free beauty abound in nature, but not in art.\n\nThe unity that we perceive in the free beauties of nature comes to us purified of all interests: it is a unity that makes reference to no definite purpose. But it reflects back to us an order that has its origin in ourselves, as purposive beings. Hence it bears the indeterminate marks of purpose. As Kant put it, aesthetic unity displays \"purposiveness without purpose.\" Aesthetic experience, which leads us to see each object as an end in itself, also leads us to a sense of the purposiveness of nature.\n\nThe perception of \"purposiveness,\" like the regulative ideas of reason (see above, p. 73), is not a perception of what is, but a perception \"as if.\" However, it is an inescapable \"as if \": we _must_ see the world in this way if we are to find our proper place in it, both as knowing and as acting creatures. Aesthetic judgment, which delivers to us the pure experience of design in nature, frees us both for theoretical insight and for the endeavors of the moral life. It also permits the transition from the theoretical to the practical: finding design in nature, we recognize that our own ends might be realized there ( _J._ 196). Moreover, and again like the ideas of reason, the concept of purposiveness is \"supersensible\": it is the idea of a transcendental design, the purpose of which we cannot know.\n\nAesthetic experience is the vehicle of many such \"aesthetic ideas.\" These are ideas of reason that transcend the limits of possible experience, while trying to represent, in \"sensible\" form, the inexpressible character of the world beyond ( _J._ 313\u201314). There is no true beauty without aesthetic ideas; they are presented to us both by art and by nature. The aesthetic idea imprints on our senses an intimation of a transcendental realm. The poet, even if he deals with empirical phenomena, \"tries by means of the imagination . . . to go beyond the limits of experience and to present [these things] to sense with a completeness of which there is no example in nature\" ( _J._ 314). This is how Kant explains the effect of aesthetic condensation. For example, when Milton expresses the vengeful feelings of Satan, his smoldering words transport us. We feel that we are listening not to this or that, as one might say, \"contingent\" emotion, but to the very essence of revenge. We seem to transcend the limitations contained in every natural example and to be made aware of something indescribable that they palely reflect. When Wagner expresses through the music of _Tristan_ the unassuageable longing of erotic love, it is again as though we had risen above our own circumscribed passions and glimpsed a completion to which they aspire. No concept can allow us to rise so far: yet the aesthetic _experience_ , which involves a perpetual striving to pass beyond the limits of our point of view, seems to \"embody\" what cannot be thought.\n\nTeleology and the Divine\n\nKant attempts, then, to move from his philosophy of beauty to an account of our relation to the world that will be free of that limitation to our own perspective that he had argued, in the first _Critique_ , to be a necessary condition of self-consciousness. In aesthetic experience we view ourselves in relation to a supersensible (that is, transcendental) reality that lies beyond the reach of thought. We become aware of our own limitations, of the grandeur of the world, and of the inexpressible good order that permits us to know and act on it. Kant has recourse to Burke's distinction between the beautiful and the sublime. Sometimes, when we sense the harmony between nature and our faculties, we are impressed by the purposiveness and intelligibility of everything that surrounds us. This is the sentiment of beauty. At other times, overcome by the infinite greatness of the world, we renounce the attempt to understand and control it. This is the sentiment of the sublime. In confronting the sublime, the mind is \"incited to abandon sensibility\" ( _J._ 246).\n\nKant's remarks about the sublime are obscure, but they reinforce the interpretation of his aesthetics as a kind of \"premonition\" of theology. He defines the sublime as \"that, the mere capacity of thinking which, evidences a faculty of mind transcending every standard of taste\" ( _J._ 250). It is the judgment of the sublime that most engages our moral nature. It thereby points to yet another justification of the \"universality\" of taste, by showing that, in demanding agreement, we are asking for complicity in a moral sentiment ( _J._ 265). In judging of the sublime, we demand a universal recognition of the immanence of a supersensible realm. A person who can feel neither the solemnity nor the awesomeness of nature lacks in our eyes the necessary sense of his own limitations. He has not taken that \"transcendental\" viewpoint on himself from which all true morality springs.\n\n**Kant draws on Anglo-Irish philosopher** and statesman Edmund Burke's (1729\u201397) concepts of the sublime and the beautiful in his discussion of aesthetic experience. Burke is shown here in a 1770 engraving by James Watson after a portrait by Sir Joshua Reynolds.\n\nIt is from the presentiment of the sublime that Kant seems to extract his faith in a Supreme Being. The second part of the _Critique of Judgment_ is devoted to \"teleology\": the understanding of the ends of things. Here Kant expresses, in a manner that has proved unsatisfactory to many commentators, his ultimate sympathy for the standpoint of theology. Our sentiments of the sublime and of the beautiful combine to present an inescapable picture of nature as created. In beauty we discover the purposiveness of nature; in the sublime we have intimations of its transcendent origins. In neither case can we translate our sentiments into a reasoned argument: all we know is that we know nothing of the transcendental. But that is not all we _feel_. The argument from design is not a theoretical proof, but a moral intimation, made vivid to us by our sentiments toward nature, and realized in our rational acts. It is realized, in the sense that the true end of creation is intimated through our moral actions: but this intimation is of an ideal, not of an actual, world. So we prove the divine teleology in all our moral actions, without being able to show that this teleology is true of the world in which we act. The final end of nature is known to us, not theoretically, but practically. It lies in reverence for the pure practical reason that \"legislates for itself alone.\" When we relate this reverence to our experience of the sublime, we have a sense, however fleeting, of the transcendental ( _T_. 446).\n\nThus it is that aesthetic judgment directs us toward the apprehension of a transcendent world, while practical reason gives content to that apprehension, and affirms that this intimation of a perspectiveless vision of things is indeed an intimation of God. This is what Kant tries to convey both in the doctrine of the aesthetic ideas and in that of the sublime. In each case we are confronted with an \"employment of the imagination in the interests of mind's supersensible province\" and a compulsion to \"think nature itself in its totality as a presentation of something supersensible, without our being able to put this presentation forward as objective\" ( _J._ 268). The supersensible is the transcendental. It cannot be thought through concepts, and the attempt to think it through \"ideas\" is fraught with self-contradiction. Yet the ideas of reason\u2014God, freedom, immortality\u2014are resurgent in our consciousness, now under the guise of imperatives of action, now transformed by imagination into sensuous and aesthetic form. We cannot rid ourselves of these ideas. To do so would be to say that our point of view on the world is all that the world consists in, and so to make ourselves into gods. Practical reason and aesthetic experience humble us. They remind us that the world in its totality, conceived from no finite perspective, is not ours to know. This humility of reason is also the true object of esteem. Only this is to be reverenced in the rational being, that he feels and acts as a member of a transcendental realm, while recognizing that he can know only the world of nature. Aesthetic experience and practical reason are two aspects of the moral: and it is through morality that we sense both the transcendence and the immanence of God.\n\n**An allegorical representation** of the _D\u00e9claration des Droits de L'homme et du Citoyen_ (Declaration of the Rights of Man and of the Citizen), a symbol of the French Revolution. Although Kant did not support the Revolution, he harbored republican sympathies and shared the Enlightenment vision of many of his contemporaries.\n[SEVEN\n\nEnlightenment and Law ](Roge_9781402782107_epub_c7_r1.html#d6e7)\n\nTHE CRITICAL PHILOSOPHY EXAMINES the structure of thought, and tells us what can and cannot be proved by it. It stands in judgment over all philosophies, and its theorems are not ordinary proofs, but proofs about the provable. In short, it is a metaphilosophy, a philosophy about philosophy. It foreshadows the metalogic and metamathematics that, in the hands of Tarski and G\u00f6del, have altered our conception of mathematical reasoning, and undermined some of the criticisms made of Kant's view of mathematics, as a body of synthetic a priori truth.\n\nThis does not mean that Kant's philosophy was without results, or unable to challenge prevailing orthodoxy. On the contrary, in the practical sphere, Kant believed, the critical method made direct contact with moral judgment, and delivered conclusions that must be accepted, even if people should have an interest in denying them. Morality is the core of practical reasoning; but it is not the whole of it, and in his last years Kant turned his attention to politics, hoping by the use of the critical method to settle some of the vexing questions of legitimacy and right.\n\nOf ever-increasing importance to Kant's contemporaries, and also to Kant himself, was the vision of human society and institutions that had been put forward by Rousseau, and that made the self-legislation of the free individual into the foundation of legitimate political order. This vision encouraged an increasing skepticism toward authority; it gave precedence to the individual conscience over the dictates of church and state; and it fed the belief in progress, as the natural condition of humanity when freed from superstitious obedience. Kant's moral philosophy, which derived a complete system of duty from the idea of self-legislation, gave the ultimate philosophical endorsement to the new vision of politics, and the Enlightenment ( _Aufkl\u00e4rung_ ), as it was to be known, is indelibly associated with Kant, as its most explicit and articulate exponent. Indeed, it was Kant who first ventured to define Enlightenment, as \"the liberation of man from his self-imposed minority,\" adding that this minority lies \"not in lack of understanding, but in a lack of determination and courage to use it without the assistance of another\" ( _What Is Enlightenment?_ , R. 54).\n\nKant's republican sympathies were well known in his day, though he was an articulate opponent of revolution, seeing it as a return to the state of nature\u2014a return that could never be rationally justified, since it rendered the rule of reason impossible. The violence of the French Revolution confirmed Kant's suspicion of the revolutionary project; but it did not diminish his admiration for Rousseau, or his commitment to the Enlightenment vision that he shared with so many of the revolutionaries.\n\n**Kant's admiration for Rousseau** did not diminish despite his opposition to the French Revolution. In this engraving, which appeared over the caption \" _C'est ainsi qu'on se venge des traitres_ \" (Thus we will avenge the traitors), French revolutionaries carry the heads of the Marquis de Launay, a governor of the Bastille, and public servant Jacques de Flesselles on pikes after the storming of the Bastille.\nThe Unwritten Critique\n\nKant's writings on politics belong to the last decade of his active life. The most important of them is generally known in English as the _Metaphysical_ _Elements of Justice_ , which forms the first part of _The Metaphysics_ _of Morals_ , published in 1797, with a second edition in 1798. The texts of both editions contain lapses and obscurities that have suggested to many commentators that Kant's powers were already failing, and that the grounds of his political philosophy had not been properly worked out. Recent scholarship, however, has tended to the view that lecture notes and other extraneous material might have been included in the text, Kant being increasingly dependent on an amanuensis when transcribing his drafts. Whatever the truth about the composition of _The Metaphysics of_ _Morals_ , one thing is certain, which is that it contains no \"Critique of Political Reasoning,\" nor anything that might be regarded as a derivation from a priori principles of the idea of legitimate political order. Equally certain, to my mind, is that there is contained in Kant's later writings the seeds of a critical philosophy of politics, and that this unwritten fourth critique is as worthy of study as the other three.\n\nUniversalism\n\nKant was the prophet of \"Enlightenment universalism.\" He believed in a single, universal human nature, to which appeal should be made when examining the legitimacy and authority of our local arrangements. Since practical reason is valid for all people everywhere, there can be no special pleading on behalf of authorities whose sole claim to obedience is that history has placed them where they are. On the other hand, it is never possible in human affairs to wipe away the legacy of history\u2014not without risking a return to the state of nature and the war of all against all. Hence, Kant thought, the appeal to universal humanity must take the form of a regulative, rather than a constitutive, idea. We should not seek to establish a single set of laws for all mankind; rather, we should judge each jurisdiction in terms of certain a priori constraints, conformity to which would be the sign of its legitimacy.\n\nFurthermore, since the source of all authority in human affairs is reason, and since reason is common to us all, we must recognize that the benefits and burdens of political order lie equally on everyone. On this basis, slavery is clearly ruled out; so too are arbitrary distinctions whose sole purpose is to give one person power over another, or one class of persons privileges that confer no benefits on those who do not possess them. Hence, in his political writings, Kant writes as though distinctions of race and class are of no intrinsic political significance, and as though citizenship is and ought to be the aspiration of all mankind.\n\nIndeed, Kant goes further than most of his contemporaries in mounting, in _Perpetual Peace_ (1795), the first systematic argument for a universal form of government\u2014one that will transgress national boundaries and erase the arbitrary divisions among people that, in Kant's view, are the true cause of hostilities between them. It is to Kant that the idea of a \"league\" or \"federation of nations\" is due; and Kant's theory has been embodied in our modern structures of transnational legislation.\n\nThe Social Contract\n\nAs we should expect, given the legalistic form of his ethical theory, Kant's political thought is jurisprudential. For Kant a political order is, first and foremost, a system of laws. At the same time he followed Hobbes, Locke, and Rousseau in proposing the social contract as the ultimate test of political legitimacy. Only if a political order can command the consent of those who are subject to it can it claim objective authority. The social contract is therefore \"a contract on which alone a civil and thus completely lawful constitution can be based and a community established\" ( _Common Saying_ , R. 79). However, there is a crucial difference between Kant and his predecessors. The social contract is not, for Kant, a constitutive but a regulative principle. We should not suppose that there is an actual contract, whether explicit or implied, at the heart of legitimate social order. Societies arise in another way, and are always burdened by a historical legacy that cannot be undone without injustice or violence. We should consider the social contract, therefore, only as a limiting idea of reason\u2014as a test that any actual legal order must pass, if it is to be legitimate. The social contract therefore takes the following form: \"if a law is so framed that a whole people _could not possibly_ give it their consent . . . the law is unjust; but if it is _at all possible_ that a people might agree on it, then the people's duty is to look on the law as just, even assuming that their present way of thinking were such that, if consulted, they would probably refuse to agree\" ( _Common Saying_ , R. 79).\n\nThe definition is critically ambiguous, shifting loosely between \"a whole people,\" \"a people,\" and \"the people.\" The most favorable construction, it seems to me, is this: each citizen has an absolute veto over every law, but he can exercise this veto only if he _could_ not agree to the law. The fact that he _does_ not agree is insufficient grounds for dissent. This is because what we _in fact_ agree to depends upon our desires, circumstances, and \"empirical conditions.\" We must discount all such conditions, if we are to address our question to reason alone. (Note the parallel with the categorical imperative.) In speaking of a social contract, therefore, we are referring not to an actual agreement between empirical selves, but to a hypothetical agreement between purely noumenal beings. Each of us, considered from the point of view of pure practical reason, is such a noumenal being.\n\nThe idea of such a hypothetical social contract, arrived at by a process of abstraction, has been revived in recent times by John Rawls, who uses it as a test for the just distribution of goods in a society. Kant was not interested in distributions, however, but only in laws, and, although he regarded all citizens as equal in their citizenship, and therefore in their rights and responsibilities, modern ideas of \"social justice\" were remote from his concerns. Although Kant believed in equality under the law, and equal rights of citizenship, he believed this to be compatible with the greatest inequality in possessions ( _Common Saying_ , R. 75).\n\nNatural Law\n\nThe system of law prevailing in Prussia was derived directly from Roman law, which Kant held in great esteem, not least because it was an attempt at a universal code, giving weight not only to local traditions and historical institutions, but also to philosophical principles applicable to all mankind. In particular, the Roman law distinguished the civil law, which was the province of earthly rulers, from natural law, which owed its authority to human nature and which stood higher than any local system of command. Thus, the civil law could uphold slavery and define the rights and duties of a slave, but the natural law, which was derived from reason alone, could recognize no such relation.\n\nMedieval jurists and philosophers had devoted much attention to the concept of natural law, believing it to be an attempt to transcribe in legal form the revealed will of God. Kant saw it in another way, as a vindication of the critical philosophy. The natural law, for him, was nothing less than a judicial translation of the categorical imperative. In the Roman law, therefore, Kant saw the proof that legal reasoning led of its own accord to the synthetic a priori principles of practical reason for which he had found the transcendental proof ( _M._ 236\u201337).\n\nPersons and Things\n\nKant's moral philosophy had also given cogency to another concept taken from Roman law: the concept of the person. Originally _persona_ meant a mask, and it described the character worn by the actor on the stage. The Roman law borrowed the term to denote the litigant in a court of law, whose rights and duties were defined by the law and who was in a certain sense the creation of the law before which he appeared. The legal person is the bearer of legal privileges (rights, freedoms, etc.) and legal burdens (duties, penalties, etc.). As Kant rightly saw, this concept is ultimately not legal but metaphysical, and assumes a distinction in reality between free _subjects_ and empirical _objects_. Rights and duties cannot attach to objects, which are bound by empirical conditions and laws. They can attach only to subjects who are free, in just the sense defined in Kant's moral philosophy. Law, by which human societies are governed, is addressed to the noumenal self.\n\nThree features of persons stand out in the context of political thinking: their accountability, their power of self-legislation, and their nature as ends in themselves. Because they are accountable for their actions, persons stand in a special relation to one another, and this relation is the basis of political order. Changes in the world can be _imputed_ to people, whenever these changes arise through action or neglect ( _M._ 239). Through law, imputation becomes a universal relation, binding each person to society, and making each person answerable for his acts and omissions.\n\nBecause persons are self-legislating members of the kingdom of ends, they are subject to two kinds of constraint\u2014internal and external. Internal legislation is demanded by the categorical imperative, and no just law can conflict with it. But there is also external legislation, in which some action that is neither demanded nor forbidden by the moral law is nevertheless commanded or forbidden by the state. The social contract (as defined above) is the test of the legitimacy of this external legislation, and it goes without saying that such legislation could never be accepted, if it set out to override the internal dictates of duty.\n\nFurthermore, law must respect the nature of persons, as ends in themselves. Slavery, fraud, and tyrannical oppression all conflict with the categorical imperative, and it is a core requirement of political order that people should be able to relate to each other and to the state as ends, and not as means only.\n\nHuman Rights\n\nLaw, for Kant, has the same metaphysical basis as morality, and is therefore grounded a priori in the idea of transcendental freedom. He identifies a single universal principle of justice, which he states in the following obscure words: \"Every action is just [ _recht_ ] if it or its maxim allows each person's freedom of choice to coexist together with the freedom of everyone in accordance with a universal law\" ( _M._ 230). It is a well-known problem for translators that the noun _Recht_ means both \"right\" and \"law,\" and the adjective _recht_ both \"just\" and \"right.\" The adjective occurs here in the context of a theory of justice, the principal purpose of which is to connect the ideas of justice and freedom: hence my translation. To put Kant's universal principle more perspicuously: an action is just if it respects the freedom of others, and does so not accidentally but on principle. Every free action must accommodate the freedom of others. For Kant this was simply another way of stating the basic requirement of justice, that all persons are to be treated as ends in themselves, and never as means only.\n\nThis universal principle is not incompatible with coercion, but coercion should always have, as its ultimate goal, the establishment of an order in which justice\u2014and therefore mutual freedom\u2014prevails. This was Kant's judicious way of amending Rousseau's notorious maxim, that, in a state founded on the social contract, the dissenter must be \"forced to be free.\"\n\nIt follows from Kant's view of justice that there is only one innate or \"natural\" right, which is that of freedom itself: \"Freedom (independence from the constraint of another person's will), in so far as it is compatible with the freedom of everyone else in accordance with a universal law, is the one sole and original right belonging to every person by virtue of his humanity\" ( _M._ 237). The qualification here is important: I do not have a natural right to do as I please; rather, I have a right to exercise such freedom as does not limit the freedom of others. This principle is the basis of \"classical liberalism\" and is still an unwritten law of many Western constitutions.\n\nIt is a small step from those a priori principles to the idea of human rights, as the one universal constraint on government. In describing rights as \"human,\" rather than \"civil,\" we invoke the natural law, and with it the universal principles that set limits to legislation in every place and time. Moreover, Kant's powerful idea of the person as an end in himself suggests the basic form of these universal rights: they are vetoes in the hands of the one who possesses them. There are certain things that you must not do to me without my consent, and this fact gives me a kind of sovereignty over my life that you cannot legitimately invade or diminish. This individual sovereignty translates into law as a system of rights: the right to life, limb, and property; the right to proceed peacefully about my own purposes; the right to engage in lawful relations with others, and so on.\n\nWhile Kant can be seen, in this way, as the greatest precursor of the modern doctrine of human rights, he was not so feebleminded as to suppose (as many people now suppose) that there can be rights without duties. Indeed, for Kant the concept of right is secondary to that of duty. What gives reality to my rights is your duty to respect them; and it is the motive of duty that is being obliquely invoked in all reference to the rights of the citizen ( _M._ 239). And I can claim rights only if I am prepared to pay the price, which is the acceptance of the very same duties that I impose upon you through my claim. In particular, I have a duty to uphold the universal principle of justice, and this means that I must accept and endorse the system of punishment without which there would be no law and no lasting community ( _M._ 331\u201332).\n\nThe Republican Ideal\n\nKant regarded legitimate political order as a legal system, founded in the respect for persons, and organized according to republican principles. Kant was influenced by French Enlightenment thinkers, notably by Montesquieu, into thinking of a republic as a distinct form of government, conferring a distinct status on its members. While you can be the _subject_ of an absolute ruler, you can be the _citizen_ only of a republic: the concepts of citizen and republic are mutually dependent parts of a single complex idea. In a republic the people themselves are the authors of the laws that govern them, and these laws express the \"general united Will\" (as Kant, echoing Rousseau, describes it) by which the consent of the governed is secured ( _M._ 313\u201314).\n\nA republic exists only if there is representative government, in which the people as a whole are represented in the legislature. The process of representation translates the inherent sovereignty of the people into an actual sovereign power, which Kant sees on the model of a legal person: a forum of collective decision making, in which the interests of the people as a whole are advanced through law, but which is itself answerable to the law that it lays down ( _M._ 341). Quite how representative government is achieved is one of the obscurities (some would say culpable obscurities) in Kant's political philosophy. At one point Kant argues that \"only fitness for voting qualifies someone to be a citizen\" ( _M._ 314). While this suggests that Kant was an advocate of democracy, he at once qualifies his remark by following the French revolutionary Abb\u00e9 Siey\u00e8s in distinguishing the active from the passive citizen\u2014the latter class enjoying no right to vote, and including all who are not fully independent. Among passive citizens he includes servants, minors, apprentices, and all women, adding that \"all such people lack civil personality,\" a fact that is, however, \"by no means opposed to the freedom and equality that such persons have as _human beings_ who together make up a people\" ( _M._ 315). Elsewhere he criticizes democracy as a form of despotism ( _PP_ , R. 101), and insists that the idea of republican government has no intrinsic connection with democratic choice. Without going into the matter in depth, the least that can be said is that Kant's conception of citizenship is a long way from the views of modern democrats, however much it may have set him at odds with the forms of government that then prevailed in Central Europe.\n\n**Charles de Secondat,** baron de Montesquieu (1689\u20131755).\n\nThe Separation of Powers\n\nThis defective account of representation is to some extent mitigated by Kant's subtle defense of the separation of powers\u2014an idea introduced into political philosophy by Locke and further developed by Montesquieu. Like Montesquieu, Kant distinguishes the legislative, judicial, and executive powers in a state, arguing that each must be considered as a \"moral person\" (that is, an accountable agent, with rights and duties of its own), while recognizing that they cannot exist in isolation, being both mutually dependent and mutually subordinate to one another ( _M._ 316). The business of government consists in maintaining the equilibrium between these powers, and so enabling the \"general united will\" of the people to find expression in the legislative assembly. Kant was clearly of the view that majority voting does not in itself produce legitimate law. What is required is a legislative _process_ , in which the competing interests of the people can be brought into equilibrium and subjected to judicial review. Only then can the rights of each citizen be guaranteed, and the threat of tyranny avoided.\n\nThis partly explains Kant's hostility to democracy. In a pure democracy, he thought, government and governed are one and the same. In such circumstances, the arbitrary desires of majorities pass immediately into law, regardless of whether the rights and freedoms of everyone are thereby respected. Respect for freedom requires a brake upon the hasty exercise of power. Laws should therefore be made by one body, applied by another, and adjudicated by a third, and certainly not made and applied by the majority.\n\n**This British cartoon** depicting the execution by guillotine of French king Louis XVI was published on January 25, 1793, four days after the king's death. Devils fly above the guillotine among the words _Vive la nation_ (Long live the nation) and _Ca ira_ , from the song \" _Ah! \u00e7a ira_ \" (Oh! it'll be fine), a song composed in 1790 and emblematic of the French Revolution.\n\nKant's republic was not incompatible with constitutional monarchy, although in many places he expresses his hostility to inherited powers, and especially to the inherited privileges of the nobility. In the end he fails to declare an alternative to democracy, and leaves the reader uncertain as to how the legislative chamber of a republic is supposed to be filled, and what conditions must be met, if it is to be representative of the people as a whole. It seems as though Kant regarded the representative republic as an Ideal of Reason, which we may approximate but never achieve. He argues powerfully against all forms of abrupt transition, and for a general attitude of obedience toward any form of sovereignty that is not plainly in violation of the categorical imperative. The recent events of the Terror had set his heart against revolution in general and regicide in particular, and his attitude to democracy can best be summarized in the words that Gandhi used of Western civilization, that \"it would be a good idea,\" with the proviso that, for Kant, an idea is always an \"idea of reason\"\u2014that is, a guide to practice, not a description of some possible reality.\n\nPrivate Property\n\nAmong the rights that belong to the citizen by virtue of the universal principle of justice, Kant identified the right of property as all-important, attempting a kind of transcendental deduction of ownership, in answer to the question \"how is it possible for an object to be mine?\" ( _M._ 249). He distinguished _sensible_ from _intelligible_ possession, the first being an empirical relation, the second a relation of right (M. 245). All general propositions about rights, he argued, are a priori, since they are laws of reason: hence their reference to the intelligible or noumenal realm, of which, from the point of view of practical reason, I myself am a part.\n\nProperty rights are not merely grounded a priori: they involve a _synthetic_ a priori imperative, like the imperatives of the moral law. To say that an object is mine is to add something positive to my sovereignty as a free being\u2014something that is not merely contained in the concept of freedom. Philosophers had striven in vain to justify this addition, but had not seen that it could be justified only by some kind of transcendental argument, which would show the possibility of ownership to be among the preconditions of practical reason. Kant does not succeed in producing this argument: but he rightly suggests that the essence of the right of property is revealed in the violation suffered when that right is abused. This violation involves a refusal to recognize the existence of the other as an end in himself, and therefore contradicts the supposition that he is a free, rational agent.\n\nKant's argument was later taken up by Hegel, who represented private property as the instrument through which the autonomy and self-identity of the rational agent is realized in an objective world. Through Hegel, Kant's conception of property became the focus of the leading political debate in nineteenth-century intellectual life: the debate between socialists and their opponents.\n\nCivil Society\n\nIn the course of his writings on politics, Kant offers a description of the \"civil society\"\u2014meaning a society organized by law, under a sovereign legislative power ( _M._ 256\u201357). In such a society, he argues, there must be property, contract, and the law that upholds them both. There must also be economic life and the means to facilitate contracts between strangers. Kant therefore offers a theory of money, and its representational nature, arguing that money is \"a thing the use of which is only possible through its being alienated\" ( _M._ 286)\u2014a peculiar feature that indicates that the possession of money must stand proxy for possession of another kind, in effect, possession of power over other people's labor. Kant therefore offers a \"real definition\" of money, as \"the universal means by which the labour of people can be exchanged\" ( _M._ 287). This definition foreshadows the nineteenth-century discussions of alienated labor, to which Kant's theory of the person gave additional force.\n\nKant was unusual among political philosophers in discussing the place of sex and sexual relations in civil society. He recognized that sexual morality presents a problem to the liberal worldview. Why should the behavior of \"consenting adults in private\" be the concern of the state? Indeed, why should it be the concern of morality, if no harm ensues to the parties, and no harm is inflicted on anyone else? How can the philosopher of freedom avoid endorsing sexual freedom, both as a political necessity and as a moral ideal?\n\nIn facing up to this question Kant drew once again on his concept of the person as end in himself. He boldly (and rightly) affirmed that there are sexual acts in which the other is treated not as a person but as a thing, in which the subject is, so to speak, eclipsed by the object. This is what we mean, or ought to mean, by perversion, and it violates a fundamental duty to the other and to oneself ( _M._ 278\u201379). The details of Kant's sexual morality (which coincide with the Christian-inspired view of marriage, as it was current in his day) may not appeal to all his modern readers. However, he made the first steps toward understanding what is at stake in sex, why someone can be jeopardized by another's desire, and why the personality is at risk in our sexual encounters. His ability to cast light on this difficult topic further confirmed his belief that he had identified, through his theory of the person, the metaphysical predicament in which we human beings stand.\n\nPerpetual Peace\n\nKant defended the republican ideal\u2014the ideal of free and equal citizenship, in which law is self-legislated by the people as a whole\u2014on two grounds. First, he considered it to be an a priori consequence of the categorical imperative. Second, he held that republican government is the precondition of peace between nations. Wars result when individual self-interest overrides the requirements of justice, and when a despotic power sees advantage to itself in the violent imposition of its will. If the nations of the world are organized as republics, so that no decision is taken except with the consent of the citizens as a whole, then the a priori laws of practical reason would have a voice in political decisions. War would be ruled out, not only as an unjust means of prosecuting collective purposes, but as an act of folly that runs counter to the common interest. In other words, it would be seen to conflict both with the categorical imperative of morality, and with the hypothetical imperative of survival.\n\nHowever, the condition of \"perpetual peace\" could not be brought about at once or by fiat. For, even if organized as republics, the nations are in a state of nature vis-\u00e0-vis one another, and therefore each nation is a \"standing offence\" to its neighbor ( _PP_ , R. 102). Positive steps must be taken to break down the barriers between the peoples of the world. Just as it is in the interests of individuals to abolish the state of nature and to enter into a civil society, so as to enjoy the freedom that is theirs by right, so it is in the interest of the nations to submit to a common rule of law, and to overcome the state of nascent belligerence that has afflicted the history of mankind. We should therefore be guided by the Idea of a \"world republic\" ( _PP_ , R. 105)\u2014a condition that is not in fact achievable but that could nevertheless regulate the dealings between individual states, and draw them insensibly into federation. Such a federation of republics, united by international law, would guarantee peace between nations. Each republic would have an interest in maintaining the law that protects it from aggression, and each would subscribe to a universal doctrine of rights that would ensure just dealings between them.\n\nLike many of the intellectual constructions that inform his political thinking, Kant's \"world republic\" is an Ideal of Reason. His theory of the ideal is expressly anti-utopian. The utopian is the one who thinks that the ideal can be realized, and who therefore sets out to destroy the obstacles that stand in its way. The Kantian believes that ideals cannot be realized, since we live in an imperfect world, impeded by empirical circumstances. Ideals must be construed as regulative principles, which guide us down the path of amelioration. Hence we must strive always to amend things, and never to tear them down.\n\nIn the course of exploring his ideal of world government, Kant makes trenchant criticisms of colonialism, and argues that it is only within a federal union of republics that politics and morality can be in agreement with each other. Until that union is achieved, the statesman is always dependent on cunning, lies, and subterfuge if he is to advance the national interest. Habits of secrecy, made necessary by the actual relations between states, violate the \"transcendental formula of public right,\" which is that an action is wrong if it is not compatible with being made public ( _PP_ , R. 126). We should work toward the condition of open government, in which the dealings between states are informed by the same integrity and honesty as the dealings between people, when these are guided by the categorical imperative. The respect for persons should be extended to the states that represent us, since they too are persons, whose rights and duties should be defined, as ours are, by a single moral law.\n\nKant's political vision has appealed to many subsequent philosophers and jurists, and has lost none of its relevance in the two centuries since his death. Subsequent wars and revolutions have made Kant's optimism unfashionable. But his appeal to natural law and human rights, and his attempt to derive a theory of justice from the premise of transcendental freedom, are of abiding relevance. Had Kant foreseen the events that led (among countless other calamities) to the destruction of his beautiful birthplace and the wholesale murder of its inhabitants, he would perhaps have lost some of the faith in human nature that shines through even his most abstract arguments about justice. But that faith remains an inspiration to those who believe, as Kant believed, that Reason can guide us both through its immediate imperatives, and through its unrealizable Ideals.\n\n**Kant's work was widely influential** and gave rise to important schools of thought. This photograph, taken on June 27, 1992, shows the inauguration ceremony for a statue of Immanuel Kant in the Russian city of Kaliningrad (formerly Kant's hometown of K\u00f6nigsberg), part of the city's recognition of the two hundredth anniversary of the German philosopher's death.\n[EIGHT\n\nTranscendental Philosophy ](Roge_9781402782107_epub_c7_r1.html#d6e8)\n\nKANT WAS REGARDED by his immediate successors as having irreversibly changed the course of philosophy. But already in Kant's lifetime the intellectual world was torn by controversy over the meaning of his critical system. Was Kant really a Leibnizian after all, as Eberhard had accused him of being ( _K._ 107)? Did he believe that the world of nature is nothing but a \"well-founded phenomenon,\" reality itself consisting in timeless, spaceless, noumenal substances whose attributes are derived from reason alone? Is the \"thing-in-itself\" the underlying substance that sustains appearances? In a series of letters to the aging Kant, his pupil Jakob Beck rehearsed this interpretation, and sought to demonstrate its untenability. But, if the transcendental philosophy is not a version of Leibnizian rationalism, why is it not, then, a repetition of the skeptical empiricism of Hume? Kant's philosophy is very much clearer in its negative than in its positive aspect, and in his day he had been called (by J. G. Hamann) \"the Prussian Hume.\" In the long peroration that concludes the \"Antinomy\" of the first _Critique_ , Kant emphasizes this negative aspect, and writes with pride of the method that has enabled him to rise above all preexisting argument in order to show that certain conclusions are not just undemonstrated, but indemonstrable.\n\nNeither the Leibnizian nor the Humean interpretation is really tenable. It is true that Kant sometimes speaks of concepts as \"rules\" for organizing our perceptions (e.g., A. 126), a conclusion that is reminiscent of Hume. It is also true that he is tempted by the \"transcendental hypothesis\" of a realm of \"things as they are\" (e.g., A. 780, B. 808), a conclusion that would align him with Leibniz. But these remarks are aberrations. Kant's true critical philosophy can be assimilated to neither of its antecedents, since it removes the grounds from both.\n\nThe first important school of thought to arise out of Kantian philosophy was the \"subjective idealism\" of Fichte (1762\u20131814), Schelling (1775\u20131854), and Hegel (1770\u20131831). According to these philosophers, the critical philosophy, in arguing away the \"thing-in-itself,\" had shown that reality is to be conceived in mental terms. Knowledge of an object is construed as \"positing\" ( _setzen_ ), rather than \"receiving.\" For Fichte, Kant's great achievement was to have shown that the mind has knowledge only through its own activity; in an important sense, therefore, the objects of knowledge are a _product_ of that activity. Thus Fichte wrote to a friend: \"I suppose I am more strictly a transcendental idealist than Kant; for he still admits a manifold of appearance, but I assert in plain terms that even this is produced by us, by means of a creative power.\" The mind is identified with the \"transcendental self,\" construed as the one noumenal object with which we are acquainted. But who, once again, are _we_? In Fichte's philosophy the transcendental self becomes a kind of universal spirit by which the separate empirical selves are constructed, along with the \"world of appearance\" in which they expend their energies, the whole depending on an unknowable synthesis that generates nature from the inexhaustible reservoir of the \"thing-in-itself.\"\n\nSchopenhauer too (1788\u20131860) was influenced by this interpretation, believing that Kant had rightly identified the \"transcendental self\" with the will (which is therefore the true \"substance\" behind appearances). For Schopenhauer, scientific concepts such as space, time, and causality apply only in appearance, imposing order on the world of appearance (or the \"veil of Maya\"\u2014the term Schopenhauer borrowed from oriental mysticism). Behind this veil the will takes its endless, unknowable, and unsatisfiable course. Hegel, by contrast, developed Fichte's idea of the known as \"posited\" by the knower. He tried to show that the objective reference justified in the transcendental deduction is but the first stage in an expanding process of self-knowledge. Mind ( _Geist_ ) comes to know itself through the positing of an ever more complex world. Hegel described this process as \"dialectical,\" meaning not to bury but to praise it. He believed that Kant's first _Critique_ had displayed, not the errors of pure reason, but the dynamic process of conjecture and refutation whereby reason constantly negates its own advances, achieving from the ruin of partial knowledge an ever more complete, more \"absolute,\" picture of reality.\n\n**Arthur Schopenhauer** (1788\u20131860).\n\nKant would have rejected that return to the Leibnizian vision. \"The light dove,\" he wrote, \"cleaving the air in her free flight, and feeling its resistance, might imagine that flight would be still easier in empty space\" (A. 5, B. 8). Thus he dismissed as insubstantial any pretence to an absolute form of knowledge, which seeks to soar above the resistant medium of experience. The notion of a transcendental object is misunderstood when considered as referring to a real thing. The idea is posited only as a \"point of view\" (A. 681, B. 710), in order to make clear that \"the principles of pure understanding can apply only to objects of the senses . . . never to things in general without regard to the mode in which we are able to apprehend them\" (A. 246, B. 303). There is no description of the world that can free itself from the reference to experience. Although the world that we know is not our creation, nor merely a synopsis of our perspective, it cannot be known except from the point of view that is ours. All attempts to break through the limits imposed by experience end in self-contradiction, and, although we may have intimations of a \"transcendental\" knowledge, that knowledge can never be ours. These intimations are confined to moral life and aesthetic experience, and, while they tell us, in a sense, what we really are, they can be translated into words only to speak unintelligibly. Philosophy, which describes the limits of knowledge, is always tempted to transcend them. But Kant's final advice to it is that given in the last sentence of Wittgenstein's _Tractatus Logico-Philosophicus_ : That whereof we cannot speak, we must consign to silence.\nFURTHER READING\n\nWritings by Kant\n\nThe standard edition of Kant's works is published by the Deutsche Akademie der Wissenschaften, as _Kants gesammelte Schriften_ (Berlin, 1900\u2013). It is now the general practice to include the page numbers of this edition in the margins of English translations. Hence, where practical, my page references are to the standard German text. However, until recently, there has been no attempt at a systematic English edition of Kant's works, and many of the writings are still most conveniently obtainable in editions that make no reference to the German pagination. In these cases I have indicated in the list of abbreviations at the beginning of this book which edition I have used.\n\nThe enterprise of providing authoritative and scholarly translations of Kant's works has been undertaken by Paul Guyer and Allen Wood, as general editors of the _Cambridge Edition of the Works of Immanuel_ _Kant_ (Cambridge, 1993\u2013), which is destined to be the standard English edition. Guyer and Wood have themselves translated the first _Critique_ for this edition, though the excellent translation by Norman Kemp Smith (London, 1929) remains the source most frequently used. Both translations contain marginal references to the page numbering of the original first and second editions (A and B). The volume of the Cambridge edition entitled _Practical Philosophy_ , translated by Mary Gregor, contains most of Kant's moral and political works in an accurate translation. There is also a good translation of the _Critique of Practical_ _Reason_ and the _Groundwork of the Metaphysic of Morals_ by L. W. Beck, in his _Kant's Critique of Practical Reason and Other Writings in_ _Moral Philosophy_ (Chicago, 1949). Translations of the _Critique of Judgment_ have been less successful. The one now most frequently referred to is that by W. Pluhar (Hackett, Indianapolis, 1987), though the previous standard edition (tr. J. C. Meredith, Oxford, 1928) is still usable. Such problems as these translations present are due more to Kant than to the translators. The first part of the _Metaphysics of Morals_ is available in a translation by John Ladd as _Metaphysical Elements of Justice_ (2nd edition, Indianapolis, 1999), while the whole work is contained in Mary Gregor's volume. Ladd rearranges Kant's text, for reasons both editorial and philosophical. Other works on political philosophy can be found in Hans Reiss (ed.), _Kant: Political Writings_ , tr. H. B. Nisbet (Cambridge, 1970).\n\nKant's writings on theology, including _Religion Within the Limits of_ _Reason Alone_ (translated as _Religion Within the Boundaries of Mere Reason_ ) are contained in _Kant: Religion and Rational Theology_ , tr. and ed. Allen Wood and George di Giovanni (Cambridge, 1996 [part of the Cambridge edition]).\n\nThere is no adequate selection in English from Kant's voluminous writings. The student cannot avoid jumping in at the deep end, with the _Critique of Pure Reason._\n\nWritings About Kant\n\nThe few biographies of Kant make unexciting reading. The fullest, although not the most accurate, is that of J. H. W. Stuckenberg, _The Life_ _of Immanuel Kant_ (London, 1882). That by Ernst Cassirer, entitled _Kant's_ _Life and Thought_ (New Haven, 1981), casts considerable light on Kant's philosophical development.\n\nCommentaries are legion. A growing interest in Kant among English-speaking philosophers has led to many works of high quality and lucidity. That by P. F. Strawson, _The Bounds of Sense_ (London, 1966), contains a thorough exposition, and partial defense, of the argument of the first _Critique_ , in its \"objective\" interpretation. The \"subjective\" rejoinder from Ralph Walker ( _Kant_ , in the series Arguments of the Philosophers, London, 1979), is clear and scholarly, although rather less persuasive. Those interested in a vigorous empiricist interpretation will enjoy Jonathan Bennett's two commentaries, _Kant's Analytic_ (Cambridge, 1966) and _Kant's Dialectic_ (Cambridge, 1974). The best short introduction in English remains that of A. C. Ewing (London, 1938) entitled _A Short Commentary_ _on Kant's Critique of Pure Reason_. More recent work has made a conscientious attempt to remain true to Kant's own intellectual concerns, and has therefore avoided reading the controversies of contemporary academic philosophy into his often obscure arguments. Exemplary in this respect are H. E. Allison, _Kant's Transcendental Idealism: An Interpretation_ _and Defense_ (New Haven, 1983), and Sebastian Gardner, _Kant_ _and the Critique of Pure Reason_ (London, 1999). Neither the second nor the third _Critique_ has received commentary of the same quality. On the ethics, I recommend H. J. Paton, _The Categorical Imperative_ (London, 1947), R. B. Sullivan, _Immanuel Kant's Moral Theory_ (Cambridge, 1989), and C. M. Korsgaard, _Creating the Kingdom of Ends_ (Cambridge, 1996). On the aesthetics there is much to be gained from Donald W. Crawford, _Kant's Aesthetic Theory_ (Wisconsin, 1974), and Paul Guyer, _Kant and the_ _Claims of Taste_ (Cambridge, Mass., 1979). Kant's political theory has been comprehensively summarized by Howard Williams in _Kant's Political Philosophy_ (Oxford, 1983), and aspects of it treated in depth in Howard Williams (ed.), _Essays on Kant's Political Philosophy_ (Cardiff, 1992).\nINDEX\n\nNote: Page numbers in _italics_ include illustrations and photographs\/captions.\n\naesthetics, \u2663, \u2666, \u2665, _\u2660_\u2013 \u2020, \u2021, \u0394,   \n\u2207\n\nantinomies, \u2663, \u2666, \u2665, \u2660, \u2020,   \n\u2021, \u0394\n\na posteriori knowledge\/truth, \u2663, \u2666\n\nappearance and reality, \u2663, \u2666\n\napperception, transcendental unity of, \u2663,   \n\u2666, \u2665, \u2660, \u2020\n\na priori knowledge\/truths   \ncausality and, \u2663  \nconcepts and, \u2666, \u2665  \nobjective deduction and, \u2660  \nprinciples as, \u2020  \nspace\/time and, \u2021  \nsynthetic, \u0394, \u2207, \u039f, \u25ca, \u2205, \u2217,   \n\u2295, \u2297  \ntranscendentalism and, \u221e, \u2202  \nunderstanding and, \u03b1, \u03b2, \u03b3\n\nAristotle, \u2663, \u2666\n\nAshley-Cooper, Anthony. _see_ Shaftesbury,   \nthird Earl of\n\nBatteux, l'abb\u00e9 Charles, \u2663\n\nBaumgarten, A. G., \u2663, \u2666, \u2665, \u2660, \u2020\n\nbeauty, \u2663, _\u2666\u2013_ \u2665, \u2660, \u2020, _\u2021_\n\nBecker, Johann Gottlieb, _\u2663_\n\nBeck, J. S., \u2663, \u2666\n\nBerkeley, G., \u2663\n\nBoethius, \u2663\n\nBritish Moralists, \u2663, _\u2666_\n\n\"Broad and Narrow Road,\" _xii_ \u2013 \u2663\n\nBurke, Edmund, \u2663, _\u2666_\n\ncategorical imperatives, \u2663, \u2666, \u2665,   \n\u2660\n\ncategories, \u2663, \u2666, \u2665, \u2660, \u2020,   \n\u2021, \u0394\n\ncausality, \u2663  \ncosmology and, \u2666, \u2665  \nexperience and, \u2660  \nfreedom and, \u2020, \u2021, \u0394  \nHume and, \u2207, \u039f, \u25ca  \nidentity and, \u2205  \nmonads and, \u2217  \nobjectivity and, \u2295\n\ncivil society, \u2663\n\n_Cogito ergo sum_ , _\u2663_\n\ncommon sense, \u2663\n\nconcepts, \u2663, \u2666, \u2665, \u2660, \u2020,   \n\u2021\n\nconservation, law of, \u2663, \u2666\n\ncontemplation, \u2663\n\nCopernican Revolution, \u2663\n\ncosmology, _\u2663_\u2013 \u2666, \u2665\n\nCrusius, C. A., \u2663\n\n_D\u00e9claration des Droits de L'homme et du_  \n _Citoyen_ , _\u2663_\u2013 \u2666\n\ndemocracy, \u2663, \u2666\n\nDescartes, Ren\u00e9, \u2663\u2013 _\u2666_, \u2665, \u2660, \u2020, \u2021,   \n\u0394, \u2207\n\ndesign arguments, \u2663, \u2666, \u2665, \u2660\n\nDoerstling, Emil, _\u2663_\n\nduty, _xii_ \u2013 \u2663, \u2666, \u2665, \u2660, \u2020, \u2021, \u0394, \u2207\n\nEberhard, J. A., \u2663, \u2666\n\nEinstein, A., \u2663\n\nempiricism   \nantinomies and, \u2663  \nconcepts and, \u2666, \u2665  \nexperience and, \u2660, \u2020  \nHume supporting, \u2021, \u0394  \nidealism and, \u2207  \nnecessities and, \u039f  \nobjectivity and, \u25ca  \nsynthetic a priori knowledge and, \u2205  \ntranscendental arguments and, \u2217\n\nEnlightenment, \u2663, _\u2666_\u2013 \u2665, \u2660, \u2020\n\nethics, \u2663, \u2666, \u2665\n\nexistentialism, \u2663\n\nexperience   \na priori knowledge and, \u2663, \u2666  \naesthetics and, \u2665, \u2660, _\u2020_  \ncausal necessities and, \u2021  \nknowledge and, \u0394, \u2207, \u039f, \u25ca, \u2205, \u2217  \nobjectivity and, \u2295, \u2297  \nobjects of, \u221e, \u2202, \u03b1, \u03b2  \nperspectives and, \u03b3, \u03ba  \nrationalism and, \u0398  \nself-consciousness and, \u03a6  \nspace\/time and, \u03b4, \u03bb  \nsynthesis and, \u03c8  \ntranscendental objects and, \u03d6  \nunderstanding and, \u03d1\n\nFichte, J. G., \u2663, \u2666\n\nform, \u2663\n\nFrederick the Great, \u2663, \u2666, \u2665\n\nFrederick William \u2663, \u2666\n\nfreedom, \u2663, \u2666, \u2665, \u2660, \u2020\n\nFrench Revolution, _\u2663_\u2013 \u2666, _\u2665_, _\u2660_\n\nFriedrich, Caspar David, _\u2663_\u2013 \u2666\n\nGandhi, M., \u2663\n\nGod, \u2663, \u2666, \u2665, \u2660  \nexistence of, \u2020, _\u2021_, \u0394, \u2207, \u039f  \njudgment of, \u25ca  \nmetaphysics and, \u2205  \nnature of, \u2217  \npractical reason and, \u2295\n\nG\u00f6del, K., \u2663\n\nGreen, Joseph, \u2663\n\nHals, Frans, _\u2663_\n\nHamann, J. G., \u2663, \u2666\n\nhappiness, _\u2663_\u2013 \u2666, \u2665\n\nharmony, \u2663\n\nHegel, G. W. F., \u2663, \u2666, \u2665, \u2660\n\nHeidegger, M., \u2663\n\nHeine, Heinrich, \u2663\n\nHerder, J. G., \u2663\n\nHobbes, T., \u2663\n\nhuman rights, \u2663\n\nHume, David, \u2663, _\u2666_, \u2665  \ncausality and, \u2660, \u2020, \u2021  \nconcepts and, \u0394  \nempiricism of, \u2207, \u039f  \nidentity and, \u25ca  \nobjective knowledge and, \u2205  \nreason and, \u2217, \u2295 _\u2013\u2297_  \nskepticism of, \u221e, \u2202, \u03b1, \u03b2, \u03b3  \ntheology and, \u03ba\n\nHussert, Edmund, \u2663\n\nHutcheson, Francis, \u2663\n\nhypothetical imperatives, \u2663\n\nidealism, \u2663, \u2666, \u2665, \u2660, \u2020, \u2021\n\n\"Ideal of Reason,\" \u2663\n\nidentity, \u2663, \u2666, \u2665\n\nillusion, \u2663, \u2666, \u2665, \u2660, \u2020\n\nimagination, \u2663, \u2666\n\ninnate ideas, \u2663, \u2666\n\nintentional objects, \u2663\n\nintuition   \nconcepts and, \u2663, \u2666  \nconsciousness and, \u2665  \nforms of, \u2660  \nimagination and, \u2020  \nknowledge and, \u2021, \u0394  \nmetaphysics and, \u2207  \nmorality and, \u039f, \u25ca  \nsensory, \u2205, \u2217, \u2295  \nspace\/time and, \u2297, \u221e\n\nJachmann, R. B., \u2663, \u2666\n\njudgment   \naesthetics and, \u2663, \u2666, \u2665, \u2660  \nconcepts and, \u2020  \ncontemplation and, \u2021, \u0394  \nempirical, \u2207  \nintellectualizing\/sensualizing and, \u039f  \nreason and, \u25ca  \nsubjective deduction and, _\u2205_\u2013 \u2217, \u2295  \nsynthetic a priori, \u2297  \nunderstanding and, \u221e, \u2202  \nwritings on, \u03b1, \u03b2, \u03b3\n\njustice, _\u2663_, \u2666, \u2665, \u2660, \u2020\n\nKant, Immanuel, _ii_ \u2013\u2663, _\u2666_, _\u2665\u2013_ \u2660,   \n_\u2020_\u2013 \u2021  \ncharacter of, \u0394  \ndeath of, _\u2207_\u2013 \u039f  \nearly life of, _\u25ca_  \nas teacher, \u2205, \u2217, \u2295, _\u2297_  \nwomen and, \u221e, \u2202\n\nKepler, J., \u2663\n\nKingdom of Ends, \u2663, \u2666\n\nK\u00f6nigsberg, Germany, \u2663, \u2666, \u2665, _\u2660_, \u2020,   \n_\u2021_\u2013 \u0394\n\nlaw and morality, \u2663\n\nlaw, role of, \u2663\n\nLeibniz, G. W., \u2663, _\u2666_, \u2665  \naesthetics and, \u2660  \nexistence and, \u2020  \ninnate ideas and, \u2021  \nintellectualizing appearances and, \u0394  \nmetaphysics of, \u2207, \u039f, \u25ca, \u2205  \nmonads and, \u2217, \u2295, \u2297  \nobjective knowledge and, \u221e  \nrationalism of, \u2202  \nreason and, \u03b1, \u03b2  \nwritings on, \u03b3, \u03ba, \u0398\u2013 _\u03a6_\n\nLessing, G. E., \u2663\n\nLichtenberg, G. C., \u2663\n\nLocke, John, \u2663, \u2666, \u2665\n\nlogical positivism, \u2663\n\nLouis \u2663, _\u2666_\n\nMagnus, Albertus, _\u2663_\u2013 \u2666\n\nmathematics, \u2663, \u2666, \u2665, \u2660, \u2020\n\nMendelssohn, Moses, \u2663, \u2666, \u2665, \u2660\n\nmetaphysics, _\u2663_\u2013 \u2666  \naesthetics and, \u2665  \ncategories and, \u2660  \ndivisions of, \u2020  \nempiricism and, \u2021  \nLeibnizian system and, \u0394, \u2207, \u039f, \u25ca  \nproblems with, \u2205  \npure reason and, \u2217  \nscientific laws and, \u2295  \nsubstances and, \u2297  \nas synthetic a priori truth, \u221e, \u2202  \nwritings on, \u03b1, \u03b2, \u03b3, _\u03ba_, \u0398\n\nMilton, John, \u2663\n\nmonads, \u2663, \u2666, \u2665, \u2660, \u2020\n\nmoney, \u2663\n\nMontesquieu, baron de, \u2663, _\u2666_, \u2665\n\nmorality, \u2663, _\u2666_\u2013 \u2665  \ncategorical imperatives and, \u2660  \nGod and, \u2020  \nintuition and, \u2021, \u0394  \nobjectivity and, \u2207  \npractical reason and, \u039f  \nself and, \u25ca  \nuniversality of, \u2205  \nwill and, \u2217  \nwritings on, \u2295, \u2297, \u221e, _\u2202_\n\nmotion, law of, \u2663, _\u2666_\n\nmusic, \u2663, \u2666\n\nnatural law, \u2663, \u2666, \u2665, \u2660\n\nnecessity, \u2663, \u2666, \u2665, \u2660\n\nNewton, Isaac, \u2663, \u2666, \u2665, _\u2660_\n\nnoumena, \u2663, \u2666, \u2665, \u2660, \u2020,\n\n\u2663, \u2666, \u2665\n\nobjective deduction, \u2663, \u2666, \u2665, \u2660, \u2020,   \n\u2021, \u0394\n\nobjectivity   \ncausality and, \u2663, \u2666  \ncontemplation and, \u2665  \nempirical, \u2660  \nillusion of, \u2020  \njudgment and, \u2021, \u0394  \nknowledge and, \u2207, \u039f  \nnatural necessity and, \u25ca  \ntranscendental idealism and, \u2205  \nvalidity and, \u2217\n\nontological argument, \u2663, \u2666\n\npeace, \u2663\n\npersons, \u2663, \u2666\n\nphenomena, \u2663, \u2666, \u2665, \u2660\n\nPhillips, G., _\u2663_\n\npietism, _xii_ \u2013 \u2663, \u2666\n\nPlato, \u2663, \u2666\n\npleasure, \u2663, \u2666, \u2665, \u2660,   \n\u2020\n\npolitics, \u2663, \u2666\n\nPrinciple of Contradiction, \u2663\n\nPrinciple of Sufficient Reason, \u2663, \u2666,   \n\u2665\n\nprinciples, \u2663, \u2666\n\n\"private language\" argument, \u2663\n\nproperty, \u2663\n\npsychology, rational, _\u2663_\u2013 \u2666\n\nPurgstall, Count, \u2663\n\npurposiveness, \u2663, \u2666, \u2665\n\nrationalism, \u2663, \u2666, \u2665, \u2660\n\nRawls, J., \u2663\n\nreason   \naesthetics and, \u2663  \nfaith and, \u2666  \nfreedom and, \u2665  \nHume and, \u2660, \u2020\u2013 _\u2021_  \nideas of, \u0394, \u2207, \u039f, \u25ca  \nillusion and, \u2205  \nknowledge and, \u2217  \nmetaphysics and, \u2295  \npassion and, \u2297  \npractical, \u221e, \u2202, \u03b1, \u03b2, \u03b3,   \n\u03ba, \u0398, \u03a6  \npure, \u03b4, \u03bb, \u03c8, \u03d6, \u03d1  \nrationalism and, \u039b  \nreality and, \u03a0  \nregulative principles of, \u03a3  \nrelations of ideas and, \u2663\u2663  \ntranscendental self and, \u2666\u2666  \nuniversalism and, \u2665\u2665  \nwritings on, \u2660\u2660, \u2020\u2020, _\u2021\u2021_\u2013 \u0394\u0394, \u2207\u2207, \u039f\u039f,   \n\u25ca\u25ca, \u2205\u2205\n\n\"Refutation of Idealism,\" \u2663, \u2666\n\nregulative principles, \u2663, \u2666, \u2665\n\nReinhold, K. L., \u2663\n\nrelations of ideas, \u2663, \u2666\n\nreligion\/theology, \u2663, \u2666, _\u2665_\u2013 \u2660, \u2020,   \n\u2021, \u0394, \u2207\n\nrepublicanism, \u2663, \u2666, \u2665, \u2660,   \n\u2020\n\nrespect for others, \u2663, \u2666,   \n\u2665\n\nReynolds, Joshua, _\u2663_\n\nRoman law, \u2663, \u2666\n\nRousseau, Jean-Jacques, _\u2663_, \u2666, \u2665, \u2660, _\u2020_,   \n\u2021, \u0394, \u2207\n\nRuhnken, David, \u2663\n\nSartre, J.-P., \u2663\n\nSchelling, F. W. J. von, \u2663\n\nSchlick, Moritz, _\u2663_\n\nSchopenhauer, Arthur, \u2663, _\u2666_\u2013 \u2665\n\nSchulze, G. E., \u2663\n\nSecondat, Charles de. _see_ Montesquieu,   \nbaron de\n\nself   \nexistence of, \u2663  \nHume doubting, \u2666  \nissues of, \u2665  \nmorality and, \u2660  \nsoul and, \u2020\n\nself-consciousness, \u2663, \u2666, \u2665, \u2660, \u2020,   \n\u2021, \u0394\n\nself-evidence, \u2663\n\nsensibility, \u2663, \u2666, \u2665, \u2660\n\nseparation of powers, \u2663\n\nsexuality, \u2663, \u2666\n\nShaftesbury, third Earl of, \u2663, _\u2666_, \u2665,   \n\u2660\n\nSiey\u00e8s, Abb\u00e9, \u2663\n\nslavery, \u2663, \u2666, \u2665\n\nsocial contract, \u2663, \u2666, \u2665\n\nsoul, \u2663, \u2666, \u2665, \u2660\n\nspace\/time, \u2663, \u2666, \u2665, \u2660  \nappearance and, \u2020  \nboundaries of, \u2021  \nempirical reality of, \u0394, \u2207  \nidentity through, \u039f  \nordering in, \u25ca\n\nSta\u00eb\u2663, Mme. de, \u2666\n\nsubjective deduction, _\u2663_\u2013 \u2666, \u2665, \u2660,   \n\u2020, \u2021\n\nsublime, \u2663, \u2666, \u2665, _\u2660_\u2013 \u2020\n\nsubstances, \u2663, \u2666, \u2665, \u2660, \u2020\n\nsynthesis, \u2663\n\nsynthetic truths, \u2663, \u2666, \u2665, \u2660\n\nTarski, A., \u2663\n\ntaste, \u2663, \u2666, \u2665, \u2660\n\n\"thing-in-itself,\" \u2663, \u2666, \u2665, \u2660, \u2020,   \n\u2021, \u0394, \u2207\n\ntranscendental   \ndeduction, \u2663, \u2666, \u2665, \u2660, \u2020,   \n\u2021  \nfreedom, \u0394, \u2207, \u039f, \u25ca, \u2205  \nidealism, \u2217, \u2295, \u2297, \u221e, \u2202  \nobjects, \u03b1, \u03b2, \u03b3, \u03ba, \u0398, \u03a6  \npsychology, \u03b4  \nself, \u03bb, \u03c8, \u03d6, \u03d1  \nunity of apperception, \u039b\n\nunconditioned knowledge, \u2663\n\nuniversalism, \u2663, \u2666, \u2665\n\nVienna Circle, \u2663, _\u2666_\n\nvon Zedlitz, H., \u2663\n\nWagner, Richard, \u2663\n\nWatson, James, _\u2663_\n\nWentzel, Johann Friedrich, _\u2663_\n\nwill, \u2663, \u2666, \u2665, \u2660, \u2020\n\nWinckelmann, H., \u2663\n\nWittgenstein, Ludwig, \u2663, \u2666, \u2665,   \n\u2660\n\nWoellner, K. von, \u2663\n\nWolff, Christian von, \u2663, \u2666\n\nwomen, \u2663\nPICTURE CREDITS\n\n**BRIDGEMAN ART LIBRARY** :\u2666; \u2660; \u2665\n\n**CORBIS** :\u2663: \u00a9 Bettmann\/CORBIS\n\n**GETTY IMAGES** :\u03b6: Imagno\/Contributor\/Getty Images;\n\n\u2020: AFP\/Getty Images\n\n**GOOGLE BOOKS** :\u03c0; \u03bb\n\n**THE GRANGER COLLECTION, NEW YORK** :\u03b4; \u03c8\n\n**LEBRECHT PHOTO LIBRARY** :\u00ac\n\n**MARY EVANS PICTURE LIBRARY** :\u0398\n\n**NATIONAL LIBRARY OF MEDICINE** :\u00b1\n\n**COURTESY OF PRINTS & PHOTOGRAPHS DIVISION, LIBRARY OF CONGRESS**:\n\n\u03b1: LC-USZ62-51093;\n\n\u03b3: LC-USZC4-5133;\n\n\u00de: LC-USZ62-61713;\n\n\u0394: LC-USZ62-48733;\n\n\u2021: LC-USZC4-12944;\n\n*: LC-USZ62-130773;\n\n\u00a5: LC-DIG-ppmsca-10742\n\n**COURTESY OF WIKIMEDIA COMMONS** :\n\n\u00a7: Charlotte Reihlen (idea), Paul Beckmann (execution), _Der breite und der schmale Weg_ ;\n\n\u00b6: Emil Doerstling, _Kant and Friends at Table_ , ca. 1900;\n\n\u03a0: Alte Universit\u00e4t Koenigsberg;\n\n\u03a3: Immanuel Kant statue, in Universidade Federal de Minas Gerais, Belo Horizonte\/Upload by Andrevruas;\n\n\u03a6: Frans Hals, _Ren\u00e9 Descartes_ (1649);\n\n\u00a4: Johann Friedrich Wentzel, _Gottfried Wilhelm_ _Leibniz_ , ca. 1700;\n\n\u25ca: Anthony Ashley-Cooper 3rd Earl of Shaftesbury, Engraving by Gribelin Sculp after a painting by J. Closterman Pinx, in _Characteristicks of Men, Manners, Opinions, Times_ , 5th ed., 1732;\n\n\u00b5: Caspar David Friedrich, _Chalk Cliffs on R\u00fcgen_ , ca. 1818;\n\n\u03ba: Representation of the Declaration of the Rights of Man and of the Citizen, 1789;\n\n\u03b7: Ludwig Sigismund Ruhl, _Arthur Schopenhauer_ , ca. 1818 \nBRIEF INSIGHTS\n\nA series of concise, engrossing, and enlightening books that explore   \nevery subject under the sun with unique insight.\n\n_Available now or coming soon:_\n\n**THE AMERICAN  \nPRESIDENCY**\n\n**ARCHITECTURE**\n\n**ATHEISM**\n\n**THE BIBLE**\n\n**BUDDHISM**\n\n**CHRISTIANITY**\n\n**CLASSICAL  \nMYTHOLOGY**\n\n**CLASSICS**\n\n**CONSCIOUSNESS**\n\n**THE CRUSADES**\n\n**ECONOMICS**\n\n**EXISTENTIALISM**\n\n**GALILEO**\n\n**GANDHI**\n\n**GLOBALIZATION**\n\n**HISTORY**\n\n**HUMAN  \nEVO LUTION**\n\n**INTERNATIONAL  \nRELATIONS**\n\n**JUDAISM**\n\n**KAFKA**\n\n**KANT**\n\n**LINCOLN**\n\n**LITERARY THEORY**\n\n**LOGIC**\n\n**MACHIAVELLI**\n\n**MARX**\n\n**MATHEMATICS**\n\n**MODERN CHINA**\n\n**MUSIC**\n\n**NELSON MANDELA**\n\n**PAUL**\n\n**PHILOSOPHY**\n\n**PLATO**\n\n**POSTMODERNISM**\n\n**PSYCHOLOGY**\n\n**THE REAGAN  \nREVOLUTION**\n\n**RELATIVITY**\n\n**RENAISSANCE ART**\n\n**RUSSIAN  \nLITERATURE**\n\n**RUSSIAN  \nREVO LUTION**\n\n**SEXUALITY**\n\n**SHAKESPEARE**\n\n**SOCIAL AND CULTURAL  \nANTHROPOLOGY**\n\n**SOCIALISM**\n\n**STATISTICS**\n\n**THE TUDORS**\n\n**THE VOID**\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by deaurider, Paul Clark and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net (This file was\nproduced from images generously made available by The\nInternet Archive)\n\n\n\n\n\n\n\n\n\n                                  THE\n                           SLAVERY QUESTION.\n\n                                  BY\n\n                            JOHN LAWRENCE,\n\n      AUTHOR OF \"PLAIN THOUGHTS ON SECRET SOCIETIES,\" AND \"BRIEF\n                    TREATIES ON AMERICAN SLAVERY.\"\n\n                            THIRD EDITION.\n\n    THE DISCUSSION OF SLAVERY WILL PROCEED, WHEREVER TWO OR\n    THREE ARE GATHERED TOGETHER--BY THE FIRESIDE, ON THE HIGHWAY,\n    AT THE PUBLIC MEETING, IN THE CHURCH. THE MOVEMENT\n    AGAINST SLAVERY IS FROM THE EVERLASTING ARM.\n\n                                                  CHARLES SUMNER.\n\n                              DAYTON, O.,\n\n     PUBLISHED BY ORDER OF THE TRUSTEES OF THE CONFERENCE PRINTING\n            ESTABLISHMENT OF THE UNITED BRETHREN IN CHRIST.\n\n                      VONNIEDA & KUMLER, AGENTS.\n\n                                 1854.\n\n\n\n\n      Entered according to act of Congress, in the year 1854, by\n\n                          VONNIEDA & KUMLER,\n\n         in the Clerk's Office of the District Court of Ohio.\n\n\n\n\nPREFACE.\n\n\nAmerican slavery is a great sin--a complicated iniquity--a gigantic\nbarbarism--and it \"is evil, only evil, and that continually.\" But the\ndepth of this wickedness is not very frequently sounded, if, indeed it\ncan be sounded. The magnitude of this crime is not often measured, if,\nindeed it is possible to determine its dimensions.\n\nSlavery has narcoticized the consciences of the American people to a\nmost alarming extent. A deep sleep has come over the moral sense, which\nit would seem cannot be broken by the cries and entreaties of three\nmillions of wretched bondmen. Are we not in imminent danger of being\ncursed with Pharaoh's hardness of heart? May we not be visited speedily\nwith judicial blindness such as was inflicted upon the doomed nations\nand cities of antiquity?\n\nThe standard of national morality has been degraded to the level of an\ninfamous lower law enacted by scheming political traders.\n\nOur national government, in all its departments--Executive, Judicial\nand Legislative--has been transformed into a pliant tool in the hands\nof an unscrupulous oligarchy.\n\nThe powerful American Churches have ceased to be asylums for the\noppressed, defenders of the down trodden, uncompromising foes of\ntyranny, and they have become, on the contrary, the apologists of\noppressors, a terror to the oppressed, and the only reliable bulwark of\nAmerican slavery.\n\nThe author has aimed to present in the following pages such a\ndiscussion of the general subject of slavery as would be calculated\nto awaken the thoughts, and feelings, especially of those who have not\nhad an opportunity of examining this question in larger and more ably\nwritten productions. There are thousands of honest people who would\ntake a decided position on a Christian anti-slavery platform, and\nthrow their whole influence in the right direction if they were made\nacquainted with slavery as it is, and with their duties religiously\nand politically in relation to it. It is with the design of benefiting\nthe common people--the people of plain sense--who are not offended at\nplain talk and plain facts, that the following work is published. If\nthe workingmen of the free and slave States can be aroused into action,\nslavery must fly from the churches and perish from the nation.\n\nWith this purpose in view, we have sketched a history of the\nAfrican slave trade, showing how slavery originated; have defined\nslavery--proving that its essential principle is property in a human\nbeing; and laws, facts and incidents have been adduced to illustrate\nthe system so that even a child may see and feel its enormity.\n\nAnd, as a corrupt moral sense has been still more corrupted by efforts\nto bring revealed religion to the support of slavery, particular pains\nhave been taken to prove that not a single word, nor precept, nor\nexample can be adduced from the Bible which sanctions any such system;\nand that the whole spirit of religion as revealed under the old economy\nand the new, is utterly and irreconcilably opposed to all slavery.\n\nIt has been thought proper to present a concise view of the position\noccupied by the American Churches upon this question. No church can\ncomplain when its ecclesiastical action on so grave a subject is\nre-published. And besides, it is quite necessary for honest people to\nknow on what platforms the religious denominations of the country stand.\n\nThe true position of a religious society or church in relation to\nslavery is exhibited. This is a point of more than ordinary importance.\nThe doctrine is maintained that the honor of the Bible, the purity,\npower, peace, and success of the Church, its duty to God, to freedom,\nto slaveholders and especially to slaves, demand that it have no\nfellowship with slaveholding.\n\nParticular pains have been taken to point out the political duties of\nChristians in relation to slavery.\n\nThe inquiry, \"how are we to get rid of slavery?\" is taken up, and the\nposition assumed and defended that it ought to be abolished immediately.\n\nThe book closes with a glance at the prospects. The watchman tells\nus that the sky from many points of observation is dark, but still\nthat there are some very encouraging indications. The uncorrupted\nconscience, reason, truth, Christianity and prayer, are on the side of\nthe oppressed; and God, who is love, is their hope, and cannot fail\nto come to their help and bring them forth with a mighty hand and an\nout-stretched arm.\n\nQuite a number of works on slavery have been consulted in the\npreparation of this discussion, among which may be mentioned, \"American\nSlave Code\" by Mr. Goodell; \"Barnes on Slavery;\" \"Bible Servitude,\" by\nE. Smith; \"Elliott on American Slavery;\" \"Slavery and the Church,\" by\nMr. Hosmer; \"Debate on Slavery by Blanchard & Rice;\" \"Non-fellowship\nwith Slaveholders,\" by Mr. Fee; \"Sermon on the Slave Trade\" by Jonathan\nEdwards; and \"Thirteenth Annual Report of the American and Foreign\nAnti-slavery Society.\"\n\nNo \"mealy words\" have been used in this book. I have only aimed to\npresent the plain truth, and shall be rewarded in whatever mite of\ninfluence it may cast on the side of liberty.\n\n\n\n\nCONTENTS.\n\n\n CHAPTER I.\n\n ORIGIN OF AMERICAN SLAVERY.\n\n THE SLAVE TRADE.\n\n Seven millions of slaves in America--Slavery originated in the\n African slave trade--Slave-trade unprovoked--Excited by lust for\n gold--Commenced by the Portuguese in 1434--Spaniards in 1511--English\n in 1556--President Edwards quoted--100,000 annually destroyed--Report\n made to the British House of Commons--Startling statistics--A slave\n ship described--Slave-trade declared to be piracy and abolished\n                                                                 page 13\n\n\n CHAPTER II.\n\n SLAVERY DEFINED.\n\n PROPERTY IN A HUMAN BEING.\n\n A slave is a chattel--Authorities quoted--Advertised and sold as\n property--Facts adduced--sale of a boy--a woman with an infant in her\n arms--a mother--American slave-code identical in principle with the\n Roman                                                           page 30\n\n\n CHAPTER III.\n\n SLAVERY ILLUSTRATED.\n\n THE CHATTEL PRINCIPLE IN PRACTICE.\n\n Slaves denied an education--Laws--Instances--Slavery disregards\n matrimonial connections--Painful facts                          page 41\n\n\n CHAPTER IV.\n\n SAME SUBJECT CONTINUED.\n\n Slavery disregards the parental and filial\n relations--Facts--Slave-mother's lament                         page 56\n\n\n CHAPTER V.\n\n SAME SUBJECT CONTINUED.\n\n Slavery utterly impoverishes its victims--Exposes them to unbridled\n lust--unrestrained passion--irresponsible tyranny--Heart-rending\n incidents!                                                      page 64\n\n\n CHAPTER VI.\n\n SAME SUBJECT CONTINUED.\n\n Severity of laws against slaves--partial--unreasonable and\n cruel--Practice worse than the laws--Burning of slaves--Horrible\n examples                                                        page 80\n\n\n CHAPTER VII.\n\n SLAVERY AND RELIGION.\n\n CURSE OF CANAAN--Doubtful authority--Did not allude to slavery--A mere\n prediction at best--Africans not the descendants of Canaan      page 88\n\n\n CHAPTER VIII.\n\n SAME SUBJECT CONTINUED.\n\n PATRIARCHAL SERVITUDE AND SLAVERY--No patriarch ever owned\n a slave--Slavery had no existence in the time of the\n patriarchs--Diodorus, Athenaeus and Rollin quoted--The Hebrew word\n SERVANT not equivalent to the English word SLAVE--Abraham's servants\n converts from idolatry                                          page 94\n\n\n CHAPTER IX.\n\n SAME SUBJECT CONTINUED.\n\n LAW OF MOSES AND SLAVERY--Levitical statutes not perfect--Allowed what\n it would now be wrong to practice--Dr. Stowe quoted--Servitude under\n the law of Moses essentially different from American slavery--Meaning\n of \"buy,\" \"heathen,\" \"bondmen,\" and \"forever,\"--Servants not\n stolen--Voluntary--Provision for religious improvement--Kind\n treatment--Could not be sold--Equal to their masters--Certain\n emancipation--Salvation of the heathen the primary design of\n introducing foreign servants                                   page 107\n\n\n CHAPTER X.\n\n SAME SUBJECT CONTINUED.\n\n NEW TESTAMENT AND SLAVERY--SERVANTS mentioned but not SLAVES--DOULOS\n does not mean SLAVE--New Testament does not regulate slavery\n because it cannot be regulated--Slaveholders not addressed by the\n Apostles--Onesimus not a slave--Character of Roman slavery--Contrary\n to the fundamental principles of revealed religion--The character of\n God--Common origin of man--General Redemption--Moral precepts--And is\n necessarily unjust and unequal                                 page 124\n\n\n CHAPTER XI.\n\n AMERICAN CHURCHES AND SLAVERY.\n\n THE POSITION THEY OCCUPY.\n\n Presbyterians (O. S. and N. S.)--Congregational--Methodist\n Episcopal, North and South--Methodist Protestant--Wesleyan Methodist\n Connection--Baptist, Regular--Freewill--Seventh Day--Evangelical\n Association--United Brethren--Various Churches--Summary View   page 149\n\n\n CHAPTER XII.\n\n SLAVERY AND THE CHURCH.\n\n NON-FELLOWSHIP WITH SLAVEHOLDERS.\n\n Scriptural view--Church must keep slaveholders out--If they get\n in, it must expel them--If the Church sanction slavery officially\n or practically, withdraw from it--Non-slaveholding required\n that it may be holy--The pillar of truth--That it may honor the\n Scriptures--Convert the world--Be faithful to slaveholders and to\n slaves--Non-fellowship required by decency--humanity--If fellowshiped,\n we shall have slaveholding preachers, and women-sellers and\n cradle-plunderers for class-mates--Cases given                 page 169\n\n\n CHAPTER XIII.\n\n SAME SUBJECT CONTINUED.\n\n OBJECTIONS ANSWERED.\n\n Kind slaveholders--Examples--Excusable slaveholders--Slavery a\n political matter--Fault of the public corruption--Fault of the\n laws--Slaveholders from necessity--Slaves their property--All right\n ONLY this one thing--Take them in to convince them of the wrong--Mr.\n Fee's opinion                                                  page 184\n\n\n CHAPTER XIV.\n\n POLITICAL DUTIES OF CHRISTIANS.\n\n EXTIRPATION OF SLAVERY FROM THE WORLD.\n\n Necessity of government--Obligation of political action--Voters\n responsible for slavery--United States Constitution does not endorse\n slavery--Founders of the Republic intended that slavery should die\n out speedily--Character of the government changed--Great work for\n Christian citizens--Slavery in the District--Territories--Slave\n States--Throughout the world                                   page 195\n\n\n CHAPTER XV.\n\n ABOLITION OF SLAVERY.\n\n IMMEDIATE EMANCIPATION.\n\n The duty plain and scriptural--Break every yoke--proclaim a year of\n Jubilee--Slavery cannot be reformed--Slaves prepared for freedom--Free\n people of color--Fugitives in Canada--West India emancipation--<DW52>\n people not dangerous when free--Amalgamation--Our fears originate\n in our guilt--Colonization scheme impracticable--Wrong--Watkins\n quoted--All objections mere excuses--We must emancipate to escape the\n judgments of God--Too long delayed--A good example             page 206\n\n\n CHAPTER XVI.\n\n WHAT OF THE NIGHT?\n\n THERE IS HOPE IN GOD ONLY.\n\n The government intensely pro-slavery--Political horizon lowering--The\n great denominations and benevolent societies heartily supporting\n slavery--Ecclesiastical heavens dark--Deep prejudices in the masses of\n the people--Douglass quoted--God is on the side of the oppressed--He\n is stirring the nation--Question cannot rest--Agitation goes on--Truth\n is on the side of the slave--Literature coming to his aid--A pure\n Church arising to plead his cause--\"TOIL AND TRUST.\"           page 219\n\n\n\n\nAMERICAN SLAVERY.\n\n\n\n\nCHAPTER I.\n\nOrigin of American Slavery.\n\nTHE SLAVE TRADE.\n\n\nOn the continent of America and adjacent Islands there are more than\nseven millions of slaves. Between three and four millions of these\nare enslaved by the most liberal, enlightened and prosperous nation\non the Globe. The American _Republic_ is a great slaveholding nation,\nand, viewed in its slaveholding character, might fitly be termed also,\nthe American _Despotism_. The highest form of freedom is here enjoyed\nby about twenty millions of persons and the lowest type of slavery\nsuffered by more than three millions. One seventh of all born under our\nDemocratic Constitution and under our world-renowned stars and stripes,\nare hereditary slaves.\n\nAmerican slavery has flourished three hundred years, being coeval with\nthe Reformation, and running back over one twentieth part of the whole\nperiod of time since Adam. Nine generations of slaves, under a crushing\nweight of despotism, have toiled and suffered on through a wretched\nlife, and have gone murmuring down to the grave.\n\nWe shall now inquire into the origin of this immense iniquity. American\nslavery originated directly in the African slave trade; a trade most\ndishonorable to human nature, bad as that nature is admitted to be,\nand most disgraceful to christian civilization. Its history, although\nnot fully written, except by heaven's recording angel, cannot be read\nby a humane person, even in its fragmentary form, without the deepest\nsorrow. It is a history of villainy, of relentless cruelty, of raging,\nhollow-hearted avarice and of unmitigated diabolism on the one side;\nand of wrongs, wretchedness and writhing anguish on the other.\n\nNothing had occurred to provoke a marauding attack upon the Africans.\nThey were a peaceable and harmless people, and had no means of exciting\neither the jealousy or the displeasure of Europeans. They had not\nviolated treaties, nor declared wars. The bloody wars among the African\ntribes, of which we hear so much from those who would palliate the\natrocities of the slave trade, were excited by the traders themselves,\nand so far from palliating, only add _blackness_ to the darkness of\ntheir crimes. The old Roman soldier, who enslaved a national enemy whom\nhe valiantly met and conquered in what is called honorable warfare,\nmight have claimed, with the semblance of plausibility, that the life\nhe had spared legitimately belonged to him. But the African slave\ntrader could not plead even this unmanly and unmerciful apology. The\nAfricans were not national enemies, and were not in arms.\n\nNo, it was not revenge, ambition, or patriotism, but CUPIDITY which\nprompted the slave trade--\n\n    \"The lust of gold, unfeeling and remorseless!\n    The last corruption of degenerate man.\"\n\nAvaricious men launched and manned the slave ship, unfurled the sails\nand stood at the helm. In their perilous voyage over the wide ocean,\namid storms and tempests, not one noble impulse swelled their bosoms;\nnot one philanthropic purpose strengthened their courage; not one\nhumane pulsation throbbed in their hearts. The slaver went on its long\nvoyage under the patronage of the Prince of darkness, for the one and\nonly purpose of making gold out of the sale of the bodies and souls of\nmen; of distilling wealth from blood and tears and agony. Montgomery\nsaid truly--\n\n    \"Cruel as death, insatiate as the grave,\n    False as the winds that round his vessel blow;\n    Remorseless as the gulf that yawns below,\n    Is he who toils upon the wafting flood,\n    A CHRISTIAN BROKER IN THE TRADE OF BLOOD!\"\n\nBut it was not avarice in the crew of the slave ship alone which\nincited and drove this iniquitous business. The prime movers were\nthe owners of the estates to be worked. Had those men been unwilling\nto grow rich upon unrewarded toil, the _slaver_ never would have\nsailed to Africa and plundered its shores. But the piratical crew and\nthe purchasers of the victims of their nefarious traffic were in a\nvillainous co-partnership.\n\nWhen the slaver had reached its destination and had anchored off the\nslave coast, the following methods were employed in securing a cargo.\n1st. Declarations of friendship were made and many of the unsuspecting\nnatives were induced, out of curiosity or for trade, to go aboard the\nvessel, and when there were suddenly confined and permitted no more to\nreturn. 2d. Parties of the crew were sent out to surprise and carry\noff innocent children and youth as they went to the fields or gathered\nin groups to play in the groves. Think of the anguish of those African\nmothers and of the distress of their affrighted children! 3d. Villages\nwere fired in the night, and as many of the defenseless inhabitants as\ncould be captured by force of arms were carried off. 4th. The chiefs\nof different tribes were hired to act as the agents of the slaver in\nprocuring slaves. Rum, of which all savages are extremely fond, was the\nprincipal incentive. Inflamed by this demon, the native chiefs made war\nupon each other, and sold the prisoners captured to the traders for a\nfresh supply of rum.\n\nThe African slave trade was commenced on a small scale a few years\nbefore the discovery of America. We learn from the Encyclopedia\nAmericana \"that, in 1434, a Portuguese captain name _Alonzo Gonzales_,\nlanded in Guinea, and carried away some  lads, whom he sold\nadvantageously to Moorish families settled in the South of Spain.\nSix years after, he committed a similar robbery, and many merchants\nimitated the practice, and built a fort to protect the traffic.\"\n\nAfter a discovery of the Gold Mines of America, quite a number of\n<DW64>s were imported, first by the Portuguese then by the Spaniards,\nto labor in those mines. In 1511 Ferdinand, King of Spain, authorized\nthe importation of a large number. About this period it is said, and\ngenerally believed that _Bartolomeo las Cas_, a Catholic Priest,\ninfluenced by a feeling of pity toward the Indians, whom the Spaniards\nwere enslaving, proposed to Ximenes the regular importation of <DW64>s.\nWhether this be true or not, Charles the V. in 1517, granted the\nprivilege to Lebresa, of importing 4000 slaves to America annually.\nLebresa sold his right to import to Genoese merchants, for about\n$25,000. These merchants now commenced the slave trade in earnest.\n\n_Sir John Hawkins_ has the honor of being the first English captain\nwho engaged in the business of stealing <DW64>s. In 1556 he made an\nunsuccessful effort at <DW64> catching near Cape Verd. He made another\neffort at a different point; and after burning the towns, was so\nbravely resisted by the inhabitants, that he lost seven men, and only\ncaptured ten. He continued his depredations until his ship was loaded\nwith human beings, which he sold in America.[1] The trade was now\nvigorously prosecuted by the christian nations of Europe. It is said\nthat Charles the V., Louis XIII. and Queen Elizabeth had some trouble\nwith their consciences about this horrible trade, but they were quieted\nby the argument that it brought the African into a good situation to be\nconverted! Pope Leo X. declared that \"not only the christian religion\nbut nature itself cried out against a State of slavery.\"\n\nThese feeble expressions of disapprobation were scarcely heard and\nthe trade went on vigorously--cupidity triumphing over conscience and\nsilencing almost, for many years, the voice of humanity and religion.\n\nAn extract from a sermon preached on the slave trade by _President\nEdwards_, in the year 1791 will now be quoted. At the time this good\nman lifted his voice against this traffic, it will be remembered that\nit was authorized by the Constitution of the United States, and was a\nsource of great profit to those engaged in it.\n\n\"The slave trade is wicked and abominable on account of the cruel\nmanner in which it is carried on. Beside the stealing or kidnapping of\nmen, women and children, in the first instance, and the instigation\nof others to this abominable practice, the inhuman manner in which\nthey are transported to America, and in which they are treated on the\npassage and in their subsequent slavery, is such as ought forever to\ndeter every man from acting any part in this business, who has any\nregard to justice or humanity. They are crowded so closely into the\nholds and between the decks of vessels, that they have room scarcely\nto lie down, and some times not room to sit up in an erect posture,\nthe men at the same time fastened together with irons, by two and two:\nand all this in the most sultry climate. The consequence of the whole\nis, that the most dangerous and fatal diseases are soon bred among\nthem, whereby vast numbers of those exported from Africa perish in the\nvoyage; others in dread of that slavery which is before them, and in\ndistress and despair from the loss of their parents, their children,\ntheir husbands, their wives, all their dear connections, and their dear\nnative country itself, starve themselves to death, or plunge themselves\ninto the ocean. Those who attempt in the former of those ways to\nescape from their persecutors, are tortured by live coals placed to\ntheir mouths. Those who attempt an escape in the latter and fail, are\nequally tortured by the most cruel beating. If any of them make an\nattempt as they sometimes do, to recover their liberty, some, and as\nthe circumstance may be, many, are put to immediate death, others,\nbeaten, bruised, cut and mangled in a most inhuman and shocking manner,\nare in this situation, exhibited to the rest, to terrify them from the\nlike attempt in future: and some are delivered up to every species\nof torment, whether by the application of the whip, or of any other\ninstrument, even of fire itself, as the ingenuity of the ship master,\nor of his crew is able to suggest, or their situation will admit; and\nthese torments are purposely continued for several days before death is\npermitted to afford relief to these objects of vengeance.\n\n\"By these means, according to the common computation, twenty-five\nthousand, which is a fourth part of those who are exported from Africa,\nand by the concession of all, twenty thousand, annually perish, before\nthey arrive at the places of their destination in America.\"\n\nThe same writer computed that of the one hundred thousand slaves\nannually exported, 60,000 were captives taken in war, and that ten\npersons were killed in the capture of one. _Sixty thousand then in the\ntime of Jonathan Edwards were slain in battle, 40,000 destroyed on\nthe voyage_ and in the seasoning, making _an annual destruction of_\n100,000 men, woman and children, in order to procure 60,000 slaves!\nThis computation may be relied upon, as Jonathan Edwards was a careful\nwriter, and no enthusiast.\n\nFor three hundred years this horrible traffic had been prosecuted\nbefore Mr. Edwards delivered the sermon from which we have quoted,\nand at that period the annual slaughter was 100,000, and the annual\nenslavement 60,000! How many perished during those three hundred years\nGod only knows. Rum had excited wars among the natives, and the whole\ncoast, and far into the interior was turned into a battle field. No\none was safe. The poor African could not lie down securely at night,\nfor men-stealers were ransacking the country watching for their prey\nlike hungry tigers; villages were burned, property destroyed, and the\nwretched inhabitants, either captured, killed, or caused to fly from\ntheir homes, and perish perhaps with famine.\n\nIn a report made to the British House of Commons, it was estimated\nthat from 1807 to 1847, including a period of only forty years, _ten\nmillions_ of persons had been made the victims of this traffic! TEN\nMILLIONS; one-half of whom were murdered in Africa; one fourth during\nthe \"middle passage;\" and the remaining fourth reduced to _property_\nand doomed, with their posterity, to a life of degradation, suffering\nand toil! And all this gigantic robbery and murder perpetrated in the\nfavored nineteenth century!\n\nPermit me to direct your attention to a single slave ship which sailed\nonly a few years ago. This ship was examined by the officers of a\nBritish man-of-war. The following is from the pen of Mr. Walsh, an eye\nwitness of what he relates.\n\n\"The ship had taken in, on the coast of Africa, 336 males and 226\nfemales, making in all 562, and had been out 17 days, during which she\nhad thrown overboard _fifty five_!\n\n\"The slaves were all enclosed under grated hatchways between decks.\nThe space was so low, that they sat between each other's legs, and\nthey were stowed so close together, that there was no possibility\nof their lying down, or at all changing their position by night or\nday. As they belonged to, and were shipped on account of different\nindividuals, they were all branded like sheep, with the owner's marks\nof different forms. These were impressed under their breasts or on\ntheir arms, and as the mate informed me, with perfect indifference,\nquiemados pelo ferro quento--burnt with the red hot iron. Over the\nhatchway stood a ferocious looking fellow with a scourge of many\ntwisted thongs in his hand, who was the slave driver of the ship; and\nwhenever he heard the slightest noise below, he shook it over them,\nand seemed eager to exercise it. As soon as the poor creatures saw us\nlooking down at them their dark and melancholy visages brightened up.\nThey perceived something of sympathy and kindness in our looks which\nthey had not been accustomed to, and feeling instinctively, that we\nwere friends, they immediately began to shout and clap their hands.\nOne or two had picked up a few Portuguese words, and cried out _Viva!\nviva!_ The women were particularly excited. They all held up their\narms; and when we bent down and shook hands with them, they could not\ncontain their delight, they endeavored to scramble upon their knees,\nstretching up to kiss our hands; and we understood that they knew we\nhad come to liberate them. Some, however, hung down their heads, in\napparently hopeless dejection, some were greatly emaciated, and some,\nparticularly children, seemed dying. But the circumstance which struck\nus most forcibly, was, how it was possible for such a number of human\nbeings to exist, packed up and wedged together as tight as they could\ncram, in low cells, three feet high, the greater part of which, except\nthat immediately under the hatchways, was shut out from light or air,\nand this when the thermometer, exposed to the open sky, was stand-in\nthe shade, on our deck at 89 deg.. The space between the decks was divided\ninto two compartments, three feet, three inches high; the size of one\nwas 16 feet by 18 feet, and of the other 40 feet by 21 feet; into the\nfirst there were crammed the women and girls, into the second the men\nand boys; 226 fellow beings were thus thrust into one space 288 feet\nsquare, and 336 into another 800 feet square, giving to the whole an\naverage of 23 inches, and to each of the women, not more than thirteen.\nThe heat of these horrid places was so great and the odor so offensive,\nthat it was quite impossible to enter them even had there been room.\nThey were measured as above when the slaves had left them. The officers\ninsisted that the poor suffering creatures should be admitted on deck\nto get air and water. This was opposed by the mate of the slaver, who,\nfrom a feeling that they deserved it, declared they would murder them\nall. The officers (of the Eng. ship,) however, persisted, and the poor\nbeings were all turned up together. It is impossible to conceive the\neffect of this eruption; 507 fellow creatures of all ages and sizes,\nsome children, some adults, old men and women, all in a state of total\nnudity, scrambling out together to taste a little pure air and water.\nThey came swarming up like bees from the aperture of a hive, till the\nwhole deck was crowded to suffocation, from stem to stern; so that it\nwas impossible to imagine where they could all have come from, or how\nthey could all have been stowed away. On looking into the places where\nthey had been crammed, there were found some children next the sides\nof the ship, in the places most remote from light and air; they were\nlying in nearly a torpid state, after the rest had turned out. The\nlittle creatures seemed indifferent as to life or death; and when they\nwere carried on deck, many of them could not stand. After enjoying, for\na short time, the unusual luxury of air, some water was brought; it\nwas then that the extent of their sufferings was exposed in a fearful\nmanner. They all rushed like maniacs toward it. No entreaties, or\nthreats, or blows could restrain them; they shrieked, and struggled,\nand fought with one another, for a drop of this precious liquid, as\nif they grew rabid at the sight of it. When the poor creatures were\nordered down again, several of them came, and pressed their heads\nagainst our knees, with looks of the greatest anguish, at the prospect\nof returning to the horrid place of suffering below.\"[2]\n\nBut the English ship was obliged to release the slaver and abandon to\ndespair those defenseless victims, as it was found upon examination\nthat it had not violated a vile privilege then allowed Brazilian ships\nto obtain slaves south of a certain line.\n\nIt is a humiliating fact that for a period of three centuries the\nwhole christian world was engaged in plundering a heathen shore of\nits inhabitants, speculating in their bodies and souls and spreading\namongst them intemperance, war and all unutterable woes. The history of\nthis wickedness will never be fully known until the general judgment.\nThen will the ocean have a tale to tell of the thousands who were\nsmothered in the slave prisons which floated upon her bosom, and of the\nmultiplied thousands who were famished and buried in her deeps. The sea\nwill send up her witnesses, and Africa, wet with tears and blood, will\nbear a testimony before God in that day which will make the ears of all\nthat hear it to tingle!\n\nBut let us glance at a more hopeful view of the subject. In 1783 a\npetition was addressed to the house of Parliament, Great Britain, for\nthe abolition of this trade. THOMAS CLARKSON was the mover, and the\ngreat champion of the cause. In 1788 Mr. Pitt presented a petition\nagainst the trade and introduced the subject of its abolition into the\nhouse of Commons. The opposition to this measure was united, powerful\nand violent. At length in 1792 the house of Commons passed a bill for\nthe abolition of the slave trade to take place in 1795. This bill\nwas rejected in the House of Lords. About this time the _National\nAssembly_ in France, declared all the slaves in the French colonies\nfree. Mr. Wilberforce brought into the British Parliament another bill\nin 1796, which provided that this trade should be abolished forever\nafter 1797--but this bill was lost also. The efforts of the friends\nof humanity were redoubled, and in \"_1806 Fox moved that the House\nof Commons should declare the slave trade inconsistent with justice,\nhumanity and sound policy, and immediately take effective measures\nfor its abolition_.\" This measure passed by a large majority--and\nJan. 1808 was fixed as the time for its abolition. _In 1824 a law\nwas passed declaring the trade to be piracy._ Portugal provided for\nthe total abolition of this trade in 1823. France in 1815--Spain in\n1820--Netherlands in 1818--Sweden in 1813--Brazil in 1830--Denmark in\n1804. The United States prohibited it by Constitution in 1809--and in\n1814 engaged by the treaty of Ghent to do all in her power for its\nentire suppression.\n\nBut, notwithstanding these praiseworthy efforts, the trade continued,\nand with increased barbarity, and is even yet carried on to some extent\nin defiance of all the navies of the world.\n\nWe have now seen that _avarice_ was at the bottom of the slave trade;\nthat it was an unprovoked and unparalleled outrage upon the Africans;\nthat it was prosecuted without the slightest regard to the comfort\nor lives of the captured; that the whole civilized world, after an\nexperience of centuries, became horrified at its terrible iniquity;\nthat now the trade is declared to be PIRACY; that the slave-ship can\nbe protected by no flag under heaven; and that all who engage in the\ntrade may be captured and hanged up by the neck as the most execrable\nwretches.\n\nThus a traffic which received the sanction of the Pope of Rome, and\nwas prosecuted under the immediate auspices of Christian kings and\ngovernments for three centuries, was attacked by CLARKSON, WILBERFORCE\nand other agitators, and, though powerfully defended by avarice and\ninterest; though hoary with age; though protected by statesmen, by\nthe commercial and planting interests, that attack was vigorously\nfollowed up until reason, religion and humanity felt outraged by it,\nand demanded in a voice which rulers dared not refuse to hear, that it\nbe at once and forever abolished. So much for agitation! Thank God for\nthis progress!\n\n\n\n\nCHAPTER II.\n\nSlavery Defined.\n\nPROPERTY IN A HUMAN BEING.\n\n\nThat we may proceed intelligently in the discussion of the subject\nupon which we have entered, it is important to understand precisely\nwhat American slavery is. Some learned men have confused this subject\nby confounding the relation of the slave with other relations from\nwhich it _essentially_ differs. An apprentice, a miner, hired laborer,\nserf or a villein is not a slave. All these relations lack, as we\nshall see, the distinguishing feature of slavery. The slave is placed\nin a condition far removed from any other class of human beings in\nenlightened, civilized, or savage society. He stands in a legal\nrelation below all others.\n\nThe American slave code describes the slave and slavery with remarkable\nprecision and horrible distinctness. According to that code a slave is\na CHATTEL. He is, body, soul and spirit, to all intents and purposes\nwhatsoever, PROPERTY--the property of the master to whom he belongs;\nand slavery is that \"peculiar institution\" which, originating in\npiracy, systematically despoils human beings of their manhood--of all\ninborn rights, degrades them to the state of chattelhood, and forcibly\ndetains them in that degradation. PROPERTY IN A HUMAN CREATURE is the\n_essential_ and _peculiar_ principle of slavery. This is the basis of\nthe system, and all laws, regulations, usages, deprivations, wrongs,\nsins, sufferings and miseries which belong to the system are built\nupon this foundation. Numerous and cruel systems of oppression have\nexisted but not one of them has ventured to lay sacrilegious hands\nupon \"the image of God,\" and convert it into a _thing_ to be bought,\nsold, executed for debt, willed, and used as an article of merchandise.\nSlavery alone has done this. Some authorities will now be cited to\nprove the correctness of this definition.\n\n\"The cardinal principle of slavery, that the slave is not to be ranked\namong sentient beings but among things, obtains in all these (slave)\nstates.\" (_Judge Stroud._)\n\n\"Slaves shall be claimed, held, taken, reputed, and adjudged in law, to\nbe chattels personal in the hands of their owners and possessors, and\ntheir executors, administrators and assigns to all intents and purposes\nwhatsoever.\" (_Law of South Carolina._)\n\n\"A slave is one who is in the power of the master to whom he belongs;\nthe master may sell him, dispose of his person, his industry, and his\nlabor; he can do nothing, possess nothing nor acquire anything but what\nmust belong to his master.\" (_Law of Louisiana._)\n\n\"A slave is in absolute bondage; he has no civil rights, and can hold\nno property, except at the will and pleasure of his master; a slave\nis a rational being, endowed with understanding like the rest of\nmankind; and whatever he lawfully acquires, and gains possession of by\nfinding or otherwise, is the acquirement and possession of his master.\"\n(_Wheeler._)\n\nA law of Mississippi reads thus: \"When any sheriff or other officer\nshall serve an attachment upon slaves, horses, or _other live stock_,\"\netc. \"Being property, slaves may be bought and sold by persons capable\nof buying and selling other property.\" (_Hon. J. K. Paulding._)\n\nHenry Clay said--\"I know that there is a visionary dogma which holds\nthat <DW64> slaves cannot be the subject of property. I shall not\ndwell on the speculative abstraction. _That is property which the\nlaw declares to be property._ Two hundred years of legislation have\nsanctified and sanctioned <DW64> slaves as property.\"\n\nAny one who will take up a southern newspaper will soon discover\nfrom the manner in which slaves are advertised for _sale_, that the\nlaws which reduce them to chattels are not _dead_ statutes. An\nadvertisement in the Richmond (Va.) Whig, is headed thus:\n\n\"Large sale of <DW64>s, horses mules and cattle.\" Among the articles\nto be sold are, 175 <DW64>s, among whom are some carpenters and\nblacksmiths, 10 horses, 33 mules, 100 head of cattle, 100 sheep and 200\nhogs. \"The <DW64>s will be sold for cash, the _other property_ on a\ncredit of nine months.\"[3]\n\nWhole volumes of such advertisements might be collected from the most\nrespectable and widely circulated southern journals, and I have seen a\nfew advertisements for the sale of men women and children, hogs, corn\nand cattle promiscuously, in respectable religious papers, sustained by\nchurches whose leading avowed object is, to \"spread scriptural holiness\nover these lands.\"\n\nAnd slaves are not only advertised but actually _sold as property\nis sold_. Raising slaves for the market, selling them, speculating\nupon them and driving them from one State to another, creates an\nextensive and lucrative trade. The _Virginia Times_ estimated that\nin 1836 the number of slaves exported from Virginia alone was _forty\nthousand_--worth $24,000,000. The _Natchez Courier_ estimated that\nin 1836 two hundred and fifty thousand slaves had been imported into\nLouisiana, Mississippi, Alabama and Arkansas, from the more Northern\nStates. The _Baltimore Register_ said, \"Dealing in slaves has become\na large business; establishments are made in several places in Md.\nand Va. at which they are sold like cattle.\" Prof. Dew said in 1831;\n\"Virginia is in fact a <DW64> raising State for the other states.\" _Judge\nUpshur_ of Va. said in the Va. Convention, 1831; \"The value of slaves\nas an article of property, depends much on the state of the market\nabroad. If it should be our lot, as I trust it will be, to acquire the\ncountry of Texas, their price will rise again.\" \"From the single port\nof _Baltimore_,\" says Mrs. Stowe \"in the last two years, a thousand and\nthirty slaves have been shipped to the southern market.\" Slaves now\nbring a very high price in cash. Only the other day a _brick-layer_ in\nS. C. sold for $1,905; three others at the same sale brought over $1000\neach.\n\nIn the prosecution of this traffic the feelings and interests, the\nparental, connubial and filial relations of slaves are utterly\ndisregarded. They are sold for the benefit of the _master_, as a\nhorse is sold, and bought _to suit the purchaser_. To all intents and\npurposes slaves are daily bought and sold like cattle. Alas, that my\npen is compelled to write this fact.\n\nA respectable gentleman (Dr. Elwood) was an eye witness to a sale of\nslaves in Petersburg, Va., in 1846. He saw some old men and women go\nupon the auctioneer's stand to be sold to the highest bidder. The case\nof a beautiful youth affected him most deeply. \"His hair,\" said Mr. E.\n\"was brown and straight, his skin exactly the hue of white persons,\nand no discernible trace of <DW64> features in his countenance. Some\nvulgar jests were passed on his color, and $200 was bid for him;\nbut the audience remarked that was not enough to begin on for such a\nlikely young <DW64>; some said a white <DW64> was more trouble than he\nwas worth. Before he was sold his mother rushed from the house upon the\nportico, crying in frantic grief, 'My son, O! my boy, they will take\naway my dear'--Here her voice was lost as she was rudely pushed back\nand the door closed. The sale was not for a moment interrupted, and\nnone of the crowd appeared to be affected by the scene. The poor boy\ntrembled and wiped the tears from his cheeks with his sleeves. He was\nsold for about 250 dollars.\"\n\nAfter this boy was sold a woman was called upon the stand. She had an\ninfant in her arms, but she dared not take it with her. \"She gave it\none wild embrace, before leaving it with an old woman, and hastened\nmechanically to obey the call; but stopped, threw up her arms, screamed\nand was unable to move!\" Those who know a mother's love can understand\nthe agony which raged in her maternal bosom.\n\nThe following is from the pen of an aged preacher, now living in\nCanada, who escaped from slavery some years since. When the master to\nwhom he belonged died, he, with his fellow slaves, were put up for\nsale. Said he--\n\n\"My brothers and sisters were bid off one by one, while my mother,\nholding my hand, looked on in an agony of grief, the cause of which I\nbut ill understood at first, but which dawned on my mind with dreadful\nclearness as the sale proceeded. My mother was then separated from me,\nand put up in her turn. She was bought by a man named Isaac R----,\nin Montgomery county, Md., and then I was offered to the assembled\npurchasers. My mother half distracted with the parting forever from all\nher children pushed through the crowd, while the bidding for me was\ngoing on, to the spot where R. was standing. She fell at his feet and\nclung to his knees, entreating him in tones that a mother only could\ncommand, to buy her _baby_ as well as herself, and spare to her _one_\nof her little ones at least.\" But this man thus appealed to \"disengaged\nhimself from her with such violent kicks and blows as to reduce her to\nthe necessity of creeping out of his reach and mingling the groan of\nbodily suffering with the sob of a broken heart.\"\n\nThese cases are presented as examples to show the meaning and intent of\nthe code which declares that a slave is _property_--and _has no rights\nor interests_; and they are not rare and extreme cases brought in here\nonly for effect, but are such as occur daily in all the slave states;\nand they are _perfectly in keeping_ with the spirit of American\nslavery. Those persons were sold precisely as _other property_ is sold.\n\nFrom these authorities and facts it is clear that a slave occupies a\nrelation as far beneath the apprentice, miner, hired laborer, or even\nthe villein of the Feudal Age, or the Russian serf, as mere _property_\nis beneath _manhood_ with all its possessions and God-like powers--as\nfar as a brute is below a man \"made in the image of God.\"\n\nThe American slave code is almost an exact copy of the old savage Roman\nslave code, which was conceived in the dark night of heathenism, and\nbrought forth reeking with blood in the unholy travail of sanguinary\nwars, before that empire had been enlightened and conquered by the\npeaceful and just Gospel of Christ. That it may be seen _where_ English\nand American law-makers obtained the spirit of the American slave code,\nthe following synopsis of the Roman law on slavery is inserted.\n\n\"By the Roman civil law, slaves were esteemed merely as chattels of\ntheir masters; they had no name but what the master was pleased to\ngive them for convenience. They were not capable of personal injuries\ncognizable by the law. They could take neither by purchase nor descent,\ncould have no heirs, could make no will. The fruits of their labor\nand industry belonged to their masters. They could not plead nor be\nimpleaded, and were utterly excluded from all civil concerns. They\nwere incapable of marriage, not being entitled to the considerations\nthereof. The laws of adultery did not (among themselves) effect them.\nThey might be sold, transferred, mortgaged, pawned. _Partus sequitur\nventrem_ was the rule indiscriminately applied to slaves and cattle.\"\n(_Harris and McHenry._)[4]\n\nAt a glance it will be seen that the Roman and American slave codes\nare identical in spirit--that the distinguishing principle of both is\nproperty in man. Our christian legislators therefore must acknowledge\nthemselves indebted to Pagan Rome for the type of slavery which they\nhave instituted and maintained in Christian America. All the main\nfeatures of cruelty, injustice and savageness, inherent in that ancient\nsystem of oppression, have been faithfully copied, and not in the\nslightest degree modified or softened.\n\nLet us recapitulate. A slave is property. His bones and sinews, genius,\nskill, virtue, mind, soul; all he is, all he may be, all he acquires\nin this life, belongs to his master and is put down in his ledger as\nworth so many dollars. He is without choice as to what he will do, what\namount of labor he will perform, or for whom he shall toil. He can own\nnothing, inherit nothing, will nothing. He cannot make a contract for\nhimself, nor claim the protection of the laws as a man. He is wholly\nin the power of his master and totally defenseless against his lusts,\navarice, or brutality. I defy human ingenuity, nay, if I may be so\nbold, I challenge Lucifer himself to invent a system of oppression\nwhich leaves a man more completely destitute, defenseless and degraded.\n\n\n\n\nCHAPTER III.\n\nSlavery Illustrated.\n\nTHE CHATTEL PRINCIPLE IN PRACTICE.\n\n\nWe will now enter more definitely into an examination of that terrible\ninstitution which practically justifies the African slave-trade by\nholding on to its victims and substituting in its stead an inter-state\nslave-trade in moral turpitude fully equaling it; which, in a land of\nfree institutions, holds in galling chains more than three millions of\nour dear fellow creatures; annually robs a hundred thousand American\nmothers of their babes; and despoils one hundred thousand children\nevery year of that precious freedom which is their birthright and\nreduces them to a level with unreasoning beasts. Our task will be\npainful, but let us proceed.\n\n1. _Slaves are denied an education._ I think it is universally admitted\nthat education and slavery are utterly incompatible, and that total\nignorance of letters and general imbecility of intellect are essential\nto its successful continuance, and indeed, its very existence in any\ncountry. Hence in the United States, where millions of dollars are\nannually expended for schools and colleges, and where it is almost\nuniversally believed that a sound education is conducive to good\nmorals, the spread of civilization, the preservation of liberty and the\nprogress of Christianity, even here nothing is done for the education\nof slaves. While millions of free children are annually gathered into\nschools and diligently instructed, the children of slaves, although\nequally capable, are permitted to grow up without the least attention\nto their mental culture. But this, though bad enough, is not the\nworst. If slaves were at liberty to follow out their own inclinations,\nthey might many of them, even without encouragement or help, acquire\na respectable education. But the laws _punish the slave with great\nseverity who, with any motive or under any circumstances_, may attempt\nto learn to read or write, and also any person who may teach him.\n\nSome of the laws and opinions relating to the education of slaves,\n(free <DW64>s generally included) will now be cited. \"Virginia Revised\nCode of 1819. That all meetings or assemblages of slaves or free\n<DW64>s, or free <DW64>s and mulattoes mixing and associating with\nsuch slaves at any meeting house or houses &c., in the night; or at\nany _school_ or _schools_ for teaching them reading or writing either\n_in the day or night_, under _whatsoever pretext_, shall be deemed and\nconsidered an UNLAWFUL ASSEMBLY; and any justice of a county wherein\nsuch assemblage shall be, shall issue his warrant, directed to any\nsworn officer or officers, authorizing him or them to enter the house\nor houses where such unlawful assemblages may be, for the purpose\nof apprehending or dispersing such slaves, and to inflict _corporal\npunishment_ on the offender or offenders, at the discretion of any\njustice of the peace not exceeding twenty lashes.\" (_Goodell's American\nSlave Code._)\n\nNo person in Virginia is allowed to open a school for the instruction\nof <DW52> persons or to teach them to read and write under a penalty\nof $100 and six months imprisonment. It may be thought that these\nlaws are not now enforced and stand as a dead letter upon the statute\nbook. But the following cool item of news published in the _Richmond\nExaminer_ under date of May 12th, 1853, will satisfy any one that they\nare enforced.\n\n\"BREAKING UP A <DW64> SCHOOL.--The officers at Norfolk made a descent\non Tuesday upon a <DW64> school, kept in the neighborhood of the\nStone Bridge, by a Mrs. Douglass and her daughter, and the teachers,\ntogether with their sable pupils, were taken before his Honor. They\nacknowledged their guilt, but pleaded ignorance of the law, and were\ndischarged, on a promise to do so no more; a very convenient way of\ngetting out of the scrape. The law of this State imposes a fine of one\nhundred dollars, and imprisonment for six months, for such offenses; is\npositive, and allows no discretion in the committing magistrate.\"\n\nIf a free <DW64> in North Carolina attempt to teach a slave to read, or\nif he give to a slave a religious tract, a spelling book or the bible,\nhe may be imprisoned or take _thirty-nine lashes_! If a white person\nattempt to teach a slave the laws subject him to a fine of $200 for\neach offense.\n\n\"In Georgia, if a white teach a free <DW64> or a slave to write he is\nfined $500, and imprisoned at the discretion of the Court; if the\noffender be a <DW52> man, bond or free, he may be fined or whipped at\nthe discretion of the Court. This law was enacted in 1829.\" (_Jay's\nInquiry._)\n\n\"In Louisiana the penalty for teaching slaves to read and write is one\nyear's imprisonment.\"\n\n\"In North Carolina, the patrols were ordered to search every <DW64>\nhouse for books or prints of every kind. Bibles and hymn books were\nparticularly mentioned.\" (_Goodell._)\n\n\"We have,\" said Mr. Berry in the Va. House of Delegates, \"_as far as\npossible closed every avenue by which light may enter their minds. If\nwe could extinguish the capacity_ to see the light, our work would be\ncompleted; they would then be on a level with the beasts of the field\nand we would be safe! I am not certain that we would not do it, if we\ncould find out the process, _and that on the plea of necessity_.\"\n\nWhen Frederick Douglass was a slave and belonged to Mr. Auld, his\nmistress, who had been lately married, manifested toward him true\nwomanly kindness and commenced to teach him the art of reading. \"But\nwhen my master heard of it,\" says Douglass in his Narrative, \"he at\nonce forbade Mrs. Auld to instruct me further, telling her among other\nthings, that it was unlawful, as well as unsafe to teach a slave to\nread. To use his own words further he said, 'If you give a <DW65> an\ninch he will take an ell. A <DW65> should know nothing but to obey\nhis master--to do as he is told to do. Learning would spoil the best\n<DW64> in the world. Now, said he, if you teach that <DW65> (speaking of\nmyself,) how to read there would be no keeping him. It would forever\nunfit him to be a slave. He would at once become unmanageable, and of\nno value to his master. As to himself it would do him no good, but a\ngreat deal of harm. It would make him discontented and unhappy.'\"\n\nIs not that a terrible institution which can only be sustained by\nenchaining the immortal mind and withholding entirely the advantages\nof education? Think of it. A slave's soul, as is often the case, is\npossessed with an unquenchable passion for improvement. He has a mind\nin constant unrest--active, elastic, aspiring. A benevolent friend\nengages to instruct him at night in the rudiments of learning, but\nwhile engaged in this good work the law seizes them, and hurries the\nslave to the whipping-post and the friend to prison. Twenty, thirty\nor forty lashes on the bare back are rather poor encouragement to the\nstudent, and a heavy fine and long imprisonment with felons hard pay\nfor a teacher. But slavery makes it a crime to learn to read even the\nbible, and a penitentiary offense to teach a slave the alphabet!\n\nThe object of this is plainly declared by Mr. Berry of Va., viz: to\nclose every avenue of light from the slave's mind--to debase him as\nlow as possible--and thus put resistance out of his power--that he may\nbecome a docile and profitable _chattel_.\n\nThese laws are a bold defiance of the Almighty who constructed the\nmarvelous powers of the human mind for improvement and activity and who\nrevealed in written language his word for the comfort and guidance of\nall his creatures. They interpose a barrier between the slave and his\nMaker and thus hinder his salvation. Even convicts in prison are taught\nto read the scriptures, and in this respect slavery is more severe with\nits victims than justice is with the worst criminals.\n\n2. _Slavery does not recognize the matrimonial connections of slaves._\nAs slaves are to be put as nearly as possible upon a level with\n\"_other_ property\" the slave code with singular meanness, but perfect\nconsistency, refuses to the slave a lawful marriage, subjects him\nto conditions which are inconsistent with that sacred relation, and\nexposes slave wives to the unbridled lust of masters and overseers!\n\n\"With the consent of their masters slaves may marry * * * but whilst in\na state of slavery it cannot produce any _civil effect_, because slaves\nare _deprived of all civil rights_.\" (_Judge Mathews._)\n\n\"A slave cannot even contract matrimony, the association which takes\nplace among slaves and is _called_ marriage, being properly designated\nby the word _contubernium_, a relation which has no sanctity, and to\nwhich no civil rights are attached.\" (_Judge Stroud._)\n\n\"A slave has never maintained an action against the violator of his\nbed.\" (_Daniel Dulany, Att'y Gen. Md._)\n\n\"Slaves were not entitled to the conditions of matrimony, and\ntherefore they had no relief in cases of adultery.\" (_Dr. Taylor._)\n\n\"Marriage is a civil ordinance they cannot enjoy. Our laws do not\nrecognize this relation as existing among them, and of course, do not\nenforce by any sanction, the observance of its duties. Indeed, until\nslavery waxeth old and tendeth to decay, there cannot be any legal\nrecognition of the marriage rite, or the enforcement of its consequent\nduties. For all the regulations on this subject _would limit the\nmaster's absolute right of property_ in the slaves. In his disposal\nof them he could no longer be at liberty to consult merely his own\ninterests. He could no longer separate the wife and the husband to suit\nthe convenience or interest of the purchaser.\" (_Address of the Synod\nof Ky._)\n\nThe laws intend to make slaves absolute property, and hence no relation\nis legalized which would detract from the value of that property. The\ninterest of the _owner_ alone is consulted. These laws, horrible as\nthey appear, are entirely consistent with _chattel slavery_. And the\ngeneral _practice_ upon these laws comes up fully to their spirit.\nWhenever the convenience, interest or passion of a master requires it,\nslaves are sold and scattered abroad without the slightest regard to\nthose dear and sacred connections, which they regard, and which God,\nno doubt, regards as marriage. In newspaper advertisements for runaway\nslaves it is frequently stated that the fugitive property was bought at\na certain place \"where he has a wife,\" and the probability is that he\nis \"lurking about that place.\" An advertisement in a New Orleans paper,\nafter describing the slave Charles, as \"six feet high,\" \"copper color,\"\nrather \"pleasing appearance,\" adds, in order that the pursuers may have\nsome clue to his whereabouts, \"it is more than probable that he will\nmake his way to Tennessee, as he has a wife now living there.\"\n\nAnother advertises the runaway \"Ned,\" of \"copper color, full forehead.\"\n\"Ned,\" continues the notice, \"was purchased in Richmond of Mr. Goodin,\nand has a wife in that vicinity.\"\n\nAnother describes a runaway woman, and suggests that she may be lurking\nabout \"in the country where her husband is owned.\"\n\nThese are very natural suggestions. A husband, though a slave, and\nbound to his wife by no legal tie, is not unfrequently to the slave\nwife all that _husband_ means, and if that wife escape from her\nunfeeling oppressors, who have carried her away to a distant State,\nit is quite natural that she should bend her steps toward the partner\nof her bosom, and subject herself to incredible hardships and dangers\nthat she might see his face once more, and unburden to him her\nsorrow-ladened heart.\n\nAnd that wife, though a slave, unprotected by the laws, driven by the\nshameful lash, insulted, disgraced and neglected, is a _wife_ still.\nAnd when \"Ned,\" as he is called, runs away, it is quite natural that\nhe should, impelled by a husband's love, seek out the hut where years\nbefore he had been suddenly separated from her. These advertisements\nfor husbands who are supposed to be \"lurking about\" in search of their\nwives, and of wives hunting for their husbands, tell a sad tale. What\nhusband or wife can read them without deep sorrow?\n\nThe following statement from the pen of an eye witness will illustrate\nscenes which are being enacted continually in the prosecution of the\ninter-state slave trade.\n\n\"As I went on board the steamboat I noticed eight <DW52> men,\nhandcuffed and chained together in pairs, four women and eight or ten\nchildren of the apparent ages of from four to ten years, all standing\ntogether in the bow of the boat, in charge of a man standing near them.\n* * * Coming near them, I perceived they were all greatly agitated;\nand, on inquiry I found they were all slaves, who had been born and\nraised in North Carolina and had just been sold to a speculator, who\nwas now taking them to the Charleston market. Upon the shore there\nwas a number of <DW52> persons, women and children, awaiting the\ndeparture of the boat; and my attention was particularly attracted\nby two  females of uncommonly respectable appearance, neatly\nattired, who stood together, a little distance from the crowd, and upon\nwhose countenance was depicted the keenest sorrow. As the last bell was\ntolling, I saw the tears gushing from their eyes, and they raised their\nneat cotton aprons and wiped their faces under the cutting anguish of\nsevered affection. They were the wives of two of the men in chains.\nThere, too, were mothers and sisters, weeping at the departure of their\nsons and brothers; and there, too, were fathers, taking the last look\nof their wives and children. My whole attention was directed to those\non shore, as they seemed to stand in solemn and submissive silence,\noccasionally giving utterance to the intensity of their feelings by a\nsigh or a stifled groan. As the boat was loosed from her moorings, they\ncast a distressed, lingering look to those on board, and turned away\nin silence. My eye now turned to those in the boat; and although I had\ntried to control my feelings amidst my sympathies for those on shore,\nI could conceal them no longer, and I found myself literally 'weeping\nwith those that wept.' I stood near them, and when one of the husbands\nsaw his wife upon the shore wave her hand for the last time, in token\nof her affection, his manly efforts to restrain his feelings gave\nway, and fixing his watery eyes upon her, he exclaimed, 'This is the\nmost distressing thing of all! My dear wife and children, farewell!'\nThe husband of the other wife stood weeping in silence, and with his\nmanacled hands raised to his face, as he looked upon her for the last\ntime. Of the poor women on board; three of them had husbands whom they\nleft behind. One of them had three children, another had two, and the\nthird had none. These husbands and fathers were among the throng upon\nthe shore, witnessing the departure of their wives and children, and\nas they took their leave of them, they were sitting together upon\nthe floor of the boat sobbing in silence, but giving utterance to no\ncomplaint. But the distressing scene was not yet ended. Sailing down\nthe Cape Fear river twenty-five miles, we touched at the little village\nof Smithport, on the south side of the river. It was at this place that\none of these slaves lived, and here was his wife and five children;\nand while at work on Monday last, his purchaser took him away from\nhis family, carried him in chains to Wilmington, where he had since\nremained in jail. As we approached the wharf, a flood of tears gushed\nfrom his eyes, and anguish seemed to have pierced his heart. The boat\nstopped but a moment, and as she left, he bid farewell to some of his\nacquaintances whom he saw upon the shore, exclaiming, 'Boys, I wish\nyou well; tell Molly (meaning his wife) and the children I wish them\nwell, and hope God will bless them.' At that moment he espied his\nwife on the stoop of a house some rods from the shore, and with one\nhand which was not in the handcuffs, he pulled off his old hat, and\nwaving it toward her, exclaimed, 'Farewell!' As he saw by the waving\nof her apron that she recognized him, he leaned back upon the railing,\nand with a faltering voice repeated, 'Farewell, forever.' After a\nmoment's silence, conflicting passions seemed to tear open his heart,\nand he exclaimed, 'What have I done that I should suffer this doom?\nOh, my wife and children, I want to live no longer!' and then the big\ntear rolled down his cheek, which he wiped away with the palm of his\nunchained hand, looked once more upon the mother of his five children,\nand the turning of the boat hid her face from him forever.\"\n\n\nANOTHER EXAMPLE.\n\n\"I shall never forget the scene which took place in the city of St.\nLouis while I was yet in slavery. A man and his wife, both slaves,\nwere brought from the country to the city for sale. They were taken to\nthe rooms of Austin & Savage, auctioneers. Several slave speculators,\nwho are always to be found at auctions where slaves are to be sold,\nwere present. The man was first put up and sold to the highest bidder.\nThe wife was next ordered to ascend the platform. I was present. She\nslowly obeyed the order. The auctioneer commenced, and soon several\nhundred dollars were bid. My eyes were intensely fixed on the face of\nthe woman, whose cheeks were wet with tears. But a conversation between\nthe slave and his new master soon arrested my attention. I drew near\nthem to listen. The slave was begging his new master to purchase his\nwife. Said he, 'Master, if you will only buy Fanny I know you will get\nthe worth of your money. She is a good cook, a good washer, and her\nmistress liked her very much. If you will only buy her how happy I will\nbe!' The new master replied that he did not want her, _but, if she\nsold cheap_, he would purchase her. I watched the countenance of the\nman while the different persons were bidding on his wife. When his new\nmaster bid you could see the smile on his countenance, and the tears\nstop, but as another would, you could see the countenance change, and\nthe tears start afresh. * * But this suspense did not last long. The\nwife was struck off to the highest bidder, who proved to be not the\nowner of her husband. As soon as they became aware that they were to\nbe separated, they both burst into tears; and as she descended from\nthe auction stand, the husband walking up to her and taking her by the\nhand, said, 'Well, Fanny we are to part forever on earth. You have been\na good wife to me. I did all I could to get my new master to buy you\nbut he did not want you. I hope you will try to meet me in heaven. I\nshall try to meet you there.' The wife made no reply but her sobs and\ncries told too well her own feelings.\" (_Narrative of William Brown._)\n\n\n\n\nCHAPTER IV.\n\nSlavery Illustrated--Continued.\n\nTHE CHATTEL PRINCIPLE IN PRACTICE.\n\n\n3. _Slavery disregards the parental and filial relations._ The family\nis a type of heaven. It is the foundation of the social system--of\nsocial order, refinement and happiness. Destroy this relation and the\nmost enlightened people will speedily relapse into barbarism. It is\na God-instituted relation, and around it Jesus Christ has thrown the\nsolemn sanction of his authority. Nature implants in the hearts of\nparents an affection for their offspring which is sweeter than life\nand stronger than death; and this affection, when associated with\nintelligence and religion, eminently fits them to care for helpless\ninfancy, to guide the feet of inexperienced youth, and to lead the\nopening heart and expanding mind to virtue and to God. Without the\nsoothing, ennobling and virtue-inspiring influences which emanate from\nthe domestic hearth, this world, I fear, would become a pandemonium.\n\nBut slavery, true to its leading principle, utterly disregards and\nruthlessly tramples upon the parental and filial relations. As soon as\na child is born of a slave-mother it is put down on the table of stock\nand is henceforth subject to the conditions of property. The father\ncannot say--\"This is my son. I will train him up in the fear of God,\nbestow upon him a liberal education and by help divine make a true man\nof him, that he may be my staff in old age.\" No, the slaveholder has\na usurped claim upon the boy, which, in the code of the \"lower law,\"\nannihilates entirely the father's claim. The mother is not permitted\nto press the new born babe to her throbbing bosom and rejoice over it,\nsaying--\"This is my daughter--I will by the assistance of grace give\nher tender mind a pious inclination, encourage her to walk in the path\nof virtue and religion, to seek the 'good part,' chosen by Mary of old,\nthat she may become an ornament of her sex.\" No, that female child is\na valuable part of the planter's stock, and the mother is encouraged\nto nurse it well that it may bring a high price in the market!\nParents have no more to say as to the disposition of their children\nthan animals have as to what shall be done with their young. There\nis not a law in any State, if we may except Louisiana, which imposes\nthe slightest restraint upon masters who may be disposed to sell the\nchildren of slaves. In Louisiana an old law prohibits the separation\nof slave children from their mothers before they are ten years of age.\nBut this law, were it not a dead letter as we are assured it is, would\nafford but a trifling mitigation of the wrong. At any time the master\nmay gather up all the saleable children on his plantation, submit\nthem to the inspection of a trader, strike a bargain for the lot, and\nthen start them off like a drove of young cattle, without saying one\nword about it to the fathers or mothers of those children. And it\noften occurs that when the slave mother returns from the field, weary\nwith the toils of the day, she finds her hut desolate. Where are my\nchildren? she asks. She calls--no answer--and is presently informed by\na fellow-slave that they are sold and gone! Yes--a christian (?) master\nhas taken advantage of her absence and sent them off without giving her\na parting word with them! They shall never more return! And yet this\ndistressed mother has no redress.\n\nMaternal love flows in a slave-mother's bosom with all its wonted\ndepth and intensity, and the total disregard of this affection is the\noccasion of the deepest sorrows recorded in the annals of slavery.\n\n\"In slaveholding States, except in Louisiana no law exists to prevent\nthe violent separation of parents from their children.\" (_Stroud._) A\nslave has no more legal authority over his child than a cow has over\nher calf. (_Jay._) John Davis, a dealer in slaves at Hamburg, S. C.,\nadvertises that he has on hands, direct from Va., \"one hundred and\ntwenty likely young <DW64>s of both sexes; among them small girls,\nsuitable for nurses, and several small boys without their mothers.\"\n\nFrederick Douglass relates that \"when he was three years old his mother\nwas sent to work on a plantation eight or ten miles distant, and after\nthat he never saw her except in the night. After her days toil she\nwould occasionally walk over to her child, lie down with him in her\narms, hush him to sleep in her bosom, then rise up and walk back again\nto be ready for her field work by daylight.\"--_Key to Uncle Tom's\nCabin._\n\nThe following incident occurred within the present year (1853.) We copy\nfrom the _Cleveland True Democrat_.\n\n\"It will be remembered by some of our citizens that about two or\nthree months since, a <DW52> man visited our city for the purpose of\nobtaining money enough to buy his child that was held as a slave in\nKentucky. Through the generosity of J. H. Smith and his congregation,\nwith some added by private individuals, the amount was raised, and the\nhappy <DW64> went on his way rejoicing. Now comes the saddest part of\nthe tale. When the poor <DW52> man arrived at his home, he immediately\nhanded the money, to obtain which had cost him so much labor, over\nto a friend, who started immediately to Kentucky. Arriving there, the\nmoney was laid before the master by the gentleman, when to the utter\nastonishment of the latter, the slaveholder burst into a fiendish\nlaugh, and said 'he'd be ---- if he would sell the boy at any price.'\nHe refused all terms, laughed at all exhortations, and finally ordered\nthe gentleman who wished to purchase the boy out of the house. He left\nsorrowfully, knowing how his bad success would affect the father, who\nwas in a delirium of joy at the idea of seeing his long lost son.\nImagine the feeling of that man when it was communicated to him that\nhis boy was lost forever. Our informant tells us that he said not a\nword, nor wept; but any one familiar with a human heart, could tell\nwhat agony that poor black man was in. He seems to have grown ten years\nolder, and it is feared, unless some change takes place, that he will\nsoon die. His life seems worse than death, and he loudly prays for the\nlatter to come.\"\n\nThe holder of that boy only did what the laws allowed him to do, and\nhis conduct was in perfect consistency with _chattel_ slavery. Men can\ndo as they like about selling the property which the law allows them.\n\nScenes of the most provoking and heart-rending character, scenes in\nwhich humanity is outraged, scenes which would bring the blood to the\ncheek of a savage, even to behold, are enacted in all the Southern\nStates from day to day, with seeming unconcern! The most bitter cries\npierce the skies and go up to heaven apparently unheard by man. \"Here\nis a man, a slave-trader, driving before him two boys with a hickory\nstick, and carrying a child under his arm. At a little distance is the\n_mother_ with chains on her wrists, stretching out her hand toward\nthe babe; but is prevented, because a strong man holds her while she\nendeavors to follow her shrieking babe and her sobbing boys. The owner\nwho sold the two boys, stands calmly, unmoved, smoking a cigar, while\nthe overseer holds the mother, and the trader whips off the boys and\ncarries with him the screaming child.\" This is precisely the way that\nother live stock is sold, and those dealers are only doing what the law\nallows. No one is surprised at them. They may be respectable citizens\nand good church members!\n\nChristian reader, pass not over these facts with a light heart. I\nbeseech you to think upon them as a man and a christian ought. You love\nhome, you esteem family relations the dearest and most sacred upon\nearth, and you would resist with all your power a tyranny which would\ninvade your own family circle and carry away your children for the\nexclusive benefit of others. For humanity's sake let your sympathies go\nout in behalf of the millions of your fellow creatures who are deprived\nof all the blessings of family and home. Have you not a heart to bleed\nfor those mothers whose children, in tender youth, are ruthlessly\ntorn away from them for no higher object than the pecuniary advantage\nof their masters? J. G. Whittier, the \"slave's poet,\" represents in\nmournful strains the Virginia slave mother's lament for her daughters,\nsold and gone to the far South.\n\n      Gone, gone--sold and gone,\n      To the rice-swamp dank and lone.\n    Where the slave-whip ceaseless swings,\n    Where the noisesome insect stings,\n    Where the fever demon strews\n    Poison with the falling dews,\n    Where the sickly sunbeams glare\n    Through the hot and misty air,--\n      Gone, gone,--sold and gone,\n      To the rice-swamp dank and lone,\n      From Virginia's hills and waters,--\n      Woe is me, my stolen daughters!\n\n      Gone, gone,--sold and gone,\n      To the rice-swamp dank and lone.\n    There no mother's eye is near them,\n    There no mother's ear can hear them;\n    Never, when the torturing lash\n    Seams their back with many a gash,\n    Shall a mother's kindness bless them,\n    Or a mother's arms caress them.\n      Gone, gone, &c.\n\n      Gone, gone,--sold and gone,\n      To the rice-swamp dank and lone.\n    O, when weary, sad, and slow,\n    From the fields at night they go,\n    Faint with toil, and racked with pain,\n    To their cheerless homes again,--\n    There no brother's voice shall greet them,\n    There no father's welcome meet them.\n      Gone, gone, &c.\n\n      Gone, gone,--sold and gone,\n      To the rice-swamp dank and lone.\n    From the tree whose shadow lay\n    On their childhood's place of play;\n    From the cool spring where they drank;\n    Rock and hill, and rivulet bank;\n    From the solemn house of prayer,\n    And the holy counsels there,--\n      Gone, gone, &c.\n\n      Gone, gone, sold and gone,\n      To the rice-swamp dank and lone;\n    Toiling through the weary day,\n    And at night the spoiler's prey.\n    O, that they had earlier died,\n    Sleeping calmly, side by side,\n    Where the tyrant's power is o'er,\n    And the fetter galls no more!\n      Gone, gone, &c.\n\n      Gone, gone,--sold and gone,\n      To the rice-swamp dank and lone.\n    By the holy love He beareth,\n    By the bruised reed He spareth,\n    O, may He to whom alone\n    All their cruel wrongs are known\n    Still their hope and refuge prove,\n    With a more than mother's love!\n      Gone, gone, &c.\n\n\n\n\nCHAPTER V.\n\nSlavery Illustrated--Continued.\n\nTHE CHATTEL PRINCIPLE IN PRACTICE.\n\n\n4. _Slavery utterly impoverishes its victims._ The earth is an\ninheritance bestowed upon man by the common Father of all; hence every\nhuman being has an indefeasible right to live upon it and to acquire a\npossession in it. This right is not simply conventional, but it belongs\nto man as _man_.\n\nNow slavery is directly opposed to this law of nature. It strips a\nslave of everything, and of the power to acquire anything. No one is\nso poor as a slave. He cannot own a coat, or a pair of shoes, a house,\nor a foot of land. No industry, economy, skill or patriotism can\nrelease him from this state of destitution, because it is a logical\nresult of the relation in which he is placed by the slave code. Being\nhimself a chattel, whatever he acquires or in any way gains possession\nof, is, as a matter of course, the acquirement and possession of his\nmaster. Hence, while living in a land of universal plenty, and toiling\nincessantly upon the fruitful earth, created and adorned for the use of\nevery man, no alms-house pauper is so wretchedly impoverished as the\nAmerican slave.\n\n\"Slaves have no legal rights in things, real or personal; but\nwhatever they may acquire, belongs in point of law to their masters.\"\n(_Stroud._) \"Slaves are incapable of inheriting or transmitting\nproperty.\" (_Civil Code._)\n\nHere is a case which will illustrate the point in hand. A slave by the\nname of Frederick enlisted and fought bravely through the American\nRevolution. In 1821 his name was found on the muster roll, and a\nwarrant was issued granting him the soldier's bounty of a thousand\nacres of land. Now whose land was that? Reason and justice would\nanswer, it belonged to the black veteran and his heirs forever. But the\nheirs of Frederick's old master understood something about slave law,\nand brought the case into court that it might be legally determined\nwho owned the bounty land. After much learned argument, Judge Catron\ndelivered the following decision:--\"Frederick, the slave of Col.\nPatton, earned this warrant by his services in the continental line.\nWHAT IS EARNED BY THE SLAVE BELONGS TO THE MASTER, by the common law,\nthe civil law, and the recognized rules of property in the slaveholding\nStates of this Union.\"\n\nThis was an extreme case, and as Pres. Blanchard observes, \"if\nShylock's bond of human flesh might have been relaxed, if ever the laws\nof slavery might have been mitigated in practice, it ought to have been\nin the case of this veteran soldier.\" But the \"pound of flesh\" was\nexacted. The law reducing slaves to utter pauperism is inexorable. Poor\nFrederick had no more claim to that land than Col. Patton's horse had.\n\n5. _Slavery authorizes the violation of the most solemn contracts._\nStrictly speaking, a slave cannot become a party to a legal contract.\nHis inability to do so arises out of his relation to society, and the\nevil genius which presides at all times over legislation for slaves\nis very careful to permit nothing to be enacted, unless from absolute\nnecessity, that can be construed into an acknowledgment that the slave\nis a man and has rights which he is authorized to maintain. Hence a\ncontract with a slave may be violated with impunity. He may suffer\nthe most flagrant wrongs, but is barred from courts of justice and can\nobtain no relief.\n\nOn this point the following authorities are quoted.\n\n\"Chancery cannot enforce a contract between a master and his slave\nthough the slave perform his part.\" (_Wheeler._) \"One principle\nprevails in all the States * * and that is that a slave cannot make a\ncontract, _not even the contract of marriage_.\" (_ib._)\n\n\"In the case of Sawney _vs._ Carter the court refused to enforce a\npromise by a master to emancipate his slave where the conditions of the\npromise had been partly complied with. The court proceeded upon the\nprinciple that it was not competent to a court in Chancery to enforce a\ncontract between a master and slave, even though the contract should be\nfully complied with on the part of the slave.\" (_Goodell._)\n\nIn numerous instances masters and other white persons have taken\nadvantage of this unjust and malicious feature of slave law. It is\nno uncommon occurrence for a slave to contract with his master for\nfreedom. He agrees to raise, by extra labor, a specified sum of money\nwhich is to be the price of his liberty. Animated with the hope of\nobtaining that precious right for which he has long sighed, he endures\nincredible hardships, toils night after night, and, at the end of\nmany weary years, lays before his master a part or the whole of the\nprice agreed upon. Now when this is done, the master may, in perfect\naccordance with American slave law, pocket the hard-earned money and\nsell the slave to the next trader, or keep him until death in his own\nservice. If the slave repine at this treatment, he may be whipped\ninto submission. If he run away, he may be pursued with revolvers and\nblood-hounds, and we are all required by the Fugitive Slave Law to help\ncatch him and carry him back to his faithless master. A case occurred\nwithin the present year in Ky., which illustrates this odious feature\nof slave law. Here is a brief statement of the facts.\n\n\"Sam Norris, a <DW52> man, has been living in Covington about five\nyears, has married a free <DW52> woman and has had by her several\nchildren. He belongs to a Mr. J. N. Patton, of Virginia, who permitted\nhim to come to Covington, and engage in whatever services he saw\nproper, on condition that Sam would pay him out of his earnings, a\nstipulated sum per annum, we believe, about $100. The surplus, whatever\nit might be, was to belong to the slave. Sam was punctual for several\nyears. He was sober and industrious, and in his humble way, very\nprosperous. About two years ago Mr. Patton came west on a visit and\nagreed with Sam that if he would pay him the sum of $400 he would give\nhim his freedom. Sam gratefully accepted the proposal, and at once paid\ndown out of his hard earnings $135 and has since given his master some\n$40 or $50 more.\n\n\"Patton now comes forward to rescind the contract and claim his slave.\nThe case was yesterday decided by the Hon. Judge Pryor, in favor of\nPatton. In delivering his decision, his Honor stated the following\nfacts:\n\n\"1st. That the laws of Kentucky recognize but two modes of liberating\nslaves, by will and by deeds of emancipation.\n\n\"2d. That a slave cannot make a contract.\n\n\"3d. That the contract was executory, and at the time fixed for the\n<DW64>'s freedom, future and contingent.\n\n\"4th. That so long as Sam was a slave, the master was entitled to his\nservices, and the money he (Patton) had received was in law his own.\n\n\"The opinion was able and elaborate, and the authorities numerous and\ndecided. His Honor characterized the case as one of great \"hardship and\ncruelty,\" and every one in the court room seemed to sympathize deeply\nwith the poor <DW64>.\"\n\nA lady at St. Louis, Mo., related to Mrs. F. D. Gage the following\ncircumstance, which transpired in that city a short time ago.\n\n\"I had, said the lady, an old <DW52> woman washing for me a few years\nago, for four or five years--one of the most faithful, truthful,\nand pious women, I ever knew--black or white. She was once a slave,\nbelonging to Davenport. But he was a kinder man than other men, and\ngave her the privilege of buying her freedom for one thousand dollars!\nThis sum the old and faithful creature earned and paid herself. Only\nthink of it!--one thousand dollars for the privilege of buying what our\nwise statesmen call the \"inalienable right of men,\" bestowed by the\nCreator. When free she stipulated for the freedom of her son, and this,\nwith years of toil, she earned; and when he came to manhood he too was\nfree.\n\n\"Think of this, fair mothers of our land! Ye who hug to your heart the\nchildren of your love, and feel a mother's love and this for them.\nYou work to clothe, to school and make comfortable those dependent\nupon your care; but which of you can measure the toil that this poor,\nstricken mother had to bear, ere she filed away the galling chains\nfrom the limbs of her child!\n\n\"Well, when the mother and son were free, they pledged themselves to\nthe owner of another plantation, to pay another thousand for the wife\nand child of the ransomed son. The master allowed the woman to come to\nthe city, and live with her husband, and work on her own hook--paying\nhim so much per month. Three hundred dollars has been paid. Some time\nin April, this oppressed class had a public tea-party and fair, to\ngather funds to furnish their church, a neat edifice on ---- St. The\nmother, son, and wife were there, returned home, or started home, about\nmidnight--the horses ran away, and George, attempting to get off the\ncarriage to assist the driver, fell, and his head was dashed to pieces\nagainst the corner of a curb-stone.\n\n\"He died instantly, and the morning papers announced the fact, and\nspoke of him as \"highly worthy and respectable, and a member of ----\nChurch.\" But no sooner had the owner of Susan, the wife, heard of\nGeorge's death, than he hurried to the city post-haste, and took the\nafflicted wife from their house, drove her to the Slave auction, and\nsold her to southern traders.\n\n\"Thus were the three hundred dollars lost to those who earned it, the\nold, toiling mother left childless; and the young wife, but yesterday\nrejoicing in the strength and hope of freedom and love, suddenly\nturned into a chattel, and sold \"away down South,\" to be a beast of\nburden--perhaps for a Legree.\"\n\n\"When did it happen inquired Mrs. Gage?\"\n\n\"Why, here, lately. I met the old mother as I came from the \"Fourth\"\nPic nic. She was dressed in deep mourning. I had not seen her for a\nlong time, for they had got them a home, and she did not wash any more.\nI asked her what had happened, and she told me all. O! Mrs. G., how it\nmade me feel! I celebrating our liberty, she, a woman--a wife--a mother\nmourning over enslaved and doubly-wronged children.\n\n\"\"I know there is a God, Mrs. Lilly,\" the poor bowed creature said to\nme, \"I know there is a good God, and a Jesus, or I should give up in\ndespair, and sometimes I do; I look up and down and all round, and\n_there is no light_!\"\"\n\n_Slavery leaves its victims a prey to unchecked avarice._ What\nprotection has a slave against the avarice of his master? Let us see. A\nlaw of South Carolina provides that slaves shall \"not labor to exceed\n_fifteen hours_\" out of twenty four. This is called protection!\n\n\"The slave is driven to the field in the morning about four o'clock.\nThe general calculation is to get them to work by day-light. The time\nfor breakfast is between nine and ten o'clock. This meal is sometimes\neaten 'bite and work,' others allow fifteen minutes, and this is the\nonly rest the slave has while in the field.\" (_G. W. Westgate._)\n\n\"In North Carolina, the legal standard of food for a slave must not be\nless than a _quart of corn per day_. In Louisiana the legal standard is\none barrel of Indian corn--or the equivalent thereof in rice, beans or\nother grain, and a pint of salt, every month.\" \"The quantity allowed by\ncustom,\" said T. S. Clay of Georgia, \"is a peck of corn per week.\"\n\nWhen they return to their miserable huts at night, they find not there\nthe means of comfortable rest, but on the cold ground they must lie,\nwithout covering, and shiver while they slumber.\n\n\"The clothing of slaves by day, and their covering by night, are\ninadequate either for comfort or decency, in any or most of the\nslaveholding States.\" (_Elliott._)\n\nIt is notorious that slaves, on large plantations especially, are\nmiserably fed, clothed and lodged, and during busy seasons of the year,\nmost unmercifully worked.\n\n6. _Slavery abandons its victims to unbridled lust._ Against a master's\nlusts a slave has no protection. It is an established principle of\nthe slave code that the _testimony_ of a slave against a white person\ncannot be received in a court of justice. A slave woman who may be\nabused cannot resort to the law. To whom can she appeal? To God only.\nThe master may torture her in any way, so that he take not her life, in\norder to _force_ a compliance with his base designs!\n\n\"A very beautiful girl belonging to the estate of John French, a\ndeceased gambler of New Orleans, was sold a few days since, for the\nround sum of _seven thousand dollars_! An ugly old bachelor, named\nGouch, was the purchaser. The _Picayune_ says that she was remarkable\nfor her beauty and intelligence; and that there was considerable strife\nas to who should be the purchaser.\" (_Elliott._)\n\nAny one can understand why that beautiful, intelligent slave girl\nbrought SEVEN THOUSAND DOLLARS! She was bought for a sacrifice to\nlust! And the law gave her no protection. It required her to submit\nunresistingly to the will of her owner and that owner was a base\nlibertine!\n\n7. _Slavery exposes its victims to the fury of unrestrained passion._\nA master in a violent passion may fall upon his slave, and beat him\nunmercifully without the slightest provocation and _the slave has no\nredress_.\n\n\"The master is not liable for an assault and battery committed upon the\nperson of his slave.\" (_Wheeler._)\n\nA Methodist minister, Rev. J. Boucher, relates the following incident:\n\n\"While on the Alabama circuit I spent the Sabbath with an old circuit\npreacher, who was also a doctor, living near 'the horse shoe,'\ncelebrated as Gen. Jackson's battle ground. On Monday morning early,\nhe was reading Pope's Messiah to me, when his wife called him out. I\nglanced my eye out of the window, and saw a slave man standing by, and\nthey consulting over him. Presently the doctor took a rawhide from\nunder his coat, and began to cut up the half-naked back of the slave.\nI saw six or seven inches of the skin turn up perfectly white at every\nstroke, till the whole back was red with gore. The lacerated man cried\nout some at first; but at every blow the doctor cried, 'won't ye hush?\nwon't ye hush?' till the slave finally stood still and groaned. As soon\nas he had done, the doctor came in panting, almost out of breath, and,\naddressing me, said, 'Won't you go to prayer with us, sir?' I fell on\nmy knees and prayed, but what I said I knew not. When I came out the\npoor creature had crept up and knelt by the door during prayer; and his\nback was a gore of blood quite to his heels.\"\n\nNow this slave could not appeal to the law for redress or protection;\nand the same cruel beating might have been repeated every week until\ndeath had come to his relief, and the poor wretch must only bear\nit--that is all. He was wholly at the mercy of the passions of his\nmaster.\n\n8. _Slavery subjects its victims to uncontrolled and irresponsible\ntyranny._ Irresponsible power cannot be safely entrusted with the\nwisest and most humane persons. It is always liable to great abuses.\nBut when all sorts of men are invested with it, when it can be\npurchased with money, terrible beyond conception are its results.\nWoe to the unhappy man who is put absolutely into the power of a\nhard hearted villain. But slaves are property and are exposed to the\nirresponsible power of their masters.\n\nA master or overseer may, with impunity inflict upon a slave, without\nthe slightest provocation, any kind of torture, which can be endured,\nand impose upon him all kinds of sufferings, hardships and insults.\n\nHe may clothe him in rags, feed him upon corn, lodge him in a mere\npen of poles, work him beyond his ability, kick him, cuff him, knock\nhim down, put him in stocks, strip him, tie him to a stake, and with\na keen lash lay on his bare back until the blood runs in a stream to\nhis heels. The laws not only allow this to be done, but it is done\ncontinually. _Women_, yes, tender, delicate women; daughters, sisters\nand mothers are unprotected by the laws. They may be, and are tied to\nthe whipping post; every day that we live, this _is_ done, and their\nquivering flesh mangled by the cow-skin.\n\nDr. Howe visited a prison in New Orleans, in which fugitive slaves are\nconfined, and to which many slaves are brought by their masters to be\nwhipped, for which punishment a small fee is paid. In a letter to Hon.\nCharles Sumner, he says:\n\n\"Entering a large paved court-yard, around which ran galleries filled\nwith slaves of all ages, sexes and colors, I heard the snap of a\nwhip, every stroke of which sounded like the sharp crack of a pistol.\nI turned my head, and beheld a sight which absolutely chilled me to\nthe marrow of my bones, and gave me, for the first time in my life,\nthe sensation of my hair stiffening at the roots. There lay a black\ngirl flat upon her face, on a board, her two thumbs tied, and fastened\nto one end, her feet tied, and drawn tightly to the other end, while\na strap passed over the small of her back, and, fastened around the\nboard, compressed her closely to it. Below the strap she was entirely\nnaked. By her side, and six feet off, stood a huge monster with a long\nwhip, which he applied with dreadful power and wonderful precision.\nEvery stroke brought away a strip of skin, which clung to the lash, or\nfell quivering on the pavement, while the blood followed after it. The\npoor creature writhed and shrieked, and in a voice which showed alike\nher fear of death and her dreadful agony, screamed to her master, who\nstood at her head, 'O, spare my life! don't cut my soul out!' But still\nfell the horrid lash; still strip after strip peeled off from the skin;\ngash after gash was cut in her living flesh, until it became a livid\nand bloody mass of raw and quivering muscle. It was with the greatest\ndifficulty I refrained from springing upon the torturer, and arresting\nhis lash; but, alas! what could I do, but turn aside and hide my tears\nfor the sufferer, and my blushes for humanity? This was in a public\nand regularly-organized prison; the punishment was one recognized and\nauthorized by the law. But think you that the poor wretch had committed\na heinous offense, and had been convicted thereof and sentenced to the\nlash? Not at all. She was brought by her master to be whipped by the\ncommon executioner, without trial, judge or jury, just at his beck or\nnod, for some real or supposed offense, or to gratify his own whim or\nmalice. And he may bring her day after day, without cause assigned, and\ninflict any number of lashes he pleases, short of twenty-five, provided\nonly he pays the fee. Or, if he choose, he may have a private whipping\nboard on his own premises, and brutalize himself there.\"\n\nAll this is done according to law. \"We cannot allow,\" said Judge\nRuffin, \"the right of the master to be brought into discussion in the\ncourts of justice. The slave, to remain a slave, must be made sensible\nthat there is NO APPEAL FROM HIS MASTER.\" The same Judge decided--that\n\"THE POWER OF THE MASTER MUST BE ABSOLUTE IN ORDER TO RENDER THE\nSUBMISSION OF THE SLAVE PERFECT.\" How dreadful is this tyranny!\n\n\n\n\nCHAPTER VI.\n\nSlavery Illustrated--Continued.\n\nSEVERITY OF THE LAWS AGAINST SLAVES.\n\n\nAs the laws provide for the degradation of the slave to a state of the\nmost stupid ignorance, it would naturally be supposed that little would\nbe required in the way of obedience, and that when a slave did trespass\na very light punishment would be meted out to him. Evidently this would\nbe the humane and just course, for where little is given little should\nbe required. In this, however, as in most other things slavery is\nprecisely contrary to nature, humanity and reason.\n\nSlaves are punished by the laws for numerous acts which are in\nthemselves perfectly right.\n\n\"For seeking liberty a slave is proclaimed an outlaw and may be\nlawfully killed.\" (_Goodell._) \"He may be punished for attending\nreligious meetings at night. He may be publicly whipped for keeping a\ngun, or a pistol. For visiting a wife or child without a written pass,\nhe may be whipped. For striking a white person, no matter how great the\nprovocation, whipping--and for the second or third offence, DEATH.\"\n(_Goodell._) These are but specimens of the cruel and vexatious laws\nby which the slave's life is embittered. He, poor wretch, must have so\nmany lashes on the bare back for almost every thing which his manhood\nprompts him to do. He must always be on the look out to act and feel\nas a mere brute--he must crouch and bend in constant abjectness or his\nback shall pay the penalty. But for _actual crimes_ the disproportion\nbetween the punishment of slaves and white persons is very great.\n\n\"In Va., by the revised code of 1819, there are seventy-one offenses\nfor which the penalty is death when committed by slaves and imprisonment\nwhen committed by the whites.\" (_Jay's Inquiry._)\n\n\"In Mississippi there are seventeen offenses punishable with death when\ncommitted by slaves, which, if committed by white persons, are either\npunished by fines or imprisonment, or punishment not provided for by\nthe statute or at common law.\" (_Goodell._)\n\nA law of Md., provides that--\"Any slave for rambling in the night,\nor riding on horseback or running away, may be punished by whipping,\ncropping and branding in the cheeks or otherwise, not rendering him\nunfit for labor.\"\n\nAnd yet, notwithstanding the extreme and unreasonable partiality and\nseverity of these laws, it is not unusual for the barbarous spirit of\nslavery to overleap them in its unmerciful punishment of the slave.\nWhen the slave commits a high crime, not unfrequently does a furious\nmob seize him, and hang him up without trial as if he were a mean dog.\nCalmness and solemnity, which should always characterize the punishment\nof the _greatest_ criminals in christian countries, give place to the\nmost violent and cruel passions. Judgment, mercy, law, humanity, God\nand Christianity, are all forgotten in the hasty and insane desire to\nhave the wretched bondman pushed out of the world. And perhaps the\ncrime which has so violently stirred up the community against him was\ncommitted under the greatest provocations. His soul may have been\nwrithing under a crushing sense of repeated wrongs. His wife may have\nbeen abused before his eyes while he was not permitted to defend her.\nHis daughter may have been dishonored, and he, without appeal for her\nprotection to church or State, compelled to suffer it in silence. And\nhis own back may have been smarting from the maddening lash--and in a\nmoment of frenzy or despair he may have smitten his oppressor to the\nearth.\n\nAnd, for this crime he is treated as a prince of criminals, is hung up\nwithout trial, or perhaps _burned alive_!\n\nOur souls have been harrowed up by a circumstance which transpired\nduring the present year (1853) in the State of Mo. Two <DW64> men for\nthe commission of murder were arrested and tied to a tree, near the\ncounty seat of Jasper co., a fire was kindled around them, and in\nthe presence of two thousand persons, they were burned to death! No\ntime for reflection or repentance was allowed. Not a word of warning\nor exhortation was permitted. Even a humane mode of being killed was\ndenied. But they were, in this year, during the Presidency of _Pierce_,\nin the State of Missouri, _burned without trial_!\n\nIn 1842 a <DW64> was burned at Union Point, Mississippi. The _Natchez\nFree Trader_ gives the following account of the horrible work.\n\n\"The body was taken and chained to a tree immediately on the bank\nof the Mississippi, on what is called Union Point. Fagots were then\ncollected, and piled around him to which he appeared quite indifferent.\nWhen the work was completed, he was asked what he had to say. He then\nwarned all to take example by him, and asked the prayers of all around;\nhe then called for a drink of water, which was handed to him; he drank\nit, and said, 'Now set fire--I am ready to go in peace!' The torches\nwere lighted and placed in the pile, which soon ignited. He watched\nunmoved the curling flame, that grew until it began to entwine itself\naround and feed upon his body: then he sent forth cries of agony\npainful to the ear, begging some one to blow his brains out; at the\nsame time surging with almost superhuman strength, until the staple\nwith which the chain was fastened to the tree (not being well secured)\ndrew out, and he leaped from the burning pile. At that moment the sharp\nringing of several rifles was heard: the body of the <DW64> fell a\ncorpse on the ground. He was picked up by some two or three, and again\nthrown into the fire and consumed--not a vestige remaining to show that\nsuch a being ever existed.\"\n\nA <DW52> man was burned in St. Louis, Mo., in 1836, in presence of an\nimmense throng of spectators. The _Alton Telegraph_ gives the following\ndescription of the scene.\n\n\"All was silent as death while the executioners were piling wood around\ntheir victim. He said not a word, until feeling that the flames had\nseized upon him. He then uttered an awful howl, attempting to sing\nand pray, then hung his head, and suffered in silence, except in\nthe following instance: After the flames had surrounded their prey,\nhis eyes burnt out of his head, and his mouth seemingly parched to\na cinder, some one in the crowd, more compassionate than the rest,\nproposed to put an end to his misery by shooting him, when it was\nreplied, \"that would be of no use, since he was already out of pain.\"\n\"No, no,\" said the wretch, \"I am not. I am suffering as much as ever;\nshoot me, shoot me.\" \"No,\" said one of the fiends, who was standing\nabout the sacrifice they were roasting, \"he shall not be shot. _I would\nsooner slacken the fire, if that would increase his misery._\"\"[5]\n\nIt may be said that we have in these illustrations of slavery,\nexaggerated. But this can not be the case, for we have given the\nlaws and the practice together, and have furnished the testimony of\neye-witnesses. And we could bring forward a thousand witnesses from\nthe midst of slavery, whose testimony would confirm all we have said.\nYea more; they would declare that half the extent of the evils of this\nhorrible institution are unknown. Hear if you please, a voice from\nNorth Carolina--Mr. Swain:\n\n\"Let any man of spirit and feeling for a moment cast his thoughts\nover this land of slavery--think of the nakedness of some, the hungry\nyearnings of others, the flowing tears and heaving sighs of parting\nrelations, the wailings of _woe_, the bloody cut of the keen lash,\nand the _frightful scream_ that rends the very skies--and all this to\ngratify ambition, lust, pride, avarice, vanity, and other depraved\nfeelings of the human heart. THE WORST IS NOT GENERALLY KNOWN. Were all\nthe miseries, the horrors of slavery, to burst at once into view, a\npeal of seven fold thunder could scarce strike greater alarm.\"\n\nHear the venerable John Rankin, a native and long resident of\nTennessee. (_See Elliot pp. 225._)\n\n\"Many poor slaves are stripped naked, stretched and tied across\nbarrels, or large bags, _and tortured with the lash during hours,\nand even whole days, till their flesh is mangled to the very bones_.\nOthers are stripped and hung up by the arms, their feet are tied\ntogether, and the end of a heavy piece of timber is put between their\nlegs in order to stretch their bodies, and so prepare them for the\ntorturing lash--and in this situation they are often whipped till\ntheir bodies are covered _with blood and mangled flesh_--and, in order\nto add the greatest keenness to their sufferings, their wounds are\nwashed with _liquid salt_! And some of the miserable creatures are\npermitted to hang in that position till they actually _expire_; some\ndie under the lash, others linger about for a time, and at length die\nof their wounds, and many survive, and endure again similar torture.\nThese bloody scenes are _constantly exhibiting in every slaveholding\ncountry--thousands of whips are every day stained in African blood_!\nEven the poor _females_ are not permitted to escape these shocking\ncruelties.\"\n\nAnd finally listen dispassionately to the _Presbyterian Synod of\nKentucky_, composed of those whose interest it was to present slavery\nin as favorable a light as possible. (_See Elliot pp. 225._)\n\n\"This system licenses and produces _great cruelty_. Mangling,\nimprisonment, starvation, every species of torture, may be inflicted\nupon him, [the slave,] and he has no redress. There are now in our\nwhole land two millions of human beings, exposed, defenseless, to\nevery insult, and every injury short of maiming or death, which\ntheir fellow-men may choose to inflict. They suffer all that can be\ninflicted by wanton caprice, by grasping avarice, by brutal lust, by\nmalignant spite, and by insane anger. Their happiness is the sport\nof every whim, and the prey of every passion that may, occasionally\nor habitually, infest the master's bosom. If we could calculate the\namount of woe endured by ill-treated slaves, it would overwhelm every\ncompassionate heart--it would move even the obdurate to sympathy.\nThere is also a vast sum of suffering inflicted upon the slave by\nhumane masters, as a punishment for that idleness and misconduct which\nslavery naturally produces. BRUTAL STRIPES and all the varied kinds of\npersonal indignities, are not the only species of cruelty which slavery\nlicenses.\"\n\n\n\n\nCHAPTER VII.\n\nSlavery and Religion.\n\n\"CURSED BE CANAAN.\"\n\n\nMany slaveholders and their apologists have sought to find authority\nfor the \"enormity and crime\" of slavery, in the Holy Bible. And we are\nnot surprised that the vile oppressor, smarting under the lashings of\na guilty conscience, and condemned by the united voice of reason and\nhumanity, should fly for refuge from public scorn and condemnation, to\na shelter, however insecure, erected by a perversion of the writings\nand example of those remarkable men, who fill a prominent place in\nsacred history. How consoling it must be to the slaveholder, while\nstanding upon the neck of an unresisting brother, and crushing his\nhumanity into the dust with heartless cruelty, to hear from a doctor\nof divinity that Noah countenanced the enslavement of a part of his\nposterity, that Abraham was an extensive slaveholder, that Moses\nincorporated the system into the only government ever instituted by\ndirect authority from Heaven, and that it received, in its very worst\nform, under the Roman government, the tacit, if not positive sanction\nof Jesus and the apostles.\n\nMy observation sustains me in saying that no class of slaveholders\nare more pertinacious and incorrigible than the religious class--the\nscripture-quoting class. If we are to believe them, slaveholding is not\na sin PER SE, but of itself is a perfectly innocent thing. The very\nbest of men hold slaves, yea, it is, they tell us, the duty of good men\nunder some circumstances to hold slaves. To be sure THEY do not hold\nslaves for \"gain,\" but from motives of pure \"charity,\" or from stern\n\"necessity.\" _They_ and _their_ slaves are ALWAYS in such peculiar\ncases that emancipation would be impolitic, impracticable, even a sin!\nStill, from all appearances, they are as careful to keep their slaves\nfrom running off as common sinners are--their slaves are fed, clothed,\nwhipped, worked, robbed and used up precisely as are the slaves of the\nmost notorious publicans.\n\nAfter having seen how slavery originated, and what it is in theory\nand practice, it may seem useless if not impious to inquire seriously\nwhether a system so manifestly unjust, cruel and diabolical, is\nsanctioned in the Bible; but the confidence with which slaveholders and\ntheir apologists quote it in defense of slavery, and the recklessness\nwith which it is denounced by a class of infidel abolitionists, impel\nus to enter into this inquiry; and in pursuing it we shall endeavor to\nexamine carefully all the arguments relied upon by the advocates of\nhuman bondage. The first passage in order is found in Genesis 9: 25.\n\"And he said, cursed be Canaan, a servant of servants shall he be to\nhis brethren.\"\n\nIt is assumed that this curse was pronounced by divine authority;\nthat the servitude here mentioned is identical with slavery; that the\nprediction of the oppression of a people justifies their oppressors;\nand finally, that American slaves are the identical posterity of Canaan.\n\n1. As it respects the authority of this curse, there is a circumstance\nintimately associated with its utterance which excites a shadow of\ndoubt with regard to its inspiration. \"And Noah awoke from his wine\"\nand pronounced this malediction. Is it not possible that these words\nwere the hasty expression of excited feeling and not the solemn\nenunciation of a divine anathema?\n\n2. But in order to prove the validity of the argument, it must be\nproved that servitude and slavery are relations of essentially the\nsame character, and this cannot be done. Neither philology nor history\naffords the slightest proof of the assumption that to be a servant of\nservants is equivalent to being a _slave of slaves_.\n\n3. But does the prediction of the oppression of a people justify that\noppression? Verily it does not. The Lord said unto Abraham that his\nseed should be afflicted in a strange land four hundred years. But who\nwill pretend to justify the Egyptian task-masters on the plea that the\naffliction of Israel had been predicted? The divine prescience sees all\nthings at one glance, and may inspire men to prophesy, but prophecy\ntouches not the moral agency of men. When our Lord was crucified, the\n\"scripture was fulfilled,\" but they who crucified him were murderers,\nnevertheless. Hence, even should we admit that the curse pronounced\non Canaan was of divine authority, and that it meant slavery, no\nstronger apology for slaveholding could be derived therefrom than\nEgyptian oppressors might have drawn from the words of Jehovah, for\nthe affliction of Israel in Egypt four hundred years. The cases are\nparallel.\n\n4. But the argument is utterly baseless because American slaves are not\nthe posterity of Canaan, upon whom the curse was pronounced, and hence\nthat anathema affords just as good an apology for the enslavement of\nEnglishmen as  Americans. Ham had four sons,--Cush, Misriam,\nPhut, and Canaan, and the curse was directed against _Canaan_ or\nCanaan's posterity. But, says one, are not the <DW64>s children of\nCanaan? By no means. No scholar has ever pretended that Canaan was the\nprogenitor of the <DW64> race.\n\nThe sacred penman is very careful to put this matter _beyond dispute_.\nHe says: \"And Canaan begat Sidon his first born, and Heth, and the\nJubisite, and the Amorite, and the Girgasite, and the Hivite, and the\nArkite, and the Sinite, and the Arvadite, and the Zemarite, and the\nHamathite; and afterward were the families of the Canaanites spread\nabroad. And the border of the Canaanites was from Sidon, as thou comest\nto Gerar, unto Gaza; as thou goest unto Sodom and Gomorrah, and Admah\nand Zeboim, even unto Lasha.\" Gen. 10: 15-19. Now these nations and\nboundaries were all located in Asia, and we have no evidence of the\nsubsequent removal of any of the posterity of Canaan to Africa except\nit be the founders of Carthage,--a city which was long mistress of\nthe sea, and the proud rival of imperial Rome. The Carthaginians were\nsupposed to be the descendants of Canaan.\n\nThis curse, therefore, did not allude to slavery, but servitude;\nand as it is a mere prediction of what would be the relation of\nCanaan's posterity it afforded no apology for the oppression of that\nposterity;[6] and finally the Africans and  Americans _are\nnot the descendants of Canaan_, and hence, the passage can have no\napplication to them; and affords _just as good authority_ for the\nenslavement of Englishmen, Dutchmen and Frenchmen as <DW64>s.\n\nHow absurd is the attempt to take this anathema, construe it to\n_mean_ and _justify_ chattel slavery, and then _stretch_ it over the\nposterity, not of Canaan, but of Cush even after the blood of the\nCushites (Moses' wife was a Cushite) has been mingled with the blood\nof the \"first families\" of Virginia, and of all the Southern states. A\nlarge number of slaves are white--much whiter than their masters and\nmistresses. The first Bible argument for slavery appears, when weighed,\n\n    \"Light as a puff of empty air.\"\n\nHave slaveholders no better? We will see.\n\n\n\n\nCHAPTER VIII.\n\nSlavery and Religion--Continued.\n\nPATRIARCHAL SERVITUDE AND SLAVERY.\n\n\nThe next Bible argument for slavery, usually adduced, is founded upon\nthe assumption that the patriarchs were slaveholders, and particular\nstress is placed upon the example of Abraham, \"the friend of God,\" who,\nit is confidently asserted, was an extensive slaveholder.\n\nThe Harmony Presbytery, South Carolina, \"_Resolved_, that slavery has\nexisted from the days of those good old _slaveholders_ and patriarchs,\nAbraham, Isaac and Jacob.\"\n\nThe Presbytery of Tombecbee said: \"In the Bible the state of slavery is\nclearly recognized. Abraham the friend of God had slaves born in his\nhouse and bought with his money.\"\n\nDr. Fuller, in his controversy with Dr. Wayland, assumed that father\nAbraham was a slaveholder, and that his example was a sufficient\nwarrant for slaveholding in all ages. The same position was taken by\nDr. Rice in his debate with Mr. Blanchard. Mr. Fletcher, author of a\nlate voluminous defense of slavery, takes the same position.\n\nIt will be perceived that in this argument two things are assumed. 1st\nThat the patriarchs did hold slaves. 2d That the example of a patriarch\nis conclusive evidence in the case. If it should appear after an\nexamination of the case, that none of the patriarchs owned slaves, or\nthat the example of a patriarch is not conclusive evidence on all moral\nquestions, and may not, in every case, be safely followed, then this\nargument will also be found wanting.\n\nNow, I assume the position that neither Abraham, nor any other\npatriarch, ever owned a slave; and as evidence in support of this\nposition submit the following facts and considerations.\n\n1. The Bible does not record such a fact. In no chapter or verse\nis Abraham, Isaac or Jacob called a slaveholder, slave-driver,\nslave-trader, or by any other name indicative of such a relation. Nor\nis any man, or woman in their employ, either in the house or field,\nor in any way associated with them, called a slave or by any name\nindicative of that relation.\n\n2. The Bible records in connection with the history of the patriarchs,\nno circumstance from which slaveholding may be legitimately inferred.\nThose inseparable concomitants of slavery, the _whip_, _coffle_,\n_chain-gang_, _whipping-post_ and _overseer_, are not named in\npatriarchal history.\n\n3. Some circumstances are recorded from which we obtain presumptive\nevidence that they did not own slaves. Take for example, an incident in\nthe life of Abraham. He was sitting in his tent door in the cool of the\nday and saw at a little distance three strangers whom he immediately\napproached and invited, in the spirit of genuine hospitality, to tarry\nwith him and partake of some refreshments. When he had obtained their\nconsent, he hastened unto the tent to Sarah and requested her to bake\nsome cakes with all possible dispatch, while he should run to the herd\nand fetch a calf tender and good and have it dressed. The repast was\nsoon provided, the guests were seated around the wholesome meal, and\nAbraham stood by them under the tree while they ate. Now, I submit, had\nthis patriarch been a slaveholder, he would have ordered \"Cuffee\" to\nthe flock after the calf, and had Sarah been a mistress of slaves she\nwould have ordered \"Dinah\" to the kneading trough. In this incident\nthere is no mention of slaves. A \"young _man_\" is respectfully noticed\nwithout the slightest hint that he was a slave. Abraham and Sarah went\nabout preparing this entertainment precisely as good people do, who\nattend to their own work, and have no slaves to order around.\n\n4. We have good reasons for believing that chattel slavery had no\nexistence in the world at the time the patriarchs referred to,\nflourished. Abraham was born only two years after the death of Noah,\nand when as yet the postdiluvian world was in its infancy, and it\nis not probable, leaving history out of view, that slavery could\nhave been instituted at so early a period. But the most ancient and\nreliable history furnishes evidence that for a period after the flood,\nreaching down far this side the patriarchal age, universal freedom was\npreserved.[7]\n\nOn the authority of _Diodorus_, Shuckford says, that \"the nations\nplanted by Noah and his descendants, _had a law against slavery; for\nno person among them could absolutely lose his freedom and become a\nbondsman_.\" (_Shuckford's Connections, Vol. II, pp. 80._)\n\n\"Athenaus, a Greek historian of great merit, observes that the\nBabylonians, Persians, as well as the Greeks, and divers other nations,\ncelebrated annually a sort of Saturnalia, or feast, instituted most\nprobably IN COMMEMORATION OF THE ORIGINAL STATE OF FREEDOM, IN WHICH\nMEN LIVED BEFORE SERVITUDE WAS INTRODUCED; AND AS MOSES REVIVED SEVERAL\nOF NOAH'S INSTITUTIONS, so there are appointments in the law to\npreserve the freedom of the Israelites.\"\n\nFrom these authorities to which others might be added, we conclude that\nslavery had no existence among the nations which arose immediately\nafter the flood. Noah, it seems was a good democrat, and gave\nexistence to institutions which secured the personal freedom of his\ndescendants; and absolutely prohibited their enslavement. And it also\nappears that those institutions were for a long period observed, and\nfinally incorporated by Moses into the Law for the preservation of the\nliberties of the Israelites. Now, Abraham was contemporary with the\nsons of Noah, and was a governor of one of the very earliest nations\nalluded to by the historians above quoted; hence it is clear, that\nslavery had no existence in his day, and consequently he could not have\nbeen a slaveholder.\n\nAgainst this view it may be urged that slavery existed in Egypt in\nthe time of Joseph, that Joseph was sold as a slave, and that the\nIsraelites were slaves when in Egypt. To this objection we answer:\n\n1. The assumption that slavery existed in Egypt in the time of the\npatriarchs is without foundation. _Herodotus_, gives a \"true and full\"\naccount of the ancient Egyptians, specifies with great care the various\nclasses of men, but does not mention slaves. _Diodorus_, gives a\ncareful statement of the ancient Egyptian constitution, but is silent\nrespecting slavery.\n\n_Rollin_ says: \"Husbandmen, shepherds, and artificers formed the\nthree lower classes of lower life in Egypt, but were nevertheless had\nin _very great esteem_, particularly husbandmen and shepherds.\" We\nhave the best of reasons, therefore, for believing that the wholesome\ninstitutions of Noah were preserved for a long time in Egypt. That\na system of servitude existed in that country is true, but absolute\nslavery was not permitted. Parents possessed great authority over their\nchildren, and might sell them or their services, for a limited time,\nbut this was not slavery. A year of release was provided for all, so\nthat no one could, as Diodorus observes, \"absolutely lose his freedom\nand become a bondsman!\"\n\n2. Joseph was not a slave. He was doubtless sold as a servant for a\nlimited period, and evidently that period had expired before he arose\nto the high station of Steward of Potipher's house.\n\n3. The Israelites were not slaves in Egypt. They maintained their\nnationality, preserved their family relations, owned property, and\nwere not distributed throughout the country, as chattel slaves are.\nTheir servitude was national. Their task masters were appointed by the\ngovernment, and they labored for the public benefit. They were not\ndomestic slaves.\n\nThe position I think is invulnerable, that in the nations which\narose and peopled the earth, immediately after the flood, slavery\nhad no existence; and as the patriarchs flourished in that period,\nthe inference is clear that they did not own slaves, and were not\nslaveholders. Those holy men would hardly be the first to violate the\nfree institutions of Noah, and disgrace the golden age of freedom, by\nthe enslavement of their brothers.\n\nBut it is asserted with a show of confidence that the word servant,\nas applied in the scriptures to a class of persons, means precisely\nwhat our word slave means. Hence, when it is said that Abraham had\n_servants_, it is assumed that he had SLAVES. Now, although what has\nbeen proved, is altogether sufficient to exculpate that good man\nand all the patriarchs from the charge of slaveholding, we deem it\nimportant that the word translated servant be well understood; and with\nthe aid of the best authorities we shall now proceed to make it plain.\n\nThe Hebrew words translated servant, service, and servants, are derived\nfrom _abadh_, meaning to labor, _to work_, to do work. This word occurs\nin the Hebrew scriptures some hundreds of times, in various forms of\nthe word, and is never rendered _slaves_. Occasionally, our translators\nhave prefixed the word _bond_, and made it read _bond_-servant, but\nthis was done without authority, as precisely the same word is used in\nthe original. The original word is used to denote the following kinds\nof service: To work for another; Gen. 29: 20. To serve or be servants\nof a king; 2d Sam., 16: 19. To serve as a soldier; 2d Sam., 2: 12, 13,\n15, 30, 31. To serve as an ambassador; 2d Sam., 10: 2, 4. It is applied\nto a worshipper of the true God; Nehemiah, 1: 10. To a minister;\nIsaiah, 49: 6. It is also applied to king Rehoboam; 1st Kings, 11: 7,\nand to the Messiah, Isaiah, 42: 1.[8]\n\nIt is used in Gen. 2: 15. And the Lord God took the man, and put him in\nthe garden of Eden to DRESS IT. Adam was put into Eden, not to _serve_\nor _dress_ the garden as a SLAVE, but as a man. The same word is used\nto express the service performed for Laban by Jacob. The relation of\nJoshua to Moses is expressed by the same word; Ex., 33: 21. It is also\nused in the fourth commandment. Six days shalt thou _labor_, etc.\n\nFrom these examples of the use of the word it is clear that the idea of\nchattel slavery is not found in it. It is used to express all kinds of\nservice--the service of God, a king, a friend, or an employer.\n\nThe word _ama_, rendered _maid-servant_, _bond-maid_, _maid_,\n_hand-maid_, and the word _shiphhah_ with similar renderings, are\napplied to Hagar, Ruth, Hannah, Abigail, Bilhah and Zilpah, and\nevidently mean no more than our English word servant in its usual\nacceptation. Those women were not slaves, they were free women. It has\nbeen very properly remarked that if _chattel_ slavery existed among the\nHebrews at any time it is not a little surprising that the language\ncontains no word which expresses the relation.\n\nSome have endeavored to force into the word translated _servant_ &c.,\nthe idea of slavery because it is said that Abraham had servants\n\"bought with money.\" But from the ancient use of the word _buy_ or\n_bought_ we are not to infer that the persons bought became slaves.\nWives were procured in the times of the patriarchs by _purchase_.\nBoaz said--\"Moreover Ruth, the Moabitess, the wife of Mahlon, have\nI purchased to be my wife.\" The same word (_kanithi_) is used here\nto express the manner in which Boaz obtained his wife, that is used\nin Gen. to show how a part of Abraham's servants were obtained. But\nthe beautiful _Ruth_ was not a _slave_. Jacob purchased his beloved\nRachel, and less beloved Leah, but those wives and mothers of the\ntwelve patriarchs could not have been _slaves_. Had they been chattels,\nwhy, then, according to an essential feature of the American slave\ncode, the twelve patriarchs would all have been born in the same\ncondition. _Partus sequitur ventrem._ A Hebrew might sell himself on\na limited time, and he might be bought by a wealthy neighbor, but no\none, I believe, has ever pretended that he became a SLAVE thereby. The\ncontract was voluntary. The employer bought the services of his fellow,\nand paid in _advance_ for the same, not to a third person, but to the\nservant himself. God is said to have purchased (_kanitha_) his people;\nPs., 75: 2.\n\nHence from the scriptural use of the word buy, or bought, we are not\nauthorized to infer that the persons purchased became slaves. Such an\ninference would do violence to the holy word.\n\nThe true state of the matter in respect to Abraham, and his case is\nmainly relied upon, was without a doubt this. Abraham, being a wise,\nwealthy and good man, gathered around him many devoted friends who,\nupon his removal to a distant location, desired to accompany him,\nto receive the benefits of his friendship and counsels, live under\nhis patriarchship, as he was a prince, (see Gen., 23: 6,) and enjoy\nthe protection of his power. Some of these may have been involved in\npecuniary embarrassments or obligations of service to other persons,\nwhich made it necessary for the benevolent patriarch to release them by\npaying them in advance for many years of service.\n\nMany of these servants were doubtless converts from idolatry, which\nhad been made in Haran. In Gen. 12: 5, the fact is recorded of the\nremoval of Abraham, Sarai, their effects, and of \"the SOULS they\nhad gotten.\" This word \"gotten\" is translated, says Mr. Carothers,\nfrom _osa_, which is used in Ezekiel 18: 31, to express the work of\nconversion. \"Cast away from you all your transgressions, and make you a\nnew heart and a new spirit.\" And this rendering of the word \"gotten\" is\nconfirmed by the Chaldee paraphrase on this passage, which reads thus:\n\"Souls they had instructed or turned from idolatry and taught in the\ntrue religion.\" \"The Hebrews have a tradition,\" says Banberg, \"that\nAbraham brought over many men, and Sarah many women from infidelity\nto the knowledge and worship of the true God; and thus _made_ them\nspiritually.\" A similar mode of expression is used by St. Paul: \"I have\n_begotten_ you through the gospel.\" The idea that Abraham and Sarah\nmade slaves of their converts is simply preposterous.\n\nFrom the foregoing facts and considerations it is perfectly clear\nto my mind, that the effort to find an apology for slaveholding\nin patriarchal servitude is a total failure. The charge that the\npatriarchs held slaves is _wholly_ without foundation,--is a\ndisingenuous attack upon their reputation, and a miserable subterfuge\nfor hard-hearted oppressors, who are seeking an apology or excuse for\nsins which loudly cry for the vengeance of heaven! Could Father Abraham\narise from the dead, visit the South, and there behold thousands of\nhis spiritual children toiling without remuneration, shut out from\nthe blessings of family and home, denied an education and all means\nof intellectual improvement, driven by the keen lash of a brutal\noverseer, and then should he hear an appeal made to the patriarchs in\njustification of this system of unmingled tyranny, he would indignantly\nrepel the appeal as a base calumny!\n\nIt is surprising with what confidence the example of the patriarchs\nis urged in justification of slavery in the absence of all proof or\nsemblance of proof, that they were implicated in this practice. But\nour surprise is increased when we consider that, even could it be made\nappear that the patriarchs did hold slaves, this fact of itself, would\nafford not the slightest apology for slaveholding now. The patriarchs,\nit is admitted, had a plurality of wives, but their example is not now\na sufficient warrant for polygamy. There is not an ecclesiastical court\nin the United States and territories, if we may except the Mormon,\nUtah, which would accept the example of the patriarchs as an apology\nfor the man who should stand up before that court with two wives\nleaning on his arms. The argument therefore appears utterly worthless\nand shallow from every point of view.\n\n\n\n\nCHAPTER IX.\n\nSlavery and Religion--Continued.\n\nLAW OF MOSES AND SLAVERY.\n\n\nIt is claimed by the advocates of human bondage that in the law\ndelivered by Moses for the government of the children of Israel, until\nthe establishment of the kingdom of Christ, slavery is distinctly\nrecognized, carefully regulated, and unequivocally sanctioned; and\nhence, that it is an institution upon which Jehovah now looks with\napprobation. We cannot believe, they argue, that it is wrong for\nchristians to practice what the law of Moses permitted or sanctioned.\nTo this argument we reply:--\n\n1. That many things were allowed by the law of Moses which are strictly\nprohibited by the law of Christ. That law was imperfect in its\ncharacter, limited in its application, and temporary in its design. It\ncontained a number of statutes which could by no means be incorporated\ninto the laws of a christian state.\n\nAmong the things commanded and allowed by the law under consideration,\nthe following may be specified:--\n\n1. It commanded a Hebrew, even though a married man, with wife and\nchildren living, to take the childless widow of a deceased brother, and\nbeget children with her; Deut., 25: 5-10.\n\n2. The Hebrews, under certain restrictions, were allowed to make\nconcubines, or wives for a limited time, of women taken in war; Deut.\n21: 10-14.\n\n3. A Hebrew who already had a wife, was allowed to take another also;\nprovided he still continued his intercourse with the first as her\nhusband, and treated her kindly and affectionately; Exodus 21: 9-11.\n\n4. By the Mosaic law, the nearest relative of a murdered Hebrew could\npursue and slay the murderer, unless he could escape to the city\nof refuge; and the same permission was given in case of accidental\nhomicide; Num. 35: 9-34.\n\n5. The Israelites were commanded to exterminate the Canaanites, men,\nwomen and children; Deut. 9: 12; 20: 16-18.\n\n\"Each of these laws, although in its time it was an ameliorating\nlaw, designed to take the place of some barbarous abuse, and to be\na connecting link by which some higher state of society might be\nintroduced, belongs confessedly to that system which St. Paul says made\nnothing perfect. They are a part of the commandment which he says was\nannulled for the weakness and unprofitableness thereof, and which, in\nthe time which he wrote, was waxing old, and ready to vanish away.\"\n(_Dr. Stowe._)\n\nNow, will any one pretend that it is proper for a christian, having\na wife, to take also the wife of a deceased brother? But the law of\nMoses authorized this as clearly as any one pretends that it authorized\nslavery. Is it allowable for a christian to take a concubine or marry\nthree or four wives? But the law of Moses allowed this as distinctly\nas any one believes that it allowed slavery. Would it be right for\na christian to pursue a neighbor who had committed accidental or\nintentional homicide, overtake and slay him? But the law of Moses\njustified the Jewish man-slayer as plainly as the most ultra defender\nof slavery maintains that it justified slaveholding. Suppose we\nadmit, for argument sake, that slavery was authorized by the law of\nMoses, does it follow as a matter of course, that the law of Christ\nauthorizes it? By no means; for we have seen that the former authorized\nconcubinage, polygamy, extermination of the heathen, and summary\nvengeance upon the unwitting murderer, all of which things are utterly\nincompatible with the precepts of the latter. And slavery might very\nproperly be placed in the category of those practices allowed by the\nlaw, but prohibited by the gospel. Thus the argument for slavery from\nthe law of Moses proves too much, and therefore proves nothing.\n\n2. But if, as is claimed, the Jews _were_ authorized to enslave their\nfellow men, which we by no means admit, it was by express authority\nfrom God, who alone may deprive any of his creatures of the rights\nwith which he has invested them. Express grants _were_ made to the\n\"chosen seed,\" as for instance, the forcible occupancy of the land of\nCanaan, and of the cities thereof. Now those grants were not made to\nAmericans, but to the ancient Israelites, and it is neither modest nor\nsensible for citizens of the United States to act under a charter which\nthey admit was made to an ancient nation, for a temporary purpose. Let\nthe American slaveholder show the same _authority_ for slaveholding\nwhich he maintains the Jew could produce. Has God ever made a grant to\n_Americans_ to enslave the Africans?\n\n3. Again, the passage mainly relied upon is found in Leviticus, 25:\n44-47; in which the Jews are authorized to procure servants of the\n_nations_, (not heathen, for _heathen_ is not in the original) round\nabout them. Now if this celebrated passage be at all to the purpose,\nit is, as Pres. Edwards has said, \"a permission to every nation under\nheaven to buy slaves of the nations round about them; to us, to buy\nof our Indian neighbors; to them, to buy of us; to the French to\nbuy of the English, and to the English to buy of the French; and so\nthrough the world. Thus according to this construction, we have here an\ninstitution of a universal slave trade, by which every man may not only\nbecome a merchant, but may rightfully become the merchandize itself of\nthis trade, and be bought and sold like a beast.\" Who is willing to\nadmit the consequences of this construction?\n\nWe might here rest the case, because these three considerations, taken\nseparately, or together, destroy entirely the whole force of the\nargument for American slavery predicated upon Levitical servitude.\n\nWe shall now inquire what kind of servitude _was_ recognized and\nregulated by the law of Moses. The particular statute upon which the\nmain reliance is placed, by the friends of slavery, and which is\nsupposed to contain the black and bloody _charter_ for the degradation\nof humanity, is found in Leviticus 25: 44-47, and reads as follows:--\n\n\"Both thy bondmen and thy bondmaids which thou shalt have, shall be of\nthe heathen that are round about you: of them shall ye buy bondmen and\nbondmaids. Moreover of the children of strangers that do sojourn among\nyou, of them shall ye buy, and of their families that are with you,\nwhich they beget in your land: and they shall be your possession. And\nye shall take them as an inheritance for your children after you, to\ninherit them for a possession, they shall be your bondmen forever.\"[9]\n\n1. The word _slave_, it will be observed, does not occur in this\npassage, nor does bondmen and bondmaids mean anything more than\nmen-servants and women-servants. The word _bond_, as we have seen, is\ngratuitously supplied by our translators, and is not in the original;\nand the word _servants_ means no more than _laborers_ or _workers_. All\nkinds of servants are described by the term here found, and hence from\nits use in this place, it cannot be inferred that the persons referred\nto were slaves. The passage clearly authorized the procurement of\nservants from adjoining nations, which was a thing perfectly right in\nitself, and that is all it did authorize.\n\n2. Nor does the fact that the passage allowed the _purchase_ of\nservants, prove that the persons purchased were slaves, or became\nslaves. Irishmen were, many of them, a few years since, \"bought\nservants.\" They were sold to pay for their passage to this country,\nbut the whole transaction was voluntary on the part of the \"sons of\nErin,\" and looked to their benefit. Jacob, as we have seen, purchased\nLeah and Rachel with fourteen years of labor. Our blessed Savior hath\npurchased us with his own blood. The idea of chattel slavery cannot\nbe associated with the word buy or bought, as used in the sacred\nwritings, without doing great violence to their meaning. The phrase,\n\"of them shall ye _buy_\" may be properly rendered, \"of them shall ye\n_get_, or _obtain_ servants.\" The word translated buy, in the passage\nbefore us, is in other places translated \"get\" or \"getteth.\" Thus, \"He\nthat beareth reproof _getteth_ understanding.\" Prov., 15: 32. \"He that\n_getteth_ wisdom, loveth his own soul.\" Prov., 19: 8. But the meaning\nof the word buy, and sell, as applied to the purchase and sale of men,\nis definitely settled by its use in the context of the passage which\nwe are examining. It is used in verse 47, \"if thy brother wax poor and\n_sell himself_\" etc. In verse 39, the reading is, \"and _be sold_.\"\nThese passages are intended to convey an idea of the same transaction,\nand that transaction was nothing more nor less than the voluntary sale\nof a poor man to a rich one, not as a slave, but as a servant. The sale\nwas made, and the _money was received by the servant who sold himself_,\nwith which he released himself and family from pecuniary embarrassment.\nIn this sale and purchase of a man, the idea of slavery is utterly\nexcluded. Now is it probable that the words buy, and sell, in this\nsame chapter, when applied to foreign servants, were used in a totally\ndifferent sense? To suppose this would be to charge Moses, as Wm. Jay\nobserves, with a fraudulent intent to render the meaning of his law\ndoubtful and unintelligible.\n\n3. Considerable stress is placed upon the phrase, \"shall be of the\n_heathen_,\" as if heathenism was a crime to be punished with a still\ndeeper degradation than idolatry can produce. \"The word heathen,\"\nsays Mr. Jay, \"is gratuitously inserted by our translators instead of\n_nations_, the meaning of the original.\"\n\n4. Permission was also given for the purchase of the \"children of\nthe strangers.\" \"'Children of the strangers' is an orientalism, for\nstrangers, as 'children of the East,' 'children of the Province,'\n'children of the Ethiopians.' Hence, the Jews, instead of buying little\nboys and girls of their parents, were to buy foreigners residing in the\ncountry; and not only foreigners, but their descendants, natives of\nPalestine.\" (_Jay._)\n\n5. \"They shall be your bondmen forever.\" In this phrase is supposed to\nbe found a charter for perpetual, hereditary, hopeless bondage. Mr.\nJay very justly remarks upon it as follows: \"The preconceived opinions\nof the translators tempted them to give such a color to this sentence\nas best accorded with their proslavery theory. Hence this strong\nexpression in the text, while in the _margin_ the _literal_ translation\nis honestly given, \"Ye shall serve yourselves with them forever.\" Not a\nword about bondmen, but merely an unlimited permission, as to time, to\nuse or employ foreigners or strangers.\"\n\nThe proslavery construction renders the permission absurd, because in\nthe first place it would be impossible for any one man literally to be\na bondman forever, unless servitude could be continued in heaven or\nhell. And, in the next place, it could not continue in the same person\nin Israel beyond the great jubilee.\n\nNow when this passage in Leviticus 25, is stripped of all the\nproslavery glosses of the translators, the following is, as the\nexcellent writer just quoted observes, its plain and obvious\nmeaning:--\"You may buy of themselves, for servants, men and women who\nare natives of the adjoining countries, just as you have already been\nauthorized to buy your own countrymen for servants. You may also buy,\nfor servants, strangers residing among you, and their descendants; and\nyour children after you may do the same. You may always employ them as\nservants.\"\n\nThe servitude permitted by the law of Moses has been most grossly\nmisrepresented, and misunderstood. It was not an institution looking\nmainly to the advantage of the rich and powerful, while it crushed the\npoor and defenseless into the dust, disregarding their interests and\ntheir sorrows, but it was a benificent arrangement intended to relieve\nthe unfortunate and open a door of hope to the Gentile inquirer of the\nway to Zion. Now observe carefully the following facts:--\n\n1. Servants were not kidnapped or stolen from the surrounding nations.\nThe stealing of a man was made a capital offense. He that stealeth a\nman and (_or_ it should be) selleth him, or if he be found in his hand,\nhe shall surely be put to death. Ex. 21: 16. Now, as all the slaves\nin America have been stolen, those who stole them, and those who hold\nthem, are worthy of death according to the law of Moses.\n\n2. All the servants obtained by the Jews from neighboring nations were\n_voluntary_ servants. This is proved in the following way. 1. Foreign\nservants, and native Hebrew servants were obtained in the same manner.\nNative Hebrews became servants (except in cases of crime) by voluntary\ncontract. 2. Obedience to the law of Moses was a condition of servitude\nin the Jewish state. An idolater was not allowed to remain in the land.\nAnd a bought servant was obliged to renounce idolatry, receive the\nrite of circumcision, and in all things conform to the law of Moses,\nas his master was required to do. Gen. 17: 10-15. Ex. 23: 15-20. Deut.\n16: 10-18. All were required to enter into the most solemn religious\ncovenant. \"Ye stand this day, all of you, before the Lord your God;\nyour captains of your tribes, your elders, and your officers, with\nall the men of Israel, your little ones, your wives, and thy stranger\nthat is in thy camp, _from the hewer of thy wood unto the drawer of\nthy water_; that _thou_ shouldest enter into covenant with the Lord\nthy God, and into his oath, which the Lord thy God maketh with thee\nthis day.\" Deut. 29. But conformity to the law of Moses was voluntary.\nWe cannot conceive that a Jew was allowed to buy a heathen servant\nagainst his will, tie him, inflict upon him the rite of circumcision,\nand then compel him to observe the great feasts ordained by the law,\nand, otherwise conform to the Jewish religion. Hence the acceptance\nof a place as a servant in a Jewish family was a matter of choice.\n3. Servants were not obliged to remain with their masters. If they\nsaw proper to change their situation, they had a perfect right to do\nso, just as laborers now have, and there was no fugitive slave law to\nprevent them from so doing. \"Thou shalt not deliver unto his master\nthe servant that is escaped unto thee. He shall dwell with thee, even\namong you, in that place he shall choose, in one of thy gates where it\nliketh him best: thou shalt not oppress him.\" Deut. 23: 15, 16. From\nthese facts, the conclusion is irresistible that servitude was not\n_forced_ upon a foreigner, but voluntarily accepted by him, and that\nhis continuance in that relation was voluntary. How great the contrast\nbetween this system and American slavery which utterly disregards the\n_will_ of slaves.\n\n3. Foreign servants were to be treated in all respects precisely as\nnative Hebrew servants were to be treated. \"Ye shall have one manner of\nlaw, as well for the stranger as for one of your own country, for I am\nthe Lord your God.\" Lev. 24: 22.\n\n4. Ample provisions were made for the religious improvement of servants\nof all classes and especially foreign servants. They were to observe\nthe sabbath, go up with their masters to the three great annual feasts\ncelebrated at Jerusalem, listen to the reading of the law, and in short\nenjoy all the advantages of the Jewish religion. Mr. Barnes estimates\nthat in a period of fifty years, not less than twenty three were\nappropriated to the exclusive benefit of servants, during which time\ntheir whole attention might be devoted to the interests of their souls.\nDoes not this indicate that the great design of the employment of\nforeign servants was religious? Is there the least similarity between\nthis system of servitude and American slavery?\n\n5. Special provisions were made to secure the kind treatment of all\nforeigners, foreign servants of course included. \"Thou shalt not vex\na stranger nor oppress him.\" \"Thou shalt not oppress a stranger, for\nye know the heart of a stranger.\" \"Cursed be he that perverteth the\njudgment of the stranger.\" \"The Lord your God regardeth not persons.\nLove ye therefore the stranger.\" But does not American slavery vex and\noppress the stranger and pervert his judgment? The wide world cannot\nproduce a class of persons who are, or ever have been oppressed, if\nAmerican slaves are not. The word oppression is too feeble to express\nthe tyranny suffered by the strangers in our land.\n\n6. Servants under the law of Moses could not be sold. No permission\nwas given for the sale of servants. They could not be taken for the\npayment of debts, or as pledges, or presents. _They never were sold_\nor given away. The reason of this is found in the fact that they were\nnot chattels,--they were recognized as men, and had made a contract for\nservice which their masters could not at pleasure annul. We have seen\nthat the trade in slaves is an extensive and lucrative business.\n\n7. The Hebrew law regarded servants as naturally equal to their\nmasters, and hence, they were allowed to marry into their master's\nfamily, and inherit, under some circumstances, their master's property.\nDeut. 21: 10-14. A slave is not regarded as a man, can own nothing,\nand inherit nothing. What a contrast! American slavery, and Hebrew\nservitude seem to be erected upon totally different foundations.\n\n8. At stated periods the mild form of servitude instituted by the law\nof Moses expired. A Hebrew who became a servant could not be required\nto continue in that relation more than six years. And every fiftieth\nyear was a grand Jubilee, at the commencement of which liberty was\nproclaimed throughout all the land unto _all the inhabitants thereof_.\nLev. 25: 10, 11. Contracts for service, under any circumstances, could\nnot hold beyond that great jubilee. It was a glorious institution, and\na type of the proclamation of the gospel. But American slavery knows no\njoyful jubilee! For three hundred years no proclamation of freedom has\nbeen made throughout all this land unto all the inhabitants thereof.\nNo, generation after generation of slaves goes down to their graves in\n_despair_! Slavery is without a jubilee.\n\n9. The grand design of the introduction of foreign servants into the\nJewish state was their salvation. From a careful examination of this\nwhole subject, we are fully satisfied that the 25th chapter of Lev.\ncontains, as Mr. Smith has said, \"the constitution of Heaven's first\nMissionary society, by which a door of mercy and salvation was opened\nto the heathen, through which they could obtain access to the altar of\nGod, find mercy and live.\"\n\nIt will be observed that a foreigner could obtain a permanent residence\nin Israel in but two ways,--1st By becoming a servant in a Jewish\nfamily, and, 2d By purchasing a house in a walled city. Now, when\nin connection with these facts, we consider that to the Jews were\ncommitted the \"lively oracles;\" that the only temple of God on earth\nwas erected on Mt. Moriah; that the divinely appointed priesthood\nand sacrifices were in Jerusalem; and also that a renunciation of\nidolatry and hearty acceptance of the God and religion of the bible\nwas absolutely required of those foreigners who desired to become\nservants; that when they did become servants they were blessed with all\nthe precious privileges of the Jewish religion, and after a few years,\nbecame, with their families, adopted members of the Jewish state,\nhaving all the rights, immunities and honors of the chosen people of\nGod; I say, when all these facts are impartially weighed, they convince\nus that the _end_ of the provision alluded to for the admission of\nforeign servants was religious--the salvation of those servants.\n\nAnd history affords a powerful argument in support of this position.\nWhat was the practical operation of the law of Moses in relation to\nforeign servants? If the pro-slavery view of that law be correct, then\nhistory would record the fact that the commonwealth of Israel was a\nslaveholding commonwealth. It would state that the Jews traded in men,\nand that this traffic was important. We should read of poor, ignorant,\nchained idolaters traveling in mournful procession to a great slave\npen at Jerusalem, situated under the shadow, perhaps of the temple of\nGod, and from thence into every part of the land. And when our Savior\nappeared, he would have come into contact with those wretched slaves,\nand would have said something about them. Do we find these facts in\nhistory? No, not one of them. Jerusalem, thank God, was a free city.\nJudea a free state. Foreigners were employed from age to age, as\nservants, but as was contemplated, they embraced the religion of God,\nbecame adopted citizens and were fully identified with the commonwealth\nof Israel. \"After circumcision they were,\" as Jahn says, \"recorded\namong the Hebrews,\" and after the jubilee they enjoyed all the\nimmunities of the children of Abraham. Such was the intention, and such\nthe results of Levitical servitude. Between that system and American\nslavery there is scarcely any thing in common. Slavery originated in\npiracy, is a system of savage tyranny, degrading to the intellect,\ndestructive of morality, blasting to hope and happiness, and tending to\nbarbarism and crime. Servitude under the law of Moses, originated in a\nbenevolent desire to open a door of hope to the heathen, was kind and\njust in its requirements, guarding with extreme jealousy the interest\nof servants, and admirably calculated to lead their minds to morality,\nvirtue and the knowledge of God. Slavery, therefore, can find no\nsanction in the law of Moses. Why, if that law were applied to American\nslavery it would abolish it. Compel slaveholders to use their slaves\nas the law of Moses required servants to be used, and you will soon see\nan end of slavery.\n\n\n\n\nCHAPTER X.\n\nSlavery and Religion--Continued.\n\nNEW TESTAMENT AND SLAVERY.\n\n\nOur Lord's New Testament is the bulwark of human freedom. Its great,\nbroad, solid truths constitute an impregnable foundation for a temple\nof liberty capacious enough to hold the entire human race. This is the\nlast book in the world to search in order to find any thing favorable\nto oppression; and oppressors have usually preferred to \"burrow amid\nthe types and shadows of the ancient economy.\" An effort has been made,\nhowever, to wrest a sanction for the abomination of slavery out of this\nlast and best revelation from heaven, and to convert some passages\nfound in the writings of the apostles into chains and fetters to bind\nin hopeless bondage those very persons for whom Christ died.\n\nWe will quote the passages usually adduced to prove that it is the duty\nof some men to be slaves, and of others to be slaveholders.\n\n\"Servants, be obedient to them that are _your_ masters according to\nthe flesh, with fear and trembling, in singleness of your heart, as\nunto Christ; not with eye-service, as men-pleasers; but as the servants\nof Christ, doing the will of God from the heart; with good will doing\nservice, as to the Lord, and not to men; knowing that whatsoever good\nthing any man doeth, the same shall he receive of the Lord, whether\nhe be bond or free. And ye masters, do the same things unto them,\nforbearing threatening: knowing that your Master also is in heaven;\nneither is there respect of persons with him.\" Eph. 6: 5, 6, 7, 8, 9.\n\"Servants, obey in all things _your_ masters according to the flesh;\nnot with eye-service, as men-pleasers; but in singleness of heart,\nfearing God.\" Col. 3: 22. \"Let as many servants as are under the yoke\ncount their own masters worthy of all honour, that the name of God and\n_his_ doctrine be not blasphemed. And they that have believing masters,\nlet them not despise _them_, because they are brethren; but rather do\n_them_ service, because they are faithful and beloved partakers of the\nbenefit. These things teach and exhort.\" 1st Tim. 6: 1, 2. \"_Exhort_\nservants to be obedient unto their own masters, _and_ to please _them_\nwell in all _things_; not answering again; not purloining, but showing\nall good fidelity; that they may adorn the doctrine of God our Saviour\nin all things.\" Titus, 2: 9, 10. \"Masters, give unto your servants that\nwhich is just and equal; knowing that ye also have a Master in heaven.\"\nCol. 4: 1.\n\nWe will inquire in the first place whether these passages teach that\nit is the duty of some persons to be _slaves_. And it may be remarked\nthat if a class of human beings _ought_ to sustain this horrible\nrelation, the law requiring them to do so, should be written in the\nplainest possible manner. If any one should claim _me_ and _my_ family\nas slaves, upon a pretense that God had authorized our enslavement,\nI would demand a warrant for so terrible a degradation, which no\nreasonable man could question. Let us see whether the scriptures cited\nprove _unquestionably_ that to live in a state of slavery is a duty\nwhich God requires.\n\n1. It will be seen at a glance that there is not a word said about\n_slaves_ in any of these quotations. The word _slave_ or _slaves_\nis not once used! And yet these passages, inculcating the duties\nof _servants_, have been rung in the ears of our poor _slaves_\nfor the last three hundred years, by hypocritical preachers and\nslaveholders, as if heaven were chiefly interested and delighted in\nthe perpetuation of an institution which degrades millions of men\nto a point as low as manhood can possibly descend. The whole gospel\npreached to slaves is mixed up with this satanic perversion. Even the\nsong of angels announcing \"peace on earth and good will to men,\" is\naccompanied to the ear of the American bondman, with the base, coarse\ncorruption,--\"_Slaves_, obey your masters.\"\n\n2. The word _servants_, used in these scriptures, is not synonymous\nwith the word _slaves_, as the preachers of oppression assume. The word\n_andrapodon_ means slave, but that word, the learned tell us, does not\noccur in the sacred writings. The word _douloi_, used in the above\nquotations, and translated servants, means precisely what our English\nword servants means, as that word is understood in free countries. \"Our\nEnglish word servant,\" says a good authority, \"is an exact translation\nof the Greek word _doulos_. And to translate it into the definite word\nslave is a gross violation of the original. Our translators of the\nscriptures have uniformly translated the word doulos into the word\nservant, _never_ into the word slave, and for the reason that it never\nmeans slave. The apostles addressed servants in general, but never\nslaves in particular; and therefore the term slave (andrapodon) is not\nfound in apostolic writings.\"\n\nThe word _doulos_ occurs in the New Testament one hundred and twenty\ntwo times,[10] and in no case has it been translated slave. To show\nthe utter fallacy of the assumption that it is synonymous with slave,\npermit us to supply _slave_ in a few passages where _doulos_ occurs,\ninstead of servant, for if slave and servant mean the same thing, they\nmay be used interchangeably without violating the sense. \"Paul and\nTimotheus the _slaves_ of Jesus Christ.\" \"These are the _slaves_ of the\nMost High God which do show unto us the way of salvation.\" \"And a voice\ncame out of the throne, saying, Praise our God all ye his _slaves_, and\nye that fear him small and great.\" \"I am thy fellow _slave_.\" We might\nextend these quotations indefinitely, but a sufficient number have been\ngiven to show the absurdity of the assumption that the words servant\nand slave describe the same relation. The pro-slavery rendering of\ndoulos, would make slaves of all the redeemed, and of the holy angels,\nand would, as Mr. Smith remarks, extend the territory of slavery over\nheaven itself.\n\n3. The phrase \"servants under the yoke\" means no more than obligation\nto perform service according to agreement or contract. He who had an\nengagement with an unbelieving master should perform his contract,\nor fulfill his obligation with scrupulous fidelity in order that the\nname of God and his doctrine be not blasphemed. The word \"yoke\" does\nnot necessarily imply slavery. Our Savior said \"take my _yoke_ upon\nyou,\" but certainly he did not invite any one to become a slave. The\nword yoke is used in the scriptures to represent the ceremonial law;\n\"dominion of Jacob over Esau, in the matter of his father's blessing;\"\npolitical subjugation of the Israelites; the authority of king David\nover his subjects, etc., etc.; but not in a single passage in the\nscriptures, unless it be in 1st Tim. 6: 1, does it describe the state\nof a domestic slave, and the assumption that it means slave in this\nplace is altogether without proof to sustain it.\n\n4. There is one passage in the New Testament addressed to servants\nwhich has not yet been quoted. \"Servants be subject to your masters\nwith all fear; not only to the good and gentle, but also to the\nfroward. For this is thank-worthy, if a man for conscience toward\nGod endure grief, suffering wrongfully.\" 1st Pet. 2: 18, 19. In this\npassage _doulos_ does not occur, but _oiketes_, which some suppose,\nmeans _slave_. But of this evidence is wanting. The same word is used\nfour times only in the New Testament, and is, in no case, translated\nslave. (See Luke 16: 13. Acts 10: 7. Rom. 14: 4. 1st Pet. as above.) In\none place it is rendered household-servant, and it seems to be used to\ndistinguish _house_-servants from others. \"The word comes from oikos, a\nhouse.\"[11]\n\n5. If the sacred writers above quoted had intended to address slaves,\nthey would, in the first place, have done so plainly by _calling_\nthem slaves. In the second place the directions would have been\napplicable to persons in a state of slavery. As to the terms used in\nthe directions, we have seen that they do not apply properly to slaves;\nand the directions themselves afford proof that they were given to\npersons who were not chattel slaves. The advice and exhortations imply\nfreedom from _absolute authority_ and a power of choice not compatible\nwith slavery. They are exhorted to perform service \"AS THE SERVANTS\nOF CHRIST, DOING THE WILL OF GOD FROM THE HEART.\" That is, they were\nto be actuated by the highest motives, and were not to toil as the\nservants of men, but of God. Again, they are advised not to \"DESPISE\"\ntheir masters. Such directions have no pertinence, if addressed to\nhuman chattels. To whom then were they addressed? We answer, to\nvoluntary laborers or servants who received a compensation for their\nwork. The relations of servant and master or laborer and employer are\nnecessary, legitimate and honorable relations. All men have not the\nskill to acquire or manage capital, and capital is essential to the\naccomplishment of great enterprises, to the march of improvement,\nand the progress of civilization. Capital invested in railroads,\ncanals, machinery, factories, ships, merchandise, etc., requires many\nlaborers to manage it; and the directions we are considering require\nthat those laborers be honest, faithful, pleasant, and industrious in\nthe discharge of the duties they engage to perform. And even though\nan employer be not a very good man, as is often the case with men of\ncapital, christian servants or laborers are instructed to attend to\ntheir duties in the fear of God and in a manner that will recommend to\nthose employers the religion which they profess. Yea, though servants\nhave an engagement with a hard-hearted, overbearing, abusive heathen\nmaster, the apostles would have them perform their part, with the\nutmost fidelity, suffering \"_wrongfully_\" if need be, for the sake of\nChrist. These directions are judicious, and their observance would work\nto the advantage of laborers in all countries.\n\nNow it is clear that those scriptures do not teach unquestionably\nthat it is the duty of some persons to be slaves. If the apostles had\nsaid, \"slaves be obedient to your masters for you are their _property_\nand they have a right to you and all you can earn, _because_ you are\nproperty,\" then the matter would have been settled. Then we should\nadmit that _some_ men ought to be slaves, but upon the heels of this\nadmission would follow a question very difficult to settle viz: _Who\nis to obey_ the command to be a slave? How is it to be determined who\nshall become a human chattel and who the owner of said chattel?\n\nBut the assertion that God requires men to be slaves is a wicked\nassertion. It charges God with folly and inconsistency. He desires\nthe elevation of man, but slavery brutalizes him. He encourages the\nenlightenment of the mind and the expansion of the understanding, but\nslavery darkens the mind and enchains the understanding. God cannot be\npleased with the ignorance, stupor, injustice and servile wretchedness\nwhich are necessary to the very existence of slavery, and hence he can\nnot make it the duty of any man to be a slave, for this would be the\nsame as to make it his duty to be stupid, ignorant and wretched. No,\nGod does not will that any man or woman should be a slave. Man was made\nin the image of God's independence and sovereignty. The instinct of\nfreedom is strong in his bosom. It has resisted oppression in all ages,\nand it will resist it, with God on its side, until it shall triumph!\n\nWe will now inquire whether the apostolic addresses to masters\nauthorize some men to sustain the relation of _slaveholders_. It\nshould be observed that there are but two places in the New Testament\nin which the duties of masters are pointed out. Permit us to repeat\nthose duties. \"And ye masters do the same things unto them, forbearing\nthreatening, knowing that your master also is in heaven.\" \"Masters give\nunto your servants that which is just and equal, knowing that ye also\nhave a master in heaven.\"\n\nIs it possible that from these words men will take license to seize\ntheir fellows and convert them into _property_; despoil them of all\ntheir rights; deny them an education; banish them from courts of\njustice; break up their homes; take their wages without compensation;\ndrive them in chain-gangs from state to state, and whip, beat, and\nabuse them until they perish from the earth? Yes, it is possible.\nThis has been done. \"Was there ever,\" said Dr. Wayland, \"such a moral\nsuperstructure raised on such a foundation? * * If the religion of\nChrist allows such a license from such precepts as these, the New\nTestament would be the greatest curse that ever was inflicted on our\nrace.\" We remark\n\n1. In these directions there is not the slightest intimation that the\nmasters addressed were _slaveholders_ and that the servants in their\nemploy were slaves. The term slaveholders (_andrapodistais_,) is not\nused in the above passages, and this term is only once found in the\napostolic writings.[12] It is found in the following text: \"Knowing\nthis that the law is not made for a righteous man, but for the lawless\nand disobedient, for the ungodly and for sinners, for the unholy\nand profane, for murderers of fathers and murderers of mothers, for\nmanslayers, for whoremongers, for them that defile themselves with\nmankind, for _andrapodistais_, (slaveholders or menstealers) for liars,\netc.\" 1st Tim. 1: 9, 10.\n\nAnd it is not only a fact that slaveholders are not addressed in these\npassages, but _the directions given are such as no slaveholder in the\nworld can observe_. How can a slaveholder give unto a _slave_ that\nwhich is _just_ and _equal_? The _slave_ can own nothing, will nothing,\ninherit nothing, and hence it is impossible, in the very nature of the\ncase, for his owner to give him a just compensation for his labor. And\nthe _slave_ has a just right to _himself_ to _liberty_ and the very\nfirst honest and enlightened effort of a slaveholder to give to his\nslave that which is just and equal would result in his _emancipation_!\n_Justice_ and _equality_ are incompatible with slaveholding.\n_Injustice_ and _inequality_ are its essential principles. Let us hear\nMrs. STOWE'S comment on what Christian legislators have seemed to\nconsider _just_ and _equal_ when making laws for slaves:--\n\n    \"First, they commence by declaring that their brother shall no\n    longer be considered as a person, but deemed, sold, taken, and\n    reputed, as a chattel personal.--This is \"just and equal!\"\n\n    This being the fundamental principle of the system, the following\n    are specified as its consequences:\n\n    1. That he shall have no right to hold property of any kind, under\n    any circumstances.--Just and equal!\n\n    2. That he shall have no power to contract a legal marriage, or\n    claim any woman in particular for his wife.--Just and equal!\n\n    3. That he shall have no right to his children, either to protect,\n    restrain, guide or educate.--Just and equal!\n\n    4. That the power of his master over him shall be ABSOLUTE, without\n    any possibility of appeal or redress in consequence of any injury\n    whatever.\n\n    To secure this, they enact that he shall not be able to enter suit\n    in any court for any cause.--Just and equal!\n\n    That he shall not be allowed to bear testimony in any court where\n    any white person is concerned.--Just and equal!\n\n    That the owner of a servant, for \"malicious, cruel, and excessive\n    beating of his slave, cannot be indicted.\"--Just and equal!\n\n    It is further decided, that by no indirect mode of suit, through a\n    guardian, shall a slave obtain redress for ill-treatment. (Dorothea\n    v. Coquillon et al, 9 Martin La. Rep. 350.)--Just and equal!\n\n    5. It is decided that the slave shall not only have no legal\n    redress for injuries inflicted by his master, but shall have no\n    redress for those inflicted by any other person, unless the injury\n    impair his property value.--Just and equal!\n\n    Under this head it is distinctly asserted as follows:\n\n    There can be no offence against the peace of the state, by the mere\n    beating of a slave, unaccompanied by any circumstances of cruelty,\n    or an intent to kill and murder. The peace of the state is not\n    thereby broken.\" (State v. Manner, 2 Hill's Rep. S. C.)--Just and\n    equal!\n\n    If a slave strike a white, he is to be condemned to death; but if a\n    master kill his slave by torture, no white witnesses being present,\n    he may clear himself by his own oath. (Louisiana.)--Just and equal!\n\n    The law decrees fine and imprisonment to the person who shall\n    release the servant of another from the torture of the iron collar.\n    (Louisiana.)--Just and equal!\n\n    It decrees a much smaller fine, without imprisonment, to the man\n    who shall torture him with red-hot irons, cut out his tongue, put\n    out his eyes, and scald or maim him. (Ibid.)--Just and equal!\n\n    It decrees the same punishment to him who teaches him to write as\n    to him who puts out his eyes.--Just and equal!\n\n    As it might be expected that only very ignorant and brutal people\n    could be kept in a condition like this, especially in a country\n    where every book and every newspaper are full of dissertations\n    on the rights of man, they therefore enact laws that neither he\n    nor his children to all generations, shall learn to read and\n    write.--Just and equal!\n\n    And as, if allowed to meet for religious worship, they might\n    concert some plan of escape or redress, they enact that \"no\n    congregation of <DW64>s, under pretence of divine worship, shall\n    assemble themselves; and that every slave found at such meetings\n    shall be immediately corrected _without trial_, by receiving on the\n    bare back twenty-five stripes with a whip, switch or cowskin.\" (Law\n    of Georgia, Prince's Digest, p. 447.)--Just and equal!\n\n    Though the servant is thus kept in ignorance, nevertheless in his\n    ignorance he is punished more severely for the same crimes than\n    freemen.--Just and equal!\n\n    By way of protecting him from over-work, they enact that he shall\n    not labor more than five hours longer than convicts at hard labor\n    in a penitentiary!\n\n    They also enact that the master or overseer, not the slave, shall\n    decide when he is too sick to work.--Just and equal!\n\n    If any master, compassionating this condition of the slave, desires\n    to better it, the law takes it out of his power, by the following\n    decisions:\n\n    1. That all his earnings shall belong to his master,\n    notwithstanding his master's promise to the contrary; thus making\n    him liable for his master's debts.--Just and equal!\n\n    2. That if his master allow him to keep cattle for his own use, it\n    shall be lawful for any man to take them away, and enjoy half the\n    profits of the seizure.--Just and equal!\n\n    3. If his master sets him free, he shall be taken up and sold\n    again.--Just and equal!\n\n    If any man or woman runs away from this state of things, and, after\n    proclamation made, does not return, any two justices of the peace\n    may declare them outlawed, and give permission to any person in the\n    community to kill then by any ways or means they think fit.--Just\n    and equal!\"\n\n                                                  (_See Key, pp. 241._)\n\nIf slaveholding is an illustration of what St. Paul meant by justice\nand equality, who can tell what is injustice and inequality? Let it be\nunderstood that a slaveholder cannot give to a slave, _while he holds\nhim as a slave_, that which is _just_ and _equal_, because the greatest\ninjustice and inequality enters into the very nature of the relation of\nslaveholder. Could a man be a _just_ robber or an _honest_ thief? No,\nbecause injustice and dishonesty enter necessarily into the business\nof robbing and stealing. Even so is it impossible for justice and\nequality to enter into slaveholding, because, it is in its very nature,\nrobbery, theft, extortion, oppression, and a complication of almost all\nvillainies.\n\nIt is clear from the examination of all the passages in the New\nTestament relating to masters and servants, that those masters were not\nslaveholders and that those servants were not slaves.\n\nBut it will be asked did not slavery exist in the apostles' days? We\nanswer it did exist. The Roman government tolerated chattel slavery.\nWhy then did not the apostles regulate it by prescribing the duties\nof slaveholders and slaves? It has been assumed, and justly too,\nthat \"slavery no more than murder _can_ be regulated. That which\nis essentially and eternally wrong has nothing in it on which the\nclaim of morality can rest. Morality requires its destruction, not\nits regulation.\"[13] The law of God does not point out the duties\nof liars, adulterers and thieves, because as such, they can have no\nduties. So God did not attempt to regulate Roman slavery which was a\nmost vile and crushing despotism. He did not intend that SLAVERY should\nbe continued, and hence it was not to be regulated but destroyed. We\nhave no evidence in the above passages that SLAVEHOLDERS were admitted\ninto the church of Jesus Christ by the apostles.\n\nSlaveholders and the upholders of the infamous Fugitive Slave Law,\nlay the case of Onesimus to their consciences as a healing unction\nwhen dogging down the fugitive slave. In their blindness they assume\nthat Philemon was a slaveholder, Onesimus a slave, and St. Paul a\nslave-catcher. But not a word of this is true.\n\n1. Onesimus was a SERVANT and not a SLAVE, and Philemon was not a\nSLAVEHOLDER. The assumption that the one was a slave and the other a\nslave-owner is altogether without support.\n\n2. Onesimus was not _forcibly_ sent back. St. Paul did not arrest\nhim, and send him in chains to Philemon, charging the expense to the\ngovernment.\n\n3. He was not sent back as a _servant_, much less a _slave_. How then?\nWhy as a \"brother beloved.\" \"Thou therefore receive him as mine own\nbowels-- * * receive him as myself.\" \"If he oweth thee ought put that\non mine account.\" These directions are wholly inconsistent with the\nidea of slavery. If Onesimus was the _property_ of Philemon, Paul\nknew that he owed the service of his whole life. But Onesimus was no\nslave. Had he been a slave Paul would have said, \"Receive him not as\na slave (_andrapodon_) but above a slave,\" instead of saying, \"not as\na servant (_doulos_) but above a servant.\" Onesimus was a relative of\nPhilemon, probably a natural brother,--brother \"in the flesh;\" as may\nbe inferred from Philem., verse 16. He was undoubtedly a young man of\ngreat promise, and was not only entrusted with the epistle of Paul to\nPhilemon, but jointly with Tychicus was the bearer of the venerable\napostle's letter to the church at Colosse. On the authority of Calmet,\nand indeed of Ignatius, it is affirmed that he succeeded Timothy as\nbishop of Ephesus.\n\nThey who affirm that the New Testament writers sanctioned Roman\nslavery, seem not to be aware of the serious imputation they cast upon\nthat book and its authors. Look at that awful despotism, that you may\nunderstand what a savage, scaly, bloody-mouthed beast was welcomed into\nthe church and baptized with a Christian baptism, if we may believe\nthe advocates of human bondage.\n\n1. \"The (Roman) slave had no protection against the avarice, rage, or\nlust of the master, whose authority was founded in absolute property;\nand the bondman was viewed less as a human being subject to arbitrary\ndominion, than as an inferior animal, dependent wholly on the will of\nhis overseer.[14]\n\n2. \"He might kill, mutilate or torture his slaves for any or no offence;\nhe might force them to become gladiators or prostitutes.\n\n3. \"The temporary unions of male with female slaves were formed and\ndissolved at his command; families and friends were separated when he\npleased.\n\n4. \"Slaves could have no property but by the sufferance of their\nmasters.\n\n5. \"While slaves turned the handmill they were generally chained, and\nhad a broad wooden collar to prevent them from eating the grain.\n\n6. \"The runaway when taken was severely punished, * * * sometimes with\ncrucifixion, amputation of a foot, or by being sent to fight as a\ngladiator with wild beasts; but most frequently by being branded on the\nbrow with letters indicative of his crime.\n\n7. \"By a decree passed by the Senate, if a master was murdered when his\nslaves might possibly have aided him, all his household within reach\nwere held as implicated and deserving of death.\"\n\nIs it possible that the holy apostles gave their sanction to a system\nbased on such laws?\n\nBut all the fundamental principles of revealed religion are against\nslavery.\n\n1. THE CHARACTER OF GOD.--God is _just_ and cannot favor a system which\ndisregards all the principles of justice. But slavery outrages every\nprinciple of justice: therefore God must be opposed to slavery. God\nis _impartial_,--no respecter of persons, and he cannot be favorable\nto a system which is based upon partiality. But slavery is a system\nof superlative partiality: hence God is opposed to slavery. God is\n_love_,--and love wills the highest happiness of the intelligent\nuniverse, and the removal of every obstruction to the progress of\nmen to that happiness. But slavery obstructs that progress. It is a\nbarbarizing system, necessarily involving millions of men in ignorance,\ncrime and misery: therefore God must will its extirpation. All the\ndivine attributes are hostile to slavery. \"Thus saith the Lord, execute\nye judgment and righteousness, and deliver the spoiled out of the hand\nof the oppressor.\" \"Learn to do well; seek judgment, relieve the\noppressed; judge the fatherless; plead for the widow.\"\n\n2. THE COMMON ORIGIN OF MAN.--The unity of the human race is admitted\nby all scientific men, and the bible plainly teaches us that \"out of\none blood hath God made all nations to dwell upon the face of the\nearth.\" Whatever difference of feature, color, intellect or stature,\nmay be found in the various parts of the globe, is attributable to\nmanners, climate, education, and the pleasure the Creator has in\nvariety. Every human being is a _man_, possessing all the rights of\na man. All men are brothers, born into the world on a common level.\nHence one man cannot claim his brother and his brother's family without\ncommitting an outrageous insult. If the right to claim belongs to\nany, it belongs to all, and now whose right shall hold? We say if\nthe right to enslave belongs to any it belongs to all, and how is it\nto be determined who will sink from the right to own slaves to the\ncondition of a SLAVE? Must the strong reduce to slavery the weak, and\nthus make might the _arbiter_? Such a conclusion would be contrary to\nthe plainest dictates of reason. If men have a common parentage, and\nare brothers, they inherit common rights, and those rights ought to be\nrespected. That system which authorizes one part of the common family\nof man to plunder another part of their dearest rights--of _all_ their\nrights, is a wrong system. But slavery authorizes this very thing:\ntherefore slavery is wrong.\n\n3. JESUS CHRIST IS THE REDEEMER OF ALL.--Jesus is the second Adam, and\nsustains a relation to the human family co-extensive with the first\nAdam. He is the Mediator, High Priest and Elder Brother of every child\nof man. All have been purchased with a priceless offering; and hence\nthe claims of Christ are paramount to all other claims, and no one can\nrightfully become the owner of a fellow-being, unless Christ as Creator\nand Redeemer first relinquish his claim. A system which should attempt\nforcibly, and without divine permission, to seize upon the Saviour's\npurchase, would be robbery--a robbery of God. But slavery does seize\nupon the purchase of a Saviour's blood without divine permission:\ntherefore slavery is robbery--robbery of God.\n\n4. THE MORAL PRECEPTS OF CHRISTIANITY.--The moral precepts\nof Christianity condemn slavery. Take for example the golden\nrule--\"Therefore all things whatsoever ye would that men should do unto\nyou, do ye even so unto them.\" Can any slaveholder obey this precept?\nIf that wealthy planter who stands at the head of a large family, were\na slave with all his household, what course would he have his owner\npursue? Would he not wish him to grant a deed of immediate manumission\nto all his family and to himself? Would he not urge the matter as\none of immense importance? Is it possible that he could desire to be\ndeprived of liberty, education, permanent family connections, and of\nthe proceeds of his toil? Could any sane man wish to have his sons and\ndaughters grow up in the stupor, ignorance, and miseries of slavery?\nNo, it is not possible. Every sound-minded man would regard the\nsubjugation of himself and family to slavery as a dreadful calamity,\nand would consider the man who should _hold_ them in that condition as\nan unfeeling, inhuman tyrant.--Therefore no sound-minded man can hold a\nslave without violating the golden rule--without doing unto others as\nhe would _not_ have others do to him.\n\n5. THE COMMANDMENTS ARE ALL AGAINST SLAVERY. \"Honor thy father and\nthy mother.\" But slavery places the master between the child and the\nparent, and makes it impossible for the child practically to obey this\ncommand, in the performance of those duties which cheer the hearts and\nlighten the burdens of parents, especially in old age. \"Thou shalt not\nkill.\" But slavery authorizes in many cases the killing of slaves.\n\"In North Carolina, any person may lawfully kill a slave who has been\noutlawed by running away or lurking in the swamps.\" \"By a law of South\nCarolina, a slave endeavoring to entice another slave to run away, if\nprovisions, etc., be prepared to aid in such running away, shall be\npunished with death.\" \"Another law of the same State, provides that if\na slave when absent from the plantation, refuse to be examined by any\nwhite person, such white person may seize and chastise him; and if the\nslave shall _strike_ such person, he may be _lawfully killed_.\"--\"Thou\nshalt not commit adultery.\" But female slaves are compelled to commit\nadultery. The law places them wholly within the power of their masters\nand overseers, and they dare not, they cannot resist their demands.\n\"Thou shalt not steal.\" But slavery exists by theft. Every slave is a\nstolen man. Every slaveholder is a man-stealer. The slave was stolen\nfrom Africa, or stolen from his rightful owner, himself, in America. No\nsophistry can make it plausible that the African slave trade is piracy,\nand that the _perpetuation_ of slavery is an innocent business. It is\ntheft as clearly to go to the <DW64> hut in Virginia and steal a babe as\nto go to a hut in Africa and do the same deed. Certainly a child born\nin our happy Republic is as free in the sight of God as one born under\nthe rule of the King of Dahomey! \"All are created free,\" hence the\nholding of any one as a slave is _theft_ persevered in. \"Thou shalt not\nbear false witness against thy neighbor.\" But slavery does bear false\nwitness against the slave, who is our neighbor. It denies his natural\nequality, his right to liberty, property--in short, his manhood.\nThis is all as _false_ as false can be. \"Thou shalt not covet.\" But\nslavery covets not only a man's property, but the man himself. We see\nthat slavery violates every commandment of the second table of the\nDecalogue, and indeed violates every precept of the first table, as\nmight readily be shown.\n\nIt is clear that slavery receives no sanction from the curse pronounced\nupon Canaan, from patriarchal servitude, from the law of Moses, nor\nfrom the law of Christ. In the light of the divine word it appears a\ngigantic barbarism, full of hate to the human brotherhood. It annuls\nthe law of God respecting the family and society. It obstructs the\nprogress of education and religion. It is condemned by the whole spirit\nof revealed religion. Only a devil could pray for its perpetuation\nand extension. It is not only a sin, but a combination of stupendous\nsins--\"the sum of all villainies,\" in the language of Wesley, \"an\nenormity and a crime, for which perdition has scarcely an adequate\npunishment,\" in the language of Clarke. \"Slavery,\" said the celebrated\nJabez Bunting, \"is always wrong, essentially, eternally, incurably\nwrong.\"\n\n\n\n\nCHAPTER XI.\n\nAmerican Churches and Slavery.\n\nTHE POSITION THEY OCCUPY.\n\n\nThe christian church ought to be a faithful exponent of the benevolent\nspirit and doctrines of Jesus Christ. Liberty, truth and humanity,\nthough insulted, betrayed and proscribed every where else, should find\nwithin its sacred enclosures a welcome, a refuge and a stronghold.\nIts watchmen ought to be faithful men, uninfluenced by flattery,\nuncorrupted by gold, unawed by the popular will. The church ought to\nbe the most independent body on earth. Standing as it does upon the\nEternal Rock, holding the promise of successful resistance against\nthe \"gates of hell,\" and of certain triumph over all the powers of\ndarkness, the oppressor ought to know that he could not intimidate it\nby menace, silence its witnesses, win its smiles, or by any means be\npermitted to set his unhallowed feet within its pale. The church ought\nto be a terror to slaveholders; and although usage, prejudice, pride,\npassion, wealth, literature, and the selfish interests of men should\nall be combined against the oppressed, they should be certain of an\nunswerving and powerful friend and advocate in the church.\n\nWe say such should be the acknowledged and indisputable character and\nconduct of that body popularly known as the church, because then it\nwould be a faithful exponent of the divine philanthropy of Jesus, of\nhis \"good will to men,\"--then it would be precisely what the church was\nwhen it acknowledged no law superior to the will of God.\n\nWe propose now to ascertain the position of the American churches in\nrelation to the slavery question. The most of them have been compelled\nto take some action on this exciting subject. We shall notice, more\nespecially, the late action of various denominations, both for and\nagainst slavery, that the reader may know precisely where each branch\nof the _Protestant_ churches of this country, may be found. We do not\ndeem it necessary to exhibit the relation of the _Catholic_ church to\nslavery. We may remark here, however, by the way, that this church, if\nit be proper to call it a church, is soundly pro-slavery, and is, in\nAmerica, as it is everywhere else, a staunch advocate of oppression.\nFew Protestant churches excel the Catholic in slaveholding.\n\n\nPRESBYTERIAN (OLD SCHOOL.)\n\nThe Presbyterian church (O.S.) stands fully and unequivocally on the\nside of the oppressor. It is true that a few earnest anti-slavery men\nmay be found in this denomination, but their influence upon it is\nscarcely felt. They are not able in the least to modify the decided,\nunfaltering pro-slavery position maintained by the General Assembly.\nSo far as I know, the most ultra friends of slavery are perfectly\nsatisfied with the late ecclesiastical action and influence of this\nchurch. It makes no pretensions to anti-slavery. The slaveholder is\nwelcomed to its communion, is authorized to preach and is elevated\nto the highest posts of honor. At the last General Assembly fifty\nslaveholding presbyteries were represented. The place of meeting was\nCharleston, South Carolina. Dr. Lord, author of a celebrated sermon in\nsupport of the fugitive slave law, was elected moderator. The General\nAssembly of 1845, by a vote of 168 to 13, \"Resolved, That the existence\nof domestic slavery, under the circumstances in which it is found in\nthe southern portion of this country, is no bar to Christian communion.\"\n\nThis church has been _progressing_ in the wrong direction. In 1818,\nbefore the excision of the Presbyteries which formed the New School\nbody, the General Assembly declared that \"the voluntary enslaving of\none part of the human race by another was a gross violation of the most\nprecious and sacred rights of human nature,\" \"utterly inconsistent\nwith the law of God,\" and \"totally irreconcilable with the spirit\nand principles of the gospel.\" This was a noble declaration, but\nslaveholders were not excluded from the church as they should have\nbeen, but continued to flock in, until in 1836, a _slaveholder_\npresided over the General Assembly who openly said--\"I draw my warrant\nfrom the Scriptures of the Old and New Testament to hold my slaves\nin bondage.\"--Since 1836 the General Assembly has been wholly under\nthe control of the pro-slavery interest. Her doctors of divinity have\nwritten learned treaties in defense of slavery, and slaveholders are\nat ease, yea, sleep undisturbed in her communion, and for all that that\nchurch is likely to say or do, will sleep on until they find themselves\nin company with Dives.\n\n\nPRESBYTERIAN (NEW SCHOOL.)\n\nWhen the New School General Assembly was organized only three\nslaveholding Presbyteries were represented. There are now about twenty.\nA very large proportion of the ministers and members of this church are\n_somewhat_ anti-slavery, and many of them _decidedly_ anti-slavery; but\nthe holding of slaves is not made a test of communion. Slaveholders\nhave been and are now flocking into it. Ministers of the sanctuary and\nmembers of the General Assembly are slaveholders. Nevertheless, the\naction of the General Assembly has been such as to keep up an agitation\nand render the southern portion of the church somewhat restless.\n\nThe following resolution was adopted by the General Assembly, which\nconvened at Detroit in 1850:\n\n    \"That the holding of our fellow-men in the condition of slavery,\n    EXCEPT in those cases where it is unavoidable by the laws of\n    the State, by the obligations of guardianship or the demands of\n    humanity, is an offense, in the proper import of that term, as\n    used in the Book of Discipline, chap. i., sec. 3, which should be\n    treated in the same manner as other offenses.\"\n\nThe _exceptions_ in this resolution are sufficient, especially when\nexplained at the south, to cover almost all cases of slaveholding.\n\nThe Assembly of 1853 adopted a report earnestly requesting the\nPresbyteries in the slaveholding States to lay before the next Assembly\ndistinct and full statements touching the following points:\n\n    \"1. The number of slaveholders in connection with the churches\n    under their jurisdiction, and the number of slaves held by them.\n\n    \"2. The extent to which slaves are held by an unavoidable\n    necessity, 'imposed by the laws of the States, the obligations of\n    guardianship, and the demands of humanity.'\n\n    \"3. Whether a practical regard, such as the Word of God requires,\n    is evinced by the Southern churches for the sacredness of the\n    conjugal and parental relations as they exist among slaves; whether\n    baptism is duly administered to the children of slaves professing\n    Christianity; whether slaves are admitted to equal privileges and\n    powers in the Church courts; and in general to what extent and in\n    what manner provision is made for the religious well-being of the\n    enslaved.\"\n\nThe debate on this report and the subsequent action of the southern\nPresbyteries prove conclusively that the Detroit resolution is utterly\nfutile, and that slaveholding goes on in the southern part of the\nchurch without interruption. On this report, Rev. Mr. McLain, of\nMississippi said:\n\n    \"We disavow the action of the Detroit Assembly. We have men in our\n    Church who buy slaves, and work them, because they can make more\n    money by it than any other way. All who can, own slaves; and those\n    who cannot, want to.\"\n\nRev. William Homes, of Mo., said:\n\n    \"The action of the Assembly of Detroit is null and void; for how\n    can any man be found, not to be included in one or the other of the\n    exceptions contained in it? All claim that their slaveholding is\n    involuntary and justifiable. He concluded by strenuously asserting\n    that the South would not submit to these inquiries.\"\n\nRev. William Terry, of Va., said:\n\n    \"He could not promise that the Virginia Presbyteries would give\n    any replies to these inquiries. There was no hope, so long as\n    slavery exists, that the church shall be free from it. If it has\n    come to be true that the feeling of the North will not suffer the\n    slaveholding ministers and members to remain in fellowship with the\n    Church, the South will not remain with you. They do not contemplate\n    a disconnection with slavery.\"\n\nSince the meeting of the Assembly the Presbyteries in the South have\nalmost unanimously protested against the action in relation to slavery\nas \"inquisitorial,\" and have resolved to disregard totally the \"earnest\nrequest\" of the General Assembly. They have also resolved that the\nagitation of the subject in the Assembly must cease as a _condition_ of\nthe continued union of the church. Whether the pro-slavery element of\nthis denomination will prevail, so as to \"bury out of sight the Detroit\nresolution, silence the General Assembly on slavery, and make the New\nSchool Presbyterian Church a quiet home for those who \"buy\" \"sell\" and\n\"work\" slaves \"because they can make money out of them,\" cannot now be\ndetermined. We hope not, but knowing the aggressive spirit of slavery,\nwe fear.[15]\n\n\nCONGREGATIONAL.\n\nIt is somewhat difficult to define with any great degree of\nprecision, the position of the Congregational churches in relation\nto slavery. Many of these churches are actively anti-slavery. The\nCongregationalists of Ohio, in a convention held at Mansfield:\n\n    \"Resolved That we regard American slavery as both a great evil and\n    a great violation of the law of God and the rights of man; and that\n    we deem it our sacred duty to protest, by every christian means,\n    against slaveholding, and against any and all acts which recognize\n    the false and pernicious principle that makes merchandise of man.\"\n\nThe largest representative body of congregationalists which has\nexpressed itself on the question of slavery recently was the Albany\nConvention which met in 1852. This body adopted the following\nresolution:\n\n    Resolved, That in the opinion of this Convention, it is the\n    tendency of the gospel, wherever preached in its purity, to correct\n    all social evils, and to destroy sin in all its forms; and that it\n    is the duty of Missionary Societies to grant aid to churches in\n    slaveholding States in the support of such ministers only as shall\n    so preach the gospel, and inculcate the principles and application\n    of gospel discipline, that, with the blessing of God, it shall\n    have its full effect in awakening and enlightening the moral\n    sense in respect to slavery, and in bringing to pass the speedy\n    abolition of that stupendous wrong; and that wherever a minister\n    is not permitted so to preach, he should, in accordance with the\n    directions of Christ in such cases \"depart out of that city.\"\n\nIt is believed that Congregationalists generally are _progressing_ in\nthe right direction.[16]\n\n\nMETHODIST EPISCOPAL CHURCH (NORTH AND SOUTH.)\n\nJohn Wesley pronounced slavery to be the \"sum of all villanies.\" The\ndiscipline of the Methodist Episcopal Church is quite positive in its\ncondemnation of slavery. Some of the early Methodist preachers gave\nno quarters to this sin. But as the church increased in numbers and\npopularity, slaveholders, who at first came in by mere sufferance,\nassumed a bolder position, and finally ruled the whole church \"with a\nrod of iron.\"\n\nThe General Conference which convened in Cincinnati in 1836, after a\nwarm discussion, adopted the following resolution:\n\n\"Resolved, By the delegates of the Annual Conferences, in General\nConference assembled, that we are decidedly opposed to modern\nabolitionism, and wholly disclaim any _right, wish, or intention to\ninterfere in the civil and political relation between the master and\nslave as it exists_ in the slaveholding States of this Union.\" Yeas\n120, nays 14.\n\nThis resolution was an offering to appease the bloody Moloch of\nslavery, which had been aroused somewhat by _Orange Scott_. At a Gen.\nConf. in 1840, held in Baltimore, a resolution was passed depriving\n<DW52> persons of the right of testifying against white persons. The\nresolution reads as follows:\n\n\"Resolved, That it is inexpedient and unjustifiable for any preacher to\npermit <DW52> persons to give testimony against white persons, in any\nState where they are denied that privilege by law.\"[17]\n\nThe division of this Church (or secession, as some call it, of the\nChurch South) has as yet resulted, so far as we can see, in no\nadvantage to the slave. The southern portion or branch is not more\npro-slavery than before; and the northern division occupies precisely\nthe ground maintained when the resolutions of 1836 and 1840 were\nadopted, and when there were embraced within her communion the owners\nof 200,000 slaves. SLAVEHOLDING IS NOT A BAR TO MEMBERSHIP IN THE\nMETHODIST EPISCOPAL CHURCH NORTH. Ten or eleven conferences are now\nslaveholding, and between 30 and 40,000 slaves are owned at the present\ntime by members of this church.\n\nThe Baltimore Conference, which belongs to the church North, passed in\n1836 the following resolution:\n\n    \"Resolved, That this Conference disclaims any fellowship with\n    abolitionism. On the contrary, while it is determined to maintain\n    its well known, and long established position, by keeping the\n    traveling preachers composing its own body, free from slavery; it\n    is also determined not to hold connexion with any ecclesiastical\n    body, that shall make non-slaveholding a condition of membership in\n    the Church.\"\n\nThis conference, so far from regarding slaveholding in the _membership_\na sin, seems to consider it a virtue, and a condition of fellowship.\n\nAn effort to introduce the slavery question into the last General\nConference was defeated, speakers were choked down, and the conference\nclosed in disorder. Since the meeting of that body a number of\nConferences have passed resolutions calling for the adoption of a rule\nwhich would exclude slaveholders from the church. Some strong men[18]\nseem determined not to rest the question until there is a semblance\nat least of consistency between the professions and practice of\nMethodism on slavery. This church has been \"as much as ever deploring\nthe evils of slavery,\"[19] for scores of years, and as _much as ever_\nstrengthening and building up the iniquity! And as a Methodist writer\nin the _Northern C. Advocate_ in a late article asks--\"Is it not high\ntime for honest and God-fearing anti-slavery ministers and members of\nthe Methodist Episcopal Church, to inquire whether in her official\nposition, her anti-slavery professions and character are not all a\n_mere sham_!\" It is to be feared that the good men of this church, who\nare laboring to effect its renovation from this foul sin, are doomed to\ndisappointment, as many others, who have preceded them, have been. The\nfact that three new slaveholding conferences will be represented in the\nnext General Conference of this body, augurs unfavorably.\n\n\nMETHODIST PROTESTANT CHURCH.\n\nThis branch of the Methodist family is fearfully involved in the sin\nof slaveholding.--Slavery has silenced the voice of the church organ.\nSlaveholders have free access to its communion. The discipline contains\na very disgraceful clause in relation to  members. Article 12,\nSec. 1st secures the right of suffrage to all male members who are\nWHITE. Article 7, Sec. 3, gives to each annual conference power to\nmake for  members of the church \"_such terms_ of suffrage\" as\nthey may think proper. In the same article the apparently neutral, but\nreally pro-slavery character of this church is seen in the following\nwords: \"But neither the General Conference nor any annual conference\nshall assume power to interfere with the constitutional powers of the\ncivil governments, _or with the operations of the civil laws_.\" The\ncivil law is the highest law recognized in this article, and where that\nmakes _chattels_ of men, this church is forbidden to interfere. In\nthese quotations the principles of _caste_ and _lower-lawism_, are most\nclearly inculcated. It is with surprise and sorrow that we find such\nodious features in the discipline of a church which boasts of MUTUAL\nRIGHTS.\n\n\nWESLEYAN METHODIST CONNECTION.\n\nThis denomination of Christians stands boldly and unequivocally upon\nthe solid bible anti-slavery platform; and although not a large body,\nits influence has already been widely felt. It comes behind in no\nanti-slavery gift or grace. Its pulpit and press _speak out_ earnestly\nand powerfully. The Syracuse Conference recently adopted the following\nresolutions, which are such as all the conferences of the connection\npass unanimously:\n\n    \"_Resolved_, That we hold--as ever--in abhorrence the _system_,\n    esteeming it as ranking first in the dark list of systematized\n    piracy, and all intelligent supporters of the abomination as being\n    nothing, less or more, than willing pirates.\n\n    \"_Resolved_, That to ask us to fraternize with any of the thousand\n    and one organized or unorganized influences, going directly or\n    indirectly to sustain the system, prominent among which are the\n    principal churches and the great political parties of the country,\n    is to offer direct insult to our sense of Christian propriety and\n    gentlemanly courtesy.\"\n\n\nBAPTISTS (REGULAR.)\n\nThe _Regular Baptist_ Church occupies a decidedly pro-slavery position.\nWhere slavery exists, it does not make slaveholding a bar to communion.\nIt is true that there is a division between the Northern and Southern\nBaptist churches in _benevolent operations_, but this division is\n\"one, not of principle, but of policy. Hence, there has been from the\nfirst, between the leaders of the Northern and Southern Associations,\na cordial fraternization.\"[20] This church is very influential in the\nSouth, and from no ecclesiastical organization has American slavery\nreceived a more powerful and hearty sanction. Many Baptists are,\nhowever, warm friends of the slave, but they have not been able to\nchange or modify in the slightest degree the pro-slavery position of\nthe general body.\n\n\nBAPTISTS (FREE-WILL.)\n\nThe _Free-will Baptist_ Church is decidedly anti-slavery. It stands\nin the front rank of those societies which are on the side of the\noppressed battling for humanity. Amongst other excellent resolutions\nsubmitted by the committee on slavery at the last General Conference\nthe following will show on what platform to look for a true Free-will\nBaptist:\n\n    \"_Resolved_, That we re-affirm our opposition to the whole system\n    of American Slavery; holding it to be absurd in the light of\n    Reason, infamous in the eye of Justice, a deadly foe to human\n    welfare, a libel on the Decalogue, and a reckless attack on the\n    religion of Christ; and the only change we would recommend in our\n    denominational attitude and policy on this subject, is, to take an\n    advanced position in our warfare against the system, and to give a\n    more open and public expression to our hostility.\"\n\n\nBAPTISTS (SEVENTH-DAY.)\n\nThe position of this branch of the Baptist family may be known from the\nfollowing resolution passed by the Eastern Association:\n\n\"_Resolved_, That we enter our solemn protest against the system of\nAmerican slavery, as a sin against God, and a libel on our national\ndeclaration, that \"all men are created free and equal.\"\"\n\n\nEVANGELICAL ASSOCIATION.\n\nThe Evangelical Association has inserted in its discipline the\nfollowing resolution which indicates its ecclesiastical position:\n\n    \"_Question._ What is to be done respecting slaveholders and the\n    slave-trade?\n\n    \"_Answer._ We have long since been convinced that the buying and\n    selling of men and women, and slavery, is a great evil, and ought\n    to be abhorred by every Christian: be it therefore known to all\n    fellow-members, that none shall be allowed, under any pretence or\n    condition whatever, the holding of slaves or the trafficking in the\n    same.\"\n\n\nTHE UNITED BRETHREN IN CHRIST.[21]\n\nThis church believes slavery to be in itself a sin. The Constitution,\nwhich can only be altered by a vote of two-thirds of all the members of\nthe society, declares that \"involuntary servitude shall in no way be\ntolerated.\" The 32d Section of Discipline reads as follows:\n\n    \"All slavery in every sense of the word is totally prohibited, and\n    shall in no way be tolerated in our Church. Should any be found in\n    our society who hold slaves, they cannot continue as members unless\n    they do personally manumit or set free such slaves. And when it is\n    known to any of our ministers in charge of a circuit, station or\n    mission, that any of its members hold a slave or slaves, he shall\n    admonish such member to manumit such slave or slaves; and if such\n    persons do not take measures to carry out the discipline, they\n    shall be expelled by the proper authorities of the church; and any\n    minister refusing to attend to the duties above described shall be\n    dealt with by the authorities to which he is amenable.\"\n\nThis section, substantially, has been in force since 1821. The United\nBrethren have congregations in Missouri, Kentucky, Virginia and\nMaryland.\n\n    At the General Conference, May 12, 1853, the _Southern delegates_\n    reported that there were twelve cases of legal connection with\n    slavery in the Church, but they were of a character so peculiar,\n    that a difference of opinion had arisen as to whether the\n    discipline _intended_ to exclude them. The opinion and advice of\n    the Conference was asked. The following answer, in substance, was\n    given:\n\n    \"All those cases reported are cases prohibited by the plain letter\n    of our Discipline. Execute papers of immediate emancipation.--The\n    sympathy of this Conference given to palliated cases of slavery\n    would be an entering wedge of slavery into our Church. The Church\n    must be disconnected with slavery in all its forms. The bishops are\n    instructed to carry out the _letter_ of Discipline.\"\n\n    The action in this case was taken without a dissenting vote, and\n    the delegates from the South assured the Conference that the\n    intention of the Discipline, as above explained, should be executed.\n\n    The General Conference made provision for the publication of a\n    monthly magazine. The following is from the Prospectus:\n\n    \"The immediate abolition of slavery; rejecting that most odious and\n    barbarous notion, that man has a right to hold property in man. The\n    position will be taken that this is a monster that can never be\n    tamed, a sin which violates every precept of the Bible. It will be\n    our object to show that slavery (by which we mean the holding of\n    property in man) is sinful, necessarily sinful, under all possible\n    and conceivable circumstances.\"\n\n\nVARIOUS CHURCHES.\n\nBesides the churches already mentioned the following are decidedly\nanti-slavery:--\"Associate Presbyterian,\" \"Reformed Presbyterian,\"\n\"Free Presbyterian,\" (of which the venerable John Rankin is a member,)\nmany local \"Independent\" churches, and the \"Friends\" or Quakers. The\nQuakers have a world-wide reputation for practical philanthropy. And\non the other hand the following large denominations are decidedly\npro-slavery:--\"German Reformed,\" \"Dutch Reformed,\" \"Cumberland\nPresbyterian,\" \"Lutheran\" and \"Disciple\" (or Campbellite.)\n\nThe following estimate made by W. G. Gephart, a Presbyterian minister,\nwill give a \"bird's eye view\" of the relation of the leading\ndenominations of this country to slavery as it stood a few years since.\nAt the present time they are only _more deeply_ involved in the trade\nin the souls of men, than they were when this estimate was made:\n\n DENOMINATIONS.                                        NO. OF SLAVES.\n   Methodists,                                             219,563\n   Presbyterian, Old and New School,                        77,000\n   Baptists,                                               125,000\n   Campbellites,                                           101,000\n   Episcopalians,                                           88,000\n   Allow for all other denominations,                       50,000\n                                                           -------\n Total number of slaves owned by ministers of the gospel\n   and members of the different Protestant churches,       660,563\n\n\"Now, suppose the average value of all these slaves be only $400 each,\nand it will give a capital of $264,225,200! invested in humanity, the\ninterests of 660,653 beings upon whom God has chartered immortality,\nand stamped it with the signet of his own image.\"\n\nFrom this review it will be perceived that the most influential\ndenominations have given their sanction to slavery. They have opened\nwide their doors to slaveholders, and have welcomed them to their\ncommunion. They have not advised nor commanded them to emancipate their\nslaves as a condition of admission to the church, to the Lord's table,\nto the pulpit, or even into heaven itself!\n\nDivines have, by a perversion of the Bible, corrupted the consciences\nof Southern, aye, even of Northern Christians, by the most subtle and\nmonstrous errors. The holy Bible has been made, in the language of\nBlanchard, a smith shop whence consecrated hands have brought fetters\nfor the feet, and manacles for the mind! \"We have,\" said Frederick\nDouglass, \"men-stealers for ministers, woman-whippers for missionaries,\nand cradle-plunderers for church-members. The man who wields the\nblood-clotted cow-skin during the week fills the pulpit on Sunday and\nclaims to be a minister of the meek and lowly Jesus. The man who robs\nme of my earnings at the end of each week, meets me as class-leader on\nSunday morning, to show me the way of life, and the path of salvation.\nHe who sells my sister, for purposes of prostitution, stands forth as\nthe pious advocate of purity. He who proclaims it a religious duty to\nread the Bible, denies me the right of learning to read the name of God\nwho made me. He who is the religious advocate of marriage, robs whole\nmillions of its sacred influence, and leaves them to the ravages of\nwholesale pollution. The warm defender of the sacredness of the family\nrelation is the same that, scatters whole families,--sundering husbands\nand wives, parents and children, sisters and brothers,--leaving the hut\nvacant, and the hearth desolate. We see the thief preaching against\ntheft, and the adulterer against adultery. We have men sold to build\nchurches, women sold to support the gospel, and babes sold to purchase\nBibles for the _poor heathen! all for the glory of God and the good of\nsouls!_ The slave auctioneer's bell and the church-going bell chime\nin with each other, and the bitter cries of the heart-broken slave\nare drowned in the religious shouts of his pious master. Revivals of\nreligion and revivals in the slave trade go hand in hand together.\nThe slave prison and the church stand near each other. The clanking\nof fetters and the rattling of chains in the prison, and the pious\npsalm and solemn prayer in the church may be heard at the same time.\nThe dealers in the bodies and souls of men, erect their stand in the\npresence of the pulpit, and they mutually help each other. The dealer\ngives his blood-stained gold to support the pulpit, and the pulpit, in\nreturn, covers his infernal business with the garb of Christianity.\"\n\n\n\n\nCHAPTER XII.\n\nSlavery and the Church.\n\nNON-FELLOWSHIP WITH SLAVEHOLDERS.\n\n\nWe shall now proceed to show what we conceive to be the true position\nof a Christian church in relation to slavery. It has been demonstrated\nthat slavery is a complicated and monstrous iniquity involving a\ndirect violation of the whole second table of the Decalogue. This being\nan established position it will not be difficult to determine the\nrelation which the church should sustain to this sin, and to those who\ncommit it.\n\nThe scriptural position of a Christian and a Christian society in\nrelation to sin, may be ascertained from the following quotations: \"But\nI have written unto you not to keep company--if any man that is called\na brother be a fornicator, or covetous, or an idolater, or railer, or\ndrunkard, or extortioner, with such an one, no, not to eat.\"\n\n\"Wherefore come out from among them, and be ye separate, saith the\nLord; and touch not the unclean thing, and I will receive you.\"\n\n\"And have no fellowship with the unfruitful works of darkness, but\nrather reprove them.\"\n\n\"Now we command you brethren, in the name of our Lord Jesus Christ,\nthat ye withdraw yourselves from every brother that walketh disorderly.\"\n\nIn these passages the duty of open and decided non-fellowship with\nsinners is unequivocally asserted. 1. Do not \"keep company\" with\ncovetous persons and extortioners. Do not \"eat\" with them at the\nsacramental table, for this would imply a sanction of their sin. 2.\n\"Come out from among them.\" Let there be between you a plain line of\ndemarcation so that the whole world will know that you are not in favor\nwith their sin, and are not a party to it. \"Have NO FELLOWSHIP.\" Be not\nunited in any associations which require it. Go not with them to the\nsacramental board. Unite not with them in benevolent efforts for the\nconversion of the world, for this would require fellowship. Have _no_\nfellowship. 4. \"In the name of the Lord Jesus withdraw yourselves\"--cut\noff all ties which imply fellowship. Do this solemnly--do it in the\nname of the blessed Jesus--do it for the glory of God--do it as an act\nof discipline--withdraw yourselves from _every_ disorderly walker--from\nevery \"darkness worker,\"--let them be unto you \"as a heathen man and a\npublican.\"\n\nNow how are these scriptures to be obeyed respecting the great sin of\nslavery? We answer: 1. The church should debar slaveholders from its\ncommunion. While they remain impenitent in relation to the monstrous\nsin of slavery and refuse to emancipate their slaves, they should be\nperemptorily refused admittance into the fellowship of saints. At the\n_door_ they ought to be met by an emphatic \"No sirs; your hands are red\nwith blood, your purses are filled with unjust gains, you rob the widow\nand the fatherless, you make merchandise of men, repent, reform, do\njustly, love mercy, or away ye men-stealers!\"\n\n2. If by any means slaveholders have obtained a place in the church,\nthey should be plainly dealt with, according to the directions given\nin such cases by the sacred writers, and in case of a refusal on\ntheir part to \"hear the church,\" they should be immediately thrust\nout--accounted as \"heathen\"--\"delivered unto Satan for the destruction\nof the flesh.\"\n\n3. But in case a church refuses to discipline slaveholders, as\nit disciplines other offenders against God, and on the contrary\npersistently retains them in its communion and officially recognizes\nthem as members of the household of faith,--as holy persons,--as good\nchristians, then a christian can do no better than to withdraw from\nthat church. He cannot remain in it without giving an expressed or\nimplied sanction to a slaveholding christianity. The whole force of\nhis piety and influence will go abroad to create the conviction that\nslavery is right and quite consistent with holiness.\n\nIn support of this view of the true position of a church and\nof christians in relation to slavery, the following additional\nconsiderations are submitted:\n\n1. The church is required to be holy. But it cannot approximate to\nholiness while welcoming into its pale sinners such as slaveholders\nare, and sanctioning such an impurity as is slavery.\n\n2. The Church is required to be the \"pillar and ground of the truth.\"\nBut a slaveholding church wofully perverts and corrupts the truth in\nmany important particulars. The truth that God hates oppression and\nrobbery for instance, is corrupted by it, for it pronounces the very\nchief oppressors and robbers the true children of God, and assures\nthe world that He approbates their conduct. It corrupts the truth in\nrelation to the true idea of a christian. It denies that justice,\nmercy and love, are essential attributes of a christian character, by\npassing off upon a deluded world a class of persons as christians who\nare pre-eminently unjust, unmerciful, and full of hate to the human\nbrotherhood.\n\n3. The church should honor the holy scriptures. But a slaveholding\nchurch necessarily dishonors them. The church is presumed to be a\nfaithful and competent expounder of the doctrines and moral precepts\nof the Bible, and hence what it approves, it is supposed, the Bible\nsanctions, and as it approves of slavery, it gives currency to the\nidea that the Bible is a pro-slavery book,--that Christianity is\nfavorable to oppression, and an enemy to equality and fraternity.\nThus a slaveholding church dishonors the Word of Truth and is an\ninfidel-making organization. Non-fellowship with slaveholding is\ndemanded as a condition of faithfulness to the Bible.\n\n4. The church is expected to convert the world to righteousness. But\nit can never do this while shielding the Leviathan of sins. Slavery is\na system of barbarism which must necessarily be destroyed in order to\nthe evangelization of America and of the world. The tyranny, injustice\nand cruelty of masters, and the ignorance, servility and general\ndegradation of slaves are inconsistent with christianity, and to\nsanction these is to sanction and sustain sin, and interpose a barrier\nto the progress of truth and righteousness. And in addition to this, a\nchurch must have a character to give it influence with men. A church\nwithout character for disinterestedness, benevolence and truth, will be\ndespised by men and forsaken of God. A slaveholding church is without\na good moral character, and hence lacks moral power. Men will be slow\nto believe that, while fiercely defending a monstrous national sin, it\nis in earnest in its opposition to lesser crimes and trivial wrongs.\nHow powerless is a body of christians whose virtue gives way under the\ntemptation of a popular and lucrative vice! How justly branded with\ncowardice and hypocrisy!\n\n5. Duty to slaveholders demands non-fellowship with slaveholding.\nThe course pursued by the popular churches involves the souls of\nslaveholders in imminent peril. Their consciences are lulled into\nquietude or narcoticized by deadly moral nostrums, skillfully prepared\nand treacherously administered by time-serving, fleece-seeking\nhirelings, who assume the sacred office of shepherds. Many of them are\nnot aware of their sin and danger, and how can they be aroused while\nhonored in the church and flattered as good christians, and imitators\nin the slaveholding business, of the good old patriarchs? To save these\nmen the church must be plain with them, and require repentance of _all_\ntheir sins, and _especially_ of the sin of slaveholding, as a condition\nof a place in the temple of God.\n\n6. Duty to the slave demands non-fellowship with slaveholding. The\noppressed have a claim upon the church, because Christ died for them,\nand they are, while enslaved, in such a situation that they can neither\nlove him with all their powers, nor do much to establish his church\nand publish his name in the earth. Hence it is the duty of christians\nand christian societies to break off the fetters which bind not only\ntheir limbs but their minds. THE AMERICAN CHURCH IS ABLE TO EMANCIPATE\nEVERY SLAVE IN THE LAND. Who doubts that it is its duty? But in order\nto do this glorious work, the principle of strict non-fellowship with\nslaveholders must be adopted. Let every church in America declare\nslavery to be a sin and exclude slaveholders from its communion, and\nthe doom of slavery will be sealed. All the laws and compromises and\ncompacts which the ingenuity of the prince of darkness could invent\nwould not preserve it. It is the church which is the bulwark of\nslavery. Not one day could it stand up in this country without the\nstrength imparted to it by a powerful but awfully corrupted church.\n\"Let all the evangelical denominations,\" says _Albert Barnes_, \"but\nfollow the simple example of the Quakers in this country, and slavery\nwould soon come to an end. There is not vital energy enough; there\nis not power of numbers and influence enough, out of the church, to\nsustain it. Let every religious denomination in the land _detach\nitself_ from all connection with slavery, without saying a word against\nothers; let the time come when, in all the mighty denominations of\nChristians, it can be announced that the evil has ceased _with\nthem_ FOREVER, and let the voice from each denomination be lifted up\nin kind, but firm and solemn testimony against the system--with no\n'mealy' words; with no attempt at apology; with no wish to blink it,\nwith no effort to throw the sacred shield of religion over so great\nan evil--and the work is done. There is no power _out_ of the church\nthat could sustain slavery an hour if it were not sustained _in_ it.\"\nHence the reasons for non-fellowship with slaveholding are as vast\nas the interests temporal and eternal of millions and millions of\nour fellow-creatures, and as vast as the treachery which leaves them\nin chains! Depend upon it the curse of God will come down upon the\nAmerican church in a storm of fiery vengeance if it arise not and do\njustice to the slave!\n\n7. If slaveholders are admitted to church-fellowship _no class of\nsinners on earth should be excluded_. The church cannot consistently\nexpel from its communion the rich man who grinds the face of the poor\nlaborer that reaps down his fields, and at the same time retain the\nslaveholder who lives _entirely_ upon the unpaid labor of the poor.\nHe who _occasionally_ cheats his neighbor out of a few dollars cannot\nconsistently be censured by the church while the man who cheats whole\nfamilies out of domestic comfort, home, education, and their all,\npasses without reproof. The occasional adulterer cannot receive church\ndiscipline in the presence of him who compels his slaves to live\ntogether without the sanction, and without the protection of the law.\nHe who steals a sheep cannot be cast out from a church in which he\nwho steals men occupies a high seat. As slaveholding is a violation\ndirectly or indirectly of every commandment of the Decalogue, if it\ncannot and must not be disciplined, then church discipline is useless;\nand all classes of sinners should be admitted and retained in this\nHoly Temple, unless the principle be established that he who commits\na _petty_ offense shall be cast out, but he who has the heart and\ncourage to commit a high offense, a daring crime, shall remain in full\nfellowship. I have wondered how slaveholding church members could try\nand expel from a religious society a poor <DW64> who, in addition to\nhis peck of corn per week, had stolen a little meat, while they were\nconscious of robbing that same <DW64> of the products of his daily toil,\nand of his own soul and body.\n\n8. To maintain its _independence_ the church must discard fellowship\nwith slaveholders.--In no case have slaveholders been willing to\noccupy an humble position in a religious body long. They assume to\nbe pre-eminently _the_ members of the church, and the press, pulpit,\nand General Assembly or General Conference, _must_, unequivocally,\nendorse, or patriarchalize their slaveholding. The history of all\nthe pro-slavery churches in America is proof of this remark. A few\nslaveholders are able to change entirely the action of a powerful\necclesiastical body--to range it on the side of oppression, to silence\nor subborn its witnesses, to shut up its sympathies and take away the\nbow of hope from the slave. How many of the hundreds of ministers in\nthe whole south are free to utter their convictions on slavery to\nday? How many religious presses are unfettered? If then the church\nwould stand upon the solid rock of truth, unawed by the popular will,\nuncorrupted with gold, the immutable friend of _man_, proclaiming\nand enforcing the _whole_ truth, it must keep out of her communion\nlegalized and practiced tyrants.\n\n9. Regard for decency, refined sensibility and common humanity, urges\nnon-fellowship with slaveholders. The members of a slaveholding church\nbecome insensible to the grossest outrages upon the better feelings\nof slaves, and they habitually commit acts, without a blush, which,\none should think, would pale the cheek of a demon. For illustration\ntake a well authenticated fact: \"A runaway slave in 1841, assigned the\nfollowing as the reason why he refused to commune with a church of\nwhich he was a member. 'The church,' said he, 'had silver furniture\nfor the administration of the Lord's supper, to procure which they\nsold _my brother!_ and I could not bear the feelings it produced to\ngo forward and receive the sacrament from the vessels which were the\npurchase of my brother's blood!'\" But the members of that church,\ngenerally, were altogether without feeling upon the subject, and were\nas little disturbed in selling a slave to purchase silver ware for the\nsacramental table, or to pay a parson, or to support a missionary, as\nin selling a mule for the same purposes.\n\n10. If slavery be fellowshiped in the church, then _slaveholding\npreachers_ will be coming around and preaching the gospel to us!\nA dealer in human flesh will undertake to teach us to be just and\nmerciful. We will be expected to receive the elements of the holy\nsacrament from hands that use the cowskin occasionally on the backs\nof slaves! It is notorious that churches which fellowship slavery\nhave an exceedingly dumb and callous ministry on the subject of\noppression. Frederick Douglass, I think it was, who said that the\nhardest master he ever served was a Methodist Protestant preacher.\nThe following incident will illustrate this thought: \"A minister of\nthe gospel owned a female slave, whose husband was owned by another\nman in the same neighborhood. The husband did something supposed to\nbe an offense sufficient to justify his master in selling him for the\nsouthern market. As he started, his wife obtained leave to visit him.\nShe took her final leave of him, and started to return to her master's\nhouse. She went a few steps and returned and embraced him again, and\nstarted a second time to go to her master's house; but the feelings of\nher heart again overcame her, and she turned about and embraced him\nthe third time. Again she endeavored to bear up under the heavy trial,\nand return; but it was too much for her--she had a woman's heart. She\nreturned the fourth time, embraced her husband--and turned about,--A\nMANIAC!\"--(Anti-slavery Record.)\n\nGood God! can any one plead for the admission of such cruelty into the\nbosom of the church and into the ministry?\n\nAnd let it be remembered that this preacher simply did what the legal\nrelation authorized, and what all slaveholding ministers may do without\necclesiastical censure.\n\n11. If slavery be fellowshiped in the church, then we shall be\ncompelled to sit in religious meetings, class-meetings and conference\nmeetings, and hear a good experience told by one who lives on the\ntoil of wretched slaves, and who would sell at public sale one of our\nown brethren in the Lord, yea, even ourselves, if the _laws_ would\nallow it. Take the following specimen of a Methodist _sister_, and ask\nyourselves how you would like to attend class with her.\n\n\"A poor woman was put in jail about a week since. It is the jail that\ncost the people of the United States nearly, or quite, $60,000. Had\nthis woman committed crime? Not the least in the world. Her mistress\nwants to sell her, and pocket the money--that's all. She puts her into\njail simply to know where she is when she finds a customer. This poor\nwoman, offered for sale, expects to be _confined_ in a few weeks. She\nhas a husband and mother, but neither of them are allowed to go into\nthe jail to visit her. The husband tried to talk with her through\nthe grated window, the other day, but was driven off by some menial\nof the establishment. Amanda, the slave-woman, is a member of the\nMethodist Church, which takes the name of Bethlehem. I hear she is in\ngood standing in the Church, and sustains a fair and good character\ngenerally. The mistress--the owner--the trader--who is she? She is\nMiss A. B., a venerable spinster, a few years ago from Virginia,\nand now residing in this city. She brought with her this woman, her\nmother, and two or three children, upon whose wages she has lived for\nyears past, and now proposes to put Amanda in her pocket. She (Miss\nA. B.) is a member of the Methodist Episcopal Church, belongs to the\nM'Kendree Chapel congregation, and attends _class_ regularly. I am\nglad to say some of the brethren are a little _stirred_ about this\ntransaction.\"--_Elliott_, _page 73._\n\n\"A _little stirred_!\" Indeed! One would think they would have stirred\nthat villainous woman out of the Church in short metre, or stirred out\nof it themselves. But no, they were _only_ \"_a little_ stirred!\"\n\n\n\n\nCHAPTER XIII.\n\nSlavery and the Church.\n\nOBJECTIONS TO THE EXCLUSION OF SLAVEHOLDERS ANSWERED.\n\n\n1. It has been objected that fellowship should not be withdrawn from\n_all_ slaveholders, because some of them are exceedingly kind to\ntheir slaves. To this it may be answered that it is impossible for a\nmaster to be really very kind to those he holds in slavery, because\nthe _holding_ of them in that relation is _extreme_ unkindness. A kind\nslaveholder? What entitles him to that character? Does he renounce the\nclaim of _property_ in his slaves? No. Does he _hire_ them to work for\nhim and _pay_ them when the work is done? No. Does he open a school on\nhis plantation for their mental and moral culture? No. Does he permit\nhis slaves to instruct each other in the rudiments of education? No.\nDoes he use his influence to have the diabolical laws enacted to crush\nthe manhood out of the <DW52> man, repealed? No. Does he secure his\nslaves against the chances of the inter-state slave trade--against\nsale at auction for his debts--against the lash of a Legree? No. What\nthen entitles him to the character of a _kind_ slaveholder? Why he\nsimply treats them as a good man treats a fine horse or a favorite dog.\nHe feeds them well, works them moderately, whips but little, but robs\nthem of _all_! We abuse language when we say--a _benevolent_ robber, a\n_gentlemanly_ pickpocket, an _honorable_ pirate or a _kind_ slaveholder.\n\nThe poet, _Longfellow_, while traveling in Va., became acquainted with\nan honest old slave owned by a fine specimen of a kind, Christian,\nPresbyterian slaveholder. Said he:\n\n\"Calling at a blacksmith's shop for a small job of work, I found the\nsmith was a slave. On inquiring to whom I should make payment, he told\nme I might to him. His practice was to receive all the money paid at\nthe shop, and pay it over to his master at night. I asked him how his\nmaster knew whether he rendered a just account. He replied, that he\nknew him too well not to trust him. That, as wrong as his master did\nby him, it was no excuse for him to do wrong by his master. He could\ndeceive his master, but he could not deceive God, to whom he must\nrender his final account. He said he was a Baptist, and had regular\nfamily prayers. His master was a Presbyterian, to whom he gave credit\nfor good usage and good training. But as he had faithfully served him\nfifty years, he did think that he ought to have the remainder of his\ndays to himself. He regretted that he could not read the Bible; and\nI was pained to hear him attempt to quote it, he made such blunders.\nThe tears started in the eyes of the poor man as he spoke of his\nhard condition, and looked forward to death only for release from\nhis bondage. He thanked God that he had no children to inherit his\nignorance and servitude.\"\n\nThe kindness of certain slaveholders might be mere favorably considered\nif it were productive of any permanent practical benefits to the slave;\nbut while it leaves him in the depth of his wretchedness,--exposed\nto all the horrors of the worst form of slavery, it is a meritless\nthing--unworthy the name of kindness. The kind slaveholder knows\nthat when he dies his slaves will be sold at auction together with\nhis horses, cattle, and plantation. What avails his fancied kindness\nwhen he knows the horrible chances to which he subjects his helpless\nvictims. And how deeply guilty is he in the sight of God for refusing\nto break every yoke when he has the opportunity! To illustrate this\nthought and show the sequel of kind slaveholding we will subjoin\na sketch of a woman's history who was the property of a _kind_\nslaveholder.\n\n\"A kind slave-master, in one of the Carolinas, had a large family,\nof various colors, some enslaved, some free. One of the slaves was\nhis favorite daughter, and much accomplished. Dying, he willed his\nheir, her brother, to provide for her handsomely, and make her free.\nBut her brother was a slave-master, and she was a slave. He kept and\ndebauched her. At the end of four years he got tired of her; and that\nnotorious slave-dealer, Woolfork, coming down to collect a drove, he\nsold his sister to him. \"There is her cottage,\" said he to Woolfork;\n\"she is a violent woman. I don't like to go near her; go and carry her\noff by yourself.\" Woolfork strode into the cottage, told her of the\nfact and ordered her to prepare. She was dreadfully agitated. He urged\nher to hasten. She arose and said, 'White man, I don't believe you. I\ndon't believe that my brother would thus sell me, and his children. I\nwill not believe unless he come himself.'--Woolfork coolly went, and\nrequired her brother's presence. The seducer, the tyrant came, and,\nstanding at the door, confirmed the slaveholder's report. 'And is it\ntrue? and have you sold me?' she exclaimed. 'Is it really possible?\nLook at this child! Don't you see in every feature the lineaments of\nits father? Don't you know that your blood flows in its veins? Have\nyou, have you sold me?' The terrible fact was repeated by her master.\n'These children,' said she, with a voice only half articulate, 'shall\nnever be slaves.' 'Never mind about _that_,' said Woolfork, 'go and get\nready. I shall only wait a few minutes longer.' She retired with her\nchildren. The two white men continued alone. They waited. She returned\nnot. They grew tired of waiting, and followed her to her chamber. There\nthey found their victims beyond the reach of human wickedness, bedded\nin their blood.\"--(Anti-Slavery Record.)\n\n2. Slaveholders ought not to be excluded from the church, it is argued,\nbecause their views and feelings on the subject of slavery have been\ncorrupted by the prevalence of this popular sin. They are not, it is\nmaintained, individually responsible--the fault--the sin, the shame\nattaches to a false public morality. Dr. McClintock offers this\nobjection in the following words: \"Their position,\" he says, \"has the\neminent unhappiness of almost necessitating a feeble or corrupt moral\nsense on this subject; they are carried along by a great movement that\nabsorbs their individuality, so to speak; the personal conscience is\nlost in the general sense of the community. The great work to be done\nis to purify that general sense; not to curse and malign individual\nslaveholders, but to break up the false public morality in which the\nsystem finds its main support.\"[22]\n\nWe answer that no man is excusable for falling in with a \"great\nmovement\" which is manifestly wicked. Noah, Lot, Abraham and Elijah\nwere not carried along with sin in this way. Their moral sense was\nneither enfeebled nor corrupted by the prevailing vices. The apostles\ndid not lose their \"personal conscience\" in the \"general sense\" of\nidolatrous communities, in the midst of which they labored. And in\n_no case_ does the Bible excuse a sinner because of the prevalence\nof sin.--Idolaters were not taken into church because that vice was\nsustained by law and prevailing custom. And he who lived in Corinth in\nthe days of St. Paul, found himself in the midst of gross, shameless\nsensuality--and it was quite easy for such a person to fall in with\nthe vices for which that city was notorious; and some Christians did\nfall in with those vices. But did St. Paul excuse them, and forbid\ntheir expulsion from the church, throwing the blame of their conduct\nupon the prevailing vice? Did he ordain that until the \"general\nsense\" were purified, the \"fornicator,\" the \"incestuous person\" and\nthe \"drunkard\" must remain in the church? By no means. He knew that\nthe _public_ conscience was made up of _individual_ consciences--that\npublic corruption was the aggregate of individual corruption--and hence\nthat the only possible method of reaching and purifying the general\nsense, was by reaching and purifying the individual sense. And hence\nindividual purity was required as a condition of church membership.\nChurches now proceed precisely upon this principle in relation to all\nsins, however prevalent, slavery excepted; and no good reason can be\noffered for making it an exception. And if slaveholders _have_ an\nenfeebled moral sense, which is certainly the case, it is because the\nministry and church have been recreant to duty and truth, and have\nsaid to them \"peace, peace, when God had not spoken peace.\" The only\nway to prevent them from being swept along by the flood tides of this\ndevastating iniquity until they launch upon the shoreless sea of wrath,\nis to sound the alarm! But alas, those watchmen who have their ear are\napt to say to them, do not be alarmed--the \"false _public_ morality\"\nwill be a satisfactory apology for your sins! When asked by the judge\nwhy you were an oppressor, you can answer, that you only followed the\nprevailing example!\n\n3. Slavery, it is objected, is a _political_ question and hence the\nchurch ought not to meddle with it. We answer, that slavery is not\nonly a political, but a moral question--it is a question concerning\nthe rights of man, and all that concerns _man_ concerns a christian.\nTemperance is made a political question, should the church therefore\nfellowship the drunkard? The observance of the Sabbath is a political\nquestion--must the church therefore drop it, lest it be entangled with\npolitics? The same may be said of gambling, perjury and theft.\n\n4. But, says one, the laws uphold slavery, and whatever of blame\nattaches to slaveholding is justly chargeable to the laws. To this\nit is answered that slaveholders are the _makers of their own laws_,\nand hence are responsible for them. But if they had no voice in the\ngovernment it would be impossible to shift the responsibility of\nslaveholding upon the laws, because, in the first place, a good man\ncannot innocently avail himself of the provisions of laws which permit\nhim to injure his fellow creatures; and in the next place, the laws\ncompel no one to hold slaves. They allow it, but do not require it.\n\n5. But some, it is urged, are slaveholders from necessity, hence they\nought not to be blamed. This cannot be. The laws do not compel people\nto buy, steal, trade for, receive as a gift, or inherit slaves. Any\none may refuse to own this kind of property unless he is an idiot or\na child. And if by any means a man finds himself in possession of\nslaves he can emancipate them. It is not far to the free states. Why do\nnot those pious Methodists and Presbyterians, who are always talking\nof the impossibility of \"_getting rid_\" of their slaves, permit the\nabolitionists to help them? They would cheerfully pilot them, or give\nthem a free passage on the Under-Ground Railroad! But all those pious\nslaveholders from necessity are ready to lynch or imprison any man who\nmay undertake to release them from the \"necessary evils\" of slavery. A\nslaveholder from necessity is one who holds slaves _because holding_\nthem is a necessary condition of _robbing_ them.\n\n6. But the church has no right to ask a man to give away his property\nand impoverish himself. Yes, the church has a right to require a man\nto restore stolen property, and this is the kind of property slaves\nare. As to impoverishing slaveholders, there is danger of that, but\npoverty is no crime and is often good for the soul. It is better to be\na Lazarus in this world with his future, than a Dives with his future.\nAnd besides, there is no law of God allowing a man to roll in wealth\nacquired by robbery.\n\n7. Nothing can be said against some slaveholders _only_ that they\nhold slaves. In every other respect they are christian-like in their\nconduct, and it seems hard to exclude such fine people from the church.\n\nAlas that any christian should speak of slaveholding as \"_only_\"\na small objection. But one sin may ruin the soul. Some men are in\nevery respect excellent persons _except_ that they are addicted\nto intemperate habits, to lying, or to licentiousness--shall they\ntherefore be excused for their besetting sin, and allowed to indulge\nit? One who has cheated a poor white neighbor out of only one year's\ntoil, ought never to be admitted into the church until he makes\nrestitution. So in the case of a slaveholder--let him be just to\n_every_ creature of God--let him give up his idol or serve it in its\nappropriate temple, and not disgrace the church of God with its image\nand worshiper.\n\n8. It has been maintained that slaveholders should be taken into the\nchurch that they may come under the direct influence of the gospel,\nthe tendency of which is to destroy slavery. We answer--_a._ The\nsame reason might be urged with equal force for the admission of the\ndrunkard, liar, thief or adulterer.\n\n_b._ Experience proves that slaveholders, when admitted to church\nfellowship, are not more likely to emancipate their slaves than others.\nThey are apt to settle down in the belief that it is right to hold\nslaves, and the height of impertinence for any one to meddle with them\nabout it. A minister in Kentucky, Rev. Mr. Fee, who is well acquainted\nwith this subject from experience and actual observation, says of the\nslaveholder--\"The way to lull his conscience on the subject is, to\nbring him into the church in the practice of his sin. I know repeated\ninstances of persons whose consciences and hearts, at the time of\ntheir awakening, seemed to be tender on the subject of slaveholding.\nBut after they had been fully received, and a few comfortable meetings\npassed over, they became wholly indifferent; and after hearing or\nreading one or two pro-slavery sermons, declaring slavery to be a Bible\ninstitution, they were almost ready to seize the torch, and apply the\nfires of persecution to the individual who would disturb their Zion.\nThe place to induce the slaveholder to give up his sin is at the\ntime, or before, he enters the door of the church; before he has been\npronounced as being in a salvable state; for 'all that a man hath will\nhe give for his life.'\"\n\nBut this is no abstruse question as \"cotton Divines\" would persuade\nus. Slaveholding is a wicked business and must be treated as such.\nIt is impossible to treat it as such while fellowship is extended to\nslaveholders. The christian is bound to refuse that fellowship. If any\nbranch of the church officially or practically sanctions slavery and\nendorses the piety of slaveholders, then, in order to be consistent and\nsafe, a christian must come out of that church, because in it, he will\nbe a partaker of its sins and a sufferer of its plagues.\n\n\n\n\nCHAPTER XIV.\n\nPolitical Duties of Christians.\n\nTHE EXTIRPATION OF SLAVERY FROM THE WORLD.\n\n\nCivil government is necessary to the preservation, prosperity and\nsafety of society. In some important sense, \"the powers that be, are\nordained of God.\" It does not appear that the Creator has established\nany specific _form_ of government, but the genius of christianity is\nevidently democratic. The leading _objects_ of government are defined\nto be \"the punishment of evil doers and the praise of them that do\nwell.\" When a government fails to protect and encourage the good and to\npunish evil doers,--when it becomes a mighty engine of oppression, the\nobject of its institution is frustrated.\n\nIn the United States the voters are responsible for the character\nof the government. The people are the sovereign rulers. The ballot\nbox controls legislation. If our country is badly governed it is the\npeople's fault.\n\nThe free white people of America are responsible for the existence\nof American slavery. They could at the ballot box break every yoke.\nThey have the power to release more than three millions of slaves and\nthereby make heaven and earth rejoice!\n\nA weighty responsibility, therefore, rests upon voters in relation to\nslavery. If it continue, it will be because they shall will it, and\nexpress that will at the ballot box. He who votes for a representative\nthat is pledged to sustain slavery, becomes responsible for that\nrepresentative's acts on the slavery question. The responsibility\ncannot be shifted or dodged. Representatives consult the will of\ntheir constituents and act as they wish them to act. They are only the\npeople's agents, the echo of the people's voice.\n\nIn the light of these facts how can a christian vote for a slaveholder\nor a friend of slavery? How can he, by his vote, say that slavery\nshall be perpetual? Every pulsation of a christian's heart beats in\nharmony with liberty; he could not have slaves in his own hands. How\nthen can he, how dare he, by his vote, chain them and deliver them over\nto the slave driver? It is mean and wicked for a strong man to beat\na weak one, but it is equally as mean and wicked to _hold_ the weak\nman so that the strong one may beat him at his leisure and with ease.\nSo it is bad to own a slave and tax his sinews, sweat and blood, to\nbeat and bruise him, but it is equally wrong to hold the slave while\nthe southern slaveholder does the same thing. Hence, he who votes for\npro-slavery representatives, votes for slavery and all its swarms of\nevils, and is indirectly a slaveholder himself.\n\nLet it be distinctly understood, then, that political power has been\nentrusted to the christian people of America by the God of nations,\nwho holds them responsible for its proper exercise; and that acting\npolitically is a serious business, affecting the interests directly,\nin this country, of twenty millions of freemen, and more than three\nmillions of slaves; and also affecting indirectly, the interests of the\nwhole human family.\n\nIf the supporters of slavery continue to control the policy of the\nAmerican government; to trample under foot the \"higher law;\" to render\nthe Declaration of Independence a nullity; to denationalize liberty;\nto nationalize slavery and perpetuate and extend it; and thus to belie\nall our professions of Democracy, and render this government a Godless\ntyrant, delighting in crushed hopes and hearts--then the whole human\nrace may weep. That our government has been progressing toward this\nterrible consummation for the last thirty years is but too evident.\n\nThe Declaration of Independence is a sound anti-slavery document. It\ndoes not regard the right of all men to liberty as an unsettled opinion\nor a question to be proved by abstruse argument, but pronounces it a\n\"SELF EVIDENT TRUTH.\"\n\nThe Constitution in _form_ if not in fact, pretty fully embodies the\nsentiments of the Declaration. The word slave is not found in it, and\nit was kept out not accidentally, but purposely. The framers of the\nConstitution carefully guarded that instrument against any endorsement\nof slavery. In the convention which formed the Constitution, Gov.\nMorris of Pennsylvania said, \"He never would concur in upholding\ndomestic slavery. It was a nefarious institution.\" Mr. Getry, of\nMassachusetts, in the same convention said, \"we had nothing to do with\nthe conduct of the States as to slavery, _but we ought to be very\ncareful not to give any sanction to it_.\" The idea that there _could_\nbe property in man was carefully excluded from the Constitution. It was\nabout to be foisted into that instrument by the adoption of a report of\na committee fixing a tax on importations. But Mr. _Sherman_ was against\n\"_acknowledging men to be property_, by taxing them as such under\nthe character of slaves.\" _Madison \"thought it wrong to admit in the\nconstitution the idea that there could be property in man.\"_ But if the\nidea of property in man was carefully excluded from the Constitution,\nthen it is clear that chattel slavery is not in form recognized, much\nless established by that instrument.\n\nIt is evident that the framers of the Constitution expected the speedy\nabolition of slavery; and hence, while providing in fact though not\nin form, for its continuance under the constitution, by virtue of\n_local_ State laws, they so framed that instrument that it would not\ncountenance slavery or deny the glorious doctrines of the immortal\nDeclaration, which contained what Mr. Sumner calls \"the national heart,\nthe national soul, the national will, and the national voice.\"[23]\n\n_Washington_ said \"That it was among his _first wishes_ to see some\nplan adopted, by which slavery may be _abolished_ by law.\"\n\n_Adams_ regarded slavery as \"a sacrilegious breach of trust.\"\n\n_Hamilton_ considered slaves, \"though free by the law of God, held in\nslavery by the laws of men.\"\n\n_Jefferson_ said that the \"abolition of domestic slavery was the\ngreatest object of desire.\"\n\n_Patrick Henry_ said--\"I will not, I cannot justify it.\"\n\n_Benj. Franklin_, when 84 years of age, came up before Congress with\na petition from the \"Abolition Society of Pennsylvania, praying that\nbody to countenance the restoration of liberty to those unhappy men,\nwho alone, in this land of freedom are degraded into perpetual bondage,\nand who, amidst the general joy of surrounding freemen are groaning in\nservile subjection.\" This petition besought Congress to \"step to the\nVERY VERGE of the power vested in them for _discouraging_ every species\nof traffic in the persons of our fellow men.\"\n\nThese facts afford conclusive evidence, that the founders of the\nAmerican Republic did not intend to fasten upon the object of their\ntoils, perils and sacrifices, a monster which would speedily eat out\nits virtue, destroy its vitality and overthrow it forever.\n\nBut the policy of the government has been reversed. Millions of\nacres of territory have been purchased and annexed to make room for\nslavery, which has become a great national pet--the god before whom\naspiring politicians must kneel and worship as a condition of political\nelevation.\n\nThe President of the United States and his Cabinet, the Supreme\nCourt, and both Houses of Congress are all under the control of the\nSlaveocracy. No man can be a President of the United States unless he\nbows the knee and swears upon the altar of this modern Baal. Zeal for\nthe infamous Fugitive Slave Law is now a particular test of political\northodoxy. A Congressman who advocates the principles of Washington,\nFranklin and Jefferson is considered as standing outside of any\n\"healthy organization\" and is not deemed worthy of a place on the\nmost insignificant Congressional committee. Our government has been\nthoroughly changed from an anti-slavery to a pro-slavery government.\n\nIn view of these facts how important that the concentrated moral and\npolitical power of every American christian be brought to the rescue of\nour great Republic from the sin and shame of its present position.\n\nChristians, in the States where slavery exists, are under obligations\nto use their whole political and moral power to bring about the speedy\nrepeal of the entire slave code. That code is a miserable barbarism and\nshould be swept away forever from the statutes of Christian States.\nMy christian brethren in Virginia, Kentucky and Missouri, are you\nprepared to use all the power, moral and political, with which you\nare entrusted, as you shall answer to God, for the emancipation of\nyour suffering fellow citizens? Your political influence must tell\nsomewhere! Remember _that_.\n\nChristians in the free States are obliged to do what is in their\npower for the repeal of all laws which bear upon the <DW52> man\nbecause he is a _colored_ man. The word \"white\" ought to be erased\nfrom the statutes of all christian States. All \"black laws\" are\nanti-democratic, anti-christian, and not only insult and annoy, but\ndiscourage the <DW52> man and obstruct his progress in the path of\nimprovement.--Christian brethren of the free States, you have not done\nyour duty toward your  brother. You have sustained laws which\ngall his neck as a heavy yoke. You have treated him as an alien and an\nenemy. Will you henceforth do him justice, as you shall answer to God?\n\nChristian citizens of all the States are directly responsible for the\nexistence of slavery in the District of Columbia, and they should not\nbe content until that foul pollution is wiped away from the Capital of\nour country. Slavery at Washington is especially a national disgrace, a\nblistering shame, a satire upon our professions.\n\nWhen the foreign minister or visitor comes to our country, and goes\nto Washington, he sees in the streets, at the hotels, and everywhere,\na poor, stupid, oppressed people, whose very speech and looks betray\ntheir ignorance and servility. Ah! Is this American freedom? Equality?\nRepublicanism? Upon inquiry, he finds that one-seventh of all the\npeople are in this state of servile wretchedness.\n\nAnd when a member of Congress from a free State goes to the proud\nCapital of his country, he beholds passing by the tall and splendid\nbuildings of the government, droves of men, women and children, chained\ntogether,--some sullenly indifferent to their fate--others weeping as\nif their hearts would break.--Who are these? American citizens!\n\nMen, as white as some members of Congress, and women as fair as their\nwives and as virtuous as their daughters, are cried off at auction to\nthe highest bidder, in Washington!\n\nThere our senators and representatives sit and legislate, in sight\nof the slave prison, and slave market--in hearing of the clanking of\nchains, and coffles,--and of the wail of slave mothers, weeping for\ntheir children, because they are\n\n    \"Gone, gone, sold and gone.\"\n\nThey are also responsible for the extension of slavery into territory\nnow free. If they go not to the utmost verge of their power to save\nthe Lord's free earth from the overspreading and blighting curse of\nslavery, they cannot but be execrated by an enlightened posterity.\n\nBut more than all this. A christian is a citizen of the world, and\nhence is required to employ the whole force of his moral and political\npower for the extirpation of slavery from every State in the Union,\nand from every country on the globe. The influence of an intelligent,\nactive christian citizen is worldwide. He cannot be the dupe or tool\nof any party; he is never shackled by party organizations; he does not\ncommit the keeping of his conscience to political leaders. He sincerely\nloves God, believes the Bible, and loves his fellow-men, because\nthey are men. Prejudice, caste, and all other relics of barbarism,\nhe has thrown away. He talks, votes and prays for universal liberty\nand righteousness. In the pulpit, in the shop, on the farm, anywhere,\neverywhere the whole weight of his influence is thrown against slavery\nin the territories, in the District of Columbia, in the States, and\nagainst it wherever it exists in the world. As he seeks for the\nphysical, intellectual and moral improvement and happiness of all men,\nhe must desire intensely the speedy extirpation of slavery from the\nearth.\n\nChristian voter, when you approach the ballot box, think of the three\nmillions of bondmen who are holding up their hands \"all manacled and\nbleeding,\" pleading to you for deliverance!\n\n\n\n\nCHAPTER XV.\n\nAbolition of Slavery.\n\nIMMEDIATE EMANCIPATION.\n\n\n    \"Long has thy night of sorrow been,\n    Without a star to cheer the scene.\n    Nay; there was One that watched and wept,\n    When thou didst think all mercy slept;\n    That eye which beams with love divine\n    Where all celestial glories shine.\n    Justice shall soon the sceptre take;\n    The scourge shall fall, the tyrant quake.\n    Hark! 'tis the voice of One from heaven;\n    The word, the high command is given,\n    'Break every yoke, loose every chain,\n    To usher in the Savior's reign.'\"\n\nMany persons, who appear to be sensible of the evils of slavery, seem\nutterly at a loss for some feasible method of abolishing it. \"It is\nhere in our midst,\" say they, \"and how are we to get rid of it?\"\n\nTo this question we have a plain scriptural answer. \"LOOSE THE BANDS\nOF WICKEDNESS,\"--\"UNDO THE HEAVY BURDENS,\"--\"LET THE OPPRESSED GO\nFREE,\"--\"BREAK EVERY YOKE,\"--\"PROCLAIM LIBERTY THROUGHOUT ALL THE LAND,\nUNTO ALL THE INHABITANTS THEREOF.\"\n\nImmediate, unconditional, universal emancipation is the only just, the\nonly reasonable and the only possible method of adjusting the slavery\nquestion. To this measure the people of the United States _must_ come.\nA general Jubilee is inevitable. Slavery is an unmitigated wrong.\nEvery element of it is at variance with the happiness of man and the\nlaw of God. It is without a single redeeming principle, and hence its\ndestruction--its total annihilation is necessary.\n\nSince the gigantic wrongs of slavery have been so generally made known\nas somewhat to arouse the public conscience from its long sleep, some\nwriters, anxious to preserve the system, have proposed to _reform_ it.\nThey say, \"Slavery, of itself, is a very innocent relation, but its\nevils are horrible. Let us correct the evils and preserve the system.\"\n\nBut slavery cannot be reformed, so as to make it a tolerable\ninstitution because its essential feature--viz, _property in a human\nbeing_, is, wherever imposed, an outrageous, an insufferable wrong. Who\nwould think of reforming robbery--of making laws to regulate robbers\nin their trade--and to prevent brutal men from engaging in it? What if\nit should be enacted by grave senators that none but gentlemen should\nrob, and that they must do it genteelly--using no _unnecessary_ cruelty\nor coercion? All the world would laugh such senators to scorn. But\nslavery is from beginning to end a system of robbery, which it is as\nimpossible to reform, so as to take away its \"evils,\" as it is to so\nreform piracy as to destroy its evils, and make it a humane, just and\nchristian trade.\n\nBut the American slaves, it is maintained, are _not prepared for\nfreedom_. This objection is without foundation. God _creates_ men free,\nand sends them forth into the world with such endowments as are needed\nin a state of freedom, and as are suited to _no other_ state. To say\nthat a race, which God has _created_ free, is unprepared for freedom is\nto reproach the Maker. Freedom is the native element of man. And\n\n    \"The heavens, the earth, man's heart and sea,\n    Forever cry, _let all be free_!\"\n\n\"Not prepared for freedom?\" This has been the watchword of oppressors\nin all ages. The \"people,\" the uninformed \"masses,\" have, in the\nestimation of tyrants, always been prepared for slavery and injustice\nof every kind, but never for freedom. And it has ever been their policy\nto render them less fit for any station or any responsibility in life.\nThey never put forth an effort to prepare their victims for any higher\nbusiness than obsequious submission to usurped authority. True to this\nspirit, those who are most noisy about the unfitness of slaves for\nfreedom, are most zealous for the maintenance of those odious laws and\nusages which shut them out from all chance of mental and moral culture.\n\nAnd if the slaves are unprepared for freedom, what is to prepare them\nfor it? Their present degradation is owing to slavery, and it is not\nlikely that the continuance of the _cause_ of their degradation will\nelevate them. Remove the cause, and the effect will cease. Emancipate\nthe <DW52> man, open to him our schools and colleges, place before him\nmotives for action such as animate freemen, and swell the hearts of\nChristians, give him an _opportunity_ and he will prove himself every\nwhit a MAN. How mean and hypocritical the objection, that slaves are\nnot prepared for freedom, when we employ the whole weight of our laws\nand prejudices to crush out their manhood, and as far as possible unfit\nthem for any condition except that of working animals.\n\nBut thousands of slaves have fled from their oppressors, and, in the\nmidst of the greatest difficulties and embarrassments, have not only\nproved themselves prepared for freedom, but also to take a position\namongst the most cultivated and honored freemen.\n\nThe half-free <DW52> people of the United States _prove_ themselves\nworthy of all the rights of American citizens.\n\nThere are now in Canada about 35,000 fugitive slaves; and no\npeople have ever entered upon the possession of freedom under more\nembarrassing circumstances. They were born in chains. The iron yoke\nhad galled their necks. Their backs had felt the keen lash. In their\nflight they were pursued by hungry blood-hounds and more hungry\nmarshals.--Naked, broken in spirit, impoverished and uneducated, they\nreached a cold, ungenial clime. But they were free! And those 35,000\nescaped slaves are rapidly improving in wealth, intelligence, and in\nevery social virtue. In the town of Buxton 130 families reside who own\na body of 9,000 acres of land. The fugitive slaves of Canada West now\nown 25,000 acres of land. Were they not prepared for freedom?\n\nImmediate emancipation worked admirably in the British West Indies. The\nmasters were not murdered by the emancipated slaves, as was predicted,\nbut good order reigned everywhere. The liberated people have been\nrapidly improving in intelligence and wealth.--The terrible wrongs and\nmiseries of slavery are no more. Rev. Mr. Richardson, a missionary in\nJamaica, speaking of the moral condition of those islands, says:\n\n\"Marriage is much more common than formerly, and the blessings of\nthe family and social relations are much more extensively enjoyed.\nThe Sabbath is also more generally observed. The means of education\nand religious instruction are better enjoyed, although but little\nappreciated and improved by the great mass of the people. It is\nalso true, that the moral sense of the people is becoming somewhat\nenlightened. But while this is true, yet their moral condition is very\nfar from being what it ought to be.\n\n\"Our brightest hopes and fondest anticipations must and will centre\naround the YOUTH of this island. I see the hand of Providence steadily\nurging onward, with resistless might, the car of Progress. Gaunt\nPrejudice and grim Superstition gradually give way; Darkness and Error\nrecede before the sunlight of Truth; and even the demon of Lust and\nthe giant Intemperance (twin brothers in Satan's family) are bereft of\ntheir power, and chained for a season. I see intelligence, purity, and\npiety supplanting ignorance, licentiousness, and irreligion, and this\nmoral waste becoming transformed until it blooms and flourishes as the\ngarden of God.\"\n\n\"Immediate emancipation?\" exclaims a fearful friend, \"that will never\ndo! Murder, amalgamation, and many other evils will be inevitable\nconsequences of such a measure. Let us colonize the slaves. Send them\nback to their own country.\" To these objections it may be answered,\n\n1. <DW52> men are not more inclined to murder than are white men.\nAfricans have the same natural dispositions which distinguish other\nraces.\n\n2. Many masters have emancipated their slaves, and thereby secured\ntheir undying affection. Liberated slaves have never turned with bloody\nhands upon their liberators.\n\n3. In the West India Islands 800,000 slaves were emancipated in one\nday, and although sixteen years have since elapsed, none of the\nterrible massacres which were predicted by the opponents of the measure\nhave occurred.\n\n4. This fear of the vengeance of emancipated slaves arises, doubtless,\nfrom a guilty conscience--or a feeling that it is richly deserved. A\nhighwayman robs a man, and then says, if I let him go he may have me\narrested and punished, therefore I will kill him. Americans say, on the\nsame principle, we have most terribly abused our slaves, and hence, if\nwe let them go they will retaliate, therefore, we must continue the\nwrong for self preservation!\n\n5. As to amalgamation we have only to say that slavery is an extensive\nsystem of forced amalgamation. In the free States this much dreaded\nevil is of rare occurrence. Immediate emancipation would speedily\narrest the very thing here deprecated.\n\n_a._ The colonization scheme is impracticable. Between three and\nfour millions of people can never be shipped off to Africa. It is\nimpracticable to send even the annual increase of the free <DW52>\npopulation. There are in America now about twelve millions of <DW52>\npeople, and there is no power, civil or ecclesiastical, which can carry\nthem away to Africa.[24] A few will go and ought to go as missionaries,\nbut the great and rapidly increasing masses are firmly planted on this\ncontinent and here they must remain.\n\n_b._ Forcible colonization is wrong. <DW52> people have the same\nright to live in America that white people have. The Creator made the\nearth for the habitation of man, and He has never surrendered his\nownership of it to any government. The <DW52> man has a right to\nlive in any country on the globe--a right derived from the Creator.\nHas God said that every race under heaven may have a home in America\nbut the African? Never. It is impertinent as well as wicked for one\npeople to say to another, \"you shall not live in this State, nor on\nthis continent.\" Such people arrogate to themselves a prerogative which\nJehovah only possesses.\n\n_c._ The present popular scheme of colonization leaves unquestioned\nthe title of the slaveholder, encourages the doctrine that the Bible\nsanctions the institution, appeals to the basest prejudices of\nthe American people to induce them to countenance the scheme, and\nencourages the enactment of such laws as now disgrace the statutes of\nseveral of the free States, in order, it would seem, to harrass the\nfree <DW52> man until he shall be compelled to flee from the land of\nhis birth to a distant shore for refuge. One who speaks what he knows,\nsays,\n\n\"I speak the words of soberness and truth when I say that the most\ninveterate, the most formidable, the deadliest enemy of the peace,\nprosperity, and happiness of the <DW52> population of the United\nStates, is that system of African colonization which originated in and\nis perpetuated by a worldly, Pharaoh-like policy beneath the dignity\nof a magnanimous and Christian people;--a system which receives much\nof its vitality from _ad captandum_ appeals to popular prejudices,\nand to the unholy, groveling passions of the canaille;--a system that\ninterposes every possible obstacle in the way of the improvement and\nelevation of the <DW52> man in the land of his birth;--that instigates\nthe enactment of laws whose design and tendency are obviously to annoy\nhim, to make him feel, while at home, that he is a stranger and a\npilgrim--nay more,--to make him 'wretched, and miserable, and poor, and\nblind, and naked;'--to make him 'a hissing and a by-word,' 'a fugitive\nand a vagabond' throughout the American Union;--a system that is so\nirreconcilably opposed to the purpose of God in making 'of _one_ blood\nall nations for to dwell on _all_ the face of the earth,' that when\nthe dying slaveholder, under the lashes of a guilty conscience, would\ngive to his slaves unqualified freedom, it wickedly interposes, and\npersuades him that 'to do justly and love mercy' would be to inflict an\nirreparable injury upon the community, and that to do his duty to God\nand his fellow-creatures, under the circumstances, he should bequeath\nto his surviving slaves the cruel alternative of _either expatriation\nto a far-off, pestilential clime, with the prospect of a premature\ndeath, or perpetual slavery, with its untold horrors, in his native\nland_.\"--_Watkins._\n\nMany objections are offered against immediate emancipation, but they\nare evidently mere _excuses_. This may be laid down as a safe rule:\n_Offer no objection to the manumission of slaves which would not\nsatisfy you were you yourselves the slaves to be manumitted._ Tried\nby this reasonable and scriptural rule all apologies, objections and\nexcuses offered for the perpetuation of human bondage, vanish away.\nThere can be no good reason advanced for the continuance of this curse\na single year longer. Too long already has it dishonored our churches\nand our country. Too many souls have been already involved by it in\nhopeless ruin. Too many generations of slaves have already gone in\nsorrow and despair down to their graves. Too long has the public\nconscience been debauched. Justice, humanity and religion with united\nvoice call for immediate emancipation.\n\nIf our free institutions are to be preserved they must be released from\nthe folds and the deadly charm of this monster serpent. Freedom cannot\nflourish in its coils nor survive in its slimy embrace.\n\nIndividual and national repentance and reformation only can avert the\nterrible judgments of an offended God. The cries of the oppressed have\ngone up into the ears of the Lord of Sabaoth, and he will be avenged\nspeedily.\n\n    \"We have offended. O! my countrymen!\n    We have offended very grievously;\n    And been most tyrannous. From east to west\n    A groan of accusation pierces heaven!\"\n\nThere are not more than one hundred and twenty thousand slaveholders in\nthe United States, and it would be easy for them to settle this whole\nquestion in one year or even in a day. Let them simply be honest, be\njust, obey the Bible, overcome their pride, avarice, prejudices and\nlusts, and the work will be done. The example of _Freeborn Garretson_\nis commended to the special attention of all slaveholders, and\nespecially of those who profess religion. This good man says:\n\n\"As I stood with a book in my hand, in the act of giving out a hymn,\nthis thought powerfully struck my mind: 'It is not right for you to\nkeep your fellow-creatures in bondage; you must let the oppressed\ngo free.' I knew it to be that same blessed voice which had spoken\nto me before. Till then I had not suspected that the practice of\nslave-keeping was wrong; I had not read a book on the subject, nor\nbeen told so by any. I paused a minute, and then replied, 'Lord, the\noppressed shall go free.' And I was as clear of them in my mind, as if\nI had never owned one. I told them they did not belong to me, and that\nI did not desire their services without making them a compensation. I\nwas now at liberty to proceed in worship. After singing, I kneeled to\npray. Had I the tongue of an angel, I could not fully describe what I\nfelt: all my dejection, and that melancholy gloom which preyed upon me,\nvanished in a moment, and a divine sweetness ran through my whole frame.\n\n\"It was God, not man, that taught me the impropriety of holding slaves:\nand I shall never be able to praise him enough for it. My very heart\nhas bled, since that, for slaveholders, especially those who made a\nprofession of religion; for I believe it to be a crying sin.\"\n\n\n\n\nCHAPTER XVI.\n\nWhat of the Night?\n\nHOPE THOU IN GOD.\n\n\nAre there any prospects that the long and dreary night of American\ndespotism will speedily end in a joyous morning?\n\nIf we turn our eye towards the political horizon we shall find it\noverspread with heavy clouds portentous of evil to the oppressed. The\ngovernment of the United States is intensely pro-slavery. The great\npolitical parties, with which the masses of the people act, vie with\neach other in their supple and obsequious devotion to the slaveocracy.\nThe wise policy of the fathers of the Republic to confine slavery\nwithin very narrow limits, so that it would speedily die out and be\nsupplanted by freedom, has been abandoned; the whole spirit of our\npolicy has been reversed--and our national government seems chiefly\nconcerned for the honor, perpetuation and extension of slavery.\n\nThe powerful religious denominations have been following in the wake\nof the state. Their ancient and bold testimony against slavery has\nbeen expurgated from their confessions and disciplines, or completely\nneutralized.--Slavery _as it is_ receives their unqualified sanction.\nThe giant Christian publication societies of the day so completely\nignore the question of slavery that a reader of all their books would\nnot suspect that millions of slaves are groaning under an iron yoke\nin this country. Dark as a starless, moonless midnight, is the aspect\npresented by the heavens of the popular religious denominations.\n\nAmerican prejudice is yet very powerful. The polite, educated, and\ntalented free  traveler is exposed, in most parts of the Union,\nto the coarsest insults from this gaunt demon. He feels everywhere its\nhellish power. One who was more than twenty years a slave presents in\nthe following eloquent language a true picture of the present anomalous\ncondition of the children of Ham in the midst of the general joy of\nfreedom:\n\n\"The Hungarian, the Italian, the Irishman, the Jew and the Gentile,\nall find in this goodly land a home; and when any of them, or all of\nthem, desire to speak, they find willing ears, warm hearts, and open\nhands. For these people, the Americans have principles of justice,\nmaxims of mercy, sentiments of religion, and feelings of brotherhood\nin abundance. But for _my_ poor people, (alas, how poor!)--enslaved,\nscourged, blasted, overwhelmed, and ruined, it would appear that\nAmerica had neither justice, mercy, nor religion. She has no scales\nin which to weigh our wrongs, and no standard by which to measure our\nrights.... Here, upon the soil of our birth, in a country which has\nknown us for two centuries, among a people who did not wait for us\nto seek them, but who sought us, found us, and brought us to their\nown chosen land,--a people for whom we have performed the humblest\nservices, and whose greatest comforts and luxuries have been won from\nthe soil by our sable and sinewy arms,--I say, sir, among such a\npeople, and with such obvious recommendations to favor, we are far less\nesteemed than the veriest stranger and sojourner.... We are literally\nscourged beyond the beneficent range of both authorities--human\nand divine. We plead for our rights, in the name of the immortal\ndeclaration of independence, and of the written constitution of\ngovernment, and we are answered with imprecations and curses. In the\nsacred name of Jesus we beg for mercy, and the slave-whip, red with\nblood, cracks over us in mockery.... We cry for help to humanity--a\ncommon humanity, and here too we are repulsed. American humanity\nhates us, scorns us, disowns and denies, in a thousand ways, our very\npersonality. The outspread wing of American christianity, apparently\nbroad enough to give shelter to a perishing world, refuses to cover us.\nTo us, its bones are brass, and its feathers iron. In running thither\nfor shelter and succor, we have only fled from the hungry bloodhound to\nthe devouring wolf,--from a corrupt and selfish world to a hollow and\nhypocritical church.\"--_Fred. Douglass._\n\nBut dark as is this picture, there is still hope. The exorbitant\ndemands of the slave power, the extreme measures it adopts, the deep\nhumiliation to which it subjects political aspirants, will produce a\nreaction. Inflated with past success it is throwing off its mask and\nrevealing its hideous proportions. It is now proving itself the enemy\nof _all_ freedom.\n\nThe extreme servility of the popular churches is opening the eyes of\nmany earnest people to the importance of taking a bolder position.\nThey are finding out that it is a duty to come out from churches which\nsanction the vilest iniquity that ever existed, or exhaust their zeal\nfor the oppressed in tame resolves, never to be executed.\n\nThe truth is gaining ground that slaveholding is a great sin, that\nslaveholders are great sinners, and that he who apologises for the\nsystem is a participator in the guilt and shame.\n\nFree mission societies, reform publication societies, and free churches\nare rising up all over the country, in the free and in the slave\nStates. They take their stand upon a solid Bible platform, and their\npower will be rapidly augmented until the strongholds of oppression\nwill tremble at their approach.\n\nLiterature is coming to the rescue of the slave, and even now is\npleading his cause with astonishing power in all the languages of\nchristendom.\n\nChristianity is on the side of the slave, and its true spirit is\nbeginning to be practically applied.\n\nThousands of devout persons are found day and night pleading with God\nfor the speedy deliverance of the captive.\n\nBut a voice from heaven is heard saying, \"HOPE THOU IN GOD.\" God is\non the side of the oppressed. He will never abandon them. He approves\ntheir cause, hears their cries, and is interested in all their\nmovements. Those millions of  Americans are now in the fiery\nfurnace, but He will bring them out. From their house of bondage they\nwill come forth, and accomplish a glorious mission on the earth.\nGod has reserved for them some of the grandest achievements in music,\npoetry, science, arts, morals, freedom and religion. Never has he\npermitted a people to be more deeply humbled, and none will in the end\nbe more highly exalted. God's ways are not as our ways. He can make the\nwrath of man to praise him.\n\nThe day of deliverance is not distant. God is stirring up the nations.\nThe slavery question is agitating the whole enlightened world. It\ncannot be put to rest. Politicians pronounce it dead and solemnly bury\nit, but it rises before the third day and confronts them in every\nassembly. Church councils resolve to let it alone, but it will not let\nthem alone. They hate agitation, and cry for peace, but are answered,\n\"_first pure, then peaceable_.\"\n\nGod of liberty! hasten the hour when the reddening East shall authorize\nthe joyful announcement to American bondsmen--\"_the morning cometh_.\"\nTill then let us \"TOIL AND TRUST.\"\n\n\nFOOTNOTES:\n\n[1] See Elliott on Slavery, p. 40.\n\n[2] R. Walsh, Encyclopedia Americana, Art. Slavery.\n\n[3] Here are a few advertisements taken from respectable southern\npapers, verbatim.\n\n    SLAVES WANTED.--We are at all times purchasing Slaves, paying the\n    highest cash prices. Persons wishing to sell will please call at\n    242 Pratt St. (Slatter's old stand.) Communications attended to.\n                                                B. M. & W. L. CAMPBELL.\n\n    A <DW64> FOR SALE.--I wish to sell a black girl about 24 years old,\n    a good cook and washer, handy with a needle, can spin and weave. I\n    wish to sell her in the neighborhood of Camden Point; if not sold\n    there in a short time, I will hunt the best market; or I will trade\n    her for two small ones, a boy and girl.\n\n    November 15, 1852\n                                                              M. DOYAL.\n\n    100 <DW64>s FOR SALE, at my depot on Commerce street, immediately\n    between the Exchange Hotel and F. M. Gilmer, Jr.'s Warehouse,\n    where I will be receiving constantly, large lots of <DW64>s during\n    the season, and will sell on as accommodating terms as any house\n    in this city. I would respectfully request my old customers, and\n    friends to call and examine my stock.\n\n    MONTGOMERY, November 2, 1852.\n                                                       JNO. W. LINDSEY.\n\n    GREAT SALE OF <DW64>s BY J. & L. T. LEVIN.--On Thursday, December\n    30, at 11 o'clock, will be sold at the Court House in Columbia, one\n    hundred valuable <DW64>s.\n\n    It is seldom such an opportunity occurs as now offers. Among them\n    are only four beyond 45 years old, and none above 50. There are\n    twenty five prime young men, between sixteen and thirty; forty of\n    the most likely young women, and as fine a set of children as can\n    be shown!\n\n    Terms, &c.\n                                                     December 18, 1852.\n\n[4] Prof. B. B. Edwards says--\"From the time of Augustus to Justinian\nwe may allow three slaves to one free man; we shall thus have a free\npopulation in Italy of 6,944,000; and of slaves 20,832,000.\"\n\nOn the treatment of Roman slaves _Guizot_ remarks that \"it would be\neasy to give the most frightful and heartrending accounts of the manner\nin which the ancient Romans treated their slaves. Entire volumes are\noccupied with the details.\" (_Hist. Civilization._)\n\n[5] These facts are well authenticated. The \"Union Point\" tragedy\ndid not occur in 1854, as reported recently, and denied by the \"Free\nTrader,\" _but it did occur_ in 1842, and we have quoted the \"Trader's\"\nown account.\n\n[6] Accepting this celebrated curse as an inspired prophecy, and we\nare inclined to receive it as such, it finds an easy fulfillment in\nthe conquests of Joshua over the Canaanites; in the oppression of\nthe Phoenicians, (who were descendants of Canaan,) by the Chaldeans,\nPersians and Greeks; and finally in the subjugation and destruction\nof the Carthaginians, by the Romans. This is the opinion of President\nEdwards, and it is entitled to respect.\n\n[7] I avail myself in what follows upon this point, of the\ninvestigations of Rev. E. Smith, who has thrown much light upon this\nsubject. See \"_Bible Servitude\" pp. 91_, for a full discussion of this\npoint.\n\n[8] For these criticisms on the Hebrew word the author is indebted to\nAlbert Barnes' 'Inquiry into the scriptural views of slavery.'\n\n[9] The passage in Ex. 21: 20, 21, applies, as all admit, mainly, if\nnot exclusively, to _native Hebrew servants_, and as no one finds in\nthe limited voluntary servitude of the native Hebrews a warrant for\nhereditary slavery, I have not thought it necessary to dwell upon it.\nIt may be observed, however, that the word \"_punished_,\" is rendered\nin the marginal reading, \"avenged;\" and the meaning of the law is that\nthe interest the master had in the life of the servant should be taken\nas _presumptive_ evidence that he did not intend to kill him, unless\nthe case was very clear, and hence that he should not be _avenged\nsummarily_, by a relative of the servant, but be regularly tried and\npunished by the appointed authorities.\n\n[10] See Barnes' Inquiry.\n\n[11] See Smith.\n\n[12] Tract of the American Reform Book and Tract Society.\n\n[13] Bible Servitude.\n\n[14] These facts are drawn by Jay from Blair's Inquiry into the state\nof slavery among the Romans.\n\n[15] Note. One little circumstance, which occurred in the General\nAssembly of 1853, indicates a remarkably conservative spirit in that\nbody. Dr. Judd, in writing the history of the division of the Old\nand New-Schools, put in _one_ chapter on slavery. This chapter made\nthe book _offensive_ to the south. To reconcile all parties, it was\nagreed that two editions of the work be printed, one for northern, the\nother for southern circulation--the latter to be _minus the chapter on\nslavery_!\n\n[16] The \"American Missionary Association,\" which has no fellowship\nwith slaveholding, and the American Reform Book and Tract Society,\nwhich is doing much for the dissemination of Christian anti-slavery\ndoctrines, are sustained mainly by Congregationalists. The main body of\nthe Congregationalists, however, adhere to the old Boards.\n\n[17] Can any one conceive of any _virtuous_ reason which prompted the\npassage of such a rule? Is there not a deep and dark iniquity among\nslaveholders which makes it not only necessary that slaves should be\nexcluded from civil, but also from ecclesiastical tribunals?\n\n[18] Amongst these, and at the head of them stands _Mr. Hosmer_, Editor\nof the Northern C. Advocate, author of \"_Slavery_ and the _Church_,\"\nand a number of other excellent books.\n\n[19] See Methodist Discipline.\n\n[20] Annual Report of American and Foreign Anti-slavery Society.\n\n[21] Distinguished from the Moravians, or old United Brethren by the\nadditional phrase--\"_in Christ_.\"\n\n[22] Methodist Quarterly.\n\n[23] Hon. Charles Sumner's speech on the Repeal of the Fugitive Slave\nBill, delivered in the Senate, August 1852, is one of the finest\nspecimens of eloquence in the English language. Its arguments too, are\nunanswerable.\n\n[24] The following estimate of their numbers and localities is taken\nfrom one of the able reports of the British and Foreign Anti-slavery\nSociety, carefully drawn up by its former Secretary, John Scoble, Esq.:\n\n United States,       3,650,000\n Brazil,              4,050,000\n Spanish Colonies,    1,470,000\n S. Amer. Republics,  1,130,000\n British Colonies,      750,000\n Hayti,                 850,000\n French Colonies,       270,000\n Dutch Colonies,         50,000\n Danish Colonies,        45,000\n Mexico,                 70,000\n Canada,                 35,000\n                     ----------\n Total,              12,370,000\n\n                               _Rep. Am. and For. Anti. Slav. Society._\n\n\n\n\n    Transcriber's Note:\n\n    Every effort has been made to replicate this text as faithfully as\n    possible. Some minor corrections of spelling and puctuation have\n    been made.\n\n    Italic text has been marked with _underscores_.\n    OE ligatures have been expanded.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Slavery Question, by John Lawrence\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n# FOOD  \nIS THE  \nSOLUTION\n\n# WHAT TO EAT TO SAVE THE WORLD\n\n# 80+ RECIPES FOR A GREENER PLANET  \nAND A HEALTHIER YOU\n\nBegin Reading\n\nTable of Contents\n\nAbout the Author\n\nCopyright Page\n\n**Thank you for buying this**\n\n**Flatiron Books ebook.**\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on the author, click here.\nThe author and publisher have provided this e-book to you for your personal use only. You may not make this e-book publicly available in any way. **Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at:<http:\/\/us.macmillanusa.com\/piracy>**.\nFor my sister, Hillary\u2014for opening my eyes and starting me on my path\n\nFor my mother, Janet\u2014for my senses of service and justice\n\nFor my wife, Lara\u2014for everything else in my world\n\n\"To ask the biggest questions, we can start with the personal: what do we eat? What we eat is within our control, yet the act ties us to the economic, political and ecological order of our whole planet. Even an apparently small change\u2014consciously choosing a diet that is good for both our bodies and the earth\u2014can lead to a series of choices that transform our whole lives.\"\n\n**\u2014FRANCES MOORE LAPP\u00c9, _DIET FOR A SMALL PLANET_**\n\n# FOREWORD\n\n# JAMES CAMERON\n\nIn 2012, my wife, Suzy, and I adopted a plant-based diet. Many people come to plant-based eating through health issues\u2014maybe they're facing a quadruple bypass, or want their family to eat better. For us, it was all about sustainability.\n\nI'd known for several years that the carbon footprint of animal agriculture is incredibly high. But I was under the impression that we need animal protein in our diets. Then I saw the film _Forks Over Knives_ and began reading into the topic. I learned that not only do we not _need_ animal proteins, but that they're actually _detrimental_ to us in many ways.\n\nSo, comfortable we were doing the right thing not just for the environment but also for ourselves, we went fully plant-based. We explored the huge variety of vegan foods that are out there. We got recipe books and checked out new restaurants. To our surprise, it was much easier than we thought it'd be.\n\nAnd we loved the food. When you start eating plant-based, your whole palate opens up\u2014your taste sensitivities change. There are countless species of plants with rich flavor profiles and aromas and nutrients. My diet is so much broader and more varied now than when I was eating a typical meat-based diet. And when we want those old comfort foods that taste like meat, there are so many good substitutes available\u2014and more coming into the market all the time. Plus, there are great ethnic options\u2014vegan ravioli and Thai food, Indian food, Mexican food, noodles\u2014things that we all recognize and love, and that always have plant-based options.\n\nThese foods are kinder to the land and kinder to our bodies. The fact is that eating more plant-based foods is the number one thing we can do for the environment\u2014for habitat loss, biodiversity loss, water pollution, and so many other issues. Almost every major environmental problem could be solved by a global shift toward plant-based eating. So the right choices to make for our health are also the right choices to make for the earth.\n\nMy recommendation? Read up on the issues. Arm yourself with the facts. There are plenty of good books on the topic of plant-based eating. The more you feel good about how you're eating\u2014that there's a higher purpose to your diet\u2014the more likely you are to stick with it. Program yourself for success. Try a twenty-one-day plant-based challenge. Figure out some recipes you want to make, try new foods and see what you like, find some great new restaurants to eat at. Plan your strategy a bit so you'll be successful with your new diet\u2014the fate of the planet depends on it.\n\n_James Cameron is an Academy Award\u2013winning director known for some of the biggest box-office hits of all time, including_ Titanic, The Terminator, Aliens, Avatar, _and more. He is also an oceanic explorer, having been the first person to reach the bottom of the Mariana Trench on a solo mission, and an environmental activist._\n\n\"Almost every major environmental problem could be solved by a global shift toward plant-based eating.\"\n\n\u2014 **JAMES CAMERON**\n\n# INTRODUCTION\n\n##\n\nFOOD IS THE SOLUTION\n\nWhen I was about ten years old, my father and I built a small backyard garden. In a sunny corner of our yard, we dug out a rectangle in the grass. We shoveled a trench along its edges, into which we laid bricks to create a border. We picked out stones and twigs from the dirt, breaking up the clumps with our hands until the soil was smooth and dropped easily through our fingers. We sowed seeds for green beans, carrots, radishes. From a garden center down the street, we bought seedlings for tomatoes, peppers, and eggplants, carefully transplanting them into the ground.\n\nRight around this time, I took an interest in cooking. I learned to make scrambled eggs\u2014albeit via microwave and mug (which tasted about as good as it sounds). I'd take great pride in organizing our spice cabinet, combining half-full jars of cinnamon and paprika, placing the most commonly used spices toward the front. I picked through my mother's favorite recipe clippings and pasted them into a homemade \"cookbook.\"\n\nAs my interest in food evolved, I progressed from microwaved eggs to stovetop cooking. I started inventing my own recipes. But my fascination with food mostly focused on growing, preparing, and eating it.\n\nThen, when I was twelve, my sister returned from school one day with an announcement: she had become a vegetarian.\n\n\"Mr. Thayer says there are antibiotics in chicken meat,\" she proclaimed. \"He says there are hormones and all kinds of other stuff in meat, too.\"\n\n_Vegetarian?_ It was the first time I'd ever even heard the word, let alone _known_ a vegetarian. No meat? Not even chicken? Fish? Bacon?! As any good little brother would do, I poked fun. At dinner, I'd gleefully place a chunk of bloody steak on the end of my fork and stick it into my sister's face while mooing loudly.\n\nMy mother took a different approach. Alongside a pot of beef chili for us, she'd have a bubbling pot of bean chili for my sister\u2014even using separate spoons for stirring. A dish of chicken enchiladas would sit side by side in the oven with a dish of veggie enchiladas.\n\nBecause I'd always loved food, I certainly wasn't averse to trying some of these mysterious new dishes. I'd eat one beef taco and one bean taco; I'd add bacon to a veggie burger. Slowly, my palate expanded and my culinary horizons broadened.\n\nLo and behold, I actually liked many of these new foods. I started ordering dishes with \"bean curd\" at our local Chinese restaurant\u2014not realizing until much later that bean curd is another term for tofu. I tried vegetarian chicken patties and, to my amazement, loved them. I'd order veggie fajitas at Mexican restaurants instead of meat fajitas. I was no vegetarian, but I liked the food. _Whodathunk it?_\n\nIt wasn't until later, in my teens, that I learned more about the social issues associated with food production. I learned that what we eat plays a central role in many of society's deepest problems, and can also help solve them.\n\nWhat I learned is that food can save the world.\n\nTake our own health, for example. Results from the U.S. Centers for Disease Control and Prevention's (CDC) _2011\u20132012 National Health and Nutrition Examination Survey_ show that nearly 35 percent of U.S. adults ages twenty and older are overweight, more than 35 percent are obese, and nearly 10 percent are extremely obese. For children and adolescents, the numbers are also staggering, with nearly one in every five individuals between two and nineteen years old being obese.\n\nThis is a full-blown crisis. The human costs\u2014increased heart disease, diabetes, stroke, and more\u2014are staggering. And the economic costs are overwhelming: an obese individual pays thousands of dollars more in healthcare costs than a non-obese individual, according to one study published in the Journal of Health Economics. All told, that study found, obesity accounts for a whopping $200 billion\u2014more than 20 percent\u2014of all U.S. health expenses.\n\nAnd even though they can offer some help, we needn't rely on healthcare providers, government, or corporations to solve this crisis. The power to reclaim our health can be found in what we put on our plates.\n\n\"A diet higher in plant-based foods,\" concludes a Scientific Report of the U.S. Dietary Guidelines Advisory Committee, \"is more health promoting... than is the current U.S. diet.\"\n\nWhen we do a deep dive into virtually any major healthcare problem plaguing our population, we see a clear correlation to food and diet.\n\nHeart disease, for example, is the number-one killer in America\u2014responsible for a full quarter of all deaths. The causes of this killer are cholesterol (which is only found in animal products) and saturated fat (high levels of which are found in animal products). When our cholesterol and saturated fat levels rise\u2014through consuming too many foods high in them\u2014our arteries become blocked, leading to heart attacks.\n\nBut countless scientific studies have confirmed that plant-based whole-food diets can prevent and reverse heart disease in patients without the use of medical intervention.\n\nFor example, in 1948, the National Heart, Lung and Blood Institute and Boston University embarked on an ambitious project to identify risk factors for heart disease. Today, their Framingham Heart Study (FHS) continues, making it the longest-running clinical study in all of medical history.\n\n\"The American Heart Association recognizes the role of plant-based foods in a healthy dietary pattern. Use Meatless Mondays as another opportunity to eat a well-balanced diet.\"\n\n\u2014NANCY BROWN, AMERICAN HEART ASSOCIATION CEO\n\n\"The research shows one thing very clearly: we all need to eat more plants and less meat.\"\n\n\u2014AMERICAN INSTITUTE FOR CANCER RESEARCH\n\n\"If Americans adopted a vegetarian diet,\" said former FHS director Dr. William Castelli in a PBS interview, the heart disease epidemic \"would disappear.\"\n\nDiabetes is another epidemic, with 1.4 million people diagnosed with the disease each year, and one in every ten healthcare dollars going toward treating it.,\n\nAnd yet we know that by eating more plant-based foods, diabetics can get their blood sugar to drop and thus reduce or eliminate their dependence on insulin. In a study published in the _American Journal of Clinical Nutrition_ , insulin dependence dropped by about 60 percent\u2014with half the study's participants able to get off of insulin entirely\u2014within just sixteen days of adopting plant-based diets.\n\nThe list goes on: cancer, stroke, obesity, autoimmune disorders like rheumatoid arthritis\u2014so many of the health problems that weigh us down as individuals, and that weigh down our nation's healthcare system, can be prevented and reversed by eating more plant-based foods.\n\nImagine a world in which we all are healthier\u2014where we're happier with our bodies and suffer fewer aches and pains. Imagine a world in which we don't have to spend as much of our money on health services, leaving more to spend on our passions and allowing us all to be more generous and charitable. Imagine a world with less disease, fewer early deaths. That world is a happier world, a better world\u2014and plant-based foods are the solution to creating it.\n\nSuch a world is also a much kinder, gentler world\u2014one filled with more empathy and compassion.\n\nToday, nearly 80 billion land animals are raised and killed for food each year. These include mammals like pigs and cows, birds like turkeys and chickens. Of course, there are family farmers out there who provide good living environments for their animals\u2014but virtually all meat, dairy, and eggs eaten today come from factory-farmed animals.\n\nIn 1950, the total number of farm animals in the United States was somewhere near 100 million; by 2015, it had mushroomed to roughly 9.2 billion. According to USDA data, during the same period that the number of farm _animals_ increased by 9,400 percent, the number of _farms_ producing those animals decreased by 60 percent.\n\nThat, in a nutshell, is factory farming\u2014and its consequences are catastrophic.\n\n# BETTER YOU, BETTER WORLD\n\nMICHAEL GREGER, MD, FACLM\n\n**At sixty-five, my grandmother was given a** medical death sentence. Suffering from end-stage heart disease, wheelchair-bound, and out of surgical heart bypass options, her life was basically over.\n\nThen she heard about Nathan Pritikin.\n\nBorn in 1915, Pritikin was an inventor, nutritionist, and longevity researcher\u2014one of our country's earliest \"lifestyle medicine\" pioneers. At forty-two, after being diagnosed with heart disease himself, Pritikin began his search for treatment, eventually publishing his findings in his (now renowned) book _The Pritikin Diet_ , and later opening the Pritikin Longevity Center, offering diet and lifestyle coaching with the goal of preventing and reversing disease.\n\nSo when my grandmother, nearly on death's door, heard of Pritikin, she reached out, eventually coming to follow his program.\n\nSo what exactly was Pritikin's plan? His extensive research centered primarily on a low-fat, plant-based diet rich in fruits and veggies, beans, nuts, legumes, and whole grains. Just as we now know that eating more plant-based foods plays a critical role in preventing and reversing environmental catastrophes, we also know plant-based diets can play a central role in overall health and wellness by preventing and reversing many of the diseases that plague us most.\n\nTopping the list for the leading cause of disability and death in the U.S. is heart disease: America's number-one killer and the same disease that caused my grandmother to reach out to Nathan Pritikin. And high cholesterol levels are thought to be a primary cause of this leading killer.\n\nThis may help explain why a plant-based diet, which is free of cholesterol and saturated animal fats, has been so successful in preventing and treating the disease. Indeed, plant-based diets are the only type of diets ever proven not only to help prevent heart disease but also to reverse it in the majority of patients.\n\nFollowing heart disease is cancer, America's number-two killer. Here, too, the balance of evidence suggests that a whole-foods, plant-based diet may help prevent, treat, slow, and in some cases even reverse cancer progression.\n\nAs the American Institute for Cancer Research concludes, \"The research shows one thing very clearly: we all need to eat more plants and less meat.\"\n\nSome reasons why plant-based diets may be so effective include lowering cancer-promoting growth hormone levels, methionine intake, inhibiting angiogenesis, intercepting carcinogens, and increasing fiber and antioxidants. Additionally, eating lots of fruits and vegetables (e.g., nine daily servings) may boost detoxifying enzymes, lower inflammation, and make for healthier bowel movements, ridding the body of excess estrogen and cholesterol. It's no wonder populations eating diets centered on whole plant foods have lower cancer rates.\n\nAnother major health threat here in the United States is diabetes. And, as is the case with heart disease and cancer, plant-based diets may successfully prevent, treat, and even reverse type 2 diabetes\u2014including in children.\n\nExcluding meat, milk, and other animal products may reduce the risk of diabetes and gestational diabetes by increasing insulin sensitivity, boosting hormone-binding proteins, helping to prevent obesity, and reducing exposure to heme iron, dioxins, and PCBs.\n\nEggs may be particularly risky: eating only one egg a week is associated with nearly double the odds of getting diabetes. And fish, especially salmon, is one of the primary sources of PCBs and other industrial toxins, which may also play a role in the development of diabetes.\n\nOn the other hand, many plant-based foods appear to be protective. Beans may be especially beneficial when replacing meat or refined carbs, such as white rice. By eating plant-based and living a healthy lifestyle, up to 95 percent of type \u0662 diabetes cases are avoidable.\n\nOf course, simply preventing disease is not the only goal of good nutrition. Everyone, whether eating a plant-based diet or not, ought to be sure to get enough (but not too much) protein, calcium, and iron, for example.\n\nVitamin B12 is one important supplement for those eating plant-based diets. B12 is made by neither animals nor plants, but by microbes. And B12 deficiency may occur without supplementation. Thankfully, there are safe, cheap, convenient sources available widely. The simplest regimen is to take 2,500 micrograms of the cyanocobalamin form of B12 once a week.\n\nOmega-3 fatty acids are also essential. Plant-based sources of omega-3 include flaxseed or chia seed, walnuts, and algae-based DHA supplements, which are bioequivalent to fish oil omega-3 but without the harmful industrial pollutants often found in fish oils.\n\nCombined with even minimal supplementation, plant-forward eating has been definitively shown to prevent, treat, and reverse some of our nation's biggest killers while improving health, wellness, and longevity.\n\nNathan Pritikin knew this decades ago\u2014and that would ultimately save my grandmother's life. Pritikin actually chronicled her story in his biography: \"Mrs. Greger had heart disease, angina, and claudication,\" he wrote. \"Her condition was so bad she could no longer walk without great pain in her chest and legs. Within three weeks, though, she was not only out of her wheelchair but was walking ten miles a day.\"\n\nThanks to Pritikin, she was able to live another thirty-one years\u2014to the grand old age of ninety-six. His plant-based diet extended her life by nearly a full third.\n\nIndeed, debilitating, life-shortening diseases and other ailments don't have be a normal part of aging\u2014and they don't have to be your future. Like many of the most pressing environmental problems, they can instead be prevented, treated, _and even reversed_ with a healthier diet rich in whole, plant-based foods. So eat up, live long, and prosper!\n\n_A founding member and Fellow of the American College of Lifestyle Medicine, Michael Greger, MD, is a physician, author, and internationally recognized speaker on nutrition, food safety, and public health. He's a graduate of the Cornell University School of Agriculture and Tufts University School of Medicine. His book_ How Not to Die _became an instant_ New York Times _bestseller. He has videos on more than two thousand health topics freely available atNutritionFacts.org._\n\n# THANKS, GIVING\n\nJESSE EISENBERG\n\n**At some point in my twenties,** my family made the switch.\n\nI'd missed a few Thanksgiving dinners because of work, and when I finally returned home, my family cavalierly told me that they now celebrated Thanksliving. That's not a typo.\n\nThanksliving is just like Thanksgiving\u2014the fun holiday in November\u2014but with an _L_ instead of that fun _G_. Thanksgiving, with the _G_ , they explained, _is brutal and sanctifies slaughter. Thanksliving_ [with an _L_ ] _celebrates life. It's vegan and we honor two turkeys we've saved by placing their photos at the center of our table._\n\nWith no time to make other plans, I trudged to my family's house in New Jersey on the final Thursday of November 2009. We took a moment of silence to reflect on the horrors of the poultry industry. We talked about what virtue we were most proud of. And the coup de grace: My sister pulled out Jonathan Safran Foer's _Eating Animals_ and opened to a dogeared passage about the slaughter of turkeys. We sat around a plate of tofu, hearing of turkey-cide. She was not only preaching to the converted, she was nauseating them.\n\nLooking around the room\u2014these familiar faces, nodding empathically along to the stories of turkey death\u2014I suddenly realized: _my family has gone nuts._\n\nThe following year, I decided to skip Thanksliving. I wasn't opposed to the food; in fact, I have spent many years as a vegetarian and, of course, know that it's a far better diet for the health of the eater, the animals, and the environment. But I couldn't bear to see my family so earnest. These were funny people! People who would have laughed at the family that made Thanksliving dinner!\n\nI was working near Los Angeles, so I asked a producer friend if he'd have me over for dinner. He told me not to expect much. \"We just have a little turkey, stuffing, cranberry sauce\u2014nothing fancy.\" _Nothing fancy_ was exactly what I needed.\n\nThis producer\u2014I will call him John Doe\u2014had three young children and a wife who was a pediatrician. I had met the family several times and I always thought they were a perfect group. They were always making inside jokes and traveling as a team. And their dinner was unremarkable, for which I was very thankful. We just sat down and made stupid jokes and ate normal American food.\n\nHalfway through, Jane Doe mentioned going to her parents' house for next year's Thanksgiving. And John\u2014normally a mild-mannered guy\u2014blew up. \"I told you a thousand times, I am not stepping foot into that psychopathic house!\" This was clearly the middle of an ongoing fight.\n\nJane threw down her fork. \"Then I guess we'll just have to get divorced next year and you can go to your dysfunctional family's house and drink with your father until you pass out again.\"\n\nI stared at the Doe kids, shocked. They stared back, as if to say, \"Welcome to our every day.\"\n\nJane stormed into the kitchen. I suddenly longed for the newfound earnestness of my family, their moral eagerness, their progressive idealism. I realized every family is insane. But at least mine cared about the world.\n\n_Jesse Eisenberg is an Academy Award\u2013nominated actor and the author of_ Bream Gives Me Hiccups and Other Stories.\n\n\"Factory farming is not only unacceptably cruel, but unsustainably inefficient.\"\n\n\u2014JOHN SAUVEN, EXECUTIVE DIRECTOR OF GREENPEACE UK\n\nAt these facilities, animals are routinely abused in ways that would be illegal if they were done to a cat or dog. Egg-laying hens are crammed into cages so small they can't spread their wings. Mother pigs are often forced to live in crates so tiny they can't even turn around. Chickens in the meat industry are bred to be Frankenbirds, growing too fat, too fast for their young bodies to keep up with and suffering crippling injuries as a result.\n\nThen there's fish. Worldwide, an estimated 2.7 _trillion_ fish are ripped from the oceans each year for food, suffering decompression as they're pulled from the water or being suffocated or crushed aboard fishing vessels. Countless more are raised on aquatic factory farms, where they're packed gill to gill in disease-ridden tanks; though no firm numbers exist for these aquaculture-raised fish, it's likely in the tens of billions.\n\nAll this suffering, on such a massive scale, is antithetical to what we believe about animals. We are a nation of animal lovers. We spend nearly $100 billion on our pets each year. We bird watch. We whale watch. We support, through donations, more than twenty thousand individual animal protection charities across the nation. Caring people in every corner of the country rescue and rehabilitate injured wildlife. People rush into flood zones and other disaster areas to rescue animals in need. We volunteer at shelters. The list of ways in which we share our lives with and help our fellow creatures is nearly limitless.\n\nOur compassion for animals extends to those caught up in the food-supply chain.\n\nThe Food Marketing Institute is the leading trade association for the grocery industry. \"When it comes to attributes beyond those that render personal benefits,\" concludes FMI in its \"Shopper Trends\" report, \"shoppers prioritize animal welfare second only to employment practices.\" And a study by the American Farm Bureau Federation\u2014a major lobbying force for factory farmers\u2014found that 95 percent of Americans believe animals on farms ought to be treated well.\n\nAnd yet, for some of us, there's a disconnect between what's on our plates and those values of compassion that we hold so dear: millions of us don't want animals to suffer and believe they should be treated well, yet also continue eating products from factory-farmed animals.\n\nIf more of us can connect those dots\u2014between what we already want for animals and how our food choices impact them\u2014we can better align our morals with our menus to create a kinder world.\n\nSuch a world would also be better for people. After all, at its root, our willingness to eat animals even when we know they've suffered comes down to our proclivity for placing animals in an \"other\" category\u2014focusing on how they may be different from we humans.\n\nOf course, fish were born with gills instead of lungs, chickens with wings instead of arms, pigs with hooves instead of feet. Humans were born with hair instead of fur and a spark of imagination that other species may lack. But while we humans are dissimilar to other animals in many ways, we also share so much in common with them. We feel pain and joy; we can suffer. So do other animal species. Anyone who's had beloved pets at home knows we share with other animals personal preferences unique to us as individuals (one of my cats is obsessed with peanut butter, while another loves spinach). Humans and other animals share a desire to live our lives according to our best interests, to engage in our natural behaviors, to have social interactions with others of our kind and even with other species.\n\nAnd if, through our diets, we can learn to focus more on those _similarities_ rather than our _differences_ , we may foster a deeper sense of empathy\u2014which could have powerful implications on how we treat one another.\n\nSay a person comes to the conclusion that simply because chickens were born with wings rather than arms we ought not to cause their suffering. Say they begin eating fewer chickens, or none at all. In making that switch, that person has widened their circle of compassion. Now, how likely would it be for that person, with their newfound empathy toward another species entirely, to lack empathy for fellow humans who were born, say, gay instead of straight, black instead of white, female instead of male, Middle Eastern instead of American?\n\nIndeed, being kind to animals can also make us kinder toward our fellow man. And conversely, people who are cruel to animals exhibit more dangerous and violent behaviors toward one another.\n\nThis is why law enforcement agencies like the FBI, police, and Department of Justice use cruelty to animals as a red flag when profiling violent criminals. One scientific study, published in the _Journal of Interpersonal Violence_ , concluded that \"animal abuse may be a red flag indicative of family violence in the home.\" In that study, a whopping 60 percent of participants who witnessed or took part in acts of animal cruelty as children also reported instances of child abuse or domestic violence.\n\nSo imagine a world where our relationships with animals were based less on abuse and suffering and more on kindness and respect. In that world, our circle of compassion would include not only the dogs and cats in our homes but also the chickens, cows, pigs, and fish on farms. Imagine that throughout the day, as we decide what to eat, we constantly flex that compassion by making kind choices.\n\nWhat impact could that constant compassion have on society? Racism, sexism, homophobia, xenophobia, war, hatred\u2014the more empathy we have and exhibit toward others, the more these poisons could begin to fade away.\n\nBut at the end of the day, even if we make ourselves healthier, in body and in spirit, we can't have the type of better world so many of us envision and hope for without a healthy planet on which to build it\u2014and it's clearer than ever that our planet is in peril.\n\nMost people are familiar with the mother of all environmental disasters: climate change. Erratic weather, rising sea levels, extreme heat and cold\u2014this catastrophe threatens nearly all life on earth. We're also losing rich, important ecosystems and biodiversity at an alarming rate, with up to one hundred thousand distinct species becoming extinct each year, according to the World Wildlife Fund. Our waterways are being impacted, too: in 2013, the U.S. Environmental Protection Agency (EPA) found that the majority of rivers and streams in America are too polluted to support healthy aquatic life, with 55 percent of our waterways rated as being in \"poor\" condition and only 21 percent of rivers considered \"healthy biological communities.\"\n\nSoil erosion, air pollution, overfishing, deforestation, water scarcity: The list of ways our earth is in jeopardy is virtually unending. For that reason, I've focused this book on the _environmental_ implications of our food choices: because what we eat is not only a major factor in deciding what kind of world we live in, but also in what kind of world we live _on_. And it's become clear that meat-heavy diets are destroying the planet.\n\n\"We all need protein,\" points out a Bill Gates video titled _The Science Behind Plant-Based Proteins_. \"But animals aren't the most efficient way to get it.\"\n\nIn fact, producing just one pound of meat means feeding an animal up to fifteen pounds of grains and other crops. That means growing massive amounts of feed, which often means destroying ancient forests to make room for farmland. It means spending huge quantities of fresh water to grow those crops.\n\nThen we take those crops (which we've put all those resources into growing), process them, and feed them to the tens of billions of farm animals used each year in the global agricultural system\u2014animals who themselves also require water to drink, land to live on, and other valuable inputs.\n\nThe caloric conversion is weak, too: according to a report produced in collaboration with the World Bank, even the most efficient sources of meat convert only around 11 percent of gross feed energy into human food. That means that even in the best cases, nearly 90 percent of what we put into turning animals into meat is wasted.\n\nIt's an inefficient system that taxes our planet's finite resources in incredible ways.\n\nSo it's no surprise that the livestock sector is now a leading greenhouse gas emitter, estimated to account for more direct emissions than the entire global transport sector, according to the United Nations Food and Agriculture Organization.\n\nOur animal-heavy diet is also a major contributor to water degradation\u2014impacting fish populations and polluting our global waterways. And it's an incredibly thirsty system, with hundreds of gallons of water needed to produce a single pound of chicken or glass of milk, and dozens of gallons needed to produce a single egg.\n\nThis makes sense, considering the inherent inefficiencies that accompany a system in which we funnel massive amounts of grain and water through animals to produce protein, rather than converting plant matter to protein more directly. And they're precisely why global development and environmental advocates encourage a diet more focused on plants and less focused on animals.\n\n\"Eating plant-forward dishes can go a long way in reducing diet-related carbon emissions.\"\n\n\u2014NATURAL RESOURCES DEFENSE COUNCIL (NRDC)\n\n\"The more meat everybody eats, the more polluting it is and the more wasteful it is.\"\n\n\u2014BARACK OBAMA\n\n\"The reality is that it takes massive amounts of land, water, fertilizer, oil and other resources to produce meat,\" points out Oxfam America in an article promoting meat reduction. \"Significantly more than it requires to grow other nutritious and delicious kinds of food.\"\n\nLess resource use, more efficient resource allocation, more compassion toward animals, spreading empathy to one another, curing and preventing our most serious health crises: plant-based eating, in so many ways, is the common denominator that can solve many of our planet's problems and create a truly better world.\n\nThis is what I came to understand so many years ago, after beginning my own journey with plant-based foods, and it's what I've been fortunate to spend my career working toward. As a professional food advocate, I've helped major food companies extend their corporate circles of compassion by demanding better treatment for farm animals in their supply chains\u2014working with major brands like Burger King, McDonald's, and Walmart to move their suppliers in a more humane direction. I've also worked with the world's largest financial firms, which have invested hundreds of billions of dollars in animal agriculture, to leverage that power as a force for change. And I've worked with the food industry, consumers, and other advocates to share the myriad of benefits of eating more plants and less meat.\n\nOf course, I'm not alone in this quest. Countless other individuals and organizations are spreading these concepts as well. And millions of people are catching on.\n\nIn its July 2015 article \"Even Carnivores Are Putting More Fake Meat on Their Plates,\" NPR reported, \"Consumers\u2014and not just vegetarians\u2014are also warming up to products like tempeh, tofu, and seitan that can stand in for meat.\"\n\nThe meat industry has noticed. \"Plant protein is in,\" reports _Meatingplace_ \u2014an industry magazine. Even Tyson Foods\u2014one of world's largest meatpackers\u2014has taken a stake in a plant-based chicken company and announced a $150 million venture capital firm to focus on plant-based proteins.\n\n\"Plant-based protein is growing almost, at this point, a little faster than animal-based,\" said Tyson's CEO in 2017. \"So I think the migration may continue in that direction.\"\n\nIndeed, Americans are now featuring plant-based foods right smack dab in the center of our plates. And not just because doing so is healthier and better for the planet, but also because these foods taste great.\n\nI've always loved ice cream, for example, but not until I adopted a plant-based diet did I know the joys of eating ice cream made from, say, coconut milk. I was floored the first time I tried buffalo wings made from cauliflower instead of chicken. The first time I sprinkled nutritional yeast\u2014little yellow flakes that have a Parmesan-like flavor\u2014on popcorn, it was like fireworks of flavor were going off in my mouth.\n\nIndeed, once I started exploring this vibrant, rich, diverse array of plant-based foods that existed all around me, my long-standing interest in food only grew. What if my sister had never learned about antibiotics? What if she'd never become a vegetarian? I could have missed out on so much culinary variety, so many incredible foods that I now love and crave and get to enjoy every day.\n\nOf course, one needn't go fully vegetarian or vegan to enjoy these foods. In addition to lessening our environmental footprint, being healthier for us, and avoiding animal abuse, simply incorporating _more_ plant-based foods into our everyday diet can help expand our culinary repertoire and introduce us to all kinds of new tastes and flavors and textures.\n\nAnd we needn't let the perfect be the enemy of the good. Eating for a better world doesn't have to be an all-or-nothing endeavor. Certainly it's true that locally grown, minimally processed, whole, plant-based foods like organic produce and lentils have the least eco-impact of any foods we might choose to eat. But we needn't eat _only_ those foods to make a difference. Of course, to keep up any kind of diet\u2014be it for health or the environment\u2014we must find that diet to be attainable and _personally_ sustainable. After all, the most effective diet is always the one we stick with.\n\nSo while frozen plant-based nuggets are certainly not quite as healthy or environmentally friendly as, say, locally grown lentils, they're far better than chicken nuggets. A prepackaged black bean burger is not a zero-footprint product by any means, but it's much lighter on the planet and our bodies than a hamburger. Yes, almond milk requires land and water and other resources to produce, but compared to cow's milk? No contest.\n\nSo do what works for you\u2014what tastes good and is attainable and sustainable in your daily life. Try \"flexitarian\" approaches like Meatless Mondays. Try doing VB6 (eating vegan before six p.m., the approach suggested by author Mark Bittman). Try simply moving meat from the center of the plate to the side of the plate.\n\nAnd try new foods! Explore the rich array of colorful produce and interesting, delicious plant-based offerings now available at every turn. On this planet-in-peril\u2014with all the ecological, biological, and sociological problems plaguing our world at so many levels\u2014the more we each recognize that the solutions to these problems can be found in our food, the better off we'll all be. The power is on our plates\u2014let's use it.\n\n\"Anchoring a plate with a massive hunk of animal protein is so last century.\"\n\n\u2014WALL STREET JOURNAL\n\n# [PART ONE:  \nOUR PLANET](toc.xhtml#part01)\n\n# \"The future is about a plant-based diet.\"\n\n\u2014JAMIE OLIVER\n\n# EARTH\n\n_Imagine the Earth, spinning in orbit: our 197-million-square-mile, four-and-a-half-billion-year-old ball of nitrogen, oxygen, and carbon amid the quiet blackness of space._\n\n_But Earth is much more than that._\n\nZoom in through the outermost thermosphere, through the stratosphere, through the clouds. Pass over the North Pole's stark ice caps. Glide over Siberia, covered in snow-capped pine and birch trees. Watch the landscape move from white to green throughout Eurasia\u2014over Kazakhstan, Uzbekistan\u2014then to browns above the Middle East and northern Africa, and to blue over the Atlantic Ocean. Pass over New York City's gray skyscrapers, through the Shenandoahs, their chestnut and red oaks' brilliant shades of orange and yellow, and across the mighty Mississippi. Now slow yourself above the Midwest, above Missouri. Finally, pause above the tiny town of Arrow Rock\u2014a pixel on the planet.\n\nIt's here in Arrow Rock, Missouri, where Day and Whitney Kerr bought a Greek Revival cottage, a fixer-upper. That was in 1980, long before they could have known that purchase would turn their own story into one of an epic battle for their land and livelihoods, for the literal earth beneath their feet.\n\n\"It was in terrible shape,\" said Day, who, along with his wife, purchased and restored the home to its former 1860s glory. \"It never had indoor plumbing or central heating. No one had lived there for thirteen years. There were times when we thought we were just crazy, but Whitney and I are both committed to historic preservation. We respect history.\"\n\nAnd Arrow Rock's got history everywhere one looks: for over a half century, the _entire town_ has been registered as a National Historic Landmark. It was in Arrow Rock that Dr. John Sappington\u2014whose home the Kerrs also restored\u2014created the quinine pill; it was there that the Santa Fe Trail and Missouri River intersected, where lauded Missouri River artist George Caleb Bingham lived and worked.\n\nIt was also where, in 2007, hog farmer Dennis Gessling proposed building a massive factory farm that would confine nearly five thousand pigs\u2014ninety animals for each of the town's fifty-six residents\u2014and produce more than two million gallons of waste each year.\n\nAnd he wanted to build it just a quarter mile north of the Kerrs' home.\n\n\"This is a historic site for the state and nation,\" said local bed-and-breakfast owner Kathy Borgman. \"This is a spot where history was made. You can't move it. Who would want to come here if the air stinks so bad you cannot breathe it?\"\n\nAll across the country, communities are threatened by confined animal feeding operations (CAFOs) like the one proposed by Gessling.\n\n\"Factory farms can negatively affect rural historic areas in a number of ways,\" writes Jennifer Sandy of the National Trust for Historic Preservation. \"CAFOs obviously have an immediate and negative impact on the historic rural landscape. They are large in scale, housing thousands of animals, and generally consist of utilitarian metal buildings and manure retaining ponds.\"\n\nTascosa Feedyard, Bushland, Texas (2013). Courtesy the artist and Bruce Silverstein Gallery, New York. CREDIT: MISHKA HENNER\n\nIn these facilities, animals are confined by the thousands, tens of thousands, or even hundreds of thousands. In the case of pig-production facilities like the one proposed in Arrow Rock, breeding animals are most often confined inside individual gestation crates\u2014metal cages that are essentially the same length and width of the pigs' own bodies, preventing them from even turning around. These animals endure this confinement day and night for four months, while pregnant. As they're about to give birth, they're moved into another, similar, type of crate\u2014after which they're reimpregnated and put back in a gestation crate for the cycle to repeat. A mother pig lives like this for her entire four years on this earth.\n\nThese types of facilities are dramatically changing the American landscape, which was once dotted with more traditional-type farms\u2014indeed, what most of us imagine when we think of a _farm_.\n\nToday, those idyllic places are mostly gone\u2014and that's no accident.\n\n\"Forget the pig is an animal\u2014treat him just like a machine in a factory,\" recommended _Hog Farm Management_ magazine in 1976. Two years later, _National Hog Farmer_ advised: \"The breeding sow should be thought of, and treated, as a valuable piece of machinery whose function is to pump out baby pigs like a sausage machine.\"\n\nAnd so it was that factory farms were born\u2014leaving a lasting mark on animals, our planet, our nation, and on individual communities.\n\n\"As small- and medium-sized producers are forced out,\" Sandy reports, \"many historic farm structures are abandoned or demolished. Areas with factory farms also often see an increase in truck traffic, which can have visual and auditory impacts.\"\n\n\"Raising meat takes a great deal of land and...has a substantial environmental impact.\"\n\n\u2014BILL GATES\n\nThe citizens of Arrow Rock, Missouri were not going to have it in their backyard.\n\n\"These CAFOs spend as little money as they have to get as much money as they can out of the animal,\" Day Kerr observes. \"In the process, they do not protect the environment and have no regard for how they impact their neighbors.\"\n\nSo the Kerrs and their neighbors banded together and stood their ground, raising a stink with local and state governments about the proposed pig factory.\n\nThey incorporated as Citizens to Protect State Parks and Historic Sites. They formed alliances with other communities being impacted by CAFOs. They hired lawyers. They demanded reviews from the U.S. Department of Interior (since Arrow Rock is a national historic landmark). They generated press coverage and statewide buzz. They filed lawsuits\u2014including against the Missouri Department of Natural Resources and its director.\n\nAnd they won. After a prolonged battle, the people of Arrow Rock\u2014resting stop for Lewis and Clark, and a town nearly crippled by a massive pig factory\u2014claimed victory when Gessling officially withdrew his plans.\n\nBut things could have easily gone the other way.\n\nIn 2016, years after the controversy had been settled, I spoke to Kathy Borgman. She expressed concern that the same threat Gessling posed to her community still exists elsewhere.\n\n\"What we accomplished in Arrow Rock was successful in one regard,\" Borgman says. \"But the CAFO problem continues all over Missouri, all over the country.\"\n\nIndeed, elsewhere across the entire _planet_ , factory farms continue wreaking havoc on the earth beneath us.\n\nIn fact, globally, animal agribusiness is by far the single largest user of land, according to the United Nations Food and Agriculture Organization (FAO) report _Livestock's Long Shadow_. \"In all, livestock production accounts for 70 percent of all agricultural land and 30 percent of the land surface of the planet,\" found the FAO.\n\n# WHAT EXACTLY _IS A_ FACTORY FARM?\n\n_Factory farm_ refers to industrial farm animal production. Other terms\u2014like _concentrated animal feeding operation_ (CAFO) and _animal feeding operation_ (AFO)\u2014have legal definitions. \"AFOs congregate animals, feed, manure and urine, dead animals, and production operations on a small land area,\" describes the EPA. An AFO may be designated as a CAFO if it's considered \"large\" or \"medium\" or through special designation by the EPA. Regardless of terminology, these are all places where thousands or tens of thousands of animals live\u2014often crammed together or confined in cages\u2014and where environmental problems persist.\n\n# FROM THE HORSE'S MOUTH...\n\n**\"I moved from Switzerland in 1983, to my part of Missouri because of the wide-open spaces and clean environment. When Premium Standard Farms purchased four thousand acres next to my farm in 1993, I knew nothing about the company. But when I heard about the type of hog facility that they were intending to build there, I immediately opposed them. The facility next to my property has an eighty thousand hog-head capacity, which is replaced 2.8 times a year. This facility and others nearby generate so much waste that they have turned our land into waste-handling facilities, which is an immoral and unethical way to use the land.**\n\n**I can't describe how terrible the odor from the lagoons, spray fields, and barns often is. We can't keep our windows open, and sometimes you can even smell the odor through the shut windows. You open the door and the smell almost knocks you over. One of the worst parts of it is that the odor hits at unpredictable times so it is a constant threat. Breathing in such a terrible stench makes you feel desperate. One time I was planting soybeans on a field and I got so sick to my stomach that I had to stop planting. My wife has allergies, which are aggravated by the odor. She is in the health field, and she believes that many of her patients are also suffering from worse allergies from the hog odor. A year or so ago, I went on vacation to a beautiful national park; when I entered my house upon my return and smelled the terrible odor, I broke down and cried.**\n\n**\"As farmers and human inhabitants of this planet, we are but caretakers of what has been given to us. The tree in front of my house will be here long after I am gone. Every tree I plant will be for my grandchildren to enjoy. Everything I destroy will be gone forever. So will it be with the land.\"**\n\n\u2014ROLF CHRISTEN, GREEN CITY, MISSOURI\n\nThat's right: nearly a third of _all the land on the planet_ goes toward producing protein from animals. And that production isn't happening in deserts and on craggy cliffs that might otherwise go unused.\n\nSo, leaving Arrow Rock and the Midwest on our global journey, we head southeast\u2014over the Gulf of Mexico and western tip of Cuba, across the brilliant blue Caribbean Sea, eventually finding ourselves in perhaps the epicenter of factory farming's earthly impact: the Amazon rain forest.\n\nSeventy percent of previously forested land in the Amazon is now pasture for farm animals, with animal feed crops covering a large part of the remainder. There's no example in Earth's history of old-growth and primary forestland being more quickly converted to human land use.\n\nIn the southeastern Amazon, more than half the woodland-savannah ecosystem of the Cerrado has already been converted to agriculture, mostly for the production of meat and animal feed. And three countries away, in Costa Rica, roughly half the small nation's tropical forest land is now used for meat production. The list goes on: in Africa, Southeast Asia, and elsewhere across the globe, we're losing rich, diverse ecosystems on account of our meat-heavy diets.,\n\n\"The consumption of animal-sourced food products by humans is one of the most powerful negative forces affecting the conservation of terrestrial ecosystems and biological diversity,\" reports a study published in _Science of the Total Environment_. That's habitat loss for wild animals and native plant species, soil loss, water and nutrient pollution, and more.\n\nIf our diets don't change, things will get worse. Some countries, that same report found, may even require up to _50 percent increases_ in land used for meat production.\n\nSo how much land does it take, exactly, to produce meat for our diets? Looking squarely at the land impact of _American_ meat consumption, researchers from Bard College, the Weizmann Institute of Science, and Yale University found that the animal-based portion of an average American's diet requires over _150,000 square feet_ of crop and pasture land annually\u2014that's nearly the area of three NFL football fields used to produce meat, eggs, and dairy for each one of us, each year.\n\nBut the good news is, there's a solution. No, we don't all need to do as Day and Whitney Kerr, Kathy Borgman, and the other citizens of Arrow Rock, Missouri, did: we don't each need to take up legal and public battles against individual factory-farm operations.\n\nBut we can certainly stop funding the perpetrators.\n\nAfter all, though Arrow Rock managed to stave off its proposed factory farm, unless we curb our reliance on animal products\u2014by eating more plant-based foods\u2014other communities across the world won't be so lucky.\n\n\"They tell you we've got more money than God and will outspend you and outlast you,\" laments Borgman, referring to the power of industrial agriculture. \"So it'll be the tide of the markets and the tide of masses to really change things.\"\n\nBut how does each of us play a part in that growing tide? One way for us to help is dietary\u2014and delicious. Many people are now practicing the \"Three Rs\" of eating: 1) reducing meat and other animal products in their diets by 2) replacing those products with plant-based foods, and 3) refining our diets so as to avoid products from factory farms.\n\n# THE THREE RS OF EATING FOR A BETTER PLANET\n\n**1 REDUCE** our consumption of meat, dairy, eggs, and other animal-based foods.\n\n**2 REPLACE** animal-based foods in our diet with plant-based foods.\n\n**3 REFINE** any animal products we consume by avoiding foods from factory farms.\n\nCutting down forest land near Ecuador's Napo River in order to grow more livestock feed. CREDIT: TOMAS MUNITA\/CIFOR\n\nWe can enjoy more black bean burgers and fewer beef burgers, bean-and-rice burritos instead of chicken burritos, BBQ seitan or tofu instead of pulled pork, chick _pea_ salad instead of chick _en_ salad. We can try almond milk, soy milk, coconut milk, or any other variety of the myriad of nondairy alternatives now available at coffee shops and grocery stores nationwide.\n\nWe can participate in programs like Meatless Mondays\u2014enjoying the time-honored tradition of a once-weekly vacation from meat. In fact, it was the U.S. Food Administration that created Meatless Mondays over a century ago to save resources during WWI. The effort was brought back during WWII, and then revived in 2003 by the Johns Hopkins School of Public Health as a wellness and sustainability initiative. Countless individuals and even entire school districts (like Houston, Los Angeles, Detroit, and more) are now participating, and the idea has spread like wildfire across colleges, universities, hospital systems, and more.\n\nThese types of small actions can make a huge difference. In its 2016 report _Shifting Diets for a Sustainable Food Future_ , the World Resources Institute found that by reducing animal protein intake in regions that currently consume the most, like the United States, per-person diet-related land use could be nearly _cut in half_.\n\nConsidering the impact factory farming has made and continues to make, the importance of a 50 percent reduction in our diet-related land use can't be underestimated. At this very moment, the Amazon continues burning to accommodate our meat-heavy diets. Right now, communities like Arrow Rock are pushing back against land grabs by would-be factory farmers. Farms are shutting down, replaced with factories. Rural landscapes across the country and world are being industrialized\u2014forever changed because of our diets.\n\nThe evidence is clear: for the earth beneath and all around us\u2014composed of rich, varied, important ecosystems and communities\u2014we've all simply got to eat less meat and more plants.\n\nDeforestation vs. the Amazon CREDIT: GREENPEACE\/RICARDO BELIEL\n\n**70%** The amount of Earth's agricultural land used for livestock production.\n\n**30%** The amount of all Earth's land mass used for livestock.\n\n**Three Football Fields** The amount of land needed per person each year to sustain the animal-based portion of a standard diet.\n\n**50%** The amount each of us can decrease our diet-related land use by eating more plant-based foods.\n\n## DIRT FIRST!  \nSHEDDING SOIL BY THE TON\n\nBY DANIEL IMHOFF\n\nAgricultural production is the largest source of soil erosion in the United States, with 90 percent of croplands shedding soil above rates that are sustainable for the long term. Since the majority of the nation's cropland is used for growing animal feed, high rates of soil erosion are a direct consequence of industrial animal food production. By squandering the very foundation of agriculture itself\u2014productive soil\u2014animal factory farming is compromising the ability of future generations to meet their needs.\n\nAccording to the United Nations Food and Agriculture Organization, the world is losing an equivalent of 10 to 15 million acres of farmland through erosion each year. This is a land area the size of Belgium and the Netherlands combined. Some experts argue that soil loss is humanity's most serious environmental challenge.\n\nEach year, the United States loses nearly two billion tons of soil, making farmlands less fertile and setting off a cascade of destructive consequences. The loss of just a few centimeters of topsoil can reduce the productivity of good soils by 40 percent and that of poorer soils by 60 percent. With the flight of bare soil through wind or rain erosion go valuable organic matter and nutrients needed to grow healthy plants. The ability for soil to absorb, retain, and filter water also drastically declines. Estimates just to replace those nutrients and pump more water onto farmlands due to soil loss are over $30 billion per year.\n\nIncreasing demands for cattle feed and corn-based ethanol are putting even more strain on already overstressed farmland. Farmers are foregoing rotations of soybeans (which provide more soil protection) to grow more corn. And to grow corn they are moving land out of the USDA Conservation Reserve Program (CRP), a decades-old economic incentive program that has been fairly successful in protecting marginal soils from the plow.\n\n\u2014FROM _CAFO: THE TRAGEDY OF INDUSTRIAL ANIMAL FACTORIES_\n\n\"The nation that destroys its soil destroys itself.\"\n\n\u2014FRANKLIN D. ROOSEVELT\n\n## LIVESTOCK & LAND\n\nTurning animals into food requires enormous amounts of land\u2014to grow the crops to feed the animals, and for the animals themselves to live on.\n\n\"Many of us feel helpless in the face of environmental challenges, and it can be hard to know how to sort through the advice about what we can do to make a meaningful contribution to a cleaner, more sustainable, healthier world. Having one designated meat-free day a week is a meaningful change that everyone can make, that goes to the heart of several important political, environmental, and ethical issues all at once.\"\n\n\u2014PAUL MCCARTNEY\n\nCows grazing on land that used to be forest in Rio Branco, Brazil. CREDIT: KATE EVANS\/CIFOR\n\n\" **Diversify Your Diet.** Try to eat an all-veggie meal... instead of one meat meal a week. More natural resources are used to provide meat than plants.\"\n\n\u2014THE UNITED NATIONS FOOD & AGRICULTURE ORGANIZATION\n\n## **IN A NUTSHELL**\n\n\u2022 Factory farms are destroying national and global landscapes and communities.\n\n\u2022 The introduction of large-scale CAFOs negatively impacts communities' livelihoods and well-being.\n\n\u2022 Historic areas and structures are being replaced with large-scale agricultural operations.\n\n\u2022 Our alarming loss of soil due to animal agriculture is making farmlands less fertile.\n\n\u2022 Producing animal proteins requires huge inputs of land\u2014for the animals, their feed, and more.\n\n\u2022 The Amazon is disappearing day by day to make room for more animal feed-crop production.\n\n\u2022 This is a major change in the landscape, and also loss of habitat for many native species.\n\n**Now for the good news!** Eating more plant-based proteins can help us each use less land for our diet, avoid destroying rain forests like the Amazon, and preserve local communities and landscapes. Of course, all types of agriculture alter landscapes, and there is no catchall solution, but plant-based food production tends to be less invasive than factory farms. A simple dietary shift can play a big role in protecting the earth beneath us.\n\n# WATER\n\nCREDIT: DANIEL WEDEKING\n\n_Continuing our global journey, imagine leaving Arrow Rock, Missouri, and heading east. Pass over the Missouri River, then the Mississippi, their muddy waters flowing south. Pass over Illinois's Carlyle Lake\u2014a 26,000-acre reservoir\u2014and Indiana's Wabash River. Head north, then east again, along the coast of Lake Erie. Take notice of all the communities, from rural towns to major cities, built near these rivers and lakes, so settlers would have clean, reliable sources of water for drinking, bathing, and farming\u2014for sustaining life. Finally, slow yourself over upstate New York's Finger Lakes region, and pause above one of those communities: the small town of Willet, nestled along the banks of the Ostelic River._\n\nIt was there in Willet, just a few days after Christmas 2008, that twenty-five-year-old Cody Carlson found himself hunched over, scraping frozen cow feces off a thick steel cable.\n\nCarlson grew up in Manhattan, though he had been living in Santa Cruz, earning his degree in community studies from the University of California. Back on the California beaches, it would have been hard to imagine his current position: an undercover factory-farm investigator working for an animal protection organization, freezing his ass off on a massive dairy complex.\n\nAfter graduating, Carlson landed a job at a Manhattan-based firm that conducts investigative research for corporations. There, he spent much of his time in the firm's pro bono division, offering free services to nonprofits. He'd been reaching out to environmental agencies he thought could use his services when, on the news, he learned about an animal cruelty expos\u00e9 conducted by the group Mercy For Animals (MFA). Carlson, a vegetarian since the age of thirteen, was intrigued. He got in touch.\n\n\"I remember e-mailing them and basically asking if they'd like some free research from us,\" recalls Carlson. But what the MFA could really use, they told him, was a full-time undercover investigator.\n\nNext thing he knew, he was quitting his job in the city, putting his belongings into storage, and driving upstate in search of work on a factory farm. After an unsuccessful attempt at getting hired at an egg-producing CAFO, Carlson saw a job posting for a \"maintenance technician\" at Willet Dairy in Locke, New York\u2014nestled in the state's picturesque Finger Lakes region.\n\nThough, as Carlson would learn, Willet Dairy was anything _but_ picturesque.\n\nThe company's 7,800 cows allowed it to ship about 40,000 gallons of milk every day. Those cows also produced roughly 314 _million_ pounds of manure in 2006, according to New York's Department of Environmental Conservation. That's equal, in weight, to almost 78,000 Ford F-150 trucks\u2014from just one dairy facility in one state in one year.\n\nIt's a lot of waste\u2014and helping manage it all was central to Carlson's role at the complex.\n\n\"When I went in for my interview, really the only substantive question they asked me was if I thought I could handle being ankle-deep in shit all day,\" he recalls. \"'It's not a job for everyone,' they said.\"\n\nDuring that interview, farm managers took Carlson on a walk-through to ensure he'd be up for the task.\n\n\"That was when I saw how these cows live\u2014basically standing in overcrowded concrete aisles all day long, hock-deep in their own waste,\" he recounts. \"Twice a day, they'd be moved out to a separate area, where they'd be hooked up to machines for milking. After that, they were brought back into the concrete aisles. That's basically their whole lives.\"\n\nIt was in those aisles where Carlson spent most of his time undercover as a farm \"technician,\" documenting this routine abuse.\n\n\"Yeah, it was a generous title,\" says Cody with a laugh. \"Basically, you've got all these cows standing there letting loose all day. There's an engine that pulls a V-shaped metal device along a steel cable to collect the manure. Eventually, it's pumped down to these massive lagoons that are basically poop ponds.\"\n\nHis job was primarily to fix the pulling and pumping system as it broke\u2014which happened a lot.\n\n\"The cable would fray and hit the cows. It'd get clogged up. It'd freeze. We'd have to scrape the frozen feces off and un-jam the thing\u2014basically like unclogging a gigantic toilet,\" said Carlson. \"It was always something different.\"\n\nAnd not just for Carlson.\n\n\"I can't even get a friggin' clean glass of water,\" said neighbor Karen Strecker, who described turning on her faucets only to find liquid manure pouring out. \"I've been taking a bath and actually had cow shit pour into the tub,\" she said. \"It's nasty.\"\n\nAnd not just to see and smell. For years, Strecker was consistently dosed with antibiotics to treat illnesses caused by her exposure to Willet's waste.\n\n\"Do people get sick when manure gets spread?\" Strecker's doctor, Ahmad Mehdi, asked rhetorically. \"Yes, it's a fact,\" he said. \"When you have ten thousand cows in one place, that's a lot of manure.\"\n\nAcross the country, factory farms house all this manure in massive lagoons, like those at Willet. And those lagoons are often close to rivers, lakes, tributaries, and other waterways, increasing the chances they'll harm local ecosystems and water supplies.\n\nNorth Carolina has been ground zero for this problem.\n\nOn September 16, 1999, Hurricane Floyd struck Cape Fear, along the state's coast, with 105 mph winds. Floyd flooded pig waste lagoons, unleashing millions of gallons of waste into nearby waterways. The county's environmental health director at the time reported that nearly one in ten private wells in the area\u2014families' drinking water\u2014tested positive for fecal bacteria in the storm's wake.\n\nAs it was, the Cape Fear area already had much to be afraid of from factory farms: a few years earlier, in New Hanover County, 2 million gallons of liquid hog waste from a lagoon spilled into a Cape Fear River tributary. Then a hole in a Duplin County pig waste lagoon spilled another 1.5 million gallons of manure and urine into a swamp adjoining the Persimmon Branch\u2014also a tributary of the river. In Jones County, 1 million gallons of liquid hog waste spilled into the Trent River. And in Sampson County, nearly 200,000 gallons spilled into Turkey Creek.\n\nTravel across the country and you see a similar trail\u2014or stream\u2014of aquatic destruction: About 1,000 miles northwest of Cape Fear\u2014near Elmwood, Illinois\u2014a dairy operator pumped 2 million gallons of waste from his 40 _million\u2013gallon_ lagoon into a ravine, draining into Kickapoo Creek and killing local fish populations;,  only 250 miles away, a massive spill in Missouri killed _all aquatic life_ in an 11-mile stretch of Mussel Fork Creek; about 1,800 miles southwest from there\u2014in Oakdale, California\u2014Pete Hetinga of 3H Dairy Farm admitted that over the course of four years, he had discharged wastewater polluted with cow urine and feces into streams that flow into the Tuolumne River and Sacramento Delta; and roughly 800 miles north of Oakdale, in Washington, two factory farms dumped the _entire contents_ of their lagoons, spilling a total of 2 million gallons of manure, near the Yakima River\u2014 _in the same week_.\n\nLike the one Cody Carlson worked at, dairy complexes often pollute waterways and threaten communities. CREDIT: JO-ANNE MCARTHUR\/WE ANIMALS\n\n\"For every pound of beef produced in the industrial system, it takes two thousand gallons of water. That is a lot of water, and there is plenty of evidence that the Earth cannot keep up with the demand.\"\n\n\u2014PRINCE CHARLES, FROM HIS _FUTURE OF FOOD_ SPEECH\n\nAnd of course, as the saying goes, all rivers flow to the sea\u2014including those flooded with toxic waste.\n\nAfter Hurricane Floyd, the tens of millions of gallons of escaped manure\u2014plus tens of thousands of dead pigs who'd drowned inside factory farms\u2014rushed into the Atlantic Ocean's Pamlico Sound, resulting in a \"dead zone\" hundreds of miles wide.\n\nA dead zone is an area of water where oxygen has been choked out, and thus oxygen-dependent life dies; they're caused by pollution, like from factory farms\u2014and they can be massive.\n\nIn 2015, the Gulf dead zone encompassed nearly 6,500 square miles; 2002's record-breaking 8,400-square-mile dead zone could have fit North _and_ South Carolina inside it.\n\nAt the same time runoff and spills from factory farms inland are crippling our nation's waterways and creeping into the ocean, much farther out at sea, our (meat-heavy) diets are also having a negative impact.\n\n\"Of all the threats facing the oceans today, overfishing takes the greatest toll on sea life,\" reports the Environmental Defense Fund. A whopping 2.7 trillion fish are estimated to be taken from our oceans each year\u2014that's more than thirty times the number of all chickens, pigs, cows, goats, sheep, and other livestock farmed globally each year combined.\n\nSo it's no surprise that one study published in the journal _Science_ predicts that by 2048, _all the world's fisheries_ will have collapsed.\n\nThis loss of fish life is devastating in its own right. These trillions of creatures may be dissimilar from pets and other animals we've come to personally know, but they're certainly smart, social beings.\n\nDr. Culum Brown at Macquarie University's Department of Biological Sciences knows this firsthand. He spent much of his childhood tagging along with his father in Southeast Asia. \"Much of that time I was in the water snorkeling,\" he told me. \"It doesn't take much to inspire a kid, and the underwater world was just awesome to me. I never looked back.\"\n\nToday, Dr. Brown is a professor at Macquarie University's Department of Biological Sciences, specializing in the behavioral ecology and evolution of fishes. \"Fish are more intelligent than they appear,\" he says. \"In many areas, such as memory, their cognitive powers match or exceed those of 'higher' vertebrates, including nonhuman primates.\"\n\nAnd even putting the fish themselves aside, the seafood trade is also complicit in killing other marine animals\u2014considered \"by-catch.\"\n\nGlobally, commercial fishing enterprises ravage the oceans for their \"target\" catch\u2014like tuna\u2014by dragging massive nets through the sea. In the process, these indiscriminating nets also scoop up innumerable other animals.\n\n\"Hundreds of thousands of marine mammals are injured or killed every year by fishermen around the world,\" reports NPR, because of \"our taste for seafood.\"\n\nA MESSAGE FROM\n\n## ENVIRONMENTAL WORKING GROUP PRESIDENT KEN COOK\n\n**Some of my earliest experiences with food** production came as a young kid when I spent summers working on my uncle's cattle ranch in Missouri. I then went on to study agriculture at the University of Missouri, where I received my master's in soil science. Since then, I've spent much of my career focused on farm and food policy.\n\nWhen I started EWG in 1993, our focus was on food. Agriculture is the leading cause of water pollution in this country, for example, and factory farms are one of the leading reasons. EWG recently mapped all the CAFO operations in North Carolina and we estimate that these CAFOs produce more than 10 billion gallons of wet animal waste alone every year.\n\nIf we're going to get serious about addressing water pollution, we're going to need to recognize that meat production, globally, is a leading factor. Here at home, Americans eat too much meat. We consume 60 percent more meat than Europeans, and we need to adjust our expectations of how much meat we should be eating way down, and eat less meat and cheese, period.\n\nThroughout my career, I've done my best to raise public awareness and outrage over the disastrous impacts large-scale agriculture has on our water systems and environment. And as each year passes, the sense of urgency only escalates.\n\nWith the rise of China and other big, emerging economies, there is a much greater demand for meat, which means more of the world's waterways will be polluted. However, there is good news to highlight, too. Never before have there been so many companies, organizations, and citizens mobilized around the various issues of food production and factory farming and how the industry adversely impacts the environment. And coupled with the power of social media and other tactics to harness and forge pressure, there is a real opportunity for organizations like EWG, its supporters and forward-thinking companies alike to bring about significant and positive changes in the way our food is produced.\n\n_Ken Cook is one of the environmental field's most influential critics of industrial agriculture. Under his leadership, EWG has pioneered digital technologies to expose the harm done by misconceived crop subsidies and runaway agricultural pollution, and empowered American families with easy-to-use tools to eat better, healthier, and more sustainably. Cook is a board member (and founding chairman) of Food Policy Action, Organic Voices, and the Amazon Conservation Team, and a former member of the board of the Organic Center._\n\nA catfish factory farm in Louisiana. CREDIT: SCOTT BAUER\/USDA\n\nA Natural Resources Defense Council study confirms that over 650,000 aquatic mammals are unintentionally killed each year in the global seafood trade\u2014including New Zealand sea lions, several different whale species, porpoises, dolphins, and more.\n\nUnfortunately, one way the industry has begun tackling this problem is, misguidedly, through aquaculture: fish factory farms.\n\nPicture a major dairy farm or a pig complex: animals packed next to one another, barely able to move, and producing huge amounts of waste. That's aquaculture\u2014and it's the fastest growing form of animal farming on the planet.\n\nAquaculture operations can either be established inland\u2014using gigantic tanks\u2014or in open waters, using a netting system instead.\n\nIn 2013, 155 _billion_ pounds of these factory-farmed fish were produced. That's the equivalent of almost 350,000 Statues of Liberty.\n\nAnd these fish factories come with their own set of ecological disasters.\n\nFor one, they often _use_ more fish than they produce\u2014by feeding fish to the fish they're farming. It sounds like a tongue twister, and is certainly twisted. It means that even if eating _farmed_ fish, one is likely still contributing to all the problems associated with _wild-caught_ fish, since wild-caught fish may be used as feed in aquaculture.\n\nThese operations can be highly damaging in other ways, too.\n\n\"Open net-cage salmon farming,\" for example, \"is currently one of the most harmful aquaculture production systems and poses environmental threats in all regions it is practiced,\" writes the Coastal Alliance for Aquaculture Reform.\n\n**When we think about water use, we tend to think of our showers. And when we think about water and _food_ , we may only think about the water we use to prepare it. But what about all the water used to _produce_ our foods?**\n\n\"They may breathe with gills instead of lungs and move with fins instead of feet, but fishes are every bit as intelligent, complex, and social as other animals we may be more familiar with, like dogs and cats.\"\n\n\u2014JONATHAN BALCOMBE, NEW YORK TIMES\u2013BESTSELLING AUTHOR OF _WHAT A FISH KNOWS: THE INNER LIVES OF OUR UNDERWATER COUSINS_\n\nThose threats include sea lice (marine parasites that can kill wild fish), algae blooms (which can release toxins into the water), high concentrations of waste\u2014which may be laced with antibiotics\u2014left on the ocean floor, and alien species being introduced into native waters.\n\nSo whether on one of the world's many fish factory farms or out in the open seas\u2014it's clear that the overconsumption of animal proteins is unleashing environmental disasters.\n\nThat is, of course, unless we continue shifting our diets toward more _plant-based_ foods\u2014which is exactly what Cody Carlson would like to see.\n\nIt's been more than seven years since Carlson ended his undercover investigation into Willet. In that time, the wind has indeed blown the buying public away from meat, seafood, and dairy\u2014and toward consuming more plants.\n\n\"Growing up in New York City, you know, I was raised on food like pizza and pastrami,\" he says. \"I've been vegetarian for almost twenty years now, and the last few years have definitely been, in my opinion, the best time for meat-free options.\"\n\nCarlson chalks up the increase in those options to the general public shifting more toward plant-forward fare. That added demand is already taking a bite out of factory farms.\n\n\"With plant-based diets all the rage in the U.S. currently, milk alternatives\u2014and particularly those from almonds\u2014are seeing strong sales growth and increased innovation, while sales of dairy milk are contracting,\" reports _Food Navigator_. \"In the past five years, sales of almond milk have grown 250 percent to more than $894.6 million... [while] the total milk market shrunk by more than $1 billion.\"\n\n\"It's inspiring,\" says Carlson, of this growth in the plant-based food sector. \"When I was working in that hellhole, it was easy to lose hope. But plant-based eating has just become so popular in the last few years. I think about everyone, everywhere, impacted by these factory farms. More people are going to continue making this shift\u2014I know it. Now I've got nothing _but_ hope for the future.\"\n\nA diver for the Philippine fishing vessel _Vergene_ at work in and around a skipjack tuna purse seine net. CREDIT: ALEX HOFFORD\/GREENPEACE\n\n\"The factory meat industry has polluted thousands of miles of America's rivers, killed billions of fish, pushed tens of thousands of family farmers off their land, sickened and killed... U.S. citizens, and treated millions of animals with unspeakable and unnecessary cruelty.\"\n\n\u2014ROBERT F. KENNEDY JR.\n\nNetted fish off the coast of South Africa. Global fish stocks are collapsing due to seafood-heavy diets. CREDIT: MATTHEW PRESCOTT\n\n## **IN A NUTSHELL**\n\n\u2022 Factory-farm waste causes massively contaminated runoff.\n\n\u2022 Storm water can mix with animal waste, carrying it into our rivers and oceans.\n\n\u2022 Growing huge quantities of grain and corn to feed animals uses vast amounts of water.\n\n\u2022 It can take hundreds of gallons of water to produce a single pound of meat.\n\n\u2022 The oceans are being destroyed by seafood overconsumption.\n\n\u2022 By 2048, all the world's fisheries could collapse.\n\n\u2022 Trawling vessels rip aquatic mammals, sea turtles, coral reefs, and more from the oceans.\n\n\u2022 Fish farms may use more fish to feed their fish than the total amount of fish they produce.\n\n\u2022 Open-net fish farms may introduce dangerous alien species to the waters in which they operate.\n\n**Now for the good news!** Eating more plant-based foods can help us tread lighter when it comes to water pollution and overuse. We of course need to use water to produce all foods, but producing plant-based foods is far less water-intensive (plus, no waste lagoons!), and eating less seafood can help us take a bite out of overfishing and destructive fish farming.\n\n# AIR\n\nLisa Inzerillo and her map showing the factory farms in her community. CREDIT: MATTHEW PRESCOTT\n\n_Leaving upstate New York on our quest, imagine heading south\u2014across Pennsylvania's Pinchot Forest, across the Delaware River\u2014to the Delmarva Peninsula: a 200-mile-long, 80-mile-wide peninsula wedged between the Chesapeake Bay and the Atlantic Ocean._\n\nThe Delmarva is almost a world unto itself. Herds of wild horses run on the peninsula's pristine beaches. Its fourteen-thousand-acre Chincoteague National Wildlife Refuge is home to elk and red foxes and ponies and pelicans. Historic farmhouses, quaint communities, cotton fields, and vineyards dot the landscape as you drive north from Virginia Beach into Maryland, then Delaware, up toward Philadelphia.\n\nBut life on the Delmarva isn't exactly what it seems.\n\n\"We're under attack,\" said resident Lisa Inzerillo. \"I'm afraid to even go outside.\"\n\nLisa and her husband, Joe, live on land that's been part of their family for four generations, since Lisa's great-grandparents first built there in 1875. Smack in the middle of the peninsula, between Assateague Island National Seashore to the east and Tangier Sound to the west, the Princess Anne, Maryland, property was idyllic.\n\nIn 2010, Lisa and Joe built a new home there. They dug a pond in the back for wildlife, built a small carriage house off to the side. They'd sit out back and relax on their little slice of paradise, breathing in the country air. It was, in a word, perfect. \"That is, until about three years ago,\" Joe told me when I visited them in 2017, \"when the chicken factories starting popping up.\"\n\nIt's those chicken factories\u2014and the harmful, putrid gases they emit into the neighborhood's air\u2014that cause Lisa and Joe to live in such fear.\n\nNowadays, Lisa and Joe are surrounded by dozens upon dozens of these industrial poultry complexes: a mile and a half south is one complex with thirty massive chicken sheds; a mile north, off Peggy Neck Road, there are another nine; three miles southwest, there are yet another nine. The list goes on.\n\nPicture driving through your average American small town\u2014modest split-level and ranch homes with front lawns, one after another, kids' toys in the front yards. But now picture that every fourth or fifth plot is not a house, but a mega poultry farm confining tens of thousands of birds\u2014mixed right in with the homes.\n\nMany of the farms, if you can call them that, are muddy and overgrown. Buzzards constantly circle overhead, ready to pick off dead birds. Huge tractor trailers\u2014carrying birds, feed, and equipment to and from the facilities\u2014now dominate the neighborhood's narrow streets. Ammonia from the birds' waste fills the air, as do flies. And the stench\u2014the stench ranges from sickening on good days to gut-churning on bad ones.\n\n\"This property's been in my family almost 150 years,\" said Lisa. \"We built this house ourselves just a few years ago. And now we'll have to sell it, probably at a loss. Who wants to buy a house in the middle of factory-farm hell, where the air smells so bad you can't even go outside?\"\n\nA glimpse inside a poultry warehouse. CREDIT: JO-ANNE MCARTHUR\/WE ANIMALS\n\nStories like Lisa's are common up and down the Delmarva. At any given time, the peninsula is home to nearly _600 million_ chickens. Kept in an estimated 4,600 massive warehouses that can reach 600 feet long and that look more like airplane hangars than farms, chickens now rule the region's roost: there are nearly 500 birds for each of the 1.4 million human residents.\n\nAnd residents are crying foul about it. \"I'm sick to death of them saying that these are family farms,\" said Somerset County's Thomas Kerchner, referring to poultry producers' proclivity for insisting that factory farms are _family_ -owned. \"They are _farms_ like a steel mill is a _blacksmith shop_. There's no comparison.\"\n\nHe and his wife, Sherri, put their home up for sale after deciding they simply could not live among the stench.\n\n\"Where you had fifty thousand chickens on a given plot of ground, you've got a half million or two million now\u2014which produces a huge problem of what to do with the manure,\" said Tom Horton, a columnist for Delmarva's _Bay Journal_.\n\nHorton, who covered environmental issues for three decades at the _Baltimore Sun_ , is concerned for the region. \"It's too much manure,\" he says. \"Too many animals.\"\n\nToo many animals indeed. Today, the United States produces nearly nine billion chickens every single year. That means that in this country, every minute of every day\u2014night and day\u2014we slaughter over seventeen thousand birds.\n\nThe vast majority of those birds have been genetically manipulated to grow unnaturally fat, unnaturally fast. In 1920, it took a chicken sixteen weeks to reach about two pounds; today's chickens may now reach nearly triple that in only _six_ weeks. According to a report in the journal _Poultry Science_ , it's the equivalent of a two-month-old human baby weighing six hundred pounds.\n\nBecause the birds are so young when they're killed\u2014at about forty-five days old\u2014their bones remain immature. So as they become more and more top-heavy throughout their short lives, countless animals suffer broken bones, unable to keep up with their rapidly growing size. They suffer heart attacks and their lungs fail\u2014also a result of their fast growth and enormous size. Nowhere else in the animal kingdom do we see baby animals several weeks old dying from heart attacks in such vast numbers.\n\nAnd their size isn't the only astronomically large thing about industrially produced farm animals: every single minute of every single day, farm animals nationwide produce 7 million pounds of waste. In Delmarva alone, chickens produce roughly 1.5 billion pounds of manure each year\u2014more than all the annual human waste from New York City, Washington, San Francisco, and Atlanta combined.\n\nLisa and Joe, Thomas and Sherri\u2014indeed, everyone else dealing with this problem on the Delmarva: they are not alone. Almost a thousand miles away in Calhoun, Kentucky, seventy-six-year-old Bernadine Edwards is surrounded by more than one hundred mega chicken factories within just a two-mile radius.\n\n\"There's a horrible odor, a stench, and I have flies and rodents digging in, trying to get into my house,\" said Edwards. \"I'm too old to start over,\" she lamented. \"I can't afford to.\"\n\nPerhaps not surprisingly, the rural areas where factory farms often set up shop are not wealthy neighborhoods full of luxury cars and new homes; rather, they tend to be concentrated in lower-income areas, particularly with non-white populations.\n\nEnvironmental justice advocates call this environmental racism. Unsavory and environmentally destructive facilities\u2014landfills, coal plants, toxic-waste dumps, and more\u2014are placed in lower-income neighborhoods, often inhabited by people of color. This goes for factory farms, too, which a University of North Carolina School of Public Health study found \"are disproportionately located in communities of color and regions of poverty.\"\n\nThat same study also found that schools with a significant number of white and higher-income students were located, on average, 10.8 miles away from CAFOs; schools with a significant number of non-white students and roughly half their students on lunch program assistance were located only 4.9 miles away.\n\n\"Jesus says of the birds of the air that 'not one of them is forgotten before God.' How then can we possibly mistreat them or cause them harm? I ask all Christians to recognize and to live fully this dimension of their conversion. May the power and the light of the grace we have received also be evident in our relationship to other creatures and to the world around us. In this way, we will help nurture that sublime fraternity with all creation which Saint Francis of Assisi so radiantly embodied.\"\n\n\u2014POPE FRANCIS\n\nThe poultry industry confines billions of birds in mega complexes nationwide. CREDIT: FRANCIS LEROY\n\n\"These companies seek rural areas where unemployment, or underemployment, is high and people are desperate,\" says Aloma Dew, a Sierra Club organizer in Kentucky. \"They assume that poor, country people will not organize or speak up, and that they will be ignorant of the impacts on their health and quality of life.\"\n\n\"Quality of life\" seems like a stretch here. Bernadine Edwards in Kentucky and the residents of the Delmarva are exposed to a litany of harmful gases in their air\u2014like hydrogen sulfide, which can cause skin, eye, and respiratory irritation; brain and heart disorders; and even death.\n\nBecause factory farms can confine hundreds of thousands of chickens or other animals at a single location, these gases may be emitted at high concentrations; that can be detrimental to nearby communities.\n\nIn fact, both North Carolina and Iowa\u2014two of our nation's largest factory-farming hubs\u2014have published controlled studies of CAFOs' ill effects on their neighbors. In Iowa, for example, increased instances of eye and upper respiratory problems persisted among people living within two miles of a factory farm.\n\nWhen I visited with Lisa and Joe on the Delmarva, they complained of respiratory problems. \"I'm an ER physician,\" Joe said. \"I'm around sickness all day but never get sick from that. It's the poultry farms that are the problem.\"\n\nThe fans attached to mega poultry facilities often blow \"CAFO dust\" that threatens communities for miles around. Lisa Inzerillo, on the Delmarva Peninsula, has felt (and breathed) these impacts firsthand. CREDIT: USDA\n\n\"Communities near industrial farm animal production facilities are subject to air emissions that... may significantly affect certain segments of the population. Those most vulnerable\u2014children, the elderly, individuals with chronic or acute pulmonary or heart disorders\u2014are at particular risk.\"\n\n\u2014THE PEW COMMISSION ON INDUSTRIAL FARM ANIMAL PRODUCTION, A PROJECT OF THE PEW CHARITABLE TRUSTS AND JOHNS HOPKINS BLOOMBERG SCHOOL OF PUBLIC HEALTH\n\nAnd with that, he ducked out to pick up his respiratory medication at the pharmacy. He grabbed his keys and jacket and left through the side door. Lisa, saddened, chimed in: \"It's because all these mega chicken houses all around us have these giant fans on them. They're blowing God knows what into the air all day long. And we're breathing it in. When it's real windy, you can't even go outside. It's either the stench or the flies. We just stay inside mostly now.\"\n\nWhat keeps Lisa and Joe inside is the particulate matter\u2014including of the fecal variety\u2014that escapes from CAFOs and is carried by the wind. What they breathe in when they go outside is referred to as CAFO dust\u2014and it's incredibly dangerous: it's a top cause of bronchitis, can cause heart disorders like arrhythmia, and can even lead to heart attacks.\n\nAnd these ill effects are nothing new: we've known of them for years.\n\nOne report, conducted for the North Dakota Attorney General's office, analyzed fifty-six socioeconomic studies about the impact CAFOs have on local communities.\n\n\"Based on the evidence,\" the study concluded, \"public concern about the detrimental community impacts of industrialized farming is warranted.\" The report continued: \"This conclusion rests on five decades of government and academic concern with this topic... five decades of social science research which has found detrimental effects of industrialized farming on many indicators of community quality of life.\"\n\nFifty years of social science and government concern over this issue; living proof from communities on the Delmarva, in Kentucky, Iowa, and North Carolina; study after study, year after year: how is it that producers have been able to continue ravaging rural communities for so long?\n\nIn short, we let them.\n\nA MESSAGE FROM\n\n## OXFAM AMERICA CAMPAIGN MANAGER BEN GROSSMAN-COHEN\n\n**Right now, one person in nine** goes to bed hungry every night\u2014not because there isn't enough food to go around, but because of deep power imbalances and improper resource allocation. The reality is that eating meat requires way more energy, land, water, and other finite resources than other delicious foods. And individuals can help rectify this imbalance with their own diets simply by eating less meat.\n\nIndeed, our diets are deeply intertwined with a complex food system that greatly affects people all over the world. We've learned over decades of responding to humanitarian crises and working in some of the poorest places on the planet that ending poverty requires major shifts in how we grow, distribute, and consume food.\n\nFor example, Oxfam is working closely with workers in poultry-processing facilities throughout the United States who face some of the most dangerous and difficult jobs for extremely low wages. Many of the workers in these plants are people of color, often undocumented immigrants who lack basic workplace protections.\n\nAnd of course, we all vote when we make choices about what we eat. The more people are willing to adjust their diets\u2014like by eating less meat\u2014and use their voice, the more we can take the growing concern over how our food is produced and create systemic change. The more we can shift incentives for the food industry to produce more plant-based foods and build greater ties with like minds in developing countries to do the same at the local level, the more we can avoid the spread of the worst sins of our current production model.\n\nThe simple truth is that we cannot sustain the current model of production and consumption without regard for the effect on resources and our climate. Faced with constrained resources it is always the poorest and most vulnerable who lose out. The more we can do to use our voices to advocate for change and our pocketbooks to support those businesses who are doing the right thing, the better everyone will be.\n\n_Ben Grossman-Cohen is campaign manager for Oxfam America. Through its international confederation of organizations working together with partners and local communities in more than ninety countries, Oxfam mobilizes the power of people against poverty._\n\nCaged chickens in India used to support a growing dependence on animal proteins. CREDIT: FRANK LOFTUS\/THE HSUS\n\n\"When the poison control center official spoke to Julie Jansen, his words were shocking: Ma'am, the only symptoms of hydrogen sulfide poisoning you're not experiencing are seizures, convulsions, and death. Leave the area immediately. Panic-stricken, Jansen grabbed her six children and her friends' two children and drove away from her home. Jansen first thought the 11-year-old, home-based day care center she owned in Olivia, Minnesota had been hit by a flu bug. In the spring of 1995, 17 children ranging in age from newborn to 13 shared a long list of symptoms\u2013diarrhea, nausea, headaches, vomiting, teary eyes, and stuffy noses. She soon noticed that it only happened when the wind blew from the south. Two factory-scale hog farms had recently located not more than a mile and a half away. It turned out the hog operations were poisoning the air with toxic wastes.\"\n\n\u2014FROM THE NRDC REPORT _CESSPOOLS OF SHAME_\n\n\"The same factory farmers who intensively confine animals and deal them never-ending privation also release massive volumes of untreated animal waste, making life miserable for people and animals.\"\n\n\u2014WAYNE PACELLE, PRESIDENT AND CEO, THE HUMANE SOCIETY OF THE UNITED STATES\n\nActually, we _made_ them: in 1960, the average American ate only 34.3 pounds of poultry each year. In 2016, that amount had climbed to more than 108 pounds per person, according to the National Chicken Council.\n\nSo compared to 1960, today we have nearly twice as many people each eating more than twice as many birds.,  No wonder production has gotten so intensified, so concentrated, so dangerously out of control.\n\nThe good news is that we can each use the power of our plates to fix this problem, simply by cutting back on chicken and other poultry products in our diets.\n\nThat's exactly what Lisa Inzerillo has done. \"I can't even eat chicken anymore,\" she told me. \"I feel like I've gone through a grieving process and I just can't get past being angry over what they've done to us. They're ruining our lives, disrespecting animals, hurting our neighbors. So yeah, I just can't eat it anymore.\"\n\nFortunately for Lisa and countless other people cutting back on poultry, there are protein-packed plant-based foods that can help us live healthier and avoid supporting the types of industrial operations that are polluting our air and making life so difficult for so many communities. There are products like tempeh and seitan. The market for tofu is exploding as more Americans enjoy the product, going from a $1 billion industry to nearly a $5 billion industry. And a huge variety of chicken-like-but-chicken-free meatless products have taken off in the market and are now available virtually everywhere.\n\nMany restaurants now offer plant-based chicken products. Tropical Smoothie Caf\u00e9 offers customers the opportunity to replace chicken with a product called Beyond Meat in nearly any item. Beyond Meat, made entirely from plant-based ingredients, has the taste and texture of chicken, yet has 20 grams of protein in each 3-ounce serving, compared to 13 grams for the same portion of chicken.\n\nBesides Beyond Meat, there are brands like Gardein (meat-free chicken tenders, scaloppini, and more), Boca (plant-based chicken patties and nuggets), and Morningstar Farms (Buffalo wings and more)\u2014to name only a few.\n\nGrocery stores now offer entire sections of these products\u2014and not just Whole Foods: at Walmart, Kroger, Safeway, Albertsons, and nearly every other supermarket chain, one can find delicious plant-based \"chicken\" products. Target has even launched its own brand of vegetarian chicken. The company's Simply Balanced store brand meat-free \"meats\"\u2014like Korean BBQ Meatless Chicken, Teriyaki Meatless Chicken, Smoky Chipotle Meatless Chicken, and Mushroom Miso Meatless Turkey\u2014are available nationwide.\n\nThese products are certainly not zero-footprint foods. But their success is the result of millions more consumers now mixing meat-free meals into their diets. Just as many people are seeking out more humane products from local producers committed to higher animal welfare and environmental standards, countless people are also shifting to more plant-focused diets.\n\nSome do it for their own health, with data clearly showing that diets lower in meat and higher in plant proteins are better for us. Some do it for the animals\u2014perhaps uncomfortable with the way these chickens are warehoused, or with knowing that seventeen thousand of them are slaughtered for our plates every minute of every day.\n\nOthers are doing it for people like Bernadine Edwards in Kentucky and Lisa Inzerillo in Maryland\u2014for all the families and communities nationwide who, with each breath, are victimized by our nation's chicken addiction. They're opting, with every dinnertime decision, to use their forks as a force for change\u2014and each of us can, too.\n\n## **IN A NUTSHELL**\n\n\u2022 Air pollution from factory farms takes a toll on local communities.\n\n\u2022 Putrid odors can prevent people from going outside or opening their windows.\n\n\u2022 Dangerous gases like methane, ammonia, and hydrogen sulfide are emitted as waste breaks down.\n\n\u2022 Exposure to these gases can cause illness and neurological problems, including depression.\n\n\u2022 Factory farms tend to be concentrated in low-income areas.\n\n\u2022 Poor people and people of color are disproportionately impacted by factory farms' air pollution.\n\n\u2022 On Maryland's Eastern Shore, the poultry industry is ravaging poor communities and negatively affecting families who've lived on the peninsula for hundreds of years.\n\n**Now for the good news!** We can all rest\u2014and breathe\u2014a little easier knowing that plant-based foods are easier on the air. Producing, say, chick _peas_ instead of chick _en_ is less polluting and gentler on local communities. So simply by enjoying more of these types of foods\u2014even if meat is still sometimes on the plate\u2014we can each help play a significant role in cleaning the air.\n\n# FIRE\n\nCREDIT: RODRIGO BALEIA\/GREENPEACE\n\n\"Climate change is no longer some far-off problem. It is happening here. It is happening now. And it becomes more dramatic with each passing year.\"\n\n\u2014PRESIDENT BARACK OBAMA\n\nFor the final stop on our worldwide voyage, we leave the Delmarva and head, this time, not to another place on the planet\u2014but to a spot far above it. Rise straight up: back through the clouds, into the stratosphere. Continue through the thermosphere until you have a clear view of the Earth below. From above, watch the erratic weather patterns\u2014the hurricanes, the tsunamis. Watch the ice caps shrink and coastal areas flood. This is the raging fire that is global warming; this is climate change. Its impacts are devastating\u2014and all around us.\n\nOne sweltering Sunday in the summer of 2014, my wife and I headed south from our home in New Orleans toward the Gulf of Mexico. We drove down state highway 90, across the Bayou des Allemands and through all the little Creole towns with names like LaRose and Galliano\u2014past Catfish Lake and Little Lake and Lake Jesse and North Lake and South Lake and Lake Palourde and Lake Laurier\u2014peppered with colorful houses built high up on wooden stilts or floating on empty fifty-five-gallon drums.\n\nThe region's so awash that on a map, it looks like a bright blue canvas painted, Jackson Pollock\u2013style, with only slight drizzles of intersecting green.\n\nSix thousand years ago, the region was entirely submerged. Melting glaciers eventually created Lake Pont-chartrain and carved out Pine Island, near present-day New Orleans. When the Mississippi changed course, heading toward the new island, it dumped sediment into the Gulf\u2014sediment that formed land and eventually connected with Pine Island, creating the land mass that, today, includes everything between the Gulf and the Big Easy.\n\nThat all played out at a snail's pace, over millennia.\n\nThen came climate change.\n\nThough Earth's climate has always been a work-in-progress, today's scientific consensus is that human activities are undoubtedly causing dangerous changes to it\u2014like progressively erratic weather patterns and increasing global temperatures.\n\nSea levels, too, are rising, which has Joann Bourg and her family gravely worried.\n\nJoann\u2014along with her sisters, mother, son, and niece\u2014are the third generation of Bourgs living on the family's land, about eighty miles south of New Orleans, in Isle de Jean Charles, which is essentially a floating town situated precariously between Lake Chien and Wonder Lake.\n\nBut it may not be afloat much longer: thanks to climate change, the region is essentially sinking back into the Gulf.\n\n\"Yes, this is our grandpa's land,\" Ms. Bourg told the _New York Times_ in a profile on the region's fate. \"But it's going under one way or another.\"\n\nIn 2016, a $48 million U.S. Department of Housing and Urban Development grant became the first use of federal funds to relocate an entire community impacted by climate change.\n\nLouisiana is under constant threat of flooding due to the climate change. CREDIT: USDA\n\n\"We're going to lose all our heritage, all our culture,\" grieved Biloxi-Chitimacha-Choctaw Chief Albert Naquin. \"It's all going to be history.\"\n\nWorldwide, families like the Bourgs and people like Chief Naquin are at war with our changing climate\u2014a war for their land and lives.\n\nTuvalu\u2014a tiny island nation in the South Pacific, north of Fiji and east of Papua New Guinea, whose highest point is merely fifteen feet above sea level\u2014may be among Earth's first national victims of that war.\n\n\"Cyclones and tropical storms have been getting much worse since the 1980s,\" says Hilia Vavae, Tuvalu's chief meteorologist. \"We had a big drought starting in 1999. Flooding from extreme high tides is increasing also. In the late 1990s, water started coming out of the ground\u2014first puddles, then a whole sea.\"\n\nIt seems Tuvalu\u2014like southern Louisiana\u2014may soon meet a watery grave.\n\n\"If, as predicted, sea levels continue to rise,\" confirms the International Federation of Red Cross and Red Crescent Societies, \"this string of low-lying islands in the southwest Pacific could gradually disappear.\"\n\nSo who's to blame? In the unending finger-pointing that often accompanies climate change discussions\u2014the few remaining deniers left misguidedly point to natural processes, scientists point to human-caused emissions, the solar industry points to coal, the coal industry to transport, and so on\u2014one thing has become inarguably clear: it's not our fingers we ought to be pointing, but our forks.\n\nThe consequences of meat overconsumption are undeniable\u2014on our land, in our waters, in our air. The Amazon being deforested for animal agriculture is a travesty in its own right, as is the oceanic devastation wrought by the seafood sector; Lisa Inzerillo on the Delmarva Peninsula certainly doesn't need any reason other than the attack on her own air quality to protest poultry production.\n\nScientists from ICESCAPE\u2014Impacts of Climate Change on the Eco-Systems and Chemistry of the Arctic Pacific Environment\u2014study the region's melting ice. CREDIT: NASA\/KATHRYN HANSEN\n\nBut making matters far worse is the fact that each of these ecological nightmares, as bad as they are individually, also plays into a much broader catastrophe: climate change.\n\n\"The livestock sector is a major stressor on many ecosystems and on the planet as a whole,\" opens _Livestock's Long Shadow_ \u2014that same UN Food and Agriculture Organization (FAO) study that reported on animal agriculture's massive land requirements. \"Globally, it is one of the largest sources of greenhouse gasses.\"\n\nLivestock, the FAO reports, contribute a whopping 14.5 percent of _all_ greenhouse gas emissions worldwide.\n\nBut in its research, the FAO didn't even count fish or shrimp as \"livestock,\" focusing instead on mammals and birds. This is a grave omission of emissions, points out Risto Isom\u00e4ki in his 2016 book, _Meat, Milk and Climate: Why It Is Absolutely Necessary to Reduce the Consumption of Animal Products_. It means emissions from animal proteins\u2014not just land-based livestock, but _all_ animal proteins\u2014in our diet are actually vastly higher than reported.\n\nThink back to all those ocean-trawling vessels. On top of depleting fish stocks and killing marine mammals like dolphins and whales, all those ship fleets scouring the oceans worldwide also burn massive quantities of fossil fuels, emitting huge amounts of greenhouse gases in the process.\n\nMangrove canopy, Indonesia. Worldwide, millions of hectares of mangrove swamps have been converted to shrimp farms, emitting enormous quantities of carbon dioxide in the process. CREDIT: SIGIT DENI SASMITO\/CIFOR\n\n\"If every American skipped one meal of chicken per week and substituted vegetables and grains, for example, the carbon dioxide savings would be the same as taking more than half a million cars off of U.S. roads.\"\n\n\u2014ENVIRONMENTAL DEFENSE FUND\n\nSOURCE: ENVIRONMENTAL WORKING GROUP\n\n\"Did you know that the meat industry is responsible for approximately 20 percent of the world's greenhouse gas emissions? In fact, producing one calorie of meat requires nearly twenty times the amount of energy as one plant calorie! With global meat consumption tripling over the last four decades, the meat industry now emits over 36 billion tons of greenhouse gases annually and is showing no signs of slowing down. If we want to make a real dent in the world's carbon footprint and reduce our own personal footprint, we need to eat less meat.\"\n\n\u2014EARTH DAY NETWORK\n\nThink back to aquaculture. Fish factory farms are to blame for much more than just localized water pollution. Millions of hectares of mangrove swamps, for example, have been converted to shrimp farms, emitting enormous quantities of carbon dioxide as they're destroyed. In fact, the World Watch Institute points out that _more than half_ the world's mangrove forests have been destroyed, with shrimp farming a main perpetrator in their destruction.\n\nAnd even with these omissions in the FAO's research, the numbers are still staggering.\n\nAs the FAO reports, livestock cause \"an even larger contribution [to emissions] than the transportation sector worldwide.\" That's right: our use of animals for food\u2014 _not even counting seafood_ \u2014contributes more to climate change than all the world's cars, trucks, trains, planes, and ships _combined_.\n\nMuch of the livestock sector's emissions are the result of the fact that all these animals we raise for food have to eat food themselves.\n\nThink of the 600 million chickens on the Delmarva, and the billions more throughout the country. Factory farms require massive amounts of feed.\n\nAs the _American Journal of Clinical Nutrition_ reports, in the United States, farm animals consume more than _700 percent_ the amount of grain than is eaten by the _entire_ American (human) population.\n\nWhere does that feed come from? Remember the Amazon and other tropical forest land being razed for meat production? Though some of that razing is done for _grazing_ , much is done to make room for _feed crops_. And during all that deforestation\u2014that converting of forest land to feed land\u2014grave quantities of greenhouse gases are spewed into the atmosphere.\n\nThen, once the land has been converted, managing those feed crops also requires profanely heavy fossil fuel consumption: running all the tractors and processors and cars and trucks and other industrial equipment deals yet another major blow to our climate.\n\nSo what about animals who eat grass instead of grain? Though \"grass-fed\" cows represent only a minuscule percentage of the entire livestock sector, they too are a significant contributor to greenhouse gas emissions.\n\nCows are very gaseous animals, and especially when they eat grass, their digestive processes emit colossal quantities of methane\u2014a greenhouse gas that's _30 times more potent_ than even carbon dioxide. Globally, farm animals are one of the most significant sources of anthropogenic methane, and are responsible for up to 40 percent of all methane emissions worldwide.,\n\nThis is why \"grass-fed meat is more, not less, greenhouse gas intensive,\" says Dr. Gidon Eshel, a climatologist at Bard College's Center for Environmental Policy.\n\nBut people need protein. And while industrial chicken, fish, pig, and cow operations of all kinds are extremely climate _un_ friendly, certainly it's true that growing, processing, and transporting food of _any_ kind (whether animal- or plant-based) requires resources.\n\nSo what would the impact on climate change be if we simply processed more plant foods into protein-packed products of their own, rather than funneling so many through animals? What if we cut out the middle _hen_?\n\nAccording to the Environmental Defense Fund, if each American replaced chicken with plant-based foods at _just one meal per week_ , the carbon dioxide savings would be the same as taking more than half a million cars off U.S. roads.\n\nAnd that's just chicken\u2014and just carbon dioxide.\n\nOn the whole, \"the production of animal-based foods,\" confirms a study published in the journal _Climate Change_ , \"is associated with higher greenhouse gas emissions than plant-based foods.\"\n\nGrass-fed cows emit huge amounts of methane, making grass-fed beef \"more, not less, greenhouse gas intensive,\" according to climatologist Dr. Gidon Eshel. CREDIT: JO-ANNE MCARTHUR\/THE GHOSTS IN OUR MACHINE\n\nA MESSAGE FROM\n\n## SIERRA CLUB DEPUTY EXECUTIVE DIRECTOR BRUCE HAMILTON\n\n**As the threat of climate disruption** becomes increasingly urgent, it makes sense that every source of greenhouse gas emissions should come under scrutiny. Both the reckless burning of fossil fuels and unsustainable agricultural practices are major contributors to greenhouse gas emissions. However, as the recent documentary film _Cowspiracy_ rightly points out, the latter are too often overlooked as a potential source of reductions.\n\nIn fact, how we farm and what we eat can make a real difference for our climate future, and that knowledge should inform not only our personal choices but also our public policies.\n\nEliminating or reducing meat consumption in your diet is one important way to reduce your contribution to climate change, since animal agriculture is the single largest source of global greenhouse gas emissions from food production. At the same time, the Sierra Club continues to support broader reforms in food production that will also help limit climate disruption.\n\nMany current agricultural practices, such as large-scale monocropping (the practice of growing a single crop year after year on the same land) and concentrated animal-feeding operations (CAFOs), consume disproportionate amounts of fossil fuels, pollute our water and air, deplete the soil, and diminish biodiversity. The good news is that we have many opportunities to improve in all of these areas.\n\nWe're calling for reform of industrial agricultural and food system practices, to minimize contributions to greenhouse gases and to maximize carbon sequestration in plants and soils. The pollution from concentrated animal-feeding operations in particular is grossly disproportionate to the amount of food produced. Growing heavily subsidized energy-intensive corn to convert to ethanol fuel makes no sense from an energy, food-supply, climate, or pollution standpoint, and it should be opposed.\n\nThe single greatest source of agricultural greenhouse gas emissions is livestock, particularly factory-raised animals. Cattle (for both beef and milk, as well as for inedible outputs like manure and draft power) are responsible for about two-thirds of livestock emissions.\n\nFortunately, we _can_ cut livestock emissions significantly not only by reducing personal meat consumption but also by following best practices and ending our reliance on concentrated animal-feeding operations. The Sierra Club continues to strongly oppose the establishment of new CAFOs and believes we should phase out existing operations as soon as possible.\n\nFurthermore, ensuring soil maintains its carbon stock is a highly effective means of carbon sequestration. Yet most agricultural soils have had their carbon stock dramatically reduced by soil loss, excessive tillage, overgrazing, erosion, and overuse of chemical nitrous fertilizers. In fact, the world's cultivated and grazed soils have lost 50 to 70 percent of their original carbon stock. In the process, billions of tons of carbon have been released into the atmosphere. That's why it's critical that we rebuild soil carbon through regenerative agricultural practices.\n\nMassive food production operations are at the root of many of these problems. Converting our natural landscapes into intensive agricultural operations can change land from carbon sinks to carbon sources. Deforestation, plowing up prairies, and filling wetlands destroys existing carbon sinks and releases that carbon into our atmosphere, increasing emissions.\n\nAs consumers, we each have a personal role to play as well, through our choices about the foods we eat:\n\n\u2022 Whenever possible, we can support locally owned and operated farms, which are generally far less destructive and far more productive. This also reduces the need for long-distance transportation of foods.\n\n\u2022 Striving to reduce food waste, through smaller serving sizes, composting, and recycling, will also reduce greenhouse gas emissions.\n\n\u2022 Organically grown foods that don't rely on chemicals are better for the soil, climate, and our health.\n\nAddressing climate disruption is important enough that we cannot afford to overlook any strategy for success. Fortunately, just as with transitioning from fossil fuels to clean energy, we can reap important collateral benefits by adopting more responsible and sustainable agricultural practices and by making smarter lifestyle choices.\n\n_Bruce Hamilton has worked for the Sierra Club for four decades. He's served on the Environmental Support Center Board of Directors, the U.S. Department of Energy Environmental Advisory Board, and the U.S. Environmental Protection Agency National Advisory Council on Sustainable Economies._\n\n\"There is much the government can do and should do to improve the environment. But even more important is the individual... for it is the individual who himself benefits, and also protects a heritage of beauty for his children and future generations.\"\n\n\u2014Lady Bird Johnson\n\nFor that study, scientists looked at more than 55,000 individuals' diets\u2014including vegans, vegetarians, and meat-eaters\u2014aged twenty to seventy-nine years old. Their conclusions? Greenhouse gas emissions of those eating meat-heavy diets \"are approximately twice as high\" as individuals eating a plant-based diet. Thus, concludes the report, reducing our meat consumption would lead to lower greenhouse gas emissions.\n\nTime and again, study after study, leading scientists have urged that in order to curb climate change, we _must_ eat more plants and less meat.\n\n\"Shifting global demand for meat and dairy produce is central to achieving climate goals,\" concludes one report from Europe's most reputable think tank, the prestigious Royal Institute of International Affairs\u2014more commonly known as Chatham House.\n\n\"Globally we should eat less meat,\" found another Chatham House study. \"We cannot avoid dangerous climate change unless consumption trends change.\"\n\nThe message is clear.\n\nThe verdict is in.\n\nThe time is now.\n\nAt each meal, we make choices. Collectively, all those choices made throughout each day impact individuals and ecosystems across the planet. Pouring cow's milk on our cereal or putting it in our coffee every morning may cause people living next to dairy factory farms to suffer. A ham sandwich for lunch every day may support a pig factory farm. A tuna sandwich may kill whales. And chicken wings for dinner may degrade the lives of people like Lisa Inzerillo on the Delmarva.\n\nMoreover, all told, it's now clear that meat-and-milk-heavy meal plans indisputably and immensely contribute to climate change, putting families like the Bourgs, in Louisiana\u2014and of course, _all_ of Earth's inhabitants and ecosystems\u2014at extreme risk.\n\nOn the other hand, a diet higher in plant-based proteins can dramatically ease agriculture's planetary pitfalls, creating better outcomes for our neighbors, our waterways, animals, our land, and our climate.\n\nThis is certainly one reason why plant-based eating has grown in popularity in recent years, with millions more among us now focusing on plant-forward fare.\n\nAnd it's part of the reason why entire governments are now starting to try to shift the tides of consumption. In 2016, China announced plans to slash the entire country's meat consumption by 50 percent\u2014a move that was lauded by climate campaigners worldwide. The same year, it was reported that Chiara Appendino\u2014the mayor of Turin, Italy\u2014planned to make the entire city vegetarian by promoting meat-free eating as a matter of public policy.\n\nMeat's impact on our climate has also helped lead major school districts\u2014Los Angeles, Detroit, Houston, and many more\u2014along with countless colleges and universities to now have entirely meat-free days in their cafeterias. And it's also helped pique the interest of Silicon Valley, where venture capital firms are now pouring money into plant-based food startups.\n\nFishing trawlers in the Western Sahara. As high as the UN figures are for livestock-related greenhouse gas emissions, they don't even include emissions from the countless fishing ships constantly trawling our world's oceans. CREDIT: WESTERN SAHARA RESOURCE WATCH\n\n\"The growing meat intensity of diets around the world is one of the issues connected to this global crisis.\"\n\n\u2014Al Gore\n\nBut we don't need to be venture capitalists, school dining directors, or mayors to take part in this shift. Whether adopting a vegan or vegetarian diet, trying Meatless Mondays, drinking almond milk instead of cow's milk, or simply replacing meat with plants more often, there are a myriad of ways for everyone to eat more sustainably.\n\nIndeed, on this spinning ball of green and blue that we all call home, with all its ecological problems, eating more plants and less meat is the most important way\u2014and the tastiest\u2014to turn things around. So to those who've decided to make their tables sustainable by enjoying more food for a better planet, here's to you!\n\n## **IN A NUTSHELL**\n\n\u2022 Erratic weather patterns, the melting of our icecaps, and more\u2014scientists agree we're in a climate catastrophe.\n\n\u2022 Island nations are facing threats, like increased storms and flooding, from climate change.\n\n\u2022 In the United States, low-lying areas like southern Louisiana are sinking, displacing families and entire communities.\n\n\u2022 Scientists are clear that animal agriculture is a major driver of climate change.\n\n\u2022 Transforming our land to grow animal feed, raising and processing those crops, then funneling the feed through animals requires huge fossil fuel inputs.\n\n\u2022 Even \"locally produced\" animal proteins can be climate unfriendly, since animals are inefficient at converting natural resources into food.\n\n\u2022 All told, industrial animal agriculture contributes more greenhouse gases than the entire transportation sector\u2014than all trains, planes, cars, trucks, and ships combined.\n\n**Now for the good news!** All the lowest-emitting foods, when it comes to greenhouse gases, are plant-based. Beans emit less than 2 kg of CO2 per serving while beef emits 27 kg; tofu emits 2 kg while tuna emits 6 kg. This means that by simply enjoying more plant-based foods, we can\u2014in a very concrete, immediate way\u2014help curb climate change.\n\n# [PART TWO:  \nOUR PLATES](toc.xhtml#part02)\n\n# \"Eat food. Not too much. Mostly plants.\"\n\n\u2014MICHAEL POLLAN\n\n## CAN LENTILS SAVE THE WORLD?\n\nWell, maybe not on their own. But along with other protein-rich legumes and grains, hearty vegetables, wholesome seeds, and even plant-based \"meats\"\u2014eating more of these foods can play a pivotal role in reversing some of our planet's most serious problems.\n\nTake the humble lentil, for example. It weighs in at less than a single kilogram of carbon emissions per serving, while beef contributes a whopping 27 kilograms. No contest.\n\nBut so what? Isn't comparing lentils and beef like comparing apples and oranges? Work your way through these recipes and you'll find out firsthand that isn't the case. Enjoy a Spicy Lentil Burger instead of a hamburger, or Spaghetti with Lentil Meatballs, and you'll see that these protein-packed pulses can offer tastes and textures that are satisfying and delicious, with a much lighter environmental footprint.\n\nPlus, lentils (and other beans and legumes) can help conserve vast quantities of water. Consider Creamy Basil-Chick _pea_ \u2014instead of chick _en_ \u2014Lettuce Cups. According to the Water Footprint Network, a pound of chicken requires more than 570 gallons of water to produce, while it takes only 5 gallons to produce chickpeas and other beans. Or try a Spicy Chocolate Milk Shake with Whipped Coconut Cream for dessert \u2014instead of cow's milk (one glass of which takes _820 gallons_ of water to produce), this tasty treat uses soy milk (20 gallons per glass). That's a 97 percent reduction in water use just from one simple swap!\n\nConcerned not just about our water _use_ , but also what we're doing to our world's oceans through overfishing? Try crab-free Crab Cakes and Rainbow Veggie Sushi. These fish-free feasts are easier on our world's oceans, not to mention the aquatic animals that call those oceans home.\n\n\"Vegetables are, without a doubt, the most flavorful items on a plate.\"\n\n\u2014CHEF MARIO BATALI\n\nEnjoy recipes that utilize tree fruits\u2014knowing that trees help purify our air\u2014like Cherry Chia Puff Pastry Breakfast Tarts or Oatmeal with Chai-Poached Pears. And delight in ingredients that can help lower our land footprint\u2014like using rice paper instead of pork to make bacon or using tofu instead of eggs to make frittata.\n\nOf course, no foods have _zero_ footprint: living and eating on this earth will always mean consuming _some_ amount of resources. But all things considered, plant-based foods\u2014whether whole grains and vegetables, proteins like tofu, or even plant-based meat products\u2014tend to help us tread lighter. They can help us emit fewer greenhouse gases, use less water, save our oceans, reduce air pollution (and even help purify the air), and use less land.\n\nPlus, enjoying more of these delicious foods doesn't have to be a zero-sum endeavor: you needn't go whole hog toward veganism to make a difference. Even simply incorporating _more_ of these recipes into your normal repertoire can help you take a big bite out of so many of the problems that plague our planet. And you may discover new foods you love, or new ways to prepare foods you already love.\n\nSo eat up and enjoy, knowing you're helping save the planet\u2014one delicious bite after the next. Bon app\u00e9tit!\n\nBREAKFAST\n\nApple Cinnamon Pancakes\n\nBiscuits and Gravy\n\nBlack Bean Breakfast Burritos\n\nBlueberry Buckwheat Waffles\n\nBreakfast Banana Split\n\nCherry Chia Puff Pastry Breakfast Tarts\n\nClassic French Toast\n\nMix 'n' Match Superfood Smoothies\n\nOatmeal with Chai-Poached Pears\n\nPB&J Power Smoothies\n\nPea and Potato Frittata\n\nRice Pudding with Coconut and Cranberries\n\nRise 'n' Shine Breakfast Sandwich\n\nSweet Potato and Parsnip Hash\n\nTofu Scramble 101\n\nSOUPS, STEWS, SALADS, AND SANDWICHES\n\nBLT Bagel Sandwich\n\nBloody Beet Burgers\n\nBuffalo Cauliflower Wraps with Blue Cheese\n\nGinger-Sesame Soba Noodle Salad\n\nLoaded Black Bean Dogs\n\nMardi Gras Gumbeaux\n\nMassaged Kale Caesar with Pumpernickel Croutons\n\nReuben Sandwich\n\nRoasted Coconut-Beet Soup\n\nRoasted Root Vegetable and Quinoa Salad\n\nSesame Seaweed Salad\n\nSlawppy Joes\n\nSourdough Panzanella Salad\n\nSpicy Lentil Burgers\n\nVietnamese B\u00e1nh M\u00ec with Sriracha Chicken\n\nWild Mushroom Chili\n\nMAIN DISHES\n\nCashew Cream Mac 'n' Cheese\n\nChicken and Cabbage Wontons\n\nCoconut-Lemongrass Curry with Rice Noodles\n\nCreamy Basil-Chickpea Lettuce Cups\n\nGrilled Veggie Pizza with BBQ Sauce\n\nJerked Jackfruit Tacos with Grilled Pineapple\n\nMeat-Free Meat Loaf with Tomato-Maple Glaze\n\nMushroom and Kale Galette\n\nNo-Frills Cheese Pizza\n\nOyster Mushrooms \u00e0 la Marini\u00e8re\n\nPittsburgh Pierogis\n\nProtein-Packed Burrito Bowl\n\nRainbow Veggie Sushi\n\nSave-the-Bay Crab Cakes\n\nSmoky Seitan Kebabs with Peanut Sauce\n\nSpaghetti and Lentil Meatballs\n\nSteamed Buns with Tofu and Mushrooms\n\nSun-Dried Tomato and Basil Sausages\n\nWild Rice Risotto with Peas and Asparagus\n\nSIDES AND EXTRAS\n\nBlackened Brussels Sprouts with Capers and Raisins\n\nCreamy Herbed Mashed Potatoes\n\nCrispy Rice Paper Bacon\n\nEasy Cheesy Garlic Bread\n\nGreen Bean Casserole\n\nLonestar BBQ Beans\n\nMint and Tofu Summer Rolls\n\nMinted Green Pea Guacamole\n\nPistachio and Sunflower Seed Dukkah\n\nPlant-Based Parmesan\n\nQuick Country Coleslaw\n\nRoasted Red Pepper Hummus\n\nRosemary Cornbread Mini Muffins\n\nSpicy Sweet Potato Fries with Ranch\n\nSweet-and-Salty Curried Popcorn\n\nDESSERTS\n\nAll-the-Nut-Butters Cups\n\nApple-Pecan Crisp\n\nAvocado Chocolate Mousse with Vanilla Bean Whip\n\nChocolate Brownies with Puffed Quinoa\n\nClassic Pumpkin Pie\n\nCoconut Basbousa\n\nLemon-Ginger Blueberry Cake with Cream Cheese Frosting\n\nLemon Meringue Pie\n\nMinted Lemon-Cucumber Sorbet\n\nMom's Noodle Kugel\n\nPeanut Butter Chocolate Chip Cookie Ice Cream Sandwiches\n\nPeanut Butter Trail Mix Cookies\n\nRaw Chocolate-Hazelnut Truffles\n\nRaw Lemon Vanilla Raspberry Slab\n\nSmoky Maple-Pecan Cheesecake\n\nSpicy Chocolate Milk Shake with Whipped Coconut Cream\nBREAKFAST\n\nApple Cinnamon Pancakes\n\nBiscuits and Gravy\n\nBlack Bean Breakfast Burritos\n\nBlueberry Buckwheat Waffles\n\nBreakfast Banana Split\n\nCherry Chia Puff Pastry Breakfast Tarts\n\nClassic French Toast\n\nMix 'n' Match Superfood Smoothies\n\nOatmeal with Chai-Poached Pears\n\nPB&J Power Smoothies\n\nPea and Potato Frittata\n\nRice Pudding with Coconut and Cranberries\n\nRise 'n' Shine Breakfast Sandwich\n\nSweet Potato and Parsnip Hash\n\nTofu Scramble 101\nAPPLE CINNAMON PANCAKES\n\nI grew up in New Hampshire, across the street from a maple-sugar shack\u2014where sap is collected from sugar maple trees and boiled down into syrup. On weekends, the Sugar Shack, as we called it, served breakfast, and many people from the neighborhood and town would gather there for pancakes and fresh syrup. My favorite? Apple cinnamon. This recipe offers a healthier, more sustainable twist on those classic pancakes: because they use applesauce instead of eggs, they cut back on water usage and carbon emissions. Eat up!\n\nSERVES 2 PREP AND COOKING TIME: 35 MINUTES\n\nINGREDIENTS\n\n1\u00bd C all-purpose flour\n\n2 Tbsp sugar\n\n1 tsp baking soda\n\n2 tsp ground cinnamon\n\nPinch of salt\n\n1 C unsweetened applesauce\n\n1 tsp apple cider vinegar\n\n\u00be C almond milk\n\n1 Tbsp coconut oil, plus additional as needed\n\n1 apple, peeled, cored, and diced\n\nFresh fruit, for serving\n\nPure maple syrup, for serving\n\nMETHOD\n\nCombine the flour, sugar, baking soda, cinnamon, and salt in a large bowl. Make a well in the center.\n\nCombine the applesauce, vinegar, and almond milk in a small bowl or measuring cup. Pour the wet ingredients into the well in the dry ingredients, whisking as you go. Continue whisking until the mixture is smooth.\n\nSet a nonstick skillet over medium heat and let it get good and hot. Melt the coconut oil in the pan and swirl it around until the pan is evenly coated.\n\nPour about 3 tablespoons of the batter into the pan. Once it firms up a bit, sprinkle some of the diced apples on top. Cook until bubbles appear. Using a spatula, flip and cook for another couple minutes on the second side. The pancake will puff up and eventually stop growing. When this happens, it's ready. Transfer the pancake to a plate or serving platter.\n\nRepeat the above steps until all the batter has been used up, adding more coconut oil to the pan as needed.\n\nPile with whatever fresh fruit you have available and drizzle with maple syrup before devouring.\nBISCUITS AND GRAVY\n\nFlaky biscuits topped with rich, creamy gravy will start any morning off right. Because these buttermilk-style biscuits use neither butter nor milk (but rather almond milk mixed with cider vinegar, which creates a similar outcome), they're more environmentally friendly\u2014with almond milk requiring less water and emitting fewer greenhouse gases than cow's milk. The gravy is satisfying, nutritious, and filled with flavor.\n\nSERVES 4 PREP AND COOKING TIME: 40 TO 50 MINUTES\n\nINGREDIENTS\n\nFor the Biscuits:\n\n\u00be C plain almond milk\n\n2 tsp apple cider vinegar\n\n2 C all-purpose flour, plus more for dusting\n\n2 tsp baking powder\n\n\u00bd tsp baking soda\n\n\u00bd tsp salt\n\n\u00bc C margarine, chilled\n\nFor the Gravy:\n\n\u00bc C margarine\n\n\u00bd C all-purpose flour\n\n2 C plain almond or soy milk\n\n1 tsp salt\n\n1 tsp freshly ground black pepper\n\nOptional Toppings:\n\nChopped scallions\n\nShredded vegan cheese\n\nFresh or pickled jalape\u00f1os, sliced\n\nMETHOD\n\nFirst, make the biscuits:\n\nPreheat the oven to 450\u00b0F.\n\nCombine the almond milk and vinegar in a small bowl and set aside.\n\nIn a large bowl, combine the flour, baking powder, baking soda, and salt. Using your (clean) hands, work in the margarine until it's evenly distributed but still a bit chunky. Pour in the almond milk\u2013vinegar mixture, stirring gently. Use clean hands to lightly knead the dough, just enough so that everything sticks together and it has a nice, silky consistency.\n\nTransfer the dough to a generously floured work surface. Form it into a rectangle about 1 inch thick. Use a cookie cutter (or drinking glass) to cut biscuits, placing them on a lightly floured baking sheet as you go.\n\nAs you cut the biscuits, press the leftover dough together and cut more, until you've used up all the dough. (Avoid waste by pressing the last dough scraps together to create an off-shape biscuit.) Bake for 10 minutes, then remove and set aside.\n\nWhile the biscuits bake, make the gravy:\n\nMelt the margarine over medium heat in a large skillet or saucepan. Add \u00bc cup of the flour and stir continuously for 3 to 4 minutes to make a roux. Add the remaining \u00bc cup flour, the milk, salt, and pepper. Increase the heat to medium-high and whisk to combine thoroughly. Heat until the gravy thickens, stirring frequently\u2014about 4 minutes more. If too thick, remove from the heat and whisk in a little water.\n\nTo assemble:\n\nCut open two biscuits and place them in a bowl. Top with as much gravy\u2014and any optional toppings\u2014as you'd like.\n\n\"If the factory farming industry does indeed unravel\u2014and it must\u2014then there is hope that we can, gradually, reverse the environmental damage it has caused.... None of this will be easy. The hardest part about returning to a truly healthy environment may be changing the current totally unsustainable heavy-meat-eating culture of increasing numbers of people around the world. But we must try. We must make a start, one by one.\"\n\n\u2014DR. JANE GOODALL\nBLACK BEAN BREAKFAST BURRITOS\n\nThese breakfast burritos are a simple and delicious way to start any day. The black beans and tofu give a boost of planet-friendly protein, while the corn, spices, and toppings add additional flavor and texture. And because the filling is baked\u2014not fried\u2014it's not only better for the planet, but for you as well.\n\nSERVES 2 PREP AND COOKING TIME: 40 MINUTES\n\nINGREDIENTS\n\n2 garlic cloves, finely chopped\n\n3 scallions, finely chopped\n\n\u00bd tsp ground turmeric\n\n\u00bd tsp ground cumin\n\n\u00bd tsp salt\n\n\u00bd tsp paprika\n\n\u00bc tsp freshly ground black pepper\n\n2\u00bd Tbsp coconut oil\n\n1 C fresh, frozen, or canned corn kernels\n\n1 (15-oz) can black beans, drained\n\n1 (8-oz) block extra-firm tofu\n\n2 Tbsp nutritional yeast\n\n2 (8- or 12-inch) flour tortillas\n\n1 avocado, halved and pitted\n\n\u00be C chopped tomatoes (optional)\n\n2 Tbsp vegan sour cream (optional)\n\nCilantro (optional)\n\nHot sauce (optional)\n\nMETHOD\n\nPreheat the oven to 350\u00b0F.\n\nPut the garlic and scallions in a very large bowl. Add the turmeric, cumin, salt, paprika, pepper, coconut oil, and corn. Add the black beans. Stir to combine. Crumble the tofu into the bowl and add the nutritional yeast. Stir to combine everything well.\n\nTransfer the mixture to an 8-inch casserole or baking dish. Bake, uncovered, until the liquid has evaporated and the top is slightly browned, 20 to 25 minutes.\n\nRemove from the oven. Put the tortillas on a baking sheet and place in the oven for 1 minute to warm them up. Remove from the oven. Spread half an avocado onto each wrap. Add as much of the tofu\u2013black bean mixture as you'd like. Add whichever other toppings you're using\u2014chopped tomatoes, sour cream, cilantro, and\/or hot sauce\u2014then roll into burritos. Enjoy immediately.\n\nWant to go the extra eco-mile?\n\nCook the black beans from scratch and cut the corn freshly off the cob to cut down on packaging.\nBLUEBERRY BUCKWHEAT WAFFLES\n\nDespite its name, buckwheat is actually not related to wheat, but rather to sorrel and rhubarb. Its seeds are the part we eat, and they're rich in complex carbohydrates\u2014making it a healthier option than white flour, and giving the foods we cook with it an earthier flavor and color. And this recipe for buckwheat waffles uses almond milk instead of cow's milk, which cuts its water footprint down by 97 percent, and uses aquafaba (chickpea juice) instead of eggs, which reduces food waste and the dish's carbon emissions.\n\nMAKES 8 TO 10 WAFFLES PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n1\u00bd C buckwheat flour\n\n2 tsp baking powder\n\n1 tsp baking soda\n\n\u00bc tsp salt\n\n1 tsp ground cinnamon\n\n6 Tbsp aquafaba (chickpea juice; see here)\n\n2 Tbsp sugar\n\n1 C almond milk\n\n1 tsp pure vanilla extract\n\n3 Tbsp unsweetened applesauce\n\n1 C plain nondairy yogurt\n\n\u00bd C margarine, melted, plus more for greasing\n\n\u00bd C fresh or frozen blueberries\n\nPure maple syrup, for serving\n\nFlaked almonds, for serving (optional)\n\nFresh berries, for serving (optional)\n\nMETHOD\n\nIn a large bowl, combine the buckwheat flour, baking powder, baking soda, salt, and cinnamon.\n\nIn the bowl of a stand mixer fitted with the whisk attachment or in a large bowl using a handheld mixer, beat the aquafaba on medium speed until it holds soft peaks, 5 to 10 minutes. Add the sugar, a little at a time, as you go.\n\nIn a separate bowl, whisk together the almond milk, vanilla, applesauce, yogurt, margarine, and \u00bc cup water.\n\nPour the wet mixture into the dry mixture, add the blueberries, and stir to combine.\n\nCarefully fold the beaten aquafaba into the batter\u2014work gently so as to not deflate it too much.\n\nTurn on the waffle iron and let the waffle batter sit for 10 minutes while the waffle iron gets hot.\n\nLightly grease the waffle iron with margarine and pour in enough batter for one waffle (the amount will depend on your iron). Close and cook for 2 to 5 minutes, until done. Repeat until all the batter has been used up.\n\nTop with warm maple syrup, flaked almonds, berries, and\/or any other favorite toppings you have on hand.\n\n**Equipment Alert:** This recipe requires a waffle iron and stand mixer or some stamina with a handheld one.\nBREAKFAST BANANA SPLIT\n\nWho says banana splits are only for dessert? This healthier version of a banana split will start your day off right\u2014with fresh fruit, raw granola, and a jolt of espresso for that morning buzz.\n\nSERVES 2 PREP AND COOKING TIME: 15 MINUTES\n\nINGREDIENTS\n\nDouble shot of espresso\n\n\u00bc C sugar\n\n\u00bc C dates, pitted and chopped\n\n\u00bc C shredded coconut\n\n\u00bc C pecans, chopped\n\n4 Tbsp nondairy yogurt, any flavor\n\n2 bananas\n\n\u00bd C fresh berries\n\nMETHOD\n\nIn a small saucepan, combine the espresso and sugar and bring to a boil over medium-high heat. Lower the heat to maintain a simmer and cook, stirring continuously, for 5 minutes. Remove from the heat and refrigerate until completely cool.\n\nIn a food processor, combine the dates, coconut, and pecans. Pulse to combine well.\n\nSpoon 2 tablespoons of the yogurt onto each plate. Slice the bananas down the middle lengthwise and place on top. Top with the granola, berries, and espresso drizzle. Enjoy immediately.\n\n**Tip:** Don't own an espresso maker? Order a shot to go from your local coffee shop!\n\n**Equipment Alert:** This recipe requires a food processor.\n\n\"Go vegetable heavy. Reverse the psychology of your plate by making meat the side dish and vegetables the main course.\"\n\n\u2014CHEF BOBBY FLAY\nA MESSAGE FROM\n\n## GREENPEACE USA EXECUTIVE DIRECTOR ANNIE LEONARD\n\n**One of the most important shifts we've seen over** the past thirty years is that more and more people are realizing that while changing our personal eating habits is critical, we must also recognize the wider impacts of what we put on our plates, and act to change our society's relationship to agriculture and food. From the overfishing of our oceans, to exploited and underage migrant farm workers, to poisonous pesticide intensive farming and more: our food and agriculture system needs an overhaul.\n\nThe connection between human consumption of meat and dairy products and damage to our planet is now more widely recognized. Livestock is the most significant contributor to nitrogen and phosphorus pollution of streams, rivers, and coastal waters worldwide. They're also major emitters of greenhouse gas emissions. In fact, the livestock sector is responsible for about 14 percent of global emissions, as much as the transportation sector.\n\nThat's not to say that animal agriculture is the only problem; there are also plant-based ingredients that are causing severe environmental destruction. Take palm oil, for instance, which comes from tropical countries like Indonesia, Malaysia, and parts of Africa. As we've tried to ditch trans fats from our diets, demand for palm oil has shot up, and these countries are now chopping down thousands of acres of lush tropical rain forests to clear the way for palm tree plantations. Indonesia has lost over a quarter of its forest\u2014an area the size of Germany\u2014since 1990, which is a tragedy for the indigenous communities and tigers, orangutans, and other endangered species that need the forest to survive.\n\nA lot's changed since I turned vegetarian in the early '80s. Back then, I ate a lot of grilled cheese sandwiches because that was the only thing on the menu. These days, eating organic, vegetarianism, and trying to live more sustainably have become much more mainstream. If you're used to having meat with every meal, start experimenting with plant proteins like lentils and beans instead. As well as being meat substitutes, they're also a whole new cuisine to explore!\n\nI also recommend trying to eat food that's produced locally, that's in season, and that's organic where possible. If you have access to a CSA or a farmers' market, they're great. I get a box of fresh veggies delivered each week. It often includes things I never saw before. Thank goodness they include recipes.\n\nAnother key piece of the puzzle is wasting as little food as possible. Rotting food waste in landfills is a major source of climate-polluting methane. Composting is a great alternative, but if we can be smart about how much we buy and what we do with our leftovers, we can avoid that waste altogether.\n\nMuch of the world has become disconnected from the food we eat and lost knowledge about who grows it and how. It's important to remember that our power to change this doesn't just come from dollars we spend. Our voices and votes are essential in helping to fix our broken food system to make sure it's healthy for all eaters, workers, wildlife, and the planet.\n\n_Annie Leonard began her career at Greenpeace in 1988 and now serves as Executive Director for Greenpeace USA. She has decades of experience investigating and explaining the environmental and social impacts of our stuff: where it comes from, how it gets to us, and where it goes after we get rid of it. Her book,_ The Story of Stuff, _is a_ New York Times _bestseller, and her film of the same name blossomed into The Story of Stuff Project, which works to empower people around the globe to fight for a more sustainable and just future._\nCHERRY CHIA PUFF PASTRY BREAKFAST TARTS\n\nFlaky, airy, gooey, sweet\u2014these puff pastry tarts are wonderful for breakfast, parties, dessert, or any time of day. Nutrient-rich chia seeds help thicken the filling in a low-calorie and sustainable way, and cherry trees can have real benefits on our air quality: in addition to trees helping purify our air, a single cherry tree can also perfume the air with more than two hundred thousand flowers.\n\nMAKES 8 PREP AND COOKING TIME: 45 MINUTES\n\nINGREDIENTS\n\n2 (10 x 15-inch) sheets vegan puff pastry\n\n2 C frozen cherries\n\n1 C granulated sugar\n\n2 tsp fresh lemon juice\n\n1 tsp ground cinnamon\n\n1 tsp pure vanilla extract\n\n\u00bc C chia seeds\n\n2 Tbsp almond milk, plus more for brushing\n\n1 C confectioners' sugar\n\nMETHOD\n\nDefrost the puff pastry sheets according to the package directions on the counter in advance.\n\nPreheat the oven to 400\u00b0F.\n\nSet two of the cherries aside for the frosting and place the remainder in a small saucepan. Add the granulated sugar, lemon juice, cinnamon, and vanilla. Stir until the sugar is dissolved. Bring to a boil, decrease the heat to low, and simmer for 5 minutes, mashing the cherries as they cook. Stir in the chia seeds and cook to thicken, 5 minutes more. Remove from the heat and let cool for several minutes.\n\nLine a baking sheet with parchment paper. Cut the puff pastry into 16 rectangles (into quarters lengthwise and then in half crosswise) and then gently transfer 8 of them to the parchment-lined baking sheet. These will be the bottoms of the tarts. (You may need to do this in batches depending on the size of the baking sheet.)\n\nSpoon about 1 tablespoon of the filling onto the center of each of the 8 pastry rectangles on the baking sheet. Spread it around, leaving a border of about \u00bd inch around the edges. Top each with a matching rectangle and then press around the edges with a fork to seal. Pierce three holes in the top of each pastry with the tip of a sharp knife. Brush lightly with almond milk and bake for 20 minutes. Remove from the oven and set aside.\n\nMeanwhile, combine the confectioners' sugar and the almond milk in a small bowl. Stir until smooth. Take those two cherries you set aside earlier and squeeze their juice into the frosting. Stir to combine.\n\nDrizzle the frosting over the pastries while they are still warm.\n\n**Tip:** Want to save some time? Though not quite as good as making it yourself, you can use store-bought jam of any flavor you'd like as the filling.\nCLASSIC FRENCH TOAST\n\nWho doesn't love a nice, hot plate of French toast in the morning? But eggs and milk aren't the most eco-friendly way to make it. For example, it takes 53 gallons of water to produce a single egg and 820 gallons of water to produce just 8 ounces of milk (think of all the feed grown for the chickens and cows, for starters). This recipe uses almond milk, and is thickened with a little tofu and chia seeds instead. It fries up well and is super satisfying\u2014all while being a little lighter on the planet.\n\nSERVES 3 PREP AND COOKING TIME: 20 MINUTES\n\nINGREDIENTS\n\n1\u00bd C almond milk, plus more if needed\n\n\u00bd C firm tofu\n\n1 Tbsp pure maple syrup, plus more for serving\n\n2 Tbsp chia seeds\n\n1\u00bd Tbsp nutritional yeast\n\n1\u00bd tsp ground cinnamon\n\n\u00bc tsp ground nutmeg\n\nPinch of salt\n\n6 slices bread, about \u00be inch thick\n\n3 Tbsp margarine, for frying, plus more as needed\n\nFresh berries, for serving\n\nConfectioners' sugar, for serving\n\nMETHOD\n\nPut the almond milk, tofu, maple syrup, chia seeds, nutritional yeast, cinnamon, nutmeg, and salt in a food processor or blender and pulse until mixed and the tofu is almost smooth but a tiny bit crumbly. You want this liquidy enough to coat the bread with; if it's not, add some more almond milk, a little at a time, and pulse.\n\nPlace the bread slices neatly inside a large casserole dish and pour the almond milk mixture over the bread. Flip the slices to coat the other side and set aside to soak in the mix.\n\nIn a large skillet, melt the margarine over medium heat. Once the pan is hot, about 3 minutes, add a few slices of bread. Cook until golden brown on the first side, about 3 minutes, then flip to cook the other side until golden brown.\n\nStack on plates and serve topped with maple syrup, fresh berries, and confectioners' sugar.\n\n**Want to go the extra eco-mile?** Use bread that's on its last leg to avoid food waste! You don't want it stale, but it needn't be totally fresh, either, since it'll soften up once coated in batter.\nMIX 'N' MATCH SUPERFOOD SMOOTHIES\n\nJam-packed with phytonutrients, vitamins, and antioxidants, greens are true superfoods. Combined with other good-for-you-and-the-planet foods\u2014like fresh fruits and vegetables, nut butters, hemp seeds, turmeric, chia seeds, and more\u2014a green smoothie a day may literally help keep the doctor away. Just mix and match, using the options I've listed here as a guide. And don't be afraid to get creative with your smoothies. Drink up!\n\nSERVES 2 PREP AND COOKING TIME: 5 MINUTES\n\nINGREDIENTS\n\n2 C Greens\n\nArugula\n\nBeet greens\n\nBok choy\n\nBroccoli\n\nCollard greens\n\nDandelion greens\n\nKale\n\nRomaine lettuce\n\nSpinach\n\nSwiss chard\n\n2 C Liquid\n\nAlmond milk\n\nCoconut milk\n\nCoconut water\n\nHemp milk\n\nHerbal tea (cooled)\n\nOrange juice\n\nSoy milk\n\nWater\n\n3 C Fruits\/Veggies\n\nApple\n\nApricot\n\nBanana\n\nBlackberries\n\nBlueberries\n\nButternut squash\n\nCarrot\n\nCherries\n\nCucumber\n\nMango\n\nMelon\n\nPeaches\n\nPears\n\nPineapple\n\nRaspberries\n\nSweet Potato\n\nOptional Boosters\n\n2 Tbsp almond butter\n\n2 Tbsp peanut butter\n\n2 Tbsp raw cashews\n\n1 tsp ground cinnamon\n\n2 Tbsp chia seeds\n\n2 Tbsp flax meal\n\n1 tsp fresh ginger\n\n2 Tbsp hemp seeds\n\n4 Medjool dates\n\n1 tsp ground turmeric\n\n1 tsp pure vanilla extract\n\n2 Tbps hulled pumpkin seeds\n\n2 shots of espresso\n\n1 tsp spirulina\n\nMETHOD\n\nFirst, pick the combination you want. Start simple if you're new to green smoothies. Spinach, pineapple, raspberries, hemp seeds, and orange juice make a fruity combo if you like sweet smoothies. For something a little lighter on the carbs, try kale, cucumber, spirulina, and almond milk. Feeling adventurous? Swiss chard, banana, pumpkin, vanilla, cinnamon, ginger, coconut milk, and Medjool dates is outrageously good.\n\nOnce you're ready to roll, place everything in the blender and blend on high until well combined. (Blending time will depend on how powerful your blender is.) Adjust the liquid amount to reach your desired thickness. Pour into glasses and enjoy immediately. It's really that simple.\n\n**Want to go the extra eco-mile?** Use in-season produce (see here).\nOATMEAL WITH CHAI-POACHED PEARS\n\nWhen I was young, I lived within walking distance of my elementary school. On snowy New Hampshire mornings, my mom would send me off down the road with a stomach full of oatmeal. \"It sticks to your ribs,\" she'd say\u2014and I'd always arrive at school still feeling warm from my breakfast. This recipe uses pears poached in chai tea (yum!) to offer an extra-earthy start to any cold morning.\n\nSERVES 2 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n2 small ripe pears\n\n\u00bc C sugar\n\n1 Tbsp pure vanilla extract\n\n2 chai tea bags\n\n1 C instant oats\n\n1 C almond milk\n\nPure maple syrup, for serving\n\nMETHOD\n\nPeel the pears, leaving the stems intact. Place in a small saucepan with the sugar, vanilla, and tea bags and add enough water so that the pears are just covered. If they're tall pears, lay them down. Bring to a boil, turn the heat down to low, and simmer for 20 minutes. (If your pears aren't ripe yet, add 5 minutes to the cooking time.)\n\nDrain the water and let the pears cool while you make the oatmeal.\n\nPlace the oats in a medium saucepan with 2 cups water and bring to a boil, then decrease the heat to low and simmer for 5 minutes or until thick. Remove from the heat and stir in the almond milk.\n\nSpoon the oatmeal into serving bowls. Slice the pears in half and place on top, then drizzle with maple syrup.\n\n\"Eating, for me, is how you proclaim your beliefs three times a day.\"\n\n\u2014NATALIE PORTMAN\nPB&J POWER SMOOTHIES\n\nI start almost every day with some variation of this simple smoothie. The chia, pumpkin, and hemp seeds\u2014as well as the peanut butter\u2014give a big ol' boost of protein, flavor, and fiber, while the flax offers a healthy dose of omega-3s.\n\nSERVES 2 PREP TIME: 5 MINUTES\n\nINGREDIENTS\n\n2 ripe bananas, frozen in chunks\n\n2 Tbsp peanut butter\n\n1 C frozen blueberries\n\n2 C almond milk\n\n2 tsp chia seeds\n\n2 tsp flax meal\n\n2 Tbsp hulled pumpkin seeds\n\n2 Tbsp hemp seeds\n\n1 Tbsp pure maple syrup (optional)\n\nMETHOD\n\nBlend all the ingredients in a blender until well combined and the seeds are liquefied. In a Vitamix or other high-speed blender on high, that's about 1 minute; in a standard blender, it may take longer. Pour into glasses and enjoy immediately.\n\nSMOOTHIE HACKS GALORE!\n\nAdd more almond milk if you prefer a smoother smoothie.\n\nIf you'd rather use store-bought protein powder (like Vega brand), you can omit the pumpkin, hemp, and chia seeds. Unless you want to be like Arnold\u2014in which case, go nuts.\n\nOr you can make your own protein powder in advance by blending chia seeds, flax meal, hulled pumpkin seeds, and hemp seeds. Just be sure to refrigerate it if you make it in advance\u2014it'll keep for up to 2 weeks.\n\nIf you've forgotten to freeze your bananas in advance, don't fret! They work well fresh, too.\n\nMaking this just for one person? Cut the amounts in half (duh).\n\n**Want to go the extra eco-mile?** Have bananas that are starting to turn brown? Avoid waste by peeling and freezing them in chunks before they go bad, then add them to your smoothies!\nPEA AND POTATO FRITTATA\n\nFrittata is an Italian egg dish\u2014similar to a quiche, but sans crust and filled with savory ingredients. Tasty, yes. Sustainable? Not so much. Enter tofu: it's protein-packed and holds vegetables well\u2014and it's more eco-friendly. (One egg may contribute to rain forests being destroyed for chicken feed, and an egg results in about 140 percent more greenhouse gas emissions than a serving of tofu does.) In this version of the classic Italian dish, arrowroot flour provides a smooth texture, nutritional yeast gives it a slightly sharp taste (like Parmesan might), and green peas add a little color and some extra protein.\n\nSERVES 4 PREP AND COOKING TIME: 1\u00bd HOURS\n\nINGREDIENTS\n\n2 Tbsp olive oil\n\n4 russet potatoes, cut into \u00bd-inch dice\n\n1 tsp black salt\n\n1 C frozen peas\n\n1 (14-oz) package silken tofu\n\n2 heaping Tbsp arrowroot flour\n\n\u00bc C nutritional yeast\n\n1 tsp onion powder\n\n\u00bd tsp garlic powder\n\n\u00bd tsp ground turmeric\n\nPinch of freshly ground black pepper\n\n\u00bc C chopped fresh parsley\n\n1 C shredded vegan cheese (optional)\n\nFresh spinach, for serving (optional)\n\nSriracha, for serving (optional)\n\nKetchup, for serving (optional)\n\nMETHOD\n\nPreheat the oven to 375\u00b0F. Pour the olive oil into an 8 x 12-inch casserole dish.\n\nAdd the potatoes, stirring to coat evenly. Sprinkle with \u00bd teaspoon of the salt and bake for 30 minutes.\n\nRemove the peas from the freezer and let them defrost while the potatoes cook.\n\nPlace the tofu, arrowroot flour, nutritional yeast, onion powder, garlic powder, turmeric, remaining \u00bd teaspoon salt, and the pepper in a food processor or blender. Puree until smooth.\n\nOnce the potatoes are cooked, loosen them from the bottom of the dish using a spatula. Add the peas and parsley. Fold in the tofu mixture. Sprinkle the top with cheese (if using) and return to the oven. Bake for 40 minutes more.\n\nRemove from the oven and let rest for 10 minutes before serving.\n\nServe the frittata on its own, or with fresh spinach, Sriracha, or ketchup.\n\n**Tip:** Want to make it even more colorful and healthy? Swap in sweet potatoes for the russet potatoes!\n\n\"We can make a commitment to promote vegetables and fruits and whole grains on every part of every menu.\"\n\n\u2014MICHELLE OBAMA\nRICE PUDDING WITH COCONUT AND CRANBERRIES\n\nRice pudding is a truly global dish, eaten in virtually every corner of the world. In Iraq, they call it _zarda wa haleeb_ , and eat it with date syrup. In India, they call it _dudhapak_ and prepare the dish with sugar, nuts, and saffron. In Italy, _budino di riso_ is made with milk and orange peel. And in Norway, _risengrynsgr\u00f8t_ is eaten around Christmas, often with cinnamon. This rice pudding is made with tart cranberries and rich coconut milk, sweetened with a bit of sugar, and rounded out with cinnamon and vanilla.\n\nSERVES 2 TO 4 PREP AND COOKING TIME: 50 MINUTES\n\nINGREDIENTS\n\n1\u00bd C short-grain white rice\n\n1 heaping tsp ground cinnamon, plus more for serving\n\n1 tsp pure vanilla extract\n\n\u00be C canned coconut milk\n\n\u2153 C sugar\n\n1 C fresh cranberries\n\n\u00bc C unsweetened coconut flakes\n\nMETHOD\n\nPlace the rice, cinnamon, vanilla, and 3 cups water into a medium saucepan. Cover and bring to a boil over high heat, then lower the heat to low and simmer for 25 minutes, or until almost all the liquid has been absorbed.\n\nAdd the coconut milk, sugar, and cranberries. Cover and bring the mixture to a boil, then lower the heat to maintain a simmer and cook, stirring occasionally, until all the coconut milk has been absorbed and the rice is thick and creamy, about 15 minutes. Be careful of the cranberries when you remove the lid to stir the pudding; they will pop as they cook!\n\nRemove from the heat and let rest for 5 minutes, uncovered. Give it another stir and then spoon into bowls and top with flaked coconut and a dusting of cinnamon.\nRISE 'N' SHINE BREAKFAST SANDWICH\n\nGo above and beyond with this delightful breakfast sandwich. A vegan sausage patty, an easy homemade egg-free patty using chickpea flour, and a slice of gooey vegan cheese all piled onto a toasted English muffin\u2014what a morning! And because chickpeas emit 60 percent less CO2 than eggs, this recipe is as environmentally conscious as it is tasty.\n\nSERVES 4 PREP AND COOKING TIME: 45 MINUTES\n\nINGREDIENTS\n\n\u00be C chickpea flour\n\n2 Tbsp nutritional yeast\n\n2 Tbsp olive oil\n\n\u00bd tsp black salt\n\n\u00bd tsp ground turmeric\n\n\u00bc tsp freshly ground black pepper\n\n1 Tbsp vegetable oil, plus more for greasing\n\n4 vegan sausage patties\n\n4 English muffins, toasted\n\nKetchup\n\n4 slices vegan cheese (optional)\n\nMETHOD\n\nWhisk together the chickpea flour, nutritional yeast, olive oil, salt, turmeric, pepper, and \u00be cup water until smooth. Let sit for 30 minutes, giving one final stir at the end. Don't be tempted to taste it yet\u2014chickpea flour tastes weird before it's cooked.\n\nIn a large skillet, heat the vegetable oil over medium-high heat. Oil the insides of four egg rings.\n\nPlace the egg rings in the pan and fill with the batter. Cook the first side for 3 minutes. Gently remove the egg rings, flip, and cook for 3 minutes more. Remove from the pan and set aside.\n\nAdd the vegan sausage patties to the pan and cook until browned, about 3 minutes per side.\n\nBuild each sandwich on an English muffin: a sausage patty, an egg patty, ketchup, and cheese!\n\n**Want to go the extra eco-mile?** Substitute a slice of the Sun-Dried Tomato and Basil Sausage for a store-bought patty, and replace the vegan cheese slices with avocado to cut down on packaging.\n\n**No egg rings? No problem.** Make DIY egg rings using round cookie cutters or the screw-on rings of mason jar lids. Just make sure they're clean and oiled before using. And when you flip the patties, use a spatula to flip the whole thing\u2014DIY ring and all\u2014rather than removing the ring first. When you go to pop the patty out at the end, carefully hold the ring using a fork and push the patty through the wide end.\n\n**Shopping Tip:** Chickpea flour is sometimes called garbanzo flour, gram flour, or _besan_.\nSWEET POTATO AND PARSNIP HASH\n\nNothing says earthy like root vegetables! Parsnips have cream-colored skin and flesh; left in the ground, they become sweeter after winter frosts. (If unharvested, they produce a stem topped by small yellow flowers.) Mixed with sweet potatoes, they make a wonderful morning hash. Some experts even say eating lots of sweet potatoes, considered a superfood, will help you live longer and healthier. How sweet is that?\n\nSERVES 2 AS A MAIN OR 4 AS A SIDE PREP AND COOKING TIME: 20 MINUTES\n\nINGREDIENTS\n\n1 Tbsp coconut oil\n\n1\u00bd C \u00bd-inch cubes sweet potato\n\n1\u00bc C \u00bd-inch cubes parsnip\n\n1 tsp Old Bay seasoning\n\n1 tsp onion powder\n\n\u00bd tsp freshly ground black pepper\n\n\u00bd tsp garlic powder\n\nMETHOD\n\nIn heavy-bottomed skilled, heat the coconut oil over medium-high heat.\n\nAdd the cubed vegetables and sprinkle with the Old Bay, onion powder, pepper, and garlic powder. Stir to coat thoroughly with the oil and seasonings. Cook for about 5 minutes, without stirring. Once the underside starts to brown, use a spatula to turn them all over and cook, stirring every minute or so, until all sides turn golden brown and soft, 5 to 10 minutes.\n\nRemove from the heat and let sit for 5 minutes before serving.\n\n**Want to go the extra eco-mile?** Make this recipe when sweet potatoes and parsnips are in season (see here).\n\n**Serving Suggestion:** I like to serve this with the Apple Cinnamon Pancakes or Blueberry Buckwheat Waffles to add a little something savory alongside something sweet.\nTOFU SCRAMBLE 101\n\nThere's nothing quite like waking up on a Sunday morning and spending a couple hours drinking coffee, listening to music, and whipping up a big breakfast. And no full breakfast would be complete without a scrumptious scramble. If you're looking for a zero-cholesterol, filling, planet-friendly recipe, this introductory scramble made with tofu instead of eggs is the perfect place to start.\n\nSERVES 2 PREP AND COOKING TIME: 20 MINUTES\n\nINGREDIENTS\n\n\u00bd tsp ground turmeric\n\n\u00bd tsp black salt\n\n\u00bc tsp garlic powder\n\n2 Tbsp nutritional yeast\n\n1 Tbsp canola oil\n\n4 scallions, finely chopped\n\n1 (16-oz) block extra-firm tofu\n\nFresh baby spinach, for serving\n\nKetchup, for serving\n\nMETHOD\n\nCombine the turmeric, salt, garlic powder, and nutritional yeast in a small bowl. Set aside.\n\nIn a large skillet, heat the canola oil over medium-high heat. Add the scallions. Cook for about 2 minutes.\n\nCrumble the tofu into the pan, making some chunks small and some larger.\n\nAdd the spice mixture, stirring to evenly coat the tofu.\n\nCook, stirring frequently to ensure it doesn't stick to the pan, until the liquid has evaporated, about 10 minutes.\n\nEnjoy immediately, served atop some fresh baby spinach and topped with ketchup.\n\n**Serving Suggestion:** Serve with some toast, Sweet Potato and Parsnip Hash, and Crispy Rice Paper Bacon alongside.\n\n**Note:** This is a basic scramble. Try adding fresh broccoli, kale, mushrooms, or all of the above! Just double the oil and add the vegetables with the scallions, cooking for a few minutes before adding the tofu.\n\n\"I don't eat meat, fish, or eggs... I eat a lot of tofu.\"\n\n\u2014SHANIA TWAIN\nSOUPS, STEWS, SALADS, AND SANDWICHES\n\nBLT Bagel Sandwich\n\nBloody Beet Burgers\n\nBuffalo Cauliflower Wraps with Blue Cheese\n\nGinger-Sesame Soba Noodle Salad\n\nLoaded Black Bean Dogs\n\nMardi Gras Gumbeaux\n\nMassaged Kale Caesar with Pumpernickel Croutons\n\nReuben Sandwich\n\nRoasted Coconut-Beet Soup\n\nRoasted Root Vegetable and Quinoa Salad\n\nSesame Seaweed Salad\n\nSlawppy Joes\n\nSourdough Panzanella\n\nSpicy Lentil Burgers\n\nVietnamese B\u00e1nh M\u00ec with Sriracha Chicken\n\nWild Mushroom Chili\nBLT BAGEL SANDWICH\n\nWhen I was growing up, my small town had one lunch spot where basically everyone would eat. We'd go every weekend, and my favorite was the BLT, which they did perfectly. But industrial pig farms take a toll on the communities where they operate. So here's a tasty, healthy, eco-friendly version using rice paper bacon instead of pork bacon.\n\nSERVES 2 PREP AND COOKING TIME: 15 MINUTES\n\nINGREDIENTS\n\n6 pieces Crispy Rice Paper Bacon\n\n2 bagels, any type\n\nVegan mayo\n\nA few lettuce leaves\n\n1 tomato, sliced\n\n1 avocado, sliced (optional)\n\n\u00bd medium cucumber, sliced (optional)\n\n2 radishes, thinly sliced (optional)\n\n1 small red onion, thinly sliced (optional)\n\n2 fresh chives, chopped (optional)\n\n2 Tbsp sprouts (optional)\n\nMETHOD\n\nBreak each piece of the rice paper bacon in half.\n\nSlice and toast the bagels.\n\nBuild your sandwiches on the bagels using the mayo, bacon, lettuce, and tomato. If desired, add the avocado, cucumber, radishes, onions, chives, and\/or sprouts.\nBLOODY BEET BURGERS\n\nBeets are healthy, versatile, and full of iron. With a single (beef-based) burger emitting a whopping 6.8 pounds of carbon, these bloody beet burgers are lighter on the planet and oh-so-tasty to boot!\n\nSERVES 4 PREP AND COOKING TIME: 50 MINUTES\n\nINGREDIENTS\n\nFor the patties:\n\n1 (15-oz) can green or black lentils, drained\n\n1 large beet, peeled and shredded (about 2 C)\n\n\u00bd C fresh parsley, finely chopped\n\n\u00bd C walnuts, coarsely chopped\n\n1 tsp onion powder\n\n1 tsp garlic powder\n\n1 tsp ground cumin\n\n1 tsp salt\n\n\u00bc tsp freshly ground black pepper\n\n\u00bd tsp liquid smoke\n\n\u00bd tsp soy sauce\n\n\u00bc C tahini\n\n\u00be C instant oats\n\n2 Tbsp tomato paste\n\nSqueeze of lemon juice\n\n4 Tbsp canola oil\n\nTo assemble:\n\n4 buns\n\nYour favorite burger condiments\n\nA few pickles, sliced (optional)\n\nFinely sliced red onion (optional)\n\n1 tomato, sliced (optional)\n\nMETHOD\n\nPut the lentils, shredded beet, parsley, and walnuts in a large bowl. Add all the remaining patty ingredients\u2014except the canola oil\u2014and stir to combine well. Let sit for 30 minutes.\n\nForm the mixture into four large balls. Flatten into patties and set aside on a plate or tray.\n\nIn a skillet, heat 2 tablespoons of the canola oil over medium heat. Cook the patties, two at a time, until golden brown, about 4 minutes per side, adding more oil as needed. (If cooking all four patties at once, use all 4 tablespoons oil.) Transfer to a paper towel\u2013lined plate.\n\nBuild the burgers on buns with your favorite condiments and any of the optional toppings.\n\n**Serving Suggestion:** Enjoy with a side of Quick Country Coleslaw and topped with the ranch dressing from the Spicy Sweet Potato Fries recipe.\n\n**Want to go the extra eco-mile?** Cook the lentils from scratch to cut down on packaging. You'll need 1\u00bd cups cooked lentils.\nBUFFALO CAULIFLOWER WRAPS WITH BLUE CHEESE\n\nMy mother was only twelve when upstate New York's Anchor Bar invented a new kind of hot wing. News of the dish spread like wildfire, and it quickly became a staple throughout her hometown of Buffalo. My mom is now a vegetarian and has made the switch from Buffalo chicken to Buffalo cauliflower. With low carbon and air footprints, cauliflower is a much more sustainable choice: all the kick without the cluck.\n\nSERVES 2 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\nFor the cauliflower:\n\nVegetable oil, for greasing\n\n1 C rice flour\n\n\u00bd tsp baking powder\n\n\u00bd tsp garlic powder\n\n1 tsp salt\n\n1 medium head cauliflower, sliced\n\n2 Tbsp margarine\n\n\u2153 C Frank's Red Hot sauce\n\nFor the blue cheese:\n\n\u2154 C plain dairy-free yogurt\n\n1 C vegan mayo\n\n2 Tbsp apple cider vinegar\n\n1 Tbsp lemon juice\n\n2 Tbsp nutritional yeast\n\n1 tsp vegan Worcestershire (optional)\n\n1 tsp garlic powder\n\n1 tsp onion powder\n\n\u00bd tsp agave nectar\n\n\u00bc tsp salt\n\n\u00bc tsp freshly ground black pepper\n\nAbout \u00bc block firm tofu\n\nFor serving:\n\n2 flour wraps\n\nLettuce\n\nCelery sticks\n\nMETHOD\n\nFirst, make the Buffalo cauliflower:\n\nPreheat the oven to 400\u00b0F. Lightly oil a rimmed baking sheet.\n\nIn a large bowl, whisk together the rice flour, baking powder, garlic powder, salt, and 1 cup plus 2 tablespoons water. Coat the cauliflower pieces in the batter and place on the prepared baking sheet. Roast for 20 minutes.\n\nMelt the margarine in a saucepan on medium heat. Whisk in the hot sauce and remove from the heat.\n\nWhile the cauliflower is cooking, make the blue cheese dressing:\n\nCombine all the dressing ingredients except the tofu in a medium bowl and mix well. Crumble in the tofu and stir until combined. Set aside in the fridge.\n\nRemove the cauliflower from the oven and transfer to a bowl. Toss with the hot sauce\u2013margarine mixture and return to the baking sheet. Place back in the oven and roast for 10 minutes more. Remove from the oven and let cool for 5 minutes.\n\nServe the Buffalo cauliflower in the wraps with lettuce and celery and a liberal dose of blue cheese.\nGINGER-SESAME SOBA NOODLE SALAD\n\nI travel a lot for work, so I spend a lot of time in airplanes and hotel rooms. I feel like I'm in a constant war with germs. Enter: ginger. This immune-boosting wonder root can help keep you strong and healthy, and it packs a spicy little bite. Combined with sesame oil for some (good) fat, umami-rich soy sauce, edamame for added protein, and maple syrup for a sweet note, ginger shines in this cold salad.\n\nSERVES 4 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n2 Tbsp toasted sesame oil\n\n1 garlic clove, finely minced\n\n1 tsp ginger, finely minced\n\n2 Tbsp rice vinegar\n\n2 Tbsp soy sauce\n\n2 Tbsp pure maple syrup\n\n2 C frozen shelled edamame\n\n2 tsp sea salt\n\n1 (10.5-oz) package buckwheat soba noodles\n\n2 large carrots, julienned\n\n1 red bell pepper, julienned\n\n1 large cucumber, julienned (optional)\n\n2 scallions, thinly sliced\n\n\u00bd C fresh cilantro, finely chopped (optional)\n\n2 Tbsp black sesame seeds\n\nMETHOD\n\nHeat the sesame oil in a small saucepan. Add the garlic and ginger and remove from the heat. Let the mixture sit for 5 minutes and then add the vinegar, soy sauce, and maple syrup. Stir until well combined and set aside.\n\nBring a medium pot of water to a boil. Add the edamame and 1 teaspoon of the salt and cook for 5 minutes. Remove the edamame with a slotted spoon or spider, transfer to a colander, and rinse under cold water. Set aside.\n\nReturn the water to a boil and cook the soba noodles according to the package instructions. Drain and rinse (to prevent sticking) and transfer to a large bowl.\n\nAdd the carrots, bell pepper, cucumber (if using), and edamame. Add the dressing and lightly toss. Sprinkle the scallions, cilantro (if using), and sesame seeds on top and serve.\n\nEnjoy at room temperature or chilled in the fridge for a few hours.\nLOADED BLACK BEAN DOGS\n\nThese black bean dogs are protein-packed and so flavorful. The liquid smoke and smoked paprika give them a nice, smoky flavor and the fresh cilantro gives a bit of a bite. Top 'em off with corn, avocado, sour cream, or really any other dog toppings you like. And because these dogs are all pup without the pork, you can enjoy them knowing your franks are easier on your waistline and the planet.\n\nSERVES 4 PREP AND COOKING TIME: 1\u00bd HOURS\n\nINGREDIENTS\n\nFor the dogs:\n\n\u00be C canned black beans\n\n2 Tbsp chopped fresh cilantro\n\n2 Tbsp olive oil\n\n2 Tbsp tomato paste\n\n1 tsp liquid smoke\n\n1\u00bd C vital wheat gluten\n\n1 heaping tsp onion powder\n\n\u00be tsp garlic powder\n\n1 tsp smoked paprika\n\n2 Tbsp nutritional yeast\n\n\u00be tsp sea salt\n\n\u00bd tsp freshly cracked black pepper\n\nFor the salsa:\n\n15 cherry tomatoes, quartered\n\n\u00bd C corn kernels, fresh or canned\n\n\u00bc C finely chopped red onion\n\n\u00bc tsp salt\n\n\u00bd tsp freshly ground black pepper\n\n\u00bd C finely chopped red bell pepper\n\n1 Tbsp distilled white or apple cider vinegar\n\nTo assemble:\n\n4 hot-dog buns\n\n2 Tbsp canola oil\n\n1 large avocado, pitted, peeled, and cut into \u00bc-inch chunks\n\n\u00bd C vegan sour cream\n\nChopped fresh cilantro\n\nMicrogreens\n\n1 jalape\u00f1o, sliced (optional)\n\nMETHOD\n\nFirst, make the dogs:\n\nPlace a metal steaming basket inside a large pot. Add 2 to 3 inches water, ensuring the water is below the steaming basket. Cut four pieces of aluminum foil, about 7 inches wide.\n\nCombine the black beans, cilantro, olive oil, tomato paste, liquid smoke, and \u00be cup plus 2 tablespoons water in a blender or food processor. Puree until smooth.\n\nPlace the vital wheat gluten, onion powder, garlic powder, paprika, nutritional yeast, salt, and pepper in a large bowl and stir with a fork to combine. Pour the black bean mixture into the wheat gluten mixture, stirring with a fork as you go, then use clean hands to knead everything into a ball.\n\nBreak the ball into four pieces and form each piece into a sausage shape. Place each sausage in the middle of a piece of foil and then roll it up like a Tootsie Roll\u2014not too tightly, as the dogs expand as they cook.\n\nPlace the foil-wrapped hot dogs in the steamer basket, cover, and bring the water to a boil. Steam for 40 minutes. (If the water level gets too low, add more as needed.) Remove from the heat and let cool for at least 10 minutes before unwrapping each hot dog.\n\nWhile the dogs are cooking, make the salsa:\n\nCombine all the salsa ingredients in a medium bowl. Stir and set aside.\n\nAssemble and enjoy:\n\nSlice and toast the hot-dog buns.\n\nIn a large skillet, heat the canola oil over medium-high heat. Add the dogs and cook, turning every 2 to 3 minutes, until golden brown all over.\n\nPlace a dog in each toasted bun. Top with the salsa, avocado, sour cream, cilantro, microgreens, and jalape\u00f1o (if using). Enjoy immediately.\n\n**Serving Suggestion:** Serve with a side of Quick Country Coleslaw or Spicy Sweet Potato Fries.\n\n**Tips:** On a hot summer day, cook the dogs on the grill instead of in a skillet. Also, want a more traditional dog? Skip the toppings and instead enjoy with classic hot-dog condiments\u2014like ketchup, mustard, and relish.\n\n**Shopping Tip:** You can final vital wheat gluten in the flour aisle of most grocery stores!\n\n\"Reduce your red meat intake to help the environment.\"\n\n\u2014LEONARDO DICAPRIO\nMARDI GRAS GUMBEAUX\n\nGumbo is a hearty stew that originated during the 1700s in southern Louisiana\u2014the same region that's now sinking back into the Gulf of Mexico in large part due to climate change. Gumbo is thickened with a roux (in this case, made from flour and oil) and always includes what Bayou Staters call the \"Holy Trinity\" of vegetables: celery, peppers, and onions. It usually includes meat or fish, but for those of us who want to get all those Louisiana flavors without contributing to the region's watery fate, this version uses beans, mushrooms, veggie chicken, and vegetables instead. As they say in the Big Easy, _laissez les bons temps rouler_!\n\nSERVES 6 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\n1 C brown rice\n\n1 (9-oz) package Beyond Meat grilled chicken strips\n\n1 (14-oz) can crushed tomatoes\n\n2 vegetable bouillon cubes\n\n\u00bd tsp fil\u00e9 powder (optional)\n\n\u00be tsp dried thyme\n\n\u00bd tsp cayenne pepper\n\n\u00bd tsp paprika\n\n3 bay leaves\n\n\u00bd tsp sea salt\n\n1\u00bd tsp vegan Worcestershire sauce\n\n1 green bell pepper, finely diced\n\n2\u00bd C sliced fresh okra\n\n1\u00bd celery stalks, finely diced\n\n4 garlic cloves, minced\n\n1 C canned red kidney beans, drained and rinsed\n\n1 C canned great northern beans, drained and rinsed\n\n\u00bc C coconut oil\n\n\u00bc C all-purpose flour\n\n1 large onion, finely diced\n\n2 C mushrooms (any kind), chopped\n\nCrystal hot sauce, for serving (optional)\n\nMETHOD\n\nCombine the rice and 2 cups water in a saucepan. Cover and bring to a boil over a high heat. Lower the heat to low and simmer until the water has been absorbed, about 20 minutes. Remove from the heat and set aside.\n\nWhile the rice is cooking, remove the chicken strips from the freezer and set aside to defrost slightly.\n\nIn a large bowl or pitcher, combine the crushed tomatoes, bouillon cubes, fil\u00e9, thyme, cayenne, paprika, bay leaves, salt, Worcestershire, and 3 cups water. Set the mixture aside.\n\nCombine the bell pepper, okra, celery, and garlic in a bowl and set aside. Combine the beans in a separate bowl and set aside as well.\n\nIn a large, wide pot with a lid (I use a 4-quart cast-iron braiser), melt the coconut oil over medium heat. Sprinkle the flour over the oil and stir or whisk continuously until the roux darkens to a caramel-brown color, about 10 minutes. To keep the roux from burning (which will ruin the dish), make sure to stir continuously and don't have the heat too high\u2014it takes a while, but is worth it.\n\nAdd the onion to the roux and stir continuously until soft, about 5 minutes. Add the vegetables and cook, stirring frequently, until all the vegetables are soft, about 10 minutes. (During this time, the sliminess of the okra will cook off.)\n\nAdd the crushed tomato mixture, mushrooms, and chicken strips and bring to a boil. Lower the heat to maintain a simmer, cover, and cook for about 15 minutes. After 15 minutes, remove the cover, add the beans, and return to a boil. Lower the heat to maintain a simmer and cook, uncovered, for 15 minutes more, until slightly thickened.\n\nServe the gumbo over brown rice and top with Louisiana's own Crystal hot sauce if you enjoy that extra kick.\n\n**Want to go the extra eco-mile?** Replace the Beyond Meat chicken strips with 1\u00bd cups extra beans to cut down on packaging.\n\n**Shopping Tip:** Fil\u00e9 powder is ground sassafras leaves and acts as a thickener and a seasoning for gumbo. Find it in the spice aisle.\nMASSAGED KALE CAESAR WITH PUMPERNICKEL CROUTONS\n\nMost anchovies are used in the unsustainable practice of turning wild-caught fish into food for farmed fish. That makes this nutty, creamy, anchovy-free Caesar both satisfying and ocean-friendly. Plus, because the recipe doesn't contain eggs, it has less of a water footprint. Vibrant kale, sweet and hearty pumpernickel croutons, creamy cashew dressing, and capers: this salad is an explosion of flavors.\n\nSERVES 2 AS A MAIN OR 4 AS A SIDE PREP AND COOKING TIME: 20 MINUTES\n\nINGREDIENTS\n\nFor the dressing:\n\n1 C cashews, soaked in water to cover for 4 hours\n\n2 Tbsp apple cider vinegar\n\n2 Tbsp olive oil\n\n\u00bd tsp garlic powder\n\n1 tsp Dijon mustard\n\n2 heaping Tbsp capers\n\n1 tsp vegan Worcestershire sauce\n\n2 Tbsp nutritional yeast\n\nGenerous pinch of salt\n\nFreshly ground black pepper\n\nFor the croutons:\n\n3 or 4 slices pumpernickel bread\n\n3 Tbsp olive oil\n\n\u00bd tsp garlic powder\n\n1 Tbsp nutritional yeast\n\n\u00bd tsp salt\n\nFor the salad:\n\n1 (16-oz) bag or bunch of kale\n\n1 tsp salt\n\n1 (14-oz) can chickpeas, drained and rinsed\n\nNutritional yeast\n\nMETHOD\n\nPreheat the oven to 400\u00b0F.\n\nMake the dressing:\n\nDrain the cashews and place all the dressing ingredients in a blender along with \u00bd cup water. Blend until smooth. Add 1 tablespoon more water at a time as needed until the desired consistency is reached. Store in the fridge until ready to use.\n\nMake the croutons:\n\nCut the bread into small, crouton-size pieces\u2014about \u00bc inch each\u2014and place in a container or bowl with the olive oil, garlic powder, nutritional yeast, and salt. Cover and shake so the bread is evenly coated. Spread the croutons out on a baking sheet and bake for 10 minutes or until totally crispy but not burnt. Set aside.\n\nMeanwhile, make the salad:\n\nRinse the kale in cool water. If not using pre-chopped kale, rip the leaves into bite-size pieces and place in a bowl, discarding the thicker, tougher parts of the stems. Sprinkle the salt over the leaves and massage them until they go limp and are taking up about half the amount of room they were taking up before.\n\nPour the dressing over the leaves and use clean hands to evenly coat the leaves with the creamy, salty goodness.\n\nTransfer to a serving bowl. Add the chickpeas. Top with the croutons and a generous sprinkle of nutritional yeast.\nREUBEN SANDWICH\n\nA Reuben sandwich is typically made with loads of unsustainable (and not the healthiest) ingredients\u2014like corned beef, cheese, and Russian dressing. This recipe uses tempeh instead of corned beef, egg-free Russian dressing, and plant-based cheese to make an eco-friendly Reuben that's meaty and filling and, of course, perfectly messy.\n\nSERVES 2 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\nFor the Russian dressing:\n\n\u00bd C vegan mayo\n\n\u00bc C ketchup\n\n2 tsp prepared horseradish\n\n\u00bd tsp vegan Worcestershire sauce\n\n2 Tbsp pickle relish\n\nFor the tempeh:\n\n2 Tbsp apple cider vinegar\n\n2 Tbsp pure maple syrup\n\n2 Tbsp soy sauce\n\n1 (8-oz) package tempeh, cut into \u00bd-inch-thick slices\n\nTo assemble:\n\n4 slices rye bread\n\n4 slices vegan cheese\n\n1 C sauerkraut, excess liquid squeezed out\n\n2 Tbsp margarine\n\nMETHOD\n\nMake the Russian dressing:\n\nCombine all the dressing ingredients in a small bowl. Set aside in the fridge.\n\nPrepare the tempeh:\n\nCombine the vinegar, maple syrup, soy sauce, and 1 tablespoon water in a small bowl.\n\nPut the tempeh in a large skillet in a single layer. Add the vinegar mixture and turn the tempeh so it's thoroughly coated in the marinade.\n\nCook over medium-high heat until the mixture starts to thicken and the tempeh is brown on both sides, 5 to 7 minutes. Remove from the heat and set aside.\n\nAssemble and enjoy:\n\nBuild the sandwiches by spreading the Russian dressing on the bread and topping with 2 slices of cheese, some tempeh, and sauerkraut.\n\nMelt the margarine in a clean pan over medium-high heat. Place the sandwiches in the pan and cook until golden on each side and the cheese has softened, about 3 minutes per side.\n\nRemove from the pan, cut in half, and serve immediately.\n\n**Serving Suggestion:** Serve with a pickle spear and potato chips or a side of Quick Country Coleslaw.\nROASTED COCONUT-BEET SOUP\n\nBeets have served people well for ages. We turn them into sugar. We've used them as a dye. In the Middle Ages, they were often used to treat digestive illnesses. In the nineteenth century, wine was sometimes colored with them. Earthy and sweet with a vibrant red color, they combine nicely with veggie stock, thyme, and coconut milk in this hearty soup.\n\nSERVES 4 PREP AND COOKING TIME: 1\u00bd HOURS\n\nINGREDIENTS\n\n6 to 8 medium beets\n\n\u00bc C olive oil\n\n2 yellow onions, finely chopped\n\n4 C vegetable stock\n\n\u00bd tsp freshly ground black pepper\n\n1 C canned coconut milk\n\n4 garlic cloves, finely chopped\n\nFresh thyme, for garnish\n\nMETHOD\n\nPreheat the oven to 400\u00b0F.\n\nTrim the greens and root ends off the beets. Place them, whole, in a roasting or cast-iron pan and roast for 1 hour.\n\nLet the beets cool a bit, then hold them under cold running water and rub off their skins. (If they're stubborn skins, you may need the help of a peeler.) Chop into \u00bd-inch cubes.\n\nIn a medium saucepan, heat the olive oil over medium-high heat. Add the onions and cook until browned, about 8 minutes. Add the beets and stir for 2 minutes, then add the stock, pepper, coconut milk, and garlic. Simmer for 5 minutes so that the garlic is cooked and all the flavors are combined.\n\nLet cool for 10 minutes and then transfer to a blender and blend until smooth. Pour into bowls and garnish with lots of fresh thyme.\n\n**Tips:** Roast the beets ahead of time and store in the fridge overnight.\n\n**Serving Suggestion:** Serve with toasted sourdough, Rosemary Cornbread Mini Muffins, or Sun-Dried Tomato and Basil Sausages.\n\n\"Most people are eating more vegetables at this point. It's a good way to live and it's a good way to have a healthy, long natural life.\"\n\n\u2014CHEF MARIO BATALI\nROASTED ROOT VEGETABLE AND QUINOA SALAD\n\nHere's a hearty, healthy salad straight from the earth. Carrots and sweet potatoes give the dish vibrant hues and a sweet taste, balanced out by the creamy parsnip, tahini, and cider vinegar. The whole thing is capped with a Middle Eastern flair thanks to the cumin, coriander, and pomegranate seeds.\n\nSERVES 4 PREP AND COOKING TIME: 50 MINUTES\n\nINGREDIENTS\n\nFor the salad:\n\n1 C quinoa\n\n2 C vegetable stock\n\n2 C \u00bd-inch cubes parsnip\n\n2 C \u00bd-inch cubes carrot\n\n2 C \u00bd-inch cubes sweet potato\n\n2 C \u00bd-inch cubes russet potato\n\n2 Tbsp olive oil\n\n1 tsp salt\n\n1 tsp ground coriander\n\n1 tsp ground cumin\n\n1 C raw almonds, chopped\n\n4 C arugula, washed and dried\n\nFor the dressing:\n\n\u00bd C tahini\n\n\u00bc C pure maple syrup\n\n\u00bc C apple cider vinegar\n\n2 tsp ground turmeric\n\nGenerous pinch of salt\n\nCracked black pepper\n\n\u00bd C pomegranate seeds, for garnish (optional)\n\nMETHOD\n\nPreheat the oven to 400\u00b0F.\n\nFirst, make the salad:\n\nPlace quinoa and stock in a medium saucepan. Bring to a boil over high heat. Lower the heat to maintain a simmer, cover, and cook until all the stock has been absorbed\u2014about 20 minutes. Remove from the heat, fluff with a fork, and set aside.\n\nPlace the parsnip, carrot, sweet potato, and russet potato cubes into a large bowl. Add the olive oil, salt, coriander, and cumin. Stir to coat thoroughly, then spread the cubed vegetables evenly over two baking sheets. Roast for 15 minutes.\n\nRemove the baking sheets, turn the vegetables with a spatula, add the chopped almonds, and return to the oven for another 10 minutes. Remove and let cool slightly.\n\nPlace the arugula in a large bowl. Top with half the quinoa, then half the vegetables, then the remaining quinoa and the remaining vegetables. Toss a couple of times to lightly combine.\n\nMake the dressing:\n\nCombine all the dressing ingredients in a small bowl and mix well.\n\nDrizzle the salad with as much of the dressing as you'd like (refrigerate any leftovers) and sprinkle pomegranate seeds over the top, if desired.\n\n**Tip:** Try this as a macro bowl, with the quinoa, vegetables, greens, and almonds served in \"clusters\" together in a bowl, rather than tossed as a salad.\n\n\"My diet had... a gradual trend toward less and less meat, even less and less fish, until now I rely so much on other sorts of protein, on many vegetables, most of them farm raised, and on fruits and pastas. My shift has been the result of many things\u2014books like Jonathan Safran Foer's _Eating Animals_ , films such as _Food, Inc._ , and my own observations of factory farms, feedlots, fish farms, and the condition of the meats and fish sold in many of our supermarkets.\"\n\n\u2014MARTHA STEWART\nSESAME SEAWEED SALAD\n\nSeaweed = super food. Wakame (edible seaweed) contains a compound that can help burn fatty tissue, as well as healthy omega-3 fatty acids. When you want that taste of the sea without all the problems associated with fish farming or trawling, this seaweed salad is poised to become your go-to.\n\nSERVES 4 PREP TIME: 5 MINUTES, PLUS 20 MINUTES FOR SOAKING\n\nINGREDIENTS\n\n1 oz (28g) dried seaweed (wakame or dulse)\n\n2 Tbsp sesame seeds\n\n1 Tbsp miso paste\n\n2 Tbsp soy sauce\n\n1 Tbsp toasted sesame oil\n\n2 Tbsp rice vinegar\n\n\u00bd tsp agave nectar\n\n\u00bd tsp grated fresh ginger\n\n1 red chile, finely sliced\n\nMETHOD\n\nPlace the seaweed in a bowl, add water to cover, and soak for 20 minutes to rehydrate. Drain and squeeze out excess water, then finely chop the seaweed.\n\nWhile the seaweed is rehydrating, toast the sesame seeds in a dry skillet over a medium heat. Stir continuously until they become fragrant and start to make a popping noise, 3 to 5 minutes. Transfer to a plate and let cool.\n\nPlace about three-quarters of the toasted sesame seeds in a mortar (setting the rest aside) and lightly grind with the pestle. (If you don't own a mortar and pestle, you can skip this step.)\n\nTransfer the ground seeds to a small bowl, add the remaining ingredients, and whisk to combine. Pour over the seaweed, tossing lightly with clean hands to distribute.\n\nCover and refrigerate for at least 10 minutes, or until ready to serve.\n\nSprinkle with the reserved toasted sesame seeds before serving.\n\n**Serving Suggestion: S** erve with Rainbow Veggie Sushi.\nSLAWPPY JOES\n\nI grew up eating the worst possible sloppy Joes\u2014courtesy of my grade school cafeteria. I revisited them as an adult and was surprised to learn that when done right, they're actually quite amazing. This grown-up version uses Beyond Meat ground beef\u2014made entirely from plants, and with more protein than beef\u2014and is topped with a big ol' pile of coleslaw to offer a healthy, eco-friendly version of that old cafeteria standby.\n\nSERVES 2 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n1 Tbsp olive oil\n\n1 yellow onion, finely chopped\n\n1 (12-oz) bag Beyond Meat beef crumbles\n\n1 C tomato puree\n\n3 garlic cloves, minced\n\n3 Tbsp tomato paste\n\n1 Tbsp vegan Worcestershire sauce\n\n2 soft buns\n\nQuick Country Coleslaw, for serving\n\nMETHOD\n\nIn a heavy-bottomed skillet, heat the olive oil over medium-high heat. Add the onion and cook until translucent, about 2 minutes. Add the beef crumbles and cook until lightly browned all over, about 5 minutes.\n\nAdd the tomato puree, garlic, tomato paste, and Worcestershire. Simmer for 5 to 10 minutes more.\n\nOnce the meat is cooked, remove from the heat.\n\nToast the buns (or don't, if you prefer them soft). Build the sloppy Joes on the buns with a pile of coleslaw topped with the meat. Enjoy\u2014and don't be afraid to get messy!\n\n\"By eating meat, we share the responsibility of climate change.\"\n\n\u2014THICH NHAT HANH\nSOURDOUGH PANZANELLA\n\nOriginally from Tuscany, panzanella\u2014or panmolle\u2014is a light, airy salad made from bread and tomatoes. Plus, you can use stale (or going-stale) bread, which can help cut back on food waste and make the dish extra-eco-friendly.\n\nSERVES 2 AS A MAIN OR 4 AS A SIDE PREP AND COOKING TIME: 20 MINUTES\n\nINGREDIENTS\n\n3 to 5 big slices sourdough bread\n\n3 Tbsp olive oil\n\n2 Tbsp apple cider vinegar\n\n1 tsp whole-grain mustard\n\n\u00bd tsp sea salt\n\n1\u00bd lbs cherry tomatoes, halved\n\n1 small red onion, thinly sliced\n\n\u00bd C baby spinach\n\n2 Tbsp capers\n\n10 to 20 fresh basil leaves\n\nFreshly cracked black pepper\n\nMETHOD\n\nToast the bread (in a toaster or grill pan, or under the broiler) until lightly blackened. Let cool.\n\nCombine the olive oil, vinegar, mustard, and salt in a medium bowl. Add the tomatoes, along with the red onion. Stir until well coated in the dressing.\n\nPlace the baby spinach in a serving bowl. Tear the cooled sourdough into bite-size chunks and place on top of the spinach. Spoon the tomatoes and onion over the sourdough, then pour the dressing remaining in the bowl over the entire salad. Top with the capers and fresh basil and then season generously with freshly cracked black pepper.\n\nServe immediately.\nSPICY LENTIL BURGERS\n\nLentils and beef both pack protein, but one is resource-intensive, requiring copious amounts of land, feed, and water while emitting huge volumes of climate-changing gases like carbon and methane. (Can you guess which one that is?) Try tossing one of these lentil burgers into the mix every once in a while. It'll make your meal more environmentally friendly plus offer some varied taste and texture to spice up that tired old burger routine.\n\nSERVES 4 PREP AND COOKING TIME: 40 MINUTES\n\nINGREDIENTS\n\nFor the patties:\n\n1 (15-oz) can green or black lentils, drained and rinsed\n\n5 or 6 sun-dried tomatoes, chopped\n\n\u00bc C almonds, chopped\n\n\u00bc C fresh parsley, chopped\n\n\u00bd tsp garlic powder\n\n1 tsp onion powder\n\n\u00bd tsp paprika\n\n\u00bd tsp ground cumin\n\n\u00bc tsp freshly ground black pepper\n\n\u00be tsp salt\n\n\u00be C instant oats\n\n1 Tbsp almond butter\n\n1 Tbsp nutritional yeast\n\n1 Tbsp Sriracha (optional)\n\n2 Tbsp canola oil\n\nTo assemble:\n\n1 avocado, quartered, pitted, and peeled\n\n4 burger buns\n\nLettuce\n\n4 slices vegan cheese (optional)\n\nSliced fresh tomato\n\nSliced red onion\n\nMETHOD\n\nFirst, make the patties:\n\nCombine all the patty ingredients except the canola oil in a large bowl. Using moist, clean hands, form the mixture into four balls and then flatten them into patties roughly 1-inch thick.\n\nIn a large skillet, heat the canola oil over medium-high heat. Gently place the patties in the oil and cook until golden brown on the underside, about 3 minutes. Flip and cook for 3 minutes more.\n\nAssemble and enjoy:\n\nCut the avocado into quarters and spread each quarter over the bottom of a burger bun. Add lettuce, vegan cheese (if using), a burger patty, a slice of tomato, red onion, and any other condiments you'd like.\n\n**Want to go the extra eco-mile?** Cook the lentils from scratch in advance to cut down on packaging. You'll need 1\u00bd cup cooked lentils for 4 burgers.\n\n**Serving Suggestion:** Try the ranch dressing from the Spicy Sweet Potato Fries recipe on these.\n\n\"Beef, barbecue, and chicken wreak havoc on the air, water, soil, and on the health and well-being of communities.\"\n\n\u2014RACHEL CARSON COUNCIL\nVIETNAMESE B\u00c1NH M\u00cc WITH SRIRACHA CHICKEN\n\nB\u00e1hn m\u00ecs use a single-serving airy baguette that's light and airy on the inside and crunchy on the outside. Less heavy than other breads, this lets all the fillings inside shine. Note that you'll need to make the pickles ahead of time, so plan accordingly.\n\nSERVES 1 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n4 Gardein Seven Grain Crispy Tenders\n\n1 baguette or French bread loaf, cut into roughly 6-inch pieces\n\nVegan mayo\n\n1 small handful of homemade pickles (recipe follows)\n\n2 sprigs cilantro, coarsely chopped\n\nFresh jalape\u00f1o slices\n\nSriracha\n\n1 wedge of lime\n\nCracked black pepper\n\nMETHOD\n\nCook the tenders according to the instructions on the package.\n\nHalve the baguette lengthwise and spread a good slathering of mayo on the cut side of each half.\n\nPlace the cooked tenders inside and top with pickles, cilantro, some jalape\u00f1o slices, and Sriracha. Squeeze some lime over everything, crack a little black pepper on top, and enjoy immediately.\n\n**Want to go the extra eco-mile?** Replace the Gardein with slabs of broiled tofu, cut into 1-inch-thick rectangles (see here for instructions) or the seitan \"meat\" from the Kebab recipe here, to cut down on packaging.\n\nPICKLES\n\nThis recipe makes enough pickles for 10 to 15 sandwiches.\n\nMAKES ABOUT 6 CUPS PREP TIME: 15 MINUTES, PLUS OVERNIGHT CHILLING\n\nINGREDIENTS\n\n\u00bd large daikon radish, peeled and julienned\n\n3 medium carrots, julienned\n\n1 large cucumber, peeled, seeded, and julienned\n\n2 C boiling water, plus more as needed\n\n2 Tbsp salt\n\n3 Tbsp sugar\n\n1 Tbsp Sriracha\n\n\u00bd C rice vinegar\n\nMETHOD\n\nDivide the daikon, carrots, and cucumber evenly between two clean 1-quart jars with lids.\n\nCombine the boiling water, salt, and sugar in a bowl and stir until the salt and sugar have dissolved. Add the Sriracha and vinegar and stir to combine. Pour into the jar and top with extra boiling water if needed, so that all the vegetables are covered.\n\nLoosely place the lid over the top and let cool to room temperature. Once cool, seal and refrigerate overnight before serving. The pickles will keep in the refrigerator for up to 2 weeks.\nWILD MUSHROOM CHILI\n\nI like to make this recipe when I know I'll be spending a cozy fall Sunday at home\u2014just so I can be immersed in its smell for as long as possible. The black beans offer a hearty protein that's far more eco-friendly than beef (who wants all the deforestation that goes along with industrial cattle ranching?), and the mushrooms add a nice earthy flavor with an additional meaty texture.\n\nSERVES: 2 TO 4 PREP AND COOKING TIME: 1\u00bd TO 3 HOURS, DEPENDING ON THE BEANS' AGE\n\nINGREDIENTS\n\n1 C dried black beans\n\n1 tsp salt\n\n18 oz assorted wild mushrooms, coarsely chopped\n\n2 medium yellow onions, finely chopped\n\n1 bunch cilantro, finely chopped\n\n4 garlic cloves, finely chopped\n\n1 (14-oz) can chopped tomatoes\n\n1 Tbsp soy sauce\n\n2 Tbsp tomato paste\n\n1 tsp crushed red pepper\n\n1 (10.75-oz) can condensed tomato soup\n\nVegan sour cream, for garnish (optional)\n\nMETHOD\n\nPlace the beans in a large pot with 5 cups water and the salt. Cover and bring to a boil, then lower the heat to maintain a simmer. Cook, half covered, until you can easily squish a few beans on the side of the pot and most but not all the water has been absorbed, 60 to 90 minutes, depending on the beans' age.\n\nOnce the beans are soft, add the mushrooms, onions, cilantro, garlic, chopped tomatoes, soy sauce, tomato paste, and crushed red pepper. Increase the heat a little and cook until the vegetables are soft, 15 to 30 minutes more. At the end, stir in the tomato soup and let warm through, 2 to 3 minutes.\n\nEnjoy immediately or remove from the heat and cover with a lid until ready to serve. Garnish with vegan sour cream, if desired.\n\n**Want to go the extra eco-mile?** Swap the sour cream for homemade cashew cream! Just combine \u00bc cup raw cashews, water to cover, a squeeze of lemon juice, and a pinch of salt in a high-speed blender and blend until very smooth.\n\n**Serving Suggestion:** Serve with a side of Rosemary Cornbread Mini Muffins.\nMAIN DISHES\n\nCashew Cream Mac 'n' Cheese\n\nChicken and Cabbage Wontons\n\nCoconut-Lemongrass Curry with Rice Noodles\n\nCreamy Basil-Chickpea Lettuce Cups\n\nGrilled Veggie Pizza with BBQ Sauce\n\nJerked Jackfruit Tacos with Grilled Pineapple\n\nMeat-Free Meat Loaf with Tomato-Maple Glaze\n\nMushroom and Kale Galette\n\nNo-Frills Cheese Pizza\n\nOyster Mushrooms \u00e0 la Marini\u00e8re\n\nPittsburgh Pierogis\n\nProtein-Packed Burrito Bowl\n\nRainbow Veggie Sushi\n\nSave-the-Bay Crab Cakes\n\nSmoky Seitan Kebabs with Peanut Sauce\n\nSpaghetti and Lentil Meatballs\n\nSteamed Buns with Tofu and Mushrooms\n\nSun-Dried Tomato and Basil Sausages\n\nWild Rice Risotto with Peas and Asparagus\nCASHEW CREAM MAC 'N' CHEESE\n\nWhy rely on boxed mac 'n' cheese when you can get a bit crafty with your mac 'n' cheese? This dish uses cashews for that rich, fatty flavor\u2014combined with nutritional yeast for a cheesy taste. It's quick, easy, healthy, and better for the planet than using cow's cheese: cashew trees are lighter on the land than cows, providing wildlife habitat and preventing erosion. So give two big, creamy thumbs up for planet-friendly pasta!\n\nSERVES 4 PREP AND COOKING TIME: 30 MINUTES, PLUS SOAKING TIME FOR CASHEWS\n\nINGREDIENTS\n\n1 C raw cashews, soaked in water to cover for 4 hours\n\n2 Tbsp lemon juice\n\n\u00bd tsp soy sauce\n\n\u00be tsp mustard\n\n2 tsp onion powder\n\n\u00bd tsp garlic powder\n\n1 tsp salt\n\n\u00bc C nutritional yeast\n\n2 Tbsp margarine\n\n1 (16-oz) box pasta (shells or macaroni)\n\nFreshly ground black pepper\n\nSmoked paprika, for garnish (optional)\n\nFresh basil, for garnish (optional)\n\nMETHOD\n\nRinse the soaked cashews. Put them in a blender. Add enough fresh water so that they are just covered, then blend until very smooth, about 1 minute. Add the lemon juice, soy sauce, mustard, onion powder, garlic powder, salt, nutritional yeast, and margarine. Blend until well combined. Add a little more water, 1 tablespoon at a time, until you have a smooth and creamy sauce.\n\nBring a large pot of water to a boil. Cook the pasta according to the package instructions. Drain in a colander and rinse thoroughly. Return the pasta to the pot and add the cashew cheese sauce.\n\nSpoon into bowls and garnish with pepper, paprika (if using), and basil (if using), or whatever other toppings you love (saut\u00e9ed onions, garlic, and cherry tomatoes work well!).\n\n**Tip:** Like baked mac? Put the mac 'n' cheese in a baking dish and bake for 10 to 15 minutes at 350\u00b0F, or until slightly browned.\n\n\"If you're driving a Prius or you're shopping green or looking for organic, you should probably be a semi-vegetarian.\"\n\n\u2014MARK BITTMAN\nCHICKEN AND CABBAGE WONTONS\n\nTravel to almost any corner of the world and you'll find a dumpling, or some variation of it: ravioli (Italy), pierogis (Poland), samosas (India), empanadas (Latin America), kibbeh (the Middle East), and of course wontons (China). These little fried wontons use meat-free chicken and fresh vegetables instead of meat to cut back on air pollution in an oh-so-delicious way.\n\nSERVES 2 PREP AND COOKING TIME: 30 TO 40 MINUTES\n\nINGREDIENTS\n\nFor the dipping sauce:\n\n1 Tbsp toasted sesame oil\n\n3 Tbsp soy sauce\n\n2 Tbsp rice vinegar\n\n\u00bd Tbsp sugar\n\n1 Tbsp Sriracha (optional)\n\nFor the wonton filling:\n\n2 C chopped Beyond Meat chicken strips\n\n1\u00bd C shredded cabbage\n\n\u00bd C chopped scallions\n\n1 garlic clove, chopped\n\n1 Tbsp chopped fresh cilantro\n\n1 Tbsp grated fresh ginger\n\n1 Tbsp toasted sesame oil\n\n1 Tbsp soy sauce\n\n1 Tbsp rice vinegar\n\n1 Tbsp sesame seeds\n\nTo assemble and garnish:\n\n30 wonton wrappers, defrosted if frozen\n\nSesame oil\n\nChili oil\n\nSesame seeds\n\nChopped, fresh cilantro\n\nMETHOD\n\nMake the dipping sauce:\n\nCombine all the dipping sauce ingredients, including the Sriracha (if using), in a small bowl. Set aside in the refrigerator.\n\nMake the filling:\n\nPlace all the filling ingredients in a food processor. Pulse until the chicken and cabbage are finely minced and everything is well combined.\n\nAssemble the wontons:\n\nFill a small bowl with water and set it to the side. You'll use this to wet the edges of the wrappers.\n\nLay out a few wrappers, keeping the rest covered with a damp paper towel so they don't dry out. Place roughly 1 to 2 teaspoons of the filling in the center of each wrapper. Make sure the filling is just in the middle (not on the edges) or it will prevent the ends from sticking together. Gently wet the edges of one wrapper with your fingertips and fold it in half, squeezing out the air and squeezing the edges together to ensure they stick together. Place the finished wonton on a plate or platter. Repeat until you have used up all the filling.\n\nPlace a heavy-bottomed skillet over a medium-high heat and lightly coat the bottom with sesame oil. Fill the pan with wontons, leaving room to flip them, and cook until the bottoms turn golden, about 3 minutes. Add about \u00bc cup water and then cover. (If you don't have a lid for your pan, you can use another pan, the lid from a large saucepan, a large plate, or aluminum foil as a cover.) Cook until all the water has evaporated, 2 to 3 minutes.\n\nRemove from the heat and garnish with chili oil, sesame seeds, and cilantro. Serve with the dipping sauce alongside.\n\n**Want to go the extra eco-mile?** Replace the meatless chicken with 1\u00bd cups crumbled pressed extra-firm tofu (see here for instructions on pressing tofu). Just add it to the other filling ingredients after you've processed them.\n\n**Equipment Alert:** This recipe requires a food processor.\nCOCONUT-LEMONGRASS CURRY WITH RICE NOODLES\n\nLemongrass is a tropical plant in the grass family. It's widely used as a culinary herb in Asian cooking and as a medicinal herb in India. Its oil is sometimes used as a \"lure\" to attract honeybees. And it's commonly used in teas, soups, and curries for its subtle, delicious, citrusy flavor. This dish combines lemongrass with meaty shiitake mushrooms, rich coconut milk, tofu, vermicelli rice noodles, and other deliciousness to create an aromatic, flavorful curry that'll leave you full and happy. Just watch out for bees!\n\nSERVES 2 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n\u00bd (14-oz) package extra-firm tofu\n\n3 stalks lemongrass\n\n1 Tbsp coconut oil\n\n1 Tbsp red curry paste\n\n2 C canned coconut milk\n\n\u00bc tsp salt\n\n1 C vegetable stock\n\n1 C thinly sliced shiitake mushrooms\n\n1 C \u00bd-inch pieces carrot\n\n1 C \u00bd-inch pieces broccoli\n\nAbout 6 oz rice vermicelli noodles\n\nFresh lime wedges, for garnish\n\nChopped fresh cilantro, for garnish\n\nSliced scallions, for garnish\n\nMETHOD\n\nCut the tofu block in half crosswise. Cut into triangles and set aside.\n\nCut the lemongrass stalks down the middle, remove the outer layer, and set aside.\n\nIn a large saucepan, melt the coconut oil over medium-high heat. Add the curry paste and lemongrass. Stir until fragrant, about 2 minutes.\n\nAdd the coconut milk, salt, stock, mushrooms, and carrots. Increase the heat to high and bring to a boil, then lower the heat to maintain a simmer. Cook for about 7 minutes and then add the tofu and broccoli. Cook for 5 minutes more.\n\nDivide the vermicelli noodles in half and place into two large serving bowls\u2014the bowls you're going to serve or eat this dish from. Remove any large pieces of lemongrass from the curry and then pour the curry over the noodles. Cover by at least an inch, or more (depending on how soupy you want it at the end). Let sit for a few minutes so the noodles soften, and then gently stir to soften the noodles even more. (You don't need to cook the rice noodles first. By the time the curry is the right temperature to eat, they will be cooked.)\n\nTop with a squeeze of fresh lime, chopped cilantro, and scallions and enjoy immediately.\n\n**Want to go the extra eco-mile?** Save the broccoli stems and use them in the Quick Country Coleslaw to cut down on food waste.\nCREAMY BASIL-CHICKPEA LETTUCE CUPS\n\nThe chickpea is a rain-fed crop with a low water footprint. It also improves soil structure, enriching it by fixing nitrogen. This favors crop rotation, contributing to a more sustainable model of agriculture. So we celebrate the humble chickpea for its earth-saving properties in this creamy, nutty lettuce cup recipe.\n\nSERVES 2 PREP AND COOKING TIME: 15 MINUTES, PLUS SOAKING TIME FOR CASHEWS\n\nINGREDIENTS\n\n1 C raw cashews, soaked in water to cover for 4 hours, plus more (unsoaked) for serving\n\n\u00bc C fresh basil leaves, plus more for serving\n\n1 Tbsp rice vinegar or distilled white vinegar\n\n1 garlic clove\n\nJuice of \u00bd lemon\n\n\u00bd tsp salt\n\n1 (14-oz) can chickpeas, drained and rinsed\n\n1 Tbsp capers\n\n1 medium cucumber, cut into \u00bd-inch cubes\n\n2 celery stalks, chopped\n\n\u00bd C chopped red onion\n\n10 romaine lettuce leaves\n\n2 Tbsp hulled pumpkin seeds\n\n1 Tbsp black sesame seeds\n\nFreshly cracked black pepper\n\nRaisins, for garnish (optional)\n\nMETHOD\n\nRinse the soaked cashews. Place them in a blender with 1 cup water. Add the basil leaves, vinegar, garlic, lemon juice, and salt. Blend until totally smooth (about 1 minute in a Vitamix or other high-speed blender, or a bit longer in a normal blender).\n\nPlace the chickpeas and capers in a large bowl. With a potato masher or a large fork, mash them a little bit. Add the cucumber, celery, and red onion. Add the sauce and stir to combine.\n\nLay the lettuce leaves out like cups or taco shells. Spoon 2 heaping tablespoons of the creamy chickpea mix into each cup. Sprinkle with pumpkin seeds, black sesame seeds, a few additional cashews, and basil leaves, season with some freshly cracked black pepper, and garnish with a few raisins, if you like. Enjoy immediately.\n\n**Tip:** Try toasting the pumpkin seeds first in a hot, dry pan until lightly browned.\n\n**Want to go the extra eco-mile?** Save the aquafaba (chickpea juice; see here) for Chocolate Brownies (here) or Lemon Meringue Pie\u2014to cut down on waste.\n\n\"If more people had vegetarian or vegan diets, especially in the developed world, the environmental impact would be significant.\"\n\n\u2014MARK TERCEK, NATURE CONSERVANCY CEO\nGRILLED VEGGIE PIZZA WITH BBQ SAUCE\n\nHomemade pizza is one of my all-time favorite things to make. I recommend inviting people over and making multiple pizzas. (Who says adults can't have pizza parties?) This pizza uses grilled eggplant, zucchini, and mushrooms along with BBQ sauce and dollops of vegan cream cheese (I like the Tofutti and Kite Hill brands) to create a smoky, fiery pie that's sure to please.\n\nSERVES 2 PREP AND COOKING TIME: 2 HOURS\n\nINGREDIENTS\n\nFor the grilled vegetables:\n\n1 medium eggplant, cut into \u00bc-inch-thick rounds\n\n1 medium zucchini, cut into \u00bc-inch-thick rounds\n\n3 large button mushrooms, cut into \u00bc-inch pieces\n\n\u00bc C balsamic vinegar\n\n\u00bc C canola oil\n\n\u00bd tsp salt\n\n1\u00bd tsp liquid smoke\n\nFor the crust:\n\n1\u00bd C all-purpose flour, plus more for dusting\n\n1 tsp salt\n\n1 tsp active dry yeast\n\n\u00bd C plain almond milk\n\n2 Tbsp olive oil\n\nCornmeal, for dusting (optional)\n\nTo assemble:\n\n\u00bd C BBQ sauce\n\n\u00bd C plus \u0661 Tbsp vegan cream cheese\n\nFresh basil leaves\n\nMETHOD\n\nFirst, prep the vegetables:\n\nPut the eggplant, zucchini, and mushrooms in a large casserole dish or container.\n\nWhisk together the vinegar, canola oil, salt, and liquid smoke in a small bowl. Pour over the vegetables and turn to coat evenly. Let sit for at least 30 minutes.\n\nCook the vegetables in a hot grill pan for about 3 minutes per side. Alternatively, lay them flat on a baking sheet and place them under the broiler for 6 to 8 minutes on one side, then flip them and broil for 4 minutes on the other. Return them to the dish they were marinating in and let cool while you prepare the rest of the pizza.\n\nNext, make the crust:\n\nPlace the flour, salt, and yeast in a large bowl and stir to combine. Make a well in the center.\n\nWarm the almond milk by microwaving it in a microwave-safe mug for 40 seconds. Add 1 tablespoon of the olive oil and stir to combine. Slowly pour the almond milk mixture into the well in the dry ingredients, gently stirring as you go. It'll start out as a liquidy paste, and then get more solid. Once the mixture is solid enough to handle, use your hand to form the dough into a ball.\n\nPlace the dough on a lightly floured surface and knead for about 3 minutes, until you have a nice, silky-smooth ball of dough. Wipe out the bowl and place a little bit of olive oil in the bottom of it. Return the dough to the bowl, turning to coat it with oil. Cover with a clean dish towel and place it somewhere warm until the dough has doubled in size. This usually takes an hour but can take up to two depending on where you live.\n\nOnce the dough has doubled, punch it to deflate it, then return it to the floured counter. Roll it into a football shape, cover with the dish towel, and let sit for 20 minutes more.\n\nAt this point, preheat the oven to 400\u00b0F while the dough rests.\n\nAfter 20 minutes, press the dough football flat and then sprinkle the top with a little flour and roll it into a pizza crust roughly 12 x 20 inches. (I find it helps to sprinkle a little cornmeal on my rolling surface, too.) Carefully transfer the dough to a large baking sheet.\n\nFinish the pizza:\n\nSpoon the BBQ sauce over the pizza dough and spread it around with the back of a tablespoon, leaving a 1-inch border along the edges.\n\nPile the grilled veggies onto the pizza and bake for 20 minutes.\n\nRemove from the oven and add the cream cheese in roughly 1-tablespoon dollops. Return the pizza to the oven for a final 5 minutes.\n\nRemove from the oven and let cool for 3 to 5 minutes. Top with fresh basil leaves and cut into 6 to 8 slices. Enjoy alone or with a side salad.\nJERKED JACKFRUIT TACOS WITH GRILLED PINEAPPLE\n\nJerk is a style of Jamaican cooking that involves a super-hot spice mixture and (usually) meat. But what's a guy or gal to do if you want that same fiery flavor without the environmental problems that often accompany meat production? Well, bring on the jackfruits! This fruit, when marinated and shredded, mimics the taste and texture of pulled pork. So this recipe offers a real culinary mashup of ingredients\u2014tropical fruit with Caribbean meat spices in a Mexican tortilla. You'll never think of fruit the same way again!\n\nSERVES 2 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n3 Tbsp agave nectar\n\n3 Tbsp apple cider vinegar\n\n2 Tbsp olive oil\n\n1\u00bd Tbsp ground cinnamon\n\n2 Tbsp freshly ground black pepper\n\n1 Tbsp dried thyme\n\n2 tsp ground allspice\n\n1\u00bd tsp cayenne pepper\n\n\u00bd tsp ground nutmeg\n\n\u00bd tsp garlic powder\n\n\u00bd tsp onion powder\n\n\u00bd tsp salt\n\n1 (20-oz) can jackfruit, drained\n\n3 pineapple rings, fresh or canned\n\n6 soft corn or flour tortillas\n\n1\u00bd C sliced red cabbage\n\n12 cherry tomatoes, quartered\n\nVegan sour cream, for serving\n\nChopped fresh cilantro, for serving\n\nFresh lime, for serving\n\nMETHOD\n\nIn a small bowl, combine the agave nectar, vinegar, olive oil and \u00bc cup water. Set aside.\n\nCombine the cinnamon, black pepper, thyme, allspice, cayenne, nutmeg, garlic powder, onion powder, and salt in a small bowl. Place a large, flat-bottomed skillet over a medium-high heat. Once hot, add the spice mixture and stir for about 2 minutes, or until fragrant. Pour the agave mixture into the skillet and heat for 30 seconds. Add the jackfruit and stir to break up the jackfruit as it softens. Heat through, about 3 minutes, and set aside.\n\nHeat a grill pan over high heat and grill the pineapple rings until lightly blackened, about 3 minutes per side. (If you don't have a grill pan, just use a regular heavy-bottomed skillet and sear them for a couple of minutes on each side.)\n\nWarm the tortillas on the stove or in the microwave. Build your tacos using the jackfruit, grilled pineapple, cabbage, cherry tomatoes, sour cream, and cilantro. Serve with wedges of lime.\nMEAT-FREE MEAT LOAF WITH TOMATO-MAPLE GLAZE\n\nMom's meat loaf was always one of my absolute favorites. But a single beef serving dishes out almost 30 kg of climate-change-causing CO2, so this recipe uses chickpeas instead. Coming in at less than 2 kg of CO2 per serving, they're a much eco-friendlier choice. And combined with carrots, celery, panko bread crumbs, and delicious flavorings, they cook up into a robust, hearty loaf that'll leave you wanting seconds.\n\nSERVES 4 PREP AND COOKING TIME: 1 HOUR 10 MINUTES\n\nINGREDIENTS\n\nFor the loaf:\n\n2 Tbsp olive oil, plus more for greasing\n\n2 (14-oz) cans chickpeas, drained and rinsed\n\n1 large yellow onion, diced\n\n2 celery stalks, finely diced\n\n2 carrots, finely diced\n\n2 garlic cloves, minced\n\n3 Tbsp chopped walnuts\n\n2 C panko bread crumbs\n\n\u00bd C plain almond milk\n\n2\u00bd Tbsp soy sauce\n\n3 Tbsp vegan Worcestershire sauce\n\n2 Tbsp nutritional yeast\n\n2 Tbsp flax meal\n\n2\u00bd Tbsp tomato paste\n\n1 tsp liquid smoke\n\n\u00bc tsp freshly ground black pepper\n\n\u00bd tsp ground sage\n\n\u00bd tsp dried thyme\n\nFor the glaze:\n\n\u00bc C tomato paste\n\n2 Tbsp pure maple syrup\n\n2 Tbsp apple cider vinegar\n\n1 Tbsp soy sauce\n\n1 tsp paprika (preferably smoked paprika)\n\n\u00bd tsp freshly ground black pepper\n\nDash of crushed red pepper\n\n2 Tbsp sesame seeds\n\nMETHOD\n\nFirst, make the loaf:\n\nPreheat the oven to 375\u00b0F. Lightly oil a 9 x 5-inch loaf pan.\n\nPlace all the meat loaf ingredients in a large bowl and mix together.\n\nIn \u00be-cup batches, place the loaf mixture into a food processor and process until most of the chickpeas are broken up (you want it smooth, but still a little chunky in the end). Place each batch into the oiled loaf pan as you go.\n\nPack the mixture into the pan and smooth out the top. Bake, uncovered, for 30 minutes.\n\nMeanwhile, make the glaze:\n\nWhisk together all the glaze ingredients, except the sesame seeds, in a bowl and set aside.\n\nAfter 30 minutes of baking, remove the loaf and top it with the glaze, smoothing it over. Sprinkle the sesame seeds on top and bake for 25 minutes more.\n\nRemove from the oven, let cool for at least 15 minutes, slice, and serve.\n\n**Equipment Alert:** This recipe requires a food processor.\n\n**Serving Suggestion:**  \nServe with a side of Massaged Kale Caesar salad.\n\n\"Reducing meat consumption is a growing trend, driven by health and environmental considerations... I feel healthier, more active, and by making my diet more varied, I never feel like I'm missing out on anything.\"\n\n\u2014SIR RICHARD BRANSON\nMUSHROOM AND KALE GALETTE\n\nIn French cooking, galette can refer to several types of round or free-form crusty cakes or pies. (In New Orleans, where I used to live, they eat a sweet version called galette des rois\u2014or King Cake\u2014around Mardi Gras each year.) This galette is savory, made with mushrooms, kale, eggplant, and capers, plus some soy yogurt for creaminess. It's perfect for dinner parties, when you want a pretty centerpiece meal that'll be sure to please.\n\nSERVES 4 PREP AND COOKING TIME: 1\u00bd HOURS\n\nINGREDIENTS\n\n2 Tbsp olive oil\n\n1 red onion, finely chopped\n\n1 small eggplant, cut into \u00bd-inch cubes\n\n6 baby bella or button mushrooms, sliced\n\n1\u00bd C chopped tomatoes\n\n6 kale leaves, stemmed and chopped\n\n2 Tbsp capers\n\n\u00bc C nondairy yogurt\n\n1 tsp dried oregano\n\n2 tsp sea salt\n\n2 C whole wheat flour, plus more for dusting\n\n\u00bd C instant oats\n\n\u00bc C chilled margarine\n\n\u00bc to \u00bd C cold water\n\nAlmond milk, for brushing\n\nPine nuts, for garnish\n\nMETHOD\n\nIn a large skillet, heat the olive oil over medium-high heat.\n\nAdd the onion and stir so that it is coated in the oil. Add the eggplant and cook for about 5 minutes. Add the mushrooms and tomatoes and cook for 2 minutes more. (If sticking, add a splash of water as you go.) Add the kale and stir until the leaves are limp, about 2 minutes. Add the capers, yogurt, oregano, and 1 teaspoon of the salt. Sprinkle 1 tablespoon of the flour on top, then stir to combine. Remove from the heat and let sit while you prepare the crust.\n\nPreheat the oven to 375\u00b0F.\n\nPlace the remaining flour, the oats, and the remaining 1 teaspoon salt in a large bowl. Stir to combine. Add the margarine and work it through with clean fingers until the mixture is crumbly. Add the water, 2 tablespoons at a time, kneading between additions, until the mixture forms a ball.\n\nLightly dust a clean work surface with flour. Knead the dough a few times until it forms a nice, compact ball. Flatten with the palm of your clean hands and then sprinkle with a little bit of flour and roll into a circle with a rolling pin. Roll back and forth a few times and then flip, sprinkle with some more flour, roll, flip, and repeat until you have a circle about 15 inches in diameter.\n\nTransfer the dough to a baking sheet and spoon the filling into the middle. Fold the edges up, pressing the dough together where the edges touch to encourage sticking. Brush the dough with almond milk and sprinkle the pine nuts over the exposed part of the filling.\n\nBake for 45 minutes. Remove and let cool for 5 minutes before cutting and serving.\n\n\"I'm eating a far more plant-based diet.\"\n\n\u2014OPRAH WINFREY\nNO-FRILLS CHEESE PIZZA\n\nWho doesn't love a classic cheese pie once in a while? But cow's-milk mozzarella comes with all kinds of devastation\u2014like runoff from factory farms into local waterways and using copious amounts of water to grow cattle feed. So once in a while, try your pie with Daiya mozzarella instead\u2014it uses a blend of tapioca flour (for meltability) and pea protein instead of cow's milk. _Molto delizioso_!\n\nSERVES 2 PREP AND COOKING TIME: 45 MINUTES\n\nINGREDIENTS\n\n2\u00bd C all-purpose flour, plus more for dusting\n\n2\u00bd tsp baking powder\n\n1 tsp salt\n\n3 Tbsp olive oil\n\n1 (6-oz) can tomato paste\n\n\u00bd tsp garlic powder\n\n1 tsp dried oregano\n\n1 (8-oz) packet Daiya mozzarella shreds\n\nMETHOD\n\nPreheat the oven to 400\u00b0F.\n\nPlace the flour, baking powder, and salt in a large bowl and stir to combine. Make a well in the center.\n\nCombine \u00be cup water and 1 tablespoon of the olive oil in a small dish. Slowly pour this into the well in the dry ingredients, stirring as you go. It'll start out as a liquidy paste. Once firm enough to handle, use clean hands to form the dough into a ball inside the bowl. If the dough isn't coming together, add up to another \u00bc cup water, a little at a time, until it forms a nice ball. (If it's too wet, add some more flour.)\n\nPlace the dough on a lightly floured work surface and knead for about 3 minutes. Divide it into two balls and roll each out into a 10-inch circle. Carefully transfer each dough round to its own baking sheet.\n\nPlace the tomato paste, remaining 2 tablespoons olive oil, the garlic powder, and the oregano in a small dish and stir to combine. Spoon the sauce onto the crusts and spread it evenly with the back of a tablespoon, leaving a 1-inch border along the edges. Top the pizzas evenly with the mozzarella.\n\nBake the pizzas until the crusts are crispy but not burnt, 10 to 12 minutes. (If your oven is too small to fit both, cook one and then the other.) Enjoy immediately.\n\n**Serving Suggestion:** Try dipping your pizza in the ranch dressing from the Spicy Sweet Potato Fries recipe.\n\n**Note:** Want some frills on your no-frills pizza? Plant-Based Parmesan, fresh basil leaves, olives, capers, sun-dried tomatoes, artichokes, mushrooms, and red onion all make tasty additions.\nA MESSAGE FROM\n\n## WHOLE FOODS MARKET FOUNDER AND CEO JOHN MACKEY\n\n**In my early twenties, I became vegetarian,** back before anyone had even heard of vegans. I thought it was just a healthier way to live and was aware it did less harm to animals. I moved into a vegetarian co-op and fell in with that lifestyle. My friends were vegetarian, and so I was a vegetarian. You might say it was the good kind of peer pressure. Then, much later in life, when I was already running Whole Foods, I was challenged by an animal protection advocate to learn more about factory farming and some of the products we were selling. That process led me to become vegan, and I haven't looked back since.\n\nIn my opinion, you can't be an environmentalist\u2014not a true environmentalist\u2014unless you're eating a plant-based diet. The environmental impact of eating so much animal-based food is terrible for our planet in so many ways. From animal-waste disposal and polluting ground waters to devastating the rain forests and increasing our carbon footprint\u2014there's no question that a plant-based diet is gentler and more environmentally sustainable.\n\nAnd yet there's so much confusion out there about what healthy, sustainable eating is. That's no accident. There are big, vested interests\u2014the egg industry, the dairy industry, the meat industry\u2014that put out a lot of dummy studies to purposely cause that confusion.\n\nBut there's a strong consensus in the legitimate science. The really _good_ science increasingly shows that there are certain things everyone agrees about. Whatever you're eating\u2014vegan or omnivore or paleo or whatever else\u2014everyone agrees we should eat more fresh fruits and veggies. And the science is clear that we should be eating, as Michael Pollan says, \"real food, mostly plants, not too much.\" The science is clear that eating too many animal-based foods correlates very closely with heart disease, obesity, cancer, diabetes, and environmental harm. Anyone who says anything to the contrary will have no legitimate studies to support their point of view.\n\nDo we all have to be vegan? I certainly am, but there's a place at the table for everyone when it comes to eating a healthier, more plant-forward, eco-friendlier diet.\n\n_John Mackey has led Whole Foods Market as it has grown from a single store, founded in 1978, to an $11 billion Fortune 300 company and a top U.S. supermarket. For fifteen consecutive years,_ Fortune _magazine has included Whole Foods Market on its \"100 Best Companies to Work For\" list. Mackey has been recognized for his work over the years by being named Ernst & Young's \"United States Entrepreneur of the Year,\"_ Institutional Investor's _\"Best CEO in America,\"_ Barron's _\"World's Best CEO,\" MarketWatch's \"CEO of the Year,\"_ Fortune's _\"Businessperson of the Year,\" and_ Esquire's _\"Most Inspiring CEO.\"_\nOYSTER MUSHROOMS \u00c1 LA MARINI\u00c8RE\n\n\u00c0 la marini\u00e8re is a French style of cooking mussels in butter and wine. This version is more sea- (and sea creature\u2013) friendly, using oyster mushrooms instead of mussels. The French lentils add protein, and the dish is served in a pool of broth with a chunk of rustic bread. It's perfect for a cool fall evening.\n\nSERVES 2 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\n\u00bd C dried French lentils\n\n2 vegetable bouillon cubes\n\n2 Tbsp raw cashews, soaked in water to cover for at least 4 hours\n\n3 Tbsp olive oil\n\n2 garlic cloves, minced\n\n\u00be C chopped shallots\n\n\u00bc C fresh parsley, chopped\n\n6 dried bay leaves\n\n1\u00bd Tbsp fresh thyme\n\n1 tsp chopped fresh rosemary\n\n\u00bc tsp salt\n\n\u00bc tsp freshly ground black pepper\n\n2 C oyster mushrooms, chopped\n\n1 C Pinot Grigio or other dry white wine\n\nA baguette or other rustic bread, for serving\n\nMETHOD\n\nPlace the lentils in a pot with 1\u00bd cups water and the bouillon cubes. Cover and bring to a boil. Lower the heat to maintain a simmer and cook until soft, about 30 minutes.\n\nMeanwhile, drain and rinse the soaked cashews. Place them in a blender. Add enough water to just cover them and blend until smooth, about 1 minute. Pour the cashew cream into a small bowl and set it aside.\n\nAfter 30 minutes, remove the lentils from the heat. Drain, reserving \u00bd cup of the cooking liquid. Set aside.\n\nIn a skillet, heat the olive oil over medium-high heat. Add the garlic and shallots and cook, stirring, until soft, 3 to 4 minutes. Add the parsley, bay leaves, thyme, rosemary, salt, and pepper and cook for 1 minute more.\n\nAdd the oyster mushrooms. As they cook, they'll release liquid. Cook until that liquid has evaporated, 5 to 8 minutes. Add the wine and cook for 3 to 4 minutes to let some of the alcohol burn off. Remove and discard the bay leaves. Add the cashew cream you made earlier and the reserved \u00bd cup lentil cooking liquid. Stir to combine. Add the cooked lentils and cook for about 1 minute more.\n\nWhen finished, you'll have a hearty dish in a light broth. Spoon this onto a plate and top it off with some of the broth, so the broth pools on the plate. Serve with a chunk of baguette or other type of rustic bread.\n\n**Tips:** If oyster mushrooms are too pricey, any kind of mushrooms will work. Also, enjoy the leftover white wine with your dinner\u2014the best kind of zero-waste dining!\nPITTSBURGH PIEROGIS\n\nMy wife is from Pittsburgh, where pierogis are basically the city's official food (and a way of life). Every time the Steelers are in a big game, she makes a massive plate of these delicious Polish dumplings for our friends and family. Here's her recipe for straight-out-of-Pittsburgh vegan pierogis.\n\nMAKES 30 TO 40 SMALL PIEROGIS PREP AND COOKING TIME: 2 HOURS\n\nINGREDIENTS\n\nFor the dough:\n\n1\u00bc C all-purpose flour\n\n\u00bc tsp salt\n\n1 Tbsp olive oil\n\n\u00bc C lukewarm water\n\n\u00bc C vegan sour cream\n\n2 Tbsp margarine, at room temperature\n\nFor the filling:\n\n\u2153 tsp sea salt, plus more for the cooking water\n\n1\u00bc lb Yukon Gold potatoes, peeled and cut into \u00bd-inch pieces\n\n1 Tbsp margarine\n\n\u00bc C plain soy milk\n\n1 tsp onion powder\n\n\u00bd tsp garlic powder\n\n\u00bc tsp freshly ground black pepper\n\nFor frying and serving:\n\nAll-purpose flour, for dusting\n\nSalt\n\n1 Tbsp canola oil\n\n\u00bd C vegan sour cream\n\n\u00bd C sauerkraut, squeezed\n\nMETHOD\n\nFirst, make the dough:\n\nCombine the flour and salt in a large bowl.\n\nWhisk the olive oil into the water and then slowly pour that into the flour, mixing as you go. Add the sour cream and margarine.\n\nWith clean hands, work the dough until it loses most of its stickiness, about 5 minutes, kneading it into a loose ball. Wrap the dough ball in plastic wrap and refrigerate for 30 minutes or up to 2 days.\n\nMeanwhile, make the filling:\n\nBring a large pot of salted water to a boil. Add the potatoes and cook until soft, about 15 minutes.\n\nRemove from the heat and drain. Rinse with cold water to cool them slightly. Return them to the pot and add the rest of the ingredients for the filling. Mash with a potato masher until smooth. Set aside.\n\nForm and fry the pierogis:\n\nFlour a clean work surface. Unwrap the dough and transfer it to the work surface. Knead for about 5 minutes, adding more flour as you go, until the dough is smooth but not sticky.\n\nRoll the dough into a large, very thin rectangle. Using a cookie cutter or thin-rimmed drinking glass roughly 3 inches in diameter, cut as many circles as possible into the rectangle of dough.\n\nLift away the remaining dough, knead it into a ball, roll that out again, and cut more circles. Repeat these steps until all the dough has been used up. You should have 30 to 40 small dough circles at the end.\n\nTake the mashed potato filling and roll it into roughly 30 to 40 small balls (one for each dough circle).\n\nBring a very large pot of salted water to a low boil. Set up a workstation with the dough circles, potato balls, a lightly floured baking sheet, and a small bowl of cold water (to wet your fingers when forming the pierogis).\n\nPlace a potato ball in the center of a dough circle, and then flatten the filling slightly. Using your clean finger, wet the edges of the dough circle. Fold the dough together to make a half circle and pinch the edges shut. Place the pierogi on the floured baking sheet and use a fork to seal the edges. Repeat with the remaining dough and filling. Your first few might be a bit weird, but you'll quickly get the hang of it.\n\nPlace about 6 pierogis into the pot of boiling water. Once they float to the surface, about 3 minutes, scoop them out with a slotted spoon and place them on a plate. Repeat until all the pierogis are boiled.\n\nIn a large nonstick skillet, heat the canola oil over medium-high heat. Add a few pierogis and fry until browned, 1 to 2 minutes per side. Repeat to fry the entire batch.\n\nTransfer the cooked pierogis to a serving plate. Top with vegan sour cream and sauerkraut. Enjoy!\n\n\"I'm eating less meat these days myself, and I'm proud of it... Twenty-eight percent of greenhouse gases come from livestock farming, so cutting back on meat consumption even slightly can have a big, positive impact.\"\n\n\u2014ARNOLD SCHWARZENEGGER\nPROTEIN-PACKED BURRITO BOWL\n\nThis is one hearty bowl of deliciousness! Gardein's plant-based chicken tenders are meaty and have a much greener footprint than real-meat chicken fingers (just think of all those Delmarva Peninsula residents dealing with poultry farm air pollution). Add quinoa and black beans, and this bowl is a protein-packed meal.\n\nSERVES 2 PREP AND COOKING TIME: 40 MINUTES\n\nINGREDIENTS\n\nFor the vinaigrette:\n\n2 Tbsp fresh lime juice\n\n1 garlic clove, minced\n\n\u00bd tsp agave nectar\n\n\u00bd tsp sea salt\n\n\u215b tsp ground cumin\n\n2 Tbsp olive oil\n\n1 Tbsp chopped fresh cilantro\n\nFor the bowl:\n\n1 (9.5-oz) packet Gardein Chipotle Lime Chicken Fingers\n\n1 C quinoa\n\nSea salt\n\n2 ears corn, husked\n\n1 Tbsp coconut oil\n\n\u00bc head purple cabbage, thinly sliced\n\n1 (15-oz) can black beans, drained and rinsed\n\n1 ripe avocado, halved and pitted, then diced, if desired\n\n10 cherry tomatoes, halved\n\n\u00bd red onion, finely sliced\n\nFresh cilantro, for garnish\n\nLime wedges, for garnish\n\nSesame seeds, for garnish\n\nMETHOD\n\nPreheat the oven to 450\u00b0F.\n\nFirst, make the vinaigrette:\n\nMix the vinaigrette ingredients together in a small bowl. Set aside in the fridge.\n\nAssemble the bowl:\n\nBake the Gardein according to the package instructions. Set aside once done.\n\nPut the quinoa in a saucepan with \u0661\u00bd cups water and a pinch of salt. Cover, bring to a boil, and lower the heat to maintain a simmer. Cook until the liquid has been absorbed, 15 to 20 minutes. Remove from the heat and fluff with a fork. Set aside.\n\nHeat a skillet over medium-high heat. Brush the corn with coconut oil, sprinkle with salt, and place in the hot pan. Cook, turning every few minutes, until lightly charred and cooked all over. Set aside.\n\nCut the corncobs into thirds and slice the chicken fingers. Assemble each bowl with half the corn, chicken fingers, cabbage, black beans, avocado, tomatoes, and onion. Sprinkle garnishes on top, add the dressing, and enjoy!\n\n**Want to go the extra eco-mile?** Replace the Gardein with tofu. Just broil a few strips (see here).\n\n**Serving Suggestion:** This also pairs well with a side of Roasted Red Pepper Hummus.\nRAINBOW VEGGIE SUSHI\n\nYou know what they say: give a man a fish and he'll eat for a day; teach a man to fish and all the world's fish will be gone by 2050. This no-fish dish uses red pepper, yellow mango, purple cabbage, green cucumber and avocado, and orange carrot to create flavorful, colorful sushi that's easy on our oceans and their inhabitants.\n\nSERVES 2 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\n2\u00bd C short-grain brown rice\n\n\u2153 C vegan mayo\n\n8 sheets roasted nori\n\n1 red bell pepper, thinly sliced\n\n1 carrot, julienned\n\n1 mango, pitted, peeled, and julienned\n\n1 cucumber, julienned\n\n1 avocado, pitted, peeled, and thinly sliced\n\n2 C thinly sliced cabbage\n\nSoy sauce, for dipping\n\nWasabi paste (optional)\n\nPickled ginger (optional)\n\nMETHOD\n\nPlace the rice and 5 cups water in a medium saucepan. Cover and bring to a boil over high heat. Turn the heat down to low and simmer for 30 minutes. Remove from the heat and let sit, covered, for 10 minutes. Remove the lid, fluff with a fork, and let cool. Once fully cool, stir in the mayo.\n\nWhile rice is cooking, if you've never made homemade sushi before, now would be a good time to watch a video of it online so you can see the rolling technique.\n\nOnce you're ready to roll (literally!), wet a sushi mat and place a nori sheet on top of it, shiny-side down. Place one-eighth of the cooked rice on the bottom two-thirds of the nori sheet and press it out into a flat, even layer.\n\nPlace one-eighth of each ingredient (or any combination thereof that you'd like!) along the middle of the rice. Roll the bottom end of the sushi mat over the ingredients to enclose them in the rice and then continue to roll, using quite firm pressure to ensure that everything holds together. Set aside and repeat with the remaining ingredients to make a total of 8 rolls.\n\nUse a slightly wet serrated knife to cut each roll crosswise into 8 equal-size pieces.\n\nEnjoy immediately with soy sauce for dipping and wasabi and pickled ginger alongside, if you like.\n\n**Tip:** Make the rice in the morning or even the night before so it's fully cooled by the time you use it. If you try to roll rice that's too warm, you'll end up with soggy sushi\u2014and no one likes soggy sushi.\n\n**Serving Suggestion:** Serve with Sesame Seaweed Salad.\nSAVE-THE-BAY CRAB CAKES\n\nI've spent much of my life near the Chesapeake Bay, where crab cakes are a local favorite. But rather than pulling crabs from the bay to make the dish, this recipe instead uses hearts of palm\u2014which are soft and hearty and pull apart like crabmeat does. It's an eco-friendlier twist on a regional favorite.\n\nSERVES 4 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\nFor the tartar sauce:\n\n\u00bc C vegan mayo\n\n2 tsp sweet relish\n\n1 tsp yellow mustard\n\n\u00bd tsp fresh lemon juice\n\nDash of salt\n\nFreshly cracked black pepper\n\nFor the crab cakes:\n\n1 (14-oz) can hearts of palm, drained\n\n1 Tbsp olive oil\n\n\u00bd C minced onion\n\n2 tsp minced garlic\n\n\u00bc C minced bell pepper\n\n1\u00bc C fresh, frozen, or canned corn kernels\n\n1 C bread crumbs\n\n2\u00bd Tbsp finely chopped fresh parsley\n\n2 tsp yellow mustard\n\n1\u00bd Tbsp Old Bay seasoning\n\n\u00bc C vegan mayo\n\n\u00bc C vegetable oil\n\nMETHOD\n\nFirst, make the tartar sauce:\n\nMix all the sauce ingredients together in a small bowl and refrigerate.\n\nMake the crab cakes:\n\nDrain the hearts of palm and lay them out on a cutting board. Thinly slice them lengthwise and then chop them. Into a large bowl, squeeze the chopped pieces between clean fingers until they break apart. Set aside.\n\nIn a large skillet, heat the olive oil over medium-high heat. Add the onion, garlic, and bell pepper. Saut\u00e9 until tender, about 4 minutes. Add the corn and saut\u00e9 for 1 minute more. Remove from the heat.\n\nPlace the vegetable mixture into the bowl with the hearts of palm, setting the skillet aside for later. Mix well to combine. Add \u2153 cup of the bread crumbs, the parsley, mustard, 1 tablespoon of the Old Bay, and the mayo. Mix well.\n\nFill the bottom of a shallow bowl or pie plate with the remaining \u2154 cup bread crumbs. Add the remaining \u00bd tablespoon Old Bay and mix.\n\nTake about \u00bc cup of the vegetable mixture and drop it into the bread crumbs. Roll it around until it's well coated and form into a small patty with clean hands. Transfer to a large plate or baking sheet. Repeat until all the vegetable mixture has been used. (You should have about 7 large crab cakes; if you want them a bit smaller, you could form 8 to 10 smaller patties instead.)\n\nGently wipe out the skillet. Add the vegetable oil and heat over medium-high heat. Once the oil is hot (about 1 minute), place the patties in the skillet, carefully\u2014so as not to splatter the oil\u2014and cook until golden brown on each side, about 2 minutes per side. Transfer to a paper towel\u2013lined plate to drain excess oil. Repeat until all the crab cakes have been fried.\n\nTop with tartar sauce and enjoy immediately, either on their own or with a side of Quick Country Coleslaw.\n\n**Tip:** As you fry all the crab cakes, the bread crumbs left over in the oil may burn. If this happens, drain the oil between batches, wipe out the pan, and heat up fresh oil.\nSMOKY SEITAN KEBABS WITH PEANUT SAUCE\n\nKebab is a general term that can apply to various types of Middle Eastern grilled meats. In America, we're most familiar with shish kebab: small pieces of meat that are skewered and grilled. But beef and lamb come with a high eco-footprint (especially when the animals eat grass, resulting in higher methane emissions). This smoky kebab recipe uses easy homemade seitan instead, to cut down on emissions. Topped with peanut sauce and served with fresh lime and herbs, it's a deliciously smoky and more sustainable way to enjoy this tasty dish.\n\nSERVES 2 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n2 Tbsp creamy peanut butter\n\n4 Tbsp soy sauce\n\n2 Tbsp pure maple syrup\n\n1 tsp Sriracha (optional)\n\n1 C vital wheat gluten\n\n1 tsp garlic powder\n\n1 tsp onion powder\n\n1 tsp smoked paprika\n\n1 tsp dried thyme\n\n2 Tbsp nutritional yeast\n\n\u00bd tsp liquid smoke\n\n1 tsp vegan Worcestershire sauce\n\n1 Tbsp olive oil\n\n1 Tbsp tomato paste\n\nFresh cilantro, for garnish\n\nLime wedges, for serving\n\nMETHOD\n\nPreheat the oven to 375\u00b0F. Line a baking sheet with parchment paper and set aside.\n\nCombine the peanut butter, 2 tablespoons of the soy sauce, maple syrup, and Sriracha, if using, in a small bowl. Stir until well combined. Set the peanut sauce aside until ready to serve.\n\nPlace the vital wheat gluten, garlic powder, onion powder, paprika, thyme, and nutritional yeast in a large bowl and stir to combine.\n\nIn another bowl, place the liquid smoke, Worcestershire, olive oil, tomato paste, remaining 2 tablespoons soy sauce, and \u00bd cup water. Stir to combine.\n\nPour the wet mixture into the dry mixture. Stir gently as you pour, until it vaguely resembles ground beef. Knead the dough in the bowl for 2 minutes. It shouldn't stick to your hands at all.\n\nOnce you have a loose ball, break it into 18 small chunks. Press 3 chunks together to form one larger piece, then flatten it out. That'll be one kebab. Repeat this until you have 6 kebabs.\n\nPlace the kebabs on the prepared baking sheet. Flatten them out and bake for 10 minutes. Remove and let cool slightly. Turn the oven to broil.\n\nWet your skewers with water, and skewer each kebab. Place them back on the baking sheet and return to the oven, broiling each side for 2 minutes. Remove and serve topped with fresh herbs, with the peanut sauce and lime wedges alongside.\n\n**Serving Suggestion:** These are also great as the meat inside a Vietnamese B\u00e1nh M\u00ec.\n\n\"Please eat less meat.\"\n\n\u2014DR. RAJENDRA PACHAURI, FORMER CHAIR OF THE UN INTERGOVERNMENTAL PANEL ON CLIMATE CHANGE (IPCC)\nSPAGHETTI AND LENTIL MEATBALLS\n\nWhat could be more classic than spaghetti and meatballs? But alas, one serving of beef creates about 27 kg of CO2 emissions. A serving of lentils? Only 0.9 kg. That's a 96.6 percent reduction. With lentils, walnuts, oats, and\u2014yes\u2014even peanut butter (trust me), this dish will win over the hearts and stomachs of even the hungriest carnivore.\n\nSERVES 4 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\nFor the meatballs:\n\n3 Tbsp olive oil\n\n\u00bd C minced onion\n\n4 garlic cloves, minced\n\n1 tsp peanut butter\n\n2 Tbsp rolled (not instant) oats\n\n5 sun-dried tomatoes\n\n\u00bd tsp dried oregano\n\n\u00bd tsp dried rosemary\n\n\u00bd tsp dried thyme\n\n\u00bc tsp crushed red pepper\n\n\u00bd tsp dried sage\n\n\u00bc tsp freshly ground black pepper\n\n\u00bd tsp sea salt\n\n\u00bc C fresh parsley\n\n1\u00bd Tbsp tomato paste\n\n1 Tbsp flax meal\n\n1 (15-oz) can green or black lentils, drained\n\n6 Tbsp Plant-Based Parmesan\n\n5 Tbsp bread crumbs\n\nFor the sauce:\n\n\u00bd tsp salt, plus more for the pasta water\n\n1 Tbsp olive oil\n\n1 yellow onion, finely chopped\n\n3 to 5 garlic cloves, finely chopped\n\n2 (14-oz) cans chopped tomatoes\n\n\u00bc C tomato paste\n\n\u00bd tsp freshly ground black pepper\n\n\u00bd tsp dried oregano\n\n\u00bd tsp dried thyme\n\n1 tsp agave nectar or sugar\n\nSmall handful of fresh basil leaves\n\nTo assemble:\n\n1 (16-oz) box spaghetti\n\nOlive oil\n\nFresh basil leaves, for garnish\n\nFresh parsley leaves, for garnish\n\nPlant-Based Parmesan\n\nMETHOD\n\nFirst, make the meatballs:\n\nPreheat the oven to 375\u00b0F. Line a baking sheet with parchment paper.\n\nHeat a large skillet (I use cast iron) over medium-high heat. Once hot, add 2 tablespoons of the olive oil, and all of the onion and the garlic. Cook for about 3 minutes, then remove from the heat.\n\nTransfer the garlic and onions to a food processor (set the skillet aside) and add the rest of the meatball ingredients and 2 tablespoons water. Pulse to combine well, but avoid pureeing entirely. (You're going to roll this into balls next, so if it's too wet, add more bread crumbs, a little at a time, and pulse to combine.)\n\nScrape the mixture into a large bowl and stir once more to combine.\n\nForm the meatball mixture into about 12 small balls, gently rolling them in the palm of your hand with your fingertips and arranging them on the prepared baking sheet as you go.\n\nSet the skillet over medium-high heat. Add the remaining 1 tablespoon olive oil and as many meatballs as you can fit, leaving room to turn them. Cook until browned on most sides, about 2 minutes per side, turning or rolling them as you go. (Don't worry\u2014you bake them next, so they don't need to be fully cooked yet.)\n\nRearrange all the cooked meatballs on the baking sheet and bake for 20 minutes. Remove from the heat and set aside.\n\nMeanwhile, make the sauce:\n\nIn a saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook for about 4 minutes. Add the remaining sauce ingredients and mix well. (Be careful of splatter when you add the chopped tomatoes.) Bring to a boil, then lower the heat to maintain a simmer and cook, uncovered, for about 10 minutes while you make the pasta.\n\nAssemble and enjoy:\n\nBring a large pot of salted water to a boil. Cook the pasta in the boiling water according to the directions on the package. Drain and mix some olive oil through to keep the pasta from sticking.\n\nServe the pasta in deep bowls topped with meatballs, sauce, fresh basil and parsley, and Plant-Based Parmesan. Mangia!\n\n**Equipment Alert:** This recipe requires a food processor.\n\n**Serving Suggestion:** Serve with a side of Easy Cheesy Garlic Bread.\n\n**Want to go the extra eco-mile?** Cook the lentils from scratch to cut down on packaging. You'll want 1\u00bd cups cooked lentils, and may want to make them in advance to save time.\nA MESSAGE FROM\n\n## NATURAL RESOURCES DEFENSE COUNCIL PRESIDENT RHEA SUH\n\n**I grew up in Boulder, Colorado,** which is surrounded by the magnificent Rocky Mountains and spectacular landscapes. When I was a kid, I kind of assumed that everyone had a backyard like mine, overlooking the Rockies. I was privileged to regularly hike, and camp, and ski, and fish, and to relax in the great outdoors. This experience was a critical part of my early years, and really influenced my vision of myself and of the world around me. That environmentalism was sort of \"baked into\" me from childhood.\n\nToday, climate change is the central environmental challenge of our time. It threatens our people, our oceans, our forests, our lands. It has the potential to cause a catastrophic collapse of species worldwide. It is a threat to the natural systems that support all life.\n\nDirectly connected to that is the growing scarcity of fresh water. Glaciers are melting worldwide, and that is already causing water shortages in Andean countries, parts of China, and elsewhere. We're seeing tens of thousands of Rocky Mountain whitefish dying in the Yellowstone River, for heaven's sake, because a microscopic parasite is thriving and fish are being stressed in water levels that are lower and temperatures that are higher than they should be.\n\nAnd finally, there is environmental injustice, with low-income communities and people of color bearing a disproportionate burden of dirty air, polluted water, and other environmental ills. We have a lot of work to do to get this right.\n\nAnd food touches on all of this\u2014air, water, wildlife, lands\u2014and impacts everything we're working to change\u2014energy, transportation, conservation. It's all wrapped up in how we produce, process, distribute, and consume our food. We won't get the equation right unless we get the food piece right.\n\nWe can eat more plants, for starters. Pound for pound, plants have a vastly smaller carbon and environmental \"food-print\" compared to animal products. Beef is the SUV of food, resulting in more than thirty times more climate-changing emissions, pound for pound, than lentils, for example.\n\nGoing vegetarian or vegan is great, but even changing your diet a little bit can make a big difference. But reducing the amount of red meat we eat\u2014even by a small amount\u2014can improve our health, save us money, and reduce our contribution to the planet's environmental burden. And don't forget that eating is an act of pleasure. Make some healthy, environmentally friendly changes to your diet that you truly enjoy, and chances are you'll want to repeat them and share with friends and family. That's making a difference.\n\n_Rhea Suh is president of the Natural Resources Defense Council, leading nearly five hundred scientists, attorneys, and policy experts at one of the country's most effective environmental-action organizations. Before joining NRDC, Suh was nominated by President Barack Obama and served as the Assistant Secretary for Policy, Management, and Budget at the U.S. Department of the Interior._\nSTEAMED BUNS WITH TOFU AND MUSHROOMS\n\nLegend says that thousands of years ago, Chinese military leader Zhuge Liang encountered a river his troops couldn't cross. He was told to sacrifice his men and toss their heads into the river, to appease the gods for safe crossing. Instead, he ordered that head-shaped buns stuffed with meat be thrown into the river. Thus was born the steamed bun. This version takes Liang's compassion further: rather than being meat-filled, they're filled with hearty tofu, mushrooms, and vegetables. You can toss them into a river, though I recommend eating them.\n\nSERVES 2 (MAKES 6 SMALL BUNS) PREP AND COOKING TIME: 1 HOUR, PLUS 1 HOUR RISING TIME\n\nINGREDIENTS\n\n1 heaping Tbsp sugar\n\n1 C lukewarm water\n\n2 tsp instant yeast\n\n2\u00bc C all-purpose flour\n\nTiny pinch of salt\n\n2 Tbsp soy sauce\n\n2 Tbsp rice vinegar\n\n1 Tbsp toasted sesame oil\n\n1 Tbsp agave nectar\n\n1 tsp Sriracha\n\n\u00bd (16-ounce) block extra-firm tofu, cut into 12 slices\n\nVegetable oil, for brushing\n\n1 tsp black sesame seeds\n\n2 large brown mushrooms, thinly sliced\n\n\u00bc large cucumber, thinly sliced\n\nChopped scallions\n\nChopped fresh cilantro\n\nMETHOD\n\nIn a measuring cup or small bowl, dissolve the sugar in the warm water. Add the yeast. Stir and let sit until it gets foamy, about 5 minutes.\n\nPlace the flour and salt in a large bowl. Once the yeast mixture has some bubbles, add it to the flour. Stir gently and knead until it's smooth and soft.\n\nReturn the dough to the bowl, cover with a dish towel, and leave in a warm place until the dough has doubled in size, about 1 hour. (If you live in a cold climate, place the bowl in a 95\u00b0F oven to rise.)\n\nWhile the dough is rising, combine the soy sauce, vinegar, sesame oil, agave, and Sriracha in a small dish. Add the tofu and turn to coat in the marinade.\n\nWhen the dough has doubled in size, punch the air out and split the dough into 6 pieces. Roll them into balls and lightly brush the bottoms with vegetable oil. Sprinkle with the sesame seeds and let sit for 10 minutes.\n\nWhile the buns are sitting, arrange bamboo steaming baskets so they are sitting in a wok or saucepan with just enough water in the bottom of the pan to not get the baskets wet.\n\nPlace the buns in the steamer baskets with enough room for them to expand a little. Bring to a boil, cover, and steam for 20 minutes. Remove from the heat and let sit for 5 minutes. Remove the lids, being careful not to drip water onto the buns.\n\nSet a skillet over medium heat. Add the mushrooms, 1 tablespoon water, and 2 tablespoons of the tofu marinade (reserving the rest). Cook until brown, about 5 minutes.\n\nMove the mushrooms to one side of the pan and add the tofu to the other side. Cook for 2 minutes per side to heat the tofu through.\n\nRemove the buns from their baskets and carefully split them down the middle. Fill with the tofu, mushrooms, cucumber slices, scallions, and cilantro. Pour the remaining marinade into a small dish to spoon over the buns as you eat them.\nSUN-DRIED TOMATO AND BASIL SAUSAGES\n\nSausage's environmental footprint can be high. Think about that mega pig farm the people of Arrow Rock, Missouri, fought against\u2014or all the pollution caused by pig factories in North Carolina. These sausages use vital wheat gluten: all protein, no pork. Mixed with white beans, sun-dried tomatoes, and basil for Italian flair, they're perfect for any occasion.\n\nSERVES 2 TO 4 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\n4 sun-dried tomatoes, chopped\n\n2 garlic cloves, minced\n\n\u00bd C canned white beans, drained and rinsed\n\n10 fresh basil leaves\n\n2 Tbsp olive oil\n\n1 Tbsp apple cider vinegar\n\n1 tsp seeded mustard\n\n1\u00bd C vital wheat gluten\n\n1 tsp onion powder\n\n\u00be tsp sea salt\n\n\u00bd tsp freshly ground black pepper\n\n2 Tbsp nutritional yeast\n\nMETHOD\n\nPlace a metal steaming basket inside a large pot. Add 2 to 3 inches of water, ensuring the water is below the basket. Cut eight pieces of aluminum foil about 7 inches wide.\n\nPlace the sun-dried tomatoes, garlic, and white beans in a blender or food processor. Add the basil leaves, olive oil, vinegar, mustard, and 1 cup water. Puree until smooth.\n\nPlace the vital wheat gluten, onion powder, salt, pepper, and nutritional yeast in a large bowl and stir with a fork to combine. Pour the puree into the dry ingredients, stirring with a fork as you go, then use clean hands to knead everything into a ball.\n\nBreak the ball into 8 pieces and form each into a sausage shape. Roll each sausage in a piece of foil and twist at either end (like a Tootsie Roll)\u2014not too tight, as the sausages will expand as they cook.\n\nPlace the sausages in the steamer basket. Cover and bring to a boil. Steam for 40 minutes. (If the water gets too low, add more.) Once done, remove from the heat and let cool for at least 10 minutes before unwrapping each sausage.\n\nYou now have sausages that you can use as you please\u2014fry them in a pan, serve with mashed potatoes, take them to a BBQ and grill them, put them in a sandwich. The sky's the limit! Use immediately or store in an airtight container in the fridge for up to 3 days.\n\n**Serving Suggestion:** Serve with a side of grilled veggies drizzled with balsamic vinegar, or a simple salad.\n\n**Shopping Tip:** You can find vital wheat gluten in the flour aisle of most grocery stores.\nWILD RICE RISOTTO WITH PEAS AND ASPARAGUS\n\nRisotto is a northern Italian rice dish slowly cooked in stock. The stock is added a little at a time and continually topped off as the rice cooks and absorbs it. This releases starches from the rice that give the final product a creamy finish. The stock can be derived from meat, but for this version, we use veggie stock. The addition of wild rice\u2014which grows in water\u2014gives a nice earthy feel, and the peas help pack a little extra protein into the dish.\n\nSERVES 4 TO 6 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\n10 C vegetable stock (see Note)\n\n2 Tbsp olive oil\n\n3 shallots, finely chopped\n\n3 garlic cloves, minced\n\n2 C Arborio rice\n\n1 C wild rice\n\n\u00bd C dry white wine\n\n2 C frozen peas\n\n1 (16-oz) bunch asparagus, cut into 1-inch pieces\n\nSalt and freshly ground black pepper\n\n\u00bd C fresh mint leaves, chopped, plus more for serving\n\n\u00bd C fresh basil leaves, chopped, plus more for serving\n\n\u00bd C pine nuts\n\nFresh lemon wedges, for serving\n\nMETHOD\n\nBring the stock to a boil in a large saucepan, then lower the heat to maintain a simmer or very low boil.\n\nMeanwhile, in a large saucepan, heat the olive oil over medium heat. Add the shallots and garlic and cook for about 3 minutes. Add the rices and stir until coated in oil and slightly translucent. Add the wine.\n\nYou now want to add just enough hot stock to cover the rice by about 1 inch. Bring it to a boil, then lower the heat to maintain a simmer and cook until the liquid has been mostly absorbed by the rice.\n\nOnce the first round of liquid has absorbed by the rice, add more hot stock\u2014about \u00bd cup\u2014and repeat until that liquid has been absorbed, too. Repeat as many times as you need to until the rice is fully cooked and very soft. (The amount of stock you need and the number of times you'll need to do this depends on how dry the rice is, how hot your kitchen is, etc. Generally, it takes about 30 minutes.)\n\nOnce the rice is cooked, stir in the frozen peas and cook for 3 minutes.\n\nAdd the asparagus to the pot and cook for 2 minutes more.\n\nRemove the risotto from the heat, cover, and let sit for 5 minutes. Season with salt and pepper. Stir the mint and basil leaves into the risotto.\n\nServe immediately, garnished with pine nuts, additional mint and basil, and a squeeze of lemon juice.\n\n**Note:** If you don't have stock, bring 10 cups water to a boil and add 5 cubes of vegetarian bouillon.\nSIDES AND EXTRAS\n\nBlackened Brussels Sprouts with Capers and Raisins\n\nCreamy Herbed Mashed Potatoes\n\nCrispy Rice Paper Bacon\n\nEasy Cheesy Garlic Bread\n\nGreen Bean Casserole\n\nLonestar BBQ Beans\n\nMint and Tofu Summer Rolls\n\nMinted Green Pea Guacamole\n\nPistachio and Sunflower Seed Dukkah\n\nPlant-Based Parmesan\n\nQuick Country Coleslaw\n\nRoasted Red Pepper Hummus\n\nRosemary Cornbread Mini Muffins\n\nSpicy Sweet Potato Fries with Ranch\n\nSweet-and-Salty Curried Popcorn\nBLACKENED BRUSSELS SPROUTS WITH CAPERS AND RAISINS\n\nThis recipe combines Brussels sprouts with vinegar and agave for a sweet-and-sour taste. Add walnuts, capers, raisins, garlic, and more, and you've got a flavorful, aromatic dish that's crave-worthy on its own, as a snack, or as a side dish.\n\nSERVES 4 TO 6 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\n2 lb Brussels sprouts\n\n\u00bc tsp sea salt\n\n\u00bd C olive oil\n\n\u00bc C red wine vinegar\n\n1\u00bd Tbsp agave nectar\n\n\u00be C walnuts, chopped\n\n\u2153 C capers\n\n3 Tbsp golden raisins\n\n2 garlic cloves, minced\n\n\u00bc tsp ground cumin\n\n1 shallot, minced\n\nPinch of freshly ground black pepper\n\nMETHOD\n\nPreheat the oven to 425\u00b0F.\n\nPeel any bad leaves from the outsides of the sprouts and cut any larger sprouts in half. Place them in a large bowl as you go. Add the salt and \u00bc cup of the olive oil. Stir to coat well.\n\nPour the coated sprouts into a large cast-iron skillet and roast for 20 minutes, turning several times so the sprouts cook evenly.\n\nWhile the sprouts are in the oven, put the remaining \u00bc cup olive oil, the vinegar, and the agave in a blender and blend until emulsified, about 20 seconds. (You can also whisk this instead of blending, but you need to whisk well to emulsify the oil.) Pour into a small bowl and add the walnuts, capers, raisins, garlic, cumin, shallot, and pepper.\n\nRemove the sprouts from the oven and evenly pour the dressing over them. Mix to coat and bake for 15 minutes more.\n\nTurn the oven to broil and let the tops of the sprouts blacken, 3 to 10 minutes, depending on your oven and how blackened you want them.\n\nRemove from the oven (carefully, as the skillet will be hot!) and enjoy immediately on their own or as a side dish.\n\n**Serving Suggestion:** Serve alongside Meat-Free Meat Loaf.\nCREAMY HERBED MASHED POTATOES\n\nIt's hard to go wrong with a big, piping hot plate of mashed potatoes. This vegan version uses almond milk instead of cow's milk, margarine instead of butter, nutritional yeast for a little cheesy flavor, and chopped chives for... well, just so we can say there's something green in there!\n\nSERVES 4 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n2 lb Yukon Gold potatoes, peeled and cubed\n\n1\u00bd tsp salt\n\n1 C plain almond milk\n\n2 Tbsp margarine\n\n2 Tbsp nutritional yeast\n\n\u00bc tsp freshly ground black pepper, plus more to taste\n\n1 tsp dried rosemary\n\n\u00bd tsp dried thyme\n\n2 Tbsp chopped fresh chives\n\nMETHOD\n\nPut the potatoes and 1 teaspoon of the salt in a large pot with enough cold water to cover them by 3 to 4 inches. Bring to a boil over high heat. Lower the heat to maintain a simmer and cook for 15 to 20 minutes, or until soft.\n\nDrain the water from the cooked potatoes and return them to the pot. Add the almond milk, margarine, nutritional yeast, remaining \u00bd teaspoon salt, the pepper, rosemary, and thyme. Mash until smooth using a potato masher. Serve, topped with chives and additional pepper.\n\n**Tip:** You can use any unsweetened, unflavored, creamy plant-based milk, such as soy, hemp, or coconut, in place of the almond milk.\n\n\"Eliminating or reducing meat consumption in our diets is one important way to reduce our contribution to climate change... So enjoy Meatless Mondays, Tofu Tuesdays, Salad Sundays and everything in between. Because a locally sourced plant-heavy diet is not only good for you, but good for our planet too.\"\n\n\u2014THE SIERRA CLUB\nCRISPY RICE PAPER BACON\n\nCrispy, salty, smoky\u2014this bacon hits the spot, without all the eco-woes that accompany industrial pig farming. It crisps and bubbles, and is marbled with shades of brown and red and white.\n\nSERVES 4 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n2 Tbsp olive oil\n\n3 Tbsp soy sauce\n\n\u00bd tsp pure maple syrup\n\n\u00bd tsp liquid smoke\n\n2 Tbsp nutritional yeast\n\n1 tsp smoked paprika\n\n\u00bd tsp garlic powder\n\n\u00bc tsp onion powder\n\nPinch of freshly ground black pepper\n\n10 sheets rice paper\n\nMETHOD\n\nPreheat the oven to 400\u00b0F.\n\nIn a small bowl, combine all the ingredients except the rice paper and whisk together.\n\nNow you're going to cut the rice paper sheets into strips. It's tricky. Lay out one sheet of rice paper and, using a sharp knife, cut it into roughly 1-inch-wide strips. It'll crack apart (especially if your knife isn't sharp). Set the intact strips aside and repeat for all 10 sheets. At the end, you should have a nice pile of cut strips. (And it doesn't matter if they aren't perfect-looking!)\n\nNext, set up a large workspace with a parchment-covered baking sheet, a wide bowl or pie dish of water, and a dish of the marinade.\n\nTake two strips of roughly the same shape and hold them together. Dip them in the water. They'll stick together. Squeegee some of the water off with your fingers.\n\nDip the doubled strip in the marinade, gently, to coat both sides. Place on the parchment-lined baking sheet. Repeat until all the strips have been dipped and you have a bunch of double-thick bacon strips.\n\nBake until crispy, about 8 minutes. Careful: They'll go from crispy to burnt quickly.\n\nEnjoy immediately, or transfer to a wire rack and let cool before transferring to an airtight container and storing in the fridge. They're great on sandwiches, as a breakfast side, and more!\n\n**Tip:** Like your bacon a little chewier? Just bake it for a minute less.\n\n**Serving Suggestion:** Serve alongside the Tofu Scramble or any other breakfast dish.\n\n\"Like most people, I don't think I can be easily fooled. But that's just what happened when I was asked to taste a chicken taco and tell whether the meat inside was real or fake. The meat certainly had the look and the smell of chicken. I took a bite and it had the taste and texture of real chicken, too. But I was surprised to learn that there wasn't an ounce of real chicken it. The 'meat' was made entirely of plants. And yet, I couldn't tell the difference. What I was experiencing was more than a clever meat substitute. It was a taste of the future of food.\"\n\n\u2014BILL GATES\nEASY CHEESY GARLIC BREAD\n\nWhen I was young, I worked in a pizzeria. I became obsessed with the cheesy garlic bread that was one of their specialties. This is my reinvention of that recipe using plant-based, planet-friendly ingredients.\n\nSERVES 2 TO 4 PREP AND COOKING TIME: 10 MINUTES\n\nINGREDIENTS\n\n2 soft sub rolls\n\n1 Tbsp margarine\n\n\u00bd tsp salt\n\n1\u00bd tsp garlic powder\n\n1 Tbsp nutritional yeast\n\n1 tsp dried oregano\n\n\u00bd tsp freshly ground black pepper\n\nDaiya mozzarella shreds\n\nMETHOD\n\nPreheat the oven to 400\u00b0F.\n\nSlice each sub roll in half lengthwise. Lay them sliced-side up on a baking sheet.\n\nOn each half, spread about half the margarine. (If it's too cold to spread, warm it in the microwave for a few seconds to soften.)\n\nIn a small bowl, mix together the salt, garlic powder, nutritional yeast, oregano, and pepper. Sprinkle this across all four roll halves equally and use a butter knife to push it into the margarine a little bit.\n\nSprinkle each half with as much or as little mozzarella as you'd like.\n\nBake until toasted and the cheese has melted, 5 to 7 minutes.\n\n**Want to go the extra eco-mile?** Cut down on packaging by replacing the vegan mozzarella shreds with an extra 1 tablespoon nutritional yeast.\n\n**Serving Suggestion:** Enjoy with Spaghetti and Lentil Meatballs or Massaged Kale Caesar salad.\nGREEN BEAN CASSEROLE\n\nThis was always a Thanksgiving must-have in the Prescott household. And a Christmas must-have. And a Hanukkah must-have. Creamy mushroom soup, fresh green beans, and indulgent French-fried onions: You can't go wrong. This recipe is a new take on our family's classic, using shiitakes instead of button mushrooms and almond milk instead of cow's milk. Perfect any time of year\u2014especially holidays.\n\nSERVES 4 TO 6 PREP AND COOKING TIME: 30 MINUTES, PLUS SOAKING TIME FOR THE MUSHROOMS\n\nINGREDIENTS\n\n\u00be C dried shiitakes\n\n1 C boiling water\n\n4 C halved green beans\n\n\u00bc tsp salt, plus a pinch\n\n1 C almond milk\n\n1 vegetable bouillon cube, crushed into powder\n\n2 Tbsp nutritional yeast\n\n\u00bc tsp freshly ground black pepper\n\n\u00bc tsp dried sage\n\n1\u00bd C French-fried onions\n\nMETHOD\n\nPlace the dried mushrooms in a heatproof bowl. Add the boiling water and let sit for 1 hour.\n\nPreheat the oven to 375\u00b0F.\n\nMeanwhile, place the beans in a pot of cold water with a pinch of salt. Cover and bring to a boil. Cook for 3 to 4 minutes. Remove from the heat, drain immediately, and rinse under cold water.\n\nPour the liquid from the mushrooms into a medium bowl. Add the almond milk, crushed bouillon cube, nutritional yeast, \u00bc teaspoon salt, pepper, and sage. Whisk to combine. Finely chop the rehydrated mushrooms and add these to the liquid, along with half the French-fried onions.\n\nPlace the beans in a casserole dish, pour the liquid over, and then toss a couple of times to make sure the onions and mushrooms are nicely distributed throughout the beans and not sitting on top. Sprinkle the remaining onions over the top of the beans and bake for 15 minutes.\n\nRemove from the heat and let sit for 5 minutes before serving.\n\n**Serving Suggestion:** Serve with Meat-Free Meat Loaf and Creamy Herbed Mashed Potatoes.\n\n**Want to go the extra eco-mile?** Make this recipe when green beans are in season (see here).\nA MESSAGE FROM\n\n## THE CENTER FOR BIOLOGICAL DIVERSITY'S STEPHANIE FELDSTEIN\n\nIf you're looking for an easy and enjoyable way to conserve water; protect our oceans, lakes, and rivers; and save an amazing array of wild animal species, it's simple: eat less meat.\n\nThere are more than 7.4 billion people on the planet. It took about two hundred thousand years for the human population to reach 1 billion, then only two hundred years for that to increase sevenfold. There's never been another large, vertebrate animal that's grown as much, as quickly, or with such devastating consequences. In the past fifty years alone, our population has doubled, while wildlife populations\u2014including animals living in our oceans\u2014have plummeted by about half.\n\nAnd it's not just our sheer numbers that are crowding out other species\u2014it's our reckless overconsumption, relying on polluting, destructive industries. And at the top of that list is factory farming, the planet's single most environmentally destructive industry.\n\nThirty-five thousand miles of U.S. rivers, which provide habitat to countless fish, amphibians, and birds, are polluted by factory farms. Orange clownfish (like the star of _Finding Nemo_ ) and many other unique, beautiful species are losing their coral reef habitat to ocean acidification caused by greenhouse gas pollution\u2014caused in large part by the meat industry.\n\nEvery time we eat, we can make a wildlife-friendly choice. Fortunately, there are abundant plant-forward foods out there to help us do this. We can choose veggie sushi over fish sushi, bean burgers over hamburgers. Almond milk is easier on the planet than cow's milk. There's a whole rainbow of produce available at local grocers nationwide that offers delicious and vibrant ways to help save the planet. It's one of the best\u2014and tastiest\u2014ways to protect the environment.\n\n_Stephanie Feldstein is Population and Sustainability Director for the Center for Biological Diversity\u2014a national nonprofit conservation organization dedicated to the protection of endangered species and wild places. Using science, law, and creative media, the Center works to secure a future for all species great and small, with a focus on protecting the lands, waters, and climate that species need to survive._\nLONESTAR BBQ BEANS\n\nI make these beans whenever I've got visitors\u2014when I want an easy and filling meal with some Welcome-to-Texas flair. And because beans have remarkably low carbon and water footprints, my visitors and I can rest easy (and full) knowing we're eating something that's healthy, great tasting, and sustainable.\n\nSERVES 4 TO 6 PREP AND COOKING TIME: 1\u00bd TO 2 HOURS\n\nINGREDIENTS\n\n2 C dried pinto beans\n\n1 vegetable bouillon cube\n\n\u00bd tsp paprika\n\n\u00bd tsp liquid smoke\n\n1 Tbsp vegan Worcestershire sauce (optional)\n\n1 Tbsp agave nectar or molasses\n\n\u00bc C ketchup\n\n1 tsp mustard\n\n\u00bc tsp cayenne pepper\n\n2 Tbsp olive oil\n\n\u00bd C chopped onion\n\n\u00bd C finely chopped celery\n\n\u00bc C finely chopped carrots\n\n2 garlic cloves, minced\n\nMETHOD\n\nPlace the beans, 4 cups water, and the bouillon cube in a medium saucepan. Cover and bring to a boil. Lower the heat to maintain somewhere between a simmer and a very low boil and cook until the beans are soft, 60 to 90 minutes.\n\nWhile the beans cook, in a small bowl, whisk together the paprika, liquid smoke, Worcestershire (if using), agave, ketchup, mustard, cayenne, and 1 tablespoon water. Set aside.\n\nOnce the beans are done, remove them from the heat and set aside\u2014but don't drain them. Also, you want the water level to be just below the beans when they're done; if it's too high, remove the cover and boil for 2 minutes or so to let some of the excess water evaporate before removing from the heat.\n\nIn a large, deep skillet, heat the olive oil over medium-high heat for 2 to 3 minutes. Add the onion, celery, carrots, and garlic. Cook, stirring occasionally, until the vegetables soften, 5 to 8 minutes.\n\nPour the beans\u2014plus their cooking liquid\u2014into the skillet. Be careful! The water hitting the hot skillet will cause splatter. The beans will kind of pile up in the center of the skillet and the water will push to the sides. Go with that (no need to stir). Cook over medium heat until most of the liquid has evaporated, about 10 minutes. Add the BBQ sauce, stir to combine, and cook to heat through, about another 1 minute. Enjoy!\n\n**Serving Suggestions:** Top with vegan sour cream and serve over rice, or alongside the Jerked Jackfruit Tacos or Sun-Dried Tomato and Basil Sausages.\nMINT AND TOFU SUMMER ROLLS\n\nUnlike their spring counterparts, summer rolls aren't fried, but rather eaten cold and fresh. These Vietnamese rolls often contain pork and shrimp. But if you want a meat-free version that avoids all the troubles wrought by industrial pig and shrimp farming (air pollution, mangrove forest destruction, and so on), try using fresh vegetables and tofu instead. It's lighter on the planet and makes this already-fresh dish even fresher tasting.\n\nSERVES 2 PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n\u00bd (8.8-oz) package rice vermicelli noodles\n\n2 Tbsp creamy peanut butter\n\n2 Tbsp soy sauce\n\n2 Tbsp pure maple syrup\n\n2 tsp Sriracha\n\n1 tsp rice vinegar\n\n1 tsp ground ginger\n\n1 medium cucumber, seeded and julienned\n\n\u00bd (16-oz) block extra-firm tofu, cut into matchsticks\n\n\u00bd head iceberg lettuce, chopped\n\n\u00bd green bell pepper, sliced (optional)\n\n\u00bd C enoki mushrooms (optional)\n\nAbout 24 fresh mint leaves\n\nAbout 24 fresh Thai basil leaves\n\n8 (10-inch) sheets rice paper\n\n2 tsp sesame seeds\n\nMETHOD\n\nCook the rice noodles according to the directions on the package. Rinse in cold water, drain, and set side.\n\nCombine the peanut butter, soy sauce, maple syrup, Sriracha, vinegar, and ginger in a small bowl. Stir to combine. Thin with water if too thick. Set the peanut sauce aside in the fridge.\n\nSet the cucumber, tofu, lettuce, bell pepper (if using), mushrooms (if using), mint leaves, and basil leaves in individual piles.\n\nFill a large bowl or saucepan\u2014something a little bigger than 10 inches in diameter\u2014with lukewarm water. Set up a station with a plate to make the summer rolls on, a plate to put finished rolls on, and all the ingredients within arm's reach.\n\nImmerse a rice paper sheet in the water for about 10 seconds and then place on the rolling plate. It will still be a little hard. This is okay\u2014it will soften on the plate and be perfect for rolling once all the ingredients are on it. Leaving it in the water until completely soft makes it incredibly difficult to work with.\n\nSprinkle some of the sesame seeds onto the rice paper sheet, near one of the edges. Add a few pieces of cucumber, mint leaves, basil leaves, a piece of tofu, a smidge of the noodles, and a piece of lettuce. Add bell pepper and enoki mushrooms, if using. (For your first few, go light on the fillings, as it'll be easier to get the technique down.) Fold the sides over the fillings and then fold the bottom flap up and over the fillings, tuck, and roll like you're rolling a burrito. Repeat until all the rolls are done and enjoy immediately, with the dipping sauce.\n\n**Want to go the extra eco-mile?** To avoid food waste, if you have leftover fillings, make them into a small salad and top with the peanut sauce or even just a splash of rice vinegar.\n\n**Tip:** Never made summer rolls before? Watch a video online of the technique first!\nMINTED GREEN PEA GUACAMOLE\n\nLiving in Texas, my guac bar is set pretty high. I'm not gonna lie, when traveling outside the Lone Star state, I find most guacamole is simply not up to snuff. But as good as the classic avocado-tomato-onion recipe is, sometimes I just want a little somethin' extra in my guac. Enter: minted green pea guacamole. It's like your classic guac, but with mint for a little added freshness and green peas for a little added protein.\n\nSERVES 4 TO 6 PREP AND COOKING TIME: 25 MINUTES\n\nINGREDIENTS\n\n1 Tbsp pine nuts\n\nHandful of fresh cilantro, chopped\n\nHandful of fresh mint leaves, chopped\n\n1 garlic clove, minced\n\n1 C frozen peas\n\n\u2153 C finely chopped red onion\n\n2 large ripe avocados\n\nJuice of 1 lime\n\n\u00bd tsp salt\n\nFreshly ground black pepper\n\nTortilla chips or crudit\u00e9s, for serving\n\nMETHOD\n\nIn a dry pan, toast the pine nuts over medium heat until browned but not burnt.\n\nPlace the toasted pine nuts in a mortar and add the cilantro, mint, and garlic. Mash together with the pestle to make a pesto, then transfer to a large bowl.\n\nPlace the peas in a small bowl and microwave for 60 seconds. Add them to the pesto.\n\nAdd the onion to the bowl, along with the avocado flesh, lime juice, and salt. Mash until somewhere between smooth and chunky, according to your preference.\n\nSeason with the pepper and stir to combine.\n\nGarnish with additional fresh mint and serve immediately, with tortilla chips or crudit\u00e9s.\n\n**Note:** No mortar and pestle? Just mix all the pesto ingredients together, finely chopping the cilantro, mint, and garlic first\u2014or process in a food processor.\nPISTACHIO AND SUNFLOWER SEED DUKKAH\n\nDukkah is an Egyptian condiment made using nuts, seeds, herbs, and spices. Blended into a crumbly powder, dukkah is used for dipping bread into, or for sprinkling atop vegetables. This version uses toasted pistachios and sunflower seeds, along with sesame seeds and spices,to create an aromatic, delicious blend that's great for parties, or just to keep on hand as a tasty addition to any meal.\n\nSERVES 6 TO 8 PREP AND COOKING TIME: 20 MINUTES\n\nINGREDIENTS\n\n\u00bd C shelled pistachios\n\n\u00bd C shelled sunflower seeds\n\n\u00bd C sesame seeds\n\n1 Tbsp coriander seeds\n\n1 Tbsp cumin seeds\n\n1 tsp sea salt\n\n1 tsp black peppercorns\n\nMETHOD\n\nPlace all the ingredients in a dry skillet over medium heat and toast, turning continuously with a spatula until fragrant and the seeds start to make a popping sound, about 5 minutes.\n\nRemove from the heat and let cool for 10 minutes.\n\nTransfer to a food processor or blender. Process in 3- to 5-second intervals, until you have a chunky powder. (Stop and check every few seconds to ensure you don't accidentally turn it into nut butter by overprocessing!)\n\nServe with bread and olive oil (dip the bread in the oil, then the dukkah), or sprinkle on roasted vegetables, salads, pizza, and more. Store leftovers in an airtight container at room temperature.\n\n**Serving Suggestion:** Along with bread and oil, Roasted Red Pepper Hummus makes a great addition.\nPLANT-BASED PARMESAN\n\nNo pasta would be complete without parm. I'm a fiend for the stuff, and like to really heap on big piles of it. Fortunately, you can use nuts and nutritional yeast to replace the fatty, tangy flavors that parmesan lends to Italian cuisine. I dare you not to eat it out of the jar by the spoonful.\n\nMAKES 1\u00bc CUPS PREP AND COOKING TIME: 15 MINUTES\n\nINGREDIENTS\n\n\u00bd C pine nuts\n\n\u00bc C chopped Brazil nuts\n\n\u00bc C shelled sunflower seeds\n\n\u00bc C nutritional yeast\n\n\u00bd tsp garlic powder\n\n1 tsp salt\n\nMETHOD\n\nPlace the pine nuts, Brazil nuts, and sunflower seeds in a dry skillet over medium heat and stir until lightly toasted. Keep an eye on them, as pine nuts can burn quickly. Once golden brown, remove from the heat and let cool.\n\nPlace all the ingredients in a food processor or blender and pulse until the mixture resembles bread crumbs. Spoon into an airtight jar and store in the fridge until ready to use (up to 2 weeks).\n\n**Tip:** Don't have Brazil nuts? Just double the sunflower seeds.\n\n**Equipment Alert:** This recipe requires a food processor or blender.\n\n**Serving Suggestion:** This shines atop pizza (see here and here), the Spaghetti and Lentil Meatballs, the Massaged Kale Caesar salad, and so many other foods.\n\n\"A plant-based diet can and will change your life.\"\n\n\u2014RYAN SEACREST\nQUICK COUNTRY COLESLAW\n\nThis slaw is perfect for any BBQ, picnic, or potluck! The tangy cider vinegar with creamy mayo and sweet agave combine for a balanced slaw just like Mama used to make. And because it uses nondairy milk and egg-free mayo, it's easier on the earth\u2014conserving water and land, and emitting fewer greenhouse gases.\n\nSERVES 8 TO 10 PREP AND COOKING TIME: 15 MINUTES\n\nINGREDIENTS\n\n1\u00bc C vegan mayo\n\n3 Tbsp apple cider vinegar\n\n2 Tbsp plain almond milk\n\n1 tsp sea salt\n\n1 Tbsp sugar\n\n1 tsp celery seed\n\n\u00bd tsp freshly ground black pepper\n\n\u00bc tsp garlic powder\n\n\u00bc tsp onion powder\n\n2 medium carrots\n\n\u00bd head green cabbage\n\n\u00bd head red cabbage\n\nMETHOD\n\nIn a small bowl, whisk or stir together the mayo, vinegar, almond milk, salt, sugar, celery seed, pepper, garlic powder, and onion powder. Set aside.\n\nShred the carrots and cabbages using a grater. Place in a large bowl and mix with clean hands.\n\nPour the dressing on top and mix well to combine.\n\nCover and refrigerate. Enjoy chilled as a side, especially with a Spicy Lentil Burger or any other sandwich.\n\n**Note:** Check out the Slawppy Joe recipe for another tasty way to enjoy this dish!\n\n**Want to go the extra eco-mile?** Next time you make broccoli, save the stems. Shred them in a food processor and then replace the cabbage with them (or add them to the cabbage). It'll cut down on food waste and is delicious!\n\n\"I am a firm believer in eating a full plant-based, whole-food diet.\"\n\n\u2014ARIANA GRANDE\nROASTED RED PEPPER HUMMUS\n\nHummus has become so popular that some farmers in the South have begun switching from tobacco to chickpeas in order to accommodate the demand. In this recipe, roasted red peppers offer a fiery addition, making it ideal as a dip, a spread, or added to any salad.\n\nSERVES 2 TO 4 PREP TIME: 10 MINUTES\n\nINGREDIENTS\n\n1 (14-oz) can chickpeas, drained and rinsed\n\n\u00bd C roasted red peppers\n\n2 garlic cloves, minced\n\n2 Tbsp tahini\n\nJuice of \u00bd lemon\n\n1 Tbsp olive oil, plus more for garnish\n\n\u00bd tsp ground cumin\n\n\u00bd tsp salt\n\nSesame seeds, for garnish\n\n1 Tbsp chopped fresh parsley, for serving\n\nMETHOD\n\nCombine the chickpeas, roasted peppers, garlic, tahini, lemon juice, olive oil, cumin, salt, and a splash of water in a blender or food processor and blend until well combined. Continue blending, adding water a little at a time until the desired consistency is achieved (water is what will make your hummus creamy).\n\nGarnish with sesame seeds, some more olive oil, and the parsley. Store in an airtight container in the fridge for up to 5 days.\n\n**Tip:** If you like your hummus tangier, double the lemon juice.\n\n**Want to go the extra eco-mile?** Save the aquafaba (chickpea juice) for the Chocolate Brownies or the Lemon Meringue Pie\u2014to cut down on food waste.\nROSEMARY CORNBREAD MINI MUFFINS\n\nThese go wonderfully with the Wild Mushroom Chili. Moist and filling, with aromatic rosemary, they'll make the whole house smell great. Bet you can't eat just one!\n\nMAKES 12 MUFFINS PREP AND COOKING TIME: 45 MINUTES\n\nINGREDIENTS\n\n1 C canned coconut milk\n\n1 tsp apple cider vinegar\n\n1 C cornmeal\n\n\u00bd C all-purpose flour\n\n\u00bd tsp baking powder\n\n\u00bd tsp baking soda\n\n\u00bc tsp salt\n\n\u00bd C plus 1 Tbsp olive oil\n\n1 small yellow onion, finely chopped\n\n2 red bell peppers, finely chopped\n\n6 cherry tomatoes, halved\n\nFresh rosemary, for garnish\n\nMETHOD\n\nPreheat the oven to 350\u00b0F. Lightly grease a 12-cup muffin pan with vegetable oil or cooking spray.\n\nIn a small bowl, combine the coconut milk and vinegar and set aside.\n\nCombine the cornmeal, flour, baking powder, baking soda, and salt in a large bowl and set aside.\n\nIn a heavy-bottomed skillet, heat 1 tablespoon of the olive oil over medium heat. Add the onion. Cook for 2 minutes and then add the bell peppers and cook until soft, about 5 minutes more. Remove from the heat and set aside.\n\nAdd the remaining \u00bd cup olive oil to the coconut milk\u2013vinegar mixture. Whisk the wet ingredients into the bowl of dry ingredients. Add the cooked onion\u2013bell pepper mixture, and then stir slightly, to combine.\n\nSpoon into muffin cups. Gently press half a cherry tomato into each top and top with fresh rosemary. Bake for 15 minutes or until a toothpick inserted into the center comes out clean. Let cool for 10 minutes and then enjoy, preferably smothered with margarine.\nSPICY SWEET POTATO FRIES WITH RANCH\n\nSweet potatoes offer all kinds of health benefits over their more common cousins. For this recipe, we take a usually-not-so-healthy favorite and offer a new spin to pack in some extra flavor (and spice) along with better health. So break out the ketchup and eat up.\n\nSERVES 2 PREP AND COOKING TIME: 45 MINUTES\n\nINGREDIENTS\n\n2 medium sweet potatoes\n\n2 Tbsp olive oil\n\n1 tsp salt\n\n\u00bd tsp crushed red pepper\n\n\u00bd C vegan mayo\n\n\u00bc tsp onion powder\n\n\u00bc tsp garlic powder\n\nPinch of freshly ground black pepper\n\n\u00bc C plain almond milk\n\n1 Tbsp nutritional yeast\n\n3 tsp dried parsley\n\nMETHOD\n\nPreheat the oven to 400\u00b0F.\n\nScrub the sweet potatoes. With a sharp knife, cut them into \u00bc-inch-thick fries. Place in a large bowl and coat well with olive oil, salt, and crushed red pepper.\n\nSpread the fries onto a rimless baking sheet. The trick to getting crispy fries is to place them far enough apart, on a baking sheet with no edges so that as they cook all the steam they are releasing can evaporate, rather than being absorbed by surrounding fries. You may need to use two baking sheets, depending on the size of your oven. Bake the fries for 15 minutes.\n\nMeanwhile, combine the mayo, onion powder, garlic powder, black pepper, almond milk, nutritional yeast, and 1\u00bd teaspoons of the dried parsley in a bowl. Stir to combine well and set aside in the fridge.\n\nOnce the fries have baked for 15 minutes, turn them all with a spatula. Return to the oven for another 15 to 20 minutes, depending on how well done you want them.\n\nSprinkle with the remaining 1\u00bd teaspoons dried parsley and serve with the ranch dressing for dipping. Enjoy alone as a snack, or with the Buffalo Cauliflower Wrap or any other sandwich or burger.\n\n**Want to go the extra eco-mile?** Make this when sweet potatoes are in season (see here).\nSWEET-AND-SALTY CURRIED POPCORN\n\nThere's nothing quite like plopping down on the couch with a movie (or three) and a huge bowl of homemade popcorn. This is one of our favorites\u2014salty, sweet, and aromatic, it's perfect for a cozy night in.\n\nSERVES 2 TO 4 PREP AND COOKING TIME: 10 MINUTES\n\nINGREDIENTS\n\n6 Tbsp coconut oil\n\n1 Tbsp curry powder\n\n3\u00bd Tbsp pure maple syrup\n\n1 tsp salt\n\n1 C popcorn kernels\n\nMETHOD\n\nMelt 4 tablespoons of the coconut oil in a small saucepan. Add the curry powder and stir for 1 minute, or until fragrant. Remove from the heat, stir in the maple syrup and salt, and set aside.\n\nIn a large pot with a lid, melt the remaining 2 tablespoons coconut oil over medium-high heat. Add 3 kernels of popcorn and cover. Once those three kernels pop, add all the unpopped kernels and cover. Shake vigorously, using a dish towel to hold the lid in place with your thumbs as you hold the pot by its handles. Place the pot back on the heat and wait for the popping to begin. It will start out slowly and get more and more intense, then slowly die down again. Once you haven't heard a pop for a few seconds, remove from the heat and carefully remove the lid to let the steam escape.\n\nIf you're nervous or unsure about the heat on your stove, you can hold the pot just above the burner, shaking it intermittently, to give it a break from being on the heat. It will take a little longer for all the kernels to pop, but is a good way to prevent burning.\n\nPour the topping over the popcorn, cover, and shake vigorously (or mix in a very large bowl) to combine. Enjoy!\n\n**Alternative Topping:** Not a sweet curry fan? No worries. Instead of curry powder and maple syrup, use 1 tablespoon Old Bay seasoning and 2 tablespoons nutritional yeast in the first stage instead, and use that as your topping.\nDESSERTS\n\nAll-the-Nut-Butters Cups\n\nApple-Pecan Crisp\n\nAvocado Chocolate Mousse with Vanilla Bean Whip\n\nChocolate Brownies with Puffed Quinoa\n\nClassic Pumpkin Pie\n\nCoconut Basbousa\n\nLemon-Ginger Blueberry Cake with Cream Cheese Frosting\n\nLemon Meringue Pie\n\nMinted Lemon-Cucumber Sorbet\n\nMom's Noodle Kugel\n\nPeanut Butter Chocolate Chip Cookie Ice Cream Sandwiches\n\nPeanut Butter Trail Mix Cookies\n\nRaw Chocolate-Hazelnut Truffles\n\nRaw Lemon Vanilla Raspberry Slab\n\nSmoky Maple-Pecan Cheesecake\n\nSpicy Chocolate Milk Shake with Whipped Coconut Cream\nALL-THE-NUT-BUTTERS CUPS\n\nWhy stick with plain peanut butter cups when you could use other nut butters? Almond, cashew, hazelnut\u2014even sunflower butter! And why use milk chocolate when you could make your own super-easy dark chocolate? Mix 'em, match 'em\u2014you can't go wrong!\n\nMAKES TEN 1-INCH CUPS PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n3 oz cocoa butter\n\n\u00bc C pure maple syrup\n\n\u00be C unsweetened cocoa powder\n\nPinch of salt\n\n2 Tbsp confectioners' sugar, plus more if needed\n\n3\u00bd Tbsp nut butter (hazelnut, almond, or peanut)\n\nMETHOD\n\nPlace ten small paper muffin cup liners or chocolate molds on a dinner plate.\n\nMake a DIY double boiler: bring 1 cup water to a boil in a small saucepan. Place a small metal bowl on top (making sure the bottom of the bowl does not touch the water).\n\nPut the cocoa butter in the hot bowl and stir with a silicone spatula; it will melt quickly. Add the maple syrup and stir to combine. Add the cocoa powder and salt and stir until smooth and well combined. Turn the heat off.\n\nSpoon 1 tablespoon of the mixture into each muffin cup or mold. (You'll have leftover chocolate; save it for later to top the cups.) Place the entire plate of muffin cups in the freezer until the chocolate is firm, about 15 minutes.\n\nWhile that's firming up in the freezer, sift the confectioners' sugar into a bowl. Add the nut butter. Roll the mixture into 10 small balls. If they're too wet to roll, add more sugar.\n\nRemove the plate from the freezer and place a nut butter ball in the center of each mold, on top of the chocolate. Press down ever so slightly, being careful not to push it all the way to the edges.\n\nSpoon 1 tablespoon of the remaining chocolate mixture on top of each, and then place back in the freezer until firm, about 15 minutes more. Once set, transfer to an airtight container and store in the fridge until ready to eat (up to 1 week).\nAPPLE-PECAN CRISP\n\nNothing says fall like spending the day picking apples and then enjoying the fruits of your labor with a warm bowl of comforting crisp. And because trees, including fruit trees, soak up CO2 and produce oxygen, they can help offset climate-changing emissions. In fact, even a single-acre apple orchard will extract about fifteen tons of CO2 from the air each year. You know what they say: an apple a day keeps global warming away!\n\nSERVES 4 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\n4 heaping Tbsp margarine, chilled, plus more for the baking dish\n\n10 medium apples (about 3 lb)\n\n2 heaping tsp ground cinnamon\n\n7 heaping Tbsp sugar\n\n1 C all-purpose flour\n\n1 C rolled oats\n\n\u00bd C pecans, coarsely chopped\n\n\u00bd tsp vanilla extract\n\n\u00bd tsp baking powder\n\nPinch of salt\n\nMETHOD\n\nPreheat the oven to 350\u00b0F. Lightly coat a 9 x 13-inch casserole dish with a tiny bit of margarine.\n\nPeel, core, and slice each apple, placing them in a large bowl as you go.\n\nSprinkle the cinnamon and 3 tablespoons of the sugar over the apples. Add 1 tablespoon of the flour and mix well to combine. Pour the apple mixture into the prepared casserole dish.\n\nIn the same bowl, combine the remaining 4 tablespoons sugar, remaining flour, the oats, pecans, vanilla, baking powder, and salt. Stir to combine. Add the chilled margarine. Using clean hands, work the margarine into the mixture until it's evenly distributed. Sprinkle the topping over the apples and bake, uncovered, until the apples are soft, 45 to 55 minutes.\n\nRemove from the oven and let cool for 5 minutes before serving.\n\n**Serving Suggestion:** Top with the Whipped Coconut Cream\n\n**Want to go the extra eco-mile?** Make this when apples are in season\u2014and even try hand picking them yourself at an orchard!\nAVOCADO CHOCOLATE MOUSSE WITH VANILLA BEAN WHIP\n\nAvocado makes a creamy, decadent replacement for heavy cream and eggs in this dish. Combined with maple syrup, vanilla, and cocoa powder, you can't taste the avocado at all\u2014and the texture is spot-on. Plus, it's more sustainable! Pound for pound, eggs take almost four hundred gallons of water to produce and cow's milk takes more than eight hundred gallons\u2014while avocados take only about 140 gallons per pound to grow. That's a huge reduction in water use from just one simple swap. How sweet is that?\n\nSERVES 4 PREP TIME: 15 MINUTES, PLUS TIME TO CHILL\n\nINGREDIENTS\n\n2 ripe avocados\n\n\u00bc C pure maple syrup\n\n3 Tbsp unsweetened cocoa powder, plus more for dusting\n\n\u00bd tsp pure vanilla extract\n\nPinch of salt\n\n1 (5.4-oz) can coconut cream, chilled\n\n2 Tbsp confectioners' sugar\n\n1 whole vanilla bean\n\nFresh raspberries, for garnish (optional)\n\nMETHOD\n\nScoop the flesh out of the avocado and place it in a blender. Add the maple syrup, cocoa powder, vanilla extract, and salt. Puree until thick and smooth. It should be sweet and rich and not taste at all like avocado. If it's too avocado-tasting for you, try adding a little more maple syrup and cocoa.\n\nTransfer to an airtight container and refrigerate for several hours, until cold.\n\nMeanwhile, place the coconut cream in a bowl. Add the confectioners' sugar. Cut the vanilla bean down the middle lengthwise and scrape the seeds into the bowl. Using a handheld mixer, whip until thick.\n\nSpoon the mousse into bowls. Top with whip and garnish with fresh raspberries (if using) and a dusting of cocoa.\n\n**Equipment Alert:** This recipe requires a handheld mixer.\nA MESSAGE FROM\n\n## PRO RACE CAR DRIVER AND ECO-CHAMPION LEILANI M\u00a8UNTER\n\nI became vegetarian (and then vegan) because I love animals and didn't want to see them harmed. It turns out that was also one of the most eco-friendly moves I could have made.\n\nWe know, for example, that communities across the country are suffering through the air pollution caused by mega factory farms. We also know that air pollution is causing global warming. Most people associate their eco-footprint with the kind of car they drive or fuel they burn; when they think of our planet's air and the problems it faces, they may just think of industrial smokestacks and smog.\n\nAt home, I drive a 100-percent electric car that I power from the solar panels on the roof of my house. But it is likely that I am making a more positive impact on our planet and air through eating plant-based foods.\n\nAnd no matter where you fall on the spectrum\u2014vegan, vegetarian, or something else entirely\u2014it's easier than ever to make sustainable choices! Look around at the grocery store and you'll see all kinds of amazing veg-forward foods. Check out menus at your local restaurants and I bet you'll see some great veggie options you'd never even noticed.\n\nAs someone who's gone through the life- (and planet-) transforming decision to focus my diet on plants rather than animals, I promise you'll be better for it! Whether you go all in on plant-based eating or just make an effort to eat plant-based foods more often, you'll discover a great variety of new foods that are satisfying, delicious, easy to prepare, and eco-friendly! That's something that you, the animals, and the planet can all feel good about.\n\n_Leilani M\u00fcnter\u2014a professional race car driver and environmental activist\u2014has been named Discovery Planet's Green number one eco-athlete in the world and was awarded an_ ELLE _magazine Genius Award. She's an outspoken advocate for solar power, electric cars, plant-based diets, and animal protection. An active lobbyist on these issues, Leilani has been a guest at the White House and the United Nations in Geneva. She also uses her race car to get environmental messages in front of America's 75 million race fans: her cars have carried messages about renewable energy, clean energy legislation, and animal protection._\nCHOCOLATE BROWNIES WITH PUFFED QUINOA\n\nPeanut butter and chocolate is my favorite flavor combo. And because this rich brownie recipe uses aquafaba instead of eggs, it's more eco-friendly than your typical brownie. (Aquafaba, after all, is a zero-waste product!) The puffed quinoa on top gives the brownie an extra airy crunch and a little zap of protein.\n\nSERVES 8 PREP AND COOKING TIME: 1 HOUR PLUS COOLING TIME\n\nINGREDIENTS\n\n7 oz dark bittersweet chocolate\n\n2 Tbsp coconut oil\n\n\u00bd tsp pure vanilla extract\n\n3 Tbsp creamy peanut butter\n\n\u00bd C plus 2 Tbsp aquafaba (chickpea juice)\n\n1 C sugar\n\n1 C all-purpose flour\n\n2 Tbsp unsweetened cocoa powder\n\n1 tsp baking powder\n\n\u00bd tsp salt\n\n2 Tbsp puffed quinoa (optional)\n\nMETHOD\n\nPreheat the oven to 350\u00b0F.\n\nMake a DIY double boiler: Bring 1 cup water to a boil in a small saucepan. Place a small metal bowl on top (be sure the bottom of the bowl does not touch the water).\n\nBreak the chocolate into smaller pieces and put them in the bowl, along with the coconut oil, vanilla, and peanut butter. Stir until everything is melted and well combined.\n\nIn the bowl of a stand mixer fitted with the whisk attachment or in a large bowl using a handheld mixer, beat the aquafaba and sugar until thick and creamy and doubled in size, about 5 minutes.\n\nAdd the melted chocolate mixture to the bowl, scraping all the goodness out with a silicone spatula. Beat until just combined, about 30 seconds. Sift the flour, cocoa powder, baking powder, and salt into the bowl. Use a sieve, as cocoa can be very lumpy. Beat again until it's nicely combined, about 30 seconds.\n\nLine a 7 x 11-inch baking pan with parchment paper. Pour the brownie batter in. It will be quite thick. Use a silicone spatula to scrape everything from the bowl into the pan and then smooth it out.\n\nSprinkle the puffed quinoa over the brownies and bake for 30 minutes. Remove and let cool before cutting into 8 squares.\n\n**Tip:** If you don't have puffed quinoa, you can sprinkle some chopped pistachios (or other kind of nuts) on top.\n\n**Equipment Alert:** This recipe requires a stand mixer or handheld mixer.\nCLASSIC PUMPKIN PIE\n\nWith no big slice of pumpkin pie to end the feast, my first Thanksgiving as a vegan was a bit rough. Whether you're fully vegan and want to make sure to end your holidays right or you're simply looking for a healthier, eco-friendlier version of this classic dessert to try out, this recipe is sure to please.\n\nSERVES 6 TO 8 PREP AND COOKING TIME: 2 HOURS PLUS OVERNIGHT COOLING TIME\n\nINGREDIENTS\n\n2 C all-purpose flour, plus more for dusting\n\n2 Tbsp sugar, plus more for sprinkling\n\n\u00bd tsp salt\n\n\u00bc cup very cold margarine\n\n\u00bd C very cold water\n\n1 (15-oz) can pure pumpkin puree\n\n1 C coconut cream\n\n\u00bd C pure maple syrup\n\n3 Tbsp arrowroot powder\n\n2 heaping tsp ground cinnamon\n\n2 heaping tsp ground ginger\n\n\u00bd tsp ground nutmeg\n\n1 tsp pure vanilla extract\n\nMETHOD\n\nPlace the flour, sugar, and \u00bc tsp of the salt in a large bowl. Stir to combine. Add the cold margarine and work it through with clean fingers until the dough resembles bread crumbs. Add the water, 1 tablespoon at a time, kneading after each addition, until the mixture forms into a ball.\n\nLightly dust a clean work surface with flour. Press the dough into a flat, round disk. Cover with an inverted bowl and let sit for 10 minutes.\n\nUncover the dough, sprinkle the top with a little flour, and then roll over it with a rolling pin about four times. Flip it over, sprinkle with more flour, and roll over it again. Repeat until the dough is roughly 12 inches in diameter\u2014large enough to cover the bottom and up the sides of a pie dish.\n\nCarefully transfer the rolled-out dough to a pie dish and press it up the sides. Any overhang can be folded down and any broken bits can be easily pressed back together with your fingers. Prick holes in the bottom with a fork and set aside.\n\nOnce the dough is made, preheat the oven to 375\u00b0F.\n\nPlace the remaining \u00bc teaspoon salt, the pumpkin, coconut cream, maple syrup, arrowroot powder, cinnamon, ginger, nutmeg, and vanilla in a blender or food processor and blend until smooth. Pour into the pie dish and smooth the top. Sprinkle a little sugar on top.\n\nBake for 1 hour.\n\nRemove from the oven and let cool fully on the counter. Cover and refrigerate overnight before slicing and serving.\n\n**Equipment Alert:** This recipe requires a blender or food processor.\nCOCONUT BASBOUSA\n\nBasbousa is a super-sweet Middle Eastern cake made from semolina or farina and soaked in simple syrup. It's so moist, you almost need to eat it with a spoon. Semolina is a by-product of wheat milling, making this sweet treat both tasty and more sustainable. And because this particular recipe uses nondairy ingredients, it cuts down on the cake's water and climate footprints too.\n\nSERVES 8 TO 10 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\n1 C vanilla nondairy yogurt\n\n\u00bd C almond milk\n\n2 C coarse semolina or farina\n\n1 tsp baking powder\n\n2\u00bd C sugar\n\n\u00bd C margarine, melted\n\n\u2153 C shredded coconut, plus more for garnish\n\n1 tsp ground cinnamon\n\nMETHOD\n\nPreheat the oven to 350\u00b0F. Lightly grease a baking dish (9 x 13 inches works well) with margarine.\n\nIn a large bowl, mix together the yogurt, almond milk, semolina, baking powder, and 1 cup of the sugar. Melt the margarine in the microwave in a mug or small bowl and add it to the mixture, stirring to combine. Let sit for 1 minute. Add the coconut and stir until well mixed.\n\nPour the mixture into the prepared baking dish, spreading it evenly. Bake for 30 to 45 minutes. (Check at 30 minutes\u2014it should be yellowish with golden brown edges when done.)\n\nWhile the cake is baking, combine 1\u00bd cup water, the cinnamon, and the remaining 1\u00bd cups sugar in a saucepan. Bring to a boil, stirring until the sugar has dissolved. Once boiling, turn down the heat and simmer until the syrup thickens slightly, 1 to 2 minutes. Set aside to cool.\n\nWhen the cake is done, remove it from the oven and cut it into pieces (leaving it in the dish). Pour the cooled syrup over the top, making sure to cover all parts. Let the syrup-coated cake rest for 20 minutes, allowing the cake to fully absorb the syrup.\n\nTo serve, scoop out the pieces, roughly, with a spatula and stack on a plate, sprinkling with extra coconut as you go. It'll crumble and the pieces will not be uniform (since it's sooooo moist)\u2014don't worry, it's supposed to be like that. Finish it off with some more coconut and eat immediately or chilled.\nLEMON-GINGER BLUEBERRY CAKE WITH CREAM CHEESE FROSTING\n\nThe tartness of fresh lemons, the zing of ginger, and the sweetness of blueberries all shine in this cake. The vegan cream cheese frosting adds a nice creaminess to the cake, without the planetary pitfalls associated with dairy production. It all comes together to form a more sustainable\u2014and highly delectable\u2014dessert.\n\nSERVES 6 TO 8 PREP AND COOKING TIME: 1 HOUR, PLUS COOLING TIME\n\nINGREDIENTS\n\n1 C almond milk\n\n8 Tbsp fresh lemon juice\n\n6\u00bd Tbsp coconut oil, melted\n\n1 C sugar\n\n2\u00bd C all-purpose flour\n\n1 tsp baking powder\n\n1 tsp baking soda\n\n1 tsp ground ginger\n\n1 tsp ground cinnamon\n\n\u00bd tsp salt\n\n1 C fresh or frozen blueberries\n\n1 (8-oz) package vegan cream cheese\n\n\u2153 C margarine, at room temperature\n\n3\u00bd C confectioners' sugar\n\n1 tsp lemon zest\n\n\u00bd tsp pure vanilla extract\n\n\u00bc tsp pure almond extract (optional)\n\nAdditional blueberries, lemon zest, and\/or edible flowers, for garnish (optional)\n\nMETHOD\n\nPreheat the oven to 350\u00b0F. Line a round 12-inch cake pan with parchment paper.\n\nCombine the almond milk and 7 tablespoons of the lemon juice in a medium bowl and set aside for 5 minutes. After 5 minutes, add the melted coconut oil and stir to combine.\n\nIn a large bowl, combine the sugar, flour, baking powder, baking soda, ginger, cinnamon, and salt. Stir to combine. Add the wet mixture, whisking or beating with a handheld mixer as you go. Add the blueberries and fold them into the cake batter with a spatula.\n\nPour the cake batter into the prepared pan. Bake until the top is lightly golden, 45 to 55 minutes. The cake is done when you can stick a toothpick or skewer into the center and it comes out clean. Remove from the oven and let cool.\n\nWhile the cake is baking, place the cream cheese and margarine in a bowl and beat with a handheld mixer until thick and smooth. Add the remaining 1 tablespoon lemon juice, the confectioners' sugar, lemon zest, vanilla, and almond extract (if using). Beat on high until you have a thick and creamy frosting, 3 to 5 minutes. Set aside.\n\nOnce cake is entirely cool, top it with the frosting and additional blueberries, lemon zest, and\/or edible flowers.\n\nIf not eating within a few hours, store in the fridge and bring to room temperature before serving.\nLEMON MERINGUE PIE\n\nThis pie brims with fresh lemon taste. Sweet and tart, it's just like Grandma used to make\u2014almost. Instead of using egg whites, it uses aquafaba\u2014the \"juice\" from canned chickpeas. In 2014, French chef Jo\u00ebl Roessel discovered that this can be whipped into foams that are functionally equivalent to egg whites but healthier. It's also more sustainable: in addition to being a zero-waste food that would otherwise end up down the drain, aquafaba helps us avoid eggs, which means avoiding all the air pollution associated with chicken farming and the climate-changing greenhouse gas emissions associated with feed production. So just like Grandma used to make? Maybe not. Better than Grandma used to make? Definitely. (Sorry, Grandma.)\n\nSERVES 6 TO 8 PREP AND COOKING TIME: 1\u00bd HOURS, PLUS COOLING TIME\n\nINGREDIENTS\n\nFor the crust:\n\n2 C all-purpose flour, plus more for dusting\n\n2 Tbsp sugar\n\n\u215b tsp salt\n\n\u00bd C very cold margarine\n\n\u00bd C ice-cold water\n\nFor the filling:\n\n\u00bd C sugar\n\nZest of 1 lemon\n\n\u2154 C fresh lemon juice\n\n\u2153 C coconut cream\n\n1 Tbsp agar-agar (powder, not flakes)\n\n2 Tbsp arrowroot powder\n\n1 (14-oz) packet silken tofu\n\n\u00bd tsp ground turmeric\n\nFor the meringue topping:\n\n\u00be C aquafaba (chickpea canning liquid)\n\n\u00bd tsp cream of tartar\n\n\u00bd tsp pure vanilla extract\n\n1 C sugar\n\nMETHOD\n\nFirst, make the crust:\n\nPreheat the oven to 350\u00b0F.\n\nPlace the flour, sugar, and salt in a large bowl and stir to combine. Add the cold margarine and work it through with clean fingers until the dough resembles bread crumbs. Add the water, 1 tablespoon at a time, kneading after each addition, until the mixture forms into a ball.\n\nLightly dust a clean work surface with flour. Place the ball of dough on top and press it into a flat, round disk. Cover with an inverted bowl and let sit for 10 minutes.\n\nUncover the dough, sprinkle the top of the disk with a little flour, and then roll it with a rolling pin about four times. Flip the dough over, sprinkle with more flour, and roll over it again. Repeat until the dough is roughly 12 inches in diameter\u2014large enough to cover the bottom of a pie dish and go up the sides.\n\nCarefully transfer to a pie dish and press it up the sides. Remove or fold down any overhang. Prick holes in the bottom with a fork. If you have dried beans or pie weights on hand, line the dough with aluminum foil and fill with a layer of beans or weights about \u00bd inch thick, to prevent it from rising while it bakes. (If you don't have this, no problem.) Bake for 30 minutes. Remove from the oven and let cool.\n\n**Timesaver:** Make the crust and filling the night before and refrigerate, covered, overnight. Then, the next day, make the meringue, bake, cool, and eat.\n\nMeanwhile, make the lemon curd filling:\n\nPlace all the filling ingredients in a food processor or blender and puree until totally smooth. Transfer to a saucepan, place over medium-high heat, and whisk continuously until the mixture has thickened, about 5 minutes. Once thick, pour the filling into the cooled piecrust and let set, at least 15 minutes.\n\nMake the meringue:\n\nIf you've made meringue using eggs, then this is much the same\u2014except you're using aquafaba (the juice from a can of beans) instead of the whites of a chicken egg. Chick _pea_ protein instead of chick _en_ protein.\n\nPour the aquafaba into a large bowl or the bowl of a stand mixer fitted with the whisk attachment. Add the cream of tartar and vanilla and beat on medium-high speed until the mixture holds soft peaks, 5 to 10 minutes. Add the sugar, a little at a time, and beat until the mixture holds stiff peaks. It will take another 10 to 20 minutes of beating, but the end result is well worth your time. If you think you have stiff enough peaks but you aren't sure, hold the bowl upside down; if the meringue stays put, it's ready.\n\nAnd for the grand finale:\n\nSpoon the meringue on top of the pie. Make it about 2 inches thick and ensure it touches all the edges. Make little peaks as you go. You can also pipe the meringue on instead, if you want something fancier.\n\nBake for 5 minutes. Remove and let fully cool, at least 2 hours. Enjoy!\n\n**Tip:** Want the peaks of your meringue a little darker? Increase the oven temperature to 400\u00b0F and place the pie under the broiler for the final minute. Or you can (safely!) blast it with a kitchen blowtorch.\n\n**Equipment Alert:** This recipe requires a stand mixer.\nMINTED LEMON-CUCUMBER SORBET\n\nI scream, you scream, we all scream for... sorbet! In this recipe, fresh mint, tart lemon juice, and that oh-so-subtle cucumber flavor join forces to create a dairy-free dessert that's refreshing and eco-friendly. Add a dash of liquor before freezing for an adult-friendly summer treat.\n\nSERVES 2 PREP TIME: 10 MINUTES, PLUS FREEZING TIME\n\nINGREDIENTS\n\n\u00bd C sugar\n\n1\u00bd Tbsp fresh lemon juice\n\n10 fresh mint leaves\n\n1 small cucumber, peeled and cut into \u00bc-inch chunks\n\nZest of \u00bd lemon\n\n1 tsp white rum or vodka (optional)\n\nMETHOD\n\nPut the sugar and lemon juice in a small saucepan and place over high heat. Cook until the lemon juice is bubbling, then remove from the heat and stir until the sugar has dissolved to create syrup. Add the mint leaves. Stir, then set aside to fully cool.\n\nPlace the cucumber, cooled syrup, lemon zest, and rum (if using) in a food processor or high-speed blender. Pulse until smooth and then pour into a freezer-safe container.\n\nFreeze for at least 4 hours or overnight, stirring once after 2 hours. When it's ready, the sorbet will easily hold its form while being scooped but will never freeze rock-hard, due to the sugar. You can therefore enjoy it as soon as it is ready, or leave it in the freezer for up to 2 weeks.\n\nServe alone, or garnished with slices of cucumber and lemon.\n\n**Want to go the extra eco-mile?** Try growing mint on your windowsill or in your yard to cut down on food miles and packaging.\n\n\"I changed my diet to a vegan diet, really just to experiment to see what it was like. And I felt better, so... I'm likely to continue it for the rest of my life.\"\n\n\u2014AL GORE\nMOM'S NOODLE KUGEL\n\nGrowing up in a mildly Jewish household, we only ate traditional Jewish foods around Passover or Hanukkah. Noodle kugel was always a must-have. Sweet, filling, and creamy, this dessert is one of my favorites. But normally, kugel is made with a laundry list of planet-unfriendly ingredients\u2014like eggs, butter, and cheese. These ingredients require large amounts of water and land, and emit huge quantities of greenhouse gases. (Just the cheese alone emits about 13.5 kg of CO2 per serving!) So this recipe uses tofu instead\u2014a much gentler choice, emitting only 2 kg of CO2 per serving\u2014along with plant-based sour cream and margarine. The first time I tried it out on my mother, you can imagine I was nervous. But she loved it, our guests loved it, and the whole dish was gobbled up. Not one person was able to pass over this tasty, eco-friendly kugel.\n\nSERVES 8 TO 10 PREP AND COOKING TIME: 1 HOUR\n\nINGREDIENTS\n\n8 oz pasta (rotini or spelt ribbons)\n\n1 apple or pear\n\n\u00bd (16-oz) package soft tofu\n\n\u00bc C pure maple syrup\n\n2 tsp pure vanilla extract\n\n1 C vegan sour cream\n\n\u00bc C margarine, melted\n\n\u00bd C sugar\n\n1 tsp ground cinnamon\n\nPinch of ground nutmeg\n\n\u00bc tsp salt\n\n1 C raisins\n\nInstructions\n\nPreheat the oven to 350\u00baF. Lightly grease an 11 x 8-inch casserole dish with margarine.\n\nBring a large pot of salted water to a boil. Cook the pasta in the boiling water according to the directions on the package. Drain, rinse, and set aside.\n\nThinly slice the apple or pear and put it in a large bowl. Finely crumble the tofu into the bowl. Stir in all the remaining ingredients, in the order listed, and mix. The tofu should be mashed and fine by the end. Add the cooked pasta and mix thoroughly to combine.\n\nTransfer the mixture to the prepared baking dish. Bake until the kugel is firm on the inside and the top turns golden brown and begins to crisp, 40 to 45 minutes.\n\nRemove from the oven, let sit for at least 10 minutes, and serve.\n\n**Serving Suggestion:** Kugel is great served cold or at room temperature!\nPEANUT BUTTER CHOCOLATE CHIP COOKIE ICE CREAM SANDWICHES\n\nWhen I first became vegan in the late '90s, these peanut butter chocolate chip cookies were one of the first desserts I learned how to bake. They're rich and creamy, soft and sweet, and total people-pleasers. This recipe combines them with delicious nondairy ice cream for a decadent dessert.\n\nSERVES 3 PREP AND COOKING TIME: 35 MINUTES, PLUS 5 HOURS FREEZING TIME\n\nINGREDIENTS\n\n1\u00bd C all-purpose flour\n\n\u00bd C granulated sugar\n\n\u00bd C brown sugar\n\n\u00be tsp baking soda\n\n1 tsp arrowroot powder\n\n\u00bc tsp salt\n\n1 C peanut butter\n\n\u00bd tsp pure vanilla extract\n\n\u00bc C margarine, room temperature\n\n\u00bd C almond milk\n\n\u00bd (12-oz) bag dark chocolate chips\n\nNondairy ice cream (any flavor you'd like)\n\nMETHOD\n\nOnce you're ready, preheat the oven to 350\u00b0F.\n\nIn a large bowl, mix the flour, granulated sugar, brown sugar, baking soda, arrowroot powder, and salt. Add the peanut butter, vanilla, and margarine. Using clean hands, mix thoroughly. It's worth the mess (using your hands gives it the right consistency). Add the almond milk, a little at a time, until it's not too wet but not too dry. Continue with your hands. Add the chocolate chips and mix some more.\n\nRoll the dough into 1-inch balls and place on a baking sheet with enough space between them so they can spread to be about 3-inch cookies. Flatten them slightly, first with the palm of your hand and then with your fingers.\n\nBake for 15 minutes, being careful to not let the bottoms burn. Remove from the oven and let cool. (They'll still be soft when removed\u2014they'll harden as they cool.)\n\nSandwich as much ice cream between two cookies as you'd like and enjoy immediately.\nA MESSAGE FROM\n\n## GOOD FOOD INSTITUTE EXECUTIVE DIRECTOR BRUCE FRIEDRICH\n\nHumans have the unique ability to improve our world and lives through blending imagination with design. Smartphones allow farmers and textile workers in the developing world to start small businesses and move out of desperate poverty. Modern air travel and the Internet have made travel and information more accessible than before.\n\nToday, each of us can bring that same spirit of innovation and improvement to our dinner plates. And we should. After all, what we choose to eat affects everything\u2014the earth, our water and air, our climate, animals. Just as modern automobiles replaced the horse and buggy, better alternatives can now replace industrial animal agriculture.\n\nWe now have plant-based meats and milks that have the taste and texture of their animal-based counterparts but that are more environmentally sustainable. These products can help us avoid antibiotics, growth hormones, and Salmonella. They can reduce our land and water costs and slash greenhouse gas emissions. They're healthier for us. They're safer. And most important, they taste fantastic.\n\nAdd these types of innovative foods to an abundance of whole grains and legumes, produce, and simple plant-based proteins like tofu and tempeh and the opportunities for better, more sustainable eating are virtually endless. In a world where we all need to eat\u2014and where we all want _what we eat_ to be good\u2014the more we capitalize on those opportunities, the better off we and our planet will be.\n\n_Bruce Friedrich is Executive Director of the Good Food Insti\u00adtute, and has been a leading food and agriculture reformer for over two decades. He holds degrees from Grinnell College, Johns Hopkins University, and the London School of Econom\u00adics._\nPEANUT BUTTER TRAIL MIX COOKIES\n\nThese cookies have all the things. Rolled oats, sunflower seeds, raisins, and peanut butter make for a filling and fun cookie. They are missing one thing, though: eggs. That means they're better for the earth (and, of course, the animals) and have zero cholesterol. Talk about a win-win!\n\nMAKES 10 COOKIES PREP AND COOKING TIME: 30 MINUTES\n\nINGREDIENTS\n\n2 C rolled oats\n\n\u00bd C all-purpose flour, plus more if needed\n\n1 tsp baking powder\n\n1 tsp ground cinnamon\n\n\u2154 C raisins\n\n\u00bc C shelled sunflower seeds\n\nPinch of salt\n\n2 Tbsp coconut oil\n\n5 heaping Tbsp peanut butter\n\n\u00bd C sugar\n\n1 tsp pure vanilla extract\n\n\u00bd C almond milk, plus more if needed\n\nMETHOD\n\nPreheat the oven to 350\u00b0F. Lightly grease a baking sheet with margarine.\n\nCombine oats, flour, baking powder, cinnamon, raisins, sunflower seeds, and salt in a large bowl and mix thoroughly.\n\nPlace the coconut oil, peanut butter, sugar, and vanilla in a medium saucepan and and set over medium heat. Stir until all the ingredients are combined and the sugar has completely dissolved.\n\nPour the warm peanut butter mixture into the bowl of dry ingredients and stir until well combined. Add the almond milk and mix thoroughly. You want it somewhere between wet and dry, and it should hold together. Add about \u00bc cup more almond milk if not wet enough, or more flour if too wet.\n\nUsing wet hands, roll 10 golf ball\u2013size balls of cookie dough, placing them on the prepared baking sheet as you go. Once you have rolled all the cookie dough into balls, press them flat, first with the palm of your hand and then with a fork, so they're about \u00bd inch thick.\n\nBake for 15 minutes. Remove from the oven and let cool completely on the baking sheet.\n\nEnjoy immediately or store in an airtight container.\n\n**Tip:** Chocolate lovers can substitute chocolate chips for half the raisins. Just be sure the mixture has cooled before adding the chocolate, and add them last (after the almond milk).\nRAW CHOCOLATE-HAZELNUT TRUFFLES\n\nEasy, decadent, and the perfect quick chocolate fix\u2014these raw truffles make a great snack.\n\nMAKES 10 TO 20 PREP TIME: 20 MINUTES\n\nINGREDIENTS\n\n10 Medjool dates, pitted\n\n\u00be C hazelnut meal, plus more for rolling\n\n\u2153 C hazelnut butter\n\n\u00bc C unsweetened cocoa powder, plus more for rolling\n\n\u00bd tsp pure vanilla extract\n\nPinch of sea salt\n\nCoconut meal\n\nMETHOD\n\nPlace the dates, hazelnut meal, hazelnut butter, cocoa powder, vanilla, salt, and 1 tablespoon water in a food processor and pulse until the dates are broken down. Process until everything is sticky and you can easily form a ball from the mixture. If your dates were a little dry and things aren't coming together so easily, add an extra tablespoon of water and process some more.\n\nUsing wet hands, roll the mixture into balls. You can go as big or as small as you like, but I find that 12 balls of about 1 ounce each are the perfect size. Roll the truffles in coconut meal, cocoa powder, or hazelnut meal (or mix and match). Store in an airtight container in the fridge until ready to enjoy (up to 1 week).\n\n**Tips:** Bob's Red Mill hazelnut meal is sold in the baking section of many grocery stores. Can't find hazelnut butter or meal? Almond butter and almond meal work well as replacements.\nRAW LEMON VANILLA RASPBERRY SLAB\n\nThis delicious dessert is made by layering flavorful cashew cream and berries on top of an almond-date base. It uses coconut milk and nuts instead of cow's milk for a lighter eco-footprint, and lemon zest and juice for a little tartness. It's creamy and sweet, tart and tasty.\n\nSERVES 8 PREP TIME: 20 MINUTES, PLUS TIME TO SET\n\nINGREDIENTS\n\n15 Medjool dates, pitted\n\n1 C raw almonds\n\n1 C shredded coconut\n\nZest of 1 lemon\n\n2 pinches of salt\n\n4 C fresh raspberries\n\n3 C raw cashews, soaked in water to cover for 4 hours\n\n1 C canned coconut milk\n\n\u00bd C agave nectar\n\n\u00bc C fresh lemon juice\n\n1\u00bd tsp pure vanilla extract\n\n\u00bd C coconut flakes, for garnish\n\nMETHOD\n\nLine an 8-inch square baking pan with parchment paper. (This will make it easier to remove the slab later.) The best way is to cut the paper to 10 inches wide, so it goes up and over two sides of the pan and you can use the excess parchment hanging over to lift the slab out once it's set.\n\nPlace the dates, almonds, shredded coconut, lemon zest, and a pinch of salt in a food processor. Pulse until the almonds are finely ground and the mixture holds together. Press the mixture into the bottom of the prepared pan and smooth the top. Place 1\u00bd cups of the raspberries on top of the base.\n\nDrain and rinse the cashews and place them in a food processor or blender, along with the coconut milk, agave, lemon juice, vanilla, and remaining pinch of salt. Process until smooth, 2 to 4 minutes.\n\nPour half the vanilla-cashew filling on top of the raspberries in the baking pan. Add 1\u00bd cups of the berries to the remaining filling in the food processor and process until well combined. Pour this over the vanilla layer. Top with the remaining berries and the coconut flakes. Freeze until set, at least 4 hours, though overnight is preferable.\n\nRemove from the freezer 10 minutes before serving. To remove from the dish, run a warm, wet knife along the sides that are in direct contact with the dish and then lift it out using the paper overhang on the other edges.\n\nHeat up a sharp knife by running hot water over it and cut the slab into 8 pieces. Enjoy immediately and keep leftovers refrigerated.\n\n**Equipment Alert:** This recipe requires a food processor.\nSMOKY MAPLE-PECAN CHEESECAKE\n\nThis sultry pecan cheesecake recipe combines the richness of coconut milk and the sweetness of maple syrup along with smoked pecans to deliver a creamy, unique, lip-smacking dessert. Because it uses tofu (instead of eggs), it's lighter on our waistlines and the planet.\n\nSERVES 6 TO 8 PREP AND COOKING TIME: 1\u00bd HOURS\n\nINGREDIENTS\n\n1 C plus 1 Tbsp pure maple syrup\n\n1 (13.5-oz) can coconut milk\n\n\u00be box of graham crackers (about 18 cracker sheets)\n\n\u00bc C sugar\n\n\u00be C margarine, melted\n\n1 (14-oz) package silken tofu\n\n\u2153 C all-purpose flour\n\n1 tsp pure vanilla extract\n\nPinch of salt\n\n2 C chopped pecans\n\n\u00bc tsp liquid smoke\n\nMETHOD\n\nPreheat the oven to 425\u00b0F.\n\nPlace 1 cup of the maple syrup and the coconut milk in a saucepan. Bring to a boil, lower the heat to maintain a simmer, and cook, whisking occasionally, for 15 minutes. Remove from the heat and place in the fridge to cool a bit.\n\nUsing a food processor, turn the graham crackers into fine crumbs. Place them in a bowl. Add the sugar and melted margarine. Stir until the ingredients are well combined and the cracker crumbs are soaked. Press into a 9-inch round cake pan or glass pie dish, working the mixture up the sides. Set aside in the fridge.\n\nRinse the food processor. Place the slightly cooled coconut milk\u2013maple syrup mixture into it. Add the tofu, flour, vanilla, and salt. Blend until smooth and well combined. Add 1 cup of the pecans and pulse to combine.\n\nPour that mixture into the chilled piecrust and bake for 10 minutes.\n\nRemove the cheesecake after 10 minutes and lower the oven temperature to 350\u00b0F.\n\nPlace the remaining 1 cup pecans in a bowl. Add the remaining 1 tablespoon maple syrup and the liquid smoke. Stir to coat, then sprinkle over the cheesecake. Bake for 40 minutes more.\n\nRemove from the oven and let cool. Cover and refrigerate for a few hours or up to overnight before serving.\n\n**Note:** Don't want this smoky? Just leave the liquid smoke out of the recipe.\nSPICY CHOCOLATE MILK SHAKE WITH WHIPPED COCONUT CREAM\n\nSpice up your chocolate with a little kick of cayenne. This spicy milk shake with whipped coconut cream is a decadent dessert that bites back. And since the recipe uses almond milk and nondairy ice cream for the shake, and coconut cream instead of dairy cream for the whipped topping, it has less of a water and climate footprint than a standard milk shake. Moo-ve over dairy, there's a new shake in town!\n\nSERVES 2 PREP TIME: 10 MINUTES\n\nINGREDIENTS\n\nFor the whipped cream:\n\n1 (5.4-oz) can coconut cream, chilled\n\n\u00bc C confectioners' sugar\n\n\u00bc tsp pure vanilla extract\n\nFor the milk shake:\n\n\u00be C almond milk\n\n2 C vegan chocolate ice cream\n\n2 Tbsp unsweetened cocoa powder\n\n\u215c tsp ground nutmeg\n\nTiny pinch of salt\n\nPinch of cayenne pepper (optional)\n\nPinch of ground cinnamon, for garnish (optional)\n\nMETHOD\n\nFirst, make the whipped cream:\n\nScoop the chilled coconut cream into a bowl, discarding the little bit of liquid left in the can. Add the confectioners' sugar and vanilla. Using a handheld mixer, whip until thick. Set aside in the fridge.\n\nNow for the shake:\n\nPlace all the milk shake ingredients in a blender and blend until smooth.\n\nPour the milk shake into glasses and top with the whipped cream and a pinch of cinnamon (if using). Enjoy!\n\n\"We should all be... focusing on enjoying more plant-based delights.\"\n\n\u2014JAMIE OLIVER\nA MESSAGE FROM\n\n## CHEF JOS\u00c9 ANDR\u00c9S\n\nOn January 12, 2010, a massive 7.0-magnitude earthquake hit Haiti, just sixteen miles west of the country's capital city of Port-au-Prince. Over the next two weeks, fifty-two major aftershocks had been recorded, with an estimated three million people affected by the quake.\n\nShortly after, I traveled to Haiti to assist in relief efforts. I was struck to see that the grinding poverty Haitians live with day-to-day had been exacerbated by dirty cooking conditions in overcrowded and unsafe tent cities. And I was also struck to learn that the problem extends far beyond Haiti.\n\nEvery day, all across the world, nearly three billion people eat food prepared using crude cookstoves or open fires in homes with poor or no ventilation. Exposure to this smoke can cause illnesses including pneumonia, lung disease, and cancer\u2014killing an estimated two million people annually. This problem has been categorized by the World Health Organization as the fifth biggest health risk in developing countries.\n\nThe fact is that how we cook and eat food can have major social implications\u2014on our own health, the health of our communities, and the health of the planet. Whether it's illness caused by unsafe cooking methods, or heart disease and climate change caused by diets that focus too little on plant-based foods, we owe it to ourselves to make sure that everyone the world over has access to healthy, safely prepared, sustainably produced food\u2014along with a knowledge about how their food choices impact the world.\n\nIn 2013, I was honored to join The United Nations Foundation's Global Alliance for Clean Cookstoves as Culinary Ambassador, helping to raise awareness about the death, sickness, and injury caused by toxic smoke from cooking stoves in developing countries. I formed World Central Kitchen to provide innovative solutions to alleviate hunger throughout the developing world, including the deployment of clean and innovative cooking solutions, because for too many women and children in Haiti and elsewhere, the danger and difficulty of cooking has reached a crisis point. And I've partnered with former U.S. Secretary of State Hillary Clinton, Global Ambassador Julia Roberts, and many others to ensure that this issue is on the agenda of advocates, chefs, business owners, and policymakers worldwide.\n\nI've also worked to ensure that my restaurants serve an array of healthy, sustainable, plant-based dishes that everyone can enjoy. My Washington, D.C., restaurant Beefsteak, for example, serves mostly plant-based dishes\u2014putting meat as a side-of-the-plate extra rather than the centerpiece.\n\nWhenever we eat, we make choices. Those choices are wrapped up in family and tradition; they're wrapped up in taste and mood. And whether we think about it or not, they're also wrapped up in what kind of global citizen we want to be. Across the world, millions of people have little or no real choice about how they eat\u2014about how much food they have, how they prepare it, how it's produced. Those of us who do have a choice share in the responsibility to make the right choices. Fortunately, countless people are doing just that, by choosing healthier, safer, more sustainably produced meals\u2014creating a better, cleaner world with every bite.\n\n_Named one of TIME 's \"100 Most Influential People\" and awarded \"Outstanding Chef\" by the James Beard Foundation, Jos\u00e9 Andr\u00e9s is an internationally recognized culinary innovator, author, educator, television personality, humanitarian, and chef\/owner of ThinkFoodGroup. He is the only chef globally who has both a two-star Michelin restaurant and four Bib Gourmands. Andr\u00e9s is also a committed advocate for food and hunger issues and is known for championing the role of chefs in the national debate on food policy. His awards and distinctions include the 2017 Lifetime Achievement Award from International Association of Culinary Professionals, the 2015 National Humanities Medal, and the Congressional Hispanic Caucus Institute's Chair's Medallion Award._\nSHOPPING FOR A BETTER WORLD\n\nAdjusting to (and sticking with) any diet can be tough if you don't have the right foods handy. By keeping a well-stocked pantry (and fridge) full of protein-packed, plant-based staples, you can ensure success in your effort to easily enjoy eco-friendlier fare.\n\nIf you're vegetarian or vegan, these plan(e)t-based pantry items will probably be your everyday staples. If you're a meat, dairy, and egg reducer, at least keep some of these items on deck to help mix up your normal culinary repertoire and ensure success in your efforts to help the planet with every plate. Happy shopping!\n\nPROTEIN 101\n\nWe all want well-balanced meals that are as filling as they are healthy. While Americans, on average, _over_ consume protein, there's no doubt that combined with a healthy helping of fiber, it helps us feel satisfied after any meal. If you want to mix things up from meat, try packing a protein punch with any of these whole, natural foods.\n\n**Tofu:** Tofu is made essentially the same way cheese is\u2014except from soybeans instead of milk. Reports suggest that Benjamin Franklin introduced it to America in the eighteenth century, after sending it home from one of his many trips abroad. Here's a hint, though: don't eat it raw! Just as you wouldn't like the taste of raw, unflavored chicken, many people dislike the taste of raw tofu. Cut it up and pop it in any stir-fry, though, and it's great. Broil it on a baking sheet and it puffs up and turns golden brown. Scramble it, mash it, slather it in BBQ sauce and grill it\u2014there are so many delicious ways to prepare tofu!\n\n**Tempeh:** Tempeh is similar to tofu in that it's typically made from soybeans as well\u2014though instead of being cultured and pressed, like cheese, they're fermented and formed into blocks. It's got a meaty, dense texture and cooks well in almost any format. Try marinating it before cooking it. Tempeh's got a potent taste, and a nice marinade will help it soak up your favorite flavors.\n\n**Seitan:** This is a magically meaty product made from vital wheat gluten, wheat's primary protein. You can buy it in a plastic tub (like tofu) or vacuum-packed in the refrigerated section of some stores, or make it at home with relative ease. And it cooks up just like meat, too, offering a hearty texture and taking on whatever flavors it's cooked with.\n\n**Beans and Legumes:** Enough can't be said about beans and legumes. Black beans, pinto beans, garbanzos (chickpeas), kidney beans, white beans, lentils of all colors and types, and more\u2014these are some of the most protein-rich and diverse foods on the planet. You can certainly buy any kind of bean or legume in a can. Want to try your hand at home preparation, though? Buy 'em dried and cook them yourself. Some (like lentils) cook as quickly and as easily as rice. Dried beans, on the other hand, take longer (overnight soaking plus a few hours of cooking)\u2014but preparing dried beans at home really brings out their taste in a big way, and gives you more control over their texture.\n\nTOFU 101\n\nIt's versatile, delicious, and protein-rich. But let's face it: to the uninitiated, tofu can be a little scary. In the same way you wouldn't eat an entirely unseasoned chicken breast, you probably won't want to eat raw tofu. And just in the same way raw chicken needs to soak up flavors to taste good, so does tofu. Same principle, different protein. Don't worry! Here are some tips to help ensure a positive tofu experience:\n\n**Press it!** Pressing your tofu before using it helps it firm up and also improves its ability to take on whatever flavors you're cooking it in. To press tofu, open the packet and pour the water down the drain. Wrap the block of tofu in a clean dish towel or some paper towels. Next, place it on a large plate or tray. Place something flat and heavy on top of it\u2014like a cutting board, heavy skillet, or another plate. (If what you put on top of it isn't that heavy, put something heavy\u2014like a jar of beans or jug of water\u2014on top of that.) Leave it there for at least 10 minutes, or even up to a few hours. If you are working with a 16-oz block of tofu, cut it in half crosswise through the middle first.\n\n**Freeze it!** Freezing your tofu before using it gives it a whole new texture: it makes it firmer and meatier. To freeze your tofu, just take the whole package and stick it in the freezer. Before you use it, thaw the tofu on your counter for several hours. Then once thawed, press a little of the excess water out with clean hands, slice, cook, and enjoy.\n\n**Broil it!** Broiling tofu before you use it makes it puff out and become nice and golden brown, almost like it's deep-fried (but not). To broil your tofu, cut the block into squares (however big or small you want). Place them on a baking sheet and broil in the oven for 15 minutes or until golden brown. Remove from the oven, let cool, then scrape the tofu off the baking sheet with a spatula. From here, you can put the tofu in salads as-is, pop it in some sauce for a stir fry, put it on a sandwich, and so much more. It's like a much healthier (and so much easier) version of deep-frying.\n\nPLANT-BASED \"MEATS\"\n\nMany people trying to eat more eco-friendly foods start by simply reducing meat and replacing it with the plant-based versions of their favorite items. It used to be that a veggie burger was one of your only options; today, we find a huge range of plant-based meats at grocery stores in virtually every town and city. Of course, like all packaged foods, they have a higher eco-footprint than do whole foods; but from an environmental standpoint, compared to their meaty counterparts, they're the clear winner. So if you want some ready-made options on hand for busy weeknight meals, these are a solid choice.\n\n**Chicken:** Gardein offers perhaps the best \"chicken\" tenders on the market (plus all kinds of other bird-free poultry). Try Beyond Meat's chicken strips (and other products)\u2014they have more protein than chicken, and are prepared exactly the same way. Morningstar Farms, Boca, store brand products by Target and Whole Foods, and Trader Joe's offer nuggets and strips as well as seasoned, marinated chicken-free bites. The list of plant-based poultry goes on and on.\n\nA MEAL FOR ALL SEASONS\n\nIn addition to eating more whole, plant-based meals, we can lessen our environmental _food_ -print by enjoying more seasonal produce, too. Eating seasonally helps cut down on food miles, as produce likely travels a shorter distance to reach your local market when it's in season. Of course, what's in season will change depending on where you live, but as you peruse the recipes, use this as a general guide to help you get started.\n\n**Burgers:** Beyond Meat's pre-made burgers and pre-formed patties (which look, taste, and cook just like ground beef) are best sellers. Gardein's beefless burgers are hard to beat, too. And then there are classics like Boca's Vegan Original, as well as burgers packed with whole grains and vegetables, like Sunshine Burgers and Dr. Praeger's. Find the brand(s) you like best!\n\n**Faux fish:** Okay, so there's not a ton of fish-free seafood out there. But Gardein makes a stellar frozen, breaded fish fillet that keeps forever in the freezer and tastes incredible. Top it with some lemon juice or a plant-based tartar sauce and you won't know the difference!\n\n**Pork without the pig:** Tofu Pups and Smart Dogs are two veggie dog brands I like. Field Roast and Tofurky both make an incredible variety of sausages made entirely from plant-based ingredients\u2014creole-flavored, apple sage, spicy chipotle, you name it! Gardein makes Sweet and Sour Porkless Bites and BBQ pulled pork from plant-based ingredients. These items are usually found in the produce sections of grocery stores (by the tofu) and in the freezer section.\n\n**Breakfast meats:** Sweet Earth makes a vegan bacon that will blow any carnivore's mind, and Lightlife Smart Bacon is another popular brand. Field Roast makes breakfast links as well that are sweet and filled with umami flavor. There are all kinds of other plant-based breakfast meats out there\u2014give them a try and see which ones you like best.\n\n**Deli meats:** Where tofu is sold, you'll often find a huge array of plant-based deli meats: bologna, ham, turkey slices of all flavors and brands, pepperoni, and much, much more. Want a sandwich in a cinch? Keep a couple of these in your fridge.\n\nFAT = FLAVOR\n\nAnimal-based fats tend to be harder on the earth than their plant-based counterparts, because of the inefficiencies associated with factory farming. On the other hand, there are loads of healthy, plant-based fats that are easier on our bodies and gentler on the earth. Try reducing your use of animal-filled fats like butter and lard by swapping for these eco-friendlier fats.\n\n**Avocado:** Avocado is a fantastic source of healthy fat. You can blend it into sauces and dressings, and even incorporate it into desserts! It's also excellent as a cheese replacement when you want something fatty, tasty, and creamy that isn't dairy on a sandwich or burger.\n\n**Oils:** Olive oil: We know it. We love it. It's a classic. You can cook virtually anything with it, and it adds a richness and flavor that's fatty and decadent and delicious. But have you tried coconut oil? It's more eco-friendly than animal fat and has a nice, soft taste; it also cooks incredibly well. Try it in place of butter or vegetable oil when cooking. It's solid when cold and liquid when warm, so keep that in mind for storing. Oh, and what about toasted sesame oil? Adds characteristic flavor to any Asian dish\u2014either as a dressing or cooking oil!\n\n**Margarine:** Try Earth Balance margarine in place of butter once in a while. It's not (by any means) the only plant-based butter on the market, but is likely the most popular. Spreads well, melts perfectly.\n\nWHAT'S THE DEAL WITH PALM OIL?\n\nPalm oil is a significant cause of deforestation, especially in Malaysia and Indonesia, where huge tracts of forests and carbon-rich swamps are being destroyed for palm plantations. This emits massive amounts of greenhouse gases and threatens biodiversity\u2014with orangutans, tigers, rhinoceros, and elephants being killed or displaced by the process.\n\n**Mayo:** Just Mayo by Hampton Creek has become one of the most popular brands of plant-based mayo\u2014and one of the most affordable. Even the world's largest food-service company, serving millions of meals a day (at universities, ballparks, and more), now uses Just Mayo for 100 percent of its mayonnaise needs nationwide! Plus, it's available everywhere\u2014even the Dollar Tree! Vegenaise is another widely available brand. Even the ubiquitous Hellmann's and Best Foods brands make plant-based mayo, branded as \"Carefully Crafted.\"\n\nMOO-FREE MILKS\n\nCountless people are now drinking nondairy milks. Try a variety of them and see which you like best. You'll find they vary in taste and consistency\u2014that you prefer one brand over another, or one type for baking and another in your coffee (just like you wouldn't bake with half-and-half or drink heavy cream). Here's a short list to get you started:\n\n**Soy creamer:** Gone are the days of putting regular ol' soy milk into coffee. Now those trying to make their morning joe healthier and more environmentally friendly have all kinds of brands (and flavors!) of soy-based creamers to choose from.\n\n**Soy milk:** Vanilla, chocolate, plain, unsweetened, sweetened\u2014there are so many types available now. Find the kind you like best. For baking, unsweetened soy milk is best; for drinking or putting on cereal, try vanilla.\n\n**Nut milk:** Nut milks are gaining in popularity in a huge way. In fact, almond milk is one of the fastest-growing beverages in the marketplace. You can even get almond milk in any espresso drink at Dunkin' Donuts nationwide, and most coffee shops now offer it, too. It's a reliable option, and far more planet-friendly than cow's milk!\n\nCOW-FREE CHEESE\n\nAside from vegans, countless people who continue eating cheese in moderation are now mixing things up with the amazing new plant-based cheeses\u2014cheeses made from almonds and cashews and coconut and tapioca and all kinds of other delicious products that are creamy and fatty and spread well and melt well... all while being more eco-friendly to produce. They're available nationwide, and there are always more coming out in the market. By the time you read this, I bet a new plant-based cheese will have hit the shelves!\n\n**Miyoko's:** Miyoko's Creamery makes higher-end vegan cheeses. Country-style herbes de Provence, aged English sharp farmhouse, high Sierra rustic alpine, and more. They're nutty and incredible.\n\n**Field Roast Chao:** Field Roast, known for their plant-based meats, dove into the cheese world with a major splash, taking coconut oil and other ingredients and turning them into slices under the Chao brand name. From tomato cayenne to creamy original, these cheese slices are sure to please.\n\n**Follow Your Heart:** For years, Follow Your Heart was known as the maker of Vegenaise. More recently, it's begun making plant-based cheese products that will knock your socks off. Want a block of cheddar without the eco-harm that comes with industrial dairy production? Follow Your Heart's got your back. Provolone slices, mozzarella shreds, and much more, these dairy-free delights are not to be missed.\n\n**Kite Hill:** Renowned plant-based chef Tal Ronnen, of Crossroads Kitchen fame, helped create Kite Hill, bringing what some call the most incredible vegan cheeses in history to the market. They make artisanal delicacies like almond milk ricotta and soft, fresh truffle dill and chive cheese, as well as a variety of schmears, like herbed cream cheese.\n\n**Daiya:** Daiya's cheese uses an innovative ingredient\u2014tapioca\u2014to help it melt and stretch just like dairy cheese. Their cheddar, pepperjack, and mozzarella shreds are especially good on pizzas and for mac 'n' cheese.\n\n**Nutritional yeast:** Okay, this is not a cheese\u2014technically. It comes as a powder or as flakes. But it has a really cheesy, nutty flavor that can't be beat. And it's a whole food, so its eco-footprint is low. You can sprinkle it on popcorn or pasta like Parmesan, or make decadent plant-based mac 'n' cheese with it. It has a million and one uses.\n\nESCHEWING EGGS\n\nWant to try egg-free baking? How about a breakfast scramble? Producing eggs is extremely resource-intensive when it comes to protein production, so keep some of these items on deck to use (at least) once in a while if you want to lower your ecological footprint.\n\n**The Vegan Egg:** From Follow Your Heart, the Vegan Egg comes as a powder that, when mixed with water, scrambles well for breakfast and can be used in any baked good. A quick note, though: if scrambling, try adding some additional flavor\u2014like pepper, sage, or rosemary\u2014to really make it pop.\n\n**Egg replacer:** Ener-G and Bob's Red Mill brand egg replacers are classic pantry items to replace eggs. Just mix with water to a certain ratio, and you can whip up cookies, cakes and other baked goods that taste and feel just like they have eggs, but are eco-friendlier and healthier.\n\n**Applesauce:** If you're looking for a whole-foods substitute for eggs in baked goods, applesauce is your best friend. Try substituting \u00bc cup of it for each egg called for in a recipe for sweet, baked desserts. Whodathunkit?!\n\n**Flaxseeds:** Full of omega-3 fatty acids, flaxseeds are a total power food. When ground up into a powder and mixed with water, they make a great egg replacer.\n\n**Aquafaba:** Aquafaba is a relatively new discovery that's taking the culinary world by storm. You basically take the liquid from a can of chickpeas and whip it for a long time to turn it into a poufy, fluffy, white substance that mimics whipped egg whites perfectly. You can even bake it into meringue! It sounds weird, but Google it and watch some videos\u2014you'll be amazed. Plus, because it's made from an ingredient that would otherwise go down the drain, it's extra eco-friendly!\n\n**Chickpea flour:** Also called garbanzo flour, gram flour, or _besan_ , chickpea flour is full of protein and can be used to mimic eggs in many recipes (like the Rise 'n' Shine Breakfast Sandwiches).\n\nGO NUTS\n\nEver tried nuts blended into a creamy sauce? Well, actually, you probably have\u2014if you've had peanut sauce at a Thai restaurant. Other nuts can also be blended into creamy deliciousness. Cashews and almonds work especially well for soup bases and dressings (soak them overnight or for a few hours and blend them up with water). Pine nuts make a rich addition to any pasta dish or salad. Then there are peanuts and macadamia nuts and pistachios and pecans and walnuts and all kinds of others. Keep a bunch in your pantry and, well, go nuts with them!\n\nGRAINS GALORE AND SEEDS-A-PLENTY\n\nWhite rice is certainly fine, but want something more nutritious and flavorful that'll pack even more protein, flavor, and other goodness into your meat-free meals? Try quinoa. It's a tiny seed that's huge on protein, full of flavor, and cooks in a jiff. Millet, barley, couscous, amaranth, brown rice, wild rice, farina, kamut\u2014these will leave you satisfied and full as you embark on eco-friendlier eating.\n\n# ACKNOWLEDGMENTS\n\nThank you to Josh Balk, Paul Shapiro, and Kristie Middleton. Thank you also to Alex Hodgkins, with whom I started my plant-based journey so many years ago. And to Michelle Geiss, my dear friend and focus group guru, as well as Caitlin Hayes, whose early edits on this book were invaluable. Thank you also, of course, to my father, Parker.\n\nThank you to Sally Ekus, and everyone else at The Lisa Ekus Group. And to Whitney Frick, Kara Rota, and the team at Flatiron Books. Without them, this book wouldn't exist.\n\nThank you to my culinary captain and photographer, Jessica Prescott. Her vision for the food and book helped make this what it is. And thanks to Amy Sly, for her on-point design.\n\nThank you to Staub, Vitamix, and Mauviel for the equipment and to Nouri Zarrugh for the delicious basbousa recipe.\n\nThanks to all those who contributed essays, blurbs, and photographs.\n\nThank you to my team of recipe testers: Ali Crumpacker, Amy Webster, Andy Cook, Ann Herbert, Ben Peterson, Bruce Mallory, Eddie Garza, Gabe Wigtil, Hillary Prescott (hey, sis!), Janet Prescott (hi, Mom!), Jim Reilly, Josie Morris, Joyana Hunt (also for the edits!), Justin Leonard, Kamber Sherrod, Karla Dumas, Karla Goodson, Karla Waples, Katie Scott, Lauren Pitts, Marsha Filion, Nicole Nuss, Paul Petersan, PJ Smith, Regan Karlson, Sam Hagio, Terry Hagio, Veronica Martin, and Wanda White.\n\nAnd finally, thank you to those who helped in other ways: Bernie Unti, Chad Sarno, Cody Carlson, Deb Olin Unferth, Erik Olson, Geoff Orme Evans, Janet Friesen, Jill Soffer, Joe Meloy, Jonathan Kaplan, Karen Bouris, Kat Clark, Lisa Inzerillo, Rachel Querry, Suzanne Barnard, Terri Depaolo, and anyone else I've forgotten here.\n\n# CITATIONS\n\n*Please note some of the links referenced through this work may no longer be active.\n\n.Lapp\u00e9, Frances Moore. _Diet for a Small Planet: 20th Anniversary Edition_. New York: Random House, 1991. Print. Page 8.\n\n.Fryar, Cheryl, Carroll, Margaret, and Ogden, Cynthia. _Prevalence of Overweight, Obesity, and Extreme Obesity Among Adults: United States, 1960\u20131962 Through 2011\u20132012_. The Centers for Disease Control and Prevention, 19 Sept. 2014, https:\/\/www.cdc.gov\/nchs\/data\/hestat\/obesity_adult_11_12\/obesity_adult_11_12.htm. Accessed 30 June 2017.\n\n.\"Childhood Obesity Facts.\" _The Centers for Disease Control and Prevention_ , 10 April 2017, <https:\/\/www.cdc.gov\/obesity\/data\/childhood.html>, Accessed 30 June 2017.\n\n.Cawley, John, Meyerhoefer, Chad. \"The medical care costs of obesity: An instrumental variables approach.\" _Journal of Health Economics_ , 2012; 31 (1): 219.\n\n.United States. USDA and U.S. Department of Health and Human Services. _Scientific Report of the 2015 Dietary Guidelines Advisory Committee: Advisory Report to the Secretary of Health and Human Service and the Secretary of Agriculture_. February 2015. Web. Accessed 6 May 2016. Available at <http:\/\/health.gov\/dietaryguidelines\/2015-scientific-report\/PDFs\/Scientific-Report-of-the-2015-Dietary-Guidelines-Advisory-Committee.pdf>. Accessed 30 June 2017.\n\n.\"Gazpacho.\" The Humane Society of the United States, n.d. Available at <http:\/\/www.humanesociety.org\/issues\/eating\/recipes\/soups\/gazpacho_aha.html?credit=web>. Accessed 30 June 2017.\n\n.\"AICR, The China Study, and Forks Over Knives.\" The American Institute for Cancer Research. n.d. Available at <https:\/\/web.archive.org\/web\/20130716220718\/http:\/www.aicr.org\/about\/advocacy\/the-china-study.html>.\n\n.\"The Hidden Epidemic: Heart Disease in America.\" _PBS_. 2007.\n\n.\"Fast Facts: Data and Statistics about Diabetes.\" _American Diabetes Association_ , Dec 2015. <http:\/\/professional.diabetes.org\/sites\/professional.diabetes.org\/files\/media\/fast_facts_12-2015a.pdf>. Accessed 30 June 2017.\n\n.Ibid.\n\n.Anderson, J. W. & Ward, K.: \"High carbohydrate, high-fiber diets for insulin-treated men with diabetes mellitus.\" _American Journal of Clinical Nutrition_ , 32: 2312, 1979.\n\n.Based on calculations from <http:\/\/www.fao.org\/faostat\/en\/#data>.\n\n.\"Farm Animal Statistics: Slaughter Totals.\" The Humane Society of the United States. 25 June 2015. Web. Accessed 6 May 2016. Available at www.humanesociety.org\/news\/resources\/research\/stats_slaughter_totals.html.\n\n.Dimitri, C., Effland, A., Conklin, N. _The 20th Century Transformation of U.S. Agriculture and Farm Policy_. United States Department of Agriculture Economic Research Service. Economic Information Bulletin Number 3. Page 5, Figure 3. June 2005.\n\n.Sauven, John. \"Why meat eaters should think much more about soil.\" _The Guardian._ 16 May 2017. Available at <https:\/\/www.theguardian.com\/commentisfree\/2017\/may\/16\/meat-eaters-soil-degradation-over-grazing>. Accessed 30 June 2017.\n\n.\"Fish Count Estimates.\" _FishCount.org.uk_ , 2014, <http:\/\/fishcount.org.uk\/fish-count-estimates>. Accessed 30 June 2017.\n\n.\"Pet Industry Market Size and Ownership Statistics.\" _American Pet Products Association_ , <http:\/\/www.americanpetproducts.org\/press_industrytrends.asp>. Accessed 30 June 2017.\n\n.\"Directory of Charities and Nonprofit Organizations.\" _GuideStar_ , <https:\/\/www.guidestar.org\/nonprofit-directory\/environment-animals\/animal-protection-welfare-services\/1.aspx>. Accessed 30 June 2017.\n\n.\"U.S. Grocery Shopper Trends 2015 Executive Summary.\" _The Food Marketing Institute_ , 2015. Available at <https:\/\/www.fmi.org\/docs\/default-source\/document-share\/fmitrends15-exec-summ-06-02-15.pdf>. Accessed 30 June 2015.\n\n.Norwood, F. B., J. L. Lusk, and R. W. Prickett. 2007. Consumer preferences for farm animal welfare: Results of a nationwide telephone survey. Working paper. Department of Agricultural Economics, Oklahoma State University, Stillwater, OK. Available at <http:\/\/cratefreefuture.com\/pdf\/American%20Farm%20Bureau-Funded%20Poll.pdf>. Accessed 30 June 2017.\n\n. _Journal of Interpersonal Violence_. 2009 Jun; 24(6):1036-56. doi: 10.1177\/0886260508319362. Epub 2008 Jun 10. Available at <https:\/\/www.ncbi.nlm.nih.gov\/pubmed\/18544751>.\n\n.\"How Many Species are we Losing?\" World Wildlife Fund for Nature. Available at http:\/\/wwf.panda.org\/about_our_earth\/biodiversity\/biodiversity. Accessed 30 June 2017.\n\n.The United States Environmental Protection Agency. (26 March 2013). _EPA Survey Finds More Than Half of the Nation's River and Stream Miles in Poor Condition_ [Press release]. Available at <https:\/\/yosemite.epa.gov\/opa\/admpress.nsf\/0\/26A31559BB37A7D285257B3A00589DDF>. Accessed 30 June 2017\n\n.The Gates Notes. \"The Science Behind Plant-Based Protein.\" Online video. _YouTube_ , 19 March 2013. Accessed 6 May 2016. Available at <https:\/\/www.youtube.com\/watch?v=jjlVq5GiHbo>.\n\n.Weise, Elizabeth. \"Eating can be energy-efficient, too.\" _USA Today_. 21 April 2009. Available at <http:\/\/usatoday30.usatoday.com\/news\/nation\/environment\/2009-04-21-carbon-diet_N.htm>.\n\n.Searchinger, T., Hanson, C., Ranganathan, J., Lipinski, B., Waite, R., Winterbottom, R., Dinshaw, A., and Heimlich, R. 2013. Creating a Sustainable Food Future: Interim Findings. Page 4. Available at <https:\/\/www.wri.org\/sites\/default\/files\/wri13_report_4c_wrr_online.pdf>.\n\n.2013. Gerber, P.J., Steinfeld, H., Henderson, B., Mottet, A., Opio, C., Dijkman, J., Falcucci, A. & Tempio, G. Tackling climate change through livestock \u2013 A global assessment of emissions and mitigation opportunities. Food and Agriculture Organization of the United Nations (FAO), Rome. Available from http:\/\/www.fao.org\/docrep\/018\/ie\/ie.pdf.\n\n.Own calculations: (1) based on U.S. total average water footprint per product from Mekonnen, M.M. and Hoekstra, A.Y. 2012. A global assessment of the water footprint of farm animal products. Ecosystems 15:401-15; and (2) whole milk and egg weights from United States Department of Agriculture Economic Research Service in cooperation with the Agricultural Marketing Service, the Agricultural Research Service, and the National Agriculture Statistics Service. 1992. Weights, measures, and conversion factors for agricultural commodities and their products, p. 28 Table 19 and p. 34 Table 21. Agricultural Handbook No. 697. www.ers.usda.gov\/media\/935958\/ah697_002.pdf. Accessed May 6, 2016.\n\n.NRDC (nrdcfood). \"Eating plant-forward dishes can go a long way in reducing diet-related carbon emissions.\" <http:\/\/bit.ly\/2q6WFl9> via @mindbodygreen. 27 April 2017, 10:00 AM. Tweet. Available at <https:\/\/twitter.com\/NRDCFood\/status\/857595205679173632>.\n\n.Lia Marie Johnson. \"Obama and I have a serious conversation...\" Online video. _YouTube_ , 7 Nov 2016. Accessed 6 February 2017. Available at <https:\/\/www.youtube.com\/watch?v=Dl1stHbyJAw>.\n\n.Grossman-Cohen, Ben. \"'Meatless Monday' Too Hot a Potato for USDA.\" _CNN_. 2 August 2012. Available at www.cnn.com\/2012\/08\/02\/opinion\/grossman-cohen-meatless-monday.\n\n.Barclay, Eliza. \"Even carnivores are putting more fake meat on their plates.\" _National Public Radio_. 15 August 2013. Available at www.npr.org\/sections\/thesalt\/2013\/08\/14\/212024490\/even-carnivores-are-putting-more-fake-meat-on-their-plates.\n\n.Scipioni, Jade and Libasi, Matthew. \"Tyson Foods CEO: The Future of Food May be Meatless.\" _FOX Business_. 7 March 2017. Available at <http:\/\/www.foxbusiness.com\/features\/2017\/03\/07\/tyson-foods-ceo-future-food-might-be-meatless.html>.\n\n.Black, Jane. \"Meat on the Side: Modern Menus Shift the Focus to Vegetables.\" _The Wall Street Journal_. 31 Oct 2014. Available at <https:\/\/www.wsj.com\/articles\/meat-on-the-side-modern-menus-shift-the-focus-to-vegetables-1414784266>. Accessed 30 June 2017.\n\n.Timothy V. \"Jamie Oliver Interview\u2014Veganism, McDonald's, Bill Gates.\" Online video. _YouTube_ , 4 Feb 2015. Accessed 21 October 2016. Available at www.youtube.com\/watch?v=gAEgSLeepOw.\n\n.Kennedy, Wally. \"Effort to keep CAFO from Arrow Rock mirrors fight to protect parks.\" _Joplin Globe_. 17 November 2007. Available at www.joplinglobe.com\/archives\/effort-to-keep-cafo-from-arrow-rock-mirrors-fight-to\/article_fe9f629a-3a98-5f27-a49d-ce47c711b6a4.html.\n\n.\"American FactFinder.\" United States Census Bureau. Accessed 5 May 2016.\n\n.Kennedy, Wally. \"Effort to keep CAFO from Arrow Rock mirrors fight to protect parks.\" _Joplin Globe_. 17 November 2007. Available at www.joplinglobe.com\/archives\/effort-to-keep-cafo-from-arrow-rock-mirrors-fight-to\/article_fe9f629a-3a98-5f27-a49d-ce47c711b6a4.html.\n\n.Ibid.\n\n.Sandy, Jennifer. \"Factory Farms: A Bad Choice for Rural America.\" _Forum Journal_. Vol. 23, No. 2. Winter 2009. Available at: www.preservationnation.org\/forum\/library\/public-articles\/factory-farms.html.\n\n.Gates, Bill. \"The Future of Food.\" _The Gates Notes_. 18 March 2013. Available at www.gatesnotes.com\/About-Bill-Gates\/Future-of-Food.\n\n.Kennedy, Wally. \"Effort to keep CAFO from Arrow Rock mirrors fight to protect parks.\" _Joplin Globe_. 17 November 2007. Available at www.joplinglobe.com\/archives\/effort-to-keep-cafo-from-arrow-rock-mirrors-fight-to\/article_fe9f629a-3a98-5f27-a49d-ce47c711b6a4.html.\n\n.Borgman, Kathy. Phone interview by Matthew Prescott. 11 October 2016.\n\n.Steinfeld, H., Gerber, P., Wassenaar, T., Castel, V., Rosales, M., and De Haan, C. 2006. \"Livestock's long shadow: environmental issues and options\" (Rome: Food and Agriculture Organization of the United Nations). Available at <ftp:\/\/ftp.fao.org\/docrep\/fao\/010\/a0701e\/a0701e.pdf>.\n\n.Marks, Robbin. \"Cesspools of Shame.\" July 2001. NRDC. Available at: www.nrdc.org\/sites\/default\/files\/cesspools.pdf.\n\n.Steinfeld, H., Gerber, P., Wassenaar, T., Castel, V., Rosales, M., and De Haan, C. 2006. \"Livestock's long shadow: environmental issues and options\" (Rome: Food and Agriculture Organization of the United Nations). Available at <ftp:\/\/ftp.fao.org\/docrep\/fao\/010\/a0701e\/a0701e.pdf>.\n\n.Walker, R., Browder, J., Arima, E., Simmons, C., Pereira, R., Caldas, M., Shirota, R., Zen, S.d., 2009. \"Ranching and the new global range: Amaz\u00f4nia in the 21st century.\" _Geoforum_ 40, 732\u2013745.\n\n.Bianchi, C., Haig, S., 2013. Deforestation trends of tropical dry forests in central Brazil. Biotropica 45, 395-400.\n\n.Morales-Hidalgo, D., 2006. Tree cover assessment: with special focus on the relative position issue. Case Studies in Open Areas in Costa Rica. Cuvillier Verlag.\n\n.Rulli, M.C., Saviori, A., D'Odorico, P., 2013. \"Global land and water grabbing.\" _Proc. Natl. Acad. Sci._ 110, 892\u2013897.\n\n.Devendra, C., Thomas, D., 2002a. \"Crop\u2013animal systems in Asia: importance of livestock and characterisation of agro-ecological zones.\" _Agric. Syst._ 71, 5\u201315.\n\n.B. Machovina et al. \"Science of the Total Environment\" 536 (2015). Pages 419\u2013431. Available at <http:\/\/dx.doi.org\/10.1016\/j.scitotenv.2015.07.022>.\n\n.Ibid.\n\n.Own calculation using: (1) 2014. Eshel, Gidon et al. Land, irrigation water, greenhouse gas, and reactive nitrogen burdens of meat, eggs, and dairy production in the United States. _PNAS_. 19 August 2014. Vol. 111, No. 33. Available at www.pnas.org\/cgi\/doi\/10.1073\/pnas.1402183111; with (2) square meters converted to square feet; and for the square footage of an NFL football field (3) Smith, Greg. \"A controversial end to an instant classic.\" 13 Jan. 2010. National Football League. Accessed 6 May 2016. Available at www.nfl.com\/news\/story\/09000d5d815b45a6\/printable\/a-controversial-end-to-an-instant-classic.\n\n.Borgman, Kathy. Phone interview by Matthew Prescott. 11 October 2016.\n\n.Roosevelt, Franklin. _Letter to all State Governors on a Uniform Soil Conservation Law_. 26 Feb 1937. Available at: www.presidency.ucsb.edu\/ws\/?pid=15373.\n\n.Ranganthan, J., Vennard, D., Waite, R., Dumas, P., Lipinksi, B., Searchinger, T., and Globagri-WRR Model Authors (2016). Shifting Diets for a Sustainable Future. Page 37. Available at www.wri.org\/sites\/default\/files\/Shifting_Diets_for_a_Sustainable_Food_Future_0.pdf.\n\n.Vaughan, Adam. \"Paul McCartney Backs Meat-Free Monday to Curb Carbon Emissions.\" _The Guardian_. 15 June 2009. Available at: www.theguardian.com\/environment\/2009\/jun\/15\/paul-mccartney-meat-free-monday.\n\n.\"Climate Actions to Change our World.\" United Nations Food and Agriculture Organization. 16 October 2016. Available at: www.fao.org\/world-food-day\/2016\/climate-actions\/en.\n\n.Carlson, Cody. Phone interview by Matthew Prescott. 12 May 2016.\n\n.Lerner, Rebecca. \"Toxic Fumes, Blisters & Brain Damage: The Cost of Doing Business?\" _The Ithaca Times_. 2 April 2008. Available at: www.organicconsumers.org\/news\/toxic-fumes-blisters-brain-damage-cost-doing-business. Note: Subsequently, Karen Strecker's father and other residents unsuccessfully sued the dairy for Clean Water Act violations.\n\n.Ross, Brian and Schecter, Anna. \"Got Milk? Got Ethics? Animal Rights v. U.S. Dairy Industry.\" _ABC News_. 26 January 2010. <http:\/\/abcnews.go.com\/Blotter\/animal-rights-us-dairy-industry\/story?id=9658866>.\n\n.(1) Lerner, Rebecca. \"Toxic Fumes, Blisters & Brain Damage: The Cost of Doing Business?\" _The Ithaca Times_. 2 April 2008. Available at: www.organicconsumers.org\/news\/toxic-fumes-blisters-brain-damage-cost-doing-business; with (2) 157,000 tons converted to pounds.\n\n.Based on curb weight of a 2015 Ford F-150 4x2. Ford Motor Company. \"F-150 Specifications.\" n.d. Available at: www.ford.com\/trucks\/f150\/specifications.\n\n.Lerner, Rebecca. \"Toxic Fumes, Blisters & Brain Damage: The Cost of Doing Business?\" _The Ithaca Times_. 2 April 2008. Available at: www.organicconsumers.org\/news\/toxic-fumes-blisters-brain-damage-cost-doing-business.\n\n.Ibid.\n\n.Hauter, Wenonah. _Foodopoly: The Battle over the Future of Food and Farming in America_. New York: The New Press, 2014. Print.\n\n.Kilborn, Peter. \"Hurricane Reveals Flaws in Farm Law.\" _The New York Times_. 22 September 1999. Available at: www.nytimes.com\/library\/national\/101799floyd-environment.html.\n\n.\"Corporate Hogs at the Public Trough.\" _Sierra Club_. San Francisco, California. 1999.\n\n.Merritt Frey, et al. \"Spills and Kills: Manure Pollution and America's Livestock Feedlots.\" _Clean Water Network_. Izaak Walton League of America and Natural Resources Defense Council. August 2000.\n\n.\"Corporate Hogs at the Public Trough.\" _Sierra Club_. San Francisco, California. 1999.\n\n.Merritt Frey, et al. \"Spills and Kills: Manure Pollution and America's Livestock Feedlots.\" _Clean Water Network_. Izaak Walton League of America and Natural Resources Defense Council. August 2000.\n\n.Hopkins, Elaine. \"Dairy Farm Ordered to Clean Up\u2014Inwood Dairy Must Cut Herd Size, Clean Ravine of 2 Million Gallons of Waste.\" _Peoria Journal Star_. 22 February 2001.\n\n.Ibid.\n\n.\"Corporate Hogs at the Public Trough.\" _Sierra Club_. San Francisco, California. 1999.\n\n.Gordon, Stuart and Lindsay, Alvie. \"Polluting Dairyman Gets Jail.\" _The Modesto Bee_. 14 March 1998.\n\n.Prince Charles, Prince of Wales. _A speech by HRH The Prince of Wales to the Future for Food Conference, Georgetown University, Washington DC_. 4 May 2011. Available at www.princeofwales.gov.uk\/media\/speeches\/speech-hrh-the-prince-of-wales-the-future-food-conference-georgetown-university.\n\n.Environmental Science, Enger and Smith, 6th ed. Fishery management: Page 201-Page 203. Available at: www.mhhe.com\/biosci\/pae\/es_map\/articles\/article_53.mhtml.\n\n.United States. EPA. _PA California Dairy Farmer Sentenced for Clean Water Violations_. Press Release. 20 March 1998. Available at <https:\/\/yosemite.epa.gov\/opa\/admpress.nsf\/0\/34ADD4BDC5B8707A852565CD005FD499>.\n\n.Environmental Defense Fund. \"Overfishing: Worse than You Might Think.\" n.d. Accessed 11 May 2016. Available at: www.edf.org\/oceans\/overfishing-worse-you-might-think.\n\n.\"Fish Count Estimates.\" <http:\/\/fishcount.org.uk\/fish-count-estimates>.\n\n.Worm, Boris, Barbier, Edward B., Beaumont, Nicola, Duffy, J. Emmett, Folke, Carl, Halpern, Benjamin S., Jackson, Jeremy B. C., Lotze, Heike K., Micheli, Fiorenza, Palumbi, Stephen R., Sala, Enric, Selkoe, Kimberley A., Stachowicz, John J., and Watson, Reg. \"Impact of Biodiversity Loss on Ocean Ecosystem Services.\" _Science_. Vol. 314, Issue 5800, pp. 787-790. 3 November 2006. Available at: <http:\/\/science.sciencemag.org\/content\/314\/5800\/787.abstract>.\n\n.Brown, Culum. \"Animal minds: Not just a pretty face.\" _New Scientist_. 12 June 2004. Available at: www.newscientist.com\/article\/mg18224515.200-animal-minds-not-just-a-pretty-face.\n\n.Oceana. \"Responsible fishing: Stopping overfishing.\" n.d. Accessed 11 May 2016. Available at http:\/\/oceana.org\/en\/our-work\/promote-responsible-fishing\/bottom-trawling\/learn-act\/more-on-bottom-trawling-gear.\n\n.Ibid.\n\n.Harris, Richard. \"Whales, Dolphins Are Collateral Damage In Our Taste For Seafood.\" NPR. 8 January 2014. Available at www.npr.org\/blogs\/thesalt\/2014\/01\/07\/260555381\/thousands-of-whales-dolphins-killed-to-satisfy-our-seafood-appetite.\n\n.Smith, Z., Gilroy, M., Eisenson, M., Schnettler, E., Stefanski, S. Natural Resources Defense Council. \"Net Loss: The Killing of Marine Mammals in Foreign Fisheries.\" January 2014. Available at www.nrdc.org\/sites\/default\/files\/mammals-foreign-fisheries-report.pdf.\n\n.\"Aquaculture.\" World Wildlife Fund for Nature. Available at <http:\/\/wwf.panda.org\/what_we_do\/footprint\/agriculture\/aquaculture.cfm>. Accessed 30 June 2017.\n\n.(1) United Nations Food and Agriculture Organization. \"Global aquaculture production statistics updated to 2013: Summary information.\" Table 1. March 2015. Available at: www.fao.org\/3\/a-i4899e.pdf; with (2) total metric tons converted to pounds.\n\n.Based on reported weight of 450,000 pounds. The Statue of Liberty \u2013 Ellis Island Foundation. \"Statue Facts.\" n.d. Available at: www.libertyellisfoundation.org\/statue-facts.\n\n.Balcombe, Jonathan. Phone interview by Matthew Prescott. 15 May 2016.\n\n.\"Salmon Fishing Problems.\" Coastal Alliance for Aquaculture Reform. _Farmed and Dangerous_. n.d. Available at: www.farmedanddangerous.org\/salmon-farming-problems.\n\n.Ibid.\n\n.Crawford, Elizabeth. \"Almond milk sales continue to surge, while dairy milk contracts, Nielsen data shows.\" _Food Navigator_. 15 April 2016. Available at: www.foodnavigator-usa.com\/Manufacturers\/Almond-milk-sales-continue-to-surge-as-dairy-milk-contracts-Nielsen.\n\n.Midkiff, Ken. _The Meat You Eat: How Corporate Farming Has Endangered America's Food Supply_. New York: St. Martin's Press, 2004. Print. Back cover.\n\n.Inzerillo, Lisa. Personal interview by Matthew Prescott. 18 February 2017.\n\n.Montgomery, Jess. \"Delaware's broiler chicken output grows.\" _The News Journal_. 29 April 2014. Available at: www.delawareonline.com\/story\/money\/business\/2014\/04\/29\/delawares-broiler-chicken-output-grows\/8475187.\n\n.Schuessler, Ryan. \"Maryland residents fight poultry industry expansion.\" _Al Jazeera America_. 23 Nov. 2015. Available at: <http:\/\/america.aljazeera.com\/articles\/2015\/11\/23\/maryland-residents-fight-poultry-industry-expansion.html>.\n\n.Kobell, Rona. \"Poultry mega-houses forcing Shore residents to flee from stench, traffic.\" _The Bay Journal_. 22 July 2015. Available at: <http:\/\/marylandreporter.com\/2015\/07\/22\/poultry-mega-houses-forcing-shore-residents-to-flee-stench-traffic>.\n\n.\"Poisoned Waters.\" _Frontline_. PBS. 21 April 2009. Web. Accessed 10 May 2016. Relevant segment available at: www.pbslearningmedia.org\/resource\/envh10.sci.life.eco.chickenwaste\/chicken-waste-and-water-pollution.\n\n.Ibid.\n\n. _Poult Sci._ 2013 Jan; 92(1):64\u201383. doi: 10.3382\/ps.2012-02745.\n\n.\"The Facts.\" _Cowspiracy_. Web. Accessed 10 May 2016. Available at: www.cowspiracy.com\/facts.\n\n.\"Poisoned Waters.\" _Frontline_. PBS. 21 April 2009. Web. Accessed 10 May 2016. Relevant segment available at: www.pbslearningmedia.org\/resource\/envh10.sci.life.eco.chickenwaste\/chicken-waste-and-water-pollution.\n\n.Parker, Suzi. \"How poultry producers are ravaging the rural south.\" _Grist_. 22 February 2006. Available at: http:\/\/grist.org\/article\/parker1.\n\n.Ibid.\n\n.Mirabelli, Maria C., Wing, Steve, Marshall, Stephen W., and Wilcosky, Timothy C., \"Race, Poverty, and Potential Exposure of Middle-School Students to Air Emissions from Confined Swine Feeding Operations.\" Available at http:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC1440786.\n\n.Pope Francis. _Encyclical Letter of the Holy Father on Care for our Common Home_. 24 May 2015. Available at: <http:\/\/w2.vatican.va\/content\/francesco\/en\/encyclicals\/documents\/papa-francesco_20150524_enciclica-laudato-si.html>.\n\n.Parker, Suzi. \"How poultry producers are ravaging the rural south.\" _Grist_. 22 February 2006. Available at: http:\/\/grist.org\/article\/parker1.\n\n.Doujaiji, B. and Al-Tawfiq, J. \"Hydrogen sulfide exposure in an adult male.\" _Annals of Saudi Medicine_. 30(1): 76\u201380. April 1010. Available at: www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC2850187.\n\n.Good, Kate. \"How Factory Farming Creates Air Pollution.\" _One Green Planet_. 10 March 2015. Available at: http:\/\/www.onegreenplanet.org\/environment\/how-factory-farming-creates-air-pollution.\n\n.The Pew Charitable Trusts. (29 April 2008). _Pew Commission Says Industrial Scale Farm Animal Production Poses \"Unacceptable\" Risks to Public Health, Environment_ [Press release]. Available at <http:\/\/www.pewtrusts.org\/en\/about\/news-room\/press-releases\/2008\/04\/29\/pew-commission-says-industrial-scale-farm-animal-production-poses-unacceptable-risks-to-public-health-environment>. Accessed 30 June 2017.\n\n.\"Air Pollution.\" Sustainable Table. n.d. Available at: <http:\/\/gracelinks.org\/media\/pdf\/air_pollution_tp_20090826.pdf>.\n\n.Curtis Stofferahn, \"Industrialized Farming and Its Relationship to Community Well-Being: an Update of the 2000 Report by Linda Labao,\" Special report prepared for the state of North Dakota, Office of Attorney General, www.und.edu\/org\/ndrural\/Lobao%20&%20Stofferahn.pdf.\n\n.Marks, Robbin. \"Cesspools of Shame.\" July 2001. NRDC. Available at: www.nrdc.org\/sites\/default\/files\/cesspools.pdf.\n\n.Pacelle, Wayne. \"HSUS 2016 annual report: Transformational progress for orcas and elephants, farm and lab animals, and others.\" Web blog post. _A Humane Nation_. The Humane Society of the United States, 15 March 2017. Web. Available at <http:\/\/blog.humanesociety.org\/wayne\/2017\/03\/hsus-annual-report-2016-transformational-progress.html>. Accessed 30 June 2017.\n\n.\"Per Capita Consumption of Poultry and Livestock, 1965 to Estimated 2016, in Pounds.\" _National Chicken Council_. 13 April 2016. Accessed 10 May 2016. Available at: www.nationalchickencouncil.org\/about-the-industry\/statistics\/per-capita-consumption-of-poultry-and-livestock-1965-to-estimated-2012-in-pounds.\n\n.\"Chart of U.S. Population: 1790 \u2013 2000.\" Census-Charts.com. n.d. Available at: www.census-charts.com\/Population\/pop-us-1790-2000.html.\n\n.United States. U.S. Census Bureau. U.S. and World Population Clock. Available at: www.census.gov\/popclock.\n\n.\"Sales and Trends.\" _Soyfoods Association of America_. Available at: www.soyfoods.org\/soy-products\/sales-and-trends.\n\n.Tropical Smoothie Caf\u00e9 menu. n.d. Available at: <https:\/\/s3.amazonaws.com\/tscws\/downloads\/TSC_Food_Smoothies_Menu.pdf>.\n\n.Beyond Meat. Lightly Seasoned Strips. n.d. Available at: <http:\/\/beyondmeat.com\/products>.\n\n.United States. USDA and U.S. Department of Health and Human Services. _Scientific Report of the 2015 Dietary Guidelines Advisory Committee: Advisory Report to the Secretary of Health and Human Service and the Secretary of Agriculture_. February 2015. Web. Accessed 10 May 2016. Available at <http:\/\/health.gov\/dietaryguidelines\/2015-scientific-report\/PDFs\/Scientific-Report-of-the-2015-Dietary-Guidelines-Advisory-Committee.pdf>.\n\n.United States. Office of the White House Press Secretary. _Remarks by the President at the GLACIER Conference\u2014Anchorage, AK_. 31 August 2015. Web. Accessed 17 October 2016. Available at www.whitehouse.gov\/the-press-office\/2015\/09\/01\/remarks-president-glacier-conference-anchorage-ak.\n\n.America's Climate Choices: Panel on Advancing the Science of Climate Change; National Research Council (2010). _Advancing the Science of Climate Change_. Washington, D.C.: The National Academies Press. ISBN 0-309-14588-0. Available at: www.nap.edu\/catalog.php?record_id=12782.\n\n.Davenport, Coral and Robertson, Campbell. \"Resettling the First American 'Climate Refugees.'\" _The New York Times_. 3 May 2016. Available at: www.nytimes.com\/2016\/05\/03\/us\/resettling-the-first-american-climate-refugees.html.\n\n.Ibid.\n\n.Allen, Leslie. \"Will Tuvalu Disappear Beneath the Sea?\" _Smithsonian Magazine_. August 2004. Available at: www.smithsonianmag.com\/travel\/will-tuvalu-disappear-beneath-the-sea-180940704\/?all.\n\n.Ibid.\n\n.\"Tuvalu: Joining forces to tackle climate change.\" International Federation of Red Cross and Red Crescent Societies. n.d. Available at: www.ifrc.org\/Global\/Case%20studies\/Disasters\/cs-tuvalu-en.pdf.\n\n.Steinfeld, H., Gerber, P., Wassenaar, T., Castel, V., Rosales, M., and De Haan, C. 2006. \"Livestock's long shadow: environmental issues and options\" (Rome: Food and Agriculture Organization of the United Nations). Available at <ftp:\/\/ftp.fao.org\/docrep\/fao\/010\/a0701e\/a0701e.pdf>.\n\n.United Nations Food and Agriculture Organization. \"Key Facts and Findings.\" n.d. Available at: www.fao.org\/news\/story\/en\/item\/197623\/icode.\n\n.\"Fighting Global Warming with Food.\" Environmental Defense Fund. 27 August 2007. Available at <https:\/\/web.archive.org\/web\/20080604070742\/edf.org\/article.cfm?contentID=6604>.\n\n.\"Limit Your Meat Consumption.\" _Earth Day Network_ , n.d. Available at <http:\/\/www.earthday.org\/limit-meat-consumption>. Accessed 30 June 2017.\n\n.Isomaki, Risto. _Meat, Milk and Climate: Why it is Absolutely Necessary to Reduce the Consumption of Animal Products_. Helsinki: Into Publishing, 2016. Page 1. Print.\n\n.Kandy, Daniel. \"The World Mangrove Atlas: Hope Amid Despair.\" _The World Watch Institute_. 20 August 2010. Available at: http:\/\/blogs.worldwatch.org\/nourishingtheplanet\/the-world-mangrove-atlas-hope-amid-despair.\n\n.Steinfeld, H., Gerber, P., Wassenaar, T., Castel, V., Rosales, M., and De Haan, C. 2006. \"Livestock's long shadow: environmental issues and options\" (Rome: Food and Agriculture Organization of the United Nations. Pp. 272). Available at <ftp:\/\/ftp.fao.org\/docrep\/fao\/010\/a0701e\/a0701e.pdf>.\n\n.2003. Pimentel, David and Marcia Pimentel. \"Sustainability of meat-based and plant-based diets and the environment.\" _American Journal of Clinical Nutrition_.78 (suppl): 660S\u20133S. Available from http:\/\/ajcn.nutrition.org\/content\/78\/3\/660S.full.\n\n.Eshel, Gidon. \"Grass-fed beef packs a punch to environment.\" _Reuters_. 8 April 2010. Available at: http:\/\/blogs.reuters.com\/environment\/2010\/04\/08\/grass-fed-beef-packs-a-punch-to-environment.\n\n.Steinfeld, H., Gerber, P., Wassenaar, T., Castel, V., Rosales, M., and de Haan, C. 2006. \"Livestock's long shadow: environmental issues and options.\" Food and Agriculture Organization of the United Nations, p. 112.\n\n.Ramanathan, V. and Xu, Y. 2010. \"The Copenhagen Accord for limiting global warming: criteria, constraints, and available avenues.\" Proceedings of the National Academy of Sciences of the United States of America 107 (18).\n\n.Eshel, Gidon. \"Grass-fed beef packs a punch to environment.\" _Reuters_. 8 April 2010. Available at: <http:\/\/blogs.reuters.com\/environment\/2010\/04\/08\/grass-fed-beef-packs-a-punch-to-environment>.\n\n.\"Fighting Global Warming with Food.\" The Environmental Defense Fund website. 24 July 2007. Archived copy available at: web.archive.org\/web\/20080923070051\/http:\/www.edf.org\/article.cfm?contentid=6604.\n\n.Pine, Joselyn. _Wit and Wisdom of America's First Ladies_. 2014. Mineola: Dover Publications. Print.\n\n.Scarborough, P., Appleby, N., Mizdrak, A., Briggs, A.D.M., Travis, R., Bradbury, K., and Key, T. \"Dietary greenhouse gas emissions of meat-eaters, fish-eaters, vegetarians and vegans in the UK.\" _Climate Change_. July 2014. Vol. 125, Issue 2, pp. 179\u2013192. Available at: <http:\/\/link.springer.com\/article\/10.1007\/s10584-014-1169-1>.\n\n.Ibid.\n\n.2014. Bailey, Rob, Froggatt, Antony, and Wellesley, Laura. _Livestock \u2013 Climate Change's Forgotten Sector_. The Royal Institute of International Affairs. Available from www.chathamhouse.org\/sites\/files\/chathamhouse\/field\/field_document\/20141203LivestockClimateChangeBaileyFroggattWellesley.pdf.\n\n.2015. Wellesley, L., Happer, C., and Froggatt, A. _Changing Climate, Changing Diets: Pathways to Lower Meat Consumption_. The Royal Institute of International Affairs. Available from www.chathamhouse.org\/publication\/changing-climate-changing-diets%20.\n\n.TED. \"The Case for Optimism on Climate Change.\" Online video. _YouTube_ , 14 March 2016. Accessed 30 June 2017. Available at <https:\/\/www.youtube.com\/watch?v=gVfgkFaswn4>.\n\n.Pollan, Michael. \"Unhappy Meals.\" _The New York Times_. 28 January 2007. Available at <http:\/\/www.nytimes.com\/2007\/01\/28\/magazine\/28nutritionism.t.html>. Accessed 5 October 2017.\n\n.Katzen, Mollie. _The Heart of the Plate: Vegetarian Recipes for a New Generation_. New York: Houghton Mifflin Harcourt. 2013. Print.\n\n.Goodall, Jane. _Harvest for Hope: A Guide to Mindful Eating_. New York: Time Warner Book Group, 2005. Print. Page 111.\n\n.Huget, Jennifer Larue. \"Bobby Flay's Recipe for Eating Healthfully.\" _The Washington Post_. 20 September 2011. Available at <https:\/\/www.washingtonpost.com\/lifestyle\/wellness\/bobby-flays-recipe-for-eating-healthfully\/2011\/09\/16\/gIQA2I8uhK_story.html>.\n\n.Thomas-Wachsberger, Beatrice. \"Natalie Portman: a rare bloom.\" _New Zealand Herald_. 5 May 2011.\n\n.The White House. (13 Sept 2010). _Remarks by the First Lady in Address to the National Restaurant Association Meeting_ [Press release]. Available at <https:\/\/obamawhitehouse.archives.gov\/the-press-office\/2010\/09\/13\/remarks-first-lady-address-national-restaurant-association-meeting>. Accessed 30 June 2017.\n\n.Nazarali, Roselina. \"Shania Twain is vegetarian.\" _Foodista_. 3 December 2012. Available at www.foodista.com\/blog\/2012\/12\/03\/shania-twain-is-vegetarian.\n\n.Armstrong, Laura and Brankin, Emma. \"Leo's Big Misteak.\" _The Sun_. 19 November 2016. Available at: www.thesun.co.uk\/tvandshowbiz\/2221787\/leonardo-dicaprio-preaches-about-meat-at-a-veggie-talk-but-is-later-caught-tucking-into-lamb-tagine-at-a-steakhouse.\n\n.\"Mario Batali Talks Meatless Monday.\" _Meatless Monday_ , 25 Jan 2016. Available at <http:\/\/www.meatlessmonday.com\/articles\/mario-batali-talks-meatless-monday-tuesday-and-wednesday-on-the-chew>. Accessed 30 June 2017.\n\n.Stewart, Martha. _Meatless: More than 200 of the Very Best Vegetarian Recipes_. New York: Random House, 2013. Print.\n\n.Jenkins, Willis, Tucker, Mary Evelyn, and Grim, John. _Routledge Handbook of Religion and Ecology_. New York: Routledge, 2016.\n\n.\"Sustainable Food.\" Rachel Carson Council, n.d. Available at http:\/\/rachelcarsoncouncil.org\/our-work\/sustainable-food. Accessed 30 June 2017.\n\n.TED. \"What's Wrong with what we Eat \u2013 Mark Bittman.\" Online video. _YouTube_ , 21 May 2008. Accessed 30 June 2017. Available at <https:\/\/www.youtube.com\/watch?v=5YkNkscBEp0>.\n\n.Bessette, Chanelle. \"Mark Tercek on Small Changes that Make a Big Difference.\" _Fortune_. 5 Nov 2013. Available at www.fortune.com\/2013\/11\/05\/mark-tercek-on-small-changes-that-make-a-big-difference.\n\n.Branson, Richard. \"Why I've Given up Eating Beef.\" _Virgin_. 9 July 2014. Available at www.virgin.com\/richard-branson\/why-ive-given-up-eating-beef.\n\n.Winfrey, Oprah. \"Conscious Eating: What I Learned on the 21-Day Cleanse.\" _O, The Oprah Magazine_. October 2008. Available at: oprah.com\/omagazine\/What-I-Know-for-Sure-Oprahs-Vegan-Cleanse.\n\n.Schwarzenegger, Arnold. \"Arnold's Opinion on Vegetarianism: The Oak's Thoughts on Going Meatless.\" _Flex_. N.d. Available at www.flexonline.com\/nutrition\/arnolds-opinion-vegetarianism.\n\n.Agence France-Presse. \"Lifestyle changes can curb climate change: IPCC chief.\" 15 January 2008. Available at www.abc.net.au\/news\/2008-01-16\/lifestyle-changes-can-curb-climate-change-ipcc\/1013982.\n\n.Lantry, Lauren. \"My Plate, My Planet.\" _Sierra Club_. Available at: www.sierraclub.org\/compass\/2015\/05\/my-plate-my-planet.\n\n.Gates, Bill. \"The Future of Food.\" _The Gates Notes_. 18 March 2013. Available at www.gatesnotes.com\/About-Bill-Gates\/Future-of-Food.\n\n.Borges, Marco. _The 22-Day Revolution Cookbook_. New York. Penguin Random House, 2016.\n\n.\"Celebrity vegans and the veganism facts you need.\" MSN. 13 June 2016. Available at: www.msn.com\/en-gb\/health\/nutrition\/celebrity-vegans-and-the-veganism-facts-you-need\/ss-BBmuG5R#image=5.\n\n.D'estries, Michael. \"Al Gore Says He'll Stay Vegan 'For Life.'\" _Mother Nature Network_. 13 March 2014. Available at: www.mnn.com\/health\/fitness-well-being\/blogs\/al-gore-says-hell-likely-stay-vegan-for-life.\n\n.Buckley, Jemma. \"I'm going vegetarian 3 times a week says Jamie Oliver.\" _Daily Mail_. 12 May 2015. Available at: dailymail.co.uk\/tvshowbiz\/article-3079084\/I-m-going-vegetarian-3-times-week-says-Jamie-Oliver-s-got-veggie-cook-book-coming-out.html.\n\n# INDEX\n\nThe index that appeared in the print version of this title does not match the pages in your eBook. Please use the search function on your eReading device to search for terms of interest. For your reference, the terms that appear in the print index are listed below.\n\n**A**\n\nAFOs (animal feeding operations)\n\n_see also_ factory farms\n\nair\n\nalmond milk\n\nClassic French Toast\n\nCreamy Herbed Mashed Potatoes\n\nGreen Bean Casserole\n\nLemon-Ginger Blueberry Cake with Cream Cheese Frosting\n\nOatmeal with Chai-Poached Pears\n\nPB&J Power Smoothies\n\nSpicy Chocolate Milk Shake with Whipped Coconut Cream\n\nalmonds:\n\nRaw Lemon Vanilla Raspberry Slab\n\nRoasted Root Vegetable and Quinoa Salad\n\nAmazon rain forest\n\nAmerican Farm Bureau Federation\n\nAmerican Heart Association\n\nAmerican Institute for Cancer Research\n\n_American Journal of Clinical Nutrition_\n\nAndr\u00e9s, Jos\u00e9\n\nanimal cruelty\n\n_see also_ factory farms\n\nAppendino, Chiara\n\napples:\n\nApple-Cinnamon Pancakes\n\nApple-Pecan Crisp\n\nMom's Noodle Kugel\n\napplesauce\n\naquaculture (fish factory farms)\n\naquafaba\n\nArrow Rock, Mo.\n\nAsparagus, Wild Rice Risotto with Peas and\n\navocados\n\nAvocado Chocolate Mousse with Vanilla Bean Whip\n\nBlack Bean Breakfast Burritos\n\nBLT Bagel Sandwich\n\nLoaded Black Bean Dogs\n\nMinted Green Pea Guacamole\n\nProtein-Packed Burrito Bowl\n\nRainbow Veggie Sushi\n\nSpicy Lentil Burgers\n\n**B**\n\nbacon, veggie\n\nBLT Bagel Sandwich\n\nCrispy Rice Paper Bacon\n\nBagel BLT Sandwich\n\nBalcombe, Jonathan\n\nbananas:\n\nBreakfast Banana Split\n\nPB&J Power Smoothies\n\nB\u00e1nh M\u00ec with Sriracha Chicken, Vietnamese\n\nBasbousa, Coconut\n\nbasil:\n\nCreamy Basil Chickpea Lettuce Cups\n\nGrilled Veggie Pizza with BBQ Sauce\n\nSourdough Panzanella\n\nSun-Dried Tomato and Basil Sausages\n\nBatali, Mario\n\nbeans\n\nBlack Bean Breakfast Burritos\n\nGreen Bean Casserole\n\nLoaded Black Bean Dogs\n\nLonestar BBQ Beans\n\nMardi Gras Gumbeaux\n\nProtein-Packed Burrito Bowl\n\nSun-Dried Tomato and Basil Sausages\n\nWild Mushroom Chili\n\n_see also_ chickpeas\n\nbeets:\n\nBloody Beet Burgers\n\nRoasted Coconut-Beet Soup\n\nBingham, George Caleb\n\nBiscuits and Gravy\n\nBittman, Mark\n\nBLT Bagel Sandwich\n\nblueberries:\n\nBlueberry Buckwheat Waffles\n\nLemon-Ginger Blueberry Cake with Cream Cheese Frosting\n\nPB&J Power Smoothies\n\nBorgman, Kathy\n\nBoston University\n\nBourg, Joann\n\nBranson, Richard\n\nBrazil\n\nbread:\n\nBiscuits and Gravy\n\nBlueberry Buckwheat Waffles\n\nCherry Chia Puff Pastry Breakfast Tarts\n\nClassic French Toast\n\nEasy Cheesy Garlic Bread\n\nRosemary Cornbread Mini Muffins\n\nSourdough Panzanella\n\nSteamed Buns with Tofu and Mushrooms\n\nbreakfast:\n\nApple-Cinnamon Pancakes\n\nBiscuits and Gravy\n\nBlack Bean Breakfast Burritos\n\nBlueberry Buckwheat Waffles\n\nBreakfast Banana Split\n\nCherry Chia Puff Pastry Breakfast Tarts\n\nClassic French Toast\n\nMix 'n' Match Superfood Smoothies\n\nOatmeal with Chai-Poached Pears\n\nPB&J Power Smoothies\n\nPea and Potato Frittata\n\nRice Pudding with Coconut and Cranberries\n\nRise 'n' Breakfast Sandwich\n\nSweet Potato and Parsnip Hash\n\nTofu Scramble 101\n\nbreakfast meats, vegan\n\nbroccoli, in Coconut-Lemongrass Curry with Rice Noodles\n\nBrown, Culum\n\nBrown, Nancy\n\nBrownies, Chocolate, with Puffed Quinoa\n\nBrussels Sprouts, Blackened, with Capers and Raisins\n\nBuckwheat Blueberry Waffles\n\nBuns, Steamed, with Tofu and Mushrooms\n\nburgers\n\nBloody Beet Burgers\n\nSpicy Lentil Burgers\n\nBurrito Bowl, Protein-Packed\n\nBurritos, Black Bean Breakfast\n\n**C**\n\ncabbage:\n\nChicken and Cabbage Wontons\n\nQuick Country Coleslaw\n\nRainbow Veggie Sushi\n\n_CAFO: The Tragedy of Industrial Animal Factories_\n\nCAFOs (confined animal feeding operations)\n\n_see also_ factory farms\n\nCake, Lemon-Ginger Blueberry, with Cream Cheese Frosting\n\nCameron, James\n\nCameron, Suzy Amis\n\ncancer\n\nCape Fear\n\nCapers, Blackened Brussels Sprouts with Raisins and\n\ncarbon dioxide\n\ncarbon sequestration\n\nCarlson, Cody\n\ncarrots:\n\nCoconut-Lemongrass Curry with Rice Noodles\n\nGinger-Sesame Soba Noodle Salad\n\nPickles\n\nQuick Country Coleslaw\n\nRainbow Veggie Sushi\n\nRoasted Root Vegetable and Quinoa Salad\n\ncashews:\n\nCashew Cream Mac 'n' Cheese\n\nRaw Lemon Vanilla Raspberry Slab\n\nCasserole, Green Bean\n\nCastelli, William\n\nCauliflower Wraps, Buffalo, with Blue Cheese\n\nCenter for Biological Diversity\n\nCenters for Disease Control and Prevention (CDC)\n\n_Cesspools of Shame_\n\nChai-Poached Pears, Oatmeal with\n\nCharles, Prince\n\nChatham House\n\ncheese, plant-based\n\nBuffalo Cauliflower Wraps with Blue Cheese\n\nCashew Cream Mac 'n' Cheese\n\nEasy Cheesy Garlic Bread\n\nLemon-Ginger Blueberry Cake with Cream Cheese Frosting\n\nNo-Frills Cheese Pizza\n\nPlant-Based Parmesan\n\nReuben Sandwich\n\nCheesecake, Smoky Maple-Pecan\n\nCherry Chia Puff Pastry Breakfast Tarts\n\nchicken, plant-based\n\nChicken and Cabbage Wontons\n\nMardi Gras Gumbeaux\n\nProtein-Packed Burrito Bowl\n\nVietnamese B\u00e1nh M\u00ec with Sriracha Chicken\n\nchickens\n\nchickpea flour\n\nchickpeas:\n\naquafaba\n\nCreamy Basil Chickpea Lettuce Cups\n\nMassaged Kale Caesar with Pumpernickel Croutons\n\nMeat-Free Meat Loaf with Tomato-Maple Glaze\n\nRoasted Red Pepper Hummus\n\nChili, Wild Mushroom\n\nChincoteague National Wildlife Refuge\n\nchocolate:\n\nAll-the-Nut-Butters Cups\n\nAvocado Chocolate Mousse with Vanilla Bean Whip\n\nChocolate Brownies with Puffed Quinoa\n\nPeanut Butter Chocolate Chip Cookie Ice Cream Sandwiches\n\nRaw Chocolate-Hazelnut Truffles\n\nSpicy Chocolate Milk Shake with Whipped Coconut Cream\n\ncholesterol\n\nChristen, Rolf\n\ncilantro:\n\nMinted Green Pea Guacamole\n\nWild Mushroom Chili\n\ncinnamon:\n\nApple-Cinnamon Pancakes\n\nApple-Pecan Crisp\n\nBlueberry Buckwheat Waffles\n\nClassic Pumpkin Pie\n\nMom's Noodle Kugel\n\nPeanut Butter Trail Mix Cookies\n\nRice Pudding with Coconut and Cranberries\n\nclimate change\n\n_Climate Change_\n\nCoastal Alliance for Aquaculture Reform\n\ncoconut:\n\nCoconut Basbousa\n\nCoconut-Lemongrass Curry with Rice Noodles\n\nRaw Lemon Vanilla Raspberry Slab\n\nRice Pudding with Coconut and Cranberries\n\nRoasted Coconut-Beet Soup\n\nSpicy Chocolate Milk Shake with Whipped Coconut Cream\n\nColeslaw, Quick Country\n\nSlawppy Joes\n\ncompassion\n\nCook, Ken\n\ncookies:\n\nPeanut Butter Chocolate Chip Cookie Ice Cream Sandwiches\n\nPeanut Butter Trail Mix Cookies\n\ncorn\n\nBlack Bean Breakfast Burritos\n\nProtein-Packed Burrito Bowl\n\nSave-the-Bay Crab Cakes\n\nCornbread Mini Muffins, Rosemary\n\nCosta Rica\n\ncows\n\n_Cowspiracy_\n\nCrab Cakes, Save-the-Bay\n\nCranberries, Rice Pudding with Coconut and\n\ncucumber:\n\nGinger-Sesame Soba Noodle Salad\n\nMint and Tofu Summer Rolls\n\nMinted Lemon-Cucumber Sorbet\n\nPickles\n\nRainbow Veggie Sushi\n\ncurry:\n\nCoconut-Lemongrass Curry with Rice Noodles\n\nSweet-and-Salty Curried Popcorn\n\n**D**\n\ndairy complexes\n\nWillet Dairy\n\ndairy products\n\nmilk\n\nDaiya\n\ndates:\n\nRaw Chocolate-Hazelnut Truffles\n\nRaw Lemon Vanilla Raspberry Slab\n\ndead zones\n\nDelmarva Peninsula\n\ndesserts:\n\nAll-the-Nut-Butters Cups\n\nApple-Pecan Crisp\n\nAvocado Chocolate Mousse with Vanilla Bean Whip\n\nCherry Chia Puff Pastry Breakfast Tarts\n\nChocolate Brownies with Puffed Quinoa\n\nClassic Pumpkin Pie\n\nCoconut Basbousa\n\nLemon-Ginger Blueberry Cake with Cream Cheese Frosting\n\nLemon Meringue Pie\n\nMinted Lemon-Cucumber Sorbet\n\nMom's Noodle Kugel\n\nPeanut Butter Chocolate Chip Cookie Ice Cream Sandwiches\n\nPeanut Butter Trail Mix Cookies\n\nRaw Chocolate-Hazelnut Truffles\n\nRaw Lemon Vanilla Raspberry Slab\n\nRice Pudding with Coconut and Cranberries\n\nSmoky Maple-Pecan Cheesecake\n\nSpicy Chocolate Milk Shake with Whipped Coconut Cream\n\nDew, Aloma\n\ndiabetes\n\nDiCaprio, Leonardo\n\n_Diet for a Small Planet_ (Lapp\u00e9)\n\ndogs, veggie\n\nLoaded Black Bean Dogs\n\nDukkah, Pistachio and Sunflower Seed\n\n**E**\n\nEarth\n\nEarth Day Network\n\n_Eating Animals_ (Foer)\n\nEcuador\n\nEdwards, Bernadine\n\neggplant:\n\nGrilled Veggie Pizza with BBQ Sauce\n\nMushroom and Kale Galette\n\neggs\n\nreplacements for\n\nEisenberg, Jesse\n\nempathy\n\nEnvironmental Defense Fund\n\nEnvironmental Protection Agency (EPA)\n\nEnvironmental Working Group (EWG)\n\nEshel, Gidon\n\n**F**\n\nfactory farms\n\nAFOs\n\nCAFOs\n\nchicken\n\ndefinitions of\n\nfish\n\ninvestigators of\n\nlagoons of\n\nsoil and\n\nTacosa Feedyard\n\nfats\n\nsaturated\n\nFeldstein, Stephanie\n\nField Roast\n\nfire\n\nfish\n\nfactory farms\n\nfish, faux\n\nflaxseeds\n\nFlay, Bobby\n\nflexitarian approach\n\nFoer, Jonathan Safran\n\nFollow Your Heart\n\nFood Marketing Institute (FMI)\n\n_Food Navigator_\n\n_Forks Over Knives_\n\nFramingham Heart Study (FHS)\n\nFrancis, Pope\n\nFrench Toast, Classic\n\nFriedrich, Bruce\n\nFries, Spicy Sweet Potato, with Ranch\n\nFrittata, Pea and Potato\n\n**G**\n\nGalette, Mushroom and Kale\n\nGarlic Bread, Easy Cheesy\n\nGates, Bill\n\nGessling, Dennis\n\ngestation crates\n\nginger:\n\nGinger-Sesame Soba Noodle Salad\n\nLemon-Ginger Blueberry Cake with Cream Cheese Frosting\n\nGoodall, Jane\n\nGood Food Institute\n\nGore, Al\n\ngrains\n\nGrande, Ariana\n\nGravy, Biscuits and\n\nGreen Bean Casserole\n\ngreenhouse gas emissions\n\nGreenpeace UK\n\nGreenpeace USA\n\nGreger, Michael\n\nGrossman-Cohen, Ben\n\nGuacamole, Minted Green Pea\n\nGumbeaux, Mardi Gras\n\n**H**\n\nHamilton, Bruce\n\nhazelnut butter:\n\nAll-the-Nut-Butters Cups\n\nRaw Chocolate-Hazelnut Truffles\n\nhealth and disease\n\nheart disease\n\nhearts of palm, in Save-the-Bay Crab Cakes\n\nHorton, Tom\n\nhot dogs, veggie\n\nLoaded Black Bean Dogs\n\nHumane Society of the United States\n\nHummus, Roasted Red Pepper\n\nHurricane Floyd\n\n**I**\n\nice cream:\n\nPeanut Butter Chocolate Chip Cookie Ice Cream Sandwiches\n\nSpicy Chocolate Milk Shake with Whipped Coconut Cream\n\nICESCAPE\u2014Impacts of Climate Change on the Eco-Systems and Chemistry of the Arctic Pacific Environment\n\nImhoff, Daniel\n\nInternational Federation of Red Cross and Red Crescent Societies\n\nInzerillo, Joe\n\nInzerillo, Lisa\n\nIsom\u00e4ki, Risto\n\n**J**\n\nJackfruit Tacos with Grilled Pineapple, Jerked\n\nJohn Hopkins Bloomberg School of Public Health\n\nJohnson, Lady Bird\n\n_Journal of Health Economics_\n\n**K**\n\nkale:\n\nMassaged Kale Caesar with Pumpernickel Croutons\n\nMushroom and Kale Galette\n\nKebabs, Smoky Seitan, with Peanut Sauce\n\nKennedy, Robert F., Jr.\n\nKerchner, Sherri\n\nKerchner, Thomas\n\nKerr, Day and Whitney\n\nKite Hill\n\nKugel, Mom's Noodle\n\n**L**\n\nlagoons\n\nLapp\u00e9, Frances Moore\n\nlegumes\n\n_see also_ beans; chickpeas\n\nlemon:\n\nLemon-Ginger Blueberry Cake with Cream Cheese Frosting\n\nLemon Meringue Pie\n\nMinted Lemon-Cucumber Sorbet\n\nRaw Lemon Vanilla Raspberry Slab\n\nLemongrass-Coconut Curry with Rice Noodles\n\nlentils\n\nBloody Beet Burgers\n\nOyster Mushrooms \u00e0 la Marini\u00e8re\n\nSpaghetti and Lentil Meatballs\n\nSpicy Lentil Burgers\n\nLeonard, Annie\n\nLettuce Cups, Creamy Basil Chickpea\n\nlivestock\n\ncows\n\nfeed for\n\nland and\n\npigs\n\n_Livestock's Long Shadow_\n\nLouisiana\n\n**M**\n\nMackey, John\n\nMac 'n' Cheese, Cashew Cream\n\nmain dishes:\n\nCashew Cream Mac 'n' Cheese\n\nChicken and Cabbage Wontons\n\nCoconut-Lemongrass Curry with Rice Noodles\n\nCreamy Basil Chickpea Lettuce Cups\n\nGrilled Veggie Pizza with BBQ Sauce\n\nJerked Jackfruit Tacos with Grilled Pineapple\n\nMeat-Free Meat Loaf with Tomato-Maple Glaze\n\nMushroom and Kale Galette\n\nNo-Frills Cheese Pizza\n\nOyster Mushrooms \u00e0 la Marini\u00e8re\n\nPittsburgh Pierogis\n\nProtein-Packed Burrito Bowl\n\nRainbow Veggie Sushi\n\nSave-the-Bay Crab Cakes\n\nSmoky Seitan Kebabs with Peanut Sauce\n\nSpaghetti and Lentil Meatballs\n\nSteamed Buns with Tofu and Mushrooms\n\nSun-Dried Tomato and Basil Sausages\n\nWild Rice Risotto with Peas and Asparagus\n\nmangrove swamps\n\nmanure\n\nmaple:\n\nMeat-Free Meat Loaf with Tomato-Maple Glaze\n\nSmoky Maple-Pecan Cheesecake\n\nmargarine\n\nmayonnaise\n\nMcCartney, Paul\n\nmeat\n\ngrass-fed\n\nland impact of\n\n_Meat, Milk and Climate: Why It Is Absolutely Necessary to Reduce the Consumption of Animal Products_ (Isom\u00e4ki)\n\nMeatballs, Lentil, Spaghetti and\n\n_Meatingplace_\n\nMeatless Mondays\n\nMeat Loaf, Meat-Free, with Tomato-Maple Glaze\n\nmeats, plant-based\n\nsee also chicken, plant-based\n\nMehdi, Ahmad\n\nMercy For Animals (MFA)\n\nmethane\n\nmilk\n\nmilks, nondairy\n\nMilk Shake, Spicy Chocolate, with Whipped Coconut Cream\n\nmint:\n\nMint and Tofu Summer Rolls\n\nMinted Green Pea Guacamole\n\nMinted Lemon-Cucumber Sorbet\n\nMiyoko's Creamery\n\nMousse, Avocado Chocolate, with Vanilla Bean Whip\n\nMuffins, Rosemary Cornbread Mini\n\nM\u00fcnter, Leilani\n\nmushrooms:\n\nCoconut-Lemongrass Curry with Rice Noodles\n\nGreen Bean Casserole\n\nGrilled Veggie Pizza with BBQ Sauce\n\nMushroom and Kale Galette\n\nOyster Mushrooms \u00e0 la Marini\u00e8re\n\nSteamed Buns with Tofu and Mushrooms\n\nWild Mushroom Chili\n\n**N**\n\nNapo River\n\nNaquin, Albert\n\nNational Heart, Lung and Blood Institute\n\nNational Trust for History Preservation\n\nNatural Resources Defense Council (NRDC)\n\n_New York Times_\n\nNhat Hanh, Thich\n\nnoodles:\n\nCoconut-Lemongrass Curry with Rice Noodles\n\nGinger-Sesame Soba Noodle Salad\n\nMint and Tofu Summer Rolls\n\nMom's Noodle Kugel\n\n_see also_ pasta\n\nNorth Carolina\n\nNPR\n\nnut butters:\n\nAll-the-Nut-Butters Cups\n\n_see also_ peanut butter\n\nnut milks\n\n_see also_ almond milk\n\nnuts\n\nnutritional yeast\n\n**O**\n\noats:\n\nApple-Pecan Crisp\n\nOatmeal with Chai-Poached Pears\n\nPeanut Butter Trail Mix Cookies\n\nObama, Barack\n\nObama, Michelle\n\noils\n\nOliver, Jamie\n\nomega-3 fatty acids\n\noverweight and obesity\n\nOxfam America\n\n**P**\n\nPacelle, Wayne\n\nPachauri, Rajendra\n\npalm oil\n\nPancakes, Apple-Cinnamon\n\nPanzanella, Sourdough\n\nParmesan, Plant-Based\n\nparsnips:\n\nRoasted Root Vegetable and Quinoa Salad\n\nSweet Potato and Parsnip Hash\n\npasta:\n\nCashew Cream Mac 'n' Cheese\n\nSpaghetti and Lentil Meatballs\n\n_see also_ noodles\n\npeanut butter:\n\nAll-the-Nut-Butters Cups\n\nPB&J Power Smoothies\n\nPeanut Butter Chocolate Chip Cookie Ice Cream Sandwiches\n\nPeanut Butter Trail Mix Cookies\n\nSmoky Seitan Kebabs with Peanut Sauce\n\npears:\n\nMom's Noodle Kugel\n\nOatmeal with Chai-Poached Pears\n\npeas:\n\nMinted Green Pea Guacamole\n\nPea and Potato Frittata\n\nWild Rice Risotto with Peas and Asparagus\n\npecans:\n\nApple-Pecan Crisp\n\nSmoky Maple-Pecan Cheesecake\n\npepper, bell:\n\nGinger-Sesame Soba Noodle Salad\n\nRainbow Veggie Sushi\n\nRoasted Red Pepper Hummus\n\nPew Charitable Trusts\n\nPew Commission on Industrial Farm Animal Production\n\nPhilippines\n\nPickles\n\nPierogis, Pittsburgh\n\npies:\n\nClassic Pumpkin Pie\n\nLemon Meringue Pie\n\nSmoky Maple-Pecan Cheesecake\n\npigs\n\nPineapple, Grilled, Jerked Jackfruit Tacos with\n\nPistachio and Sunflower Seed Dukkah\n\npizza:\n\nGrilled Veggie Pizza with BBQ Sauce\n\nNo-Frills Cheese Pizza\n\nPollan, Michael\n\nPopcorn, Sweet-and-Salty Curried\n\nPortman, Natalie\n\npotatoes:\n\nCreamy Herbed Mashed Potatoes\n\nPea and Potato Frittata\n\nPittsburgh Pierogis\n\nRoasted Root Vegetable and Quinoa Salad\n\npoultry\n\nPritikin, Nathan\n\nprotein\n\nPudding, Rice, with Coconut and Cranberries\n\nPumpernickel Croutons, Massaged Kale Caesar with\n\nPumpkin Pie, Classic\n\n**Q**\n\nquinoa\n\nChocolate Brownies with Puffed Quinoa\n\nProtein-Packed Burrito Bowl\n\nRoasted Root Vegetable and Quinoa Salad\n\n**R**\n\nRachel Carson Council\n\nraisins:\n\nBlackened Brussels Sprouts with Capers and Raisins\n\nMom's Noodle Kugel\n\nPeanut Butter Trail Mix Cookies\n\nRanch, Spicy Sweet Potato Fries with\n\nRaspberry Slab, Raw Lemon Vanilla\n\nReduce, Replace, Refine\n\nrice:\n\nMardi Gras Gumbeaux\n\nRainbow Veggie Sushi\n\nRice Pudding with Coconut and Cranberries\n\nWild Rice Risotto with Peas and Asparagus\n\nrice paper:\n\nBLT Bagel Sandwich\n\nCrispy Rice Paper Bacon\n\nMint and Tofu Summer Rolls\n\nRio Branco\n\nRisotto, Wild Rice, with Peas and Asparagus\n\nRocky Mountains\n\nRoosevelt, Franklin D.\n\nrosemary:\n\nCreamy Herbed Mashed Potatoes\n\nOyster Mushrooms \u00e0 la Marini\u00e8re\n\nRosemary Cornbread Mini Muffins\n\nRoyal Institute of International Affairs\n\n**S**\n\nsalads:\n\nGinger-Sesame Soba Noodle Salad\n\nMassaged Kale Caesar with Pumpernickel Croutons\n\nQuick Country Coleslaw\n\nRoasted Root Vegetable and Quinoa Salad\n\nSesame Seaweed Salad\n\nSourdough Panzanella\n\nsandwiches:\n\nBloody Beet Burgers\n\nBLT Bagel Sandwich\n\nBuffalo Cauliflower Wraps with Blue Cheese\n\nLoaded Black Bean Dogs\n\nReuben Sandwich\n\nRise 'n' Breakfast Sandwich\n\nSlawppy Joes\n\nSpicy Lentil Burgers\n\nVietnamese B\u00e1nh M\u00ec with Sriracha Chicken\n\nSandy, Jennifer\n\nSappington, John\n\nsausage, vegan\n\nRise 'n' Breakfast Sandwich\n\nSun-Dried Tomato and Basil Sausages\n\nSauven, John\n\nSchwarzenegger, Arnold\n\n_Science_\n\n_Science Behind Plant-Based Protein, The_\n\n_Science of the Total Environment_\n\nSeacrest, Ryan\n\nseasonal produce\n\nSeaweed Salad, Sesame\n\nseeds\n\nseitan\n\nSmoky Seitan Kebabs with Peanut Sauce\n\nsesame:\n\nGinger-Sesame Soba Noodle Salad\n\nSesame Seaweed Salad\n\n_Shifting Diets for a Sustainable Food Future_\n\nshopping\n\nshrimp\n\nsides and extras:\n\nBlackened Brussels Sprouts with Capers and Raisins\n\nCreamy Herbed Mashed Potatoes\n\nCrispy Rice Paper Bacon\n\nEasy Cheesy Garlic Bread\n\nGreen Bean Casserole\n\nLonestar BBQ Beans\n\nMint and Tofu Summer Rolls\n\nMinted Green Pea Guacamole\n\nPistachio and Sunflower Seed Dukkah\n\nPlant-Based Parmesan\n\nQuick Country Coleslaw\n\nRoasted Red Pepper Hummus\n\nRosemary Cornbread Mini Muffins\n\nSpicy Sweet Potato Fries with Ranch\n\nSweet-and-Salty Curried Popcorn\n\nSierra Club\n\nSilicon Valley\n\nSlawppy Joes\n\nsmoothies:\n\nMix 'n' Match Superfood Smoothies\n\nPB&J Power Smoothies\n\nSoba Noodle Salad, Ginger-Sesame\n\nsoil\n\nSorbet, Minted Lemon-Cucumber\n\nsoups:\n\nRoasted Coconut-Beet Soup\n\nWild Mushroom Chili\n\nSourdough Panzanella\n\nsoybeans\n\nsoy creamer\n\nsoy milk\n\nSpaghetti and Lentil Meatballs\n\nStewart, Martha\n\nstews:\n\nMardi Gras Gumbeaux\n\nWild Mushroom Chili\n\nStrecker, Karen\n\nSuh, Rhea\n\nSummer Rolls, Mint and Tofu\n\nsunflower seeds:\n\nPeanut Butter Trail Mix Cookies\n\nPistachio and Sunflower Seed Dukkah\n\nPlant-Based Parmesan\n\nSushi, Rainbow Veggie\n\nsweet potatoes:\n\nRoasted Root Vegetable and Quinoa Salad\n\nSpicy Sweet Potato Fries with Ranch\n\nSweet Potato and Parsnip Hash\n\n**T**\n\nTacos, Jerked Jackfruit, with Grilled Pineapple\n\nTacosa Feedyard\n\nTarts, Cherry Chia Puff Pastry Breakfast\n\ntempeh\n\nReuben Sandwich\n\nTercek, Mark\n\nThree Rs\n\ntofu\n\nBlack Bean Breakfast Burritos\n\nCoconut-Lemongrass Curry with Rice Noodles\n\nLemon Meringue Pie\n\nMint and Tofu Summer Rolls\n\nMom's Noodle Kugel\n\nPea and Potato Frittata\n\nSmoky Maple-Pecan Cheesecake\n\nSteamed Buns with Tofu and Mushrooms\n\ntips for preparing\n\nTofu Scramble 101\n\ntomatoes:\n\nBLT Bagel Sandwich\n\nJerked Jackfruit Tacos with Grilled Pineapple\n\nMardi Gras Gumbeaux\n\nMeat-Free Meat Loaf with Tomato-Maple Glaze\n\nMushroom and Kale Galette\n\nNo-Frills Cheese Pizza\n\nProtein-Packed Burrito Bowl\n\nRosemary Cornbread Mini Muffins\n\nSlawppy Joes\n\nSourdough Panzanella\n\nWild Mushroom Chili\n\ntomatoes, sun-dried:\n\nSpaghetti and Lentil Meatballs\n\nSun-Dried Tomato and Basil Sausages\n\ntortillas:\n\nBlack Bean Breakfast Burritos\n\nJerked Jackfruit Tacos with Grilled Pineapple\n\nTrail Mix Peanut Butter Cookies\n\nTruffles, Raw Chocolate-Hazelnut\n\nTuvalu\n\nTwain, Shania\n\nTyson Foods\n\n**U**\n\nUnited Nations (UN)\n\nFood and Agriculture Organization (FAO)\n\nIntergovernmental Panel on Climate Change (IPCC)\n\nUniversity of North Carolina School of Public Health\n\nUSDA\n\nConservation Reserve Program (CRP)\n\nUS Department of Housing and Urban Development\n\nU.S. Dietary Guidelines Advisory Committee\n\n**V**\n\nvanilla:\n\nAvocado Chocolate Mousse with Vanilla Bean Whip\n\nRaw Lemon Vanilla Raspberry Slab\n\nVavae, Hilia\n\nVegan Egg\n\nvitamin B12\n\n**W**\n\nWaffles, Blueberry Buckwheat\n\n_Wall Street Journal_\n\nwater\n\n_What a Fish Knows: The Inner Lives of Our Underwater Cousins_ (Balcombe)\n\nwheat gluten:\n\nLoaded Black Bean Dogs\n\nSmoky Seitan Kebabs with Peanut Sauce\n\nSun-Dried Tomato and Basil Sausages\n\nWhole Foods Market\n\nWild Rice Risotto with Peas and Asparagus\n\nWillet Dairy\n\nWinfrey, Oprah\n\nWontons, Chicken and Cabbage\n\nWorld Bank\n\nWorld Resources Institute\n\nWorld Watch Institute\n\nWorld Wildlife Fund\n\nWraps, Buffalo Cauliflower, with Blue Cheese\n\n**Y**\n\nyeast, nutritional\n\nyogurt, nondairy:\n\nBlueberry Buckwheat Waffles\n\nCoconut Basbousa\n\n**Z**\n\nzucchini, in Grilled Veggie Pizza with BBQ Sauce\n\n# [PRAISE FOR  \nFOOD IS THE SOLUTION](toc.xhtml#praise)\n\n\"This book is an important call to action for anyone who eats.\"\n\n\u2014EMERIL LAGASSE\n\n\"As global citizens, we all share in the responsibility of creating a better future. With this book, Matthew Prescott shows us how our food choices can do just that.\"\n\n\u2014JOS\u00c9 ANDR\u00c9S\n\n\" _Food Is the Solution_ shares easy ways to look and feel great while also respecting the Earth. Talk about a win-win!\"\n\n\u2014ELLE MACPHERSON\n\n\"Individual action can lead to collective action, and collective action opens the door for system wide, positive change. So please do all you can, starting at home\u2014where Matthew Prescott's inspiring book can guide you to some healthy and delicious choices.\"\n\n\u2014FRED KRUPP, PRESIDENT OF THE ENVIRONMENTAL DEFENSE FUND\n\n\" _Food Is the Solution_ will give you the tools you need to incorporate plant-based eating into your everyday life. Start now, and help the planet.\"\n\n\u2014MARK R. TERCEK, PRESIDENT AND CEO OF THE NATURE CONSERVANCY\n\n\"Devour this book. Eat it up. It might just save your life and the world.\"\n\n\u2014MICHAEL GREGER, MD, FACLM, BESTSELLING AUTHOR OF _HOW NOT TO DIE_\n\n\"Food plays a big part of our identity, and this book will help us align what we eat with who we are.\"\n\n\u2014KEEGAN-MICHAEL KEY\n\n\"There's nothing sexier than saving the Earth\u2014and this book shows you how you can do that through the power of your plate.\"\n\n\u2014PAMELA ANDERSON\n\n\"Eat more plants for a better planet: Easy peasy. Bon app\u00e9tit!\"\n\n\u2014PAUL \"PEE-WEE HERMAN\" REUBENS\n\n\"Food is power, and this book will help you use it.\"\n\n\u2014DAVID CHANG, FOUNDER OF MOMOFUKU\n\n\"Matthew Prescott proves that food can be delicious while also helping mend the social and ecological fabric of our world.\"\n\n\u2014MICHAEL SYMON, HOST OF ABC'S _THE CHEW_\n\n\"Though diet is but one of many things we need to change, we won't be able to save the world without dramatically changing our food and agriculture practices. Matthew Prescott has assembled a detailed and delicious road map to survival.\"\n\n\u2014MICHAEL BRUNE, EXECUTIVE DIRECTOR OF THE SIERRA CLUB\n\n\" _Food Is the Solution_ is the guide I wish I had as a young mom. This is the blueprint for anyone who eats.\"\n\n\u2014LAURIE DAVID, ACADEMY AWARD\u2013WINNING PRODUCER OF _AN INCONVENIENT TRUTH_\n\n\"A superb and thought-provoking book about the long-term impact of what we eat on the environment and our health. We need to learn how to vote with our mouths and this is a great guide to help us do so.\"\n\n\u2014THEODORE ROOSEVELT IV, CHAIRMAN, CENTER FOR CLIMATE AND ENERGY SOLUTIONS\n\n\"Eating with the Earth in mind has never been easier, thanks to this important and eye-opening book.\"\n\n\u2014JOHN MACKEY, CEO OF WHOLE FOODS MARKET\n\n\"More people than ever are enjoying plant-based foods as a way to help themselves and the planet. Matthew Prescott's book makes it easy and accessible for anyone to lend a hand\u2014and fork.\"\n\n\u2014T\u00c9A LEONI\n\n\"The planet needs us, and this book can help.\"\n\n\u2014MOBY\n\n\"There's a place at the table for everyone when it comes to environmental eating, and this book can help you find yours.\"\n\n\u2014BOB PERCIASEPE, FORMER DEPUTY ADMINISTRATOR OF THE U.S. EPA\n\n\"The fact is, the more plant-based we eat, the better we feel, the better we are. No more alternative facts.\"\n\n\u2014ROSIE O'DONNELL\n\n\"Each one of us can help protect the planet when we choose what to eat. It's simpler than you may think! This book will give you all the tools you need.\"\n\n\u2014VANI HARI, BESTSELLING AUTHOR OF _THE FOOD BABE WAY_\n\n\"Want to save the world while enjoying great food? This marvelous book shows you how.\"\n\n\u2014JOHN ROBBINS, BESTSELLING AUTHOR OF _DIET FOR A NEW AMERICA_\n\n\"This book is the best thing since sliced bread.\"\n\n\u2014BRUCE FRIEDRICH, EXECUTIVE DIRECTOR OF THE GOOD FOOD INSTITUTE\n\n\"Matthew Prescott has cut through all the confusing information on the food supply to bring us great recipes that are good for us and the planet. A must-read!\"\n\n\u2014KRISTIN DAVIS\n\n\" _Food Is the Solution_ offers moving stories and helpful tips so we can all live more sustainably.\"\n\n\u2014KEN COOK, PRESIDENT OF ENVIRONMENTAL WORKING GROUP\n\n\"The message for the future is really very simple: Get as close to a whole-food, plant-based diet as possible.\"\n\n\u2014DR. T. COLIN CAMPBELL, AUTHOR OF _THE CHINA STUDY_\n\n\"Great cause, great recipes, great read!\"\n\n\u2014EVAN SHARP, CO-FOUNDER OF PINTEREST\n\n\"This book will help you celebrate Earth Day every day!\"\n\n\u2014KATHLEEN ROGERS, PRESIDENT OF EARTH DAY NETWORK\n\n\"Matthew Prescott provides a compelling case that most foods that are good for you are also good for the world. Whether you're a carnivore or vegan, you'll benefit from reading this book.\"\n\n\u2014LAWRENCE WILLIAMS, CEO OF THE U.S. HEALTHFUL FOODS COUNCIL\n\n\"We impact the world at each meal, and this book can help us use our plates as a force for good.\"\n\n\u2014RICHARD LINKLATER\n\n\"There's a crisis in our oceans and on our planet. This book offers concrete, easy, and delicious ways to help.\"\n\n\u2014LOUIE PSIHOYOS, ACADEMY AWARD\u2013WINNING DIRECTOR OF _THE COVE AND RACING EXTINCTION_\n\n\"We can heal our bodies through the right types of food, and by consuming and producing the right ingredients, food also becomes a medicine for our planet.\"\n\n\u2014PETRA N\u011aMCOV\u00c1\n\n\"Great book, great food, better world\u2014what more could we ask for?\"\n\n\u2014TAL RONNEN, CROSSROADS KITCHEN\n\n\"The world\u2014literally, planet Earth\u2014needs this book!\"\n\n\u2014KATHY FRESTON, BESTSELLING AUTHOR OF _QUANTUM WELLNESS_\n\n\"For those of us already enthralled by Matthew Prescott's perspective and insight into how what we eat affects who we are and the fate of the world, this book comes at exactly the right time.\"\n\n\u2014MAYIM BIALIK\n\n\"This is one of the most crucial books of our time. It offers amazing, healthy food while being environmentally conscious and informative. A must-read!\"\n\n\u2014HIS ROYAL HIGHNESS KHALED BIN ALWALEED, PRINCE OF SAUDI ARABIA\n\n\"Use this book as an important tool to help you turn your diet into a force for good.\"\n\n\u2014KIER\u00c1N SUCKLING, EXECUTIVE DIRECTOR OF THE CENTER FOR BIOLOGICAL DIVERSITY\n\n\"Protecting our environment takes all hands\u2014and mouths\u2014on deck. _Food Is the Solution_ shows how simple it can be to adopt a more Earth-friendly lifestyle, which is a critical first step to changing our habits as a society.\"\n\n\u2014MARGIE ALT, EXECUTIVE DIRECTOR OF ENVIRONMENT AMERICA\n\n\"Thought-provoking resources and recipes to help you kick-start a more environmentally friendly diet.\"\n\n\u2014IAN ANDERSON, FRONT MAN FOR JETHRO TULL\n\n\" _Food Is the Solution_ will help you save the Earth and live more compassionately.\"\n\n\u2014GENE BAUR, FOUNDER OF FARM SANCTUARY\n\n\"Today's chefs are spending more time than ever before on plant-forward foods. As this book proves, culinarians can activate positive change to help us all eat better and diversify our food choices, while also preserving our planet and its precious resources.\"\n\n\u2014SCOTT GIAMBASTIANI, GLOBAL PROGRAM CHEF AT GOOGLE\n\n\"This provocative and valuable book reminds us that a smart diet is good for our health, our soul, and our planet.\"\n\n\u2014GENE KARPINSKI, PRESIDENT OF THE LEAGUE OF CONSERVATION VOTERS\n\n\"When it comes to food and its impact on the environment, animal welfare, and public health, we've reached a fork in the road\u2014and Matthew Prescott's book will help you use yours to choose the right way.\"\n\n\u2014WAYNE PACELLE, PRESIDENT AND CEO OF THE HUMANE SOCIETY OF THE UNITED STATES\n\n\"Matthew Prescott has created an outstanding resource to support global change.\"\n\n\u2014JULIANAA SATIE, FOUNDER OF THE SCHOOL OF NATURAL COOKERY\n\n\"Matthew Prescott has offered a significant tool for anyone who wants to take action for the planet every time they eat.\"\n\n\u2014LINDSEY ALLEN, EXECUTIVE DIRECTOR OF RAINFOREST ACTION NETWORK\n\n\"A menu of ideas to help us eat in a more environmentally responsible and healthy way.\"\n\n\u2014JANET RANGANATHAN, WORLD RESOURCES INSTITUTE\n\n\"For your health, the environment, and animals\u2014this book has everything you need to make a difference!\"\n\n\u2014DORON PETERSAN, FARE WELL AND STICKY FINGERS BAKERY\n\n\"We can use our plates to help the planet. This book is a giant lightbulb switched on to help us.\"\n\n\u2014JOHN SCHLIMM, AUTHOR OF _FIVE YEARS IN HEAVEN_\n\n\"A timely resource to help us all eat better and tread lighter on the planet.\"\n\n\u2014JAMIE RAPPAPORT CLARK, PRESIDENT AND CEO OF DEFENDERS OF WILDLIFE\n\n\"For anyone wondering what they can do for the climate, this important book proves our everyday choices can have powerful consequences\u2014not to mention, some seriously delicious results.\"\n\n\u2014KEN BERLIN, PRESIDENT AND CEO OF THE CLIMATE REALITY PROJECT\n\n\"As this book shows, even slight changes in diet can help improve water quality.\"\n\n\u2014WILLIAM C. BAKER, PRESIDENT OF CHESAPEAKE BAY FOUNDATION\n\n\"We can fight for the planet and public health in the courts or when we eat. Matthew Prescott shows us how to make a difference with every meal.\"\n\n\u2014TRIP VAN NOPPEN, PRESIDENT OF EARTHJUSTICE\n\n\"We can stand up for the planet when we sit down to eat, and Matthew Prescott has given us straightforward, easy ways to do exactly that.\"\n\n\u2014TERRY TAMMINEN, CEO OF THE LEONARDO DICAPRIO FOUNDATION AND FORMER SECRETARY OF THE CALIFORNIA ENVIRONMENTAL PROTECTION AGENCY\n\n\"I get asked all the time how to do more to help animals, the planet, and people. I always say to start with what you do every day\u2014and we all eat! This book will help you feel better while helping others in an easy, great tasting way.\"\n\n\u2014KRISTIN BAUER VAN STRATEN\n\n\"Food tastes so much better when you know it is giving back to the world. This book highlights how and why.\"\n\n\u2014CHAD SARNO, COAUTHOR OF _CRAZY SEXY KITCHEN_\n\n\" _Food Is the Solution_ can help you invest in your health and the health of the Earth at every meal.\"\n\n\u2014LESLIE SAMUELRICH, PRESIDENT OF GREEN CENTURY CAPITAL MANAGEMENT\n\n\"This book is a great guide to a simpler, healthier, and more compassionate way to live.\"\n\n\u2014WENDIE MALICK\n\n# ABOUT THE AUTHOR\n\nMatthew Prescott is a leading figure in the global movement to reform how we farm and eat. He's Senior Food Policy Director for the Humane Society of the United States and advisor to the Good Food Institute. A sought-after speaker and thought leader, Prescott has spent over a decade and a half sharing his ideas with Ivy League universities, Fortune 500 companies, consumers, and more. His efforts have directly led to sweeping changes in the supply chains of hundreds of major food companies, impacted countless individuals' diets, and have been covered extensively by the media: his work has been featured by CNN, in the pages of the _New York Times_ , the _Wall Street Journal_ , the _Boston Globe_ , and countless more; his writings have appeared in _USA Today_ , the _Washington Post_ , _Barron's_ , and others; his photographs have been featured in _Rolling Stone_ and _Food & Wine_; he's been a guest on national television news programs; and he was even once a guest on NPR's _Wait Wait... Don't Tell Me_. He lives in Austin, Texas, with his wife, the writer Lara Prescott. You can sign up for author updates here.\n\n**Thank you for buying this**\n\n**Flatiron Books ebook.**\n\nTo receive special offers, bonus content,\n\nand info on new releases and other great reads,\n\nsign up for our newsletters.\n\nOr visit us online at\n\nus.macmillan.com\/newslettersignup\n\nFor email updates on the author, click here.\n\n# CONTENTS\n\nTitle Page\n\nCopyright Notice\n\nDedication\n\nFOREWORD\n\nby James Cameron\n\nINTRODUCTION\n\nFood Is the Solution\n\n[PART ONE:  \nOUR PLANET](part01.xhtml)\n\n**Earth**\n\n**Water**\n\n**Air**\n\n**Fire**\n\n[PART TWO:  \nOUR PLATES](part02.xhtml)\n\n**Breakfast**\n\n**Soups, Stews, Salads, and Sandwiches**\n\n**Main Dishes**\n\n**Sides and Extras**\n\n**Desserts**\n\nShopping for a Better World\n\nAcknowledgments\n\nCitations\n\nIndex\n\nPraise for Food is the Solution\n\nAbout the Author\n\nCopyright\nFOOD IS THE SOLUTION. Copyright \u00a9 2018 by Matthew Prescott. Foreword copyright \u00a9 2018 by James Cameron. All rights reserved. For information, address Flatiron Books, 175 Fifth Avenue, New York, N.Y. 10010.\n\nwww.flatironbooks.com\n\n\"Thanks, Giving\" by Jesse Eisenberg. Copyright \u00a9 2018 Jesse Eisenberg.\n\nCover designed by Amy Sly\n\nFood and cover photographs by Jessica Prescott\n\nOur eBooks may be purchased in bulk for promotional, educational, or business use. Please contact the Macmillan Corporate and Premium Sales Department at 1-800-221-7945, extension. 5442, or by e-mail at MacmillanSpecialMarkets@macmillan.com.\n\nThe Library of Congress Cataloging-in-Publication Data is available upon request.\n\neISBN 978-1-250-14446-1\n\nFirst Edition: March 2018\n\n# Contents\n\n  1. Cover\n  2. Title Page\n  3. Copyright Notice\n  4. Dedication\n  5. Foreword\n  6. Introduction\n    1. Food is the Solution\n  7. Part One: Our Planet\n    1. Earth\n    2. Water\n    3. Air\n    4. Fire\n  8. Part Two: Our Plates\n    1. Breakfast\n    2. Soups, Stews, Salads, and Sandwiches\n    3. Main Dishes\n    4. Sides and Extras\n    5. Desserts\n  9. Shopping for a Better World\n  10. Acknowledgments\n  11. Citations\n  12. Index\n  13. Praise for Food is the Solution\n  14. About the Author\n  15. Newsletter Sign-up\n  16. Contents\n  17. Copyright\n\n## Guide\n\n  1. Start Reading\n  2. Cover\n  3. Copyright\n  4. Table of Contents\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nEAT WELL ON $4\/DAY\n\nGOOD\n\nAND\n\nCHEAP\n\nLEANNE BROWN\n\nWorkman Publishing \u2022 New York\nContents\n\nIntroduction\n\nBreakfast\n\nTomato Scrambled Eggs\n\nOmelet\n\nEgg Sandwiches with Mushroom Hash\n\nPeanut Butter and Jelly Granola Bars\n\nOatmeal\n\nBreakfast Quinoa\n\nBanana Pancakes\n\nWhole-Wheat Jalape\u00f1o Cheddar Scones\n\nChocolate Zucchini Muffins\n\nBroiled Grapefruit\n\nSoup and Salad\n\nDal\n\nCorn Soup\n\nFrench Onion Soup\n\nLightly Curried Butternut Squash Soup\n\nTomato Soup and Grilled Cheese\n\nSweet or Savory Pineapple Salad\n\nKale Caesar Salad\n\nBroiled Eggplant Salad\n\nBeet and Chickpea Salad\n\nEver-Popular Potato Salad\n\nSpicy Panzanella\n\nCold (and Spicy?) Asian Noodles\n\nTaco Salad\n\nCharred Summer Salad\n\nWilted Cabbage Salad\n\nBroccoli Apple Salad\n\nSnacks, Sides, and Small Bites\n\nBrussels Sprout Hash and Eggs\n\nMexican Street Corn\n\nMy Dad's Baked Beans\n\nSmoky and Spicy Roasted Cauliflower\n\nCrispy Chickpeas and Pumpkin Seeds\n\nSpicy Green Beans\n\nPopcorn!\n\nMashed Beets\n\nMashed Cauliflower\n\nWinter Squash Puree\n\nMashed Celery Root\n\nBubble and Squeak\n\nJacket Sweet Potatoes\n\nRoasted Potatoes with Chiles\n\nPoutine\n\nThings on Toast\n\nCornmeal-Crusted Veggies\n\nGreen Chile and Cheddar Quesadillas\n\nDinner\n\nFilipino Chicken Adobo\n\nRoast Chicken\n\nPeanut Chicken and Broccoli with Coconut Rice\n\nBeef Stroganoff\n\nSpicy Broiled Tilapia with Lime\n\nCreamy Zucchini Fettuccine\n\nPasta with Eggplant and Tomato\n\nTomato and Tuna Spaghetti\n\nCauliflower Cheese\n\nSavory Summer Cobbler\n\nBarley Risotto with Peas\n\nVegetable Jambalaya\n\nSpicy, Crunchy, Creamy Polenta\n\nTofu Hot Pot\n\nDeconstructed Cabbage Rolls\n\nRoasted Vegetables\n\nChana Masala\n\nBlack-Eyed Peas and Collards\n\nVegetable Quiche, Hold the Crust\n\nBroccoli, Egg, and Cheddar Empanadas\n\nPotato and Kale Rolls with Raita\n\nStuff on Hot Dogs\n\nPotato Leek Pizzas\n\nBroccoli Rabe and Mozzarella Calzones\n\nHalf-Veggie Burgers\n\nBig Batch\n\nBest Tomato Sauce\n\nChorizo and White Bean Rag\u00f9\n\nDark and Spicy Chili\n\nSpicy Pulled Pork\n\nHummus\n\nDeviled Eggs\n\nPierogi\n\nDumplings\n\nPantry\n\nPeanut Sauce\n\nSalsa\n\nTzatziki\n\nRaita\n\nFlavor\n\nSpice Oil\n\nRoti\n\nFlour Tortillas\n\nPizza Dough\n\nCroutons or Breadcrumbs\n\nFresh Pasta\n\nRicotta\n\nRainbow Rice\n\nHow to Cook Dried Beans\n\nPickle Primer\n\nDrinks and Desserts\n\nAgua Fresca\n\nSmoothies\n\nAvocado Milkshake\n\nFast Melon Sorbet\n\nRice Pudding\n\nCoconut and Lime Brown Rice Pudding\n\nCaramelized Bananas\n\nCoconut Chocolate Cookies\n\nPeach Coffee Cake\n\nConversion Tables\n\nAcknowledgments\n\nAbout the Author\nIntroduction\n\nEating is one of life's greatest pleasures. In a perfect world, healthy and delicious food would be all around us. It would be easy to choose and easy to enjoy.\n\nBut of course, it's not a perfect world. There are thousands of barriers that can keep us from eating in a way that nourishes our bodies and satisfies our tastes. Money just shouldn't be one of them.\n\n**Kitchen skill, not budget, is the key to great food.** This cookbook is a celebration of the many delicious meals available to those on even the strictest of budgets.\n\nEating on a limited budget is not easy, and there are times when a tough week can turn mealtime into a chore. As one woman told me, \"I'm weary of the 'what's for dinner?' game.\" I hope the recipes and techniques in this book can help make those times rare and the tough choices a little more bearable.\n\nAt the same time, this book is not a meal plan\u2014those are much too individual to share on a wide scale. Every person and every family has specific needs and unique tastes. We live in different regions, different neighborhoods, and with varying means. One book cannot account for all of that, but I hope it can be a spark, a general strategy, a flexible set of approachable and cheap recipes. The rest is up to you.\n\nI think you'll find (or perhaps have already found) that learning to cook has a powerfully positive effect. **If you can become a more skilled, more conscious cook, then you'll be able to conjure deliciousness in any kitchen, anytime.** Good cooking alone can't solve hunger in America, but it can make life happier\u2014and that is worth every effort.\n\nJust as a good meal is best shared with others, so is a good recipe. I may not be able to share a meal with you, but I'd love to offer a few ideas. **What's for dinner? Here's my answer.**\n\nAbout This Book I created an earlier version of this book as the capstone project for my master's degree in food studies at New York University. After I posted a free PDF on my website (leannebrown.com), it went viral on Reddit, Tumblr, and elsewhere, racking up almost 100,000 downloads in the first few weeks. That support gave me the courage to launch a Kickstarter campaign to get printed copies of _Good and Cheap_ into the hands of people who don't have computers or who wouldn't otherwise see it. Thousands of generous supporters contributed to the campaign, donating more than 9,000 free copies of the printed book and sponsoring twenty new recipes. That first printed edition of the book was self-published, and sold out within a few months. The PDF was downloaded about 500,000 times within six months after it was first posted. What you hold in your hands is a second edition, including 30 new recipes. The experience changed my life. I hope this book changes yours.\n\nMy Philosophy The best health advice is simple: Eat fruits and vegetables. Many American cookbooks rely on meat as the central feature of a meal. **My recipes celebrate the vegetables rather than the meat.** My intent was to create satisfying food that doesn't require you to supplement your meals with cheap carbohydrates to stave off hunger. **I strove to create recipes that use money carefully, without being purely slavish to the bottom line.** For example, many recipes use butter rather than oil. Butter is not cheap, but it creates flavor, crunch, and richness in a way that cheap oils never can.\n\nI'm not a dietitian, and this isn't a diet book. I'm just a home cook, like you. If you have dietary restrictions, some recipes won't work for you as written, but that's fine\u2014you can adapt them to your needs, or just turn the page and keep looking for inspiration.\n\n**More than a book of recipes, this is a book of ideas.** I want you to tailor things to your taste. Improvisation is the soul of great cooking! If it doesn't work out every time, I hope you'll forgive me. More important, forgive yourself, and try again.\n\nA Note on $4\/Day\n\nI designed these recipes to fit the budgets of people living on SNAP, the US federal program that used to be called food stamps. If you're on SNAP, you already know that the benefit formulas are complicated, but the rule of thumb is that you end up with $4 per person, per day, to spend on food.\n\nT his book isn't challenging you to live on so little\u2014it's a resource in case that's your reality. In May 2014, there were 46 million Americans on food stamps. Untold millions more\u2014in particular, retirees and students\u2014live under similar constraints.\n\nIf you're in Canada, or anywhere else outside the United States, it might seem like this book won't apply to you. While the specifics may differ, learning to cook and take joy in simple, real food is great for anyone, anywhere.\n\nT he costs for each recipe are based on two sources. For the list of grocery ideas on page xiii, I collected prices from four grocery stores in a diverse neighborhood in New York. For specific spices and a wider variety of fruits and vegetables, I looked at online grocery stores or nationwide averages collected by the Bureau of Labor Statistics. The prices for fruits and vegetables assume that they're roughly in season, when you can get the best deals. This means, unfortunately, that you'll pay a lot more if you want to make peach coffee cake in February. I talk more about shopping in season on the following pages.\n\nThe estimates are, by necessity, a snapshot of place and time. Costs will vary in other cities, other neighborhoods, even just other stores. Please think of the numbers as a guideline, not a guarantee.\n\nMore than in most cookbooks, my recipes are flexible and encourage substitution based on availability, price, and personal tastes. A strict budget requires flexibility and a willingness to say, \"That's a good deal this week, so it's what I'll be cooking!\" Don't worry, you'll pick up the tricks quickly.\n\nA few recipes call for fancy kitchen equipment, but in my work with low-income families in New York, I've found that items like blenders, food processors, and electric mixers are fairly common. I did not, however, attempt to tackle the very real situation of people who have no kitchen, no equipment, and no space to prepare food. I simply cannot hope to do those issues justice within the bounds of one cookbook. Let's all agree that we need to keep striving to address all the issues that make it difficult for so many people to eat well.\n\nTips for Eating and Shopping Well\n\n1 buy foods that can be used in multiple meals\n\nVersatile ingredients save meals. If you buy flour, you can make Tortillas, Roti, Scones, and Pancakes. If you buy canned tomatoes, you can make soup, sauce, even chili. Need I even mention the versatility of garlic or lemons? If you always keep them around, you can make anything else taste fantastic.\n\n2 buy in bulk\n\nBuying larger amounts of one item can usually bring down the price per unit. When you're working within a tight budget, you won't always be able to afford to shop for the future, but you should do it when you can. And, of course, keep storage in mind: If the item will go bad before you can finish it, get the smaller size. Only buy what you can eat. If you buy versatile ingredients in slightly larger amounts, then you'll be able to use them quickly but still make diverse meals.\n\n3 start building a pantry\n\nIf possible\u2014and admittedly this can be difficult for people living on their own\u2014reserve part of your budget to buy one or two semi-expensive pantry items each week or month. Things like olive oil, soy sauce, and spices are pricey at first, but if you use just a little with each recipe, they go a long way. With turmeric, coriander, cumin, and chili powder, you'll suddenly have a world of flavor on your shelf. For specific advice.\n\n4 think weekly\n\nEach week, mix things up by buying different varieties of staple foods like grains and beans. This week, you might have omelet every morning with Dark and Spicy Chili later in the day, but next week you'll have yogurt for breakfast and Hummus or Chana Masala for lunch and dinner. If you have time to shop frequently, pick up smaller amounts of produce every couple of days to ensure everything is fresh. It's a lot more inspiring to pull crisp greens out of the fridge than to unstick a wilted mess from the bottom of the veggie drawer. If you can't shop as often, consider getting canned or frozen versions of the vegetables you won't use immediately.\n\n5 think seasonally\n\nDuring their local growing season, fruits and vegetables are generally cheaper and definitely tastier than outside of season. You'll notice that orange prices shoot up during the summer, yet what's available is drab and flavorless. But oranges are abundant in December and January, the peak of their season, and that's reflected in the price. At the end of summer, you can get bags of zucchini for next to nothing. Brussels sprouts are also very seasonal, coming on sale around Thanksgiving. Enjoy as much of the summer and fall produce as possible, because you'll be more limited in the winter. Then again, simmering and roasting winter vegetables is a fine way to warm up your house, and tough winter roots are easy to store. In addition, winter is a great time to search for deals on canned and frozen produce. Seasons for fruits and vegetables vary depending on where you live, so consult a local guide to growing seasons (or the chart) and use it to shop for the best deals.\n\n6 more vegetables = more flavor\n\nNothing livens up a bowl of rice like summer squash and corn! Vegetables make the best sauces: They're earthy, bright, tart, sweet, bitter, and savory. Give them a treasured spot at the top of your grocery list and you'll never be bored.\n\n7 always buy eggs\n\nWith these babies in your fridge, you're only minutes away from a satisfying meal. Scramble an egg with leftovers or drop an egg on top of a salad or a plate of stir-fried vegetables, and deliciousness is guaranteed.\n\n8 buy expensive eggs if you can\n\nFree-range or organic eggs are usually worth the money\u2014they taste so much better than regular eggs. Even at $4 a dozen, you're still only paying 33 cents per egg. Really fresh eggs, like those from a farmers' market, make a big difference in flavor.\n\n9 be careful with undercooked eggs\n\nVery rarely, raw eggs can be infected with salmonella. Many classic recipes, from mayonnaise to eggnog to Caesar dressing, are prepared with raw egg yolk, but technically only a fully cooked egg is guaranteed to be free of salmonella. Consequently, raw or runny eggs are not recommended for infants, the elderly, pregnant women, or anyone with a weakened immune system.\n\n10 buy fresh bread\n\nTry to buy fresh loaves of interesting bread from an independent bakery or the bakery in your grocery store. Although fresh loaves don't last as long as sliced bread, they're much more enjoyable, and you can use the old stuff to make Panzanella, or Croutons or Breadcrumbs to top other dishes. Later in the day, many independent bakeries offer deep discounts on bread they would otherwise have to throw out.\n\n11 don't buy drinks\n\nAll the body needs drink-wise is water. Except for milk, most packaged drinks are overpriced and deliver a lot of sugar without filling you up the way a piece of fruit or a bowl of yogurt would. If you want a special drink, make Agua Fresca, a smoothie, or tea.\n\n12 get creative with wilted vegetables\n\nSometimes you forget a pepper or bunch of spinach in the back of the fridge. Although wilted veggies might not remain fit for a salad, they'll still be wonderful in any dish that calls for saut\u00e9ed, grated, or baked vegetables. Just cut off any actual rot. You can also use them in broth.\n\n13 make your own broth\n\nIn almost any savory recipe that calls for water, homemade broth would be better. To make broth, start by saving any vegetable bits that you chop off and would normally throw away, like onion tops, the seedy parts of peppers, and the ends of carrots. Store them in the freezer until you have a few cups, then cover them with water, bring to a boil, and simmer over low heat for a few hours. Add salt to taste, and you have broth! To make a heartier broth, do the same with leftover bones or scraps of meat (preferably all the same kind of meat). Because you're using stuff you'd otherwise throw away, broth is effectively free.\n\n14 treat your freezer with respect\n\nA freezer can be a great friend for saving time by letting you prepare large batches of food at once. For instance, cooking dried beans takes a while, so make more than you need, then freeze the rest. Another great trick I learned from a reader is to dice a whole package of bacon, fry it, and then freeze it in small parcels. This makes it easy to add a small amount of bacon to a dish without the temptation of using the whole package or the fear of rancid meat.\n\n15 turn chicken skin into schmaltz\n\nSchmaltz is rendered chicken fat that you can use like butter. Buy chicken that still has its skin, then trim the skins and lay them in a pan over low heat. Add a cup or so of water and simmer until the fat releases from the skin and the water cooks off. Let the fat cool, then throw away the skins and pour the fat into a glass jar. Store in the fridge.\n\n16 buy a pepper grinder\n\nSeriously, banish pre-ground pepper from your life\u2014it loses all flavor when it sits around. Fresh pepper creates pops of intensity on the tongue and lights up bland dishes. One of the most popular dishes in Rome is just pasta with butter and pepper: Give it a try!\n\n17 buy yogurt in bulk\n\nThere are so many types of yogurt in the grocery store: some low in fat and high in sugar, some with cute animal pictures. Some are Greek. Some have chocolate shavings and candy. Some have names like \"Key lime pie.\" Now forget about all of that. The buckets of plain yogurt are the best value for your money. If you start with plain yogurt, you can make your favorite flavors in your own kitchen, where you know exactly what's going into it. The fat content is your choice.\n\nIf you have kids, ask them what flavors they can imagine and go make them! It's a lot more fun than letting the supermarket choose for you. Try something new and smash it in! If you want to make thicker Greek-style yogurt, all you have to do is strain the standard variety through cheesecloth to remove the extra water.\n\nYogurt's versatility makes it a great staple to keep in the fridge. And when you consider that there are savory options\u2014like Tzatziki or Raita\u2014the possibilities expand even more.\n\n1 Bananas\n\n2 Peanut butter\n\n3 Frozen or dried fruit\n\n4 Yogurt\n\n5 Lime\n\n6 Honey\n\n7 Coconut\n\n8 More yogurt!\n\n9 Fresh berries\n\n10 Kiwi fruit\n\n11 Red and green grapes\n\n12 Jam or jelly\n\nsupermarket strategies\n\n**With these commonly available items in your pantry, you can have a wide variety of meals on the table within minutes.** Keeping a well-stocked pantry is the key to easy, fast cooking at home. When you're living on a budget, building up supplies does take time, but just keep adding each week and you'll get there.\n\n1 protein\n\nMeat isn't the only protein! Items like dried beans, nuts, and eggs are cheap, store easily, and have multiple uses. Be aware that most fish at the grocery store has previously been frozen and was merely thawed for display. There's no harm in buying it frozen and thawing it yourself.\n\n2 dairy\n\nButter is just as good to cook with as it is on toast. The cheeses listed opposite are the ones I like best, but buy what your taste, budget, and local availability allow.\n\n3 vegetables\n\nVegetables can (and should!) be the base of most meals. Other than greens, which should be used quickly, they can be stored for a few days to a few weeks. Try to snag each vegetable as it hits peak season.\n\n4 fruits\n\nCitrus fruits are cooking essentials and they keep well. The zest and juice can liven up just about any dish, and they always make a great dressing. Bananas, apples, and melons are great quick snacks, but try every fruit you can afford! Remember, almost all fruits and vegetables have a season, so savor them at their freshest and cheapest.\n\n5 grains\n\nFlour is so inexpensive, and once you have a few basics at hand, most baked goods are a cinch to make. There's great variety in whole grains. Substitute them for rice, toss them in a salad, or add them to soup.\n\n6 canned vegetables\n\nPlenty of vegetables are good when canned, so remember to compare prices between fresh, frozen, and canned. The canned versions are fantastic in sauces. Just be aware that canned foods are often very salty, so you might want to rinse them, except for canned tomatoes.\n\n7 frozen fruits and vegetables\n\nFresh berries can be expensive, but the frozen ones often go on sale and are great for smoothies. Frozen veggies are quick to add to soups and rice dishes. Compare prices to see whether frozen is the best value.\n\n8 flavor\n\nYou can explore an extraordinary number of cuisines with these items. They add depth and excitement to the simplest dishes.\n\n9 treats that go a long way\n\nAlthough specialty items can be expensive, a little goes a long way. When you can, pick up an item or two and enjoy the results.\n\n10 spices\n\nSpices are expensive to buy and can often be a sticking point: no caloric value and they often have a high price tag. But because you use such small amounts, they end up costing pennies per recipe. If you're able to shop around, look for inexpensive spices in bulk at ethnic markets. I've suggested (below) the spices you should start with. I use these a lot in these recipes and at home. If you want to branch out further, take a look at page  to be inspired by flavor combinations.\n\nGroceries you won't regret buying\n\nprotein\n\neggs\n\ndried beans\n\nlentils\n\ntofu\n\nnuts\n\npeanut butter\n\ndairy\n\nbutter\n\nmilk\n\nyogurt\n\nqueso fresco\n\nRomano or Parmesan cheese\n\nsharp Cheddar cheese\n\nmozzarella\n\nvegetables\n\ngarlic\n\nonions\n\ncarrots\n\ncelery\n\npeppers\n\nbroccoli\n\ntomatoes\n\nhot peppers\n\nhardy greens\n\nsalad greens\n\npotatoes\n\nsweet potatoes\n\ncauliflower\n\nwinter squash\n\nfruits\n\napples\n\nmelons\n\noranges\n\nlimes\n\nlemons\n\nbananas\n\ngrains\n\nbread\n\ntortillas\n\npasta\n\nall-purpose flour\n\nwhole-wheat flour\n\noats\n\npopcorn\n\nshort-grain rice\n\nlong-grain rice\n\nbrown rice\n\ncornmeal\n\nother dried whole grains\n\ncanned vegetables\n\nwhole tomatoes\n\ntomato paste\n\ncorn\n\nfrozen fruits and vegetables\n\nberries\n\npeas\n\ngreen beans\n\ncorn\n\nflavor\n\nolive oil or vegetable oil\n\nwine vinegar\n\nanchovies\n\nsardines\n\nolives\n\nfish sauce\n\ncoconut milk\n\nmiso paste\n\nmustard\n\nsoy sauce\n\nchili sauce\n\nbrown sugar\n\nfresh herbs\n\ntreats that go a long way\n\ndried fruits\n\ndried mushrooms\n\nfrozen shrimp\n\nmaple syrup\n\nbacon\n\nvanilla extract\n\ncocoa powder\n\nspices\n\nchile flakes\n\ncinnamon\n\ncumin or cumin seeds\n\npaprika and smoked paprika\n\ncurry powder\n\noregano\n\nthyme\n\nLeftovers\n\n**Leftovers may be convenient,** but they can seem unappealing, limp, and cold after sitting in the fridge for a couple of days. That's why the sandwich, the wrap, and the taco are your friends. Here are just a few ideas for how to give leftovers a makeover very quickly to invent a whole new meal!\n\n1 chana masala wrap\n\nSounds strange, but trust me: Spread herbed mayo on the wrap and pile in the Chana Masala.\n\n2 black-eyed peas and collards wrap\n\nFold the Black-Eyed Peas and Collards into a wrap with a little hot sauce or some Tzatziki.\n\n3 tomato scrambled eggs wrap\n\nThrow the Tomato Scrambled Eggs into a wrap and add some roasted potatoes or rice for bulk.\n\n4 vegetable jambalaya burrito\n\nAdd some salsa or any leftover beans to the Vegetable Jambalaya and wrap in a flour tortilla.\n\n5 cauliflower cheese sandwich\n\nStart with cauliflower cheese, and add crunchy greens, toast, and mustard. Yum!\n\n6 Roasted Vegetables sandwich\n\nAdd some extra spices or sauces to liven up the vegetables and grill the bread for some crunch.\n\n7 Roasted Potatoes and chile taco\n\nThis dish is great in a taco: Just add a little salsa and grated cheese. I like green salsa here.\n\n8 tilapia taco\n\nFor a makeshift fish taco, add the Tilapia, Wilted Cabbage, and cilantro to a tortilla and enjoy.\n\n9 cauliflower tacos\n\nCombine Smoky and Smoky and Spicy Roasted Cauliflower with salsa and grated Cheddar or cotija cheese in a warm tortilla.\n\n10 fancied-up poutine\n\nPretend you're at a super-modern poutinery and make up some crazy toppings for your baked French fries. Black-Eyed Peas and Collards would be great, as would chili, Baked Beans, or Filipino Chicken Adobo.\n\n11 jacket sweet potatoes spread\n\nMash up leftover baked Jacket Sweet Potatoes, then spread them in a bacon sandwich for a sweet counterpoint.\n\n12 rethink toast toppings\n\nAny of the recipes that can top toast would also be great over rice or any other grain, wrapped in a tortilla, tossed with pasta, or even on a pizza. Or, take a pack of ramen noodles, skip the flavor packet, and throw in one of the toast toppings instead.\n\n13 chili three ways\n\nTake some chili, stuff into Jacket Sweet Potatoes, serve over roasted vegetables, or top hot dogs.\n\n14 reinvigorate veggies\n\nGot some veggies that look past their prime? Try them in any of the soups (pages 21\u201329) or Dark and Spicy Chili, Bubble and Squeak, in scrambled eggs, or cooked into a tomato sauce. The Broiled Eggplant Salad is lovely tossed with noodles.\n\nMashed winter vegetables make a great pierogi filling, can be added to scrambled eggs, or used as a sandwich filling.\n\nseasonal chart\n\n**Here's a chart** of some common fruits and vegetables with shading that represents the months when it's best to buy them. Generally, in-season produce is likely to be less expensive. Of course, where you live also makes a difference in cost, so make sure to check the specials when you go to the grocery store.\n\nThe print edition of this book includes a chart. Please download a PDF of this chart here: workman.com\/ebookdownloads\n\nKitchen Equipment\n\n**It might seem like** a daunting task to stock your kitchen with equipment, but it doesn't need to be scary\u2014or expensive. The tools below will help you make any recipe in this book.\n\n1 good knives\n\n\u2022 A good chef's knife is the most important tool in your kitchen. Make sure it's big and sharp.\n\n\u2022 You'll need a paring knife for smaller tasks, like peeling and coring an apple.\n\n\u2022 Get a serrated knife to cut bread and tomatoes easily.\n\n2 box grater\n\n_Use it to:_\n\n\u2022 Grate cheese\n\n\u2022 Shred potatoes\n\n\u2022 Make quick work of tough vegetables\n\n3 measuring cups and spoons\n\n4 essential pots and pans\n\n\u2022 Large cast-iron or nonstick pan\n\n\u2022 Medium-size saucepan\n\n\u2022 Large soup pot\n\n5 stirring utensils\n\n\u2022 You can stir anything with a long-handled wooden spoon.\n\n\u2022 A ladle is essential for soups, stews, and sauces.\n\n\u2022 While you can often use a fork instead, get a whisk if you're serious about sauces or desserts.\n\n6 roasting\n\n\u2022 Use oven-safe dishes and baking pans\u2014glass, ceramic, etc.\u2014for broiling, baking, and roasting.\n\n\u2022 Casserole dishes are handy for storing leftovers.\n\n7 sieve\n\n_Use it to:_\n\n\u2022 Drain pasta or boiled veggies\n\n\u2022 Sift flour\n\n\u2022 Strain extra whey from yogurt\n\n8 microplane\n\n_Use it to:_\n\n\u2022 Zest lemons and limes\n\n\u2022 Grate hard cheeses and garlic\n\n\u2022 Shred soft vegetables into a sauce\n\n9 specialty baking\n\n\u2022 Muffin tins can be used for baking in small portions (not just muffins!).\n\n\u2022 It's hard to make a cake without a cake pan.\n\n\u2022 Rimmed baking sheets can also be used to roast veggies or broil fish.\n\n10 immersion blender\n\n\u2022 Use it to puree soups and smoothies.\n\n\u2022 It's more versatile and quicker to clean than traditional blenders.\n\n\u2022 If you need power, consider investing in a food processor or full-size blender.\n\n11 flipping and shuffling utensils\n\n\u2022 Flat spatulas are for flipping pancakes.\n\n\u2022 Rounded spatulas are good to scrape bowls.\n\n\u2022 Use tongs for salads, or moving hot foods without hurting yourself.\n\n12 cutting board\n\n\u2022 Wood is longest lasting and, contrary to what you might think, the most sanitary surface for preparing raw meat. There's a reason it's called butcher block!\n\n\u2022 Cheap plastic boards are fine and easy to wash.\n\n\u2022 Don't get glass. Just don't.\n\nHow to Use This Book\n\n_Good and Cheap_ is a strategy guide, not a typical cookbook. Sure, we have _breakfast, dinner,_ and _snacks and sides._ But there is also _big batch_ for feeding a crowd or planning ahead. There is the _pantry_ section filled with the staples we eat every day and the sauces that make those basics sing. The _drinks and desserts_ are delicious and worth the effort, while making use of everyday ingredients you'll buy for other meals. The _ideas_ pages show just how much variety there is in simple things like oatmeal or popcorn. And the _methods_ are meant to teach you a process that you can use over and over again.\n\nOnce you embrace cooking, you learn that there are no rules for the best oatmeal, there's just _your_ best oatmeal. More practically, you realize that many ingredients are used in similar ways and can be easily substituted. If there is a sale on red lentils or a neighbor gives you a bag of zucchini, I want you to be armed with the skills to take advantage of that, not shackled to an inflexible recipe. The recipes in this book are a starting point. My hope is that with _Good and Cheap_ as a foundation, you'll learn to cook without recipes and be empowered to cook for your own pleasure.\nBREAKFAST\n\nTomato Scrambled Eggs\n\n$1.80 \/ serving\n\n$3.60 total\n\nThis pile of fluffy, creamy eggs holds together a mass of tangy, juicy, sweet tomatoes. This dish is best enjoyed when tomatoes are in season. Serve the eggs on, in, or alongside some toast or a tortilla, if you have any on hand. serves 2\n\n\u00bd tablespoon butter\n\n4 small or 2 large, chopped fresh tomatoes, or 2 cups chopped canned tomatoes\n\n4 eggs\n\nsalt and pepper, to taste\n\nadditions\n\nsprinkling of chopped fresh basil or other herbs\n\n1 Melt the butter in a small or medium nonstick pan over medium heat, swirling it around to coat the pan. Add the tomatoes and cook until they release their juice and most of it evaporates, 5 to 7 minutes.\n\n2 Meanwhile, crack the eggs into a bowl and add a generous sprinkling of salt and pepper. Beat the eggs lightly with a fork.\n\n3 Turn the heat down to low and add the eggs to the pan. Using a spatula, gently mix the eggs with the tomatoes and stir carefully and continuously, to keep the eggs from forming chunks. Turn the heat down as low as possible; the slower your eggs cook, the creamier they'll be.\n\n4 Once the eggs are cooked to your desired consistency, turn off the heat and add any chopped herbs. Basil is the best with tomatoes. I like my scrambled eggs loose and juicy with the eggs forming very soft curds. You can let them cook a little longer if you prefer your eggs drier.\n\nOmelet\n\n$0.80 \/ serving\n\n$1.60 total\n\nI make this omelet at least once a week. It's insanely delicious, whether laden with veggies or kept simple. I love it with dill, but it's good with almost any herb, or with scallions. I tend to use Cheddar and Romano, since they go well with most vegetables. But other great combos are goat cheese with zucchini, and cauliflower with sausage and chile flakes.\n\nIf I'm serving two people, I usually cut one large omelet in half rather than making two. However, if you feel like being fancy, make a pair of two-egg omelets simply by using half the ingredients for each. For a French style omelet, roll it up in Step 4 instead of folding it. The result will be quite thin and tender. serves 2\n\n4 eggs\n\n2 tablespoons finely chopped fresh dill\n\nsalt and pepper, to taste\n\nbutter, for the pan\n\n1 shallot or \u00bd small red onion, finely diced\n\n\u00bc cup grated cheese\n\n1 Place the eggs, dill, salt, and pepper in a bowl, and beat with a fork.\n\n2 Melt a small pat of butter in a big frying pan over medium-high heat. Once it's sizzling, add the shallot and saut\u00e9 for about 2 minutes, until it's translucent and smells great.\n\n3 Add the egg to the hot pan and swirl it to coat the surface evenly. If the center of the omelet cooks more quickly than the edge, use a spatula to pull any raw egg into the middle. Then stop touching it.\n\n4 After about 30 seconds, toss the cheese on top of the egg, along with any other raw or cooked vegetable you feel like adding. Once none of the egg remains translucent, 30 seconds to 2 minutes, fold the omelet in half with your spatula and lift it out of the pan. You don't want any brown on your eggs.\n\nEgg Sandwiches with Mushroom Hash\n\n$1.80 \/ sandwich\n\n$3.60 total\n\nEgg sandwiches are a mainstay of every corner deli in New York City, and for good reason: They're cheap and easy, fast, and delicious. I knew I had to include one when Charlene, one of my early supporters, asked for a recipe with eggs and mushrooms. (I'm thankful she did! I don't really like mushrooms, so they're scarce in this book, even though plenty of people love them.) Like most sandwiches, this recipe is really flexible. In particular, you can change the hash to use whatever you have around. Sad leftovers take on new life when turned into a hash and matched with the rich fattiness of a runny egg. serves 2\n\n2 teaspoons butter\n\n1 small potato, diced\n\nsalt and pepper, to taste\n\n8 ounces mushrooms, sliced\n\n2 cloves garlic, finely chopped\n\n2 rolls or English muffins, split, or 4 slices of bread\n\n2 eggs\n\nadditions\n\nsliced tomato\n\npitted, peeled, and sliced avocado\n\ncheese\n\nvariations\n\npotato and onion\n\npotato and pea\n\ncollards and bacon\n\nzucchini\n\nchorizo and green chile\n\n1 Melt 1 teaspoon of the butter in a pan over medium heat. Throw in the potato pieces and cook, stirring minimally, until they just start to brown and soften, about 5 minutes. Season them with salt and pepper.\n\n2 Add the mushrooms and garlic and cook, stirring, until the mushrooms are brown and have shrunk down, another 5 minutes. If the potato pieces are getting stuck to the pan, add a splash of water. Pierce one piece of potato with a fork to test it. If it goes through easily, you're done. If not, cook for a few more minutes. (The smaller the potatoes are chopped, the quicker they'll cook.) Taste and adjust the seasoning to your preferences.\n\n3 Place the rolls or bread in the toaster. Meanwhile, melt the other teaspoon of butter in a medium, nonstick skillet over medium heat. Crack the eggs in and dust with salt and pepper.\n\n4 _If you like your eggs sunny-side up_ , place a lid over the pan to ensure that the whites will cook through without making the yolks hard. Once the whites are no longer translucent, turn off the heat and remove the eggs from the pan.\n\n_If you like eggs over easy (my favorite)_ , wait until the yolks are cooked but still look runny, then flip each egg with a spatula and let the other side cook for about 15 seconds. That'll get your whites fully cooked, but keep the yolks runny\u2014the best. If you prefer hard yolks (please, no!), then cook for another 30 seconds.\n\n5 Moving quickly so everything stays nice and hot, assemble the ingredients into a sandwich, layering the veggies first, topping with the egg, and using any condiments you like. Way better than what you'll find at the corner deli!\n\nPeanut Butter and Jelly Granola Bars\n\n$0.30 \/ bar\n\n$3.60 total\n\nTired of endless PB+J sandwiches? Give these bars a try instead! I designed them for my friend Alex, the best long-distance runner I know. They are a little more crumbly than a store-bought granola bar, so be careful when eating these on the go\u2014you'll probably leave a trail of crumbs on the sidewalk and down your shirt. As a bonus, you can find the ingredients in any corner store or food pantry. Any kind of jam or jelly will do\u2014I used blueberry, but grape, strawberry, or any flavor would be tasty. You can use quick-cooking oats if they're all you have, but I prefer the bite and chew of rolled oats. For a little more crunch along with your chew, you can also substitute a cup of Rice Krispies for a cup of the oats. makes 12 bars\n\nbutter or vegetable oil, for the pan\n\n3 cups rolled oats, or 2 cups rolled oats and 1 cup Rice Krispies\n\n\u00bd cup peanut butter\n\n\u00bd cup jelly or jam\n\n\u00bc cup hot water\n\n\u00bc teaspoon salt\n\nadditions\n\n\u00bd cup chopped nuts\n\n\u00bd cup shredded coconut\n\n\u00bd cup chopped dried fruit\n\n\u00bd cup honey (instead of the jelly)\n\n1 Preheat the oven to 350\u00b0F.\n\n2 Grease an 8- by 11-inch baking pan. If you have a different size pan, that's fine\u2014it'll just change how thick the bars are, so you'll want to adjust the cooking time a little. A larger pan may take less time to cook through and get crunchy, while a smaller pan will take a few more minutes\u2014just keep an eye on it in the oven.\n\n3 Pour the oats into a large bowl.\n\n4 Place a small saucepan over low heat and add the peanut butter, half of the jelly, the water, the salt, and any other additions. Stir until smooth, about 2 minutes.\n\n5 Pour the peanut butter and jelly concoction into the oats and mix until all the oats are coated and you have a sticky mass. Dump the mixture into the buttered baking pan and press it into an even layer. Spread the remaining jelly over the top.\n\n6 Pop the baking dish into the oven for 25 minutes. The bars are done when the edges are toasty and brown. Mmm. Crunchy.\n\n7 Leave the bars in the pan until they cool completely, about an hour, then slice them into 12 bars.\n\nideas\n\nOatmeal\n\n$0.15 \/ serving\n\n$0.30 total\n\n**Oatmeal is a hot and comforting breakfast** that will give you energy for a great morning. It's also extremely inexpensive, so it will allow you to spend a bit more on lunch and dinner. Basic oatmeal has a reputation for being dull, but this recipe can be dressed up in so many ways, you'll never get bored.\n\nThink of basic oatmeal as a foundation. Make it your own with the ideas on the next pages. Whether it's milky and sweet or savory and salty, I'm sure you can find a favorite way to enjoy a hot bowl of oats in the morning!\n\nBasic Oatmeal\n\nserves 2\n\n1 cup rolled oats\n\n2 cups water\n\n\u00bc teaspoon salt\n\n1 Add the oats, water, and salt to a small pot and bring to a boil over medium heat. Immediately turn the heat to low and place a lid on the pot.\n\n2 Cook for 5 minutes, until the oats are soft and tender and most of the water has evaporated. You can add more water if you like your oatmeal smooth and thin, or use slightly less to make it thick and creamy.\n\n1 Pumpkin\n\n$0.75 serving \/ $1.50 total\n\n\u00bd cup canned pumpkin\n\n\u00be cup milk (or almond milk or soy milk)\n\n2 tablespoons brown sugar, plus more to taste\n\n1 teaspoon ground cinnamon\n\nadditions\n\n\u00bc teaspoon ground ginger\n\n\u00bc teaspoon ground cloves\n\ndrizzling of maple syrup\n\nWhisk the pumpkin, milk, and 1\u00bc cups water (not the full 2 cups in the basic recipe) in a pot. Add the oats, salt, brown sugar, and spices. Cook over medium-low heat until the mixture just comes to a boil, 2 to 5 minutes. Turn to low for 5 more minutes. Add maple syrup or more sugar to taste.\n\n2 Savory\n\n$0.75 serving \/ $1.50 total\n\n2 or 3 scallions, white and green parts, finely chopped\n\n\u00bc cup grated sharp Cheddar cheese\n\n1 teaspoon butter\n\n2 eggs\n\nCook the oatmeal as directed in basic oatmeal (left), adding the scallions in Step 1. Just before it's done, stir in the cheese. While the oatmeal cooks, melt the butter in a pan over medium heat. Crack in the eggs, then cover the pan and fry until the yolks are runny but the whites are cooked, 2 to 3 minutes. Top each bowl of oats with one fried egg!\n\n3 Coconut and Lime\n\n$0.75 serving \/ $1.50 total\n\n\u00bc cup unsweetened coconut flakes\n\n2 tablespoons sugar\n\njuice of \u00bd lime\n\nAdd the coconut and sugar to the oatmeal, and cook as directed in Basic Oatmeal. Turn off the heat and squeeze the lime juice over the top.\n\n4 Fruity\n\n$0.55 serving \/ $1.10 total\n\n\u00bd cup berries, or chopped fruit, fresh or frozen\n\n1 tablespoon sugar\n\nCook the oatmeal as directed in Basic Oatmeal, but 2 minutes before it's ready, add the berries and sugar and stir to combine. It's surprising how many variations you can come up with just by trying a new fruit or combining several varieties.\n\n5 Apple Cinnamon\n\n$1 serving \/ $2 total\n\n2 cups apple juice or cider\n\n1 teaspoon ground cinnamon\n\n1 apple, cored and chopped\n\nCook the oats as directed in Basic Oatmeal, using the apple juice instead of the water and adding the cinnamon. Top with the apple. If you want the apple to be soft and warm, cook it along with the oats.\n\n6 Baklava\n\n$0.75 serving \/ $1.50 total\n\n1 teaspoon ground cinnamon\n\n1 tablespoon finely grated orange zest\n\n4 tablespoons honey\n\n2 tablespoons chopped almonds or pistachios\n\nAdd the cinnamon, orange zest, and 2 tablespoons of the honey before cooking the oatmeal as directed in Basic Oatmeal). Once it's done, top each bowl with another tablespoon of honey and a tablespoon of nuts.\n\n7 Chocolate\n\n$0.50 serving \/ $1 total\n\n1 cup milk\n\n1 tablespoon cocoa\n\n1 tablespoon sugar\n\nA brilliant suggestion from a wonderful reader, Karen Lofstrom. Who needs Cocoa Puffs when you can have cocoa oatmeal?\n\nReplace one of the two cups of water with 1 cup of milk. Stir the cocoa and sugar into the milk and water before you add it to the pan\u2014this will keep the lumps away! Then proceed as usual with the oatmeal.\n\nBreakfast Quinoa\n\n$1.25 \/ serving\n\n$2.50 total\n\nQuinoa makes a great alternative to oatmeal from time to time. It contains more naturally occurring protein than oats do, so it keeps you full longer. Unfortunately, it also comes with a higher price tag, but if you buy in bulk, it can still be affordable. serves 2\n\n1 cup white quinoa\n\n2 cups water\n\n\u00bc teaspoon salt\n\n1 tablespoon sugar\n\n1 cup berries or chopped fruit, fresh or frozen\n\n1 Add the quinoa, water, and salt to a small pot and bring to a boil over medium heat. Turn the heat down to low. Place a lid on the pot, slightly askew, to let steam escape.\n\n2 After about 10 minutes, add the sugar and half the fruit, stirring to combine. The quinoa should take about 20 minutes total, but keep an eye on it and add more water if it becomes too dry. It's ready when the grains are translucent, have doubled in size, and you can see an opaque ring around each grain.\n\n3 Fluff up the quinoa with a fork and scoop it into bowls. Top with the rest of the fruit.\n\nBanana Pancakes\n\n$0.75 \/ Serving\n\n$3 total\n\nWith the creamy texture and delicious flavor of bananas, these pancakes are stunningly good. You will be seriously popular if you feed these to your family or friends. Another plus: This is a great way to get rid of mushy bananas that doesn't involve making banana bread. serves 4, makes 10 to 15 pancakes\n\n2 cups all-purpose flour\n\n\u00bc cup brown sugar\n\n2 teaspoons baking powder\n\n1 teaspoon baking soda\n\n1 teaspoon salt\n\n4 bananas\n\n2 eggs\n\n1\u00bd cups milk\n\n1 teaspoon vanilla extract\n\nbutter, for cooking and serving\n\nsyrup, for serving\n\n1 Preheat the oven on its lowest setting.\n\n2 Combine the flour, brown sugar, baking powder, baking soda, and salt in a medium-size bowl. Mix thoroughly with a spoon.\n\n3 In another medium bowl, mash 2 of the bananas with a fork. Add the eggs, milk, and vanilla, and mix well to combine.\n\n4 Add the dry mixture to the bananas, stirring with a spoon until everything just comes together. Tender pancakes come from not overmixing the batter. If there are still a few pockets of flour, don't worry about it.\n\n5 Let the mixture sit for 10 to 15 minutes. Meanwhile, slice the 2 remaining bananas.\n\n6 Place a nonstick or cast-iron skillet or griddle over medium heat. Once it's hot, melt a small amount of butter, about \u00bd teaspoon, in the skillet and ladle some pancake batter into the center of the pan. A normal amount of batter is \u00bc to \u2153 cup, but you can make your pancakes as large or small as you like. If it's your first time making pancakes, make them smaller: They'll be easier to flip.\n\n7 As soon as the batter is in the pan, place 3 to 4 banana slices atop the uncooked side of the pancake. Once the edges of the pancake start to dry up and you can see the middle start to bubble, flip the pancake over. Cook until it is browned on both sides, about 30 seconds to 1 minute per side.\n\n8 Stack the finished pancake on a plate in the warm oven and repeat Steps 6 and 7 until you run out of batter. Serve hot, with butter and syrup.\n\nIf you're in a hurry, use a larger skillet or griddle and cook a few pancakes at a time. Making batches of three or four is much faster than one pancake at a time!\n\nWhole-Wheat Jalape\u00f1o Cheddar Scones\n\n$0.75 \/ scone\n\n$4.50 total\n\nSpicy, cheesy, flaky\u2014these scones are at their best straight out of the oven. They're delicious for breakfast, or with a plate of beans or a pile of vegetables, or crumbled into stew or chili. Cutting the cheese into cubes rather than grating it means you'll have pockets of gooey cheese that contrast nicely with the crumb of the scone. If you want the heat of the jalape\u00f1o, leave the seeds and membrane; if you like it milder, remove them and chop up only the pepper itself. makes 6 scones\n\n\u00bd cup (1 stick) butter (see box)\n\n2\u00bd cups whole-wheat flour, plus more for the work surface\n\n1 tablespoon baking powder\n\n1 teaspoon salt\n\n1 jalape\u00f1o pepper, finely chopped (see headnote)\n\n4 ounces sharp Cheddar cheese, diced to about \u00bc inch\n\n2 eggs, lightly beaten\n\n\u00bd cup milk\n\negg wash (optional)\n\n1 egg\n\nsalt and pepper, to taste\n\nuse the wrapper from the stick of butter to grease the baking sheet.\n\n1 Place the butter in the freezer for 30 minutes.\n\n2 Preheat the oven to 400\u00b0F. Line a baking sheet with parchment paper or lightly grease it with butter.\n\n3 Combine the flour, baking powder, and salt in a large bowl.\n\n4 Remove the butter from the freezer and grate it directly into the flour mixture. (Use a cheese grater\u2014it's the best way to break up butter without melting it.) Using your hands, gently squish the butter into the flour until everything is incorporated but not smooth. The chunks of butter will create flaky scones.\n\n5 Add the jalape\u00f1o, cheese, beaten eggs, and milk to the bowl, then use your hands to mix everything gently until it just comes together. The dough will probably be a little shaggy, but that's fine.\n\n6 Sprinkle flour on a clean countertop and dump the dough onto it. Gently shape the dough into a disk about 1\u00bd inches thick. Cut the disk into 6 triangular wedges, like a pizza, and move them to the baking sheet.\n\n7 If you want to make the egg wash (recommended!), lightly beat the egg and brush it over the scones (you'll have more egg wash than you need to use). Sprinkle salt and pepper over each scone. Bake for 25 minutes or until golden brown.\n\nChocolate Zucchini Muffins\n\n$0.20 \/ muffin\n\n$4.80 total\n\nWhen my friend Michael challenged me to create a recipe that used dark chocolate, I got a little worried: Dark chocolate is expensive! But then I remembered that cocoa powder is deeply, darkly chocolaty, without the expense. I thought of the chocolate zucchini cake my mother made when I was growing up and knew I had something.\n\nThis is a great breakfast treat that uses staples you should generally have on hand, such as flour, oats, and yogurt. The yogurt and zucchini make these muffins super moist and yummy, but still a reasonably nutritious (if slightly sugary) choice for breakfast. Make them in midsummer, during the height of zucchini season, when larger zucchini are really cheap. Big zucchini are generally a bit woodier, but they're great for baking. makes 24 small muffins\n\nbutter or vegetable oil, for the muffin tin\n\n2 cups grated zucchini (start with 1 large or 2 small zucchinis)\n\n1\u00bd cups all-purpose flour\n\n1\u00bd cups rolled oats\n\n1\u00bd cups sugar\n\n\u00bd cup cocoa powder\n\n2 teaspoons baking soda\n\n1 teaspoon salt\n\n4 eggs\n\n1 cup plain yogurt\n\nadditions\n\n1 tablespoon ground cinnamon\n\n\u00bd cup dark chocolate chips\n\n1 Preheat the oven to 350\u00b0F.\n\n2 Grease 24 muffin cups with butter, or, if you have paper liners on hand, line the muffin cups.\n\n3 Cut off the round end of the zucchini (which is a little tough), keeping the stem to use as a handhold. Shred the zucchini with a box grater into a large bowl, stopping before you get to the stem.\n\n4 Dump the flour, oats, sugar, cocoa powder, baking soda, salt, and cinnamon and chocolate chips, if using, into a medium bowl.\n\n5 Mix the eggs and yogurt with the grated zucchini. Add the dry ingredients, mixing until just combined.\n\n6 With a spoon, dollop the batter into the muffin tins until each cup is about three-quarters full. Transfer to the oven and bake for 20 minutes.\n\n7 Pull the muffins out and poke them in the middle with a toothpick or knife. If it comes out wet, bake the muffins for 5 more minutes. If it comes out clean, they're done.\n\n8 Let the muffins cool in their tins for 20 to 30 minutes, then eat warm!\n\nBroiled Grapefruit\n\n$1 \/ serving\n\n$2 total\n\nIf your oven has a broiler, this is a fast and fun way to liven up a standard, healthy breakfast of grapefruit, turning it into a hot and sticky treat. It might just make a grapefruit lover out of you! If you have maple syrup on hand, use it instead of sugar for even more flavor. serves 2\n\n2 grapefruits\n\n2 tablespoons light or dark brown sugar\n\nsalt\n\n1 Turn your oven's broiler up to high.\n\n2 Cut each grapefruit in half at the equator and place the halves, cut sides up, on a rimmed baking sheet or another ovenproof pan. Sprinkle the halves evenly with brown sugar and top with just a tiny bit of salt to bring out the flavor.\n\n3 Place the grapefruit halves under the broiler until they turn bubbly and a little brown (or even black) around the edges. This usually takes me about 3 minutes, but every broiler is different, so monitor it\u2014yours may take up to 8 minutes. Don't get distracted! Black on the edges is okay; black sticky mess in the center is not. Overbroiling ruins a good meal fast.\n\n4 Let the grapefruit cool down for a minute or two before serving.\nSOUP AND SALAD\n\nDal\n\n$0.60 \/ serving\n\n$2.40 total\n\nThis thick lentil soup is a flavor-packed staple of the Indian table. There are a ton of ways to prepare dal, but the core\u2014beyond the lentils themselves\u2014is usually fresh ginger, garlic, and chile, along with some dry spices. You can use any type of lentil you like. If you're using larger lentils (like chana dal, French lentils, or split mung beans), soak them for 30 minutes. If you're using the small orange lentils, then don't bother soaking them\u2014they cook very quickly. serves 4\n\n1 tablespoon butter\n\n1 medium onion, finely chopped\n\n1 teaspoon cumin seeds\n\n1 teaspoon black mustard seeds\n\n1 teaspoon turmeric\n\n2 cloves garlic, finely chopped\n\n1 jalape\u00f1o or serrano pepper, finely chopped (see box)\n\n\u00bd-inch piece of ginger, grated\n\n2 cups lentils\n\nsalt and pepper, to taste\n\nadditions\n\nheavy cream\n\nsprinkling of chopped fresh cilantro\n\nsprinkling of chopped scallions\n\n1 Melt the butter in a medium-size pot over medium heat. Add the onion and cook for 1 minute, then add the cumin seeds and mustard seeds and stir them until they sizzle. Toss in the turmeric, garlic, and jalape\u00f1o and cook until the vegetables soften, 3 to 4 more minutes. Add the ginger and stir-fry for about 30 seconds.\n\n2 Add the lentils along with enough water to cover them, then cook, covered, until the lentils are tender, about 20 minutes for split red lentils, 30 for green or brown, or 45 for other whole varieties. Plan according to what lentil you are using (package directions can be helpful).\n\n3 Sample the dal and add salt and pepper to taste. You'll probably need a fair bit of salt to bring out all the flavors\u2014a teaspoon or so.\n\n4 Top the dish with a splash of cream, some chopped fresh cilantro or scallions, and serve.\n\nIf you prefer mild dishes, use a quarter of the jalape\u00f1o to start and add more if you decide you like it. Serrano peppers are hotter than jalape\u00f1os, so only use those if you like your food very spicy. Like salt, the amount of chile pepper should be to your taste. And you can always seed and devein your peppers to reduce their heat.\n\nCorn Soup\n\n$1.25 \/ serving\n\n$5 total\n\nThis thick, sweet, satisfying soup is a favorite of kids and adults. It's wonderful to make in late summer to early autumn when corn on the cob is at its peak, but canned or frozen corn can also serve as a warm reminder of summer in the depths of winter. If you're making this soup with corn on the cob, the first step is to make corn broth out of the cobs once you've removed the kernels. If you're using canned or frozen corn, you'll need to substitute chicken or vegetable broth instead. Pair it with a slice of garlic bread or add a hard-boiled egg for extra protein. serves 4 to 6\n\n1 tablespoon butter\n\n1 onion, finely chopped\n\n2 stalks of celery, finely chopped\n\n1 green or red bell pepper, stemmed, seeded, and finely chopped\n\n1 small potato, diced\n\n4 cloves garlic, finely chopped\n\n1 chile pepper, finely chopped (optional)\n\n4 cups corn, fresh (from 4 to 8 ears), canned, or frozen\n\n1 tablespoon cornmeal or all-purpose flour\n\n5 cups corn broth (recipe below), vegetable broth, or chicken broth\n\nsalt and pepper, to taste\n\n1 Melt the butter in a large pot with a lid over medium heat. Add the onion, celery, bell pepper, and potato and stir. Cover the pot and let everything cook until the onion is translucent, about 5 minutes.\n\n2 Remove the lid and add the garlic and the chile pepper, if using. Stir the vegetables, adding a splash of water or broth to free any that get stuck to the bottom of the pot.\n\n3 Let the vegetables cook, stirring occasionally, until they are lightly browned and soft, 5 minutes more. The potatoes should not yet be fully cooked.\n\n4 Add the corn and cornmeal to the pot and stir. Pour in the broth and bring to a boil, then turn the heat down to low and simmer until the broth thickens and becomes opaque, about 30 minutes.\n\n5 Add salt and pepper. If you made your own corn broth, you'll probably need at least a teaspoon of salt; if you used store-bought broth, you'll need less.\n\nCorn Cob Broth\n\nMakes 5 cups\n\n4 to 8 corn cobs, kernels reserved for the soup (see box)\n\n2 bay leaves (optional)\n\nsalt and pepper, to taste\n\nPlace the corn cobs and the bay leaves, if using, in a large stockpot and cover with water. Bring to a boil over high heat, then turn the heat down to medium and let boil for about 30 minutes. Taste the broth and add salt and pepper until it has a nice corn flavor. Measure out 5 cups of broth for the soup and store the rest. If you don't want to have leftovers, boil the broth longer, until it is reduced to 5 cups. This will make the flavor of the broth stronger. The broth will keep for several months in the freezer, or for a few weeks in the refrigerator.\n\nFrench Onion Soup\n\n$1.50 \/ serving\n\n$9 total\n\nBest if you accept it now: You are going to cry making this recipe, because the first step is to chop a mountain of onions. But crying is good for us from time to time. Soon you will be on to the magical part, watching a colossal pile of onions shrink and caramelize to make a sweet, flavorful, wonderful soup. Save this recipe for the winter, when other vegetables are out of season and you want to fill your home with rich aromas. As my friend Marilyn, who suggested this recipe, says, \"The smell in your kitchen is absolute heaven.\" serves 6\n\n4 pounds onions, any type\n\n4 cloves garlic\n\n2 tablespoons butter\n\n2 bay leaves\n\n3 teaspoons salt, plus more to taste\n\n2 teaspoons pepper, plus more to taste\n\n6 slices bread\n\n1\u00bd cups grated Gruy\u00e8re or Cheddar cheese\n\nadditions\n\nbeef broth or chicken broth instead of water in Step 4\n\n\u00bd cup red wine, or 1 tablespoon vinegar, any type\n\n1 teaspoon chile flakes\n\nsprinkling of fresh thyme leaves\n\n1 Chop each onion in half from tip to tail, peel them, then slice them into approximately \u00bc-inch-thick (or smaller) half-moon slices. Big slices are fine since you're cooking the onions for so long. Slice the garlic as well.\n\n2 Melt the butter in a large pot over medium heat. Add the onions, garlic, and bay leaves. Cover the pot with a lid and leave it until the onions release a lot of moisture, about 10 minutes. Give them a stir.\n\n3 Cook for 1 to 1\u00bd hours, stirring every 20 minutes. When the onions at the bottom start to stick and turn dark, add a splash of water to unstick them. Don't worry, they aren't burning, just caramelizing. The water helps lift off the sticky, delicious, sweet part!\n\n4 Once the onions are very dark and about a quarter the volume they once were, add 8 cups of water, the salt, and the pepper. Bring to a boil and cover the pot again, turn the heat down to low, and let it simmer for another hour.\n\n5 Taste and adjust with salt and pepper as needed, then ladle the soup into bowls. Discard the bay leaves.\n\n6 Now it's time to make cheese toast! If you want classic French onion soup\u2014with the toast directly in the soup, which makes it a bit soggy\u2014place a piece of bread on top of each (ovenproof!) bowl of soup, sprinkle with cheese, then heat the bowls under the broiler until the cheese is bubbly, 2 to 5 minutes. If you don't like soggy toast, just broil the cheese toast on its own and serve it on the side to dunk.\n\nIf you're using any of the additions, add them with the water (or broth) in Step 4. The beef broth is traditional and will make the soup richer. The vinegar adds a nice hit of acidity, which provides some depth. The chile flakes or thyme add another dimension to the soup.\n\nLightly Curried Butternut Squash Soup\n\n$1.50 \/ serving\n\n$6 total\n\nSquash is a wonderful vegetable for soup: It's flavorful and has a divinely smooth texture when cooked and pureed. Serve this soup to people who think they don't like squash or curry, and you'll change some minds. You can substitute any winter squash for the butternut\u2014I just like butternut because it's faster to peel and chop than its many cousins. serves 4\n\n1 butternut squash or equivalent amount of other winter squash (about 2 pounds)\n\n1 tablespoon butter\n\n1 medium onion, chopped\n\n1 green bell pepper, stemmed, seeded, and chopped\n\n3 cloves garlic, finely chopped\n\n1 teaspoon ground cumin\n\n1 teaspoon ground coriander\n\n1 teaspoon ground turmeric\n\n1 teaspoon cayenne pepper\n\n1 can coconut milk, 2 tablespoons reserved for garnish\n\nsalt and pepper, to taste\n\nadditions\n\nsour cream\n\nchopped scallions\n\nchopped fresh cilantro\n\n1 To prepare the squash, peel off the tough skin with a potato peeler. Cut the squash in half lengthwise with a sharp chef's knife, then scoop out the seeds and goop. (You can save the seeds for a tasty snack later, if you like\u2014see page .)\n\n2 Next, slice off the stem and the very bottom of the squash and throw them away. Place each half of the squash flat side down on a cutting board. Chop each half into \u00bd-inch slices, then cut each slice into cubes.\n\n3 Melt the butter in a large pot over medium heat. Add the onion, bell pepper, and garlic and saut\u00e9 until translucent, about 2 minutes.\n\n4 Add the cubed squash, cumin, coriander, turmeric, and cayenne, and stir it all together. Put a lid on the pot and cook for 2 minutes more. Add the coconut milk and 3 cups of water and stir.\n\n5 Bring the soup to a boil, then turn down the heat to low and let it cook until the squash is tender, about 30 minutes. Taste the soup and add salt and pepper as needed. Soup usually needs a fair bit of salt to bring out the flavors, so be generous.\n\n6 If you have an immersion blender, you can puree the soup in the pot. If you have a stand blender, wait until the soup has cooled before transferring it to the blender. Puree until smooth, then taste again and add any more salt and pepper if needed. You can enjoy the soup as is or serve it with a drizzle of the reserved coconut milk or a dollop of sour cream, plus some chopped scallions or cilantro.\n\nTomato Soup and Grilled Cheese\n\n$2.40 \/ serving\n\n$12 total\n\nIf I get to choose, this will be my last meal on earth. Some of the first \"cooking\" I did for myself, at age ten or so, was plopping out a can of tomato soup. I felt very adult. Now that I know how easy it is to make from scratch, I never bother with the canned stuff. And if your family is anything like mine, making a grilled cheese sandwich will get you more glory and appreciation than a four-course dinner you slaved over for hours. To me, everything that is good and honest, delicious and comforting, is contained in velvety tomato soup and a crusty grilled cheese sandwich.\n\nThe sandwich recipe makes five sandwiches, so adjust according to your needs. Use any good melting cheese, or a blend of cheeses. If you have just the ends of a few cheeses left over from other meals, this is a great way to use them. For the bread, I like rye, but anything will do. serves 5\n\nThe Soup\n\n1 tablespoon butter\n\n2 medium onions, chopped\n\n4 cloves garlic, finely chopped\n\n2 cans (28 ounces each) pureed tomatoes (see box)\n\n6 cups vegetable broth (homemade is great, but a bouillon cube dissolved in water works just fine)\n\nsalt and pepper, to taste\n\nadditions\n\n\u00bd cup heavy cream, to make cream of tomato soup\n\nleaves from 2 sprigs fresh thyme\n\n2 tablespoons chopped fresh basil\n\nzest of 1 lemon\n\n1 Melt the butter in a large pot over medium heat. Add the onions, stir them to coat, then place a lid on the pot and leave it for 5 minutes. Take the lid off and stir the onions. Add the garlic and cover again until the onions are soft and just starting to brown, another 2 minutes.\n\n2 Add the pureed tomatoes and vegetable broth to the pot and stir, being sure to scrape any sticky onions off the bottom to keep them from burning. Bring the soup to a boil, then turn it down to low to simmer for about 10 minutes.\n\n3 Taste and add salt and pepper as needed. Add the cream, herbs, or lemon zest, if using. For smooth soup, use an immersion blender to quickly puree the onions into the tomato mixture. If you're using a standard blender, wait for the soup to cool before you blend it.\n\nI prefer to puree canned whole tomatoes in a blender or food processor\u2014the quality is better than the ones already pureed in the can.\n\nThe Sandwiches\n\nsoftened butter, for the bread\n\n10 slices bread\n\nDijon mustard (optional)\n\n2\u00bd cups grated cheese (see headnote)\n\n1 Spread the butter on one side of each slice of bread, all the way to the edges.\n\n2 Flip over 5 slices, buttered side down, and top each with mustard, if using, and then with approximately \u00bd cup of the cheese. Top each of your cheesified slices with another piece of bread, buttered side up.\n\n3 Place a pan over medium heat for about a minute (see box). Stuff as many sandwiches into the pan as you can fit. I usually don't do more than 3 at a time so there's room to flip them.\n\n4 Fry the sandwiches until they turn golden brown, about 2 minutes. Press down gently on the top of each sandwich for even browning. Flip the sandwiches over with a spatula and repeat on the other side. Press down again to make sure that the cheese in the middle is melting. If either side is not as golden brown as you would like, flip that side down and cook it some more.\n\n5 Once the sandwich is golden and the cheese is melted in the middle, serve it up with the soup. Dunking is optional.\n\nIf your pan is too hot on the bottom, it will burn the sandwich before the cheese melts, so start with medium heat and lower the temperature as necessary. Placing a lid over your pan will trap some steam and melt the cheese faster.\n\nSweet or Savory Pineapple Salad\n\n$1.40 \/ serving\n\n$2.80 total\n\nPineapple is so sweet, juicy, and crunchy that its citric acid punch can sneak up on you\u2014suddenly, you realize you can't feel your lips or the tip of your tongue anymore! Canned pineapple is rarely as strongly flavored and acidic as its fresh cousin, but it sure is convenient and affordable\u2014plus, you get juice. Here are two quick and simple ways to add flavor to a good old can of pineapple. The sweet version makes a great dessert, snack, or even a topping for yogurt. The savory version is lovely on its own, almost like a pineapple salsa. Enjoy it on seafood or pork, or with beans in a taco. serves 2 as a side\n\nSweet\n\n1 can pineapple, in juice\n\n1 tablespoon sugar\n\nzest of \u00bd lime\n\nsalt, a wee pinch\n\nSavory\n\n1 can pineapple, in juice\n\n1 tablespoon finely chopped red chile\n\n2 tablespoons chopped fresh cilantro\n\nsalt, to taste\n\n1 Open the can of pineapple. Drain the juice into a glass and drink it!\n\n2 If the pineapple is cut into chunks, simply scoop them into a bowl. If you have rings, chop them into bite-size pieces first.\n\n3 Add the remaining ingredients for the sweet or savory salad. Stir and taste. For the sweet, don't forget the salt\u2014it brings out the sweetness in the acidic pineapple. Adjust to taste, and serve.\n\nKale Caesar Salad\n\n$2.25 \/ serving\n\n$4.50 total\n\nThis kale is treated like romaine lettuce in a classic Caesar salad, not the creamy modern version. The bitterness of the greens is delicious alongside the rich, fatty dressing. You could also use Swiss chard. If you're worried about the raw egg yolk in the dressing, simply omit it. And make the croutons first\u2014they'll take longest. serves 2, or 4 as a side\n\n1 raw egg yolk, from a high-quality fresh egg\n\n2 teaspoons lemon juice\n\n2 teaspoons Dijon mustard\n\n3 tablespoons olive oil\n\nsalt and pepper, to taste\n\n1 large bunch kale\n\nCroutons, about 2 cups\n\nfreshly grated Romano cheese\n\nadditions\n\n1 clove garlic, grated\n\n1 anchovy fillet, chopped\n\n1 Drop the egg yolk into a large mixing bowl. Add the lemon juice, mustard, and garlic and anchovy, if using. Whisk briskly until the dressing is light and frothy. Slowly add the olive oil, whisking the whole time. Once everything is incorporated, add the salt and pepper, then adjust to your taste. I like it very lemony.\n\n2 Remove the large stem from the center of the kale leaves. (Lacinato kale, sometimes called Tuscan or dinosaur kale, has the easiest stems to remove.) Chop the leaves in half lengthwise, then cut into thin ribbons. This method disguises the kale's tough texture.\n\n3 Toss the kale in the bowl to coat it with dressing. Set it aside for 10 minutes or leave it in the fridge for up to 4 hours. The kale will become tender as it marinates.\n\n4 Before serving, toss in the croutons and a sprinkling (or more!) of Romano cheese, according to your taste.\n\nBroiled Eggplant Salad\n\n$2.25 \/ serving\n\n$4.50 total\n\nEggplant haters often cite mushy texture as their biggest complaint. Well, this salad has you covered. Broiled eggplant has a crunchy and meaty texture, and the tahini dressing makes the salad rich and creamy. To turn it from a side into a lunchtime salad, throw in some roasted potatoes or chickpeas. serves 2 as a side\n\n1 medium eggplant, sliced into circles\n\n1 tablespoon tahini\n\n1 tablespoon lemon juice\n\nsalt and pepper, to taste\n\nchopped scallions\n\nadditions\n\nsprinkling of chile flakes\n\nfinely chopped fresh dill\n\nroasted potatoes\n\nchickpeas\n\n1 Turn your oven's broiler to high. Arrange the slices of eggplant on a baking sheet, then place them under the broiler for 3 to 8 minutes, depending on the power of your broiler. (If you want the eggplant to get some char, position it as close to the broiler as possible.) Watch carefully. Once the eggplant begins to blacken, remove it from the oven and flip the slices over.\n\n2 Repeat the process on the other side. Once your eggplant is nicely charred, chop it into bite-size pieces.\n\n3 In a medium-size bowl, mix the tahini, lemon juice, and chile flakes, if using, and toss in plenty of salt and pepper. Add the eggplant and stir. Add more salt or lemon juice according to your taste. Top it with scallions, or dill, if available, and serve!\n\nBeet and Chickpea Salad\n\n$1.75 \/ serving\n\n$3.50 total\n\nThis dish is spicy, crunchy, and almost certainly the pinkest salad you'll ever eat! Don't be scared. serves 2 as a side\n\n2 or 3 beets\n\n1 cup chickpeas, cooked or canned, drained\n\n3 tablespoons peanuts\n\ndressing\n\n1 tablespoon fresh lime juice\n\n1 teaspoon chili sauce\n\n1 tablespoon olive oil\n\nsalt and pepper, to taste\n\n1 Peel the raw beets, removing the stems if necessary, then shred them with a box grater. Place the shredded beets in a medium-size bowl along with the chickpeas and peanuts.\n\n2 Stir the lime juice, chili sauce, olive oil, and salt and pepper together in a small bowl to make the dressing. Taste and adjust the salt and pepper to your liking.\n\n3 Add the dressing to the bowl with the beets and mix to combine. Let everything sit about 5 minutes so that the flavors can soak into the vegetables and the beet juices can mingle with the dressing.\n\nEver-Popular Potato Salad\n\n$0.75 \/ serving\n\n$3 total\n\nI developed this recipe because I'm not a big fan of mayonnaise-based potato salads. It's really the simplest thing: just potatoes in a regular vinaigrette. Smaller potatoes are best, but whatever you have will be fine! You can add all kinds of extras to make it more festive, but people always rave about the salad as is. The secret is that potatoes actually have really nice flavor\u2014all you have to do is season them properly. Let potatoes be potatoes, no need to disguise 'em!\n\nIf you have leftover roasted potatoes or other root vegetables, the same idea works well. Just skip the cooking part and go straight to the dressing. serves 4 as a side\n\n2 pounds potatoes\n\n2 tablespoons olive oil\n\n2 tablespoons lemon juice, lime juice, or vinegar\n\n2 teaspoons Dijon mustard\n\nsalt and pepper, to taste\n\n3 or 4 scallions\n\nadditions\n\nchopped fresh dill\n\nchopped fresh parsley\n\npaprika\n\nfinely chopped fresh chiles\n\nfinely chopped pickles\n\n1 If you're using very large potatoes, chop them into halves or quarters to speed up the cooking\u2014or dice them into bite-size pieces if you're really in a hurry. Otherwise, keep the potatoes whole.\n\n2 Put the potatoes in a large pot with a lid and cover with water. Bring to a boil over medium-high heat, then turn the heat down to medium and set the lid askew so that steam can escape.\n\n3 After boiling about 25 minutes, stab the largest potato with a fork. If the fork pierces the potato easily, it's fully cooked. If not, boil for 5 more minutes. It's fine if they're a little overcooked, but undercooked potatoes are awful.\n\n4 Drain the potatoes. Once they're cool enough to handle safely (but still warm), roughly chop them into bite-size pieces (unless you did that in Step 1).\n\n5 While the potatoes are cooling, mix the olive oil, lemon juice, mustard, salt, and pepper in a large bowl. Whisk it briskly until the liquid is blended. If you don't have a whisk, use a fork.\n\n6 Throw the potatoes into the bowl and toss them in the dressing to coat. Add a generous amount of salt as you stir. Potatoes are very bland without salt! Let them marinate for 10 minutes.\n\n7 Meanwhile, remove the outer layer and any wilty top bits from the scallions, chop them, and sprinkle them over the top. Toss the salad once more (along with any additions you are using), then taste and adjust the salt, pepper, and citrus juice or vinegar as you see fit. This keeps very well, covered, in the fridge for up to a week, and it travels nicely to a picnic or potluck. Have fun!\n\nSpicy Panzanella\n\n$1.30 \/ serving\n\n$5.20 total\n\nA former classmate of mine, George, likes salads with a little kick. (As you can maybe tell, I do, too!) For inspiration, I turned to panzanella, a classic Italian bread-and-tomato salad. The Italians are true masters of making leftovers delicious. Here, old hard bread soaks up tomato juice and dressing for a super-flavorful and filling salad. You can toss in any vegetable or fruit as long as it's juicy. Bell peppers or carrots won't work so well, but peaches, grapes, and zucchini all do. If you don't like spicy salads as much as George and I do, feel free to seed and stem the jalape\u00f1o to remove its fierce heat, or replace it entirely with garlic or shallot. serves 4 as a side\n\n2 small field cucumbers or 1 English cucumber\n\n2 medium tomatoes, chopped\n\nsalt and pepper, to taste\n\n4 slices day-old bread\n\ndressing\n\n2 tablespoons olive oil, plus a few drops for the pan\n\n1 jalape\u00f1o, finely chopped\n\n2 tablespoons chopped tomato\n\nsalt and pepper, to taste\n\njuice of 1 lime\n\nadditions\n\nchopped fresh herbs\n\nchopped peaches, nectarines, or plums\n\nfinely chopped red onion\n\nchopped zucchini or summer squash\n\npitted olives\n\ngrapes\n\n1 If you're using field cucumbers\u2014usually cheaper than English\u2014peel them to remove the tough skin. (A little leftover peel is not a problem.) For English cucumbers, there's no need to peel.\n\n2 Reserve about 2 tablespoons of the chopped tomatoes to use in the dressing, but throw the rest of the tomatoes and all of the cucumbers into a large bowl. Sprinkle generously with salt and pepper; the salt helps draw out the juices. Toss the vegetables and set aside.\n\n3 Place a small saucepan over medium heat and add a few drops of olive oil. Add the jalape\u00f1o and saut\u00e9 until it sizzles and smells good, about a minute, then add the rest of the chopped tomato and a tablespoon of water. Cook until the tomato juices release, another 2 minutes. Sprinkle liberally with salt and pepper.\n\n4 Once the water has evaporated, turn off the heat and dump the jalape\u00f1o-tomato mixture on your cutting board. Chop it up very finely, then throw it back into the pan\u2014with the heat off\u2014with the lime juice and 2 tablespoons of olive oil. Stir to combine, taste, and add salt and pepper as needed. You've got dressing!\n\n5 Chop or tear the bread into bite-size pieces, then toast it in a skillet over medium heat, tossing occasionally, until the bread chunks are toasty on all sides. Alternatively, just toast full slices of bread in a toaster and tear them up afterward, or skip the toasting if the bread is already super-hard.\n\n6 Add the bread and dressing to the vegetables and stir to combine. Taste and adjust the salt and pepper once more. Let the salad sit for a few minutes so that the bread can soak up the juices, then serve!\n\nCold (and Spicy?) Asian Noodles\n\n$2.50 \/ serving\n\n$5 total\n\nNo question\u2014cold but spicy food is refreshing and delicious on a hot summer day. Here's a recipe that you can really make your own. Use whatever sauce or dressing you like and whatever vegetables you have around, or just a few scallions. If you have some Spice Oil on hand, be sure to add it. It's amazing in this. serves 2, or 4 as a side\n\n12 ounces dried spaghetti, soba, or any Asian noodles\n\n2 tablespoons soy sauce\n\n1 bunch scallions, chopped\n\n1 cucumber, finely chopped\n\nsalt and pepper, to taste\n\nadditions\n\nSpice Oil\n\nPeanut Sauce\n\ngrated carrot\n\nchopped hard-boiled egg\n\n1 Prepare the noodles according to the package instructions. Rinse them under cold water and drain in a colander.\n\n2 Put the noodles in a bowl and add the soy sauce, Spice Oil if you have it, scallions, and cucumber (and any other additions), and mix with a fork or tongs. Taste it and add salt and pepper or more Spice Oil as needed.\n\n3 Let the noodles sit in the fridge for about an hour if you can. The flavors will mingle and become more intense. The finished dish should keep for up to 3 days, covered, in the fridge. A great make-ahead lunch.\n\nTaco Salad\n\n$2.60 \/ serving\n\n$5.20 total\n\nThis salad is a great use for leftover beans or pulled pork\u2014crunchy, fresh, yet satisfying enough to be a whole meal. I like to make taco salad for a weekday lunch in the summertime. serves 2, or 4 as a side\n\n4 cups chopped lettuce\n\n1 cup cooked beans, pulled pork, or ground beef\n\n2 small tomatoes, chopped\n\n\u00bd cup corn kernels, canned or fresh\n\n2 or 3 scallions, finely chopped\n\n1 cup tortilla chips, roughly crushed\n\n\u00bc cup shredded sharp Cheddar cheese or queso fresco\n\ndressing\n\n\u00bc cup sour cream or yogurt\n\njuice of 1 lime\n\nsalt and pepper, to taste\n\nadditions\n\nchopped cucumber\n\nchopped jalape\u00f1o (remove seeds for less heat)\n\nbell peppers, stemmed, seeded, and chopped\n\ngrated carrots\n\nSalsa\n\n1 Mix the lettuce, beans, tomatoes, corn, scallions, tortilla chips, cheese, and any additions in a large bowl.\n\n2 In a small bowl, stir together the sour cream, lime juice, salt, and pepper. Taste it and adjust the salt, pepper, and lime to your liking.\n\n3 Just before serving, pour the dressing over the salad and toss to coat. Eat immediately, maybe with a few extra chips on the side.\n\nCharred Summer Salad\n\n2.75 \/ serving\n\n$5.50 total\n\nOne of the early supporters of this book, Gina, can't eat gluten and wanted more Mexican-inspired options. I designed this spicy summer salad for her, topped with popcorn for the kind of crunch that croutons would provide. Use smaller zucchini, and save the big ones for Chocolate Zucchini Muffins. If you own a grill, use it instead of the broiler! serves 2, or 4 as a side\n\n3 small or 2 medium zucchini\n\n2 ears corn\n\n1 tablespoon olive oil or vegetable oil\n\nsalt and pepper, to taste\n\n2 ounces crumbled cotija or feta cheese\n\n1 cup popped Popcorn\n\ndressing\n\njuice of 1 lime\n\n1 tablespoon olive oil\n\n\u00bd teaspoon chili powder, plus more for dusting\n\nsalt and pepper, to taste\n\n1 Turn your oven's broiler to high.\n\n2 Chop off both ends of the zucchini, then slice each into four long sticks. Shuck the corn. Lay the zucchini and corn on a baking sheet, then rub them with the oil, making sure they're well coated. Sprinkle with salt and pepper.\n\n3 Broil the zucchini and corn for 2 to 5 minutes, depending on how powerful your broiler is. The zucchini should start to blacken in some spots. This is good! Turn the corn over as needed (but not the zucchini!) to make sure it cooks evenly. Broil until the vegetables are lightly charred, another 2 to 5 minutes. Let them cool.\n\n4 To make the dressing, mix the lime juice, olive oil, chili powder, salt, and pepper in a large bowl. Taste it and adjust the seasoning, if necessary.\n\n5 Chop the zucchini into bite-size pieces and slice the corn kernels from the cobs (see box).\n\n6 Transfer the vegetables into the bowl with the dressing. Add the crumbled cotija or feta and mix well. Sprinkle the popcorn over the top, then dust with a little extra chili powder, salt, and pepper.\n\nto cut the kernels off a cob of corn, hold the cob by the stem, vertically, inside a large bowl. Rest the other end of the corn on the bottom of the bowl. Using a sharp knife, slice the kernels off the cob as far down as you can go until your wrist hits the lip of the bowl. Slice all the way around the cob. Flip the cob over vertically and slice off the kernels that you missed closer to the pointy end.\n\nWilted Cabbage Salad\n\n$1.05 \/ serving\n\n$4.20 total\n\nThe idea for this recipe came from a wonderful reader, Karen. Wilted cabbage keeps its crunch and freshness for several days, whereas a regular lettuce salad turns gross soon after being dressed. (Kale behaves similarly to cabbage\u2014it's one of the reasons I love the kale salad!) This salad is best if left to marinate in the tart dressing overnight. Just be sure to add the peanuts shortly before serving to prevent sogginess. serves 4 as a side\n\n1 medium-size cabbage, finely chopped\n\n1 tablespoon salt\n\n\u00bd cup raw peanuts\n\n\u00bd bunch scallions, finely chopped\n\ndressing\n\n2 tablespoons olive oil\n\n2 tablespoons rice vinegar or lemon juice\n\nsalt and pepper, to taste\n\nadditions\n\ngrated carrot\n\nfinely chopped apple\n\nsesame seeds\n\na few drops of sesame oil\n\n1 Toss the cabbage and salt in a large bowl. Place something heavy, like a pot (any size that fits in the bowl), on top of the cabbage. The weight, along with the salt, will encourage the cabbage to expel its moisture. Leave it for 2 hours. This method will take away some bitterness, leaving the crunchy texture of raw cabbage.\n\n2 Roast the peanuts in a single layer in a skillet over medium heat, occasionally tossing them and moving them, until they are lightly brown all over, about 5 minutes. Alternatively, spread the peanuts on a baking sheet and broil them about 2 minutes. Keep an eye on them so they don't burn. You want them nice and golden. Sprinkle a bit of salt on the roasted peanuts and set them aside.\n\n3 Combine the olive oil, rice vinegar, salt, and pepper in a small bowl. Mix it up and taste. Adjust the salt and pepper as you like. Remember that the cabbage is already salted, so you won't need too much salt in the dressing.\n\n4 Once the 2 hours have passed, toss the cabbage again with your hands. Cabbage treated in this way will last for several days. Before serving, add the scallions, peanuts, and dressing. Toss, taste, and adjust the seasoning as you see fit.\n\nBroccoli Apple Salad\n\n$0.80 \/ serving\n\n$3.20 total\n\nThe bitterness of broccoli is delicious paired with the sweet tartness of apples. Plenty of crunch, too! For a slightly richer and creamier dish, try the yogurt dressing. serves 4 as a side\n\n1 large crown and stem of broccoli\n\n2 apples\n\ndressing\n\njuice of 1 lemon\n\n1 tablespoon olive oil\n\nsalt and pepper, to taste\n\nalternative yogurt dressing\n\n1 tablespoon yogurt\n\n1 teaspoon olive oil\n\n1 teaspoon lemon juice\n\n1 teaspoon chopped fresh dill\n\nsalt and pepper, to taste\n\n1 Slice the stem of the broccoli into \u215b-inch disks. If you can't get them that thin, don't worry, but if you have the patience, thinner is better! Once you reach the crown of the broccoli, cut the florets off and slice them as thin as you can. Set the broccoli in a bowl.\n\n2 Halve and core the apples, then place the apples flat side down on your cutting board to make them easier to slice. Slice the apples into \u215b-inch pieces, and dump them into the bowl with the broccoli.\n\n3 Mix the lemon juice, olive oil, salt, and pepper in a small bowl. Taste it and season with more salt and pepper as you see fit.\n\n4 Pour the dressing over the broccoli and apples and mix it all together.\n\nIf you put your plates in the fridge for 10 minutes before serving, the salad will stay crisp slightly longer. For the best presentation, pile the salad as high and tight as you can manage.\nSnacks, Sides, and Small Bites\n\nBrussels Sprout Hash and Eggs\n\n$1.75 \/ serving\n\n$3.50 total\n\nThis simple concoction is a great dish for brunch, a light lunch, or a side. The Brussels sprouts get salty and tangy from the olives and lemon, then they crisp and caramelize on the bottom. Mix in the little bit of fat from the egg yolk, and wow is this delicious. serves 2, or 4 as a side\n\n1 pound Brussels sprouts\n\nsalt and pepper, to taste\n\n1 tablespoon butter\n\n3 cloves garlic, finely chopped\n\n6 olives, finely chopped\n\n2 eggs\n\nlemon juice, to taste\n\n1 Chop off the ends of the sprouts. Slice them in half, then finely chop each half. Place the shreds in a bowl\u2014you should end up with about 4 cups\u2014and sprinkle with salt and pepper.\n\n2 Melt the butter in a medium-size nonstick pan with a lid over medium-high heat, swirling to coat the pan. Add the Brussels sprouts and the garlic, then let them cook until just a little wilted, about 1 minute. Toss the mixture. Add the olives and toss again.\n\n3 Crack the eggs into the pan so they aren't touching. Sprinkle with salt and pepper. Pour in 2 tablespoons water and cover with the lid. Let the eggs steam, undisturbed, until the whites are cooked through but the yolks are still runny, about 2 minutes.\n\n4 Turn off the heat and squeeze lemon juice over everything. Serve immediately.\n\nMexican Street Corn\n\n$1 \/ serving\n\n$4 total\n\nThis recipe takes fresh, sweet summer corn\u2014already amazing\u2014and adds salt, tang, and spice to the experience. If you have an outdoor grill, prepare the corn that way, but for those without, a broiler is a great shortcut! Use any chile powder you like, such as ground ancho powder or cayenne pepper. serves 4 as a side\n\n4 ears corn\n\n4 tablespoons mayonnaise\n\n\u00bd cup grated cotija, queso blanco, feta, Romano, or Parmesan cheese\n\ndusting of chile powder\n\n1 lime, sliced into wedges\n\n1 Turn your oven's broiler to high.\n\n2 Peel off the corn husks and clean off all the silk. Leave the green ends attached for a convenient handhold.\n\n3 Place the corn on a baking sheet and stick them under the broiler for 2 to 3 minutes. Rotate and repeat until they're brown and toasty all the way around\u2014no more than 10 minutes total.\n\n4 Working quickly, spread a tablespoon of mayonnaise over each ear of corn, lightly coating every kernel. Next, sprinkle the cheese over the corn. It should stick fairly easily to the mayonnaise, but you'll probably get a little messy.\n\n5 Sprinkle chile powder on top, but not too heavily, or it'll be gritty.\n\n6 Last, squeeze lime juice all over the corn, and serve hot!\n\nMy Dad's Baked Beans\n\n$1.50 \/ serving\n\n$3 total\n\nMy dad loves beans in basically any form. This is his formula for the quickest, easiest way to get beans on his plate without missing out on great flavor. Dad's beans rely on a can of baked beans as the base, while my version uses dried beans you might have left over from another meal. Mine requires a little more cooking and chopping to create the sauce, but it ends up less expensive because of the dried beans. Both versions taste great, so go with what works best for you: super quick and cheap, or quick and cheaper. serves 2, or 4 as a side\n\n2 teaspoons chipotle chiles in adobo with a bit of their sauce, or any chile sauce\n\n2 cans (13.5 ounces each) baked beans\n\n2 tablespoons mustard\n\n2 tablespoons molasses or brown sugar\n\nvariations\n\nspicy mustard instead of plain\n\nomit the chipotle\n\ntoppings\n\nsalsa\n\nscallions\n\nfresh cilantro\n\navocado\n\ntomato\n\ncrumbled bacon\n\nchunks of ham\n\n1 If you're using the chipotle in adobo, chop it finely to be sure the spice will be evenly distributed.\n\n2 Mix the chiles, beans, mustard, and molasses in a pot and place over medium heat. Cook until the beans are warmed through, 2 to 5 minutes. Give it a stir and serve. Or do it all in the microwave\u2014that works just as well!\n\n3 For a side dish, serve as is. To make a full meal, serve with leftover rice, over toast, in a burrito, scrambled with eggs, or stir-fried with onions and bell peppers.\n\nAnd Mine\n\n2 teaspoons chipotle chiles in adobo with a bit of their sauce, or any chile sauce\n\n3 cups cooked pinto, red, or black beans\n\n\u00bd cup pureed or chopped canned tomatoes, with juice\n\n\u00bc onion, finely chopped\n\n2 tablespoons mustard\n\n2 tablespoons molasses or brown sugar\n\nCook everything in a pot over medium heat until the juices thicken, about 5 minutes. That's it!\n\nSmoky and Spicy Roasted Cauliflower\n\n$0.85 \/ serving\n\n$3.40 total\n\nRoasted veggies are always delicious, but something magical happens to cauliflower in the oven. It gets so crispy and nutty, and that flavor is brought out even more with the spices here. I'm happy to just eat a bowl of this for dinner, maybe with an egg on top. serves 4 as a side\n\n1 head cauliflower, stem and florets, cut into small pieces\n\n2 cloves garlic, unpeeled\n\n1 tablespoon butter, melted\n\n1 teaspoon smoked paprika\n\n\u00bd teaspoon cayenne pepper\n\nsalt and pepper, to taste\n\n1 Preheat the oven to 400\u00b0F.\n\n2 Arrange the cauliflower and the garlic in a large roasting pan. Drizzle the butter over the cauliflower and sprinkle the paprika, cayenne, and generous amounts of salt and pepper over the top. Use your hands to coat the cauliflower with the butter and spices.\n\n3 Bake until the cauliflower can be easily pierced with a fork and the florets begin to brown, 45 minutes to 1 hour. If you like things extra crispy and dark brown, bake for the full hour. To serve, squeeze the roasted garlic among the florets and trash the skins.\n\nCrispy Chickpeas and Pumpkin Seeds\n\n$0.50 \/ serving\n\n$1 total\n\nSave the seeds from your winter squash or pumpkins: Just remove them from the juicy bits inside the squash and rinse them off. Spread them out to dry on a clean surface or in an oven on its lowest setting for 15 minutes. Once dry, you can store them for several weeks. After you've built up a good supply, toast them up and enjoy! Just like with popcorn, these are great with different spice combinations. My favorite is a half teaspoon each of ground coriander, turmeric, cumin, and cayenne pepper. serves 2 as a snack\n\n1\u00bd cups cooked chickpeas, drained\n\n\u00bd cup pumpkin or winter squash seeds\n\n1 teaspoon butter, melted\n\n1 teaspoon salt\n\n2 teaspoons any combination of ground spices (see headnote)\n\n1 Preheat the oven to 400\u00b0F.\n\n2 Put the chickpeas, pumpkin seeds, butter, salt, and spices in a bowl. Mix to coat every chickpea and seed with the spices.\n\n3 Spread the chickpeas and pumpkin seeds on a baking sheet in a single layer.\n\n4 Bake for 20 minutes.\n\n5 Remove the baking sheet from the oven and turn the chickpeas and seeds with a spatula, making sure they aren't sticking too much. Put the baking sheet back in the oven until everything is crusty and golden, 10 more minutes.\n\n6 Let the chickpeas and seeds cool for 10 minutes. Scoop into a bowl and serve.\n\nSpicy Green Beans\n\n$0.65 \/ serving\n\n$1.30 total\n\nWhenever I make these green beans, I think, \"Why don't I eat this every day?\" Throw a fried egg on top, serve with rice, and you have a delicious meal. Try to find the chile paste called sambal oelek; it's commonly available and you'll love having it in your flavor arsenal. It's my favorite Asian chile paste for this dish. It has a lovely, pure flavor, and (usually) no added vinegar or garlic. serves 2 as a side\n\n1 teaspoon vegetable oil\n\n8 ounces green beans, ends trimmed and chopped into bite-size pieces\n\n2 cloves garlic, finely chopped\n\n1 teaspoon soy sauce\n\n1 teaspoon sambal oelek\n\nadditions\n\n1 teaspoon grated ginger\n\n1 teaspoon lemon juice\n\n1 Place a frying pan over medium heat and add the vegetable oil. Once it's hot, add the green beans. Let them cook undisturbed for about 1 minute.\n\n2 Mix the garlic, soy sauce, sambal oelek, and any additions in a small bowl. This is your sauce!\n\n3 After 1 minute, the beans should have turned bright green. Add about \u00bc cup of water to the pan. Cook for another 2 minutes, until the water is mostly gone.\n\n4 Pour the sauce into the pan and toss gently to coat. Cook until everything is fragrant and most of the liquid is gone, another 2 minutes. Poke the beans with a fork: If it goes through easily, they're done.\n\n5 Add more chile sauce or soy sauce to taste if you want the beans hotter or saltier.\n\nideas\n\nPopcorn!\n\n$0.25 \/ serving\n\n$1 total\n\n**Popcorn is such a great snack.** It's easy to forget how simple and cheap it is to prepare at home when there are so many prepackaged varieties available at ten times the price.\n\nIt's also fun to get kids involved since the transformation is so dramatic and explosive. Science! Make the popcorn your own by adding your favorite flavors to it. My family went for Parmesan and black pepper on Sunday nights when I was growing up.\n\nBe sure to experiment and figure out what suits you best. I've suggested a few toppings on the opposite page. This recipe makes 10 to 12 cups of popped popcorn\u2014enough for four people. Eat it while it's hot!\n\nBasic Popcorn\n\nserves 4 as a snack\n\n2 tablespoons vegetable oil\n\n\u2153 cup unpopped popcorn kernels\n\n2 tablespoons butter, melted\n\nsalt, to taste\n\n1 Place a large pot with a tight-fitting lid on the stove. Pour in the vegetable oil, then the popcorn kernels. Put the lid on and turn the heat to medium.\n\n2 Using oven mitts, gently shake the pot from side to side to make sure the kernels are evenly distributed in the oil. Once the popcorn begins to pop, turn the heat down to medium low and shake again.\n\n3 Once the popping slows to 5 to 10 seconds between pops, turn off the heat. Wait until you're sure the popping has stopped, and remove the lid.\n\n4 Dump the popcorn into a large bowl, pour the butter over the top, and sprinkle with salt and any other toppings. Gently toss to coat.\n\n1 Minced scallion and cilantro\n\n2 Turmeric and coriander\n\n3 Spice Oil\n\n4 Parmesan and black pepper\n\n5 Cayenne pepper and smoked paprika\n\n6 Brown sugar and orange zest\n\n7 Chili powder and lime\n\n8 Parmesan and dried oregano\n\nMashed Beets\n\n$0.75 \/ serving\n\n$3 total\n\nMaking a mash or puree out of beets is a little different than making a mash out of another kind of root. Beets are much less starchy, so they don't naturally fluff up as much as other winter vegetables. For the smoothest results, use a food processor. The puree is electric magenta and delicious (although it's just as good when chunky). serves 4 as a side\n\nsalt, to taste\n\n4 medium-size beets\n\n1 tablespoon butter\n\n3 cloves garlic, finely chopped\n\n\u00bc cup chicken or vegetable broth\n\npepper, to taste\n\nadditions\n\norange juice instead of broth\n\ndill\n\nyogurt or sour cream\n\nvinegar\n\n1 Bring a pot of salted water to a boil. Add the beets, in their skins, and cook until a knife can pierce them easily, about 40 minutes.\n\n2 Drain the water and let the beets cool for 5 minutes.\n\n3 Meanwhile, melt the butter in a pan over medium heat. Saut\u00e9 the garlic until it smells great and turns translucent, but not brown, about 2 minutes. Remove from the heat and set aside.\n\n4 Once the beets are cool, chop off the tough stems and peel off the skin. It should be quite easy, because the cooking will have softened the exterior.\n\n5 Dice the beets and add them to a bowl or a food processor, along with the saut\u00e9ed garlic, the broth, and any additions. Process or mash until the mixture is very smooth. Taste and add salt and pepper as needed.\n\nMashed Cauliflower\n\n$1 \/ serving\n\n$4 total\n\nMashed cauliflower has a very light, nutty taste that soaks up the flavors you add to it, much like potatoes. Unlike potatoes, it's lighter in your belly and more interesting in texture. serves 4 as a side\n\nsalt, to taste\n\n1 head cauliflower, chopped\n\n1 tablespoon butter\n\n3 cloves garlic, finely chopped\n\npepper, to taste\n\nadditions\n\nspices\n\ngrated cheese\n\nyogurt or sour cream\n\nchiles, finely chopped\n\nginger, finely grated\n\nscallions, chopped\n\n1 Bring a large pot of salted water to a boil. Add the cauliflower and cook until tender, 5 to 7 minutes.\n\n2 Drain the water and remove the cauliflower.\n\n3 Place the pot back over medium heat and add the butter. Let it melt, add the garlic, and saut\u00e9 for about 2 minutes. Put the cauliflower back into the pot. Stir to coat, and cook until it is just heated through, or for a little longer to let it get golden and crispy. Your choice!\n\n4 Turn off the heat and let the mixture cool for 5 minutes. Use a potato masher to smash the cauliflower roughly. Add salt, pepper, and any other additions, and mash and mix some more. Taste and adjust as desired.\n\nWinter Squash Puree\n\n$0.60 \/ serving\n\n$2.40 total\n\nThe easiest way to cook winter squash is to roast it whole. The inside becomes soft and smooth and you can scoop it out of the skin with ease. serves 4 as a side\n\n1 tablespoon butter, plus more for the pan\n\n1 butternut, kabocha, acorn, delicata, or other winter squash (except spaghetti squash)\n\n3 cloves garlic\n\nsalt and pepper, to taste\n\nadditions\n\nyogurt or sour cream\n\nbrown sugar and cinnamon\n\nfinely chopped chiles\n\ncurry powder\n\nraisins\n\nsage\n\n1 Preheat the oven to 400\u00b0F. Butter a baking sheet.\n\n2 Slice the squash in half using a big, sharp knife. Scoop out the innards. Set the halves facedown on the sheet.\n\n3 Bake in the oven until a knife poked into the squash goes through easily, 30 to 40 minutes.\n\n4 Melt the butter in a pan over medium heat. Add the garlic and saut\u00e9 about 2 minutes. Remove from the heat.\n\n5 Scoop the squash into a large bowl with the garlic, the butter from the pan, and any other additions. Stir until smooth. Taste and add salt and pepper as needed.\n\nMashed Celery Root\n\n$0.75 \/ serving\n\n$3 total\n\nCelery root is just what it sounds like. It's the root from which the stalks of celery grow, and yes, you can eat it! Knobbly and ugly as it may appear, it tastes as if a potato and celery had a baby. serves 4 as a side\n\nsalt, to taste\n\n1 medium-size celery root, peeled and diced\n\n1 tablespoon butter\n\n3 cloves garlic\n\n\u00bd cup broth or water, warm\n\npepper, to taste\n\nadditions\n\nchopped fresh cilantro\n\ngrated ginger\n\nyogurt or sour cream\n\nchile flakes\n\n1 Bring a pot of salted water to a boil.\n\n2 Add the celery root to the pot and let the water return to a boil. Reduce the heat to a simmer and cook until the celery root is easy to pierce with a fork, 20 minutes. Drain the water and put the celery root in a bowl.\n\n3 Place the pot back over medium heat and add the butter. Let it melt, add the garlic, and saut\u00e9 until fragrant, about 2 minutes. Put the celery root back in the pot. Stir to coat, and cook until the celery root begins to get a bit mushy, about 5 minutes.\n\n4 Turn off the heat and add the broth along with any other additions. Use a potato masher, food processor, or electric mixer to smash the celery root until smooth. (Get it smooth enough and you'll almost think it's a potato.) Season with salt and pepper to taste.\n\nmethod\n\nBubble and Squeak\n\n**Bubble and Squeak is a traditional British weekend breakfast food** meant to use up the leftovers from the night before. Because it's British, of course the recipe includes mashed potatoes! It's basically a big potato pancake with stuff mixed into it. Chop up any other stray vegetables you have lying around. Find a carrot? Grate it or finely chop it and add it to the bowl. Cabbage, peas, corn, broccoli, Brussels sprouts\u2014all these things are great. You won't want to add really watery vegetables like tomato, zucchini, or cucumber, though, so stick to precooked or tougher ones.\n\nThe adorable name comes from the sound this dish makes in the pan, the squeak coming from the cabbage, which is the most common mix-in for the potato. The method I'm giving you is for one big pancake, but you could make smaller pancakes for a neater presentation. You can also forget the pancakes entirely and make a hash out of the mixture. This method is more of a technique than a traditional recipe, so just try to embrace the concept: Crispy and hot, you can't go wrong!\n\n2 cups mashed potatoes\n\n1 cup mashed root vegetables or roasted vegetables\n\nsalt and pepper, to taste\n\n1 tablespoon butter\n\nadditions\n\n\u00bd cup peas\n\n\u00bd cup cabbage\n\n2 to 3 scallions, finely chopped\n\n1 **Combine the mashed potatoes and mashed root vegetables or roasted vegetables in a large bowl**. The above measurements are a general guideline, but you can tweak the ratios so long as you have enough potato to hold it all together. (If it's falling apart, add more potato!)\n\n2 **Sprinkle with salt and pepper and mix everything together into a big thick kind of dough.** Taste it and see if it needs more salt and pepper. Since you are using leftovers that are already seasoned, you probably won't need to add much salt. This may seem haphazard, but it's a very forgiving recipe and as long as the mixture is at least half mashed potato it should work out just fine.\n\n3 Melt the butter in a cast-iron or nonstick skillet over medium heat. **Add the mixture to the pan and press it into a flat pancake** that fills the pan all the way to the edge. Let it cook for 5 to 7 minutes without touching it.\n\n4 Use a spatula to check the underside of the pancake. **When it's golden, it's time to flip it**. (If it's not, wait another minute or two.) Now comes a decision. You can try to flip the pancake all in one go\u2014something I usually fail at\u2014or you can simply flip it bit by bit. No need to worry if you break it up. After you flip the pancake, squish it back together again with the spatula.Brown the other side of the pancake for another 10 minutes. Once you have a nice golden crust around the whole pancake, turn off the heat. Let cool in the pan for 10 minutes.\n\n5 **Slice into wedges to serve.**\n\nJacket Sweet Potatoes\n\n$1.20 \/ serving\n\n$4.80 total\n\nI like to serve these with all kinds of toppings, usually leftovers from other meals. Try filling them with roasted chicken, beans and cheese, corn and tomatoes\u2014whatever you have around (but avoid veggies that can't stand up to residual heat, like lettuce or cucumbers). serves 4\n\n4 large sweet potatoes\n\nsalt and pepper, to taste\n\n\u00bc cup sour cream\n\n\u00bd bunch scallions, finely chopped\n\n1 Preheat the oven to 400\u00b0F.\n\n2 Scrub the sweet potatoes and stab them with a fork a few times. Lay them on a baking sheet.\n\n3 Bake for 60 to 75 minutes. Because sweet potatoes vary greatly in size, stick them with a long knife after an hour to check them. If it goes through easily, they're ready. If not, bake longer.\n\n4 Let the potatoes cool for 15 minutes. Make a long cut along the top of each potato and open them gently. Beat the soft, orange middle with a fork to fluff it up.\n\n5 Sprinkle with salt and pepper. Let each person add sour cream and scallions (or more salt and pepper) to their taste.\n\nRoasted Potatoes with Chiles\n\n$0.75 \/ serving\n\n$3 total\n\nIt doesn't get much simpler or more satisfying than roasted potatoes. You can use any pepper you like\u2014from large, dark poblanos to Hungarian wax chiles to bell peppers. When you chop the peppers, be sure to get rid of the seeds and white ribs inside. In addition to being a great side dish, this makes a delicious taco filling. Alternatively, try it alongside some black beans and rice or piled high on a plate with an egg on top. serves 4 as a side\n\n4 medium-size potatoes, chopped into bite-size pieces\n\n4 medium-size chile peppers or 2 bell peppers, chopped into bite-size pieces\n\n2 cloves garlic, unpeeled\n\n1 tablespoon butter, melted\n\nsalt and pepper, to taste\n\n1 Preheat the oven to 400\u00b0F.\n\n2 Tumble the potatoes, peppers, and garlic together in a large roasting pan.\n\n3 Pour the butter over the top and sprinkle liberally with salt and pepper. Potatoes need quite a bit of salt! Use your hands to mix everything up.\n\n4 Roast until you can spear the potatoes easily with a fork and everything is a little crispy, about 1 hour. Squish the garlic cloves, spread the roasted garlic throughout, and discard the skins.\n\nPoutine\n\n$1.75 \/ serving\n\n$7 total\n\nPoutine, a French-Canadian dish of French fries, gravy, and cheese curds, isn't an everyday meal, but it's a favorite. Since I don't like deep-frying at home, I bake the fries\u2014they still get crispy without the fuss. Montreal-style poutine is made with vegetable gravy, as in this recipe, but you can also use beef or turkey gravy. Of course, proper poutine uses cheese curds, and if you can find them do use them, but fresh mozzarella works almost as well. It has the same spongy quality, just maybe with a little less squeak. However, using fresh mozzarella makes this recipe come out a little more on the expensive side than you would think. If you use less or skip the cheese entirely, you can cut the price in half. serves 4 as a side\n\n2 tablespoons vegetable oil\n\n2 to 3 medium russet potatoes, cut into sticks\n\nsalt and pepper, to taste\n\n6 ounces fresh mozzarella cheese, diced\n\n1 to 2 scallions, chopped\n\ngravy\n\n2 tablespoons butter\n\n1 shallot or 3 scallions, finely chopped\n\n3 cloves garlic, finely chopped\n\n6 leaves fresh sage, finely chopped (optional)\n\n2 tablespoons all-purpose flour\n\n1\u00bd cups vegetable broth\n\n1 teaspoon soy sauce\n\n\u00bd teaspoon cayenne pepper\n\nsalt and pepper, to taste\n\n1 Preheat the oven to 400\u00b0F.\n\n2 Spread 1 tablespoon of the vegetable oil on a baking sheet, then add the potato. Drizzle the remaining tablespoon of oil over the potatoes and sprinkle them generously with salt and pepper. Toss the potatoes with your hands and evenly space them across the baking sheet. Bake for 20 minutes.\n\n3 Meanwhile, melt the butter in a small saucepan over medium heat. Swirl it around to coat the pan. Add the shallot, garlic, and sage, if using, and cook until the shallots are translucent but not brown, about 2 minutes. Quickly stir in the flour. Add a little vegetable broth if the mixture gets too lumpy.\n\n4 Let the mixture cook until it turns light brown. Add the vegetable broth, soy sauce, and cayenne pepper. Bring the gravy to a boil, then turn down the heat and let it cook until it thickens to a gravy-ish consistency, about 5 minutes, stirring occasionally. Taste it, adding salt and pepper as needed. Turn down the heat to its lowest setting, just enough to keep the gravy warm.\n\n5 After the fries have baked for 20 minutes, remove them from the oven. Shuffle them with a spatula and test with a fork. If it goes through easily, the fries are ready. If you want them a little crispier, flip them over and put them back in the oven for a bit.\n\n6 Once they're done, pile one layer of fries onto a plate. Top with the diced mozzarella and the gravy. Repeat with a second layer and sprinkle with scallions and more freshly ground black pepper.\n\nideas\n\nThings on Toast\n\n$0.75 \/ serving\n\n$1.50 total\n\n**I love bread, and toast in particular is my comfort food.** Here, I suggest that you take some toast and put something tasty on it. That's it!\n\nI like these recipes for times when I'm on my own and want a quick meal or snack. They're a great way to use leftovers or turn a side dish into a full meal. What makes this more like a special dinner than a quick snack is the way you treat the bread\u2014toasting it in the pan like a nice piece of fish.\n\nA pile of saut\u00e9ed or raw veggies over buttered, crusty toast is the perfect meal for one or two and a great way to try a new vegetable. I've suggested a few topping variations on the following pages, but you can use pretty much any veggie dish from this book or invent your own. Go with raw vegetable salads or, as I most often do, saut\u00e9 veggies or cooked beans in oil with flavors like garlic and chiles, olives and dill, ginger and turmeric, or any other classic combination (page ). It only takes a few minutes, and the result is an aromatic, attractive, satisfying meal.\n\nBasic Toast\n\nserves 2\n\n4 slices bread\n\n2 tablespoons butter\n\nsalt and pepper, to taste\n\n2 fried eggs (optional)\n\n1 Melt \u00bd tablespoon of the butter in a pan over medium heat. Place 2 slices of bread in the pan, let them cook for about 2 minutes, then lift them with a spatula to see if they're golden brown underneath. When they are, flip 'em over.\n\n2 Add another \u00bd tablespoon of the butter to the pan to make sure the second side of the toast becomes just as golden as the first. Sprinkle the toast with salt and pepper. Once the second side is golden, set the bread on a plate to await its topping. Repeat with the remaining bread and butter.\n\n3 Pile on some toppings (see the following pages for inspiration) and finish each with a fried egg, if you have it.\n\n1 Peas and Lemon\n\n$1 serving \/ $2 total\n\nA less salty, more rustic version of the British classic mushy peas.\n\n1 teaspoon olive oil\n\n2 cloves garlic, finely chopped\n\n1 cup peas, fresh or frozen\n\n1 teaspoon lemon juice\n\nfreshly grated Romano or Parmesan cheese\n\nsalt and pepper, to taste\n\nPlace a pan on the stove over medium heat and add the olive oil. Drop in the garlic and peas along with 2 tablespoons water so that the peas can steam a bit. Leave them until they turn bright green. Sprinkle with the lemon juice, cheese, salt, and pepper, then remove the peas from the heat and mash with the back of a fork, either in the pan or in a bowl. Pile onto warm toast and enjoy!\n\n2 Asian Greens Gra Prow\n\n$2.50 serving \/ $5 total\n\nTry any Asian green you can get your hands on, from bok choy to tatsoi to gai lan. _Gra prow_ means stir-fried with Thai basil.\n\n1 teaspoon vegetable oil\n\n2 cloves garlic, chopped\n\n1 teaspoon grated ginger\n\n2 teaspoons soy sauce\n\n1 bunch Asian greens, stems and leaves chopped separately\n\n1 handful Thai basil\n\nsalt and pepper, to taste\n\nHeat the oil in a pan over medium heat. Saut\u00e9 the garlic until fragrant, 2 minutes, then add the ginger, soy sauce, and the chopped stems of the greens. Cook until almost tender, 4 to 5 minutes. Add the rest of the greens and cook for 2 more minutes. Turn off the heat and mix in the Thai basil, salt, and pepper.\n\n3 Caramelized Onions and Cheddar\n\n$1 serving \/ $2 total\n\nCaramelizing onions isn't quick, but if you make a large batch it's worth it. Sweet onion and salty Cheddar are a drool-inducing pair!\n\n1 tablespoon butter\n\n1 red onion, thinly sliced\n\nsharp Cheddar cheese, thinly sliced\n\nsalt and pepper, to taste\n\nMelt the butter in a pan over low heat. Add the onion and cook slowly, about 20 minutes. As the onions darken, stir occasionally, adding water to loosen the sticky bits. When the onions are sweet and caramelized, spread them over toast, and top with Cheddar, salt, and pepper. Put the toast back in the pan and cook, covered, until the cheese is bubbly.\n\n4 Roasted Vegetables\n\n$1 serving \/ $2 total\n\nOne of my favorite ways to eat leftover roasted vegetables.\n\nRoasted Vegetables\n\nfreshly grated Romano or Parmesan cheese\n\npepper, to taste\n\nSimply create a ridiculously tall pile of vegetables like the winter squash and leeks pictured here, then sprinkle with grated Romano and fresh pepper. You can also add any sauce you have on hand, or sprinkle crushed nuts on top.\n\n5 Salty Broccoli\n\n$1.50 serving \/ $3 total\n\nMost broccoli that enters my home meets its fate as this dish.\n\n1 teaspoon olive oil\n\n3 cloves garlic, chopped\n\n1 teaspoon chile flakes\n\n1 anchovy fillet, chopped\n\n1 crown and stem of broccoli, chopped\n\nfreshly grated Romano or Parmesan cheese\n\nsalt and pepper, to taste\n\nWarm the oil in a pan over medium heat. Add the garlic and chile flakes and cook 2 minutes. Add the anchovy and cook for another minute. Add the broccoli and \u00bc cup water. Cover the pan, steam for 3 minutes, then toss and cook for 2 minutes, until the broccoli is tender and the water is gone. Spoon onto warm toast. Top with cheese, salt, and pepper.\n\n6 Broiled Eggplant Salad\n\n$1.75 serving \/ $3.50 total\n\nYet another use for leftovers\u2014or just a way to make a great salad more substantial.\n\nBroiled Eggplant Salad\n\nchopped fresh herbs or greens\n\nany cheese, crumbled or grated\n\nSimply dollop the Broiled Eggplant Salad onto toast, then add some herbs or greens to the top for a fresh counterpoint, along with a bit of cheese.\n\n7 Peas and Collards\n\n$1 serving \/ $2 total\n\nOh man, is there anything more comforting than beans (or these bean-like peas) on toast? Everyone will be delighted even if you're just using up leftovers. To make the meal a little more fancy, use scones instead of toast.\n\nBlack-Eyed Peas and Collards\n\nWhole-Wheat Jalape\u00f1o Cheddar Scones\n\nHeat up the peas and collards in a pan on the stovetop or simply in the microwave. Spoon over warm toast and enjoy. If you're using the scones, slice them in half and toast them in a toaster oven or under your broiler before dolloping each half with the pea mixture.\n\n8 Spinach and Chickpea\n\n$4.50 serving \/ $2.25 total\n\nA popular tapas dish in Spain. This recipe makes a little more topping than you need for four pieces of toast\u2014but hey, leftovers are tasty.\n\n1 teaspoon butter\n\n2 cloves garlic, chopped\n\n1 cup cooked chickpeas\n\n1 bunch spinach, washed, thicker stems removed\n\nsalt and pepper, to taste\n\nsmoked paprika (optional)\n\nMelt the butter in a pan on medium heat. Add the garlic and cook for 2 minutes. Add the chickpeas and spinach, then cook for 2 to 5 minutes, until the spinach cooks down but is still bright green. Taste and add salt and pepper, then spoon over warm toast. If you have it, sprinkle with smoked paprika.\n\n9 Avocado\n\n$0.90 serving \/ $1.80 total\n\nYears ago, my partner and I arrived in Sydney, Australia, after cycling hundreds of miles. Our hosts gave us a mountain of toast with avocado. I may have eaten an entire loaf of bread and maybe three avocados. It is one of the most memorable meals of my life.\n\n1 avocado\n\nsalt and pepper, to taste\n\nchile flakes (optional)\n\nlemon or lime wedge (optional)\n\nSlice the avocado and remove the pit\u2014it should pop out easily. Scoop the avocado flesh out of each side, and slice in half again so you have 4 equal portions. Mash the avocado with a fork and spread onto the toast. Top with salt and pepper, a few chile flakes, and a squeeze of lemon or lime juice, if using.\n\n10 Apple Cheddar\n\n$1.40 serving \/ $2.80 total\n\nHave you ever seen anyone eat apple pie with a little Cheddar on it? This is the same, but even better! Try using an aged white Cheddar if you can get it, but your favorite standard Cheddar will do just as well. Crisp apple plus luscious, salty, crumbly Cheddar equals one of my favorite late-night snacks (even minus the toast).\n\n2 ounces Cheddar cheese\n\n1 apple\n\nsalt and pepper, to taste\n\nThinly slice the Cheddar and slice the apple. I prefer to layer the apples and then slide small slices of Cheddar in between them, but do it in whatever way makes sense to you.\n\n11 Saut\u00e9ed Mushrooms\n\n$1.75 serving \/ $3.50 total\n\nRich and earthy, the smell of mushrooms and garlic will have neighbors knocking on your door. These are also an incredible hamburger topping, so consider making enough for leftovers.\n\n1 tablespoon butter\n\n1 pound mushrooms, sliced\n\n4 cloves garlic, chopped\n\nsalt and pepper, to taste\n\nMelt the butter in a pan over medium heat. Add the mushrooms and saut\u00e9 until they have shrunk and expelled some of their liquid, about 2 minutes. Mix in the garlic and cook until the mushrooms are light brown all over, about 5 more minutes. Add salt and pepper to taste. Spoon onto warm toast.\n\n12 Korean-Style Spinach\n\n$1.75 serving \/ $3.50 total\n\nThis spinach, saut\u00e9ed with sesame and garlic, is one of my favorites.\n\n1 teaspoon olive oil\n\n4 cloves garlic, chopped\n\n1 bunch spinach, washed, thicker stems removed\n\n1 teaspoon soy sauce\n\n\u00bd teaspoon toasted sesame oil\n\nsalt, to taste\n\n1 teaspoon sesame seeds\n\nHeat the olive oil in a pan over medium heat. Add the garlic and cook, 2 minutes. Add the spinach and soy sauce and cook for 2 minutes, until the spinach has wilted and shrunk. Turn off the heat and add the sesame oil and salt. Mix and taste. Remove the spinach from the pan and squeeze out any excess moisture. Serve over hot slices of toast. Sprinkle sesame seeds on top.\n\nmethod\n\nCornmeal-Crusted Veggies\n\n**These are kind of like veggie French fries.** The cornmeal makes them super crunchy, and they're great with a dipping sauce. Might I suggest Peanut Sauce? Almost any vegetable works well with this method\u2014some of my favorites are zucchini wedges, bell pepper strips, and wedges of cooked winter squash. (Pictured are bell peppers and green beans.) It's sort of like fried green tomatoes or okra, but this baked version skips the expense and mess of the oil while keeping the crunch.\n\noil or butter, for the baking sheet\n\n\u00bd cup all-purpose flour\n\n2 eggs\n\n\u00bc cup milk\n\n1 cup cornmeal\n\n1 teaspoon salt\n\n1 teaspoon black pepper\n\n1 teaspoon paprika\n\n\u00bd teaspoon garlic powder\n\n8 ounces green beans, stemmed\n\nvariations\n\nzucchini wedges\n\nbell pepper strips\n\ncooked winter squash wedges\n\ncauliflower florets\n\nbroccoli florets\n\nwhole okra\n\ncarrot or parsnip sticks\n\nasparagus, whole if very thin\n\n1 **Preheat the oven to 450\u00b0F.**\n\n2 **Grease a baking sheet** with a small amount of oil or butter.\n\n3 **Set up your breading station!** Spread the flour on a plate. Crack both eggs into a bowl, add the milk, and beat lightly with a fork. On another plate, spread the cornmeal, salt, black pepper, paprika, and garlic powder, and mix them up with your fingers.\n\n4 A few at a time, **dredge the green beans in the flour and transfer them to the egg mixture.** Coat the beans lightly with the egg, being careful to shake off any excess. Then **transfer to the cornmeal mixture** and coat them evenly.\n\n5 **Spread the veggies on the baking sheet.** Repeat until you've done them all. If you run out of any of the three mixtures, just mix up a bit more.\n\n6 **Bake for 10 to 15 minutes,** until golden and crispy. Enjoy hot with your favorite dipping sauce!\n\nGreen Chile and Cheddar Quesadillas\n\n$1.75 \/ serving\n\n$3.50 total\n\nThese are a great snack or a quick meal, and you can add pretty much anything to them! To make 'em cheaper, use fresh tortillas made from scratch. serves 2\n\n4 Tortillas\n\n\u00bd cup chopped green chiles, canned or fresh\n\n\u00bd cup grated sharp Cheddar cheese\n\n1 tablespoon chopped fresh cilantro\n\nto serve\n\nSalsa, for serving\n\nsour cream, for serving\n\n1 Place 1 tortilla on a cutting board and spread half of the green chiles evenly over the top. Sprinkle half of the cheese over the chiles, then top with half the cilantro. Place another tortilla on top of each prepared tortilla to form a quesadilla. Repeat!\n\n2 Place a large, nonstick pan over medium heat. Once it's hot, add a quesadilla and toast until the tortilla browns, about 1 minute. Flip it over and brown the second side, then do the same for the other quesadilla. Slice into triangles and enjoy with some fresh salsa and sour cream. \nDinner\n\nFilipino Chicken Adobo\n\n$1.30 \/ serving\n\n$10.40 total\n\nThis ultra-adaptable recipe comes to us courtesy of Tony Pangilinan, who grew up on food stamps after his family immigrated from the Philippines \"with nothing but four suitcases and a lot of dreams.\" After several decades of struggle, Tony achieved those dreams and can now help family members who remain in poverty in the Philippines. He says that, despite their hard circumstances, his family still feels blessed.\n\nFilipino adobo\u2014very different from Spanish adobo\u2014is basically anything cooked in vinegar, soy sauce, and garlic. Although this version is made with chicken, you can use any meat or vegetables you like. Because it's vinegar-based, it keeps well in the fridge! serves 8\n\n8 chicken thighs\n\n\u00be cup rice vinegar or distilled white vinegar\n\n\u00bc cup soy sauce\n\n2 cloves garlic, minced\n\n\u00bd teaspoon black pepper\n\n2 bay leaves\n\n2 tablespoons vegetable oil\n\n2 medium-size potatoes, chopped\n\n4 medium-size carrots, sliced\n\n2 cups white rice\n\n2 pinches of salt\n\n2 teaspoons cornstarch mixed with 1 tablespoon water\n\nadditions\n\n4 jalape\u00f1o peppers, sliced (remove seeds for less heat)\n\nginger, grated\n\nvariations\n\n1\u00bd pounds pork shoulder or butt\n\nsubstitute 1 can (13.5 ounces) coconut milk for the water\n\n1 Cut off the chicken fat. Don't get rid of every last bit, just trim what seems excessive.\n\n2 Stir together the vinegar, soy sauce, garlic, pepper, and bay leaves in a large, non-aluminum pan. Add the chicken, coating each piece. Cover and marinate for 30 minutes (overnight is great).\n\n3 Remove the chicken from the marinade and pat each piece dry with paper towels. (Don't throw out the marinade! You'll use it later.)\n\n4 Pour the oil into a large pot over medium heat. Once it's hot, add enough chicken to cover the bottom. Cook until one side of the chicken is browned, just a few minutes, then flip it over and repeat. When the first batch of chicken is done, remove it from the pot and repeat.\n\n5 After all the chicken is browned, put it back in the pot along with the marinade, potatoes, carrots, and \u00be cup of water. Turn the heat up, bring the liquid to a boil, then decrease the heat to low. Simmer until the meat is cooked through and the carrots and potatoes are soft, about 45 minutes. Cut into the meat. If it's no longer pink, it's done.\n\n6 When the dish is almost ready, pour the rice into a medium-size pot with 4 cups of water and the salt. Bring to a boil over medium heat. Turn the heat down and cover with a lid placed slightly askew. Cook until the water is gone, about 20 minutes.\n\n7 Remove the bay leaves from the adobo and stir in the cornstarch and water mixture. Let it thicken until the chicken and vegetables are well glazed. Serve over the rice.\n\nRoast Chicken\n\n$1.50 to $2.75 \/ serving\n\n$8 to $15 total\n\nAlthough chicken ranges widely in prices, a whole chicken is usually less expensive than buying single pieces like breasts or thighs\u2014plus you can make broth later from the bones and any meat too difficult to get off. The leftovers can be used in sandwiches, in tacos, over a salad, or tossed with sauce and mixed into pasta. This is a base recipe: Add spices to the butter or sprinkle them over the surface of the chicken to change up the flavor in any way you like. serves 6\n\n1 whole chicken, about 4 pounds\n\n1 tablespoon butter, melted\n\nsalt and pepper, to taste\n\n2 cloves garlic\n\n1 lemon\n\n1 Preheat the oven to 400\u00b0F.\n\n2 Remove the giblets from the inside of the chicken and chop off the neck. Keep them for broth later. You can either freeze them or just leave them in the fridge if you plan to make broth in the next couple of days. Rub the entire bird with the melted butter, then sprinkle it with salt and pepper.\n\n3 Smash the garlic cloves with the side of a knife blade, remove the peel, and slice the lemon in half. Stuff the garlic and lemon into the body cavity.\n\n4 Place the chicken in a roasting pan or an ovenproof skillet. Transfer to the oven and cook for 1 hour. If you have a meat thermometer, check to make sure the chicken is at 165\u00b0F, the temperature at which chicken is completely safe to eat, but an hour in a 400\u00b0F oven should be long enough to cook it fully.\n\n5 Let the chicken rest for at least 10 minutes before you carve it to make sure you don't lose any of the tasty juices. Remove the garlic and lemon before carving.\n\nAfter you carve away all the meat, make chicken broth from the carcass, giblets, and neck. Simmer them for several hours in a pot full of water along with vegetable scraps like the ends of onions and carrots, plus a generous helping of salt.\n\nPeanut Chicken and Broccoli with Coconut Rice\n\n$1.50 \/ serving\n\n$9 total\n\nThis recipe uses peanut sauce to elevate a pretty plain chicken and broccoli stir-fry to something you'll want to serve to your favorite guests. Make a full batch of the Peanut Sauce and use some for this and the rest for dipping veggies, dressing salads, or smothering your favorite protein. serves 6\n\n1\u00bd cups long-grain rice\n\n1 can (13.5 ounces) coconut milk\n\n\u00bd teaspoon salt, plus more to taste\n\n1\u00bd pounds chicken (any part), chopped into bite-size pieces\n\npepper, to taste\n\n2 teaspoons vegetable oil\n\n6 cups chopped broccoli, stems and florets separated (about 1 large bunch)\n\n\u00bd cup Peanut Sauce\n\nchopped fresh cilantro\n\nvariation\n\n10 ounces tofu, cut into cubes and marinated in \u00bc cup soy sauce, instead of chicken\n\n1 Rinse the rice. Add it, along with the coconut milk, salt, and 1\u00bd cups water, to a pot over medium heat. Bring to a boil, then turn the heat to low. Let the rice simmer, covered, with the lid askew, until the liquid is gone, about 20 minutes. If the rice is done before the stir-fry, remove it from the heat, fluff it a bit with a fork so it doesn't stick to the pot, and cover to keep it warm.\n\n2 While the rice cooks, sprinkle the chicken with salt and pepper and set aside.\n\n3 Place a large pan or wok over medium-high heat and add 1 teaspoon of the vegetable oil. Let it get hot and add the broccoli stems. Cook, stirring occasionally, to soften the stems, about 3 minutes. Add the tops of the broccoli and \u00bc cup of water and cover the pan. It will steam and sizzle a lot, so watch out! Let the broccoli cook until the water evaporates, about 3 more minutes. Test a piece of broccoli with a fork. It should be just barely tender, but not soft. Turn off the heat and remove the broccoli from the pan.\n\n4 Add the remaining teaspoon of oil to the pan and put it back over medium heat. Add the chicken and cook, stirring occasionally, until it's no longer pink, about 5 minutes. Add another \u00bc cup water and stir occasionally until the chicken is cooked all the way through, another 2 minutes.\n\n5 Add the peanut sauce and stir to coat the chicken. Don't worry if the sauce seems too thick at first. It will blend with the water to become a glaze.\n\n6 Once the chicken is coated with sauce, put the broccoli back into the pan and stir it all together. Taste and add salt as needed.\n\n7 Scoop the coconut rice onto plates and top with the broccoli, chicken, and cilantro.\n\nBeef Stroganoff\n\n$2.75 \/ serving\n\n$16.50 total\n\nBeef stroganoff is one of my husband's favorites, so I make it as a treat for him\u2014and one of my early readers, Dave, says his mother made it for him growing up. It's a classic winter meal from Eastern Europe that warms up a cold house and fills the air with rich aromas. You can use any cut of beef, but adjust the cooking time based on the toughness. If you aren't sure about which cuts of beef to use here, ask your butcher. serves 6\n\n1 pound beef chuck or other cut\n\nsalt and pepper, to taste\n\n2 tablespoons butter\n\n2 onions, chopped\n\n2 large carrots, chopped\n\n1 tablespoon flour\n\n2 teaspoons paprika\n\n1 pound egg noodles or any pasta\n\n3 cloves garlic, finely chopped\n\n1 pound mushrooms, chopped\n\n\u00bd cup sour cream\n\n3 tablespoons mustard\n\nadditions\n\n\u00bd cup red wine (add with the water in Step 3)\n\npotatoes and 1 bell pepper, chopped (add with the carrots and onions in Step 3)\n\nfresh dill (for garnish)\n\n1 Chop the raw beef into bite-size pieces and season them generously with salt and pepper.\n\n2 Melt 1 tablespoon of the butter in a large saucepan over medium heat. Toss in enough beef to cover the bottom of the pan. You may need to cook the meat in two batches, depending on the size of your pan. Brown the meat on all sides, then set it aside on a plate.\n\n3 Add the onions and carrots to the pan and cook until the onions become translucent. Sprinkle the flour and paprika over the top, then cover everything with 4 cups of water. Put the meat back into the pot. Place a lid slightly askew on the pot so steam can escape. Let cook over medium-low heat until the beef is tender and the water has turned into broth, about 2 hours.\n\n4 Bring a large pot of salted water to a boil. Cook the noodles according to the package instructions. Try to time it to coincide with finishing the stew.\n\n5 Meanwhile, melt the remaining tablespoon of butter in another pan over medium heat. Add the garlic and cook until fragrant, about a minute. Add the mushrooms, toss them to coat, sprinkle with salt and pepper, and cook, stirring occasionally, until they shrink and turn brown, about 5 minutes. Turn off the heat, taste, and add salt and pepper as needed.\n\n6 Check on the beef. If the water has reduced to about a cup of thick, flavorful liquid and the beef is tender, it's done! If not, let it cook a little longer and keep checking on it. Once it's ready, turn down the heat and stir in the mushrooms, sour cream, and mustard. Taste and add more salt, pepper, and paprika if needed.\n\n7 Serve the stew over the noodles and sprinkle with more paprika.\n\nIf you're using a pricier, more tender cut of beef, like round steak or sirloin tip, you don't need to cook it nearly as long. Simply brown the meat, substitute 1 cup of beef broth for the water, and cook for 20 to 30 minutes.\n\nSpicy Broiled Tilapia with Lime\n\n$4.50 \/ serving\n\n$9 total\n\nThis meal comes together so quickly it's astonishing. Broiled fish is crispy on the outside and flaky and moist on the inside. Serve with rice, a vegetable mash, or a favorite side dish like Spicy Green Beans. If you saut\u00e9 some vegetables while the fish cooks, dinner will be on the table in minutes. serves 2\n\n1 teaspoon vegetable oil\n\n1 teaspoon salt\n\n\u00bd teaspoon black pepper\n\n1 teaspoon cayenne pepper\n\n1 teaspoon ground cumin\n\n\u00bd teaspoon garlic powder\n\n\u00bd teaspoon dried oregano\n\n2 fillets tilapia or other white fish\n\n\u00bd lime\n\n1 Turn your oven's broiler to high. Line a baking sheet with aluminum foil and coat it evenly with the vegetable oil to keep the fish from sticking.\n\n2 In a small bowl, mix together the salt, pepper, cayenne, cumin, garlic powder, and oregano. Sprinkle them over both sides of the fish and massage gently with your fingers until the fish is covered thoroughly.\n\n3 Place the fish on the prepared baking pan. If you are using a fillet that still has its skin, make sure it is skin-side up so it will get crispy.\n\n4 Broil the fillets for 4 to 7 minutes. The fish will cook very quickly, so check after 4 minutes by gently inserting a butter knife into the thickest part. If it goes through easily and the fish flakes apart, then it's done. If the knife meets resistance and the fish stays together, put the fillets back under the broiler for another few minutes. Once you've made this recipe once or twice, you'll be able to tell at a glance when your fish is done.\n\n5 When the fish is done, squeeze lime juice over it.\n\nSome broilers are not strong enough to get the skin crispy\u2014if that's the case for you, pull it off just before serving. Or leave it. Crispy or not, fish skin is delicious.\n\nCreamy Zucchini Fettuccine\n\n$1.80 \/ serving\n\n$3.60 total\n\nZucchini and summer squash are so abundant in the summer months. This simple pasta is like a lighter, brighter fettuccine alfredo. It also comes together in no time\u2014the veggies will be ready by the time your pasta is cooked. You'll love it, I promise. serves 3, or 2 very hungry people\n\nsalt, to taste\n\n8 ounces fettuccine\n\n4 tablespoons butter\n\n4 cloves garlic, finely chopped\n\n\u00bd teaspoon chile flakes\n\n2 small zucchini, finely diced\n\nzest of 1 lemon\n\n\u00bc cup heavy cream\n\n\u00bd cup grated Romano or Parmesan cheese\n\npepper, to taste\n\nsprinkling of finely chopped fresh basil (optional)\n\n1 Bring a pot of water to a boil over high heat. Salt the water liberally. This is how pasta gets its flavor, so don't be shy! Most of the salt won't end up in the pasta.\n\n2 Cook the pasta according to the package directions. I drain the pasta just before it's finished so it doesn't get mushy when I add it to the vegetable pan and it gets cooked a little more.\n\n3 Meanwhile, melt 1 tablespoon of the butter in a pan over medium heat. Add the garlic and chile flakes. Let them sizzle for 30 seconds to a minute, then add the zucchini. Stir the vegetables to coat. Cook, stirring occasionally, until some of the water has cooked off and the veggies are tender, 5 to 7 minutes. Young summer zucchini doesn't need much cooking. Add the lemon zest. Stir!\n\n4 Drain the fettuccine and add it to the pan along with the remaining 3 tablespoons of butter, the cream, and most of the cheese. Toss the fettuccine around the pan, and add salt and lots of freshly ground pepper to taste. Top with a bit more cheese and a sprinkling of basil, if using, and serve immediately.\n\nPasta with Eggplant and Tomato\n\n$2.50 \/ serving\n\n$5 total\n\nThis dish is similar to a traditional pasta alla norma, but without anchovies and ricotta salata. It's a perfect weeknight dinner in the summertime. I like to use a tubular pasta for this dish, but you can use anything, even spaghetti. The eggplant and tomatoes come together into a sauce that is thick, jammy, and savory. serves 3, or 2 very hungry people\n\nsalt\n\n8 ounces rigatoni\n\n2 tablespoons olive oil\n\n1 large eggplant, cubed\n\n4 cloves garlic, finely chopped\n\n\u00bd teaspoon chile flakes\n\n2 cups finely diced canned tomatoes\n\n\u00bc cup freshly grated Romano or Parmesan cheese\n\nsprinkling of finely chopped fresh basil (optional)\n\npepper, to taste\n\n1 Put a pot of water over high heat and add a good shake of salt. Bring it to a boil and cook the pasta according to the package instructions.\n\n2 Meanwhile, set a wide pan over medium-high heat and splash in the olive oil. Let it get hot, then add the eggplant cubes, sprinkle with salt, and cook until the cubes start to brown, about 5 minutes. If the eggplant starts to look too dry, add a bit of water.\n\n3 Once the cubes are a little brown on all sides, add the garlic and chile flakes and stir. Add the tomatoes and cook, stirring occasionally, about 15 minutes. Again, if it looks too dry, add a bit of water. Everything will shrink up and become a sort of loose, thick sauce.\n\n4 Add half the cheese and half the basil, if using, and stir to combine.\n\n5 Once the pasta is cooked, drain it and add it to the saucepan. Toss everything together, then turn off the heat. Add salt and pepper to taste, and serve in bowls sprinkled with the remaining Romano and basil.\n\nTomato and Tuna Spaghetti\n\n$2.25 \/ serving\n\n$4.50 total\n\nWe already know that a can of tuna can be great in a salad or sandwich for a quick, satisfying lunch, but those uses are just scratching the surface. Canned tuna has a light flavor and a lovely, flaky texture that goes well in all kinds of dishes where you'd like more protein. This traditional Italian pasta dish brings together common pantry items for a beautiful and quick weeknight dinner. serves 3, or 2 very hungry people\n\nsalt, to taste\n\n8 ounces spaghetti\n\n1 tablespoon olive oil\n\n4 cloves garlic, finely chopped\n\n1 cup pureed or chopped canned tomatoes (see box)\n\n1 can (5 ounces) tuna (see box)\n\n\u00bd cup grated Romano cheese\n\npepper, to taste\n\nadditions\n\n2 anchovy fillets, finely chopped\n\n2 tablespoons chopped olives\n\n\u00bc cup Breadcrumbs\n\n1 tablespoon capers\n\nYou can use either oil- or water-packed tuna here. The oil-packed tuna will slip and slide over the noodles a little more easily, but the water-packed can work just fine if you add another tablespoon of olive oil along with it.\n\n1 Boil a large pot of heavily salted water and cook the spaghetti according to the package instructions.\n\n2 Meanwhile, make your sauce. Place a pan over medium heat and add the olive oil. Let it get warm, then add the garlic and saut\u00e9 until it smells great, about 1 minute.\n\n3 Add the pureed tomatoes to the garlic and swirl around the pan. Let the sauce warm up for 2 minutes before adding the tuna. Use a wooden spoon to break up the tuna, and cook until it's heated through, about 3 more minutes.\n\n4 Add \u00bc cup of the grated Romano, reserving the rest to sprinkle on top. Taste and add salt and pepper as needed. Turn down the heat as low as possible and place a lid over the pan to keep it warm while you finish boiling the spaghetti.\n\n5 Once the spaghetti is ready, drain most of the water from the pot, reserving a small amount. Dump the spaghetti into the pan with the sauce and toss everything together. If the sauce isn't moving nicely and coating the noodles, add a little bit of the reserved pasta water and toss it some more.\n\n6 Once the spaghetti is coated, serve it topped with the remaining Romano.\n\nIf you want to create a more complex dish, include one or more of the additions. The anchovies, olives, and capers can be added to the sauce at the beginning, but use the breadcrumbs at the end with the reserved Romano for a little crunch.\n\nCauliflower Cheese\n\n$1.65 \/ serving\n\n$6.60 total\n\nThis is a classic side dish in Great Britain: creamy, cheesy sauce over cauliflower, baked in the oven until the edges get crunchy and bubbly. It's like a healthier and more flavorful version of macaroni and cheese. Try substituting broccoli or cooked winter squash for the cauliflower\u2014everyone will love it. With broccoli, this dish becomes an interpretation of the classic sleazy broccoli with cheese sauce\u2014but a little less sleazy. You can also add some breadcrumbs to the top of the dish before baking if you like extra crunch. Enjoy with a green salad. serves 4\n\n2 teaspoons salt, plus more to taste\n\n1 head cauliflower, cut into bite-size pieces\n\n1 tablespoon butter, plus more for the baking dish\n\n3 cloves garlic, finely chopped\n\n\u00bd teaspoon chile flakes\n\n1 bay leaf\n\n1 tablespoon all-purpose flour\n\n1\u00bd cups milk\n\n6 ounces sharp Cheddar cheese, grated\n\npepper, to taste\n\nadditions\n\n1 tablespoon Dijon mustard\n\n4 scallions, finely chopped\n\nzest of 1 lemon\n\n1 teaspoon smoked paprika\n\n\u00bd teaspoon dried thyme\n\nBreadcrumbs\n\nsprinkling of finely chopped fresh basil\n\n1 Preheat the oven to 400\u00b0F.\n\n2 Bring a large pot of water to a boil over high heat. Add the salt and the cauliflower, then leave it for 4 minutes.\n\n3 Meanwhile, butter a baking dish large enough to comfortably accommodate all the cauliflower. I usually use a pie dish. Drain the cauliflower and add it to the dish.\n\n4 Melt the butter in a medium-size saucepan over medium heat. Add the garlic, chile flakes, and bay leaf and cook for about 1 minute. Add the flour and stir quickly. The flour-butter mixture is called a roux. You want the roux to get just a little brown\u2014this will probably take another minute. Slowly add the milk to the pot, stirring all the while to incorporate the roux and make a creamy sauce.\n\n5 Continue cooking the sauce, stirring occasionally, until it just comes to a boil, about 5 to 7 minutes. Once a couple of bubbles appear, turn off the heat and stir the cheese into the sauce. Include any additions at this point (except breadcrumbs). Taste the sauce and add salt and pepper as needed. Remove the bay leaf. The sauce should be creamy, smooth, and savory.\n\n6 Pour the sauce over the cauliflower, sprinkling with breadcrumbs if desired. Place the dish in the oven and bake until the top is brown and bubbly, about 40 minutes.\n\nSavory Summer Cobbler\n\n$2 \/ serving\n\n$8 total\n\nCelebrate summer's most ubiquitous vegetables\u2014tomatoes and zucchini\u2014with a crunchy, peppery, Southern biscuit topping. For a variation, swap the zucchini for eggplant. Chop the eggplant into bite-size pieces, salt them, and set them aside for 30 minutes before using them in Step 2 as you would with the zucchini. serves 4\n\n1 tablespoon olive oil, plus more for the pan\n\n3 or 4 medium-size zucchini or summer squash, chopped into bite-size pieces\n\n3 or 4 large tomatoes, canned or fresh, chopped into bite-size pieces (see box)\n\n3 cloves garlic, finely chopped\n\n4 scallions, finely chopped\n\nzest of 1 lemon\n\n\u00bc cup fresh basil (optional)\n\nsalt and pepper, to taste\n\ntopping\n\n\u00bd cup (1 stick) unsalted butter\n\n1\u00bd cups all-purpose or whole-wheat flour\n\n\u00bd cup cornmeal\n\n1 tablespoon baking powder\n\n\u00bd teaspoon salt\n\n1 teaspoon black pepper\n\n1 teaspoon smoked paprika\n\n\u00bd cup grated sharp Cheddar cheese, plus more for sprinkling\n\n1 cup milk\n\nsprinkling of fresh chopped herbs or scallions\n\n1 Put the butter for the topping in the freezer for 30 minutes. Preheat the oven to 425\u00b0F.\n\n2 Lightly oil an 8- by 10-inch baking dish (or any baking dish that will accommodate the vegetables) and pile in the zucchini, tomatoes, garlic, scallions, lemon zest, and basil, if using. Pour the olive oil over the top, scatter a generous amount of salt and pepper over everything, and mix it up with your hands. Bake the vegetables for 25 minutes while you prepare the biscuit topping.\n\n3 Combine the flour, cornmeal, baking powder, salt, pepper, paprika, and cheese in a bowl.\n\n4 Once the butter is frozen, use a box grater to flake it into the flour mixture. Gently massage the butter into the flour with your fingers until the mixture is crumbly but still clumpy. Add the milk and quickly bring the dough together. Don't knead it: Lumpiness is fine and results in a flaky topping. Put it in the fridge until the vegetables come out of the oven.\n\n5 Once the vegetable mixture has cooked for 25 minutes, remove it from the oven and quickly top it with small clumps of biscuit dough. The vegetables should still be visible in some areas.\n\n6 Bake until the vegetables are bubbly and the topping is lightly browned, 20 to 25 minutes. Top with some more Cheddar and some chopped herbs or scallions.\n\nIf you have very large and juicy tomatoes, you might want to remove some of the seeds and juice so that the final dish is less watery. If a little juice doesn't bother you, just leave them as is.\n\nBarley Risotto with Peas\n\n$2 \/ serving\n\n$6 total\n\nBarley is sadly underused\u2014other than in the odd soup, it rarely makes an appearance. What a shame! It has such a chewy, satisfying texture and nutty flavor. In this play on risotto, barley takes the place of Arborio rice for a dish that is less creamy, but more hearty and requires much less stirring. The pops of chewy barley and peas go well together, but as with regular risotto, you can use any vegetable\u2014just adjust the cooking time depending on what you choose. For a more traditional risotto flavor, substitute Romano cheese for the ricotta. serves 3\n\n1 cup pearl barley\n\n5 cups broth (see box)\n\n1 tablespoon butter\n\n1 onion, chopped\n\n3 cloves garlic, finely chopped\n\nzest and juice of 1 lemon\n\n2 cups frozen peas\n\n\u00bd cup Ricotta\n\nsalt and pepper, to taste\n\n2 slices bacon, chopped (optional\u2014see box)\n\nUse homemade broth if possible. I like vegetable, but chicken is fine. Beef or fish might be overwhelming, but could work if you use different vegetables. Even a bouillon cube dissolved in water is fine. Just don't buy boxed broth\u2014it's a total rip-off.\n\n1 Preheat the oven to 350\u00b0F.\n\n2 Pour the barley onto a rimmed baking sheet, spreading it into an even layer. Place it in the oven and bake until the barley is golden brown, 10 minutes. If you're short on time, skip this step, but you'll lose out on the toasty flavor.\n\n3 Place a small pot over low heat and add the broth. It just needs to heat up and stay warm.\n\n4 Melt the butter in a large, heavy-bottomed pan over medium heat. Add the onion and cook until it becomes translucent, about 3 minutes. Add the garlic and cook for another minute.\n\n5 Add the toasted barley and lemon zest, then stir to coat with the butter and onion. Add a ladleful (about \u00bd cup) of broth and stir. Cook for about 30 minutes, stirring occasionally and adding another ladleful of broth whenever the barley looks like it needs it. Reduce the heat if you notice the broth disappearing quickly.\n\n6 At the 20-minute mark you should have used up about 4 cups of broth. Add the frozen peas and another \u00bd cup broth. Stir until the broth is absorbed. Try a piece of barley to see whether it is fully cooked. There should be no hard center and the barley should be soft, but still chewy and whole. If the barley is not quite cooked, add the remaining \u00bd cup broth, stir, and keep cooking. Once it's ready, add the lemon juice and ricotta and stir. Taste and add salt and pepper as needed. You should be generous with the pepper, but the salt will depend on the saltiness of your broth, so be mindful.\n\nIf you want to use bacon, add it in Step 4 instead of the butter. Let the bacon get crispy and release its fat. Continue with Step 4.\n\nVegetable Jambalaya\n\n$0.65 \/ serving\n\n$3.90 total\n\nI don't make jambalaya exactly the way they do down South, but this vegetable-heavy version is faster and just as good\u2014a great, throw-everything-in-the-pot kind of meal. It's spicy, savory, and deeply satisfying. The leftovers are great for burritos or warmed up with a fried egg on top. serves 6\n\n2 tablespoons vegetable oil or butter\n\n1 medium-size onion, chopped\n\n1 green bell pepper, stemmed, seeded, and chopped\n\n3 stalks of celery, chopped\n\n3 cloves garlic, finely chopped\n\n\u00bd small green chile, finely chopped\n\n2 large tomatoes, chopped\n\n2 bay leaves\n\n1 teaspoon paprika\n\n1 teaspoon garlic powder\n\n1 teaspoon cayenne pepper\n\n\u00bd teaspoon dried thyme\n\n\u00bd teaspoon dried oregano\n\n1 teaspoon Worcestershire sauce or soy sauce\n\n\u00be cup long-grain rice\n\n3 cups vegetable or chicken broth\n\nsalt and pepper, to taste\n\nadditions\n\nslices of fried sausage\n\nshrimp\n\nleftover meat, tofu, or beans\n\n1 Place a large, heavy-bottomed pot over medium-high heat and add the oil. After it gets hot, add the onion, bell pepper, and celery and cook for about 5 minutes, until they become translucent but not brown.\n\n2 Add the garlic, chile, tomatoes, bay leaves, paprika, garlic powder, cayenne, thyme, oregano, salt, pepper, and Worcestershire sauce. Let everything cook until some of the tomato juice releases, about 1 minute.\n\n3 Add the rice and slowly pour in the broth. Lower the heat to medium and let the dish cook until the rice absorbs all the liquid, 20 to 25 minutes. If you're using any of the additions, throw them in to cook with the rice after 15 minutes have passed.\n\n4 Taste and adjust the salt, pepper, and any other spices.\n\nSpicy, Crunchy, Creamy Polenta\n\n$1.75 \/ serving\n\n$3.50 total\n\nPolenta + vegetable + egg = satisfying and delicious. You can also add a can of corn to the polenta for bursts of flavor. Or add frozen peas, scallions, olives, or (my favorite) green chiles to do something a little different. Or skip the Romano cheese and add \u00bc cup of grated Cheddar for maximum cheese factor. serves 2\n\n\u00bd teaspoon salt, plus more to taste\n\n\u00bd cup polenta or cornmeal\n\n4 cups fresh spinach or 1 cup thawed frozen spinach\n\n3 cloves garlic\n\n1 anchovy (optional)\n\n1 tablespoon olive oil or butter\n\n\u00bd teaspoon chile flakes or 1 chopped fresh chile\n\n2 eggs\n\nsprinkling of freshly grated Romano or Parmesan cheese\n\npepper, to taste\n\n1 Add 2 cups of water to a medium-size pot and bring it to a boil over medium-high heat. Add the salt, then turn the heat down to low and slowly pour in the polenta, stirring briskly with a wooden spoon. Stirring while pouring is crucial to make creamy, lump-free polenta. Once it's smooth and thick, leave the spoon in the pot and place a lid on it, slightly askew, so that steam can escape.\n\n2 Let the polenta cook while you prepare the rest of the meal, checking in occasionally to give it a stir. The total cooking time should be 25 to 30 minutes, but if you're in a rush, you can eat it after 15.\n\n3 Meanwhile, roughly chop the spinach. Finely chop the garlic and anchovy, if using, and set aside.\n\n4 Place a pan over medium heat and add half the olive oil. Let the pan heat up until it sizzles when you flick it with water. Add the garlic, anchovy, and chile flakes. Let them cook until you can smell them, about 1 minute. Add the spinach and toss it around with tongs, or just swirl the pan to coat the spinach with the garlic mixture. Let everything cook until the spinach is wilted, 3 to 5 minutes. Turn off the heat and transfer the mixture to a bowl to wait for the polenta and eggs.\n\n5 When the polenta is about 2 minutes from being done, start the eggs. Carefully wipe off the pan you just used, and put it back over medium heat. Add the remaining olive oil. When the oil is hot, crack the eggs into the pan and cover with a lid to steam them, 1 to 2 minutes. You'll have sunny-side-up eggs with fully cooked whites.\n\n6 Scoop the polenta into two bowls. Add some Romano and lots of salt and pepper. Layer the spinach mixture over the polenta.\n\n7 Once the whites have cooked, remove the eggs from the pan with a spatula and lay them over the spinach. Top with another sprinkling of cheese and a little more salt and pepper.\n\nTofu Hot Pot\n\n$1.80 \/ serving\n\n$7.20 total\n\nI got really excited when my friend Iva asked for a recipe that featured the Chinese flavors she grew up with. After all, Chinese cooking depends on the same general principles as Good and Cheap: Build bright flavors from key ingredients, and use lots of veggies with just a little meat or fish. The ginger-garlic broth in this hot pot is spectacular! The effect of a small amount of toasted sesame oil is remarkable\u2014an investment, but a transformative flavor. Use whatever vegetables you have around, but mushrooms help create an earthy broth. serves 4\n\n1 tablespoon finely grated ginger (see box)\n\n4 cloves garlic, finely grated\n\n\u00bd pound mushrooms, chopped\n\n1 teaspoon chile paste\n\n2 tablespoons soy sauce\n\n2 teaspoons toasted sesame oil\n\n1 pound firm tofu\n\n4 medium-size carrots, chopped\n\n4 scallions, white and green parts, separated and chopped\n\n8 ounces dried spaghetti, soba, or any Asian noodles\n\nhandful of bean sprouts (optional)\n\nadditions\n\n1 pound chicken, pork, or beef instead of tofu\n\nhandful of peanuts, chopped\n\ncabbage\n\nchile peppers, finely chopped\n\nsprinkling of chopped fresh cilantro\n\nkimchi, for topping\n\n1 daikon radish, sliced, for topping\n\n1 Place the grated ginger and garlic in a pot over medium heat. A few seconds later, once you start to smell the garlic, pour in 8 cups of water. Bring to a boil, then decrease the heat to low. Add the mushrooms, chile paste, soy sauce, and toasted sesame oil. Place a lid on the pot and let simmer for 20 minutes.\n\n2 Cut the tofu into 4 slices, then cut each slice into 8 squares. (Or just chop it up however you like.)\n\n3 Add the tofu, carrots, and the white parts of the scallions to the broth. Cook until the carrots are tender, about 10 minutes more.\n\n4 Add the noodles and boil until they soften, usually just a few minutes, although it depends on the type of noodle. Check the package directions for specific cooking times.\n\n5 Taste the broth. If it isn't salty enough, splash in more soy sauce. Adjust the sesame oil and chile paste to your taste as well.\n\n6 Ladle the soup into bowls. For a little crunch, top with the bean sprouts, if using, and the green parts of the scallions.\n\nWhen you buy ginger, just store it in the freezer. It's much easier to grate when frozen! You don't need to peel it\u2014just grate and the skin will flake off. (It'll last for a couple of months.)\n\nIf you have leftovers, you'll find you like this soup even more the next day. Overnight, the flavors will infuse into the tofu, as well as combining with each other. You might want to store the noodles separately, though, because otherwise they'll get mushy.\n\nDeconstructed Cabbage Rolls\n\n$1.50 \/ serving\n\n$9.00 total\n\nCasseroles are a great way to stretch your cooking-without-a-recipe muscles. As one reader, Carolie, reminded me, they require little prep time, yield many meals, and the leftovers will keep nicely in the fridge or freezer. So here's my adaptation of one of Carolie's favorite casseroles, itself a play on cabbage rolls, a traditional Eastern European dish that is delicious but labor-intensive. This version is a good way to use up leftover rice or grains and lentils, including leftover Rainbow Rice. For the sausage, I use fresh chorizo because it's easy to find in my neighborhood and I love the spicy, smoky flavor, but you should use whatever you like. serves 6\n\n1 tablespoon butter\n\n1 fresh sausage, about 4 ounces\n\n1 onion, chopped\n\n4 cloves garlic, finely chopped\n\n1 small or \u00bd large cabbage, cored and chopped\n\nsalt and pepper, to taste\n\n3 cups cooked rice\n\n4 cups cooked lentils\n\n3\u00bd cups pureed canned tomatoes or Best Tomato Sauce\n\nadditions\n\nBreadcrumbs, for topping\n\nolives\n\npeas or corn\n\ncheese\n\nany spice combination\n\nvariations\n\nground beef, turkey, or pork instead of lentils and sausage\n\nSwiss chard or collards instead of cabbage\n\n1 Preheat the oven to 350\u00b0F. Lightly oil a large casserole dish.\n\n2 Melt the butter in a large pan over medium heat. Slide the casing off the sausage and crumble the raw meat into the pan. Saut\u00e9 the meat until it's no longer pink, about 5 minutes, then transfer to a large bowl.\n\n3 Add the onion and garlic to the pan with the sausage drippings and saut\u00e9. Once the onion turns translucent, about 3 minutes, add the cabbage and saut\u00e9 until it's tender enough to jab easily with a fork, 5 to 7 minutes. Season generously with salt and pepper.\n\n4 While the cabbage cooks, mix the rice and lentils with the sausage in the bowl. Add salt, pepper, and any other spices or additions you'd like. Make sure you taste the mixture as you season it. If both parts of the casserole are tasty, you'll end up with a delicious meal. If they aren't seasoned well, it'll be bland.\n\n5 Spread half of the lentil-rice-sausage mixture in an even layer in the casserole dish. Next, spread half of the cabbage mixture on top. Then, as evenly as possible, pour half of the pureed tomatoes over everything. Repeat the layers and sprinkle with salt and pepper. If adding breadcrumbs, sprinkle over the top.\n\n6 Bake until the casserole is hot and bubbly, about 30 minutes.\n\nmethod\n\nRoasted Vegetables\n\n**When the weather turns cool, I want to eat warm, flavorful food.** Roasting is easy, it warms up the kitchen, and it makes the house smell like the holidays. If you're uncertain how to prepare a new vegetable, you usually can't go wrong with roasting\u2014most things end up sweeter, with nice crunchy bits. If you roast a bunch of vegetables at the beginning of the week, you can eat them throughout the week in various ways: with eggs at breakfast, folded into an omelet, as a side dish, in a taco or sandwich, on toast, or with any grain.\n\nolive oil or butter\n\nsalt and pepper, to taste\n\nroots\n\npotatoes, sweet potatoes, beets, turnips, onions, parsnips, carrots, sunchokes, kohlrabi, fennel bulbs\n\nnon-roots\n\nbell peppers, winter squash, broccoli, Brussels sprouts, cauliflower, asparagus, eggplant\n\nadditions\n\nwhole, unpeeled garlic cloves\n\nlemon slices or lemon zest\n\nanything you would pair with roast chicken\n\ntough herbs like sage, oregano, thyme, or bay leaves\n\nany spice combination\n\nfavorite sauce, a soft cheese, or mayonnaise, for serving\n\n1 **Preheat the oven to 400\u00b0F.**\n\n2 **Clean your vegetables.** Generally, I prefer to leave the skin on. Skin tastes nice and gets crispy, there's a lot of nutrition in the skin, and peeling is fussy! Just be sure to wash the vegetables thoroughly.\n\n3 **Chop them up.** Many vegetables are nice roasted whole, like new potatoes or little sunchokes or turnips\u2014they will be crispy and salty on the outside and bursting with fluffy, starchy goodness inside. However, the general rule is that the smaller you chop things, the faster they cook, so try to keep everything about the same size.\n\n4 **Dump your vegetables into a roasting pan**. Drizzle everything with olive oil or melted butter\u2014about 2 tablespoons per standard-size roasting pan. Season generously with salt and pepper and add any other additions from the list. Use your hands to coat the vegetables thoroughly with the oil and spices.\n\n5 **Pop the pan in the oven.** Root vegetables generally need to bake for at least 1 hour or longer, but check on them after 30 minutes. Non-roots need only 25 to 30 minutes, so check on those earlier. Poke them with a knife to test their doneness. If the knife meets no resistance, they're finished; if not, let them cook longer (you don't want undercooked root vegetables in particular\u2014they're awful). Don't worry about leaving them in too long. Unlike vegetables overcooked through boiling or steaming, overcooked roasted vegetables may dry out a bit, but they still retain their shape and flavor.\n\n6 After you pull the vegetables out of the oven, **push them around with a spatula to free them from the pan.** Remove any garlic cloves and smash them into a fine paste (remove the skins at this point), then put the garlic back in the pan and mix together. Squeeze the juice out of any lemons and discard the woody bits of any cooked herbs.\n\n7 **Add a little more butter, a bit of a favorite sauce, or a little soft cheese or mayonnaise, and serve.**\n\nChana Masala\n\n$1.50 \/ serving\n\n$3 total\n\nThis Indian chickpea dish is a beloved staple in my home. If you don't have cooked chickpeas, you can use canned, but it will cost about $1 more. For a full meal, serve the chana masala over rice or with Roti. serves 2\n\n1 tablespoon ground coriander\n\n1 teaspoon ground turmeric\n\n\u00bc teaspoon cayenne powder\n\n\u00bd teaspoon garam masala\n\n1 teaspoon smoked paprika (optional)\n\n\u00bd teaspoon salt, plus more to taste\n\n\u00bd tablespoon ghee (see box) or \u00bd tablespoon butter plus a splash of olive oil\n\n1 teaspoon cumin seeds\n\n1 small onion, chopped\n\n3 cloves garlic, finely chopped\n\n1 teaspoon grated ginger\n\n\u00bd jalape\u00f1o pepper, finely chopped (remove seeds for less heat)\n\n1 cup pureed canned tomatoes (see box)\n\n2\u00bd cups cooked chickpeas, drained\n\ngarnishes\n\nchopped fresh cilantro\n\nRaita or yogurt\n\n1 Combine the coriander, turmeric, cayenne, garam masala, smoked paprika, if using, and salt in a small bowl.\n\n2 Melt the ghee in a small saucepan over medium-low heat. Once it begins to sizzle, add the cumin seeds and stir for about 5 seconds until you can smell them. Add the onion and saut\u00e9 until it's a little soft, 2 to 3 minutes. Add the garlic and cook for 2 minutes. Add the ginger and jalape\u00f1o and cook for 1 minute longer. Add the spice mixture you made in Step 1, then the pureed tomatoes. Mix, then put a lid on the pan and let everything cook down, 5 to 10 minutes.\n\n3 Once the tomato has reduced and the ghee starts to separate from the sauce, add the chickpeas and \u00bd cup of water. Stir, bring it to a boil, and then decrease the heat to a simmer. Cook for 10 minutes, and then squish a few chickpeas with the back of a spoon to thicken the sauce before serving. Garnish with cilantro and Raita.\n\nGHEE, a traditional staple in Indian cooking, is just butter with the milk solids removed. It can withstand higher temperatures than butter without burning. You can make it at home by melting butter and letting the white milk solids rise to the top. Scoop them out and discard them. The leftover golden liquid is ghee. Store it in the fridge.\n\nBlack-Eyed Peas and Collards\n\n$1.25 \/ serving\n\n$5 total\n\nThis is similar to the Southern classic Hoppin' John, but streamlined into a one-pot meal, where you cook the collards with the beans instead of on their own. If you have them, you can add more vegetables to the base along with the onion\u2014celery, carrot, bell pepper, and some canned tomatoes would all be great in this. If you want to skip the bacon, just add smoked paprika to replace the smoky flavor. This dish pairs well with rice, any other grain, or with some toast or flatbread. serves 4\n\n1 cup dried black-eyed peas\n\n1 tablespoon butter\n\n1 large onion, finely chopped\n\n3 cloves garlic, finely chopped\n\n3 strips bacon, cut into small pieces\n\n1 bay leaf\n\n1 large bunch collard greens\n\n1 teaspoon salt, plus more to taste\n\npepper, to taste\n\n1 Place the black-eyed peas and 4 cups of water in a bowl and soak overnight.\n\n2 Melt the butter in a large saucepan over medium heat. Add the onion, garlic, bacon, and bay leaf. Cover the pan with a lid and leave it for 2 minutes. Stir occasionally and cook until the onion is translucent and the bacon is starting to crisp, about 5 minutes.\n\n3 Drain the black-eyed peas and pour them into the saucepan. Cover them with water and turn the heat down to medium-low. Cook for 30 minutes to 2 hours. The cooking time will depend on how old the peas are, which is difficult to predict. The peas are done when you can easily squish them on the countertop with the back of a spoon. Check on them every half hour or so, and if water boils off, add more water (hot, preferably) to cover them.\n\n4 While the peas cook, line up several leaves of collard greens on a cutting board and slice the tough central stem away from the leaves. Discard the stems. Thoroughly wash the collards and chop them into bite-size pieces. Alternatively, use your hands to tear the collards into small pieces.\n\n5 Once the peas are cooked, add the collards to the pot and put the lid back on. Add the salt and some freshly ground pepper, then stir. Taste the liquid and peas and add more salt as needed. Cover the pan with a lid and leave until the collards are tender, 10 to 15 minutes. Remove from the heat and serve.\n\nVegetable Quiche, Hold the Crust\n\n$1.50 \/ serving\n\n$6 total\n\nAs much as I love this quiche hot, I like it even better cold out of the fridge the next day. It makes a great fast breakfast or lunch (paired with a side salad). The quiche in the picture uses broccoli, but you can make it with pretty much any kind of vegetable. Some of my favorites are roasted green chiles and Cheddar, winter squash with goat cheese, zucchini and tomato, or spinach and olive. Spreading out onions on the bottom of the quiche adds a crust-like layer and a bit of crunch. serves 4\n\n1 tablespoon butter\n\n1 large onion, sliced into half-moons\n\n1 teaspoon salt, plus more to taste\n\n\u00bd teaspoon pepper, plus more to taste\n\n3 to 4 cups chopped vegetables (see box)\n\n8 eggs\n\n1 cup milk\n\n1 cup grated Cheddar or other cheese\n\n1 Preheat the oven to 400\u00b0F.\n\n2 Melt the butter in a cast-iron or ovenproof skillet over medium heat. (If your skillet isn't ovenproof, transfer everything to a pie plate in Step 3 to bake it.) Add the onion slices and sprinkle a bit of salt and pepper over them. Cook the onions until they are golden brown and starting to caramelize, about 10 minutes.\n\n3 Remove the pan from the heat and spread the onions evenly across the bottom. Spread the vegetables evenly over the onions. The dish or pan should look fairly full.\n\n4 In a bowl, use a fork to beat the eggs lightly with the milk, cheese, 1 teaspoon of salt, and \u00bd teaspoon of pepper, just enough to break up the yolks and whites. This is a savory custard mixture. Pour the custard over the vegetables and onions and enjoy watching it fill in all the open spaces.\n\n5 Transfer the quiche to the oven and bake for 1 hour. Once the surface is lightly brown all the way across, it's fully cooked.\n\n6 Let the quiche cool for about 20 minutes, then slice into wedges.\n\nFor hardier vegetables like broccoli, cauliflower, or winter squash, I suggest steaming or cooking them before adding them to the quiche to ensure they'll be fully cooked. For tomatoes, zucchini, spinach, or any other quick-cooking vegetable, just use them fresh.\n\nBroccoli, Egg, and Cheddar Empanadas\n\n$0.60 \/ empanada\n\n$7.20 total\n\nEvery culture has some version of dough stuffed with tasty filling\u2014it's a perfect combination. So empanadas, dumplings, pierogi, and calzones in one book isn't that crazy\u2014right? My friend Barb felt the same way, so I created this recipe for her. The cornmeal isn't traditional in empanadas\u2014I just like the extra crunch it gives. You can substitute more flour for the cornmeal if you like. makes 12 empanadas\n\noil or butter, for the baking sheet\n\ndough\n\n\u00bc cup (\u00bd stick) butter\n\n2 cups all-purpose or whole-wheat flour, plus more for rolling the dough\n\n\u00bd cup cornmeal\n\n\u00bd teaspoon salt\n\n1 egg\n\n\u00bd cup cold water\n\nfilling\n\n4 cups chopped broccoli, florets and stems\n\n8 eggs\n\n2 cloves garlic, finely chopped\n\n\u00bd teaspoon chile flakes\n\nsalt and pepper, to taste\n\n1 cup grated sharp Cheddar cheese\n\nIf you own a pastry brush, an egg wash will make the empanadas shinier. Since it only affects appearance, this step is optional. To make the egg wash, beat an egg with a fork in a small bowl, then brush the tops of the empanadas.\n\n1 Preheat the oven to 400\u00b0F. Lightly oil or butter 2 baking sheets.\n\n2 Make the dough: Place the \u00bc cup butter in the freezer for 10 minutes. Meanwhile, mix the flour, cornmeal, and salt in a large bowl. Grate the cold butter directly into the flour mixture. Use clean hands to squish the butter gently into the flour until it looks like breadcrumbs.\n\n3 Make a crater in the flour mixture. Crack 1 egg into it and pour in the cold water. Mix with your hands until the dough comes together into a smooth ball. If you're using whole-wheat flour and the dough seems dry, add another tablespoon of water. Wrap the ball of dough in plastic or cover it with a moist towel.\n\n4 Make the filling: Place a large pan over medium heat and add the broccoli and 1 cup water. Cover it with a lid and cook until the water is gone and the broccoli is tender, 5 to 7 minutes.\n\n5 Meanwhile, crack the 8 eggs into a bowl. Add the garlic, chile flakes, salt, and pepper, and beat lightly to combine.\n\n6 Once the broccoli is tender, pour the eggs into the pan with the broccoli and stir until the eggs are just scrambled, about 2 minutes. Turn off the heat, add the cheese, and stir again.\n\n7 Dust a clean countertop lightly with flour. Unwrap the dough and divide it into 12 equal pieces. Roll each piece into a ball with your hands, then use a rolling pin to flatten each ball into a thin circle, a little bigger than a coaster. Place some filling\u2014about a heaping \u2153 cup\u2014on one side of the circle, then fold over the other side to form a half-moon. Pinch the edges together and place the empanada on the prepared baking sheet. Repeat!\n\n8 Bake until the empanadas turn golden brown, about 20 minutes. Cool for 10 minutes before serving.\n\nPotato and Kale Rolls with Raita\n\n$0.70 \/ roll\n\n$5.60 total\n\nThese are a great meal to make when you have leftover Roti and Raita. The possibilities for fillings are endless\u2014this quick version with potatoes and greens is tasty and satisfying. serves 4\n\n1 tablespoon ghee or butter (see box)\n\n1 teaspoon cumin seeds\n\n1 small onion, finely chopped\n\n3 cloves garlic, finely chopped\n\n1 tablespoon finely grated ginger\n\n1 teaspoon ground turmeric\n\n1 teaspoon ground coriander\n\n1 teaspoon cayenne pepper\n\n1 teaspoon salt, plus more to taste\n\n2 large or 4 medium-size potatoes, chopped\n\n1 bunch kale or spinach, stems removed, chopped\n\n8 Roti\n\nsprinkling of chopped fresh cilantro\n\nRaita\n\n1 Place a skillet over medium heat and add the ghee. Once it is hot, add the cumin seeds and let them sizzle for 5 seconds before adding the onion. Let the onion cook for 2 minutes, stirring occasionally.\n\n2 In a small bowl, combine the garlic, ginger, turmeric, coriander, cayenne pepper, salt, and 1 tablespoon water.\n\n3 Add the spices to the onion mixture and cook, stirring, for another 2 minutes. It will smell strongly aromatic. This step is important because the spices will toast and release their flavor.\n\n4 Next, add the potatoes and stir them with the onion and spice mixture to coat. Add about a cup of water and cover the pan with a lid. Let the potatoes cook until tender, about 10 minutes, stirring occasionally. Add more water as needed. The water helps everything cook evenly, but you want the final mixture to be dry potatoes with just enough moisture to keep them from sticking. When you add water, be sure to let it cook off.\n\n5 Test the potatoes with a fork: If you can easily pierce them, they're ready. Once they are, add the kale and stir until the kale is wilted, 1 to 2 minutes. Taste and add more salt if needed.\n\n6 To assemble the rolls, scoop \u215b of the mixture into the center of a roti, distributing it in an even line. Roll it up.\n\n7 Serve 2 roti per person with cilantro and a generous dollop of raita, either inside the wrap or on the side.\n\nideas\n\nStuff on Hot Dogs\n\n**Although you might eat hot dogs** with great delight on a sunny day, they don't feel so exciting when they're all you have in the fridge, or all a picky child will eat. But don't get stuck thinking hot dogs are boring. Whether pork, beef, kosher, or veggie, hot dogs are livelier with a generous vegetable topping. Here are some ideas to get you going.\n\n1 Quick Teriyaki Carrots\n\n$0.75 total\n\nThis quick teriyaki sauce is great on all kinds of vegetables. Try it with carrots and then experiment from there.\n\n2 tablespoons soy sauce\n\n1 teaspoon brown sugar\n\n1 clove garlic, grated\n\n2 to 3 carrots, grated\n\n1 Add the soy sauce, brown sugar, and garlic to a hot pan over medium heat. Let it sizzle.\n\n2 Once the sugar is dissolved, toss the carrots in and cook until they absorb the sauce, about 2 minutes.\n\n2 Salt and Vinegar Cucumbers with Dill\n\n$1.20 total\n\nThese tangy cucumbers are like a quick form of pickles. Add a tablespoon of dill or mustard seeds for a more pickley flavor. Store leftovers in a sealed container in the fridge and they'll keep about a week.\n\n1 field cucumber\n\n2 tablespoons vinegar\n\n1 teaspoon salt\n\n1 Thinly slice the cucumber.\n\n2 Scoop the cucumber into a bowl with the vinegar and salt and toss. Marinate for 20 minutes.\n\n3 Mexican Street Corn\n\nSimply cut the corn off the cob and mix the mayo, chili powder, cheese, and lime into the kernels.\n\n4 Wilted Cabbage Salad\n\nChop the cabbage finely so that it can be distributed evenly over your hot dog.\n\n5 Salsa\n\nTry to drain a little bit of the juice so it doesn't make your bun too soggy. Crumble tortilla chips on top for some crunch.\n\n6 Sweet or Savory Pineapple Salad\n\nThis is a classic combo, especially with pork! Chop the pineapple finely so it won't fall off.\n\nPotato Leek Pizzas\n\n$2.25 \/ pizza\n\n$9 total\n\nObviously you should just make all kinds of pizza. Seriously, do it. Make it a Thursday-night tradition and an excuse to use up leftovers. This pizza is a fun variation that confounds expectations\u2014proof that, indeed, anything is good on pizza! makes 4 personal pizzas\n\n2 tablespoons olive oil\n\n1 large russet potato or 3 small potatoes, sliced into thin circles\n\nsalt and pepper, to taste\n\n3 leeks, trimmed, washed, and sliced into circles\n\nall-purpose flour, for shaping the dough\n\n1 recipe Pizza Dough\n\n1 pound fresh mozzarella cheese, shredded\n\n1 Preheat the oven to 500\u00b0F.\n\n2 Place a large pan over medium heat and add 1 tablespoon of the olive oil. Once the oil is hot, add as many potato slices as will fit in the pan, spacing them out to make sure each slice is touching the bottom. (If you slice them thinly enough, they'll turn out almost like little chips.)\n\n3 Let the potatoes cook until they start to crinkle around the edges and turn brown, about 2 minutes. Flip them over and brown the other side, another minute or so, then move them to a bowl. Continue in batches, as needed. Sprinkle with salt and pepper, then (after they cool down!) toss with your hands to make sure they're evenly coated.\n\n4 Heat the remaining tablespoon of oil in the same pan, then throw in the leeks, stirring occasionally until they're soft, about 5 minutes. Toss them in the bowl with the potato slices, add a bit more salt and pepper, and stir.\n\n5 Sprinkle flour on a clean countertop. Divide the pizza dough into 4 equal pieces and place one piece on the countertop. Using your hands or a rolling pin, stretch the dough into crust. I like to make mine really thin and big, but it's up to you how thick to make it.\n\n6 Once the crust is the desired shape and thickness, dust the back of a baking sheet with flour to keep the crust from sticking, then place the crust on the sheet.\n\n7 Layer a quarter of the potato and leek mixture on top of the crust and sprinkle with a quarter of the shredded mozzarella. Bake for 5 to 8 minutes. If it's your first time, simply keep an eye on the oven to see when the pizza's done. The crust should be light brown and the cheese melted. Repeat the process until you've baked all your pizzas. If your oven is big enough, you can, of course, do more than one pizza at a time.\n\nBroccoli Rabe and Mozzarella Calzones\n\n$1.50 \/ calzone\n\n$6 total\n\nCalzones are pizza in a slightly different form\u2014a form that lets you stuff in more filling without weighing down the crust. Broccoli rabe is great, but you can use any bitter green, or even broccoli or cauliflower. You'll love these crusty pockets full of oozy goodness! makes 4 calzones\n\nall-purpose flour or cornmeal, for shaping the dough\n\n1 tablespoon olive oil\n\n1 large bunch broccoli rabe, chopped\n\n4 cloves garlic, finely chopped\n\n1 teaspoon chile flakes\n\n2 anchovy fillets, finely chopped (optional)\n\nsalt and pepper, to taste\n\n1 recipe Pizza Dough\n\n2 cups grated mozzarella cheese\n\nIf you have Italian sausage in your fridge, crumble it into the pan with the broccoli rabe in Step 2. Sausage and broccoli rabe are a classic combination.\n\n1 Preheat the oven to 500\u00b0F (or as hot as your oven gets). Sprinkle a small amount of flour over a baking sheet and set it aside.\n\n2 Place a large pan over medium heat and add the olive oil. Once the oil is hot, add the tough stem ends of the broccoli rabe and cook for 2 minutes. Next, add the rest of the broccoli rabe, including the leafy parts, along with the garlic, chile flakes, and the anchovies, if using. Cook, stirring occasionally, until the stems are tender, about 5 minutes. Add salt and pepper, and set the filling aside.\n\n3 Sprinkle flour on a clean countertop. Divide the pizza dough into 4 equal pieces and place one piece on the countertop. Using your hands or a rolling pin, roll out the dough as you would for pizza, until it is quite thin.\n\n4 Pile a quarter of the broccoli rabe mixture and \u00bd cup of the mozzarella onto one side of the circle, leaving a lip around the edge.\n\n5 Gather up the half of the dough that isn't weighed down with filling and fold it over to create a half-moon shape. Pinch the edges of the dough together. Place the calzone carefully on the prepared baking sheet and repeat Steps 3 to 5 until you have 4 calzones.\n\n6 Bake until the calzones are golden brown on the outside, 6 to 8 minutes. Be careful when you bite into them\u2014they'll be hot!\n\nHalf-Veggie Burgers\n\n$0.90 \/ serving\n\n$7.20 total\n\nWhen a reader named Quinn suggested a recipe that used both lentils and meat, I started thinking about how veggie burgers and beef burgers each have their own strengths. Why not combine the two ideas to create a burger with meaty flavor but the lean protein and low cost of lentils? And so I offer you the half-veggie burger. May it rest a little lighter in your belly.\n\nYou can use almost any vegetable to make these burger patties, except for lettuce and other greens, or super-watery vegetables like tomato or cucumber. Make sure the vegetables are either small to begin with (like corn or peas) or finely chopped so that they cook evenly. I went for a bell pepper this time. Vegetables like potato, squash, or eggplant, which are inedible raw, should be fully cooked before you add them to the patty. serves 8\n\n3 cups cooked lentils or beans\n\n1 cup finely chopped bell pepper or other vegetable\n\n1 pound ground beef or other ground meat\n\n1 egg (optional)\n\nsalt and pepper, to taste\n\n8 buns\n\n1 Roughly mash the lentils with the back of a large spoon.\n\n2 Mix the lentils, bell pepper, and ground beef with your hands in a large bowl. If you're grilling, add an egg to keep the patties from crumbling. Season with salt and pepper and form into 8 patties.\n\n3 Place a large skillet over medium-high heat (or fire up the grill, if you have one), and add the patties. Sear them until they're dark brown on one side, about 5 minutes, then flip 'em and do the same on the other side. If you want cheeseburgers, lay cheese on the patties after flipping them once.\n\n4 Serve on toasted buns with your favorite condiments and fresh vegetables. Burgers are a great place to be adventurous!\n\nIf you won't eat all of the burgers at once, wrap any leftover raw patties in plastic. They will keep in the refrigerator for a few days or in the freezer for up to 2 weeks.\nBig Batch\n\nBest Tomato Sauce\n\n$1 \/ cup\n\n$7 total\n\nThere are many ways to make tomato sauce. I don't find that the more complex recipes taste any better\u2014this one is boldly tomatoey and works on just about anything. It also takes 5 minutes to make. Can't beat that. MAKES 7 CUPS\n\n2 tablespoons olive oil\n\n6 cloves garlic, finely chopped\n\n1 teaspoon chile flakes\n\n2 cans (28 ounces each) tomatoes, crushed or diced\n\nzest of 1 lemon (optional)\n\nsalt and pepper, to taste\n\n1 Add the olive oil to a saucepan over medium heat.\n\n2 Saut\u00e9 the garlic until it smells great and becomes translucent, 1 minute. Add the chile flakes and cook for 30 seconds.\n\n3 Add the cans of tomatoes, mix, and cook until warmed through.\n\n4 Add the lemon zest, if using, then salt and pepper to taste. Because canned tomatoes are often already salted, you may not need to add any.\n\nIf you want a thicker sauce that will stick to pasta better, cook it a little longer to evaporate more of the liquid, 10 to 20 minutes. Use immediately, keep in a jar in the fridge for up to a week, or portion it out into containers and store in the freezer for a month.\n\nChorizo and White Bean Rag\u00f9\n\n$1.25 \/ cup\n\n$7.50 total\n\nAfter my friend Chris told me he loves a good rag\u00f9, I developed a version that is as hearty as a meaty tomato sauce without the cost and heaviness of a traditional rag\u00f9. A batch of this is probably enough for eight people, served with grated Romano or Parmesan cheese over pasta, polenta, or grits. MAKES 6 CUPS\n\n2 tablespoons butter or vegetable oil\n\n2 onions, chopped\n\n6 cloves garlic, finely chopped\n\n2 tablespoons finely chopped jalape\u00f1o (optional)\n\n1 pound fresh chorizo, casing removed (or other fresh sausage)\n\n3 cups canned or fresh tomatoes, pureed\n\n3 cups cooked cannellini, navy, or butter beans\n\nsalt and pepper, to taste\n\n1 Melt the butter in a pan over medium heat, swirling it to coat the pan. Add the chopped onions and cook until they turn translucent, 3 to 4 minutes.\n\n2 Toss in the garlic, jalape\u00f1o (if using), and fresh chorizo, then saut\u00e9 for about a minute. Add the tomatoes and beans, then simmer until the sauce is thick and the sausage is cooked, about 5 minutes. Taste and add salt and pepper as needed.\n\nFreeze this sauce if you don't intend to eat it within a few days. It won't keep very long in the fridge.\n\nDark and Spicy Chili\n\n$1.65 \/ serving\n\n$19.80 total\n\nChili is such a crowd-pleaser. If you don't already have your own award-winning recipe, give this one a try. To stretch this recipe even further, serve it with rice. SERVES 12\n\n2 tablespoons ground cumin\n\n2 tablespoons dried oregano\n\n2 tablespoons ground coriander\n\n1 tablespoon ground cinnamon\n\n2 tablespoons cocoa powder\n\n1 to 4 canned chipotle chiles in adobo, finely chopped\n\n1 pound ground beef or turkey\n\n1 pound Mexican chorizo (fresh, casings removed), or 1 pound more ground beef or turkey\n\n2 medium-size onions, chopped\n\n6 cloves garlic, finely chopped\n\n2 bell peppers, stemmed, seeded, and chopped\n\n2 carrots, chopped\n\n6 cups cooked black beans\n\n2 cans (28 ounces each) diced or crushed tomatoes\n\n1 tablespoon salt, plus more to taste\n\nto serve\n\nchopped scallions\n\nchopped fresh cilantro\n\nsour cream\n\ncheddar cheese\n\n1 In a small bowl, combine the cumin, oregano, coriander, cinnamon, cocoa powder, and chiles, and stir well. If it's your first time making this chili, use one or two chipotles. You can always add more, but you can't take away!\n\n2 Place a large pot over medium heat and add the ground beef and chorizo. Cook the meat, breaking it up and stirring, until it is no longer pink. The fat from the meat should be enough to keep everything from sticking to the pot.\n\n3 Add the spice mixture to the meat and stir until you can smell the spices, about 20 seconds. Add the onions, garlic, peppers, and carrots, and stir. Place a lid on the pot and cook, stirring occasionally, until the onion is translucent, about 10 minutes.\n\n4 Add the black beans, tomatoes, and 4 cups of water, then stir. Bring the mixture to a boil, then turn the heat to low and simmer for 1\u00bd hours with the lid askew to allow steam to escape. Taste and season with salt and pepper.\n\n5 Serve in big bowls topped with scallions and cilantro, if using, or freeze in small portions for later.\n\nTo make vegetarian chili, use a bit of oil to brown the onions and substitute more beans and veggies for the meat. Most of the flavor comes from the spices and vegetables, so you won't miss much.\n\nYou can use \u00bd cup chili powder instead of the cumin, oregano, coriander, and chipotles in the spice mixture\u2014just add the cinnamon and cocoa for a hint of sweetness.\n\nTo make in a slow cooker, follow the instructions up to Step 3, then transfer the mixture to the slow cooker, and add the rest of the ingredients. Cook on low for 8 to 10 hours.\n\nSpicy Pulled Pork\n\n$1.40 \/ serving\n\n$14 total\n\nPulled pork is incredibly flavorful, rich, spicy, and remarkably versatile. Although it seems expensive, it's quite a bargain when you look at the price per serving. As with many special meals, this one takes quite a long time to prepare. Most of the time, however, is just spent waiting for it to cook \"low and slow.\"\n\nMy favorite way to serve pulled pork is over squishy hamburger buns or in tacos with crunchy vegetables. Pulled pork sandwiches are great with cabbage slaw, so try using the Wilted Cabbage Salad. Serve with a simple green salad, corn on the cob, steamed green beans, or any other summery vegetables. SERVES 10\n\n\u2153 cup brown sugar\n\n2 tablespoons ground coffee\n\n2 tablespoons kosher salt\n\n4 teaspoons smoked paprika\n\n3 teaspoons sweet paprika\n\n2 teaspoons ground cumin\n\n1 teaspoon ground coriander\n\n1 teaspoon ground cloves\n\n1 teaspoon garlic powder\n\n1 teaspoon black pepper\n\n1 pork shoulder (about 5 pounds)\n\n1 Make a dry rub by mixing all of the ingredients except the pork in a small bowl.\n\n2 Apply the rub liberally to the pork shoulder, pressing it gently into the meat until you've covered every side. Set any leftover rub aside for later.\n\n3 Place the pork shoulder in a large pot with a tight-fitting lid. Leave it in the fridge for 2 hours or overnight to let the flavors seep in.\n\n4 Preheat the oven to 200\u00b0F.\n\n5 Pour enough water into the pot to cover the bottom. Put the lid on and place the pot in the oven for 10 to 12 hours. The rule of thumb is 1\u00bd to 2 hours per pound of pork, but I find it usually takes a little longer than that. You want the internal temperature to reach 200\u00b0F.\n\nIf you don't have a meat thermometer, figuring out the internal temperature is trickier, but you can test it by feel. Poke the meat with a finger: When it's so soft that it falls apart on its own, take it out of the oven. It's hard to overcook it at such a low temperature, so don't be too concerned about that.\n\n6 Remove the meat from the juices and gently tear the pork apart with two forks or your hands. Discard any larger bits of fat. If any section is hard to tear apart, the meat hasn't cooked enough. If you have the time to spare, put it back in the oven for another couple of hours.\n\n7 Once you've pulled all of the pork, mix in any remaining rub and transfer it to a casserole dish or a large plate. If you aren't eating the meat right away, stash it in the fridge, covered, for 3 to 4 days.\n\nto make a sauce from the pot full of drippings, bring them to a gentle boil in a pot over medium-high heat and let the juices thicken for 20 to 30 minutes. The fat will rise to the top: It's the clear, thick layer, not the thin, red liquid below. Skim off as much of the fat as possible. Mix a few spoonfuls of the pan drippings with the pork before serving.\n\nideas\n\nHummus\n\n$0.40 \/ serving\n\n$1.60 total\n\n**Hummus is my go-to snack when hunger hits midafternoon.** I often double or triple this recipe and freeze it. A food processor makes things easier, but without one all is not lost. After all, hummus was invented long before food processors.\n\nIf you have dried chickpeas, cook them using the instructions. Canned chickpeas will increase the cost of the recipes. (I also think the flavor of dried beans is better.)\n\nUnless you are extremely thorough and patient with your handmade hummus, you will have a chunkier texture than the super-smooth and creamy store-bought hummus. It will still taste great, though!\n\nBasic Hummus\n\nserves 4\n\n2 cups cooked chickpeas (see box)\n\n1 tablespoon tahini\n\n1 tablespoon lemon juice\n\n1 clove garlic, finely chopped\n\n1 tablespoon olive oil, plus more for serving\n\nsalt and pepper, to taste\n\n1 _If you are making the hummus by hand_ , warm the chickpeas for about 30 seconds in the microwave.\n\n2 Mash the tahini, lemon juice, garlic, olive oil, salt, and pepper in a bowl.\n\n3 Slowly add \u00bc cup water a bit at a time, mashing as you add it, until the mixture is smooth, creamy, and light, about 5 minutes. Taste it and adjust the seasoning. Add more oil and tahini if you want it richer.\n\n1 _If you are using a food processor_ , add all the ingredients to the processor along with \u215b cup water.\n\n2 Once the mixture is a smooth paste, taste it to check the consistency. For a smoother and lighter hummus, add a bit more water. Taste it and adjust the seasoning. Add more oil and tahini if you want it richer.\n\n1 Roasted Garlic\n\n$0.50 serving \/ $2 total\n\n1 Preheat the oven to 350\u00b0F.\n\n2 Slice the top off of a head of garlic so that you can see the cloves (leave the peel). Drizzle the cloves with olive oil. Wrap in tinfoil and place on a tray in the oven. Roast for 1 hour.\n\n3 Remove the garlic from the oven. When it is cool enough to touch, gently squeeze all but 3 of the garlic cloves into the hummus and mix according to the basic recipe.\n\n4 Finely chop the leftover garlic cloves and sprinkle on top of the hummus with extra olive oil.\n\n2 Lemon\n\n$0.45 serving \/ $1.80 total\n\nReplace the \u00bc cup water with 3 tablespoons lemon juice and 1 tablespoon water. Add the zest of 1 lemon.\n\n3 Chipotle\n\n$0.45 serving \/ $1.80 total\n\nAdd 1 or 2 chipotles in adobo to the hummus. If you're not using a food processor, make sure the chipotle is finely chopped. Mix a bit of adobo sauce with olive oil and drizzle over the top.\n\nideas\n\nDeviled Eggs\n\n$0.15 \/ half egg\n\n$3.60 total\n\n**Deviled eggs are my favorite party food and the perfect recipe to dedicate to my friend Camilla.** At parties, I often eat too much random junk food and end up feeling gross. These eggs are a great antidote: festive and delicious without the empty calories. Although they're a little fussy, they aren't actually difficult to make. Here's the formula: Make a basic deviled egg and add one of the 8 flavors (or create your own!).\n\n12 eggs\n\nsalt and pepper, to taste\n\n2 scallions, finely chopped (optional)\n\ndash of paprika (optional)\n\nHard-boiled eggs are easier to peel if the eggs you boil aren't quite fresh, so try making these with eggs that have been sitting in the fridge a week or two.\n\nBasic Deviled Eggs\n\nMakes 24 half eggs\n\n1 Place a layer of eggs at the bottom of a pot that is large enough to fit them all with a bit of wiggle room. If you can't fit all your eggs, don't stack them\u2014they might crack. Hard-boil them in batches instead.\n\n2 Cover the eggs with cold water. Place the pot over medium heat and bring to a boil. As soon as the water is boiling, turn off the heat and cover the pot with a tight-fitting lid. Set a timer for 10 minutes.\n\n3 When the timer goes off, carefully pour out the hot water and cover the eggs with very cold water. The cold water stops the cooking process so that you don't end up with that slightly icky blue-green skin around your yolk.\n\n4 Peel the eggs. Everyone has his or her own technique, but I like to gently roll each egg across the counter to crack the shell. Roll the egg around until it looks like a cracked desert landscape, then peel it starting from the bottom (where the air pocket is). Once peeled, rinse the egg and set it aside. Repeat until you have peeled all the eggs.\n\n5 Slice each egg in half lengthwise. Pop the yolks out and put them in a medium-size bowl. Don't worry if you leave a little yolk behind. Set the whites aside on a plate.\n\n6 Sprinkle the yolks with salt and pepper, then add the other ingredients of your choice to the bowl. Mash with a fork until you have a relatively smooth paste.\n\n7 Spoon the yolk mixture back into each egg. Pile the filling high! Alternatively, scoop the filling into a plastic sandwich bag. Cut off the corner of the sandwich bag and squeeze the yolk mixture into the whites.\n\n8 Sprinkle with the scallions and some paprika for color, if you have it.\n\n1 Classic\n\n2 tablespoons mustard\n\n2 tablespoons mayonnaise\n\n2 tablespoons water, pickle brine, or lemon juice\n\n2 Chile and Lime\n\n2 tablespoons mayonnaise\n\n2 tablespoons lime juice\n\n1 jalape\u00f1o pepper, finely chopped (remove seeds for less heat)\n\n3 Ramen-Inspired\n\n2 tablespoons mayonnaise\n\n2 tablespoons soy sauce\n\n1 tablespoon rice vinegar\n\nchili sauce, to taste\n\n4 Curried\n\n2 tablespoons mayonnaise\n\n2 tablespoons water\n\n4 teaspoons curry powder, or 1 teaspoon each of turmeric, cayenne, coriander, and cumin\n\n5 Tomato\n\n2 tablespoons mayonnaise\n\n\u00bc cup finely chopped fresh or canned tomatoes or Best Tomato Sauce\n\n6 Chile and Cheese\n\n2 tablespoons mayonnaise\n\n2 tablespoons chopped green chiles\n\n2 tablespoons grated cheese\n\n7 Chorizo\n\n2 tablespoons mayonnaise\n\n2 tablespoons cooked, minced fresh chorizo\n\n1 teaspoon paprika\n\n8 Feta and Dill\n\n2 tablespoons mayonnaise\n\n2 tablespoons crumbled feta\n\n1 tablespoon chopped dill\n\nPierogi\n\n$0.20 \/ pierogi\n\n$14 total\n\nThis recipe is huge and will feed you for days. It takes time and effort, but the results are worth it. The best approach is to invite a couple of friends over for a pierogi-making party. Everyone takes home a bag or two for the freezer, and it's a great time! For the filling, you should play around with some of your favorite things\u2014there aren't many flavors that don't work in potatoes. I usually use several additions and a strong, aged cheddar. MAKES 60 TO 72 PIEROGI\n\ndough\n\n4\u00bd cups all-purpose flour, plus more for shaping the dough\n\n2 teaspoons salt\n\n2 cups yogurt or sour cream\n\n2 eggs\n\nfilling\n\n5 russet potatoes, roughly cubed\n\nsalt and pepper, to taste\n\n1\u00bd cups shredded sharp Cheddar cheese\n\nadditions\n\n2 scallions, chopped\n\n4 cloves roasted garlic\n\n2 tablespoons Dijon mustard\n\n1 teaspoon cayenne pepper\n\n1 teaspoon paprika\n\nto serve\n\n1 tablespoon butter\n\nscallions, chopped\n\nsour cream\n\n1 In a large bowl, mix the flour and salt. Pour in the yogurt, eggs, and 1 tablespoon of water. Mix everything slowly and carefully with clean hands until it comes together into a smooth dough. The dough will be quite sticky. Cover it with a towel or plastic wrap while you make the filling.\n\n2 Put the chopped potatoes in a pot. Cover with water, then add a bit of salt. Cover with a lid and bring to a boil over medium-high heat. Remove the lid and cook the potatoes until tender, about 20 minutes. Test them with a fork. If it goes through easily, they're done.\n\n3 Drain the potatoes and add the shredded cheese, salt, pepper, and any additions you might enjoy.\n\n4 Mash the potatoes with an electric mixer or two forks. Once the filling is ready, gather some friends\u00ad\u00ad\u2014shaping takes time.\n\n5 Flour your countertop liberally. Split the dough in half. Keep one half covered, but place the other half on the floured surface. Use a rolling pin to flatten the dough about \u215b-inch thick. Punch out as many 3-inch dough circles as possible, using a round cookie cutter or a small glass. Squish the scraps into the remaining covered half of the dough.\n\n6 Place about a tablespoon of filling in the center of one circle of dough. Fold the dough over the filling and press the edges to create a dumpling. The stickiness should ensure a tight seal. Lay the pierogi on a floured surface and use a fork to squish the edges together. Keep going until you run out of circles, then repeat with the remaining dough and filling.\n\n7 Once you've formed all your pierogi, bring a pot of water to a boil over high heat. Add 12 pierogi to the water and let them cook until they rise to the surface, about 1 minute. Pull out the boiled pierogi with a spoon, bring the water back to a boil, and repeat with batches of the remaining pierogi.\n\n8 If you're planning to freeze some of the pierogi, let them cool down and then put them in freezer bags with the air squeezed out. I usually do 12 to a bag, but you can portion them out in whatever way suits you. They will keep for at least 6 months.\n\n9 You can eat the pierogi just boiled, but I prefer them fried afterward. Melt the butter in a pan over medium heat, then fry up as many pierogi as you want. (Six per person is plenty.) Flip them every few minutes until they're browned on all sides. Serve with scallions and a dollop of sour cream.\n\nDumplings\n\n$0.12 \/ dumpling\n\n$7.20 total\n\nMy friend Raffaella comes from a huge family and fondly recalls making dumplings with her sisters growing up. (Her brothers only helped eat them.) Dumplings are a great way to use up veggies that aren't fresh anymore. Minced and stuffed inside a dumpling, they come back to life! MAKES 60 DUMPLINGS\n\ndough\n\n4 cups all-purpose flour, plus more for shaping the dough\n\n1 teaspoon salt\n\n2 eggs\n\nveggie filling\n\n3 cups finely chopped broccoli\n\n2 cups grated carrot\n\n8 ounces firm tofu, crumbled\n\n2 tablespoons soy sauce\n\n1 teaspoon toasted sesame oil\n\n2 scallions, chopped\n\n2 eggs\n\npork filling\n\n1 pound ground pork or sausage, cooked or raw\n\n3 cups finely chopped collards, chard, spinach, or scallions\n\n2 tablespoons soy sauce\n\n1 teaspoon toasted sesame oil\n\n2 scallions, chopped\n\n2 eggs\n\n1 Mix the flour and salt in a large bowl. Make a crater in the middle, crack in the eggs, and add 1 cup of water. Use one hand like a shovel to mix the dough into a shaggy mass. If it seems too dry, add a few drops of water. Knead the dough for a minute, then cover it with plastic wrap or a damp towel and let it rest for 30 minutes to 2 hours.\n\n2 In a large bowl, mix your desired filling ingredients.\n\n3 Once the dough has rested, split it into 4 chunks. Dust your countertop with flour, then roll the first piece of dough into a log. Cover the other pieces so they don't dry out.\n\n4 Cut the log into 15 equal slices, then form one of the slices into a flat disk with your hands. With a rolling pin, flatten the disk into an almost paper-thin circle about the size of a drink coaster.\n\n5 Place a heaping tablespoon of filling in the center of the dough. Lift all the edges to meet in the middle, then pinch it closed like a little parcel. If the dough won't stick to itself, wet your fingertips and dab the edges.\n\n6 Repeat until you run out of either filling or dough. Ask for help from family or friends\u2014one person can roll while the others fill and cook.\n\n7 Now, a tough decision: Steam, fry, or boil?\n\n_To steam them_ , spread a small amount of oil in a large pan. Fill the pan with dumplings\u2014as many as you can fit without them sticking to each other. Turn the heat to medium and let them sizzle for a minute. Once the dough has absorbed most of the oil, add about \u00bd cup of water, then quickly cover with a lid. The water will splatter and sizzle. Leave the lid on for about a minute to steam the dumplings, then turn the heat to low and remove the lid. Cook until the water evaporates, then turn off the heat. Your dumplings should be steamed on top with crispy, brown bottoms.\n\n_To pan-fry them_ , follow the technique above, but use more oil. Skip the water and the lid entirely. Just keep frying! Once the dumplings are golden on one side, flip them to fry on the other side. This method is awkward with parcel-style dumplings but works well for other shapes (such as a pierogi shape), so plan accordingly.\n\n_To boil them,_ drop the dumplings into a pot of boiling water. When they rise to the top, they're ready to eat, usually in 1 or 2 minutes.\nPantry\n\nPeanut Sauce\n\n$3 \/ cup\n\n$3 total\n\nThis is my go-to dipping sauce for everything from veggies (especially the cornmeal-crusted) to flatbread, to shrimp. But it's versatile enough to make a great marinade or coating for chicken or fish. My fridge is rarely without it. Try it, you'll see. makes 1 cup\n\n1 jalape\u00f1o pepper or other chile (remove seeds for less heat), or 2 tablespoons chile paste\n\n3 cloves garlic\n\n1 shallot or small onion\n\n1 teaspoon vegetable oil\n\n\u00bd to 1 cup coconut milk\n\n\u00bd cup sugar-free peanut butter\n\n1 tablespoon soy sauce\n\nadditions\n\n1 teaspoon ground turmeric\n\n1 tablespoon brown sugar\n\n\u00bd teaspoon sesame oil\n\n1 Finely chop the jalape\u00f1o, garlic, and shallot, or use a food processor to make them into a paste. (If you're using chile paste instead of a fresh pepper, add it in Step 2.)\n\n2 Add the oil to a saucepan over medium heat. Once it's warm, saut\u00e9 the pepper and garlic until fragrant, about 2 to 3 minutes. Add the \u00bd cup of coconut milk, turmeric, and chile paste, if using.\n\n3 Let everything come to a boil, then turn the heat down to low. Stir in the peanut butter, soy sauce, and brown sugar and sesame oil, if using. If the sauce is too thick, add more coconut milk to thin it out. Once the mixture is well combined, taste it and add whatever you think it needs, concentrating on the salt and spices in particular.\n\nSalsa\n\n$0.75 \/ cup\n\n$2.25 total\n\nSummertime salsas combine a load of fresh tomatoes with smaller amounts of choice vegetables and fruit. Apart from its usual use on tortilla chips and tacos, this salsa is a wonderful topping for fish or chicken, as a sauce for cold noodles, or as a finishing touch on a savory breakfast. If you aren't a fan of cilantro, substitute another herb; mint, savory, or lemon balm work well. makes 3 cups\n\n\u00bd medium onion, finely chopped\n\n2 cups chopped tomatoes\n\n1 jalape\u00f1o pepper, finely chopped (remove seeds for less heat)\n\njuice of 1 lime\n\n\u00bc cup finely chopped fresh cilantro\n\nsalt and pepper, to taste\n\nadditions\n\nchopped mango, peach, plum, or pineapple\n\nbeans\n\ncorn\n\nfinely chopped garlic\n\nchipotle chiles in adobo instead of the jalape\u00f1o\n\n1 If you like raw onion, skip ahead to Step 2. Otherwise, take the edge off by saut\u00e9ing the onion with a bit of water in a pan over medium heat. The onion is ready once the water has boiled off.\n\n2 Mix the onion, tomato, pepper, lime juice, cilantro, salt, and pepper in a bowl. Be sure to add enough salt and pepper!\n\n3 Taste the salsa. You're looking for a balance of spicy from the jalape\u00f1o pepper, sweet from the tomatoes, and bright and fresh from the herbs and lime juice. If something's out of balance, add more of the appropriate ingredient to bring it back into balance.\n\n4 Store in an airtight container in the fridge. Fresh salsa won't last as long as store-bought salsa because it doesn't have any preservatives, but it's so tasty that I'm sure you'll finish it fast!\n\nFor a winter version, use canned tomatoes and heat everything but the lime juice on the stovetop for 5 minutes to marry the flavors. Finish with the lime juice and store.\n\nTzatziki\n\n$1.75 \/ cup\n\n$3.50 total\n\nThis classic yogurt sauce uses cucumbers in a way you might not have thought of before. Straining the yogurt and cucumber intensifies the flavor but, if you're in a hurry, skip the straining steps and just mix the ingredients together. makes 2 cups\n\n1 large cucumber\n\n1 teaspoon salt\n\n2 cups yogurt\n\n2 tablespoons chopped fresh dill\n\n2 scallions, finely chopped\n\n1 clove garlic, finely chopped (optional)\n\nsalt and pepper\n\n1 Grate the cucumber and place it in a sieve over a large bowl. Add the salt and mix it around. Leave it for 30 minutes to 2 hours, occasionally pressing the cucumber gently into the sieve to get the liquid out. The salt will help leach the water out of the cucumber.\n\n2 Line another sieve (or the same one, cleaned) with a paper towel or cheesecloth. Place it over a large bowl and pour the yogurt into it. You can leave it for as little as 1 hour on the counter or overnight in the fridge. The longer you leave it, the thicker it will get. This is how Greek yogurt is made!\n\n3 Mix the yogurt with the strained cucumber, dill, scallions, and garlic, if using, and taste. Season with salt and pepper and add more dill or scallions to taste.\n\n4 Enjoy on sandwiches, as a dip, with pita or chips, or over meatballs, kebabs, or anything spicy.\n\nRaita\n\n$1.25 \/ cup\n\n$2.50 total\n\nThis traditional Indian yogurt sauce is simple and surprisingly tasty. Spoon it onto Chana Masala, Potato and Kale Rolls with Raita, or anything spicy to cool things down. This recipe is extremely loose\u2014just stir some of your favorite chopped vegetables into yogurt and add salt and pepper. Use this as a stepping-stone to develop your signature raita. makes 2 cups\n\n1 cup yogurt\n\n1 cup chopped cucumber\n\n\u00bd cup chopped tomato\n\n\u00bc cup chopped red onion\n\n1 teaspoon ground cumin\n\n\u00bd teaspoon cayenne pepper\n\n2 tablespoons chopped fresh cilantro\n\nsalt and pepper\n\nadditions\n\n1 tablespoon grated ginger\n\n2 tablespoons mint\n\n\u00bc cup chickpeas\n\ncooked spinach\n\n1 Stir all the ingredients together in a medium-size bowl. Add salt and pepper to taste.\n\n2 Store the raita in a covered container in the fridge until you're ready to use it. It will keep for up to 3 days.\n\nideas\n\nFlavor\n\n**Many of the recipes in this collection can be easily modified to your taste.** Learning to cook with different spices, herbs, and aromatics will instantly elevate your cooking and open up new and interesting possibilities. Although spices last a reasonably long time, they do grow stale and lose their intensity after a couple of months. When you experiment, buy small amounts of new spices in the bulk aisle and go for larger quantities of the spices you use most often.\n\nTry the flavor combinations below on bland basics like rice, roasted chicken, vegetables, and potatoes. Starchy potatoes are almost the perfect flavor sponge.\n\nMix spices into butter, add them to popcorn, or sprinkle them on toast. In short, experiment!\n\ngarlic\n\nGarlic! You should always have this around. Cheap and flavorful, it goes well with almost everything. Try saut\u00e9ing vegetables, especially bitter green ones like broccoli and Brussels sprouts, with the following flavors, or pack the flavors into a creamy dressing over arugula or mustard greens. Any would make a great base for a pasta sauce, or bring a spark to plain rice or barley. Oh, and shellfish\u2014did I mention shellfish?\n\n\u2022 garlic and lemon zest\n\n\u2022 garlic and fresh basil, fresh parsley, or rosemary\n\n\u2022 garlic, onion, and ginger (heat with olive oil as the base for a stir-fry)\n\n\u2022 garlic, anchovy, and chile flakes\n\ncitrus\n\nThe strongest lemon flavor comes not from the juice, but from the essential oils trapped in the bright, top layer of skin. Ditto for limes and oranges. If you want something to really taste like citrus, try adding just a touch of finely grated zest. It's very strong, though, so be careful with it. I add lemon zest to my tomato sauce, lime zest to pork tacos, and orange zest to dressing. Citrus brings out other flavors almost as well as salt.\n\n\u2022 orange, lemon, and lime zests\n\n\u2022 lime zest, coconut, and chile flakes\n\n\u2022 orange zest, scallions, and fresh cilantro\n\n\u2022 lime zest and chipotle powder\n\n\u2022 lemon zest and fresh mint (amazing with cucumbers or watermelon)\n\nherbal\n\nTry sprinkling these classic combinations on roasted root vegetables or rub them all over a chicken before you roast it.\n\n\u2022 sage, rosemary, and thyme\n\n\u2022 sage and garlic (incredible with winter squash)\n\n\u2022 paprika and fresh dill\n\n\u2022 fennel seeds and fresh parsley\n\nFennel seeds\u2014sweet and a little licorice-y\u2014will be a familiar flavor to Italian sausage lovers, but it's not especially common in North America. Try them in a salad or in a blend with other spices. They bring out the flavors around them.\n\nearthy\n\nYou want to get serious about spices? Try these South Asian combinations and don't be shy with them!\n\n\u2022 ginger, cinnamon, and black pepper\n\nExcept for the pepper, the flavors above might seem like sweet spices, but they are wonderful in rice pilaf and with winter root vegetables. Add these to tea along with some cardamom for a traditional masala chai.\n\n\u2022 cardamom, coriander, and bay leaf\n\n\u2022 cumin seeds, coriander seeds, and mustard seeds\n\nFor one of the best stir-fries you'll ever have, heat a teaspoon each of the above in a dry skillet until you smell them, then add vegetables.\n\nsweet\n\nIf you're a baker, get yourself some of these lovely sweet spices to enliven all your standard favorite breads, cakes, and cookies. You can even add them to tea if you get adventurous. Try to get whole nutmeg rather than pre-ground. You'll be amazed at the difference that just a tiny grating of the fresh stuff will add to your food.\n\n\u2022 cinnamon, ginger powder, nutmeg, and clove\n\n\u2022 lemon zest and thyme (savory flavors that are incredible in a simple shortbread cookie)\n\n\u2022 orange zest and cocoa powder\n\n\u2022 lime and coconut\n\n\u2022 green cardamom and vanilla\n\nTry to get whole cardamom pods, smash the green pods and remove the black seeds in the center. Their aroma is much stronger when fresh rather than pre-ground. However, pre-ground cardamom is still fine in baked goods\u2014you may just need to use a little more of it to get the full flavor. The whole pods are lovely in rice and tea as well, so track some down if you're a cardamom fan.\n\nmake your own spice blends\n\nSome common spice blends can be made easily at home. For instance, instead of buying pre-made chili powder, make your own blend of cumin, cayenne pepper, garlic powder, oregano, and paprika. The spices included in the ingredients for Dark and Spicy Chili (with the exception of cocoa powder) make a chili powder that can be substituted for the pre-made stuff.\n\nCurry powder is a bunch of South Asian spices blended together, and there are many, many varieties of curry blends\u2014usually, some combination of turmeric, coriander, cumin, and chile powder.\n\nGaram masala, another spice blend that is essential to Indian cooking, is most often made with similar spices, but rarely includes chile powder. Try pre-made curry and garam masala blends to start, and work your way up to making your own.\n\nDon't buy pumpkin pie spice. It's usually stale and bland. Just hit your pumpkin pie or cookies with a mix of cinnamon, nutmeg, ground ginger, and a little clove.\n\navoid some dried herbs\n\nDried oregano, dried thyme, and dried rosemary add a lot of flavor to chili, soups, and roasted vegetables. However, don't bother with dried basil, dried cilantro, dried dill, and definitely not dried mint, unless you particularly like them and can't afford to get them fresh. They lose almost all their magic when dried.\n\nsalt\n\nSalt is essential to bringing out the natural flavor in any food and should be used alongside any other flavors you choose. If something doesn't taste quite right, the answer most of the time is to add a bit more salt. Granted, those with health conditions where sodium levels can cause harm should watch their intake! If you can't use much salt in your diet, try replacing it with spice, a squeeze of citrus juice, or a sprinkling of citrus zest.\n\nIt's easy to make a basic salad dressing, so you should never have to buy it pre-made. Simply follow the instructions for the basic vinaigrette on page , and add any of the flavor combinations suggested here to experiment. The ones under citrus are the best place to start.\n\nSpice Oil\n\n$3.50 \/ cup\n\n$3.50 total\n\nUse this spice oil on salads, in cold noodle dishes, or on roasted or saut\u00e9ed vegetables. You can get all of the spices at most Asian grocery stores. makes 1 cup\n\n1 clove garlic\n\n1 cup olive or vegetable oil\n\n2 tablespoons chile flakes or chopped dried red chiles\n\n1 teaspoon Sichuan or regular peppercorns\n\n1 star anise\n\n\u00bd teaspoon cumin seeds\n\n\u00bc teaspoon salt\n\n1 Use the side of a knife to crush the garlic clove, and peel it when it cracks open.\n\n2 Place the crushed garlic in a small pot and add the olive oil, chile flakes, peppercorns, star anise, cumin seeds, and salt. Warm the mixture over low heat, until it starts to bubble gently and you can hear a bit of a sizzle, about 10 minutes. Turn off the heat. You don't want it to be so hot that the spices start to cook or fry.\n\n3 Put the covered pot in the fridge for 4 to 8 hours.\n\n4 Taste the oil. If it isn't strongly spicy, let it infuse for a few more hours. Once it's ready, strain the oil through a sieve to remove the spices. Store in a jar in the fridge for up to a week.\n\nIf you don't have a sieve, place a lid over the pot and carefully pour the oil out sideways to strain it. Allow a small opening between the lid and the pot so the oil can escape while the spices are left behind.\n\nRoti\n\n$0.03 \/ roti\n\n$0.50 total\n\nThese flatbreads are a staple in many parts of India. They're quick to make and very tasty when fresh. Enjoy them with a curried filling, dip them into soups or stews, or wrap them around eggs at breakfast. makes 16 roti\n\n2 cups whole-wheat flour (see box)\n\n1 teaspoon salt\n\n1 cup water\n\nvegetable oil, for coating your hands\n\n1 Mix the flour, salt, and water in a small bowl using one clean hand. It should form a fairly moist dough. Knead until smooth\u2014anywhere from 2 to 5 minutes\u2014and form into a ball. Cover with a damp towel or paper towel and set aside for 10 minutes to an hour.\n\n2 Rub a small amount of oil onto your hands to make it easier to handle the dough. Divide the dough into 16 small balls.\n\n3 Sprinkle a countertop with flour and place one piece of dough in the middle. Cover the ball with flour so it doesn't stick to the surface, then gently roll it out with a rolling pin (or a bottle if you're in a pinch) until it's thin and flat, about \u215b inch thick. As you roll the dough, unstick it from your counter and flip it over. To make it round, roll straight in front of you, then turn the dough 90 degrees and roll out again.\n\n4 Place a nonstick skillet over medium heat (cast-iron is best). Once the pan is hot, add the roti and cook until the dough lifts away from the pan around the edges and small bubbles form, 1 to 2 minutes. Flip the bread over and cook the other side. You want to see light brown bubbles all over the dough. Don't let it get too dark, though, as this will make the roti too crunchy to use for rolls. Repeat until you run out of dough.\n\n5 Keep the roti under a towel on the counter or in a warm oven until ready to serve.\n\nWhole-wheat durum flour, sometimes called _roti_ or _chapati_ flour, can be found in Indian grocery stores and is the best flour to use for this recipe. You can use regular whole-wheat flour as well, but if you want to make this recipe often, consider a special trip to get roti flour!\n\nFlour Tortillas\n\n$0.07 \/ tortilla\n\n$1.70 total\n\nHomemade tortillas are a bit of work, but they're totally worth it. With practice, you'll get quicker and enjoy the process as much as the results. makes 24 small tortillas\n\n1\u00bc cups all-purpose flour, plus more for shaping the dough\n\n1\u00bc cups whole-wheat flour\n\n2\u00bd teaspoons baking powder\n\n1 teaspoon salt\n\n\u2153 cup clarified butter or lard, at room temperature (see box)\n\n1 cup hot water\n\nLard is more traditional, but I prefer clarified butter (which is just butter with the milk solids removed). Even regular butter will work. You can also buy ghee from an Indian grocer and use that.\n\n1 Whisk together the flours, baking powder, and salt in a large bowl. Add the clarified butter. Using your fingers, work the butter into the flour until the mixture looks like moist crumbs. Add the hot water\u2014not boiling, just hot\u2014and form the dough with your hands. Leave it in the bowl for an hour, covered by plastic wrap or a moist towel.\n\n2 Roll the dough into 24 small balls, then cover them again.\n\n3 Lightly flour your countertop. Gently flatten one dough ball with your palm, then roll it out with a rolling pin. Flip it over to make sure it doesn't stick to the counter; add more flour if it does stick. Once the dough is a nice, thin, 5- to 6-inch circle, set it aside under a moist towel.\n\n4 Once you've rolled out one or two tortillas, put a nonstick or cast-iron pan over medium-high heat. Let it get nice and hot. Place a tortilla in the pan. Once it starts to dry up around the edges, flip it over with a spatula, then gently press it down to give it some color underneath.\n\n5 Once the tortilla has brown spots on both sides, remove it from the pan and continue with the next. Work quickly! As you wait for each tortilla to cook, roll out more. You'll get better at this part with practice.\n\n6 If you're serving the tortillas soon, place them in a warm oven to keep them pliable. If they're for later in the day, pile them under a cloth while you finish making them. Once you're done, wrap them in aluminum foil and put them in the fridge. Heat them in the oven before serving.\n\nPizza Dough\n\n$0.20 \/ crust\n\n$0.80 total\n\nThere are two ways to make pizza dough: the fast way and the slow way. They're the same amount of work, just with different wait times. The slow method is convenient for a weekday with a little preparation\u2014make it the night before you plan to make pizza, pop it in the fridge, and then pull it out to rise a few hours before dinner. If you're organized enough to make the slow dough, I recommend taking the extra time: It tastes best, and is much softer and easier to stretch. makes 4 individual pizzas\n\n3 cups all-purpose flour or bread flour, plus more for shaping the dough\n\n1\u00bd teaspoons salt\n\n\u00bd to 1 teaspoon instant yeast\n\n1 tablespoon olive oil, plus more for coating the bowl\n\n1\u00bc cups water, at room temperature\n\nFast Method\n\n1 Measure out the flour, salt, and 1 teaspoon of yeast into a big bowl. Mix in the oil with your hands, crumbling until the texture is a bit sandy, then add the water. Keep mixing until the dough comes together.\n\n2 Lightly flour your countertop. Gently stretch and fold the dough to knead it, pushing against the countertop over and over. Massage the dough until it becomes smooth and elastic, about 5 to 7 minutes. The dough will be smooth but quite wet.\n\n3 Add a small amount of oil to a bowl. Place your dough ball in the bowl and cover with plastic wrap. Let it rise for 1\u00bd to 3 hours, depending on the warmth of your kitchen. It's done rising when it has doubled in size. Now you're ready to make pizza! For rolling instructions, see page .\n\nSlow Method\n\n1 Follow Steps 1 and 2 of the fast method, but add only \u00bd teaspoon of yeast to the flour mixture. Use very cold water instead of room temperature.\n\n2 Add a small amount of oil to a bowl. Place your dough ball in the bowl and cover with plastic wrap. Put it into the fridge overnight. Letting the yeast work overnight creates a better flavor; it also makes the dough more elastic and easy to work with.\n\n3 The next day, 2 to 3 hours before you want to bake your pizzas, remove the dough from the fridge to return to room temperature. Now you're ready to make pizza! See page  for how to roll it out.\n\nmethod\n\nCroutons or Breadcrumbs\n\n**I am constantly haunted by crusts of hard, several-days-old bread that I have neglected.** Luckily, there are plenty of delicious solutions that help those crusts avoid the trash can. Croutons and breadcrumbs will keep for ages in a sealed container on the counter, and when you have them around, you'll find yourself using them everywhere\u2014and perhaps even finding excuses to make a salad. You can make this with any amount of bread.\n\nbread\n\nbutter or vegetable oil, as needed\n\nsalt and pepper, to taste\n\n1 **Prepare the bread.** For croutons, cut the bread into cubes. For breadcrumbs, mince the loaf with a knife, tear it apart, or throw small chunks of bread into a food processor. If the bread is too hard to cut, wrap it in a kitchen towel, sprinkle some water on the towel, and microwave for 20 to 30 seconds. This will restore just enough moisture to let you cut the bread easily.\n\n2 **Place a large pan on the stove over medium heat.** Add enough butter\u2014usually at least a tablespoon\u2014to coat the bottom of the pan. **Melt the butter.**\n\n3 **Working in batches, add the bread and toss it gently until it's coated in butter.** Let the bread sit for 2 minutes, then flip the pieces over. Keep tossing and turning, leaving it for a minute or so at a time, until the bread is brown all over. Add more butter as needed. The bread cubes may get kind of dry, and a little more fat will help them brown more evenly. Sprinkle with salt and pepper. It is basically impossible, unless you are very patient (which I am not), to get every side of the cubes browned, so just get them generally looking good and toasty and then take them off the heat. Taste one and add more salt and pepper, if you need to.\n\n4 **Use the breadcrumbs or croutons immediately, or store them in a sealed container after letting them cool off.**\n\nFor breadcrumbs, if you like, you can go oil free: just toast whole slices of bread and then crush or process them into small pieces.\n\nFresh Pasta\n\n$0.35 \/ serving\n\n$0.35 total\n\nWhen a reader, Jeanne, asked for a good pasta dish, I decided to show her how to create it from scratch. Sure, making pasta by hand requires elbow grease and a good rolling pin, but you'll be surprised at how simple, cheap, and tasty it is. If an Italian grandmother can do it, so can you! Because fresh pasta is so wonderful, the sauce you pair it with doesn't need to be complicated. Try it with Best Tomato Sauce or Chorizo and White Bean Rag\u00f9 and a little cheese. serves 1 very hungry person\n\nMultiply this recipe by the number of people you are serving. The stated quantities are a useful ratio, but they produce big portions.\n\n\u00be cup all-purpose or bread flour, plus more for shaping the dough\n\n1 egg\n\nolive or vegetable oil\n\nsalt\n\n1 Put the flour in a bowl. Make a crater in the center of the flour and crack the egg into it. Mix with your hands. The egg takes a while to release all its moisture, so don't panic if things are dry at first. If, after mixing for about a minute, the dough still seems excessively dry, add a teaspoon of water. Keep mixing until you develop a stiff dough that is quite dry. The dryness makes it easier to roll out and keeps the noodles from sticking together when you cook them.\n\n2 Add a small amount of oil to a bowl. Place the dough in the bowl, cover with a moist towel or plastic wrap, and let rest for 1 to 2 hours.\n\n3 Once an hour (or more) has passed, you should notice a marked change in the dough. The egg will have released its moisture to form a pale yellow, smooth, pliable dough. Knead it again to create a smooth ball.\n\n4 Tear or slice the dough into manageable pieces\u2014usually as many as the number of people you're feeding. Dust your countertop heavily with flour, then use a rolling pin to make the dough as thin as you can. Rolling it out will take a while because it's tough and stretchy. Try to get it thin enough to see light through it. The thinner the dough, the quicker it will cook, but don't make the dough so thin that it tears.\n\n5 By the time the pasta is rolled out, it should be dry enough to avoid sticking to itself. If it's still moist, let it sit for a few minutes.\n\n6 Boil a pot of heavily salted water.\n\n7 Slice the dough into whatever size noodles you like. If you fold the dough over itself a few times, it's easy to make the noodles a consistent size. Shake the cut noodles on a tray with a bit of flour to keep them from sticking.\n\n8 When the pot of water has come to a boil, add the pasta. Fresh pasta takes as little as 30 seconds to cook, if the noodles are thin. It's ready when it changes color and starts to float. Taste a noodle after they start to float to see if it's cooked. If not, let it cook a little longer.\n\n9 Drain and serve immediately. Uncooked pasta will keep, wrapped in plastic or spread out on a covered baking sheet, for up to 2 days in the fridge. Dust it with enough flour to keep it from sticking to itself.\n\nRicotta\n\n$4 \/ Cup\n\n$6 total\n\nTo transform regular milk into something truly special, make your own ricotta. It's a neat recipe to do with kids because it's simple, yet the transformation of the milk is dramatic. Try this in lasagna or anywhere else you would usually use ricotta, but be sure to enjoy a little of it on its own to appreciate its full deliciousness. I love it best on toast (shocking!) with a sprinkling of pesto or chopped herbs on top to cut the richness of the cheese. Save the whey to use in soups instead of broth. makes 1\u00bd cups\n\n\u00bd gallon whole milk\n\n1 cup heavy cream\n\n1 teaspoon salt\n\n3 tablespoons distilled white vinegar or lemon juice\n\nadditions\n\nfresh herbs\n\ndried herbs\n\nany spice\n\n1 Place the milk, cream, and salt in a large, heavy-bottomed pot and bring just to a boil. When just one bubble comes up to the top, add the vinegar, give it a stir, and turn off the heat. The vinegar will create a disturbance and the whey and cream will begin to separate.\n\n2 Meanwhile, line a sieve with cheesecloth or two layers of paper towels and place over a large bowl.\n\n3 Leave the mixture in the pot for 5 minutes, then stir to separate the cheese and the whey completely. They should be two distinct parts of the mixture. If some of it is still milky, add another tablespoon of vinegar, stir, and let it sit for a few more minutes.\n\n4 Pour your ricotta mixture into the cheesecloth-lined sieve with the bowl underneath to catch the whey. Leave for 30 minutes to drain.\n\n5 The cheesecloth will be full of a crumbly white cheese. That's your ricotta! Taste it and add more salt if it needs it, or add herbs or spices.\n\n6 Eat immediately, or pack into an airtight container and store in the fridge. The ricotta will keep for about 3 days.\n\nideas\n\nRainbow Rice\n\n$0.20 \/ serving\n\n$0.40 total\n\n**Here are three quick ways to make plain rice a little more exciting.** An early reader, Charles, said he loves rice with vegetables, but these treatments work for grains other than rice as well\u2014everything from quinoa to barley to farro. Vegetables are a great way to liven up the usual rice and beans.\n\nPlain Rice\n\nserves 2\n\n2 cups water\n\n1 cup rice\n\n2 pinches of salt\n\nTo make plain rice, pour the water into a pot and add the uncooked rice and salt. That'll be enough for two generous portions, or three or four smaller servings. With the lid off, bring to a low boil over medium heat. Turn down the heat to low and put the lid on slightly askew, so that the steam can escape. Cook for about 20 minutes, until the water is all gone and the rice is fluffy.\n\n1 Green Rice\n\n$1 serving \/ $2 total\n\n1 cup frozen spinach, beet greens, chard, or fresh parsley\n\nChop the spinach as much as you like. The more finely chopped, the more it will disperse into the rice. Cook as directed for plain rice, but after cooking for 15 minutes, mix the spinach into the rice. Cook with the lid off for the last 5 minutes. Adding the spinach at the end keeps it lush and bright, rather than sad and overcooked.\n\n2 Orange Rice\n\n$1 serving \/ $2 total\n\n1 cup canned winter squash, pumpkin, or sweet potato puree\n\nStir the squash with 1\u00bd cups of water in a bowl, then pour it into a pot with the rice and salt. (You can also use frozen, boiled, or saut\u00e9ed squash.) Cook as directed for plain rice.\n\n3 Red Rice\n\n$1 serving \/ $2 total\n\n1 cup canned tomatoes, pureed\n\nStir the tomatoes with 1\u00bd cups of water, then pour into a pot with the rice and salt. Cook as directed for plain rice.\n\nmethod\n\nHow to Cook Dried Beans\n\n**This method works for cooking any kind of beans in any amount.** Make a large batch on a Sunday and use them in meals throughout the week. The only thing to keep in mind is that the older and bigger they are, the longer they take to cook. Some may take as long as 4 hours. If you eat a lot of beans, consider investing in a pressure cooker, which will cut down on the boiling time.\n\nadditions\n\nbay leaf\n\nbouquet garni of favorite tough herbs\n\ndried herbs and spices\n\nonion\n\ngarlic\n\nchiles\n\nginger, sliced\n\n1 The best way to prepare dried beans is to **soak them in water overnight**. The next day, drain and rinse the beans thoroughly before cooking them in fresh water. If you don't have the opportunity to soak the beans ahead of time, you can make up for it: Cover the beans with water, then bring them to a boil in a large pot. After 10 minutes, take them off the heat and drain them.\n\n2 **Place the drained beans in a large pot and cover with fresh water.** Include any additions you desire.\n\n3 **Bring to a boil over medium heat, then turn down the heat so that the beans simmer gently.** Put a lid on the pot, but leave it askew so the water doesn't boil over.\n\n4 **Check on the beans every half hour or so,** being sure to keep them covered with water if it boils away.\n\n5 **Beans take vastly different lengths of time to become tender.** The older and bigger they are, the longer they take to cook. Very old, very large beans can take as long as 4 hours. If you are making refried beans or beans for a soup or stew, don't worry about overcooking them\u2014it's fine if they're mushy. If you want to maintain their shape and integrity, however, monitor them closely once they're nearly done, after a couple of hours.\n\n6 **Once the beans are tender, you can drain them or leave them in their cooking water,** depending on what you're using them for. Remove and discard the additions, if necessary, and add salt to taste\u2014they will need a fair bit!\n\nmethod\n\nPickle Primer\n\n**Pickles made from cucumbers are undeniably America's favorite kind of pickle.** But pickled carrots, beets, asparagus, onions, and cauliflower are delicious as well. Pickled vegetables are not only incredibly delicious\u2014they are also a great way to preserve vegetables when you have an unexpected bounty that you don't want to go to waste. A friend left five pounds of zucchini on your doorstep? Pickle it! You might also want to consider pickles for holiday gifts. I get requests for them every year!\n\nThe form of pickling I'm describing isn't proper, long-term preserving or canning. That process is worth learning, but it requires specialized equipment. Here we're learning the quick method to create tangy, flavorful, quick pickles that blow any store-bought pickles away.\n\nessentials\n\nvegetables (cucumbers, zucchini, cauliflower, carrots, beets, asparagus, onions, peppers, or string beans)\n\n1-quart mason jars (see box)\n\nadditions to the jar\n\nsprigs of dill\n\ngarlic cloves, quartered\n\nbrine\n\n1 cup distilled white vinegar\n\n1 cup water\n\n1 tablespoon salt\n\n1 teaspoon mustard seeds\n\n1 teaspoon dill seeds\n\n1 bay leaf\n\nadditions to the brine\n\n1 or 2 dried chiles, crushed\n\ncinnamon stick\n\n2 whole cloves\n\n\u00bd teaspoon coriander seeds\n\n\u00bd teaspoon allspice berries\n\n\u00bd teaspoon fennel seeds\n\n\u00bd teaspoon celery seeds\n\n1 tablespoon sugar\n\nMultiply this recipe by the number of quarts of brine you want to end up with. The quantities below are a ratio\u2014they will make roughly enough brine to fill a 1 quart mason jar (depending on how tightly packed it is with vegetables). You can adjust the ratio according to how many vegetables you have and how many jars you want to fill. I suggest making at least 6 quarts at once if you have the space in the fridge. It's more efficient to make a large batch and, trust me, they will be popular.\n\n1 **Cut up the vegetables however you like.** Smaller pieces will pickle faster because the brine takes less time to penetrate them. I generally go with bite-size pieces, or long strips if making cucumber pickles. You can also add any other vegetables or herbs for flavor.\n\n2 **Fill the jars with the vegetables.**\n\n3 **Make the brine:** Pour the vinegar, water, and salt into a pot. Put in any other additions. If you don't maintain a spice collection, you can buy pickling mix in the spice aisle of most grocery stores. Instead of the seeds and bay leaf, add 1 tablespoon of pickling mix to the water, vinegar, and salt. And for sweet pickles, add about 1 tablespoon of sugar.\n\n4 Bring the brine to a boil and turn the heat down to low. **Simmer for10 minutes.**\n\n5 **Pour the hot liquid over the vegetables in the jars.** If you don't have enough, don't panic. Simply make more using the ratio in the brine ingredients list. If you've run out of spices, just skip them.\n\n6 **Place the lids over the jars.** Don't screw them on tightly until the jars have come to room temperature. Once the jars have cooled completely, screw on the lids and put the jars in the fridge.\n\n7 **Leave the pickles for 2 weeks to let the brine do its work.** After 2 weeks, eat them!\n\nIdeally, you'll have glass jars with tight-fitting screw-top lids. You can usually find jars for canning at hardware stores or many grocery stores, but another kind of glass jar or an old pickle jar will do in a pinch. Mason jars can be used over and over and I use them to house things like dried beans and grains, so they are very useful and inexpensive storage.\nDrinks and Desserts\n\nmethod\n\nAgua Fresca\n\n**Refreshing, hydrating, and beautiful, these drinks are great at a party,** and they can help you use up fruit before it goes bad. Use this as a starting point for developing your own favorite.\n\n2 cups chopped fruit\n\n4 cups water\n\nadditions\n\n1 teaspoon vanilla extract\n\nsqueeze of citrus juice\n\nsugar\n\nmint leaves\n\nleaves of other herbs\n\nvariations\n\nblueberry and lemon\n\ncucumber and lemon\n\nmango and lime\n\nmelon\n\norange\n\npapaya and pineapple\n\npeach and vanilla\n\nstrawberry and mint\n\n1 **For a very lightly flavored agua fresca, just mix the fruit and water together.** Done! Obviously, if you want more fruit flavor, then use less water; for a more subtle flavor, use more water.\n\n2 **Serve over ice.** Try some of the variations I've suggested or whatever fruits you like! If you store this in the fridge, it'll keep for about 3 days or up to a week, depending on the fruit.\n\nI usually run my agua fresca through the blender. If you want the drink to be clear, strain the pulpy leftovers of the fruit after blending. If you're using blueberries or oranges or other fruit with a skin, you'll almost certainly want to strain it. For some fruits, you can also choose to leave the pulp: in the case of melons, the pulp purees so smoothly that you barely notice it.\n\nideas\n\nSmoothies\n\n**I have four types of smoothies** here, but, of course, once you get the hang of it, there are many more flavor combinations to be made. Frozen fruit is perfect for smoothies. Or, use overripe fruit that you wouldn't eat otherwise. Add a teaspoon of vanilla extract to any of these and they will seem incredibly professional. The frozen melon drinks, in particular, are a refreshing slushy-like treat on a hot summer day. Two\u2014watermelon and honeydew\u2014are pictured at right. (The mango lassi is not pictured.) serves 2\n\n1 Drinkable Yogurt\n\n$0.50 serving \/ $1 total\n\n\u00bd cup yogurt, not Greek\n\n\u00bd cup fruit juice\n\nIf you like the grocery store's yogurt drinks, try making these at home for less! You don't even need to blend them\u2014just add the yogurt and juice to a jar, then shake.\n\n2 Berry Smoothie\n\n$1 serving \/ $2 total\n\n\u00bd cup yogurt, not Greek\n\n1 cup frozen berries\n\nmilk or juice to thin as needed\n\n1 teaspoon vanilla extract\n\nBlend the yogurt, berries, a bit of milk, and vanilla until smooth, then adjust with more berries or milk to your taste.\n\n3 Melon Smoothie\n\n$0.50 serving \/ $1 total\n\n1 cup chopped frozen melon\n\n\u00bd cup water or juice\n\n1 teaspoon vanilla extract\n\nWhen you buy a melon, dice and freeze whatever you don't eat. Pull it out a cup at a time and blend it with a bit of water or juice and vanilla to thin it out. It's like a slushy, but better!\n\n4 Mango Lassi\n\n$1 serving \/ $2 total\n\n1 mango, peeled, pitted, and diced\n\n1 cup yogurt, not Greek\n\nmilk to thin as needed\n\nBlend the mango and yogurt together. If it's too thick to drink with a straw, add some milk to thin it out. A ripe and juicy mango combined with yogurt is often all you need. Be warned: If you make this for children, they will request it over and over.\n\nAvocado Milkshake\n\n$1.15 \/ serving\n\n$2.30 total\n\nJohn, the reader who introduced me to the silky magic of this milkshake, lives in California, where avocados are often less than a dollar. If you can find a similar deal, whip up a batch of these! serves 2\n\n1 avocado, peeled and pitted\n\n2 cups milk, coconut milk beverage, almond milk, or rice milk\n\n1 teaspoon vanilla extract\n\n1 tablespoon lime juice\n\npinch of salt\n\n2 tablespoons sugar\n\nToss the avocado, milk, vanilla, lime juice, salt, and sugar into a blender and whiz them up! Let it go for a while so that the avocado breaks down and blends with the milk. Once the liquid is Kermit the Frog green, it's ready. Taste it and add more sugar or lime juice as needed.\n\nIf your avocado isn't quite ripe, a bit more lime juice will bring out its flavor.\n\nFast Melon Sorbet\n\n$0.60 \/ serving\n\n$2.40 total\n\nWhen you see lovely watermelons, honeydews, and cantaloupes on sale, buy them up. Eat half, then cube and freeze the other half. When you want a quick dessert or smoothie, pull out a bag of frozen melon and whip this up. serves 4\n\n2 cups chopped frozen melon\n\n\u00bd cup plain yogurt\n\n\u00bc cup sugar\n\n1 teaspoon vanilla extract or lime juice (optional)\n\nAdd the melon, yogurt, sugar, and vanilla, if using, to a food processor or blender. Process or blend until just smooth. Don't blend too much, or the sorbet will become overly soft. Serve immediately or store in the freezer to enjoy later.\n\nideas\n\nRice Pudding\n\n$0.35 \/ serving\n\n$1.40 total\n\n**Rice pudding is so simple to make\u2014it's almost as easy as cooking plain rice.** It's a wonderful make-ahead dessert if you have guests, and it can easily be dressed up with berries, nuts, or Caramelized Bananas. I've included a few variations, but please do experiment: Milk, rice, and sugar are a great canvas. Try switching out the sugar for honey, or stirring in a spoonful of cocoa powder, peanut butter, or coconut flakes. serves 4\n\nBasic (Vanilla) Pudding\n\nserves 4\n\n\u00bd cup long-grain rice\n\n2 cups milk\n\n\u00bd cup sugar\n\n\u215b teaspoon salt\n\n2 teaspoons vanilla extract or \u00bd vanilla bean, seeds scraped\n\n1 Place a pot over medium heat and add the rice, milk, sugar, and salt. Stir until the sugar is dissolved and then bring to a boil. Once the milk is just boiling, turn the heat down to low and simmer for 20 minutes. Place a lid slightly askew on the pot so the steam can escape.\n\n2 Remove a grain of rice and taste it to see whether it's cooked. If it is soft and chewy all the way through, then it's done. If not, let it cook for a few more minutes. Once the rice is cooked, add the vanilla, and stir.\n\n3 Remove the pudding from the heat and let it cool to room temperature. Let it chill in the fridge for 2 hours before serving. You can also eat it hot, but if you wait, the pudding will thicken in the fridge and have a more pleasing texture.\n\n1 Indian-Style\n\n$0.65 serving \/ $2.60 total\n\n\u00bd cup basmati rice\n\n2 cups milk\n\n\u00bd cup sugar\n\n\u215b teaspoon salt\n\n1 teaspoon cardamom\n\n\u00bc cup chopped almonds or pistachios\n\nFollow the instructions for vanilla rice pudding, but use basmati rice and add cardamom in place of the vanilla in Step 2. Top with chopped nuts.\n\n2 Pumpkin Pie\n\n$0.60 serving \/ $2.40 total\n\n\u00bd cup long-grain rice\n\n\u00be cup canned or fresh pumpkin puree\n\n2 cups milk\n\n\u00bd cup brown sugar\n\n\u215b teaspoon salt\n\n1 teaspoon ground cinnamon\n\n\u00bd teaspoon ground cloves\n\n\u00bd teaspoon ground ginger\n\nFollow the instructions for vanilla rice pudding, but substitute brown sugar and add pumpkin, cinnamon, cloves, and ginger, and stir together before adding to the pot. Simmer for 25 minutes (instead of 20), stirring occasionally.\n\nCoconut and Lime Brown Rice Pudding\n\n$0.60 \/ serving\n\n$2.40 total\n\nTangy, rich, creamy, and just a little chewy, this dessert may be a healthier alternative to the rice puddings on page 175, but it sure doesn't feel like a compromise. serves 2\n\n\u00bd cup short-grain brown rice\n\n2 cups water\n\n\u215b teaspoon salt\n\n1 can (13.5 ounces) full-fat coconut milk\n\n\u00bd cup sugar\n\nzest of 1 lime or orange\n\n4 slices ginger, added with the coconut milk then removed before serving (optional)\n\n1 Place a pot over medium heat. Add the rice, water, and salt. Bring to a boil, turn the heat down to low, and cook the rice for 35 minutes. Place a lid slightly askew on the pot so the steam can escape.\n\n2 Dump the rice into a sieve to drain. It should be almost cooked, but not quite.\n\n3 Add the coconut milk and sugar to the now-empty pot over medium heat. Stir and let cook until the sugar is dissolved.\n\n4 Put the brown rice back in the pot and bring it to a boil over medium heat. Turn the heat down to low and simmer with the lid on for 15 more minutes. Test to make sure the rice is tender\u2014if it's not, let it simmer for a few more minutes. Add the lime or orange zest and stir.\n\n5 Remove the pudding from the heat and let it cool to room temperature. You can also eat it hot, but if you chill it for 2 hours before serving, it will thicken and have a more pleasing texture.\n\nCaramelized Bananas\n\n$0.35 \/ serving\n\n$0.70 total\n\nThese bananas\u2014cooked in just a bit of caramel\u2014are crispy and gooey on the outside and almost like a soft pudding inside. Sweet, messy, and irresistible. serves 2\n\n1 tablespoon butter\n\n2 tablespoons brown sugar\n\n2 bananas, peeled and split in half lengthwise\n\n1 Melt the butter in a nonstick or cast-iron pan over medium-high heat. Add the sugar and let it melt into the butter for about 2 minutes. Stir once to create a smooth mixture.\n\n2 Place the bananas flat-side down in the butter-sugar mixture, then cook until they become brown and sticky, about 2 minutes. Carefully flip them over and do the same on the other side.\n\n3 Serve them as is or cut them into quarters. Drizzle any caramel left in the pan over the bananas. Serve with ice cream or on their own.\n\nCoconut Chocolate Cookies\n\n$0.25 \/ cookie\n\n$10 total\n\nThis is a just-chewy-enough, just-crispy-enough, just-gooey-enough cookie that's perfect for a special treat. makes approximately 40 cookies\n\n\u2154 cup shaved unsweetened coconut\n\n1 cup (2 sticks) unsalted butter\n\n2 cups all-purpose flour\n\n1 teaspoon salt, plus extra for sprinkling\n\n1 teaspoon baking soda\n\n1\u00bd cups firmly packed dark brown sugar\n\n2 eggs\n\n2 teaspoons vanilla extract\n\n1\u00bd cups chocolate chips\n\n1 Preheat the oven to 350\u00b0F.\n\n2 Spread the coconut in a thin, even layer on a baking sheet. Place it in the oven and bake until it's light brown, toasty, and aromatic, 5 to 8 minutes.\n\n3 Melt the butter in a heavy-bottomed saucepan over low heat. Allow the melted butter to cool in the pan for a few minutes.\n\n4 Stir together the flour, salt, and baking soda in a medium-size bowl.\n\n5 Beat the brown sugar and melted butter together in a large bowl until smooth, about 2 minutes. Add the eggs and vanilla and beat until the mixture lightens in color, about 5 minutes. Add the flour mixture to the brown sugar mixture, a third at a time, until it forms a dark brown, homogeneous mass. Add the chocolate chips and toasted coconut and stir until just combined.\n\n6 Place the dough in the fridge for 20 minutes. You can skip this step if you're in a hurry, but resting the dough in the fridge will help your cookies bake more evenly.\n\n7 Lightly butter a baking sheet and scoop heaping tablespoons of dough onto it, leaving large spaces between each cookie so they have room to spread out. I usually do about 6 cookies per sheet. Just before putting the cookies into the oven, sprinkle them with salt.\n\n8 Let the cookies bake until they're golden, 8 to 10 minutes. After you take them out of the oven, leave them on the baking sheet to set for 2 minutes, then move them to a wire rack to cool further. The cookies are very soft when they first come out of the oven, but they'll firm up quickly, so if you leave them on the sheet for 2 minutes, they'll transfer more easily. Don't stack the cookies until they're fully cooled.\n\n9 Repeat Steps 7 and 8 until all the dough is used. Store the finished cookies in an airtight container for up to a week.\n\nPeach Coffee Cake\n\n$0.75 \/ serving\n\n$9 total\n\nAdapted from the apple cake often served during Rosh Hashanah, this coffee cake is simple and wonderful for dessert, with tea, or as a sweet breakfast. The juicy peaches add a ton of flavor. If you buy peaches in season, the cost can be quite reasonable. serves 12\n\n1 cup (2 sticks) unsalted butter, at room temperature\n\n6 peaches, pitted and cut into 8 slices each\n\njuice of \u00bd lemon\n\n1 teaspoon ground cinnamon\n\n2 cups all-purpose flour\n\n2 teaspoons baking powder\n\n1\u2153 cups plus 1 tablespoon firmly packed brown sugar\n\n\u215b teaspoon salt\n\n1 teaspoon vanilla extract\n\n2 large eggs\n\nThis cake is a great base for all kinds of fruit, from other stone fruit like plums, to apples, to berries. Anything but citrus or melons will work. Try frozen fruit if what you want is not in season.\n\n1 Preheat the oven to 350\u00b0F.\n\n2 Use the paper wrapping from the butter to lightly grease an 8- by 11-inch glass baking dish or 9-inch springform pan. Any shape will do as long as it is large enough. This cake doubles in size when it bakes.\n\n3 Mix the peach slices, lemon juice, and cinnamon with your hands in a large bowl, making sure the peaches are well coated in cinnamon.\n\n4 Stir together the flour and baking powder in a medium-size bowl, getting rid of any lumps.\n\n5 Beat the butter, 1\u2153 cups brown sugar, and salt in another large bowl, either with a wooden spoon or an electric mixer. Stop when the mixture is fluffy and has slightly lightened in color, about 5 minutes. Add the vanilla, then the eggs one at a time, fully mixing in the first before adding the second.\n\n6 Add the flour mixture to the butter mixture, gently incorporating until it's smooth. (If you're using an electric mixer, switch to a wooden spoon.) The batter will be quite thick.\n\n7 Spread half the batter over the bottom of the buttered pan. Evenly distribute 24 of the peach slices over the top. (There should be 48 in total.) Spread the other half of the batter over the peaches, then top with the rest of the peaches. Sprinkle with the remaining 1 tablespoon brown sugar and place the cake in the oven.\n\n8 Bake until a knife inserted into the center comes out clean, about 1 hour. \nConversion Tables\n\n**Please note that all conversions are approximate but close enough to be useful when converting from one system to another.**\n\nOven Temperatures\n\n---\n\nFahrenheit | Gas Mark | Celsius\n\n250 | \u00bd | 120\n\n275 | 1  | 140\n\n300 | 2  | 150\n\n325 | 3 | 160\n\n350 | 4 | 180\n\n375 | 5 | 190\n\n400 | 6 | 200\n\n425 | 7 | 220\n\n450 | 8 | 230\n\n475 | 9 | 240\n\n500 | 10 | 260\n\nNOTE: Reduce the temperature by 20\u00b0C (68\u00b0F) for fan-assisted ovens.\n\nApproximate Equivalents\n\n1 stick butter = 8 tbsp = 4 oz = \u00bd cup = 115 g\n\n1 cup all-purpose presifted flour = 4.7 oz\n\n1 cup granulated sugar = 8 oz = 220 g\n\n1 cup firmly packed brown sugar = 6 oz = 220 g to 230 g\n\n1 cup honey or syrup = 12 oz\n\n1 cup grated cheese = 4 oz\n\n1 cup dried beans = 6 oz\n\n1 large egg = about 2 oz or about 3 tbsp\n\n1 egg yolk = about 1 tbsp\n\n1 egg white = about 2 tbsp\n\nLiquid Conversions\n\n---\n\nUS  | Imperial | Metric\n\n2 tbsp | 1 fl oz | 30 ml\n\n3 tbsp | 1\u00bc fl oz | 45 ml\n\n\u00bc cup | 2 fl oz | 60 ml\n\n\u2154 cup | 2\u00bd fl oz | 75 ml\n\n\u2153 cup + 1 tbsp | 3 fl oz | 90 ml\n\n\u2153 cup + 2 tbsp | 3\u00bd fl oz | 100 ml\n\n\u00bd cup | 4 fl oz | 125 ml\n\n\u2154 cup | 5 fl oz | 150 ml\n\n\u00be cup | 6 fl oz | 175 ml\n\n\u00be cup + 2 tbsp | 7 fl oz | 200 ml\n\n1 cup | 8 fl oz | 250 ml\n\n1 cup + 2 tbsp | 9 fl oz | 275 ml\n\n1\u00bc cups | 10 fl oz | 300 ml\n\n1\u2153 cups | 11 fl oz | 325 ml\n\n1\u00bd cups | 12 fl oz | 350 ml\n\n1\u2154 cups | 13 fl oz | 375 ml\n\n1\u00be cups | 14 fl oz | 400 ml\n\n1\u00be cups + 2 tbsp | 15 fl oz | 450 ml\n\n2 cups (1 pint) | 16 fl oz | 500 ml\n\n2\u00bd cups | 20 fl oz (1 pint) | 600 ml\n\n3\u00be cups | 1\u00bd pints | 900 ml\n\n4 cups | 1\u00be pints | 1 liter\n\nWeight Conversions\n\n---\n\nUS\/UK  | Metric\n\n\u00bd oz | 15 g\n\n1 oz | 30 g\n\n1\u00bd oz | 45 g\n\n2 oz | 60 g\n\n2\u00bd oz | 75 g\n\n3 oz | 90 g\n\n3\u00bd oz | 100 g\n\n4 oz | 125 g\n\n5 oz | 150 g\n\n6 oz | 175 g\n\n7 oz | 200 g\n\n8 oz | 250 g\n\n9 oz | 275 g\n\n10 oz | 300 g\n\n11 oz | 325 g\n\n12 oz | 350 g\n\n13 oz | 375 g\n\n14 oz | 400 g\n\n15 oz | 450 g\n\n1 lb | 500 g\nAcknowledgments\n\nFirst, special thanks to my husband, Dan. Your support and expertise were critical. But thank you most of all for never letting me give up or give in, even when I really wanted to. You make me a better person and without you, _Good and Cheap_ would be just a shadow of what it is.\n\nThank you to Mum, Dad, Emily, Hannah, and Papa for your unconditional love and support. Thank you to my incredible friends who believed in this idea before anyone else. Your excitement was infectious! Your advice and support kept me sane. Thanks particularly to Jhen for jumping in, Claire for making me look good, and Sarah and Matt for making me feel powerful.\n\nThanks to my exceptionally patient and understanding editor, Liz, a great advocate and friend. And to Selina, John, Suzie, Jenny, Emily, Janet, Gordon, Orlando, and Amanda, who have given and continue to give so much to make this project a success, I cannot thank you enough. To everyone at Workman: Thank you for taking a chance with me. I'm positive that no other first-time author has ever benefitted from so much devotion from her publisher.\n\nAnd thanks to all of you who shared your food stories: joyful, painful, incredible. You remind me every day why this work is important. The outpouring of love and support I've received is enough for several lifetimes, and I couldn't be more grateful. To those who told me this book has given them hope, inspired them, or otherwise brought them pleasure: I don't deserve so much gratitude for so little, but doing work that matters is all I have ever wanted. \nAbout the Author\n\nAuthor photo by Jordan Matter Photography. \u00a9 Workman Publishing.\n\n**Leanne Brown** is a very silly person who lives in New York City with her husband, Dan Lazin. She grew up in Edmonton, Canada, where her awesome family and many of her favorite people still reside. Before she moved to NYC, she liked to ride her bicycle everywhere. Now she rides sometimes, but takes the subway or walks other times. She likes to run in the park until she is really tired, eat pastries on the street, and go to bed late. She loves fairness, sci-fi and fantasy, tasty food, and laughing. She has been delighted by cooking and baking ever since she realized that they are the closest things we have to magic.\nSPECIAL DISCOUNTS for NON-PROFITS\n\nA special edition of Good and Cheap is available for non-profit organizations wishing to give books away to those with limited incomes. Deep discounts are available on bulk purchases of 10 or more copies. Please email goodandcheap@workman.com for more information.\n\nCopyright \u00a9 2015 by Leanne Brown\n\nPhotographs copyright \u00a9 by Leanne Brown and Dan Lazin\n\nAll rights reserved. A portion of this book was previously published by the author under a Creative Commons license. Terms of the Creative Commons license relate to portions of the book from that edition only. No portion of this book may be reproduced\u2014mechanically, electronically, or by any other means, including photocopying\u2014without written permission of the publisher. Published simultaneously in Canada by Thomas Allen & Son Limited.\n\nAn earlier version of this book was distributed under a Creative Commons Attribution-NonCommercialShareAlike 4.0 license. For more information, visit creativecommons.org\/licenses\/by-nc-sa\/4.0.\n\nLibrary of Congress Cataloging-in-Publication Data is available.\n\nISBN 978-0-7611-8499-7\n\nCover photo by Leanne Brown Additional Photo Credits: page 16 (from top to bottom): pixelrobot\/fotolia, lester120\/fotolia, arinahabich\/fotolia, komodoempire\/fotolia, simmittorok\/fotolia, taiftin\/fotolia; page 17 (from top to bottom): Tarzhanova\/fotolia, Workman Publishing, Viktor\/fotolia, photos777\/fotolia, pioneer\/fotolia, Andy Lidstone\/fotolia.\n\nWorkman books are available at special discounts when purchased in bulk for premiums and sales promotions as well as for fund-raising or educational use. Special editions or book excerpts also can be created to specification. For details, contact the Special Sales Director at the address below, or send an email to specialmarkets@workman.com.\n\nWorkman Publishing Co., Inc.  \n225 Varick Street  \nNew York, NY 10014-4381  \nworkman.com\n\nWORKMAN is a registered trademark of Workman Publishing Co., Inc.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n\n\nProduced by Martin Robb and Ted Robb.  HTML version by Al Haines.\n\n\n\n\n\n\n\n\nIn Freedom's Cause\n\nG. A. Henty\n\n\n\n\nCONTENTS\n\n       I  Glen Cairn\n      II  Leaving Home\n     III  Sir William Wallace\n      IV  The Capture of Lanark\n       V  A Treacherous Plot\n      VI  The Barns of Ayr\n     VII  The Cave in the Pentlands\n    VIII  The Council at Stirling\n      IX  The Battle of Stirling Bridge\n       X  The Battle of Falkirk\n      XI  Robert The Bruce\n     XII  The Battle of Methven\n    XIII  The Castle of Dunstaffnage\n     XIV  Colonsay\n      XV  A Mission to Ireland\n     XVI  An Irish Rising\n    XVII  The King's Blood Hound\n   XVIII  The Hound Restored\n     XIX  The Convent of St. Kenneth\n      XX  The Heiress of the Kerrs\n     XXI  The Siege of Aberfilly\n    XXII  A Prisoner\n   XXIII  The Escape from Berwick\n    XXIV  The Progress of the War\n     XXV  The Capture of a Stronghold\n    XXVI  Edinburgh\n   XXVII  Bannockburn\n\n\n\n\nPREFACE.\n\n\nMY DEAR LADS,\n\nThere are few figures in history who have individually exercised\nso great an influence upon events as William Wallace and Robert\nBruce. It was to the extraordinary personal courage, indomitable\nperseverance, and immense energy of these two men that Scotland\nowed her freedom from English domination. So surprising were the\ntraditions of these feats performed by these heroes that it was at\none time the fashion to treat them as belonging as purely to legend\nas the feats of St. George or King Arthur.  Careful investigation,\nhowever, has shown that so far from this being the case, almost\nevery deed reported to have been performed by them is verified by\ncontemporary historians. Sir William Wallace had the especial bad\nfortune of having come down to us principally by the writings of\nhis bitter enemies, and even modern historians, who should have\ntaken a fairer view of his life, repeated the cry of the old English\nwriters that he was a bloodthirsty robber. Mr. W. Burns, however,\nin his masterly and exhaustive work, The Scottish War of Independence,\nhas torn these calumnies to shreds, and has displayed Wallace as\nhe was, a high minded and noble patriot. While consulting other\nwriters, especially those who wrote at the time of or but shortly\nafter the events they record, I have for the most part followed\nBurns in all the historical portions of the narrative. Throughout\nthe story, therefore, wherein it at all relates to Wallace, Bruce,\nand the other historical characters, the circumstances and events\ncan be relied upon as strictly accurate, save only in the earlier\nevents of the career of Wallace, of which the details that have\ncome down to us are somewhat conflicting, although the main features\nare now settled past question.\n\nYours sincerely,\n  G.A. HENTY.\n\n\n\n\nChapter I\n\nGlen Cairn\n\n\nThe village of Glen Cairn was situated in a valley in the broken\ncountry lying to the west of the Pentland Hills, some fifteen miles\nnorth of the town of Lanark, and the country around it was wild\nand picturesque.  The villagers for the most part knew little of\nthe world beyond their own valley, although a few had occasionally\npaid visits to Glasgow, which lay as far to the west as Lanark was\ndistant to the south. On a spur jutting out from the side of the\nhill stood Glen Cairn Castle, whose master the villagers had for\ngenerations regarded as their lord.\n\nThe glory of the little fortalice had now departed. Sir William\nForbes had been killed on his own hearthstone, and the castle had\nbeen sacked in a raid by the Kerrs, whose hold lay to the southwest,\nand who had long been at feud with the Forbeses. The royal power\nwas feeble, and the Kerrs had many friends, and were accordingly\ngranted the lands they had seized; only it was specified that Dame\nForbes, the widow of Sir William, should be allowed to reside in\nthe fortalice free from all let or hindrance, so long as she meddled\nnot, nor sought to stir up enmity among the late vassals of her\nlord against their new masters.\n\nThe castle, although a small one, was strongly situated.  The spur\nof the hill ran some 200 yards into the valley, rising sharply\nsome 30 or 40 feet above it. The little river which meandered down\nthe valley swept completely round the foot of the spur, forming a\nnatural moat to it, and had in some time past been dammed back, so\nthat, whereas in other parts it ran brightly over a pebbly bottom,\nhere it was deep and still. The fortalice itself stood at the\nextremity of the spur, and a strong wall with a fortified gateway\nextended across the other end of the neck, touching the water on\nboth sides.  From the gateway extended two walls inclosing a road\nstraight to the gateway of the hold itself, and between these walls\nand the water every level foot of ground was cultivated; this garden\nwas now the sole remains of the lands of the Forbeses.\n\nIt was a narrow patrimony for Archie, the only son of Dame Forbes,\nand his lady mother had hard work to keep up a respectable state,\nand to make ends meet. Sandy Grahame, who had fought under her\nhusband's banner and was now her sole retainer, made the most of the\ngarden patches. Here he grew vegetables on the best bits of ground\nand oats on the remainder; these, crushed between flat stones,\nfurnished a coarse bread.  From the stream an abundance of fish could\nalways be obtained, and the traps and nets therefore furnished a\nmeal when all else failed. In the stream, too, swam a score and more\nof ducks, while as many chickens walked about the castle yard, or\nscratched for insects among the vegetables. A dozen goats browsed\non the hillside, for this was common ground to the village, and\nDame Forbes had not therefore to ask for leave from her enemies,\nthe Kerrs. The goats furnished milk and cheese, which was deftly\nmade by Elspie, Sandy's wife, who did all the work indoors, as her\nhusband did without. Meat they seldom touched.  Occasionally the\nresources of the hold were eked out by the present of a little\nhill sheep, or a joint of prime meat, from one or other of her old\nvassals, for these, in spite of the mastership of the Kerrs, still\nat heart regarded Dame Mary Forbes as their lawful mistress, and\nher son Archie as their future chief.  Dame Mary Forbes was careful\nin no way to encourage this feeling, for she feared above all things\nto draw the attention of the Kerrs to her son. She was sure that\ndid Sir John Kerr entertain but a suspicion that trouble might ever\ncome from the rivalry of this boy, he would not hesitate a moment\nin encompassing his death; for Sir John was a rough and violent\nman who was known to hesitate at nothing which might lead to his\naggrandizement. Therefore she seldom moved beyond the outer wall\nof the hold, except to go down to visit the sick in the village.\nShe herself had been a Seaton, and had been educated at the nunnery\nof Dunfermline, and she now taught Archie to read and write,\naccomplishments by no means common even among the better class in\nthose days.  Archie loved not books; but as it pleased his mother,\nand time often hung heavy on his hands, he did not mind devoting\ntwo or three hours a day to the tasks she set him. At other times\nhe fished in the stream, wandered over the hills, and brought in\nthe herbs from which Dame Forbes distilled the potions which she\ndistributed to the villagers when sick.\n\nOften he joined the lads of the village in their games.  They\nall regarded him as their leader; but his mother had pressed upon\nhim over and over again that on no account was he to assume any\nsuperiority over the others, but to treat them strictly as equals.\nDoubtless the Kerrs would from time to time have news of what was\ndoing in Glen Cairn; and while they would be content to see him\njoining in the sports of the village lads, with seemingly no wish\nbeyond that station, they would at once resent it did they see\nany sign on his part of his regarding himself as a chief among the\nothers.\n\nNo inconsiderable portion of Archie's time was occupied in acquiring\nthe use of arms from Sandy Grahame. His mother, quiet and seemingly\nresigned as she was, yet burned with the ambition that he should\nsome day avenge his father's death, and win back his father's lands.\nShe said little to him of her hopes; but she roused his spirit by\ntelling him stories of the brave deeds of the Forbeses and Seatons,\nand she encouraged him from his childhood to practise in arms with\nSandy Grahame.\n\nIn this respect, indeed, Archie needed no stimulant.  From Sandy\neven more than from his mother he had heard of his brave father's\ndeeds in arms; and although, from the way in which she repressed any\nsuch utterances, he said but little to his mother, he was resolved\nas much as she could wish him to be, that he would some day win\nback his patrimony, and avenge his father upon his slayers.\n\nConsequently, upon every opportunity when Sandy Grahame could spare\ntime from his multifarious work, Archie practised with him, with\nsword and pike. At first he had but a wooden sword. Then, as his\nlimbs grew stronger, he practised with a blunted sword; and now\nat the age of fifteen Sandy Grahame had as much as he could do to\nhold his own with his pupil.\n\nAt the time the story opens, in the springtime of the year 1293,\nhe was playing at ball with some of the village lads on the green,\nwhen a party of horsemen was seen approaching.\n\nAt their head rode two men perhaps forty years old, while a lad of\nsome eighteen years of age rode beside them.  In one of the elder\nmen Archie recognized Sir John Kerr.  The lad beside him was his\nson Allan. The other leader was Sir John Hazelrig, governor of\nLanark; behind them rode a troop of armed men, twenty in number.\nSome of the lads would have ceased from their play; but Archie\nexclaimed:\n\n\"Heed them not; make as if you did not notice them.  You need not\nbe in such a hurry to vail your bonnets to the Kerr.\"\n\n\"Look at the young dogs,\" Sir John Kerr said to his companion.\n\"They know that their chief is passing, and yet they pretend that\nthey see us not.\"\n\n\"It would do them good,\" his son exclaimed, \"did you give your\ntroopers orders to tie them all up and give them a taste of their\nstirrup leathers.\"\n\n\"It would not be worth while, Allan,\" his father said.  \"They will\nall make stout men-at-arms some day, and will have to fight under\nmy banner. I care as little as any man what my vassals think of\nme, seeing that whatsoever they think they have to do mine orders.\nBut it needs not to set them against one needlessly; so let the\nvarlets go on with their play undisturbed.\"\n\nThat evening Archie said to his mother, \"How is it, mother, that\nthe English knight whom I today saw ride past with the Kerr is\ngovernor of our Scottish town of Lanark?\"\n\n\"You may well wonder, Archie, for there are many in Scotland\nof older years than you who marvel that Scotsmen, who have always\nbeen free, should tolerate so strange a thing.  It is a long story,\nand a tangled one; but tomorrow morning I will draw out for you\na genealogy of the various claimants to the Scottish throne, and\nyou will see how the thing has come about, and under what pretence\nEdward of England has planted his garrisons in this free Scotland\nof ours.\"\n\nThe next morning Archie did not forget to remind his mother of her\npromise.\n\n\"You must know,\" she began, \"that our good King Alexander had three\nchildren--David, who died when a boy; Alexander, who married a\ndaughter of the Count of Flanders, and died childless; and a daughter,\nMargaret, who married Eric, the young King of Norway. Three years\nago the Queen of Norway died, leaving an only daughter, also named\nMargaret, who was called among us the 'Maid of Norway,' and who,\nat her mother's death, became heir presumptive to the throne, and\nas such was recognized by an assembly of the estates at Scone. But\nwe all hoped that the king would have male heirs, for early last\nyear, while still in the prime of life, he married Joleta, daughter\nof the Count of Drew. Unhappily, on the 19th of March, he attended\na council in the castle of Edinburgh, and on his way back to his\nwife at Kinghorn, on a stormy night, he fell over a precipice and\nwas killed.\n\n\"The hopes of the country now rested on the 'Maid of Norway,' who\nalone stood between the throne and a number of claimants, most of\nwhom would be prepared to support their claims by arms, and thus\nbring unnumbered woes upon Scotland. Most unhappily for the country,\nthe maid died on her voyage to Scotland, and the succession therefore\nbecame open.\n\n\"You will see on this chart, which I have drawn out, the lines by\nwhich the principal competitors--for there were nigh upon a score\nof them--claimed the throne.\n\n\"Before the death of the maid, King Edward had proposed a marriage\nbetween her and his young son, and his ambassadors met the Scottish\ncommissioners at Brigham, near Kelso, and on the 18th of July, 1290,\nthe treaty was concluded. It contained, besides the provisions of\nthe marriage, clauses for the personal freedom of Margaret should\nshe survive her husband; for the reversion of the crown failing\nher issue; for protection of the rights, laws, and liberties of\nScotland; the freedom of the church; the privileges of crown vassals;\nthe independence of the courts; the preservation of all charters\nand natural muniments; and the holding of parliaments only within\nScotland; and specially provided that no vassal should be compelled\nto go forth of Scotland for the purpose of performing homage or\nfealty; and that no native of Scotland should for any cause whatever\nbe compelled to answer, for any breach of covenant or from crime\ncommitted, out of the kingdom.\n\n\"Thus you see, my boy, that King Edward at this time fully recognized\nthe perfect independence of Scotland, and raised no claim to any\nsuzerainty over it. Indeed, by Article I it was stipulated that\nthe rights, laws, liberties, and customs of Scotland should remain\nfor ever entire and inviolable throughout the whole realm and its\nmarches; and by Article V that the Kingdom of Scotland shall remain\nseparate and divided from England, free in itself, and without\nsubjection, according to its right boundaries and marches, as\nheretofore.\n\n\"King Edward, however, artfully inserted a salvo, 'saving the rights\nof the King of England and of all others which before the date of\nthis treaty belong to him or any of them in the marches or elsewhere.'\nThe Scottish lords raised no objection to the insertion of this\nsalvo, seeing that it was of general purport, and that Edward\npossessed no rights in Scotland, nor had any ever been asserted\nby his predecessors--Scotland being a kingdom in itself equal to\nits neighbour--and that neither William the Norman nor any of his\nsuccessors attempted to set forward any claims to authority beyond\nthe Border.\n\n\"No sooner was the treaty signed than Edward, without warrant\nor excuse, appointed Anthony Beck, the warlike Bishop of Durham,\nLieutenant of Scotland, in the name of the yet unmarried pair; and\nfinding that this was not resented, he demanded that all the places\nof strength in the kingdom should be delivered to him. This demand\nwas not, however, complied with, and the matter was still pending\nwhen the Maid of Norway died. The three principal competitors--Bruce,\nBaliol, and Comyn--and their friends, at once began to arm; but\nWilliam Fraser, Bishop of St. Andrews, a friend of Baliol, wrote to\nKing Edward suggesting that he should act as arbitrator, and more\nthan hinting that if he chose Baliol he would find him submissive\nin all things to his wishes.  Edward jumped at the proposal, and\nthereupon issued summonses to the barons of the northern counties\nto meet him at Norham on the 3d of June; and a mandate was issued\nto the sheriffs of Northumberland, Cumberland, Westmoreland, York,\nand Lancaster, to assemble the feudal array at the same rendezvous.\n\n\"Now, you know, my son, that, owing to the marriages between royal\nfamilies of England and Scotland, there has been a close connection\nbetween the countries. Many Scotch barons have married English\nheiresses, and hold lands in both countries, while Scottish maidens\nhave married English knights. Thus it happens that a great number\nof the Scotch nobility are as much Englishmen as Scotchmen, and are\nvassals to England for lands held there. Four of the competitors,\nJohn Baliol, Robert Bruce, John Comyn, and William Ross, are all\nbarons of England as well as of Scotland, and their lands lying\nin the north they were, of course, included in the invitation. In\nMay, Edward issued an invitation to the Bishops of St. Andrews,\nGlasgow, and other Scotch nobles to come to Norham, remain there,\nand return, specially saying that their presence there was not to\nbe regarded as a custom through which the laws of Scotland might\nin any future time be prejudiced. Hither then came the whole power\nof the north of England, and many of the Scotch nobles.\n\n\"When the court opened, Roger Brabazon, the king's justiciary,\ndelivered an address, in which he stated that Edward, as lord\nparamount of Scotland, had come there to administer justice between\nthe competitors for the crown, and concluded with the request that\nall present should acknowledge his claim as lord paramount.  The\nScottish nobles present, with the exception of those who were\nprivy to Edward's designs, were filled with astonishment and dismay\nat this pretension, and declared their ignorance of any claim of\nsuperiority of the King of England over Scotland. The king, in a\npassion, exclaimed:\n\n\"'By holy Edward, whose crown I wear, I will vindicate my just\nrights, or perish in the attempt.'\n\n\"However, he saw that nothing could be done on the instant, and\nadjourned the meeting for three weeks, at the end of which time the\nprelates, nobles, and community of Scotland were invited to bring\nforward whatever they could in opposition to his claim to supremacy.\n\n\"At the time fixed the Scotch nobles again met, but this time on\nthe Scottish side of the Border, for Edward had gathered together\nthe whole of the force of the northern counties.\n\n\"Besides the four claimants, whose names I have told you, were Sir\nJohn Hastings, Patrick Dunbar, Earl of March, William de Vesci,\nRobert de Pinkeny, Nicholas de Soulis, Patrick Galythly, Roger de\nMandeville, Florence, Count of Holland, and Eric, King of Norway.\nWith the exception of Eric, the Count of Holland, Dunbar, and\nGalythly, all of these were of Norman extraction, and held possessions\nin England.  When the meeting was opened the prelates and nobles\npresent advanced nothing to disprove Edward's claim to supremacy.\nThe representatives of the commons, however, did show reason against\nthe claim, for which, indeed, my son, as every man in Scotland\nknows, there was not a shadow of foundation.\n\n\"The king's chancellor declared that there was nothing in these\nobjections to Edward's claim, and therefore he resolved, as lord\nparamount, to determine the question of succession. The various\ncompetitors were asked whether they acknowledged Edward as lord\nparamount, and were willing to receive his judgment as such; and\nthe whole of these wretched traitors proceeded to barter their\ncountry for their hopes of a crown, acknowledged Edward as lord\nparamount, and left the judgment in his hands.\n\n\"Bruce and Baliol received handsome presents for thus tamely\nyielding the rights of Scotland. All present at once agreed that\nthe castles and strongholds of Scotland should be surrendered into\nthe hands of English commanders and garrisons. This was immediately\ndone; and thus it is, Archie, that you see an English officer\nlording it over the Scotch town of Lanark.\n\n\"Then every Scotchman was called upon to do homage to the English\nking as his lord paramount, and all who refused to do so were\nseized and arrested. Finally, on the 17th of November last, 1292--the\ndate will long be remembered in Scotland--Edward's judgment\nwas given at Berwick, and by it John Baliol was declared King of\nScotland.\n\n\"Thus for eighteen months Scotland was kept in doubt; and this was\ndone, no doubt, to enable the English to rivet their yoke upon our\nshoulders, and to intimidate and coerce all who might oppose it.\"\n\n\"There were some that did oppose it, mother, were there not?--some\ntrue Scotchmen who refused to own the supremacy of the King of\nEngland?\"\n\n\"Very few, Archie. One Sir Malcolm Wallace, a knight of but small\nestate, refused to do so, and was, together with his eldest son,\nslain in an encounter with an English detachment under a leader\nnamed Fenwick at Loudon Hill.\"\n\n\"And was he the father of that William Wallace of whom the talk was\nlately that he had slain young Selbye, son of the English governor\nof Dundee?\"\n\n\"The same, Archie.\"\n\n\"Men say, mother, that although but eighteen years of age he is of\ngreat stature and strength, of very handsome presence, and courteous\nand gentle; and that he was going quietly through the streets when\ninsulted by young Selbye, and that he and his companions being set\nupon by the English soldiers, slew several and made their escape.\"\n\n\"So they say, Archie. He appears from all description of him\nto be a remarkable young man, and I trust that he will escape the\nvengeance of the English, and that some day he may again strike\nsome blows for our poor Scotland, which, though nominally under\nthe rule of Baliol, is now but a province of England.\"\n\n\"But surely, mother, Scotchmen will never remain in such a state\nof shameful servitude!\"\n\n\"I trust not, my son; but I fear that it will be long before we\nshake off the English yoke. Our nobles are for the most part of\nNorman blood; very many are barons of England; and so great are the\njealousies among them that no general effort against England will\nbe possible. No, if Scotland is ever to be freed, it will be by\na mighty rising of the common people, and even then the struggle\nbetween the commons of Scotland and the whole force of England aided\nby the feudal power of all the great Scotch nobles, would be well\nnigh hopeless.\"\n\nThis conversation sank deeply into Archie's mind; day and night\nhe thought of nothing but the lost freedom of Scotland, and vowed\nthat even the hope of regaining his father's lands should be\nsecondary to that of freeing his country. All sorts of wild dreams\ndid the boy turn over in his mind; he was no longer gay and light\nhearted, but walked about moody and thoughtful. He redoubled his\nassiduity in the practice of arms; and sometimes when fighting with\nSandy, he would think that he had an English man-at-arms before him,\nand would strike so hotly and fiercely that Sandy had the greatest\ndifficulty in parrying his blows, and was forced to shout lustily\nto recall him from the clouds. He no longer played at ball with the\nvillage lads; but, taking the elder of them aside, he swore them\nto secrecy, and then formed them into a band, which he called the\nScottish Avengers. With them he would retire into valleys far away\nfrom the village, where none would mark what they were doing, and\nthere they practised with club and stake instead of broadsword\nand pike, defended narrow passes against an imaginary enemy, and,\ndivided into two parties, did battle with each other.\n\nThe lads entered into the new diversion with spirit.  Among the\nlower class throughout Scotland the feeling of indignation at the\nmanner in which their nobles had sold their country to England was\ndeep and passionate. They knew the woes which English domination\nhad brought upon Wales and Ireland; and though as yet without a\nleader, and at present hopeless of a successful rising, every true\nScotchman was looking forward to the time when an attempt might be\nmade to throw off the English yoke.\n\nTherefore the lads of Glen Cairn entered heart and soul into\nthe projects of their \"young chief,\" for so they regarded Archie,\nand strove their best to acquire some of the knowledge of the use\nof sword and pike which he possessed. The younger lads were not\npermitted to know what was going on--none younger than Archie\nhimself being admitted into the band, while some of the elders were\nyouths approaching man's estate. Even to his mother Archie did not\nbreathe a word of what he was doing, for he feared that she might\nforbid his proceedings. The good lady was often surprised at the\ncuts and bruises with which he returned home; but he always turned\noff her questions by muttering something about rough play or a\nheavy fall, and so for some months the existence of the Scottish\nAvengers remained unsuspected.\n\n\n\nChapter II\n\nLeaving Home\n\n\nOne day when \"the Avengers\" were engaged in mimic battle in a glen\nsome two miles from the village they were startled with a loud\nshout of \"How now, what is this uproar?\" Bows were lowered and\nhedge stakes dropped; on the hillside stood Red Roy, the henchman\nof Sir John Kerr, with another of the retainers. They had been\ncrossing the hills, and had been attracted by the sound of shouting.\nAll the lads were aware of the necessity for Archie's avoiding the\nnotice of the Kerrs, and Andrew Macpherson, one of the eldest of\nthe lads, at once stepped forward:  \"We are playing,\" he said, \"at\nfighting Picts against Scots.\"\n\nThis was the case, for the English were so hated that Archie had\nfound that none would even in sport take that name, and the sides\nwere accordingly dubbed Scots and Picts, the latter title not being\nso repugnant, and the companies changing sides each day.\n\n\"It looks as if you were fighting in earnest,\" Roy said grimly,\n\"for the blood is streaming down your face.\"\n\n\"Oh, we don't mind a hard knock now and again,\" Andrew said\ncarelessly. \"I suppose, one of these days, we shall have to go out\nunder Sir John's banner, and the more hard knocks we have now, the\nless we shall care for them then.\"\n\n\"That is so,\" Roy said; \"and some of you will soon be able to handle\narms in earnest. Who are your leaders?\" he asked sharply, as his\neye fixed on Archie, who had seated himself carelessly upon a rock\nat some little distance.\n\n\"William Orr generally heads one side, and I the other.\"\n\n\"And what does that young Forbes do?\" Red Roy asked.\n\n\"Well, he generally looks on,\" Andrew replied in a confidential\ntone; \"he is not much good with the bow, and his lady mother does\nnot like it if he goes home with a crack across the face, and I\ndon't think he likes it himself; he is but a poor creature when it\ncomes to a tussle.\"\n\n\"And it is well for him that he is,\" Red Roy muttered to himself;\n\"for if he had been likely to turn out a lad of spirit, Sir John\nwould have said the word to me before now; but, seeing what he is,\nhe may as well be left alone for the present.  He will never cause\ntrouble.\" So saying, Red Roy strolled away with his companion, and\nleft the lads to continue their mimic fight.\n\nNews travelled slowly to Glen Cairn; indeed, it was only when\na travelling chapman or pedlar passed through, or when one of the\nvillagers went over to Lanark or Glasgow, carrying the fowls and\nother produce of the community to market, that the news came from\nwithout.\n\nBaliol was not long before he discovered that his monarchy was but\na nominal one. The first quarrel which arose between him and his\nimperious master was concerning the action of the courts. King Edward\ndirected that there should be an appeal to the courts at Westminster\nfrom all judgments in the Scottish courts. Baliol protested that it\nwas specifically agreed by the Treaty of Brigham that no Scotchman\nwas liable to be called upon to plead outside the kingdom; but\nEdward openly declared, \"Notwithstanding any concessions made before\nBaliol became king, he considered himself at liberty to judge in\nany case brought before him from Scotland, and would, if necessary,\nsummon the King of Scots himself to appear in his presence.\" He\nthen compelled Baliol formally to renounce and cancel not only the\nTreaty of Brigham, but every stipulation of the kind \"known to\nexist, or which might be thereafter discovered.\" Another appeal\nfollowed, and Baliol was cited to appear personally, but refused;\nhe was thereupon declared contumacious by the English parliament,\nand a resolution was passed that three of the principal towns of\nScotland should be \"seized,\" until he gave satisfaction. All this\nwas a manifest usurpation, even allowing Edward's claims to supremacy\nto be well founded.\n\nAt this moment Edward became involved in a quarrel with his own\nlord superior Phillip, king of France, by whom he was in turned\nsummoned to appear under the pain of contumacy. Edward met this\ndemand by a renunciation of allegiance to Phillip and a declaration\nof war, and called upon Baliol for aid as his vassal; but Baliol\nwas also a vassal of the French king, and had estates in France\nliable to seizure. He therefore hesitated. Edward further ordered\nhim to lay an embargo upon all vessels in the ports of Scotland,\nand required the attendance of many of the Scottish barons in his\nexpedition to France. Finding his orders disobeyed, on the 16th\nof October Edward issued a writ to the sheriff of Northampton,\n\"to seize all lands, goods, and chattels of John Baliol and other\nScots.\"\n\nThe Scotch held a parliament at Scone. All Englishmen holding office\nwere summarily dismissed. A committee of the estates was appointed\nto act as guardian of the kingdom, and Baliol himself was deprived\nof all active power; but an instrument was prepared in his name,\nreciting the injuries that he and his subjects had sustained at the\nhands of the English king, and renouncing all further allegiance.\nFollowing this up, a league was concluded, offensive and defensive,\nbetween the French king and Scotland, represented by the prelates,\nnobles, and community. Edward Baliol, the king's son, was contracted\nto marry the French king's niece. Phillip bound himself to assist\nScotland against any invasion of England, and the Scotch agreed to\ncross the Border in case Edward invaded France.\n\nIn making this alliance the Scots took the only step possible; for\nthey had no choice between fighting England with France as their\nally, or fighting France as the subjects of King Edward. The contest\nwhich was approaching seemed all but hopeless. The population\nof England was six times as large as that of Scotland, and Edward\ncould draw from Ireland and Wales great numbers of troops. The\nEnglish were trained to war by constant fighting in France, Ireland,\nand Wales; while the Scots had, for a very long period, enjoyed\na profound peace, and were for the most part wholly ignorant of\nwarfare.\n\nEdward at once prepared to invade Scotland; in January he seized\nthe lands owned by Comyn in Northumberland and sold them, directing\nthe money to be applied to the raising and maintenance of 1000\nmen-at-arms and 60,000 foot soldiers, and in February issued a writ\nfor the preparation of a fleet of 100 vessels.\n\nOn the 25th of March he crossed the Tweed with 5000 horse and 30,000\nfoot. The Scotch leaders were, of course, aware of the gathering\nstorm, and, collecting their forces, attempted a diversion by\ncrossing the Border to the west and making a raid into Cumberland.\nKing Edward, however, marched north and besieged Berwick, the richest\nand most flourishing of the towns of Scotland. With the exception\nof the castle, it was weakly fortified. The attack was commenced\nby the fleet, who were, however, repulsed and driven off. A land\nassault, led by the king in person, was then made; the walls were\ncaptured, and the town completely sacked. The inhabitants were\nbutchered without distinction of age, sex, or condition, and even\nthose who fled to the churches were slain within the sanctuary.\nContemporary accounts differ as to the numbers who perished on this\noccasion. Langtoff says 4000; Hemingford, 8000; Knighton, another\nEnglish writer, says 17,000; and Matthew of Westminster, 60,000.\nWhichever of these writers is correct, it is certain that almost\nthe whole of the men, women, and children of the largest and most\npopulous Scottish town were butchered by the orders of the English\nking, who issued direct orders that none should be spared.  From\nthis terrible visitation Berwick, which was before called the\nAlexandria of the West, never recovered.  The castle, which was held\nby Sir William Douglas, surrendered immediately; and Sir William,\nhaving sworn fealty to the English king, was permitted to depart.\n\nThe English army now marched north. Patrick, Earl of Dunbar,\nwas with King Edward; but his wife, a noble and patriotic woman,\nsurrendered the castle to the Scots. The Earl of Surrey, with\na powerful army, sat down before it. The Scotch nobles and people\nmarched in great numbers, but with little order and discipline,\nto raise the siege. They were met by Surrey, whose force, inured\nto arms, easily routed the Scotch gathering, no fewer than 10,000\nbeing killed in the conflict and retreat. The English army was\njoined by 15,000 Welsh and 30,000 from Ireland, and marched through\nScotland, the castles and towns opening their gates to Edward as\nhe came, and the nobles, headed by James the Stewart, coming in and\ndoing homage to him. Baliol was forced to appear in the churchyard\nof Strath-Cathro, near Montrose, arrayed in regal robes, and to\nresign his kingdom to the Bishop of Durham as Edward's representative,\nand to repeat the act a few days afterwards at Brechin in presence\nof the king himself. He was then, with his son, sent a prisoner to\nLondon, where they were confined in the Tower for several years.\nFrom Brechin Edward marched through the whole of Scotland, visiting\nall the principal towns. He had now dropped the title of Lord\nParamount of Scotland, the country being considered as virtually\npart of England. Garrisons were placed in every stronghold in the\ncountry, and many new castles were raised to dominate the people.\nThe public documents were all carried away to England, the great\nseal broken in pieces, and the stone of Scone--upon which, for\nfive hundred years, every Scotch monarch had been crowned--was\ncarried away to Westminster, where it has ever since formed the\nseat of the thrones upon which English monarchs have been crowned.\n\nThe tide of war had not passed near Glen Cairn; but the excitement,\nas from time to time the news came of stirring events, was very\ngreat. The tidings of the massacre of Berwick filled all with\nconsternation and grief. Some of the men quitted their homes and\nfought at Dunbar, and fully half of these never returned; but great\nas was the humiliation and grief at the reverses which had befallen\nthe Scotch arms, the feeling was even deeper and more bitter at the\nreadiness with which the whole of the Scotch nobles flocked in to\nmake their peace with King Edward.\n\nIt seemed so incredible that Scotland, which had so long successfully\nresisted all invaders, should now tamely yield without a struggle,\nthat the people could scarce believe it possible that their boasted\nfreedom was gone, that the kingdom of Scotland was no more, and the\ncountry become a mere portion of England.  Thus, while the nobles\nwith their Norman blood and connections accepted the new state of\nthings contentedly enough, well satisfied to have retained rank and\nland, a deep and sullen discontent reigned among the people; they\nhad been betrayed rather than conquered, and were determined that\nsome day there should be an uprising, and that Scotland would make\na great effort yet for freedom.  But for this a leader was needed,\nand until such a one appeared the people rested quiet and bided\ntheir time.\n\nFrom time to time there came to Glen Cairn tales of the doings of\nthat William Wallace who had, when the English first garrisoned the\nScottish castles, while Edward was choosing between the competitors\nfor her throne, killed young Selbye at Dundee, and had been outlawed\nfor the deed. After that he went and resided with his uncle, Sir\nRonald Crawford, and then with another uncle, Sir Richard Wallace\nof Riccarton. Here he gathered a party of young men, eager spirits\nlike himself, and swore perpetual hostility to the English.\n\nOne day Wallace was fishing in the Irvine when Earl Percy, the\ngovernor of Ayr, rode past with a numerous train.  Five of them\nremained behind and asked Wallace for the fish he had taken. He\nreplied that they were welcome to half of them. Not satisfied with\nthis, they seized the basket and prepared to carry it off. Wallace\nresisted, and one of them drew his sword. Wallace seized the staff\nof his net and struck his opponent's sword from his hand; this he\nsnatched up and stood on guard, while the other four rushed upon\nhim.  Wallace smote the first so terrible a blow that his head was\ncloven from skull to collarbone; with the next blow he severed the\nright arm of another, and then disabled a third.  The other two\nfled, and overtaking the earl, called on him for help; \"for,\" they\nsaid, \"three of our number who stayed behind with us to take some\nfish from the Scot who was fishing are killed or disabled.\"\n\n\"How many were your assailants?\" asked the earl.\n\n\"But the man himself,\" they answered; \"a desperate fellow whom we\ncould not withstand.\"\n\n\"I have a brave company of followers!\" the earl said with scorn.\n\"You allow one Scot to overmatch five of you! I shall not return\nto seek for your adversary; for were I to find him I should respect\nhim too much to do him harm.\"\n\nFearing that after this adventure he could no longer remain in\nsafety with his uncle, Wallace left him and took up his abode in\nLag Lane Wood, where his friends joining him, they lived a wild\nlife together, hunting game and making many expeditions through the\ncountry. On one occasion he entered Ayr in disguise; in the middle\nof a crowd he saw some English soldiers, who were boasting that they\nwere superior to the Scots in strength and feats of arms. One of\nthem, a strong fellow, was declaring that he could lift a greater\nweight than any two Scots. He carried a pole, with which he offered,\nfor a groat, to let any Scotchman strike him on the back as hard\nas he pleased, saying that no Scotchman could strike hard enough\nto hurt him.\n\nWallace offered him three groats for a blow. The soldier eagerly\naccepted the money, and Wallace struck him so mighty a blow that\nhis back was broken and he fell dead on the ground. His comrades\ndrew their swords and rushed at Wallace, who slew two with the\npole, and when it broke drew the long sword which was hidden in\nhis garments, and cut his way through them.\n\nOn another occasion he again had a fracas with the English in Ayr,\nand after killing many was taken prisoner.  Earl Percy was away,\nand his lieutenant did not venture to execute him until his return.\nA messenger was sent to the Earl, but returned with strict orders\nthat nothing should be done to the prisoner until he came back.\nThe bad diet and foul air of the dungeon suited him so ill, after\nhis free life in the woods, that he fell ill, and was reduced\nto so weak a state that he lay like one dead--the jailer indeed\nthought that he was so, and he was carried out to be cast into the\nprison burial ground, when a woman, who had been his nurse, begged\nhis body. She had it carried to her house, and then discovered that\nlife yet remained, and by great care and good nursing succeeded\nin restoring him. In order to prevent suspicion that he was still\nalive a fictitious funeral was performed.  On recovering, Wallace\nhad other frays with the English, all of which greatly increased\nhis reputation throughout that part of the country, so that more\nadherents came to him, and his band began to be formidable. He\ngradually introduced an organization among those who were found to\nbe friendly to the cause, and by bugle notes taken up and repeated\nfrom spot to spot orders could be despatched over a wide extent of\ncountry, by which the members of his band knew whether to assemble\nor disperse, to prepare to attack an enemy, or to retire to their\nfastnesses.\n\nThe first enterprise of real importance performed by the band was\nan attack by Wallace and fifty of his associates on a party of\nsoldiers, 200 strong, conveying provisions from Carlisle to the\ngarrison of Ayr.  They were under the command of John Fenwick,\nthe same officer who had been at the head of the troop by which\nWallace's father had been killed.  Fenwick left twenty of his men\nto defend the wagons, and with the rest rode forward against the\nScots. A stone wall checked their progress, and the Scotch, taking\nadvantage of the momentary confusion, made a furious charge upon\nthem with their spears, cutting their way into the midst of them\nand making a great slaughter of men and horses. The English rode\nround and round them, but the Scots, defending themselves with spear\nand sword, stood so staunchly together that the English could not\nbreak through.\n\nThe battle was long and desperate, but Wallace killed Fenwick with\nhis own hand, and after losing nigh a hundred of their number the\nEnglish fled in confusion. The whole convoy fell into the hands of\nthe victors, who became possessed of several wagons, 200 carriage\nhorses, flour, wine, and other stores in great abundance; with\nthese they retired into the forest of Clydesdale.\n\nThe fame of this exploit greatly increased the number of Wallace's\nfollowers. So formidable did the gathering become that convoys by\nland to Ayr were entirely interrupted, and Earl Percy held a council\nof the nobility at Glasgow, and consulted them as to what had best\nbe done. Finally, Sir Ronald Crawford was summoned and told that\nunless he induced his nephew to desist from hostilities they should\nhold him responsible and waste his lands. Sir Ronald visited the\nband in Clydesdale forest, and rather than harm should come upon\nhim, Wallace and his friends agreed to a truce for two months. Their\nplunder was stowed away in places of safety, and a portion of the\nband being left to guard it the rest dispersed to their homes.\n\nWallace returned to his uncle's, but was unable long to remain\ninactive, and taking fifteen followers he went with them in disguise\nto Ayr.  Wallace, as usual, was not long before he got into a\nquarrel. An English fencing master, armed with sword and buckler,\nwas in an open place in the city, challenging any one to encounter\nhim. Several Scots tried their fortune and were defeated, and then\nseeing Wallace towering above the crowd he challenged him. Wallace\nat once accepted, and after guarding himself for some time, with\na mighty sweep of his sword cleft through buckler, arm, headpiece,\nand skull.  The English soldiers around at once attacked him; his\nfriends rallied round him, and after hard fighting they made their\nway to the spot where they had left their horses and rode to Lag\nLane Wood.\n\nWhen Earl Percy heard that Wallace had been the leader in this\nfray, and found on inquiry that he had slain the sword player in\nfair fight after having been challenged by him, he refused to regard\nhim as having broken the truce, for he said the soldiers had done\nwrong in attacking him. Earl Percy was himself a most gallant\nsoldier, and the extraordinary personal prowess of Wallace excited\nin him the warmest admiration, and he would fain, if it had been\npossible, have attached him to the service of England.\n\nAs soon as the truce was over Wallace again attacked the English.\nFor a time he abode with the Earl of Lennox, who was one of the\nfew who had refused to take the oath of allegiance, and having\nrecruited his force, he stormed the stronghold called the Peel\nof Gargunnock, near Stirling. Then he entered Perth, leaving his\nfollowers in Methven Wood, and hearing that an English reinforcement\nwas upon the march, formed an ambush, fell upon them, and defeated\nthem; and pressing hotly upon them entered so close on their heels\ninto Kincleven Castle, that the garrison had no time to close the\ngate, and the place was captured. Great stores and booty were found\nhere; these were carried to the woods, and the castle was burned\nto the ground, as that of Gargunnock had been, as Wallace's force\nwas too small to enable him to hold these strongholds. Indignant\nat this enterprise so close to their walls the English moved out\nthe whole garrison, 1000 strong, against Wallace, who had with him\nbut fifty men in all. After a desperate defence, in which Sir John\nButler and Sir William de Loraine, the two officers in command,\nwere killed by Wallace himself, the latter succeeded in drawing off\nhis men; 120 of the English were killed in the struggle, of whom\nmore than twenty are said to have fallen at the hands of Wallace\nalone. Many other similar deeds did Wallace perform; his fame grew\nmore and more, as did the feeling among the Scotch peasantry that\nin him they had found their champion and leader.\n\nArchie eagerly drank in the tale of Wallace's exploits, and his soul\nwas fired by the desire to follow so valiant a leader. He was now\nsixteen, his frame was set and vigorous, and exercise and constant\npractice with arms had hardened his muscles. He became restless\nwith his life of inactivity; and his mother, seeing that her quiet\nand secluded existence was no longer suitable for him, resolved\nto send him to her sister's husband, Sir Robert Gordon, who dwelt\nnear Lanark.  Upon the night before he started she had a long talk\nwith him.\n\n\"I have long observed, my boy,\" she said, \"the eagerness with\nwhich you constantly practise at arms; and Sandy tells me that he\ncan no longer defend himself against you. Sandy, indeed is not a\nyoung man, but he is still hale and stout, and has lost but little\nof his strength. Therefore it seems that, though but a boy, you may\nbe considered to have a man's strength, for your father regarded\nSandy as one of the stoutest and most skilful of his men-at-arms.\nI know what is in your thoughts; that you long to follow in\nyour father's footsteps, and to win back the possessions of which\nyou have been despoiled by the Kerrs. But beware, my boy; you are\nyet but young; you have no friends or protectors, save Sir Robert\nGordon, who is a peaceable man, and goes with the times; while\nthe Kerrs are a powerful family, able to put a strong body in the\nfield, and having many powerful friends and connections throughout\nthe country. It is our obscurity which has so far saved you, for\nSir John Kerr would crush you without mercy did he dream that you\ncould ever become formidable; and he is surrounded by ruthless\nretainers, who would at a word from him take your life; therefore\nthink not for years to come to match yourself against the Kerrs.\nYou must gain a name and a following and powerful friends before\nyou move a step in that direction; but I firmly believe that the\ntime will come when you will become lord of Glencairn and the hills\naround it. Next, my boy, I see that your thoughts are ever running\nupon the state of servitude to which Scotland is reduced, and have\nmarked how eagerly you listen to the deeds of that gallant young\nchampion, Sir William Wallace. When the time comes I would hold\nyou back from no enterprise in the cause of our country; but at\npresent this is hopeless.  Valiant as may be the deeds which Wallace\nand his band perform, they are as vain as the strokes of reeds upon\narmour against the power of England.\"\n\n\"But, mother, his following may swell to an army.\"\n\n\"Even so, Archie; but even as an army it would be but as chaff before\nthe wind against an English array.  What can a crowd of peasants,\nhowever valiant, do against the trained and disciplined battle of\nEngland.  You saw how at Dunbar the Earl of Surrey scattered them\nlike sheep, and then many of the Scotch nobles were present. So\nfar there is no sign of any of the Scottish nobles giving aid or\ncountenance to Wallace, and even should he gather an army, fear\nfor the loss of their estates, a jealousy of this young leader,\nand the Norman blood in their veins, will bind them to England,\nand the Scotch would have to face not only the army of the invader,\nbut the feudal forces of our own nobles. I say not that enterprises\nlike those of Wallace do not aid the cause, for they do so greatly\nby exciting the spirit and enthusiasm of the people at large, as\nthey have done in your case. They show them that the English are\nnot invincible, and that even when in greatly superior numbers\nthey may be defeated by Scotchmen who love their country. They keep\nalive the spirit of resistance and of hope, and prepare the time\nwhen the country shall make a general effort. Until that time\ncomes, my son, resistance against the English power is vain. Even\nwere it not so, you are too young to take part in such strife, but\nwhen you attain the age of manhood, if you should still wish to\njoin the bands of Wallace--that is, if he be still able to make\nhead against the English--I will not say nay.  Here, my son,\nis your father's sword. Sandy picked it up as he lay slain on the\nhearthstone, and hid it away; but now I can trust it with you. May\nit be drawn some day in the cause of Scotland! And now, my boy,\nthe hour is late, and you had best to bed, for it were well that\nyou made an early start for Lanark.\"\n\nThe next morning Archie started soon after daybreak.  On his back\nhe carried a wallet, in which was a new suit of clothes suitable\nfor one of the rank of a gentleman, which his mother had with great\nstint and difficulty procured for him.  He strode briskly along,\nproud of the possession of a sword for the first time. It was in\nitself a badge of manhood, for at that time all men went armed.\n\nAs he neared the gates of Lanark he saw a party issue out and ride\ntowards him, and recognized in their leader Sir John Kerr. Pulling\nhis cap down over his eyes, he strode forward, keeping by the side\nof the road that the horsemen might pass freely, but paying no heed\nto them otherwise.\n\n\"Hallo, sirrah!\" Sir John exclaimed, reining in his horse, \"who\nare you who pass a knight and a gentleman on the highway without\nvailing his bonnet in respect?\"\n\n\"I am a gentleman and the son of a knight,\" Archie said, looking\nfearlessly up into the face of his questioner. \"I am Archie Forbes,\nand I vail my bonnet to no man living save those whom I respect\nand honour.\"\n\nSo saying, without another word he strode forward to the town. Sir\nJohn looked darkly after him.\n\n\"Red Roy,\" he said sternly, turning to one who rode behind him,\n\"you have failed in your trust. I told you to watch the boy, and\nfrom time to time you brought me news that he was growing up but\na village churl. He is no churl, and unless I mistake me, he will\nsome day be dangerous. Let me know when he next returns to the\nvillage; we must then take speedy steps for preventing him from\nbecoming troublesome.\"\n\n\n\n\nChapter III\n\nSir William Wallace\n\n\nArchie's coming had been expected by Sir Robert Gordon, and he was\nwarmly welcomed. He had once or twice a year paid short visits to\nthe house, but his mother could not bring herself to part with him\nfor more than a few days at a time; and so long as he needed only\nsuch rudiments of learning as were deemed useful at the time, she\nherself was fully able to teach them; but now that the time had come\nwhen it was needful that he should be perfected in the exercises\nof arms, she felt it necessary to relinquish him.\n\nSir Robert Gordon had no children of his own, and regarded his\nnephew as his heir, and had readily undertaken to provide him with\nthe best instruction which could be obtained in Lanark. There was\nresident in the town a man who had served for many years in the\narmy of the King of France, and had been master of arms in his\nregiment. His skill with his sword was considered marvellous by\nhis countrymen at Lanark, for the scientific use of weapons was as\nyet but little known in Scotland, and he had also in several trials\nof skill easily worsted the best swordsmen in the English garrison.\n\nSir Robert Gordon at once engaged this man as instructor to Archie.\nAs his residence was three miles from the town, and the lad urged\nthat two or three hours a day of practice would by no means satisfy\nhim, a room was provided, and his instructor took up his abode in\nthe castle. Here, from early morning until night, Archie practised,\nwith only such intervals for rest as were demanded by his master\nhimself. The latter, pleased with so eager a pupil, astonished at\nfirst at the skill and strength which he already possessed, and\nseeing in him one who would do more than justice to all pains that\nhe could bestow upon him, grudged no labour in bringing him forward\nand in teaching him all he knew.\n\n\"He is already an excellent swordsman,\" he said at the end of\nthe first week's work to Sir Robert Gordon; \"he is well nigh as\nstrong as a man, with all the quickness and activity of a boy. In\nstraightforward fighting he needs but little teaching. Of the finer\nstrokes he as yet knows nothing; but such a pupil will learn as\nmuch in a week as the ordinary slow blooded learner will acquire\nin a year. In three months I warrant I will teach him all I know,\nand will engage that he shall be a match for any Englishman north\nof the Tweed, save in the matter of downright strength; that he will\nget in time, for he promises to grow out into a tall and stalwart\nman, and it will need a goodly champion to hold his own against\nhim when he comes to his full growth.\"\n\nIn the intervals of pike and sword play Sir Robert Gordon himself\ninstructed him in equitation; but the lad did not take to this so\nkindly as he did to his other exercises, saying that he hoped he\nshould always have to fight on foot. Still, as his uncle pointed\nout that assuredly this would not be the case, since in battle\nknights and squires always fought on horseback, he strove hard to\nacquire a firm and steady seat.  Of an evening Archie sat with his\nuncle and aunt, the latter reading, the former relating stories of\nScotch history and of the goings and genealogies of great families.\nSometimes there were friends staying in the castle; for Sir Robert\nGordon, although by no means a wealthy knight, was greatly liked,\nand, being of an hospitable nature, was glad to have guests in the\nhouse.\n\nTheir nearest neighbour was Mistress Marion Bradfute of Lamington,\nnear Ellerslie. She was a young lady of great beauty. Her father had\nbeen for some time dead, and she had but lately lost her mother,\nwho had been a great friend of Lady Gordon. With her lived as\ncompanion and guardian an aunt, the sister of her mother.\n\nMistress Bradfute, besides her estate of Lamington, possessed\na house in Lanark; and she was frequently at Sir Robert's castle,\nhe having been named one of her guardians under her father's will.\nOften in the evening the conversation turned upon the situation\nof Scotland, the cruelty and oppression of the English, and the\nchances of Scotland some day ridding herself of the domination.\n\nSir Robert ever spoke guardedly, for he was one who loved not strife,\nand the enthusiasm of Archie caused him much anxiety; he often,\ntherefore, pointed out to him the madness of efforts of isolated\nparties like those of Wallace, which, he maintained, advanced in\nno way the freedom of the country, while they enraged the English\nand caused them to redouble the harshness and oppression of their\nrule.  Wallace's name was frequently mentioned, and Archie always\nspoke with enthusiasm of his hero; and he could see that, although\nMistress Bradfute said but little, she fully shared his views. It\nwas but natural that Wallace's name should come so often forward,\nfor his deeds, his hairbreadth escapes, his marvellous personal\nstrength and courage, were the theme of talk in every Scotch home;\nbut at Lanark at present it was specially prominent, for with his\nband he had taken up his abode in a wild and broken country known\nas Cart Lane Craigs, and more than once he had entered Lanark and\nhad had frays with the English soldiers there.\n\nIt was near a year since the defeat of Dunbar; and although the\nfeats of Wallace in storming small fortalices and cutting off English\nconvoys had excited at once hope amongst the Scotch and anger in the\nEnglish, the hold of the latter on the conquered country appeared\nmore settled than ever.  Wallace's adherents had indeed gained in\nstrength; but they were still regarded as a mere band of outlaws\nwho might be troublesome, but were in no degree formidable.\n\nEvery great town and hold throughout Scotland was garrisoned by\nEnglish in force deemed amply sufficient to repress any trouble\nwhich might arise, while behind them was the whole power of England\nready to march north in case it should be needed. It seemed, indeed,\nthat Scotland was completely and for ever subjugated.\n\nOne afternoon, when Archie had escorted Mistress Bradfute to\nLamington, she said to him as he bade her farewell:\n\n\"I think you can keep a secret, Master Forbes.\"\n\n\"I trust so,\" Archie replied.\n\n\"I know how much you admire and reverence Sir William Wallace. If\nyou will come hither this evening, at eight o'clock, you shall see\nhim.\"\n\nArchie uttered an exclamation of delight and surprise.\n\n\"Mind, Archie, I am telling you a secret which is known only to\nSir William himself and a few of his chosen followers; but I have\nobtained his permission to divulge it to you, assuring him that\nyou can be fully trusted.\"\n\n\"I would lay down my life for him,\" the lad said.\n\n\"I think you would, Archie; and so would I, for Sir William Wallace\nis my husband!\"\n\nArchie gave a gasp of astonishment and surprise.\n\n\"Yes,\" she repeated, \"he is my husband. And now ride back to your\nuncle's. I left the piece of embroidery upon which I was working on\nyour aunt's table. It will be a good excuse for you to ride over\nwith it this evening.\" So saying, she sprang lightly from the\npillion on which she had been riding behind Archie. The lad rode\nback in wild excitement at the thought that before night he was\nto see his hero whose deeds had, for the last three years, excited\nhis admiration and wonder.\n\nAt eight o'clock exactly he drew rein again at Lamington.  He was\nat once admitted, and was conducted to a room where the mistress\nof the house was sitting, and where beside her stood a very tall\nand powerfully built young man, with a singularly handsome face\nand a courteous and gentle manner which seemed altogether out of\ncharacter with the desperate adventures in which he was constantly\nengaged.\n\nIn Scotland the laws of chivalry, as they were strictly observed\nin the courts of England and France, did not prevail. Sir William\nWallace had not received the order of knighthood; but in Scotch\nfamilies the prefix of Sir descended from father to eldest son, as\nit does in the present day with the title of Baronet. Thus William\nWallace, when his father and elder brother were killed, succeeded\nto the title.  Knighthoods, or, as we should call them, baronetcies,\nwere bestowed in Scotland, as in England, for bravery in the field\nand distinguished services. The English, with their stricter laws\nof chivalry, did not recognize these hereditary titles; and Sir\nWilliam Wallace and many of his adherents who bear the prefix of\nSir in all Scotch histories, are spoken of without that title in\ncontemporary English documents. Archie himself had inherited the\ntitle from his father; and the prefix was, indeed, applied to the\nheads of almost all families of gentle blood in Scotland.\n\n\"This, Sir William,\" Marion said, \"is Sir Archibald Forbes, of whom\nI have often spoken to you as one of your most fervent admirers.\nHe is a true Scotsman, and he yearns for the time when he may draw\nhis sword in the cause of his country.\"\n\n\"He is over young yet,\" Sir William said smiling; \"but time will\ncure that defect. It is upon the young blood of Scotland that our\nhopes rest. The elders are for the most part but half Scotchmen, and\ndo not feel shame for their country lying at the feet of England;\nbut from their sons I hope for better things. The example of my\ndear friend, Sir John Grahame, is being followed; and I trust that\nmany young men of good family will soon join them.\"\n\n\"I would that the time had come when I too could do so, sir,\" Archie\nsaid warmly. \"I hope that it will not be long before you may think\nme capable of being admitted to the honour of fighting beside you.\nDo you not remember that you yourself were but eighteen when you\nslew young Selbye?\"\n\n\"I am a bad example to be followed,\" Sir William replied with a\nsmile; \"besides, nature made an exception in my case and brought\nme to my full strength and stature full four years before the time.\nMistress Marion tells me, however, that you too are strong beyond\nyour years.\"\n\n\"I have practised unceasingly, sir, with my weapons for the last\ntwo years; and deem me not boastful when I say that my instructor,\nDuncan Macleod of Lanark, who is a famous swordsman, says that\nI could hold my own and more against any English soldier in the\ngarrison.\"\n\n\"I know Duncan by report,\" Sir William replied, \"and that he is a\nfamous swordsman, having learned the art in France, where they are\nmore skilled by far than we are in Scotland. As for myself, I must\nown that it is my strength rather than my skill which gives me an\nadvantage in a conflict; for I put my trust in a downright blow,\nand find that the skill of an antagonist matters but little, seeing\nthat my blow will always cleave through sword as well as helm.\nNevertheless I do not decry skill, seeing that between two who\nare in any ways equally matched in strength and courage the most\nskilful swordsman must assuredly conquer.  Well, since that be the\nreport of you by Master Duncan, I should think you might even take\nto arms at the age that I did myself and when that time comes,\nshould your intentions hold the same, and the English not have made\nan end of me, I shall be right glad to have you by my side. Should\nyou, in any of your visits to Lanark--whither, Marion tells me, you\nride frequently with Sir Robert Gordon--hear ought of intended\nmovements of English troops, or gather any news which it may concern\nme to know, I pray you to ride hither at once. Marion has always\nmessengers whom she may despatch to me, seeing that I need great\ncare in visiting her here, lest I might be surprised by the English,\nwho are ever upon the lookout for me. And now farewell! Remember\nthat you have always a friend in William Wallace.\"\n\nWinter was now at hand, and a week or two later Mistress Marion\nmoved into her house in Lanark, where Archie, when he rode in,\noften visited her. In one of her conversations she told him that\nshe had been married to Sir William nigh upon two years, and that\na daughter had been born to her who was at present kept by an old\nnurse of her own in a cottage hard by Lamington. \"I tell you this,\nArchie,\" she said, \"for there is no saying at what time calamity\nmay fall upon us. Sir William is so daring and careless that I\nlive in constant dread of his death or capture; and did it become\nknown that I am his wife, doubtless my estate would be forfeited\nand myself taken prisoner; and in that case it were well that my\nlittle daughter should find friends.\"\n\n\"I wonder that you do not stay at Lamington,\" Archie said; \"for\nSir William's visits to you here may well be discovered, and both\nhe and you be put in peril.\"\n\n\"I would gladly do so,\" she said; \"but as you may have heard, Young\nHazelrig, the governor's son, persecutes me with his attentions;\nhe is moved thereto methinks rather by a desire for my possessions\nthan any love for myself.  He frequently rode over to Lamington\nto see me, and as there are necessarily many there who suspect, if\nthey do not know, my secret, my husband would be more likely to be\nsurprised in a lonely house there, than he would be in the city,\nwhere he can always leave or enter our abode by the passage into\na back street unseen by any.\"\n\nA few days later Archie had ridden into Lanark bearing a message\nfrom his uncle; he had put up his horse, and was walking along the\nprincipal street when he heard a tumult and the clashing of swords;\nhe naturally hurried up to see what was the cause of the fray, and\nhe saw Sir William Wallace and a young companion defending themselves\nwith difficulty against a number of English soldiers led by young\nHazelrig, the son of the governor, and Sir Robert Thorne, one of\nhis officers. Archie stood for a few moments irresolute; but as\nthe number of the assailants increased, as fresh soldiers hearing\nthe sound of the fray came running down the street, and Sir William\nand his friend, although they had slain several, were greatly\novermatched, he hesitated no longer, but, drawing his sword, rushed\nthrough the soldiers, and placing himself by the side of Wallace,\njoined in the fray. Wallace recognized him with a nod.\n\n\"It is sooner than I bargained for, Sir Archie; but you are very\nwelcome. Ah! that was well smitten, and Duncan did not overpraise\nyour skill,\" he exclaimed, as Archie cut down one soldier, and\nwounded another who pressed upon him.\n\n\"They are gathering in force, Sir William,\" the knight's companion\nsaid, \"and if we do not cut our way through them we shall assuredly\nbe taken.\" Keeping near the wall they retreated down the street,\nArchie and Sir John Grahame, for it was he, clearing the way, and\nWallace defending the rear. So terrific were the blows he dealt\nthat the English soldiers shrank back from attacking him.\n\nAt this moment two horsemen rode up and reined in their horses to\nwitness the fray. They were father and son, and the instant the\neyes of the elder fell upon Archie he exclaimed to his son:\n\n\"This is good fortune. That is young Forbes fighting by the side\nof the outlaw Wallace. I will finish our dispute at once.\"\n\nSo saying he drew his sword, and urged his horse through the\nsoldiers towards Archie; the latter equally recognized the enemy of\nhis family. Sir John aimed a sweeping blow at him.  The lad parried\nit, and, leaping back, struck at the horse's leg. The animal fell\ninstantly, and as he did so Archie struck full on the helm of Sir\nJohn Kerr, stretching him on the ground beside his horse.\n\nBy this time the little party had retreated down the street until\nthey were passing the house of Marion Bradfute. The door opened,\nand Marion herself cried to them to enter. So hemmed in were they,\nindeed, that further retreat was now impossible, and there being\nno time for hesitation, Wallace and his companions sprang in before\ntheir assailants could hinder them, and shut the door behind them.\n\n\"Marion,\" Wallace exclaimed, \"why did you do this? It mattered\nnot were I killed or taken; but now you have brought danger upon\nyourself.\"\n\n\"But it mattered much to me. What would life be worth were you\nkilled? Think not of danger to me. Angry as they may be, they will\nhardly touch a woman. But waste no time in talking, for the door\nwill soon yield to their blows. Fly by the back entrance, while\nthere is time.\"\n\nSo saying, she hurried them to the back of the house, and without\nallowing them to pause for another word almost pushed them out, and\nclosed the door behind them. The lane was deserted; but the shouts\nand clamour of the English soldiers beyond the houses rose loud in\nthe air. \"Quick, Sir William,\" Sir John Grahame said, \"or we shall\nbe cut off!  They will bethink them of the back way, and send\nsoldiers down to intercept us.\"\n\nSuch, indeed, was the case, for as they ran they heard shouts behind,\nand saw some English soldiers entering the other end of the lane.\nIn front, however, all was clear, and running on they turned into\nanother street, and then down to the gate. The guard, hearing the\ntumult, had turned out, and seeing them running, strove to bar\ntheir way. Wallace, however, cleared a path by sweeping blows with\nhis sword, and dashing through the gates into the open country\nthey were safe. For some distance they ran without checking their\nspeed, and then as they neared a wood, where they no longer feared\npursuit, they broke into a walk.\n\n\"My best thanks to you,\" Wallace said to Archie. \"You have indeed\nproved yourself a staunch and skilful swordsman, and Duncan's opinion\nis well founded.  Indeed I could wish for no stouter sword beside\nme in a fight; but what will you do now? If you think that you were\nnot recognized you can return to your uncle; but if any there knew\nyou, you must even then take to the woods with me.\"\n\n\"I was recognized,\" Archie said in a tone of satisfaction.  \"The\narmed knight whom you saw attack me was Sir John Kerr, the slayer\nof my father and the enemy of my house.  Assuredly he will bring\nthe news of my share in the fray to the ears of the governor.\"\n\n\"I do not think that he will carry any news for some time,\" Sir\nWilliam replied; \"for that blow you gave him on the head must have\nwell nigh brought your quarrel to an end.  It is a pity your arm\nhad not a little more weight, for then, assuredly you would have\nslain him.\"\n\n\"But the one with him was his son,\" Archie said, \"and would know\nme too; so that I shall not be safe for an hour at my uncle's.\"\n\n\"In that case, Sir Archie, you must needs go with me, there being\nno other way for it, and truly, now that it is proved a matter of\nnecessity, I am glad that it has so chanced, since I see that your\nyouth is indeed no drawback; and Sir John Grahame will agree with\nme that there is no better sword in my company.\"\n\n\"Yes, indeed,\" the young knight said. \"I could scarce believe my\neyes when I saw one so young bear himself so stoutly. Without his\naid I could assuredly have made no way through the soldiers who\nbarred our retreat; and truly his sword did more execution than\nmine, although I fought my best. If you will accept my friendship,\nyoung sir, henceforth we will be brothers in arms.\" Colouring with\npleasure, Archie grasped the hand which the young knight held out\nto him.\n\n\"That is well said, Sir John,\" Wallace assented. \"Hitherto you and\nI have been like brothers; henceforth there will be three of us,\nand I foresee that the only difficulty we shall have with this\nour youngest relation will be to curb his courage and ardour. Who\nknows,\" he went on sadly, \"but that save you two I am now alone in\nthe world! My heart misgives me sorely as to the fate of Marion; and\nwere it not for the sake of Scotland, to whom my life is sworn, I\nwould that I had stopped and died outside her door before I entered\nand brought danger upon her head. Had I had time to reflect, methinks\nI would have done so; but I heard her call, I saw the open door,\nand without time for thought or reflection I leapt in.\"\n\n\"You must not blame yourself, Sir William,\" Grahame said, \"for,\nindeed, there was no time for thought; nor will I that it should\nhave been otherwise, even should harm, which I cannot believe,\nbefall Mistress Marion.  It is on you that the hopes of Scotland\nnow rest. You have awakened her spirit and taught the lesson of\nresistance. Soon I hope that the fire now smouldering in the breast\nof every true Scotsman will burst into flame, and that Scotland\nwill make a great effort for freedom; but were you to fall now,\ndespair would seize on all and all hope of a general rising be at\nan end.\"\n\nWallace made no reply, but strode silently forward. A short distance\nfarther they came to the spot where three of Wallace's followers\nwere holding horses, for he had on his entry into Lanark, been\naccompanied by another of his party, who had been slain at the\ncommencement of the fray. Wallace bade Archie mount the spare horse,\nand they then rode to Cart Lane Craigs, scarce a word being spoken\non their journey.\n\nWallace's headquarters were upon a narrow shelf of rock on the face\nof a steep and craggy hill. It was well chosen against surprise,\nand could be held against sudden attack even by a large force,\nsince both behind and in front the face of the hill was too steep\nto be climbed, and the only approach was by a steep and winding\npath which two men could hold against a host. The ledge was some\n50 feet long by 12 wide.  At the back a natural depression in the\ncrags had been deepened so as to form a shallow cave just deep\nenough to afford a defense against the weather; here a pile of\nheather served as a bed for Wallace, Grahame, and one or two others\nof the leaders of his company, and here Wallace told Archie that\nhis place was to be. On the ledge without were some low arbours of\nheather in which lay ten of Wallace's bravest companions; the rest\nof his band were scattered among the surrounding hills, or in the\nwoods, and a bugle note repeated from place to place would call\nall together in a short space of time.\n\nOf stores and provisions there was no lack, these having been\nobtained in very large quantities from the convoys of supplies and\nthe castles that had been captured. Money, too, was not wanting,\nconsiderable amounts having fallen into their hands, and the\npeasantry through all the country round were glad in every way to\nassist the band, whom they regarded as their champions.\n\nArchie sat down by Sir John Grahame, who gave him particulars\nregarding the strength of the various bands, their position, the\nrules which had been laid down by Wallace for their order, the system\nof signals and other particulars; while Wallace paced restlessly up\nand down the narrow shelf, a prey to the keenest anxiety. Towards\nnightfall two of the men were despatched towards Lanark to endeavour\nto find out what had taken place there; but in an hour they returned\nwith a woman, whom both Sir William and Archie recognized as one of\nthe female attendants of Marion. A single glance sufficed to tell\nher tale. Her face was swollen with crying, and wore a look of\nhorror as well as of grief.\n\n\"She is dead!\" Wallace exclaimed in a low voice.\n\n\"Alas!\" the woman sobbed, \"that I should have to tell it.  Yes, my\ndear mistress is dead; she was slain by the orders of the governor\nhimself, for having aided your escape.\"\n\nA groan burst from Wallace, a cry of horror and indignation from\nhis followers. The former turned, and without a word strode away\nand threw himself upon the heather. The others, heart struck at\nthe cruel blow which had befallen their chief, and burning with\nindignation and rage, could only utter oaths of vengeance and curses\non the English tyrants.\n\nAfter a time Grahame went to the cave, and putting his hand on\nWallace's shoulder strove to address a few words of consolation to\nhim.\n\nSir William rose: \"I have done with weeping, Grahame, or rather I\nwill put off my weeping until I have time for it.  The first thing\nto think of is vengeance, and vengeance I swear that I will have.\nThis night I will strike the first blow in earnest towards freeing\nScotland. It may be that God has willed it that this cruel blow,\nwhich has been struck at me, shall be the means of bringing this\nabout. Hitherto, although I have hated the English and have fought\nagainst them, it has been but fitfully and without order or method,\nseeing that other things were in my heart. Henceforth I will live\nbut for vengeance and Scotland. Hitherto the English have regarded\nme as an outlaw and a brigand. Henceforth they shall view me as an\nenemy to be dreaded. Sound the signal of assembly at once. Signify\nthat as many as are within reach shall gather below in two hours.\nThere will be but few, for, not dreaming of this, the bands but\ntwo days since dispersed. But even were there none but ourselves\nit would suffice. Tonight we will take Lanark.\"\n\n\n\n\nChapter IV\n\nThe Capture of Lanark\n\n\nA low shout of enthusiasm rose from Wallace's followers, and they\nrepeated his words as though it had been a vow: \"Tonight we will\ntake Lanark.\" The notes of a bugle rang through the air, and Archie\ncould hear them repeated as by an echo by others far away in the\nwoods.\n\nThe next two hours were spent in cooking and eating a meal; then\nthe party on the ledge descended the narrow path, several of their\nnumber bearing torches. At a short distance from its foot some other\ntorches were seen, and fifteen men were found gathered together.\n\nIn a few words the sad news of what had taken place at Lanark was\nrelated to them and the determination which had been arrived at,\nand then the whole party marched away to the west. Archie's heart\nbeat with excitement as he felt himself engaged in one of the adventures\nwhich had so filled his thoughts and excited his admiration. An\nadventure, too, far surpassing in magnitude and importance any in\nwhich Wallace had hitherto been engaged.\n\nIt seemed almost like an act of madness for twenty-five men to\nattack a city garrisoned by over 500 English troops, defended by\nstrong walls; but Archie never doubted for a moment that success\nwould attend the enterprise, so implicit was his confidence in his\nleader. When at some little distance from the town they halted,\nand Wallace ordered a tree to be felled and lopped of its branches.\nIt was some eight inches in diameter at the butt and thirty feet\nlong. A rope had been brought, and this was now cut into lengths\nof some four feet. Wallace placed ten of his men on each side of\nthe tree, and the cords being placed under it, it was lifted and\ncarried along with them.\n\nBefore they started Wallace briefly gave them his orders, so that\nno word need be spoken when near the town.  The band were, when\nthey entered, to divide in three. Sir John Grahame, with a party,\nwas to make for the dwelling of Sir Robert Thorne. Auchinleck, who\nhad arrived with the party summoned by the bugle, was to arouse\nthe town and attack any parties of soldiers in the street, while\nWallace himself was to assault the house of Hazelrig. He bade Archie\naccompany him.\n\nKnowing the town well Wallace led the party to the moat at a spot\nfacing a sally port. They moved without a word being spoken. The\nmen bearing the tree laid it noiselessly to the ground. Wallace\nhimself sprang into the moat and swam across. The splash in the\nwater attracted the attention of a sentry over the gate, who at\nonce challenged. There was no answer, and the man again shouted,\npeering over the wall to endeavour to discover what had caused\nthe splash. In a few vigorous strokes Wallace was across, hauled\nhimself up to the sill of the door, and with his heavy battleaxe\nsmote on the chains which held up the drawbridge. Two mighty blows\nand the chains yielded, and the drawbridge fell with a crash across\nthe moat.\n\nInstantly the men lifted the tree, and dashing across swung it\nlike a battering ram against the door--half a dozen blows, and\nthe oak and iron yielded before it. The door was burst in and the\nparty entered Lanark.  The sentry on the wall had fled at once to\narouse the garrison. Instantly the three leaders started to perform\nthe tasks assigned to them.  As yet the town lay in profound sleep,\nalthough near the gate windows were opening and heads were being\nput out to ascertain the cause of the din. As the Scots ran forward\nthey shouted \"Death to the English, death to the bloody Hazelrig!\"\nThe governor had long been odious for his cruelty and tyranny, and\nthe murder of Marion Bradfute had that day roused the indignation\nof the people to the utmost. Not knowing how small was the force\nthat had entered the town, but hoping only that deliverers had\narrived, numbers of the burghers rose and armed themselves, and\nissued forth into the streets to aid their countrymen. Wallace soon\narrived at the governor's house, and with a few blows with his axe\nbroke in the door; then he and his followers rushed into the house,\ncutting down the frightened men as they started up with sudden\nalarm, until he met Sir John Hazelrig, who had snatched up his arms\nand hurried from his chamber.\n\n\"Villain!\" Wallace exclaimed, seizing him by his throat; \"your time\nhas come to make atonement for the murder of my wife.\"\n\nThen dragging him into the street he called upon the burghers,\nwho were running up, to witness the execution of their tyrant,\nand stepping back a pace smote off his head with his sword. Young\nHazelrig was also killed, as were all soldiers found in the house.\nThe alarm bells were ringing now, and in a few minutes the armed\nburghers swarmed in the street. As the English soldiers, as yet\nbut scarce awake, and bewildered by this sudden attack, hurried\nfrom their houses, they were fallen upon and slain by Wallace and\nthe townspeople. Some of those in the larger houses issuing forth\ntogether were able to cut their way through and to make their\nescape by the gates; many made for the walls, and dropping in the\nmoat swam across and escaped; but two hundred and fifty of their\nnumber were left dead in the streets. The town, once cleared of the\nEnglish, gave itself up to wild rejoicings; bonfires were lighted\nin the streets, the bells were rung, and the wives and daughters\nof the citizens issued out to join in their rejoicing and applaud\ntheir liberators.\n\nWallace held council at once with the chief burghers.  Their talk\nwas a grave one, for though rejoicing in the liberation of the city,\nthey could not but perceive that the situation was a serious one.\nBy the defeat and destruction of the garrison, and the slaying of\nthe governor, the town would bring upon itself the terrible wrath\nof King Edward, and of what he was capable the murdered thousands\nat Berwick sufficiently attested. However, the die was cast and there\nwas no drawing back, and the burghers undertook to put their town\nin a state of full defence, to furnish a contingent of men-at-arms to\nWallace, and to raise a considerable sum of money to aid him in the\ncarrying on of the war; while he on his part undertook to endeavour,\nas fast as possible, to prevent the English from concentrating their\nforces for a siege of the town, by so harassing their garrisons\nelsewhere that none would be able to spare troops for any general\npurposes.\n\nProclamations were immediately made out in the name of Wallace,\nand were sent off by mounted messengers throughout the country.\nIn these he announced to the people of Scotland that he had raised\nthe national banner and had commenced a war for the freeing of the\ncountry from the English, and that as a first step he had captured\nLanark. He called upon all true Scotchmen to rally round him.\n\nWhile the council was being held, the wives of the burghers had\ntaken the body of Marion from the place where it had been cast,\nand where hitherto none had dared to touch it, and had prepared it\nfor burial, placing it in a stone coffin, such as were in use in\nthose days, upon a car which was covered with trappings of white and\ngreen boughs.  Soon after daybreak a great procession was formed,\nand accompanied by all the matrons and maids of Lanark the body\nwas conveyed to the church at Ellerslie, and there buried with\nthe rites of the church. This sad duty ended, Wallace mounted his\nhorse and rode for Cart Lane Craigs, which he had named as the\nrendezvous where all who loved Scotland and would follow him, were\nto assemble. Archie rode first to Sir Robert Gordon's. His uncle\nreceived him kindly.\n\n\"Ah! my boy,\" he said, \"I feared that your wilful disposition would\nhave its way. You have embarked young on a stormy course, and none\ncan say where it will end. I myself have no hope that it can be\nsuccessful. Did the English rule depend solely on the troops which\ngarrison our towns and fortresses, I should believe that Wallace\nmight possibly expel them; but this is as nothing. Edward can march\na hundred and fifty thousand trained soldiers hither, and how will\nit be possible for any gathering of Scotchmen to resist these?\nHowever, you have chosen your course, and as it is too late to\ndraw back now, I would not dispirit you. Take the best of my horses\nfrom the stable, and such arms and armour as you may choose from\nthe walls. Here is a purse for your own private needs, and in this\nother are a hundred pounds, which I pray you hand to Sir William\nWallace. Fighting never was in my way, and I am too old to begin\nnow. Tell him, however, that my best wishes are with him. I have\nalready sent word to all my tenants that they are free, if they\nchoose, to follow his banner.\"\n\n\"You have plenty of pikes and swords in the armoury, uncle; weapons\nwill be very useful; can I take some of them?\"\n\n\"Certainly, Archie, as many as you like. But your aunt wants you\nto ride at once to Glen Cairn, to ask your mother to come over here\nand take up her abode till the stormy times are over. The news of\nlast night's doings in Lanark will travel fast, and she will be\nterribly anxious. Besides, as the Kerrs are heart and soul with the\nEnglish faction, like enough they will take the opportunity of the\ndisturbed times, and of your being involved in the rising, to destroy\nthe hold altogether, seeing that so long as it stands there it is\na sort of symbol that their lordship over the lands is disputed.\"\n\n\"The very thing that I was going to ask you, uncle. My mother's\nposition at Glen Cairn would always be on my mind.  As to the\nKerrs, let them burn the castle if they will. If the rising fail,\nand I am killed, the line will be extinct, and it matters little\nabout our hold. If we succeed, then I shall regain my own, and shall\nturn the tables on the Kerrs, and will rebuild Glen Cairn twice as\nstrong as before. And now can I take a cart to convey the arms?\"\n\n\"Certainly, Archie; and may they be of service in the cause. You\nwill, I suppose, conduct your mother hither?\"\n\nArchie replied that he should do so, and then at once made his\npreparations for the start. His uncle's armoury was well supplied,\nand Archie had no difficulty in suiting himself. For work like\nthat which he would have to do he did not care to encumber himself\nwith heavy armour, but chose a light but strong steel cap, with a\ncurtain of mail falling so as to guard the neck and ears, leaving\nonly the face exposed, and a shirt of the same material. It was\nof fine workmanship and of no great weight, and did not hamper\nhis movements.  He also chose some leg pieces for wearing when on\nhorseback. He had already his father's sword, and needed only a\nlight battleaxe and a dagger to complete his offensive equipment.\nThen he took down from the racks twenty swords and as many short\npikes, and bonnets strengthened with iron hoops, which, although\nlight, were sufficient to give much protection to the head. These\nwere all placed in a light cart, and with one of his uncle's\nfollowers to drive, he took his seat in the cart, and started for\nCart Lane Craigs.\n\nHere he concealed the arms in a thicket, and then went up to speak\nto his leader.\n\n\"May I take ten men with me to Glen Cairn, Sir William? I am going\nto fetch my mother to reside with my uncle until the storm is\nover. He has sent you a hundred pounds towards the expenses of the\nstruggle. I want the guard because it is possible that the Kerrs\nmay be down there. I hear Sir John was carried away, three hours\nafter the fight, in a litter; it was well for him that he was not\nin Lanark when we took it. But like enough this morning, if well\nenough to give orders, he may be sending down to Glen Cairn to see\nif I have returned, and may burn the hold over my mother's head.\"\n\n\"Certainly,\" Sir William replied. \"Henceforth I will put twenty\nmen under your special orders, but for today Sir John Grahame shall\ntell off some of his own party. Of course they will go well armed.\"\n\nHalf riding in the cart and half walking by turns, the party reached\nGlen Cairn late in the afternoon. The news of the fall of Lanark\nhad already penetrated even to that quiet village, and there was\ngreat excitement as Archie and his party came in. One of Wallace's\nmessengers had passed through, and many of the men were preparing\nto join him. Dame Forbes was at once proud and grieved when Archie\ntold her of the share which he had had in the street fray at Lanark,\nand in the capture of the town. She was proud that her son should\nso distinguish himself, grieved that he should, at so young an\nage, have become committed to a movement of whose success she had\nbut little hope. However, she could not blame him, as it seemed as\nif his course had been forced upon him. She agreed to start early\nthe next morning.\n\nIt was well for Archie that he had brought a guard with him,\nfor before he had been an hour in the hold a boy ran in from the\nvillage saying that a party of the Kerrs was close at hand, and\nwould be there in a few minutes. Archie set his men at once to pile\nup a barricade of stones breast high at the outer gate, and took\nhis position there with his men. He had scarcely completed his\npreparations when the trampling of horses was heard and a party\nof ten men, two of whom bore torches, headed by young Allan Kerr,\nrode up. They drew rein abruptly as they saw the barricade with\nthe line of pikes behind it.\n\n\"What want you here, Allan Kerr?\" Archie said.\n\n\"I came in search of you, little traitor,\" young Kerr replied\nangrily.\n\n\"Here I am,\" Archie said; \"why don't you come and take me?\"\n\nAllan saw that the number of the defenders of the gate exceeded\nthat of his own party, and there might, for aught he knew, be more\nwithin.\n\n\"I will take you tomorrow,\" he said.\n\n\"Tomorrow never comes,\" Archie replied with a laugh.  \"Your father\nthought to take me yesterday. How is the good knight? Not suffering,\nI trust, greatly either in body or temper?\"\n\n\"You shall repent this, Archibald Forbes,\" Allan Kerr exclaimed\nfuriously. \"It will be my turn next time.\"\n\nAnd turning his horse he rode off at full speed, attended by his\nfollowers.\n\n\"We had best start at once, Master Archie,\" Sandy Graham said:\n\"it is eight miles to the Kerrs' hold, and when Allan Kerr returns\nthere you may be sure they will call out their vassals and will be\nhere betimes in the morning. Best get another cart from the village,\nfor your men are weary and footsore, seeing that since yesterday\neven they have been marching without ceasing. Elspie will by this\ntime have got supper ready.  There was a row of ducks and chickens\non the spit when I came away.\"\n\n\"That were best, Sandy. Do you see to their comforts, and aid my\nmother pack up such things as she most values, and I will go myself\ndown to the village for the cart, for I wish to speak with some\nthere.\"\n\nArchie had no difficulty in engaging two carts, as he thought that\none would be needed for his mother and what possessions she might\ntake. Then he went from house to house and saw his old companions,\nand told them of his plans, which filled them with delight. Having\ndone this he returned to the hold, hastily ate the supper which\nhad been put aside for him, and then saw that his mother's chests,\nwhich contained all her possessions save a few articles of heavy\nfurniture, were placed in one of the carts. A bed was then laid\non its floor upon which she could sit comfortably. Elspie mounted\nwith her. Archie, Sandy, and the men took their places in the other\ncarts, and the party drove off. They had no fear of interruption,\nfor the Kerrs, ignorant of the number who had arrived with Archie\nat Glen Cairn, would not venture to attack until they had gathered\na considerable force, and would not be likely to set out till\nmorning, and long before that time Dame Forbes would have arrived\nat her sister's.\n\nThe journey was indeed performed without incident, the escort\nleaving them when within two or three miles of Lanark, and making\ntheir way direct to the craigs, whither Archie, the moment he had\nseen his mother safely at Sir Robert Gordon's, returned. He did\nnot mount the craig, but wrapping himself in his cloak lay down at\nits foot.\n\nAs soon as it was daylight he walked out a mile on the road towards\nGlen Cairn. He soon saw a party approaching in military order. They\nhalted when they reached him. They were twenty in number, and were\nthe lads of his band at Glen Cairn, ranging between the ages of\nsixteen and eighteen.  They had originally been stronger, but some\nof the elders had already joined Wallace's followers.\n\n\"Now,\" Archie said, \"I can explain matters farther than I did last\nnight. I have procured arms for you all, and I hope that you will\nhave opportunities of using them. But though some of you are old\nenough to join Wallace's band, there are others whom he might not\ndeem fit to take part in such desperate enterprises.  Therefore\nat first make but little show of your arms. I shall present you to\nSir William, telling him that I have brought you hither to serve\nas messengers, and to enter towns held by the English and gather\nnews, seeing that lads would be less suspected than men. But I\npropose farther, what I shall not tell him, that you shall form a\nsort of bodyguard to him. He takes not sufficient care of himself,\nand is ever getting into perils.  I propose that without his\nknowing it, you shall be ever at hand when he goes into danger of\nthis sort, and may thus prevent his falling into the hands of his\nenemies. Now, mind, lads, this is a great and honourable mission.\nYou must be discreet as well as brave, and ready all of you to give\nyour lives, if need be, for that of Scotland's champion. Your work\nas messengers and scouts will be arduous and wearisome. You must\nbe quiet and well behaved--remember that boys' tricks and play\nare out of place among men engaged in a desperate enterprise. Mingle\nnot much with the others. Be active and prompt in obeying orders,\nand be assured that you will have opportunities of winning great\nhonour and credit, and of having your full share of hard knocks.\nYou will, as before, be divided into two companies, William Orr and\nAndrew Macpherson being your lieutenants in my absence. You will\nobey their orders as implicitly as mine. Cluny, you have, I suppose,\nbrought, as I bade you last night, some of your sister's garments?\"\n\n\"Yes, Sir Archie,\" the boy, who was fair and slight, said, with a\nsmile on his face.\n\n\"That is right. I know you are as hearty and strong as the rest;\nbut seeing that your face is the smoothest and softest of any,\nyou will do best should we need one in disguise as a girl. And now\ncome with me. I will show you where your arms are placed; but at\npresent you must not take them. If I led you as an armed band to\nWallace he might deem you too young. I must present you merely as\nlads whom I know to be faithful and trustworthy, and who are willing\nto act as messengers and scouts to his force.\"\n\nSo saying Archie led the band to the thicket where he had placed\ntheir arms, and the lads were pleased when they saw the pikes,\nswords, and head pieces. Then he led them up the craig to Wallace.\n\n\"Why, whom have you here?\" Sir William exclaimed in surprise. \"This\nwill not do, Sir Archie. All lads are not like yourself, and were\nI to take such boys into my ranks I should have all the mothers in\nScotland calling out against me.\"\n\n\"I have not brought them to join your ranks, Sir William, although\nmany of them are stout fellows who might do good service at a\npinch. I have brought them to act as messengers and scouts. They\ncan carry orders whithersoever you may have occasion to send. They\ncan act as scouts to warn you of the approach of an enemy; or if\nyou need news of the state of any of the enemy's garrisons, they\ncan go thither and enter without being suspected, when a man might\nbe questioned and stopped. They are all sons of my father's vassals\nat Glen Cairn, and I can answer for their fidelity. I will take\nthem specially under my own charge, and you will ever have a fleet\nand active messenger at hand when you desire to send an order.\"\n\n\"The idea is not a bad one,\" Sir William replied; \"and in such a\nway a lad may well do the work of a man.  Very well, Sir Archie,\nsince you seem to have set your mind upon it I will not say nay.\nAt any rate we can give the matter a trial, understanding that you\ntake the charge of them and are responsible for them in all ways.\nNow, lads,\" he said turning, \"you have heard that your lord, for\nhe is your rightful lord, and will, if Scotland gains the day, be\nyour real lord again, has answered for you. It is no boys' play\nin which you have taken service, for the English, if they conquer\nus, will show no further mercy to you than to others of my band. I\nunderstand then that you are all prepared, if need be, to die for\nScotland. Is this so?\"\n\n\"We are, sir,\" the lads exclaimed together.\n\n\"Then so be it,\" Sir William said. \"Now, Sir Archie, do you fix\na place for their encampment, and make such other arrangements\nas you may think fit. You will, of course, draw rations and other\nnecessaries for them as regular members of the band.\"\n\nArchie descended with his troop from the craigs, and chose a spot\nwhere they would be apart from the others. It was a small piece of\nground cut off by the stream which wound at the foot of the craigs,\nso that to reach it it was necessary to wade knee deep through the\nwater. This was no inconvenience to the lads, all of whom, as was\ncommon with their class at the time, were accustomed to go barefoot,\nalthough they sometimes wore a sort of sandal. Bushes were cut\ndown, and arbours made capable of containing them. The spot was\nbut a little distance from the foot of the path up the craigs, and\nany one descending the path could be seen from it.\n\nArchie gave orders that one was always to be above in readiness to\nstart instantly with a message; that a sentry was to be placed at\nthe camp, who was to keep his eyes upon the path, and the moment\nthe one on duty above was seen to leave, the next upon the list was\nto go up and take his place.  None were to wander about the wood,\nbut all were to remain in readiness for any duty which might be\nrequired. The two lieutenants were charged to drill them constantly\nat their exercises so as to accustom them to the weight and handle\nof their arms. Two were to be sent off every morning to the depot\nwhere the provisions were issued, to draw food for the whole for\nthe day, and four were to be posted five miles away on the roads\nleading towards the craigs to give warning of the approach of any\nenemies. These were to be relieved every six hours. They were to\nbe entirely unarmed, and none were to issue from the camp with arms\nexcept when specially ordered.\n\nHaving made these arrangements, and taking with him one of the band\nas the first on duty above, he rejoined Wallace at his post on the\ncraigs.\n\nWallace's numbers now increased fast. On hearing of the fall of\nLanark, and on the receipt of the proclamation calling upon all\ntrue Scotchmen to join him in his effort to deliver their country\nfrom its yoke, the people began to flock in in great numbers. Richard\nWallace of Riccarton and Robert Boyd came in with such force as\nthey could collect from Kyle and Cunningham, among whom were not\nless than 1000 horsemen. Sir John Grahame, Sir John of Tinto, and\nAuchinleck assembled about 3000 mounted troops and a large number\nof foot, many of whom, however, were imperfectly armed. Sir Ronald\nCrawford, Wallace's uncle, being so close to Ayr, could not openly\njoin him, but secretly sent reinforcements and money. Many other\ngentlemen joined with their followers.\n\nThe news of the fall of Lanark and of the numbers who were flocking\nto join Wallace paralysed the commanders of the English garrisons,\nand for a time no steps were taken against him; but news of the\nrising was instantly sent to King Edward, who, furious at this\nfresh trouble in Scotland, which he had deemed finally conquered,\ninstantly commenced preparations for another invasion. A body of\ntroops was at once sent forward from England, and, being strengthened\nby bodies drawn from all the garrisons, assembled at Biggar.  The\narmy was commanded by the Earl of Kent. Heralds were sent to Wallace\noffering him not only pardon but an honourable post if he would\nsubmit, but warning him that if he refused this offer he should,\nwhen taken, be treated as a rebel and hung.\n\nWallace briefly refused submission, and said that he should be\nready to give battle on the following morning.\n\nAt daybreak the army set forth, divided into three parts.  Wallace,\nwith Boyd and Auchinleck, commanded one; Sir John Grahame, with\nWallace of Riccarton, the second; Sir Walter of Newbigging, with his\nson David and Sir John Clinto, the third. The cavalry were placed\nin front. The footmen, being imperfectly armed and disciplined,\nand therefore unable to withstand the first charge of the English,\nfollowed the cavalry.\n\nBefore marching forward Wallace called the commanders round him\nand charged them earnestly to restrain their men from plunder until\nthe contest was decided, pointing out that many a battle had been\nlost owing to the propensity of those who gained the first advantage\nto scatter for plunder. Just as the Scotch were moving, a body of\n300 horsemen, well armed and equipped, from Annandale and Eskdale,\nled by Halliday, Kirkpatrick, and Jardine, joined them; and with\nthis accession of strength they marched forward confidently against\nthe enemy.\n\n\n\n\nChapter V\n\nA Treacherous Plot\n\n\nSo rapid was the advance of Wallace's army that the English had\nscarce time to form when they were upon them. The Scotch charged\nwith extreme impetuosity among the English ranks, directing the\nonslaught principally against the centre, commanded by the Earl of\nKent.\n\nThe English resisted stoutly; but the Earl of Kent was struck down\nby Wallace himself, and was with difficulty borne off the field;\nand after severe fighting, the whole English army was thrown into\ndisorder and took to flight. Some hundreds were killed in action,\nand many more in the pursuit which followed; this, however, Wallace\nwould not allow to be pushed too far lest the fugitives should\nrally and turn. Then the victorious Scots returned to the English\ncamp. In this was found a great abundance of provisions, arms,\nand other valuable booty. Many of the cattle were killed, and a\nsumptuous feast prepared. Then Wallace had the whole of the spoil\ncarried off into a place of safety in the heart of a neighbouring\nbog, and he himself fell back to that shelter.\n\nIn the morning the English, who had rallied when the pursuit had\nceased, again advanced, hoping to find Wallace unprepared. They\nwere now commanded by the Earl of Lancaster, and had received some\nreinforcements in the night.  They passed over the scene of the\nprevious day's battle, and at last came in sight of the Scotch\narmy. Wallace at first advanced, and then, as if dismayed at their\nsuperior strength, retired to the point where, in order to reach\nthem, the English would have to cross a portion of the bog. The\nsurface was covered with moss and long grass, and the treacherous\nnature of the ground was unperceived by the English, who, filled\nwith desire to wipe out their defeat of the preceding day, charged\nimpetuously against the Scotch line. The movement was fatal, for\nas soon as they reached the treacherous ground their horses sunk\nto the saddle girths. The Scotch had dismounted on firmer ground\nbehind, and now advanced to the attack, some working round the\nflanks of the morass, others crossing on tufts of grass, and so\nfell upon the struggling mass of English. The Earl of Westmoreland\nand many others of note were killed, and the Earl of Lancaster,\nwith the remains of his force, at once retreated south and recrossed\nthe Border.\n\nArchie had taken no part in the first battle. Wallace had asked\nhim whether he would fight by his side or take command of a body\nof infantry; and he chose the latter alternative. Almost all the\nknights and gentlemen were fighting on horse with their followers,\nand Archie thought that if these were repulsed the brunt of the fray\nwould fall upon the infantry. On this occasion, then, he gathered\nwith his band of lads a hundred or so pikemen, and formed them in\norder, exhorting them, whatever happened, to keep together and to\nstand stoutly, even against a charge of horse. As the victory was\nwon entirely by the cavalry he had no opportunity of distinguishing\nhimself. Upon the second day, however, he did good service,\nas he and his lightly armed footmen were able to cross the bog in\nplaces impracticable to the dismounted men-at-arms in their heavy\naccoutrements.\n\nThe victory of Biggar still further swelled Wallace's forces. Sir\nWilliam Douglas joined him, and other gentlemen. A great meeting\nwas held at Forest Kirk, when all the leaders of Wallace's force\nwere present; and these agreed to acknowledge him as general of\nthe Scottish forces against England, with the title of Warden of\nScotland.\n\nKing Edward was at this time busied with his wars in France, and\nwas unable to despatch an army capable of effecting the reconquest\nof that portion of Scotland now held by Wallace; and as the English\nforces in the various garrisons were insufficient for such purpose,\nthe Earl of Percy and the other leaders proposed a truce. This\nwas agreed to. Although Wallace was at the head of a considerable\nforce, Sir William Douglas was the only one among the Scottish\nnobles of importance who had joined him; and although the successes\nwhich he had gained were considerable, but little had been really\ndone towards freeing Scotland, all of whose strong places were\nstill in the hands of the English, and King Edward had not as yet\nreally put out his strength.\n\nThe greater portion of the army of Wallace was now dispersed.\n\nShortly afterwards the governor of Ayr issued a notice that a great\ncouncil would be held at that town, and all the Scotch gentlemen\nof importance in the district were desired to attend. Wallace was\none of those invited; and deeming that the governor might have\nsome proposition of Edward to lay before them, he agreed to do so.\nAlthough a truce had been arranged, he himself with a band of his\nmost devoted followers still remained under arms in the forest,\nstrictly keeping the truce, but holding communications with his friends\nthroughout the country, urging them to make every preparation, by\ncollecting arms and exercising their vassals, to take the field with\na better appointed force at the conclusion of the truce. Provisions\nand money were in abundance, so large had been the captures effected;\nbut Wallace was so accustomed to the free life of the woods that he\npreferred to remain there to taking up his abode in a town. Moreover,\nhere he was safe from treachery; for he felt sure that although the\nEnglish nobles and leaders would be incapable of breaking a truce,\nyet that there were many of lower degree who would not hesitate at\nany deed of treachery by which they might gain reward and credit\nfrom their king. Archie's band were found of the greatest service\nas messengers; and although he sometimes spent a few days at Sir\nRobert Gordon's with his mother, he generally remained by the side\nof Wallace. The spot where the Scottish leader was now staying lay\nabout halfway between Lanark and Ayr.\n\nArchie heard with uneasiness the news of the approaching council,\nand Wallace's acceptance of the invitation. The fact that the Earl\nof Percy, a very noble knight and gentleman, had been but lately\nrecalled from the governorship of Ayr and had been replaced by\none of somewhat low degree, Arlouf of Southampton, still further\nincreased his doubts. It seemed strange that the governorship of so\nimportant a town--a post deemed fitting for Earl Percy--should\nbe bestowed on such a man, were it not that one was desired who\nwould not hesitate to perform an action from which any honourable\nEnglish gentleman would shrink.\n\nTwo days before the day fixed for the council he called Cluny\nCampbell and another lad named Jock Farrel to him.\n\n\"I have a most important mission for you,\" he said. \"You have heard\nof the coming council at Ayr. I wish to find out if any evil is\nintended by the governor. For this purpose you two will proceed\nthither. You Cluny will put on the garments which you brought with\nyou; while you Jock had best go as his brother. Here is money. On\nyour way procure baskets and buy chickens and eggs, and take them\nin with you to sell.  Go hither and thither among the soldiers\nand hear what they say. Gather whether among the townspeople there\nis any thought that foul play may be intended by the English. Two\nof the band will accompany you to within a mile of Ayr, and will\nremain there in order that you may from time to time send news by\nthem of aught that you have gathered. Remember that the safety of\nWallace, and with it the future of Scotland, may depend upon your\ncare and vigilance. I would myself have undertaken the task; but\nthe Kerrs are now, I hear, in Ayr, and a chance meeting might ruin\nall; for whatever the truce between English and Scotch, they would\nassuredly keep no truce with me did they meet me. Mind, it is a\ngreat honour that I have done you in choosing you, and is a proof\nthat I regard you as two of the shrewdest of my band, although the\nyoungest among them.\"\n\nGreatly impressed with the importance of their mission, the lads\npromised to use their utmost vigilance to discover the intentions\nof the governor; and a few minutes later, Cluny being attired in\nhis sister's clothes, and looking, as Archie laughingly said, \"a\nbetter looking girl than she was herself,\" they started for Ayr,\naccompanied by two of their companions. They were to remain there\nuntil the conclusion of the council, but their companions would be\nrelieved every six hours. Upon their way they procured two baskets,\nwhich they filled with eggs and chickens; and then, leaving their\ncomrades a mile outside Ayr, fearlessly entered the town.\n\nThe council was to take place in a large wooden building some short\ndistance outside the town, which was principally chosen because it\nwas thought by the governor that the Scotch gentlemen would have\nless reluctance to meet him there than if they were asked to enter\na city with a strong garrison of English.\n\nThe first day the lads succeeded in finding out nothing which could\ngive any countenance to suspicion that treachery was intended. They\nhad agreed to work separately, and each mingled among the groups\nof citizens and soldiers, where the council was the general topic\nof conversation.  There was much wonder and speculation as to the\nobject for which the governor had summoned it, and as to the terms\nwhich he might be expected to propound, but to none did the idea\nof treachery or foul play in any way occur; and when at night they\nleft the town and sent off their message to Archie, the lads could\nonly say that all seemed fair and honest, and that none either of\nthe townspeople or soldiers appeared to have the least expectation\nof trouble arising at the council.  The following morning they\nagreed that Jock should hang round the building in which the council\nwas to be held, and where preparations for the meeting and for a\nbanquet which was afterwards to take place were being made, while\nCluny should continue his inquiries within the walls. Jock hid away\nhis basket and joined those looking on at the preparations. Green\nboughs were being carried in for decorating the walls, tables, and\nbenches for the banquet. These were brought from the town in country\ncarts, and a party of soldiers under the command of an officer\ncarried them in and arranged them. Several of the rustics looking\non gave their aid in carrying in the tables, in order that they\nmight take home to their wives an account of the appearance of the\nplace where the grand council was to be held. Jock thrust himself\nforward, and seizing a bundle of green boughs, entered the barn.\nCertainly there was nothing here to justify any suspicions. The\nsoldiers were laughing and joking as they made the arrangements;\nclean rushes lay piled against a wall in readiness to strew over\nthe floor at the last moment; boughs had been nailed against the\nwalls, and the tables and benches were sufficient to accommodate\na considerable number.  Several times Jock passed in and out, but\nstill without gathering a word to excite his suspicions. Presently\nArlouf himself, a powerful man with a forbidding countenance, rode\nup and entered the barn. He approached the officer in command of\nthe preparations; and Jock, pretending to be busy in carrying his\nboughs, managed to keep near so as to catch something of their\nconversation.\n\n\"Is everything prepared, Harris?\"\n\n\"Yes, sir; another half hour's work will complete everything.\"\n\n\"Do you think that is strong enough?\" the governor asked.\n\n\"Ay; strong enough for half a dozen of these half starved Scots.\"\n\n\"One at a time will do,\" the governor said; and then, after a few\nmore words, left the barn and rode off to Ayr.\n\nJock puzzled his head in vain over the meaning of the words he had\nheard. The governor had while speaking been facing the door; but\nto what he alluded, or what it was that the officer had declared\nstrong enough to hold half a dozen Scots, Jock could not in the\nslightest degree make out. Still the words were strange and might\nbe important; and he resolved, directly the preparations were\nfinished and the place closed, so that there could be no chance of\nhis learning more, to return himself to Archie instead of sending\na message, as much might depend upon his repeating, word for word,\nwhat he had heard, as there was somehow, he felt, a significance in\nthe manner in which the question had been asked and answered more\nthan in the words themselves.\n\nCluny had all day endeavoured in vain to gather any news. He had\nthe day before sold some of his eggs and chickens at the governor's\nhouse, and towards evening he determined again to go thither and\nto make an attempt to enter the house, where he had heard that the\nofficers of the garrison were to be entertained that evening at a\nbanquet. \"If I could but overhear what is said there, my mind would\nbe at rest. Certainly nothing is known to the soldiers; but it may\nwell be that if treachery is intended tomorrow, the governor will\nthis evening explain his plans to his officers.\"\n\nHe had, before entering the town, again filled up his basket with\nthe unsold portion of Jock's stock, for which the latter had no\nfurther occasion. The cook at the governor's, when he had purchased\nthe eggs on the previous day, had bade him call again, as Cluny's\nprices were considerably below those in the market. It was late\nin the afternoon when he again approached the house. The sentry at\nthe gate asked no question, seeing a girl with a basket, and Cluny\nwent round again to the door of the kitchen.\n\n\"How late you are, girl!\" the cook said angrily. \"You told me you\nwould come again today, and I relied upon you, and when you did\nnot come it was too late, for the market was closed.\"\n\n\"I was detained, sir,\" Cluny said, dropping a curtsey; \"my mother\nis ill, and I had to look after the children and get the dinner\nbefore they went away.\"\n\n\"There, don't waste time talking,\" the cook said, snatching the\nbasket from him. \"I have no time to count the eggs now; let me know\nthe tale of them and the chickens at the same price as you charged\nyesterday, and come for your money tomorrow; I have no time to pay\nnow. Here,\" he called to one of the scullions, \"take out these eggs\nand chickens quickly, but don't break any, and give the basket to\nthe girl here.\"\n\nSo saying he hurried off to attend to his cooking.\n\nCluny looked round. But three paces away a half open door led into\nthe interior of the house. His resolution was taken in a moment.\nSeeing that none were looking at him he stole through the door,\nhis bare feet falling noiselessly on the stones. He was now in\na spacious hall. On one side was an open door, and within was a\nlarge room with tables spread for a banquet. Cluny entered at once\nand looked round for a place of concealment; none was to be seen.\nTablecloths in those days were almost unknown luxuries. The tables\nwere supported by trestles, and were so narrow that there was\nno possibility of hiding beneath them; nor were there hangings or\nother furniture behind which he could be concealed. With a beating\nheart he turned the handle of a door leading into another apartment,\nand found himself in a long and narrow room, used apparently as\nthe private office of the governor.  There were many heavy chairs\nin the room, ranged along the wall, and Cluny crouched in a corner\nby the window beside a chair standing there. The concealment was a\npoor one, and one searching would instantly detect him; but he had\nno fear of a search, for he doubted not that the cook, on missing\nhim, would suppose that he had left at once, intending to call\nfor his money and basket together the next morning. It was already\ngrowing dusk, and should no one enter the room for another half\nhour he would be hidden in the shadow in the corner of the room;\nbut it was more probable still that no one would enter.\n\nThe time passed slowly on, and the darkness rapidly increased. Through\nthe door, which Cluny had drawn to but had not tightly closed on\nentering, he could hear the voices of the servants as they moved\nabout and completed the preparations in the banquet hall. Presently\nall was quiet, but a faint light gleaming in through the crack\nof the door showed that the lights were lit and that all was in\nreadiness for the banquet. Half an hour later and there was a heavy\ntrampling of feet and the sound of many voices. The door was suddenly\nclosed, and Cluny had no doubt that the dinner was beginning. Rising\nto his feet he made to the door and listened attentively.\n\nA confused din met his ears, but no distinct words were audible.\nHe could occasionally faintly hear the clattering of plates and\nthe clinking of glasses. All this continued for nigh two hours, and\nthen a sudden quiet seemed to fall upon the assembly. Cluny heard\nthe door close, and guessed that the banquet was at an end and the\nservitors dismissed. Now, if ever, would something of importance\nbe said within, and Cluny would have given his life to be able to\nhear it. Many times he thought of turning the handle and opening\nthe door an inch or two. Locks in those days were but roughly made;\nthe slightest sound might attract attention, and in that case not\nonly would his own life be forfeited, but no news of the governor's\nintentions--no matter what they might be--could reach Wallace;\nso, almost holding his breath, he lay on the ground and listened\nwith his ear to the sill of the door. The silence was succeeded by\na steady monotonous sound as of one addressing the others. Cluny\ngroaned in spirit, for no word could he hear. After some minutes\nthe murmur ceased, and then many voices were raised together; then\none rose above the rest, and then, distinct and clear, came a voice\nevidently raised in anger.\n\n\"As you please, Master Hawkins; but if you disobey my orders,\nas King Edward's governor here, you will take the consequences. I\nshall at once place you in durance, and shall send report to the\nking of your mutinous conduct.\"\n\n\"Be that as it may,\" another voice replied; \"whatever befall me, I\ntell you, sir, that Thomas Hawkins will take no part in an act of\nsuch foul and dastardly treachery. I am a soldier of King Edward.\nI am paid to draw my sword against his enemies, and not to do the\nbloody work of a murderer.\"\n\n\"Seize him!\" the governor shouted. \"Give him in charge to the guard,\nto lay in the castle dungeon.\"\n\nThere was a movement of feet now heard, but Cluny waited no\nlonger. The angry utterances had reached his ear, and knowing that\nhis mission was accomplished he thought only now of escape before\ndetection might take place. He had noticed when he entered the room\nthat the windows were, as was usually the case with rooms on the\nlower floors, barred; but he saw also that the bars were wide enough\napart for a lad of his slimness to crawl through. The banqueting\nroom was raised three steps above the hall, and the room that he\nwas in was upon the same level; the window was four feet from the\nfloor, and would therefore be probably seven or eight above the\nground without, which would account for its not being more closely\nbarred. He speedily climbed up to it and thrust himself through the\nbars, but not without immense difficulty and great destruction to\nhis feminine garments.\n\n\"Poor Janet!\" Cluny laughed to himself as he dropped from the\nwindow to the ground. \"Whatever would she say were she to see the\nstate of her kirtle and petticoats!\"\n\nThe moon was young, but the light was sufficient to enable Cluny\nto see where he was. The window opened into a lane which ran down\nby the side of the governor's house, and he was soon in the principal\nstreet. Already most of the citizens were within their houses. A\nfew, provided with lanterns, were picking their way along the uneven\npavement.  Cluny knew that it was impossible for him to leave the\ntown that night; he would have given anything for a rope by which\nhe might lower himself from the walls, but there was no possibility\nof his obtaining one. The appearance of a young girl wandering in\nthe streets alone at night would at once have attracted attention\nand remarks. So Cluny withdrew into a dark archway, and then sat\ndown until the general silence told him that all had retired to\nrest. Then he made his way along the street until he neared the\ngateway, and there lying down by the wall he went to sleep.\n\nWhen the gate was opened in the morning Cluny waited until a few\npersons had passed in and out and then approached it.  \"Hallo! lass,\"\nthe sergeant of the guard, who was standing there, said. \"You are\na pretty figure with your torn clothes!  Why, what has happened to\nyou?\"\n\n\"If you please, sir,\" Cluny said timidly, \"I was selling my eggs\nto the governor's cook, and he kept me waiting, and I did not know\nthat it was so late, and when I got to the gates they were shut,\nand I had nowhere to go; and then, please sir, as I was wandering\nabout a rough soldier seized me and wanted to kiss me, and of\ncourse I would not let him, and in the struggle he tore my clothes\ndreadfully; and some burghers, who heard me scream, came up and the\nman left me, and one of the burghers let me sleep in his kitchen,\nand I don't know what mother will say to my clothes;\" and Cluny\nlifted the hem of his petticoat to his eyes.\n\n\"It is a shame, lass,\" the sergeant said good temperedly; \"an I\nhad been there I would have broke the fellow's sconce for him; but\nanother time, lass, you should not overstay the hour; it is not good\nfor young girls to be roaming at night in a town full of soldiers.\nThere, I hope your mother won't beat you, for, after all, it was\nthe fault of the governor's cook rather than yours.\"\n\nCluny pursued his way with a quiet and depressed mien until he was\nfairly out of sight of the gates. Then he lifted his petticoats to\na height which would have shocked his sister Janet, to give free\nplay to his limbs, and at the top of his speed dashed down the road\ntoward Lanark. He found his two companions waiting at the appointed\nspot, but he did not pause a moment.\n\n\"Are you mad, Cluny?\" they shouted.\n\nAnd indeed the wild figure, with its tucked up garments, tearing\nat full speed along the road, would have been deemed that of a mad\ngirl by any who had met it.\n\n\"Come on!\" he shouted. \"Come on, it is for life or death!\" and\nwithout further word he kept on at full speed.  It was some time\nbefore his companions overtook him, for they were at first too\nconvulsed by laughter at Cluny's extraordinary appearance to be able\nto run. But presently, sobered by the conviction that something of\nextreme importance must have happened, they too started at their\nbest speed, and presently came up with Cluny, upon whose pace the\nmile he had already run told heavily.\n\n\"For the sake of goodness, Cluny, go slower,\" one of them panted\nout as they came to him. \"We have nine miles yet to run, and if we\ngo on like this we shall break down in another half mile, and have\nto walk the rest.\"\n\nCluny himself, with all his anxiety to get on, was beginning to\nfeel the same, and he slackened his pace to a slinging trot, which\nin little over an hour brought them to the wood.\n\n\n\n\nChapter VI\n\nThe Barns of Ayr\n\n\nArchie was anxiously awaiting the arrival of his messenger, for the\nthree lads were met two miles out by another who had been placed on\nwatch, and had come on ahead at full speed with the news of their\napproach.  The report brought in by Jock Farrell of the words that\nhe had overheard in the barn prepared for the meeting, had been\nreported by Archie to Wallace. Sir John Grahame and the other\ngentlemen with him all agreed that they were strange, and his friends\nhad strongly urged their leader not to proceed to the meeting.\nWallace, however, persisted in his resolution to do so, unless\nhe received stronger proofs than those afforded by the few words\ndropped by the governor and his officer, which might really have\nno evil meaning whatever. He could not throw doubt upon the fair\nintentions of King Edward's representative, for it might well be\nsaid that it was the grossest insult to the English to judge them\nas guilty of the intention of a foul act of treachery upon such\nslight foundation as this. \"It would be a shame indeed,\" he said,\n\"were I, the Warden of Scotland, to shrink from appearing at\na council upon such excuse as this.\" The utmost that Archie could\nobtain from him was that he would delay his departure in the morning\nuntil the latest moment, in order to see if any further news came\nfrom Ayr.\n\nThe meeting was to be held at ten o'clock, and until a little before\nnine he would not set out. He was in the act of mounting his horse\nwhen Cluny Campbell arrived.\n\n\"What are your news, Cluny?\" Archie exclaimed, as the lads, panting\nand exhausted, ran up.\n\n\"There is treachery intended. I overheard the governor say so.\"\n\n\"Come along with me,\" Archie exclaimed; \"you are just in time,\nand shall yourself tell the news. Draw your bridle, Sir William,\"\nhe exclaimed as he ran up to the spot where Sir William Wallace,\nGrahame, and several other gentlemen were in the act of mounting.\n\"Treachery is intended--my messenger has overheard it. I know\nnot his tale, but question him yourself.\"\n\nImportant as was the occasion, the Scottish chiefs could not resist\na smile at the wild appearance of Archie's messenger.\n\n\"Is it a boy or a girl?\" Wallace asked Archie, \"for it might be\neither.\"\n\n\"He is one of my band, sir. I sent him dressed in this disguise as\nit would be the least suspected. Now, Cluny, tell your own story.\"\n\nCluny told his story briefly, but giving word for word the sentences\nthat he had heard spoken in anger by the governor and his officer.\n\n\"I fear there can be no doubt,\" Wallace said gravely when the\nlad had finished--\"that foul play of some kind is intended, and\nthat it would be madness to trust ourselves in the hands of this\ntreacherous governor.  Would that we had had the news twenty-four\nhours earlier; but even now some may be saved. Sir John, will you\ngallop, with all your mounted men, at full speed towards Ayr. Send\nmen on all the roads leading to the council, and warn any who may\nnot yet have arrived against entering.\"\n\nSir John Grahame instantly gave orders to all those who had horses,\nto mount and follow him at the top of their speed; and he himself,\nwith the other gentlemen whose horses were prepared, started at\nonce at full gallop.\n\n\"Sir Archie, do you cause the 'assembly' to be sounded, and send\noff your runners in all directions to bid every man who can be\ncollected to gather here this afternoon at three o clock. If foul\nplay has been done we can avenge, although we are too late to save,\nand, by Heavens, a full and bloody revenge will I take.\"\n\nIt was not until two in the afternoon that Sir John Grahame returned.\n\n\"The worst has happened; I can read it in your face,\" Wallace\nexclaimed.\n\n\"It is but too true,\" Sir John replied. \"For a time we could obtain\nno information. One of my men rode forward until close to the Barns,\nand reported that all seemed quiet there. A guard of soldiers were\nstanding round the gates, and he saw one of those invited, who had\narrived a minute before him, dismount and enter quietly. Fortunately\nI was in time to stop many gentlemen who were proceeding to the\ncouncil, but more had entered before I reached there. From time\nto time I sent forward men on foot who talked with those who were\nstanding without to watch the arrivals. Presently a terrible rumour\nbegan to spread among them--whether the truth was known from some\ncoarse jest by one of the soldiers, or how it came out, I know not.\nBut as time went on, and the hour was long past when any fresh\narrivals could be expected, there was no longer motive for secrecy,\nand the truth was openly told. Each man as he entered was stopped\njust inside the door. A noose was dropped over his neck, and he\nwas hauled up to a hook over the door. All who entered are dead.\"\n\nA cry of indignation and rage broke from Wallace and those standing\nround him, and the Scottish leader again repeated his oath to take\na bloody vengeance for the deed.\n\n\"And who are among the murdered?\" he asked, after a pause.\n\n\"Alas! Sir William,\" Grahame said, \"your good uncle, Sir Ronald\nCrawford, the Sheriff of Ayr, is one; and also Sir Richard Wallace\nof Riccartoun; Sir Bryce Blair, and Sir Neil Montgomery, Boyd,\nBarclay, Steuart, Kennedy, and many others.\"\n\nWallace was overwhelmed with grief at the news that both his uncles,\nto whom he was greatly attached, had perished. Most of those around\nhad also lost relatives and friends, and none could contain their\ngrief and indignation.\n\n\"Was my uncle, Sir Robert Gordon, among the victims?\" Archie\ninquired.\n\n\"No,\" Sir John replied; \"happily he was one of the last who came\nalong the road.\"\n\n\"Thank God for that!\" Archie said earnestly; \"my uncle's slowness\nhas saved his life. He was ever late for business or pleasure, and\nmy aunt was always rating him for his unpunctuality. She will not\ndo so again, for assuredly it has saved his life.\"\n\nThe men came in but slowly, for the bands had all dispersed to\ntheir homes, and it was only those who lived within a few miles\nwho could arrive in time. Little over fifty men had come in by the\nhour named. With these Wallace started at once towards Ayr. Archie's\nband fell in with their arms, for they too burned to revenge the\nmassacre, and Wallace did not refuse Archie's request that they\nmight join.\n\n\"Let them come,\" he said; \"we shall want every sword and pike\ntonight.\"\n\nThis was the first time that Wallace had seen the band under arms,\nfor at the battle of Biggar, Archie had kept them from his sight,\nfearing that he might order them from the field.\n\n\"They look well, Sir Archie, and in good military order.  Hitherto\nI have regarded them but as messengers, and as such they have done\ngood service indeed; but I see now that you have them in good order,\nand that they can do other service on a pinch.\"\n\nOne member of Wallace's band was left behind, with orders to wait\nuntil seven o'clock, and then to bring on as fast as they could\nmarch all who might arrive before that hour. The band marched to\nwithin a mile of the barns. They then halted at a stack of straw,\nand sat down while one of Archie's band went forward to see what was\nbeing done. He reported that a great feast, at which the governor\nand all the officers of the garrison, with other English dwelling\nin town, were present, was just beginning in the great barn where\nthe massacre had taken place.\n\nSoon after nine o'clock the man who had been left behind, with ten\nothers, who had come in after Wallace had marched, came up. Each\nman, by Wallace's directions, drew a great truss of straw from the\nstack, and then the party, now eighty in all, marched toward the\nbarn. Wallace's instructions were that so soon as the work had\nfairly begun, Grahame, with Archie and half the band, was to hurry\noff to seize the gate of Ayr, feigning to be a portion of the guard\nat the barn.\n\nWhen they approached the spot they saw that the wooden building was\nbrightly lit up with lights within, and the English guard, some\nfifty in number, were standing carelessly without, or, seated\nround fires, were carousing on wine which had been sent out by the\nrevellers within.\n\nThe Scotch stole up quietly. Wallace's party, composed of half the\nstrength, handed their bundles of straw to the men of Grahame's\ncompany; then with a sudden shout they fell upon the English\nsoldiers, while Grahame's men, running straight to the door of the\nbarn, threw down their trusses of straw against it, and Sir John,\nsnatching down a torch which burned beside the entrance, applied\nfire to the mass, and then, without a moment's delay, started at a\nrun towards the town. Taken wholly by surprise the English soldiers\nwere slain by Wallace and his men almost before they had time\nto seize their arms. Then the Scots gathered round the barn.  The\nflames were already leaping up high, and a terrible din of shouts\nand cries issued from within. The doors had been opened now, but\nthose within were unable to force their way across the blazing mass\nof straw. Many appeared at the windows and screamed for mercy, and\nsome leapt out, preferring to fall by the Scottish swords rather\nthan to await death by fire within.\n\nThe flames rose higher and higher, and soon the whole building\nwas enveloped, and ere many minutes all those who had carried out,\nif not planned, the massacre of Ayr had perished. In the meantime\nGrahame and his party had reached the gate of Ayr. Bidding others\nfollow him at a distance of about a hundred yards, he himself, with\nArchie and ten of his followers, ran up at full speed.\n\n\"Quick!\" he shouted to the sentry on the gate. \"Lower the bridge\nand let us in. We have been attacked by Wallace and the Scots, and\nthey will speedily be here.\"\n\nThe attention of the guard had already been attracted by the sudden\nburst of light by the barns. They had heard distant shouts, and\ndeemed that a conflagration had broken out in the banqueting hall.\nNot doubting for an instant the truth of Grahame's story, they\nlowered the drawbridge instantly, and Sir John and his companions\nrushed across.\n\nThe guard were only undeceived when Grahame and his followers fell\nupon them with their heavy broadswords.  They had left their arms\nbehind when they had assembled on the walls to look at the distant\nflames, and were cut down to a man by the Scots. By this time the\nrest of Grahame's band had arrived.\n\nSo short and speedy had been the struggle that no alarm had been\ngiven in the town. The inmates of a few houses near opened their\nwindows and looked out.\n\n\"Come down as quickly as you may,\" Sir John said to them; \"we have\ntaken Ayr.\"\n\nSeveral of the burghers were soon in the street.\n\n\"Now,\" Sir John said, \"do two of you who know the town well go\nwith me and point out the houses in which the English troops are\nquartered; let the others go from house to house, and bid every\nman come quickly with his sword to strike a blow for freedom.\"\n\nSir John now went round the town with the guides and posted two or\nmore men at the door of each house occupied by the English. Soon\nthe armed citizens flocked into the streets, and when sufficient\nwere assembled the blowing of a horn gave the signal. The doors of\nthe houses were beaten in with axes, and, pouring in, the Scotch\nslew the soldiers before they had scarce awakened from sleep. Very\nfew of the English in the town escaped to tell of the terrible\nretaliation which had been taken for the massacre of Ayr.\n\nOne of the few who were saved was Captain Thomas Hawkins. Archie,\nmindful of the part which he had taken, and to which, indeed, the\ndiscovery of the governor's intention was due, had hurried direct to\nthe prison, and when this was, with the rest of the town, taken,\ndiscovered the English officer in chains in a dungeon, and protected\nhim from all molestation.\n\nThe next morning he was brought before Wallace, who expressed to\nhim his admiration of the honourable course which he had adopted,\ngave him a rich present out of the booty which had been captured,\nand placed him on a ship bound for England.\n\nA week after the capture of Ayr one of Archie's band came into his\nhut. Tears were running down his cheeks, and his face was swollen\nwith weeping.\n\n\"What is it, Jock?\" Archie asked kindly.\n\n\"Ah! Sir Archie! we have bad news from Glen Cairn.  One has come\nhither who says that a few days since the Kerrs, with a following\nof their own retainers, came down to the village. Having heard\nthat some of us had followed you to the wars, they took a list of\nall that were missing, and Sir John called our fathers up before\nhim. They all swore, truly enough, that they knew nought of our\nintentions, and that we had left without saying a word to them.\nSir John refused to believe them, and at first threatened to hang\nthem all.  Then after a time he said they might draw lots, and\nthat two should die. My father and Allan Cunninghame drew the evil\nnumbers, and Kerr hung them up to the old tree on the green and put\nfire to the rooftrees of all the others. Ah! but there is weeping\nand wailing in Glen Cairn!\"\n\nArchie was for a while speechless with indignation. He knew well\nthat this wholesale vengeance had not been taken by the Kerrs because\nthe sons of the cottagers of Glen Cairn had gone to join the army\nof Wallace, but because he deemed them to be still attached to their\nold lord; and it was to their fidelity to the Forbeses rather than\nto Scotland that they owed the ruin which had befallen them.\n\n\"My poor Jock!\" he said, \"I am grieved, indeed, at this misfortune.\nI cannot restore your father's life, but I can from the spoils of\nAyr send a sufficient sum to Glen Cairn to rebuild the cottages\nwhich the Kerrs have destroyed. But this will not be enough--we\nwill have vengeance for the foul deed. Order the band to assemble\nat dusk this evening, and tell Orr and Macpherson to come here to\nme at once.\"\n\nArchie had a long consultation with his two young lieutenants,\nwhose fathers' cottages had with the others been destroyed.\n\n\"What we have to do,\" Archie said, \"we must do alone.  Sir William\nhas ample employment for his men, and I cannot ask him to weaken\nhis force to aid me in a private broil; nor, indeed, would any aid\nshort of his whole band be of use, seeing that the Kerrs can put\nthree hundred retainers in the field. It is not by open force that\nwe must fight them, but by fire and harassment. Fighting is out\nof the question; but we can do him some damage without giving him\na chance of striking a blow at us. As he has lighted Glen Cairn,\nso shall he see fires blazing round his own castle of Aberfilly.\nWe will not retaliate by hanging his crofters and vassals; but if\nhe or any of his men-at-arms falls into our hands, we will have\nblood for blood.\"\n\nIn the course of the afternoon Archie saw his chief and begged\nleave to take his troop away for some time, telling Sir William of\nthe cruel treatment which the Kerrs had dealt at Glen Cairn, and\nhis determination to retaliate for the deed.\n\n\"Aberfilly is a strong castle, Archie,\" Wallace said; \"at least so\npeople say, for I have never seen it, so far does it lie removed\nfrom the main roads. But unless by stratagem, I doubt if my force\nis strong enough to capture it; nor would I attack were I sure of\ncapturing it without the loss of a man.  The nobles and landowners\nstand aloof from me; but it may be that after I have wrested some\nmore strong places from the English, they may join me. But I would\nnot on any account war against one of them now. Half the great\nfamilies are united by ties of blood or marriage. The Kerrs, we\nknow, are related to the Comyns and other powerful families; and\ndid I lift a hand against them, adieu to my chance of being joined\nby the great nobles. No; openly hostile as many of them are, I must\nlet them go their way, and confine my efforts to attacking their\nfriends the English. Then they will have no excuse of personal\nfeud for taking side against the cause of Scotland. But this does\nnot apply to you. Everyone knows that there has long been a blood\nfeud between the Forbeses and the Kerrs, and any damage you may\ndo them will be counted as a private feud. I think it is a rash\nadventure that you are undertaking with but a handful of boys,\nalthough it is true that a boy can fire a roof or drive off a\nbullock as well as a man. However, this I will promise you, that\nif you should get into any scrape I will come with what speed I\ncan to your rescue, even if it embroil me with half the nobles of\nScotland. You embroiled yourself with all the power of England in\nmy behalf, and you will not find me slack in the hour of need. But\nif I join in the fray it is to rescue my friend Archie Forbes, and\nnot to war against John Kerr, the ally of the English, and my own\nenemy.\"\n\nArchie warmly thanked his leader, but assured him that he had no\nthought of placing himself in any great peril.\n\n\"I am not going to fight,\" he said, \"for the Kerr and his retainers\ncould eat us up; we shall trust to our legs and our knowledge of\nthe mountains.\"\n\nAfter dark Archie and his band started, and arrived within ten miles\nof Aberfilly on the following morning.  They rested till noon, and\nthen again set out. When they approached one of the outlying farms\nof the Kerrs, Archie halted his band, and, accompanied by four of\nthe stoutest and tallest of their number, went on to the crofter's\nhouse. The man came to the door.\n\n\"What would you, young sir?\" he said to Archie.\n\n\"I would,\" Archie said, \"that you bear a message from me to your\nlord.\"\n\n\"I know not what your message may be; but frankly, I would rather\nthat you bore it yourself, especially if it be of a nature to anger\nSir John.\"\n\n\"The message is this,\" Archie said quietly: \"tell him that Archibald\nForbes bids him defiance, and that he will retort upon him and his\nthe cruelties which he has wrought in Glen Cairn, and that he will\nrest not night nor day until he has revenge for the innocent blood\nshed and rooftrees ruthlessly burned.\"\n\n\"Then,\" the crofter said bluntly, \"if you be Archibald Forbes, you\nmay even take your message yourself. Sir John cares not much upon\nwhose head his wrath lights, and I care not to appear before him\nas a willing messenger on such an errand.\"\n\n\"You may tell him,\" Archie said quietly, \"that you are no willing\nmessenger; for that I told you that unless you did my errand your\nhouse should, before morning, be a heap of smoking ashes. I have\na following hard by, and will keep my word.\"\n\nThe crofter hesitated.\n\n\"Do my bidding; and I promise you that whatever may befall the\nother vassals of the Kerrs, you shall go free and unharmed.\"\n\n\"Well, if needs must, it must,\" the crofter said; \"and I will\ndo your bidding, young sir--partly because I care not to see my\nhouse in ruins, but more because I have heard of you as a valiant\nyouth who fought stoutly by the side of Wallace at Lanark and\nAyr--though, seeing that you are but a lad, I marvel much that you\nshould be able to hold your own in such wild company. Although\nas a vassal of the Kerrs I must needs follow their banner, I need\nnot tell you, since you have lived so long at Glen Cairn, that the\nKerrs are feared rather than loved, and that there is many a man\namong us who would lief that our lord fought not by the side of the\nEnglish. However, we must needs dance as he plays; and now I will\nput on my bonnet and do your errand. Sir John can hardly blame me\ngreatly for doing what I needs must.\"\n\nGreat was the wrath of Sir John Kerr when his vassal reported to\nhim the message with which he had been charged, and in his savage\nfury he was with difficulty dissuaded from ordering him to be hung\nfor bringing such a message. His principal retainers ventured,\nhowever, to point out that the man had acted upon compulsion, and\nthat the present was not the time, when he might at any moment\nhave to call upon them to take the field, to anger his vassals, who\nwould assuredly resent the undeserved death of one of their number.\n\n\"It is past all bearing,\" the knight said furiously, \"that an insolent\nboy like this should first wound me in the streets of Lanark, and\nshould then cast his defiance in my teeth--a landless rascal,\nwhose father I killed, and whose den of a castle I but a month ago\ngave to the flames. He must be mad to dare to set his power against\nmine. I was a fool that I did not stamp him out long ago; but woe\nbetide him when we next meet!  Had it not been that I was served\nby a fool\"--and here the angry knight turned to his henchman, Red\nRoy--\"this would not have happened. Who could have thought that\na man of your years could have suffered himself to be fooled by a\nboy, and to bring me tales that this insolent upstart was a poor\nstupid lout! By Heavens! to be thus badly served is enough to make\none mad!\"\n\n\"Well, Sir John,\" the man grumbled, \"the best man will be sometimes\nin error. I have done good service for you and yours, and yet ever\nsince we met this boy outside the gates of Lanark you have never\nceased to twit me concerning him.  Rest secure that no such error\nshall occur again, and that the next time I meet him I will pay him\nalike for the wound he gave you and for the anger he has brought\nupon my head. If you will give orders I will start at daybreak\nwith twenty men.  I will take up his trail at the cottage of John\nFrazer, and will not give up the search until I have overtaken and\nslain him.\"\n\n\"Do so,\" the knight replied, \"and I will forgive your having\nbeen so easily fooled. But this fellow may have some of Wallace's\nfollowers with him, and contemptible as the rabble are, we had best\nbe on our guard.  Send round to all my vassals, and tell them to\nkeep good watch and ward, and keep a party of retainers under arms\nall night in readiness to sally out in case of alarm.\"\n\nThe night, however, passed quietly. The next day the knight sallied\nout with a strong party of retainers, and searched the woods and\nlower <DW72>s of the hill, but could find no signs of Archie and\nhis followers, and at nightfall returned to the castle in a rage,\ndeclaring that the defiance sent him was a mere piece of insolent\nbravado. Nevertheless, he kept the horses again saddled all night\nready to issue out at the slightest alarm.  Soon after midnight\nflames suddenly burst out at a dozen of the homesteads. At the\nwarder's shout of alarm Sir John Kerr and his men-at-arms instantly\nmounted. The gate was thrown open and the drawbridge lowered, and\nSir John rode out at the head of his following.  He was within a\nfew feet of the outer end of the drawbridge when the chains which\nsupported this suddenly snapped. The drawbridge fell into the moat,\nplunging all those upon it into the water.\n\nArchie, with his band, after detaching some of their number to fire\nthe homesteads, had crept up unperceived in the darkness to the\nend of the drawbridge, and had noiselessly cut the two projecting\nbeams upon which its end rested when it was lowered. He had intended\nto carry out this plan on the previous night, but when darkness set\nin not a breath of wind was stirring, and the night was so still\nthat he deemed that the operation of sawing through the beams could\nnot be effected without attracting the attention of the warders\non the wall, and had therefore retreated far up in the recesses of\nthe hills. The next night, however, was windy, and well suited to\nhis purpose, and the work had been carried out without attracting\nthe attention of the warders. When Kerr and his men-at-arms rode\nout, the whole weight of the drawbridge and of the horsemen crossing\nit was thrown entirely upon the chains, and these yielded to a\nstrain far greater than they were calculated to support.\n\nThe instant the men-at-arms were precipitated into the moat, Archie\nand his companions, who had been lying down near its edge, leapt to\ntheir feet, and opened fire with their bows and arrows upon them.\nIt was well for Sir John and his retainers that they had not stopped to\nbuckle on their defensive armour. Had they done so every man must\nhave been drowned in the deep waters. As it was, several were killed\nwith the arrows, and two or three by the hoofs of the struggling\nhorses. Sir John himself, with six of the eighteen men who had\nfallen into the moat, succeeded in climbing up the drawbridge and\nregaining the castle. A fire of arrows was at once opened from the\nwalls, but Archie and his followers were already out of bowshot;\nand knowing that the fires would call in a few minutes to the spot\na number of the Kerr's vassals more than sufficient to crush them\nwithout the assistance of those in the castle, they again made for\nthe hills, well satisfied with the first blow they had struck at\ntheir enemies.\n\nThe rage of Sir John Kerr was beyond all expression.  He had himself\nbeen twice struck by arrows, and the smart of his wounds added to\nhis fury. By the light of the burning barns the garrison were enabled\nto see how small was the party which had made this audacious attack\nupon them; and this increased their wrath.  Men were instantly set\nat work to raise the drawbridge from the moat, to repair the chains,\nand to replace the timbers upon which it rested; and a summons was\ndespatched to the whole of the vassals to be at the castle in arms\nby daybreak.\n\nAgain the woods were searched without success, and the band then\ndivided into five parties, each forty strong.  They proceeded to\nexplore the hills; but the Pentlands afforded numerous hiding places\nto those, like Archie and most of his band, well acquainted with\nthe country; and after searching till nightfall the parties retired,\nworn out and disheartened, to the castle. That night three of the\noutlying farms were in flames, and the cattle were slaughtered in\ntheir byres, but no attack was made upon the dwelling houses.  The\nfollowing night Sir John distributed the whole of his vassals among\nthe farms lying farthest from the castle, putting twenty men in\neach; but to his fury this time it was five homesteads nearer at\nhand which were fired. The instant the first outburst of flame was\ndiscovered the retainers hurried to the spot; but by the time they\nreached it no sign of the assailants was visible; the flames had\nhowever taken too good a hold of the various barns and outbuildings\nto be extinguished.\n\n\n\n\nChapter VII\n\nThe Cave in the Pentlands\n\n\nJohn Kerr was well nigh beside himself with fury.\n\nIf this was to go on, the whole of his estate would be harried,\nhis vassals ruined, and his revenues stopped, and this by a mere\nhandful of foes. Again he started with his vassals to explore the\nhills, this time in parties of ten only, so as to explore thoroughly\na larger space of ground. When at evening the men returned, it was\nfound that but two men of one of the parties, composed entirely of\nmen-at-arms from the castle, came back.  They reported that when\nin a narrow ravine showers of rocks were hurled down upon them from\nboth sides.  Four of their number were killed at once, and four\nothers had fallen pierced by arrows from an unseen foe as they fled\nback down the ravine.\n\n\"Methinks, Sir John,\" Red Roy said, \"that I know the place where\nthe Forbeses may have taken up their abode.  When I was a boy I\nwas tending a herd of goats far up in the hills, and near the pass\nwhere this mischance has today befallen us I found a cave in the\nmountain's side. Its entrance was hidden by bushes, and I should\nnot have found it had not one of the goats entered the bush and\nremained there so long that I went to see what he was doing. There\nI found a cave. The entrance was but three feet high, but inside\nit widened out into a great cavern, where fifty men could shelter.\nPerchance Archie Forbes or some of his band may also have discovered\nit; and if so, they might well think that no better place of\nconcealment could be found.\"\n\n\"We will search it tomorrow,\" the knight said. \"Tell the vassals\nto gather here three hours before daybreak.  We will start so as\nto be there soon after sunrise. If they are on foot again tonight\nthey will then be asleep.  Did you follow the cave and discover\nwhether it had any other entrances beyond that by which you entered?\"\n\n\"I know not,\" the henchman replied; \"it goes a long way into the\nhills, and there are several inner passages; but these I did not\nexplore, for I was alone and feared being lost in them.\"\n\nThe next night some more homesteads were burnt, but this time the\nvassals did not turn out, as they had been told to rest until the\nappointed hour whatever might befall.\n\nThree hours before daybreak a party of fifty picked men assembled\nat the castle, for this force was deemed to be ample.  The two men\nwho had escaped from the attack on the previous day led the way\nto the ravine, and there Red Roy became the guide and led the band\nfar up the hillside. Had it been possible they would have surrounded\nthe cave before daylight, but Roy said that it was so long since\nhe had first found the cave, that he could not lead them there\nin the dark, but would need daylight to enable him to recognize\nthe surroundings. Even when daylight came he was for some time at\nfault, but he at last pointed to a clump of bushes, growing on a\nbroken and precipitous face of rock, as the place where the cave\nwas situated.\n\nRed Roy was right in his conjecture. Archie had once, when wandering\namong the hills, shot at a wild cat and wounded it, and had followed\nit to the cave to which it had fled, and seeing it an advantageous\nplace of concealment had, when he determined to harry the district\nof the Kerrs, fixed upon it as the hiding place for his band. Deeming\nit possible, however, that its existence might be known to others,\nhe always placed a sentry on watch; and on the approach of the Kerrs,\nCluny Campbell, who happened to be on guard, ran in and roused the\nband with the news that the Kerrs were below. Archie immediately\ncrept out and reconnoitred them; from the bushes he could see that\nhis foes were for the present at fault. Sir John himself was standing\napart from the rest, with Red Roy, who was narrowly scrutinizing\nthe face of the cliff, and Archie guessed at once that they were\naware of the existence of the cavern, though at present they could\nnot determine the exact spot where it was situated. It was too late\nto retreat now, for the face of the hill was too steep to climb\nto its crest, and their retreat below was cut off by the Kerrs. He\ntherefore returned to the cave, leaving Cluny on guard.\n\n\"They are not sure as to the situation of the cave yet,\" he said,\n\"but they will find it. We can hold the mouth against them for any\ntime, but they might smoke us out, that is our real danger; or if\nthey fail in that, they may try starvation.  Do half a dozen of you\ntake brands at once from the embers and explore all the windings\nbehind us; they are so narrow and low that hitherto we have not\ndeemed it worth while to examine them, but now they are really our\nonly hope; some of them may lead round to the face of the hill,\nand in that case we may find some way by which we may circumvent\nthe Kerrs.\"\n\nSix of the lads at once started with flaming pine knots, while\nArchie returned to the entrance. Just as he took his place there\nhe saw Red Roy pointing towards the bushes. A minute or two later\nSir John and his followers began to advance. Archie now called out\nthe rest of his band, who silently took their places in the bushes\nbeside him. Led by Sir John and his personal retainers, the assailants\napproached the foot of the rocks and began to make their way up,\nusing the utmost precaution to avoid any noise. There was no longer\nany need for concealment, and as the foremost of the assailants\nbegan to climb the great boulders at the foot of the precipice,\na dozen arrows from the bush above alighted among them; killing\nthree and wounding several others. Sir John Kerr shouted to his\nmen to follow him, and began to clamber up the hill. Several arrows\nstruck him, but he was sheathed in mail, as were his men-at-arms,\nand although several were wounded in the face and two slain they\nsucceeded in reaching the bushes, but they could not penetrate\nfurther, for as they strove to tear the bushes aside and force an\nentry, those behind pierced them with their spears, and as but four\nor five assailants at a time could gain a footing and use their\narms they were outnumbered and finally driven back by the defenders.\nWhen Sir John, furious at his discomfiture, rejoined his vassals\nbelow, he found that the assault had already cost him eight of his\nbest men. He would, however, have again led them to the attack,\nbut Red Roy said:\n\n\"It were best, my lord, to send back and bid fifty of the vassals\nto come up hither at once, with bows and arrows.  They can so riddle\nthose bushes that the defenders will be unable to occupy them to\nresist our advance.\"\n\n\"That were a good step,\" Sir John said; \"but even when we gain\nthe ledge I know not how we shall force our way through the hole,\nwhich you say is but three feet high.\"\n\n\"There is no need to force our way in,\" Red Roy replied; \"each\nman who climbs shall carry with him a <DW19> of wood, and we will\nsmoke them in their holes like wolves.\"\n\n\"'Tis well thought of, Roy; that assuredly is the best plan.  Send\noff at once one of the most fleet footed of the party.\"\n\nArchie, watching from above, saw the assailants draw back out of\nbowshot, and while one of their number started at full speed down\nthe hillside, the others sat down, evidently prepared to pass some\ntime before they renewed the attack.  Leaving two of the party on\nguard, Archie, with the rest, re-entered the cavern.  The searchers\nhad just returned and reported that all the various passages came\nto nothing, save one, which ascended rapidly and terminated in a\nhole which looked as if it had been made by rabbits, and through\nwhich the light of day could be seen.\n\n\"Then it is there we must work,\" Archie said. \"I will myself go\nand examine it.\"\n\nThe passage, after ascending to a point which Archie judged to\nbe nigh a hundred feet above the floor of the cave, narrowed to a\nmere hole, but two feet high and as much wide. Up this he crawled\nfor a distance of four or five yards, then it narrowed suddenly\nto a hole three or four inches in diameter, and through this, some\nthree feet farther, Archie could see the daylight through a clump\nof heather. He backed himself down the narrow passage again until\nhe joined his comrades. \"Now,\" he said, \"do four of you stay here,\nand take it by turns, one after the other, to enlarge the hole\nforward to the entrance. As you scrape the earth down you must past\nit back handful by handful. Do not enlarge the outer entrance or\ndisturb the roots of the heather growing there. Any movement might\nbe noticed by those below. It is lucky, indeed, that the rock ends\njust when it gets to its narrowest, and that it is but sandy soil\nthrough which we have to scrape our way.  It will be hard work,\nfor you have scarce room to move your arms, but you have plenty of\ntime since we cannot sally out till nightfall.\"\n\nThe hours passed slowly, and about noon the lookout reported that\na number of bowmen were approaching.\n\n\"They are going to attack this time under cover of their fire,\"\nArchie said, \"and as I do not wish to hazard the loss of any lives,\nwe will keep within the cave and let them gain the ledge. They can\nnever force their way through the narrow entrance. The only thing\nI fear is smoke. I purpose that if they light a fire at the mouth\nof the cave, we shall retire at once up the passage where we are\nworking, and block it up at a narrow place a short distance after\nit leaves this cavern, with our clothes. You had best take off some\nof your things, scrape up the earth from the floor of the cavern,\nand each make a stout bundle, so that we can fill up the hole\nsolidly.\"\n\nThis was soon done, and the bundles of earth were laid in readiness\nat the point upon which their leader had fixed.  In the meantime\nArchie had rejoined the lookout.\n\n\"They have been scattered for some time,\" the guard said, \"and have\nbeen cutting down bushes and making them into <DW19>s.\"\n\n\"Just what I expected,\" Archie exclaimed. \"The bowmen are joining\nthem now. We shall soon see them at work.\"\n\nSir John Kerr now marshalled his retainers. He and his men-at-arms\ndrew their swords, and the rest, putting the bundles of <DW19>s on\ntheir shoulders, prepared to follow, while the bowmen fitted their\narrows to the string.\n\n\"Fall back inside the cave,\" Archie said; \"it is of no use risking\nour lives.\"\n\nThe band now gathered in a half circle, with level spears, round\nthe entrance. Soon they heard a sharp tapping sound as the arrows\nstruck upon the rock, then there was a crashing among the bushes.\n\n\"Come on!\" Sir John Kerr shouted to the vassals. \"The foxes have\nslunk into their hole.\" Then came low thuds as the <DW19>s were\ncast down. The light which had streamed in through the entrance\ngradually became obscure, and the voices of those without muffled.\nThe darkness grew more intense as the <DW19>s were piled thicker\nand thicker; then suddenly a slight odour of smoke was perceived.\n\n\"Come along now,\" Archie said; \"they have fired the pile, and there\nis no fear of their entrance.\"\n\nTwo of their number, with blazing pine knots, led the way. When\nthey reached the narrow spot all passed through, Archie and Andrew\nMacpherson last; these took the bundles of earth, as the others\npassed them along from behind, and built them up like a wall across\nthe entrance, beating them down as they piled them, so as to make\nthem set close and fill up every crevice. Several remained over\nafter the wall was completed; these were opened and the earth crammed\ninto the crevices between the bags. The smell of smoke had grown\nstrong before the wall was completed, but it was not too oppressive\nto breathe. Holding the torch close to the wall, Archie and his\ncomrade stopped closely the few places through which they saw that\nthe smoke was making its way, and soon had the satisfaction of\nseeing that the barrier was completely smoke tight.\n\nThere was plenty of air in the passage to support life for some\ntime, but Archie called back to those who were labouring to enlarge\nthe exit, in order to allow as much fresh air as possible to enter.\nA strong guard, with spears, was placed at the barrier, although\nArchie deemed that some hours at least would elapse before the Kerrs\ncould attempt to penetrate the cave. The fire would doubtless be\nkept up for some time, and after it had expired it would be long\nbefore the smoke cleared out sufficiently from the cave to allow\nof any one entering it. After a time, finding that there was no\ndifficulty in breathing, although the air was certainly close and\nheavy, Archie again set the lads at work widening the entrance,\ngoing up himself to superintend the operation.  Each in turn crept\nforward, loosened a portion of the earth with his knife, and then\nfilling his cap with it, crawled backward to the point where the\npassage widened.  It was not yet dark when the work was so far done\nthat there now remained only a slight thickness of earth, through\nwhich the roots of the heath protruded, at the mouth of the passage,\nand a vigorous push would make an exit into the air.  The guard at\nthe barrier had heard no movement within.  Archie withdrew one of\nthe bags; but the smoke streamed through so densely that he hastily\nreplaced it, satisfied that some hours must still elapse before\nthe assailants would enter the cave. They watched impatiently\nthe failing light through the hole, and at last, when night was\ncompletely fallen, Archie pushed aside the earth and heather, and\nlooked around. They were, it seemed to him, on the side of the hill\na few yards from the point where it fell steeply away. The ground\nwas thickly covered with heather. He soon made his way out and\nordered Andrew Macpherson, who followed him, to remain lying at\nthe entrance, and to enjoin each, as he passed out, to crawl low\namong the heather, so that they might not show against the skyline,\nwhere, dark as it was, they might attract the attention of those\nbelow. Archie himself led the way until so far back from the edge\nas to be well out of sight of those in the valley. Then he gained\nhis feet, and was soon joined by the whole of his band.\n\n\"Now,\" he said, \"we will make for Aberfilly; they think us all\ncooped up here, and will be rejoicing in our supposed deaths. We\nwill strike one more blow, and then, driving before us a couple of\nscore of oxen for the use of the army, rejoin Wallace. Methinks we\nshall have taken a fair vengeance for Kerr's doings at Glen Cairn.\"\n\nThe consternation of the few men left in the castle was great when,\nthree hours after sunset, eight homesteads burst suddenly into\nflames. They dared not sally out, and remained under arms until\nmorning, when Sir John and his band returned more furious than ever,\nas they had penetrated the cavern, discovered the barrier which\nhad cut off the smoke, and the hole by which the foe had escaped;\nand their fury was brought to a climax when they found the damage\nwhich had been inflicted in their absence. Many a week passed before\nthe garrison of Aberfilly and the vassals of the Kerrs were able\nto sleep in peace, so great was the scare which Archie's raid had\ninflicted upon them.\n\nThe truce was now at an end. The indignation excited by the\ntreachery of the English spread widely through Scotland, and the\npeople flocked to Wallace's standard in far greater numbers than\nbefore, and he was now able to undertake operations on a greater\nscale. Perth, Aberdeen, Brechin, and other towns fell into his\nhands, and the castle of Dundee was invested. In the south Sir\nWilliam Douglas captured the castles of Sanquhar, Desdeir, and\nothers, and the rapid successes of the Scots induced a few of the\ngreater nobles to take the field, such as the Steward of Scotland,\nSir Andrew Moray of Bothwell, Sir Richard Lundin, and Wishart,\nBishop of Glasgow.\n\nWallace was one day lamenting to Archie and his friend Grahame\nthat the greater nobles still held aloof.  \"Above all,\" he said, \"I\nwould fain see on our side either Comyn or the young Bruce. Baliol\nis a captive in London, and it is to Comyn or Bruce that Scotland\nmust look for her king. So long as only I, a poor knight, am at the\nhead of this rising, it is but a rebellion against Edward, and its\nchances are still so weak that but few men, who have aught to lose,\njoin us; but if Bruce or Comyn should raise his banner all would\nreceive him as our future king. Both are lords of wide territories,\nand besides the forces they could bring into the field, they would\nbe joined by many of the principal nobles, although it is true that\nthe adherents of the other would probably arm for Edward. Still\nthe thought of a king of their own would inflame the popular mind,\nand vast numbers who now hesitate to join a movement supported by\nso little authority, would then take up arms.\"\n\n\"Which of the two would you rather?\" Archie asked.\n\n\"I would rather the Bruce,\" Wallace said. \"His father is an inert\nman and a mere cypher, and the death of his grandfather, the\ncompetitor, has now brought him prominently forward. It is true\nthat he is said to be a strong adherent of England and a personal\nfavourite of Edward; that he spends much of his time in London; and\nis even at the present moment the king's lieutenant in Carrick and\nAnnandale, and is waging war for him against Sir William Douglas.\nStill Comyn is equally devoted to England; he is older, and less\ncan be hoped from him. Bruce is young; he is said to be of great\nstrength and skill in arms, and to be one of the foremost knights\nin Edward's court. He is, I hear, of noble presence, and is much\nloved by those with whom he comes in contact.  Did such a man\ndetermine to break with Edward, and to strive to win the crown\nof Scotland as a free gift of her people, instead of as a nominee\nof Edward, and to rule over an independent kingdom instead of an\nEnglish province, he would attract all hearts to him, and may well\nsucceed where I, as I foresee, must sooner or later fail.\"\n\n\"But why should you fail when you have succeeded so far?\" Archie\nasked.\n\n\"Because I have with me but a small portion of the people of\nScotland. The whole of the northern lords hold aloof, and in the\nsouth Carrick and Annandale and Galloway are hostile. Against me\nI have all the power of England, Wales, and Ireland; and although\nI may for a time win victories and capture towns I am certain,\nArchie, in the end to be crushed.\"\n\n\"And will all our efforts have been in vain?\" Archie said, with\ntears in his eyes.\n\n\"By no means, my brave lad; we shall have lighted the fire of a\nnational resistance; we shall have shown the people that if Scotland,\ndivided against herself, and with all her great nobles and their\nvassals standing sullenly aloof, can yet for a long time make head\nagainst the English, assuredly when the time shall come, and she\nshall rise as one man from the Solway to Caithness, her freedom\nwill be won. Our lives will not have been thrown away, Archie, if\nthey have taught this lesson.\"\n\nWallace had by this time returned from his expedition farther\nnorth, and his force was in camp near Lanark, which town, when not\nengaged in distant enterprises, was regarded as the centre of the\nmovement. That evening Archie said, that as his leader purposed to\ngive his troops rest for a week or two, he should go to his uncle's\nfor a short time.\n\n\"And if you can spare them, Sir William, I would fain let my band\ngo away for the same time. They have now been six months from home.\"\n\n\"Certainly,\" Wallace said, \"they need a rest after their hard work.\nThey are ever afoot, and have been of immense service.\"\n\nHaving obtained this permission, Archie went to the spot where his\nband were encamped. \"I have another expedition for you,\" he said,\n\"this time all together; when that is over you will be able to go\nhome for a few days for a rest. They will all be glad to see you,\nand may well be proud of you, and I doubt not that the spoil which\nyou gathered at Ayr and elsewhere will create quite a sensation at\nGlen Cairn. There are some of you who are, as I remember in the old\ndays, good shots with the bow and arrow. Do ten of you who were\nthe best at home get bows and arrows from the store. Here is an\norder for you to receive them, and be all in readiness to march at\ndaylight.\"\n\nThe next morning the band set out in a southwesterly direction,\nand after a long day's march halted near Cumnock.  In the morning\nthey started at the same time, observing more caution as they went,\nfor by the afternoon they had crossed the stream and were within the\nboundaries of Carrick. They halted for the night near Crossraguel\nAbbey. Here for the first time Archie confided to his followers\nthe object of their march.\n\n\"We are now,\" he said, \"within a few miles of Turnberry Castle, the\nresidence of Bruce. Sir William has a great desire to speak with\nhim; but, seeing that Bruce is at present fighting for King Edward\nagainst Douglas, there is little chance of such a meeting coming\nabout with his goodwill. He has recently returned from Douglasdale.\nHere, in the heart of his own country, it is like enough that he\nmay ride near his castle with but a few horsemen. In that case we\nwill seize him, without, I trust, having to do him hurt, and will\nbear him with us to Lanark. We may have to wait some time before\nwe find an opportunity; but even if the ten days for which I have\nasked, lengthen to as many weeks, Sir William will not grudge the\ntime we have spent if we succeed. Tomorrow morning let those who\nhave bows go out in the forest and see if they can shoot a deer;\nor failing that, bring in a sheep or two from some of the folds.\nAs each of you has brought with you meal for ten days, we shall be\nable to keep an eye on Turnberry for some time.\"\n\nThe next day Archie, with Andrew Macpherson and Cluny Campbell,\nmade their way through the woods until within sight of the castle,\nwhich was but a mile distant. The strongholds of the lords of\nCarrick stood on a bold promontory washed by the sea.\n\n\"It would be a hard nut to crack, Sir Archie,\" his lieutenant said.\n\"Unless by famine, the place could scarce be taken.\"\n\n\"No,\" Archie replied, \"I am glad that our mission is rather to\ncapture the earl than his castle. It is a grand fortalice.  Would\nthat its owner were but a true Scotchman! This is a good place on\nwhich we are standing, Andrew, to place a scout. Among the trees\nhere he can watch the road all the way from the castle to the point\nwhere it enters the forest.  Do you, Cluny, take post here at once.\nMark well all that passes, and what is doing, and all bodies of men\nwho enter or leave the castle. There is no occasion to bring news\nto me, for it would be unlikely that we should meet in the forest;\nyou have therefore only to watch. Tomorrow I shall return with the\nband, and encamp in the woods farther back. Directly we arrive,\nyou will be relieved of your guard.\"\n\nThe following day the band moved up to a spot within half a mile\nof the seaward edge of the forest, and a few hundred yards from the\nroad to Crossraguel Abbey.  It was only on this road that Archie\ncould hope to effect a capture; for the country near the coast was\nfree of trees, and no ambush could be set. The lords of Carrick\nwere, moreover, patrons of the abbey; and Bruce might ride over\nthither with but a small party, whereas, if journeying south, or\nsoutheast towards Douglasdale, he would probably be marching with\na strong force. For several days they watched the castle; bodies of\nmounted men entered and departed. Twice parties, among whom ladies\ncould be seen, came out with their hawks; but none came within\nreach of their lurking foes.\n\nOn the fifth morning, however, the lad on watch ran into the glade\nin which they were encamped and reported that a small body of\nseemingly two or three knights, with some ladies, followed by four\nmounted men, had left the castle and were approaching by the route\ntowards the abbey.\n\nNot a moment was lost. Archie placed six of his company, with pike\nand sword, close to the road, to form across it when he gave the\norder, and to bar the retreat of any party who had passed. Another\nparty of equal strength he placed 100 yards further on, and with\nthem himself took post; while he placed four, armed with bows and\narrows, on either side, near the party which he commanded. Scarcely\nhad his preparations been made when a trampling of horses was heard,\nand the party were seen approaching. They consisted of Robert Bruce,\nhis brother Nigel, and three of his sisters--Isabel, Mary, and\nChristina. Behind rode four men-at-arms.  From the description which\nhe had heard of him Archie had no doubt that the elder of the two\nknights was Robert Bruce himself, and when they approached within\nthirty yards he gave a shout, and, with his band, with levelled\nspears, drew up across the road. At the same moment the other party\nclosed in behind the horsemen; and the eight archers, with bent\nbows and arrows drawn to the head, rose among the trees. The party\nreined in their horses suddenly.\n\n\"Hah! what have we here?\" Bruce exclaimed. \"An ambush--and on\nall sides too!\" he added as he looked round.  \"What means this?\nAre you robbers who thus dare attack the Bruce within a mile of\nTurnberry?  Why, they are but lads,\" he added scornfully. \"Rein\nback, girls; we and the men-at-arms will soon clear a way for you\nthrough these varlets. Nay, I can do it single handed myself.\"\n\n\"Halt! Sir Robert Bruce,\" Archie exclaimed in a loud clear voice.\n\"If you move I must perforce give the word, and it may well be that\nsome of the ladies with you may be struck with the arrows; nor,\nyoung though my followers may be, would you find them so easy a\nconquest as you imagine. They have stood up before the English ere\nnow; and you and your men-at-arms will find it hard work to get\nthrough their pikes; and we outnumber you threefold. We are no\nrobbers. I myself am Sir Archibald Forbes.\"\n\n\"You!\" exclaimed Robert Bruce, lowering his sword, which he\nhad drawn at the first alarm and held uplifted in readiness for a\ncharge; \"you Sir Archibald Forbes! I have heard the name often as\nthat of one of Wallace's companions, who, with Sir John Grahame,\nfought with him bravely at the captures of Lanark, Ayr, and other\nplaces, but surely you cannot be he!\"\n\n\"I am Sir Archibald Forbes, I pledge you my word,\" Archie said\nquietly; \"and, Sir Robert Bruce, methinks that if I, who am, as\nyou see, but yet a lad--not yet having reached my seventeenth\nyear--can have done good service for Scotland, how great the\nshame that you, a valiant knight and a great noble, should be in\nthe ranks of her oppressors, and not of her champions! My name will\ntell you that I have come hither for no purpose of robbery. I have\ncome on a mission from Wallace--not sent thereon by him, but\nacting myself in consequences of words which dropped from him.  He\nsaid how sad it was that you, who might be King of a Scotland free\nand independent, by the choice of her people, should prefer the\nchance of reigning, a mere puppet of Edward, over an enslaved land.\nHe spoke in the highest terms of your person, and held that, did\nyou place yourself at its head, the movement which he commands\nwould be a successful one. Then I determined, unknown to him, to\nset out and bring you to him face to face--honourably and with\ncourtesy if you would, by force if you would not. I would fain it\nshall be the former; but believe me, you would not find it easy to\nbreak away through the hedge of pikes now around you.\"\n\nBy this time the whole party had gathered round the horsemen. Bruce\nhesitated; his mind was not yet made up as to his future course.\nHitherto he had been with England, since upon Edward only his chances\nseemed to depend; but latterly he had begun to doubt whether even\nEdward could place him on the throne in despite of the wishes of\nhis countrymen. His sisters, who, taking after their mother, were\nall true Scotchwomen, now urged upon him to comply with Archie's\nrequest and accompany him to Lanark. Their hearts and wishes were\nentirely with the champion of their country.\n\n\"Go with him, Robert,\" Isabel, the eldest, exclaimed.  \"Neither\nI nor my sisters fear being struck with the arrows, although such\nmight well be the case should a conflict begin; but, for your own\nsake and Scotland's, go and see Wallace.  No harm can arise from\nsuch a journey, and much good may come of it. Even should the\nnews of your having had an interview with him come to the ears of\nEdward, you can truly say that you were taken thither a captive,\nand that we being with you, you were unable to make an effort to\nfree yourself.  This young knight, of whose deeds of gallantry we\nhave all heard\"--and she smiled approvingly at Archie--\"will\ndoubtless give you a safeguard, on his honour, to return hither\nfree and unpledged when you have seen Wallace.\"\n\n\"Willingly, lady,\" Archie replied. \"One hour's interview with my\nhonoured chief is all I ask for. That over, I pledge myself that\nthe Earl of Carrick shall be free at once to return hither, and\nthat an escort shall be provided for him to protect him from all\ndangers on the way.\"\n\n\n\n\nChapter VIII\n\nThe Council at Stirling\n\n\nArchie had been mounted on the march from the camp, and his horse\nbeing now brought, he started with Bruce, young Nigel and the ladies\nsaluting him cordially.\n\n\"I trust,\" the former said, \"that Wallace will succeed in converting\nmy brother. I am envious of you, Sir Archie. Here are you, many\nyears younger than I am, and yet you have won a name throughout\nScotland as one of her champions; while I am eating my heart out,\nwith my brother, at the court of Edward.\"\n\n\"I trust it may be so, Sir Nigel,\" Archie answered. \"If Sir Robert\nwill but join our cause, heart and soul, the battle is as good as\nwon.\"\n\nThe journey passed without adventure until they arrived within two\nmiles of Lanark, where Archie found Wallace was now staying. On\nthe road Bruce had had much conversation with Archie, and learned\nthe details of many adventures of which before he had only heard\nvaguely by report. He was much struck by the lad's modesty and\nloyal patriotism.\n\n\"If ever I come to my kingdom, Sir Archie,\" he said, \"you shall\nbe one of my most trusted knights and counsellors; and I am well\nassured that any advice you may give will be ever what you think\nto be right and for the good of the country, without self seeking\nor in the interest of any; and that is more than I could look for\nin most counsellors. And now methinks that as we are drawing near\nto Lanark, it will be well that I waited here in this wood, under\nthe guard of your followers, while you ride forward and inform\nWallace that I am here. I care not to show myself in Lanark, for\nbusy tongues would soon take the news to Edward; and as I know not\nwhat may come of our interview, it were well that it should not be\nknown to all men.\"\n\nArchie agreed, and rode into the town.\n\n\"Why, where have you been, truant?\" Sir William exclaimed as Archie\nentered the room in the governor's house which had been set apart\nfor the use of Wallace since the expulsion of the English. \"Sir\nRobert Gordon has been here several times, and tells me that they\nhave seen nought of you; and although I have made many inquiries I\nhave been able to obtain no news, save that you and your band have\ndisappeared.  I even sent to Glen Cairn, thinking that you might\nhave been repairing the damages which the fire, lighted by the\nKerrs, did to your hold; but I found not only that you were not\nthere yourself, but that none of your band had returned thither.\nThis made it more mysterious; for had you alone disappeared I should\nhave supposed that you had been following up some love adventure,\nthough, indeed, you have never told me that your heart was in any\nway touched.\"\n\nArchie laughed. \"There will be time enough for that, Sir William,\nten years hence; but in truth I have been on an adventure on my\nown account.\"\n\n\"So, in sober earnest, I expected, Archie, and feared that your\nenterprise might lead you into some serious scrape since I deemed\nthat it must have been well nigh a desperate one or you would not\nhave hidden it from my knowledge.\"\n\n\"It might have led to some blows, Sir William, but happily it did\nnot turn out so. Knowing the importance you attached to the adhesion\nof the cause of Scotland of Robert the Bruce, I determined to fetch\nhim hither to see you; and he is now waiting with my band for your\ncoming, in a wood some two miles from the town.\"\n\n\"Are you jesting with me?\" Wallace exclaimed. \"Is the Bruce really\nwaiting to see me? Why, this would be well nigh a miracle.\"\n\n\"It is a fact, Sir William; and if you will cause your horse to\nbe brought to the door I will tell you on the road how it has come\nabout.\"\n\nIn another five minutes Sir William and his young follower were\non their way, and the former heard how Archie had entrapped Robert\nBruce while riding to Crossraguel Abbey.\n\n\"It was well done, indeed,\" the Scottish leader exclaimed; \"and\nit may well prove, Archie, that you have done more towards freeing\nScotland by this adventure of yours than we have by all our months\nof marching and fighting.\"\n\n\"Ah! Sir William, but had it not been for our marching and fighting\nBruce would never have wavered in his allegiance to Edward. It was\nonly because he begins to think that our cause may be a winning\none that he decides to join it.\"\n\nThe meeting between Wallace and Bruce was a cordial one. Each\nadmired the splendid proportions and great strength of the other,\nfor it is probable that in all Europe there were no two more\ndoughty champions; although, indeed, Wallace was far the superior\nin personal strength while Bruce was famous through Europe for his\nskill in knightly exercise.\n\nArchie withdrew to a distance while the leaders conversed. He\ncould see that their talk was animated as they strode together up\nand down among the trees, Wallace being the principal speaker. At\nthe end of half an hour they stopped, and Wallace ordered the horses\nto be brought, and then called Archie to them.\n\n\"Sir Robert has decided to throw in his lot with us,\" he said, \"and\nwill at once call out his father's vassals of Carrick and Annandale.\nSeeing that his father is at Edward's court, it may be that many\nwill not obey the summons. Still we must hope that, for the love\nof Scotland and their young lord, many will follow him. He will\nwrite to the pope to ask him to absolve him for the breach of his\noath of homage to Edward; but as such oaths lie but lightly on men's\nminds in our days, and have been taken and broken by King Edward\nhimself, as well as by Sir William Douglas and other knights who are\nnow in the field with me, he will not wait for the pope's reply,\nbut will at once take the field. And, indeed, there is need for\nhaste, seeing that Percy and Clifford have already crossed the\nBorder with an English army and are marching north through Annandale\ntowards Ayr.\"\n\n\"Goodbye, my captor,\" Bruce said to Archie as he mounted his horse;\n\"whatever may come of this strife, remember that you will always\nfind a faithful friend in Robert Bruce.\"\n\nWallace had, at Archie's request, brought six mounted men-at-arms\nwith him from Lanark, and these now rode behind Bruce as his escort\nback to his castle of Turnberry. There was no time now for Archie\nand his band to take the rest they had looked for, for messengers\nwere sent out to gather the bands together again, and as soon as\na certain portion had arrived Wallace marched for the south. The\nEnglish army was now in Annandale, near Lochmaben. They were far\ntoo strong to be openly attacked, but on the night following his\narrival in their neighbourhood Wallace broke in upon them in the\nnight. Surprised by this sudden and unexpected attack, the English\nfell into great confusion. Percy at once ordered the camp to be\nset on fire.  By its light the English were able to see how small\nwas the force of their assailants, and gathering together soon\nshowed so formidable a front that Wallace called off his men, but\nnot before a large number of the English had been killed. Many of\ntheir stores, as well as the tents, were destroyed by the conflagration.\nThe English army now proceeded with slow marches towards Ayr.  At\nIrvine the Scotch leaders had assembled their army--Douglas,\nBruce, The Steward, Sir Richard Loudon, Wishart, Bishop of Glasgow,\nand others.  Their forces were about equal to those of the English\nmarching against them. Wallace was collecting troops further north,\nand Archie was of course with him.\n\n\"I fear,\" the lad said one day, \"that we shall not be able to reach\nIrvine before the armies join battle.\"\n\n\"Sir William Douglas and Bruce are there, and as it lies in their\ncountry it were better to let them win the day without my meddling.\nBut, Archie, I fear there will be no battle.  News has reached\nme that messengers are riding to and fro between Percy's army and\nthe Scots, and I fear me that these half hearted barons will make\npeace.\"\n\n\"Surely that cannot be! It were shame indeed to have taken up the\nsword, and to lay it down after scarce striking a blow.\"\n\n\"Methinks, Archie, that the word shame is not to be found in the\nvocabulary of the nobles of this unhappy land. But let us hope for\nthe best; a few days will bring us the news.\"\n\nThe news when it came was of the worst. All the nobles, headed by\nWishart, Douglas, and Bruce, with the exception only of Sir Andrew\nMoray of Bothwell, had made their submission, acknowledging their\nguilt of rebellion, and promising to make every reparation required\nby their sovereign lord.  Percy, on his part, guaranteed their lives,\nlands, goods, and chattels, and that they should not be imprisoned\nor punished for what had taken place.\n\nSir William Douglas and Bruce were ordered to find guarantees for\ntheir good conduct; but Sir William Douglas, finding himself unable\nto fulfil his engagements, surrendered, and was thrown into prison\nin Berwick Castle, and there kept in irons until he died, his death\nbeing attributed, by contemporary historians, to poison.\n\nThe surrender of the leaders had little result upon the situation.\nThe people had won their successes without their aid, and beyond\nthe indignation excited by their conduct, the treaty of Irvine did\nnothing towards ensuring peace, and indeed heightened the confidence\nof the people in Wallace.  The movement spread over the whole\nof Scotland. Skirmishes and unimportant actions took place in\nall quarters. The English were powerless outside the walls of the\nfortresses, and in Berwick and Roxburgh alone was the English power\nparamount. Most of the great nobles, including Comyn of Buchan,\nComyn of Badenoch, and twenty-six other powerful Scottish lords,\nwere at Edward's court, but many of their vassals and dependants\nwere in the field with Wallace.\n\nAbout this time it came to the ears of the Scotch leader that Sir\nRobert Cunninghame, a Scotch knight of good family, who had hitherto\nheld aloof from any part in the war, had invited some twelve others\nresident in the counties round Stirling, to meet at his house in\nthat city that they might talk over the circumstances of the times.\nAll these had, like himself, been neutral, and as the object of\nthe gathering was principally to discover whether some means could\nnot be hit upon for calming down the disorders which prevailed,\nthe English governor had willingly granted safe conducts to all.\n\n\"Archie,\" Sir William said, \"I mean to be present at the interview.\nThey are all Scotch gentlemen, and though but lukewarm in the cause\nof their country, there is no fear that any will be base enough\nto betray me; and surely if I can get speech with them I may rouse\nthem to cast in their lot with us.\"\n\n\"It were a dangerous undertaking, Sir William, to trust yourself\nwithin the walls of Stirling,\" Archie said gravely.  \"Remember how\nmany are the desperate passes into which your adventurous spirit\nhas brought you, and your life is of too great a consequence to\nScotland to be rashly hazarded.\"\n\n\"I would not do it for a less cause,\" Sir William said; \"but the\ngain may be greater than the risk. So I shall go, Archie, your wise\ncounsel notwithstanding, and you shall journey with me to see that\nI get not into scrapes, and to help me out of them should I, in\nspite of your care, fall into them.\"\n\n\"When is the day for the meeting?\" Archie asked.\n\n\"In three days' time. The day after tomorrow we will move in that\ndirection, and enter the town early the next day.\"\n\nNo sooner had he left Wallace than Archie called his band together.\nThey still numbered twenty, for although three or four had fallen,\nArchie had always filled up their places with fresh recruits, as\nthere were numbers of boys who deemed it the highest honour to be\nenrolled in their ranks.  Archie drew aside his two lieutenants,\nAndrew Macpherson and William Orr.\n\n\"I have an enterprise on hand,\" he said, \"which will need all your\ncare, and may call for your bravery. Sir William Wallace purposes\nto enter Stirling in disguise, to attend a meeting of nobles to be\nheld at the residence of Sir Robert Cunninghame. I am to accompany\nhim thither. I intend that the band shall watch over his safety,\nand this without his having knowledge of it, so that if nought comes\nof it he may not chide me for being over careful of his person. You\nwill both, with sixteen of the band, accompany me. You will choose\ntwo of your most trusty men to carry out the important matter of\nsecuring our retreat. They will procure a boat capable of carrying\nus all, and will take their place in the bend of the links of\nForth nearest to the castle, and will hoist, when the time comes,\na garment on an oar, so that we may make straight for the boat. The\nground is low and swampy, and if we get a fair start even mounted\nmen would scarce overtake us across it. I think, William, that the\nlast recruit who joined was from Stirling?\"\n\n\"He was, Sir Archie. His parents reside there. They are vendors of\nwood, as I have heard him say.\"\n\n\"It could not be better,\" Archie replied; \"and seeing that they\nhave allowed their son to join us, they must surely be patriots. My\npurpose is, that on the morning of the interview you shall appear\nbefore the gates with a cart laden with firewood, and this you shall\ntake to the house of Campbell's father. There you will unload the\nfirewood, and store the arms hidden beneath it, placing them so\nthat they may be readily caught up in case of necessity. In twos\nand threes, carrying eggs, fowls, firewood, and other articles,\nas for sale, the rest of the band will come into the town, joining\nthemselves with parties of country people, so that the arrival of\nso many lads unaccompanied will not attract notice. James Campbell\nwill go with you, and will show you the way to his father's house.\nHe will remain near the gate, and as the others enter will guide\nthem there, so that they will know where to run for their arms should\nthere be need. You must start tomorrow, so as to enter Stirling on\nthe next day and arrange with his father for the keeping of the\narms.  His mother had best leave the town that evening. Should\nnought occur she can return unsuspected; but should a tumult arise,\nand the arms have to be used, his father must leave the town with\nus. He shall be handsomely rewarded, and provision made for him\nin the future. When you see me enter with Sir William, bid Jock\nFarrell follow me at a little distance; he will keep me always in\nsight, and if he see me lift my hand above my head he will run with\nall speed to give you the news. On his arrival, you, Andrew, with\nthe half you command, will hurry up to my assistance; while you,\nWilliam, with the others, will fall suddenly upon the guard at the\ngate, and will at all hazards prevent them from closing it, and so\ncutting off our retreat, until we arrive. Seize, if you can, the\nmoment when a cart is passing in or out, and slay the horse in the\nshafts, so that as he falls the cart will prevent the gate from\nbeing closed, and so keep the way open, even should you not be able\nto resist the English until we come up. Have all the band outside\nStirling on the night before, so that you will be able to make every\narrangement and obtain a cart in readiness for taking in the wood\nand arms in the morning. Let all bring their bows and arrows, in\naddition to pike and sword, for the missiles may aid us to keep the\nsoldiers at bay. Now, Andrew, repeat all my instructions, so that\nI may be sure that you thoroughly understand my wishes, for any\nsmall error in the plan might ruin the whole adventure.\"\n\nOn the morning of the day fixed for the meeting Sir William Wallace,\naccompanied by Archie, entered the gates of Stirling. Both were\nattired as young farmers, and they attracted no special attention\nfrom the guards. For a time they strolled about the streets. They\nsaw the gentlemen who had been invited by Sir Robert Cunninghame\narrive one by one. Others, too, known as being specially attached\nto the English party, rode in, for the governor had invited those\nwho assembled at Cunninghame's to meet him afterwards in the castle\nin order that he might hear the result of their deliberations; and\nhe had asked several others attached to the English party to be\npresent.\n\nWhen most of the gentlemen invited had entered Sir Robert Cunninghame's\nWallace boldly followed them; and Archie sat down on a doorstep\nnearly opposite. Presently he saw two figures which he recognized\nriding up the street, followed, as the others had been by four\narmed retainers.  They were Sir John Kerr and his son. Archie rose\nat once, and turned down at a side street before they came up, as\na recognition of him would be fatal to all their plans. When they\nhad passed up the street to the castle he returned and resumed\nhis seat, feeling more uneasy than before, for the Kerrs had seen\nWallace in the affray at Lanark, and a chance meeting now would\nbetray him. An hour and a half passed, and then Archie saw the\nKerrs riding down the street from the castle. Again he withdrew\nfrom sight, this time down an archway, whence he could still see\nthe door on the opposite side. Hitherto he had been wishing to see\nit open and for Wallace to appear; and now he dreaded this above\nall things.  His worst fears were realized, for just as the horsemen\nreached the spot the door opened, and Wallace stepped out. His\nfigure was too remarkable to avoid notice; and no sooner did Sir John\nKerr's eye fall upon him than he exclaimed, \"The traitor Wallace!\nSeize him, men; there is a high reward offered for him; and King\nEdward will give honour and wealth to all who capture him.\"\n\nAs Sir John spoke Archie darted across the street and placed himself\nby Wallace's side, holding his hand high above his head as he did\nso; and at the instant he saw Jock Farrell, who had been lounging\nat a corner a few yards away, dart off down the street at the top\nof his speed.\n\nSir John and his retainers drew their swords and spurred forward;\nbut the horses recoiled from the flashing swords of Wallace and\nhis companion.\n\n\"Dismount,\" Sir John shouted, setting the example; \"cut them both\ndown; one is as bad as the other. Ten pounds to the man who slays\nthe young Forbes.\"\n\nWallace cut down two of the retainers as they advanced against\nthem, and Archie badly wounded a third.  Then they began to retreat\ndown the street; but by this time the sound of the fray had called\ntogether many soldiers who were wandering in the streets; and these,\ninformed by Sir John's shouts of \"Down with Wallace! Slay! Slay!\"\nthat the dreaded Scotch leader was before them, also drew and joined\nin the fight.  As they came running up from both sides, Wallace\nand Archie could retreat no further, but with their backs against\nthe wall kept their foes at bay in a semicircle by the sweep of\ntheir swords.\n\nThe fight continued by two or three minutes, when a sudden shout\nwas heard, and William Orr, with eight young fellows, fell upon the\nEnglish soldiers with their pikes. The latter, astonished at this\nsudden onslaught, and several of their number being killed before\nthey had time to turn and defend themselves, fell back for a moment,\nand Wallace and Archie joined their allies, and began to retreat,\nforming a line of pikes across the narrow street. Wallace, Archie,\nWilliam Orr, and three of the stoutest of the band were sufficient\nfor the line, and the other five shot between them. So hard and\nfast flew their arrows that several of the English soldiers were\nslain, and the others drew back from the assault.\n\nAndrew Macpherson's sudden attack at the gate overpowered the guard,\nand for a while he held possession of it, and following Archie's\ninstructions, slew a horse drawing a cart laden with flour in the\nact of entering.  Then the guard rallied, and, joined by other\nsoldiers who had run up, made a fierce attack upon him; but his\nline of pikes drawn up across the gate defied their efforts to break\nthrough. Wallace and his party were within fifty yards of the gate\nwhen reinforcements from the castle arrived. Sir John Kerr, furious\nat the prospect of his enemies again escaping him, headed them in\ntheir furious rush. Wallace stepped forward beyond the line and\nmet him. With a great sweep of his mighty sword he beat down Sir\nJohn's guard, and the blade descending clove helmet and skull, and\nthe knight fell dead in his tracks.\n\n\"That is one for you, Archie,\" Wallace said, as he cut down a\nman-at-arms.\n\nIn vain did the English try to break through the line of pikes.\nWhen they arrived within twenty yards of the gate, Wallace gave\nthe order, and the party turning burst through the English who were\nattacking its defenders and united with them.\n\n\"Fall back!\" Wallace shouted, \"and form without the gates. Your\nleader and I will cover the retreat.\"\n\nPassing between the cart and the posts of the gates, the whole\nparty fell back. Once through, Wallace and Archie made a stand, and\neven the bravest of the English did not venture to pass the narrow\nportals, where but one could issue at a time.\n\nThe band formed in good order and retreated at a rapid step. When\nthey reached a distance of about 300 yards, Wallace and Archie,\ndeeming that sufficient start had been gained, sprang away, and\nrunning at the top of their speed soon rejoined them.\n\n\"Now, Archie, what next?\" Sir William asked; \"since it is you who\nhave conjured up this army, doubtless your plans are laid as to what\nshall next be done. They will have horsemen in pursuit as soon as\nthey remove the cart.\"\n\n\"I have a boat in readiness on the river bank, Sir William. Once\nacross and we shall be safe. They will hardly overtake us ere we\nget there, seeing how swampy is the ground below.\"\n\nAt a slinging trot the party ran forward, and soon gained the\nlower ground. They were halfway across when they saw a large body\nof horsemen following in pursuit.\n\n\"A little to the right, Sir William,\" Archie said; \"you see that\ncoat flying from an oar; there is the boat.\"\n\nAs Archie had expected, the swampy ground impeded the speed of\nthe horsemen. In vain the riders spurred and shouted, the horses,\nfetlock deep, could make but slow advance, and before they reached\nthe bank the fugitives had gained the boat and were already halfway\nacross the stream.  Then the English had the mortification of seeing\nthem land and march away quietly on the other side.\n\n\n\n\nChapter IX\n\nThe Battle of Stirling Bridge\n\n\nUpon rejoining his force Sir William Wallace called the few knights\nand gentlemen who were with him together, and said to them:\n\n\"Methinks, gentlemen, that the woes of this contest should not fall\nupon one side only. Every one of you here are outlawed, and if you\nare taken by the English will be executed or thrown in prison for\nlife, and your lands and all belonging to you forfeited. It is time\nthat those who fight upon the other side should learn that they\ntoo run some risk. Besides leading his vassals in the field against\nus, Sir John Kerr twice in arms has attacked me, and done his best\nto slay me or deliver me over to the English. He fell yesterday by\nmy hand at Stirling, and I hereby declare forfeit the land which\nhe held in the county of Lanark, part of which he wrongfully took\nfrom Sir William Forbes, and his own fief adjoining.  Other broad\nlands he owns in Ayrshire, but these I will not now touch; but the\nlands in Lanark, both his own fief and that of the Forbeses, I,\nas Warden of Scotland, hereby declare forfeit and confiscated, and\nbestow them upon my good friend, Sir Archie Forbes. Sir John Grahame,\ndo you proceed tomorrow with five hundred men and take possession\nof the hold of the Kerrs. Sir Allan Kerr is still at Stirling, and\nwill not be there to defend it. Like enough the vassals will make\nno resistance, but will gladly accept the change of masters. The\nKerrs have the reputation of being hard lords, and their vassals\ncannot like being forced to fight against the cause of their country.\nThe hired men-at-arms may resist, but you will know how to make\nshort work of these. I ask you to go rather than Sir Archibald\nForbes, because I would not that it were said that he took the\nKerr's hold on his private quarrel.  When you have captured it he\nshall take a hundred picked men as a garrison. The place is strong.\n\n\"Your new possessions, Archie, will, as you know, be held on\ndoubtful tenure. If we conquer, and Scotland is freed, I doubt in\nno way that the king, whoever he may be, will confirm my grant.\nIf the English win, your land is lost, be it an acre or a county.\nAnd now let me be the first to congratulate you on having won by\nyour sword and your patriotism the lands of your father, and on\nhaving repaid upon your family's enemies the measure which they\nmeted to you. But you will still have to beware of the Kerrs. They\nare a powerful family, being connected by marriage with the Comyns\nof Badenoch, and other noble houses.  Their lands in Ayr are as\nextensive as those in Lanark, even with your father's lands added\nto their own.  However, if Scotland win the day the good work that\nyou have done should well outweigh all the influence which they\nmight bring to bear against you.\n\n\"And now, Archie, I can, for a time, release you. Ere long Edward's\narmy will be pouring across the Border, and then I shall need every\ngood Scotchman's sword. Till then you had best retire to your new\nestates, and spend the time in preparing your vassals to follow\nyou in the field, and in putting one or other of your castles in\nthe best state of defence you may. Methinks that the Kerr's hold\nmay more easily be made to withstand a lengthened siege than Glen\nCairn, seeing that the latter is commanded by the hill beside it.\nKerr's castle, too, is much larger and more strongly fortified. I\nneed no thanks,\" he continued, as Archie was about to express his\nwarm gratitude; \"it is the Warden of Scotland who rewards your\nservices to the country; but Sir William Wallace will not forget\nhow you have twice stood beside him against overwhelming odds, and\nhow yesterday, in Stirling, it was your watchful care and thoughtful\nprecaution which alone saved his life.\"\n\nArchie's friends all congratulated him warmly, and the next morning,\nwith his own band, he started for Glen Cairn.  Here the news that\nhe was once more their lawful chief caused the greatest delight.\nIt was evening when he reached the village, and soon great bonfires\nblazed in the street, and as the news spread burned up from many\nan outlying farm.  Before night all the vassals of the estate came\nin, and Glen Cairn and the village was a scene of great enthusiasm.\n\nMuch as Archie regretted that he could not establish himself in\nthe hold of his father, he felt that Wallace's suggestion was the\nright one. Glen Cairn was a mere shell, and could in no case be\nmade capable of a prolonged resistance by a powerful force. Whereas,\nthe castle of the Kerrs was very strong. It was a disappointment\nto his retainers when they heard that he could not at once return\namong them; but they saw the force of his reasons, and he promised\nthat if Scotland was freed and peace restored, he would again make\nGlen Cairn habitable, and pass some of his time there.\n\n\"In the meantime,\" he said, \"I shall be but eight miles from you,\nand the estate will be all one. But now I hope that for the next\nthree months every man among you will aid me--some by personal\nlabour, some by sending horses and carts--in the work of\nstrengthening to the utmost my new castle of Aberfilly, which I\nwish to make so strong that it will long resist an attack. Should\nScotland be permanently conquered, which may God forfend, it could\nnot, of course, be held; but should we have temporary reverses we\nmight well hold out until our party again gather head.\"\n\nEvery man on the estate promised his aid to an extent far beyond\nthat which Archie, as their feudal superior, had a right to demand\nfrom them. They had had a hard time under the Kerrs, who had raised\nall rents, and greatly increased their feudal services. They were\nsure of good treatment should the Forbeses make good their position\nas their lords, and were ready to make any sacrifices to aid them\nto do so.\n\nNext morning a messenger arrived from Sir John Grahame, saying that\nhe had, during the night, stormed Aberfilly, and that with scarce\nan exception all the vassals of the Kerrs--when upon his arrival\non the previous day they had learned of his purpose in coming,\nand of the disposition which Wallace had made of the estate--had\naccepted the change with delight, and had joined him in the assault\nupon the castle, which was defended only by thirty men-at-arms.\nThese had all been killed, and Sir John invited Archie to ride\nover at once and take possession. This he did, and found that the\nvassals of the estate were all gathered at the castle to welcome\nhim.  He was introduced to them by Sir John Grahame, and they\nreceived Archie with shouts of enthusiasm, and all swore obedience\nto him as their feudal lord.  Archie promised them to be a kind\nand lenient chief, to abate any unfair burdens which had been laid\nupon them, and to respect all their rights.\n\n\"But,\" he said, \"just at first I must ask for sacrifices from you.\nThis castle is strong, but it must be made much stronger, and must\nbe capable of standing a continued siege in case temporary reverses\nshould enable the English to endeavour to retake it for their\nfriend, Sir Allan Kerr. My vassals at Glen Cairn have promised an\naid far beyond that which I can command, and I trust that you also\nwill extend your time of feudal service, and promise you a relaxation\nin future years equivalent to the time you may now give.\"\n\nThe demand was readily assented to, for the tenants of Aberfilly\nwere no less delighted than those of Glen Cairn to escape from the\nrule of the Kerrs. Archie, accompanied by Sir John Grahame, now\nmade an inspection of the walls of his new hold. It stood just where\nthe counties of Linlithgow and Edinburgh join that of Lanark. It\nwas built on an island on a tributary of the Clyde. The stream was\nbut a small one, and the island had been artificially made, so that\nthe stream formed a moat on either side of it, the castle occupying\na knoll of ground which rose somewhat abruptly from the surrounding\ncountry. The moat was but twelve feet wide, and Archie and Sir John\ndecided that this should be widened to fifty feet and deepened to\nten, and that a dam should be built just below the castle to keep\nback the stream and fill the moat. The walls should everywhere be\nraised ten feet, several strong additional flanking towers added,\nand a work built beyond the moat to guard the head of the drawbridge.\nWith such additions Aberfilly would be able to stand a long siege\nby any force which might assail it.\n\nTimber, stones, and rough labour there were in abundance, and\nWallace had insisted upon Archie's taking from the treasures which\nhad been captured from the enemy, a sum of money which would be\nample to hire skilled masons from Lanark, and to pay for the cement,\niron, and other necessaries which would be beyond the resources\nof the estate.  These matters in train, Archie rode to Lanark and\nfetched his proud and rejoicing mother from Sir Robert Gordon's\nto Aberfilly. She was accompanied by Sandy Graham and Elspie:  the\nformer Archie appointed majordomo, and to be in command of the\ngarrison whenever he should be absent.\n\nThe vassals were as good as their word. For three months the work\nof digging, quarrying, cutting, and squaring timber and building\nwent on without intermission. There were upon the estates fully\nthree hundred ablebodied men, and the work progressed rapidly. When,\ntherefore, Archie received a message from Wallace to join him near\nStirling, he felt that he could leave Aberfilly without any fear\nof a successful attack being made upon it in his absence.\n\nThere was need, indeed, for all the Scotch, capable of bearing\narms, to gather round Wallace. Under the Earl of Surrey, the high\ntreasurer Cressingham, and other leaders, an army of 50,000 foot\nand 1000 horse were advancing from Berwick, while 8000 foot and\n300 horse under Earl Percy advanced from Carlisle.  Wallace was\nbesieging the castle of Dundee when he heard of their approach,\nand leaving the people of Dundee to carry on the siege under the\ncommand of Sir Alexander Scrymgeour, he himself marched to defend\nthe only bridge by which Edward could cross the Forth, near Stirling.\n\nThus far Surrey had experienced no resistance, and at the head\nof so large and well appointed a force he might well feel sure\nof success. A large proportion of his army consisted of veterans\ninured to service in wars at home, in Wales, and with the French,\nwhile the mail clad knights and men-at-arms looked with absolute\ncontempt upon the gathering which was opposed to them. This consisted\nsolely of popular levies of men who had left their homes and taken\nup arms for the freedom of their country. They were rudely armed and\nhastily trained. Of all the feudal nobles of Scotland who should have\nled them, but one, Sir Andrew Moray, was present. Their commander\nwas still little more than a youth, who, great as was his individual\nvalour and prowess, had had no experience in the art of war on\na large scale; while the English were led by a general whose fame\nwas known throughout Europe.\n\nThe Scots took up their station upon the high ground north of the\nForth, protected from observation by the precipitous hill immediately\nbehind Cambuskenneth Abbey and known as the Abbey Craig. In a bend\nof the river, opposite the Abbey Craig, stood the bridge by which\nthe English army were preparing to cross.  Archie stood beside\nWallace on the top of the craig, looking at the English array.\n\n\"It is a fair sight,\" he said; \"the great camp, with its pavilions,\nits banners, and pennons, lying there in the valley, with the old\ncastle rising on the lofty rock behind them. It is a pity that such\na sight should bode evil to Scotland.\"\n\n\"Yes,\" Wallace said; \"I would that the camp lay where it is, but\nthat the pennons and banners were those of Scotland's nobles, and\nthat the royal lions floated over Surrey's tent.  Truly that were\na sight which would glad a Scot's heart. When shall we see ought\nlike it? However, Archie,\" he went on in a lighter tone, \"methinks\nthat that will be a rare camp to plunder.\"\n\nArchie laughed. \"One must kill the lion before one talks of dividing\nhis skin,\" he said; \"and truly it seems well nigh impossible that\nsuch a following as yours, true Scots and brave men though they\nbe, yet altogether undisciplined and new to war, should be able to\nbear the brunt of such a battle.\"\n\n\"You are thinking of Dunbar,\" Wallace said; \"and did we fight in\nsuch a field our chances would be poor; but with that broad river\nin front and but a narrow bridge for access, methinks that we can\nrender an account of them.\"\n\n\"God grant it be so!\" Archie replied; \"but I shall be right glad\nwhen the day is over.\"\n\nThree days before the battle the Steward of Scotland, the Earl of\nLennox, and others of the Scotch magnates entered Surrey's camp\nand begged that he would not attack until they tried to induce the\npeople to lay down their arms.  They returned, however, on the third\nday saying that they would not listen to them, but that the next\nday they would, themselves, join his army with their men-at-arms.\nOn leaving the camp that evening the Scotch nobles, riding homeward,\nhad a broil with some English soldiers, of whom one was wounded by\nthe Earl of Lennox. News being brought to Surrey, he resolved to\nwait no longer, but gave orders that the assault should take place\non the following morning. At daybreak of the 11th of September,\n1297, one of the outposts woke Wallace with the news that the English\nwere crossing the bridge. The troops were at once got under arms,\nand were eager to rush down to commence the battle, but Wallace\nrestrained them. Five thousand Welsh foot soldiers crossed the\nbridge, then there was a pause, and none were seen following them.\n\"Were we to charge down now, Sir William,\" Archie said, \"surely we\nmight destroy that body before aid could come to them.\"\n\n\"We could do, Archie, as you say,\" Wallace replied, \"but such\na success would be of little worth, nay, would harm rather than\nbenefit us, for Surrey, learning that we are not altogether to be\ndespised, as he now believes, would be more prudent in future and\nwould keep his army in the flat country, where we could do nought\nagainst it. No, to win much one must risk much, and we must wait\nuntil half Surrey's army is across before we venture down against\nthem.\"\n\nPresently the Welsh were seen to retire again. Their movement had\nbeen premature. Surrey was still asleep, and nothing could be done\nuntil he awoke; when he did so the army armed leisurely, after which\nSurrey bestowed the honour of knighthood upon many young aspirants.\nThe number of the Scots under Wallace is not certainly known; the\nmajority of the estimates place it below twenty thousand, and as\nthe English historian, who best describes the battle, speaks of it\nas the defeat of the many by the few, it can certainly be assumed\nthat it did not exceed this number.\n\nOnly on the ground of his utter contempt for the enemy can the\nconduct of the Earl of Surrey, in attempting to engage in such a\nposition, be understood. The bridge was wide enough for but two,\nor at most three, horsemen to cross abreast, and when those who had\ncrossed were attacked assistance could reach them but slowly from\nthe rear.\n\nThe English knights and men-at-arms, with the Royal Standard and\nthe banner of the Earl of Surrey, crossed first.  The men-at-arms\nwere followed by the infantry, who, as they passed, formed up on\nthe tongue of land formed by the winding of the river.\n\nWhen half the English army had passed Wallace gave the order to\nadvance. First Sir Andrew Moray, with two thousand men, descended\nthe hills farther to the right, and on seeing these the English\ncavalry charged at once against them.  The instant they did so\nWallace, with his main army, poured down from the craig impetuously\nand swept away the English near the head of the bridge, taking\npossession of the end, and by showers of arrows and darts preventing\nany more from crossing. By this maneuver the whole of the English\ninfantry who had crossed were cut off from their friends and inclosed\nin the narrow promontory.\n\nThe English men-at-arms had succeeded in overthrowing the Scots,\nagainst whom they had charged, and had pursued them some distance;\nbut upon drawing rein and turning to rejoin the army, they found\nthe aspect of affairs changed indeed. The troops left at the head\nof the bridge were overthrown and destroyed.  The royal banner and\nthat of Surrey were down, and the bridge in the possession of the\nenemy.  The men-at-arms charged back and strove in vain to recover\nthe head of the bridge. The Scots fought stubbornly; those in front\nmade a hedge of pikes, while those behind hurled darts and poured\nshowers of arrows into the English ranks.  The greater proportion\nof the men-at-arms were killed. One valiant knight alone, Sir\nMarmaduke de Twenge, with his nephew and a squire, cut their way\nthrough the Scots, and crossed the bridge. Many were drowned in\nattempting to swim the river, one only succeeding in so gaining\nthe opposite side.\n\nThe men-at-arms defeated, Wallace and the chosen band under him,\nwho had been engaged with them, joined those who were attacking the\nEnglish and Welsh, now cooped up in the promontory. Flushed with\nthe success already gained the Scots were irresistible, and almost\nevery man who had crossed was either killed or drowned in attempting\nto swim the river. No sooner had he seen that the success in this\nquarter was secure than Wallace led a large number of his followers\nacross the bridge. Here the English, who still outnumbered his army,\nand who had now all the advantage of position which had previously\nbeen on the side of the Scots, might have defended the bridge, or\nin good order have given him battle on the other side.  The sight,\nhowever, of the terrible disaster which had befallen nearly half\ntheir number before their eyes, without their being able to render\nthem the slightest assistance, had completely demoralized them,\nand as soon as the Scotch were seen to be crossing the bridge they\nfled in terror. A hot pursuit was kept up by the fleet footed and\nlightly armed Scots, and great numbers of fugitives were slain.\n\nMore than 20,000 English perished in the battle or flight, and the\nremainder crossed the Border a mere herd of broken fugitives.\n\nThe Earl of Surrey, before riding off the field, committed the charge\nof the Castle of Stirling to Sir Marmaduke de Twenge, promising him\nthat he would return to his relief within ten weeks at the utmost.\nAll the tents, wagons, horses, provisions, and stores of the English\nfell into the hands of their enemies, and every Scotch soldier\nobtained rich booty.\n\nCressingham was among the number killed. It was said by one\nEnglish historian, and his account has been copied by many others,\nthat Cressingham's body was flayed and his skin divided among the\nScots; but there appears no good foundation for the story, although\nprobably Cressingham, who had rendered himself peculiarly obnoxious\nand hateful to the Scots, was hewn in pieces. But even were it\nproved that the ill story is a true one, it need excite no surprise,\nseeing the wholesale slaying, plundering, and burning which had\nbeen carried on by the English, and that the Scottish prisoners\nfalling into their hands were often mutilated and tortured before\nbeing executed and quartered. The English historians were fond of\ncrying out that the Scotch were a cruel and barbarous people whenever\nthey retaliated for the treatment which they suffered; but so far\nfrom this being the case, it is probable that the Scotch, before\nthe first invasion of Edward, were a more enlightened and, for\ntheir numbers, a more well-to-do people than the English. They had\nfor many years enjoyed peace and tranquillity, and under the long\nand prosperous reign of Alexander had made great advances, while\nEngland had been harassed by continuous wars and troubles at home\nand abroad. Its warlike barons, when not engaged under its monarchs\nin wars in Wales, Ireland, and France, occupied themselves in quarrels\nwith each other, or in struggles against the royal supremacy; and\nalthough the higher nobles, with their mailclad followers, could\nshow an amount of chivalrous pomp unknown in Scotland, yet the\ncondition of the middle classes and of the agricultural population\nwas higher in Scotland than in England.\n\nArchie, as one of the principal leaders of the victorious army,\nreceived a share of the treasure captured in the camp sufficient to\nrepay the money which he had had for the strengthening of the Castle\nof Aberfilly, and on the day following the battle he received\npermission from Sir William to return at once, with the 250 retainers\nwhich he had brought into the field, to complete the rebuilding of\nthe castle. In another three months this was completed, and stores\nof arms and munition of all kinds collected.\n\nImmediately after the defeat at Stirling Bridge, King Edward summoned\nthe Scottish nobles to join Brian Fitzallan, whom he appointed\ngovernor of Scotland, with their whole forces, for the purpose of\nputting down the rebellion. Among those addressed as his allies were\nthe Earls Comyn of Badenoch, Comyn of Buchan, Patrick of Dunbar,\nUmfraville of Angus, Alexander of Menteith, Malise of Strathearn,\nMalcolm of Lennox, and William of Sutherland, together with James\nthe Steward, Nicholas de la Haye, Ingelram de Umfraville, Richard\nFraser, and Alexander de Lindsay of Crawford. From this enumeration\nit is clear that Wallace had still many enemies to contend with at\nhome as well as the force of England. Patrick of Dunbar, assisted\nby Robert Bruce and Bishop Anthony Beck, took the field, but was\ndefeated.  Wallace captured all the castles of the earl save Dunbar\nitself, and forced him to fly to England; then the Scotch army\npoured across the Border and retaliated upon the northern counties\nfor the deeds which the English had been performing in Scotland\nfor the last eight years. The country was ravaged to the very walls\nof Durham and Carlisle, and only those districts which bought off\nthe invaders were spared.  The title which had been bestowed upon\nWallace by a comparatively small number was now ratified by the\ncommonalty of the whole of Scotland; and associated with him was\nthe young Sir Andrew Moray of Bothwell, whose father had been the\nonly Scotch noble who had fought at Stirling, and it is notable\nthat in some of the documents of the time Wallace gives precedence\nto Andrew Moray.\n\nThey proceeded to effect a military organization of the country,\ndividing it up into districts, each with commanders and lieutenants.\nOrder was established and negotiations entered into for the mutual\nsafeguard of traders with the Hanse towns.\n\nThe nobles who ventured to oppose the authority of Wallace and his\ncolleague were punished in some cases by the confiscation of lands,\nwhich were bestowed upon Sir Alexander Scrymgeour and other loyal\ngentlemen, and these grants were recognized by Bruce when he became\nking. In these deeds of grant Wallace and Moray, although acting as\ngovernors of Scotland, state that they do so in the name of Baliol\nas king, although a helpless captive in England. For a short time\nScotland enjoyed peace, save that Earl Percy responded to the raids\nmade by the Scots across the Border, by carrying fire and sword\nthrough Annandale; and the English writers who complain of the\nconduct of the Scots, have no word of reprobation for the proclamation\nissued to the soldiers on crossing the Border, that they were free\nto plunder where they chose, nor as to the men and women slain,\nnor the villages and churches committed to the flames.\n\n\n\n\nChapter X\n\nThe Battle of Falkirk\n\n\nWhile Wallace was endeavouring to restore order in Scotland, Edward\nwas straining every nerve to renew his invasion. He himself was\nupon the Continent, but he made various concessions to his barons\nand great towns to induce them to aid him heartily, and issued writs\ncalling upon the whole nobility remaining at home, as they valued\nhis honour and that of England, to meet at York on January 20th,\n\"and proceed under the Earl of Surrey to repress and chastise the\naudacity of the Scots.\" At the same time he despatched special\nletters to those of the Scottish nobles who were not already in\nEngland, commanding them to attend at the rendezvous.\n\nThe call upon the Scotch nobles was not generally responded to.\nThey had lost much of their power over their vassals, many of whom\nhad fought under Wallace in spite of the abstention of their lords.\nIt was clear, too, that if they joined the English, and another\ndefeat of the latter took place, their countrymen might no longer\ncondone their treachery, but their titles and estates might be\nconfiscated. Consequently but few of them presented themselves at\nYork. There, however, the English nobles gathered in force. The\nEarls of Surrey, Gloucester, and Arundel; the Earl Mareschal and\nthe great Constable were there; Guido, son of the Earl of Warwick,\nrepresented his father. Percy was there, John de Wathe, John de\nSeagrave, and very many other barons, the great array consisting\nof 2000 horsemen heavily armed, 1200 light horsemen, and 100,000\nfoot soldiers.\n\nSir Aymer de Vallance, Earl of Pembroke, and Sir John Sieward, son\nof the Earl of March, landed with an army in Fife, and proceeded\nto burn and waste. They were met by a Scotch force under Wallace\nin the forest of Black Ironside, and were totally defeated.\n\nSurrey's army crossed the Border, raised the siege of Roxburgh,\nand advanced as far as Kelso. Wallace did not venture to oppose\nso enormous a force, but wasted the country on every side so that\nthey could draw no provisions from it, and Surrey was forced to\nfall back to Berwick; this town was being besieged by a Scottish\nforce, which retired at his approach. Here the English army halted\nupon receipt of orders from Edward to wait his coming. He had hastily\npatched up a peace with France, and, having landed at Sandwich,\nsummoned the parliament, and on the 27th of May issued writs to\nas many as 154 of his great barons to meet him at Roxburgh on the\n24th of June. Here 3000 cavalry, men and horses clothed in complete\narmour; 4000 lighter cavalry, the riders being armed in steel but\nthe horses being uncovered; 500 splendidly mounted knights and\nmen-at-arms from Gascony; and at least 80,000 infantry assembled\ntogether, with abundance of materials and munition of war of all\nkinds. This huge army marched from Roxburgh, keeping near the coast,\nreceiving provisions from a fleet which sailed along beside them.\nBut in spite of this precaution it was grievously straitened, and\nwas delayed for a month near Edinburgh, as Wallace so wasted the\ncountry that the army were almost famished, and by no efforts were\nthey able to bring on a battle with the Scots, whose rapid marches\nand intimate acquaintance with the country baffled all the efforts\nof the English leaders to force on an action.\n\nEdward was about to retreat, being unable any longer to subsist his\narmy, when the two Scottish Earls of Dunbar and Angus sent news to\nthe king that Wallace with his army was in Falkirk forest, about\nsix miles away, and had arranged to attack the camp on the following\nmorning. The English at once advanced and that evening encamped at\nLinlithgow, and the next morning moved on against the Scots.\n\nLate in the evening Archie's scouts brought in the news to Wallace\nthat the English army was within three miles, and a consultation\nwas at once held between the leaders. Most of them were in favour\nof a retreat; but Comyn of Badenoch, who had lately joined Wallace,\nand had been from his rank appointed to the command of the cavalry,\nwith some of his associates, urged strongly the necessity for\nfighting, saying that the men would be utterly dispirited at such\ncontinual retreats, and that with such immensely superior cavalry\nthe English would follow them up and destroy them. To these arguments\nWallace, Sir John Grahame, and Sir John Stewart, yielded their own\nopinions, and prepared to fight. They took up their position so\nthat their front was protected by a morass, and a fence of stakes\nand ropes was also fixed across so as to impede the advance or\nretreat of the English cavalry.  The Scotch army consisted almost\nentirely of infantry. These were about a third the number of those\nof the English, while Comyn's cavalry were a thousand strong.\n\nThe infantry were formed in three great squares or circles, the\nfront rank kneeling and the spears all pointing outwards. In the\nspace between these squares were placed the archers, under Sir John\nStewart.\n\nThe English army was drawn up in three divisions, the first commanded\nby the Earl Marechal, the Earl of Lincoln and Hereford; the second\nby Beck, the warlike Bishop of Durham, and Sir Ralph Basset;\nthe third by the king himself.  The first two divisions consisted\nalmost entirely of knights and men-at-arms; the third, of archers\nand slingers.\n\nWallace's plan of battle was that the Scottish squares should first\nreceive the brunt of the onslaught of the enemy, and that while\nthe English were endeavouring to break these the Scotch cavalry,\nwhich were drawn up some distance in the rear, should fall upon\nthem when in a confused mass, and drive them against the fence or\ninto the morass.\n\nThe first division of the English on arriving at the bog made a\ncircuit to the west. The second division, seeing the obstacle which\nthe first had encountered, moved round to the east, and both fell\nupon the Scottish squares. The instant they were seen rounding\nthe ends of the morass, the traitor Comyn, with the whole of the\ncavalry, turned rein and fled from the field, leaving the infantry\nalone to support the whole brunt of the attack of the English. So\nimpetuous was the charge of the latter that Sir John Stewart and\nhis archers were unable to gain the shelter of the squares, and\nhe was, with almost all his men, slain by the English men-at-arms.\nThus the spearmen were left entirely to their own resources.\n\nEncouraged by Wallace, Grahame, Archie Forbes, and their other\nleaders, the Scottish squares stood firmly, and the English cavalry\nin vain strove to break the hedge of spears.  Again and again the\nbravest of the chivalry of England tried to hew a way through. The\nScots stood firm and undismayed, and had the battle lain between\nthem and the English cavalry, the day would have been theirs. But\npresently the king, with his enormous body of infantry, arrived on\nthe ground, and the English archers and slingers poured clouds of\nmissiles into the ranks of the Scots; while the English spearmen,\npicking up the great stones with which the ground was strewn,\nhurled them at the front ranks of their foes. Against this storm\nof missiles the Scottish squares could do nothing. Such armour\nas they had was useless against the English clothyard arrows, and\nthousands fell as they stood.\n\nAgain and again they closed up the gaps in their ranks, but at last\nthey could no longer withstand the hail of arrows and stones, to\nwhich they could offer no return. Some of them wavered. The gaps\nin the squares were no longer filled up, and the English cavalry,\nwho had been waiting for their opportunity, charged into the midst\nof them. No longer was there any thought of resistance. The Scots\nfled in all directions. Numbers were drowned by trying to swim the\nriver Carron, which ran close by. Multitudes were cut down by the\nhost of English cavalry.\n\nSir Archie Forbes was in the same square with Wallace, with a few\nother mounted men. They dashed forward against the English as they\nbroke through the ranks of the spearmen, but the force opposed them\nwas overwhelming.\n\n\"It is of no use, Archie; we must retire. Better that than throw\naway our lives uselessly. All is lost now.\"\n\nWallace shouted to the spearmen, who gallantly rallied round him,\nand, keeping together in spite of the efforts of the English cavalry,\nsucceeded in withdrawing from the field.  The other squares were\nentirely broken and dispersed, and scarce a man of them escaped.\n\nAccounts vary as to the amount of the slaughter, some English\nwriters placing it as double that of the army which Wallace could\npossibly have brought into the field, seeing that the whole of the\ngreat nobles stood aloof, and that Grahame, Stewart, and Macduff of\nFife were the only three men of noble family with him.  All these\nwere slain, together with some 25,000 infantry.\n\nWallace with about 5000 men succeeded in crossing a ford of the\nCarron, and the English spread themselves over the country. The\ndistricts of Fife, Clackmannan, Lanark, Ayr, and all the surrounding\ncountry were wasted and burnt, and every man found put to the sword.\nThe Scotch themselves in retreating destroyed Stirling and Perth,\nand the English found the town of St. Andrew's deserted, and burnt\nit to the ground.\n\nNo sooner had Wallace retreated than he divided his force into\nsmall bands, which proceeded in separate directions, driving off the\ncattle and destroying all stores of grain, so that in a fortnight\nafter the battle of Falkirk the English army were again brought\nto a stand by shortness of provisions, and were compelled to fall\nback again with all speed to the mouth of the Forth, there to obtain\nprovisions from their ships. As they did so Wallace reunited his\nbands, and pressed hard upon them. At Linlithgow he fell upon their\nrear and inflicted heavy loss, and so hotly did he press them that\nthe great army was obliged to retreat rapidly across the Border,\nand made no halt until it reached the fortress of Carlisle.\n\nThat it was compulsion alone which forced Edward to make his\nspeedy retreat we may be sure from the fact that after the victory\nof Dunbar he was contented with nothing less than a clean sweep\nof Scotland to its northern coast, and that he repeated the same\nprocess when, in the year following the battle of Falkirk, he again\nreturned with a mighty army. Thus decisive as was the battle of\nFalkirk it was entirely abortive in results.\n\nWhen the English had crossed the Border, Wallace assembled the few\ngentlemen who were still with him, and announced his intention of\nresigning the guardianship of Scotland, and of leaving the country.\nThe announcement was received with exclamations of surprise and\nregret.\n\n\"Surely, Sir William,\" Archie exclaimed, \"you cannot mean it. You\nare our only leader; in you we have unbounded confidence, and in\nnone else. Had it not been for the treachery of Comyn the field of\nFalkirk would have been ours, for had the horse charged when the\nEnglish were in confusion round our squares they had assuredly been\ndefeated. Moreover, your efforts have retrieved that disastrous\nfield, and have driven the English across the Border.\"\n\n\"My dear Archie,\" Wallace said, \"it is because I am the only leader\nin whom you have confidence that I must needs go. I had vainly hoped\nthat when the Scottish nobles saw what great things the commonalty\nwere able to do, and how far, alone and unaided, they had cleared\nScotland of her tyrants, they would have joined us with their\nvassals; but you see it is not so. The successes that I have gained\nhave but excited their envy against me. Of them all only Grahame,\nStewart, and Macduff stood by my side, while all the great earls\nand barons either held aloof or were, like Bruce, in the ranks of\nEdward's army, or like Comyn and his friends, joined me solely to\nbetray me. I am convinced now that it is only a united Scotland can\nresist the power of England, and it is certain that so long as I\nremain here Scotland never can be united. Of Bruce I have no longer\nany hope; but if I retire Comyn may take the lead, and many at\nleast of the Scottish nobles will follow him. Had we but horsemen\nand archers to support our spearmen, I would not fear the issue;\nbut it is the nobles alone who can place mounted men-at-arms in\nthe field.  Of bowmen we must always be deficient, seeing that our\npeople take not naturally to this arm as do the English; but with\nspearmen to break the first shock of English chivalry, and with\nhorsemen to charge them when in confusion, we may yet succeed, but\nhorsemen we shall never get so long as the nobles hold aloof. It\nis useless to try and change my decision, my friends. Sore grief\nthough it will be to me to sheathe my sword and to stand aloof\nwhen Scotland struggles for freedom, I am convinced that only by my\ndoing so has Scotland a chance of ultimate success in the struggle.\nDo not make it harder for me by your pleadings. I have thought long\nover this, and my mind is made up. My heart is well nigh broken by\nthe death of my dear friend and brother in arms, Sir John Grahame,\nand I feel able to struggle no longer against the jealousy and\nhostility of the Scottish nobles.\"\n\nWallace's hearers were all in tears at his decision, but they felt\nthat there was truth in his words, that the Scottish nobles were\nfar more influenced by feelings of personal jealousy and pique than\nby patriotism, and that so long as Wallace remained the guardian\nof Scotland they would to a man side with the English. The next day\nWallace assembled all his followers, and in a few words announced\nhis determination, and the reasons which had driven him to take\nit. He urged them to let no feelings of resentment at the treatment\nhe had experienced, or any wrath at the lukewarmness and treachery\nwhich had hitherto marked the Scottish nobles, overcome their feeling\nof patriotism, but to follow these leaders should they raise the\nbanner of Scotland, as bravely and devotedly as they had followed\nhim.\n\nThen he bade them farewell, and mounting his horse rode to the\nseacoast and passed over to France.\n\nAlthough he had retired from Scotland, Wallace did not cease from\nwar against the English; but being warmly received by the French\nking fought against them both by sea and land, and won much renown\namong the French.\n\nAfter returning to England, Edward, finding that the Scottish leaders\nstill professed to recognize Baliol as king, sent him to the pope\nat Rome, having first confiscated all his great possessions in\nEngland and bestowed them upon his own nephew, John of Brittany;\nand during the rest of his life Baliol lived in obscurity in Rome.\nA portion of the Scotch nobles assembled and chose John Comyn of\nBadenoch and John de Soulis as guardians of the kingdom. In the\nautumn of the following year Edward again assembled a great army\nand moved north, but it was late; and in the face of the approaching\nwinter, and the difficulty of forage, many of the barons refused\nto advance. Edward himself marched across the Border; but seeing\nthat the Scots had assembled in force, and that at such a season\nof the year he could not hope to carry his designs fully into\nexecution, he retired without striking a blow. Thereupon the castle\nof Stirling, which was invested by the Scots, seeing no hope of\nrelief, surrendered, and Sir William Oliphant was appointed governor.\n\nThe next spring Edward again advanced with an army even greater\nthan that with which he had before entered Scotland. With him were\nAlexander of Baliol, son of the late king, who was devoted to the\nEnglish; Dunbar, Fraser, Ross, and other Scottish nobles. The vast\narmy first laid siege to the little castle of Carlaverock, which,\nalthough defended by but sixty men, resisted for some time the\nassaults of the whole army, but was at last captured. The Scots\nfell back as Edward advanced, renewing Wallace's tactics of wasting\nthe country, and Edward could get no further than Dumfries. Here,\nfinding the enormous difficulties which beset him, he made a pretence\nof yielding with a good grace to the entreaties of the pope and the\nKing of France that he would spare Scotland; he retired to England\nand disbanded his army, having accomplished nothing in the campaign\nsave the capture of Carlaverock.\n\nThe following summer he again advanced with the army, this time\nsupported by a fleet of seventy ships.  The Scots resorted to their\nusual strategy, and, when winter came, the invaders had penetrated\nno further than the Forth. Edward remained at Linlithgow for a\ntime, and then returned to England. Sir Simon Fraser, who had been\none of the leaders of the English army at Carlaverock, now imitated\nComyn's example, and, deserting the English cause, joined his\ncountrymen.\n\nThe greater part of the English army recrossed the Border, and the\nScots captured many of the garrisons left in the towns. Sir John\nSeagrave next invaded Scotland with from 20,000 to 30,000 men, mostly\ncavalry.  They reached the neighbourhood of Edinburgh, when Comyn\nand Fraser advanced against them with 8000 men, chiefly infantry.\nThe English army were advancing in three divisions, in order\nbetter to obtain provisions and forage. After a rapid night march\nthe Scotch came upon one of them, commanded by Seagrave in person;\nand conceiving himself sufficiently strong to defeat the Scots unaided\nby any of the other divisions, Sir John Seagrave immediately gave\nbattle.\n\nAs at Falkirk, the English cavalry were unable to break through\nthe Scottish pikes. Great numbers were killed or taken prisoners,\nSeagrave himself being severely wounded and captured, with\ntwenty distinguished knights, thirty esquires, and many soldiers.\nScarcely was the battle over when the second English division, even\nstronger than the first, arrived on the field. Encumbered by their\nprisoners, the Scots were at a disadvantage; and fearing to be attacked\nby these in the rear while engaged in front, they slaughtered the\ngreater portion of the prisoners, and arming the camp followers,\nprepared to resist the English onslaught. This failed as the first\nhad done; the cavalry were defeated with great loss by the spearmen,\nand many prisoners taken--among them Sir Ralph Manton.\n\nThe third English division now appeared; and the Scots, worn out\nby their long march and the two severe conflicts they had endured,\nwere about to fly from the field when their leaders exhorted them\nto one more effort. The second batch of prisoners were slaughtered,\nand the pikemen again formed line to resist the English charge.\nAgain were the cavalry defeated, Sir Robert Neville, their leader,\nslain, with many others, and the whole dispersed and scattered.\nSir Robert Manton, who was the king's treasurer, had had a quarrel\nwith Fraser, when the latter was in Edward's service, regarding\nhis pay; and Fraser is said by some historians to have now revenged\nhimself by slaying his prisoner. Other accounts, however, represent\nManton as having escaped.\n\nThe slaughter of the prisoners appears, although cruel, to have\nbeen unavoidable; as the Scots, having before them a well appointed\nforce fully equal to their own in number, could not have risked\nengaging, with so large a body of prisoners in their rear. None of\nthe knights or other leaders were slain, these being subsequently\nexchanged or ransomed, as we afterwards find them fighting in the\nEnglish ranks.\n\nSeeing by this defeat that a vast effort was necessary to conquer\nScotland, King Edward advanced in the spring of 1303 with an army\nof such numbers that the historians of the time content themselves\nwith saying that \"it was great beyond measure.\"  It consisted of\nEnglish, Welsh, Irish, Gascons, and Savoyards. One division, under\nthe Prince of Wales, advanced by the west coast; that of the king,\nby the east; and the two united at the Forth. Without meeting any\nserious resistance the great host marched north through Perth and\nDundee to Brechin, where the castle, under the charge of Sir Thomas\nMaille, resisted for twenty days; and it was only after the death\nof the governor that it surrendered.\n\nThe English then marched north through Aberdeen, Banff, and Moray\ninto Caithness, carrying utter destruction everywhere; towns and\nhamlets, villages and farmhouses were alike destroyed; crops were\nburned, forests and orchards cut down. Thus was the whole of Scotland\nwasted; and even the rich abbeys of Abberbredok and Dunfermline,\nthe richest and most famous in Scotland, were destroyed, and the\nwhole levelled to the ground. The very fields were as far as possible\ninjured--the intention of Edward being, as Fordun says, to blot\nout the people, and to reduce the land to a condition of irrecoverable\ndevastation, and thus to stamp out for ever any further resistance\nin Scotland.\n\nDuring the three years which had elapsed since the departure\nof Wallace, Archie had for the most part remained quietly in his\ncastle, occupying himself with the comfort and wellbeing of his\nvassals. He had, each time the English entered Scotland, taken the\nfield with a portion of his retainers, in obedience to the summons\nof Comyn. The latter was little disposed to hold valid the grants\nmade by Wallace, especially in the case of Archie Forbes, the Kerrs\nbeing connections of his house; but the feeling of the people in\ngeneral was too strongly in favour of the companion of Wallace for\nhim to venture to set it aside, especially as the castle could not\nbe captured without a long continued siege. Archie and many of the\nnobles hostile to the claims of Comyn obeyed his orders, he being\nthe sole possible leader, at present, of Scotland. Edward, however,\nhad left them no alternative, since he had, in order to induce\nthe English nobles to follow him, formally divided among them the\nlands of the whole of the Scotch nobles, save those actually fighting\nin his ranks.\n\nArchie was now nearly three-and-twenty, and his frame had fully\nborne out the promise of his youth. He was over the average height,\nbut appeared shorter from the extreme breadth of his shoulders;\nhis arms were long and sinewy, and his personal strength immense.\n\nFrom the time of his first taking possession of Aberfilly he had\nkept a party of men steadily engaged in excavating a passage from\nthe castle towards a wood a mile distant. The ground was soft and\noffered but few obstacles, but the tunnel throughout its whole\nlength had to be supported by massive timbers. Wood, however, was\nabundant, and the passage had by this time been completed. Whenever,\nfrom the length of the tunnel, the workmen began to suffer from\nwant of air, ventilation was obtained by running a small shaft\nup to the surface; in this was placed a square wooden tube of six\ninches in diameter, round which the earth was again filled in--a\nfew rapidly growing plants and bushes being planted round the\norifice to prevent its being noticed by any passerby.\n\n\n\n\nChapter XI\n\nRobert The Bruce\n\n\nAt the last great invasion by Edward, Archie did not take the field,\nseeing that Comyn, in despair of opposing so vast a host, did not\ncall out the levies.  Upon the approach of the English army under\nthe Prince of Wales he called the whole of his tenants into the\ncastle. Great stores of provisions had already been collected. The\nwomen and children were sent away up into the hills, where provisions\nhad also been garnered, and the old men and boys accompanied them.\nAs the Prince of Wales passed north, bands from his army spreading\nover the country destroyed every house in the district. Archie was\nsummoned to surrender, but refused to do so; and the prince, being\non his way to join his father on the Forth, after himself surveying\nthe hold, and judging it far too strong to be carried without\na prolonged siege, marched forward, promising on his return to\ndestroy it. Soon afterwards Archie received a message that Wallace\nhad returned. He at once took with him fifty men, and leaving the\ncastle in charge of Sandy Graham, with the rest of his vassals, two\nhundred and fifty in number, he rejoined his former leader. Many\nothers gathered round Wallace's standard; and throughout Edward's\nmarch to the north and his return to the Forth Wallace hung upon\nhis flanks, cutting off and slaying great numbers of the marauders,\nand striking blows at detached bands wherever these were in numbers\nnot too formidable to be coped with.\n\nStirling was now the only great castle which remained in the hands\nof the Scotch, and King Edward prepared to lay siege to this. Save\nfor the band of Wallace there was no longer any open resistance in\nthe field. A few holds like those of Archie Forbes still remained\nin the hands of their owners, their insignificance, or the time\nwhich would be wasted in subduing them, having protected them from\nsiege.  None of the nobles now remained in arms.\n\nBruce had for a short time taken the field; but had, as usual,\nhastened to make his peace with Edward.  Comyn and all his adherents\nsurrendered upon promise of their lives and freedom, and that they\nshould retain their estates, subject to a pecuniary fine. All the\nnobles of Scotland were included in this capitulation, save a few\nwho were condemned to suffer temporary banishment. Sir William\nWallace alone was by name specially exempted from the surrender.\n\nStirling Castle was invested on the 20th of April, 1304, and for\nseventy days held out against all the efforts of Edward's army.\nWarlike engines of all kinds had been brought from England for\nthe siege. The religious houses of St. Andrews, Brechin, and other\nchurches were stripped of lead for the engines. The sheriffs of\nLondon, Lincoln, York, and the governor of the Tower were ordered\nto collect and forward all the mangonels, quarrels, and bows and\narrows they could gather; and for seventy days missiles of all\nkinds, immense stones, leaden balls, and javelins were rained upon\nthe castle; and Greek fire--a new and terrible mode of destruction--was\nalso used in the siege. But it was only when their provisions\nand other resources were exhausted that the garrison capitulated;\nand it was found that the survivors of the garrison which had\ndefended Stirling Castle for upwards of three months against the\nwhole force of England numbered, including its governor, Sir William\nOliphant, and twenty-four knights and gentlemen, but a hundred and\ntwenty soldiers, two monks, and thirteen females.\n\nDuring the siege Wallace had kept the field, but Archie had, at\nhis request, returned to his castle, which being but a day's march\nfrom Stirling, might at any moment be besieged.  Several times,\nindeed, parties appeared before it, but Edward's hands were too\nfull, and he could spare none of the necessary engines to undertake\nsuch a siege; and when Stirling at length fell he and his army\nwere in too great haste to return to England to undertake another\nprolonged siege, especially as Aberfilly, standing in a retired\nposition, and commanding none of the principal roads, was a hold\nof no political importance.\n\nA short time afterwards, to Archie's immense grief, Sir William\nWallace was betrayed into the hands of the English.  Several\nScotchmen took part in this base act, the principal being Sir John\nMenteith. Late historians, in their ardour to whitewash those who\nhave for ages been held up to infamy, have endeavoured to show that\nSir John Menteith was not concerned in the matter; but the evidence\nis overwhelming the other way. Scotch opinion at the time, and\nfor generations afterwards, universally imputed the crime to him.\nFordun, who wrote in the reign of Robert Bruce, Bowyer, and Langtoft,\nall Scotch historians, say that it was he who betrayed Wallace, and\ntheir account is confirmed by contemporary English writings.  The\nChronicle of Lanercost, the Arundel MSS., written about the year\n1320, and the Scala Chronica, all distinctly say that Wallace was\nseized by Sir John Menteith; and finally, Sir Francis Palgrave has\ndiscovered in the memoranda of the business of the privy council\nthat forty marks were bestowed upon the young man who spied out\nWallace, sixty marks were divided among some others who assisted\nin his capture, and that to Sir John Menteith was given land of\nthe annual value of one hundred pounds--a very large amount in\nthose days.\n\nThe manner in which Wallace was seized is uncertain; but he was at\nonce handed by Sir John Menteith to Sir John Seagrave, and carried\nby him to London. He was taken on horseback to Westminster, the\nmayor, sheriffs, and aldermen, with a great number of horse and\nfoot, accompanying him.  There the mockery of a trial was held,\nand he was in one day tried, condemned, and executed. He defended\nhimself nobly, urging truly that, as a native born Scotsman, he\nhad never sworn fealty or allegiance to England, and that he was\nperfectly justified in fighting for the freedom of his country.\n\nEvery cruelty attended his execution. He was drawn through the\nstreets at the tails of horses; he was hung for some time by a\nhalter, but was taken down while yet alive; he was mutilated and\ndisembowelled, his head then cut off, his body divided in four,\nhis head impaled over London Bridge, and his quarters distributed\nto four principal towns in Scotland. Such barbarities were common\nat executions in the days of the Norman kings, who have been\ndescribed by modern writers as chivalrous monarchs.\n\nA nobler character than Wallace is not to be found in history. Alone,\na poor and landless knight, by his personal valour and energy he\naroused the spirit of his countrymen, and in spite of the opposition\nof the whole of the nobles of his country banded the people in\nresistance against England, and for a time wrested all Scotland from\nthe hands of Edward. His bitter enemies the English were unable to\nadduce any proofs that the epithets of ferocious and bloodthirsty,\nwith which they were so fond of endowing him, had even a shadow\nof foundation, and we may rather believe the Scotch accounts that\nhis gentleness and nobility of soul were equal to his valour. Of\nhis moderation and wisdom when acting as governor of Scotland there\ncan be no doubt, while the brilliant strategy which first won the\nbattle of Stirling, and would have gained that of Falkirk had not\nthe treachery and cowardice of the cavalry ruined his plans, show\nthat under other circumstances he would have taken rank as one of\nthe greatest commanders of his own or any age.\n\nHe first taught his countrymen, and indeed Europe in general, that\nsteady infantry can repel the assaults even of mailclad cavalry.\nThe lesson was followed at Bannockburn by Bruce, who won under\nprecisely the same circumstances as those under which Wallace had\nbeen defeated, simply because at the critical moment he had 500\nhorse at hand to charge the disordered mass of the English, while\nat Falkirk Wallace's horse, who should have struck the blow, were\ngalloping far away from the battlefield. Nor upon his English\nconquerors was the lesson lost, for at Cressy, when attacked by\nvastly superior numbers, Edward III dismounted his army, and ordered\nthem to fight on foot, and the result gave a death blow to that\nmailed chivalry which had come to be regarded as the only force\nworth reckoning in a battle. The conduct of Edward to Wallace,\nand later to many other distinguished Scotchmen who fell into his\nhands, is a foul blot upon the memory of one of the greatest of\nthe kings of England.\n\nEdward might now well have believed that Scotland was crushed for\never. In ten years no less than twelve great armies had marched\nacross the Border, and twice the whole country had been ravaged\nfrom sea to sea, the last time so effectually, that Edward had\ngood ground for his belief that the land would never again raise\nits head from beneath his foot.\n\nHe now proceeded, as William of Normandy after Hastings had done,\nto settle his conquest, and appointed thirty-one commissioners, of\nwhom twenty-one were English and ten so called Scotch, among them\nSir John Menteith, to carry out his ordinances. All the places of\nstrength were occupied by English garrisons.  The high officers and\na large proportion of the justiciaries and sheriffs were English,\nand Edward ruled Scotland from Westminster as he did England.\n\nAmong the commissioners was Robert Bruce, now through the death\nof his father, Lord of Annandale and Carrick; and Edward addressed\na proclamation to him, headed, \"To our faithful and loyal Robert\nde Bruce, Earl of Carrick, and all others who are in his company,\ngreeting;\" and went on to say that he possessed the king's fullest\nconfidence. But though Scotland lay prostrate, the spirit of\nresistance yet lingered in the hearts of the commonalty. Although\nconquered now the memory of their past success still inspired them,\nbut until some leader presented himself none could stir. It was in\nAugust that Wallace had been executed. Archie had received several\nsummonses from the English governors of Stirling and Lanark to\ncome in and do homage to Edward, but he had resolutely declined,\nand the task of capturing his castle was too heavy a one to\nbe undertaken by any single garrison; still he saw that the time\nmust come, sooner or later, when he would have to choose between\nsurrender and death. When matters settled down it was certain that\na great effort would be made to root out the one recalcitrant south\nof the Forth. For some time he remained gloomy and thoughtful,\na mood most unusual to him, and his mother, who was watching him\nanxiously, was scarcely surprised when one day he said to her:\n\n\"Mother, I must leave you for a time. Matters can no longer continue\nas they are. Surrender to the English I will not, and there remains\nfor me but to defend this castle to the last, and then to escape\nto France; or to cross thither at once, and enter the service of\nthe French king, as did Wallace. Of these courses I would fain take\nthe latter, seeing that the former would bring ruin and death upon\nour vassals, who have ever done faithful service when called upon,\nand whom I would not see suffer for my sake. In that case I should\npropose that you should return and live quietly with Sir Robert\nGordon until times change.\"\n\nDame Forbes agreed with her son, for she had long felt that further\nresistance would only bring ruin upon him.\n\n\"There is yet one other course, mother, and that I am about to take;\nit is well nigh a desperate one, and my hopes of success are small,\nyet would I attempt it before I leave Scotland and give Aberfilly\nback again to the Kerrs. Ask me not what it is, for it were best\nthat if it fail you should not know of it. There is no danger in\nthe enterprise, but for a month I shall be absent. On my return\nyou shall hear my final resolve.\"\n\nHaving attired himself as a lowland farmer, Archie proceeded to\nEdinburgh, and there took ship for London; here he took lodgings\nat an inn, which he had been told in Edinburgh was much frequented\nby Scotchmen who had to go to London on business. His first care\nwas to purchase the garments of an English gentleman of moderate\nmeans, so that he could pass through the streets without attracting\nattention.\n\nHe was greatly impressed with the bustle and wealth of London.\n\n\"It is wonderful,\" he said to himself, \"that we Scots, who were\nafter all but an army of peasants, could for nigh ten years have\nsupported a war against such a country as this, and it seems madness\nto adventure farther in that way. If my present errand fails I will\nassuredly hold firm to my resolve and seek a refuge in France.\"\n\nArchie ascertained that Robert the Bruce lodged at Westminster,\nand that great gaieties were taking place at the court for joy at\nthe final termination of hostilities with Scotland, now secured by\nthe execution of Wallace.  He despatched a letter to the earl by\na messenger from the inn, saying that one who had formerly known\nhim in Scotland desired earnestly to speak to him on matters of\ngreat import, and begging him to grant a private interview with him\nat his lodging at as early an hour as might be convenient to him.\nThe man returned with a verbal reply, that the earl would see the\nwriter at his lodging at nine o'clock on the following morning.\n\nAt the appointed time Archie presented himself at the house inhabited\nby Bruce. To the request of the earl's retainer for his name and\nbusiness he replied that his name mattered not, but that he had\nreceived a message from the earl appointing him a meeting at that\nhour.\n\nTwo minutes later he was ushered into the private cabinet of Robert\nBruce. The latter was seated writing, and looked up at his unknown\nvisitor.\n\n\"Do you remember me, Sir Robert Bruce?\" Archie asked.\n\n\"Methinks I know your face, sir,\" the earl replied, \"but I cannot\nrecall where I have seen it.\"\n\n\"It is five years since,\" Archie said, \"and as that time has changed\nme from a youth into a man I wonder not that my face has escaped\nyou.\"\n\n\"I know you now!\" the earl exclaimed, rising suddenly from his\nseat. \"You are Sir Archibald Forbes?\"\n\n\"I am,\" Archie replied, \"and I have come now on the same errand I\ncame then--the cause of our country.  The English think she is\ndead, but, though faint and bleeding, Scotland yet lives; but there\nis one man only who can revive her, and that man is yourself.\"\n\n\"Your mission is a vain one,\" Bruce replied. \"Though I honour you,\nSir Archibald, for your faith and constancy; though I would give\nmuch, ay all that I have, were my record one of as true patriotism\nand sacrifice as yours, yet it were madness to listen to you. Have\nI not,\" he asked bitterly, \"earned the hatred of my countrymen?\nHave I not three times raised my standard only to lower it again\nwithout striking a blow?  Did I not fight by Edward at the field\nof Falkirk? Ah!\" he said in a changed tone, \"never shall I forget\nthe horror which I felt as I passed over the field strewn with\nScottish corpses. Truly my name must be loathed in Scotland; and\nyet, Sir Archibald, irresolute and false as I have hitherto proved\nmyself, believe me, I love Scotland, the land of my mother.\"\n\n\"I believe you, sir,\" Archie said, \"and it is therefore that I\nimplore you to listen to me. You are now our only possible leader,\nour only possible king. Baliol is a captive at Rome, his son a courtier\nof Edward.  Wallace is dead. Comyn proved weak and incapable, and\nwas unable to rally the people to offer any opposition to Edward's\nlast march. Scotland needs a leader strong and valiant as Wallace,\ncapable of uniting around him a large body, at least, of the Scotch\nnobles, and having some claim to her crown. You know not, sir, how\ndeep is the hatred of the English. The last terrible incursion of\nEdward has spread that feeling far and wide, and while before it was\nbut in a few counties of the lowlands that the flame of resistance\nreally burnt, this time, believe me, that all Scotland, save perhaps\nthe Comyns and their adherents, would rise at the call. I say not\nthat success would at once attend you, for, forgive me for saying\nso, the commonalty would not at first trust you; but when they saw\nthat you were fighting for Scotland as well as for your own crown,\nthat you had, by your action, definitely and for ever broken with\nthe English, and had this time entered heart and soul into the cause,\nI am sure they would not hold back. Your own vassals of Carrick and\nAnnandale are a goodly array in themselves and the young Douglas\nmight be counted on to bring his dalesmen to your banner. There\nare all the lords who have favoured your cause, and so stood aloof\nfrom Comyn. You will have a good array to commence with; but above\nall, even if unsuccessful at first, all Scotland would come in\ntime to regard you as her king and champion. Resistance will never\ncease, for even Wallace was ever able to assemble bands and make\nhead against the English, so will it be with you, until at last\nfreedom is achieved, and you will reign a free king over a free\nScotland, and your name will be honoured to all time as the champion\nand deliverer of our country. Think not, sir,\" he went on earnestly\nas Bruce paced up and down the little room, \"that it is too late.\nOther Scotchmen, Fraser and many others, who have warred in the\nEnglish ranks, have been joyfully received when at length they\ndrew sword for Scotland.  Only do you stand forth as our champion,\nbelieve me, that the memory of former weakness will be forgotten\nin the admiration of present patriotism.\"\n\nFor two or three minutes Bruce strode up and down the room; then\nhe paused before Archie.\n\n\"By heavens,\" he said, \"I will do it! I am not so sanguine as you,\nI do not believe that success can ever finally attend the enterprise,\nbut, be that as it may, I will attempt it, win or die. The memory\nof Robert Bruce shall go down in the hearts of Scotchmen as one\nwho, whatever his early errors, atoned for them at last by living\nand dying in her cause. My sisters and brothers have long urged me\nto take such a step, but I could never bring myself to brave the\npower of England. Your words have decided me. The die is cast.\nHenceforward Robert Bruce is a Scotchman. And now, Sir Archibald,\nwhat think you my first step should be?\"\n\n\"The English in Scotland are lulled in security, and a sudden blow\nupon them will assuredly at first be wholly successful. You must\nwithdraw suddenly and quietly from here.\"\n\n\"It is not easy to do so,\" Bruce replied. \"Although high in favour\nwith Edward, he has yet some suspicions of me--not,\" he said\nbitterly, \"without just cause--and would assuredly arrest me did\nhe know that I were going north. My only plan will be to appear\nat court as usual, while I send down relays of horses along the\nnorthern road. You will ride with me, Sir Archie, will you not?\nBut I must tell you that I have already, in some degree, prepared\nfor a movement in Scotland. Comyn and I have met and have talked\nover the matter.  Our mutual claims to the crown stood in the way,\nbut we have agreed that one shall yield to the other, and that\nwhoso takes the crown shall give all his lands to be the property\nof the other, in consideration of his waiving his claim and giving\nhis support. This we have agreed to, and have signed a mutual bond\nto that effect, and though it is not so writ down we have further\nagreed that I shall have the crown and that Comyn shall take Carrick\nand Annandale; but this was for the future, and we thought not of\nany movement for the present.\"\n\n\"It were a bad bargain, sir,\" Archie said gravely; \"and one that I\ntrust will never be carried out. The Comyns are even now the most\npowerful nobles in Scotland, and with Carrick and Annandale in\naddition to their own broad lands, would be masters of Scotland,\nlet who would be called her king. Did he displease them, they\ncould, with their vassals and connections, place a stronger army\nin the field than that which the king could raise; and could at any\nmoment, did he anger them, call in the English to his aid, and so\nagain lay Scotland under the English yoke.\"\n\n\"I will think of it, Sir Archie. There is much in what you say, and\nI sorely doubt the Comyns. Henceforth do not fear to give me your\nadvice freely. You possessed the confidence of Wallace, and have\nshown yourself worthy of it. Should I ever free Scotland and win\nme a kingdom, believe me you will not find Robert Bruce ungrateful.\nI will give orders tomorrow for the horses to be privately\nsent forward, so that at any hour we can ride if the moment seem\npropitious; meanwhile I pray you to move from the hostelry in the\ncity, where your messenger told me you were staying, to one close\nat hand, in order that I may instantly communicate with you in case\nof need. I cannot ask you to take up your abode here, for there\nare many Scotchmen among my companions who might know your face,\nor who, not knowing, might make inquiry of me as to your family;\nbut among the crowd of strangers who on some business or other at\nthe court throng the inns of the city of Westminster, one figure\nmore or less would excite neither question nor comment.\"\n\nThat afternoon Archie took up his abode at Westminster.  A week\nlater one of Bruce's retainers came in just as Archie was about to\nretire to bed, and said that the Earl of Carrick wished immediately\nto see Master Forbes. Sir Archie had retained his own name while\ndropping the title. He at once crossed, to Bruce's lodging.\n\n\"We must mount at once!\" the earl exclaimed as he entered. \"What\nthink you? I have but now received word from a friend, who is\na member of the council, to say that this afternoon a messenger\narrived from the false Comyn with a letter to the king, containing\na copy of the bond between us.  Whether the coward feared the\nconsequences, or whether he has all along acted in treachery with\nthe view of bringing me into disgrace, and so ridding himself of\na rival, I know not; but the result is the same, he has disclosed\nour plans to Edward. A council was hastily called, and it has but\njust separated. It is to meet again in the morning, and the king\nhimself will be present. I am to be summoned before it, being, as\nit is supposed, in ignorance of the betrayal of my plans. It was\nwell for me that Edward himself had pressing engagements, and was\nunable to be present at the council. Had he been, prompt steps would\nhave been taken, and I should by this time be lying a prisoner in\nthe Tower. Even now I may be arrested at any moment. Have you aught\nfor which you wish to return to your inn?\"\n\n\"No,\" Archie replied. \"I have but a change of clothing there, which\nis of no importance, and we had best lose not a moment's time. But\nthere is the reckoning to discharge.\"\n\n\"I will give orders,\" the earl said, \"that it shall be discharged\nin the morning. Now let us without a moment's delay make to the\nstables and mount there. Here is a cloak and valise.\"\n\nThe earl struck a bell, and a retainer appeared.\n\n\"Allan, I am going out to pay a visit. Take these two valises to\nthe stable at once, and order Roderick to saddle the two bay horses\nin the stalls at the end of the stables. Tell him to be speedy, for\nI shall be with him anon. He is not bring them round here. I will\nmount in the court.\"\n\nFive minutes later Bruce and Archie, enveloped in thick cloaks\nwith hoods drawn over their faces, rode north from Westminster. At\nfirst they went slowly, but as soon as they were out in the fields\nthey set spur to their horses and galloped on in the darkness.\n\nThe snow lay thick upon the ground, and the roads were entirely\ndeserted.\n\n\"Farewell to London!\" Bruce exclaimed. \"Except as a prisoner I\nshall never see it again. The die is cast this time, Sir Archie,\nand for good; even if I would I can never draw back again. Comyn's\ntreachery has made my action irrevocable--it is now indeed death\nor victory!\"\n\nAll night they rode without drawing rein, save that they once\nchanged horses where a relay had been provided. They had little\nfear of pursuit, for even when Bruce's absence was discovered none\nof his household would be able to say where he had gone, and some\ntime must elapse before the conviction that he had ridden for\nScotland, in such weather, would occur to the king. Nevertheless,\nthey travelled fast, and on the 10th of February entered Dumfries.\n\n\n\n\nChapter XII\n\nThe Battle of Methven\n\n\nBruce had, during the previous week, sent messages saying to several\nof his friends in Annandale and Carrick that he might at any time\nbe among them, and at Dumfries he found many of them prepared to\nsee him. The English justiciaries for the southern district of the\nconquered kingdom were holding an assize, and at this most of the\nnobles and principal men of that part were present.  Among these\nwere, of course, many of Bruce's vassals; among them also was John\nComyn of Badenoch, who held large estates in Galloway, in virtue\nof which he was now present.\n\nAs soon as the news that Bruce had arrived in the town spread, his\nadherents and vassals there speedily gathered round him, and as,\naccompanied by several of them, he went through the town he met\nComyn in the precincts of the Grey Friars. Concerning this memorable\nmeeting there has been great dispute among historians. Some have\ncharged Bruce with inviting Comyn to meet him, with the deliberate\nintention of slaying him; others have represented the meeting as\naccidental, and the slaying of Comyn as the result of an outburst\nof passion on the part of Bruce; but no one who weighs the facts,\nand considers the circumstances in which Comyn was placed, can feel\nthe least question that the latter is the true hypothesis.\n\nBruce, whose whole course shows him to have been a man who acted\nwith prudence and foresight, would have been nothing short of mad had\nhe, just at the time when it was necessary to secure the goodwill\nof the whole of the Scotch nobles, chosen that moment to slay Comyn,\nwith whom were connected, by blood or friendship, the larger half\nof the Scotch nobles. Still less, had he decided upon so suicidal\na course, would he have selected a sanctuary as the scene of the\ndeed. To slay his rival in such a place would be to excite against\nhimself the horror and aversion of the whole people, and to enlist\nagainst him the immense authority and influence of the church.\nTherefore, unless we should conclude that Bruce--whose early\ncareer showed him to be a cool and calculating man, and whose future\ncourse was marked throughout with wisdom of the highest character--was\nsuffering from an absolute aberration of intellect, we must\naccept the account by those who represent the meeting as accidental,\nand the slaying as the result of an outburst of passion provoked\nby Comyn's treachery, as the correct one.\n\nWhen Bruce saw Comyn approaching he bade his followers stop where\nthey were and advanced towards Comyn, who was astonished at his\npresence.\n\n\"I would speak with you aside, John Comyn,\" Bruce said; and the\ntwo withdrew into the church apart from the observation of others.\n\nThen Bruce broke into a torrent of invective against Comyn for his\ngross act of treachery in betraying him by sending to Edward a copy\nof their agreement.\n\n\"You sought,\" he said, \"to send me to the scaffold, and so clear\nthe way for yourself to the throne of Scotland.\"\n\nComyn, finding that dissimulation was useless, replied as hotly.\nThose without could hear the voices of the angry men rise higher\nand higher; then there was a silence, and Bruce hurried out alone.\n\n\"What has happened?\" Archie Forbes exclaimed.\n\n\"I fear that I have slain Comyn,\" Bruce replied in an agitated\nvoice.\n\n\"Then I will make sure,\" Kirkpatrick, one of his retainers, said;\nand accompanied by Lindsay and another of his companions he ran in\nand completed the deed.\n\nScarcely was this done than Sir Robert Comyn, uncle of the earl,\nran up, and seeing what had taken place, furiously attacked Bruce\nand his party. A fierce fray took place, and Robert Comyn and\nseveral of his friends were slain.\n\n\"The die is cast now,\" Bruce said when the fray was over; \"but\nI would give my right hand had I not slain Comyn in my passion;\nhowever, it is too late to hesitate now. Gather together, my\nfriends, all your retainers, and let us hurry at once to attack\nthe justiciaries.\"\n\nIn a few minutes Kirkpatrick brought together those who had\naccompanied him and his companions to the town, and they at once\nmoved against the courthouse. The news of Bruce's arrival and of\nthe fray with the Comyns had already reached the justiciaries, and\nwith their retainers and friends they had made hasty preparations\nfor defence; but seeing that Bruce's followers outnumbered them,\nand that a defence might cost them their lives, they held parley\nand agreed to surrender upon Bruce promising to allow them to\ndepart at once for England. Half an hour later the English had left\nDumfries.\n\nBruce called a council of his companions.\n\n\"My friends,\" he said, \"we have been hurried into a terrible strife,\nand deeply do I regret that by my own mad passion at the treachery\nof Comyn I have begun it by an evil deed; but when I tell you of the\nway in which that traitor sought to bring me to an English block,\nyou will somewhat absolve me for the deed, and will grant that,\nunhappy and unfortunate as it was, my passion was in some degree\njustified.\"\n\nHe then informed them of the bond into which he and Comyn had\nentered, and of its betrayal by Comyn to Edward.\n\n\"Thus it is,\" he said, \"that the deed has taken place, and it\nis too late to mend it. We have before us a desperate enterprise,\nand yet I hope that we may succeed in it.  At any rate, this time\nthere can be no drawing back, and we must conquer or die. It was\ncertain in any case that Comyn and his party would oppose me, but\nnow their hostility will go to all lengths, while Edward will never\nforgive the attack upon his justiciaries. Still we shall have some\nbreathing time. The king will not hear for ten days of events here,\nand it will take him two months at least before he can assemble\nan army on the Border, and Comyn's friends will probably do nought\ntill the English approach. However, let us hurry to Lochmaben\nCastle; there we shall be safe from any sudden attack by Comyn's\nfriends in Galloway. First let us draw out papers setting forth\nthe cause of my enmity to Comyn, and of the quarrel which led to\nhis death, and telling all Scotchmen that I have now cut myself\nloose for ever from England, and that I have come to free Scotland\nand to win the crown which belongs to me by right, or to die in\nthe attempt.\"\n\nMany of these documents being drawn out, messengers were despatched\nwith them to Bruce's friends throughout the country, and he and\nhis followers rode to Lochmaben.\n\nArchie Forbes went north to his own estate, and at once gave\nnotice to his retainers to prepare to take the field, and to march\nto Glasgow, which Bruce had named as the rendezvous for all well\ndisposed towards him. From time to time messages came from Bruce,\ntelling him that he was receiving many promises of support; the\nwhole of the vassals of Annandale and Carrick had assembled at\nLochmaben, where many small landowners with their retainers also\njoined him.  As soon as his force had grown to a point when he\nneed fear no interruption on his march toward Glasgow, Bruce left\nLochmaben.  On his way he was joined by the first influential\nnobleman who had espoused his cause; this was Sir James Douglas,\nwhose father, Sir William, had died in an English prison. At the\ntime of his capture his estates had been bestowed by Edward upon\nLord Clifford, and the young Douglas, then but a lad, had sought\nrefuge in France. After a while he had returned, and was living\nwith Lamberton, Bishop of St. Andrews, who had been one of Wallace's\nmost active supporters.\n\nThe young Douglas, on receiving the news that Bruce was marching\nnorth, at once mounted, rode off, and joined him. He was joyfully\nreceived by Bruce, as not only would his own influence be great\namong his father's vassals of Douglasdale, but his adhesion would\ninduce many others to join. Receiving news of Bruce's march,\nArchie moved to Glasgow with his retainers. The English garrison\nand adherents in Glasgow fled at his approach. Upon arriving there\nBruce solemnly proclaimed the independence of Scotland, and sent\nout notices to all the nobles and gentry, calling upon them to join\nhim.\n\nFortunately the Bishop of St. Andrews, and Wishart, Bishop of\nGlasgow, another of Wallace's friends, at once declared strongly\nfor him, as did the Bishop of Moray and the Abbot of Scone. The\nadhesion of these prelates was of immense importance to Bruce, as\nto some extent the fact of their joining him showed that the church\nfelt no overwhelming indignation at the act of sacrilege which he\nhad committed, and enabled the minor clergy to advocate his cause\nwith their flocks.\n\nMany of the great nobles hostile to the Comyn faction also joined\nhim; among these were the Earls of Athole, Lennox, Errol, and\nMenteith; Christopher Seaton, Sir Simon Fraser, David Inchmartin,\nHugh de la Haye, Walter de Somerville, Robert Boyd, Robert Fleming,\nDavid Barclay, Alexander Fraser, Sir Thomas Randolph, and Sir\nNeil Campbell. Bruce's four brothers, Edward, Nigel, Thomas, and\nAlexander, were, of course, with him. Bruce now moved from Glasgow\nto Scone, and was there crowned King of Scotland on the 27th of\nMarch, 1306, six weeks after his arrival at Dumfries.  Since the\ndays of Malcolm Canmore the ceremony of placing the crown on the\nhead of the monarch had been performed by the representative of\nthe family of Macduff, the earls of Fife; the present earl was in\nthe service of the English; but his sister Isobel, wife of Comyn,\nEarl of Buchan, rode into Scone with a train of followers upon the\nday after the coronation, and demanded to perform the office which\nwas the privilege of the family. To this Bruce gladly assented,\nseeing that many Scotchmen would hold the coronation to be irregular\nfrom its not having been performed by the hereditary functionary, and\nthat as Isabel was the wife of Comyn of Buchan, her open adhesion\nto him might influence some of that faction. Accordingly on the\nfollowing day the ceremony was again performed, Isobel of Buchan\nplacing the crown on Bruce's head, an act of patriotism for which\nthe unfortunate lady was afterwards to pay dearly. Thus, although\nthe great majority of the Scotch nobles still held aloof, Bruce was\nnow at the head of a considerable force, and he at once proceeded\nto overrun the country. The numerous English who had come across\nthe Border, under the belief that Scotland was finally conquered,\nor to take possession of lands granted them by Edward, were all\ncompelled either to take refuge in the fortified towns and castles\nheld by English garrisons, or to return hastily to England.\n\nWhen the news of the proceedings at Dumfries and the general\nrising in the south of Scotland reached Edward he was at the city\nof Winchester. He had been lately making a sort of triumphant\npassage through the country, and the unexpected news that Scotland\nwhich he had believed crushed beyond all possibility of further\nresistance was again in arms, is said for a time to have driven\nhim almost out of his mind with rage.\n\nNot a moment was lost. Aylmer de Valence, Earl of Pembroke, was at\nonce commissioned to proceed to Scotland, to \"put down rebellion\nand punish the rebels,\" the whole military array of the northern\ncounties was placed under his orders, and Clifford and Percy were\nassociated with him in the commission. Edward also applied to the\npope to aid him in punishing the sacrilegious rebels who had violated\nthe sanctuary of Dumfries.  As Clement V was a native of Guienne,\nand kept his court at Bordeaux within Edward's dominions, his\nrequest was, of course, promptly complied with, and a bull issued,\ninstructing the Archbishop of York and the Bishop of Carlisle to\nexcommunicate Bruce and his friends, and to place them and their\npossessions under an interdict. It was now that the adhesion of\nthe Scottish prelates was of such vital consequence to Bruce. Had\nthe interdict been obeyed, the churches would have been closed,\nall religious ceremonies suspended, the rites of the church would\nhave been refused even to dying men, and the dead would have been\nburied without service in unconsecrated ground. So terrible a weapon\nas this was almost always found irresistible, and its terrors had\ncompelled even the most powerful monarchs to yield obedience to\nthe pope's orders; but the Scotch prelates set the needs of their\ncountry above the commands of the pope, and in spite of repeated bulls\nthe native clergy continued to perform their functions throughout\nthe whole struggle, and thus nullified the effect of the popish\nanathema.\n\nKing Edward was unable himself to lead his army against the Scots,\nfor he was now sixty-seven years old, and the vast fatigues and\nexertions which he had undergone in the course of a life spent almost\ncontinually in war had told upon him.  He had partially lost the\nuse of his limbs, and was forced to travel in a carriage or litter;\nbut when he reached London from Winchester a grand ceremony was\nheld, at which the order of knighthood was conferred by the king\nupon the Prince of Wales, and three hundred aspirants belonging to\nthe principal families of the country, and orders were given that\nthe whole military array of the kingdom should, in the following\nspring, gather at Carlisle, where Edward himself would meet them\nand accompany them to Scotland.  The Earl of Pembroke, with Clifford\nand Percy, lost no time in following the orders of Edward, and with\nthe military power of the northern counties marched into Scotland.\nThey advanced unopposed to the Forth, and crossing this river proceeded\ntowards Perth, near which town the Scottish army were gathered.\nArchie Forbes, who stood very high in favour with Bruce, had urged\nupon him the advantage of carrying out the tactics formerly adopted\nby Wallace, and of compelling the enemy to fall back by cutting\noff all food supplies, but Bruce would not, in this instance, be\nguided by his counsel.\n\n\"When the king advances next spring with his great army, Sir Archie,\nI will assuredly adopt the course which you point out, seeing\nthat we could not hope to withstand so great an array in a pitched\nbattle; but the case is different now. In the first place all the\ncastles and towns are in the hands of the English, and from them\nPembroke can draw such provision as he needs. In the second place\nhis force is not so superior to our own but that we may fight him\nwith a fair hope of victory; and whereas Wallace had never any\ncavalry with him, save at Falkirk when they deserted him at the\nbeginning of the battle, we have a strong body of mounted men-at-arms,\nthe retainers of the nobles with me, therefore I do not fear to\ngive them battle in the open field.\"\n\nIn pursuance of this determination Bruce sent a challenge to Pembroke\nto meet him with his army in the open field next day. Pembroke\naccepted the challenge, and promised to meet his opponent on the\nfollowing morning, and the Scotch retired for the night to the\nwood of Methven, near Perth. Here many of them set out on foraging\nexcursions, the knights laid aside their armour, and the army\nprepared for sleep.\n\nArchie Forbes was much dissatisfied at the manner in which Bruce had\nhazarded all the fortunes of Scotland on a pitched battle, thereby\nthrowing away the great advantage which their superior mobility and\nknowledge of the country gave to the Scots. He had disarmed like\nthe rest, and was sitting by a fire chatting with William Orr and\nAndrew Macpherson, who, as they had been his lieutenants in the\nband of lads he had raised seven years before, now occupied the\nsame position among his retainers, each having the command of a\nhundred men. Suddenly one who had been wandering outside the lines\nin search of food among the farmhouses ran hastily in, shouting\nthat the whole English army was upon them.\n\nA scene of the utmost confusion took place. Bruce and his knights\nhastily armed, and mounting their horses rode to meet the enemy.\nThere was no time to form ranks or to make any order of battle.\nArchie sprang to his horse. He bade his lieutenants form the men\ninto a compact body and move forward, keeping the king's banner\never in sight, and to cut their way to it whenever they saw it was\nin danger.  Then, followed by his two mounted squires, he rode after\nthe king. The contest of Methven can scarce be called a battle, for\nthe Scots were defeated before it began. Many, as has been said,\nwere away; great numbers of footmen instantly took flight and\ndispersed in all directions. Here and there small bodies stood and\nfought desperately, but being unsupported were overcome and slain.\nThe king with his knights fought with desperate bravery, spurring\nhither and thither and charging furiously among the English\nmen-at-arms.  Three times Bruce was unhorsed and as often remounted\nby Sir Simon Fraser. Once he was so entirely cut off from his\ncompanions by the desperation with which he had charged into the\nmidst of the English, that he was surrounded, struck from his horse,\nand taken prisoner.\n\n\"The king is taken!\" Archie Forbes shouted; \"ride in, my lords,\nand rescue him.\"\n\nMost of the Scotch knights were so hardly pressed that they could\ndo nothing to aid the king; but Christopher Seaton joined Archie,\nand the two knights charged into the midst of the throng of English\nand cut their way to Bruce. Sir Philip Mowbray, who was beside\nthe captured monarch, was overthrown, and several others cut down.\nBruce leapt into his saddle again and the three for a time kept at\nbay the circle of foemen; but such a conflict could have but one\nend. Archie Forbes vied with the king in the strength and power of\nhis blows, and many of his opponents went down before him.  There\nwas, however, no possibility of extricating themselves from the\nmass of their foes, and Bruce, finding the conflict hopeless, was\nagain about to surrender when a great shout was heard, and a close\nbody of Scottish spearmen threw themselves into the ranks of the\nEnglish horse.  Nothing could withstand the impetuosity of the\nassault. The horsemen recoiled before the levelled spears, and the\npikemen, sweeping onward, surrounded the king and his companions.\n\n\"Well done, my brave fellows!\" Archie cried; \"now keep together in\na close body and draw off the field.\"\n\nThe darkness which had at first proved so disastrous to the Scots\nwas now favourable to them. The English infantry knew not what was\ngoing on. The cavalry tried in vain to break through the ranks of\nthe spearmen, and these, keeping closely together, regained the\nshelter of the wood, and drew off by way of Dunkeld and Killiecrankie\nto the mountains of Athole. On their way they were joined by Edward\nBruce, the Earl of Athole, Sir Neil Campbell, Gilbert de la Haye,\nand Douglas, and by many scattered footmen.\n\nTo his grief Bruce learned that Randolph, Inchmartin, Somerville,\nAlexander Fraser, Hugh de la Haye, and others had been captured,\nbut the number killed had been small.  When once safe from pursuit\na council was held. It was agreed at once that it was impossible\nthat so large a body could find subsistence in the mountains of\nAthole, cooped up as they were by their foes. The lowlands swarmed\nwith the English; to the north was Badenoch, the district of their\nbitter enemies the Comyns; while westward lay the territory of\nthe MacDougalls of Lorne, whose chieftain, Alexander, was a nephew\nby marriage of the Comyn killed by Bruce, and an adherent of the\nEnglish.\n\nBeyond an occasional deer, and the fish in the lochs and streams,\nthe country afforded no means of subsistence, it was therefore\ndecided to disband the greater portion of the force, the knights\nand nobles, with a few of their immediate retainers, alone remaining\nwith the king, while the main body dispersed and regained their\nhomes. This was done; but a few days later a messenger came saying\nthat the queen, with the wives of many of the gentlemen, had arrived\nat Aberdeen and sought to join the king. Although an accession\nof numbers was by no means desirable, and the hardships of such\na life immense for ladies to support, there was no other resource\nbut for them to join the party, as they would otherwise have speedily\nfallen into the hands of the English. Therefore Bruce, accompanied\nby some of his followers, rode to Aberdeen and escorted the queen\nand ladies to his mountain retreat.\n\nIt was a strange life that Bruce, his queen, and his little court\nled. Sleeping in rough arbours formed of boughs, the party supported\nthemselves by hunting and fishing.\n\nGins and traps were set in the streams, and Douglas and Archie\nwere specially active in this pursuit; Archie's boyish experience\nat Glen Cairn serving him in good stead. Between him and Sir James\nDouglas a warm friendship had sprung up. Douglas was four years\nhis junior. As a young boy he had heard much of Archie's feats with\nWallace, and his father had often named him to him as conspicuous\nfor his bravery, as well as his youth. The young Douglas therefore\nentertained the highest admiration for him, and had from the time\nof his joining Bruce become his constant companion.\n\nBruce himself was the life and soul of the party. He was ever\nhopeful and in high spirits, cheering his followers by his gaiety,\nand wiling away the long evenings by tales of adventure and chivalry,\ntold when they were gathered round the fire.\n\nGradually the party made their way westward along Loch Tay and\nGlen Dochart until they reached the head of Strathfillan; here, as\nthey were riding along a narrow pass, they were suddenly attacked\nby Alexander MacDougall with a large gathering of his clansmen.\nSeveral of the royal party were cut down at once, but Bruce with\nhis knights fought desperately. Archie Forbes with a few of the\nothers rallied round the queen with her ladies, and repelled every\neffort of the wild clansmen to break through, and continued to draw\noff gradually down the glen. Bruce, with Douglas, De la Haye, and\nsome others, formed the rearguard and kept back the mass of their\nopponents. De la Haye and Douglas were both wounded, but the little\nparty continued to show a face to their foes until they reached\na spot where the path lay between a steep hill on one side and\nthe lake on the other.  Then Bruce sent his followers ahead, and\nhimself covered the rear.  Suddenly three of the MacDougalls, who\nhad climbed the hillside, made a spring upon him from above.  One\nleapt on to the horse behind the king, and attempted to hold his\narms, another seized his bridle rein, while the third thrust his\nhand between Bruce's leg and the saddle to hurl him from his horse.\nThe path was too narrow for Bruce to turn his horse, and spurring\nforward he pressed his leg so close to the saddle that he imprisoned\nthe arm of the assailant beneath it and dragged him along with\nhim, while with a blow of his sword he smote off the arm of him\nwho grasped the rein. Then, turning in his saddle, he seized his\nassailant who was behind him and by main strength wrenched him round\nto the pommel of the saddle and there slew him. Then he turned and\nhaving cut down the man whose arm he held beneath his leg, he rode\non and joined his friends.\n\nIn the course of the struggle the brooch which fastened his cloak\nwas lost. This was found by the MacDougalls and carried home as\na trophy, and has been preserved by the family ever since, with\napparently as much pride as if it had been proof of the fidelity\nand patriotism of their ancestors, instead of being a memento of\nthe time when, as false and disloyal Scotchmen, they fought with\nEngland against Scotland's king and deliverer.\n\n\n\n\nChapter XIII\n\nThe Castle of Dunstaffnage\n\n\nBruce's party were now more than ever straitened for provisions,\nsince they had to depend almost entirely upon such fish as they\nmight catch, as it was dangerous to stray far away in pursuit of\ndeer. Archie, however, with his bow and arrows ventured several\ntimes to go hunting in order to relieve the sad condition of the\nladies, and succeeded two or three times in bringing a deer home\nwith him.\n\nHe had one day ventured much further away than usual.  He had not\nsucceeded in finding a stag, and the ladies had for more than a\nweek subsisted entirely on fish. He therefore determined to continue\nthe search, however long, until he found one. He had crossed several\nwooded hills, and was, he knew, leagues away from the point where\nhe had left his party, when, suddenly emerging from a wood, he came\nupon a road just at the moment when a party some twenty strong of\nwild clansmen were traversing it. On a palfrey in their centre was\na young lady whom they were apparently escorting. They were but\ntwenty yards away when he emerged from the wood, and on seeing him\nthey drew their claymores and rushed upon him.  Perceiving that\nflight from these swift footed mountaineers would be impossible,\nArchie threw down his bow and arrows, and, drawing his sword, placed\nhis back against a tree, and prepared to defend himself until the\nlast.\n\nParrying the blows of the first two who arrived he stretched them\ndead upon the ground, and was then at once attacked by the whole of\nthe party together. Two more of his assailants fell by his sword;\nbut he must have been soon overpowered and slain, when the young\nlady, whose cries to her followers to cease had been unheeded in\nthe din of the conflict, spurred her palfrey forward and broke into\nthe ring gathered round Archie.\n\nThe clansmen drew back a pace, and Archie lowered his sword.\n\n\"Desist,\" she cried to the former in a tone of command, \"or my uncle\nAlexander will make you rue the day when you disobeyed my orders.\nI will answer for this young knight.  And now, sir,\" she said,\nturning to Archie, \"do you surrender your sword to me, and yield\nyourself up a prisoner. Further resistance would be madness; you\nhave done too much harm already. I promise you your life if you\nwill make no further resistance.\"\n\n\"Then, lady,\" Archie replied, handing his sword to her, \"I willingly\nyield myself your prisoner, and thank you for saving my life from\nthe hands of your savage followers.\"\n\nThe young lady touched the hilt of his sword, and motioned him to\nreplace it in its scabbard.\n\n\"You must accompany me,\" she said, \"to the abode of my uncle Alexander\nMacDougall. I would,\" she continued, as, with Archie walking beside\nher palfrey, while the Highlanders, with sullen looks, kept close\nbehind, muttering angrily to themselves at having been cheated by\nthe young lady of their vengeance upon the man who had slain four\nof their number, \"that I could set you at liberty, but my authority\nover my uncle's clansmen does not extend so far; and did I bid them\nlet you go free they would assuredly disobey me. You are, as I\ncan see by your attire, one of the Bruce's followers, for no other\nknight could be found wandering alone through these woods.\"\n\n\"Yes, lady,\" Archie said, \"I am Sir Archibald Forbes, one of the\nfew followers of the King of Scotland.\"\n\nThe lady gave a sudden start when Archie mentioned his name, and\nfor some little time did not speak again.\n\n\"I would,\" she said at last in a low voice, \"that you had been\nany other, seeing that Alexander MacDougall has a double cause of\nenmity against you--firstly, as being a follower of Bruce, who\nslew his kinsman Comyn, and who has done but lately great harm to\nhimself and his clansmen; secondly, as having dispossessed Allan\nKerr, who is also his relative, of his lands and castle. My uncle\nis a man of violent passions, and\"--she hesitated.\n\n\"And he may not, you think,\" Archie went on, \"respect your promise\nfor my life. If that be so, lady--and from what I have heard of\nAlexander MacDougall it is like enough--I beg you to give me back\nmy surrender, for I would rather die here, sword in hand, than be\nput to death in cold blood in the castle of Dunstaffnage.\"\n\n\"No,\" the lady said, \"that cannot be. Think you I could see you\nbutchered before mine eyes after having once surrendered yourself\nto me?  No, sir. I beseech you act not so rashly--that were certain\ndeath; and I trust that my uncle, hostile as he may be against you,\nwill not inflict such dishonour upon me as to break the pledge I\nhave given for your safety.\"\n\nArchie thought from what he had heard of the MacDougall that his\nchance was a very slight one. Still, as the young ever cling to hope,\nand as he would assuredly be slain by the clansmen, he thought it\nbetter to take the chance, small as it was, and so continued his\nmarch by the side of his captor's palfrey.\n\nAfter two hours' journey they neared the castle of Alexander\nof Lorne. Archie could not repress a thrill of apprehension as he\nlooked at the grim fortress and thought of the character of its\nlord; but his bearing showed no fear, as, conversing with the young\nlady, he approached the entrance.  The gate was thrown open, and\nAlexander of Lorne himself issued out with a number of retainers.\n\n\"Ah! Marjory!\" he said, \"I am glad to see your bonny face at\nDunstaffnage. It is three months since you left us, and the time\nhas gone slowly; the very dogs have been pining for your voice.\nBut who have we here?\" he exclaimed, as his eye fell upon Archie.\n\n\"It is a wandering knight, uncle,\" Marjory said lightly, \"whom\nI captured in the forest on my way hither. He fought valiantly\nagainst Murdoch and your followers, but at last he surrendered to\nme on my giving him my pledge that his life should be safe, and\nthat he should be treated honourably.  Such a pledge I am sure,\nuncle,\" she spoke earnestly now, \"you will respect.\"\n\nAlexander MacDougall's brow was as black as night, and he spoke in\nGaelic with his followers.\n\n\"What!\" he said angrily to the girl; \"he has killed four of my\nmen, and is doubtless one of Bruce's party who slipped through my\nfingers the other day and killed so many of my kinsmen and vassals.\nYou have taken too much upon yourself, Marjory. It is not by you\nthat he has been made captive, but by my men, and you had no power\nto give such promise as you have made. Who is this young springall?\"\n\n\"I am Sir Archibald Forbes,\" Archie said proudly--\"a name which\nmay have reached you even here.\"\n\n\"Archibald Forbes!\" exclaimed MacDougall furiously.  \"What!  the\nenemy and despoiler of the Kerrs! Had you a hundred lives you\nshould die. Didst know this, Marjory?\" he said furiously to the\ngirl. \"Didst know who this young adventurer was when you asked his\nlife of me?\"\n\n\"I did, uncle,\" the girl said fearlessly. \"I did not know his name\nwhen he surrendered to me, and afterwards, when he told me, what\ncould I do? I had given my promise, and I renewed it; and I trust,\ndear uncle, that you will respect and not bring dishonour upon it.\"\n\n\"Dishonour!\" MacDougall said savagely; \"the girl has lost her senses.\nI tell you he should die if every woman in Scotland had given her\npromise for his life. Away with him!\" he said to his retainers;\n\"take him to the chamber at the top of the tower; I will give him\ntill tomorrow to prepare for death, for by all the saints I swear\nhe shall hang at daybreak. As to you, girl, go to your chamber,\nand let me not see your face again till this matter is concluded.\nMethinks a madness must have fallen upon you that you should thus\nventure to lift your voice for a Forbes.\"\n\nThe girl burst into tears as Archie was led away. His guards took\nhim to the upper chamber in a turret, a little room of some seven\nfeet in diameter, and there, having deprived him of his arms, they\nleft him, barring and bolting the massive oaken door behind them.\n\nArchie had no hope whatever that Alexander MacDougall would change\nhis mind, and felt certain that the following dawn would be his\nlast. Of escape there was no possibility; the door was solid and\nmassive, the window a mere narrow loophole for archers, two or\nthree inches wide; and even had he time to enlarge the opening he\nwould be no nearer freedom, for the moat lay full eighty feet below.\n\n\"I would I had died sword in hand!\" he said bitterly; \"then it\nwould have been over in a moment.\"\n\nThen he thought of the girl to whom he had surrendered his sword.\n\n\"It was a sweet face and a bright one,\" he said; \"a fairer and\nbrighter I never saw. It is strange that I should meet her now\nonly when I am about to die.\" Then he thought of the agony which\nhis mother would feel at the news of his death and at the extinction\nof their race. Sadly he paced up and down his narrow cell till\nnight fell. None took the trouble to bring him food--considering,\ndoubtless, that he might well fast till morning. When it became\ndark he lay down on the hard stone, and, with his arm under his head\nwas soon asleep--his last determination being that if possible\nhe would snatch a sword or dagger from the hand of those who came\nto take him to execution, and so die fighting; or if that were\nimpossible, he would try to burst from them and to end his life by\na leap from the turret.\n\nHe was awakened by a slight noise at the door, and sprang to his\nfeet instantly, believing that day was at hand and his hour had\ncome. To his surprise a voice, speaking scarcely above a whisper,\nsaid:\n\n\"Hush! my son, make no noise; I am here as a friend.\" Then the\ndoor closed, and Archie's visitor produced a lighted lantern from\nthe folds of his garments, and Archie saw that a priest stood before\nhim.\n\n\"I thank you, father,\" he said gratefully; \"you have doubtless come\nto shrive me, and I would gladly listen to your ministrations.  I\nwould fain intrust you, too, with a message to my mother if you\nwill take it for me; and I would fain also that you told the Lady\nMarjory that she must not grieve for my death, or feel that she is\nin any way dishonoured by it, seeing that she strove to her utmost\nto keep her promise, and is in no way to blame that her uncle has\noverriden her.\"\n\n\"You can even give her your message yourself, sir knight,\" the\npriest said, \"seeing that the wilful girl has herself accompanied\nme hither.\"\n\nThus saying, he stepped aside, and Archie perceived, standing\nbehind the priest, a figure who, being in deep shadow, he had not\nhitherto seen. She came timidly forward, and Archie, bending on\none knee, took the hand she held out and kissed it.\n\n\"Lady,\" he said, \"you have heard my message; blame not yourself,\nI beseech you, for my death. Remember that after all you have\nlengthened my life and not shortened it, seeing that but for your\ninterference I must have been slain as I stood, by your followers.\nIt was kind and good of you thus to come to bid me farewell.\"\n\n\"But I have not come to bid you farewell. Tell him, good Father\nAnselm, our purpose here.\"\n\n\"'Tis a mad brain business,\" the priest said, shrugging his shoulders;\n\"and, priest though I am, I shall not care to meet MacDougall in\nthe morning. However, since this wilful girl wills it, what can I\ndo? I have been her instructor since she was a child; and instead\nof being a docile and obedient pupil, she has been a tyrannical\nmaster to me; and I have been so accustomed to do her will in all\nthings that I cannot say her nay now. I held out as long as I could;\nbut what can a poor priest do against sobs and tears? So at last\nI have given in and consented to risk the MacDougall's anger, to\nbring smiles into her face again. I have tried in vain to persuade\nher that since it is the chief's doing, your death will bring no\ndishonour upon her. I have offered to absolve her from the promise,\nand if she has not faith in my power to do so, to write to the\npope himself and ask for his absolution for any breach that there\nmay be; but I might as well have spoken to the wind. When a young\nlady makes up her mind, stone walls are less difficult to move; so\nyou see here we are.  Wound round my waist are a hundred feet of\nstout rope, with knots tied three feet apart. We have only now to\nascend the stairs to the platform above and fix the rope, and in\nan hour you will be far away among the woods.\"\n\nArchie's heart bounded with joy with the hope of life and freedom;\nbut he said quietly, \"I thank you, dear lady, with all my heart for\nyour goodness; but I could not accept life at the cost of bringing\nyour uncle's anger upon you.\"\n\n\"You need not fear for that,\" the girl replied. \"My uncle is\npassionate and headstrong--unforgiving to his foes or those he\ndeems so, but affectionate to those he loves. I have always been his\npet; and though, doubtless, his anger will be hot just at first,\nit will pass away after a time. Let no scruple trouble you on that\nscore; and I would rather put up with a hundred beatings than live\nwith the knowledge that one of Scotland's bravest knights came to\nhis end by a breach of my promise. Though my uncle and all my people\nside with the English, yet do not I; and I think the good father\nhere, though from prudence he says but little, is a true Scotsman\nalso. I have heard of your name from childhood as the companion\nand friend of Wallace, and as one of the champions of our country;\nand though by blood I ought to hate you, my feelings have been very\ndifferent. But now stand talking no longer; the castle is sound\nasleep, but I tremble lest some mischance should mar our plans.\"\n\n\"That is good sense,\" Father Anselm said; \"and remember, not a\nword must be spoken when we have once left this chamber. There is\na sentry at the gate; and although the night is dark, and I deem\nnot that he can see us, yet must we observe every precaution.\"\n\n\"Holy father,\" Archie said, \"no words of mine can thank you for\nthe part which you are playing tonight.  Believe me, Archie Forbes\nwill ever feel grateful for your kindness and aid; and should you\never quit Dunstaffnage you will be welcomed at Aberfilly Castle. As\nto you, lady, henceforth Archie Forbes is your knight and servant.\nYou have given me my life, and henceforth I regard it as yours. Will\nyou take this ring as my token? Should you ever send it to me, in\nwhatever peril or difficulty you may be, I will come to your aid\ninstantly, even should it reach me in a stricken battle.  Think not\nthat I speak the language of idle gallantry. Hitherto my thoughts\nhave been only on Scotland, and no maiden has ever for an instant\ndrawn them from her. Henceforth, though I fight for Scotland, yet\nwill my country have a rival in my heart; and even while I charge\ninto the ranks of the English, the fair image of Marjory MacDougall\nwill be in my thoughts.\"\n\nFather Anselm gave a slight start of surprise as Archie concluded,\nand would have spoken had not the girl touched him lightly. She\ntook the pledge from Archie and said, \"I will keep your ring, Sir\nArchibald Forbes; and should I ever have occasion for help I will\nnot forget your promise. As to your other words, I doubt not that\nyou mean them now; but it is unlikely, though I may dwell in your\nthoughts, that you will ever in the flesh see Marjory MacDougall,\nbetween whose house and yours there is, as you know, bitter enmity.\"\n\n\"There! there!\" Father Anselm said impatiently; \"enough, and more\nthan enough talk. Go to the door, Sir Archibald, and prepare to\nopen it directly I have blown out the light.  The way up the stairs\nlies on your right hand as you go out.\"\n\nNot another word was spoken. Noiselessly the little party made their\nway to the roof; there one end of the rope was quickly knotted round\nthe battlement. Archie grasped the good priest's hand, and kissed\nthat of the girl; and then, swinging himself off the battlement,\ndisappeared at once in the darkness. Not a sound was heard for\nsome time, then the listening pair above heard a faint splash in\nthe water. The priest laid his hands on the rope and found that it\nswung slack in the air; he hauled it up and twisted it again round\nhis waist. As he passed the door of the cell he pushed it to and\nreplaced the bars and bolts, and then with his charge regained the\nportion of the castle inhabited by the family.\n\nA few vigorous strokes took Archie across the moat, and an hour\nlater he was deep in the heart of the forest. Before morning broke\nhe was far beyond the risk of pursuit; and, taking the bearings of\nthe surrounding hills, he found himself, after some walking, at the\nspot where he had left the royal party. As he had expected, it was\ndeserted; he, however, set out on the traces of the party, and that\nnight overtook them at their next encampment.\n\nWith the reticence natural to young lovers Archie felt a disinclination\nto speak of what had happened, or of the services which Marjory\nMacDougall had rendered him. As it was naturally supposed that he\nhad lost his way in the woods on the previous day, and had not reached\nthe encampment in the morning, until after they had started, few\nquestions were asked, and indeed the thoughts of the whole party\nwere occupied with the approaching separation which the night\nbefore they had agreed was absolutely necessary. The ladies were\nworn out with their fatigues and hardships, and the Earl of Athole,\nand some of the other elder men, were also unable longer to support\nit. Winter was close at hand, and the hardships would increase ten\nfold in severity. Therefore it was concluded that the time had come\nwhen they must separate, and that the queen and her companions,\naccompanied by those who could still be mounted, should seek shelter\nin Bruce's strong castle of Kildrummy. The Earl of Athole and the\nking's brother Nigel were in charge of the party.\n\nBruce with his remaining companions determined to proceed into\nKintyre, the country of Sir Neil Campbell, and thence to cross for\na time to the north of Ireland. Sir Neil accordingly started to\nobtain the necessary vessels, and the king and his company followed\nslowly. To reach the Firth of Clyde it was necessary to cross Loch\nLomond. This was a difficult undertaking; but after great search\nSir James Douglas discovered a small boat sunk beneath the surface\nof the lake. On being pulled out it was found to be old and leaky,\nand would hold at best but three. With strips torn from their garments\nthey stopped the leaks as best they could, and then started across\nthe lake. There were two hundred to cross, and the passage occupied\na night and a day; those who could not swim being taken over in the\nboat, while the swimmers kept alongside and when fatigued rested\ntheir hands on her gunwales. They were now in the Lennox country,\nand while Bruce and his friends were hunting, they were delighted\nto come across the Earl of Lennox and some of his companions,\nwho had found refuge there after the battle of Methven.  Although\nhimself an exile and a fugitive the earl was in his own country,\nand was therefore able to entertain the king and his companions\nhospitably, and the rest and feeling of security were welcome indeed\nafter the past labours and dangers.\n\nAfter a time Sir Neil Campbell arrived with the vessels, and,\naccompanied by the Earl of Lennox, Bruce and his companions embarked\nat a point near Cardross.  They sailed down the Clyde and round\nthe south end of Arran, until, after many adventures and dangers,\nthey reached the Castle of Dunaverty, on the south point of the Mull\nof Kintyre, belonging to Angus, chief of Islay. Here they waited\nfor some time, but not feeling secure even in this secluded spot\nfrom the vengeance of their English and Scottish foes, they again\nset sail and landed at the Isle of Rathlin, almost midway between\nIreland and Scotland.  Hitherto Robert Bruce had received but little\nof that support which was so freely given to Wallace by the Scotch\npeople at large; nor is this a matter for surprise. Baliol and\nComyn had in turn betrayed the country to the English, and Bruce\nhad hitherto been regarded as even more strongly devoted to the\nEnglish cause than they had been. Thus the people viewed his attempt\nrather as an effort to win a throne for himself than as one to free\nScotland from English domination. They had naturally no confidence\nin the nobles who had so often betrayed them, and Bruce especially\nhad, three or four times already, after taking up arms, made his\npeace with England and fought against the Scots. Therefore, at first\nthe people looked on at the conflict with comparative indifference.\nThey were ready enough to strike for freedom, as they had proved\nwhen they had rallied round Wallace, but it was necessary before\nthey did so that they should possess confidence in their leaders.\nSuch confidence they had certainly no cause whatever to feel in\nBruce. The time was yet to come when they should recognize in him a\nleader as bold, as persevering, and as determined as Wallace himself.\n\nThe people of Rathlin were rude and ignorant, but simple and\nhospitable. The island contained nothing to attract either adventurers\nor traders, and it was seldom, therefore, that ships touched there,\nconsequently there was little fear that the news of the sojourn of\nthe Scotch king and his companions would reach the mainland, and\nindeed the English remained in profound ignorance as to what had\nbecome of the fugitives, and deemed them to be still in hiding\nsomewhere among the western hills.\n\nEdward had in council issued a proclamation commanding \"all the\npeople of the country to pursue and search for all who had been in\narms and had not surrendered, also all who had been guilty of other\ncrimes, and to deliver them up dead or alive, and that whosoever\nwere negligent in the discharge of his duty should forfeit their\ncastles and be imprisoned.\"\n\nPembroke, the guardian, was to punish at his discretion all who\nharboured offenders. Those who abetted the slayers of Comyn, or who\nknowingly harboured them or their accomplices, were to be \"drawn\nand hanged,\" while all who surrendered were to be imprisoned during\nthe king's pleasure. The edict was carried out to the letter, and\nthe English soldiery, with the aid of the Scotch of their party,\nscoured the whole country, putting to the sword all who were found\nin arms or under circumstances of suspicion.\n\n\n\n\nChapter XIV\n\nColonsay\n\n\nArchie, having little else to do, spent much of his time in fishing.\nAs a boy he had learned to be fond of the sport in the stream of\nGlen Cairn; but the sea was new to him, and whenever the weather\npermitting he used to go out with the natives in their boats. The\nIrish coast was but a few miles away, but there was little traffic\nbetween Rathlin and the mainland. The coast there is wild and\nforbidding, and extremely dangerous in case of a northerly gale\nblowing up suddenly. The natives were a wild and savage race, and\nmany of those who had fought to the last against the English refused\nto submit when their chiefs laid down their arms, and took refuge\nin the many caves and hiding places afforded in the wild and broken\ncountry on the north coast.\n\nThus no profitable trade was to be carried on with the Irish\nmainland. The people of Rathlin were themselves primitive in their\nways. Their wants were few and easily satisfied.  The wool of their\nflocks furnished them with clothing, and they raised sufficient grain\nin sheltered spots to supply them with meal, while an abundance of\nfood could be always obtained from the sea. In fine weather they\ntook more than sufficient for their needs, and dried the overplus\nto serve them when the winter winds kept their boats from putting\nout. Once or twice in the year their largest craft, laden with dried\nfish, would make across to Ayr, and there disposing of its cargo\nwould bring back such articles as were needed, and more precious\nstill, the news of what was passing in the world, of which the\nsimple islanders knew so little.  Even more than fishing, Archie\nloved when the wind blew wildly to go down to the shore and watch\nthe great waves rolling in and dashing themselves into foam on\nthe rocky coast. This to him was an entirely new pleasure, and he\nenjoyed it intensely. Perched on some projecting rock out of reach\nof the waves, he would sit for hours watching the grand scene,\nsometimes alone, sometimes with one or two of his comrades. The\ninflux of a hundred visitors had somewhat straitened the islanders,\nand the fishermen were forced to put to sea in weather when they\nwould not ordinarily have launched their boats, for in the winter\nthey seldom ventured out unless the previous season had been\nunusually bad, and the stores of food laid by insufficient for winter\nconsumption. Archie generally went out with an old man, who with\ntwo grownup sons owned a boat. They were bold and skilful fishermen,\nand often put to sea when no other boat cared to go out.\n\nOne evening the old man, as usual before going to sea, came into\nthe hut which Archie and Sir James Douglas inhabited, and told him\nthat he was going out early the next morning. \"Fish are scarce,\"\nhe said, \"and it would be a disgrace on us islanders if our guests\nwere to run short of food.\"\n\n\"I shall be ready, Donald,\" Archie replied, \"and I hope we shall\nhave good sport.\"\n\n\"I can't see what pleasure you take, Sir Archie,\" the young Douglas\nsaid, when the fisherman had left, \"in being tossed up and down on\nthe sea in a dirty boat, especially when the wind is high and the\nsea rough.\"\n\n\"I like it best then,\" Archie replied; \"when the men are rowing\nagainst the wind, and the waves dash against the boat and the spray\ncomes over in blinding showers, I feel very much the same sort of\nexcitement as I do in a battle. It is a strife with the elements\ninstead of with men, but the feeling in both cases is akin, and\nI feel the blood dancing fast through my veins and my lips set\ntightly together, just as when I stand shoulder to shoulder with\nmy retainers, and breast the wave of English horsemen.\"\n\n\"Well, each to his taste, I suppose,\" Douglas said, laughing; \"I\nhave not seen much of war yet, and I envy you with all my heart the\nfights which you have gone through; but I can see no amusement in\ngetting drenched to the skin by the sea.  I think I can understand\nyour feeling, though, for it is near akin to my own when I sit on\nthe back of a fiery young horse, who has not yet been broken, and\nfeel him battle with his will against mine, and bound, and rear,\nand curvet in his endeavours to throw me, until at last he is\nconquered and obeys the slightest touch of the rein.\"\n\n\"No doubt it is the same feeling,\" Archie replied; \"it is the joy\nof strife in another form. For myself, I own I would rather fight\non foot than on horseback; I can trust myself better than I can\ntrust my steed, can wheel thrice while he is turning once, can defend\nboth sides equally well; whereas on horseback, not only have I to\ndefend myself but my horse, which is far more difficult, and if he\nis wounded and falls I may be entangled under him and be helpless\nat the mercy of an opponent.\"\n\n\"But none acquitted them better on horseback at Methven than you\ndid, Sir Archie,\" the young fellow said, admiringly. \"Did you not\nsave the king, and keep at bay his foes till your retainers came\nup with their pikes and carried him off from the centre of the\nEnglish chivalry?\"\n\n\"I did my best,\" Archie said, \"as one should always do; but I felt\neven then that I would rather have been fighting on foot.\"\n\n\"That is because you have so much skill with your weapon, Sir\nArchie,\" Douglas said. \"On horseback with mace or battleaxe it is\nmainly a question of sheer strength, and though you are very strong\nthere are others who are as strong as you. Now, it is allowed that\nnone of the king's knights and followers are as skilful as you\nwith the sword, and even the king himself, who is regarded as the\nsecond best knight in Europe, owns that on foot and with a sword\nhe has no chance against you. That we all saw when you practiced\nfor the amusement of the queen and her ladies in the mountains of\nLennox.  None other could even touch you, while you dented all our\nhelmets and armour finely with that sword of yours. Had we continued\nthe sport there would not have been a whole piece of armour among\nus save your own harness.\"\n\nArchie laughed. \"I suppose, Douglas, we all like best that in which\nwe most excel. There are many knights in the English army who would\nassuredly overthrow me either in the tilting ring or in the field,\nfor I had not the training on horseback when quite young which is\nneeded to make a perfect knight, while I had every advantage in the\nlearning of sword playing, and I stick to my own trade. The world\nis beginning to learn that a man on foot is a match for a\nhorseman--Wallace taught Europe that lesson. They are slow to believe it,\nfor hitherto armed knights have deemed themselves invincible, and\nhave held in contempt all foot soldiers.  Stirling, and Falkirk,\nand Loudon Hill have taught them the difference, but it will be a\nlong time before they fairly own a fact so mortifying to chivalry;\nbut the time will come, be well assured, when battles will be\nfought almost with infantry alone. Upon them the brunt of the day\nwill fall, and by them will victory be decided, while horsemen\nwill be used principally for pursuing the foe when he is broken,\nfor covering the retreat of infantry by desperate charges, or by\ncharging into the midst of a fray when the infantry are broken.\"\n\n\"All the better for Scotland,\" James Douglas said, cheerfully.\n\"We are not a nation of horsemen, and our mountains and hills, our\nforests and morasses, are better adapted for infantry than cavalry;\nso if ever the change you predict come to pass we shall be gainers\nby it.\"\n\nAt daybreak next morning Archie went down to the cove where his\nfriend the fisherman kept his boat. The old man and his two sons\nwere already there, but had not launched their craft.\n\n\"I like not the look of the weather,\" the fisherman said when\nArchie joined him. \"The sky is dull and heavy, the sea is black\nand sullen, but there is a sound in the waves as they break against\nthe rocks which seems to tell of a coming storm.  I think, however,\nit will be some hours before it breaks, and if we have luck we may\nget a haul or two before it comes on.\"\n\n\"I am ready to go or stay,\" Archie said; \"I have no experience in\nyour weather here, and would not urge you against your own judgment,\nwhatever it be; but if you put out I am ready to go with you.\"\n\n\"We will try it,\" the fisherman said, \"for food is running short;\nbut we will not go far from the shore, so that we can pull back if\nthe weather gets worse.\"\n\nThe boat was soon launched, the nets and oars were already on\nboard, and they quickly put out from the shore.  The boat carried\na small square sail, which was used when running before the wind.\nIn those days the art of navigation was in its infancy, and the art\nof tacking against the wind had scarcely begun to be understood;\nindeed, so high were the ships out of water, with their lofty poops\nand forecastles, that it was scarce possible to sail them on a\nwind, so great was the leeway they made. Thus when contrary winds\ncame mariners anchored and waited as patiently as they might for\na change, and voyage to a port but two days' sail with a favouring\nwind was a matter of weeks when it was foul.\n\nAfter rowing a mile from land the nets were put out, and for some\ntime they drifted near these. From time to time the old fisherman\ncast an anxious eye at the sky.\n\n\"We must get in our nets,\" he said at last decidedly; \"the wind is\nrising fast, and is backing from the west round to the south. Be\nquick, lads, for ere long the gale will be on us in its strength,\nand if 'tis from the south we may well be blown out to sea.\"\n\nWithout a moment's delay the fishermen set to work to get in the nets,\nArchie lending a hand to assist them.  The younger men thoroughly\nagreed in their father's opinion of the weather, but they knew too\nwell the respect due to age to venture upon expressing an opinion\nuntil he had first spoken. The haul was a better one than they had\nexpected, considering that the net had been down but two hours.\n\n\"'Tis not so bad,\" the fisherman said, \"and the catch will be right\nwelcome--that is,\" he added, as he looked toward the land, \"if\nwe get it safely on shore.\"\n\nThe wind was now blowing strongly, but if it did not rise the boat\nwould assuredly make the land. Archie took the helm, having learned\nsomewhat of the steering on previous excursions, and the three\nfishermen tugged at the oars. It was a cross sea, for although the\nwind now blew nearly in their teeth, it had until the last half\nhour been from the west, and the waves were rolling in from the\nAtlantic. The boat, however, made fair progress, and Archie began\nto think that the doubts of the fishermen as to their making the\nshore were in no wise justified, when suddenly a gust, far stronger\nthan those they had hitherto met, struck the boat. \"Keep her head\nstraight!\" the fisherman shouted. \"Don't let the wind take it one\nside or the other.  Stick to it, boys; row your hardest; it is on\nus now and in earnest, I fear.\"\n\nThe three men bent to their oars, but Archie felt that they were\nno longer making headway. The boat was wide and high out of the\nwater; a good sea boat, but very hard to row against the wind.\nAlthough the men strained at the oars, till Archie expected to see\nthe tough staves crack under their efforts, the boat did not seem\nto move. Indeed it appeared to Archie that in the brief space when\nthe oars were out of the water the wind drove her further back than\nthe distance she had gained in the last stroke. He hoped, however,\nthat the squall was merely temporary, and that when it subsided\nthere would still be no difficulty in gaining the land. His hope\nwas not realized. Instead of abating, the wind appeared each moment\nto increase in force.  Clouds of spray were blown on the top of\nthe waves, so that at times Archie could not see the shore before\nhim. For nearly half an hour the fishermen struggled on, but\nArchie saw with dismay that the boat was receding from the shore,\nand that they had already lost the distance they had gained before\nthe squall struck them. The old fisherman looked several times over\nhis shoulder.\n\n\"It is of no use,\" he said at last; \"we shall never make Rathlin,\nand must even run before the gale. Put up the helm, young sir, and\ntake her round. Wait a moment till the next wave has passed under\nus--now!\" In another minute the boat's head was turned from land,\nand she was speeding before the gale.\n\n\"In with your oars, lads, and rig the mast, reef down the sail to\nthe last point; we must show a little to keep her dead before the\nwind; we shall have a tremendous sea when we are once fairly away\nfrom the shelter of the island. This gale will soon knock up the\nsea, and with the cross swell from the Atlantic it will be as much\nas we can do to carry through it.\"\n\nThe mast was stepped and a mere rag of sail hoisted, but this was\nsufficient to drive the boat through the water at a great speed.\nThe old fisherman was steering now, and when the sail was hoisted\nthe four men all gathered in the stern of the boat.\n\n\"You will go between Islay and Jura, I suppose,\" one of the younger\nmen said.\n\n\"Ay,\" his father said briefly; \"the sea will be too high to windward\nof Islay.\"\n\n\"Could we not keep inside Jura?\" Archie suggested; \"and shelter in\nsome of the harbours on the coast of Argyle?\"\n\n\"Ay,\" the old man said; \"could we be sure of doing that it would\nbe right enough, but, strong as the wind is blowing her, it will\nbe stronger still when we get in the narrow waters between the\nislands and the mainland, and it would be impossible to keep her\neven a point off the wind; then if we missed making a harbour we\nshould be driven up through the Strait of Corrievrekan, and the\nbiggest ship which sails from a Scottish port would not live in the\nsea which will be running there. No, it will be bad enough passing\nbetween Islay and Jura; if we get safely through that I shall try\nto run into the narrow strait between Colonsay and Oronsay; there\nwe should have good and safe shelter. If we miss that, we must\nrun inside Mull--for there will be no getting without it--and\neither shelter behind Lismore island far up the strait, or behind\nKerara, or into the passage to Loch Etive.\"\n\n\"It will not be the last, I hope,\" Archie said, \"for there stands\nDunstaffnage Castle, and the lands all belong to the MacDougalls.\nIt is but two months back I was a prisoner there, and though I then\nescaped, assuredly if I again get within its walls I shall never\ngo out again. As well be drowned here.\"\n\n\"Then we will hope,\" the fisherman said, \"that 'tis into some other\nharbour that this evil wind may blow us; but as you see, young sir,\nthe gale is the master and not we, and we must needs go where it\nchooses to take us.\"\n\nFiercer and fiercer blew the gale; a tremendous cross sea was now\nrunning, and the boat, stout and buoyant as she was, seemed every\nmoment as if she would be engulfed in the chaos of water. Small as\nthe sail had been it had been taken down and lashed with ropes to\nthe yard, so that now only about three square feet of canvas was\nset.\n\n\"We can show a little more,\" the fisherman shouted in Archie's ear,\n\"when we get abreast of Islay, for we shall then be sheltered from\nthe sea from the west, and can run more boldly with only a following\nsea; but till we get out of this cross tumble we must not carry\non, we only want steerage way to keep her head straight.\"\n\nNever before had Archie Forbes seen a great gale in all its strength\nat sea, for those which had occurred while at Rathlin were as nothing\nto the present; and although on the hillside round Glen Cairn the\nwind sometimes blew with a force which there was no withstanding,\nthere was nothing to impress the senses as did this wild confusion\nand turmoil of water. Buoyant as was the boat, heavy seas often broke\non board her, and two hands were constantly employed in bailing;\nstill Archie judged from the countenance of the men that they did\nnot deem the position desperate, and that they believed the craft\nwould weather the gale.  Towards midday, although the wind blew\nas strongly as ever, there was a sensible change in the motion of\nthe boat. She no longer was tossed up and down with jerky and sudden\nmotion, as the waves seemed to rise directly under her, but rose\nand fell on the following waves with a steady and regular motion.\n\n\"We are well abreast of Islay,\" the old fisherman said when Archie\nremarked on the change to him. \"There!  do you not see that dark\nbank through the mist; that is Islay. We have no longer a cross sea,\nand can show a little more sail to keep her from being pooped. We\nwill bear a little off toward the land--we must keep it in sight,\nand not too far on our left, otherwise we may miss the straits and\nrun on to Jura.\"\n\nA little more sail was accordingly shown to the gale, and the boat\nscudded along at increased speed.\n\n\"How far is it to Colonsay?\" Archie asked.\n\n\"Between fifty and sixty miles from Rathlin,\" the fisherman said.\n\"It was eight o'clock when we started, ten when the squall struck\nus, it will be dark by four, and fast as we are running we shall\nscarcely be in time to catch the last gleam of day. Come, boys,\"\nhe said to his sons, \"give her a little more canvas still, for it\nis life and death to reach Colonsay before nightfall, for if we\nmiss it we shall be dashed on to the Mull long before morning.\"\n\nA little more sail was accordingly shown, and the boat tore through\nthe water at what seemed to Archie to be tremendous speed; but she\nwas shipping but little water now, for though the great waves as\nthey neared her stern seemed over and over again to Archie as if\nthey would break upon her and send her instantly to the bottom,\nthe stout boat always lifted lightly upon them until he at length\nfelt free from apprehension on that score. Presently the fisherman\npointed out a dark mass over their other bow.\n\n\"That is Jura,\" he said; \"we are fair for the channel, lads, but\nyou must take in the sail again to the smallest rag, for the wind\nwill blow through the gap between the islands with a force fit to\ntear the mast out of her.\"\n\nThrough the rest of his life Archie Forbes regarded that passage\nbetween Islay and Jura as the most tremendous peril he had ever\nencountered. Strong as the wind had been before, it was as nothing\nto the force with which it swept down the strait--the height of\nthe waves was prodigious, and the boat, as it passed over the crest\nof a wave, seemed to plunge down a very abyss. The old fisherman\ncrouched low in the boat, holding the helm, while the other three\nlay on the planks in the bottom. Speech was impossible, for the\nloudest shouts would have been drowned in the fury of the storm. In\nhalf an hour the worst was over. They were through the straits and\nout in the open sea again, but Islay now made a lee for them, and\nthe sea, high as it was, was yet calm in comparison to the tremendous\nwaves in the Strait of Jura. More sail was hoisted again, and in\nan hour the fisherman said, \"Thank God, there are the islands.\"\nThe day was already fading, and Archie could with difficulty make\nout the slightly dark mass to which the helm pointed.\n\n\"Is that Colonsay?\" he asked.\n\n\"It is Oronsay,\" the fisherman said. \"The islands are close together\nand seem as if they had once been one, but have been cleft asunder\nby the arm of a giant. The strait between them is very narrow, and\nonce within it we shall be perfectly sheltered. We must make as\nclose to the point of the island as we can well go, so as not to\ntouch the rocks, and then turn and enter the strait. If we keep\nout any distance we shall be blown past the entrance, and then our\nonly remaining chance is to try and run her on to Colonsay, and\ntake the risk of being drowned as she is dashed upon the rocks.\"\n\nThe light had almost faded when they ran along at the end of Oronsay.\nArchie shuddered as he saw the waves break upon the rocks and fly\nhigh up into the air, and felt how small was the chance of their\nescape should they be driven on a coast like that. They were but\nfifty yards from the point when they came abreast of its extremity;\nthen the fisherman put down the helm and turned her head towards\nthe strait, which opened on their left.\n\n\"Down with the sail and mast, lads, and out with your oars; we must\nrow her in.\"\n\nNot a moment was lost, the sail was lowered, the mast unstepped,\nand the oars got out, with a speed which showed how urgent was\nthe occasion. Archie, who did not feel confidence in his power\nto manager her now in such a sea, took his seat by the man on the\nstroke thwart, and double banked his oar. Five minutes desperate\nrowing and they were under shelter of Oronsay, and were rowing more\nquickly up the narrow strait and towards the shore of Colonsay,\nwhere they intended to land. A quarter of an hour more and they\nstepped ashore.\n\nThe old fisherman raised his hat reverently. \"Let us thank God\nand all the saints,\" he said, \"who have preserved us through such\ngreat danger. I have been nigh fifty years at sea, and never was\nout in so wild a gale.\"\n\nFor a few minutes all stood silent and bare headed, returning\nfervent thanks for their escape.\n\n\"It is well,\" the old man said, as they moved inland, \"that I have\nbeen so far north before; there are but few in Rathlin who have\neven been north of Islay, but sometimes when fish have been very\nplentiful in the island, and the boat for Ayr had already gone,\nI have taken up a boatload of fish to the good monks of Colonsay,\nwho, although fairly supplied by their own fishermen, were yet\nalways ready to pay a good price for them. Had you been in a boat\nwith one who knew not the waters, assuredly we must have perished,\nfor neither skill nor courage could have availed us. There! do you\nsee that light ahead? That is the priory, and you may be sure of\na welcome there.\"\n\nThe priory door was opened at their ring, and the monk who unclosed\nit, greatly surprised at visitors on such a night, at once bade\nthem enter when he heard that they were fishermen whom the storm\nhad driven to shelter on the island.  The fishermen had to lend\ntheir aid to the monk to reclose the door, so great was the power\nof the wind. The monk shot the bolts, saying, \"We need expect no\nfurther visitors tonight;\" and led them into the kitchen, where a\nhuge fire was blazing.\n\n\"Quick, brother Austin,\" he said to the monk, who acted as cook,\n\"warm up a hot drink for these poor souls, for they must assuredly\nbe well nigh perished with cold, seeing that they have been wet\nfor many hours and exposed to all the violence of this wintry gale.\"\n\nArchie and his companions were, indeed, stiff with cold and exposure,\nand could scarce answer the questions which the monks asked them.\n\n\"Have patience, brother! have patience!\" brother Austin said. \"When\ntheir tongues are unfrozen doubtless they will tell you all that\nyou want to know. Only wait, I pray you, till they have drunk this\nposset which I am preparing.\"\n\nThe monk's curiosity was not, however, destined to be so speedily\nsatisfied, for just as the voyagers were finishing their hot drinks\na monk entered with a message that the prior, having heard that\nsome strangers had arrived, would fain welcome and speak with them\nin his apartment. They rose at once.\n\n\"When the prior has done questioning you,\" brother Austin said,\n\"return hither at once. I will set about preparing supper for you,\nfor I warrant me you must need food as well as drink. Fear not but,\nhowever great your appetite may be, I will have enough to satisfy\nit ready by the time you return.\"\n\n\"Welcome to Colonsay!\" the prior said, as the four men entered his\napartment; \"but stay--I see you are drenched to the skin; and it\nwere poor hospitality, indeed, to keep you standing thus even to\nassure you of your welcome. Take them,\" he said to the monk, \"to\nthe guest chamber at once, and furnish them with changes of attire.\nWhen they are warm and comfortable return with them hither.\"\n\nIn ten minutes Archie and his companions re-entered the prior's\nroom. The prior looked with some astonishment at Archie; for in\nthe previous short interview he had not noticed the difference in\ntheir attire, and had supposed them to be four fishermen. The monk,\nhowever, had marked the difference; and on inquiry, finding that\nArchie was a knight, had furnished him with appropriate attire.\nThe good monks kept a wardrobe to suit guests of all ranks, seeing\nthat many visitors came to the holy priory, and that sometimes the\nwind and waves brought them to shore in such sorry plight that a\nchange of garments was necessary.\n\n\"Ah!\" the prior said, in surprise; \"I crave your pardon sir knight,\nthat I noticed not your rank when you first entered. The light is\nsomewhat dim, and as you stood there together at the door way I\nnoticed not that you were of superior condition to the others.\"\n\n\"That might well be, holy prior,\" Archie said, \"seeing that we\nwere more like drowned beasts than Christian men.  We have had a\nmarvellous escape from the tempest--thanks to God and his saints!--seeing\nthat we were blown off Rathlin, and have run before the\ngale down past Islay and through the Straits of Jura. Next to the\nprotection of God and His saints, our escape is due to the skill\nand courage of my brave companions here, who were as cool and calm\nin the tempest as if they had been sitting by the ingle fires at\nhome.\"\n\n\"From Rathlin!\" the prior said in surprise, \"and through the strait\n'twixt Islay and Jura! Truly that was a marvellous voyage in such\na gale--and as I suppose, in an open boat.  But how comes it,\nsir knight--if I may ask the question without prying into your\nprivate affairs--that you, a knight, were at Rathlin? In so wild\nand lonely an island men of your rank are seldom to be found.\"\n\n\"There are many there now, holy prior, far higher in rank than\nmyself,\" Archie replied, \"seeing that Robert the Bruce, crowned King\nof Scotland, James Douglas, and others of his nobles and knights,\nare sheltering there with him from the English bloodhounds.\"\n\n\"The Bruce at Rathlin!\" the prior exclaimed, in surprise.  \"The\nlast ship which came hither from the mainland told us that he was\na hunted fugitive in Lennox; and we deemed that seeing the MacDougalls\nof Lorne and all the surrounding chiefs were hostile to him, and\nthe English scattered thickly over all the low country, he must\nlong ere this have fallen into the hands of his enemies.\"\n\n\"Thanks to Heaven's protection,\" Archie said devoutly, \"the king\nwith a few followers escaped and safely reached Rathlin!\"\n\n\"Thou shouldst not speak of Heaven's protection,\" the prior said,\nsternly, \"seeing that Bruce has violated the sanctuary of the\nchurch, has slain his enemy within her walls, has drawn down upon\nhimself the anathema of the pope, and has been declared excommunicated\nand accursed.\"\n\n\"The pope, holy father,\" Archie replied, \"although supreme in\nall holy things, is but little qualified to judge of the matter,\nseeing that he draws his information from King Edward, under whose\nprotection he lives.  The good Bishops of St. Andrews and Glasgow,\nwith the Abbot of Scone, and many other dignitaries of the Scottish\nchurch, have condoned his offense, seeing that it was committed\nin hot blood and without prior intent. The king himself bitterly\nregrets the deed, which preys sorely upon his mind; but I can answer\nfor it that Bruce had no thought of meeting Comyn at Dumfries.\"\n\n\"You speak boldly, young sir,\" the prior said, sternly, \"for one\nover whose head scarce two-and-twenty years can have rolled; but\nenough now. You are storm staid and wearied; you are the guests of\nthe convent.  I will not keep you further now, for you have need\nof food and sleep. Tomorrow I will speak with you again.\"\n\nSo saying, the prior sharply touched a bell which stood on a table\nnear him. The monk re-entered. The prior waved his hand: \"Take these\nguests to the refectory and see that they have all they stand in\nneed of, and that the bed chambers are prepared. In the morning I\nwould speak to them again.\"\n\n\n\n\nChapter XV\n\nA Mission to Ireland\n\n\nFather Austin was as good as his word, and it was long indeed\nsince Archie had sat down to such a meal as that which was spread\nfor him. Hungry as he was, however, he could scarce keep his eyes\nopen to its conclusion, so great was the fatigue of mind and body;\nand on retiring to the chamber which the monks had prepared for\nhim, he threw himself on a couch and instantly fell asleep.  In\nthe morning the gale still blew violently, but with somewhat less\nfury than on the preceding evening. He joined the monks at their\nmorning meal in the refectory, and after their repast they gathered\nround him to listen to his news of what was doing in Scotland; for\nalthough at ordinary times pilgrims came not unfrequently to visit\nthe holy isle of Colonsay, in the present stormy times men stirred\nbut little from home, and it was seldom that the monks obtained news\nof what was passing on the mainland. Presently a servitor brought\nword that the prior would see Archie.\n\n\"It was ill talking last night,\" the prior said, \"with a man hungry\nand worn out; but I gathered from what you said that you are not\nonly a follower of Bruce, but that you were with him at that fatal\nday at Dumfries when he drew his dagger upon Comyn in the sanctuary.\"\n\n\"I was there, holy father,\" Archie replied, \"and can testify that\nthe occurrence was wholly unpremeditated; but Bruce had received\nsufficient provocation from the Comyn to afford him fair reason for\nslaying him wheresoever they might meet. But none can regret more\nthan he does that that place of meeting was in a sanctuary. The\nComyn and Bruce had made an agreement together whereby the former\nrelinquished his own claims to the throne of Scotland on condition\nthat Bruce, on attaining the throne, would hand over to him all\nhis lordships in Carrick and Annandale.\"\n\n\"It were a bad bargain,\" the prior said, \"seeing that Comyn would\nthen be more powerful than his king.\"\n\n\"So I ventured to tell the Bruce,\" Archie replied.\n\n\"Thou?\" the prior said; \"you are young, sir, to be in a position\nto offer counsel to Robert Bruce.\"\n\n\"I am young, holy prior,\" Archie said modestly; \"but the king is\ngood enough to overlook my youth in consideration of my fidelity\nto the cause of Scotland. My name is Archibald Forbes.\"\n\n\"Sir Archibald Forbes!\" the prior repeated, rising; \"and are you\nreally that loyal and faithful Scottish knight who fought ever by\nthe side of Wallace, and have almost alone refused ever to bow the\nknee to the English?  Even to this lonely isle tales have come of\nyour valour, how you fought side by side with Wallace, and were,\nwith Sir John Grahame, his most trusty friend and confidant. Many\nof the highest and noblest of Scotland have for centuries made\ntheir way to the shrine of Colonsay, but none more worthy to be\nour guest. Often have I longed to see so brave a champion of our\ncountry, little thinking that you would one day come a storm driven\nguest. Truly am I glad to see you, and I say it even though you may\nhave shared in the deed at Dumfries, for which I would fain hope\nfrom your words there is fairer excuse to be made than I had hitherto\ndeemed. I have thought that the Bishops of St. Andrews and Glasgow\nwere wrong in giving their countenance to a man whom the holy\nfather had condemned--a man whose prior history gives no ground\nfor faith in his patriotism, who has taken up arms, now for, now\nagainst, the English, but has ever been ready to make terms with\nthe oppressor, and to parade as his courtier at Westminster. In\nsuch a man I can have no faith, and deem that, while he pretends\nto fight for Scotland, he is in truth but warring for his\nown aggrandizement. But since you, the follower and friend of the\ndisinterested and intrepid champion of Scotland, speak for the\nBruce, it maybe that my judgement has been too severe upon him.\"\n\nArchie now related the incident of his journey to London to urge\nBruce to break with Edward and to head the national movement. He\ntold how, even before the discovery of his agreement with Comyn,\nbrought about by the treachery of the latter, Bruce had determined\ndefinitely to throw in his cause with that of Scotland; how upon\nthat discovery he had fled north, and, happening to meet Comyn at\nDumfries, within the limits of the sanctuary, had, in his indignation\nand ire at his treachery, drawn and slain him. Then he told the\ntale of what had taken place after the rout of Methven, how bravely\nBruce had borne himself, and had ever striven to keep up the hearts\nof his companions; how cheerfully he had supported the hardships,\nand how valiantly he had borne himself both at Methven and when\nattacked by the MacDougalls of Lorne.\n\n\"Whatever his past may have been,\" Archie concluded, \"I hold that\nnow the Bruce is as earnest in the cause of Scotland as was even\nmy dear leader Wallace. In strength and in courage he rivals that\nvaliant knight, for though I hold that Wallace was far more than\na match for any man of his time, yet Bruce is a worthy second to\nhim, for assuredly no one in Scotland could cross swords with him\non equal chances. That he will succeed in his enterprise it were\nrash to say, for mighty indeed are the odds against him; but if\ncourage, perseverance, and endurance can wrest Scotland from the\nhands of the English, Robert Bruce will, if he lives, accomplish\nthe task.\"\n\n\"Right glad am I,\" the prior replied, \"to hear what you have told\nme. Hitherto, owing to my memory of his past and my horror at his\ncrime--for though from what you tell me there was much to excuse\nit, still it was a grievous crime--I have had but little interest\nin the struggle, but henceforth this will be changed. You may\ntell the king that from this day, until death or victory crown his\nefforts, prayers will be said to heaven night and day at Colonsay\nfor his success.\"\n\nIt was four days before the storm was over and the sea sufficiently\ncalmed to admit of Archie's departure.  During that time he remained\nas the honoured guest of the priory, and the good monks vied with\nthe prior in their attentions to the young knight, the tales of\nwhose doings, as one of Scotland's foremost champions, had so often\nreached their lonely island. At the end of that time, the sea being\nnow calm and smooth, with a light wind from the north, Archie bade\nadieu to his hosts and sailed from Colonsay.\n\nLight as the wind was, it sufficed to fill the sail; and as the boat\nglided over the scarce rippled water Archie could not but contrast\nthe quiet sleepy motion with the wild speed at which the boat had\ntorn through the water on her northern way. It was not until the\nfollowing morning that Rathlin again came in sight.\n\nAs the boat was seen approaching, and was declared by the islanders\nto be that which they had regarded as lost in the storm a week\npreviously, the king, Douglas, and the rest of his followers made\ntheir way down to the shore; and loud was the shout of welcome\nwhich arose when Archie stood up and waved his hand.\n\n\"Verily, Archie Forbes,\" the king said as he warmly embraced the\nyoung knight, \"I shall begin to think that the fairies presided\nat your birth and gave you some charm to preserve your life alike\nagainst the wrath of men and of the elements. Never assuredly did\nanyone pass through so many dangers unscathed as you have done.\"\n\n\"I hope to pass through as many more, sire, in your service,\" Archie\nsaid smiling.\n\n\"I hope so, indeed,\" Bruce replied; \"for it were an evil day for\nme and for Scotland that saw you fall; but henceforth I will fret\nno more concerning you. You alone of Wallace's early companions\nhave survived.  You got free from Dunstaffnage by some miracle\nwhich you have never fully explained to me, and now it would seem\nthat even the sea refuses to swallow you.\"\n\n\"I trust,\" Archie said more gravely, \"that the old saying is not\ntrue in my case, and that hanging is not to be my fate.  Assuredly\nit will be if I ever fall into the hands of Edward, and I shall\nthink it a cruel fate indeed if fortune, which has spared me so\noften in battle, leads me to that cruel end at last.\"\n\n\"I trust not indeed, Sir Archie,\" the king said, \"though hanging now\nhas ceased to be a dishonourable death when so many of Scotland's\nbest and bravest have suffered it at the English hands. However, I\ncannot but think that your fairy godmother must have reserved for\nyou the fate of the heroes of most of the stories of my old nurse,\nwhich always wound up with 'and so he married, and lived happily\never after.' And now, Archie, tell me all that has befallen you,\nwhere you have been, and how you fared, and by what miraculous chance\nyou escaped the tempest. All our eyes were fixed on the boat when\nyou laboured to reach the shore, and had you heard the groans\nwe uttered when we saw you give up the effort as hopeless and fly\naway to sea before the wind you would have known how truly all\nyour comrades love you. We gave you up as assuredly lost, for the\nislanders here agreed that you had no chance of weathering the\ngale, and that the boat would, ere many hours, be dashed to pieces\neither on Islay or Jura, should it even reach so far; but the most\nthought that you would founder long ere you came in sight of the\nland.\"\n\nAccompanying the king with his principal companions to the hut\nwhich he occupied, Archie related the incidents of the voyage and\nof their final refuge at Colonsay.\n\n\"It was a wonderful escape,\" the king said when he finished, \"and\nthe holy Virgin and the saints must assuredly have had you in their\nespecial care. You have cost us well nigh a fortune, for not one\nof us but vowed offerings for your safety, which were, perchance,\nthe more liberal, since we deemed the chances of paying them so\nsmall. However, they shall be redeemed, for assuredly they have\nbeen well earned, and for my share I am bound, when I come to my\nown, to give a piece of land of the value of one hundred marks a\nyear to the good monks of St. Killian's to be spent in masses for\nthe souls of those drowned at sea.\"\n\nSome days later the king said to Archie, \"I have a mission for you;\n'tis one of danger, but I know that that is no drawback in your\neyes.\"\n\n\"I am ready,\" Archie said modestly, \"to carry out to the best of\nmy power any errand with which your majesty may intrust me.\"\n\n\"I have been thinking, Sir Archie, that I might well make some sort\nof alliance with the Irish chieftains.  Many of these are, like\nmost of our Scotch nobles, on terms of friendship with England;\nstill there are others who hold aloof from the conquerors. It would\nbe well to open negotiations with these, so that they by rising\nmight distract Edward's attention from Scotland, while we, by our\nefforts, would hinder the English from sending all their force\nthither, and we might thus mutually be of aid to each other. At\npresent I am, certes, in no position to promise aid in men or money;\nbut I will bind myself by an oath that if my affairs in Scotland\nprosper I will from my treasury furnish money to aid them in carrying\non the struggle, and that if I clear Scotland of her oppressors\nI will either go myself or send one of my brothers with a strong\nforce to aid the Irish to follow our example. The mission is, as\nyou will see, Sir Archie, a dangerous one; for should any of the\nEnglish, or their Irish allies, lay hands on you, your doom would\nbe sealed. Still you may do me and Scotland great service should\nyou succeed in your mission. Even minor risings would be of much\nutility, seeing that they would at any rate prevent Edward from\nbringing over troops from Ireland to assist in our conquest. I have\nthought the matter over deeply, and conclude that, young as you are,\nI can intrust it to you with confidence, and that you are indeed\nthe best fitted among those with me to undertake it.  Douglas is\nbut a boy; my brother Edward is too hot and rash; Boyd is impatient\nand headstrong, trusty and devoted to me though he is; but I am\nsure that in you there is no lack either of prudence or courage.\nHence, Sir Archie, if you will undertake it I will intrust it to\nyou.\"\n\n\"I will willingly undertake it, sire, since you think me fitting\nfor it, and deem it a high honour indeed that you have chosen me.\nWhen will you that I start?\"\n\n\"It were best to lose no time,\" the king replied, \"and if you have\nno reason for delay I would that you should embark tonight, so that\nbefore daybreak you may have gained the Irish shore. They tell me\nthat there are many desperate men in refuge among the caves on the\ncoast, and among these you might choose a few who might be useful\nto you in your project; but it is not in this part that a rising\ncan be effected, for the country inland is comparatively flat and\nwholly in the hands of the English. It is on the west coast that\nthe resistance to the English was continued to the last, and here\nfrom time to time it blazes out again. In those parts, as they tell\nme, not only are there wild mountains and fastnesses such as we\nhave in Scotland, but there are great morasses and swamps, extending\nover wide tracts, where heavy armed soldiers cannot penetrate,\nand where many people still maintain a sort of wild independence,\ndefying all the efforts of the English to subdue them. The people\nare wild and savage, and ever ready to rise against the English.\nHere, then, is the country where you are most likely to find chiefs\nwho may enter into our plans, and agree to second our efforts for\nindependence.  Here are some rings and gold chains, which are all\nthat remain to me of my possessions. Money I have none; but with\nthese you may succeed in winning the hearts of some of these savage\nchieftains. Take, too, my royal signet, which will be a guarantee\nthat you have power to treat in my name. I need not tell you to be\nbrave, Sir Archie; but be prudent--remember that your life is of\nthe utmost value to me. I want you not to fight, but simply to act\nas my envoy. If you succeed in raising a great fire in the west\nof Ireland, remain there and act as councillor to the chiefs,\nremembering that you are just as much fighting for Scotland there\nas if you were drawing sword against her foes at home. If you find\nthat the English arm is too strong, and the people too cowed and\ndisheartened to rise against it, then make your way back here by\nthe end of three months, by which time I hope to sail hence and to\nraise my standard in Scotland again.\"\n\nOn leaving the king Archie at once conferred with Duncan the fisherman,\nwho willingly agreed that night to set him ashore in Ireland.\n\n\"I will land you,\" he said, \"at a place where you need not fear\nthat any English will meet you. It is true that they have a castle\nbut three miles away perched on a rock on the coast.  It is called\nDunluce, and commands a wide seaward view, and for this reason it\nwere well that our boat were far out at sea again before morning\ndawned, so that if they mark us they will not suppose that we have\ntouched on the coast; else they might send a party to search if\nany have landed--not even then that you need fear discovery, for\nthe coast abounds in caves and hiding places. My sons have often\nlanded there, for we do a certain trade in the summer from the island\nin fish and other matters with the natives there. If it pleases\nyou my son Ronald, who is hardy and intelligent, shall land with\nyou and accompany you as your retainer while you remain in Ireland.\nThe people there speak a language quite different to that which you\nuse in the lowlands of Scotland and in England, but the language\nwe speak among ourselves closely resembles it, and we can be easily\nunderstood by the people of the mainland. You would be lost did\nyou go among the native Irish without an interpreter.\"\n\nArchie thankfully accepted the offer, and that night, after bidding\nadieu to the friends and his comrades, started in Duncan's boat.\n\n\"'Tis a strange place where I am going to land you,\" the fisherman\nsaid; \"such a place as nowhere else have my eyes beheld, though they\nsay that at the Isle of Staffa, far north of Colonsay, a similar\nsight is to be seen. The rocks, instead of being rugged or square,\nrise in close columns like the trunks of trees, or like the columns\nin the church of the priory of Colonsay. Truly they seem as if\nwrought by the hands of men, or rather of giants, seeing that no\nmen could carry out so vast a work. The natives have legends that\nthey are the work of giants of old times. How this may be I know\nnot, though why giants should have engaged in so useless a work\npasses my understanding. However, there are the pillars, whosoever\nplaced them there.  Some of them are down by the level of the sea.\nHere their heads seem to be cut off so as to form a landing place,\nto which the natives give the name of the Giant's Causeway. Others\nin low rows stand on the face of the cliff itself, though how any\ncould have stood there to work them, seeing that no human foot can\nreach the base, is more than I can say. 'Tis a strange and wonderful\nsight, as you will say when the morning light suffers you to see\nit.\"\n\nIt was fortunate that Duncan knew the coast so well, and was able\nby the light of the stars to find a landing place, for quiet as the\nsea appeared a swell rose as they neared the shore, and the waves\nbeat heavily on the wild and rocky coast.  Duncan, however, steered\nhis boat to the very foot of the Causeway, and then, watching his\nopportunity, Archie sprang ashore followed by Ronald. A few words\nof adieu were spoken, and then the boat rowed out to sea again,\nwhile Archie and Ronald turned away from the landing place.\n\n\"It were best,\" the young fisherman said, \"to find a seat among the\nrocks, and there to await the dawn, when I can guide you to some\ncaves hard by; but in the darkness we might well fall and break a\nlimb did we try and make our way across the coast.\"\n\nA niche was soon found, and Archie and his companion sat down for\na while. Archie, however, soon discovered that the sides and back\nof his seat were formed of the strange columns of which Duncan had\nspoken, and that he was sitting upon the tops of others which had\nbroken off. Eagerly he passed his hands over the surface of these\nstrange pillars, and questioned his companion as to what he knew\nabout them; but Ronald could tell him no more than his father\nhad done, and Archie was forced to await the dawn to examine more\nclosely the strange columns. Daylight only added to his wonder.\nOn all sides of him stretched the columns, packed in a dense mass\ntogether, while range above range they stood on the face of the\ngreat cliffs above him. The more he examined them the more his\nwonder grew.\n\n\"They can neither be the work of men nor giants,\" he said, \"but\nmust have been called up by the fantastic freak of some powerful\nenchanter. Hitherto I have not believed the tales of these mysterious\nbeings of old times; but after seeing these wonderful pillars I\ncan no longer doubt, for assuredly no mortal hand could have done\nthis work.\"\n\nRonald now urged that they had better be moving, as it was possible,\nalthough unlikely enough, that one passing along the top of the\ncliffs might get sight of them. They accordingly moved along the\nshore, and in a quarter of a mile reached the mouth of a great\ncave. The bottom was covered with rocks, which had fallen from the\nroof, thickly clustered over with wet seaweed, which, indeed, hung\nfrom the sides far up, showing that at high tide the sea penetrated\nfar into the cave.\n\n\"The ground rises beyond,\" Ronald said, \"and you will find recesses\nthere which the tide never reaches.\" They moved slowly at first\nuntil their eyes became accustomed to the darkness; then they kept\non, the ground getting more even as they ascended, until they stood\non a dry and level floor.\n\n\"Now I will strike a light,\" Ronald said, \"and light the torch\nwhich I brought with me. We are sure to find plenty of driftwood\ncast up at the highest point the tide reaches. Then we can make a\nfire, and while you remain here I will go out and find some of the\nnatives, and engage a guide to take us forward tonight.\"\n\nTaking out his flint and steel, Ronald proceeded to strike a light,\nand after several efforts succeeded in doing so and in igniting\nsome dried moss which he had brought with him, carefully shielded\nfrom damp in the folds of his garment. As a light flame rose\nhe applied his torch to it; but as he did so, came an exclamation\nof astonishment, for gathered in a circle round them were a dozen\nwild figures. All were armed and stood in readiness to strike down\nthe intruders into their hiding place. They were barefooted, and\nhad doubtless been asleep in the cave until, when awakened by the\napproaching footsteps and voices, they had silently arisen and\nprepared to fall upon the intruders.\n\n\"We are friends,\" Ronald said in the native language when he\nrecovered from his start of surprise. \"I am Ronald, a fisherman\nfrom Rathlin, and was over here in the summer exchanging fish for\nsheep.\"\n\n\"I recollect you,\" one of the men said; \"but what do you here so\nstrangely and secretly? Are the English hunting you too from your\nisland as they have done us?\"\n\n\"They have not come to Rathlin yet,\" Ronald said.\n\n\"Doubtless they would do so, but 'tis too poor to offer any\ntemptation for their greed. But they are our enemies as they are\nyours. I am here to guide this Scottish knight, who is staying at\nRathlin, a fugitive from their vengeance like yourself, and who is\ncharged with a mission from the King of Scotland to your chiefs,\nwhom he would fain induce to join in a rising against the power of\nthe English.\"\n\n\"He is welcome,\" the man who appeared to be the leader of the party\nreplied, \"and may he succeed in his object; but,\" he continued\nbitterly, \"I fear that the chance is a small one.  The Norman foot\nis on our necks, and most of those who should be our leaders have\nbasely accepted the position of vassals to the English king. Still\nthere are brave hearts yet in Ireland who would gladly rise did they\nsee even a faint chance of success. Hundreds are there who, like\nus, prefer to live the lives of hunted dogs in caves, in mountain\nfastnesses, or in the bogs, rather than yield to the English yoke.\nTell me your plans and whither you would go; and I will give you\nguides who know every foot of the country, and who can lead you to\nthe western hills, where, though no open resistance is made, the\nEnglish have scarce set foot.  There we generally find refuge;\nand 'tis only at times, when the longing to see the homes of our\nchildhood becomes too strong for us, that I and those you see--all\nof whom were born and reared between this and Coleraine--come\nhither for a time, when at night we can issue out and prowl round\nthe ruins of the homes of our fathers.\"\n\nWhile this conversation had been going on, the others, seeing that\nthe visit was a friendly one, had set to work, and bringing up\ndriftwood from below, piled it round the little blaze which Ronald\nhad commenced, and soon had a great fire lighted. They then produced\nthe carcass of a sheep which they had the evening before carried\noff. Ronald had brought with him a large pile of oaten cakes, and\na meal was speedily prepared.\n\nArchie could not but look with surprise at the wild figures around\nhim, lit up by the dancing glare of the fire. Their hair lay in\ntangled masses on their necks; their attire was of the most primitive\ndescription, consisting but of one garment secured round the waist\nby a strap of untanned leather; their feet and legs were bare.\nTheir hair was almost black; their eyes small and glittering, with\nheavy overhanging brows; and they differed altogether in appearance\neven from the wildest and poorest of the Scottish peasantry. In\ntheir belts all bore long knives of rough manufacture, and most of\nthem carried slings hanging from the belt, in readiness for instant\nuse. In spite of the wildness of their demeanour they seemed kindly\nand hospitable; and many were the questions which they asked Ronald\nconcerning the King of Scotland and his knights who were in refuge\nat Rathlin.\n\nWhen the meal was over all stretched themselves on the sand like so\nmany animals, and without further preparation went off to sleep.\nArchie, knowing that nothing could be done until nightfall,\nfollowed their example. The fire had by this time burned low, and\nsoon perfect stillness reigned in the great cavern, save that far\naway at its mouth the low thunder of the waves upon the rocks came\nup in a confused roar.\n\n\n\n\nChapter XVI\n\nAn Irish Rising\n\n\nWhen night came on Archie started for the west, accompanied by\nRonald and two of the Irish as guides.  They crossed the country\nwithout question or interference, and reached the wild mountains\nof Donegal in safety. Archie had asked that his conductors should\nlead him to the abode of the principal chieftain of the district.\nThe miserable appearance of the sparsely scattered villages through\nwhich they had passed had prepared him to find that the superiors\nof such a people would be in a very different position from the\nfeudal lords of the Highlands of Scotland. He was not surprised,\ntherefore, when his attendants pointed out a small hold, such as\nwould appertain to a small landowner on the Scottish Border, as the\nresidence of the chief. Around it were scattered a number of low\nhuts composed of turf, roofed with reeds. From these, when the\napproach of strangers was reported, a number of wild looking figures\npoured out, armed with weapons of the most primitive description.\nA shout from Archie's guides assured these people that the newcomer\nwas not, as his appearance betokened him, a Norman knight, but\na visitor from Scotland who sought a friendly interview with the\nchief.\n\nInsignificant as was the hold, it was evident that something like\nfeudal discipline was kept up.  Two men, armed with pikes, were\nstationed on the wall, while two others leant in careless fashion\nagainst the posts of the open gate. On the approach of Archie an\nelderly man, with a long white beard, came out to meet them.  Ronald\nexplained to him that Archie was a knight who had come as an emissary\nfrom the King of Scotland to the Irish chieftains, and desired to\nspeak with the great Fergus of Killeen. The old man bowed deeply\nto Archie, and then escorted him into the house.\n\nThe room which they entered occupied the whole of the ground\nfloor of the hold, and was some thirty feet wide by forty long. As\napparently trees of sufficient length to form the beams of so wide\nan apartment could not be obtained, the floor above was supported\nby two rows of roughly squared posts extending down from end to\nend. The walls were perfectly bare. The beams and planks of the\nceiling were stained black by the smoke of a fire which burned in\none corner; the floor was of clay beaten hard. A strip some ten\nfeet wide, at the further end, was raised eighteen inches above the\ngeneral level, forming a sort of dais. Here, in a carved settle of\nblack wood, sat the chief. Some females, evidently the ladies of\nhis family, were seated on piles of sheepskins, and were plying\ntheir distaffs; while an aged man was seated on the end of the dais\nwith a harp of quaint form on his knee; his fingers touched a last\nchord as Archie entered, and he had evidently been playing while\nthe ladies worked. Near him on the dais was a fire composed of\nwood embers, which were replenished from time to time with fresh\nglowing pieces of charcoal taken from the fire at the other end of\nthe room, so that the occupants of the dais should not be annoyed\nby the smoke arising close to them.\n\nThe chief was a fine looking man about fifty years old.  He was\nclad in a loose fitting tunic of soft dark green cloth, confined at\nthe waist by a broad leathern band with silver clasp and ornaments,\nand reaching to his knees. His arms were bare; on his feet he wore\nsandals, and a heavy sword rested against the wall near his hand.\nThe ladies wore dresses of similar material and of somewhat similar\nfashion, but reaching to the feet. They wore gold armlets; and the\nchief's wife had a light band of gold round her head. The chief\nrose when Archie entered; and upon the seneschal informing him of\nthe rank and mission of his visitor he stepped from the dais, and\nadvancing, greeted him warmly. Then he led him back to the dais,\nwhere he presented to him the ladies of his family, ordering the\nretainers, of whom about a score were gathered in the hall, to\nplace two piles of sheepskins near the fire. On one of these he sat\ndown, and motioned to Archie to take his place on the other--his\nown chair being removed to a corner. Then, through the medium of\nRonald, the conversation began.\n\nArchie related to the chief the efforts which the Scotch were\nmaking to win their freedom from England, and urged in the king's\nname that a similar effort should be made by the Irish; as the\nforces of the English, being thereby divided and distracted, there\nmight be better hope of success. The chief heard the communication\nin grave silence. The ladies of the family stood behind the chief\nwith deeply interested faces; and as the narrative of the long\ncontinued struggle which the Scots were making for freedom continued\nit was clear, by their glowing cheeks and their animated faces,\nhow deeply they sympathized in the struggle.\n\nThe wife of the chief, a tall and stately lady, stood immediately\nbehind him with her two daughters, girls of some seventeen or\neighteen years of age, beside her. As Ronald was translating his\nwords Archie glanced frequently at the group, and thought he had\nnever seen one fairer or more picturesque. There was a striking\nlikeness between mother and daughters; but the expression of staid\ndignity in the one was in the others replaced by a bright expression\nof youth and happiness. Their beauty was of a kind new to Archie.\nTheir dark glossy hair was kept smoothly in place by the fillet\nof gold in the mother's case, and by purple ribbons in that of the\ndaughters. Their eyebrows and long eyelashes were black, but their\neyes were gray, and as light as those to which Archie was accustomed\nunder the fair tresses of his countrywomen.  The thing that struck\nhim most in the faces of the girls was their mobility, the expression\nchanging as it seemed in an instant from grave to gay--flushing\nat one moment with interest at the tale of deeds of valour, paling\nat the next at the recital of cruel oppression and wrong. When Archie\nhad finished his narrative he presented to the chief a beautifully\nwrought chain of gold as a token from the King of Scotland.\n\nThe chief was silent for some time after the interpreter concluded\nArchie's narrative; then he said:\n\n\"Sir knight, it almost seems to me as if I had been listening to\nthe tale of the wrongs of Ireland, save that it appears that the\nmastery of the English here has been more firmly established than\nwith you. This may be from the nature of the country; our hills\nare, for the most part, bare, while yours, you say, are covered\nwith forest. Thus the Normans could more easily, when they had once\ngained the upper hand, crush out the last vestiges of opposition\nthan they could with you. As I judge from what you say, the English\nin Scotland hold all the fortresses, and when the people rise they\nremain sheltered in them until assistance comes from England. With\nus it is different. First they conquer all the country; then from\na wide tract, a third perhaps of the island, they drive out the whole\nof the people, and establish themselves firmly there, portioning the\nland among the soldiery and repeopling the country with an English\nrace. Outside this district the Irish chieftains, like myself,\nretain something of independence; we pay a tribute, and are in the\nposition of feudatories, being bound to furnish so many men for\nthe King of England's wars if called upon to do so.  The English\nseldom come beyond their pale so long as the tribute is paid, and\nthe yoke, therefore, weighs not so heavy upon us; but were we to\nrise, the English army would pour out from its pale and carry fire\nand sword throughout the country.\n\n\"We, like you, have been without one who would unite us against the\ncommon enemy. Our great chiefs have, for the most part, accepted\nEnglish titles, and since their power over the minor chiefs is\nextended, rather than decreased by the changed circumstances, they\nare well content, for they rule now over their districts, not only\nas Irish chieftains, but as English lieutenants. You have seen,\nas you journeyed here, how sparse is the population of our hills,\nand how slight would be the opposition which we could offer, did\nthe Earl of Ulster sweep down upon us with trained English soldiers.\n\n\"Were there a chance of success, Fergus of Killeen would gladly\ndraw the sword again; but I will not bring ruin upon my family\nand people by engaging in a hopeless enterprise.  Did I raise\nmy standard, all Donegal would take up arms; but Donegal alone is\npowerless against England. I know my people--they are ready for\nthe fray, they would rush to battle and perish in thousands to win\nvictory, but one great defeat would crush them. The story of the\nlong fight which your Wallace, with a small following, made against\nthe power of England, will never be told of an Irish leader. We\nhave bravery and reckless courage, but we have none of the stubborn\nobstinacy of your Scottish folk. Were the flag raised the people\nwould flock to it, and would fight desperately; but if they lost,\nthere would be utter and complete collapse. The fortitude to support\nrepeated defeats, to struggle on when the prospect seems darkest,\ndoes not belong to my people.\n\n\"It is for this reason that I have no hope that Ireland will ever\nregain its independence. She may struggle against the yoke, she\nmay blaze out again and again in bloody risings, our sons may die\nin tens of thousands for her; but never, I believe, as long as the\nmen of the two countries remain what they are, will Ireland recover\nher independence, for, in the long run, English perseverance and\ndetermination will overcome the fitful courage of the Irish.  I\ngrieve that I should say it.  I mourn that I feel it my duty to\nrepress rather than to encourage the eager desire of my people to\ndraw the sword and strike for freedom; but such is my conviction.\n\n\"But understand, sir knight, that whatever I may think, I shall\nnot be backward in doing my part. If Ireland again rises, should\nthe other native chieftains determine to make one more effort to\ndrive the English across the channel, be sure that Fergus of Killeen\nand the men of Donegal will be in the front of the battle. No heart\nbeats more warmly for freedom than mine; and did I stand alone I\nwould take to the bogs and join those who shelter there, defying\nthe might of England. But I have my people to think of. I have seen\nhow the English turn a land to desolation as they sweep across it,\nand I will not bring fire and sword into these mountain valleys\nunless all Ireland is banded in a common effort. You have seen\nScotland wasted from sea to sea, her cities burned, her people\nslain by thousands, her dales and valleys wasted; and can you tell\nme that after these years of struggle you have gained any such\nadvantage as would warrant your advising me to rise against England?\"\n\nArchie was silent. Thinking over the struggle in which he had\ntaken part for so many years, and remembering the woes that it had\nbrought on Scotland, and that, after fighting so long, Bruce and\nthe handful of fugitives at Rathlin were the sole survivors of the\npatriotic party, he could not but acknowledge at heart the justice\nof the chiefs words. His sole hope for Scotland now rested in the\nperseverance and personal valour of the king, and the stubborn\ncharacter of the people, which he felt assured would lead them\nto rise again and again, in spite of disaster and defeat, until\nfreedom was won. The Irish possessed no Bruce; their country was\nless defendible than Scotland; and if, as Fergus said, they had none\nof that indomitable perseverance which enabled the Scotch people\nagain and again to rise against the yoke, what hope could there\nbe of final success, how could he be justified in urging upon the\nchieftain a step which would bring fire and sword into those quiet\nvalleys! For some time, therefore, after Ronald had translated the\nchief's speech he remained silent.\n\n\"I will not urge you further, sir,\" he said, \"for you are surely\nthe best judge of what is good for your people, and I have seen\nsuch ruin and desolation in Scotland, so many scores of ruined\ntowns and villages, so many thousands of levelled homesteads, that\nI will not say a single word to urge you to alter your resolution.\nIt is enough for me that you have said that if Ireland rises you\nwill also draw the sword. I must carry out my instructions, and\nhence shall travel south and visit other chiefs; they may view\nmatters differently, and may see that what Ireland cannot do alone\nshe may do in conjunction with Scotland.\"\n\n\"So be it!\" Fergus said. \"Believe me, if you raise a flame through\nthe west the north will not hang back.  And now I trust that you\nwill remain here for a few days as my guest. All that I have is\nyours, and my wife and daughters will do their best to make the\ntime pass pleasantly for you.\"\n\nArchie remained three days at the chiefs hold, where the primitive\nlife interested him greatly. A lavish hospitality was exercised.\nSeveral sheep were killed and roasted each day, and all comers were\nfree to join the repast. The chief's more immediate retainers, some\ntwenty in number, ate, lived, and slept in the great hall; while\ntables were spread outside, at which all who came sat down without\nquestion. The upper rooms of the hold were occupied by the chief,\nthe ladies of his family, and the female domestics. Here they retired\nwhen they felt disposed, but their meals were served on the dais.\nIn the evening the harper played and sang legends of deeds of bravery\nin the day of Ireland's independence; and as Ronald translated the\nsongs to him Archie could not but conclude privately that civil war,\nrapine, strife, and massacre must have characterized the country\nin those days.\n\nAt the conclusion of his stay Fergus appointed two of the retainers\nto accompany Archie south, and to give assurance to the various\nwild people through whom he might pass, that Archie's mission was\na friendly one to Ireland, and that he was an honoured friend and\nguest of the chief of Killeen.\n\nOn his arrival in Mayo Archie found matters more favourable to his\nmission. An insurrection had already broken out, headed by some of\nthe local chieftains, originating in a broil between the English\nsoldiers of a garrison and the natives. The garrison had been\nsurprised and massacred, and the wild Irish were flocking to arms.\nBy the chieftains here Archie, on explaining his mission, was warmly\nwelcomed.  As they were already in arms no urging on his part was\nneeded, and they despatched messengers throughout the country,\nsaying that an emissary from Scotland had arrived, and calling upon\nall to rise and to join with the Scotch in shaking off the yoke of\nEngland.\n\nArchie had therefore to travel no farther, and decided that he\ncould best carry out his mission by assisting to organize and lead\nthe Irish forces. These he speedily discovered were beyond all\ncomparison inferior, both in arms, in discipline, and in methods\nof fighting, to the Scots.  For a dashing foray they would be\nexcellent. Hardy, agile, and full of impetuosity, they would bear\ndown all resistance instantly, were that resistance not too strong;\nbut against stubborn and well armed troops they would break like\na wave against a rock.  Archie saw that with such troops anything\nlike regular war would be impossible, and that the struggle must\nbe one of constant surprises, attacks, and forays, and that they\ncould succeed only by wearing out and not by defeating the enemy.\nWith such tactics as these they might by long perseverance succeed;\nbut this was just what Fergus had warned him they would not practise,\nand that their courage was rather of a kind which would lead them\nto dash desperately against the line of levelled spears, rather\nthan continue a long and weary struggle under apparently hopeless\ncircumstances.\n\nThe chiefs, hearing from Archie that he had acted as one of Wallace's\nlieutenants in battles where the English had been heavily defeated,\nwillingly consented that he should endeavour to instil the tactics\nby which those battles had been won into their own followers; but\nwhen they found that he proposed that the men should remain stationary\nto withstand the English charges, they shook their heads.\n\n\"That will never do for our people,\" they said. \"They must attack\nsword in hand. They will rush fearlessly down against any odds, but\nyou will never get them steadily to withstand a charge of men-at-arms.\"\n\nArchie, however, persuaded them to allow him to organize a band of\ntwo hundred men under his immediate orders. These were armed with\nlong pikes, and were to form a sort of reserve, in order that if\nthe wild charge of the main body failed in its object these could\ncover a retreat, or serve as a nucleus around which they could\nrally. The army swelled rapidly; every day fresh chiefs arrived\nwith scores of wild tribesmen.  Presently the news came that an\nEnglish force was advancing from the Pale against them.  A council\nwas held at which Archie was present. Very strongly he urged his\nviews upon the chieftains, namely: that they should altogether\ndecline a pitched battle; but that, divided into numerous parties,\nthey should enter the Pale, destroying weak garrisons and ravaging\nthe country, trying to wear out the English by constant skirmishes\nand night attacks, but refusing always to allow themselves to be\ntempted into an engagement.\n\n\"The English cannot be everywhere at once,\" he urged.  \"Let them\nhold only the ground on which their feet stand.  As they advance\nor retire, close ever in on their rear, drive off their cattle and\ndestroy their crops and granaries in the Pale; force them to live\nwholly in their walled towns, and as you gain in strength capture\nthese one by one, as did we in Scotland. So, and so only, can you\nhope for ultimate success.\"\n\nHis advice was received with a silence which he at once saw betokened\ndisapproval. One after another of the Irish chieftains rose and\ndeclared that such a war could not be sustained.\n\n\"Our retainers,\" they said, \"are ready to fight, but after fighting\nthey will want to return to their homes; besides, we are fifteen\nthousand strong, and the English men-at-arms marching against us\nare but eight hundred; it would be shameful and cowardly to avoid a\nbattle, and were we willing to do so our followers would not obey\nus. Let us first destroy this body of English, then we shall be\njoined by others, and can soon march straight upon Dublin.\"\n\nArchie saw that it was hopeless to persevere, and set out the\nfollowing day with the wild rabble, for they could not be termed\nan army, to meet the English. The leaders yielded so far to his\nadvice as to take up a position where they would fight with the best\nchance of success. The spot lay between a swamp extending a vast\ndistance, and a river, and they were thus open only to an attack\nin front, and could, if defeated, take refuge in the bog, where\nhorsemen could not follow them.\n\nOn the following morning the English were seen approaching. In\naddition to the 800 men-at-arms were 1000 lightly equipped footmen,\nfor experience had taught the English commanders that in such a\ncountry lightly armed men were necessary to operate where the wide\nextending morasses prevented the employment of cavalry. The English\nadvanced in solid array: 300 archers led the way; these were\nfollowed by 700 spearmen, and the men-at-arms brought up the rear.\nThe Irish were formed in disordered masses, each under its own\nchieftain. The English archers commenced the fight with a shower\nof arrows. Scarcely had these began to fall when the Irish with a\ntremendous yell rushed forward to the assault. The English archers\nwere swept like chaff before them. With reckless bravery they threw\nthemselves next upon the spearmen. The solid array was broken by\nthe onslaught, and in a moment both parties were mixed up in wild\nconfusion.\n\nThe sight was too much for Archie's band to view unmoved, and these,\nin spite of his shouts, left their ground and rushed at full speed\nafter their companions and threw themselves into the fight.\n\nArchie was mounted, having been presented with a horse by one of\nthe chiefs, and he now, although hopeless of the final result, rode\nforward. Just as he joined the confused and struggling mass the\nEnglish men-at-arms burst down upon them. As a torrent would cleave\nits way through a mass of loose sand, so the English men-at-arms\nburst through the mass of Irish, trampling and cutting down all in\ntheir path. Not unharmed, however, for the Irish fought desperately\nwith axe and knife, hewing at the men-at-arms, stabbing at the\nhorses, and even trying by sheer strength to throw the riders to\nthe ground. After passing through the mass the men-at-arms turned\nand again burst down upon them. It was a repetition of the first\ncharge. The Irish fought desperately, but it was each for himself;\nthere was neither order nor cohesion, and each man strove only to\nkill a foe before being himself slain.  Archie and the chiefs, with\nthe few mounted men among the retainers, strove in vain to stem\nthe torrent. Under the orders of their leaders the English kept\nin a compact mass, and the weight of the horses and armour bore\ndown all opposition. Four times did the men-at-arms burst through\nthe struggling mass of Irish. As they formed to charge the fifth\ntime the latter lost heart, and as if acting under a simultaneous\ninfluence they turned and fled.\n\nThe English horse burst down on the rear of the mass of fugitives,\nhewing them down in hundreds. Those nearest to the river dashed in,\nand numbers were drowned in striving to cross it. The main body,\nhowever, made for the swamp, and though in the crush many sank in\nand perished miserably here, the great majority, leaping lightly\nfrom tuft to tuft, gained the heart of the morass, the pursuing\nhorse reining up on its edge.\n\nRonald had kept near Archie in the fight, and when all was lost\nran along by the side of his horse, holding fast to the stirrup\nleather. The horsemen still pressed along between the river and\nthe morass, and Archie, following the example of several of the\nchiefs, alighted from his saddle, and with his companion entered\nthe swamp. It was with the greatest difficulty that he made his\nway across it, and his lightly armed companion did him good service\nin assisting several times to drag him from the treacherous mire\nwhen he began to sink in it. At last they reached firmer ground in\nthe heart of the swamp, and here some 5000 or 6000 fugitives were\ngathered.  At least 4000 had fallen on the field. Many had escaped\nacross the river, although numbers had lost their lives in the\nattempt. Others scattered and fled in various directions. A few\nof the chiefs were gathered in council when Archie arrived. They\nagreed that all was lost and there was nothing to do but scatter\nto their homes. Archie took no part in the discussion. That day's\nexperience had convinced him that nothing like a permanent and\ndetermined insurrection was possible, and only by such a movement\ncould the Scottish cause be aided, by forcing the English to send\nreinforcements across St.  George's Channel. After seeing the\nslaughter which had taken place, he was rejoiced at heart that the\nrising had commenced before he joined it, and was in no way the\nresult of his mission, but was one of the sporadic insurrections\nwhich frequently broke out in Ireland, only to be instantly and\nsternly repressed.\n\n\"We have failed, Sir Knight,\" one of the chiefs said to him, \"but\nit was not for want of courage on the part of our men.\"\n\n\"No, indeed,\" Archie replied through his interpreter; \"never did I\nsee men fight more fiercely, but without discipline and organization\nvictory is well nigh impossible for lightly armed footmen against\nheavy mailclad cavalry.\"\n\n\"The tactics you advised were doubtless good,\" the chief said; \"I\nsee their wisdom, but they are well nigh impossible to carry out\nwith such following as ours. They are ever impatient for the fray,\nbut quickly wearied by effort; ready to die, but not to wait; to\nthem prudence means cowardice, and their only idea of fighting is\nto rush full at a foe. See how they broke the English spearmen!\"\n\n\"It was right well done,\" Archie replied, \"and some day, when well\ntrained and disciplined, Irish soldiers will be second to none in\nthe world; but unless they will submit to training and discipline\nthey can never hope to conquer the English.\"\n\n\"And now, Sir Knight, what do you propose doing?\" the chief said.\n\n\"I shall make my way north,\" Archie replied, \"and shall rejoin my\nking at Rathlin.\"\n\n\"I will send two of my men with you. They know every foot of the\nmorasses of this neighbourhood, and when they get beyond the point\nfamiliar to them will procure you two others to take their places.\nIt will need all your prudence and courage to get through, for\nthe English men-at-arms will be scouring the country in groups of\nfour, hunting all those they come across like wolves. See, already!\"\nand he pointed to the horizon; \"they are scattering round the edge\nof the morass to inclose us here; but it is many miles round, and\nbefore tomorrow is gone not a man will be left here.\"\n\nWhen darkness fell, Archie, accompanied by Ronald and his guides,\nset out on his journey. Alone he could never have found his way\nthrough the swamps, but even in the darkness his guides moved along\nquickly, following tracks known to them with the instinct of hounds;\nArchie kept close on their heels, as a step only a few inches from\nthe track might plunge him in a deep morass, in which in a few\nseconds he would sink out of sight. On nearing the edge of the\nbog the guides slackened their pace. Motioning to Archie to remain\nwhere he was, they crept forward noiselessly into the darkness.\nNot far off he could hear the calls of the English horsemen. The\nsounds were repeated again and again until they died away in the\ndistance, showing that a cordon had been drawn round the morass so\nas to inclose the fugitives from the battle of the previous day.\n\nIn a quarter of an hour the guides returned as noiselessly as they\nhad departed, and Archie continued the march at their heels. Even\ngreater caution than before was now necessary in walking, for the\nEnglish, before darkness had set in, had narrowly examined the edge\nof the morass, and had placed three or four men wherever they could\ndiscover the slightest signs of a track. Thus Archie's guides were\nobliged to leave the path by which they had previously travelled.\nTheir progress was slow now, the party only moving for a few yards\nat a time, and then halting while the guides searched for ground\nsolid enough to carry their weight. At last Archie felt the ground\ngrow firmer under his foot, and a reconnaissance by the guides\nhaving shown them that none of the English were stationed opposite\nto them, they left the morass, and noiselessly made their way across\nthe country until far beyond the English line.\n\nAll night they walked, and at daybreak entered another swamp, and\nlay down for the day in the long coarse grass growing on a piece of\nfirm ground deep in its recesses. In the evening one of the guides\nstole out and returned with a native of the neighbourhood, who\nundertook to show Archie the way on his further journey.\n\nTen days, or rather nights, of steady journeying brought Archie\nagain to the rocky shore where he had landed.  Throughout he had\nfound faithful guides, whom he had rewarded by giving, as was often\nthe custom of the time, in lieu of money, a link or two of one of\nhis gold chains. He and Ronald again took refuge in the cave where\nthey had passed the first night of their landing. It was untenanted\nnow.\n\nHere they abode for a fortnight, Ronald going up every two or three\ndays to purchase provisions at the scattered cottages. On Saturday\nnight they lit a great fire just inside the mouth of the cave, so\nthat while the flames could be seen far out at sea the light would\nbe unobserved by the garrison of Dunluce or any straggler on the\ncliff above. It had been arranged with Duncan that every Saturday\nnight, weather permitting, he should sail across and look for\na signal fire. The first Saturday night was wild and stormy, and\nalthough they lit the fire they had but slight idea that Duncan\nwould put out. The following week, however, the night was calm and\nbright, and after piling up the fire high they proceeded to the\ncauseway, and two hours later saw to their joy a boat approaching.\nIn a few minutes they were on board, and by the following morning\nreached Rathlin.\n\nThe king and his companions welcomed Archie's return warmly,\nalthough the report which he made showed that there was no hope of\nobtaining any serious diversion of the English attack by a permanent\nrising in Ireland; and the king, on hearing Archie's account of\nall that had passed, assured him that he felt that, although he had\nfailed, no one, under the circumstances, could have done otherwise.\n\n\n\n\nChapter XVII\n\nThe King's Blood Hound\n\n\nThe only other event which occurred throughout the winter was the\narrival of a fishing boat with a messenger from one of the king's\nadherents, and the news which he brought filled them with sorrow\nand dismay.  Kildrummy had been threatened with a siege, and the\nqueen, Bruce's sisters Christine and Mary, his daughter Marjory,\nand the other ladies accompanying them, deemed it prudent to leave\nthe castle and take refuge in the sanctuary of St. Duthoc, in Ross\nshire.\n\nThe sanctuary was violated by the Earl of Ross and his followers,\nand the ladies and their escort delivered up to Edward's lieutenants\nand sent to England. The knights and squires who formed the escort\nwere all executed, and the ladies committed to various places\nof confinement, where most of them remained in captivity of the\nstrictest and most rigorous kind until after the battle of Bannockburn,\neight years later. The Countess of Buchan, who had crowned Bruce\nat Scone, and who was one of the party captured at St.  Duthoc,\nreceived even fouler treatment, by Edward's especial orders,\nbeing placed in a cage on one of the turrets of Berwick Castle so\nconstructed that she could be seen by all who passed; and in this\ncruel imprisonment she was kept like a wild beast for seven long\nyears by a Christian king whom his admirers love to hold up as a\nmodel of chivalry.\n\nKildrummy had been besieged and taken by treachery.  The king's\nbrother, Nigel Bruce, was carried to Berwick, and was there hanged\nand beheaded. Christopher Seaton and his brother Alexander, the\nEarl of Athole, Sir Simon Fraser, Sir Herbert de Moreham, Sir David\nInchmartin, Sir John Somerville, Sir Walter Logan, and many other\nScotchmen of noble degree, had also been captured and executed,\ntheir only offence being that they had fought for their country.\n\nIn all the annals of England there is no more disgraceful page than\nthat which chronicles the savage ferocity with which King Edward\nbehaved to the Scottish nobles and ladies who fell into his hands.\nThe news of these murders excited the utmost fury as well as grief\namong the party at Rathlin, and only increased their determination\nto fight till the death against the power of England.\n\nThe spring was now at hand, and Douglas, with Archie Forbes and\na few followers, left in a boat, and landed on the Isle of Arran.\nIn the bay of Brodick was a castle occupied by Sir John Hastings\nand an English garrison. The Scots concealed themselves near the\ncastle, awaiting an opportunity for an attack. A day or two after\ntheir arrival several vessels arrived with provisions and arms for\nthe garrison. As these were being landed Douglas and his followers\nsallied out and captured the vessels and stores. The garrison of\nthe castle made a sortie to assist their friends, but were driven\nin with slaughter, and the whole of the supplies remained in the\nhands of the Scots, causing great rejoicing to the king and the\nrest of the party when a few days later they arrived from Rathlin.\n\nBruce now proposed an immediate descent upon Carrick, there, in the\nmidst of his family possessions, to set up his banner in Scotland.\nThe lands had been forfeited by Edward and bestowed upon some of\nhis own nobles.  Annandale had been given to the Earl of Hereford,\nCarrick to Earl Percy, Selkirk to Aymer de Valence. The castle of\nTurnberry was occupied by Percy with three hundred men.  Bruce sent\non his cousin Cuthbert to reconnoitre and see whether the people\nwould be ready to rise, but Cuthbert found the Scots sunk in\ndespair. All who had taken up arms had perished in the field or\non the scaffold. The country swarmed with the English, and further\nresistance seemed hopeless.  Cuthbert had arranged to light a beacon\non a point at Turnberry visible at Lamlash Bay in Arran, where the\nking, with his two hundred men and eighty-three boats, awaited the\nsight of the smoke which should tell them that circumstances were\nfavourable for their landing.\n\nCuthbert, finding that there was no chance of a rising, did not\nlight the bonfire; but as if fortune was determined that Bruce\nshould continue a struggle which was to end finally in the freedom\nof Scotland, some other person lit a fire on the very spot where\nCuthbert had arranged to show the signal. On seeing the smoke the\nking and his party at once got into their boats and rowed across\nto the mainland, a distance of seventeen miles. On reaching land\nthey were met by Cuthbert, who reported that the fire was not of his\nkindling, and that the circumstances were altogether unfavourable.\nBruce consulted with his brother Edward, Douglas, Archie, and his\nprincipal friends as to what course had better be pursued.  Edward\ndeclared at once that he for one would not take to sea again; and\nthis decision settled the matter.\n\nThe king without delay led his followers against the village\noutside the castle, where a considerable portion of the garrison\nwere housed. These were assailed so suddenly that all save one\nwere slain. Those in the castle heard the sounds of the conflict,\nbut being unaware of the smallness of the assailant's force, did\nnot venture to sally out to their assistance.\n\nPercy, with his followers, remained shut up in the castle, while\nBruce overran the neighbouring country; but an English force under\nSir Roger St. John, far too powerful to be resisted, advanced to\nTurnberry, and Bruce and his followers were obliged to seek refuge\nin the hills.  Thomas and Alexander, the king's brothers, with Sir\nReginald Crawford, had gone to the islands to beat up recruits, and\nreturning in a vessel with a party who had joined them, landed at\nLoch Ryan. They were attacked at once by Macdowall, a chieftain\nof Galloway, and routed. The king's brothers, with Sir Reginald\nCrawford, were carried to Carlisle severely wounded, and delivered\nover to King Edward, who at once sent them to the scaffold.\n\nThese wholesale and barbarous executions saddened the Scots, and,\nas might be expected, soon roused them to severe reprisals. Bruce\nhimself, however, although deeply stirred by the murder of his\nthree brothers and many dear friends, and by the captivity and\nharsh treatment of his wife and female relatives, never attempted\nto take vengeance for them upon those who fell into his hands,\nand during the whole of the war in no single instance did he put a\nprisoner to death.  He carried magnanimity, indeed, almost to the\nextent of impolicy; for had the nobles of England found that those\nof their number who fell into Bruce's hands suffered the penalty\nof death, which Edward inflicted upon the Scotch prisoners, they\nwould probably have remonstrated with the king and insisted upon\nhis conducting the war in a less barbarous and ferocious fashion.\n\nSir James Douglas was so stirred by the murder of the three Bruces and\nso many of his friends and companions, that he resolved henceforth\nto wage an exterminating war against the English, and by the recapture\nof his own stronghold, known as Castle Douglas, began the series\nof desperate deeds which won for him the name of the Black Douglas,\nand rendered his name for generations a terror among the English on\nthe Border. The castle had been conferred by Edward on Sir Robert\nde Clifford, and was occupied by an English garrison. Douglas\nrevealed his intention only to Archie Forbes, who at once agreed\nto accompany him. He asked leave from the king to quit their hiding\nplace for a time, accompanied by Archie, in order to revisit Douglas\nHall, and see how it fared with his tenants and friends. The king\nacquiesced with difficulty, as he thought the expedition a dangerous\none, and feared that the youth and impetuosity of Douglas might lead\nhim into danger; before consenting he strongly urged on Archie to\nkeep a strict watch over the doings of the young noble.\n\nAccompanied by but one retainer, the friends set out for Douglasdale.\nWhen they arrived there Douglas went to the cottage of an old and\nfaithful servant named Thomas Dickson, by whom he was joyfully\nreceived. Dickson went out among the retainers and revealed to such\nas could be most surely depended upon the secret of their lord's\npresence, and one by one took them in to see him. The friends\nhad already determined upon their course, and the retainers all\npromised to take part in the scheme. They were not numerous enough\nto assault the castle openly, but they chose the following Sunday\nfor the assault. This was Palm Sunday and a festival, and most of\nthe garrison would come to the Church of St. Bride, in the village\nof the same name, a short distance from the castle.\n\nDickson with some of his friends went at the appointed time, with\narms concealed under their clothes, to the church; and after the\nservice had commenced Douglas and some of his followers gathered\noutside.  Unfortunately for the plan, some of those outside set\nup the shout, \"A Douglas!\" prematurely before the whole party had\narrived and were ready to rush into the church. Dickson with his\nfriends at once drew out their arms and attacked the English; but\nbeing greatly outnumbered and for a time unsupported, most of them,\nincluding their leader, were slain. Sir James and his followers then\nfought their way in, and after a desperate fight all the garrison\nsave ten were killed.\n\nThe party then proceeded to the castle, which they captured without\nresistance. Douglas and his companions partook of the dinner which\nhad been prepared for the garrison; then as much money, weapons,\narmour, and clothing as they could carry away was taken from the\ncastle. The whole of the vast stores of provisions were carried\ninto the cellar, the heads struck out of the ale and wine casks,\nthe prisoners were slain and their bodies thrown down into the mass,\nand the castle was then set on fire. Archie Forbes in vain begged\nDouglas to spare the lives of the prisoners, but the latter would\nnot listen to him. \"No, Sir Archie,\" he exclaimed; \"the King of\nEngland held my good father a prisoner in chains until he died;\nhe has struck off the heads of every one of our friends who have\nfallen into his hands; he has wasted Scotland from end to end with\nfire and sword, and has slain our people in tens of thousands. So\nlong as this war continues, so long will I slay every prisoner who\nfalls into my hands, as King Edward would slay me did I fall into\nhis; and I will not desist unless this cruel king agrees to show\nquarter to such of us as he may capture. I see not why all the\nmassacreing and bloodshed should be upon one side.\"\n\nArchie did not urge him further, for he too was half beside himself\nwith indignation and grief at the murder of the king's brothers\nand friends, and at the cruel captivity which, by a violation of\nthe laws of sanctuary, had fallen upon the ladies with whom he had\nspent so many happy hours in the mountains and forests of Athole.\n\nDouglas and Archie now rejoined the king. For months Bruce led\nthe life of a hunted fugitive. His little following dwindled away\nuntil but sixty men remained in arms. Of these a portion were\nwith the king's brother in Galloway, and with but a handful of men\nBruce was lying among the fastnesses of Carrick when Sir Ingram\nde Umfraville, with a large number of troops sent by the Earl of\nPembroke from Edinburgh, approached. Wholly unable to resist so\nlarge a force, Bruce's little party scattered, and the king himself,\nattended only by a page, lay hidden in the cottage of a peasant.\nThe English in vain searched for him, until a traitorous Scot went\nto Umfraville and offered, for a reward of a grant of land to the\nvalue of 40 pounds annually, to slay Bruce.\n\nThe offer was accepted, and the traitor and his two sons made their\nway to Bruce's place of concealment.  As they approached, Bruce\nsnatched his bow from his page and shot the traitor through the\neye. One son attacked him with an axe, but was slain with a blow\nfrom the king's sword. The remaining assailant rushed at him with\na spear; but the king with one blow cut off the spearhead, and\nbefore the assailant had time to draw his sword, stretched him\ndead at his feet.  After this the king with his adherents eluded\nthe search of the English and made their way into Galloway. The\npeople here who were devoted to the English cause determined to hunt\nhim down, and two hundred men, accompanied by some blood hounds,\nset off towards the king's retreat; but Bruce's scouts were on\nthe watch and brought him news of their coming. The king with his\nparty retired until they reached a morass, through which flowed a\nrunning stream, while beyond a narrow passage led through a deep\nquagmire.\n\nBeyond this point the hunted party lay down to rest, while the\nking with two followers returned to the river to keep watch. After\nlistening for some time they heard the baying of the hounds coming\nnearer and nearer, and then, by the light of a bright moon, saw\ntheir enemies approaching.\n\nThe king sent his two followers to rouse the band. The enemy,\nseeing Bruce alone, pressed forward with all haste; and the king,\nknowing that if he retired his followers would be attacked unprepared,\ndetermined alone to defend the narrow path. He retired from the\nriver bank to the spot where the path was narrowest and the morass\nmost impassable, and then drew his sword. His pursuers, crossing\nthe river, rode forward against him; Bruce charged the first, and\nwith his lance slew him; then with a blow with his mace he stretched\nhis horse beside him, blocking the narrow passage. One by one his\nfoes advanced, and five fell beneath his blows, before his companions\nran up from behind. The Galloway men then took to flight, but nine\nmore were slain before they could cross the ford.\n\nThe admiration and confidence of Bruce's followers were greatly\naroused by this new proof of his courage and prowess. Sir James\nDouglas, his brother Edward, and others soon afterwards returned\nfrom the expeditions on which they had been sent, and the king\nhad now 400 men assembled.  This force, however, was powerless to\nresist an army of English and Lowland Scots who marched against\nhim, led by Pembroke in person. This force was accompanied by John,\nson of Alexander MacDougall of Lorne, with 800 of his mountaineers.\nWhile the heavy armed troops occupied all the Lowlands, Lorne and\nhis followers made a circuit in the mountains so as to inclose the\nroyal fugitive between them.\n\nBruce, seeing that resistance was impossible, caused his party to\nseparate into three divisions, and Douglas, Edward Bruce, and Sir\nArchibald Forbes were charged to lead their bands, if possible,\nthrough the enemy without fighting. The king tried to escape by a\ndifferent route with a handful of men. John of Lorne had obtained\nfrom Turnberry a favourite blood hound belonging to Bruce, and\nthe hound being put upon the trace persistently followed the king's\nparty.  Seeing this, Bruce ordered them all to disperse, and,\naccompanied only by his foster brother, attempted to escape by\nspeed.\n\nAs they sped along the mountain side they were seen by Lorne, who\ndirected his henchman, with four of his bravest and swiftest men,\nto follow him. After a long chase the MacDougalls came up with\nBruce and his foster brother, who drew their swords and stood on the\ndefence. The henchman, with two of his followers, attacked Bruce,\nwhile the other two fell on his foster brother. The combat was a\ndesperate one, but one by one the king cut down his three assailants,\nand then turned to the assistance of his foster brother, who\nwas hardly pressed. The king's sword soon rid him of one of his\nassailants, and he slew the other.  Having thus disembarrassed\nthemselves of the whole of their immediate assailants, Bruce and his\ncompanion continued their flight. The main body of their hunters,\nwith the hound, were but a short distance away, but in a wood the\nfugitives came upon a stream, and, marching for some distance down\nthis, again landed, and continued their flight.\n\nThe hound lost their scent at the spot where they had entered the\nwater, and being unable to recover it, Lorne and his followers\nabandoned the chase. Among the king's pursuers on this occasion\nwas his nephew Randolph, who had been captured at the battle of\nMethven, and having again taken the oath of allegiance to Edward\nhad been restored to that monarch's favour, and was now fighting\namong the English ranks.\n\nThe search was actively kept up after Bruce, and a party of three\nmen-at-arms came upon him and his foster brother.  Being afraid to\nattack the king, whom they recognized, openly, they pretended they\nhad come to join him.\n\nThe king suspected treachery; and when the five lay down for the\nnight in a cottage which they came upon he and his companion agreed\nto watch alternately. Overcome by fatigue, however, both fell asleep,\nand when they were suddenly attacked by the three strangers, the\nfoster brother was killed before he could offer any resistance.\nThe king himself, although wounded, managed to struggle to his\nfeet, and then proved more than a match for his three treacherous\nassailants, all of whom, after a desperate struggle, he slew.\n\nThe next morning he continued his way, and by nightfall succeeded\nin joining the three bands, who had safely reached the rendezvous\nhe had appointed.\n\nA few hours after this exploit of Bruce, Archie with two or three\nof his followers joined him.\n\n\"This is indeed a serious matter of the hound,\" Archie said when\nBruce told him how nearly he had fallen a victim to the affection\nof his favourite. \"Methinks, sire, so long as he remains in the\nEnglish hands your life will never be safe, for the dog will always\nlead the searchers to your hiding places; if one could get near\nenough to shoot him, the danger would be at an end.\"\n\n\"I would not have him shot, Archie, for a large sum. I have had him\nsince he was a little pup; he has for years slept across my door,\nand would give his life for mine. 'Tis but his affection now that\nbrings danger upon me.\"\n\n\"I should be sorry to see the dog killed myself,\" Archie said, \"for\nhe is a fine fellow, and he quite admitted me to his friendship\nduring the time we were together. Still, sire, if it were a question\nbetween their lives and yours, I would not hesitate to kill any\nnumber of dogs. The whole future of Scotland is wrapped up in you;\nand as there is not one of your followers but would gladly give\nhis life for yours, it were no great thing that a hound should do\nthe same.\"\n\n\"I cannot withstand you in argument, Archie,\" the king said smiling;\n\"yet I would fain that my favourite should, if possible, be spared.\nBut I grant you, should there be no other way, and the hound should\ncontinue to follow me, he must be put to death. But it would grieve\nme sorely. I have lost so many and so dear friends in the last\nyear, that I can ill spare one of the few that are left me.\"\n\nArchie was himself fond of dogs, and knowing how attached Bruce\nwas to his faithful hound he could quite understand how reluctant\nhe was that harm should come to him.  Still, he felt it was necessary\nthat the dog should, at all hazards, be either killed or taken\nfrom the English, for if he remained in their hands he was almost\ncertain sooner or later to lead to Bruce's capture. He determined\nthen to endeavour to avert the danger by abstracting the dog from\nthe hands of the English, or, failing that, by killing him. To do\nthis it would be absolutely necessary to enter the English camp.\nThere was no possibility of carrying out his purpose without running\nthis risk, for when in pursuit of the king the hound would be held\nby a leash, and there would be many men-at-arms close by, so that\nthe difficulty of shooting him would be extremely great, and Archie\ncould see no plan save that of boldly entering the camp.\n\nHe said nothing of his project to Bruce, who would probably have\nrefused to allow him to undertake it; but the next morning when\nhe parted from him--for it was considered advisable that the\nfugitives should be divided into the smallest groups, and that only\none or two of his retainers should remain with Bruce--he started\nwith his own followers in the direction of Pembroke's camp. He\npresently changed clothes with one of these, and they then collected\na quantity of firewood and made it into a great <DW19>. Archie gave\nthem orders where they should await him, and lifting the <DW19> on\nhis shoulders boldly entered the camp. He passed with it near the\npavilion of Pembroke. The earl was standing with some knights at\nthe entrance.\n\n\"Come hither, Scot,\" he said as Archie passed.\n\nArchie laid his bundle on the ground, and doffing his bonnet strode\nwith an awkward and abashed air toward the earl.\n\n\"I suppose you are one of Bruce's men?\" the earl said.\n\n\"My father,\" Archie replied, \"as well as all who dwell in these\ndales, were his vassals; but seeing that, as they say, his lands\nhave been forfeit and given to others, I know not whose man I am\nat present.\"\n\n\"Dost know Bruce by figure?\"\n\n\"Surely,\" Archie said simply, \"seeing that I was employed in the\nstables at Turnberry, and used to wash that big hound of his, who\nwas treated as a Christian rather than a dog.\"\n\n\"Oh, you used to tend the hound!\" Pembroke said. \"Then perhaps\nyou could manage him now. He is here in camp, and the brute is so\nsavage and fierce he has already well nigh killed two or three men;\nand I would have had him shot but that he may be useful to us. If\nhe knows you he may be quieter with you than others.\"\n\n\"Doubtless he would know me,\" Archie said; \"but seeing that I have\nthe croft to look after, as my father is old and infirm, I trust\nthat you will excuse me the service of looking after the hound.\"\n\n\"Answer me not,\" Pembroke said angrily. \"You may think yourself\nlucky, seeing that you are one of Bruce's retainers, that I do not\nhave you hung from a tree.\n\n\"Take the fellow to the hound,\" he said to one of his retainers,\n\"and see if the brute recognizes him; if so, put him in charge of\nhim for the future. And see you Scot, that you attempt no tricks,\nfor if you try to escape I will hang you without shrift.\"\n\nArchie followed the earl's retainer to where, behind his pavilion,\nthe great dog was chained up. He leapt to his feet with a savage\ngrowl on hearing footsteps approaching. His hair bristled and he\ntugged at his chain.\n\n\"What a savage beast it is!\" the man said; \"I would sooner face\na whole company of you Scots than get within reach of his jaws.\nDickon,\" he went on as another soldier, on hearing the growl, issued\nfrom one of the smaller tents which stood in rear of the pavilion,\n\"the earl has sent this Scot to relieve you of your charge of the\ndog; he is to have the care of him in future.\"\n\n\"That is the best turn the earl has done me for a long time,\" the\nman replied. \"Never did I have a job I fancied less than the tending\nof that evil tempered brute.\"\n\n\"He did not use to be evil tempered,\" Archie said; \"but was a quiet\nbeast when I had to do with him before.  I suppose the strangeness\nof the place and so many strange faces have driven him half wild.\nBeside, he is not used to being chained up. Hector, old fellow,\"\nhe said approaching the dog quietly, \"don't you know me?\"\n\nThe great hound recognized the voice and his aspect changed\nat once. The bristling hair lay flat on his back; the threatening\njaws closed. He gave a short deep bark of pleasure, and then began\nleaping and tugging at his chain to reach his acquaintance. Archie\ncame close to him now. Hector reared on his hind legs, and placed\nhis great paws on his shoulders, and licked his face with whines\nof joy.\n\n\"He knows you, sure enough,\" the man said; \"and maybe we shall get\non better now. At any rate there may be some chance of sleep, for\nthe brute's howls every night since he has been brought here have\nkept the whole camp awake.\"\n\n\"No wonder!\" Archie said, \"when he has been accustomed to be petted\nand cared for; he resents being chained up.\"\n\n\"Would you unchain him?\" the man asked.\n\n\"That would I,\" Archie replied; \"and I doubt not that he will stay\nwith me.\"\n\n\"It may be so,\" the man replied; \"but you had best not unchain him\nwithout leave from the earl, for were he to take it into his head\nto run away, I would not give a groat for your life. But I will go\nand acquaint the earl that the dog knows you, and ask his orders\nas to his being unchained.\"\n\nIn two or three minutes he returned.\n\n\"The earl says that on no account is he to be let free. He has told\nme to have a small tent pitched here for you. The hound is to be\nchained to the post, and to share the tent with you. You may, if\nyou will, walk about the camp with him, but always keeping him in\na chain; but if you do so it will be at your peril, for if he gets\naway your life will answer for it.\"\n\nIn a short time two or three soldiers brought a small tent and\nerected it close by where the dog was chained up.  Archie unloosed\nthe chain from the post round which it was fastened, and led\nHector to the tent, the dog keeping close by his side and wagging\nhis tail gravely, as if to show his appreciation of the change, to\nthe satisfaction of the men to whom hitherto he had been a terror.\nSome heather was brought for a bed, and a supply of food, both\nfor the dog and his keeper, and the men then left the two friends\nalone. Hector was sitting up on his haunches gazing affectionately\nat Archie, his tail beating the ground with slow and regular strokes.\n\n\"I know what you want to ask, old fellow,\" Archie said to him; \"why\nI don't lead you at once to your master? Don't you be impatient,\nold fellow, and you shall see him ere long;\" and he patted the\nhound's head.\n\nHector, with a great sigh expressive of content and satisfaction,\nlay down on the ground by the side of the couch of heather on which\nArchie threw himself--his nose between his forepaws, clearly\nexpressing that he considered his troubles were over, and could now\nafford to wait until in due time he should be taken to his master.\nThat night the camp slept quietly, for Hector was silent. For the\nnext two days Archie did not go more than a few yards from his tent,\nfor he feared that he might meet some one who would recognize him.\n\n\n\n\nChapter XVIII\n\nThe Hound Restored\n\n\nOn the third day after his arrival at the camp Archie received\norders to prepare to start with the hound, with the earl and a large\nparty of men-at-arms, in search of Bruce. A traitor had just come\nin and told them where Bruce had slept the night before. Reluctantly\nArchie unfastened the chain from the pole, and holding the end in\nhis hand went round with Hector to the front of the pavilion.  He\nwas resolved that if under the dog's guidance the party came close\nup with Bruce, he would kill the dog and then try to escape by\nfleetness of foot, though of this, as there were so many mounted\nmen in the party, he had but slight hope.  Led by the peasant they\nproceeded to the hut, which was five miles away in the hills.  On\nreaching it Hector at once became greatly excited. He sniffed\nhere and there, eagerly hunted up and down the cottage, then made\na circuit round it, and at last, with a loud deep bay he started\noff with his nose to the ground, pulling so hard at the chain that\nArchie had difficulty in keeping up with him. Pembroke and his\nknights rode a little behind, followed by their men-at-arms.\n\n\"I pray you, Sir Earl,\" Archie said, \"keep not too close to my\ntraces, for the sound of the horse's hoofs and the jingling of the\nequipments make him all the more impatient to get forward, and even\nnow it taxes all my strength to hold him in.\"\n\nThe earl reined back his horse and followed at a distance of some\nfifty yards. He had no suspicion whatever of any hidden design\non Archie's part. The fact that the hound had recognized him had\nappeared to him a sure proof of the truth of his tale, and Archie\nhad put on an air of such stupid simplicity that the earl deemed\nhim to have but imperfect possession of his wits. Moreover, in any\ncase he could overtake him in case he attempted flight.\n\nArchie proceeded at a trot behind the hound, who was with\ndifficulty restrained at that pace, straining eagerly on the chain\nand occasionally sending out his deep bay. Archie anxiously regarded\nthe country through which he was passing. He was waiting for an\nopportunity, and was determined, whenever they passed near a steep\nhillside unscaleable by horsemen, he would stab Hector to the heart\nand take to flight.  Presently he saw a man, whose attire showed\nhim to be a Highlander, approaching at a run; he passed close by\nArchie, and as he did so stopped suddenly, exclaiming, \"Archibald\nForbes!\" and drawing his broadsword sprang at him. Archie, who was\nunarmed save by a long knife, leapt back. In the man he recognized\nthe leader of the MacDougall's party, who had captured him near\nDunstaffnage. The conflict would have terminated in an instant had\nnot Hector intervened. Turning round with a deep growl the great\nhound sprang full at the throat of the Highlander as with uplifted\nsword he rushed at Archie.  The impetus of the spring threw the\nMacDougall on his back, with the fangs of the hound fixed in his\nthroat. Archie's first impulse was to pull the dog off, the second\nthought showed him that, were the man to survive he would at once\ndenounce him. Accordingly, though he appeared to tug hard at Hector's\nchain, he in reality allowed him to have his way. Pembroke and his\nknights instantly galloped up. As they arrived Hector loosed his\nhold, and with his hair bristly with rage prepared to attack those\nwhom he regarded as fresh enemies.\n\n\"Hold in that hound,\" Pembroke shouted, \"or he will do more damage.\nWhat means all this?\" For a minute Archie did not answer, being\nengaged in pacifying Hector, who, on seeing that no harm was\nintended, strove to return to his first foe.\n\n\"It means,\" Archie said, when Hector was at last pacified, \"that\nthat Highlander came the other day to our cottage and wanted to\ncarry off a cow without making payment for it.  I withstood him,\nhe drew his sword, but as I had a stout cudgel in my hand I hit him\non the wrist ere he could use it, and well nigh broke his arm. So\nhe made off, cursing and swearing, and vowing that the next time\nhe met me he would have my life.\"\n\n\"And that he would have done,\" Pembroke said, \"had it not been\nfor Bruce's dog, who has turned matters the other way. He is dead\nassuredly. It is John of Lorne's henchman, who was doubtless on\nhis way with a message from his lord to me. Could not the fool have\npostponed his grudge till he had delivered it? I tell you, Scot,\nyou had best keep out of the MacDougalls' way, for assuredly they\nwill revenge the death of their clansman upon you if they have\nthe chance, though I can testify that the affair was none of your\nseeking. Now let us continue our way.\"\n\n\"I doubt me, Sir Earl, whether our journey ends not here,\" Archie\nsaid, \"seeing that these hounds, when they taste blood, seem for\na time to lose their fineness of scent; but we shall see.\"\n\nArchie's opinion turned out correct. Do what they would they could\nnot induce Hector again to take up his master's trail, the hound\nagain and again returning to the spot where the dead Highlander\nstill lay.  Pembroke had the body carried off but the hound tugged\nat his chain in the direction in which it had gone, and seemed to\nhave lost all remembrance of the track upon which he was going.\nAt last Pembroke was obliged to acknowledge that it was useless to\npursue longer, and, full of disappointment at their failure, the\nparty returned to camp, Pembroke saying: \"Our chase is but postponed.\nWe are sure to get tidings of Bruce's hiding place in a day or two,\nand next time we will have the hound muzzled, lest any hotheaded\nHighlander should again interfere to mar the sport.\"\n\nIt was some days before further tidings were obtained of Bruce.\nArchie did not leave his tent during this time, giving as a reason\nthat he was afraid if he went out he should meet some of Lorne's\nmen, who might take up the quarrel of the man who had been killed.\nAt length, however, another traitor came in, and Pembroke and his\nparty set out as before, Hector being this time muzzled by a strap\nround his jaw, which would not interfere with his scent, but would\nprevent him from widely opening his jaws.\n\nThe scent of Bruce was again taken up at a lonely hut in the hills.\nThe country was far more broken and rough than that through which\nthey had followed Bruce's trail on the preceding occasion. Again\nArchie determined, but most reluctantly, that he would slay the\nnoble dog; but he determined to postpone the deed to the latest\nmoment. Several places were passed where he might have succeeded\nin effecting his escape after stabbing the hound, but each time his\ndetermination failed him. It would have been of no use to release\nthe dog and make himself up the hillside, for a blood hound's pace\nwhen on the track is not rapid, and the horsemen could have kept\nup with Hector, who would of course have continued his way upon\nthe trail of the king. Presently two men were seen in the distance;\nthey had evidently been alarmed by the bay of the hound, and were\ngoing at full speed. A shout of triumph broke from the pursuers,\nand some of the more eager would have set spurs to their horses\nand passed the hound.\n\n\"Rein back, rein back,\" Pembroke said, \"the country is wild and\nhilly here, and Bruce may hide himself long before you can overtake\nhim. Keep steadily in his track till he gains flatter country, where\nwe can keep him in sight, then we shall have no more occasion for\nthe hound and can gallop on at full speed.\"\n\nArchie observed, with satisfaction, that Bruce was making up an\nextremely steep hillside, deeming probably that horsemen would be\nunable to follow him here, and that he would be able to distance\npursuers on foot. Ten minutes later his pursuers had reached the\nfoot of the hill. Pembroke at once ordered four knights and ten\nmen-at-arms to dismount.\n\n\"Do you,\" he said, \"with the dog, follow hard upon the traces of\nBruce. When you reach the top signal to us the direction in which\nhe has gone. Follow ever on his track without stopping; he must at\nlast take to the low country again.  Some of my men shall remain\nhere, others a mile further on, and so on round the whole foot of\nthe hills. Do you, when you see that, thinking he has distanced\nyou, which he may well do being more lightly armed and flying for\nhis life, he makes for the low country again, send men in different\ndirections to give me warning. The baying of the dog will act as\na signal to us.\"\n\nWhile the men had been dismounting and Pembroke was giving his\norders Archie had proceeded up the hill with the hound. The path\nwas exceedingly steep and difficult.\n\n\"Do not hurry, sirrah,\" Pembroke called; \"hold in your hound till\nthe others join you.\" But Archie paid no attention to the shout,\nbut kept up the steep path at the top of his speed. Shouts and\nthreats followed him, but he paused not till he reached the top\nof the ascent; then he unfastened Hector's collar, and the dog,\nrelieved from the chain which had so long restrained him, bounded\naway with a deep bay in pursuit of his master, whose scent was now\nstrong before him.  As Archie looked back, the four knights and\ntheir followers, in single file, were, as yet, scarce halfway up\nthe ascent. Lying round were numbers of loose boulders, and Archie\nat once began to roll these down the hillside. They went but slowly\nat first, but as they reached the steeper portion they gathered\nspeed, and taking great bounds crashed down the hillside. As these\nformidable missiles burst down from above the knights paused.\n\n\"On!\" Pembroke shouted from below; \"the Scot is a traitor, and he\nand the hound will escape if you seize him not.\" Again the party\nhurried up the hill. Three of them were struck down by the rocks,\nand the speed of all was impeded by the pauses made to avoid the\ngreat boulders which bounded down toward them.  When they were\nwithin a few yards of the top Archie turned and bounded off at full\nspeed. He had no fear of being himself overtaken. Lightly clad and\nunarmed, the knights and men-at-arms, who were all in full armour,\nand who were already breathed with the exertions they had made,\nwould have no chance of overtaking him; indeed he could safely have\nfled at once when he loosed Hector, but he had stopped to delay the\nascent of his pursuers solely to give the hound as long a start\nas possible. He himself could have kept up with the hound; the\nmen-at-arms could assuredly not do so, but they might for a long\ntime keep him in sight, and his baying would afterwards indicate\nthe line the king was taking, and Bruce might yet be cut off by\nthe mounted men.  The delay which his bombardment had caused had\ngiven a long start to the hound, for it was more than five minutes\nfrom the time when it had been loosed before the pursuers gained\nthe crest of the hill. Archie, in his flight, took a different\nline to that which the dog had followed.  Hector was already out\nof sight, and although his deep baying might for a time afford an\nindex to his direction this would soon cease to act as a guide, as\nthe animal would rapidly increase his distance from his pursuers,\nand would, when he had overtaken the king, cease to emit his warning\nnote. The pursuers, after a moment's pause for consultation on the\ncrest of the hill, followed the line taken by the hound.\n\nThe men-at-arms paused to throw aside their defensive armour,\nbreast, back, and leg pieces, and the knights relieved themselves\nof some of their iron gear; but the delay, short as it was, caused\nby the unbuckling of straps and unlacing of helms, increased the\ndistance which already existed between them and the hound, whose\ndeep notes, occasionally raised, grew fainter and fainter. In a\nfew minutes it ceased altogether, and Archie judged that the hound\nhad overtaken his master, who, on seeing the animal approaching\nalone, would naturally have checked his flight. Archie himself\nwas now far away from the men-at-arms, and after proceeding until\nbeyond all reach of pursuit, slackened his pace, and breaking into\na walk continued his course some miles across the hills until he\nreached a lonely cottage where he was kindly received, and remained\nuntil next day.\n\nThe following morning he set out and journeyed to the spot, where,\non leaving his retainers more than a week before, he had ordered\nthem to await his coming. It was another week before he obtained\nsuch news as enabled him again to join the king, who was staying at\na woodcutter's hut in Selkirk Forest. Hector came out with a deep\nbark of welcome.\n\n\"Well, Sir Archie,\" the king said, following his dog to the door,\n\"and how has it fared with you since we last parted a fortnight\nsince? I have been hotly chased, and thought I should have been\ntaken; but, thanks to the carelessness of the fellow who led my\nhound, Hector somehow slipped his collar and joined me, and I was\nable to shake off my pursuers, so that danger is over, and without\nsacrificing the life of my good dog.\"\n\nArchie smiled. \"Perchance, sir, it was not from any clumsiness that\nthe hound got free, but that he was loosed by some friendly hand.\"\n\n\"It may be so,\" the king replied; \"but they would scarcely have\nintrusted him to a hand friendly to me. Nor would his leader, even\nif so disposed, have ventured to slip the hound, seeing that the\nhorsemen must have been close by at the time, and that such a deed\nwould cost him his life. It was only because Hector got away, when\nthe horsemen were unable to follow him, that he escaped, seeing\nthat, good dog as he is, speed is not his strong point, and that\nhorsemen could easily gallop alongside of him even were he free.\nWhat are you smiling at, Sir Archie? The hound and you seem on\nwondrous friendly terms;\" for Hector was now standing up with his\ngreat paws on Archie's shoulder.\n\n\"So we should be, sire, seeing that for eight days we have shared\nbed and board.\"\n\n\"Ah! is it so?\" Bruce exclaimed. \"Was it you, then, that loosed\nthe hound?\"\n\n\"It was, sir,\" Archie replied; \"and this is the history of it;\nand you will see that if I have done you and Hector a service in\nbringing you together again the hound has repaid it by saving my\nlife.\"\n\nEntering the hut, Archie sat down and related all that had happened,\nto the king.\n\n\"You have done me great service, Sir Archie,\" Bruce said when he\nconcluded his tale, \"for assuredly the hound would have wrought my\nruin had he remained in the hands of the English. This is another\nof the long list of services you have rendered me. Some day, when\nI come to my own, you will find that I am not ungrateful.\"\n\nThe feats which have been related of Bruce, and other personal\nadventures in which he distinguished himself, won the hearts of\ngreat numbers of the Scotch people. They recognized now that they\nhad in him a champion as doughty and as valiant as Wallace himself.\nThe exploits of the king filled their imaginations, and the way in\nwhich he continued the struggle after the capture of the ladies of\nhis family and the cruel execution of his brothers and so many of\nhis adherents, convinced them that he would never desist until he\nwas dead or a conqueror. Once persuaded of this, larger numbers\ngathered round his banner, and his fortunes henceforth began steadily\nto rise.\n\nLord Clifford had rebuilt Douglas Castle, making it larger and\nmuch stronger than before, and had committed it to the charge of\nCaptain Thirlwall, with a strong garrison. Douglas took a number\nof his retainers, who had now joined him in the field, and some\nof these, dressing themselves as drovers and concealing their\narms, drove a herd of cattle within sight of the castle toward an\nambuscade in which Douglas and the others were laying in ambush.\nThe garrison, seeing what they believed a valuable prize within\ntheir grasp, sallied out to seize the cattle. When they reached the\nambuscade the Scots sprang out upon them, and Thirlwall and the\ngreater portion of his men were slain. Douglas then took and destroyed\nthe castle and marched away. Clifford again rebuilt it more strongly\nthan before, and placed it in charge of Sir John Walton. It might\nhave been thought that after the disasters which had befallen\nthe garrison they would not have suffered themselves to be again\nentrapped. Douglas, however, ordered a number of his men to ride\npast within sight of the castle with sacks upon their horses,\napparently filled with grain, but in reality with grass, as if\nthey were countrymen on their way to the neighbouring market town,\nwhile once more he and his followers placed themselves in ambush.\nHeaded by their captain, the garrison poured out from the castle,\nand followed the apparent countrymen until they had passed the\nambush where Douglas was lying.  Then the drovers threw off their\ndisguises and attacked them, while Douglas fell upon their rear,\nand Walton and his companions were all slain. The castle was then\nattacked, and the remainder of the garrison being cowed by the\nfate which had befallen their leader and comrades, made but a poor\ndefence.  The castle was taken, and was again destroyed by its\nlord, the walls being, as far as possible, overthrown.\n\nShortly after the daring adventures of Bruce had begun to rouse\nthe spirit of the country Archie Forbes found himself at the head\nof a larger following than before.  Foreseeing that the war must be\na long one he had called upon his tenants and retainers to furnish\nhim only with a force one third of that of their total strength.\nThus he was able to maintain sixty men always in the field--all\nthe older men on the estate being exempted from service unless\nsummoned to defend the castle.\n\nOne day when he was in the forest of Selkirk with the king a body\nof fifty men were seen approaching.  Their leader inquired for Sir\nArchibald Forbes, and presently approached him as he was talking\nto the king.\n\n\"Sir Archibald Forbes,\" he said, \"I am bidden by my mistress, the\nlady Mary Kerr, to bring these, a portion of the retainers of her\nestates in Ayrshire, and to place them in your hands to lead and\ngovern.\"\n\n\"In my hands!\" Archie exclaimed in astonishment. \"The Kerrs are all\non the English side, and I am their greatest enemy. It were strange,\nindeed, were one of them to choose me to lead their retainers in\nthe cause of Scotland.\"\n\n\"Our young lord Sir Allan was slain at Methven,\" the man said, \"and\nthe lady Mary is now our lady and mistress.  She sent to us months\nago to say that she willed not that any of her retainers should any\nlonger take part in the struggle, and all who were in the field\nwere summoned home. Then we heard that no hindrance would be offered\nby her should any wish to join the Bruce; and now she has sent by\na messenger a letter under her hand ordering that a troop of fifty\nmen shall be raised to join the king, and that it shall fight under\nthe leading and order of Sir Archibald Forbes.\"\n\n\"I had not heard that Sir Allan had fallen,\" Archie said to the\nking as they walked apart from the place where the man was standing;\n\"and in truth I had forgotten that he even had a sister. She must\nhave been a child when I was a boy at Glen Cairn, and could have\nbeen but seldom at the castle--which, indeed, was no fit abode\nfor so young a girl, seeing that Sir John's wife had died some\nyears before I left Glen Cairn.  Perhaps she was with her mother's\nrelations. I have heard that Sir John Kerr married a relation of\nthe Comyns of Badenoch. 'Tis strange if, being of such bad blood\non both sides, she should have grown up a true Scotchwoman--still\nmore strange she should send her vassals to fight under the banner\nof one whom she must regard as the unlawful holder of her father's\nlands of Aberfilly.\"\n\n\"Think you, Sir Archie,\" the king said, \"that this is a stratagem,\nand that these men have really come with a design to seize upon\nyou and slay you, or to turn traitors in the first battle?\"\n\nArchie was silent. \"Treachery has been so much at work,\" he said\nafter a pause, \"that it were rash to say that this may not be a\ntraitorous device; but it were hard to think that a girl--even\na Kerr--would lend herself to it.\"\n\n\"There are bad women as well as bad men,\" the king said: \"and if\na woman thinks she has grievances she will often stick at nothing\nto obtain revenge.\"\n\n\"It is a well appointed troop,\" Archie said looking at the men,\nwho were drawn up in order, \"and not to be despised.  Their leader\nlooks an honest fellow; and if the lady means honestly it were\nchurlish indeed, to refuse her aid when she ventures to break with\nher family and to declare for Scotland. No; methinks that, with\nyour permission, I will run the risk, such as it may be, and will\njoin this band with my own. I will keep a sharp watch over them at\nthe first fight, and will see that they are so placed that, should\nthey mean treachery, they shall have but small opportunity of doing\nharm.\"\n\n\n\n\nChapter XIX\n\nThe Convent of St. Kenneth\n\n\nBruce, as the result of his successes, was now able to leave\nhis fastnesses and establish himself in the districts of Carrick,\nKyle, and Cunningham. Pembroke had established himself at Bothwell\nCastle, and sent a challenge to Bruce to meet him with his force at\nLoudon Hill.  Although his previous experience of such challenges\nwas unfortunate, Bruce accepted the offer. He had learned much\nsince the battle of Methven, and was not likely again to be caught\nasleep; on the 9th of May he assembled his forces at Loudon Hill.\n\nIt was but a small following. Douglas had brought 100 men\nfrom Douglasdale, and Archie Forbes had as many under his banner.\nBruce's own vassals had gathered 200 strong, and as many more of\nthe country people had joined; but in all, the Scotch force did\nnot exceed 600 men, almost entirely on foot and armed with spears.\nBruce at once reconnoitred the ground to discover a spot where his\nlittle force might best withstand the shock of Pembroke's chivalry.\nHe found that at one place near the hill the road crossed a level\nmeadow with deep morasses on either side. He strengthened the position\nwith trenches, and calmly awaited the approach of his enemy.  Upon\nthe following day Pembroke's army was seen approaching, numbering\n3000 knights and mounted men-at-arms, all in complete armour. They\nwere formed in two divisions. The battle was almost a repetition\nof that which had been fought by Wallace near the same spot.  The\nEnglish chivalry levelled their spears and charged with proud\nconfidence of their ability to sweep away the rabble of spearmen\nin front of them. Their flanks became entangled in the morasses;\ntheir centre tried in vain to break through the hedge of Scottish\nspears, and when they were in confusion, the king, his brother\nEdward, Douglas, Archie Forbes, and some twenty other mounted men\ndashed through a gap in the spearmen and fell upon them. The second\ndivision, seeing the first broken and in confusion, turned and took\nto flight at once, and Pembroke and his attendants rode, without\ndrawing rein, to Bothwell Castle.\n\nA few days later Bruce encountered and defeated Ralph de Monthermer,\nEarl of Gloucester, and compelled him to shut himself up in the\nCastle of Ayr.\n\nArchie Forbes was not present at the second battle, for upon the\nmorning after the fight at Loudon Hill he was aroused by his servant\nentering his tent.\n\n\"A messenger has just brought this,\" he said, handing him a small\npacket. \"He bids me tell you that the sender is a prisoner in the\nconvent of St. Kenneth, on Loch Leven, and prays your aid.\"\n\nArchie opened the packet and found within it the ring he had given\nto Marjory at Dunstaffnage. Without a moment's delay he hurried\nto the king and begged permission to leave him for a short time on\nurgent business, taking with him twenty of his retainers.\n\n\"What is your urgent business, Sir Archie?\" the king asked. \"A lady\nis in the case, I warrant me. Whenever a young knight has urgent\nbusiness, be sure that a lady is in question. Now mind, Sir Archie,\nI have, as I have told you, set my heart upon marrying you to\nMistress Mary Kerr, and so at once putting an end to a long feud\nand doubling your possessions. Her retainers fought well yesterday,\nand the least I can do to reward so splendid a damsel is to bestow\nupon her the hand of my bravest knight.\"\n\n\"I fear, sire,\" Archie said laughing, \"that she must be content\nwith another. There are plenty who will deem themselves well paid\nfor their services in your cause by the gift of the hand of so rich\nan heiress. But I must fain be excused; for as I told you, sire,\nwhen we were together in Rathlin Island, my heart was otherwise\nbestowed.\"\n\n\"What! to the niece of that malignant enemy of mine, Alexander of\nLorne?\" the king said laughing. \"Her friends would rather see you\non the gibbet than at the altar.\"\n\n\"I care nought for her friends,\" Archie said, \"if I can get herself.\nMy own lands are wide enough, and I need no dowry with my wife.\"\n\n\"I see you are hopeless,\" the king replied. \"Well, go, Archie; but\nwhatever be your errand, beware of the Lornes.  Remember I have\nscarce begun to win Scotland yet, and cannot spare you.\"\n\nA quarter of an hour later Archie, with twenty picked men, took\nhis way northward. Avoiding all towns and frequented roads, Archie\nmarched rapidly north to the point of Renfrew and crossed the Firth\nof Clyde by boat; then he kept north round the head of Loch Fyne,\nand avoiding Dalmally skirted the head of Loch Etive and the <DW72>s\nof Ben Nevis, and so came down on Loch Leven.\n\nThe convent stood at the extremity of a promontory jutting into the\nlake. The neck was very narrow, and across it were strong walls,\nwith a gate and flanking towers. Between this wall and the convent\nwas the garden where the inmates walked and enjoyed the air free\nfrom the sight of men, save, indeed, of fishers who might be passing\nin their boats.\n\nOutside the wall, on the shore of the lake, stood a large village;\nand here a strong body of the retainers of the convent were always\non guard, for at St. Kenneth were many of the daughters of Scotch\nnobles, sent there either to be out of the way during the troubles\nor to be educated by the nuns.  Although the terrors of sacrilege\nand the ban of the church might well deter any from laying hands\nupon the convent, yet even in those days of superstition some were\nfound so fierce and irreverent as to dare even the anger of the\nchurch to carry out their wishes; and the possession of some of these\nheiresses might well enable them to make good terms for themselves\nboth with the church and the relations of their captives. Therefore a\nnumber of the retainers were always under arms, a guard was placed\non the gate, and lookouts on the flanking towers--their duty\nbeing not only to watch the land side, but to shout orders to keep\nat a distance to any fisherman who might approach too closely to\nthe promontory.\n\nArchie left his party in the forest under the command of William\nOrr. He dressed himself as a mountaineer, and, accompanied by Cluny\nCampbell, and carrying a buck which they had shot in the forest,\nwent boldly down into the village.  He soon got into conversation\nwith an old fisherman, and offered to exchange the deer for dried\nfish. The bargain was quickly struck, and then Archie said:\n\n\"I have never been out on the lake, and would fain have a view of\nthe convent from the water. Will you take me and my brother out\nfor a row?\"\n\nThe fisherman, who had made a good bargain, at once assented, and\nrowed Archie and Cluny far out into the lake.\n\nAs they passed along at some distance Archie saw that the shore was\nin several places smooth and shelving, and that there would be no\ndifficulty in effecting a landing. He saw also that there were many\nclumps of trees and shrubs in the garden.\n\n\"And do the nuns and the ladies at the convent often walk there?\"\nhe asked the fisherman.\n\n\"Oh yes,\" he answered; \"of an evening as I come back from fishing\nI can see numbers of them walking there. When the vesper bell rings\nthey all go in. That is the chapel adjoining the convent on this\nside.\"\n\n\"It is a strong building,\" Archie said as when past the end of\nthe promontory they obtained a full view of it.  \"It is more like\na castle than a convent.\"\n\n\"It had need be strong,\" the old man said; \"for some of the\nrichest heiresses in Scotland are shut up there.  On the land side\nI believe there are no windows on the lower storey, and the door\nis said to be of solid iron.  The windows on that side are all\nstrongly barred; and he would have hard work, indeed, who wanted\nby force or stratagem to steal one of the pretty birds out of that\ncage.\"\n\nArchie had no idea of using force; and although he had been to some\nextent concerned in the breach of sanctuary at Dumfries, he would\nhave shrunk from the idea of violating the sanctuary of St. Kenneth.\nBut to his mind there was no breach whatever of that sanctuary in\naiding one kept there against her will to make her escape. Having\nascertained all that he wished to know, he bade the boatman return\nto shore.\n\n\"Keep a lookout for me,\" he said, \"for I may return in a few days\nwith another buck, and may bring a comrade or two with me who would\nlike an afternoon's fishing on the lake. I suppose you could lend\nme your boat and nets?\"\n\n\"Assuredly,\" the fisherman replied. \"You will not mind taking into\nconsideration the hire of the boat in agreeing for the weight of\nfish to be given for the stag?\"\n\nArchie nodded, secretly amused at the old man's covetousness, for\nhe knew that the weight of fish he had given him for the stag which\nhe had brought down was not one fourth the value of the meat.\n\nHe then returned with Cluny to the band. Some time before daybreak\nhe came down to the place again, and, entering the water quietly,\nat a distance from the promontory, swam noiselessly out, and landed\nat the garden, and there concealed himself in a clump of bushes.\nDaylight came. An hour later some of the nuns of the second order,\nwho belonged to poor families and acted as servants in the convent,\ncame out into the garden, and busied themselves with the cultivation\nof the flowers, vegetables, and herbs. Not till the afternoon did\nany of the other inmates appear; but at about four o'clock the\ngreat door of the convent opened, and a number of women and girls\nstreamed out. The former were all in nuns' attire, as were a few\nof the latter, but their garb was somewhat different from that of\nthe elder sisters; these were the novices.  The greater number,\nhowever, of the girls were dressed in ordinary attire, and were the\npupils of the convent. While the nuns walked quietly up and down\nor sat on benches and read, the pupils scattered in groups laughing\nand talking merrily together. Among these Archie looked eagerly\nfor Marjory. He felt sure that her imprisonment could be detention\nonly, and not rigorous seclusion. Presently he espied her. She\nwas walking with two of the nuns and three or four of the elder\nresidents at the convent, for many of these were past the age of\npupildom; and were there simply as a safe place of refuge during\ntroublous times. The conversation appeared to be an animated one.\nIt was not for some time that the group passed within hearing of\nArchie's place of concealment. Then Archie heard the voice of one\nof the nuns raised in anger:\n\n\"It is monstrous what you say, and it is presumptuous and wicked\nfor a young girl of eighteen to form opinions for herself. What\nshould we come to if every young woman were to venture to think and\njudge for herself? Discord and disorder would be wrought in every\nfamily. All your relations and friends are opposed to this sacrilegious\nmurderer, Robert Bruce. The church has solemnly banned him, and\nyet you venture to uphold his cause.\"\n\n\"But the Bishop of Glasgow,\" Marjory said, \"and many other good\nprelates of our church side with him, and surely they must be good\njudges whether his sins are unpardonable.\"\n\n\"Do not argue with me,\" the sister said angrily. \"I tell you this\nobstinacy will be permitted no longer. Had it not been that Alexander\nof Lorne begged that we would not be harsh with you, steps would\nlong since have been taken to bring you to reason; but we can no\nlonger permit this advocacy of rebellion, and the last unmaidenly\nstep which you took of setting at defiance your friends and relatives,\nand even of sending messages hence, must be punished. The abbess\nbade me reason with you and try and turn your obstinate will. Your\ncousins of Badenoch here have appealed to you in vain. This can no\nlonger be tolerated. The lady abbess bids me tell you that she gives\nyou three days to renounce the rebel opinions you have so frowardly\nheld, and to accept the husband whom your uncle and guardian has\nchosen for you, your cousin John of Lorne, his son. During that\ntime none will speak to you. If at the end of three days you are\nstill contumacious you will be confined to your cell on bread and\nwater until better thoughts come to you.\"\n\nWhile the conversation had been going on, the little group had\nhalted near the bushes, and they now turned away, leaving Marjory\nstanding by herself. The girl sat down on a bench close to where\nshe had been standing, exclaiming to herself as she did so, \"They\nmay shut me up as a prisoner for life, but I will never consent to\ntake sides against the cause of Scotland or to marry John of Lorne.\nOh! who is there?\" she exclaimed, starting suddenly to her feet as\na man's voice behind her said:\n\n\"Quite right, Mistress Marjory, well and bravely resolved; but pray\nsit down again, and assume an attitude of indifference.\"\n\n\"Who is it that speaks?\" the girl asked in a tremulous voice,\nresuming her seat.\n\n\"It is your true knight, lady, Archibald Forbes, who has come to\nrescue you from this captivity.\"\n\n\n\"But how can you rescue me?\" the girl asked after a long pause. \"Do\nyou know the consequences if you are found here within the bounds\nof the convent?\"\n\n\"I care nothing for the consequences,\" Archie said. \"I have in the\nwoods twenty stout followers. I propose tomorrow to be with three\nof them on the lake afishing.  If you, when the bell rings for your\nreturn in the evening, will enter that little copse by the side of\nthe lake, and will show yourself at the water's edge, we will row\nstraight in and take you off long ere the guards can come hither\nto hinder us. The lake is narrow, and we can reach the other side\nbefore any boat can overtake us. There my followers will be awaiting\nus, and we can escort you to a place of safety. It is fortunate\nthat you are ordered to be apart from the rest; none therefore will\nmark you as you linger behind when the bell rings for vespers.\"\n\nMarjory was silent for some time.\n\n\"But, Sir Knight,\" she said, \"whither am I to go? for of all my\nfriends not one, save the good priest, but is leagued against me.\"\n\n\"I can take you either to the Bishop of Glasgow, who is a friend of\nthe Bruce and whom I know well--he will, I am sure, take charge\nof you--or, if you will, lady, I can place you with my mother,\nwho will receive you as a daughter.\"\n\n\"But what,\" the girl said hesitatingly, \"will people say at my\nrunning away from a convent with a young knight?\"\n\n\"Let them say what they will,\" Archie said. \"All good Scots, when\nthey know that you have been in prison here solely from the love\nof your country, will applaud the deed; and should you prefer it,\nthe king will, I know, place you in charge of the wife of one of\nthe nobles who adheres to him, and will give you his protection\nand countenance. Think, lady, if you do not take this opportunity\nof gaining your freedom, it may never occur again, for if you are\nonce shut up in your cell, as I heard threatened, nothing save an\nattack by force of arms, which would be sheer sacrilege, can rescue\nyou from it. Surely,\" he urged, as the girl still remained silent,\n\"you can trust yourself with me. Do I not owe my life to you? and\nI swear that so long as you remain in my charge I will treat you\nas my sister in all honour and respect.\"\n\nFor some minutes the girl made no answer. At length she said,\nstanding up, and half turning toward the bushes:\n\n\"I will trust you, Sir Archie. I know you to be a brave and honourable\nknight, and I will trust you. I know 'tis a strange step to take,\nand the world will blame me; but what can I do? If I refuse your\noffer I shall be kept a prisoner here until I consent to marry John\nof Lorne, whom I hate, for he is as rough and cruel as his father,\nwithout the kindness of heart, which, save in his angry moments,\nthe latter has ever had toward me.  All my relations are against\nme, and struggle against my fate as I may, I must in the end bend\nto their will if I remain here. 'Tis a hard choice to make; but\nwhat can I do? Yes, I will trust to your honour; and may God and\nall the saints punish you if you are false to the trust! Tomorrow\nevening, as the vespers are chiming, I will be at the water's edge,\nbehind yonder clump of bushes.\"\n\nThen, with head bent down and slow steps, Marjory returned to\nthe convent, none addressing her as she passed through the groups\nof her companions, the order that she was to be shut out from the\nrest having been already issued.  Archie remained in his place of\nconcealment until the gardens were deserted and night had fallen.\nThen he left his hiding place, and, entering the lake, swam quietly\naway, and landed far beyond the village. An hour's walk brought\nhim to the encampment of his comrades.\n\nAt daybreak next morning the band, under the command of William\nOrr, started for their long march round the head of the lake to\nthe position which they were to take up on the opposite side facing\nthe convent, Archie choosing three of the number most accustomed\nto the handling of oars to remain with him. With these he set out\non a hunt as soon as the main body had left, and by midday had\nsucceeded in killing a stag.  With this swung on a pole carried by\nhis followers Archie proceeded to the village. He speedily found\nthe fisherman with whom he had before bargained.\n\n\"I did not expect you back again so soon,\" the old man said.\n\n\"We killed a buck this morning,\" Archie said carelessly, \"and my\nfriends thought that the afternoon would be fine for fishing.\"\n\n\"You can try if you like,\" the fisherman said, \"but I fear that\nyou will have but little sport. The day is too bright and clear,\nand the fish will be sulking at the bottom of the lake.\"\n\n\"We will try,\" Archie said, \"nevertheless. Even if the sport is\nbad it will be pleasant out on the lake, and if we catch nothing we\nwill get you to give us some fresh fish instead of dry. The folks\nin the hills will be no wiser, and it will not do for us to return\nempty handed.\"\n\nThe fisherman assented, and placed the oars and nets in the boat,\nand Archie and his companions entering rowed out into the middle\nof the lake, and then throwing over the nets busied themselves with\nfishing.\n\nAs the old man had predicted, their sport was but small, but this\nconcerned them little. Thinking that they might be watched, they\ncontinued steadily all the afternoon casting and drawing in the\nnets, until the sun neared the horizon.  Then they gathered the\nnets into the boat and rowed quietly towards the shore. Just as\nthey were abreast the end of the promontory the bell of the chapel\nbegan to ring the vespers.  A few more strokes and Archie could\nsee the clump of bushes.\n\n\"Row quietly now,\" he said, still steering toward the village.\n\nHe was about a hundred yards distant from the shore of the convent\ngarden. Just as he came abreast of the bushes the foliage was parted\nand Marjory appeared at the edge of the water. In an instant the\nboat's head was turned toward shore, and the three rowers bent to\nthe oars.\n\nA shout from the watchman on the turret showed that he had been\nwatching the boat and that this sudden change of its course had\nexcited his alarm. The shout was repeated again and again as the\nboat neared the shore, and just as the keel grated on the sand the\nouter gate was opened and some armed men were seen running into the\ngarden, but they were still two hundred yards away. Marjory leapt\nlightly into the boat; the men pushed off, and before the retainers\nof the convent reached the spot the boat was speeding away over the\nlake. Archie gave up to Marjory his seat in the stern, and himself\ntook an oar.\n\nLoch Leven, though of considerable length, is narrow, and the boat\nwas nearly a third of the way across it before two or three craft\nwere seen putting out from the village in pursuit, and although\nthese gained somewhat, the fugitives reached the other shore a long\ndistance in advance. William Orr and his men were at the landing\nplace, and soon the whole party were hurrying through the wood.\nThey had no fear of instant pursuit, for even in the fast gathering\ngloom those in the boats would have perceived the accession of\nforce which they had received on landing, and would not venture\nto follow. But before morning the news of the evasion would spread\nfar and wide, and there would be a hot pursuit among the mountains.\n\nScarce a word had been spoken in the boat. Marjory was pale and\nagitated, and Archie thought it best to leave her to herself. On\nthe way through the wood he kept beside her, assisting her over\nrough places, and occasionally saying a few encouraging words. When\ndarkness had completely set in three or four torches were lit, and\nthey continued their way until midnight. Several times Archie had\nproposed a halt, but Marjory insisted that she was perfectly able\nto continue her way for some time longer.\n\nAt midnight, however, he halted.\n\n\"We will stop here,\" he said. \"My men have been marching ever since\ndaybreak, and tomorrow we must journey fast and far. I propose that\nwe keep due east for some time and then along by Loch Rannoch, then\nacross the Grampians by the pass of Killiecrankie, when we can make\ndown to Perth, and so to Stirling.  The news of your escape will\nfly fast to the south, and the tracks to Tarbert and the Clyde\nwill all be watched; but if we start at daybreak we shall be far on\nour way east before they begin to search the hills here; and even\nif they think of our making in this direction, we shall be at\nKilliecrankie before they can cut us off.\"\n\n\n\n\nChapter XX\n\nThe Heiress of the Kerrs\n\n\nWhile Archie was speaking Marjory had sat down on a fallen tree. She\nhad not slept the night before, and had been anxious and agitated\nthe whole day. The excitement had kept her up; but she now felt\ncompletely worn out, and accepted without protest Archie's decision\nthat a halt must be made.\n\nThe men were already gathering sticks, and a bright fire soon blazed\nnear the spot where she had seated herself. Ere long some venison\nsteaks were broiled in the flames. At Archie's earnest request\nMarjory tried to eat, but could with difficulty swallow a few\nmorsels. A bower of green boughs was quickly made for her, and the\nground thickly piled with fresh bracken, and Marjory was in a very\nfew minutes sound asleep after the fatigue and excitement of the\nday.\n\nWith the first dawn of morning the men were on their feet. Fresh\nsticks were thrown on the fire and breakfast prepared, for the\nmarch would be a long and wearisome one.\n\n\"Breakfast is ready, Mistress Marjory,\" Archie said, approaching\nthe bower.\n\n\"And I am ready too,\" the girl said blithely as she appeared at\nthe entrance. \"The sleep has done wonders for me, and I feel brave\nand fresh again. I fear you must have thought me a terrible coward\nyesterday; but it all seemed so dreadful, such a wild and wicked\nthing to do, that I felt quite overwhelmed. Today you will find me\nready for anything.\"\n\n\"I could never think you a coward,\" Archie said, \"after you faced\nthe anger of that terrible uncle of yours for my sake; or rather,\"\nhe added, \"for the sake of your word. And now I hope you will eat\nsomething, for we have a long march through the forest and hills\nbefore us.\"\n\n\"Don't fear that I shall tire,\" she said. \"I am half a mountaineer\nmyself, and, methinks, can keep on my feet as long as any man.\"\n\nThe meal was hastily eaten, and then the party started on their\nway.\n\n\"I have been wondering,\" the girl said, as with light steps she\nkept pace with Archie's longer strides, \"how you came to know that\nI was in the convent.\"\n\nArchie looked surprised.\n\n\"How should I know, Mistress Marjory, but through your own messenger?\"\n\n\"My own messenger!\" Marjory exclaimed. \"You are jesting, Sir Archie.\"\n\n\"I am not so, fair lady,\" he said. \"Surely you must remember that\nyou sent a messenger to me, with word that you were captive at St.\nKenneth and needed my aid?\"\n\nThe girl stopped for a moment in her walk and gazed at her companion\nas if to assure herself that he was in earnest.  \"You must be surely\ndreaming, Sir Archie,\" she said, as she continued the walk, \"for\nassuredly I sent you no such message.\"\n\n\"But, lady,\" Archie said, holding out his hand, \"the messenger\nbrought me as token that he had come from you this ring which I\nhad given you, vowing that should you call me to your aid I would\ncome immediately, even from a stricken field.\"\n\nThe blood had rushed into the girl's face as she saw the ring.\nThen she turned very pale. \"Sir Archibald Forbes,\" she said in\na low tone, after walking for a minute or two in silence, \"I feel\ndisgraced in your eyes.  How forward and unmaidenly must you have\nthought me thus to take advantage of a vow made from the impulse\nof sudden gratitude.\"\n\n\"No, indeed, lady,\" Archie said hotly. \"No such thought ever entered\nmy mind. I should as soon doubt the holy Virgin herself as to deem\nyou capable of aught but what was sweet and womanly. The matter\nseemed to me simple enough.  You had saved my life at great peril\nto yourself, and it seemed but natural to me that in your trouble,\nhaving none others to befriend you, your thoughts should turn to\none who had sworn to be to the end of his life your faithful knight\nand servant.  But,\" he went on more lightly, \"since you yourself\ndid not send me the ring and message, what good fairy can have\nbrought them to me?\"\n\n\"The good fairy was a very bad one,\" the girl said shortly, \"and I\nwill rate him soundly when I see him for thus adventuring without\nmy consent. It is none other than Father Anselm; and yet,\" she\nadded, \"he has suffered so much on my behalf that I shall have to\nforgive him. After your escape my uncle in his passion was well\nnigh hanging the good priest in spite of his holy office, and drove\nhim from the castle. He kept me shut up in my room for many weeks,\nand then urged upon me the marriage with his son. When he found\nthat I would not listen to it he sent me to St. Kenneth, and there\nI have remained ever since. Three weeks ago Father Anselm came to\nsee me. He had been sent for by Alexander of Lorne, who, knowing\nthe influence he had with me, begged him to undertake the mission\nof inducing me to bend to his will. As he knew how much I hated\nJohn of Lorne, the good priest wasted not much time in entreaties;\nbut he warned me that it had been resolved that unless I gave way\nmy captivity, which had hitherto been easy and pleasant, would be\nmade hard and rigorous, and that I would be forced into accepting\nJohn of Lorne as a husband. When he saw that I was determined not\nto give in, the good priest certainly hinted\" (and here she \nagain hotly) \"that you would, if sent for, do your best to carry\nme off. Of course I refused to listen to the idea, and chided him\nfor suggesting so unmaidenly a course. He urged it no further, and\nI thought no more of the matter.  The next day I missed my ring,\nwhich, to avoid notice, I had worn on a little ribbon round my\nneck. I thought at the time the ribbon must have broken and the\nring been lost, and for a time I made diligent search in the garden\nfor it; but I doubt not now that the traitor priest, as I knelt\nbefore him to receive his blessing on parting, must have severed\nthe ribbon and stolen it.\"\n\n\"God bless him!\" Archie said fervently. \"Should he ever come to\nAberfilly the warmest corner by the fire, the fattest capon, and\nthe best stoop of wine from the cellar shall be his so long as\nhe lives. Why, but for him, Lady Marjory, you might have worn out\nmonths of your life in prison, and have been compelled at last to\nwed your cousin. I should have been a miserable man for life.\"\n\nThe girl laughed.\n\n\"I would have given you a week, Sir Archie, and no more; that\nis the extreme time which a knight in our days can be expected to\nmourn for the fairest lady; and now,\" she went on, changing the\nsubject, \"think you we shall reach the pass across the Grampians\nbefore night?\"\n\n\"If all goes well, lady, and your feet will carry you so far,\nwe shall be there by eventide. Unless by some chance encounter we\nneed have no fear whatever of pursuit.  It will have been daylight\nbefore the news of your flight fairly spread through the country,\nthough, doubtless, messengers were sent off at once in all directions;\nbut it would need an army to scour these woods, and as they know\nnot whether we have gone east, west, north, or south, the chance is\nfaint indeed of any party meeting us, especially as we have taken\nso straight a line that they must march without a pause in exactly\nthe right direction to come up with us.\"\n\nAt nightfall the party camped again on the <DW72> of the Grampians,\nand the following morning crossed by the pass of Killiecrankie and\nmade toward Perth.\n\nThe next night Marjory slept in a peasant's cottage, Archie and his\ncompanions lying down without.  Wishing to avoid attention, Archie\npurchased from the peasant the Sunday clothes of his daughter, who\nwas about the same age and size as Marjory.\n\nWhen they reached Perth he bought a strong horse, with saddle and\npillion; and with Marjory behind him, and his band accompanying\nhim on foot, he rode for Stirling. When he neared the town he heard\nthat the king was in the forest of Falkirk, and having consulted\nMarjory as to her wishes rode directly thither.\n\nBruce, with his followers, had arrived but the day before, and\nhad taken up his abode at the principal house of a village in the\nforest. He came to the door when he heard the trampling of a horse.\n\n\"Ah! Sir Archie, is it you safely returned, and, as I half expected,\na lady?\"\n\n\"This, sire,\" Archie said, dismounting, \"is Mistress Marjory\nMacDougall, of whom, as you have heard me say, I am the devoted\nknight and servant. She has been put in duress by Alexander of Lorne\nbecause in the first place she was a true Scots woman and favoured\nyour cause, and because in the second place she refused to espouse his\nson John.  I have borne her away from the convent of St. Kenneth,\nand as I used no force in doing so no sacrilege has been committed.\nI have brought her to you in all honour and courtesy, as I might a\ndear sister, and I now pray you to place her under the protection\nof the wife of one of your knights, seeing that she has no friends\nand natural protectors here. Then, when she has time to think, she\nmust herself decide upon her future.\"\n\nThe king assisted Marjory to dismount.\n\n\"Fair mistress,\" he said, \"Sir Archibald Forbes is one of the bravest\nand truest of my knights, and in the hands of none might you more\nconfidently place your honour. Assuredly I will do as he asks me,\nand will place you under the protection of Dame Elizabeth Graham,\nwho is now within, having ridden hither to see her husband but this\nmorning.  But I trust,\" he added, with a meaning smile, \"that you\nwill not long require her protection.\"\n\nThe king entered the house with Marjory, while Archie, with his\nband, rejoined the rest of his party, who were still with the king.\nAfter having seen that the wants of those who had accompanied him\nhad been supplied he returned to the royal quarters. The king met\nhim at the door, and said, with a merry smile on his face:\n\n\"I fear me, Sir Archie, that all my good advice with regard to\nMistress Mary Kerr has been wasted, and that you are resolved to\nmake this Highland damsel, the niece of my arch enemy Alexander of\nLorne, your wife.\"\n\n\"If she will have me,\" Archie said stoutly, \"such assuredly, is\nmy intent; but of that I know nothing, seeing that, while she was\nunder my protection, it would have been dishonourable to have spoken\nof love; and I know nought of her sentiments toward me, especially\nseeing that she herself did not, as I had hoped, send for me to come\nto her aid, and was indeed mightily indignant that another should\nhave done so in her name.\"\n\n\"Poor Sir Archie!\" the king laughed. \"Though a man, and a valorous\none in stature and in years, you are truly but a boy yet in these\nmatters. It needed but half an eye to see by the way she turned\npale and red when you spoke to her that she loves you. Now look\nyou, Sir Archie,\" he went on more seriously; \"these are troubled\ndays, and one knows not what a day may bring forth. Graham's tower\nis neither strong nor safe, and the sooner this Mistress Marjory\nof yours is safely in your stronghold of Aberfilly the better for\nboth of you, and for me also, for I know that you will be of no\nmore good to me so long as your brain is running on her. Look you\nnow, she is no longer under your protection, and your scruples on\nthat head are therefore removed; best go in at once and ask her\nif she will have you. If she says, 'Yes,' we will ride to Glasgow\ntomorrow or next day. The bishop shall marry you, and I myself will\ngive you your bonny bride.  This is no time for wasting weeks with\nmilliners and mantua makers. What say you?\"\n\n\"Nothing would more surely suit my wishes, sire,\" Archie said; \"but\nI fear she will think me presumptuous.\"\n\n\"Not a bit of it,\" the king laughed. \"Highland lassies are accustomed\nto sudden wooing, and I doubt not that when she freed you last\nautumn from Dunstaffnage her mind was just as much made up as yours\nis as to the state of her heart.  Come along, sir.\"\n\nSo saying, the king passed his arm through that of Archie, and\ndrew him into the house. In the room which they entered Marjory\nwas sitting with Lady Graham. Both rose as the king entered.\n\n\"My Lady Graham,\" the king said, \"this my good and faithful knight\nSir Archie Forbes, whose person as well as repute is favourably\nknown to you, desires to speak alone with the young lady under\nyour protection. I may say he does so at my special begging, seeing\nthat at times like these the sooner matters are put in a straight\ncourse the better.  Will you let me lead you to the next room while\nwe leave the young people together?\"\n\n\"Marjory,\" Archie said, when he and the girl were alone, \"I fear\nthat you will think my wooing rude and hasty, but the times must\nexcuse it. I would fain have waited that you might have seen more\nof me before I tried my fate; but in these troubled days who can\nsay where I may be a week hence, or when I can see you again were\nI once separated from you!  Therefore, dear, I speak at once. I\nlove you, Marjory, and since the day when you came like an angel\ninto my cell at Dunstaffnage I have known that I loved you, and\nshould I never see you again could love none other. Will you wed\nme, love?\"\n\n\"But the king tells me, Sir Archie,\" the girl said, looking up with\na half smile, \"that he wishes you to wed the Lady Mary Kerr.\"\n\n\"It is a dream of the good king,\" Archie said, laughing, \"and he\nis not in earnest about it. He knows that I have never set eyes on\nthe lady or she on me, and he was but jesting when he said so to\nyou, having known from me long ago that my heart was wholly yours.\"\n\n\"Besides,\" the girl said hesitating, \"you might have objected to\nwed Mistress Kerr because her father was an enemy of yours.\"\n\n\"Why dwell upon it?\" Archie said a little impatiently.  \"Mistress\nKerr is nothing in the world to me, and I had clean forgotten her\nvery existence, when by some freak or other she sent her retainers\nto fight under my command. She may be a sweet and good lady for what\nI know; she may be the reverse. To me she is absolutely nothing;\nand now, Marjory, give me my answer. I love you, dear, deeply and\ntruly; and should you say, 'Yes,' will strive all my life to make\nyou happy.\"\n\n\"One more question, Archie, and then I will answer yours. Tell me\nfrankly, had I been Mary Kerr instead of Marjory MacDougall, could\nyou so far forget the ancient feud between the families as to say\nto me, 'I love you.'\"\n\nArchie laughed.\n\n\"The question is easily answered. Were you your own dear self it\nwould matter nought to me were your name Kerr, or MacDougall, or\nComyn, or aught else. It is you I love, and your ancestors or your\nrelations matter to me not one single jot.\"\n\n\"Then I will answer you,\" the girl said, putting her hand in his.\n\"Archie Forbes, I love you with my whole heart, and have done\nso since I first met you; but,\" she said, drawing back, as Archie\nwould have clasped her in his arms, \"I must tell you that you have\nbeen mistaken, and that it is not Marjory MacDougall whom you would\nwed, but Mary, whom her uncle Alexander always called Marjory,\nKerr.\"\n\n\"Marjory Kerr!\" Archie repeated, in astonishment.\n\n\"Yes, Archie, Marjory or Mary Kerr. The mistake was none of my\nmaking; it was you called me MacDougall; and knowing that you had\nreason to hate my race I did not undeceive you, thinking you might\neven refuse the boon of life at the hands of a Kerr. But I believed\nthat when you thought it over afterwards you would suspect the\ntruth, seeing that it must assuredly come to your ears if you spoke\nof your adventure, even if you did not already know it, that Sir\nJohn Kerr and Alexander of Lorne married twin sisters of the house\nof Comyn. You are not angry, I hope, Archie?\"\n\n\"Angry!\" Archie said, taking the girl, who now yielded unresistingly,\nin his arms. \"It matters nothing to me who you were; and truly I\nam glad that the long feud between our houses will come to an end.\nMy conscience, too, pricked me somewhat when I heard that by the\ndeath of your brother you had succeeded to the estates, and that\nit was in despite of a woman, and she a loyal and true hearted\nScotswoman, that I was holding Aberfilly. So it was you sent the\nretainers from Ayr to me?\"\n\n\"Yes,\" Marjory replied. \"Father Anselm carried my orders to them.\nI longed to know that they were fighting for Scotland, and was sure\nthat under none could they be better led.\"\n\n\"And you have told the king who you are?\" Archie asked.\n\n\"Yes,\" the girl said, \"directly we entered.\"\n\n\"And you agree that we shall be married at once at Glasgow, as the\nking has suggested to me?\"\n\n\"The king said as much to me,\" Marjory said, colouring; \"but oh!\nArchie, it seems dreadful, such an unseemly bustle and haste, to\nbe betrothed one day and married the next!  Whoever heard of such\na thing?\"\n\n\"But the circumstances, Marjory, are exceptional. We all carry our\nlives in our hands, and things must be done which at another time\nwould seem strange. Besides, what advantage would there be in\nwaiting? I should be away fighting the English, and you would see\nno more of me. You would not get to know me better than you do\nnow.\"\n\n\"Oh! it is not that, Archie.\"\n\n\"Nor is it anything else,\" Archie said smiling, \"but just surprise.\nWith the King of Scotland to give you away and the Bishop of Glasgow\nto marry you, none can venture to hint that there is anything that\nis not in the highest degree orthodox in your marriage. Of course\nI shall have to be a great deal away until the war is over and\nScotland freed of her tyrants. But I shall know that you are safe\nat Aberfilly, which is quite secure from any sudden attack. You will\nhave my mother there to pet you and look after you in my absence,\nand I hope that good Father Anselm will soon find his way there and\ntake up his abode. It is the least he can do, seeing that, after\nall, he is responsible for our marriage, and having, as it were,\ndelivered you into my hands, ought to do his best to make you happy\nin your captivity.\"\n\nMarjory raised no further objection. She saw, in truth, that,\nhaving once accepted Archie Forbes as her husband, it was in every\nway the best plan for her to marry him without delay, since she had\nno natural protectors to go to, and her powerful relations might\nstir up the church to view her evasion from the convent as a defiance\nof its authority.\n\nUpon the following day the king moved with his force to Glasgow,\nwhich had already been evacuated by the English garrison, and\nthe next morning Marjory--for Archie through life insisted upon\ncalling her by the pet name under which he had first known her--was\nmarried to Sir Archibald Forbes. The Bruce gave her away, and\npresented her with a splendid necklet of pearls. His brother Edward,\nSir James Douglas, and other companions of Archie in the field also\nmade the bride handsome presents. Archie's followers from Aberfilly\nand the contingent from Marjory's estates in Ayr were also present,\ntogether with a crowd of the townspeople, for Archie Forbes, the\ncompanion of Wallace, was one of the most popular characters in\nScotland, and the good city of Glasgow made a fete of his marriage.\n\nSuddenly as it was arranged, a number of the daughters of the wealthiest\ncitizens attired in white attended the bride in procession to the\naltar. Flowers were strewn and the bride and bridegroom were heartily\ncheered by a concourse of people as they left the cathedral.\n\nThe party then mounted, and the king, his brother, Sir James Douglas,\nand some other knights, together with a strong escort, rode with\nthem to Aberfilly. Archie had despatched a messenger to his mother\nwith the news directly the arrangements had been made; and all\nwas prepared for their coming. The tenants had assembled to give a\nhearty welcome to their lord and new mistress. Dame Forbes received\nher as she alighted from the pillion on which she had ridden behind\nArchie, and embraced her tenderly.\n\nIt was the dearest wish of her life that Archie should marry; and\nalthough, when she first heard the news, she regretted in her heart\nthat he should have chosen a Kerr, still she saw that the union\nwould put an end to the long feud, and might even, in the event\nof the final defeat of Bruce, be the means of safety for Archie\nhimself and security for his possessions.\n\nShe soon, however, learned to love Marjory for herself, and to be\ncontented every way with her son's choice. There was high feasting\nand revelry at Aberfilly that evening. Bonfires were burned in the\ncastle yard, and the tenants feasted there, while the king and his\nknights were entertained in the hall of the castle.\n\nThe next morning the king and his companions again mounted and\nrode off. Sir James Douglas was going south to harry Galloway and\nto revenge the assaults which the people had made upon the king.\nThere was a strong English force there under Sir Ingram Umfraville\nand Sir John de St. John.\n\n\"I will give you a week, Sir Archie, to take holiday, but can spare\nyou no longer. We have as yet scarce begun our work, for well nigh\nevery fortress in Scotland is in English hands, and we must take as\nmany of them as we can before Edward moves across the Border again.\"\n\n\"I will not outstay the time,\" Sir Archie said. \"As we arranged\nlast night, I will march this day week with my retainers to join\nSir James Douglas in Galloway.\"\n\n\n\n\nChapter XXI\n\nThe Siege of Aberfilly\n\n\nPunctual to his agreement, Archie Forbes marched south with his\nretainers. He was loath, indeed, to leave Marjory, but he knew well\nthat a long time indeed must elapse before he could hope to settle\ndown quietly at home, and that it was urgent to hurry on the work\nat once before the English made another great effort to stamp out\nthe movement. Marjory did not attempt to induce him to overstay\nhis time. She was too proud of his position as one of the foremost\nknights of Scotland to say a word to detain him from the field.\nSo she bade him adieu with a brave face, reserving her tears until\nafter he had ridden away.\n\nIt had been arranged that Archie should operate independently\nof Douglas, the two joining their forces only when threatened\nby overwhelming numbers or when any great enterprise was to be\nundertaken. Archie took with him a hundred and fifty men from his\nestates in Lanark and Ayr.  He marched first to Loudon Hill, then\ndown through Cumnock and the border of Carrick into Galloway. Contrary\nto the usual custom, he enjoined his retainers on no account to\nburn or harry the villages and granges.\n\n\"The people,\" he said, \"are not responsible for the conduct of\ntheir lords, and as I would not see the English harrying the country\nround Aberfilly, so I am loath to carry fire and sword among these\npoor people. We have come hither to punish their lords and to capture\ntheir castles. If the country people oppose us we must needs fight\nthem; but beyond what is necessary for our provisions let us take\nnothing from them, and show them, by our conduct, that we hold\nthem to be Scotchmen like ourselves, and that we pity rather than\nblame them, inasmuch as by the orders of their lords they are forced\nto fight against us.\"\n\nArchie had not advanced more than a day's march into Galloway when\nhe heard that Sir John de St. John was marching with four hundred\nmen-at-arms to meet him.\n\nThere were no better soldiers in the following of Bruce than the\nretainers of Aberfilly and Glen Cairn. They had now for many years\nbeen frequently under arms, and were thoroughly trained to fight\ntogether. They had the greatest confidence in themselves and their\nleader, and having often with their spears withstood the shock\nof the English chivalry, Archie knew that he could rely upon them\nto the fullest. He therefore took up a position on the banks of\na river where a ford would enable the enemy to cross. Had he been\nless confident as to the result he would have defended the ford,\nwhich could be only crossed by two horsemen abreast. He determined,\nhowever, to repeat the maneuver which had proved so successful at\nStirling Bridge, and to let half of the enemy cross before he fell\nupon them.\n\nThe ground near the river was stony and rough. Great boulders,\nwhich had rolled from the hillside, were thickly scattered about\nit, and it would be difficult for cavalry to charge up the somewhat\nsteeply sloping ground in anything like unbroken order.\n\nWith eighty of his men Archie took up a position one hundred yards\nback from the stream. With great exertions some of the smaller\nboulders were removed, and rocks and stones were piled to make a\nwall on either flank of the ground, which, standing two deep, he\noccupied. The remaining seventy men he divided equally, placing one\ncompany under the command of each of his two faithful lieutenants,\nAndrew Macpherson and William Orr. These took post near the river,\none on each side of the ford, and at a distance of about one hundred\nyards therefrom. Orr's company were hidden among some bushes growing\nby the river. Macpherson's lay down among the stones and boulders,\nand were scarce likely to attract the attention of the English,\nwhich would naturally be fixed upon the little body drawn up to\noppose them in front.  The preparations were scarcely completed\nwhen the English were seen approaching. They made no halt at the\nriver, but at once commenced crossing at the ford, confident in\ntheir power to overwhelm the little body of Scots, whose number\nhad, it seemed to them, been exaggerated by the fears of the country\npeople. As soon as a hundred of the men-at-arms had passed, their\nleader marshalled them in line, and with level spears charged up\nthe <DW72>s against Archie's force. The great boulders broke their\nranks, and it was but in straggling order that they reached the\nnarrow line of Scottish spears.  These they in vain endeavoured to\nbreak through. Their numbers were of no avail to them, as, being\non horseback, but twenty men at a time could attack the double row\nof spearmen. While the conflict was at its height Archie's trumpet\nwas sounded, for he saw that another hundred men had now crossed\nthe ford.\n\nAt the signal the two hidden parties leapt to their feet, and with\nlevelled pikes rushed towards the ford. The English had no force\nthere to resist the attack, for as the men-at-arms had passed, each\nhad ridden on to join the fray in front. The head of the ford was\ntherefore seized with but little difficulty. Orr, with twenty men,\nremained here to hold it and prevent others from crossing, while\nMacpherson, with fifty, ran up the hill and fell upon the rear of\nthe confused masses of cavalry, who were striving in vain to break\nthe lines of Archie's spears.\n\nThe attack was decisive; the English, surprised and confused by\nthe sudden attack, were unable to offer any effectual resistance to\nMacpherson's pikemen, and at the same moment that these fell upon\nthe rear, Archie gave the word and his men rushed forward upon the\nstruggling mass of cavalry. The shock was irresistible; men and\nhorses fell in numbers under the Scottish spears, and in a few\nminutes those who could manage to extricate themselves from the\nstruggling mass rode off in various directions. These, however, were\nfew in number, for ninety were killed and seventy taken prisoners.\nSt. John himself succeeded in cutting his way through the spearmen,\nand, swimming the river below the ford, rejoined his followers,\nwho had in vain endeavoured to force the passage of the ford. With\nthese he rapidly retired.\n\nA detachment of fifty men were sent off with the prisoners to\nBruce, and Archie, with the main body of his followers, two days\nlater joined the force under Sir James Douglas.\n\nUpon the following morning a messenger from Aberfilly reached\nArchie.\n\n\"My lord,\" he said, \"I bring you a message from the Lady Marjory.\nI have spent five days in searching for you, and have never but\nonce laid down during that time, therefore do not blame me if my\nmessage is long in coming.\"\n\n\"What is it, Evan? nought is wrong there, I trust?\"\n\n\"The Lady Marjory bade me tell you that news has reached her, that\nfrom each of the garrisons of Ayr, Lanark, Stirling and Bothwell,\na force is marching toward your hold, which the governor of Bothwell\nhas sworn to destroy. When I left they were expected hourly in\nsight, and this is full a week since.\"\n\n\"Aberfilly can hold out for longer than that,\" Archie said, \"against\naught but surprise, and the vassals would have had time to gather.\"\n\n\"Yes,\" the man replied, \"they were flocking in when I came away; the\nmen of Glen Cairn had already arrived; all the women and children\nwere taking to the hills, according to the orders which you gave.\"\n\n\"And now, good Evan, do you eat some supper, and then rest. No\nwonder you have been so long in finding me, for I have been wandering\nwithout ceasing. I will start at once with my followers here for\nAberfilly; by tomorrow evening we will be there.\"\n\nArchie hurried to the hut occupied by Douglas, told him the news,\nand said he must hurry away to the defence of his castle.\n\n\"Go, by all means, Archie,\" Douglas replied. \"If I can gather a\nforce sufficient to relieve you I will myself march thither; but\nat present I fear that the chances of my doing so are small, for\nthe four garrisons you have named would be able to spare a force\nvastly larger than any with which I could meet them in the field,\nand the king is no better able to help you.\"\n\n\"I will do my best,\" Archie said. \"The castle can stand a stout\nsiege; and fortunately I have a secret passage by which we can\nescape.\"\n\n\"Never mind the castle,\" Douglas replied. \"When better days come\nwe will rebuild it again for you.\"\n\nA few notes on a horn brought Archie's little band of followers\ntogether. Telling them the danger which threatened Glen Cairn,\nArchie placed himself at their head, and at a rapid step they\nmarched away. It was five-and-forty miles across the hills, but\nbefore morning they approached it, and made their way to the wood in\nwhich was the entrance to the subterranean passage leading to the\ncastle. Archie had feared that they might find the massive doors\nwhich closed it, a short distance from the entrance, securely\nfastened as usual.  They were shut, indeed, but as they approached\nthem they heard a challenge from within.\n\n\"It is I, Sir Archie Forbes.\"\n\nThe door was opened at once. \"Welcome, Sir Archie!\" the guard said.\n\"The Lady Marjory has been expecting you for the last five days,\nand a watch has been kept here constantly, to open the doors should\nyou come.\"\n\n\"The messenger could not find me,\" Archie said. \"Is all well at\nthe castle?\"\n\n\"All is well,\" the man replied. \"The English have made two attacks,\nbut have been beaten back with loss.  This morning some great\nmachines have arrived from Stirling and have begun battering the\nwalls. Is it your will that I remain here on guard, now that you\nhave come?\"\n\n\"Yes,\" Archie answered. \"It were best that one should be always\nstationed here, seeing that the entrance might perchance be\ndiscovered by one wandering in the wood, or they might obtain the\nsecret of its existence from a prisoner. If footsteps are heard\napproaching retire at once with the news.  There is no danger if\nwe are warned in time, for we can turn the water from the moat into\nit.\"\n\nArchie and his followers now made their way along the passage until\nthey entered the castle. As they issued out from the entrance a\nshout of joy rose from those near, and the news rapidly flew through\nthe castle that Archie had arrived.  In a moment Marjory ran down\nand threw herself into his arms.\n\n\"Welcome back, Archie, a thousand times! I have been grievously\nanxious as the days went on and you did not return, and had feared\nthat some evil must have befallen you.  It has been a greater anxiety\nto me than the defence of the castle; but I have done my best to\nbe hopeful and bright, to keep up the spirits of our followers.\"\n\n\"It was no easy task for your messenger to find me, Marjory, for\nwe are ever on the move. Is my mother here?\"\n\n\"No, Archie, she went a fortnight since on a visit to Lady Gordon.\"\n\n\"It is well,\" Archie said, \"for if in the end we have to leave the\ncastle, you, who have proved yourself so strong and brave, can,\nif needs be, take to the hills with me; but she could not support\nthe fatigues of such a life. And now, dear, we have marched all\nnight and shall be glad of food; while it is preparing I will to\nthe walls and see what is going on.\"\n\nAs Archie reached the battlement a loud cheer broke from the\ndefenders gathered there, and Sandy Grahame hurried up to him.\n\n\"Welcome back, Sir Archie; glad am I to give up the responsibility\nof this post, although, indeed, it is not I who have been in command,\nbut Lady Marjory. She has been always on the walls, cheering the\nmen with her words and urging them to deeds of bravery; and, indeed,\nshe has frightened me sorely by the way in which she exposed herself\nwhere the arrows were flying most thickly, for as I told her over\nand over again, if the castle were taken I knew that you would be\nsure that I had done my best, but what excuse should I be able to\nmake to you if I had to bear you the news that she had been killed?\"\n\n\"And what did she say to that, Sandy?\"\n\n\"Truth, Sir Archie, she's a woman and wilful, and she just laughed\nand said that you would know you could not keep her in order\nyourself, and could not therefore expect me to rule her.\"\n\n\"That is so, Sandy,\" Archie laughed; \"but now that I am back I\nwill for once exert my authority, and will see that she runs into\nno further danger. And now, how goes the siege?\"\n\n\"So far they have done but little damage, Sir Archie; but the\nmachines which they brought up yesterday will, I fear, play havock\nwith our walls. They have not yet begun their work, for when they\nbrought them up yesterday afternoon our men shot so hotly that they\nhad to fall back again; but in the night they have thrown up high\nbanks of earth, and have planted the engines under their shelter,\nand will, ere long, begin to send their messengers against our\nwalls. Thrice they assaulted the works beyond the drawbridge and\ntwice we beat them back; but last night they came on with all their\nforce. I was myself there, and after fighting for a while and seeing\nthey were too strong for us, I thought it best to withdraw before\nthey gained footing in the work, and so had time to draw off the\nmen and raise the drawbridge.\"\n\n\"Quite right, Sandy! The defenders of the post would only have\nbeen slaughtered, and the assailants might have rushed across the\ndrawbridge before it could have been raised.  The post is of little\nimportance save to defend the castle against a sudden surprise, and\nwould only have been a source of constant anxiety and loss.  How\nmany do you reckon them?  Judging by their tents there must be\nthree or four thousand.\"\n\n\"About three thousand, Sir Archie, I make it; and as we had no time\nto get the tenants in from my lady's Ayrshire estate, we have but\ntwo hundred men in the castle, and many of these are scarce more\nthan boys.\"\n\n\"I have brought a hundred and fifty with me, Sandy, so we have as\nmany as we can use on the walls, though I could wish I had another\nhundred or two for sorties.\"\n\nHalf an hour later the great machines began to work, hurling vast\nstones with tremendous force against the castle wall. Strongly\nas this was built, Archie saw that it would ere many days crumble\nbefore the blows.\n\n\"I did not reckon on such machines as these,\" he said to Sandy.\n\"Doubtless they are some of the huge machines which King Edward\nhad constructed for the siege of Stirling, and which have remained\nthere since the castle was taken. Fortunately we have still the\nmoat when a breach is made, and it will be hard work to cross that.\"\n\nAll day the great stones thundered against the wall. The defenders\nwere not idle, but kept up a shower of arrows at the edge of the\nmound behind which the machines were hidden; but although many of\nthose working there were killed, fresh relays came constantly up,\nand the machines never ceased their work. By nightfall the face\nof the wall was bruised and battered. Many of the stones in front\nhad fallen from their places.\n\n\"Another twenty-four hours,\" Archie said to Marjory, as he joined\nher in the great hall, \"and the breach will be begun, forty-eight\nand it will be completed. They will go on all night, and we may\nexpect no rest until the work is done. In an hour's time I shall\nsally out from the passage into the wood and beat up their camp.\nExpecting no attack from the rear, we shall do them rare damage\nere they can gather to oppose us. As soon as they do so we shall\nbe off again, and, scattering in various directions, gather again\nin the wood and return here.\"\n\nAn hour later Archie, with two hundred men, started.  No sooner had\nhe left than Marjory called Sandy Grahame and Andrew Macpherson,\nwhom he had left in joint command during his absence.\n\n\"Now,\" she said, \"I am not going to remain quiet here while\nSir Archie does all the fighting, therefore do you gather all the\ngarrison together, leaving only twenty to hold the gate.  See that\nthe wheels of the drawbridge are well oiled, and the hinges of the\ngate. Directly we see that the attack has begun upon the camp we\nwill lower the drawbridge quietly, open the gates, and sally out.\nThere is no great force in the outer work. When we have cleared\nthat--which, if we are quick, we can do without alarming the\ncamp, seeing what a confusion and uproar will be going on there--we\nwill make straight along to the point where the machines are\nplaced. Let some of the men take axes and cut the ropes, and let\nothers carry <DW19>s well steeped in oil, we will pile them round\nthe machines and light them, and thus having ensured their destruction,\nwe will fall back again.\"\n\n\"But, Lady Marjory--\" Sandy began.\n\n\"I will have no buts, Sandy; you must just do as I order you, and\nI will answer to Sir Archie. I shall myself go forth with you and\nsee that the work is properly done.\"\n\nThe two men looked doubtfully at each other.\n\n\"Now, Andrew,\" Marjory said briskly, \"let us have no hesitation or\ntalk, the plan is a good one.\"\n\n\"I do not say that it is not a good one,\" Sandy replied cautiously,\n\"or that it is not one that Sir Archie might have carried out if\nhe had been here.\"\n\n\"Very well, Andrew, then that is quite enough. I give you the\norders and I am responsible, and if you and Sandy do not choose to\nobey me, I shall call the men together myself and lead them without\nyou.\"\n\nAs Sandy and Andrew were quite conscious that their lady would be\nas good as her word, they at once proceeded to carry her orders into\neffect. The wheels of the portcullis and drawbridge were oiled, as\nwere the bolts and hinges of the gate. The men were formed up in\nthe courtyard, where presently they were joined by Marjory who had\nput on a light steel cap and a shirt of mail, and who had armed herself\nwith a light sword. The men gathered round her enthusiastically,\nand would have burst into cheers had she not held up her hand to\ncommand silence.\n\n\"I will to the wall now,\" she said, \"to watch for the signal. The\ninstant the attack begins and the attention of those in the outwork\nis called that way, draw up the portcullis noiselessly and open\nthe gate, oil the hinges of the drawbridge and have everything\nin readiness; then I will join you. Let the drawbridge be lowered\nswiftly, and as it falls we will rush across. You have, I suppose,\ntold off the men who are to remain behind.  Tell them that when\nthe last of us have crossed they are to raise the drawbridge a few\nfeet, so that none can cross it until we return.\"\n\nThen, accompanied by Macpherson, she ascended the walls. All was\nquiet in the hostile camp, which was about a quarter of a mile\ndistant, and only the creaking of the wheels of the machines, the\norders of those directing them, and the dull crash as the great\nstones struck the wall, broke the stillness of the night. For half\nan hour they watched, and then a sudden uproar was heard in the\ncamp.  The Scottish war cry pealed out, followed by shouts and\nyells, and almost instantly flames were seen to mount up.\n\n\"My lord is at work,\" Marjory said, \"it is time for us to be doing\nalso.\" So saying she ran down to the courtyard. Sandy Grahame,\nMacpherson, and a few picked men took their place around her, then\nthe drawbridge was suddenly run down, and the Scots dashed across\nit. As Marjory had anticipated, the English in the outwork had\ngathered on the farther side and were watching the sudden outbreak\nin the camp.  Alarmed at the prospect of an attack, perhaps by the\nBruce, in that quarter, they were suddenly startled by the rush\nof feet across the drawbridge, and before they had time to recover\nfrom their surprise the Scots were upon them. The latter were\nsuperior in numbers, and the English, already alarmed by the attack\nupon their camp, offered but a feeble resistance. Many were cut\ndown, but the greater part leapt from the wall and fled towards\nthe camp. The moment resistance ceased the outer gate was thrown\nopen, and at full speed the Scotch made for the machines. The party\nhere had suspended their work and were gazing towards the camp,\nwhere the uproar was now great. The wind was blowing briskly and\nthe fire had spread with immense rapidity, and already half the\ncamp was in flames. Suddenly from the bank above the Scots poured\ndown upon them like a torrent. There was scarcely a thought of\nresistance. Stricken with dismay and astonishment at this unexpected\nattack, the soldiers working the machines fled hastily, only a few\nfalling beneath the swords of the Scots.  The men with axes at once\nfell upon the machines, cutting the ropes and smashing the wheels\nand levers which worked them, while those with the <DW19>s piled\nthem round. In less than two minutes the work was done, lighted\ntorches were applied to the <DW19>s, and the flames soon shot up\nhotly.\n\nThe Scots waited but a minute or two to see that the work was\nthoroughly done and that the flames had got fair hold, and then,\nkeeping in a close body, they retired to the castle. Not a soul\nwas met with by the way, and leaving Andrew Macpherson with fifty\nmen to hold the outwork until Archie should return and decide\nwhether it should be occupied, Marjory, with the rest, re-entered\nthe castle.\n\nShe at once ascended to the walls again, where Sandy also posted\nthe men to be in readiness to open fire with their arrows should\nthe English return and endeavour to extinguish the flames round\nthe machines.  The sound of fighting had ceased at the camp. By the\nlight of the flames numbers of the English could be seen pulling\ndown the tents which the fire had not yet reached and endeavouring\nto check the conflagration, while a large body of horse and foot\nwere rapidly advancing toward the castle.\n\nAs soon as they came within bowshot range the archers opened fire,\nand the English leaders, seeing that it was already too late to\nsave the machines, which were by this time completely enveloped in\nflames, and that men would only be sacrificed to no good purpose,\nhalted the troops. They then moved towards the outwork, but finding\nthis in possession of the Scots, they fell back again to the camp\nto take council as to the next steps to be adopted.  Archie's attack\nhad been crowned with complete success. Apprehending no danger\nfrom behind, the English had neglected to place sentries there,\nand the Scots were already among the tents before their presence\nwas discovered. Numbers of the English were cut down and the tents\nfired, and as soon as the English recovered from their first surprise\nand began to form, Archie gave the word for a retreat. This was\neffected without molestation, for the first thought of the English\nwas to save the camp from total destruction. The reports of the\nmen who escaped from the castle outwork and the outburst of flames\naround the machines added to the confusion which reigned, and the\nleaders, who had by the light of the flames ascertained that the\nassault upon the camp had been made by a small body of the enemy,\ndeemed it of the first importance to move at once to save the\nmachines if it were still possible.\n\nThe Scots regained the entrance to the passage without the loss\nof a single man, and passing through, soon re-entered the castle.\nMarjory had laid aside her warlike trappings and awaited her\nhusband's return at the inner entrance of the passage.\n\n\"We have had good success, Marjory,\" Archie said as he greeted\nher, \"as you will have seen from the walls.  The greater part of\nthe English camp is destroyed; we have killed great numbers, and\nhave not lost a man.\"\n\n\"That is good news indeed, Archie. We, too, have not been quite\nidle while you have been away.\"\n\n\"Why, what have you been doing, Marjory?\" Archie asked in surprise.\n\n\"Come up to the walls and I will show you.\"\n\nArchie mounted with her, and gave a start of surprise as he looked\ntowards the machines. The great body of fire had died down now, but\nthe beams of the machines stood up red and glowing, while a light\nflickering flame played round them.\n\n\"You see we have not been idle, Archie. We have destroyed the\nmachines, and retaken the outwork, which is now held by Andrew\nMacpherson with fifty men.\"\n\n\"Why, what magic is this, wife?\"\n\n\"No magic at all, Sir Knight. We have been carrying out the work\nwhich you, as a wise and skilful commander, should have ordered\nbefore you left. We have taken advantage of the confusion of the\nenemy by the fire in their camp, and have made a sortie, and a\nsuccessful one, as you see.\"\n\n\"I am delighted, indeed,\" Archie said; \"and the destruction of\nthose machines is indeed a great work. Still Sandy and Macpherson\nshould not have undertaken it without orders from me; they might\nhave been cut off and the castle stormed before I came back.\"\n\n\"They had orders from me, sir, and that was quite sufficient. To\ndo them justice, they hesitated about obeying me, and I was well\nnigh ordering them to the dungeon for disobedience; and they only\ngave way at last when I said they could stop at home if they liked,\nbut that I should lead out the retainers. Of course I went in your\nplace with armour and sword; but perhaps it was as well that I had\nno fighting to do.\"\n\n\"Do you mean, Marjory, that you really led the sortie?\"\n\n\"I don't think I led it, Archie; but I certainly went out with it,\nand very exciting it was. There, dear, don't look troubled.  Of\ncourse, as chatelaine of the castle, I was bound to animate my\nmen.\"\n\n\"You have done bravely and well, indeed, Marjory, and I am proud\nof my wife. Still, dear, I tremble at the thought of the risk you\nran.\"\n\n\"No more risk than you are constantly running, Archie; and I am\nrather glad you tremble, because in future you will understand my\nfeelings better, left here all alone while you are risking your\nlife perpetually with the king.\"\n\nThe success of the sally and the courage and energy shown by Marjory\nraised the spirits of the garrison to the highest pitch; and had\nArchie given the word they would have sallied out and fallen upon\nthe besiegers.  Two days later fresh machines arrived from Stirling,\nand the attack again commenced, the besiegers keeping a large body\nof men near the gate to prevent a repetition of the last sally.\nArchie now despatched two or three fleet footed runners through\nthe passage to find the king, and tell him that the besiegers were\nmaking progress, and to pray him to come to his assistance.  Two\ndays passed, and the breach was now fairly practicable, but the\nmoat, fifty feet wide, still barred the way to the besiegers. Archie\nhad noticed that for two or three days no water had come down from\nabove, and had no doubt that they had diverted the course of the\nriver. Upon the day after the breach was completed the besiegers\nadvanced in great force up the stream from below.\n\n\"They are going to try to cut the dam,\" Archie said to Sandy; \"place\nevery man who can draw a bow on that side of the castle.\"\n\nAs the English approached a rain of arrows was poured into them,\nbut covering themselves with their shields and with large mantlets\nformed of hurdles covered with hides they pressed forward to the\ndam. Here those who had brought with them picks and mattocks set\nto work upon the dam, the men with mantlets shielding them from\nthe storm of arrows, while numbers of archers opened fire upon the\ndefenders.  Very many were killed by the Scottish arrows, but the\nwork went on. A gap was made through the dam. The water, as it rushed\nthrough, aided the efforts of those at work; and after three hours'\nlabour and fighting the gap was so far deepened that the water in\nthe moat had fallen eight feet. Then, finding that this could now\nbe waded, the assailants desisted, and drew off to their camp.\n\nA council was held that evening in the castle as to whether\nthe hold should be abandoned at once or whether one attack on the\nbreach should be withstood. It was finally determined that the\nbreach should be held. The steep sides of the moat, exposed by the\nsubsidence of the water, were slippery and difficult. The force in\nthe castle was amply sufficient at once to man the breach and to\nfurnish archers for the walls on either side, while in the event\nof the worst, were the breach carried by the English, the defenders\nmight fall back to the central keep, and thence make their way\nthrough the passage. Had it not been for the possibility of an\nearly arrival of the king to their relief all agreed that it would\nbe as well to evacuate the castle at once, as this in the end must\nfall, and every life spent in its defence would thus be a useless\nsacrifice. As, however, troops might at any moment appear, it was\ndetermined to hold the castle until the last.\n\nThe next morning a party of knights in full defensive armour\ncame down to the edge of the moat to see whether passage could be\neffected. They were not molested while making their examination,\nas the Scottish arrows would only have dropped harmless off their\nsteel harness. Archie was on the walls.\n\n\"How like you the prospect, Sir Knights?\" he called out merrily.\n\"I fear that the sludge and slime will sully your bright armour and\nsmirch your plumes, for it will be difficult to hold a footing on\nthose muddy banks.\"\n\n\"It were best for you to yield, Sir Archibald Forbes, without giving\nus the trouble of making our way across your moat.  You have made\na stout resistance, and have done enough for honour, and you must\nsee that sooner or later we must win our way in.\"\n\n\"Then I would rather it should be later,\" Archie replied.  \"I\nmay have done enough for honour, but it is not for honour that I\nam fighting, but for Scotland. Your work is but begun yet, I can\nassure you. We are far from being at the end of our resources yet.\nIt will be time enough to talk about surrendering when you have\nwon the breach and the outer walls.\"\n\nThe knights retired; and as some hours passed without the besiegers\nseeing any preparation for an assault they judged that the report\ncarried back to camp was not an encouraging one. Large numbers of\nmen were, however, seen leaving the camp, and these toward sunset\ncame back staggering under immense loads of brushwood which they\nhad cut in the forest.\n\n\"They intend to fill up the moat,\" Archie said; \"it is their wisest\ncourse.\"\n\nHe at once directed his men to make up large trusses of straw, over\nwhich he poured considerable quantities of oil.  Early the next\nmorning the English drew out of their camp, and advanced in martial\narray. Each man carried a great <DW19>, and, covering themselves\nwith these as they came within bowshot, they marched down to the\nmoat. Each in turn threw in his <DW19>, and when he had done so\nreturned to the camp and brought back another. Rapidly the process\nof filling up the moat opposite to the breach continued. The besiegers\nkept up a rain of arrows and darts, and many of the English were\nkilled. But the work was continued without intermission until well\nnigh across the moat a broad crossway was formed level with the\nouter bank, but a narrow gap remained to be filled, and the English\nleaders advanced to the front to prevent the Scots on the breach\nrushing down to assault those placing the <DW19>s.\n\nSomewhat to the surprise of the English the defenders remained\nstationary, contenting themselves with hurling great stones at their\nbusy enemy. Suddenly there was a movement.  Archie and a party of\nhis best men dashed down the breach, and, climbing on the causeway,\nfor a moment drove the workers and their guards back. They were\nfollowed by twenty men carrying great trusses of straw. These were\npiled against the <DW19>s forming the end of the causeway. Archie\nand his band leapt back as a torch was applied to the straw. In a\nmoment the hot flames leapt up, causing the knights who had pressed\nafter the retreating Scots to fall back hastily. A shout of triumph\nrose from the garrison and one of dismay from the besiegers.\nSaturated with oil, the trusses burnt with fury, and the <DW19>s\nwere soon alight. A fresh wind was blowing, and the flames crept\nrapidly along the causeway.  In a few minutes this was in a blaze\nfrom end to end, and in half an hour nothing remained of the great\npile save charred ashes and the saturated <DW19>s which had been\nbelow the water in the moat, and which now floated upon it.\n\nThe besiegers had drawn off when they saw that the flames had\ngained a fair hold of the causeway. The smoke had scarcely ceased\nto rise when a great outcry arose from the English camp, and the\nlookout from the top of the keep perceived a strong force marching\ntoward it. By the bustle and confusion which reigned in the camp\nArchie doubted not that the newcomers were Scots. The garrison were\ninstantly called to arms.  The gates were thrown open, and leaving\na small body only to hold the gates, he sallied out at the head of\nhis men and marched toward the English camp.  At the approach of\nthe Scottish force the English leaders had marched out with their\nmen to oppose them. Bruce had been able to collect but three hundred\nand fifty men, and the English, seeing how small was the number\nadvancing against them, prepared to receive them boldly. Scarcely\nhad the combat begun when Archie with his band entered the English\ncamp, which was almost deserted. They at once fired the tents, and\nthen advanced in a solid mass with level spears against the rear\nof the English. These, dismayed at the destruction of their camp,\nand at finding themselves attacked both front and rear, lost heart\nand fell into confusion.  Their leaders strove to rally them,\nand dashed with their men-at-arms against the spearmen, but their\nefforts to break through were in vain, and their defeat increased\nthe panic of the footmen. Archie's party broke a way through their\ndisordered line and joined the body commanded by the king, and the\nwhole rushed so fiercely upon the English that these broke and fled\nin all directions, pursued by the triumphant Scots.\n\n\"I am but just in time I see, Sir Archie,\" Bruce said, pointing\nto the breach in the wall; \"a few hours more and methinks that I\nshould have been too late.\"\n\n\"We could have held out longer than that, sire,\" Archie replied.\n\"We have repulsed an attack this morning and burnt a causeway of\n<DW19>s upon which they attempted to cross the moat; still, I am\ntruly glad that you have arrived, and thank you with all my heart\nfor coming so speedily to my rescue, for sooner or later the hold\nmust have fallen; the great machines which they brought with them\nfrom Stirling proved too strong for the wall.\"\n\n\"And how has the Lady Marjory borne her during the siege?\" the king\ninquired.\n\n\"Right nobly,\" Archie replied; \"ever in good spirits and showing a\nbrave face to the men; and one night when I made a sortie through\nmy secret passage, and fell upon the English camp from the other\nside, having left the castle in her charge, she headed the garrison\nand issuing out, recaptured the outworks, and destroyed the machines\nby fire.\"\n\n\"Bravely done,\" the king said, \"and just what I should expect from\nyour wife. You did well to take my advice in that matter.\"\n\n\"We shall never agree there, sire, for as you know I followed my\nown will and wed the bride I had fixed upon for myself.\"\n\n\"Well, well, Sir Archie, as we are both satisfied we will e'en let\nit be; and now, I trust that you have still some supplies left,\nfor to tell you the truth I am hungry as well as weary, and my men\nhave marched fast and far.\"\n\n\"There is an abundance,\" Archie replied; \"to last them all for a\nmonth, and right willingly is it at their service.\"\n\nThe king remained a week at Aberfilly, his men aiding Archie's\nretainers in repairing the gap in the dam and in rebuilding the\nwall; and as five hundred men working willingly and well can effect\nwonders, by the time Bruce rode away the castle was restored to\nits former appearance. Archie marched on the following day, and\nrejoined Douglas in Galloway.\n\n\n\n\nChapter XXII\n\nA Prisoner\n\n\nAfter some consultation between the leaders, it was agreed to make\nan attempt to capture the castle of Knockbawn. It was known to\npossess a garrison of some sixty men only, and although strong,\nArchie and Sir James believed that it could be captured by assault.\nIt was arranged that Archie should ride to reconnoitre it, and\ntaking two mounted retainers he started, the force remaining in the\nforest some eight miles distant. The castle of Knockbawn stood on\na rocky promontory, jutting a hundred and fifty yards into the sea.\nWhen he neared the neck of the point, which was but some twenty\nyards wide, Archie bade his followers fall back a short distance.\n\n\"I will ride,\" he said, \"close up to the castle walls. My armour\nis good, and I care not for arrow or crossbow bolt. It were best\nyou fell back a little, for they may have horses and may sally out\nin pursuit. I am well mounted and fear not being overtaken, but it\nwere best that you should have a good start.\"\n\nArchie then rode forward toward the castle. Seeing a knight\napproaching alone the garrison judged that he was friendly, and it\nwas not until it was seen that instead of approaching the drawbridge\nhe turned aside and rode to the edge of the fosse, that they\nsuspected that he was a foe. Running to the walls they opened fire\nwith arrows upon him, but by this time Archie had seen all that\nhe required. Across the promontory ran a sort of fissure, some ten\nyards wide and as many deep. From the opposite edge of this the\nwall rose abruptly. Here assault would be difficult, and it was\nupon the gateway that an attack must be made. Several arrows had\nstruck his armour and glanced off, and Archie now turned and quietly\nrode away, his horse being protected by mail like himself. Scarce\nhad he turned when he saw a sight which caused him for a moment\nto draw rein. Coming at full gallop toward the promontory was a\nstrong body of English horse, flying the banner of Sir Ingram de\nUmfraville. They were already nearer to the end of the neck than\nhe was.  There was no mode of escape, and drawing his sword he\ngalloped at full speed to meet them. As he neared them Sir Ingram\nhimself, one of the doughtiest of Edward's knights, rode out with\nlevelled lance to meet him. At full gallop the knights charged\neach other. Sir Ingram's spear was pointed at the bars of Archie's\nhelmet, but as the horses met each other Archie with a blow of his\nsword cut off the head of the lance and dealt a tremendous backhanded\nblow upon Sir Ingram's helmet as the latter passed him, striking\nthe knight forward on to his horse's neck; then without pausing a\nmoment he dashed into the midst of the English ranks.\n\nThe horsemen closed around him, and although he cut down several\nwith his sweeping blows he was unable to break his way through them.\nSuch a conflict could not last long.  Archie received a blow from\nbehind which struck him from his horse. Regaining his feet he\ncontinued the fight, but the blows rained thick upon him, and he\nwas soon struck senseless to the ground.\n\nWhen he recovered he was in a room in the keep of the castle. Two\nknights were sitting at a table near the couch on which he was lying.\n\"Ah!\" exclaimed one, on seeing Archie open his eyes and move, \"I\nam glad to see your senses coming back to you, sir prisoner. Truly,\nsir, I regret that so brave a knight should have fallen into my\nhands, seeing that in this war we must needs send our prisoners\nto King Edward, whose treatment of them is not, I must e'en own,\ngentle; for indeed you fought like any paladin. I deemed not that\nthere was a knight in Scotland, save the Bruce himself, who could\nhave so borne himself; and never did I, Ingram de Umfraville, come\nnearer to losing my seat than I did from that backhanded blow you\ndealt me.  My head rings with it still. My helmet will never be\nfit to wear again, and as the leech said when plastering my head,\n'had not my skull been of the thickest, you had assuredly cut\nthrough it.' May I crave the name of so brave an antagonist?\"\n\n\"I am Sir Archibald Forbes,\" Archie replied.\n\n\"By St. Jago!\" the knight said, \"but I am sorry for it, seeing that,\nsave Bruce himself, there is none in the Scottish ranks against\nwhom King Edward is so bitter. In the days of Wallace there was no\none whose name was more often on our lips than that of Sir Archibald\nForbes, and now, under Bruce, it is ever coming to the front. I had\nthought to have asked Edward as a boon that I should have kept you\nas my prisoner until exchanged for one on our side, but being Sir\nArchibald Forbes I know that it were useless indeed; nevertheless,\nsir knight, I will send to King Edward, begging him to look mercifully\nupon your case, seeing how bravely and honourably you have fought.\"\n\n\"Thanks for your good offices, Sir Ingram,\" Archie replied, \"but\nI shall ask for no mercy for myself. I have never owed or paid him\nallegiance, but, as a true Scot, have fought for my country against\na foreign enemy.\"\n\n\"But King Edward does not hold himself to be a foreign enemy,\" the\nknight said, \"seeing that Baliol, your king, with Comyn and all\nyour great nobles, did homage to him as Lord Paramount of Scotland.\"\n\n\"It were an easy way,\" Archie rejoined, \"to gain a possession to\nnominate a puppet from among the nobles already your vassals, and\nthen to get him to do homage. No, sir knight, neither Comyn nor\nBaliol, nor any other of the Anglo-Norman nobles who hold estate\nin Scotland, have a right to speak for her, or to barter away her\nfreedom. That is what Wallace and thousands of Scotchmen have fought\nand died to protest against, and what Scotchmen will do until their\ncountry is free.\"\n\n\"It is not a question for me to argue upon,\" Sir Ingram said\nsurlily. \"King Edward bids me fight in Scotland, and as his knight\nand vassal I put on my harness without question.  But I own to\nyou that seeing I have fought beside him in Gascony, when he, as\na feudal vassal of the King of France, made war upon his lord, I\ncannot see that the offence is an unpardonable one when you Scotchmen\ndo the same here.  Concerning the lawfulness of his claim to be\nyour lord paramount, I own that I neither know nor care one jot.\nHowever, sir, I regret much that you have fallen into my hands,\nfor to Carlisle, where the king has long been lying, as you have\ndoubtless heard, grievously ill, I must forthwith send you. I must\nleave you here with the governor, for in half an hour I mount and\nride away with my troop. He will do his best to make your sojourn\nhere easy until such time as I may have an opportunity of sending\nyou by ship to Carlisle; and now farewell, sir,\" he said, giving\nArchie his hand, \"I regret that an unkind chance has thrown so\ngallant a knight into my hands, and that my duty to the king forbids\nme from letting you go free.\"\n\n\"Thanks, Sir Ingram,\" Archie replied. \"I have ever heard of you\nas a brave knight, and if this misfortune must fall upon me, would\nsooner that I should have been captured by you than by one of less\nfame and honour.\"\n\nThe governor now had a meal with some wine set before Archie, and\nthen left him alone.\n\n\"I am not at Carlisle yet,\" Archie said to himself. \"Unless\nI mistake, we shall have Sir James thundering at the gate before\nmorning. Cluny will assuredly have ridden off at full speed to carry\nthe news when he saw that I was cut off, and e'en now he will be\nmarching towards the castle.\" As he expected, Archie was roused\nbefore morning by a tremendous outburst of noise. Heavy blows were\ngiven, followed by a crash, which Archie judged to be the fall of\nthe drawbridge across the fosse. He guessed that some of Douglas's\nmen had crept forward noiselessly, had descended the fosse, and\nmanaged to climb up to the gate, and had then suddenly attacked\nwith their axes the chains of the drawbridge.\n\nA prodigious uproar raged in the castle. Orders were shouted, and\nthe garrison, aroused from their sleep, snatched up their arms\nand hastened to the walls. Outside rose the war cry, \"A Douglas! A\nDouglas!\" mingled with others of, \"Glen Cairn to the rescue!\" For\na few minutes all was confusion, then a light suddenly burst up\nand grew every instant more and more bright.\n\n\"Douglas has piled <DW19>s against the gates,\" Archie said to\nhimself. \"Another quarter of an hour and the castle will be his.\"\n\nThree or four minutes later the governor with six soldiers, two\nof whom bore torches, entered the room.  \"You must come along at\nonce, sir knight,\" the governor said. \"The attack is of the fiercest,\nand I know not whether we shall make head against it, but at any\nrate I must not risk your being recaptured, and must therefore\nplace you in a boat and send you off without delay to the castle\nat Port Patrick.\"\n\nIt was in vain for Archie to think of resistance, he was unarmed\nand helpless. Two of the soldiers laid hands on him and hurried\nhim along until they reached the lower chambers of the castle. The\ngovernor unlocked a door, and with one of the torch bearers led\nthe way down some narrow steps.  These were some fifty in number,\nand then a level passage ran along for some distance. Another door\nwas opened, and the fresh breeze blew upon them as they issued\nforth. They stood on some rocks at the foot of the promontory on\nwhich the castle stood. A large boat lay close at hand, drawn to\nthe shore. Archie and the six soldiers entered her; four of the\nlatter took the oars, and the others seated themselves by their\nprisoner, and then the boat rowed away, while the governor returned\nto aid in the defence of the castle.\n\nThe boat was but a quarter of a mile away when on the night air\ncame the sound of a wild outburst of triumphant shouts which told\nthat the Scots had won their way into the castle. With muttered\ncurses the men bent to their oars and every minute took them further\naway from Knockbawn.\n\nArchie was bitterly disappointed. He had reckoned confidently on\nthe efforts of Douglas to deliver him, and the possibility of his\nbeing sent off by sea had not entered his mind. It seemed to him\nnow that his fate was sealed. He had noticed on embarking that\nthere were no other boats lying at the foot of the promontory, and\npursuit would therefore be impossible.\n\nAfter rowing eight hours the party reached Port Patrick, where Archie\nwas delivered by the soldiers to the governor with a message from\ntheir commander saying that the prisoner, Sir Archibald Forbes,\nwas a captive of great importance, and was, by the orders of Sir\nIngram de Umfraville who had captured him, to be sent on to Carlisle\nto the king when a ship should be going thither. A fortnight passed\nbefore a vessel sailed. Archie was placed in irons and so securely\nguarded in his dungeon that escape was altogether impossible. So\nharsh was his confinement that he longed for the time when a vessel\nwould sail for Carlisle, even though he was sure that the same fate\nwhich had attended so many of Scotland's best and bravest knights\nawaited him there.\n\nThe winds were contrary, and the vessel was ten days upon the\nvoyage. Upon reaching Carlisle Archie was handed to the governor\nof the castle, and the next morning was conducted to the presence\nof the king himself. The aged monarch, in the last extremity of\nsickness, lay upon a couch. Several of his nobles stood around him.\n\n\"So,\" he said as the prisoner was brought before him, \"this is\nArchibald Forbes, the one companion of the traitor Wallace who has\nhitherto escaped my vengeance. So, young sir, you have ventured\nto brave my anger and to think yourself capable of coping with the\nLion of England.\"\n\n\"I have done my utmost, sir king,\" Archie said firmly, \"such as\nit was, for the freedom of my country. No traitor am I, nor was my\nleader Wallace. Nor he, nor I, ever took vow of allegiance to you,\nmaintaining ever that the kings of England had neither claim nor\nright over Scotland. He has been murdered, foully and dishonourably,\nas you will doubtless murder me, and as you have killed many nobler\nknights and gentlemen; but others will take our places, and so the\nfight will go on until Scotland is free.\"\n\n\"Scotland will never be free,\" the king said with angry vehemence.\n\"Rather than that, she shall cease to exist, and I will slay till\nthere is not one of Scottish blood, man, woman, or child, to bear\nthe name. Let him be taken to Berwick,\" he said; \"there let him be\nexposed for a week in a cage outside the castle, that the people\nmay see what sort of a man this is who matches himself against the\nmight of England. Then let him be hung, drawn, and quartered, his\nhead sent to London, and his limbs distributed between four Scotch\ncities.\"\n\n\"I go, sir king,\" Archie said, as the attendants advanced to seize\nhim, \"and at the end of the week I will meet you before the throne\nof God, for you, methinks, will have gone thither before me, and\nthere will I tax you with all your crimes, with the slaughter of\ntens of thousands of Scottish men, women, and children, with cities\ndestroyed and countries wasted, and with the murder in cold blood\nof a score of noble knights whose sole offence was that they fought\nfor their native country.\"\n\nWith these words Archie turned and walked proudly from the king's\npresence. An involuntary murmur of admiration at his fearless bearing\nescaped from the knights and nobles assembled round the couch of\nthe dying monarch.\n\nWhen, two days later, Archie entered the gates of Berwick Castle\nthe bells of the city were tolling, for a horseman had just ridden\nin with the news that Edward had expired on the evening before,\nbeing the 6th day of July, 1307, just at the moment when he was\non the point of starting with the great army he had assembled to\ncrush out the insurrection in Scotland.\n\nSo deep was his hate for the people who had dared to oppose his will\nthat when dying he called before him his eldest son, and in the\npresence of his barons caused him to swear upon the saints that so\nsoon as he should be dead his body should be boiled in a cauldron\nuntil the flesh should be separated from the bones, after which the\nflesh should be committed to the earth, but the bones preserved,\nand that, as often as the people of Scotland rebelled, the military\narray of the kingdom should be summoned and the bones carried at\nthe head of the army into Scotland. His heart he directed should\nbe conveyed to and deposited in the Holy Land.\n\nSo died Edward I, a champion of the Holy Sepulchre, King of England,\nLord of Ireland, Duke of Aquitaine, conqueror of Wales, and would\nbe conqueror of Scotland. In many respects his reign was a great\nand glorious one, for he was more than a great conqueror, he was,\nto England, a wise and noble king; and taken altogether he was\nperhaps the greatest of the Plantagenets.\n\nHistorians have striven to excuse and palliate his conduct toward\nScotland. They have glossed over his crimes and tried to explain\naway the records of his deeds of savage atrocity, and to show that\nhis claims to that kingdom, which had not a shadow of foundation\nsave from the submission of her Anglo-Norman nobles, almost all of\nwhom were his own vassals and owned estates in England, were just\nand righteous.  Such is not the true function of history. Edward's\nsole claim to Scotland was that he was determined to unite under his\nrule England, Scotland, Wales, and Ireland, and he failed because\nthe people of Scotland, deserted as they were by all their natural\nleaders, preferred death to such a slavery as that under which\nIreland and Wales helplessly groaned. His dying wishes were not\nobserved. His body was laid in rest in Westminster Abbey, and on\nthe tomb was inscribed, \"Edward I the mallet of the Scots.\"\n\n\n\n\nChapter XXIII\n\nThe Escape from Berwick\n\n\nOn entering the castle Archie was at once conducted to a sort of\ncage which had been constructed for a previous prisoner. On the\noutside of a small cell a framework of stout beams had been erected.\nIt was seven feet in height, six feet wide, and three feet deep.\nThe bars were four inches round, and six inches apart.  There was\na door leading into the cell behind. This was closed in the daytime,\nso that the prisoner remained in the cage in sight of passersby,\nbut at night the governor, who was a humane man, allowed the door\nto remain unlocked, so that the prisoner could enter the inner cell\nand lie down there.\n\nThe position of the cage was about twenty-five feet above the\nmoat. The moat itself was some forty feet wide, and a public path\nran along the other side, and people passing here had a full view\nof the prisoner.  There were still many of Scottish birth in the\ntown in spite of the efforts which Edward had made to convert it\ninto a complete English colony, and although the English were in\nthe majority, Archie was subject to but little insult or annoyance.\nAlthough for the present in English possession, Berwick had always\nbeen a Scotch town, and might yet again from the fortune of war\nfall into Scottish hands. Therefore even those most hostile to them\nfelt that it would be prudent to restrain from any demonstrations\nagainst the Scottish prisoners, since in the event of the city\nagain changing hands a bloody retaliation might be dealt them.\nOccasionally a passing boy would shout out an epithet of contempt\nor hatred or throw a stone at the prisoner, but such trifles were\nunheeded by him. More often men or women passing would stop and\ngaze up at him with pitying looks, and would go away wiping their\neyes.\n\nArchie, after the first careful examination of his cell, at once\nabandoned any idea of escape from it. The massive bars would have\ndefied the strength of twenty men, and he had no instrument of any\nsort with which he could cut them.  There was, he felt, nothing\nbefore him but death; and although he feared this little for\nhimself, he felt sad indeed as he thought of the grief of Marjory\nand his mother.\n\nThe days passed slowly. Five had gone without an incident, and but\ntwo remained, for he knew that there was no chance of any change\nin the sentence which Edward had passed, even were his son more\ndisposed than he toward merciful measures to the Scots, which Archie\nhad no warrant for supposing. The new king's time would be too\nclosely engaged in the affairs entailed by his accession to rank,\nthe arrangement of his father's funeral, and the details of the\narmy advancing against Scotland, to give a thought to the prisoner\nwhose fate had been determined by his father.\n\nAbsorbed in his own thoughts Archie seldom looked across the moat,\nand paid no heed to those who passed or who paused to look at him.\n\nOn the afternoon of the fifth day, however, his eye was caught by\ntwo women who were gazing up at the cage. It was the immobility of\ntheir attitude and the length of time which they continued to gaze\nat him, which attracted his attention.\n\nIn a moment he started violently and almost gave a cry, for in\none of them he recognized his wife, Marjory.  The instant that the\nwomen saw that he had observed them they turned away and walked\ncarelessly and slowly along the road.  Archie could hardly believe\nthat his eyesight had not deceived him. It seemed impossible that\nMarjory, whom he deemed a hundred miles away, in his castle at\nAberfilly, should be here in the town of Berwick, and yet when he\nthought it over he saw that it might well be so. There was indeed\nample time for her to have made the journey two or three times while\nhe had been lying in prison at Port Patrick awaiting a ship.  She\nwould be sure, when the news reached her of his capture, that he\nwould be taken to Edward at Carlisle, and that he would be either\nexecuted there or at Berwick. It was then by no means impossible,\nstrange and wondrous as it appeared to him, that Marjory should be\nin Berwick.\n\nShe was attired in the garment of a peasant woman of the better\nclass, such as the wife of a small crofter or farmer, and remembering\nhow she had saved his life before at Dunstaffnage, Archie felt that\nshe had come hither to try to rescue him.\n\nArchie's heart beat with delight and his eyes filled with tears at\nthe devotion and courage of Marjory, and for the first time since\nhe had been hurried into the boat on the night of his capture a\nfeeling of hope entered his breast.  Momentary as the glance had\nbeen which he had obtained of the face of Marjory's companion,\nArchie had perceived that it was in some way familiar to him. In\nvain he recalled the features of the various servants at Aberfilly,\nand those of the wives and daughters of the retainers of the estate;\nhe could not recognize the face of the woman accompanying Marjory\nas belonging to any of them. His wife might, indeed, have brought\nwith her some one from the estates at Ayr whom she had known from\na child, but in that case Archie could not account for his knowledge\nof her. This, however, did not occupy his mind many minutes; it\nwas assuredly one whom Marjory trusted, and that was sufficient\nfor him. Then his thoughts turned wholly to his wife.\n\nAny one who had noticed the prisoner's demeanor for the last few\ndays would have been struck with the change which had come over\nit. Hitherto he had stood often for hours leaning motionless, with\nhis arms crossed, in the corner of his cage, with head bent down and\nlistless air, his thoughts only being busy; now he paced restlessly\nup and down his narrow limits, two steps each way and then a turn,\nlike a caged beast; his hands were clenched, his breast heaved,\nhis breath came fast, his head was thrown back, often he brushed\nhis hand across his eyes, and rapid words came from his lips.\n\nThe sun sank. An hour later a jailer brought his jug of water and\npiece of bread, and then, without a word, retired, leaving, as usual,\nthe door into the cell open, but carefully locking and barring the\ninner door.  Archie had a longer walk now, from the front of the\ncage to the back of the cell, and for three hours he paced up and\ndown. Sometimes he paused and listened attentively. The sounds in\nthe town gradually died away and all became still, save that he\ncould hear the calls of the warder on the battlement above him.\nThe night was a very dark one and he could scarcely make out the\ngleam of water in the moat below.\n\nSuddenly something struck him a sharp blow on the face and fell at\nhis feet. He stooped and picked it up, it was an arrow with a wad\nof wool fastened round its point to prevent it from making a noise\nshould it strike the wall or cage; to the other end was attached a\npiece of string. Archie drew it in until he felt that it was held\nfirmly, then after a moment the hold relaxed somewhat, and the\nstring again yielded as he drew it. It was now, he felt, taut from\nthe other side of the moat. Presently a stout rope, amply sufficient\nto bear his weight, came into his hands. At the point of junction\nwas attached some object done up in flannel. This he opened, and\nfound that it was a fine saw and a small bottle containing oil. He\nfastened the rope securely to one of the bars and at once commenced\nto saw asunder one of the others. In five minutes two cuts had\nbeen noiselessly made, and a portion of the bar five feet long came\naway. He now tried the rope and found that it was tightly stretched,\nand evidently fixed to some object on the other side of the moat.\nHe grasped it firmly with his arms and legs and slid rapidly down\nit.\n\nIn another minute he was grasped by some strong arms which checked\nhis rapid progress and enabled him to gain his feet without the\nslightest noise. As he did so a woman threw her arms round him,\nand he exchanged a passionate but silent embrace with Marjory. Then\nshe took his hand and with noiseless steps they proceeded down the\nroad. He had before starting removed his shoes and put them in his\npockets.  Marjory and her companion had also removed their shoes,\nand even the keenest ears upon the battlements would have heard\nno sound as they proceeded along the road. Fifty yards farther and\nthey were among the houses. Here they stopped a minute and put on\ntheir shoes, and then continued their way. Not a word was spoken\nuntil they had traversed several streets and stopped at the door\nof a house in a quiet lane; it yielded to Marjory's touch, she and\nArchie entered, and their follower closed and fastened it after\nthem.\n\nThe moment this was done Marjory threw her arms round Archie's neck\nwith a burst of tears of joy and relief.  While Archie was soothing\nher the third person stirred up the embers on the hearth and threw\non a handful of dry wood.\n\n\"And who is your companion?\" Archie asked, after the first transports\nof joy and thankfulness were past.\n\n\"What! don't you recognize Cluny?\" Marjory asked, laughing through\nher tears.\n\n\"Cluny! of course,\" Archie exclaimed, grasping his follower's hand\nin his. \"I only caught a glimpse of your face and knew that it was\nfamiliar to me, but in vain tried to recall its owner. Why, Cluny,\nit is a long time since you went dressed as a girl into Ayr!  And\nso it is my good friend who had shared my wife's dangers.\"\n\n\"He has done more than that, Archie,\" Marjory said, \"for it was\nto him that I owe my first idea of coming here.  The moment after\nthe castle was taken and it was found that you had been carried\noff in a boat by the English, Cluny started to tell me the news.\nYour mother and I were beside ourselves with grief, and Cluny, to\ncomfort us, said, 'Do not despair yet, my lady; my lord shall not\nbe killed by the English if I can prevent it. The master and I\nhave been in a good many dangers, and have always come out of them\nsafe; it shall not be my fault if he does not slip through their\nhands yet.'  'Why, what can you do, Cluny?' I said. 'I don't know\nwhat I can do yet,' he replied; 'that must depend upon circumstances.\nMy lord is sure to be taken to Carlisle, and I shall go south to\nsee if I cannot get him out of prison. I have often gone among the\nEnglish garrisons disguised as a woman, and no one in Carlisle is\nlikely to ask me my business there.' It was plain to me at once that\nif Cluny could go to your aid, so could I, and I at once told him\nthat I should accompany him. Cluny raised all sorts of objections,\nbut to these I would not listen, but brought him to my will by saying,\nthat if he thought my being with him would add to his difficulties\nI would go alone, but that go I certainly would.  So without more\nado we got these dresses and made south.  We had a few narrow\nescapes of falling into the hands of parties of English, but at last\nwe crossed the frontier and made to Carlisle. Three days later we\nheard of your arrival, and the next morning all men were talking\nabout your defiance of the king, and that you had been sent to Berwick\nfor execution at the end of the week. So we journeyed hither and\ngot here the day after you arrived. The first step was to find\na Scotchwoman whom we might trust.  This, by great luck, we did,\nand Mary Martin, who lives in this house, is a true Scotchwoman,\nand will help us to the extent of her power; she is poor, for her\nhusband, who is an Englishman, had for some time been ill, and died\nbut yesterday. He was, by what she says, a hard man and cruel, and\nhis death is no grief to her, and Mary will, if she can, return\nwith her daughter to Roxburgh, where her relations live, and where\nshe married her husband, who was a soldier in the English garrison\nthere.\"\n\n\"But, Marjory,\" Archie said, \"have you thought how we are to escape\nhence; though I am free from the castle I am still within the walls\nof Berwick, and when, tomorrow, they find that I have escaped, they\nwill search every nook and corner of the town. I had best without\ndelay try and make my way over the walls.\"\n\n\"That was the plan Cluny and I first thought of,\" Marjory replied;\n\"but owing to the raids of the Douglas on the border, so strict\na watch is kept on the walls that it would be difficult indeed to\npass. Cluny has tried a dozen times each night, but the watch is\nso vigilant that he has each time failed to make his way past them,\nbut has been challenged and has had several arrows discharged at\nhim. The guard at the gates is extremely strict, and all carts that\npass in and out are searched. Could you have tried to pass before\nyour escape was known you might no doubt have done so in disguise,\nbut the alarm will be given before the gates are open in the morning,\nand your chance of passing through undetected then would be small\nindeed. The death of the man Martin suggested a plan to me.  I\nhave proposed it to his wife, and she has fallen in with it. I\nhave promised her a pension for her life should we succeed, but I\nbelieve she would have done it even without reward, for she is a\ntrue Scotchwoman. When she heard who it was that I was trying to\nrescue, she said at once she would risk anything to save the life\nof one of Scotland's best and bravest champions; while, on the other\nhand, she cares not enough for her husband to offer any objection\nto my plans for the disposal of his body.\"\n\n\"But what are your plans, Marjory?\"\n\n\"All the neighbours know that Martin is dead; they believe that Cluny\nis Mary's sister and I her niece, and she has told them that she\nshall return with us to Roxburgh. Martin was a native of a village\nfour miles hence, and she is going to bury him with his fathers\nthere. Now I have proposed to her that Martin shall be buried\nbeneath the wood store here, and that you shall take his place in\nthe coffin.\"\n\n\"It is a capital idea, Marjory,\" Archie said, \"and will assuredly\nsucceed if any plan can do so. The only fear is that the search\nwill be so hot in the morning that the soldiers may even insist\nupon looking into the coffin.\"\n\n\"We have thought of that,\" Marjory said, \"and dare not risk it.\nWe must expect every house to be searched in the morning, and have\nremoved some tiles in the attic. At daybreak you must creep out\non the roof, replace the tiles, and remain hidden there until the\nsearch is over.  Martin will be laid in the coffin. Thus, even\nshould they lift the lid, no harm will come of it. Directly they\nhave gone, Cluny will bring you down, and you and he dig the grave\nin the floor of the woodshed and place Martin there, then you\nwill take his place in the coffin, which will be placed in a cart\nalready hired, and Cluny, I, Mrs. Martin, and her daughter will\nthen set out with it.\"\n\nSoon after daybreak the quick strokes of the alarm bell at the\ncastle told the inhabitants of Berwick that a prisoner had escaped.\nArchie at once betook himself to his place of concealment on the\nroof. He replaced the tiles, and Cluny carefully obliterated all\nsigns of the place of exit from within.  A great hubbub had by\nthis time arisen in the street. Trumpets were blowing, and parties\nof soldiers moving about in all directions.  The gates remained\nunopened, orders being given that none should pass through without\na special order from the governor.\n\nThe sentries on the wall were doubled, and then a house to house\nsearch was commenced, every possible place of concealment being\nrummaged from basement to attic. Presently the searchers entered the\nlane in which Mrs. Martin lived.  The latch was ere long lifted,\nand a sergeant and six soldiers burst into the room.  The sight\nwhich they beheld quieted their first noisy exclamations.  Four\nwomen in deep mourning were kneeling by a rough coffin placed on\ntrestles. One of them gave a faint scream as they entered, and Mary\nMartin, rising to her feet, said:\n\n\"What means this rough intrusion?\"\n\n\"It means,\" the sergeant said, \"that a prisoner has escaped from\nthe castle, one Archibald Forbes, a pestilent Scotch traitor. He\nhas been aided by friends from without, and as the sentries were\nwatchful all night, he must be hidden somewhere in the town, and\nevery house is to be searched.\"\n\n\"You can search if you will,\" the woman said, resuming the position\non her knees. \"As you see, this is a house of mourning, seeing\nthat my husband is dead, and is today to be buried in his native\nvillage, three miles away.\"\n\n\"He won't be buried today,\" the sergeant said; \"for the gates are\nnot to be opened save by a special order from the governor. Now,\nlads,\" he went on, turning to the men, \"search the place from top\nto bottom, examine all the cupboards and sound the floors, turn over\nall the wood in the shed, and leave not a single place unsearched\nwhere a mouse could be hid.\"\n\nThe soldiers scattered through the house, and were soon heard\nknocking the scanty furniture about and sounding the floors and\nwalls. At last they returned saying that nothing was to be found.\n\n\"And now,\" the sergeant said, \"I must have a look in that coffin.\nWho knows but what the traitor Scot may be hid in there!\"\n\nMrs. Martin leaped to her feet.\n\n\"You shall not touch the coffin,\" she said; \"I will not have the\nremains of my husband disturbed.\" The sergeant pushed her roughly\naside, and with the end of his pike prised up the lid of the coffin,\nwhile Mrs.  Martin and the other three mourners screamed lustily\nand wrung their hands in the greatest grief at this desecration of\nthe dead.\n\nJust as the sergeant opened the coffin and satisfied himself that\na dead man really lay within, an officer, attracted by the screams,\nentered the room.\n\n\"What is this, sergeant?\" he asked angrily. \"The orders were to\nsearch the house, but none were given you to trouble the inmates.\"\n\nMrs. Martin began volubly to complain of the conduct of the soldiers\nin wrenching open the coffin.\n\n\"It was a necessary duty, my good woman,\" the officer said, \"seeing\nthat a living man might have been carried away instead of a dead\none; however, I see all is right.\"\n\n\"Oh, kind sir!\" Mrs. Martin said, sobbing, \"is it true what this\nman tells me, that there is no passage through the gates today? I\nhave hired a cart to take away my husband's body; the grave is dug,\nand the priest will be waiting. Kind sir, I pray of you to get me\na pass to sally out with it, together with my daughter, sister,\nand niece.\"\n\n\"Very well,\" the officer said kindly, \"I will do as you wish.  I\nshall be seeing the governor presently to make my report to him;\nand as I have myself seen the dead body can vouch that no ruse\nis intended. But assuredly no pass will be given for any man to\naccompany you; and the Scot, who is a head and shoulders taller\nthan any of you, would scarcely slip out in a woman's garment. When\nwill the cart be here?\"\n\n\"At noon,\" the woman replied.\n\n\"Very well; an hour before that time a soldier will bring out the\npass. Now, sergeant, have you searched the rest of the house?\"\n\n\"Yes, sir; thoroughly, and nothing suspicious has been found.\"\n\n\"Draw off your men, then, and proceed, with your search elsewhere.\"\n\nNo sooner had the officer and men departed than Cluny ran upstairs,\nand removing two of the tiles, whispered to Archie that all was\nclear. The hole was soon enlarged, and Archie re-entering, the pair\ndescended to the woodshed which adjoined the kitchen, and there,\nwith a spade and mattock which Cluny had purchased on the preceding\nday, they set to work to dig a grave. In two hours it was completed.\nThe body of John Martin was lowered into it, the earth replaced\nand trodden down hard, and the wood again piled on to it.\n\nAt eleven o'clock a soldier entered with the governor's pass\nordering the soldier at the gate to allow a cart with the body of\nJohn Martin, accompanied by four women, to pass out from the town.\n\nAt the appointed time the cart arrived. Archie now took his place\nin the coffin. His face was whitened, and a winding sheet wrapped\nround him, lest by an evil chance any should insist on again\nlooking into the coffin. Then some neighbours came in and assisted\nin placing the coffin in the cart. The driver took his place beside\nit, and the four women, with their hoods drawn over their heads,\nfell in behind it weeping bitterly.\n\nWhen they arrived at the gate the officer in charge carefully read\nthe order, and then gave the order for the gate to be opened. \"But\nstop,\" he said, \"this pass says nothing about a driver, and though\nthis man in no way resembles the description of the doughty Scot,\nyet as he is not named in the pass I cannot let him pass.\" There\nwas a moment's pause of consternation, and then Cluny said:\n\n\"Sister Mary, I will lead the horse. When all is in readiness, and\nthe priest waits, we cannot turn back on such a slight cause.\" As\nthe driver of the cart knew Mary Martin, he offered no objection,\nand descended from his seat. Cluny took the reins, and, walking by\nthe side of the horse's head, led him through the gates as these\nwere opened, the others following behind. As soon as they were\nthrough, the gate closed behind them, and they were safely out of\nthe town of Berwick.\n\nSo long as they were within sight of the walls they proceeded at\na slow pace without change of position, and although Cluny then\nquickened the steps of his horse, no other change was made until two\nmiles further they reached a wood.  Then Cluny leapt into the cart\nand wrenched off the lid of the coffin. It had been but lightly\nnailed down, and being but roughly made there were plenty of crevices\nthrough which the air could pass.\n\n\"Quick, Sir Archie!\" he said, \"let us get this thing out of the\ncart before any person happen to come along.\"\n\nThe coffin was lifted from the cart, and carried some short\ndistance into the wood. A few vigorous kicks separated the planks\nwhich composed it. These were taken and thrust separately among\nbushes at some little distance from each other. Cluny then unrolled\nthe bundle which he had brought from the cart, and handed to Archie\na suit of clothes fitted for a farmer. These Archie quickly put\non, then he returned to the cart, which he mounted, and took the\nreins. The others got up behind him and seated themselves on the\nstraw in the bottom of the cart. Then Archie gave the horse a smart\ncut with his whip, and the cart proceeded at a steady trot along\nthe road to the west.\n\n\n\n\nChapter XXIV\n\nThe Progress of the War\n\n\nA mile or two after leaving Berwick the cart had left the main road\nrunning by the coast through Dunbar to Edinburgh, and had struck\nwest by a country track.  But few houses were met with, as the\nwhole of the country within many miles of the sea had been harried\nand devastated by the various English armies which had advanced\nfrom Berwick. After proceeding for some miles they came to a point\nwhere the track they had been following terminated at a little hamlet\namong the hills. Here they left the cart, making an arrangement with\none of the villagers to drive it back on the morrow into Berwick.\nThey were now beyond all risk of pursuit, and need fear nothing\nfurther until they reached the great north roads running from\nCarlisle to Edinburgh and Stirling. Cluny therefore resumed male\nattire. They had no difficulty in purchasing a couple of swords\nfrom the peasants of the village, and armed with these they started\nwith Marjory and the two women over the hills. It was early autumn\nnow; the weather was magnificent, and they made the distance in\nquiet stages, and crossing the Pentlands came down upon Aberfilly\nwithout meeting with a single danger or obstacle.\n\nIt needs not to describe the joy of Archie's mother at his return.\nThe news spread like lightning among the tenantry, and in an hour\nafter the wayfarers reached the castle men and women could be seen\nflocking over the hills at the top of their speed to express their\ndelight and enthusiasm at their lord's return. By nightfall every\ntenant on the estate, save those prevented by age or illness, had\nassembled at the castle, and the rejoicings which had taken place\nat the marriage of their lord were but tame and quiet beside the\nboisterous enthusiasm which was now exhibited.\n\nAlthough Marjory had at first been welcomed for the sake of her\nhusband, the fact that she was a Kerr had excited a deep though\nhidden hostility to her in the minds both of those who had been\nher father's vassals at Aberfilly, and the old retainers of the\nForbeses at Glen Cairn. The devotion and courage which she had shown\nin the defence of the castle and in the enterprise for the rescue\nof their lord swept away every vestige of this feeling, and henceforth\nMarjory ranked in their affections with Archie himself, and there\nwas not a man upon the estate but felt that he could die for her\nif needs be.\n\nAfter a week's stay at home Archie rode away and joined the king,\ntaking, however, but four or five retainers with him.  Bruce received\nhim with extreme warmth. He had heard of his capture, and the news\nthat he was condemned to die at Berwick had also reached him, and\nhe had no doubt but Archie had shared the fate which had befallen\nhis own brothers and so many of his bravest friends. His pleasure,\ntherefore, equalled his surprise when his brave follower rode into\nhis camp. Many of Archie's friends assembled as soon as it was\nknown that he had arrived; and after the first greetings the king\nasked him for a recital of the means by which he had escaped from\nthe fate decreed him by Edward. Archie related the whole story,\nand at its conclusion the king called to his attendants to bring\ngoblets and wine.\n\n\"Sirs,\" he said, \"let us drink to the health of Mistress Marjory\nForbes, one of the bravest and truest of Scotch women. Would to\nHeaven that all the men of our country were animated by as noble and\ncourageous feelings! Our friend, Sir Archibald Forbes, has indeed\nwon a jewel, and I take no small credit to myself that I was the\nfirst who advised him to make Mistress Kerr his wife.\"\n\nThe toast was given with enthusiasm; but Archie afterwards protested\nagainst the king assuming any credit to himself in the matter, since,\nalthough it was true that he had advised him to marry Mistress Mary\nKerr, he had wished him to abandon, for her sake, Mistress Marjory,\nthe niece of Alexander MacDougall, who had set him free from her\nuncle's hold of Dunstaffnage.\n\n\"Now, Archie,\" the king said, when they were again alone together,\n\"I suppose, seeing that you have come hither without your following,\nthat you wish for a time to remain quiet at home, and seeing that\nyou have suffered severe imprisonment and a grievous risk of death\nin my cause, methinks you have well earned the right to rest quiet\nfor a while with your brave lady. At present I can dispense with\nthe services of your retainers. Most of the low country is now in\nmy hands, and the English garrisons dare not venture out of their\nstrong places. The army that the King of England collected to crush\nus has been, I hear, much disorganized by his death, and the barons\nwill doubtless wring concessions and privileges from his son before\nthey spread their banners to the wind again.  From all reports the\nnew king has but little of his father's ability and energy, and\nmonths may elapse before any serious effort is made against us.\nI am despatching my brother Edward to join Douglas in subduing\nGalloway, and during his absence I shall be content to remain here\nin the field with a small following, for the English governors\nof the towns will, methinks, stand only on the defensive, until a\nstrong army marches north from England. When Galloway is subdued\nthe lowlands will be all in my hands save for the English garrisons,\nand I shall on Edward's return set myself to punish the Comyns and\nthe other traitor nobles of the north, who are well nigh all hand\nand glove with the English. So long as Scotland has such powerful\nenemies in her midst she cannot hope to cope with the forces which\nEngland can send against her. Alone and united the task is one\nwhich will tax her strength to the utmost, seeing that England is\nin wealth and population so far her superior, and Edward disposes\nof the force of Ireland, of Wales, and of Gascony; therefore my\nfirst task must be to root out these traitor nobles from among us.\nWhen I move north I shall need your company and your strength; but\nuntil Edward has cleared the English out of Galloway, captured the\nstrongholds, and reduced it to obedience, you can stop in Aberfilly,\nand there at times, when I have no enterprise on hand and can take\na few days, I will come and rest if you will give me hospitality.\"\n\nSo until the following spring Archie Forbes remained quietly and\nmost happily at home. Several times the king came and stayed a few\ndays at Aberfilly, where he was safe against surprise and treachery.\nNot long after Archie's return home, Father Anselm arrived, to\nArchie's satisfaction and the great joy of Marjory, and took up\nhis abode there.\n\nIn the spring Archie, with his retainers, joined the king, who was\ngathering his army for his march into the north.  During the winter\nGalloway had been subdued, and Douglas being left in the south as\ncommander there, Edward Bruce joined his brother, around whom also\ngathered the Earl of Lennox, Sir Gilbert de la Haye, and others.\nThe position in Scotland was now singular: the whole of the\ncountry south of the Forth was favourable to Bruce, but the English\nheld Roxburgh, Jedburgh, Dumfries, Castle Douglas, Ayr, Bothwell,\nEdinburgh, Linlithgow, Stirling, and Dumbarton. North of the Forth\nnearly the whole of the country was hostile to the king, and the\nfortresses of Perth, Dundee, Forfar, Brechin, Aberdeen, Inverness,\nand many smaller holds, were occupied by English garrisons.\n\nThe centre of hostility to Bruce, north of the Forth, lay in the two\ngreat earls, the Comyns of Badenoch and Buchan, and their allies.\nBetween them and Bruce a hatred existed beyond that caused by\ntheir taking opposite sides. Comyn of Badenoch was the son of the\nman Bruce had slain at Dumfries, while Buchan hated him even more,\nsince his wife, the countess, had espoused the cause of Bruce and\nhad crowned him at Scone, and was now shamefully imprisoned in the\ncage at Berwick. It must be supposed that Buchan's anger against\nhis countess was as deep and implacable as that of Edward himself,\nfor, as the English king's most powerful ally in Scotland, he could\nsurely have obtained the pardon and release of his wife had he\ndesired it. On the other hand, Bruce had a private grudge against\nComyn, for upon him had been conferred Bruce's lordship of Annandale,\nand he had entered into possession and even occupied the family\ncastle of Lochmaben.\n\nThe king and his army marched north, and were joined by Alexander\nand Simon Frazer, with their followers. They marched to Inverness,\nwhich, with various other castles in the north, they captured. All of\nthese castles were, when taken, destroyed, as Bruce had determined\nto leave no strongholds in the land for the occupation of his\nenemies.  He himself could not spare men to hold them, and their\ncapture was useless if upon his retirement they could again be\noccupied by the enemy. Returning southward they were encountered\nby an army under Buchan, composed of his own retainers and a party\nof English. This force was completely defeated.\n\nTo the consternation of his followers Bruce was now attacked by a\nwasting illness, which so enfeebled him that he was unable to sit\non his horse; it was the result of the many privations and hardships\nwhich he had undergone since the fight at Methven. His brother,\nLennox, the Frazers, and Archie Forbes held a council and agreed\nthat rest for some time was absolutely necessary for the king, and\nthat sea air might be beneficial to him. They therefore resolved\nto move eastward to the Castle of Slaines, on the sea coast\nnear Peterhead. That such a step was attended by great peril they\nwell knew, for the Comyns would gather the whole strength of the\nHighlands, with accessions from the English garrisons, and besiege\nthem there. The king's health, however, was a paramount consideration;\nwere he to die, the blow might be fatal to Scotland, accordingly\nthe little force marched eastward. They reached Slaines without\ninterruption, and as they expected the castle was soon surrounded and\nbesieged by the forces of Buchan, who had been joined by Sir John\nMowbray and Sir David de Brechin, nephew of the King of England. For\nsome time the siege went on, but the assailants gained but little\nadvantage, and indeed trusted rather to famine than force to reduce\nthe castle.\n\nWeeks passed on, and although his followers thought that he was\nsomewhat better, the king's health improved but slowly. Provisions\nnow began to run very short. When they had come nearly to an end\nthe Scots determined to sally out and cut their way through the\nvastly superior strength of the enemy. The king was placed in a\nlitter, his mounted knights and followers surrounded him, and round\nthese the footmen formed a close clump of pikes; the hundred men\nfrom Aberfilly formed the front rank, as these could be best relied\nupon to withstand the charge of the English horse. The gates were\nthrown open, and in close ranks the garrison sallied out, forming,\nas soon as they passed through, in the order arranged. So close\nand serried was the hedge of spears, so quiet and determined the\nattitude of the men, that, numerous as they were, the men of Buchan\nand the English lords shrank from an encounter with such adversaries,\nand with the banner of the king and his knights flying in their\ncentre the little band marched on through the lines of the besiegers\nwithout the latter striking a blow to hinder their way.\n\nWithout interruption the royalists proceeded to Strathbogie. The\nsatisfaction of the king at the daring exploit by which he had been\nrescued from such imminent peril did more for him than medicine or\nchange of air, and to the joy of his followers he began to recover\nhis strength. He was then moved down to the river Don. Here Buchan\nand his English allies made a sudden attack upon his quarters,\nkilling some of the outposts. This attack roused the spirit and\nenergy of the king, and he immediately called for his war horse\nand armour and ordered his men to prepare for action. His followers\nremonstrated with him, but he declared that this attack by his\nenemies had cured him more speedily than medicine could have done,\nand heading his troops he issued forth and came upon the enemy\nnear Old Meldrum, where, after a desperate fight, Buchan and his\nconfederates were defeated with great slaughter on Christmas day,\n1307.  Buchan and Mowbray fled into England. Brechin took refuge\nin his own castle of Brechin, where he was afterwards besieged and\nforced to surrender.\n\nBruce now marched into the territory of Comyn, where he took a terrible\nvengeance for the long adhesion of his hated enemy to England. The\nwhole country was wasted with fire and sword, the people well nigh\nexterminated, and the very forests destroyed. So terrible was the\ndevastation that for generations afterwards men spoke of the harrying\nof Buchan as a terrible and exceptional act of vengeance.\n\nThe castle of Aberdeen was next invested. The English made great\nefforts for its succour, but the citizens joined Bruce, and a\nunited attack being made upon the castle it was taken by assault\nand razed to the ground. The king and his forces then moved into\nAngus. Here the English strongholds were all taken, the castle\nof Forfar being assaulted and carried by a leader who was called\nPhillip, a forester of Platane.  With the exception of Perth, the\nmost important fortress north of the Forth, and a few minor holds,\nthe whole of the north of Scotland, was now in the king's hands.\nIn the meantime Sir James Douglas, in the south, had again taken\nhis paternal castle and had razed it to the ground. The forests of\nSelkirk and Jedburgh, with the numerous fortresses of the district,\nwere brought under the king's authority, and the English were several\ntimes defeated. In the course of these adventures Sir James came\nacross Alexander Stewart, Thomas Randolph, the king's nephew,\nwho, after being taken prisoner at Methven, had joined the English\nparty, and Adam O'Gordon. They advanced with a much superior force\nto capture him, but were signally defeated.  O'Gordon escaped into\nEngland, but Stewart and Randolph were taken.\n\nThis was a fortunate capture, for Randolph afterwards became one of\nthe king's most valiant knights and the wisest of his counsellors.\nAfter this action Douglas marched north and joined the king. The\nlatter sternly reproached Randolph for having forsworn his allegiance\nand joined the English.  Randolph answered hotly and was committed\nby his uncle to solitary confinement, where he presently came to\na determination to renew his allegiance to Bruce, and henceforward\nfought faithfully and gallantly under him.\n\nGalloway had risen again, and Edward Bruce, with Sir Archie Forbes,\nwas detached to reduce it. It was a hard task, for the local\nchiefs were supported by Sir Ingram de Umfraville and Sir John de\nSt. John; these knights, with 1200 followers, met the Scots on the\nbanks of the Cree, which separates the countries of Kirkcudbright\nand Wigton, and although greatly superior in numbers, were completely\ndefeated by the Scottish pikemen, and compelled to take refuge in\nthe castle of Butele. Edward Bruce and Archie continued the task\nof subjugating the country; but St. John having retired to England,\nreturned with fifteen hundred men-at-arms, and with this strong force\nset out in pursuit of the small body of Scots, of whom he thought\nto make an easy capture. Then occurred one of the most singular and\nbrilliant feats of arms that took place in a war in which deeds of\ndaring abounded.  Edward Bruce having heard from the country people\nof the approach of his adversaries, placed his infantry in a strong\nposition, and then, with Archie Forbes and the fifty men-at-arms\nwho constituted his cavalry, went out to reconnoitre the approach\nof the English. The morning was thick and misty. Ignorant of each\nother's position, the two forces were in close vicinity, when the\nfog suddenly lifted, and Edward Bruce and Archie beheld close to\nthem the overwhelming force of St.  John, within bowshot distance.\nIt was too late to fly. Edward Bruce exclaimed to Archie:\n\n\"There is nothing for it but to charge them.\"\n\n\"Let us charge them,\" Archie replied.\n\nThe two leaders, setting spurs to their horses, and closely followed\nby their fifty retainers, dashed like a thunderbolt upon the mass\nof the English men-at-arms, before these, taken equally by surprise,\nhad time to form, and burst clean through them, overthrowing and\nslaying many, and causing the greatest confusion and surprise.\nRiding but a short distance on, the Scots turned, and again burst\nthrough the English lines.  Numbers of the English were slain,\nand many others turned rein. A third time the Scots charged, with\nequally fatal effect. The English were completely routed. Many\nwere killed and many taken prisoners, and the rest rode for England\nat their best speed. History scarcely recalls another instance of\n50 men routing in fair fight 1500. This extraordinary success was\nfollowed by a victory over Sir Roland of Galloway and Donald of\nthe Isles on the banks of the Dee, the Lord of the Isles being made\nprisoner; and eventually the whole country was reduced to obedience,\nwith the exception of one or two garrisons, no less than thirteen\ncastles being captured, in addition to the victories gained in the\nfield.\n\nGalloway being restored to order, Archie Forbes returned home, and\nremained for two or three months with his wife and mother. He was\nthen summoned by the king to join him again, as he was about to\nmarch to reduce the region over which his deadly foes Alexander\nand John of Lorne held sway.  The country into which the royal army\nnow penetrated was extremely mountainous and difficult, but they\nmade their way as far as the head of Loch Awe, where Alexander and\nJohn of Lorne, with 2000 men, were gathered to dispute the passage.\nThe position was an extremely strong one, and the Lornes were\nconfident that it could not be forced. Immediately to the north\nof the head of the lake rises the steep and lofty mountain Ben\nGruachan.  From the head of the lake flows the river Awe connecting\nit with Loch Etive, and the level space between the foot of the\nmountain and the river is only wide enough for two to ride abreast.\nThis passage was known as the Pass of Brander, and the Lornes might\nwell believe that their position was unassailable.\n\nBefore advancing into the pass Bruce detached Douglas, with Sir\nAlexander Frazer, Sir William Wiseman, and Sir Andrew Grey, with\na body of lightly armed infantry and archers. These, unnoticed by\nthe enemy, climbed the side of the mountain, and going far up it,\npassed along until they got behind and above the enemy. The king\nordered his main body to lay aside all defensive armour so that\nthey could more easily climb the hill and come to a hand to hand\nconflict with the enemy. Then he moved along towards the narrow\npass.  As they approached it the men of Lorne hurled down a torrent\nof rocks from the hillside above.\n\nWith a few heavy armed men Bruce pushed forward by the water side,\nwhile Archie Forbes led the main body up the hillside. The climb was\nstiff and difficult, and many were swept down by the rocks hurled\nby the enemy; but at last they came to close quarters with the foe,\nand a desperate struggle ensued.\n\nIn the meantime Douglas and his party had attacked the defenders\nfrom the other side, at first showering arrows among them, and\nthen falling upon them with sword and battleaxe. Thus attacked in\nfront and rear, the men of Lorne lost heart and gave way. On both\nsides the royalists pressed them hotly, and at last they broke\nfrom the hillside and fled down to the river, intending to cross\nby a wooden bridge and destroy it behind them, but before many had\npassed Douglas with his followers arrived upon the spot and seized\nthe bridge, cutting off their retreat. Great numbers of the men of\nLorne were slain, and the survivors made their escape up the mountain\nside again. The Lornes themselves were on board some galleys on\nLoch Awe, their intention having been to land in Bruce's rear when\nhe was fairly entangled in the narrow pass.  On witnessing the utter\ndiscomfiture of their followers they rowed rapidly away, and landed\nfar down the lake.  Alexander fled to England, where he ended his\nlife.\n\nBruce now advanced through the country of Lorne, which, having\nnever suffered from the English raids that had over and over again\ndevastated the rest of Scotland, was rich and flourishing, and large\nquantities of booty were obtained.  Dunstaffnage was besieged and\ncaptured, and having received hostages from all the minor chiefs\nfor their good behaviour the king and his army returned to Glasgow.\n\nIn the following spring a truce was negotiated by the intervention\nof the King of France between the belligerents; but its duration was\nbut short, for so long as English nobles held estates and occupied\ncastles in Scotland breaches of the peace would be constantly\noccurring. Bruce besieged the castle of Rutherglen, near Glasgow;\nbut Edward despatched the Earl of Gloucester to raise the siege,\nand as Bruce's army was still small he was forced to retire at his\napproach.\n\nIn February, 1309, the clergy of Scotland assembled in a provincial\ncouncil at Dundee, and issued a declaration in favour of Bruce\nas lawful king of Scotland. In this document they set forth that\nalthough Baliol was made king of Scotland by the King of England,\nBruce, the grandfather of the king, was always recognized by the\npeople as being nearest in right; and they said: \"If any one, on\nthe contrary, claim right to the aforesaid kingdom in virtue of\nletters in time passed sealed, and containing the consent of the\npeople and the commons, know ye that all this took place in fact\nby force and violence, which could not at the time be resisted,\nand through multiplied fears, bodily tortures, and various terrors.\"\n\nThis document was sealed by all the bishops, as representing the\nclergy. A similar document was drawn up and signed by the estates\nof Scotland. Therefore, henceforth Bruce could claim to be the king\nnot only as crowned and by right, but by the approval and consent\nof the clergy and people of Scotland. A few months afterwards James,\nthe Steward of Scotland, whose course had ever been vacillating, died,\nand his son Walter, a loyal Scotsman, succeeded him. He afterwards\nmarried the king's daughter Marjory, and became the founder of the\nroyal line of Stuart.\n\n\n\n\nChapter XXV\n\nThe Capture of a Stronghold\n\n\nWhile Bruce had by his energy and courage been wresting Scotland,\nstep by step, from the English, no serious effort had been made by\nthe latter to check his progress. Small bodies of troops had from\ntime to time been sent from the north; but the king had made no\ngreat efforts, like those of his father, to reduce the country to\nobedience by the exercise of the whole strength of England.  Edward\nII differed widely from his father in disposition. At times he was\nroused to fits of spasmodic energy, but for the most part he was\nsunk in sloth and supineness. He angered and irritated his barons\nby his fondness for unworthy favourites, and was engaged in constant\nbroils with them.\n\nSo called governors of Scotland were frequently appointed and as\noften superseded, but no effectual aid was given them to enable\nthem to check the ever spreading insurrection.  But Perth was now\nthreatened by Bruce; and the danger of this, the strongest and most\nimportant northern fortress, roused Edward from his lethargy.  A\nfleet was fitted out for the Tay. Troops, under the Earl of Ulster,\nwere engaged to be transported by an English fleet of forty ships,\nsupplied by the seaports, and intended to cooperate with John of\nLorne in the west. Edward himself, with a powerful army, accompanied\nby the Lords Gloucester, Warrenne, Percy, Clifford, and others,\nadvanced into Scotland as far as Renfrew.  Bruce could oppose no\neffectual resistance in the field to so large a force, but he used\nthe tactics which Wallace had adopted with such success. The country\nthrough which the English were advancing was wasted. Flocks and\nherds were driven off, and all stores of grain burned and destroyed.\nHis adherents, each with their own retainers, hung upon the skirts\nof the English army, cutting off small parties, driving back bodies\ngoing out in search of provisions or forage, making sudden night\nattacks, and keeping the English in a state of constant watchfulness\nand alarm, but always retiring on the approach of any strong force,\nand avoiding every effort of the English to bring on an engagement.\n\nThe invaders were soon pressed by want of provisions, and horses\ndied from lack of forage. The great army was therefore obliged to\nfall back to Berwick without having struck a single effective blow.\nAfter this Edward remained inactive at Berwick for eight months,\nsave that he once again crossed the Border and advanced as far as\nRoxburgh, but only to retreat without having accomplished anything.\nThe Earls of Gloucester and Warrenne reduced the forest of Selkirk\nand the district, and restored the English power there; while the\nking's favourite, Piers Gaveston, Earl of Cornwall, went by sea to\nPerth and tried to reduce the surrounding country, but the Scotch,\nas usual, retired before him, and he, too, after a time, returned\nto Berwick. The efforts of the defenders to starve out the invading\narmies of England were greatly aided by the fact that at this time\na great famine raged both in England and Scotland, and the people\nof both countries were reduced to a condition of want and suffering.\nNot only did the harvest fail, but disease swept away vast numbers\nof cattle and sheep, and in many places the people were forced to\nsubsist upon the flesh of horses, dogs, and other animals.\n\nDuring the years which had elapsed since the battle of Methven,\nBruce had never been enabled to collect a force in any way worthy\nof the name of an army. His enterprises had been a succession of\ndaring feats performed by small bodies of men. Even now, when the\nnobles dared no longer openly oppose him, they remained sullenly\naloof, and the captures of the English strongholds were performed\neither by the king or his brother Edward, with their retainers from\nAnnandale and Carrick; by Douglas with the men of Douglasdale; or\nby some simple knights like Archie Forbes, the Frazers, Boyle, and\na few others, each leading their own retainers in the field.  The\ngreat mass of the people still held aloof, and neither town nor\ncountry sent their contingents to his aid. This was not to be wondered\nat, so fearfully had all suffered from the wholesale vengeance of\nEdward after the battle of Falkirk.\n\nGreat successes had certainly attended Bruce, but these had been\nrendered possible only by the absence of any great effort on the\npart of England, and all believed that sooner or later Edward would\narouse himself, and with the whole strength of England, Ireland,\nand Wales again crush out the movement, and carry fire and sword\nthrough Scotland. Still the national spirit was rising.\n\nArchie Forbes divided his time pretty equally between the field and\nhome, never taking with him, when he joined the king, more than a\nthird of the entire strength of his retainers; thus all had time\nto attend to their farms and the wants of their families, and\ncheerfully yielded obedience to the call to arms when the time\ncame.\n\nOne day while the king was stopping for a few days' rest at Aberfilly,\na horseman rode in.\n\n\"I have great news, sire,\" he said. \"Linlithgow has been captured\nfrom the English.\"\n\n\"That were good news indeed,\" the king said; \"but it can scarce be\npossible, seeing that we have no men-at-arms in the neighbourhood.\"\n\n\"It has been done by no men-at-arms, my liege,\" the messenger said;\n\"but as Forfar was taken by Phillip the Forester and his mates,\nso has Linlithgow been captured by a farmer and his comrades, one\nWilliam Bunnock.\"\n\nIt was indeed true. The castle of Linlithgow, forming as it did\na link between the two strongholds of Edinburgh and Stirling, was\na place of great importance and was strongly garrisoned by the\nEnglish.  Naturally the whole country round suffered severely from\nthe oppressions of the garrison, who supplied themselves by force\nwith such provisions and stores as were needful for them. Payment\nwas of course made to some extent, as the country otherwise would\nspeedily have been deserted and the land left untilled; but there\nwas almost necessarily much oppression and high handedness.  Bunnock,\nhearing of the numerous castles which had been captured by the\nking and his friends with mere handfuls of followers, determined at\nlast upon an attempt to expel the garrison of Linlithgow. He went\nabout among his friends and neighbours, and found many ready to\njoin his enterprise.  These one night placed themselves in ambush\namong some bushes hard by the castle gate. Bunnock himself concealed\neight chosen men with arms in a wagon of hay.  The horses were\ndriven by a stout peasant with a short hatchet under his belt,\nwhile Bunnock walked carelessly beside the wagon. As he was in the\nhabit of supplying the garrison with corn and forage, the gate was\nreadily opened on his approach. As soon as the wagon was exactly\nbetween the gate posts Bunnock gave the signal and struck down the\nwarder at the gate; the driver with his hatchet cut the traces, the\nmen leapt up from their concealment in the hay, and the main body\nlying in ambush close by rushed up, and, taken wholly by surprise,\nunarmed and unprepared, the garrison was speedily overpowered and\nthe castle taken.\n\nIt was in the spring of 1311 that this important capture took place.\nBruce, as usual, had the castle levelled to the ground. Bunnock was\nrewarded by a grant of land which still bears his name, softened\ninto Binney.  Again the English made preparations for a renewed\ninvasion, but the barons were too much occupied by their private\nbroils and their quarrels with the king to assemble at his order,\nand nothing came of it. Bruce's position at home was so established\nthat he resolved upon a counter invasion, and accordingly, having\nassembled a larger force than had hitherto gathered under his\nbanner, crossed the Border near the Solway, burnt and plundered the\ndistrict round Gilsland, ravaged Tynedale, and after eight days'\nhavock returned with much booty to Scotland. In the following\nmonth he again entered England, carried fire and sword through the\ncountry as far as Corbridge, swept Tynedale, ravaged Durham, and\nafter levying contributions for fifteen days returned with much\nbooty to Scotland.\n\nAlthough the English made much outcry at this invasion, the English\nauthor of the Chronicle of Lanercost, whose monastery was occupied\nby the king during the raid, distinctly states that he slew none\nsave in actual conflict; and again, that though \"all the goods of\nthe country were carried away, they did not burn houses or slay\nmen.\" Thus, though Bruce's wife and daughter were still prisoners\nin England, though his brothers had been executed in cold blood,\nhe conducted his warfare in England in a manner which contrasts\nstrongly indeed with the conduct of the English in Scotland.\n\nAfter this Bruce marched north again and laid siege to Perth. For\nsix weeks he invested the town, but without making any impression.\nThen he retired his forces as if abandoning the attempt. At night,\nhowever, he returned, ladders were placed in the ditches against\nthe walls, and with his knights he led his followers on to the\nassault. The garrison were carousing in honour of their successful\ndefence and the defeat of the enemy, and taken wholly by surprise\nwere unable to oppose a vigorous resistance, and all were killed\nor captured. Some accounts say that the English soldiers were made\nprisoners, and the renegade Scots fighting with them were put to\nthe sword; while others affirm that all who were taken prisoners\nwere spared.\n\nAnother incursion into England followed the fall of Perth. Hexham,\nCorbridge, and Durham were destroyed.  Douglas penetrated as far as\nHartlepool and an immense spoil was carried off, until the people\nof the bishopric purchased a truce for the sum of 2000 pounds, and\nthose of Northumberland, Cumberland, and Westmoreland bought off\nthe invaders at a like price.\n\nCarlisle was assaulted by Douglas, but unsuccessfully.  He also\nattempted to surprise Berwick by a night attack, and had placed his\nscaling ladders against the wall, when the garrison was alarmed by\nthe barking of a dog, and the assailants were repulsed. The Scots\nrecrossed the frontier laden with an enormous booty.\n\nThe king himself now entered Galloway and reduced the four remaining\nstrongholds held by the English there--the castles of Butele,\nDalswinton, Lochmaben, and Tibbers.  He then proceeded to Dumfries,\nwhich he forced to surrender, and entered it as the victorious King\nof Scotland, just seven years after the time when he had commenced\nthe war by expelling the English justiciary.\n\nArchie Forbes did not accompany the king in this campaign. He\nhad indeed been summoned, but just before the army started on its\nraid into England Bruce was lamenting, in Archie's hearing, that\nthe continued possession of the strong castle of Dunottar on the\neast coast still afforded the English an opportunity for creating\ndiversions in the north, by landing troops there.\n\n\"If you will permit me, sire,\" Archie said, \"I will undertake its\ncapture with my retainers. It is doubtless too strong to be captured\nby open assault with such a strength, but as Douglas has thrice\ntaken Castle Douglas by stratagem, 'tis hard if I cannot find some\nway for capturing Dunottar.\"\n\n\"Be it so, Sir Archie,\" the king said. \"If you succeed you will have\ndone good service indeed; and as I know that though ever ready to\nbuckle on your armour when I need you, you would yet rather live\nquiet at Aberfilly with your fair wife, I promise you that if you\ncapture Dunottar, for a year and a day you and your retainers shall\nhave rest, except if the English cross the Border in such force\nthat the arm of every Scotchman able to wield a sword is needed in\nits defence.\"\n\nHaving chosen a hundred of his most active and experienced men\nArchie set out for the north.  Crossing the Forth above Stirling,\nhe marched through Perth and across the Carse of Gowrie through\nForfar on to Montrose. Here he left his band, and taking with him\nonly William Orr, both being attired in peasants' dress, followed\nthe coast till he reached Dunottar.\n\nThe castle, which was of great strength, stood in a little bay\nwith a fishing village nestled beside it.  \"'Tis a strong place,\nWilliam, and, if well provisioned, might hold out against an army\nfor months, and as supplies could be thrown in by sea it could only\nbe captured by battering down its solid walls by machines.\"\n\n\"'Tis indeed a strong place, Sir Archie,\" William Orr replied, \"and\nit were assuredly better to slip in by the gates than to climb over\nthe walls; but after the captures of so many of their strongholds\nby sudden surprise, we may be sure that a careful watch will be\nkept.\"\n\n\"Doubtless they are shrewdly on guard against surprise,\" Archie\nsaid; \"but as they know that the king and his host are just now\ncrossing the Border into Cumberland, they may well think that for\na time they are safe from disturbance. 'Tis in that that our best\nchance lies.\"\n\nEntering the village they purchased some fish from the fishermen,\nand asking a few careless questions about the garrison, found\nthat it was composed of 150 men, and that extreme precautions were\ntaken against surprise. The gates were never opened save to allow\nparties to pass in and out, when they were instantly closed and the\ndrawbridge raised.  Only ten of the garrison at a time were ever\nallowed to leave the castle, and these must go out and come in\ntogether, so that the gates should not be opened more than twice a\nday.  \"They generally come out,\" the man said, \"at eleven o'clock\nand go in at four; at eleven o'clock all with corn, wood, and\nother stores for the castle must present themselves, so that the\ndrawbridge need only be lowered at those times.  The governor,\nSir John Morris, swears that he will not be caught asleep as were\nthose of Linlithgow and Castle Douglas. I fear,\" he concluded,\n\"that we of Dunottar will be the last in Scotland to be free from\nthe English yoke.\"\n\n\"That is as it may be. Other castles have been captured, and maybe\nthe lion of Scotland may float on those walls ere long.\"\n\nThe man looked keenly at him.\n\n\"Methinks there is meaning in your words,\" he said, \"and your\nlanguage does not accord with your attire. I ask no questions; but\nbe sure that should an attempt be made, there are a score of strong\nfellows among us who will be ready to strike a blow for freedom.\"\n\n\"Is that so?\" Archie replied; \"then, man, taking you to be a true\nScot, I will tell you that the attempt will be made, and that\nsoon, and that, if you will, you can aid the enterprise. I am Sir\nArchibald Forbes, of whom, perhaps, you have heard.\"\n\n\"Assuredly,\" the man said in a tone of deep respect, \"every Scotsman\nknows the name as that of one of the king's truest and bravest\nknights.\"\n\n\"My purpose is this,\" Archie said. \"On a dark night some ninety-five\nof my men will march hither; I need a faithful friend to meet them\noutside the village to lead them in, and to hide them away in the\ncottages, having already arranged beforehand with their owners to\nreceive them. I, myself, with four of my men will come hither in a\nfishing boat well laden with fish; we will choose a time when the\nwind is blowing, and will seem to have been driven here by stress\nof weather and disabled. Then I shall try to sell our cargo for the\nuse of the garrison. As we carry it in we shall attack the guard,\nand at the signal those hidden will rush out and cross the drawbridge.\"\n\n\"The plan is a good one,\" the fisherman said; \"its difficulty mainly\nlies in the fact that the drawbridge will be raised the moment you\nhave crossed it, and long before your followers could arrive it\nwould be high in the air, and you would be cut off from all aid. It\nnever remains down for an instant after men have passed over it.\"\n\n\"That adds to the difficulty,\" Archie said thoughtfully; \"but\nI must think of some plan to overcome it. Do you quietly go about\namong those you can surely trust and arrange for them to be ready\nto open their doors and take my men in without the slightest noise\nwhich might attract the sentries on the walls. So long as the wind\nis quiet and the sea smooth we shall not come, but the first day\nthat the wind blows hard you may expect us. Then do you go out on\nthe south road and wait for my party half a mile from the village.\nIf they come not by midnight, return home and watch the following\nnight.\"\n\n\"I understand,\" the fisherman said, \"and will do as you bid me; and\nwhen the time comes you can rely upon twenty stout fellows here in\naddition to your own force.\"\n\n\"'Tis nigh eleven,\" Archie said, looking at the sun, \"and we will\nbe off at once, as the soldiers will soon be coming out, and it\nwere best the governor did not hear that two strangers were in the\nvillage. Vigilant as he is, a small thing might excite his suspicion\nand add to his watchfulness.\"\n\nArchie and William Orr returned to Montrose, and there the former\nmade an arrangement with the master of a large fishing boat to keep\nhis vessel ready to put to sea at any moment.\n\nThree weeks passed without any change in the weather; then the wind\nbegan to rise and the aspect of the sky betokened a storm. William\nOrr at once set out with ninety-five men for Dunottar. Archie went\ndown to the port and purchased a large quantity of fish which had\nbeen brought in that morning in various boats, and had it placed\non board the craft that he had hired. Then he with four of his\nfollowers, the strongest and most determined of his retainers,\ndressed as fishermen, went on board and the boat at once put to sea,\nhaving, besides Archie and his men, the master and his two hands.\nThe main body had started on foot at ten in the morning, but it was\nlate in the afternoon before the boat put out, as Archie wished to\narrive in broad daylight next morning.\n\nThe wind was on the shore, and the boat was sorely tossed and\nbuffeted. Ere next morning, showing but a rag of sail, she ran into\nDunottar harbour. They had had great difficulty in keeping off the\ncoast all night, and the play had nigh turned into a tragedy, so\nnarrow had been their escape of being cast ashore. The bulwarks\nwere washed away, and the boat was in a sore plight as it drew\nalongside the little quay.  Assuredly no suspicion would occur to\nany who saw her enter that aught save stress of weather had driven\nher in.\n\nIt was twelve o'clock in the day when they reached the port. Most\nof the inhabitants had come down to the water side to see the\nstorm beaten craft enter, and among them were some soldiers of the\ngarrison. Archie bade four of his men remain below, so that the\nunusual number of hands should attract no attention. One of the first\nto come on board was the fisherman with whom Archie had spoken.\n\n\"Your men are all here,\" he said in a low tone to Archie, \"and are\nstowed away in the cottages. Everything went well, and there was\nnot the slightest noise.\"\n\nArchie now went on shore and entered into conversation with one of\nthe soldiers.\n\n\"Think you,\" he said, \"that the governor would buy my cargo of\nfish. I have a great store on board, for I had good luck before\nthe storm suddenly broke upon me just as I was leaving the fishing\ngrounds for Montrose.  The gale may last for some days, and my boat\nwill need repairs before I put to sea, therefore my fish will be\nspoiled before I can get them to market, and I will make a good\nbargain with the governor if he will take them from me.\"\n\n\"I should think that he will do so gladly,\" the soldier said, \"for\nhe can salt them down, and they make a pleasant change. How much\nhave you got?\"\n\n\"About ten baskets full,\" Archie replied, \"of some hundred pounds\neach.\"\n\n\"I will go with you to the castle,\" the soldier said. \"The governor\nwill lower the drawbridge for no man, but you can speak with the\nwarder across the moat and he will bear your message to the governor,\nand should he agree, you must present yourself with your men with\nthe fish at four o'clock, at which time the drawbridge will be\nlowered for us to return to the castle.\"\n\nArchie accompanied the soldier to the end of the drawbridge, and\nparleyed with the warder. The latter acquainted the governor that\nthe master of the fishing boat which had been driven in by stress\nof weather would fain dispose of his cargo of fish on cheap terms,\nand returned for answer that the governor would give sixpence for\neach basket of a hundred pounds. Archie grumbled that he should\nreceive thrice that sum at Montrose; still that as he must sell\nthem or let them spoil, he accepted the offer, and would be there\nwith the fish at four o'clock.\n\nHe then returned to the boat, his ally, the fisherman, taking word\nround to the cottages that at four o'clock all must be in readiness\nto sally out on the signal, and that William Orr was to dress half\na dozen of his men in fishermen's clothes and saunter up carelessly\nclose to the castle, so as to be able to rush forward on the instant.\n\nAt the appointed hour Archie, accompanied by his four followers,\neach of whom carried on his shoulder a great basket filled with\nfish, stepped on to the quay and made their way to the castle. By\nthe side of the moat facing the drawbridge the ten English soldiers\nwho had been out on leave for the day were already assembled.\n\n\"Are you all there?\" the warder asked.\n\n\"Yes,\" Archie said, \"but I shall have to make another two trips\ndown to the boat, seeing that I have ten baskets full and but four\nmen to carry them.\"\n\n\"Then you must bring another load,\" the warder said, \"when the\ndrawbridge is lowered tomorrow. You will have to stop in the castle\ntonight, and issue out at eleven tomorrow, for the governor will\nnot have the drawbridge lowered more than twice a day.\"\n\n\"I would fain return to my boat,\" Archie said, \"as I want to be at\nwork on the repairs; but if that be the rule I must needs submit\nto it.\"\n\nThe drawbridge was now lowered. The soldiers at once stepped on to\nit. The four pretended fishermen had set down their baskets, and\nnow raised them on their shoulders again.  One of them apparently\nfound it a difficult task, for it was not until Archie and his\ncomrades were half across the drawbridge that he raised it from\nthe ground. As he did so he stumbled and fell, the basket and its\ncontents rolling on to the ground.\n\n\"You must wait until the morning,\" the warder called; \"you are too\nlate to enter now.\"\n\nThe man lay for a moment where he had fallen, which was half on the\ndrawbridge, half on the ground beyond it.  \"Now, then,\" the warder\ncalled sharply, \"make haste; I am going to raise the drawbridge.\"\n\nThe man rose to his feet with a shout just as the drawbridge began\nto rise. He had not been idle as he lay.  As he fell he had drawn\nfrom underneath his fisherman's frock a stout chain with a hook\nat one end and a large ring at the other. This he had passed round\none of the chains by which the drawbridge was raised, then under\nthe beam on which it rested when down, and had fastened the hook\nin the ring.\n\nSurprised at the shout, the warder worked the windlass with extra\nspeed, but he had scarcely given a turn when he found a sudden\nresistance. The chain which the fisherman had fixed round the end\nprevented the bridge from rising.  As the man had shouted, Archie\nand his three comrades were entering the gate.  Simultaneously they\nemptied their baskets before them. Concealed among the fish were\nfour logs of wood; two were three feet long, the full depth of the\nbaskets, two were short wedge shaped pieces.  Before the soldiers\nin front had time even to turn round, the two long pieces were\nplaced upright in the grooves down which the portcullis would fall,\nwhile the two wedge shaped pieces were thrust into the jamb of the\ngate so as to prevent it from closing.  Then the four men drew long\nswords hidden beneath their garments and fell upon the soldiers.\n\n\n\n\nChapter XXVI\n\nEdinburgh\n\n\nSo vigilant was the watch in the castle of Dunottar that the instant\nthe cry of alarm rose almost simultaneously from the warder above\nand the soldiers at the gate, the portcullis came thundering down.\nIt was caught, however, by the two upright blocks of wood, and\nremained suspended three feet above the sill.  The armed guards\nat the gate instantly fell upon Archie and his companions, while\nothers endeavoured in vain to close the gates. Scarcely had the\nswords clashed when the man who had chained down the drawbridge\njoined Archie, and the five with their heavy broadswords kept at\nbay the soldiers who pressed upon them; but for only a minute or\ntwo did they have to bear the brunt of the attack unsupported, for\nWilliam Orr and the five men who had been loitering near the moat\ndashed across the bridge, and passing under the portcullis joined\nthe little band.\n\nThe alarm had now spread through the castle, and the governor\nhimself, followed by many of his men, came rushing down to the\nspot, shouting furious orders to the warder to raise the drawbridge,\nbeing in ignorance that it was firmly fixed at the outer end.\n\nArchie and his followers were now hotly pressed, but soon a thunder\nof steps was heard on the drawbridge, and the whole of the band,\ntogether with some twenty or thirty of the fishermen, passed under\nthe portcullis and joined them. Archie now took the offensive, and\nbearing down all opposition burst with his men into the courtyard.\n\nThe combat was desperate but short. The governor with some of his\nsoldiers fought stoutly, but the suddenness of the surprise and\nthe fury and vigour with which they were attacked shook the courage\nof many of the soldiers.  Some, instead of joining in the fray, at\nonce threw away their arms and tried to conceal themselves, others\nfought feebly and half heartedly, and the cries of \"A Forbes!  A\nForbes!  Scotland!  Scotland!\" rose louder and louder as the\nassailants gradually beat down all resistance. In ten minutes from\nthe falling of the portcullis all resistance was virtually over.\nThe governor himself fell by the hand of Archie Forbes, and at\nhis death those who had hitherto resisted threw down their arms\nand called for quarter. This was given, and the following day the\nprisoners were marched under a strong guard down to Montrose, there\nto be confined until orders for their disposal were received from\nthe king. For the next fortnight Archie and his retainers, aided by\nthe whole of the villagers, laboured to dismantle the castle. The\nbattlements were thrown down into the moat, several wide breaches\nwere made in the walls, and large quantities of straw and wood piled\nup in the keep and turrets. These were then fired, and the Castle\nof Dunottar was soon reduced to an empty and gaping shell. Then\nArchie marched south, and remained quietly at home until the term\nof rest granted him by the king had expired.\n\nTwo girls and a son had by this time been born to him, and the\nmonths passed quietly and happily away until Bruce summoned him to\njoin, with his retainers, the force with which Randolph had sat down\nbefore Edinburgh Castle. Randolph was delighted at this accession\nof strength. Between him and Douglas a generous rivalry in gallant\nactions continually went on, and Douglas had scored the last\ntriumph. The castle of Roxburgh had long been a source of trouble\nto the Scots.  Standing on a rocky eminence on the margin of the\nTeviot, just at its junction with the Tweed and within eight miles\nof the Border, it had constituted an open door into Scotland, and\neither through it or through Berwick the tides of invasion had ever\nflowed.  The castle was very strongly fortified, so much so that\nthe garrison, deeming themselves perfectly safe from assault, had\ngrown careless. The commandant was a Burgundian knight, Gillemin\nde Fienne. Douglas chose Shrove Tuesday for his attack. Being a\nfeast day of the church before the long lenten fast the garrison\nwould be sure to indulge in conviviality and the watch would be\nless strict than usual. Douglas and his followers, supplied with\nscaling ladders, crept on all fours towards the walls. The night\nwas still and they could hear the sentries' conversation. They had\nnoticed the objects advancing, but in the darkness mistook them for\nthe cattle of a neighbouring farmer. Silently the ladders were\nfixed and mounted, and with the dreaded war cry, \"A Douglas! A\nDouglas!\" the assailants burst into the castle, slaying the sentries\nand pouring down upon the startled revellers. Fienne and his men\nfought gallantly for a time, but at length all surrendered, with\nthe exception of the governor himself and a few of his immediate\nfollowers, who retired into a tower, where they defended themselves\nuntil the following day; then Fienne being seriously wounded, the\nlittle party also surrendered. As Douglas had no personal quarrel\nwith the garrison of Roxburgh such as he bore with those who occupied\nhis ancestral castle, he abstained from any unnecessary cruelties,\nand allowed the garrison to withdraw to England, where Fienne soon\nafterwards died of his wounds.\n\nThe castle was as usual levelled to the ground, and as the stronghold\nof Carlaverock soon afterwards surrendered, the districts of Tweeddale\nand Galloway were now completely cleared of the English, with the\nexception of the Castle of Jedburgh, which they still held.\n\nRandolph had been created Earl of Moray, and after establishing\nhimself in his new earldom he had returned with his feudal followers\nand laid siege to Edinburgh, whose castle was considered all but\nimpregnable. It had been in the possession of the English ever since\nit was captured by Edward I in 1296, and was strongly garrisoned\nand well provisioned.\n\nEven when joined by Archie Forbes and his retainers Randolph felt\nthat the castle could not be captured by force.  The various attempts\nwhich he made were signally foiled, and it was by stratagem only\nthat he could hope to carry it.  The news of the capture of Roxburgh\nby Douglas increased his anxiety to succeed.  Accompanied by Archie\nhe rode round the foot of the steep rock on which the castle stands,\neagerly scanning its irregularities to see if by any possibility\nit could be scaled.\n\n\"I would give a brave reward,\" he said to Archie, \"to any who could\nshow us a way of climbing those rocks, which, methinks, even a goat\ncould scarcely manage to ascend.\"\n\n\"I can tell you of a way,\" a Scotch soldier who was standing a few\npaces off when he made the remark, said, saluting the earl.  \"It\nneeds a sure foot and a stout heart, but I can lead a score of\nmen with such qualifications to the foot of yonder walls;\" and he\npointed to the castle rising abruptly from the edge of the rocks.\n\n\"If you can make good your word, my brave fellow,\" Randolph said,\n\"you may ask your own reward, and I pledge you my word, that if it\nbe aught in reason it shall be granted.  But who are you, and how\ndid it come that you know of a way where none is supposed to exist?\"\n\n\"My name is William Francus,\" the soldier said. \"I was at one time,\nbefore the king took up arms, a soldier in the castle there. I had\na sweetheart in the town, and as my turn to go out from the castle\ncame but slowly I used at night to steal away to visit her. I found\nafter a great search that on the face of yonder wall where it looks\nthe steepest, and where in consequence but slight watch is kept,\na man with steady foot and head could make shift to climb up and\ndown, and thus, if you please, will I guide a party to the top of\nthe rock.\"\n\n\"It looks impossible,\" Randolph said, gazing at the precipice;\n\"but as you tell me that you have done it others can do the same.\nI will myself follow your guidance.\"\n\n\"And I,\" Archie said.\n\n\"What, Sir Archie, think you is the smallest number of men with\nwhom, having once gained footing on the wall, we may fight our way\nto the gates and let in our friends.\"\n\n\"I should think,\" Archie replied, \"that with thirty men we might\nmanage to do so. The confusion in the garrison will be extreme\nat so unexpected a surprise, and if we divide in two parties and\npress forward by different ways they will think rather of holding\ntogether and defending themselves than of checking our course, and\none or other of the parties should surely be able to make its way\nto the gates.\"\n\n\"Thirty let it be then,\" Randolph said. \"Do you choose fifteen\nactive and vigilant men from among your retainers; I will pick as\nmany from mine, and as there is no use in delaying let us carry\nout the enterprise this very night; of course the rest of our men\nmust gather near the gates in readiness to rush in when we throw\nthem open.\"\n\nAs soon as it was dark the little party of adventurers set out\non their way. Francus acted as guide, and under his leading they\nclimbed with vast difficulty and no little danger up the face of\nthe precipice until they reached a comparatively easy spot, where\nthey sat down to recover their breath before they prepared for the\nfinal effort.\n\nThey could hear the sentries above speaking to each other, and\nthey held their breath when one of them, exclaiming suddenly, \"I\ncan see you!\" threw down a stone from the battlement, which leapt,\ncrashing down the face of the rock close beside them. Great was\ntheir relief when a loud laugh from above told them that the sentry\nhad been in jest, and had but tried to startle his comrade; then\nthe two sentries, conversing as they went, moved away to another\npart of the walls.\n\nThe ascent was now continued, and proved even more difficult than\nthat which they had passed. They were forced continually to halt,\nwhile those in front helped those following them, or were themselves\nhoisted up by the men behind.  At last, panting and breathless, they\nstood on the summit of the rock, on a narrow ledge, with the castle\nwall rising in front of them. They had, with enormous difficulty,\nbrought up a light ladder with them. This was placed against the\nwall. Francus was the first to mount, and was followed by Sir Andrew\nGrey, whom Randolph had invited to be of the party, by Archie Forbes,\nand by the earl. Just as the latter stepped on to the battlements\nthe sentries caught sight of them and shouted:\n\n\"Treason! treason! to arms!\" An instant stir was heard in the\ncastle. Rapidly the thirty men followed each other up the ladder,\nand so soon as the last had gained the battlements they divided in\nthree bodies, each headed by one of the leaders. One party descended\nstraight into the castle and there attacked the soldiers who were\nhurrying to arms, while the others ran along the wall in opposite\ndirections, cutting down the sentries and brushing aside all\nopposition until together they met at the gate. This was thrown\nopen, and the Scots outside running up at the top of their speed\npoured into the castle. At first Randolph's party, which had\ndescended into the courtyard, had been hotly pressed, and had with\ndifficulty defended themselves; but the attention of the startled\ngarrison was distracted by the shouts upon the walls, which told\nthat other parties of their assailants had gained footing there.\nAll sorts of contradictory orders were issued. One commanded them\nto cut down the little party opposed to them, another ordered them\nto hurry to the walls, a third to seize the gate and see that it\nwas not opened. The confusion reached its height as the Scots poured\nin through the open gate. The garrison, surprised and confounded\nas they were at this, to them, almost magical seizure of the castle\nby their foes, fought bravely until the governor and many of the\nofficers were killed. Some of the men threw down their arms, and\nothers, taking advantage of their knowledge of the castle, made\ntheir way to the gate and escaped into the open country.\n\nThe news of the capture was immediately sent to the king, by whose\norders the castle and walls were razed to the ground, and thus\nanother of the strongholds, by whose possession the English were\nenabled to domineer over the whole of the surrounding country, was\ndestroyed.\n\nWhile Douglas and Randolph were thus distinguishing themselves\nEdward Bruce captured the castle of Rutherglen, and afterwards the\ntown of Dundee; and now, save Stirling Castle, scarcely a hold in\nall Scotland remained in English hands. Thus was Scotland almost\ncleared of the invader, not by the efforts of the people at large,\nbut by a series of the most daring and hazardous adventures by the\nking himself and three or four of his knights, aided only by their\npersonal retainers. For nine years they had continued their career\nunchecked, capturing castle by castle and town by town, defeating\nsuch small bodies of troops as took the field against them, England,\nunder a supine and inactive king, giving itself up to private\nbroils and quarrels, while Scotland was being torn piecemeal from\nher grasp.\n\nAfter Edward Bruce had captured Dundee he laid siege to Stirling.\nAs this castle had for many months resisted Edward I backed by the\nwhole power of England, Bruce could make little impression upon\nit with the limited appliances at his disposal. From February till\nthe 24th of June the investment continued, when the governor, Sir\nPhilip Mowbray, becoming apprehensive that his provisions would\nnot much longer hold out, induced Edward Bruce to agree to raise\nthe siege on condition that if by the 24th of June next, 1314, the\ncastle was not effectually relieved by an English force, it should\nthen be surrendered.\n\nNo satisfactory explanation has ever been given of the reasons which\ninduced Edward Bruce to agree to so one sided a bargain.  He had\nalready invested the place for four months, there was no possibility of\nan army being collected in England for its relief for many months\nto come, and long ere this could arrive the garrison would have\nbeen starved into surrender. By giving England a year to relieve\nthe place he virtually challenged that country to put forth all\nits strength and held out an inducement to it to make that effort,\nwhich internal dissension had hitherto prevented. The only feasible\nexplanation is that Edward Bruce was weary of being kept inactive\nso long a time before the walls of the fortress which he was unable\nto capture, and that he made the arrangement from sheer impatience\nand thoughtlessness and without consideration of the storm which he\nwas bringing upon Scotland. Had it been otherwise he would surely\nhave consulted the king before entering upon an agreement of such\nextreme importance.\n\nBruce, when he heard of this rash treaty, was highly displeased,\nbut he nevertheless accepted the terms, and both parties began at\nonce their preparations for the crowning struggle of the war. The\nEnglish saw that now or never must they crush out the movement\nwhich, step by step, had wrested from them all the conquests which\nhad been won with such vast effort under Edward I; while Bruce saw\nthat a defeat would entail the loss of all that he had struggled\nfor and won during so many years.\n\nKing Edward issued summonses to the whole of the barons of England\nand Wales to meet him at Berwick by the 11th of June with all their\nfeudal following, while the sheriffs of the various counties and\ntowns were called upon to supply 27,000 foot soldiers. The English\nof the settlements in Ireland were also summoned, besides O'Connor,\nPrince of Connaught, and twenty-five other native Irish chiefs,\nwith their following, all of whom were to be under the command of\nRichard de Burgh, Earl of Ulster.\n\nThe Prince Bishop of Constance was requested to furnish a body\nof mounted crossbowmen. A royal fleet of twenty-three vessels was\nappointed to assemble for the purpose of operating on the east\ncoast, while the seaports were commanded to fit out another fleet\nof thirty vessels. A third fleet was ordered to assemble in the\nwest, which John of Lorne was appointed to command under the title\nof High Admiral of the Western Fleet of England. From Aquitaine\nand the French possessions the vassals were called upon to attend\nwith their men-at-arms, and many knights from France, Gascony, and\nGermany took part in the enterprise.\n\nThus, at the appointed time over 100,000 men assembled at Berwick,\nof whom 40,000 were men-at-arms, and the rest archers and pikemen.\nFor the great armament the most ample arrangements were made in the\nway of warlike stores, provisions, tents, and means of transport,\ntogether with the necessary workmen, artificers, and attendants.\n\nThis army surpassed both in numbers and equipments any that Edward\nI had ever led into Scotland, and is considered to have been the most\nnumerous and best equipped that ever before or since has gathered\non English ground. Of the whole of the great nobles of England only\nfour were absent--the Earls of Warrenne, Lancaster, Arundel, and\nWarwick--who, however, sent their feudal arrays under the charge\nof relations.\n\nAmong the leaders of this great army were the Earls of Gloucester,\nPembroke, Hereford, and Angus, Lord Clifford, Sir John Comyn, Sir\nHenry Beaumont, Sir John Seagrave, Sir Edmund Morley, Sir Ingram\nde Umfraville, Sir Marmaduke de Twenge, and Sir Giles de Argentine,\none of the most famous of the Continental knights.\n\nWhile this vast army had been preparing, Bruce had made every\neffort to meet the storm, and all who were loyal and who were able\nto carry weapons were summoned to meet at Torwood, near Stirling,\nprevious to the 24th of June. Here Edward Bruce, Sir James Douglas,\nRandolph, Earl of Moray, Walter the Steward, Angus of Isla, Sir\nArchibald Forbes, and a few other knights and barons assembled with\n30,000 fighting men, besides camp followers and servants.  It was\na small force indeed to meet the great army which was advancing\nagainst it, and in cavalry in particular it was extremely weak.\nThe English army crossed the Border, and marched by Linlithgow and\nFalkirk toward the Torwood.\n\nEach army had stirring memories to inspire it, for the English in\ntheir march crossed over the field of Falkirk, where sixteen years\nbefore they had crushed the stubborn squares of Wallace; while from\nthe spot which Bruce selected as his battleground could be seen\nthe Abbey Craig, overlooking the scene of the Scottish victory of\nStirling Bridge. On the approach of the English the Scotch fell\nback from the Torwood to some high ground near Stirling now called\nthe New Park.  The lower ground, now rich agricultural land called\nthe Carse, was then wholly swamp. Had it not been so, the position\nnow taken up by Bruce would have laid the road to Stirling open to\nthe English.\n\nThe Scotch army was divided into four divisions. The centre was\ncommanded by Randolph. Edward Bruce commanded the second, which\nformed the right wing.  Walter the Steward commanded the left wing,\nunder the guidance of Douglas, while the king himself took command\nof the fourth division, which formed the reserve, and was stationed\nin rear of the centre in readiness to move to the assistance of\neither of the other divisions which might be hard pressed.  The camp\nfollowers, with the baggage and provisions, were stationed behind\nthe Gillies Hill.\n\nThe road by which the English would advance was the old Roman\ncauseway running nearly north and south. The Bannock Burn was fordable\nfrom a spot near the Park Mill down to the village of Bannockburn.\nAbove, the banks were too high and steep to be passed; while below,\nwhere ran the Bannock through the carse, the swamps prevented\npassage.  The army was therefore drawn up, with its left resting\non the sharp angle of the burn above the Park Mill, and extended\nwhere the villages of Easterton, Borestine, and Braehead now stand\nto the spot where the road crosses the river at the village of\nBannockburn. In its front, between it and the river, were two bogs,\nknown as Halberts Bog and Milton Bog, while, where unprotected by\nthese bogs, the whole ground was studded with deep pits; in these\nstakes were inserted, and they were then covered with branches and\ngrass. Randolph's centre was at Borestine, Bruce's reserve a little\nbehind, and the rock in which his flagstaff was placed during the\nbattle is still to be seen. To Randolph, in addition to his command\nof the centre division, was committed the trust of preventing any\nbody of English from passing along at the edge of the carse, and\nso making round to the relief of Stirling.\n\nOn the morning of Sunday, the 23d of June, immediately after\nsunrise, the Scotch attended mass, and confessed as men who had\ndevoted themselves to death. The king, having surveyed the field,\ncaused a proclamation to be made that whosoever felt himself unequal\nto take part in the battle was at liberty to withdraw. Then, knowing\nfrom his scouts that the enemy had passed the night at Falkirk, six\nor seven miles off, he sent out Sir James Douglas and Sir Robert\nKeith with a party of horsemen to reconnoitre the advance.\n\nThe knights had not gone far when they saw the great army advancing,\nwith the sun shining bright on innumerable standards and pennons,\nand glistening from lance head, spear, and armour. So grand and\nterrible was the appearance of the army that upon receiving the\nreport of Douglas and Keith the king thought it prudent to conceal\nits full extent, and caused it to be bruited abroad that the enemy,\nalthough numerous, was approaching in a disorderly manner.\n\nThe experienced generals of King Edward now determined upon making\nan attempt to relieve Stirling Castle without fighting a pitched\nbattle upon ground chosen by the enemy. Had this attempt been\nsuccessful, the great army, instead of being obliged to cross\na rapid stream and attack an enemy posted behind morasses, would\nhave been free to operate as it chose, to have advanced against\nthe strongholds which had been captured by the Scots, and to force\nBruce to give battle upon ground of their choosing.  Lord Clifford\nwas therefore despatched with 800 picked men-at-arms to cross the\nBannock beyond the left wing of the Scottish army, to make their\nway across the carse, and so to reach Stirling.  The ground was,\nindeed, impassable for a large army; but the troops took with\nthem <DW19>s and beams, by which they could make a passage across\nthe deeper parts of the swamp and bridge the little streams which\nmeandered through it.\n\nAs there was no prospect of an immediate engagement, Randolph,\nDouglas, and the king had left their respective divisions, and had\ntaken up their positions at the village of St. Ninians, on high\nground behind the army, whence they could have a clear view of the\napproaching English army.  Archie Forbes had accompanied Randolph,\nto whose division he, with his retainers, was attached.  Randolph\nhad with him 500 pikemen, whom he had withdrawn from his division\nin order to carry out his appointed task of seeing that the English\ndid not pass along the low ground at the edge of the carse behind\nSt. Ninians to the relief of Stirling; but so absorbed were knights\nand men-at-arms in watching the magnificent array advancing against\nthe Scottish position that they forgot to keep a watch over the\nlow ground. Suddenly one of the men, who had straggled away into\nthe village, ran up with the startling news that a large party of\nEnglish horse had crossed the corner of the carse, and had already\nreached the low ground beyond the church.\n\n\"A rose has fallen from your chaplet, Randolph,\" the king said\nangrily.\n\nWithout a moment's loss of time Randolph and Archie Forbes set off\nwith the spearmen at a run, and succeeded in heading the horsemen\nat the hamlet of Newhouse. The mail clad horsemen, confident in\ntheir numbers, their armour, and horses, laid their lances in rest,\nstruck spurs into their steeds, and, led by Sir William Daynecourt,\ncharged down upon the Scotch spearmen. Two hundred of these consisted\nof Archie Forbes' retainers, all veterans in war, and who had more\nthan once, shoulder to shoulder, repelled the onslaught of the\nmailed chivalry of England. Animated by the voices of their lord\nand Randolph, these, with Moray's own pikemen, threw themselves\ninto a solid square, and, surrounded by a hedge of spears, steadily\nreceived the furious onslaught of the cavalry. Daynecourt and many\nof his men were at the first onslaught unhorsed and slain, and those\nwho followed were repulsed. Again and again they charged down upon\nthe pikemen, but the dense array of spears was more than a match\nfor the lances of the cavalry, and as the horses were wounded and\nfell, or their riders were unhorsed, men rushed out from the square,\nand with axe and dagger completed the work. Still the English\npressed them hard, and Douglas, from the distance, seeing how hotly\nthe pikemen were pressed by the cavalry, begged the king to allow\nhim to go to Randolph's assistance. Bruce, however, would suffer no\nchange in his position, and said that Randolph must stand or fall\nby himself. Douglas, however, urged that he should be allowed to\ngo forward with the small body of retainers which he had with him.\nThe king consented, and Douglas set off with his men.\n\nWhen the English saw him approach they recoiled somewhat from the\nsquare, and Douglas, being now better able to see what was going\non, commanded his followers to halt, saying that Randolph would\nspeedily prove victorious without their help, and were they now\nto take part in the struggle they would only lessen the credit of\nthose who had already all but won the victory. Seeing the enemy in\nsome confusion from the appearance of the reinforcement, Randolph\nand Archie now gave the word for their men to charge, and these,\nrushing on with spear and axe, completed the discomfiture of the\nenemy, killed many, and forced the rest to take flight.  Numbers,\nhowever, were taken. Randolph is said to have had but two men killed\nin the struggle.\n\n\n\n\nChapter XXVII\n\nBannockburn\n\n\nAfter the complete defeat of the party under Lord Clifford, and the\nfailure of their attempt to relieve Stirling, Randolph and Douglas\nreturned together to the king. The news of their success spread\nrapidly, and when Randolph rode down from St. Ninians to his\ndivision, loud cheers broke from the whole Scottish army, who were\nvastly encouraged at so fair a commencement of their struggle with\nthe English.\n\nThe English army was still advancing slowly, and Bruce and his\nleaders rode down to the front of the Scottish line, seeing that\nall was in order and encouraging the men with cheering words. When\nthe English army approached the stream King Edward ordered a halt\nto be sounded for the purpose of holding a council, whether it was\nbest to encamp for the night or at once to advance against the\nenemy. The Earls of Gloucester and Hereford, who commanded the\nfirst division, were so far ahead that they did not hear the sound\nof the trumpet, and continuing their onward march crossed the Bannock\nBurn and moved on toward the Scotch array.  In front of the ranks\nof the defenders the king was riding upon a small palfrey, not\nhaving as yet put on his armour for the battle. On his helmet he\nwore a purple cap surmounted by a crown. Seeing him thus within\neasy reach, Sir Henry de Bohun, cousin of the Earl of Hereford,\nlaid his lance in rest and spurred down upon the king. Bruce could\nhave retired within the lines of his soldiers; but confident in his\nown prowess, and judging how great an effect a success under such\ncircumstances would have upon the spirits of his troops, he spurred\nforward to meet his assailant armed only with his axe. As the\nEnglish knight came thundering down, the king touched his palfrey\nwith his spur, and the horse, carrying but a light weight, swerved\nquickly aside; De Bohun's lance missed his stroke, and before he had\ntime to draw rein or sword, the king, standing up in his stirrups,\ndealt him so tremendous a blow with his axe as he passed, that it\ncleft through helmet and brain, and the knight fell dead to the\nground.\n\nWith a shout of triumph the Scotch rushed forward and drove\nthe English advance guard back across the stream; then the Scotch\nleaders led their men back again to the position which they had\nquitted, and reformed their array. Douglas, Edward Bruce, Randolph,\nand Archie Forbes now gathered round the king and remonstrated with\nhim on the rashness of an act which might have proved fatal to the\nwhole army.  The king smiled at such remonstrances from four men\nwho had, above all others, distinguished themselves for their rash\nand daring exploits, and shrugging his shoulders observed only that\nit was a pity he had broken the shaft of his favourite axe.  The\nEnglish array now withdrew to a short distance, and it became evident\nthat the great battle would be delayed till the morrow. The Scotch\narmy therefore broke its ranks and prepared to pass the night\non the spot where it stood.  The king assembled all his principal\nleaders round him, and after thanking God for so fair a beginning\nof the fight as had that day been made, he pointed out to them how\ngreat an effect the two preliminary skirmishes would have upon the\nspirits of both armies, and expressed his confidence in the final\nresult. He urged upon them the necessity for keeping their followers\nwell in hand, and meeting the charges of the enemy's horse steadily\nwith their spears; and especially warned them, after repulsing\na charge, against allowing their men to break their array, either\nto plunder or take prisoners, so long as the battle lasted, as the\nwhole riches of the English camp would fall into their hands if\nsuccessful. He pledged himself that the heirs of all who fell should\nhave the succession of their estates free from the usual feudal\nburdens on such occasions.\n\nThe night passed quietly, and in the morning both armies formed\ntheir array for battle. Bruce, as was customary, conferred the\nhonour of knighthood upon several of his leaders. Then all proceeded\nto their allotted places and awaited the onset. Beyond the stream\nand extending far away towards the rising ground were the English\nsquadrons in their glittering arms, the first division in line,\nthe others in heavy masses behind them. Now that the Scotch were\nfairly drawn up in order of battle, the English could see how\nsmall was their number in comparison with their own, and the king\nin surprise exclaimed to Sir Ingram de Umfraville:\n\n\"What! will yonder Scots fight us?\"\n\n\"That verily will they,\" the knight replied, for he had many a\ntime been engaged in stout conflict with them, and knew how hard\nit was even for mail clad knights to break through the close lines\nof Scottish spears. So high a respect had he for their valour, that\nhe urged the king to pretend to retire suddenly beyond the camp,\nwhen the Scots, in spite of their leaders, would be sure to leave\ntheir ranks and flock into the camp to plunder, when they might be\neasily dispersed and cut to pieces. The king, however, refused to\nadopt the suggestion, saying, that no one must be able to accuse\nhim of avoiding a battle or of withdrawing his army before such\na rabble.  As the armies stood confronting each other in battle\narray a priest passed along the Scottish front, crucifix in hand,\nexhorting all to fight to the death for the liberty of their country.\nAs he passed along the line each company knelt in an attitude of\nprayer. King Edward, seeing this, exclaimed to Sir Ingram:\n\n\"See yonder folk kneel to ask for mercy!\"\n\n\"Ay, sire,\" the knight said, looking earnestly at the Scots, \"they\nkneel and ask for mercy, but not of you; it is for their sins they\nask mercy of God. I know these men, and have met and fought them,\nand I tell you that assuredly they will win or die, and not even\nwhen death looks them in the face will they turn to fly.\"\n\n\"Then if it must be so,\" said the king, \"let us charge.\"\n\nThe trumpet sounded along the line. First the immense body of\nEnglish archers crossed the burn and opened the battle by pouring\nclouds of arrows into the Scottish ranks.  The Scotch archers, who\nwere in advance of their spearmen, were speedily driven back to\nshelter beyond their line, for not only were the English vastly more\nnumerous, but they shot much further and more accurately. And now\nthe knights and men-at-arms, on their steel clad horses, crossed\nthe burn.  They were aware of the existence of Milton Bog, which\ncovered the Scottish centre, and they directed their charge upon\nthe division of Edward Bruce on the Scottish right. The crash as\nthe mailed horses burst down upon the wood of Scottish spears was\ntremendous. Bruce's men held firm, and the English in vain strove\nto break through their serried line of spears. It was a repetition\nof the fight of the previous day, but on a greater scale. With\nlance and battleaxe the chivalry of England strove to break the\nranks of the Scotch, while with serried lines of spears, four deep,\nthe Scotch held their own. Every horse which, wounded or riderless,\nturned and dashed through the ranks of the English, added to the\nconfusion. This was much further increased by the deep holes into\nwhich the horses were continually falling, and breaking up all order\nin their ranks. Those behind pressed forward to reach the front,\nand their very numbers added to their difficulty.\n\nThe English were divided into ten divisions or \"battles,\" and\nthese one by one crossed the stream with banners flying, and still\navoiding the centre, followed the line taken by the first, and\npressed forward to take part in the fray.\n\nRandolph now moved with the centre to the support of the hardly\npressed right, and his division, as well as that of Edward Bruce,\nseemed to be lost among the multitude of their opponents. Stewart\nand Douglas moved their division to the right and threw themselves\ninto the fray, and the three Scottish divisions were now fighting\nside by side, but with a much smaller front than that which they\nhad originally occupied. For a time the battle raged furiously\nwithout superiority on either side. The Scotch possessed the great\nadvantage that, standing close together in ranks four deep, every\nman was engaged, while of the mounted knights and men-at-arms who\npressed upon them, only the front line was doing efficient service.\nNot only, therefore, was the vast numerical superiority of the\nEnglish useless to them, but actually a far larger number of the\nScottish than of themselves were using their weapons in the front\nrank, while the great proportion of the English remained helplessly\nbehind their fighting line, unable to take any part whatever in\nthe fight. But now the English archers came into play again, and\nfiring high into the air rained their arrows almost perpendicularly\ndown upon the Scottish ranks.  Had this continued it would have\nbeen as fatal to the Scots at Bannockburn as it was at Falkirk; but\nhappily the Scottish horse told off for this special service were\nhere commanded by no traitors, and at the critical moment the king\nlaunched Sir Robert Keith, the mareschal of Scotland, against the\narchers with 500 horsemen. These burst suddenly down upon the flank\nof the archers and literally swept them before them. Great numbers\nwere killed, others fell back upon the lines of horsemen who were\nranged behind, impatient to take their share in the battle; these\ntried to drive them back again, but the archers were disheartened,\nand retreating across the stream took no further part in the battle.\nThe charge of the Scottish horses should have been foreseen and\nprovided against by placing strong bodies of men-at-arms on the\nflanks of the archers, as these lightly armed troops were wholly\nunable to withstand a charge by cavalry.\n\nThe Scottish archers, now that their formidable opponents had\nleft the field, opened a heavy fire over the heads of the pikemen\nupon the horsemen surrounding the squares, and when they had shot\naway their arrows sallied out and mingled in the confused mass of\nthe enemy, doing tremendous execution with their axes and knives.\nHitherto the king had kept his reserve in hand; but now that the\nEnglish archers were defeated and their horsemen in inextricable\nconfusion, he moved his division down and joined in the melee, his\nmen shouting his well known battle cry.\n\nEvery Scotch soldier on the field was now engaged. No longer did the\nbattle cries of the various parties rise in the air. Men had no\nbreath to waste in shouting, but each fought silently and desperately\nwith spear or axe, and the sound of clanging blows of weapons, of\nmighty crash of sword or battleaxe on steel armour, with the cries\nand groans of wounded men were alone heard. Over and over again the\nEnglish knights drew back a little so as to gain speed and impetus,\nand flung themselves on the Scottish spears, but ever without effect,\nwhile little by little the close ranks of the Scotch pressed forward\nuntil, as the space between their front and the brook narrowed, the\nwhole of the English divisions became pent up together, more and\nmore incapable of using their strength to advantage. The slaughter\nin their front divisions had already been terrible. Again and\nagain fresh troops had taken the places of those who had formed the\nfront ranks, but many of their best and bravest had fallen.  The\nconfusion was too great for their leaders to be able to direct them\nwith advantage, and seeing the failure of every effort to break\nthe Scottish ranks, borne back by the slow advance of the hedge of\nspears, harassed by the archers who dived below the horses, stabbing\nthem in their bellies, or rising suddenly between them to smite\ndown the riders with their keen, heavy, short handled axes, the\nEnglish began to lose heart, and as they wavered the Scotch pressed\nforward more eagerly, shouting, \"On them! on them! They give way!\nthey give way!\"\n\nAt this critical moment the servants, teamsters, and camp followers\nwho had been left behind Gillies Hill, showed themselves. Some of\ntheir number from the eminence had watched the desperate struggle,\nand on hearing how their soldiers were pressed by the surrounding\nhost of English men-at-arms they could no longer remain inactive.\nAll men carried arms in those days. They hastily chose one of their\nown number as leader, and fastening some sheets to tent poles as\nbanners, they advanced over the hill in battle array, and moved\ndown to join their comrades.  The sight of what they deemed a fresh\ndivision advancing to the assistance of the Scotch brought to\na climax the hesitation which had begun to shake the English, and\nensured their discomfiture. Those in rear turned bridle hastily,\nand crossing the Bannock Burn, galloped away. The movement so begun\nspread rapidly, and although those in front still continued their\ndesperate efforts to break the line of Scottish spears, the day was\nnow hopelessly lost. Seeing that this was so, the Earl of Pembroke\nseized the king's rein and constrained him to leave the field with\na bodyguard of 500 horse. Sir Giles de Argentine, who had hitherto\nremained by the king's side, and who was esteemed the third best\nknight in Europe--the Emperor Henry of Luxemberg and Robert\nBruce being reckoned the two best--bade farewell to the king as\nhe rode off.\n\n\"Farewell, sire,\" he said, \"since you must go, but I at least must\nreturn; I have never yet fled from an enemy, and will remain and\ndie rather than fly and live in disgrace.\"\n\nSo saying, the knight spurred down to the conflict, and charged\nagainst the array of Edward Bruce, and there fell fighting valiantly.\nThe flight of the king and his attendants was the signal for a\ngeneral rout. Great numbers were slain, many men were drowned in\nthe Forth, and the channel of the Bannock was so choked with the\nbodies of dead men and horses that one could pass over dry shod. The\nscattered parties of English were still so numerous that Bruce held\nhis men well in hand until these had yielded themselves prisoners.\nDouglas was charged to pursue the king, but he could only muster\nsixty horsemen. A short distance from the field he met a Scottish\nbaron, Sir Laurence Abernethy, with twenty-four men-at-arms,\non his way to join the English, for even as yet but few of the\nScottish nobles were on the side of the king. Upon hearing what had\nhappened, Sir Laurence, with the easy facility which distinguished\nthe Scottish nobles of the period, at once changed sides, swore\nfealty to Bruce, and joined Douglas in the pursuit of his late\nfriends.  They overtook the king's party at Linlithgow, but Pembroke\nkept his men well together, and while still retiring, showed so\nbold an appearance that Douglas did not venture to charge. Finally\nthe English reached the Castle of Dunbar, where the king and his\nimmediate attendants were received by his ally, Earl Patrick of\nDunbar. So cowed were the fugitives that they left their horses\noutside the castle gate, and these were captured by their pursuers.\nThe main body of the king's bodyguard continued their way in good\norder, and reached Berwick in safety. Edward gained England in\na fishing boat from Dunbar.  Eighteen years had elapsed since his\nfather had entered Scotland with an army deemed sufficient for its\nentire subjugation; had sacked and destroyed the rich and prosperous\ntown of Berwick, routed the army of Baliol, marched through Scotland,\nand, as he believed, permanently settled his conquest.  Now the\nson had lost all that his father had won.\n\nAmong the fugitive remains of the English army were a considerable\nbody of Welsh, who, being lightly armed, fled at full speed toward\nthe Border, but being easily distinguished by their white dresses\nand the absence of defensive armour, almost all were slain by\nthe peasantry. The Earl of Hereford, the Earl of Angus, Sir John\nSeagrave, Sir Anthony Lucy, Sir Ingram de Umfraville, with a great\nnumber of knights, 600 men-at-arms, and 1000 infantry, keeping\ntogether, marched south toward Carlisle.\n\nAs they passed Bothwell Castle, which was held by the governor for\nEngland, the earls and knights entered the castle, their followers\nremaining without; but the governor, on hearing the result of the\nbattle, closed the gates and took all who had entered prisoners,\nand, changing sides, handed them over to Bruce. Their followers\ncontinued their march south, but were for the most part slain or\ntaken prisoners before they reached the Border.\n\nWhen all resistance had ceased on the field the victors collected\nthe spoil. This consisted of the vast camp, the treasures intended\nfor the payment of the army, the herds of cattle, and stores of\nprovisions, wine, and forage; the rich wearing apparel and arms\nof the knights and nobles killed or made prisoners, many valuable\nhorses, and the prisoners who would have to be ransomed, among whom\nwere twenty-two barons and sixty knights.\n\nThe spoil was estimated at 200,000 pounds, equal to 3,000,000\npounds of money in these days. The king refused to take any share\nin this plunder, dividing it wholly among his troops. 30,000 English\nlay dead on the field, including 200 knights and 700 esquires, and\namong the most distinguished of the dead were the Earl of Gloucester,\nSir Giles de Argentine, Lord Robert Clifford, Sir Edmund Manley,\nseneschal of England, Sir William de Mareschal, Sir Payne Tybtot,\nand Sir John Comyn. Sir Marmaduke de Twenge was among the prisoners.\n\nBruce's conduct to his prisoners was even more honourable to himself\nthan was the great victory that he had won. In spite of his three\nbrothers, his brother in law Seaton, his friends Athole and Frazer,\nhaving been executed by the English, and the knowledge that their\nmangled remains were still exposed over London Bridge and the\ngates of Carlisle and Newcastle--in spite of the barbarous and\nlengthened captivity of his wife, his sister and daughter, and his\nfriend the Countess of Buchan--in spite of the conviction that\nhad he himself been made prisoner he would at once have been sent\nto the scaffold--Bruce behaved with a magnanimity and generosity\nof the highest kind. Every honour was paid to the English dead, and\nthe bodies of the chief among these were sent to their relatives in\nEngland, and the prisoners were all either ransomed or exchanged.\nSir Marmaduke de Twenge was dismissed free of ransom and loaded\nwith gifts, and even the Scotch nobles, such as Sir Philip Mowbray,\nwho were taken fighting in the ranks of their country's enemy, were\nforgiven.  This noble example exercised but little influence upon\nthe English. When Edward Bruce was killed four years afterwards\nat Dundalk in Ireland, his body was quartered and distributed, and\nhis head presented to the English king, who bestowed upon Birmingham--who\ncommanded the English and sent the gift to him--the dignity\nof Earl of Louth.\n\nAmong the prisoners was Edward's poet laureate, Baston, a Carmelite\nfriar, who had accompanied the army for the purpose of writing\na poem on the English victory.  His ransom was fixed at a poem on\nthe Scotch victory at Bannockburn, which the friar was forced to\nsupply.\n\nWith Bannockburn ended all hope on the part of the English of\nsubjugating Scotland; but the war continued fitfully for fourteen\nyears, the Scotch frequently invading England and levying heavy\ncontributions from the northern counties and towns, and the English\noccasionally retaliating by the same process; but at length peace\nwas signed at Northampton.\n\nIn 1315 a parliament assembled at Ayr for the purpose of regulating\nthe succession to the throne. It was then agreed that in case of the\nking's death without male issue his brother Edward should succeed\nto it, and that if Edward left no heirs, the children of Marjory,\nthe king's daughter, should succeed.  Shortly afterwards Marjory was\nmarried to Walter the Steward. Edward Bruce was killed unmarried.\nA son was afterwards born to the king, who reigned as David II,\nbut having died without issue, the son of Marjory and the Steward\nbecame king. The hereditary title of Steward was used as the surname\nfor the family, and thus from them descended the royal line of\nStewart or Stuart, through which Queen Victoria at present reigns\nover Great Britain, Ireland, and their vast dependencies.\n\nAfter Bannockburn Archie Forbes went no more to the wars. He was\nraised to the dignity of Baron Forbes by the king, and was ever\nrewarded by him as one of his most trusty councillors, and his\ndescendants played a prominent part in the changing and eventful\nhistory of Scotland; but the proudest tradition of the family was\nthat their ancestor had fought as a patriot by the side of Bruce\nand Wallace when scarce a noble of Scotland but was leagued with\nthe English oppressors of their country.\n\n\n\n\nTHE END\n\n\n\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of In Freedom's Cause, by G. A. Henty\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n#  **ACKNOWLEDGEMENTS**\n\nMany thanks to R\u00f3is\u00edn Magill for her advice and assistance in the creation of this book. Thanks also to Peter Sanger of the _Antigonish Review_ , Boris Castel of _Queen's Quarterly_ , Karen Mulhallen of _Descant_ , and Charise Foster of _CV2_ \u2013 all of whom published excerpts from _Nevermore_ in their magazines. And to the late Robert Wagner, who broadcast a sequence of these poems on _CBC Radio_ ; as well as to the _Canada Council for the Arts_ , the _Ontario Arts Council_ , and the _Writer's Trust George Woodcock Fund_ for their generous financial assistance.\n\nI also wish to acknowledge my editor Allan Briesmaster for his wise counsel and patience; my designer Julie McNeill for an inspired and creative transformation of the manuscript into this beautiful book; and to the editors and staff of Quattro Books: Beatriz Hausner, John Calabro, Luciano Iacobelli, Sonia Di Placido, Lisa Young and Maddy Curry, for their enthusiastic commitment to this publication.\n\n#  **ILLUSTRATION CREDITS**\n\nMaurice Wilson \u2013 pages 52, 82, 92, 106, 138, 144, 152, 158\n\nMick Loates \u2013 pages 5, 58, 82, 100, 178\n\nTim Bramfitt \u2013 pages 20, 26, 32, 64, 70, 132, 166\n\nPeter Hayman \u2013 pages 120, 126\n\nAugsburg Artist \u2013 page 40\n\nRichard Lydeckker \u2013 page 46\n\nJohn James Audubon \u2013 page 112\n\n# **BIBLIOGRAPHY**\n\nALLEN, G. M. 1947 Extinct and Vanishing Mammals of the Western Hemisphere. Spec. Publ. Am. Comm. Internat. Wildlife Protection, 11. New York.\n\nAMADON, D. 1950 The Hawaiian honeycreepers (Drepaniidae). Bull. Amer. Mus. Nat. Hist. 95, 157-257.\n\nANONYMOUS [probably Tafforet] 1726 Relation de l'\u00cele de Rodrigue. [Ms. In Minist\u00e8re de la Marine, Paris]. Quoted in Hacisuka, op. cit. and in Strickland & Melville, and Vinson, op. cit. infra.\n\nARNOLD, E. N. 1979 Indian Ocean giant tortoises: their systematica and island adaptations. Phil. Trans. Roy. Soc. Lond. B 286 127-146.\n\nARNOLD, E. N. 1980 Recently extinct reptile populations from Mauritius and R\u00e9union, Indian Ocean. Jour. Zool., London 191 33-47.\n\nAUDUBON, J. J. L. & BACHMAN, J. 1854 Quadrupeds of North America, 3 vol. New York.\n\nBALES, S. L. & TANNER, N. 2007 Ghost Birds: Jim Tanner and the Quest for the Ivory-billed Woodpecker, 1935-1941. Knoxville: University of Tennessee.\n\nBEECHEY, F. W. & VIGORS, N. A. 1839 The zoology of Captain Beechey's vogage to the Pacific... in the Blossom... 1825 1828. London.\n\nBERGER, A. J. 1972 Hawaiian Birdlife. Honolulu: University Press of Hawaii.\n\nBODDAERT, P. 1785 Elenchus Animalium, Rotterdam.\n\nBOWES, A. J. 1787-89 A journal of voyage from Portsmouth to New South Wales and China in the Lady Penrhyn.\n\nBROWN, R. 1973 Has the Thylacine really vanished? Animals, 15.\n\nBRYDEN, H. A. (Ed.) 1899 Great and Small Game of Africa. London.\n\nCHAMBERS, P. 2004 A Sheltered Life: the Unexpected Life of the Giant Tortoise. London: John Murray.\n\nDELACOUR, J. 1954 The Waterfowl of the World. 4 vol. London: Country Life.\n\nDUBOIS, LE SIEUR. 1669 Journal. Proc. Zool. Soc. Lond. 1884.\n\nDU TERTRE, J. B. 1667-71 Histoire g\u00e9n\u00e9rale des Antilles habit\u00e9es par les Francois, 4 vol. (in 3). Paris.\n\nFINN, F. 1915 Wild Animals of Yesterday and To-Day. London.\n\nFISHER, J., SIMON, N. & VINCENT, J. (Eds) 1969 The Red Book. [IUCN data]. London. Collins.\n\nFORSHAW, J. M. 1973 Parrots of the World. Melborne: Landsdowne. New York: Doubleday.\n\nFORSTER, J. R. 1778 Observations made during a voyage round the world... London. [German edition, slightly expanded: 1779 Berlin.]\n\nFULLER, E. 2001 Extinct Birds. Ithaca, New York: Cornell University Press.\n\nG\u00c9RARD, J. 1861 Lion Hunting and Sporting Life in Algeria. (Gustave Dor\u00e9, illus.) London.\n\nGIBSON, G. 2009 The Bedside Book of Beasts. Toronto: Doubleday.\n\nGMELIN, S. G. 1774 Reise durch Russland, St. Petersburgh.\n\nGOODWIN, D. 1970 Pigeons and Doves of the World. London: British Museum (Natural History).\n\nGOULD, J. 1840-48 The Birds of Australia, 7 vol. [Supplements 1-5, 1851-69]. London. 1845-63 The Mammals of Australia, 3 vol. London.\n\nGRADY, W. 2007 Bringing Back the Dodo. Toronto: McClelland and Stewart.\n\nGRANT, C. 1801 History of Mauritius. [Deals with all Mascarene Islands]. London.\n\nGREENWAY, J. C. Jr. 1967 Extinct and Vanishing Birds of the World, second, revised edition. New York: Dover.\n\nGRIEVE, S. 1885 The great auk, or a garefowl... its history, archaeology, and remains. London.\n\nGUILDER, E. R. 1961 The former Distribution and Decline of the Thylacine. Aust. Hour. Sci. 23 (7). 1966 In pursuit of the Thylacine. Oryx, August 1966.\n\nGUNTHER, A. 1877 The Gigantic Land-Tortoises (Living and Extinct) London: Taylor & Francis.\n\nGUNTHER, A. & HUME, J. 2008 Lost Land of the Dodo London: A. C. Black.\n\nHACHISUKA, M. 1953 The Dodo and Kindred Birds. London: Witherby.\n\nHALL, R. L. & SHARP, H 1978 Wolf and Man, evolution in parallel. New York: Academic Press.\n\nHALLIDAY, T. 1978 Vanishing Birds: Their Natural History and Conservation. London: Sidgwick & Jackson. New York: Holt, Rinehart & Winston.\n\nHARMANSEN, HEEMSKERCK, MATALEIF, SCHOUTEN, SPILBERBEN, VAN\n\nDEN ROECKE, VAN DER HAGEN & NECKE. 1702-06 Receuil des Voiages qui ont servi \u00e0 l'\u00eatablissement... de la Compagnie des Indes Orientales... [French trans of Dutch Journals], 5 vol. Amsterdam.\n\nHARPER, F. 1945 Extinct and Vanishing Mammals of the Old World. Spec. Publ. Am. Comm. Internat. Wildlife Protection, 12. New York.\n\nHERBERT, T. 1638 Some Yeares' Travaile into divers parts of Asia and Afrique... Revised and enlarged by the Author. London.\n\nHINDWOOD, K. A. 1932 An historic diary. Emu, 32, 17-29.\n\nHINDWOOD, K. A. 1938 The Extinct Birds of Lord Howe Island. Austr. Mag. 6, 319-324.\n\nHONEGGER, R. E. 1981 List of Amphibians and Reptiles Either Known or Thought to Have Become Extinct since 1600. Biol. Conserv. 19, 141-158.\n\nHOOSE, P. 2004 Race to Save the Lord God Bird. New York: Farrar, Strauss.\n\nIUCN (International Union for Conservation of Nature and Natural Resources) Derniers Refuges. (Atlas). Brussels: Elsevier.\n\nIUCN Red Data Books (Fish, Reptiles & Amphibians, Birds Plants and Mammals). Continuously updated SCMU (Species Conservation Monitoring Unit), 219c Huntingdon Rd., Cambridge, UK.\n\nJOHNSGAARD, P. A. 1973 Grouse and Quails of North America. Lincoln: University of Nebraska Press.\n\nJONES, F. W. 1924 The Mammals of South Australia. Adelaide.\n\nLEGUAT, F. 1708 Voyages et Aventures de Francois Leguat, & de ses Compagnons, en deux isles d\u00e9sertes de Indes Orientales... 2 vol. London. [English, 1 vol., trans: A New Voyage to the East Indies by Francis Leguat and his companions, 1708. London].\n\nLOPEZ, B. H. 1978 Of Wolves and Men. London: Dent. 1979. New York: Scribner.\n\nLYDEKKER, R. 1912 The Horse and its Relatives, London: George Allen and Unwin.\n\nMACFARLAND, C. G., VILLA, J.& TORO, B. 1974 The Galapagos giant tortoises (Geochelone elephantopus). Part 1: Status of the surviving populations. Biol. Conserv. 6 118\u2013133.\n\nMECH, D. 1970 The Wolf. New York: Doubleday.\n\nNORTH-COOMBES, A. 1971 The Island of Rodigues. Port Louis, Mauritius: privately printed.\n\nOLIVER, W. R. B. 1949 Moas of New Zealand. Dominion Museum Bulletin, 15. Wellington.\n\nOLIVER, W. R. B. 1955 New Zealand Birds. Wellington: Reed.\n\nOWEN, C. 2001 The Wrecks of Eden. Toronto: Wolsak and Wynn.\n\nPADDIE, R. 2000 The Last Tasmanian Tiger. Cambridge.\n\nPOCOCK, R. I. 1913 The affinities of the Antarctic Wolf. Proc. Zool. Soc. London, 382-393.\n\nPOLIAKOV, I. S. 1881 Isvestia Russki Geographischeski. St. Peterburg 17: 1-20.\n\nRADDE, G. 1893 On the present range of the European Bison in the Caucasus. Proc. Zool. Soc. London.\n\nRIDE, W. D. I. 1970 Guide to the Native Mammals of Australia. London: Oxford University Press.\n\nRIPLEY, S. D. 1952 Vanishing and extinct bird species of India. Journ. Bombay Nat. Hist. Soc. 50 (4) 902-906.\n\nRIPLEY, S. D. 1977 Rails of the World. Toronto: Feheley.\n\nROTHCHILD, L. W. R. 1907 Extinct birds. London: Hutchinson and Co.\n\nSCARLETT, R. J. 1974 Moa and Man in New Zealand. Notornis, 21 (1).\n\nSCHORGER, A. W. 1955 The Passenger Pigeon. Madison: University of Wisconsin.\n\nSHELDON, C. 1912 Wilderness of the North Pacific Coast Islands, London.\n\nSHOEMAKER, H. W. 1915 A Pennsylvania bison hunt. Middleburg, Pa.\n\nSILVERBURG, W. S. 1969 The Auk, the Dodo and the Oryx: vanished and vanishing creatures. Tadworth, Surrey: World's Work.\n\nSMITH, C. H. 1841 The Natural History of Horses, Edinburgh.\n\nSTEJNEGER, L. 1936 Georg Wilhelm Steller, the pioneer of Alaskan Natural History. Cambridge, Mass. Harvard University Press.\n\nSTREET, P. 1961 Vanishing animals: preserving nature's rarities. London: Faber.\n\nSTRESEMAN, E. 1950 Birds collected during Capt. James Cook's last expedition (1776-1780) Auk, 67, 66-88.\n\nSTRICKLAND, H. E. & MELVILLE, A. G. 1848 The Dodo and its kindred... London.\n\nSWENK, M. H. 1916 The Eskimo Curlew and its disappearance. Ann. Rep. Smithsonian Inst. For 1915, 325-340.\n\nTANNER, J. T. 1942 The ivory-billed woodpecker. Reprint. New York: Dover.\n\nTEMPLE, S. A. 1974 Last Chance to Save Round Island. Wildlife, Summer 1974.\n\nTHOMPSON, A. L. 1964 A New Dictionary of Birds. London: Nelson.\n\nVAN DENBURGH, J. 1914 The gigantic land tortoises of the Galapagos archipelago. Proc. Calif. Acad. Sci. 2, 203-374.\n\nVINSON, J. & VINSON, J. M. 1969 The Saurian Fauna of the Mascarene Islands.\n\nMaur. Inst. Bull. 6 (4), 2103-320.\n\nWALTERS, M. 1980 The Complete Birds of the World. London: David & Charles.\n\nWARNER, R. E. 1968 The Role of Introduced Diseases in the Extinction of the Endemic Hawaiian Avifauna. Condor, 70.\n\nWATERHOUSE, G. R. & DARWIN, C. 1839 Zoology of the voyage of the Beagle. (Mammalia). London.\n\nZISWILLER, U. 1967 Extinct and Vanishing Animals, revised English edition, F. & P. Bunnell, London: Longmans Green.\n\n# **ABOUT THE AUTHOR**\n\nDavid Day was born and raised in Victoria, British Columbia. His first book _The Cowichan_ (based on his timber camp journals) was published in 1975. In 1978, Day published _A Tolkien Bestiary_ , the first of his six internationally best-selling books on the works of J. R. R. Tolkien. Day has travelled extensively, and lived in England, Greece and Spain. He currently lives in Toronto.\n\nDavid Day's landmark book on animal extinction, the _Doomsday Book of Animals_ \u2013 with an introduction by the Duke of Edinburgh \u2013 was selected in 1981 as a \"Book of the Year\" by _Time Magazine, The Christian Science Monitor, The Los Angeles Times, The Observer_ , and _New Scientist_. This was followed by the _Whale War_ (1987), _Eco-Wars_ (1988), _Encyclopedia of Vanished Species_ (1989), and _Noah's Choice_ (1990). Day has also been an environmental columnist for Britain's _Daily Mail, Evening Standard, Mail on Sunday, Sunday Times_ , and _Punch Magazine_. His _Punch_ columns became the books _True Tales of Environmental Madness_ in 1990 and _The Complete Rhinoceros_ in 1994.\n\nIn 1996 he wrote the _Lost Animals_ British Channel Four and Japanese NHK TV series of one hundred five-minute documentaries on extinct species. It was narrated by the Oscar-winning actress Greta Scacchi, and later was translated into 20 languages. His _Whale War_ was the basis of a British ITV documentary.\n\nDavid Day has also written six illustrated books of animal stories and three illustrated books of animal poems for children. His _Emperor's Panda_ was runner-up for both the Governor General's Award and the National Library Award. His CBC and National Magazine award-winning poems have been praised by both the Canadian poet Al Purdy and the British poet laureate Ted Hughes.\n\n##  **THE DODO AND THE PHOENIX**\n\n###  **F IRST WATCH 6 A. M. PRIME**\n\n####  **DODO \u2013 1680 \u2013 _Raphus cucullatus_**\n\n#####  Sir Thomas Herbert \u2013 1629  \n_**Some Yeares Travaile,** _ **Mauritius**\n\nFirst, here and here only is generated the Dodo, which for shape and rareness may antagonize the Phoenix of Arabia: her body is round and fat, few weigh less than fifty pound, are reputed of more for wonder then for food. Her visage darts forth melancholy, as sensible of Nature's injurie in framing so great a body to be guided with complementall wings, so small and impotent, that they serve only to prove her a bird. The halfe of her head is naked seeming couered with a fine vaile, her bill is crooked downwards, in midst is the thrill, from which part to the end tis of a light greene, mixt with pale yellow tincture; her eyes are small and like to Diamonds, round and rowling; her clothing downy feathers, her traine three small plumes, short and inproportionable, her legs suting to her body, her punces sharpe, her appetite strong and greedy.\n\n> Sir Thomas Herbert was appointed by King Charles I to the first English ambassadorial mission to the court of the Shah of the Persian Empire. This mission (1627-29) was an attempt to negotiate a silk trade route (bypassing the Ottoman Turkish Empire) through lands and waters controlled by the Persians. It was on his return journey \u2013 via India and Africa \u2013 that Herbert's ship _The Hart_ stopped in Mauritius and he wrote this first account of the Dodo in English.\n> \n> The Dodo was the world's largest member of the pigeon family: a giant flightless dove the weight of two large domestic turkeys. The Portuguese were the first Europeans to briefly encounter the Dodo in 1507, but the Dutch were the first to record its existence in 1581, and name it _Dodoor_ because it resembled a gigantic version of the Dutch Little Grebe or _Dodaers_ , meaning \"plump-arse.\" Sometimes called the \"Devil's Chicken,\" the Dodo's desirability as food for sailors varied with the seasons. When fruit was abundant, it was succulent, but during the lean season, the flesh was very tough and ill-tasting.\n> \n> The claim that \"dodo\" is derived from the Old Portuguese _duodo_ meaning \"fool\" appears to be without foundation. A more recent theory suggests the name was simply imitative of its 2-note pigeon-like call: \"doo-doo.\" The Dodo's reputation as a creature \"unfit\" for survival seems odd considering its evolution and survival over five million years; or twice as long as humans have existed.\n\n####  **WHITE DODO \u2013 1770 \u2013 _Victorianis imperialis_**\n\n#####  James Tatton \u2013 1631  \n_**Voyage of Captain Carleton,** _ **Reunion**\n\nThere is a store of Land-fowl, both small and great, plenty of Doves, great Parrots and such like; and a great fowl of the bigness of a Turkey, very fat, and so short-winged that they cannot fly, being white, and in a manner tame; and so are all other fowles, as having not been troubled or feared with shot.\n\n> Dodos were originally believed to have inhabited all three of the Mascarenes: those remote Indian Ocean islands of Mauritius, Rodriguez and Reunion. Fought over by the Portuguese, Dutch, French and British, Rodriguez was eventually successfully colonized by the French East Indian Company. An island with an area of just 110 sq. km., it was inhabited by 60,000 plantation slaves. Like all the Mascarenes, Reunion originally had no land mammals and virtually no predators, but was rapidly invaded by feral cats, dogs, rats, monkeys, pigs and goats, all of whom fed on the indigenous species. The result has been that virtually all of the Mascarenes' endemic animal species and nearly half of its endemic plant species are either endangered or extinct.\n> \n> Just as the Dodo is on the Mauritius coat of arms, the White Dodo has become the emblematic bird of Reunion. Indeed, in the 1990's the national television network adopted it as the hero of a major animated television series. However, recent archaeological excavations have proved that the legendary White Dodo of Reunion was not a Dodo at all.\n> \n> It was, in fact, a White Flightless Ibis \u2013 _Threskiornis solitarius_ \u2013 which became extinct in 1770. This large white flightless bird of Reunion was related to the Sacred Ibis of Egypt. A remarkable and unique species, but it was not a member of the Rhaphidae family of the Mauritius Dodo and the Rodriguez Solitaire.\n\n####  **DODO \u2013 1680 \u2013 _Raphus cucullatus_**\n\n#####  Sir Hamon L'Estrange \u2013 1638  \n_**L'Estrange Diaries,** _ **London, England**\n\nUpon a street there was a house near Lincoln Fields where I discovered the picture of a strange fowle hung out upon a clothe. Upon paying for entry, there I saw on display a great fowl, bigger than the largest Turkey Cock, but stouter and thicker and of a more erect state. The keeper called it a Dodo, and in the end of a chimney in the chambers there lay a heape of large pebble stones, whereof hee gave it many in our sight, some as big as nutmegs and the keeper told us she eats them conducing to digestion.\n\n> Sir Hamon L'Estrange was an English theologian who while walking in central urban London stopped at a house to examine a live captive Dodo. L'Estrange's interest in the Dodo may also have been provoked by Herbert's account. His _Some Yeares Travaile_ was a popular book of the day.\n> \n> Curiously, upon its demise, this Dodo was acquired as a stuffed exhibit by John Tradescant (creator of Charles II's \"physics garden\") for his \"cabinet of curiosities.\" This was eventually integrated with the collection of the antiquarian Jacob Ashmole to form the basis of the Ashmolean Museum in Oxford. Here, the world's only intact Dodo was kept, until the deteriorating specimen was accidentally incinerated, although fortunately the head and one foot were rescued. The excavations of the dry swamp bed of the Mare Aux Songes in Mauritius resulted in the discovery of numerous new skeletal remains of Dodos in 1865; and in 2005.\n> \n> Still one might argue that there is a phoenix-like aspect to this story, as this famous specimen in Oxford was the inspiration for the creation of Lewis Carroll's fictional Dodo in his _Alice in Wonderland_ (1865). The Dodo was a private joke the author shared with Alice about himself. Carroll (a.k.a. Charles Dodgson) suffered from stuttering, and often introduced himself as Mr. Do-Do-Dodgson.\n> \n> The Dodo was the subject of a number of paintings by European artists: most notably Roeland Savery's iconic bird of 1626. However, the Mughul artist, Mansur of Naggash \u2013 in the court of the Emperor Jehangir in Goa \u2013 created the most well-observed painting of a Dodo drawn from life, sometime between 1610 to 1624.\n\n####  **SOLITAIRE DODO \u2013 1780 \u2013 _Pezohaps solitarius_**\n\n#####  Francois Leguat of Bresse \u2013 1708  \n_**A New Voyage to the East Indies,** _ **Rodriguez**\n\nThe Females are wonderfully beautiful, some fair, some brown; I call them fair, because they are the colour of fair Hair. They have a sort of Peak, like a Widow's upon their Beak, which is of a dun colour. No one feather is straggling from the other. The Feathers on their Thighs are round like shells at one end, and being there very thick, have an agreeable effect. They have two Risings on the Craws, and Feathers are whiter there than the rest, which lively represents the fine bosom of a Beautiful Woman. They walk with so much stateliness and good Grace, that one cannot help admiring and loving them.\n\n> During the reign of Louis XIV, Francois Leguat was forced to flee to Holland; where in 1691, along with other prominent Huguenots, he was persuaded to attempt to establish a new French Protestant Refuge Colony in the \"New Eden\" of Reunion. They were promised two ships to service the colony of L'ile Eden, but instead they found themselves on a single small frigate that deserted them on Rodriguez Island.\n> \n> After two years in the failed Rodriguez colony, Leguat and the others escaped on a small boat to Mauritius, and were imprisoned by the Dutch both in Mauritius and Batavia for several years. At the age of 70 the wandering Leguat found sanctuary in London, and published his _Voyages \u2013 1689-98_ in French, English and Dutch editions. Leguat's account of his harrowing desert island adventures included descriptions of the Dodo that were entirely at odds with any previous tales; however, they have proved to be extremely accurate. Leguat recorded their mating rituals. Noting that these birds mated for life, he also remarked how they were far from passive when it came to protecting their young.\n> \n> The Solitaire was the intelligent, beautiful Cinderella of the Dodo (Raphidae) family. Nonetheless, these characteristics did not save it from sharing the fate of its less charming cousin. Nor did it save at least two dozen other indigenous Mascarene species (pigeons, parrots, rails, tortoises, snakes, and skinks) from extinction.\n\n####  **LA MARE AUX SONGES**\n\n##### Dodo \u2013 1680\n\nThe first ships appear off these shores\n\nJust as Galileo's telescope reveals\n\nThe dry seas of the moon\n\nNow, we dig and search for relics: a beak or a bone\n\nOf a creature that will be nevermore forever\n\nSomething in the Dodo's foolish Mona Lisa smile\n\nProvoked Mogul artists and Dutch masters\n\nAnd provokes us still, as we discover\n\nA human skull with an equally enigmatic grin\n\nNext to the bones of this flightless giant dove\n\nIt is impossible to stave off thoughts\n\nOf our own eventual eclipse\n\nHow will we be seen: tragic or comic?\n\nHubristic, or an evolutionary mistake?\n\nIn what arid sea and by whom will our relics be found\n\nWhen our monuments crumble away?\n\nI try to imagine Alexander or Ozymandias\n\nBut instead, as darkness falls\n\nOn the edge of this dry \"Sea of Dreams\"\n\nThe moon with the bone-white face of Pagliacci\n\nLooms up before me\n\nOnly then do I recall his murdered wife\n\n_Columbina,_ meaning \"Dove\"\n\nAnd hear: _\"Le commedia e finita!\"_\n\nGod's judgement, delivered in Caruso's terrible\n\nPerfect voice\n\n##  **SINBAD AND THE RUKH**\n\n###  **S ECOND WATCH 7 A. M. PRIME**\n\n####  **ELEPHANT BIRD or RUKH \u2013 1700**  \n_**Aepyornis maximus** _\n\n#####  Marco Polo \u2013 1298  \n_**Travels: Concerning the Isle of Madastar,** _ **Madagascar**\n\nThe people of this island report that at a certain season of the year, an extraordinary giant bird, which they call a Rukh, makes an appearance in the southern regions. The Grand Khan having heard this extraordinary relation, sent messengers to the island to enquire about these curious matters.\n\n> Marco Polo's account of Madagascar was written in 1294 after his return by ship via the Indies and the coast of Africa from the court of the Great Khan. Arab traders had for centuries before Europeans traded along the coast of Africa and long believed Madagascar to be the home of this legendary giant Rukh or Roc. Anyone familiar with the stories of Sinbad and Aladdin from the _Thousand and One Nights_ was acquainted with this extraordinary bird; but its celebrity is not confined to that work.\n> \n> _\"Rukh,\"_ says one Arabic and Persian Dictionary, \"is the name of a monstrous bird, which is said to have powers sufficient to carry off a live rhinoceros.\" Its existence seems to have been universally credited in the East; and those Arabian navigators with whom Marco Polo conversed did not hesitate to attest to its existence.\n> \n> However, it was not until 1866 that a complete skeleton of this bird was discovered. And it was not until then that the bird's size was calculated. It may have not been the size of the mythical Roc, but with a height of ten feet and weight of half a ton, it was certainly the largest and heaviest bird to ever walk the earth.\n> \n> The name Elephant Bird seems somewhat peculiar, as Madagascar has never had an indigenous elephant population. It does serve to indicate the bird's huge size, or perhaps is a reference to Sinbad's legend, wherein it was so large it was capable of carrying off elephants. Its scientific name, _Aepyornis,_ means \"tall or lofty bird.\"\n\n#####  Sieur Etienne de Flacourt \u2013 1658  \n_**History of the Great Island of Madagascar** _\n\nThe Vouron Patra is a giant bird that lives in the country of the Amphrates people in the south of Madagascar. The great eggs of these birds are used to store water. So that people of these places may not catch it, the Vouron Patra seeks the loneliest places.\n\n> France took possession of Madagascar in 1642, and Flacourt was the island's first governor and a keen naturalist. Shortly after completing his _History,_ Flacourt was killed by Algerian pirates while on a return voyage to France, and there were no other written accounts of this bird for nearly two centuries.\n> \n> Then, in 1832, Victor Sganzin, a French artillery officer stationed in Madagascar, found one native family using as a water bowl half an eggshell a foot in diameter. Upon querying, he was told that such eggs were quite common in certain parts of the island. Sganzin purchased the egg and sent it to Jules Verreaux, a French collector of curiosities in Cape Town. Verreaux excitedly sailed for Paris with his discovery. Sadly, a terrible storm struck and Verreaux, the ship and the Roc's egg all sank beneath the waves.\n> \n> The Elephant Bird's egg is the largest that any egg could possibly grow. It is estimated that any egg larger would have to have a shell so thick that it could not be broken by a hatching chick. Over the last century a considerable number of these eggs have been discovered, the largest measuring a metre in circumference with a fluid capacity of two imperial gallons \u2013 equal to 200 chicken eggs, or three times the size of the largest dinosaur eggs.\n\n####  Ship Surgeon John Joliffe \u2013 1848  \n_**Voyage to the Spice Islands,** _ **Indian Ocean**\n\nMonsieur Dumarele, a merchant of Reunion, spoke of having seen at Port Leven the shell of an enormous egg, the production of an unknown bird inhabiting the wilds of the country, which held the almost incredible quantity of _thirteen wine quart bottles of fluid!!!_ He having himself carefully measured the quantity. Dumarele offered to purchase the egg from the natives, but they declined selling it, stating that it belonged to their chief, and that they could not dispose of it without his permission. The natives said that the egg was found in the jungle, and observed that such eggs were very rarely met with, and that the bird which produces them is still more rarely seen.\n\n> John Joliffe was the surgeon on a British ship that stopped in Madagascar in 1848, where he was befriended by Dumarele, a trader from Reunion, who traded extensively along the Madagascar coast. Joliffe was somewhat sceptical, but reported his account to the British ornithologist, Hugh Strickland. Lacking physical evidence, Strickland replied: \"The sight of one sound egg would be worth a thousand theories.\"\n> \n> Two years later, in 1850, an awe-struck Geoffroy Saint-Hilaire, the director of the Paris Zoo, received not one, but three \"sound eggs\" along with a crate of massive bones of this bird. These were collected by Captain Abadie, master of a French merchant ship recently returned from Madagascar. The largest measured approximately 13\" by 16\" in diameter and had a fluid capacity of nine litres.\n> \n> Arab and Malay settlers came to settle Madagascar about a millennium before the French, and began to cut down its virgin forest and invade the nesting grounds that once were the sandy shores of Madagascar. When the final blow fell \u2013 whether by gun, spear or fire \u2013 is not known, although by 1700 the Elephant Bird was almost certainly extinct.\n> \n> The discovery of Elephant Bird eggs in the late 19th century prompted H. G. Wells to write a little-known pre _-Jurassic Park_ short story entitled \"Aepyornis Island.\"\n\n#####  **THE RUKH'S EGG**\n\n###### Elephant Bird or Rukh \u2013 1700\n\n1.\n\nO Kings of the Ages \u2013 what deeds have been done?\n\nWhat taboos broken and whirlwinds unleashed?\n\nFor Al\u00e1 al-Din's wish to possess the Rukh's Egg\n\nThe mighty Slave of the Lamp turned\n\nOn his Master with the threat of annihilation:\n\n_\"O Ignorant Man, Know ye not what ye ask?_\n\n_Know ye not from its outward form_\n\n_What is held within the Rukh's Egg?_\n\n_For this wish I should slaughter you!\"_\n\nAnd then came the Destroyer of Worlds,\n\nThe Plunderer of Palaces, the Despoiler of Cities.\n\n2.\n\nAnd yet, for all else the great Jin\n\nAnswered Al\u00e1 al-Din, with a flourish:\n\n_\"To hear is to obey!\"_\n\nIn a single night, the Slave of the Lamp\n\nRaised a grand pavilion all built of alabaster,\n\nOf Sumaki-marble, of jasper and jet and jade.\n\nAnd beneath the great dome was a belvedere\n\nWith the latticed casements of four and twenty windows\n\nGlazed and glittering with rubies and with emeralds.\n\nAnd therein, Al\u00e1 al-Din was placed upon a golden throne\n\nAnd guarded by mamuluks and eunuchs\n\nAnd served by handmaids and dancers\n\nAnd eight and forty slave girls.\n\nAnd all were covered in raiments of silk\n\nAnd gold brocades and or frayed cloths\n\nEmbroidered with pearls and precious gems\n\nLit with candles of camphor and ambergris\n\nAnd filled with the sounding of trumpets\n\nAnd beating of kettle drums.\n\nBut all this ended with Al\u00e1 al-Din's fatal wish.\n\nFor at that utterance, from the abyss a voice thundered:\n\n_\"By Allah, ye deserve that I reduce you to ashes_\n\n_This very moment and scatter you upon the wind._\n\n_\"Better ye wish to hang the mother of the Prophet_\n\n_From the pavilion dome.\"_\n\n3.\n\nOnce prized above all things,\n\nWhat was the meaning of the Rukh's Egg?\n\nWhat made it the one great taboo?\n\nLike Al\u00e1 al-Din, we are still unknowing\n\nOf its inner from its outer significance.\n\nWhat was the secret of this talisman\n\nKnown only to the Wise?\n\nWhat is here to be discovered?\n\nWhat fatal prophecy is to be revealed?\n\nThe Rukh's Egg is like a planet unto itself,\n\nBut bone white, barren as the moon\n\nAnd now hollow, bereft of all life:\n\nVision of our world to come.\n\n##  **LA GRANDE CARAPACE**\n\n###  **T HIRD WATCH 8 A. M. PRIME**\n\n####  **MAURITIAN GIANT TORTOISES \u2013 1700**  \n_**Cylindraspis inepta; Cylindraspis trisserata** _\n\n#####  Sir Thomas Herbert \u2013 1630  \n_**Some Yeares Travaile,** _ **Mauritius**\n\nThe Ile has no humane inhabitants. Those creatures that possesse it, have it on condition to pay tribute to such ships as famine, or foule weather force to anchor there. Here are land Tortoyses so great that they will creepe with two men's burthen, and serve more for sport than service or solemne Banquet.\n\n> Sir Thomas Herbert, who gave us the first descriptions of the Dodo in English, also wrote this early account of the similarly doomed Mauritius Giant Tortoises. There were two species of giant tortoise on Mauritius, but sadly Herbert was quite wrong about them being unpalatable. The French and Portuguese slaughtered tens of thousands for meat for their crews on their Indian Ocean ships. On Mauritius, one Abb\u00e9 Pengr\u00e9 noted that, while on the island, his diet consisted entirely of _\"soupe de tortue, tortue enficasee, tortes en doube, tortue en godiveau, oeufs de tortue, foie de tortue, tells etaient Presque nos unique ragouts.\"_ Predictably, both species soon became extinct.\n> \n> Curiously, after his return to England, Sir Thomas Herbert became embroiled in the civil war (on the side of Cromwell), and found himself watching over another endangered species: the British monarchy. Herbert was appointed servant and jailer to the imprisoned Charles I. During the King's last months, Herbert was the monarch's sole attendant, and accompanied the King to the scaffold. Later, after the phoenix-like resurrection of the monarchy, Herbert was made a baronet by Charles II for his kindness toward Charles I, and wrote his compassionate _Threnodia Carolina \u2013 Last Days of Charles I._ The _Threnodia_ was published in 1678, a date that almost exactly coincides with the extinction of Herbert's Dodo, and is within two decades of the same fate befalling the Mauritius Tortoises.\n\n####  **REUNION GIANT TORTOISES \u2013 1773**  \n_**Cylindraspis indica; Cylindraspis borboica** _\n\n#####  Marquis Henri Du Quesne \u2013 1689  \n_**Memoire \u2013 Projet de Republique a L'ile Eden,** _ **Reunion**\n\nThere are vast numbers of Tortoises: their flesh is very delicate and the fat better than butter or the best oil, for all kinds of sauces which is also a good remedy for many ills. The biggest ones can carry a man with greater ease than a man can carry them. Each animal usually makes two pots of oil and twenty people can be fed from one of these turtles.\n\n> Here the Marquis Henri Du Quesne writes glowingly of one of the many bounteous natural treasures awaiting colonists he is commissioned to transport to the L'ile Eden \u2013 or Reunion \u2013 the largest of the three Mascarene islands. As on Mauritius, the two species of Giant Tortoise on Reunion were slaughtered in the thousands to supply meat for traders.\n> \n> However, this new Eden for the exiled and endangered Huguenots failed. In fact, Du Quesne was the naval commander of the frigate _La Hirondelle,_ whom Francois Leguat accuses of abandoning the colonists (including himself) on the uninhabited Rodriguez.\n> \n> To give some idea of the number of giant tortoises once inhabiting these islands, it should be observed that in the widely scattered array of islands in the Seychelles Group, there is the tiny islet of Aldabra Atoll. Measuring less than eighteen by seven miles (most of which is a lagoon), it was inhabited by over one hundred thousand tortoises. Uninhabited by humans and without a safe harbour or sufficient fresh water, Aldabra was the habitat of the only surviving species of Indian Ocean Tortoise ( _Aldabrachelys gigantean_ ).\n\n####  **RODRIGUEZ GIANT TORTOISES \u2013 1800**  \n_**Cylindraspis vosmaeri; Cylindraspis peltastes** _\n\n#####  Francois Leguat of Bresse \u2013 1708  \n_**A New Voyage to the East Indies,** _ **Rodriguez**\n\nThere are such plenty of Land Turtles on this Isle, that sometimes you see 2-3000 of them in a Flock; so that you can go above a hundred paces on their backs without setting foot on the ground. They meet together in the evening in shady places, and lie so close, that one would think those spots were paved with them.... We all unanimously agreed 'twas better than the best Butter in Europe. To anoint one's self with this Oil is an excellent Remedy for Surfeits, Colds, Cramps, and several other Distempers. The Liver of the animal is extraordinarily delicate, 'tis so Delicious that one may say of it, it always carries its own Sauce with it, dress it how you will.\n\n> During the reign of Louis XIV, Francois Leguat was forced to flee to Holland, where, in 1691, along with other Huguenots, he was persuaded to establish a new Protestant Colony in the \"New Eden\" of Reunion. They were promised two ships, but instead were deserted on Rodriguez.\n> \n> In 1693, after two years as the only inhabitants on an island 40 miles square, they built a boat and sailed to Mauritius, where they were imprisoned before they escaped to Batavia, where they were once again imprisoned. When finally released by the Dutch after seven years, Leguat and two others were the only survivors of L'ile Eden.\n\n#####  Mah\u00e9 de Labourdonnais \u2013 1730  \n_**French East India Company Decree** _ **, Rodriguez**\n\nAll ship's captains must be forbidden to send out their boats to collect tortoises without informing the island commandants and stating the numbers they require.\n\n> Though pirates and occasional Dutch naval ships had been taking tortoises from Rodriguez for some time, it was not until Leguat's memoirs were published in 1708 that the island came to be regarded as a meat reservoir for the French and English navies. By 1730, Reunion Tortoises were becoming scarce and the Mauritian herds had entirely vanished. This decree was made \u2013 not so much to preserve the island tortoises \u2013 as to attempt to enforce a monopoly on their harvesting. In any case, the decree was ignored and the tortoise populations continued to rapidly plummet.\n> \n> Rodriguez was later re-colonized by the Mauritius governor Mah\u00e9 de Labourdonnais with a small group of soldiers, lascars and slaves who were required to gather live tortoises. They sent 10,000 tortoises off annually. Some shiploads were of 6,000 tortoises and on several occasions three quarters of the cargo perished. The last big haul was in January 1768 when L'Heureux took off 1215 \"carosses\" (the largest size tortoises).\n> \n> In 1791, the last overseer, Jean de Valgny, died on Rodriguez, a virtual castaway dependent for food for himself and two slaves largely on the generosity of visiting ships. For the tortoises were gone and, with them, went France's interest in Rodriguez. The last tortoises ever seen were two down at the bottom of an inaccessible ravine in 1795.\n\n####  **MARION'S GIANT TORTOISE \u2013 1929** _**Cylindraspis sumeri** _\n\n#####  Albert Gunther \u2013 1875  \n**Letter to _The Times,_ London**\n\nAs to the origin of the large tortoise \u2013 known as _Marion's Tortoise_ \u2013 living in Mauritius... I am afraid positive evidence will be obtained only when, after the death of the animal, the bones can be compared with those of the other Mascarene tortoises. But I trust that great care will be taken in prolonging the existence of one of the oldest terrestrial creatures and, probably, the last of its race.\n\n> Marion's Tortoise was named after Marc-Joseph Marion du Fresne who was a renowned naval officer, trader and explorer. In 1766, the Chevalier Marion brought this captive tortoise from Reunion to Mauritius. There in Port Louis the captain served as harbour master, and upon his departure, placed the tortoise in the care of the French Garrison. Marion's Tortoise remained there until 1810, when the British bombarded Port Louis and forced a French surrender. During the attack, Marion's Tortoise was wounded \u2013 scarring its shell \u2013 but fortunately recovered to serve as the British Royal Artillery mascot for another half century before it came to the attention of Albert Gunther of the zoology department of the British Museum through correspondence with Captain Samuel Pasfield Oliver who served as an artillery officer on Mauritius.\n> \n> In the months following his letter to _The Times_ , Albert Gunther received photographs and numerous reliable accounts that proved beyond doubt that Marion's Tortoise was indeed the last living Mascarene Tortoise \u2013 and probably the only one still living after 1800. It was largely due to Gunther's persistence in gathering the public support of Walter Baron Rothschild, Sir Joseph Hooker, Richard Owen and Charles Darwin that any Giant Land Tortoises elsewhere in the world survived into the twentieth century.\n> \n> Marion's Tortoise lived on as the Royal Artillery mascot until 1929, when he died as the result of a fall down a well. He had been in captivity for 163 years. His actual age is unknown, but as he was brought to Mauritius as a large mature adult, he must have survived for at least two entire centuries.\n\n####  **METHUSELAH'S TURTLE**\n\n##### Marion's Giant Tortoise \u2013 1929\n\nIn the year 1929, Marion's Tortoise\n\nSuffered a fatal fall\n\nHe was two hundred years old\n\nIf he was a day, when he toppled\n\nFrom the Mauritian garrison's parapet\n\nWho among us could imagine\n\nWhat it was like?\n\n_Death is before me today_\n\n_Like the sweet scent of myrrh_\n\n_Rising from a thorn tree in the desert_\n\nIn 1766, a decade before the American Revolution \u2013\n\nAlready full-grown and the last of his race \u2013\n\nHe was brought from the Isle of Rodriguez\n\nTo the garrison by the Chevalier de Marion\n\nHe outlived the last of the Bourbon Kings,\n\nThe Emperor Napoleon on his lonely isle,\n\nBismarck, and Queen Victoria\n\nHe kept an even keel\n\nEven as the Titanic turned turtle\n\nAnd survived the years of the Great War\n\nThat ended the power of the Kaiser,\n\nThe Ottomans and the Russian Czars\n\n_Death is before me today_\n\n_Like a strong wind filling_\n\n_The sails of a great ship_\n\nTo the French and English in the garrison\n\nHe was the soldier's mascot: the embodiment\n\nOf steadfastness and endurance\n\nAnd the island's Hindu lascars and slaves\n\nHad an ancient legend ready-made for him:\n\nHe was Chukwa, the Great Tortoise\n\nUpon whose back was heaped\n\nThe weight of the whole world\n\nAnd perhaps from their own enslavement\n\nThrough generations of exile and captivity,\n\nThey understood something of his suffering\n\n_Death is before me today_\n\n_Like the desire of an ancient prisoner_\n\n_To see his home again_\n\nThe Tortoise's fall\n\nWas an unlikely accident\n\nBlind or not, after a century and a half\n\nThe garrison's corridors were no mystery\n\nHis own time had long passed\n\nThe weight of the world\n\nDid indeed rest upon his back\n\nThe air above his carapace was empty\n\nYet heavier for all that with the weight of time\n\nNot quite the immortal Chukwa\n\nHe was not the Spiritus Mundi\n\nNot the footstool of Venus\n\nNot Methuselah's Pet\n\nBut only a lonely mortal tortoise\n\nIt was time to join the others\n\nThis was the end of his race\n\nIt was time to see his home again\n\n##  **THE BULL OF HEAVEN**\n\n###  **F IRST WATCH 9 A. M. TERCE**\n\n####  **AUROCHS OR WILD OX \u2013 1627 \u2013 _Bos primigenius_**\n\n#####  Gilgamesh Poet \u2013 2100 BC  \n_**Epic of Gilgamesh,** _ **Sumeria**\n\nIshtar led the Bull of Heaven\n\nDown into the world.\n\nWhen the Bull of Heaven bellowed\n\nThe earth shook and quaked:\n\nLakes and streams were emptied\n\nAnd the Euphrates stopped its flow.\n\nWhen the Bull of Heaven bellowed again\n\nAnd its great hoof struck the ground:\n\nThe earth cracked open\n\nAnd into that deadly abyss fell\n\nThe armies of the King of Uruk.\n\n> The Aurochs or Wild Ox first appears in literature in that first epic of the human race, _Gilgamesh_ (Mesopotamia \u2013 2100 BC), as the \"Bull of Heaven\" belonging to Anu the King of the Gods is lent to his daughter Ishtar, who wishes to use it to destroy the hero Gilgamesh, the King of Uruk. The Aurochs is also referred to as the _Reem_ in the Bible, and as the British traveller Canon Tristram wrote in 1884, \"on the Assyrian monuments its chase is represented as the greatest feat of hunting in the time of the dynasties of Nineveh.\"\n> \n> In Greek mythology, the Aurochs was the avatar of the thunderbolt god, Zeus, and the god of earthquakes, Poseidon. In the 5th century BC, the Greek historian Herodotus, in his _The Persian Wars,_ wrote: \"That whole region [of Thrace] is full of lions and wild bulls with gigantic horns which are brought into Greece.\" In the earlier Minoan civilization of Crete many images of these massive wild bulls being used in the spectacular sport of bull jumping were painted on the walls of the palace of Knossos.\n\n#####  Julius Caesar \u2013 65 BC  \n_**De Bello Gallico,** _ **Gaul**\n\nThey are but little less than Elephants in size, and are of the species, colour and form of a bull. Their strength is very great, and also their speed. They spare neither man nor beast that they see. They cannot be brought to endure the sight of men, nor can they be tamed, even when taken young. The people who take them in pitfalls assiduously destroy them: and young men harden themselves in this labour, and exercise themselves in this kind of chase; and those who have killed a great number \u2013 the horns being publicly exhibited in evidence of the fact \u2013 obtain great honour. The horns in amplitude, shape and species, differ much from the horns of our oxen. They are much sought after; and after having been edged with silver at their mouths they are used for drinking vessels at great feasts.\n\n> The Aurochs or Wild Ox was known to Caesar as the _Ur_ or _Uras._ Measuring six and a half feet at the shoulder, the Aurochs was the archetypal wild bull, as its Latin name _Bos primigenius_ suggests. It was from the Aurochs that all our domestic cattle are derived. It is the source of what was the single most important animal domestication in the evolution of agriculture-based civilizations.\n> \n> Recent DNA research indicates there were at least three subspecies of Aurochs. Two were the Indian Aurochs from which the \"humped\" cattle of India are descended, and the North African Aurochs from which the \"humpless\" cattle of the Near East are descended. What survived in a wild state into historic times in Europe was this terrifying, primeval, and untameable Giant Wild Ox.\n\n#####  Pliny the Elder \u2013 65 AD  \n_**Natural History,** _ **Rome**\n\nScythia produces but very few animals in consequence of the scarcity of shrubs. Germany, which lies close adjoining it, has not many animals, though it has some very fine kinds of wild oxen: the Bison, which has a mane, and the Urus.\n\n> The hunting of the Aurochs seems to have been regarded as the exclusive right of kings and aristocrats among the Germanic peoples. In the 9th century, Charlemagne hunted Aurochs in the forests near Aix-la-Chapelle; while in the German epic the _Niebelungenlied_ , we learn that the hero Siegfried slaughtered four of these mighty beasts near Worms in the 12th century.\n> \n> The Aurochs was sometimes confused with that other giant of the European forests: the Wisent or European Bison. Both of these beasts are famously and frequently portrayed in hunting scenes in prehistoric cave paintings. It is instructive to realize that although populations of both these massive quadrupeds were hunted by man for at least 40,000 years, it took the advent of firearms to result in their extinction.\n\n#####  Benedyct Fulinski \u2013 1627  \n_**Fulinski Manuscript XVII,** _ **Poland**\n\nA century ago, King Sigismund III Vasa, seeing the imminent danger of a quick and total extermination of the Aurochs, proclaimed orders with the object of protecting the feeding grounds of these animals, the number of which at this time did not amount to more than some ten pieces. Unluckily the enactment of those orders came too late and in consequence the Aurochs disappeared from the lands over the next century.\n\n> By 1400, two royal preserves in Poland were the last refuges of the Aurochs. By 1564, there were only 30 surviving animals in the Jaktorowska forest preserve. By 1620, this last herd had dwindled to one last animal. With her death in 1627, after nearly four millennia, the \"Bull of Heaven\" finally enters the annals of literature \u2013 along with the dragon and the griffin \u2013 as an entirely (non-existent) mythical creature.\n> \n> Like the Aurochs, the Wisent was becoming rare by the end of the Middle Ages. By the 1800s, Wisent populations surviving in the wild had dwindled to small herds in only two regions and in two forms. Both the Caucasian Wisent ( _Bison bonasus caucasicus_ ) and the Lithuanian Wisent ( _Bison bonasus bonasus_ ) populations survived because they lived in substantial park reserves under the protection of the Czar of Russia. However, after the Great War and the Russian Revolution, these last animals were no longer protected in imperial reserves, and by 1923, both the Caucasian and Lithuanian Wisent became extinct in the wild. Then, in 1925, an aging bull called Kaukasus, the last of the Caucasian race, died in the Hamburg Zoo.\n> \n> However, a remarkable captive breeding program by the European Bison Society in the 1920's gathered just 6 captive animals from various zoos, and miraculously saved the Lithuanian race from extinction. Since the end of the Second World War, new parklands were established, and today there are several thousand European Bison of the Lithuanian subspecies roaming woodland reserves in Europe.\n\n####  **TAURUS**\n\n##### Aurochs or Wild Ox \u2013 1627\n\nThe last black bull's wounding\n\nWas like the bursting\n\nOf an old sun in the belly\n\nHis roar betrayed the centuries\n\nOf refusing to tread warily in this world\n\nFlutes were made from his bones\n\nThey play them even now\n\nAnd hunting horns from his ivory crown\n\nIn his veins the singing of a hundred\n\nRapacious birds\n\nHe was strength without malice\n\nHis forehead was a crushing millstone\n\nFearful, lordly beast of all the mythologies\n\nFather of the minotaur\n\nA small part of his stubborn blood\n\nIs yet in the fighting bulls of the ring\n\nWho break the bodies of horses and men\n\nHis legend is as ancient as the pale moon\n\nBut his life now is only a fish's song\n\nAnd his spirit is steel and flint striking\n\nOn a dark and empty plain\n\n##  **THE HORSES OF DAWN**\n\n###  **S ECOND WATCH 10 A. M. TERCE**\n\n####  **TARPAN \u2013 1887 \u2013 _Equus feras feras_**\n\n#####  Herodotus \u2013 445 BC  \n_**History of the Persian Wars,** _ **Greece**\n\nThe third great river in Scythia is Hypanis. This stream rises within the limits of Scythia, and has its source in another vast lake, around which wild white horses graze. The lake is called properly enough, the Mother of the Hypanis.\n\n> The Tarpan is the Eurasian Wild Horse or \"True Wild Horse\" ( _Equus feras_ or _Equus feras feras_ ) from which the many breeds of the Domestic Horse ( _Equus caballus_ ) are descended. The name \"Tarpan,\" meaning \"Wild Horse,\" is from a Turkic language (Kyrgyz or Kazakh). It was also known as the _Dzerlik-adu_ by the Mongols (and the _Yeh-Ma_ by the Chinese). The Tatars and Cossacks, like these others, distinguished the Tarpan wild horse from the feral horse, which they called the _Takja_ or _Muzin._ The oldest human records we have of the existence of Tarpans are to be found in cave paintings 20-40,000 years old. The wild horses on the walls of the caves of Lascaux are perfectly-detailed portraits of the Tarpan with its distinctive \"Mohawk\" mane.\n> \n> Herodotus in Book Four of _The Persian Wars_ concerns himself with Scythia, which he defines as all the lands between the Danube and the Don Rivers. Herodotus credits the nomadic tribesmen there with being the finest horsemen in the world, and Scythia as the one part of the world where true wild horses still roamed in abundance. The region where wild white horses were to be found grazing (and survived until the early 19th century) was probably the region known as the Pripyat Marshes on today's Polish-Belarus-Ukraine border.\n\n#####  Max Toppen \u2013 1870  \n_**History of Mascovia** _ **, Danzig**\n\nIn the time of the Teutonic Knights, wild horses and other game were hunted for the sake of their skins. In 1543 Duke Albert sent an order to command at Lyck, bidding him take measures for the preservation of wild horses. Proofs of the horse being an object of the chase in Poland and Lithuania are found far into the seventeenth century.\n\n> In the forests of Bia\u0142owie\u017ca, Poland, records of the hunting of the Tarpan go back to 1409, when King W\u0142adyslaw Jagiello arranged a royal chase in honour of his cousin, Witold of Lithuania. In these immense forests the Tarpan, along with European Bison (Wisent), survived in substantial numbers through most of the 18th century. However, by 1800, the Tarpan had vanished everywhere except on the steppes of Tauria and Cherson. The last herds died out in the Ukraine around 1850, and the last known pure-bred Tarpan was killed near Askamia Nova in Russia on Christmas Day in 1879. Eight years later, the last captive Tarpan died in Poland.\n\n####  Harold J. E. Peake \u2013 1933  \n_**On the Domestication of the Horse,** _ **London**\n\nThe first mention of the horse is documented before 2000 BC in Babylon, where it is called the 'Ass from the East', but it does not seem to have been introduced into Mesopotamia before the arrival of the Kassite conquerors in 1746 BC. The horse was also well known to the Hittites, who arrived in Asia Minor about 1900 BC from the Northwest. All this evidence tends to show that the horse was used as a means of transport both in Persia and upon the Russian steppe well before 2000 BC. It seems likely that it was first tamed in that part of the world, or still farther east in Mongolia, as early as 3000 BC.\n\n> Samuel Gotlieb Gmelin, a German physician, botanist, and explorer of Russia in the days of Catherine the Great, was the first to collect specimens of the Tarpan and make it known to science. In 1766, Gmelin was appointed professor of botany at St. Petersburg, and the next year he was sent on an expedition to study the natural history of the Russian Empire. In his four-volume _Reise Durch Russland_ , Gmelin recorded seeing the animals in 1769 in the Brobrovsk region near Voronezh. Gmelin's hunters killed a stallion and two mares and a foal. He described the stallion as \"hardly as large as the smallest Russian pony.\" Gmelin explored the Don and Volga Rivers and the shores of the Caspian Sea. In the Caucasus he was taken hostage by Usmey Khan of Khaitakes, and died in captivity at the age of 29.\n> \n> In 1784 Pieter Boddaert named the species _Equus ferus,_ based on Gmelin's description. This was to distinguish it from its descendants, the 300 or so breeds of the Domestic Horse ( _Equus caballus,_ as named by Linnaeus in 1758).\n\n####  Otto Antonius \u2013 1938  \n_**On the Recent Equidae** _ **, Vienna**\n\nIn times not long before the beginning of historical days there were true wild horses or Tarpan spread over the whole Eurasiatic continent from the Atlantic to the Pacific. The Tarpan was a mouse-dun horse, which the medieval philosopher Albertus Magnus means when he calls the colour of the wild horse 'cinereus', as in ash-coloured. The first Duke of Prussia, Albert von Hohenzollern, sent wild horses as highly esteemed gifts to the Emperor, and also to the Archduke Ferdinand, so there can be no doubt that these horses were truly royal game \u2013 like the Urus and the Bison. The last refuge of the Tarpanis in Poland was the great game park of Count Zamoyski. Here they were strictly protected until the civil conflict and severe winter of 1812 made feeding and survival impossible. Franz von Falz-Fein, the owner of the matchless Ascania Nova Zoo, has told the life-history of that last wild horse of Europe, an old one-eyed mare, lingering for years around the feral horses, covered by domestic he-horses, captured, escaped with its filly, and some years later hunted and killed on the ice by the peasants of Agaiman.\n\n> Otto Antonius was an Austrian zoologist and palaeontologist who became director of the Tiergarten Sch\u00f6nbrunn in Vienna. An active member of the Zoological Society of London, Antonius was co-founder of modern zoological biology. He was a pioneer in creating modern humane zoo environments and redefining zoos as places to protect and breed endangered species. A key figure in saving the Wisent from extinction, he was also one of the first to attempt \"rebreeding\" domestic animals back to their wild ancestral stock. At the close of World War II, the Sch\u00f6nbrunn Zoo was nearly destroyed in bombing raids, and with the approach of the Russian Army on 9 April 1945, Antonius and his wife committed suicide.\n> \n> Antonius' methods were adapted by others and resulted in the successful captive-breeding of the Przewalski Wild Horse ( _Equus przewalski_ or _Equus feras przewalski_ ) \u2013 the only surviving form of Wild Horse \u2013 despite its total extinction in the wild. His concept of \"rebreeding\" has also resulted in the controversial \"reconstruction\" of \"Tarpan-like\" animals that can now be seen in a number of zoos and wildlife refuges.\n\n####  **THE CENTAURS**\n\n##### Tarpan \u2013 1887\n\nLittle wild horses of the steppes\n\nStraggling tribal men\n\nWhat has the Emperor\n\nBehind the Great Wall to fear?\n\nHe feeds on peacock tongues\n\nHis people cover the lands of the world\n\nHis power changes the course of rivers\n\nAnd shapes the face of the earth\n\nHe is disturbed by a terrible vision\n\nHe fears the image of a demon\n\nWith eight limbs\n\nThat is fierce and swift\n\nThe Tartar horn blows\n\nAnd the vision\n\nIn a whirlwind comes\n\nAt the Dragon Gate\n\nAt the Ivory Gate\n\nThe bowmen mount the towers\n\nBut the demons pass by, not against the wall\n\nBut around, their brave hearts pounding distance away\n\nIt is the frenzy of the little horses\n\nAnd the savage tribal men\n\nTwo creatures made one\n\nThe Tartar horn blows\n\nAnd the Tarpan carries the tribesmen on\n\nThe wall is left to the west wind\n\nThe yellow axes of the Imperial Guard are untried\n\nThe horsemen pass by, pass around\n\nThe cities are pyres\n\nThe jade images are shattered\n\nThe silk of the Imperial banners\n\nLies tattered beneath the conqueror's hooves\n\n##  **THE LIONS OF ROME**\n\n###  **T HIRD WATCH 11 A. M. TERCE**\n\n####  **ATLAS GOLDEN LION \u2013 1922 \u2013 _Panthera leo leo_**\n\n#####  Herodotus \u2013 410 BC  \n_**The Persian Wars** _ **, Greece**\n\nThat whole region [of Thrace] is full of lions... upon Xerxes' march the camels that carried the provisions of the Persian army were set upon by lions, which left their lairs and came down by night, but spared the men and the beasts of burden, while they made the camels their prey. I marvel what may have been the cause which compelled the lions to leave the other animals untouched and attack the camels, when they had never seen that beast before, or had any experience of it.\n\n> The Barbary or Atlas Golden-Mane Lion, as its Latin name _Panthera leo leo_ suggests, was the remnant population of the archetypal lion species that once roamed throughout the entire Mediterranean basin. As this curious observation by the Greek historian Herodotus in his _The Persian Wars_ (410 BC) confirms, lions were still numerous in Europe in Classical Greek times, and they were certainly still extant in Macedonia in the days of Alexander the Great. The Mediterranean Lion was probably extinct in its northern range of Europe by the time of Julius Caesar's Rome, but remained relatively numerous for most of the next two millennia throughout North Africa \u2013 from Egypt to Morocco \u2013 until the advent of modern firearms.\n> \n> Also known as Nubian Lions, these are the same ferocious animals portrayed in Eug\u00e8ne Ferdinand-Victor Delacroix's famous _Lion Hunting in Morocco_ (1854), painted two decades after his visit to North Africa. Richard Lydecker, in his _The Game of Africa_ (1908), describes the Atlas or Barbary Lion as \"very large, dusky ochery, with the mane very thick, long, and extending to the middle of the back, and a thick and heavy mane on the under parts. The males may weigh 500 lbs and measure 10 feet from nose to tip of tail.\"\n\n#####  Pliny the Elder \u2013 65 AD  \n_**Natural History,** _ **Rome**\n\nQuintus Scaevola, the son of P. Scaevola, when he was _curule aedile_ , was the first to exhibit at Rome a combat of a number of lions; and L. Sylla, who was afterwards Dictator, during his praetorship, gave the spectacle of a fight of one hundred lions with manes. After him, Pompeius Magnus exhibited six hundred lions in the Circus, three hundred and fifteen of which had manes; Caesar, the Dictator, exhibited four hundred.\n\n> As H. A. Bryden wrote in his _Great and Small Game of Africa_ (1899): \"There is little doubt that the Romans drew their chief supply of Lions for the arena and gladiatorial combats from Mauretania and Numidia. Pliny speaks of hundreds at a time being shown by Pompey and Caesar in the Roman arena. This bespeaks a great abundance of Lions in North Africa. Lions still linger here and there in the southeast and southwest of Algeria.\"\n> \n> John Bostock, the 1855 translator of Gaius Plinius Secundus' _Natural History_ , noted: \"Seneca gives an account of this exhibition; he says that the lions were turned loose into the Circus, and that spearmen were sent by King Bocchus, who killed them with darts.\" Scaevola's title of _curule aedile_ was that of a chief magistrate (consul or praetor) of Rome's public works, buildings and games.\n> \n> Also in North Africa, the Atlas Brown Bear ( _Ursus arctos crowtheri_ ) suffered an almost identical history and fate to that of the Atlas Golden Lion. Herodotus and the Roman writers Virgil, Juvenal and Martial refer to this animal as the \"Libyan Bear,\" while Pliny tells us that in 61 BC Domitius Ahenobarbus brought \"one hundred Numidian Bears\" to Rome's arenas. As late as 1830, there were live captive Atlas Brown Bears in Morocco and Marseille. In 1841, it was investigated by the Zoological Society and pronounced a rare and unique species or subspecies \"very different from any other bear.\" It became extinct about 1870.\n\n####  Eliakim Littell \u2013 1882  \n_**English Mechanic and World of Science and Art** _\n\nFew of those who daily pass Sir Edwin Landseer's lions in Trafalgar Square know that not far beneath their feet... in the Pleistocene gravels there are bones of the cave-lion (Felis leo spelaea).\n\n> Golden Atlas Lions from the Regent's Park Zoological Gardens were the models for Sir Edwin Landseer's famous Trafalgar Square bronze sculptures, but curiously enough, in the 1830's excavations of the foundations of Trafalgar Square numerous fossil bones of extinct prehistoric animals were unearthed. Among them were those of the original \"British Lion\" ( _Felis leo spelaea_ ) and prehistoric ancestor of the Atlas Lion ( _Felis leo leo_ ). In _The Living Age_ (1887), Littell further reported that medieval Londoners had earlier discovered \"the fossil bones of cave lions... [which] have lain so securely stored for untold ages beneath Charing Cross and Trafalgar Square... [but] were long known as dragon bones.\"\n> \n> Atlas or Barbary Lions have had a long historical association with British monarchy. Captive Atlas Lions were kept in the Tower of London as early as the 12th century during the reign of Henry II and Richard the Lionheart. Two Barbary Lion skulls unearthed during excavations of the Tower were radio-carbon dated during the time of Edward I (c. 1300) and Henry V (c. 1420), respectively.\n> \n> At the time of the Trafalgar Square excavations in the 1830's, the Duke of Wellington had the Atlas Lions kept in cramped cages in the Tower removed to more humane enclosures in London's Regent's Park, where Sir Edwin Landseer came to view them in 1858. One famous Victorian pure-bred Barbary Lion named Sultan lived in the London Zoo until 1896.\n\n####  Alfred Edward Pease \u2013 1899  \n_**African Game** _ **, Algeria**\n\nThe North African Lion has today become so rare that it may be said to be nearing extinction. It lingers only in the country that might be described as the Mediterranean littoral zone, though an occasional lion is shot in the interior. Before the French came, the Turks had encouraged the Arabs to destroy them by freeing the two great lion-hunting tribes, the Ouled Meloul and Ouled Cessi, from all taxes and paying liberally for their skins. The French gave 50 francs for a skin. There are now a few lions still in the Province de Constantine, in the thick Atlas forests between Soukarras and La Calle.\n\n> Alfred Edward Pease's prediction only took a couple of decades to realize. This remnant population was ruthlessly extinguished within the next two decades. The last wild Barbary Golden Lion was shot in the remote forests of the Atlas Mountains in Morocco in 1922.\n> \n> Remarkably, the Barbary Golden Lion was not the first Lion to become extinct on the African continent in modern times. It was another distinctive subspecies at the other end of the animal's range in South Africa. This was the Cape Black-Mane Lion ( _Panthera leo melanochaitus_ ) which was almost as large as the Barbary Lion. It also exhibited a black mane as large and luxuriant as the gold mane of its cousin. Captive animals of this race were exhibited live in Amsterdam, where they may be found in the drawings of Rembrandt. The last Black-Mane Lion in the Cape was shot in 1858, while a small number survived to the north in Natal for a few more years. The last recorded sighting of this Lion was a large male hunted down in Natal and shot by General Bisset in 1865.\n> \n> By 1900, the only surviving non-sub-Saharan race, the once numerous Asiatic or Persian Lion ( _Panthera leo persica_ ) \u2013 so famously portrayed in art since Biblical times \u2013 survived only because the Nawab of Junagadh gave the beast a sanctuary in India's Gir Forest. By 1908, it was claimed there was only a single pride of 13 Lions remaining in this preserve. Fortunately, a great deal of effort has recently been made to save this subspecies and there are now several hundred Asiatic Lions in existence.\n\n####  **THE LION'S GATE**\n\n##### Atlas Golden Lion \u2013 1920\n\nAt the gate was the god of Lions\n\nCut in a single massive stone\n\nAnd raised above the desert sand\n\nIn the time of empires\n\nThis Lion's eye was a brilliant swastika\n\nA sunwheel on the desert wind\n\nHis voice was a stone\n\nRolling over the roof of the world\n\nEven the great parapet of his head\n\nIs broken now\n\nThe foundation of the temple\n\nIs cracked and wasted\n\nUpon the wall\n\nThe carved images are all but gone:\n\nVanished men pursue vanished beasts\n\nIn that ancient dream of a green\n\nAnd fruitful land\n\n##  **THE SEA ELEPHANT**\n\n###  **F IRST WATCH 12 A. M. SEXT**\n\n####  **STELLER'S SEA COW \u2013 1768 _\u2013 Hydrodamalis gigas_**\n\n#####  Georg Wilhelm Steller \u2013 1741  \n_**Journal of a Voyage with Bering from Kamchatka to America** _\n\nTheir capture was affected by a large iron hook, the point of which resembled the fluke of an anchor, the other end being fastened by means of an iron ring to a very long stout rope, held by thirty men on shore. A strong sailor took this hook and with five other men stepped into the boat, and one of them taking the rudder, the other four rowing, quietly hurried towards the herd. The harpooner stood in the bow of the boat with the hook in his hand and struck as soon as he was near enough to do so, whereupon the men on shore, grasping the other end of the rope, pulled the desperately resisting animal laboriously toward them. Those in the boat, however, made the animal fast by means of another rope and wore it out by continuous blows, until tired and completely motionless, it was attacked with bayonets, knives and other weapons and pulled up on the land. Immense slices were cut from the still living animal, but all it did was shake its tail furiously and make such resistance with its forelimbs that big strips of cuticle were torn off. In addition it breathed heavily, as if sighing. From the wounds in the back the blood spurted upward like a fountain.\n\n> Georg Wilhelm Steller was the only trained naturalist ever to see this animal alive. And Steller's accounts were of the first-ever encounter in the very brief bloody history of human-sea cow relations, as the animal became extinct just twenty-six years later. The only species of its genera and family, the Steller Sea Cow was distantly related to the two other surviving (but endangered) _Sirendae_ : the Dugongs and the Manatees. With a maximum weight of over seven tons, it was 14 times the size of a Manatee: the marine equivalent of a full-size African Elephant. Indeed, \"Sea Elephant\" would have been a far more appropriate name, as both the Elephant and the Sea Cow were descended from a common prehistoric land-dwelling ancestor.\n> \n> Steller was the ship's naturalist on the famous expedition of the Danish explorer Vitus Bering. During the summer of 1741, Bering sailed eastward from Kamchatka to Alaska. On the return voyage, they were shipwrecked on the desolate island that now carries Bering's name.\n\n#####  Georg Wilhelm Steller \u2013 1741  \n_**Journal of a Voyage with Bering from Kamchatka to America** _\n\nWhen an animal was caught with the hook, those nearest in the herd began to stir also and feel the urge to bring succour. To this end some tried to upset the boat with their backs, while others pressed down the rope and endeavoured to break it, or strove to remove the hook from the wound by blows of their tail, in which they actually succeeded several times... . It is most remarkable proof of their conjugal affection, that the male, after having tried with all his might, although in vain, to free the female caught by the hook, and in spite of the beating we gave him, nevertheless followed her to the shore, and that several times, even after she was dead, he shot unexpectedly up to her like a speeding arrow. Early the next morning, when we came to cut up the meat and bring it in the dugout, we found the male again standing by the female, and the same I observed once more on the third day when I went there by myself for the sole purpose of examining the intestines.\n\n> The Bering Expedition ship, the _St. Peter_ , ran aground and was wrecked while attempting to find safe anchorage on Bering Island. A total of 32 crewmen did not survive the winter, including Vitus Bering himself. Georg Wilhelm Steller was the only officer healthy enough to take command and assumed leadership over the ten crew members not crippled by illness. Steller doctored the rest and oversaw the building of shelters and the rebuilding of the wrecked _St. Peter_. Somehow, he also managed time to record the amazing assortment of wildlife on the island that had never before been seen by man.\n> \n> Among the many species discovered by Bering was this massive sea mammal which proved to be the primary food source for the survivors of the _St. Peter._ As Steller observed, the Sea Cow was their salvation: \"This great beast is 28 to 35 feet long and 22 feet thick about midsection... . Each Sea Cow provided more than seven thousand pounds of meat and fat, and the red flesh tasted much like good beef... . All of us who had partaken of it, soon found out what a salutary food it was, as we soon felt a marked improvement in strength and health.\"\n\n####  Georg Wilhelm Steller \u2013 1741  \n_**Journal of a Voyage with Bering from Kamchatka to America** _\n\nThese animals love shallow and sandy places along the seashore, but they spend their time more particularly about the mouths of the gullies and brooks, the rushing fresh water of which always attracts them in herds. They keep the half-grown and young in front of them when pasturing, and are very careful to guard them in the rear and on the sides when travelling, always keeping them in the middle of the herd... . In the spring they mate like human beings, particularly towards evening when the sea is calm. Before they come together many amorous preludes take place. The female, constantly followed by the male swims leisurely to and fro eluding him with many gyrations and meanderings, until, impatient of further delay, she turns on her back as if exhausted and coerced, whereupon the male, rushing violently upon her, pays tribute of his passion, and both give themselves over in mutual embrace.\n\n> Despite the hardships of survival on the island, Steller took time to closely observe the nature and habits of the Sea Cows and many other species. After eight months wintering on Bering Island, Steller and his crew built a new craft from the wrecked _St. Peter_ and in August 1742 sailed back to Kamchatka with news of their discovery. This rapidly resulted in Russian exploitation and monopoly of the whole Northwest Pacific coast of Alaska \u2013 especially in the enormously lucrative trade in fur-bearing sea mammals \u2013 for over a century.\n> \n> However, instead of being rewarded for his heroic efforts in bringing the survivors of the _St. Peter_ safely home, and bringing the momentous news of the discovery of Alaska, Steller found himself embroiled in disputes with the petty jealousies of corrupt Russian officials and bureaucrats in Kamchatka for the next four years. Worn out by a series of arrests, trials and imprisonments, in November 1756, Steller died of fever at age 37. His _Journals_ and his _De Bestiis Marinis_ ( _On the Beasts of the Sea_ ) were published posthumously in St. Petersburg, establishing him as one of the greatest naturalists of his time.\n> \n> Besides Steller's Sea Cow, five other species carry the name of this extraordinary naturalist: Steller's Sea Lion ( _Eumetopius jubatus_ ), the largest species of sea lion; Steller's Sea Eagle ( _Haliaeetus pelagicus_ ), the world's heaviest eagle; Steller's Eider ( _Polysticta sterlleri_ ), the smallest form of Sea Duck; the Steller's Jay ( _Cyanocitta stelleri_ ); and the Steller's Spectacled Cormorant ( _Phalacrocorax perspicillatus_ ), a large flightless bird that sadly suffered the same fate as Steller's Sea Cow and became extinct in 1850.\n\n####  Petr Yakovlev \u2013 1764  \n_**Chancellery Report,** _ **Bering Island, Kamchatka**\n\nFrom 1743 until 1763, hardly a winter passed without one or more parties spending eight or nine months hunting fur animals on Bering Island, during which time the crews lived almost exclusively on the meat of the sea cow. But that was not all, for more than half of the expeditions which wintered there did so for the express purpose of laying in stores of sea cow meat for the further journey, which usually lasted two to three years more. From 1763 the visits to Bering Island seem to grow scarcer, probably due to the very fact that sea cows had now become so nearly exterminated that the few left were insufficient to maintain any wintering and foraging expeditions.\n\n> The Russian mining engineer Petr Yakovlev wintered on Bering Island in 1754-1755. Displaying an unusually astute awareness of the severe environmental damage being inflicted on this newly discovered island, Yakovlev prophetically anticipated the extinction of the Sea Cow if over-exploitation of the species was allowed to continue. As he recorded, because of the animal's sheer size, it was estimated that four animals were critically wounded and abandoned for every one that was successfully beached and killed for food. Petr Yakovlev unsuccessfully petitioned Kamchatkan authorities and requested a _ukase_ or edict from the Bol'sheretsk Chancellery prohibiting \u2013 or at least limiting \u2013 the wholesale slaughter of this species. This appeal fell upon deaf ears and the last Sea Cow was killed around 1768 on Bering Island.\n\n####  **SIREN SONG**\n\n##### Steller's Sea Cow \u2013 1768\n\nSome still nights\n\nOn the shores of Bering's sea\n\nYou may imagine them\n\nHuge as the hull\n\nOf an overturned ship\n\nMoaning in the rolling surf\n\nFountain of hot blood pulsing\n\nFurnace of the deep heart\n\nWave-worn giants, idle lovers\n\nOn the swell of the sea\n\nBigger than elephants\n\nSkin like the bark of an ancient oak\n\nSnorting like horses\n\nPawing the kelp meadows\n\nWith their tough hooves\n\nLike bulls in pasture\n\nTide riders, storm biders\n\nSlow to lust as elephants\n\nPassionate as whales\n\nBeauty here in a thing not itself beautiful\n\nAs delight in the play of light\n\nOn a mountain or a great rock\n\nYet something vastly alive\n\nAs these once were alive\n\nSome still nights\n\nOn the shores of Bering's sea\n\nYou may imagine them\n\nThe great breath-song\n\nThrough the sighing night\n\n##  **THE TERROR BIRD**\n\n###  **S ECOND WATCH 1 P. M. SEXT**\n\n####  **GIANT MOA \u2013 1850 \u2013 _Dinornis maximus_**\n\n#####  Joel S. Polack \u2013 1838  \n_**History of New Zealand** _\n\nAmong the Maori there were traditions of supernatural beings in the form of birds, having waylaid native travellers, among the forest wilds, vanquishing them with an overpowering strength, killing and devouring them. The traditions are reported with an air of belief that carries conviction.\n\n> New Zealand was an island-world that was a time-capsule of the planet forty million years ago. It was a world inhabited primarily by birds; except for bats, there were no mammals (not even marsupials) in New Zealand until the legendary arrival of the Great Fleet of the Polynesian Maoris, about 1300 AD. The first European to \"discover\" these islands was Dutch explorer Abel Tasman in 1642, who named it after the small island of Zealand in Holland. However, the extreme hostility of the warlike Maoris prevented further incursions by Europeans until the arrival of Captain James Cook's ship _Endeavour_ in 1769. Three years later, a French expedition arrived in the Bay of Islands, but rapidly withdrew after the Maoris killed and ate 27 of its officers and men. Among those suffering this fate was the expedition's commander, Captain Marc-Joseph Marion du Fresne, after whom Mauritius' Marion's Tortoise was named.\n> \n> Although both the British and French expeditions recorded hundreds of new species, the Giant Moas were not among them. Joel S. Polack's account is the first published acknowledgement of the existence of these giant birds. Polack was a trader in New Zealand between 1834 and 1837. In 1838, Polack published his two-volume _History of New Zealand_ , in which he said he had found bones of a large ostrich-like bird, and added that the Maori claimed that in times not long past \"very large birds existed, but the scarcity of animal food, as well as the easy method of entrapping them, has caused their extermination.\"\n\n#####  Reverend William Williams \u2013 1839  \n_**Williams' Journal** _ **, Cloudy Bay, New Zealand**\n\nThe natives there had mentioned to an Englishman of a whaling party that there was a bird of extraordinary size to be seen only at night on the side of a hill near there; and that he, with the native and a second Englishman, went to the spot; that after waiting some time they saw the creature at some little distance, which they describe as being fourteen or sixteen feet high. One of the men proposed to go nearer and shoot, but his companion was so exceedingly terrified, or perhaps both of them, that they were satisfied with looking at him, when in a little time he took alarm and strode up the mountain.\n\n> The Reverend William Williams was a Protestant missionary to the Bay of Islands in New Zealand who had translated the New Testament into Maori. Williams also compiled a Maori-English dictionary and built the first church in Poverty Bay. In the summer of 1838, Williams spent many months travelling in the East Cape district of the North Island (with another amateur naturalist, and printer, William Colenso) and recorded this incident near Waipapu.\n> \n> In 1839 a traveller named John Rule obtained a 10-inch section of thigh bone of some huge creature in New Zealand and brought it back to Britain. He gave the bone to Richard Owen, an authority on prehistoric animals and the man who coined the word \"Dinosaur.\" The bone was so massive that it looked like an ox's thigh bone. Owen initially thought this was a joke, but further examination convinced him that this was the leg bone of an unknown giant ostrich-like bird. In November 1839, Owen delivered a lecture to the Zoological Society of London in which he boldly asserted that New Zealand was once inhabited by an undiscovered species, a giant flightless bird.\n> \n> Remarkably, only weeks after Owen's prophetic lecture, a shipment of Moa bones collected by the Reverend William Williams and fellow missionary the Reverend Richard Taylor reached England. These bones immediately confirmed Owen's analysis and made him famous overnight. Owen gained royal recognition and became one of the chief forces behind the creation of the British Natural History Museum at South Kensington.\n\n####  William Colenso \u2013 1842  \n_**Journal of Natural History,** _ **New Zealand**\n\nI heard from the natives of a certain monstrous animal; while some said it was a bird, and others a person, all agreed that it was called a Moa; and that in general appearance it somewhat resembled an immense domestic cock, with the difference, however, that if anyone ventured to approach the dwelling of this wonderful creature, he would be invariably trampled on and killed by it.\n\n> In 1841, William Colenso returned to New Zealand to search for more Moa relics. He went down the east coast of the North Island, asking Maoris to help him find evidence of these birds. He recovered five Moa thigh bones and a large number of Maori myths about these great birds and returned with them to Britain in 1842. Colenso was the first to record the name \"Moa,\" although there is some evidence to suggest the authentic Maori name may have been \"Tarepo.\"\n> \n> In 1844, the British Governor FitzRoy spoke with an 85-year-old Maori named Haumatangi who claimed to remember the second visit of Captain Cook in 1773, and stated that the last Moa in his part of New Zealand had been seen two years before Cook's arrival. Another Maori, Kawana Papai, said he had himself taken part in Moa hunts when he was a boy about 1790. He said the birds were rounded up and killed with spears. He claimed they were dangerous to hunt, as the Moa defended itself fiercely with vigorous kicks; but as they had to stand on one leg to kick, Maori hunters would strike at the supporting leg and bring it to ground.\n> \n> Excavations of other early tribal settlements have uncovered charred Moa bones and Moa eggs utilized as water bottles by native peoples. And in 1859, miners made the remarkable discovery of a Maori tomb containing a skeleton of a chieftain placed in a sitting position with its skeletal hands clutching a large Moa egg measuring 10 inches by 7 inches. Later still, in 1939 \u2013 a century after the first European acknowledgement of these semi-mythical birds \u2013 excavations by New Zealanders Roger Duff and Robert Falla in Pyramid Valley on the South Island resulted in the discovery of over 140 complete skeletons of six distinct Moa species.\n\n####  Reverend Richard Taylor, M. A., F. G. S \u2013 1855  \n_**Te Ika A Mauri \u2013 New Zealand and Its Inhabitants** _\n\nOf all the birds that have once existed in New Zealand, by far the most remarkable is the Moa. Perhaps it is the largest bird which ever had existence, at least during the more recent period of the earth's history; and it is by no means certain that it is even now extinct! I first discovered its remains in 1839, at Tauronga, and now Waipau. These bones were of recent but undeterminable age. However, Mr Meurant, employed by the Government as Native interpreter, stated to me that in the latter end of 1823, he saw the flesh of the Moa in Molyneux harbour; since that period, he has seen feathers of the same bird in the Native's hair. They were a black or dark colour, with purple edge, having quills like those of the Albatross in size, but much coarser. He saw a Moa bone which reached four inches above his hip from the ground and as thick as his knee, with flesh and sinews upon it. The flesh looked like beef. The slaves who were from the interior, said that it was still to be found inland. A man named George Pauley, now living in Fareaux Straits told him he had seen the Moa, which he described as being an immense monster, standing about twenty feet high. He saw it near a lake in the interior. It ran from him and he also from it. He saw its footmarks before he came to the river Tairi, and the mountains.\n\n> The Reverend Richard Taylor arrived as a missionary on the East Coast and the Bay of Islands in New Zealand with the Reverend William Williams. When Taylor made the voyage to England with the manuscript of his history of New Zealand in 1855, he brought with him the Maori Chief Hoani Wiremu Hipango and gained an audience with Queen Victoria and Prince Albert.\n> \n> Over the next decade, Sir Richard Owen classified, reconstructed, and named many of the Moa species. Owen gave these giant birds a scientific name to match their size: _Dinornis_ or \"Terror Bird\" \u2013 which is comparable to his name for prehistoric reptiles: \"Dinosaur\" or \"Terror Lizard.\" He named six of the dozen species now known. These were: _Dinornis robustus, D. elephantopus, D. crassus, D. giganteus_ and _D. gracilis_ (i.e., robust, elephant-footed, fat, giant, and slender Moa). Finally there was the _Dinornis maximus_ \u2013 the Giant Moa measuring 12 to 14 feet tall and a quarter ton in weight. This was taller than the _Aepyornis maximus_ of Madagascar, but at a quarter ton, only about half the weight of the Elephant Bird.\n\n####  **THE EXCAVATION**\n\n##### Giant Moa \u2013 1850\n\nThe earth is pulled up like a coffin lid\n\nBones like unstrung bows\n\nAnd a battlefield of broken spearshafts\n\nGiant shin bones cracked\n\nLike diviners' yarrow stalks\n\nBeaks like shattered spearheads\n\nThe hexagrams of this dry graveyard swamp\n\nI saw the broad clawed footprint\n\nOf the giant bird in clay\n\nBeneath the hump of the hill\n\nThat shadows the valley\n\nLater we found a ring of Moa skulls\n\nWith a hunched chieftain's skeleton at its centre\n\nClutched in his spidery grip:\n\nA great Moa egg\n\nLike the dragon pearl\n\nThat is the Maori moon\n\nSap of the veronica wood\n\nThey called \"Moa's blood\"\n\nThey roasted Moa flesh on its fires\n\nWhen an enemy was defiant at the death\n\nThey called him \"fierce as the Moa\"\n\nAnd when the white man came\n\nAnd the Maoris fell\n\nWhen pestilence and war broke them\n\nAnd all their world was ending\n\nThey cried out, \" _Ka ngaro I te ngaro a te Moa_ \"\n\nAlas, we are \"lost as the Moa is lost\"\n\nIn this valley I am shaken\n\nBy a vision rising\n\nLike the dragon pearl moon\n\nEclipsing, for a dark hour\n\nThe bright noonday sun\n\n##  **GALAPAGOS MUTTON**\n\n###  **T HIRD WATCH 2 P. M. SEXT**\n\n####  **FLOREANA ISLAND GALAPAGOS TORTOISE \u2013 1876**  \n_**Chelonoidis nigra nigra** _\n\n#####  William Dampier \u2013 1684  \n_**New Voyage Round the World** _\n\nThe Tortoises on these Isles are so numerous that 500 or 600 men might subsist on them alone for several months without any other sort of provision. They are extraordinarily large and fat, and so sweet that no pullet eats more pleasantly. One of the largest of these creatures will weigh 150-200 lb and some of them are two foot or two foot six inches over the carapace or belly.\n\n> The Galapagos Archipelago is a group of strange volcanic islands unvisited by humans until their discovery by the Spanish in 1535. Thereafter, they were exploited for food and water by passing ships. However, for centuries they remained largely unoccupied except by adventurers and pirates who used the islands as a base for attacking Spanish treasure ships. From the end of the 17th to the middle of the 18th centuries, the Galapagos Islands served as a haven for English buccaneers like William Dampier.\n> \n> Dampier must have dined upon very immature specimens of Galapagos Tortoises in his travels, unless he was describing the weight and size of the meat alone taken from these animals. Mature male Galapagos Tortoises were commonly 300 kilograms or 700 lbs in weight and females around 180 kilos or 400 lbs. The largest recorded specimens weighed in excess of 400 kilos or nearly 900 lbs and measured 1.9 metres or over six feet in length. The original Tortoise population of these islands was approximately one quarter of a million, but by 1900 this was reduced to less than 3,000 in the entire archipelago.\n> \n> Dampier was the first man to sail around the world three times, and his _New Voyage Round the World_ was the first round the world travelogue and sold four editions in its first two years. It made him a celebrity in British literary and intellectual circles. On 6 August 1698, John Evelyn wrote: \"I dined with Mr Samuel Pepys, where was Captain Dampier, who had been a famous buccaneer, and printed a relation of his very strange adventure, and his observations. He seemed a more modest man than one would imagine by the relation of the crew he had assorted with.\" One of the classics of maritime writing, William Dampier's _New Voyage Round the World_ was acknowledged as a great inspiration for explorers like Captain James Cook and Captain George Vancouver, as well as authors such as Daniel Defoe, Robert Louis Stevenson, and Joseph Conrad.\n\n####  **SANTA FE ISLAND GALAPAGOS TORTOISE \u2013 1890**  \n_**Chelonoidis nigra barringtoni** _\n\n#####  Amasa Delano \u2013 1801  \n_**Voyages of Commerce and Discovery** _\n\nThe Terrapin, or Land Tortoise found here is by far, the largest, best and most numerous of any place I have visited. Some of the largest weigh three or four hundred pounds; but their common size is between fifty and one hundred pounds. They have a very long neck, which, together with their head, has a very disagreeable appearance, very much resembling a large serpent. I have seen them with necks between two and three feet long, and when they saw anything new to them, or met each other, they would raise their heads as high as they could, their necks being nearly vertical, and advance with their mouths wide open, appearing to be the most spiteful of any reptile whatever. I was put in the same kind of fear that I felt at the sight or near approach of a snake when I first saw a very large one. I had a musket in my hand at the time, at the touch of the muzzle of which, he dropped himself upon the ground and instantly secured all his limbs within his shell. They are perfectly harmless, as much so as any animal I know of, notwithstanding their threatening appearance. They have not teeth, and of course they cannot bite very hard. They take their food into their mouths by the assistance of the sharp edge of the upper and under jaw, so as to nip grass, flowers, berries, or shrubbery \u2013 the only food they eat.\n\n> Amasa Delano was the Bostonian captain of the _Perseverance_ who wrote of his stay in Galapagos in his _Voyages of Commerce and Discovery_ (1789-1807). As Delano acknowledged, exploitation of Giant Galapagos Tortoises for food or oil was practiced by every passing ship. Today four Galapagos Land Tortoise subspecies are listed as extinct. However, recent genetic analysis has brought many classifications into dispute. It also brings into question just when a species is considered extinct. For example, the Floreana Island Tortoise definitely became extinct on Floreana Island by 1876. However, in January 2012, genetic fingerprinting of Isabella Island Tortoises revealed a number of surviving mixed-race or hybrid Floreana-Isabella Tortoises.\n\n####  **FERNANDINO ISLAND GALAPAGOS TORTOISE \u2013 1906**  \n_**Chelonaidis nigra phantastica** _\n\n#####  Admiral David Porter \u2013 1812  \n_**A Cruise to the Pacific Coast** _\n\nBefore engaging in battle, the British ship cleared her decks by heaving overboard her cargo of live elephant tortoises. A few days afterwards we were so fortunate to find ourselves surrounded by about 50 of them which we picked up and brought on board, as they had been lying in the same place where they had been thrown over, incapable of any exertion in that element, except that of stretching out their long necks.\n\n> David Porter, the Admiral of the American Pacific Fleet, reported this event in the aftermath of an engagement with a British ship in the Pacific waters north of the Galapagos during the War of 1812. In 1823, Captain Benjamin Morrel wrote an account which explains why a British man-of-war in the Pacific would take on such a large cargo of live tortoises: \"I have taken many tons of these animals on board my own vessels from 5-6 months without once taking on food or water. They have been known to live unfed on some whaling ships for 14 months without any apparent diminution of health or weight.\"\n> \n> The maximum lifespan of these Giant Land Tortoises is unknown. However, the longest-lived Pacific Galapagos Tortoise \u2013 affectionately known as \"Harriet\" \u2013 was a Santa Cruz Tortoise ( _Chelonaidis nigra porteri_ ) taken from the island in 1835 by Charles Darwin. Harriet was later given to the _HMS Beagle_ 's First Lieutenant John Clement Wickham and brought to Australia. Eventually, Harriet found a home in the Brisbane Botanic Gardens until her death at the age of 176 on 19 May 2006.\n> \n> Strangely enough, the oldest Indian Ocean Tortoise died just one month later. This was the famous Aldabra Tortoise ( _Aldabrachelys gigantean_ ), known as _Adwaita_ (Sanskrit for the \"One and Only\"). Taken captive in 1750 for the menagerie of \"Clive of India,\" One-and-Only died in the Alipore Zoo in Kolkata, India, on 23 June 2006 at the age of 250 years.\n> \n> Another ancient Tortoise was the 188 year old Tonga specimen, named Tu'i Malila or \"Little King.\" Tu'I Malila was presented to the King and Queen of Tonga by Captain James Cook in 1777. Tu'i Malila lived in Tonga's royal gardens until his death on 19 May 1965. Although considered one of the most beautiful of Tortoises, with its golden star-patterned shell, the Little King was not a giant, but a medium-sized (35 lb) Madagascar Radiated Tortoise ( _Geochelone radiate_ ).\n\n####  **PINTA ISLAND GALAPAGOS TORTOISE \u2013 1971**  \n_**Chelonoidis nigra abingdoni** _\n\n#####  Charles Darwin \u2013 1835  \n_**Voyage of the Beagle** _\n\nI will first describe the habits of the tortoise, which has been so frequently alluded to. These animals are found, I believe, on all the islands of the Archipelago. The inhabitants believe these animals are absolutely deaf; certainly they do not hear a person walking close behind them. I was always amused when overtaking one of the great monsters, as it was quietly pacing along, to see how suddenly, the instant I passed, it would draw in its head and legs, and uttering a deep hiss fall to the ground with a heavy sound, as if struck dead. I frequently got on their backs, and then giving a few raps on the hinder part of their shells, they would rise up and walk away; but I found it very hard to keep my balance.\n\n> Charles Darwin found in the Galapagos a perfect natural laboratory of evolution when he visited in 1835. His observations there enabled him to put forward in _The Origin of the Species_ his revolutionary theory of evolution by natural selection. Darwin wrote concerning the tortoises' immense size: \"Mr Lawson, an Englishman, and vice-governor of the colony, told us that he had seen several so large, that it required six or eight men to lift them from the ground; and that some had afforded as much as two hundred pounds of meat... . The tortoises when thirsty are obliged to travel long distances. Hence broad and well-beaten paths branch off in every direction from the wells down to the sea coast; and the Spaniards by following them up, first discovered the watering-places.\"\n> \n> The Pinta Tortoise was believed to be extinct in the early twentieth century, but one last specimen was discovered in 1971 and taken into captivity. It is now listed as \"extinct in the wild\" and \"functionally extinct\" in captivity, as no female exists. Two attempts at mating this animal with females of other subspecies have resulted in the production of hybrid eggs. However, in both cases the eggs have failed to hatch. Known as \"Lonesome George,\" this animal is famously listed on Galapagos tourist advertisements as the \"world's rarest living creature.\" He is estimated to be a spry and healthy one hundred years of age. He is the Galapagos Islands' most celebrated tourist attraction and lives at the Darwin Biological Research Station on Santa Cruz Island.\n\n####  **LA TORTUGA NEGRA**\n\n##### Floreana Galapagos Tortoise \u2013 1876\n\nKnown to the Spaniards as \" _La Tortuga Negra_ \"\n\nThis ancient giant carefully paced out its life\n\nMoving just fast enough to avoid the lava flow\n\nAnd slow enough to endure the volcanic heat\n\nA thousand millennia passed peacefully\n\nAs the island's lava slowly cooled in the tropic sea\n\nThen one evening, something appeared on the horizon:\n\nLittle floating isles with lights drifted ashore\n\nCompare this encounter\n\nTo the tale of Aeschylus' Turtle:\n\nA tortoise scooped up and lifted\n\nInto the air by an eagle\n\nFrom a great height \u2013 we are told\n\nThat bird mistook the great man's bald pate\n\nFor a large stone, and\n\nWishing to crack open his lunch\n\nLet the turtle drop\n\nWho was the more surprised, then\n\nUpon hearing that fatal crunch:\n\nTortoise or Man?\n\nOne can only speculate, of course\n\nBut, most probably, with one last gasp \u2013\n\nLike the last _Tortuga Negra_ dropped by sailors\n\nInto a boiling pot \u2013\n\nBoth died of astonishment\n\n##  **THE SHOGUN'S WOLF**\n\n###  **F IRST WATCH 3 P. M. NONE**\n\n####  **SHAMANU OR JAPANESE WOLF \u2013 1904**  \n_**Canis lupus hodophilax** _\n\n#####  Hirata Atsutane \u2013 1810  \n_**The Superiority of the Ancients** _ **, Japan**\n\nThere are many wolves which are called the messengers of the gods of the mountains, and people from other parts of the country come and, applying through the guards of these mountains, choose and borrow one of these wolves as a defence against fire. That is to say they only arrange to borrow it and do not take a wolf to their place. And from the day of borrowing they offer daily food to the spirit of the wolf.\n\n> The Shinto shrine Mitsumine Jinga was an important site of wolf worship in Japan. By tradition it was built by Prince Yamato Takeru, the legendary unifier of Japan in the 4th century, who, during his campaign of pacification of central Honshu, became lost in the mountain passes of the Karisaka. He was rescued and guided out of the mountains by a supernatural white wolf.\n> \n> The Shamanu or Yamainu (meaning \"Mountain Dog\") was the world's smallest wolf: weighing about 14 lbs, it measured from 12-16 inches at the shoulder, and just a little over 3 1\/2 feet in length (including its one-foot-long tail). The Shamanu was endemic to the Japanese islands of Honshu, Shikoku and Kyushu.\n> \n> By the Ainus, the aboriginal Japanese people, the Shamanu was called the \"Howling God\" because it so often howled for hours from hilltops and mountains. There is an Ainu myth origin about a white wolf that mates with a goddess; the offspring of this union became the ancestors of the Ainu.\n\n#####  Coenraad Jacob Temminck \u2013 1839  \n_**Fauna Japonica,** _ **Leiden, Holland**\n\nThe Shamanu or Japanese Wolf is chiefly distinguished from the European Wolf by its smaller size and shorter legs. It is as much dreaded by the Japanese as the European Wolf is in its range.\n\n> The Shamanu or Honshu Miniature Wolf was first described to Western science in 1839 as _Canis hodophilax_ by Coenraad Temminck, the first director of the Dutch Museum of Natural History. This was based on specimens from the first zoological expedition to Japan by Philipp Franz von Siebold and Heinrich Burger, both Germans, employed by the Dutch at a trading post near Nagasaki.\n> \n> The Shamanu was just one of several hundreds of species classified by Coenraad Temminck, twenty of which carry his name. Although there is some dispute over the matter, today the Shamanu is commonly classified as a subspecies of the Grey Wolf: _Canis lupus hodophilax._\n\n####  C. P. Hodgson \u2013 1860  \n_**A Residence in Nagasaki and Hakodate** _ **, Japan**\n\nWolves were brought to the doors of the omnivorous Europeans and offered for sale. Every Japanese house in the north had, as well as a notice giving street number and family details, a charm to keep wolves away from their doors.\n\n> Small as it was, the Shamanu seems to have been greatly feared by the Japanese. However, there was once one other indigenous wolf on the island of Hokkaido. This was the Ezo Wolf ( _Canis lupis hattai_ ), which became extinct in Japan by 1888. It was a much larger animal than the Shamanu: heavier and longer, with a bigger head and more powerful teeth and jaws. Already rare by the mid-19th century, bounties of 7 to 10 yen ensured its extinction. Between 1878 and 1888, more than 1,500 Ezo Wolves were brought in for bounty. Since then, no sign of this animal was seen again in Japan.\n\n####  N. P. Anderson \u2013 1905  \n_**Anderson's Report \u2013 British Zoological Expedition** _ **, Japan**\n\nThis Wolf was purchased in the flesh, and I can learn but little about it. It is rare, some say almost extinct.\n\n> N. P. Anderson was a collector for a British expedition to Japan. This skin and skull are now to be found in the British Natural History Museum in South Kensington, London. It is one of the only two or three specimens in existence outside Japan. There is a fully mounted specimen from the Fukushima Prefecture, Hondo which is preserved in the Tokyo Science Museum.\n> \n> Anderson was correct about it being rare, but incorrect about it being \"almost extinct.\" The animal killed for the British collectors on 23 January 1905, near the village of Washikaguchi, Nara Prefecture, Honshu, was the very last the world would ever see of the Shamanu: the little Howling God that was the Japanese Wolf.\n\n####  **THE HOWLING GOD**\n\n##### Shamanu or Japanese Wolf \u2013 1905\n\nBroken wolf's tooth\n\nA charm, set in pounded gold\n\nAncient blood in a bronze bowl\n\nShogun hunted Shamanu, the little wolf,\n\nThrough forests and stony fields\n\nHorses and dogs chased him to mountains\n\nBeneath exhausted stars\n\nIn ancient days\n\nLegends gave him hypnotic power\n\nGeese swooned before\n\nThe glowing of his eyes\n\nBeneath the willow tree\n\nTravellers and pale maidens\n\nSuccumbed to his voice\n\nShamanu, the little Howling God,\n\nAscends the Nine Heavens\n\nWanders the land of nine jade rivers\n\nAnd nine jade mountains\n\nShamanu's blood is red amber now\n\nHis eye is sapphire\n\nHis flesh the cinnabar ore\n\nThat gives up bright quicksilver\n\nAnd deadly vapour\n\n##  **THE KANGAROO WOLF**\n\n###  **S ECOND WATCH 4 P. M. NONE**\n\n####  **THYLACINE or TASMANIAN TIGER \u2013 1936**  \n_**Thylacinus cynocephalus** _\n\n#####  Reverend Robert Knopwood \u2013 1805  \n_**Pastor's Diary** _ **, Risdon, Tasmania**\n\nAm engaged all the morn, upon business examining 5 prisoners that went into the bush. They informed me that on 2 May when they were in the wood they see a large Tyger that the dog they had with them went nearly up to it and when the Tyger see the men which were about 100 yards away, it went away. I make no doubt but here are many wild animals which we have not yet seen.\n\n> The name Tasmanian Tiger dates back to 1642, when the island had been \"discovered\" by Dutch explorer Abel Tasman who named the island Van Diemen's Land after his patron, the Governor of Batavia. That year, Francoys Jacobz, Tasman's pilot, led an expedition inland and reported \"the footing of wild beasts having claws very like a tiger.\" Later reports by Dutch East India Company officers also mentioned \"tiger\" footprints. However, there were no recorded sightings until 1805.\n> \n> Robert Knopwood's 18 June 1805 diary entry is the first recorded actual sighting of a Thylacine. Knopwood was the pastor for the first settlement on Tasmania: a penal colony at Derwent River founded in 1803.\n> \n> The indigenous Palowi people, of Adnyamathanha Aboriginal stock, had occupied Tasmania for over ten thousand years after the end of the last ice age when the land bridge to Australia sank beneath the sea. They knew the Thylacine as the _Marrukurli_ or _Inarru-kurli_ and had many legends concerning the animal.\n\n#####  William Oxley \u2013 1810  \n_**Van Diemen's Land Company Report,** _ **Tasmania**\n\nSettlements here are free from the Native Dog \u2013 the Dingo, the dread of the Stock Holders in New South Wales. The only Animal unknown on the Continent is the Hyena Opossum, but even here is rarely seen. It flies at the approach of Man, and has not been known to do any Mischief.\n\n> In 1806, Tasmania's Deputy Surveyor-General George Harris wrote an official description of this \"newly discovered\" creature and assigned it a genus and species: _Dideiphis cynocephala_ (or \"dog-headed opossum\"). He sent an illustrated report describing both the Thylacine and Tasmanian Devil to Sir Joseph Banks, President of the Royal Society and former naturalist on Captain Cook's first voyage.\n> \n> The Thylacine was the world's largest carnivorous marsupial. Related to the kangaroo, it was often described as a marsupial wolf or (because of its stripes) a marsupial tiger. As it was primarily a nocturnal hunter, it also had eyes with strange elliptical pupils. Although it resembles the placental wolf, it is one of the most striking examples of \"convergent evolution\" \u2013 the evolution of a body shape suited to its role and resembling unrelated animals occupying similar ecological niches.\n\n####  John Gould \u2013 1863  \n_**Mammals of Australia** _ **, Australia**\n\nA price is already put upon the head of the native Tiger, as it is called; but the fastnesses of the Tasmanian rocky gullies, clothed with impenetrable forests, will, for the present, preserve it from destruction.\n\n> John Gould was Australia's greatest and most informed naturalist painter of his time; however, his confidence in the survival of the Tasmanian Tiger in remoter parts of Tasmania did not prove to be correct. Thylacines were not only given an unreasonable reputation as sheep-killers, but despite all evidence to the contrary, some settlers even revived those old European superstitions about wolves, and insisted that the Tasmanian wolf killed only for blood (both animal and human) like a vampire.\n> \n> By 1820 Hobart was the second-largest town in Australia. It had shifted from whaling and sealing to a farming and sheep-ranching settlement. The Thylacine soon became a scapegoat for sheep killings, although most killings were the work of feral dogs, descendants of dogs taken to the island in 1798.\n> \n> Accounts of supposed Thylacine attacks on sheep are very dubious and were an excuse for mismanagement by the notoriously inefficient stockholders. As early as 1828 and 1829, livestock had been released into land not ready for them and without shelter. Hundreds died, compounded by unusually harsh weather in 1829. For those facing charges of incompetence, a ready scapegoat was found in the form of the Thylacine. In 1840, Van Diemen's Land Company raised the bounty to 10 shillings per head.\n\n####  George Smith \u2013 1909  \n_**Royal Natural History Magazine,** _ **Australia**\n\nThe shepherds wage incessant war on the creature, in summer laying traps and hunting it with dogs, in winter following up its tracks through the snow. A reward of a pound is given for the head by the Government, but the shepherd generally rides round with the head to several sheep-owners in the district, and takes toll from them all before depositing it at the police station. In consequence a large reward must be offered for the carcass of a Tiger, and an offer of \u00a310 during a year for a live Tiger to be delivered was unsuccessful. It pays a shepherd very much better just to hack off the head and take it round on his rides.\n\n> The year 1884 saw the setting up of local groups such as the \"Buckland and Spring Bay Tiger and Eagle Extermination Society.\" It is unlikely that it ever existed in large numbers, but there were ridiculous stories of Thylacines hunting in packs and killing up to a hundred sheep in a night just for sport. In 1888, a bill was passed offering a \u00a31-per-head bounty on Thylacines, a substantial amount in those days, and the animal was hunted in its own habitat far from any farms. Between 1888 and 1914 at least 2,268 Thylacines were known to have been killed and turned in. In 1910 a distemper epidemic (possibly brought in by domestic dogs) further depleted the population.\n> \n> The first live captive Thylacines displayed were at Regent's Park, London in 1850. Another was displayed in a menagerie in Hobart from 1854. By the 1860s several zoos around the world had Thylacines, but they were not often seen in the wild. In 1911, London Zoo paid \u00a368 for a single Thylacine. The last captive Thylacine in America died in the Bronx Zoo, New York, in 1919. In September 1911, the earliest film footage of a live Thylacine was made in the Beaumaris Zoo, which later became the Hobart Zoo. Here five of the seven brief silent film appearances of this animal were made: one in 1911, three in 1928 and one in 1933. The other two films were of the last living Thylacine outside of Australia in the London Zoo before its death in 1931.\n\n####  H. S. Mackay \u2013 1914  \n_**Memoirs of a Tiger-Hunting Man,** _ **Tasmania**\n\nA bull terrier once set upon a Wolf and bailed it up in a niche in some rocks. There the Wolf stood with its back to the wall, turning its head from side to side, checking the terrier as it tried to butt in from alternate and opposite directions. Finally the dog came in close and the Wolf gave one sharp, fox-like bite, tearing a piece of the dog's skull clean off, and it fell with the brain protruding, dead.\n\n> It was risky to hunt Thylacines with dogs; the creatures had no fear of them, and the dogs were often unwilling to tackle a trapped Thylacine, even if they outnumbered it. Even when cornered by the largest kangaroo-hunting hounds, they seemed to show little fear, and many of these valuable dogs were killed when attempting to chase them down. Thylacines could defend themselves well enough against dogs, but were no match for bullets, snares or poison. Strangely enough, there are no accounts of any even moderately successful attacks on humans, even when cornered or trapped. Despite its harmlessness to humans, the Thylacine none the less suffered the same persecution as the wolf suffered throughout the world.\n> \n> Sir Ray Lankester in _Harmsworth Natural History_ (1910) wrote: \"When one watches the Tasmanian Wolf, one comes to the conclusion that its appearance, ways and movements suggest the fancy that it is a Kangaroo Masquerading as a Wolf, though not very successfully.\"\n\n####  R. Boswell \u2013 1937  \n_**Australian Museum Magazine,** _ **Australia**\n\nThe former range of the Tasmanian Tiger must have been very great as I know of one Tasmanian, who with his brothers, killed as many as twenty-four of these animals during one day, and received a reward of \u00a31 per head for each animal. Fortunately, the Tasmanian Tiger is now wholly Protected.\n\n> The last authenticated shooting of a Thylacine in the wild was on 13th May 1930 by Wilf Batty on his farm in the Mawbanna district. The last one captured was in 1933, and kept in the Hobart Zoo. This animal was filmed by David Fleay in 1933, who argued forcefully for the animal to be kept in a better facility. This animal can be seen opening its formidable gaping jaws to about a 140 degree angle.\n> \n> The Tasmanian government gave the Thylacine full legal protection on 14 July 1936. While a committee considered further methods of protection, another government department was still issuing hunting permits. Two months later, on 2 September 1936, the last Thylacine in existence, the animal filmed by David Fleay and named \"Benjamin\" \u2013 died as the result of near-starvation and neglect in the Hobart Zoo. Absurdly, that year the image of two rampant Thylacines were placed on the official Tasmanian Coat of Arms.\n\n####  Arthur Mee \u2013 1947  \n_**The Children's Encyclopedia,** _ **London, England**\n\nA sort of nightmare wolf: its home is Tasmania, its lair is a dark cave or cleft in the rocks; its habits are those of our own wolves, reinforced with an acid tincture of peculiar savagery. It seems to be the Caliban of the wolf tribe, making up in ferocity and blank savagery what it lacks in refined cunning like other marsupials. It carries its young in a pouch, and the whimper of young Hopeful in that furry cradle has sounded in the ears of many a sheep as it has fallen a helpless victim to the fangs of the mother.\n\n> Arthur Mee perpetuated the misinformation about the supposedly bloodthirsty, sheep-killing Thylacine in his _Children's Encyclopedia._ This is from an edition circa 1946. Over a decade after the animal became extinct, the myth of this supposedly ferocious vampire-like killer persisted.\n> \n> The same year, there was a recording of a popular ballad about Tasmania with a verse that conjured up the bloodthirsty image of the Tasmanian Tiger that was published in O'Lochlainn's _Folk Songs of Ireland:_\n> \n> Our cots we fence with firing\n> \n> And slumber while we can\n> \n> To keep the wolves and tigers from us\n> \n> In Van Dieman's Land.\n> \n> Ironically, in 1966, Tasmanian officials declared a huge game reserve as an area of preservation for the Thylacine, which had not been seen for 30 years. A number of expeditions have been launched, but it is now agreed the Tasmanian Thylacine is extinct.\n> \n> As for the original human population of Tasmania: the last of the Palowi tribal aboriginal people were hunted down and exterminated within forty years of the 1803 colonization. In the 1840's the last Tasmanians were housed by missionaries in the original convict colony buildings at Derwent River. Thirty years later, Trugenaner (Truganini), the last full-blood Tasmanian, died there in 1876.\n\n#####  **THE NIGHT STALKER**\n\n###### Thylacine or Tasmanian Tiger \u2013 1936\n\n1.\n\n_Tasmanian Tiger, Kangaroo Wolf_\n\n_Pouched Dog, Hyena Opossum_\n\nNo name seemed to fit\n\nThis creature of the night\n\nHead and teeth of a wolf\n\nEyes and stripes of a tiger\n\nGaping jaws of a crocodile\n\nIt had the torso of a hyena\n\nThe backward pouch of an opossum\n\nFront legs of a dog\n\nHind legs and tail of a kangaroo\n\nResorting to the Lego-language\n\nOf the ancient Greeks\n\nWe came up with _\"Thylacine\"_\n\nWhich is \"a pouched something\"\n\nShy and elusive\n\nNever a danger to man\n\nYet, somehow we made it\n\nA nightmare beast\n\n2.\n\nStill struggling toward definition,\n\nWe gave it a second Greek name:\n\n_\"Cynocephalus\"_ or \"Dog-head\"\n\nLike a pagan underworld god\n\nThe creature didn't know\n\nIt was a chimera\n\nThought it was just itself\n\nNot a composite riddling beast\n\nSome kind of sphinx\n\nIn a land of the kangaroo and platypus\n\nWho could say what was strange?\n\nIn fact, in its own dreamtime\n\nWe were the nightmare\n\n3.\n\nIn this creature's world\n\nWhat was truly monstrous\n\nWhat was absolutely deadly\n\nWas us\n\nFor all the First Tasmanians\n\nBoth men and beasts\n\nNothing so strange\n\nHad entered their world\n\nSince the end of the ages of ice\n\nNothing so sudden\n\nAnd dangerous\n\nSince the beginning\n\nWhen molten fire poured down\n\nFrom the night sky into the boiling sea\n\n##  **THE TIGER'S TALE**\n\n###  **T HIRD WATCH 5 P. M. NONE**\n\n####  **BALI TIGER \u2013 1937 \u2013 _Panthera tigris balica_**\n\n#####  Pliny the Elder \u2013 65 AD  \n_**Natural History** _ **, Rome**\n\nHyrcania and India produce the Tiger, an animal of tremendous swiftness and ferocity. The late Emperor Augustus was the first person who exhibited at Rome a Tiger in the arena. This was in the consulship of Quintus Tubero and Fabius Maximus, at the dedication of the theatre of Marcellius, on the fourth day before the nones of May: the late Emperor Claudius exhibited four at one time.\n\n> Augustus Caesar's introduction of the first Tigers into the Roman arena is also mentioned by Suetonius in his _Life of Augustus_ ; while Lampridius informs us that the mad teenage Emperor Heliogabalus yoked Tigers to his chariot in imitation of the god Bacchus. Martial, who lived a little after Pliny, speaks of Tigers in considerable numbers exhibited in Rome by Domitian.\n\n#####  Christian Brothers \u2013 1150 AD  \n_**The Book of Beasts** _ **, Cambridge**\n\nTigris the Tiger gets his name from his rapid pace for the Persians, Greeks and Medes used to call the arrow 'tygris'. The beast can be distinguished by his manifold markings, by his courage and by his wonderful velocity. And after him the River Tigris is named because it is the swiftest of all rivers.\n\n> This description is from the Cambridge _Book of Beasts_ (1150 AD), one of the many famous Latin bestiaries which were medieval books of natural (and supernatural) history. In large part, they were encyclopaedic gleanings from classical authors like Lucretius and Pliny combined with traveller's tales and folklore. This _Book of Beasts_ was translated from the Latin by T. H. White, the medievalist and author of the Arthurian novel _The Once and Future King._ White's translation begins with a quotation by T. H. Huxley which perhaps explains something of the bestiary's eternal appeal: \"Ancient traditions, when tested by severe processes of modern investigation, commonly enough fade away into mere dreams; but it is singular how often the dream turns out to have been a half-waking one, presaging a reality.\"\n\n####  Baron Oscar Vojnick \u2013 1911  \n_**On the East India Group of Islands** _ **, Bali**\n\nIn the western part of Bali Island, along the northern shore, in the mountains of Groendoel, we discovered tiger footprints. On November 2nd while collecting twigs to be used for the construction of a fence around the traps, the carcass of a freshly killed kidang [roe deer] was encountered by our people. The trap was set in front of the kidang in the thicket. Minaut was almost certain that the tiger would be caught in another day. I was much less convinced, as the many human tracks could have warned the tiger. But no, it came in to feed, and the trap caught one of its forelegs, just below the wrist.\n\n> In 1912, Ernst Schwarz was able to write in _Big Game Hunting_ that the Tiger was \"fairly common in Bali.\" Two decades later it was gone. The Bali Tiger was the smallest of eight subspecies of Tigers, and the first to become extinct. Today, three (Bali, Java, Caspian) of the eight subspecies of Tiger are extinct, and another four are critically endangered. The Tiger had survived millennia of hunting by man with impunity, but when modern high-powered rifles became widespread, Tiger populations rapidly declined to near extirpation everywhere.\n\n####  Arthur de Carlo Sowerby \u2013 1923  \n_**\"China's Fur Trade,\" China Journal** _ **, Hong Kong**\n\nThe Tiger is the most dreaded and highly prized of the carnivores. Not only is his skin of value, but his whole carcass; for the Chinese believe that the bones, blood, heart and even the flesh of the tiger have medicinal properties of rare power, and will pay a goodly price for concoctions brewed by the apothecary that contain such ingredients as powdered tiger's knee-cap, or clotted tiger's blood. The heart of the tiger is supposed to impart to the consumer the courage and strength of the tiger itself. On this account the tiger has been hunted till he is almost extinct.\n\n> Sir James Frazer in _The Golden Bough_ in 1920 wrote concerning Tiger fetishes: \"The Miris of Assam prize tiger's flesh as food for men; it gives them strength and courage. But 'it is not suited for women; it would make them too strong-minded'. In Korea the bones of tigers fetch a higher price than those of leopards as a means of inspiring courage. A Chinaman in Seoul bought and ate a whole tiger to make himself brave and fierce.\"\n\n####  George Jennison \u2013 1928  \n_**Noah's Cargo** _ **, Java**\n\nThe Rampoc, or 'Tiger fight' waged against men, is a Court ceremonial in which the Royal bodyguard, in four ranks and armed with long pikes forms a square of fifty yards, in the centre of which the tiger cage is deposited covered with straw. The door is held in place by a rope of equally inflammable material. The Court being assembled, two officials of high rank walk in sedately to the sound of music, fire the straw, and return equally composedly back to the ranks that open to receive them. Meanwhile, the tiger, maddened by burns and fear, is bounding frantically in the cage until the door falls, when it is sure to leave at once. Outside it may, and often does, hesitate; it looks round for a foe on which to vent its fury, and dashes, as a rule, for the two men whose isolation marks them for attack. It is received on the spears, and not uncommonly is impaled by its own leap and weight.\n\n> The second smallest species, the Javan Tiger ( _Panthera tigris sondaica_ ), was also hunted to extinction some time around 1988. Curiously, it was on Java's Trini Island that one of the oldest human skeletal remains was discovered in 1891: the famous 700,000 year old Java Man ( _Pithecanthropus erectus\/Homo erectus_ ). It was also here, just over a century later, that the oldest Tiger skeletal remains were discovered: the 1,200,000 year old Trini Tiger ( _Panthera tigris trinilensis_ ).\n\n####  VOS Company Report \u2013 1931  \n_**Dutch East Indian Company Survey** _ **, Bali**\n\nIn Bali the Tiger is one of the five traditional totemic Barong dancers: Tiger, Boar, Dragon, Elephant and Lion. Each represents an area of Bali that has its own protective spirit modelled on a different animal. Barong is the King of Spirits and the leader of this host of good spirits. Barong is in mortal combat with the Rangda Witch of Black Magic and her evil graveyard spirits. Despite being numbered among the protective spirits, the Tigers of Bali have enjoyed no popular protection in return. A few of this race yet live in West Bali but are having a hard time. They are much sought after by hunters from Java, so they will certainly disappear within a few years.\n\n> This Dutch East Indies Survey reported that uncontrolled hunting by Dutch colonial trophy hunters on excursions from Java was rife and fashionable. The report's prediction of extinction proved correct: the last known Bali Tiger was a female shot at Sumbar Kima in West Bali on 27 September 1937.\n> \n> The third Tiger extinction was an animal over twice the size of the Bali and Java subspecies. The Caspian Tiger ( _Panthera tigris virgata_ ) measured over 10 feet in length and weighed in excess of 500 pounds. A government and military policy of tiger eradication in Russia, combined with uncontrolled hunting in the Caspian regions of Turkey, Iran and Afghanistan, so depleted the population by the 1940's, that there were thought to be only a dozen left in the mountains of northern Iran by the late 1950s. Sometime between 1960 and 1980, the Caspian Tiger became extinct.\n> \n> With the exception of the Indian Tiger ( _Panthera tigris tigris_ ) in protected reserves in India, all other surviving tiger subspecies are critically endangered and number only in the hundreds in the wild. These are the Siberian Tiger ( _Panthera tigris altaica_ ), the Chinese Tiger ( _Panthera tigris amoyensis_ ), the Sumatran Tiger ( _Panthera tigris sumatrae_ ) and the Indo-Chinese Tiger ( _Panthera tigris corbetti_ ).\n\n####  **LAST RITES**\n\n##### Bali Tiger \u2013 1937\n\nThe little prince of tigers\n\nIs dead\n\nThe chime of temple bells\n\nAn army of votive arrows is offered\n\nA long rope of prayer tethers the sun\n\nMaster of the emerald forest\n\nLord of the crystal waterfall\n\nThat feeds the sacred lake\n\nDread foe of the Monkey King\n\nAnd the Dragon Master\n\nHe was like a flame in the forest\n\nHis fearful laughter was in the land\n\nNow he is the small flickering light\n\nOf an altar candle\n\nThe yellow tallow of memory\n\nSlowly flowing away\n\n##  **VIKINGS AND GAREFOWL**\n\n###  **F IRST WATCH 6 P. M. VESPERS**\n\n####  **GREAT AUK OR GAREFOWL \u2013 1844**  \n_**Alca (Pinguinus) impennis** _\n\n#####  Jacques Cartier \u2013 1534  \n_**Journals of Jacques Cartier,** _ **Isle of Birds, Newfoundland**\n\nSome of these birds are as large as geese, being black and white with a beak like a crow's. They are always in the water, not being able to fly in the air, inasmuch as they have only small wings about the size of half one's hand, yet they move as quickly along the water as other birds fly through the air. And these birds are so fat that it is marvellous. In less than an hour we filled two boats full of them, as if they had been stones, so that besides them which we did eat fresh, each of our ships did powder and salt five or six barrels of them.\n\n> Jacques Cartier's account of this unequal meeting of Great Auk and Man on the Isle of Birds (Funk Island) off the coast of Newfoundland in 1534 is the first written record of Europeans hunting these birds in North America: \"And on the twenty-first of the said month of May we set forth from this [Catalina] harbour with a west wind, and sailed north, northeast of Cape Bonavista as far as the Isle of Birds, which island was completely surrounded and encompassed by a cordon of loose ice, split up into cakes. In spite of this belt of ice our two longboats were sent off to the island to procure some of the birds, whose numbers are so great as to be incredible, unless one has seen them; for although the island is about a league in circumference, it is so exceedingly full that one would think they had been stowed there. In the air and round about are an hundred times as many more as on the island itself.\"\n> \n> A curious theory links the Great Auk with the discovery of North America. It proposes that the Icelandic Vikings, already familiar with Auks and aware in a general sense of bird migrations, would have noticed the seasonal disappearances and departures of vast rafts of birds in their western waters. It would have been a small step for the Vikings to conclude that these Garefowl were travelling to other, similar islands in the west. Consequently, it has been suggested they could have followed the migrating legions of Auks, who would have led them first to Iceland, and later to Newfoundland and Labrador.\n> \n> The Great Auk was the strongest and swiftest of northern swimming and diving birds, and almost immune to marine predators; while its rocky nesting sites in the rough North Atlantic were for millions of years inaccessible to men. It numbered in the tens of millions until recent historic times.\n\n#####  Anthonie Parkhurst \u2013 1578  \n_**Hakluyt's Voyages,** _ **Isle of Penguins, Newfoundland**\n\nThere are sea Gulls, Murres, Duckes, wild Geese, and many other kind of birdes store, too long to write, especially on one island named Penguin, where we may drive them on a planke into our ship as many as shall take her. These birds are also called, Penguins and cannot flie; there is more meate in one of these then in a goose; the Frenchmen that fish neere the grand baie, doe bring small store of flesh with them, but victuall themselves always with these birdes.\n\n> Cartier's \"Isle of Birds\" of 1534, Parkhurst's \"Isle of Penguins\" of 1578, and Cartwright's \"Funk Island\" in 1792 are one and the same rookery visited by all three voyagers, and by countless others over nearly three centuries. The summer \u2013 the nesting season \u2013 was the only time that Great Auks could be taken, for the rest of the year they dispersed and lived at sea, where they were safe from human hunters. So each summer, the men came to their rookeries and set up camps where they waited for the birds to come to them. Stone corrals were constructed on the islets, and the birds were driven into them and slaughtered.\n> \n> Slaying the Auks during the nesting season compounded the destruction; as did egg collection, as each nesting pair only produced a single egg. This slaughter and depredation continued unabated for nearly three centuries. The general presumption of most observers was summed up in one hunter's account written in 1622: \"God made the innocence of so poor a creature to become such an admirable instrument for the sustentation of man.\"\n> \n> By the Vikings, this bird was known as the _Geirfugl_. And by the Irish it was known as _Gearrabhul_ \u2013 meaning \"the strong stout bird with the spot\" \u2013 which was corrupted through usage to \"Garefowl.\" Among the Welsh, who once hunted the bird on islets off Britain's shores, they were called _Pingouins_ , which means \"white head.\" The Great Auk was the original \"Penguin: \" the \"penguins\" of the southern hemisphere having derived their name from British explorers who presumed these flightless Antarctic birds were a southern variety of the North Atlantic bird.\n\n####  George Cartwright \u2013 1792  \n_**Journal of Transactions and Events,** _ **Funk Island, Newfoundland**\n\nInnumerable flocks of sea-fowl breed upon this isle every summer, which are of great service to the poor inhabitants of Fogo when the water is smooth, they make their shallop fast to the shore, lay their gang-boards from the gunwale of the boat to the rocks, and then drive as many penguins on board, as she will hold: for, the wings of those birds being short, they cannot fly, nor escape. The birds which people bring thence, they salt and eat, in lieu of salt pork.\n\n> Fishermen and Newfoundland colonists flocked to these rookeries every nesting season to kill birds and steal eggs. Also, many new industrial uses were found for these birds: feather beds, meat for bait, oil for lighting lamps, oil for fuel for stoves, and collar bones for fish hooks; and dried Auks (full of oil) were used as torches. The annual wholesale gathering of eggs was an especially destructive practice. Just one vessel commanded by a Captain Mood took 100,000 eggs in a single day.\n> \n> By 1810 Funk Island was the only West Atlantic rookery left, and these constant summer raids by oil and feather merchants' crews soon finished off even this last vast North American nesting site as well. The only place Great Auks existed after Funk Island rookery's demise was a group of islands on the southwestern tip of Iceland. Most Auks withdrew to a lonely outcropping of rock called Geirfuglasker, or \"Auk Rocks\" \u2013 which was a lonely and dangerous outcropping of rock so rugged, it was safe from all but the most determined and foolhardy of hunters.\n> \n> In 1830, the already near-extinct species suffered a further cataclysmic disaster. A volcano erupted under the sea near Iceland and caused a seaquake that resulted in the destruction of Geirfugl Island. The last rookery of the Great Auk sank beneath the sea, and the colony was destroyed and scattered.\n> \n> It became apparent that about 60 Great Auks survived the sinking of Geirfuglasker. These birds took refuge on an even smaller and equally dangerous rocky outcrop, known as Eldey Island. It is from this last station that nearly all of the skins and eggs now found in European collections have been obtained, and during the fourteen years (1830-1844) that the Garefowl frequented this rock, one bird after another was hunted down for museum bounties, until there were just two left.\n\n####  Professor A. Newton \u2013 1861  \n_**The Last Garefowl Hunt \u2013 4 June 1844,** _ **Eldey Island, Iceland**\n\nAs the three men leapt from the boat and clambered up on the rocks they saw two Garefowls sitting among numberless other rock-birds, and at once gave chase. The Garefowls showed not the slightest disposition to repel the invaders, but immediately ran along under the high cliff, their heads erect, their little wings somewhat extended. They uttered no cry of alarm, and moved with their short steps, about as quickly as a man could walk. Jon Brandsson, with outstretched arms, drove one into a corner, where he soon had it fast. Sigurdr Islefsson and Ketil Ketilsson pursued the second, and the former seized it close to the edge of the rock, here risen to a precipice some fathoms high, the water being directly below it. Ketil then returned to the sloping shelf whence the birds had started, and saw an egg lying on the lava slab, which he knew to be a Garefowl's. He took it up, but finding it was broken, put it down again. All this took place in much less time than it takes to tell it.\n\n> Newton gives this account as being recorded in 1861, just 14 years after the event of the slaying of this last pair of Great Auks on the Icelandic skerry of Eldey. Another oral account confirms that Brandsson and Islefsson each killed a bird, but Ketilsson, frustrated at returning empty-handed, smashed the last intact egg with his boot. A third account, by James Wolley, was published in _Ibis_ in 1861: \"The capture of these two birds was effected through the efforts of an expedition of fourteen men, led by Vilhjaimur Hakonarsson; but only three were able to land on the rock, and they at great risk, namely, Sigurdr Islefsson, Ketil Ketilsson, and Jon Brandsson. Only two Great Auks were seen, and both were taken \u2013 Jon capturing one, and Sigurdr the other. It appears that this event took place between the 2nd and 5th June 1844. It appears that this expedition was undertaken at the instigation of Herr Carl Siemsen, who was anxious to obtain the specimens... the birds were sold for eighty rigsbank-dollars, or about \u00a39.\"\n> \n> It is notably ironic that Great Auk relics have continued to appreciate in value over the years since its extinction. In 1934, in the midst of the Depression, one stuffed Great Auk sold for nearly $5,000, or five years' wages for a fisherman; and a single egg for two years' wages. Then, on 4 March 1971, the Icelandic Natural History Museum paid \u00a39,000 for a single mounted specimen of a Great Auk.\n\n####  **ALPHA AND OMEGA**\n\n##### Great Auk or Garefowl \u2013 1844\n\nIn the year 1844, Isambard Kingdom Brunel\n\nStood on the iron deck of the first transatlantic passenger liner\n\nAs the age of sail gave way to the age of steam\n\nIn that year, Samuel Morse\n\nSent a coded message singing through the steel lines\n\nAnd electric relays of America's first telegraph\n\nIts chosen path was alongside the railway line\n\nBetween Baltimore and Washington\n\nThe message was biblical and portentous\n\nIt read: \"What hath God wrought?\"\n\nIn the year 1844, three daring men climbed\n\nUp a precarious chimney of volcanic rock\n\nIn the wave lashed North Atlantic\n\nHere the last lonely pair of Great Auks\n\nWere chased down and slaughtered\n\nTheir last egg crushed\n\nThis too heralded the birth of our modern age\n\n\"What hath God wrought?\"\n\n##  **DARWIN' S DOGS**\n\n###  **S ECOND WATCH 7 P. M. VESPERS**\n\n####  **WARRAH or ANTARCTIC WOLF-FOX \u2013 1876** _**Disicyon australis** _\n\n#####  Lord John Byron \u2013 1741  \n_**Adventures of Midshipman John Byron** _ **, Falkland Islands**\n\nFour creatures of great fierceness resembling wolves, ran up to their bellies in the water to attack the people in the boat and so we were forced to put out to sea again to find a safer haven further along the sandy shore. But soon as we saw the beasts again in a state of alarm we set fire to the tussock to get rid of them. But in this we had greater effect than expected as the country was ablaze as far as the eye could reach for several days, and we could see them running in great numbers.\n\n> The _Adventures of Midshipman John Byron_ included the wrecking of his ship _The Wager_ in the notorious Straits of Magellan. Midshipman John Byron eventually became Captain John Byron, the fourth Lord Byron and grandfather of the great Romantic poet George Gordon, the sixth Lord Byron. This was one of the most famously harrowing adventures of the time. With the survivors of the wreck, John Byron arrived on the Falklands where he encountered these seemingly ferocious \"Wolves.\" Like his grandson, Byron was a controversial character who earned the name \"Foul Weather Jack\" after his promotion to rank of commodore after he completed a record-breaking twenty-two month circumnavigation of the globe. As commander of the _Dolphin_ and _Tamar_ he claimed the Falklands for George III in 1764, despite France's Marquis de Bougainville's simultaneous efforts at colonization of those islands that same year.\n> \n> Curiously, the first description of this \"Wolf-Fox\" on record came from Richard Simson, who sailed on the British ship _The Welfare_ in 1690: \"We saw foxes twice as big as in England, we caught a young one, which we kept on board for some months.\" However, Simson's ship became engaged in a battle in the first Falklands War in which the Wolf-Fox became the first casualty: deciding a battleship was no fit place for man or beast, the poor creature panicked after the first cannon volley and leapt overboard.\n\n#####  Dom Pernetty \u2013 1764  \n_**A Voyage to the Malouin Islands** _ **, Falkland\/Malouin Islands**\n\nOfficers of M de Bougainville's suite were, so to speak, attacked by a sort of wild dog; this is, perhaps, the only savage animal and quadruped which exists on the Malouin Islands. But, perhaps too, this animal is not actually fierce, and only came to present itself and approach us, because it had never before seen men. The birds do not fly from us: they approached us as if they had been tame.\n\n> Dom Pernetty, also the author of _Dissertation Upon America,_ was a member of the Marquis de Bougainville's famous expedition. These uninhabited islands in the South Atlantic were called the Malouin Islands by the French, the Malvinas by the Spanish, and the Falkland Islands by the British, and were seen by all three powers as a key safe port for ships attempting to passage from the Atlantic to the Pacific.\n> \n> The Marquis Louis-Antoine de Bougainville also described this animal and its habits in his _Voyage Round the World_ (1770): \"The Wolf-Fox, so called, because it digs itself in earth and because its tail is longer and more fully furnished than that of a wolf, lives in the dunes of the sea shore. It follows the game and plans its trails intelligently, always by the shortest route from one bay to another. On our first landing we quite believed that they were the paths of human inhabitants.\"\n> \n> The Marquis de Bougainville's historic circumnavigation of the globe in 1766-69 in the frigate _La Boudeuse_ and the store-ship _L'Etoile_ just preceded Captain James Cook's first voyage in the _HMS Endeavour_ to the South Pacific, where both men were ordered to observe the transit of Venus in 1778. Bougainville's was only the 20th circumnavigation of the world since Magellan's 250 years previously, in 1519, and since the second by Sir Francis Drake 190 years before, in 1580. It was also the first circumnavigation by the French, and the first that could be said to have a true \"natural science\" agenda.\n\n####  Charles Darwin \u2013 1833  \n_**Zoology of the Voyage of the Beagle** _ **, Falkland Islands**\n\nTheir numbers have been greatly decreased by the singular facility with which they are destroyed. I was assured by several of the Spanish countymen, who are employed in hunting cattle which run wild on these islands, that they have repeatedly killed them by means of a knife held in one hand and a piece of meat to tempt them to approach in the other. The number of these animals during the past fifty years must have been greatly reduced: already they are entirely banished from half of East Falkland. It cannot, I think, be doubted, that as these islands are now being colonized, before the paper is decayed on which this animal is figured, it will be ranked among those species which have perished from the earth.\n\n> Charles Darwin, while on the islands, observed the \"Falkland Island Wolf-Fox\" or \"Antarctic Wolf\" (that was locally known as the Warrah) and collected three skins, two of which were later presented to the London Zoological Society. What interested Darwin most about the Warrah was that it was the only predator on the islands and, apart from a small variety of mouse, the only land mammal. The Warrah seemed to live on an unlikely diet of mice, sea birds, eggs, and (perhaps) small sea mammals.\n> \n> Just how the Warrah could have evolved on these isolated islands without any substantial prey species, and with no other related species on the islands at all, was something of a mystery to Darwin. As Darwin's prophecy of extinction quite rapidly came to pass, the mystery of the Warrah's evolution is not likely to be resolved.\n\n####  Lt. Colonel Charles Hamilton-Smith \u2013 1839  \n_**The Dog Tribe** _ **, Edinburgh**\n\nThe Falkland Island Aquara Dog I discovered in the fur stores of Mr Jacob Astor in New York, a large collection of peltry which came from the Falkland Islands, where, according to reports that gentleman received, his hunters had nearly extirpated the species.\n\n> Artist-naturalist-antiquarian-spy, Hamilton-Smith, a veteran of the Napoleonic and 1812-14 Wars, was the author of many books on natural history and an authority on the history of military uniforms and costumes. He was only slightly incorrect, in that the Astor hunters did not quite extirpate the \"Antarctic Wolf-Fox\" or \"Aquara Dog.\" A few stragglers hid out in the hills for another few decades, and remarkably enough, in 1868 one captive Warrah was acquired by the London Zoo in Regent's Park. There it survived for a number of years, narrowly missing out on being the last of its species. That dubious honour was bestowed upon a wild Warrah at Shallow Bay in the Falkland's Hill Cove Canyon, where it was shot and skinned in 1876.\n> \n> The first North American wolf to become extinct was the Newfoundland White Wolf in 1911. This animal's scientific name was _Canis lupus beothucus_ in ironic recognition of the similarly extinct Beothuk Indians of Newfoundland. Other North American wolves exterminated during the 20th century were the Kenai Wolf ( _Canis lupus alces_ ) of the Kenai Peninsula in Alaska in 1915; the Texas Grey Wolf ( _Canis lupus monstrabilis_ ) and the New Mexican Mogollon Mountain Wolf ( _Canis lupus mogollonensis_ ), both in 1920; the Great Plains Lobo Wolf ( _Canis lupus nubilus_ ) in 1926; the Southern Rocky Mountain Wolf ( _Canis lupus youngi_ ) in 1940; and the Cascade Mountains Brown Wolf ( _Canis lupus fuscus_ ) in 1950.\n> \n> Besides these lost North American forms of _Canis lupus_ (Grey Wolf), all three forms of _Canis rufus_ (Red Wolf) became extinct in the wild (or so interbred with coyotes as to cease to exist as a pure species) by 1980. These were: the Texas Red Wolf ( _Canis rufus rufus_ ), the Mississippi Red Wolf ( _Canis rufus gregoryi_ ), and the Florida Black Wolf ( _Canis rufus floridanis_ ).\n\n####  **WARRAH'S LAST SONG**\n\n##### Warrah or Antarctic Wolf-Fox \u2013 1876\n\nEverybody heard singing in the night\n\nOne went to the valley to listen\n\nOne climbed the hill to see\n\nA wolf was sitting far off\n\nOn a high ridge beneath the stars\n\nHe was singing\n\nHe was an old wolf\n\nHis teeth worn and broken\n\nHe was an old wolf\n\nOn a high ridge beneath the stars\n\nHe was an old wolf\n\nBut he sang a young wolf's song\n\n_At daybreak I go_\n\n_At daybreak I go I run I go_\n\n_At daybreak I go I run I go_\n\n_At daybreak I go_\n\nHe was an old wolf\n\nOn a high ridge beneath the stars\n\nHe was an old wolf\n\nBut he sang a young wolf's song\n\n_At daybreak I go_\n\n_At daybreak I go I go I go_\n\n##  **COLUMBUS ' CURLEWS**\n\n###  **T HIRD WATCH 8 P. M. VESPERS**\n\n####  **ESKIMO CURLEW \u2013 1985 \u2013 _Numenius Borealis_**\n\n#####  Christopher Columbus \u2013 1492  \n**De Las Cassas, _Columbus' Journal,_ Sargasso Sea**\n\nImmense flocks of birds, far more than we have seen before, passed overhead all day long, coming always from the north and heading always towards the southwest. Which led the admiral to believe that they were going to sleep on land or were, perhaps, flying from the winter which was about to come to the lands from which they came. On this account the admiral decided to abandon the westward course and to steer west-south-west, with resolve to proceed in that direction for two days but after two days, more birds were sighted and all that night his men heard birds passing so for two days again they sailed west-south-west, whereupon the Pinta being the swifter ship went on ahead of the Santa Maria and was the first to sight land.\n\n> In 1961, the ornithologist James Tooke wrote: \" _Columbus' Journal_ tells us the birds the sailors snared from these immense flocks were plainly field birds that could not possibly find rest on water... . Consequently, Columbus decided to change course and follow the birds in hope of finding land. Five days later, Columbus set foot on the island of San Salvador... . Although the mariners did not know what the non-stop birds were, the date and direction of flight \u2013 at that time and in that place \u2013 identify them as Eskimo curlews and golden plovers, making their oversea flight one of 2,500 miles.\"\n\n#####  John James Audubon \u2013 1833  \n_**Ornithological Biography** _ **, Labrador**\n\nDuring a thick fog, the Esquimaux Curlews made their appearance near the harbour of Bras d'Or. They evidently came from the north, and arrived in such dense flocks as to remind me of the Passenger Pigeons. They continued to arrive for several days, in flocks which seemed to me to increase in number. They flew in close masses, sometimes high, at other times low, but always with remarkable speed, and performing beautiful evolutions in the air. The appearance of man did not intimidate them, for they would alight so near us, or pass over our heads at so short a distance, that we easily shot them.\n\n> The legendary John James Audubon was America's greatest pioneering ornithologist and artist. Born Jean-Jacques Foug\u00e8re Audubon in Haiti, he was the illegitimate son of a French naval officer and privateer, and a Spanish Creole mistress from Louisiana. The Haiti 1788 slave rebellion forced Audubon's father to return to France, where the boy was raised and educated. To avoid conscription in the Napoleonic Wars, the young Audubon travelled to America on a false passport where he became a frontiersman and merchant. Eventually he settled on his life's passion as an ornithologist and wildlife artist. He was a trailblazer in his portrayal of life-size birds \"drawn from life,\" and in publishing _Birds of America_ with its magnificent octavo colour plates, which was subscribed to by 1100 patrons and earned him $36,000.\n> \n> The Eskimo Curlew was one of the champion long-distance migrants: flying from the shores of the Arctic Ocean and the Bering Sea to Labrador and the Gulf of St. Lawrence, then making a 4,000 kilometre non-stop ocean crossing to Guiana and Brazil, then overland to southern Argentina and Chile \u2013 and as far south as Tierra del Fuego. The return route was up the Pacific side of South and Central America before reaching Texas and Nebraska, and northward back to the High Arctic in an enormous elliptical loop over the entire North and South American continents. This was an overall migratory flight of over 30,000 km.\n\n#####  Dr. A. S. Packard \u2013 1861  \n_**Ornithological Journals** _ **, Labrador**\n\nThe Curlews appeared in great numbers. We saw a flock a mile long and nearly as broad; the sum total of their notes sounded at times like the wind whistling through the rigging of a thousand ton vessel; at others the sound seemed like the jingling of multitudes of sleigh-bells.\n\n> The noted Massachusetts ornithologist Edward Howe Forbush, author of _Birds of New England,_ first observed vast flocks of Passenger Pigeons as a child in the 1860's. When he wrote about them in 1916, he explained the manner of their demise: \"When the Passenger Pigeon began to decrease in numbers, about 1880, the marksmen looked about for something to take its place in the market in the spring. They found a new supply in the great quantities of Plover and Curlews in the Mississippi valley that season... . They were shot largely for western markets at first; they began to come into the eastern markets in numbers about 1886... . These markets were: Halifax, Montreal, Philadelphia, Boston, New York, Wichita, St. Louis, Chicago and Detroit. Boston shipments alone amounted to tens of thousands of birds every spring through 1887 to 1896.\"\n\n#####  Myron H. Swenk \u2013 1915  \n_**Procedures of the Nebraska Ornithologists Union** _\n\nIn the 1880's, hunters would drive out from Omaha and shoot until the slaughtered birds literally filled wagonloads up with the sideboards on. Sometimes when the flight was unusually heavy and the hunters were well supplied with ammunition their wagons were too quickly and easily filled, so whole loads of the birds would be dumped on the prairie, their bodies forming piles as large as a couple of tons of coal where they would be allowed to rot while the hunters proceeded to refill their wagons with fresh victims.\n\n> The first president of the Nebraska Ornithologists Union, Dr. Lawrence Bruner, wrote about the Eskimo Curlew in 1896: \"These flocks reminded the settlers of the flights of passenger pigeons and the curlews were given the name of 'Prairie Pigeons'. They contained thousands of individuals and would often form dense masses of birds extending for a quarter to a half mile in length and a hundred yards or more in width. When the flock would alight the birds would cover 40 or 50 acres of ground.\"\n> \n> Myron Swenk also noted: \"In the Midwest states the fields where they were known to gather were patrolled regularly and scanned with binoculars. Sometimes a gunner would put himself in a line of flight where he had only to shoot one bird from a flock to cause the remaining birds to circle again and again until most or all had been shot. He then waited for the next flock.\"\n> \n> The vast migratory flocks of Eskimo Curlew had virtually vanished everywhere by 1900. The last birds were sighted and shot in Nebraska in 1911. The last record of the appearance of Eskimo Curlews in the state of Massachusetts was on August 28, 1929 when they appeared on a dinner table at a hotel at Schoodic Point.\n> \n> The birds last seen in South America were shot in 1925. Less than a dozen sightings of individuals were recorded over the next century, and many of these have proven to be similar curlew-like species. The last confirmed sighting was in 1963. The last \"possible\" sightings of more than a single bird were in Texas in 1981 and Kendall Island, Northwest Territories, in 1985.\n\n####  **CRY OF THE CURLEW**\n\n##### Eskimo Curlew \u2013 1985\n\n1.\n\nKnud Rasmussen among the Inuit\n\nClaimed the word for poem\n\nWas _Anertsa_ , meaning \"a breath\"\n\nNeither speech nor song\n\nBut fixed musical diction\n\nA flow of beautiful smooth words\n\nRecited in a low voice\n\nDelivered at a moderate pace\n\nA crystallized form of natural language\n\nThat must be worked and shaped\n\nAs smoothly and carefully as soapstone\n\nWord images and word charms\n\nPolished like walrus ivory\n\nOld pieces are learned from the elders\n\nThey are passed round and admired\n\nLike fine carvings or well-made weapons\n\nBut do not use them too often\n\nDo not wear them out\n\nThese charms were made for protection\n\nAgainst disease, against the evil effects\n\nOf shooting stars, of cunning foes, of envy\n\nWhen death occurs, strict mourning taboos\n\nAre observed, but when the taboo is lifted\n\nThe bereaved ones use these charms\n\nMurmuring them each to the other\n\nAnd each to those now gone\n\n2.\n\nRasmussen also records another Inuit word\n\nFor poem: _Taigdlia_ , meaning \"to name\"\n\nThere have been 40 common names\n\nFor the Eskimo Curlew\n\nOne Inuit name meant \"Swiftwing\"\n\nA Cree meant \"Sweetgrass\"\n\nAn Abnaki name meant \"Beaked One\"\n\nMany other names were made\n\nIn imitation of their various cries\n\nAmong the Inuit they were called\n\n_Pi-pi-pi-uk, Tura-Tura_ and _Ak-ping-ak_\n\nTo the Cree _Wee-kee-me-nase-su,_\n\n_Wee-kee-me-new, We-Ke-wa-ne-so_\n\n3.\n\nThis recitation of their many names\n\nLinks them together like prayer beads on a rosary\n\nGrants the passage of their many souls\n\nAcross that Other World's continents\n\nIn the outports of Labrador\n\nThey were the _Gallou Bird_ and the _C'lew_\n\nAmong the Qu\u00e9b\u00e9cois\n\nThey were _Corbigeau des Esquimaux_\n\nWhile the farmers of Ontario\n\nCalled them _Little Sicklebills_\n\nMarket hunters named them _Doughbirds_\n\nAnd _Futes_ in New England and New York\n\nAnd _Prairie Pigeons_ in the American West\n\nThe Islanders of the Caribbean greeted them\n\nAs _Wood Snipe, Chittering_ and _Shivering Curlew_\n\nWhile to the Latinos of Venezuela and Columbia\n\nThey were known as _Zarapito esquimal_\n\nIn Uruguay as _Chorlo polar_\n\nAnd _Chorlo campino_\n\nIn the pampas of the Argentine\n\n4.\n\nSo this poem is my word charm,\n\nSmooth and polished as I can make it\n\nLet us hear your names again:\n\nLittle Sicklebill, Sweetgrass, Swiftwing\n\nAnd your voice carried\n\nLike your wings on the north wind:\n\n_Pi-pi-pi-uk, Tura-Tura_\n\n_Wee-kee-me-nase-su_\n\n_Wee-kee-me-new_\n\n_We-Ke-wa-ne-so_\n\n##  **THE GOLDEN AGE**\n\n###  **F IRST WATCH 9 P. M. COMPLINE**\n\n####  **GIANT WHITE SWAMPHEN \u2013 1830 \u2013 _Porphyrio albus_**\n\n#####  Surgeon Arthur Bowes \u2013 1788  \n_**Ship's Log: Lady Penrhyn** _ **, Lord Howe Island, Tasman Sea**\n\nI could not help but picturing to myself the Golden Age as described by Ovid \u2013 to see Fowls and Coots and Reels walking totally fearless and unconcerned in all parts round us so we had nothing more to do than stand a minute or two and knock down as many as we pleased. If we missed them they would never run away. The Pidgeons also were as tame and would sit upon the branches till you might go and take them off with your hands. Many hundreds of all sorts together with Parrot and Paroquets, Magpies and other Birds were caught and carried on board our ship with the Charlotte.\n\n> The Lord Howe Island Giant Swamphen \u2013 also known as the White Gallinule \u2013 was just one of over a hundred species of birds that inhabited this isolated island in the Tasman Sea between New Zealand and Australia. No human had set foot on this small island (measuring five miles by one mile) and its tiny satellite islet. Indeed, its avian inhabitants had no contact with any sort of mammals (except bats) and no predators of any sort. However, everything changed on 13 March 1788, when the British ship _HMS Supply_ arrived with the _Scarborough, Lady Penrhyn_ and _Charlotte,_ and the island was named Lord Howe, in honour of the First Lord of the Admiralty.\n> \n> The red-billed, white-feathered and flightless Lord Howe Island Giant Swamphen was the largest representative of its family of \"Coots and Reels\" (that is: _Coots_ and _Rails_ of the _Rallidae_ family) in Polynesia. Being large, prominent and unafraid of humans, it rapidly became the island's first and most obvious candidate for extermination. It is known from only three existing skin specimens and two illustrations. Two were made on that first encounter in 1788 by Thomas Watling and John White; and by George Raper on a second visit in 1790. The Giant White Swamphen was certainly extinct by 1830.\n\n#####  Captain Thomas Gilbert \u2013 1788  \n_**Ship's Log: Charlotte** _ **, Lord Howe Island, Tasman Sea**\n\nSeveral of these Birds I knocked down, and their legs being broken, I placed them near me as I sat under a tree. The pain they suffered caused them to make a doleful cry which brought 5 or 6 dozen of the same kind to them and by that means I was able to take nearly the whole of them.\n\n> Early European observers never seemed to tire of their own amazement at the tameness of birds which had never before seen men. Yet their reactions to such unwariness were always predictably and brutally the same. This kind of unbridled slaughter of huge numbers of the entirely tame birds by the crews of ships which made regular stops on voyages out of Australia's Sydney Harbour resulted in the dramatic reduction of the once abundant bird populations on this small island. There were once well over a hundred bird species on the island; most were vagrant or migrating species that existed elsewhere. However, a dozen endemic species had nowhere else to go, and eight of these are now extinct; the other four are critically endangered. The Giant Swamphen in 1830, Lord Howe Island Blue Pigeon ( _Columba vitiensis godmanae_ ) in 1853, and the Red-Fronted Green Parakeet ( _Cyranoramphus novaezelandiae subflavescens_ ) in 1869 were the first to be exterminated.\n> \n> These first ships were part of one of the world's greatest sea expeditions: the First Fleet of 11 ships that on 13 May 1787 sailed from Britain to establish the first European colony in New South Wales, Australia. A naval escort consisting of the flagship _HMS Sirius_ and the armed tender _HMS Supply_ accompanied six convict transport ships (including two former slave ships) and two food transport ships. The journey of over 24,000 km of the 11 ships and 1400 people travelled 252 days without loss of a ship and fewer than 70 deaths \u2013 considering the conditions \u2013 was something of a triumph of the tactical planning and navigational skills of Captain (later Admiral) Arthur Phillip. The fleet landed at Botany Bay on 18 January 1788, just 18 years after Cook's first landing; but finding this unsuited to settlement sailed into what Phillip described as \"the finest harbor in the world, in which a thousand sail of the line may ride in the most perfect security.\" He named it Sydney Harbour in honour of the British Home Secretary, Lord Sydney. There Phillip established the first British settlement in Australia on 26 January 1788 \u2013 now annually celebrated as Australia Day.\n\n####  Dr. James Foulis \u2013 1847  \n_**Resident's Journal** _ **, Lord Howe Island, Tasman Sea**\n\nThere are but few birds that belong to this island \u2013 the only valuable kind being a large pigeon.\n\n> For half a century, Lord Howe Island was a stopping-over place on the journey between Sydney and Norfolk Island. It also was used by whalers for water and food. A small settlement of three men with Maori wives and two children was established in 1834, subsisting on trade with passing ships. In 1851, during Dr. James Foulis' stay on the island, there was a population of only sixteen colonists.\n> \n> Just how vulnerable flightless birds can be to human incursion was dramatically demonstrated by the case of New Zealand's Stewart Island Wren ( _Xenicus [Traversia] lyalli_ ). At the northern tip of New Zealand's South Island, the one mile square wooded Stephens Island had a human population of exactly one. The following is a British Ornithological Society Report published in 1895: \"At a recent meeting of the Ornithologist's Club in London, the Hon. W. Rothschild, the well-known collector, described this veritable rara avis, specimens of which he had obtained from Mr Henry Travers of Wellington, who, we understand, got them from the lighthouse-keeper at Stephens Island, who in his turn is reported to have been indebted to his cat for this remarkable ornithological 'find'. As to how many specimens Mr Travers, the lighthouse-keeper and the cat managed to secure between them we have no information, but there is very good reason to believe that the bird is no longer to be found on the island, and as it is not known to exist anywhere else, it has apparently become quite extinct. This is probably a record performance in the way of extermination. The English scientific world will hear almost simultaneously of its discovery and its disappearance.\"\n> \n> Another endemic New Zealand bird species was also rapidly extinguished in an equally arbitrary manner. The Huia ( _Heteralocha acutirostris_ ) was a totally unique and beautiful bird with long black and white tail feathers which were traditionally only worn by Maori Chiefs. A hat with Huia feathers was presented to the future George V of England on a royal visit in 1900. The result was an immediate European craze for Huia feathers. A brief seven years later, in 1907, the Huia was declared extinct. In 2010, a single Huia feather sold at auction for NZ $8,000.\n\n####  Alan McCulloch \u2013 1921  \n_**Birds of Lord Howe Island** _ **, Tasman Sea**\n\nThis paradise of birds has become a wilderness and the quietness of death reigns where all once was melody. The very few birds remaining are driven from their nests by rats, and their eggs eaten.\n\n> The June 1918 grounding of the _SS Makambo_ at Ned's Beach allowed black rats onto the island previously protected by an anchorage which only allowed landing by small boats. With the wreck of the _Makambo_ , the last pockets of the island's endemic bird populations \u2013 already critically endangered through hunting and habitat destruction from humans and introduced goats, pigs and cats \u2013 suffered their final fatal blow. Within 2 to 7 years, the Doctor Bird or L. H. I. Vinous-tinted Blackbird ( _Turdus xanthopus vinitinctus_ ) in 1920, the L. H. I. Big Grinnel Flycatcher ( _Gerygone igata insularis_ ) in 1920, the L. H. I. White Eye ( _Zosterops strenua_ ) in 1923, the Rainbird or L. H. I. Fantail Warbler ( _Ripidura fuliginosa cervina_ ) in 1924, and the L. H. I. Red-Eye Starling ( _Aplonis fuscus hullianus_ ) in 1925 \u2013 all joined the Swamphen, Green Parakeet and Blue Pigeon on the list of extinct species from this tiny island.\n> \n> As mentioned, the remarkable Giant White Swamphen was a member of the _Rallidae_ family, which has a 70 million year old ancestral line, and consists of Rails, Crakes, Coots and Gallinules: the order _Gruiformes_ , which include Bustards and Cranes. This order is largely made up of ground-feeding, ground-nesting, walking birds that rarely fly, and many are today critically endangered: especially on remote Pacific isles. Besides the Lord Howe Island Giant Swamphen, at least ten other Pacific Island Rails are known to have become extinct: the Chatham Island Modest Rail ( _Gallirallus modestus_ \u2013 1900), Chatham Island Dieffenbach's Rail ( _Gallirallus dieffenbachia_ \u2013 1840), Macquarie Island Banded Rail ( _Gallirallus philippensis macquariensis_ \u2013 1880), Fiji Barred-Winged Rail ( _Nesoclopeus poeciloptera_ \u2013 1965), Tahiti Rail ( _Gallirallus ecuadata_ \u2013 1900), Samoan Wood Rail ( _Pareudiastus pacificus_ \u2013 1873), Carolina Island Kittlitz's Rail ( _Porzana monasa_ \u2013 1850), Iwo Jima Rail ( _Poliolimnas cinereus brevipes_ \u2013 1924), Laysan Island Rail ( _Porzana palmeri_ \u2013 1944), and the Wake Island Rail ( _Gallirallus wakensis_ \u2013 1945).\n> \n> Introduced cats, pigs, goats and rats were frequently the reasons for the final extinction of Rail species on small islands. The Laysan Island Rail suffered a similar fate to the birds of Lord Howe Island when during the Second World War a wrecked American landing craft drifted ashore and black rats escaped onto the island. The Wake Island Rail also failed to survive the war, not because of rats, but rather gruesomely because a besieged garrison of starving Japanese soldiers hunted down and ate the entire population.\n\n####  **OVID'S ISLAND**\n\n##### Giant White Swamphen \u2013 1830\n\n_\"It was in the murderous age of iron_\n\n_That men spread sails before the wind_\n\n_And came to live on plunder\"_\n\nSo it was on the Tasman Sea\n\nWhen the convict ships came to this isle\n\nWhere no man had ever trod before\n\nThese iron men swiftly brought an end\n\nTo the last earthly relic of Ovid's fragile _Age of Gold_\n\n_\"And the quietness of death reigns_\n\n_Where once was melody\"_\n\nHere the pale ghost of the whistling Swamphen\n\nStalks the pandanus groves and palm glades\n\nAnd the shades of the Rain Bird and the Fantail\n\nThe Doctor Bird, the White Eye and the Red-Eye Starling\n\nAnd those many others who no longer sing\n\nUpon the branches of the flowering white cedar\n\nNor tremble as the sea wind blows upon the petals\n\nOf the wedding lily or the mountain rose\n\n##  **THE FEATHERED GODS**\n\n###  **S ECOND WATCH 10 P. M. COMPLINE**\n\n####  **HAWAIIAN OAHU O-O \u2013 1837 \u2013 _Moho apicalis_**\n\n#####  Captain James Cook \u2013 1779  \n_**The Third Voyage of Captain James Cook** _ **, Hawaii**\n\nThe islanders have another dress appropriated to their Chiefs, and used on ceremonious occasions, consisting of a yellow feathered cloak and helmet, which, in point of beauty and magnificence, is perhaps nearly equal to that of any nation in the world. These feathered cloaks are made of different lengths, in proportion to the rank of the wearer, some of them reaching no lower than the middle, others trailing on the ground. The inferior Chiefs have also a short cloak, resembling the former, made of the long tail-feathers of the cock, the tropic and man of war birds, with a border of the small red and yellow feathers, and a collar of the same. Others again are made of feathers entirely white, with variegated borders. These feathered dresses seem to be exceedingly scarce, appropriated to persons of the highest rank, and worn by the men only.\n\n> Lt. James King of the _HMS Discovery_ later added to this account with a passage describing Cook's welcome by the Hawaiians in Kealakekua Bay on 26 January 1779: \"The next day, about noon, the king, in a large canoe, attended by two others, set out from the village in great state. Their appearance was grand and magnificent. In the first canoe was Terreeoboo and his chiefs, dressed in their richest feathered cloaks and helmets, and armed with long spears, and daggers; in the second, came the venerable Kaoo, the chief of the priests, and his brethren, with their idols displayed on red cloth. These idols were busts of gigantic size, made of wicker-work, and curiously covered in the small feathers of various colours, wrought in the same manner with their cloaks. Their eyes were of large pearl oysters... . As soon as I saw them approaching, I ordered out our little guard to receive the king; and Captain Cook... arrived nearly at the same time... when the king rose up, and in a very graceful manner threw over the Captain's shoulders the cloak he himself wore, put a feathered helmet upon his head, and a curious fan in his hand. He also spread at his feet five or six other cloaks, all exceedingly beautiful, and of the greatest value.\"\n> \n> The magnificent feathered cloaks and helmets that were presented to Captain Cook were eventually brought to Britain and are still held as part of the British Museum's collection. And as Captain Cook himself recorded: \"A more rich and elegant Dress than this perhaps the Arts of Europe have not yet been able to supply.\"\n\n####  **MOLOKAI O-O \u2013 1904 \u2013 _Moho bishopi_**\n\n#####  Captain George Vancouver \u2013 1794  \n_**'The Discovery' Journal of Captain George Vancouver,** _ **Hawaii**\n\nKing Kamehameha was garbed in a striking yellow cloak made principally of bright yellow feathers which reached from his shoulders to the bottom of the canoe. On his head he wore a very handsome feathered helmet, and made altogether a magnificent appearance.\n\n> The cloak of King Kamehameha described by Captain George Vancouver was constructed over a period of eight generations of kings, and was believed to have been made from the plumes of over 80,000 birds. The thousands of golden plumes that made up these royal cloaks were taken almost entirely from the four O-O species ( _Moho apicalis, M. bishopi, M. nobilis, M. braccata_ ) and the single Mamo species ( _Drepanis pacifica_ ) \u2013 all of which are now extinct. This cloak was the feathered equivalent of the crown jewels of the Hawaiian kings. However, it doesn't appear as if this was the ultimate cause of their extinction, as it was forbidden for commoners to hunt these birds or possess their feathers except on behalf of the royal family. Also according to some accounts, many of these birds were captured and stripped of their gold plumes rather than killed.\n> \n> Goats, sheep and cattle put ashore by Captain Cook and Captain Vancouver between 1778 and 1794 caused extensive damage to native forests. Subsequently, much forest has been cut down by Europeans in order to plant sugar, pineapples and coffee. Imported rats, cats and mongoose took a terrible toll on native birds, while imported domestic pigeons resulted in a plague of avian malaria that exterminated all Honeycreepers and Honeyeaters below 4,000 feet in altitude \u2013 which is the range of its deadly imported carrier, the Night Mosquito.\n> \n> Nor were the islands' birds the only victims of the European invasions that followed Captain Cook. The native Polynesian population at the time of Cook's visit was estimated between 400,000 and 800,000. By 1900, the native population was 40,000, and by 1950 it was less than 10,000, or a decrease of 98% of their original population, and only 2% of the Hawaiian non-native population of 500,000. By 2010 it had increased to 250,000 or nearly 20% of Hawaii's 1\u00bc million population.\n\n####  **HAWAII O-O \u2013 1934 \u2013 _Moho nobilis_**\n\n#####  Coenraad Jacob Temminck \u2013 1838  \n_**Manuel D'Ornithology,** _ **Leiden, Holland**\n\nThe most striking characteristic of _Moho noblis_ is a tuft of beautiful long silky feathers decorating each flank; the bright yellow colour of this waving plume makes a pleasing contrast to the otherwise dark colouring of its plumage. These are the feathers Sandwich Islanders use to make their cloaks. The English traveller Dixon tells us they catch these birds with ease, pull out bunches of feathers on the flanks, and then let the birds go.\n\n> The first O-O Honeyeater to become extinct was the Oahu O-O ( _Moho aplicalis_ ) in 1837. This was undoubtedly because Oahu was the most densely populated island and the location of the port of Honolulu. The second extinction was the related but non-Moho species, the Kioea ( _Chaetoptila angustipluma_ ) in 1850; and the third was the Molokai O-O ( _Moho bishopi_ ) in 1904. The Hawaiian O-O ( _Moho noblis_ ) was the most numerous species of Hawaiian Islands Honeyeaters because its habitat on the Big Island of Hawaii was the most extensive. Because of this, it was nearly the last of the Honeyeaters to become extinct, in 1934. The only exception was a tiny population of Kuaia O-O ( _Moho braccata_ ) which miraculously survived until 1987 in an isolated patch of forest in a high-altitude volcanic cone within the island. This was above the range of the Night Mosquito and, as the site of the world's highest recorded rainfall, of little interest to feral cats.\n> \n> In 2008 DNA analysis revealed that Hawaiian Honeyeaters did not evolve from Australasian Honeyeaters ( _Meliphagidae_ ), and that they belonged to the entirely distinct family of Mohoidae, consisting of two genera and five species. This is the only known extinction of an entire avian family in historic times.\n\n######  **KAUAI O-O \u2013 1987 \u2013 _Moho braccata_**\n\n#####  J. C. Greenway \u2013 1967  \n_**Extinct and Vanishing Birds of the World** _ **, New York**\n\nFive forms of the Honeyeater family \u2013 known as the O-O \u2013 were once to be found on as many of the Hawaiian Islands. These were all strikingly beautiful, either brown or jet black, with metallic reflections and with bright yellow ornamental plumes. All of these are now extinct. Because the natives captured them for their yellow plumes it has often been said that they were extirpated by the Hawaiians, but it is not probable that this was the cause of their disappearance. As in the case of the extinction of twenty forms of the Hawaiian Honeycreepers, it is probable that destruction of the native forests is the primary cause.\n\n> Honeycreepers were the youngest of all bird families and among the most beautiful and varied. All descended from one ancestor species that a century ago had evolved into 9 genera, twenty-two species and sixty-four subspecies. All measured from 10 to 21 cm and were brilliantly coloured with specialized beak forms for feeding on flowering trees.\n> \n> Ninety-seven percent of all Hawaiian plant and tree species are endemic and unique to the islands. By 1950, three quarters of the natural forests were lost to cultivation, cattle browsing or fire. Over 270 plant species, subspecies and varieties are known to be extinct and 800 more are endangered. In fact, an astonishing sixty percent of the 68 known forms of native land birds are probably extinct.\n> \n> As noted, all five endemic Hawaiian Honeyeaters are now extinct; and forty percent of all endemic Hawaiian Honeycreepers are also gone. These include: the Mamo ( _Drepanis pacifica_ ) _,_ Black Mamo ( _Drepanis funereal_ ) _, Ula-Ai-Hawane_ ( _Ciridops anna_ ) _,_ Laysan Apapane ( _Himatione sanguinea freethi_ ) _,_ Great Amakihi ( _Loxops sagittiriostris_ ) _,_ Molokai Alauwahio ( _Loxops maculata flammea_ ), Lanai Alauwahio ( _Loxops maculate montana_ ) _,_ Oahu Akepa ( _Loxops coccinea rufa_ ) Hawaiian Akiola ( _Hemingnathus obscurus obscurus_ ) _,_ Lanai Akiola ( _Hemingnathus obscurus lanaiesis_ ) _,_ Oahu Akiola ( _Hemingnathus obscurus ellisianus_ ) _,_ Kauai Akiola ( _Hemingnathus obscurus procerus_ ), Oahu Nukupuu ( _Hemingnathus lucidus lucidus_ ) _,_ Kauai Nukupuu ( _Hemingnathus lucidus hanapepe_ ) _,_ Maui Nukupuu ( _Hemingnathus lucidus affinis_ ) _,_ Greater Koa Finch ( _Psittirostra palmeri_ ) _,_ Lesser Koa Finch ( _Psittirostra flaviceps_ ) _,_ and Kona Finch ( _Psittirostra kona_ ) _._\n\n####  **THE ANCESTORS**\n\n##### Hawaiian Oahu O-O \u2013 1837\n\nThe Ancestors prepare to sail\n\nThey raise the mast of the outrigger\n\nAnd pray to Kanaloa\n\nFor a calm sea and swift passage\n\nAs a fair wind rises, the Ancestors\n\nDepart for the isle of the Feathered Gods\n\nWhere still the lush Koa forests bloom\n\nAnd the winged jewel-and-gem birds\n\nSing to the glory of flowering trees\n\nWhere fly the O-O and the Mamo\n\nWhose whirring wings are fashioned\n\nOf glittering gold and the sheen of jet\n\nWhere they feed on the honey flower\n\nWith the emerald Amakihi\n\nAnd the ruby bright Alauwahio\n\nAnd others with plumes the colour\n\nOf citrine, tourmaline, amethyst and pearl\n\nFlit through forests heavy with flower and fruit\n\nFilled with bird song and sunlight\n\n##  **THE VAN DYKE**\n\n###  **T HIRD WATCH 11 P. M. COMPLINE**\n\n####  **IVORY-BILLED WOODPECKER \u2013 1951 \u2013 _Campephilus principalis_**\n\n#####  Mark Cateby \u2013 1731  \n_**Natural History of Carolina, Florida and Bahama Islands** _\n\nThe Bill is white as Ivory, three inches long, and channelled from basis to the point: the Iris of the eye is yellow: the hind-part of the Head adorned with a large peaked crest of scarlet-feathers: a crooked stripe runs from the eye on each side the Neck, towards the Wing: the lower part of the Back and Wings (except the large Quill-feathers) are white, all the rest of the Bird is black. The Bills of these Birds are much valued by the _Canada Indians_ , who make coronets of them for their Princes and great warriors by fixing them round a Wreath, with their points outward. The _Northern Indians,_ having none of these Birds in their cold country purchase them from the Southern people at the price of two, and sometimes three buck-skins a bill.\n\n> The Ivory-billed Woodpecker was always a rather rare bird. As early as 1785, the Welsh naturalist Thomas Pennant reported that the bird was \"scarce.\" Its territory was broad but it was very specialized and thinly populated. Before the heyday of the timber industry (1800-1900) the Ivory-bill inhabited all mature low-lying timberland in south-eastern North America. It needed considerable tracts of mature timberland to be able to find sufficient food to live on; each breeding pair would cover a territory of more than 2000 acres. And with their conspicuous markings and loud, resonant calls these red-crested birds were once relatively easy to find in their habitats.\n> \n> The pre-eminent pioneer of American natural history of the time was the English artist-naturalist Mark Cateby with his landmark _Natural History of Carolina, Florida and Bahama Islands: Containing the Figures of Birds, Beasts, Fishes, Serpents, Insects and Plants._ Nearly a century before Audubon, Cateby was the first naturalist to create folio-size colour engraved plates.\n> \n> Cateby was the first ornithologist to publish a description of this bird which he called the \"Large White-Billed Woodpecker, \" but he mentions that \"the Spaniards call 'em Carpentaros\" because of their rapid destruction of wood. It is best known as the Ivory-billed Woodpecker, but over the years became known by a variety of other names: Pearl Bill or Pearly-billed Woodpecker, King Woodpecker, Log-Cock, King Wood Cock, and Lord God Bird. In Louisiana, the Cajun French knew the bird as _Grand pique-bois, Poule de bois,_ and _Grand Pic Noir a bec blanc._\n\n#####  Alexander Wilson \u2013 1812  \n_**American Ornithology** _ **, North Carolina**\n\nI had secured a wounded specimen of this remarkable bird and saw no harm in keeping it for a short time in my hotel room. In less than an hour I returned, and, on opening my door he set up a distressing shout, which appeared to proceed from grief that he had been discovered in his attempt to escape. He had mounted along the side of the window, nearly as high as the ceiling, a little below which he began to break through. The bed was covered with large pieces of plaster, the lath was exposed for at least fifteen inches square, and a hole large enough to admit a fist, opened to the weatherboards; so that, in less than another hour he would certainly have succeeded in making his way out.\n\n> Alexander Wilson was a Scottish-American poet, naturalist, illustrator \u2013 and the most important pre-Audubon figure in American ornithology. Indeed, Audubon's encounter with the older artist-naturalist in Louisville, Kentucky in 1810 may have inspired him to consider publishing his own work. Wilson began his encyclopaedic _American Ornithology_ in 1805. It was published in 9 volumes from 1808 to 1814 and included 268 species. Wilson died in 1813 before the last volume could be printed.\n> \n> The Ivory-billed Woodpecker was certainly the most famous of all Woodpeckers. At 20 inches (50 cm) and over one pound in weight (500 gm) it is among the most spectacular; however, it was actually only the second largest species. The largest \u2013 measuring a remarkable 23 inches (58 cm) \u2013 was the Imperial Woodpecker ( _Campephilus imperialis_ ), which once inhabited the mature pine forests of Mexico's Sierra Madres. Similar in size to the Raven, the Imperial was nearly twice the weight of the Ivory-bill. Only one ornithologist had observed this bird while it was alive. The first specimen was acquired in 1832, and the last in 1950.\n> \n> Not a single photograph of a live Imperial Woodpecker was known to exist. Astonishingly, in 2011 Cornell University released newly acquired film footage of a living Imperial Woodpecker. This remarkable vivid 16mm colour film was taken by an amateur ornithologist and dentist from Pennsylvania, named William L. Rhein. In 1956, Dr. Rhein travelled to the district of Durango in the Sierra Madres where, while riding on a mule, he spotted and filmed an adult female Imperial Woodpecker in some ancient pine trees. Dr. Rhein died in 1999, apparently unaware of the historic significance of his home movie.\n\n####  John James Audubon \u2013 1826  \n_**Ornithological Biography** _ **, Kentucky**\n\nI have always imagined, that in the plumage of the beautiful Ivory-billed Woodpecker, there is something very closely allied to the style of colouring of the great Van Dyke. The broad extent of its dark glossy body and tail, the large and well-defined white markings of its wings, neck and bill, relieved by the rich carmine of the pendent crest of the male, and the brilliant yellow of its eye, have never failed to remind me of some of the boldest productions of that inimitable artist's pencil. The flight of this bird is graceful in the extreme. The transit from one tree to another, even should the distance be as much as a hundred yards, is performed by a single sweep, and the bird appears as if merely swinging itself from the top of the one tree to that of the other, forming an elegantly curved line. At this moment all the beauty of the plumage is exhibited, and strikes the beholder with pleasure. Its notes are clear, loud, yet rather plaintive. They are heard at a considerable distance, perhaps half a mile, and resemble the false high note of a clarinet they are usually repeated three times in succession, and may be represented by the monosyllable _pait, pait, pait._\n\n> John James Audubon is the third and greatest of America's pioneering ornithologist-artists. The Ivory-bill was one of his favourite subjects, and became the most sought-after and investigated of all rare birds. It has been estimated that not more than 500 Ivory-bills survived into the first half of the 20th century, yet during that time 400 traceable specimens were collected by museums: 140 in just two American museums alone. The last verified sighting of an American Ivory-billed Woodpecker was in 1951.\n> \n> For a time, it seemed that the last hope for the Ivory-billed Woodpecker resided in a sole surviving subspecies: _Campephilus principalis bairdii_ in Cuba. In 1960, there were theoretical estimates that a small population of twenty Cuban Ivory-billed Woodpeckers were surviving in a remote Cuban forest reserve. Sadly, by 1974, this population had vanished; but then in March of 1986, hope was again renewed with a possible sighting of a pair of birds in Cuba's Jaguani Forest Reserve. However, since that time nothing more has been reported.\n\n####  President Theodore Roosevelt \u2013 1907  \n_**Theodore Roosevelt: An Autobiography** _ **, Louisiana**\n\nIn stature, in towering majesty, the Louisiana cypress are unsurpassed by any trees of our eastern forest; lordlier kings of the green-leaved world are not be found until we reach the sequoias and redwoods of the Sierras... . The most notable birds and those which most interested me were the great Ivory-billed woodpeckers. Of these I saw three, all of them in groves of giant cypress; their brilliant white bills contrasting finely with the black of their general plumage. They were noisy but wary, and they seemed to me to set off the wilderness of the swamp as much as any of the beasts of the chase.\n\n> In 1907, Theodore Roosevelt set out on a bear hunt in the ancient cypress forests (in what became known as the Singer Forest Reserve) near Tallulah, Louisiana. Roosevelt's hunting ground was the last great swath of virgin Mississippi River hardwood bottomland forest left in America \u2013 and consequently the last refuge of the nation's Ivory-billed Woodpecker. Over the next two decades there were very few sightings of a bird that many believed was extinct.\n> \n> In 1935, however, seven pairs of America's rarest bird were discovered on the reserve. Then, from 1937 to 1939, James Tanner managed (at the last possible moment) to study and film a nesting pair in the reserve and produce his authoritative study _The Ivory-billed Woodpecker_. However, by 1939, the Singer population declined to just six birds, and Tanner estimated the entire population of Ivory-bills in America was not more than twenty-two birds.\n> \n> After the destruction of the old growth habitat on the Singer Reserve by large scale logging operations, there was only one fully confirmed sighting of the Ivory-bill in America in 1951. However, there have been many unconfirmed sightings and a multitude of second-hand accounts throughout rural southern United States. Most of these appear to have been sightings of that other large red-headed Woodpecker, the Pileated Woodpecker ( _Drocopus pileatus_ ).\n> \n> In 2005, there was believed to be a valid sighting of the bird in the \"Big Woods\" Forest Reserve in Arkansas. This was followed up by video evidence of a living Ivory-billed Woodpecker. However, the video has proved to be highly controversial, and despite a massive multi-million dollar search and monitoring operation, nothing has subsequently been seen or heard of the Lord God Bird.\n\n####  **THE LORD GOD BIRD**\n\n##### Ivory-billed Woodpecker \u2013 1951\n\nForged in the deep brilliant mind\n\nOf some savage god\n\nThen hidden away, in the twisting of river bends\n\nMazing channels, sloughs and levees\n\nAnd in the swelling waters of these ancient forests\n\nOf giant cypress, blackgum and red ash\n\nMohawk red crest mounted\n\nOn a flickering jackhammer head\n\nThe _\"thump, thump, thump\"_\n\nBeating heart of the bayou\n\nThe swooping fiery spirit\n\nOf this twilight world\n\nFlame we can no longer see\n\nSong we can no longer hear\n\nSearching for this bird, we find\n\nIn the lost language of the Natchez\n\nOnly the word _\"Pu-kup\"_ meaning \"red\"\n\nWe learn its _\"yap yap yap\"_ call\n\nWas followed by a plaintive cry\n\nLike the sound of a tin horn,\n\nOr a cracked clarinet\n\nNow at one with the vanished\n\nBiloxi, Muskogee and Quapaw\n\n_\"Downstream People\"_\n\nFlame we can no longer see\n\nSong we can no longer hear\n\n##  **SCREAM OF THE QUAGGA**\n\n###  **F IRST WATCH 12 P. M. MIDNIGHT**\n\n####  **QUAGGA \u2013 1883 \u2013 _Equus quagga quagga_**\n\n#####  George Edwards \u2013 1758  \n_**Gleanings of Natural History,** _ **London**\n\nThis curious animal was brought alive, together with the male, from the Cape of Good Hope: the male dying before they arrived at London, I did not see it; but this female lived several years at a house of his Royal Highness, the Prince of Wales, at Kew. The noise it made was much different from that of an ass, resembling more the confused barking of a mastiff-dog. It seemed to be savage and fierce in nature: no one would venture to approach it, and could not mount on its back. I saw it eat a large paper of tobacco, paper and all; and I was told, it would eat flesh, or any kind of food they would give it. I suppose that proceeding from necessity, or habit, in its long sea voyage; for it undoubtedly feeds naturally much as other horses and asses do, I mean on vegetables.\n\n> George Edwards was an aristocratic artist-naturalist who recorded the arrival of this first Quagga to reach Britain from the Cape of Good Hope. On the recommendation of Sir Hans Sloane, Edwards had been appointed the librarian of the Royal College of Physicians. Besides _Gleanings_ , Edwards also published his earlier _History of Birds_ , in which he described and illustrated more than 600 species never before described or delineated. He was later to become known as the \"Father of British Ornithology.\"\n> \n> The Quagga was often mistaken for some sort of cross-bred animal: half-zebra, half-horse. It was, in fact, a distinctive type of Zebra. Its head and forequarters were striped like a zebra while its hindquarters were a solid rufous brown. Alternatively, the Edwards animal was for a time believed to be the female variation of the fully striped white-legged Burchell's Cape Zebra. In fact, Burchell's Zebra ( _Equus quagga burchelli_ ) is now classified as a subspecies of the nominate Quagga species ( _Equus quagga quagga_ ) and both suffered the same fate.\n> \n> Zebras from \"Ethiopia\" were known to the Romans when the historian Dio Casio described them as \"horses of the sun resembling tigers.\" However, it was William Dampier, the famous adventurer and buccaneer, who, upon visiting the Dutch Colony of the Cape of Good Hope in 1691, first gave a description of Burchell's Cape Zebra: \"a very beautiful sort of wild ass whose body is curiously striped with equal lists of white and black.\"\n\n#####  William John Burchell \u2013 1811  \n_**Travels in the Interior of Southern Africa** _ **, London**\n\nI could compare the clatter of their hooves to nothing but to the din of a tremendous charge of cavalry, or rushing of a mighty tempest. I could not estimate the accumulated number at less than fifteen thousand, a great extent of the country being actually chequered black and white with their congregated masses. As the panic caused by the report of our rifles extended, clouds of dust hovered over them, and the long necks of troops of ostriches were also to be seen, towering above the heads of their less gigantic neighbours, and sailing past with astonishing rapidity. Groups of purple Sassaybes and brilliant red and yellow Hartebeests likewise lent aid to complete the picture, which must have been seen to be properly understood, and which beggars all attempt at description.\n\n> William John Burchell was a remarkable English explorer and naturalist who from 1810-15 gathered over 50,000 specimens in the most extensive collection of African plant and animal species ever amassed \u2013 before or since. From 1825-30, he made a similarly extraordinary collection in Brazil which included 16,000 insects, and 1,000 animals. As an authority of Southern Africa \u2013 having travelled over 7,000 kilometres of largely unexplored territory \u2013 Burchell was instrumental in the parliamentary decision in favour of British emigration to the Cape in 1820. In later years much angered by disputes with the British Museum, and feeling unappreciated after having totally exhausted his own personal fortune in creating these astounding collections, Burchell took his own life in 1863.\n> \n> The \"purpleSassaybes\" andthe \"redHartebeests\" colourfullydescribed by Burchell in his _Travels_ as part of these vast herds of Quaggas and Burchell's Zebras were sadly also all hunted to extinction. _Sassaby_ was the Saan Bushmen's name for a \"goat-horned\" antelope with a blue-grey coat that the Boers called _Blaauwbok_ and the English called Blue Buck ( _Hippotragus leucophaeus_ ). Burchell's \"red Hartebeest\" was the nominate Cape Red Hartebeest ( _Alcelaphus caama caama_ ).\n> \n> The remarkable and unique Blue Buck was actually the first African animal to become extinct in historic times: the last specimen being shot about eighty years after its discovery by Europeans in 1799. By the 1830's, all of the Cape Red Hartebeest herds had been reduced to a handful of animals that were kept in a private reserve until they too were killed off by poachers by 1940.\n\n####  Jared Diamond \u2013 1997  \n_**Guns, Germs, and Steel,** _ **New York**\n\n\"Efforts at domestication of zebras went as far as hitching them to carts: they were tried out as draft animals in 19th century South Africa. Alas zebras become impossibly dangerous as they grow older. Zebras have the unpleasant habit of biting a person and not letting go. They thereby injure even more American zoo-keepers each year than do tigers. Zebras are also virtually impossible to lasso with a rope because of their unfailing ability to watch the rope noose fly toward them and then to duck their head out of the way. Hence it has rarely (if ever) been possible to saddle or ride a zebra, and South Africans' enthusiasm for their domestication waned.\"\n\n> In England in the 1830's there was a brief vogue for Quaggas as harness animals after Sheriff Parkins became famous in London society for his pair of exotic imports. A few decades later, Lord Rothschild rode in a carriage pulled by two Burchell's Zebras. Eventually, however, both the Quagga and the Zebra proved to be too unpredictable. Quaggas in particular were prone to dangerous fits of rage. In 1860, the Regent's Park Zoo's only attempt to breed captive Quaggas failed when the stallion beat itself to death in its enclosure.\n> \n> The last wild Quagga stallion was hunted down in the Cape in 1878; and the last captive Quagga died in the Artis Magistra Zoo in Amsterdam on the 12 August 1883. Only 23 specimens of the Quagga now exist in museums anywhere in the world.\n> \n> Burchell's Zebra, known to the Boers as the _Bontequagga_ (meaning \"painted Quagga\"), managed to avoid extinction for a few decades longer than its cousin. The last known Bontequagga died in captivity in London's Regent's Park Zoo in 1910.\n\n#####  **HORSES OF THE SUN**\n\n###### Quagga \u2013 1883\n\n1.\n\nBushmen called them _Quay-Hay_\n\nIn imitation of their \"barking neigh\"\n\nMemories of them gather like thunderclouds\n\nOn the edge of this landscape of the mind:\n\nA high veldt with kloofs and koppies\n\nWhere thoughts like strong winds gust\n\nThrough the tall grass, acacia\n\nWhistling thorn and quiver trees\n\n2.\n\nKnowing they have no concept of heaven\n\nOr hell or soul or resurrection\n\nI summon them anyway\n\nAnd for a moment, the gates are forced\n\nBy the rolling thunder of thousands\n\nUpon thousands of pounding hooves\n\nLegions of stallions and mares\n\nBurst onto the veldt\n\n_\"Horses of the sun resembling tigers\"_\n\nAccording to Dio Cassio\n\nNostrils flared, _\"savage and fierce by nature\"_\n\nMagnificent in their defiance\n\nWhile the Boer hunters wait\n\nOver water with long guns\n\n3.\n\nThe Orion constellation\n\nIs read differently here\n\nStill the Celestial Hunter\n\nBut the sword is no sword\n\nThis one carries a spear\n\nAnd the three stars\n\nOf the belt are no belt\n\nBut three Quagga\n\nThe hunter's prey\n\nFor millennia after stars have gone out\n\nWe still marvel at the beauty of their light\n\nThe brilliant memory of what once was\n\n##  **XENOPHON'S ONAGERS**\n\n###  **S ECOND WATCH 1 A. M. MIDNIGHT**\n\n####  **ASSYRIAN ONAGER \u2013 1930 \u2013 _Equus hemionus hemippus_**\n\n#####  Xenophon \u2013 401 BC  \n_**Anabasis: The March of Cyrus,** _ **Mesopotamia**\n\nCyrus now advanced through Arabia, having the Euphrates on the right, five days' march through the desert, a distance of thirty-five parasangs. In this region the ground was entirely a plain, level as the sea. There were wild animals, however, of various kinds; the most numerous were wild asses; there were also many ostriches, as well as bustards and antelopes; and these animals the horsemen of the army sometimes hunted. The wild asses, when any one pursued them, would start forward a considerable distance, and then stand still; and again, when the horse approached, they did the same; it was impossible to catch them, unless the horsemen, stationing themselves at intervals, kept up the pursuit with a succession of horses. The flesh of those that were taken resembled venison, but more tender.\n\n> Travelling through Mesopotamia in 1884, Canon Tristram wrote of the Assyrian Onager: \"This is the Wild Ass of Scripture and the Ninevite sculptures mentioned by Strabo, Eratosthenes, Artemidorus, Homer, and Pliny. Measuring just one metre at the shoulder, it was the smallest and swiftest of all the horse family.\"\n> \n> Xenophon's adventures are among the most famous in all of military history. He commanded an elite force of ten thousand Greek mercenary soldiers who supported Cyrus, a pretender to the throne of the Persian Empire. The Greeks completely routed the opposing army, but disastrously Cyrus was killed in the battle, so Xenophon's Greeks had to fight their way over thousands of miles back to Greece.\n> \n> Xenophon's observations were precisely confirmed over twenty-two centuries later by the British explorer Blandford in 1876: \"It is said the Onager can not be caught by a single horseman in the open. But at other times they are caught in relays of horsemen and greyhounds. Some say their meat is prized above all other venison.\"\n\n#####  Sir Austen Layard \u2013 1850  \n_**Nineveh and Its Remains** _ **, Mesopotamia**\n\nA great herd of Wild Asses are seen in the Sinjar region west of Mosul. Those mentioned by Xenophon must have been seen in these very plains. The Arabs sometimes catch the foals during the spring, and bring them up with milk in their tents. They are of light fawn colour \u2013 almost pink. The Arabs still eat their flesh.\n\n> Sir Austen Layard's remarkable travels to Mesopotamia were made expressly to uncover the lost civilizations of Assyria. He succeeded in discovering the lost cities of Nineveh and Nimrod, and the palaces of Ashurbanipal and Sennacherib. It was his knowledge of Xenophon and the Scriptures that led to his success. Here, while camped on the desolate site of ancient Nineveh, Layard remembers the biblical passages relating to wrath of Jehovah being unleashed upon Sennacherib and quotes Zephaniah (ii, 13-15): \"And He will stretch out His Hand against the north, and destroy Assyria; and will make Nineveh a desolation, dry like a wilderness... a place for beasts to lie down in\": an accurate description of Assyria's fate.\n\n####  J. E. T. Aitchison \u2013 1885  \n_**British Botanical Expedition** _ **, Persia**\n\nMy guide took me to a slight elevation, above the 'Plain of Wild Asses'. For some time I could see nothing; at last whilst using my glasses, I noticed clouds of dust, like a line of smoke left in the track of steamers. These several lines of dust-cloud were caused by herds of Asses, galloping in various directions over the great plain. One herd came well within a mile's distance; from its extent, I am even now of the opinion that the herd consisted of at least 1000 animals. I counted sixteen of these lines of dust-cloud at one time on the horizon.\n\n> J. E. T. Aitchison conducted botanical surveys of Persia and Afghanistan for the Royal Botanical Gardens at Kew for Sir William Hooker. Wild Asses are of two distinct types: the African Wild Ass ( _Equus africanus_ ), ancestor of the domestic donkey, and the irascible and never domesticated Asian Onager ( _Equus hemionus_ ). Besides the Assyrian Onager of Syria and Mesopotamia there was one other subspecies in the Near East: the Persian Onager ( _Equus hemionus onager_ ) \u2013 now critically endangered with only about 500 animals in two reserves in Iran. Aitchison's account testifies to the immense numbers that, before the introduction of modern firearms, once inhabited these now nearly empty deserts.\n\n####  William Ridgeway \u2013 1905  \n_**Contributions to the Study of Equidae** _ **, London**\n\nThat the Onager was regularly captured and domesticated in Assyria in ancient times is clearly established by one of the bas-reliefs discovered by Sir Austen Layard at Nineveh. The relief, which is one of a series of slabs recording scenes in the life and hunting expeditions of Ashurbanipal (668-626 BC), represents two of the king's attendants lassoing a wild ass. The other two asses are seen running away.\n\n> William Ridgeway is correct here. Sir Austen Layard was not only aware that he was following in the footsteps of Xenophon, but interestingly enough, this is the same animal Layard discovered precisely observed by a sculptor over 2400 years earlier in the royal hunt on the walls of Nineveh that he was in the process of excavating. And furthermore, besides the Onager, there are found the images of the Persian Lion and the Aurochs \u2013 all three suffering the same fate as the Assyrians.\n> \n> Austen Layard later presented the British Museum \u2013 along with these massive relief wall sculptures of the royal hunt portraying the Assyrian Onager \u2013 a specimen of this same species: the only wild-caught example in any British collection.\n> \n> This region of Old Mesopotamia between the Tigris and the Euphrates rivers has been the setting for many other military adventures for over five thousand years. It has seen the rise and fall of the Sumerians, the Babylonians, the Hittites, the Assyrians, the Medes, the Persians, the Greeks, the Romans, the Mongols, the Turks, and the British. And Mosul seems to have been the seat of power for extravagantly ruthless dictators from the days of Sennacherib to those of Saddam Hussein.\n\n####  Otto Antonius \u2013 1938  \n_**On the Recent Equidae** _ **, Vienna**\n\nThe little Hemippus of Mesopotamia and Syria, domesticated by the ancient Sumers before the introduction of the horse became totally extinct in recent years. It could not resist the power of modern guns in the hands of the Anazeh and Shammar nomads, and its speed, great as it may have been, was not sufficient always to escape from the velocity of the modern motor car which more and more is replacing the Old Testament Camel-Caravan.\n\n> The social anthropologist Jared Diamond writes extensively about the how and why of the process of domestication of animals. He wonders why, out of the world's 148 possible big wild herbivorous mammal candidates \u2013 although a number have been \"tamed\" \u2013 only a dozen have been successfully domesticated, and only six have been universally adopted: sheep, goat, pig, cow, horse, and donkey.\n> \n> Interestingly enough, Diamond points out that, while there are eight distinctive species of wild equids (horses and relatives) that would all appear to be suitable candidates, only two of them \u2013 the Tarpan as the ancestor of the Horse and the North African Ass as the ancestor of the Donkey \u2013 were successfully domesticated. He then specifically observes the case of the Assyrian Onager. \"Closely related to the North African ass is the Asiatic ass, also known as the Onager. Since its homeland includes the Fertile Crescent, the cradle of Western civilization and animal domestication, ancient peoples must have experimented extensively with onagers. We know from Sumerian and later depictions that onagers were regularly hunted, as well as captured and hybridized with donkeys and horses. Some ancient depictions of horse-like animals used for riding or for pulling carts may refer to onagers. However, all writers about them, from Romans to modern zookeepers, decry their irascible temper and their nasty habit of biting people. As a result, although similar in other respects to ancestral donkeys, onagers have never been domesticated.\"\n> \n> The last wild Assyrian Onager was shot in 1927 in Arabia, while the last captive animal died in the Sch\u00f6nbrunn Zoo in Vienna in 1930. This, despite the efforts of the Zoo's director Otto Antonius, one of the co-founders of modern zoological biology, and one of the leading lights in the rescue of such endangered species as the Wisent.\n\n#####  **THE WALLS OF NINEVEH***\n\n###### Assyrian Onager \u2013 1930\n\n\"Onagers are among the beasts of the royal hunt on the walls\n\nof Ashurbanipal's palace at Nineveh.\" \u2013 Sir Austen Layard, 1850\n\n1.\n\nWhen those in the heavens\n\nHad not been named\n\nWhen those beneath the earth\n\nWere without a name\n\nWe raced alive and free\n\nWith no need of names\n\nThe speeding world was turning\n\nBeneath our bright hooves\n\nAs whirlwinds and sandstorms\n\nArose in our wake\n\nAnd chased the sun's fire\n\nAcross the desert sky\n\n2.\n\nThen later came those Others\n\nWith their names and words\n\nWith their dark incantations\n\nAnd even darker spells\n\nBuilding kingdoms and empires\n\nBy the power of words\n\nWith names to own and suppress\n\nAnd spells to enslave\n\nMastering the art of slaughter\n\nMade a paradise\n\nWatered with the tears of slaves\n\nAnd the blood of beasts\n\n3.\n\nIn these passageways of word\n\nAnd image cut in stone\n\nIn these corridors of power\n\nIn the great king's house\n\nWith lion-slaying arrows\n\nWith axes and spears\n\nWith horse-slaves and chariots\n\nThe lord of the hunt reigns\n\nHere we too are captives\n\nIn this counterfeit world\n\nWhere we forever flee\n\nThe king's deadly wrath\n\n4.\n\nWhen shall we come back\n\nFrom this eternal night?\n\nWhen shall we come back\n\nTo the world of light?\n\nWhen shall we come back\n\nAnd these shackles sever?\n\nWhen shall we come back\n\nTo our life forever?\n\nThen with a voice like thunder\n\nThis dreadful wonder\n\nEnds our soul's endeavour\n\nWith its answer: \"Never!\"\n\n* \"The Walls of Nineveh\" elegy is a curious literary exercise. It is an attempt at an ancient Sumerian verse stanza form. It may be the first new poem to be composed in this form in about three thousand years.\n\n##  **OUT OF DREAMTIME**\n\n###  **T HIRD WATCH 2 A. M. MIDNIGHT**\n\n####  **TOOLACHE WALLABY \u2013 1939 \u2013 _Wallabia greyi_**\n\n#####  Sir Joseph Banks \u2013 1770  \n_**The Endeavour Journals of Joseph Banks** _ **, Queensland, Australia**\n\nQuadrupeds we saw but few and were able to catch few of them that we did see. The largest was calld by the natives _Kangaru_. It is different from any European and indeed any animal I have heard or read of except the Gerbua of Egypt, which is not larger than a rat when this is as large as a midling Lamb; the largest we shot weighd 84 lb. It may however be easily known from all other animals by the singular property of running or rather hopping upon only its hinder legs carrying its fore bent close to its breast; in this manner however it hops so fast that in the rocky bad ground where it is commonly found it easily beat my grey hound, who tho he was fairly started at several killd only one and that quite a young one.\n\n> Sir Joseph Banks was an aristocratic gentleman and natural scientist who, in _The Endeavour Journals of Joseph Banks_ , recorded his discoveries as part of Captain James Cook's first expedition to the South Pacific. The Cook expedition was nominally a scientific one sent to observe the transit of Venus in the South Pacific, but sealed orders opened at sea revealed to Cook that his secret mission was to seek out \"Terra Australis Incognita\" \u2013 and claim it for the British Crown.\n> \n> On 20 April 1770, Captain James Cook sighted the east coast of Australia. Landing at Botany Bay, Cook then proceeded up the east coast of the continent. His ship ran aground on the Great Barrier Reef and he had to spend seven weeks on the coast of what now is Queensland repairing the damage. It was here that Banks made an extensive collection of unique species of Australian flora and fauna, and became the first European to observe one of the many \"Kangaroo\" species (which includes Wallabies). The name _Gangarru_ comes from the language of the Aboriginal Guugu Yimithirr people.\n> \n> Banks was the President of the Royal Society for 43 years and was the intellectual force behind the settlement of Australia and the historic launching of the First Fleet of colonists, soldiers and convicts in 1787-88, less than two decades after Cook's discovery of the continent.\n\n#####  John Gould \u2013 1840  \n_**The Mammals of Australia** _ **, Melbourne**\n\nI never saw anything so swift of foot as the Toolache Wallaby. It does not appear to hurry itself until the dogs have got pretty close, when it bounds away like an antelope with first a short jump, and then a long one, leaving the dogs far behind. One hunter complained bitterly after a week in the bush: 'I have had twenty runs a day with four swift dogs and not succeeded in getting one.' Another hunter claimed to have pursued one animal on horseback for over three miles without success.\n\n> John Gould, the artist-naturalist, was the most meticulous of Australia's 19th century wildlife illustrators. Son of a Royal Gardener at Windsor, he initially assisted his father as a taxidermist, and eventually was appointed to the Royal Zoological Society. His expedition to Australia in 1838-40 culminated in the monumental publication of his _Birds of Australia_ (1840-48) in 7 volumes, and _Mammals of Australia_ (1845-63) in 3 volumes. Gould became the foremost authority on Australian birds and mammals.\n> \n> The Toolache Wallaby was considered by all to be the swiftest and most beautiful of all the Kangaroo family. It measured about five feet, nose to end of tail, and was highly valued both for its fur and as a game species. It was capable of speeds of over 45 mph. The Toolache Wallaby inhabited the open plains of South Australia, and favoured open ground \"intersected by extensive salt lagoons and bordered by pine-ridges.\" Relying on its speed to outrun dingoes or aboriginal hunters, its choice of open ground made it easy prey for long range rifles of white hunters. Sadly, these skin hunters with rifles were extremely successful. The animal's magnificent skins were still being marketed in great numbers in Melbourne even in 1923 when there were only a couple of bands of a dozen left in the wild.\n> \n> The Toolache or Grey's Wallaby scientific appellation _greyi_ was given in honour of Sir George Grey KCB, who led two disaster-ridden expeditions into previously unexplored regions of Western Australia. He later became governor of Western Australia, then New Zealand and Cape Colony \u2013 and eventually Prime Minister of New Zealand.\n\n####  Professor F. Wood Jones \u2013 1924  \n_**Preservation of the Toolache Wallaby** _ **, Robe, South Australia**\n\nBy far the fleetest of all the Wallabies, and the fairest. There cannot be more than a few score of these animals left in existence; including the 14 animals protected by Mr Brown here on the Konetta Station. And yet, the poaching continues. Their beautiful pelts have been marketed in very large numbers in the sales rooms of Melbourne. Something must be done, and soon.\n\n> Between John Gould's observations (1852) and F. Wood Jones' (1924), the Toolache population had dwindled from \"swarms\" to packs of \"five or six individuals.\" In 1923, there was a small band of 14 Toolache Wallabies on the Konetta Station. In May of that year, Professor Wood Jones desperately attempted to capture and transfer the animals to a safe sanctuary on Kangaroo Island. However, this failed because these spirited creatures literally ran themselves to death trying to escape their well-intentioned pursuers.\n> \n> In 1936, Bernard C. Cotton travelled to Robe, South Australia, where he made a 16mm motion film of the Toolache Wallaby kept in a fenced paddock by Mr. J. Brown. Cotton described the animal as \"probably the last living representative of its species.\" The remarkable film footage shows the animal running at an estimated speed of 42 miles per hour, while the voice of the narrator comments: \"Note the lithe and graceful movement as portrayed by the slow motion camera.\"\n> \n> This film footage is indeed that of the last Toolache Wallaby in existence. A handwritten note attached \"in Mr Allan Rau's hand who was the South Australian Museum Taxidermist states: 'This animal died 10 July 1939 and the skin is in South Australia Museum'.\"\n> \n> Other Australian marsupials' extinctions included: the Eastern Hare-Wallaby ( _Lagorchestes leporides_ ) in 1890, Gilbert's Potoroo ( _Potorous gilberti_ ) in 1900, Pig-footed Bandicoot _(Chaeropus ecaudatus)_ in 1907, Western Barred Bandicoot ( _Perameles mysura_ ) in 1910, Greater Rabbit Bandicoot or Bilby ( _Macrotis lagatis grandis_ ) in 1930, and Eastern Barred Bandicoot ( _Perameles fasciata_ ) in 1940.\n> \n> Australia's smallest Kangaroo, the miniature (one foot tall) Parma White-Fronted Wallaby, was believed to have been exterminated in 1932. Its rediscovery in 1966 was greeted with some celebration. In the wake of this discovery, a local politician made a well-meaning, but somewhat puzzling, pledge on behalf of the Australian Government. \"The Parma Wallaby,\" he declared, \"will never be allowed to become extinct again.\"\n\n#####  **PRAYER OF THE WINABARAKU**\n\n###### Toolache Wallaby or Winabaraku \u2013 1940\n\n1.\n\nNow the All-Father thought\n\nThere should be sky\n\nThere should be mountains\n\nAnd there should be sea\n\nThere should be beasts on the plain\n\nAnd birds in the trees\n\nThere should be fish in the lakes\n\nAnd creatures in the sea\n\nSo He made them\n\nAnd for this we give thanks\n\n2.\n\nNow the All-Father thought\n\nIn this time and in this place\n\nThere should be the Winabaraku\n\nSo He made us \u2013 and we arose\n\nLike a desert spring in flood time\n\nIn the land of salt marsh\n\nAnd the stringy bark tree\n\nAnd for this we give thanks\n\n3.\n\nNow the All-Father thought\n\nIn this time and in this place\n\nAll He did and made\n\nMust come to an end\n\nIn the land of salt marsh\n\nAnd the stringy bark tree\n\nSo like a desert spring in drought time\n\nWe descend to deep caverns far below\n\nAnd for this we give thanks\n\n##  **THE GREAT BUFFALO HUNT**\n\n###  **F IRST WATCH 3 A. M. VEIL**\n\n####  **AMERICAN BLACK BISON \u2013 1825**  \n_**Bison bison pennsylvanicus** _\n\n#####  Hernando Cortez \u2013 1581  \n_**De Solis: History of Conquest** _ **, Mexico**\n\nIn the second Square of the same House were the Wild Beasts, captured by Montezuma's Hunters, in strong Cages of Timber, ranged in good Order, and under Cover: Lions, Tygers, Bears, and all others of the savage Kind which New-Spain produced; among which the greatest Rarity was the Mexican Bull; a wonderful composition of divers animals. It has crooked Shoulders, with a Bunch on its Back like a Camel; its Flanks dry, its Tail large, and its Neck covered with Hair like a Lion. It is cloven footed, its Head armed like that of a Bull, which it resembles in Fierceness, with no less strength and Agility.\n\n> Hernando Cortez's 1521 sighting of the Bison \u2013 or American Buffalo \u2013 in Montezuma's menagerie is the first by a European as recorded by the historian De Solis in 1724. It was probably captured in the northern Mexican state of Coahuila, some 500 miles north of Montezuma's palace in what is now Mexico City. In 1530, another Spaniard, Alvar Nunez Cabeza, was wrecked on the Gulf coast, west of the Mississippi delta, from whence he wandered westward into what is now Texas. He appears to be the first European to sight bison herds in their wild state.\n> \n> In 1542, the conquistador Coronado was the third European to record the sighting of American Buffalo. In the region of the Texas panhandle he saw for the first time what he described as \"crooked-backed oxen.\" It was also a notable first encounter between bison and horses. The event is described by one of Coronado's soldiers, Castenada: \"The first time we encountered these beasts, all the horses took to flight on seeing them, for they were a horrible sight. They have a broad and short face, eyes two palms from each other, and projecting in such a manner sideways that they can see a pursuer. Their beard is like that of goats, and so long that it drags the ground when they lower the head. They have, on the anterior portion of the body, a frizzled hair like sheep's wool; it is very fine upon the croup, and sleek like a lion's mane. Their horns are very short and thick, and can scarcely be seen through the hair. They always change their hair in May, and at this season they really resemble lions. Their tail is very short, and terminates in a great tuft. When they run they carry it in the air like scorpions.\"\n\n#####  Samuell Argoll \u2013 1612  \n_**Voyage to the New World** _ **, Virginia Colony**\n\nAnd then marching into the Countrie, I found great store of Cattle as big as Kine, of which the Indians that were my guides killed a couple, which we found to be very good and wholesome meate, and are very easie to be killed, in regard they are heavy, slow, and not so wild as other beasts of the wilderness.\n\n> The English navigator Samuell Argoll wrote of this encounter with an Eastern bison that saved his party from starvation. It is the earliest recorded sighting of the Eastern Black Bison and took place in 1612 on the banks of the Potomac River, near the present-day site of America's capitol, Washington, D. C.\n> \n> There were once four subspecies of Bison or American Buffalo which numbered in the hundreds of millions. The tallest and largest was the now extinct Black Bison ( _Bison bison pennsylvanicus_ ). Found in all the American states east of the Mississippi River, it was the first to be hunted to extinction by 1825. The second species to suffer extinction was the far western Oregon Bison ( _Bison bison oreganos_ ) around 1850 \u2013 shortly after the Lewis and Clark expedition established the Oregon Trail, and opened the region (Oregon to California) west of the Rocky Mountains for settlement. The two other races, the Great Plains Bison ( _Bison bison bison_ ) of America and the Wood Bison ( _Bison bison athabasca_ ) of Canada, would certainly have been entirely extinguished by 1890 had it not been for an 11th hour rescue effort by the American and Canadian governments at the instigation of the handful of influential conservationists who formed the American Bison Society.\n\n####  Colonel Daniel Boone \u2013 1770  \n_**Frontier Journals** _ **, Kentucky**\n\nVast herds grazed over these lands. The buffaloes were more frequent than I have seen cattle in the settlements, browzing on the leaves of the cane, or cropping the herbage of those extensive plains, fearless because ignorant of the violence of man. Sometimes we saw thousands in a drove, and the numbers about the salt springs were amazing. Their ways are as beaten as our great roads, and no herb grows therein.\n\n> The wildlife artist James Audubon wrote of the Black Bison of the Eastern American states in his _Quadrupeds of North America_ : \"In the days of our boyhood and youth, Buffaloes roamed over the small and beautiful prairies of Indiana and Illinois, and herds of them stalked through the open woods of Kentucky and Tennessee; but they had dwindled down to a few stragglers, which resorted chiefly to the 'Barrens', towards the years 1808 and 1809, and soon after entirely disappeared.\n> \n> \"Their range has since that period gradually tended westward, and now you must direct your steps 'to the Indian country', and travel many hundreds of miles beyond the fair valleys of the Ohio, towards the great rocky chain of mountains that forms the backbone of North America, before you can reach the Buffalo, and see him roving in his sturdy independence upon the vast elevated plains, which extend to the base of the Rocky Mountains.\"\n> \n> Another contemporary account of the hunting of the Eastern Black Bison by M. Ashe records the obvious reason for their demise: \"The carnage of these beasts was everywhere the same. I met with a man who had killed two thousand buffaloes with his own hand, and others no doubt have done the same thing. In consequence of such proceedings not one buffalo is at this time to be found east of the Mississippi.\"\n\n####  Colonel Richard Irving Dodge \u2013 1871  \n_**Plains of the Great West,** _ **Arkansas**\n\nIn May of 1871, I drove a light wagon from Old Fort Zara to Fort Larned, on the Arkansas. At least 25 miles of this distance was through one immense herd, composed of countless smaller herds of buffalo. The whole country appeared one great mass of buffalo, moving slowly to the northward. When I had reached a point where the hills were no longer more than a mile from the road, the buffalo on the hills, turned, stared an instant, then started at full speed directly towards me, stampeding and bringing with them the numberless herds through which they passed, and pouring down upon me all the herds, in one immense compact mass of plunging animals, mad with fright, and as irresistible as an avalanche. This situation was by no means pleasant.\n\n> In later years, Colonel Dodge reflected on the reasons for the rapid extinction of those vast migrating herds: \"It was, then, the hide-hunters, who wiped out the great southern herd in four short years. The prices received for hides varied considerably, according to circumstances, but for the green or undressed article it usually ranged from 50 cents for the skins of calves to $1.25 for those of adult animals in good condition. Such prices seem ridiculously small, but when it is remembered that, when buffaloes were plentiful it was no uncommon thing for a hunter to kill from forty to sixty head a day, it will readily be seen that the chances of making very handsome profits were sufficient to tempt hunters to make extraordinary exertions.\n> \n> \"Moreover, even when the buffalo were nearly gone, the country was overrun with men who had absolutely nothing else to look as a means of livelihood, and so, no matter whether the profits were great or small, so long as enough buffaloes remained to make it possible to get a living by their pursuit, they were hunted down with the most determined persistency and pertinacity.\"\n> \n> Colonel Dodge's estimate of buffalo slaughtered in southern herd for the years 1872 to 1874 was 3,698,730 (of which more than half were killed and wasted, without taking hides or meat). By 1875, less than 10,000 survived, and these were soon gone.\n\n####  General Philip H. Sheridan \u2013 1878  \n_**Sheridan's Memoirs** _ **, Indian Territories**\n\nThe Buffalo Hunters have done more in the last two years to settle the vexed Indian Question than the entire regular army in the last thirty years. They have destroyed the Indian's commissary. Send them powder and lead, if you will, and let them kill, skin and sell until they have exterminated the buffalo.\n\n> General Sheridan voices the deliberate policy of the American military and government of promoting the extermination of the buffalo herds of the west as a means of starving the native Indian tribes into submission.\n> \n> Congressman McCormick wrote in _The New Mexican_ newspaper in Santa Fe that a United States federal surveying commission kept careful records of the slaughter and reported that there were two thousand hunters on the plains killing these animals for their hides. One party of sixteen hunters was reported having killed twenty-eight thousand buffaloes in the summer of 1872 in supposedly Indian Territories.\n> \n> Sheridan absolutely encouraged the poaching of buffalo and the invasion of the Indian Territories. He suggested Congress mint a medal of honour for the hunters with a dead buffalo on one side and a depressed looking Indian on the other. He was widely attributed with the saying, \"The only good Indian is a dead Indian.\"\n> \n> The other ex-Civil War soldier assigned to oversee Indian Affairs policy was General William Tecumseh Sherman. In a letter to President Grant, Sherman wrote: \"We are not going to let a few thieving Indians check or stop progress. We must act with vindictive earnest against the Sioux, even to their extermination, men, women and children.\" Sherman also saw killing off the buffalo as the means of starving Indians into submission.\n\n####  **CITIES OF GOLD**\n\n##### American Black Bison \u2013 1825\n\nCoronado searching for the Seven Cities of Gold\n\nMarched through Arizona and New Mexico\n\nSeeking the legendary Cibolo\n\nWith its streets and roofs of gold\n\nIts walls studded with gems\n\nDiscovered instead Zuni and Pueblo\n\nVillages of rough-cut stone and adobe\n\nEmbittered and raging with greed\n\nCoronado waded across the Rio Grande\n\nStubbed his steel toe on the Grand Canyon\n\nButchered and burned his way across Texas\n\nDodging Apache and Comanche arrows\n\nThe Spaniard stamped on into Kansas\n\nIn his relentless quest for Gran Quivera\n\nThat other chimeric promised land\n\nWith its mother lode of fabled gold\n\nCoronado looking for shining spires\n\nFound instead a vast plain crowded with beasts\n\n\"Horned, huge and monstrously ugly\"\n\nA dark satanic host \u2013 the very scent\n\nOf which terrorized his horses\n\nDenied the spectacular glory\n\nOf Aztec gold, of Inca silver\n\nBloodied Coronado limped home in shame\n\nWith a wagon train heaped with the skins\n\nOf those \"crook-backed oxen\"\n\nThose same buffalo robes that were to become\n\nThe true currency of the wilderness\n\nTriggering a stampede into the long darkness\n\nThat would eclipse all life on that great wide plain\n\nAnd from the bones of those millions and millions\n\nSown like dragons teeth into the parched earth\n\nThere arose seven cities; and seven cities more\n\nRicher than the greediest dreams\n\nOf Coronado's conquistadors\n\nThese Cities of Gold, these New Eldorados\n\nShimmer in the sun\n\nAnd yet, when darkness falls\n\nOn the empty plains beyond city lights\n\nFor some, there is this lingering memory\n\nOf the distant thunder of those endless herds\n\nAnd the last of the Ghost Dancers\n\nDressed in buffalo robes, chanting:\n\n_Nothing lives forever_\n\n_Nothing lives forever_\n\n_Except the earth and the sky_\n\n_Nothing lives forever_\n\n##  **THE BLUE METEOR**\n\n###  **S ECOND WATCH 4 A. M. VEIL**\n\n####  **PASSENGER PIGEON OR MIGRATING DOVE \u2013 1914**  \n_**Ectopistes migratorius** _\n\n#####  Jacques Cartier \u2013 1534  \n_**Voyages de Jacques Cartier** _ **, New France**\n\nOn 1 July 1534, Jacques Cartier in his ship's log \u2013 as reported in his _Voyages de Jacques Cartier_ \u2013 recorded the first written European sighting of a vast flock of Passenger Pigeons just off the coast of what is now Prince Edward Island. Cartier's description is here whimsically translated in the form of a haiku:\n\n> \"The sky darkens\n> \n> with an infinite multitude\n> \n> of wild blue pigeons.\"\n\n> In 1605, another French explorer, Samuel de Champlain, in his ship's log \u2013 as reported in his _Voyages de Samuel de Champlain_ \u2013 made the second French sighting of a vast flock of Passenger Pigeons just off the coast of Maine. Champlain's log entry is similarly succinct, and like Cartier's is here ironically translated as a haiku:\n> \n> \"There are countless doves\n> \n> whereof we knock down and take\n> \n> a goodly number.\"\n> \n> There followed a number of early descriptions of these birds in \"New France\" written by the missionaries. Among them, Gabriel Sagard-Th\u00e9odat, in his _Le Grand Voyage du pays des Hurons_ of 1632: \"There are here an endless multitude of doves, which the Huron call 'Orittey' that feed in part on acorns which they easily swallow whole. In the beginning they were so stupid that they allowed themselves to be knocked down by blows of stones and poles from beneath the trees, but at present they are a little more wary.\"\n> \n> And again from \"New France,\" in 1663, in the records known as the _Jesuit Relations and Allied Documents_ , we find: \"Among the birds of every variety to be found here, it is to be noted that Pigeons abound in such numbers that this year one man killed a hundred and thirty-two at a single shot. They passed continually in flocks so dense, and so near the ground, that sometimes they were struck down with oars.\"\n\n#####  William Strachey \u2013 1612  \n_**Letters** _ **, Virginia Colony**\n\nA kind of wood-pidgeon we see here in the winter time, and of such numbers, as I should drawe the creditt of my relation concerning all the other in question, yf I should expresse what extended flocks, and how manie thousands in one flock, I have seen in one daie, wondering (I must confesse) at their flight, when, like so many thickned clowdes, they (having fed to the northward in the day tyme) retourne againe more sowardly towards night to their roust; but there be manie hundred witnesses, who maie convince this my report, yf herein yt testifieth an untruth.\n\n> In 1614, two years after William Strachey, we have another record of Passenger Pigeons in the letters of the Governor of the Virginia Colony, Sir Thomas Dale: \"There are wilde Pidgeons in Winter beyond number or imagination, my selfe have scene three or four houres together flockes in the Aire, so thicke that even they have shadowed the Skie from us.\"\n> \n> A similar observation was made in 1625 in Nicolas-Jean de Wassenaer's _First Settlement of New Netherlands_ (in what is now Manhattan, New York City): \"The Birds most common are wild Pigeons; these are so numerous that they shut out the sunshine.\"\n\n####  William Wood \u2013 1634  \n_**Wood's New England Prospect** _\n\nThese Birds come into the Countrey, in the beginning of our Spring, at which time (if I may be counted worthy, to be believed in a thing that is not so strange as true) I have seen them fly as if the Ayerie regiment had been Pigeons; seeing neyther beginning nor ending, in length, or breadth of these Millions of Millions. The shouting of people, the ratling of Gunnes, and the pelting of small shotte could not drive them out of their course, but so they continued for foure or five houres together. Many of them build amongst the Pine-trees, thirty miles to the North-east of our plantations; joyning nest to nest, and tree to tree by their nests, so the Sunne never sees the ground in that place, from whence the Indians fetch whole loades of them.\n\n> In 1648, the Governor of New England, John Winthrop, recorded in his _The History of New England_ : \"This month, when our first harvest was near had in, the pigeons came again all over the country, but did no harm, (harvest being just in, ) but proved a great blessing, it being incredible what multitudes of them were killed daily. It was ordinary for one man to kill eight or ten dozen in half a day, yea five or six dozen at one shoot, and some seven or eight. Thus the Lord showed us, that he could make the same creature, which formerly had been a great chastisement, now become a great blessing.\"\n> \n> Nearly a century after Strachey's observations, the numbers of Passenger Pigeons do not seem to have diminished in the New England Colonies, as suggested in Cotton Mather's account in his _The Christian Philosopher_ : \"I will add a Curiousity relating to the Pidgeons, which annually visit my own Country in their Seasons, in such incredible numbers, that they have commonly been sold for Two-pence a dozen; Yea, one Man has at one time surprised no less than two hundred dozen in his Barn, into which they have come for Food, and by shutting the door, he has had them all.\"\n\n####  John James Audubon \u2013 1813  \n_**Ornithological Biography** _ **, Kentucky**\n\nI cannot describe to you the extreme beauty of their aerial evolutions, at once, like a torrent, and with noise like thunder, they rushed into a compact mass, pressing each other towards the centre. In these almost solid masses, they darted forward in undulating and angular lines, descended and swept close over the earth with inconceivable velocity, mounted perpendicularly so as to resemble a vast column, and, when high, were seen wheeling and twisting with their continued lines, which then resembled the coils of a gigantic serpent.\n\n> In his Louisville, Kentucky _Journals_ , Audubon wrote: \"In autumn of 1813, I left my house at Henderson, on the banks of the Ohio, on my way to Louisville. The air was literally filled with Pigeons; the light of the sun was obscured as by an eclipse; and the continued buzz of wings had a tendency to lull my senses to repose... . Before sunset I reached Louisville, distant from Henderson fifty-five miles, the Pigeons were still passing in undiminished numbers, and continued to do so for three days in succession.\" Audubon estimated the size of this flock over Louisville as \"one billion, one hundred and fifteen million, one hundred and thirty-six thousand pigeons.\"\n\n####  John James Audubon \u2013 1814  \n_**Ornithological Biography,** _ **Kentucky**\n\nLet us now, kind Reader, inspect their place of nightly rendezvous. I arrived there nearly two hours before sunset. Few Pigeons were then to be seen, but a great number of persons, with horses and wagons, guns and ammunition, had already established encampments on the borders. Many trees two feet in diameter, I observed broken off at no great distance from the ground; and the branches of many of the largest and tallest had given way, as if the forest had been swept by a tornado. As the period of their arrival approached, their foes anxiously prepared to receive them. Some were furnished with iron-pots containing sulphur, others were with torches of pine-knots, many with poles, and the rest with guns. Suddenly there burst forth a general cry of: \"Here they come!\" The noise they made, though yet distant, reminded me of a hard gale at sea, passing through the rigging of a close-reefed vessel. As the birds arrived and passed over me, I felt a current of air that surprised me. Thousands were soon knocked down by the pole-men. The birds continued to pour in. The fires were lighted, and a magnificent, as well as wonderful and almost terrifying sight presented itself. The Pigeons, arriving by thousands, alighted everywhere, one above another, until solid masses as large as hogsheads were formed on the branches all round. Here and there the perches gave way under the weight with a crash, and falling to the ground, destroying hundreds of birds beneath, forcing down the dense groups with which every stick was loaded. It was a scene of uproar and confusion. I found it quite useless to speak, or shout to those who were nearest to me. Even the reports of the guns were seldom heard, and I was made aware of the firing only by seeing the shooters reloading.\n\n> Audubon describes the aftermath of slaughter in this 1814 nesting site: \"No one dared venture within the line of devastation. The Pigeons were constantly coming, and it was past midnight before I perceived a decrease in the number of those that arrived. The uproar continued the whole night. Towards the approach of day, the noise in some measure subsided, long before objects were distinguishable, pigeons began to move off, and at sunrise all that were able to fly had disappeared.\"\n\n####  James Fenimore Cooper \u2013 1823  \n_**The Pioneers** _ **, New York**\n\nIf the heavens were alive with pigeons, the whole village seemed equally in motion, with men, women, and children. Every species of fire-arms from the French ducking-gun with a barrel near six feet in length, to the common horseman's pistol, was in the hands of the men and boys; while bows and arrows, and others in rude imitation of cross-bows, were carried by many. So prodigious was the number of the birds, that the scattering fire of the guns, with the hurling of missiles, and the cries of the boys, had no other effect than to break off small flocks from the immense masses that continued to dart along the valley, as if the whole of the feathered tribe were pouring through that one pass. None pretended to collect the game, which lay scattered over the fields in such profusion as to cover the very ground with the fluttering victims.\n\n> The Massachusettes State Ornithologist Edward Forbush recorded this account of Chief Pokagon, the last Pottawottomi chief, describing an approaching flock of Passenger Pigeons while camping on the Manistee River in Michigan: \"One morning I was startled by hearing a gurgling, rumbling sound, as though an army of horses laden with sleigh bells was advancing through the deep forests toward me. As I listened intently, I concluded that instead of the tramping of horses it was distant thunder; and yet the morning was clear, calm and beautiful. Nearer and nearer came the strange commingling of sleigh bells mixed with an approaching storm. While I gazed in wonder and astonishment I beheld moving before me in an unbroken front millions of pigeons.\"\n\n####  William Dunlop \u2013 1832  \n_**A Backwoodsman** _ **, Toronto**\n\nFor three or four days the town resounded with one continuous roll of firing, as if a skirmish were going on in the streets. Every gun, pistol, musket, blunderbuss, and firearm of whatever description, was put in requisition. The constable and police were on the alert, and offenders without number were pulled up. Among them were honourable members of the executive and legislative counsel, crown lawyers, respectable staid citizens, and last of all, the sheriff of the county; till at last it was found that pigeons, flying within easy shot, were a temptation too strong for human virtue to withstand; and so the contest was given up and a sporting jubilee proclaimed to all and sundry.\n\n> Some professional hunters used huge net traps baited with decoy birds, known as \"stool pigeons.\" These captured birds had their eyes sewn shut and their legs pinned to a post or \"stool.\" Acorns were then strewn about the netted area, and when a flock passed over, the stool was pulled out from under the blind bird, which would flutter its wings to land. The fluttering wings made it appear that the bird was landing and safely feeding. Birds from the passing flock would be lured into landing, and thousands of birds could be caught at once in the huge spring-loaded butterfly nets and slaughtered.\n\n####  John Frost \u2013 1848  \n_**Game Birds of America** _ **, New York**\n\nNone of the names bestowed upon this species are sufficiently descriptive of it. ' _Passenger_ ', an English expression, and ' _migratoria_ ', the Latin name, fall equally short, inasmuch as every known pigeon is to a greater or less extent migratory. The ' _swarm_ ' pigeon, the ' _flood_ ' pigeon, or even the ' _deluge_ ' pigeon would be a more appropriate appellation; for the weight of their numbers breaks down the forest with scarcely less havoc than if the stream of the Mississippi were poured upon it.\n\n> By 1860, pigeon hunting became a full time occupation for several thousand men. With the advent of the telegraph and the railroad, hunters were able to follow and slaughter the migrating birds wherever they landed. By 1896, virtually the entire surviving population of Passenger Pigeons came together in one last great nesting outside Bowling Green, Ohio near Mammoth Caves. The entire nesting was slaughtered and loaded in boxcars, but due to a derailment, the hunters' efforts were wasted: rotting carcasses of all 200,000 birds had to be dumped in a deep ravine a few miles from the railway loading yard.\n> \n> On 24 March 1900 in Pike County, Ohio, the last wild Passenger Pigeon was shot by a young boy. On 1 September 1914 in the Cincinnati Zoo, \"Martha,\" a passenger pigeon born in captivity, died aged 29 years. She was the last of her species. Four years later \u2013 on 21 February 1918 \u2013 in the same pagoda aviary in the Cincinnati Zoo, the last Carolina Parakeet ( _Conuropsis carolinensis_ ), named Incas, died at age 32.\n\n####  **TROJANS**\n\n##### Passenger Pigeon or Migratory Dove \u2013 1914\n\n1.\n\nThere is no epic poem for the Passenger Pigeon\n\nNo Iliad or Aeneid. No requiem. No lament\n\nYet, billions in flight once eclipsed the sun\n\nBrought darkness and chaos at midday\n\nTo New York, Boston and Philadelphia\n\nTo Montreal, Chicago and New Orleans\n\nThe longest and swiftest of the world's pigeons\n\nIt measured a foot and a half \u2013 beak to tail\n\nBeautifully stream-lined and steel blue\n\nMile-a-minute flash across the open sky\n\nThe whole of forested America was its empire\n\nFrom the Arctic tree line to the Caribbean Sea\n\nThe greatest natural wonder of the New World\n\nIts annual migrations \u2013 the greatest congregations\n\nOf birds ever gathered on the planet\n\nThe extinction of this feathered tribe\n\nIs as much our brutal legacy\n\nAs the extinction of the Trojans\n\nWas the savage heritage of the Greeks\n\nAnd after their annihilation, came the destruction\n\nOf the nation's beech forests by axe and fire\n\nAn act akin to the levelling of the walls of Troy\n\n2.\n\nHad there been a valiant Hector among their number?\n\nCertainly there was an Astyanax among the millions\n\nUpon millions that were hurled to their death\n\nBut there was no Aeneas to carry his father\n\nAnd lead his child out of the flames\n\nThere was to be no hope of the survival\n\nOf a royal bloodline in some distant future age\n\nOnly a pathetic huddled gathering\n\nLike those last few enslaved Trojan Women\n\nOne last captive flock held in a cage\n\nAnd displayed for the conquerors' amusement\n\nIn the Cincinnati Zoo\n\nAnd yes, among them was a Hecuba\n\nThe last great queen of her race\n\nThere, in a pen open to wind and rain\n\nIll-fed and neglected by their keepers\n\nThe captives fell away one by one\n\nUntil only the queen remained\n\nWithout the least awareness of irony\n\nOr intent at a final insult\n\nHer keepers gave her the name \"Martha\"\n\nAfter Martha Washington \u2013 mother of the nation\n\nThat exterminated her species\n\nNor did her humiliation end with her death\n\nHer carcass encased in a block of ice\n\nWas transported to the capital\n\nWhere she was eviscerated and embalmed\n\nDenied the dignity and respect of a proper burial\n\nHer carcass is displayed in the Smithsonian\n\nLike a severed head on a spike\n\nIn her passing came the end\n\nOf this great wonder of our world\n\nAs significant to us as the obliteration\n\nOf Troy was to the Greeks\n\nShould the memory of \"Martha\" haunt us\n\nAs the shade of Hecuba haunted the Greeks?\n\nThose conquerors who \u2013 for centuries thereafter \u2013\n\nWould hear, in the howling desolate wind\n\nSweeping over the plains of the Hellespont,\n\nThe tortured wailing of that last great queen\n\n##  **THE GODLESS GAME**\n\n###  **T HIRD WATCH 5 A. M. VEIL**\n\n####  **HEATH HEN \u2013 1933 \u2013 _Tympanuchus cupido cupido_**\n\n#####  Alexander Wilson \u2013 1811  \n_**American Ornithology** _ **, Wilmington, North Carolina**\n\nA sharp focus is brought to bear on the state of mind of the people of those days* when conservation was discussed. It is recorded that when the chairman read the name of the bill \" _An Act for the Preservation Of the Heath-Hen and Other Game_ \", the northern members were astonished and could not see the propriety of preserving \" _Indians or any other heathen\"_.\n\n> * State Legislature, New York \u2013 1791\n\n> Heath Hens inhabited nearly the whole of the open prairie from Maine to Pennsylvania. They bred and fed in groupings of several thousand strong and the plain was alive in springtime with the characteristic uproar of hundreds of males \"booming\" their mating dance. This made them easy targets for market hunters, and Wilson notes that the price of a brace of grouse over 20 years after 1800 went up from $1 to $5. Despite its obvious immanent obliteration, no measures were taken to preserve it, and it became extinct in New York State, Pennsylvania, and mainland New England by 1844. A sternum bone in New York's American Museum of Natural History is the only evidence of the bird's having once lived in New York State.\n\n#####  Thomas Nuttal \u2013 1840  \n_**An Ornithologist's Journal** _ **, Boston, Massachusetts**\n\nNot long ago, the Heath Hen had been so common on the ancient brushy site of Boston, that servants stipulated with their employers not to have Heath Hen brought to the table oftener than a few times a week. In western Massachusetts the last bird was shot in 1830, and is all but extinct on the mainland. It is now only protected on the island of Martha's Vineyard.\n\n> In 1890, there were an estimated 200 surviving Heath Hens \u2013 all on the island of Martha's Vineyard. However, by 1908, these were down to 50 when a 1600 acre reserve was set aside for them, and their numbers rose dramatically by 1915 to 2000. It appeared as if the species would survive, but in 1916 the population dropped to 150; and by 1925 there were only 20 remaining. By 1930, there was only a single male until, at the age of 8, it vanished forever on the evening of 11 March 1933.\n> \n> New Zealand's only indigenous game bird was the once numerous New Zealand Quail or Koreke ( _Coturnix novaezelandiae_ ). This bird suffered depletion of numbers through over-hunting for food and sport, but the final extirpation was only achieved through the efforts of Sir Walter Buller, New Zealand's foremost ornithologist. Keen to acquire specimens of the near-extinct species, in 1868 near Blueskin Bay, Buller discovered a covey of twenty birds. In the name of science, Buller shot every one of them: the last twenty Koreke in existence.\n> \n> The most remarkable and unique species of game bird to become extinct in the last couple of centuries has to be the Bengali Pink-Headed Duck ( _Rhodensessa caryophyllacea_ ). In Bengali it was called the _Saknal_ ; in Hindustani _Golabi-sir_ ; in Tirhoot _Umar_ ; and in Nepalese _Dumrar_. Once thought to be a true freshwater duck (genus _Anas_ ), it later was assigned a genus unto itself, _Rhodenessa._ It was not only unique in having a pink head and neck, but was also the only duck to lay perfectly spherical eggs. Extinguished in the wild by 1935, the last Pink-Headed Duck died in captivity in Fitzwarren Park, England in 1942.\n> \n> Another unique species of waterfowl and game bird was a sea-duck that once inhabited the Atlantic coast of Canada and America. This was the strikingly parti-coloured Labrador Duck ( _Camptorhynchus labradorius_ ), which was also called the Pied Duck and Skunk Duck because of its distinctive black and white markings. Like the Pink-Headed Duck, it was the only bird of its genus, _Camptorhynchus_. The Labrador Duck was last seen and shot on Long Island in 1875.\n\n####  Arthur Beetle Hough \u2013 1933  \n_**Vineyard Gazette** _ **, Martha's Vineyard, Massachusetts**\n\nThe Heath Hen failed to adapt to changing conditions and fell victim to the laws of natural selection. This is a curious thing, for until the white men took over the land, the Heath Hen had achieved an admirable adaptation, embodying such fine distinctions of nature that scientists appreciate their nicety and would like to understand them better. Even if you knew where a Heath Hen was, against a background of twigs and brush, you could not see it unless it moved. Failed to adapt! Why, no creature was ever more at home, more nicely adjusted to place and time than the Heath Hen on the Vineyard plains! The whole trouble lay in the fact that the Heath Hen was a bird man could kill, and so it had to die. The extinction of the Heath Hen has taken away part of the magic of Martha's Vineyard. This is the added loss of the island. There is a void in the April dawn, there is an expectancy unanswered, there is a tryst not kept.\n\n> This moving obituary for the Heath Hen was published in the _Vineyard Gazette_ on April 21, 1933 by the newspaper's owner, Arthur Beetle Hough.\n\n#####  **THE STREAM OF LIFE***\n\n###### Heath Hen \u2013 1933\n\n**Arthur Beetle Hough**\n\n_**Vineyard Gazette** _\n\nNow we know there are degrees\n\nEven in death.\n\nAll around us nature is full of casualties,\n\nBut they do not interrupt the stream of life.\n\nYet, to the Heath Hen something\n\nMore than death has happened,\n\nOr, rather, a different kind of death.\n\nThere is no survivor, there is no future,\n\nThere is no life to be created in this form again.\n\nWe are looking upon the uttermost finality\n\nWhich can be written,\n\nGlimpsing the darkness which will not know\n\nAnother ray of light.\n\nWe are in touch\n\nWith the reality of extinction.\n\n* As an elegy to the Heath Hen, I found I could do no better than construct a \"found poem\" from final lines of Arthur Beetle Hough's 1933 obituary of the Heath Hen. This is the only found poem chosen to serve as an elegy in this collection, and is as emotionally moving as it is profoundly thoughtful about \"the reality of extinction.\"\n\n_History is not the past. It is the consciousness_\n\n_of the past used for present purposes._\n\n_\u2013_ E. H. Carr\n**INDEX**\n\n**Principal Animals and Reportage Authors**\n\nAitchison, J. E. T. \u2013\n\n**American Black Bison \u2013 **\n\n**see **Black Bison****\n\nAnderson, N. P. \u2013\n\n**Antarctic Wolf Fox \u2013 **\n\n**see **Warrah****\n\nAntonius, Otto \u2013 ,\n\nArgoll, Samuell \u2013\n\n**Assyrian Onager \u2013 Equus hemionus hemippus (1930) \u2013**\n\n**Atlas Golden Lion \u2013 Panthera leo leo (1922) \u2013**\n\nAtsutane, Hirata \u2013\n\nAudubon, John James \u2013 ,,,\n\n**Auk \u2013 **\n\n**see **Great Auk****\n\n**Aurochs \u2013 Bos primogenius (1627) \u2013**\n\n**Bali Tiger \u2013 Panthera tigris balica (1937) \u2013**\n\nBanks, Sir Joseph \u2013\n\n**Bison \u2013 **\n\n**see **Black Bison****\n\n**Black Bison \u2013 Bison bison pennsylvanicus (1825) \u2013**\n\nBoone, Daniel \u2013\n\nBoswell, R. \u2013\n\nBowes, Surgeon Arthur \u2013\n\nBurchell, William John \u2013\n\nByron, Lord John \u2013\n\nCaesar, Julius \u2013\n\nCartier, Jacques \u2013 ,\n\nCartwright, George \u2013\n\nCateby, Mark \u2013\n\nColenso, William \u2013\n\nColumbus, Christopher \u2013\n\nCook, Captain James \u2013\n\nCooper, James Fenimore \u2013\n\nCortez, Hernando \u2013\n\n**Curlew \u2013 **\n\n**see **Eskimo Curlew****\n\nDampier, William \u2013\n\nDarwin, Charles \u2013 ,\n\nde Carlo Sowerby, Arthur \u2013\n\nde Flacourt, Sieur Etienne \u2013\n\nde Labourdonnais, Mah\u00e9 \u2013\n\nDelano, Amasa \u2013\n\nDiamond, Jared \u2013\n\nDodge, Colonel Richard Irving \u2013\n\n**Dodo \u2013 Raphus cucullatus (1680) \u2013**\n\nDu Quesne, Marquis Henri \u2013\n\nDunlop, William \u2013\n\n**Eastern American Bison \u2013 **\n\n**see **Black Bison****\n\nEdwards, George \u2013\n\n**Elephant Bird \u2013 Aepyornis maximus (1700) \u2013**\n\n**Eskimo Curlew \u2013 Numenius borealis (1985) \u2013**\n\n**Floreana Island Galapagos Tortoise \u2013 Chelonoidis nigra nigra (1876) \u2013**\n\nFoulis, Dr. James \u2013\n\nFrost, John \u2013\n\nFulinski, Benedyct \u2013\n\n**Galapagos Tortoise \u2013 **\n\n**see **Floreana Island Galapagos Tortoise****\n\n**Garefowl \u2013 **\n\n**see **Great Auk****\n\n**Giant Moa \u2013 Dinornis maximus (1850) \u2013**\n\n**Giant Swamphen \u2013 Porphyrio albus (1830) \u2013**\n\nGilbert, Captain Thomas \u2013\n\nGilgamesh Poet \u2013\n\nGould, John \u2013 ,\n\n**Great Auk \u2013 Alca (pinguinus) impennis (1844) \u2013**\n\nGreenway, J. C. \u2013\n\nGunther, Albert \u2013\n\nHamilton-Smith, Lt. Colonel \u2013\n\n**Hawaiian Oahu O-O \u2013 Moho aplicalus (1837) \u2013**\n\n**Heath Hen \u2013 Tympanuchus cupido cupido (1933) \u2013**\n\nHerbert, Sir Thomas \u2013 ,\n\nHerodotus \u2013 ,\n\nHodgson, C. P. \u2013\n\n**Honeyeaters \u2013 **\n\n**see **Hawaiian Oahu O-O****\n\n**Horse, Wild \u2013 **\n\n**see **Tarpan****\n\nHough, Arthur Beetle \u2013\n\n**Ivory-billed Woodpecker \u2013 Campephilus principalis (1972) \u2013**\n\nJennison, George \u2013\n\nJoliffe, Surgeon John \u2013\n\n**Kangaroo Wolf \u2013 **\n\n**see **Thylacine****\n\nKnopwood, Reverend Robert \u2013\n\nL'Estrange, Sir Hamon \u2013\n\nLayard, Sir Austen \u2013\n\nLeguat, Francois \u2013 ,\n\n**Lion \u2013 **\n\n**see **Atlas Golden Lion****\n\nLittell, Eliakim \u2013\n\nMackay, H. S. \u2013\n\n**Mauritian Tortoise \u2013 Geochelone inepta (1700) \u2013**\n\nMcCulloch, Alan \u2013\n\nMee, Arthur \u2013\n\n**Migratory Dove \u2013 **\n\n**see **Passenger Pigeon****\n\n**Moa \u2013 **\n\n**see **Giant Moa****\n\nNewton, Professor A. \u2013\n\nNuttal, Thomas \u2013\n\n**Onager, Syrian \u2013 **\n\n**see **Assyrian Onager****\n\n**O-O \u2013 **\n\n**see **Hawaiian Oahu O-O****\n\n**Ox \u2013 **\n\n**see **Aurochs****\n\nOxley, William \u2013\n\nPackard, Dr. A. S. \u2013\n\nParkhurst, Anthonie \u2013\n\n**Passenger Pigeon \u2013 Ectopistes migratorius (1914) \u2013**\n\nPeake, Harold J. E. \u2013\n\nPease, Alfred Edward \u2013\n\nPernetty, Dom \u2013\n\n**Pigeon \u2013 **\n\n**see **Passenger Pigeon****\n\nPliny the Elder \u2013 , ,\n\nPolack, Joel S. \u2013\n\nPolo, Marco \u2013\n\nPorter, Admiral David \u2013\n\n**Quagga \u2013 Equus quagga quagga (1883) \u2013**\n\nRidgeway, William \u2013\n\nRoosevelt, Theodore \u2013\n\n**Rukh \u2013 **\n\n**see **Elephant Bird****\n\n**Sea Cow \u2013 **\n\n**see **Steller's Sea Cow****\n\n**Shamanu \u2013 Canis lupus hodophilax (1904) \u2013**\n\nSheridan, General Philip H. \u2013\n\nSmith, George \u2013\n\nSteller, Georg Wilhelm \u2013 , ,\n\n**Steller's Sea Cow \u2013 Hydrodamalis gigas (1767) \u2013**\n\nStrachey, William \u2013\n\n**Swamphen \u2013 **\n\n**see **Giant Swamphen****\n\nSwenk, Myron H. \u2013\n\n**Tarpan \u2013 Equus feras feras (1887) \u2013**\n\nTatton, James \u2013\n\nTaylor, Reverend Richard \u2013\n\nTemminck, Coenraad Jacob \u2013 ,\n\n**Thylacine \u2013 Thylacinus cynocephalus (1936) \u2013**\n\n**Tiger, Bali \u2013 **\n\n**see **Bali Tiger****\n\n**Tiger, Tasmanian \u2013 **\n\n**see **Thylacine****\n\n**Toolache Wallaby \u2013 Wallabia greyi (1940) \u2013**\n\nToppen, Max \u2013\n\n**Tortoise, Galapagos \u2013 **\n\n**see **Floreana Galapagos Tortoise****\n\n**Tortoise, Mauritian \u2013 **\n\n**see **Mauritian Tortoise****\n\nVancouver, Captain George \u2013\n\nVojnick, Baron Oscar \u2013\n\n**Wallaby, Toolache \u2013 **\n\n**see **Toolache Wallaby****\n\n**Warrah Wolf-Fox \u2013 Disicyon australis (1876) \u2013**\n\n**Wild Horse \u2013 **\n\n**see **Tarpan****\n\n**Wild Ox \u2013 **\n\n**see **Aurochs****\n\nWilliams, Reverend William \u2013\n\nWilson, Alexander \u2013 ,\n\n**Wolf, Falkland \u2013 **\n\n**see **Warrah****\n\n**Wolf-Fox \u2013 **\n\n**see **Warrah****\n\n**Wolf Japanese \u2013 **\n\n**see **Shamanu****\n\n**Wolf, Tasmanian \u2013 **\n\n**see **Thylacine****\n\nWood, William \u2013\n\nWood Jones, Professor F. \u2013\n\n**Woodpecker, Ivory-billed \u2013 **\n\n**see **Ivory-billed Woodpecker****\n\nXenophon \u2013\n\nYakovlev, Petr \u2013 \nText \u00a9 David Day, 2012\n\nIllustration \u00a9 Tim Bramfitt, Peter Hayman, Mick Loates, Maurice Wilson, 2012\n\nNo part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, without the prior written consent of the publisher or a license from The Canadian Copyright Licensing Agency (Access Copyright). For an Access Copyright license, visit www.accesscopyright.ca or call toll free to 1-800-893-5777.\n\nThe engraving of the Dutch in Mauritius (1601) is by the celebrated Flemish engraver Johann Theodor de Bry of Liege (1561-1623).\n\nCover and Book Design: Julie McNeill, McNeill Design Arts\n\nEditor: Allan Briesmaster\n\n_To Allan Thornton and the Agents of the E. I. A., Watchmen of the Wilderness_\n\n**Library and Archives Canada Cataloguing in Publication**\n\nDay, David, 1947-\n\nNevermore: a book of hours \/ written by David Day and illustrated by Maurice Wilson... [et al.].\n\nIssued also in electronic format.\n\nISBN 978-1-926802-68-8 ISBN EPUB 978-1-926802-69-5\n\nI. Wilson, Maurice, 1914-1987 II. Title.\n\nPS8557. A93N482012 C813'.54 C2011-906253-4\n\nFourfront Editions\n\nQuattro Books\n\n720 Bathurst Street, Suite 200\n\nToronto, Ontario M5S 2R4\n\n#  **INTRODUCTION**\n\n_**NEVERMORE: A Book of Hours**_ has been conceived in the form of a twenty-four hour vigil. Each hour is dedicated to one now-vanished animal species. And each of these hourly meditations is divided into four parts: _Illustration, Reportage, Commentary,_ and _Elegy._\n\n_Illustration:_ an original commissioned portrait of each species by one of this book's four distinguished wildlife artists: Tim Bramfitt, Peter Hayman, Mick Loates and Maurice Wilson.\n\n_Reportage:_ testimonies by witnesses \u2013 who were often the first (or last) to see these animals alive \u2013 in the form of monologues taken from naturalists' journals, explorers' ships' logs, travellers' memoirs.\n\n_Commentary:_ notations on the testimonies, the witnesses and the animal species which attempt to give them a scientific and historic context.\n\n_Elegy:_ original elegies composed in poetic conventions and forms related to each animal's geographic and cultural environment.\n\nIn these epicedia, I am at one with Gary Snyder, who in his _Turtle Island,_ wrote: \"I am a poet who has preferred not to distinguish in poetry between nature and humanity.\" In this book, human history and natural history are not separate disciplines: humans and animals share a common and interactive history.\n\nThrough the reportage and commentaries here, the reader will see how the fates of these animals are directly linked to individuals and events in human history: Julius Caesar with the Aurochs, Xenophon with the Assyrian Onager, Marco Polo with the Elephant Bird, Christopher Columbus with the Eskimo Curlew, Hernando Cortez with the American Bison, Jacques Cartier with the Great Auk, Samuel de Champlain with the Passenger Pigeon, Captain Cook with the Hawaiian O-O, Vitus Bering with Steller's Sea Cow, Charles Darwin with the Antarctic Wolf-Fox.\n\nIn the reportage, I have extracted what might almost be seen as \"prose poems\" from the chronicles of historic witnesses because these authentic voices often convey that sense of the astonishment experienced in those first encounters with these newly discovered creatures; and \u2013 as often \u2013 a sense of pathos with their ultimate passing.\n\n_NEVERMORE: A Book of Hours_ is an adaptation of the _Agrega,_ a Coptic \u2013 or Ancient Egyptian \u2013 word meaning \"Book of Hours.\" The _Agrega_ is primarily used today by the Coptic Orthodox Church as a schedule for observing hours of prayer, instruction and meditation \u2013 and is comparable to the medieval Roman Catholic _Horae,_ a Latin word meaning \"Book of Hours.\"\n\nBooks of Hours were the most common type of surviving medieval illuminated manuscripts and personal devotional manuals. They were variable in content but all contained a schedule of hourly readings and prayers adopted from the daily rounds of monastic devotion known as the \"canonical hours,\" as well as being a calendar of prayers for saints and martyrs and a vigil for the souls of the dead.\n\nIn _NEVERMORE: A Book of Hours_ , the day is divided into eight Coptic canonical hours, and each of these is divided into three watches. Each watch is then assigned an \"animal martyr\" as the focus of meditation in order to cover the full twenty-four hours of this vigil.\n\nThe Coptic _Agrega_ was chosen in preference to the Latin _Horae_ largely because the \u2013 now extinct \u2013 Coptic language is actually a demotic form of Egyptian hieroglyphics, and is linked to the Ancient Egyptian _Book of the Dead._ These \"coffin texts\" used a similar structure of hours for the passage of the souls of the dead through the afterlife.\n\nThis book is a requiem to vanished animals which I have tracked down in the museums and institutes around the world. Over the years I have touched the Aurochs horn, the feathers of the Passenger Pigeon, the pelt of the Shamanu, and the amulet of the Dodo's skull. I have seen mounted specimens and preserved skins of the Bali Tiger, the Tarpan and the Thylacine. I have observed trophy heads of the Barbary Lion, the Blue Buck and the Quagga. And I have been amazed by the gigantic thigh bones of the Moa, the massive rib cage of the 7-ton Steller's Sea Cow, and the phenomenal two-gallon Elephant Bird egg.\n\nThese are now as much mythical animals as the dragon and the unicorn, but here I have attempted to give them a form of life in the only place they may now exist \u2013 the human imagination.\n\n#  **Other Books by David Day**\n\n**Natural History and Ecology**\n\nThe Doomsday Book of Animals\n\nThe Whale War\n\nThe Eco Wars\n\nThe Encyclopedia of Vanished Species\n\nTrue Tales\n\nNoah's Choice\n\nGreen Penguin Book Guide\n\nThe Complete Rhinoceros\n\n**Poetry**\n\nThe Cowichan\n\nMany Voices:\n\nCanadian Indian Poetry (Editor)\n\nThe Scarlet Coat Serial\n\nThe Animals Within\n\nGothic\n\nVisions of St. Louis the M\u00e9tis\n\nJust Say 'No' to Family Values\n\n**History**\n\nHistory & Literature:\n\nHeritage Anthology (Editor)\n\nMyth and Mountains:\n\nHeritage Anthology (Editor)\n\nMen of the Forest:\n\nHeritage Anthology (Editor)\n\n**Children's Fiction and Poetry**\n\nThe Emperor's Panda\n\nThe Swan Children\n\nThe Sleeper\n\nAska's Animals\n\nThe Big Lie\n\nThe Walking Catfish\n\nAska's Birds\n\nThe Wolf Children\n\nThe King of the Woods\n\nTippu\n\nAska's Sea Creatures\n\n**Fantasy and Mythology**\n\nThe Burroughs Bestiary\n\nA Tolkien Bestiary\n\nCastles\n\nTolkien: The Illustrated Encyclopedia\n\nTolkien's Ring\n\nThe Tolkien Companion\n\nA to Z of Tolkien\n\nThe Quest for King Arthur\n\nThe Hobbit Companion\n\nTolkien's World\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n#\n\n# CONTENTS\n\n_Foreword_\n\n1. Stroking Bees\n\n2. A House, Three Schools and A War\n\n3. Stirring It Up\n\n4. Rural Pursuits\n\n5. The Honey-Barrel\n\n6. Treading the Boards\n\n7. Rattling the Bars\n\n8. Starting Again\n\n9. Smallfilms\n\n10. Taking Time\n\n11. Mainstream\n\n12. Colouring In\n\n13. Looking Outwards\n\n14. Looking Inwards\n\n15. The Window\n\n16. Africa and After\n\n17. Decade of Dementia\n\n18. Changing Direction\n\n_Afterword_\n\n# FOREWORD\n\nThere is a period from early middle-age onwards when one is prone to become nostalgic about the childhood brand of sweets one ate in the playground and the kind of toys one played with in the bedroom: ooh, those Spangles, foaming shrimps and rice paper flying saucers \u2013 aah, good old Mousetrap, Etch-a-Sketch and Slinky. We remember too the intensity of concentration and ecstasy with which we watched television during what my generation is convinced was the Golden Age of children's broadcasting: _Rentaghost, Robinson Crusoe, Blue Peter_ and \u2013 above all, supremely above all \u2013 the masterpieces of Oliver Postgate, _Pogle's Wood, Noggin the Nog, The Clangers, Bagpuss_ and _Ivor the_ _Engine._\n\nFrom today's perspective, when the smallest amount of success is recognised with instant celebrity and riches, it seems extra ordinary that Postgate and his partner at Smallfilms, Peter Firmin, could have penetrated, stimulated and entranced the minds and imaginations of so many generations of children and yet themselves have remained relatively anonymous. From what I know of Oliver Postgate, riches and celebrity were never his goal.\n\nThe story of how he and Firmin started Smallfilms and began their thirty year journey as storytellers is well told by Oliver in this wonderful autobiography. Few of his early contemporaries might have guessed that Oliver would become a children's writer, puppeteer, artist and narrator and even fewer would have guessed that his career would provide such a contribution to the richness, comity and joy of Britain. He was either too modest or too unaware of the reach and importance of his programmes ever to vaunt his achievements, but they were inestimable.\n\nThe levels of charm, narrative pleasure, characterisation, wit and complete lack of condescension apparent in all of Postgate's work were rare enough then; today they are all but extinct. During bouts of childhood theism, I always supposed that if God had a voice it would be that of Oliver Postgate, the same matchless blend of authority, kindliness and humour. And if Oliver was God then we were all inhabitants of the planet 'far, far away where the Clangers live', where we could also find the Soup Dragon, Noggin, Olaf the Lofty, Ivor, Professor Yaffle and Jones the Steam, not to mention that 'old, saggy cloth cat, baggy, and a bit loose at the seams' who starred in what was voted the best children's programme of all time in a 1999 poll.\n\nThere are all kinds of ways of thinking about service: there is the kind Oliver never had truck with, military service, but, as he proved throughout his long, fulfilled life, it is possible to serve your country by inspiring its children and enriching its culture. There may be no medals struck for that, but there is the award of the abiding love and gratitude of millions.\n\nStephen Fry\n\n# DISCLAIMER\n\nThis is not a story, this is a life.\n\nHaving spent much of my life writing stories, I am used to being in control of the material I have thought up. I can alter it, mould it to my fancy and fit it tidily into a plot with a beginning, a proper middle and an end.\n\nI tried to do the same with this and of course it didn't work. My life was already there, already fully inconsistent, irrevocably tangled, ill-timed and beset with incongruous events.\n\nI suppose if I had been a dedicated person, one like my grandfather, who had a single guiding purpose throughout his life, there might have been a significant narrative on which to thread the events. But, although I have committed myself fairly passionately to various causes and purposes, these have come and gone as randomly as the circumstances that gave rise to them.\n\nEven the main work of my life, making films for children's television \u2013 _Noggin the Nog, Ivor the Engine, The Clangers, Bagpuss_ and the others \u2013 was not something I deliberately trained for and set out to do. It was something I slipped into almost by accident, because of some trouble with magnets.\n\nSo, in the end, I simply looked through my memory for the pieces that have, for reasons of their own, stayed with me, perhaps because they once made sense, illuminated some perception, caused grief or joy, or were just fun. I have assembled the incidents in more or less chronological order and have put together what seems to be a sort of travel book.\n\nThe engine wasn't reliable, the track was badly laid, it led through dark tunnels and into weedy sidings, and as to where it was heading... your guess is as good as mine!\n\nBut I hope you will enjoy the scenery.\n\nOliver Postgate\n\n_December_ _1999_\n\n# STROKING BEES\n\n## __I. Going back.__\n\nOn a dull day in the early 1990s, I took the number 13 bus to Hendon, got off at the corner of Shirehall Lane and walked along it towards the house where I was born.\n\nShirehall Lane, a quiet suburban street, was definitely familiar. The big elm trees had gone but the same houses were there, though they seemed smaller and closer together than I remembered. But as well as that, something was different, something was missing. Then I saw what it was: people. Nobody was coming or going, nothing was happening. The street was deserted.\n\nAs I walked I tried to conjure up the people who used to be about. For a start there were two sorts of ice-cream man, Wall's and Eldorado. They would be coming along on their box-tricycles, pinging their bells. Errand boys on their heavy bikes would whistle as they passed. The dustman's cart had two big horses. The ragand-bone man had one very small horse which pulled a small cart loaded with strange articles. The ice-man had a noisy black lorry which dripped. He carried a huge gleaming block of ice on a sacking pad on his shoulder, holding it with a pair of fearsome black tongs \u2013 I was afraid of him. Hopeful people with suitcases were going from door to door selling things \u2013 brushes, laces, insurance, salvation \u2013 to the housewives in the houses.\n\nI turned the corner into the cul-de-sac and saw a line of small, shabby, semi-detached houses. In 1925, in one of these, number four, I was born and had spent the first ten years of my life. I walked up to number four and stood in front of it. No wave of recognition came over me, no golden memories flooded back into my heart. The only feeling I had about that poky pebble-dashed house was that it was quite extraordinarily small.\n\nThen, gradually, I began to realize what was wrong. In those days I had been much nearer the ground. So, rather creakily, I lowered myself on to all fours, leaned forwards, nudged open the garden gate with my head and peered in at child, or perhaps large dog, level.\n\nThat did the trick. The first thing I saw, just inside the gate, was a piece of thick metal pipe sticking up out of the ground. On top of it was a small metal box with a curved top and a grille in its side. I don't know what it was \u2013 a drain-ventilator perhaps \u2013 but sixty-five years before I had known it well and had enjoyed its company. I now looked at it with great affection, recalling very clearly the feel of its sun-warmed metal under my hand.\n\nBeyond the pipe-thing, under the skinny privet hedge, I could feel again the black beetly earth between the roots and, behind that, at the foot of the house, I saw with a shudder the violent texture of the edge of the rendering. There the bottom of the thick stony skin of the house had curled up and broken off, leaving a jagged edge of powdery cement to moulder away and suppurate shiny brown pebbles. The mixture had fallen as a loose gritty scree against the side of the house. It was still there.\n\nFeeling slightly foolish, I hauled myself to my feet and dusted off my knees. So, yes, that was the house. I now recognized the raised brick path which wended its way from the gate to the front doorstep. But it was only three paces from end to end, and where was the wide lawn where the tall horn-poppies grew? I had roamed the flower-beds lifting off their pointed hats so that the bright yellow-gold flowers could spread out in the sun, and when a bee landed on a flower I would stroke its back with my finger.\n\nMy mother had been quite alarmed when I told her about stroking the bees. She seemed to think it was the wrong thing to do, but I was very close to them and I was sure they didn't mind. I think she may have told me not to do it any more but I don't remember whether or not I obeyed her. I wore a pale blue sort of blouse and padded pants, and in the sunshine the front garden was my kingdom.\n\nI could remember stroking the bees quite clearly but equally clearly I could see that the garden in my memory had absolutely no connection with the garden in front of me. So, rather dispirited, I turned away. Going back hadn't really been a lot of help. I realised that if I wanted to see how things were all those years ago I would have to rely on the pictures that were already in my head \u2013 if I could find them.\n\nAs I walked away I tried to look at the inside of the house as it had been when I was young. The first thing I saw, on my mother's dressing-table, was a jar shaped like a small casserole. That was magical. Except for the knob on the top which was a clear mauvecoloured marble, it was made of pale purple translucent glass. When I lifted the lid, very carefully as it was crisp and fragile, it would ring like a bell. The jar was completely filled by an ephemeral ball of swan's down which, if I lifted it by its ribbon-tag, would spring softly out and release a cloud of marvellously smelling pink powder, so fine that it floated in the air. Sometimes the cloud would float into a shaft of sunlight where it would light up like a flame.\n\nMost of my parents' bedroom was occupied by an immensely tall four-poster bed. This had fluted pillars of shiny dark wood. The flutes bulged slightly but were held together by a tight concave band of the same wood, amazingly smooth and delicate of form. I would wrap my arms around these pillars, stroke them and gaze into the rich darkness of the wood. That great bed was a different place from the ordinary world. The deep pillows and the wide eiderdown were a haven of luxury and indulgence. One of the privileges of being ill was to be moved from my thin-skinned iron bed to lie in splendour in the big bed, to be cosseted and made special. There I would float in absolute happiness, knowing that I was in the safe centre of life.\n\nMy brother John was three years older than me. We slept in the back bedroom, which we shared with a tall linen-press like an open-fronted chest of drawers and a large wickerwork laundry skip which crackled and clicked to itself during the night.\n\nOther sensations came back: the rich taste of the leather pram strap which I chewed, the delicious cold flavour of a Wall's penny Sno-frute, the smell of roses. Daisy had a wide blue bowl in which she would float big roses from the garden. This stood on a table just below the height of my nose, so I could lean on it and gaze across the wide landscape of roses, watch some luckless insect clambering from petal to petal, and sniff my heart full of the glorious scent.\n\nThere was music too. In the downstairs back room stood a tiny piano-like instrument called a Dulcitone. Our housekeeper Elsie would sit at it and press the keys with one finger. For her it would play 'Baa, Baa, Black Sheep' in delightful tinkly tones, like a carillon of chiming clocks. It wouldn't do it for me. However hard I hit the keys and shouted the words, it refused to play that song or any other. I hated it.\n\nThe front room, which we weren't always allowed to go into, had a powerful smell of its own. It was a darkish room, lined entirely with books, and it contained my father's intriguing roll-top desk, which had pigeonholes and sliding drawers and strange interlocking containers. I remember the day my brother and I discovered, third along in the bookshelf to the right of the fireplace, a heavy German book called _Die Erotik in der Photographie_. The text was of course incomprehensible but the many photographs of interestingly undressed ladies were deeply fascinating. Some of the photographs were printed side by side in pairs. I discovered that if I held the book up and allowed my eyes to look right through these photographs, beyond them and away into the distance, the images would merge until at last a voluptuous three-dimensional form materialized on the paper. I was proud of this accomplish ment and with the right glasses on I can do it to this day, but sadly the allure of the coy Teutonic lovelies has diminished.\n\nOn a tall table in the window stood a gramophone with a magnificent polished-wood horn. Sometimes, late at night, this could be heard playing scratchy German tangos.\n\nThe smell of that front room lingers to this day in the drawers of my father's desk, which is now in my study. It is the smell of briar pipes and tobacco, mixed with the rich scent of cigar boxes. It brings back clear memories of the place, but not of the people.\n\nBut there is one incident in which I can recall the people \u2013 both myself and another person \u2013 with dreadful clarity. I was standing on the doorstep of the house next door where John's friend, Russell Wright, lived. I banged on the door as usual. Mrs Wright opened it. As usual I said: 'Can I come in?' Mrs Wright looked down on me sternly from a great height and, unexpectedly, said: 'No. You can't.' Then she added: 'You don't come knocking on other people's doors saying \"Can I come in?\" That's rude! That's what rude, naughty children do.' Then she slammed the door in my face.\n\nI stood there, profoundly astonished.\n\nFrom that moment on I think I began to be more aware, or perhaps more wary, of people; certainly so of Mrs Wright. I think my wariness may have taken the form of an anxiety to please. I know I liked to try and find things to say that would interest people and so cause them to feel friendly towards me. Little pieces of information would lodge in my otherwise unoccupied mind and when what seemed to me an appropriate moment came up, I would bring them out.\n\nOn Saturday mornings the milkman always came to collect money. He yodelled, 'Milk-oh,' as he came up to the door and clanked his metal basket on the step. I ran to open the door and, while Elsie went to fetch his money, I leaned nonchalantly against the door-post to have a chat.\n\nI said: 'You don't look like a dog.'\n\nHe said: 'I'm not a dog.'\n\nI said: 'Muvver says you are.'\n\n'Does she?'\n\n'Yes. Muvver said the milkman's a dirty dog because of the short change, and you're the milkman.'\n\nThis incident caused a lot of noise, and later a lot of headshaking and glaring was done at me. I felt profoundly guilty and very ashamed, a feeling that was made worse by the fact that I didn't actually know what it was that I had done wrong.\n\nPerhaps if I had known that this pattern was going to be repeated throughout my life I would have taken more notice, but at the time I think I put it out of my mind quite quickly and went to watch the milkman's horse, whose behaviour was more reliable.\n\nWhile he did his deliveries along the street, the milkman would leave his horse and cart outside our house. He would put a nosebag on the horse's head. I would watch the horse sniff and snort into the bag of bran and oats that was fixed over its nose and then, suddenly, throw its head and the bag high up into the air in a cloud of dusty chaff. Then it would lower it gently to the ground and go on munching.\n\nI have since learned that horses do this for a reason. They do it in order to bring the heavy oat-grains up from the bottom of the bag so that they can eat them. That was a disappointment: I thought it was just _joie de vivre._\n\n## _II. Wonders._\n\nEven though I can remember their names, Elsie, Amy and Peggy, and I know I was fond of them, I can't recall exactly what our housekeepers looked like. But \u2013 even though she wasn't always there and when she was it wasn't always easy to attract her attention \u2013 it was my mother Daisy who was my refuge and haven. I would like to be able to describe her as she was in those very early days, but it is quite impossible. As I try to picture her I find myself folding softly into her safety, which was a place from which no comparisons could be made, a place where it was good to be, a place where there were good times.\n\nBathtime was particularly good. It was presided over by a large copper-coloured geyser at the far end of the bath. This had a sort of swivel tap which Daisy would turn on and light with a match. A bright finger of blue and yellow flame would spring from it. Daisy would turn on the water tap and then deftly swing the flame round into the interior of the geyser. There came a moment of thrilled anticipation and then the whole geyser went 'WOOF' and seemed for a second to be so full of flame that it must burst. After that it settled down to a gentle roar as the hot water poured out of its spout and splashed into the bath.\n\nIn those days the bath seemed much larger. There were lots of waves and drifts of foam, a rubber duck, elusive soap to slither after, a long scratchy loofah and Daisy's huge sponge which drank up so much water that you could hardly lift it. My brother would be at the other end making even more waves and ill-treating the duck. Then, after the bath, there would be big warm bath towels and running about with nothing on and after that sitting up in bed in flannelette pyjamas being told a story.\n\nTaking a bath was never a particularly private activity in our house. John and I would wander in and out of the bathroom while Daisy was in the bath, to talk of domestic matters or make enquiries about anatomy. This was quite ordinary.\n\nI think my father, Ray, may have been a bit less hospitable. I can only remember once seeing him in the bath. He was very hairy and his body seemed to fill the bath almost completely. It occurred to me at the time that he didn't need a lot of hot water for his bath and that if he had been truly bath-shaped a couple of jugfuls would probably have been enough to cover him quite adequately.\n\nRay is almost as hard to describe as Daisy. He was very large and very warm, with a roughish pelt that smelt faintly of tobacco smoke. Being an author, he was usually busy in his study but even so he did observe certain ceremonies. Every Saturday I was given my 'Saturday Penny'. Ray would solemnly make the presentation and then he and I would put on our coats and walk along to the sweetshop to spend it.\n\nRay had a large woolly overcoat and a large furry-backed hand. When walking beside him I liked to get my head between his hand and his coat and hang on to two of his fingers. This was a very comfortable arrangement because his hand was as warm as a hotwater bottle and it also kept the wind off.\n\nThe sweetshop had a marvellous array of cheap goodies. It was amazing what you could buy for a penny \u2013 four round gobstoppers which changed colour and flavour as you sucked them, or four liquorice bootlaces, or two blocks of solid lemonade or one chocolate-covered ice-cream cone, which didn't have ice-cream in it but pink marshmallow, or even, I think, a yellow sherbet-fountain with a liquorice tube in the top. This looked a bit like a firework and behaved in much the same way when you sucked at the tube and suddenly found yourself with a mouthful of explosive, prickly tasting sherbet.\n\nSometimes we would be taken shopping, to Golders Green on the top of a rattling, roaring open-topped bus, an experience I always hugely enjoyed because I was fascinated by all forms of transport. We would get off at Hoop Lane and walk along the wide pavement of the shopping parade. There we would often see a three-piece brass band: three strong-looking men wearing medals and playing golden instruments while another capered and saluted and smiled, rattling a box for pennies. I expect they were out-of-work exsoldiers, but they made a lovely noise.\n\nIn the draper's shop with its broad mahogany counters all was calmness, efficiency and respectful servility, but high up, close to the ceiling, the money-jars were whizzing backwards and forwards like frenzied bats to and from the cashier, a lady in a high round desk at the centre of a web of wires.\n\nOne amazing shop had fires burning in the window with perforated tins rotating on the top. They made smoke, but this was no ordinary smoke, it was the smoke of roasting coffee; powerful, pungent stuff. I wasn't sure I liked the smell but it was devilish exciting.\n\nThe fish shop had a cold smell which came from a marble slab where wet dead fish were laid out, and also from a deep metal tray in which live eels were squiggling.\n\nIn those days quite a lot of ladies wore foxes around their necks. These animals were very thin, with sparse hair and eyes as bright as beads. Eventually I decided that they weren't really alive but I still didn't want to touch one.\n\nOccasionally we travelled on from Golders Green to visit our cousins the Coles, who lived in a tall house in West Hampstead.\n\nTo do this we travelled on the queen of all public transport \u2013 the tram.\n\nTrams were tall and thin and had a special smell. This wasn't smoky and oily like the smell of buses, it was clean and almost fresh. As it started to move, a tram made a rich metallic grinding noise, _Gerdoing \u2013 gerdoing, gerdoing, gerdoing_ _..._ which rose in pitch as it speeded up until it became a steady song, accompanied by clangs, clunks, creaks and graunching noises as the tram swayed and pitched on its narrow rails. When it came to a sharp corner the tram would jerk sideways and lurch as if it had been barged by an elephant. To this day I don't understand why trams didn't just fall flat on their sides at every corner.\n\nOne evening, on our way back from the Coles', I saw something quite unexpected. It was dusk, almost dark, which was the best time to travel on the open top of a tram. Being above the lampposts the deck was dark except when the tram-pole spluttered and gave off white sparks as it rolled, hissing, along its wire. You could hold on to the rail and look down on the stream of cars and vans and see the people moving busily on the pavements, lit by the bright shop windows. It was like being high in the rigging of a tall ship. Standing there, scanning the view from side to side, I suddenly observed, or perhaps I should say, noted the predicament of, what I could only conclude must be a hitherto unknown section of society.\n\nAlong the Finchley Road, as along many other big roads in London, there were parades of shops, all brightly lit and thronged with people. From the top of the tram I could see that the shops themselves had dark flat roofs. Behind these, on a level with me but invisible from below, was a line of windows. Many of these were brightly lit and as the tram passed I could see straight into each room from corner to corner. I saw families having tea, people sitting reading, a boy doing his homework, a mum ironing on the table, even a thin man and a fat lady playing cellos \u2013 all life was there for me to look at, set out in single bright boxes.\n\nThis was quite fascinating, but gradually an awful thought occurred to me. These upstairs people had no front doors. They were totally isolated. They had to live their upstairs lives in an upstairs world which was completely separate from ours.\n\nAmy, our housekeeper, was in the cabin below. I ran down to tell her about it. I explained to her how the rooms were set side by side for the whole length of the shopping parade, so there was no way out on that level, and there was no way in or out downwards because the dark flat roof was in the way and underneath was all shops, and shops had counters which you weren't allowed to go behind and shops were often closed. I was alarmed about this but Amy didn't seem to think it was very serious. She said she was quite sure the upstairs people had ways to get in and out if they wanted to.\n\nEventually I had to agree that this might be the case. But even so the anxiety must have remained in my mind because, years later, I noticed some unobtrusive front doors squashed between the shops on that part of the Finchley Road, and I remember feeling distinctly relieved.\n\nIn 1930 I was five and it was time for me to go to school. I wasn't enthusiastic about this idea, mainly because it meant I had to master the intricacies of dressing in school clothes. I had to put on a thick vest with short sleeves and a grey tuck-in shirt. The rather long and complicated underpants had loops on the side through which the leather tabs of the braces had to be threaded before they were buttoned to the grey flannel short trousers. If you forgot to do this, one side of the pants came down, which made walking difficult. But I mastered the procedures and, to my surprise, found I quite liked school.\n\nThis was Woodstock School, a private school with about two hundred pupils in a spacious house and grounds on the Golders Green Road. The school was run, and perhaps owned, by a Dutch couple, Mr and Mrs de Vries.\n\nThe atmosphere of the school was friendly, gentle, respectful and firm. In general we sat still and learned things, like how to read and write and do sums, a grounding for which I never cease to be grateful. The teachers, and sometimes even Sir (Mr de Vries), were genuinely pleased if I did something well, and awarded stars for good work, stars which Sir liked to draw all over the page in elaborate patterns and many coloured crayons till they looked like bursting fireworks. The teachers were equally unhappy if the work wasn't good and didn't hesitate to sling it back and have it done again if it wasn't up to scratch. I didn't really mind that either. I liked to please.\n\nReading my first school reports I see that: '... his drawings are interesting as they frequently deal with the mechanical workings of things, particularly with water works', that I had a bad stammer and was nervous and easily upset and also, rather nicely, that: 'He is a lovable wee fellow and has delightful manners.'\n\nThe member of the staff that I remember most clearly was a lady who came in to teach us Art. I think her name was Miss Horrocks. She looked quite old, wore a long dress and beads, and spoke in a slightly wistful manner with a dark, croaky voice. The fascinating thing about her was that her hair grew in the form of intricately woven circular pads which covered each ear like earphones.\n\nMiss Horrocks showed us pictures of people with straight noses and lots of floating hair who wore lovely flowing clothes and lived in dreamy decorated landscapes. These, she told us, were Art. I found it truly wonder-full, deeply imbued with rich but unknown significance.\n\nWhat Miss Horrocks liked us to do for her was make copies of designs which she called 'tile patterns'. She had stacks of cards of these which she would hand out for us to look at and copy. As far as I can recall the patterns were mostly of formal entwinings of leaves and flowers, but the one I remember most clearly, perhaps because I thought it was a bit sinister, looked something like a partially deflated ace of spades surrounded by limp black snakes, which seemed to be fainting in coils on to the floor.\n\nLooking back I have the impression that Miss Horrocks may have been a genuine surviving disciple of William Morris, or even of the Pre-Raphaelites, and I feel rather honoured to have met her.\n\nFor the first year or so we were taken to school and collected by Amy. Of all the various housekeepers who looked after us at Hendon, she is the one I remember with particular affection. We had even been to stay at her parents' tiny house in Leiston in Suffolk. The house and the neighbourhood were fascinating, especially the water supply which was brought from a well in shiny buckets, but I saw little of the place because I almost immediately came down with measles.\n\nMy body became very pink and knobbly and I was put to bed. I think Ray and Daisy may have been away in Paris at the time, so my grandfather, George Lansbury, who was a member of the Cabinet in the newly elected Labour Government, was alerted. Perhaps as a result of this a uniformed policeman was sent to the house to ask after my health. This event caused great excitement in the neighbourhood but it was nothing to the excitement that came a day or two later when Grandad himself turned up, really just to pat me and wish me well because by then I was over the worst of the measles. Being in bed I didn't witness his arrival but apparently the reception was fairly rapturous. These were poor people and they all knew George Lansbury.\n\n## _III. Grandad._\n\nPoor people were George Lansbury's life. He was a founder member of the Labour Party, a militant pacifist, a lifelong campaigner for social justice who had become a much loved and revered figure, known as the uncrowned king of London's East End. He it was who, accompanied by a brass band, had led the whole of the Poplar Borough Council to the High Court and on to prison, for refusing to implement the Means Test.\n\nMy mother Daisy was George's seventh child (he had twelve in all) and she was brought up in 'the movement', where there was exciting work for all to do, including, for her, the task of impersonating, and being arrested for, Dorothy Pankhurst, the Suffragette leader.\n\nMy father, Raymond Postgate, who came from a formidably academic family, had turned against the academic life and had embraced socialism. He became a conscientious objector in the 1914\u201318 war, and was imprisoned and disowned by his father. He went to work for George Lansbury when he was editor of the _Daily Herald_ , and married his secretary, and daughter, Daisy.\n\nGrandad played an important part in our lives. Well, no, he didn't exactly play a part, he was just there, a godlike personage hovering somewhere above us, likely to appear unexpectedly at any moment. I can see him standing in the doorway. He was huge. His bowler hat over his big white-whiskered face and massive black overcoat completely filled the door. He also had the largest ears I have ever seen. They didn't stick out, they just covered a large area. I distinctly remember sitting on his knee and admiring them.\n\nI already knew that Grandad was an important man, both in the family and in the world, but I didn't know that he was a truly great man, nor that he would one day stand up in the House of Commons, reprove Winston Churchill for behaving like God Almighty and tell him to hold his tongue.\n\nOnce or twice Daisy took us to the House of Commons to see Grandad at work. The great palace was guarded by policemen but they each greeted Daisy by name and had a joke with her as we passed. Daisy walked through the sombrely magnificent halls and corridors of the House as if she lived there, and at every corner the officers smiled at her and spoke. I felt very proud.\n\nGrandad was pleased to see us and took us all to tea on the terrace that overlooked the Thames. From there we watched the tugboats pulling lines of barges loaded high with planks of yellow wood. Grandad knew the names of the timber merchants and where the cargoes were going. I was very impressed by that but I also remember feeling a bit anxious about my important grandfather, sitting on the terrace having tea with us. It occurred to me that he was supposed to be in Parliament making laws, not wasting his time having tea with children. I raised this point with him and he took me into an inner hall of the tearoom and showed me a machine that was making ticking noises in short bursts and issuing paper tape. He lifted a piece of the tape and read it to me. Apparently it told him what was going on in the Chamber. He explained that he was allowed to go wherever he liked whenever he liked so long as he came and looked at the tape now and then to see what was going on, and if he was really needed in the Chamber to vote, a Division Bell would be rung to tell him it was time for him to go back in.\n\nOnce we actually went into the Commons Chamber to hear part of a debate. Daisy took us up into the Visitors' Gallery, from where we looked down on the lines of green leather benches and saw a few Members sitting here and there. We saw the golden mace on the table and the Speaker on his throne, wearing a wig.\n\nDaisy shushed us because something was happening. A voice muttered: 'Mr Lansbury' and I saw Grandad rise quite slowly and look around. I was so nervous for him I felt quite sick, but he seemed perfectly at ease as he began to speak. At first he spoke quietly, but then he gradually warmed to his work and his voice swelled to a sort of musical baying, which caught and carried my imagination along and would perhaps have moved me greatly if I had had the slightest idea what he was on about. Then, quite suddenly, he stopped and sat down. Amazingly, there was no applause or cheering. A few Members shifted the way they were sitting, and grunted. Then one of them, who seemed a bit angry, jumped up and began to yap like a terrier. That caused several of the Members to get up and, rather rudely I thought, walk straight out. The debate proceeded, but we didn't stay.\n\nMy grandfather's work in Parliament, making laws, seemed to me very august and remote from our own lives, but at other times and places we had a real part to play. In his capacity as First Commissioner of Works Grandad often had to attend ceremonial engagements. As these tended to be a bit solemn he liked to bring along a grandchild to lighten the proceedings and do small tasks, like cutting tapes.\n\nI had the honour of assisting him at the opening of Lansbury's Lido, a section of the Serpentine lake in Hyde Park which he had had adapted for the people of London to swim in. On this occasion I became a shade over-excited and inaugurated the swimming season myself by falling in, a gesture which was much appreciated by the onlookers, less so by my mother who had to borrow a towel from the changing-rooms to dry me.\n\nBy far the most alarming thing that had ever happened in my short life took place by accident on May Day in, I guess, 1931. There was, as always, a great rally in Hyde Park, at which, as always, George gave one of his huge rousing speeches that was received rapturously by an enthusiastic, loving crowd. At some point, after he had finished speaking and was leaving, I became separated from the party. The situation was that Daisy and Grandad were in a taxi with the windows open and I was on the grass about twenty feet away from them. This would have been no problem if I hadn't been stuck in the middle of a solid cheering crowd that was mobbing the taxi to shake his hand. The crowd was shouting: 'Good Old George!' George was shouting: 'Thank you, Brothers and Sisters!' and: 'Keep up the good work!' My mother was shouting: 'That's my son over there!' and I was shouting: 'Help!' I was only at waist-height in a tight press of people that was moving as a single cheering mass, carrying me with it. I had to hang on to the arm of the man beside me to save myself from going under and being trampled.\n\nBy pure good luck the man saw me and caught my mother's eye. In an instant I was lifted bodily into the air and passed from hand to hand like a long parcel over the heads of the cheering crowd, to be posted head-first in through the window of the taxi, amid laughter and cheers of relief from the crowd, and tears of fury from my mother.\n\n## _IV. The Green Bench._\n\nAs I was looking at these memories my feet had taken me away from the house in Hendon and along old familiar streets to what Ray had called 'the Recreation Ground', a small park about a quarter of a mile away.\n\nI sat on a particular green bench in the top corner of the park and looked around. The view from it was no more inspiring than the sight of the house had been \u2013 I saw a small, fairly drab area of worn grass divided by smooth paths and clumps of blackishlooking trees.\n\nSo this was the wide space where John and I and our friends used to play. We played football with coats for goalposts, rounders or one-ended hit-and-run cricket. We had quite big teams, so we must have played with other boys as well but, apart from Russell Wright and Jimmy Thwaites and Kenneth Collins, I couldn't bring any of them to mind. I do know one thing for certain \u2013 we never played with girls!\n\nGirls, I remembered, were different. John and I had girl cousins and of course we treated these more or less as people, but girls as a race were creatures apart, strange and unpredictable \u2013 as I had already found out.\n\nWhen I was quite young, about four or five, there had lived, in the house opposite, a golden curly-haired Shirley Temple called, I think, Desire\u00b4e. One day she expressed a desire to have me over to play with. So, through the good offices of go-betweens (their maid and ours), I was duly procured, brushed down and delivered into her amazingly pink and frilly house. There seemed to be nobody else there. I remember myself as being dumb with embarrassment. Desire\u00b4e may also have been embarrassed, but she was voluble. Her discourse was a continuous paean of elaborate self-praise, punctuated with prods and interspersed with small items of scorn about me. I believe at one point she proposed an activity which at the time struck me as being a bit rude. I didn't cooperate and quite soon, when grown-ups reappeared, she had me sent home for being, in some serious but unspecified way, naughty. I was deeply ashamed but very, very relieved to get away. We never spoke again.\n\nI don't think I minded about that. Her paths were not mine. John and I now had important projects that she probably wouldn't have enjoyed \u2013 like damming the River Brent. This was no more than a broad, shallow brook, with a bed of black slime-covered things, some of which were stones. Wearing shiny Wellington boots we dislodged these and used them for building dams and diversions, a fascinating task which we did well and enjoyed a lot, even though we often got into trouble for coming home smelly. But for me, personally, the main attraction of the River Brent was that it was edged with patches of untamed jungle, lost land which had been left by the developers and had gone to waste.\n\nHere the terrain was generally steep, with scrubby trees and bushes, bramble patches and rocks of broken concrete. In places there were branches to swing on and small smelly waterfalls that glugged out of the ends of pipes and flowed down muddy gorges to the brook below. But there were also precipitous paths that led to dark leafy bowers where, in summer, one could sense stillness, feel oneself far from civilization and even hope to see a rabbit. These sylvan glades, so near to home but so different, were awesome and full of magic.\n\nSitting there on the bench, I wondered what sort of boys we were. Were we wild? No, I think we were pretty tame. We didn't scratch the paint of the cars or let down the tyres or break off the wipers. We didn't steal, we didn't break-and-enter, didn't mug old ladies.\n\nWe went to the Saturday-morning cinema, of course, and at home we could sometimes be quiet. There was no such thing as television to watch or home computers to play with, but we did listen to the wireless and we played games, cards, board games, paper games, or made models with Meccano. We also read books, _Alice in Wonderland, The Wind in the Willows, The Midnight Folk_ , _The Amulet_ , etc., but these were not as essential to our lives as the comics and adventure weeklies \u2013 like _The Wizard and Hotspur_ \u2013 which my father called 'penny dreadfuls' and deplored. So if necessary we had to read these under the bed-covers, by torchlight.\n\nAs the years passed John and I gradually became larger, louder, more enterprising and more of a nuisance. This was made worse by the fact that we didn't really get on.\n\nAfter some initial reluctance he had become, with reservations, reconciled to the fact of my existence and was even willing to use me, occasionally, as a menial accomplice, but not as a companion.\n\nI was untiring in my efforts to procure his acceptance and approval, while he was not only determined not to give it himself but was also anxious to prevent my getting any from anybody else.\n\nIn our dealings with grown-ups John and I were strictly in competition and loudly jealous of what we thought were our rights. I also remember spending a lot of time whining at whoever was there, trying to get them to _make_ John play with me, while John was probably whining at them to get me off his back. The rest of the time we spent noisily bickering.\n\nI have the feeling that 'they' \u2013 Elsie, Amy, Peggy, even Daisy \u2013 came to see the pair of us as a single agglomeration of squabble, nuisance and overcrowding, and became thoroughly fed up with it. So it must have been a great relief, if the weather was reasonable, to be able to send us to play in the Recreation Ground.\n\nAs to our characters, what John and I were like personally, there is a better reference than my cloudy memory. In about 1934 Ray wrote a novel called _No Epitaph_ , which was, as he admitted, largely autobiographical. Included in it were thumbnail sketches of the hero's two sons, James and Richard. He wrote:\n\n... They were good-looking children, in fact as well as in his prejudiced eyes. James' dark colouring was unlike either of his parents. His complexion was olive, his hair black, his eyes bright and dark brown. His most attractive feature was his bright, white-toothed grin. Inattentive, rampageous and greedy, he got himself forgiven innumerable escapades by his cheerful smile and instant assumption of a false penitence... James held a larger place in his father's affection than Richard, a year younger. Richard was not indeed so apparently lovable a child... With one year's gap in everything, Richard had had to strain always, and learned to ascribe his failure to cheating by his brother, to ill-luck, to insufficient self-advertisement, to anything rather than his natural inferiority... Because his elder brother grabbed all the attention that he could, Richard had formed the habit of talking almost incessantly, so that he might break into any pause in the conversation and for a minute secure the centre of the stage. Once started on his way, it had become an obsession with the child: he thought aloud like an old man. 'Vere is,' he would say \u2013 he had retained one or two childish mispronunciations \u2013 'a man in ve park who gives sixpences if you run fastest in races. Yesterday I ran faster van James and he gave me sixpence. I know he is vere and and it is true. He always gives sixpences and he wears a brown hat. I can run faster van James. When I have forty sixpences I shall buy a bicycle. Arfur, James' friend at school who doesn't like me, was bought a bicycle by his faver...' Anne was too conscientious a mother to rebuke or punish Richard for 'lying', but his incessant stream of small imaginings was a minor nuisance, battering on her half-attentive ears day after day. It obscured Richard's sweet and generous nature and his puppylike affection. He was far more unselfish than his brother: give him a quarter of chocolates and he would scatter them around his friends (or rather his brother's friends, who would condescend from their attitude of contempt to him just long enough to rob him completely) while James would stuff himself silently in a corner...*\n\nAlthough the age gap is different I'm sure that James was John. I also have evidence to confirm that Richard was me, and there is no reason to doubt that the piece is a fairly accurate account of how our parents, or at least our father, saw us. When I look back at myself the picture I get is very similar to the one Ray gives, but seen from a different angle. It is that of a child constantly scrambling for acceptance, like somebody frantically trying to clamber on to a crowded raft, hoping somehow to be allowed a place. Fortunately, I could also see that the child knew nothing of this. He never lost hope, never gave up trying, bore no grudges and took whatever small mercies came his way with innocent delight.\n\nThat is all ancient history but there is one small piece of evidence with which I would like to set the record straight. The green bench I was sitting on was the one which the man in a brown hat had sat on when he _did_ , once, give me sixpence for winning a race. His hat was brown and, dammit, I wasn't lying.\n\nI got up and walked away from these disagreeable memories. There were more pleasant ones to entertain.\n\nJohn and I were not stuck in that poky house for all our time. Ray and Daisy were great travellers and they took us to France almost every year, and once on an epic journey to Spain.\n\nThey would often go for short visits to Paris on their own. Daisy had even flown there in an airliner, a Handley-Page Heracles, from Croydon Aerodrome. The adventure had frightened her terribly but I, with my passion for vehicles, thought it must have been terribly exciting.\n\nDaisy's younger brother, our uncle Eric, had had a different aeronautical adventure. He was to fly in an airship, but he was lucky enough to miss his flight because his car broke down. The airship was the ill-fated _R101_ which, that night, crashed into a hill in France and burst into flames.\n\nI saw the newspaper pictures of the crumpled skeleton of the airship and heard the reports on the wireless, and I know I wondered what I would do if I were on an airliner which was about to crash. It occurred to me that the obvious thing to do at, or just before, the moment of impact, was to jump vigorously into the air. In that way one would be going upwards, not downwards, and would land gently, as one would from such a jump. I wondered whether anybody had thought of doing this. The idea stayed in the back of my mind, along with the car which had small wheels in front and large wheels at the back so that it would be going downhill all the time, thus eliminating the need for an engine.\n\n## _V. Another Place._\n\nWhen they went to Paris Ray and Daisy usually left John and me at home with the housekeeper or, quite often, they would send us with her to a left-wing holiday camp called Treetops. Once or twice we stayed with Grannie, Ray's mother, sometimes with other relations. Only once did we spend a long time in a completely strange environment.\n\nThis happened at the beginning of 1933 when Daisy had arranged to become a film star. No, that's an exaggeration. What happened was that Rudolph Messel, who was a very rich young man as well as a socialist, had formed an organization called the Socialist Film Council, which was going to make _The Road to Hell_ , a film about the evils of unemployment and the Means Test. Everything was set up for this and shooting was about to start when Amy, our housekeeper and the king-pin of the family's life, had to go back to her home in Leiston. This was a serious crisis because somebody had to look after John and me.\n\nSo, towards the end of March, we were sent to Woodstock School as boarders.\n\nWoodstock School only took about half a dozen boarders at a time but that was quite enough because Mrs de Vries, the headmistress, saw to them herself. We slept in tiny rooms high up under the roof, we had our meals and were left to play in a large, uncomfortable, bare room with a floor of brown linoleum, and each weekday morning at a quarter to nine we simply went to school, but through a green-baize door in the hall, not through the main entrance.\n\nI was lonely and miserable so I was quite glad to be taken over by a slightly older girl whose name was, I think, Laura. She was very shocked to discover that I had not been taught to pray because she knew it was essential to 'pray for God', as she put it, every morning and evening. She explained that if I didn't do this God wouldn't know I was there and consequently wouldn't be able to look after me. She showed me how to kneel at the foot of the bed and told me what to say. I remember I felt a bit dubious about the whole procedure but this didn't interest Laura. Her only purpose was to get me to do what had to be done. What I thought about it was irrelevant.\n\nIn fact I didn't think anything about it. I mean, as far as I know, I had no previous acquaintance with the subject. Ray and Daisy were simply not religious, they didn't talk about God. Nor, as far as I can remember, did Amy. So Laura had a clear field.\n\nI asked Laura a lot of questions about God and also about Jesus who, she said, was also God but not quite. Her answers weren't really very clear but I was able to piece together a picture of a white-haired old man, a bit like Grandad, but cross. He was sitting on a white cloud surrounded by glory, which was goldcoloured and shiny. I wondered where He was. Laura said He was everywhere, which didn't seem very likely, and that He could do anything and see everything, and that He moved in a mysterious way.\n\nI knew about that. I had once seen my parents' friend, the novelist Naomi Mitchison, 'move in a mysterious way'. She was playing charades I think, wearing a wide hat and a cloak. It had been very impressive.\n\nI observed the rituals Laura required and also consented to do a ballet dance with her in front of everybody, or, to tell the truth, half a ballet dance, because she had dressed me in a sort of frill, like a vestigial tutu, and after the gramophone ran down in the middle of the dance I refused to go on with it on the grounds that I was making an idiot of myself. After that she gave up her charge of my life and let me be.\n\nBut the question of the existence or otherwise of God remained in my mind. As far as I could tell He didn't seem to be taking a very active part in my life because cause and effect seemed to be proceeding according to what I assumed to be natural laws. However, it was worth while finding out, so, roaming through the empty school one chilly afternoon, I decided to confront my maker and test His mettle.\n\nI looked out of a window and said: 'All right then, try this. If that bird gets to the telegraph pole before the red bus reaches the pillar-box there _is_ a God. If it doesn't there _isn't._ '\n\nThis was a very fair test because the bird and the bus were going at about the same speed and, as far as any human could estimate, were going to pass the pole and the pillar-box at about the same moment; so it would only require a minute twitch of omnipotence to slow the bus down a shade and let the bird win. Nobody else would notice.\n\nIn the event the bus and the bird arrived exactly simultaneously. This told me that if God was there He probably wasn't paying attention. So I repeated the tests a few times. God scored one, Non-existence two, with three draws.\n\nThen it began to snow, which was unusual in April. I watched the big white flakes swing slowly down and melt away on the wet ground. They fell so thickly that I could hardly see the road, let alone the pillar-box. It suddenly occurred to me that God, if He was there, might have decided He'd had more than enough of this carry-on. So I rather quickly turned away and ran back down the empty brown lino corridor to find the others. I was cold and I wanted, more than anything in the world, to go home.\n\n## _VI. Bloomsbury_ _-on-_ _the-Marsh._\n\nFor all its pokiness, our house in Hendon was far more full of life than the bleak impersonal school house. Although John and I didn't see all that much of our parents, even I could sense that this was because Ray and Daisy were busy people, joining enthusiastically in with, if not an actual 'movement', some common social purpose which they shared with their friends.\n\nHowever, we were sometimes able to take more of a part in their world because Daisy had obtained a car, a second-hand bullnosed Morris, with an open top and grey-green oilcloth seats. Its gearbox sang a different note for each gear and its bulb-hooter didn't just go 'HONK' when you squeezed it, it went 'SqueeeAHONK' like an outraged sea lion. Most of the trips we took in it were fairly local but occasionally we took longer journeys, real holiday journeys with luggage, to stay with their friends in the country.\n\nI guess the first time Daisy drove us to Bloomsbury-on-the-Marsh must have been in about 1931. She took us up the Cambridge road, turned into Essex and drove through Toppesfield to Bradfields, which was the country house of Francis and Vera Meynell.\n\nI had never consciously seen a really old house before and I was astonished. It looked as if a giant had sat on it. All the walls were crooked, either leaning in or leaning out, and the roof was bent and dented, like an old hat.\n\nThe Meynells' garden was large and glorious, the grassy side of a hill, rolling down to a stream with a bridge at the bottom. Under some trees at the top of the garden was a large pond. Beside this was a patch of lawn with a wooden diving platform, and it was all surrounded by a modest hedge. But, once I was sure it wasn't going to fall in on me, it was the house that I found really fascinating. None of the doors was quite door-shaped and they didn't have handles but latches. The floors were particularly alarming because they sloped in all directions and the boards were odd-shaped, but they smelled most sweetly of beeswax.\n\nAlso, while they were there, our parents and their friends seemed to be so full of fun that the house rang with laughter. I hoped we would be able to go there often, which indeed we did, for many years.\n\nNeedless to say there is no such place as Bloomsbury-on-the-Marsh. The name was probably invented much later by some wit, but it could be taken to refer to an undefined area covering part of the south of Suffolk and the north of Essex in which some of the better-off, cultured, vaguely left-wing lot seemed to have bought themselves country houses.\n\nOur host and master of ceremonies, Francis Meynell, was a large, loud, exuberant man. Born in 1891 to a family of distinguished poets \u2013 his mother was Alice Meynell, his godfather Francis Thompson \u2013 he grew up into the world of the literary avant-garde in Bayswater, in a house where well-known poets and authors came and went, where there was much singing and poetry reading and endless literary talk by his mother's devoted admirers.\n\nFrancis's political life was forged in the same revolutionary fire as that of Ray and Daisy. George Lansbury was his hero and, later, his employer. Like Ray he had been a conscientious objector in the First World War. He worked on the _Daily Herald_ , and later smuggled diamonds into the country in jars of butter for the Soviet trade delegation. Once, to Lansbury's embarrassment, he brought in a Soviet contribution of several thousand pounds to the funds of the _Daily Herald_ , a gift which, to everybody's regret, had to be declined.\n\nAfter that their paths diverged. Ray went on to his life in history and politics, Francis to his lives in poetry, typography and publishing and to his loves, which were cricket, tennis, sociability, games, fun and women. Although Daisy could be a bit sardonic about the last of these, he remained my parents' close friend throughout their lives.\n\nIt was said that Francis knew everybody. That wasn't meant to be taken literally, I think it meant that Francis knew everybody who was anybody in the many fields in which he was interested. That was probably fairly true and it was also true that a lot of them lived in the neighbourhood, or turned up there as weekend guests.\n\nThus it happened that the short wide frenzied man with a squeaky voice, who bullied people to play games and hated losing, turned out to be H. G. Wells, and the elderly man with a thin ratty voice who was treated with a certain cautious respect was, I was told, Bertrand Russell. I drop those names here out of pure retrospective snobbery, but at the time they didn't mean a thing to me. I just saw them as fellow guests who were there, like everybody else, to have a good time.\n\nI had already got the message that my parents had come here to have fun and enjoy themselves in their way, and that I was welcome to do the same, in my way, which could be to join in the fun, or not, as I wished. I found I had no quarrel with this situation and was, I believe, fairly sparing in my demands for their attention.\n\nI do remember being a bit of a spoilsport about the nude bathing at the pond. I decided to wear bathing trunks and was ridiculed for my genteel modesty, but I persisted, not because of modesty but because of water-beetles. I had been to murky ponds like that before. I had caught water-beetles in my net and I had noticed that some of them had jaws like bolt-croppers. There was no way I was going to take my unprotected private parts, vestigial though they may have been, within their reach.\n\nBut, apart from necessary reservations like that, I was in a very neutral situation and quite happy to be an unnoticed spectator. Francis seemed to take the healthy view that other people's children were OK to have about so long as they didn't get in the way and spoil the grown-ups' games; which was fine by me. His own son, who was a year or two younger than me, didn't get off so lightly and was constantly subjected to applause and admiration. Surprisingly perhaps, I didn't envy him.\n\nLooking back, I think I can see why that time and place has left such a powerful flavour in my memory. It may have been because, although London was the place where Ray and Daisy and their friends were serious, anxious for the world and busy at their work, the Meynells' was where they laughed, put aside doubt, were frivolous and played \u2013 and maybe one gets a clearer picture of what people are like and how they feel when they are talking spontaneously together and having fun.\n\nThe people we met there were altogether motley, with no shared characteristics except that of being individuals. If they had anything in common it was that they seemed to have in their manner a certain underlying confidence. When they talked they talked as thinkers and innovators. I can see that they were people who quite naturally set the agenda for their own lives. In fact I think they saw themselves as setting the agenda for the future of the world, a future which they were already helping to bring about, a future in which agriculture was going to eradicate hunger, in which Socialism was going to make personal greed a thing of the past and there would be no more rich and no more poor, in a golden age that was already on its way.\n\nI sensed that there was a lot of gladness about, and quite right too! The dragons of moralism, snobbery and conformity had been vanquished. Love, life and liberty were all new, wrested by their own courage from the iron grip of Queen Victoria. They had come into their kingdom and taken their places as part of the new cultural elite, which indeed they were, because the cultural elite was much smaller in those days and quite a lot of it lived in the neighbourhood. So there was no shortage of genuine literary giants to lead them prancing in sandals and Liberty prints through a magic William Morris forest, ringing with licentious laughter. And perhaps the pranciest and most licentious of them all was Francis Meynell.\n\nOf course the future didn't work out as they had hoped, and I dare say that away from the Meynells' they could already see that it wasn't going to. Stalin was turning the Soviet dream into a bloody nightmare, Hitler was rising to rekindle a different, perhaps even more bloody, ambition. But all the same, I think that to have lived at that time, with those dreams, must have been very glorious and exciting.\n\n* R. W. Postgate, _No Epitaph_ (Hamish Hamilton, 1932), p. 98.\n\n# A HOUSE, THREE SCHOOLS AND A WAR\n\n## _I. Finchley._\n\nIn 1935 we moved from Hendon to Finchley, a distance of about four miles. Our new house, which had conker trees in the front garden, was immense. Well, it was only immense to my eyes which compared it with our little box-house in Hendon. Nor was it new. It was Victorian, made of dirty greenish-brown bricks.\n\nRay unlocked the wide front door and led us into the house. The hall was dark and dusty and smelt damp. He led the way up the wide, handsome, creaking staircase. The main landing was as big as a whole room in our Hendon house. He opened a door and showed me into a very large empty room with walls of mottled brown and mauve. There was nothing on the floor except some bits of whitish clinker that had once spilled from the broken fireplace.\n\n'This will be your room,' he said.\n\nThen John and I were let loose to roam through the great house from the two big attic rooms to the coal-smelling cellars. I found the idea of living in a house that I could actually run about in very exciting. The same was true of the back garden. In Hendon the back garden was a small patch closely overlooked by other people's windows. This one was large and secluded. Quite a lot of it was metalled over with cracked asphalt because it had been the playground of the school next door, but there was a large grassy hump right across the back of it and it was lined all round with old pear trees and overhung by a very big walnut tree.\n\nHaving more space, John began to collect pets: white mice, white rats, rabbits, small snakes and lizards, even a very large goldfish that had cost five shillings and lived in a sink in the garden until some predator got it. After it had disappeared, Queagh, our cat, helped us to look for that goldfish. He made a great show of turning over the dead leaves and searching the shrubbery all around the sink. That didn't fool us. He had eaten it.\n\nQueagh was by far the most clear-cut character in the family. He was surly, with crumpled ears and an evil sense of humour. His pleasure was to sit in front of the vivarium where the snakes and lizards lived and gently stroke the glass front with his paw. This would cause the glass to inch itself along its groove until the reptiles could squeeze out. He didn't try to catch them; he was content just to sit and watch them slither and scuttle across the garden, knowing that the neighbours would soon be ringing up in a state of panic to demand the removal of our revolting creatures from their nice rockery.\n\nTaken all in all the new house was a great improvement, partly because it was so spacious but also because it was convivial. Ray and Daisy had given dinner parties at Hendon but the pokiness of the house had been a constraint. The Finchley house was broad and welcoming. I can see the fire in the long room, bright with a mountain of Coalite, and I can hear the dining room ringing with laughter, not embarrassed, strained or forced laughter but the happy, genuine laughter of people fully at ease with each other in an atmosphere where all undertones of suspicion or anxiety had been dispelled and replaced by whole-hearted trustful enjoyment.\n\nAs a neighbourhood, Finchley suited us much better than Hendon because we had family friends and relations nearby. In fact the suggestion that we should move had originally come from H. L. (Lance) Beales and his wife Taffy, who lived in an even larger house about a hundred yards away. Their children, Mary and Michael, also kept a lot of pets, mainly rabbits and guinea-pigs. John used to take his rabbits to visit Mary's and they would give them the freedom of the grass tennis court for their meeting, or Jamboree, as they called it. John and Mary would run about the court trying, for some reason, to prevent the rabbits copulating \u2013 an absurd ambition!\n\nDown Hendon Avenue and across the brook lived Douglas and Margaret Cole with their children, Jane, Ann and Humphrey. They had moved from West Hampstead to Freelands, a wide rambling house with a couple of acres of garden around it. A new hard tennis court had been built beside the house and in summer we used to go there almost every Sunday afternoon to left-wing tennis parties where there was good intellectual conversation, indifferent tennis and an unending supply of China tea.\n\nTennis was _the_ game in the thirties. Unfortunately our asphalt back garden wasn't big enough for a court, so Ray and Daisy marked it out with silver paint and bought a badminton set. Ray also bought a small pickaxe and a wheelbarrow so that we could have the pleasure \u2013 and it was a pleasure \u2013 of prising up parts of the crisp asphalt skin to expose the earth beneath. As time passed the garden became very fertile, in a wild sort of way, with a forest of giant, very delicious blackberries at the far end. The pride of the garden was a climbing rose that Daisy had bought in Woolworth's for sixpence. Its name was 'Albertine', it grew to enormous size and I worked out that over the years it must have given us a million flowers.\n\nI can see that garden in the summertime, with the crumbly yellowish bricks of the house, the badminton patch deeply embow ered in the surrounding trees and the squirrels dropping walnut shells on the players. It is a picture of mellow ease and contentment.\n\nBy 1936 I had graduated to long trousers, at least for ceremonial occasions, and John and I had both become very sartorial and particular about what we wore; details which, looking back, seem to be no more than nuances of drabness. Clothes were very dull in those days, but we wore sock-suspenders and put Brylcreem on our hair.\n\nOnce or twice a year Daisy would take us to visit the Gents' Outfitters in the Charing Cross Road. The high point of the outings would be Chinese lunch with chopsticks at the Shanghai Restaurant in Greek Street, where the waitress was called Doris and had been no further east than Clacton. Between the Outfitters in Charing Cross Road there were seedy little shops which advertised DAMAROIDS and sold strangely shaped trusses. These were puzzling, but the amusement arcades, where the illuminated bagatelle games thumped and pinged and whistled, were fascinating. Daisy found them boring, so she tended to stay outside and look at the shops. When this happened I was able to nip to the back of the arcade and furtively put a penny into one of the machines marked ADULTS ONLY. These were always disappointing. You peered in and saw an old scratchy photograph. When you turned the handle the photographs tipped over and tottered past like a flicker-book. The pictures were faint and blurred and seemed to concern minor happenings in the lives of wellcorseted ladies and rolly-eyed gentlemen in suits.\n\nI didn't really need to be furtive about watching the not-veryrude pictures because Daisy was not in any way a prude \u2013 far from it! Perhaps from the music halls, where she had listened to Marie Lloyd belting out her hilarious, ribald songs, she had developed a healthily bawdy view of life and a very vulgar laugh. I recall one spectacular outing on which she took us to see the Crazy Gang at the Victoria Palace. The Crazy Gang were renowned for the high quality of their rude jokes and _double entendres_ , many of which were so sophisticated that their import was lost on the nice-minded in the audience. However, that afternoon there was one member of the audience who not only saw every aspect of every gag but was also laughing uproariously at them. John and I got the distinct impression that the Crazy Gang were enjoying Daisy's appreciation so much that they were slipping in particularly fruity jokes especially for her, so it sometimes seemed as if she was the only member of the audience who was laughing. John and I were so embarrassed that we were practically hiding under the seats, but Daisy really enjoyed herself.\n\nSometimes, if one of us had a birthday, or had Christmas money to spend, we would go to Hamley's toy shop in Regent Street. It was from there that I chose, by grace of Ray and Daisy's friend, Rudolph Messell, who was rich and liked to give big presents, something that was destined to become my most prized possession. This was a Super-de-luxe Conjuring Set by Ernest Sewell. It came in a big box that cost nearly two pounds and contained some very sophisticated pieces of equipment with which I looked forward to impressing rapturous audiences.\n\nRapturous audiences were not readily available to small boys of eleven, but there would soon be a 'window of opportunity' \u2013 Grandad's party.\n\nGeorge Lansbury's seventy-seventh birthday party might have been a rather muted occasion because he was at what must have been the lowest point in his long political career. At the Labour Party Conference in 1935 he had suffered a vicious personal attack from Ernest Bevin, who had, in effect, called him a traitor and accused him of stabbing the Labour Party in the back. The Party faithful had not rallied to his support and he had resigned from the leadership.\n\nSuch a shattering blow would have broken most politicians and left them bitter. It is a measure of George's greatness that his spirit seemed to be quite undamaged and he was his usual jovial, grandfatherly self.\n\nGrandad's birthday party had always been quite an elaborate affair which was attended by all, or certainly most, of his large family. Usually thirty or forty people turned up and I am told the occasions could be a bit tense, often for reasons that were more political than personal. Ray couldn't abide Party-line Communists and every year of Stalin's reign made him more contemptuous of them. Equally he had little time for the far right wing of the Labour Party, which our uncle Ernest Thurtle represented. He could also be less than respectful of Labour activists who affected what they thought were working-class accents in order to show that they were down-to-earth trade-unionists. To cap it all, Daisy's eldest sister, Annie, was a Christian fundamentalist. She wasn't really on speaking terms with any of them except perhaps her sister Nellie, who worked at Collet's bookshop and wasn't quite a Communist, though her youngest sister Violet very definitely was. So at one level the party was festive and full of fun, while on another it was fraught with submerged implications, turned backs and barbed references. Apparently John was sensitive to these tensions and found them a bit alarming. I was totally oblivious to them and so, I believe, was Grandad. He and I simply enjoyed the party and joined in with gusto.\n\nAt this particular party I was especially excited because, after a deal of campaigning, it had been established that I would be permitted to entertain Grandad and the party with one, just one, of the tricks from my amazing new conjuring set \u2013 but I wasn't to muck about and waste time. That was fine by me and I set it all up most carefully. The moment came and my parents, sighing indulgently, apologized to the assembled guests and craved their indulgence while their tedious son showed off with his conjuring trick.\n\nEverybody sat down and I began the trick, which was the best one in the box, involving a tall magic goblet. I didn't waste time. I borrowed Grandad's engraved gold watch which had been presented to him by the Borough of Poplar. I lowered it carefully into the goblet on to a bed of cotton wool and after a few deft passes transformed it into a half a pint of hot coffee. I had in fact improved the trick. In order to add verisimilitude I had earlier slipped a lump of yellow soap into the coffee with the result that as I was pouring it proudly from the goblet into a mug, something round and yellow slurped in with it.\n\nGrandad gave a bellow and shouted: 'Cripes! The little tyke's done it again!' \u2013 or words to that effect.\n\nSuddenly there was a lot of noise. Grandad was demanding his watch back, Daisy was shouting for a kitchen towel, Nellie was denouncing me, John was looking on the floor. Everybody was talking very loudly and urgently, while I, the only sensible person there, was fumbling for the soap in the mug of coffee in order to get it out and show that it was only soap. I recovered it successfully but, for some reason, Grandad didn't seem interested in soap and went on shouting for his watch. I could make his watch reappear but not by dismantling the trick and showing how it worked \u2013 no proper conjurer would ever do that. I consented to bring it back, but only by going through the whole trick again in reverse.\n\nSo then everybody had to sit down again and be still and quiet while I poured the soap and coffee carefully back into the goblet, placed over it the various covers and false tops, intoned the necessary incantations and finally, with a proud gesture, revealed the watch, still ticking, on its bed of cotton wool. It had been a good trick, and I, personally, thought it had gone rather well.\n\n## _II. Time Out._\n\nYou might think that settling down in a large comfortable house might have reduced Ray and Daisy's love of travel. Not a chance. We had hardly been there a year when the time came for us to pack our rucksacks and set off into the unknown.\n\nOur journey started at dead of night. We took a taxi all the way from Finchley to Millwall Docks on the River Thames. There in the darkness we stepped down on to the glistening cobbles of the quay and, clutching our bags, gingerly made our way up the gangplank towards a square of yellow light in the middle of a black shape that was the SS _Margharita_ , a small cargo ship bound for Denmark on the morning tide.\n\nI was put to bed in a bunk in a cabin made of painted iron. When I touched the metal walls they were warm and felt as if they were alive with the faint vibrations of the ship. Nothing special seemed to be happening and I may have slept. Then, much later I think, the judder of the engines starting up nearly shook me out of the bunk, and later still, after I had got used to the gentle thudding, the ship gave one blast on its hooter, a blast which sounded in my ear as if it had been blown through the pipe beside my pillow. That woke me with a jolt.\n\nI was too excited to try and sleep any more, so, as grey light had begun to be visible through the porthole, I put on some clothes and, climbing down past my sleeping brother, stepped out of the cabin. I made my way along the iron passage which served the eight or so passenger cabins, climbed a companionway, pushed open a small mahogany door and felt the sharp cold air on my face. The world outside was dark grey \u2013 uniform, uninterrupted grey. I shivered. The air didn't seem to be foggy because I could see blurred lights not far away, but it could have been misty. It was too dark to be sure.\n\nHanging on to the dewy rail I inched my way along the deck. Above me a door opened and a benign face with a peaked cap over it looked down. It was the Captain. He was on the bridge.\n\n'Yerwll ber cerld,' he said, 'erts werm erp her. Cmern erp.'\n\nI must explain that although, like most Danes, Captain J\u00f8rgensen spoke good English, he also followed their habit of using only one vowel.\n\nI accepted his invitation and joined him on the bridge. This was a pleasantly warm small summer house with large windows. The only occupants at that early hour were Captain J\u00f8rgensen and his Queagh-like cat, who was sitting on the window sill looking out at the greyness.\n\nKeeping one hand on the wheel the Captain reached out with the other and pulled a high stool to the window. 'Yerurly,' he said, 'sert dern.' I sat on the stool, put my elbows on the window sill beside the cat and we sat there, watching the morning arrive.\n\nFor no reason that I can identify, this was to be one of the happiest mornings in my short life, and the memory of it has stayed with me ever since. Perhaps this was a special time because at that moment there was nothing else I had to do and no other place where I should be. Of course I could have stayed in my bunk in the oily-smelling cabin, but if I had done so I would have missed the very gradual painting in of form and light that was the coming of the day.\n\nFirst, very slowly, the slate-grey flatness which was the water took shape. Then I began to notice that a sort of glowing was filling the air, as if it were being lit from the inside, and against it ripples and reflections began to take light and shade and become discernible as moving water. Soon, like slow magic, a dark cut-out shape with silver lines beyond it began to materialize in the distance. It seemed remote and mythical, like a fairy-tale fortress creating itself from nothing at the very edge of time. It was, I believe, a gasworks near Gravesend. As we glided slowly on other shapes began to form on the bank of the river, ethereal factories, groups of silent cranes, long flat walls with bollards and, between them, shining arms of mud and reeds swept away into emptiness as the ship passed a side river.\n\nMore scenes materialized. One by one still groups of grey houses, immobile clumps of colourless trees, flat grey fields, broken fences, and complex rusty machines as high as churches, some surrounded by pyramids of gravel, appeared from the mist. These softly offered themselves to our view and slipped away past. They were just suggestions, none of them was real. What was real was the warm rumbling of the ship's engines.\n\nSometimes we passed tall thin poles sticking up out of the water, each faithfully reflected by another pole, pointing downwards. I suddenly saw a large fruit-like object clear its way through the mist and float quite swiftly towards us. Well, no, I realized it was anchored; we were approaching it. The thing was about the size of an elephant but fat, ugly and very rusty. It seemed infinitely forlorn, alone in the dirty water.\n\n'S'berl-burj,' said the Captain. 'It'll clurng.'\n\nHe was right. After we had passed the poor thing and were leaving it behind, the ship's wash caught it and the bell-buoy did indeed go _clurng_... and then _clurng_... again; a rusty reproachful sound which did nothing but cause a handful of roosting gulls to lift off and swoop around a few times before settling again on the now-silent buoy.\n\nA small man wearing plimsolls brought me a mug of sweet tea. The cat turned up its nose at this and moved to another window but I stayed watching as a single line of gold was pencilled across the grey sea and, like a curtain, the mist slowly and graciously moved itself aside to reveal that it was a vertical wall of cloud sitting on the water under a wide, cold, very slightly blue sky.\n\nIn the sudden clear light our little ship sailed along below this tall, incongruous cliff. The estuary was now so wide that only the far horizon was edged with dark knobbly land and streaks of heavy cloud. I looked up at the cliff of cloud beside us and noticed that the very top of it was shining with pale gold light. I wondered at this for a moment and then, from straight in front of us, the sun, hoisting itself up from behind the streaky clouds, unleashed a shaft of light, a shaft which lit the whole ship and the sea and all the world around us with glorious sunshine, even reflecting it up into the ceiling of the wheelhouse, where it lit, with dappled moving light, the high cobwebby shelf where dusty bottles and packets were stored.\n\n'Smerning,' observed the Captain. 'Brerkfirst?'\n\n'Oh, yes, please!' I said. Then, realising that I hadn't been very polite, I added: 'And, er, thank you very much, sir, I hope you didn't mind me...'\n\n'Nerw,' said the Captain. 'Gerw.'\n\nNobody had told me that Copenhagen, or Kj\u00f8benhavn as I was expected to call it, was full of canals. Well, no, not full, but there were several, with little motor boats that were buses. Nor did they tell me that it was full of bicycles, very full.\n\nIn my memory I can see a wide park with railings. It is sunny and I am standing on the pavement beside a crossroads. The traffic policeman turns and puts out an arm. At once the traffic moves forward, but silently, because it consists almost entirely of persons on bicycles, respectable middle-aged persons as well as young ones. A well set-up matron, with her broad hat pinned securely on, pedals sedately past. The back wheel of her old-fashioned sit-upand-beg bicycle has a fan of strings stretched from hub to mudguard to save her skirt from tangling with the spokes. In a basket on the carrier sits a poodle. The rider looks slightly Edwardian and very ladylike, except that she is puffing a cigar.\n\nAt another time I saw a well-dressed gentleman come past riding a very superior-looking bike. He was smiling and people were touching their hats to him in a friendly way. A short way behind him came two soldiers in uniform, cycling side by side and chatting together. Later somebody told me that the gentleman was the king, but of course I didn't believe it.\n\nI liked Denmark a lot. I remember it as being a gentle, courteous place with people who were always kind and really friendly. Perhaps the local courtesy had rubbed off on us because I don't remember doing much squabbling with John. No doubt I was still very touchy about protocol and 'fairness' but I don't remember being quite as grizzly as usual. If so it must have been a relief to Ray and Daisy and I hope they enjoyed that holiday as much as I did.\n\n## _III. Conventional Education._\n\nAcademically, Woodstock School had turned out to be something of a disappointment so, late in 1936, it was decided that if I was to have a chance of passing the Entrance Examination to a secondary school I should attend a small school a couple of miles away in Friern Barnet which specialized in preparing pupils for that exam.\n\nThe school, which I shall call Crumlin House, was run by a North Country family, the Springbottoms, all of whom did their stint of teaching. They may not have been highly qualified but the curriculum was devoted solely to stocking up the minds of the Entrants, as they were called, with the facts and figures required for passing the Secondary Entrance Examination. Mr Springbottom, the headmaster, did occasionally allow himself to take a broader view and tell us that fixed-wheel bicycles were better than free-wheel ones if you were, as he was, a proper cyclist and that it's the damp muggy weather that gives you colds, not proper cold weather, wise saws that we duly wrote down and committed to memory. He also taught us the arcane language of Business Letters, which included passages like:\n\nDear Sirs,\n\nYours of the 12th ult. to hand in re of which we acknowledge safe receipt of same and...\n\nAssuring you of our best attention at all times,\n\nBeg to remain,\n\nYours faithfully\n\n... which I found quite baffling. This was mainly because Mr Springbottom never explained anything. He and the school saw their task as being to impart information and instruction, not to encourage understanding. So to the question: 'Why...?' he would answer: 'Because that's what you write.'\n\nThe school was also pretentious in a half-hearted way, with rather elaborate uniforms, a school song and long-winded speechdays which dwelt on the Spirit of the School. All this was a bit incongruous in a 'sausage-machine' crammer. I suspect it irritated my parents a bit and this may have caused them not to treat the Springbottoms with quite the respect they thought was their due. Mrs Springbottom in particular would make scathing remarks about the stuck-up children of clever-clever parents, which she made sure everybody knew were directed at me \u2013 though I hadn't the slightest idea why.\n\nThings like that weren't important. Life was real and earnest. We had to be ready, fully primed, for the momentous Examination, at which the Entrants from Crumlin House School would sweep the board and honour their Alma Mater by excelling in all ways. As the day came closer the tension rose to fever pitch. Time after time we sat mock exams using previous years' papers. I was very frightened and convinced that I would fail but, when at last we were taken to another school to sit the exam and the paper was laid before me, it had written on it:\n\nI had no idea what this had to do with the kings and queens and British Empire geography I had stuffed into my head but, as all the questions were as straightforward as that one, I nipped through the paper and waited for the exam proper to start.\n\nIt didn't start. That was it. We went home. The rest of the week was free, perhaps for us to recover from the rigours of the Exam. On the following Monday Mrs Springbottom addressed Assembly. She told us that the exam had been a great success and thanks to the valiant efforts of the school nearly all the pupils had passed. She told us that she had had letters of thanks from the parents of all the pupils, except one.\n\n'Except one!' she repeated, glaring at me... 'Postgate!'\n\nI went forward and stood before her and the school.\n\n'Well?' she demanded... 'Well?'\n\n'Er, er... thankyouverymuch.'\n\n'I should think so too!'\n\nI crept back to my place and that afternoon when I returned home I told Daisy that Mrs Springbottom had been a bit cross because she hadn't written to thank her for getting me through the exam and that I had had to say a public thankyou on her behalf.\n\nDaisy seemed a bit cross about this too, but the next morning she did give me a letter to take to Mrs Springbottom. I gave it to her at Assembly and directly it was over she ordered me to collect up my belongings and go home. There wasn't room in my satchel for all my things so she grudgingly gave me a paper bag. Although it split as soon as I was out of the school I didn't feel like going back to ask for another. I found a piece of string in the gutter and tied the things together with that.\n\nI can see, now, that the change from a formal academic examination to a simple IQ test must have come as a terrible blow to the school, but even so I don't see why they had to take it out on me \u2013 but then I don't know what was in Daisy's letter.\n\nI had passed the exam and was offered a place at Woodhouse County Secondary School in North Finchley.\n\nWoodhouse School was, I think, a quite ordinary middle-ofthe-road school, typical of England in the 1930s. It had a House system comprising four houses, Gordon, Livingstone, Nightingale and Scott, and its school motto was 'Cheerfulness with Industry'. I was a lowly member of Scott House, which was of course the best.\n\nIn general we behaved respectfully to the teachers, held up a hand when we wished to speak, attended to what was being said and did our work. In due course I found that teachers could be cheeked and would usually take it in good part, providing one was careful to preserve respect, calling them 'Sir' or 'Miss', and didn't go on too long. All of them would, if approached courteously at an appropriate moment, be willing to give a careful answer to a seriously asked question, and be quite companionable about it, particularly if the question had to do with their subject.\n\nEssentially there were two social orders in the school. The main one was the formal hierarchy, with the headmaster at the top, then the teachers and house-captains. Below them came the prefects, the school and finally us, the newcomers. There were recognized heroes in the hierarchy, mostly distinguished achievers in sport or music, whom we were expected to look up to. The other was the social life of the pupils, lived during unsupervised moments in the classrooms, in the playground, the cycle sheds and on the playing fields during the dinner-hour. As a school, Woodhouse was thought to be a bit 'rough', by which I mean that the coinage of our lives together seemed to be a sort of formalized thuggery, in which the most important thing was to be seen to be 'tough'. Very occasionally there would be real fights between potential top persons in the class but for the rest of us it was a ritual by which we confirmed our position in the pack with small arm-twistings and shoves. Girls, as always, were different. They had a social hierarchy of their own from which boys were excluded, not that we would have wanted to have anything to do with girls anyway.\n\nOf course with a complex institution like a school there were occasional failures of communication. It was as a result of one of these that I could be said to have dived for the House. One of the house-prefects came up to us in the field and said: 'Any of you rabbits dive?' So I, foolish but, as always, anxious to help, admitted that when I was on holiday I had once dived off some rocks.\n\n'Right,' he said. 'You're diving for the House,' and went away.\n\nI was proud, but only for a minute. When I read the list of awesome types of dives that I had been entered for I went, cap in hand, to the Head of House to tell him I couldn't dive and that it was all a mistake. He told me not to be modest and that it was my duty to the House to do my best. I went to the Sports master but he was too busy to talk and directed me back to the Head of House. He again refused to hear that I couldn't dive, told me not to be a ninny and said that they were all depending on me to put up a good show.\n\nThe Swimming Sports Day came. The diving was announced and duly took place. Six times I was called and six times I managed to find the courage to drop myself off the end of the board into the water, gaining for my House a total of nine points out of a possible sixty and for myself a red patch all down my front where my body had slapped the water. I was an object of ridicule and for most of my life I have been deeply ashamed of that disgrace. Only very recently have I been able to revise that view and look on the incident with something like pride. I had been ordered to dive for the House and I bloody did it. The fact that it was a complete fiasco was not my fault. I had told them it was going to be \u2013 and it was!\n\n## _IV. Mocking Providence._\n\nFor me the spring of 1938 was full of excitement; I had been promised a brand-new full-size bike for my birthday. I had pored over the coloured catalogue of bicycles for hours and, with the expert advice of John and his friends, had eventually chosen a green Raleigh with half-drop handlebars, cable brakes and a Sturmey-Archer three-speed gear. It was priced at _\u00a3_ 5 19s 6d.\n\nThis was quite a large sum of money for Ray and Daisy to fork out but it was all part of a Grand Plan which they were hatching. We were to undertake a valedictory tour of France, on bicycles. Even for Ray and Daisy, this was an enterprising idea. They were not in the bloom of youth. Ray was ponderous and overweight. Daisy was in her late forties and rheumaticky. Neither of them was in the habit of taking exercise and neither of them had been on a bicycle for decades. But, with the war-clouds looming, they both felt that this might be the last time they would see their beloved France.\n\nThat was one serious reason for the trip. The other was that the franc stood at 1,750 to the pound, a rate of exchange that was not so much advantageous as mythically bountiful. A practical difficulty arose from the fact that our parents were very short of money at the time. Although once we had got to France the living would be very cheap, getting there was going to be expensive. So, obviously, bicycles were the logical choice.\n\nBicycles were obtained, saddlebags and panniers were bought. Our cousin, Peter Thurtle, and his wife-to-be, Kitty, joined the project and Daisy made divided skirts with box-pleats for her and Kitty to wear. Ray enrolled us all in the Cyclists' Touring Club.\n\nThe start of the tour proper was a triumph worthy of Jacques Tati. We left the small hotel in La Ferte\u00b4 Alais, just south of Paris, and proceeded ' _en crocodile_ ' along a quiet wide road. At the edge of the town we came over a gentle hill and rolled down to a roundabout.\n\nRay, in the lead, put out his arm to signal his intent and rode steadily round the roundabout, to the left. John, knowing better, put his other arm out and rode steadily round it, to the right. I, confused, swung round and rode back towards the others, seeking instructions, but in vain. Kitty followed Ray to the left. Peter followed John to the right. Daisy jumped off and pushed her bike across the middle of the roundabout, shouting. I rode across behind her. The gendarme, his Gauloise hanging from his lip, stood with his hands on his hips observing the spectacle, but in the end chose not to comment.\n\nThat holiday has left me with the most glorious pictures. We rode along a set of stone bridges across a wide-spread river and in through the gates of a medieval town. Everything \u2013 the sun, the sky, the glittering of the leaves in the lines of trees, the lazy flat river rippling gently over gravel \u2013 was part of a fairy tale. Every roof, every dark shadow, every patch of sun-warmed stone in that town was full of glory and magic. I was drunk with amazement and delight and that night, when I went to bed filled with excellent dinner, I wished with all my heart that I could take the feeling of that place with me for the rest of my life.\n\nA few years later I came across the work of Alfred Sisley. He too had seen Moret-sur-Loing and had spent much of his life painting the very scenes that I had taken in that day. When I saw his paintings I jumped for joy, not so much because they were great paintings but because, for me, they were like a magic snapshot album \u2013 each painting filled to overflowing with the light and essence of places that I had already seen and loved. In fact I found out later that the whole of that holiday was being painted by the Impressionist and Post-impressionist schools. We rode along the towpaths of canals under the poplar trees and passed slow barges with washing on their lines, gently turning over the brown water as a plough turns soft earth. Camille Pissarro must have been there with his easel. The long straight Roman roads were lined with stalky plane trees leaving dabs of shadow for us to ride through, singing at the tops of our voices as the warm wind blew in our hair. There must have been half a dozen painters there. But I, being pig-ignorant, knew nothing of art. I was scooping in, through my eyes, ears and nose, the impressions that those great painters had immortalized, but I was taking them in neat, for the first time, just as they must once have done.\n\nFor John and me the riding was easy. We were both experienced cyclists, light and wiry. Each morning our bicycles were like young horses, circling the yard, eager to be away on the road. We developed a habit of whizzing on ahead, leaving the others to trundle after us at their own speed. Then we would stop at a wayside estaminet, order _vin rouge-grenadine a 'l'eau de selz_ , and wait for the elders to catch up and pay the bill, which would only be about five francs. They couldn't very well refuse such a small sum and the slight tint of rosy inebriation that was my happy condition on those mornings was soon burned away in leg-work.\n\nLunch was bought \u2013 a long loaf, some p\u00e2t\u00e9, tomatoes, cheese, wine and water \u2013 in a village, and eaten, with the bottles chilling in the shallows and the toes fingering cool sand, on the green bank of some shady stream.\n\nTowards evening we would come to the designated town. There John and I would drop back, leaving Ray, riding slow and high on his high, slow bicycle, to make his entrance and sniff the atmosphere, but circumspectly, as at dusk a bull elephant approaches a waterhole, leaving the herd to await his signal to follow.\n\nRay would come to a hotel and heave to outside it. The herd would follow and an important discussion about price and suitability would ensue. At St-Valerien, madame of the hotel came out to greet us. She spoke most highly of her establishment and intimated that if we were to take dinner as well as bed and breakfast all would be felicitous and of good price. The herd acquiesced and madame led us in, shouting orders as she did so to bring the dormant hotel to life. Small children with baskets, on foot and on bicycles, were despatched to the shops. A boiler was lit, bags were unpacked, baths were taken, and after a short cool stroll through the village we came back to find, not just dinner but a proper menu with choices. Later, the sheets were coarse but clean and I slept, as always in those times, still as a stone angel.\n\nWaking up in a French village was always the same and always surprising. The sun would be up first, trying to find a way in through the shutters. It would reflect blurred green and gold stripes on to the ceiling, which would move and change with the passing shadows. Then, as if by a signal, somebody outside would drop a bucket, _clang_ , on to the stones. A door would creak, a chain rattle, and then a shrill voice would let fly with a short complicated yell which would echo among the resonant walls. What was it, a greeting or an imprecation? Whatever it was it would be answered by more buckets being thrown and other voices from other directions giving answering yells until it seemed the whole town was out there, greeting the day.\n\nThat morning, after a French breakfast of croissants and hot chocolate we said goodbye to madame, and Ray paid the bill which, when converted, worked out at one and ninepence (9p) a head.\n\nRay, for all his atheism, was fascinated by churches. During the tour we must have seen and examined quite a number of them, but the one which has left the strongest impression with me was the Abbey at Ve\u00b4zelay. There the tops of the pillars and all the odd corners of the stonework are infested with carvings of ribald persons and unlikely creatures doing unexpected things. I noticed them and searched them out, wondering what they were up to, wondering indeed what religion was up to that it gave house-room to such manifestly disreputable denizens.\n\nIt soon became clear that we had not come to France for the architecture. Gradually, as we travelled westwards, I began to recognize the names of some of the towns and villages. I wondered why this was, and then I realized where I had seen them before \u2013 on the labels of wine bottles. Sleepy little towns like Nuits-St-Georges, Puligny-Montrachet, Volnay and Mersault were approached by Ray with the reverence of a pilgrim \u2013 which indeed he was, for in these places the nectar of the gods might still be found and taken. Here the _vin maison was premier cru_ and the 'good bottle' of the evening was the stuff of dreams.\n\nAlthough they had made the pilgrimage in a spirit of dread, fearing that the impending war might sweep it all away, Ray and Daisy were not so foolish as to allow the wine or food to be tainted with foreboding. It was a very happy time and I suspect that, if anything, the thought that he and Daisy and the rest of us might be mocking Providence simply added to Ray's delight.\n\n## _V. When War Comes._\n\nThe family returned to Finchley and the real world, a bit saddlesore but content, to find that the international situation had taken a turn for the worse. At school we no longer said: 'If war comes...'\n\nWe said: 'When war comes...'\n\nThen, at the end of September, Neville Chamberlain flew back from Munich waving the absurd document that, he said, ensured peace in our time. Herr Hitler had confirmed that the Sudetenland had been his last territorial ambition in Europe. The threat of war was said to be over.\n\nI remember the enormous feeling of euphoria that swept through us. Hearts were lighter for many days, but the underlying sense that war was still somewhere in the offing was not fully dispelled and I don't think anybody, even among the pupils of Woodhouse School, was really surprised when, in March 1939, Hitler's army marched into Prague and took the whole of Czechoslovakia.\n\nPolitics, the international situation, the prospect of war, had never seemed fully real to me, mainly because I hadn't bothered to pay any attention to them, but gradually things began to happen which brought them home.\n\nI watched Anderson shelters being delivered. These were the corrugated iron bolt-holes that householders would assemble and bury in their back gardens as a refuge from bombing. There had been an 'air raid drill' at school and I had been given a gas mask in a cardboard box, which I had tried on and didn't like. There had also been talk of evacuation; of taking children away and sending them to stay with strangers in safe places. All these things concentrated my attention. I still had the idea, somewhere in my head, that war was supposed to be exciting, even heroic, but these preparations seemed peculiarly unexciting, inconvenient and boring. They were also frightening, in a numb sort of way, because they didn't seem to be preparations for anything positive, like riding into battle, but more like preparations for some overall but unspecified disaster that would soon befall us all.\n\nRay seemed to be fairly calm about it all but Daisy was very frightened indeed. She told me that 'they' now had bombs that could wipe out half a town at one blast, and guns that could fire round corners and seek out their targets, and gas that was invisible but deadly. Her picture of the power of modern armaments was, fortunately for the world, premature; such things were yet to come. She didn't speak of it again but fears for our safety must have been in her mind because she and Ray had made plans for our evacuation.\n\nThey had met W. B. Curry, the headmaster of Dartington Hall School in Devon, and he had generously offered to take John and me as day-pupils at reduced fees if the war came. Their old friend Kay Starr was the secretary of Leonard Elmhirst, one of the owners of Dartington Hall. She offered to provide accommodation and look after us. So that was settled; the moment war came John and I were to go straight to Devon.\n\nMeanwhile we made ready for war. We stuck strips of brown paper across the windows. We upholstered the thick-walled cupboards between the kitchen and the hall with mattresses and we listened to the wireless, which told us that crowds of children were waiting at railway stations to be evacuated, carrying their gas masks in cardboard boxes.\n\nOn September the first Hitler invaded Poland and Ray said it was time for us to go. Philip and Valerie Beales (Lance and Taffy's son and daughter-in-law) offered to give us a lift in their car. They would be going to somewhere near Cirencester, which was more than halfway to Devon. The offer was gratefully accepted.\n\nIt was a nightmare journey. The main roads were packed and the drivers very agitated. The police and the air raid wardens were out in force, shouting at motorists to put their lights out, which they did, and as a result couldn't see where they were going.\n\nWe reached our destination at about 4 a.m. and slept on the floor of a mill belonging to the Beales' uncle, who had a beard. His name was Percy and he was, I think, a weaver. Then, quite late the next morning, after mugs of tea and chunks of bread and jam, John and I got on our bikes and rode together into the future.\n\nWell, we didn't exactly ride _together_. John was seventeen and a proper cyclist. I was thirteen and beginning to be a bit overweight, so he tended to be a long way in front of me, getting impatient. My rucksack was heavy and my gas mask case banged against my leg. We stayed at 'bed and breakfast' houses and at a cafe\u00b4 we heard the wireless say '... no such undertaking has been received and that consequently this country is at war with...'\n\nThen, one sunny morning, we rode up the long leafy drive to Dartington Hall, the grand medieval house and estate that was the great centre of art and progress. Entering the ancient courtyard through the arch, I felt as if I was stepping into a film about knights in armour and maidens locked in towers. I expected a richly caparisoned white horse to ride up to us, bearing a bearded serjeant-at-arms on its broad back. In fact a young man in corduroy trousers and sandals came and asked if he could help us. He directed us to Kay Starr's office, which was through the Great Hall.\n\nThe Great Hall was amazing. It was huge and echoey but bright with sunlight flooding in through the tall windows. The walls were hung with bright banners and the floor strewn with rush mats. It smelled sweetly of beeswax and flowers, and had clearly just been made ready for the arrival of a king.\n\nWe knocked on a door in the far corner. A voice said, 'Come in.' We stepped down into Kay Starr's office: a tall rather embarrassed-looking youth and a shorter, fattish one, who grinned, both quite grimy and dishevelled.\n\nKay was a tall, tidy, tweedy stick of a lady, very upright, clean and exact. I shall never forget the look on her face when she saw us. It was dismay, but a dismay so intense and overwhelming that for a few seconds she couldn't speak.\n\nThen she said: 'Oh, so you've arrived, have you?'\n\n'Yes,' I replied, still grinning.\n\n## _VI. Progressive Education._\n\nWhen I told people I might be going to Dartington Hall School they would sometimes go 'Ooooh!' and look at me coyly, as if there was something slightly risqu\u00e9 about the idea, because Dartington Hall was what was called a 'progressive' school. They had heard tell, they said, that there was no uniform and people sometimes went about with nothing on! From some I heard that there were no rules and no proper classes, that the pupils were in charge of the staff and there was fun and freedom all the time. From others I heard that it was licentious, and surely a sink of sin. Everything that was said about the school was charged with strong feeling. It may seem a bit odd today but in 1939 the very idea of educating people in an unconventional way was slightly outrageous.\n\nNow it was the first day of term and as I rode down the hill towards the school I realised that I had absolutely no idea what I was going to find when I arrived. It might of course be a quite ordinary place like Woodhouse School, with uniforms, prefects and an Assembly. I think I was rather hoping it would be.\n\nI couldn't see anywhere to put my bike so I leaned it against the wall, opened the front door and looked in.\n\nI heard a lot of noise and saw a lot of people milling about. They weren't all children, some of them seemed to be grown-ups but none of these were wearing academic gowns. In fact they were dressed in all sorts of different clothes and were just chattering away together and looking at notices.\n\nI stepped in and stood just inside the door. People smiled at me as they brushed past but didn't say anything. After a few minutes of standing about I realised that nobody was coming to find me, so I had better find somebody myself. I moved along a corridor to the right and saw a door marked 'Office'. I knocked but nobody answered. So in the end I pushed the door open and walked in. A lady was sitting at a typewriter.\n\n'Hallo,' she said, 'I'm Sarah.'\n\n'Postgate,' I said, 'Oliver Postgate. I am here.'\n\n'Yes, I can see. What can I do for you?'\n\nI'm sorry. I don't know what to do.'\n\nSarah sighed with resignation, arose and led me back into the foyer. There she pointed to a gigantic timetable. 'Look,' she said, 'you're in C Group. In half an hour you have Art.'\n\n'Yes, but what do I do?'\n\n'Why, whatever you like! You are free, absolutely free! Aren't you lucky!' She lifted her arms in joyful freedom and swirled away, back to her office.\n\nI stood there, stunned. I looked at the board. I turned and looked at the doors. I looked at the people milling busily about, oblivious of my existence. I have never felt quite so lonely and baffled in my life.\n\nHow long I stood there I don't know. I remember finding a man looking out of a window and asking him if he knew where Art was. He must have told me because I came at last to the Art Room. It was deserted. I sat at a table.\n\nAfter about twenty minutes a large girl in trousers walked in. She dumped her things on a table and said: ' 'Lo, who're you?'\n\n'Postgate,' I said.\n\n'What?'\n\n'Postgate.'\n\n'Poached egg? You a poached egg?'\n\n'No...'\n\nFour or five more people came in, all dressed differently; then a thin young man wearing a brown jersey.\n\n'Hallo,' he said in quite a friendly tone, 'I'm Mark. Who are you?'\n\n'He's a poached egg!' shouted the girl.\n\n'No, I'm not,' I whispered.\n\nThe Art class didn't seem to begin at any particular moment. Some people had rummaged around and found paper and powderpaints but the rest just chatted. The teacher chatted with them. Nobody spoke to me so I found a piece of paper and a crayon and drew a fish. The class must have ended at some point because everybody left.\n\nFor the rest of the day I kept my head down and did my best to follow on behind the others. I clung on to the hope that somebody official would come along to acknowledge my existence, but it didn't happen.\n\nLate in the afternoon I noticed that people were going into the dining room to eat. I wondered whether I was supposed to join them. I didn't know who to ask so I went to the Office to ask the typing lady if it would be all right to go home. The Office was locked, so, feeling cold with misery, I got on my bike and rode back to Kay Starr's house.\n\nJohn had had a different but equally daunting time. Notice had been taken of him and he had been told to wait for some sort of 'tutor', who simply didn't turn up. So where I was feeling baffled and lost, he was healthily angry. He noted in his diary that it was a silly school and he asked to be taken away from it as soon as possible. Very soon he was staying with our uncle Richmond in Exeter and going to a school there.\n\n*\n\nMy first day at Dartington Hall School had not only been very different from my first day at Woodhouse School, it had been quite different from anything I had been able to imagine. In all my conjectures about what Dartington might be like I had always wondered what sort of _school_ it would be, how we, the pupils, would be controlled and directed. It had never occurred to me that there mightn't be a _school_ there at all, that I would have no desk, no place to go, and that nobody would tell me what to do next. But that's how it was.\n\nFrom the moment I had arrived at Woodhouse School, I had been fully involved in the organized life of the school and had felt quite secure in the place I held in that benevolent authoritarian hierarchy. At Dartington there seemed to be no benevolent authority, nobody to notice what I did or didn't do. I might as well not have existed.\n\nAlthough I didn't realise it at the time, I was completely desperate, frantically trying to find a place for myself in a social order that didn't in fact exist. In this I was of course unsuccessful but my behaviour must have been observed, because it was mentioned in my first report:\n\nOliver... started off by showing off in ways that did not endear himself to either children or staff... both in class and in his written work his chief motive seems to be to gain admiring attention.\n\nRay passed that report to me with a contemptuous gesture and I remember it hitting me like a body-blow, not so much because it was so casually damning as because it was the _only_ comment that had been made. During the term I had been given no clue as to whether or not my behaviour was acceptable. I had spent my time in a sort of ongoing panic because I hadn't been able to get anybody to _acknowledge_ my existence, let alone _admire it._\n\nLooking back, I can get some really embarrassing pictures of the obnoxious way I carried on in that first term. During the first weeks I think I went in for a bit of the formal thuggery and armtwisting that had been customary at Woodhouse. That simply didn't work. I remember trying to be funny in class. That was a waste of time. Answering back at the teachers provided no thrills and was just ignored. I can see myself clowning and nobody taking any notice. I pretended to be clever and know everything but all that brought was mockery \u2013 all these were the standard reactions to a smart-arsed git trying to grab attention. That was fair enough but it was the response, or rather the lack of response, of the teachers to this carry-on that I found completely incomprehensible. At Woodhouse they wouldn't have put up with it. I would have been put in my place in no time (and so would have known my place). Here they took no notice at all. They suffered the inconvenience and looked away. Needless to say, this discouragement simply caused me to redouble my efforts.\n\nIn contrast to this rather frantic school life my current home life was very quiet, if not exactly homely. Kay Starr was, I think, kindly disposed towards me, but I knew that she felt very keenly the intrusion and disruption that my presence caused in her life. She was also, in herself, a righteous person, very conscious of the shortcomings of the rest of the world, which she did not hesitate to condemn.\n\nIn the evenings Kay would often say: 'Don't let's bother with proper supper, let's just have some scrambled egg on our knees.' Then there would be small pieces of thin toast covered with reconstituted dried egg and perhaps an apple to follow. I would sit very carefully and eat it neatly and when I had finished she would look at me anxiously and her face would tell me that she couldn't think what she was going to do with me for the rest of the long evening. Neither could I.\n\nAt weekends I was, as Kay's prot\u00e9g\u00e9, allowed to partake of her life in Dartington Hall itself. This was a completely different world from the school. This was an essentially civilized, artistically sensitive foundation which had been created by Mr and Mrs Elmhirst to provide a centre of excellence, where individuals and groups engaged in the arts could be encouraged and supported in their work. There were painters, musicians, sculptors, silversmiths, even a ballet company, all busy working away at their arts, sustained by the loving generosity of Leonard and Dorothy Elmhirst.\n\nThe collection of people drawn to that cultural magnet was amazingly varied and interesting. The atmosphere seemed to be very free and easy but Kay, being perhaps conscious of her position as Leonard Elmhirst's private secretary, was at pains to emphasize the need for me to be as unobtrusive as possible. So I walked on tiptoe through that part of Kay's world, as I tiptoed through every other part of her neat house and her neat life, because I was steeped in long-term apology for being there and being a nuisance and because I was totally intimidated by her, even though she had told me, quite firmly, that we were close friends.\n\nGetting used to life in Kay's cold house was not easy, but it was my life at school that was causing me the most bafflement.\n\nLooking back I can see that what I was having trouble with was the freedom. Freedom was assumed to be the aim and joy of life, and I was sure it was, but I didn't seem to be very good at it. Freedom was presenting itself to me as a succession of choices that I had to make. These were not just choices between set alternatives like: 'Will you have jam or marmalade?' These were more basic choices like: 'Do I want to bother to get out of this chair?' or 'Do I want to find out what the next class is and go to it?'\n\nThis unsettling responsibility was complicated by the fact that I truly believed that I was _supposed_ to make decisions that reflected my fancy. If I asked somebody which option I should take, I was asked in reply: 'Well, which do you _want_ to take?' \u2013 a question to which I had no answer, because I had no fancy and no purpose of my own. Today I can see quite clearly that all I really wanted was somebody to be there and mind enough to tell me what to do \u2013 and that, alas, was not on the menu.\n\nSo I think I just soldiered on and, as children and chameleons do, began to take on the colour of my surroundings. I started to affect the arrogant unconcern that I imagined was the convention, disdained the timetable and tried to follow what I hoped was my fancy. I must emphasize that I was not doing this as a real affirmation of personal independence. Quite the contrary, I was simply doing my best to conform, to do what I imagined was expected of me.\n\nAbsurd as it may seem, I think that may have worked. It seems that I did become what was (for Dartington Hall School) a socially acceptable person, because my next report said that I had _'settled_ _down'_ , was _'becoming more popular '_ and that my time was _'for_ _the most_ _part being profitably occupied'._\n\nThere was some truth in the last of these observations because I found a microscope in the biology lab and became fascinated by the underwater life that seemed to appear by itself in ponds and puddles when they became stagnant. I would cycle around the estate collecting smelly jars of it and bring them back to the biology lab. I sometimes think that could have become 'my subject' if I had managed, or even been encouraged, to organize my studies a bit, but that didn't happen. At one time I became very interested in pottery and spent some months doing very little else, but here again my activity was quite solitary, random and disorganized. For a few months a craze for small theatricals swept through the school. At another time people collected pieces of tubing and blew bullets made of modelling-clay at each other.\n\nThese peripheral activities were engrossing enough to ensure that the academic side of school life was neglected. I was supposed to spend half an hour each week with my tutor, Boris, to talk about my work. He was the Chemistry master and, like me, an inveterate small-joke-maker. At our tutorial meetings he was icily polite and punctilious but I'm sorry to say he was unable to conceal from me the fact that he disliked me intensely and found the task of discussing my educational future distasteful. I didn't care a lot for it myself because I, being fifteen years old, had found other interests, so I tended to miss tutorials. When this happened Boris would remind me of the possible academic consequences of neglecting my studies, but he did so more in resignation than in hope.\n\nThe other interest I had found was not pottery, pond-life or play-acting. It was sex, or rather, the distant prospect of sex.\n\nMany people tended to assume that because Dartington Hall School was 'progressive' and free of restraints, the life of the pupils would inevitably be an ongoing orgy of sexual indulgence. That wasn't true.\n\nQuite a lot of fairly heavy petting went on but, as far as I remember, not much else.\n\nThe general policy of the school seemed to be one of noninterference. There were no organized house-based activities, so the pupils were mostly left to stew in their juice, which probably explains why the social life was so obsessively sex-centred. Looking back I can see that although it may all have been a bit overwrought, I don't think it was unhealthy, just children growing up.\n\nMy position in this social life was marginal. Being a day-pupil meant that I was always a sort of visitor in the boarders' houses. This made me slightly uncomfortable. Also I was still very much in awe of girls and knew that if they were not treated carefully they could give you a nasty bite, though their bites were of scorn and derision.\n\nAt Dartington, girls began to take on a new light. They became objects of desire and they glowed with thrilling but unspecific allure. I became infatuated with girls in general but I was also in love with one girl in particular. This girl shone in my sky like the full moon. Every ordinary movement she made was steeped in sensual magic and as I watched her I knew that my sole aim in life was to approach her and be as near to her as possible. My infatuation was made all the more intense by the fact that I had no clear idea where my feelings were supposed to lead. I had hardly reached puberty and, although the rabbits had taught me that the sole purpose of courtship was sexual intercourse, the very idea of having to do anything so ungainly seemed to me quite unthinkable and I quickly put such thoughts from my mind. Doing this did not diminish my ardour, it allowed it to be pure, which was a relief. It also allowed it to be private, something which nobody else knew about. Thus it came as a shock to discover that everybody, including the beloved, knew all about my feelings, and that her very natural efforts to avoid my cowlike gaze were being aided and abetted by most of her friends, who were watching the farcical carry-on with derisive amusement.\n\nThis was a disappointing introduction to romance, but not surprising. The idea that a girl might value my attention and be pleased to receive it had simply never crossed my mind. I knew for certain that girls were by nature always hostile to advances and I assumed that my role as a suitor could only be that of the plaintive lovelorn swain, ever hopeful but always unworthy. The assumption may sometimes have been accurate, but it was never attractive.\n\n## _VII. The Home Front._\n\nReturning home from school for the Christmas holiday in 1939 was a happy anticlimax. In September we had fled, as refugees, from the terrors of the war that was to come. The war had come and had brought no terrors, at least not to 45 Hendon Lane in Finchley. Like Mole in _The Wind in the Willows_ I returned to my home and was deeply moved to find it miraculously unchanged, exactly as I had left it. The windows were still striped with brown paper and at night fully covered by heavy blackout pads made of cardboard and black cloth.\n\nRation books had arrived but were not yet to be used. We looked at them closely and tried to divine what amounts of the various foods we might be going to receive but to no avail. Otherwise there was not much war to be seen. The only bombs that went off in London were put there by the IRA. So I painted huge frogs and Chinese dragons on the blackout pads and settled down to enjoy Christmas.\n\nAlthough the house was unchanged, the atmosphere was quite different. Dread had gone, or been shelved for the moment, and there was gaiety about. War had not, after all, wiped everything away and it was good to be alive and able to enjoy ordinary things.\n\nOnly at dusk, as darkness fell and the blackout was complete, did a sense of menace return. Then no man-made light could be shown, in case the German eyes lurking in the sky should see it. Even the air-raid warden's torch was sparingly used \u2013 flashed on to see something and then off again at once. There were very few cars about and their headlamps were intricately shielded so as to shine a very little light down on to the road. Without any streetlights or house lights the streets were in a world separate from people, a place where everything was still and sharp-edged. Even when it was absolutely black, at times when there was no moon, the stars were clouded and I was truly blind, I discovered that I could tell where I was by interpreting the subtle echoes of my own footsteps. I had first noticed this faculty at Dartington, where every night I walked back from Foxhole over the hill to Kay's house near the Hall. I had found that on pitch-black nights I could walk back almost as briskly as on clear nights, so long as I didn't think about it too much and remembered to sing.\n\nIn the spring of 1940 I heard that Grandad was very seriously ill and, in May, that he had died. I would have liked to go to his funeral but for some reason this wasn't possible, perhaps Dartington was too far from London, I don't know.\n\nDartington was also a long way from the war. Kay and I listened to the news on the wireless and saw the maps in the newspapers, all covered with arrows, but the reality of the war only really came home to me when I returned to Finchley in the summer of 1940.\n\nThere the sky was very noisy and was often decorated with circles of smoke as the aeroplanes fought out the Battle of Britain. The air-raid sirens sang at all hours and from time to time the earth would shudder underfoot and a heavy _crump_ would be heard. The kitchen at 45 Hendon Lane was dark because the windows were heavily sandbagged. The cellar had been shored up to make an air-raid shelter and a long escape-way had been dug under the long room to a trapdoor in the garden. As ration books were now in full use and food was short, the honorary post of head of household had fallen to Daisy. She held the books and took to the task of feeding the flock in the midst of shortages with a mixture of enthusiasm and dismay.\n\nWith her father's death an era of great activity in Daisy's life had suddenly come to an end, so I think she found this challenge to her very considerable skill as an organizer quite rewarding. Her originality and resourcefulness as a cook was certainly a great joy to the rest of us.\n\nI returned to Dartington at the end of the summer holiday fairly confident that our lives were charmed and that although war would probably be interesting, it was unlikely to be dangerous. I was wrong about that. A couple of days later a bomb fell in Hendon Lane, blowing out all the windows and damaging the house.\n\nI came back to Finchley for the Christmas holidays of 1940 to find that life there was different. It had become very busy. Daisy was now fully in charge of what was, in effect, an extended-family boarding house. Kate (who used to be called Kitty), lived there, as did Peter when he was on leave from the RAF, also our uncle Eric with his wife Emily and their three boys. Daisy's younger sister Nellie had gone back to her little flat in Gray's Inn after a nightshift and had found it not there, just a hole and a heap of charred things, so she was in residence. John was back at his old school at Kingsbury, studying for Oxford Entrance, and I was back from Devon.\n\nThe house itself had changed quite a lot. The window glass had been replaced by a sort of oiled buckram which let in some light but you couldn't see through it. The cellars had become two bedrooms with camp beds. Even Matilda the cat had given up nocturnal outings and spent her nights lying on Daisy's feet.\n\nOur night-time procedure was fairly standardized. John and I would go to bed in our own bedrooms as usual but as soon as the air-raid siren went or, when I became more blase\u00b4, as soon as I heard the sound of bombers or gunfire, I would jump out of bed, roll up the bedding like a Swiss roll with the pillow in the middle, pick it up and prance lightly down the stairs into the cellarage, where I would unroll the still warm bed on a camp bed and climb in. Once I pranced very lightly indeed because a bomb fell quite near and I took the last flight of eighteen stairs at a single bound and landed squarely on the bedding. I didn't hang about to check for broken bones, I was down and unrolled and had my head under the covers in a matter of seconds.\n\nI was frightened by the bombing, but so was everybody else, and as it was quite obvious that there was nothing to be done about it, people just went on with life as usual and hoped one wouldn't drop too close. I was also slightly thrilled by the danger and excitement of it all. This was a bit childish I suppose and the feeling would probably have evaporated if I had been really close to a disaster and seen people killed or lying dead, but, perhaps fortunately, I was lucky enough to miss that experience. I did see other unnerving sights, like bombed houses with their insides on the outsides. I remember looking up and seeing flowered wallpaper and tiled fireplaces on the outsides of walls four storeys up. One fireplace had a picture of a stag over it and the ornaments were still in place. Above it a large bath hovered, hanging on its pipes in midair, like a hippopotamus diving off a cliff.\n\nTo come home on a late tube-train through the London stations with their lines of steel bunks was a strange and moving experience. Each family brought small living-items with them: cushions, bedding, books, torches, cans of water, bowls, all personal bedroom things, and made their bunks into a piece of home. Coming alongside and waiting quietly, I was like a ghost passing through their private lives. I felt enormous affection and care for them and prayed silently for their safety, but I was also glad they took no more notice of my presence than they did of the bright trains that roared in and out, stopping less than a yard from their beds.\n\nOne night we heard the distant sound of many planes and gunfire. We went outside into a world lit by what seemed to be a sunset in the south-east. On the horizon the distant houses and trees were silhouetted black against a sky glowing golden red and flickering softly. That was the city of London being burned by incendiary bombs.\n\nThe war moved eastward. Hitler postponed his plans to invade Britain and started on the second part of his great war plan: the invasion of Russia.\n\nI was sixteen and the School Certificate Exams were looming, but even so I remained fairly uninterested in academic study. I had already found out, by reading my reports, that I was not thought to be ' _higher education material_ ' and that an art or craft career was recommended. Simply in order to get shot of the task of choosing, I settled on stage design, which I thought might interest me if I was ever going to have a career.\n\nThe school had no facilities to teach me stage design and no wish to try. The Art teacher had already dismissed my efforts in Art as ' _showing a lack of clear thinking_ ', and being ' _aimless work that_ _could be described as a slow disintegration of an idea into a pleasant fac\u00b8ade'._ This was a valid comment which might have been very useful if it had been made to me at the time, but it wasn't. As usual it was just sent to my parents in an end-of-term report.\n\nSince then I have often wondered about this phenomenon: the contrast between the outspoken and generally damning reports sent to my parents and the carefully noncommittal and evasive attitude of the teachers. Some years later I had an opportunity to talk about this with an ex-member of the school staff, one who was, incidentally, the author of one or two of my reports. He explained that, because there were no rules or any discipline to support them, the teachers were inevitably vulnerable and at a disadvantage in their dealings with their less-than-endearing pupils. These could be as foul as they liked without any fear of reprisal, while the teachers, bound by the theory-driven education method, could not answer back. This situation did not allow them to be spontaneous in their responses to their pupils, nor did it encourage them to form genuine relationships with them. Luckily, he explained, they didn't have to. Freedom was seen as being the 'cure-all'. So long as the pupil had freedom, the teacher could look away, remain cool and detached, and leave them to their fate, saving their real comments for the parents' reports.\n\nI could understand that, even sympathize with it, and it went some way to explaining why, at Dartington, I had sometimes felt I was in a zoo rather than a school. In the light of it I now cherish the thought that, although at the time I had slunk shamefully away, I did, once, cause W. B. Curry, the headmaster, to lose his cool.\n\nCurry gave evening seminars in Philosophy for interested pupils at his house. At one of these he told us that we would be discussing 'good'.\n\n'Good!' I said enthusiastically. 'Good what?'\n\n'What d'you mean, good what?'\n\n'That's what I mean,' I answered. 'What are you going to talk about that is good?'\n\n'No,' he explained. 'Not good anything... The Good. Good as a quality in its own right.'\n\nW. B. Curry went on to discourse on Plato's Essences, on the nature of Good and Truth and Beauty, and all the time I found myself becoming more and more agitated and, no doubt, huffing and puffing and tearing my hair. In the end Curry turned to me and said: 'All right, then. What's biting _you?_ '\n\nI said: 'I'm sorry, but it doesn't work.'\n\n'What do you mean?'\n\n'I don't think what we are talking about is there. I mean, I don't think I can think about Good as if it were a thing because I don't _think_ it is.'\n\n'Well, what _do_ you think it is?'\n\n'I don't know. I don't see how a quality can exist all on its own.\n\nI think it's the name of an adjective, an aspect of whatever it is describing, an observation _about_ something.'\n\n'Is that really what you think?' asked the headmaster.\n\n'Yes, I think so.'\n\n'Right, then!' he snapped, his eyes blazing. ' _FUCK OFF_ _!_ '\n\n1. If A comes before P in the alphabet, put a cross in the square marked Y.\n\n# STIRRING IT UP\n\n## _I_ _. Growing Pains._\n\nBack in London in the spring of 1942 I noticed that the bomb sites were a riot of colour. A flower, I think it was willowherb, had invaded the tops of the broken walls, coated the heaps and was springing out from between the floors where the torn wallpapers flapped, softening the hard edges and making everything part of a garden.\n\nI had left Dartington having, by some miracle, passed my School Certificate Exams. Well, it wasn't quite a miracle. A friend and I had spent the last fortnight before the exams in the school library going through the whole syllabus. After they were over I had joined the Home Guard \u2013 a gesture that had caused much amusement at the school. When I returned to Finchley I was assigned to what was known as the 'Offensive Section', though a milder band of shopkeepers it would be hard to find.\n\nI got on well with the Offensive Section but in myself I was feeling glum and uncomfortable. Well, worse than that \u2013 I was angry. I don't think I was angry with any particular person, not all the time anyway, I think it was more that I was violently discontented.\n\nIf there was a rationale to what was happening I think it was that I was slowly becoming aware of myself. Throughout my early life I had been glad enough to be allowed just to be there. I had been innocently happy when loved, innocently miserable when oppressed, bearing no grudges and keeping no records. I hadn't questioned my place at the bottom of the pile and had grown into, and later even played into, the picture of myself that I had got back from the others \u2013 that of the clever\/stupid, facetious showoff who had to be kept down.\n\nThere was a very simple reason for this. Burned right into my bones was the basic knowledge, learned perhaps at my brother's knee, that anything I said or thought was automatically, solely by virtue of the fact that it was _I_ who was saying or thinking it, certain to be stupid and therefore unacceptable. As a result I spent my time frantically searching for something to say that would be _more_ interesting or _more_ funny than what I would ordinarily have said, because only that would stand a chance of being worth listening to.\n\nThis is a fairly commonplace condition, one felt in some degree, I'm sure, by most of the human race. The trouble was that I'd had enough of it. I'd had enough of being at the bottom of the pile. I had had enough of being sneered at and enough of playing the idiot.\n\nOf course my unfortunate parents weren't really sneering at me, not on purpose anyway. I think they were actually a bit nervous of me. Daisy hadn't much liked the effect that Dartington had been having on me and her manner towards me had become guarded and slightly defensive. She would say significant things like: 'I don't expect going up to the shop to get me some carrots would be the sort of thing that would interest you.'\n\nThese were small, very forgettable, barbs which up to then I hadn't bothered to notice. Now they really got to me. I became touchy and darkly explosive. I noticed again that my Aunt Nellie (who in her day had been firmly installed at the bottom of the pile in George Lansbury's family) possessed an absolutely reliable reflex action. The family would all be at table, talking. If, at a lull in the conversation, I should start to say something, she would recognize this as a signal that the floor was clear, and cut straight into the middle of my first sentence. She did this every single bloody time and, as far as I ever knew, nobody (but me) even noticed. But I had been, and still was, too half-baked to object.\n\nNeedless to say the joker that hung over everything at that time, making nonsense of everybody's personal purposes, mine included, was the war. In about a year, when I reached the age of eighteen, I would be due for call-up. So there really wasn't any point in doing anything connected with a long-term future, because even if I was going to have one \u2013 which was by no means certain \u2013 there was no telling what it might be.\n\nStill hanging on to the possibility that I might one day become a stage designer, I enrolled at the Kingston College of Art.\n\nThis meant that I had to travel daily from Finchley to Kingston, by tube, Southern Railway and number 65 bus. This journey, an hour and a half each way, could have been boring and a burden, but I remember it as a delight. It was time out of life, time in which there was nothing I had to do but sit there and let the train carry me on. Sometimes I read, but mostly I just sat and thought. Since those days I have looked forward to long train journeys just because, as the world slides busily past outside, one can be alone inside one's head and free to roam.\n\nGoodness knows what I was thinking about on those rumbling journeys but the process of thinking was itself a great joy. I suppose I was looking for, finding and maybe fitting together, the pieces of that immense, unfinishable 3D jigsaw puzzle of the spirit, called _How things_ _a_ _re_. My mind would go ranging away, swinging from branch to branch, yodelling like Tarzan through marvellous forests of huge ideas and wide vistas of glorious realisation. Nowadays, alas, I usually go to sleep.\n\nKingston College of Art was run as a college in that the students were regarded as being adults, but I think many of the young people who started with me in the Foundation Class must have come from quite authoritarian schools, because they appeared to be absolutely in awe of the teachers and took the comments and advice they received as direct instructions, to be followed exactly. I, on the other hand, had no automatic respect for the teachers' comments and didn't know I was supposed to have. I regarded what they had to say as a contribution to my own thinking about what I was doing and acted accordingly. Some members of the staff didn't seem to mind this, others seemed a bit put out. But, unexpectedly, my fellow students hated it. They were outraged, not so much by the fact that I had the temerity to argue with or even disregard the teacher's instructions, as by the fact that doing so didn't seem to require any daring on my part. To them the very thought of rebelling against the teacher's authority called for outstanding courage. My being able to do just that, apparently quite casually, showed that I either possessed superhuman courage or was some sort of monster, so grossly arrogant and conceited that I had no grasp of the situation.\n\nThe latter view seemed to be the one that prevailed, the one that caused them to flick ink-pellets at me when the teacher was out of the room and, once or twice, to scrawl nasties on my drawing when I wasn't there.\n\nThat was the first really clear signal the outside world had given me that there was something about me that was seriously unacceptable. It wasn't just that I didn't show proper respect to the teachers. It was more fundamental than that. I could sense that there was something in the way I walked, in the way I spoke, in the way I looked, which instantly proclaimed that I was a bit 'off'; that I was peculiar. But I had no idea what it was. I found this very disturbing, so much so that I mentioned it to Daisy, who said something like: 'Well what do you expect? It's all that stuff you picked up at Dartington, isn't it?'\n\nOh. Was it? I didn't know what she was talking about. I was still convinced that Dartington was irreproachable. I had had the privilege of receiving its enlightened form of education. The idea that the effects of this could be anything but wholly beneficial was unthinkable.\n\nHowever, the hostile signals I was receiving were so clear that I had to consider the possibility that something might have been wrong with, or missing from, my education. That set the seed in my mind, but it has taken me many years to get a clear picture of the after-effects of my time at Dartington.\n\nI had left the school in 1942, happily assuming that my personal acceptability in the social circles in which I would find myself would be decided by the relevance of my thinking and the value of my opinions and contributions. I thought, in fact I knew, that my intentions were friendly. I was careful to be, in my fashion, courteous. I liked to be helpful and was glad to contribute useful ideas to any discussion and would occasionally go out of my way to tell people useful things \u2013 like, for instance, how to do their jobs.\n\nIn fact my happy assumption could not have been further from the truth. Acceptance had to be earned. Without knowing it I had been let out into a world which, in 1942, was far more rigid and intolerant than it is today, a world in which whether or not I was going to be allowed into the pack was going to be decided by how diligently I observed certain infinitely delicate nuances of behaviour with which I was supposed to signal my appreciation of the worth and position of my betters and thereby, in their charity, be accorded my place.\n\nThis was a world which any well brought-up wolf or chimpanzee would, by virtue of the early conditioning of its pack, recognize without hesitation, but, unlike Woodhouse School, Dartington had no hierarchy in which one could learn to 'know one's place', and consequently that essential piece of education had been missed out.\n\nW. B. Curry, the headmaster of Dartington, once described the education he purveyed as 'education for discontent', and as far as I was concerned he was right about that. Dartington not only omitted to teach me any manners, it gave me precious little proper education. It also, and this is something I have reason to resent to this day, _gave me no practice or training in doing things I don't_ _want to do_ _!_ This I believe to be one of the basic purposes of education because, after all, one spends most of one's life doing things one doesn't really want to do simply in order to clear the space to do what one _does want to do_. Dartington ignored this as part of a deliberate, theory-led policy, and it did so without asking me or telling me. So I had no way of knowing what was going on.\n\nNo doubt I am being unjust as well as tiresome in my comments. I have friends who were there with me who felt, and still feel, that the time they spent there was valuable and idyllic, and that (although perhaps they are too courteous to put it so crudely) if it wasn't like that for me it was probably my own fault. I am sure there is some truth in that and I dare say many of the difficulties I experienced there arose from regrettable characteristics that were probably well formed by the time I arrived there. Nevertheless, at a different school I might have learned or been taught my way out of them, but, apart from sending sneering reports to my parents, Dartington Hall School took no interest in them, or in me.\n\nHowever peculiar my life at school may have been, and however unwelcome and uncomfortable I may have been feeling at the College of Art, they were as nothing compared to the shemozzle that was going on at home.\n\nDisdaining the family rooms, I had made a kingdom of my bedroom in which I spent my time frantically drawing and painting pictures. This was a waste of time because in the house I was too overwrought to think properly about what I was doing. I declined to show the pictures to my parents and threw them away. I remember Ray observing that I threw them away because: '... they are unworthy of his genius!' That was probably meant as a joke and, perhaps because it was a joke, it stung, and made me frantically hostile.\n\nBut I can also see that I wasn't really feeling hostile towards them as they were at that moment, I was trying to get away from something else, a collection of ghosts that, over the years, had occupied the inside of my head and had made me what I was. Ray was in there, with his back to me. John was there too, flapping me disdainfully aside and looking over my head. Kay Starr was there, tall and aghast. And Daisy was very much there, her heavy Lansbury eyes drilling rays of loving anxiety that could cripple at fifty yards.\n\nMy behaviour was fearsomely overwrought, an effect, no doubt, of the quantities of new hormones that were pounding through my seventeen-year-old veins. I suspect that, had I but known it, what I most needed to do was say I was sorry, and maybe ask something like a blessing for the changes that had to happen.\n\nBut that would have taken a rare courage. Too bad we have to do our growing up when we are young.\n\n## _II. Playing the Joker._\n\nThe number 65 bus was my salvation. Faithfully every morning I had boarded it at Surbiton Station and climbed to the front seat of the upper deck while it took me along the accustomed route, past houses and shops, to the stop by the bridge over a small dirty river called the Hogsmill, a couple of hundred yards from the undistinguished block that was the Kingston-upon-Thames College of Art. There I would get off.\n\nOnly one day I didn't. I looked across out of the nearside window and saw the College waiting there with its Foundation Class beating out mock-jazz on the table tops, and Mr Kisby, for whom we had to draw tiny squares of a million fine lines to make patterns for pin-stripe shirt material, and I said: 'Er... no. Not today, thank you. Today is different.'\n\nThat said, I sat back in the captain's seat and looked out at the world ahead. The engine revved up, the gears clanked. The bus began to move. Two trumpeters beside the portal of the County Court blew a long fanfare in our honour, and the day began.\n\nI can't tell you what a glorious lifting of the spirit came from that small decision. For a moment the ghosts of conformity peered reproachfully up at me, but I was firm. I said: 'No, I am not playing truant... all right, yes, I am, but I am not running away. I am a free person and I have just decided not to go to College today. I may decide to go there tomorrow, in fact I'm sure I will, but today... today I shall go to...' By peering through a tiny oval window in the top of the destination-box I could just see the word. '... Leatherhead. I wonder where Leatherhead is.'\n\nThe number 65 was an enterprising bus. It didn't just pootle around the suburbs. Once it was clear of Surbiton it stretched its wheels and headed out into the country. The houses dropped away behind and the road went on through open common-land. One by one the passengers got off and when all the fares were taken the conductor came up and sat on the other front seat across the aisle.\n\n'S'all right down this end,' he said.\n\n'Yeh,' tis,' I replied.\n\nIt certainly was. In fact it was marvellous. It was the eighteenth of February 1943, with the sun heading up the southern sky like midsummer and the air as clear as ice.\n\nLeatherhead turned out to be a proper small country town, neat and compact. I left the bus in its yard and wondered what to do.\n\nMy legs told me they wanted to walk.\n\n'OK,' I said, 'which way?'\n\nThey said: 'Towards the sun.'\n\n'All right, then. Get on with it!'\n\nOh! What it was to have seventeen-year-old legs. My goodness, they were full of life! We strode off towards the sun into a piece of country the like of which I had never imagined.\n\nTo the right lay a broad meadow below which a small river accompanied the road \u2013 well no, actually it was flowing the other way. Beside it was a modest railway line and behind that rose a broad sweep of brown field in which the young green corn was beginning to sprout. Just a field it was, but its extremities were edged with tall leafless trees, their twiggy outlines spread sharply against the sky. Among them jackdaws, black as commas on a page, swooped and played, clunking conversationally to each other.\n\nTo the left the road led towards steep wooded chalk hills, and over the top of it all arched a bright blue sky, singing with larks.\n\nThe February air was cold and in dark places under the trees and hedgerows a faint bloom of white told of last night's frost \u2013 there's nothing in the telling of this that could impart the huge sense of freedom and escape that I and my legs felt that morning.\n\nI said: 'It'll be a long walk.'\n\nThey replied: 'Just leave it to us. Leave it to us, and sing.'\n\nSo I sang, and on they went, rejoicing.\n\nThe valley narrowed in as we reached the wooded hills. There the houses were stacked right into the valley wall and behind them patches of white showed places where the hillside had slipped and become a cliff. The main road swung to the right here, following the river, but I found a pub and bought two cheese rolls and a bottle of beer. My legs wouldn't let me stay and sit, so we took a small road to the left. This led into and along a tree-lined valley past old houses set deep in their gardens and on upwards, into the folds of the downs, following the lines of their smooth shapes in curves and swings, past clumps of dark evergreens sheltering in valleys, and copses of tall bare beech trees clinging to the slopes. Then we took a broad, steep, well-worn track, white chalk in close velvety turf, that led on up the smooth back of the hill, through a clump of mixed beech and evergreens and out on to the top, where three roe deer were standing, gazing at me in rapt astonishment before bounding away.\n\nThis was the top of the world. There I stood face to face with the sun and could look down on a miniature England and see a silver railway-line snaking through the shadowy valley where fields and hedges and villages squatted in the hazy light, quite still among the smooth hills.\n\nBeyond, beneath the sun, the haze was luminous as on a summer's evening, with only subtle tones to show more hills and plains lying in the distance. But the sun itself was quite clear of the haze and bright as summer... hot!\n\n'This is it,' said my legs, 'make the most of it.'\n\nSo I did, took off my shirt and vest and lay on a soft southfacing slope of turf. I chewed my cheese rolls, drank my beer and with the sun lighting up my closed eyelids, I slept, my arms spread wide, surrounded, held and warmed by the golden light.\n\nNot for very long. It was February. A small wind which had scooped a little air from some dark frosted corner sent me no more than a single breath.\n\nI shivered and sat up. The sun was already shifting apologetically down into the haze. 'Don't worry,' I said. 'You did well.'\n\nI scrambled into my vest, shirt, pullover and jacket and was for a stately saunter, but my legs wanted to run. And they did, straight down the hill, bounding from hump to hillock, whizzing through small copses and slithering down the skiddy bits where strangely shaped white pebbles with holes through them lay among the chalk chips.\n\nBy the time we reached the black and white hotel beside the river the valley was already in shadow, and the rest of that day has fallen from my memory. No doubt the Southern Railway provided a branch line to take me to Waterloo.\n\nPerhaps the place was magic. I know that Box Hill spurred Keats to write the last five hundred lines of _Endymion_ in record time. I wouldn't be surprised if the same Spirits and Speculations that inspired him in 1817 were still about in 1943.\n\nMagic or not, something was changed in me that day. Perhaps I had turned away from worry and badgering and not being sure, and had found broad space and clarity. There was no going back on that and if the world didn't suit? Well then, I would make another one.\n\nThe next day I went to college and found that everything was going on as usual in the Foundation Class, including the shirtpatterns. I was pleased to find that my absence had not been noticed.\n\nThen, one morning, as I was sitting at a table in the canteen drinking tea, a rather worried-looking young man came up to me and said: 'Good morning, my name is Bruce. I am a painting student.' I rose, we shook hands. I invited him to sit down and we had a perfectly ordinary conversation. He was a painter. His mind and heart were deeply involved in the art of painting, in form, space and colour. He told me his father was a taxi driver and that he was an anarchist, a great admirer of the work of Herbert Read.\n\nBruce seemed to be a careful, baffled sort of chap, but passionate and definitely his own man, answerable finally to himself.\n\nNext time we met, at the same table, other members of the painting class came and talked, including a tall attractive girl whose name was Dorothy. We met many times and although I can't remember what we all talked about, I know the conversation was both respectful and emotional, in that everything was real and everything mattered enough to be savoured and thought about carefully.\n\nIt was a time for such things. My season ticket allowed me to break my journey in the centre of London, so I haunted the National Gallery where the War Artists' paintings were on show, and the bookshops, where I remember buying copies of T. S. Eliot's newly published _Four Quartets_ (paper, one shilling (5p) each) and being deeply but incomprehensibly moved by them. In those days my response to everything I saw or read was immediate, intense and complete, not clouded by doubt or muddled by fashion.\n\nI discovered music. Big music, like Beethoven's third symphony, which sang of glory and tragedy, swept through me like a forest fire. I bought a set of second-hand 78rpm records of it and played them at full volume on the huge mahogany electric radiogram in the long room, sometimes with my head close to the loudspeaker as if to get inside the music, sometimes leaping about the room like a balletic orang utan. I found ancient, scratchy onesided (80rpm) records of Enrico Caruso that my parents had bought new when they were young. These filled me with a heartrending excitement and delight that no later recordings by other great singers would ever match. Dvor\u00e1k's ninth symphony, hackneyed perhaps today, sang to me then of breezes combing the cornfields, of birds swooping in sunlight and of life, rolling, ongoing, joyous life. I knew every note by heart and it could be heard, somewhere in my head, through all that golden idiotic time.\n\nThere really must have been something golden about the spring of 1943, at least in Kingston and along the banks of the Thames, because, and I remember this quite clearly, the sun shone all the time.\n\nDorothy, who was just as uncomfortable in her ordinary world as I was in mine, became my companion and girlfriend (though, sensibly perhaps, not my lover) and we went to find our other world. We would turn and walk softly out of the unseeing College, slink down the path by the Hogsmill stream, cross the main road and walk hand in hand along the banks of the Thames. The pictures I have of that time are full of ease and forgetfulness. We bought cherries in Kingston market and flicked the stones from the bridge. Sometimes we hired a punt and poled up the river to lie shady under the overhanging willows. We fed ducks and when the water was high we tickled the edge of it until small fishes rose to nibble our fingers.\n\nDreaming, we wandered through royal parks and along riverside paths, invaded timber barges and explored strange unused grounds where broken walls were lost in thickets of alder. For long sunsoaked hours we would lie in private clumps of tall grass, kissing and cuddling and talking of brave, absurd ambitions in the making of our world, a world that would unplug itself from civilization so that it could start again. There was no aggression in these dreams, some derision perhaps but no hate against authority, only a gentle wish to be away from it.\n\nTo show that I was serious, I formally left home. With a small tent, a sleeping bag and a book of poems, we took up residence under some trees beside the River Mole. A small boy in a boat rowed in to talk to us and share our tin of baked beans, heated up on a tiny bonfire of broken boxes. It was nice to have our own place where we could entertain friends. Later Dorothy went home and I slept there in the tent. But, though I packed everything together into a bundle and hid the bundle high in the branches of a tree before leaving for College the next morning, by nightfall it was all stolen. So I went home.\n\nOnce or twice I took Dorothy to Finchley for weekends to meet the family. This wasn't a very welcome gesture. Daisy did her best to have chats with her but Ray was icily polite, visibly, and I suspect intentionally, failing to hide his contempt behind a charade of over-elaborate courtesy. I think this must have made me angry because I remember telling Ray and Daisy that Dorothy and I were thinking of getting married. Daisy said: 'Don't be ridiculous.' So I told Dorothy I was thinking of asking her to marry me and she said: 'Don't be ridiculous.' So I didn't raise the subject again.\n\nTerm ended. Ray and Daisy, gluttons for punishment, took their two adolescent sons to the Lake District for a wet summer holiday in a boarding house. The holiday was uneasy and we were glad to return to Finchley, where I found an order to attend a Medical Board examination prior to my being called up. This I did and was pronounced A1; entirely fit for service.\n\nNothing further happened, so I returned to the art college and tried to pick up where I had left off, but somehow, for both Dorothy and myself, the steam seemed to have gone out of our hopes for new worlds.\n\nAt about the same time my college work had become more comfortable, or at least more interesting. I had been promoted from the Foundation Class and the shirt patterns to the Drawing Class with real people to draw. There, somewhat to my surprise, I very much enjoyed the life-drawing sessions and found the process fascinating and deeply concentrating. I'm sorry to say I came eventually to disagree quite profoundly with Mr Hockey, the drawing teacher. His recommended method of drawing figures was to analyse the separate forms of the model's body and fit them together like pieces of subtle dough, with all the angles and curves carefully shaped. The end product was a very correct-looking body which seemed to me to have about as much life as a piece of shaped dough. I wanted to do a drawing that had some life to it. For a long time I was turning out stilted, unsatisfactory work and then, one afternoon, it all fell into place.\n\nThis happened in a rather unexpected way. I liked to draw in ink but, in the middle of the war, I couldn't find a pen which exactly suited my purpose, so I made one. It was carved from bamboo and it worked quite well, giving a line which, as the nib dried out, became lighter in tone. Unfortunately I couldn't find an ink which had any richness of colour, so in the end I collected some pebbles, an old caviare-pot, a handful of apple-wood soot and some gum-arabic, and made a crude grinder, driven by an old gramophone motor. After several hours of grinding and straining I achieved about a thimbleful of very black ink that thinned out to a scrumptious walnutty tone.\n\nThat afternoon I was really only trying out the new ink, concentrating on the way it was working, when I suddenly noticed that the drawings which my hand was doing \u2013 a set of quick sketches of a mountainous Frenchman called M. Letailleur \u2013 were good. They were full of life, exactly the sort of drawing I had been hoping and trying to do all along. Mr Hockey, bless him, cheered me on and for the rest of the afternoon my hand was directly connected to my heart and could do no wrong.\n\nThose drawings told me something important. This was that on that afternoon the capacity to draw had been _liberated_ as an effect of the fact that my anxious unconfident conscious mind was otherwise engaged in technical matters. This showed that my capacity to draw did in fact exist and had been waiting there, and also that it had worked well _simply because_ I had spent so many hours painstakingly training my hand until it had become physically capable of putting down on paper exactly what my heart was telling it to \u2013 without having to consult my head on the way. That is a marvellous feeling, something I expect musicians come to when, after much practice, they no longer have to think about what their fingers are doing and can just play the music.\n\nI could see that in the past, when I had been drawing or painting, there had always been a temptation waiting beside me, a temptation to latch on to spontaneous effects that were in fact accidental (and had really only happened as a result of my failing to do exactly what I had intended) and, because they were attractive, keep them and pretend to myself that I had meant them. That made nonsense of the original intention while at the same time allowing me to get away with slovenly work. I began to understand what the art master at Dartington had meant when he wrote in my report that my efforts were ' _aimless work that could be_ _described as a slow disintegration of an idea into a pleasant fac\u00b8ade'._\n\nThat afternoon and those few, not really all that marvellous, drawings repaid all the frustration and muddled thinking that had gone before. I felt that I had begun to understand what I was doing at the College and could, at last, begin to feel at ease there.\n\nIt was a bit late to be feeling at ease anywhere.\n\nI went to the canteen for a cup of tea and told Bruce that I was waiting to be called up. He thought for a few moments before answering. Then he said: 'You aren't going, are you?'\n\nI heard myself answer: 'No.'\n\n'They'll put you in clink,' he said.\n\n'I expect so.'\n\nThere didn't seem to be a lot more to be said at that moment, so I finished my tea, said goodbye and went away.\n\nThere was in fact a great deal to be said and it was being said, loudly, inside my head. I had heard myself answer 'No' and from that moment a thundering row had been going on. There was no doubt that I had meant what I said, but why had I just blurted it out like that without proper consultation?\n\n'What d'you mean, proper consultation? We've been thinking about it for weeks.'\n\n'Yes, but you know what our thinking is like. It's all muddle and fancy. You know it is.'\n\n'This isn't. And anyway, it's true. I'm not going to join the army.'\n\n'But they'll call you a coward; shoot you.'\n\n'I don't think so.'\n\n'They'd have every right to.'\n\n'That's true. That's very true.'\n\n'So what are you going to do?'\n\n'I suppose I had better tell Ray and Daisy.'\n\nTelling my parents turned out not to be as difficult as I had feared. Daisy's reactions were essentially practical. She simply said: 'But you can't. You've already had the Medical Examination.' I replied that I didn't think that would make a lot of difference, just make things a bit more complicated.\n\nRay's reactions were also quite straightforward. Speaking from the point of view of a conscientious objector in the First World War, he told me why he wasn't one in this war, giving reasons which I respected. I tried to tell him what I was thinking but I remember feeling that the case I was making sounded pretty muddled and lame.\n\nRay made no attempt to argue with me and we simply discussed my situation. He said he thought there would be some 'procedures for the treatment of conscientious objectors' laid down, and he recommended me to consult the Quakers, who had helped him a lot in the First World War. I was relieved and grateful for this and very touched because he had, perhaps for the first time, actually listened to what I had to say. I felt I had been treated as a grownup.\n\nWhat he and Daisy really felt I don't know. Neither of them showed any sign of ordering me out of the house, never to darken their doors again, but I hadn't seriously thought they would. The tradition of dissent in both their lives was too well established for them to bother with gestures like that.\n\nJohn observed, almost in passing, that in a country whose survival depended on food brought in by ship and whose ships were regularly being torpedoed by U-boats, there was no reason to assume that society would feel obliged to feed people who were not contributing \u2013 a thought which was never out of my head, and probably never will be.\n\nThe aunts were, I think, fairly noncommittal, though Nell, I remember, denounced me, not for being a conscientious objector but for being fanciful and changing my mind.\n\nMy call-up papers came in November, instructing me to report to the Household Cavalry at Combermere Barracks in Windsor on the ninth of December.\n\nI collected a leaflet from the Peace Pledge Union which outlined the procedure. I would have to join the unit and refuse to put on uniform. The army would then court-martial me and send me to a civil prison. From there I would go before a tribunal.\n\nI wrote to the Commanding Officer of the Household Cavalry at Windsor, politely letting him know that I would not be joining his regiment on the day appointed but that in order to avoid inconvenience I would present myself for arrest at the gates of Combermere Barracks at 11.30 in the morning of 13 December 1943.\n\nOn the thirteenth of December I packed a small bag, said quite a fond goodbye to my parents, and took the North London Railway from Finchley Road to Windsor.\n\nMy goodness, that was a slow railway! I sat in an empty unheated compartment in an antique train that must have crawled into twenty little stations in cold dismal suburbs I had never heard of, and at each one I said to myself: 'Oh please, may I get off here and just go home?'\n\nEventually the train reached Windsor and, on leaden legs, I made my way to Combermere Barracks, where, at 11.30 a.m.precisely, wearing a brown raincoat, I stood outside the gates and waited.\n\n## _III. Uncertainties._\n\nNothing happened. I stood at the gates of the barracks, duly making myself available for arrest, by appointment, as arranged, and, dammit, nobody came to arrest me! I was affronted. I had travelled a long dreary way and I was cold. Also the hope that it might all be a mistake, that I had got the day wrong, that they had decided not to do it after all, was bounding about inside me like an unruly dog. It was all a bit bloody much!\n\nThen I saw an armoured car, a vehicle like a small tank with wheels, coming towards me. The officer in charge was standing high in the turret looking around him.\n\n'That's a bit elaborate!' I thought.\n\nThe armoured car drove straight past me. 'Thank you very much,' I said.\n\nSo then I did what I should have done in the first place: just walked in through the gates. Nobody stopped me. The nearest building looked like a guardhouse. I peered in through a window and saw a soldier sitting at a desk.\n\nI tapped on the window. He looked up and waved to me to come in through the door, which was open.\n\n'Hallo,' I said, 'I've arrived.'\n\n'Yes?'\n\n'I've come to give myself up.'\n\nHe looked up. 'To give yourself what?' he asked.\n\n'Up.'\n\n'Up what?'\n\n'No, not exactly.'\n\n'You've come to join up?'\n\n'No, not really.'\n\n'Got any papers?'\n\n'Papers?'\n\n'Look, you can't join the army if you haven't got the proper papers.'\n\n'No, I'm not here to join the army.'\n\n'What then?'\n\n'I'm here _not_ to join the army.'\n\nThe soldier thought about this for a moment. Then he said: 'You know, I'd have thought you would have been better off doing that somewhere else.'\n\n'Oh, dear,' I said. 'May I sit down?'\n\n'Help yourself.'\n\nI sat down on a folding chair and tried to marshal my thoughts.\n\n'I have come here to be arrested,' I said, quite slowly.\n\nThe soldier looked at me in silence. Then he picked up the telephone. 'Got a right one here, sir,' he said. 'Civvy. Says he's come to be arrested... Sir... Sir!' He put his hand over the mouthpiece. 'Duty Officer's compliments, and by whom was you wishing to be arrested?'\n\n'The Household Cavalry... I thought. I thought it was all arranged. I wrote to...'\n\n'... Hang on, hang on!' The soldier spoke into the telephone. 'Household Cavalry, sir... Right, sir... Sir!'\n\nHe put down the receiver. 'Orders is to arrest you, if you insist, only Major Whatton-Burby says why don't you just push off out of it?'\n\nThat certainly was a temptation, but... 'No,' I said, regretfully, 'I think I'd better stay.'\n\n'Please yourself. Perhaps you'd come this way.'\n\nThe soldier ushered me into a cell at the end of the guardroom. There were three truckle-beds in it, each with a mattress, grey blanket and dark-grey shiny pillow.\n\n'I'll leave the door unlocked, if you don't mind. Less formal, if you know what I mean.'\n\n'Thank you.'\n\n'Cup of tea?'\n\n'Ta! That'd be nice.'\n\nI sat on one of the beds. The soldier brought me a mug of very sweet tea. 'Make yourself at home. There's plenty of time.'\n\nHe was right about that. Nothing happened. Well, no. At dinner time he brought me bangers and mash with Brown Sauce on an enamel plate and, later, he rather diffidently accompanied me to the door of the WC out at the back. 'Regulations,' he said.\n\n'Persons in custody.'\n\n'Of course,' I said. 'Got to be careful.'\n\n'S'right.'\n\nAfter lunch I put my feet up. The shiny dark-grey patina on the pillow seemed most likely to be impacted Brylcreem or something of the sort. I decided it wasn't dangerous and put my head on it.\n\nI hoped I would doze off.\n\n'Warner's pissed off again!'\n\nI sat up with a jerk. Gales of laughter were coming from the guardroom.\n\n'He's not! How'd 'e do it?'\n\n'Slipped off detail. Spud-bashing. Hopped it.'\n\n'Be down the Dilly by now!'\n\n'What's doing? Sending after him?'\n\n'No such luck. Major says: leave it to the Redcaps.'\n\n'Pity. Could have done with an airing.'\n\n'Yeh.'\n\nThen quiet again for about three hours. The day grew dark.\n\nThe sound of steps outside. Stamping, muffled shouts. The guard detail was called to attention. Louder stamping noises. The soldier marched into the cell, followed by an officer.\n\n'Pris'ner.. Shun!' he bawled.\n\nI began to get up off the bed.\n\n'Oh, no need for that,' said the officer kindly.\n\n'No trouble,' I said, 'no trouble at all.'\n\nI stood up, attentively.\n\n'Ah, er...' The officer seemed embarrassed. 'Could I perhaps have your name, please?'\n\n'Richard Oliver Postgate.'\n\n'Oh.' He looked at a notebook.\n\n'I wrote to the Commanding Officer,' I said.\n\n'Did you?... Colonel Sir Alastair Mews?'\n\n'I don't know the name.'\n\n'He'll be in the Cairngorms.'\n\n'Ah,' I said.\n\n'Well, er. Can you sort of put me in the picture?' he suggested. 'Please, do sit down.'\n\nWe sat on separate beds and I told him that I had been supposed to report on the 9th but didn't because I was a conscientious objector, and had been advised that in order to save inconvenience I should present myself for arrest at a later date.\n\n'That's extremely civil,' he said, and laughed slightly, perhaps at the thought of its not being military.\n\n'What exactly are you to be arrested for?'\n\n'Being absent without leave?' I suggested.\n\n'Oh, yes. Yes, of course. Well, that should be no problem. You were supposed to be here last week, then?'\n\n'That's right.'\n\n'So you'll be on the intake roster for the 9th and posted as notarrived, absent without leave?'\n\n'I dare say.'\n\n'Well, that's fine, then,' said the officer, standing up. 'We'll just do the paperwork for that and I'll, er, leave you to it.'\n\n'Thank you,' I said.\n\n'Carry on, Corpr'l-o'-th'Horse.'\n\n'Sir!' The guard detail stood to attention as he left.\n\nLater there was shepherd's pie and baked beans. Later still there was more stamping and a tiny officer, accompanied by a huge Corporal of the Horse, appeared. The officer couldn't have been more than about fifteen years old.\n\nHe looked up at me. 'Any complaints?' he piped.\n\n'No, no, I don't think so, everything's fine, really...' I began, but he had already turned round and walked out before I'd even begun to speak.\n\nI was furious. A great rage welled up inside me. The rudeness! The appalling impertinence of that fancy-dressed twerp! I marched to the other end of the cell and viciously kicked one of the beds, hurting my foot. That gave me something to swear about.\n\nNothing more happened, except a mug of cocoa and a quick supervised wash out at the back. There was absolutely nothing wrong, everybody had been most courteous. That tiny officer was probably a boy from the OTC of one of the local schools, like for instance Eton. I couldn't think why I had got into such a rage at him. Looking back I can see why. I was just petrified.\n\nBack in the cell I used my spare vest as a pillowcase, and kept my underclothes and socks on under my pyjamas. Just as well I did, it was a cold night. I doubled the blanket on one of the beds and climbed under it.\n\n'You'd better sleep,' I said to myself, without much hope of it happening. I was wrong about that. I must have fallen asleep almost at once.\n\nThrough my dreams I heard stamping feet. A door slammed open. A voice shouted: 'Pris'ner SHUN!'\n\nA bright electric light bulb was shining in the cell.\n\n'Pris'ner SHUN!' roared a soldier standing by the doorway.\n\nI sat up. 'Yes, right-ho, hang on a minute.'\n\nA different tiny officer marched in. 'Any complaints?' he piped.\n\n'Oh, er well, no. I've only just woken up as a matter of fact. You don't happen to know what time it is, do you?'\n\nBut he was gone.\n\nThe soldier remained. I didn't recognize him.\n\n'Good morning,' I said, 'if it is morning?'\n\n'ABLUTIONS!' he shouted.\n\n'Oh yes?'\n\n'C'mon! UP! UP! UP! UP!' he punctuated this by smacks on the blanket with a short leather-covered stick he was carrying. 'UP! UP! UP! UP!'\n\nI jumped out of bed.\n\n'Right! Out the BACK! One-two, one-two, one-two!'\n\nI grabbed my towel and walked quickly out to the wash-house. The soldier followed close behind shouting 'One-two, one-two... Come on! Get at it now!'\n\n'I'll have a pee first, if you don't mind.'\n\n'There'll be no Pee-if-you-don't-mind where you're going! Get at it now! I haven't got all day.'\n\nI washed as quickly as I could, which was very quickly because it was dark, outside, December, cold water and no soap. The soldier started shouting again and I walked back to the cell, taking care not to walk in time to his shouts. This was quite a different soldier from the one who had been there the day before. This one was podgy-fat with a shiny white face and a mean-looking moustache.\n\nI went into the cell. He slammed the door shut and locked it.\n\nThe cell was dark. There was no sound from the guardroom. I had no idea what the time was. I felt for my clothes and got dressed. I stood there in the dark, feeling very cold and wondering what to do. I got back into bed again.\n\nThis treatment was more like what I had been expecting. There was absolutely no reason why the army should be courteous and accommodating to me and quite a lot of reason why it should feel hostile and unaccommodating. The important thing, I thought, was that I should behave courteously and accommodatingly to the army, while making it clear that I was doing so by choice, not because I had given way to the impulse to obey when shouted at. It was all very well my deciding what I had to do, quite another thing to do it.\n\n'The bugger's gone back to bed!'\n\nThe door slammed open. 'UP! UP! UP!'\n\nI scrambled out of bed.\n\n'He's got all his bloody clothes on! Slovenly! That's what you are! You're a disgrace, lying in bed all day!'\n\n'What else was I supposed to do?'\n\n'Stand to attention when you speak to me!'\n\n'... No,' I said, and sat down on the bed.\n\n'Oh!' said the Corporal of the Horse. 'Like that, is it? Having a tantrum, are we? Well, you'll see. You'll see.'\n\nHe turned on his heel and marched out, slamming the door. I sat exactly where I was, trembling, but at the same time feeling that making the gesture of defiance hadn't really been very difficult.\n\nThe day grew lighter. I grew hungry. Through a spy-hole in the door I could see a table with a plate of fried bread with baked beans on top. Eventually I heard the Corporal of the Horse say he was going out. Then one of the other soldiers nipped in with the plate and a mug of tea.\n\n'He's a hard one,' explained the soldier, 'Corp'ral-o'-th'-Horse Morris. You want to keep out of his way.'\n\n'Fat chance I'll have of doing that.' 'Fat chance any of us have. Mind out!'\n\nHe slipped out and locked the door as the Corporal returned.\n\nCold baked beans on cold fried bread was not my favourite breakfast but I ate every scrap. Then I lay on, but not in, the bed.\n\nApart from a supervised visit to the WC, nothing happened for several hours. The day began to darken.\n\nThen the door slammed open.\n\n'Pris'ner Shun!'\n\nI sat on the edge of the bed, waiting for somebody to appear.\n\n'On your feet!'\n\n'What for?'\n\n' 'Cause you're insanitary. That's what for.'\n\n'Insanitary?'\n\n'On your feet! You're going to be dealt with. You're a disgrace. Come on. One-two, one-two, one-two. Right turn.'\n\nI walked out of the cell, through the guardroom and into the cold air of the yard. It was a dismal day.\n\n'Straight ahead. One-two, one-two...'\n\n'Yes, all right. Don't keep shouting.'\n\n'You'll be shouting, where you're going.'\n\nWe came to a door in a wall.\n\n'Open the door and enter,' ordered the corporal.\n\nI opened the door and entered; into what looked like a hairdresser's saloon without the advertisements.\n\n'No nonsense now! Sit in the chair.'\n\n'What if I say no?' I asked.\n\n'In the chair, one-two, one-two.'\n\n'What if you were to say please?'\n\n'Don't come that stuff with me. You're a disgrace. Lying about all day with your hair down over your ears. A bit of self-respect, that's what you need.'\n\nWhile saying this the Corporal of the Horse did not lay hands on me. Holding his arms tight to his sides, he bumped me with his belly towards and into the chair. I sat down.\n\nThe hairdresser deftly covered me with the white apron.\n\n'How do you like it, sir?' he asked.\n\n'Oh, high on top, I think, with a blue rinse.' We both roared with laughter. The Corporal glared. The hairdresser snipped.\n\nThe wind was cold about my shorn ears as we walked back to the cell. Later another midget officer came asking for complaints. I mentioned to him the hippo in the WC but he was on his way.\n\nThen all was quiet for what seemed like several hours. Lunch having been overlooked, I was very hungry. I was also agitated, pacing up and down. I wished something would happen, like for instance, supper.\n\nSupper didn't appear, Warner did. Having been recaptured by the Redcaps, he was marched into the cell by Corporal of the Horse Morris.\n\n'Oh, look!' he cried, 'And there's this gorgeous boy to keep me company. Oooh, you are _darlings!'_ The door was slammed.\n\nWarner flopped down on one of the beds. 'I have had the most _a_ _w_ _ful_ day,' he announced. 'There was this lovely man, rich as Christmas cake and not a bit kinky, well, except... you know. And you see, there we were, in this rather pretty bar, him in his Homburg hat and twinkly eyes and me with my rubber fronts in, and we were doing fine, just sort of getting around to the real questions, when in came these four Redcaps and literally dragged me out, without a word! I mean, it was humiliating. All I could do was wave!'\n\nWarner stopped speaking and looked at me. He was a fairly ordinary-looking young man wearing a crumpled uniform. He had a smooth face with traces of mascara around the eyes.\n\nHe said: 'I'm rabbiting on again. Lionel Warner, Trooper, of this regiment, God help it.'\n\n'Oliver Postgate.' We shook hands.\n\nHe said: 'You're in a right two-and-eight, aren't you?'\n\nI nodded, unable to speak.\n\nWarner said: 'Listen. Don't take any notice of me. I'm all mouth! I mean, I'm, well... you know.'\n\nI nodded.\n\n'Right,' he said. 'Well, anyway, what I'm saying is...' He seemed embarrassed. 'Er... you don't have to worry.'\n\n'Oh... thanks.'\n\n'You've got other things to worry about, I dare say.'\n\n'A bit, yes.'\n\n'Tell me.'\n\nI gave him a brief rundown on my situation and prospects.\n\nHe listened in silence, then he said: 'Hmm. It sounds all right. Quite a long journey, though. You could hit trouble on the way. But I'll tell you one thing: you're in the right place, here.'\n\n'Am I?'\n\n'Oh, yes. This lot, they're ornamental. All spit and polish and do it by the book. They're all into careers and they are dead scared 'cause if they put a foot wrong, just once, they'll be posted to the Second Regiment in North Africa. You've only got to threaten them with the \"proper procedures\" and you'll have them eating out of your hand. So don't worry! Come on, smile!'\n\nI smiled. 'I could do with eating out of something. I'm ravenous.'\n\n'Now that,' said Warner, 'will be attended to. I have friends in high places. I have but to rub the lamp and behold... Oh dear! There should be a puff of green smoke and a crash of cymbals.'\n\nNothing happened, but a few minutes later the door opened softly and a soldier slipped in carrying two plates of dinner.\n\n'Sorry I'm late,' he said. 'Morris was everywhere. I'm afraid they're a bit untidy, the dinners, but I got three from the cookhouse and sort of amalgamated them. Guessed you'd be hungry.'\n\n'You were right,' I said. 'Ta!'\n\nI don't remember what the food was but I wolfed it down.\n\nLater there was cocoa and decorous, supervised ablutions.\n\n'Ugh!' said Warner, with feeling. 'That's the one thing I can't stand about this place, the black pillows.'\n\n'I've got a hanky you could have,' I suggested.\n\n'No, don't bother.'\n\n'It's no trouble. At least it's clean.'\n\nI gave him my spare clean handkerchief. He laid it carefully on his grimy pillow.\n\n'That's nice,' he said. 'Be of good cheer.'\n\n'You too. Good night, Lionel.'\n\n'Good night, Oliver.'\n\nIn the morning he wasn't there any more. Vanished as if he had never existed. But he had. The hanky had been folded and was tucked under my pillow.\n\nCorporal of the Horse Morris opened the door.\n\n'Where's Warner gone?' I asked.\n\n'Mind your own business,' he replied.\n\nI could have hit him. But I didn't, which was probably just as well.\n\nThis was the beginning of another day, one which was interesting only because I was taken to the cook-house to do some spudbashing and had a pleasant chat with a nice lady cook called Mrs Murch.\n\nI was taken back to the guardhouse and in the evening I told yet another tiny officer about the wallaby under the washbasin, but he didn't seem to mind.\n\nThe next day the Duty Officer I had met on the first day popped in to sit on the bed and check a few details. He said that absent without leave wasn't a sufficient charge on its own. I explained that that was just the hors d'oeuvre. The next move would be to charge me with refusing to put on uniform. He seemed to get the hang of that and pottered away.\n\nAnother day passed and the following afternoon Corporal of the Horse Morris and another soldier marched in.\n\n'Come on! Smarten up a bit! Can't you do something about your personal appearance?'\n\n'What for?'\n\n'Interview.'\n\n'What interview?'\n\n'Never you mind. Do something about your hair.'\n\nI combed my hair and shook out the jacket I had been lyingon. I retied my tie. 'How do I look?'\n\n'Horrible. Have to do. Come on, outside! One-two, one-two...'\n\nMorris and the other soldier marched one each side of me, across the yard and into the main door of a big building. I found I had great difficulty in not walking in step with them, but I managed it.\n\nThey marched me up the stairs and into a room where a very senior-looking officer was sitting writing at a big desk. I finished up standing in front of the desk with Morris directly behind me.\n\nThe officer looked up. He was elderly, with full walrus whiskers.\n\n'You are 306346 Trooper Postgate, R. O.'\n\n'No.'\n\n'Stand tattention! Address supiorofficers SIR!' barked Morris, prodding me in the kidneys with his stick.\n\n'My name is Oliver Postgate.'\n\n'Address supiorofficers SIR!' prodded Morris.\n\nColonel Walrus and I looked at each other balefully for a moment.\n\n'I would be very happy to address you as sir, if you'd like me to,' I said, 'but it's a bit difficult to think straight while that chap behind me keeps prodding me with his stick.' (Or words to that effect. Let me not give the impression that I was cool and calm. I was practically speechless.)\n\nColonel Walrus flicked a glance at Morris. Then he spoke.\n\n'Trooper 306346 Postgate, R. O. You have been medically examined by the appropriate Medical Board and found fit for combatant service with His Majesty's Forces. Consequently you are now deemed to be a soldier. Do you understand that?'\n\n'I hear what you say but I don't agree that it is so.'\n\n'Whether or not you agree is immaterial. You are deemed to be a soldier and you are required to wear the appropriate uniform. Consequently I order you to go into another room and don the said uniform.'\n\n'Pris'ner SHUN!'\n\nI was conducted to an adjoining room which had four chairs and a table in it. A uniform was laid out on the table.\n\n'Come along, then! Get it on!'\n\nI sat on one of the chairs. The two soldiers stood over me.\n\n'Oh, come on,' said Morris quite kindly, 'make it easy for yourself. Put it on.'\n\n'No, I don't think I will, thanks.'\n\nI sat for about five minutes. Then Morris said: 'Do you refuse, then?'\n\n'That's right.'\n\nThey led me back to Colonel Walrus. He repeated the order, with variations. I was led away again. This time we all sat in the chairs.\n\nThe third time, as we entered the room with the uniform, Morris asked: 'Are you going to?'\n\n'No.' So we just turned round and came out again.\n\n'Trooper Postgate,' said Colonel Walrus, 'you have three times been given a legitimate order by a superior officer and three times you have refused to obey that order. So you are charged with insubordination and will be dealt with accordingly. You understand?'\n\n'Er... yes.'\n\n'Dismiss!'\n\n'Pris'ner SHUN! About turn. One-two, one-two.'\n\n'... Oh and by the way,' said Colonel Walrus.\n\nWe stopped in the doorway.\n\n'It'll take a day or two to set up, but I think we've got the drift of it. Did you do anything about a Friend?'\n\n'Friend?'\n\n'The prisoner may have either a defending officer or a Prisoner's Friend.'\n\n'Oh, I see. I haven't thought about that. I can't think of anybody I would actually think of as a friend. Trooper Warner is a pleasant sort of chap but, you know... sir.'\n\nAt this Corporal of the Horse Morris gave a snort of laughter and nipped smartly out of sight behind the door.\n\nColonel Walrus said: 'It should be somebody from outside, don't you think? You chappies have some sort of organization, don't you? Isn't there a society of some sort?'\n\n'There's the Society of Friends, sir. They're Quakers.'\n\n'Society of _Friends_ , eh? Sounds ideal.'\n\n'I think they have a Meeting in Windsor. Or there's Friends' House, that's their sort of Head Office, in the Euston Road.'\n\n'Well, that'll be it, then, won't it? Off you go.'\n\n'Pris'ner SHUN!'\n\nThe next day was a Saturday, which was uneventful. Then on Sunday afternoon a rather nervous young man came to call. He was a Friend from Windsor Meeting. He had come to offer to be my Prisoner's Friend for the forthcoming court martial. We went through the procedures together. It all seemed fairly cut and dried and there wouldn't be a lot for him to do if all went according to plan.\n\nThe court martial went quite smoothly. Four officers sat behind a table, Colonel Walrus and two other oldish ones and a young one who was very quiet and seemed anxious. The charge was read out. I pleaded guilty and was sentenced to three months' imprisonment.\n\nIt was all over in a few minutes and we were led away. Corporal of the Horse Morris, almost affable, gave my Friend and me mugs of tea by the guardroom stove. Then we shook hands and our Meeting was over. The Corporal and I watched him go.\n\n'By the way,' I said, 'what did happen to Warner?'\n\n'Pissed off again,' said the Corporal.\n\n## _IV. Absurdities._\n\nI had worked out what prison was going to be like. After the uncertainties of the Guards' barracks I thought it would be a pleasure to have a quiet cell of my own and live a well-ordered life with perhaps a few mailbags to sew.\n\nI was wrong about that. My time in the nick turned out to be far from quiet. I was delivered to Feltham Juvenile Prison, where the inmates, of any age, wore short trousers and were paraded outside in the freezing wind. I was taken to a place called West Tech., which was a large disused classroom containing thirteen bed-boards. This was where the conscientious objectors were housed.\n\nThe 'conchies' were thirteen young men of markedly different characters and backgrounds. As far as I remember there were two Quakers, two Seventh-Day Adventists, two Plymouth Brethren, one Church of England evangelist, three Jehovah's Witnesses and three non-religious persons, one of whom was an educational psychology student, another a cinema projectionist from Swansea and the last was me.\n\nEach of the religious sects was visited once a week by its minister, who conducted whatever service was appropriate in the side room. The Quaker visitor was a very kind, warm-hearted man who quaffed his mug of prison cocoa with gusto and said: 'Ah! that was good!' This was a hearty, well-meant gesture, but unconvincing. The cocoa was indisputably foul.\n\nThe other services took place in private, though I did once inadvertently go into the side room where the Jehovah's Witnesses were conferring with their minister, who wore a trilby hat. I was intrigued to observe that they were huddled together, underlining selected passages in their Bibles in red ink, and chuckling with glee.\n\nOne evening, the Prison Governor came into West Tech. He said: 'Anybody here know anything about the theatre?'\n\nLike a fool, I said: 'I do, a bit.'\n\nThe next morning I found myself in the Governor's Office, accompanied by a principal officer, who was a sort of sergeantmajor of the 'screws'. To his disgust the Governor invited me to take a seat in his nicely furnished office while he outlined his scheme.\n\nThe Governor, a crisp-voiced upright man with a decisive manner, told me he had invited a local dramatic society to come to the prison in about three weeks' time to perform its current play, _Badger's Green_ , by R. C. Sherriff, in the assembly hall of the prison, and that I was to prepare the stage.\n\n'Ah, I see... Yes. But I'm not exactly a free agent, sir.'\n\n'No worry about that. You shall have a blue armband. That gives you access to all premises. The Institution has ample workshop facilities and services. My officers have orders to give you every cooperation... understood?\n\n'Yes, sir,' I said.\n\n'Understood, Mr Saunders?'\n\n'Yessir!' barked the outraged Principal Officer.\n\n'Right, then. Off you go!'\n\nThe Principal Officer marched me out of the office and along the corridor. 'What am I going to do about that, then?' I asked. 'I'll tell you what you're going to do, lad. You're going to watch it. That's what!'\n\n'I'll have to do the job, though.'\n\n'Frigging stupid if you ask me. Always coming up with these fart-arsed fancies. Gives me the squitters. Gives us all the squitters! All right, then. You can't bloody help it. I can't bloody help it. It's got to be bloody done.'\n\n'Yes, I know that. But how do I set about bloody doing it... sir?'\n\n'That's your bloody problem, not mine. I give you a blue band, OK? I put the word about among my officers that you've got a charmed life, that you're carrying out the Governor's direct instructions and can dance through the place like the queen of the frigging fairies, OK?'\n\nThe officer took me to a store, where I received a blue armband.\n\n'That's it then,' he said, 'I've done my bit, so if you come a cropper, don't come bloody snivelling to me.'\n\nMr Saunders walked away down the corridor, leaving me, an inmate, standing alone and free to follow my fancies, in the middle of one of His Majesty's Prisons, an institution created and administered solely for the purpose of preventing such activity.\n\nAs always there was no alternative but to have a go and, in the event, it worked out quite well. I was able to get some lighting rigged up and, apart from a couple of disasters, like blowing a substation fuse and getting myself accidentally locked in the assembly hall for the night (a fate which I avoided by ringing the Riot Bell), things came together very satisfactorily, and members of the prison staff, once they came to terms with the incongruity of it, enjoyed joining in with the project. I can see the Governor, in full dungarees, painting in a grove of cypresses and complaining that the house-painting brushes from the store were no good for detail work. I did suggest that I could nip into town and get some from the art shop. He was already putting his hand in his pocket for the money when he remembered there were reasons why that idea wasn't altogether practical.\n\n_Badger's Green_ was a pleasant play, but it was about the politics of village cricket. In the evening, when it was performed before the assembled felons, all my fears were realized. From the moment the curtain went up the audience greeted the play with cat-calls and made lewd contributions to the dialogue, some of which caused the jeune premie 're to blush quite prettily \u2013 much to their delight. Later in the performance a semblance of order was restored, but only because the Governor and Principal Officers were silently patrolling the aisles.\n\nAfter it was over I was back on hoeing carrots and laying tiles, which was almost a relief, but my sentence was nearly up and quite soon I was sitting in the Reception Block waiting for the van to take me to Wandsworth Prison, from where I would be taken to the Conscientious Objectors' Tribunal.\n\nMr Saunders, the Principal Officer, came past. He looked at me sitting on a bench. I was wearing handcuffs and not enjoying it at all.\n\n'Ah,' he said. 'Given up being a genius, have we?'\n\n'None of my doing,' I said.\n\n'That's true.'\n\nHe spoke to the Transport Officer: 'This gentleman here's not a villain, he's a genius. Take the cuffs off.'\n\nWandsworth Prison was more like the sort of prison I had imagined \u2013 old, fairly dirty, fairly uncomfortable, quite impersonal, but peaceful. I did sew one or two mailbags but not very well. In any case I was only going to be there for the inside of a week in order that I could be escorted, in a taxi, wearing my own clothes (such luxury!) to the tribunal.\n\nThe tribunal was a fairly perfunctory occasion. In a large panelled room sat a number of people on folding chairs in front of three elderly people sitting at a table on a small raised platform.\n\nWhen my turn came, one of them asked after my aunt Margaret, with whom he had been at Cambridge. I promised to give her his good wishes. The tribunal conferred and announced that I was to be discharged from His Majesty's Forces and sent to work underground in a coal-mine.\n\nI didn't hear any of the next case because I was taken straight back to Wandsworth Prison and later that day put into a prison van, a Black Maria. It isn't possible to see where you are going in a Black Maria, so when I was finally set down I hadn't the slightest idea where I was.\n\nHowever, with sinking heart, I recognized the place. It was Combermere Barracks in Windsor.\n\n'What the hell are you doing here?' cried Corporal of the Horse Morris.\n\n'Buggered if I know!'\n\n'Cripes! I thought we'd seen the last of you!'\n\n'Wish you had.'\n\nMy disappointment was as nothing to the embarrassment of the Household Cavalry. The Duty Officer came rushing over at once to explain to me that they were very sorry but they couldn't accommodate me in the guardroom cell because I had committed no offence. They couldn't put me in the dormitories with the men because I wasn't in uniform and, you know, not one of them, as you might say... They couldn't send me back to prison because I had served my sentence and they couldn't discharge me because the papers hadn't come through.\n\n'What am I supposed to do?'\n\n'Jiggered if I know,' said the Duty Officer.\n\nIn the end they consulted the Quartermaster's storeman, who was an expert on improvisation. He put up a camp bed for me in the storeroom. As to what I should do with my time, nobody had any idea, nor any suggestions. I think they hoped I would dematerialize. So I just wandered about the barracks or did a bit of potato peeling for Mrs Murch. I don't think I was any trouble to anybody but I did feel a bit of a gooseberry standing about in my old mackintosh.\n\nI think it was the next day that I came face to face with Colonel Walrus. For a moment I thought he was going to explode.\n\n'What the blazes are you doing here?' he roared.\n\n'Er... Jiggered if I know,' I said. 'I seem to be sort of left over.'\n\n'You're not being left over here!' said the Colonel. 'Why don't you push off out of it?'\n\n'I'm not allowed to, am I?'\n\n'Rubbish! Can't have you hanging about the place making it untidy. You're going on leave.'\n\n'Good... when?'\n\n'Now! Immediately! This minute! Come along!'\n\nWe marched together, this time in step, to the guardroom. There the Colonel signed dockets.\n\n'Here's an open rail warrant and a ten-bob note.'\n\n'Thank you sir,' I said, and held out my hand.\n\nHe shook it firmly. 'Now push off!' he said.\n\nI walked out of the gates and across the road to a phone box. I rang Daisy.\n\n'I'm out,' I said.\n\n'Oh, are you?' she replied. 'Will you be in to supper? It's only corned-beef hash, I'm afraid.'\n\nI can sit here, fifty years on, and look back at that muddled, wilful young man, or rather, overgrown boy, of eighteen, trundling eagerly back to Finchley on that dreadful suburban railway, and I confess that as well as embarrassment (which really only comes from the fact that we still share the same life), I have a sneaking admiration for the oaf. He may have been wrong in what he did, and it is easy to say with hindsight that he was, but he saw what he thought was true and in a morass of doubt, guilt and fear, he stood by it. Oddly enough the times when he was most confused and unhappy came when he didn't know what to do next. Once he was committed and clear about the path he was going to take he was surprised to find in himself a previously unknown serenity of mind.\n\nHis view of the situation, one which I recognize and would, with some dull reservations, support, was much the same as that of his grandfather, George Lansbury. He saw that Hitler, though a monster, was a product of history \u2013 if you put a nation into the state of penury and shame that Germany was put into by the Treaty of Versailles, then a Hitler will rise to rekindle its anger and wreak vengeance on the world. Like his grandfather he could foresee no end-product of the war but another round of the cycle of oppression, retribution, aggression and devastation.\n\nAlthough he was called, and sometimes even called himself, a pacifist, and although he would, I believe, have died rather than kill, he recognized in himself the instincts that fire the blood to battle and understood the general feeling of the time \u2013 that Hitler and fascism were an evil that, for the sake of freedom and democracy, must be fought and destroyed at any cost, even if that cost was the outright destruction of everything and everybody, by modern weapons used in the rage of battle. He wholeheartedly respected society's right to choose that course and did not cam paign against it, scorn it, or harbour the illusion that pacifism could be seen as a popularly acceptable policy. Not being encumbered with absolute principles he had even admitted that if somebody had put a rifle in his hand and Hitler in the sights he would probably have pulled the trigger.\n\nWhen, as I have often done, I confront him with this and say: 'If you would have done that why would you not fight?', I think he replies: 'Because of arithmetic. There are good ways to die and bad ways to die, some can give their lives gladly, some, perhaps most, are wholly innocent, not even knowing what the war is about. They have their lives unjustly torn from them. But in the end death is death and can be counted. Fifty million people were to die in that war. If I had killed Hitler personally many of those millions might have lived. If I had gone out and joined in the killing, then even more would have died, and for what? Was there ever a political cause that was worth the death of fifty million people and the destruction of an entire continent \u2013 with, as far as I could see, every likelihood that nothing would come of it at the end but another turn of the wheel and the next generation slaughtered?'\n\nBut I am putting words into his mouth. Perhaps he would say something simpler, like: 'You are old and you aren't remembering right. I was eighteen. I was not _sure_ of anything except that I was being commanded to take a particular course which involved my having to go somewhere and kill people I had never met and had no quarrel with. I did have a choice. I could either do as I was told and kill them, or I could, if I was willing to take the consequences, refuse to do that. As an immediate decision was required I chose the option that did not involve killing people.'\n\nIn the event, as we have seen, the consequences I took were far from unpleasant. Even though I was frightened out of my wits, the procedure was conducted in a way that was unexpectedly considerate. That was a relief but, nevertheless, there was not a moment in that time when I was not aware, somewhere in my heart, that I had defied my country, had refused to come to its aid, and by that act had become a stranger.\n\nBy virtue of its inherent goodwill and kindness my country had allowed me to continue to live and eat. It had not only allowed me to stand by my convictions, it had respected them. After that I knew I could not, in good conscience, refuse to reciprocate that generosity; that I would always have to respect the convictions of others even though I might disagree with them. I don't expect I have managed to do that all my life, but I have never forgotten that I am living here as a guest, a welcome guest perhaps, but none the less a visitor. I don't regret this. It sets a good curb on righteousness.\n\nThere would be time for thoughts like that later. For now I was a free man, with ten bob and an open rail pass in my pocket, and I was heading for home and corned-beef hash.\n\n# RURAL PURSUITS\n\n## _I. The Shires._\n\nRay and Daisy were glad to see me in one piece and the cornedbeef hash was lovely. Corned-beef hash always is lovely, but in 1944 it was a rare delicacy.\n\nFor me the pleasure of being home and at liberty were tinged with uncertainty about what might be going to happen next. My guess was that the Ministry of Labour would now come after me quite swiftly and try to send me to the coal-mines.\n\nThis suspicion was alerted when, within days of my return, I received an instruction from the Ministry of Labour to attend for a medical examination but, as it turned out, I was mistaken about the purpose of the examination. The Medical Board kept me waiting all day, which was not unusual, but when they did examine me they did so very thoroughly for a long time, which _was_ unusual. It seemed as if they were determined to find something  wrong with me. In the end I grew weary and drew the doctor's attention to some slight adhesions under my shoulderblades.\n\n'Got it!' shouted the doctor. 'That will do. Thank you. Please get dressed now. Good afternoon.'\n\nI later received from Dr Dunlop, our GP, who was associated with the Medical Board, the intriguing information that the Ministry of Labour had clipped a note to my file to the effect that it would be obliged if they would find something, anything, wrong with me which would confirm that I was unfit for work underground in the coal-mines.\n\nI was grateful for that unexplained mercy but even so there was no certainty that the Ministry of Labour had really given up on me. I decided that my best move would be to get myself a job on a farm.\n\nDaisy's brother Eric knew a family who kept a dairy farm in Dorset. He very kindly approached them and the farmer offered me a job as a general farm labourer for 35 shillings a week.\n\nSo, sometime in April 1944, I took my bike and my rucksack on the train to Sherborne in Dorset and started on my new career as a farm worker. As usual I had allowed myself to look into the future and had, in my mind's eye, seen the peaceful idyllic life of the English countryside with the lowing herd winding slowly o'er the lea, the ploughman plodding homeward in tune with the eternal rhythm of the seasons.\n\nI was mistaken about that.\n\nBob, the farmer I had come to work for, was a thin tight-lipped man who ran everywhere and wouldn't look me in the eye. He was without humour and failed to respond to attempts at conversation. I persevered, but there was always an atmosphere of electric tension in the air, the sort that comes between people who have quarrelled. If he had to speak to me I saw his hands trembling as if with terror. It was clear that he genuinely and sincerely hated me. I was quite relieved when I discovered that he also hated almost everybody else.\n\nI hoped to find some other friends, but the local people, though not actually hostile, didn't want to have anything to do with me.\n\nAfter a few months it suddenly occurred to me that I was a free man, that I didn't really have to go on with the wretched business if I didn't want to. So I gave Bob a week's notice and left. He didn't even bother to say goodbye, but I didn't expect him to.\n\nI went home to Finchley for a few days and approached the Quakers to find out whether they would accept me for one of the Friends' Ambulance Units. They wouldn't, so I looked for another farm job. Once again I made the contact through friends of friends and found an informal job with Giles, a well-bred young man who was calling himself a farmer and growing some market-garden vegetables on some land near Oxford. Giles's heart wasn't in farming. It was in celebrating his manhood at sessions of rural dalliance with various young and not-so-young women who moved in to help with the farming.\n\nI myself received offers of dalliance from one or two of the lady guests. These were flattering but I declined them, partly out of nervousness but mainly on the grounds that one stud bull on the premises was enough.\n\nI think Giles took the same view, because he gave me a task which kept me well out of the way \u2013 clearing some acres of woodland below the house. He told me to go to the horse-sale and buy the cheapest strong-looking horse I could find and an old collar and chain that would just about last for the job of hauling out the cut wood.\n\nThe horse I chose looked fine to me but he was named, ominously, Rocket. I noticed, as I was walking him away from the sale, that the men in flat caps who had sold him to me were chuckling together, gleefully satisfied that they had found a sucker and unloaded a practically useless animal on to him.\n\nThey turned out to be wrong about that. Rocket soon became a good friend and we worked well together. He was happy to munch peacefully for as long as it took me to assemble a bundle of cut saplings and brushwood, but, once harnessed to it, he remembered his days with Ben Hur and charged off through the undergrowth at a wild gallop. The only way I could keep up was to ride the bundle, hauling at the reins and shouting. At the edge of the wood he would stop and hang his head, panting heavily while I undid the chain and carefully looped it over his back. The first couple of times I waited for him to recover his breath before walking back into the wood, but after that I left him there, telling him to follow me when he felt up to it \u2013 which he did, a few minutes later.\n\nRocket was also great at pulling out trees. I climbed as high as I could into the tree and hauled the chain up with a piece of washing line. I fixed it and led Rocket out until the chain was tight. Then, hauling on the washing line with all my might, I called him to do his bit. I think he took this as a personal challenge because he whinnied and, with nostrils flared, reared up like a charger, dropping his whole weight forward on to the collar and following it with all the power of his legs. The tree didn't stand a chance. It came over with a crunch and a crackle and he dragged it nonchalantly away.\n\nI can see that with his temperament Rocket wouldn't have been much use for all-day plodding work as a farm horse, but I'm sure he could have had a great career in B-movies.\n\nThe year moved on. Without telling me Giles sold Rocket, so I left.\n\nI tried the Friends' Ambulance Unit again, but without success. November was not a good month in which to look for agricultural work and as I had collected a small income-tax rebate \u2013 something like \u00a38 \u2013 I decided to stay at home until the weather improved.\n\nFor years, perhaps for all my life, I had enjoyed thinking. When doing a traditional job, one which called for a lot of manual effort, I would often go off into a dream in which I would try to conjure up a tool or a machine that would do the job for me.\n\nSugar-beet lifting was probably the most back-breaking job on the farm. I designed a machine for doing that but it was rather complicated. Muck-spreading came a close second. The dung had to be forked off the cart on to the field in heaps. Then the heaps had to be spread with heavy long-handled forks. The machine for that was simple. I have somewhere a sketch of a cart that has long screws holding a loose plate that goes right across the interior of the cart. As the cart moves across the field the screws are slowly turned (by being connected to the wheels), which causes the plate to shove the dung slowly off the back of the cart, where it falls on to spinning paddles which slap the clods of dung and hurl them in all directions.\n\nI showed this sketch to the farmer in Dorset. It caused him to give one of his rare laughs and gleefully tell me that it was obviously ridiculous and that I was completely mad.\n\nHowever, back in London in the winter, I discovered the fountainhead and true depository of all such fancies \u2013 the Patent Office Library. In a huge room, the size of a small cathedral, were thousands of volumes of the most marvellous testaments to human ingenuity, ranging from the absurd to the sublime. I would go there quite early in the morning and look for previous Specifications of whatever device I was thinking about. I had learned how to use the catalogue and codes, seeking out _Vehicles, agricultural,_ _adapted for specific purposes_ _... etc_. And there it was: my muckspreader. OK, so somebody had already thought of it. That meant it was a good idea.\n\nApart from the pleasures of the Patent Office Library, the winter of 1944\u20135 was fairly uneventful. I spent quite a lot of it mooching around the streets of Soho and drinking endless cups of tea in the Swiss cafe\u00b4. So when spring came I was quite glad to get my third farm job.\n\nThis was with two Czech brothers named Drexler, who ran a large market garden near Kidlington in Oxfordshire. The work there was fairly boring but in my spare time I was able to go into Oxford on the bus and haunt John and his friends at Balliol College.\n\nIt didn't last. In June the gods sent a thunderstorm with hailstones the size of golf-balls. With great courage I put on a steel helmet and rounded up the Drexlers' pigs and horses, who were squealing under the bombardment but too stupid to take cover. Alas, my heroism went unrewarded. The pigs and horses were all right but the uninsured greenhouses were totally shattered, which meant that the Drexlers were bankrupt and I was out of work again.\n\nAt my aunt Emily's suggestion I moved to Essex and took lodgings in Wethersfield with their friend, Son Wicks, who was a builder. There I got my fourth farm job, as a general farm labourer on a large local farm.\n\nRay and Daisy were fond of the country. I knew that they would have liked to be able to afford a country retreat, especially one near Bloomsbury-on-the-Marsh. I mentioned this to Son Wicks and he immediately suggested a vacant cottage at Penny Pot, which was only about four miles from Bradfields, the Meynells' house near Toppesfield. Ray and Daisy bought their cottage for under \u00a3600, including renovations, and were very happy indeed.\n\nOne day I had an unexpected stroke of luck \u2013 I found a motorbike. It was blocking up a gap in somebody's hedge. I was able to buy it for \u00a33 and an old iron bed-head, which went into the hedge in its place. Under the dirt it was a 1926 New Imperial 350cc side-valve chain-drive motor-cycle with a square tank.\n\nIt was a good bike, in its fashion. It never went faster than about 40 m.p. h., it always did at least 120 miles to the gallon and it had absolutely no springs. Riding it was like riding a pneumatic drill, but it was my own and I loved it.\n\nMy memories of that summer are hazy: mainly of work, of hot harvesting, of stooking the sheaves of wheat after the reaper and binder had passed, threshing out horse-beans, curry-combing the two big Suffolk horses. And riding blissfully along deserted lanes in the cool of the evening, the engine going sweetly, sounding like a clapped-out sewing-machine with asthma.\n\nOne memory is very clear. It was a morning early in August. I was in Son Wicks's yard getting my motorbike out when his wife, Olga, came out with a newspaper.\n\nShe said: 'We've dropped something nasty on Japan.'\n\nFor some reason the news that an atomic bomb had been dropped on Hiroshima filled me with an absolute physical terror, as if, for a moment, I had seen the end of the world. Reading about it was like hearing a scream inside myself, a scream which silenced everything else in my head and left it numb. After it came an extraordinary sense, a conviction that somewhere in my mind I had known about it and had been waiting for it to happen. That was completely impossible, of course. The only prior connection I could recall was Daisy's remark, made six years earlier, that 'they' now had bombs that could wipe out half a town. I hadn't forgotten that. They didn't have them then. Now they did.\n\nThere was nothing to be done except get on my bike and go to work as usual, wondering what was to become of us.\n\n## _II. Engines._\n\nThe cottage in Essex celebrated my parents' accession by producing a magnificent crop of sloes and greengages, which Ray and Daisy carried back to Finchley and made into wine.\n\nThe sloe wine was particularly delicious, dark red and crystalclear, strong and not too sweet, with a long, mellow, exotic aftertaste of, well, tiny bitter plums. The family wines were a solace and delight in the dark years of deprivation. And for Ray a source of reassurance and hope for the future was to be found in the very last page of the Ministry of Food Handbook on Home Winemaking, whereon it was written... A GOOD WINE CAN ALSO BE MADE FROM GRAPES.\n\nBuck's Farm was sold and I was sacked, but I still hoped it might be possible for me to join either the Friends' Ambulance Unit or one of the relief teams that were being set up to try to help reorganize life in the ruins of Europe. I asked the Red Cross and other organizations but was turned down, on the perfectly reasonable grounds that only people with specific skills were being sought. So I was back in London for the winter of 1945\u20136, mooching about and looking for something to do, or make.\n\nDaisy suffered a lot from fibrositis in her arms and shoulders. Any heavy activity would cause her pain, so I decided to make her a washing machine. I knew more or less how they worked. A washing machine was a round metal tub on legs that had in it a sort of three-cornered paddle which rotated backwards and forwards, sloshing the clothes about in the sudsy water.\n\nIt occurred to me that if the whole tub were to be rotated backwards and forwards, the paddle could be part of the tub. This would simplify the design and avoid the problem of how to make a leak-proof hole in the tub for the paddle spindle to go through.\n\nI pictured a large deep container, a milk-churn in fact, with a smooth piece of wood three inches (75mm) square and about eighteen inches (45cm) long, bolted to the wall of the churn on the inside. When the churn was rotated backwards and forwards with suds and clothes in it, the piece of wood acted like a breakwater, creating the amount of sloshing necessary to loosen the dirt from the clothes.\n\nThe rest of the design was purely mechanical. I made a strong four-legged wooden frame with an open top in which the churn could sit. I balanced it on a pivot at the centre of its base and held it upright with four small rubber wheels fixed to the top of the frame. Thus the churn could be rotated in an upright position inside its frame. After that all I had to do was buy a large electric motor with a gearbox from the army surplus shop in New Oxford Street and fix it to the frame, coupling the churn to a crank plate on the motor-shaft by a connecting rod. It functioned correctly; when the motor was switched on the churn rotated backwards and forwards a quarter of a turn each way at a rate of about sixty sloshes a minute.\n\nIt also worked triumphantly as a washing-engine. It washed the clothes really well, beautifully clean and quite undamaged. It would even rinse them if you ran water through it with a hose from the tap.\n\nDaisy was absolutely dismayed by it. Just looking at the thing, standing like a home-made cannon in a corner of the scullery, filled her with dread. Hearing it in action alarmed her terribly, though both Ray and I thought its noise magnificent, like the sound of tramcars copulating. This simile, though apt and evocative, did nothing to improve Daisy's confidence.\n\nHowever, Daisy was nothing if not loyal to the cause. She screwed up her courage and consented to use the washing-engine.\n\nShe was, perhaps understandably, cautious in her approach to it. She would drop the clothes in at arm's length, fill it with hot water from a hose, reach out and pour in the soap-flakes, drop on the lid and then, with the door to the kitchen held open, flick on the switch and run out, locking the door behind her. I wondered why she locked it. It was almost as if she was afraid the engine, which was going _'Yerk_ - _graunch-_ _slop, yerk_ - _graunch_ - _slop, yerk-_ graunch-slop' to itself in the scullery, was going to clamber out and chase after her.\n\nDaisy admitted that this was precisely what she was afraid of and, I have to confess, her fears, although exaggerated, were not altogether unfounded. The washing-engine did have an unforeseen peculiarity. Due to an inevitable degree of flexibility in the construction of the frame it had a tendency, while working, to walk, or rather shuffle, about the scullery in an idle inquisitive sort of way.\n\nDaisy said that was the worst part; coming back after the engine had been going for ten or fifteen minutes and not knowing for sure where in the scullery she would find it, though often she would hear it nudging softly at the other side of the door as if mutely pleading with her to open it.\n\nOne day, as she came back, she heard a different, far more ominous, sound. The healthy ' _Yerk_ - _graunch_ - _slop_ ' had changed to ' _Clonk_ - _scrindle-_ _yaaha, clonk_ - _scrindle-_ _yaaha'_ the last sound being a desperate bubbling sigh. She opened the door an inch and peered in. She saw a terrible sight. The washing-engine had tripped over and fallen. The poor thing had spread its contents on to the floor and was lying on its side, writhing and reciprocating in a heaving pool of foam and underclothes.\n\nDaisy closed the door and ran to fetch me but, even as she did so, the soapsuds reached its electrical works and the ill-fated engine blew the fuses and expired. All the lights in the house went out.\n\nAs I removed the debris and cleared up the scullery, I could see that the project hadn't been altogether successful.\n\nAnother device, a high-speed food-mixer built into a screw-top coffee-jar, had turned out to be very acceptable. Daisy was pleased with it, even proud of it on my behalf, but alas, that too eventually succumbed to fate. One Sunday morning, perhaps because she hadn't screwed it in properly, the jar came adrift in full flight and distempered the scullery from floor to ceiling with Yorkshire pudding batter.\n\nApart from a bird-scarer which trapped my aunt Emily and roared at her at two o'clock in the morning, a fan-heater, an inspired toilet-roll holder, the first electric rotary lawn-mower and a simple but appallingly dangerous electric circular saw on a kitchen table in the back garden, I am fairly sure I didn't make any more useful engines that winter, but even so I dare say Ray and Daisy were secretly relieved when, early in 1946, I decided it was time for me to be off into another year of 'work of national importance'.\n\nThe year started well. I visited Dartington Hall and had a meeting with Leonard K. Elmhirst, who, with his American wife Dorothy, was the owner of the hall and founder of the estate companies. It was a good meeting. L. K. listened to me rabbiting on about my pet subject, the feeling that the men coming back from the wars were not going to be content to step back into laborious archaic working practices. Power tools and labour-saving methods were going to be needed. L. K. agreed enthusiastically and we got on famously.\n\nAs a result of that meeting I got my fifth rural job: as a winchtractor driver for the Dartington Hall Woodlands Department, with a private brief from L. K. to report in due course on the potential for mechanization in the forestry work of the estate.\n\nFor the first time in my life I felt really well set up, as if I was at the start of what could perhaps become a career. I didn't know exactly what a winch-tractor driver did, and I had no inkling that it would turn out to be the most spectacular job I had ever taken on.\n\nThe work wasn't being done on the Dartington estate itself but in the woods near their sawmill at Moretonhampstead, a small town high up on Dartmoor. There I took lodgings with a Miss Overdown, who lived in a small house near the mill.\n\nMy tractor was a petrol-driven Fordson with a large simple winch on the back which had cast-steel spades attached. These could be dropped down so that when the winch was pulling the spades would dig into the ground and stop the tractor being dragged back. The machine was immensely strong. With the wirerope hooked around the trimmed trunks of the felled oak trees and using the stumps as pulley-blocks, the tractor was able to haul them across the rocky precipitous terrain and stack them by the track. That was the main part of the job and it was delicate as well as brutal in that one false move could have sent a loose five-ton tree trunk bounding down into the river below. But perhaps the greatest test of the tractor's strength came when I had to pull one of the really big trees out by the roots. This only happened once or twice \u2013 I am glad to say \u2013 when the tree in question was directly in the way of the track. Once the wire rope was firmly hooked around a high branch and the winch began to pull, the tractor was dragged back like a toy until the spades were firmly dug in. Then the real tug-of-war began. The wire pulled so taut that it thrummed like the string of a double-bass. The huge tree leaned over but held firm. Then, as I opened the throttle, the front of the tractor was lifted clean off the ground and slowly rose until all four wheels were in the air and it was standing on the dug-in spades. I had discovered the trick of playing it on the clutch, lifting and dropping the tractor in rhythm so that the tree started rocking on its roots. I did this until, at last, with a series of cracks like pistol shots, the great tree came away. The winch pulled it half over and then its own weight brought it down, with a swoosh of air sweeping through the branches and an earth-shaking thud as it hit the ground. Rocket would have been proud of us.\n\n## _III. Roy's World._\n\nHome life at Miss Overdown's residence in Moretonhampstead was, to put it simply, interesting. She was a gross, florid lady, probably in her mid-fifties, who had been, in her day, a much admired and sought-after prostitute in Plymouth. She was not in the least ashamed of her choice of profession and was happy to tell me how she came to choose it.\n\nShe said: 'When I were a little girl, my da sat me on his knee and he said to me: \"Chloe\u00a8, my girl, when you'm grown up, you'll choose what you'm a going a do. Now you make sure you do some-en you enjoy!\" Well...' she giggled, put her finger to her cheek and cast her eyes coyly down, '... there bain't naught I enjoy more nor fucking. So there weren't really no choosing to it.'\n\nMiss Overdown's house was small and dirty and, even though it contained very little in the way of furniture, it was a tip. This didn't matter really because although she took in lodgers, these were mostly muddy labourers like me. She had a son, a pastyfaced youth in his late teens, surly and vaguely hostile. When he saw me and was told I would be staying there he said: 'You're stupid staying here. I'm not.' That was partly true. He had obtained a broken-down saloon car and had had it dumped in front of the house as if it was parked there. He spent all his spare time sitting in it, listening to the wireless and eating lumps of bread and jam.\n\nSometimes Miss Overdown would go to the front door and shout to him: 'Eric, you coming in to tea, then?' To which he would, sometimes, reply: 'I'm not eating any of that old muck!'\n\nThat was overstating it a bit but he did have a point. The wartime sausage, the basis of Miss Overdown's cuisine, was never very palatable at the best of times but when boiled in a big saucepan with plenty of water and a few handfuls of macaroni it would swell up into a pallid water-filled bulb. This, when punctured on the plate, would sully the surrounding macaroni with muddy liquid and gristly fragments that imparted to it a very faint flavour, like the smell of wet cardboard. One had to be hungry to eat that.\n\nThe only other lodger, Roy, was about my age. He was a contract worker with a gang which did ploughing for the War Agricultural Committee. Roy made a great impression on me because he was the most absolutely ignorant person I had ever met. I use the word 'ignorant' here in the old-fashioned bucolic sense as meaning 'without courtesy or consideration'. Until I met Roy I hadn't realised how simple life could be if you really, sincerely, had no interest in or concern for anybody else at all.\n\nWhen Roy first barged in through the door of Miss Overdown's front room I knew at once, from the way he moved, that he was a man totally committed to one purpose: getting what he wanted.\n\nRoy didn't seem to have noticed me. He peeled off his grimy shirt and dropped it on the floor. Then he flopped down into the broken easy chair, unlaced and hauled off his muddy boots, revealing shiny dirty bare feet that filled the room with an abominable stench. He sighed with relief and wiggled his toes. Then he shouted: 'Missus! Where's my tea?'\n\nThere came no reply. He stood up, dropped his trousers off, followed them with grey underpants, stepped out of them and walked, bollock-naked and smelly, out of the room and up the stairs.\n\nMiss Overdown came in carrying the big saucepan of boiled sausages and macaroni. She looked at the clothes on the floor and shouted, 'What's this?'\n\n'That's yer washing,' came the reply from upstairs.\n\nMiss Overdown kicked the dirty clothes into the hall.\n\n'His rent hasn't come for two weeks,' she said reproachfully.\n\n'Oh dear, I _am_ sorry,' I replied.\n\nRoy came in wearing tweed plus-fours, a dinner jacket and no shirt.\n\nHe dropped more dirty clothes on the floor.\n\n'What's that?' asked Miss Overdown.\n\n'Yer washing. I'm going out.'\n\n'Your rent's not come for a fortnight.' she said.\n\n'Fuck-all to do with me.'\n\nRoy sat down at the table, slopped most of the macaroni stuff on to his plate and started shovelling it in.\n\n'You can't stay if your rent's not paid,' said Miss Overdown.\n\n'Can't help that. You'll just have to write to the \"War Ag\" and put in for it, won't you?'\n\nHe reached out for the saucepan.\n\nI don't know what I was doing, just standing there. Perhaps I was waiting to be introduced or something. I suddenly came to and, realising that if I wanted any food I would have to act, grabbed the handle of the saucepan and whisked it away.\n\n'Yur!' growled Roy.\n\n'Oliver Postgate,' I replied. 'Pleased to meet you.' I poured what was left of the macaroni stuff on to my plate and started to eat.\n\nRoy looked hard at me as if he was trying to work out what I was.\n\n'You live here?' he asked.\n\nMy mouth was full so I couldn't reply.\n\n'I said, d'you live here?' he growled.\n\n'Yesh.'\n\nRoy went on looking at me. Then he belched loudly and stood up, knocking over the chair.\n\n'I'll want the boots in the morning, Missus,' he said and walked out.\n\nMiss Overdown and I looked at each other. She picked up the chair.\n\nI was suddenly angry. 'But... but... but, is he always like that?'\n\n'Oh, yes.' She looked at me as if she was surprised by the question. 'He's a pig. They'm all like that; whole family. They'm ignorant.'\n\n'Ah, well, I suppose his mother loves him.'\n\n'I doubt that,' said Miss Overdown.\n\nRoy showed his true mettle a week or two later, when Miss Overdown, who had been limping and sighing for some days, consented to show me what she called 'the place on her leg'. This was a very large ulcer or something that had gone septic. The patch covered half her thigh. It was red and throbbing. She had all the symptoms of advanced septicaemia. I ordered her to bed and ran to the doctor's surgery. It was closed. I stuck an urgent note through the door. I ran to the call-box and rang the hospital in Exeter but they wouldn't send an ambulance without a doctor's say-so. I went to the bakery where her son Eric worked and asked to see him. He came out. I told him his mother was very ill and asked him to go back to the house and wait for the doctor. He looked at me as if I was mad and went back into the bakery.\n\nI went to the grocer and bought some food. When I got back to the house Roy was there. He saw my basket. 'You doing tea, then?' he asked.\n\nI saw a piece of paper on the floor and picked it up.\n\n'What's this?'\n\n'Dunno. Doctor or somebody came in to look at Missus upstairs. He gave me that. Prescription or something.'\n\n'You going to get it made up?' I asked.\n\nRoy looked blank.\n\n'She'll probably die if she doesn't have the medicine.'\n\n'That's what he said.'\n\n'Right. So are you going to get it?'\n\n'Fuck-all to do with me,' said Roy.\n\nI looked at the prescription. It was for M&B, one of the new wonder drugs, like penicillin.\n\n'I'd better get this made up,' I said.\n\n'Haven't had my tea yet,' said Roy.\n\n'Fuck-all to do with me,' I said and ran off up the road.\n\nThe chemist-cum-hairdresser's was closed but I banged on the door until somebody came to open it. The chemist didn't have the medication in stock but he realised the urgency and told me to go round to the vet. The vet made up the prescription and charged me half-a-crown for it.\n\nI returned to Miss Overdown's house. Eric was outside in his car with bread and jam. Roy was sitting at the table in the front room, glowering. Miss Overdown was in her bed, snoring in a heavy, sinister way. I managed to wake her up and got her to take four pills and a glass of water. She fell back to sleep.\n\nI went downstairs. As I went into the front room I said: 'You doing tea then, Roy?' He had opened his mouth to ask me the same question but I had got there first. So he just glared at me.\n\nThen I made a suggestion. I said: 'If you speak to Eric very nicely, he might just let you have a piece of his bread and jam.'\n\nRoy got up at once and walked out to Eric's car. I followed a minute or two later. Roy had forced himself halfway in through the car window and seemed to be shouting in Eric's ear, but I didn't think it likely that he would get any bread and jam.\n\n'Fuck-all to do with me,' I said to myself and walked down the town to buy some fish and chips from the van that came round once a week. I was learning the rules of ignorance. As far as I was concerned Roy could feed himself or bugger off. In the event he buggered off. I never saw him again.\n\n## _IV. Beatrice._\n\nFor the next week or two I did what I could for Miss Overdown. I cooked food when I came home, made breakfast, set out her pills and even bullied Eric into coming home at midday to give her bread and jam. Miss Overdown seemed embarrassed by this attention and so, strangely, were some of the local people, who watched me, nudged each other and sniggered. I found this unnerving and began to wonder whether I had committed some social gaffe. Perhaps, for that time and place, Roy's attitude would have been more acceptable.\n\nAt weekends it was a great joy to escape from Roy's world and drive down on my motorbike to visit friends at Dartington. Going there was like stepping out of the dark ages into friendliness and light, from agitation and suspicion into quiet and acceptance.\n\nI had several friends at Dartington but it was a special pleasure to visit Cecily Martin and her family, who lived at Yarner Barn.\n\nOn one of those happy weekends I unexpectedly encountered my uncle, G. D. H. Cole, who was also staying with Cecily Martin while he attended a learned conference at Dartington Hall.\n\nCecily Martin was relieved to see me because my uncle Douglas, though a much-loved friend, was so dry and formidably intellectual that she had completely run out of conversation. She hoped I might be able to think of something to do or talk about that would take his attention.\n\nI was flattered, and of course accepted the invitation but, as I felt the challenge would be more to my ingenuity than my intellect, I suggested to Cecily that Douglas and I should make a muchneeded kennel for their dog.\n\nAfter he had recovered from the initial shock of being appointed carpenter's labourer, the Chichele Professor of Social and Political Theory and Fellow of All Souls at the University of Oxford put on a leather apron and knuckled down with a will. He took instructions well and was neat and careful in his work, though a shade slow.\n\nHowever, as the kennel began to take shape it became clear that something was bothering him.\n\nHe said: 'It's all very well making a kennel for a dog, but how are you going to explain to the dog what it is?'\n\nI laughed the question aside but he persisted: 'Look!' he said, quite firmly. 'This is a serious situation. The dog is just sitting there. It's a dumb animal. We can't explain things to it. So what are we going to do?'\n\nIndeed the dog, whose name I believe was Digby, was sitting there, but he was not _just_ sitting there. He was watching us closely as we worked.\n\nI looked at Digby and said: 'D'you want to try it, then?'\n\nWithout a flicker of hesitation Digby jumped up on to the bench, climbed into the box, turned around a few times and then curled up, thus giving us a comprehensive demonstration of the settling-down procedure.\n\n'All right, is it?'\n\nDigby stood up, wagged his tail and jumped down. It was all right.\n\nI mention this simple incident not so much because I found it surprising but because of the effect it had on Douglas. He was, if one can say such a thing of anyone so rigorously erudite, gob-smacked. He stood there, still as a crane, for some minutes while his mind was coming to terms with the implications of this impossible event. Then he gave a loud _cronk_ of laughter and, apparently, accepted it wholeheartedly and without argument. From then on he was more than happy to discourse to Digby on the fine details of his kennel, asking his advice about the positioning of the door and roof. This puzzled the dog because he really had no idea what Douglas was talking about. But he understood perfectly when Douglas invited him to assess the exact positioning of the precious aged blanket that was his bed, and was happy to try it several times before they were both satisfied. In fact they were more than satisfied: both Digby and Douglas were really happy and pleased with their day's work.\n\nOn another of those slightly magical weekends I made one of the most exciting discoveries in my life.\n\nRooting through the outhouses at Cecily Martin's farm, I came across something large covered with a cobwebby tarpaulin. I lifted it carefully and found underneath it the car which Daisy had owned and driven so many years before. Of course it wasn't the same car but it was identical, a bull-nosed Morris Cowley tourer with greygreen oil-cloth seats. I squeezed the bulb of the horn. Probably for the first time in years it went ' _SqueeeeeA-OINK_ ', just like Daisy's used to. Wildly excited I ran in and told Cecily what I had found. 'Oh dear,' she said. 'Is that old thing still there? Do me a favour, would you? Get rid of it for me.'\n\nI gave a whoop of delight, jumped on my bike and drove down to the garage on the Plymouth road.\n\nThe estimate for putting the car in order came to \u00a337, a small enough charge for the work involved but beyond my means at the time. I telephoned Daisy, long-distance, and told her I wanted to buy something on her behalf for \u00a337 but I didn't want to tell her what it was, yet. She was puzzled, then intrigued and in the end consented.\n\nI told the garage to go ahead. The proprietor warned me that, what with getting parts and so on, the work might take a few weeks. So I swallowed my excitement and returned to Moretonhampstead.\n\nAs requested by L. K. Elmhirst, I wrote a tactfully worded report about the prospects for mechanization at Moretonhampstead. There wasn't a lot of scope there. The main problem was with the lorries and the timber-carriage, which were all road vehicles, with worn-smooth road tyres. These either couldn't get up the rough hills because of wheel-spin or else they got bogged down. This meant that I had to spend a lot of my time using the tractor to pull them out instead of getting the trunks stacked ready for the sawmill.\n\nI sent the report privately to Mr Elmhirst. I received no direct reply but a day or two later I was summarily transferred to a subsidiary company that was planting seedling trees near Newton Abbot.\n\nI was a bit puzzled by this but, on the whole, relieved to be leaving the gruesome realities of life on the moor.\n\nI returned to Dartington and stayed at one of the houses in the junior school which had been turned into an agricultural hostel and general lodging-house. After Miss Overdown's establishment this was a really pleasant place to live, especially because the company there was so varied that conversation could be quite lively and interesting.\n\nThe work I was now doing was much more varied than the logging work on the moor. As well as planting seedling trees we put up fences and gates, and trimmed the weeds from young plantations and the dead branches from the trunks of larch trees in the dark woods.\n\nThere was plenty of scope for labour-saving devices in these activities and I sketched several out, including a post-hole borer, two fence-post drivers and a steel-wire strimmer. None of these were world-changing innovations but they were all worth looking into. But first there was a small question in my mind that needed clearing up.\n\nMy informal mandate from L. K. Elmhirst was to report on the potential for mechanization in the forestry operations with a view to suggesting some ideas. That was fine, but I had noticed, in the small print of the work-contract I had been given, a clause to the effect that any inventions, ideas or patents that an employee might develop would become the sole property of the Company.\n\nI wrote a note to the head of the Woodlands Department mentioning that, as requested, I had been working on some ideas, but before discussing them I would like to have a chat with him to clarify the situation about the ownership of the patents, etc.\n\nThe immediate effect of this was similar to that of my report about Moretonhampstead but more final. I received no answer, just a week's wages in lieu of notice and my cards.\n\nI was very rattled by this. I couldn't have been sacked because my ideas were no good, because nobody had even looked at them. I asked to see Mr Elmhirst but was told he was still in America. It occurred to me that perhaps his idea of putting me in to report (spy?) on the Woodlands Department had not been popular. Perhaps, who could say? The only thing to do was write off that great career and go.\n\nSo, early one midsummer morning, under a blue sky, I threw my bag on to the back seat of Daisy's open car and wound the starting-handle. As always the engine started on the first pull, and the gearbox sang as I drove away, glad to be gone, happy to be starting another life, but somewhere else.\n\nAs I drove I looked back over the recent past. For most young people the years between seventeen and twenty-one are important and formative. Those are the years in which one's character is formed, the years in which lifelong friendships are made, in which one finds out who one is and where one is going. Above all they are the years for passionate relationships and love affairs.\n\nFor me \u2013 and without a doubt for thousands of others whose lives were complicated by the war \u2013 those years didn't work out like that.\n\nI looked back on my time as a farm worker and saw no rompings in haylofts with buxom milkmaids. I would have liked a bit of that and one or two had been very romp-withable, but I was diffident and different and didn't really belong. I usually got on quite well with the people I worked with but, not speaking quite the same language, I never felt fully at ease in their world.\n\nEven so, I didn't regret the years, didn't feel they had been wasted. I was strong, I knew how to work, I did it well, I enjoyed it and I loved the smell of hay and cows and rain. Also, even if I was occasionally tripped into patches of confusion and anxiety, I could think quite well. As for love affairs? There would be time, when there was a place.\n\nI had some money saved up. It was only a few pounds, but I could take life easy for a month or two. Then I would see what the world had to offer, or maybe what I had to offer the world.\n\nBut, most of all, as I was bowling along on that sunny day, I was looking forward to bringing Daisy her new-old car.\n\nDaisy was delighted. When she was introduced to the car, she asked me politely what its name was and I heard myself reply: 'Beatrice.' There was no argument about the name and Beatrice was duly welcomed into the family. Daisy and I bought canvas and made her a new hood. We bought plastic sheets and webbing to repair her drop-in side windows. We poured woodworm-killer into the interstices of the bodywork to discourage the death-watch beetles, painted it and polished her brass nose.\n\nDaisy was nervous of driving Beatrice in the London streets so, a couple of weeks later, my cousin Peter, my uncle Eric and I took turns driving her to the cottage in Essex.\n\nHaving Beatrice with us in Essex was a great luxury. We no longer had to wait for rare, unpredictable buses or have to get out the old bicycles, which Ray and Daisy were finding hard work. As well as visiting friends we could go shopping or go swimming in Gosfield lake, where the water was warm and brown as Windsor soup.\n\nFor Ray and Daisy that summer must have been like coming back to a much-loved time after a long absence. With their own cottage and their own car they were once again fully equipped for Bloomsbury-on-the-Marsh visiting. The question that remained in my mind was whether the Bloomsbury-on-the-Marsh that they had known and loved in the twenties and thirties was still there.\n\nOddly enough, it was.\n\nDuring the war Ray and Francis Meynell, along with many other capable, well-educated, middle-aged persons, had become civil servants \u2013 and very good at it they were too. Like everybody else they needed somewhere to relax so, albeit more rarely and with some austerity, the weekend parties had continued. Indeed I think Francis felt that to manage to carry on with fun in the face of dreadfulness was almost a patriotic gesture, it showed that war could not defeat them. So, give or take the effects of age and changes in their personal lives, the dream, with variations, had survived.\n\nIn 1946 the possession of any sort of car conferred great distinction on the owner, and owning Beatrice, who was a reincarnation of the one she had driven along the same lanes so many years ago, was for Daisy a nostalgic pleasure which she loved to share with her friends.\n\nI remember how excited she was the day she drove us over to a lunch party at the handsome house of a well-known politician, because our host was known to be a lover of old cars. He happened to be in the garden when we appeared through the gates and he came to greet us, rejoicing at the venerable beauty of ours.\n\n'That is a genuine bull-nosed Morris!' he cried.\n\n'Her name is Beatrice,' announced Daisy with pride.\n\n'Ah,' he said thoughtfully. 'Is it? Well, then... Excuse me. Would you mind just waiting there a moment?'\n\nHe walked briskly back to the house and disappeared around the side. I heard the sound of large doors being closed and then he came out and waved us in.\n\nLater, after much lunch and laughter, I asked our host what had been happening when we arrived. He led me through passages and cloakrooms to a door that opened into a garage. There stood a long beautiful Lancia sports car, low, sleek and grey.\n\n'That's it,' he said.\n\n'It certainly is!' I replied. 'But why...?'\n\n'The name,' he explained. 'His name is Dante, you see. They mustn't meet, of course.'\n\n'Oh, I see! Of course not,' I said, nodding sagely, and we came away.\n\nOf course I was lying. I didn't see. Being twenty-one and ignorant I didn't know the legend of Dante and Beatrice. When I told my father what had happened he chortled very happily, but the old so-and-so declined to explain to me why it was they mustn't meet.*\n\n*\n\nThat happy, silly, nostalgic, summer flowed by. We returned to Finchley and reality. There, in October 1946, I heard that the Save the Children Fund had accepted my application and was offering me a job as an ambulance-driver for one of the Relief Teams they had set up in Germany. So at last I was off to what was left of the wars, or rather, to what the wars had left of Germany.\n\n* Beatrice: a nine-year-old girl glimpsed by the poet Dante Alighieri when he was also nine. She became the love of his life and the inspiration of his work but it seems they never actually met. I still don't know why they mustn't meet. OP.\n\n# THE HONEY-BARREL\n\nWe crossed the Channel by night, waited on a cold quay in Ostend and then boarded an old decrepit train which bumped along on uneven rails.in I was wearing standard army-issue battledress. This, although physically comfortable, was mentally quite a shock, I having refused to put the uniform on during the war. I was also slightly embarrassed by the fact that as a civilian in uniform I had officer status and bore the honorary rank of major, even though the post I was about to take up was that of a lowly driver attached to a Relief Team in the Hamelin town of Brunswick \u2013 or rather, Braunschweig.\n\nI must have slept on the train because I remember waking up and looking out of the window to get my first sight of devastated Europe. I saw fields and hedgerows, neat houses and one or two slow-moving people, well wrapped up against the chill November wind, who were pulling tiny four-wheeled carts, rather like doll's cots on wheels, along the rough roads. The countryside didn't look the least bit devastated and the people seemed to be doing what they had been doing for centuries. I saw an ox-cart loaded with dung, the steam from it hung in the air like a mist.\n\nThen the train passed through a town (I don't know its name) and that was flat; rubble. It looked as if it had been trampled by giants. There was no sign of habitation, no sign of any standing building. Every single brick and stone had been broken away and become part of the heap. In places the debris had been pushed aside to uncover streets, showing shallow dry canals with beds of broken cobblestones. Except for one or two slow people pulling their carts along these, the place was deserted. I had seen quite a lot of bomb damage in England, but nothing quite as absolute as that crushed town.\n\nThe train didn't stop. There would have been no point.\n\nI laid my cheek on the dirty window and watched the landscape pass, wondering what exactly I had come for, what a Relief Team would do. Where, I wondered, were the people who had lived in that town? Would they all be standing, huddled in some huge pen, waiting for me to come along and give them food, clothes, houses and hope? Possibly... I didn't know. I reminded myself that I was here to play a simple part in a huge, highly organized emergency exercise. At that moment I felt relieved that I was just going to be a driver.\n\nOur journey ended at the Red Cross headquarters in Vlotho. There, in the dull light of a overcast November afternoon, I was able to walk out into the street.\n\nThe town didn't seem to be particularly damaged, but I had an almost overwhelming feeling that it was near to death.\n\nHow could a town be near to death? I stood and looked at it. Everything I saw was old, worn out, or broken. The road was pot-holed, with cobbles lying loose. The street-lamps were rusted and bent. Guttering hung from the chipped eaves. The cracked and tilted pavements and the crumbling house walls were worn and grimy, as if dirty things had been dragged against them for years. But there were no things there; there was nothing to show that this was a place where people lived, just bare steps, bare paintpeeled doors, hinge-pins sticking out of cracked gate-posts to show where there might once have been wrought-iron gates.\n\nA tall white-faced man was walking in the gutter. He was handsomely dressed in a leather cap and a leather greatcoat with wide lapels and carved bone buttons. As he came closer I saw that from the knee down his coat was in tatters and that string was wound around his polished boots to keep the soles on. In his little cart was a large, ornately carved barometer. He stopped and gave me an emaciated smile.\n\n' _Zigaretten_ _?_ ' he suggested.\n\n'By all means,' I replied, taking out a packet of cigarettes and offering him one. He took four, smiled, bowed, put them carefully into his pocket and then, absurdly, picked up the barometer and tried to push it into my hands. I remember thinking it was peculiarly ugly.\n\n'Oh, no, no, no, I already have a barometer,' I explained, untruthfully.\n\nThe man looked distressed and thoughtful. From an inside pocket he slowly withdrew a rectangular object wrapped in faded cloth. It was about the size of a small book, but when he unwrapped it I saw that it was a very thick picture-frame, moulded in the form of fat leaves and fruits. These were brightly painted and embellished with gilded edges and purple spraywork. In the centre of it was an oval glass covering a faded sepia photo of a whiskery man wearing a helmet with a spike on the top. I had seen pictures like that before.\n\n'But that's come off somebody's grave!' I said.\n\nThe man stood up very straight. ' _Finckelglockner!_ ' he said proudly, and clicked his heels.\n\n'Oh, is it?' I said. 'A relative, perhaps?'\n\n' _Zigaretten?_ ' he suggested.\n\nI held out the packet. He took four more cigarettes, looked long and fondly at the portrait and then, turning his head away, reached out as if to hand it to me.\n\nI didn't take it. I fled. Muttering good wishes I backed off and made my way briskly to the Red Cross headquarters where there was laughter and chatter in the canteen.\n\nThat was my first encounter with ' _die grosse Knappheit'_ (the great shortage), the period just after the end of the war when money lost its value and cigarettes became the only real currency. I had heard about this in England and, although I didn't smoke, I had been advised always to carry cigarettes because sometimes that was the only way to pay for things.\n\nThe Save the Children Fund Relief Section that I was attached to was accommodated in a large house in a pleasant suburban street which had somehow escaped the bombing. Braunschweig was a patchwork of almost undamaged areas and areas of total devastation \u2013 one just walked, or drove, through the dead bits. The team drivers, myself and a young man named Chesley, were billeted in a couple of rooms over the team's garage on the outskirts of the town.\n\nThe Relief Team consisted of a team leader, who was absent when I arrived, and about eight other individuals, each one with a special interest \u2013 youth groups, the blind, children's homes, etc.\n\nThe team's transport fleet had been provided by the army \u2013 two heavy lorries, two 2-ton Army ambulances and two trucks, all in rather poor condition, without heaters. These may have been standard issue for units on active service, but they were not ideal for taxiing individual social workers about on their visits. Driving in a 3-ton army lorry when it is well loaded can be quite comfortable but with no load it gets thrown about like an empty box. But that's what we had. My job was to drive the members of the team. Their job was to hold on tight and put up with the jolting.\n\nAbout once a fortnight a delivery of relief materials came. Shortly after I arrived an HQ lorry dumped a large wooden barrel containing honey, and a steel drum about the same size, labelled cod-liver oil, along with a small sack of louse-combs, into the elegant front garden.\n\nAs nobody in the team had any idea what to do about these massive items I found myself with the task of working out how to deal with them.\n\nI enlisted the help of our mechanic, Giff. We manhandled them into a truck and took them out to the team's garage. On the way we stopped at the Army Medical Centre where I told them I needed five hundred empty small medicine bottles and a roll of plastic tubing. These were fetched and handed over without question.\n\nAt the garage we rolled the drum of cod-liver oil on to a stand, chopped off ten lengths of plastic tubing and, at carefully timed one minute intervals, set up ten separate siphons into ten bottles.\n\nIn principle that was an excellent system. In operation it would also have been if the oil had consented to flow at an even rate. Unfortunately some bottles filled faster than others, so we had to watch very closely and pounce on each one as it filled. Sometimes three or four would overflow at the same moment, so if either of us lost concentration for a second there would be a slow flood. Changing the bottles was slippery and messy and the tray we had put under them was soon half-full of spilt oil which had to be baled out with a tablespoon. But we persevered, slurping tubes and bottles in the smelly mess, non-stop for the next eleven hours. When we had screwed on the last bottle-top we tipped all five hundred bottles into the bath to wash them. The whole place stank of bad fish.\n\nCompared with that the task of dealing with the barrel of honey was simplicity itself. Giff and I stood the barrel upright in the back of one of the 3-ton lorries, tied it on securely and smashed in the top with an axe. As I expected, the honey had set hard. I borrowed a strong garden spade, scrubbed it thoroughly, put it in the back of the lorry with a carton of cod-liver oil bottles and the bag of louse-combs, and we were ready to move out and relieve Germany.\n\nGiff and I went back to the house to tell our proud news to the team, but there was nobody there. However, that evening I mentioned to Helen, a nice middle-aged lady whose interest was in Children's Homes, that the goods were now ready for distribution. She was pleased and suggested that we take them out the next day.\n\nWisely Helen brought one of the sofa cushions with her to ease the journey. That meant we could drive the lorry at a fair speed without causing her to yelp.\n\nThe Children's Homes (perhaps we would call them orphanages) were mostly run by the various Churches, Evangelical, Catholic, Lutheran, etc., in conditions of utter poverty and surprising cheerfulness. The first Home we visited was in a large, part-ruined bungalow deep in the country. It stood in an unkempt but welltrodden garden with the front door wide open. We walked in. The inside of the house was smelly, almost completely devoid of furniture and fittings, and full of stick-skinny barefoot children dressed in drab worn-out clothes, who were laughing and singing and running about as if happiness was the only fuel they needed for their lives.\n\nOddly, they didn't crowd around us. We stood in the hall for a moment and then a small girl came to us, curtsied, said good morning, and skipped away to fetch the Home-keeper.\n\nThe Home-keeper was a thin, old, wispy-haired man wearing a dark suit. He looked pale and serious but when he saw the honeybarrel he clapped his hands with delight and sent children scurrying for a container. They brought him a long oval saucepan, a fishkettle I think, into which, with great difficulty, I managed to scrape three or four spadefuls of sticky honey. The children bore it away in triumph.\n\nAs I was sorting out some bottles of cod-liver oil a woman came out of the kitchen to watch us. She was young but her face was hard. She looked at me fixedly with such implacable hatred that I could only turn my eyes away. The man had noticed. He spoke to me softly in English: 'It is not bad. Her man has been in Russia,' he explained. 'It is often so.' He touched my arm and smiled very gently. 'Please do not worry.'\n\nI stood still for a moment. Then, in my school German, I said: 'It is good here.'\n\n'Oh yes,' he said, 'we live well.' And he laughed.\n\n'Is there anything you miss?' I asked.\n\n'Oh yes, brandy, and coffee made from beans. But that is so long ago I have forgotten.'\n\nHelen came back from explaining to the woman about using the louse-combs in wet hair. 'We must be off,' she said.\n\n'Oh no!' cried the man. 'We must sing to you. We know a very correct English Christmas song. Come!'\n\nHe led us into the chilly sparse hall where he sat at a creaking harmonium and played the introduction to the song.\n\n'What's the tune?' I whispered to Helen.\n\n'I know it, but I can't place it,' she muttered.\n\nThen the cheerful reedy voices of the children came in with the words:\n\n'I'm dreamink of a vite Crischmas...'\n\nI stood there and wept like a river.\n\nThere seemed to be a multitude of Children's Homes in the area so, after a few more bumpy excursions, I was entrusted with the task of distributing the honey and cod-liver oil and louse-combs on my own. I had also looted some coffee beans and small paper bags from the army stores, and poured glasses of my Officer's Brandy allocation into some of the unused cod-liver oil bottles. So, fully equipped, I set off each morning with my list of visits, and threw my 3-ton lorry along the pitted roads. One or two of the Homes I visited were gloomy and sorry for themselves, not without reason, but most were like the first one we went to, completely light-hearted, full of fun and genuinely happy to receive whatever was going, even the louse-combs. I could see why. In the fairly recent past times had been very bad, not only because of the physical shortages of food and fuel but also because while the Third Reich had been bleeding to death there had been no future to look towards, only the prospect of even greater sacrifices being imposed in order to maintain the delusion that there was hope of victory. The spadefuls of honey that I brought were not just sweet food, they were also a symbol, a precursor, of better times that were coming, lighting up for them the possibility of a future without war, starvation and fear.\n\nI was entirely infatuated with everything that was happening, buzzing with energy and delight. My school German rushed back into my head, flourishing long-forgotten grammatical tables, whole pages of which would pop up behind my eyes when needed, and in no time I found myself talking fluently to the people I met in a language that may, in part at least, have been German. Apparently the language was serviceable, but I did notice that after a while the people I was talking to tended to dissolve into fits of giggles, in which I usually joined.\n\nHaving another, necessarily simple, language to use with my new friends was not in any way a drawback; quite the contrary. My ridiculous German was not encumbered by my past. It was free of the anxieties, inhibitions and social nuances that silted up my use of my own language. I could express simple direct thoughts and feelings in a way that would, if I had been speaking in English to a group of English people (like for instance the Relief Team), simply have caused embarrassment to everybody, especially me.\n\nLet me say at once that I have every respect for English reticence and understatement, and for the exactness and subtlety of the English language. I also know the danger, and the power, of the sweeping simple expressions of emotion that German can deliver \u2013 Hitler was a master of these \u2013 but even so, at that moment in my life, the German language, such of it as I could use, helped to liberate my silly diffident spirit and allowed me to join wholeheartedly with the happiness that was all around.\n\nI felt I could have spent the whole of my life, like some mythological spirit, bounding along broken roads in the aftermath of wars, dispensing honey and hope with a spade. Alas it had to end. The barrel was suddenly empty and I was back with the team, doing my bit as a driver.\n\nWell, not quite. The Team Leader came back off leave and immediately fell out with the Transport Officer, who stamped his foot and resigned. As a result his very undemanding job fell to me and I became Acting Transport Officer. In that capacity I was able to do a magnificent deal with the Transport Officer of the Control Commission by which I borrowed two Volkswagen cars and a motorbike in return for the use of our 3-ton lorries. This was strictly against orders but it made travelling much more comfortable, especially as the winter was now setting in.\n\nNow that the team had a leader I looked forward to seeing the huge highly organized emergency exercise that I was playing my humble part in swing into action. The Team Leader had brought back some excellent gramophone records from the UK. We listened to these with pleasure but otherwise everything went on as before.\n\nThat was disappointing but at the same time it wasn't easy to see what else the members of the team could do. What was needed in Germany at that time was food and clothes, shelter and, urgently, fuel. The Relief Team did not have such things to offer.\n\nI reminded myself that mine was not to reason why. The team members needed to come and go from the house and I, who knew nothing of their purposes, was there to drive them. However, Bob, who was a very large man connected with the Boy Scout organization, would often talk about his work. He was enthusiastic about reviving, or recreating, the Christian Youth Groups that had existed in the 1920s and early 30s, before such things were taken over by the Nazis. He felt that for the great task of reconstruction that the German people had before them it was important to foster a spirit of cooperation, comradeship and common purpose among young people.\n\nI remember going along with Bob to an evening get-together of the _Evangelische Jugendgruppe_ (Evangelical Youth Group).\n\nThis was quite a noisy occasion, but with a certain formality about it. The young men would walk briskly up to each other and, each gazing sternly into the eyes of the other, clasp hands to forearms in a Roman handshake, stamp a foot and shout: ' _FREUNDSCHAFT!_ ' (Friendship!)\n\nThe girls seemed watchful and subdued. I noticed that one of them was obviously quite ill, flushed with fever and hardly able to stand, so I offered to drive her home at once. She was very grateful but the chief young man said no. He said she could not depart because it was not proper that she should do so.\n\nI said that she was ill and that I was going to drive her home. The young man \u2013 no older than me \u2013 glared. I glared back. But I was not only the guest, I was also, in his eyes, the conqueror. In the end he turned and walked away, sat down in a corner with his back to the room.\n\nI helped the girl into the truck and drove her to her home. The night air did her no good at all and by the time we got to her parents' makeshift house she was practically unconscious. So I put her over my shoulder and carried her in. I remember thinking how surprisingly light she was.\n\nI drove back to the hall. As I drew up I heard the sound of a guitar and voices singing. Looking through a window I saw the young men sitting together. Flushed and glazed with nostalgia, they were singing one of those rousing gut-wrenching songs of glory and conquest that had fired the blood of the Wehrmacht and led it to its death. The words may have been changed but the song was still the same.\n\nAs I drove Bob home I wondered whether the same rousing songs had been sung in the 1920s. Probably they had, and certainly they were being sung in the 1930s while Herr Hitler was fostering a spirit of cooperation, comradeship and common purpose among young people for the glorious task that he had in mind for the German people \u2013 I shivered and hoped Bob knew what he was doing.\n\nAlthough I met many German people and made some good friends while I was there, we rarely spoke of Herr Hitler, or of the recently ended war. Only once did anybody deliberately set out to inform me about what had happened. This was a middle-aged lady who, having accompanied her sister to a meeting with the Team Leader, found herself waiting in the sitting room with only me as a companion.\n\nThe lady told me that the war had been very terrible. With that I agreed.\n\nShe told me that the German people had done terrible things during the war, very terrible things, but that for her the most terrible had been the betrayal of the vision. This was the pure, glorious vision that she and her friends had all together felt in their souls at the great rallies in N\u00fcrnberg. That had been a time of truth.\n\nI ventured to wonder whether Herr Hitler's vision had really been so true and pure if it had led to such terrible happenings.\n\nThe lady looked at me with affection tinged with pity. She reminded me that she had heard the F\u00fchrer speak. She had been there with him. For a moment she had clasped his hand and looked deep into his eyes. Oh yes, the vision was pure. She held the light of it holy in her memory, separate from all that had followed. The failures and the betrayals were the work of lesser men and also of other, dirty interferences.\n\nWere these other interferences to do with Communism, I wondered.\n\nShe shuddered and told me that Communism was a darkness that spread in the shadows, bringing doubt and subversion to make all things uncertain. To guard the light it had to be swept away.\n\n'And the Jews?'\n\nShe was silent for a moment. Then she said. 'Oh yes. Oh yes. Terrible things were done. Hurtful, hateful things by evil people. But they too were done to guard the light. They were wrong to do and for that we have guilt, but the light itself was pure. It is still so.'\n\nAnd abruptly she rose and went to sit in the hall, not wishing me to see her tears.\n\nI have no idea how typical that lady's feelings were. Most people I spoke to were far less forthcoming and usually confined themselves to asking how soon I thought we would be going to start fighting the Russians. That, they said, was the inevitable next move, and once we realised who our true enemies were, we would be looking to Germany for help... wouldn't we?... Yes?\n\nI tended to duck out of that sort of conversation.\n\nAt the beginning of December the temperature dropped to minus eighteen degrees Celsius. I had never been in cold like that before; it bit my face and numbed my extremities through layers of clothing. The streets seemed to ring with cold as if the walls and pavements, even the trees, had frozen hard as glass. Even the antifreeze in the vehicle radiators froze to a mush.\n\nThe team house and my room over the garage were warm because our fuel was supplied through the army but I wondered how the people of the town were managing. When I asked Giff if he knew how fuel was delivered to the people, he laughed and said it wasn't delivered.\n\n'Do you mean there are no fuel supplies to the town?' I asked.\n\n'Oh yes, coal comes in but it doesn't get delivered.'\n\n'What happens to it?'\n\n'It gets pinched. Come tonight, I will show you the _Kohlenklau_ (coal pinch).'\n\nJust after midnight, in the pitch dark, we drew up quietly beside the railway yards at the other side of the town. I could see nothing except the black silhouette of the sheds. Giff turned the 15cwt so that its nose was pointing into the yard. Then he turned the headlights on.\n\nI saw a train of coal trucks on the siding. The chutes had been opened so that the coal had slipped in heaps on to the track. Clambering over the heaps were literally hundreds of people, furtively scooping the coal into baskets and bags. What impressed me was the fact that so many of them were well dressed and that the rucksacks they were stuffing with coal were expensive framed types, the ones that mountain climbers use. Fearing the light they scuttled around to the dark side. So we came away.\n\nSoon after that the snow began to fall. About two and a half feet fell in the night. It brought complete silence and, for a while, immobility. Very efficient snowploughs came out and soon the town streets were navigable. Then we could get about, but only locally. Life in general slowed down and the members of the team spent quite a lot of time sitting about in the house trying not to get on each other's nerves.\n\nI took the opportunity to take a few days' leave. I had borrowed a pair of skis from the Military Police quartermaster and I was eager to try them. They were terrible skis and my first attempt, in Bad Harzburg, was a disaster, but later, in the ski-resort, I managed to master the skis and had a marvellous, invigorating holiday. I also recognized, and had a drink with, a young Guards officer who turned out to have been on my court martial. 'I was very nervous,' he said. 'It was my first court martial.'\n\n'Mine too,' I replied.\n\nI returned, full of zeal, and went to offer my services to an American religous relief team which was looking after refugees coming in from East Prussia. To my surprise I was roundly denounced by them for 'doing God's Work' when I was not a 'True Believer'. They appeared to assume that as I had rejected the Love of God I had a duty to live a life of sin and debauchery and not go pinching the heavenly Brownie Points that were rightly theirs. I got away in time, just as they were about to forgive me and welcome me into the brotherhood of repentence.\n\nI had an even more alarming encounter a few days later.\n\nPolitics had started up again in Germany and, because it was proving to be a vehicle for passions, grievances and antagonisms that must have been brewing up under the surface for years, we had been warned to keep out of the way of any demonstrations.\n\nI remembered the political meetings and demonstrations at which my grandfather had spoken. They had been moving, powerful occasions but essentially warm-hearted and enthusiastic. I wondered what a post-war German demonstration would be like.\n\nI took the rather asthmatic motorbike I had borrowed from the Control Commission and pottered through the town towards the park where the demonstrations were likely to be held. As I was passing through one of the bombed-out areas where the road was lined with ruins and rubble, I saw coming towards me a mass of angry-looking people. They were shouting and waving their arms. They were singing as they marched. They were throwing stones and turning over parked cars. Like an eruption of red-hot lava from some volcano of hatred the riot was flowing towards me, glowing with rage and roaring some huge muddled song.\n\nI turned the bike for a quick getaway and sat astride it, knowing that I could kick-start it and escape, but daring myself to stay as long as possible. Then, as I saw the whites of the eyes and the shining ecstatic faces of the young men leading the crowd, I kickstarted the engine.\n\nIt didn't start.\n\nI kicked and kicked at the starter pedal and the engine rotated, but it didn't fire.\n\nThe riot came closer. They were only a few yards away. I turned to the crowd. 'The damn thing won't start!' I shouted.\n\n'English shit-bike!' shouted one of the men.\n\n'You should get a BMW!' shouted another.\n\n'Jump on, we'll give you a shove!' shouted a third.\n\nI jumped on and four of them ran forward and pushed the bike. The engine fired at last and they gave a shout of triumph.\n\n'Ride on to England!' shouted one.\n\n'I shall!' I replied, and at that moment I meant it.\n\nOne or two of them may have thrown the odd stone, just as a sort of token, but I was away like a bat out of hell.\n\nI couldn't drive properly because my head was too confused. So once I was far enough away I stopped the bike and sat on it, shaking from head to foot, trying to sort out what had happened. What was confusing me was the realisation that from the instant I had spoken directly to them, the young men had, in my eyes, completely changed. They had ceased to be the fangs of a gigantic dragon of rage and had turned into a crowd of excited youngsters having a great time throwing their weight about. Something had been seriously wrong with my perceptions. They had not been reliable and that had rattled me almost as much as the encounter.\n\nEventually I recovered my composure and, freezing cold, kickstarted the bike. This time the damn thing started first kick and carried me back to the team house where I sat with my hand on one of the radiators to thaw out my blood.\n\nThen, in February 1947, the thaw came and we had to deal with the river behind the team house. This had become jammed with ice-floes and was threatening to flood, so we all had to go out on to the bridge and pick at the floes with long bamboo poles tipped with heavy spikes. That was tremendous fun, especially when the floes finally cracked and sidled away under the arches of the bridge in a gurgling rush of grey water.\n\nNot long after that, at the end of February, the spring arrived. You could smell it in the air. People smiled and soon they had food in their baskets, sometimes even, I was told, jars of honey! A thought that had been in the back of my mind for some time now came to the surface. Germany's own economy had begun to recover. I could see that its progress would benefit far more by my absence than by my continued presence.\n\nI resigned, wrapped three bottles of gin in my bed-roll and returned to England and the future, whatever that was to be.\n\n# TREADING THE BOARDS\n\n## _I. The Academy._\n\nIn 1942 I had been severely admonished by Ray and Daisy's friend, the painter Stella Bowen. At the time she had been painting a portrait of me holding one of John's rabbits on my lap. This meant I had to stay very still and quiet, concentrating on the hope that the rabbit, whose name was Snowball, was not going to pee on me again. In this situation one does feel rather vulnerable. Stella said: 'You'd better make up your mind what you're going to _b_ _e_. If you fritter away your life on fancies you'll end up being a player at all trades and master of none!' A few years later I might have argued, but I was seventeen and the condemnation in her voice was so absolute that I took it on the chin, vowing that if I should ever come to know my mind I would be sure to make it up, and _be_... something or other.\n\nSo, in 1947, coming back from frittering my life in Germany, I repeated my vow and determined to make up my mind and decide on a career... or something of the sort... I wasn't sure what.\n\nI had definitely concluded that stage design was not for me but I was still attracted by the idea of working in the theatre, perhaps as a stage manager. Then I thought: why not go the whole hog and be an actor?\n\nAn actor?... Yes, it had a simple decisive ring to it. I firmly proclaimed this intention to my friends and family. Ray, taking the cue, consulted his friends at the Savile Club and was told by a distinguished knight of the theatre that the only place to go for training was the London Academy of Music and Dramatic Art in Earl's Court. I duly wrote to the Principal and was offered a place at the Academy.\n\nSo I came to another first day at school. Carrying a pad and pencil and a bound volume of Shakespeare's plays, I diffidently entered the shabby portals of an imposing house at the corner of the Earl's Court Road and Cromwell Road. I crossed the brown lino of the hall and sat on a folding chair in what would have been a very elegant large drawing room if there had been any decoration or furniture in it.\n\nMy fellow students were a motley collection, ranging from exquisite, languid starlets being groomed for fame by the studios that owned them, through serious-minded teachers of drama in the making and fairly mature ex-service people, to rumbustious teenagers just out of school.\n\nThe Academy was run economically. The Principal was not an actor, nor was he directly connected with the theatre. He did no teaching and as far as we could tell had no views on how drama should be taught. In the main his policy seemed to be to invite professional people, usually out-of-work actors, to give lectures on acting, or whatever they chose to call it \u2013 and let them get on with it.\n\nThe actor\/lecturers would come to the Academy, play their parts as lecturers (which were often entertaining and instructive to watch), and then go.\n\nThe interesting, and to my mind rewarding, characteristic of this system was that although what the lecturers had to tell us about the art of acting was usually very clear and definitive, the instruction they gave was often incompatible with, and sometimes contradictory to, what we had been told by another lecturer the day before.\n\nIt would have been good to have had an opportunity to discuss this phenomenon with the lecturers themselves, but that wasn't what they were being paid to do, and in any case they were legging it out of the building the moment the session was over.\n\nI don't see this as a having been a deprivation. On the contrary I see it as a benefit, because it meant that we, the students, were forced to take on board the contradictory instructions we received and then have to work out for ourselves, among ourselves, where the truth lay; or rather, were able to find out that there was no absolute truth, that we each had to find our own way to deliver the goods. It may be a novel approach to education but I would recommend it.\n\nNot all the lecturers were passers-by. Beatrice Forbes-Robertson, of the great theatrical family, who was in her eighties but still light as a feather, brought Shakespeare to life for us through her own delight. Trevennen Peters, bearded and sardonic, was as likely to talk on philosophy as on acting and had once, I remember, thanked me for bringing a note of proper disrespect to his lectures.\n\nBut the empress of Friday afternoon was a really great lady: Frieda Hodgson. In her presence one was lifted and carried forward on a deep warm tide of encouragement and possibility. She had a close personal relationship with every one of her students, a relationship that was far from indulgent. Her eye was clear and critical but at the same time she gave each student the sense of being both supported and appreciated, and from that sense came the courage they needed to lift themselves over the threshold of diffidence.\n\nTechnically I'm not sure that Frieda was a very exact teacher, at least I found that if I listened too closely to the words she was saying they didn't always make a lot of sense, but that was not her role. As an inspirer Frieda was without peer and many distinguished actors and actresses in the theatre today have reason to cherish her belief in them, and pay tribute to the part she played in bringing them to success.\n\nAt LAMDA I played a love-sick fairy and the king of the ants in Karel Capek's _Insect_ _Play_ , and provided shiny carapaces for the dung-beetles by turning my father's overcoats inside out. We played a scene from Shakespeare's _As_ _You Like_ _It_ in an imaginary railway station, complete with hand-luggage and tickets. I also, to my shame, played a piece of delicate, elegant, Restoration comedy (from _The Rivals_ , I think) wearing American army boots.\n\nThe time I spent at LAMDA was, in some ways, one of the happiest in my life. In that acceptive community of individuals I was able to be at ease and make some long-lasting friendships, as well as learning something of the craft of acting.\n\nMy heart, which had seemed quiescent for the last few years, was moved again. I got to know a nervous, tearful girl named Ro, who happened to live near me in North London. We became close friends and I slowly fell in love with her. Our relationship was not easy, mainly because she didn't feel the same for me, but it was to prove durable.\n\nThe course at LAMDA was for two years. At the end of the first year I took the Gold Medal Examination, for which I did my piece from _The Rivals_ (in the boots). The adjudicator that year was Peter Ustinov, which was slightly confusing because at the time he was married to one of my relatives and we had met on family occasions. We greeted each other warmly, he overlooked the boots, and awarded me the Gold Medal for Acting, which I treasure to this day.\n\nWhen we saw the curriculum for the next year at LAMDA, I and some of the other post-war-service students noticed that it was to be an exact repeat of the first year. This fact was confirmed by the Principal, who saw no fault in it. Several of us did see fault in it, and promptly left the Academy to seek our fortunes in the real world of the theatre.\n\n## _II. The Playhouse._\n\nAs I turned to go, the agent said: 'Do yourself a favour, lad. Don't go flashing that gong about. It doesn't signify.'\n\n'Oh, right-ho,' I said lightly, putting my Gold Medal for Acting back in my pocket, 'you'll be in touch, then, will you?'\n\nThe agent looked at me sadly without answering, and I left.\n\nBeing an actor in 1948 didn't often involve acting. Mainly it involved climbing many flights of narrow smelly stairs to importune bored theatrical agents. It involved endlessly poring over the small advertisements in _The Stage_ and it also involved something else, something invisible and imponderable: having inside knowledge of where to go, what to do and who to be seen hobnobbing with in order to be at the centre of the theatrical scene \u2013 the place where, one assumed, the jobs were being handed out.\n\nThere were no jobs being handed out in London, so I was lucky to spot an ad in _The Stage_ from a big production company that was casting actors and staff for a group of first-class theatres in the south of England. I applied and was called to an interview with the producer. He warned me that he was taking a great risk in doing so but he was going to appoint me actor\/stage-director for their Brighton company \u2013 at \u00a33 10s a week.\n\nI was over the moon. I bought the necessary dinner jacket and bow-tie from the Fifty-shilling Tailors. I travelled to Brighton, found myself cheap lodgings and, as instructed, presented myself at the first-class theatre. This was called the Playhouse and was situated in a back street in Kemptown, the poor end of the town.\n\nThe door of the theatre was open, so I went in. The auditorium was small, but long and thin, presumably because it had been built as a cinema. The stage was a bit shallow, which was OK, but behind it and on each side of it there was hardly any space, just a wall at the back and a wall on each side with an ordinary door in it. I saw a pencilled note stuck to the box-office window: 'QW rehearsal Sunday 2.30 p. m.'. It was now 11.30 a.m.\n\nI wandered about Brighton for the next hour or two, feeling very keyed-up and wondering what QW meant. Then at 2.20 p.m.\n\nI entered the theatre and walked straight into a loud row that was going on between the producer and a large middle-aged lady with a fat, sad face. The producer saw me coming and, to end the row perhaps, turned his anger on to me. 'Oh, there you are. About bloody time. We're relying on you, you know! Listen everybody, this is Oliver Post... er, thing. Our new stage-director. So from now on all questions go to him. Come on, let's get on with reading this play. From the top... Oh, just mime the telephone for now, dear. We'll need a telephone for rehearsals, Oliver, see to it. Come on everybody. Settle down now, we've only got a week!'\n\nThey started to read the play \u2013 _Quiet Wedding_. I stood there, stunned with agitation. What? Just _what_ was I supposed to be doing?\n\nI survived, and managed to do my job. Some of the plays were fun but the romantic lead was always played by a middle-aged lady called Mona. This was because the 'big production company with a group of first-class theatres' was a fiction. Mona's husband, a chemical manufacturer, owned the Playhouse.\n\nWe put on some surprisingly elaborate productions, including Noe\u00a8l Coward's trilogy of one-act plays, _Tonight at Eight-thirty._ One of these was _Brief Encounter_ , in which a young actor named John Teagen and I played the comic relief \u2013 two half-drunk soldiers in the station tea-room. After the show a fast-talking gentleman called Mr Jobson came round to see us. He said we were a great act and he offered us a spot in his forthcoming pantomime, _Aleddin_ , which would be touring number one dates in the south of England at Christmas.\n\nWhat a piece of luck that was! And providential too, because after _Brief Encounter_ the company was dismissed. Gossip had it that Mona's encounters with the leading man, both on and off the stage, had been briefer than she would have wished and as a result she had decided she didn't want her toy any more. So the chemical manufacturer closed it and the Kemptown Playhouse reverted to its original function \u2013 showing rude films.\n\n## _III. Pantomime._\n\nMr Jobson was as good as his word. He came to tea in John Teagen's room in Earls Court and talked tremendously fast and enthusiastically about his Christmas ventures. We gave him hot buttered toast to eat but that didn't slow down the torrent of words, it just gave them substance.\n\nMr Jobson was by trade a scenery contractor. He had, he said, huge warehouses full of gorgeous scenery and full set-pieces, from which he supplied all the most glittering shows in the south. For his own ventures he had the pick of the best scenery and the greatest talent in the world at his fingertips. He told us of the marvellous acts that he had booked for the show, how he had managed to engage Tubby Blackwell and Zelda! We had heard of them of course \u2013 best 'Big Wheels' in the business! And the Great Ernest Watson, whose dog, Lily, did those amazing card tricks. And did we know Armand and Helo\u00efse, Apache dancers extraordinaires? We must! The verve, the vitality, the passion! Oh, hot stuff, strong stuff for a panto, a bit sexy, but WOW! And the dancers, COR! The Mavis Bewley Dancers... tap, ballet, speciality, Lat'n American, they could do it all, and sing too, big chorus numbers, marvellous! But we were to have real music as well, with a real musical director, young but really talented, and a full threepiece orchestral ensemble.\n\n'What about _Aladdin_?' I asked.\n\nUninterrupted the flow went on. He had got the best Principal Boy there ever was and a great Abanazer. The Emperor was a wow and the Princess... Oh! The thought of her beauty almost silenced him for a second. She was a stunner, gorgeous, but so delicate, so vulnerable, so pure. And we, we two were going to be a smash hit. We did _real_ acting. He was, he declared, a great respecter of real acting. He had seen at once that we had the true spark. Not just the tinsel glitter of show-biz but the real thing, the clear light of great art.\n\nWe found this outpouring of praise, flattery and buttery toast crumbs quite irresistible. We readily agreed take the part of a pair of Chinese policemen at \u00a315 a week for the Act, which was twice what we had been getting at the Playhouse.\n\nThe rehearsal room above the Rose and Crown in Tooting was large, with some chairs set out and some stacked by a piano at one end. As in a doctor's waiting room a number of ordinary-looking people were sitting around the edge of the room. John Teagen and I entered, smiled and took seats next to the door.\n\nAfter a while I had the feeling that we were being scrutinized. People would look at us, then turn to each other and quietly say something. Then they would look at us again.\n\nA tall man with slightly military bearing rose and came over. He was holding a paper.\n\n'Teagen and Post?' he said.\n\n'I beg your pardon?'\n\n'It says here Teagen and Post, on the cast list. Chinese policemen.'\n\n'Oh, right. This is John Teagen. I'm Oliver Postgate.'\n\n'So you are Teagen and Post. Got your music?'\n\n'Music?'\n\nThe man looked at us in silence and then marched out of the room. He could soon be heard talking angrily on the telephone on the landing. The other people watched us anxiously.\n\nSitting together at the other side of the room was a middleaged couple, she knitting, he reading a magazine. To break the silence I got up and walked over to them. 'Good morning,' I said. 'My name is Oliver Postgate. This is John Teagen.'\n\nThe lady smiled. 'Pleased to meet you. This is Gerald Worple. I'm Amy.'\n\n'May we join you?' I asked.\n\n'Oh, please do.'\n\nSo John and I sat down and we had a quiet chat. Amy and Gerald Worple were in fact Armand and Helo\u00efse, Apache dancers extraordinaires.\n\n'Do you know what's going on?' I asked.\n\n'We'll just be doing the moves, I expect,' said Amy. 'In fact I think we'll do ours now. Come on, Gerald.'\n\nAmy took a sheaf of music out of her bag and gave it to a young woman who took it to the piano.\n\nThen the place seemed to come to life. The Apache dancers' music was exciting. The real Parisian angry-passionate stuff. Gerald and Amy checked the tempo, got into the beat, and took their places for the dance. 'Right. From the top, please,' said Amy.\n\nWhat Gerald and Amy did next was extraordinaire. They walked briskly forward to meet each other, held hands for a second and then turned away. She waved her arms and stepped back. He walked around behind her and lightly tapped her on the head. She stooped slightly, turned around, and they walked backwards together across the room, took hands and circled slowly, nodding to the music. She touched the top of his head, he stooped and stood still while she walked across the room.\n\nWe watched, fascinated. The music swooped to its fiery end.\n\n'Lovely,' said Amy. 'Have to watch out for the rallentando. We can't hold it for ever. How was it for time?'\n\n'Thirteen eleven,' said the pianist.\n\n'I think it'll stretch a bit,' said Amy.\n\nShe and Gerald came and sat down, not even out of breath.\n\n'We're thinking of opening a guest-house in St Leonards,' said Amy.\n\n'Are you? I've got an aunt lives in Hastings.'\n\n'Can't keep young and fit for ever!' laughed Gerald. 'Got to make provision for our old age. By the way, how d'you reckon it went?'\n\n'Fantastic,' I said. 'Marvellous music. Strong stuff. Tell me, though, what were the taps on the head? You know, just now and then.'\n\n'Oh, they're the trays,' said Amy. 'We do the trays, you know.'\n\n'The trays?'\n\n'Yes. Gerald's the waiter. I'm the tart. In this French cafe\u00b4. We slam trays about. We use those cheap tin trays. They just crumple up. You must have seen it done!'\n\n'No, never,' I said. 'Sounds fantastic. I can't wait.'\n\n'We can't wait, neither,' said the tall man, who had been standing over us. 'We'll do your bit now, OK?'\n\n'Er... right-ho.'\n\nJohn Teagen and I stood up, feeling as if we were about to be sentenced. We stood in silence while the company looked at us.\n\n'What's your act, then?' asked the tall man.\n\n'Well, er... nothing worked out.'\n\n'What do you do? Rolls and falls? Stage-slaps?'\n\n'A bit... at the drama school.' That got a laugh.\n\n'Do you dance? Tap? Soft shoe? Speciality?'\n\nI shook my head.\n\n'Can you put across a number?'\n\n'A what?'\n\n'Can you sing?'\n\n'I don't know.'\n\n'So we're going to have to teach you, are we?'\n\n'I don't know.' I was near to tears. 'May I go home, please?'\n\n'No. Sorry. You two have got work to do. Take off your jackets and put these on.' The tall man reached behind the table and brought out two toy policeman's helmets.\n\n'Just a minute,' I said.\n\n'Yes, what?'\n\n'Who are you?'\n\n'Me?' The man seemed puzzled.\n\n'Yes, you. Do you by any chance have a name? What do you do?'\n\n'Trevor Prangley. I play Abanazer. I also produce the show. OK?'\n\n'Ah, thank you. Very pleased to meet you, Mr Prangley. My name is Oliver Postgate. This is John Teagen.'\n\n'I know that. It's written on the cast list. You're Teagen and Post. Well now, that's settled. I wonder if you would do us a favour and put on the helmets. Then we'll do some slaps.'\n\nAbanazer was sounding almost conciliatory so John and I rolled up our sleeves and put on the helmets.\n\nTo do a stage-slap, one person slaps the other, but just stops short or just misses. At the apparent moment of impact the person slapped secretly claps his hands to create the sound of the slap. It's easy once you've rehearsed it, and very convincing. Rolls and falls are something else. The essence is to appear to be knocked over but in fact to propel oneself, relying on the orchestra to provide the noise. Neither of us was exactly athletic, or supple. We were both clumsy, deeply embarrassed and nervous.\n\n'Anybody would think,' observed the Principal Boy, a podgy lady who chain-smoked, 'that we had performing seals.'\n\nShe wasn't referring to our clumsy falls but to the lino where, at every fall, my sweaty hands had left a pair of slithery flippermarks.\n\nSoon, thank goodness, the helmets were casualties. Trying to do a forward roll, I had landed squarely on the point of mine and rammed it so hard on to my head that the papier-m\u00e2ch\u00e9 had to be peeled off in pieces.\n\nAbanazer was scathing about our knockabout act, but he said that with the help of the orchestral effects, the whizzes, bangs and claps, we might just about get away with it.\n\nThat evening we went our ways bruised, aching and disconsolate.\n\nRehearsals continued and after a while Abanazer turned his attention to the song and dance routines. There, he said, we had first-class material to work with.\n\nThe first dance routine was classical; Egyptian. We had to stand dead sideways, holding strange gestures to imitate the figures in the hieroglyphs of Ancient Egypt, and shuffle across the stage, whipping round at the end of each line and shuffling back in a different posture. The music was weird, there were no words and I couldn't really see what any of it had to do with Chinese or policemen. When I asked Abanazer he said it was all Oriental and anyway what difference did it make?\n\nThe rehearsing of _Aladdin_ itself, the play, was fairly low-key and perfunctory, partly because most of the cast had played it before, and partly because the ones doing speciality acts, being otherwise engaged, didn't turn up till the last day.\n\nThe Princess, whose beauty and delicacy had so moved Mr Jobson, also turned up on the last day of rehearsals. Although her acting was limp and her manner lethargic, she was indeed strikingly beautiful and, strangely, type-cast for the role of the cloistered princess whom none may set eyes on, because she was squired by a heavy man in a double-breasted suit who couldn't keep his hands off her. He watched her, and glared at everybody else, throughout the rehearsal and whisked her away in a big car the moment it was over.\n\nEventually the time came to move. I went ahead and fixed up cheap lodgings for John and myself in Ryde. Then I walked out to find the cinema, where the glittering show was to open. It seemed sad, with only a stand-up board outside and some curled photos of the cast pinned up in the foyer.\n\nLooking at these I was intrigued to notice that the Great Ernest Watson wore full evening tails and a top hat, and his dog, Lily, a frilly collar for their act. Tubby Blackwell was, it seemed, a fixedbar acrobat and Zelda wore skin-tights covered with sequins. I didn't recognize Gerald and Amy from their photograph. At that moment the picture of the handsome, sullen waiter, Armand, and the perky, flaunting Parisian tart, Helo\u00efse, posing against the Impressionist caf\u00e9 background seemed more alive and interesting than _Aladdin_ or, come to that, Ryde. I was really looking forward to seeing their act.\n\nOn the afternoon before the show was due to open there was a run-through of the play, but most of the time was spent sorting out the scenery and props for the speciality acts. Gerald and Amy had their own caf\u00e9 cloth. The Great Ernest had a massive chromium and black set-piece like a nineteen-thirties cocktail bar. Tubby and Zelda's fixed bar had strong guy-wires that had to be hooked into holes which Zelda had drilled into the corners of the floor of the stage with a brace and bit. _Aladdin_ 's scenery, the street in Old Peking beside the Emperor's Palace, included various wingflats, one of which was a fish and chip shop and another a piece of Baroque church. I mentioned this to Gerald who explained that, as Mr Jobson was a scenery contractor, all the good stuff went out at Christmas, to paying companies. His own productions had to make do with what was left over.\n\nThe Mavis Bewley Dance Troupe, a collection of local schoolgirls aged between eleven and fourteen years, was everywhere, winking and flaunting and squeaking. Mavis Bewley played for them herself. She chose a position out of the audience's sight behind the corner of the proscenium from which to play her accordion. As it was a very large accordion, her left arm and half the instrument would move in and out of the audience's view. As this was happening in time to the music the effect wasn't particularly distracting.\n\nAs the afternoon wore on and curtain-up time came closer I felt the familiar first-night excitement rising in my blood. Everything seemed to be in hand. There was one moment of emergency: Gerald and Amy's trays hadn't arrived with their skip, so somebody had to go out and buy some \u2013 but that was not my job.\n\nThe half was called. As I put on the costume and the make-up I could hear the rustling and chatter of the audience coming in. John and I wished each other good luck and we moved to our positions, one on each side at the back of the stage.\n\nThe girls were already on. The orchestra \u2013 piano, drums and trombone \u2013 was doing a good job with the plinky-plonky overture.\n\nThe end chord came. The curtain hissed as it went up its wires.\n\n'Sing up, Gels!' shrilled Mavis as she hauled at her accordion.\n\nThe clear untutored voices of the girls sang out:\n\n'Hail mighty Emperor of China\n\nOf noble birth and ancestreee.\n\nGladly we bow before thee,\n\nVoicing our love and loyalteee...'\n\nIt had started. They were doing well.\n\nAs the song ended the Emperor in his fine robes strolled in, followed by his Grand Vizier. He strolled a bit slowly because for some reason the girls were being a bit slow moving out. The orchestra vamped a few bars until most of them were off and then the Emperor spoke his first line:\n\n'Well, Chung-k'ing, what entertainment have you prepared for me this lovely morning?'\n\nThat was the first time I had heard the Emperor use his full voice. It was very impressive.\n\nThe Grand Vizier did his imitation of Winston Churchill, which the Emperor (rightly, I thought) rejected, demanding stronger stuff. That was our cue. We fell across the stage. John did his snatch of cod-opera. We did our knockabout act, but as the chap in the orchestra doing the whizzes, bangs and claps couldn't actually see us, he did them at the wrong times. That didn't help. Then we went straight into our number. The Egyptian piece was very mysterious and, I suspect, baffling.\n\nI felt mounting despair as the music went on and on, but, eventually, the number ended. It was received with respectful silence by the audience. The Emperor, as he was supposed to, ordered us away. We chased off and he was able to start his ominous song about what would happen to anybody who caught sight of his beautiful daughter, the Princess.\n\n'... to compensate,\n\nHow sad his fate,\n\nHe'll perish,\n\nmis'rableeee...'\n\nThe last note was rich and deep. Awesome stuff!\n\nThe play was doing well. The girls were still having difficulty getting on and off the stage for their chorus numbers. This was because the scenery had been set too close and they had to hop, one at a time, over the struts. So they were queuing at the back of the stage, giggling, but that didn't seem to matter. The acting was very strong and the drama unfolded. Aladdin met the Princess. They fell in love. They kissed and were discovered. The Policemen were called in to arrest Aladdin. He was about to be dragged off to execution when Abanazer stalked in (my goodness he was frightening!) and brought the thunderous power of his magic to bear on the situation:\n\n'... and under my magic spell,\n\nyou shall all stand where you are,\n\nand let the boy and I\n\ngo WHERE we WILL!... AHA!'\n\nAbanazer snarled and threw the magic sequin dust. We all froze in absurd positions waiting for the thunderclap, the flash and the blackout, during which the halfway cloth of the next scene, the Road to the Cave, would be lowered in.\n\nUnfortunately none of these things happened. We stood there bathed in golden light for, it seemed, minutes on end.\n\nThen the Emperor exercised his authority. 'Well, sod that!' he said loudly and walked off. We all relaxed. At that moment the halfway cloth dropped in like a bomb. Luckily Abanazer and Aladdin were on the front side of it. Book-wings in the shape of rocks were shoved into each side and the scene was set. Abanazer and Aladdin were at last able to go on with the play, at least until the end of that scene, when they discovered that they had been boxed in by the book-wings. There was no way out.\n\nAbanazer dealt with that small problem very promptly. He reached down, picked up the whole bottom-batten of the cloth and, lifting it high in the air, courteously ushered Aladdin underneath it. He followed through and dropped the lot behind him with a crash. For this he received tumultuous applause.\n\nThe curtain came down for the end of the first act.\n\nThe second act opened with Tubby and Zelda's fixed-bar number. Tubby was an amazing athlete. He wasn't tall or handsome; he was like a tennis-ball made of solid muscle. He swung and circled and somersaulted in perfect time to the music. Watching from the wings I could see what Mr Jobson had meant when he said he had the best 'Big Wheels' in the business. Zelda curtsied and posed as he did it, looking very glittery in her sequins.\n\nWatching from the wings I noticed something else. The boards which the guy-wires were hooked into were lifting. Each time Tubby performed a grand circle they seemed to spring a bit more. I looked across and caught John's eye. He nodded and we both stepped boldly on to the stage and stood one on each side with our arms folded, looking like policemen.\n\nZelda saw us and skipped lightly across. She gave me a flashing smile and hissed: 'Piss off out of it!' I opened my mouth to speak but she pirouetted and fluttered across to John: 'Sod off! This is our act!' she whispered and kicked his ankle.\n\nMeanwhile Tubby, apparently unaware of any problems, was whirling in the air like an indiarubber Catherine-wheel. Zelda crossed in front of me and jabbed me in the solar-plexus with her elbow.\n\n'I'm holding the floor down!' I gasped.\n\n'Fuck off! Go on! Do it now!'\n\nI caught John's eye and we both stepped back. Ten seconds later, with a crack like a pistol shot, one of the guy-wires broke out. The fixed bar came unfixed. The audience gasped but Tubby was a genius. He let the bar go, whipped into an airborne somersault and landed on his feet beside Zelda. They smiled and took their bows.\n\nAs they came off Zelda was weeping. She glared accusingly at John and me. 'He could have broke his poor neck!' she wailed, and ran to her dressing room.\n\nThe audience took that disaster, as it took everything else, in its stride. But as the debris of Tubby's equipment was whisked away in the blackout and the drama went ahead, I prayed that the spirit of the Marx Brothers that had somehow got into the performance would at least be gentle from now on.\n\nFor a while it seemed that it might be so. Widow Twankey, a skinny old man with an ingratiating manner, tried fervently to persuade the children to believe in fairies but they, equally fervently, told him to get stuffed. The cocktail bar, with the Great Ernest in it, was found incongruously in the streets of Old Peking. His dog Lily, though obviously anxious to please, muffed a couple of her card-tricks, but the audience forgave her and, far from blaming, loved her for it. All seemed to be well.\n\nThe last act opened with the Apache dance extraordinaire. That was what I had come to see. I posted myself in Mavis's place to have a good view. Their backcloth was old but very striking. In the pools of light the white bamboo caf\u00e9 table and chairs appeared to glow with an inner light. The years had fallen from Gerald and Amy and, in their costumes and make-up, he truly was the lustful jealous Armand and she the teasing coquette, Helo\u00efse.\n\nThe fierce music brought them suddenly to life and they swung into their routine. It was recognizable as being the one that we had seen in Tooting, but now, where they had walked, they flew, where he had lifted, he threw. They clinched and whirled, fled and were recaptured. Helo\u00efse seized a tray from the pile, swung it and brought it down on his head with a crash. He staggered, pretended to fall and then leapt up again, grabbed her, swung her round and over his head and, taking another tray, clouted her with that. The dance went on, full of fiery lust and primal savagery, but slowly, something \u2013 could it be age? \u2013 began to tell. Their movements faltered. They staggered. They wavered and recovered, swung their trays, missed and, as the music swooped to its climactic end, they sank to the floor and collapsed, together at last in each other's arms. It was a very original and moving performance.\n\nThe audience was in raptures. It rose and cheered, but it was several seconds before Armand and Helo\u00efse were able to stand up to acknowledge the applause. They smiled and bowed, but as they turned I saw that it was Gerald Worple who was taking Amy gently by the arm and leading her away from the stage.\n\nQuite late the next morning I was sitting in the Cosy Caf\u00e9 with the Worples. Amy was explaining.\n\n'It's the rage does it,' she said. 'You can't do the act properly if you don't let it get to you a bit. I mean... it wouldn't look right. But you see, the proper tin trays don't count. They don't hurt, they just crumple up. Isn't that right, dear?'\n\nGerald nodded. 'That's it. The moment you belted me with that stainless-steel job I just saw red. From then on it was all for real.'\n\n'It certainly looked real to me,' I said.\n\n'Oh it was,' said Amy. 'I was that furious, if I'd had a club I'd have laid him out cold. I would have, really!'\n\nShe put her hand over Gerald's and they looked at each other. She patted it. 'Don't worry, love. The proper trays have come. We'll be all right now.' She turned to me.\n\n'What about you two, then? Your whole act has been cut. That really is a shame.'\n\nI said: 'No, and yes... No, I don't mind its being cut. I'm glad to be off the hook. But also... yes. It is a shame you haven't got a proper knockabout act for _Aladdin_. I'm sorry about that. I feel a bit ashamed.'\n\n'Not your fault,' said Gerald. 'Jobson should never have booked you, not without telling you what you were in for. He's all fancies, and stupid with it. And he's a crook as well.'\n\nAmy patted his hand. 'Now don't say things, dear. We just have to keep an eye open, that's all.'\n\n'Even so,' I went on, 'I feel I've let the company down and they have every right to feel hostile towards us.'\n\n'What company?' asked Gerald. 'There isn't any company. There's a dozen or so people stuck on the Isle of Wight doing their own thing and getting paid for it. They don't care tuppence for you or the play, or for anything except the money.'\n\nThat was a comfort. I felt better about the next fortnight. I would be a respectable inscrutable Chinese policeman, join in the songs, arrest Aladdin and be part of the scenery.\n\nTime passed slowly but fairly easily. Audiences dropped away until towards the end we were playing to half-filled houses. Mr Jobson turned up on the last day. He was full of praises and invited us all to join him in the big hotel after the show for a celebratory glass.\n\nWe waited in the hotel bar for about a quarter of an hour, feeling a bit embarrassed at not buying any drinks. Then suddenly the Emperor looked at Abanazer, who looked at Widow Twankey. He looked at the Principal Boy who shouted: 'Christ! He's fucking off with the takings!'\n\nWithout another word the cast streamed out of the hotel and set off for the ferry terminal to head Mr Jobson off. 'Thank God we're on an island.' sighed Widow Twankey as he puffed up the steps. 'If it'd been on the mainland we'd have had to fish for it.'\n\n## _IV. Thronging._\n\nThe pantomime was perhaps the pinnacle of my theatrical career. From then on it was all downhill.\n\nI got a good engagement as actor\/stage manager with an excellent company at the Margate Hippodrome, but that only lasted a few weeks.\n\nI did an audition for the Royal Shakespeare Company, which was embarrassing. I applied for a General Audition for BBC Radio. This took place in an empty studio with a microphone. Perhaps by mistake the people listening in another room had left their microphone on, so I could softly hear them sneering and mocking from the start.\n\nThat was depressing and I was feeling sorry for myself when I met John Arlott. He was an old friend who was well on his way to becoming a famous cricket commentator.\n\nJohn was very sympathetic when he heard about the General Audition and he immediately gave me a part in a programme he was doing about Irish Railways for the BBC Education Department.\n\nI don't think I gave a very good performance. I was appallingly nervous, while the rest of the cast, all full-time members of the BBC Repertory Company, were so appallingly relaxed that they were mucking about and playing practical jokes. One of the members of the cast, a big sporting chap named Greg, played a particularly disturbing trick. While I was reading a rather long paragraph, he stood on the other side of the microphone and vigorously chewed the sleeve of his Harris tweed jacket. That had a very unwelcome effect on my salivary glands. It was all meant in fun of course and as soon as he saw that because I was so nervous the jokes were really putting me off, he stopped and apologized at once. Perhaps to make amends, he invited me to come back to his house for a cup of tea after the recording.\n\nGreg's house was sumptuously furnished, like a Victorian club, with brass fittings, dark wood and deep leather chairs. He also had, surprisingly, a butler. Greg handed the butler his coat and said we would like tea as soon as poss., with toast.\n\nGreg and I sat in armchairs beside a blazing fire. We talked about the hazards of broadcasting and radio work in general. After a while the butler appeared with a large silver tray which he carried to the table beside Greg. He lowered the tray to a point about two inches above the polished table and then, quite deliberately, dropped it. The tray fell with a loud clatter. The butler tossed his head and flounced out of the room. Greg took no notice at all, just went on talking.\n\nLuckily none of the china was broken. Greg poured the tea. The hot buttered toast was excellent. We had a long friendly chat and then I went home. As the butler held the front door open for me to leave, he bowed and said something softly under his breath, but I didn't catch it.\n\nThe next time I met John Arlott I mentioned what had happened. He laughed and told me that Greg was, as people now say, 'gay' and that he had a habit, very distressing to his boyfriend the butler, of bringing home attractive young men for his pleasure.\n\n'Oh!' I said. 'I never guessed.'\n\n'Of course not!' laughed John. 'You were perfectly safe. There is no armour like innocence.'\n\nI have since wondered how many times my innocence or, more accurately, ignorance, has guarded my honour. I have no way of knowing.\n\nMy next engagement, at a dismal theatre in Southport, was mercifully very short. The producer, a sly spiteful man, had engaged twice as many actors as he needed, with the intention of sacking half of them almost immediately.\n\nLater in the year, it must have been 1947, I mingled with the mighty. I was given the part of First Fisherman in a production of _Pericles_ at the Rudolph Steiner Theatre. Pericles was being played by Paul Scofield, and Second Fisherman was played by a young man whose name was Donald Sinden. Unfortunately the production was just a showcase and only ran for a couple of performances.\n\nMy theatrical career ended with my last engagement \u2013 one day as a film extra. I can't remember the name of the film or anything about the director except that he was Hungarian.\n\nI went to the film studios very early in the morning and was given a nineteenth-century jacket to wear over my ordinary clothes and a cloth cap with a knob on the top. After waiting some hours, I and several dozen other friendly people, equally oddly dressed, were invited on to the film set, which seemed to be a square in some European city at about the turn of the century. In the absence of any direct instructions I think most of us assumed that we were there to give the illusion that the place was populated. So we walked about and met each other and engaged in imaginary conversations, even made purposeful journeys and generally tried not to look like a load of extras standing about on the set.\n\nAfter a few minutes of this the director addressed us through a loudhailer: 'No, no. It says here please zat you are a _throng_. Now please will you please commence to throng... Action!'\n\nWe moved about rather faster and showed urgency by speaking briskly to one another and pointing at imaginary things.\n\n'No, no, no! Zis is not a cattle-market. Please you are here to throng. Garry, love, I ask you please, why do zey not throng?'\n\nGarry, a nice young man with wavy hair, took the loudhailer and had a word with us: 'Now come along, please. Let's get this together. You aren't just wandering about, you are a throng. Now you lot on the left, you throng in from that side and the ones on the right throng in from this side. OK?... Action!'\n\nSo I thronged in from the right and met a girl I had known at the Drama School who was thronging the other way. 'Did we do thronging?' she asked. 'Not _as such_ ,' I replied as I passed on.\n\n'No, no, no! Zis is not country-dancing, please. Garry, please, they must _throng_!'\n\nGarry had another word with us and next time we thronged in from the four corners, meeting in the middle in a polite apologetic scrum. I raised my cap to my friend.\n\nShe said: 'We did do self-realization in music and movement.'\n\n'Not quite the same thing perhaps,' I suggested.\n\n'Is no good!' wailed the director. 'Cut, please. For zis we must have a throng, not a football Rugby. Now please Garry if zese people cannot, send zem away thank you and fetch me people who will properly throng.'\n\nGarry took the loudhailer and said: 'Right, then. We'll break now for coffee. Thank you, folks. Please call in at the office as you leave.'\n\nI came away with _\u00a3_ 3 in my pocket and it was only a quarter to eleven in the morning \u2013 not a bad rate of pay for unskilled thronging.\n\n# RATTLING THE BARS\n\n## _I. The White Knight._\n\nBy the beginning of 1950 life in our house at Finchley had changed completely. John and his girlfriend from Oxford days, Mary Stewart, were married in October 1948 and had gone to live in Twickenham, and all the wartime lodgers had moved on. Having been captain of a busy ship all through the war, feeding the hungry relatives and dealing with every emergency as it arose, Daisy now found that the emergencies, apart from the boring restrictions of food rationing, were over. Except for my intermittent presence, she and Ray were alone and, as she put it, '... rattling about in that damn great house'.\n\nI had things to do. I had long ago realised that the one essential thing an aspiring actor needs is another source of income. So I was trying to find a market for my designs for agricultural machinery. Nobody would look at my mechanical sugar-beet lifter but one firm showed interest in my bouncing fence-post driver, the corkscrew post-hole digger and the over-arm jib which I had worked out for lifters and excavators. They were fascinated by the overarm jib but in the end they decided that all these things were too outlandish ever to take on, and returned them, but with a small cheque, which was gratefully received.\n\nI could see that I should be offering simpler ideas, ones for which capital costs would be low and the financial returns fairly swift. That obviously ruled out my favourite inventions \u2013 various rotary engines, the jet-lifted hovercraft and the giant silent toc-toc boat.\n\nOn the other hand there was hope for the alarm wristwatch that didn't buzz or ring but silently tapped on the wearer's wrist. And of course for the windscreen wipers for motor-cyclists' goggles, a pair of which I had actually made and worn while riding my push-bike \u2013 much to the amusement of passers-by. I think it was these which caused my father to observe, slightly ruefully, that he was surprised to find that he was the father of the White Knight.*\n\nIt occurred to me that one field in which small ideas would probably be welcome would be toy-design. So, quite deliberately, I settled down with a pad of blank paper and a pencil to try and think of new ideas for toys.\n\nI was attracted by the idea of remotely controlled mechanical toys. Radio controls already existed for model aircraft but they were expensive and limited in scope. It seemed to me that toys could be controlled from a distance but not necessarily by invisible means. An unobtrusive wire would do.\n\nThere were some tiny cheap reversible electric motors on the market. I bought some of these and used them to make some wirecontrolled toy vehicles, which could be steered like tanks. The toy I finally made was very peaceful \u2013 a remotely driven fork-lift truck. This would buzz about, picking up small things like cigarette packets or cups and saucers, lifting them up and stacking them in neat piles. I loved it and played with it for hours, becoming so skilled that I could work all the systems at once, so that it behaved almost as if it were alive.\n\nI took this one to the managing director of a major toy manufacturer. I didn't say much. I simply put it on his desk and we watched it begin to gather his papers and pens and scissors and thing-holders, carry them to the edge and drop them one by one into the waste-paper basket. He pressed the intercom and said into it: 'Everybody in here.' Then he sat back in his chair and watched the procedure with great satisfaction. His colleagues filed quietly into the office and they too watched in respectful silence. I felt rather proud. When the desk was clear I called it on to my hand.\n\n'No office should be without one,' said the managing director.\n\nThe demonstration had been very successful. Everybody was full of fun, but in the end the firm didn't take the toy.\n\nNone of this was making me any money. I realised that if I wanted a quick return I would do better to be a craftsman, making and selling finished articles.\n\nLike what, for example? There followed a period of frenzied enterprises, many of which I have forgotten. I spent a lot of time making Batik scarves and hand-painted materials. These were very attractive and might have gone well today in Carnaby Street and the craft markets, but in 1950 the buyers in the big stores sneered at them.\n\nIt occurred to me that there might be some future in making animations for television, so I made a flat pig which played 'Parlezmoi d'amour' on a violin, interspersed with squeals and grunts. The effect of this on the BBC Television Office at Alexandra Palace was totally hysterical \u2013 far more devastating than I had dared to imagine. They howled and wept and begged me to stop. They fell completely in love with the gross article and insisted that I leave it with them, being sure that it would be the inspiration of great things.\n\nI heard no more about the virtuoso pig, or about anything else, so I concluded that there was no future for me in television. By the autumn of 1950 I also had to admit that the theatre, or rather my ambitions to have any part in it, had receded so far that I could no longer, with any honesty, describe myself as an actor. This distancing was not so much due to my not being any good as an actor \u2013 there hadn't been much chance to find out about that \u2013 as to my lack of commitment. I didn't live for the theatre. My sole interest and raison d'\u00eatre was not centred in the West End. I didn't spend the whole of my time badgering agents, mingling with the hopeful and exchanging theatre gossip. That wasn't in my blood and, in the theatre of 1950, if that wasn't in your blood there was little chance of work coming your way.\n\nI had also had a problem with the language. I had an unforgivable habit of saying what I thought was true before I had really thought about what would be wise, or kind, to say. The language of theatrical flattery does have meanings but these are coded and cushioned by elaborate tact, simply because, in a world where life is being sustained only by far-fetched hopes and carefully tended illusions, words of cold reality are just poison. I am ashamed not to have been able to learn that language, but perhaps that too was in my blood.\n\nHaving concluded that I wasn't, after all, cut out to be an actor, I couldn't go on pretending that the casual projects I had been dabbling in were just an auxiliary sideline which I followed while I was 'resting' between engagements. They were my only work \u2013 and very potty they seemed to be.\n\nI see that I have been writing about this period now, forty-five years on, almost as if, at the time, I had been coolly and singlemindedly pursuing my prospects and purposes. That is not how it was. As I lurched desperately from one project to another I was also lurching, inside myself, from one picture of myself to another, from confidence to contempt. It was a very uncomfortable and overwrought period in my life.\n\nIn 1950 there was a vogue for mechanical shop-window displays. These were often mysterious, being based on optical or mechanical illusions.\n\nI made several of these, mostly based on magnets. My favourite was a box on which, with no visible means of propulsion, a small reclining figure (with a hole through it) was busily pedalling itself round and round on a sort of trolley, and holding up a placard with the words unfair to henry moore written on it.\n\nI took it along to Harvey and Goodman Ltd., who specialized in point-of-sale displays. The man there looked at it for a long time. Then he said: 'OK, so who is this guy Henry Moore?' I tried to explain, but he had lost interest.\n\nDepressed, I returned to the theatre, took a job as a flyman at the King's Hammersmith and later did some scene painting in Harrow. Neither job lasted more than a couple of weeks.\n\nTowards the end of 1950 my sister-in-law Mary suggested I should show some of my gadgets to the husband of one of her University friends. This was Tony, a young man who ran a firm which specialized in designing and setting up exhibitions and displays and was interested in the idea of offering his clients animated displays for their exhibition stands.\n\nTony invited me to join his team of freelance designers who were working from his new offices in the Cromwell Road. Having been at a loose end for so long I eagerly accepted his invitation.\n\nThe office was an empty room with loose papers and bits of handbag fittings left by the previous tenants when they fled, and for a while it was plagued by debt-collectors and bailiffs. Tony's 'team' was a bit sparse. It consisted of himself, a rather tense young lady who had to be addressed as Miss Page, me and, sometimes, a man who drove a van. Also, he explained, the business was starting from scratch, so any money that came in initially would have to be ploughed back in, though of course we would have a share of the grand final profits of each project. Although he didn't put it so crudely, this meant that for the moment we would be working for nothing. That fact did not stop Tony representing his organization as being substantial, fully equipped and well established, a habit that led to some embarrassing misunderstandings.\n\nQuite soon a real break came. Tony was offered a contract to supply and set up the interiors of a set of showcases in one of the pavilions at the great Festival of Britain that was due to be built on the south bank of the Thames in 1951.\n\nThese showcases, which were intended to extol the virtues of plastic materials, included no less than four animated displays \u2013 three fairly simple rotary devices and one bubble-machine. The bubble-machine was, in essence, a large diagram depicting a flow of materials, the flow being marked out by thin glass tubes through which coloured liquid, regularly interspersed with air-bubbles, would travel slowly along \u2013 rather like a moving dotted line. Bubble-machines were quite frequently installed at exhibitions but, although I had seen them, I didn't know how they worked.\n\nThis sudden access of business greatly enhanced Tony's view of the status of his company and caused him to invite a designer to join the team. The designer's name was Robert Auf-der-Mauer. He was a first-rate graphic artist and a sensible chap. He was to be a great companion in the difficult times that were to come and a great friend for the rest of his life. I remember telephoning to arrange to meet him at Gloucester Road tube station. 'How shall I recognize you?' I asked. 'You can't miss me,' he replied. 'I look like a friendly goose.' Which indeed he did.\n\nNever having worked in the field of Exhibition and Display before, I hadn't appreciated the remarkable delaying and stultifying effect that having to deal with important design-persons would have. The job we had to do could have been worked out and finished in a few weeks if we had been given the necessary data and the opportunity to do it. Unfortunately, each single item of visual design had to be discussed, conceptually pondered and elaborately negotiated by a hierarchy of pretentious self-important overseers, each determined to stamp his own personality on the work. Tony, who didn't actually do anything, was quite at home in the posy world of meetings and discussions but, for Bob Auf-der-Mauer and me, who were eventually going to have to deliver the goods, the sheer absence of decisions or workable instructions was a nightmare.\n\nI had another nightmare of my own. I was committed to supplying the bubble-machine and had blithely assumed that I could either find an existing pump mechanism or at least get some information on how they worked. My efforts to locate either a machine or an expert were a complete failure and I was left with the prospect of either admitting that I couldn't do the job \u2013 thus jeopardizing the whole contract \u2013 or settling down to work out from first principles how to make one. I had to choose the latter.\n\nIt seemed to me that the mechanism necessary to provide the flow of alternate bubbles and fluid along the tubes could possibly be fairly simple in that, given the right situation, the division of the flow might occur automatically. I set up a number of different rigs in the hope of making this happen, but they weren't successful.\n\nOne morning, after having spent the whole of the previous day failing to produce anything but small floods, Bob suggested that we should take a more basic and mechanistic approach to the problem: that we should make up a machine that would take hold of alternate doses of air and water and just stuff them into the tubing so that they would shove each other along. So we resorted to force.\n\nI made an air-pusher from a sawn-off bicycle pump and found a small gear-pump to deliver small amounts of coloured water.\n\nWe fitted a glass T-junction tube to the top of the circuit in the specimen display panel we had made and connected the air-pump and the water-pump to that. Then Bob and I knelt in front of it, he with the water-pump and I with the air-pump, and we took turns squeezing small bursts of air and water into the system. In a sense it worked. Air and water did go into the tubes and after we got into a rhythm the air and water did go into the circuit in approximately equally spaced doses, but these did not proceed in a stately fashion along the tube. As the pumps pushed and pulled, they hiccuped back and forth with such violence that the bubbles broke up. After some hours of trying different speeds and manipulations of pressure we had to admit that force was not going to work.\n\nFeeling thoroughly fed up, Bob and I hung up the various bits we had been holding, left the thing to stew in its juice and turned to go out for a beer and a sandwich.\n\nAs we reached the door Bob put a hand on my arm. 'Turn around very slowly,' he whispered.\n\nI turned. Travelling with stately dignity along the top tube of the display was a procession of alternate one-inch red and white bubbles. We watched them turn the corner and busily proceed back along the next section with the bubbles perfectly even and intact.\n\nWe tiptoed back to the display and once again knelt before it, watching fascinated and praying that we would be able to divine the mystery of its functioning.\n\nWe gradually came to see that the secret lay in a delicate setting of the flows of air and water into the T-junction at the top of the circuit. These had, purely by chance, been left exactly right.\n\nSo that was it! We went out for a jubilant beer and sandwich and then came back. It was still working, and it burbled away contentedly for the rest of the day.\n\nThat was a glorious end to one of our problems, but more were to come. The main characteristic of work for the Festival was that nothing that was supposed to happen happened when it was supposed to. Our material was finished and ready on time but the building it was to go into, the Power and Production Pavilion, was, to put it simply, not there. It was eventually made available to us exactly a week before the Festival was due to open, whereupon we discovered that, for reasons we knew nothing of, the showcases had not been made according to the plans we had been working to. T here was no time to argue about this and no point in doing so because nobody was taking responsibility. Bob and I just set to and, sustained by Benzedrine and knobs of sugar, worked, nonstop, night and day, for the whole week.\n\nEven then I didn't quite finish. As the King and Queen and the two Princesses came through the pavilion, viewing the exhibits on the formal Opening Day, they might have thought that all the ingenious animated displays were electrically driven. Not so; I was lying on my back underneath one of them, winding it by hand.\n\nOnce I was sure the Royal Legs were heading towards the exit I overcame the temptation to drop off to sleep where I was, scrambled out, staggered to the tube and found my way back to Finchley, where I fell into bed. I remained there for the next fortyeight hours, lifting my head at regular intervals to eat large meals brought by Daisy and then falling back into sleep.\n\nWhen, eventually, I woke up properly and found out what day it was, I hurried back to the Festival to finish installing the animated display and check the rest of the devices.\n\nEverything seemed to be in order, so I wandered off through the Festival to view the wonders that were on offer. I marvelled at the Skylon and the Dome of Discovery and enjoyed the impressive water-scuttle fountain that looked, behaved and sounded like a heap of giant cooking pans tumbling off a wet draining-board.\n\nThe exhibit which, as Quakers say, spoke most strongly to my condition, was in the Lion and Unicorn Pavilion. In this was a full-size effigy of Lewis Carroll's White Knight, who muttered encouragement to himself and carried with him certain inventions that had clearly been chosen for their absurdity. Among these, I noticed, was a pair of goggles with windscreen wipers. Perhaps Ray had been right after all.\n\n## _II. Silver Linings._\n\nThe work was completed and the Festival of Britain was under way but, strangely, no profit came to Tony's team. After a disagreement about the cost of a display clock I told Tony I was resigning.\n\n'What? After all I've taught you!' he cried.\n\n'No. After all the money you've never paid me,' I replied.\n\nI went on to make some animated displays on my own account. These were profitable but not frequent enough to constitute a profession. What I needed was a proper job.\n\nOnce again I was lucky. By a sequence of coincidences I fell into a job. One of Ro's friends from her ballet-school days was a South African girl named Clementina. She had met and married a young comedian who was, at that time, taking the world of showbusiness by storm with his brilliant use of a sink-pump and a chair-back \u2013 his name was Michael Bentine.\n\nOne day when Michael and Clem came to see Ro and her parents at their house in Hendon, I happened to be there and happened to have with me the toy fork-lift truck that I had made. Michael was delighted with this and said at once that I must show it as soon as possible to a fellow member of a gliding club that he belonged to.\n\nThat being so, a meeting was arranged. I took the fork-lift truck and one or two other devices to meet Ralph Furness in the office of his firm, Kays Industries Ltd., which was a small room over a shop in Ealing.\n\nMy animations were of little use to Ralph in his business, which was perspex fabrication, but his reaction to them was refreshing. He took them seriously. He respected them, and me, and gave serious thought to the question of how they could be developed.\n\nHe suggested that if I could design and draw the parts for a perspex prototype of my toy fork-lift truck he could have it made up by a model-maker.\n\nRalph also told me about a new process that he was thinking of taking up. This was a method by which small plastic articles like buttons and brooches could be coated with metal \u2013 copper, silver or brass \u2013 by electroplating.\n\nWell, there was just one thing I knew about electroplating and that was that the object to be electroplated had to be able to conduct electricity. It was quite impossible to electroplate plastic articles.\n\nRalph handed me a large gold button. It felt very light in weight but it was, undoubtedly, gold.\n\n'This can't be solid gold,' I said.\n\n'No, it's brass, electroplated on to copper.'\n\nRalph took a pair of pliers and cut the button in half.\n\n'Brass-plate over copper-plate over a silver lining on pink plastic,' he said, handing me half of the button. The metal plating was as thin as card on the pink plastic but it was definitely metal.\n\n'So you are a true alchemist,' I said. 'You can turn base materials into gold.'\n\n'Not quite, but I hope to be able to coat base plastic with chemically deposited silver and then plate it with copper and turn it into antique-finish copper, silver or golden brass, for the rag trade. There's a big fashion for metal-finish buttons coming up and I've not only bought the process but also the aged chief chemist who originally developed it.'\n\n'How far have you got with it?' I asked.\n\n'I've rented an empty pub in Baker's Lane.'\n\n'Is that all?'\n\n'We know that it has already been done. We know the exact sequence and nature of the processes and treatments the buttons must go through to get their coats. We have the expert to brew up the chemicals and the button-manufacturers will give us all the buttons we need. What I need next is somebody to design and put together a production line big enough for us to do bulk sample runs.'\n\n'When do I start?'\n\n'Now.'\n\n'How much?'\n\n'Twelve quid a week.'\n\nSo that was that.\n\nTurning from putting together fiddly animated displays to designing a complete manufacturing process was like turning out of a bumpy side-lane into a broad main road, getting into gear and opening the throttle.\n\nThough I didn't know it at the time, what I was helping to put together was not just a process. It would grow and become, for several people as well as me, a place of work and a way of life.\n\nSetting up a button-plating plant for Kay's Industries was in many ways similar to putting together an animated display. There were processes that needed to be done and I was going to work out how to do them. Then I was going to prepare the place in which they were to be done, in the sequence in which they had to be done, in a way that would be appropriate for the people doing it \u2013 that last consideration was in some ways the most interesting because people, unlike machines, could become bored or tired and consequently careless, as well as unhappy.\n\nThe process of coating the plastic buttons with metal was complex and involved the use, in exact sequence, of various bulkhandling machines: rougheners, washers, chemical dips, sensitizers, jar-rollers and some very subtle electroplating barrels. Also, for finishing, burnishing barrels, lacquer-dips and a spinner.\n\nThe unit we put together was relatively small, just enough to provide a good flow of samples for the commercial travellers to take to the button-dealers in search of orders.\n\nThe orders came in so fast that Ralph had to face the moment of truth that comes to all such projects \u2013 the prospect of having to raise working capital in order to invest in a large-scale factory plant designed and properly made by competent engineers. Fortunately Ralph didn't do anything so unwise. The prototype installation was working perfectly well, so we simply replicated it, turning it from a one-barrel to a six-barrel plant, still using the same sort of makeshift devices that I had put together for the first one but on a larger scale. At the same time I moved imperceptibly from being the creative designer to being the general manager, master button-silverer and occasional tea-boy for a very unorthodox factory.\n\nThis evolution took place over several months, a period during which my own life also evolved and changed. I became quietly contented, mainly, I think, because my mind was engrossed in the work. There was so much hard thinking to do, so many designs and methods to work out and so many problems to overcome, with the growing satisfaction of seeing the scheme slowly come to fruition. My greatest excitement was to collect and carry away the open boxes of finished work, boxes brim-full of gold and silver buttons, bright, rich and numerous as an _Arabian Nights_ treasure. When I brought them to the office where they would be packed up and despatched, I would lift a box and pour a glistening stream into the weighing machine. I would dip my hands into the shining pool, letting the polished buttons flow through my fingers, like a miser rejoicing in his wealth.\n\nBy about 1953 the button-plating had settled down, and work was flowing through in a steady stream. At the workshops, a converted stable in Hanwell, a suburb to the west of London, we employed an experienced electroplater, whose name was Harry, and his assistant, whose name was Eric. Although nineteen years old and quite fit, Eric was 'a bit slow' and lived in a local 'home'. There was also Wally, a policeman who turned up when he was off duty to do odd jobs, and Claude Gillett who was Welsh and a window-cleaner. He did some odd jobs but more often just came in for a chat. Our chats were a great part of my life and we became good friends.\n\nI remember that place and those times with great affection. The work was dirty and sometimes dangerously smelly but there seemed to be no need for hassle or shop-floor politics, life just flowed along at its own pace.\n\nI bought a little car, a 1930 Austin Seven tourer. I named her Albertine after the sixpenny Woolworth's rose that flowered so profusely in the garden at Finchley. I painted her lemon yellow and shiny black. She was neat and comely and very strong. I loved her dearly. Sometimes after work I would drive down to Weybridge to spend the weekend with Bob and Margery Auf-der-Mauer and their ever-growing family. Working at Kays Industries left me so filthy that Margery always kept two towels for me. One old one that didn't matter for the first wash and an ordinary one for the second. My memory of those visits is misty \u2013 just days of forgotten ease.\n\nAll these things should have been enough to keep me quiet, and in most ways they were, but once the plating works was complete and running there was no urgent thinking to do, no more problems to be solved and nothing more to make. So I suppose my mind began to wander again \u2013 not that it had ever stopped. Thoughts, possibilities, ideas, grew on me like mildew. Although the plating process was all right, I didn't much care for the patterns of the buttons that were going through, so I put together a crude injection-moulding machine with which I could make plastic articles using home-made metal moulds. I then began to produce buttons and brooches designed especially for electroplating \u2013 medallions, and replicas of antique coins which I thought would make very distinguished fashion accessories. I metallized these and finished them by hand. They looked really handsome and authentic. I showed them to Ralph's travellers and to one or two of our contacts in the button-trade. They all hated them, were certain there was no market for them and that there never would be.\n\nThey were probably right. I tried a couple of other patterns and then gave the idea up. Oddly enough, once I was quite sure something wasn't going to work out I never hesitated to scrap it \u2013 so long as nobody else was involved. I didn't mind wasting my own time on uncertain follies but I was embarrassed if I wasted other people's. So as far as possible I worked on my own. The only person exempt from this reservation was Keith Hutcheon.\n\nI think I met Keith at one of the club theatres, perhaps the Watergate, where he was doing some wiring and I was helping Ro. He was about my age but, unlike me, he was a proper, qualified engineer. Fortunately he was tolerant of ignorance and shared my appreciation of elegance and originality in mechanical objects and designs. We started talking about this and that, about the insides of rotary pumps, about the similarities between conventional and gas-turbine engines, about alternative sources of energy, etc. \u2013 all the sorts of things that swilled about in my head but which would unfailingly put most of my friends to flight if I talked about them. Keith and I talked for several hours that day and have continued our conversation without interruption, except by time and space, ever since.\n\n## _III. Public Stomach._\n\nMy having a proper job had a happy effect on my life at home. I became noticeably less agitated and was able to enjoy the company of my parents in a way that hadn't been possible before. In their eyes I had ceased to be 'a problem' and, in certain fields, had even become worthy of respect. I was able to look at them with affection and pay a bit of attention to their lives.\n\nAs I have mentioned, Daisy had felt the ending of her active role in family life very keenly. She was now in her late fifties and her rheumatism was painful. We all encouraged her to do something independent. We even managed to persuade her to start on what would have been some really exciting memoirs, but she lost momentum very easily and didn't keep it up. Having been a firstrate secretary all her life, I don't think Daisy really was an originator \u2013 she was a coper.\n\nRay, on the other hand, had been going from strength to strength. With the war over and a Labour government in power, he now felt able to move away from active political work. He was fast becoming an expert on wine and was addressing what he saw as a major injustice of the time \u2013 the wilful ill-treatment of food.\n\nRay's contention was that although during the war, and the years of austerity that followed it, housewives and home cooks had learned to be enterprising and innovative with the sparse, monotonous food that was available, commercial caterers had more or less given up. Their standards, which had never been high even before the war, had sunk to a level where the restaurants were, in Ray's view, cheating the public by spoiling perfectly good food through careless, slovenly cooking while at the same time fending off criticism with excuses about austerity and shortages which were often long out of date.\n\nRay wrote articles about this in _The Leader magazine_ and _Lilliput_. These brought into existence the Good Food Club, an institution which, as he explained, had no existence except in the minds of those who were moved to write to him with recommendations and comments on their experiences of eating out.\n\nLetters poured in. _Lilliput_ couldn't cope with them, so in 1952 Ray brought the lot home to Finchley where the spare room became the office of the President of the Good Food Club.\n\nDaisy, complaining loudly, waded straight into the task of managing the office. She marshalled the helpers, organized the complex filing system, even keeping a separate, confidential 'Bad Food Guide' containing denunciations, some so withering as to be unpublishable, that unlucky members had sent in. She also, as always, fed everybody.\n\nMeanwhile Ray would be at his desk, evaluating the reports and composing the concise, witty entries that were to become the hallmark of the _Good Food Guide_. He also, as always, poured wine.\n\nThere wasn't a lot I could do to help because I was in full-time work but, being on the sidelines, watching Ray becoming celebrated and lionized in the slightly precious world of gourmets and winewriters, was a great pleasure. Daisy, remembering perhaps the revolutionary commitments of their youth, remained affectionately sardonic about these new distinctions and this may have saved him from becoming too carried away. I, rather carelessly, coined for him the title 'Public Stomach Number One'. He affected displeasure at this piece of disrespect but couldn't resist quoting it.\n\nBeatrice, the old bull-nosed Morris car, had been sold into retirement, so when the Good Food Club needed inspections of restaurants in the home counties, Daisy commissioned me to buy her a replacement. The car I found was a 1938 Hillman Minx tourer (it had to be a tourer because Daisy liked to drive with the hood down). She named the car Miranda, on account of the need to remember which inn she and Ray were heading for.*\n\nSometimes, on summer days when Ray was in town at some gastronomic 'do', Daisy and I would take Miranda out to lunch. We didn't bother her with inns, we would raid the fridge and the larder, purloin a bottle of Ray's rather-better-than- _ordinaire_ claret and, with tablecloth, glasses, table napkins, plates, knives and forks all heaped together in a basket, take ourselves off for a very excellent picnic under willow trees beside a small river somewhere in Hertfordshire.\n\nThese were happy outings for both Daisy and myself. I have the impression that we both felt deeply affectionate towards each other but also, in the service of that affection, respectful of each other's privacy. My fearsome behaviour as an adolescent had no doubt made Daisy wary of enquiring into the frustrations of my private life, while I, though well aware that her current situation \u2013 that of being a spare part at the margins of Ray's hyped-up career \u2013 was less than satisfactory, was careful not to badger her about doing something about it. We just sat in the grass and enjoyed the wine. For two outspoken, tactless people, renowned for always saying the first thing that came into their heads, we did rather well.\n\n## _IV. Feasts._\n\nThe rituals that most fully caught the spirit of those Bacchanalian times were the bottling parties. Ray, through his connections with the wine trade, would select a hogshead of wine. He couldn't afford, let alone drink, a hogshead of wine, but he only had to mention his intent to friends at the Savile Club for them to apply eagerly to subscribe,\n\nThe barrel would be shipped and delivered to the out-house, where it would be jacked up on to a specially made stand called a scantling. Meanwhile the subscribers would be gathering up used wine bottles, removing old labels and washing them out with bottlebrushes. Ray would confer with his typographical cronies, elegant labels would be printed and, on the appointed day, corks would be soaked and Daisy would oversee the setting out of a magnificent cold collation on the big dining table. In the evening the subscribers would arrive in their handsome cars and when all were present and attentive the barrel would be ceremonially broached.\n\nTasks were allocated among the subscribers, as fetchers, fillers, swigger-corker, carriers, labellers, packers etc., and bottling would commence.\n\nRay had obtained a massive antique bottle-corker made of solid cast iron. It weighed as much as an early printing-press and had a similar handle. If carelessly used, this could shatter bottles like eggshells, so I, being used to it, usually manoeuvred myself into the post of swigger-corker \u2013 the swigging part of the job being the necessary adjustment of the level of wine in each bottle before fitting it into the corker and slamming in the cork.\n\nThe system usually ran well for most of the evening but gradually a certain confusion, bordering on hilarity, would creep into the proceedings. This would be due to various causes. A certain amount of innocent tasting had been going on and the alcoholic fumes of open and spilt wine were quite affecting, but the main cause of jollity was simply general bonhomie, the happiness of good friends engaged together in a common joyful task.\n\n*\n\nRay's bottling parties were very jolly occasions, but for sheer magnificence there was nothing to compare with the ceremonies which he attended in France. In 1951 Ray had been honoured by the Jurade de St-E\u00b4 milion, a prestigious guild of wine-growers whose function was to judge and maintain the quality of the wines of the region. He was appointed its first overseas _pair_ (peer) and was thereby entitled to assist at its ceremonies.\n\nThe town of St-E\u00b4 milion is ideally suited to such purposes because it spreads along the curved slope at the end of a valley like an amphitheatre with, at its centre point, a tall square solid-looking tower, Le Tour du Roi. The town also has two churches. The new one in the square dates from the thirteenth century. The old one underneath it, whose belfry tower sprouts incongruously out of the ground, was originally a cave in the solid rock which had been inhabited by the saint in the eighth century.\n\nThe F\u00eate de Printemps of 1954, which Daisy and I had the good fortune to attend with Ray, did not just involve the use of a hall or a hotel, it involved the use of the whole town. It started in the morning with the Jurats, a body of portly gentlemen in scarlet robes trimmed with white, assembling before the War Memorial to lay a wreath. They were followed by a similar number of other well-built gentlemen, including Ray, wearing good suits and _\u00e9pitoges_ , which were ornamental shoulder-stoles to show that they were _pairs_. After them came the _prud'hommes_ who, as far as I remember, wore no badge of office. From the Memorial they walked in procession through the town, accompanied by Les Compagnons du Bon Temps de Sarthe, a group of tall men in hunting caps and pink jackets carrying, or rather wound into, golden French hunting horns. These blasted rousing fanfares at each corner as the procession made its way through the town and down steep paths to the cavern of the Old Church, where, by the light of flaming torches, the wines of the year before were poured from bottles without labels and, in the presence of the spirit of the saint, judged and pronounced upon. After that the procession broke up and moved to the banquet.\n\nMy memory of the banquet is confused. It was laid on whitecovered trestle tables with folding chairs in a wide stone room, arched but with a low ceiling. Perhaps it was an undercroft or crypt. What assailed me in that echoing place was the ear-shattering noise generated by a couple of hundred happy French people engaged in ordinary conversation. The banquet itself was magnificent but, I must say, not very well organized, so that the food was served slowly and arrived rather cold. However there were some lively speeches, including one in which the Premier Jurat welcomed, as Guest of Honour, the Paris Chief of Police, and hoped he would soon have time to enjoy a long and well-deserved holiday away from the bustle of the city. As the Chief of Police was at that moment under investigation for alleged corruption, this invitation was taken to be particularly apt and was rapturously applauded.\n\nAs I didn't know a lot of French, much of this hilarity went over my head. So after some hours I slipped away to get some air and have a look at the Tour du Roi where the judgement would be promulgated. On closer inspection the tall tower turned out to be less solid than it had seemed. It was in fact completely hollow. A staircase with a flimsy hand-rail was built against the inside of the walls, and this led up to a narrow walk-way around the top of the tower, the centre void being lightly roofed with tiles.\n\nI heard the sound of hunting-horns and, realising that the banquet must be ending, I turned to walk back. After a few paces I came face to face with Daniel Querre, the Procurer Syndic du Jurade, panting down the hill in advance of the procession. Daniel was our host, a short man but bulky, indeed almost spherical. His eyes lit up with delight as he saw me.\n\n'Ah! What chance!' he cried. 'One pushes!'\n\nKnowing my duty I got behind him and, with as much courtesy as the procedure would allow, literally shoved him up the narrow staircase and out into the cold air at the top. I turned to go back down again.\n\nThere was no going down. A single line of scarlet persons could already be seen puffing up the awesome staircase, led by the vast M. Chaillot. As he neared the top I saw his face, grey-pale against the scarlet of his robes, and I reached a hand down to help him up the last few steps. He took it in a vice-like grip and, using both hands, clambered up me like a ladder.\n\n'One excuses oneself,' he muttered humbly, 'You know, the vertigo!' Indeed I did know the vertigo.\n\nMore and more rotund persons were emerging on to the narrow parapet and slowly filling it all the way round. Finally the Compagnons with their golden horns emerged and took their positions. From a distance the tower must have looked as if it was wearing a soft crown of scarlet and white bobbles, trimmed with spots of bright gold.\n\nI, quite terrified, had backed into a corner and was crouched low. Through the battlements I could see that the whole town was out there watching. Lining walls and balconies, streets, parapets and windows, the people of St-E\u00b4 milion were eagerly awaiting their moment of truth.\n\nThe tower had been fully wired with loudspeakers so, when he came to make his speech, Daniel Querre's magnificent voice rolled out into the amphitheatre of the valley and echoed away into the distance.\n\nHe told them he was there to bring them the great good news that the wine of the year before had been appraised and found truly worthy to bear the Great Seal of the _Jurade_ on its barrels and so take its place among the prestigious vintages that bore the holy name of the saint, to whose glory and to the glory of God this vintage was now dedicated, with all the honour and love of the people there assembled, '... and of our friends the dead, beneath the stones... and even the birds of the air...'\n\nAs he said these last words, hampers at the foot of the tower were thrown open and a multitude of white doves released. These circled the tower and rose around it in a white cloud. As they did so the evening sun caught them and their flashing wings shone white-gold against the darkening blue sky. The town cheered and laughed and wept. Les Compagnons sped them on their way with golden fanfares and, as the shining birds flew high and turned away towards their lofts, the ceremony ended at last \u2013 apart from the delicate and nerve-racking task of getting the Jurats down the dreadful stairs.\n\n## _V. Friends._\n\nEver since the days at LAMDA I had remained close to Frieda Hodgson and her family, who also lived in Finchley. Frieda held open house every Sunday evening, gatherings to which any and all of her friends from LAMDA were always welcome. I really enjoyed those evenings because they were filled with the warmth and ease that had characterized her Friday afternoon sessions at the Academy. There diffidence was suspended and joyful appreciation prevailed. There people sang and danced, recited their poems, told stories. There I could make an exhibition of myself without any hint of embarrassment, because the attention was always entirely on what was being offered and never on the temerity of the person offering it. I was even, occasionally, persuaded to sing the blues, God forgive me. There I met several gentle people who became lifelong friends \u2013 though where life has taken most of them by now I don't know.\n\nAmong those friends the closest, I think, was Jill. She was light, moved delicately and was full of laughter \u2013 most of the time. Not all of the time, though. I remember sitting with her at one of Frieda's soir\u00e9es, both of us feeling glum from the sorrows of our love lives. She was in love with Frieda's eldest son, Charles, and I was, as ever, in thrall to Ro. Things were not going well for either of us.\n\nI said: 'My car isn't all that fit at the moment but next Tuesday I shall have the firm's van. How about we go to the Norfolk Broads?'\n\n'Yes... let's.'\n\nSo we did. I collected Jill at six in the morning in the mists of what was going to become a bright spring day. The van ran sweetly, chewing up the miles till we came to Newmarket. There we saw shifty-looking men in flat caps leaning on hedges, peering at distant racehorses through gigantic binoculars the size of quart beer-bottles. These, I heard later, were tipsters' men, 'spying for form'.\n\nI can recall very few details of that day. We ran along raised paths between wide expanses of water. We lay in long grass looking at the sky and listening to the larks. We laughed a lot and gladly resisted the temptation to pour out all our troubles to each other, because time was too good to waste on boring things like that. We were simply happy to have that day to enjoy together without expecting, needing or looking for anything from one another except to feel the sunshine and be able to share the pleasure of being alive.\n\nIn the evening we drove quietly home again. Nothing special happened that day, but it was a day that neither of us have forgotten or ever will forget.\n\nI treasure a slightly puzzled astonishment about the happiness of that day-out-of-life. Certainly it bore no resemblance to the tense anxieties of what, at the time, I thought of as my love life.\n\nI must say I feel a bit diffident writing about my love life, not because it was dramatic but because it was uneventful. As I mentioned earlier, I had met Ro at the Drama School in 1947 and we had formed a rapport which, on my part, soon became an infatuation, a state of feeling which Ro did not share.\n\nSuch relationships aren't particularly unusual, but they are usually ended fairly quickly. For some reason this one went on for several years. I confess I don't know for sure why Ro put up with the nuisance of my importunate presence for so long, but I suspect the answer was simple \u2013 she was engrossed in her own important purposes and sadnesses which took priority over all other considerations. Also she found me useful, especially once I had got a car.\n\nAs for me, I think I remained attendant because I chose to. I couldn't shake off the infatuation, so I put up with it. The situation was ridiculous but in a way it suited us, because Ro was always anxious to receive service and I was a chap who was always glad to drop whatever he was doing and help. Also, although attending Ro in her purposes was unrewarding in some ways, it was not in itself a burden. Quite the contrary, her world was interesting.\n\nRo and her friend Hilary had teamed up to offer a property-making and costume service to the small shows and 'intimate' revues that were being put on in the many little club-theatres that in those days existed in London. This was a fascinating episode in the history of the theatre because the shows, produced on a shoestring budget, were able to attract a more sophisticated and intellectual audience than the big West End theatres could.\n\nThe little theatre I remember best was the Watergate Theatre Club. This was a series of rather grim converted cellars with a tiny auditorium and stage, which could become an uproariously delighted place when a really good revue was on. Most of these were written by Peter Myers. Peter was a plump and particularly single-minded man. In fact he was a power-house of drive and purpose, all centred on the theatre and the productions he was preparing. Like a general, he commanded everybody and, as if he was a general, they obeyed. He and his wife Prue and their three children lived a few doors away from us in Hendon Lane. So, quite apart from meeting them at the theatre, they were our neighbours.\n\nPrue, in particular, became a family friend. She was a tall, slightly ungainly looking young woman, a year or two older than me. Ray liked her a lot, and would often suggest that I should invite her to dinner parties \u2013 'to make up the numbers' as he put it. This I was happy to do because I also liked her, though I confess I was a bit puzzled by the picture I got of her family life. Peter, her husband, seemed to treat her like a servant, while she was quite open and vocal in her condemnation of the overbearing and completely inconsiderate way in which he carried on.\n\n## _VI. Serious Events._\n\nIn February 1956 a late, bitter frost hit the whole Bordeaux region, including St-E\u00b4 milion. Hundreds of acres of vines were frozen; vines which for decades had produced some of the greatest wines in the world were now pieces of frozen wood. The exact damage could not be assessed at once but it seemed likely that most of the older plants were done for. This was a source of grief and anxiety for Ray \u2013 until something came to wipe it from his mind.\n\nOne morning in March, Daisy came down to breakfast fully dressed. Ray and I didn't comment on this, assuming that she was going out somewhere. She was: to see the doctor. She was diagnosed as having cancer of the cervix and was taken straight to St Thomas's Hospital to undergo a serious operation.\n\nRay and I walked about that big house like zombies, having slightly stilted conversations about trivial matters. The wary affection that had grown between us in recent years did not, alas, allow us to share grief and fear, only to respect it in the other. However, there was plenty to do, including buzzing backwards and forwards to the hospital.\n\nThe operation went well and was declared a success. In due course Daisy came home to recuperate. Amid rejoicing a barrel of wine was bought and bottled with a special label celebrating her recovery. Daisy was dubious about the gesture but enjoyed the wine. Before long life was more or less as usual again, apart from the unspoken shadow of a recurrence that always follows such an operation.\n\nI think we were all, in our ways, very rattled by this brush with mortality. I know I found myself looking quite closely at my own life, wondering, once again, what I was doing with it. One thing was quite clear: Kays Industries was going into decline. The metallized button trade had passed its peak and was beginning to drop away. Before long my job would come to an end. The time was coming when I would once again have to find something else to do.\n\nFor some years I had been interested in photography. I loved the work of photographers like Cartier-Bresson and Robert Doisneau, whose pictures had a clear intensity, capturing a single instant of life in a simple composition. Typically I wasn't content just to enjoy their work, I wanted to have a go. I bought a 35mm camera, took lots of photographs of people and places, some of which I got paid for, and was considering the possibility of doing some freelance photo-journalism for picture-papers like _Picture Post_ and _The Leader_.\n\nAlthough the frost in Bordeaux had been a serious tragedy, I had noticed that the event had not been widely reported in the English press, and I wondered about doing some sort of article on it.\n\nThat idea came up again in the late summer of that year, when Daisy was fairly well recovered. Without mentioning that it might be their last chance to do so, Ray proposed that we should take a trip to Bordeaux to visit their friends. I suggested that it might be easier for Daisy if we hired a car in Bordeaux so that I could drive them about on their visits. Then, at other times, I could use the car to take some photographs of the frozen vines and perhaps put together an article that I could offer to one of the picture papers.\n\nWe had a very good holiday and I was also able to travel around in the hired car interviewing the farmers, who were using oxen to grub out the dead vines. I took lots of photographs, some of which looked fine to me as illustrations to the article, but in the event the editor of _The Leader_ turned them and the article down, saying, 'Very nice, but not quite up to our standard.'\n\nAfter that I started a new round of doomed projects. I did some specimen book jackets and record sleeves, all of which were admired before being turned down. To Ray's contacts in the office machinery trade which he had overseen during the war I offered a hydraulic typewriter \u2013 a concept that filled them with horror.\n\nKeith Hutcheon and I proposed and discussed several ideas for cars including a 'fuel-consumption warning indicator' and an 'average speed indicator' to go on the dashboard. The first of these would warn the driver when he was using the accelerator extravagantly and the second would show, on a dial, how many minutes late or early one was on a pre-determined schedule for a particular journey. As usual we enjoyed the thinking but didn't succeed in developing the ideas to a point where they could be offered for sale. All these ideas are long extinct because nowadays such things would be done with electronics.\n\nJust to see if I could do it, I wrote some short stories. I wrote one about a cat that dreamed it was a tiger. I liked it and read it to people, who also liked it, so I wrote another about a milkman who had a magic day. It had no real plot and nothing much happened, but it wrote itself, the words running down my arm to the pen in my hand, with my head racing along behind trying to keep up. It was the same sort of feeling that I had had in the art college when I was doing those drawings of M. Letailleur in 1943. It was as if I couldn't go wrong.\n\nI was nervous of showing the stories to Ray but I needed his opinion, which I knew would be absolutely honest. So I left the typescripts on the chest in the hall. They vanished and the next evening were back again with a note pinned to the second page.\n\nIt read: 'These are a lot better written than most of the rubbish I have to read nowadays, but watch out for clich\u00e9s. You know how to recognize a clich\u00e9?...' Ray went on to discourse on ways to ferret out these dreaded forms.\n\nI asked about 'style' \u2013 was the style of my writing OK?\n\nRay said: 'Style isn't something else. If you write what you have to write as clearly and readably as you can, that's enough. That will be your style.'\n\nMuch elated, I wrote more stories and sent them to magazines like _Argosy_ and _Reader's Digest_. Then, suddenly, I had to stop. Another, far more urgent call on my time had come. My car had just fallen apart.\n\nThis wasn't my first car, the wild sweet Albertine. She had died a year earlier and I had written an elegiac poem in her honour (which I still cannot read without tears). This was another Austin Seven, a Ruby, whose bodywork had split in half, making it impossible to close the doors.\n\nI rebuilt it as a sort of jeep, with a high windscreen and an open body made from sixteen-gauge aluminium. It was very ungainly and had the air of being, well... accidental, as if it had been bashed together out of aluminium sheet in somebody's front garden. It did have one magnificent piece of equipment: a mechanical Klaxon hooter from the ship's chandlers in Shaftesbury Avenue. This had a brass knob on the top which, if you thumped it with your fist, would give a squawk like a gigantic chicken, a noise so astounding that it would stop an _Evening Standard_ van in full flight.\n\nThis mucking about with home-made cars was all very well, but it wasn't going to provide me with a way to earn my living.\n\n_Reader's Digest_ and _Argosy_ sent back my stories. I didn't mind too much about that. I had had a go at writing stories and knew I could do it. In fact I looked at one or two of those first stories quite recently and they are among the best things I ever wrote. But at that time I couldn't see that story-writing could ever earn me a living. So, almost as a last resort, I turned back to the display trade.\n\n## _VII. Hitting the Buffers._\n\nI had never really severed my connections with the display trade. I was still occasionally making and installing bubble-machines and other animated displays. Now, rather than looking for individual display jobs, I decided to aim for some small-scale manufacturing. I had in mind to develop one absolutely basic component which I could manufacture, assemble, sell and deliver more or less by myself, without any intermediaries. This was quite a sensible idea, or would be if we could find the right article to manufacture. I first thought of a simple iron-cogwheel electric motor that could be used to power shop-window displays. In consultation with Keith Hutcheon I made a couple of prototypes and they worked quite well but then, one day, entirely by chance, I came across something much more immediately useful.\n\nThis was a tiny battery-operated magnetic pendulum that would do something simple \u2013 like, for instance, moving the arm of a pig playing a violin \u2013 for three weeks nonstop on one torch battery. This seemed to fill the bill exactly. We could manufacture the pendulums, incorporate them in runs of shop-window displays and also, possibly, sell them as a product. I checked the patent situation and it was clear, so Keith and I designed a plastic version of the pendulum and in due course drew a moulding tool in which the impression could be milled out of a solid block. That would be cheap and the parts could be moulded on the injection-moulding machine I had made for the ill-fated antique-button project. We went ahead and turned out some samples. I offered them to a display firm in Pimlico, which gave me an order for fifty, to be delivered the next week.\n\nSo life suddenly became real and earnest. With Keith's help I set to and moulded a batch of pendulum parts. We set up a kitchentable assembly line to wind the coils and fix on the metal bits, and at one point, I remember, we had a dozen of the little clickers ticktocking away. The order was delivered and another one taken. Great industrial empires had grown from such tiny beginnings. I felt I might be on the brink of becoming a real businessman, owning the means to produce simple articles for which there would be an assured market, prospering at last in the world of commerce!\n\nI remember the feeling very well. I was walking down Victoria Street carrying a batch of pendulums to deliver to the display firm. I came to Horseferry Road and stood there, waiting to cross. As I stood there my legs sent up a message: 'We aren't going.'\n\nI said: 'Cross the road. I'm going to cross the road.'\n\n'No you're not,' they said, 'not without us.'\n\nThat was true. So I backed off and sat on a bit of wall.\n\nAll this sounds absurd, but it is as near as I can get to describing what happened. I couldn't make myself cross the road. Something was wrong.\n\nI had to sit there and conduct meaningful negotiations with my body in the hope of persuading it to resume normal working. The points it was making were quite simple: 'You are going down a blind alley. The path you are taking is not for you. You would be useless and miserable as a commercial person. You are a lousy craftsman, a hopeless salesman, a rotten work-getter and an even worse businessman. Also, plying a trade like that \u2013 as an itinerant vendor of devices that customers don't even know they want \u2013 calls for a level of single-mindedness, persistence and self-confidence that you just don't have. And, above all else, you aren't really interested! You don't want to be tied to producing potty pendulums through all eternity. Pack it in!'\n\n'What shall I do?' I asked, much chastened by the truth of those observations.\n\n'You've got to do what you said you would do, of course. So we deliver the pendulums, finish out any other orders, do what has to be done, even new things if necessary, but you, you get out of your head any delusion that this is the field you want to work in because it just isn't. OK?'\n\n'OK,' I said, and we crossed Horseferry Road. I must say the legs felt a lot better for the confrontation and my step was light.\n\n* The White Knight is a character in Lewis Carroll's Alice _Through the Looking Glass_ who invents particularly unlikely articles.\n\n* 'Do you remember an inn, Miranda, do you remember an inn?...' ('Tarantella', a poem by Hilaire Belloc.)\n\n# STARTING AGAIN\n\n## _I. Stopping and Looking._\n\nTo be forced to realise that in a major part of one's life one has been working one's way into a dead end is, to put it mildly, unsettling.\n\nI could see that I was no stranger to dead ends. My involvement with Ro had long since reached and passed the point where I was more trouble to her than I was worth, there was no doubt about that. But also the last few years were strewn with ideas and projects that I had taken just so far and then, at the first hint of discouragement, abandoned, telling myself that there was no future in them. Sometimes that had been true, but had it been true in every case? Certainly it wasn't true of the pendulum project. That was surely viable.\n\nDidn't I want to be an individual entrepreneur? Didn't I want to be a success and show the world I was right after all? If, now that the knitting up of ideas, setting up of methods and costing out of the work was done, my heart wasn't in the task of taking the project ahead and becoming a prosperous manufacturer of small pendulums, then what the hell was it in? Did I harbour some concealed reluctance? Had I always secretly hoped my projects would fail? Why couldn't I do what other people did \u2013 just decide what I was going to do and get on with it?\n\nI had no answer to these questions. Asking them made my head ache.\n\nI had another difficulty: I wasn't interested. I didn't want to have anything to do with any of it. All the ideas I had had seemed to have been solitary schemes, private flights of fancy which might or might not be successful but which were all, in the end, directed towards the advancement of me. And at that time I was, to put it bluntly, thoroughly fed up with me. I was fed up with what might be called my 'personality', with the ingratiating, overwrought jokey chap, who was full of far-fetched notions and fancies, and in an almost constant state of submerged agitation. I can see that I didn't so much _dislike_ the image I wore, it was more that I was profoundly _bored_ by it.\n\nYears earlier I had overheard Mrs Hodgson say, almost in passing: '... but of course you never really begin to live until you are committed...' At the time I hadn't understood what she meant, but the thought had stayed in my mind as something that might one day turn out to be important.\n\nAt that moment, however, one thing was very clear to me. The prospect of committing my life to the single-minded pursuit of the trappings of my boring image \u2013 personal success, wealth, dignity, respectability, membership of the Round Table, ulcers, a Mercedes \u2013 was not just uninteresting, it was frightening and peculiarly lonely, like spending one's life decorating one's tomb.\n\nMeanwhile, life shuffled on. I went on with what was there \u2013 doing the silvering, installing a bubble-machine at Olympia, making up pendulums. I don't think anybody, except perhaps Daisy, noticed that my customary gloom level had deepened and had begun to rumble.\n\nSometime in October or November 1956, Prue Myers and I arranged to go to the Lyric Theatre in Hammersmith to see _Look_ _Back in Anger_ , the new play by John Osborne.\n\nI don't recall the details of the play but I do remember that I found it very moving. In the interval Prue and I talked about our lives, comparing them with those of the people in the play, and I took the opportunity to ask her something that had been in my mind about her life with Peter. She had, as I mentioned earlier, been fairly outspoken about her dislike of the way he carried on. I now asked if I was correct in assuming that, like many married couples who on the surface appeared to be hostile towards each other, she and Peter really shared some deep-rooted bond that held them together.\n\nShe thought about this before answering. Then she said: 'No.'\n\nI waited for her to say something more and after a pause she added: 'No. The children hold me, but not to Peter. I would get away if I could.'\n\nThe bell went and we returned to our seats. After the play we had a drink in the bar. I looked straight into her eyes, which was something I don't think I had ever done before.\n\nThen we went back to our houses in Finchley. I knew something extraordinary had happened but I wasn't at all sure what it was.\n\n## _II. Marriage_.\n\nAbout a year later, in November 1957, Prue and I were married.\n\nIt had been an eventful year, difficult and complicated, especially so for Prue, and for me it had been explosive. Most of what I thought I knew about love and life was turned upside down.\n\nI mentioned earlier that a level of agitation had always been present in my mind. That wasn't strictly true. There had been isolated occasions and circumstances in which I had been completely quiet. Some of these had been internal: for instance, when I knew I had made my decision to refuse to join the army I had been very frightened but, inside myself, incongruously serene. I had been quiet too, on brief occasions like the outing to the Norfolk Broads with Jill. There had been other, similar times and, although there was no doubt that they had something to do with gender, in that my companions had usually been women, these times seemed to me to have had nothing to do with sex and everything to do with friendliness and acceptance.\n\nIn my small experience, sex was something quite different, an alluring minefield into which, under the spell of the loved one's attractions, one was inexorably drawn towards some deeply fraught encounter, which just might turn out to be a lustful consummation but was far more likely to be a scornful rejection.\n\nI had got that wrong. With Prue I was completely free of agitation, safe in a world which not only had everything to do with affection and acceptance, but also, just as strongly, with love and sex. In this world I knew something for certain \u2013 that it was where, in due course, if it could be arranged, I would like to spend my life, or rather, we would like to spend our lives. And that knowledge, in itself, brought me complete serenity. Whether it was going to work out that way or not didn't really matter, what we felt for each other at that time was a simple warmth, a roaring tenderness that would take whatever came.\n\nFor a while I wasn't sure what Prue's ambitions were. Although we were quite clear about our feelings for each other, I was not asking her to break her marriage. I would go along with whatever future she chose. Nothing else would have been appropriate.\n\nIn the end she chose; asked me if I could really want to take on a rather ugly woman and three rumbustious children and I replied: 'If you can disentangle yourselves and come to me free and freely, there is nothing else I want in my life. And besides, you're not ugly, or not very.' So she hit me, but not very hard, and that was that. As I walked home in the moonlight that night, I was thirty feet tall and I plucked the street-lamps like daffodils. Fourteen years earlier, as I had slithered down Box Hill, I had said that if I didn't like the world as it was I would make a new one. Now I was ready to start.\n\nThere was certainly a lot to be done. First I had to part company with Kays Industries, the display industry and Ro. I also left home and went to stay with Denzil, my old friend from Drama School, in a flat in Earl's Court. My emotions, which had been crouching for so long in a glum rut, were suddenly riding a roller coaster, swooping from tears to laughter, from joy to despair. Sometimes a tide of self-doubt would wash into me \u2013 what on earth did I imagine I was doing, dragging Prue and her children into my follies? How could I possibly imagine... etc.?\n\nPrue had her times of anxiety, of course. Naturally she was centred on the welfare of her children and anything that might put that in jeopardy was dangerous. In particular she refused to have any truck with my occasional bouts of self-doubt, dismissing them out of hand as ridiculous. Sometimes our anxieties coincided and then tensions and misunderstandings could brew up. But always, underneath the troubles, lay the still certainty that we only wanted to work through them and be together, and with that certainty came, for some reason, time. There seemed to be plenty of it about. We had all our lives to come. We could be calm.\n\nWe needed to be calm, or at least firm of purpose. Prue had a very difficult job to do and it was one in which I could only play a peripheral part. Having decided quite slowly, over a period of years, that if the opportunity should arise she was going to leave Peter, she was now doing it, which meant that she had to negotiate a divorce and generally clear a path for herself and her three children to get away.\n\nFirst there was a pantomime to play. Although Peter had been far from faithful, Prue was legally the guilty party, so she and I were required formally to furnish 'evidence of adultery'. The scene was set by Peter's solicitors who required us to have a romantic assignation in a certain room at a particular hotel in Russell Square. There, in the morning, we were to be discovered, in flagrante delicto, by a private detective.\n\nThe assignation was more embarrassing than romantic and, I'm sorry to say, neither of us was able to take our role seriously. We caught a glimpse of the private detective as he crept in behind the maid with the breakfast trolley and immediately nipped into the en-suite bathroom where he lurked, breathing loudly, perhaps to make sure we knew he was there.\n\nThe maid left. We wondered what we ought to do next. Our poached eggs were on the trolley, on warm plates under round silver domes. It seemed a pity to let them get cold, so we sat on the edge of the bed and ate them. They were excellent. Then we had toast and marmalade. I think the detective may have taken the crunching of toast as his cue because he very slowly appeared round the bathroom door and removed his bowler hat.\n\n'Aha!' he said significantly.\n\n'Yes, indeed,' said Prue. 'Cup of tea?'\n\nHe gave her a reproachful look and took out a notebook. He gazed severely around the room, licked his pencil and began to write. Then he pointed the pencil at us and told us to identify ourselves.\n\nWe suggested various names which he found difficult to accept so, to prove that she really was Br\u00fcnhilde, Prue jumped up in her nightie and, holding the two silver domes to her bosom, sang him a few bars from Wagner's opera.\n\nThis behaviour seemed to cause the detective great distress. Hurriedly assuring us that no further identification was required, he literally ran from the room. We were a bit worried by his sudden departure but it turned out that in spite of our levity the legal requirements had been fulfilled and the divorce proceedings were able to grind on.\n\nFortunately the proceedings were not complicated by strife. Prue's two elder children, her son Kevan, who was twelve, and her daughter Kerris, who was ten, were fairly enthusiastic about the idea of leaving Peter. He hadn't allowed their existence to inconvenience him and he definitely wasn't popular with them. Although they may have had some reservations about my becoming their stepfather, I believe they took the general view that it was likely to be preferable to the status quo.\n\nThe younger boy, Krispian, known as Pip, was four. He had more complex reservations. He said he was entirely in favour of Prue leaving Peter, and he never wavered in this, but he didn't want her to marry me, or anybody. He wanted her to wait and marry him \u2013 a perfectly understandable ambition with which I had every sympathy.\n\nI had known Prue's children for some years as neighbours, so we weren't strangers and, although it took Pip quite a long time to become fully reconciled to my presence, we all managed to get along quite well, largely I think because I didn't attempt to take on the emotional role of father.\n\nNevertheless, I had taken on a role. I had moved imperceptibly from being alone and at odds with myself and the world to being, if not the actual father of a family, at least its loving protector; a position to which I found I was entirely and joyfully committed. This commitment was not the result of a conscious decision on my part, it was just there. My attention had moved away from myself and I could just get on with doing whatever had to be done next. I had begun to live.\n\nI recall one particular incident that showed how far my state of mind had changed during 1957. It was in September, I think. Ray and Daisy had been to France and on their return Prue and I went down to meet them at Charing Cross Station.\n\nAs we walked down the platform I saw Daisy coming towards us. I looked at her face and, as I did so, what felt like a bow-wave of pure agitation seemed to sweep down the platform and roll over and through me. The sensation lasted a second or two and was gone, leaving behind the old familiar state of generalized worriedness, the urgent sense that something, somewhere, had either gone wrong or was going to go wrong, and that it was my fault.\n\nNothing had gone wrong. Greetings were exchanged. I took the bags and we all walked towards the forecourt where Miranda was parked. As we walked I realised that the wave of anxiety I had felt had originated as much in myself as in Daisy. I also realised that whatever anxiety she may have felt about me came from her affection for me and that my vulnerability to her anxiety was a measure of my affection for her. So that was OK. If necessary I would look after Daisy and Ray the same as I looked after Prue and her children. There was nothing to worry about there.\n\nBy that time the divorce situation had more or less sorted itself out. The outrages had been roared, the recriminations hurled, the collusions plotted, the resignations shrugged. Having left the family home, I was living in Earl's Court, and Prue had moved, with the children, into half a house in North Finchley.\n\nI think the one who most enjoyed that move was the dog. A highly intelligent, socially perceptive mongrel bitch of no breeding, Bach had overseen Prue's bringing up of the children. She had always kept out of Peter's way and although at first she had been suspicious of my presence, the day after Prue told Peter what was going to happen she had bounded up to me, full of welcome. We became great friends and after the move I had the pleasure of meeting her special friend, who was an elderly single lady called Doris who lived in the council flats round the back. On Sunday afternoons Bach would call on her and invite her back to the house for a cup of tea and a chat with Prue, and then see her safely home again.\n\nIt was a comparatively quiet time. All we could do now was wait for the divorce decree and hope that it would come through in time for the wedding. It very nearly didn't. On the morning of the wedding Prue showed great presence of mind by borrowing a fiver from Ray and taking a taxi to Somerset House. There she demanded the 'decree absolute', obtained it, and raced with it to the Kensington Register Office. Thus, in the nick of time, she saved us from the embarrassment of having to attend the rather sumptuous reception which Ray and Daisy had laid on at Finchley, _before_ the wedding.\n\n## _III. Small Time_.\n\nIt was all very well my feeling ready and eager to do whatever had to be done next. The trouble was finding out _what_ to do. I had to earn money. It didn't matter a sausage what I did so long as it had a pay packet.\n\nQuite early in 1957, just to keep the wolf from the door, I had found a source of casual work as what was called a stage manager in an independent television company \u2013 Associated Rediffusion Ltd. A stage manager was a sort of programme director's labourer, a spare body to do odd jobs and attend to small matters as they arose. Stage managers were engaged by the day, for which they received \u00a33, not a lot, even in 1957, but if one did the job fairly well one could be sure of some continuity.\n\nOn the whole I think I was quite a good stage manager \u2013 once I had learned the two basic rules, both of which were completely alien to my temperament. One was: Never make suggestions. The director has enough to do without being lumbered with other people's bright ideas. The other was: Don't bump into people. This was more subtle. Bumping into people wasn't necessarily a physical act. From my days as a stage manager in small theatres I had brought a primal sense of responsibility to the production and a watchful eye for potential disasters. If the scenery looked as if it was going to fall down I would run and hold it up. When I did this in the television studio the scene-shifters went on strike. I had to learn to stand back.\n\nThose were the straightforward parts of a stage manager's job. The unexpected parts were, naturally enough, the results of situations that had not been foreseen like, for instance, trying to arrange for the removal of Sylvia, who was a camel, or the more difficult task of obliging a temperamental choreographer who stamped his foot and ordered me to go and punch the nasty costume-designer on the nose. Oddly enough I couldn't afford to refuse the order because stage managers were day-labourers and if I had I wouldn't have been booked again next day. So I went and asked the costume designer for permission, and he very kindly inclined his nose my way so that I could gently carry out my duty. He also, not unexpectedly, suggested other intimacies, which I declined.\n\nFor a while after starting work with the ITV company I entertained an ambition to become something in television \u2013 a director, perhaps. I even considered going to night school to study for a qualification, but I soon realised that I was temperamentally unsuited to a job which depended on decisiveness, coolness and confidence. However, other qualities in my character must have been observed, because when it came to a real crisis the programme makers knew who to turn to.\n\nA director I had never heard of rang me at home one day.\n\nHe said: 'Sorry to trouble you, but they tell me you can supply us with a collapsible souffl\u00e9.'\n\n'Do they?' I asked, playing for time.\n\n'That's right.'\n\n'Who are \"they\", may I ask?'\n\n'Oh, you know, people... directors and so on.'\n\n'Ah.' I felt a small flush of pride.\n\nI said: 'But of course... Certainly... I hope so... Wait a minute! What the blazes is a collapsible souffl\u00e9?'\n\n'Well, what it is... a souffl\u00e9 that collapses, you know, like they do. It's for Bebe Daniels in _Life with the Lyons_. She makes this marvellous souffl\u00e9. When she takes it out of the oven we can see that it has risen beautifully. But then, as she carries it in triumph to the table where everybody is waiting, it slowly sinks, you know, like they do.'\n\n'Oh, yes, I know. When do you want it?'\n\n'Tomorrow afternoon.'\n\n'OK.'\n\nThank goodness a collapsible souffl\u00e9 is quite easy to make. I pinched one of Daisy's baking tins, glued a piece of soft rubber sheet into it so that it lay loosely across its top, and pulled the sheet about till it was really out of shape. Then I drilled a hole in the base of the tin and fitted a rubber-sheet flap inside it to act as a valve to keep the air in. When I blew into this hole the rubber top rose up like a souffl\u00e9. I painted it and rubbed sand on to it until it looked very like a risen souffl\u00e9. The paint and the sand also made the rubber sheet slightly heavy so that all Bebe Daniels had to do at the right moment was secretly push her little finger into the hole in the base and press in the flap to let the air out. The weight of the sand caused the souffl\u00e9 to collapse most convincingly. I was very proud of the article, especially when I saw the entry in the Property Master's stock-list:\n\n'Souffl\u00e9, collapsible... qty: one only. For Bebe Daniels.'\n\nOne Thursday afternoon when I was at my friend Tyrer Copple's joinery works in Battersea, a phone call came through from Geoffrey Hughes, the director of a series of science programmes called _New Horizon_ , which was presented by Dr Bronowski.\n\nGeoffrey said: 'Listen carefully... You take a cube and stand it on one of its points... OK?'\n\n'Yes. I can picture that.'\n\n'Right, so holding it dead vertical you cut it horizontally across the middle, exactly halfway up. Can you picture that?'\n\n'Yes, I think so.'\n\n'Good,' said Geoffrey. 'So what shape are the faces of the two halves that you have cut?'\n\n'Cor!... Buggered if I know.'\n\n'That's fine,' said Geoffrey. 'I want one, about the size of a football, tomorrow afternoon.'\n\nI asked Tyrer Copple the question. He couldn't visualize it but being a practical chap with his own joinery works, he sorted out a ten-inch cube of timber and we blocked it up on the bed of the band-saw so that we could saw it through the middle, as requested. The cut faces \u2013 as if you hadn't already worked it out \u2013 were perfect hexagons.\n\n_New Horizon_ was a rich vein. I was engaged for the series as stage manager, which provided a small but regular wage. Also most of the productions called for some special visual devices, many of which I provided. I remember a rotating hydrogen atom whose electrons (torch bulbs) jumped (switched) across and turned it into a helium atom (or was it the other way around?). I remember several sets of optical illusions and a very baffling collection of wheels behind slots which illustrated something a visiting Belgian professor called 'Causalit\u00e9'. My favourite item was a plastic dodecahedron which could be assembled in front of the cameras from a heap of magnetized pentagons.\n\nThis small trade \u2013 supplying unlikely articles \u2013 continued as an occasional sideline for some years. On 4 November 1963 I was asked to supply a full-size effigy of General de Gaulle to be burned on the bonfire in David Frost's firework-night programme. It meant working all night but it looked magnificent. The ITV company sent a large car to collect it and I heard later that as they passed the House of Commons a policeman recognized the General sitting in the passenger seat and saluted him.\n\nThat was perhaps the most prestigious article I supplied. I think the most spectacular came from a cubic metre of fresh seaweed, urgently culled from Whitstable beach and stuffed into black dustbin bags. The programme, grateful as it was for the prompt service, said it could have done without the several worried crabs that emerged from the weed because they ran about the studio, molested the scantily clad sea-nymphs lying languidly on the plaster rocks, and sent them shrieking off the set.\n\n## _V. Of Mice and Magnets_.\n\nThis work I was doing for television \u2013 itinerant stage manager and occasional property supplier \u2013 was just about as peculiar and inconsequential as anything I had been doing before, but it seemed to be providing enough for us to live on from day to day. That was OK but precarious. I definitely needed either a full-time job or a freelance way to earn a reasonable living.\n\nWhile I had been doing a stint on children's programmes, I had come to see that quite a lot of the material that was being sent out was pretty thin stuff, just about adequate to fill the time.\n\nSo, on the basis that I probably couldn't do a lot worse, I sat down one morning in 1958 and took pencil and paper to write a story-series for children's television.\n\nIn the event I was lucky. A title occurred to me almost at once: _Alexander the Mouse_. That sounded promising. I wondered what it could be about. Obviously a mouse, but a mouse of distinction, royal perhaps? Yes, but perhaps he didn't know it. Aha! There's a fairy-tale: _The Mouse Born to be King_. Yes, but king of where? Where is a mouse kingdom? Under the floorboards and in similar lost places. OK, so that's the bones of it, but what's the story? What happens? How does it begin?... Well, with his name, of course. His brothers and sisters have proper mouse names like _Eeek, Gleek_ and _Pleep_. How come he is lumbered with a mouthful like _Alexander_? So he wants to know. That is the quest.\n\n'All right, that'll do for now,' I said to myself. 'Start writing. We'll see what turns up.'\n\nI hadn't quite expected that. I thought I would have to work everything out to the last detail and then go through the mundane task of writing it down. I hadn't realised that at the same time as I was doing the mundane task with the front of my mind, another far more efficient department in the background could get on with brewing up and fitting together the story. It was a good morning, indeed a good week. The words came running down my arm as fast as I could write them and by the end of it I had the six episodes of the story on paper.\n\nI took it to Michael, who was Head of Light Entertainment. He called me in the next day and told me that he liked it and that he thought it would make an excellent vehicle.\n\n'Vehicle for what?' I asked.\n\n'For our new, marvellous, magnetic animation system!' he cried, his eyes lighting up with enthusiasm. 'The most exciting development in motion-portrayal since film itself! We have the inventor under contract. He will oversee the production. You will like Shamus.'\n\nFor some reason Michael's enthusiasm was not instantly infectious, and I couldn't help thinking he might have got the 'vehicle' bit the wrong way around, but I was delighted that my story had been accepted and was genuinely enthusiastic about that.\n\n'You get on with it,' said Michael. 'You liaise with Shamus, get yourself an artist to do your pictures. You tell the story on air. Find some music in the library or see if Steve Race has got something that will do.'\n\nReeling slightly but highly elated, I came away.\n\nShamus was an eager Irishman, full to bursting with the wonderful potential of his invention. The basic idea was to have a flat picture, laid on a horizontal table with a very thin, nonmagnetic surface, like aluminium. Lying on this picture, as part of it, would be a flat figure made of thin card which had a thin magnet sellotaped to its underside. If you held a magnet under the table so that it locked onto the magnet on the flat figure, you could move the figure across the picture by pulling the magnet across. There would be two special tables, each with its own skilled animator. Each table would have a mirror above it which would be set at forty-five degrees, so that the television camera could look directly into the mirror and so down onto the table. That all seemed pretty straightforward.\n\nFinding an artist willing to do over twenty different backgrounds as well as several cut-out figures for a fee of \u00a330 an episode, seemed likely to be a more difficult task. Then I remembered my friend Maurice Kestelman, who was now Head of Fine Art at the Central School of Art and Design in Southampton Row. I went to see him and he suggested one of his lecturers.\n\nThus it was that I came to meet Peter Firmin and his wife Joan and their three girls, Charlotte, Hannah and Josie (who was only a few days old and very pink). They lived in a ground-floor flat in Battersea.\n\nPeter was a bit dubious about working for television. It didn't seem like proper illustration, and there was rather a lot of it, for very little money. He wasn't so much bothered about the lack of money as about whether or not he would be able to get through the work in time. But in the end I persuaded him and he agreed to have a go.\n\nIn the event Peter drew the first batch of pictures quite quickly. We laid out the mice and the camera rehearsals for the first episode of _Alexander the Mouse_ began. They were a nightmare.\n\nThe system worked, in the sense that the animator could reach under the table with a magnet and, once that magnet had locked on to the mouse's magnet, move the mouse along. That was fine but if the magnet didn't hit the exact spot (which was invisible underneath the table), the mouse would zip across the picture to meet it. Or worse, if the magnet was the wrong way around, the mouse would execute a somersault to bring the north and south poles together, or worse still, if the animator's magnet was the wrong way around but it came up exactly in the right place, the mouse would be lifted bodily off the table by magnetic repulsion, turn over in the air and come down again back to front and underside upwards. Then all the animator could do was reach a hand into the picture in full view of the camera and turn it the right way up. Also the magnets sometimes just let go, leaving the mouse stranded. Then the animator would have to back-track with the magnet and cast about to try and pick it up again.\n\nAt the first rehearsals we regarded these difficulties as being teething troubles, but they were not. Hardly a programme went out without some incongruity \u2013 a hand coming into shot or a mouse coming adrift. I sometimes felt really relieved that once they were transmitted the pictures were gone for ever.\n\nIn spite of all these vicissitudes the programmes were well received by the company. The story was OK, a bit muddled, without much action, but full of good pictures of a mouse-angled world. The last episode ended with the coronation of Alexander, which took place in the mouse-cathedral under a warehouse down by the London Docks. It was a glittering occasion in which the choir of poor church-mice sang the anthem 'Lo, the mouse in triumph comes, see him in his glory', and I gave a solemn commentary in the manner of Richard Dimbleby. After it was transmitted Michael reeled into the studio smiling and full of congratulations.\n\nHe said: 'That was one of the nicest programmes I have ever seen and if we aren't arrested for lese-majesty, we'll be done for blasphemy!'\n\nWe were all very chuffed by that, and I was even more chuffed when, a few days later, the Contracts Department rang me and ordered twenty more episodes of _Alexander_. We were in business.\n\nPeter and I started on the next batch of _Alexanders_. He now had plenty of work because his skill had been noticed by other producers and he had been offered similar contracts for other programmes, for which he had rounded up several assistants, so their flat in Battersea was fairly buzzing.\n\nI wasn't doing too badly either but I had nothing in my head to follow Alexander. So I was quite relieved when Michael asked me to suggest a programme for deaf children. That narrowed the field a bit and the limitation triggered off an idea. I suggested a slow Chinese story about a small boy and a water buffalo.\n\nMichael liked the idea but he and I were both fed up with the magnets. A different form of animation was needed.\n\nI suggested making the programmes on 16mm film by singleframe animation. Michael rejected that because the budget of \u00a3150 for each ten-minute episode wouldn't begin to cover the cost of filming the series.\n\nWouldn't it? I wondered. Making films was usually quite costly, but the money went on all sorts of things I wouldn't need, like film crews, lights, locations, technicians and sound-recording facilities.\n\nIf I was going to make on film the same sort of programme as we were making with the magnetic animation, I wasn't going to need much more than a camera capable of taking single frames (i. e., able to expose one frame of the film at a time), and a table on which I could move the flat card figures in single frames (i. e., moving the figure a tiny distance, shooting one frame and then moving it again), which was the basic process of single-frame filming.\n\nI consulted the Company's film department about the project. They laughed it off, saying that even if I _could_ raise the capital costs of setting up a studio, all the profit would go on paying the interest.\n\nEven so the idea wouldn't go away. I could make an animation table. I could put together an editing device, I didn't need a studio, there was a spare room in the half-house in North Finchley where we were living. The arithmetic of the project was a bit daunting, but possible. I located a producer friend from the BBC who had become an honorary Chinaman and was doing authentic Chinese paintings. He liked the story and said he would be willing to do the artwork for \u00a330 an episode and wait for payment until I was paid. There was no need for sound equipment because the film, being for the deaf, was to be silent. So it looked as if, for that film at least, I would need almost no capital.\n\nI did the sums over and over and then, one day, went to Michael and said: 'If you can find \u00a3175 for each episode and pay on delivery, I can make _The Journey of Master Ho_ for that.'\n\n'All right,' he said, 'go away and do it.'\n\nI came away from Michael's office feeling very alarmed indeed. Just what had I been idiotic enough to let myself in for now?\n\nThe answer was simple: a lot of hard work that had to be done very quickly.\n\nMeanwhile Peter, equally disenchanted with the magnets, had moved in a different direction. For certain shots in _Alexander_ , the ones that had to be reliable, he had made cut-card mechanical animations with sliders and moving panels worked from the back. These were only ever seen once, which was rather a waste. Peter felt that the principle had a lot of potential and he set about developing it.\n\nAt I believe Joan's suggestion, he made some card-pictures with whole nursery rhymes on them. One had a cat actually playing the fiddle in one corner, a cow which jumped clean over the moon, a dog falling about with laughter and a roguish dish, who grabbed the coy spoon and ran away with her \u2013 on little rotary legs! These card-pictures were real works of art, especially when seen from the back, where all the intricate, ingenious sliders, handles, springs, cottons and rubber bands can be seen working.\n\nThe television company was delighted and they invited a young Australian singer called Rolf Harris to bring his accordion and do a programme in which he would sing the nursery rhyme while Peter worked the reveals and sliders in time to the song. Peter also made smaller, even more complicated animations in which he could cause the characters to sing and play instruments to prerecorded, sometimes speeded-up, songs, for which Rolf persuaded Peter to sing the harmonies. Later Rolf sang a song about a kangaroo and hit the big time, but a young Country and Western singer, Wally Whyton, took his place and the programme, which was called _Musical Box_ , continued for some years.\n\n# SMALLFILMS\n\n## _I. Small Miracles_.\n\nThe animation table and camera gantry went up in the spare room. The Chinese paintings arrived and with them the moment of truth. My assertion that making animated films was simply a matter of pushing pieces of cardboard about under the camera was about to be tested. Was it true?\n\nThe answer, as is often the case, was more complicated than the question. Yes, with great patience you could reconstruct motion in terms of tiny movements each taking one twenty-fifth of a second, there was no doubt about that, but whether or not I was going to be able to do it convincingly was yet to be seen. The only way to find out was to do it. So I just got on with it.\n\nI got through it. The films were delivered, accepted and paid for. I settled my accounts and owned the equipment. The film I had made was really quite bad. It was a great improvement on the magnetic animation but it was juddery and uncertain, poorly paced and ragged. However, as the filming went on, the technique seemed to settle down until, in the later episodes, the slightly stilted formal movements of the characters appeared to be quite congruous to the Chineseness of it all.\n\nAs well as being transmitted on television _The Journey of Master_ _Ho_ was shown to deaf children in their special schools and, I was told, they enjoyed the films. So that was something done.\n\nBut there was more to it than that. I have indicated that the process of making a cartoon film was slow and laborious. What I omitted to mention was that it was also tremendously exciting.\n\nI remember the very first test-piece of film I ever made. I loaded the camera and chose one of the Chinese backgrounds. I assembled the card-pieces for a side view of Master Ho, the Chinese boy, and jointed the flat arms and legs to the body with bits of cotton and sticky-tape. I did the same for the pieces of the water buffalo and laid the two of them in position on the Chinese landscape background so that they made a good picture.\n\nI placed the picture on the table, lined it up under the camera and shot about five seconds of film. Then I sat on my stool and, using a small pointed screwdriver as a pusher, moved the legs and body of Master Ho a millimetre at a time across the picture in what I imagined might be an imitation of the way he could have walked \u2013 remembering to shoot one frame of film between each movement. After about an hour of this Master Ho had reached the water buffalo, so I stopped moving him along and, frame by frame, let him bow to the buffalo. For good measure I let the buffalo bow to him. Then they both straightened up.\n\nI had the film processed, brought it home and laced the print into my Moviscop editor, which had a screen like a tiny television set. As I wound the film through by hand I saw the Chinese landscape with the boy and the water buffalo standing in it. It did look very good. Then, suddenly, by God!... the boy actually walked across the picture and bowed to the water buffalo, and then, look!... the buffalo bowed back! I wound the film in reverse and they did it backwards. Then I wound it forwards again quite slowly and watched them do it again.\n\nThat was all, but I was fizzing with excitement. It didn't matter what the picture was of. It didn't matter that Master Ho had stumbled rather than walked... I had done something momentous. I had opened up another dimension to the still picture. I had given it the extra dimension of time. I had made it come to life.\n\nSeeing Alexander the Mouse being pulled along by an invisible magnet had been interesting and fun, but seeing Master Ho walking across was different, and astonishing, because it had never happened in real time. I had built the happening piecemeal by hand, using tiny building-bricks of time, each one twenty-fifth of a second in length \u2013 three hundred and forty-three of them to be exact \u2013 and I had done it by myself, in the spare-room, on a home-made animation table, using a camera worked with pieces of Meccano and a string.\n\nMy body, which always understood much better what was going on than my head, was dancing about, sending up waves of idiotic delight and conjecture. 'This way,' it shouted, 'you could make whole worlds, complete places full of people, and with your love you can give them life!'\n\n'Now take it easy.'\n\n'And you have absolute power, absolute control, over everything in those worlds. Pigs can fly, forests can walk. The river can sing and flow uphill. At your command the world could blow itself to dust and ice.'\n\n'That'll do, that'll do. Any moment now you'll be inventing dragons.'\n\n'Oh, any amount of dragons. Dragons are two a penny. Peter will draw them and you will make them roar and blow great blasts of paper fire, for which, by the way, you'll need a double-exposure.'\n\nAll that exuberance may seem a bit overstated today, but forty-two years and two hundred films later, if I were to lace the rushes into the projector, and see for the first time the work I had done the week before spring suddenly into life, I would still feel exactly the same thrill.\n\nAnd in a sense it was all true. Peter Firmin and I did make worlds, nearly a dozen of them, down to the last nut and bolt.\n\nBut that was all to come. In 1958 all I knew was that I had the tool to do a particular sort of work. Now I was faced with the task of finding, from somewhere in the sky, the happenings and places, the stories, that I would use it for \u2013 and sell them to the television company.\n\nLet me say now that anybody who thinks that making a living by writing original stories is an easy happy sort of life is, to put it politely, mistaken. There is nothing so numbingly terrifying as a blank piece of paper on to which you are committed, by contract, to write a whole complex piece of yet-to-be-created life.\n\nSometimes a brainwave would come, but more often I found myself scouring the inside of my head trying to imagine the _sort of_ _thing_ that heads of TV departments might fancy.\n\nI remember asking myself if there was one particular sort of creature that everybody could feel unashamedly sentimental about. Was there something somewhere that would surely touch their heartstrings? Not cuddly bunnies or cute dollies, that was for sure. Dogs? Horses? Hmmm... Cars? Motorbikes? Boats? What did every boy of my generation want to be when he grew up? An engine driver, of course. I knew an engine driver, or rather an engine driver's mate.\n\nDenzil Ellis, my friend from Drama School, had been an engine-fireman. He had fired the _Royal Scot_ (or was it the _Flying_ _Scotsman_?), shovelling coal into its firebox as it roared along the track. He once told me how his day used to start. At some unearthly hour in the morning he would have to go and wake up the engine, clear out the firebox, take paper and wood and light its fire, just as you or I would light a stove or a kitchen range, so that the engine would be boiling nicely by the time it was due to set off for Edinburgh. I had been touched by that story because it seemed to give a human dimension to the glowing hundred-ton steel power-machine thundering up the main line.\n\nSo, all right. A railway engine. What sort? Where would it be?\n\nWhere?... That wasn't too difficult. Wales was the place. In Wales the railways don't run straight across flat ground at high speeds. They wander up and down the valleys and along the tops of the hills, and in my mind Wales was already a magic place, a place of glorious eccentricities and unlikely logic. I knew this because I was intoxicated by the writings of Dylan Thomas and had been carrying a copy of _Under Milk Wood_ in my pocket for years.\n\nAll right, then. A Welsh railway engine. So what does a Welsh railway engine do for a living? Pulls trucks with things in. A bit dull, really. Hardly the stuff of legend. Wasn't there something a bit more interesting for it to do?\n\nThe answer came \u2013 wait for it \u2013 while I was in the bath. I muttered 'Eureka,' climbed out dripping wet, and telephoned Peter. I said: 'He wants to sing in the choir.'\n\n'Of course,' said Peter, as if it was a matter of course that in Wales a railway engine would want to sing in the choir.\n\nThat single thought provided the touchstone for the world of _Ivor the Engine_ , that ordinary world in which the extraordinary was just going to be part of the landscape.\n\n'What landscape?' I asked myself.\n\n'Well, you know, the pictures I'm going to put in front of the camera and make a film of.'\n\n'Now wait a minute! You don't know the first thing about making _films_! All you've done up to now is cobble together a story to replace a barbaric form of magnetic animation. Now you're proposing to commit yourself to making a set of proper films, real films, which will be judged by experienced film-makers who have been on courses, who are qualified, knowing about camera angles and style and pace and fashion. The Art of Film-making is something people study and write books about. You don't know anything about any of it. Why do you imagine you can just walk in and do it?'\n\nWhy indeed? I took this very real anxiety to my friend Mary Beales (now Mary Orrom) and her husband Michael, who were both experienced professional film-makers. I sat with them in their flat and told them, in a muddled way, with hoots and puffs, the basic story of _Ivor the Engine_.\n\nThey enjoyed the story, so I asked them the question: 'Could I seriously think of myself as being qualified to take on a contract to make a film of it?'\n\nFor a moment they didn't seem to understand the question \u2013 then they laughed.\n\nMichael said: 'You've got the \"visuals\" in your head and you know how they follow each other and know who is going to say what to whom, and where. In fact, you can see this film in your head, so you can make it.'\n\nMary and Michael went on to give me some advice that I have never forgotten. They told me not to bother about what anybody else in the film world was doing or had done. I should just concentrate on what I myself was doing, which was to show the story as clearly as I could. At the end of the day the film might not be the most stunningly impressive example of the film-maker's art, but it would do its job, which was to bring the story to life.\n\n'But,' I said, 'what about the Art of Film-making?'\n\nMary and Michael had no opinion of the Art of Film-making. They said there were only really four things to know about making films and they are just common sense \u2013 and I can't for the life of me remember what they were! I do remember that they told me that the camera, the film, the equipment and the whole pretentious mystique of the Art of Film-making were there for one purpose and one purpose only, to serve me and help me to tell the story as clearly as I could. And they added: 'So you might as well get on with it.'\n\nHeartened by this encouragement I plunged straight into writing the story.\n\nI clearly recall sitting down and writing the first lines:\n\nNot very long ago, in the top left-hand corner of Wales, there was a railway. It wasn't a very long railway or a very important railway, but it was called the Merioneth and Llantisilly Rail Traction Company Limited, and it was all there was. And in a shed, in a siding, at the end of the railway, lived the Locomo tive of the Merioneth and Llantisilly Rail Traction Company Limited, which was a long name for a little engine, so his friends just called him Ivor.\n\nNow in the morning, Jones the Steam the engine driver would come down over the hill, jump the stream beside the shed and open the door.\n\n'Morning, Ivor! Jumping cold this morning...'\n\nThe story unfolded quite slowly: Jones lit Ivor's fire and when the boiler boiled, made himself a pot of tea from it. He took Ivor out of the shed and down to the water tower, where he filled his tank with water, and then on to the signal box, where Owen was still asleep. They woke him with a whistle and soon they were on their way along the top of the hill above the sea towards Llaniog, where Dai Station the stationmaster was waiting for them, looking fed up as usual...\n\nThen, just as suddenly, I stopped. In spite of Mary and Michael's enthusiasm I felt I wanted to complete one small piece of film for us to look at and maybe show to the ITV company as a sample. I did some sketches of what I imagined would be the backgrounds for the shots and actions, put together a sort of storyboard alongside the piece of script and took them straight down to Peter in Battersea.\n\nApart from small details like the height of Ivor's funnel, Peter's vision of Ivor and his country was perfectly in tune with my own. When, after a day or two, Peter came up with the drawings for the first two minutes of film, I found them completely satisfying and was delighted to have my first view of the place and people with whom my imagination was going to live and work, on and off, for many years.\n\nI filmed that first two minutes and, somewhat nervously, took the print along to the television company, where we showed it to the head of the department, the senior producer and some of their staff, in one of their viewing-studios. The film was of course silent so I told the story as it appeared on the screen, played all the parts and included all the puffs, hisses, clonks, scrapes and whistles that I could manage. I am happy to say that they were not only highly delighted with the film but also falling about with laughter, partly, I suspect, in response to my impromptu pantomime of vocal sound effects.\n\nThe ITV company bought the series on the spot, so I was in business and going to be busy. Fortunately, now that I had the pictures in my head, the people, Jones the Steam, Owen the Signal, Dai Station, simply turned up for work. I didn't have to invent them, they came off my pen as if they had been there all along, waiting to be called. It was the same with the places: Llaniog, Tan-y-Gwlch, Grumbly Town, Llanmad,* Tewyn Beach. I could see them all, clear and complete, as I came to them. And, as had happened with _Alexander_ , somewhere in my head the story was forming itself while I was writing it.\n\nThe actual filming still presented some technical problems, mainly arising from the fact that unlike _The Journey of Master Ho, Ivor_ _the Engine_ was to have sound. My impromptu performance had been fine for an audition but the final film would have to have a complete sound-track recorded on to it. I would have to make the film and then take it along to a sound studio and record the sound while watching the picture, more or less as I had done at the audition.\n\nThat seemed reasonable enough but I suddenly realised that it wasn't going to work. _Without a sound-track, how was I going to know_ _the right number of frames to shoot for each incident?_ This was an absolute problem. Unless I knew exactly how many twenty-fifths-of-asecond frames were available in the sound-track for a particular piece of action, I simply couldn't begin to film it!\n\nI eventually dealt with this by making a rough sound recording of the whole film, leaving time for the various actions and incidents to take place. Then I timed each sentence and section of it so that I knew how many seconds they took to happen. Then I converted the seconds to numbers of frames and wrote the numbers on to the script.\n\nThat process sounds laborious and complicated, and it was,\n\nEven so I really enjoyed those final recording sessions. Reading the script, bearing in mind the character I was speaking for, watching the picture in order to anticipate the animation so that the words and actions would fit together, called for intense concentration. But then, after it was done, to sit back and watch the section being re-run with the sound, was (if I had got it right) another small miracle, because where there had been cardboard figures waving their arms about, now they were alive and talking, their gestures punctuating and expressing their words. And when the cotton wool puffed up out of Ivor's funnel it went _Psst-koff,_ _psst-koff, psst-koff_ and was real steam. That was a special pleasure.\n\nThe last, but certainly not the least, small miracle was still to come. That was the music.\n\nSteve Race, who had composed a piece for _Alexander the Mouse_ , suggested a bassoon for a small Welsh engine, and he put me in touch with Vernon Elliott, who was a professional bassoonist and also a composer.\n\nVernon Elliott was a bit diffident about composing music for a Welsh railway engine because he had never done anything like it before. I explained that I had never made a film about a Welsh railway engine before, so we were both in the same boat. I gave Vernon a script and he promised to try and think of something.\n\nA day or two later he rang me to say that he thought he had something that might do. I went over to his house to listen to it.\n\nThe tune was short and simple and absolutely right, a beautiful, marvellously evocative theme that brought Ivor and his world instantly to mind.\n\nThen, suddenly I remembered something \u2013 The Grumbly and District Choral Society! 'Vernon, do you happen to have a Welsh choir?'\n\n'Well, yes,' said Vernon, 'as a matter of fact I do. There's a choir in North London I work with sometimes. They aren't all Welsh but I'm sure they'd love to sing for a Welsh railway engine.'\n\nThat was a marvellous piece of luck. Vernon recorded the bassoon solos at the sound studios and a week or two later I recorded the choir singing 'Cwm Rhondda' in a Catholic Church Hall in Hampstead. The choir wanted to do the recording for nothing but Vernon, being a professional, persuaded them to accept a modest fee. I was more than glad to pay it because the sound was heartrendingly beautiful.\n\nI came away with two reels of tape and a small feeling of pride, which came from the sense that I had caused and helped some glorious music to exist. Little did I know how significant the effect of that recording of 'Cwm Rhondda' was to be, nor did I realise that Vernon and I had just begun a collaboration that would bring us both years of delight.\n\n## _II. Mass Production_.\n\nLooking back at 1959 I find myself wondering how on earth we got through the sheer volume of happenings.\n\nFirst I had the six _Ivor the Engine_ films to make. How long this was going to take was uncertain, because although a particular shot might take an hour or more to film in single frames, it could take even longer to set up initially. The background would have to be chosen and lined up with the camera, the figures would need to be assembled and have their limbs attached with tiny pieces of sticky-tape and cotton. Also, although making the film was essentially a simple task, it was also subtle. Objects, especially living things, don't move at constant speed. A body has weight. If you move the object exactly the same distance in each frame, it stops and starts suddenly and looks as if it was made of cardboard, but if you graduate the movement correctly it can look like a person walking or a railway engine moving off.\n\nI also found myself thinking very carefully about the choice and sequence of the shots.\n\nIn the same way that writing a story is not simply a matter of writing lines of words, but calls on the writer to assemble the sentences in such a way that the reader receives them in the right order for stacking in the mind, so too the task of filming the story in pictures calls on the film-maker to present the pictures in such a way that the feeling of what is going on is delivered and the viewer has the sense that it is actually happening. Luckily I was usually able to visualize the shots in my head and make my choice before filming and, even though my choice was never consciously analytical, the sequences were usually pretty satisfactory. But even so it all took time, as did the occasions when I had to nip down to Battersea on the tube to collect the odd cardboard arm or leg or railway carriage that Peter and I had forgotten to include in the list.\n\nPiecing together my diary for the spring of 1959 I see that although Prue was in the late stages of pregnancy, so I was holding the fort, I was also still doing odd days of stage-managing and providing props for ITV. Apparently I was also designing a gear-train for a new off-set litho printer for Ralph Furness and a Mr Groak (I've no idea how I came to be doing that). There was also litigation on the children's maintenance to sort out and Bach's latest litter of puppies to find homes for \u2013 and all the time I was wondering what I was going to do after I finished filming _Ivor the Engine_.\n\nThat time was fast approaching, so I went to see Mary Orrom's friend Violet Drummond, who wanted films made of some delicious children's books which she had written and illustrated about a rather posh young girl called Little Laura, who wore a Mouflon hat and was looked after by a nanny.\n\nA few days later, while I was at the sound studio recording the sound-track of the last episode of _Ivor_ , a call came through from Barnet Hospital to tell me that Prue had given birth to twin boys. I dropped everything and hared off to the hospital. It wasn't proper visiting time, but I stole into the ward and saw Prue sitting up looking very pleased with herself, with two small basket-cots, one on each side of her bed.\n\nI looked in one of these and saw a small perfect young person who was, miraculously, my son. I touched his face with the back of one finger, but very softly so as not to disturb his important sleep, and felt an enormous emotion, of pride, awe and affection. I then walked round to the other side and saw my other son. I touched his face as well and felt exactly the same emotion all over again.\n\n'But they're exactly the same,' I said.\n\n'Of course they are, silly,' said Prue. 'They're twins. Anyway they're not quite the same. Stephen is six ounces heavier than Simon.'\n\nI didn't have a chance to ask which was which because the fierce ward sister chased me out, ordering me not to return until the proper time. I hared back to the sound studio and, completely drunk with elation and confusion, finished off the sound-track for the last episode of _Ivor the Engine_ \u2013 which was the end of my work for the series.\n\nIt was just as well I had finished the filming because the arrival in our flat of the twins and all their paraphernalia meant that the camera and animation table were summarily banished from the bedroom, the cots were set up and a stream of visits and celebrations began.\n\nGradually the pattern of family life settled into orbit around the timetable of its newest members. For a while I became incongruously maternal and would drink cups of coffee and have chats with other young mums about the details of looking after babies. Prue said I caused some puzzlement because I had a habit of referring to babies in the plural, as if everybody always had twins.\n\nBy now money really was running out so, having negotiated another corner for the animation table and a contract with Violet Drummond, I was able to make a start on the films of _Little Laura_.\n\nAt first I was worried about the technicalities of these films because Violet Drummond's characters and her style of drawing, though artless and delightful on the page, were a bit inconsistent and wayward. In reality I was being too particular. The strange filming was perfectly in keeping with her style, and the whole effect was delightful and hilarious.\n\nMeanwhile the _Ivor_ films had been delivered and were being transmitted. As Prue and I sat and watched it on our rather blurred television set I found myself wondering what the film was really like. I was so conscious of its faults \u2013 the clumsy cuts, the missed sound effects, the juddery movements \u2013 that I was quite unable to enjoy the story or gauge its worth.\n\nTelevision is a notoriously unresponsive medium. There is no audience present at the performance. Nobody claps. Small programmes like ours were rarely noticed by the press. The first inkling I had that our film was being appreciated came when I heard that because the transmission of _Ivor the Engine_ coincided with the ITV company's weekly board meeting, the directors had arranged for a television set to be wheeled into the boardroom so that they could watch it. That was very gratifying, but the real moment of glory was yet to come.\n\nLater that year the ITV company assembled a compilation of its best children's programmes to show at a rather luxurious reception for the celebrities of the television and entertainment world. Among the items being shown was the last episode of _Ivor_ _the Engine_. It came fairly near the end of the programme which, up till then, had been received politely by the audience. As soon as Ivor came on the screen I could feel the audience go quiet. As it went on I felt them warm up and begin to chuckle and chortle. Then at last, after Ivor had trundled into the siding at Grumbly Gasworks where the choir was waiting for him, came the coup de gr\u00e2ce... 'Cwm Rhondda'. The singing rose in a rich wave which no heart, however hard, could have resisted. Vernon Elliott's North London Welsh choir, with Vernon playing Ivor's pipes on his bassoon, had done us proud, and as the film ended the celebrities were laughing and crying and clapping at the same time. It had all worked. I muttered a prayer of thanks to the late Reverend John Hughes, to whose heirs I had just paid a substantial fee for the use of his hymn, and felt very happy.\n\nThere was one interesting fly in the ointment. All the other items shown in the programme had been introduced by a presenter who duly mentioned the names and gave credit to the people who had made them. However, when it came to Ivor's turn, the presenter just said that the next item was all the work of one person (which wasn't true), without mentioning any name. Also, at the end of the piece, the music suddenly stopped short because the last picture, which was the end-caption with Peter's, Vernon's and my name on it, had been cut off \u2013 so somebody had gone to a lot of trouble to ensure that the audience was not told who had made that particular item.\n\nThis omission had not passed unnoticed by that sophisticated audience. Ursula Eason, who was Deputy Head of Children's Programmes at the BBC, came over at the end of the show to congratulate us on the film and commiserate about the absence of credits. She said she realised that the omission of our names was obviously an act of petty meanness, but she made the unexpected suggestion that we should perhaps look on it as a sort of compliment, because it showed that we had made something that was worth being seriously jealous of.\n\n## _III. Moving On_.\n\nAs we were to discover would often happen, we came upon our next project as the result of an apparently accidental coincidence.\n\nAt different times both Peter Firmin and I had visited the Edward VII Gallery at the British Museum, where we had both noticed a set of Norse chessmen from the island of Lewis. What had impressed us was that, far from being fierce and warlike, it was quite clear that these were essentially kindly, non-belligerent characters, who were thoroughly dismayed by the prospect of contest. Neither Peter nor I had any idea that this impression was going to be significant.\n\nAmong the characters in one of the programmes which Peter had been involved with was a wicked baron for whom he had made animated moustaches that twirled most evilly.\n\nOne day, while he was travelling through Neasden on his way to the studios at Wembley, it occurred to Peter that the chessmen in the Edward VII Gallery could well have been called Nogs, that their Prince was a Noggin and that the wicked baron with the twirly moustaches could be their wicked uncle, perhaps a Nogbad.\n\nPeter thought of a tale from the Land of Nog. It told of the death of a much-loved king and of his son, Prince Noggin, who had to choose a bride within six weeks, because if he did not the crown would go to his wicked uncle, Nogbad the Bad. Noggin could not choose from among the busty maidens of the Northlands, but a strange bird with green feathers, whose name was Graculus, brought him a walrus tooth carved with the likeness of Nooka, daughter of Nan of the Nooks, the ruler of the Land of the Midnight Sun. (This part of the plot was suggested by Peter and Joan's daughter, Hannah, who was four. She had seen the famous film _Nanook of the North_ which was about the home life of the Eskimos and had decided she was a Nook and could not therefore be expected to wear pyjamas in bed.)\n\nThe Nogs made a longship and sailed beyond the black ice at the edge of the world to the Land of the Midnight Sun. There Noggin and Nooka plighted their troth and amid great feasting Graculus flew back to the Northlands with the glad news. On the way he was captured by Nogbad the Bad and imprisoned in his castle. Nogbad was on his way to claim the crown when a robin took a green feather to Noggin's mother, Queen Grunhilda. She recognized the feather and sent guards to release Graculus while she went to town to sort Nogbad out. He was duly banished and, as the square sail of Noggin's longship appeared over the horizon, Nogbad slunk away over the high pass to stay with his granny in Finland.\n\nPeter wrote a synopsis and a shooting-script for a couple of episodes, not with film in mind but intended to be shown 'live' with complex mechanical animations. He offered this to the ITV company as an alternative to the magnetic animation but, for some reason, it was turned down. Rather disappointed, he passed it over to me to consider as a film. I thought it was great. We rejigged the pictures to suit the film treatment and I wrote one or two patches of slightly heroic prose to set the tone. We took _The Saga of Noggin_ _the Nog_ to the BBC, who accepted it, offering us the princely sum of \u00a3100 for each ten-minute episode.\n\nWe accepted the offer. Peter agreed that I should write the shooting scripts from his episodes and synopses and we set to work.\n\nWorking on the Noggin saga was a bit complicated because in 1959 both families moved house. Peter and Joan and their daughters moved into a farmhouse in Blean, just north of Canterbury, in Kent. This had a spacious farmyard with a range of outbuildings, including a cow-shed, some pigsties and a massive, leaky, thatched barn.\n\nPrue said: 'What about moving down there?'\n\nThat was obviously the most sensible thing to do, so we made an offer for a tall thin house in a terrace-block right on the seafront at Whitstable. The house was four storeys high, it had two scruffy bathrooms, a derelict service-lift, a small rough back garden and wide views over the glistening estuary of the Thames with, occasionally, a grandstand view of the yacht-racing from the first-floor balcony. Just the other side of a narrow private road in front of the house was the sea wall and the beach. After the dull flat in Finchley we all thought it was a really exciting place to live.\n\nThe move was a turning-point in all our lives. Prue and I now had a place of our own. The children had room to move and grow. The air was good.\n\nBy now I was fairly confident that the business was viable. I could see that, with the help of Peter and others, I could make films that would be acceptable to the TV companies, at a price they would be willing to pay, and earn our living by doing it.\n\nThe obvious next move would be to expand. I could take on capital and form a proper film company, with a studio, employees and new, state-of-the-art equipment. That was the conventional approach and I could see that if I had personal ambition, if I wanted to make a name for myself in the field of film-making, if I wanted my work to become recognized as world-leading in its excellence and win Oscars, that was the way I would have to go.\n\nDid I want these things? Well, yes, I wouldn't mind, one day, perhaps. But for the moment all I wanted was to be an author, a writer of stories, who just happened to have an extension to his pen, a facility which allowed him to write the stories on to film and so bring them to life. That was ambition enough to go on with.\n\nSo all right, what did I need to maintain this facility and make a living from it? It would have been nice to have a Sinclair singleframe rostrum camera and rotary table, a Steenbeck editing suite, a double-headed projector and a host of other glorious devices, but they would have set me back several thousand pounds and I would have had to spend the rest of my life paying off the loan.\n\nOne item was absolutely essential. That was a good camera with a rock-steady picture. I had one, a Bolex. The other optical equipment, like the projector and the editing viewer, were auxiliary, things which I could buy secondhand and adapt to my purposes. Everything else I could probably put together myself. The most important thing was to own everything and not be saddled with debt.\n\nThe great advantage of the move to Kent was that Peter and I now had a proper cow-shed to work in. There would be no more tube journeys across London to fetch a cardboard leg or half a dozen tiny hands. Also I now had somewhere to work which was away from home.\n\nI had my equipment in the dairy part and Peter had his studio and desk in the end where the cows had lived. Using a piece of scaffold-pole we bashed a hole in the wall overlooking the duckpond so that he could have a window to work by.\n\nAt my end of the cow-shed I built a new, improved animation table, mainly out of scaffold poles and pieces of bicycle, which had a camera gantry that would go up and down. This had several sophisticated facilities, like a piece of dressmaker's tape-measure glued to the back wall to show the height of the camera, side rollers to carry very long backgrounds and a new Meccano drive mechanism for the camera that had an electrically operated release and gave me an exposure of about three-quarters of a second for each frame. This meant that I no longer needed the very bright lights which had tended to overheat the card figures and make them curl.\n\nThen I made the one technical change which completely revolutionized my way of working. I thought again about the sound-recording procedures and realised that I had been working back to front. The business of making rough guide tracks, timing them with a stopwatch and converting the time to frames was not just boring, it was a nonsense.\n\nI could see that if I made the complete, final, finished soundtrack with all the music and effects already on it, first, before I even started to film, I could have this dubbed on to 16mm film. Then I could run the 16mm sound-track through the sound-head of my projector, listen to each sound and read off its frame numbers from the frame-counter. Then I would know exactly how many frames each movement would take and could film whole sequences which would match the sound-track exactly!\n\nI was tremendously pleased with myself for inventing this way of working, probably because at the time I didn't know that it had been standard practice in the cartoon film industry for years.\n\nAnother, unforeseen, advantage of this method was that I could have a cassette recording of the finished sound-track which Peter and I could listen to before starting work. In order to sort out the shots and pictures, we would sit at a table and run the tape backwards and forwards until we could see exactly what would be needed.\n\nAll these technicalities are the commonplace details of small film-making, but for us they were crucially significant to the progress of our enterprise. The price we were getting for the films was so low that for us success did not lie in the expansion of our facilities or in the increase of our reputation, it lay in streamlining the financial shoestring on which we worked, honing each piece of procedure, both in terms of time and cost, so that at the end there would be enough money over for us to live on.\n\nIt was a close thing. What with the twins and the move and the reorganization of the way we were going to work, I had forgotten to look for any more work.\n\nQuite suddenly, in the spring of 1960, I discovered that I was practically out of money.\n\nAlthough the BBC had liked _The Saga of Noggin the Nog_ they hadn't yet asked for any more, but I had offered them a set of six films called _The Seal of Neptune_ , which was a sort of underwater fairy-tale about the adventures of a sea horse and a shrimp. Ursula Eason was kind enough to confirm their intention to take the series, so, on the strength of that and other suggestions that were in the offing, I was able to borrow some money from Ray to tide us over. However it was a salutory experience, one which reminded me forcefully how precarious our living was. It also served very effectively to concentrate the mind.\n\n## _IV. Sea Horses and Penguins_.\n\nThe BBC offered a slightly higher price for _The Seal of Neptune_ : \u00a3125 an episode. That was still far from generous even in 1960 but it meant that there was a little more money to spend on music. That was good because the story needed music, gentle magic underwater music and dark sinister music to go with the sunken city and the evil sharks.\n\nThe opening lines were written with music in mind. You can almost hear it in the background...\n\nFar, far away, a long way from here in the Western Sea, where the palm trees wave behind the golden sand and the coral reefs grow like gardens under the water, where the sun keeps the sea warm and clear and it is always summer, there was a rock. It was a rock made of old sandstone, and the gentle ocean, flowing across and around it for thousands and thousands of years, had worn it into shelves and caves, crevices and crannies...\n\n...but of course I had no idea what the music was going to be like. I just marked up the script with the places where I thought music would be appropriate and handed it over to Vernon.\n\nVernon wrote some pieces for harp, clarinet and bassoon that, once again, were beautiful and exactly right. They fitted my hopes and imaginings precisely. Peter's drawings were ideal, soft and watery, in tones of grey with glints of reflection (colour television was still years away). My animation could have been better \u2013 inadvertent juddery movements are the bane of hand-done animation. They don't notice too much ordinarily but with characters that are supposed to be floating, fishlike, in an underwater world, they do jar the senses. I was very conscious of them when I saw the films but maybe I was being overcritical because the BBC liked them and paid up very promptly. Indeed the whole project went through very swiftly, partly, I suppose, because underwater persons, seahorses, shrimps, fish, etc. are simpler to animate than people with legs and arms, or railway engines with wheels and puffs of cotton wool.\n\nThe films were finished and delivered in a matter of weeks, and soon I was ranging around for the next project. Not that I was completely out of work. Small pieces of film for current affairs programmes were being ordered. Most of these were simple animated films, which I could often do more or less overnight, some were more elaborate and some involved live filming. But these were just odds and ends. I still needed to find the next film series. The inside of my head seemed quite empty. There was no new world waiting there.\n\nI didn't actually think of something for our next series, I happened to walk past it in the Firmins' back garden. Joan had put out some washing, a line of wet clothes dripping on to the grass. At the end of the line, pegged to it by its woolly beak, was a fullsize knitted penguin, also dripping wet. I had walked past before I realized what it was but I turned back to take a closer look. I thought how uncomfortable it must be to be pegged out on the line by one's beak with one's stuffing sopping wet. A stuffed knitted penguin would hardly be seaworthy. It would have to be a land bird, unless the wool was very special. Wasn't there some French knitting wool named _Pingouin_? Perhaps this uncomfortable bird was a pingwing. The Firmins kept goats, chickens, ducks, all sorts of livestock in their farmyard-garden. Maybe they also kept pingwings, or possibly they didn't keep them, they just lived there. The barn and outhouses were so conglomerated with junk that a family of pingwings could live there without anybody noticing.\n\nI looked over towards the open door of the barn and imagined another pingwing, a smaller one, Penny Pingwing perhaps, peering nervously round the door. She seemed very agitated. Then she spotted Mr Pingwing hanging on the line and ran straight across the yard, flapping her short wing-arms and shouting: 'Papa! Papa! Come quickly! Come quickly!'\n\nMr Pingwing replied, through his nose: 'I carn't cub. I've just bid wodged!'\n\n'You must! You must come at once, quickly! _Mama has laid an_ _egg!_ '\n\nMr Pingwing swung his body so that he could reach the peg and unclip it. He fell to the ground with a squelch. As they ran off towards the barn he left a wet trail on the gravel.\n\nNone of this took more than a few seconds to happen in my head but I thought it would do. I borrowed the washed pingwing from Josie, dried it and took a set of photographs of it in conversation with other members of the farmyard. I also made some enquiries about its origin. Apparently that was the only pingwing. Gloria, Peter Firmin's sister, had knitted it for Josie. I rang Gloria and invited her to knit a family of five pingwings \u2013 Mr and Mrs, their children, Paul and Penny, and their fluffy (Angora wool?) baby.\n\nGloria agreed, so I felt confident enough to take the little book of photographs and Josie's pingwing to the Senior Producer of Children's Programmes at the ITV company to ask if they would like a set of nature films about the home life of these unusual creatures.\n\nThe Senior Producer said: 'How are you going to film them?'\n\nI said: 'In single frames. The same way as I've been doing the cartoons only I'll be filming puppets, not card figures.'\n\n'Good luck!' he replied, dubiously. 'But yes. I like them. Show us some scripts, of course, and a piece of film, and if they're OK I'm pretty sure we'll buy them.'\n\nI came away from that meeting with the now-familiar feeling that once again I had let myself in for doing something I knew nothing about.\n\nCould I film puppets in the same way that I had filmed cartoons? There didn't seem to be any reason why not but I wouldn't know for sure till I had tried.\n\nSo while Gloria was making the first pingwing skin, I considered its internal parts. I could see that they stood upright but, like penguins, they had flat flipper-feet. For my purposes these didn't need to be attached to the bodies. The pingwings would travel on their bottoms and I could tuck flat feet underneath as appropriate. So they needed very heavy bottoms. I started with a 2lb (about 1kg) weight from a weighing-machine and screwed a length of threaded rod into it. I would fix the skeleton to that.\n\nI tried making a wire skeleton. It was too springy. It would be impossible to gauge the movements exactly. I needed a skeleton that was jointed so that the limbs could be moved into position and stay exactly where they were without any spring-back. The joints would have to be made.\n\nThe Clerkenwell Screw Company in Hatton Garden kept every imaginable shape and size of screw and screw-related object in stock. There I found some 3\u20448in (1cm) diameter brass balls with 1\u20443in (3mm) threaded screw-holes in them. I have no idea what they were originally meant for, but for me they were to be the artificial hips, elbows, shoulder and neck-joints of the pingwings. I made each joint by clamping the ball between holes drilled in flat steel strips (Meccano) and I made the limbs and body and beak out of wood.\n\nWhen the first knitted pingwing skin arrived from Gloria, I delicately inserted the skeleton into it, stuffing it inch by inch with chopped soft-plastic foam. I sewed it together under the base and it was made. It felt good. Best of all the limbs stayed _exactly_ where I put them.\n\nI set the camera on a tripod. I stood the pingwing on a table, lit it with a spotlight and settled down to an hour or two of singleframe waddling, posturing and peering at things.\n\nSeeing the film when it came back was yet another small miracle. The squidgy woolly chap waddled and peered about in a very determined and purposeful manner. I must say he seemed a shade puzzled about what he was there for, but I knew how he felt and was pleased with his performance.\n\nSo was the Senior Producer. I got the contract.\n\nAll I had to do now was film a set of single-frame puppets, out of doors, in Peter and Joan's cluttered yard.\n\nJust in case you should ever be tempted to do so, may I give a word of advice: _do not try to make single-frame films out of doors._ The reason is simple: _because the light keeps changing_. While you are moving the puppets between frames a cloud moves across the sun. That means the next frame is darker than the one before. The effect on the film, as you can imagine, is weird, like neon lights flashing on and off at irregular intervals.\n\nLuckily for me the weather turned out to be fairly dull and drab, which gave me a fairly even-toned film, but there were some days when I stood like Job, holding up my exposure-meter and hurling curses at the sky.\n\nThere were other unforeseen perils. I remember filming a conversation between Paul and Penny Pingwing by a low grey wall. It was quite a complicated conversation which took the best part of an hour to complete. What I hadn't noticed was a woolly-bear caterpillar on top of the wall. It wasn't doing anything except slowly looping its way along, looking perhaps for a way down, but of course the single-frame camera only saw it at irregular intervals. So on the film the caterpillar dominated the scene by performing a frenzied fandango. It had to be re-shot.\n\nSome of the shots revealed the presence of the omnipotent ghost that causes everything to happen. The first time I saw this puzzling manifestation I couldn't think what it was. As the Pingwings walked away from the camera they left behind them a wake of crushed, heaving grass. Then I realised what it was \u2013 my footprints, which I had left as I walked backwards and forwards between the camera and the puppets.\n\nVernon's music more or less rescued the Pingwings series. He wrote a 'Pingwings' Walk' which was so apt and hilarious that it imbued their waddle with glorious absurdity, an absurdity so irresistible that it made the faults of animation look like part of the comedy. Vernon also wrote an 'English Country Air' to go over the opening sequence that was, and indeed is, quite exquisite.\n\nThe munificence of the ITV company, which was paying \u00a3175 per episode, enabled me, at last, to employ an actor. Our friend Stephen Sylvester, who worked in the ITV company, suggested that Olwen Griffiths would be ideal as a pingwing.\n\nI telephoned Olwen and we arranged to meet in the foyer of Television House. 'You can't miss me,' she said, 'I'm vast!'\n\nOlwen was a handsome bright-eyed Welsh girl with red hair and a face full of life and laughter. She wasn't exactly vast; rotund, perhaps. She was also a really professional actress with a quick understanding and an almost unlimited range of voices at her instant command.\n\nOur first recording session at Brent Laboratories sound studio went very smoothly. Olwen got on well with Ron Hussey and Colin Bennett, the recording engineers, and we recorded some good episodes. The only slight difficulty came during the late morning when a sound like faint distant thunder began to be audible on some takes. This was eventually identified and dealt with by Colin, who nipped out to the caf\u00e9 to fetch Olwen a cheese roll and a doughnut to stop her tummy rumbling.\n\nStephen's choice had been inspired. Olwen was a glorious lady. She was to be my referee, collaborator and critic for many years, during which she shared with me, and our other actor friends, the happy task of giving the first breath of life to the characters who inhabited our imaginary worlds.\n\nThe Pingwing films were fun. They were light-hearted, homely and not violent, and the music was delicious. The TV company liked them and, although I didn't know it at the time, were all set to buy more of the same. What I do remember is that as we came away from showing them the last episode the Senior Producer said to me: ' _Pingwings_ are lovely but I am commanded by \"The Powers that Be\" to order thirteen more _Ivor the Engine_ films from you.'\n\n'But that's absurd,' I said. 'The story of _Ivor the Engine_ is over. He now sings in the choir. That's it.'\n\n'Well, _Ivor_ is their first love and the offer is there. I suggest you think about it.'\n\n'But I've got nothing in my head. It would take time.'\n\n'Listen. \"The Powers that Be\" want more _Ivors_. They don't mind waiting. So take your time.'\n\n'Oh, all right,' I said, rather ungraciously. 'I'll see what I can think of.'\n\n* Ray suggested 'Llanmad' because Dylan Thomas's town in _Under Milk Wood_ was named 'Llareggub'.\n\n# TAKING TIME\n\n## _I. The Dogwatch_.\n\nIt was all very well the Senior Producer of the TV company telling me to 'take my time' to think about the next _Ivor the Engine_ series. I didn't have any time to take! I had a living to earn and, unlike him, I didn't have a salary.\n\nI did get one job, an acting job, two days' filming. In this I was a Viennese professor of high repute and great erudition, wearing an old-fashioned bathing-costume and gumboots, giving, in a heavy accent, a lecture\/demonstration on the English Art of Boatpunting.\n\nI will expound to you the theme of this lecture, which is concerned mainly with the poling, the method of propulsion. In it I depict what might be called the _classical approach_ , in which the punt is loaded with many poles. The method is for the poler to take a pole to the front of the punt, push it down into the mud and then walk back along the boat while clinging to the pole, thus causing the punt to move forwards through the water, the poler remembering always to let go of the pole as he reaches the back end of the boat. He or she will then pick up another pole, walk to the front and repeat the procedure.\n\nFor reasons of economy this method has been largely superseded by the _standard approach_ , in which the poler does not always abandon the pole when he or she reaches the back end of the punt but deftly recovers it by hauling it up out of the mud and carrying it again to the front for a second usage. This method is far less costly in poles, using very few, sometimes only one, for the whole journey.\n\nFinally there is the _modern approach_ , in which the poler does not even walk along the boat but stands at the back end and walks with the hands up the pole itself. This is the most skilled method and has the great advantage for passengers in the punt in that they do not have to be walked upon.\n\nIt was a long, lugubrious joke, very Germanic, but lightened a little by the failures and misfortunes of the demonstrator whose actions were often ironical to the commentary and who suffered many of the stock jokes about punting, most of them ending up with me falling in.\n\nThis was filmed by Patrick Dowling, who was later to become famous as the producer of _Vision On_ , a celebrated programme for the deaf, and it was great fun to do, in a damp sort of way.\n\nAfter the transmission, which I had to attend because some shots were 'live', an extraordinary thing happened. I was invited to the Reception Room where there were several engineers, some of whom I knew, and also Eamonn Andrews, the famous presenter of _This is Your Life_!\n\nFor a moment I wondered whether I might be going to be the subject of the presentation, that 'This' was going to be 'my life' \u2013 and I got ready to run.\n\nThank goodness, it was not. It turned out that I was to receive a presentation. Eamon Andrews ceremoniously handed me a film-can with a label on it. The label was inscribed as THE BBC TELECINE AWARD, presented by the Telecine Department for the best children's programme of the year 1959: _The Saga of Noggin the_ _Nog_ \u2013 which was the first film that Peter Firmin and I had made for the BBC.\n\nI was completely amazed and delighted, particularly so because this was an appreciation by technicians, by the people whose job it was to watch programmes all the time. They were usually the most detached and cynical audience of all. To have won their approval was a tremendous compliment.\n\n'Thank you. Thank you indeed!' I stammered. 'This is a singular honour.'\n\n'You're right there,' said Eamonn, 'it is unique. The BBC Telecine Award has never been given before.'\n\n'Hasn't it? Why not?'\n\n'Because they just invented it for this occasion!'\n\nThat was a proud moment.\n\nBut you can't live on proud moments. I needed work.\n\nI was envious of the situation that Peter Firmin and Wally Whyton were in. They had their own weekly programme, _Musical_ _Box_ , which was bringing them in a regular wage. I wondered whether Peter and I might be able to do a similar programme. I couldn't sing of course, but I could make films, live films as well as animated films, and I was set up to do them very quickly. Could we perhaps put together a sort of magazine-type programme with stories and pictures, some of Peter's card-animations and pieces of film?\n\nWe could. I proposed the Grumbles Light, which was an offshore lighthouse with me as its keeper. I would need a companion \u2013 a seagull? a seal? No something more domestic, a dog perhaps. Yes.\n\nOK, so what would a dog do on a lighthouse? Watch television?\n\nWhy not? He could be a watch-dog and spend the dog-watch watching dog-television. Of course there was dog-television. The dog and I would watch it on our steam television set.\n\nPeter and I went through our stock. We found some clips of film and mapped out one or two programmes which really seemed quite possible, even amusing. So we put the idea together and offered it. The ITV company accepted it without demur and we found ourselves committed to conjuring up thirteen weekly programmes called _The Dogwatch_. Starting in November, these were scheduled to be recorded in pairs once a fortnight on the new AMPEX videotape system.\n\nPeter began by making the dog. It was a shaggy cockney mongrel with yellowish mop hair and an eager, puzzled expression. Its name was Fred Barker.\n\nNext we had to find somebody to work the puppet and be Fred Barker. I looked out a friend from Drama School, Ivan Owen, who had already been playing various parts in children's programmes. He quite liked the character of Fred Barker but wasn't sure he wanted to spend his time sitting under a table with his arm up in the air, working a puppet. Still, like the rest of us, he needed the money, so he agreed to have a go.\n\nOn the whole I think Ivan made a sensible choice. He gave great life and character to Fred Barker and, long after _The Dogwatch_ was over, he was to spend many profitable years working a puppet, but this was not Fred Barker, this was another of Peter's creations, Basil Brush.\n\nBasil Brush took Ivan over completely and became a hilarious TV personality. Ivan wisely maintained anonymity and was only occasionally referred to by Basil Brush as 'the chap who gives me a hand'.\n\nHowever, all this was yet to come. In the winter of 1960 we had _The Dogwatch_ to prepare.\n\nThe programmes were very rough and ready. Peter and I would spend the week finding bits and pieces to show on dog-television \u2013 cut-out caption stories, glove puppets and model-making. Sometimes I would go out with the camera and take pieces of film, of riding on a miniature railway or going out on an oyster yawl.\n\nEach episode began with Ivan singing a sea-dog shanty, but after that we didn't bother with a script, just made it up as we went along. The dog-serial, which I filmed with cut-out card figures, was a Chicago-type gangster film starring Wilbur Dog who hailed from Doggeville Flats in Utah. It was written in \u2013 brace yourself \u2013 shaggy doggerel! Awesome stuff!\n\nThe effort and stress of putting a new _Dogwatch_ programme together each week was very wearing and everybody was relieved when, in the spring of 1961, after twenty-six episodes of makeshift mayhem, _The Dogwatch_ was taken off.\n\nThat had certainly succeeded in 'taking time' and both Peter and I were looking forward to some less frenzied occupation, like making films.\n\nI did have one moment of anxiety for our future. Just as we were about to record the last two episodes, one of the engineers came to speak to me. He told me that he was the representative of the union, the ACTT, and that they had had a meeting about me and my work the day before.\n\nMy heart sank. I was sure I had been doing all sorts of things that only union members were allowed to do, like sound-recording, live filming, electrical work, building scenery. Anything that needed doing I had just done for myself without a second thought.\n\n'I dare say I've been stepping over the mark a bit,' I suggested.\n\nHe laughed. 'You can say that again.'\n\n'What's going to happen? Are you going to black the programmes?'\n\n'No, no point in doing that. We like the programmes. It was just a matter of establishing categories, that's all. It was formally moved and resolved that if you stick to doing the sort of things you do, things that nobody in their right mind would ever dream of doing, then as far as the union is concerned you can get on with it.'\n\n'Oh,' I said, surprised and profoundly relieved. 'That's good. Thank you. But tell me, what is this category you have established for me?'\n\n'Madman,' he replied.\n\n# MAINSTREAM\n\n## _I. Settling In_.\n\nOn a bright June day in 1961 I said to the booking clerk at Whitstable station: 'I want a period return to London, please.'\n\nHe replied: 'You must be out of your mind!'\n\nWe laughed and I bought the ticket, but he was quite right. As I sat in the bumpy train looking out at the sunshine on the green marsh fields and the grey-blue sea beyond, I saw that Whitstable on a bright June day was a lovely place to be and that London was a hole. I didn't regret having moved. Our work could perfectly well be done from Kent.\n\nDuring the _Dogwatch_ period the supply of peripheral work had thickened up. Peter and I had been commissioned to prepare an _Ivor_ book. TV Publications Ltd. had started a new comic (I beg your pardon, 'children's weekly'), called _TV LAND_ , 'featuring all our favourite TV characters and programmes'. It wanted an eightframe picture story about _Ivor the Engine_ every week. The other London jobs, like stage-managing and prop-making, didn't earn much and were drying up.\n\nThese were all sidelines. What I wanted to do was get back to making films. The BBC hadn't yet asked for any more _Noggin_ films but the ITV company that had put down its marker for another thirteen _Ivor the Engine_ films was still waiting. My own feeling had been that the story of _Ivor the Engine_ had finally ended when he sang in the choir, but this had turned out to be untrue because, in order to write the weekly stories for _TV LAND_ , I had had to take my imagination to the corner of Wales where Ivor worked. Peter and I had surveyed this area quite fully for the first film series and he had made a map of the line with the stations and main features marked, so in our minds it was a completely real place. When I came to look at the neighbourhood more closely I saw that there were plenty of new people to meet and plenty of events going on, some of which had happened since Ivor had joined the Choral Society. They had won a silver laurel at the Eisteddfod, which Ivor wore on his funnel. The Society's annual outing had been to the seaside at Tewyn Beach, where the railway engineers had laid on facilities for Ivor to have a bathe, with disastrous results. Mrs Porty, the slightly tipsy lady who lived in the big house, had given refuge to a nest of brown birds in a disused cuckoo-clock. Her donkey, Bluebell, an old friend of Ivor's, had been given a private truck to ride on. As the result of a regrettable misunderstanding, Old Nell, the sheepdog, had chased the Rev. Poultry up a tree. The gold-miner, Mr Dinwiddy, had put his gold back into the ground and Mrs Porty, who was very rich, had bought the whole railway, lock, stock and barrel, to save it from being taken over by the Welsh National Railways.\n\nSo, having duly reported these events each week in _TV LAND_ I came to realise that there was absolutely no reason why we couldn't go ahead and make the next lot of thirteen films right away. The stories were waiting. We took the contract for the films and started work.\n\nI suppose the person who was made most busy was Vernon Elliott. I asked him for several pieces for the Ivor films, including three speeds of travelling music \u2013 main, cruising and fast \u2013 a donkey-trot and some more songs from the choir.\n\nFor the recording sessions Vernon would collect some of his associates, usually professional players from the Philharmonia Orchestra, and we would spend an afternoon at some suitable venue, rehearsing and recording all the music for the series on to quarter-inch tapes. I never ceased to marvel at the players' professionalism, at their ability to look at Vernon's pencil-written music and grasp instantly, not just the notes and the tune, but also the feel and purpose of the piece, before they had even played it.\n\nI always came away from those recording sessions ringing with excitement, clutching the reels of recorded tape and feeling marvellously elated at having had some part to play in the creation of these wonders.\n\nAs time passed and we became more practised, the voicerecording sessions became nearly as efficient as the music ones. For the Ivor voices I had two actors to help me, David Edwards and Olwen Griffiths. Olwen was of course Welsh, so her portrayal of Welsh people was always authentic, though her tour de force, the queenly, inebriated Mrs Porty, was extremely English.\n\nWe all tried to speak for our characters as simply and spontaneously as we could, avoiding as far as possible using consciously artificial voices and, perhaps the most important of all, avoiding adding extra drama.\n\nBy this time Peter and I had had enough of sharing a cow-shed and he had raised and refurbished four pigsties behind the barn to accommodate one taller pig. This was my new studio, and very spacious it was until I started filling it with mechanical junk.\n\nFrom then on the system worked quite smoothly and, with the production of the second _Ivor the Engine_ series, Peter and I knew we were entering into the main part of our lives' work.\n\nI think 1962 may have been our most prolific year. We were feeding the hungry television really well. I remember noticing in the _Radio_ and _TV Times_ that in one week Peter and I between us had a total of five different programmes listed... _Noggin the_ _Nog_ and _The Seal of Neptune_ on the BBC, and on ITV, _Ivor the_ _Engine_ , Peter and Wally Whyton's _Musical Box_ and their puppet show _The Three Scampies_ , which was hosted by Howard Williams \u2013 not to mention Peter's puppets, Ollie Beak and Fred Barker, who were likely to appear at almost any time on children's programmes.\n\nOne incident in 1962 heralded the approach, not of riches but at least of financial solvency and a bit over. We signed an agency agreement with a nice lady called Angela who worked for Talbot Television Ltd., who would sell showings of our films abroad. She started at once and sold them to the most unlikely places, countries like Cyprus, Israel, Rhodesia and Spain \u2013 but the money was very welcome.\n\nAnother incident, late in 1962, heralded the realisation of something more subtle. I was standing in Piccadilly Circus tube station buying a ticket. A small boy came up to me pushing an effigy in a doll's pram. He plucked at my trouser pocket and said: 'Penny for the guy, Mister?'\n\nI replied: 'Penny? I haven't got a penny to my name!'\n\n'Course you have!' he cried. 'You've got pots of money. You're _Noggin the Nog_!'\n\nHow he knew who I was I don't know, I suppose he had heard me speaking to the booking-clerk or something, but it shook me. For years, like dwarves under the hill, Peter and I had been working away, turning out fodder for the television. I knew children would have watched our films, but it had never really occurred to me that anybody would remember them.\n\nI gave the boy half a crown and, for the first time in my life, felt the taste of fame.\n\nA couple of weeks later the BBC asked for a second _Saga of_ _Noggin the Nog_.\n\n## _II. The Land of Nog_.\n\n'In the lands of the North, where the black rocks stand guard against the cold sea, in the dark night that is very long the men of the Northlands sit by their great log fires and they tell a tale...'\n\nSo began each epic of _The Saga of Noggin the Nog_ and nobody who watched those films in the 1960s will ever hear those words without remembering the slow icy chords with which Vernon Elliott (bassoon) and his daughter Bridget (clarinet) heralded and accompanied that opening speech. The piece of music that Vernon had composed was not only just right for its purpose but thrillingly evocative of the Land of Nog, which was now going to be our next world.\n\nWhen the request for a second saga came through, it came as a bit of a surprise. It was nearly two years since we had made the first one and the sort of everyday incidents I was accustomed to look for in Ivor's Wales would be quite inappropriate for a saga. I was going to need more heroic stuff, a legendary tale, full of dangers and adventures, one which had a beginning in which disaster would befall, a middle in which a quest would be ventured, and an end in which, by some near-miracle, success would be wrested from the jaws of failure \u2013 or something of the sort.\n\nThe Nogs themselves were essentially peaceable folk. The only reliable villain was Nogbad the Bad, who was the fountain of all evil. He had been banished at the end of the first saga but who knew what villainy he had been up to since then? Who indeed? I had absolutely no idea. I sat in front of a piece of blank paper and drew pictures of dragons. I liked dragons and I had promised myself some. Dragons are always useful in sagas and epics even though they have a bad reputation for wreaking havoc. Unfortunately there didn't seem to be any about in the Land of Nog. Perhaps they had retired. Perhaps they had become friendly and had formed the Dragons' Friendly Society, an institution with assets of untold wealth, a massive treasure of gold and jewels, locked in some icy cave. Perhaps that had attracted Nogbad's attention. He always had a sharp nose for other people's treasure. Mind you, such a treasure would have a guardian, a dragon of course, but one accustomed to the cold, perhaps an ice dragon.\n\nI don't know in what order I had these random thoughts, but it came to pass, eventually, that a small angry man from the Hot Water Valley, in the far north of the kingdom beyond the Glass Mountains, arrived at Noggin's castle to seek his help in getting rid of a huge fierce dragon that was rampaging through their valley. Noggin, with Thor Nogson and the great green bird, Graculus, travelled to the Hot Water Valley to confront the dragon, who was an ice dragon and not fierce at all. He was of course the treasurer of the Dragons' Friendly Society, who had been smoked out of his cold cave. How Nogbad was discovered and how he nearly escaped with sacks of treasure on Noggin's own sledge and how he was finally apprehended by the now angry dragon and turned into a snowman, are now a part of television folk history.\n\nTo tell the truth I had felt very dubious about the basic plot of _Noggin and the Ice Dragon_ \u2013 it seemed to me quite spurious and worthless \u2013 but, as I was later to realise, this was the effect of a sort of mental fatigue induced by anxiety. Once I had outlined the plot to Prue and Peter and Joan, and seen their eyes light up, I was able to see that it would do, and then the details came flooding in by themselves. Even the voice of the Ice Dragon, which was a mixture between Prue's brother Capel and the actor, the late Wilfred Lawson, came by itself, as did Graculus's finding of his long-lost family home, a place that he had never seen but had remembered from the time when he was an egg. Discovering these details, piecing them together and writing them down was a time of great joy in my life and I feel enormously privileged to have been able to bring them to light.\n\nFor _Noggin and the Ice Dragon_ we were going to need a lot of incidental music. Vernon had written plenty of glorious music to accompany Ivor on his travels, including a delicious donkey-trot, but he was a bit baffled by the Noggin music because this seemed to need to follow the action in a more detailed way. I found a way to help him with this by drawing a sketch of each piece of music. This took the form of an illustrated graph, with a base-line of the overall time, say: three minutes. On this I would draw pictures of the actions and happenings in sequence at approximately the time they would happen. I would give the sketch to Vernon and, as soon as he saw the pictures, he found he began to think of the music. The sketches, as he put it, started him off.\n\nAt Prue's suggestion I asked Ronnie Stevens, an old friend from the club-theatre days, to share the Noggin voices with me. His voice was lighter than mine, which was an advantage. He was also very relaxed and professional in his approach.\n\nThe sound-track of _Noggin and the Ice Dragon_ was a joy to make. When it was finished Peter and I sat with the cassette recorder, listening to it and seeing the pictures. By now our imaginations had become attuned to the medium we were using, so that we actually thought in terms of flat card cut-outs, and animation procedures which we knew were possible. This was in no way a limitation to Peter's imagination \u2013 the pictures he produced were magnificent. The music fitted with them really well and the end result was very satisfying. I remember taking the first prints along to the showing at the BBC and thinking: 'These they will enjoy!' They did, quite a lot, so much so that they more or less gave us carte blanche to supply them with Noggin sagas whenever we, or rather they, wished.\n\nIf we had followed 'good business practice' that would have heralded the start of a constant stream of Noggin-type programmes 'created' according to a formula.\n\nEven if we had wanted to do that, we couldn't have, because Peter and I had other work to do and, in any case, thinking up reasonable Noggin sagas took time. I needed to be patient and let the story brew up somewhere in my head. The main trouble was that, for a saga, one needs villainy and, after his chilly encounter with the Ice Dragon, Nogbad the Bad seemed to have gone to ground, perhaps in some warmer clime.\n\nHowever, I was fortunate enough to come across one of the great thinkers of his time, a tall, somewhat obsessive man who was Court Inventor to Prince Noggin. His name was Olaf the Lofty and, it must be said, he was in many ways a thorough nuisance because many of his inventions were more trouble than they were worth. Some, however, were brilliant. Among these was a clockwork-driven flying machine in the form of a longship with leather wings. This, he claimed, would flap up into the sky and bear the Nogs away to the ends of the earth, if they should happen to want to go there.\n\nFor most of _Noggin and the Flying Machine_ Nogbad is nowhere to be seen, but the small man who arrives rather spectacularly in a jar, bringing a magic carpet, is in the same league as, if not in league with, that evil uncle.\n\nHe steals the crown of the Northlands and flies off on the magic carpet. Naturally the Nogs give chase in the flying machine, which of course is wrecked by a magic storm thrown by the small man's snuff.\n\nThe flying machine is mended with some stolen feathers. They take off again and come to the Land of Silver Sand. There they are captured by the Emir, another wicked uncle, and at last meet up with their own \u2013 Nogbad the Bad, the author of all the villainy\n\nAll is not lost. Just as the Nogs are about to be eaten by the Emir's vast black Genie, Grolliffe, the Ice Dragon, is called to their rescue. Complaining bitterly of the heat, the Ice Dragon makes short work of the genie, rolls him into a puff of grey smoke and blows him into an ice-cube.\n\nAfter I had written _Noggin and the Flying Machine_ I found myself wondering whether by including the Land of Silver Sand I had been intruding on other people's territory \u2013 taking other people's settings and conventions and using them as if they were my own. I thought about this and came to the conclusion that it was really only a matter of being geographically accurate.\n\nWhen they, inadvertently, travelled to the Land of Silver Sand, the Nogs entered a country that was just as fictional as the Land of Nog but much better known, being recognizable as the setting for _The Arabian Nights_ , and Ali Baba and the Forty Thieves (as well as that terrible dance we did for the pantomime in the Isle of Wight). Once the Nogs had arrived there it was more or less inevitable that they would become involved with such cliche\u00b4s as genies, magic carpets, impersonations and palace intrigues, simply because such things are known to be the stuff of ordinary life in that imaginary country.\n\nSo, unwittingly, perhaps I had been derivative, but I suspect that most plots have been used many times over and most of the incidents in them have been derived from, or could be traced to, things that have happened in life or literature.\n\nThat is one of the reasons why I have always resisted the suggestion that I have been 'creative'. Very occasionally, ideas that seemed to be wholly original would turn up in my head. When these were relevant I would grab them with delight, but the idea that I had _created_ them is, I am sure, quite misplaced. When it was looking for stories, my mind would go trawling through a mud of memory, a morass of experience, life and literature, to see what it could find. It is true that I usually rearranged what came up, perhaps showed it from an unusual angle, but I have always had to respect the fact that _novelty_ is not, in itself, either interesting or valuable. Its interest and value lies in the degree to which it is able to give a particular view of the condition that we, as human beings, find interesting and valuable, i. e., being alive. That condition doesn't need to be _created_ because it is always there \u2013 even if we aren't noticing it.\n\nI am glad to say that while I was writing _Noggin and the Flying_ _Machine_ none of these nervous considerations entered my head. What I was looking for was a usable story and, though I say it myself, I got a rattling good one.\n\nVernon wrote some really evocative music for the flight of the flying-machine and a tremendously exciting accompaniment to the magic storm, but it fell to me to provide the mechanical sound effects \u2013 the sound of the huge clockwork engine starting up and the great wings beating faster and faster as they bore the incongruous vessel through the water and up into the sky, where the sound would merge seamlessly into the glorious flying theme that Vernon had composed.\n\nPeter and I spent some hours collecting mechanical bits and pieces and recording their noises. I think we settled on a broken wooden toy lorry as a prime mover. When its wheel was turned it went: _Ker-clonk.. bonk_. This was a bit basic and woody, so we screwed a tin to it with a strip cut down to catch against a knob on the wheel. Now it went _Ker-clonk.. cherrrrnk, tweek-bonk_ , the _cherrrnk_ being a nice metallic spring-sound. This was still a bit slight for such a large machine so we hung a piece of lavatory chain in the tin and mounted the whole thing on a hinged padded board which lifted and fell as it rotated. Now it went:\n\n_Ker plink-plink.. Chernk-tweek plinkaplank FLOMP_.\n\nThe FLOMP was the flap of the great wings. It still sounded a bit small, like somebody turning a mechanical toy, but when we recorded it and played it back at half-speed it was magnificent. Then we could see the absurd machine surging forward and lifting off into the sky.\n\nThe films of _Noggin and the Flying Machine_ took off well with the BBC and they let it be known that they would appreciate it if we would kindly produce another saga as soon as possible, like later in the year.\n\nThis wasn't easy to arrange because, while I had been filming _Noggin and the Flying Machine_ with my hands, in my head I had been putting together stories for the third set of _Ivor the Engine_ films.\n\nLuckily I had made frequent trips to our imaginary Wales looking for stories to go in _TV LAND_ Weekly, so I was able to report that life there had become quite eventful. A dragon's egg had been found in Ivor's firebox. It hatched out during choir practice and the dragon sang rather well. His presence in Ivor's firebox, while welcome, did present some problems and eventually, after one or two unfortunate incidents and the arrival of another dragon, the mythical beasts decided to avoid publicity and retire to a local volcano. Then Mr Brangwyn's pigeons escaped and, on his way back from helping to lure them down off Miss Price's roof, Ivor nearly ran into an elephant lying on the line. The elephant, which had a bad foot, proved even more difficult to house than the dragons but once again, after some difficulties, help arrived in the form of the elephant's keeper, Mr Bani Moukerjee, who had spotted an elephant's boot on Ivor's truck. He was reunited with his elephant, Mr Banger's Circus arrived, and gave them all a free show.\n\nThis was only a small part of what had been going on but it was enough for the thirteen episodes of film that had been asked for.\n\nThe Northlands, meanwhile, had gone quiet again. Life there was so peaceful and complete that Olaf the Lofty, walking on the hill, was bewailing the fact that, in Noggin's city, Nog civilization had reached its final peak. Everything was perfect and there could be nothing more to invent. At this the whole side of the hill blew up, throwing out a cloud of steam and a very small man whose eyes were not used to the light. Olaf rescued the man, who was an Omrud, one of the little people who lived under the hill. His name was Groont and he was Court Inventor to the Kingunder-the-Hill. In return Olaf was given the secret of Groont's latest invention, a method of harnessing the powers of fire and water. This inspired Olaf to build a handsome steam engine to pull Noggin's royal carriage. Needless to say it ran amok and jumped over a cliff.\n\n_Noggin and the Omruds_ is not about a quest, it is about an invasion. By Olaf's alchemical folly Nogbad's crows become the size of cart-horses and they fall on Noggin's castle like vultures on their prey. Once again Nogbad nearly achieves his goal, to take the crown of the Northland, but this time he is thwarted by the ingenuity of the Omruds, and by the time they have finished with him he is too small to do any harm \u2013 or so they think.\n\nThe garden which the Nogs make for the Kingunder-the-Hill is beautiful, as is the music which Vernon Elliott wrote for it, calm, elegant and glorious. It seemed very final and appropriate, not only for the ending of _Noggin and the Omruds_ but perhaps also for the end of all the _Noggin_ sagas. I remember feeling almost hopeful that it might be so. Like Olaf the Lofty, I was beginning to think there might be nothing left to invent.\n\nThere followed a period of relative quiet. I say 'relative' because as well as turning out our weekly stories for _TV LAND_ and contributions to its Annual, I made a set of films for the BBC programme _Blue Peter_. Ronnie Stevens and I, with Vernon and his musicians, made a Noggin gramophone record. Peter and I also, somewhat incongruously, made some filmed inserts for _The Telegoons_ , a strange adaptation of the radio programme, _The Goon Show_. I can recall nothing of this job except a Christmas pudding falling from the top of the Kremlin and cracking the pavement of a Moscow street.\n\nThen the BBC became impatient and said it wanted the next _Noggin_ saga.\n\nIn _Noggin and the Firecake_ we moved into a darker area of myth and magic. There was no quest, no travelling dangerously to far lands, only unknown magic powers, locked by the Sword of the Sorcerer deep in the earth. In this Nogbad's simple villainy is completely outclassed by the power of a stone giant who befriends Noggin's young son, Knut, who had heard it singing under the stone.\n\nCertainly Nogbad's villainy takes him nearly to his finest hour, but even as he reaches it the power that he has unleashed closes in on him and is set to crush him.\n\nBut in all this the most dangerous material, _firecake_ , was invented by the silliest character, Olaf the Lofty \u2013 but that has a more terrible moral than any story.\n\nWhen I showed her the script Ursula Eason, the Deputy Head of Children's Programmes at the BBC, looked at me quizzically. 'Social comment now, is it?' she growled.\n\n'No, not especially,' I replied, lying. 'Just another saga, like the others.' Ursula laughed and commissioned the films.\n\n## _III. Houses_.\n\nThe tall terrace of houses known as Wavecrest, of which ours was number six, stood high above the sea wall, breasting the storms. There was, people used to say, nothing between it and the North Pole, and when the wind hit it at night you could feel the house go _thud_ inside itself. On such nights I would sometimes dream that I was standing with my feet on the sea wall and my hands under the first-floor balcony, either gripping the rail to hold the houses down or pushing at the balcony to hold them up. Prue would turn over and say: 'It's all right, darling.' And I would reply: 'It's not bloody all right, you stupid woman. The sea's coming over any moment!'\n\nThe sea never did come over the wall, but it didn't need to, it could come under it. The sea wall, like the houses, had been built on the shingle, so once the tide rose high enough to reach the wall, the water just oozed underneath and rose gently through the basement floor. Fortunately that only happened once.\n\nOn good days, when the sun shone and the sea-breeze smelt fresh, the place was idyllic. The children would run straight out on to the beach and Prue and I would carry tea down on a tray and sit against a breakwater. Bach especially loved the beach and at low tide she would scamper away, splashing through the puddles to chase small seabirds at the distant water's edge.\n\nIt certainly was an exciting place to live but, in the end, too tall and thin. Prue had varicose veins and hauling the shopping up four flights of stairs did them no good. So, late in 1962, we bought an ex-pub on the Canterbury to Whitstable road.\n\nRed Lion House, once the Red Lion Public House, had been built in about 1580, an ancient 'pre-fab' with oak beams that were mortised together and numbered with chisel-marks. It had sloping floors and crooked windows and a slightly disreputable air, as if it had once been a highwayman's haunt. The front yard had been enclosed, so the house now stood a little way back from the road and was set in a large rough garden with a dilapidated garage-cumbarn, two wells, a small copse and a stream with a rope on which to swing across it. The interior layout \u2013 public bar, private bar, snug, tap-room etc. \u2013 was a bit unusual for a dwelling-house but as we looked around the house, Prue and I knew that this was where we were going to live our lives.\n\nTo my great delight I found that the Essex-boarding in the main room was covering up the original varnished tongue-and-grooved boarding of the pub and also that the ugly suburban firesurround  had been set in front of a huge open fireplace big enough to roast an ox. I ripped out the Essex-boarding and the firesurround and got the builder to lay in a tiled hearth. We did some decorating and then, on December 22nd 1962, moved in.\n\nThat was the day the snow began to fall. I remember the family shivering disconsolately in the main room while Prue's son Kevan and I tried to light a fire. We set newspaper and kindling wood in the middle of the tiled hearth and then lit it. The smoke rose lazily and began to roam about the room as if looking for something. We flapped at it with newspapers to drive it back towards the fireplace and eventually it seemed to recognize the chimney and began to move into it. After an hour or so the fire was burning really well, so well that the chimney was drawing in air from all directions, freezing the backs of our legs with icy draughts while the fire was roasting our fronts. We finished up putting paraffin stoves in the kitchen and on the landing to warm the incoming draughts.\n\nIn that first bitter winter, when the sea froze and the snow stayed until March, we lived an almost medieval life, centred around the great wood fire. Kevan and I made regular sorties into the neglected woodland at the top of the field behind the house and brought back thick branches and sometimes whole dead trees on our shoulders. The cat, Perdita, who loved these outings, would ride, growling, on the wood. Bach would have loved them too but, sadly, she had died of old age and heart-trouble while we were still in Whitstable.\n\nWe sawed the branches into long logs which the fire burned away like tinder. We made a fine sledge-run in the field behind the house and used the metal top of an old dead fridge as a sledge.\n\nRay and Daisy came down to visit us, taking the precaution of booking in at the Chaucer Hotel in Canterbury. Daisy was a bit dubious about the primitive facilities of Red Lion House but she was delighted with the sledge run. I can see the old lady now, crouching in the lid-sledge, her figure muffled and dark against the white snow and her cigar flaming cherry red as she hurtled down the hill, straight towards the brambles. She was saved from certain scratching by young Will Gollop, who heroically threw himself in the path of the sledge before it reached them. She was tipped out and wasn't a bit grateful; insisted on having another go.\n\nThe snow eventually melted and by the time spring came we were well settled in our new old house. I commandeered the Private Bar as my study. That room had the advantage of being relatively soundproof, which was a godsend because although the four-acre field behind the house provided ample rough chasing for the boys, their friends and the dog, the house always seemed to be well filled and brimming with noise.\n\nDuring the holidays Kevan was with us. Kerris, his sister, was in residence, attending school at Faversham, living the spectacularly inconsiderate life of a vigorous, highly emotional fifteen-year-old girl. Pip, their young brother, was going to the Primary School in Blean and the twins, Simon and Stephen, who were four and a half, were going to nursery-school in the mornings. We had also collected a schoolfriend of Kerris's, Fiona Larkin. She came in to help Prue and look after the twins, who seemed to have a genius for amiable destructiveness. She became an honorary member of the family and has been ever since.\n\nProbably the most joyful occasion for Prue and me was our decision to have another baby. Prue, at forty-one, was thought to be a bit old to be having babies but our GP took the view that having had five already she could probably take having another one in her stride.\n\nDaniel was born in February 1964 at the maternity hospital in Whitstable. I met him when he was about twenty minutes old and yelling. He wore a wristband with his name on it: 'Spontaneous D. Postgate'. That, I remember, puzzled me.\n\n## _IV. Small Persons_.\n\nIn spite of the troubles with outdoor light I had enjoyed making the _Pingwing_ films. Filming single-frame puppets was less fiddly  than filming cut-out cartoons. There were no backgrounds to set up, no figures to assemble and joint, no cotton-wool puffs of steam to move. The puppets were just there. I could choose the shots and camera angles as I went along.\n\nSo now I was looking for a new puppet series, preferably one that I wouldn't have to film out of doors too much. I was sure there were other small persons about, perhaps not in Peter and Joan's farmyard, but somewhere. People had gnomes in their gardens, of course, but they were a bit sedentary and rarely seemed to do anything except fish.\n\nMaybe there were low-angle persons in the forest. The trees in the wood behind Red Lion House were mostly scrubby oaks with brambly undergrowth \u2013 not a very congenial habitat. But among them, relics of an earlier forest, were several massive beech trees. These had wide spreading canopies of leaves under which no brambles grew and wide boles whose gnarled roots spread out to grip the ground. A hollow beech tree would make a well-appointed residence for some woodland person, or maybe persons, a family perhaps. I wondered what they would be. Not gnomes, garden or otherwise, I disliked gnomes. Hobgoblins? No, a bit malicious. So were hobyars. How about leprechauns? They were wily and Celtic and had to do with tricks and magic. I thought the small persons who would live in the roots of these trees would be ordinary. They would be quiet people who wouldn't want to have anything to do with magic. I tried different names in my head and eventually came up with one that sounded all right: _Pogles_.\n\nRight then. Mr and Mrs Pogle live in the root of a tree and grow beans in their little garden. I looked all through the wood to find a beech tree with a hole suitable for a Pogle's front door, but found none. The only thing to do, in fact the obvious thing to do, was to get Peter to make me one in the studio.\n\nI proposed _The Pogles_ to Peter. He liked the project in general and said he would be happy to make the people and the tree-root house, but, like me, he had no idea what the story would be about.\n\nI had only one clue about this: Mr and Mrs Pogle were not magic-type people. They didn't hold with magic and didn't want to have anything to do with it. But suppose magic wanted to have something to do with them. How would small ordinary respectable people fare when faced with magic?\n\nOnce again I started writing and left it to chance. Mr Pogle went out with his barrow to look for some old dried beans in the shed of the empty cottage. He didn't find any there but under the hearthstone he found one perfect bean, a bean so large and shiny that he could hardly lift it into his barrow. He wheeled it home to Mrs Pogle, who was not pleased because she wanted to plant a line of dwarf beans in her garden, not one giant bean.\n\n'But look at it, wife! It's a lovely bean!'\n\n'Listen to it, you mean.'\n\nThe bean was fast asleep and snoring.\n\nWondering whether it would wake up one day, the Pogles planted it. Expecting that it would take some while to grow, they turned to go indoors, but before they reached the front door the bean was up and awake, stretching its branches and shouting for its breakfast.\n\nThe only breakfast the Pogles could offer the Plant was some bilberry wine. They poured it around the roots and the Plant was much revived and somewhat inebriated. It was happy to oblige Mrs Pogle's wish for dwarf beans by waving a magic leaf and causing them to appear, already planted and growing, in her garden.\n\nMr Pogle was delighted with this encounter with magic, but Mrs Pogle was dubious. Neither of them knew what troubles their involvement with it might bring. Come to think of it, nor did I.\n\nObviously I had wandered into another well-worn, well-stocked world, the world of rural fairy-tales. There would be good and evil, probably fairies and witches. The Pogles would have to have a real part to play, however inadvertently, in the drama that was to unfold.\n\nThe story, when I finally got it together, was quite straightforward. It involved a baby in a cot, which the Pogles found in the flower of the Plant, some fairies who did little more than twitter, a tiny silver crown of unknown significance and a witch, who wanted the crown and ill-treated the Pogles in a quite inexcusable way in her efforts to obtain it, tying Mrs Pogle up in a small sack and imprisoning Mr Pogle in a cage of twigs.\n\nThen, after the Pogles had escaped from her with the help of a magic flower from the Plant, they barred the door.\n\nThe witch laid siege to their house. First she appeared on the doorstep as a jug of milk, left there, apparently, by Mr Burgess the cow-man. This ruse was thwarted by a clockwork singing bird which, along with a tiny silver crown, the fairies had given the Pogles to calm the baby. As Mr Pogle opened to door to collect the jug the bird flew out, picked it up and dropped it on a stone. The witch ran screaming.\n\nIn her second attempt the witch was less subtle: she appeared as an old boot and simply kicked the door in. The Pogles still refused to tell her where the crown was hidden and she was about to turn them into toads when the clockwork bird took the silver crown from its hiding-place and dropped it on Mr Pogle's head.\n\nMr Pogle, who by this time disliked magic more than ever, now had to have a contest of spells with the witch. As he was wearing the crown of the king of the fairies, Mr Pogle naturally won it hands down. But it seemed that there was nothing he or Mrs Pogle could think of to turn the witch into, without her being able to turn herself back again into a witch. So that was what Mr Pogle turned her into; a nothing: 'no thing at all, no shadow, no thought, no terror, no memory of evil, no touch of cold to foul the night, no thing at all... be!'\n\nAnd there she was, or rather, wasn't. The bird, who was of course the king of the fairies in disguise, thanked the Pogles for their care for his infant son, congratulated Mr Pogle on his choice of destiny for the witch, greeted his old friend the Plant and commanded the fairies to stop twittering and bring them a barrel of best bilberry wine. The Plant grew instruments and they danced a merry jig.\n\nThe house-in-the-tree-root that Peter built for the Pogles in my studio was particularly realistic and handsome. It had a stout wooden door with a bell beside it which the hedgehog who called in the mornings would ring, and an upstairs window with shutters. Real ivy was growing up it and the garden was greengrocer's grass. Like all of their kind, Mr and Mrs Pogle were short and stocky and wore very large boots with hinged toe-caps. These contained lumps of lead to stop them falling over.\n\nThe Pogles were a joy to work with. For them I developed a very economical and telling way of animating conversations. They would strike postures to express and punctuate whatever they were saying and simply hold them until the next phrase.\n\nI made the films and, as always, sent prints to Talbot Television Ltd., our foreign sales agents, and this prompted Paul Talbot, president of the parent Corporation, Fremantle International Inc., to fly in and pay us a visit.\n\nPaul was a classic American captain of industry, large, bluff, warm-hearted and full of entrepreneurial drive. He and I had a long, wide-ranging discussion in which it transpired that he was interested in our capitalization. His idea was to finance Smallfilms so that we could set up a proper studio and go forward from these one-horse shoestring films and move into making broad-appeal family-viewing programmes. That, he said, was where real money could be made.\n\nI had to think about this. It was probably the best chance to become really rich that we would ever be offered. But I had reservations.\n\nOne was that Peter and I were perfectly all right as we were. We had a comfortable market for our modest films, and it was pretty clear that so long as we could find the ideas to feed it, British TV would feed us \u2013 with the foreign sales as a bonus.\n\nThe other reservation was personal. I couldn't see myself being a captain of industry, even on a small scale. As things were, Peter and I weren't even in partnership. When there was work to do on my films we would negotiate a contract for it on the back of an envelope or something. This established that we each had our own territory, so I didn't tell him how to do the drawings and he didn't tell me how to make the films. That saved a lot of potentially fraught compromises, accommodations and cumbersome policy meetings.\n\nI suppose the real virtue of the situation, as far as I was concerned, was the knowledge that if, one day, I should have no films to make, I could close the door of the studio and walk away. I owed it nothing, except the \u00a31.50 a week which I paid Peter towards the rates, etc., which I usually forgot to pay anyway. No, we had a good working set-up already in place. The job of thinking up the stories loaded my small muse with quite enough anxiety for me. I didn't want to add any more burdens to it.\n\nPaul took the point. We laughed and he bore no grudge. His company went on to sell our films diligently for many years and he and his wife Susan have been our close personal friends ever since.\n\n## _V. Pogles' Wood_.\n\nIn the middle of 1965 Doreen Stephens, who was Head of Family Programmes at the BBC, asked me to come in for a meeting about _The Pogles_. She told me the Corporation's view, which was that Mr and Mrs Pogle were very welcome at the BBC, and that friendly magic, as practised by the Plant, was perfectly acceptable, but that witches screaming in the back garden were definitely _non_ _grata_. I wasn't particularly surprised at this, in fact I rather felt the same.\n\nHowever, Doreen's good news was that the BBC would like us to offer it a set of thirteen Pogle films for the _Watch with Mother_ programme for small children and their mums, which went out at lunchtime. Her suggestion was that the Pogles could visit real places and watch rural activities actually taking place. I liked the idea very much but I doubted whether Mr and Mrs Pogle would bother to go and watch such things on their own. I thought they should have a larger family, with young persons who could watch the activities for the first time and ask questions.\n\nPeter and I thought about this. We chose a boy-person aged about six, whose name was Pippin, and a small animal-person, halfway between a squirrel and a rabbit, who was a Tog. I knew in my own mind that Pippin was actually the son of the king of the fairies, now in the care of the Pogles and, consequently, if the witch should ever find a way to cease being nothing, he could become vulnerable to witch-type magic, but I didn't mention this to the BBC. About the ancestry of Tog I had no information at all. He was a bright, sensible chap, even though his speech wasn't very intelligible.\n\nPeter and I put together some synopses and sketches of stories, including visits to the dairy, the beekeeper, the sheep-shearer, the farrier, woodcutter, road-builders, etc., and took them along to Doreen and her associate, Joy Whitby. They were very pleased, said the stories were exactly the sort of thing they had been hoping for, and commissioned the films at once, offering about double the money that they had been paying for Noggin films. That was absolutely fine with us. The only worrisome thought came from Joy Whitby's parting words, which were: 'We'd really rather like it if you would make the Pogles a bit more _quirky_.'\n\nI came away completely flummoxed by this request. I tried to imagine myself going up to Mr and Mrs Pogle like some sort of producer and saying: 'Well yes, darlings, everything's quite lovely, but, and I know it's not easy, but I do have to ask you to be a bit more _quirky_... Yes, you know... _quirky_. I'm sure you understand.' Of course they wouldn't understand. They wouldn't know what I was talking about. I know that as the writer I had some powers, but these didn't include rebuilding people who were already fully alive and in working order. Wisely perhaps I said nothing and just got on with making the films.\n\nPeter had had his big barn refurbished, so he was able to build me a really interesting piece of woodland terrain on a big table, about two metres wide and three long, in the middle of the barn, with a couple of movable leafy flats for backgrounds. This made a really versatile stage because the camera could look at it from any of four different directions. Every few days Peter and I would cut fresh-leaved branches and lay in tufts of grass, so the set provided me with an almost infinite number of different Pogle walks. The great joy was that the light was artificial and everything was at waist-height, so I never had to get down on to hands and knees or shake my fist at the sky again.\n\nThat first set of thirteen films of _Pogles' Wood_ kept me busy for over a year. They were closely followed by a second set, _Pogles'_ _Wood II_ , which were less rural in content, relying mainly on stories and happenings that could be filmed in the studio. These included a set of pipe-cleaners which came magically alive and were extraordinarily good at making folded paper animals and birds. These duly took life and walked about. There was a documentary film on the growing of umbrellas, and a tale about a queen and a carpenter in which the pieces of wood did their own carpentry. This was cast entirely from the families. Simon and Stephen, our twins, both played the carpenter, Katy Firmin played the queen, Daniel and Matthew White were soldiers and David White was a farmer. There was also a story about a foolish tiger and one about a beautiful princess who stole things. These were all delivered magically by the benevolent Plant.\n\nThe films were well received by the BBC and, being a new feature of the prestigious _Watch With Mother_ programmes, they attracted some attention from the media. Two very pleasant young people came down from one of the tabloid newspapers and spent a jolly day with us collecting material for a feature.\n\nI saw the headline next day:\n\nCALLING MRS POGLE\n\nShe's the homely little woman who hopes with her family to put Andy Pandy's nose out of joint... ANDY PANDY and Bill and Ben the Flowerpot men have new TV rivals \u2013 the Pogles!\n\nI rang the newspaper at once to confirm that it was only a joke. It wasn't. I tried to explain that we were not in any way in competition with Andy Pandy or anyone else, but the editor I spoke to didn't understand. The piece was good, straightforward, incisive journalism. Didn't we _want_ to win, didn't we _want_ to be  seen as triumphant? Didn't we even want the publicity, for goodness' sake?\n\nWell, actually, no.\n\n## _VI. Red Lion Cottage_.\n\nA day or two before Christmas in 1964 I telephoned Ray and Daisy to wish them well and find out what their plans were. I rang two or three times but received no reply. This was odd because I knew they weren't away on holiday. It was worrying too. So, in the end, I had to get into the car and drive to Finchley.\n\nThey were all right. Daisy was sitting in the square armchair in the long room with a glass of gin. Ray was upstairs in his study, sitting at his desk reading a detective novel. They were pleased to see me and each blamed the other for not answering the telephone. Daisy took the view that as the calls were always for Ray he could jolly well answer them himself, and Ray took the view that as he was busy the least Daisy could do was answer the telephone.\n\nI took the view that having to pound seventy miles up the old A2 on a nasty evening simply because the old things were too quarrelsome to answer the phone was a bit bloody much, but I didn't say so out loud. We had a pleasant evening and, very late that night, I drove back to Kent, thinking.\n\nRay and Daisy had always had their quarrels, and these had sometimes been spectacular, but they had been battles between equals and had always been repaired. This time, seeing them in their separate rooms in that great empty house, I felt that some slow change had been taking place.\n\nI could see that, as well as being physically in poor shape (she suffered a lot of pain from her fibrositis), Daisy was also deeply unhappy. It wasn't difficult to see why. Ever since the war, when she had been queen of the house, her role in life had been slowly diminishing and becoming more marginal. It's true that _The Good_ _Food Guide_ had given her quite a lot of work to do, but that had mainly been filing and clerking. Then in 1962 _The Good Food Guide_ had been taken over by the Consumers' Association and its office had moved into London. This had saved her some painful filing work but it had ended her usefulness.\n\nDaisy, who had been so confident and independent in her youth, who had taken an active and responsible part in so much political life and struggle, who had made a modern marriage with Ray, a marriage in which both would honour the other's freedom of spirit and neither would ever be a chattel of the other, now found that her life had slowly ground to a halt in a siding while Ray's life, as a celebrated expert on food and wine, had gone puffing away into clouds of fame and flattery.\n\nAs I drove home I could see that she was, quite simply, lonely. I could also see that none of this was anybody's fault. Time had just moved on.\n\nI was unhappy about their being so far away. Perhaps I could persuade them to move to somewhere closer to hand \u2013 not in the house, but within reach.\n\nIn the end Prue and I decided to offer them the outbuildings: the dilapidated garage\/barn in the corner of the garden, which they could, if they wished, and if building permission were to be forthcoming, rebuild as a sort of mews-bungalow for themselves over a store-room and garage.\n\nSlightly to my surprise, they jumped at the offer and we went ahead with the project. Ray and Daisy were very clear about what they wanted in the way of accommodation so, after a disastrous run-in with a local architect (who proposed a building which not only ignored their wishes but was also ugly), they asked me to design it for them.\n\nThis was an enormously exciting idea. I had always wanted to design a house \u2013 not necessarily an extraordinary, innovative sort of house but one for people to live in, a house that would make the best of the necessary limitations and considerations and sit comfortably in the landscape.\n\nI now think that Red Lion Cottage did just about fill the bill. The only serious design fault was that Daisy and Ray had specified a spacious open-plan living room and kitchen, with two smallish bedrooms. That was a mistake because they had become so accustomed to having their own space that they got under each other's feet. They would have been more comfortable with two large bed-sitters and a small kitchen-diner. However, this problem was eased shortly before they finally moved in by our turning the store-room under their house into a library and study for Ray.\n\nOnce Red Lion Cottage was built and furnished, Ray let it be known that he wasn't actually intending to move into it until the work he had to do was completed. However, fate took a hand here. He suffered a minor heart attack in 1965, as a result of which he was ordered to walk three miles a day. So he and Daisy took to coming down for quite long periods during which he was able to set up a good working arrangement with Fred.\n\nI should explain that Fred was our second dog, a brown mongrel bitch, and also that Ray utterly abhorred dogs. His fear and disgust of them amounted to such a phobia that he wouldn't allow a dog into the house, or stay in the same room with one. Nevertheless, Fred would appear at a certain time each day and sit at the top of Ray and Daisy's garden, gazing meaningfully at the French windows. Ray would see her and put on his boots. Then they would walk, a respectful three paces apart, up the field, through the wood and along the fire-break to the old railway line. On the way back Ray would sit on the bench at the top of the field and Fred would sit some feet away. Then they would walk down the field together and part company at the fence.\n\nIn a guarded way Ray was deeply touched by this arrangement, especially so by Fred's perfect respect for his reservations. As they walked he would talk to her on many subjects and she would wag her tail and smile, but never come within reach.\n\nWe too seemed to have hit on approximately the right degree of association between the establishments. Although our houses were adjacent, we visited by invitation or necessity. There were some slight inconveniences, like our 'shared' telephone line which enabled Daisy to cut in and read me her shopping-list while I was talking to the BBC, but in general I have very happy memories of Ray and Daisy's time here in the late 1960s. I can see them ambling across the grass on their way to take Sunday lunch with us in the garden, she carrying a board of excellent cheese and he carrying an even more excellent bottle of claret. On these occasions I think they were quite careful not to be too grandparental but at the same time they seemed perfectly at ease. Daisy would complain fulsomely, but without rancour, about Ray's behaviour, while he would sit happily and smile, or have a separate conversation, glad perhaps that she had another audience. Lunch over, we would sit in the shade with the last of the wine while the boys went chasing about in the field. They would come running in every few minutes, steaming like horses, to tell us something important, take a swig of our wine and then charge away again.\n\nMy main concern in those years was for Prue's health. She had developed chronic bronchitis, which would arrive during the winter and stay, wheezing and coughing, until the spring. It would leave her fairly low in stamina until the summer began. Not that Prue didn't lead a very active life. She was a pillar of the Canterbury Meeting of the Society of Friends (Quakers), which she and Kerris had originally joined in Finchley, and much of her spare time was taken up with events and activities connected with the Meeting. I didn't have nearly such an active social life. Being fairly fully engrossed with film-making, I seemed to have little time or headspace to spare. I did have one or two outside interests. I had always loved new visions. In fact I feel really lucky to have shared the wonder of many of the innovations that we now take for granted. At that time I was astonished by Hi-Fi stereophonic sound reproduction. I bacame an enthusiast (and a bore) and made my own loudspeakers. Also, enticed by my bank manager, I helped to found a squash club and, more or less by default, became its chairman.\n\n## _VII. Serious Considerations_.\n\nWhen he was nearly four Daniel came to me with a serious complaint. This was about the way I treated the puppets.\n\nHe said: 'You hurt Tog's nose.'\n\n'Tog's nose?'\n\n'Yes, in the film. You fell him on his nose and the end of his nose is hurtable, very hurtable, falling on it.'\n\n'But, Dan,' I said, 'you've seen Tog. He's just a puppet made of fibreglass and bits of fur. You know that.'\n\n'Yes, of course I do, but the end of his nose is hurtable. Don't fall him on it again.'\n\n'Oh, all right,' I said. 'But, Dan, did you see that Tom and Jerry film yesterday? The one where he gets sliced up with a circular saw?'\n\n'Yes!' cried Dan, his eyes lighting up. 'Wasn't that funny!'\n\nThat puzzled me. Why should slicing up Tom and Jerry with a circular saw be uproariously funny when Tog falling on his nose is a piece of grievous bodily harm? I still don't know the answer. Perhaps it is a question for psychologists.\n\nCertainly, for some, the _Pogles' Wood_ films seemed to have strong psychological overtones. Tony Gruner, our European sales agent, later sold them to a German television station. He read me a letter he had received through the station. It came from a Consortium of Educational Television Viewers which had noted that the educational approach of our _Pogles' Wood_ series did not accord with that of either of the two main schools of Educational Psychology. It wished to know the name, credentials and affiliation of the educational psychologist that we had retained for the making of the films and would we also be so kind as to let them know our interpretation of the symbolical significance of the erect Plant standing beside the grassy hole in the root of the tree?\n\nThe enquiry was both hilarious and unsettling. On the one hand my reaction was to treat it as a joke. I wrote a careful reply in kind but Tony Gruner refused to forward it on the grounds that the German market was too valuable to risk offending. He pointed out that as the Consortium was obviously Freudian, it wouldn't have a sense of humour. On the other hand I found myself wondering whether or not I really ought to have paid some attention to the education theorists. The only criteria Peter and I used when deciding whether or not a particular idea or piece of material was suitable for filming were:\n\n  1. Did we enjoy it? Was it fun?\n  2. Did we think children would enjoy it? Were we sure it wasn't nasty, frightening, of evil example?\n  3. Did we think grown-ups would like it \u2013 especially the heads of TV departments who would, we hoped, like it enough to buy it?\n\nI had never given conscious thought to sociological-type considerations like the need for positive racial discrimination, the equitable distribution of stereotypes and the elimination of potential implications of social inequality. Nor had I given thought to psychology, though I can remember hearing people who had studied such things speaking of 'strategies for capturing the cognitive', the 'importance of relevant role-model examples' and the need to 'reinforce the acceptability of established demographic trends'. In fact Peter and I had been personally reproved by a sociologist from the local university who told us we _shouldn't_ be making films like _Pogles' Wood_ , which showed life in the country as being pleasant, because in future people and their children were going to have to live in conurbations. So we _should_ be making films about how nice it is to live in towns.\n\nFortunately, in 1967, the pottiness of 'political correctness' had not yet infected our thinking. It was to be twelve years before the _Ivor the Engine_ books would be banned by Brent Council Libraries for racial discrimination (this happened, believe it or not, not because the stories mocked the Welsh, which indeed they did, relentlessly, but because in one of the books there was an Indian elephant-keeper, an absolutely delightful loving man, who spoke with an accent that might be called 'Bombay Welsh' and consequently his portrayal might have been thought to be in some way less than respectful to persons of delicate racial sensibility).\n\nI think our books received more of that sort of scrutiny than the films, probably because a book could be held in the hand, studied and condemned. This danger wasn't something Peter and I had seriously considered, the books having, as it were, grown out of the television programmes.\n\nThe book side of our work was now growing rather well. Eric Marriott, of Edmund Ward Ltd., who were already publishing the Noggin First Readers, told us that he was a great fan of the _Noggin_ sagas which, he said, had 'conceptual integrity'.\n\nOf course Peter and I were pleased and flattered by this piece of praise, but I must confess that I had reservations. I didn't want to enquire too closely into what Eric meant by 'conceptual integrity'. I felt that if I started clambering down into the wells of my imagination to try and find out which bits had 'conceptual integrity' and which bits were 'psychologically or sociologically acceptable', I would muddy them beyond recovery and probably never be able to use them again.\n\nSo I smiled and waited for Eric to come to the point, which was to tell us that he wanted to publish books of _Noggin sagas_ , at the rate of two a year, for the foreseeable future.\n\nThat answered the question of whether or not Nogbad had escaped from Roundhead Valley at the end of _Noggin and the_ _Firecake_. He was going to have to have escaped (if you will pardon the tense) because we were going to need him for the books. He was going to be busy.\n\nNogbad was going to make a steam-driven island, and a terrible Trojan pie. He was going to grow the seeds of Grundelstein's Greater Gripewort to break Noggin's castle. He would rediscover the game of _Hnefetafl_ and use it for his evil purposes, he would steal Olaf's invention and displease a sea monster, he would turn time backwards and make morning follow afternoon. He even threatened to melt the polar ice-cap and drown the world. There was no villainy too vile to serve his dread purposes. Nogbad was a great chap and I'm pleased to have known him. He was popular too. When we did book-signing sessions Peter liked to draw sketches on the fly-leaf for the customers. Most of them asked for portraits of Nogbad.\n\nPeter and I were going to be busy. It was fortunate that we didn't have any film work to do at the time, because there was book work to be done. Peter was in the middle of illustrating _Ivor's_ _Outing_ , a second _Ivor the Engine_ book, and Paul Hamlyn Ltd. had asked us to make some _Pogles' Wood_ picture books. Then, as well as continuing with our weekly contributions, the publishers of _TV_ _LAND_ asked us to do a complete _Pogles Annual_ and left it to Peter and myself to produce all the finished material.\n\nWe put together a _Pogles Annual_ for each year up to 1974 and I must say I enjoyed them enormously. They were full, rich confections but, like all Annuals, they had their year and were gone. I must say I slightly regret this. There was good stuff in them.\n\nSo, by 1967, work was going well. Films had been made and were being sold all over the world. Comics, Annuals and books were growing from them and there was even a Noggin gramophone record \u2013 what more could we hope for?\n\nOne day Peter came to me and said: 'What exactly is \"merchandising?\"'\n\n'I don't know. Isn't it something to do with selling things?'\n\nMerchandising turned out to be the selling of consumer products packaged so as to connect them with television characters. Today it is an enormous industry.\n\nStrictly speaking the exploitation of the characters which he had made was Peter's department, not mine. He also had a share in the copyright of several other puppets, including Basil Brush, whose pelt he had made. The BBC had just started its own merchandising department so we made a deal with them and looked forward to riches from the sale of _Ivor the Engine_ toothbrush holders and _Pogle_ pencil boxes.\n\nUnfortunately, for reasons I have never quite fathomed, our characters didn't really take off into the big time of merchandising.\n\nBut the ribald fox, Basil Brush, bless his furry tail, earned for Peter a shower of small cheques which continued for many years.\n\n## _VIII. Prue and Me_.\n\n1968 started badly for Prue. The year came in, as usual, with her bronchitis, and that was, as usual, dreadful. Then, in February, when it was just beginning to ease off, she was called into hospital for a long-awaited operation for her varicose veins.\n\nPrue was now forty-five and I was forty-two. We had been married for ten years, years that had passed with the speed of an express train, years in which there had been so much to do, so many mouths to feed, so many small disasters to attend to, that we really hadn't had a lot of time to take notice of each other.\n\nPrue would occasionally mention, a shade ruefully, that we seemed to have missed out a bit on 'romance', that we hadn't really had the proper honeymoon period, as prescribed in Mills and Boon literature and the women's magazines. In a sense that was true because the circumstances of our coming together had been a bramble-patch of external emergencies. However, the affection and commitment that we had found and forged in that complicated time had proved entirely durable and even without the red roses and champagne our time together was easy and full of fun.\n\nThe basic pattern of our life was slightly different from the traditional one. During our early years, when I had been working at home in the flat in Finchley, my work had had to take its chance among the imperatives of looking after new babies. I was quite able to change nappies and I could cook so, when Prue was ill, I had always taken over the running of the house as a matter of course.\n\nI was quite happy to do this when necessary but I sometimes wonder whether, at other times, I may have been a bit too capable. Having spent much of her first marriage frantically having to cope with her first husband's total detachment from practical or financial matters, Prue was happy not so much to delegate responsibility, as to abandon it. Even after we had moved to Kent and I was fully engaged in the production of films, I still didn't have to work regular hours, so it was more or less assumed that I could take time off to deal with household difficulties. I sometimes got the impression that as a result of this the whole family had come to the conclusion that the easiest way to deal with problems was to save them until I came back from the studio and then bombard me with them. So, as I came through the door I would be greeted with cries of: 'Oliver! The washing-machine has been making a funny noise,' or 'There's still a horse in the garden,' or 'Dad, why is there a pond in the dining-room?'\n\nThese were small things in themselves and, being by nature and instinct a coper, I naturally attended to them but, and I expect this is probably the lot of all copers, they piled up and I gradually found that because I had had to make the decisions, I was not only held to be responsible for everything that happened but also, by some generally held but unstated assumption, potentially to blame for whatever might be going to happen.\n\nThis feeling could be daunting and there were times when old fears would come crowding back. Then I would be reminded very forcibly that everything I had been managing to get away with during the past few years had been part of a dangerously extended confidence trick, a sandcastle based on nothing but my original arrogant presumptions and sustained by nothing but a piece of unwarranted good luck, which was that, _so far_ , I had managed to find and write some fairly acceptable stories. Then my spirit would drop like a stone and I would sit, numb, in front of my piece of blank paper.\n\nThat piece of paper was, of course, the main well-spring of my anxieties \u2013 I was committed to trawling next year's income out of my imagination or down from the sky, or from somewhere. There was no certainty that I would find it. But if I didn't, we were done for.\n\nAs I mentioned earlier, Prue had always refused to entertain the idea that I could possibly have any doubts about my abilities. She dismissed such thoughts out of hand and blandly assumed that delivery was automatic. She would say: 'Don't worry, darling, you'll think of something, but if you're passing Cawleys, could you get half a pound of ground almonds? I'm doing macaroons for the Bring and Buy.' This was the opposite of reassuring and it could leave me feeling depressed and isolated.\n\nHowever, the ship of our life sailed on. My agitations were just part of the weather and, on the whole, I was glad that this was so, because once the particular worries that had given rise to them were dealt with, the anxieties would evaporate and I would be my usual, fairly ordinary, self.\n\nLife in our house was always pretty chaotic, with everybody getting on with whatever they were doing and amiably bawling out  those in their way. There seemed to be plenty of time \u2013 time for tears to be wiped, time for laughter, time for a cuddle \u2013 in fact I only had to sit down for a moment and I would become a climbing-frame. There was room for everybody and there didn't seem to be any need to bother with indulgences, sentimentalities, special togethernesses (keeping the others out), or any of the sad luxuries of emotional competition. Of course there were arguments; egos could be bruised as easily as knees, but it all washed past, was mended and forgotten.\n\nI dare say our life with our children was far from being 'Educationally Correct', but I remember that time, when the family was young, as being a period of broad, easy happiness.\n\n## _IX. Ray and Daisy_.\n\nOn a dull day in November 1968 I was in my car on the way to Finchley, listening to a radio programme that I had recorded earlier. The programme was _Desert Island Discs_ and Ray was the subject. So he was, rather diffidently, choosing the pieces of music he would take to his desert island. The time was gruesomely apt because while I was driving up the A2 listening to him, Ray was actually undergoing an operation for cancer at St Mark's Hospital in London.\n\nThe operation was deemed a success. Ray took it all very lightly and although the doctors hadn't told him or Daisy that he had had cancer, I'm fairly sure he knew, but preferred to maintain the fiction that he had been operated on for, as he put it, 'warts on the bowel'.\n\nRay recovered very swiftly and was soon in good health. The only significant difference the operation made to his life was to persuade him finally to relinquish the house in Finchley and, having had his study and library installed in the lower part of Red Lion Cottage, move down here for good.\n\nRay took to life in the country very well and before long he became a member of Peter Firmin's Blean Footpath Group, with whom he spent every Sunday morning treading out the paths and, armed with a walking stick to which I had fixed a cutting edge, slashing away at intrusive nettles. He seemed in general to be content, busy rewriting and bringing up to date the last few years of H. G. Wells's _Outline of History_ , a task he had been engaged on sporadically since the mid-fifties.\n\nDaisy had feared that the move out of London would put an end to their social life but in practice the opposite seemed to happen. A steady stream of visitors appeared \u2013 according to Daisy, many more than they had seen at Finchley. Daisy enjoyed this but found it exhausting, so she was quite pleased when I sometimes agreed to be their commis-chef and waiter. This meant that after she had done the main part of the cooking \u2013 a task which she would not have delegated \u2013 I would come in, make the gravy, dish up the vegetables and serve the meal, while she made merry with the guests. It was a pleasure for all concerned, particularly me.\n\nSo, keeping our fingers crossed that Ray's recovery would be as complete as Daisy's had been fourteen years earlier, the families seemed ready to settle down to a tranquil future. Ray celebrated the move by negotiating a barrel of wine, which was bottled with due celebration on the terrace at Red Lion Cottage.\n\nDaisy wasn't the only 'Lansbury girl' in this corner of Kent. My Communist aunt Violet lived with her husband in a bungalow near Folkestone but, perhaps for ideological reasons, we didn't see her very often. However, we did see quite a lot of my aunt Nell. A couple of years earlier, at her request, Prue and I had obtained a small fisherman's cottage in Whitstable for her to retire to. Nell was Daisy's younger sister, one of the aunts who had lodged at 45 Hendon Lane during the war. As a child she had always been at loggerheads with Daisy and, although they had managed to get along quite well during the war, I think she may have been a bit daunted by Ray and Daisy's arrival in Blean, because their childhood antagonism seemed to reassert itself. Nevertheless she felt it her duty to call on Daisy fairly frequently and often came in to see us as well, usually to complain to Prue about the awfulness of her elder sister, who would, in her turn, pop in to tell her about how unforgivably irritating Nell had been. I think Prue got the feeling that the number of my elderly relatives in the neighbourhood had increased to a rather unwelcome extent.\n\n# COLOURING IN\n\n## _I. Outer Space_.\n\nIn 1968 the BBC issued a decree which put paid to our ambitions to make any more _Pogles' Wood_ type films. In future all bought films were to be in colour \u2013 not only 'in colour' but 'colourful'. They wanted something completely new and quite astonishing.\n\nOh, yes. Well, that was a challenge. For me it meant another piece of paper to sit and look at. Once again we were going to have to pull our living out of the sky.\n\nOut of the sky?... What about that?\n\nIn my youth \u2013 the 1930s \u2013 the conquest of the air and the development of the aeroplane had been in the forefront of progress. Now, in the 1960s, it was the conquest of outer space and rockets that held the public's imagination. So maybe we could, literally, pull our next film out of the sky.\n\nSo far we hadn't had anything to do with outer space, but science fiction had been out there for years and whole mythologies had been found, or perhaps planted, in space. I did know a bit about what it was like. I had read H. G. Wells and some of the 'sci-fi' writers and, several years earlier when they were about three years old, our twins, Simon and Stephen, had told me about the moon. Apparently there was a giant called Edward on the other side of the moon who lived on soup, hot soup. I asked how he obtained the soup and they explained that as the moon was quite full of soup, all he had to do was unscrew a volcano and suck it out through a straw. I mentioned that I thought the other side of the moon was thought to be very, very cold and they pointed out that the soup was very, very hot. That was all useful information.\n\nWhile Peter and I were ranging through our stock, looking for old ideas to recycle, I realised that I had been mistaken about our not having had anything to do with outer space. One of the Noggin 'First Readers' was called _Noggin and the Moonmouse_. In this a small spacecraft about the size of a pumpkin landed in the new horse trough. A small mouselike character wearing a duffel coat climbed out of it. Children and dogs chased it but eventually Nooka made friends with it and found out what it had come for: fuel for the craft. So Nooka and the moonmouse collected oil and vinegar and soap-flakes from the grocer's, poured them into the craft and then, borne aloft on a massive cloud of oily, vinegary soap-bubbles, the craft hurtled away into space.\n\nThat wasn't really a lot to go on, but it was worth investigating. It was unlikely that the visitor would have come from our moon because Edward lived there, but there were other moons about and there was no reason why they shouldn't contain suitable soup. Also, because nowadays there was so much space-programme debris whizzing about in the sky, they were unlikely to be living on the outside of their moon. They had probably taken to living in caves. The entrances to the caves would have metal covers over them and when they came out the covers would tip over and clang open like dustbin lids.\n\nPeter and I wondered about the name of this extraterrestrial life form. The Nogs had just called the visitor a moonmouse, but that was a description rather than a name. If we were going to make films about them they would have to have a generic name like... what?\n\nI remember the conversation quite well. We tried out names... _'Hopkins?.. Flomps?... Clangers?.. Hambrils?.. Winglooms?'_\n\n'Hang on!' I said. 'Go back.'\n\n'Go back to what?'\n\n'One of those... _Clangers._ I like _Clangers_?'\n\n'Oh? I don't know. It sounds a bit rude to me. Why _Clangers_?'\n\n'Because that's the noise the lids make as they tip over.'\n\n'Oh... Oh I see. Yes, all right, then.'\n\nSo there they were, somewhere out in space, the Clangers, a small tribe or extended family of civil mouse-like persons living their peaceful lives on, in and around a small, undistinguished moon. No doubt during the several hundred years that had passed since Noggin's time the breed had evolved a bit. They no longer had long tails with tufts on the end, perhaps because the tufts kept getting into the soup. Now they were plump and shocking pink, with noses that were long, perhaps for sucking up the soup. They had round ears which drooped when they were sad and big flat feet with holes in. Joan Firmin knitted them and she dressed the ladies, Mother Clanger, Tiny Clanger and the aunts, in patchwork coats, and Peter made suits of armour plates for the men, Major Clanger, Small Clanger and the uncles.\n\nThe chance of the Clangers' lives being quiet and peaceful was not very great. In space anything could exist and anything could happen. Even so, I knew that home life was home life wherever it was, even when it was being confused and assaulted by the unlikely. On a moon which is not protected by a blanket of atmosphere, as our Earth is, any of the various objects that float, whizz or hurtle about in space can simply drop in \u2013 and have to be entertained, dealt with or disposed of according to the Clangers' standards of hospitality.\n\nWe knew what the moon would look like and we knew that the Clangers lived in caves, but we had no idea what the inside of the caves was like. We didn't have any details about how the Clangers lived, with what furniture, what beds, what food? Here again the page was blank. Faced with it Peter said: 'I just can't see it. I can't see what it is like.' So I drew a very rough sketch of a cavern with a path and a shelf and some bed-holes with covers and a tableblock. These must have triggered off Peter's imagination because the places he eventually built, with things that could be put together from existing materials but set in incongruous situations \u2013 wheelbushes, copper-leaf trees, and small thickets of drinking-straws \u2013 far outstripped, in originality and completeness of conception, anything I could have thought of.\n\nOnce we had established the main scenario in our minds \u2013 the interstellar family in its self-contained economy, with the Soup Dragon serving green soup and blue-string pudding \u2013 the pattern of their lives became clear and stories began to materialize. A skyrocket accidentally shot down an iron chicken. The Clangers reassembled it and made it welcome but it had, by Clangers' standards, rather bad manners. However Tiny Clanger was kind to it and so, before flying away to its iron nest, the bird laid an iron egg and gave it to Tiny Clanger as a present. The egg was heard to be playing music to itself. When it rolled away it split in half, revealing that it was full of loose music notes. Most of these were eaten by the Soup Dragon, but Small and Tiny Clanger managed to save two of them and plant them outside. The notes germinated and grew into music trees, which flowered and formed more notes. The music they played was very sweet but it was as a means of levitation and propulsion that the Clangers found music most useful. It was from the music-boat, high in the sky, that Small Clanger was able to do his best magnet-fishing. There he caught the top hat which contained the Froglets, who were conjurers. The Soup Dragon did battle with an engine that manufactured plastic articles. That was very exciting as well as untidy.\n\nThen, one day, one seed from an unknown plant floated down on to the surface. The Clangers nurtured it and it grew abundantly, so abundantly that the whole moon was covered with verdure, and the sky-moos with their great wing-ears had to be summoned to eat it. And of course there was the love affair of the Gladstone Bags \u2013 that was true romance. So much was going on that we needed do no more than piece it together into episodes and film it.\n\nWhen we got the go-ahead from the BBC the first thing we had to do was place the Clangers' moon somewhere in outer space. Although, as I said, I tend to reject the accusation of being 'creative', I suppose Peter and I did come near to being the authors of a genesis when we brought the Clangers' universe into existence.\n\nI remember describing the process to a conference which Peter and I attended in Germany in 1984:\n\nIn the beginning was the void and the void was dark and without form, being \"eight by five\" sheets of battened hardboard painted midnight blue. And on the first day took we a bucket of white emulsion and big floppy brushes and threw white stars thereon, even unto the extremities thereof. And we looked upon it and saw that it was terrible.\n\nAnd on the second day we painted it out and started again...\n\nThat was rather a heavy joke but it was also a fairly accurate description of what we were doing. In the event we managed to make the firmament more convincing, or at least more fanciful, by hanging some strange heavenly bodies, mostly assembled from old Christmas tree decorations, in front of it. Peter built the moon in plaster on a plastic football, with shaped craters and silver painted buttons for the lids. It was pale blue in colour and, as the camera moved slowly across and the majestic orb came into view, floating serenely in the night sky, with its craters sharp-edged in the clear golden light, the impression was truly awesome, especially when accompanied by Vernon's evocative space-music.\n\nWhile Peter had already started building the sets I had a moment of anxiety: the BBC suddenly decided that it didn't like the scripts.\n\nI should explain that the Clangers' conversation had presented a technical problem. They spoke a language of very articulate whistling squeaks, which needed to be translated from its natural medium of nuclear-magnetic-resonance (there being no air to carry sound) into audible terms. The nearest I could get to that was to write out the script in full and then persuade Stephen Sylvester to help me record the dialogue. We did this as we had done for the Pogle films by reading it, or rather playing the inflections of it, on a selection of Swannee whistles. In this way I was hoping to make a sort of wild-life film in which, by listening carefully, the viewer would be able to understand what was being said and work out what was going on.\n\nRather nervously I asked Ursula Eason of the BBC what they objected to in the scripts. She replied: 'The bad language.'\n\nI could only think of one piece of bad language. One of the episodes begins with Major Clanger trying to open the big sliding doors of the main cave-mouth. It jams and his first line is:\n\n'Oh sod it! The bloody thing's stuck again!'\n\n'That's it,' said Ursula. 'You know quite well we can't say things like that on children's programmes.'\n\n'But...' I said, 'they don't say it. They whistle it.'\n\n'But surely people will know?'\n\n'If they have nice minds they will hear him say \"Oh dear me. The naughty thing is jammed again.\"'\n\n'Oh, all right, then, I suppose so, but please keep the language moderate.'\n\nI moderated the expletives and hoped the scripts would now be acceptable. However, the BBC still had some reservations, and these were in fact more sensible. They weren't happy about our relying solely on the whistles to tell the story. They said they didn't think young children would listen carefully. They would just turn away and ask their mums to tell them what the Clangers were saying.\n\nSo, rather reluctantly, I made a separate voice-over tape, a sort of intermittent running-commentary on what was going on. It worked quite well but I have always wondered how the films would go in their original form.\n\nI did try it once. I took an episode of _The Clangers_ to the 1984 conference in Germany and showed it to the participants without my voice-over. Afterwards I asked them whether they had been able to understand what the Clangers were saying.\n\n'But of course,' said some. 'They are speaking perfect German.'\n\n'But no,' said another. 'That is not so. They spoke only Swedish.'\n\nOnce the tensions of the internal ball and clamp joints of their skeletons were set right, the Clangers were a pleasure to animate. To support them when they were out of balance I spiked their toes into the moon rock, which had been specially made for that purpose from a mixture of plaster and polystyrene balls. Spiking in the inch-long tintacks did tend to make my thumbs sore but in general the Clangers were easy to work and their movements quite exact. The thing which really took the time was the flying.\n\nOn the Clangers' moon objects were continually landing or taking off. This wasn't too difficult to film if they were coming straight down or going straight up, you just pulled them up or let them down on a thread. That was no problem, except when the firework rockets on the human-type space module set fire to the scenery.\n\nThe difficult flyers were the ones that were supposed to come in or take off at an angle, like aeroplanes. I remember I came to the point of filming the first of these and realised that I hadn't actually worked out how to do it. This was a problem that would have to be solved because the Iron Chicken, the Sky-moos, the music boat, a flock of tin teapots and lots of other denizens of deep space were going to arrive and depart like that, not to mention the Cloud, which hovered and wandered idly about overhead.\n\nThe basic crane was not too difficult to assemble. I had a strong tripod with a 'screw-pan', which meant that, by turning a handle, the head of it could be moved around a tiny distance at a time. I bolted a long piece of wood to this and hung the music-boat on the end of it by a length of fine fishing-trace. With the tripod-head slightly tilted I found I could screw it slowly round and so bring the boat in to land in a wide arc. That was all right, but what I hadn't borne in mind was that things hanging on threads swing about, not only from side to side and back to front but also round and round. I tried various ways of steadying the motion so that I could film it in single frames, ending up with a wooden cross-piece clipped to the end of the wooden jib from which I attached up to eight different threads to various parts of the object to be flown, each of which had to be delicately adjusted and then painted with photographers' matt-black paint to render it invisible. This method of suspension succeeded in stopping the wilder swings but still allowed the object to vibrate slightly. The only way to deal with that was simply to wait. After each tiny movement I had to allow the vibration to die away before pressing the button on the camera. It was only a matter of seconds but it meant that a shot of, say, eight seconds' duration, which was two hundred frames, could take the best part of a day to film.\n\nFilming _The Clangers_ was the most challenging, absorbing and difficult work I had ever undertaken. I found it immensely satisfying, probably because it took all my concentration. By then I had, more or less, come to understand how to do single-frame filming, so that although I was working in tiny increments, moving the characters according to frame numbers marked on the script, I could mentally translate this into live motion and in my mind's eye see pretty exactly what the final film was going to look like.\n\nBy that time I had also learned a lot more about manufacturing sound effects and, using two tape-recorders and a collection of noise-makers, including bells, scrapers, whistles, crashers and a tin violin, I had found out how to build up some very creditable echoes and eerie resonances, effects that really helped to evoke the infinite vastness of space \u2013 in Peter's barn.\n\nThe _Clangers_ films were duly shown and seemed, in general, to go down well. One or two commentators hated them quite fiercely but Owen Reed, the Head of Children's Television, enjoyed them enormously and said he thought the Clangers' moon was the most complete and convincing imaginary world he had ever seen. Coming from a man the main part of whose job was looking at films of imaginary worlds, that was a real compliment. Praise came from further afield as well. An engineer from NASA is reputed to have described _The Clangers_ as 'a valiant attempt to bring a note of realism to the fantasy of the Space Programme'.\n\nI shall always remember, with a certain dread, a note of realism that turned up in one episode. In it a human astronaut arrives in a space module and attempts to collect rocks from the Clangers' moon. The Clangers try to help but he is alarmed. Many mishaps befall him. He slips into a soup-well, is rescued, runs away so fast that he goes into low orbit and has to be captured with a fishingmagnet, and is finally stuffed back into his module. There is nothing unusual about any of that. What was unusual was the fact that it was scheduled to be transmitted on the same day as one of the Apollo missions was due to land on the moon. I pleaded with the BBC to show something else, but in vain. I am still haunted by the thought that if the mission had failed and the astronauts had perished (as well they might have done), that _Clangers_ transmission would have been one of the blackest jokes in television history.\n\n## _II. Realities_.\n\nOne evening in 1970 our old Quaker friends John and Nora Starke came to dinner. At the time Prue had had a basin-full of committee meetings, school functions, housekeeping, and coping with my touchy relatives, and was feeling thoroughly fed up.\n\nJohn and Nora recognized this condition and said: 'You should be in Wales.' We asked what they meant and they told us about a crummy cottage which they leased for a tiny rent from a farmer in mid-Wales. Whenever they had had enough of civilization they would drive down there and vanish. That part of Wales was a foreign country where the hills were very large and the natives didn't speak English unless they had to. John and Nora were quite certain it would do us good to go there and they offered to lend us the cottage at any time.\n\nPrue and I took up the invitation. The cottage was crummy, primitive and damp but the air was exhilarating, clean and shining, and the isolation complete. Prue found it very peaceful. She said she was deeply envious of the Starkes' good fortune, though she added that if the cottage was hers she would do something about the pervasive damp and the unknown creatures in the roof. I reminded her that the cottage didn't really belong to the Starkes, they rented it for something like ten shillings a week from a Mr Jones.\n\n'I wish it was ours,' said Prue.\n\nOne of Prue's most endearing qualities, one which was most precious to me, was her certainty about her own feelings. She knew, fully and wholeheartedly, exactly how she felt about anything she saw or heard. I, by contrast, was always uncertain, inconsistent and plagued by doubts and reservations.\n\nThe German TV station which had bought the rights to _Pogles'_ _Wood II_ had paid us slightly more than the BBC had originally paid. So, with money burning my pockets, I spent the best part of a week with John and Nora Starke, driving along narrow lanes and hill tracks looking at cottages. We had collected sheaves of estate agents' leaflets but, as so often seems to happen, there was something unsatisfactory about all the properties on offer.\n\nThen, as we drove up one of the mountain roads, I spotted a solid-looking cottage on the side of the hill immediately below the road. It seemed to be empty so we stopped to have a look. The garden, or rather the patch of field that dropped away below the front door, was crowded with nettles and spotted with curls of rusty corrugated iron, barbed wire and broken things, including a squashed kettle and a rusty milk-churn. Peering through the dirty windows we could see that the inside was a tip. On top of a few bits of broken mouldy furniture a 1930s tin kitchen unit was lying on its side with a hole in it. There was a big fireplace into whose chimney generations of jackdaws had dropped so many twigs, bits and dry dead things that the pile had filled it to the top and spread out in a fan-shape across the rotting floorboards. In the field, buried deep among the nettles, we found a fallen estate agent's 'For Sale' board.\n\nCould this be the place we were looking for?\n\nTwo days later I was waiting at the same cottage while John fetched Prue from the station. She got out of his car a few yards down the road and as she walked up towards the cottage she threw her arms wide and shouted: 'Yes! Yes! This is it!'\n\nThen, as if for the first time, I was able to see the view \u2013 down the valley towards Ffarmers, up the valley towards where the tiny River Twrch rose in the folding hills. I saw the stately beech trees that sheltered the cottage and appreciated the way it was set, comfortably, with its back to the hill, looking down on its world of shaped fields marked out with stone walls, of bunched copses and reedy meadows where here and there the wandering river flashed as the sun caught it. Below us we saw the backs of a pair of buzzards, wheeling in the air above the meadow, their cries punctuating the distant bleating of the sheep.\n\n'The silence!' cried Prue. 'Just listen to the silence!'\n\nIt was certainly one of the noisiest silences I had ever heard. Jackdaws were clunking, cows were lowing, and somewhere not far away a woodpecker was firing its machine-gun beak, but even so I knew what Prue meant; silence, like beauty, is in the ear of the beholder.\n\nWe sat on stones against the warm front wall and ate bread and cheese and bara brith, which is Welsh currant bread, with butter. No food was ever so delicious.\n\nThen Prue looked in through the window, saw the debris and laughed.\n\n'Are you sure?' I asked.\n\n'Don't be ridiculous!' she replied, and her eyes were so full of excitement and happiness that I knew the question was ridiculous. This was her place.\n\nAfter that there was nothing to be done except buy it. We drove down to Evans the Estate Agents in Lampeter, who succeeded in finding a file for the cottage, we put down a deposit and went home, buzzing inside like bees.\n\n'It sounds quite lovely,' said Daisy, 'really beautiful, and I think your buying it is a marvellous idea. I hope you'll enjoy it as much as we all enjoyed the cottage in Essex.'\n\n'I hope you and Ray will enjoy it as well,' I said.\n\nDaisy laughed: 'Don't be daft. We shan't see it.'\n\nI had to admit to myself that this could be true, if only because the journey would be too long and bumpy for her painful frame and it was going to be many months before the cottage would be able to provide the sort of comforts that Daisy now found essential to life. It also turned out to be true for another more serious reason, a reason that put all other thoughts from our minds.\n\nRay had been suffering from headaches. These were becoming more frequent and severe and to them were added occasional bouts of giddiness and loss of balance. Other more disturbing symptoms began to appear. We, that is Daisy, Prue and I, had a strong suspicion these were recurrences of his cancer. So perhaps had Ray, but that was a closed subject. After several weeks of deterioration our GP was at last able to bring himself to admit that this might be the case, but he said nothing to Ray.\n\nRay, aware that he needed nursing, insisted on being put in a local nursing-home but, after a few days, he was admitted to the Kent and Canterbury Hospital where, in a quiet private room, he very gently went out of his mind. His discourses remained as closely reasoned and perceptive as ever but what he was perceiving was not always reality.\n\nDaisy rallied her powers. She commanded the services of Jack, a particular taxi driver working for a local firm, and he took her each morning to the hospital. There she sat with Ray, fed him, tended him and, as she later told me, they repaired their love.\n\nDaisy mentioned to me that even though Ray's faculties were failing fast, he was still alert enough to express strong reservations about being fed the pink hospital blancmange. That day Prue and I were taking Ray's sister Margaret to lunch at Giovanni's Restaurant in Whitstable. Giovanni asked kindly after Ray and I told him about the blancmange. As we left he presented us with a basket containing six pots of freshly made cre 'me caramel.\n\n'Such a great man must not have to eat blancmange,' he said. 'These he will enjoy.'\n\nGiovanni was right. When Daisy fed them to him Ray sat up and took notice. His appreciation of cre 'me caramel was unimpaired.\n\nAnother of Ray's faculties that seemed very durable was his scholarship. He had the good luck to be attended by one of the senior consultants of the hospital, who was himself a classical scholar. Daisy told me that whenever Dr John came in to examine Ray they would deal lightly with medical matters and then launch themselves into a serious discussion about the metres of Aeschylus or some similarly abstruse classical subject, about which Ray still seemed to have complete understanding.\n\nEven that understanding soon passed. His eyes became blank and, on the twenty-ninth of March 1971, Ray died.\n\n'Nobody is to be nice to me,' said Daisy as she climbed slowly out of the taxi and paid off Jack for the last time. 'Nobody is to be kind,' she ordered as she marched up the steps of Red Lion Cottage and closed the door.\n\nPrue and I understood and respected her resolution, but there were matters to attend to. After a while I went into Daisy's house through the French windows and we faced each other across the table and talked woodenly about sending notices and arranging the funeral.\n\nWhile she had been looking after Ray I think Daisy's own ailments had been in abeyance. Now they were back and I could see that she was in considerable distress, particularly from the virtually permanent condition of shingles that afflicted the lower half of her body. Nevertheless she was determined to stay in charge and deal with whatever had to be dealt with herself. She did not speak about the future.\n\nThe funeral passed and the family came back to Red Lion Cottage to drink the Tokay. This was a special bottle of Tokay Essenz (1866), a legendary wine which Ray had kept because it was reputed to revive persons near to death. It hadn't done so and consequently, as instructed, the family finished the bottle.\n\nAfter the funeral Daisy kept busy, writing and receiving letters and dealing with Ray's estate. I could see that these tasks would soon be done but she still refused to speak of the future and when I, rather tentatively, suggested that she and I might go away for a few days to a hotel in Brighton which she was fond of, she fended the suggestion off quite briskly.\n\nShe said: 'Don't be daft. I've got much too much to do. But listen, why don't you and Prue go away to your blessed cottage for a few days and let me get used to all this in my own way? You could leave the twins and Dan with John and Mary. They would be happy to have them. I shall be all right here.'\n\nPrue and I appreciated that Daisy could do with some time without us under her feet and that her suggestion was practically a command. So it was arranged and we went to Wales.\n\nPrue and I had a fairly rough few days. It was raining heavily and early April is not a good time to camp in a near-derelict cottage on a mountain, so we didn't try. We slept in a local hotel.\n\nAs we returned and I drove in through the gate of Red Lion House at about 2.30 in the morning, I saw that the light was on in Daisy's bedroom in the cottage.\n\nI walked up the steps to Daisy's front door knowing, or perhaps fearing, what I was going to find. Her tear-stained note was in the typewriter. She was in her bed, still breathing, but very heavily. For a few moments I paced up and down, not knowing what to do. Then, as the law required, I rang the doctor. After a while he turned up and immediately summoned an ambulance. I went with the ambulance and spoke to the young doctor in the hospital casualty ward.\n\nI said: 'Whatever else you do, please don't bring her back to life. She would be so angry!'\n\nHe looked at me as if I was mad.\n\nI said: 'She made her decision. Let her go.'\n\nAs if I had uttered an obscenity he turned on his heel and walked away.\n\nLate the next day a ward sister rang me and gently broke the news that Daisy had died without regaining consciousness.\n\n'I am glad to hear that,' I said.\n\n'Yes,' she replied, and her voice told me that she knew exactly what I meant.\n\n'What do I do now?' I asked.\n\n'Nothing for your mother,' said the sister. 'You could choose an undertaker tomorrow and ask them to ring the hospital. They will look after everything.'\n\n'Thanks,' I said.\n\nThe next day I got up very early and went out into the field behind the house. It was a glorious morning. The first of the sun was lighting up the dew on the young green leaves. The air was new and bright, every bird in Kent was singing its part in the dawn chorus, and I asked: 'Is it true? Did you really want no more of this?' Her answer came back into my head as clearly as if she had been standing beside me: 'Yes, yes. I wanted no more of it. I had had enough and it was time to go. That's all. It doesn't matter. Don't mind.'\n\nThen the tears came, washing through me in a flood. These were plain tears of ordinary grief, tears for the sorrow of how things are, where there is nothing to be done except weep and weep and wish it were not so.\n\n## _III. Afterwards_.\n\nRay's passing had been, on the whole, fairly well organized. He had acquiesced to fate and, in its time, fate had taken him fairly gently. There had been deep sadness at his funeral but it had all been, as I have heard Quakers say, 'in right ordering'.\n\nDaisy's gesture was of a different order. Her defiance of fate, her rejection of her own decay, her taking charge of her own destiny, were typical of her and, in their way, glorious. But the effect was like a hammer-blow. I was numb and dazed, but in this I was luckier than Prue, who was shattered. During her last few months Daisy had become, to put it mildly, a bit 'difficult'. Prue had borne the brunt of this and had kept her temper, but in herself she had been feeling hostile towards the interfering old bat. She was actually very fond of Daisy and could have done with a chance to make her peace with her, but now it was too late. We both wept, but Prue's tears were full of a pain that was near to anger.\n\nKay Starr, now infirm and in her late eighties, wrote from her nursing home: '... I wish I'd had the guts to do it years ago... So dry your eyes my darlings, salute true courage when you see it and get on with your lives while you can.'\n\nSo we did as she said, turned back to our lives. Then I came to see how important it is to allow oneself to mourn, how necessary and proper it is, in spirit if not in apparel, to wear black. I walked on through my life, doing whatever came next, comforting Prue as best I could, loving our children, but without exuberance. I found I didn't want to do anything that I would ordinarily have enjoyed because to do it without enjoying it would be to waste it, and that would have been disrespectful. I wasn't miserable about this, it was appropriate, and I was thankful that our friends were able to see that this was so.\n\nIn June 1971 Prue and I attended a delightful party given as a wake for Ray and Daisy by Margaret and Bill Lacy at their restaurant in London. This was a tremendously happy and loving occasion which they would both have enjoyed enormously. In fact, as I mingled with the guests, I kept half expecting to come across Ray smiling expansively or Daisy recounting some exciting, if difficult to follow, reminiscence. The only speech was made by their old and dear friend, Gip Wells. He gave a brief, but beautiful, evocation of their lives, and we said goodbye.\n\nIt was time for me to go back to work. I still had _Clanger_ films to finish but I wasn't exactly full of enthusiasm or ideas for the future, in fact my head was completely numb. So I confess I was relieved when I heard from the Rural District Council in Wales that, having been alerted to the existence of the cottage by my solicitor's enquiries, it had inspected the property and condemned it as unfit for human habitation. This meant that I would have to drive down to Wales immediately to try and negotiate a reprieve.\n\nI saw the council people, who were quite friendly and willing to allow us time to carry out the required improvements, which we were going to do anyway.\n\nI spent a cold night on a camp bed among the debris in the clammy cottage, listening to the sheep coughing in the lee of the cottage wall.\n\nBut the next morning was glorious. I cooked bacon and eggs over a camping-stove and ate them out of the pan. Then my legs, which had been docile and well behaved for years, demanded an outing.\n\nWe took off straight up the hill behind the cottage, over springy nibbled turf to the long ridge known as Craig Twrch, where sensible prehistoric people had once made their round houses.\n\nThere, where the morning wind blew clean through my body and the morning sun warmed the back of my neck and my hands, I could feel the bones of the living world close under my feet. My legs commanded me to sing and we bounded up along the ridge to the knot of rocks at the top. I stood on the highest rock and held open my coat. The wind took it and filled it like a sail. I leaned into it and for a moment I knew that all I had to do was hop up on my toes and I would lift off like a bird. I would ride the wind, circling out over the valley, uttering the shrill cry of the buzzard. My toes, being part of my legs and sensible, did no such thing. They knew quite well that I would just fall on my face and probably roll off down the hill like a loosened boulder.\n\nSo I just stood there, breathing in the air of Wales, looking around at the rolling hills, half-lit by the morning sun, and seeing the dark woody valleys which were slotted into their crevices, each with its own swift stream and leaping waterfalls. Below me I saw the pattern of stone-walled fields filling the low land where the river looped its way along, and I knew that Prue had chosen well.\n\nThen it was time to go. My legs switchbacked down the hill and slid me the last steep twenty feet or so down a slide of bracken to deliver me with a thump on to the lane behind the cottage.\n\nLater that year we would have a family holiday with camping gear and Kerris's boyfriend, who was a PE instructor, would help me move tons of wall stones to make a fortified terrace in front of the cottage. Later still I would find a builder, Mr Evans, who called himself an agricultural mason and mixed his cement in the middle of the sitting-room floor. He would put in doors and windows and proper stairs. After that I would build all the upstairs rooms in T&G boarding and bring a pipe from a waterfall half a mile up the hill so that Mr Evans could install a sumptuous avocado-coloured bathroom. And I would, eventually, make a hydro-electric station out of a washing-up bowl and a potato-peeler, and this would give us glorious home-made light. But all these joys were yet to come. On that day I was to drive back to Kent and really come to grips with the question of what we were going to do with the rest of our lives.\n\nThe first thing I had to do was finish filming the last of the _Clangers_ films. It was soon done. I gave them back to Peter to put in his puppet-chest, dismounted the camera and turned off the lights. We would keep everything for a while in case it was needed again.\n\nAs I packed up the _Clangers_ material I felt that another era was over. I realised that, far from overlapping the work, I had been giving no thought at all to what our next filming project might be, and also that I felt a strong disinclination to do so.\n\nHowever, in the late summer, some work came looking for us.\n\nThe Birmingham Repertory Theatre Company had just received its brand new architect-designed theatre and for its first Christmas production it wanted to put on a Noggin pantomime. This was to be a huge, no-expense-spared extravaganza with songs and music, dancers and a proper orchestra (and, they promised, no men in evening dress doing card tricks, no grand-circlers and no Apache dancers clouting each other with steel trays). A bright young man called Michael, who was to produce the show, came down to see us. He described the marvellous new theatre and told us that it could provide _absolutely_ any technical facilities \u2013 film backprojection, mixed multimedia and live action, reveals, transformations, explosions, storms, magic carpets, flying machines. We had but to ask and whatever it was it would be provided, as if by magic.\n\nThis was a very exciting prospect and it triggered my imagination to come up with a number of first-rate multimedia ideas which were promptly turned down as being too difficult. That cleared the air a bit and we were able to begin thinking, or at least I was, because I was going to have to write the play.\n\nThe play I eventually put together was called _The Rings of Nudrug_ and it was unashamedly derivative of the Noggin sagas, which was exactly what the theatre was hoping for. As well as a wicked Emir with his cringing lackey and a most evil and plaintive Nogbad, it included a complete, working flying-machine, a magic carpet, an exploding barrel and some exploding dungeon-bars, a family of damp oppressed trolls who, like the Firecake giants, finally defeat Nogbad and save the day, thus gaining their freedom. It was all good standard Nogstuff.\n\nPeter had never designed a stage set before but his friend, the distinguished theatre designer Brian Currah, came down to Blean to give him a crash course in theatre and costume design. The sets and costumes which Peter designed were magnificent, the play was cast and went into rehearsal. The technical department took the challenge and started to build the amazing devices.\n\nNorman Dannatt, the composer, an old friend of Prue's from the Watergate Revue Days, came up with some marvellous songs to go with Vernon Elliott's Nog music and all in all the _Noggin_ _Christmas Show_ was loud, rumbustious, hilarious and astonishing. The children loved it all and joined in with gusto, especially when the anti-heroes, Paul Chapman as Nogbad and Paul Henry as the Emir, being villains, cursed and swore at the audience when they were applauded, but bowed and curtsied their appreciation when they were hissed.\n\nI enjoyed it a lot, and by the time it was due to close I was ready to start thinking about our next lot of films, or I would have been if there hadn't been something more important to do.\n\n## _IV. Cotobro_.\n\n1972 came in with Kerris emigrating to Australia and Prue taking to her bed with her worst ever bout of bronchitis.\n\nAfter the grim reality of the recent past I was deeply anxious to take care of her and determined to do something to try and stave off the annual illness. A few years earlier she and I had taken a trip to Italy, a marvellous, magical trip to the Cinque Terre which could only be reached by mule or a railway tunnel that Mussolini had caused to be driven through the rocky shore. Prue had loved that trip and the warm Mediterranean air had made her bloom.\n\nI suggested that as soon as she felt well enough to travel we should leave wet England and find the sun. Prue had always had a fondness for southern Spain. So, feeling devilish affluent, I went to a high-class agency and rented a villa on the Costa del Sol for a month.\n\nPrue and I and our three boys left England on a really chilly night in early March. As I stepped out of the cramped charter flight on to Malaga airport at about one o'clock in the morning, I breathed the air and found it fresh, warm and welcoming, like an English night in June.\n\nWe collected the keys of a neat little car from a yawning man in a grey suit and, by the light of a silvery moon, set off for our forty-mile drive, eastwards along the coast road. We were the swinging holidaymakers of the seventies, heading into our dream-time of sun-drenched leisure in a rented villa on the Costa del Sol.\n\nThe drive was astonishing, especially so because, driving on the right, our car was on the seaward side of the road, overlooking the water. The road itself was not very wide, or very well kept, in fact recent rains had brought down small deltas of mud and stones, but the views, of bays and inlets, of rocky outcrops covered with pines, olive groves and tiny white houses, with every detail sharp and calm in the moonlight, were breathtakingly beautiful.\n\n'Look down there!' I would say.\n\n'Watch the bloody road!' Prue would cry, cringing away from the sheer drop beside her.\n\nOur instructions from the travel agency had been very exact. Just before Almu\u00f1ecar we turned right down a wide newly metalled road towards Cotobro. This road snaked and looped down the steep slope towards the sea, gradually becoming narrower and more gravelly until by the time it reached Cotobro, a small glittering bay with a line of houses beside the beach, it was the dry bed of a stream.\n\nThe instructions said there would be a steep lane on the left leading up to the villa. It was there all right, but it had a line of barbed wire stretched across it. I stopped and inspected the wire. It was firmly stapled in, not meant to be moved, but the air was soft and full of the scent of growing herbs, and the light was magical.\n\nIn the car the boys were asleep and Prue was exhausted. In front of me the handsome villas in their lush gardens stepped, one above the other, up the steep slope. I took the keys from Prue and set off like James Bond, climbing swiftly, hand over hand, up terraced walls, over lawns and paved patios, past sleeping chaiseslongues and cold barbecues, softly creeping up to front doors and silently trying the keys. Either the villas were empty or I was a good burglar \u2013 not a bark was heard.\n\nIn one of them the key fitted. I pushed the door, it opened. Full of excitement I ran down a side-path to the road and woke Prue. We took a bag each and I led her, yawning, up to the villa.\n\n'Here it is!' I whispered and pushed the door open. We stepped into our villa.\n\n'Bloody hell, what's this?' cried Prue.\n\n'Water, I think.' I flapped my foot in it. Yes, the floor was two inches deep in water.\n\n'Where's the light?' she asked.\n\n'No,' I said. 'Don't touch the light. You could be electrocuted.'\n\n'Oh, sod it!'\n\nThe villa was neat and clean, the beds had been made up ready for us with large Spanish candlewick bedspreads over them. These were eagerly wicking up the water into the beds. We grabbed them and pulled them off at once.\n\n'Can't you stop the water?'\n\nThere was a faint hissing noise. I traced it to a plywood box in the garden. I kicked off the lid and turned off the tap. The hissing stopped.\n\nLuckily the villa had been built a bit crooked so the beds in one of the rooms were still dry. We went down to the car and fetched the sleepy boys. Before they had time to wake up completely and enjoy the disaster we had tucked them into bed.\n\nThe water was seeping away. We found some dry bedding and got some sleep ourselves. In the morning I rang the agent's representative who sent a plumber. The plumber wept when he saw the mess and then took a big hammer and smashed the marble floor that he had laid with such care a year or two earlier.\n\nAfter a couple of days Prue and I were sitting on the terrace drinking coffee. The boys had vanished.\n\nI said: 'How's your bronchitis?'\n\nShe said: 'What bronchitis?' So that was something done.\n\nAfter a few days we hardly saw the boys at all except at mealtimes. They had teamed up with the sons of a Belgian couple who lived in a small block of apartments built in the crook of the valley. The Belgian boys spoke good English and, being resident, knew all the local amenities. Prue and I made a convenient arrangement with Guido and Le\u00b4a, their parents: at meal-times we would feed whatever crowd of boys happened to be about.\n\nThe apartments in the block where Guido and Le\u00b4a lived were very cool and spacious, with deep covered balconies opening the whole width of the living rooms. I remember sitting with them one evening, admiring the way they had been planned.\n\nI told them how Prue's bronchitis had vanished without trace and I wondered whether there were any more apartments as nice as theirs for sale and how much they would cost.\n\nLe\u00b4a said: 'You're sitting in one.'\n\nGuido said: 'Four thousand pounds in used notes.'\n\nSo we bought it.\n\n## _V. A Grievous Gift_.\n\nBy now our ownership of property was getting a bit out of hand. Our newest acquisition, the _apartamento_ in Cotobro, definitely put paid to Prue's annual bouts of bronchitis, so that was a good investment. Our friends Joyce and Alec Veitch and their family bought one of the other apartments in the block. Guido's building hadn't been too good and the electricity tended to go weak, which meant that the lights would dim. Spanish law was draconian and unpredictable and Guido turned out to be technically illegal and non-existent, as well as being something of a megalomaniac, but for several years we had marvellous family holidays there, times full of fun and easy living. We wouldn't have missed that for anything.\n\nThe cottage in Wales was also a great joy, even though it was constantly under attack from the insidious weather. My aunt Nell had decided not to live in Whitstable any more so, as a result, we still owned her now empty cottage and half of a flat in Hove which John and I had bought for her use. We still of course owned and lived in Red Lion House.\n\nAs well as all that it had gradually come home to me that Red Lion Cottage, the house that, a few years earlier, I had specifically designed for Ray and Daisy to live in, was now returned to me.\n\nThat grievous gift affected me more deeply than I had anticipated. Now that I stood in their place and it was empty, I knew that Ray and Daisy were really dead. And that with them many of the hopes that they had lived in, and the movement that they had lived for, had also passed away or become modified out of recognition.\n\nThe simple truths that George Lansbury had seen around him in the 1880s \u2013 the inherent injustice of the capitalist system, the insolence of privilege, the wickedness of exploitation, the wretchedness of poverty and the absolute wrongness of war \u2013 had long since ceased to be simple, and the clarion call to revolution that those realities had inspired had become muted. The bright vision that had lit the eyes of Ray and Daisy's friends at Bloomsbury-on-the-Marsh in the 1930s \u2013 the prospect of a just and loving social order based on the principles of true socialism \u2013 had not materialized.\n\nRay and Daisy had not been blind to these changes. They had lived through them and their grasp of political situations had always been clear and definite. I suppose, without knowing it, I had thought of Ray as being the referee of our safety in the world and had more or less left politics to him. I remember that whenever I wanted to know something about what was going on in the world I would usually start by asking him and, more often than not, I would simply accept his view. From now on, if I wanted to know what was happening, I was going to have to find out for myself.\n\nIt was too early for me to be sure what effect Daisy's absence had had on me because the echoes of the manner of her death were still too strong. That had brought home to me the simple fact that, like it or not, things are the way they are, and once you have checked them, tested them and can see for sure that there is no way they can be otherwise, then you've got to do what has to be done. It reminded me that I was now the keeper of several other people's safety and if, in my complacency, I should fail to act when action was needed, then whatever happened as a result would be, in some degree, my doing. These observations didn't come to me as a piece of rational argument, they came as a single cold shudder.\n\nRay and Daisy's death marked the end of an era in my own life. I was comfortably set in our home and family, and was running a successful and rewarding business. That could have continued peacefully for many years, and indeed it did continue for as long as fate was going to allow, but not quite so peacefully. I think some underlying assumption of stability must have been lost, because I became more aware of the world and of our situation in it and, perhaps as a result, my interests, anxieties and obsessions began to range more widely and this was going to lead me into other, less convenient, paths. But that was yet to come.\n\n## _VI. The Cat_.\n\nLife went on. Peter and I had a _Pogles' Annual_ to do and the BBC was waiting for us to complete some single-frame inserts for _Sam_ _on Boff's Island_ for its Education Department.\n\nThen the BBC let us know what we wanted next, and thus we came at last to what was perhaps the largest, and certainly the most multifarious of our productions \u2013 the saggy old cloth cat.\n\nOnce again the chief character was not born (or created) for the part. We had to look for it. The BBC had told us that it wanted a colourful, all-singing, all-dancing, musical extravaganza for _Watch_ _with Mother_ but, as usual, had no suggestions to offer as to what should be in it.\n\nAs usual, Peter and I thought we had nothing in our heads. Also as usual, this turned out not to be quite true.\n\nFor a long time, in some far-flung corner of his mind, Peter had been harbouring an Indian Army cat. Retired, of course, and definitely pukka, this cat spent a lot of time at a children's hospital somewhere in the hills of the Raj. His visits were particularly popular because he told the children extraordinary stories. These were extraordinary because he could, if he so wished, cause his thoughts to appear, as if by magic, in a bubble above his head.\n\nThis was good news. A cat like that could be very useful to us, very useful indeed. But there were problems. Even though the cat could have been a glove-puppet like Basil Brush, and his thoughts could easily have been conjured with electronic magic, there was no way we could accommodate a hospital ward full of live argumentative children in Peter's barn. We were strictly a singleframe cartoon and puppet outfit.\n\nPeter and I pushed ideas backwards and forwards for quite a long time. The cat, whose name, we concluded, was Bagpuss, was English, certainly retired, and sedentary, preferring to sleep on his own cushion. All cats, given the opportunity, will choose to sleep in a shop window, where it is warm and they can be admired without being disturbed.\n\nRight then. A cat in a shop window. What sort of shop? What else was in it? What did it sell?\n\nOnce Bagpuss was firmly in place, the answers to the other questions came fairly quickly. Peter's youngest daughter, Emily, would own both cat and shop. There were various pieces of bric-\u00e0-brac in the shop window, some of which would, in certain circumstances, come to life. Like what, for instance? We scouted around and found a rag doll that Joan Firmin had made for a Pingwing film. Her name was surely Madeleine. Peter still had a felt toad that played a banjo. His name was Gabriel. By now we knew that Bagpuss had a set of 'familiars': mice, who lived on a mouse organ.\n\nMice? Cats and mice don't mix. Yes, but these mice were 'family' \u2013 Charliemouse who wore a check suit, Lizziemouse in a navy-blue dress, Eddiemouse, who had red trousers I think, and numerous others, whose names I made up as we went along. They acted as a disrespectful chorus, doing odd jobs, singing songs and setting up deceptions.\n\nAll right. So what was a mouse organ? It was a Marvellous Mechanical device, worked by a bellows \u2013 something between a pianola and a television set.\n\nThe place was filling up well, but we still felt the need for another, more serious, character. We thought of a dark, shadowy Professor Bogwood. We weren't sure we cared for him. Nor was the BBC, mainly because he was a human-type character. Also he had no sense of humour and wasn't a bit ridiculous. But all the same I still liked the idea of a distinguished academic personage who could claim to know everything; one who was dry and thinvoiced and would go ' _Nerp, nerp, nerp_ ,' in a birdy way. I saw him as being a mixture of Professor Bertrand Russell and my uncle, G. D. H. Cole. That was it! A bookish bird, a bird bookend, a wooden woodpecker called, naturally, Professor Yaffle.\n\nGood. That was the cast. What was the plot? Or rather, since this was to be a set of thirteen self-contained fifteen-minute programmes, what would be the pattern of activity into which the stories would fit?\n\nOrdinary shops sold things. That was a bit too ordinary, and we couldn't cope with real live people coming in and out. So this shop didn't _sell_ anything. All right, but what exactly would happen in Emily's shop?\n\nThen the pattern began to emerge. Emily would bring things to the shop: broken, lost things. She would put them down in front of Bagpuss, say a poem to wake him up and leave them there. Then Bagpuss and his friends would wake up. They would look at whatever Emily had brought and decide how to mend it. They would do whatever was necessary to mend the article and then they would put it in the shop window so that whoever it belonged to could come in and collect it. That was all there was to it.\n\nFine. But what did they mend things with?\n\nWell, certainly not pins and glue. The mice would clean whatever it was quite carefully, but the real mending would have to be done with songs and stories and pictures, with anything that the magic imagination could bring to bear upon its predicament.\n\nThat seemed like a reasonable scenario. The shop itself was probably antique. In fact the whole scene would be set at the turn of the century, using some early sepia vignetted photographs as an introduction. I remember Emily Firmin wasn't all that keen about the idea of being photographed as a Victorian child and appearing, even in a set of still vignettes, at the start of each episode. She felt it would cause her embarrassment at school (which indeed it did, for a while), but she agreed to do it.\n\nNow we had to find the people to play the characters. In this I was doubly fortunate. The introduction to _Sam on Boff 's Island_ had been a very bright piece of music. It had caught my ear, as one might say, and I made enquiries about its origin. It had been played by a couple of folk-musicians, Sandra Kerr and John Faulkner. Between them they could play every sort of instrument from a mountain dulcimer to an Irish fiddle. They knew and could sing every tune in the world and didn't bother with written music, except as a last resort. They were exactly suited to be Gabriel the Toad and Madeleine the Rag Doll and in those roles were happy to play whatever music and sing whatever songs would be needed.\n\nSandra and John took a very relaxed attitude to the composing and recording of the music. Once I had worked out a few episodes I would make a very rough list of the bits where I thought music might be appropriate. I would send it to them to think about. Then we would borrow a fairly silent room in a remote house and, taking the various articles that we intended to celebrate with us, would spend a happy day with the tape-recorder, thinking up and recording whatever songs and tunes came to mind. I had had a set of variable-speed spindles made for the tape machine, so we were able to record the accompaniment to the mouse songs at normal speed and then play it back at a slow speed while we all sang for the mice at the slow speed but in high voices. This, when played back at normal speed, finally produced the elegant fluting tones for which they became famous. (Incidentally, the mouse singing out-of-tune was me.)\n\nI had to get started on the actual scripts. I put together the first story about a particularly intriguing article, a ship in a bottle. Bagpuss was extremely puzzled by this. 'A _ship_ , in a _bottle_?' he muttered. 'Who could sail it? Where would it sail to?' We discussed this predicament, remembered a song about a duck, and a poem about some mermaids and recorded it.\n\nWe all listened to it and it seemed to be OK.\n\nSo then, while Peter was making the shop window set and jointing the characters, I started trying to think of other articles which would give scope for some interesting songs and stories. I couldn't possibly just _think_ of such things. I had to find them, or rather, somebody had to find them or, better, make them. Joan Firmin, who made embroidered cushions, very kindly made one with an Athenian owl on it. She also embroidered the pictures for the story of the owls of Athens who were so proud of their beautiful singing that they demanded treasure and mocked the moon. So proud they were that the moon created nightingales to put them to shame. Joan also made the enamel pieces that were the jewels for the frog princess. She wasn't a frog to start with, just an ordinary bored princess, but when she was kissed by a frog she, naturally, turned into a frog princess, and, I believe, lived happily ever after.\n\nThe pattern worked surprisingly well. Once we had a suitable article, the ideas about it seemed to come by themselves. The Ballet Shoe was found in a cupboard. It was obviously old and worn. Also, as Professor Yaffle pointed out, _one_ ballet shoe was not a lot of use except perhaps to a one-legged ballet-dancer. Madeleine suggested that shoes could be lived in \u2013 there was an old woman who lived in a shoe. That was the cue for the mice to pump up the Mouse Organ to show pictures of the old woman who lived in a shoe, while Madeleine sang the song. Professor Yaffle pointed out that a dirty old ballet shoe was much too small for anybody to live in. The mice contested this assertion and, being magically numerous, demonstrated that nearly twenty mice could comfortably occupy it. Yaffle was not amused, insisted that there was nothing you could do with one old shoe. Bagpuss wondered what could be done with it but Charliemouse knew exactly what to do. The mice would sail it. They would row, row, row it gently down the stream. That was the cue for another traditional song.\n\nGabriel played his banjo and sang the song, though the words he sang were not the traditional ones because they concerned the adventures of two mice in their quest for Stilton cheese.\n\nHere is a snatch of it:\n\n_Tune_ : 'Row, Row, Row the Boat'\n\n_Verse_ :\n\nThey rowed and they rowed without a care   \nand didn't look where they went.\n\nWell you might ask how you would fare,  \ntravelling backwards without a care,   \nwhen you suddenly came to the top of the stair...\n\nWell over the stairs they went, they went,   \nover the stairs they went!\n\n_Mouse chorus_ :\n\nBing, bang, bongo-bump, heavily down the stairs,   \nbinglety, banglety, bongelty, bumpety.\n\nDown the apples and pears!\n\n_Verse_ :\n\nThey picked them up from their dreadful fall   \nand dusted off their knees,\n\nAnd down the hall by the kitchen wall  \nthey rowed and you could hear them call.   \nor you could if they hadn't been quite so small...\n\n'Where is that loverly cheese, that cheese,   \nwhere is that loverly cheese?'\n\nThe song, which, as Yaffle agrees, is a lot of bosh, does nothing for the condition of the ballet shoe. Madeleine reproves the mice for singing silly songs when they should be cleaning and mending the shoe. The mice show solidarity and threaten to strike if they are not allowed to sing. Negotiations take place and they agree to work so long as they can sing at the same time. They sing as they wash and polish and fit new ribbons on to the shoe, until it is clean and beautiful and fit for a prima ballerina.\n\nProfessor Yaffle unexpectedly betrays his ignorance by admitting that he doesn't know what a prima ballerina is. So the mice have to bring in a clockwork dancing ballerina to show him. They wind it up and she dances for him. The Professor is enchanted. Besotted by her beauty he dances and falls head-over-heels in love with her, and head-over-heels into the ballet shoe where he gets wound up in the new ribbons. The mice untangle him and push the shoe into the window, so that if a one-legged ballerina should happen to come past she could come in and collect it.\n\nThat all fell into place quite neatly, as did the story of the Hamish, a tartan cloth frog-like object that had lain on our mantelpiece for years. His was a sad tale of Scottish solitude told to an eerie skirling of the bagpipes, played backwards. Professor Yaffle insisted that the thing was really a porcupine that had lost its spikes and was intended as a porcupin-cushion. That reminded us of a poem that Stephen Sylvester had written for us to go in a _Pogle Annual_ about the adventures of a porcupine who was foolish enough to go up in a balloon, with a predictably unfortunate ending. Sandra and John set that to music and sang it for Madeleine and Gabriel. In the chords of that tune we found something that we had been looking for: the introductory music for the series, the opening theme that led through the antique vignettes to Emily's poem:\n\nBagpuss, dear Bagpuss, old fat furry cat-puss,   \nWake up and look at this thing that I bring.   \nWake up, be bright, be golden and light,  \nBagpuss, oh hear what I sing...\n\nAnd of course Bagpuss woke up, gave his celebrated yawn, and the story began.\n\nBut of course the essential key to unlock each episode was the article to be found. To locate these I would range about the place, picking things up and putting them down, scraping my brain for ideas about their history. This was quite hard work, but in fact far less daunting than sitting in front of a blank piece of paper with absolutely nothing in one's head. At least the things we found were tangible and their very existence stimulated speculation about their provenance.\n\nWe didn't always have to find or make real things, we could look into our memories. I remembered having once seen an ivory Chinaman that rocked. I knew that would have a story, so Peter made me one, and a turtle to go with it. A raffia elephant that had been in our toy-box for ever and had lost its ears in some longforgotten incident. In this case the most difficult task for Peter was to make a repaired replica of him for when he was mended. Peter couldn't mend the scrappy ear-less dirty one that Emily found in case I had to refilm it, so he had to make a completely shiny new one. Peter also made what is for me the _pi\u00e8ce de r\u00e9sistance_ of the series, a beautiful folding Mouse Mill, which made chocolate biscuits out of breadcrumbs and butterbeans \u2013 a procedure which Professor Yaffle had great difficulty in coming to terms with.\n\nThe smallest giant of all was an imitation Staffordshire figurine. It had to be an imitation because it starts off broken and is magically mended, and nobody in their senses would deliberately break a real Staffordshire figure, even for a film. In the event I filmed the end of the story first, and when the film came back from the labs and we were sure it was all right, Peter and I ceremonially broke the figure in order to film the beginning of the story. The piece of Old Man's Beard simply came out of a hedge. We looked at that for a long time before we realised that it was the silvery royal beard of a king, a material that would make priceless carpets and do wonders for the weaving trade. The fiddle was an ordinary Irish leprechaun's fiddle which of course has better tone than most full-size fiddles even though it is as small as a finger. That was found in a bucket. Flying was just a trick with a basket and paper fans; the mice were responsible for that, but it did bring us the story of Uncle Henry and Auntie Ada, poor church mice, who flew up to scrape some of the gold off the sun with a teaspoon. Uncle Feedle was made for the song about the tailor and his mouse, 'Hi-diddle-unkum-feedle' (or words to that effect). That story didn't please the mice, so we had to think of a different one, so Peter's daughter Charlotte made Uncle Feedle a rag doll in a cloth country with an inside-out house.\n\nThe amount of material there was in those thirteen Bagpuss films was colossal. I'm not surprised it took me several months to get half of the programmes together. I remember I was filming number five, _The Hamish_ , and I hadn't thought of anything for the last three \u2013 eleven, twelve and thirteen. Bagpuss himself did nothing to help. He was just a saggy old cloth cat, baggy and a bit loose at the seams \u2013 but Emily loved him.\n\nAnd we loved making them. Giving them life was a splendid, if somewhat hectic, episode in our lives, and whenever I see the films again I feel very happy.\n\nThe BBC liked them too and for the next thirteen years \u2013 from 1974 to 1987 \u2013 they were shown regularly twice a year and became part of the world of a whole generation of children, who love them dearly to this day.\n\nFrom a distance of the twenty-five years which have passed since we made the _Bagpuss_ films, I have begun to be able to see how they earned that affection. They are simple, and they are well founded in a safe place. They are full of fun and, like a good meal, they are rich and satisfying. They also stretch the mind and flex the imagination. And the songs and the pictures are marvellous.\n\nLooking back at that time, when I was busily making those safe gentle films in our make-believe world, it seems ironical to recall that even while I was doing it, I, personally, was feeling deeply disturbed and agitated about the safety and gentleness of the real world that I and my family were living in.\n\nThat was a very different sort of story.\n\n# LOOKING OUTWARDS\n\n## _I. Non-politics_.\n\nIn the early 1970s, along with, I suspect, quite a number of other people in the kingdom, I began to get frightened. I began to sense that something was going seriously wrong with the way our country was being run \u2013 if indeed it was being 'run' at all. I had an uneasy feeling that it was sliding slowly out of control.\n\nI had a vague and probably rather simplistic picture of what was going on: some of the trade unions had grown large enough and powerful enough to take a proper capitalist market-type view of their role and screw the bosses for all they could get. The big employers and even the government, itself a very big employer, couldn't come to terms with that situation and, to all intents and purposes, seemed to have declared war on the unions.\n\nNowadays the relationship between the trade unions and industry seems to be fairly calm. Both sides have come to see that they are partners in the process of production, but in 1972 the doctrinaire enmities between Labour and Capital were too deeply entrenched for their relationship to be seen as anything but one of direct confrontation. The 'Captains of Industry' _knew_ that the unions were led by 'Revolutionaries', whose sole purpose was to bring about, by fair means or foul, the destruction of capitalism, privilege and the very fabric of the nation. On the other hand the union leaders _knew_ that their task was to bring about the end of the exploitation of the working classes and restore justice to their situation, which they saw as being one of near-slavery.\n\nI remember Edward Heath's Conservative government, from June 1970 to March 1974, as a chronicle of disputes, deadlocks and crises. The states of emergency that were declared, the three-day week, the ninety-day wage freeze, and the mid-winter power cuts, were all symptoms of a system falling into what looked like terminal decline. I remember being surprised that society didn't collapse, surprised that there didn't come a day when nobody felt able to take the future for granted any more, a day when suddenly it would be 'every man for himself' with, perhaps, gangs of the dispossessed looting the new supermarkets, followed by a general breakdown of services, chaos and finally starvation. I don't know how near we came to that but in the dark times when it was cold and the electricity was off, the sense of impending disaster grew very strong. I dare say we came nowhere near that point but when I was in Germany in 1947 I had seen the _Kohlenklau_ , seen the respectable people of Braunschweig scrabbling over the coalwagons to fill their posh rucksacks with nutty slack, and I knew that civilized life _could_ come apart at the seams.\n\nAs well as that sense of dread, what I felt most strongly in 1973 was a feeling of hopeless exasperation. Listening to the wrangles, the obviously vain assertions of authority, the obdurate declarations of solidarity and the emollient reassurances of politicians \u2013 which were perhaps the most frightening \u2013 I had the feeling that we, the public, were being treated as if we were stupid. I found myself becoming more and more dejected by the sheer irrelevance of what was going on in the House of Commons. It seemed to me that, at a time when the country most needed real government, Parliament had simply been hi-jacked by two parties of doctrinaire zealots, Labour and Conservative, whose sole interest was to defeat each other at any cost. That cost was already being borne by the country as a whole, in terms of lost markets, flat investment, rising unemployment and rampant inflation. Something had to be done about it, and quickly \u2013 but I had no clear idea what.\n\nLife went on as usual, of course. Just thinking that something had to be done didn't make any difference to that, neither did talking about it, which I did rather a lot.\n\nThere followed an unusually muddled episode in my life, one in which I was alternately chilled with embarrassment at my own vehemence and at other times filled with a crusading zeal for some end which I couldn't really see clearly, let alone articulate. And underneath it all simmered an incongruous anger with party politics in general and party-political Members of Parliament in particular.\n\nPrue disliked thinking about politics but she even more disliked my getting involved with it.\n\n'It _irks_ me,' she said, 'you going about haranguing people!'\n\n'Why?'\n\n'Because it's not _you_. It's not the sort of thing you do. You're not even a member of a Party, so nobody knows which side you're on.'\n\n'You know quite well I'm not on either side.'\n\n'There! You see what I mean!'\n\n'Look, you know exactly what I think, and you know I've got to do something about it.'\n\n'Oh all right, I suppose, but it you're going to play Winston Churchill and save the nation, why don't you just get on with it and stop talking about it all the time?'\n\nPrue had a point there. I was being a bore.\n\nSo I wrote lots of letters to newspapers, some of which were published. Then, as the first General Election of 1974 was coming up, I wrote to Wyn Knowles, the editor of the radio programme _Woman's Hour_ , and asked her whether I might be permitted to make a Non-party Political Broadcast on behalf of the Poor Bloody Electors (or words to that effect). Somewhat to my surprise she invited me to come in and record it.\n\nThat called my bluff. But, 'in for a penny, in for a pound', I overruled embarrassment and put together a four-minute speech. In it I made an impassioned, if somewhat confused, appeal to the Members of Parliament who were about to take office to set aside the squabbles of party politics and do the job which they were being paid to do \u2013 work together to seek agreement on the best way to save the country from committing industrial suicide. The speech was well received and repeated.\n\nAfter it had washed away and the election was over I felt calmer in spirit. I had actually done something, though nobody would ever know what effect, if any, it had had.\n\nNevertheless, the more I thought about it, the more clearly I saw that although political groupings were a natural phenomenon of the democratic process, they were not essential to it, and if their wrangling reached the point where it took over Parliament, so that its attention to its proper function, _the exercise of government_ , was diverted and replaced with _an inter-party power struggle_ , they became a betrayal of that function, a corruption that could destroy the credibility of the institution itself, bring government into disrepute and eventually make it impossible.\n\nAs the second General Election of 1974 was drawing near, I persuaded Monica Sims, the Head of Children's Programmes at BBC Television, to let me make a very short _Clangers_ film which would expose some of the absurdities of political electioneering.\n\nI made _Vote for Froglet_ in record time \u2013 three days. I delivered it and it was shown, I believe, twice. It was all right, a pleasant little Morality Play, but not something that would bring the world to its senses.\n\nThen, on September 7th 1974, the _Guardian_ published an article by Professor S. E. Finer called 'In Defence of Deadlock'. When I read it I cheered, because here was the Gladstone Professor of Government and Public Administration, a fellow of All Souls, Oxford, saying, albeit in a more erudite and better-informed way, exactly the same sort of things as I had been thinking and muttering about! I can't tell you what a power of good that article did me. After reading it everything fell into place and I could at last see which side I was going to have to take in the coming election.\n\nThe burden of Professor Finer's argument was that the only way out of the aggressive stalemate in which we were stuck was going to be the election of a hung parliament, one in which neither of the massive doctrinaire parties had an overall majority. Consequently they would have no option but to compromise and negotiate a consensus. This would enable MPs to do their job, i. e., to get together and work for the common good. The key to this change would be to elect as many members as possible of the third party, the Liberals.\n\nPolling day was drawing near. I broke with my family's socialist tradition and threw in my lot with the Liberal Party. I wrote and distributed a leaflet for the local Liberal candidate. I'm not sure she liked, or even understood, the message of it, which was that everybody should vote for the Liberal Party not because they supported its policies but because that was the only way to escape from the destructive wrestling of the two doctrinaire dinosaur parties \u2013 and bring back proper democracy.\n\nThe General Election came and went. Labour was returned again, this time with a tiny majority. The local Liberal candidate didn't get in.\n\nFor my own part, I was disappointed but not surprised.\n\nThe economy ground on. By the end of 1974 the spiral of prices and wage increases had pushed the cost of living up by 20 per cent, but civilization didn't come apart. The big unions went on using their muscle to get their way and so consolidated the climate of antagonism in which the Conservative Party would be moved to appoint a lady gladiator as its leader.\n\nI came away with a firm conviction that government was too important and delicate a process to be left to politicians, especially those filled with a zeal to pursue doctrinaire ends. I could see that the only hope for civilization lay with the moderate, sensible people who, while longing for a quiet life, needed to realise that moderation and common sense must be a passion as real as any fanaticism, as it was the only light that would guide us through the dark rockery of economic decay.\n\n## _II. Sun Worship_.\n\nWhen I look back at my life in the early 1970s I am astonished at the quantity and diversity of the projects and follies that I was engaged in. Where did I find the time?\n\nThe main project was, of course, work. During 1973 and 1974, while I was deep in non-politics, I wrote and made thirteen fifteenminute _Bagpuss_ films, wrote two _Pogle Annuals_ , five _Bagpuss_ books, two _Bagpuss Annuals_ , two _Noggin_ books and a stream of contributions to _TV LAND_ weekly.\n\nAs if that wasn't enough, I had conceived and designed a handsome new studio which I proposed to build, myself, in the field beside Red Lion House. This was to have solar heating, a method which, in the wake of the great leap in the price of crude oil, was very much in the public interest.\n\nBuilding the studio was a fairly long-term ambition, but during the preparations for it an incident occurred which was to take my attention and propel it like a rocket in a completely new direction.\n\nI had thought out a very simple solar-heating system for the proposed studio, I made a small-scale model of a unit with a thermometer-recorder in it to measure the temperature of the air passing through, and propped it up in the corner of the garden, just to see if it would get warm. It did. It melted. I found it lying in the grass, a treacle-pie of congealed polystyrene and warped perspex. Although I was later to discover that this was not such a significant phenomenon as it had seemed, it was quite spectacular and I was tremendously excited. I was already persuaded that solar heating could be a really important and useful form of renewable energy, one which didn't use up fossil fuel, issue noxious hot vapours or breed quantities of eternally poisonous material, and it now seemed that I might have stumbled on a collecting system that could bring solar energy into service at a particularly reasonable cost.\n\nBut let me begin at the beginning. On a fairly dull afternoon in February I was sitting in a greenhouse. The greenhouse faced south and had been built as a lean-to against a solid brick wall. The temperature outside was cold, about seven degrees Celsius. The temperature inside the greenhouse was very pleasant, about sixteen degrees. In fact it had been much warmer in the greenhouse earlier on in the day, when the sun came out for about an hour. Then it had reached about twenty-five degrees, but that had dropped when the sun went in and now most of the warmth was radiating from the brick wall at the back, which had been heated by the sunlight.\n\nSitting there I felt very comfortable, especially so because there was nothing artificial about the warm atmosphere. It occurred to me that a greenhouse would be a very pleasant place to live in... well, perhaps, providing certain obvious disadvantages could be overcome, like it wouldn't be advisable to throw stones, and there would be a certain lack of privacy. It would also become deucedly cold at night and, in sunlight, boiling hot.\n\nHere was material for a really enjoyable think. Unlike nonpolitics, this subject did not involve people and prejudices, doctrines and destinies, only some generally benign facts. Just thinking about the sun's warmth warmed the spirit and I happily settled down to conjure different propositions from my imagination.\n\nIt was quite obvious to me that people wouldn't want to live like cucumbers, under glass, so I wondered about having a very thin greenhouse as part of the roof. How deep did a greenhouse need to be in order to collect solar energy? I imagined a fairly flat box about 4in (10cm) deep with a glass top to let the sun shine in. It would be well insulated and inside it there would be a plate of some dark lightweight material that would heat up in the sun and give up its heat to air which was being passed through the box.\n\nThe warmed air would then be blown into the house by a ducted fan or, if the house was already warm enough, it would be blown through a heat-storage block like, for instance, a deep layer of pebbles under the floor. This would soak up the heat and slowly release it to the house during cooler times.\n\nAfter some experimenting I chose concertina-pleated aluminium foil as the collecting material. This was so thin that it would heat up almost instantly when there was any solar radiation coming in, even from a cloudy sky \u2013 so full sunlight was not essential. I also found that it heated fresh air almost as well as recycled warm air. This meant that the house could have a plentiful supply of fresh air \u2013 a great blessing because so many state-of-the-art, energyefficient, optimally insulated houses smelt of cooked cabbage.\n\nIn summer the system could provide hot water and also contribute to cooling the house. In full sunshine the air in the roof collectors would become very hot. It would pass over a water tank to heat the water in it and then be vented to the sky. This would remove much of the 'solar gain' received by the roof (a major cause of overheating), and also draw 'used' air out of the house, allowing fresh air to flow into it from outside, thus helping to keep it cool.\n\nEven though by the end of 1974 my plan to build a new studio was defunct, I still wanted to go ahead with developing the system.\n\nSolar heating had in fact been about for many years but mainly in places where there was plenty of sunshine and, because in such places there wasn't usually a need for house-heating, most of the systems were small and designed only to provide hot water. These worked really well in sunny climates and, perhaps as a consequence of this, it was generally assumed that 'water-transfer' systems were the best and therefore, in all circumstances, the most desirable.\n\nThe market responded to the current public interest in solar energy with a rash of new companies selling these standard waterheating units for installation in the UK. Unfortunately there is simply not enough bright sunshine in the UK to provide a worthwhile amount of really hot water except on particularly sunny days, so they turned out not to be a lot of use.\n\nIn the system I was proposing, any air temperature in the collector which was above room temperature \u2013 about twenty degrees Celsius \u2013 would make a welcome contribution. Even with cloudy skies that temperature would be reached quite often in the house-heating season and this meant that the air-transfer system would go on working for longer into the winter and start again earlier in the spring. It also had the extra advantage of warming up very swiftly, which would allow it to collect the heat from short bursts of sunshine. This was something the heavier water-filled units couldn't do. On top of that my system had no pipes and radiators, header-tanks, pumps, plumbing or water-plates, but only some pleated foil, a fan and a duct. This meant that the mechanical parts would cost less than one-third of the price of a water-filled system and the collector area could be the whole or a large part of the roof. This was very encouraging.\n\nEventually the moment of truth arrived. I took out patents and approached some of the big house-building companies. They had all heard of solar heating but the myth that water-transfer systems were inherently the best had become such an article of faith that they wouldn't even think about an air-transfer system.\n\nHowever, I didn't give up. Although at that particular time a new lot of film work had turned up, I decided to go on with my research, and did so, in a quiet way, for several years.\n\nLooking back over that period, during which I did succeed in making contact with some large companies, I can see that although solar heating was felt to be fashionable and virtuous, the final deciding factor was always going to be whether or not it would make money.\n\nNevertheless it was an interesting and enjoyable project. I made enough money from option licences and consultancy fees to build a half-size solar house in the field beside Red Lion House. This was made of wood and insulated with rock-wool but its thermal characteristics were much the same as those of an ordinary house. It had a large pebble-block under the floor to store the heat and a whole roof of aluminium-foil type collectors.\n\nI received gifts of technical equipment and a great deal of much appreciated encouragement from various academics, including Dr Strange at the University of Kent and Professor Pat O'Sullivan at the University of Wales. I think they both quite enjoyed my rather ham-fisted approach to scientific research, as shown by my tendency to bypass all but the most essential mathematics and, having thought my way to a picture of how something might work, just try it out. I think they may have seen this as being a different style of philosophy rather than what it really was \u2013 sheer necessity.\n\nThe house worked very well indeed, keeping itself warm from early spring to late autumn and delivering hot water on request all through the summer.\n\nUnfortunately this radiant demonstration of the efficiency of the system was not sufficient to convince the sponsor company that it would be profitable. The deciding factor, they told me, was that the artefact did not have a large enough 'maintenance component'. When asked what this was they explained that, to be worth marketing, a system needed to require regular maintenance, which the purchaser would regularly have to pay for.\n\nIn essence, what I was offering was a few boxes and a fan which, once installed, would probably work indefinitely. However wondrously efficient it might be, such a device could not command a high initial purchase price nor would it require much future attention. Thus its long-term profit potential was low. Also, as a high-profile company at the leading edge of technical innovation in the field of energy conservation, they had their reputation to consider and, however well it might perform, they really didn't think it would be appropriate for them to offer anything quite so primitive.\n\nThat was disappointing but, in a grisly way, understandable.\n\nFor a while I was discouraged but then, quite unexpectedly, I was approached by a big company that produced, among other things, canisters of household gas. Their research department had been impressed by the figures in one of my 'presentations', and they offered me a retainer of _\u00a3_ 2,000 a year to consult with them about their solar projects and help to develop new ones, using, where appropriate, my solar house as a test bed. I was very delighted about this and tried to join in with energy and gusto.\n\nQuite soon I began to see that energy and gusto were inappropriate and that my attempts to carry the project forward were misplaced. They seemed slightly embarrassed by my eagerness to help and had some difficulty in finding work for me to do. After a while the penny dropped and I realised that they were really only interested in retaining the idea and reserving the patent in case they might want it later. Once I had grasped this life became much easier. I no longer felt I had to devote a lot of my time to earning the consultancy fee and was able to start on the essay that I had been intending to write for some years, a rather-less-than-learned paper outlining the general principles of the system I had designed.\n\nIn April 1979 I was slightly alarmed, as well as honoured, to be asked to deliver this paper at the International Solar Energy Society's conference at the Royal Institution. I became slightly less nervous when Julian Keable, who had instigated the invitation, told me he was hoping my lecture would serve to lighten the occasion. I think it may have done that, a bit. Certainly there were some good laughs.\n\nThat lecture was my swan-song for the solar project. The consultancy continued in a rather half-hearted way until 1981 and then, at a time when public interest in solar energy had almost completely evaporated, it was quietly dropped.\n\nI was content to let it go. The ideas had been practical. I had made a solar house which had kept itself warm from early February to late November 1977. The concept \u2013 of using sunlight instead of burning up the fuel \u2013 was valid and, in real terms, globally important. It still is, but for now, in a world ruled by convenience and profit, it can only hope for praise, not action. Maybe one day, in the face of some yet-to-be-recognized imperatives, somebody will see the necessity and simply invent it over again \u2013 goodness knows it's simple enough!\n\nI hope they will, and good luck to them.\n\n# LOOKING INWARDS\n\n##  _I. Prue_.\n\nPrue was not irked by my enthusiasm for solar energy. It's true she was a bit put out when, at the beginning of 1974, I made off with our only hot-water bottle and cut it up to make a measured-flow bellows to pump air through a plastic box in the garden, but the hot air duct in the solar house was marvellous for drying things and on the whole she felt it was a change for the better after my entanglement with non-politics. At least I wasn't getting into a lather and making an exhibition of myself.\n\nThat is not to say that Prue was particularly interested in what I was doing. She wasn't, but that was fine with me because she had plenty of interests and purposes of her own. She was an Elder and Overseer of the Society of Friends (Quakers). She also helped to run the International Club, the Cyrenians, the Council for the Advancement of State Education, the Local Government Liberal Association and the house, as well as going to art, yoga and Spanish classes.\n\nThe house was rather empty. My stepchildren had flown. Kevan Myers, the eldest, was now in India, walking the hippie trail, Kerris was still in Australia, where she was to become a television producer, and Krispian was at university in Norwich reading American Studies.\n\nIn 1972, when they were thirteen years old, we had sent our twins, Simon and Stephen, to a Quaker boarding school. We had done this because they had ceased to be children and part of a pack, and had become leggy boys with quite distinct personalities, and we felt it was time for them to lead individual lives. The Quaker Headmaster had given us a solemn undertaking that Simon and Stephen would be recognized and treated respectfully as separate individuals, not indistinguishable identical twins as had so often happened in the past. So, with their approval, Prue and I decided to let them go there.\n\nNow they had suddenly become ardently sartorial. With their hair bobbed, they wore shirts with very long pointed collars, tight jackets, strangely flared trousers and massive platform shoes that looked like leather buns. To our eyes the effect was more comical than elegant but, for the twins, conforming to the dernier cri of fashion was, apparently, utterly crucial to their social standing. We found it all a bit incongruous to the Quaker principle of simplicity and said so. The school found our comments meaningful, but did no more than join us in deploring the trend. It also found meaningful the fact that no serious attempts to distinguish between Simon and Stephen seemed to have been made. ' _Mea culpa_ ,' sighed the Principal, smiling indulgently.\n\nSo all we had left of the tribe was Daniel, who was still at the local school contemplating the eleven-plus examinations, a black Labrador named Maud and a visiting cat called Woodie. Even so the house seemed to be quite busy, with people coming and going and things to do and plan all the time. Over the years we had coopted part-time members into the family. Apart from Fiona, who had been a member for years, Prue and I were always happy to see Dominique, who was French and had once been Kevan's girlfriend. From Australia, Kerris sent us a pair of star-crossed lovers who were fleeing from the oppressive conventionality of suburban Sydney.\n\nOur lives just moved along, chaotically but contentedly, on quite a broad front.\n\nAnd like that, I assumed, it would continue. And perhaps it would have done if Prue hadn't come to me one morning and said: 'None of it is any good.'\n\n'None of what?' I asked.\n\n'None of any of it. Nothing is any good.'\n\nI looked at her eyes. They were different from usual but I couldn't say in what way. They looked heavy and dark, like dates.\n\n'What has happened?' I asked.\n\n'Nothing has happened. Everything is changed.'\n\n'Changed?' I asked\n\n'Yes... no. Don't question me. I don't know. Something has fallen on me. On my head and neck, it's heavy and everything is going wrong and it's all my fault.'\n\n'What fell on you?... Was it some sort of worry?'\n\nPrue took a long time to answer that. Then she said: 'That and blame and uselessness. I am useless. It has all been useless.'\n\n'That's ridiculous.'\n\n'I don't know what to do. Tell me what to do.'\n\n'Well, you could put out the things for breakfast,' I said, rather crossly.\n\nVery carefully, thinking very hard, Prue put out the mugs and plates, the marmalade and muesli.\n\n'Is that all right?' she asked.\n\n'I expect so,' I replied, looking the other way.\n\n'No! Say!' she snapped.\n\n'Yes, it's fine,' I said. But the penny had begun to drop. 'That's absolutely OK. Could you bring me an apple and an orange so I can cut them up?'\n\nShe brought them at once. 'Could I make the tea?' she asked.\n\n'Yes. That's a good idea.'\n\nShe made the tea and we sat together to eat our ordinary breakfast.\n\n'What are you doing today?' I asked.\n\n'Oh God...' she muttered, and crouched down.\n\n'Let's make a list,' I suggested.\n\nWe made a list and looked at it. We discussed it and agreed that everything seemed possible.\n\n'Don't try to remember it all. Take the list and do the things one at a time.'\n\nWithout another word she left the table and went away. I sat there, forcing down my anger. Was it anger or violent impatience? Whatever it was I knew it was completely out of place, but even so it was scudding around inside me. I think it was probably fear. I didn't know what was going on. I didn't understand. This was not Prue.\n\nThen I heard the Mini start up and I suddenly wondered whether she ought to be driving. Too late, she was out of the gate and heading into Whitstable.\n\nWhen she returned several hours later Prue was still in darkness but slightly proud because she had done everything on the list and, better still, she had been cheerful at people.\n\n'Nobody noticed,' she said. 'I can do things, but you must tell me.'\n\n'OK,' I said.\n\nSo that became the pattern we followed.\n\nLater, because Prue had refused to do so, I went to see our doctor, Dennis, about her condition.\n\n'Depression,' he said. 'Probably endogenous.'\n\n'So what do we do about it?'\n\n'Just what you are doing \u2013 live through it.'\n\n'Is that all?'\n\n'Get her to come and see me.'\n\nPrue still wouldn't agree to see the doctor, so we pottered on until, quite soon, it was time to go to Wales. There she found life much easier to cope with and we all had quite a peaceful holiday. I felt rather nervous about what her state of mind would be after we got back to Red Lion House, but I needn't have worried. By some miracle all her sadness and doubt had blown away and she sang continuously under her breath. She upbraided me roundly for being a sluggard and a kill-joy and immediately began to repaint the bedroom, starting in the bottom left-hand corner.\n\n'Fetch me more paint,' she called.\n\n'What colour?'\n\n'This colour, you fool.'\n\n'What sort of paint? Oil or emulsion? Undercoat or gloss?'\n\n'Oh, paint! Just paint! Fetch it!'\n\nOn the way to the paint shop I called into the surgery. Dennis was there. I told him what was happening.\n\n'Oh, she's just manic,' he said. 'Endogenous _manic_ -depression. It's quite common. The mood tends to go down fairly slowly over a period of weeks and then suddenly it shoots up into mania.'\n\n'What do I do?'\n\n'Same as before. Live through it. Ask her to come and see me.'\n\n'I'll ask her.'\n\nPrue had no time to waste on seeing doctors. She had lots of more important things to do. She asked me what the hell I was doing shopping when I should be working on films. I reminded her that I hadn't any films to make at the moment and with great difficulty managed to stop her ringing the BBC in order to tell them to commission more films at once. Then I moved the furniture out of the bedroom so that she could get at the walls with her paintbrush.\n\nAfter a few days the ebullience seemed to wane. I was relieved to see the wild flights of fancy being forgotten but I dreaded the thought of the depression coming back. In fact it didn't come back, or rather it did so slowly enough for us to adjust our way of life to it. We did this mainly by sharing jobs \u2013 so that when there was cooking to do one of us would cut up and the other cook. This was a very companionable activity, which I enjoyed a lot. It brought us close together and while we were doing it she was sometimes able to laugh. It didn't drive away the depression but it meant she could take part in life, and that would quieten her spirit.\n\nPrue's confidence was at its lowest in January 1975, when she was due to fly to India. There she had arranged to stay, or rather walk, with Kevan for some weeks and then fly on to Western Australia to visit Kerris. This project, which she had earlier anticipated with great eagerness, had now become an impossible task, one so full of inevitable disasters that she couldn't even think of attempting it. I remember telling her that she was going to India even if I had to bring the Jumbo jet round to the front door, because I was sure that once she reached India she would be all right.\n\nAnd so it was. The next day I received a telegram:\n\n'ARRIVED SAFELY SANE LOVE PRUE'\n\nPrue's time in India was glorious. I read about it when the letters began to arrive but in my bones I had felt it happening from the moment her telegram came. The marvels and pities of India touched her and greeted her with joy and as she rode in an ancient bus painted with brightly coloured gods and loaded with laughing people and their livestock, her heart sang.\n\nBy 1975 the hippie caste was well established in India and Kevan knew the rules and social observances \u2013 which side of the fire to sit and how to greet the Swami. He also knew the walks and was able to take Prue to see palaces with walls of air and palaces carved from the solid rock, and show her decorated elephants. They played Frisbee with a holy man and met a beggar who, being a holy beggar, gave them small bread rolls and honey, and recited the poem 'Twinkle twinkle little star' to them.\n\nWhen they walked in the hills together, word had gone before and people came from their houses to smile at her, saying: 'Look! Here is the mother. Look! Here is the one who walks with his mother.' Perhaps this was notable because in those days most of the hippies were young, and older persons, especially parents, were rare. Prue, at fifty-three, was definitely old by Indian standards but she walked and laughed like a girl. It was a great wonder, and she loved it.\n\n*\n\nAs I sensed Prue's happiness in India, a cloud was lifted from me and I became full of life. Daniel and I bought ourselves a colour TV and organized our time very agreeably.\n\nThat was all very well but in 1975, nearly a year after finishing the _Bagpuss_ films, I still hadn't thought of any new ones. It was time to pay attention to earning our living.\n\nI mentally summoned my creative Muse and had a few strong words with her, telling her in no uncertain terms to come up with a brand-new film series of astonishing novelty and genius.\n\nHer reply was equally forthright: 'I have made you something like twelve complete worlds, all of them real and fully conceived down to the last nut and bolt, truck, shovel, sword, helmet and flipper. Barring miracles, mate, that is your lot.'\n\nMonica Sims, the Head of Children's Television at the BBC, chose that moment to ring me and enquire lightly about the next film series we were going to offer her. I went to see her, rather shamefaced, to confess that as far as I could see there wasn't one. Slightly to my surprise she seemed almost pleased to hear it. She told me that what the BBC wanted, and had wanted for years, was _Ivor the Engine_. She didn't just want brand-new films. If I could make them again from scratch in colour, she would be delighted to have all the old black and white films, as well as any new ones I could think of.\n\nThis was an absolutely enormous relief \u2013 almost like a reprieve. The only problem was that I would have to locate and buy back the rights to the original _Ivor the Engine_ films.\n\nI managed to track down a firm called Rediffusion Holdings Ltd. which, I discovered, held these rights. They were delighted to hear that Ivor might one day be resurrected and insisted on giving me back all the rights, at once, for nothing.\n\nI was very moved by this piece of unexpected generosity and, full of enthusiasm, set to and rewrote the films. Once again I asked Olwen Griffiths to join me in reading the voices. She introduced me to Tony Jackson for the other male voices and the three of us settled down to record the voice-tracks for the first batch of twenty films.\n\nTony is an absolutely first-rate actor, with a wide command of voices. He had an immediate rapport with the inhabitants of Llaniog and brought them to life most accurately and hilariously. We spent some very happy days at the recording studios and I came away delighted with the results.\n\nSo, with the next couple of years' filming lined up and ready to go, I went to collect Prue from the airport.\n\n## _II. Time Off_ _._\n\nDo you remember the summer of 1976? It was hot. It was so hot and dry that the top leaves of the trees shrivelled and the high branches began to droop.\n\nOn the afternoon of the first of September I was alone, driving back from Wales in my new 2CV motor car. In truth I was far from alone; several thousand other people were with me, sitting in their boiling vehicles on the arterial road at Chiswick, on the way into London. The whole traffic jam shimmered in the heat as the merciless sun beat down on it. This was not the heat of England, this was of the Sahara. There was no breath of wind. The air we had to breathe was so hot and dry and overused that I had really begun to wonder whether it was still serviceable. Certainly on that stretch of traffic jam this was a serious question because, for some reason, people were keeping their car engines running.\n\nAs I sat there sweating I dreamed of walking to a gate in a high brick wall, opening it and finding myself in some magic garden of deep shade, where the air was cool and the only sound to be heard would be birdsong and the gentle rippling of water.\n\nSo, not wasting any more time on idle conjecture, I drove my car up on to the grass verge, got out, locked it and walked over to an open gate in the high wall beside the road.\n\nI went through the gate into a garden of deep shade, where the air was cool under a grove of immense trees overhanging a dark, formal pool on which birds were singing, or rather quacking. This garden could not be magic but it was, without doubt, mythical, for, on a grassy mound, amid the dark greenery surrounding a tiny perfect Greek temple, I caught glimpses of nymphs and satyrs, posing and capering on their plinths. That they were made of stone did not detract from their beauty any more than the sign beside me, warning that my dogs must be kept under control, did anything to diminish my delight at this extraordinary escape from reality.\n\nI walked across to a gravel path and saw, down an avenue of statues, a perfect Palladian villa, where a man in a peaked cap was slowly issuing tickets to people sitting in deck chairs. To the right of the villa I saw a long ornamental lake, like a reach of some great imaginary river, spreading across the green. In fact it _was_ the reach of some great river because these were the gardens of Chiswick House, where the ornamental lake is fed with water through a conduit from the River Thames. And the water was indeed spreading coolly across the green because on that day the Thames' tide was exceptionally high and its water was flooding back into the lake.\n\nThe gravel on which I was standing was scorching hot, blazing the sun's heat back up at me like a fire. My shirt was sticking to my back and through its thatch of hair the top of my head was beginning to cook. There was only one thing to do, only one place to be.\n\nI took off my shoes and socks, rolled up my trousers and picked up a deckchair. Then, like a stout heron, I stepped delicately into the water and, feeling the blades of grass with my wet toes, walked across the deliciously flooded lawn until I came to a place where the water lay about four inches deep and the trees overhead arched their branches like the nave of a green cathedral. There I set up my chair, sat myself carefully down, looked around and saw that it was good.\n\nIt was more than good, it was cheap. The man in the peaked cap had been watching me balefully as he pondered his duty, which was of course to walk on the water to my side and claim twenty pence for the use of the chair. Then he must have come to a decision, because he waved a hand to me and turned away. I was excused the fee. Not that I would have grudged it, because it was clear to me that I had come to the chosen place. It was for this day that the gardens had been laid out. It was for this day that the great trees had grown and spread their leaves. It was to cool my ankles on this God-given day that the long lake had been dug with such labour, that the River Thames had chosen its tides. To have the honour to take part in its culmination was a privilege I did not deserve but of which I was deeply appreciative. I joined my hands behind my head and settled down for a long cool think.\n\nThis was a time out of life so I could let my mind wander away from immediate concerns. I decided it would be a good moment to take stock of my life, so I called up the memory of my parents' friend, the artist Stella Bowen, delivering her stern injunction as I sat in her studio holding the rabbit, in... when would it have been?... about 1942. She had said: 'You'd better make up your mind what you're going to be. If you fritter away your life on fancies you'll end up being a player at all trades and master of none!'\n\nNow, thirty-four years later, I could see that I had indeed done quite a lot of frittering. To that extent her prophecy had come to pass, and I certainly hadn't managed to _be_ anything. I had often been able to say that I was _doing_ something \u2013 display-making, fabric-dyeing, photographing, solar-consulting, stage-managing, etc., etc. \u2013 you name it, I had had a go at it. But as for _being_ something... that was different. If I had taken up a career and had made myself a place in society, it might have given me more confidence. I could then have said: 'Speaking as a bookmaker (or a barrister, or a bishop)...' The identification would have given my utterances substance and respectability, even perhaps dignity. All I could say was: 'Speaking as a person...' or at bad times: 'Speaking as a nobody...'\n\nDid I mind that?\n\nWell, no. Come to think of it I was happy with it, partly because it was simply true, and partly because if I had had the trappings of an image to carry about, my imagination would have been confined to its boundaries. I mightn't have been able to slope off into unseemly projects, mightn't have been able to travel light and fast, mightn't have been able to take my heart with me.\n\nAs for 'being master of a trade...'? I had mastered a few procedures. During what I had thought of as the wasted years of frittering I had picked up experience and skills. I had learned to think from, and then to work from, first principles. I had learned to see a situation as it was and how to look closely and persistently for ways to do things that needed to be done. I had learned not to expect things and, above all, I had found the courage to get off my butt and have a go. That had been a rigorous apprenticeship.\n\nSo when, quite by chance, I had seen, as the solution to a problem with magnets, the idea of making small films, I was able to work it out and have a go at it. But I know I wouldn't have carried it off, indeed I wouldn't have dared to try, if I hadn't already had that training, if I hadn't worked for and earned an MA in both Pig-headed Self-reliance and Applied Improvisation.\n\nSo that injunction which had been haunting me through the years had been a nonsense. I was certainly not a master of the Art of Animation. Lots of people, from Walt Disney down, produced better animation than I did, but none of them could match my speed and economy. What I was master of was the Art of making small films at a price that would give us a good life \u2013 and, dammit, that would do.\n\nMy mind slipped away from wrangling with the long-dead artist. Life had been good, the day was cooler now and there were more recent, happier memories to enjoy.\n\nI stretched out and thought of some.\n\nThere was this railway engine in Wales which Dan and I went to name. We wished luck to all who sailed in her and then Dan swung the bottle of brown ale to smash on her bows. It didn't smash, it dented the engine, so he had to do it again. The second time the bottle exploded like a hand-grenade but luckily Dan was at the far end of the rope. Later I had a chance to drive the engine. Have you ever driven a locomotive...?\n\n'Excuse us,' said my ankles.\n\n'What?' I sat up. I must have been asleep because the sun was low in the sky and the tide had ebbed away, leaving wet grass through which a cold draught was sneaking.\n\n'We are cold,' they said.\n\n'We are late,' I replied, and we hurried back to the car.\n\nThe arterial road was fairly busy but no longer jammed. When I got into the car I found the air inside stiflingly stale and hot, a legacy of the afternoon that we had dodged.\n\n'Take us home,' I said.\n\n## _III. Procedures_.\n\nAs I drove home the mood of self-satisfaction that I had felt in the park began to evaporate. I was a fool and my folly was dangerous. I was also fanciful and self-indulgent, not paying attention to the real problem in our lives, which was Prue's predicament. Her depression would fall in front of her like a black mirror. At its worst this would take her whole attention, reflecting only her guilt for what she saw as her wilful inadequacy, and blocking or colouring her view of everything else in life.\n\nOn the other hand, although they took away the despair, I found her sudden excursions into manic exuberance far more disturbing and difficult to deal with. Then a mirror was still there, still blocking her view of reality, but the reflection it showed her was mighty and full of driving confidence.\n\nThe first sign of this would be the singing under her breath. This would continue all day, through every moment when she wasn't actually speaking. For me it was sharply reminiscent of the imaginary guitar which my brother John used to play when he wanted it to be known that he wasn't interested in what I was saying to him. With Prue it was different, of course, because she didn't know she was doing it, but even so I found the sound unnerving, as it was almost always the precursor of some exciting but impossible project \u2013 about which I was going to have to be beastly and try to get her to see that it wasn't on. This was a Catch-22 situation in that when she was exuberant, Prue would have no truck with critical comment, brushing it aside either without attention or with anger. When she was depressed, any word of criticism would reduce her to complete despair and that, in love, I could not face. So it seemed there was no middle ground except, we found, in time. I gradually discovered that our salvation lay in forgetting, in my capacity not to hold on to or force issues, but to let time take things clear away, so that when the moment came to do so we could put our arms around each other and still be there.\n\nThe rest of 1976 and the beginning of 1977 turned out to be pretty ordinary, as ordinary went in our lives \u2013 we bought another house! Prue had been very fond of the tall house we had once had on the seafront at Whitstable, so she decided to sell the shares which her father had left her and buy another one two doors away in the same block. Her idea was to make a small income from letting the rooms to students. She also hoped that having some independent interest and responsibility might prove to be an antidote to her bouts of depression. In this she was mistaken. Letting lodgings turned out to be a commitment to on-going aggravation which, far from giving her satisfaction, left her either in limp cowering misery or fierce avenging wrath, according to her underlying state of mind.\n\nIn 1977 Prue decided to take up the promise of university education which the war had postponed thirty-eight years earlier. She enrolled at the new University of Kent at Canterbury as a mature student, to study Theology.\n\nFor some reason I found this move profoundly frightening. I can't say why. Indeed I had very little idea what Theology was likely to be about but, none the less, the prospect of her studying it filled me with dread. I could find no rational explanation for this feeling so I minded my own business.\n\nHaving finished the remakes of _Ivor the Engine_ , I once again found myself sitting in front of a blank piece of paper. Once again I was rescued by a piece of good luck. Frank Muir, who, with his colleague Dennis Norden, had written many of the funniest and most original radio scripts in the last two or three decades, asked me to prepare a set of films based on his stories about a small Afghan hound called What-a-Mess \u2013 a very pleasant prospect!\n\nAnother pleasant prospect came a little later \u2013 _Ivor the Engine_ was nominated for a BAFTA award. The award ceremony was to take place at a glittering banquet at the Wembley Conference Centre. I invited everybody connected with the films but unfortunately Prue wasn't well enough to come. She suggested that I should invite our old friend Margaret Polmear in her place.\n\nIn the event _Ivor the Engine_ didn't win an award, but we all had a good party. After the banquet was over Margaret and I noticed a 'pop-disco' going on in one of the side-halls. We wandered into this darkened place and saw psychedelic spotlights roving across a mass of writhing, jigging dancers and felt the bass beat behind the agonized vocal thumping into our heads like a soft hammer.\n\nWe stood and watched, I an overweight boy and Margaret a tall beautiful woman, regal and full of fun \u2013 but both of us well into our fifties.\n\n'This stuff makes me feel really old,' I said. 'I can't dance to that sort of music.'\n\n'Nonsense,' said Margaret firmly. 'They just don't know what sort of dance it is... listen!'\n\nI listened. To me the music seemed quite formless.\n\n'That's a veleta!' cried Margaret. 'An old-fashioned waltz. Can't you pick up the beat \u2013 _TUM TUM-ti, tum_...?'\n\n'Yes, but it's a bit fast, isn't it?'\n\n'That's all right! Come on!'\n\nMargaret took my arm and we swung swiftly into the dance. Chanting ' _TUM, TUM-ti, tum_.. _TUM, TUM-ti, tum_...' in time to the music, we sailed and swirled across the floor. The other dancers moved smartly aside to clear a path for us. Twice around the ballroom we made our majestic progress and the floor was ours alone.\n\n'That'll do!' panted Margaret. 'We must stop before we drop.'\n\nSo we twirled, bowed and, laughing happily, left. Margaret Polmear was a truly glorious lady. Prue and I loved her most dearly.\n\nBut that was to be the end of fun for quite a time.\n\nEarly in 1978 Prue went to see the top Harley Street consultant for endogenous depression. He put her on to a new medication which, he said, would deal with all her symptoms. It dealt with some of them but its main effect was to exacerbate the almost constant slight diarrhoea that she had been suffering for two or three years as what was called a hysterical side-effect of her depression.\n\nThen, at the beginning of March, she came to me after visiting the doctor.\n\nShe said: 'I'm sorry. I've got cancer.'\n\nThe surgeon, a Mr C., interviewed Prue and was quite lighthearted about her situation. He said there was a small tumour in her colon, but it was very treatable. The procedure would be routine. Taking out a section of bowel was a fairly major operation but quite run-of-the-mill.\n\nPrue was called into hospital on the eleventh of March and the operation was carried out on the fifteenth. Prue came through it well and was in good spirits, though very weak.\n\nMy turn came while Prue was still in hospital. On the instructions of our GP I had to go for an inspection of my interior by barium X-ray. This revealed a small object about the size of a pea, swinging on what looked like a sort of thread attached to the wall of my colon. Mr C. told me it was a polyp. He said it wasn't, yet, malignant but all the same he would 'deal with it surgically'. It would be a minor procedure. The hospital would call me in when a bed became available.\n\nOn the twenty-sixth of March Prue came out of hospital. Mr C. told us that the operation had been very successful and it was now simply a matter of recuperation.\n\nI would have liked to have taken Prue to Spain for a while to soak up the sun, but she wouldn't let me because I was liable to be called into hospital myself at any moment and, the NHS being what it was, one couldn't afford to miss one's place in the queue. If we had known how long we were going to have to wait, we could have gone to Australia!\n\n# THE WINDOW\n\n## _I. Operations_.\n\nIn 1978 the local hospital had a very sensible system for accommodating patients coming in for operations. They would ring up on the Thursday to tell you to be ready to go in on the Saturday, but they would also advise you to telephone the ward on Saturday morning just to make sure the bed was available. It usually wasn't and after half a dozen of these near-misses I was in a right state of nerves, and by the time, late in October, I heard that I really was going to be admitted, I was a jelly of apprehension.\n\nPrue and I came trooping into the rather impersonal-looking ward. There were several other men there, in beds, but we avoided each other's eyes. The nurse suggested that I should take off my clothes, put on my pyjamas and get into bed. This seemed to be an odd thing to do in the middle of the afternoon, but I complied.\n\nPrue went home. I sat up in bed and read a book, or tried to. It was only a small non-malignant polyp in the colon, hardly larger than a pea. I had seen it on the X-ray fluoroscope. Mind you, as Mr C. the surgeon had explained, they would have to get to it, and that would entail what he called a laporotomy \u2013 but he hadn't seemed a bit worried about that.\n\nI couldn't read, so I fished my briefcase out of the locker and tried to do some of the bits of work I had brought with me. There was absolutely nothing in there that wouldn't keep. Everybody had been alerted about the operation. All the deadlines had been openended. There was money in the bank... etc. Everything could perfectly well wait until the larger anxiety had been dealt with and that anxiety was, for once, something that definitely did not call for any action on my part. With a certain feeling of relief I closed the briefcase and shoved it back into the locker.\n\nI sat up and looked around. It occurred to me that being in hospital could become a bit boring. Perhaps I was in for a dull time. I had no idea that my time there would turn out to be absolutely extraordinary.\n\nThe chap in the bed opposite had had his operation and was due to leave in a day or two. He was looking well so I put on my dressing-gown and slippers and ambled over to have a chat with him. His name was Reg and by trade he was a master-baker. I had always wanted to know how to make bread, and once he sensed this he was happy to tell me all about his early apprenticeship. For my part, having nothing on my mind except the impending operation, I enjoyed this enormously.\n\nApart for the fact that I was given nothing to eat, the days passed very peacefully but I remember that, when the day of the operation came, my legs insisted on my getting out of bed and pacing up and down in the corridor outside the ward, in order to use up the nervous energy. The sister came out and laughed at me. She held me still by the arms and told me that she had had dozens of patients far worse than me in her charge and they were all well and away. So there was nothing to worry about. I agreed... and added: 'But even so, if it's all the same to you, I'd like to go on pacing up and down for a bit.'\n\nTime passed. Eventually I was ordered back to bed and given an injection to sedate me. It didn't do that. I was wheeled into the operating theatre, feeling quite drunk. Mr C. and his team were there, very jolly and full of fun. Then... time telescopes. There must have been darkness, I could sense that. Now there was pain and raging thirst. I was back in the ward with my belly wrapped up in sticking-plaster. Time telescopes again.\n\nThe surgeon's second-in-command, Mr H., appeared at the foot of the bed, surrounded by a group of young persons in white coats, student surgeons perhaps. He told me proudly that the operation had been very successful. They had found nothing there. There was no polyp. Wasn't that good news?\n\nWell, no. It wasn't. 'Wait a minute,' I said. 'Does that mean it was all unnecessary?'\n\n'There... was... no... polyp,' he repeated, as if to a child.\n\nI was confused. 'But I saw it! I saw it on the fluoroscope. The radiologist saw it too.'\n\n'Oh, that was just photography \u2013 technical stuff. You wouldn't understand that.'\n\n'I've been a photographer for twenty-five years.'\n\nThe surgeon waved airily. 'Very well, if you're the expert I'm sure you know best,' he said, and turned away.\n\n'But I saw it,' I said, to his departing back. He waved again and went on walking away.\n\nThen, apparently, something must have gone wrong. I was aware that people in white coats kept coming with stethoscopes to listen to my belly. I don't remember a lot but Mr C. himself appeared and light-heartedly told me that the gut system hadn't started up again. He told me he was going to, as he put it, unzip me and do it all again... which I believe he did, that afternoon.\n\nThe next is darkness, pain and thirst, dreadful thirst and wild panicky anger. In a steel-barred cot-cage, tilted so that lights shone straight into my eyes, I fought a desperate battle with two nurselike ogresses for the possession of a wet sponge. I lost and felt a sharp jab, then more darkness.\n\nThen, suddenly, I opened my eyes and saw daylight. I was back in the ward. That was a blessed relief but I seemed to keep dipping out of consciousness, not into dream but into darkness.\n\nA mixture of utter panic, made hopeless by total physical weakness occupied me. I was spread-eagled on some wet steep roof with my hands splayed out flat on the wet slates to try and stop myself slipping down and falling. The slightest movement and I would start to slip, so all the time my attention was racing from corner to corner of my body, checking and checking again.\n\nTwo nice child-nurses looked down on me and smiled. Then they fiddled with the drip that was piped into my neck. Apparently it wasn't dripping properly. 'Could I help them?' 'Certainly.' It seemed that the drip would only drip properly if I kept my head turned away to the left. 'Could I please do that?' 'Yes, yes of course.' They turned to go. One of them added: 'But keep an eye on it, would you? Let us know if it stops again.'\n\n'Very dicey, these drips,' they said as they walked away.\n\nThe problem lay upon me. To see whether or not the drip was working I would have to turn my head away from the left. If I did that it would stop working. But I had to do so because if it didn't drip I wouldn't stay on the roof. I had to tell them, but I mustn't turn.\n\nLooking back it seems such a trivial problem. I could have called out or something, but I was weak and all my energy was using itself in panic-racing.\n\nI had finally failed. I couldn't do one simple thing to help, the one easy task I had said I would do. I was no use. It was my fault for pretending, for showing off. I had pretended to have a polyp. I had defied and contradicted the surgeon. I had made them operate all over again, I had fought the nurses and now... the last bloody straw! I couldn't even perform the one simple task that would save my life. 'Typical! Once again Bloody Oliver has bitten off more than he can chew, only this time it's for real, and final!'\n\nBlame and electric panic, black shame and despair kept swinging across and thudding into my limp body as if it was a punching bag. How long it went on I don't know, but eventually it was night-time and the rest of the ward was dim and settled. I knew at last that with my own hands I had hauled the stone across my own grave and that I was underneath it. My own voice sneered silently: 'You put it there. You lift it off.' There was no lifting it off. I was done for.\n\nThe light was bright. Sister stood in the middle of the ward looking slowly around at her patients. Then she was holding me in her arms like a baby. I was weeping and incoherent with confusion and shameful apology for something... being a nuisance. A nuisance about something... the drip, yes. She held me and heard me. I had her whole attention and knew that I could have it for as long as I needed it. The drip was not important and it had been working. It had been regularly checked. 'There is nothing,' she said, holding me very tight, 'there is nothing, nothing, nothing, that you have to do about anything, except just be there and let us look after you. That is not being a nuisance because looking after you is what we are here for, and we like doing it.' Then she laid me gently on to the pillow and went on with her rounds. Nurses came and washed me, smoothed me, tidied me up. Perhaps, if I kept very still, I would not slip off the roof.\n\nTwo tall young men in white coats, New Zealanders I think, came to tell me that at the second operation they had found the polyp and had taken it away. They had used a new, highly technical device, rather like a snake with a television-camera in its nose. Amazing! I was pleased for them but I rather hoped they would go away soon. I couldn't hold my attention to them.\n\nTime telescopes again. Was it the same day or the next? I had lain there while my eyes walked the landscapes of the mouldings on the ceiling and I had kept very still, my mind still clinging to the slippery roof, hoping against hope that I would not slip into oblivion.\n\nIt was early morning. From my bed I could see a small triangular section of glass at the top corner of the main window. It was lit by the pale purplish light of dawn. I had seen the light of dawn many times but there was something about it today, as if I was seeing it for the first time. I turned my head towards to the ward and there, implicit, was the silent roar of pain, panic and terror.\n\nI turned my head back to the light. What was there about that light? I saw it as it was, part of the chilly November morning outside, just light, but it reminded me... no, I recognized in it, a beauty. Cold it was, like an icicle, but all beauty, a single clear essence, untouched by detail. I can feel it now, as I write, cold as the frost on heather where the morning sun, creeping down the hillside, touches it with sunlight and turns it into dew.\n\nI turned my head back to the ward again. Panic surged in.\n\nI turned to the window. There it was, strong as ever, cold, clear, breathtakingly calm and beautiful. It was no passing glimpse, it was steady as a lamp post. All time stood still there. I could lie for ever and for ever I would be still.\n\nI asked: 'Is this death?'\n\nI could ask the question so it seemed it wasn't.\n\nDay came. People were busy. Prue turned up with clean pyjamas. She saw me and said: 'Oh, you are upset, my darling.'\n\n'Oh yes, I am upset! Please, please, let me be upset. Put your silly head close to mine and hear what I have to say. It is important. Listen!'\n\nWords came: gibberish almost, dragged from under scraps of sobs... 'A terrible beauty... somewhere north of that dreadful hotel near Ross... we will find it. There is a tree, sort of coneshaped. We are there... bare feet on cold grass but you are there, one day... soon!'\n\nThese were the words of dreams through which I was groping for the feeling, trying to tell the huge future, the wide, rolling, clear-singing delight inherent in that cold simple light.\n\n'Don't be upset!'\n\n'Oh, yes! Please let me be upset at last!'\n\nClose to her cheek my tears rolled down and made us damp. I was empty. Something was done.\n\nLater still, maybe several days later, Mr C. himself came to see me. He looked puzzled. My charts, it seemed, were impeccable. 'Take out the pipes and feed him,' he commanded.\n\nSo, bit by bit, over the next few hours, the various pipes were taken out. I was allowed to drink. I was able to move about. I was almost whole, almost myself. I ate food. I tried standing up \u2013 and sat down again fast. I was very weak but all the time inside me I felt an excitement, not a taut violin-string excitement but an easygoing, calm looking-forwardness.\n\nIron-fingered physiotherapists pincered my chest and commanded me to cough. I stood up and was able to take steps. I was full of something, expectancy perhaps, or was it just that I was looking forward to being better?\n\nNo, not just that. Something more... something was new.\n\nI walked very carefully to the big window at the end of the ward. Although the ward was full of people, I knew they were my friends. I knew I was with them but I was also alone, with a low growling of anticipation inside me.\n\nAs I reached the window and looked out, a vast wave of joy met me and bowled into me. It came, not through the usual channels of eye, brain, mind etc., but welled up in every part of me as a fountain-burst of delight... hands, feet, ears and sewn-up belly!\n\nIt felt as if every cell in my body ran out into the streets of my blood and chased singing through my veins: 'Come on, fellows! The old bugger has let us out at last!'\n\nAnd that was just my body.\n\nMy eyes?\n\nThey saw an ordinary windy day in November. The wind was scudding dead chestnut leaves across the roof of the maternity ward and slapping them on to the wet trunks of the trees. A few seagulls swooped and turned, fingering the wind. Heavy clouds were jostling in the sky. This was not a day one would take much notice of in the ordinary way, but I think I was seeing it all for the first time in my life. No more than that did I see, but what I saw there was so clear and complete, with a beauty so simple and essential, that I was poleaxed with amazement.\n\nNow, hang on! This was not in the book. This was not anything I had ever experienced or expected. In my life beauty had lurked in the corner of the eye. Look straight at it and it is not there. Only the memory can hold it. (Let me go slowly now. Pray for my words because I am groping.)... I was there for, maybe, five minutes, but those five minutes held eternity in every direction.\n\nFirst there was this beauty. Everything I saw was no more than what it was, but, as it was, it possessed, was part of, a single clear essence of itself, and of everything else. (Oh dear, words!)\n\nI felt a single flavour, I saw a single beauty which was the essence of what was there... the seagullness of the seagulls, the flatness of the leaves, the shine of wet on the tree trunks... these were among the multitude of joys that made up a whole joy that encompassed not only what I saw, but me and everything beyond it... on, on and out of sight!\n\nWhat hit me then was a realisation that this joy in life that drives through all things is the life that drives all things. I felt the huge engine, the driving, rolling river of life and death, of happiness and sadness, a river of which my dark fumbling life was only a tiny part, a leaf on the rapids, yet, in my realisation of it, I was part of the river itself, both a part of it and it a part of me.\n\nThough I saw only what was there, that was what I saw in it, and I knew beyond doubt that this was not any sort of illusion. On the contrary, I knew that I was seeing clearly for the first time in my life... seeing how things are.\n\nAnd that was just my eyes!\n\nMy body was already singing, but my mind, the Head Office where nosy clerks add up doubts and consider reservations, was still trying to sort out what the blazes was going on. For a few seconds they held on to who I was and where I was, but the beauty was so clear, so complete and safe, and the relief so profound, that with an almost physical release I let go and gave myself entirely to the enjoyment, to the sheer delight of seeing.\n\nThe realisation that filled my mind and heart was not exactly a new revelation. It was more as if I had remembered something forgotten, or was finding something that had been lost, something I had known all along but in some fear or confusion had mislaid... 'Of course! Yes. That is how things are. Oh that! Yes.'\n\nI knew I had found the central truth that, without knowing it, my life had been spent seeking. I knew that in that realisation all happiness lies, all confirmation and all safety is found. That it had no words, that it was without a worked-out rationale was, in that huge beauty, quite irrelevant. But still, almost as a parting gift, the nosy clerks ran up a pattern of the glories. It went:\n\nFirst there is this beauty which by some accident of fate you can now see. That beauty is the seeing of it. The seeing of it fills you with delight. That delight is love for it.\n\nThat which you are looking at you are loving with all your heart and yet that loving is not separate from what you are. You are the seeing and all you are doing is paying attention to what is in front of you now. The more carefully, the more clearly, the more fully you lay your mind and heart open to what you are seeing, the more you will appreciate, the more you are appreciating, the nature of what you see... for that is what love is, just allowing yourself, or perhaps being allowed, to see. Love is a part of seeing and it is all happiness.\n\n'And that!' laughed the nosy clerks, 'is your lot!' And, tossing aside their biros, they threw their arms around each other and ran out into the streets of my happiness.\n\nIs that picture too far-fetched? Perhaps, but I know that there did come a moment when everything fell into place and the anxious, needling, self-conscious, wary 'self' that I carry about, stopped marking up, judging, evaluating, categorizing, and just switched off. It wasn't there. I wasn't there.\n\nWell, no. I was there. I hadn't vanished. I had just stopped paying attention to myself. My self was there but I sensed that it had become complete. I had the feeling that it had always been trying to reach around to form a ring. Now that ring had clicked in. Now I was ready for use. Thin as paper, yet strong as steel, I was a fine ring through which all life roared like a blowlamp.\n\nOf course I was there! I rejoiced in my own existence as I rejoiced in everything else I saw, but this self-awareness was something quite different from the self-consciousness of my ordinary times. This did not take my attention away from what I was seeing. This _was_ the attention.\n\nThen what happened?\n\nNothing, really. Something astonishing had happened inside my head, but there was absolutely no need to _do_ anything about it. I remember flopping on to my bed, hoisting my happy legs up on to the flat and lying there on the half-shell, still as sleep but marvellously awake in a wide golden lake of pure peace that stretched away in every direction.\n\nThat lake was time. Something had happened to time. It had stopped gibbering and jabbing at me. I was no longer tied to its tail like a tin can. It lay still and receptive all around me. Because the last vestige of the feeling that I should be anywhere other than exactly where I was had gone, I was able to enjoy and pay attention to what was here, now, without any hurry or sense of urgency. I had all the time in the world.\n\nIt would be some hours before anybody came visiting and I knew I was going to enjoy every moment of that time to the full. I had time for everything, time for everybody, time to walk through the garden of my mind, just enjoying it.\n\nI remember turning to Jim in the next bed. I was lucky to have him for a companion. He was a quiet, worried man. We knew we were old friends the moment we met and we treated each other with the elaborate discourtesy that is the mark of true respect and affection.\n\nI commanded him to tell me the story of his life, but slowly.\n\nThis, after some formal diffidence, he did. It was a saga that unfolded layer upon layer of amazing adventure and infamy in the far corners of the Empire.\n\nHow a Warrant Officer (Regular) came to be running two casinos and a cinema, while being blackmailed ever so politely by the High Commissioner of a central African state, is a tale that Kipling would have enjoyed. I enjoyed it. I lay in my lake and soaked it all up, occasionally darting in the odd question to get something clear.\n\nThe next few days were new to my life. Although the scene was always the same impersonal ward, every day had its own particular flavour. Even now, as I write, I have to keep reminding myself that each day, which feels so different in my memory, happened in exactly the same place. They each have a realness that is so individual and clear that it is almost a colour... blue days, greenish days, golden days.\n\nEverything I saw or did was a joy. Even pain, though painful, didn't matter. The sun shining on the chairs at the window end of the ward made a fairyland. I hoisted myself over there and sat in the sun. It seemed to bless me with a warmth that was love itself, something miraculous, the first sunlight I had ever really felt.\n\nI remember thinking that it would be good to be old and be so peaceful. Friends might come and apologize for having left me for so long, and I would say: 'No, no. Time is my friend. We enjoy each other's company now. I am here and this is now. As here is here now and now is now here, I don't have to go running after either of them and they can't be left behind. We drift along together and among us time is still.'\n\nThe state of mind did not fade away. I did not fall back into my usual dubious self. I thought of the possibility but it seemed to be quite laughable. I saw myself as one sees a friend: warts and all. I remembered some of the legion of dreadful, shameful faux-pas that I had committed in my life. Usually these would wake me rigid with embarrassment... ' _Eeeek!_ Did I really say that!'... but now I laughed at them and at myself, not derisively but affectionately... 'The fool!' I had been forgiven.\n\nPrue would come swinging in wearing her steel-worker's shirt and trousers, strong and sure as a thoroughbred racing camel. 'Style!' I said to myself. 'She has style!' and I held out my arms.\n\nI mended miraculously fast. Incidentally, just to make sure, I asked Sister if there was anything in the drugs I was taking to bring about such a happiness. She laughed and said there wasn't. I was on one Valium a day just to stop me floating out through the window, but that was all... and why didn't I just enjoy being happy while I had the chance? Good advice!\n\nFriends came, to bring me sympathy in my hours of suffering, and were assaulted by my friendliness. Lots of them came, far more than I expected or deserved, but never for a second was I anything but delighted to see them. Never once did I have to wonder what we were going to talk about. All I remember is the pleasure of seeing them. I seemed to have a stillness in my seeing which allowed me to look into people without intruding or disturbing, and then I seemed able to see out again, but through their eyes. It was as if I had no substance. I was a seeing, but my seeing lit them.\n\nOne of the effects of this single-mindedness was what seemed to be a marked improvement in the quality of my thinking, and also a very basic change in the _way_ I was doing it. Certainly, at that time, and for quite a while afterwards, my mind was unexpectedly calm and confident. It was accurate, unbiased, unhurried and, quite simply, more efficient. I knew that while thinking I could set up literally any number of different qualifying considerations as sidecircuits and be completely certain that, without my having to think about them again, if they became relevant they would punch themselves up.\n\nI remember noticing how different that was from the way I usually thought. Usually I had to waste time dashing around on my mental scooter, checking and rechecking that everything I might be going to need was still there.\n\nMy mind also seemed to be capable of a different sort of thinking. This seems to have been more like a pondering, not a directed thought-process whose end is a conclusion, but a general contemplation of whole situations with the purpose of clarifying and relating observations. I was thinking in immensely broad terms whose forms were not words, categories or even conceived entities, but what seemed to be the essential defining flavours, the ' _therenesses_ ' of the propositions and ideas that I was looking at \u2013 and both the thoughts and the thinking were parts of the same delight, a process for which I can only find one word: beauty!\n\nOr, to put it simply, I was minding. I could appreciate the exact nature of whatever I was looking at with an emotional commitment that was very far from my usual grudging response. My personal views and opinions were just not entering into the perception, and yet, as a sort of side-effect, I had a far stronger sense of being alive and of belonging to life than I had ever had before.\n\n## _II. Coming Home_.\n\nA word of advice: if ever you have a major abdominal operation, go home in an ambulance or hire a Rolls-Royce.\n\nI had mended surprisingly fast and was soon well enough to leave hospital. Corseted like a duchess, dewy-eyed at parting from my friends, I was helped to the Mini and Prue drove me home. That five-mile journey from the local hospital to Red Lion House was excruciatingly painful. It felt as if I and my battered belly were being dragged over a rockery in a cardboard box.\n\nWe arrived home. The house was much as usual. The boys had slid the furniture about so that it blocked the way across the room. They were draped about on the chairs watching the television while the room slowly filled with smoke. Prue said she thought I should go straight on up to bed, but the smoke was really my responsibility. I had made the counterbalanced cowl over the fire-basket. The boys just hadn't adjusted it right. I reset it quite easily and then staggered up to bed. That incident may seem trivial but I think that was the moment I met myself again.\n\nAs I grew stronger I noticed that I was making a point of telling people about what had happened. I had been telling them about it all along, because I had been so full of happiness that anybody could have a slice. Now I noticed a subtle shift of purpose. Now I was talking about it in order to make sure it was still there. It was, but my old self was edging back. Like rats returning to a ship that didn't sink after all, the anxieties and urgencies of my ordinary life crept on board and began to gnaw.\n\nTimes were patchy. I could still lie flat on my bed, breathe out, smile and float away into clear certainty and calm. Kevan, who had been in India and had spent time in an _ashram_ , came to see me. He was deeply delighted and glad for me, telling me that I had inadvertently fallen into the state of non-being that is the goal of spiritual development. We had a very happy time together, a time for which I was particularly grateful because I was beginning to need some reassurance. Prue had let it be known, in the nicest possible way, that she had heard quite enough about ecstasy \u2013 I had every sympathy with that. Other people were also reacting with less enthusiasm to my discourses, and I saw that I was becoming a bore. This thought chilled me and I began to feel guilty, as if I had lost, or let slip, something infinitely precious.\n\nThe doctor came and looked at me. He said how glad he was to see that I was beginning to be depressed at last. He informed me that a state of mild elation is quite usual after a serious operation, but if it persists it may require treatment. No doubt I would soon be normal again.\n\nThank you. Thank you very much. That was a shattering grief.\n\nI duly fell in upon myself, into a standard depression, into self-obsession, self-pity, unconfidence, withdrawal, panic \u2013 all the usual stuff, just what the doctor had ordered.\n\nAnd yet, somewhere inside me \u2013 and if anything it made me more angry with myself \u2013 I knew that none of this turmoil had anything to do with what I had seen. I knew that I had seen _how_ _things are_ and I knew that that is how they would still be if and when I would be able to see clearly again.\n\nThe doctor had been right. I did recover from the depression. I did become normal again. I was once again the old maverick, alone with his outlandish fancies but at the same time quite friendly and serviceable, busy doing whatever needed to be done.\n\nFor Prue, the lodging house at 8 Wavecrest continued to be a soap opera of disasters, but at least she was in good health and not too depressed. In fact she was full of fun and excitement because Kerris had come back from Australia in August to tell her that she was pregnant and that her baby was due in late March 1979. So Prue had already booked her flight for 5 March and was ticking off the days before she could take off.\n\nThen the day came. We took her to the airport and launched her towards Australia and grandmotherhood.\n\nAs I watched Prue walk away I remember feeling deeply affectionate towards her. We had walked together through so much: through difficulties, disappointments, misunderstandings, incomprehensions, through poverty and plenty, through peevishness, boredom, exasperation, despair, laughter and long slabs of pure unremembered contentment, and underneath it all had lain the bedrock of our friendship, the shared foothold that took no account of passing change.\n\nMore recently, we had both been close to death. That would, in its time, bring change. But meanwhile I knew she would be safe and happy with Kerris and her friends in Australia, and I turned away, contented, to attend to my thoughts.\n\nThere was thinking to be done. I was still profoundly astonished by what had happened to me at the hospital window and I needed to think it through. For a time I had lived in a state of mind that was altogether unlike what I had hitherto regarded as being normal and ordinary. The experience had been astonishingly joyous and it seemed to me that if that sort of thing could happen as part of ordinary life, then ordinary life could well be something very different from what I had thought it was.\n\nSome of the friends I spoke to about this were impatient with my desire to understand what had happened. They were adamant that it had not been part of 'ordinary' life, that what I had had was a _religious_ experience, that by God's grace I had been granted a Revelation of His love. So why did I not just accept it and leave it at that?\n\nWhy not indeed? The joy that I had experienced was so overwhelming and so obviously a part of love that I had to open my heart to the possibility that it could in some degree have been divine.\n\nI could also see that it had dealt very effectively with two human concerns that are usually thought of as being in the province of religion.\n\nI had experienced forgiveness. This hadn't been so much a forgiveness of my sins as a realisation that the faults which I had felt to be sins were not in reality sins at all, but follies, often misplaced eagernesses. Also I knew that the forgiveness had not been given by some outside agency, it was I who had done the realising.\n\nThe other concern was immortality, the hope of the Life to Come. I had always held to an intellectual rationale about death; I could see that it was perfectly possible that I might not be anywhere at all after the end of my life. But, all the same, life would go on, other people would have the discourtesy to go on living after I was dead and I would be missing the fun.\n\nSuch thoughts are deeply uncomfortable, but all my life I had felt even more uncomfortable with the alternative that was being offered by the Church when I was young \u2013 that if I was very good, obedient and worshipful throughout my life I might, at some unspecified time after I was dead, be resurrected and allowed into Heaven. That had seemed, on the whole, to be so far-fetched as to be discountable.\n\nThen this happened. I looked out of the window and felt the river of life. Everything, all the love that I had ever loved, all the love that had ever been loved, was there, flowing on in that river, and I knew that I was part of that river and that it was a part of me. And from that simple perception came a safety in which all joy and all knowledge were one and there was no more seeking to be done. I was a leaf on that river and there was no way I could ever wish it to stop on my account and there would be no death in my not being there. Others would feel the same joys in life that I had felt and in their feeling their lives would be just as real as mine had been. Nothing would have been changed, nothing lost.\n\nThe rationale that I had so glumly held to was not altered. It was just realised \u2013 made real. And, not to put too fine a point on it, that disposed of any ambitions for immortality that I might have been harbouring. I would quite like to have another glimpse of that moment before I die, that would make dying a part of the joy but, if not, it doesn't matter, it will always be there, a part of how things are, not of who I am.\n\nDuring the weeks that followed I made a number of clumsy forays into religious-type conjecture, but I began to notice that the picture of a vision having been granted to me, personally, by a divine parental deity, was becoming incongruous. I think this was happening because the experience itself had radically changed my own internal situation. In the same way that a starving person, who has been dreaming longingly of food, once he is fed suddenly finds that he has no more hunger and therefore dreams no more, so I, who had lived in a shadow of anxiety in which the presence of God's continuing love might have been a source of comfort and reassurance, had suddenly come into a light in which all personal anxiety had dissolved, and with it had gone the separateness of my 'self'. So, ungracious as it may seem, I no longer felt the need of anything, even God's love, for myself, simply because at that moment there was no self there to need anything.\n\nLet me say at once that I (being strictly agnostic) could never reject the possibility that my experience could have been divinely instigated. God may well move in mysterious and unexpected ways, possibly by exerting influence on the secretions of this chemical computer I call my mind \u2013 who can say?\n\nBut there was one thing I could say for sure, if only because it had been staring me in the face all along. This was that what came to me at the window was not a vision of something 'other', not part of a separate 'spiritual dimension'. Nothing extra had been added to what I was seeing. Quite the opposite \u2013 something had gone. Something had been wiped away, my window had been cleaned and I had seen no more than the glory of what had been there all the time.\n\n## _III. Flights_.\n\nPrue returned, tanned and fit, full of happy stories of her time in Australia, of the good friends she had made there, and of the joys of having a granddaughter.\n\nThe photos of the infant Catherine were duly adored and in no time at all we were back into our ordinary lives, with Prue buzzing about in her Mini and me getting on with films and projects.\n\nIn due course Prue went for a check-up at Mr C.' s clinic. When she came back she said: 'I'm sorry. I've still got cancer.'\n\nThat which we had both feared, but had not spoken of for some months, had come to pass. Mr C. had told her that there was a recurrence on the suture-line and had announced that he was going to operate again immediately. This time, he said, she would end up with something called a colostomy, which, he added lightheartedly, would be no trouble. We both knew what a colostomy was \u2013 a plastic bag attached to her belly \u2013 and we weren't stupid enough to imagine it would be no trouble. However, we could cope with it.\n\nPrue and I sat together and looked at the situation. Prue had the option of refusing another operation. This option had not been mentioned by the surgeon but, in theory at least, the decision was for her to make. She could either go along with the set medical procedure or stand against it, defy the doctors, and be allowed to die.\n\nIn the end Prue decided to let the procedure go ahead as announced. We waited, but Mr C.' s idea of 'operating immediately' turned out to be a bit slack. We had to wait several weeks.\n\nImmediately after the operation Prue was very weak, but she was soon well enough for the specialist nurses to be able to teach her how to deal with the colostomy. Then, after about another week, she was well enough to walk along the corridor on my arm.\n\nIn the corridor we met a bright young man with a beard, one of Mr C.' s registrars. He had assisted at the operation and was delighted to see her looking so well.\n\n'Excellent operation!' he said, beaming. 'Absolutely first-class.'\n\nThus reassured and light of heart, we returned to the ward. Prue went to her bed and I went to the sister's office to ask about something. The duty doctor was there and I told him how happy Prue and I were to hear that the operation had been such a success.\n\n'Oh, yes,' he said. 'We only wish we could have got it all out.'\n\n'You mean she still has cancer?'\n\n'Oh, yes.'\n\n'And she's going to bloody die!'\n\nThe doctor looked shocked. 'Well, if you must put it like that... yes.'\n\n'So what was that bog-brush-faced tit doing telling us what an excellent operation it was?'\n\n'It was an excellent operation, as operations go. It just wasn't completely successful, that's all.'\n\n'So now, five minutes after that grinning oaf has allowed us to believe that everything is marvellous, I'm going to have to go and tell Prue that she is, after all, going to die.'\n\n'It's just a bit of misunderstanding,' said the doctor, a shade impatiently.\n\nI fell into a chair, my head reeling. The doctor left the room. One of the nurses, sensing an emergency, ran to fetch me a cup of tea.\n\nI sat there, gripping the seat of the chair and glaring at the wall. Grief and rage were boiling inside me. After a while the doctor looked in again. 'Oh, you're still here,' he said.\n\n'How can I?' I growled.\n\n'Oh, I'll tell her if you don't want to,' he said, and trotted away.\n\nWhen I had pulled myself together, I went to see Prue. She was sitting up in bed, weeping quietly. She took my hand and smiled.\n\n'That was rotten for you,' she said. Then tears came and my anger blew away. There was no point in being angry.\n\nA day or two later Prue was discharged. She had wanted to talk to Mr C. about her situation but she was told that Dr O'C. was now in charge of her treatment. He would be prescribing whatever was necessary and she could have a proper talk with him when she came in to see him the next day. In the event that didn't work out because Dr O'C. had a long queue of patients and only had time to mark crosses on her belly for radiotherapy. Prue asked him if they could have a talk about the future and he replied that an appointment had been made for her to have radiotherapy but if she decided she didn't want radiotherapy she didn't have to turn up. That was the talk.\n\nPrue went for the radiotherapy the next day and it made her alarmingly sick, with agonizing pains which lasted for hours. She said she couldn't live with that.\n\nThat evening, it was the second of July 1979, Kerris phoned from Australia. She told Prue that there was a Dr Holt in Perth who had a new treatment for cancer tumours called 'Tronado'. She had spoken to him and he had said he thought he could treat her, but only if she came to Perth immediately, bringing the surgeon's notes.\n\nI found Dr O'C.' s telephone number and told him about this. His reply was brief: 'If you can get to see Dr Holt in Perth, do so at once.'\n\nThe next forty-eight hours was a stampede. I bought flights for us, leaving at 9.30 a.m. on the fifth of July from Heathrow, wrote a letter to Mr C. asking for copies of his notes, rushed up to London to negotiate visas and renew my passport, wept when I discovered that the Passport Office couldn't do it in time and was reprieved when they discovered I was Noggin the Nog.\n\nThat was the good news. The bad news came the next evening when, after a day of rushing wildly about, packing, organizing, leaving messages and apologies \u2013 all the preparation for summary flight \u2013 a second-class letter came from Mr C.' s secretary, informing me that he had decided against Prue's travelling to Australia. I managed to find Mr C.' s telephone number and more or less ordered him to provide the notes. With little grace he authorized me to collect them from the hospital, which we did on our way to Wimbledon to stay the night with Prue's twin brother Cay and his wife Jeanne. Then, apart from leaving all our travel documents behind in their flat \u2013 luckily Jeanne had the presence of mind to come after us to Heathrow in a taxi \u2013 everything went fairly smoothly.\n\nOnce on the Jumbo jet and in the air, I crumpled up completely. But Prue, who had sprung into a full manic phase a day or two earlier, was full of energy, singing to herself and buttonholing the cabin staff for serious chats. I had never thought I would be glad to see her in that state, but I think she rode through those panic days on a bow-wave of indomitable exuberance, and for that mercy I shall always be profoundly grateful.\n\nWhen Kerris and some of her friends met us at Perth airport and drove us out to her house in the suburb of Subiaco, Prue was in fine fettle, ready for a party, but I wasn't. I think Kerris must have noticed this because I found myself being politely led away by her friends Faith and Alan. Apparently I was to stay with them for a day or two. I didn't argue, I literally didn't know what time of day it was.\n\nFaith was a very distinguished and successful Australian actress. Alan, who was an academic at the University of Western Australia, was still very English after living in Western Australia for many years. I shall never forget their gentle unintrusive kindness and the sheer relief of being able to let go of my charge.\n\nThe next morning I walked out into a soft misty drizzle. Faith thought I was mad because July is their mid-winter, but by English standards the drizzle felt warm and welcoming, and I wandered, feeling very relaxed, along the wide suburban streets, marvelling at the amazing, strange flowers that grow in Australian gardens and the way the eucalyptus trees wear their smooth bark \u2013 hanging loosely about their trunks, like old socks.\n\nWhen Alan came back that afternoon we went for a walk along the rocky edge of the Swan River estuary. Alan was a psychologist \u2013 not a practising psychotherapist but an academic. I mentioned to him my experience in hospital and, slightly to my surprise, he became intensely interested. This was due to the fact that 'the mind' was his subject, and he took a close professional interest in any reports or observations about its behaviour and capacities. He suggested, quite seriously, that as soon as possible I should try to find the time to write it all down.\n\nThis was very exciting but, happy though I was, and grateful to Faith and Alan for their hospitality and the much-needed respite that it had given me, this was not a time for writing. I had other, more urgent matters to attend to\n\nThe next day I went with Prue for her first interview with Dr Holt. He told us about the Tronado. This was essentially a very special sort of low-temperature microwave oven which fits around the patient and gently induces an exact amount of heat, enough to kill cancer cells but leave healthy cells unharmed.\n\nDr Holt didn't promise that the Tronado would _cure_ Prue of cancer but he did say he thought it would take away the remaining growths and restore her to good health.\n\nThe Tronado treatment was not unpleasant but at first it didn't seem to be having any effect. Then, after a few weeks, her health began to improve. She became stronger by the day and quite soon the scans and X-rays showed no signs of any tumours at all. The treatment seemed to be working.\n\nThe course of treatment ended but Dr Holt suggested that Prue should stay on in Perth for a few weeks so that he could monitor her progress. Dan had flown out to join us. Stephen was already there, deeply involved in the fringe theatres. Kerris was now a television producer at the ABC, with a wide circle of friends and acquaintances, most of whom were actors and actresses who looked to her for work.\n\nTime passed. Prue's emergency seemed to be over and she was feeling well, but I suddenly realised that I had run out of money. Although I had so far been able to do very little work on the films I was to make for him, I wired Frank Muir to ask for an advance. Frank instantly sent me \u00a31,000, so we were solvent again, but even so I could see that it was time for me to go home and get on with some work.\n\nSo in the autumn of 1979 Daniel and I flew back to England, he to school and impending exams, I to make films and see to the future, whatever that was to be.\n\nI brought with me some tapes of conversations with Dr Holt and details of the Tronado system. I had in mind to try and interest the UK medical establishment in the system, so I put together a presentation and took it to the top cancer research units.\n\nMy approaches were received with complete courtesy but no success. As far as the Tronado was concerned I was knocking at closed doors. That was furiously disappointing.\n\nHowever, for me and Prue at least, the campaign did have one worthwhile side-effect. When Professor Peckham, of the Royal Marsden Hospital, heard about Prue's treatment he asked if she would consent to become his patient \u2013 almost as if she would be doing him a favour!\n\nWe were highly delighted and relieved, not only because we were both dreading the prospect of returning to our local hospital and the mercies of Mr C. and Dr O'C., but also because the Royal Marsden was the top cancer hospital in this part of the country and Professor Peckham was the top man in it. Whatever the final outcome would be, we could now feel sure that Prue would be receiving the best possible attention. We could do no better.\n\nThe Royal Marsden did a series of tests which indicated that as far as they could detect there was no sign of cancer in her body. However, just to be on the safe side, we were asked to come back every couple of months for a check-up.\n\nApart from these visits, each one of which was a journey into fear, Prue and I went on with our lives as usual. On top of all her other activities, Prue was now involved with looking after a hostel for Vietnamese boat-people near Maidstone, while I was mainly busy catching up with my film work. In her spare time Prue wrote essays on comparative religion for her University course, while I, in mine, tried to do as Alan had suggested, write down what had been happening to me.\n\n## _IV_ _._ Agap\u00e9.\n\nWhen Alan and I talked beside the Swan River in Australia, he had recognized my episode at the hospital window as being characteristic of phenomena known to psychologists as 'peak experiences'. He told me that research into these had revealed that such events were far more common than had been anticipated. Apparently something like thirty per cent of the population had similar experiences in their lives, but most of them were acutely diffident about talking or writing about something that was not only very precious to them but also completely out of the ordinary, and therefore very difficult to describe in words. As a result, he said, there was a shortage of first-hand evidence and consequently anything I could offer in that line would be welcome.\n\nSo I set to and started to try and put something together. I immediately found out that Alan had been quite right about how difficult it would be to find the right words to describe, in the linear mode of language, something that was not only completely unlike anything that had happened before but had also occurred as a single explosion of realisation.\n\nI also came to understand the diffidence that people had felt about talking of something so precious and incongruous. Some of the perceptions I was coming to, although they were true and illuminating to myself, were so radical as to appear absurd, if not outrageous.\n\nThe first of these was that I don't exist.\n\nThat does sound a bit radical, I admit, but what I had realized was that the person I think I am, the self I am self-conscious of, has no actual, continuous existence. I live in this self-image, I work for it, serve it, have ambitions for it and, above all, eagerly seek approval and acceptance of it in the world and the seeking eyes of others. But it is, none the less, an illusion, an artefact of my anxiety which is being created and recreated from moment to moment as it sends up pictures of my potential inadequacy according to the degree of danger that it senses.\n\nI'm sure this is perfectly ordinary, but self-consciousness also (by definition) diverts one's attention away from one's surroundings and in towards oneself. In so doing it acts like a sort of censor to the perceptions. At its worst, self-consciousness can make one idiotically embarrassed and overwrought or, in my case, grimly self-obsessed, miserable and oblivious to anything around me. At its best, in periods of intense concentration, I can become so unaware of myself that self-conscious is suspended. Then I am fully alive and my creative muse, or whatever it is, can just get on with its job. But that suspension is temporary. As soon as I look up worry clicks back. I think this happens because there is always a silting of anxiety lurking in one's mind, not enough to cause trouble but enough to cause some of one's appreciation to be reserved and one's attention to remain wary. I don't think there is anything wrong with that. It is the way thinking beings live and it encompasses a great range of happiness and delight as well as times of ordinary contentment.\n\nI suspect that it is only in very exceptional circumstances, times when, perhaps by some brutality of fate, one's heart has been swept entirely clean of anxiety, that one may stumble inadvertently into a world which, although it is actually still the same one, seems, to one's eyes, to be utterly changed; a world in which one is both fully aware of oneself but at the same time completely given away to, and identified with, everything that one is seeing. That's what I meant when I said 'I don't exist'. Our language seems to have no way to express the perception that one might be a verb rather than a noun. What I am, what I exist _as_ , is the attention. I am the process proceeding, the seeing and hearing of a time-river of perceptions which my mind-body is taking in as they pass and is being minutely changed from moment to moment by the seeing that is love. In that becoming is my life and the confirmation of my existence.\n\nI finished the essays, showed them to a few friends, one of whom, an established scientist and a strict and particular atheist, said he thought they should be burned. I sent copies to Alan and asked him a crucial question \u2013 whether he thought that what had happened to me was peculiar to this particular eccentric person or was in some degree applicable to most people. He replied that he thought the latter was true. So I decided to offer the essays for publication as a book.\n\nThis was not a completely ridiculous idea. The 1970s was a great decade for those seeking fresh insights and new lifestyles. Zen Buddhism was very much in vogue and plenty of books on what might be called 'alternative philosophies' were being published. Most of these, it's true, originated in America but none the less I hoped there might be publishers in England who would find what I had written worthwhile bringing out.\n\nI was wrong about that. They turned it down.\n\nPublishers are not usually willing to give their reasons for rejecting work, but one was kind enough to tell me that because the book was difficult to categorize by way of subject, it would 'present marketing problems'.\n\nA more robust rejection came from a fringe publisher specializing in books on 'self-fulfilment'. He just said the essays didn't deliver the goods. They lacked what he called 'spiritual inspirationality'. They did not provide, as he put it, a 'Do-it-Yourself Nirvana kit' \u2013 which was what he said the souls of his readers were seeking.\n\nI couldn't argue with any of that. My essays were never intended to provide a recipe for bliss, salvation or self-fulfilment and they were definitely not a work of academic scholarship. They had simply reported a fortuitous change that had taken place in the way I see things, and had discussed some of the implications that had arisen as a result. I had hoped that they might provide some food for thought. I was disappointed to find that none of the publishers shared that hope.\n\nI shoved the manuscripts into a drawer.\n\nTo put away the essays was a disappointment, but probably a useful one. Not before time it reminded me of a truth which, in my exuberance, I had persistently overlooked. It is that you can tell people about a marvellous experience until the cows come home, but, unless they themselves happen to be in a state of mind which will allow them to empathize with it, they simply can't be expected to get the hand of it. At best, they might take an academic interest in the phenomenon. At worst they, quite understandably, could become thoroughly bored and irritated.\n\nHowever, for myself, the situation was relatively simple. The episode had made a permanent difference to the way I saw myself, for which, in the years to come, I was to be grateful. And they had also brought some important and long-lasting changes in my feelings towards other people and my picture of the pattern of human life.\n\nIn the past I had gone along with the conventional view that people's actions and attitudes could be 'good' or 'bad', according to a balance between two separate opposing qualities or forces: _good_ and _evil_.\n\nI had now come to see that \u2013 in the same way that in physics there are not two opposing forces, _hot_ and _cold_ , but only one force, _heat_ , and the absence of it (which we call cold) \u2013 there was, as a general rule, only one scale in human attitudes: _fear_ , which is the blinder that closes our eyes to hurt and unleashes aggression, and the absence of it, _safety_ , which can open our eyes to love, life and the possibility of change.\n\nThis meant that some of the actions and attitudes that I had thought of as morally 'bad' could well have been no more than a side-effect of fear, perhaps not of physical fear but of a more subtle fear of personal isolation \u2013 anxiety. So I was reminded, not for the first time, that the key to the clearing of perceptions and the consequent elimination of potential conflict lies in the giving and taking of the safety that mutual acceptance can bring.\n\nJesus Christ, for whose philosophy I have the greatest respect, put it more succinctly. He said: 'Love thy neighbour as thyself.'\n\nI had always seen the truth of that injunction, but it had left me with a problem: where do I find, in myself, the _love_ that I am to feel for the ugly sod? I can command myself to love him but, unfortunately, the only genuine emotion I feel is a sort of guilty inadequacy born of a muted exasperation with him (or her) for being so persistently and wilfully unlovable.\n\nSo love is not to be commanded. It is essentially an effect, the result of some prior reassurance. Consequently I can't expect myself to be able to feel genuine love until _after_ I have received some confirmation of my own acceptedness, in fact until after I myself have received love and been made welcome by it.\n\nThat seemed like a ring with no starting-point and yet I was sure there must be some way into it, sure there was a simple possibility that I had missed.\n\nThe answer came to me in a slightly extraordinary manner, which I would like to tell you about.\n\nIt was early in 1979, Prue was still in Australia and I was staying in London. Sensing that the clarity I had felt was ebbing away, I had become frantically overwrought and impatient with myself, and was also having such frightening dreams that I was almost afraid to go to bed and be alone inside my head.\n\nThen, one night, I dreamed that I had made a boat. I launched it and, somewhat unexpectedly, it floated. Still floating in it, I woke up clear-headed, single-minded and bursting with life and understanding. I ran to our friend Dommi and commanded her coffee and her attention.\n\nMy discourse was probably as confused as it was excited, I don't remember the words, but the gist of it was that somewhere inside myself I had been grappling with a problem that appeared to have no solution, but now at last I had managed to turn round and look at what I had been trying to solve it _with_!\n\nI had been using the conventional idea of 'love' as the only possible starting-point and had been trying to engender it in myself.\n\nThat had been a nonsense! There was no need for me to _love_ anybody! Indeed there was no need for me even to like them, no need to admire, condemn, compare or make any judgement of them at all! All I needed to do was see them, recognize their presence and make room for them.\n\nWas that really all that was necessary?\n\nAlmost certainly not _all_ , but it was a starting-point, and I had just seen, as clear as daylight, how it worked.\n\nI knew, from myself, that most of the anxieties I feel are caused by a hunger for something \u2013 a simple _acceptance_ of me as a person.\n\nThat acceptance may be something which I need. But it is also something which I can give, and give quite easily, because giving it only calls on me to _do_ something. It doesn't call on me to _be_ anything more than I am or to feel any emotions I don't feel. All I have to do is see you and greet the simple fact of your existence, acknowledging that it is the same as my existence, saluting the 'thereness' of you.\n\nI can, by this expression of a perception that is so obvious to me, but which may be far from obvious to you, make room for you in the world, room enough for you to stand up and see how things are.\n\nBut then, if you accept my acceptance of you, I receive a greater gift because that is, in its turn, an acceptance of me, a confirmation of the reality of _my_ existence. Then, as fear and hostility evaporate, one's attention turns away from oneself, life is good and one can do anything that needs to be done. 'Who one is' becomes irrelevant.\n\nBut isn't that what _love_ is about?\n\nYes, of course, but, unlike what we call 'love', _acceptance_ is not necessarily special, precious or particularly personal. It is just a part of ordinary seeing!\n\nLater I came to see that the confusion probably arose from an old crudeness in our language. We tend to use one word, _love_ , for a group of feelings for which the Greeks used at least three different words: _philos_ \u2013 a particular devotion to something or somebody, _eros_ \u2013 sexual attraction, and another word, _agap\u00e9_ , which I would use in its oldest sense ( _agapa\u014d_ : to treat with affection) for the attitude that I was trying to describe \u2013 friendly acceptance.\n\nI see _agap\u00e9_ as being simpler, more profound and much less disturbing than the emotion which we think of as 'love'. But I am sure it is the seed-bed in which love grows.\n\nIn my exuberance I wrote a poem about it. Here it is:\n\n_Agap\u00e9_\n\nI will not lumber you with love\n\nnor climb on you to measure you for sins\n\nnor wipe you over with forgivenesses\n\nnor kick your shins.\n\nI know your eyes do not see out of mine\n\nnor are your tears the tears I shed\n\nbut I don't care,\n\nfor I will take your hand and make a place for you,\n\nbecause you're there.\n\nNot for some complicated ploy\n\nof pity, piety or private greeds\n\nbut for an older, simpler joy that,\n\nnothing wanting, nothing needs,\n\nexcept to live.\n\nFor, as I see you feel the rain\n\nand breathe the air,\n\nso just to know the sun that shines on you\n\nshines on me too\n\nconfirms the sunlight,\n\nmakes it sure,\n\ntells us we live, are there,\n\nthat now will do,\n\nthat here is here,\n\nand asks no more.\n\n# AFRICA AND AFTER\n\n## _I. Elephant_.\n\nIt's easy to find an elephant in England. You just go to the zoo, where, in London at least, they have their own ziggurat-type house and are, apparently, quite well disposed towards visitors. Elephants had always held a special place in Prue's heart and imagination, but the ones in the zoo did not quite fill it. She wanted to see a real one.\n\nFinding an elephant in Africa was obviously going to be a more complicated project, but luck was on our side. Our dear friends Joyce and Alec had been posted to Nairobi where he held the position of Vice-president in charge of Africa for a multinational company. They made us a very generous offer: 'You find your way to our house and we'll set up the safari!' That sort of offer only comes once in a lifetime, so we didn't hesitate. We flew to Nairobi, taking Daniel and their son, James, with us.\n\nIn the event we had a marvellous holiday. We saw multitudes of strange animals, passed lakes rimmed with millions of pink flamingos and travelled immense distances through magnificent countryside, but although we saw recently broken trees and dinnerplate footprints, both of which told of the passing of elephants, it was nearly a fortnight before, after much stalking, we were able to stop our Range-Rover silently in the middle of a dark wood.\n\nThe elephant was directly ahead, standing beside the track, looking at us with dark noncommittal eyes. As it stood it idly lifted pieces of the chocolate-coloured earth with its trunk and flicked them, with studied nonchalance, over its back.\n\nThere was no doubt in my mind that the elephant was waiting for us. We had badgered it and pestered it from all directions and now it had had enough. Alec stopped the car about thirty yards away and we sat and stared.\n\nApart from feeling seriously nervous I felt embarrassed and intrusive. This was the elephant's place, not mine. I had no business to be there. If it wanted to it could walk across and tip the Range-Rover over like a dinner table but, fortunately for us, at that moment, it was choosing not to do so.\n\nI felt the presence of a thinking mind \u2013 not a set of reflex responses but a large intelligence, capable of slow, powerful elephant-type feelings, appreciations that were quite different from ours and perhaps more subtle, which its breed had developed during the millions and millions of years of complex evolution that had brought it to its present peculiar grandeur. I took out my camera and took a shaky picture of it through the windscreen.\n\n'Right,' said Prue, 'that'll do.'\n\nAlec slipped the car into reverse and, without revving the engine, backed quietly off and headed away. This time, as we slithered along the tracks, we kept catching sight of elephants.\n\nAt one point, as we turned a corner, a massive head appeared right beside the car and blasted a trumpet-call through the side window. That put the wind up us. And later, two or three times, we noticed elephants moving alongside us through clearings beside our track, almost as if they were escorting us out of their wood. I was happy to be obliging them.\n\nLater I said to Prue: 'Have you seen your elephant?'\n\n'Oh yes,' she replied, smiling. 'Yes, I have!' And she added: 'I can die now.'\n\n'No, not yet,' I said, and turned aside. Tears were never far away.\n\n## _II. Other Places_.\n\nAfter the warmth and brightness of Africa, England felt chill and sombre. Prue and I returned to Red Lion Cottage, which we had moved into, and took up the threads of our ordinary lives, I to finish the films for Frank Muir and start some _Noggin_ films in colour which the BBC had asked for, and Prue to her various Quaker activities.\n\nThe year moved on. I had finished my films for Frank and was well into the _Noggin_ films. Prue continued with her studies at university. At my sister-in-law Mary's request a kind policeman had climbed through the window of my aunt Nell's little flat in Hove and found her in her bed, quite dead, holding a cup of tea. Mary was very shaken indeed, but perhaps more by the suddenness of the discovery than the sad fact. If asked, I am sure Nell would have chosen just such a death.\n\nThen, in the spring of 1981, I suddenly received an invitation to return to Australia in August to be Artist-in-Residence for six months at the Western Australian Institute of Technology, a large campus just outside Perth. This had come through Peter Efford, who worked there for the School of Art and Design. Prue and I had met him and his wife Georgia when we were in Perth in 1979, and had formed a happy friendship with them.\n\nI was very flattered by the invitation but I felt a bit confused because I had no clear idea what being an Artist-in-Residence would entail. I also didn't feel that I should put such a distance between myself and Prue at this stage of her illness. So I was for turning the job down. Prue said she thought I should take it. In the end we decided to ask the advice of her consultant at the Royal Marsden.\n\nThe consultant was in no doubt about what was best to do. 'Go to Australia,' she said. 'Take Prue with you and let her have another course of the Tronado treatment with Dr Holt.'\n\n'Oh, right-ho,' we said, and I took the job.\n\nAs I drove Kerris's old Datsun through the sunny, spacious suburbs of Perth and headed west towards the Western Australian Institute of Technology, the anxieties I had felt about taking up the post of Artist-in Residence came flooding back. I had written and asked, politely but persistently, what exactly I was coming there to do and had been answered, courteously and enthusiastically, with the assurance that I was exactly the sort of person they would want for the post.\n\nAs I turned into the hundred acre car park I met the same uncertainty as I had felt as I rode down to Dartington Hall School in 1939. And my reception turned out to be much the same as on that occasion \u2013 nobody noticed.\n\nI walked in through what must have been one of the main gates and saw before me a broad, gently rising grassy hill, on the landscaped slopes of which were stacked a number of quite colossal buildings, reminiscent of the massive bunkers which the Germans built as municipal air-raid shelters during the war. Lots of brightly dressed people were there, walking, talking, going about their business. I consulted signposts and in due course found my way to the School of Art and Design, which was my destination.\n\nThere I met Peter Efford running along the corridor. He was obviously busy but he insisted on taking me for a cup of coffee and told me to make myself at home. 'You'll soon get into the swing of things,' he said.\n\nI didn't seem to get into the swing of anything that day. I spent it walking about the campus, smiling at people and doing my best to look at ease.\n\nThe next morning Peter introduced me to the diploma students, half a dozen pleasant young people who, he said, were engaged in individual projects. He told them that I was Artist-in-Residence and they seemed pleased to hear it. I stood about for a while but, as they didn't seem to have any further use for me, I made my excuses and did some more walking about.\n\nAfter a couple of days of this I cornered Peter and said I was sure there must be more to life than walking about and smiling. 'Of course!' he cried. 'You must have an office!' He found me an office, or rather half an office because I shared it with a chap who was in charge of the plumbing, but, Peter said, it would do as a base for operations.\n\nI sat in the office and wondered what operations it might do as a base for. Suddenly Peter put his head round the door, his face wreathed in enthusiastic smiles, and said: 'Film would like you to come and see their presentation.' I jumped up. 'Lovely!' I said. 'What's it about?'\n\n'They're very innovative,' said Peter, 'they do some really ground-breaking work. You may recognize the location. I believe it was filmed in London.'\n\nSeveral people were sitting in the viewing-studio. We joined them, the lights went down and we watched the film, which was in black and white. Although the picture was lurching about because the camera was being hand-held, I could certainly recognize the first shot. It was taken from the end of a carriage in a moving tube train on, I think, the Piccadilly Line, with the camera pointing down the interior of the carriage, looking at the passengers. The shot was held for quite a long time, during which I noticed that the passengers were, understandably, becoming a bit embarrassed by the watching camera. I also noticed that the sound was completely out of synchronization, so that the noise of doors opening and the station announcements could be heard while the train was travelling, and while we were seeing the doors being opened, we heard the roaring of the train through the tunnel. I wondered what the next shot was going to be, in fact I was eagerly looking forward to it because I shared the embarrassment of the passengers. I really didn't want to go on staring at them in that intrusive way.\n\nHowever, this was not to be. That shot continued for nearly half an hour and only ended when a grinning young man stepped into the carriage with a clapper-board and clapped it in front of the camera. The picture went black. The film was over.\n\nThere was some clapping from the audience and after a few moments the same young man, still grinning, came over to us. I was so angry I could have hit him, but I didn't. I said: 'I think I must be a bit slow, but what was the film about?'\n\n'Ah,' he said, conspiratorially, 'it was about _sound_.'\n\n'Sound?'\n\n'Yes! Sound has not received the recognition it deserves. Sound has always been regarded as the poor relation of the other components of film \u2013 pictures, lighting, production, etc. \u2013 those are the ones that take all the credit.'\n\n'I see, but tell me something. This film, the film that you have made, is it intended to be watched by _people_?'\n\nHe looked at me blankly for a moment. Then he spoke quite sharply: 'It's like I said. We're trying to rectify an unfair situation. We are striking a blow on behalf of a neglected faculty. People have watched the film today. People will watch it again.' He turned on his heel and walked away.\n\n'Terrific!' said Peter. 'The passion they have. It really gives one hope! Tell you what... there's a show this afternoon by the Transcendental Realism group. They're a splinter from Film but they use projectors.'\n\n'What? They use projectors instead of cameras?'\n\n'Not exactly, I'm not sure. Shall we find out?'\n\n'Yes, all right.'\n\n'Exciting, isn't it!'\n\nTranscendental Realism came about in four empty rooms in which four people carrying 8mm projectors on their shoulders were walking about projecting films of the interiors of other empty rooms on to the walls and into corners. As quite a number of people had come to the show, most of the images were being projected on to them rather than the walls so that they had to scuffle aside, doing their best to avoid tripping over the trailing cables. As this wasn't easy to do, the projected images tended to lurch about in a rather sea-sickly way and some bad feeling began to arise between the projectionists and their audience. I waited to see whether other transcendental aspects were going to be revealed, but when one of the projectionists fell over and pulled out his plug, so that the place was suddenly in blessed darkness, I slipped away, right away, out of the building, off the campus and back to the flat.\n\nThe next day I made an appointment to see Derek, the Director of the Centre for Communication and the Arts. He it was who had invited me. He could tell me what I was to do.\n\nDerek told me how tremendously excited everybody was to know that I was there and what a privilege it was for the Institute to have me on the campus, but he brushed aside with polite laughter my idea that I should stoop to actually _doing_ anything. It was enough for them that I should _be_ Oliver Postgate. He was sure my very presence would prove to be an invaluable inspiration to the students.\n\nI came away dazed with flattery but still none the wiser about my role. However, I think the penny must have dropped with Derek because members of the staff began to seek me out in order to ask intelligent questions in connection with my subject.\n\nSome of these questions were quite baffling. I had no idea what I felt about the influence of Wladislaw Kotzk on the Trinka school of animation. Did I know where Smallfilms stood in the field of Animation? I didn't know where the field of Animation was, and I had no opinion about towards where the Art of Animation was progressing.\n\nAll this brought home to me with salutory force the fact that my sense of being a fraud was not just natural nervousness, it was well founded. There was another whole world out there, the world of study, a world in which people's reputations were being categorized in some hierarchy of comparison and esteem, a hierarchy administered by its priests, the critics and pundits, a hierarchy about which I knew absolutely nothing.\n\nNevertheless there was no doubt that this was the world into which I had been invited and which I was being paid to serve. But it was, as Derek had indicated, a world in which there was no call for any action \u2013 it was sufficient simply to _be_. Under my laurel wreath I could wear modesty as an ornament to my perfection and float about in a pool of veneration, just being _the_ Oliver Postgate, the doyen of the Art of Animation. The idea was potty, particularly so because none of the students and very few of the staff had ever heard of me or seen my work. What the hell was I doing in this place?\n\nA man in dungarees put his head round the door of my office. He said: 'Cup of tea in the basement.'\n\nIn the basement store-room, neatly cornered behind some folding screens and stacked equipment I found an old kitchen table covered with newspaper, four wooden chairs, and a gas ring. Frank came from Bristol and his colleague, Jan, was Dutch. They were the technicians.\n\nFrank poured me a mug of tea.\n\n'How're you feeling?' he asked.\n\n'Totally buggered. Don't know whether I'm on my arse or my elbow!'\n\n'Thought as much,' said Frank.\n\n'Don't take it too seriously.' said Jan.\n\n'Artist-in-Residence can be a dog's life,' said Frank. 'There was one who was a poet. Ended up standing by the door of the English Department shouting his poems at the people who came past.'\n\n'How come?'\n\n'Students didn't want to know.'\n\n'The students? But didn't the School arrange lectures or anything?'\n\n'That's not how it works here,' explained Jan. 'The curriculum is democratic. It's like a supermarket. The courses are named and their attractions are advertised and they're put up on the noticeboards along with the name of the lecturer who's taking them. Then the students can look them over and sign up for the ones they fancy. If you're a lecturer and not enough students sign up for your course, the course gets chopped and you get chopped with it.'\n\n'So where do I fit into all this?'\n\nJan and Frank looked at each other.\n\n'I don't think you do,' explained Jan. 'No Credits come from you because you're not a course. Your presence is an extra facility. I expect you'll find it advertised on the noticeboards and any student who wants to can come and make use of you.'\n\n'Great! Perhaps they should wrap me in cling-film and set me out on a shelf.'\n\n'I should say that's more or less what they have done.'\n\n'That's nice.'\n\n'I thought you'd like it,' said Frank.\n\n'Thanks for the tea.'\n\n'Any time. That's your mug.'\n\nI was grateful for the company as well as the tea. I was at last beginning to get the hang of my situation and it made me feel slightly sick. I went back to my office.\n\nPeter's smiling face came round the door. 'Coming to the pictures?' he asked, his eyes glowing.\n\nMuch as I loved Peter Efford, and he is one of the nicest men I have ever met, I couldn't quite manage to share his instant enthusiasms.\n\n'Well... which pictures?' I asked, cautiously.\n\n'Oh, a proper film \u2013 _Indiana Jones and the Temple of Doom_. It's on in town. I'm taking a group of second-year media students.'\n\n'What? As part of their course?'\n\n'Oh, yes. We have to keep up with the products of the commercial cinema, and Indiana Jones is a very significant work. It's got Harrison Ford in it.'\n\n'Fine! I'm on.'\n\nSo we just went to the pictures. The students treated it as a hilarious outing and I found it a great relief to be able just to sit there and enjoy the film. Perhaps when it was over I would consider the question of whether and in what way Indiana Jones needed to be regarded as significant.\n\nIn the event that question came up sooner than expected. I felt a tap on my shoulder and one of the students leaned forward to whisper into my ear.\n\n'Excuse me, can I ask you something?'\n\n'Sure.'\n\n'Is it good?'\n\n'What?'\n\n'The film... is it good?'\n\n'Well... er, do you like it?'\n\n'I don't know. I want to like it if it's good.'\n\n'Then I expect it is good.'\n\n'Right,' he said, and sat back, a satisfied customer.\n\nTime passed. I did a lot more walking about and smiling but I didn't get to see the students' work until one day I was invited to attend a showing of one of the students' films.\n\nThis was a live film, not a cartoon, but it wasn't easy to follow because things didn't seem to have worked out very well for the producer. In fact the showing was accompanied by a detailed running-commentary describing the difficulties and failures which they had encountered.\n\nI went to two other showings of students' films and was disturbed to find that each time the same thing happened. The films were muddled and the students gave running-commentaries of excuses.\n\nLater I asked Jan: 'Do they really have such a terrible lot of difficulties?'\n\nHe said: 'They have difficulties, of course, but that's not what it's about. The disasters are a sort of insurance.'\n\n'Insurance?'\n\n'Yes.\n\n'You mean they deliberately make the film badly?'\n\n'Oh no, not deliberately. A disaster happens but I think the producer unconsciously welcomes it because it lets the film off the hook.'\n\n'How do you mean?'\n\n'It shows that the film itself doesn't really matter. It is just a sort of token of intent. What matters and is valuable is not what a thing _is_ , but what it is _said to be_. Hours of conceptual discussion will have gone into that project and the student will have had to prepare a long and complicated spiel setting out its artistic purpose and significance. That is the work of art. And if the film itself can't be seen properly because the production didn't quite work out... well, that's a pity perhaps, but it can't be helped.'\n\n'But that's absurd!'\n\n'I thought you'd like it,' said Frank.\n\nPerhaps, after a while, my lack of employment began to be noticed, because a couple of staff-members, people I had never met before, came by and invited me to join them at an informal seminar in one of the reception rooms. I was happy to accept the invitation and went along. Everybody helped themselves to canape\u00b4s and glasses of wine \u2013 the Institute was always very generous in its hospitality \u2013 and we all sat down. In the silence I suddenly realised that everybody was looking at me.\n\n'It is quite informal,' said one lady reassuringly.\n\n'You mean you're waiting for _me_ to say something.'\n\n'Oh, yes. We're really looking forward to it.'\n\n'But what do you want me to talk _about_?'\n\nShe waved her arms in wide, fulsome gestures: 'Oh, anything, anything, whatever you like!' she cried. 'That's up to _you_.'\n\n'Oh.'\n\nI should have done a runner but, perhaps because I felt myself to be under some sort of obligation, I didn't. I stood up and opened my mouth, but my mind was a complete blank.\n\nTo this day I remember nothing about that afternoon except a dreadful churning of confusion and panic.\n\nEventually I escaped and went back to my office. There I found a printed hand-bill on my desk. It advertised a film made by an Aboriginal group which was to be shown that evening. Pinned to it was a pencilled memo, apparently from the Director of the English Department.\n\n'Oliver! Thought you might like to come to this. G.'\n\nI remembered that when I had talked to Derek at the beginning of term he had told me how eager the English Department was to meet me and have the benefit of my attention. I had been vaguely waiting to hear from them but so far nothing had turned up. Perhaps this was it.\n\nSo that evening I drove back to the campus and made my way to the film and lecture theatre. The lights were on but the big auditorium was empty. I looked into the projection room and asked if this was right for the Aboriginal film.\n\n'S'right!' said the projectionist.\n\nI took a seat and, after a while, the lights went down and the show began. The film wasn't very comprehensible. It was a bit like a home movie that hadn't been cut. After about twenty minutes I got up and went round to the projection room.\n\n'I think I've seen enough, thanks.'\n\n'You the audience?'\n\n'That's me.'\n\n'Good on yer, sport,' he said and pulled a switch.\n\nI drove home rather pensively. I decided that I could take a hint, but I wasn't quite sure what the hint was. Until then I had felt that the Institute's lack of attention had been fortuitous.\n\nNow, however, the Institute had noticed my presence and had twice made use of me. Once it had assumed that I was available as an impromptu side-show, which was a bit crass, and once it had invited me to a film-show which absolutely nobody among the fourteen thousand inhabitants, including the person who had asked me to come, had had any intention of attending. That was contemptuous. I resolved that from that time forth I would be somewhat scanter of my presence.\n\nI think I may have attended other happenings connected with the Institute, but I can't bring them to mind. The last clear memory I have is of attending a lecture on a subject rather intriguingly called Semiotics. The word meant something like 'the science of signalling', but the lecture wasn't about beacons or waving flags or traffic-lights, it was about the subtle way in which, the lecturer claimed, film directors deliberately choose and compose their shots and camera-movements in order to create particular atmospheres \u2013 sinisterness, impending danger, sexuality, and other less definable moods \u2013 and infiltrate them subliminally into the unconscious of the viewer. To illustrate this procedure he showed shots from Hollywood films. Each shot was examined and elaborate psychological motives imputed to its director.\n\nThe lecturer waxed eloquent about how essential it was to analyse the semiotics of any picture or film before one could hope to appreciate the motivation and purposes of the artist or director. He told us how vital it was for everybody to realise that not only a grasp of that discipline, but also the ability to put it into practice, was an absolutely essential prerequisite to any attempt to venture into visual communication technology, how...\n\nAt this point I stood up.\n\nThe lecturer stopped. 'Yes?' he asked.\n\n'May I please say something?'\n\n'Er... yes, I suppose so.'\n\nI addressed my remarks to the audience. I was too angry to be sure exactly what I was saying but I think it went something like this: 'If there are any people here who have any ambition actually to make films or take part in the making of films, may I beg you, please, to be very wary indeed of the highly dubious procedures we have just been hearing about. I have been making films for over twenty years and I am quite certain that if I had allowed myself to become bogged down in the intellectual quagmire which our friend here has just been prescribing I would never have made anything worthwhile in my whole life, basically because it attempts to use the intellect to do something that is the business of the heart.\n\n'I know how I choose the shots I take. I know how all the directors I have worked with chose their shots. They chose them because they _looked right_. What I am trying to say is that I am absolutely sure that to study and collect analyses and then attempt to fit them together and make them into a film would be the kiss of death. You can't reconstitute art from intellectual data. It is a part of love.'\n\nMy outburst was received in silence, which was finally broken by soft solo clapping from the lecturer.\n\n'Spoken like a true Semiotist!' he said.\n\nI went home. The outburst had surprised me as much as it had the audience. I realised that I hadn't so much been sounding off about semiotics \u2013 which is almost certainly a fairly harmless speculation, probably innocent of the charges I was making against it \u2013 as about the whole of my time at the Institute.\n\nUp till then I had remained fairly docile. But, now that I had heard myself say it, I could see that ' _trying to reconstitute art from_ _intellectual data_ ' was a very fair description of what I had seen going on around me for the last four months.\n\nHowever, I also had to accept the possibility that because I had never become fully involved with the Institute's work, my impressions might not have been typical. Consequently I felt it was part of my job to find out whether or not any of its members shared them. So I put together a carefully worded paper and sent a note to Derek asking if I might be allowed to deliver a short lecture one evening and perhaps follow it with an opportunity for questions and answers.\n\nIn reply I received a simple memo-slip to the effect that this was 'not thought to be appropriate'. So my period of engagement drew quietly to its close.\n\nAfter term had ended and I had left the Institute I received a very appreciative and praiseful letter from Derek in which he said he thought I might not realise just how much I had contributed.\n\nHe could have been right about that.\n\n## _III. Departures_.\n\nI don't think my impressions of life at the Institute were in any way false, but possibly my disenchantment with its unrealities was made more poignant by the realities that Prue and I were living with.\n\nPerhaps the worst of these was the black jigsaw puzzle that presented itself each morning. First came the ordinary waking up. Then came the realisation of where I was. Close behind came the sense that something was dreadfully wrong. Numb with reluctance I would force myself to take up the black pieces and fit them together. Although nobody had yet said so, Prue and I both knew in our hearts that the Tronado treatment was not working. Scans and X-rays had indicated this. But could we be wrong about it? The final piece would click into place and be still. No. Prue is on her way to death. Then a small different thought would move in: this is another day. Perhaps it will be a good one, perhaps it won't, but either way it has to be lived.\n\nOf course there were days that could only be bad. The treatment could make her feel weak and nauseous, and for that there was only bed, quiet and some pills. There were also days when a black desolation, quite different from her depression, would fill her heart. Then I could sense that she was out of my reach in some deep, old, not to be mentioned blame, for which she felt she was now paying the price. I can also remember one or two bursts of sudden rage. In one of these she threw her lunch at the wall because I was not her mother and what she wanted was her mother to be there and cook her a proper dinner, with brown gravy, and treacle pudding for afters. That shattered my composure and I fled to Peter Efford's office, where I sat, shaking uncontrollably, for what seemed like several hours. Peter took this in his stride, and so, in due course, did I. There was no issue to be settled, nothing to be explained. That evening I made a good dinner which Prue and I both enjoyed, even though there was no treacle pudding \u2013 which neither of us particularly liked anyway \u2013 and the incident was gone.\n\nPrue and I rarely spoke of death, but I think our hearts must have understood that time was precious and needed to be enjoyed, because all our perceptions and appreciations seemed to be sharpened. The flavours were good and we were given many, so many, good days. Peter and Georgia Efford were unhesitatingly welcoming, so that when a good day came, Prue could simply announce that she would like to spend it in their rambling garden at Gooseberry Hill where Georgia grew twenty different sorts of mint. I would drive her over, plant her in a shady chaise-longue, and leave her, knowing that when I came back from my day at the Institute I would find her cool and content in their kindness, and we would spend a happy evening together, talking, drinking good wine and eating the most excellent food, Peter being a superb and dedicated cook.\n\nFor our wedding anniversary we bought a delicious, soft, light, blue and white dress with a belt of blue glass beads. Prue looked lovely as she twirled in front of the mirror, glowing with pleasure and laughing like a girl. Then, still laughing, she told me to be sure to give it to Carole after she was dead.\n\nLater she fell into another bout of black despair, but this time she dealt with it by ceremoniously collecting up the papers of her Theology course and packing them away at the bottom of her suitcase. I think she put something else away with them. Perhaps it was something fairly simple, like her ambition to gain a BA from the University, or it may have been something more deep-rooted, like her lifelong association with the doctrine of original sin. I don't know but, whatever it was, her spirit became lighter and calmer and we sat together talking, quietly and easily, for several hours. We talked of the past and of my and our children's future, about her faith and my unfaith, about truth and hope, about our understandings and our misunderstandings, about our love, and we made between us a lasting peace.\n\nChristmas came, a confusing time in which the thermometer rose to well over a hundred degrees Fahrenheit. I simply couldn't believe the heat, it seemed to blaze in from all directions. Nor could I believe in the tinselled snow or the plastic icicles in the shop windows. I saw Father Christmas, drunk and scarlet-faced, in full red gear with cotton-wool whiskers, come reeling down the street ringing his bell, his flip-flop sandals squelching into the melting tarmac of the road at each step.\n\nAfter Christmas Prue spent a lot of time in bed. She was much weaker, and soon the doctors told us what we already suspected \u2013 that they could do no more for her than provide ever-stronger painkillers, and wait.\n\nIt was time to go home.\n\nOnce we were back in Red Lion Cottage and Prue was in her own bed, she rallied a lot. Mac, our GP, came to see her. She told him she was going to die soon and asked him for his best painkillers. Mac produced Fred, a new medical gadget which fed tiny amounts of morphia into her system. He looked me over and put me in charge of Fred. He also appointed me Keeper of the Medication Records.\n\nFor a while we put on a grand show. High on morphia, Prue held court from her bed while I ushered in a stream of friends, many of whom came soberly to bring her comfort in her darkest hour and left on a gale of laughter.\n\n'I think you are ever so brave!' said one.\n\n'Of course I am,' she replied, 'I've got no fucking alternative.'\n\nAt first Prue's energy had seemed inexhaustible but gradually it flagged until, on Mac's orders, I had to limit the flow of audiences. This I did, to a degree, but some of Prue's close friends, like Mary Hinshelwood, and Anne, the sister who had been in charge of the ward where I had had my operation some years earlier, continued to be welcome at any time because Prue didn't find their presence tiring, and their help and capacity to cope were invaluable.\n\nDaniel and Simon were already with us and soon Stephen came back from his travels. They took the pandemonium in their stride. When the need arose they would look after Prue. They would clean and tidy her with fond affection and no sign of queasiness or melancholy, do what they could to make her comfortable and just get on with their lives. Their unobtrusive practicality was a great relief to us both.\n\nThe cancer followed its inevitable course and before long finding food that Prue could keep down became difficult. Then her exuberance could switch to anger. I recall that in the middle of one such outburst she suddenly stopped and looked at me.\n\n'You're completely done for,' she said. 'Get in the professionals.'\n\nSo I engaged a regular supply of nurses who would come in and take over. We were glad of their help, which was very necessary, but they also jollied Prue along in a brisk, dismissive way that she found stifling and I found infuriating because, however ill she might have been, Prue was not stupid. So I was glad I was there, glad that I was able to hold her with my strong arms when the sickness tore at her as if to break her in pieces, that I was ready to take the anger and rejection as it welled up, to be winded by it of course, but still able to come back, so that when her arms reached out for me I would be there, to laugh or cry with her and give fate the Liverpool salute.\n\nThe rest of the family began to assemble. Kerris and her daughter Kitty flew in from Australia and Kevan turned up. By then Prue's energy was waning fast. A new nurse appeared. Her name was Pat and she was a specialist. Like the other nurses, she took charge, but she did so with a tender sureness that Prue and the rest of us found profoundly comforting. By now Prue was sleeping a lot of the time, but her waking time could easily turn into an ever-increasing frenzy from which the only escape would be sedation and more sleep.\n\nLate on Thursday the twenty-fifth of March 1982 she suddenly sat up and said: 'Yes, but who is getting the lunch?'\n\nI replied: 'I am, of course.'\n\n'Oh, yes,' she said. 'Yes, of course, so you are, yes.'\n\nShe settled down on to her pillows and went back to sleep. At about ten o'clock the next morning, while we were all sitting around her bed, she stopped breathing.\n\n'Whiskey,' said Pat. She went to the cupboard and poured out stiff doses. I sat with Prue for a few minutes longer. I had my eyes shut, I remember, as if I was trying to see into her darkness, but I soon began to feel _de trop_ , like somebody who has stayed in a lift after it has reached its floor and the gates have been opened. So, rather gingerly, I undid my fingers, got up and walked out into the morning and the rest of my life.\n\nAnd what a glorious spring morning it was, with clean new air and a perfect blue sky.\n\nAs we stood on the terrace in the sunshine Daniel noticed a commotion going on in the wood at the top of the field. A flock of small birds was harassing another, larger bird.\n\nThe bird rose from the trees and headed down across the field. As it passed over our heads the feathers of its face were shining in the sunshine, and we saw that it was a white owl. It circled the house once and then turned back towards the wood.\n\nSeeing it reminded me that, many years before, Ray had told me the Greeks believed that the souls of the dead pass into the bodies of owls. 'Perhaps...?' I thought, but then I put the thought aside. I have always had every respect for the supernatural but never any wish to enquire into it.\n\n## _IV. Rites_.\n\nMy own first, overwhelming, reaction to Prue's death was a flood of blessed relief, relief that the unwinnable battle with so much bitter pain and hopelessness had ended, and had ended so gently.\n\nI felt some pride too, in the knowledge that in the darkness of anger and despair, when blame, rejection, obstinacy and madness had been bashing between us, we had both held to the truth of our friendship and had known the other would not turn away \u2013 so there was no unfinished business, nothing left unsaid or unforgiven.\n\nNow that time had ended for her, Prue was no longer trapped in the racked decrepitude of her last days. Now she was the whole of her time. She was the girl whose warmth, so many years ago, had been my passport to life, whose joy had been my joy and whose certainties had been my strength. She was the loving pivot at the centre of a rich rolling rumbustious family life, and she was the fighter who had defied the death sentence of the doctors and won from fate an extra three years of life. I was proud to have been with her.\n\nWe were all sad, in our fashions, but for me this was nothing like the raging regret that had torn through me \u2013 and Prue \u2013 when Daisy had committed suicide some years earlier. This sadness was quiet, and in it we were quiet together. In fact it felt as if Prue's love for us all had not gone but was still with us, like an affection in the air, lighting our feelings, dissolving antagonisms and drawing us together.\n\nPrue had organized her funeral in great detail. There were to be no flowers and no ceremony, just a simple Quaker funeral \u2013 a Meeting for Worship. This can sometimes be completely silent, so, as many of those present would not be Quakers, she had chosen some short readings.\n\nI'm afraid I disobeyed her in one respect. There was to be one flower, a single rose that I would place on the coffin before it was carried into the crematorium chapel.\n\nWe waited in the car park at the crematorium. The hearse arrived, everybody was ushered in and I came forward and placed on the coffin, not the single rose but only its silver-paper wrapper \u2013 the rose must have fallen out somewhere.\n\nThe undertakers rose to the occasion. The bearers waited in the porch while I and their chief, in black tails and top hat, scuttled through the car park and found Prue's rose, damp and a bit crumpled.\n\nI laid it in place and, chuckling decorously, the bearers carried her into the chapel.\n\nThe Meeting was quiet and full of thoughtful happiness. Kerris read a poem by Dylan Thomas: '... and death shall have no dominion', and Kevan read a piece from T. S. Eliot's _Four Quartets_ , '... in the end is my beginning', which Prue had loved. As I was reading a piece that she had loved even more, Pat the nurse looked up and silently said: 'Prue, if you're watching, take a look at the undertaker's face!' The story I was reading was from _Winnie the_ _Pooh_ by A. A. Milne, 'Eeyore's Birthday', the one in which the donkey is given a 'useful pot' \u2013 and the undertaker had chosen that moment to put his head round the door in order to find out what was causing the unseemly mirth.\n\nOnce again my strongest memory is of the place. We trooped from the bleak impersonal chapel out on to a terrace, and there I saw a sweep of green lawn that led down to a broad vista of fields, woodlands and nesting farmhouses where a tracery of great trees was lit to glory by the hazy sun, and I was overwhelmed by the beauty of it; the peaceful grandeur of the English countryside that had been formed by centuries of summers and winters, of lives and deaths. That unleashed my tears, and, drunk with love and laughter, I put my arms around everybody and we all went home for a party.\n\nA few days later, the children and I, still very much _en famille_ , took Prue's ashes in a tin jar to the cottage in Wales and scattered them around her tree. This was a magnificent ash that stood by itself, looking down the valley which, ten years earlier, she had chosen, or perhaps recognized, as being the place where she would be happy.\n\nThere Kerris, rather diffidently I noticed, played Prue's last trump card \u2013 Easter eggs with small presents for us all. They had plotted this together a few days before she died. It was a lovely idea but, for some reason, I found it slightly unnerving.\n\nWe returned to Canterbury and went our ways \u2013 Kerris and Kitty to a round of visits before going back to Australia, Kevan to some excursion to do with the school where he was teaching, and Krispian back to Italy, I think. I can't remember exactly. Nor can I remember what the twins and Dan were doing. I know there were some films in the offing for me, but, before attending to them, I had to fly to Geneva.\n\n# DECADE OF DEMENTIA\n\n## _I. The Unthinkable_.\n\nSo why was I flying to Geneva so soon after Prue's death? The short answer was: because the Quakers in Geneva had invited me. But to explain why they had done this I have to go back to that morning in 1945 when Olga Wicks handed me the morning paper and said: 'We've dropped something nasty on Japan.'\n\nI remember the thunderous mixture of emotions that came over me at that moment \u2013 horror mixed with an overwhelming pity, but tinged with something almost like relief: the thought that now at last the war which had taken so many millions of lives must end. And behind that came, more slowly, the seed of a deeper and longer-lasting dread. The realisation that the world was now a different place, that what had happened once could happen again, that Pandora's Box had been opened.\n\nAs I rode my rattletrap motorcycle along the quiet lanes of Essex, as I stooked the sheaves of wheat and led the strong brown horses in the harvest of that glowing summer, I couldn't tell myself that anything had changed. Life was going on as before, and it went on peacefully, from day to day, until gradually that dread faded into the back of my mind.\n\nIt stayed there, unnoticed, for many years, in fact through the main part of my life. I remember walking into Trafalgar Square on the fringes of a march to Ban the Bomb and later reading in the evening paper that an earl of my acquaintance, Bertrand Russell, had been arrested for sitting down and inciting a breach of the peace. But, although I applauded and admired his action, I wasn't really sure whether or not I agreed with him.\n\nMany years later, in the 1970s, while I was working on solar energy, I had begun to feel uncomfortable about the government's nuclear power programme. It seemed to be unable, or unwilling, to entertain the well-established fact that its nuclear power-stations were producing quantities of eternally poisonous material which they, quite simply, didn't know what to do with. My confidence in the government's honesty was rattled and I found myself seriously wondering what other pieces of 'necessary' self-deception it might be engaged in.\n\nAs far as nuclear weapons were concerned, I had more or less gone along with the conventional view that the government's policy of 'deterrence' was probably the most sensible way to ensure that no nation would ever be tempted to use, or threaten to use, its atomic bombs for military conquest or blackmail. Deterrence was what made nuclear war unthinkable, and if the price the world had to pay for that security was the nuclear arms race, then it would have to be paid \u2013 or so I told myself.\n\nI had been wrong about that. Not so much about the logic of the policy as about the assumption that the government was following it. There were, to put it mildly, some inconsistencies in the policy literature it was issuing.\n\nFor example: in paragraph 4 of one of the leaflets outlining its defence policy the government declared that:\n\n... NATO's range of deterrents prevent the Soviet Union from even threatening to use the kind of force against the West which it has used against Eastern Europe... The Soviet Union knows that any military conflict between East and West at whatever level could lead to devastating damage from which neither side could escape.\n\nwhile in the next paragraph, 5, it stated:\n\n... NATO's nuclear weapons exist for preventing wars \u2013 not for fighting them. Talk of fighting a nuclear war is dangerous nonsense, because there would be no winners in such a holocaust...*\n\nThat sounded very plausible and a casual reader, or one anxious to be reassured, might have taken it on board without any further thought, but in essence what it was saying was that while on the one hand the government wanted the Russians to know that if they were to step out of line ' _at whatever level_ ' we would not flinch from destroying both them and us in a nuclear exchange, on the other hand it wanted _us_ to know that it didn't really mean it and wouldn't do it.\n\nIt was clear that the government was telling one thing to us and something different and essentially contradictory to the Russians. So the question was: 'Which of us are they lying to?'\n\nThat was an appalling question. Even to think about such a possibility was deeply unsettling. Nevertheless the evidence was there and the question had to be addressed.\n\nOnce again I found myself in a confrontation between complacency and concern, between my desire to ignore it (or find an easy comforting answer) and a chilling suspicion that something might have gone seriously wrong \u2013 that the unthinkable had begun to become thinkable.\n\nThe least I could do, I said to myself, was pay proper attention to the subject. I knew I was in favour of nuclear disarmament \u2013 that went almost without saying \u2013 but, using the clich\u00e9s of the time, I told myself to decide whether I was a Unilateralist, one who\n\nAlthough I, personally, dearly wished the nasty things would go away, and would by nature be a Unilateralist, I could see that they could not be disinvented. So the government's contention that for us unilaterally to abandon our nuclear 'deterrent' would leave us open to nuclear blackmail did make some sense.\n\nMultilateral nuclear disarmament also made sense, and the government maintained that it was pursuing an overall policy of multilateral nuclear disarmament but, it explained, it was doing so in the context of what it called 'necessary modernisation'. As this involved inviting in nuclear cruise missiles and spending billions of pounds on submarines filled with multi-headed nuclear missiles, this was not any sort of disarmament. It was a farce, like crawling slowly down an escalator that was moving swiftly upwards.\n\nThe difficulty was that the public arguments had become polarized between these two extreme views. My impression was that most people felt that if they chose which they would support, they could just pop it into a box labelled 'My Attitude to Nuclear Matters' and get on with ordinary life.\n\nI tried to do that, but it didn't work. The box ticked.\n\nI remember it ticked particularly loudly one evening when I was watching a discussion about defence strategy on the TV. I can't remember the details but they were discussing important considerations like 'nuclear parity', 'windows of opportunity', 'monitored tactical escalation' and 'acceptable collateral costs'.\n\nI tried to follow the discussion but what gradually welled up inside me like a rage was the realisation that the discussion itself was an obscenity, that it could not, _should not_ , be happening.\n\nMy reaction was partly an effect of the language, the sanitized language of nuclear carnage which was allowing them to evaluate nuances of unimaginable horror as if they were 'Best Buys' in some gruesome consumer magazine. I could see that this 'underspeak' was not just technical language, it had been invented to distance its true meaning from the very people who were using it. I suddenly realised that if the participants had seen and felt the reality behind the label-words they were thinking with, the words would have stuck in their throats. I also saw the clear and present possibility that this unwitting self-blinding could lead ordinary people and nations to commit acts of unbelievable inhumanity, simply because they hadn't fully understood what they were talking _about_. As I paced about the room, swearing, I also knew that, quite apart from the terrible language, something else had gone dangerously wrong and, furthermore, that I was going to have to get off my butt and do something about it. I didn't know what, but I was going to have to do something.\n\nI looked again at the government's defence policy \u2013 that of threatening to 'use' nuclear weapons in the face of a conventional military attack, while at the same time maintaining that to even think of doing such a thing was 'dangerous nonsense' \u2013 and tried to see what it implied.\n\nI had already seen that the policy was mendacious \u2013 deliberately ambiguous. But that was nothing new in international politics and if it deterred the Russians...\n\nNow wait a minute. Would it deter the Russians?\n\nThen I realised what had been bugging me.\n\nSeen as a deterrent that policy was completely unconvincing and, further, simply _because_ it was unconvincing it was incredibly dangerous in that it set a booby-trap which could be tripped off at any moment and precipitate the world into self-destruction.\n\nToday this may seem like dry history, but at that time it was likely to be our future and, believe me, if you couldn't block it out of your head, it was terrifying. Let me try to describe the situation:\n\nIf the government was 'misleading' the Russians and wouldn't really unleash Armageddon in the face of a conventional attack, then the Russians would read paragraph 5 and not feel any need to take any notice of the threat.\n\nIf the government was 'misleading' us, and really would unleash Armageddon, then the Russians still couldn't be relied on to take any notice, because the threat to commit national suicide was so absolutely insane that it must be seen as a bluff.\n\nWas the government bluffing? It wasn't likely to say, but the trouble with a bluff is that you have to pretend that you mean to carry it out. The means to do so must be fully installed and on instant automated readiness. Everybody must believe that whatever is threatened will if necessary be carried out.\n\nIn a celebrated speech, the Secretary of State for Defence, Mr Michael Heseltine, told us that we, the people of Britain, must show the world that we have nerves of steel, that we were ready to make any sacrifice in order to preserve our institutions and way of life. His view was that if we were resolute in our determination to combat any aggression with _all_ the weapons at our command, then, at the nuclear brink, we could be certain that our enemies would pull back.\n\nMeanwhile our 'enemies', being, one assumes, of similar mind, would also be approaching the nuclear brink with nerves of steel, resolutely certain that at the brink, we were going to pull back.\n\nSo, with both sides marching resolutely into suicide, what would have been likely to happen?\n\nI don't think either nation would have failed to recognize the danger of such a situation. But, although it is just conceivable that one or the other _might_ have pulled back, in the climate of that time it was by no means certain that they _would_ have done so, or indeed that they would have had time to do so. It was far more likely that both sides would have activated the already threatened response and ordered a _retaliatory first strike pre-emption_. This would have been to launch a massive _first strike_ of nuclear missiles targeted on their enemy's nuclear installations, not so much as a deliberate attack but as a necessary _retaliation_ for the first strike which the enemy might not yet have launched \u2013 but, as they were certainly thinking of doing the same thing, it was obviously essential to _pre-empt_ it and get our first strike in first. This would have been done in the hope (quite certainly vain) that our missiles would have managed to wipe theirs out before they could be launched \u2013 in a sort of mutual nuclear Pearl Harbor attack.\n\nOf course neither side would ever have admitted to having any such intention, in fact NATO devised a policy called 'flexible response' in which it said it would only send relatively small nuclear missiles at first, saving the bigger ones till later, but it seemed obvious to me that that just wouldn't happen. The moment any nation felt that it was in imminent danger of receiving any sort of nuclear attack it would react with a scream of terror that would have had no rational component, and press all the buttons it had.\n\n## _II. Into Gear_.\n\nToday, looking back at the 1980s with its strategies, threats and counterthreats of universal destruction, it all seems like a bad dream.\n\nIt is difficult to bring oneself to comprehend that we, the human race, have just lived through a period during which somebody might quite easily have slipped on a diplomatic banana skin, or perhaps just misread a radar signal, and thereby have unleashed an automated sequence of events which could have wiped all life from the face of the earth, that could have burned off the ozone layer, the fragile skin that protects it from the realities of space, and left it like most other stars, a dead rock, spinning unnumbered years in the void, a place where to all intents and purposes the human race never happened.\n\nBut that was how it was, and, however much we may tell ourselves that because it didn't happen it wouldn't have happened, the inescapable fact remains that it very easily could have, and we are very lucky to be alive. Bruce Kent, the Secretary of CND, called the 1980s 'the decade of dementia', and he was quite right.\n\nIn 1980 the government was not showing a lot of sanity, even in support of its own cause. After nuclear cruise missiles came to Britain it issued a well-meant pamphlet entitled _Protect and Survive_. This told us how to protect ourselves from the effects of a nuclear attack by reinforcing the stairs and hiding underneath them with a mattress, some sandwiches and a chamber-pot, for a fortnight. I think this absurdity must have triggered off an appropriate response in many people's minds, because CND, the Campaign for Nuclear Disarmament, which had almost faded away, was overnight besieged by new members.\n\nI also remember hearing a government spokesman, speaking on television. On our behalf he was balancing the total devastation of a nuclear war against the possible injustices of a Russian occupation and in the end he decided for us '... Yes, better dead than Red.'\n\nI think that famous utterance may have served to concentrate a lot of minds because small Peace Groups began to spring up all over the country.\n\nMy mind had also concentrated quite a lot. In particular I had come to see that my pacifism, my personal intention not to have anything to do with war or weapons, was completely irrelevant to the situation simply because the situation had absolutely nothing to do with either war or weapons.\n\nI had also come to see that the feeling of obscenity which had hit me so forcibly earlier was both accurate and justified. Behind all the sanitized jargon lay the inescapable fact that mankind had made for itself instruments of such power and potential for mass destruction that they could not be 'used' for any purpose except to commit universal suicide.\n\nWhat we and the world were faced with was not the prospect of _war_ in any recognizable sense, but of mindless mutual extermination brought about by what would be no more than a failure of imagination on the part of the world's leaders. To see that this was so one did not have to embrace pacifism or renounce patriotism, or do anything out of the ordinary. One just had to be a person with enough sense and courage to grasp the folly of what was going on. Any old warmonger could do this if he or she was given the chance.\n\nThe key perception, the one that, for me, made everything fall into place, was the simple realisation that the things were not, and are not, _weapons_. It was also clear to me that, although the formal possession of one or two of the nasty things might be necessary (basically because, as they couldn't be disinvented, there had to be some safeguard against some power-mad despot making one and holding the world to ransom), the main bulk of the overkill arsenals held by the super-powers were not in any sense of the word _weapons_ , and consequently the illusion that possessing them gave military power according to the quantities held was not only infinitely dangerous but also, quite simply, insane.\n\nThis showed that the _Uni_ -versus _Multi_ -lateral nuclear disarmament controversy was essentially bogus. It was not concerned with the nuclear dis- _arm_ -ament but with nuclear dis- _junk_ -ment, getting rid of material which was, in military terms, useless, dangerous junk. This process, if it could be seen clearly, would present no difficulties and could be effected quite easily in a way that would pose very little danger.\n\nI talked to people, I put forward these perceptions and found that, in most cases, people either didn't understand them or found them unacceptable. I soon came to realize, with some dismay, that I had got into a position very like the one I was in in 1974. Once again I was 'Bloody Oliver' the isolated zealot, obsessed and driven by propositions that were often alien to the conventional wisdom of most of the parties involved and anathema to the comfortable assumptions on which most people's thinking was based.\n\nAs I had in 1974, I turned to _Woman's Hour_ on BBC Radio and, to my relief and surprise, was once again made welcome. On 30 March 1981 I met Kay Evans at Broadcasting House, where we recorded a short but deeply felt speech to the nation. This, like the one seven years earlier, ended with me in a state of complete emotional exhaustion that called for the administration of comforting words and whiskey.\n\nThe broadcast caused quite a stir. One well-known broadcaster roundly condemned it as being subjective and naive, and its socalled 'political bias' was later balanced with a bland talk by a government spokesman (which, to my mind, tended to reinforce my message). It was included in _Pick of the Week_ and, I believe, _Pick_ _of the Year_.\n\nFor me the main effect of making the broadcast was that having screwed my courage to the sticking-point, I was ready and able to cope with the numerous invitations to speak to Peace Groups that came to me as a result of it.\n\nDuring June and July 1981 I addressed meetings in various parts of south-east England. What impressed me about the audiences, sometimes a dozen, sometimes a hundred or more, was that the people who had given up their evenings to come and listen were all different \u2013 bankers, bookies, housewives, hippies, teachers and tradesmen; ordinary people who had seen the same dangers as I had and had been sufficiently jolted to come out of their comfortable lives and get together to form these small Peace Groups. For a while I seemed to have become a one-man CND, and I would certainly have continued with this but for the fact that in August I had to return to Australia with Prue; she for treatment and I to start my stint with the Western Australian Institute of Technology.\n\nWhile I was in Australia I found the time to put together a pamphlet. I called it _Thinking it Through: The Plain Man's Guide to the_ _Bomb_ , and offered it to the Menard Press. The publisher, Anthony Rudolf, accepted it and, at the beginning of 1982, just as I was bringing Prue home from Australia, it was published. That should have been an exciting event for me, and in a way it was, but of course at the time I had, to put it mildly, other things on my mind.\n\nShortly after returning from Australia I received a phone call from Peter Whittle, at the Quaker United Nations Office in Geneva. He asked me if I would gather up about fifty copies of my pamphlet and, when I felt able to, bring them to Geneva so that we could distribute them to the various delegates and diplomats who were there preparing for the United Nations' Second Special Session on Disarmament, which was due to take place in New York in June.\n\nI eventually reached Geneva in early April. Peter Whittle and I scuttled through the halls and corridors of the Palace of the United Nations, delivering copies of the pamphlet to his many important acquaintances. We also arranged a couple of meetings at the Quaker House at which my words gave rise to anger (in an American diplomat), loud approval (from two Polish delegates), despair (in a Swedish diplomat) and, unexpectedly (from a glorious lady who was First Secretary to the UK Mission) \u2013 tears. She said: 'I may not come to agree with your conclusions in the end, but one thing I know is that you must walk through the world saying what you have to say, because only that way will we come to know what we are talking _about_.'\n\nIn Geneva I stayed with a Quaker couple, Ruth and Nicholas Gillett, who were working at the Quaker House. Nicholas came from a family of 'birthright' Quakers. His father, Roger Gillett, had known my father when he was a conscientious objector in 1916. My grandfather, George Lansbury, had had a lifelong friendship with the Quakers, so he and Ruth decided that I was a 'Birthright Attender'. This was a very Quakerly sort of joke.\n\nNico and Ruth drove a small clapped-out car, and I mean that _they_ drove it, in that one would be at the wheel while the other offered constructive suggestions about whether or not it would be advisable in the circumstances to engage another gear, or mused aloud about the possibility that this might perhaps be a suitable occasion on which one could consider applying the brakes. Travelling with this 'Comit\u00e9 de Conduire', as I called it, was a hilarious as well as hair-raising experience, but we went to some marvellous places.\n\nI remember, with startling clarity, driving up a narrow hairpin road hanging over precipices, to arrive, in the bright midday sunshine, at an ancient hamlet high in the Alps. The steep wooden houses and the wooden sledges, the barns, the cattle in the byres and the steaming stamping horses, were part of a nineteenthcentury painting, as were the weatherbeaten people, wearing swaddled boots and dark bundled clothes. The winter snow was still standing there, more than a metre deep, but from narrow paths and tracks, from around the yards and even from some tiny pieces of garden, the snow had been carved away, so that one walked between cliffs of snow which stood all around, in high, flat-topped blocks. But what hit me, as I climbed out of the overheated car, was the air \u2013 the glorious new air that flowed icily into my lungs, filled my heart with delight and sent the blood tingling into my fingers. In that moment all things were clear and bright, lovely and filled with life and, as was regarded as being altogether proper in the circumstances, I laughed and wept for joy.\n\nI arrived back from Geneva just in time to see the colour films of _Noggin and the Ice Dragon_ and _Noggin and the Pie_ on television. I had finished and delivered them \u2013 when was it? Only a few months earlier, but it seemed like a lifetime ago. Would I ever be able to think of and make films like that again? I had no idea.\n\nPeter and I had also been asked to make a new set of films based on a classic book, _The Doll's House_ , by the famous author Rumer Godden. The start of this was delayed because our contract was not to be with the BBC but with Goldcrest Ltd.., a large film company. They employed a firm of prestigious solicitors who acted very slowly and, I believe, at great expense. I was quite glad of the delay because it meant that I could get ready to go to New York to attend the United Nations' Second Special Session on Disarmament, which was to take place at the UN in New York at the beginning of June.\n\nIn New York I stayed in the apartment of a distant cousin, Peggy Postgate Sage, who had generously moved out and lent it to me while she stayed with friends.\n\nAs a holiday my stay in New York was delightful. Peggy took me around and showed me the sights, and New York itself was a wondrous place where it was exciting just to wander about the streets, watching the people and carrying very little money \u2013 in my socks in case I was mugged. But as a contribution to my campaign the outing was not a success. I had no access to the conference itself but I did manage to persuade a young man at the Quaker office to smuggle the five hundred pamphlets I had brought with me into the building. I also walked in the famous March of a Million to Central Park, but didn't reach it because there wasn't room.\n\nI returned to Red Lion Cottage feeling rather despondent, to find a pile of letters from Peace Groups asking me to come and speak. So I got my act together and went out on the road.\n\nDuring the next three years I spoke to over seventy meetings, wrote three more pamphlets, and issued a stream of leaflets. My contention, that nuclear 'weapons' were not really weapons at all, seemed to be fairly easy for ordinary people to get the hang of, and we had some excellent meetings, but it was still anathema to the proponents of the conventional view on both sides of the argument. I became very used to being called a traitor by Queen-and-Country patriots but it was less easy to accept being rejected by friends who, for want of a better word, I would call die-hard Pacifists. At one meeting, organized by the Quakers, the audience was made uncomfortable by my willingness even to entertain the concept of deterrence. In fact one of them, a man of about my own age, was outraged. He denounced my disloyalty and hypocrisy, and demanded to know how I, a pacifist, who had been to prison for his principles and had renounced all war and violence, could be so despicable as to be seen to condone any situation, any situation at all, in which even one of those filthy things would be allowed to continue to exist.\n\nI replied that the filthy things were there, couldn't be disinvented.\n\nHe told me I was a double-thinker and a turncoat, that I should be ashamed to mix with such vileness. Why could not I, like them, stand up and be counted as one whose life bears witness to the simple truth of love?\n\nThat riled me and I heard myself answer: 'Friend, you are whiter than white! Your purity puts me, and perhaps the world, to shame. But tell us just one thing \u2013 how dirty will you get to save the world?'\n\nI was fairly angry myself but for a moment I feared I was to become the first Quaker martyr \u2013 I mean the first to be martyred _by_ the Quakers \u2013 but his principles prevailed and he stormed out.\n\nAlthough the government usually tended to honour pure pacifists on the grounds that they were harmless, it was less indulgent towards those who opposed its nuclear policies on other grounds. One of its lines was to take every opportunity to allege (quite falsely) that the peace movement was political in purpose and was being funded by the Russians. In this it was greatly aided by members of some of the far-left political parties, known in general as the Trots. They saw the peace movement as having a political use and they joined in enthusiastically, using the meetings and marches as a platform for their own purpose, which was to proclaim that the capitalists were using the arms race for their own profit and that the only way to peace was to overturn the system.\n\nThis meant that each peace movement march, the body of which would consist of peaceful middle-class people just wanting to show their commitment to nuclear disarmament, would be led and accompanied by a fringe of prancing zealots chanting: 'Thatcher Out! Ban the Bomb! Thatcher Out!' That was usually enough to give the passers-by the opportunity they sought to conclude that the peace movement was part of the Loony Left and not worth bothering with \u2013 thus totally and tragically wasting the purpose of the march.\n\nBut, for me, the most difficult thing to cope with was the effect that what I was saying had on some of the pundits of the peace movement itself. Quite early on I had been taken aside and warned, fairly severely, that my 'emotional approach' and my tendency to dwell on unpleasant subjects like the realities of nuclear warfare was counter-productive because it disturbed people. The aim, my aim, should be to take a cool academic approach. In particular I should avoid coming out with radical observations like nuclear weapons not being weapons at all. I was told that people couldn't be expected to take any notice of notions like that because they undermined the established debate.\n\nI was forcibly reminded of this at a forum about tactics which I attended. A leading member of one of the movements ('The Freeze', I think) was in the chair. He said: 'Yes, yes, Oliver, we all know you have some ideas of your own about the things not being weapons, but that's a sideline we haven't time for now. The main thrust of our campaign _must_ be to find a way to persuade the military that it is time to put aside these terrible weapons.'\n\nSometimes it felt as if the whole world had gone mad.\n\n## _III. Indian Summer?_\n\nAnd now, as Monty Python was wont to say, for something completely different.\n\nWhen Prue and I had talked together that day in Australia, she had said: '... but do remember not to be old.'\n\n'What do you mean?'\n\n'Well... you've been kind and faithful and you've had a pretty thin time of it these last few years. Don't waste time being faithful to my memory.'\n\n'Oh, shut up,' I said, and I meant it. Things were the way things were, we both had our work cut out getting from day to day. I had no time to bother with disturbing conjectures about something so irrelevant as my future.\n\n'You're only fifty-six,' she said, 'just a boy!' And we laughed and said no more.\n\nI never consciously followed, or even remembered, her injunction but I do know that my grieving for Prue took what some might think a rather unorthodox form. This was because we had, in our fashion, come to terms with her death and had shared our time of mourning while she was alive. So remembering her and weeping for her was a part of happiness, part of the tide of sharpened appreciation that had flowed into me after her death. That incongruous state of spirit brought with it a sharpened appreciation of, and susceptibility to, the attractiveness of women.\n\nFor some reason I had come to think that loving was lighthearted. What I didn't know, or more likely in my exuberance had forgotten, was that it could also be a key to jealous ownership, emotional manipulation, the imposition of unknown obligations and, in one case, revenge \u2013 in which I found I had been booked in to atone for all the hurt that men had inflicted on women through the ages. After one or two regrettable experiences I met a girl who seemed to be on the same wavelength.\n\nWell no, she wasn't a girl but a woman in her mid-thirties, and I didn't meet her exactly. Prue and I had met her several times and we were already, in a sense, friends. But now we did, as they say, catch each other's eyes.\n\nWe had a marvellous, exciting, non-secret affair, of weekends at the cottage in Wales, at my house or in hotels. Obviously the affair was preposterous, but I enjoyed it absolutely, without reservation, and it lasted, well, three years or a bit less. When the time came for it to end I was seriously sad, but, as requested, I opened my hands and let go. She was warmly, lovingly grateful. It had been wonderful but it was over \u2013 and we went our ways.\n\nBut it wasn't over, not quite. A few days later she rang me and, coldly and carefully, rubbished the lot. The gist of her discourse was that our time together had all been a pretence, a pantomime of pity, that she had been lying to me all along, that anybody who was properly sensitive would have _known_ that she had _had_ to tell me lies.\n\nI was profoundly confused, not so much because she had rubbished _me_ \u2013 anybody could do that \u2013 as because she had rubbished so much glorious wholeheartedly-shared happiness. She had rubbished joy itself, and that I found deeply peculiar and creepy, so much so that, for some time, whenever incidents of amorous affection were shown on the television, I would instantly jump up and turn it off.\n\nTo this day I have no clear idea what inspired that apparently superfluous attack, but my friends told me it was nothing out of the ordinary, told me I was a fool to believe everything I was told \u2013 which indeed I was, and am \u2013 and anyway she might still have been lying. How could one tell?\n\nWhat was abundantly clear to me was that I was woefully ignorant and ill equipped to take part in the battle of the sexes, whatever that may be, and had better stop being a fool and start being my age, which was, I suppose, sixty.\n\n## _IV. Ends of Roads_.\n\nMy dear friends Ruth and Nico Gillett had been on at me ever since Geneva to make a film for the peace movement. I had said that I hadn't got a film in my head, but I would think about the idea. At last, in the early part of 1984, I worked out a script for a film and told Nico that I would make it 'at cost' if anybody could find the money and, just as important, could provide an outlet for the film when it was made \u2013 because there was no point in making a film if it was just going to sit on a shelf in my studio.\n\nNico took the point and almost immediately came up with the United Nations Association, which would be willing to commission a film and, presumably, show it through the many local branches it had in the country, and the Rowntree Trust, which, Nico said, might be willing to fund it.\n\nSo, having finished the first series of _The Doll's House_ and not yet having a contract for the second series, I cleared out my studio and started work on a rough film to be called _Life on Earth_ _Perhaps_.\n\nIt turned out quite well, a brisk exposition of a situation and its implications, with some fairly positive suggestions about what could be done, including, implicitly, a message to the UNA itself about the role it could (and should) be playing in implementing the 1978 UN Resolution on Disarmament. I finished it early in 1985 and showed it at the UNA Annual General Meeting, where it and a speech I made were accorded an unexpectedly rapturous reception. Later I offered it to the television companies, but none of them would show it because it was thought to be 'political'. However, I believe it was seen fairly frequently at UNA branch meetings and I showed it several times at Peace Group meetings.\n\nMeanwhile, ordinary life went on. Red Lion House had been sold, very happily, to a delightful couple, Doris and David, with whom we soon became close, respectful friends.\n\nDan and I continued to live at Red Lion Cottage. _Tottie II_ , the sequel to the first series about _The Doll's House_ , was ready to start filming and I began another year-long stint of concentrated work. This was interrupted when Peter and I were invited to take our films and ourselves as 'key-note' speakers for the BBC at a European Broadcasting Union conference on children's television programmes which was to take place at Tu\u00a8tzing in Bavaria.\n\nWe had some good times there and gave a fairly rumbustious talk, throwing the unfortunate cloth cat, _Bagpuss_ , from one to the other as a 'conch'. That was good fun.\n\nAnother welcome excursion was a holiday at a painting school in an old mill south of Bordeaux, which Mary Hinshelwood and I attended. I had always wanted to be C\u00e9zanne or, preferably, Alfred Sisley, who had painted Moret-sur-Loing so deliciously. So I set up my easel and slapped down about four paintings one after the other, in a frenzy of concentration which completely blotted out everything except what I was seeing.\n\nI wasn't C\u00e9zanne, or Sisley. Looking at the paintings now that my delight in what I saw has faded, I can see that they probably have little connection with what might be called 'art'. What they do have, although the manner of their painting is rough and impressionistic, is a startling faithfulness to the feeling and look of the places. From the photographs I took from where I was standing I can see that I was successfully doing what I was intending to do, putting down on the canvas board, as clearly as I could, exactly what was in front of me \u2013 and loving every moment of it.\n\nThen came the next film project, as usual in an unexpected way.\n\nHaving watched _Tottie II_ on television, a lady in, I think, Cornwall, had sent Peter an absolutely minute Dutch doll, about 1in (25mm) long. It was broken but she expressed the hope that Peter would be able to repair it. While it was with him the doll fell on the floor and was swallowed by the vacuum-cleaner. Peter and Joan had to turn out the bag and sort through the dust and fluff to locate its tiny limbs. Peter duly mended it and made another one for himself. He also made a companion sailor-doll whose name was Victor, who lived in a tiny toy boat on the mantelpiece, while Pinny lived in a miniature china house next to it.\n\nThere was obviously a film series there, but I jibbed at the thought of trying to animate such tiny persons as single-frame puppets. So the films of _Pinny's House_ , which Peter wrote for the BBC, were made, like _Ivor the Engine_ , as cartoons. For thirteen episodes the unfortunate dolls faced every miniature disaster that their world brought, from being picked up by a thrush and fitted into her nest, to sheltering from the giant snow-flakes in an empty snail-shell.\n\nI enjoyed the Pinny films but, even as I was making them, I knew in my heart that this was the end of the road for single-frame filming, the end of my having to sit in the shed for hours on end pushing pieces of card about with a pin. The ship of the family had come apart, the crew had taken to the boats. Keeping me afloat in my rubber dinghy didn't call for such a large income. Maybe the sales of showings of the films we had already made would suffice. For a while I wondered whether to sell Red Lion Cottage and buy myself a smaller, less demanding house, investing the profit to provide a small pension.\n\nHowever, I made no decision and went on addressing meetings and writing pamphlets. I was honoured, and rather intimidated, by receiving an invitation to give a talk at the Yearly Meeting of the Society of Friends (Quakers) which, in 1986, was to take place at Exeter University. I took the precaution not only of writing out a special speech in full but also of writing and printing a leaflet to go with it. This was entitled, in true Quakerly tones, _Towards a New_ _Peace Testimony_.\n\nIn it I set out the situation as I saw it \u2013 the threat of self-inflicted annihilation inherent in the nations' military strategies \u2013 and I ended with an extraordinary suggestion, that on this subject the Society should end its quietism and return to one of its own guiding principles, that of 'speaking truth to power', that it was time for the Society to get angry, time to 'answer to the wrath of love'.\n\nIt went down a treat, as my grandfather used to say. I think Old George would have been proud of me, and, I hope, of what I had had to say.\n\nIn the robust and fairly hilarious informal session that followed, several Friends declared that I had laid a challenge before the Society, a challenge that it must not fail to take up and go forward with.\n\nFor various reasons, which I should have known about, the Society did not go forward with the challenge. My efforts to reenliven its interest foundered and I was later reproved for suggesting unseemly behaviour.\n\nThe world went on being mad. After the Chernobyl nuclear disaster I heard the government's energy spokesman assure us that the inquiry had assessed the _unforeseen_ risks that had caused it and that this had _proved_ that our own nuclear power stations were perfectly safe! President Reagan's answer to the 'evil empire' of Communism \u2013 the Strategic Defence Initiative, known as Star Wars \u2013 grew and flowered. That was a far-fetched nonsense, but it was none the less frightening because for a while it almost seemed as if the American Dreamers were going to believe their own bullshit.\n\nI soldiered on, talked to MPs, lobbied conferences, wrote letters to high-ranking military persons, even wrote another pamphlet, _The_ _Emperor's New Clothes_. Also, at a conference, I met a number of Russian military scientists who seemed to feel much as I did, that the nuclear arms race was, and always had been, a nonsense.\n\nThen, one day, a friend from one of the first peace groups rang me. She said: 'Have you noticed? Gorbachev has been reading your pamphlets!'\n\nI don't suppose Mikhail Gorbachev had been doing anything of the sort, but what he was saying was certainly very like what I had been saying for years, and of course, coming from him the words carried weight.\n\nBy now the Iron Curtain was crumbling and, tacitly encouraged by Mikhail Gorbachev, the people of Eastern Europe were coming out on to the streets _en masse_ to assert their independence, not, I felt, so much for doctrinaire political reasons as because, after years of sparseness, they were moved by a deep universal longing for McDonald's Quarter-Pound Burgers and the Good Life which they symbolized.\n\nThe Western governments doggedly continued to claim that Russia had been _forced_ to the conference table by the success of their policy of resolutely maintaining the 'power' of their deterrents, but I don't think anybody was listening any more. I turned away at last and accepted an invitation to visit the twelfth century.\n\n* _The Balanced View_ , a C. O. I. leaflet issued in December 1981.\n\n# CHANGING DIRECTION\n\n## _I. The Cathedral_.\n\nOne bright morning in the autumn of 1984, the Keeper of the Printed Books of Canterbury Cathedral came to call on me at Red Lion Cottage. She was golden-haired and smiling, far too small a lady for such a grand-sounding post but, as I later rather rudely put it, as bright as a marigold in a school play. Her name was Naomi and she had come to ask me to lend her my celebrated voice to record a tape-cassette guide for blind people to listen to as they walked through the cathedral.\n\nI thought this was a grand idea so we sat at the table in the garden while I looked at the script she had prepared.\n\nAfter a while I asked: 'Are you proud about your prose?'\n\n'Not particularly. Why?\n\n'Well, this stuff is... er, unspeakable.'\n\nI meant that literally; that it wasn't suitable for speaking, but the word was ill-chosen.\n\nNaomi received the blow without flinching and replied with a smile. 'You can rewrite it yourself, then.'\n\n'All right.'\n\nI showed Naomi my holiday snaps \u2013 five large paintings of France \u2013 and we drove laughing back to the cathedral in my old 2CV rubbish-bin car. I don't think it occurred to either of us that this was the start of a long and happy collaboration.\n\nI rewrote parts of the guide and we recorded it within a week or so, and it was done, or nearly done. The cathedral's expert blind man, Dennis Eldridge, tried it out and reported that if he had followed the instructions on the tape he would have gone head-over-heels down the stone steps into the crypt. So minor adjustments were made and it was delivered. As far as I know no casualties were reported.\n\nThe Printed Books which Naomi kept were glorious and included, among other treasures, a veritable zoo of huge tomes, many the size of paving-stones, full of splendid, if somewhat fanciful, hand-coloured engravings of birds and beasts. Some of these, like Gesner's _Animalium_ , dated from the sixteenth century and had to be handled with great care as well as great strength.\n\nThe invitation to visit the twelfth century came in 1986 after Naomi had written, and I had recorded, a short sound-tape on the life and death of Thomas Becket, the 'holy blissful martyr' who was murdered in the cathedral in 1170.\n\nThe cathedral was pleased with the tape and suggested that we might find some pictures to go with it for an audio-visual show. Unfortunately we could find very few contemporary pictures.\n\nIt was observed, by both Naomi and my son Daniel, who was already becoming a professional cartoonist, that if we had been living in the twelfth century we would probably have settled down and spent the rest of our lives sewing a long tapestry, like the Bayeux Tapestry. It also occurred to us that if, in the twelfth century, they had had good quality paper and acrylic paint, they might well have done something a bit less laborious, something that might take no more than about a year.\n\nAs usual, the idea stuck in my mind, and soon I was sellotaping pieces of writing-paper into long strips and doing rough sketches.\n\nIt was obvious that painting a Bayeux Tapestry-type 'illumination' would require more funding than the cathedral was likely to offer, but fortunately my son Stephen's girlfriend, Sarah, was the daughter of Dan, who was Kingfisher Books, which published large illustrated young people's books of an entertaining and educational nature. So, in early 1987, I gathered up the sketches, the painted samples and a synopsis, and took them along to show him. I asked: 'Book?'\n\nDan said: 'Yes, we'll have it. There will be money. Go and do it.'\n\nAs I reeled out of Dan's office I felt a wave of emotion that was both familiar and alarming, a mixture of elation and panic, through which I heard an old warning voice mutter: 'Done it again, done it again!'\n\n'Done what?'\n\n'Taken on something you don't know how to do. Committed yourself to delivering inside a year the sort of thing that other people have taken a lifetime to complete, and which you may not be able to do at all because you don't know the first thing about it. That's what!'\n\n'I can find out.'\n\n'But you're not the right _sort_ of person. You're not an _artist_. It isn't _your_ sort of thing. People are supposed to go to college and do courses, learn about Composition, study the history of Art, get diplomas, before they even _think_ about doing things like that.'\n\n'So what's new?'\n\n'Oh nothing, I suppose,' the voice muttered, and shut up.\n\nThe journey was marvellous. Naomi, being a historian, began sorting out original sources for the life-story of Thomas Becket while I read everything I could find. Together we lived slowly through that time, distilling from it in words and pictures a clear and awful impression not only of the lives of the people and their fervent concern for their future in the afterlife but also of the two pig-headed autocrats: the wilful priest Thomas Becket and Henry II, King of England (and quite a lot of France), loving friends and raging enemies by turns \u2013 who fought out their personal squabbles on the field of history and government, wrecking the day-today administration of the nation and making a thorough nuisance of themselves.\n\nThe story was long, complex, exciting, passionate and radiantly unnecessary. It was a struggle which Henry very nearly won, and he probably would have if Thomas hadn't chosen to play his final trump cards: violent martyrdom and a pair of hair-drawers infested with fleas and lice. These finally confirmed his absolute holiness, brought him to sainthood and gave the Church its victory over the state.\n\nDoing the painting was a real pleasure, not only because it was a great relief to be back at work, proper commissioned work, but also because I could see a thin time coming up.\n\nOur films were definitely going out of fashion.\n\nThe reasons for this were kindly explained to me by a member of the department at the BBC. Apparently an edict had been issued by the powers-that-be to the effect that the _viability_ of programmes (i. e., their worth and chance of continuing), was henceforth to be _ratings-led_ (i. e., judged by the number of people watching). As the purpose of television was to entertain (as opposed to instruct or educate), the basic policy was to give the children exactly the sort of thing that they were already known to enjoy and deliver it in a form and manner that was especially exciting.\n\nI remember it occurred to me that if one was seriously determined to eliminate a whole culture and wipe out everything that had gone before, the way to set about it would be to rot it from the inside. This could best be done by ensuring that the next generation of our children, whose minds were new, empty and waiting for knowledge, were given nothing except what they _already_ knew, or had been told, was what they liked \u2013 in effect forcefeeding them with a diet of manic jelly-babies.\n\nBut, quite apart from that, it was explained to me that impeccable American educational sociologists had established that in order to prevent a child switching channels (and thus transferring the rating to another channel), a programme had to have a _hook_ (i. e., an incident sufficiently violent to re-attract the attention) every three and a half seconds. Our programmes did not have this characteristic and consequently, whatever other qualities they may or may not have had, they were not to be considered suitable for television transmission.\n\nSo the new Head of BBC Children's TV 'had no plans' to show _Bagpuss_ or any of our films, nor did they wish to commission any new ones.\n\nNevertheless, whatever the TV companies felt about them, our films were not quite dead.\n\nQuite unexpectedly, the University of Kent at Canterbury invited Peter and me to receive honorary MA degrees. The Vice-Chancellor let it be known, in the most tactful way, that the degrees were being awarded to us ' _in loco Bagpusis_ ', because the true recipient, being a cloth cat and a fiction, was unable to receive it personally. However, it was intimated that Bagpuss would be welcome to attend the Degree Congregation at the cathedral if he were appropriately robed. Joan Firmin promptly made him a perfect mortar-board, hood and gown. Apparently one or two senior members of the University found the prospect of a cloth cat in academic dress taking part in the ceremony an affront to the dignity of the Foundation, but I understand that most of them entered into the absurdity with relish.\n\nIn the event the occasion passed off with perfect decorum. Naomi brought the cat into the cathedral in a covered basket and, once she had convinced the Head Verger that it was not a bomb, the basket was placed under the lectern and she took her place in the academic procession.\n\nBagpuss took no part in the ceremony until the time came for me to make a speech of acknowledgement on his and our behalf. Then, as I lifted him from his basket, there came from the assembled multitude a great roar of cheers and applause which continued for some moments.\n\nJoking apart I was deeply moved, honoured and privileged by this piece of appreciation, not so much of myself personally, as of the work that Peter and I had for decades quietly churned out and fed to the voracious television. To see and feel, in that great cheer, a recognition that the pleasure which we had felt in bringing it into the world had in fact communicated itself to a whole generation, was an enormous, heart-warming joy. Over the years the surly old cloth cat had become the object of a great tide of affection. It made me very happy, and I returned to the twelfth century much refreshed in spirit.\n\nOctober 1987 brought two disasters. One was for everybody \u2013 the hurricane. This came in the night, buffeting the house like a gigantic boxing glove, bringing down trees and tearing off roofs.\n\nThe other disaster, which came at midday on October 30th, was for Naomi alone. She turned to her colleagues and said: 'I feel terrible. I must go home.' She had been stricken with a heavy tiredness that would not go away. My bright, swift, glowing friend was still there but, like a car with a dud battery, her energy would fade, her face would turn grey and she would have to flop down.\n\nNaomi tried to go on working but it was impossible. After many tests she was diagnosed as having myalgic encephalitis, known as ME. This was a relatively new illness which was not recognized, and was indeed resented and dismissed, by quite a number of GPs because the causes of it were not known. However there was absolutely no doubt in our minds that what Naomi had was a real illness, a genuine clinical condition that could not be overruled or written off as hysterical. Fortunately her GP was of the same mind. He supported the diagnosis and helped her to arrange her retirement. He did all he could for her, and we set about rearranging our lives around her condition.\n\nOur working partnership had long since grown into a loving friendship. I set up a bed for her in my study next to the Becket easel so that she could rest while I worked. Her husband, Geoffrey, was a kind man but not as used to looking after people as I was. When Naomi was rotten he encouraged her to stay with me.\n\nWe discovered quite early, but alas not early enough, that trying to force oneself to soldier on through ME is the most damaging approach. Naomi had tried to do this and had become totally exhausted and shattered. Doing this is probably what set the illness firmly in place and has prevented her recovery.\n\nNevertheless, life was possible. Once she had managed to recognize and accept her limitations and allowed herself to have a ticket to be an invalid, we were able to arrange our lives quite well. She would spend part of her time with Geoffrey and Fred, their ancient and honourable black cat, and part of her time with me and Thomas.\n\nWe finished Becket, slew him, entombed him, surrounded him with pilgrims buying phials of his blood and even found a corner for Erasmus to stand and look at the gold and jewel-encrusted tomb nearly four hundred years later and ask a passing angel what was Christian about all those heaped-up riches.\n\n'Search me,' replies the angel. 'Nothing to do with us.'\n\nNaomi put the finishing touches to the text and we delivered the work to the publishers.\n\nSadly, when eventually it was published, the book was badly printed. Naomi's and my year's work appeared washed-out and fuzzy. That was deeply disappointing.\n\nHowever, we now had the Illumination \u2013 fifty-four feet of handsome epic.\n\nThe first thing I did was set it up in the studio and make a film of it, a film which would later be made into videos. Then I had a set of portable stands and lights made to display the Illumination as an exhibition and looked forward to setting it up and showing it in some suitable setting, like the Chapter House of Canterbury Cathedral. Unfortunately the cathedral authorities declined to allow it to be shown, so that put an end, for the moment, to our ambitions.\n\nEarly in 1989 a small theatre company in the Midlands wanted to do a play about _Ivor the Engine_. I put together quite a good play but the company, which had engaged a pair of budding pop stars as leading actors, didn't really get the hang of it, so that apart from my friend Olwen Griffiths, and a young actor called Peter Robbins, who was a magnificent Dai Station and hilarious as Bani the elephant-keeper, the production was sad and ineffectual.\n\nAlthough by then Naomi was not very mobile, often using crutches to walk any distance, she really enjoyed travelling by train. So in May, once we had discharged our responsibilities to the Ivor play, we left it to stew in its juice in the Midlands and took the train to Granada, a journey which took two whole days. My son Simon, who with his Spanish girlfriend Dolors was living and working there, had found us a delightful apartment in a tiny narrow street. From this we made sorties into the centre of that intense city, where every day seemed to be a fiesta and, on platforms set up in the squares, fiery tiny tots in full costume were twirling and clicking among the stamping ferocious flamenco dancers.\n\nThen, suddenly, in July 1989, came the break we had been waiting for: a suitable setting for the Becket Illumination \u2013 two months at the ancient Blackfriars Priory beside the river in the centre of Canterbury.\n\nAt first very few people came to see it. That was definitely depressing, but gradually things changed. More people, many of them local, began to come in. By the time our booking was up and we had to close, the visitors were queuing on the steps. We could have continued through the rest of the summer and probably made a bomb, but the Blackfriars had to return to its usual function as Art Room for the King's School.\n\nFortunately, during the exhibition, Naomi and I had received another invitation. This one was to the fifteenth century.\n\n## _II. Triumphant Failure_.\n\nThe fifteenth and early sixteenth centuries were an era of great explorations and discoveries in which the boundaries of the known world were much extended, and, as everybody knows, the greatest of these explorations took place in 1492 when Christopher Columbus sailed across the Western Ocean and located at last the direct sea-route to Japan, where the roofs were said to be made of gold. That was his opinion anyway. The fact that, before he was anywhere near halfway to Japan, America got in the way and so undoubtedly saved him and his crews from starvation, did not alter his view of the achievement. He knew he had, near enough, discovered Japan, and to his dying day he never recognized the now obvious fact that what he had stumbled on was a completely unknown continent.\n\nFour hundred and ninety-seven years later the publishers of _Becket_ rang Naomi and me at the Blackfriars exhibition and asked us to do another book, in the same format, to depict the life of that enterprising mariner and his epic journey. This would be published in 1992 to celebrate the five hundredth anniversary of the event.\n\nWe readily agreed, but on condition that it would be properly printed, that we would make it our next job and start almost immediately. That was agreed and settled, except that the contract didn't come through for four more months, during which time I was out of work.\n\nThat was a nuisance, but there were other things to be done. Naomi's condition had deteriorated and she had at last consented occasionally to be pushed about in a wheelchair. Also she had already had enough of sharing a room with Thomas Becket and wasn't looking forward to sharing it with Columbus, so I cleared out the basement of Red Lion Cottage, removing several decades of stored treasures, and with hired help converted it into a rather pleasant flat with its own French doors to the garden.\n\nOne morning, some months after my last peace pamphlet, _The_ _Emperor's New Clothes_ , had been published and forgotten, the news came through that the Berlin Wall had been broken and was coming down.\n\nI remember the feeling of relief and exultation that slowly grew inside me, bringing back the fire that had driven me in the 1980s. But this time the fire that filled my veins was not of dread but of gladness, the stone of fear-full misunderstanding that had lain across the world for so long had been rolled aside and the Cold War had been revealed as an absurdity. I rang Nico Gillett and laughed and wept over the telephone. Neither of us knew what the future would bring, nor could we foresee the spiteful worms of petty national strife that would crawl out from under that stone, but for us that day marked the end of the decade of dementia and, we dared to hope, the return of common sense to international affairs. I knew my work was done. It had not been done by me, that was for sure, but it was over. I could turn to Columbus with a lighter heart and a cleared conscience.\n\nAnd what a marvellous idiot the old mariner must have been. A man truly after my own heart, he pursued his vision through all adversity with an absolute persistence that was entirely oblivious to the intrusion of facts. In those days theological correctness was a far more decisive consideration than scientific accuracy or common sense, but even so the theological case he made for his project was far from sound. He was also shameless in his 'adaptation' of mathematics and unbridled in his indulgence in what amounted to pure wishful thinking. This was because he knew, as in his day Thomas had known, that his destiny had been given to him by God and that in following his Great Enterprise in faith and fortitude he was serving no other master.\n\n_Columbus, the Triumphant Failure_ was delivered in good time to be published for his quincentenary. This time the publishers had the plates made more professionally and the proofs looked really good. Unfortunately the books were printed in some far-flung colony and appeared with an all-over brownish tinge as if they had been flavoured with soy sauce. Once again I wondered why I had spent so much time and care on getting the colours just right. I resolved to make no more books like that.\n\nThere was more painting to be done. Shortly after the Becket Illumination closed, yet another invitation had arrived. This came from Dr Shirley Barlow, the Master of Eliot College at the University of Kent at Canterbury, who asked us whether, when we had finished paddling about in the Western Ocean, we would care to prepare and execute a mural for the college.\n\nThis, she suggested, could be a light-hearted depiction of the history of East Kent in general and Canterbury in particular, from the end of the last ice-age to the building of Sainsbury's new superstore.\n\nI must say I was more than a little dubious about the project, partly because of the size \u2013 twenty feet wide and eight feet high \u2013 and partly because I knew absolutely nothing about the relevant history. However, as usual, the idea took root.\n\nNaomi and I raided the library and consulted her expert friends. We uncovered a rich vein of pleasantly irreverent incidents and dramatic situations, including the fishmonger who was so irate with the insolence of the clergy that he 'smote a monk with an halybut'.\n\nSo, while I was finishing Columbus I put together a sketch for the mural with Moses in the top left-hand corner receiving tablets from a faceless Jehovah, and Karl Marx in the top right-hand one, hovering over the University, flourishing _Das Kapital_. With these deities in place I arranged a complex tableau of incidents, more or less historically consecutive, around a central group of pilgrims heading for the tomb of the Holy Blissful Martyr.\n\nThe University liked this and, at my request, commissioned a quarter-size colour cartoon. It also agreed, at my request, to be patient, because I was growing old and slow and less able to keep half a dozen projects in the air at the same time.\n\nBy this time Peter and I had more or less accepted the conclusion that our films had had their day, so we were quite surprised to hear from the Children's Channel, a satellite and cable company, that they wanted to buy showings of all our colour films. That was a marvellous piece of good luck and it did wonders for the bank account, but it also meant that I had to chase around to find the original films, check and assemble each one of them, have them cleaned and, eventually, transferred to master videotape. As well as working on Columbus, Naomi and I were giving talks on Becket and overseeing the Illumination exhibition as it travelled around the country. I was also writing eight books about _The_ _Clangers_ and, as if that wasn't enough, Naomi and I had started on a detective novel.\n\nThis was called _St Badwyne's Tower_ , and it was set in a strictly fictitious cathedral city where there was financial skulduggery in the precincts. This was duly thwarted by the discovery of a corpse, but unfortunately the corpse was dry and about three hundred years old. It turned out that today's publishers were only in the market for fresh corpses with ample gore, and sex every fifty pages, which hadn't been our recipe and didn't fit with the plot. So that had to be abandoned.\n\nI was also invited to give a couple of informal seminars to the Animation School at the Royal College of Art. These were so informal that they could hardly have been said to happen, but they taught me more than I really wanted to know about the way in which our simple direct craft had been inflated into a manic pretentious pseudo-art. This was probably more than the students learned about my work, ordinary story-telling, because that subject didn't seem to interest them at all.\n\nThey wanted me to see how visually stunning their work was and how well it fitted with the current state-of-the-art fashion, in fact how 'good' it was.\n\nWhat they didn't want to know about and seemed quite unable to take on board was my perception, gently, perhaps too gently, put, that you can't really ask how 'good' a piece of work is by itself. You can only ask how well it does what it is setting out to do. A film is a communication and a communication has to communicate something. It doesn't matter how glossy, clever or avant-garde the pictures are, if they aren't _about_ anything, they are about nothing, and consequently their success can't be judged.\n\nThat was what I had been trying to say ten years earlier in Australia: that there are, in every production, two components, _what_ the work is about and _how_ it is made. They are both necessary but the _how_ is essentially the servant of the _what_ \u2013 if only because a marvellously made film about nothing is still a nothing.\n\nThe _what_ of our films were their simple, ordinary stories, simply, but beautifully pictured by Peter. The _how_ was the animation, the simple, even crude, method by which they were made, a method which was essentially congruous in that it conducted the eyes and ears of the viewer through the story with the minimum of distraction. It is true that we couldn't have made them any other way because we couldn't have afforded to, but I don't think we would have wanted to. They were, and are, all-of-a-piece as they are. In fact, over the years, I have come to see that much of the enjoyment and affection that our films are held in came from the direct, unaffected direction. They tell their stories simply and do not bombard the viewer's attention with zappy distractions.\n\n## _III. Winding Down_.\n\nWorking on the mural, which was being painted on four huge plywood panels set up in Peter and Joan's barn, meant that I had to get up every morning and go out to work. Being back in harness again, doing visible, concentrated work in the place where I had made so many films, seemed to make my mind relax and I found I was able to stop panicking about my failing memory and take time over what I was doing.\n\nNaomi, who had been so active and energetic before the onset of ME, was now dependent for most things on the services of either Geoffrey or myself. This she found frustrating, so she reorganized her finances and bought a small town-flat overlooking the sea at Broadstairs. From this she could potter round to the friendly corner shop on her wheeled walking-frame and, for a few days at a time, be independent. She still loved to travel by train, so every year we went in style, by autorail and wagon-lit to La Roche Canillac, a small steep village in the Corr\u00e8 ze, where we always took the same crummy g\u00eete, owned by an aged but astute pair of country people. There we made good friends, first a warm and hospitable, if a shade formal, Parisian couple, Line and Ren\u00e9, who had a most perfect little house and a garden full of roses. Later we met a German couple, Liza and Charlie, whose house was perched above a terraced garden as steep as a giant's staircase. In Charlie I found a great friend, a fellow madman, an Anglophile who loved original thought and ideas. We all got on famously together in a random mixture of French, German and English. I also discovered, by accident, signs of Le Tacot, a long-extinct railway, and spent many happy hours tracing its route.\n\nAs I approached the completion of _The Canterbury Chronicle_ , as the mural was to be called, I knew that this was the last large-scale painting that I was going to attempt. The panels were occupied by over three hundred people, most of them up to no good, and I was weary.\n\nThe University Works Department made a complete pig's ear of setting the mural panels up. Roistering students flicked spoonfuls of brandy butter on to it, which damaged the paint. I arranged to revarnish the painting. The offer was accepted but the Works Dept. didn't put up the staging for me to stand on. Eight years on I am still waiting for that, but I believe the people concerned have now retired. Theirs is another world, one in which people have salaries.\n\nThe fate of the Becket Illumination was slightly different. In association with the Victoria and Albert Museum, the Canterbury City Museums Department bought it to show in their grand new gallery. The grand new gallery didn't materialize, so the Illumination is now buried, possibly for ever, in the vaults.\n\nI paused and asked myself why I needed to keep throwing myself into these large projects. Why couldn't I be quiet in my mind and enjoy my retirement? I was _the_ Oliver Postgate. Could I not pin medals on my chest, put on dignity and rest on my laurels? For some reason I found this thought quite appalling, but at the same time I could see that the unquietness which drove me was both unnecessary and regrettable. I vowed that I would avoid committing myself to large projects and turn my attention to small things.\n\nNaomi's hands had lost their strength and had become painful. I saw the need for two articles. One was a wrench or something for opening jars, turning taps and doorhandles. I tried out several contrivances but the one I finished up with was a simple strapwrench with a shaped handle. This was so elegant and simple that for a moment I thought of manufacturing them. The other article was a plastic hook to open ring-pull cans of fish and roll off the lid without covering everything with oil. That was also quite elegant, being a simple shaped tool with no moving parts. Both of them were already patented, so I couldn't market them. This, I have to admit, came as something of a relief.\n\nThe next project was a bit larger. Our adopted village, La Roche, was built on a steep hillside around which the narrow roads writhed like serpents. Naomi was nervous of being pushed up and down the village, partly out of care for my still willing but goutridden knees but mainly for fear that I might accidentally let go of a handle or even that my dubious heart might pack up under the strain and she would be unleashed, like a loose roller coaster, to hurtle off down the hill.\n\nSo I turned my attention to designing a battery-driven outboard motor to fix on to her ordinary folding wheelchair. I set up a workshop in the garage and put together a fairly robust prototype made out of old plywood, luggage elastic and standard engineers' bits.\n\nThis one drove directly down on to the big wheels of the chair and, as the drive could be disengaged from each wheel separately, it had, in effect, power-steering. It worked really well and we took it to La Roche, where it proved to be a great wonder to the populace and a great blessing to us.\n\nIn 1995, my seventieth year, we lived quietly. I formally dismantled my much-loved, worn out, gimcrack filming equipment, cleared out my studio and divided it in half so that a neighbouring horse could live there. Simon and Daniel had both come back to live in small rented houses in Whitstable, Daniel with his new girlfriend, Rochelle. Now and then they came up to visit or for a meal, but they didn't take up residence.\n\nSome time earlier, I had bought the flat below Naomi's in Broadstairs so that if I were to become old I might one day retire to the seaside. Although I had high blood-pressure, angina, dicey kidneys and various other boring old person's ailments, I was in fairly good nick and didn't intend to be old for quite a long time.\n\nHowever, fate had other plans for me. In 1996 it set up something of a demolition campaign. This started with a small operation to remove a carbuncle from the middle of my back. That left a hole which had to be dressed every day by visiting nurses. That healed itself and in April we were able to take ourselves to a village called Fressin, near Hesdin in the Pas de Calais, where we stayed in a rather luxurious g\u00eete owned by an English lady.\n\nThe country around there is dotted with small graveyards, tiny gardens of headstones marking the passing of groups of soldiers who died in the 1914\u201318 war. These were not unexpected but they were, none the less, a sharp reminder of that grey, bitter time.\n\nIn contrast we stumbled on the site of a single occasion of colourful carnage, the field of Agincourt. This was a long ploughed field set between two clumps of woodland, an entirely unremarkable piece of countryside on which one would not expect anything out of the ordinary ever to have happened. But in the village we found a small museum which recounted, in startlingly graphic terms, the details of the pageant of pure slaughter that had taken place on that dull ground over five hundred years before. It also told how, after it was over, the victorious English stripped the dead of their armour and weapons and left. Then the monks of a local monastery dug a great pit and without discrimination tipped the thousands of naked corpses, the flower of France's youth, into it. They filled in the pit, said a prayer, planted a cross and left it. A cross, perhaps the same one, is still there.\n\nWhat came to both Naomi and me, that day and in the days that followed, was an overwhelming sense of the folly of it all. Agincourt had been the ultimate cock-up, in which wave after wave of the proud unthinking valorous French trampled their own dead and dying into the mud on their way to commit suicide against the new armour-piercing arrows of the English archers.\n\nWhat came to us too was an inescapable perception of something which we seem to have lost \u2013 an appreciation of the simple, absolute, God-awful painfulness of pain. That appreciation is gently honoured in those small gardens of headstones for the soldiers of World War I, but otherwise it seems to be almost completely missing from much of our lives.\n\nThen, in May, as I was driving back from Wales in a hired car, I was involved in the first major car accident of my life \u2013 head-on with a lorry. I was lucky to escape with my life and eventually returned home like a refugee, shaky and hung about with my possessions in plastic bags.\n\nA few days later, by appointment, I presented myself at the Day Surgery Ward of the Kent and Canterbury Hospital to have my ancient gall-stones removed by a procedure called laporoscopy. This was a very modern, simple operation done by remote control which, I was told, would only require one day's stay in hospital.\n\nThey were wrong about that.\n\n## _IV. The End of the Operation_.\n\nI regained consciousness and immediately regretted it. My belly was vastly distended and was filled with a pain so violent and absolute that I had to concentrate with all my might on the task of living from one moment to the next. My body was incessantly crying: 'This is not possible. This pain is not bearable.' My mind was replying: 'This is pain. You have no option.' People loomed over and spoke to me but I couldn't take in what they said or cause my mouth to make words. The surgeon came to tell me things but I couldn't find the attention to hear, holding the pain and nausea at bay was taking every moment of my consciousness. In the end a nurse gave me a huge injection which released my attention. The surgeon then told me that for some unknown reason my blood-pressure had dropped drastically during the operation and they had had to abandon it. But there was nothing to worry about. There was a slight swelling, a haematoma, in my side, but everything would be normal in a few days.\n\nThey were wrong about that too. It was a massive abscess and I had acute peritonitis. For this I was treated with cocktails of antibiotics for six weeks, during which time I was shunted in and out of hospital, had another major operation and ended up being sent to a terrible and, it turned out, terribly expensive nursinghome.\n\nAs had happened in 1978, the experience of moving close to death and coming back again caused in me an emotional upheaval which overturned my habits of thought and for a while allowed me to see again with the eyes of my childhood.\n\nSmall, as if through the wrong end of a telescope, I looked back at myself, or rather at my selves. I saw the affectionate, anxious-to-please infant becoming the smart-arsed show-off who, having been definitively labelled as contemptible, strove to show himself to be cleverer of funnier than he was, in the hope of just being allowed to be there. And in the same skin was a boy who had been born with a clear eye and an eager, impetuous, uncompromising imagination which, if anything, made him even more of a pest. He merges into the schoolboy, a facetious, posturing prat who simply wouldn't be _told_ anything but would, persistently, keep questioning the questions.\n\nThat was all old stuff and, I suspect, very ordinary, but those children are still within me, muttering even as I write.\n\nI saw the young man, a sheep, uneasy in ill-fitting wolf's clothing, stumbling confused through a life in which the only positive actions seemed to be blind leaps in the dark, gestures of solitary bravado in which I had held to my own perception of an idea or project and just shoved it through. Sometimes I got away with it, more often it had foundered. Once I got away with it beyond my wildest dreams and found I had set myself up to receive the greatest gift I could ever have hoped for, that of becoming an agent of delight, the means by which a river of fun and sheer enjoyment would be unleashed.\n\nI meant it when I say I was the _means_. The twelve or so worlds which Peter Firmin and I put together, peopled and gave life to, are other places, with an existence of their own. I am always delighted when people come and tell me how much they enjoyed them, and I love their joy, but I am not just being modest when I say that I did not _create_ that joy. I was just the cook. I also know, beyond a shadow of doubt, that if my primal purpose in making the films had been 'me-based', i.e., to enhance my personal glory, I would have produced crud, formula crud perhaps, but without integrity or life.\n\nNot all my perceptions were simple and full of fun. Sometimes what I saw was so clear and dreadful that I had had to do the unthinkable \u2013 break faith with complacency \u2013 make a public exhibition of myself and, as one kind friend put it: 'become a member of the ranting class'.\n\nBut I could also see that I hadn't broken a faith. I had held to one: not to faith in God or ultimate goodness, but to faith in reality, nasty reality, the reality of danger. I had made the connections and held to them.\n\nNow, as I clambered out of the valley of the shadow and the fear of death began to fall away, another feeling, something far more powerful, lifted me. This was an enormous elation, an overarching feeling of relief and delight in being alive, which was, and was expressed as, a profound love for all things and, in particular, for the long-suffering people nearest to me. It blew away all the baleful reserves and cautious inhibitions and bade me reach out my arms to them.\n\nI came away from that episode several years older. Something, perhaps the antibiotics, had snarled up my nervous system so that I couldn't write properly. My pencil moved the opposite way and sometimes wrote wrong letters. Naomi was also recovering. For the first couple of weeks she had, with the occasional help of the District Nurse, tried to cope with me on her own. Because of the ME she had become so shredded with exhaustion that the Senior District Nurse, the grand and glorious Cynthia, had brought her Sister-boss to see her, not me, and had commanded her to rest. So she stayed with Geoffrey and Fred and was able to come and sit with me for long times of quiet in the busy hospital ward.\n\nI recovered, perhaps even took back a year or two, but the violence of that year was not quite over. For my spirit at least, there was more to come.\n\nA big new movie had been released. I didn't see it all but I saw excerpts from it and read parts of the script. The film was called _Crash_ and it was about people getting their kicks from deliberately driving dangerously and deriving exquisite sexual pleasure from causing crashes and tangling with wrecked cars and torn bodies.\n\nIt may seem irrational to have been so affected by something so ephemeral, but I could not escape being overwhelmed by a crying rage of disgust and despair. Disgust because few human tragedies are so boring, pointless and totally regrettable as a road accident. To set aside the God-awful reality of such an incident, to glamorize it and imbue the blood and oil with sexy sleaze in order to provide a thrill seemed to me to show a peculiarly nauseous degradation of the imagination. Despair for the people, whose connection with reality and life had become so dulled and debased and...\n\n'Cool it!' I said. 'It's only a film, for goodness' sake!'\n\n'But the pain!'\n\n'It's only make-believe. Just don't let it get to you. Don't make the connections. Nobody does!'\n\nThat was easily said, but I couldn't hold it off. Since then, I have to confess, gratuitous violence on television just turns me up. There is no entertainment there, only pain, and I switch off.\n\nAm I right to do that? In a make-believe world should I not make an effort not to make the connections?\n\nMy mother Daisy had certainly 'made the connections', but hers was not a world of make-believe. She had been born into a time of hope, a time of great brightness for the human spirit. It was a fairly uncomfortable time perhaps for the human body, one in which poverty and oppression, hunger and disease, were for real and all around. But she saw them for what they were, saw the pity of them and with their own hands she and her comrades worked to make those evils a thing of the past. She had made the connections and knew that what needed to be done had to be done, if necessary by her.\n\nToday many of the ideals that Daisy strove for have been realised, and we live in a time of great material ease. This 'Western' world which we have made for ourselves is a comfortable, wellregulated place in which almost everything is found. That, on the face of it, seems fine, but one of the effects of having made it seems to be that we now live in a world which no longer needs people, a world in which many people have no meaningful part to play in their own lives and have gradually become passengers, inert consumers, farmed, nurtured and given identity by supermarkets, spin-doctors and the bright, brittle media.\n\nWhat I needed to remember was that in a world where very little is experienced first-hand, nothing needs to be taken for real. So we can choose as truth whatever we fancy, because it makes no difference, because we are just the audience. We take our pleasures and passions vicariously and, to keep us amused, our entertainers obviously have to keep pushing the boundaries of obscenity ever wider in order to bring us new and more extreme types of titillation. That is their job. That's how they make their money. It may not matter, much.\n\nBut there is perhaps a more dangerous side to this disconnection of the imagination. Nations and governments naturally have, as their priority, a duty to keep their people confident in them, and to keep their voters' complacency untroubled. So, led by the very real shortterm imperatives of commercial advantage and electoral expediency, they have gone on doing apparently advantageous, but actually harmful, things \u2013 to the land and the sea, to the forests, the ice caps and the air, and to the animals and the people living on the rest of the planet \u2013 long, long after it has become obvious to anybody who cared to make the connection that they should have stopped.\n\nOf course the most obvious of these dates back to Agincourt, where the new _bodkin-tipped_ armour-piercing arrow gave the English archers a disproportionate killing-power and so brought King Henry his victory. Today the offspring of that development, modern armaments, give any Tom, Dick or madman the power to do disproportionate damage on a terrible scale, and give any tinpot tyrant the potential ability to hold the world to ransom.\n\nSo, overall, what I have come to see is that most of the things which happen in the world today are not so much the results of policies which successive governments have adopted and carried through, as a slow invisible tide of come-uppances \u2013 the inadvertent effects of long-term mistakes, neglects and injustices, unnoticed by us perhaps but nevertheless done in our name \u2013 coming home to roost.\n\nIn Daisy's time the future was out in front. They could see it coming and they would build it themselves. Today, I suspect, the real future is what is slowly coming up behind us, but we can't see it clearly because somebody has stuck smiley faces over the wingmirrors.\n\nBy the end of 1996 I had some come-uppances of my own to cope with. My energy and my powers of concentration had become, to put it kindly, somewhat limited.\n\nSo it was time for us to tidy up our lives. Red Lion Cottage was too remote for Naomi and it was large and expensive to run. I decided to sell it and move into the basement of the house in Broadstairs.\n\nI have replaced the staircase that had been removed when the house was converted into flats, so Naomi's neat and comely flat is upstairs on the ground floor while my untidy badger's residence is below, though I have quarried out a tiny garden from some wasteground at the back. Nothing much wants to grow in it except ivy and snails, but in the afternoon the sun touches the patch of grass. Perhaps one day I shall grow some Hendon horn-poppies there.\n\nBut in the morning the sun comes singing into Naomi's front room. There we sit in the bow window and have breakfast, watching the seagulls savaging the refuse bags and seeing strange craft sailing across the German Ocean (as an 1895 brochure described it).\n\nWendy and Richard, who keep the corner delicatessen, bake a French-bread roll and can put bacon and egg-mayonnaise in it.\n\nWith one of these, my lunch, in a paper bag, I walk down to the cliff-top promenade and sit on one of the park-benches (placed there in memory of past loved ones), watch the boats bobbing in the harbour, feel the sun and the gentle breeze, and sink my teeth into the crisp blissful baguette. There, in the clear air, it begins to seem possible that the world will be able to get by without me. My legs, so long the arbiters of my conscience, agree without reserve \u2013 it has managed to do so for quite a long time.\n\nLike a diminutive Boadicea, Naomi buzzes about the town and along the windy cliff-tops in her three-wheeled electric chariot. Alas, her ME diet cannot include baguettes, or several other staple items of ordinary fare. So I do most of the cooking, not wisely nor too well. In fact I tend to use the gas cooker rather impetuously and, as a result, sometimes burn the handles of saucepans.\n\nI thought of something to deal with that \u2013 a two-inch length of aluminium tube off an old suction-cleaner pipe. I split it down the side and slid it on to the saucepan handle. It works a treat.\n\nI said: 'That's good. You could easily manufacture those and sell them \u2013 make a fortune!' And I thought: 'No, not again. No.'\n\n# FURRY TAILPIECE\n\nStill wearing his academic cap and gown, Bagpuss looks down from the high basket where he lives. His eyes are glass and have no truck with age or mortality. Perhaps he has always known that he was to be immortal, known that last year, 1999, after a decade of being spurned by the programmers as old-fashioned, bumbly and slow, his films would suddenly win the poll for THE MOST POPULAR BBC CHILDREN'S FILM EVER!\n\nPeter and I got wind of this possibility when we took him with us to the Animation Festival at Cardiff in 1998. There he was f\u00eated, hugged and cuddled, thrown from embrace to embrace by joyful young executives who shed nostalgic tears for that long-lost haven of warmth, safety and sense \u2013 Emily's shop.\n\nSuddenly new showings of the films are being booked, new videos produced. His image has been stabilized and digitized for the new century and he, with his friends at the shop, as well as _Ivor_ _the Engine_ and _The Clangers_ , are poised at the brink of a new life.\n\nI may invent no more saucepan-handle-savers, but I'm sure it will be my pleasure (and I hope profit) to spend my declining years attending upon His Feline Excellency.\n\n# AFTERWORD\n\nI sit and write this in my workshed, which was once merely a depository for my own cherished debris, but is now a resting place for Dad's Smallfilms things too. I have great plans to itemise, compartmentalise and generally organise the bags, boxes and computer discs, but I've yet to do it. I have made a start though, shuffling the storybooks and annuals into chronological order \u2013 _Pingwings, Pogles, Ivor,_ _Noggin, Clangers, Bagpuss_ \u2013 and placing them on a specially-made shelf. They look down at me peacefully, their work done, as I struggle to make stories of my own.\n\nUnder the desk old boxes sit, filled with a lifetime's worth of scripts such as _The Clangers_ \u2013 typed in hazy, old typewriter font, scribbled over with afterthoughts and instructions and ready for translation from English into Swanee-whistle. To my right proudly stands Dad's old Bolex camera, with the bits of Meccano he used to cleverly convert it into a stop-frame camera still attached. And lurking behind me are boxes of figurines, mugs and soft toys, samples of the brandnew merchandise which still find loving homes some twenty-five years after the last Smallfilms' series was made. As archives go, it's a shambles, but perhaps it's just as it should be, coming from a place of cow-sheds, blue-tac and twisted wire.\n\nOn the night Dad passed away I went to bed wondering how the news of his death would be received. Over the following days, the tributes and condolences flooded in \u2013 kind words for a man who had obviously touched the hearts of many, many people. Perhaps the nicest came from the _Guardian's_ Charlie Brooker, who set aside his joyfully acerbic onslaught on modern TV to deliver a remarkably touching tribute, describing Dad and Peter's work as something 'sewn into being like an enchanted tapestry made from 16mm film'. My brother Krispian read it out at Dad's funeral.\n\nFor me, his sudden disappearance somehow made his presence even greater. It was a curious feeling and it was difficult to consider him gone, but the reaction from the public was an unexpected distraction which helped to cushion the sense of loss. I'm sure Dad would have enjoyed it too, had he been here.\n\nBeyond his career as a children's storyteller he was, of course, a husband, father, step-father and companion, too. And his family knew him as a complex man. To try and sum him up would be like taking a clock apart to see how it worked: you end up with a pile of cogs and springs and none the wiser. And for me, the job is even harder; he stood too close to be in focus.\n\nHowever, one thing I can safely say about Dad is that, above all, he was an inspired inventor. Whether grappling with the issue of the destruction of our planet, which he felt so passionately about, or conjuring up a new device to help old hands open difficult jam jars, he met each and every challenge with the same earnest concern and attention; stripping the problem down to its essence then carefully working on it, and through it, until he found an answer. In all parts of his life he never settled for assumptions or received belief; everything had to be taken apart, scrutinised and put back together in what he considered to be a better way.\n\nI remember coming home from school one day to find him sitting in front of the telly wearing what seemed to be a pair of sunglasses. On closer inspection they turned out to be his reading glasses with bits of black card Sellotaped to the lenses. He explained to me that he had made a pin-prick hole in each piece of card and when he looked through them they improved his eyesight considerably.\n\nAt an adult-studies art class, the nude model was intrigued by the humming noise coming from a man at the back of the room. The man was more than happy to show her his home-made erasing tool \u2013 a hand-held fan with the propeller removed and replaced with a piece of rubber, which was perfect for fine-detail erasure. Needless to say, that man was my father.\n\nPerhaps my warmest memory of Dad comes from way back in the past, when I was seven or eight years old, and we would dance together to 'our song'. It was 'Lily the Pink' by The Scaffold, first discovered on a juke-box in a bar while on holiday in Spain in the early seventies. Dad would dig deep into the pockets of his copious white shorts, and a peseta later we would be careering across the tiles. He, with his arms static at his side and his fingers splayed outwards while his legs took him wherever; and I, jigging and scampering about in what I hoped would pass for a kind of dance. Occasionally we'd meet at the middle, hooking arms and swinging by way of greeting before going off on our merry way. Our dance would be repeated over the next few years, where chance would permit it, until the inevitable day came when he approached me with open arms, and \u2013 to my shame \u2013 frozen with adolescent self-consciousness, I declined.\n\nMostly though, I remember him as a very busy man, always at work on his next great project. He despaired at the sight of my brothers and me languishing in front of the telly as he hurriedly made his way through the sitting room from one project to another, and would let out a particularly withering sigh if he found us watching _Trumpton_ or something similar. But when a Smallfilms production graced the screen we'd gather as a family to enjoy the fruits of his labour, finally finding out what had taken up so much of his time.\n\nI remember asking him, long after his last Smallfilms series, if he fancied doing another. He said he had an idea to do a series on 'The Babushkas'. Set in Soviet times, it would be about the mothers of Mother Russia \u2013 a heavy and caring matriarchy \u2013 who, as it turned out, were the people who really ran the Republic. It was, perhaps, a strange idea for a children's programme. But looking at his track record, the chances are it would have been delightful.\n\nIt was his inventive skill, coupled with a wistful sentiment for childhood concerns, which made such places as the land of Nog, the small, blue, Clangers' planet and Pogles' Wood into very real worlds.\n\nI look about my shed, at the various bits and bobs from my father's life, and feel the same sort of melancholy that flavoured his work. Perhaps, if I listen very carefully, when it's very quiet, I'll hear the faint sound of a comforting yawn, the yawn of a saggy, old, cloth cat as he settles down to sleep for the very last time...\n\nThough, I'm sure, in the hearts of many, Bagpuss and all of Oliver and Peter's wonderful creations will live on forever.\n\nDaniel Postgate\n\n_Whitstable 2009_\n\n# Copyright\n\nThis edition first published in 2009 by Canongate Books Ltd, 14 High Street, Edinburgh, EH1 1TE\n\nFirst published in Great Britain in 2000 by Sidgwick & Jackson, an imprint of Macmillan Publishers Ltd, 25 Eccleston Place, London SW1W 9NF\n\nThis digital edition first published by Canongate Books in 2009\n\nCopyright \u00a9 The Estate of Oliver Postgate, 2000   \nIllustrations copyright \u00a9 Peter Firmin, 2000   \nForeword copyright \u00a9 Stephen Fry, 2009   \nAfterword copyright \u00a9 Daniel Postgate, 2009\n\nThe moral right of the author has been asserted\n\n_British Library Cataloguing-in-Publication Data_  \nA catalogue record for this book is available on request from the British Library\n\nISBN 978 1 84767 829 8\n\nwww.meetatthegate.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTable of Contents\n\nTitle Page\n\nCopyright Page\n\nDedication\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nChapter 34\n\nChapter 35\n\nChapter 36\n\nChapter 37\n\nChapter 38\n\nChapter 39\n\nChapter 40\n**CAPTURED!**\n\nAs she pushed her way through the snow-laden bushes, becoming wetter and wetter in the process, she felt a surge of pleasure. The first snare in the line held the body of a small ground-foraging bird. Evanlyn smiled grimly as she thought how once she might have objected to the cruelty of the bird's death. Now, all she felt was a sense of satisfaction as she realized that they would eat well today.\n\nAmazing how an empty belly could change your perspective, she thought, removing the noose from the bird's neck and stuffing the small carcass in her makeshift game bag. She reset the snare, sprinkling a few seeds of corn on the ground beyond it, then rose to her feet, frowning in annoyance as she realized that the melting snow had left two wet patches on her knees as she'd crouched.\n\nEvanlyn sensed, rather than heard, the movement in the trees behind her and began to turn.\n\nBefore she could move, she felt an iron grip around her throat, and as she gasped in fright, a fur-gloved hand, smelling vilely of smoke, clapped over her mouth and nose, cutting off her cry for help.\nDon't miss any of the adventures of\n\n**RANGER'S**   \n**APPRENTICE**\n\nTHE RUINS OF GORLAN   \nTHE BURNING BRIDGE   \nTHE ICEBOUND LAND   \nTHE BATTLE FOR SKANDIA   \nTHE SORCERER OF THE NORTH\n\nPUFFIN BOOKS   \nPublished by the Penguin Group   \nPenguin Young Readers Group, 345 Hudson Street, New York, New York 10014, U.S.A.   \nPenguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario, Canada M4P 2Y3   \n(a division of Pearson Penguin Canada Inc.)   \nPenguin Books Ltd, 80 Strand, London WC2R 0RL, England   \nPenguin Ireland, 25 St Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd)   \nPenguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia   \n(a division of Pearson Australia Group Pty Ltd)   \nPenguin Books India Pvt Ltd, 11 Community Centre, Panchsheel Park, New Delhi - 110 017, India   \nPenguin Group (NZ), 67 Apollo Drive, Rosedale, North Shore 0632, New Zealand   \n(a division of Pearson New Zealand Ltd)   \nPenguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue,   \nRosebank, Johannesburg 2196, South Africa\n\nRegistered Offices: Penguin Books Ltd, 80 Strand, London WC2R 0RL, England\n\nCopyright \u00a9 John Flanagan, 2006\n\nAll rights reserved\n\nTHE LIBRARY OF CONGRESS HAS CATALOGED THE PHILOMEL BOOKS EDITION AS FOLLOWS:   \nFlanagan, John (John Anthony).   \nThe battle for Skandia \/ John Flanagan.\u20141st American ed.   \np. cm.\u2014(Ranger's apprentice ; bk. 4)   \nSummary: After Ranger's apprentice Will battles Temujai warriors to rescue Evanlyn, Will's kingdom   \nof Araluen joins forces with rival kingdom Skandia to defeat a common enemy.\n\nISBN: 9781101019870\n\nThe publisher does not have any control over and does not assume   \nany responsibility for author or third-party Web sites or their content.\n\n<http:\/\/us.penguingroup.com>\n\nVersion_2\n_To Leonie, for always believing._\n**1**\n\nIT WAS A CONSTANT TAPPING SOUND THAT ROUSED WILL FROM his deep, untroubled sleep. He had no clear idea at what point he first became aware of it. It seemed to slide unobtrusively into his sleeping mind, magnified and amplified inside his subconscious, until it crossed over into the conscious world and he realized he was awake, and wondering what it might be.\n\nTap-tap-tap-tap . . . It was still there, but not as loud now that he was awake and aware of other sounds in the small cabin.\n\nFrom the corner, behind a small curtain of sacking that gave her a modicum of privacy, he could hear Evanlyn's even breathing. Obviously, the tapping hadn't woken her. There was a muted crackle from the heaped coals in the fireplace at the end of the room and, as he became more fully awake, he heard them settle with a slight rustling sound.\n\nTap-tap-tap . . .\n\nIt seemed to come from nearby. He stretched and yawned, sitting up on the rough couch he'd fashioned from wood and canvas. He shook his head to clear it and, for a moment, the sound was obscured. Then it was back once more and he realized it was coming from outside the window. The oiled cloth panes were translucent\u2014they would admit the gray light of the pre-dawn, but he couldn't see anything more than a blur through them. Will knelt on the couch and unlatched the frame, pushing it up and craning his head through the opening to study the small porch of the cabin.\n\nA gust of chill entered the room and he heard Evanlyn stir as it eddied around, causing the sacking curtain to billow inward and the embers in the fireplace to glow more fiercely, until a small tongue of yellow flame was released from them.\n\nSomewhere in the trees, a bird was greeting the first light of a new day, and the tapping sound was obscured once more.\n\nThen he had it. It was water, dripping from the end of a long icicle that depended from the porch roof and falling onto an upturned bucket that had been left on the edge of the porch.\n\nTap-tap-tap . . . tap-tap-tap.\n\nWill frowned to himself. There was something significant in this, he knew, but his mind, still fuddled with sleep, couldn't quite grasp what it was. He stood, still stretching, and shivered slightly as he left the last warmth of his blanket and made his way to the door.\n\nHoping not to wake Evanlyn, he eased the latch upward and slowly opened the door, holding it up so that the sagging leather hinges wouldn't allow the bottom edge to scrape the floor of the cabin.\n\nClosing the door behind him, he stepped out onto the rough boards of the porch, feeling them strike icy cold against his bare feet. He moved to the spot where the water dripped endlessly onto the bucket, realizing as he went that other icicles hanging from the roof were also dripping water. He hadn't seen this before. He was sure they usually didn't do this.\n\nHe glanced out at the trees, where the first rays of the sun were beginning to filter through.\n\nIn the forest, there was a slithering thump as a load of snow finally slid clear of the pine branches that had supported it for months and fell in a heap to the ground below.\n\nAnd it was then that Will realized the significance of the endless tap-tap-tap that had woken him.\n\nBehind him, he heard the door creak and he turned to see Evanlyn, her hair wildly tousled, her blanket wrapped tight around her against the cold.\n\n\"What is it?\" she asked him. \"Is something wrong?\"\n\nHe hesitated a second, glancing at the growing puddle of water beside the bucket.\n\n\"It's the thaw,\" he said finally.\n\nAfter their meager breakfast, Will and Evanlyn sat in the early morning sun as it streamed across the porch. Neither of them had wanted to discuss the significance of Will's earlier discovery, although they had since found more signs of the thaw.\n\nSmall patches of soaked brown grass were showing through the snow cover on the ground surrounding the cabin, and the sound of wet snow sliding from the trees to hit the ground was becoming increasingly common.\n\nThe snow was still thick on the ground and in the trees, of course. But the signs were there that the thaw had begun and that, inexorably, it would continue.\n\n\"I suppose we'll have to think about moving on,\" Will said, finally voicing the thought that had been in both their minds.\n\n\"You're not strong enough yet,\" Evanlyn told him. It had been barely three weeks since he had thrown off the mind-numbing effects of the warmweed given to him as a yard slave in Ragnak's Lodge. Will had been weakened by inadequate food and clothing and a regimen of punishing physical work before they had made their escape. Since then, their meager diet in the cabin had been enough to sustain life, but not to restore his strength or endurance. They had lived on the cornmeal and flour that had been stored in the cabin, along with a small stock of vegetables and the stringy meat from whatever game Evanlyn and he had been able to snare.\n\nThere was little enough of that in winter, and what game they had managed to catch had been in poor condition itself, providing little in the way of nourishment.\n\nWill shrugged. \"I'll manage,\" he said simply. \"I'll have to.\"\n\nAnd that, of course, was the heart of the problem. They both knew that once the snow in the high passes had melted, hunters would again begin to visit the high country where they found themselves. Already, Evanlyn had seen one such\u2014the mysterious rider in the forest on the day when Will's senses had returned to him. Fortunately, since that day, there had been no further sign of him. But it was a warning. Others would come, and before they did, Will and Evanlyn would have to be long gone, heading down the far side of the mountain passes and across the border into Teutlandt.\n\nEvanlyn shook her head doubtfully. For a moment, she said nothing. Then she realized that Will was right. Once the thaw was well and truly under way, they would have to leave whether she felt he was strong enough to travel or not.\n\n\"Anyway,\" she said at last, \"we have a few weeks yet. The thaw's only just started, and who knows? We may even get another cold snap.\"\n\nIt was possible, she thought. Perhaps not probable, but at least it was possible. Will nodded agreement.\n\n\"There's always that,\" he said.\n\nThe silence fell over them once more like a blanket. Abruptly, Evanlyn stood, dusting off her breeches. \"I'll go and check the snares,\" she said, and when Will began to rise to accompany her, she stopped him.\n\n\"You stay here,\" she said gently. \"From now on, you're going to have to conserve your strength as much as possible.\"\n\nWill hesitated, then nodded. He recognized that she was right.\n\nShe collected the hessian sack they used as a game bag and slung it over her shoulder. Then, with a small smile in his direction, the girl headed off into the trees.\n\nFeeling useless and dispirited, Will slowly began to gather up the wooden platters they had used for their meal. All he was good for, he thought bitterly, was washing up.\n\nThe snare line had moved farther and farther from the cabin over the past three weeks. As small animals, rabbits, squirrels and the occasional snow hare had fallen prey to the snares that Will had built, the other animals in that area had become wary. As a consequence, they had been compelled to move the snares into new locations every few days\u2014each one a little farther away from the cabin than the one before.\n\nEvanlyn estimated that she had a good forty minutes' walking on the narrow uphill track before she would reach the first snare. Of course, if she'd been able to move straight to it, the walk would have been considerably shorter. But the track wound and wandered through the trees, more than doubling the distance she had to cover.\n\nThe signs of the thaw were all around her, now that she was aware of it. The snow no longer squeaked dryly underfoot as she walked. It was heavier, wetter and her steps sank deeply into it. The leather of her boots was already soaked from contact with the melting snow. The last time she had walked this way, she reflected, the snow had simply coated her boots as a fine, dry powder.\n\nShe also began to notice more activity among the wildlife in the area. Birds flitted through the trees in greater numbers than she'd previously seen, and she startled a rabbit on the track, sending it scurrying back into the protection of a snow-covered thicket of blackberries.\n\nAt least, she thought, all this extra activity might increase the chances of finding some worthwhile game in the snares.\n\nEvanlyn saw the discreet sign that Will had cut into the bark of a pine and turned off the track to find the spot where she and Will had laid the first of the snares. She recalled how gratefully she had greeted his recovery from the warmweed drug. Her own survival skills were negligible and Will had provided welcome expertise in devising and setting snares to supplement their diet. It was all part of his Ranger training under Halt, he had told her.\n\nShe remembered how, when he had mentioned the older Ranger's name, his eyes had misted for a few moments and his voice had choked slightly. Not for the first time, the two young people had felt very, very far from home.\n\nAs she pushed her way through the snow-laden bushes, becoming wetter and wetter in the process, she felt a surge of pleasure. The first snare in the line held the body of a small ground-foraging bird. They had caught a few of these previously and the bird's flesh made excellent eating. About the size of a small chicken, it had carelessly poked its neck through the wire noose of the snare, then become entangled. Evanlyn smiled grimly as she thought how once she might have objected to the cruelty of the bird's death. Now, all she felt was a sense of satisfaction as she realized that they would eat well today.\n\nAmazing how an empty belly could change your perspective, she thought, removing the noose from the bird's neck and stuffing the small carcass in her makeshift game bag. She reset the snare, sprinkling a few seeds of corn on the ground beyond it, then rose to her feet, frowning in annoyance as she realized that the melting snow had left two wet patches on her knees as she'd crouched.\n\nEvanlyn sensed, rather than heard, the movement in the trees behind her and began to turn.\n\nBefore she could move, she felt an iron grip around her throat, and as she gasped in fright, a fur-gloved hand, smelling vilely of smoke, sweat and dirt, clapped over her mouth and nose, cutting off her cry for help.\n**2**\n\nTHE TWO RIDERS EMERGED FROM THE TREES AND INTO A CLEAR meadow. Down here in the foothills of Teutlandt, the coming spring was more apparent than in the high mountains that reared ahead of them. The meadow grasses were already showing green and there were only isolated patches of snow, in spots that usually remained shaded for the greater part of the day.\n\nA casual onlooker might have been interested to notice the horses that followed behind the two mounted men. They might even have mistaken the men, at a distance, for traders who were hoping to take advantage of the first opportunity to cross through the mountain passes into Skandia, and so benefit from the high prices that the season's first trade goods would enjoy.\n\nBut a closer inspection would have shown that these men were not traders. They were armed warriors. The smaller of the two, a bearded man clad in a strange gray and green dappled cloak that seemed to shift and waver as he moved, had a longbow slung over his shoulders and a quiver of arrows at his saddle bow.\n\nHis companion was a larger, younger man. He wore a simple brown cloak, but the early spring sunshine glinted off the chain mail armor at his neck and arms, and the scabbard of a long sword showed under the hem of the cloak. Completing the picture, a round buckler was slung over his back, emblazoned with a slightly crude effigy of an oakleaf.\n\nTheir horses were as mismatched as the men themselves. The younger man sat astride a tall bay\u2014long-legged, with powerful haunches and shoulders, it was the epitome of a battlehorse. A second battlehorse, this one a black, trotted behind him on a lead rope. His companion's mount was considerably smaller, a shaggy barrel-chested horse, more a pony really. But it was sturdy, and had a look of endurance to it. Another horse, similar to the first, trotted behind, lightly laden with the bare essentials for camping and traveling. There was no lead rein on this horse. It followed obediently and willingly.\n\nHorace craned his neck up at the tallest of the mountains towering above them. His eyes squinted slightly in the glare of the snow that still lay thickly on the mountain's upper half and now reflected the light of the sun.\n\n\"You mean to tell me we're going over that?\" he asked, his eyes widening.\n\nHalt looked sidelong at him, with the barest suggestion of a smile. Horace, however, intent on studying the massive mountain formations facing them, failed to see it.\n\n\"Not over,\" said the Ranger. \"Through.\"\n\nHorace frowned thoughtfully at that. \"Is there a tunnel of some kind?\"\n\n\"A pass,\" Halt told him. \"A narrow defile that twists and winds through the lower reaches of the mountains and brings us into Skandia itself.\"\n\nHorace digested that piece of information for a moment or two. Then Halt saw his shoulders rise to an intake of breath and knew that the movement presaged yet another question. He closed his eyes, remembering a time that seemed years ago when he was alone and when life was not an endless series of questions.\n\nThen he admitted to himself that, strangely, he preferred things the way they were now. However, he must have made some unintentional noise as he awaited the question, for he noticed that Horace had sealed his lips firmly and determinedly. Obviously, Horace had sensed the reaction and had decided that he would not bother Halt with another question. Not yet, anyway.\n\nWhich left Halt in a strange quandary. Because now that the question was unasked, he couldn't help wondering what it would have been. All of a sudden, there was a nagging sense of incompletion about the morning. He tried to ignore the feeling but it would not be pushed aside. And for once, Horace seemed to have conquered his almost irresistible need to ask the question that had occurred to him.\n\nHalt waited a minute or two but there was no sound except for the jingling of harness and the creaking of leather from their saddles. Finally, the former Ranger could bear it no longer.\n\n\"What?\"\n\nThe question seemed to explode out of him, with a greater degree of violence than he had intended. Taken by surprise, Horace's bay shied in fright and danced several paces sideways.\n\nHorace turned an aggrieved look on his mentor as he calmed the horse and brought it back under control.\n\n\"What?\" he asked Halt, and the smaller man made a gesture of exasperation.\n\n\"That's what I want to know,\" he said irritably. \" _What?\"_\n\nHorace peered at him. The look was all too obviously the sort of look that you give to someone who seems to have taken leave of his senses. It did little to improve Halt's rapidly rising temper.\n\n\"What?\" said Horace, now totally puzzled.\n\n\"Don't keep parroting at me!\" Halt fumed. \"Stop repeating what I say! I asked you 'what,' so don't ask me 'what' back, understand?\"\n\nHorace considered the question for a second or two, then, in his deliberate way, he replied: \"No.\"\n\nHalt took a deep breath, his eyebrows contracted into a deep V, and beneath them his eyes sparked with anger. But before he could speak, Horace forestalled him.\n\n\"What 'what' are you asking me?\" he said. Then, thinking how to make his question clearer, he added, \"Or to put it another way, why are you asking 'what'?\"\n\nControlling himself with enormous restraint, and making no secret of the fact, Halt said, very precisely: \"You were about to ask a question.\"\n\nHorace frowned. \"I was?\"\n\nHalt nodded. \"You were. I saw you take a breath to ask it.\"\n\n\"I see,\" said Horace. \"And what was it about?\"\n\nFor just a second or two, Halt was speechless. He opened his mouth, closed it again, then finally found the strength to speak.\n\n\"That is what I was asking you,\" he said. \"When I said 'what,' I was asking you what you were about to ask me.\"\n\n\"I wasn't about to ask you 'what,'\" Horace replied, and Halt glared at him suspiciously. It occurred to him that Horace could be indulging himself in a gigantic leg pull, that he was secretly laughing at Halt. This, Halt could have told him, was not a good career move. Rangers were not people who took kindly to being laughed at. He studied the boy's open face and guileless blue eyes and decided that his suspicion was ill-founded.\n\n\"Then what, if I may use that word once more, were you about to ask me?\"\n\nHorace drew breath once more, then hesitated. \"I forget,\" he said. \"What were we talking about?\"\n\n\"Never mind,\" Halt muttered, and nudged Abelard into a canter for a few strides to draw ahead of his companion.\n\nSometimes the Ranger could be confusing, and Horace thought it best to forget the whole conversation. Yet, as happens so often, the moment he stopped trying to consciously remember the thought that had prompted his question, it popped back into his mind again.\n\n\"Are there many passes?\" he called to Halt.\n\nThe Ranger twisted in his saddle to look back at him. \"What?\" he asked.\n\nHorace wisely chose to ignore the fact that they were heading for dangerous territory with that word again. He gestured to the mountains frowning down upon them.\n\n\"Through the mountains. Are there many passes into Skandia through the mountains?\"\n\nHalt checked Abelard's stride momentarily, allowing the bay to catch up with them, then resumed his pace.\n\n\"Three or four,\" he said.\n\n\"Then don't the Skandians guard them?\" Horace asked. It seemed logical to him that they would.\n\n\"Of course they do,\" Halt replied. \"The mountains form their principal line of defense.\"\n\n\"So how did you plan for us to get past them?\"\n\nThe Ranger hesitated. It was a question that had been taxing his mind since they had taken the road from Chateau Montsombre. If he were by himself, he would have no trouble slipping past unseen. With Horace in company, and riding a big, spirited battlehorse, it might be a more difficult matter. He had a few ideas but had yet to settle on any one of them.\n\n\"I'll think of something,\" he temporized, and Horace nodded wisely, satisfied that Halt would indeed think of something. In Horace's world, that was what Rangers did best, and the best thing a warrior apprentice could do was let the Ranger get on with thinking while a warrior took care of walloping anyone who needed to be walloped along the way. He settled back in his saddle, contented with his lot in life.\n**3**\n\nERAK STARFOLLOWER, WOLFSHIP CAPTAIN AND ONE OF THE senior war jarls of the Skandians, made his way through the low-ceilinged, wood-paneled lodge to the Great Hall. His face was marked with a frown as he went. He had plenty to do, with the spring raiding season coming on. His ship needed repairs and refitting. Most of all, it needed the fine-tuning that only a few days at sea could bring.\n\nNow this summons from Ragnak boded ill for his plans. Particularly since the summons had come through the medium of Borsa, the Oberjarl's hilfmann, or administrator. If Borsa were involved, it usually meant that Ragnak had some little task for Erak to look after. Or some not-so-little task, the wolfship skipper thought wryly.\n\nBreakfast was long since finished, so there were only a few servants cleaning up the Hall when he arrived. At the far end, seated at a rough pine table off to one side of Ragnak's High Seat\u2014a massive pinewood chair that served in place of a throne for the Skandian ruler\u2014sat Ragnak and Borsa, their heads bowed over a pile of parchment scrolls. Erak recognized those scrolls. They were the tax returns for the various towns and shires throughout Skandia. Ragnak was obsessed with them. As for Borsa, his life was totally dominated by them. He breathed, slept and dreamed the tax returns, and woe betide any local jarl who might try to shortchange Ragnak or claim any deduction that wouldn't pass Borsa's fine-tooth comb inspection.\n\nErak put two and two together and sighed quietly. The most likely conclusion that he could draw from the two facts of his summoning and the pile of tax returns on the table was that he was about to be sent off on another tax-collecting mission.\n\nTax collecting was not something that Erak enjoyed. He was a raider and a sea wolf, a pirate and a fighter. As such, his inclination was to be more on the side of the tax evaders than the Oberjarl and his eager-fingered hilfmann. Unfortunately, on those previous occasions when Erak had been sent out to collect overdue or unpaid taxes, he had been too successful for his own good. Now, whenever there was the slightest doubt about the amount of tax owing from a village or a shire, Borsa automatically thought of Erak as the solution to the problem.\n\nTo make matters worse, Erak's attitude and approach to the job only added to his desirability in Borsa's and Ragnak's eyes. Bored with the task and considering it embarrassing and belittling, he made sure he spent as little time on the job as possible. The tortuous arguments and recalculation of amounts owing after all deductions had been approved and agreed were not for him. Erak opted for a more direct course, which consisted of seizing the person under investigation, ramming a double-headed broadax up under his chin and threatening mayhem if all taxes, every single one of them, were not paid immediately.\n\nErak's reputation as a fighter was well known throughout Skandia. To his annoyance, he was never asked to make good on his threat. Those recalcitrants whom he visited invariably coughed up the due amount, and often a little extra that had never been in contention, without the slightest argument or hesitation.\n\nThe two men at the table looked up as he made his way through the benches toward the end of the room. The Great Hall served more than one purpose. It was where Ragnak and his close followers took their meals. It was also the site of all banquets and official gatherings in Skandia's rough and ready social calendar. And the small, open annex where Ragnak and Borsa were currently studying tax returns was also Ragnak's office. It wasn't particularly private, since any member of the inner or outer council of jarls had access to the hall at any time of day. But then, Ragnak wasn't the sort to need privacy. He ruled openly and made all his policy statements to the world at large.\n\n\"Ah, Erak, you're here,\" said Borsa, and Erak thought, not for the first time, that the hilfmann had a habit of stating the bleeding obvious.\n\n\"Who is it this time?\" he asked in a resigned tone. He knew there was no use trying to argue his way out of the assignment, so he might as well just get on with it. With luck, it would be one of the small towns down the coast, and at least he might have a chance to work up his crew and wolfship at the same time.\n\n\"Ostkrag,\" the Oberjarl told him, and Erak's hopes of salvaging something useful from this assignment faded to nothing. Ostkrag lay far inland, to the east. It was a small settlement on the far side of the mountain range that formed the rugged spine of Skandia and was accessible only by going over the mountains themselves or through one of the half dozen tortuous passes that wound their way through.\n\nAt best, it meant an uncomfortable journey there and back by pony, a method of transport that Erak loathed. As he thought of the mountain range that reared above Hallasholm, he had a quick memory of the two Araluen slaves he had helped to escape several months ago. He wondered what had become of them, whether they had made it to the small hunting cabin high in the mountains and whether they had survived the last months of winter. He realized abruptly that Borsa and Ragnak were both waiting for his reaction.\n\n\"Ostkrag?\" he repeated. Ragnak nodded impatiently.\n\n\"Their quarterly payment is overdue. I want you to go and shake them up,\" the Oberjarl said. Erak noticed that Ragnak couldn't quite hide the avaricious gleam that came into his eyes whenever he talked about tax and payments. Erak couldn't help giving vent to an exasperated sigh.\n\n\"They can't be overdue by more than a week or so,\" he temporized, but Ragnak was not to be swayed and shook his head violently.\n\n\"Ten days!\" he snapped. \"And it's not the first time! I've warned them before, haven't I, Borsa?\" he said, turning to the hilfmann, who nodded.\n\n\"The jarl at Ostkrag is Sten Hammerhand,\" Borsa said, as if that were explanation enough. Erak stared blankly at him. \"He should be called Sten Gluehand,\" he elaborated with heavy sarcasm. \"The tax payments have stuck to his fingers before this, and even when they're paid in full, he always makes us wait long past the overdue date. It's time we taught him a lesson.\"\n\nErak smiled with some irony at the small, sparsely muscled hilfmann. Borsa could be an extremely threatening figure, he thought\u2014when someone else was available to carry out the threats.\n\n\"You mean it's time _I_ taught him a lesson?\" he suggested, but Borsa didn't notice the sarcasm in his voice.\n\n\"Exactly!\" he said, with some satisfaction. Ragnak, however, was a little more perceptive.\n\n\"It's my money, after all, Erak,\" he said, and there was an almost petulant note in his voice. Erak met his gaze steadily. For the first time, he realized that Ragnak was growing old. The once flaming red hair was duller and turning gray. It came as a surprise to Erak. He certainly didn't feel that he was growing older, yet Ragnak didn't have too many years on him. He could notice other changes in the Oberjarl now that he had become aware of the fact. His jowls were heavier and his waistline thickening. He wondered if he was changing too.\n\n\"It's been a severe winter,\" he suggested. \"Perhaps the passes are still blocked. There was a lot of late snow.\"\n\nHe moved to the large scale map of Skandia that was displayed on the wall behind Ragnak's table. He found Ostkrag and, with one forefinger, traced the way to the closest pass.\n\n\"The Serpent Pass,\" he said, almost to himself. \"It's not impossible that all that late-season snow and the sudden thaw could have led to landslides in there.\" He turned back to Ragnak and Borsa, indicating the position on the map to them.\n\n\"Maybe the couriers simply can't get through yet?\" he suggested. Ragnak shook his head and again Erak sensed the irritability, the irrational annoyance that seemed to grip Ragnak these days whenever his will was thwarted or his judgment questioned.\n\n\"It's Sten, I know it,\" he said stubbornly. \"If it were anyone else, I might agree with you, Erak.\" Erak nodded, knowing full well that the words were a lie. Ragnak rarely agreed with anyone if it meant changing his own position. \"Get up there and get the money from him. If he argues, arrest him and bring him back. In fact, arrest him even if he doesn't argue. Take twenty men with you. I want him to see a real show of strength. I'm sick of being taken for a fool by these petty jarls.\"\n\nErak looked up in some surprise. Arresting a jarl in his own lodge was not something to be lightly contemplated\u2014particularly for such a petty offense as a late tax payment. Among the Skandians, tax evasion was considered to be almost obligatory. It was a form of sport. If you were caught out, you paid up and that was the end of it. Erak could not remember anyone being submitted to the shame of arrest on that count.\n\n\"That might not be wise,\" he said quietly, and Ragnak glared up at him, his eyes searching for Erak's over the scattered accounts on the table before him.\n\n\"I'll decide what's wise,\" he grated. \"I'm Oberjarl, not you.\"\n\nThe words were offensive. Erak was a senior jarl and by long-established custom he was entitled to air his opinion, even though it might be contrary to his leader's. He bit back the angry retort that sprang to his lips. There was no point provoking Ragnak any further when he was in this mood.\n\n\"I know you're the Oberjarl, Ragnak,\" he said quietly. \"But Sten is a jarl in his own right and he may well have a perfectly valid reason for this late payment. To arrest him in those circumstances would be unnecessarily provocative.\"\n\n\"I'm telling you he won't have what you call a 'valid reason,' damn it!\" Ragnak's eyes were narrowed now and his face was suffused with his anger. \"He's a thief and a holdout and he needs to be made an example to others!\"\n\n\"Ragnak . . . ,\" Erak began, trying to reason one last time. This time it was Borsa who interrupted.\n\n\"Jarl Erak, you have your instructions! Now do as you are ordered!\" he shouted, and Erak turned angrily to face him.\n\n\"I follow the Oberjarl's orders, hilfmann. Not yours.\"\n\nBorsa realized his mistake. He backed away a pace or two, making sure the substantial bulk of the table was between him and Erak. His eyes slid away from the other man's and there was an ugly silence. Finally, Ragnak seemed to realize that some form of back-down might be necessary\u2014although not too much. He said, in an irritated tone: \"Look, Erak, just go and get those taxes from Sten. And if he's been holding out on purpose, bring him back here for trial. All right?\"\n\n\"And if he has a valid reason?\" Erak insisted.\n\nThe Oberjarl waved a hand in surrender. \"If he has a valid reason, you can leave him alone. Does that suit you?\"\n\nErak nodded. \"Under those conditions, all right,\" he agreed.\n\nHe had the loophole he'd been looking for. As far as he was concerned, the fact that Ragnak was an insufferable pain in the buttocks was a more than valid reason for not paying taxes on time. Mind you, he might have to find another way of phrasing it when he returned without arresting Sten.\n**4**\n\nWILL CAME AWAKE WITH A JERK. He HAD BEEN SITTING ON THE edge of the porch in the sun and he realized that he must have nodded off. Ruefully, he thought about how much of his time he spent sleeping these days. Evanlyn said it was only to be expected, as he was regaining his strength. He supposed she was right. But that didn't mean he had to like it.\n\nThere was also the fact that there was so little to do around the hut where they had spent their time since escaping from the Skandian stronghold. Today he had cleared away and washed their breakfast dishes, then made the beds and straightened the few pieces of furniture in the cabin. That had taken barely half an hour, so he had groomed the pony in the lean-to behind the cabin until its coat shone. The pony looked at him, and at itself, with mild surprise. He guessed nobody had ever spent so much care on its appearance in the past.\n\nAfter that, Will had wandered aimlessly around the cabin and the small clearing, inspecting those patches where the damp brown grass was beginning to show through the snow cover. He had idly considered making some more snares, then discarded the idea. They had more than they needed already. Feeling bored and useless, he had sat down on the porch to wait for Evanlyn's return. At some stage, he must have nodded off, affected by the warmth of the sun.\n\nThat warmth was long gone now, he realized. The sun had traveled fully across the clearing and the pines were throwing long shadows across the cabin. It must be midafternoon, he estimated.\n\nA frown creased his forehead. Evanlyn had left well before noon to check the snares. Even allowing for the fact that they had moved the trapline farther and farther away from the cabin, she should have had time to reach the line, check the snares and return by now. She must have been gone for at least three hours\u2014possibly more.\n\nUnless she had already returned and, seeing him sleeping, had decided not to wake him. He stood now, his stiff joints protesting, and checked inside the cabin. There was no sign that she had returned. The game bag and her thick woolen cloak were missing. Will's frown deepened and he began to pace the small clearing, wondering what he should do. He wished he knew exactly how long she had been gone and silently berated himself for falling asleep. Deep down in the pit of his stomach, a vague uneasiness stirred as he wondered what could have become of his companion. He reviewed the possibilities.\n\nShe could have lost her way, and be wandering through the thickly growing, snow-covered pines, trying to find her way back to the cabin. Possible, but unlikely. He had blazed the paths leading to their trapline with discreet marks and Evanlyn knew where to look for them.\n\nPerhaps she had been injured? She could have fallen, or twisted an ankle. The paths were rough and steep in places and that was a definite possibility. She might be lying now, injured and unable to walk, stranded in the snow, with the afternoon drawing on toward night.\n\nThe third possibility was the worst: she had encountered someone. Anyone that she ran into on this mountain was likely to be an enemy. Perhaps she had been recaptured by the Skandians. His pulse raced for a moment as he considered the thought. He knew they would show little mercy to an escaped slave. And while Erak had helped them before, he would be unlikely to do so again\u2014even if he had the opportunity.\n\nAs he had been considering these possibilities, he had begun moving around the cabin, collecting his things in preparation for setting out to look for her. He had filled one of their water skins from the bucket of creek water that he brought to the cabin each day, and crammed a few pieces of cold meat into a carry sack. He laced on his thick walking boots, winding the thongs rapidly around his legs, almost up to the knee, and unhooked his sheepskin vest from the peg behind the door.\n\nOn the whole, he thought, the second possibility was the most likely. The chances were that Evanlyn was injured somewhere, unable to walk. The chance that she might have been retaken by the Skandians was very slim indeed, he realized. It was still too early in the season for people to be moving around the mountain. The only reason for doing so would be to hunt game. And it was still too scarce to be worth the trouble of fighting through the thick drifts of snow that blocked the way in so many parts of the mountain. No, on the whole, it was most likely that Evanlyn was safe, but incapacitated.\n\nWhich meant his next logical move would be to put a bridle and saddle on the pony and lead him along as he tracked her, so that she could ride back to the cabin once he found her. He had no doubt that he _would_ find her. He was already a skilled tracker, although nowhere near the standard of Halt or Gilan, and tracking the girl through snow-covered territory would be a relatively simple matter.\n\nAnd yet . . . he was reluctant to take the pony with him. The little horse would make unnecessary noise and a nagging doubt told Will he should proceed with caution. It was unlikely that Evanlyn had encountered strangers, but it wasn't impossible.\n\nIt might be wiser to travel unobtrusively until he found out the real state of affairs. As he came to this decision, he stripped the beds of their blankets, tying them into a roll that he slung over his shoulder. It might prove necessary to spend the night in the open and it would be better to be prepared. He picked up a flint and steel from near the fireplace and dropped it into one of his pockets.\n\nFinally, he was ready to go. He stood at the door, taking one last look around the cabin to see if there was anything else he might need. The small hunting bow and a quiver of arrows leaned by the doorjamb. On an impulse, he picked them up, slinging the quiver across his back with the roll of blankets. Then another thought struck him and he crossed back to the fireplace, selecting a half-burned stick from the ashes.\n\nOn the outside of the door, he printed in crude letters: \"Looking for you. Wait here.\"\n\nAfter all, it was possible that Evanlyn might turn up after he had left and he wanted to make sure she didn't go blundering off, trying to find him while he was trying to find her.\n\nHe took a few seconds to string the bow. Halt's voice echoed in his ears: \"An unstrung bow is just something extra to carry. A strung bow is a weapon.\" He looked at it disdainfully. It wasn't much of a weapon, he thought. But that and the small knife in his belt were all that he had. He moved to the edge of the clearing, picking up the clear trail of Evanlyn's footprints in the snow. They were blurred after a morning of spring sunshine, but they still showed up. Maintaining a steady trot, he moved off into the forest.\n\nHe followed her trail easily as it wound up into the higher reaches of the mountain. Before too long, his pace had dropped from the steady jog and he was walking, breathing hard as he went. He realized that he was in poor condition. There had been a time when he could have maintained that ground-eating lope for hours. Now, after barely twenty minutes, he was puffing and exhausted. He shook his head in disgust and continued to follow the footprints.\n\nOf course, following the trail was made easier by the fact that he already had a good idea of the direction Evanlyn had been heading. He had helped her relocate the snares a few days earlier. At that time, he recalled, they had gone at an easier pace, resting frequently so as not to tire him out.\n\nEvanlyn had been reluctant to allow him to walk so far, but had given in to the inevitable. She had no real idea how to place the snares where they might have the best chance of trapping small game. That was one of Will's areas of expertise. He knew how to look for and recognize the small signs that showed where the rabbits and birds moved, where they were most likely to poke their unsuspecting heads into the looped snares.\n\nIt had taken Evanlyn about forty minutes to reach the trapline that morning. Will covered the distance in an hour and a quarter, stopping more frequently as the time went on to rest and recover his breath. He resented the stops, knowing they were costing him daylight. But there would be no point pushing himself until he was utterly exhausted. He had to keep himself in condition to give Evanlyn any assistance she might need when he found her.\n\nThe sun had dropped over the crest of the mountain by the time he reached the blazed tree that marked the beginning of the trapline. He touched one hand to the cut bark, then turned to head off the track into the pines when he saw something out of the corner of his eye. Something that froze his heart in midbeat.\n\nThere was the clear imprint of a horse's hooves in the snow\u2014and they overlay the tracks that Evanlyn had left. Someone had followed her.\n\nForgetting his weariness, Will ran, half crouching, through the thick pines to the spot where the first snare had been laid. The snow there was disturbed and scuffed. He fell to his knees, trying to read the story that was written there.\n\nThe empty snare first: he could see where Evanlyn had reset the noose, smoothing the snow around it and scattering a few grains of seed. So there had been an animal in the snare when she'd arrived.\n\nThen he cast wider, seeing the other set of footprints moving into position behind her as she had knelt, engrossed in the task of resetting the snare and probably jubilant at the fact that they had caught something. The horse's tracks had stopped some twenty meters away. Obviously, the animal was trained to move silently\u2014much as Ranger horses were. He felt an uneasy sense of misgiving at that. He didn't like the idea of an enemy who had those sorts of skills\u2014and by now he knew he was dealing with an enemy of some kind. The signs of the struggle between Evanlyn and the enemy were all too clear to his trained eye. He could almost see the man\u2014he assumed it was a man\u2014moving quietly behind her, grabbing her and dragging her back through the snow.\n\nThe wild disturbance of the ground showed how Evanlyn had kicked and struggled. Then, suddenly, the struggling had stopped and two shallow furrows in the snow led back to where the horse waited. Her heels, he realized, as her unconscious body had been dragged away.\n\nUnconscious? Or dead, he thought. And a chill hand seized his heart at the thought. Then he shook it away determinedly.\n\n\"No sense in carrying her away if he'd killed her,\" he told himself. And he almost believed it. But he still had that gnawing uncertainty in the pit of his belly as he followed the horse's tracks back to the main trail, and then in the opposite direction of the trail that led back to the cabin.\n\nHe was glad he'd thought to bring the blankets. It was going to be a cold night, he thought. He was also glad that he'd thought to bring the bow, although he found himself wishing that he still had the powerful recurve bow that he had lost at the bridge in Celtica. It was a far superior weapon to the low-powered Skandian hunting bow. And he had the uncomfortable certainty that he was going to need a weapon in the very near future.\n**5**\n\nTHE WORLD WAS UPSIDE DOWN AND BOUNCING. GRADUALLY, As Evanlyn's eyes came into focus, she realized that she was hanging, head down, her face only centimeters away from the front left shoulder of a horse. The inverted position made the blood pound painfully in her head, a pounding that was accentuated by the steady, bouncing trot that the horse was maintaining. He was a chestnut, she noted, and his coat was long and shaggy and badly in need of grooming. The small area she could see was matted with sweat and dried mud.\n\nSomething hard ground into the soft flesh of her belly with every lurching step the horse took. She tried to wriggle to relieve the pressure and was rewarded for her efforts with a sharp blow to the back of her head. She took the hint and stopped wriggling.\n\nTurning her head to face toward the rear, she could make out her captor's left leg\u2014clad in a long, skirt-like fur coat and soft hide boots. Below her, the churned snow of the trail passed rapidly by. She realized her unconscious body had been slung unceremoniously across the front of a saddle. That projection stabbing dully into her stomach must be the pommel.\n\nShe remembered now: the slight noise behind her, the blur of movement as she started to turn. A hand, stinking of sweat and smoke and fur, clamped over her mouth to prevent her screaming. Not that there had been anyone within earshot to hear, she thought regretfully.\n\nThe struggle had been brief, with her assailant dragging her backward to keep her off balance. She had tried to fight her way free, tried to kick and bite. But the man's thick glove made her attempts at biting useless, and her kicks were ineffective as she was dragged backward. Finally, there had been an instant of blinding pain, just behind her left ear, and then darkness.\n\nAs she thought of the blow, she became aware that the area behind her left ear was another source of throbbing, another source of pain. The discomfort of being carried along helplessly like this was bad enough. But the inability to see anything, to get a look at the man who had taken her prisoner, was, if anything, worse. From this doubled-over, facedown position, she couldn't even see any features of the land they were passing through. So if she did eventually escape, she would have no memory of any landmarks that might help her retrace her steps.\n\nUnobtrusively, she tried to twist her head to the side, to get a look at the rider mounted behind her. But he obviously felt the movement, minimal as she tried to keep it, and she felt another blow on the back of her head. Just what she needed, she thought ruefully.\n\nRealizing that there was no future in antagonizing her captor, Evanlyn slumped down, trying to relax her muscles and ride as comfortably as possible. It was a fairly unsuccessful attempt. But at least when she let her head hang down, her cramping neck and shoulder muscles felt some relief.\n\nThe ground went by below her: the snow churned up by the horse's front hooves, showing the sodden brown grass that lay underneath. They were making their way downhill, she realized, as the rider reined in the horse to negotiate a steeper than normal part of the trail at a walk. She felt the rider lean back away from her as she slid forward, saw his feet pushing forward against the stirrups as he leaned back to compensate and help the horse balance.\n\nJust ahead of them, visible from her facedown position, was a patch of snow that had melted and refrozen. It was slick and icy and the horse's hooves went onto it before she could sound any warning. Legs braced, the horse slid downward, unable to check its progress. She heard a startled grunt from the rider and he leaned farther back, keeping the reins taut to still the horse's panic. They slid, scrabbled, then checked. Then they were across the icy patch and the rider urged the horse back into its steady trot once more.\n\nEvanlyn noted the moment. If it happened again, it might give her a chance to escape.\n\nAfter all, she wasn't tied onto the horse, she realized. She was merely hanging either side like a bundle of old clothes. If the horse fell, she could be off and away before the rider regained his feet. Or so she thought.\n\nPerhaps fortunately for her\u2014for she couldn't see the bow slung over the rider's back, nor the quiver full of arrows that hung at his right side\u2014the horse didn't fall. There were a few more steep sections, and a couple of other occasions when they slid, legs locked forward and rear hooves scrabbling for purchase, for several meters down the slope. But on none of those occasions did the rider lose control or the horse do more than whinny in alarm and concentration.\n\nFinally, they reached their destination. The first she knew of it was when the horse slid to a stop and she felt a hand on her collar, heaving her up and over, to send her sprawling in the wet snow that covered the ground. She fell awkwardly, winding herself in the process, and it was several seconds before she could regain her presence of mind and take the time to look around her.\n\nThey were in a clearing where a small camp had been set up. Now she could see her captor as he swung down from the saddle. He was a short, stocky man, dressed in furs\u2014a long, wide-skirted fur coat covered most of his body. On his head he wore a strange, conical fur hat. Beneath the skirts of the coat he wore shapeless trousers made from a thin kind of felt, with soft hide boots pulled up over them, about knee high.\n\nHe walked toward her now, rolling slightly with the bowlegged walk of a man who spent most of his time in the saddle. His features were sharp\u2014almond-shaped eyes that slitted to almost nothing from years of looking across long distances into the wind and the glare of a hard land. His skin was dark, almost nut brown from exposure to the sun, and the cheekbones were high. The nose was short and wide, and the lips were thin. Her first impression was that it was a cruel face. Then she amended the thought. It was simply an uncaring face. The eyes showed no signs of compassion or even interest in her as the rider reached down and grabbed her collar, forcing her to her feet.\n\n\"Stand,\" he said. The voice was thick and the accent guttural, but she recognized the single word in the Skandian tongue. It was basically similar to the Araluen language and she had spent months with the Skandians in any event. She allowed herself to be raised to her feet. She was nearly as tall as the man, she noticed, with a slight feeling of surprise. But, small as he was, the strength in the arm that dragged her upright was all too obvious.\n\nNow she noticed the bow and the quiver, and was instinctively glad that no chance had arisen for her to try to escape. She had no doubt that the man shoving her forward was an expert shot. There was something totally capable about him, she realized. He seemed so confident, so much in control. The bow might have simply marked him as a hunter. The long, curved sword in a brass-mounted scabbard on his left hip said that he was a warrior.\n\nHer study of the man was interrupted by a chorus of voices from the camp. Now that she had the time to look, she saw another five warriors, similarly dressed and armed. Their horses, small and shaggy-coated, were tethered to a rope slung between two trees, and there were three small tents placed around the clearing, made from a material that appeared to be felt. A fire crackled in a small circle of stones set in the center of the clearing and the other men were grouped around it. They rose to their feet in surprise as she was pushed toward them.\n\nOne of them stepped forward, a little apart from the others. That fact, and the commanding tone in his voice, marked him as the leader of the small group. He spoke rapidly to the man who had captured her. She couldn't understand the words, but the tone was unmistakable. He was angry.\n\nWhile he was obviously the leader of the small party, it was equally obvious that the man who had brought her here was also relatively senior. He refused to be cowed by the other man's angry words, replying in equally strident tones and gesturing toward her. The two of them stood, nose to nose, becoming louder and louder in their disagreement.\n\nShe stole a quick glance at the other four men. They had resumed their seats around the fire now, their initial interest in the captive having subsided. They watched the argument with interest, but with no apparent concern. One of them went back to turning a few green twigs with fresh meat spitted on them over the fire. The fat and juices ran off the meat and sizzled in the coals, sending up a cloud of fragrant smoke.\n\nEvanlyn's stomach growled softly. She hadn't eaten since the meager breakfast she had shared with Will. From the position of the sun, it must be late afternoon by now. She calculated that they had been traveling some three hours at least.\n\nFinally, the argument seemed to be resolved\u2014and in favor of her captor. The leader threw his hands in the air angrily and turned away, walking back to his place by the fire and dropping to a cross-legged position. He looked at her, then waved dismissively to the other man. Her fate, it appeared, was in his hands.\n\nThe horseman took a length of rawhide rope from his saddle bow and quickly ran two loops around her neck. Then he dragged her toward a large pine at the edge of the clearing and fastened the rope to it. She had room to move, but not too far in any direction. He turned her around, shoving her roughly, and grabbed her hands, forcing them behind her back and crossing the wrists over each other. She resisted. But the result was another stinging blow across the back of her head. After that, she allowed her hands to be roughly tied behind her, with a shorter piece of rawhide. She winced and muttered a protest as the knots were drawn painfully tight. It was a mistake. Another blow across the back of her head taught her to remain silent.\n\nShe stood uncertainly, hands bound and tied by her neck to the tree. She was considering the best way to sit down when the problem was solved for her. The horseman kicked her feet out from under her and sent her sprawling in the snow. That, at least, brought a couple of low chuckles from the men around the fire.\n\nFor the next few hours she sat awkwardly, her hands gradually growing numb from the pressure of the bonds. The six men now seemed content to ignore her. They ate and drank, swigging what was obviously a strong spirit from leather bottles. The more they drank, the more boisterous they became. Yet she noticed that, even though they seemed to be drunk, their vigilance didn't relax for a second. One of them was always on guard, standing outside the glare of the small fire and moving constantly to monitor the approaches to the camp from all directions. The guard changed at regular intervals, she noticed, without any dissension or need for persuasion. All of them seemed to take an equal turn too.\n\nAs it grew to full night, the men began to retire into the small felt tents. They were dome shaped and barely waist high, so their occupants had to crawl into them through a low entrance. But, she thought enviously, they were probably a lot warmer than she would be, sitting out here.\n\nThe fire died down and one of the men\u2014not the one who had captured her\u2014walked in that same bandy-legged stride toward Evanlyn and tossed a heavy blanket over her. It was rough and carried the rank smell of their horses, but she was grateful for the warmth. Even so, it was not really enough for comfort. She huddled against the tree, shrugging the blanket higher around her shoulders, and prepared for a supremely uncomfortable night.\n**6**\n\nHALT LEANED BACK AND SURVEYED HIS HANDIWORK WITH A satisfied sigh.\n\n\"There,\" he said. \"That should do the trick.\"\n\nHorace looked at him doubtfully, his eyes moving from Halt's pleased expression to the official-looking document that he had just completed forging.\n\n\"Whose seal is that at the bottom?\" he asked finally, indicating the impression of a rampant bull that was set in a large splodge of wax in the bottom right-hand corner of the parchment. Halt touched the wax gently, checking to see if it had hardened completely.\n\n\"Well, I suppose if it's anyone's it's mine,\" he admitted. \"But I'm hoping that our Skandian friends will think it belongs to King Henri of Gallica.\"\n\n\"Is that what his royal seal looks like?\" Horace asked, and Halt studied the symbol impressed in the wax a little more critically.\n\n\"Pretty much,\" he replied. \"I think the real one may be a trifle leaner in the body, but the forger I bought it from had a pretty indistinct impression to work from.\"\n\n\"But . . .\" Horace began unhappily, then stopped.\n\nHalt looked at him, one eyebrow raised quizzically. \"But?\" he repeated, making the word into a question.\n\nHorace merely shook his head. He knew Halt would probably laugh at his objection if he voiced it. \"Oh, never mind,\" he said at last. Then, realizing that the former Ranger was still waiting for him to speak, he changed the subject.\n\n\"I thought you said there was no ruling court in Gallica,\" he said. Halt shook his head.\n\n\"There's no effective ruling court,\" he told the young man. \"King Henri is the hereditary king of the Gallicans, but he has no real power. He maintains a court in the southern part of the country and lets the local warlords do as they please.\"\n\n\"Yes. I noticed some of that,\" Horace said meaningfully, thinking about the encounter with the warlord Deparnieux that had delayed their progress through Gallica.\n\n\"So old King Henri is something of a paper tiger,\" Halt continued. \"But he has been known to send envoys into other countries from time to time. Hence this.\" He gestured at the sheet of parchment that he was waving gently in the air so that the ink might dry and the wax seal might harden. The sight of the seal brought back all of Horace's misgivings.\n\n\"It just doesn't seem right!\" he blurted out, before he could stop himself. Halt smiled patiently at him, blowing gently on some damp patches of ink.\n\n\"It's as right as I can get it,\" he said mildly. \"And I doubt that the average border guard in Skandia will see the difference\u2014particularly if you're dressed in that fine suit of Gallican armor you took from Deparnieux.\"\n\nBut Horace shook his head doggedly. Now that his concern was out in the open, he was determined to plow on.\n\n\"That's not what I meant,\" he said, then added, \"and well you know it.\"\n\nHalt grinned easily at the young man's troubled expression. \"Sometimes your sense of morality amazes me,\" he said gently. \"You do understand that we have to get past the border guards if we're to have any chance of finding Will and the princess?\"\n\n\"Evanlyn,\" Horace corrected him automatically. Halt waved the comment aside.\n\n\"Whoever.\" He knew that Horace tended to refer to Princess Cassandra, the daughter of the Araluen King, by the name she had assumed when Will and Horace had first encountered her. He continued: \"You do realize that, don't you?\"\n\nHorace heaved a deep sigh. \"Yes, I suppose so, it's just that it seems so . . . dishonest, somehow.\"\n\nHalt's eyebrows rose in a perfect arch. \"Dishonest?\"\n\nHorace went on, awkwardly. \"Well, I was always taught that people's seals and crests were sort of . . . I don't know, sacrosanct. I mean . . .\" He gestured toward the figure of the bull impressed in red wax. \"That's a _king's_ signature.\"\n\nHalt pursed his lips thoughtfully. \"He's not much of a king,\" he replied.\n\n\"That's not the point. It's a principle, don't you see? It's like . . .\" He paused, trying to think of a reasonable parallel, and finally came up with: \"It's like tampering with the mail.\"\n\nIn Araluen, the mail was a service controlled by the Crown and there were dire penalties proscribed for anyone who tried to interfere with it. Not that such penalties had ever stopped Halt in the past when he'd needed to do a little tampering in that direction. He decided that it wouldn't be wise to mention that to Horace right now. Obviously, the morality code taught in Castle Redmont's Battleschool was a good deal more rigid than the behavior embraced by the Ranger Corps. Of course, the knights of the realm were entrusted with the protection of the Royal Mail, so it was logical that they should have such an attitude ingrained in them from an early part of their training.\n\n\"So how would you suggest that we deal with the problem?\" he asked at last. \"How would you get us past the border?\"\n\nHorace preferred simple solutions. \"We could fight our way in,\" he suggested with a shrug. Halt raised his eyes to heaven at the thought.\n\n\"So it's immoral to bluff our way past with an official document\u2014\" he began.\n\n\"A false document,\" Horace corrected. \"With a forged seal at the bottom.\"\n\nHalt conceded the point.\n\n\"All right\u2014a forged document if you like. That's reprehensible. But it would be perfectly all right for us to go through the border post hacking and shooting down everyone in sight? Is that the way you see it?\"\n\nNow that Halt put it that way, Horace had to admit there was an anomaly in his thinking. \"I didn't say we should kill everyone in sight,\" he objected. \"We could just fight our way through, that's all. It's more honest and above board, and I thought that's what knights were supposed to be.\"\n\n\"Knights may be, but Rangers aren't,\" Halt muttered. But he said it below his breath so that Horace couldn't hear him. He reminded himself that Horace was very young and idealistic. Knights _did_ live by a strict code of honor and ethics and those factors were emphasized in the first few years of an apprentice knight's training. It was only later in life that they learned to temper their ideals with a little expediency.\n\n\"Look,\" he said, in a conciliatory tone. \"Think about it this way: if we just barged on through and headed for Hallasholm, the border guards would send word after us. The element of surprise would be totally lost and we could find ourselves in big trouble. If we decide to fight our way in, the only way to do it is by leaving nobody alive to spread the word. Understand?\"\n\nHorace nodded, unhappily. He could see the logic in what Halt was saying. The Ranger continued in the same reasoning tone. \"This way, nobody gets hurt. You pose as an emissary from the Gallican court, with a dispatch from King Henri. You wear Deparnieux's black armor\u2014it's obviously Gallican in style\u2014and you keep your nose stuck in the air and leave the talking to me, your servant. That's the sort of behavior they'd expect from a self-important Gallican noble. There's no reason for any word to be sent informing Ragnak that two outlanders have crossed the border\u2014after all, we're supposed to be going to see him anyway.\"\n\n\"And what's in the dispatch that I'm supposed to be taking?\" Horace asked.\n\nHalt couldn't resist a grin. \"Sorry, that's confidential. You don't expect me to breach the secrecy of the mail system, do you?\" Horace gave him a pained look and he relented. \"All right. It's a simple business matter, actually. King Henri is negotiating for the hire of three wolfships from the Skandians, that's all.\"\n\nHorace looked surprised. \"Isn't that a little unusual?\" he asked, and Halt shook his head.\n\n\"Not a bit. Skandians are mercenaries. They're always hiring out to one side or another. We're just pretending that Henri wants to subcontract a few ships and crews for a raiding expedition against the Arridi.\"\n\n\"The Arridi?\" Horace said, frowning uncertainly.\n\nHalt shook his head in mock despair. \"You know, it might be more useful if Rodney spent less time teaching you people ethics and a little more time on geography. The Arridi are the desert people to the south.\" He paused and saw that this made no impression on the young man. Horace continued to look at him with a blank expression. \"On the other side of the Constant Sea?\" he added, and now Horace showed signs of recognition.\n\n\"Oh, them,\" he said dismissively.\n\n\"Yes, them,\" Halt replied, mimicking the tone. \"But I wouldn't expect you to think about them too much. There are only millions of them.\"\n\n\"But they never bother us, do they?\" Horace said comfortably. Halt gave a short laugh.\n\n\"Not so far,\" he agreed. \"And just pray they don't decide to.\"\n\nHorace could sense that Halt was on the verge of delivering a lecture on international strategy and diplomacy.\n\nThat sort of thing usually left Horace's head spinning after the first few minutes, while he tried to keep up with who was aligned with whom and who was conspiring against their neighbors and what they hoped to gain from it. He preferred Sir Rodney's type of lecture: right, wrong, black, white, out swords, hack and bash. He thought it might be expedient to head off Halt's incipient harangue. The best way to do that, he had learned from past experience, was to agree with him.\n\n\"Well, I suppose you're right about the forgery,\" he admitted. \"After all, it's only the Gallican's seal we're forging, isn't it? It's not as if you're forging a document from King Duncan. Even you wouldn't go as far as that, would you?\"\n\n\"Of course not,\" Halt replied smoothly. He began to pack away his pens and ink and his other forger's tools. He was glad he'd laid hands on the forged Gallican seal in his pack so easily. It was as well that he hadn't had to tip them all out and risk Horace's seeing the near-perfect copy of King Duncan's seal that he carried, among others. \"Now may I suggest that you climb into your elegant tin suit and we'll go and sweet-talk the Skandian border guards.\"\n\nHorace snorted indignantly and turned away. But another thought had occurred to Halt\u2014something that had been on his mind for some time.\n\n\"Horace . . .\" he began, and Horace turned back. The Ranger's voice had lost its former light tone and he sensed that Halt was about to say something important.\n\n\"Yes, Halt?\"\n\n\"When we find Will, don't tell him about the . . . unpleasantry between me and the King, all right?\"\n\nMonths ago, denied permission to leave Araluen in search of Will, Halt had devised a desperate plan. He had publicly insulted the King and, as a result, was banished for a period of one year. The subterfuge had caused Halt a great deal of mental anguish in the past months. As a banished person, he was automatically expelled from the Ranger Corps. The loss of his silver oakleaf was possibly the worst punishment of all, yet he bore it willingly for the sake of his missing apprentice.\n\n\"Whatever you say, Halt,\" Horace agreed. But Halt seemed to think, for once, that further explanation was necessary.\n\n\"It's just that I'd prefer to find my own way to tell him\u2014and the right time. All right?\"\n\nHorace shrugged. \"Whatever you say,\" he repeated. \"Now let's go and talk to these Skandians.\"\n\nBut there was to be no talking. The two riders, trailed by their small string of horses, rode through the pass that zigzagged between the high mountains until the border post finally came into sight. Halt expected to be hailed from the small wooden stockade and tower at any moment, as the guards demanded that they dismount and approach on foot. That would have been normal procedure. But there was no sign of life in the small fortified outpost as they drew nearer.\n\n\"Gate's open,\" Halt muttered as they came closer and could make out more detail.\n\n\"How many men usually garrison a place like this?\" Horace asked.\n\nThe Ranger shrugged. \"Half a dozen. A dozen maybe.\"\n\n\"There don't seem to be any of them around,\" Horace observed, and Halt glanced sideways at him.\n\n\"I'd noticed that part myself,\" he replied, then added, \"What's that?\"\n\nThere was an indistinct shape apparent now in the shadows just inside the open gate. Acting on the same instinct, they both urged their horses into a canter and closed the distance between them and the fort. Halt already felt certain what the shape was.\n\nIt was a dead Skandian, lying in a pool of blood that had soaked into the snow.\n\nInside there were ten others, all of them killed the same way, with multiple wounds to their torsos and limbs. The two travelers dismounted carefully and moved among the bodies, studying the awful scene.\n\n\"Who could have done this?\" said Horace in a horrified voice. \"They've been stabbed over and over again.\"\n\n\"Not stabbed,\" Halt told him. \"Shot. These are arrow wounds. And then the killers collected their arrows from the bodies. Except for this one.\" He held up the broken half of an arrow that had been lying concealed under one of the bodies. The Skandian had probably broken it off in an attempt to remove it from the wound. The other half was still buried deeply in his thigh. Halt studied the fletching style and the identification marks painted at the nock end of the arrow. Archers usually identified their own shafts in such ways.\n\n\"Can you tell who did this?\" Horace asked quietly, and Halt looked up to meet his gaze. Horace saw an expression of deep concern in the Ranger's eyes. That fact alone, more than the carnage around them, sent a wave of uneasiness through him. He knew it took a lot to worry Halt.\n\n\"I think so,\" said the Ranger. \"And I don't like it. It looks like the Temujai are on the move again.\"\n**7**\n\nTHE TRACKS LED TO THE EAST. AT LEAST, THAT WAS THE general direction Will had discerned from them. As the unknown horseman had made his way down the mountain, the track wound and twisted on itself, of necessity, as he followed the narrow, circuitous trails through the thick pine. But always, whenever there was a fork in the trail, the horseman chose the one that would eventually take him eastward once more.\n\nExhausted before the first hour was out, Will kept doggedly on, stumbling in the snow from time to time and, on occasions too numerous to count, falling full length to lie groaning.\n\nIt would be so easy, he thought, to just stay here. To let the aches in his unfit muscles slowly ease, to let the pounding of the pulse in his temples calm down and to just . . . rest.\n\nBut each time the temptation seized him, he thought of Evanlyn: how she had hauled him up the mountain. How she had helped him escape from the stockade where the yard slaves waited for their eventual death. How she had nursed him and cured him of the mind-numbing addiction to warmweed. And as he thought of her and what she'd done for him, somehow, each time, he found a tiny, hidden reservoir of strength and purpose. And somehow he dragged himself to his feet again and staggered on in pursuit of the tracks in the snow.\n\nWill kept dragging one foot after another, his eyes cast down to the tracks. He saw nothing else, noticed nothing else. Just the impressions of the hooves in the snow.\n\nThe sun dropped behind the mountain and the instant chill that accompanied its disappearance ate through his clothes, damp with the sweat of his exertions, and gnawed deep into his flesh. Dully, he reflected that he was lucky he had thought to bring the blankets with him. When he finally stopped for the night, the damp clothes would become a potential death trap. Without the warmth and dryness of the blankets to cocoon him, he could freeze to death in his damp clothes.\n\nThe shadows deepened and he knew nightfall wasn't far away. Still he kept on, keeping going as long as he could distinguish the scuffed hoofmarks in the trail. He was too exhausted to notice the variations in the tracks\u2014the deep troughs dug by the horse's locked-up front legs as it had slid down the steeper sections of the path. Those areas were only remarkable to him for the fact that he fell down them himself, more often than not. He could read none of the subtleties and secret messages that he had been trained to see. It was enough that there was a clear trail to follow.\n\nIt was all he was capable of.\n\nIt was long after dark and he was beginning to lose sight of the tracks now. But he continued as long as there was no possible deviation, no fork in the trail where he might have to choose one direction over another. When he came to a place where he must choose, he told himself, he would stop and camp for the night. He would wrap himself in the blankets. Perhaps he might even risk a small, well-shielded fire where he could dry his clothes. A fire would bring warmth. And comfort.\n\nAnd smoke.\n\nSmoke? He could smell it, even as he thought of a fire. Pine smoke\u2014the all-pervading smell of life in Skandia, the scented fragrance of the burning pine gum as it oozed from the wood and crackled in the flames. He stopped, swaying on his feet. He had thought of fire and, instantly, he could smell smoke. His tired mind tried to correlate the two facts, then realized there was no correlation, only coincidence. He could smell smoke because, somewhere near at hand, there was a fire burning.\n\nHe tried to think. A fire meant a camp. And that almost certainly meant that he had caught up with Evanlyn and whoever it was who had abducted her. They were somewhere close by, stopped for the night. Now all he had to do was find them and . . .\n\n\"And what?\" he asked himself in a voice thickened by fatigue. He took a long swallow from the water skin that he'd hung from his belt. He shook his head to clear it. For hours now, his entire being had been focused on one task\u2014to catch up with the unseen horseman. Now that he had almost accomplished that, he realized he had no plan as to what to do next. One thing was certain: he wouldn't be able to rescue Evanlyn by brute force. Swaying with fatigue, almost unconscious, he barely had the strength to challenge a sparrow.\n\n\"What would Halt do?\" he wondered. It had become his mantra over the past months whenever he found himself uncertain over a course of action. He would try to imagine his old mentor beside him, eyeing him quizzically, prompting him to solve the problem at hand by himself. To think it through, then to take action. The well-remembered voice seemed to sound in his ear.\n\n_Look first_ , Halt had been fond of saying. _Then act_.\n\nWill nodded, content that he had solved the problem for the time being.\n\n\"Look first,\" he repeated thickly. \"Then act.\"\n\nHe gave himself a few minutes' rest, hunkered down and leaning against the rough bole of a pine, then he stood erect once more, his muscles groaning with stiffness. He continued on the track, moving now with extra caution.\n\nThe smell of smoke grew stronger. Now it was mixed with something else and he recognized the smell of meat roasting. A few minutes later, moving carefully, he could discern an orange glow up ahead. The firelight reflected from the whiteness of the snow all around him, bouncing and magnifying in intensity. He realized that it was still some way ahead and continued along the trail. When he judged he was within fifty meters of the source of light, he moved silently off into the trees, fighting his way through the thick snow that came knee deep or higher.\n\nThe trees began to thin out, revealing a small clearing and the camp set around the fire. He lowered himself to his belly and inched forward, staying concealed in the deep shadows under the pines. He could make out dome-shaped tents now, three of them, arranged in a semicircle around the fire. He could see no sign of movement. The smell of roasting meat must have hung in the still, clear air long after the meal had been eaten, he realized. He started to edge forward when a movement behind the tents stopped him. He froze, absolutely still, as a man stepped forward into the fringe of the firelight. Stocky, dressed in furs, his face was hidden in the shadow cast by the fur hat he wore. But he was armed. Will could see the curved sword hanging at his waist and the slender lance that he held in his right hand, its butt planted in the snow.\n\nAs Will looked, he made out more detail. Horses, six of them, tethered among the trees to one side. He supposed that meant six men. He frowned, wondering how he could possibly get Evanlyn away from here, then realized that, so far, he hadn't seen her. He cast his gaze around the camp, wondering if perhaps she was inside one of the tents. Then he saw her.\n\nHuddled under a tree, a blanket pulled up to her shoulders. Peering more closely, he made out the bonds that kept her fastened in place. His eyes ached and he rubbed the back of one hand across them, then pinched the bridge of his nose between two fingers, trying to force his eyes to stay focused. It was a losing battle. He was exhausted.\n\nHe began to wriggle back into the forest, looking for a place where he could hide and rest. They weren't going anywhere this evening, he realized, and he needed to rest and recover his strength before he could accomplish anything. Tired as he was, he couldn't even begin to formulate a coherent plan.\n\nHe would rest, finding a spot far enough away to give him concealment, but not so far that he wouldn't hear the camp stirring in the morning. Ruefully, he realized that his earlier plans for a fire were now thwarted. Still, he had the blankets; that was something.\n\nHe found a hollow under the spreading branches of a massive pine and crawled into it. He hoped that the horsemen wouldn't patrol around their camp in the morning and find his tracks, then understood there was nothing he could do to prevent it. He untied the rolled-up blankets and hauled them tight around him, leaning against the bole of the massive tree.\n\nHe was never sure that he didn't fall asleep before his eyes actually closed. If not, it was certainly a close-run thing.\n\nSometime after midnight, Evanlyn woke, groaning in agony. The tight bonds were restricting blood flow and her shoulder muscles were badly cramped. The sentry, annoyed by the noise, loosened the bonds for a few minutes, then refastened her hands in front of her to take the strain off her shoulder muscles. It was a small improvement and she managed to sleep fitfully, until the sound of raised voices woke her.\n\nEvanlyn had sensed the antagonism between the two warriors the night before. But in the morning, it reached crisis point.\n\nShe wasn't to know it, but this was just the latest in a series of arguments between the two men. The small scouting party was one of many that had crossed the border into Skandia. Some weeks previously, Evanlyn had actually seen a member of an earlier party, near the hut where she and Will had spent the winter.\n\nThe man who had captured her, Ch'ren, was the son of a high-ranking Temujai family. It was the Temujai custom to have their young nobles serve a year as common soldiers before they were promoted to the officer class. At'lan, the commander of the scouting party, was a long-term soldier, a sergeant with years of experience. But, as a commoner, he knew he would never rise above his present rank. It galled him that the arrogant, headstrong Ch'ren would soon outrank him, just as it galled Ch'ren to take orders from a man he considered to be his social inferior. The day before, he had ridden off into the mountains on his own to spite the sergeant.\n\nHe had taken Evanlyn prisoner on a whim, without any real thought of the consequences. It would have been better had he remained unseen and allowed her to go on her way. The scouting party was under strict orders to avoid discovery and they had no orders to take prisoners. Nor was there any provision for holding or guarding them.\n\nThe simplest solution, At'lan had decided, was that the girl must be killed. As long as she was alive, there was the chance that she would escape and spread the word of their presence. If that happened, At'lan knew he would pay with his own life. He felt no sympathy for the girl. Nor did he feel any antagonism. His feelings about her were neutral. She was not of the People and so barely qualified as a human being.\n\nNow, he ordered Ch'ren to kill her. Ch'ren refused\u2014not out of any regard for Evanlyn, but simply to infuriate the sergeant.\n\nEvanlyn watched anxiously as they argued. Like the previous night, it was obvious to her that she was the reason for their disagreement. It was equally obvious, as their argument became more and more heated, that her position was becoming increasingly precarious. Finally, the older of the two drew back his hand and slapped the younger man across the face, sending him staggering a few paces. Then he turned and strode toward Evanlyn, drawing his curved saber as he came.\n\nShe looked from the sword in his hand to the totally matter-of-fact expression on his face. There was no malice, no anger, no expression of hatred there. Just the determined gaze of someone who, without the slightest qualm or hesitation, was about to end her life.\n\nEvanlyn opened her mouth to scream. But the horror of the moment froze the sound in her throat and she crouched, openmouthed, as death approached her. It was odd, she thought, that they had dragged her here, left her overnight and then decided to kill her.\n\nIt seemed such a pointless way to die.\n**8**\n\nHALT CAST AROUND, EXAMINING THE CONFUSED MASS OF TRACKS in the soft snow, frowning to himself as he tried to make sense of the clues there. Horace waited, bursting with curiosity.\n\nFinally, Halt stood up from where he had been kneeling, examining a particularly torn-up patch of ground.\n\n\"Thirty of them at least,\" he muttered. \"Maybe more.\"\n\n\"Halt?\" Horace asked experimentally. He didn't know if there were more details that Halt was about to reveal, but he couldn't wait any longer. The Ranger was moving away from the small stockade now, though, following another set of tracks that led into the mountains beyond the pass.\n\n\"A small party, maybe five or six, went on into Skandia. The rest of them went back the way they'd come.\"\n\nHe traced the directions with the tip of his longbow. He was speaking more to himself than to Horace, confirming in his own mind what the signs on the ground had told him.\n\n\"Who are they, Halt?\" Horace asked quickly, hoping to break through the Ranger's single-minded concentration. Halt moved a few paces further in the direction taken by the smaller party.\n\n\"Temujai,\" he said briefly, over his shoulder.\n\nHorace rolled his eyes in exasperation. \"You already said that,\" he pointed out. \"But who exactly are the Temujai?\"\n\nHalt stopped and turned to look back at him. For a moment, Horace was sure he was about to hear another comment on the sad state of his education. Then a thoughtful look crossed the Ranger's face and he said, in a milder tone than usual, \"Yes, I suppose there's no reason why you should have ever heard of them, is there?\"\n\nHorace, loath to interrupt, merely shook his head.\n\n\"They're the Riders from the Eastern Steppes,\" the Ranger said. Horace frowned, not understanding.\n\n\"Steps?\" he repeated, and Halt allowed a slight smile to show through.\n\n\"Not steps that you walk up and down,\" he told him. \"Steppes\u2014the plains and grasslands to the east. Nobody knows exactly where the Temujai originated. At one stage, they were simply a disorganized rabble of smaller tribes until Tem'gal welded them into one band and became the first Sha'shan.\"\n\n\"Sha'shan?\" Horace interrupted hesitantly, totally unaware of what the word might mean. Halt nodded and went on to explain.\n\n\"The leader of each band was known as the Shan. When Tem'gal became the overlord, he created the title Sha'shan\u2014the Shan of Shans, or the leader of leaders.\"\n\nHorace nodded slowly. \"But who was Tem'gal?\" he asked, adding hastily, \"I mean, where did he spring from?\"\n\nThis time Halt shrugged. \"Nobody really knows. Legend is that he was a simple herd boy. But somehow he became leader of one tribe, then united them with another, and another. The upshot was, he turned the Temujai into a nation of warriors\u2014probably the best light cavalry in the world. They're fearless, highly organized and absolutely pitiless when it comes to battle. They've never been defeated, to my knowledge.\"\n\n\"So what are they doing here?\" Horace asked, and Halt regarded him gravely, gnawing at his lower lip as he considered a possible answer.\n\n\"That's the question, isn't it?\" he asked. \"Perhaps we should follow this smaller group and see what we can find out. At least as long as they're heading in the direction we want to go.\"\n\nAnd slinging his bow over his left shoulder, he walked to where Abelard stood patiently, reins trailing loosely on the ground. Horace hurried after him, swinging up astride the black battlehorse he had been riding to impress the border guards. All at once, the finery that he had donned to play the role of a Gallican courier seemed a little incongruous. He nudged the black with his heel and set out after Halt.\n\nThe other two horses followed, the battlehorse on its lead rein, and Tug trotting quietly along without any need for urging or direction.\n\nHalt leaned down from the saddle, studying the ground.\n\n\"Look who's back,\" he said, indicating a trail in the snow. Horace nudged his horse closer and peered at the ground. To him there was nothing evident, other than a confusion of hoofprints, rapidly losing definition in the soft, wet snow.\n\n\"What is it?\" he asked finally.\n\nHalt replied without looking up from the track. \"The single rider who went off on his own has come back.\"\n\nSome way back, the trail had split, with one rider leaving the group and heading deeper into Skandia, while the main party had circled to the north, maintaining the same distance from the border. Now, apparently, that single rider had rejoined the group.\n\n\"Well, that makes it easier. Now we don't have to worry about his coming up behind us while we're trailing the others,\" Halt said. He started Abelard forward, then stopped, his eyes slitted in concentration.\n\n\"That's odd,\" he said, and slid down from the saddle to crouch on one knee in the snow. He studied the ground closely, then peered back in the direction from which the single rider had rejoined the group. He grunted, then straightened up, dusting wet snow from his knees.\n\n\"What is it?\" Horace asked. Halt screwed his face into a grimace. He wasn't totally sure of what he was seeing, and that bothered him. He didn't like uncertainties in situations like this.\n\n\"The single rider didn't rejoin the group here. They went this way at least a day before he did,\" he eventually said. Horace shrugged. There was a logical reason for that, he thought.\n\n\"So he was heading after them to a rendezvous,\" he suggested. Halt nodded agreement.\n\n\"More than likely. They're obviously a reconnaissance group and he may have gone scouting by himself. The question is, who followed him when he came back?\"\n\nThat raised Horace's eyebrows. \"Someone followed him?\" he asked. Halt let go a deep breath in frustration.\n\n\"Can't be sure,\" he said briefly. \"But it looks that way. The snow's melting quickly and the tracks aren't totally clear. It's easy enough to read the horse's tracks, but this new player is on foot . . . if he's really there,\" he added uncertainly.\n\n\"So . . . ,\" Horace began. \"What should we do?\"\n\nHalt came to a decision. \"We'll follow them,\" he said, mounting once more. \"I won't sleep comfortably until I find out what's going on here. I don't like puzzles.\"\n\nThe puzzle deepened an hour later when Tug, following quietly behind the two riders, suddenly threw back his head and let go a loud whinny. It was so unexpected that both Halt and Horace spun in their saddles and stared at the little horse in amazement. Tug whinnied again, a long, rising tone that had a note of anxiety in it. Horace's spare battlehorse jerked at its lead rope and whinnied in alarm as well. Horace was able to quell an incipient response from the black that he was riding, while Abelard, naturally, remained still.\n\nAngrily, Halt made the Ranger hand signal for silence and Tug's whinny cut off in midnote. The others gradually quieted as well.\n\nBut Tug continued to stand in the trail, forelegs braced wide apart, head up and nostrils flaring as he sniffed the frigid air around them. His body trembled. He was on the brink of giving vent to another of those anguished cries and only the discipline and superb training of all Ranger horses was preventing him from doing so.\n\n\"What the devil . . . ,\" Halt began, then, sliding down from the saddle, he moved quietly back to the distressed horse, patting Tug's neck gently.\n\n\"Hush now, boy,\" he murmured. \"Settle now. What's the trouble with you then?\"\n\nThe quiet voice and the gentle hands seemed to soothe the little horse. He put his head down and rubbed his forehead against Halt's chest. The Ranger gently fondled the little horse's ears, still speaking to him in a soft croon.\n\n\"There you are . . . if only you could talk, eh? You know something. You sense something, isn't that right?\"\n\nHorace watched curiously as the trembling gradually eased. But he noticed the little horse's ears were still pricked and alert. He might have been quieted, but he wasn't at ease, the apprentice realized.\n\n\"I've never seen a Ranger horse behave like that before,\" he said softly, and Halt looked up at him, his eyes troubled.\n\n\"Neither have I,\" he admitted. \"That's what has me worried.\"\n\nHorace studied Tug carefully. \"He seems to have calmed down a little now,\" he ventured, and Halt laid a hand across the horse's flank.\n\n\"He's still taut as a bowstring, but I think we can keep going. There's only an hour or so till dark and I want to see where our friends are camped for the night.\"\n**9**\n\nDeep IN THE SHELTER OF THE PINE TREE, WRAPPED IN THE inadequate warmth of the two blankets, Will spent a fitful night, dozing for short periods, then being woken by the cold and his racing thoughts.\n\nForemost in his mind was his sense of utter inadequacy. Faced with the need to rescue Evanlyn from her captors, he had absolutely no idea how he might accomplish the task. They were six men, well armed and capable-looking. He was a boy, armed only with a small hunting bow and a short dagger. His arrows were good only for small game\u2014with points made by hardening the end of the wood in a fire and then sharpening them. They were nothing like the razor-sharp broadheads that he had carried in his quiver as an apprentice Ranger. \"A Ranger wears the lives of two dozen men on his belt,\" went the old Araluen saying.\n\nHe racked his brain again and again throughout the long periods of sleeplessness. He thought bitterly that he was supposed to have a reputation as a thinker and a planner. He felt that he was letting Evanlyn down with his inability to come up with an idea. And letting down others too. In his mind's eye, half asleep and dozing, he saw Halt's bearded face, smiling at him and urging him to come up with a plan. Then the smile would fade, first to a look of anger, then, finally, of disappointment. He thought of Horace, his companion on the journey through Celtica to Morgarath's bridge. The heavily built warrior apprentice had always been content to let Will do the thinking for the two of them. Will sighed unhappily as he thought how misplaced that trust had become. Perhaps it was an aftereffect of the warmweed to which he had been addicted. Perhaps the drug rotted a user's brain, making him incapable of original thought.\n\nTime and again through that unhappy night, he asked himself the question, \"What would Halt do?\" But the device, so useful in the past for providing an answer to his problems, was ineffectual. He heard no answering voice deep within his subconscious, bringing him counsel and advice.\n\nThe truth was, of course, that given the situation and the circumstance, there was no practical action that Will could take. Virtually unarmed, outnumbered, on unfamiliar ground and sadly out of condition, all he could do would be to keep watching the strangers' encampment and hope for some change in the circumstances, some eventuality that might provide him with an opportunity to reach Evanlyn and get her away into the trees.\n\nFinally abandoning the attempt to rest, he crawled out from under the pine tree and gathered his meager equipment together. The position of the stars in the heavens told him that it was a little over an hour before he could expect to see the first light of dawn filtering through the treetops.\n\n\"At least that's one skill I've remembered,\" he said miserably, speaking the words aloud, as had become his custom during the night.\n\nHe hesitated, then came to a decision and moved off through the trees toward the campsite. There was always a chance that something might have changed. The sentry might have fallen asleep or gone off into the forest to investigate a suspicious noise, leaving the way clear to rescue Evanlyn.\n\nIt wasn't likely but it was possible. And if such an opportunity arose, it was essential that Will be present to take advantage of it. At least it was a definite course of action for him to follow, so he moved as quietly as possible, keeping one of the blankets draped around his shoulders as a cloak.\n\nIt took him ten minutes to find his way back to the small camp. When he did, his hopes were dashed. There was still a sentry patrolling and, as Will observed, the watch had changed, with a fresh man taking over the post, wide-awake and rested. He moved around the perimeter of the camp on a regular patrol, coming within twenty meters of the spot where the boy crouched hidden behind a tree. There was no sign of slackness or inattention. The man kept his point of vision moving, continually searching the surrounding forest for any sign of unusual movement.\n\nWill looked enviously at the recurve bow slung, ready strung, over the man's right shoulder. It was very similar to the one Halt had given him when he had first taken up his apprenticeship with the grim-faced Ranger. Vaguely, he recalled Halt had said something about learning how to make such a bow from the warriors of the Eastern Steppes. He wondered now if these men were some of those warriors.\n\nThe sentry's bow was a real weapon, he thought, unlike the virtual toy that he carried. Now, if he had a bow like that in his hands, and a few of the arrows that showed their feathered tips in the sentry's back quiver, he might be able to accomplish something. For a while, he played with the idea of overpowering the sentry and taking his bow, but he was forced to reject the idea.\n\nThere was no way he would get within reach of the man without being seen or heard. And, even if he could accomplish that, there was little chance of his being able to overpower an armed warrior. Pitting the small dagger he carried against the man's saber would be suicide. He could chance a throw of the knife, of course, but it was a poorly balanced weapon and ill suited for throwing, without sufficient weight in the hilt to drive the blade home into the target.\n\nAnd so he huddled in the snow at the base of the tree, watching and waiting for an opportunity that never came. He could see Evanlyn's crumpled shape to one side of the camp. The tree she was tied to was surrounded by clear space. There was no way he could approach her without the sentry seeing him. It all seemed hopeless.\n\nHe must have dozed off, lulled by the cold and by the restless night he had spent, for he was awoken by the sound of voices.\n\nIt was just after dawn and the early morning light struck obliquely through the gaps in the trees, throwing long shadows across the clearing. Two of the group of warriors were standing, a little apart from the others, arguing. The words were indecipherable to Will, but the subject of their debate was obvious, as one of them kept gesturing toward Evanlyn, still tied to the tree, huddled in the blanket she had been given, and now wide-awake and watchful.\n\nAs the discussion progressed, the men became increasingly angrier, their voices louder. Finally, the older man seemed goaded beyond restraint. He slapped the other man, sending him staggering. He nodded once, as if satisfied, then turned toward Evanlyn, his hand dropping to the hilt of his sword.\n\nFor a moment, Will remained frozen. The warrior's manner was so casual as he drew the sword and approached the girl that it seemed impossible to believe that he meant her any harm. There was a callousness about the entire scenario that seemed to belie any hostile intent. Yet it was that same callousness and casualness that created a growing sense of horror in Will. The man raised the sword above the girl. Evanlyn's mouth opened but no sound came and Will realized that killing her meant nothing, absolutely nothing at all, to the small, bowlegged warrior.\n\nActing under their own volition, Will's hands had drawn and nocked an arrow as the warrior dropped his hand to the sword hilt. The curved blade went up and Evanlyn crouched in the snow, one hand raised in a futile attempt to ward off the killing stroke. Will stepped out clear of the tree, bringing the bow to full draw as his mind rapidly weighed the situation.\n\nHis arrow wouldn't kill. It was little more than a pointed stick, even though that point had been hardened in a fire. The chances were that, if he aimed at the warrior's body, the thick furs and leather jerkin that he wore would stop the arrow before it even broke the skin. There was only one vulnerable point where the man was unprotected and that was, coincidentally, one that gave Will's shot the best chance of stopping the sword stroke. The man's wrist was exposed as his arm went up, the bare flesh showing at the end of the thick fur sleeve. All of this Will registered in the time it took him to bring the arrow's crude fletching back to touch his cheek. His aim shifted smoothly to the man's wrist, the tip of the arrow rising slightly to allow for drop. He checked his breath automatically, then released.\n\nThe bow gave a slight twang and the light arrow leapt away, arcing swiftly across the intervening space and burying its point into the soft flesh of the warrior's wrist.\n\nWill heard the strangled shout of pain as his hands moved in the well-remembered sequence, nocking another arrow and sending it after the first. The sword had dropped from the man's grasp, falling noiselessly into the thick snow and causing Evanlyn to shrink back as its razor-sharp blade just missed her arm. The second arrow slapped against the man's thick sleeve and hung there harmlessly as he grasped his right wrist, blood pouring down over his hand.\n\nShocked and caught unaware as he was, the man had still turned instinctively in the direction from which the arrow had come and now, seeing the movement as Will fired the second time, he made out the small figure across the clearing. With a snarl of anger, he released his injured wrist and clawed a long dagger from his belt with his left hand. For a moment, Evanlyn was forgotten as he pointed in Will's direction to his men, shouting for them to follow him, then began to run toward his attacker.\n\nWill's third arrow slowed the man down as it flashed past his face, causing him to jerk to one side to avoid it. But then he was coming again and two of his men were following. At the same time, Will saw a fourth man heading toward Evanlyn and his heart sank as he realized he had failed. He sent another shaft zipping toward him, knowing the effort was in vain. Turning to face the oncoming warrior, Will dropped the useless bow and reached for the knife in his own belt.\n\nAnd then he heard a sound from the past, a sound eerily familiar from hours spent in the forest around Castle Redmont.\n\nA deep thrum came from somewhere behind him, then the air-splitting hiss of a heavy shaft traveling at incredible speed, with enormous force behind it. Finally, Will heard the solid smack as it struck home.\n\nThe arrow, black-shafted, gray-feathered, seemed to appear in the center of the approaching warrior's chest. He fell backward in the snow. Another thrum-hiss-smack and the second man went down as well. The third turned and ran for the horses tethered on the far side of the camp. Galloping hoofbeats told Will that the remaining two men had already made their escape, unwilling to face the uncanny accuracy of the longbow.\n\nWill hesitated, his mind in a turmoil. Instinctively, he knew what had happened. Logically, he had no idea how it had come about. He turned and saw the barely visible, gray-cloaked figure some thirty meters behind him, the huge longbow still held at the ready, another arrow already drawn.\n\n\"Halt?\" he cried, his voice breaking. He started to run toward the figure, then remembered. Evanlyn! She was still in danger. As he turned, he heard the scrape of steel on steel and saw that she had managed to grab the fallen saber and ward off the first attack.\n\nBut it could only be a momentary respite as her hands were still tied in front of her and she was tethered firmly to the tree. He pointed toward her and yelled inarticulately, desperately urging Halt to shoot, then realized that the Ranger's view of the scene was blocked by the trees.\n\nThen another figure was bounding toward the struggling girl and her attacker. A tall, well-built figure who looked strangely familiar, wearing chain mail and a white surcoat with a strange emblem that resembled a stylized oakleaf.\n\nHis long, straight sword intercepted the curved blade as it swung down. Then he had interposed himself between Evanlyn and the man who was trying to kill her and, in a series of flashing sword strokes that bewildered the eye, he drove the other man back away from the girl. He obviously had the better of the exchange and his opponent retreated before him, his parries and strokes growing more desperate as he realized that he was totally outmatched. The man lunged clumsily with his curved blade and it was deflected easily so that his momentum carried him forward, off balance, wide open to the retaliatory backhanded cut that was already on its way\u2014\n\n\"Don't kill him!\" Halt shouted, just in time, and Horace twisted his wrist so that the flat of his blade, not the razor edge, slammed into the side of the man's head. The man's eyes rolled up and he sagged to the ground, unconscious.\n\nAnd very lucky.\n\n\"We want a prisoner,\" the Ranger finished mildly. Then he was driven back by the impact of a small body running headlong into him, and a pair of arms that wrapped around his waist, and Will was sobbing and babbling mindlessly as he embraced his teacher and mentor and friend. Halt patted his shoulder gently, and was surprised to find a single tear sliding down his own cheek.\n\nHorace sliced through Evanlyn's bonds with the edge of his sword and gently assisted her to her feet.\n\n\"Are you all right?\" he asked anxiously, then, satisfied that she was, he couldn't help a huge grin of relief breaking out across his face.\n\n\"Oh, Horace, thank God you're here!\" the girl sobbed, and throwing her arms around his neck, she buried her face in his chest. For a moment, Horace was nonplussed. He went to embrace her in return, realized he was still holding his sword, and hesitated awkwardly. Then, coming to a decision, he planted it firmly, point first in the ground, and put his arms around her, feeling the softness of her and smelling the fragrance of her hair and skin.\n\nHis grin grew wider, which he wouldn't have thought was possible. He decided there were definite advantages to being a hero.\n**10**\n\n\"YOU REALLY MEAN HORACE Is SOME KIND OF HERO IN GALLICA?\" Will asked incredulously, not totally sure that Halt and Horace weren't pulling off some kind of enormous practical joke. But the grizzled Ranger was nodding his head emphatically.\n\n\"A regular figure of respect,\" he said. Evanlyn turned to the muscular young warrior and leaned forward to touch his hand lightly.\n\n\"I can believe it,\" she said. \"Did you see the way he took care of that Temujai soldier who was trying to kill me?\" Her eyes were alight with an unusual warmth and Will, noticing it, felt a sudden stab of jealousy for his old friend. Then he pushed the unworthy thought aside.\n\nHalt had been unwilling to remain too close to the Temujai campsite. There was no telling how far away the main force might be and there was always the possibility that the two men who had escaped might lead others back to the spot.\n\nThey had retraced the path Halt and Horace had followed, moving back toward the border crossing where they had discovered the first evidence of the Temujai assault. Around the middle of the day, they found a spot on a hilltop with a good view of the surrounding terrain and a saucer-shaped depression that would keep them hidden from sight. Here, they could see without being seen, and Halt decided to camp there while he made up his mind as to their next move.\n\nThey had built a small fire, screened by a grove of young pines, and prepared a meal.\n\nEvanlyn and Will fell ravenously on the savory stew that the Ranger had prepared and for a while there was silence, broken only by the sound of dedicated eating.\n\nThen the old friends began to catch up on the events that had taken place since the final confrontation with the Wargal army on the Plains of Uthal. Will's jaw had dropped with amazement as Halt described how Horace had defeated the terrifying Lord Morgarath in single combat.\n\nHorace looked suitably embarrassed and Halt, sensing this, described the combat in a lighthearted tone, jokingly implying that the boy had stumbled clumsily and fallen under the oncoming hooves of Morgarath's battlehorse, rather than choosing to do so as a deliberate last throw of the dice to unseat his opponent. The apprentice warrior blushed and pointed out that his final ploy\u2014the double knife defense\u2014had been taught to him by Gilan and that he and Will had spent hours practicing the skill on their trip through Celtica. He made it sound as if, somehow, Will deserved some share of the credit for his victory. As he spoke, Will leaned back comfortably against a log and thought how much Horace had changed. Once his sworn enemy when they were both growing up as castle wards, Horace had since become his closest friend.\n\nWell, one of his closest friends, he thought, as he felt a shaggy head butt insistently against his shoulder. He twisted around, reaching out one hand to stroke Tug's ears and scratch the spot between them the way the little horse enjoyed. Tug let go a low snuffle of pleasure at the touch of his master's hand. Since they had been reunited, the horse had refused to stray more than a meter or two from Will's presence.\n\nHalt looked at the two of them now, across the campfire, and smiled inwardly. He felt an enormous sense of relief now that he had finally found his apprentice. A weight of self-blame had lifted from him, for he had suffered greatly in the long months since he had watched the wolfship sailing away from the Araluen coast with Will on board. He felt he had failed the youngster, that he had somehow betrayed him. Now that the boy was safely back in his care, he was filled with a deep sense of well-being. Admittedly, the events of the past day had also left a new worry gnawing at the back of his mind, but for the moment, that could wait while he enjoyed the reunion.\n\n\"Do you think you could persuade that horse of yours to stay with the other horses for a minute or two?\" he said with mock severity. \"Otherwise he'll wind up believing that he's one of us.\"\n\n\"He's been driving Halt crazy since we first found your tracks,\" Horace put in. \"He must have picked up your scent and known it was you we were following, although Halt didn't realize it.\"\n\nAt that, Halt raised an eyebrow. \"Halt didn't realize it?\" he repeated. \"And I suppose you did?\"\n\nHorace shrugged. \"I'm just a warrior,\" he replied. \"I'm not supposed to be a thinker. I leave that to you Rangers.\"\n\n\"I must admit it had me puzzled,\" Halt said. \"I've never seen a Ranger horse behave like that. Even when I ordered him to calm down and be silent, I could tell there was something on his mind. When you first stepped out of the trees to shoot, I thought he was going to take off after you.\"\n\nWill continued to rub the shaggy head as it leaned down to him. He smiled broadly around the campsite. Now that Halt was here and he was surrounded by his closest friends, he felt safe and secure once more\u2014a sensation he hadn't enjoyed in over a year. He smiled at the Ranger, relieved that Halt had been pleased with his actions. Evanlyn had described their journey across the Stormwhite Sea, and the series of events that had led to their arrival at Hallasholm.\n\nHorace had looked at Will with open admiration as she described the way he had humbled the wolfship captain Slagor in the drafty, smoky cabin on the barren island where they had sheltered from the Stormwhite's worst excesses. Halt had merely studied his apprentice with a keen glance and nodded once. That single movement meant more to Will than volumes of praise from anyone else\u2014particularly since he wasn't terribly proud of the way things had turned out at Hallasholm, and his subsequent addiction to warmweed. He had been fearful that Halt would disapprove, but when Evanlyn had spoken of her near despair when she had found him in the yard slaves' compound, mindless and unthinking, the Ranger had merely nodded once more and uttered a curse under his breath at people who would inflict such a substance on others. His eyes had met Will's anxious gaze across the fire and Will had seen a deep, deep sadness there.\n\n\"I'm sorry you had to go through that,\" his master said softly, and Will knew that everything would be all right.\n\nEventually, they had talked their fill. There would be details that could be filled in over the coming weeks, and there were items that they had forgotten. But in general terms, they were up to date with one another.\n\nThere was, however, one aspect of Halt's story that hadn't been revealed. Neither Will nor Evanlyn had learned of Halt's banishment, or his expulsion from the Ranger Corps.\n\nAs the shadows lengthened, Halt moved once more to the spot where their captive was tied hand and foot. He loosened the bonds for a few minutes, first the hands, then the feet, retying the hands before he released the second set of bonds. The Temujai warrior grunted a brief appreciation of the temporary relief. Halt had already done this several times during the afternoon, ensuring that the man wasn't permanently disabled by the restriction of the flow of blood to his hands and feet.\n\nIt also gave him an opportunity to make sure the man's bonds were tight and that he hadn't managed to loosen them or wriggle free. Knowing he would receive no reply, Halt asked the man for his name and his military unit. Although he spoke the Temujai tongue with reasonable fluency, having spent several years among the People, as they called themselves, he saw no reason to apprise the prisoner of that fact. As a consequence, Halt used the trader's language common to all the people of the Hemisphere\u2014a m\u00e9lange of Gallic, Teuton and Temujai words in a simple, pidgin-language structure that took no notice of grammar or syntax.\n\nAs he had expected, the Tem'uj simply ignored his overtures. Halt shrugged and moved away, deep in thought. Horace was sitting by the fireplace, carefully cleaning and oiling his sword. Evanlyn was in the sentry position at the brow of the hilltop, keeping watch over the hillside below them. She would be due to be relieved in another half hour, he thought idly. As Halt paced back and forth, turning over the problem that taxed his mind, he became aware of another presence beside him. He glanced around and smiled to see Will pacing with him, wrapped in the gray mottled Ranger cloak that Halt had carried with him, along with the bow he'd made and a saxe knife. The double-knife scabbards were a Ranger-issued item of equipment and Halt, expelled from the Corps, had been unable to find one for the boy. As yet, Will hadn't remarked on the fact.\n\n\"What's the problem, Halt?\" the young man asked now.\n\nHalt stopped pacing to face him, his eyebrow arcing in an expression that was familiar to Will.\n\n\"Problem?\" he repeated. Will grinned at him, refusing to be put off, refusing to be diverted. He's grown up a lot in the past year, Halt thought, remembering how that response would once have left the boy confused and disconcerted.\n\n\"When you pace back and forth like a caged tiger, it usually means you're trying to think through a problem of some kind,\" Will said. Halt pursed his lips thoughtfully.\n\n\"And I suppose you've seen so many tigers in your time?\" he asked. \"Caged and otherwise?\"\n\nWill's grin widened a little. \"And when you try to distract me from my question by asking a question back, I _know_ you're thinking over some problem,\" he added. Halt finally gave in. He had no idea that his habits had become so easy to interpret. He made a mental note to change things, then wondered if he wasn't getting too old to do so.\n\n\"Well, yes,\" he replied. \"I must admit I do have something on my mind. Nothing major. Don't let it worry you.\"\n\n\"What is it?\" said his apprentice bluntly, and Halt cocked his head sideways.\n\n\"You see,\" he explained, \"when I say 'don't let it worry you,' I mean, there's no real need for us to discuss it.\"\n\n\"I know that,\" said his apprentice. \"But what is it anyway?\"\n\nHalt drew a deep breath, then let it out in a sigh. \"I seem to remember that I once had much more authority than I seem to have these days,\" he said to no one in particular. Then, realizing that Will was still waiting expectantly, he relented.\n\n\"It's these Temujai,\" he said. \"I'd like to know what they're up to.\" He glanced across their campsite to where the Tem'uj was sitting, securely bound. \"And I've got a snowball's chance in a forest fire of finding out from our friend there.\"\n\nWill shrugged. \"Is it really any of our concern?\" he asked. \"After all, surely we can leave them and the Skandians to fight it out.\"\n\nHalt considered this, scratching at his chin with forefinger and thumb. \"I take it you're thinking along the lines of the old saying 'The enemy of my enemy is my friend'?\" he said. Will shrugged once again.\n\n\"I wasn't thinking of it in those words exactly,\" he said. \"But it does sum the situation up pretty well, don't you think? If the Skandians are kept busy fighting these Temujai, then they won't be able to bother us with their coastal raids, will they?\"\n\n\"That's true, up to a point,\" Halt admitted. \"But there is another old saying: 'Rather the devil you know.' Have you ever heard that one?\"\n\n\"Yes. So you're saying that these Temujai could be a lot more of a problem than the Skandians?\"\n\n\"Oh yes indeed. If they defeat the Skandians, there's nothing to stop them from moving on Teutlandt, Gallica, and finally Araluen.\"\n\n\"But they'd have to beat the Skandians first, wouldn't they?\" Will said. He knew, from firsthand experience, that the Skandians were fierce, fearless warriors. He could see them forming an effective buffer between the invading Temujai and the other western nations, with both sides ending up severely weakened by the war and neither presenting a threat in the near future. It was a perfect strategic position, he told himself comfortably. Halt's next words made him feel considerably less comfortable.\n\n\"Oh, they'll defeat them, all right. Make no mistake about that. It will be a savage, bloody war, but the Temujai will win.\"\n**11**\n\nAFTER THE EVENING MEAL, HALT CALLED THE SMALL GROUP together. The wind had risen with the onset of night and it whistled eerily through the branches of the pines. It was a clear night, and the half-moon shone brilliantly above them as they huddled in their cloaks around the remnants of the fire.\n\n\"Will and I were talking earlier,\" he told them. \"And I've decided that, since our discussion concerns all of us, it's only fair to tell you what I've been thinking.\"\n\nHorace and Evanlyn exchanged puzzled looks. They had both simply assumed that the master and the apprentice were catching up on lost time together. Now, it appeared, there was something else to consider.\n\n\"First and foremost,\" Halt continued, seeing he had their undivided attention, \"my aim is to get you, Will, and the Pr\u2014\" He hesitated, stopping before he used Evanlyn's title. They had all agreed that it would be safer for her to continue under her assumed name until they returned home. He corrected himself. \"Will and Evanlyn, and Horace, of course, across the border and out of Skandia. As escaped prisoners, you're in considerable danger if the Skandians recapture you. And, as we all know, that danger is even greater for Evanlyn.\"\n\nThe three listeners nodded. Will had told Halt and Horace about the risk to Evanlyn should Ragnak ever discover her real identity as King Duncan's daughter. The Oberjarl had sworn a blood vow to the Vallas, the trio of savage gods who ruled the Skandian religion, in which he promised death to any relative of the Araluen King.\n\n\"On the other hand,\" Halt said, \"I am deeply worried about the presence of the Temujai here on the borders of Skandia. They haven't come this far west in twenty years\u2014and the last time they did, they put the entire western world at risk.\"\n\nNow he really had their attention, he saw. Horace and Evanlyn sat up straighter and leaned a little closer to him. He saw the puzzled look on the young warrior's face in the firelight.\n\n\"Surely, Halt, you're exaggerating?\" Horace asked.\n\nWill looked sideways at his friend. \"That's what I thought too,\" he said quietly, \"but apparently not.\"\n\nHalt shook his head firmly. \"I wish I were,\" he said. \"But if the Temujai are moving in force, it's a threat to all our countries, Araluen included.\"\n\n\"What happened last time, Halt?\" It was Evanlyn who spoke now, her voice uncertain, the concern obvious in it. \"Were you there? Did you fight them?\"\n\n\"I fought with them and, eventually, against them,\" he said flatly. \"There were things we wanted to learn from them and I was sent to do so.\"\n\nHorace frowned. \"Such as?\" he asked. \"What could the Rangers hope to learn from a bunch of wild horsemen?\" Horace, it must be admitted, had a somewhat inflated idea of the extent of the Ranger Corps' knowledge. To put it simply, he thought they knew just about everything that was worth knowing.\n\n\"You wanted to learn how they made their bows, didn't you?\" said Will suddenly. He remembered seeing the bows carried by the horsemen and thinking how similar they were to his own. Halt looked at him and nodded.\n\n\"That was part of it. But there was something more important. I was sent to trade with them for some of their stallions and mares. The Ranger horses we ride today were originally bred from the Temujai herds,\" he explained. \"We found their recurve bows interesting, but when you consider how difficult and time-consuming they are to make, they offered no significant improvement in performance over the longbow. But the horses were a different matter.\"\n\n\"And they were happy to trade?\" asked Will. As he spoke, he turned to study the shaggy little horse standing a few paces behind him. Tug, seeing him turn to look, nickered a soft greeting. Now that Halt mentioned it, there was a distinct resemblance to the horses he had seen in the Temujai camp.\n\n\"They were not!\" Halt replied with a heartfelt shake of the head. \"They guarded their breeding stock jealously. I'm probably still wanted among the Temujai nation as a horse thief.\"\n\n\"You stole them?\" Horace asked, in a mildly disapproving tone.\n\nHalt hid a smile as he replied.\n\n\"I left what I considered a fair price,\" he told them. \"The Temujai had other ideas about the matter. They weren't keen to sell at any price.\"\n\n\"Anyway,\" Will said impatiently, dismissing the matter of whether the horses had been bought or stolen, \"what happened when their army invaded? How far did they come?\"\n\nHalt stirred the small pile of embers between them with the end of a charred stick until a few tongues of flame flickered in the red coals. \"They were heading farther south that time,\" he said. \"They overran the Ursali nation and the Middle Kingdoms in no time at all. There was no stopping them. They were the ultimate warriors\u2014fast moving, incredibly brave, but most of all, highly disciplined. They fought as a large unit, always, whereas the armies facing them almost always ended up fighting in small groups of perhaps a dozen at a time.\"\n\n\"How could they do that?\" Evanlyn asked. She had been around her father's armies enough to know that the biggest problem facing any commander once battle started was staying in effective control and maintaining communication with the troops under him. Halt looked at her, sensing the professional interest behind her question.\n\n\"They've developed a signaling system that lets their central commander direct all his troops in concerted maneuvers,\" he told her. \"It's a very complex system relying on colored flags in different combinations. They can even operate at night,\" he added. \"They simply substitute colored lanterns for the flags. Quite frankly, there was no army capable of stopping them as they drove on toward the sea.\n\n\"They'd cut through the northeast corner of Teutlandt, then on through Gallica. Every army that faced them, they defeated. Their superior tactics and discipline made them unbeatable. They were only three days' riding from the Gallican coast when they finally stopped.\"\n\n\"What stopped them?\" Will asked. A noticeable chill had fallen over the three young listeners as Halt had described the inexorable advance of the Temujai army. At the question, the Ranger gave a short laugh.\n\n\"Politics,\" he said. \"And a dish of bad freshwater clams.\"\n\n\"Politics?\" Horace snorted in disgust. As a warrior, he had a healthy contempt for politics and politicians.\n\n\"That's right. This was when Mat'lik was the Sha'shan, or supreme leader. Now, among people like the Temujai, that's a highly unstable position. It's taken by the strongest contender and very few Sha'shans have died in their beds. Although Mat'lik did, as it turned out,\" he added as an afterthought, before continuing.\n\n\"As a result, it's normal practice for anyone who might contest the position to be assigned tasks that keep them a long way from home. In this case, Mat'lik's brother, nephew and second cousin were the most likely candidates, so he made sure they were kept busy with the army. That way, not only could they not get up to mischief around him, but they could all keep an eye on one another as well. Naturally, they distrusted each other totally.\"\n\n\"Wasn't it dangerous to give them control over the army?\" Will asked. Halt signified that the question was a good one.\n\n\"Normally, it might be. But the command structure was designed so that none of them had absolute control. Mat'lik's brother Twu'lik was the strategic commander. But his nephew was the pay-master and his cousin was the quartermaster. So, one led them, one fed them and one paid them. They all had pretty equal claims on the loyalty of the soldiers. That way, they could keep one another in check.\"\n\n\"So where did the clams come in?\" Horace asked. Food was always a matter of interest to him. Halt resettled himself by the fire, leaning back against a log.\n\n\"Mat'lik was partial to freshwater clams,\" he told them. \"So much so that he very unwisely had his wife prepare him a big dish when they were out of season. It seems that some of them were tainted and he was taken by a terrible fit while eating. He screamed, tore at his throat, fell down and went into a deep coma. It was obvious that he was very close to death.\n\n\"Naturally, when news reached the army, the three main contenders for the top job couldn't get back to the Sha'shan's court fast enough. The succession would be decided by an election among the senior Shans and they knew if they weren't back there to hand out the bribes and buy votes, someone else would get the prize.\"\n\n\"So they simply abandoned the invasion?\" Will asked. \"After they'd come so far?\"\n\nHalt made a dismissive gesture. \"They were a pragmatic bunch,\" he said. \"Gallica wasn't going to go away. They'd fought their way through there once, they could always do it again. But there was only going to be one chance to get the top job.\"\n\n\"So the western hemisphere was saved by a dish of bad clams?\" Evanlyn said. The grizzled Ranger smiled grimly.\n\n\"It's surprising how often history is decided by something as trivial as bad shellfish,\" he told her.\n\n\"Where were you while this was all going on, Halt?\" Will asked his master.\n\nHalt smiled again at the memory. \"I suppose it's one of those moments you never forget,\" he said. \"I was hightailing it for the coast, with a small herd of . . .\" He hesitated, glancing sidelong at Horace. \". . . fairly purchased horses, and a Temujai fighting patrol was right behind me. They were gaining on me too. Suddenly, one morning, they reined in and watched me gallop away. Then they simply turned around and started trotting back east\u2014all the way to their homeland.\"\n\nThere was a brief silence as he finished the tale. Halt could have wagered that it would be Will who would come up with the next question, and he was not disappointed.\n\n\"So who became the Sha'shan?\" he asked. \"The brother, the nephew or the cousin?\"\n\n\"None of them,\" Halt replied. \"The election went to a dark horse candidate who had designs on the countries to the east of the Temujai homelands. The other three were executed for abandoning their mission in the west.\" He stirred the fire again, thinking back to that well-remembered day when the pursuing riders had suddenly given up the chase and left him to escape.\n\n\"And now they're back again,\" he said thoughtfully.\n**12**\n\nTHEY BROKE CAMP EARLY THE FOLLOWING MORNING AND started down toward the pass that would take them across the border once more. Horace had offered Evanlyn the black battlehorse that had belonged to Deparnieux. When she had protested that this was a far superior animal to the bay he rode, he smiled shyly.\n\n\"Maybe so. But I'm used to Kicker. He knows my ways.\" And that was the end of the matter. The prisoner rode one of the horses they had taken from the Temujai camp. A second was carrying the packs and supplies that, up until now, had been carried by Tug. Naturally, the little Ranger horse was now the proud bearer of his long-lost master.\n\nAs they came closer to the treeline at the bottom of the hill, Tug showed his happiness once more, tossing his head and whinnying. Halt turned in the saddle and smiled.\n\n\"I'm glad he's happy,\" he said. \"But I do hope he's not planning on keeping that up all the way home.\"\n\nWill grinned in reply and leaned forward to pat the little horse's shaggy neck.\n\n\"He'll settle down soon enough,\" he said. At the touch, Tug danced a few paces and tossed his head again. Surprisingly, Abelard copied the actions.\n\n\"Now he's got my horse doing it too,\" Halt said, more than a little surprised. He calmed Abelard with a quiet word, then turned to Will again. \"You seem to be popular among the horses of this world, anyway. I thought . . .\" His voice trailed away and he didn't finish the sentence. Will saw his body stiffen to attention and the gray-cloaked Ranger twisted in his saddle, peering into the trees, which were now close on either side.\n\n\"Damn!\" he muttered quietly. He turned to Horace and Evanlyn, riding behind them and leading the prisoner's horse, but before he could speak, there was a scuffle of movement in the trees and a party of armed warriors stepped out into the open behind them, blocking their retreat.\n\nHalt swung quickly to the front once more, as a second group emerged from the trees, fanning out to the sides and moving to cut them off in all directions.\n\n\"Skandians!\" exclaimed Will, as he recognized the horned helmets and round wooden shields carried by the silent warriors. Halt's shoulder slumped in a gesture of disgust with himself.\n\n\"Yes. The horses have been trying to warn us, only I didn't realize it.\"\n\nA burly figure, wearing an enormous horned helmet and with a double-bladed battle-ax laid negligently over his right shoulder, stepped forward. Behind them, Halt heard the sinister whisper of steel on leather as Horace drew his sword. Without turning, he said:\n\n\"Put it away, Horace. I think there are too many of them, even for you.\"\n\nAs Horace had moved, the huge ax had risen instantly to the ready position. The Skandian wielded it as if it were a toy. Now he spoke, and Will started at the familiar voice.\n\n\"I think we'll have you down from those horses, if you don't mind.\"\n\nUnable to stop himself, Will blurted out: \"Erak!\" and the man took a pace closer, peering at the second cloaked figure in front of him. The cowl had obscured Will's face so that the jarl hadn't recognized him. Now he could make out the boy's features and he frowned as he realized that there was something familiar about another of the riders. He hadn't recognized Evanlyn, swathed in a cloak against the cold. Now, however, he was sure that it must be she. He cursed quietly under his breath, then recovered.\n\n\"Down!\" he commanded. \"All of you.\"\n\nHe motioned the circle of men back as the four riders dismounted. The fifth, he noticed with some interest, was tied to his horse and couldn't comply. He gestured for two of his men to get the prisoner down from his saddle.\n\nHalt threw back the hood on his cloak and Erak studied the grim, bearded face. Now that he was dismounted, the man looked surprisingly small, particularly measured against Erak's own burly form. Will went to throw back his own cowl, but Erak stopped him with a hand gesture.\n\n\"Leave it for the moment,\" he said in a lowered voice. He didn't know how many of his men might recognize the former slave who had escaped from Hallasholm months ago, but for now, something told him that the fewer who made the connection, the better it would be. He looked warningly at Evanlyn.\n\n\"You too,\" he ordered, and she inclined her head in agreement. Erak turned his gaze back to Halt.\n\n\"I've seen you before,\" he said. Halt nodded.\n\n\"If you're Jarl Erak, we saw each other briefly on the beach by the fens,\" he said, and recognition dawned in the jarl's eyes. It wasn't the man's face that had struck a chord of memory, rather his bearing\u2014the way he held himself and the massive longbow that he carried still. Halt continued: \"There was quite a distance between us, as I recall.\"\n\nErak grunted. \"I seem to remember that we were well within bowshot,\" he said. Halt nodded, acknowledging the point. The Skandian's face darkened with anger as he looked once more at the bow and the quiver of arrows slung at Halt's belt.\n\n\"And now you've been up to the same foul business,\" he said. \"Although what these two have to do with it is beyond me.\" He added the last in a puzzled tone, jerking a thumb at Will and Evanlyn.\n\nNow it was Halt's turn to look puzzled. \"What foul business?\"\n\nErak gave a disgusted snort. \"I've seen you with that bow, remember? I know what you can do. And I've just seen more of your handiwork at Serpent Pass.\"\n\nUnderstanding dawned on Halt. He remembered the forlorn sight of the bodies at the small fort on the border. That must be the pass this Skandian was referring to. Since the garrison had been killed by archers and Erak knew Halt's skill with a bow, he had jumped to a rapid, if not too logical, conclusion.\n\n\"Not our work,\" he said, shaking his head. Erak stepped closer to him.\n\n\"No? I saw them there. All shot. And we followed your tracks from there.\"\n\n\"So you may have,\" Halt said calmly, \"but if you're any sort of tracker, you'd know that there were only two of us. We found the garrison at the pass dead. And we followed the tracks of a larger party\u2014the ones who killed them.\"\n\nErak hesitated. He wasn't a tracker. He was a sea captain. But one of the men who had come with him was an occasional hunter. While he didn't have the uncanny skills that the Rangers had developed in interpreting tracks, Erak now remembered that his man had said something about the possibility of there being two groups.\n\n\"Then,\" he said, bewildered by this turn of events, \"if you didn't do it, who did?\"\n\nHalt jerked a thumb at the bound prisoner. \"Him\u2014and his friends,\" he said. \"He was in a Temujai scouting party we ran into yesterday. There was a larger band who attacked the border garrison, then six of them came on into Skandia.\"\n\n\"Temujai, you say?\" Erak asked him. He knew of the warlike people from the east, of course, but it had been decades since they had come this way in any numbers.\n\n\"We killed a couple of them,\" Halt told him. \"Two got away and we captured this one.\"\n\nErak stepped to where the prisoner stood, hands tied in front of him, glaring fiercely at the big northerners who surrounded him. He studied the flat-featured, brown-skinned face and the furs the man wore.\n\n\"He's a Tem'uj, all right . . . but what were they doing here?\" he asked, almost to himself.\n\n\"That's the question I was asking,\" Halt replied.\n\nErak glanced at him with a flash of anger. He hated being confused. He preferred a simple, straightforward problem\u2014the kind he could solve with his broadax. \"For that matter,\" he snapped, \"what are you doing here?\"\n\nHalt faced him evenly, uncowed. \"I came for the boy,\" he said quietly. Erak looked at him, then at the smaller figure beside him, his face still largely concealed by the gray mottled hood. His anger faded as quickly as it had flared.\n\n\"Yes,\" he said, in a calmer tone. \"He said you would.\"\n\nLike most Skandians, Erak valued loyalty and courage. Another thought struck him\u2014something he'd wondered about for some time.\n\n\"At the beach,\" he said. \"How did you know to find us there?\"\n\n\"You left one of your men behind,\" Halt said. \"He told me.\"\n\nThe disbelief was plain on Erak's face.\n\n\"Nordal? He'd have spat in your eye before he told you anything.\"\n\n\"I think he thought he owed me,\" Halt said quietly. \"He was dying and he'd lost his sword, so I gave it back to him.\"\n\nErak went to speak, then hesitated. Skandians believed that if a man died without a weapon in his hand, his soul was lost forever. It seemed the Ranger knew about the belief.\n\n\"Then I'm in your debt,\" he said finally. Then, after another pause: \"I'm not sure how that affects this current situation, however.\" He rubbed his beard thoughtfully, looking at the fierce little Temujai warrior, for all the world like a tethered hawk. \"And I'd still like to know what this lad and his bunch are up to.\"\n\n\"That's what I had in mind,\" Halt told him. \"I was planning to get my companions here across the border into Teutlandt. Then I thought I might come back with our friend here and find the rest of the Temujai\u2014and see how many of them there are.\"\n\nErak snorted. \"You think he'll tell you?\" he asked. \"I don't know too much about the Temujai, but I know this much: you can torture them to death and they'll never tell you anything they don't want to.\"\n\n\"Yes. I've heard that too,\" Halt said. \"But there might be a way.\"\n\n\"Oh, might there?\" the jarl asked scornfully. \"And what might 'that way' be?\"\n\nHalt glanced at the horse warrior. He was following their discussion with some interest. Halt knew he spoke the trading language but he had no idea how much of the common tongue he might understand. As a member of a scouting party, it was probable that he had some command of the language. He took the jarl's arm and led him a few paces away, out of earshot.\n\n\"I rather thought I might let him escape,\" he said mildly.\n**13**\n\nTHE TWO MEN STOOD OVER THE TANGLE OF DISCARDED ROPES lying in the snow. Erak pursed his lips, then turned to Halt. \"Well, so far, you're right,\" he said. \"The little beggar escaped once Olak pretended to fall asleep on guard duty.\" He glanced sideways at the large Skandian who had been assigned to the last watch. \"You did pretend to fall asleep, didn't you?\" he added, with a touch of sarcasm.\n\nThe warrior grinned easily at him. \"I was wonderful, Jarl Erak,\" he said. \"You've never seen such a lifelike impersonation of a sleeping man. I should have been a traveling player.\"\n\nErak grunted skeptically. \"So what now?\" he asked Halt.\n\n\"Now, I follow him while he leads me to the main body of Temujai,\" the Ranger said. \"As we discussed last night.\"\n\n\"I've been thinking about that,\" Erak replied. \"And I've decided we're going to make a change. I'm going with you.\"\n\nHalt had been walking toward the spot where the horses were tethered. He stopped and turned to face the Skandian leader, a determined look on his face. \"We discussed this last night. We agreed that I would be quicker and less noticeable if I went alone.\"\n\n\"No. We didn't agree that. You agreed that,\" Erak corrected him. \"And even if you're right, you're just going to have to settle for being slower and noisier, and make allowances for the fact.\"\n\nHalt drew in breath to begin a protest, but Erak forestalled him.\n\n\"Be reasonable,\" he said. \"We've agreed that circumstances seem to make us temporary allies\u2014\"\n\n\"Which is why you'll keep my three companions here as hostages,\" Halt put in sarcastically, and Erak simply shrugged.\n\n\"Of course. They're my surety that you'll come back. But put yourself in my shoes. If there is a Temujai army out there somewhere, I don't want to take a secondhand report to my Oberjarl. I want to see it for myself. So I'm coming with you. I may need you to track the prisoner, but I can do my own looking.\"\n\nHe paused, waiting to see Halt's reaction. The Ranger said nothing, so Erak continued: \"After all, the hostages might ensure that you come back. But they're no guarantee that you'll give me an accurate report\u2014or even an honest one.\"\n\nHalt seemed to weigh the statement for a few seconds. Then he saw a possible advantage.\n\n\"All right,\" he agreed. \"But if you're coming with me, there's no need to keep my companions as hostages to guarantee my return. Let them go back across the border while you and I go find the Temujai.\"\n\nErak smiled at him and shook his head slowly. \"I don't think so,\" he replied. \"I'd like to think that I can trust you, but there's really no reason why I should, is there? If you know my men are holding your friends, it might make you less likely to stick one of those knives in me the minute we're out of sight over the hill there.\"\n\nHalt spread his hands in a an innocent gesture. \"Do you really think an undersized little runt like me could get the better of a big, hulking sea wolf like you?\"\n\nErak smiled grimly at him. \"Not for a moment,\" he said. \"But this way I'll be able to sleep nights and turn my back on you without worrying.\"\n\n\"Fair enough,\" Halt agreed. \"Now, could we get going while these tracks are still fresh, or would you prefer to argue until the snow melts?\"\n\nErak shrugged. \"You're the one who's doing all the arguing,\" he told him. \"Let's go.\"\n\nHalt glanced over his shoulder as Abelard set his hooves more securely against the steep slope. Behind him, Erak was swaying in-securely on the back of the Temujai horse. The captive had made his escape on foot, and Halt had decided that the small, shaggy and sure-footed steppes pony would be a better mount for Erak than either of Horace's battlehorses. The Skandian warriors, as was their custom, had been traveling on foot.\n\n\"I thought you said you could ride,\" he challenged as the jarl grabbed nervously at his mount's shaggy mane, holding himself in the saddle more by brute strength than any inherent sense of balance.\n\n\"I did,\" Erak replied through gritted teeth. \"I just didn't say I could ride well.\"\n\nThey had been following the escaped Temujai warrior's trail all day. After making their way through the Serpent Pass, their trail had swung back in an arc from the Teutlandt border and they were some thirty kilometers into Skandian territory once more. Halt shook his head, then went back to peering at the ground in front of them, looking for the faint traces that the fleeing Tem'uj had left behind him.\n\n\"He's very good,\" he said quietly.\n\n\"Who's that?\" Erak asked, the last word being torn from him as his horse lurched and slid a few steps. Halt indicated the trail he was following. The Skandian looked but couldn't see a thing.\n\n\"The Tem'uj,\" Halt continued. \"He's covering his tracks as he goes. I don't think your man would have been able to follow him.\"\n\nWhich was the crux of the matter. When Halt and Erak had agreed to join forces the previous night, it had been the result of their mutual need. Halt's natural inclination had been to see what the Temujai were up to. Erak had the same need. But he also had need of Halt's tracking skills. He was only too aware of his own men's limitations.\n\n\"Well,\" he said jerkily, \"that's why you're here, isn't it?\"\n\n\"Yes.\" Halt smiled grimly. \"The question is, why are you?\"\n\nErak wisely said nothing. He concentrated his efforts into staying astride the shaggy horse as it struggled up the steep slope, under the unaccustomed weight of the bulky Skandian sea captain.\n\nThey came to the crest with a sudden rush, their horses scrambling the last few meters through the wet snow. They found themselves looking down on a deep, wide valley, and beyond that, another range of hills.\n\nBelow them on the vast plain, a mass of campfires sent columns of smoke spiraling into the late-afternoon air, spreading as far as the eye could see\u2014thousands of them, surrounded by more thousands of dome-shaped felt tents. The smell of the smoke reached them now. Not heady and scented, like pine smoke, but acrid and sour smelling. Erak wrinkled his nose in disgust.\n\n\"What are they burning?\" he asked.\n\n\"Dried horse dung,\" Halt replied briefly. \"They carry their fuel source with them. Look.\"\n\nHe pointed to where the Temujai horse herd could be seen, a giant, amorphous mass that seemed to flow across the valley floor as the horses sought fresh grazing.\n\n\"Gorlog's teeth!\" Erak exclaimed, stunned at the numbers. \"How many are there?\"\n\n\"Ten thousand, maybe twelve,\" Halt replied briefly. The Skandian let out a low whistle.\n\n\"Are you sure? How can you tell?\" It wasn't a sensible question, but Erak was overwhelmed by the size of the horse herd and he asked the question more for something to say than for any other reason. Halt looked at him dryly.\n\n\"It's an old cavalry trick,\" he said. \"You count the legs and divide by four.\"\n\nErak returned the look. \"I was just making conversation, Ranger,\" he said. Halt seemed singularly unimpressed by the statement.\n\n\"Then don't,\" he replied shortly. There was silence as they studied the enemy camp.\n\n\"Are you saying there are ten to twelve thousand warriors down there?\" Erak asked finally. The number was a daunting one. At best, Skandia could put a force of fifteen hundred warriors in the field to face them. Perhaps two thousand, at the outside. That meant odds of six or seven to one. But Halt was shaking his head.\n\n\"More like five to six thousand,\" he estimated. \"Each warrior will have at least two horses. There are probably another four to five thousand personnel in the baggage train and supply columns, but they wouldn't be combatants.\"\n\nThat was a little better, thought Erak. The odds had reduced to around three or four to one. A little better, he thought. Not a lot.\n\nNot a lot by a long way.\n**14**\n\n'WAIT HERE,\" HALT SAID BRIEFLY. \"I'M GOING DOWN FOR A closer look.\"\n\n\"To hell with waiting here,\" Erak told him. \"I'm coming with you.\"\n\nHalt looked at the big Skandian, knowing it would be useless to argue. Still, he made the attempt. \"I suppose it will make no difference if I point out I'm going to have to be as inconspicuous as possible?\"\n\nErak shook his head. \"Not in the slightest. I'm not taking back a secondhand report to my Oberjarl. I want to get a closer look at these people, get some idea of what we're up against.\"\n\n\"I can tell you what you're up against,\" Halt said grimly.\n\n\"I'll see for myself,\" the jarl said stubbornly, and Halt shrugged, finally giving in.\n\n\"All right. But move carefully, and try not to make too much noise. The Temujai aren't idiots, you know. They'll have pickets out in the trees around the camp, as well as sentries on the perimeter.\"\n\n\"Well, you just tell me where they are and I'll avoid them,\" Erak replied, with a little heat. \"I can be inconspicuous when I need to.\"\n\n\"Just like you can ride, I suppose,\" Halt muttered to himself. The Skandian ignored the comment, continuing to glare stubbornly at him. Halt shrugged. \"Well, let's get on with it.\"\n\nThey tethered their horses on the reverse side of the crest, then began to work their way down through the trees to the valley below them. They had gone a few hundred meters when Halt turned to the Skandian.\n\n\"Are there bears in these mountains?\" he asked.\n\nHis companion nodded. \"Of course. But it's a bit early in the year for them to be moving around. Why?\"\n\nHalt let go a long breath. \"Just a vague hope, really. There's a chance that when the Temujai hear you crashing around in the trees, they might think you're a bear.\"\n\nErak smiled, with his mouth only. His eyes were as cold as the snow.\n\n\"You're a very amusing fellow,\" he told Halt. \"I'd like to brain you with my ax one of these days.\"\n\n\"If you could manage to do it quietly, I'd almost welcome it,\" Halt said. Then he turned away and continued to lead the way down the hill, ghosting between the trees, sliding from one patch of shadow to the next, barely disturbing a branch or a twig as he passed.\n\nErak tried, unsuccessfully, to match the Ranger's silent movement. With each slither of his feet in the snow, each whip of a branch as he passed, Halt's teeth went more and more on edge. He had just determined that he would have to leave the Skandian behind once they got within striking distance of the Temujai camp when he glimpsed something off to their left in the trees. Quickly, he held up his hand for Erak to stop. The big Skandian, not understanding the imperative nature of the gesture, kept moving till he was alongside Halt.\n\n\"What is it?\" he asked. He kept his voice low, but to Halt it seemed like a bellow that echoed among the trees.\n\nHe placed his own mouth next to the Skandian's ear and breathed, in a barely audible voice, \"Listening post. In the trees.\"\n\nIt was a familiar Temujai technique: whenever a force camped for the night, they threw out a screen of concealed, two-man listening posts to give early warning of any attempt at a surprise attack. He and Erak had just passed such a post, so that it now lay to their left and slightly behind them. For a moment, Halt toyed with the idea of continuing down the hill, then he discarded it. The screen was usually deployed in depth. Just because they had passed one post didn't mean there weren't others ahead of them. He decided it might be best to cut their losses and extract themselves as quietly as possible, trusting the gathering darkness to conceal them. It would mean abandoning the idea of getting a closer look at the Temujai force, but it couldn't be helped. Besides, with Erak along, it was unlikely they would get much closer without being seen\u2014or, more likely, heard. He leaned close to the other man and spoke softly once more.\n\n\"Follow me. Go slowly. And watch where you put your feet.\"\n\nThe snow under the trees was strewn with dead branches and pinecones. Several times as they'd made their way downhill, he had winced as Erak had trod, heavy-footed, on fallen branches, breaking them with seemingly earsplitting cracks.\n\nSilently, Halt flitted between the trees, moving like a wraith, sliding into cover after he'd gone some fifty paces. He looked back and waved the Skandian on, watched for a moment with mounting apprehension as the big man moved, swaying awkwardly as he placed his feet with exaggerated care. Finally, unable to watch him any longer, Halt looked anxiously to the left, to see if there was any sign that the men in the listening post had seen or heard them.\n\nAnd heard a ringingly loud crack, followed by a muffled curse, from the hill below him. Erak was poised in midstride, a rotten branch snapped in half on the snow in front of him.\n\n\"Freeze,\" muttered Halt to himself, in the desperate hope that the big man would have the sense to stay motionless. Instead, Erak made the vital blunder that untrained stalkers nearly always made. He dashed for cover, hoping to substitute speed for stealth, and the sudden movement gave him away to the Temujai in the listening post.\n\nThere was a shout from above them and a flight of arrows slammed into the tree behind which the Skandian had taken cover. Halt peered around his tree. He could see two shapes in the gloom. One was moving away, sounding a horn as he went. The other was poised, an arrow on the string of his bow, eyes riveted on Erak's hiding place.\n\nWaiting for the Skandian to move. Waiting to let the deadly shaft fly at him.\n\nSomehow, Halt had to give Erak a chance to get clear. He called softly, \"I'll step out and distract him. As soon as I do, you make for the next tree.\"\n\nThe Skandian nodded. He crouched a little, preparing to make a run for it. Halt called again.\n\n\"Just to the next tree. No farther,\" he said. \"That's all you'll have time for before he's back on you. Believe me.\"\n\nAgain, the Skandian nodded. He'd seen the speed and accuracy with which the Temujai sentry got the first shot away. He wondered how he would get any farther than the next tree. Halt's ploy of distracting the sentry would only work once. He hoped that the Ranger had something else in mind. Fading away now, he could hear the braying notes of the horn sounding the alarm as the other sentry raced downhill, calling for reinforcements. Whatever Halt did, he thought, he'd better do it soon.\n\nErak saw the dim form of the Ranger as he stepped into the clear from behind the tree. Erak waited a heartbeat, then ran, his legs pumping in the snow, finally diving full length and sliding behind the thick pine trunk as an arrow hissed by, just over his head. His heart was racing, even though he had covered no more than ten meters in his wild, scrambling rush up the hill. He glanced across at Halt and saw the Ranger, back in cover and some five meters farther away. He had his own longbow ready now, an arrow nocked to the string. His face was knotted in a frown of concentration. He felt the Skandian's eyes on him and called across the intervening space.\n\n\"Take a look. Carefully\u2014don't give him enough of a target to shoot at. See if he's in the same position.\"\n\nErak nodded and edged one eye around the bole of the tree. The Temujai warrior was still where he had been standing, his bow ready and half drawn. As matters stood, he held the upper hand, standing ready to shoot if either of them moved. Halt, on the other hand, would have to step into the clear, sight the man, aim and then shoot. By the time he had accomplished the first two actions, he would be dead.\n\n\"He hasn't moved,\" Erak called to the Ranger.\n\n\"Tell me if he does,\" Halt called softly in return. Lying belly-down in the snow, with just a fraction of his face protruding around the tree, Erak nodded.\n\nBehind his tree, Halt leaned back against the rough bark and closed his eyes, breathing deeply. This was going to have to be an instinctive shot. He pictured again the dark figure of the Tem'uj, silhouetted against the lighter background of the snow. He remembered the position, setting it in his brain, letting his mind take over the control of his hands, willing the aiming and release to become an instinctive sequence. He forced his breathing to settle into a calm, slow, unhurried rhythm. The secret of speed was not to hurry, he told himself. In his mind's eye, he watched the flight of the arrow as he would fire it. He pictured it over and over again until it seemed to be a part of him\u2014a natural extension of his own being.\n\nThen, in an almost trancelike state, he moved.\n\nSmoothly. Rhythmically. Stepping out into the clear, turning in a fluid motion so that his left shoulder was toward the target, the right hand pulling back on the string, left hand pushing the bow away until it was at full draw. Aiming and shooting at a memory. Not even seeing the dark figure in the trees until the arrow was already loosed, already splitting the air on its way to the target.\n\nAnd, when he finally did see the bowman in his conscious vision, knowing that the shot was good.\n\nThe heavy shaft went home. The Tem'uj fell backward in the snow, his own shot half a second too late, sailing high and harmless into the tops of the pines.\n\nErak scrambled to his feet, regarding the small, gray-cloaked figure with something close to awe.\n\nHe realized that there was already a second arrow nocked to the longbow's string. He hadn't even seen the Ranger do that.\n\n\"By the gods,\" he muttered, dropping a heavy hand on the smaller man's shoulder. \"I'm glad you're on my side.\"\n\nHalt shook his head briefly, refocusing his attention. He glared angrily at the big Skandian.\n\n\"I thought I told you to watch where you put your feet,\" he said accusingly. Erak shrugged.\n\n\"I did,\" he replied ruefully. \"But while I was busy watching the ground, I hit that branch with my head. Broke it clean in two.\"\n\nHalt raised his eyebrows. \"I assume you're not talking about your head,\" he muttered. Erak frowned at the suggestion.\n\n\"Of course not,\" he replied.\n\n\"More's the pity,\" Halt told him, then gestured up the hill. \"Now let's get out of here.\"\n**15**\n\nWHEN THEY REACHED THE CREST OF THE HILL, HALT PAUSED TO look back. Erak stopped beside him, but he grabbed the bigger man's arm and shoved him roughly toward the two tethered horses.\n\n\"Keep going!\" he yelled.\n\nIn the valley below them, he could hear alarm horns sounding and, faintly, the sound of shouting. Closer to hand, on the slope of the hill below, he could see movement among the trees as those Temujai who had been concealed in listening posts around the hillside now broke cover and headed uphill in pursuit of the two intruders.\n\n\"Damned hornets' nest,\" he muttered to himself. He estimated that there must be at least half a dozen riders on the hill below him, heading upward. A larger party was obviously forming in the camp itself, with a view to heading around the base of the hill and catching him and Erak between two pursuing forces.\n\nAlone, and mounted on Abelard, he was confident that he could outrun them easily. But burdened by the Skandian, he wasn't so sure. He'd seen the man's skill as a rider\u2014which was virtually non-existent. Erak seemed to stay in the saddle by virtue of an enormous amount of willpower and precious little else. Halt knew that he would have to come up with some kind of delaying tactic, to slow the pursuit down and give him and Erak time to make it back to the larger Skandian force.\n\nStrangely, although they had been nominal enemies up until now, the thought of abandoning the Skandian to the pursuing Temujai riders never occurred to him.\n\nHe looked back to where they had tethered Erak's horse\u2014Abelard, of course, needed no tethering. He saw with some slight satisfaction that the wolfship skipper had managed to clamber into the saddle and was sitting clumsily astride his small, shaggy mount. Halt waved a hand now in an unmistakable gesture to him.\n\n\"Get going!\" he yelled. \"Go! Go! Go!\"\n\nErak needed no second bidding. He wheeled the horse to face downhill, swaying dangerously out to one side as he did so and managing to retain his seat only by grabbing at the mane and gripping with his powerful legs around the horse's barrel of a body. Then, half in and half out of the saddle, he drove the former Temujai mount down the slope, skidding and sliding in the soft wet snow, swerving dangerously among the trees. At one stage, Erak neglected to duck as the horse drove under the snow-laden lower branches of a huge pine. There was an explosion of snow and both horse and rider emerged coated in thick white powder.\n\nHalt swung smoothly into Abelard's saddle and the little horse spun neatly, moving at a dead gallop almost before he could draw breath. Halt sat easily as Abelard slid, checked, skidded and regained his footing, gaining on the other horse and rider with every stride.\n\nHe'll be lucky to survive another fifty meters, Halt thought as Erak's mount, half out of control, swerved and skidded and slipped among the trees. It seemed only a matter of time before both horse and rider collided full tilt with one of the large pine trunks.\n\nHe urged Abelard to a greater effort and the horse responded instantly. They drew level with the plunging horse and rider and Halt, leaning down to one side, was able to grab the trailing reins. Erak had long since abandoned them and was clinging for dear life to the saddle bow.\n\nNow, at least, Halt could exercise some small control over the headlong plunging of the other horse. Abelard, sure-footed and agile, led them through the trees and Halt left the choice to him entirely. The lead rein jerked and tugged at his arm but he clung to it desperately, forcing the other horse to follow in Abelard's tracks. Abelard, as he had been trained to do, chose the most direct and, at the same time, the clearest path down the mountain. They were two-thirds of the way down now and Halt was beginning to feel more positive about their chances of escape when he heard shouting and the sound of those damned horns from the hill crest behind them. He glanced quickly back but the thickly growing trees obscured his view. Nonetheless, he knew that the sudden burst of sound heralded the appearance of the pursuing Temujai at the top of the mountain.\n\nAnd he knew that it was only a matter of time before they would overhaul him, just as he had overhauled the bulky Skandian on the small horse.\n\nA thin branch whipped across his face, bringing tears to his eyes and punishing him for taking his attention from the direction he was heading. He shook his head to get rid of the accompanying shower of snow that the branch had brought with it, then, seeing the way ahead was clear, he turned briefly again to call encouragement to Erak.\n\n\"Keep hanging on!\" he yelled and the Skandian promptly did exactly the opposite, releasing his grip with one hand so that he could wave an acknowledgment.\n\n\"Don't worry about me!\" he yelled. \"I'm doing fine!\"\n\nHalt shook his head. Frankly, he'd seen sacks of potatoes that could sit a horse better than Erak. He wondered how the Skandian ever managed to keep his feet on the heaving deck of a wolfship. The trees were thinning around them now, he noticed. Then he heard the braying note of one of the Temujai horns out to their left and realized that the first of the parties coming around the base of the mountain from the encampment must be close to heading them off. It would be a near-run thing, he thought grimly. His slight increase in knee pressure sent Abelard bounding even faster. From behind he heard a startled yell from Erak as he nearly lost his seat again. Another quick glance told him that the Skandian was still mounted, and they broke out onto the level ground between the hills.\n\nHe had been right. It _was_ a close-run race. The leading riders of the Temujai party swept into sight on the flat ground between the hills. They were barely two hundred meters away. Halt dragged Erak's horse around brutally, touched Abelard with his heels and set the two horses galloping back along the track they had followed earlier in the day. On clearer ground now, he could look behind him more easily. He made out at least a dozen riders chasing them. For a moment, the grizzled Ranger had a distinct sense of d\u00e9j\u00e0 vu, his mind racing back across the years to the time when he had been driving a herd of stolen horses with another party of Temujai howling for his blood close behind him. He grinned mirthlessly. Of course the horses had been stolen. He simply couldn't bear to disappoint Horace any further when he had told him of his previous encounter with the eastern horsemen. He'd felt at the time that the boy had been disillusioned enough for one day.\n\nNow he eased Abelard fractionally, allowing the other horse to come level with them, and tossed the reins to the Skandian jarl, who bumped and lurched in the saddle beside him. Surprisingly, Erak caught them. There was nothing wrong with his reflexes, at any rate, Halt thought.\n\n\"Keep going!\" he yelled at the Skandian.\n\n\"What . . . you . . . got . . . in . . . mind?\" Erak replied jerkily, the words lurching out of him as he was tossed and bumped in the saddle.\n\n\"Going to slow them down,\" Halt replied briefly. \"Don't stop to watch. Just keep going as hard as you can!\"\n\nErak gritted his teeth as he came down heavily on the saddle. \"This is as hard . . . as . . . I can!\" he replied. But Halt was already shaking his head. The Ranger had unslung his longbow from across his shoulders and was brandishing it in his right hand. Erak saw what was coming, a moment too late to do anything about it.\n\n\"No!\" he began. \"Don't you\u2014!\"\n\nBut then the bow whipped down across his horse's rump with a resounding crack and the beast leapt forward, stung.\n\nThe profanity that Erak was preparing for Halt was lost in his drawn-out howl as he grabbed at the saddle bow once more to keep his seat. For a second or two he was furious. Then he realized that he was still in the saddle, that he could keep his seat even at this accelerated pace. So, when the horse began to slow down to a more comfortable speed, he slapped his big hand across its backside several times, driving it on.\n\nHalt watched in satisfaction as his companion went on ahead, urging the horse on to greater efforts. In a few seconds, Erak swept around a curve in the trail that was formed between two of the hills and was out of sight.\n\nThen, in response to a well-learned knee signal, Abelard reared and pirouetted on his hind legs, spinning in a half circle so that he came to a stop at right angles to the direction they had been following.\n\nIn an instant, the horse had gone from a dead run to a full stop. Now he stood rock steady as his master stood in the stirrups, an arrow nocked to the string of his massive longbow.\n\nHe knew that the longbow outranged the smaller, flat-shooting recurve bows of the Temujai. He allowed them to close in a little farther, gauging the pace at which they were eating up the distance between him and them, estimating when he would need to release so as to have the arrow arrive at a given point just as the lead rider did. He did this without thinking, allowing the ingrained instincts and habits of years of endless practice to take over for him. Almost without realizing it, he released and the arrow sped away, sailing in a shallow arc toward the pursuers.\n\nThey were one hundred and fifty meters from him when the arrow struck the lead rider from his saddle. He slid sideways to the ground, trying to maintain his hold on the reins and bringing his horse down with him as he did. The rider directly behind him, taken totally by surprise, had no chance to avoid his leader's fallen horse. He and his horse came crashing down as well, adding to the tangle of legs and arms and bodies that rolled in a welter of thrown snow.\n\nThe riders behind them were thrown into utter confusion, with riders sawing savagely at the reins to drag their horses away from the tangle ahead of them. Horses plunged and reared, getting in each other's way, sliding stiff-legged to a halt in the snow, heading in all directions to avoid the crash. As they milled in confusion, Halt was already galloping away, rounding the bend and heading after Erak.\n\nSlowly, the Temujai regained order. The leader's horse had regained its feet and limped in a circle, blowing and snorting wildly. Its rider lay in the snow in the center of a widening circle of red. Now the others could see the cause of all the trouble: the heavy, black-shafted arrow that had arced down to take him. Accustomed to using the bow themselves with deadly skill, they were unfamiliar with the feeling of being on the receiving end\u2014and at such an extreme range. Perhaps, they realized, a headlong pursuit of the two fleeing riders wasn't such a good idea. The Temujai weren't cowards. But they weren't fools either. They had just seen clear evidence of their quarry's uncanny accuracy. They sorted themselves out and set off in pursuit again\u2014but not quite so eagerly this time, and not quite so quickly.\n\nBehind them, the second rider, who had collided with the fallen leader, was left in a vain attempt to catch the leader's horse. His own had broken its neck in the fall. He didn't seem in too much of a hurry to resume the chase.\n**16**\n\nHALT STOPPED TWICE MORE TO SLOW DOWN THE RIDERS BEHIND them. Both times, he dismounted, allowing Abelard to trot around the next bend in the trail so that he was out of sight. Then Halt waited, standing in the deep shadows thrown by the pine trees, almost invisible in the gray and green mottled cloak.\n\nWhen the Temujai riders appeared around a bend in the trail behind him, Halt launched two arrows at maximum range, on a high parabolic flight. Each time, the horsemen weren't even aware that they were being fired on until two of their number threw up their hands and tumbled from their saddles into the snow.\n\nHalt chose his ambush positions carefully. He selected places where there was a clear sight of the trail behind him, but he didn't choose every such section. After the third attack, every time the Temujai approached a bend in the trail, they slowed their pursuit, fearing they would be riding into another volley of black-shafted arrows arcing down out of the sky at them.\n\nOn the last two occasions they didn't even see Halt before he moved to remount Abelard. They soon began to rationalize, arguing that there was no real need to capture the two men who had been spying on their camp. There was, after all, little that two men could do to harm them and if they alerted the Skandian forces, well, the Temujai had come here prepared to fight anyway.\n\nThis was the result Halt had been hoping for. After stopping twice, he urged Abelard into a steady gallop, soon overtaking Erak as he lurched and swayed on the saddle of his now cantering horse. Erak heard the muffled pounding of hooves behind him and swung awkwardly in the saddle, half expecting to see a group of Temujai coming up behind. He relaxed as he recognized the gray-cloaked figure of the Ranger. His horse, without anyone to continue urging it on, slackened its pace as Abelard pounded up alongside. Halt checked him for a few strides, matching the Temujai mount's pace.\n\n\"Where have . . . you been?\" Erak asked, in that same jerky manner.\n\nHalt gestured to the trail behind him. \"Buying us some time,\" he replied. \"Can't you keep that nag of yours running faster than that?\"\n\nErak looked insulted. He'd thought he was doing rather well.\n\n\"I'll have you know I'm an excellent rider,\" he said stiffly. Halt glanced over his shoulder. There was no sign of any pursuit, but there was no knowing how long the Temujai would take to realize that he wasn't waiting for them at every corner. If they continued at this gentle, ambling pace, the riders behind them would make up the lost distance in no time.\n\n\"You may believe you're an excellent rider,\" he called, \"but there are a score or so of Temujai back there who actually are. Now get moving!\"\n\nErak saw the longbow rise, and begin to fall on his horse's rump once more. This time, he didn't waste breath or time yelling at Halt not to do it. He grabbed a handful of mane and hung on for dear life as the horse bolted away underneath him. Bouncing and jouncing in exquisite pain, he consoled himself with the thought that, when this was over, he would separate the Ranger from his head.\n\nThey swept on, Halt urging the Temujai horse on to greater efforts whenever he began to flag. The landmarks around them began to take on a familiar appearance, then they had galloped into the head of Serpent Pass, coming up to the deserted border post. There, camped outside the log walls of the small fort, Erak's twenty Skandian warriors and Evanlyn and the two apprentices were waiting for them. The Skandians came to their feet quickly, reaching for their weapons, as the two horses entered the pass at a dead run.\n\nHalt brought Abelard skidding to a stop beside his three companions. Erak tried to emulate the action, but his horse pounded on for another twenty meters or so and he had to swing it awkwardly around, swaying and slipping in the saddle as it turned, and inevitably falling in a heap in the snow as the horse finally decided to stop.\n\nTwo or three of the Skandians, unwisely, let go short bellows of laughter as Erak picked himself up. The jarl's eyes swept over them, cold as glacier ice, marking them down for later reference. The laughter died as quickly as it had sprung up.\n\nHalt threw his leg over the pommel and slid to the ground. He stroked Abelard's neck in gratitude. The little horse was barely breathing hard. He was bred to run all day if necessary. The Ranger saw the inquisitive looks of those around him.\n\n\"Did you find the main party?\" Will finally asked.\n\nHalt nodded grimly. \"We found them all right.\"\n\n\"Thousands of them,\" Erak added, and the Skandians reacted with surprise at the news. Erak silenced them with a gesture.\n\n\"There are maybe five or six thousand of them out there, probably heading this way right now.\" Once again, there were murmurs of surprise and consternation as he mentioned the numbers. One of the Skandians stepped forward.\n\n\"What do they want, Erak?\" he asked. \"What are they doing here?\"\n\nBut it was the Ranger who answered the question: \"They want what they always want,\" he said grimly. \"They want your land. And they're here to take it from you.\"\n\nHis audience looked from one to the other. Then Erak decided it was time he took command of the situation.\n\n\"Well, they'll find we're a tough nut to crack,\" he declared. He swept his battle-ax in a small arc to indicate the fort behind them. \"We'll hold the fort here and delay them while one of us takes word back to Hallasholm,\" he said. \"There may be five thousand of them, but they can only come at us in small numbers through the pass. We should be able to hold them for four or five days at least.\"\n\nThere was a growl of assent from the Skandians, and several of them swept their axes through the air in experimental patterns. The jarl was growing in confidence now that he had a definite plan of action. And it was the sort of plan that appealed to the Skandian mind: simple, uncomplicated, easy to put into effect and with a degree of mayhem involved. He looked at Halt, who was watching him in silence, leaning on the man-high longbow.\n\n\"We'll trouble you for the use of the horse again,\" he said. \"I'll send one of my men back to Hallasholm on it to raise the alarm. The rest of us will stay here and fight.\" Again, there was a savage growl from the Skandians in response. The jarl continued: \"As for you, you can stay and fight with us or go on your way. It's of no consequence to me.\"\n\nHalt shook his head, a look of bitter disappointment on his face.\n\n\"It's too late for us to go now,\" he said simply. He turned to his three young companions and shrugged apologetically. \"The Temujai main force lies right across our path back to Teutlandt. We've no choice but to stay here.\"\n\nWill exchanged glances with Evanlyn and Horace. He felt a sinking in the pit of his stomach. They had been so close to escaping, so close to going home.\n\n\"It's my fault,\" Halt continued, addressing his words to the two former captives. \"I should have got you out straightaway instead of going to see what the Temujai were up to. I thought, at worst, it would be a reconnaissance in force. I had no idea it was an invasion.\"\n\n\"It's all right, Halt,\" Will told him. He hated to see his mentor apologizing or blaming himself. In Will's eyes, Halt could do no wrong. Horace hurried to agree with him.\n\n\"We'll stay here and hold them back with the Skandians,\" he said, and one of the sea wolf warriors close by him slapped him heartily on the back.\n\n\"That's the spirit, boy!\" he said, and several others chorused their approval of Horace's intentions. But Halt shook his head.\n\n\"Nobody should stay here. There's no point.\"\n\nThat brought howls of anger and derision from the Skandians. Erak silenced them and stepped forward, staring down at the slight figure in the gray cloak.\n\n\"Yes, there's a point,\" he said, in an ominously quiet tone. \"We'll hold them here until Ragnak can muster the main force to relieve us. There are twenty of us. That should be more than enough to hold the little beggars off for a while. It won't be like when they slaughtered the garrison here. There were only a dozen men here then. We'll hold them off, or we'll die in the attempt. It's of no consequence to us as long as we delay them for three or four days.\"\n\n\"You won't last three or four hours,\" Halt said flatly, and an ugly silence fell over the small group. The Skandians were too shocked by the enormity of his insult to reply. Erak was the first to recover.\n\n\"If you believe that,\" he said grimly, \"then you have never seen Skandians fight, my friend.\" The last two words carried an enormous weight of sarcasm and dismissal. Now the other Skandians found their voices and an angry chorus grew up. The Ranger waited for the shouting to die down. He was uncowed by the Skandians' anger at his words. Finally they fell silent.\n\n\"You know that I have,\" he said, not taking his eyes from Erak's.\n\nThe Skandian leader frowned. He knew Halt's reputation, as a fighting man and a tactician. The man was a Ranger, after all, and Erak knew enough about the mysterious Ranger Corps to know that they weren't prone to issuing pointless insults or making ill-considered remarks.\n\n\"The question is,\" Halt continued, \"have you seen the Temujai fight?\"\n\nHe allowed the question to hang in the cold air between them. There was a moment of silence from the Skandians. None of them had, of course. Seeing that he had their attention, Halt continued.\n\n\"Because I have. And I'll tell you what I'd do if I were the Temujai general.\"\n\nHe swept his arm up to encompass the steep sides of the pass where they towered above the little fort. Pines grew there, clinging to the almost vertical sides of the pass, managing to find some foot-hold in the rocks and the snow.\n\n\"I'd send a party of men up onto the walls of the pass there above us. Say, two hundred or so. And from there, I'd have them direct a killing fire on anyone foolhardy enough to show his face in the open inside the fort.\"\n\nThe eyes of the group followed the direction of his pointing arm. One of the Skandians snorted scornfully.\n\n\"They'd never get up there. Those walls are impassable!\"\n\nHalt turned to face him, looking him straight in the eye, willing the man to understand and believe what he was saying by the sheer force of his conviction.\n\n\"Not impassable. Very difficult. But they will do it. Believe me, I've seen these men and what they can achieve. It may cost them fifty or so lives in the attempt, but they'll count the cost cheap.\"\n\nErak studied the cliffs above the fort, squinting to see more clearly in the rapidly fading light of the late afternoon. Maybe, he thought, the Ranger was right. He figured he might be able to scramble around up there, with ropes and tackle and a small group of hand-picked sailors\u2014the ones who tended the big square sails on the wolfships, who could slip up and down the mast as easy as walking. But the Temujai were cavalry, he thought. He voiced the objection.\n\n\"They'll never get their horses up there.\"\n\n\"They won't _need_ their horses up there,\" Halt countered. \"They'll simply sit up there and direct a plunging fire on you. The fort may command the pass, but the heights there command the fort.\"\n\nErak was silent for a long moment. He looked again up at the walls of the pass. If the trees could find footholds there, he reasoned, so could men\u2014determined men. And he was ready to believe that these Temujai were determined.\n\n\"Face it,\" Halt continued, \"this fort was never meant as a real defensive position. It's a checkpoint for people crossing the border, that's all. It's simply not designed or placed to hold an invading army at bay.\"\n\nErak studied the Ranger. The more he thought about it, the more sense Halt was making. He could picture the dangers of being caught inside the fort with a hundred or so archers perched on the cliffs above him\u2014and no way to reply to their attack.\n\n\"I think you may be right,\" he said slowly. He was honest enough to admit that Halt's experience of these eastern riders was far greater than his own. Reluctantly, he made the final decision\u2014to pass control over to Halt.\n\n\"What do you suggest we do?\" he asked. His men looked at him in surprise and he glared them to silence. Halt nodded once, acknowledging the difficulty of the decision the jarl had just reached.\n\n\"You were right about one thing,\" he said. \"Ragnak has to be warned. There's no point in our wasting any more time here. It'll take the Temujai at least half a day to get the whole army on the move. Longer for them to come through this narrow pass. Let's use the time we have. We'll ride\u2014and run\u2014like hell back to Hallasholm.\"\n**17**\n\nFULL NIGHT FELL SHORTLY AFTER THEY HAD SET OUT ON THeIR way back to Hallasholm. But they continued to move, their way lit by a brilliant three-quarter moon that sailed above them in the clear sky.\n\nHalt, Evanlyn and the two apprentices rode, while the Skandians maintained a steady jog, led by the jarl. Halt had suggested that Erak ride the captured Temujai horse again, but he had declined the offer, with a certain amount of alacrity. It seemed now that he had his feet firmly back on the ground, he was determined to keep them that way. His thighs and calves ached from the hours he had spent in the saddle that day, and his backside seemed to be one massive bruise. He was glad of the chance to walk the cramps out of his muscles.\n\nEven allowing for the fact that the Skandians were traveling afoot, Halt was content with the pace they were maintaining. The sea wolves were in superb condition. They could keep up their steady jog all night, with only brief rest periods every hour.\n\nHorace urged Kicker up beside Halt.\n\n\"Shouldn't we walk as well?\" he suggested. Halt raised an eyebrow at him.\n\n\"Why?\" he asked. The big youth shrugged, not quite sure how to articulate his thought.\n\n\"As a gesture of comradeship,\" he said finally. \"It will give them a feeling of camaraderie.\"\n\nCamaraderie, Halt knew, was something that was stressed in the early years of Battleschool training. It was part of that inconvenient knightly code. Sometimes he wished that Sir Rodney, the head of Castle Redmont's Battleschool, would give his charges a short course in practicality as well.\n\n\"Well, it will give _me_ a feeling of sore legs,\" he replied at last. \"There's no point to it, Horace. The Skandians don't care whether we walk or ride. And when there's no point to something, the best idea is not to do it.\"\n\nHorace nodded several times. Truth be told, he was relieved that Halt had rejected his suggestion. He was far more at home in the saddle than tramping through the snow. And, now that he thought about it, the Skandians didn't seem to resent the fact that the four Araluens were riding while they walked.\n\nDuring one of the brief rest stops, Halt caught Will's eye and made an almost imperceptible gesture for the boy to follow him. They walked a short distance from the rest of the party, who were sprawled at ease in the snow. A few of the Skandians watched them with mild interest, but most ignored them.\n\nWhen he judged that there was no one within earshot, Halt drew Will closer to him, his hand on the boy's shoulder.\n\n\"This fellow Erak,\" he said. \"What do you make of him?\"\n\nWill frowned. He thought back over how Erak had treated them since he had captured them at the bridge in Celtica. In the first place, he had shielded them from Morgarath, refusing to hand them over to the rebel warlord. Then, on the trip across the Stormwhite Sea, and during their stay on Skorghijl, he had shown a certain rough kindness, and even a regard, toward him and Evanlyn. Finally, of course, he had been instrumental in their escape from Hallasholm, providing clothes, food and a pony, and giving them directions to the hunting cabin in the mountains.\n\nThere was only one possible answer.\n\n\"I like him,\" he replied. Halt nodded.\n\n\"Yes,\" he said. \"So do I. But do you trust him? That's a different matter to liking.\"\n\nThis time, Will opened his mouth immediately to reply, then paused, wondering if his response might not be too impulsive. Then he realized that trust was always impulsive, and went ahead.\n\n\"Yes,\" he said. \"I do.\"\n\nHalt rubbed his chin with his forefinger and thumb. \"I must say, I agree with you.\"\n\n\"Well, he did help us to escape, you know, Halt,\" Will pointed out, and the Ranger nodded his recognition of that point.\n\n\"I know,\" he said. \"That's what I was thinking about.\"\n\nHe was conscious of the boy's curious glance, but he said no more. As the members of the small party resumed their progress toward the coast, Halt struggled with the problem of how to protect Will and Evanlyn when they returned to Hallasholm. They might be regarded as allies for the moment, merely from force of circumstance. But once they were back in the Skandians' stronghold, things could go badly for the two escaped slaves. Things could become even worse for Evanlyn should her real identity become known to the Skandian Oberjarl.\n\nYet, try as he might, the gray-haired Ranger could think of no possible alternative to their present course. The way south was barred by thousands of Temujai warriors and there was no chance that he could make it through their lines with the three young people. He and Will might manage it. But it was a big might. And he knew enough about the Temujai to know that with Horace and Evanlyn along, they would never avoid detection.\n\nSo, for the time being, at least, they had no choice but to head toward Hallasholm. In the back of his mind there was a partly formed idea that they might be able to steal a boat. Or even prevail upon Erak to transport them down the coast to the south, leapfrogging the line of advance of the Temujai army. Somehow, sometime, he would have to reach some kind of an accommodation with the Skandian jarl, he knew.\n\nThe opportunity came at the next rest stop. And it came from the jarl himself. As the Skandians allowed themselves to sprawl on the ground under the pines, Erak, seemingly casually, approached the spot where Halt was pouring water from his canteen into a collapsible canvas bucket for Abelard. The horse drank noisily as the wolfship commander stood by and watched. Fully aware of his presence, Halt continued with what he was doing.\n\nThen, when the horse stopped drinking, he said, without looking up: \"Something on your mind?\"\n\nThe jarl shifted awkwardly from one foot to another.\n\n\"We need to talk,\" he said finally, and Halt shrugged.\n\n\"We seem to be doing that.\" He kept his voice neutral. He could sense that the Skandian leader wanted something from him and he felt this might be his opportunity to gain some kind of bargaining advantage.\n\nErak glanced around, making sure that none of his men was in earshot. He knew they wouldn't like the idea he was about to propose. But, all the same, he knew that the idea was a good one. And a necessary one.\n\n\"It was you, wasn't it, at the battle of the Thorntree?\" he said at last. Halt turned to face him.\n\n\"I was there,\" he said. \"And so were a couple of hundred others.\"\n\nThe Skandian made an impatient gesture. \"Yes, yes,\" he said. \"But you were the leader\u2014the tactician\u2014weren't you?\"\n\nHalt shrugged diffidently.\n\n\"That's right, I suppose,\" he said carefully. The battle at Thorntree forest had been a defeat for the Skandians. He wondered now if Erak might be looking for some kind of revenge over the man who had led the Araluen forces. It didn't seem in character with what he knew of the Skandian, but you never could tell.\n\nErak, however, was nodding thoughtfully to himself. He hunkered down in the snow, picking up a pine twig and making random marks on the ground with it.\n\n\"And you know these Temujai, don't you?\" he said. \"You know how they fight\u2014how they organize their army?\"\n\nIt was Halt's turn to nod. \"I told you. I lived among them for a while.\"\n\n\"So . . .\" Erak paused and Halt knew that he was reaching the crucial part of their conversation. \"You'd know their strengths, and their weaknesses?\"\n\nThe Ranger barked a short, humorless laugh. \"There aren't many of those,\" he said, but Erak persisted, stabbing the twig deeper into the snow as he talked.\n\n\"But you'd know how to fight them? How to beat them?\"\n\nNow Halt began to have a glimmer of where this conversation was leading. And, with that, he felt a slight surge of hope. He might just be about to be handed the bargaining tool that he would need to protect Will and Evanlyn.\n\n\"We fight as individuals,\" the jarl said softly, seeming to talk almost to himself. \"We aren't organized. We have no tactics. No master plan.\"\n\n\"You Skandians have won your share of battles,\" Halt pointed out mildly. Erak looked up at him and Halt could see how much the sea wolf disliked what he was about to say.\n\n\"In a straight confrontation. One on one. Or even against odds of two to one. A straightforward conflict with no complications. Just a simple trial of arms. That sort of thing we can handle. But this . . . this is different.\"\n\n\"The Temujai are probably the most efficient fighting force in the world,\" Halt told him. \"With the possible exception of the Arridi in the southern deserts.\"\n\nThere was silence between them. Halt willed the Skandian to take that one last step that lay in front of him. He saw the intake of breath, then Erak said:\n\n\"You could show us how to beat them.\"\n\nIt was out in the open now\u2014exactly what Halt had begun to hope for. Carefully, like a man playing a trout that was yet to be hooked, he answered, making sure no hint of the eagerness he felt showed in his voice.\n\n\"Even if I could, I doubt I'd be given that opportunity,\" he said, trying to sound as dismissive as possible. Erak's head jerked up, a little flare of anger in his eyes.\n\n\"I could give it to you,\" he said. Halt met the other man's gaze, refusing to be intimidated by the anger there.\n\n\"You're not the Oberjarl,\" he said flatly. Erak shook his head, acknowledging the statement.\n\n\"That's right,\" he said. \"But I am a senior war leader. I carry a certain amount of weight in our War Council.\"\n\nHalt appeared unconvinced. \"Enough to convince the others to accept an outlander as leader?\"\n\nErak shook his head decisively.\n\n\"Not as a leader,\" he said. \"Skandians would never follow your direct orders. Nor any other foreigner's. But as a counselor\u2014a tactician. There are others on the council who know we need tactics. Who will understand that we need to fight as a cohesive unit, not as a thousand individuals. Borsa, for one, will agree with me.\"\n\nHalt raised an eyebrow. \"Borsa?\" He knew some of the Skandian leaders' names. This one was unfamiliar.\n\n\"The hilfmann\u2014Ragnak's chamberlain,\" Erak told him. \"He's no warrior himself, but Ragnak respects his opinions, and his brain.\"\n\n\"Let me get this straight,\" Halt said slowly. \"You're asking me to come aboard as a tactical adviser and help you find a way to beat the Temujai. And you think you can convince Ragnak to go along with the idea\u2014and not simply kill me on the spot.\"\n\nErak looked a question at him. Halt continued.\n\n\"I know he has no love for Araluens. His son died at Thorntree, after all.\"\n\n\"You'd be under my protection,\" Erak said finally. \"Ragnak would have to respect that, or fight me. And I don't think he'll be quite ready to do that. Whether I can convince the council or not\u2014and I believe I will be able to\u2014you'll be safe while you're in Hallasholm.\"\n\nAnd there, all at once, was the opportunity Halt had been waiting for.\n\n\"What about my companions?\" he asked. \"Will and the girl are escaped slaves.\"\n\nErak waved the matter aside, dismissively. \"That's a small matter compared to the fact that we're about to be invaded,\" he replied. \"Your friends will be safe as well. You have my word.\"\n\n\"No matter what?\" Halt insisted. He wanted the Skandian to commit totally. He knew that no jarl would ever go back on a sworn vow of protection.\n\n\"No matter what,\" Erak replied, and held out his hand to the Ranger. They clasped hands firmly, sealing the bargain.\n\n\"Now,\" said Halt, \"all I have to do is work out a way of beating these horse-riding devils.\"\n\nErak grinned at him. \"That should be child's play,\" he said. \"The hard part will be convincing Ragnak about it.\"\n**18**\n\nAs IT TURNED OUT, THAT TASK WAS A LOT EASIER THAN either Erak or Halt would have thought possible. Ragnak was many things, but he was no fool. When the small party returned to Hallasholm, bringing news that an army of close to six thousand Temujai horsemen was in the process of invading his country, he did the same mental arithmetic that Erak had done. He knew as well as Erak that he could muster a force of no more than fifteen hundred warriors\u2014possibly less, considering that some of the outlying settlements close to the border had probably been overrun and defeated already.\n\nLike most Skandians, Ragnak wasn't afraid of dying in battle. But he also didn't believe that one should seek such an end without first trying all other alternatives. If there were a way of defeating the invaders, he would examine it. Consequently, when Erak told him of Halt's knowledge of the Temujai, and his agreement to lend his services, and when Borsa and several other council members welcomed the idea, he accepted their arguments with no more than token resistance. As for the matter of the recaptured slaves, he dismissed the matter entirely. In normal times, he might seek to punish runaways, as a way of discouraging further escapes. But these weren't normal times, and with an invading army on his doorstep, the matter of two recaptured slaves was of slight interest to him at best.\n\nHe did, however, demand to see Halt in his private quarters, with no one else present.\n\nHe knew enough about Rangers to respect their abilities and their courage as a group. But he wanted the chance to assess this man as an individual. Ragnak's ability to form such evaluations of men had been one of his principal qualities as leader of the Skandians. Evidence of his skill was the fact that he habitually chose Erak to handle the more difficult tasks that went with ruling a nation of independent-minded, argumentative warriors.\n\nHalt was shown to the low-ceilinged, timber-lined room where Ragnak spent his private hours\u2014and these days, the Oberjarl noted ruefully, there were precious few of those. The room was like all the senior Skandians' quarters\u2014warmed by a pine log fire, with bearskins furnishing the pinewood-carved furniture, decorated with the polyglot results of years of plundering coastal villages and other ships.\n\nThe centerpiece of the room was an immense crystal chandelier, taken from an abbey on the coast of the Constant Sea years ago. With no high ceiling to hang it from, Ragnak had chosen to leave it resting on a rough pine table. It dominated the room and was more than a little awkward in the confined space. Furthermore, in its tabletop position it was totally incapable of performing its designed intention. There was no way that the fifty small oil lamps could be lit and kept burning safely.\n\nBut Ragnak loved the piece. To him, it represented art at its highest. It was an object of rare beauty, incongruous as it might be in this setting, and so he left it there.\n\nHe looked up from a scroll he was reading as Halt knocked at the door and entered, as he had been told to do. Ragnak frowned. He equated prowess in battle with physical strength and size. The man before him looked wiry enough, but his head would barely come past the Oberjarl's shoulder if both were standing. There were no two ways about it. He was a small man.\n\n\"So, you're Halt,\" he said, not sounding too interested in the fact. He saw the little man's right eyebrow rise momentarily.\n\nThen the man repeated, in exactly the same tone: \"So, you're Ragnak.\"\n\nRagnak's heavy brows came closer together in an expression of anger. But inwardly, he felt a quick flicker of respect for the man in front of him. He liked Halt's instant reply, liked the way the Ranger was showing no sign of being cowed.\n\n\"People address me as 'Oberjarl,'\" he said in an ominous tone.\n\nHalt gave just the slightest suggestion of a shrug.\n\n\"Very well, Oberjarl,\" he replied. \"I'll do the same.\"\n\nHalt studied the Oberjarl with a keen eye. He was huge, but that was fairly normal for Skandians. He didn't have the classic, sculptured musculature that a person such as Horace would achieve in the next few years, with broad shoulders and narrow hips. Rather, like all Skandians, he was bulky throughout his entire body, built like a bear.\n\nThe arms and legs were massively muscled and the face was bearded, with the long beard lovingly separated into two sweeping masses. The hair had been red originally, but now the onset of age was turning it the color of ashes in a cold fireplace.\n\nThere was a faded scar on one cheek, stretching from just under the left eye down to the point of the man's chin. Halt guessed it to be an old injury. Again, there was little to remark on in this. The Skandians chose their leaders from the ranks of warriors, not administrators.\n\nMost of all, Halt noted the eyes. He recognized the dislike that he saw there. He had been expecting to see that. But the eyes were deep-set and he could read an intelligence and cunning there as well. For that, he was grateful.\n\nIf Ragnak had been a stupid man, Halt's position might well have become untenable here. He knew of the Oberjarl's ingrained dislike for Rangers, and knew the reasons behind it. But an intelligent leader would be aware of Halt's usefulness to him, and might be prepared to set aside his personal dislike for the greater good of his people.\n\n\"I have no love of your kind, Ranger,\" the Oberjarl said. His mind was obviously running on lines similar to Halt's.\n\n\"You have little reason to,\" Halt agreed. \"But you might well find a use for me.\"\n\n\"So I'm told,\" the Skandian leader replied, once again finding himself admiring the Ranger's forthrightness.\n\nWhen he'd first heard of his son's death at Thorntree, Ragnak had been overcome with grief and rage\u2014at Araluens, Rangers and, in particular, at King Duncan.\n\nBut that had been an immediate and spontaneous reaction to his grief. A realist, he knew that his son had risked death by joining the ill-fated adventure with Morgarath's forces and, indeed, death in battle was commonplace among the Skandians, who lived to raid and pillage. As a result, over the intervening months, Ragnak's anger, if not his grief, had faded. His son had died honorably, with a weapon in his hands. That was all any Skandian could ask. That wasn't to say that he felt any affection for Rangers, but he could respect their abilities and their courage, and their worth as opponents.\n\nOr even, possibly, as allies.\n\nRagnak's vow against King Duncan and his family was another matter altogether. Chances are, had he waited, his hatred might well have abated and a more reasonable attitude might have prevailed. But, acting on impulse, he had sworn a vow to the Vallas, the triple deity who ruled Skandian religion, and that vow was inviolable.\n\nRagnak might be able to accept Halt as an ally. He might be able to recognize that those same qualities that made the Ranger a dangerous opponent could also render him a useful confederate in the upcoming battle against the Temujai invaders. That would be his personal choice. But his Vallasvow against Duncan was irrevocable.\n\n\"So,\" Ragnak said abruptly. \"Can you help us?\"\n\nHalt answered without any hesitation. \"I'm willing to do whatever I can,\" he said. \"What that might be, I have no idea as yet.\"\n\n\"No idea!\" Ragnak repeated scornfully. \"I was told that Rangers are always full of ideas.\"\n\nHalt shook his head. \"I need to assess your strengths and weaknesses first. And then I'll need maps of the surrounding countryside,\" he said. \"We'll have to find a spot that will offset their superiority of numbers as far as possible. Then I'm going to ride out for another look at the Temujai. Last time I saw them, I had my hands full keeping your senior jarl alive. Then, after I've done all that, I might be able to answer your question.\"\n\nRagnak chewed on one end of his mustache, taking in what the Ranger had said. He was impressed, in spite of himself. His ability to plan for a battle usually amounted to the words \"Everyone ready? Follow me!\" before he led the way in a frontal assault.\n\nPerhaps, he thought, this Ranger might be useful after all.\n\n\"Be aware of one thing, however, Oberjarl,\" Halt continued. Ragnak looked up at him, surprised at the tone of uncompromising command in his voice.\n\n\"I'm going to be asking you questions about your establishment, your fighting men, your numbers. They're questions that might give me an advantage in any future disagreement between our two countries.\"\n\n\"I see . . . ,\" said Ragnak slowly. He didn't like the direction the conversation was taking.\n\n\"You'll be tempted to lie to me. To exaggerate your numbers and your abilities. Don't do it.\"\n\nOnce more, the Oberjarl was taken aback at the peremptory tone of command. But Halt's gaze was unwavering.\n\n\"If I am to help you, you'll need to be honest with me. And so will your jarls.\"\n\nRagnak considered the statement for a moment or two, then nodded ponderously.\n\n\"Agreed,\" he said. \"Mind you,\" he added, \"that ax cuts two ways. You'll also be showing us how you think and plan for a battle.\"\n\nAnd once more, that trace of a smile hovered around Halt's mouth as he acknowledged the Oberjarl's point.\n\n\"That's true,\" he said. \"I guess if we want to win, we both have to be willing to lose a little.\"\n\nThe two men studied each other once more. Each decided that he liked what he saw in the other's eyes. Abruptly, Ragnak gestured to one of the massive pinewood armchairs.\n\n\"Sit down!\" he said, indicating a flagon of Gallican wine on the table between them, almost lost in the glittering crystal fittings of the chandelier.\n\n\"Have a drink and tell me this. Why do you think these Temujai have chosen to make themselves a nuisance in Skandia? Surely the way would have been easier for them to move south, through Teutlandt and Gallica.\"\n\nHalt poured himself a glass of the brilliant red wine and drank deeply. He raised an eyebrow in appreciation. Ragnak certainly knew the right wines to steal, he thought.\n\n\"I've been wondering that myself,\" he said at last. He wished the chair he was sitting on was made for someone smaller than the normal massive Skandian build. His feet barely brushed the floor as he sat there and he felt like a small boy in his father's study. \"Even if they win here, they must know that you'll be a tough nut to crack. Certainly tougher than the Teutlanders.\"\n\nRagnak snorted in derision at the mention of the unorganized, squabbling race to the immediate south. Riddled by factions and internecine distrust, the Teutlanders were at the mercy of any would-be conquerors. In fact, if Skandian ambitions had lain in that direction, Ragnak would have felt confident that he could have subjugated the country with his small army of warriors.\n\n\"And the Gallicans are nearly as bad,\" Halt continued. \"They'd be almost incapable of agreeing on one overall leader to take command. So I wondered what it was that made the Temujai swing north and risk a bloody nose here in Skandia.\"\n\n\"And?\" the Oberjarl prompted. Halt took another swallow of wine and pursed his lips thoughtfully.\n\n\"I asked myself what you had that would make the risk worthwhile,\" he said. \"And there was only one thing I could think of.\"\n\nHe paused. It was a theatrical thing to do, he knew, but he couldn't resist it. As he felt sure would happen, the Oberjarl leaned forward.\n\n\"What was it? What are they after?\"\n\n\"Ships,\" replied Halt. \"The Temujai want control of the seas. And that means their ambitions don't stop here. They're planning to invade Araluen as well.\"\n**19**\n\nEVANLYN WAS WATCHING WILL PRACTICING HIS SHOOTING. IT was something that Halt had insisted on, once they had reached the relative safety of Hallasholm. Will's speed and accuracy had fallen far below the levels that Halt found acceptable and he wasted no time making his apprentice aware of the fact.\n\n\"Remember the golden rule?\" he'd said after he'd watched Will shoot a dozen arrows at different targets set up in a semicircle in front of him, at ranges varying from fifty meters out to two hundred. Most of Will's arrows flew wide of the more distant targets, and it took him far too long to fire the set of twelve shots.\n\nWill had looked up at his mentor, knowing how badly he'd shot. Halt was frowning and shaking his head slightly. It made matters worse that Horace and Evanlyn had chosen that moment to come and watch.\n\n\"Practice?\" he'd replied glumly, and Halt had nodded.\n\n\"Practice,\" he affirmed. As they'd walked out to collect the arrows he'd fired, Halt had dropped a consoling arm around the boy's shoulders.\n\n\"Don't feel too bad about it,\" he told him. \"Your technique is still good. But you can't expect to spend the winter making snowmen in the mountains and retain your edge.\"\n\n\"Making snowmen?\" Will replied indignantly. \"I'll have you know things were pretty rough up in the mountains . . .\" He stopped as he realized that Halt had been pulling his leg. He had to admit that the Ranger was right, however. The only way to attain the almost instinctive accuracy and speed with the bow that were the hall-marks of a Ranger was to practice, constantly and assiduously.\n\nOver the following days, he took himself to the practice area and gave himself over to the task of perfecting his skills once more. As his old skill returned, along with his strength and fitness, a small crowd would follow and watch. Even though Will couldn't boast the skill levels of a full-fledged Ranger, his ability was far above that of normal archers and he was regarded by Skandians and some of the slaves with a deal of respect.\n\nEvanlyn and Horace, however, seemed to find plenty of other things to fill their days\u2014riding and hiking in the nearby woods, or sometimes taking a small skiff out on the bay. Of course, they had asked Will to join them, but each time, he had replied that he had to attend to his practice.\n\nThere were times when he could have gone. But even on these occasions, his feelings injured, he begged off, claiming the need for extra work sessions.\n\nThe practice sessions were intensified when Erak produced the double knife scabbard that Will had been wearing when he and Evanlyn had been captured by the Skandians. Erak, a true hoarder, had kept the weapons and now saw fit to return them to their rightful owner. A word from Halt let Will know that he would soon be tested for his knife-throwing skills as well. Experience had taught Will by now that the long months without practice would have eroded his abilities in this area too. So he set about restoring them. The township of Hallasholm soon rang to the repetitive thud of his throwing knife and saxe knife striking point first into a target of soft pinewood.\n\nAs each day passed, his accuracy and speed improved with both the bow and the knives. He was beginning to recapture that smooth, flowing action that Halt had drilled into him over so many hours in the forest outside Castle Redmont.\n\nNow he switched easily from target to target, his arm raising or lowering the bow to adjust for the variations in distance, his eyes wide open, seeing a total sighting picture that included the bow, the arrow and the eventual target. He was pleased that Evanlyn had chosen today to come and watch his practice session. He felt a savage exultation as arrow after arrow thudded into the targets, striking either in the center or close enough to make no difference.\n\n\"So,\" he said casually as he released two arrows at two widely varying targets in quick succession. \"Where's Horace today?\"\n\nThe arrows thudded, one after another, into their respective targets and he nodded to himself, turning ninety degrees to loose another at one of the targets set closer in.\n\nAnother hit. Another thud.\n\nThe girl shrugged. \"I think you made him feel guilty,\" she replied. \"He thought he'd better get some practice in. He's working out with some of the Skandians from Erak's crew.\"\n\n\"I see,\" replied Will, then paused to put an arrow into one of the farthest targets, watching it arc smoothly through the air before burying its point in the center ring.\n\n\"And why didn't you go along to watch him?\" He felt a little pleased that Evanlyn had chosen, finally, to see how proficient he was becoming and hadn't bothered to watch her constant companion of the past few days. Her next words dashed that small glow of pleasure, however.\n\n\"I did,\" she replied. \"But after you've seen two people whack at each other for several minutes, you develop a sense of d\u00e9j\u00e0 vu. I thought I'd come and see if you'd improved since the other day.\"\n\n\"Oh, really?\" Will replied, a little stiffly. \"Well, I hope you don't feel you've wasted your time.\"\n\nEvanlyn looked up at him. He was facing away from her, firing a sequence of shots at three targets\u2014one at fifty meters, one at seventy-five and one at a hundred. She could hear the stiff tone in his voice and wondered what was bothering him. She decided not to answer the question. Instead, she commented on the three-shot sequence, as all three arrows found their marks.\n\n\"How do you do that?\" she asked. Will stopped and turned toward her. There was a genuine note of inquiry in her voice.\n\n\"Do what?\"\n\nShe gestured toward the three targets.\n\n\"How do you know how far to lift the bow for each distance?\" she asked. For a moment the question left him nonplussed. Finally, he shrugged.\n\n\"I just . . . feel it,\" he replied uncertainly. Then, frowning, he tried to elaborate. \"It's a matter of practice. When you do it over and over again, it becomes sort of . . . instinctive, I suppose.\"\n\n\"So, if I took the bow, could you tell me how high to hold it for that middle target, for instance?\" she asked, and he cocked his head to one side, thinking the question through.\n\n\"Well . . . it's not just that. I suppose I could, but . . . there are other factors.\"\n\nShe leaned forward, her face querying, and he continued.\n\n\"Like your release . . . it has to be smooth. You can't snatch at it or the arrow goes off line. And your draw weight would probably vary.\"\n\n\"Draw weight?\"\n\nHe indicated the tension on the bowstring as he pulled it back to full draw. \"The longer your draw, the more weight you put behind the arrow. If you didn't draw exactly the same distance as I did, the result would vary.\"\n\nShe thought about the answer. It seemed logical. She pursed her lips pensively and nodded once or twice.\n\n\"I see,\" she said. There was a slight tone of disappointment in her voice.\n\n\"Is there some kind of problem?\" Will asked, and she sighed deeply.\n\n\"I was kind of hoping that maybe you could show me how to shoot so that I could actually do something when the Temujai turn up here,\" she replied, a little downcast.\n\nWill laughed. \"Well, maybe I could\u2014if we had a year to spare.\"\n\n\"I don't want to be an expert,\" she said. \"I thought maybe you could just show me one or two basic things so I could . . . you know . . .\" She tailed off uncertainly.\n\nWill shook his head apologetically, regretting the fact that he'd laughed at her.\n\n\"I'm afraid the real secret is a whole lot of practice,\" he said. \"Even if I showed you the basics, it's not something you can just learn in a week or two.\"\n\nShe shrugged again.\n\n\"I suppose not.\" She realized that her request had been unrealistic. She felt foolish now and seized the opportunity to change the subject. \"Is that when Halt thinks they'll get here\u2014a week or two?\"\n\nWill fired the last arrow in the set and laid his bow down.\n\n\"He said they could be here then. But he thinks they'll take a little longer. After all, they know the Skandians aren't going anywhere.\" He gestured to her to accompany him as he collected his arrows and they started across the practice field together.\n\n\"Did you hear his theory?\" she asked him. \"About attacking here because they want the Skandians' ships?\"\n\nWill nodded. \"It makes sense when you think about it. They can overrun Teutlandt and Gallica almost as they choose. But they'd be leaving a dangerous enemy behind them. And the Skandians could raid them anywhere along the coast, hitting them where and when they choose.\"\n\n\"I can see that,\" Evanlyn replied, tugging one of the arrows from the fifty-meter target. \"But don't you think his theory about invading Araluen is a little far-fetched?\"\n\n\"Not at all,\" Will replied. \"Hold them closer to the head as you pull them out,\" he said, indicating the next arrow as she reached for it. \"Otherwise you'll break the shaft, or warp it. There's no reason why the Temujai should stop at the Gallican coast. But if they tried to transport their army by ship without taking care of the Skandians first, they could be in big trouble.\"\n\nEvanlyn was silent for a few seconds. \"I suppose so,\" she said eventually.\n\n\"It's only a theory, after all,\" Will replied. \"Maybe they're just making sure their flanks are secure before they move into Teutlandt. But Halt says you should always plan for the worst-case scenario. Then you can't be disappointed.\"\n\n\"I guess he's right about that,\" she replied. \"Where is he, anyway? I haven't seen him around for a few days.\"\n\nWill nodded his head toward the southeast. \"He and Erak have gone to scout the Temujai advance,\" he said. \"I think he's looking for a way to slow them down.\"\n\nHe collected the last of his arrows and stowed them in his quiver. Then he stretched and flexed his arms and fingers.\n\n\"Well, I guess I'll shoot another set,\" he said. \"Are you staying to watch?\"\n\nEvanlyn considered for a moment, then shook her head. \"I might go see how Horace is doing,\" she said. \"I'll try to spread the encouragement around.\" She smiled at him, waggled her fingers in farewell and strode off across the field, back toward the palisade. Will watched her slim, upright figure as she walked away.\n\n\"You do that,\" he muttered to himself. Once more, he felt a flutter of jealousy as he thought of her watching Horace. Then he shook the feeling off, as a duck shakes water away. Head down, he began to mooch back to the firing line.\n\n\"Women,\" he muttered to himself. \"They're nothing but trouble.\"\n\nA shadow fell across the ground beside him and he glanced up, thinking for a moment that Evanlyn might have changed her mind. After all, the prospect of watching two muscle-bound hulks whacking each other with practice weapons was a little boring, he thought. But it wasn't Evanlyn, it was Tyrelle\u2014blond, pretty, fifteen years old and the niece of Svengal, Erak's first mate. She smiled shyly at him. Her eyes were amazingly blue, he realized.\n\n\"Can I carry your arrows back for you, Ranger?\" she asked, and he shrugged magnanimously, unclipping the quiver and handing it to her.\n\n\"Why not?\" he said, and her smile widened.\n\nAfter all, he thought, it would have been churlish to refuse.\n**20**\n\nTHE PINE HAD FALLEN SEVERAL YEARS BACK, FINALLY DEFEATED by the weight of snow in its branches, the insidious rot at the heart of its massive trunk and one too many seasons of gale force winter winds. Even in death, however, its neighbors had tried to support it, keeping it from the ignominy of the ground, holding it in the grip of their tangled branches so that it lay at an angle of thirty degrees to the horizontal, seemingly supported between heaven and earth by its closely packed fellows.\n\nHalt leaned now on the rough bark that still coated the dead trunk and peered down into the valley below, where the Temujai column moved slowly past.\n\n\"They're taking their time,\" Erak said, beside him. The Ranger turned to look at him, one eyebrow raised quizzically.\n\n\"They're in no hurry,\" he replied. \"It's going to take them some time to get their wagons and supply train through the passes. Their horses don't like confined spaces. They're used to the open plains of the steppes.\"\n\nThe cavalry army continued its slow advance. There seemed little order to their march, Halt thought, frowning. There were no outriders, no patrols screening the flanks of the mob of men, horses and wagons as they made their way toward Hallasholm, ninety kilometers to the north.\n\nHalt, Erak and a small party of Skandians had come southeast, moving over the mountains along steep, narrow paths where the Temujai cavalry found it more difficult to move, to scout the invaders' progress. Now, as Halt watched them, a thought struck him.\n\n\"Mind you, we could make sure they move a little slower,\" he said softly. Erak shrugged impatiently at the idea.\n\n\"Why bother?\" he asked bluntly. \"The sooner we come to grips with them, the sooner we settle this.\"\n\n\"The longer they take, the more time we have to prepare,\" Halt told him. \"Besides, it bothers me to see them just ambling along, taking no precautions, riding in no order. It's too damned arrogant.\"\n\n\"I thought you said they were smart?\" the Skandian queried, and it was Halt's turn to shrug.\n\n\"Maybe it's because they expect you to simply come at them head-on when they finally reach Hallasholm,\" he suggested. The Skandian war leader considered the thought, looking a little offended by it.\n\n\"Don't they give us any credit for strategy?\"\n\nHalt tried to hide a grin. \"How _would_ you plan to fight them?\"\n\nThere was a pause, then Erak replied reluctantly, \"I suppose I'd simply wait till they reached our position, then . . . attack them head-on.\" He looked carefully at the shorter man, but Halt was being very obvious about not saying anything further. Finally, Erak added, in an injured tone: \"But there's no need for them to simply _assume_ that.\"\n\n\"Exactly,\" Halt replied. \"So perhaps we should give them something to think about. Something to put them a little off balance\u2014and maybe put a little doubt in their minds.\"\n\n\"Is that good strategy?\" Erak asked. The Ranger grinned at him.\n\n\"It's good therapy for us,\" he replied. \"And besides, an enemy with a worm of doubt working away at his mind is less likely to make bold and unexpected moves. The more we can dissuade them from doing the unexpected, the better it will be for us.\"\n\nErak thought about the point. It seemed logical. \"So what do you want to do?\" he asked.\n\nHalt looked around at the twenty warriors who had accompanied them.\n\n\"This Olgak,\" he said, indicating the young leader of the troop. \"Is he capable of following orders, or is he a typical Skandian berserker?\"\n\nErak pursed his lips. \"All Skandians are berserkers, given the right conditions,\" he replied. \"But Olgak will follow orders if I give them.\"\n\nHalt nodded his understanding. \"Let's talk to him then,\" he said.\n\nErak beckoned the broad-shouldered younger man to join them. Olgak, seeing the signal, moved forward, his ax swinging easily in his right hand, his large circular shield on his left arm. He looked expectantly at Erak, but the jarl gestured toward Halt.\n\n\"Listen to what the Ranger has to say,\" he ordered, and the young man's eyes turned to Halt. The Ranger studied him for a few moments. His clear blue eyes were guileless and straightforward. But he saw a light of intelligence there. Halt nodded to himself, then gestured to the Temujai army below them.\n\n\"See that rabble down there?\" he asked, and when the younger man nodded, he continued, \"They're riding with no formation, with no covering scouts, and with their supply wagons and support personnel mixed up with their warriors. They don't usually travel that way. Do you know why they're doing it now?\"\n\nOlgak hesitated, then shook his head, frowning slightly. Not only didn't he know, but he didn't know why it should be important for anyone to know such a thing.\n\n\"They're doing it because they feel safe,\" Halt continued. \"Because they believe you Skandians are simply going to wait for them and meet them head-on.\"\n\nOlgak nodded now. They had reached a point that he understood. \"We are . . . aren't we?\"\n\nHalt exchanged a glance with Erak. The jarl shrugged. Skandians took a simple view of things.\n\n\"Well, yes, you are,\" Halt admitted. \"Eventually. But for now, it might be nice to make them a little less comfortable, mightn't it?\" He paused, then added, with a slight edge in his voice, \"Or do you enjoy seeing them swan through your country as if they're on holiday?\"\n\nOlgak pursed his lips, looking down at the invaders. Now that the Ranger had mentioned it, they did appear to be having an altogether too easy time of things, he thought.\n\n\"No,\" he replied. \"I can't say I enjoy seeing that. So what are we going to do about it?\"\n\n\"Erak and I are going back to Hallasholm,\" Halt told him, feeling the Skandian leader stiffen beside him as he said it. Obviously the jarl had been looking forward to a little skirmish and he wasn't thrilled to hear he was going to miss it. \"But you and your men are going to raid their lines tonight and burn those wagons.\"\n\nHe pointed with the end of his longbow to half a dozen supply wagons, trundling carelessly along at the edge of the army. Olgak grinned and nodded his approval of the idea.\n\n\"Sounds good to me,\" he said. Halt reached out and laid a firm grip on his muscular forearm, compelling the younger man to meet his steady gaze.\n\n\"But listen to me, Olgak,\" he said intensely. \"You are going to hit and run. Don't get tangled up in an extended fight, understand?\"\n\nThe young Skandian was less pleased with that command. Halt shook his arm fiercely for emphasis.\n\n\"Understand?\" he repeated. \"We do not want you and these twenty men to go down in a blaze of glory when you burn those wagons. And do you know why?\"\n\nOlgak shook his head\u2014a small, reluctant movement. Halt continued.\n\n\"Because tomorrow night, I want you to move along the column and burn more wagons\u2014and kill a few more Temujai while you're at it.\"\n\nThe idea was beginning to appeal to the younger man now.\n\n\"And if you're all killed on the first attempt, no matter how glorious it may seem at the time, by tomorrow the Temujai will simply continue on as they are, won't they?\" the Ranger asked him. Olgak nodded his understanding.\n\n\"Then each night, I want you to hit a different part of the column. Burn their supplies. Set their horses loose. Kill their sentries. Get in and out fast and don't let them trap you into a standing battle. Stay alive and keep harassing them. Got the picture?\"\n\nOlgak nodded again, now more convinced of the good sense behind the plan. \"They'll never know where we're going to hit them next,\" he said enthusiastically.\n\n\"Exactly,\" Halt said. \"Which means they'll have to set guards along the entire column. They'll have to post extra sentries at night. And all of that will slow them down.\"\n\n\"It's like coastal raiding, isn't it?\" the young Skandian said, thinking how the wolfships would appear from over the horizon without warning on an enemy coast and attack unprepared settlements. \"Do you only want us to do it at night?\" he added.\n\nHalt thought for a minute.\n\n\"For the first couple of days, yes. Then pick a spot where you can withdraw quickly into the trees and uphill\u2014somewhere their horses won't follow easily\u2014and hit them in daylight. Maybe toward the end of the day\u2014or the beginning.\"\n\n\"Keep them guessing?\" Olgak said, and Halt patted his arm approvingly.\n\n\"You've got the idea,\" he said, smiling at the younger man. \"And remember the golden rule: hit them where they aren't.\"\n\nOlgak pondered that. \"Hit them where they aren't?\" he asked finally, sounding uncertain.\n\n\"Attack in those places where their troops are spread thinnest. Make them come to you. Then fade away before they really make contact. Remember that part. It's the most important of all. Survive.\"\n\nHe could see the younger man understood. Olgak repeated the word to himself. \"Survive,\" he said. \"I understand.\"\n\nHalt turned and looked at Erak, raising an eyebrow. \"Is there any reason why you should make it an order to Olgak that he's not to get tied down in a fight, Jarl?\" he asked. Erak turned the question to the younger man.\n\n\"Well, Olgak, is there?\" he said, and the troop leader shook his head.\n\n\"I understand what you have in mind, Ranger,\" he said. \"Trust me. It's a good idea.\"\n\n\"Good man,\" Halt said quietly, then he turned to face the question he knew was coming from Erak.\n\n\"And what will we be doing while Olgak and his men are having all the fun?\" the Jarl asked.\n\n\"We're going back to Hallasholm to start preparing a reception for our friends down there,\" Halt told him. \"And while we're at it, we might send another half dozen parties out to harass the column the way Olgak will be doing. Everything we can do to slow them down will help us.\"\n\nErak shuffled his feet in the snow. He looked, Halt thought, remarkably like a child who has been told he must hand over his favorite toy.\n\n\"You could do that,\" he said finally. \"Maybe I should stay and give Olgak and his men a hand.\" But Halt shook his head, the ghost of a smile touching the corners of his mouth.\n\n\"I need you back with me,\" he said simply. \"I need your authority behind me if I'm going to be able to get things organized.\"\n\nErak opened his mouth to reply, but Olgak interrupted.\n\n\"The Ranger's right, Jarl,\" he said. \"You'll be more valuable at Hallasholm. And besides, you're getting a little long in the tooth for this sort of work, aren't you?\"\n\nErak's eyes widened with anger and he started to say something. Then he noticed that Olgak was grinning broadly and realized that the younger man was joking. He shook his head warningly, glancing at his own broadax.\n\n\"One of these days, I might just show you how long in the tooth I am,\" he said meaningfully. Olgak's grin widened. Halt regarded the two of them for a moment, then, slinging his longbow over his right shoulder, he turned and led the way back to where Abelard was tethered, along with the pony that Erak had reluctantly ridden when they came on this scouting expedition. He gathered Abelard's reins in one hand and turned back to the troop leader.\n\n\"I'm sure you'll do a good job, Olgak,\" he said. Then, glancing sidelong at the still indignant jarl, he added quietly: \"You're obviously a very brave young man.\"\n**21**\n\nGENERAL HAZ'KAM, COMMANDER OF THE TEMUJAI INVASION force, looked up from his meal as his deputy entered the tent. Even though Nit'zak was by no means a tall man, he had to stoop as he came through the low opening. The general gestured to the cushions that were scatted on the felt rug floor and Nit'zak lowered himself to sit on one of them, uttering a sigh of relief. He had been in the saddle the past five hours, checking up and down the length of the Temujai column.\n\nHaz'kam shoved the fragrant bowl of meat stew that he had been eating across to the other man and indicated for him to help himself. Nit'zak nodded his thanks, took a smaller bowl from the rug between them and scooped several handfuls into it, wincing slightly as his hand made contact with the hot food. He selected a large chunk and scooped it into his mouth, chewing heartily and nodding his appreciation.\n\n\"Good,\" he said finally. Haz'kam's concubine\u2014the general never brought any of his three wives on campaign with him\u2014was an excellent cook. The general considered that ability of far greater importance during a campaign than any physical beauty. He nodded now, belched softly and pushed his own eating bowl away. The woman moved quickly forward to remove it, then returned to her position against the curved felt wall of the tent.\n\n\"So,\" the general asked. \"What did you find?\"\n\nNit'zak screwed his face into an expression of distaste\u2014not at the next morsel of food, but at the subject matter he was about to report.\n\n\"They hit us again this evening,\" he replied. \"This time in two places. Once at the tail of the column. They stampeded a small herd of horses there. It'll take half the day tomorrow to recover them. Then another group came in from the coastal side and burned half a dozen supply wagons.\"\n\nHaz'kam looked up in surprise. \"From the coast?\" he asked, and his deputy nodded confirmation. Up until now, the nuisance raids mounted by the Skandians had been launched from the thickly wooded hills inland from the narrow coastal flatlands. The raiders would dash out, strike an undefended part of the column, then retreat into the cover of the forests and the hills where pursuit would be too risky. This new eventuality complicated things.\n\n\"They seem to have several of their ships at sea,\" the deputy told him. \"They stay out of sight during the day, then steal in after dark and land troops to hit us. Then they retreat to sea once more.\"\n\nHaz'kam probed with his tongue at a piece of meat wedged between two back teeth. \"Where, of course, we can't follow them,\" he said.\n\nNit'zak nodded. \"It means now that we'll have to cover both sides of the column,\" he said.\n\nHaz'kam muttered a low curse. \"It's slowing us down,\" he said.\n\nEach morning, hours were wasted as the massive column formed up in disciplined ranks for the day's march. And, of course, once the march began, the pace was limited by the slowest sections of the column\u2014which were the supply carts and the baggage train. It had been much faster simply moving as one vast mass.\n\nNit'zak agreed. \"So is the problem of having to screen the camp each night.\"\n\nHaz'kam took a deep swig of the fermented barley drink that the Temujai favored, then handed the leather drinking skin to Nit'zak.\n\n\"It's not what I expected,\" he said. \"They're far more organized than our intelligence had led us to believe.\"\n\nNit'zak drank deeply and gratefully. He shrugged. In his experience, intelligence was usually inaccurate at best and dead wrong at worst.\n\n\"I know,\" he said. \"Everything we'd heard about these people led me to believe that they would simply attack us in a frontal assault, without any overall strategy. I'd half expected that we'd be finished with them by now.\"\n\nHaz'kam pondered. \"Perhaps they're still gathering their main force. I suppose we have no option but to continue as we are. I imagine they'll finally make a stand when we reach their capital. Although now we'll take longer to do that.\"\n\nNit'zak hesitated for a moment with the next suggestion. Then he said: \"Of course, General, we could simply continue as we were, and accept the losses their raids are causing. They're quite sustainable, you know.\"\n\nIt was a typically callous Temujai suggestion. If the loss of lives or supplies could be balanced out by greater speed, it might well be worthwhile opting for that course. Haz'kam shook his head. But not through any sense of care for the people under his command.\n\n\"If we don't respond, we have no way of knowing that they won't hit us with a major raid,\" he pointed out. \"They could have hundreds of men in those mountains and if they chose to change from pinprick attacks to a major assault, we'd be in big trouble. We're a long way from home, you know.\"\n\nNit'zak nodded his acquiescence. That idea hadn't occurred to him. Still, he demurred slightly.\n\n\"That isn't the sort of thing we've been led to believe they're capable of,\" he pointed out, and Haz'kam's eyes met his and locked onto them.\n\n\"Neither is this,\" he said softly, and when the younger man's eyes dropped from his, he added, \"Have the men keep forming into their sixties for each day's march. And I suppose now we'd better put sentries out on the seaward side at night too.\"\n\nNit'zak muttered his assent. He hesitated a few seconds, wondering if this were one of those times when his commander wanted to continue to talk and pass the drinking skin back and forth for a few hours. But Haz'kam waved him away with a small hand gesture. Nit'zak thought that the general looked tired. For a moment, he thought about the years they had spent on campaign together and realized that Haz'kam was no longer a young man. Neither was he, he thought, as the ache in his knees testified. He bowed his head in a perfunctory salute, rose to his feet with another barely suppressed groan and went, crouching, out through the felt hanging that covered the tent doorway.\n\nIn the distance, he heard men shouting. Looking in the direction from which the noise came, he saw a bright flare of flame against the night sky. He cursed softly. The damned Skandians were raiding again, he thought.\n\nA troop of horsemen clattered by him, heading for the site of the attack. He watched them go, tempted for a moment to join them, but resisting the temptation as he realized that by the time they reached the point of the attack, the enemy would be long gone.\n**22**\n\nTHE SKANDIAN WAR COUNCIL WAS MEETING IN THE GREAT Hall. Will sat to one side, listening as Halt addressed the Skandian leader and his principal advisers. Borsa, Erak and two other senior jarls, Lorak and Ulfak, flanked the Oberjarl as they clustered around the table where Halt had spread an immense map of Skandia. The Ranger tapped a spot on the map with the point of his saxe knife.\n\n\"As of last night,\" he said, \"the Temujai were here. Maybe sixty kilometers away from Hallasholm. The delaying raids are having exactly the sort of effect we wanted. The advance has gone from thirty kilometers a day to less than twelve.\"\n\n\"Shouldn't cavalry move faster than that?\" asked Ulfak. Halt perched one leg on the bench beside the table and shook his head.\n\n\"They'll move fast enough when they're fighting,\" he told them. \"But right now, they're conserving their horses' strength, letting them feed and move easily. Besides, now that we've reinforced Olgak's men with another half dozen raiding groups, it's taking them half the day to simply form up, then set up camp again in the evening.\"\n\nHe glanced up at Erak as he added: \"Your idea of sending a few wolfships to raid their seaward flank was a good one.\"\n\nThe jarl nodded. \"It seemed logical,\" he replied. \"It's what we're good at, after all.\"\n\nRagnak thumped one massive fist on the pine planks that formed the table.\n\n\"Raids and skirmishes, nuisance attacks! They achieve nothing! It's time we hit them with our main force and settle this once and for all,\" he declared, and three of his council growled agreement.\n\n\"There'll be plenty of time for that,\" Halt cautioned. \"The most important thing is to engage them in a place that suits us\u2014one that we choose ourselves.\"\n\nAgain, the Oberjarl snarled. He knew he'd agreed to listen to Halt's advice. But these damned invaders had been flaunting themselves in his country now for several weeks. It was an affront to him and to every Skandian and he wanted to wipe the affront out, or die in the attempt. \"What's the difference where we fight them?\" he said. \"A fight is a fight. We win or we lose. But if we do lose, we'll take plenty of them with us!\"\n\nHalt removed his foot from the bench and stood straight, ramming the saxe knife back into its scabbard.\n\n\"Oh, don't worry,\" he said icily. \"There's every chance that we'll lose. But let's make sure we take as many of them with us as possible, shall we?\" The Skandians, used to bluster and boasting, were taken aback by his cold assessment of their chances for survival\u2014as he had intended them to be.\n\n\"They're cavalry,\" he continued. \"They outnumber us at least four to one. They can outmaneuver us, outrun us. And they'll look for the widest possible front to engage us on. That way, all the advantages are with them. They'll flank us, surround us and draw us out if they can.\" He saw that he had their attention. They weren't happy about the situation, but at least they were prepared to listen.\n\n\"How will they do that?\" Erak asked. He and Halt had discussed this briefing the day before. Halt wanted certain questions to be asked, and Erak was to ask them if none of the others seemed prepared to do so. The Ranger glanced quickly at Erak, but directed his answer to all of the group.\n\n\"It's a standard tactic of theirs,\" he said. \"They'll attack on a wide front, probing, hitting and retiring. Then they'll appear to become fully engaged at one or two given points. They'll stop their hit-and-run tactics and fight a pitched battle\u2014just the sort of thing that will suit your men,\" he added, glancing at Ragnak. The Oberjarl nodded.\n\n\"Then,\" Halt continued, \"they will begin to lose. Their attack will lose its cohesion and they will try to withdraw.\"\n\n\"Good!\" said Borsa, and the two other jarls grunted agreement. Ragnak, however, sensed that there was more to come. He didn't comment for the moment, but gestured for Halt to continue. The Ranger obliged.\n\n\"They'll give ground. Slowly at first, then faster and faster as panic seems to set in. Somehow they'll never move so fast that your men lose contact with them. Gradually, more and more of your warriors will be drawn out of our line, away from the shield wall, away from our defenses. As they pursue the enemy, the Temujai will become more and more desperate. At least, they'll seem to. Then, at the right moment, they'll turn.\"\n\n\"Turn?\" said the Oberjarl. \"How do you mean?\"\n\n\"They'll stop retreating when your men are strung out and in the open\u2014the strongest and fastest well ahead of their comrades. Suddenly, they'll find themselves cut off, surrounded by the Temujai cavalry. And remember, every one of their cavalrymen is an expert archer. They won't bother coming to close quarters. They can pick your men off at their leisure. And the more they kill the leaders, the more enraged those behind will become. They'll stream out to save their friends\u2014or avenge them. They'll be surrounded in turn. And wiped out.\"\n\nHe paused. The five Skandians all looked at him, struck silent. They could imagine the scenario he described. They knew the temper of their men and could see how easily such a stratagem could succeed against them.\n\n\"This is how they fight?\" Ragnak asked finally.\n\n\"I've seen it, Oberjarl. Time and time again, I've seen it. They aren't concerned with glory in battle. Only efficient killing. They'll challenge our warriors to single combat, then ambush them with ten or twenty warriors at a time. If they can't shoot to kill immediately, they'll shoot to disable. Even your strongest warriors can't continue with ten to fifteen arrow wounds in the legs. Then, when they're helpless, the Temujai will kill them.\"\n\nHe swept his gaze around the table. Satisfied that they could all see the danger that faced them, he sat down, straddling the bench. Finally, it was Borsa, the hilfmann, who broke the long silence that had fallen in the room.\n\n\"So . . . where do you want to engage them?\" he asked. Halt spread his hands wide in a questioning gesture.\n\n\"Why engage them at all?\" he asked. \"We have time to withdraw before they arrive. We could move into the hills and the forest and keep hitting them as they come farther and farther along the coastal plain here.\"\n\n\"Run away, you mean?\" Ragnak asked, his tone angry.\n\nHalt nodded several times. \"Yes. Run away. But continue to hit them at twenty or thirty or fifty points along their column. Kill them. Burn their supplies. Harass them. Make their life one long, insufferable misery until they realize that this invasion was a bad idea. Then harass them back to the border until they're gone.\"\n\nHe paused. He knew there was little chance of winning this one. But he had to try. It was the best course open to them. His heart sank as Ragnak shook his head. Even Erak's lips were compressed into a thin, disapproving line.\n\n\"Abandon Hallasholm to them?\" asked Ragnak.\n\nHalt shrugged. \"If necessary. You can always rebuild.\"\n\nBut now all the Skandians were shaking their heads and he knew what was behind it.\n\n\"Abandon everything in Hallasholm to them?\" Ragnak persisted. This time Halt made no answer. He waited for the inevitable.\n\n\"Our booty\u2014the results of hundreds of years of raiding\u2014leave that to them?\" Ragnak asked.\n\nAnd that, Halt knew, was the crux of the matter. No Skandian would ever abandon the loot he had stored up over the years\u2014the gold, the armor, the tapestries, the chandeliers, the thousand and one items that they hoarded and kept and gloated over in their storehouses. He caught Will's eye and shrugged slightly. He'd tried. Halt moved to the map once more and indicated the flatlands outside Hallasholm with his knife point.\n\n\"Alternatively,\" he said, \"we stop them here, where the coastal plain contracts to its narrowest point.\"\n\nThe Skandians craned to look again. They nodded cautious approval, now that Halt had withdrawn the suggestion that they should abandon Hallasholm and its contents to the invaders.\n\n\"This way, they can't attack on a wide front. They'll be cramped. And we can conceal men in the trees here\u2014and even in the out-buildings along the shore.\"\n\nLorak, older of the two jarls, frowned at the suggestion. \"Won't that weaken our shield wall?\"\n\nHalt shook his head. \"Not noticeably. We'll have more than enough men to form a solid defensive position here where the land is narrowest. Then, when the Temujai try their trick of falling back and bringing our men along with them, we'll appear to go along with it.\"\n\nErak moved forward to inspect the narrow neck of land that Halt was indicating.\n\n\"You mean we'll do as they want?\" he asked. Halt pushed out his bottom lip and cocked his head to one side.\n\n\"We'll appear to,\" he admitted. \"But once they stop withdrawing to counterattack, we'll bring our ambush forces out of hiding and hit them from behind. If we time it correctly, we could make life very unpleasant for them.\"\n\nThe Skandians stood, staring down at the map. Borsa, Lorak and Ulfak had blank looks as they tried to visualize the movement. Erak and Ragnak, Halt was glad to see, were slowly nodding as they understood the idea.\n\n\"Our best chance,\" he continued, \"is to force them into the sort of engagement that suits your men best\u2014close quarters, hand to hand, every man for himself. If we can catch them that way, your axmen will take a heavy toll on them. The Temujai rely on speed and movement for protection. They're only lightly armed and armored. If we had even a small force of archers, it could make an enormous difference,\" he added. \"But I suppose we can't have everything.\" Halt knew that the bow wasn't a Skandian weapon. It was no use wishing for things that couldn't be. But in his mind's eye, he could see the devastation that an organized party of bowmen could cause. He shrugged, pushing the thought aside.\n\nErak looked up at the gray-cloaked Ranger. He's small, he thought, but by the gods, he's a warrior to reckon with.\n\n\"We have to depend on our men keeping their heads,\" he said. \"Then we have to time it just right when we spring our trap\u2014otherwise the men coming from the forest and the out-buildings will be exposed themselves. It's a risk.\"\n\nHalt shrugged. \"It's war,\" he replied. \"The trick is to know which risks to take.\"\n\n\"And how do you know that?\" Borsa asked him, sensing that the small, bearded foreigner had gained the trust and the acceptance of the Oberjarl and his War Council. Halt smiled wolfishly at him.\n\n\"You wait till it's over and see who's won,\" he said. \"Then you know those were the right risks to take.\"\n**23**\n\n\"HALT,\" WILL SAID THOUGHTFULLY As He WALKED AWAY FROM the council with Halt and Erak. \"What did you mean when you said that about archers?\"\n\nHalt looked sideways at his apprentice and sighed. \"It could make a big difference to the outcome,\" he said. \"The Temujai are archers themselves. But they rarely have to face an enemy with any particular skill with the bow.\"\n\nWill nodded. The longbow was traditionally an Araluen weapon. Perhaps because of the island kingdom's isolation from the countries on the eastern landmass, it had remained peculiar to Araluen. Other nationalities might use bows for hunting or even sport. But only in the armies of the Araluens would you find the massed groups of archers that could provide a devastating rain of arrows on an attacking force.\n\n\"They understand the value of the bow as a strategic weapon,\" he said. \"But they've never had to cope with facing it themselves. I got some inkling of that when Erak and I were running from them near the border. Once I'd put a few arrows close to them, they were decidedly reluctant to come dashing around any blind corners.\"\n\nThe jarl laughed quietly at the memory. \"That's true enough,\" he agreed. \"Once you'd emptied a few saddles, they slowed down remarkably.\"\n\n\"You know, I've been thinking . . . ,\" said the boy, and hesitated. Halt grinned quietly to himself.\n\n\"Always a dangerous pastime,\" he said gently.\n\nBut Will continued: \"Maybe we should try to put together a force of archers. Even a hundred or so could make a difference, couldn't they?\"\n\nHalt shook his head. \"We haven't the time, Will,\" he replied. \"They'll be on us within two weeks. You can't train archers in that short a time. After all, the Skandians have no skill with the bow to begin with. You'd have to teach them the very basics\u2014nocking, drawing, releasing. That takes weeks, as you know.\"\n\n\"There are plenty of slaves here,\" Will persisted. \"Some of them would know the basics. Then all we'd have to do is control their range.\"\n\nHalt looked at his apprentice again. The boy was deadly serious, he could see. A small frown creased Will's forehead as he thought through the problem.\n\n\"And how would you do that?\" the Ranger asked. The frown deepened for a few seconds as Will gathered his thoughts.\n\n\"It was something Evanlyn asked me that suggested it,\" he said. \"She was watching me shoot and she was asking how I knew how much elevation to give to a particular shot and I told her it was just experience. Then I thought maybe I _could_ show her and I was thinking, if you created\u2014say\u2014four basic positions. . .\"\n\nHe stopped walking and raised his left arm as if it were holding a bow, then moved it through four positions\u2014beginning horizontally and ultimately raising it to a maximum forty-five degree angle. \"One, two, three, four, like that,\" he continued. \"You could drill a group of archers to assume those positions while someone else judged the range and told them which one to go to. They wouldn't need to be very good shots as long as the person controlling them could judge range,\" he finished.\n\n\"And deflection,\" Halt said thoughtfully. \"If you knew that at the second position your shafts would travel, say, two hundred meters, you could time your release so that the approaching enemy would reach that spot just as the arrow storm did.\"\n\n\"Well, yes,\" Will admitted. \"I hadn't taken it that far. I was just thinking of setting the range and having everyone release at the same time. They needn't aim for individual targets. They could just fire away into the mass.\"\n\n\"You'd need to anticipate,\" Halt said.\n\n\"Yes. But essentially, it would be the same as if I were firing one arrow myself. It's just that, as I released, I could call a hundred others to do the same.\"\n\nHalt rubbed his beard. He glanced at the Skandian. \"What do you think, Erak?\"\n\nThe jarl merely shrugged his massive shoulders. \"I haven't understood a word you've been saying,\" he admitted cheerfully. \"Range, defraction . . .\"\n\n\"Deflection,\" Will corrected him, and Erak shrugged.\n\n\"Whatever. It's all a puzzle to me. But if the boy thinks it might be possible, well, I'd tend to think he might be right.\"\n\nWill grinned at the big war leader. Erak liked to keep things simple. If he didn't understand a subject, he didn't waste energy wondering about it.\n\n\"I tend to think the same way,\" Halt said quietly, and Will looked at him in surprise. He'd been waiting for his mentor to point out the fundamental flaw in his logic. Now he saw that Halt was considering his proposal seriously. Then he noticed the look of exasperation that grew on Halt's face as he found the flaw.\n\n\"Bows,\" the Ranger said, disappointment in his voice. \"Where would we find a hundred bows in time to let people train with them? There probably aren't twenty in all of Skandia.\"\n\nWill's heart sank. Of course. There was the problem. It took weeks to shape and craft a single longbow, trimming the bowstave just so, providing just the right amount of graduated flex along both arms. It was a craftsman's job and there was no way they would have time to make the hundred bows they would need. Disconsolately, he kicked at a rock in his path, then wished he hadn't. He'd forgotten that he was wearing soft-toed boots.\n\n\"I could let you have a hundred,\" Erak said in the depressed silence that followed Halt's statement. Both the others turned to look at him.\n\n\"Where would you find a hundred longbows?\" Halt asked him. Erak shrugged.\n\n\"I captured a two-masted cob off the Araluen coast three seasons ago,\" he told them. He didn't have to explain that when a Skandian said _season_ he meant the raiding season. \"She had a hold full of bows. I kept them in my storeroom until I could find a use for them. I was going to use them as fence palings,\" he continued. \"But they seemed a little too flexible for the job.\"\n\n\"Bows tend to be that way,\" Halt said slowly, and when Erak looked at him, uncomprehending, he added: \"More flexible than fence palings. It's one of the qualities we look for in a bow.\"\n\n\"Well, I suppose you'd know,\" Erak said casually. \"Anyway, I've still got them. There must be thousands of arrow shafts as well. I thought they'd come in handy one day.\"\n\nHalt reached up and laid a hand on the massive shoulder. \"And how right you were,\" he said. \"Thank the gods for the Skandian habit of hoarding everything.\"\n\n\"Well, of course we hoard,\" Erak explained. \"We risk our lives to take the stuff in the first place. There's no sense in throwing it away. Anyway, do you want to see if you could use them?\"\n\n\"Lead on, Jarl Erak,\" Halt said, shaking his head in wonder and lifting an eyebrow at Will.\n\nErak set out toward the large, barnlike storehouse by the docks where he kept the bulk of his plunder.\n\n\"Excellent,\" he said happily, rubbing his hands together. \"If you decide to use them, I'll be able to charge Ragnak.\"\n\n\"But this is war,\" Will protested. \"Surely you can't charge Ragnak for doing something that will help defend Hallasholm?\"\n\nErak turned his delighted smile on the young Ranger. \"To a Skandian, my boy, all war is business.\"\n**24**\n\nEVANLYN HAD BEEN WAITING FOR HALT AND WILL TO LEAVE Ragnak's War Council. As the two gray-cloaked figures, in company with the burly Jarl Erak, emerged from the Great Hall and walked across the open ground that fronted it, she started forward to intercept them. Then she stopped, uncertain how to proceed. She had been hoping that Will might come out by himself. She didn't want to approach him in front of Erak and Halt.\n\nEvanlyn was bored and miserable. Worse, she was feeling useless. There was nothing specific she could do to contribute to the defense of Hallasholm, nothing to keep her mind occupied. Will had obviously become part of the inner circle of the Skandian leadership, and even when he wasn't in meetings with Halt and Erak, he was off practicing with his bow. It sometimes seemed that he used his practice sessions to avoid her. She felt a little flare of anger as she recalled his reaction when she asked him to teach her to shoot. He'd laughed at her!\n\nHorace was no better. Initially, he'd been happy to keep her company. But then, seeing Will constantly practicing, he'd felt guilty and began spending time on the practice field himself, honing his own skills with a small group of Skandian warriors.\n\nIt was all Will's fault, she thought.\n\nNow, as she watched him talking with his old teacher, and saw the two of them stop as Will made a point, she realized with a sense of sadness that there was a part of Will's life from which she would always be excluded. Young as he was, he was already a part of the mysterious, close-knit Ranger clan. And Rangers, she had been told since she was a small child, kept themselves to themselves. Even her father the King had been frustrated from time to time by the closemouthed nature of the Ranger Corps. As the realization hit home, she turned sadly away, leaving the two Rangers, master and apprentice, to their discussion with the Skandian jarl.\n\nMorosely, she kicked at a stone on the ground in front of her. If only there were something for her to do!\n\nShe stood uncertainly, undecided about where to go next. She turned abruptly to see if Will and Halt were still where she'd last seen them. They had moved on, but her sudden turn brought her into unexpected eye contact with a familiar, though unwelcome, figure.\n\nSlagor, the thin-lipped, shifty-eyed wolfship captain whom she had first seen on the rocky, windswept island of Skorghijl, had just emerged from one of the smaller buildings that flanked Ragnak's Great Hall. He stood now, staring after her. There was something in his look that made her uncomfortable. Something knowing, something that boded ill for her. Then, as he realized she had seen him, he turned away, walking quickly into the dark-shadowed alleyway between the two buildings. She frowned to herself. There had been something suspicious about the Skandian's manner, she thought. Half because she wanted to know more, and half because she was bored, with nothing constructive to do, she set out after him.\n\nThere had been something in the way he looked at her that told her it might be better if he didn't know she was following him. She moved to the end of the alley and peered cautiously around, just catching sight of him as he turned right at the rear of the building. She paralleled his path, moving cautiously to the next alley, pausing, then peering around again. Once more, she caught a quick glimpse of Slagor and she guessed from his general direction that he was heading for the quays, where the wolfships docked. Realizing that her own actions might appear highly suspicious, she glanced quickly around to see if anyone might be watching her. Apparently not, she decided. Still, she crossed back to the far side of the street before following in the pursuit of the wolfship skirl.\n\nAs she slid unobtrusively from building to building, she saw him several more times, confirming her first impression that he was heading for the docks. That was logical. Presumably his ship was among the fleet moored there. Probably Slagor had some ship's business to attend to, she thought. The suspicious manner that she had noticed was probably nothing more than his normal shifty-eyed demeanor.\n\nThen she cast the doubts aside. There had been something else: something knowing. Something calculating.\n\nEvanlyn was, naturally, constantly aware of her precarious position here in Hallasholm. Ragnak might have no interest in punishing a recaptured slave. But if her real identity were to become known, his reaction was a foregone conclusion. He had vowed to kill any member of the Araluen royal family. Now it seemed important to her to find out what had been behind Slagor's look. She quickened her pace and hurried down one of the narrow connecting alleys, emerging in the broad waterfront thoroughfare that Slagor had taken.\n\nHe was twenty meters ahead of her as she peered cautiously around the end of the building. His back was turned and she realized that he had no idea that she had been following him. To the left, the masts of the moored wolfships formed a forest of bare poles, bobbing and swaying with the movement of the water. On the right of the street were a series of waterfront taverns. It was toward one of these that Slagor was hurrying now, she realized.\n\nSome instinct made her ease into a doorway as the skirl reached the tavern entrance. It was as well she did, for he turned and looked back the way he had come, apparently checking to see if anyone had followed him. She frowned to herself as she shrank into the shadows of the doorway. Why should Slagor be nervous, here in the middle of Hallasholm? Certainly he was one of the less popular wolfship captains, but it was unlikely that anyone would actually do him harm. There was obviously something going on, she thought, and she determined to get to the bottom of it. Close by, moored to one of the timber quays, she saw Slagor's ship, _Wolf Fang_. She recognized it by the distinctive carved figurehead. No two wolfships had the same figurehead and she remembered this one all too well from the day when _Wolf Fang_ had come limping into the anchorage at Skorghijl. With it had come the news of Ragnak's Vallasvow against her father and herself, so she had good reason to remember the grotesquely carved icon.\n\nFor a moment, she hesitated in the doorway. Then, the door behind her opened and two Skandian women emerged, shopping baskets in hand. They stared at the stranger on their doorstep and she hurriedly apologized and moved away. Behind her, she heard the angry comments of the women as they headed for the market square. She was too obvious here, she realized. Any moment, Slagor might emerge from the tavern and see her. She glanced uncertainly at the ship, then came to a decision and, moving at a half run, she made her way down the waterfront to the quay where _Wolf Fang_ was moored. It was reasonable to assume that Slagor might come here eventually, and then she might get an inkling of what he was up to.\n\nThere was an anchor watch aboard, of course. But it was just one man and he was at the stern, leaning on the bulwark and staring at the harbor and the sea beyond. Crouching below the level of the high prow, she approached the ship and vaulted lightly over the railing, her soft-shod feet making virtually no sound as she landed on the planks of the deck. She dropped immediately into the rowing well, set below the main deck, where the rowing crew would normally sit to wield their heavy, white oak oars. The area was deserted at the moment, and she was concealed from the sight of the solitary guard at the stern. But it was only a temporary hiding place and she looked now for a better one.\n\nRight at the prow of the ship was a small triangular space, screened by a canvas flap. It was large enough to accommodate her if she crouched, and she moved quickly into it now, letting the canvas screen fall back into place behind her. She found herself sitting on coils of stiff, coarse rope, and something hard jabbed into her side. Shifting to a better position, she realized that it had been the fluke of the anchor, and the coils of heavy rope were the anchor cable. With the ship moored alongside the quay, they weren't in use. This would be as good a hiding place as any, she thought. Then she wondered if she might not be wasting her time here. Odds were that Slagor had simply come this way to visit the tavern and that after he'd drunk his fill of the harsh spirits the Skandians favored, he'd probably head on back to his lodge.\n\nShe shrugged morosely. She had nothing better to do with her time. She might as well give it an hour or so and see if anything transpired. What that _anything_ might be, she really had no idea. She'd followed Slagor on an impulse. Now, following the same impulse, she was crouched here, waiting to see what she might overhear if and when he came aboard.\n\nIt was warm in the confines of the forepeak and, once she'd moved a few of the coils, the rope made a relatively comfortable resting place. She wriggled herself into a better position and rested her chin on her elbows, peering through a small gap in the canvas to see if anything was happening outside. She felt the footsteps of the sentry as he crossed to the landward side of the ship, giving up his scrutiny of the harbor, and heard him call to someone on the shore. There was an answering voice but the words were too muffled for her to make out. Probably just a casual greeting to a passing friend, she reasoned. She yawned. The warmth was making her drowsy. She hadn't slept well the night before, thinking about Will and how their friendship seemed to be eroding with every passing day. She tried to dislike Halt, blaming him for the sudden estrangement between them. But she couldn't. She liked the small, roughly bearded Ranger. There was a dry sense of humor about him that appealed to her. And after all, he had rescued her from the Temujai reconnaissance party. She sighed. It wasn't Halt's fault. Nor Will's. It was just the way things were, she guessed. Rangers were different to other people. Even princesses.\n\n_Especially_ princesses.\n\nShe woke suddenly, thinking she was falling. She hadn't realized that she'd drifted off to sleep, lying here on the coils of rope. But she knew what had woken her. The deck beneath her had dropped suddenly as _Wolf Fang_ heaved herself into a short head sea. Now she could hear the creak and thump of the oars in their rowlocks and she realized, with a terrible sinking feeling, that _Wolf Fang_ had put to sea and she was trapped on board.\n**25**\n\nBETWEEN THEM, HALT AND WILL HAD FOUND A HUNDRED SLAVES who claimed to have some level of skill with the bow. Finding them had been one matter. Convincing them that they should volunteer to help defend Hallasholm was something else.\n\nAs a burly Teutlander forester, who seemed to have assumed the role of spokesman for them, told the two Rangers, \"Why should we help the Skandians? They've done nothing except enslave us, beat us and give us too little food to eat.\"\n\nHalt eyed the man's ample girth speculatively. If some of the slaves were underfed, this one could hardly claim to be one of them, he thought. Still, he decided to let that matter pass.\n\n\"You might find it more agreeable to be a slave of the Skandians than to fall into the hands of the Temujai,\" he told them bluntly.\n\nAnother of the assembled men spoke up. This one was a southern Gallican and his outlandish accent made his words almost indecipherable. Will finally pieced the sounds together in sufficient order to know that the man had asked: \"What do the Temujai do with their slaves?\"\n\nHalt turned a steely gaze on the Gall. \"They don't keep slaves,\" he said evenly, and a buzz of expectation ran through the assembled men. The big Teutlander stepped forward again, grinning.\n\n\"Then why would you expect us to fight against them?\" he asked. \"If they beat the Skandians, they'll set us free.\"\n\nThere was a loud mumble of consent among the others behind him. Halt held up a hand and waited patiently. Eventually, the hubbub died away and the slaves looked at him expectantly, wondering what further inducement he could offer them\u2014what he would consider to be more attractive to them than the prospect of freedom.\n\n\"I said,\" he intoned clearly, so that everyone could hear him, \"they don't keep slaves. I didn't say they set them free.\" He paused, then added, with a slight shrug of his shoulders, \"Although the religious ones among you may consider death to be the ultimate freedom.\"\n\nThis time, the commotion among the slaves was even louder. Finally, the self-appointed spokesman stepped forward again and asked, with a little less assertion, \"What do you mean, Araluen? Death?\"\n\nHalt made a careless gesture. \"The usual, I suppose: the sudden cessation of life. The end of it all. Departure for a happier place. Or oblivion, depending upon your personal beliefs.\"\n\nAgain a buzz ran through the crowd. The Teutlander studied Halt closely, trying to see some indication that the Ranger was bluffing.\n\n\"But . . .\" He hesitated, not sure whether to ask the next question, not sure that he wanted to know the answer. Then, urged by his companions, he went on: \"Why should these Temujai want to kill us? We've done nothing to them.\"\n\n\"The truth of the matter is,\" Halt told them all, \"you _mean_ nothing to them either. The Temujai consider themselves a superior race. They'd kill you out of hand because you can do nothing for them\u2014but left behind their backs, you could constitute a threat.\"\n\nA nervous silence settled over the crowd now. Halt let them digest what he had said, then he spoke again.\n\n\"Believe me, I've seen what these people are like.\" He looked into the faces of the crowd. \"I can see there are some Araluens among you. I'll give you my word as a Ranger that I'm not bluffing. Your best chance of survival is to fight with the Skandians against these Temujai. I'll leave you for half an hour to consider what I've said. You Araluens might tell the others what a Ranger's word means,\" he added. Then, beckoning for Will to follow, he turned on his heel and walked some distance away, out of earshot.\n\n\"We're going to have to offer them more,\" he said when the others couldn't hear him. \"Reluctant recruits will be almost useless to us. A man's got to have something worth fighting for if he's going to do his best. And that's what we're going to need from this bunch\u2014their best effort.\"\n\n\"So what are you going to do?\" Will asked, almost jogging to keep pace with his teacher's urgent stride.\n\n\"We're going to see Ragnak,\" Halt told him. \"He's going to have to promise to free every slave who fights for Hallasholm.\"\n\nWill shook his head doubtfully. \"He won't like that,\" he said. Halt turned and looked at him, a faint grin touching the corner of his mouth.\n\n\"He'll hate it,\" he agreed.\n\n\"Freedom?\" Ragnak exploded. \"Give them their freedom? A hundred slaves?\"\n\nHalt shrugged disdainfully. \"Probably closer to three hundred,\" he replied. \"A lot of them will have women and children they'll want to take with them.\"\n\nThe Oberjarl gave an enormous snort of incredulous laughter. \"Are you mad?\" he asked the Ranger. \"If I give three hundred slaves their freedom, we'll have virtually no slaves left. What will I do then?\"\n\n\"If you don't, you may find you have no country left,\" Halt replied. \"As to what you would do next, you could try paying them. Make them servants instead of slaves.\"\n\n\"Pay them? To do the work they're doing now?\" Ragnak spluttered indignantly.\n\n\"Why not? The gods know you can afford it well enough. And you might find they do a better job if they've got something more than a beating to look forward to at the end of the day.\"\n\n\"To hell with them!\" Ragnak said. \"And to hell with you, Ranger. I agreed to listen to you, but this is ridiculous. You'll turn me into a beggar if I let you have your way. First you want me to abandon Hallasholm to this rabble of horsemen. Now you want me to send all my slaves off back to where they came from. To hell with you, I say.\"\n\nHe glared at the Ranger for a few seconds, then, with a contemptuous wave of his hand, he turned away, refusing even to make eye contact. Halt waited a few seconds, then spoke to Erak, who was standing by his Oberjarl, an uncomfortable look on his face.\n\n\"I'm telling you, we need these men,\" he said forcefully. \"Even with them, we can still lose. But with them fighting willingly for us, we'll have a chance.\" He jerked a thumb in the direction of the Oberjarl. \"Tell him,\" he said finally, then turned on his heel and left the council room, Will hurrying behind him as he went.\n\nAs they left the hall, Halt said, almost to himself, but loud enough for Will to hear, \"I wonder if it occurs to them that if the slaves agree unwillingly to fight for them, and if, by some mad mischance, we do win, there's nothing to stop the slaves from turning their weapons on the Skandians.\" That thought had occurred to Will. He nodded agreement. \"That's why,\" Halt continued, \"we've got to give them something worth fighting for.\"\n\nThey waited at the training field for over an hour. The slaves had come to a decision, agreeing to fight against the Temujai. However, a few shifty eyes among the group told Halt and Will that, once the battle was over, the newly armed men were not going to return meekly into slavery.\n\nThere was a buzz of expectation as Erak arrived. He walked up to Halt and Will, who were standing a little apart from the archers.\n\n\"Ragnak agrees,\" he said quietly. \"If they fight, he'll free them.\"\n\nHalt nodded his head gratefully. He knew where the real impetus for Ragnak's decision had come from.\n\n\"Thank you,\" he said simply to Erak. The Skandian shrugged and Halt turned to Will. \"They'll be your men. They need to get used to taking orders from you. You tell them.\"\n\nWill hesitated, surprised. He had assumed that Halt would do the talking. Then, at an encouraging nod from his master, he stepped forward, raising his voice.\n\n\"Men!\" he called, and the low murmur of conversation among the group died instantly. He waited a second or two to make sure he had their full attention, then continued.\n\n\"Ragnak has decided. If you fight for Skandia, he'll set you free.\"\n\nThere was a moment of stunned silence. Some of these men had been slaves for ten years or more. Now, here was this slightly built youth telling them that the end to their suffering was in sight. Then a mighty roar of triumph and jubilation swept through them, at first wordless and inchoate, but rapidly settling into a rhythmic chant of one word from one hundred throats:\n\n\"Free-dom! Free-dom! Free-dom!\"\n\nWill let them celebrate for a while longer. Then he climbed onto a tree stump where he could be seen by all of them and waved his arms for silence. Gradually, the chant died away and they crowded closer around him, eager to hear what else he had to tell them.\n\n\"That's all very well,\" he said when they had quieted down. \"But first, there's the small matter of beating the Temujai. Let's get to work.\"\n\nHalt and Erak watched as Will supervised the issuing of arrows to the men. Unconsciously, both men nodded their approval of the boy. Then Erak turned to Halt.\n\n\"I nearly forgot, Ragnak had a further message for you. He said if we lose this battle and he loses his slaves as well, he's going to kill you for it,\" he said cheerfully.\n\nHalt smiled grimly. \"If we lose this battle, he may have to get in line to do it. There'll be a few thousand Temujai cavalrymen in front of him.\"\n**26**\n\nWILL CALLED THE LAST GROUP OF TEN MEN FORWARD TO THE firing line. The preceding group moved to the rear of the waiting ranks and sat down to watch. He was working the men in small groups at this stage. That gave him a manageable group to work with as he tested their ability to follow his orders and shoot at a predetermined elevation.\n\n\"Ready!\" he called. Each man took an arrow from the bin in front of him and nocked it to the string. They stood ready, their heads turned toward him, waiting for his next order.\n\n\"Remember,\" he said, \"don't try to judge the shot yourself. Just go to the position I call, make a full draw and a smooth release when I call it.\"\n\nThe men nodded. Initially, they hadn't liked the idea of having their shooting controlled by someone as young as Will. Then, after Halt had encouraged his apprentice to give a demonstration of highspeed pinpoint shooting, they had reluctantly agreed to the system Will had devised.\n\nWill took a deep breath, then called firmly: \"Position three! Draw!\"\n\nTen arms holding bows rose to a position approximately forty degrees from the horizontal. Will quickly glanced down the line to see that each man had remembered the correct position. He'd been drilling the four different elevations into them all day. Satisfied, and before the strain of holding the bows at full draw became too great, he called:\n\n\"Shoot!\"\n\nAlmost as one, there was a rapid slither of released bowstrings and a concerted hiss of arrows arcing through the air.\n\nWill watched the small flight of shafts as they arced upward, then nosed over and plunged down to bury themselves up to half their length in the turf. Again he called to the waiting line of men: \"Position three, ready!\"\n\nAs before, the ten men nocked arrows to the strings, waiting for Will's next call.\n\n\"Draw . . . shoot!\"\n\nAgain there was the slithering slap of released bowstrings hitting the archers' arm guards, and the sound of the wooden shafts scraping past the bows as they were hurled into the air. This time, as the arrows came down, Will changed his command.\n\n\"Position two . . . ready!\"\n\nThe line of left arms holding the bows extended and tilted up to a thirty-degree angle.\n\n\"Draw . . . shoot!\"\n\nAnd another ten-shaft volley was on its way. Will nodded to the ten men, who were watching him expectantly.\n\n\"All right,\" he said. \"Let's see how you did.\"\n\nHe began to pace across the open field, followed by the ten men who had just shot. There were markers set out down the middle of the field, marking 100, 150 and 200 meter distances. Position three, with the bow arm elevated forty degrees from the horizontal, should have equated to the 150 meter marker. As they approached that marker, Will nodded with satisfaction. There were sixteen arrows slanting up from the turf within a ten-meter tolerance of the mark. Two had gone long, he noticed, and two more had dropped short. He studied the long shots. The shafts were numbered so that he could assess how each member of the shooting line had performed. He saw now that the two overshoots belonged to two different archers.\n\nMoving back to the arrows that had undershot the target, he frowned slightly. The arrows were both marked with the same number. That meant the same archer had dropped his shot short of the mark both times. Will took note of the number, then moved back to view the results of the final volley. The frown deepened as he saw that nine arrows were well grouped, with one falling short by the same margin. He didn't really need to check, but a quick glance showed him that, once again, the same archer had undershot the distance.\n\nHe grunted thoughtfully.\n\n\"All right!\" he called. \"Recover your arrows.\" Then he led the way back to the firing point, the ten men following behind him.\n\n\"Who was at number four position?\" he asked.\n\nOne of the archers stepped forward, hesitantly holding up a hand and looking like a nervous pupil in school. He was a heavyset bearded man of about forty, Will noticed, yet his demeanor showed that he was totally in awe of the young Ranger facing him.\n\n\"That was me, your honor,\" he said. Will beckoned him closer.\n\n\"Bring your bow and two or three arrows,\" he said. The man picked up his bow, and selected two arrows from the bin that stood by his firing position. He was nervous at being singled out and promptly dropped the arrows, scrabbling awkwardly to retrieve them.\n\n\"Relax,\" Will told him. \"I just want to check your technique.\"\n\nThe man tried to smile in return. He'd seen they were his arrows that had fallen short and he assumed he was about to be punished. That was the way life went for a slave in Hallasholm. If you were told to do something and you didn't do it, you were punished. Now the brown-haired youth who was directing the session was grinning at him and telling him to relax. It was a novel experience.\n\n\"Take a stance,\" Will told him, and the man stood side-on to the firing range, left foot extended, left hand holding the bow at waist height.\n\n\"Position three,\" Will said quietly, and the man assumed the position that had been drilled into him all the previous day, his left arm holding the bow at forty degrees\u2014almost maximum distance. Will studied him. There seemed to be little wrong with the man's stance.\n\n\"All right,\" he said. \"Draw, please.\"\n\nThe man was using too much arm muscle and not enough of his back muscles to draw the bow, Will thought. But that was a minor fault and the result of long habit. There would be no way of changing that in the time they had left.\n\n\"And . . . shoot.\"\n\nThere it was, Will thought. A fraction of a second before the man released his shot, he relaxed the draw length slightly\u2014letting the arrow ease down a little before actually letting his fingers slip from the string. That meant that at the moment of release, the arrow was at something less than full draw, which in turn meant it was receiving less than the full power of the bow behind its flight. Halt and Will had tested all the bows to make sure they were similar in draw weight and the arrows were all exactly the same length to ensure results were as consistent as possible. The main cause for variation would be little technical errors like this one.\n\nHe looked down the range to where the colored flights of the arrow were visible against the brown, sodden grass of the spring thaw. As he had suspected, it was short again.\n\nWill explained the reason for the problem to the man, seeing from the surprised expression that he had no idea that he was relaxing the draw at the crucial moment.\n\n\"Work on it,\" he told him, giving him an encouraging slap on the shoulder. Halt had impressed on him the fact that a little encouragement in matters like these went a great deal further than scathing criticism. Will had been surprised when Halt had put him in charge of the archers' training. Even though he knew he'd be directing the archers during the battle, he'd assumed that Halt would supervise their training. But the Ranger had repeated his earlier sentiment.\n\n\"You're the one who'll be directing them once we're fighting. It's as well they get used to following your orders from the start.\"\n\nWill remembered another piece of advice the Ranger had given him. \"Men work better when they know what you have in mind,\" he told the young apprentice. \"So make sure you tell them as much as possible.\"\n\nHe stepped up onto a raised platform that had been placed here for the purpose of addressing the entire group.\n\n\"We'll break for today,\" he said in a raised voice. \"Tomorrow we'll shoot as one group. So if I've picked any technical faults in your shooting today, practice getting rid of them before the evening meal. Then get a good night's rest.\" He started to turn away, then turned back, remembering one thing more. \"Good work, all of you,\" he said. \"If you keep this up, we're going to give those Temujai a very nasty surprise.\"\n\nA growl of pleasure rose from the hundred men. Then they broke off, heading back for the warmth of the halls and lodges. Will realized that it was later than he'd thought. The sun was touching the tops of the hills beyond Hallasholm and the shadows were lengthening. The evening breeze was chilly and he shivered, reaching for the cloak that he'd hung from the platform railing as he'd directed the shooting.\n\nA half dozen boys had been assigned to help and without orders from him they gathered the arrow bins and arrows, putting them under cover in one of the store sheds that fronted the practice field. Will couldn't help noticing the admiring glances they cast his way as they went about their work. He was only a few years older than they were, yet here he was, directing a force of one hundred archers. He smiled to himself. He wouldn't have been human if he hadn't enjoyed their hero worship.\n\n\"You look pleased with yourself,\" said a familiar voice. He turned and realized Horace must have approached while he had been talking to the men. He shrugged, trying to act diffident.\n\n\"They're coming along quite well,\" he said. \"It's been a good day's work.\"\n\nHorace nodded. \"So I noticed,\" he said. Then, in a worried tone, he continued, \"Evanlyn hasn't been here with you, has she?\"\n\nWill looked up at him, instantly on the defensive. \"What if she has been?\" he asked, an argumentative tone creeping into his voice. Instantly, he saw the worried look clear from Horace's face and realized he'd misinterpreted the reason for his friend's question.\n\n\"Then she has been here?\" Horace said. \"That's a relief. Where is she now?\"\n\nNow it was Will's turn to frown. \"Just a moment,\" he said, putting a hand on Horace's muscular forearm. \"Why is it a relief? Is something wrong?\"\n\n\"Then she hasn't been here?\" Horace asked, and his face fell again as Will shook his head.\n\n\"No. I thought you were being . . . you know . . .\" Will had been about to say _jealous,_ but he couldn't quite manage it. The idea that Horace might have something to be jealous about had too much of a sense of boasting about it. He saw instantly that such thoughts were far from Horace's mind. The apprentice warrior had hardly seemed to notice Will's hesitation.\n\n\"She's missing,\" he said, in that same worried tone. He cast his hands out and looked around the empty practice field, as if he somehow expected to see her appear there. \"Nobody's seen her since midmorning yesterday. I've looked everywhere for her, but there's no sign.\"\n\n\"Missing?\" Will repeated, not quite understanding. \"Missing where?\"\n\nHorace looked up at him with a sudden flare of asperity. \"If we knew that, she wouldn't be missing, would she?\"\n\nWill put up his hands in a peacemaking gesture.\n\n\"You're right!\" he said. \"I didn't realize. I've been a little tied up trying to get these archers organized. Surely somebody must have seen her last night. Her room servants, for example?\"\n\nHorace shook his head miserably. \"I've asked them,\" he said. \"I was out on patrol most of yesterday myself, keeping an eye on the Temujai approach. We didn't get back in to Hallasholm till well after supper time, so I didn't realize she wasn't around. It was only this morning when I went to find her that I found out she hadn't been in her room last night and that nobody had seen her today. That's why I was hoping that maybe you'd . . . \" The sentence tailed off and Will shook his head.\n\n\"I haven't seen hide nor hair of her,\" he told his friend. \"But it's ridiculous!\" he exclaimed after a short silence. \"Hallasholm isn't a big enough place for someone to go missing. And there's nowhere else she could have gone. Let's face it, she can't have simply disappeared . . . can she?\"\n\nHorace shrugged. \"That's what I keep telling myself,\" he said morosely. \"But somehow, it looks as if she has.\"\n**27**\n\nUNITED NOW IN THEIR CONCERN FOR EVANLYN, THE TWO apprentices headed for Halt's quarters. All of the Araluen party had been assigned rooms in the main hall. As Halt was their leader, he had been given a small suite of three rooms. At the door, Will knocked perfunctorily and heard Halt's gruff reply: \"Come.\"\n\nAs they entered, he took in the fact that Erak was in the room with Halt. It was hard to miss the bulky Skandian. He seemed to fill most spaces he occupied. He was sprawled in one of the comfortable, carved wood armchairs that decorated the room\u2014doubtless liberated on some wolfship raid down the coast. Halt was standing by the window, framed against the low-angled light of the late afternoon. He looked quizzically at the doorway as the two boys entered hurriedly.\n\n\"Halt,\" Will began urgently, \"Horace says Evanlyn's disappeared. She's\u2014\"\n\n\"Safe and sound and back in Hallasholm.\" A familiar voice finished the sentence for him. Both boys turned to the speaker. Standing a little back, in the shadows of the room, she hadn't been evident as they'd entered.\n\n\"Evanlyn!\" Horace exclaimed. \"You're all right!\"\n\nThe girl smiled. Now that his eyes were accustomed to the darker part of the room, Will could make out that her face and clothes were smeared with grease and dirt. Her eyes met his and she smiled at him, a little wistfully. Then she upended the flask of juice that she had in her hand and drank greedily from it.\n\n\"Apparently,\" she said, setting the flask down. \"Although I have a thirst on me that I doubt I'll ever quench. All I've had to drink in the last eighteen hours was a little rainwater that made its way through the canvas covers over the . . .\" She hesitated and looked to Erak to supply the word she was after. The jarl obliged.\n\n\"Forepeak,\" he said, and Evanlyn repeated the word.\n\n\"Forepeak, exactly, of Slagor's ship,\" she said. Will and Horace exchanged puzzled glances.\n\n\"What in the devil's name were you doing there?\" Will asked. Halt answered for her.\n\n\"The devil's name is right,\" he said. \"It seems our friend Slagor has sold out to the Temujai\u2014and he's planning to betray Hallasholm to them.\"\n\n\"What?\" asked Will, his voice cracking with surprise. He looked at Evanlyn. \"How do you know?\"\n\nThe girl shrugged her slim shoulders. \"Because I heard him discussing it with the Temujai leader. They were barely two meters away from me.\"\n\n\"It seems,\" Halt put in, by way of explanation, \"that your old friend Slagor sailed down the coast yesterday to a rendezvous with the Temujai Shan\u2014one Haz'kam. And since our traitor obviously didn't trust his new allies too far, he insisted on all negotiations being carried out on board his ship\u2014just to keep Haz'kam's retainers at a distance.\"\n\n\"Which is how I came to hear it,\" Evanlyn finished. But now Horace was scratching his head in bewilderment.\n\n\"But . . . what were you doing on the ship?\" he said.\n\n\"I told you,\" Evanlyn replied. \"Eavesdropping on Slagor and the Temujai.\"\n\nHorace made an impatient gesture. \"Yes, yes, so you've said. But why were you there in the first place?\"\n\nEvanlyn went to answer, hesitated, then stopped altogether. All eyes in the room were on her now and she realized she didn't really have a logical answer to that question.\n\n\"I . . . don't know,\" she said finally. \"I was bored, I guess. And feeling useless. I was looking for something to do. And besides, Slagor looked sort of . . . shifty.\"\n\n\"Slagor always looks sort of shifty,\" Erak put in, helping himself to fruit from a bowl on the table in front of him. Evanlyn thought about it, then conceded the point.\n\n\"Well, that's true, I suppose. But he looked even shiftier than usual,\" she said. \"So I decided someone had better keep an eye on him and see what he was up to.\"\n\nTruth be told, Evanlyn was quite enjoying herself now. She had gone from feeling useless and unnecessary to being the bearer of important, even vital news to Halt and Erak. She couldn't help preening, just a little. Horace's next reaction was exactly what she'd hoped for.\n\n\"But . . . you could have been spotted! What if they'd found you there? They would have killed you,\" he said, his concern for her evident in the worried tone of his voice.\n\nThat thought had occurred to Evanlyn on more than one occasion as she'd crouched in the damp space in the bow of the wolfship. Once she had fully realized the situation she was in, her skin had crawled with the fear of discovery with every second. But now she affected a nonchalant air about the entire episode.\n\n\"I suppose so. But let's face it, someone had to do it.\"\n\nShe was delighted to notice that Horace was looking at her with something approaching awe. She glanced quickly at Will, hoping to see the same look of admiration there. His next words dashed that hope.\n\n\"All very well,\" he said dismissively. \"But the important thing is that Slagor is planning to betray us. How is he aiming to do it?\"\n\n\"That's the point, of course,\" Halt agreed. He indicated a chart of the Skandian coast that he and Erak had spread on the table between them. \"Apparently, friend Slagor plans to put to sea quietly the day after tomorrow and make for the same rendezvous point down the coast. Only this time, there'll be one hundred and fifty Temujai warriors waiting. He'll take them aboard and ferry them back here to Hallasholm\u2014\"\n\n\"He'll never fit a hundred and fifty men into one wolfship!\" Will interrupted.\n\nHalt nodded. \"Apparently, he has another two ships waiting for him out behind this island, halfway to the rendezvous.\"\n\n\"They left a week ago,\" Erak put in. \"Supposedly, they were going to raid behind the Temujai lines. It seems the skirls are in league with Slagor and they're waiting at this prearranged point.\" He tapped the map with his dagger, with which he'd been peeling fruit. A few spots of apple juice fell onto the parchment. Halt raised an eyebrow at him and wiped them away as the Jarl continued. \"With three ships, they'll carry one hundred and fifty men easily.\"\n\n\"Then what?\" Horace asked. Evanlyn, piqued that attention had been diverted from her and that Will had ignored the danger she'd been in, leapt back into the conversation.\n\n\"They'll be able to attack our forces from the rear,\" she explained. \"Think of it, one hundred and fifty men, with the element of surprise, suddenly appearing behind our lines!\"\n\n\"That could be very nasty indeed,\" Horace said thoughtfully. \"So what do we do?\"\n\n\"We've already taken the first step,\" Erak told him. \"I've sent Svengal with two of my ships out to Fallkork Island here.\" Again he tapped the juice-stained knife on the map and again Halt raised his eyes at him. \"To make sure Slagor's other two ships don't keep any rendezvous.\"\n\n\"Two against two?\" Will asked. \"Is that enough?\"\n\nThe jarl cocked his head to one side and smiled at him. \"Count yourself lucky that Svengal wasn't here to hear you say that,\" he replied. \"He'd consider his crew alone to be more than a match for two ships full of Slagor's followers. But in fact, Slagor's ships will have only rowing crews. They need all the space they have to cram those Temujai on board with them.\"\n\n\"But what do we do about Slagor?\" Will asked, and this time it was Halt who answered.\n\n\"That's the problem. If he gets wind that we know what he's up to, he'll simply abandon the plan. We'll be able to prove nothing. It'll be his word against the word of a former slave\u2014and an escaped one at that.\" He smiled at Evanlyn to show he meant no insult, but was merely stating the facts. She nodded her understanding.\n\n\"But if Slagor finds the other two ships at this island, surely that's proof?\" Horace interjected. Halt shook his head.\n\n\"Proof of what? The crews will hardly admit they were waiting to go fetch the Temujai,\" he said.\n\nHorace sat back, frowning. This was getting too complicated for him.\n\n\"Then what can we do?\" Will asked. But at that moment there was a heavy knock at the door. They all looked at each other in surprise. The clandestine nature of their discussion had made them speak in lowered tones and the sudden interruption had made them all start guiltily, as if discovered.\n\n\"Anyone expecting visitors?\" Halt asked, and as the others shook their heads, he called once more: \"Come.\"\n\nThe door opened to admit Hodak, one of Erak's younger followers. He glanced about the room, noting the identities of all present. He looked uncomfortable as he noticed Evanlyn.\n\n\"Thought I might find you here,\" he said to Erak. \"Ragnak's calling a special council in the Great Hall. He wants you there, Jarl.\" He indicated Evanlyn. \"And you'd better bring the girl with you.\"\n\n\"Evanlyn? Why should she go?\" Halt asked. He saw the girl shrink back from the young Skandian. Maybe she had some premonition of what was to come.\n\n\"The council's about her,\" Hodak said awkwardly. \"Slagor has invoked Ragnak's Vallasvow. He says the girl is really Princess Cassandra, daughter of King Duncan.\"\n**28**\n\n\"BRING HER FORWARD!\" RAGNAK'S MASSIVE VOICE, USED TO dominating the howling gales of the Stormwhite, boomed painfully in the low-ceilinged Hall. Evanlyn shrank back instinctively, then recovered as Halt touched her arm and met her eyes with a reassuring smile. She straightened her shoulders and drew herself up to full height. Will watched in admiration as she walked down the cleared space in the center of the hall. Halt, Erak and the two apprentices followed close behind her. Horace, Will noticed, was continually easing his sword in its scabbard, lifting it to free the blade, then allowing it to drop back again. Will's own hand strayed to the hilt of his throwing knife. If things went as badly as they all feared, he decided that knife was for Slagor, who was standing beside and slightly behind Ragnak. Once before, on Skorghijl, Will had demonstrated his skill with the knife to Erak's and Slagor's crews, throwing it across the room and skewering a small wooden keg next to Slagor's hand. This time, there would be no keg.\n\nThe room watched in utter silence as Evanlyn stopped before Ragnak's raised dais.\n\nShe met the Oberjarl's glower with a calm, composed expression on her face. Again, Will found himself almost overwhelmed by her courage and her composure. Slagor signaled to a pair of attendants by a side door.\n\n\"Bring in the slave,\" he called. His voice was soft and silky, totally unlike Ragnak's forceful bellow. He sounded very pleased with the current turn of events, Will thought. The two men, rowers from Slagor's crew, opened the door and dragged in a protesting, weeping figure. She was a middle-aged woman, her hair graying and her face lined before its time with the strain of unending labor, poor food and the threat of constant punishment that was the lot of a slave in Hallasholm. The sailors dragged her forward and cast her down on the floor in front of Evanlyn. She crouched there miserably, her eyes down.\n\n\"Look up, slave,\" Slagor told her in that same quiet voice. Her sobbing continued and she shook her head, her eyes still cast down at the floor. Slagor moved quickly, stepping down from the platform and drawing his saxe knife in one smooth movement. He held the razor-sharp blade below the woman's chin, pressing it into the flesh of her neck with not quite sufficient force to break the skin.\n\n\"I said, 'look up,'\" he repeated, and applied pressure to the knife to raise her eyes until she was gazing at Evanlyn. As she saw the girl, the woman began sobbing even louder.\n\n\"Shut up,\" Slagor told her. \"Shut up that noise and tell the Oberjarl what you told me.\"\n\nThere were angry welts across the woman's face. Obviously, she had been recently beaten. Her ragged shift was torn in several spots as well, and more red marks were visible on her body through the gaps. In some places, blood had soaked through the thin material. Her tear-filled eyes pleaded with Evanlyn.\n\n\"I'm sorry, my lady,\" she said, her voice breaking. \"They beat me until I told.\"\n\nEvanlyn took an involuntary step toward her. But Slagor's knife swung up and around to confront her and stop her from coming closer. Beside him, Will heard Horace's quick intake of breath and saw his hand fall to the sword hilt once more. He placed his own hand over Horace's, stopping him from drawing the sword. The heavily built apprentice looked at him, surprised. Will shook his head slightly. He realized that Horace's movement had been a reflex reaction and he knew that in this tinderbox atmosphere, if his friend ever drew that sword it could mean the end of all of them.\n\n\"Not yet.\" He mouthed the words. If the time came, he was willing to join Horace in an attack on Slagor and Ragnak. But first, he thought, they should see if Halt couldn't talk their way out of this situation.\n\n\"Leave the talking to me,\" the Ranger had told them before they left his apartment. \"And don't do anything until I tell you. Clear?\"\n\nThe two boys had nodded. Then Halt had added: \"This puts an altogether different slant on our accusing Slagor, of course.\"\n\n\"But surely you're still going to tell Ragnak?\" Will had burst out. Halt shook his head doubtfully.\n\n\"The problem is, he's got in first. If we make a counter accusation now, it will look as if we're simply doing it to save Evanlyn. Chances are, Ragnak will ignore it altogether.\"\n\n\"But you can't let him get away with . . . ,\" Will began, but Halt held up a hand to silence him.\n\n\"I'm not letting him get away with anything,\" he reassured them. \"We'll just have to pick the right time to bring the matter up, that's all.\"\n\nNow Slagor turned back to the woman on the floor. \"Tell the Oberjarl,\" he repeated.\n\nThe woman said nothing and Slagor turned to Ragnak in exasperation. \"My head slave overheard her talking to some of the others,\" he explained. \"She's Araluen originally and she said she recognized this girl here\"\u2014he jerked a thumb in Evanlyn's direction\u2014\"as the Princess Cassandra\u2014Duncan's daughter.\"\n\nRagnak's eyes narrowed and he turned slightly to inspect Evanlyn. Her chin went up and she stood a little taller under his gaze.\n\n\"She does have something of the look of Duncan about her,\" he said suspiciously.\n\n\"No! No! I was mistaken!\" the slave burst out suddenly. On her knees, she stretched her hands out to Slagor in supplication. \"Now I see her close to, I realize I was wrong, Lord Slagor. I was mistaken!\"\n\n\"You called her 'my lady,'\" Slagor reminded her.\n\n\"It was a mistake, that was all. A mistake. Now I see her properly, I can tell it's not her,\" the woman insisted.\n\nSlagor regarded her with a pained expression on his face. He turned to Ragnak again. \"She's lying, Oberjarl,\" he said. \"I'll have my men beat the truth out of her.\"\n\nHe made a signal to the two men again and one of them came forward, uncoiling a short, thick whip as he came. The woman cringed away from him.\n\n\"No! Please, my lord, please!\" Her voice was shrill with fear as she tried to crawl away. Slagor's man grabbed a handful of her hair to stop her and she cried out again, in pain as well as fear. He raised the vicious-looking whip over his head, ready to bring it down.\n\n\"Leave her alone!\" Evanlyn cried, and her voice froze the sailor where he stood. He looked uncertainly to Slagor for direction, but the wolfship captain was watching Evanlyn, waiting for her to say more.\n\n\"All right,\" she said quietly, \"There's no need to torture her further. I'm Cassandra.\"\n\nThe silence in the room was almost a physical force. Then an excited buzz broke out among the assembled crowd. Will distinctly heard the word _Vallasvow_ from several different sources.\n\n\"Silence!\" roared Ragnak, and instantly the noise ceased. He rose and moved forward to confront Evanlyn, glaring down at her. \"You are Duncan's daughter?\"\n\nShe hesitated, then replied.\n\n\"I am King Duncan's daughter,\" she said, with a slight emphasis on his title. \"Cassandra, Princess of Araluen.\"\n\n\"Then you are my enemy,\" he said, spitting the words out. \"And I've sworn that you should die.\"\n\nErak stepped forward. \"And I've sworn that she will be safe here, Oberjarl,\" he said. \"I gave my word when I asked the Ranger to help us.\"\n\nRagnak looked up angrily. Again there was a buzz of conversation through the room. Erak was a popular jarl among the Skandians and Ragnak hadn't reckoned on having to contend with him over this matter. With an invading army only days away from his stronghold, he knew he couldn't afford a split with his senior war leader.\n\n\"I am Oberjarl,\" he said. \"My vow is of greater importance.\"\n\nErak folded his arms across his chest. \"Not to me it isn't,\" he said, and there was a chorus of agreement from the crowd.\n\n\"Erak cannot defy you like this! You are Oberjarl!\" Slagor suddenly interjected. \"Have him imprisoned! He is defying your vow to the Vallas!\"\n\n\"Shut up, Slagor,\" Erak told him in an ominously calm voice. Then he readdressed himself to Ragnak. \"I didn't ask you to take your death vow, Ragnak,\" he said. \"But if you want to carry it out, I'm afraid you'll have to go through me to do it.\"\n\nNow Ragnak stepped down from his podium and walked closer to where Erak stood. They were of equal height, both massively built. He faced his old companion, the anger burning in his eyes.\n\n\"Erak, did you know? Did you know who she was when you brought her here?\"\n\nErak shook his head.\n\nSlagor snorted in disgust. \"Of course he knew!\" he cried, then stopped suddenly as the point of Erak's dagger appeared under his nose.\n\n\"I'll allow that once,\" Erak told him. \"Say it again and you're a dead man.\"\n\nWordlessly, Slagor backed away from the bigger man, putting a safe distance between himself and the point of the knife. Erak sheathed the dagger and turned back to Ragnak. \"I didn't know,\" he said. \"Otherwise I would never have brought her here, knowing of your vow. But the fact remains, I vouched for her safety and my word is all-important to me\u2014as is yours to you.\"\n\n\"Damn and blast it, Erak!\" Ragnak shouted. \"The Temujai are only three or four days' march from here! We can't afford to be fighting amongst ourselves now!\"\n\n\"It would be a shame if you had to face the Temujai with at least one, and possibly both, of your best leaders dead,\" Halt put in mildly, and the Oberjarl rounded on him in a fury.\n\n\"Shut up, Ranger! I'm of half a mind to believe that this is all your doing! No good ever came of dealing with your kind!\"\n\nHalt shrugged, unimpressed by the Skandian's fury. \"Be that as it may,\" he said, \"it occurs to me that there might be a solution to your problem\u2014for the time being, at least.\"\n\nThe buzz of conversation through the room was cut short as Ragnak swung his gaze around angrily. He watched Halt with narrowed eyes, expecting some trick or some kind of subterfuge.\n\n\"What are you talking about? My vow is binding upon me,\" he said. Halt nodded agreement.\n\n\"I understand that. But is there any time factor involved?\" he asked. Now Ragnak looked puzzled as well as suspicious.\n\n\"Time factor? How do you mean?\"\n\n\"If we accept that you plan to do your best to kill Evanlyn, knowing that Erak will try to stop you when you do\u2014not to mention the fact that if he doesn't, I most certainly will\u2014have you vowed that you'll do it at any particular time?\" Halt continued.\n\nThe puzzled expression on Ragnak's face grew more intense.\n\n\"No. I didn't specify any time. I just made the vow,\" he said finally, and Halt nodded several times.\n\n\"Good. So, as far as these Vallas are concerned, they don't care whether you try to fulfill your vow today or if you choose to wait until, say, after we've sent the Temujai packing?\"\n\nUnderstanding was beginning to dawn on the Oberjarl's face. \"That's right,\" he said slowly. \"As long as the intent is there, the Vallas will be satisfied.\"\n\n\"No!\" A shrill voice cut across them. It was Slagor, the silky, self-satisfied tones gone from his voice now. \"Can't you see, Oberjarl, he's trying to trick you? He has something in mind. The girl must die and she must die now! Otherwise your sworn word is worthless!\" Slagor's anger and his long-held desire for revenge on Evanlyn for the events that had occurred on Skorghijl had caused him to go too far. Ragnak turned on him now, a flame of anger burning in his eyes.\n\n\"Slagor, I would advise you to get rid of this reckless habit of telling your peers that they are liars,\" he said, and instantly the wolfship captain retracted his accusation.\n\n\"Of course, Oberjarl. I didn't mean\u2014\"\n\nRagnak cut him off.\n\n\"My first concern is for the safety of Skandia. With these Temujai on our doorstep, Erak and I cannot afford to be fighting. If he'll agree to postpone our differences until after we've settled with them, then I will too.\"\n\nErak nodded agreement instantly. \"It sounds like a good compromise to me.\"\n\nThere was still one thread of suspicion in Ragnak's mind. He turned back to Halt, his heavy brows knitted together in a frown.\n\n\"I can't help wondering what's in it for you, Ranger. All you've done is win a postponement.\"\n\nHalt inclined his head slightly to one side as he considered the matter. \"True,\" he replied. \"But a lot can happen in the next few days. You might be killed in the battle. Or Erak. Or me. Or all three of us. Besides that, my immediate priority is the same as yours: to see these Temujai driven back. After all, if they win here, it won't be long before they're invading Araluen as well. I have a sworn duty to try to prevent that.\" He smiled grimly. \"That's another of those vows that we all seem to rush around taking. Damned nuisances, aren't they?\"\n\nRagnak turned and stepped back up on the dais to his massive council chair.\n\n\"We're agreed then,\" he said. \"We'll settle the Temujai question first. Then we'll come back to this problem.\"\n\nErak and Halt exchanged glances, then both men nodded. Only Slagor seemed to be in disagreement with the compromise. He muttered a curse under his breath. Halt took Evanlyn's arm and began to guide her from the Great Hall, followed by the two apprentices and Erak. They hadn't gone half a dozen paces when Halt turned back to Ragnak.\n\n\"Of course, there is one more question that I'd like to hear Slagor answer,\" he said. As he hoped, at the mention of his name, everyone in the room involuntarily glanced at Slagor. Then, when all eyes were on him, Halt continued.\n\n\"Perhaps he could tell us what his ships are doing at Fallkork Island?\"\n**29**\n\nEVERYONE SAW SLAGOR'S GUILTY START OF SURPRISE WHEN Halt mentioned the name of the island. Slagor recovered quickly, but the moment had been there and it had been witnessed.\n\n\"I'm not here to answer to you, Ranger!\" he blustered angrily. \"You have no authority in this council!\"\n\nErak stepped forward, rocking on his heels, his face only centimeters from Slagor's. \"But I have,\" he told the other man. \"And I'd like to hear your answer.\"\n\n\"What's this about, Erak?\" Ragnak interrupted before Slagor could reply. Erak kept his gaze fixed on Slagor.\n\n\"Two of Slagor's ships are currently at Fallkork Island,\" he replied. \"In another day, he plans to rendezvous with them and sail down the coast to Sand Creek Bay.\"\n\nErak saw the color draining from Slagor's face as he realized that his plans had been discovered. He continued inexorably, his voice rising in volume as Slagor attempted to speak, drowning the other man out. \"There, he plans to embark one hundred and fifty Temujai warriors and land them behind our lines to attack us from the rear.\"\n\nThe room erupted as people began to shout all at once. In vain, Slagor spat abuse at Erak and protested his innocence. His followers in the hall, and there were more than a few, roared their protests, while those who favored Erak roared back, calling for Slagor's head. The bedlam continued for a full minute until Ragnak rose from his seat.\n\n\" _Silence!\"_ he bellowed.\n\nIn the ensuing quiet, you could almost hear a pin drop.\n\n\"How do you know this?\" the Oberjarl asked. He disliked Slagor. Many of the Skandians did. But the concept of such treachery was so absolutely abhorrent to the simple Skandian code of conduct that Ragnak found it impossible to believe it of anyone, even Slagor.\n\n\"His plans were overheard, Ragnak,\" Erak told him.\n\nInstantly Slagor was screaming his innocence. \"This is lies! It's a pack of filthy lies! Who heard me? Who claims I'm a traitor? Let them face me now!\"\n\n\"As a matter of fact, Ragnak,\" said Halt, raising his voice so that he was heard clearly in every corner of the room, \"the informant is here with us.\"\n\nThat piece of news stilled Slagor's protests immediately. Ragnak eyed the Ranger with distaste. Since he had arrived in Hallasholm, the comfortable, established order of things had been continually disturbed.\n\n\"Then let's hear from him,\" the Oberjarl said.\n\n\"Not him, Ragnak. Her. The informant is Evanlyn. Perhaps that's why Slagor is so keen to have her discredited and killed.\"\n\nUproar once again filled the room and Will realized how cleverly Halt had played this hand. In the confusion of the moment, nobody asked the obvious question: how could Slagor have known that Evanlyn had discovered his plan? For if he didn't know, he would have no reason to try to discredit the girl. But now that Halt had planted the seed, the Skandians would all half believe that Slagor's actions were intended to forestall Evanlyn, rather than the other way around. In that light, her accusation could not be dismissed out of hand. It had to be investigated.\n\n\"Proof!\" Slagor was shouting now, and some of his followers, realizing their own necks were close to the heads-man's ax, were shouting it too. \"Anyone can accuse me! But where's the proof?\"\n\nRagnak silenced the shouting with a gesture. \"Well, Ranger,\" he asked Halt, \"can you offer us proof of these accusations?\"\n\nErak hurriedly stepped into the breach, before Halt had to answer. \"Svengal is bringing in the two ships from Fallkork,\" he said. \"He should be in port by tomorrow.\"\n\nBut now Slagor saw the way out, saw there was no concrete evidence of the plan. \"So two of my ships are waiting at Fallkork?\" he cried, his voice shrill once more. \"What does that prove? How does that make me a traitor? It doesn't, does it, Erak?\"\n\nA few of those in the hall started to echo the thought\u2014and not just his own followers. As Halt had pointed out earlier, the mere presence of the ships at the rendezvous was no proof of Slagor's treachery. Emboldened now, Slagor stepped toward the crowd, addressing them and not the Oberjarl.\n\n\"They accuse me of treachery! They slander me! They take the word of an enemy of this country, the sworn enemy of our Oberjarl! Yet they can show no way to prove their vile claims! Is this Skandian justice? Let them find a way to prove it, I say.\"\n\nA growing chorus of voices agreed with him. Then, as if he were conducting a choir, Slagor signaled for silence and turned back to Halt.\n\n\"Can you, Ranger?\" he said, spitting the last word out as if it were an insult. \"Can you show some kind of proof?\"\n\nHalt hesitated, knowing they'd lost the momentum and the sentiment of the crowd. Knowing they'd lost. Then Will pushed forward to stand beside his mentor and friend.\n\n\"There is a way,\" he said.\n\nIt took a lot to silence a noisy crowd of Skandians, but Will's statement managed to do the trick. The voices died away as if cut by a knife and all eyes turned to the small figure, standing now between Halt and Erak. As Will might have guessed, it was Ragnak himself who broke the silence.\n\n\"How?\" he said simply.\n\n\"Well, Slagor's ships at this island, taken on their own, may be no proof of his intention to sell out to the Temujai,\" Will said carefully, thinking through his words before he spoke them aloud, knowing that all their safety hung by a hairsbreadth on the way he expressed his idea. He saw Ragnak draw breath to speak and hurried on before the Oberjarl could interrupt him. \"But . . . if Erak took _Wolfwind_ to this Sand Creek Bay, and if they happened to find, say, a hundred and fifty Temujai warriors waiting there to embark, it's a fair indication that someone is planning to betray you, isn't it?\"\n\nThere was a murmur of agreement among the assembled crowd. Ragnak frowned as he thought through the idea. Beside Will, Erak muttered: \"Good thinking, boy.\"\n\n\"That's true,\" Ragnak said finally. \"It shows there's treachery been planned. But who's to say Slagor's involved?\"\n\nWill chewed his lip as he thought over that one. But now Halt spoke up.\n\n\"Oberjarl, there's a simple way to find out. Let Erak take not one ship, but three. After all, that's the number the Temujai are expecting to see. Then he can speak with the leader of any Temujai who might happen to be there and tell them that Slagor has been detained and has sent him in his place. If the Temujai leader responds with words along the lines of 'Who the devil is Slagor?,' then our friend here is as innocent as he claims to be.\" He paused and saw that Ragnak was nodding as he considered the idea. Then he added, more deliberately, \"On the other hand . . . if the name Slagor seems familiar to the enemy, then there is all the proof you need.\"\n\n\"This is ridiculous!\" Slagor burst out. \"I swear to you, Oberjarl, that I am no traitor to Skandia! This is a plot cooked up by these Araluens.\" He gestured contemptuously at Halt and Will. \"And somehow they seem to have tricked Erak into believing it.\"\n\n\"If you're innocent,\" Ragnak said heavily, \"then you have nothing to fear from all this, do you?\" He was gazing steadily at Slagor now, noting the sheen of perspiration on the other man's forehead, noting the shrill tone that pervaded all his statements now. Slagor was scared, he thought. The more he saw that, the more he was prepared to believe that the man was a traitor.\n\n\"I don't see any reason why\u2014\" Slagor began, but Ragnak cut him off with a gesture.\n\n\"I do!\" he snapped. \"Erak, take three ships to Sand Creek Bay immediately and do as the Ranger suggests. Once you've established whether or not Slagor is involved in this plot, get back here and report. As for you . . .\" He turned to Slagor, who was beginning to edge toward the side door of the room. \"Don't try to go anywhere. I want you where I can see you until Erak returns. Ulfak, see to it!\" He addressed this last comment to one of his other senior jarls, who nodded and moved to stand beside Slagor, laying a hand on his arm.\n\n\"One thing, Oberjarl,\" Erak said, and the Skandian leader turned to him again. \"Once I've established that Slagor is involved, is it all right if we reduce the Temujai numbers a little? That'll be a few less we have to fight here, at least.\"\n\n\"Good idea,\" Ragnak said. \"But don't take any risks. I need to know the traitor's identity and you can't tell me that if the Temujai kill you.\"\n\n\"Why not go ahead with the plan they're expecting?\" Will said, before he could stop himself. The Skandian leader regarded him as if he were mad.\n\n\"Are you out of your mind?\" he said. \"Are you suggesting that Erak actually brings the Temujai back here as prisoners? We'd have to subdue them and guard them and that would take men away from our own battle line.\"\n\n\"Not back here,\" Will said, turning to appeal to Erak. \"But couldn't you find some pretext to make them get off the ships at this Fallkork Island\u2014then just leave them there?\"\n\nAgain a silence, broken this time by a deep, throaty chuckle from Erak. \"Oh, what a prize idea!\" he said, grinning fondly at Will. \"If we take these . . . horsemen . . . through the Vulture Narrows, I'm sure we can have them begging to get ashore for a few hours. The seas there are terrible at this time of year\u2014guaranteed to make any inexperienced sailors seasick!\"\n\nRagnak rubbed his chin thoughtfully. \"I take it these Temujai are unused to sailing?\" he asked Halt.\n\nThe Ranger nodded. \"Totally, Oberjarl.\"\n\nRagnak looked from Halt to his young apprentice. \"This boy of yours shows a certain talent for the sort of devious thinking we expect from you Rangers.\"\n\nHalt dropped one hand lightly on Will's shoulder, and said, with a completely straight face, \"We're very proud of him, Oberjarl. We think he'll go far.\"\n\nRagnak shook his head wearily. This sort of plot and counter-plot was beyond him. He waved one dismissing hand at Erak.\n\n\"Get your ships ready and go,\" he said. \"Then dump these Temujai on Fallkork Island and get back here.\" The matter was done as far as he was concerned, but Slagor had one last, desperate objection.\n\n\"Oberjarl! These are the people who accuse me! They're all in it together! You can't send them to verify their own charges!\"\n\nRagnak hesitated. \"Fair point.\" He turned to his hilfmann. \"Borsa, you go with them as an independent witness.\" Then, returning his gaze to Slagor, he concluded, \"As for you, you'd better hope there are no Temujai at Sand Creek Bay.\"\n**30**\n\nERAK LOOKED AT THE FIGURE STANDING BESIDE HIM IN THE stern of the wolfship and, for the hundredth time, was unable to prevent a broad grin from breaking out across his face.\n\nHalt noticed the look, and the grin, and said in a sour tone, \"It must lose its fascination after a while, surely?\"\n\nThe jarl shook his head, his grin broadening. \"Not for me,\" he replied cheerfully. \"Every time, it's just as fresh as the first.\"\n\n\"I'm so glad that Skandians have such a lively sense of humor,\" the Ranger said, scowling. It didn't serve his ill temper any better to see that several of the other Skandians were grinning as well. In truth, he was a comical figure. He had forsaken his Ranger's cloak and garb and was dressed in Skandian clothing\u2014sheepskin vest, a short fur cloak and woolen breeches, wound around with leather bindings from the knees down. At least they should have been wound from the knees down. In fact, since Halt was considerably smaller in stature than any of the adult Skandians, the leggings were bound from his thighs down, the breeches sagged alarmingly at the crutch and the sheepskin vest hung loosely on him, seemingly with room for another person of his own size inside.\n\n\"It's your own fault,\" Erak replied. \"For deciding to try to disguise yourself as one of us.\"\n\n\"I told you,\" Halt muttered. \"The Temujai got a good look at me when they were chasing us near the border\u2014and even without that, they have no reason to love anyone dressed as a Ranger.\"\n\n\"So I've heard,\" Erak said, still grinning. He bent to the sighting ring before him, checked the position of the floating lodestone and adjusted the sight ring to conform with it. Then he read off the bearing to the next headland.\n\n\"A little east to east of south,\" he said to himself, then, raising his voice, he called to his men: \"Look alive now! Sand Creek Bay lies beyond that next headland!\"\n\nThere was an expectant shuffle on the decks of the wolfship as the Skandians made sure their weapons were close to hand\u2014although not obviously so. At a nod from Erak, the masthead lookout relayed the message to the other two wolfships sailing in close company with them. Very obviously making an effort not to grin, the wolfship skipper nudged Halt in the ribs with a not too gentle elbow.\n\n\"You'd better put on your helmet,\" he told the Ranger, whose countenance darkened even further than before as he reached for the huge horned helmet that every Skandian warrior wore.\n\nThis had been the most contentious piece of equipment. Erak had maintained that no Skandian would ever appear in public without a helmet, and that there was no question of Halt's not wearing one. Yet the sizes were immense compared to what Halt considered to be his own perfectly normal head size. Even the very smallest helmet that Erak could find wobbled loosely on Halt, and came down over his ears and eyes. By dint of much padding with cloths, they had finally managed to get the helmet to sit more or less firmly on his head. But it still gaped amazingly all around.\n\nThe Skandians looked on with ill-concealed amusement as Halt carefully placed the helmet on his head. Borsa, who had joined the expedition on Ragnak's orders, shook his head and chuckled. The unwarlike hilfmann, who'd never seen a day of battle in his life, knew he looked more the part than Halt did.\n\n\"Even if this turns out to be a wild-goose chase,\" he said cheerfully, \"it will have been worth it to see this.\"\n\nHalt turned away angrily. It was a mistake. With the rapid head movement, his helmet became dislodged and tipped down over his eyes. He cursed quietly to himself, straightened the ridiculous headgear and resigned himself to the smothered laughter of the Skandians.\n\nThey had been running before a quartering wind, but now, as Erak prepared to bring _Wolfwind_ around the headland and across the wind, there was a flurry of activity on board as the big square sail was gathered in and furled to the cross yard. The long, heavy oars clattered in their tholes as the crew ran them out, and before the ship had time to lose way, they began their smooth, rhythmic stroking. Glancing behind, Halt saw the other ships had followed suit. Once again, the helmet tilted awkwardly on his head and, with a gesture of disgust, he ripped it off and dropped it to the deck. He glared at Erak, daring the big Skandian to make some comment. The jarl merely shrugged his shoulders and smiled.\n\nThey were almost around the last promontory now and those without any duties involved in keeping the ship moving and on course craned eagerly to see whether the beach would be empty\u2014or whether there would be a war party of Temujai warriors waiting for them. With tantalizing slowness, the boat crept past the headland, gradually revealing the strip of sandy beach beyond. Halt felt a sinking feeling in the pit of his stomach, as the first sight of the beach showed no sign of any Temujai. But they were only looking at the southern end of the beach, and as they came farther around, there was a soft sigh from those watching and the sinking feeling in Halt's stomach turned to a flame of fierce exultation.\n\nThere, drawn up at the center of the beach, were three squadrons of Temujai cavalry.\n\nTheir dome-shaped felt tents were pitched in neatly ordered rows. Horses were tethered on a grass sward where the beach ended. There were sixty men to a squadron, Halt knew. He presumed each squadron would be leaving ten men to tend the horses, which, of course, couldn't travel on the wolfships. The discordant blare of a Temujai horn from the beach told them that they had been sighted.\n\nBorsa shook his head sadly at the evidence of Slagor's treachery. \"I'd been hoping that this would be an empty quest,\" he said bitterly. \"The thought of any Skandian turning traitor is a bitter one to face.\"\n\nHe moved away from Halt and Erak and the two men exchanged glances. Erak shrugged. His was a more cynical temperament than the hilfmann's, and he had better knowledge of Slagor's character.\n\n\"Time to make absolutely sure,\" he said quietly, and heaved on the steering oar to bring _Wolfwind_ 's prow heading straight toward the beach. As arranged, the other two ships hove to, the rowers maintaining a slow, relaxed stroke to hold them in position against wind and tide, some two hundred meters off the beach. They were still within bowshot there, but the huge, circular Skandian shields that were ranged along the bulwarks gave the sailors protection against any Temujai attack.\n\nThose on _Wolfwind_ weren't so fortunate. They were heading straight inshore, every stroke of the oars making them more vulnerable to a sudden volley of Temujai arrows.\n\n\"Keep your heads down,\" Erak growled at his rowers. It was an unnecessary warning. They were hunched down as far as they could be, trying to prevent any part of their persons from showing above the oak bulwarks. Halt noticed that the jarl's right hand strayed from the steering oar from time to time, and brushed almost unconsciously against the haft of the massive battle-ax that leaned close by.\n\nActivity on the beach was growing now, and a party of half a dozen Temujai had moved to the water's edge. Behind them, orders were being shouted and squads were forming as troop leaders prepared their men to embark on the three wolfships.\n\nThe deep water continued in quite close to the beach. Of course, the wolfships were designed to beach in water as shallow as one meter, but the Temujai weren't aware of the fact and Halt and Erak had agreed that it made better sense to keep the enemy at a distance. Twenty meters from the water's edge, Erak gave a brief command and the oars on one side of the ship backed while the others went ahead, swinging the narrow craft through ninety degrees, virtually in her own length.\n\nErak nodded to his second in command, who hurried to the tiller. Then the jarl stepped to the shoreward side of the ship and raised his voice in his familiar storm-quelling bellow.\n\n\"Ahoy the beach!\" he called, and Halt, standing close by, hastily moved a few paces farther away.\n\nThe Tem'uj standing in the center of the small group on the beach cupped his hands and called back.\n\n\"I am Or'kam, commander of this force,\" he called. \"Where is Slagor?\"\n\nBehind him, Halt heard a quick intake of breath and turned to see Borsa shaking his head sadly, his eyes downcast. Several of the other Skandians also exchanged glances at this incontrovertible confirmation that Slagor had been involved in the plan.\n\n\"Keep still!\" Halt warned them, and the men hurriedly masked their reactions. Erak was answering now, with the story that he, Borsa and Halt had agreed upon.\n\n\"Oberjarl Ragnak was growing suspicious of our movements. It was too dangerous for Slagor to come on this expedition. He will join us at Fallkork Island.\"\n\nThere was a hurried consultation between the Temujai leaders.\n\n\"They don't like it,\" Erak muttered out of the side of his mouth.\n\n\"They don't have to like it. They just have to believe it,\" Halt told him in the same undertone. After several minutes' discussion, Or'kam stepped away from the group and called again.\n\n\"We expected Slagor. How can we be sure we can trust you? Did he give any message? Any password?\"\n\nOn the ship, the men exchanged worried glances. This was the one eventuality they had feared. If Slagor had arranged a password with the Temujai, then their plan was spoiled. Of course, their main aim had already been achieved. They had proved Slagor's complicity in the plot. But now that they were here, the chance of taking 150 men out of the enemy's battle line, without any loss to their own forces, was tempting in the extreme.\n\n\"Bluff it out,\" Halt said quickly. \"He already said he was expecting Slagor, so they didn't need a password.\" Erak nodded. It made sense.\n\n\"Look, horseman,\" Erak bellowed again. \"I don't need a password, do I? I'm here to pick you up. And I'm risking my neck to do it! Now if you choose to come aboard, then do so. If not, I'm going raiding and leaving you and Ragnak to your little war. Now you choose!\"\n\nOnce again there was an urgent consultation on the beach. They could see Or'kam's reluctance in his movements, but equally, they could see him weighing his options, and after a long, searching glance at the wolfship, he obviously decided he had nothing to fear from the skeleton crews of rowers on the three ships.\n\n\"Very well!\" he called. \"Bring your ships in and we'll board.\"\n\nBut now Erak shook his head.\n\n\"We'll bring you out on the skiffs,\" he called. \"We can't beach here.\"\n\nOr'kam made an angry gesture. Obviously he didn't like it when things didn't go precisely according to his wishes.\n\n\"What are you talking about?\" he yelled. \"Slagor beached his ship right here. I saw him do it!\"\n\nErak moved to the bulwark and stood up on it, completely exposed to any possible fire from the beach.\n\n\"Careful,\" Halt muttered, trying not to let his lips move.\n\n\"And tell me, horseman,\" Erak said, his voice heavy with sarcasm, \"did Slagor then load fifty men aboard his ship and take her _off_ the beach?\"\n\nThere was a pause as the Temujai leader thought through the reasoning in what Erak had said. Erak saw the hesitation and pressed on.\n\n\"If I beach now and load your men aboard, we'll never get her off again. Particularly with the tide falling the way it is.\"\n\nThat seemed to clinch it. Or'kam reluctantly signaled his agreement.\n\n\"Very well!\" he called. \"How many can you take at a time?\"\n\nErak resisted the temptation to heave a sigh of relief.\n\n\"Three skiffs, eight men each,\" he called. \"Twenty-four at a time.\"\n\nOr'kam nodded. \"All right, Skandian, send in the skiffs.\"\n**31**\n\n\"POSITION TWO . . . SHOOT!\" CALLED WILL, AND THE HUNDRED archers' arms rose to the same angle, drew and released, more or less simultaneously. The slithering hiss of the release was magnified a hundred times, and Will and Horace watched in satisfaction as a dark cloud of arrows arced across the intervening space to the target that had suddenly popped up.\n\nEvanlyn was sitting on an old broken cart a few meters behind the line of archers, watching the scene with interest.\n\nThey could hear the distinctive soft thudding of arrows striking into the turf around the target, and the harder, clearer smack of those arrows that actually hit it.\n\n\"Shields!\" bellowed Horace. Beside each archer, a foot soldier stepped forward with a rectangular wooden shield held on his left arm, positioned to cover both himself and the archer as he reloaded. It had been an idea the warrior apprentice had come up with while he'd been watching an earlier practice shoot. Will had readily adopted the improvement. With only one hundred archers, he couldn't afford to lose any to the return fire the Temujai were sure to mount once they saw his men in action.\n\nWill glanced quickly around to make sure his men were ready for the next shot. Then he turned back to the practice field, searching for the next target to appear.\n\nThere! As the team of men behind him hauled on a set of ropes, another flat board swung up out of the grass. But he had nearly missed the movement, waiting to see if the archers were ready. He felt a slight twinge of panic. Things were moving too fast.\n\n\"Clear!\" he called, wishing his voice wouldn't tend to break when he did this, and the shield bearers stepped clear.\n\n\"Half right! Position three . . . shoot!\"\n\nAgain they heard the slithering hiss. Another cloud of arrows cast its fleeting shadow across the field and riddled the area around the target. Already, another target was rising out of the grass, much closer in this time.\n\n\"Shields!\" Horace called again and once more the archers were hidden from return fire. As he ordered his men to do this, Horace performed the same action, concealing Will behind one of the large shields.\n\n\"Come on, come on,\" Will muttered, shifting from one foot to the other as he watched the men select new arrows and nock them to the string. The archers sensed his urgency and hurried their reloading. The extra haste made for clumsiness. Three of them dropped the arrows they were about to nock; others fumbled like beginners.\n\nFrustrated, Will realized he'd have to go with the men who were ready. He swung his gaze back to the target. But the men on the ropes were hauling it in, so that it slid toward them on its sled-like runners, matching the speed of an enemy advance. The range had closed too quickly for him to make an instant assessment. In the time that he'd been watching his men, he'd lost his concentration and his sense of the battlefield.\n\nHe stepped down angrily from his command position, a low platform built at the end of the line of archers.\n\n\"Stand down!\" he called. \"Everyone take a break.\"\n\nHe realized he'd been sweating freely with the tension and wiped a corner of his cloak across his forehead. Horace set the large shield down and joined him.\n\n\"What's the trouble?\" he asked.\n\nWill shook his head, defeated. \"It's hopeless,\" he said. \"I can't keep track of the targets and the men at the same time. I lose my perspective. You'll have to watch the men and tell me when they're ready.\"\n\nHorace frowned.\n\n\"I could,\" he agreed. \"But on the day, I think I'm going to be a little busy shielding you from any return shots. I really need to keep my eye on the enemy too. Unless you want to be turned into a pincushion.\"\n\n\"Well, someone's going to have to do it!\" Will said angrily. \"We haven't even begun to practice against the Kaijin and the whole thing's falling apart already!\"\n\nHalt had told them about the Kaijin. They were specialist marksmen and each group of sixty Temujai riders would have one with them. The Kaijin were assigned to pick off the leaders in any enemy group. It would be Will's task to counteract them and he'd devised a drill for it, with additional, smaller targets set in the field, ready to rise into view unexpectedly. But if Will was dividing his attention between his own archers and the enemy, his chances of nullifying the enemy marksmen would be low indeed.\n\nOn the other hand, his chances of being shot by one of them were considerably higher.\n\n\"I could do it,\" said Evanlyn, and both boys turned toward her. She saw the doubt in their expressions. \"I could do it. I could keep an eye on the archers and call when they're ready.\"\n\n\"But that'll put you in the battle line!\" Horace objected instantly. \"It'll be dangerous!\"\n\nEvanlyn shook her head. She noticed Will hadn't objected so far. She could see he was at least considering her idea. She hurried on before he could veto the suggestion.\n\n\"The archers aren't actually in the front line. You'll be behind it, and protected by a trench and an earth mound. You could build me a kind of a dugout at the end, beneath your command position. I'd be safe from arrows there. After all, I don't need to see the enemy, just our men.\"\n\n\"But what if the Temujai break through our line?\" Horace said. \"You'll be right in the middle of it then!\"\n\nEvanlyn shrugged. \"If the Temujai break through, it won't matter where I am. We'll all be dead. Besides, if everyone else is taking a risk, why shouldn't I?\"\n\nHorace was wise enough not to reply _Because you're a girl._ And he had to admit that she had a point. But he wasn't convinced. He turned to Will.\n\n\"What do you think, Will?\" he said. He expected the apprentice Ranger to agree with him and he was a little surprised when Will didn't answer immediately.\n\n\"I think,\" Will said slowly, \"she may be right. Let's try it.\"\n\n\"Ready,\" Evanlyn said calmly. She was crouched below the platform where Will and Horace stood.\n\n\"Clear!\" That was Horace. The shield bearers dropped to one knee beside the archers.\n\n\"Left left! Position one . . . shoot!\"\n\nThe volley was ragged and Will knew that was his fault. He'd called the order to shoot a fraction too quickly and some of the men hadn't reached full draw. He mentally kicked himself. He heard Horace calling for the shields again and saw the arrow strikes on the target\u2014as well as those that missed and fell short.\n\nBut now another danger reared its head. As the next large target swung up and began moving toward them, another, smaller one swung out from the target they had just engaged. This was a man-sized figure and it was Will's responsibility. He drew and loosed and saw his arrow slam into the target, just as Evanlyn called \"ready\" once more. He turned his attention quickly to the main target as Horace ordered the shield bearers down.\n\n\"Left! Position three . . .\" He waited, then added a correction. \"Down a half . . .\"\n\nHe forced himself to wait the full term, then called: \"Shoot!\"\n\nThis time, the volley flew truly, with the majority of arrows slamming into the target or close around it. If it had been a charging group of horsemen, the volley would have taken a severe toll.\n\n\"Shields!\" bellowed Horace, and the pattern began to repeat itself. But now Will waved a weary hand.\n\n\"Stand down,\" he said, and Horace repeated the order in a louder voice. The archers and shield bearers, who had been working at this drill for the past two hours with only a few short breaks, dropped gratefully to the grass to rest. Horace grinned at Will.\n\n\"Not bad,\" he said. \"I make it twenty out of twenty-five of those targets peppered pretty solidly. And you hit every one of the Kaijin.\"\n\nThe smaller targets attached to each large board represented the Kaijin. Freed from the need to check on both his own men and the enemy, Will had coped easily with them.\n\n\"True,\" Will said in response to Horace's comment. \"But they weren't shooting back.\"\n\nSecretly, he was pleased with his performance. He had shot well, in spite of the distractions involved in estimating range and trajectory for the larger group.\n\nHe grinned at Horace and Evanlyn. It was good to feel some of the old camaraderie back.\n\n\"Nice work, everyone,\" he said, then, raising his voice: \"Let's take a break for half an hour.\"\n\nThere was a murmur of satisfaction from the archers and they moved to the side of the practice area, where barrels of drinking water were available. Behind Will, a familiar voice spoke.\n\n\"Take a break for the rest of the day. You've done enough for the moment.\"\n\nThe three young Araluens turned at the sound of Halt's voice. Instantly, Will felt reinvigorated, bursting with curiosity about events at Sand Creek Bay.\n\n\"Halt!\" he cried eagerly. \"What happened? Were the Temujai there? Did you manage to fool them?\"\n\nBut Halt held up a hand to stop the flow of questions he knew he was about to face. He was troubled by what he had just seen as he approached.\n\n\"Why have you got Evanlyn involved in this, Will?\" he asked. He saw the hesitation in the young man's eyes, then saw his jaw set in a determined line.\n\n\"Because I need her, Halt. I need someone to keep track of the men, to let me know when they're ready. Without that, the system won't work.\"\n\n\"Couldn't someone else do that?\"\n\n\"I can't think of anyone else I can trust. I want someone who won't panic. Someone who'll keep her head.\"\n\nHalt scratched his beard thoughtfully. \"How do you know Evanlyn won't panic?\"\n\nThe answer came immediately.\n\n\"Because she didn't in Celtica\u2014at the bridge.\"\n\nHalt looked at the three young faces before him. All set. All determined. He knew Will was right. He would need someone he could trust.\n\n\"All right then,\" he said, then added, as the three beamed at him, \"But don't look so happy about it. I'm the one who'll have to explain to her father if she's shot.\"\n\n\"Now what about the Temujai?\" Will asked. \"Did you find them at Sand Creek Bay?\"\n\nAt the mention of Slagor's plot, the smile on Evanlyn's face faded, replaced by a look of anxiety.\n\n\"They were there,\" Halt said quickly, dispelling her worst fears. \"And they made it clear that they were expecting to see Slagor.\" He nodded at the girl as she let go a pent-up breath in relief. \"It puts a different complexion on things as far as you're concerned, Princess,\" he said.\n\n\"Ragnak still has his vow,\" she said dully.\n\nHalt nodded. \"True. But at least he's agreed not to act on it until after we've driven off the Temujai.\" Evanlyn made an uncertain little gesture with her hands.\n\n\"It's just postponing things,\" she said.\n\n\"Problems postponed have a habit of solving themselves, more often than not,\" Halt told her, putting an arm around her slim shoulders. Evanlyn smiled at him. But it wasn't much of a smile.\n\n\"If you say so,\" she replied. \"But Halt, don't address me as 'Princess' if you would. No point in reminding Ragnak about me at every opportunity.\"\n\nThe Ranger nodded. \"I stand corrected,\" he said. Then he added, in a lower tone that only Evanlyn heard, \"By the way, there's no need to mention it to him, but don't be too surprised if Erak's wolfship is standing by to get you out of here the minute we've seen these damned Temujai off.\"\n\nShe looked up at him then, hope in her eyes. He met her gaze and nodded meaningfully. She looked from him to the burly Skandian Jarl, who was now approaching over the field, then she leaned forward to kiss Halt lightly on the cheek.\n\n\"Thanks, Halt,\" she said softly. \"At least now I know there is an alternative.\"\n\nThe Ranger shrugged and grinned at her. \"That's what I'm here for,\" he said, pleased to see the light of hope back in her eyes. She smiled at him again and slipped away, heading back to her quarters. All at once, overwhelmed by her sense of relief that Halt had contrived a possible way out of her predicament, she felt the need to be alone for a while.\n\nSome of the Skandians who had been working the targets were calling to Erak now as he came closer, wanting to know how events had turned out at Sand Creek Bay. As the jarl confirmed Slagor's treachery, there were angry mutterings and dark looks cast toward the lodge, where Slagor was being held under guard.\n\n\"What about the Temujai, Erak?\" Will asked. \"How did you convince them to go ashore on Fallkork Island?\"\n\nErak's laughter rang around the practice field. \"We would have had to fight to stop them!\" he told the assembled audience. \"They were scrambling over each other to get back on solid land.\"\n\nThe Skandians in the crowd standing around him echoed his laughter as he continued: \"I managed to find a spot where we had the wind from astern, a steep head sea on our starboard quarter and the tide race through the narrows at the same time. A few hours of that and our fierce horse soldiers were like little lambs\u2014sick little lambs.\"\n\n\"They weren't the only ones,\" Halt replied with some feeling. \"I've been through some rough seas in my time, but I've never felt anything like the plunging and leaping you had us doing.\"\n\nOnce again Erak bellowed with laughter. \"Your master here went nearly the same shade of green as his cloak,\" he told Will. Halt raised one eyebrow.\n\n\"At least I finally found a use for that damned helmet,\" he said, and the smile disappeared from Erak's face.\n\n\"Yes. I'm not sure what I'm going to tell Gordoff about that,\" he said. \"He made me promise I'd look after that helmet. It's his favorite\u2014a real family heirloom.\"\n\n\"Well, it certainly has a lived-in feel to it now,\" Halt told him, and Will noticed there was a hint of malicious pleasure in his eye. The Ranger nodded at the group of archers who were standing by.\n\n\"You seem to have this group working pretty well,\" he said.\n\nWill felt absurdly pleased at his mentor's praise.\n\n\"Oh,\" he said, trying to sound casual. \"We're not doing too badly.\"\n\n\"Better than that from what I saw,\" Halt told him. Then he repeated his earlier suggestion. \"I meant what I said, Will. Give them the rest of the day off. Yourself too. You've earned a break. And unless I miss my guess, we're going to need all the rest we can get in the next few days.\"\n**32**\n\nIT WAS A MUTED SOUND\u2014SURF ON A BEACH A LONG WAY AWAY, or maybe the rolling of distant thunder, Will thought. Except no thunder had ever sounded like this. This sound never seemed to start and never seemed to end. It just continued, over and over, repeating itself constantly.\n\nAnd, gradually, growing louder. It was the sound of thousands of horses cantering slowly toward them.\n\nWill flexed the string on his bow a couple of times, testing the feel and the tension. His eyes were fixed on the point where they all knew the Temujai army would appear\u2014a kilometer away, where the narrow coastal strip between the hills and the sea jutted out in a promontory, temporarily blocking their view of the approaching army. His mouth was dry, he realized, as he tried, unsuccessfully, to swallow.\n\nHe reached down for the water skin that was hanging by his quiver and missed the first sight of the Temujai horsemen as they swept around the bend.\n\nThe men around him let out an involuntary cry. The horsemen rode stirrup to stirrup, in one long extended line, each horse cantering easily, matching the pace of the horse beside it.\n\n\"There must be thousands of them!\" one of the archers said, and Will could hear the fear in his voice. It was echoed in another dozen places along the line. From the ranks of Skandian warriors beyond them, there was not a sound.\n\nNow, above the dull rumble of the hooves, they could hear the jingle of harness as well, a lighter counterpoint to the rumbling hoofbeats. The horsemen came on, moving closer to the waiting ranks of silent Skandians. Then, at the single blaring note of a bugle, they reined in and came to a halt.\n\nThe silence, after the rumbling beat of their approach, was almost palpable.\n\nThen a massive roar rose from the throats of the Skandian warriors who stood by their defenses. A roar of defiance and challenge, accompanied by the ear-shattering clash of axes and broadswords on shields. Gradually, the sound died away. The Temujai sat their horses silently, staring at their enemies.\n\n\"Keep still!\" Will called to his archers. Now that he saw the Temujai front rank, his force seemed ridiculously small. There must have been six or seven hundred warriors riding side by side in that first rank. And behind them were another five or six times that number. At the center of the army, where the commander sat his horse, a sequence of colored signal flags waved. Others answered from positions in the line of horsemen. There was another horn blast\u2014a different note this time\u2014and the front rank began to walk their horses forward. The jingling of harness was apparent once more\u2014then a massive metallic slithering sound filled the air and the weak sun gleamed on hundreds of saber blades as they were drawn.\n\n\"They're going to fight close in,\" Horace said softly beside him.\n\nWill nodded. \"Remember what Halt told us? Their first move will be a feint\u2014an attack and then a false withdrawal to draw the Skandians out from behind their breastworks. They won't commit to their real attack until they have the Skandians strung out in pursuit.\"\n\nThe eighteen hundred Skandians were drawn up in three ranks on a narrow strip of flatland between the sea and the heavily timbered hills. They waited behind carefully constructed earthen breastworks. The sloping ramparts facing the Temujai were thick with sharpened stakes of various lengths, designed to impale the enemy's horses.\n\nHalt had located their main defensive position at the spot where the strip was narrowest, with their flanks protected by the steep, wooded mountains on the left and the sea on the right. Hallasholm itself was barely two hundred meters behind their line. Will's force of archers were on an earthwork berm on the right, some meters behind the main defensive line. At the moment, earth-covered wicker ramparts kept the archers hidden as they crouched behind them.\n\nHalt, Erak and Ragnak were in the command position, more or less in the center of the Skandian line, on a small knoll.\n\nNow, more signal flags were seen and the advancing cavalry broke into a trot, beginning to wheel slightly toward the Skandian left flank.\n\nThere was a stir among the archers crouched behind the breastworks. Several of them reached for the arrow bins in front of them, instinctively feeling the need to arm themselves.\n\n\"Stay down!\" Will called, wishing, as ever, that his voice wouldn't crack. Halt didn't want him revealing the presence of the archers until the Skandians had made several of their usual probing attacks.\n\n\"Wait till they're committed to a full attack, then we'll surprise them,\" he had told his apprentice.\n\nThe line of archers turned now to look at their young commander. Will forced himself to smile at them, then, feigning a casualness he certainly didn't feel, leaned his bow against the breastworks in front of him, signifying that there would be no action required of the archers for some time yet.\n\nSome of the other men copied the action.\n\n\"Nice work,\" Horace said quietly beside him. \"How can you stay so calm?\"\n\n\"It helps if you're terrified,\" Will replied, speaking out of the corner of his mouth. He was surprised at the warrior apprentice's question. Horace himself seemed to be the epitome of calm, totally unworried and seemingly unconcerned. His next statement dispelled that idea.\n\n\"I know what you mean,\" he said. \"I nearly dropped my sword when they rode around the bend there.\"\n\nThe Temujai charge was gathering pace now, breaking into a fast canter, then a gallop. As they neared the Skandian line, a major part of the force swung away, seemingly deterred by the fortifications and the sharpened stakes. They wheeled their horses to run parallel to the Skandian line for a few seconds, then began to curve back toward their own army. The Skandians yelled abuse and scorn at them. A shower of spears, rocks and other missiles erupted from the Skandian line. Most of them fell short of the galloping horsemen.\n\nA smaller group, maybe less than a hundred, continued to close on the left wing of the Skandian line. Leaning forward in their stirrups, shouting their war cries, they forced their shaggy mounts up the earth breastworks, ignoring the screams of those horses who were struck by the stakes. About two-thirds of their numbers made it to the Skandian line and they leaned down from their saddles, striking left and right with their long, curved sabers.\n\nThe Skandian defenders joined the battle eagerly. Huge axes rose and fell and more horses came down, with tortured screams. Will tried to shut his ears to the sound of horses in agony. The small, shaggy Temujai mounts were nearly identical to Tug and Abelard and it was all too easy to imagine his own horse bleeding and terrified, just as the Temujai horses were. Obviously, the Temujai thought of their horses as a means to an end, and had little affection for them.\n\nThe seething battle occupied one corner of the Skandian line. For some minutes, there seemed to be no clear picture of what was happening. Then, gradually, with cries of panic, the Temujai began to give ground, backing down the sloped earthworks, wheeling their horses and moving away, and letting the Skandians come after them with increasing eagerness.\n\nYet, to the more distant observers, it was obvious that the retreating enemy wasn't moving as fast as they might. Even those still mounted made no real effort to gallop clear. Rather, they withdrew gradually, maintaining contact with the foremost of their pursuers, drawing them farther and farther from the defensive positions they occupied and into the open ground.\n\n\"Look!\" said Horace suddenly, pointing with his sword. In response to more flag signals, and unseen by the defenders on the left flank, several hundred riders from the original Temujai charge had now completed a full circle and were wheeling back to the aid of their embattled companions.\n\n\"Just as Halt said they would,\" Horace muttered, and Will nodded wordlessly.\n\nIn the command post near the center of the Skandian line, Erak was saying much the same thing.\n\n\"Here they come, Halt, just as you said,\" he muttered. Ragnak, standing beside him, peered anxiously over the breastworks at his exposed men. Nearly a hundred Skandians had streamed out of the defenses now and were engaged with the Temujai.\n\n\"You called it correctly, Ranger,\" he agreed. From this remote position, he could see the trap about to be sprung. Had he taken his normal place, at the thick of the fighting, he would have been totally unaware of the tactic.\n\n\"Can Kormak be trusted to keep his head out there, and not let his men get out of control?\" Halt asked the Oberjarl. Ragnak scowled at the question.\n\n\"I'll kill him if he doesn't,\" he said simply. The Ranger raised one eyebrow.\n\n\"You won't have to,\" he said. Then, turning, he gestured to one of Ragnak's signalers, who stood nearby with a huge ram's horn in his hand. \"Get ready,\" he said, and the man raised the horn to his lips, pursing his mouth to form the right shape to create the mournful but penetrating note.\n\nIt was a game of cat and mouse. The smaller group of Temujai were pretending to retreat, all the while managing to stay engaged with the leading elements of the pursuing Skandians. For their part, they were simulating a wild and undisciplined pursuit, and getting farther and farther from their own lines. And all the while, the first Temujai force were circling back to fall on the exposed Skandians.\n\nThere was only one more element in the game, which was unknown to the Temujai leaders. Before dawn, Halt had directed a hundred Skandian axmen to take up positions in the fringe of the wooded slope bordering the valley. Concealed in hastily dug shallow trenches and behind fallen logs, they waited now for the signal that would tell them to make a surprise attack on the Temujai who were planning to surprise their comrades.\n\n\"Signal one,\" Halt said quietly, and the ram's horn sounded a single, extended note that echoed across the valley.\n\nInstantly, the pursuing Skandians, strung out in a long line behind the retreating Temujai riders, broke contact with the enemy and ran to form a defensive circle, their round shields forming an impenetrable wall. They were none too soon, as the second wave of Temujai horsemen was nearly upon them. As the eastern riders swept in, they were surprised to find an enemy already in a defensive formation and obviously awaiting them. The charge broke against the shield wall and another seething, struggling skirmish formed, with the hundred Skandians defending desperately against at least five times their number of horsemen.\n\nHaz'kam, commanding general of the Temujai invasion force, frowned from his command position as he watched the well-rehearsed, coordinated movement of the Skandians as they formed their shield wall.\n\n\"I don't like the look of this,\" he muttered to his second in command. \"This is not how these savages are supposed to react.\" And then the ram's horn rang out again, this time sounding three short, staccato notes that seemed to punch the air. A signal of some kind, he realized. But for what? And to whom?\n\nThe answer wasn't long in coming. There was a roar from the main Skandian ranks as a group of foot soldiers broke from the cover of the trees and ran to fall upon the encircling riders from the rear. The Skandian battle-axes took a terrible toll of the surprised Temujai, who found themselves suddenly and unexpectedly caught between the hammer of the new attacking force and the anvil of the shield wall. Surprised and confused, and with the momentum of their charge long since spent, the horsemen were easy marks for the savage northerners. In a matter of a few seconds, Haz'kam estimated that he had lost at least a quarter of his engaged force. It was time to cut his losses, he knew. He turned to his bugler.\n\n\"Retreat,\" he said quickly. \"Disengage and retreat.\"\n\nThe silver notes of the bugle spilled over the battlefield, cutting through the consciousness of the highly disciplined Temujai cavalry. This time, as they withdrew, they made no pretense of staying in contact with the Skandians. Their rapid disengagement showed how false their previous feigned retreat had been. In a matter of a few minutes, the riders were streaming back toward their own lines.\n\nFor a moment, it looked as if discipline and reason had forsaken the Skandians. Ragnak realized that, in the heat of the moment, they were on the verge of pursuing the retreating Temujai back to their own lines\u2014and to certain death for the Skandians. He quickly jumped up on the breastworks and bellowed, in his loudest storm-quelling voice: \"Kormak! Back here! Now!\"\n\nThere was no need for the ram's horn to reinforce the order. The Oberjarl's voice carried clearly to the Skandians and, as one, they ran for the shelter of the fortifications. Realizing what was happening, some of the Temujai sheathed their sabers and turned back to send a volley of arrows sailing after the Skandians.\n\nBut it was too little and too late. Apart from a few minor flesh wounds, there were no injuries.\n\nWill and Horace exchanged glances. So far, things had gone pretty well as Halt had predicted. But they didn't think the Temujai would be trying that particular trick again.\n\n\"Next time,\" said Will, \"it'll be our turn.\"\n**33**\n\nGENERAL HAZ'KAM TROTTED HIS HORSE ALONG THE FRONT rank of his army, watching as the first skirmish party made their way back to his lines. He had lost perhaps two hundred men, killed and wounded in that first encounter, he estimated. And perhaps half that number of horses. With an army of six thousand combat troops, of course, the numbers in themselves weren't terribly significant.\n\nWhat was significant, however, was the behavior of the Skandians. That first attack had been designed to reduce _their_ numbers by several hundred, not his own. In fact, there had even been the slight hope that the majority of the Skandians might have been drawn out from behind their defensive positions, into the exposed ground where they would have been easy meat for his mounted archers.\n\nHe reined in as he came level with a group of his officers. Among them, he recognized Colonel Bin'zak, his head of intelligence. The colonel was looking decidedly uncomfortable, he saw. As well he might be.\n\nHaz'kam caught his eye now and jerked his head toward the Skandian defenses.\n\n\"That was not what I was led to expect,\" he said. His voice was deceptively mild. The colonel urged his own horse forward a few paces and saluted as he came level with his commander.\n\n\"I don't know what happened, Shan Haz'kam,\" he replied. \"Somehow, they seemed to see through the trap. It's not the way I expected them to react. It's . . .\" He searched for the right words, finally saying weakly, \"It totally un-Skandian behavior.\"\n\nHaz'kam nodded several times. He held in his anger with an effort. It was undignified for a Temujai commander to show emotion on the field of battle.\n\n\"Does it occur to you, perhaps,\" he said eventually, when he was sure he could keep control of his voice, \"that the Skandians may have someone with them who knows our way of fighting?\"\n\nBin'zak frowned as he turned this thought over. In truth, it hadn't occurred to him. But now that the Shan mentioned it, it seemed the logical conclusion. Except for one factor.\n\n\"It would be unlike the Skandians to give field command to a foreigner,\" he said thoughtfully. Haz'kam smiled at him. But it was a smile without the faintest touch of humor in it.\n\n\"It was unlike them to break off their pursuit, form a shield wall and then hit us with a surprise attack from the woods too,\" he pointed out. The colonel said nothing to that. The truth of the statement was self-evident.\n\n\"There have been reports,\" the Shan continued, \"that a foreigner has been seen with the Skandians . . . one of those cursed Atabi.\"\n\n_Atabi_ , literally meaning \"the green ones,\" was the Temujai term for Rangers. In the years since Halt had made his successful horse raid, the Temujai leaders had attempted to gather as much knowledge as they could about the mysterious force of men who wore green and gray cloaks and seemed to meld into the forest. In the past few years, in preparation for this campaign, spies had even reached as far as Araluen itself, asking questions and seeking answers. They had learned little. The Rangers guarded their secrets jealously and the ordinary Araluens were reluctant to discuss the Ranger Corps with foreigners. There was a strong undercurrent of belief among Araluens that Rangers dabbled in magic and the black arts. Nobody was too keen to discuss such matters.\n\nNow, at this mention of an Atabi among the enemy, Colonel Bin'zak shrugged.\n\n\"They were rumors only, Shan,\" he protested. \"None of my men could confirm the fact.\"\n\nThe general's gaze locked on his. \"I think we've just had it confirmed,\" he said, holding the colonel's eyes until the officer looked down and away.\n\n\"Yes, Shan,\" he said bitterly. He knew his career was finished. Haz'kam now raised his voice, addressing the other officers gathered around and dismissing the matter of the disgraced intelligence colonel.\n\n\"It might also explain why our own planned surprise attack from the ocean failed to materialize,\" he said, and there were a few assenting grunts. The plot with Slagor had also been hatched by Bin'zak. Now, it seemed, the 150 men who had embarked on the Skandian ships four days ago had simply vanished into thin air.\n\nThe general came to a decision. \"No more subterfuge. We've wasted enough time here. We've been delayed by three weeks already. Standard attack from now on: rolling arrow storm until we create a weakness, then we drive through their line.\"\n\nHis commanders nodded their assent. He looked around at them, seeing their determination, their grim confidence. The Temujai were about to do what they did best, using their mobility and the devastating force of their mounted archers to probe and weaken the enemy line. Then, when the moment was right, they would drive in with their sabers and lances and finish the job. There was no shouting of battle cries, no histrionics from these men.\n\nThis was a normal day at work for them.\n\n\"Give your orders,\" Haz'kam told them. \"Watch for my commands.\"\n\nHe wheeled his horse, ready to ride back to the knoll where he had set up his command position. Already, signal flags were beginning to order the standard assault. A voice from behind made him pause.\n\n\"General!\" It was Bin'zak. He had forsaken the social honorific of \"Shan,\" Haz'kam noticed, and addressed him by his military title. The general faced his intelligence colonel now, waiting for his next words.\n\n\"Permission to ride with one of the Ulans, sir,\" Bin'zak said, his head held high. _Ulan_ was the Temujai word for the formation of sixty riders that was the basic unit of the Temujai force. Haz'kam considered the request. Normally, field grade officers were kept out of the close contact part of battles. They had no need to prove their courage or dedication. The general finally nodded permission.\n\n\"Granted,\" he said, and spurred his horse back to the command position.\n\n\"Now what?\" said Ragnak irritably as he watched the Temujai cavalry forming into groups.\n\nHalt watched too, his eyes narrowed. \"Now, I think, it's the end of the opening gambits. Now they're going to hit us in earnest.\" He pointed with his bow, sweeping it along the line of mounted horsemen facing them. \"They'll fight in their Ulans, sixty men in each unit, hitting us all along the line and wheeling away before we can respond. The idea is to pick off as many of our men as possible with arrows before launching a concentrated attack at a selected spot.\"\n\n\"Which is where?\" Erak asked. This tactical talk was making him increasingly cross. All he wanted was a dozen or so Temujai within reach of his ax. Now it appeared he would have to continue waiting for that eventuality.\n\nHalt turned to the signaler with the horn.\n\n\"Give the 'ready' call for the archers,\" he said, and as the man blew a series of long short, long short notes, he replied to Erak's question: \"Wherever their general decides they've created a weakness in our line.\"\n\n\"So what do we do while we're waiting for him to make up his mind?\" Ragnak asked irritably. Halt grinned to himself. Patience certainly wasn't high on the Skandian list of virtues, he thought.\n\n\"We surprise them with our own archers,\" he said. \"And we try to kill as many of them as we can before they become used to the fact that someone's shooting back at them.\"\n\nAll of Will's hundred archers heard the horn signal and there was an instant stirring among them. He held up a hand to calm them.\n\n\"Stay down!\" he called. He took his time and was pleased that his voice didn't crack. Maybe that was the answer for the future, he thought. He climbed up on the raised step that had been built into his command position. Horace, his shield ready, stood beside him. The wicker breastworks still concealed the archers but, when the time came, they would be pushed aside and the shield bearers would have the responsibility of protecting them from the answering storm of arrows that the Temujai would send their way.\n\nBelow Horace and Will's more exposed position, protected by earthworks and a wicker overhang, Evanlyn crouched in her position, with a clear sight of the line of archers.\n\nThe assembled troops of horsemen began to move now, cantering slowly at first, then at increasing speed. Will could see that this time, each man was armed with a bow.\n\nThey thundered toward the Skandian line\u2014not in one extended line as they had before, but in a dozen separate groups. Then, a hundred meters from the Skandians, each group wheeled so they were heading in a dozen different directions and sending volley after volley of arrows arcing up and over the Skandian lines.\n\nWill drummed his fingers nervously on the breastworks before him. He wanted to see the Temujai pattern before he committed his men. The first surprise would have the maximum potential to disrupt the enemy and he wanted to make sure he didn't waste it.\n\nNow there was a continuous rattle as the raised Skandian shields caught the majority of the arrows that the Temujai were pouring in. But not all. Men were falling along the Skandian lines and being dragged back out of the battle line by those behind them, who then stepped in to replace them. Now the second and third ranks of Skandians held their shields high, to protect them against plunging fire, while the front rank presented their shields to the more direct frontal fire.\n\nIt was an effective ploy. But it left the men blinded to the approach of the Temujai. Now, as Will watched, one group of sixty quickly slung their bows, drew sabers and darted into the Skandian line in a slashing attack, killing a dozen men before the Skandians even realized they were there. As the Skandians re-formed and moved to counterattack, the Temujai withdrew rapidly and another Ulan, waiting for this exact opportunity, poured a deadly hail of arrows into the disrupted shield wall.\n\n\"We'd better do something,\" Horace muttered. Will held his hand up for silence. The seemingly random movements of the Temujai Ulans actually had a complex pattern to them, and now that he had seen it, he could predict their movements.\n\nThe horsemen were wheeling again, galloping away from the Skandian line and back to re-form. Behind them, more than fifty Skandians lay dead, victims of either the arrows or the slashing Temujai sabers. Half a dozen Temujai bodies lay around the breastworks where the Ulans had made their lightning attack.\n\nThe Temujai riders were back in their own lines now. They would rest their horses, letting them recover their wind, while another ten Ulans took up the attack. It would be the same pattern, forcing the Skandians to cover up behind their shields, then attacking with sabers when they were blinded and, finally, pouring in volley after volley of arrows as their own men withdrew, leaving a gap in the shield wall. It was simple. It was effective. And there was a deadly inevitability about it.\n\nNow the Ulans began their wheeling, galloping dance once again. Will fixed his attention on a troop at the middle of the line, knowing that it would curve and turn and eventually come at them on a diagonal. He muttered to Horace.\n\n\"Get those breastworks down.\"\n\nHe heard the muscular apprentice bellow: \"Shields! Uncover!\" The shield bearers rushed to shove the wicker walls down, leaving the archers behind a waist-high earth berm and with a clear field.\n\n\"Ready!\" called Evanlyn, indicating that each man in the line of archers had an arrow nocked to the string. Then it was up to Will.\n\n\"Half left!\" he called, and the archers all turned to the same direction.\n\n\"Position two!\"\n\nA hundred arms raised to the same angle as Will watched the approaching group of riders, seeing in his mind's eye the galloping Temujai and the flight of arrows converging to meet at the same point in time and space.\n\n\"Down a half . . . draw!\"\n\nThe elevation corrected and one hundred arrows came back to full draw. He paused, counted to three to make sure he wasn't too soon, then yelled:\n\n\"Shoot!\"\n\nThe slithering, hissing sound told him that the arrows were on their way. Already, the archers were reaching for their next shafts.\n\nHorace, about to call for the shield bearers, waited. They were under no direct attack at the moment and there was no need to disrupt the sequence of shooting and reloading at this stage.\n\nThen the first volley struck home.\n\nMaybe it was luck. Maybe it was the result of the weeks of practice, hour after hour, but Will had directed that first volley almost perfectly. One hundred shafts arced down to meet the galloping Ulan and at least twenty of them found targets.\n\nMen and horses screamed in pain as they crashed to the ground. And instantly, the disciplined, structured formation of the Ulan was shattered. Those who were unhurt by the arrows were confronted by their comrades and their horses tumbling and rolling headlong. And as each stricken man fell, he took another with him, or caused his neighbor to swerve violently, reining his horse in, sawing on the reins until the tight formation was a milling mass of plunging horses and men.\n\n\"Ready!\" called Evanlyn. From her position, she couldn't see the result. Quickly, Will realized he had the chance now to deal a devastating blow to the enemy.\n\n\"Same target. Position two. Draw . . .\" He heard the scrape of arrows against bows as the men drew back their right hands until the feathered ends of the shafts were just touching their cheeks.\n\n\"Shoot!\"\n\nAnother volley hissed away at the tangle of men and horses. Already, Will was yelling for his men to reload. In their haste, some of them fumbled, dropping the arrows as they tried to nock them. Wisely, Evanlyn decided not to wait until they had recovered.\n\n\"Ready!\" she called.\n\n\"Same target. Position two. Draw . . .\"\n\nThey had the range and the direction now and the Temujai troop was stalled, caught in the one spot, losing their most valuable protection\u2014their mobility.\n\n\"Shoot!\" yelled Will, not caring that his voice cracked with excitement, and a third volley was on its way.\n\n\"Shields!\" bellowed Horace, shoving his own shield forward to cover himself and his friend. He had seen that some of the other Ulans had finally noticed what was happening and were riding to return fire. A few seconds later, he felt the drumming of arrows against the shield, heard the rattle as they struck other shields along the line of archers.\n\nThere was no way that the Temujai could send a squad with sabers in toward the archers. Halt had placed Will and his men to one side and behind the Skandian main line of defense. To reach them, the Temujai would have to fight their way through the Skandian axmen.\n\nThe troop that Will had engaged had taken three carefully aimed volleys\u2014nearly three hundred arrows\u2014in quick succession. Barely ten men of the original Ulan remained alive. The bodies of the others littered the ground. Their riderless horses were galloping away, neighing in panic.\n\nNow, as the other riders wheeled away toward their own lines, Will saw a further opportunity. Another two Ulans were riding in close proximity and still well within range.\n\n\"Shields down,\" he said to Horace, and the warrior passed the message along.\n\n\"Target: right front. And a half . . . Position three . . . draw . . .\" Again, he made himself wait, to be sure. \"Shoot!\"\n\nThe arrows, dark against the clean blue of the sky, arced after the withdrawing cavalry.\n\n\"Shields!\" Horace called as the arrows struck home and another dozen or so Temujai tumbled from their saddles. Behind the shelter of the big, rectangular shield, he and Will exchanged grins.\n\n\"I think that went rather well,\" said the apprentice Ranger.\n\n\"I think it went rather well indeed!\" the apprentice warrior agreed with him.\n\n\"Ready!\" called Evanlyn once more, her gaze fixed on the archers as they fitted arrows to their bowstrings. The call reminded Will, a little belatedly, that she had no way of knowing how successful their first action had been.\n\n\"Stand down!\" Will called. There was no point keeping the men tensed up while the Temujai were re-forming. He gestured to Evanlyn.\n\n\"Come on up and see the results,\" he told her.\n**34**\n\nIT TOOK SEVERAL MINUTES FOR THE TEMUJAI COMMANDER TO realize that something had gone badly wrong\u2014for the second time. There was a gap in his line as the riders returned, he realized. Then, as he cast his glance over the battlefield, he saw the tangled bodies of men and horses and frowned. He had been watching the overall action and had missed the four rapid volleys that had destroyed the Ulan.\n\nHe pointed with his lance at them. \"What's happened there?\" he demanded of his aides. But none of them had seen the destruction as it took place. His question was greeted with blank stares.\n\nA single horseman was pounding toward them, calling his name.\n\n\"General Haz'kam! General!\"\n\nThe man was swaying in the saddle and the front of his leather vest was slick with blood from several wounds. Blood stained the flanks of his horse as well, and the Temujai command staff were startled to see that the horse had been hit by at least three arrows.\n\nHorse and rider skidded to a stop in front of the command position. For the horse, it was the final effort. Weakened by loss of blood, it sank slowly to its knees, then rolled over on its side, its injured rider only managing to escape being pinned at the last moment. Haz'kam frowned as he peered at the wounded man, then recognized Bin'zak, his former chief of intelligence. True to his word, the colonel had taken his place in the front line of one of the Ulans. It had been his incredible misfortune that he'd chosen the one destroyed by Will's archers.\n\n\"General,\" croaked the dying man. \"They have archers . . .\"\n\nHe staggered a few paces toward them and now they could see the broken-off stubs of arrows in two of his wounds. On the ground beside him, the horse heaved a gigantic, shuddering sigh and died.\n\n\"Archers . . .\" he repeated, his voice barely audible, and he sank to his knees.\n\nHaz'kam tore his gaze away from the stricken colonel and scanned the enemy ranks. There was no sign of archers there. The Skandians stretched in three ranks across the narrowest part of the valley, behind their earthworks. On the seaward side, and a little behind the main force, another group stood\u2014also behind earthworks and holding large rectangular shields. But he could see no sign of archers.\n\nThere was one sure way to find them, he thought. He gestured toward his next ten Ulans.\n\n\"Attack,\" he said briefly, and the bugler sounded the call. Once more, the valley filled with the jingle of harness and the thunder of hooves as they drove forward.\n\nIn front of him, the colonel slumped forward, facedown in the sodden grass. Haz'kam made the Temujai gesture of salute, raising his left hand to his lips, then extending it out to the side in an elaborate, flowing movement. His staff did likewise. Bin'zak had redeemed himself, he thought. In the end, he had brought his general a vital piece of intelligence, even if it had cost him his own life.\n\nWill watched the approaching cavalry as, once again, they began their wheeling, circling dance. Horace stirred beside him, but some sense warned the young Ranger not to expose his men yet.\n\n\"Wait,\" he said quietly. He had half expected that a concerted attack would be launched toward their position, in an attempt to wipe them out. But this attack was like the previous one, launched along the entire front. That could mean only one thing: the Temujai leaders hadn't pinpointed the archers' position.\n\nArrows began falling on the Skandian lines and once more, the three ranks covered up with their shields. As before, a troop of Temujai broke off their maneuvering and drew sabers to launch a lightning attack on the unsighted Skandians. This time, however, Will was looking beyond them, to identify the support group who would open fire on the Skandians as their comrades withdrew. He saw them: a Ulan that had drawn to a halt some fifty meters from the Skandian front rank.\n\n\"Load!\" Will yelled down his line. Then, in an aside to Horace: \"Keep the shields up.\" He had felt the larger youth draw breath to call his next order. But Will wanted to keep his men hidden as long as possible.\n\n\"Ready!\" Evanlyn called as the last arrow nocked onto the string.\n\n\"Face left half left again!\" he called, and the archers, luckily, understood his meaning. As one, they all turned to face the direction he had picked. He had varied their drill by calling direction first but they seemed to understand what he wanted.\n\n\"Position three!\" he yelled. The arms came up to maximum elevation, the hundred of them moving as one.\n\n\"Shields down,\" he muttered to Horace and heard him repeat the order.\n\n\"Draw!\"\n\nBeneath his breath, he told himself, \"Count to three as each arm brings back its arrow to the full draw.\"\n\nThen, aloud: \"Shoot!\" and instantly, he screamed: \"Shields! Up shields!\" As Horace took up the cry, the shields swung back into position to conceal the archers from return fire\u2014and, hopefully, from observation.\n\nAgain the wait, then the volley of arrows slammed down into the Temujai Ulan, just as they were on the point of firing into the gap their comrades had forced in the shield wall. Once more, men and horses went down in screaming, tangled heaps. Grouped together as they were, and not moving, the Ulan made a perfect target for the massed arrows.\n\nAt least twenty of them were down, including their commander. Now their sergeants were yelling at the survivors to get moving. To get out of this killing ground.\n\nHaz'kam never saw the volley that struck his men. But he did see, in his peripheral vision, the concerted movement of the hundred shields as they swung back and forth like so many gates opening and closing. A few seconds later, he saw one of his foremost Ulans collapse and disintegrate.\n\nAnd then the shields moved again and he saw the archers. At least a hundred of them, he estimated, working smoothly and in unison as they launched another volley at the retreating Ulan that had attacked the Skandian line. The shields swung closed to cover the archers as more Temujai riders went down.\n\nAgain, the shields swung aside in unison, and this time he saw the solid flight of arrows, black against the sky, as they arced up and struck into another of his galloping Ulans. He turned and caught the eye of his third son, a captain on his staff. He pointed with his lance to the line of shields on the slight rise behind the Skandian ranks.\n\n\"There are their archers!\" he said. \"Take an Ulan and investigate. I want information.\"\n\nThe captain nodded, saluted and clapped spurs to the barrel-shaped body of his horse.\n\nHe was shouting commands to the leader of the nearest troop of sixty as he galloped down the front line of the Temujai army.\n\nIn their raised position behind the Skandian lines, Will and Horace were working smoothly together, pouring volley after volley into the wheeling riders. Inevitably now, they were beginning to take casualties as individual Temujai saw them and returned fire. But the shield drill worked smoothly and their improvised method of exposing the men to return fire for only a few seconds at a time was paying dividends.\n\nWhat was more, the Skandians were beginning to see the effect of the disciplined, concentrated fire on their enemies. As each volley hissed down, as arrows found their marks and Temujai saddles emptied, the waiting axmen roared their approval.\n\nFor the first time, Will had seen the Kaijin sharpshooters attached to each Ulan as they attempted to take him and Horace under fire. He had just dueled with two of them and watched in satisfaction as the second slumped sideways out of his saddle. Horace nudged his arm and pointed.\n\n\"Look,\" he said, and Will, following the line he indicated, saw a Ulan galloping from the Temujai lines and heading straight for them. There was no wheeling and turning for these riders. They were coming straight on at a dead run. And it was obvious where they were heading.\n\n\"We've been spotted,\" he said. Then, calling to his men: \"Face front half right. Load!\"\n\nHands reached for arrows, nocked them firmly to strings.\n\n\"Ready!\" That was Evanlyn once more. He grinned as he thought of how Halt had questioned the need for her to be here. Suddenly, he was glad the grizzled Ranger had lost that argument. He shook the thought aside, estimating the speed of the oncoming riders. Already, they were shooting, and arrows were rattling on the shields along the line. But all the advantages lay with Will and his men. Shooting from a stable, unmoving, elevated position, and from behind cover, they held the upper hand in any exchange.\n\n\"Position two!\" he called. \"Draw!\"\n\n\"Shields down!\" Horace yelled, giving Will just the right pause.\n\n\"Shoot!\" shouted Will.\n\n\"Shields up!\" roared Horace, covering his friend as he did so.\n\nThe archers were exposed to return fire for no more than a few seconds. Even so, under the constant barrage of arrows from the Temujai, they took a few casualties. Then their volley hit the on-rushing Ulan and wiped out the front rank of twelve, sending men and horses tumbling yet again. The riders in the following ranks tried to avoid their fallen comrades, but in vain. More horses came down; more riders tumbled out of their saddles. Some managed to leap their horses over the tangle of bodies and they were the ones who rode clear. As the others tried to reorganize, another volley, ten seconds behind the first, fell on them.\n\nHaz'kam's son, with one arrow through his right thigh and another in the soft flesh between neck and shoulder, lay across the body of his horse. He watched as the shields opened and shut and the arrows poured out in a constant, disciplined stream. He saw the two heads moving in the fortified position at the end of the archer's line.\n\nThat was what his father needed to know. He watched as another two volleys hissed into the sky. Thankfully, these were directed at another Ulan as it galloped past. He could actually hear the commands as the two men in the command position called them. One of the voices sounded absurdly young.\n\nIt was growing dark early, he reflected, and promptly realized that it could be no later than midmorning. He craned painfully to look at the sky. But it was a brilliant blue and, with a sudden thrill of fear, he realized he was dying. He was dying, with urgent information that he must pass on to his father. Groaning in pain, he dragged himself to his feet and began to stumble back toward the Temujai lines, picking his way through the tangle of fallen bodies.\n\nA riderless horse cantered past him and he tried to catch it but was too weak. Then he heard a thunder of hooves behind him and a strong hand gripped the back of his sheepskin jacket and hauled him up and over a saddlebow, where he gasped and moaned with the pain in his neck and leg.\n\nHe angled around to see his savior. It was a sergeant from one of the other Ulans.\n\n\"Take me . . . to General Haz'kam . . . urgent message,\" he managed to croak, and the sergeant, recognizing the staff insignia on his shoulders, nodded and wheeled his horse toward the command post.\n\nThree minutes later, the mortally wounded captain told his father all that he had seen.\n\nFour minutes later, he was dead.\n**35**\n\nFROM THE CENTRAL COMMAND POSITION, HALT AND ERAK watched as the smooth drill of the archers caused havoc among the Temujai ranks. Now that the attacking force was aware of them, Will's men had no chance to repeat the devastating casualties of those first three volleys that had all but wiped out a complete Ulan. But the regular, massed fire of one hundred archers, and Will's accurate direction, was breaking up attack after attack.\n\nIn addition, the Temujai now realized that their own favorite tactic had been effectively countered. If they sent one group into close combat while another stood off to provide covering fire during the withdrawal, they knew that the second group would instantly come under fire from the archers on the Skandian right flank. It was a new experience for the Temujai. Never before had they encountered such disciplined and accurate return fire.\n\nBut they were no cowards, and some of the commanders were now substituting raw courage and ferocity for tactical ploys. They began to storm toward the Skandian line, abandoning their bows and drawing sabers, trying to break through in close-in fighting, determined to bury the Skandians under sheer numbers if necessary.\n\nThey were brave and skillful fighters, and against most adversaries they might have faced, their ploy would probably have succeeded. But the Skandians reveled in hand-to-hand fighting. To the Temujai it was a matter of skill. To the northerners, it was a way of life.\n\n\"This is more like it!\" Erak bellowed cheerfully as he moved forward to intercept three Temujai scrambling over the earthen bulwark. Halt felt himself shoved to one side as Ragnak rushed to join his comrade, his own battle-ax causing terrible havoc among the small, stocky warriors who were swarming over their position.\n\nHalt stood back a little, content to let the Skandians take on the brunt of the hand-to-hand fighting. His gaze roamed outside the area of immediate engagement until he saw what he was looking for: one of the Temujai marksmen, recognizable by the red insignia on his left shoulder, was searching the milling crowd of men for the Skandian leaders. His eyes lit on Ragnak as the Oberjarl called more of his men into the breach the Temujai had forced. The Temujai's recurve bow came up, the arrow already sliding back to full draw.\n\nBut he was two seconds behind Halt's identical movement, and the Ranger's huge longbow spat its black-painted shaft before the Temujai had reached full draw. The rider never knew what hit him as he tumbled backward over the withers of his horse.\n\nSuddenly, the savage little battle was over and the surviving Temujai were scrambling back down the earth slope, capturing any horses they could and hauling themselves into the saddles.\n\nRagnak and Erak exchanged grins. Erak slapped Halt on the back, sending him reeling.\n\n\"That's better,\" he said, and the Oberjarl growled agreement. Halt picked himself up from the dirt.\n\n\"I'm so glad you're enjoying yourself,\" he said dryly. Erak laughed, then became serious as he nodded his head toward the right flank and the small group of archers, still pouring steady fire into the attackers.\n\n\"The boy has done well,\" he said. Halt was surprised to hear there was a note of pride in his voice.\n\n\"I knew he would,\" he replied quietly, then turned as Ragnak dropped a ponderous arm around his shoulders. He wished the Skandians didn't have to be quite so touchy-feely in expressing their feelings. Built the way they were, they put normal people at risk of serious damage.\n\n\"I've got to admit it, Ranger, you were right,\" the Oberjarl said. He swept his arm around the fortifications. \"All of this, I didn't think it was necessary. But I can see now that we would never have stood a chance against those devils in an open conflict. As for your boy and his archers,\" he continued, gesturing toward Will's position, \"I'm glad we looked after him when we first caught him.\"\n\nErak raised one eyebrow at that. It had caused him considerable anger that Will had been assigned to the freezing conditions of labor in the yard\u2014an assignment that should have meant almost certain death. He said nothing, however. He assumed that being supreme leader gave one a license to forget uncomfortable events from the past.\n\nHalt was studying Will's position with a critical eye. The defensive line in front of the archers was still well manned. Of all the Skandian positions, it seemed to have suffered the lowest number of casualties. Obviously, he thought, the Ulans were avoiding direct confrontation at that point. They'd seen what had happened to the troop that had charged directly at the archers.\n\nBut he knew that the Temujai general couldn't allow this situation to continue. He was losing too many men\u2014both to the constant volleys of arrows and in the desperate hand-to-hand fighting with the Skandians. Soon, he would have to do something to nullify the unexpected problem posed by the archers.\n\nHe would have been interested, but not surprised, to know that Haz'kam's thoughts were running on pretty much the same lines.\n\nThe general cursed softly as he studied the casualty reports brought in by his staff.\n\nHe turned to Nit'zak, his deputy commander, and indicated the sheet of parchment in his hand.\n\n\"We cannot go on like this,\" he said softly. His deputy leaned toward him, turning the sheet of hastily scribbled casualty figures so that he could read it. He shrugged.\n\n\"It's bad,\" he agreed. \"But not disastrous. We still have the numbers to defeat them, archers or no archers. They can't stand against us indefinitely.\"\n\nBut Haz'kam shook his head impatiently. Nit'zak had just confirmed what he had always suspected. His deputy was a capable leader in the field, but he lacked the overview necessary to make him a commanding general.\n\n\"Nit'zak, we've lost almost fifteen hundred men\u2014either killed or wounded. That's nearly a quarter of our effective force. We could easily lose that many again if we keep on like this.\"\n\nNit'zak shrugged. Like most Temujai senior officers, he cared little for the size of his casualty reports, as long as he won the battle. If Temujai warriors died in battle, he thought, that was their role in life. Haz'kam saw the gesture and correctly interpreted the thinking behind it.\n\n\"We're two thousand kilometers from home,\" he told his deputy. \"We are supposed to be subjugating this frozen little corner of hell so that we can mount an invasion of the Ara-land. How do you propose that we do that with less than half the force we started with?\"\n\nAgain, Nit'zak shrugged. He really didn't see the problem. He was accustomed to victory after victory and the idea of defeat never occurred to him.\n\n\"We knew we'd take casualties here,\" he protested, and Haz'kam let go a string of curses in an unaccustomed display of temperament.\n\n\"We thought this would be a _skirmish_!\" he spat angrily. \"Not a major engagement! Think about it, Nit'zak: a victory here could cost us so much that we might not even make it home again.\"\n\nThat was the uncomfortable truth. The Temujai had two thousand kilometers to cover before they reached their homeland on the steppes once more. And all two thousand were across hostile, temporarily conquered territories\u2014territories whose inhabitants might welcome the opportunity to rise up against a weakened Temujai force.\n\nNit'zak sat his horse in silence. He was angry at the tone of rebuke in his commander's voice, particularly in front of the other staff officers. It was a gross breach of Temujai behavior for Haz'kam to speak to him in such a fashion.\n\n\"So . . . what do you propose?\" he asked finally.\n\nFor a long time, the general didn't answer. He gazed across the intervening space to the Skandian lines, looking from the command position in the center to the line of archers drawn up on his left\u2014the Skandian right wing. Those two positions, he knew, held the key to this battle.\n\nFinally, he turned to his deputy, his mind made up.\n\n\"Strip the first fifty Ulans of their Kaijin,\" he ordered. \"And assemble them here as a special force. It's time we got rid of those damned archers.\"\n**36**\n\n\"HERE THEY COME AGAIN,\" HORACE SAID, AND WILL AND Evanlyn both turned to look toward the Temujai forces. The riders were cantering forward again, and this time it looked like a major attack. Haz'kam had committed nearly two thousand men to a frontal assault on the Skandian lines. They rode forward, their hoofbeats echoing in the valley, formed in a wedge shape that was aimed at the Skandian center and the command post where Halt, Erak and Ragnak directed the Skandian defense.\n\nWill and Evanlyn had taken advantage of the lull in the fighting to take a quick bite to eat, and a welcome drink of water. Will's throat was parched, from both the tension and the nonstop shouting of orders. He guessed Evanlyn felt the same. Horace, who had already eaten, had been keeping watch. Now, at his call, Evanlyn slipped down into her sheltered position and the archers, who had been sprawled comfortably against the earthworks, came to their feet, bows in hand. The shield bearers, who had also been relaxing, took their positions beside them.\n\nSilently, they waited. In the lull, the arrow bin in front of each archer had been replenished with new shafts. Even now, the women of Hallasholm were gathered in the Great Hall, making fresh arrows for the battle.\n\nWill studied the mass of riders. He had seventy-five archers still standing in the line, several of them lightly wounded. They had lost eleven men, killed by Temujai arrows, and a further fourteen had been wounded too seriously to continue fighting. As the Temujai force advanced, Will estimated that he could manage four volleys before they reached the Skandian line. Maybe five. That would be three hundred arrows raining down on the tightly packed mass of horsemen, and in that formation, the incidence of hits would be high. If Will aimed for the center of the mass, even his undershoots and overshoots would be effective.\n\n\"Left front, position three!\" he called, and the machine swung into action again.\n\n\"Ready!\" called Evanlyn.\n\n\"Draw . . . shoot!\" shouted Will. He gestured for Horace not to call the shields into position. As yet, they were not under attack. The more time he had to do damage to that mass of Temujai horsemen, the better chance he would give Halt and Erak to repel the Temujai's main thrust.\n\n\"Reload!\" he called, and waited for Evanlyn's call once more. When it came, he sent another volley on its way. As it started its upward trajectory, the first volley came down and he saw horsemen falling once again.\n\n\"Left half left!\" he called, swinging the aiming point to match the progress of the horsemen as they moved from right to left across his front. He called the elevation again, shortening this time, then another seventy-five shafts soared away with that now-familiar slithering sound of arrows scraping across bows. Now the horsemen were galloping and he adjusted the angle once more.\n\n\"Left left! Position two,\" he called. Evanlyn's call told him that the men had reloaded.\n\n\"Draw . . . shoot!\"\n\nAnd now he heard the first sounds of close combat as the leading ranks of horsemen made contact with the Skandian lines. It would be too risky to try to shoot into the Temujai front ranks now, but he could still interdict the ranks behind them.\n\n\"Left half left!\" he called, and the archers swung their aim point back to the right by twenty degrees. Then suddenly, the air around him was alive with the hissing sound of arrows and all along the line his archers were falling, some crying out in pain and shock and others, more ominously, silent.\n\n\"Shields! Shields!\" Horace was yelling and the shield bearers moved into position\u2014but not before more archers went down. Desperately, Will swung around and saw, for the first time, the smaller group that had moved forward to attack his position while he had been busy engaging the main force. There were about fifty archers, he estimated, all mounted, pouring steady, accurate shots into his position. Behind them rode another, larger group armed with lance and saber.\n\n\"Target front!\" he called, and muttered an aside to Horace: \"Be quick with those shields when we need them.\"\n\nThe warrior apprentice nodded, watching anxiously as the fifty riders continued to shoot. Now arrows were thudding into his own shield, and into the earth rampart in front of them.\n\n\"Position one!\" Will called. This was straight and level\u2014point-blank range. \"Draw!\"\n\n\"Ready!\" he heard Evanlyn call. Then Horace yelled for the shields to open and Will, almost on top of him, called for the release.\n\nAs the volley hissed on its way, Horace was already calling for the shields to come back into position again. But even in that short time, another half dozen of their men went down to the Temujai arrows.\n\nNow Will noticed the red insignia on the Temujai shoulders and he realized why the standard of enemy archery had picked up in accuracy and rate of fire.\n\n\"They're all Kaijin!\" he said to Horace. As he spoke, he raised his own bow and, shooting rapidly, emptied three saddles before Horace dragged him behind the shelter of his shield again. Half a dozen shafts slammed into it as he did so.\n\n\"Are you mad?\" Horace cried, but Will's eyes were wild with pain as he looked up at his friend.\n\n\"They're killing my men!\" he replied, and went to lunge out into the open once more, obsessed with the idea of stopping the Temujai specialists from picking his men off one at a time. Horace's big hand stopped him.\n\n\"It won't help if they kill you!\" he yelled and, slowly, the sense of it all sank into Will's brain.\n\n\"Ready!\" called Evanlyn. He realized that it was the third time she had given the call. She was prompting him to action. Still covered by Horace's shield, he assessed the position.\n\nThe lancers and swordsmen, unhampered by any harassing fire from the archers, were already closing with the Skandians in front of his position. Hand-to-hand fighting was breaking out along the line. Farther to his left, the main body of Temujai were engaged in a savage battle with the center of the Skandian line. The position was too confused to see who was winning if, indeed, anyone was.\n\nMeanwhile, to his front, the Temujai marksmen, gathered by Haz'kam into a special unit, were cantering parallel to the Skandian defensive line, widely dispersed so as not to offer a massed target to his volleys, and engaging his archers with accurate, aimed shots as they were exposed. He knew that if he attempted to direct another volley at the Temujai, he would lose half his men in the exchange. There was only one solution now, he realized. He leaned over his parapet, yelling to the line of archers below him\u2014a line that was now severely depleted, he saw.\n\n\"Individual shots!\" he yelled, pointing to the cantering lines of Temujai Kaijin. \"Shoot whenever you're ready and aim for their bowmen!\"\n\nIt was the best he could do. At least this way the Temujai would not be presented with an open line of shields as his men fired. They would have to react to individuals firing irregularly. It would give his men a better chance of survival. It would also lessen the effectiveness of their shooting, he knew. Without central direction, their accuracy would fall away.\n\nThere was, however, one more thing he could do. He glanced down to make sure that the arrow bin in front of him was fully charged and quickly plucked four shafts out, nocking one and holding the others ready between the fingers of his bow hand.\n\n\"Keep that shield up and ready,\" he said to Horace, and stepped forward to the parapet, still concealed by his friend's large shield. He took a deep breath, then stood clear and let the four shafts go in rapid succession, spinning back behind the cover of the shield as the first Temujai shafts whistled around their ears in reply. Horace, watching, saw two of the shooters go down to Will's arrows. A third took an arrow in the fleshy part of his calf and the fourth arrow missed entirely. He whistled in admiration. It was remarkable shooting. He was about to say something to that effect when he noticed the look of total concentration on his friend's face and decided to say nothing. Again, Will took a deep breath, nocking another arrow, then spun out into the open, loosed again and lunged back into cover.\n\nNow Horace began to truly appreciate the uncanny accuracy that had been drilled into his friend in the woods and fields around Castle Redmont, as Will spun in and out of cover, loosing off shots\u2014sometimes one, sometimes two or three\u2014and hitting mark after mark. The other archers in the Skandian force added their contributions as well, but none of them possessed the speed and accuracy of the apprentice Ranger. And as more of them were struck by counterfire from the patrolling Kaijin, the survivors became more and more nervous and arrow-shy, more likely to shoot without aiming, then dive back behind cover again.\n\n\"Change sides,\" Will ordered him briefly, gesturing for Horace, who had been standing to his left, to step across to the right. Horace shifted the shield to his right arm and Will ducked below the breastworks level and moved to Horace's left side. He had been varying his shooting pattern, sometimes shooting just one arrow, and at others letting go a rapid volley, to keep the Temujai guessing. Now he decided that they were accustomed to seeing him appearing to the right of the big shield. He selected another four arrows and stepped to his left, shooting as he came clear. Two more saddles emptied and he darted back into cover again. The change in sides had worked for him. Not a single arrow had come near him in reply.\n\nHe stepped left again, snapped off another shot and then, not knowing what instinct prompted him do it, dropped immediately to his hands and knees behind the earthworks. A vicious hiss split the air directly above him as he did so and he felt his mouth go dry with fear. Horace, seeing him drop, thought he was hit and went to his knees beside him.\n\n\"Are you all right?\" he asked urgently. Will tried a weak grin but didn't really think it came off.\n\n\"I'm fine,\" he managed to croak around the dryness in his mouth. \"Just scared to death is all.\"\n\nThey stood again, sheltering behind the shield and feeling the rattle of Temujai arrows against it. Will realized that the pattern had changed once more and the majority of the Temujai archers were concentrating on his position. It was a chance for his men to release another massed volley, he realized. But if the Temujai saw or heard him preparing them for it, the element of surprise would be lost.\n\n\"Evanlyn!\" he called to the girl, sheltering in her covered position below him. She glanced up at him, a question in her eyes, and he continued: \"Relay my directions! We'll get in another volley!\"\n\nShe waved her hand, indicating that she understood.\n\nUnwittingly, as they had concentrated on his position, trying to hit the elusive figure who darted in and out of cover and peppered them with a deadly hail of arrows, the enemy had begun to bunch up. There had been little in the way of effective fire from the other archers for some minutes now and the Kaijin had all moved toward the command position to get a shot at Will.\n\n\"Front right!\" he called softly, and heard Evanlyn relay the order. Nothing happened for a moment or so, then he heard her berating the men below her, shaming them into compliance. Gradually, one after another, they turned to face the direction she had given them.\n\n\"Ready,\" she called back, and he gave her the elevation: position one. The bows came up to the horizontal, then steadied.\n\n\"Draw,\" he said, and heard the order relayed once more. Then, taking a deep breath, he yelled: \"Shields down! Shoot!\"\n\nAnd a fraction of a second after, as the volley was still on its way, he heard Horace call: \"Shields up!\"\n\nRealizing that attention would be focused for some seconds on the line of archers, Will darted into the clear and poured arrow after arrow into the Temujai ranks. His archers' volley struck home\u2014he had less than fifty men firing now, but still a mass of arrows slammed into the Temujai riders, sending a dozen of them sprawling into the dust. Then another five went down under the hail of arrows that Will had let loose before Horace, diving at him, dragged him below the level of the earthworks before the Temujai arrows could find him.\n\nA storm of arrows thudded into the earthworks behind them. Horace rolled clear of his friend, dusting the dirt from his knees and elbows.\n\n\"Do you have a death wish?\" he asked. Will grinned at him.\n\n\"I'm just relying on your judgment,\" he replied. \"I can't keep track of everything in my head.\"\n\nThey stood behind the shield again and saw that the Kaijin, what remained of them, were edging away to longer range. They still fired at the Skandian ranks, but with far less effect than before. Will frowned as he assessed angles and positions, then pointed to the center of the Skandian line, where the main battle was still raging.\n\n\"We can start shooting volleys again,\" he told Horace. \"If the shield bearers shift their shields to their right arms, and our archers stand to their left, they'll be covered from return shots.\"\n\nHorace studied the position and nodded agreement. The remaining Kaijin were directly to their front now, so that the line of archers could shoot diagonally toward the rear of the main Skandian army without having to move from behind the cover of the shields.\n\nHastily, they called their idea down to Evanlyn, who relayed the directions to the men. The depleted line of archers looked at their young commander and nodded their understanding. Then a thought occurred to him.\n\n\"Evanlyn!\" he called, and she looked up at him, her eyes questioning. \"Once we've started, you call the volleys. Keep them at position three and just keep them shooting. I'll keep those damned Kaijin honest.\"\n\nShe grinned at him and waved a hand in reply. There would be no need to change angle or elevation once they began. The plunging volleys would be directed at the mass of the Temujai rear ranks. It might give Halt, Erak and Ragnak the respite they needed.\n\n\"Face half left!\" called Will. \"Position three!\"\n\nThe forty-odd remaining men brought their bows to maximum elevation.\n\n\"Draw . . . shoot!\"\n\nHe waited this time to assess the effect of the volley, making sure the men's angle and elevation were correct. He saw arrows striking the support ranks of Temujai, saw the panic caused by the suddenly renewed arrow storm.\n\n\"Keep them shooting!\" he called to Evanlyn.\n\nHe turned and shot at the thin line of Temujai shooters, drawing a brisk shower of arrows in return. Behind him, he heard the thrumming noise of another volley arcing away toward the main battle. He shot again, picking a target and seeing him fall. Then he felt a surge of excitement in his chest as the small group of riders began to move.\n\n\"Horace! They're pulling back!\" he yelled excitedly. He pointed wildly to the line of shooters. Less than twenty of them remained and they were gradually falling back from their exposed position. Gradually at first, anyway\u2014as they moved farther, they moved faster and faster, none of them wanting to be the last one exposed to the accurate shots from the Skandian lines.\n\nHe gripped his big friend's arm and shook him with excitement. \"They're turning it in!\" he yelled. Horace nodded soberly, jerking his thumb toward the hard-pressed line of Skandian defenders below them.\n\n\"Just as well they are,\" he said. \"Because these ones aren't.\"\n\nBelow them, the Temujai swordsmen, dismounted now, were pouring through a gap they had forced in the Skandian lines.\n**37**\n\nNIT'ZAK, FIELD COMMANDER OF THE TEMUJAI FORCE ATTACKING Will's position, had poured his men into the attack with reckless disregard. As the Kaijin engaged the archers, his lancers and swordsmen hurled themselves against the line of Skandian axmen protecting them.\n\nNit'zak had sensed that this attack was a final throw of the dice for his commander. If they couldn't break through this time, he knew Haz'kam would order a general withdrawal, unwilling to take further casualties in this campaign. The thought of withdrawal, of failure, was anathema to Nit'zak. He urged his men on now, willing them to break through the Skandian line and destroy the small but highly effective force of archers who sheltered behind it.\n\nThe ground in front of the Skandian defenses was littered with the bodies of his men and horses. But gradually, they were driving the wild northerners back as their numbers were depleted and the defensive line became more fragile. Dismounted now, the Temujai swarmed up the earth slope, slashing and stabbing with their long-bladed sabers. Grimly, the Skandians fought back.\n\n\"General!\" One of his staff grabbed his arm and pointed to a small group of riders angling away from the battle. \"The Kaijin are withdrawing.\"\n\nNit'zak cursed them as they rode away. Pampered and privileged, he thought. He knew they regarded themselves as elite members of the Temujai force. Kaijin shooters were excused the dangers of direct combat so they could sit back and pick off enemy commanders in relative safety. Now, faced with accurate and deadly return shooting for the first time in their lives, they had broken and deserted him. He made a vow that he would see them all die for their cowardice.\n\nBut that would have to wait. Now, he realized, the Skandian archers were launching flight after flight of arrows into the rear ranks of the main attack once more. They had to be stopped. The sudden resumption of the deadly volleys could well tip the balance of the battle.\n\nHaz'kam had remarked that his deputy had no sense of the bigger picture when it came to warfare. But Nit'zak had an ability that made him a superb tactical commander. He could sense the crucial moment in a battle\u2014the moment when everything hung in the balance and a determined effort from either side could make the difference between victory and defeat. He sensed such a moment now, watching his men struggling with the Skandians, seeing, for the first time, an element of uncertainty in the enemy. He drew his saber from its scabbard and turned to his own personal bodyguard, a half-Ulan of thirty seasoned troopers.\n\n\"Come on!\" he yelled, and led them in a charge toward the Skandian line.\n\nNit'zak's instincts were accurate. The Skandians, exhausted and bleeding, their numbers depleted, were hanging on with their last reserves of strength and will. The Temujai numbers seemed never-ending. For every one who fell before the Skandian axes, it seemed another two rushed to fill his place, screaming their war cries and slashing and stabbing with their sabers. Now, as a fresh force drove into the line, dismounting and scrambling up the earth berm, the balance tipped. First one, then another Skandian gave way. Then they were retreating in groups, as the Temujai drove through the gap they had finally forced, striking down the fleeing Skandians as they tried to escape.\n\nNit'zak waved his saber toward the line of archers, still pouring volley after volley at the main attack.\n\n\"The archers! Kill the archers!\" he ordered his men, and started toward them.\n\nIn the command position, Horace threw down the bulky, clumsy shield he had been using and grabbed up his own round buckler. His sword slid from its sheath with an expectant hiss as he swung his legs over the parapet.\n\n\"Stay here,\" he told Will, then headed down the slope to meet the first group of Temujai as they clambered up toward him. Now it was Will's turn to watch in awe as his friend went on the attack. His sword moved in bewildering patterns, flicking in and out, overhead, backhand, forehand, thrust, as he cut down the attackers. The first attack was driven back and now a larger group of Temujai moved toward the tall warrior. Again there was the clash of steel on steel, but now, as they threatened to encircle him, Horace was forced to give ground. Will looked down at his arrow bin. There were five arrows left and he began shooting: steady, deliberate shots to pick off the Temujai who tried to surround his friend.\n\nHe glanced toward the archers. The shield bearers had grabbed up their own weapons and were moving to protect them. In addition, some of the retreating Skandians had regrouped at the archers' position. Evanlyn was still calling the volleys, he noticed.\n\n\"Keep it up!\" he yelled, and she glanced around, nodded and turned back to her task.\n\nHorace was almost back to the elevated command position now, still fighting off the determined attacks from the Temujai. He was fighting alone, however, and vulnerable from the rear. Will, his stock of arrows finally exhausted, drew his two knives and moved to protect his friend's back.\n\nIn the center of the Skandian line, Erak sensed a similar moment of opportunity. The Temujai were fighting hard, but the savage intensity had gone from their attacks.\n\nWeakened and demoralized by the regular downpour of arrows from the right flank, their support ranks were withdrawing, and leaving those troops engaged with the Skandian line without the regular reinforcements that they needed to maintain the rhythm of their attack.\n\nHe cut down a Temujai captain who had come screaming over the earthworks, and turned to look for Halt. The Ranger was positioned behind him, standing on a parapet and coolly picking off the Temujai as they came forward.\n\nHaz'kam's tactic of stripping his Ulans of their shooters was working against the Temujai here. For a change, it was they who were losing their commanders to accurate, aimed shots, while the Skandian leaders continued to devastate anyone who came within range of their whirling axes.\n\nDisengaging himself, Erak vaulted up beside Halt. He gestured to the Skandian left wing, so far uncommitted.\n\n\"I'm thinking if we hit them from the flank, we might finish them,\" he said. Halt considered the idea for a moment. It was a risk. But battles were won by taking risks, he knew. Or lost. He came to a decision.\n\n\"Do it,\" he agreed, and Erak nodded. Then he looked beyond Halt and cursed. The Ranger swung around to look in the same direction and together they watched the Temujai breaking through the line below Will's position. They both knew that if the rain of arrows stopped, the Temujai rear ranks might well recover their cohesion and their moment might be lost.\n\nNow was the time to act.\n\n\"Bring the left flank in,\" Halt said briefly. He grabbed up a spare quiver of arrows and started to run toward Will's command post. Erak watched him go, knowing that one man wouldn't make any difference. He looked around desperately, his gaze lighting on Ragnak, standing in the middle of a circle of fallen Temujai. The Oberjarl's eyes were wild and staring. He had discarded his shield and was swinging his massive ax two-handed. Blood streamed from half a dozen wounds on his body, but he seemed oblivious. He was on the point of berserking, Erak knew. And he also knew that one man like that might make all the difference in the world.\n\nErak cut his way through to the Oberjarl, winning a brief respite as the Temujai fell back from the two huge warriors. Ragnak looked up, recognized him and showed his teeth in a triumphant, savage grin.\n\n\"We're destroying them, Erak!\" he yelled, his eyes still wild. Erak grabbed him by the arm, shaking him to make him focus his attention.\n\n\"I'm bringing in the left flank!\" he yelled, and the Oberjarl smiled and shrugged.\n\n\"Good! Let them have some fun too!\" he bellowed. Erak pointed to the battle raging on the seaward side.\n\n\"The right wing is in trouble. They've broken through. The Ranger needs help there.\"\n\nIt seemed odd to be giving orders to his supreme commander. But then he realized Ragnak was incapable of directing the flank attack in this state. He was good for only one thing\u2014a devastating, crushing attack on any enemy who stood in his way.\n\nNow, as he heard Erak's words, Ragnak nodded repeatedly.\n\n\"That sarcastic little know-all needs help, does he? Then I'm his man!\"\n\nAnd with a roar, he charged off after Halt, followed by his retinue of a dozen axmen.\n\nErak breathed a quick prayer to the Vallas. A dozen men might not be a lot, but with Ragnak in this near-berserk mode, it could be enough. Then he shoved the troubles of the right flank to the back of his mind and began yelling for a messenger. The right flank would have to look after itself for a few more minutes. Right now, he needed the left flank to hit the enemy from the side.\n**38**\n\nHORACE SENSED THE PRESENCE OF SOMEONE DIRECTLY BEHIND him and pivoted rapidly, his sword swinging back, ready to cut side-handed. Seeing the slightly built form of his friend there, grimly engaging a Temujai swordsman with his two knives, he widened his stroke and laid open the Tem'uj's forehead with the point of his sword. The trooper staggered away, hands to his face, sinking to his knees.\n\n\"What do you think you're up to?\" Horace yelled, in between parrying another attack from the front.\n\n\"I'm watching your back,\" Will told him, as he blocked a thrust from another Tem'uj trying to take Horace from the rear.\n\n\"Well, next time let me know,\" Horace said, grunting as he side-stepped a lance and hammered the hilt of his sword into its surprised owner's skull. \"I nearly cut you in half just then!\"\n\n\"There won't be a next time,\" Will replied. \"I'm not enjoying myself here.\"\n\nHorace flicked a rapid glance over his shoulder. Will was using the Ranger's double-knife defense to parry and block the Tem'uj's saber. But it wasn't a form of fighting he was particularly skilled in. Besides, it had been over a year since he and Horace had practiced the moves in the hills of Celtica. The Temujai swordsman was having the better of the exchange and in that quick glance, Horace had seen blood seeping through the left arm of Will's shirt.\n\n\"When I tell you, drop to your knees,\" Horace said.\n\n\"Fine,\" Will replied grimly. \"I may even do it before you give the word.\"\n\nIn spite of himself, Horace grinned. Then, as he drove two attackers back, he called over his shoulder: \"Now!\"\n\nHe sensed that Will had dropped to the ground and, flicking the sword into a reversed grip, he thrust backward and heard a startled cry.\n\n\"You all right?\" he called, reversing the sword again and deflecting that persistent lance once more. For a moment, there was no answer, and he felt a sudden jolt of fear that he had just stabbed his friend. Then Will answered him.\n\n\"Very impressive. Where did you learn that?\"\n\n\"Made it up just now,\" Horace said, then grunted in satisfaction as the lancer stepped a little too close and took the point of his sword in the shoulder. As the man sank to the ground, Horace withdrew the sword, flicking it into a whirling overhand cut at another Tem'uj. The cavalryman's thick felt helmet saved his life as the sword crashed down on it. But there was still enough force in the blow to knock him to his knees, concussed and cross-eyed.\n\nFor a moment, they had a brief respite. Horace stepped back and studied his friend.\n\n\"Is that arm troubling you?\" He nodded toward the widening seep of blood on Will's sleeve. Looking down, Will seemed to notice it for the first time.\n\n\"I didn't even feel it,\" he said in some surprise. Horace allowed himself a grim smile.\n\n\"You will later,\" he told him. Will shook his head doubtfully.\n\n\"If there is a later,\" he said. Then, from the lines behind them, they heard the thrum of bowstrings and the hissing flight of another volley. They looked at one another in amazement.\n\n\"It's Evanlyn,\" said Will. \"She's still got them firing!\"\n\nHorace gestured to the swarming Temujai, surrounding the thin line of defenders who were keeping them out of the archer's redoubt.\n\n\"She won't for much longer,\" he said. The Skandian line was already beginning to buckle. \"Come on! Watch my back and yell if you get in trouble.\" And with that, he bounded down the slope, his sword rising and falling as he drove his attack into the rear of the Temujai. Startled at the ferocity of his assault, they gave ground for a few seconds. Then, seeing that the new assault consisted of only two men\u2014one of them armed only with knives and small enough to be a boy\u2014they rallied and drove forward again.\n\nHorace fought grimly, gathering the few remaining defenders around him. But the enemy numbers were beginning to tell and now individual Temujai were bypassing the small knot of defenders and dropping into the trench itself, where the archers were still sending their volleys into the main Temujai force.\n\nThe two boys heard Evanlyn's voice raised in urgent tones as she directed some of the archers to fire point-blank at the attackers. They knew it was a matter of minutes before the Temujai overran the trench and killed everyone in it.\n\n\"Come on!\" said Will, leading the way toward the trench. Horace followed close behind him.\n\nA Temujai warrior barred his way and he struck at the man with his saxe knife, feeling the blow jar all the way up his arm as it struck home. A warning cry from Horace alerted him to danger and he turned just in time to block a savage saber cut with his crossed knives. Then Horace was by his side, slashing at the man who had attacked him, and the three others with him. The two friends fought side by side, but there were too many of the Temujai. Will's heart sank as he realized that they were not going to reach the trench in time. He could see Evanlyn, not twenty meters away, with a group of archers around her, facing a still larger group of Temujai as they advanced up the trench\u2014moving slowly, held back only by the threat of the bows.\n\n\"Look out, Will!\" It was Horace again, and once more they were fighting for their lives as more of the Temujai swarmed toward them.\n\nNit'zak led a party of men into the trenches that had sheltered the Skandian archers. His other men could take care of the two young warriors who had counterattacked so effectively. His task was to silence the archers once and for all.\n\nHis men poured into the trench behind him, striking out at the unarmored, virtually unarmed bowmen. They retreated down the line of the earthworks, some of them scrambling up and over and running to the rear. Grimly, Nit'zak followed until, rounding an angle in the trench, he stopped in surprise.\n\nThere was a young girl facing him, a long dagger in her hand and a look of total defiance in her eyes. The remaining archers gathered protectively around her. Then, on her command, they brought their bows up to the present position.\n\nThe two groups faced each other. There were at least ten bows aimed at him, Nit'zak saw\u2014at a range of barely ten meters. If the girl gave the order, there was no way the archers could miss. Yet, once that first volley was released, the girl and her archers would be helpless.\n\nHe flicked his eyes sideways. His men were level with him, and there were more behind. He had no intention of dying under the Skandian volley. If it might serve a purpose, he would do so willingly. But he had a job to do and he didn't have the right to die until that job was done. On the other hand, he had no qualms about sacrificing ten or twelve of his own men, if necessary, to get that job done. He gestured them forward.\n\n\"Attack,\" he said calmly, and his men surged forward in the constricted space of the trench.\n\nThere was a second's hesitation, then he heard the girl's command to shoot and the instant thrum of the bowstrings. The arrows tore into his men, killing or wounding seven of them. But the others kept on, joined by more men from behind him, and the archers broke and ran, leaving only the girl to face him. Nit'zak stepped forward, raising the saber in both hands. Curious, he studied her eyes for some sign of fear and saw none there. It would be almost a shame to kill one so brave, he thought.\n\nOff to one side, he heard an agonized cry\u2014a young man's voice that broke with fear and pain.\n\n\"Evanlyn!\"\n\nHe assumed it must be the girl's name. He saw her eyes flick away from his, and then she smiled sadly at someone out of his view. It was a smile of farewell.\n\nWill had witnessed it all. Helpless to intervene, fighting desperately to protect Horace's back and his own life, he had seen the Temujai move up the trench, saw the archers threaten them with a point-blank volley, and then watched, horrified, as the Temujai calmly moved forward once more, oblivious to the danger. The final volley stopped them for a second or two, then they charged, sweeping the archers away before them.\n\nHorace's urgent warning brought him back to his own situation and he darted sideways to avoid a saber, jabbing with the saxe to drive the off-balance Tem'uj back a few paces. He turned to look again and saw a Temujai officer poised over Evanlyn, his sword held in two hands as he raised it.\n\n\"Evanlyn!\" he cried in torment. And, hearing him, she turned, met his agonized gaze and smiled at him\u2014a smile that remembered all they had been through together in the past eleven months.\n\nA smile that remembered all they had ever meant to each other.\n\nAnd in that moment, he knew he couldn't let her die. He spun the saxe knife in his hand, catching it by the point and feeling the balance, then brought his arm back, then forward in one fluid movement.\n\nThe big knife took Nit'zak under the left arm just before he began his downward cut.\n\nHis eyes glazed and he crumpled slowly to one side, lurching against the earth wall of the trench, then sliding down to the hard-packed earthen floor. The saber fell from his hands and he plucked with weakened fingers at the heavy knife in his side. His last thought was that now Haz'kam would probably abandon the invasion after all, and he was angry about that.\n\nWill, now unarmed except for his small throwing knife, was under attack once more. He leapt forward to grapple with a Tem'uj and they rolled down the earthen slope together, with Will clinging desperately to the man's sword arm, while he, in his turn, tried to avoid the ineffectual slashing attacks Will made with the small knife.\n\nHe saw Horace overwhelmed by four warriors attacking him at once and he realized that, finally, it was all over.\n\nAnd then he heard a blood-chilling roar and a huge figure was standing over him, literally plucking his adversary from the ground and throwing him a dozen meters down the slope, to send another three men sprawling under the impact.\n\nIt was Ragnak, terrifying in his berserker rage. His shirt had been torn to ribbons and he wore no armor save his massive horned helmet. The horrifying roaring noise came constantly from his throat as he plunged into the midst of the Temujai attackers, the huge double-bladed ax whirling in giant circles as he struck his enemies down on either side.\n\nHe made no effort to protect himself and he was cut and wounded over and again. He simply ignored the fact and cut and hacked and beat at the men who had invaded his country\u2014who had dared to awaken the berserker rage in his blood.\n\nHis personal guard followed him, each man in the same awful killing rage. They drove a wedge into the Temujai force, implacable, irresistible. A dozen men who didn't care if they lived or died. Who cared about one thing and one thing only: getting close to their enemies and killing them. As many as possible. As quickly as possible.\n\n\"Horace!\" Will croaked, and tried to scramble to his feet, remembering that last image of Horace desperately holding off four attackers. And then he heard another sound\u2014a familiar one this time. It was the deep-throated thrum of a longbow. As he watched, Horace's attackers seemed to fade away like snow in the sunshine, and he knew that Halt had arrived.\n\nOn a knoll a kilometer away, Haz'kam, general of the army and Shan of the People, watched his attack fail. The enemy's left flank had curled around to crash into his main force, buckling them and driving them back, causing severe losses. On the enemy's right flank, Nit'zak and his men had finally managed to silence the Skandian archers. In his heart, he had always known that his old friend would succeed in the task.\n\nBut he had taken too long over it. The success had come too late, after his main force had been demoralized and disorganized by the constant hail of arrows. After they had been driven back in confusion by that flanking attack.\n\nIt was just one failed attack, of course, and he knew he could still win this battle, if he chose to. He could regroup his Ulans, commit his fresh reserves to drive these damned Skandians out from behind their defenses and send them scattering into the hills and the trees. For a moment, he was tempted to do it\u2014to have a savage revenge on these people who had thwarted his plans.\n\nBut the cost would be too high. He had lost thousands of men already and another attack, even a successful one, would cost him more than he could afford. He turned in his saddle and beckoned the bugler forward.\n\n\"Sound the general withdrawal,\" he said calmly. His face gave no hint of the seething fury, the bitter rage of failure that burned in his heart.\n\nIt was not polite for a Temujai general to allow his emotions to show.\n**39**\n\nRAGNAK'S BODY WAS CREMATED THE DAY AFTER THE BATTLE. The Oberjarl had died in the final moments, before the Temujai had begun their withdrawal. He had died battling a group of eighteen Temujai warriors. Two of them survived\u2014so badly injured they could barely crawl away from the terrifying figure of the Skandian leader.\n\nThere was no way of knowing who had struck the fatal blow, if, indeed, there had been one. They counted over fifty separate wounds on the Oberjarl, half a dozen of which could have caused death under ordinary conditions. As was the Skandian custom, the body was laid on his cremation pyre as it was\u2014without any attempt to clean away the blood or the mire of battle.\n\nThe four Araluens were invited to pay their last respects to the dead Oberjarl and they stood silently for a few moments before the massive pile of pitch-soaked pine logs, gazing up at the still figure. Then, politely but firmly, they were informed that the funeral of an Oberjarl, and the subsequent election of his successor, was a matter for Skandians only and they returned to Halt's apartment to await events.\n\nThe funeral rituals went on for three days. This was a tradition that had been established to allow jarls from outlying settlements time to reach Hallasholm and participate in the election of the next Oberjarl. Obviously, there were few jarls expected from the areas that the Temujai had already passed through, and the majority of the others had already been summoned to repel the invasion. But tradition called for a three-day period of mourning\u2014which, in Skandia, took the form of a lot of drinking and much enthusiastic recounting of the deceased's prowess in battle.\n\nAnd tradition, of course, was sacred to the Skandians\u2014particularly tradition that involved a lot of drinking and carousing late into the night. It was noticeable that the amount of liquor consumed and the degree of enthusiasm in the recounting of Ragnak's prowess seemed to be in direct correlation.\n\nOn the second night, Evanlyn frowned at the sound of drunken voices raised in song, counterpointed by the splintering sounds of furniture breaking as a fight got under way.\n\n\"They don't seem very sad about it,\" she pointed out, and Halt merely shrugged.\n\n\"It's their way,\" he said. \"Besides, Ragnak died in battle, as a berserker, and that's a fate that every true Skandian would envy. It gains him instant entry to the highest level of their version of heaven.\"\n\nEvanlyn twisted her mouth in a disapproving pout. \"Still,\" she said, \"it seems so disrespectful. And he did save our lives, after all.\"\n\nThere was an awkward silence in the room. None of the other three could think of a tactful way of pointing out that had Ragnak survived, he was sworn to kill Evanlyn.\n\nFinally, the period of mourning was over, and the senior jarls gathered in the Great Hall to elect their new Oberjarl. Will said hopefully, \"Do you think Erak has a chance?\" But Halt shook his head.\n\n\"He's a popular war leader, but he's only one of four or five. Add to that the fact that he's no administrator. And he's certainly no diplomat either,\" he added with some feeling.\n\n\"Is that important?\" Horace asked. \"From what I've seen, diplomacy is very low on the list of required skills in this country.\"\n\nHalt acknowledged the point with a nod. \"True,\" he admitted. \"But a certain amount of buttering up is necessary when there's an election among peers like this. Nobody gives their vote because you're the best candidate. They vote for you because you can do something for them.\"\n\n\"I guess the fact that Erak's spent the last few years as Ragnak's chief tax collector isn't going to help either,\" Will chipped in. \"After all, a lot of the people voting are the ones he's threatened to brain with an ax.\"\n\nAgain Halt nodded. \"Not a good career move if you hope to be Oberjarl one day.\"\n\nIn truth, the Ranger was indulging in a mild form of personal superstition by talking down Erak's chances in the election. There were still issues to be settled between Skandia and Araluen and he would have preferred to be settling them with Erak as the Skandian supreme leader. Still, the more they talked, the slimmer Erak's chances became. He hadn't known about the tax collecting until Will mentioned it. That would seem to put the final stopper on the jarl's chances.\n\n\"He probably wouldn't make a good Oberjarl anyway,\" Horace decided. \"What he really wants to do is get back to sea in his wolfship and go raiding somewhere.\"\n\nThe others agreed with this statement. It was reasonable and logical.\n\nBut reason and logic have little to do with politics. On the fifth day, a stunned-looking Erak stepped into Halt's apartment. He looked around at the four expectant faces and said:\n\n\"I'm the new Oberjarl.\"\n\n\"I knew it,\" said Halt instantly, and the other three looked at him, totally scandalized.\n\n\"You did?\" Erak asked, his voice hollow, his eyes still showing the shock of his sudden elevation to the highest office in Skandia.\n\n\"Of course,\" said the Ranger, shrugging. \"You're big, mean and ugly and those seem to be the qualities Skandians value most.\"\n\nErak drew himself up to his full height, trying to muster the sort of dignity that he felt an Oberjarl should assume.\n\n\"Is that how you Araluens speak to an Oberjarl?\" he asked, and Halt finally grinned.\n\n\"No. That's how we speak to a friend. Come in and have a drink.\"\n\nOver the next few days, it began to appear as if the council of jarls had chosen wisely. Erak quickly moved to end old feuds with other jarls, particularly those he had visited in his role as tax collector. And, surprisingly, he kept Borsa in the role of hilfmann.\n\n\"I thought he couldn't stand Borsa,\" Will said, puzzled. But Halt merely nodded his head in acknowledgment of Erak's choice.\n\n\"Borsa's a good administrator, and that's what Erak's going to need. A good leader is someone who knows what he's bad at, and hires someone who's good at it to take care of it for him.\"\n\nWill, Horace and Evanlyn had to think that through for a few seconds before they saw the logic in it. Horace, in fact, was still pondering it some time after the others had nodded and moved on to discuss other matters.\n\nAs Oberjarl, Erak would no longer be able to go on his annual raiding cruises at the helm of _Wolfwind_ , and that fact tinged his sudden elevation with a certain amount of regret. But he announced that he would be making one last voyage before he handed the ship over to the care of Svengal, his longtime first mate.\n\n\"I'll be taking you lot back to Araluen,\" he announced. \"Seems only fair, since I was the one responsible for your being here in the first place.\"\n\nWill was quietly pleased with the news. Now that the time was almost here to return home, he realized that he would be sad to farewell the big, boisterous pirate. With some surprise, he recognized the fact that he had come to regard Erak as a good friend. Anything that delayed the moment of parting found favor in his eyes.\n\nSpring had come, the geese were returning from the south and there were deer back in the hills, so there was plenty of fresh meat in place of the dried and salted provisions that had formed the bulk of the winter fare in Hallasholm.\n\nWhen he saw the first hunting parties returning from the high reaches inland of the Skandian capital, Will remembered one debt he still owed. Early one morning, he slipped quietly away on Tug and headed up the trail that he and Evanlyn had followed so many months ago, in a freezing blizzard.\n\nAt the little cabin where they had sheltered through the winter, he found the uncomplaining, shaggy little pony who had saved his life. The patient creature had broken the light tether holding him in the lean-to stable behind the cabin, and was quietly cropping the new season's grass in the clearing when Will arrived.\n\nTug looked a little askance at his master when Will unfastened a small sack of oats, indicating that it was for the pony alone. Will consoled his horse with a quiet pat on the muzzle.\n\n\"He's earned it,\" he told Tug, and the Ranger horse shrugged\u2014insofar as any horse is capable of shrugging. The nondescript pony may well have earned the sack of oats, but that didn't stop Tug's mouth from salivating at the sight and smell of them. When the pony had finished the oats, Will remounted Tug and, holding on to the lead rein, rode back down to Hallasholm, where he quietly returned the pony to Erak's stable.\n\nThe night before they were due to leave, Erak threw a farewell banquet in their honor. The Skandians were eager to show their appreciation of the efforts of the four Araluens in defending their land against the invaders. And with the shadow of the Vallasvow lifted from Evanlyn, they paid particular attention to her\u2014repeatedly toasting her bravery and resourcefulness in continuing to direct the fire of the archers as their position was being overrun.\n\nHalt, Borsa and Erak sat in a quiet huddle at the head table, discussing outstanding matters such as the repatriation of the slaves who had served in the archers' corps. Sadly, many of them hadn't survived the battle, but the promise of freedom had been made to their dependants as well, and the details had to be thrashed out. When the subject was finally closed, Halt judged the moment right and said quietly:\n\n\"So what will you do when the Temujai come back?\"\n\nThere was a deafening moment of silence at the head table. Erak pushed his bench back and stared at the small, grim-faced man next to him.\n\n\"Come back? Why should they come back? We beat them, didn't we?\"\n\nBut Halt shook his head slowly. \"As a matter of fact,\" he said, \"we didn't. We simply made it too costly for them to continue\u2014this time.\"\n\nErak thought about what he had said and glanced at Borsa for his opinion. The hilfmann nodded, a little reluctantly.\n\n\"I think the Ranger is right, Oberjarl,\" he admitted. \"We couldn't have held out much longer.\" Then he shifted his eyes to Halt's and asked him: \"But why should they come back?\"\n\nHalt took a sip of the rich Skandian beer before he answered. \"Because it's their way,\" he answered simply. \"The Temujai don't think in terms of this season or this year, or next year. They think of the next ten or twenty years and they have a long-term plan to dominate this part of the world. They need your ships. So they'll be back.\"\n\nErak considered the point, twisting one end of his mustache in his fingers. \"Then we'll beat them off again,\" he said.\n\n\"Without archers?\" Halt asked quietly. \"And without the element of surprise next time?\"\n\nAgain there was a silence. Then Erak said, half hopefully, \"You could help us train archers. You and the boy?\" But Halt shook his head immediately. And very definitely.\n\n\"I'm not prepared to provide Skandia with such a potent weapon,\" he said. \"Once you learned those skills, I'd never know when they might be turned against us in the future.\"\n\nErak had to admit the logic in the Ranger's statement. Skandia and Araluen were traditional enemies, after all. But Borsa, with his negotiator's ear, had caught an overtone in Halt's refusal.\n\n\"But you do have a suggestion?\" he said keenly, and Halt almost smiled at him. He'd hoped the hilfmann would see where he was heading.\n\n\"I was thinking,\" he said, \"that a force of, say, three hundred trained archers might be stationed here on a regular basis. They could spend the months of spring and summer here, then be rotated back home during the winter.\"\n\n\"Araluens?\" Erak said, beginning to catch on. Halt nodded.\n\n\"We could supply you with an archery force that way. But if it ever came to hostilities between our countries, I'd feel a lot more secure knowing you wouldn't be turning them against us. We'd need to stipulate that in the treaty,\" he added casually.\n\nErak looked cautiously at his hilfmann now. The word _treaty_ seemed to have appeared on the table in front of them without his seeing it arriving. Borsa caught his eye and shrugged thoughtfully.\n\n\"I'm proposing that we have a mutual defense treaty for a period of . . .\" Halt seemed to think and Erak suddenly had the distinct impression that he had weighed every word he was going to say well in advance of this moment. \"Five years, let's say. You get a viable force of archers\u2014\"\n\nErak decided it was time that someone else made the running. \"And you get what?\" he asked abruptly.\n\nHalt smiled at him. \"We get a peace treaty that says Skandia won't be launching any surprise attacks on our country during that period. And that in the event that hostilities become inevitable, our archers would be allowed free passage back home.\"\n\nErak shook his head abruptly. \"I'll never convince my men not to raid,\" he said indignantly. \"I'd be thrown out on my ear if I proposed that.\" But Halt held up a hand to calm him down.\n\n\"I'm not talking about individual raids,\" he said. \"We can cope with them. I'm saying no more massed attacks, like the one with Morgarath.\"\n\nThere was another long pause while Erak considered the offer. The more he thought about it, the more attractive the idea seemed. As well as any of them, he knew how close they had come to being overwhelmed by the Temujai. Three hundred trained archers would provide a powerful defensive force to Skandia, particularly if they were deployed in the narrow passes and twisting defiles at the border. He realized, with a shock, that he was beginning to think like a tactician. Maybe he'd been spending too much time around the Ranger, he thought.\n\n\"You have the authority to sign a treaty like that?\" he asked, and for the first time, Halt hesitated. In fact, he had no authority at all. As a member of the Rangers, he would have been empowered to sign, but he had been dismissed from the Corps when Duncan had banished him. He could brazen it out now, of course. He was reasonably sure that Crowley or Duncan himself would ratify such a treaty. But when that happened, Erak would know that he had acted falsely and he didn't think that was a good start to any relationship.\n\n\"I have,\" said a quiet voice from behind him, and the three men looked up in some surprise. Evanlyn, slipping away from the enthusiastic toasting and tributes, had been an interested audience to their conversation for the past few minutes.\n\n\"As Princess Royal of Araluen, I have authority to sign on my father's behalf,\" she told them, and Halt heaved an unseen sigh of relief.\n\n\"I think it's best if we do it that way,\" he said. \"After all, the princess does outrank me, just a little.\"\n**40**\n\n_WOLFWIND_ FOLLOWED THE RIVER SEMATH ALL THE WAY FROM the Narrow Sea to Castle Araluen itself. It was an astounding sight for the locals, to see a wolfship gliding, unmolested and peaceful, past their fields and villages, so far inland. The many river forts and strongpoints, which would normally have denied such progress to a Skandian ship, now deferred to the fact that Princess Cassandra's personal standard, a stooping red hawk, flew from the masthead. A message had been sent ahead of the wolfship's progress to make sure that local commanders recognized the standard and the fact that the voyagers traveled upriver in peace.\n\nIt was also something of a novelty for Erak and his crew.\n\nFinally, they rounded the last bend in the river and there before them were the soaring spires and turrets of Castle Araluen. Erak drew breath in wonder at the sight of it. Halt, watching him, was sure that, as well as the sheer admiration the castle inspired, Erak's old plundering instincts were at work, estimating just how much treasure the castle could contain. He stepped close to the Oberjarl and said softly:\n\n\"You'd never make it past the moat.\"\n\nErak started in surprise and looked at the Ranger.\n\n\"How did you know what I was thinking?\" he asked. Halt raised an eyebrow.\n\n\"You're a Skandian,\" he said.\n\nThere was a landing stage jutting out into the river, bedecked with flags and bunting. And a large crowd was awaiting their arrival. At the sight of the wolfship, they began sounding horns and cheering.\n\n\"That's a first,\" Erak said mildly, bringing a grin to Halt's face.\n\n\"And there's another,\" he said, pointing discreetly to a tall, bearded figure standing a little way back from the landing stage, surrounded by an expensively dressed retinue of knights and ladies. \"That's the King himself, come down to welcome you, Erak.\"\n\n\"More likely he's here for his daughter,\" the Skandian replied. But Halt noticed that he did look a little pleased with himself.\n\nEvanlyn had seen the tall man now and was standing in the prow of the wolfship, waving excitedly. The cheers from the shore redoubled at the sight of her and now Duncan was leading the way down the landing stage, lengthening his stride so that he was almost running, not content to stand back and preserve his royal dignity.\n\n\"Oars!\" called Erak, and the rowers raised their oars, dripping, from the water as the wolfship glided smoothly alongside the landing stage.\n\nThe Skandian crew passed mooring lines to those on shore, the two parties regarding each other with deep interest. It was the first time in memory that Araluens and Skandians had been face-to-face without weapons in their hands. Will, his face alight with the joy of the moment, leapt onto the wolfship's railing as Evanlyn hurried to the entry port in the ship's waist. She and her father, their hearts too full for words, simply smiled at each other over the decreasing gap as the line handlers hauled the ship in to the landing stage. Then the wickerwork fenders bumped and groaned and the ship was fast alongside. Svengal, grinning broadly at her, unlatched the entry port in the ship's rail and she leapt into her father's arms, burying her face in his chest.\n\n\"Dad!\" she cried once, her voice muffled by his shirt and by the sobs that welled up in her throat.\n\n\"Cassie!\" he murmured\u2014his pet name for her from when she was a toddler\u2014and the cheering intensified. Duncan was a popular king and the people knew how much pain the loss of his daughter had caused him. Even the Skandians were grinning at the scene.\n\nIn the midst of all that joy and celebration, only Halt stood apart. His face was a mask of pain and misery and he remained unobtrusively by the steering oar at the stern of the ship as the others surged forward to the waist.\n\nDuncan and Evanlyn\u2014or Cassandra, as her father knew her\u2014stood in each other's embrace, oblivious to those around them. Will, scanning the crowd, saw a heavily built form in the ranks behind the King: a middle-aged man who was waving enthusiastically at him, shouting his name.\n\n\"Will! Welcome home, boy! Welcome home!\"\n\nFor a moment, Will was puzzled, then he recognized Baron Arald\u2014a man who for years had been a stern-faced figure of authority. Now here he was, waving and yelling like a schoolboy on holiday. Will dropped lightly to the planks of the landing stage and made his way through the crowds of well-wishers to the Baron. He began to make a formal bow when the Baron grabbed his hand and started pumping it enthusiastically.\n\n\"Never mind that! Welcome home, lad! And well done! Well done! My god, I thought we'd never see you again! Wasn't that right, Rodney?\"\n\nHe spoke this last to the mail-clad knight beside him and Will recognized Sir Rodney, head of the Battleschool at Castle Redmont. He realized that the knight was anxiously scanning the faces on the deck of the wolfship.\n\n\"Yes, yes, my lord,\" he agreed distractedly. Then he seized Will's other arm and said urgently, \"Will, I thought Horace was with you. Don't tell me something's happened to him?\"\n\nPuzzled, Will looked to where Horace was shaking hands with the Skandian crew, farewelling friends among them before he came ashore.\n\n\"That's him there.\" He pointed Horace out for Sir Rodney, and had the satisfaction of seeing the knight's jaw drop in surprise.\n\n\"My god! He's turned into a giant!\" he gasped. Then Horace recognized his mentor and marched briskly through the crowd, coming to attention and saluting, his fist to his right breast.\n\n\"Apprentice Horace reporting, Battlemaster. Permission to return to duty, sir?\" he said crisply.\n\nComing to attention himself, Rodney returned the salute.\n\n\"Permission granted, apprentice.\"\n\nThen, formalities over, he seized the muscular apprentice in a bear hug and danced him around a few undignified steps, all the while crying:\n\n\"Damn me, boy, but you've done us all proud! And when the devil did you get so tall?\"\n\nOnce again, the crowd cheered with delight. Then, all at once, a silence fell over them and Will turned to see the reason. Erak Starfollower, Oberjarl of the Skandians, was stepping ashore.\n\nInstinctively, those nearest him drew back a little. Old habits died hard. Will, not wishing to see his friend insulted, started forward impulsively, but there was one other in the crowd who was quicker off the mark. Duncan, King of Araluen, stepped forward to greet his Skandian counterpart, his hand extended in friendship.\n\n\"Welcome to Araluen, Oberjarl,\" he said. \"And thank you for bringing my daughter safely home.\" And with that, the two leaders shook hands.\n\nThen the cheering started again, this time for Erak and his crew so that the Skandians looked about them with delight. And that, thought Will, was going to make it a little harder for them to raid here again in the years to come. Duncan let the cheering go on for a little while, then held up his hand for silence. He scanned the faces on the dock. Then, not seeing the one he looked for, he let his gaze switch to the wolfship.\n\n\"Halt,\" he said softly, finally seeing him, wrapped as ever in his Ranger's cloak and standing alone by the great steering oar. The King held out a hand and gestured to the dock.\n\n\"Come ashore, Halt. You're home.\"\n\nBut Halt stood awkwardly, unable to mask the sadness that he felt. His voice broke as he began to speak, and he gathered himself and started again.\n\n\"Your . . . your majesty, the year of banishment still has three weeks to run,\" he said at last.\n\nA low buzz of comment ran through the crowd. Will, unable to restrain himself, reacted in total surprise.\n\n\"Banishment? You were banished?\" he said incredulously. \"Why?\" he said. The word hung in the air. Duncan shook his head, dismissing the matter.\n\n\"A few incautious words, that was all. He was drunk and we've all forgotten what he said and I forgive him, so for god's sake, man, come ashore.\"\n\nBut Halt remained where he was. \"Your majesty, nothing would make me happier. But you must uphold the law,\" he said in a low voice. Then another speaker chimed in: Lord Anthony, the King's chamberlain.\n\n\"Halt is right, your majesty,\" he said. Anthony was a well-meaning man, but he tended to be a little pedantic when it came to interpreting the law. \"After all, he did say you were the issue of an encounter between your father and a traveling hatcha-hatcha dancer.\"\n\nThere was a gasp of horror from the crowd.\n\nDuncan, smiling thinly, said through gritted teeth: \"Thank you for reminding us all, Anthony.\"\n\nBut then a peal of helpless laughter rang out and Princess Cassandra doubled over, hooting in a most unroyal fashion. Every eye turned to her, and slowly, she recovered enough to speak.\n\n\"I'm so sorry, everyone. But if you ever knew my grandmother, you'd understand why my grandfather might have been tempted! Grandma had a face like a robber's dog\u2014and a temperament to match it!\"\n\n\"Cassie!\" her father said in his most disapproving tone, but she was holding her sides and laughing again and he couldn't keep a smile from forming at his lips. Then he felt Lord Anthony's disapproving stare on him and he recovered, nudging Cassandra until her laughter subsided into a series of choked snuffles and snorts. The laughter had been infectious, however, and it took a while for the assembled crowd to come to order. Throughout all this, Halt remained standing stiffly on the deck of the wolfship.\n\nDuncan turned to his chamberlain and said, in his most reasoning tone: \"Surely, Anthony, it's within my powers to pardon Halt for the last three weeks of his sentence?\"\n\nBut Anthony frowned and shook his head. \"It would be most irregular, your majesty,\" he said heavily. \"Such a thing would set unfortunate precedents in law.\"\n\n\"King Duncan!\" boomed Erak, and instantly he had the attention of everyone there. He realized he'd spoken a little more forcibly than he'd intended\u2014he was still getting the hang of these formal occasions. Now he continued at a more moderate level.\n\n\"Perhaps I could request that you grant this pardon\u2014as a gesture of goodwill to seal the treaty between our two countries?\"\n\n\"Good thinking!\" muttered Duncan. He turned quickly to Lord Anthony. \"Well?\" he said. The chamberlain pursed his lips thoughtfully. It was never his wish to deny the King what he wanted. He merely tried to do his duty and uphold the law. Now he saw a loophole and seized upon it gratefully.\n\n\"Such a request wouldn't set any precedents, your majesty,\" he said. \"And this is a very special occasion, after all.\"\n\n\"So be it!\" said Duncan quickly, and turned to face the figure on the wolfship. \"All right, Halt, you're pardoned\u2014so for god's sake, come ashore and let's have a drink to celebrate!\"\n\nHalt, tears in his eyes, set foot on Araluen soil once more, after eleven months and five days of banishment. As he came ashore to the renewed cheers of the crowd, those around him saw another man dressed in a gray-green cloak, who slipped forward and pressed something into his hand.\n\n\"You might be needing this again,\" said Crowley, Commandant of the Ranger Corps.\n\nAnd when Halt looked down, he saw a thin chain in his hand, with a silver oakleaf insignia on it.\n\nAnd then he knew he was really home.\n\nSomething was afoot, Will knew. After the first round of celebrations, and after Erak and his crew had set sail once more for Skandia, with the administrative details of the Araluen archery force deployment agreed for the following spring, there had been much consultation and discussion between the King and his advisers, including Halt, Crowley, Baron Arald and Sir Rodney.\n\nDuring this period, Will and Horace were left pretty much at a loose end, although there was no shortage of admirers who would greet them as friends and sit spellbound as they told the story of their time in Skandia and their fierce battle against the Temujai. But even such adulation palled after a while.\n\nHorace, now that his adventures as the Oakleaf Knight were over, had reverted to the plain white surcoat of a warrior apprentice.\n\nEvanlyn, of course, had reverted to her true identity as Princess Cassandra. She was whisked away to the royal family's apartments in one of the towers of Castle Araluen, and whenever Will saw her, she was surrounded by a retinue of knights and ladies-in-waiting. She was also, he realized, a beautiful young woman, immaculately dressed and at ease among the young nobles and ladies who surrounded her.\n\nSaddened, he felt the distance between them growing wider as he came to terms with the fact that his companion through so many adventures and dangers was, in reality, the highest-born woman in the kingdom, whereas he was the orphan child of a sergeant in the army and his farm girl wife. On those increasingly rare occasions when he did speak to Cassandra, he became awkward and stilted. He was tongue-tied in her presence and tended to mumble formulaic replies to her attempts at conversation.\n\nHis reaction frustrated and infuriated Cassandra. She was making a genuine attempt to restore their friendship to its former basis, but she was too young to realize that all the trappings of royalty and wealth, things she took for granted and gave no account to, could only serve to distance Will from her.\n\n\"Doesn't he see that I'm the same person I always was?\" she asked her mirror in frustration. But, in fact, she wasn't. Evanlyn had been a frightened girl, her life at constant risk, reliant for months on the wits and courage of her young companion to keep her safe. Then she in turn had become the savior, the one who nursed a confused, frightened boy back to health.\n\nCassandra, on the other hand, was a beautiful, perfectly groomed princess, whose station in life was so far above Will's as to be unattainable. One day, he realized, she would rule as Queen, in her father's place. It wasn't her personality that had changed. It was her position. And both she and Will were too young and inexperienced to overcome the inevitable strain that such a social gulf put upon their relationship.\n\nOddly enough, at the same time, she found herself becoming more closely aligned to Horace. Accustomed to the formality of life as an apprentice knight and the strictures and protocols of court life at Castle Redmont, Horace was unfazed by Cassandra's rank. Of course, he deferred to her and treated her with respect. But then, he always had done so. Horace's simplistic and uncomplicated approach to life led him to accept things as they were and not seek complications. Evanlyn had been his friend. Now, Princess Cassandra was too. There were certain differences in the way he might be expected to approach her and address her, but this sort of formality had been part of his training.\n\nWhen she finally broached the subject of the widening gap between herself and Will, Horace merely counseled patience.\n\n\"He'll get used to the way things are,\" he told her. \"He's a Ranger, after all, and they're sort of . . . different . . . in their ways. Give him time to adjust.\"\n\nSo Cassandra bided her time. But Horace's comment about Rangers stayed with her and she determined to do something about that situation.\n\nAnd there was, she knew, a perfect opportunity for that in the very near future.\n\nDuncan had declared a formal banquet to celebrate the safe return of his only daughter, and invitations had been carried to the fifty baronies in the kingdom. It would be a massive event.\n\nIt took a month for the invited guests to assemble, and then the immense dining hall in Castle Araluen saw an evening unrivaled since Duncan's coronation, twenty years prior.\n\nThe feasting went on for hours, with the castle servants laboring under trays of roasted meat, huge savory pastries, steaming fresh vegetables and confectioneries designed to dazzle the eyes as much as the taste. Master Chubb, the Kitchenmaster at Castle Redmont and one of the finest chefs in the kingdom, had traveled to the capital to oversee the affair. He stood in the kitchen doorway, watching in satisfaction as the nobles and their ladies devoured and destroyed the fruits of the kitchen staff's labors for the past week, and idly cracking his ladle on the head of any unwary waiter or kitchen worker who came within reach.\n\n\"Not bad, not bad,\" he muttered to himself, then directed another servant to take yet another special dish for the enjoyment of \"young Ranger Will,\" as he termed him.\n\nEventually, the massive feast was over and the entertainment was due to begin. The King's harper was nervously tuning his strings\u2014the heat of the packed dining hall had caused them to stretch unevenly\u2014and mentally reviewing the lyrics to the heroic ode he had written, celebrating the rescue of the Princess Royal from the jaws of death by three of the kingdom's worthiest heroes. He was still wishing that he had managed a better rhyme for \"Halt.\" The best he had come up with so far was to affirm that he was a man \"well worth his salt,\" which seemed, in the face of things, to be underselling the value of the legendary Ranger.\n\nBefore he was called upon, however, King Duncan rose from his seat to address the huge crowd. As ever, the vigilant Lord Anthony was on hand, and at his monarch's signal, he pounded his steel-shod staff on the flagstones of the dining hall.\n\n\"Silence before the King!\" he bellowed, and instantly, the babble of talk and laughter in the huge room fell away to nothing. All eyes turned expectantly to the top table.\n\n\"My lords and ladies,\" Duncan began, his deep voice carrying seemingly without effort to every corner of the hall, \"this occasion is one of great pleasure for me. For a start, we are here to celebrate the safe return of my daughter, Princess Cassandra\u2014an eventuality that brings me more joy than you could possibly comprehend.\"\n\nThe hall rang with cries of \"Hear! Hear!\" and enthusiastic applause.\n\n\"The other source of pleasure to me tonight is the opportunity to reward those who were responsible for her safe return.\"\n\nThis time, the applause was louder and more prolonged. The audience was delighted to see Cassandra safely back with her father. But they knew the main business of the evening was the rewarding of the three companions who had brought her there.\n\n\"First,\" said Duncan, \"would the Ranger Halt please step forward.\"\n\nThere was a murmur of interest in the crowd as the slightly built figure, for once without the anonymity of his gray-and-green cloak, stood before the King. Several of those at the rear of the hall stood to get a better view. Halt's reputation was known throughout the kingdom, but relatively few of those present had ever seen him in the flesh. That was due in no small part to the Ranger predilection for secrecy, of course. Now there were more than a few expressions of surprise at the legendary Ranger's diminutive size. Most of those present had formed a mental picture of a longbow-wielding hero of majestic build who stood just under two meters high.\n\nNow, he bowed his head to the King. Not for the first time, Duncan found himself studying the Ranger's shaggy, uneven hair-cut. It had obviously been recently trimmed in honor of the event, but Duncan couldn't help grinning. Halt had been at Castle Araluen for over a month, surrounded by servants, valets and, above all, skilled barbers. Yet apparently, he still chose to cut his own hair with his saxe knife. Duncan realized the crowd was waiting while he appraised Halt's tonsorial efforts. He gathered his thoughts and continued.\n\n\"Halt has already stated that his restoration to the ranks of the Ranger Corps is sufficient reward,\" Duncan said, and once again there was a murmur of surprise.\n\n\"As on so many occasions before this, I stand in debt of one of my most loyal officers and I accede to his wishes in this matter. Halt, I owe you more than any King ever owed a man. I will never forget all you have done.\"\n\nAnd at that, Halt inclined his head once more and slipped back to his seat, moving so quickly and unobtrusively that most of those present didn't realize he was gone, and their startled applause died stillborn.\n\n\"Next,\" Duncan said, raising his voice slightly to still the buzz of conversation that had broken out, \"let the warrior apprentice Horace stand forward.\"\n\nWill slapped his friend on the back as Horace, an apprehensive look on his face, rose from his seat and moved forward to stand at attention before the King. The crowd waited expectantly.\n\n\"Horace,\" Duncan began, straight-faced but with a hint of laughter in his eyes, \"it has come to our attention that you traveled throughout Gallica in the guise of a fully qualified knight . . .\" He made a show of consulting a note on the table before him, then added, \"The _Chevalier de Feuille du Ch\u00eane\u2014_ the Oakleaf Knight.\"\n\nHorace gulped nervously. He knew, of course, that the tale of his exploits had been told. But he had hoped that officialdom would turn a blind eye to the fact that he had no right to pose as a knight.\n\n\"Your majesty, I'm sorry . . . I sort of felt that it was necessary at the . . .\"\n\nHe realized that Duncan was eyeing him coolly, one eyebrow raised, and then it dawned on him that he had committed a grave breach of etiquette by interrupting the King. Belatedly, he stopped, and came to attention once more as the King resumed.\n\n\"As you know, I'm sure, it is highly irregular for an apprentice to bear an insignia or to pose as a knight, so now it is necessary that we rectify this irregularity.\" He paused.\n\nHorace was about to say, \"Yes sir,\" then realized he'd be interrupting again and said nothing.\n\nDuncan continued. \"I've conferred with your Baron, your Battlemaster and the Ranger Halt, and we all agree that the best solution is to regularize the situation.\"\n\nHorace wasn't sure what that meant, but it didn't sound good. Duncan made a signal and Horace heard heavy footsteps approaching from behind. Glancing sideways, he saw Battlemaster Rodney coming to a stop beside him, holding a sword and shield before him. In a daze, Horace saw the device on the shield\u2014a green oakleaf on a field of white. He watched in awe as Duncan stepped down from his dais, took the sword and touched him lightly on the shoulder with it.\n\n\"Kneel,\" Rodney hissed out of the corner of his mouth, and Horace did so, then heard the next words ringing in his ears.\n\n\"Arise, Sir Horace, Knight of the Oakleaf, and ensign in the Royal Guard of Araluen.\"\n\nThis caused bedlam in the crowd. It was virtually unheard of for an apprentice to be knighted in his second year and then to be appointed as an officer in the Royal Guard\u2014the elite force who garrisoned Castle Araluen. The nobles and their ladies went wild with delight.\n\n\"Get up,\" Rodney hissed again. Slowly, a huge grin spreading over his face, Horace rose and took the sword from the King's hand.\n\n\"Well done, Horace,\" the King said quietly. \"You've more than earned it.\"\n\nThen he shook the hand of his newest knight and indicated that he might return to his seat. Horace did so, the faces around him in a blur. He saw only the huge, delighted grin on Will's face as his friend pounded him on the back in congratulation. Then the crowd was hushed again and this time both boys heard the King's voice:\n\n\"Would the Ranger apprentice Will stand forward.\"\n\nEven though he had assumed that such a thing might happen, Will was caught unprepared. He hurried from his seat, stumbling as he went, and finally regained his balance to stand before the King.\n\n\"Will, your Ranger Corps have their own ways and their own regulations. I've spoken to your mentor, Halt, and to the Corps Commandant, and unfortunately it's beyond my power to rescind your period of training and declare you a fully qualified Ranger. Halt and Crowley insist that you must complete your full period of training and assessment.\"\n\nWill swallowed nervously and nodded. He knew that. There was still so much he had to learn about his craft, so many skills he had to develop. Horace's natural talent was sufficient for the King to waive his further training. But Will knew that could never be the case for him.\n\n\"However,\" Duncan continued, \"I can offer an alternative. It is within my power to appoint you as a lieutenant in the Royal Scouts. Your masters have agreed that you are totally qualified for such an appointment and will release you from your apprenticeship if that is your wish.\"\n\nThe assembled people gave one concerted gasp of surprise. Will was speechless. The Royal Scouts were an elite force of light cavalry, tasked with the responsibility of training the kingdom's archers and scouting ahead of the King's army in battle. Scout officers and recruits generally came from the ranks of the nobility and the appointment was virtually the equivalent of a knighthood.\n\nIt meant honor, prestige, rank and recognition, compared to another three years of grinding study and application as an apprentice.\n\nAnd yet . . .\n\nIn his heart of hearts, Will knew it was not for him. It was tempting, to be sure. But he thought of the freedom of the green forests, of the days spent with Tug and Halt and Abelard, of the fascination of learning and perfecting new skills and the intrigue of always being at the heart of events. That was a Ranger's life, and when he compared it with the protocol and etiquette, the formality and restrictions of life in Castle Araluen, he knew, for the second time in the space of a few years, what he really wanted.\n\nHe turned to look for some hint of advice from Halt, but his master was sitting, eyes cast down to the table, as was Crowley, a few places away. Then, his voice seeming unnaturally loud in the expectant silence of the room, he replied:\n\n\"You do me great honor, your majesty. But my wish is to continue my training as an apprentice.\"\n\nAnd now the babble of surprise rose to fever pitch in the room. Rangers were, as everyone agreed, different. And most people present simply could not understand Will's choice. Duncan, however, could. He gripped Will's shoulder and spoke to him alone.\n\n\"For what it's worth, Will, I think you've chosen wisely. And for your ears alone, your Craftmasters tell me that they believe you will be one of the greatest of the Rangers in the years to come.\"\n\nWill's eyes widened. To him, that knowledge was sufficient reward. He shook his head.\n\n\"Not as great as Halt, surely, your majesty?\"\n\nThe King smiled. \"I'm not sure anyone could be that great, wouldn't you agree?\"\n\nAnd with his hand still on his shoulder, he turned the lad around, to where Crowley and Halt were smiling warmly at him, making a space between them for him. The applause as he sat down was polite but a little confused. Nobody could really understand Rangers, after all.\n\nThere was one small pang of sadness in Duncan's heart as he turned toward the place where his daughter was sitting. His lips were already forming the words \"I tried,\" but when he looked, Cassandra was gone from the room.\n\nTwo days later, Will and Halt rode out from Castle Araluen, heading for the cottage by Castle Redmont. From time to time, Halt glanced fondly at his young friend. He knew Will had made a big decision and he knew his mind was troubled. He suspected it was to do with the Princess. Since the banquet, Will had tried to see her several times, to explain his decision. But she had been unavailable.\n\nHe sensed that Will wanted to be alone with his thoughts as they rode to the southwest, so he kept his peace, resolving to plunge the boy into a regimen of unremitting hard work and training that would give him no time to ponder his heartbreak.\n\nBehind the riders, two figures on a terrace of the huge castle stood watching, dwarfed by the soaring turrets and buttresses. Evanlyn raised a hand in farewell and Horace put a comforting arm around her shoulders.\n\n\"He's a Ranger,\" the newly made knight told her sympathetically. \"And people like us can never understand Rangers. There's always a part of them they keep to themselves.\"\n\nShe nodded, unable to speak. The early-morning mist that was cloaking the riders seemed to be thickening for a moment, then she blinked rapidly, and realized that it was tears misting her eyes. As they watched, the sun finally broke through and washed Castle Araluen in a pale golden light.\n\nBut Will was riding to the south, and he didn't notice.\n**Looking for more?**\n\nVisit Penguin.com for more about this author and a complete list of their books.\n\n**Discover your next great read!**\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nCONTENTS\n\nTHE LIGHTNING TOWER\n\nTHE DARK KING\nTHE HORUS HERESY\n\nTHE LIGHTNING TOWER\n\n_by Dan Abnett_\n\n_What are you afraid of? What are you really afraid of?_\n\nTHERE WAS ONCE a fine palace, and it sat like a crown of light upon the top of the world. This was in the latter days, when mankind left his birth rock for the second time, to chase a destiny denied him in a previous epoch.\n\nThe artisan masters of the many rival Masonic guilds had raised the palace up, block by gilded block, to be a statement of unity, regal and unequivocal. After a dreary, lightless Age of Strife, the warring tribes and creeds of Terra had been alloyed under one rule, and the palace was intended to symbolise that staggering achievement. All the petty dynasts and ethnarchs, all the clan-nations and gene-septs, all the despots and pan-continental tyrants, had been quelled or crushed, overthrown or annexed, Some, the smartest and most prescient, had offered terms and been embraced to the bosom of the new rule. Better fealty than the wrath of the warriors in thunder armour.\n\nBetter submission than the enmity of the world's new master.\n\nIt was said that once you had seen him, or heard him speak, you were never in doubt again. He was the one, and had always been the one. He had been the Emperor long before there was any such office to take. No one knew his birth name, because he had always, naturally, been the Emperor.\n\nEven the artisan masters of the Masonic guilds, famous for their sanctimonious craft wars and vainglorious quarrels, shut up and, in concert, built the palace for him.\n\nIt was monumental. It was not so much an edifice as a handcrafted landmass. The artisan masters built it upon Terra's greatest mountain range, and transformed the monstrous peaks into its bulwarks. It towered above a world laid to waste by centuries of war and perdition, and though that world was being rebuilt, with wondrous cities and architectural marvels blooming in the new age of Unity, nothing could match its magnificence.\n\nFor it was beautiful, a euphoric vision of gold and silver. It was said that, when they had finished their task, the artisan masters of the Masonic guilds set down their tools and wept.\n\nBy the time it was complete, it was the largest single man-made structure in known space. Its footings sank deep into the planet's mantle, its towers probed the airless limits of the atmosphere. It owned the words 'the palace' wholly, without any need for qualification, as if no other palaces existed.\n\nHe had blemished that glory. He had raised dark curtain walls around the golden halls, and cased the soaring towers in skins of armour ten metres thick. He had stripped away the jewelled facades and the crystelephantine ornamentation, the delicate minarets and the burnished cupolas, and in their places he had implanted uncountable turrets and ordnance emplacements. He had dug mighty earthworks out of the surrounding lowlands, and fortified them with a million batteries. He had yoked platforms into synchronous orbits to guard from above, their weapon banks armed and trained, day and night. He had put his men upon the walls, armoured in gold and set for the coming war.\n\nHis name was Dorn, and he was not proud of his work.\n\nVADOK SINGH, THE warmason, had a habit of stroking architectural plans as he laid them out, as if they were a beloved pet.\n\n'Necessity,' he said, his favourite word, stroking out the revised schemata of the Dhawalagiri elevation.\n\n'It's ugly,' said Dorn. He stood away from the table, leaning against one of the planning chamber's thick columns, his arms folded across his broad chest.\n\n'Ugly is what they will do if they find the Annapurna Gate weak and flimsy,' Singh replied. He stood back and lit his _boc_ pipe from a taper, allowing his flock of slaves to finish laying out the designs and adjusting the brass armature of the viewing lenses that would magnify details and project them onto the chamber wall for closer examination.\n\nDorn shrugged. 'It's still ugly. The orbis and lazulite work encrusting that gate took Menzo of Travert thirty years to complete. Pilgrims flock here simply to see it. They say it surpasses even the Eternity Gate in its aesthetic.'\n\n'Aesthetic, now?' Singh smiled. He began to pace, trailing blue smoke from the bowl of his long-stemmed pipe. His slaves followed him up and down the chamber, like a timorous litter of young following their mother. Singh was a tall man, taller than the primarch, but skeletally thin. His guild gene-bred their bloodline to favour height for purposes of surveying and overseeing. 'I do so love our conversations, Rogal. They are quite contrary. You, the warrior, and me, the craftsman, and you lecture me on aesthetics.'\n\n'I'm not lecturing,' Dorn replied. He was aware of Sigismund and Archamus in the corner of the great room, stiffening at the warmason's use of his forename. Dorn would hear about 'proper respect and protocol' again later.\n\n'Of course you're not,' said Singh, 'but it is a necessity. How many Legions does the Upstart have with him now?'\n\nDorn heard Sigismund rise to his feet. He turned and stared at the first captain of the Imperial Fists. Sigismund glowered back for a second, then left the chamber.\n\nDorn glanced back at the warmason. 'Too many,' he said.\n\nSingh held out a long, spindly arm in the direction of the schemata. 'So?'\n\n'Begin work tomorrow at sunrise. Dismantle the gate with care, and store the dismantled elements in the vaults. We will put the work back when this is done.'\n\nSingh nodded.\n\n_We will put everything back,_ thought Dorn. _When this is done, we will put everything back the way it was._\n\nA KATABATIC WIND was coming in off the lower bulwarks that night. The palace was so immense, the precipice walls bred their own microclimate. Greasy stars swam in the heat ripple of the palace's new reactors. The void shields were being tested again.\n\n_Not a palace. Not the palace anymore, a fortress._\n\nSome of those sullen stars were orbital platforms, catching the last backscatter of the sunlight as Terra turned. Dorn put on a fur-edged robe that had been in his possession since his adolescence on Inwit, and went out to walk the parapets of the Dhawalagiri prospect, to dwell upon its beauty one last time. It was one of the last sections of the palace that remained untouched. Adamantium armour plates, drab prestressed rockcrete and auto-turrets had yet to blight its ethereal lines.\n\nSoon, though. From the wall, Dorn could see the half a million campfires of the Masonic host, the labour army that would invade the prospect come sunrise with their mallets and chisels and cranes.\n\nThe robe had been his grandfather's, though Dorn had long since understood that no ties of blood linked him to the Inwit ice-caste that had raised him. He had been created from another genetic line, that most singular line, in a sterile vault deep beneath him in the buried core of the palace.\n\n_Not a palace. Not the palace anymore, a fortress._\n\nDorn had been built to rule, built to assist in his father's tireless ambitions, built to make the hard decisions. He had been made as a primarch, one of only twenty in the galaxy, engineered by the master architect of mankind, the arch-mason of genetic code.\n\n_The Imperium needs many things, but foremost it needs the ability to protect itself, to attack when necessary. That's why I gave it twenty strong teeth in its mouth._\n\nAttacking was a remarkably easy thing to do. Dorn's physical prowess humbled all but twenty human beings in creation, and those twenty were his father and his nineteen brothers. In Dorn's opinion, the real art was knowing when not to attack. His grandfather, the old Inwit sire, patriarch of the ice-hive clan, had taught him that.\n\nDorn had been the seventh lost son to be reclaimed. By the time his father's forces found him, he had become a system warlord in his own right, ruling the Inwit Cluster as the head of the House of Dorn. His grandfather had been dead forty winters, but still the warlord had slept with the fur-edged robe across his body at night. His people had called him 'emperor' until the true meaning of that title had been demonstrated by a thousand warships in the Inwit sky. Dorn had gone out to meet his father aboard _Phalanx_ , one ship against thousands, but what a ship: a fortress. His father had been impressed. Dorn had always excelled in the construction of fortresses.\n\nThat was why Dorn had returned to Terra with his gene-sire. Out of love, out of devotion, out of obedience, yes, but most of all, out of necessity, damn Singh. The stars had turned over, and Chaos had spilled out from under them. The brightest of all had fallen and the unthinkable, the heretical, had become fact.\n\nThe Imperium was attacking itself. The Warmaster, for reasons Dorn was quite at a loss to fathom, had turned upon their father, and was committing his forces to all-out war. That war would come to Terra. There was no question.\n\nIt would come. Terra needed to be ready. The palace needed to be ready. His father had asked him, as a personal boon, to return to Terra and fortify it for war.\n\nNo better man for the task. No better master of defences. Dorn and his Fists, appointed the Emperor's praetorians, could fend off any attack.\n\nBelow him, the halls of Terra were silent, and the walls deep. The only sound was the distant, eternal hum of the Astronomican. The Palace Dorn had armoured and defaced sat like a dark crown on the top of the world.\n\nRogal Dorn had built many of the finest strongholds in creation: the city fortresses on Zavamunda, the pylon spire of Gallant, the donjons along the Ruthan Marches. Impregnable bastions all, palaces for governor lords to rule from. None of them had been so essential as this fortification. None of them had been as painful to accomplish. It had been like blotting out the light or draining a sea. The bright glory of his father's triumph, the enduring monument to Unity, had been entombed inside a crude shell of utilitarian defence.\n\nAll because of Horus, because of the brightest bastard son, the bringer of new strife.\n\nDorn heard stone splinter. He looked down. He had punched his fist, his Imperial fist, through a block of stone in the parapet. He had barely registered the impact. The block was pulverised.\n\n'My lord, is everything all right?'\n\nArchamus had shadowed him from the planning chamber. Never so volatile as Sigismund, Archamus was the master of Dorn's huscarl retinue.\n\nThere was a worried look on Archamus's face.\n\n'Just venting my emotions,' Dorn said.\n\nArchamus regarded the splintered block. 'Making work for Singh's artisans, then?'\n\n'Something like that.'\n\nArchamus nodded. He hesitated, and looked out over the high walls towards the distant earthworks of the Mahabarat. 'You have wrought a wonder, you know.'\n\n'I have ruined one.'\n\n'I know you hate it, but it had to be so. And no one could have done it better.'\n\nDorn sighed. 'You're kind, old friend, but my heart is lead. This should never have been necessary. I search the limits of my imagination, and still I can conceive of nothing that begins to explain this war. Pride and ambition, insult, jealousy? They are not enough, not nearly enough, not for a primarch. They are too petty and mortal to drive a primarch to such extremity. They might provoke an argument, a feud at the worst. They would not split the galaxy in half.'\n\nDorn looked up at the night sky. 'And yet, against all reason, he comes.'\n\n'Guilliman will stop him.'\n\n'Roboute is far away.'\n\n'Russ, then. The Lion. The Khan.'\n\nDorn shook his head. 'I don't think they'll stop him either. I think he'll roll on until he reaches us.'\n\n'Then we'll stop him,' said Archamus. 'Won't we, sir?'\n\n'Of course we will. I just wish\u2014'\n\n'What?'\n\n'Nothing.'\n\n'You wish what, sir?' Archamus asked.\n\n'Nothing.'\n\nThe wind suddenly pulled at Dorn's fur-edged robe. Above them, the shields went out and then test fired again.\n\n'Can I ask you a question, sir?' asked Archamus.\n\n'Of course.'\n\n'Who are you really afraid of?'\n\n_C_ _ONSIDER_ ___THE_ ___QUESTION_ _,_ _Rogal Dorn. The first axiom of defence is to understand who you defend against. What are you afraid of? Who are you afraid of?_\n\nDorn paced the halls of the Kath Mandau Precinct where the organs of the Adeptus Terra did their work. The Precinct, an entire city contained within the terraced compounds of the inner palace, never slept. Robed clerks and burnished servitors bustled along the broad concourses. Ministers and ambassadors conducted business beneath the kilometre-high roof of the Hegemon. The great mechanism of the Imperium whirred about him, its relentless function like a ticking timepiece. This was what Unity had brought, this and the near measureless expanse of worlds and dominions that it guided and administered.\n\nFor two hundred years, the Emperor and his primarchs had fought to create the Imperium. They had waged the Great Crusade from star to star, to forge the empire of man, an epic undertaking they had all made without hesitation, because they believed, with utter conviction, in the bright destiny it would shape for their species. They had all believed. _All_ of them.\n\n_What was he afraid of? Who was he afraid of? Angron? Not him. Dorn would split his head without compunction if they came face to face. Lorgar? Magnus? There had always been a foetid whiff of sorcery about those two, but Dorn felt nothing towards them he could describe as fear. Fulgrim? No. The Phoenician was a singular foe, but not on object of terror. Perturabo? Well now, their rivalry was old, the spiteful scrapping of two brothers who fought for a father's attention._\n\nDorn smiled despite his mood. His years of exchanged insults with Perturabo seemed almost comical compared to this. They were too much alike, too jealous of one another's oh-so-similar abilities. Dorn knew it was a weakness for him to have risen to the Iron Warrior's baiting. But competition had always been a motivating force amongst the primarch brothers. It had been encouraged as a factor to drive them on to greater and yet greater accomplishments.\n\n_No, he was not afraid of Perturabo._\n\n_Horus-Lupercal, then?_\n\nDorn's aimless wanderings had taken him to the Investiary. In that broad space, an amphitheatre open to the night sky, statues of the twenty stood on ouslite plinths in a silent ring.\n\nThere was no one around. Even the Custodian Guard was absent. Lumen orbs glowed on black iron poles. The Investiary was two kilometres in diameter. Under the glittering stars, it felt like an arena, where twenty warriors had gathered to make their combat.\n\nThe second and eleventh plinths had been vacant for a long time. No one ever spoke of those two absent brothers. Their separate tragedies had seemed like aberrations. Had they, in fact, been warnings that no one had heeded?\n\nSigismund had urged that the effigies of the traitors also be removed from the Investiary. He had offered to do the work himself. This, Dorn recalled, had made the Emperor laugh.\n\nFor the time being, the traitors had been shrouded. Their towering, draped forms seemed like phantoms in the blue darkness.\n\n_Horus, then? Was it Horus?_\n\n_Perhaps. Dorn knew that Horus was the greatest of them, which made him the gravest foe. Could any one of them hope to best Lupercal on the field of war?_\n\nMartial prowess was hardly the point. Dorn had never feared an adversary in his life because of how strong he was or how hard he fought. Combat was only ever a test.\n\nWhat mattered, what engendered fear, was why an adversary fought. What _made_ him fight.\n\n_Oh, now we have it. Now the truth dawns._ He felt the hairs on his skin rise. _I'm not afraid of Horus. I'm afraid of finding out why he has turned against us. I cannot conceive of any justification for this schism, but Horus must have his reasons. I am afraid that when I know them, when they we explained to my baffled mind, I might... agree._\n\n'Would you tear them all down?'\n\nDorn turned at the sound of the voice. For a moment, it had sounded like the soft growl of his father.\n\nBut it was just a man, a cloaked and cowled man scarcely half Dorn's height. His robes were those of a simple palace administrator.\n\n'What did you say?' asked Dorn.\n\nThe man walked out into the circle of the Investiary to face Dorn. He greeted him with the old salute of Unity rather than the sign of the aquila. 'You were staring at the statues of your kin,' he observed. 'I asked... would you tear them all down?'\n\n'The statues or my kin, Sigillite?' Dorn replied.\n\n'Both. Either.'\n\n'The statues, perhaps. I believe Horus is doing a fine job with the men themselves.'\n\nMalcador smiled and looked up at Dorn. Like Dorn's, his hair was white. Unlike Dorn's, it was long like a mane. Malcador was an exceptional being, He had been with the Emperor from the inception of the Unification Wars, serving as aide, confidant and advisor. He had risen to become the master of the Council of Terra. The Emperor and the primarchs were genetically advantaged post-humans, but Malcador was just a man, and that was what made him exceptional. He stood on a par with the post human masters of the Imperium, and he was just a man.\n\n'Will you walk with me, Rogal Dorn?'\n\n'Are there not matters of state that require your attention, even at this hour, sir? The Council will bemoan your absence from the debating table!\n\n'The Council can manage for a while without me,' Malcador replied. 'I like to take the air at this time of night. The Imperium never rests, but at night, up here in the thin air of the old Himalazia, I find there is at least an illusion of rest, a time to think and free the mind. I walk. I close my eyes. The stars do not go out because I am not looking at them.'\n\n'Not yet,' said Dorn.\n\nMalcador laughed. 'No, not yet.'\n\nTHEY SAID LITTLE at first. They left the Investiary and walked along the beige stones of the Precinct's highest terraces, between the weeping fountains. They walked as far as Lion's Gate, onto the platforms that overlooked the docking rings and landing fields of the Brahmaputra Plateau. The Gate had once been a thing of magnificence, two gilded beasts rising up to lock claws in a feral dispute. Dorn's order of works had replaced them with giant grey donjons stippled with casemates and macro-gun ports. A curtain wall of bleak rockcrete encircled the gate, its edge fletched with void field vanes like the spines of some prehistoric reptile.\n\nThey stood and considered it for a long time.\n\n'I am not a subtle man,' Malcador said, at length.\n\nDorn raised his eyebrows.\n\n'Oh, all right,' said Malcador, 'perhaps I am. Guile comes easily to a politician. I know I am considered cunning.'\n\n'An old word, with no more meaning than \"wise\",' Dorn replied.\n\n'Indeed. I will accept that as a compliment. All I meant to say was, I will not attempt to be subtle now.'\n\n'No?'\n\n'The Emperor has expressed his concerns.'\n\n'Meaning?' Dorn asked.\n\nMalcador answered with a slight sigh. 'He understands you are filled with misgivings.'\n\n'Only natural, I would think, given the circumstances,' said Dorn.\n\nThe Sigillite nodded. 'He trusts you to undertake the defence. He counts on you. Terra must not fall, no matter what Horus brings. This palace must not fall. If it is to end here, then it must end in our triumph. But he knows, and I know, and you know, that any defence is only as strong as its weakest part: faith, belief, trust.'\n\n'What are you telling me?'\n\n'If there is doubt in your heart, then that is our weakness.'\n\nDorn looked away. 'My heart is sad because of what I have been made to do to this place. That's all it is.'\n\n'Is it? I don't think so. What are you really afraid of?'\n\nMALCADOR RAISED HIS hand and the lights in his chambers came on. Dorn looked around. He had never entered the Sigillite's private apartments before. Ancient images hung on the walls: flaking, fragile things of wood, canvas and decomposing pigments, preserved in thin, blue fields of stasis; the smoke pale portrait of a woman with the most curious smile; garish yellow flowers rendered in thick paint; the unflinching, rheumy gaze of an old fleshy man, cast in shadow, tobacco brown.\n\nAlong another wall hung old tattered banners showing the thunderbolt and lightning strike sigil of the Pre-Unity armies. Suits of armour \u2013 perfect, glinting thunder armour \u2013 were mounted in shimmering suspension zones.\n\nMalcador offered Dorn wine, which he refused, and a seat, which he accepted.\n\n'I have made a certain peace with myself,' Dorn said. 'I understand what I am afraid of.'\n\nMalcador nodded. He had pulled back his cowl and the light shone on his long white hair. He sipped from his glass. 'Enlighten me.'\n\n'I do not fear anyone. Not Horus, not Fulgrim, none of them. I fear the cause. I fear the root of their enmity.'\n\n'You fear what you don't understand.'\n\n'Exactly. I am at a loss to know what drives the Warmaster and his cohorts. It is an alien thing to me, quite defying translation. A strong defence relies on knowing what you are defending against. I can raise all the bulwarks and curtain walls and cannon-bastions I like, and I still won't know what it is I'm fighting.'\n\n'Perceptive,' said Malcador, 'and true of us all. I fancy even the Emperor doesn't fully understand what it is that drives Horus against us so furiously. Do you know what I think?'\n\n'Tell me.'\n\nMalcador shrugged. 'I believe it is better that we don't know. To understand it would be to understand insanity. Horus is quite mad. Chaos is inside him.'\n\n'You say that as if Chaos is a... thing.'\n\n'It is. Does that surprise you? You've known the warp and seen its corrupting touch, that's Chaos. It has touched humanity now, twisted our brightest and best. All we can do is remain true to ourselves and fend it off, deny it. Trying to understand it is a fool's errand. It would claim us too.'\n\n'I see.'\n\n'Don't see, Rogal Dorn, and you will live longer. All you can do is acknowledge your fear. That's all any of us can do. Recognise it for what it is: your pure, human sanity rocked by the sight of the warp's infecting, suffocating madness.'\n\n'Is this what the Emperor believes?' asked Dorn.\n\n'It's what he knows. It's what he _knows_ he doesn't know. Sometimes, my friend, there is salvation in ignorance.'\n\nDorn sat still for a while. Malcador watched him, occasionally sipping from his glass.\n\n'Well, I thank you for your time, sir,' said Dorn eventually. 'Your candour too. I should\u2014'\n\n'There is one other thing,' said Malcador, setting his glass down and rising to his feet. 'Something I want to show you.'\n\nMalcador crossed the chamber, and took something from a drawer in an old bureau. He walked hack to Dorn, and spread that something out on the low table between them.\n\nDorn opened his mouth but no sound issued. Fear gripped him. 'You recognise these, of course!\n\nOld cards, worn and fraying, discoloured and liver-spotted with time. One by one, Malcador laid them out.\n\n'The Lesser Arcanoi, just gaming trinkets, really, but used widely before the coming of Old Night for divination. This deck __ was made on Nostramo Quintus.'\n\n' _He_ used them,' Dorn breathed.\n\n'Yes, he did. He relied on them. He believed in cartomancy. He dealt his fate out, night after haunted night, and watched how the cards fell.'\n\n'Oh Holy Terra...'\n\n'Are you all right, sir?' Malcador asked, looking up. 'You are quite pale.'\n\nDorn nodded. 'Curze.'\n\n'Yes, Curze. Had you forgotten him, or simply blocked him out? You have bickered and sparred with many of your brothers over the years, but only Konrad Curze ever hurt you.'\n\n'Yes.'\n\n'He nearly killed you.'\n\n'Yes.'\n\n'On Cheraut, long ago.'\n\n'I remember it well enough!'\n\nMalcador looked up at Dorn. The primarch had risen to his feet. 'Then sit back down and tell me, because I wasn't there.'\n\nDorn sat. 'This is so long ago or like another life. We had brought the Cheraut system to compliance. It was hard fought. The Emperor's Children, the Night Lords and my Fists, we affected compliance. But Curze didn't know when to stop. He never knew when to stop.'\n\n'And you rebuked him?'\n\n'He was an animal. Yes, I rebuked him. Then Fulgrim told me.'\n\n'Told you what?'\n\nDorn closed his eyes. 'The Phoenician told me what Curze had told him: the fits, the seizures that had plagued Curze since his childhood on Nostramo, the visions. Curze said he had seen the galaxy in flames, the Emperor's legacy overthrown, Astartes turning on Astartes. It was all lies, an insult to our creed!'\n\n'You confronted Curze?'\n\n'And he attacked me. He would have killed me, I think. He is insane. That's why we drove him out, sick of his bloodletting. That's why he burned his home world and took his Night Lords off into the darkest parts of the stars.'\n\nMalcador nodded, and continued to deal the cards. 'Rogal, he is what you are truly afraid of, because he is fear incarnate. No other primarch uses terror as a weapon like Curze does. You are not afraid of Horus and his sallow heretics. You are afraid of the fear that sides with him, the night terror that advances alongside the traitors.'\n\nDorn sat back and breathed out. 'He has haunted me, I confess. All this time, he has haunted me.'\n\n'Because he was right. His visions were true. He saw this Heresy coming in his visions. That is the truth you fear. You wish you had listened.'\n\nDorn looked down at the cards laid out on the table before him. 'Do you believe in this divination, Sigillite?'\n\n'Let's see,' said Malcador, turning the cards over one by one: the Moon, the Martyr and the Monster, the Dark King askew across the Emperor.\n\nOne other card, the Lightning Tower.\n\nDorn groaned. 'A bastion, blown out by lightning. A palace brought to ruin by fire. I've seen enough.'\n\n'The card has many meanings,' said Malcador. 'Like the Death card, it is not as obvious as it seems. In the hives of Nord Merica, it symbolised a change in fortune, an overturning of fate. To the tribes of Franc and Tali, it signified knowledge or achievement obtained through sacrifice. A flash of inspiration, if you will, one that tumbles the world you know down, but leaves you with a greater gift.'\n\n'The Dark King lies across the Emperor,' said Dorn, pointing.\n\nMalcador sniffed. 'It's not exactly a science, my friend.'\n\nTHEY HAD BLOWN their way through the massive earthwork defences at Haldwani and Xigaze. The sky at the top of the world was on fire. Despite the bombardments of the orbital platforms and the constant sorties of the Stormbirds and the Hawkwings, the Traitor Legions advanced, up through the Brahamputra, along the delta of the Karnali. Continental firestoms raged across Gangetic Plain.\n\nAs they entered the rampart outworks of the palace, the streaming, screaming multitudes and the striding war machines were greeted by monsoons of firepower. Every emplacement along the Dhawalagiri prospect committed its weapons. Las reached out in neon slashes, annihilating everything it touched. Shells fell like sleet. Titans exploded, caught fire, collapsed on their faces and crushed the warriors swarming around their heels. Still they came. Lancing beams struck the armour-reinforced walls like lightning, like lighting smiting a tower.\n\nThe walls fell. They collapsed like slumping glaciers. Gold-cased bodies spilled out, tumbling down in the deluge.\n\nThe palace began to burn. Primus Gate fell; Lion's Gate, subjected to attack from the north; Annapurna Gate. At the Ultimate Gate, the Traitors finally sliced into the palace, slaughtering everyone they found inside. Around every broken gate, the corpses of Titans piled up in vast, jumbled heaps where they had fallen over each other in their desire to break in. The heretic host clambered across their carcasses, pouring into the palace, yelling out the name of their-\n\n'End simulation,' said Dorn.\n\nHe gazed down at the hololithic table. At his command, the forces of the enemy withdrew, unit by unit, and the palace rebuilt itself. The smoke cleared. 'Reset parameters to Horus, Perturabo, Angron and Curze.'\n\n'Opposition?' the table queried.\n\n'Imperial Fists, Blood Angels, White Scars. Resume and replay scenario.' The map flickered. Armies advanced. The palace began to burn again.\n\n'Play it out, simulation after simulation, if you like,' said the voice behind him. 'Simulations are just simulations. I know you won't fail me when the time comes.'\n\nDorn turned. 'I would never knowingly fail you, Father,' he said.\n\n'Then don't be afraid. Don't let fear get in your way.'\n\n_What are you afraid of? What are you really afraid of?_\n\n_The Lightning Tower, thought Rogal Dorn. I understand its meaning: achievement obtained through sacrifice. I'm just afraid of what that sacrifice might be._\nTHE HORUS HERESY\n\nTHE DARK KING\n\n_by Graham McNeill_\n\nWHERE BEFORE THERE had been light, now there was only darkness. The hot, urgent pulse of near death surged in his veins, the bitter flavour of betrayal fully expected, yet wholly unwelcome. This was what it would come to he knew, this was the inevitable result of na\u00efve belief in the goodness of the human heart. Death filled his senses, blood coating his teeth and the sharp reek of it thick in his nostrils.\n\nAs though it were yesterday, long buried memories of years spent on the night world of Nostramo emblazoned themselves on the forefront of his thoughts: haunted darkness punctured by stuttering lumen strips that fizzed in the shimmering, rain-slick streets and the stillness of a population kept quiescent with fear.\n\nFrom out of this foetid darkness had come illumination and hope, the promise of a better future. But now that hope was dashed as the bright lance of the future seared itself into his thoughts...\n\n_...the death of a world and a great eye of black and gold watching it burn..._\n\n_...Astartes fighting to the death beneath a red-lit sky..._\n\n_...a golden eagle cast from the heavens..._\n\nHe screamed in pain as images of destruction and the end of all things paraded before his mind's eye. Voices called out to him. He heard his name, the name his father had bestowed upon him and the one his people had given him, in the fearful watches of the dark.\n\nHe opened his eyes and let the visions fade from his mind as the sensations of the physical world returned to him. Blood and salty tears stung his eyes and he looked over to the sound of voices calling his name.\n\nHorrified faces stared at him in fear, but that was nothing new, Babble spewed from their mouths, but he could make nothing of it, the sense of the words lost in the screaming white noise filling his skull.\n\nWhat sight could be so terrible? What could evoke such horror?\n\nHe looked down as he realised he squatted atop another, living, breathing figure. A giant in torn golden robes, his bone-white hair spattered with gleaming ruby droplets.\n\nA mantle of red velvet trimmed with golden weave spread out beneath him like a bloodstain.\n\nTanned, iron flesh. Opened and bleeding.\n\nHe took in the destruction wrought on the body beneath him, raising his hands, balled into fists. Blood dripped from his fingertips and he could taste the warm richness of the genetic mastery encoded into every molecule upon his teeth.\n\nHe knew this giant.\n\nHis name was legendary, his stony heart and mastery of war unmatched. His name was Rogal Dorn.\n\nHe looked up again as he heard his own name, given voice by a warrior in the golden plate armour of the Imperial Fists who bore the black and white heraldry of its First Captain.\n\nHe knew this warrior too...\n\n'Curze!' cried Sigismund. 'What have you done?'\n\nTHE EMPTINESS OF space shimmered in the glow of distant suns beyond the armoured glass, faraway planets and unknown systems turning in their prescribed arcs without thought for the dramas being played out on the stage of human endeavour. What did those who lived beneath these suns know of the Cheraut system and the blood that had been shed to pacify it in the name of the emergent Imperium of Mankind?\n\nCurze stifled the anger such questions provoked, staring into his reflection with cold, obsidian eyes that resembled empty sockets in his pallid, sunken features. Lank hair hung to his neck like black ropes and spilled across his wide, powerful shoulders. He turned from his reflection, uncomfortable with the dreadful disappointment he saw there.\n\nGlinting metal caught his sullen gaze: his armour, standing in a shadowed alcove on the far wall. He crossed the chamber and placed his hand on the skull-faced helmet. The gem-like facets of its lenses winked in the low light and the sweeping dark wings rose from its sides like the pinions of some avenging angel of night. The burnished plates were dark, as befitted the Primarch of the Night Lords, each one contoured perfectly to his form and worked with gold edging that caught the starlight.\n\nTurning from his battle armour, he paced the hard, metallic floor of the gloomy, cavernous chamber that confined him. Thick steel columns supported a great vaulted ceiling, its upper reaches lost in shadow, and the hum of the mighty starfort's reactor beat like a pulse in the metal.\n\nThis aesthetic of functional austerity was typical of the Imperial Fists, whose artifice had constructed this mighty orbital fortress as a base of operations with which to begin the compliance of the Cheraut system.\n\nThe Emperor's Children had held their traditional victory feast before the first shot had been fired and together with Fulgrim's Legion and the Night Lords, Rogal Dorn's Imperial Fists had broken open the defences of the belligerent human coalition that resisted the coming of the Imperium. Within eight months of hard, bloody fighting, the eagle flew above the smoking ruins of the last bastion, but where Dorn lauded Fulgrim's Legion, the conduct of the Night Lords had earned only his ire.\n\nMatters had finally come to a head amid the silver ruins of Osmium.\n\nPyres of the dead stained the skies black and Curze had watched his chaplains orchestrating the executions of defeated prisoners when Dorn marched into his camp, his lean face thunderous. 'Curze!'\n\nNever once had Rogal Dorn called him by his forename.\n\n'Brother?' he had replied.\n\n'Throne! What are you doing here?' demanded Dorn, his normal, affable tone swallowed in the depths of his outrage. A phalanx of gold-armoured warriors followed their lord and Curze had immediately sensed the tension in the air.\n\n'Punishing the guilty,' he had answered coolly. 'Restoring order.'\n\nThe Primarch of the Imperial Fists shook his head. 'This not order, Curze, it is murder. Order your warriors to stand down. My Imperial Fists will take over this sector.'\n\n'Stand down?' said Curze. 'Are they not the enemy?'\n\n'Not anymore,' said Dorn. 'They are prisoners now, but soon they will be a population and part of the Imperium. Have you forgotten the Emperor's purpose in declaring the Great Crusade?'\n\n'To conquer,' said Curze.\n\n'No,' said Dorn, placing a golden gauntlet on his shoulder guard. 'We are liberators, not destroyers, brother. We bring the light of illumination, not death. We must govern with benevolence if these people ar ever to recognise our authority in this galaxy.'\n\nCurze flinched at the touch, resenting the easy friendship Dorn pretended. Bilious anger bubbled invisibly beneath his skin, but if Dorn was aware of it, he gave no sign.\n\n'These people resisted us and must pay the penalty for that crime,' said Curze. 'Obedience to the Imperium will come from the fear of punishment, you know that as well as anyone, Dorn. Kill those that resisted and the others will learn the lesson that to oppose us is to die.'\n\nDorn shook his head, taking his arm to lead him away from the curious stares their heated discussion was attracting. 'You are wrong, but we should speak of this in private.'\n\n'No,' said Curze, angrily shrugging off Dorn's grip. 'You think these people will bend the knee meekly to us because we show compassion? Mercy is for the weak and foolish. It will only breed corruption and eventual betrayal. Fear of reprisals will keep the rest of this planet in check, not benevolence.'\n\nDorn sighed. 'And the hatred planted in those you leave __ alive will pass from one generation to the next until this world is engulfed in a war the cause of which none of those fighting will remember. It will never end, don't you see that? Hate only breeds hate and the Imperium cannot be built upon such bloody foundations.'\n\n'All empires are forged in blood,' said Curze. 'To pretend otherwise is na\u00efve. The rule of law cannot be maintained by the blind hope that human nature is inherently good. Haven't we seen enough to know that ultimately the mass of humanity must be forced to into compliance?'\n\n'I cannot believe I am hearing this,' said Dorn. 'What has got into you, Curze?'\n\n'Nothing that has not always been there, Dorn,' said Curze, striding away from the mighty, golden figure and hauling one of the few remaining prisoners upright in front of his tunic. He scooped up a fallen bolter and thrust the heavy gun into the prisoner's trembling hands.\n\nCurze leaned down and said, 'Go ahead. Kill me.'\n\nThe terrified man shook his head, the oversized weapon shaking in his hands as though his limbs were palsied.\n\n'No?' said Curze. 'Why not?'\n\nThe prisoner tried to speak, but so awed by the terrifying proximity of the primarch that his words were unintelligible.\n\n'Are you afraid you will be killed?'\n\nThe man nodded and Curze addressed his warriors, 'No one harms this man. No matter what happens, he is not to be punished.'\n\nCurze had turned and walked back towards Dorn with his arms stretched out to either side of him and presenting his back to the prisoner.\n\nNo sooner had he turned away from the armed man than the gun had been raised and the hard crack of a bolter shot split the air. Sparks flew as the explosive shell ricocheted from Curze's armour and he spun on his heel to smash the prisoner's skull to splinters with his fist.\n\nThe headless corpse swayed for a moment before dropping slowly to its knees and pitching onto its chest.\n\n'You see,' said Curze, his fingers dripping blood and bone fragments.\n\n'And what was that supposed to prove?' asked Dorn, his features curled in distaste.\n\n'That any chance mortals get they will choose the path of dissent. When he thought he would be punished, he dared not shoot, but the moment he believed himself free from consequence, he acted.'\n\n'That was an unworthy deed,' said Dorn and Curze had turned away from him before he could elaborate, but the Imperial Fists' primarch caught his arm. 'Your warriors will stand down and withdraw, Curze. That is an order, not a request. Leave this planet. Now.'\n\nDorn's eyes were hard as granite and Curze knew enough of his brother's resolve to realise he had pushed him far enough. 'When this campaign is won, you and I will have words, Curze. You have crossed the line and I will no longer countenance your barbarous methods of war. Your way is not the way of the Imperium.'\n\n'I think you might be right...' whispered Curze.\n\nAnd he had led his warriors from the field of battle, their dark armour rendering them as shadows in the ruins.\n\nHe wondered what might have happened had he taken the debate to its logical conclusion.\n\nCurze shied away from the violence inherent in such a line of reasoning and ran a hand through his dark hair, feeling like a caged animal as the door to his chamber \u2013 his prison \u2013 slid open and a warrior in gleaming, midnight-blue armour entered. Through the door, he could see the purple-armoured figures of Fulgrim's Phoenix Guard, their golden halberds and copper scale cloaks glittering in the wan light of the starfort.\n\nDorn and Fulgrim were taking no chances with his confinement.\n\nThe newcomer's head was shaven bald, pale and angular, with hooded eyes of jet beneath a prominent brown and pugnacious jawline.\n\nCurze nodded in acknowledgement at the sight of his equerry, Captain Shang, and beckoned him in with an impatient wave of his hand.\n\n'What news?' asked Curze as Shang bowed curtly before him.\n\nShag said, 'The Master of the Fists recovers, my lord. A lesser being than a primarch would he dead thrice over with the wounds you dealt him.'\n\nCurze returned his gaze to the tracts of stars beyond the skin of the station, all too aware of the severity of Dorn's wounds, having clawed them with his bare hands and teeth.\n\n'Then I must await the judgement of my peers, is that it?'\n\n'With respect, my lord, you _did_ draw the blood of a brother Primarch.'\n\n'And for that they will demand blood in return, no doubt...'\n\nHe remembered Dorn coming to his chambers, enraged by the slaughters on Cheraut and incensed at what Fulgrim had told him \u2013 secrets Curze had told Fulgrim in confidence some days earlier. The fit had come upon Curze as the Phoenician had told him tales of Chemos, pitching him to the floor and wracking his mind with terrifying visions of a nightmare future of death and unremitting darkness.\n\nMoved by Fulgrim's apparent concern, Curze had confided in his old tutor, telling him of the visions that had plagued him since his earliest days on Nostramo.\n\n_A galaxy at war._\n\n_Astartes turning on one another._\n\n_Death awaiting him at his father's hands..._\n\nFulgrim's pale, aquiline features had remained stoic, but Curze had seen the unease that flickered in his eyes. He had hoped Fulgrim would keep his confession in confidence, but when Dorn had appeared at his door, he knew he was betrayed.\n\nIn truth he had little memory of what had occurred after Dorn's storming accusations of insult to the Emperor... the present had faded and the future had seized his mind with agonising visions of a galaxy locked in a cycle of unending war where the alien, the mutant and the rebel arose to feast on the rotting carcass of the Imperium.\n\n_This_ then was the future the Emperor was creating? This was the ultimate destiny of a galaxy where the fear of punishment was not the agent of control. This was the inevitable result of allowing weak men to craft the destiny of Mankind and Curze knew that, of all the primarchs, only one had the force of will required to mould the new Imperium from the soft clay of its present form.\n\n'The time has come to forge our own path, Shang,' said Curze.\n\n'Then this is the moment you foresaw?'\n\n'Yes. My brothers will seize this opportunity to be rid of us.'\n\n'I believe you are correct,' agreed Shang. 'My sources tell me there is talk, and not idle talk, of recalling the Legion to Terra to account for our methods of war.'\n\n'I knew it. Since they cannot kill me, the cowards choose to strike at me through my legion. You see, Shang? They have been waiting for this opportunity for decades. They are weak fools who have not the stomach to do what must he done, but I do, oh yes, I do indeed.'\n\n'Then what is our course, my lord?' asked Shang.\n\n'Fulgrim and Dorn may have betrayed me, but we are not without friends amongst the other Legions,' said Curze. 'But first we must put our own house in order. Tell me, what news of Nostramo?'\n\n'It is as we feared, my lord,' said Shang. 'The regime of Administrator-regent Balthius has failed. Corruption is rife, criminals govern from the ruined spires of Nostramo Quintus and lawlessness is endemic.'\n\n'Then I have no time to waste while small minded fools decide my fate as though I am a lowly menial to be chastised.'\n\n'What are your orders, my lord?' asked Shang.\n\n'Ready our ships, captain,' said Curze. 'We return to Nostramo.'\n\n'But you have been ordered to remain in seclusion, my lord,' pointed out Shang. 'Lord Fulgrim's praetorians and Dorn's Templars guard your chambers.'\n\nCurze grinned crookedly and said, 'Leave them to me...'\n\nCURZE LIFTED THE last piece of his armour from the shadowed alcove and raised   \nit above his head. He turned towards the door of his chamber and lowered his helmet until the skull-faced visor connected to his gorget with a hiss of pressurisation. His vision shifted subtly and his perceptions broadened as he blended with the shadows of the dimly lit chamber.\n\nHe slowed his breathing and stretched out his senses, the darkness a second home to him after so many years spent in its embrace as a predator on the weak and guilty. He felt a moment's regret that it had come to this, but he quashed such notions viciously. Doubt, regret and hesitancy were weaknesses others might suffer from, but not Konrad Curze.\n\nHis breathing deepened and the tenebrous chamber came alive to him.\n\nCurze felt power in the darkness; the cold intellect of hunters and creatures of the night that killed beneath its cloak. Lethal instincts honed on a thousand battlefields were now heightened to undreamed of levels and would now serve him equally well on this one.\n\nHe spread his arms wide and a ripple of psychic force pulsed like the blast wave of an explosion with Curze at its epicentre. The hanging glow strips filling the chamber exploded in quick succession, detonating one after another in showers of pellucid sparks. Broken glass tinkled musically to the steel deck in a glass rain.\n\nSputtering power cables swayed from the ceiling, hissing and fizzing like angry snakes as electric discharge strobed blue across the room.\n\nHostile red warning lights blinked. Cold light eased inside as the door opened and a handful of armoured warriors stood silhouetted.\n\nCurze leapt straight up, gripping the open lattice structure of the nearest column and swinging himself up into the deeper darkness of the chamber before the light could reach him. His legs swung around the column and he climbed higher as the warriors spread out with their halberds extended before them.\n\nHe heard them call his name, their voices echoing in the darkness.\n\nA twist of muscle and he was airborne, a glimmering shadow of dead stars and extinction. The warriors below would have the senses of their battle plate to penetrate the darkness, but they paled in comparison to those of the Night Lords' primarch. Where others saw only light and dark, Curze saw all the myriad hues and shades that were invisible to those who had not become one with its fuliginous depths.\n\nOne of the Phoenix Guard stood directly beneath him, scanning the chamber for its captive occupant, unaware that his doom lurked in the shadows above.\n\nCurze spun around the column, looping lower with each revolution and holding his hand out like an axe blade. The warrior died with his head sliced cleanly from his shoulders, the iron flesh of the primarch smashing through his armoured gorget. No sooner was the blow delivered than Curze was in motion, swooping through the darkness like a shadow.\n\nCries of alarm echoed as his gaolers realised he was amongst them, stabbing beams of helmet lamps crisscrossing madly as they sought to pinpoint his location. With skill borne of decades spent as a murderous hunter of men, Curze ghosted invisibly between the beams of light.\n\nAnother warrior fell with his torso ripped open, blood squirting from torn arteries like ruptured pressure hoses. Gunfire split the darkness, starbursts of muzzle flashes, as the warriors opened fire on their unseen attacker. None came close, for wherever they fired, Curze was already far from harm's way, spinning through the air like a malignant phantom and twisting between the bolts and wildly slashing blades.\n\nOne of Dorn's Templars backed towards a pool of light and Curze slid through the darkness towards him, moving impossibly silently for an armoured warrior. A sensation unlike anything he had felt previously danced in his blood and Curze savoured it as he understood it for what it was.\n\nContrary to Guilliman's rash pronouncement, it seemed Astartes _could_ know fear.\n\nThis fear \u2013 such as it was \u2013 was something to be treasured. Mortal fear was a rancid, sweaty thing, but this... this was caged lightning in the marrow.\n\nCurze pounced towards the armoured Templar, one of Dorn's best and bravest.\n\nVeteran or not, he died as any other man did \u2013 in blood and agony.\n\n'Death haunts the darkness,' shouted Curze. 'And he knows your names.'\n\nHe could hear frantic calls for reinforcements, but the superior systems of his own armour easily jammed them as he took to the air once more and vaulted from shadow to shadow.\n\n'No one is coming,' he said. 'You are going to die alone here.'\n\nSpraying blasts of gunfire followed his pronouncements as the warriors sought to pinpoint his location in the darkness.\n\nBut Curze _owned_ the darkness and no matter what light or senses these warriors depended upon, they were not nearly enough to stop him from killing them. He could see the survivors \u2013 a Templar and two of the Phoenix Guard \u2013 backing towards the door. They now realised this was a fight they could not win, but had made the mistake of thinking that a fight with Konrad Curze was one you could walk away from.\n\nLaughing with the joy of the hunt, a pleasure he had forgotten without worthy prey to test him, he soared through the air and dropped into their midst like an assassin.\n\nHis fist punched through the armour of the first Phoenix Guard, and Curze wrenched his victim's spinal column out. Leaving the bloody curve of crushed bone protruding from the gaping wound, he snatched the dead warrior's halberd and dropped to the floor as the other warriors turned towards the agonised scream.\n\nBefore they could react, Curze swept the halberd out in a wide, circular arc, the blade twice the width of a handspan above the deck, The energised edge cut through battle plate, meat and bone with a searing, electric tang.\n\nBoth warriors fell to the deck, grunting in pain as they collapsed onto the bloody stumps of their legs. Curze hurled his stolen halberd aside and blocked a return strike from the fallen Phoenix Guard,\n\nHe snapped his enemy's weapon in two and jammed the splintered ends through his chest.\n\nThe Templar roared in anger, managing to get off a shot before Curze was upon him. He ripped the weapon from his victim's grip and planted one knee on his chest, the other on his left arm.\n\nThe pinned warrior reached up with his free arm to strike at him. Cruze caught the blow and ripped the arm from its socket.\n\nEmergency lights began kicking in with a rising hum and thump of relays, and the chamber was suddenly illuminated with a harsh, white glow that dispelled the shadows and banished the darkness.\n\nWhere before there had been darkness, now there was only light.\n\nAnd what had once been a place of imprisonment was now an abattoir. Curling arcs of blood spray coated the walls and floor, and shattered, headless, limbless bodies lay strewn about like spilled surgical waste.\n\nCurze smiled at the scene of slaughter and the persona he had worn like a disguise since he had first knelt before his father fell away like a discarded mask.\n\nNow he was no longer Konrad Curze.\n\nNow he was the Night Haunter.\n\nNight Haunter turned over the last card and his jawline tightened as the   \nfamiliar pattern emerged once more. The strategium of his flagship was kept dark, the faint blue light of consoles and hololithic displays islands of light in the darkness. The Primarch of the Night Lords paid no attention to his surroundings, ignoring the pregnant pressure. of anticipation that bristled from every member of his bridge crew.\n\nA deck of worn cards sat on the softly glowing lectern before him, their edges scuffed and curled from decades of shuffling and dealing. Little more than a parlour game played by the indolent rich of Nostramo Quintus, he had since discovered that variations of these cards had been employed in the hives of Medea and by the tribes of the Franc as a means of divination in the time before Old Night had descended.\n\nThe cards apparently corresponded to the stratification of society at the time, with the various suits representing warriors, priests, merchants and workers. Ancient belief held that the future could be read in the patterns of cards known as the Lesser Arcanoi, but such traditions were outmoded concepts in this colourless, secular galaxy...\n\nExcept that no matter how thoroughly he shuffled the cards and dealt them on the polished glass of the lectern, the pattern was always the same.\n\nThe Moon, the Martyr and the Monster lay in a triangular pattern. The King lay reversed at the feet of the Emperor on one side of the pattern, and on the other, also reversed, was the Dove \u2013 a card academics postulated was a symbol of hope. The card he had just dealt sat at the top of the pattern, a card that had changed little over the centuries and the meaning of which, though often misinterpreted, was unmistakable.\n\nDeath.\n\nHe heard footsteps and looked up to see Captain Shang approaching, clad in his battle plate and wrapped in his ceremonial black cape of gleaming patagium. His helmet's flaring wings framed a death mask of an alien skull, its tusked lower jaw thrust beyond his throat.\n\nBehind his equerry, Night Haunter could see the gently rotating orb of Nostramo displayed on the viewscreen. Thick clouds of pollutants ringed the grey planet, shot through with emphysemic yellows and leprous browns. The radiation-blasted moon of Tenebor was just visible as a sickly orb emerging from the stained-lung corona of Nostramo's dying sun.\n\n'What it is, captain?' asked Night Haunter.\n\n'Word from the Choir chambers, my lord.'\n\nNight Haunter chuckled mirthlessly. 'My brothers?'\n\n'It would appear so, my lord.' said Shang. 'The astropaths sense a psychic bow-wave that appears to indicate a great many vessels approaching through the Empyrean.'\n\n'Dorn,' said Night Haunter, returning his attention to the cards before him.\n\n'Undoubtedly. What are your orders, my lord?'\n\nLooking once again at the world of his youth, Night Haunter felt the ever-present anger seething under his skin like hot magma beneath the fragile crust of a dying planet.\n\n'Nostramo was once the very model of a pacified planet, Shang,' said Night Haunter. 'Its populace was kept compliant through fear of the harsh punishment I would mete out to any who broke my laws. Every citizen knew his place and to break the law was death.'\n\n'I remember, my lord.'\n\n'And now we return to this...' said Night Haunter, sweeping the cards from the lectern to reveal a slowly scrolling list of text. 'A murder every eleven seconds, a rape every nine seconds, violent crimes increasing exponentially every month, suicide rates doubling every year. Within a decade, there will be nothing left of the ordered world I left behind.'\n\n'Without fear of reprisal, humanity reverts to its basest instincts, my lord.'\n\nNight Haunter nodded. 'This is it, Shang, the ultimate proof that the Emperor's belief in the goodness of mankind is folly of the worst kind.'\n\nShang hesitated before speaking again. 'Then you intend to go through with the attack?'\n\n'Of course,' said Night Haunter, staring at the doomed planet. 'Only the most extreme measures will serve as an example of our strength of will. Nostramo is dead to us now. We have come for you all...'\n\nThe primarch marched along the central walkway of the strategium to stand beneath the image of Nostramo. The moon was emerging more fully from behind the planet and reflected light glinted on the hulls of the Night Lords' fleet \u2013 a half century of vessels arrayed in battle formation above the diseased, corrupt boil that was the labyrinthine, crime-ridden spires of Nostramo Quintus.\n\nFar below was a great wound in the surface, a plunging chasm his fiery arrival had smashed in the planet's crust. Since he had emerged from its hellish depths he had known pain and suffering the likes of which others could not even guess. He had borne the pain of his tortured growth and lived with the awful self-knowledge of his own death.\n\nAnd his brothers wondered why he appeared moribund...\n\nHe heard a commotion beside him and even before the word went out. Night Haunter could feel the tearing pressure of scores of ships emerging from the gates of the Empyrean with senses beyond the five possessed by his minions.\n\n'Too late, my brothers...' he whispered. 'I will be gone before you can stop me.'\n\nNight Haunter took one last look at Nostramo and said, 'All ships. Open fire.'\n\nINCANDESCENT SPEARS OF blinding white light leapt from the barrels of uncounted batteries, stabbing down at the world below. Converging and multiplying their energies, the power of a thousand caged stars coalesced into a pillar of light thicker than the widest spire of Nostramo Quintus.\n\nThe great beam dispelled the darkness that shrouded Nostramo, the skies bathed in light and fire blooming into life as the awful heat of the Night Lord's bombardment ignited the air for kilometres in all directions.\n\nThe blinding lance of pure energy penetrated the impermeable adamantium crust of Nostramo through the ancient fault line torn by the primarch's arrival. Unimaginable energies tore downwards through the planet's layers until they reached the core in a cataclysmic explosion the likes of which the galaxy had rarely seen.\n\nNIGHT HAUNTER WATCHED the death of Nostramo with calm detachment, feeling the enormity of the action he had just taken settle upon him like a dark shroud. Strangely, it was not the burden he had expected. As he watched tectonic plates split apart and the molten heart of the planet ooze up to swallow the landscape and away burn the atmosphere, the only sensation of which he was conscious was intense relief.\n\nThe past was dead and he had shown that the creed he lived by was more than just empty words. The shockwave of this terrifying act would reverberate around the Imperium and come to the attention of those who, like him, understood the sacrifices needed to preserve the galaxy for humanity.\n\nNostramo burned and Night Haunter said, 'I take this burden of this evil upon myself and I will not fear it, for I am fear incarnate...'\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \n# Also by Julian Barnes\n\n# FICTION\n\nMetroland\n\nBefore She Met Me\n\nFlaubert's Parrot\n\nStaring at the Sun\n\nA History of the World in 10\u00bd Chapters\n\nTalking It Over\n\nThe Porcupine\n\nCross Channel\n\nEngland, England\n\nLove, Etc.\n\nThe Lemon Table\n\nArthur & George\n\nPulse\n\nThe Sense of an Ending\n\nThe Noise of Time\n\n# NONFICTION\n\nLetters from London, 1990\u20131995\n\nSomething to Declare\n\nThe Pedant in the Kitchen\n\nNothing to Be Frightened Of\n\nThrough the Window\n\nLevels of Life\n\nKeeping an Eye Open: Essays on Art\n\n# TRANSLATION\n\nIn the Land of Pain  \n _by Alphonse Daudet_\n\nTHIS IS A BORZOI BOOK PUBLISHED BY ALFRED A. KNOPF\n\nCopyright \u00a9 2018 by Julian Barnes\n\nAll rights reserved. Published in the United States by Alfred A. Knopf, a division of Penguin Random House LLC, New York.\n\nOriginally published in hardcover in Great Britain by Jonathan Cape, an imprint of Penguin Random House Ltd., London, in 2018.\n\nwww.aaknopf.com\n\nKnopf, Borzoi Books, and the colophon are registered trademarks of Penguin Random House LLC.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nNames: Barnes, Julian, author.\n\nTitle: The only story \/ by Julian Barnes.\n\nDescription: First American edition. | New York : Knopf, 2018. | \"This is a Borzoi book.\"\n\nIdentifiers: LCCN 2017047844 | ISBN 9780525521211 (hardcover) | ISBN 9780525521297 (ebook)\n\nSubjects: LCSH: Man-woman relationships\u2014Fiction. | Psychological fiction. | BISAC: FICTION \/ Literary. | FICTION \/ Coming of Age. | FICTION \/ Psychological. | GSAFD: Bildungsromans.\n\nClassification: LCC PR6052.A6657 O55 2018 | DDC 823\/.914\u2014dc23 LC record available at https:\/\/lccn.loc.gov\/\u200b2017047844\n\nEbook ISBN 9780525521297\n\n_Cover design by Carol Devine Carson_\n\nv5.2_r1\n\nep\n\n# Contents\n\nCover\n\nAlso by Julian Barnes\n\nTitle Page\n\nCopyright\n\nDedication\n\nEpigraph\n\nOne\n\nTwo\n\nThree\n\nA Note About the Author\n\nReading Group Guide\nto Hermione\n> Novel: A small tale, generally of love.\n> \n> \u2014Samuel Johnson, _A Dictionary of the English Language_ (1755)\n\n# ONE\nWould you rather love the more, and suffer the more; or love the less, and suffer the less? That is, I think, finally, the only real question.\n\n* * *\n\n\u2014\n\nYou may point out\u2014correctly\u2014that it isn't a real question. Because we don't have the choice. If we had the choice, then there would be a question. But we don't, so there isn't. Who can control how much they love? If you can control it, then it isn't love. I don't know what you call it instead, but it isn't love.\n\n* * *\n\n\u2014\n\nMost of us have only one story to tell. I don't mean that only one thing happens to us in our lives: there are countless events, which we turn into countless stories. But there's only one that matters, only one finally worth telling. This is mine.\n\n* * *\n\n\u2014\n\nBut here's the first problem. If this is your only story, then it's the one you have most often told and retold, even if\u2014as is the case here\u2014mainly to yourself. The question then is: Do all these retellings bring you closer to the truth of what happened, or move you further away? I'm not sure. One test might be whether, as the years pass, you come out better from your own story, or worse. To come out worse might indicate that you are being more truthful. On the other hand, there is the danger of being retrospectively anti-heroic: making yourself out to have behaved worse than you actually did can be a form of self-praise. So I shall have to be careful. Well, I have learned to become careful over the years. As careful now as I was careless then. Or do I mean carefree? Can a word have two opposites?\n\n* * *\n\n\u2014\n\nThe time, the place, the social milieu? I'm not sure how important they are in stories about love. Perhaps in the old days, in the classics, where there are battles between love and duty, love and religion, love and family, love and the state. This isn't one of those stories. But still, if you insist. The time: more than fifty years ago. The place: about fifteen miles south of London. The milieu: stockbroker belt, as they called it\u2014not that I ever met a stockbroker in all my years there. Detached houses, some half-timbered, some tile-hung. Hedges of privet, laurel and beech. Roads with gutters as yet unencumbered by yellow lines and residents' parking bays. This was a time when you could drive up to London and park almost anywhere. Our particular zone of suburban sprawl was cutely known as \"The Village,\" and decades previously it might possibly have counted as one. Now it contained a station from which suited men went up to London Monday to Friday, and some for an extra half-day on Saturday. There was a Green Line bus stop; a zebra crossing with Belisha beacons; a post office; a church unoriginally named after St. Michael; a pub, a general store, chemist, hairdresser; a petrol station which did elementary car repairs. In the mornings, you heard the electric whine of milk floats\u2014choose between Express and United Dairies; in the evenings, and at weekends (though never on a Sunday morning) the chug of petrol-driven lawnmowers.\n\nVocal, incompetent cricket was played on the Village green; there was a golf course and a tennis club. The soil was sandy enough to please gardeners; London clay didn't reach this far out. Recently, a delicatessen had opened, which some thought subversive in its offerings of European goods: smoked cheeses, and knobbly sausages hanging like donkey cocks in their string webbing. But the Village's younger wives were beginning to cook more adventurously, and their husbands mainly approved. Of the two available TV channels, BBC was watched more than ITV, while alcohol was generally drunk only at weekends. The chemist would sell verruca plasters and dry shampoo in little puffer bottles, but not contraceptives; the general store sold the narcoleptic local _Advertiser &_ _Gazette,_ but not even the mildest girlie mag. For sexual items, you had to travel up to London. None of this bothered me for most of my time there.\n\n* * *\n\n\u2014\n\nRight, that's my estate agent's duties concluded (there was a real one ten miles away). And one other thing: don't ask me about the weather. I don't much remember what the weather has been like during my life. True, I can remember how hot sun gave greater impetus to sex; how sudden snow delighted, and how cold, damp days set off those early symptoms that eventually led to a double hip replacement. But nothing significant in my life ever happened during, let alone because of, weather. So if you don't mind, meteorology will play no part in my story. Though you are free to deduce, when I am found playing grass-court tennis, that it was neither raining nor snowing at the time.\n\n* * *\n\n\u2014\n\nThe tennis club: Who would have thought it might begin there? Growing up, I regarded the place as merely an outdoor branch of the Young Conservatives. I owned a racket and had played a bit, just as I could bowl a few useful overs of off-spin, and turn out as a goalkeeper of solid yet occasionally reckless temperament. I was competitive at sport without being unduly talented.\n\nAt the end of my first year at university, I was at home for three months, visibly and unrepentantly bored. Those of the same age today will find it hard to imagine the laboriousness of communication back then. Most of my friends were far-flung, and\u2014by some unexpressed but clear parental mandate\u2014use of the telephone was discouraged. A letter, and then a letter in reply. It was all slow-paced, and lonely.\n\nMy mother, perhaps hoping that I would meet a nice blond Christine, or a sparky, black-ringleted Virginia\u2014in either case, one of reliable, if not too pronounced, Conservative tendencies\u2014suggested that I might like to join the tennis club. She would even sub me for it. I laughed silently at the motivation: the one thing I was not going to do with my existence was end up in suburbia with a tennis wife and 2.4 children, and watch them in turn find their mates at the club, and so on, down some echoing enfilade of mirrors, into an endless, privet-and-laurel future. When I accepted my mother's offer, it was in a spirit of nothing but satire.\n\n* * *\n\n\u2014\n\nI went along, and was invited to \"play in.\" This was a test in which not just my tennis game but my general deportment and social suitability would be quietly examined in a decorous English way. If I failed to display negatives, then positives would be assumed: this was how it worked. My mother had ensured that my whites were laundered, and the creases in my shorts both evident and parallel; I reminded myself not to swear, burp or fart on court. My game was wristy, optimistic and largely self-taught; I played as they would have expected me to play, leaving out the shit-shots I most enjoyed, and never hitting straight at an opponent's body. Serve, in to the net, volley, second volley, drop shot, lob, while quick to show appreciation of the opponent\u2014\"Too good!\"\u2014and proper concern for the partner\u2014\"Mine!\" I was modest after a good shot, quietly pleased at the winning of a game, head-shakingly rueful at the ultimate loss of a set. I could feign all that stuff, and so was welcomed as a summer member, joining the year-round Hugos and Carolines.\n\nThe Hugos liked to tell me that I had raised the club's average IQ while lowering its average age; one insisted on calling me Clever Clogs and Herr Professor in deft allusion to my having completed one year at Sussex University. The Carolines were friendly enough, but wary; they knew better where they stood with the Hugos. When I was among this tribe, I felt my natural competitiveness leach away. I tried to play my best shots, but winning didn't engage me. I even used to practise reverse cheating. If a ball fell a couple of inches out, I would give a running thumbs up to the opponent, and a shout of \"Too good!\" Similarly, a serve pushed an inch or so too long or too wide would produce a slow nod of assent, and a trudge across to receive the next serve. \"Decent cove, that Paul fellow,\" I once overheard a Hugo admit to another Hugo. When shaking hands after a defeat, I would deliberately praise some aspect of their game. \"That kicker of a serve to the backhand\u2014gave me a lot of trouble,\" I would candidly admit. I was only there for a couple of months, and did not want them to know me.\n\n* * *\n\n\u2014\n\nAfter three weeks or so of my temporary membership, there was a Lucky Dip Mixed Doubles tournament. The pairings were drawn by lot. Later, I remember thinking: Lot is another name for destiny, isn't it? I was paired with Mrs. Susan Macleod, who was clearly not a Caroline. She was, I guessed, somewhere in her forties, with her hair pulled back by a ribbon, revealing her ears, which I failed to notice at the time. A white tennis dress with green trim, and a line of green buttons down the front of the bodice. She was almost exactly my height, which is five feet nine if I am lying and adding an inch.\n\n\"Which side do you prefer?\" she asked.\n\n\"Side?\"\n\n\"Forehand or backhand?\"\n\n\"Sorry. I don't really mind.\"\n\n\"You take the forehand to begin with, then.\"\n\nOur first match\u2014the format was single-set knockout\u2014was against one of the thicker Hugos and dumpier Carolines. I scampered around a lot, thinking it my job to take more of the balls; and at first, when at the net, would do a quarter-turn to see how my partner was coping, and if and how the ball was coming back. But it always did come back, with smoothly hit groundstrokes, so I stopped turning, relaxed, and found myself really, really wanting to win. Which we did, 6\u20132.\n\nAs we sat with glasses of lemon barley water, I said,\n\n\"Thanks for saving my arse.\"\n\nI was referring to the number of times I had lurched across the net in order to intercept, only to miss the ball and put Mrs. Macleod off.\n\n\"The phrase is, 'Well played, partner.' \" Her eyes were grey-blue, her smile steady. \"And try serving from a bit wider. It opens up the angles.\"\n\nI nodded, accepting the advice while feeling no jab to my ego, as I would if it had come from a Hugo.\n\n\"Anything else?\"\n\n\"The most vulnerable spot in doubles is always down the middle.\"\n\n\"Thanks, Mrs. Macleod.\"\n\n\"Susan.\"\n\n\"I'm glad you're not a Caroline,\" I found myself saying.\n\nShe chuckled, as if she knew exactly what I meant. But how could she have?\n\n\"Does your husband play?\"\n\n\"My husband? Mr. E.P.?\" She laughed. \"No. Golf's his game. I think it's plain unsporting to hit a stationary ball. Don't you agree?\"\n\nThere was too much in this answer for me to unpack at once, so I just gave a nod and a quiet grunt.\n\nThe second match was harder, against a couple who kept breaking off to have quiet tactical conversations, as if preparing for marriage. At one point, when Mrs. Macleod was serving, I tried the cheap ploy of crouching below the level of the net almost on the centre line, aiming to distract the returner. It worked for a couple of points, but then, at 30\u201315, I rose too quickly on hearing the thwock of the serve and the ball hit me square in the back of the head. I keeled over melodramatically and rolled into the bottom of the net. Caroline and Hugo raced forward in a show of concern while from behind me came only a riot of laughter, and a girlish \"Shall we play a let?,\" which our opponents naturally disputed. Still, we squeaked the set 7\u20135, and were into the quarter-finals.\n\n\"Trouble up next,\" she warned me. \"County level. On their way down now, but no free gifts.\"\n\nAnd there weren't any. We were well beaten, for all my intense scurrying. When I tried to protect us down the middle, the ball went wide; when I covered the angles, it was thumped down the centre line. The two games we got were as much as we deserved.\n\nWe sat on a bench and fed our rackets into their presses. Mine was a Dunlop Maxply; hers a Gray's.\n\n\"I'm sorry I let you down,\" I said.\n\n\"No one let anyone down.\"\n\n\"I think my problem may be that I'm tactically naive.\"\n\nYes, it was a bit pompous, but even so I was surprised by her giggles.\n\n\"You're a case,\" she said. \"I'm going to have to call you Casey.\"\n\nI smiled. I liked the idea of being a case.\n\nAs we went our separate ways to shower, I said, \"Would you like a lift? I've got a car.\"\n\nShe looked at me sideways. \"Well, I wouldn't want a lift if you haven't got a car. That would be counterproductive.\" There was something in the way she said it that made it impossible to take offence. \"But what about your reputation?\"\n\n\"My reputation?\" I answered. \"I don't think I've got one.\"\n\n\"Oh dear. We'll have to get you one then. Every young man should have a reputation.\"\n\n* * *\n\n\u2014\n\nWriting all this down, it seems more knowing than it was at the time. And \"nothing happened.\" I drove Mrs. Macleod to her house in Duckers Lane, she got out, I went home, and gave an abbreviated account of the afternoon to my parents. Lucky Dip Mixed Doubles. Partners chosen by lot.\n\n\"Quarter-finals, Paul,\" said my mother. \"I'd have come along and watched if I'd known.\"\n\nI realised that this was probably the last thing in the history of the world that I wanted, or would ever want.\n\n* * *\n\n\u2014\n\nPerhaps you've understood a little too quickly; I can hardly blame you. We tend to slot any new relationship we come across into a preexisting category. We see what is general or common about it; whereas the participants see\u2014feel\u2014only what is individual and particular to them. We say: how predictable; they say: what a surprise! One of the things I thought about Susan and me\u2014at the time, and now, again, all these years later\u2014is that there often didn't seem _words_ for our relationship; at least, none that fitted. But perhaps this is an illusion all lovers have about themselves: that they escape both category and description.\n\n* * *\n\n\u2014\n\nMy mother, of course, was never stuck for a phrase.\n\nAs I said, I drove Mrs. Macleod home, and nothing happened. And again; and again. Except that this depends on what you mean by \"nothing.\" Not a touch, not a kiss, not a word, let alone a scheme or a plan. But there was already, just in the way we sat in the car, before she said a few laughing words and then walked off up her driveway, a complicity between us. Not, I insist, as yet a complicity to _do_ anything. Just a complicity which made me a little more me, and her a little more her.\n\nHad there been any scheme or plan, we would have behaved differently. We might have met secretly, or disguised our intentions. But we were innocent; and so I was taken aback when my mother, over a supper of stultifying boredom, said to me,\n\n\"Operating a taxi service now, are we?\"\n\nI looked at her in bewilderment. It was always my mother who policed me. My father was milder, and less given to judgement. He preferred to allow things to blow over, to let sleeping dogs lie, not to stir up mud; whereas my mother preferred facing facts and not brushing things under the carpet. My parents' marriage, to my unforgiving nineteen-year-old eye, was a car crash of clich\u00e9. Though I would have to admit, as the one making the judgement, that a \"car crash of clich\u00e9\" is itself a clich\u00e9.\n\nBut I refused to be a clich\u00e9, at least this early in my life, and so I looked across at my mother with blank belligerence.\n\n\"Mrs. Macleod will be putting on weight, the amount you're ferrying her around\" was my mother's unkindly elaboration of her original point.\n\n\"Not with all the tennis she plays,\" I answered casually.\n\n\"Mrs. Macleod,\" she went on. \"What's her first name?\"\n\n\"I don't actually know,\" I lied.\n\n\"Have you come across the Macleods, Andy?\"\n\n\"There's a Macleod at the golf club,\" he answered. \"Short, fat guy. Hits the ball as if he hates it.\"\n\n\"Maybe we should ask them round for sherry.\"\n\nAs I winced at the prospect, my father replied, \"There isn't enough call for that, is there?\"\n\n\"Anyway,\" continued my mother, tenacious of subject, \"I thought she had a bicycle.\"\n\n\"You suddenly seem to know a lot about her,\" I replied.\n\n\"Don't you start getting pert with me, Paul.\" Her colour was rising.\n\n\"Leave The Lad alone, Bets,\" said my father quietly.\n\n\"It's not _me_ who should be leaving him alone.\"\n\n\"Please may I get down now, Mummy?\" I asked with an eight-year-old's whine. Well, if they were going to treat me like a child...\n\n\"Maybe we _should_ ask them round for sherry.\" I couldn't tell if my father was being dense, or whimsically ironic.\n\n\"Don't _you_ start as well,\" my mother said sharply. \"He doesn't get it from me.\"\n\n* * *\n\n\u2014\n\nI went to the tennis club the next afternoon, and the next. As I started hacking away with two Carolines and a Hugo I noticed Susan in play on the court beyond. It was fine while I had my back to her game. But when I looked past my opponents and saw her rocking gently sideways on the balls of her feet as she prepared to receive serve, I lost immediate interest in the next point.\n\nLater, I offer her a lift.\n\n\"Only if you've got a car.\"\n\nI mumble something in reply.\n\n\"Whatski, Mr. Casey?\"\n\nWe are facing one another. I feel at the same time baffled and at ease. She is wearing her usual tennis dress, and I find myself wondering if its green buttons undo, or are merely ornamental. I have never met anyone like her before. Our faces are at exactly the same height, nose to nose, mouth to mouth, ear to ear. She is clearly noticing the same.\n\n\"If I were wearing heels, I could see over the net,\" she says. \"As it is, we're seeing eye to eye.\"\n\nI can't work out if she is confident or nervous; if she is always like this, or just with me. Her words look flirty, but didn't feel so at the time.\n\nI have put the top of my Morris Minor convertible down. If I am operating a bloody taxi service, then I don't see why the bloody Village shouldn't see who the bloody passengers are. Or rather, who the passenger is.\n\n\"By the way,\" I say, as I slow and put the car into second. \"My parents might be asking you and your husband round for sherry.\"\n\n\"Lordy-Lordy,\" she replies, putting her hand in front of her mouth. \"But I never take Mr. Elephant Pants anywhere.\"\n\n\"Why do you call him that?\"\n\n\"It just came to me one day. I was hanging up his clothes and he's got these grey flannel trousers, several pairs of them, with an eighty-four-inch waistline, and I held up one pair and thought to myself, that looks just like the back half of a pantomime elephant.\"\n\n\"My dad says he hits a golf ball as if he hates it.\"\n\n\"Yes, well. What else do they say?\"\n\n\"My mother says you'll be getting fat, what with all the lifts I'm giving you.\"\n\nShe doesn't reply. I stop the car at the end of her driveway and look across. She is anxious, almost solemn.\n\n\"Sometimes I forget about other people. About them existing. People I've never met, I mean. I'm sorry, Casey, maybe I should have...I mean, it isn't as if...oh dear.\"\n\n\"Nonsense,\" I say firmly. \"You said a young man like me should have a reputation. It seems I've now got a reputation for operating a taxi service. That'll do me for the summer.\"\n\nShe remains downcast. Then says quietly, \"Oh Casey, don't give up on me just yet.\"\n\nBut why would I, when I was falling smack into love?\n\n* * *\n\n\u2014\n\nSo what words might you reach for, nowadays, to describe a relationship between a nineteen-year-old boy, or nearly-man, and a forty-eight-year-old woman? Perhaps those tabloid terms \"cougar\" and \"toy boy\"? But such words weren't around then, even if people behaved like that in advance of their naming. Or you might think: French novels, older woman teaching \"the arts of love\" to younger man, _ooh la la._ But there was nothing French about our relationship, or about us. We were English, and so had only those morally laden English words to deal with: words like scarlet woman, and adulteress. But there was never anyone less scarlet than Susan; and, as she once told me, when she first heard people talking about adultery, she thought it referred to the watering-down of milk.\n\nNowadays we talk about transactional sex, and recreational sex. No one, back then, had recreational sex. Well, they might have done, but they didn't call it that. Back then, back there, there was love, and there was sex, and there was a commingling of the two, sometimes awkward, sometimes seamless, which sometimes worked out, and sometimes didn't.\n\n* * *\n\n\u2014\n\nAn exchange between my parents (read: my mother) and me, one of those English exchanges which condenses paragraphs of animosity into a pair of phrases.\n\n\"But I'm _nine-teen._ \"\n\n\" _Exactly_ \u2014you're _only_ nineteen.\"\n\n* * *\n\n\u2014\n\nWe were each other's second lover: quasi-virgins, in effect. I had had my sexual induction\u2014the usual bout of tender, anxious scuffle-and-blunder\u2014with a girl at university, towards the end of my third term; while Susan, despite having two children and being married for a quarter of a century, was no more experienced than me. In retrospect, perhaps it would have been different if one of us had known more. But who, in love, looks forward to retrospect? And anyway, do I mean \"more experienced in sex\" or \"more experienced in love\"?\n\nBut I see I'm getting ahead of myself.\n\n* * *\n\n\u2014\n\nThat first afternoon, when I had played in with my Dunlop Maxply and laundered whites, there was a huddle in the clubhouse over tea and cakes. The blazers were still assessing me for suitability, I realised. Checking that I was acceptably middle class, with all that this entailed. There was some joshing about the length of my hair, which was mostly contained by my headband. And almost as a follow-on to this I was asked what I thought about politics.\n\n\"I'm afraid I'm not remotely interested in politics,\" I replied.\n\n\"Well, that means you're a Conservative,\" said one committee member, and we all laughed.\n\nWhen I tell her about this exchange, Susan nods and says, \"I'm Labour, but it's a secret. Well, it was until now. So what do you make of that, my fine and feathered friend?\"\n\nI say that it doesn't bother me at all.\n\n* * *\n\n\u2014\n\nThe first time I went to the Macleod house, Susan told me to come in the back way and walk up through the garden; I approved such informality. I pushed open an unlocked gate, then followed an unsteady brick path alongside compost heaps and bins of leaf mould; there was rhubarb growing up through a chimney pot, a quartet of raggedy fruit trees and a vegetable plot. A dishevelled old gardener was double-digging a square patch of earth. I nodded to him with the authority of a young academic approving a peasant. He nodded back.\n\nAs Susan was boiling the kettle, I looked around me. The house was similar to ours, except that everything felt a bit classier; or rather, here the old things looked inherited rather than bought secondhand. There were standard lamps with yellowing parchment shades. There was also\u2014not exactly a carelessness, more an insouciance about things not being orderly. I could see golf clubs in a bag lying in the hallway, and a couple of glasses still not cleared away from lunch\u2014perhaps even the previous night. Nothing went uncleared-away in our house. Everything had to be tidied, washed, swept, polished, in case someone called round unexpectedly. But who might do so? The vicar? The local policeman? Someone wanting to make a phone call? A door-to-door salesman? The truth was that nobody ever arrived without invitation, and all that tidying and wiping was performed out of what struck me as deep social atavism. Whereas here, people like me called round and the place looked, as my mother would no doubt have observed, as if it hadn't seen a duster for a fortnight.\n\n\"Your gardener's jolly hardworking,\" I say, for want of a better conversational opener.\n\nSusan looks at me and bursts out laughing. \"Gardener? That's the Master of the Establishment, as it happens. His Lordship.\"\n\n\"I'm terribly sorry. Please don't tell him. I just thought...\"\n\n\"Still, I'm glad he looks up to snuff. Like a real gardener. Old Adam. Precisely.\" She hands me a cup of tea. \"Milk? Sugar?\"\n\n* * *\n\n\u2014\n\nYou understand, I hope, that I'm telling you everything as I remember it? I never kept a diary, and most of the participants in my story\u2014my story! my life!\u2014are either dead or far dispersed. So I'm not necessarily putting it down in the order that it happened. I think there's a different authenticity to memory, and not an inferior one. Memory sorts and sifts according to the demands made on it by the rememberer. Do we have access to the algorithm of its priorities? Probably not. But I would guess that memory prioritises whatever is most useful to help keep the bearer of those memories going. So there would be a self-interest in bringing happier memories to the surface first. But again, I'm only guessing.\n\n* * *\n\n\u2014\n\nFor instance, I remember lying in bed one night, being kept awake by one of those stomach-slapping erections which, when you are young, you carelessly\u2014or carefreely\u2014imagine will last you the rest of your life. But this one was different. You see, it was a kind of generalised erection, unconnected to any person, or dream, or fantasy. It was more about just being joyfully young. Young in brain, heart, cock, soul\u2014and it just happened to be the cock which best articulated that general state.\n\n* * *\n\n\u2014\n\nIt seems to me that when you are young, you think about sex most of the time, but you don't reflect on it much. You are so intent on the who, when, where, how\u2014or rather, more often, the great if\u2014that you think less about the why and the whither. Before you first have sex, you've heard all sorts of things about it; nowadays far more, and far earlier, and far more graphically, than when I was young. But it all amounts to the same input: a mixture of sentimentality, pornography and misrepresentation. When I look back at my youth, I see it as a time of cock-vigour so insistent that it forbade examination of what such vigour was for.\n\nPerhaps I don't understand the young now. I'd like to talk to them and ask how things are for them and their friends\u2014but then a shyness creeps in. And perhaps I didn't even understand the young when I was young. That could be true too.\n\n* * *\n\n\u2014\n\nBut in case you're wondering, I don't envy the young. In my days of adolescent rage and insolence, I would ask myself: What are the old for, if not to envy the young? That seemed to me their principal and final purpose before extinction. I was walking to meet Susan one afternoon, and had reached the Village's zebra crossing. There was a car approaching, but with a lover's normal eagerness, I started to cross anyway. The car braked, harder than its driver had evidently wanted to, and hooted at me. I stopped where I was, right in line with the car's bonnet, and stared back at the driver. I admit I was perhaps an annoying sight. Long hair, purple jeans, and young\u2014filthy, fucking young. The driver wound down his window and swore at me. I strolled round to him, smiling, and keen on confrontation. He was old\u2014filthy, fucking old, with an old person's stupid red ears. You know those sorts of ears, all fleshy, with hairs growing on them inside and out? Thick, bristly ones inside; thin, furry ones outside.\n\n\"You'll be dead before I will,\" I informed him, and then dawdled off as irritatingly as I could manage.\n\nSo, now that I am older I realise that this is one of my human functions: to allow the young to believe that I envy them. Well, obviously I do in the brute matter of being dead first; but otherwise not. And when I see pairs of young lovers, vertically entwined on street corners, or horizontally entwined on a blanket in the park, the main feeling it arouses in me is a kind of protectiveness. No, not pity: protectiveness. Not that they would want my protection. And yet\u2014and this is curious\u2014the more bravado they show in their behaviour, the stronger my response. I want to protect them from what the world is probably going to do to them, and from what they will probably do to one another. But of course, this isn't possible. My care is not required, and their confidence insane.\n\n* * *\n\n\u2014\n\nIt was a matter of some pride to me that I seemed to have landed on exactly the relationship of which my parents would most disapprove. I have no wish\u2014certainly not at this late stage\u2014to demonise them. They were products of their time and age and class and genes\u2014just as I am. They were hardworking, truthful and wanted what they thought was the best for their only child. The faults I found in them were, in a different light, virtues. But at the time...\n\n\"Hi, Mum and Dad, I've something to tell you. I'm actually gay, which you probably guessed, and I'm going on holiday next week with Pedro. Yes, Mum, that Pedro, the one who does your hair in the Village. Well, he asked me where I was going for my holidays, and I just said, 'Any suggestions?' and we took it from there. So we're off to a Greek island together.\"\n\nI imagine my parents being upset, and wondering what the neighbours would say, and going to ground for a while, and talking behind closed doors, and theorising difficulties ahead for me which would only be a projection of their own confused feelings. But then they would decide that times were changing, and find a little quiet heroism in their ability to accommodate this unanticipated situation, and my mother would wonder how socially appropriate it would be to let Pedro carry on cutting her hair, and then\u2014worst stage of all\u2014she would award herself a badge of honour for her newfound tolerance, all the while giving thanks to the God in whom she did not believe that her father hadn't lived to see the day...\n\nYes, that would have been all right, eventually. As would another scenario then popular in the newspapers.\n\n\"Hi, Parents, this is Cindy, she's my girlfriend, well, actually a little bit more than that, as you can see, she's going to be a 'gymslip mum' in a few months' time. Don't worry, she was dead legal when I swooped at the school gates, but I guess the clock's ticking on this one, so you'd better meet her parents and book the registry office.\"\n\nYes, they could have coped with that too. Of course, their best-case scenario, as previously noted, was that down at the tennis club I would meet a nice Christine or Virginia whose emollient and optimistic nature would have been to their taste. And then there could have been a proper engagement followed by a proper wedding and a proper honeymoon, leading to proper grandchildren. But instead I had gone to the tennis club and come back with Mrs. Susan Macleod, a married woman of the parish with two daughters, both older than me. And\u2014until such time as I shrugged off this foolish case of calf love\u2014there would be no engagement or wedding, let alone patter of tiny feet. There would only be embarrassment and humiliation and shame, and prim looks from neighbours and sly allusions to cradle-snatching. So I had managed to present them with a case so far beyond the pale that it could not even be admitted, much less sensibly discussed. And by now, my mother's original idea of inviting the Macleods round for sherry had been definitively junked.\n\n* * *\n\n\u2014\n\nThis thing with parents. All my friends at university\u2014Eric, Barney, Ian and Sam\u2014had it in varying amounts. And we were hardly a pack of stoned hippies in shaggy Afghan coats. We were normal\u2014normalish\u2014middle-class boys feeling the irritable rub of growing up. We all had our stories, most of them interchangeable, though Barney's were always the best. Not least because he gave his parents so much lip.\n\n\"So,\" Barney told us, as we reassembled for another term, and were exchanging dismal tales of Life at Home. \"I'd been back about three weeks, and it's ten in the morning and I'm still in bed. Well, there's nothing to get up for in Pinner, is there? Then I hear the bedroom door open, and my mum and dad come in. They sit on the end of my bed, and Mum starts asking me if I know what time it is.\"\n\n\"Why can't they learn to knock?\" asked Sam. \"You might have been in mid-wank.\"\n\n\"So, naturally I said that it was probably morning by my reckoning. And then they asked what I was planning to do that day, and I said I wasn't going to think about it till after breakfast. My dad gave this sort of dry cough\u2014it's always a sign that he's starting to boil up. Then my mum suggests I might get a holiday job to earn a little pin money. So I admit that it hadn't exactly crossed my mind to apply for temporary employment in some menial trade.\"\n\n\"Nice one, Barney,\" we chorused.\n\n\"And then my mum asked if I was planning to idle away my whole life, and you know, I was beginning to get annoyed\u2014I'm like my father in that, slow burn, except I don't give that little warning cough. Anyway, my dad suddenly loses it, stands up, rips open the curtains and shouts,\n\n\" 'We don't want you treating this place like a fucking hotel!' \"\n\n\"Oh, _that_ old one. We've all had that. So what did you say?\"\n\n\"I said, 'If this _was_ a fucking hotel, the fucking management wouldn't burst into my room at ten in the morning and sit on my fucking bed and bollock me.' \"\n\n\"Barney, you ace!\"\n\n\"Well, it was very provoking, I thought.\"\n\n\"Barney, you ace!\"\n\n* * *\n\n\u2014\n\nSo the Macleod household consisted of Susan, Mr. E.P., and two daughters, both away at university, known as Miss G. and Miss N.S. There was an old char who came twice a week, Mrs. Dyer; she had poor eyesight for cleaning but perfect vision for stealing vegetables and pints of milk. But who else came to the house? No friends were mentioned. Each weekend, Macleod played a round of golf; Susan had the tennis club. In all the times I joined them for supper, I never met anyone else.\n\nI asked Susan who their friends were. She replied, in a casually dismissive tone I hadn't noted before, \"Oh, the girls have friends\u2014they bring them home from time to time.\"\n\nThis hardly seemed an adequate response. But a week or so later, Susan told me we were going to visit Joan.\n\n\"You drive,\" she said, handing me the keys to the Macleods' Austin. This felt like promotion, and I was fastidious with my gear-changing.\n\nJoan lived about three miles away, and was the surviving sister of Gerald, who donkey's years previously had been sweet on Susan, but then had died suddenly from leukaemia, which was beastly luck. Joan had looked after their father until his death and had never married; she liked dogs and took an afternoon gin or two.\n\nWe parked in front of a squat, half-timbered house behind a beech hedge. Joan had a cigarette on when she answered the door, embraced Susan and looked at me inquisitively.\n\n\"This is Paul. He's driving me today. I really need my eyes testing, I think it's time for a new prescription. We met at the tennis club.\"\n\nJoan nodded, and said, \"I've shut the yappers up.\"\n\nShe was a large woman in a pastel-blue trouser suit; she had tight curls, brown lipstick, and was approximately powdered. She led us into the sitting room and collapsed into an armchair with a footstool in front of it. Joan was probably about five years older than Susan, but struck me as a generation ahead. On one arm of her chair was a face-down book of crosswords, on the other a brass ashtray held in place by weights concealed in a leather strap. The ashtray looked precariously full to me. No sooner had Joan sat down than she was up again.\n\n\"Join me in a little one?\"\n\n\"Too early for me, darling.\"\n\n\"You're not exactly driving,\" Joan replied grumpily. Then, looking at me, \"Drink, young sir?\"\n\n\"No thank you.\"\n\n\"Well, suit yourselves. At least you'll have a gasper with me.\"\n\nSusan, to my surprise, took a cigarette and lit up. It felt to me like a friendship whose hierarchy had been established long ago, with Joan as senior partner and Susan, if not subservient, at any rate the listening one. Joan's opening monologue told of her life since she'd last seen Susan, which seemed to me largely a catalogue of small annoyances triumphantly overcome, of dog-talk and bridge-talk, which resolved itself into the headline news that she had recently found a place ten miles away where you could get her favourite gin for some trifling sum less than it cost in the Village.\n\nBored out of my skull, half-disapproving of the cigarette Susan appeared to be enjoying, I found the following words coming out of my mouth:\n\n\"Have you factored in the petrol?\"\n\nIt was as if my mother had spoken through me.\n\nJoan looked at me with interest verging on approval. \"Now, how would I do that?\"\n\n\"Well, do you know how many mpg you get from your car?\"\n\n\"Of course I do,\" Joan replied, as if it were outrageous and spendthrift not to know. \"Twenty-eight on average around here, a bit over thirty on a longer trip.\"\n\n\"And how much do you pay for petrol?\"\n\n\"Well, that obviously depends on where I buy it, doesn't it?\"\n\n\"Aha!\" I exclaimed, as if this made the matter even more interesting. \"Another variable. Have you got a pocket calculator, by any chance?\"\n\n\"I've got a screwdriver,\" said Joan, laughing.\n\n\"Pencil and paper, at least.\"\n\nShe fetched some and came to sit next to me on the sofa, reeking of cigarettes. \"I want to see this in action.\"\n\n\"So how many off-licences and how many petrol stations are we talking about?\" I began. \"I'll need the full details.\"\n\n\"Anyone would think you're from the Inland Fucking Revenue,\" said Joan with a laugh and a thump on my shoulder.\n\nSo I took down prices and locations and distances, identifying one case of spurious false economy, and came up with her two best options.\n\n\"Of course,\" I added brightly, \"this one would be even more advantageous if you walked into the Village rather than drove.\"\n\nJoan gave a mock shriek. \"But walking's bad for me!\" Then she took my table of calculations, went back to her chair, lit up another cigarette, and said to Susan, \"I can see that he's a very useful young man to have around.\"\n\nAs we were driving away, Susan said, \"Casey Paul, I didn't know you could be so wicked. You had her eating out of your hand by the end of it.\"\n\n\"Anything to help the rich save money,\" I replied, carefully shifting gear. \"I'm your man.\"\n\n\"You _are_ my man, strange as it may seem,\" she agreed, slipping her flattened hand beneath my left thigh as I drove.\n\n\"By the way, what's wrong with your eyes?\"\n\n\"My eyes? Nothing, as far as I know.\"\n\n\"Then why did you go on about having them tested?\"\n\n\"Oh, that? Well, I have to have a form of words to cover you.\"\n\nYes, I could see that. And so I became \"the young man who drives me\" and \"my tennis partner,\" and later, \"a friend of Martha's\" and even\u2014most implausibly\u2014\"a kind of prot\u00e9g\u00e9 of Gordon's.\"\n\n* * *\n\n\u2014\n\nI don't remember when we first kissed. Isn't that odd? I can remember 6\u20132; 7\u20135; 2\u20136. I can remember that old driver's ears in foul detail. But I can't remember when or where we first kissed, or who made the first move, or whether it was both of us at the same time. And whether perhaps it was not so much a move as a drift. Was it in the car or in her house, was it morning, noon or night? And what was the weather like? Well, you certainly won't expect me to remember _that._\n\nAll I can tell you is that it was\u2014by the modern speed of things\u2014a long time before we first kissed, and a long time after that before we first went to bed together. And that between the kissing and the bedding I drove her up to London to buy some contraception. For her, not me. We went to John Bell & Croyden in Wigmore Street; I parked round the corner while she went in. She returned with a brown, unbranded bag containing a Dutch cap and some contraceptive jelly.\n\n\"I wonder if there's a book of instructions,\" she says lightly. \"I'm a bit out of practice with all this.\"\n\nIn my mood\u2014a kind of sombre excitement\u2014I'm momentarily unsure if she's referring to sex, or to putting in the cap.\n\n\"I'll be there to help,\" I say, thinking that this covers both interpretations.\n\n\"Paul,\" she says, \"there are some things it's better for a man not to see. Or to think about.\"\n\n\"OK.\" This definitely means the second option.\n\n\"Where will you keep it?\" I ask, imagining the consequences of its discovery.\n\n\"Oh, somewhere-somewhere,\" she replies. None of my business, then.\n\n\"Don't expect too much of me, Casey,\" she goes on rapidly. \"Casey. That's K.C. King's Cross. You won't be a crosspatch, will you? You won't get all ratty and shirty with me, will you?\"\n\nI lean across and kiss her, in front of whatever interested pedestrians Wimpole Street contains.\n\nI know already that she and her husband have separate beds, indeed separate rooms, and their marriage has been unconsummated\u2014or rather, sex-free\u2014for almost twenty years; but I haven't pressed her for reasons or particulars. On the one hand, I am deeply curious about almost everybody's sex life, past, present and future. On the other, I don't fancy the distraction of other images in my head when I am with her.\n\nI am surprised that she needs contraception, that at forty-eight she is still having periods, and that what she refers to as The Dreaded has not yet arrived. But I am rather proud that it hasn't. This is nothing to do with the possibility that she might get pregnant\u2014nothing could be further from my thoughts or desires; rather it seems a confirmation of her womanliness. I was going to say girlishness; and perhaps that's more what I mean. Yes, she is older; yes, she knows more about the world. But in terms of\u2014what shall I call it? the age of her spirit, perhaps\u2014we aren't that far apart.\n\n* * *\n\n\u2014\n\n\"I didn't know you smoked,\" I say.\n\n\"Oh, just the one, occasionally. To keep Joan company. Or I go out into the garden. Do you disapprove terribly?\"\n\n\"No, it just came as a surprise. I don't disapprove. I just think\u2014\"\n\n\"It's stupid. Yes it is. I just take one of his when I'm fed up. Have you noticed the way he smokes? He lights up and puffs away as if his life depends upon it, and then, when he's halfway through, he stubs it out in disgust. And that disgust lasts until he lights up the next one. About five minutes later.\"\n\nYes, I have noticed, but I let it go.\n\n\"Still, it's his drinking that's more annoying.\"\n\n\"But you don't?\"\n\n\"I hate the stuff. Just a glass of sweet sherry at Christmas, so as not to be accused of being a spoilsport. But it changes people. And not for the better.\"\n\nI agree. I have no interest in alcohol, or in people getting \"merry,\" or \"whistled,\" or \"half seas over\" and all the other words and phrases which make them feel better about themselves.\n\nAnd Mr. E.P. was no exemplar of the virtues of drinking. While waiting for his dinner, he would sit at the table surrounded by what Susan called \"his flagons and his gallons,\" pouring from them into his pint mug with an increasingly unsteady hand. In front of him was another mug, stuffed with spring onions, on which he would munch. Then, after a while, he would belch quietly, covering his mouth in a pseudo-genteel manner. As a consequence, I have loathed spring onions for most of my life. And never thought much of beer either.\n\n* * *\n\n\u2014\n\n\"You know, I was thinking the other day that I haven't seen his eyes for years. Not really. Not for years and years. Isn't that strange? They're always hidden behind his glasses. And of course I'm never there when he takes them off at night. Not that I want to see them especially. I've seen enough of them. I expect it's the same for a lot of women.\"\n\nThis is how she tells me about herself, in oblique observations which don't require a response. Sometimes, one leads on to another; sometimes, she lets drop a single statement, as if clueing me in to life.\n\n\"The thing you have to understand, Paul, is that we're a played-out generation.\"\n\nI laugh. My parents' generation don't seem at all played out to me: they still have all the power and money and self-assurance. I wish they _were_ played out. Instead, they seem a major obstacle to my growing up. What's that term they use in hospitals? Yes, bed-blockers. They were spiritual bed-blockers.\n\nI ask Susan to explain.\n\n\"We went through the war,\" she says. \"It took a lot out of us. We aren't much good for anything anymore. It's time your lot took charge. Look at our politicians.\"\n\n\"You aren't suggesting I go into politics?\" I am incredulous. I despise politicians, who all strike me as self-important creeps and smoothies. Not that I've ever met a politician, of course.\n\n\"It's exactly because people like you don't go into politics that we're in the mess we are,\" Susan insists.\n\nAgain, I am baffled. I'm not even sure who \"people like me\" might be. For my school and university friends, it seemed like a badge of honour _not_ to be interested in all the matters which politicians endlessly discussed. And then their grand anxieties\u2014the Soviet threat, the End of Empire, tax rates, death duties, the housing crisis, trade union power\u2014would be endlessly regurgitated at the family hearth.\n\nMy parents enjoyed television sitcoms, but were made uneasy by satire. You couldn't buy _Private Eye_ in the Village, but I would bring it back from university and leave it provocatively around the house. I remember one issue whose cover had a floppy 33 rpm disk loosely attached to it. Peeling off the record revealed the photo of a man sitting on the lavatory, trousers and pants round his ankles, shirttails keeping him decent. On to the neck of this anonymous squatter was montaged the head of the Prime Minister, Sir Alec Douglas-Home, with a bubble coming out of his mouth saying, \"Put that record back at once!\" I found it supremely funny, and showed it to my mother; she judged it stupid and puerile. Then I showed it to Susan, who was overcome with laughter. So that was everything decided, in one go: me, my mother, Susan and politics.\n\n* * *\n\n\u2014\n\nShe laughs at life, this is part of her essence. And no one else in her played-out generation does the same. She laughs at what I laugh at. She also laughs at hitting me on the head with a tennis ball; at the idea of having sherry with my parents; she laughs at her husband, just as she does when crashing the gears of the Austin shooting brake. Naturally, I assume that she laughs at life because she has seen a great deal of it, and understands it. \"By the way,\" I say, \"what's 'whatski'?\"\n\n\"What do you mean, 'What's whatski'?\"\n\n\"I mean, What's 'whatski'?\"\n\n\"Oh, do you mean, 'Whatski's whatski'?\"\n\n\"If you like.\"\n\n\"It's what Russian spies say to one another, silly,\" she replies.\n\n* * *\n\n\u2014\n\nThe first time we were together\u2014sexually, I mean\u2014we each told the necessary lies, then drove across to the middle of Hampshire and found two rooms in a hotel.\n\nAs we stand looking down at an acreage of magenta candlewick bedspread, she says,\n\n\"Which side do you prefer? Forehand or backhand?\"\n\nI have never slept in a double bed before. I have never slept a whole night with someone before. The bed looks enormous, the lighting bleak, and from the bathroom comes a smell of disinfectant.\n\n\"I love you,\" I tell her.\n\n\"That's a terrible thing to say to a girl,\" she replies and takes my arm. \"We'd best have dinner first, before we love one another.\"\n\nI already have an erection, and there is nothing generalised about this one. It is very, very specific.\n\nShe has a shyness to her. She never undresses in front of me; she is always in bed with her nightdress on by the time I come into the room. And the light would be out. I couldn't care less about any of this. I feel I can see in the dark, anyway.\n\n* * *\n\n\u2014\n\nNor does she \"teach me the arts of love,\" that phrase you read in books. We are both inexperienced, as I said. And she comes from a generation in which the assumption is made that on the wedding night the man \"will know what to do\"\u2014a social excuse to legitimise any previous sexual experience, however squalid, the man might have had. I don't want to go into the specifics in her case, though she does occasionally drop hints.\n\nOne afternoon, we are in bed at their house, and I suggest I ought to be going before \"Someone\" comes home.\n\n\"Of course,\" she replies musingly. \"You know, when he was at school, he always preferred the front half of the elephant, if you catch my meaning. And maybe after school. Who knows? Everyone's got a secret, haven't they?\"\n\n\"What's yours?\"\n\n\"Mine? Oh, he told me I was frigid. Not at the time. But later, after we'd stopped. When it was too late to do anything about anything.\"\n\n\"I don't think you're remotely frigid,\" I say, with a mixture of outrage and possessiveness. \"I think you're...very warm-blooded.\"\n\nShe pats my chest in reply. I know little about the female orgasm, and somehow assume that if you manage to keep going long enough, it will at some point be automatically triggered in the woman. Like breaking the sound barrier, perhaps. As I am unable to take the discussion further, I start to get dressed. Later, I think: she is warm, she is affectionate, she loves me, she encourages me into bed, we stay there a long time, _I_ don't think she's frigid, what's the problem?\n\n* * *\n\n\u2014\n\nWe talk about everything: the state of the world (not good), the state of her marriage (not good), the general character and moral standards of the Village (not good) and even Death (not good).\n\n\"Isn't it strange?\" she muses. \"My mother died of cancer when I was ten and I only ever think of her when I'm cutting my toenails.\"\n\n\"And yourself?\"\n\n\"Whatski?\"\n\n\"Yourself\u2014dying.\"\n\n\"Oh.\" She goes silent for a bit. \"No, I'm not afraid of dying. My only regret would be missing out on what happens afterwards.\"\n\nI misunderstand her. \"You mean, the afterlife?\"\n\n\"Oh, I don't believe in _that,_ \" she says firmly. \"It would all cause far too much trouble. All those people who spent their lives getting away from one another, and suddenly there they all are again, like some dreadful bridge party.\"\n\n\"I didn't know you played bridge.\"\n\n\"I don't. That's not the point, Paul. And then, all those people who did bad things to you. Seeing them again.\"\n\nI leave a pause; she fills it. \"I had an uncle. Uncle Humph. For Humphrey. I used to go and stay with him and Aunt Florence. After my mother died, so I would have been eleven, twelve. My aunt would put me to bed and tuck me in and kiss me and put out the light. And just as I would be getting off to sleep, there was a sudden weight on the side of the bed and it would be Uncle Humph, stinking of brandy and cigars and saying he wanted a goodnight kiss too. And then one time he said, 'Do you know what a \"party kiss\" is?' and before I could reply he rammed his tongue into my mouth and thrashed it around like a live fish. I wish I'd bitten it off. Every summer he did it, till I was about sixteen. Oh, it wasn't as bad as for some, I know, but maybe that's what made me frigid.\"\n\n\"You're _not,_ \" I insist. \"And with a bit of luck the old bastard will be in a very hot place. If there's any justice.\"\n\n\"There isn't,\" she replies. \"There isn't any justice, here or anywhere else. And the afterlife would just be an enormous bridge party with Uncle Humph bidding six no trumps and winning every hand and claiming a party kiss as his reward.\"\n\n\"I'm sure you're the expert,\" I say teasingly.\n\n\"But the thing _is,_ Casey Paul, it would be dreadful, entirely dreadful, if in some way that man was still alive. And what you don't wish for your enemies, you can hardly expect for yourself.\"\n\n* * *\n\n\u2014\n\nI don't know when the habit developed\u2014early on, I'm sure\u2014but I used to hold her wrists. Maybe it began in a game of seeing if I could encompass them with my middle fingers and thumbs. But it rapidly became something I did. She extends her forearms towards me, fingers making gentle fists, and says, \"Hold my wrists, Paul.\" I encompass them both, and press as hard as I can. What the exchange was about didn't need words. It was a gesture to calm her, to pass something from me to her. An infusion, a transfusion of strength. And of love.\n\nMy attitude to our love was peculiarly straightforward\u2014though I suspect a peculiar straightforwardness is characteristic of all first love. I simply thought: Well, that's the certainty of love between us settled, now the rest of life has to fall into place around it. And I was entirely confident that it would. I remembered from some of my school reading that Passion was meant to Thrive on Obstacles; but now that I was experiencing what I had only previously read about, the notion of an Obstacle to it seemed neither necessary nor desirable. But I was very young, emotionally, and perhaps simply blind to the obstacles others would find in plain sight.\n\nOr perhaps I didn't go by way of previous reading at all. Perhaps my actual thought was more like this: Here we are now, the two of us, and there is where we have to get to; nothing else matters. And though we did in the end get somewhere near to where I dreamed, I had no idea of the cost.\n\n* * *\n\n\u2014\n\nI said I couldn't remember the weather. And there's other stuff as well, like what clothes I wore and what food I ate. Clothes were unimportant necessities back then, and food was just fuel. Nor do I remember things I'd expect to, like the colour of the Macleods' shooting brake. I think it was two-tone. But was it grey and green, or perhaps blue and cream? And though I spent many key hours on its leather seats, I couldn't tell you their colour. Was the fascia panel made of walnut? Who cares? My memory certainly doesn't, and it's memory which is my guide here.\n\nOn top of this, there are things I can't be bothered to tell you. Like what I studied at university, what my room there was like, and how Eric differed from Barney, and Ian from Sam, and which one of them had red hair. Except that Eric was my closest friend, and continued so for many years. He was the gentlest of us, the most thoughtful, the one who put most trust in others. And\u2014perhaps because of these very qualities\u2014he was the one who had most trouble with girls and, later, women. Was there something about his softness, and his inclination to forgive, which almost provoked bad behaviour in others? I wish I knew the answer to that, not least because of the time I let him down badly. I abandoned him when he needed my help; I betrayed him, if you will. But I'll tell you about this later.\n\nAnd another thing. When I gave you my estate agent's sketch of the Village, some of it might not have been strictly accurate. For instance, the Belisha beacons at the zebra crossing. I might have invented them, because nowadays you rarely see a zebra crossing without a dutiful pair of flashing beacons. But back then, in Surrey, on a road with little traffic...I rather doubt it. I suppose I could do some real-life research\u2014look for old postcards in the central library, or hunt out the very few photos I have from that time, and retrofit my story accordingly. But I'm remembering the past, not reconstructing it. So there won't be much set-dressing. You might prefer more. You might be used to more. But there's nothing I can do about that. I'm not trying to spin you a story; I'm trying to tell you the truth.\n\n* * *\n\n\u2014\n\nSusan's tennis game comes back to me. Mine\u2014as I may have said\u2014was largely a self-taught business, relying on wristiness, ill-prepared body position and a deliberate, last-minute change of shot which sometimes bamboozled me as much as my opponent. When playing with her, this structural laziness often compromised my intense desire for victory. Her game had schooling behind it: she got into the correct position, hit fully through her groundstrokes, came to the net only when circumstances were propitious, ran her socks off and yet laughed equally at winning and losing. This had been my first impression of her, and from her tennis I naturally extrapolated her character. I assumed that in life too she would be calm, well ordered and reliable, hitting fully through the ball\u2014the best possible backcourt support for her anxious and impulsive partner at the net.\n\nWe entered the mixed doubles in the club's summer tournament. There were about three people watching our first-round match against a couple of old hackers in their mid-fifties; to my surprise, one of the spectators was Joan. Even when we changed ends and she was out of my eyeline, I could hear her smoker's cough.\n\nThe old hackers hacked us to death, playing like a married couple who could instinctively read one another's next move, and never needed to speak, let alone shout. Susan played solidly, as ever, whereas my game was stupidly erratic. I went for overambitious interceptions, took balls I should have left, and then fell into a lethargic sulk as the hackers closed out set and match 6\u20134.\n\nAfterwards, we sat with Joan, two teas and a gin between the three of us.\n\n\"Sorry I let you down,\" I said.\n\n\"That's all right, Paul, I really don't mind.\"\n\nHer even temper made me more irritated with myself. \"No, but I do. I was trying all sorts of stupid stuff. I wasn't any help. And I couldn't get my first serve in.\"\n\n\"You let your left shoulder drop,\" said Joan out of the blue.\n\n\"But I serve right-handed,\" I replied rather petulantly.\n\n\"That's why the left shoulder has to be kept high. Holds you in balance.\"\n\n\"I didn't know you played.\"\n\n\"Played? Ha! I used to win the fucking thing. Until my knees went. You need a few lessons, young Master Paul, that's all. But you've got good hands.\"\n\n\"Look\u2014he's blushing!\" Susan observed unnecessarily. \"I've never seen him do that before.\"\n\nLater, in the car, I say, \"So what's Joan's story? Was she really a good player?\"\n\n\"Oh yes. She and Gerald won pretty much everything, up to county level. She was a strong singles player, as you can probably imagine, until her knees let her down. But she was even better at doubles. Having someone to support and be supported by.\"\n\n\"I like Joan,\" I say. \"I like the way she swears.\"\n\n\"Yes, that's what people see and hear, and like or don't like. Her gin, her cigarettes, her bridge game, her dogs. Her swearing. Don't underestimate Joan.\"\n\n\"I wasn't,\" I protest. \"Anyway, she said I had good hands.\"\n\n\"Don't always be joking, Paul.\"\n\n\"Well, I _am_ only nineteen, as my parents keep reminding me.\"\n\nSusan goes quiet for a bit, then, seeing a lay-by, turns into it and stops the car. She looks ahead through the windscreen.\n\n\"When Gerald died, I wasn't the only one who was hard hit. Joan was devastated. They'd lost their mother when they were little, and their father had to work every day in that insurance company, so they were thrown into depending on one another. And when Gerald died...she went off the rails a bit. Started sleeping with people.\"\n\n\"There's nothing wrong with that.\"\n\n\"There is and there isn't, Casey Paul. Depends on who you are and who they are. And who's robust enough to survive. Usually, that's the man.\"\n\n\"Joan seems pretty robust to me.\"\n\n\"That's an act. We all have an act. You'll have an act one day, oh yes you will. So Joan was a bad picker. And at first it didn't seem to matter, as long as she didn't get pregnant or anything like that. And she didn't. Then she fell like a ton of bricks for...his name doesn't matter. Married of course, rich of course, other girlfriends of course. Set her up at a flat in Kensington.\"\n\n\"Good Lord. Joan was...a kept woman? A...mistress?\" These were words, and sexual functions, I'd only come across in books.\n\n\"Whatever you want to call it. The words don't fit. They rarely do. What do you call you? What do you call me?\" I don't reply. \"And Joan was completely gone on the old bastard. Waiting for his visits, believing his promises, going off on the occasional weekend abroad. He strung her along like that for three years. Then at last, as he'd always promised, he divorced his wife. And Joan thought her ship had come home. She'd proved us all wrong, what's more. 'My ship's coming home,' she kept repeating.\"\n\n\"But it hadn't?\"\n\n\"He married another woman instead. Joan read the announcement in the papers. Piled up all the clothes he'd bought her in the sitting room of the flat, poured lighter fuel over them, lit a match, walked out, slammed the door, put the keys through the letterbox, went back to her father. Turned up on his doorstep. Smelt a bit singed, I expect. Her father didn't say anything or ask any questions, just hugged her. It took her months even to tell him. The only luck\u2014if there was luck\u2014was that she didn't set the whole block of flats on fire. Just burnt a hole in an expensive carpet. She could have ended up in prison for manslaughter.\n\n\"After that she took care of her father devotedly. Became interested in dogs. Had a go at breeding them. Learned how to pass the time. That's one of the things about life. We're all just looking for a place of safety. And if you don't find one, then you have to learn how to pass the time.\"\n\nI don't think this will ever be my problem. Life is just too full and always will be.\n\n\"Poor old Joan,\" I say. \"I'd never have guessed.\"\n\n\"She cheats at crosswords.\"\n\nThis seems a non sequitur to me.\n\n\"What?\"\n\n\"She cheats at crosswords. She does them out of books. She once told me that if she gets stuck, she fills in any old word, as long as it's the right number of letters.\"\n\n\"But that defeats the whole purpose...and anyway, all the answers are in the back of those books.\" I am at a loss, so just repeat, \"Poor old Joan.\"\n\n\"Yes and no. Yes and no. But don't ever forget, young Master Paul. Everyone has their love story. Everyone. It may have been a fiasco, it may have fizzled out, it may never even have got going, it may have been all in the mind, that doesn't make it any less real. Sometimes, it makes it more real. Sometimes, you see a couple, and they seem bored witless with one another, and you can't imagine them having anything in common, or why they're still living together. But it's not just habit or complacency or convention or anything like that. It's because once, they had their love story. Everyone does. It's the only story.\"\n\nI don't answer. I feel rebuked. Not rebuked by Susan. Rebuked by life.\n\n* * *\n\n\u2014\n\nThat evening, I looked at my parents and paid attention to everything they said to one another. I tried to imagine that they too had had their love story. Once upon a time. But I couldn't get anywhere with that. Then I tried imagining that each had had their love story, but separately, either before marriage or perhaps\u2014even more thrillingly\u2014during it. But I couldn't make that work either, so I gave up. I found myself wondering instead if, like Joan, I would one day have an act of my own, an act designed to deflect curiosity. Who could tell?\n\nThen I went back and tried to imagine how it might have been for my parents in those years before I had existed. I picture them starting off together, side by side, hand in hand, happy, confident, strolling down some gentle, soft, grassy furrow. All is verdant and the view extensive; there seems to be no hurry. Then, as life proceeds, in its normal, daily, unthreatening way, the furrow very slowly deepens, and the green becomes flecked with brown. A little further on\u2014a decade or two\u2014and the earth is heaped higher on either side, and they are unable to see over the top. And now there is no escape, no turning back. There is only the sky above, and ever-higher walls of brown earth, threatening to entomb them.\n\nWhatever happened, I wasn't going to be a furrow-dweller. Or a breeder of dogs.\n\n* * *\n\n\u2014\n\n\"What you have to understand is this,\" she says. \"There were three of us. The boys got the education\u2014that's how it was. Philip's took him all the way, but the money for Alec ran out when he was fifteen. Alec was the one I was closest to. Everyone adored Alec, he was just the best. Naturally, he joined up as soon as he could, that's what the best ones did. The Air Force. He ended up flying Sunderlands. They're flying boats. They used to go out on long patrols over the Atlantic, looking for U-boats. Thirteen hours at a time. They gave them pills to help them keep going. No, that's nothing to do with it.\n\n\"So, you see, on his last leave he took me to supper. Nowhere posh, just Corner House. And he took my hands in his and said, 'Sue darling, they're complicated beasts, those Sunderlands, and I often don't think I'm up to it. They're too bloody complicated, and sometimes, when you're out there over the water, and it all looks the same, hour after hour, you've no idea where you are, and sometimes even the navigator doesn't either. I always say a prayer at takeoff and landing. I don't believe, but I say a prayer nevertheless. And every time I'm just as bloody scared as the time before. Right, I've got that off my chest. Corners up from now on. Corners up in the Corner House.'\n\n\"That was the last time I saw him. He was posted missing three weeks later. They never found a trace of his plane. And I always think of him out there, over the water, being scared.\"\n\nI put my arm around her. She shakes it loose, frowningly.\n\n\"No, that's not all. There always seemed to be these men around. It was wartime and you'd think they'd all be off fighting, but there was a jolly lot of them around at home, I can tell you. The lesser men. So there was Gerald, who couldn't pass the medical, even though he tried twice, and then Gordon, who was in a reserved occupation, as he liked to say. Gerald was sweet-tempered and nice-looking, and Gordon was a bit of a crosspatch, but anyway I just preferred dancing with Gerald. Then we got engaged, because, well, it was wartime and people did things like that then. I don't think I was in love with Gerald, but he was a kind man, that's for sure. And then he went and died of leukaemia. I told you that. It was beastly luck. So I thought I might as well marry Gordon. I thought it might make him less of a crosspatch. And _that_ part of things didn't work out, as you may have observed.\"\n\n\"But\u2014\"\n\n\" _So,_ you see, we're a played-out generation. All the best ones went. We were left with the lesser ones. It's always like that in war. That's why it's up to your generation now.\"\n\nBut I don't feel part of a generation, for a start; and, moved as I am by her story, her history, her prehistory, I still don't want to go into politics.\n\n* * *\n\n\u2014\n\nWe were driving somewhere in my car, a Morris Minor convertible in a shade of mud-green. Susan said it looked like a very low-level German staff car from the war. We were at the foot of a long hill, with no traffic in sight. I was never a reckless driver, but I pushed hard down on the accelerator pedal to get a good run at the gradient. And after about fifty yards I realised something was seriously wrong. The car was accelerating at full throttle, even though I'd now taken my foot off the pedal. Instinctively, I rammed it on the brake. That didn't help much. I was doing two things at the same time: panicking, and thinking clearly. Don't ever believe those two states are incompatible. The engine was roaring, the brakes were screaming, the car was beginning to slew across the road, we were going between forty and fifty. It never occurred to me to ask Susan what to do. I thought, this is my problem, I've got to fix it. And then it came to me: take the car out of gear. So I put in the clutch, and moved the gear stick to neutral. The car's hysteria decreased and we coasted to a halt on the verge.\n\n\"Well done, Casey Paul,\" she says. Giving me both names was usually a sign of approval.\n\n\"I should have thought of that earlier. Actually, I should have just switched off the bloody ignition. That would have done it. But it didn't cross my mind.\"\n\n\"I think there's a garage over the hill,\" she says, getting out, as if such an event were entirely routine.\n\n\"Were you scared?\"\n\n\"No. I knew you'd sort it out, whatever it was. I always feel safe with you.\"\n\nI remember her saying that, and me feeling proud. But I also remember the feel of the car as it raced out of control, as it resisted the brakes, as it bucked and slewed across the road.\n\n* * *\n\n\u2014\n\nI must tell you about her teeth. Well, two of them, anyway. The middle front ones at the top. She called them her \"rabbit teeth\" because they were perhaps a millimetre longer than the strict national average; but that, to me, made them the more special. I used to tap them lightly with my middle finger, checking that they were there, and secure, just as she was. It was a little ritual, as if I was taking an inventory of her.\n\n* * *\n\n\u2014\n\nEveryone in the Village, every grown-up\u2014or rather, every middle-aged person\u2014seemed to do crosswords: my parents, their friends, Joan, Gordon Macleod. Everyone apart from Susan. They did either _The Times_ or _The Telegraph;_ though Joan had those books of hers to fall back on while waiting for the next newspaper. I regarded this traditional British activity with some snootiness. I was keen in those days to find hidden motives\u2014preferably involving hypocrisy\u2014behind the obvious ones. Clearly, this supposedly harmless pastime was about more than solving cryptic clues and filling in the answers. My analysis identified the following elements: 1) the desire to reduce the chaos of the universe to a small, comprehensible grid of black-and-white squares; 2) the underlying belief that everything in life could, in the end, be solved; 3) the confirmation that existence was essentially a ludic activity; and 4) the hope that this activity would keep at bay the existential pain of our brief sublunary transit from birth to death. That seemed to cover it!\n\nOne evening, Gordon Macleod looked up from behind a cigarette smokescreen and asked,\n\n\"Town in Somerset, seven letters, ends in N.\"\n\nI thought about this for a while. \"Swindon?\"\n\nHe made a tolerant tut-tut. \"Swindon's in Wiltshire.\"\n\n\"Is it really? That's a surprise. Have you ever been there?\"\n\n\"Whether I have or not is hardly relevant to the business in hand,\" he replied. \"Look at it on the page. That might help.\"\n\nI went and sat next to him. Seeing a lineup of six blank spaces followed by an N didn't help me any the more.\n\n\"Taunton,\" he announced, putting in the answer. I noticed the eccentric way he did his capital letters, lifting the pen from the page to make each stroke. Whereas anyone else would produce an N from two applications of pen to paper, he made three.\n\n\"Continue mocking Somerset town. That was the clue.\"\n\nI thought about this, not very hard, admittedly.\n\n\"Taunt on\u2014continue mocking. Taunt on\u2014TAUNTON. Get it, young fellermelad?\"\n\n\"Oh, _I see,_ \" I said nodding. \"That's clever.\"\n\nI didn't mean it, of course. I was also thinking that Macleod must certainly have got the answer before he asked me. So then I added an extra clause to my analysis of the crossword\u2014or, as Macleod preferred to call it, The Puzzle: 3b) false confirmation that you are more intelligent than some give you credit for.\n\n\"Does Mrs. Macleod do the crossword?\" I asked, already knowing the answer. Two could play at this game, I thought.\n\n\"The Puzzle,\" he replied with some archness, \"is not really a female domain.\"\n\n\"My mum does the crossword with my dad. Joan does the crossword.\"\n\nHe lowered his chin and looked at me over his spectacles.\n\n\"Then let us posit, perhaps, that The Puzzle is not the domain of the womanly woman. What do you say to that?\"\n\n\"I'd say I don't have enough experience of life to come to a conclusion on that one.\" Though inwardly I was reflecting on the phrase \"womanly woman.\" Was it uxorious praise, or some kind of disguised insult?\n\n\"So that gives us an O in the middle of twelve down,\" he went on. Suddenly there was an \"us\" involved.\n\nI gazed at the clue. Something about an arbiter in work and a leaf.\n\n\"TREFOIL,\" Macleod muttered, writing it in, three strokes of his pen on the R, where others would construct it from two. \"You see, it's REF\u2014arbiter\u2014in the middle of TOIL\u2014work.\"\n\n\"That's clever too,\" I falsely enthused.\n\n\"Standard. Had it before a few times,\" he added with a touch of complacency.\n\n2b) the further belief that once you have solved something in life, you will be able to solve it again, and the solution will be exactly the same the second time around, thus offering assurance that you have reached a pitch of maturity and wisdom.\n\nMacleod decided, without my asking, to teach me the ins and outs of The Puzzle. Anagrams, and how to spot them; words hidden inside other combinations of words; setters' shorthand and their favourite tricks; common abbreviations, letters and words drawn from chess annotation, military ranks, and so on; how a word may be written upwards in the solution to a down clue, or backwards in an across clue. \" 'Running west,' you see, that's the giveaway.\"\n\nCorrection to 4). To begin: \"the hope that this arse-bendingly boring activity would keep...\"\n\nLater, I tried making an anagram out of WOMANLY WOMAN. I didn't get anywhere, of course. WANLY MOWN LOOM and other bits of nonsense were all I turned up.\n\nFurther addition: 1a) a successful means of taking your mind off the question of love, which is all that counts in the world.\n\nNonetheless, I continued to keep Macleod company while he puffed away at his Players and filled his grids with strangely mechanical pen strokes. He seemed to enjoy explaining clues to me, and took my occasional half-meant whistles and grunts as applause.\n\n\"We'll make a Puzzle-solver of him yet,\" he remarked to Susan over supper one evening.\n\nSometimes, we did things together, he and I. Nothing major, or for a long time, anyway. He asked me for my help with some rope-and-pin instrument in the garden, designed to ensure that the cabbages he was planting out were in regimental lines. A couple of times, we listened to a test match on the radio. Once, he took me with him to fill up the car with what he referred to as \"petroleum.\" I asked which garage he was intending to patronise. The nearest, he told me, unsurprisingly. I told him that I had done an analysis of price versus distance in the matter of Joan's gin, and what my findings had been.\n\n\"How incredibly boring,\" he commented, and then smiled at me.\n\nI realised that I had seen his eyes on more than one occasion recently. Whereas Susan hadn't seen them in years. Maybe she was exaggerating. Or maybe she hadn't been looking too hard in the first place.\n\nNOW ONLY MMWAA...no, that wasn't any good either.\n\n* * *\n\n\u2014\n\nHere is something I often thought at this time: I've been educated at school and university, and yet, in real terms, I know nothing. Susan barely went to school, but she knows so much more. I've got the book-learning, she's got the life-learning.\n\nNot that I always agreed with her. When she was talking about Joan, she'd said: \"We're all just looking for a place of safety.\" I pondered these words for a while afterwards. The conclusion I came to was this: maybe so, but I'm young, I'm \"only nineteen,\" and I'm more interested in looking for a place of danger.\n\n* * *\n\n\u2014\n\nLike Susan, I had euphemistic phrases to describe our relationship. We just seem to have this rapport across the generations. She's my tennis partner. We both like music and go up to London for concerts. Also, art exhibitions. Oh, I don't know, we just get on somehow. I have no idea who believed what, and who knew what, and how much my pride made it all flauntingly obvious. Nowadays, at the other end of life, I have a rule of thumb about whether or not two people are having an affair: if you think they might be, then they definitely are. But this was decades ago, and perhaps, back then, the couples you thought might be, mainly weren't.\n\n* * *\n\n\u2014\n\nAnd then there were the daughters. I wasn't much at ease with girls at that time of my life, neither the ones I met at university, nor the Carolines at the tennis club. I didn't understand that they were mostly just as nervous as I was about...the whole caboodle. And while boys were good at coming up with their own, homemade bullshit, girls, in their understanding of the world, often seemed to fall back on The Wisdom of Their Mothers. You could sniff the inauthenticity when a girl\u2014knowing no more than you did about anything\u2014said something like: \"Everyone's got twenty-twenty vision with hindsight.\" A line which could have issued word for word from the mouth of my mother. Another piece of appropriated maternal wisdom I remember from this time was this: \"If you lower your expectations, you can't be disappointed.\" This struck me as a dismal approach to life, whether for a forty-five-year-old mother or a twenty-year-old daughter.\n\nBut anyway: Martha and Clara. Miss G. and Miss N.S. Miss Grumpy and Miss Not So (Grumpy). Martha was like her mother physically, tall and pretty, but with something of her father's querulous temperament. Clara was plump and round, but entirely more equable. Miss Grumpy disapproved of me; Miss Not So was friendly, even interested. Miss Grumpy said things like, \"Haven't you got a home of your own to go to?\" Miss Not So would ask what I was reading and once, even, showed me some poetry she'd written. But I wasn't much of a judge of poetry, then or now, so my response probably disappointed her. This was my preliminary assessment, for what it was worth.\n\nIf I was uneasy with girls generally, I was the more so with ones who were a bit older than me, let alone ones whose mother I was in love with. This awkwardness of mine seemed to emphasise the insouciance with which they moved about their own house, appeared, disappeared, spoke or failed to speak. My reaction to this was possibly a bit crude, but I decided to be no more interested in them than they were in me. This seemed to amount to less than a passing 5 percent. Which was fine by me, because more than 95 percent of my interest was in Susan.\n\nSince Martha disapproved of me the more, it was to her, in a spirit of either challenge or perversity, that I said:\n\n\"I think I should explain. Susan's a kind of mother-substitute for me.\"\n\nNo, it wasn't very good, in any way. It probably sounded false, a slimy attempt at ingratiation. Martha took her time about replying, and her tone was acerbic.\n\n\"I don't need one, I've already got a mother.\"\n\nDid I mean any part of my lie? I can't believe that I did. Strange as it may seem I never reflected on our age difference. Age felt as irrelevant as money. Susan never seemed a member of my parents' generation\u2014\"played-out\" or not. She never pulled any sort of rank on me, never said, \"Ah, when you're a bit older, you'll understand\" and stuff like that. It was only my parents who harped on about my immaturity.\n\nAha, you might say, but surely the fact that you told her own daughter that for you she was a mother-substitute is a complete giveaway? You may claim it was insincere, but don't we all make jokes to allay our inner fears? She was almost exactly the same age as your mother, and you went to bed with her. So?\n\nSo. I see where you're going\u2014bus number 27 to a crossroads near Delphi. Look, I did not want, at any point, on any level, to kill my own father and sleep with my own mother. It's true that I wanted to sleep with Susan\u2014and did so many times\u2014and for a number of years thought of killing Gordon Macleod, but that is another part of the story. Not to put too fine a point on it, I think the Oedipus myth is precisely what it started off as: melodrama rather than psychology. In all my years of life I've never met anyone to whom it might apply.\n\nYou think I'm being naive? You wish to point out that human motivation is deviously buried, and hides its mysterious workings from those who blindly submit to it? Perhaps so. But even\u2014especially\u2014Oedipus didn't _want_ to kill his father and sleep with his mother, did he? Oh yes he did! Oh no he didn't! Yes, let's just leave it as a pantomime exchange.\n\nNot that prehistory doesn't matter. Indeed, I think prehistory is central to all relationships.\n\n* * *\n\n\u2014\n\nBut I'd much rather tell you about her ears. I missed my first sight of them at the tennis club, when she had her hair pulled back by that green ribbon which matched the piping and buttons of her dress. And normally, she wore her hair down, curling over her ears and descending to mid-neck. So it wasn't until we were in bed and I was rummaging and rootling around her body, into every nook and cranny, every overexamined and underexamined part of her, that, crouched above, I swept back her hair and discovered her ears.\n\nI'd never thought much about ears before, except as comic excrescences. Good ears were ears you didn't notice; bad ears stuck out like bat wings, or were cauliflowered from a boxer's punch, or\u2014like those of that furious driver at the zebra crossing\u2014were coarse and red and hairy. But her ears, ah, her ears...from the discreet, almost absent lobe they set off northwards at a gentle angle, but then at the midpoint turned back at the same angle to return to her skull. It was as if they had been designed according to aesthetic principle rather than the rules of auditory practicality.\n\nWhen I point this out to her, she says, \"It's probably so all that _rubbish_ scoots past them and doesn't go inside.\"\n\nBut there was more. As I explored them with the tips of my fingers, I discovered the delicacy of their outer rim: thin, warm, gentle, almost translucent. Do you know the word for that outermost whorl of the ear? It's called the helix. Plural: helices. Her ears were part of her absolute distinctiveness, expressions of her DNA. The double helix of her double helices.\n\nLater, turning my mind to what she might have meant by the \"rubbish\" that scooted past her astonishing ears, I thought: well, being accused of frigidity, that's a major piece of rubbish. Except that this word had gone straight into her ears and thence her brain and was lodged there, permanently.\n\n* * *\n\n\u2014\n\nAs I said, money had no more relevance to our relationship than age. So it didn't matter that she paid for things. I had none of that foolish masculine pride in such circumstances. Perhaps I even felt my lack of money made my love for Susan the more virtuous.\n\nAfter a few months\u2014maybe longer\u2014she announces that I need a running-away fund.\n\n\"What for?\"\n\n\"For running away. Everyone should have a running-away fund.\" Just as every young man should have a reputation. Where had this latest idea come from? A Nancy Mitford novel?\n\n\"But I don't want to run away. Who from? My parents? I've more or less left them anyway. Mentally. You? Why should I want to run away from you? I want you to be in my life forever.\"\n\n\"That's very sweet of you, Paul. But it's not a specific fund, you see. It's a sort of general fund. Because at some point everyone wants to run away from their life. It's about the only thing human beings have in common.\"\n\nThis is all way above my head. The only running away I might contemplate is running away with her rather than from her.\n\nA few days later, she gives me a cheque for \u00a3500. My car had cost me \u00a325; I lived for a term at university on under \u00a3100. The sum seemed both very large and also meaningless. I didn't even think it \"generous.\" I had no principles about money, either for or against. And it was entirely irrelevant to our relationship\u2014that much I knew. So when I got back to Sussex, I went into town, opened a deposit account at the first bank I came to, handed over the cheque and forgot about it.\n\n* * *\n\n\u2014\n\nThere's something I probably should have clarified earlier. I may be making my relationship with Susan sound like a sweet summer interlude. That's what the stereotype insists, after all. There is a sexual and emotional initiation, a lush passage of treats and pleasures and spoilings, then the woman, with a pang but also a sense of honour, releases the young man back into the wider world and younger bodies of his own generation. But I've already told you that it wasn't like this.\n\nWe were together\u2014and I mean together\u2014for ten or a dozen years, depending on where you start and stop counting. And those years happened to coincide with what the newspapers liked to call the Sexual Revolution: a time of omni-fucking\u2014or so we were led to believe\u2014of instant pleasures, and loose, guilt-free liaisons, when deep lust and emotional lightness became the order of the day. So you could say that my relationship with Susan proved as offensive to the new norms as to the old ones.\n\n* * *\n\n\u2014\n\nI remember her, one afternoon, wearing a print dress with flowers on it, going over to a chintz sofa and plumping herself down on it.\n\n\"Look, Casey Paul! I'm disappearing! I'm doing my disappearing act! There's nobody here!\"\n\nI look. It is half-true. Her stockinged legs show clearly, as do her head and neck, but all the middle parts are suddenly camouflaged.\n\n\"Wouldn't you like that, Casey Paul? If we could just disappear and nobody could see us?\"\n\nI don't know how serious, or how merely skittish, she is being. So I don't know how to react. Looking back, I think I was a very literal young man.\n\n* * *\n\n\u2014\n\nI told Eric that I had met this family and fallen in love. I described the Macleods, their house and their way of life, relishing my characterizations. It was the first grown-up thing that had happened to me, I told him.\n\n\"So which of the daughters have you fallen in love with?\" Eric asked.\n\n\"No, not one of the daughters, the mother.\"\n\n\"Ah, the mother,\" he said. \"We like that,\" he added, giving me marks for originality.\n\n* * *\n\n\u2014\n\nOne day, I notice a dark bruise on her upper arm, just below where the sleeve of her dress ends. It is the size of a large thumbprint.\n\n\"What's that?\" I ask.\n\n\"Oh,\" she says carelessly, \"I must have knocked it against something. I bruise easily.\"\n\nOf course she does, I think. Because she's sensitive, like me. Of course the world can hurt us. That's why we must look after one another.\n\n\"You don't bruise when I hold your wrists.\"\n\n\"I don't think the wrists bruise, do they?\"\n\n\"Not if I'm holding you.\"\n\n* * *\n\n\u2014\n\nThe fact that she was \"old enough to be my mother\" did not go down well with my mother. Nor my father; nor her husband; nor her daughters; nor the Archbishop of Canterbury\u2014not that he was a family friend. I cared no more about approval than I did about money. Though disapproval, whether active or theoretical, ignorant or informed, did nothing but inflame, corroborate and justify my love.\n\nI had no new definition of love. I didn't really examine what it was, and what it might entail. I merely submitted to first love in all its aspects, from butterfly kisses to absolutism. Nothing else mattered. Of course there was \"the rest of my life,\" both present (my degree course) and future (job, salary, social position, retirement, pension, death). You could say that I put this part of my life on hold. Except that's not right: she _was_ my life, and the rest wasn't. Everything else could and must be sacrificed, with or without thought, as and when necessary. Though \"sacrifice\" implies loss. I never felt a sense of loss. Church and state, they say, church and state. No difficulty there. Church first, church always\u2014though not in a sense the Archbishop of Canterbury would have understood it.\n\nI wasn't so much constructing my own idea of love as first doing the necessary rubble clearance. Most of what I'd read, or been taught, about love, didn't seem to apply, from playground rumour to high-minded literary speculation. \"Man's love is of man's life a thing apart \/ 'Tis woman's whole existence.\" How wrong\u2014how gender-biased, as we might now say\u2014was that? And then, at the other end of the spectrum, came the earthy sex-wisdom exchanged between profoundly ignorant if yearningly lustful schoolboys. \"You don't look at the mantelpiece while poking the fire.\" Where had that come from? Some bestial dystopia full of nocturnal, myopic grunting?\n\nBut I wanted her face there all the time: her eyes, her mouth, her precious ears with their elegant helices, her smile, her whispered words. So: I would be flat on my back, she would be lying on top of me, her feet slipped between mine, and she would place the tip of her nose against the tip of mine, and say,\n\n\"Now we see eye to eye.\"\n\nPut it another way. I was nineteen, and I knew that love was incorruptible, proof against both time and tarnish.\n\n* * *\n\n\u2014\n\nI have a sudden attack of\u2014what?\u2014fear, propriety, unselfishness? I say to her, thinking she will know more,\n\n\"You see, I haven't been in love before, so I don't understand about love. What I'm worried about is that, if you love me, it will leave you less for the other people you love.\"\n\nI don't name them. I meant her daughters; and perhaps even her husband.\n\n\"It's not like that,\" she answers at once, as if it is something she too has thought about, and has solved. \"Love's elastic. It's not a question of watering down. It adds on. It doesn't take away. So there's no need to worry about that.\"\n\nSo I didn't.\n\n* * *\n\n\u2014\n\n\"There's something I need to explain,\" she begins. \"E.P.'s father was a very nice man. He was a doctor. He collected furniture. Some of these things were his.\" She points vaguely at a heavy oak coffer and a grandfather clock I have never yet heard strike the hour. \"He actually hoped E.P. would become a painter, so he gave him the middle name of Rubens. Which was a bit unfortunate because some of the boys at school assumed he must be Jewish. Anyway, he did the usual schoolboy sketches, which everyone said were promising. But he never became more than promising, so was a disappointment to his father in that department. Jack\u2014the father\u2014was always very kind to me. He used to twinkle at me.\"\n\n\"I can't say I blame him for that.\" I wonder what might be coming next. Surely not some intergenerational imbroglio?\n\n\"We'd only been married a couple of years when Jack got cancer. I'd always thought he would be someone I could go to if I got in any trouble, and now he was going to be taken from me. I used to go and sit with him, but I would get so upset that it usually ended up with him consoling me rather than the other way round. I asked him once what he thought about it all, and he said, 'Of course I'd prefer it otherwise, but I can't complain that I haven't had a fair crack of the whip.' He liked me being with him, maybe because I was young and didn't know very much, and so I stayed there till the end.\n\n\"That day\u2014the last day\u2014the doctor\u2014the one looking after him, who was a good friend as well\u2014came in and said quietly, 'It's time to put you under, Jack.' 'You're right,' came the reply. He'd been in terrible pain for too long, you see. Then Jack turned to me and said, 'I'm sorry our acquaintance has been so brief, my dear. It's been wonderful knowing you. I'm aware that Gordon can be a difficult row to hoe, but I'll die happy knowing that I leave him in your safe and capable hands.' And then I kissed him and left the room.\"\n\n\"You mean, the doctor killed him?\"\n\n\"He gave him enough morphine to put him to sleep, yes.\"\n\n\"But he didn't wake up?\"\n\n\"No. Doctors used to do that in the old days, especially among themselves. Or with a patient they'd known a long time, where there was trust. Easing the suffering is a good idea. It's a terrible disease.\"\n\n\"Even so. I'm not sure I'd want to be killed.\"\n\n\"Well, wait and see, Paul. But that's not the point of the story.\"\n\n\"Sorry.\"\n\n\"The point of the story is 'safe and capable.' \"\n\nI think about this for a while. \"Yes, I see.\" But I'm not sure that I did.\n\n* * *\n\n\u2014\n\n\"Where do you usually go for your holidays?\" I ask.\n\n\"Paul, that's such a hairdresser's question.\"\n\nIn reply, I lean over and tuck her hair behind her ears, stroking the helices gently.\n\n\"Oh dear,\" she goes on. \"All these conventional expectations people have of one. No, not you, Casey Paul. I mean, why does everyone have to be the same? We did have a few holidays once, when the girls were young. About as successful as the Dieppe Raid, I'd say. E.P. was not at his best on holidays. I don't see what they're for, really.\"\n\nI wonder if I shouldn't press any further. Perhaps something catastrophic had happened on one of their holidays.\n\n\"So what do you say when hairdressers ask you that question?\"\n\n\"I say, 'We're still going to the usual place.' And that makes them think we've talked about it before and they've forgotten, so they usually let me off after that.\"\n\n\"Maybe you and I should have a holiday.\"\n\n\"You might have to teach me what they're for.\"\n\n\"What they're _for,_ \" I say firmly, \"is for being with someone you love a few hundred miles away from this sodding Village where we both live. Being with them all the time. Going to bed with them and waking up with them.\"\n\n\"Well, put like that, Casey...\"\n\nSo you see, there were some things I knew and she didn't.\n\n* * *\n\n\u2014\n\nWe are sitting in the cafeteria of the Festival Hall before a concert. Susan has noticed early on that when my blood sugar drops I become, in her words, \"a bit of a grumpus,\" and she is now feeding me up to prevent this. I am probably having a something-and-chips; she will content herself with a cup of coffee and a few biscuits. I love these escapes we make up to London, just for a few hours, being together, away from the Village, from my parents and her husband and all that stuff, in the noise and crush of the city, waiting for the silence and then the sudden floatingness of music.\n\nI am about to say all this when a woman comes and sits at our table without a pretence of asking if we mind. A woman of middle age, by herself; that is all she was, though in memory I might have translated her into some version of my mother\u2014at any rate, a woman who could be counted upon to disapprove of my relationship with Susan. And so, after a couple of minutes, knowing exactly what I am doing, I look across and say to Susan, in a clear, exact voice,\n\n\"Will you marry me?\"\n\nShe blushes, covers her ears and bites her lower lip. With a bang and a push and a stomp, the invader picks up her cup and makes for another table.\n\n\"Oh, Casey Paul,\" says Susan, \"you are mighty wicked.\"\n\n* * *\n\n\u2014\n\nI was having supper at the Macleods'. Clara was there too, back from university. Macleod was at the head of the table with his flagon of whatever, a mugful of spring onions in front of him like a jar of tulips.\n\n\"You might be aware,\" he said to Clara, \"that this young man appears to have joined our household. So be it.\"\n\nI couldn't tell from his tone whether he was being pedantically welcoming or slyly indicating his disdain. I looked across at Clara, but got no help with interpretation.\n\n\"Well, we shall see, shan't we?\" he continued, appearing to contradict his first pronouncement. He took in a mouthful of spring onion and shortly afterwards burped gently.\n\n\"One of the things the young man is kindly, if belatedly, addressing is the question of your mother's musical education. Or, should I say, lack thereof.\"\n\nThen, turning to me: \"Clara was named after Clara Schumann, which was perhaps a little ambitious on our part. She has never, alas, displayed much aptitude for the pianoforte, has she?\"\n\nI couldn't tell if the question was addressed to mother or daughter. As for me, I had never heard of Clara Schumann, so felt at even more of a disadvantage.\n\n\"Maybe, if your mother had begun her musical education earlier, she might have been able to pass on to you some of her now late-flowering enthusiasm.\"\n\nI had never before been in a household in which the male presence was so overbearing and yet so ambiguous. Perhaps this happens when there is only one man around: his understanding of the male role can expand unchallenged. Or perhaps this was just what Gordon Macleod was like.\n\nStill, my inability to grasp tone was a lesser matter that evening. The greater problem was that, at nineteen, I was unskilled at knowing how to behave socially at the table of a man whose wife I was in love with.\n\nDinner and conversation proceeded. Susan seemed half-absent; Clara was quiet. I asked a few polite questions and answered some rather more direct ones in return. As I had told the tennis club's high representatives, I had absolutely no interest in politics; though I did follow current events. So this would have been a few years after the Sharpeville massacre, to which I must have alluded; and doubtless my words contained some element of pious condemnation. Well, I did think it was wrong to massacre people.\n\n\"Do you even know where Sharpeville is?\" The Head of the Table had evidently identified me as a mewling pinko.\n\n\"It's in South Africa,\" I replied. But as I did so, I suddenly wondered if this was a trick question. \"Or Rhodesia,\" I added; then thought again. \"No, South Africa.\"\n\n\"Very good. And what is your considered judgement on the political scene there?\"\n\nI said something about being against shooting people.\n\n\"And what might you advise the police forces of the world to do when confronted with a rioting mob of Communists?\"\n\nI hated the way adults asked you questions in a way which implied that they already knew the answer you were going to give, and that it would always be a wrong or stupid one. So I said something, perhaps facetious, to the effect that just because they were dead, this didn't prove they were Communists.\n\n\"Have you ever _been_ to South Africa?\" Macleod roared at me.\n\nSusan stirred at this point. \"We've none of us been to South Africa.\"\n\n\"True, but I think I know more about the situation there than the two of you put together.\" It seemed that Clara was excused from complicity in ignorance. \"If we were to pile his knowledge on top of your knowledge\u2014Pelion on Ossa, as it were\u2014it still wouldn't amount to a hill of beans.\"\n\nThe long silence was broken by Susan asking if anyone wanted any more to eat.\n\n\"Have you got any beans, Mrs. Macleod?\"\n\nYes, I could be a cheeky bastard, I now realise. Well, I was only nineteen. I hadn't a clue who or what Pelion and Ossa might be; I was more struck by the notion of piling my knowledge on top of Susan's. That was what lovers did, after all: they added to one another's understanding of the world. Also, to \"know\" someone, in the Bible anyway, meant to have sex with them. So I had already piled my knowledge on top of hers. Even if it didn't amount to more than a hill of beans. However tall a hill of beans might be.\n\n* * *\n\n\u2014\n\nShe told me that her father had been a Christian Science practitioner with many adoring female acolytes. She told me that her brother who had disappeared in the war had gone to a prostitute a few weeks before his last flight because he wanted to \"find out what it was all about.\" She told me that she couldn't swim because she had heavy bones. Things like this tumbled out of her in no particular order, and in response to no particular enquiry on my part, other than the tacit one of wanting to know everything about her. So she laid them out, as if expecting me to make sense, to make order, of her life, and of her heart.\n\n\"Things aren't what they look like, Paul. That's about the only lesson I can teach you.\"\n\nI wonder if she is talking about the sham of respectability, the sham of marriage, the sham of suburbia, or...but she carries on.\n\n\"Winston Churchill, did I tell you about seeing him?\"\n\n\"You mean, you went to Number Ten?\"\n\n\"Silly, no. I saw him in a backstreet in Aylesbury. What was I doing there? Not that it matters. He was sitting in the rear seat of an open-topped car. And his face was all covered in makeup. Red lips, bright pink face. He looked bizarre.\"\n\n\"You're sure it was Churchill? I didn't realise he was...\"\n\n\"...one of them? No, it's nothing like that, Paul. You see, they were waiting to drive him through the city centre\u2014it was after we won the war, or maybe it was the General Election, and he was made up for the cameras. Path\u00e9 News and all that.\"\n\n\"How weird.\"\n\n\"It was. So quite a few people saw this strange painted mannequin in the flesh, but far more saw him on the newsreels, when he looked like they expected him to.\"\n\nI think about this for a while. It strikes me as a comic incident, rather than a general principle of life. Anyway, my interests are elsewhere.\n\n\"But you're what you look like, aren't you? _You're_ exactly what you look like?\"\n\nShe kisses me. \"I hope so, my fine and feathered friend. I hope so for both our sakes.\"\n\n* * *\n\n\u2014\n\nI used to prowl the Macleod house, part anthropologist, part sociologist, wholly lover. At first I naturally compared it to my parents' house, which I therefore found wanting. Here there was style, and ease, and none of that absurd house-pride. My parents had better, more up-to-date kitchen equipment, but I gave them no credit for this; nor for the fact that their car was cleaner, their gutters recently sluiced, their soffits regularly painted, their bathroom taps buffed to a shine, their lavatory seats hygienically plastic rather than warmingly wooden. In our house, the television was taken seriously, and stood centrally; at the Macleods', they called it the goggle-box and hid it behind a firescreen. They owned no such thing as a fitted carpet or a fitted kitchen, let alone a three-piece suite or a bathroom set in matching colours. Their garage was so full of tools, discarded sports equipment, gardening implements, old motor mowers (one working) and unwanted furniture that there was no room in it for the Austin. At first all this seemed stylish and idiosyncratic. I was initially seduced, then slowly disenchanted. My soul no more belonged in a place like this than in my parents' house.\n\nAnd, more importantly, I believed that Susan didn't belong here either. It was something I felt instinctively, and only understood much later, over time. Nowadays, when more than half the country's children are born out of wedlock (wed-lock: I've never noticed the two parts of that term before), it's not so much marriage that ties couples together as the shared occupation of property. A house or a flat can be as beguiling a trap as a wedding certificate; sometimes more so. Property announces a way of life, with a subtle insistence on that way of life continuing. Property also demands constant attention and maintenance: it's like a physical manifestation of the marriage that exists within it.\n\nBut I could see, all too well, that Susan had not been the recipient of constant attention and maintenance. And I'm not talking about sex. Or not just.\n\n* * *\n\n\u2014\n\nHere's something I need to explain. In all the time Susan and I were lovers, I never thought that we were \"deceiving\" Gordon Macleod, Mr. E.P. I never thought of him as being represented by that peculiar old word \"cuckold.\" Obviously, I didn't want him to _know._ But I thought that what took place between Susan and me had nothing to do with him; he was irrelevant to it all. Nor did I have any contempt for him, any young-buck superiority because I was sexually active with his wife and he wasn't. You may think this is just a normal lover's normal self-delusion; but I don't agree. Even when things...changed, and I felt differently about him, this aspect didn't change. He had nothing to do with us, do you see?\n\n* * *\n\n\u2014\n\nSusan, perhaps thinking that I was undervaluing her friend Joan, had told me, in a gently admonitory tone, that everyone had their own love story. I was happy to accept this, happy for everyone else to be or have been blessed, even if confident that they couldn't possibly be as blessed as I was. But at the same time, I didn't want Susan to tell me whether she had had her love story with Gerald, or with Gordon, or was having it with me. Whether there were one, two or three stories to her life.\n\n* * *\n\n\u2014\n\nI am round at the Macleods' one evening. It is getting late. Macleod has already gone to bed, and is snoring away his flagons and his gallons. She and I are on the sofa; we have been listening to some music we recently heard at the Festival Hall. I look at her in a way which makes my attentions and desires plain.\n\n\"No, Casey. Kiss me hardly.\"\n\nSo I kiss her hardly, just a brush on the lips, nothing to raise her colour. We hold hands instead.\n\n\"I wish I didn't have to go home,\" I say self-pityingly. \"I hate home.\"\n\n\"Then why do you call it home?\"\n\nI haven't thought of this.\n\n\"Anyway, I wish I could stay here.\"\n\n\"You could always pitch a tent in the garden. I'm sure there's some spare tarpaulin in the garage.\"\n\n\"You know what I mean.\"\n\n\"I know exactly what you mean.\"\n\n\"I could always climb out of a window afterwards.\"\n\n\"And be arrested for burglary by a passing copper? That _would_ land us in the _Advertiser &_ _Gazette._ \" She paused. \"I suppose...\"\n\n\"Yes?\" I hope she is coming up with a master plan.\n\n\"This thing actually turns into a sofa bed. We could put you up here. If E.P. finds you before he goes to work, we'll say\u2014\"\n\nBut just at that moment the phone rings. Susan picks it up, listens, looks at me, says, \"Yes,\" pulls a solemn face and places her hand over the mouthpiece.\n\n\"It's for you.\"\n\nIt is, of course, my mother, demanding to know where I am, which I find an otiose question, given that my current address would be right next to the number in the phone book which she must have just consulted. Also, she wants to know when I shall be back.\n\n\"I'm a bit tired,\" I say. \"So I'm staying here on the sofa bed.\"\n\nMy mother has recently had to put up with a certain amount of insolent lying from me; but insolent truth-telling is pushing things too far.\n\n\"You'll be doing no such thing. I'll be outside in six minutes.\" And then she puts the phone down.\n\n\"She'll be outside in six minutes.\"\n\n\"Lawks-a-mercy,\" says Susan. \"Do you think I should offer her a glass of sherry?\"\n\nWe giggle away the next five-and-three-quarter minutes until we hear a car out in the road.\n\n\"Off you go now, you dirty stop-out,\" she whispers.\n\nMy mother was behind the wheel in her pink dressing gown over her pink nightdress. I didn't check to see if she was driving in bedroom slippers. She was halfway down a cigarette, and before putting the car into gear, flicked the glowing stub out onto the Macleods' driveway.\n\nI got in, and as we drove my mood switched from pert indifference to furious humiliation. An English silence\u2014one in which all the unspoken words are perfectly understood by both parties\u2014prevailed. I got into my bed and wept. The matter was never referred to again.\n\n* * *\n\n\u2014\n\nSusan's innocence was the more surprising because she never tried to hide it. I'm not sure she ever tried to hide anything\u2014it was against her nature. Later\u2014well, what came later, came later.\n\nBut, for instance\u2014and I can't remember how the subject came up\u2014she once said that she wouldn't necessarily have gone to bed with me if it hadn't been for the known fact that it was bad for a man not to have \"sexual release.\" This is all that remains of the words spoken between us, that simple phrase.\n\nPerhaps it was more ignorance than innocence. Or call it folk wisdom; or patriarchal propaganda. And it set me wondering. Did this mean that she didn't desire me as much as I desired her\u2014constantly, naggingly, utterly? That sex for her meant something different? That she was only going to bed with me for therapeutic reasons, because I might explode like a hot-water cylinder or car radiator if I didn't have this necessary \"release\"? And was there no equivalent of this in female sexual psychology?\n\nLater, I thought: But if that's how she imagines male sexuality to operate, what about her husband? Did she never wonder about his need for \"release\"? Unless, of course, she had seen him explode and so realised the consequences. Or perhaps E.P. went to prostitutes in London\u2014or to the front half of some pantomime elephant? Who knew? Perhaps this explained his oddity.\n\nHis oddity, her innocence. And of course I didn't tell her in reply that young men\u2014all young men in my experience\u2014when deprived of female company, didn't have a problem with \"sexual release,\" for the simple reason that they are, were and always would be wanking away like jackhammers.\n\n* * *\n\n\u2014\n\nHer innocence, my overconfidence; her naivety, my crassness. I was going back to university. I thought it would be funny to buy her a large fat carrot as a farewell present. It would be a joke; she would laugh; she always laughed when I laughed. I went to a greengrocer's and decided a parsnip would be funnier. We went for a drive and stopped somewhere. I gave her the parsnip. She didn't laugh at all, just threw it over her shoulder, and I heard it thump against the back seat of the shooting brake. I have remembered this moment all my life, and though I haven't blushed for many years, I would blush, if I could, about that.\n\n* * *\n\n\u2014\n\nWe managed a brief holiday. I can't remember what lies we told in order to have a few days of truth together. It must have been out of season. We went somewhere near the south coast. I can't remember a hotel, so perhaps we rented a flat. What we said, thought, discovered about one another\u2014all gone. I do remember a broad, empty beach somewhere. Perhaps it was Camber Sands. We photographed one another with my camera. I did handstands on the beach for her. She is wearing a coat and the wind is whipping her hair back, and her hands, holding her coat closed at the neck, are enclosed in large, black false-fur gloves. Behind her is a distant row of beach huts, and a one-storey, shuttered caf\u00e9. No one else is in sight. You could, if you wanted, look at these photographs and deduce the season; also, no doubt, the weather. At this distance, both are meaningless to me.\n\nI was wearing a tie, that's another detail. I had taken off my jacket to do handstands for her. The tie falls straight down the middle of my upturned face, obscuring my nose, dividing me into two halves. Backhand and forehand.\n\n* * *\n\n\u2014\n\nI didn't get much post in those days. Cards from friends, letters from the university reminding me about stuff, bank statements.\n\n\"Local postmark,\" said my mother, handing me an envelope. The address was typed, and there was a heartening \"Esq.\" after my name.\n\n\"Thanks, Mum.\"\n\n\"Aren't you going to open it?\"\n\n\"I shall, Mum.\"\n\nShe huffed off.\n\nThe letter came from the secretary of the tennis club. He was informing me that my temporary membership had been terminated with immediate effect. Further, that \"due to the circumstances,\" none of the membership fee I had paid was refundable. The \"circumstances\" were not specified.\n\nSusan and I had arranged to meet at the club for a pick-up foursome. So after lunch I took my racket and sports bag and set off as if for the courts.\n\n\"Was the letter interesting?\" my mother asked impedingly.\n\nI waved my racket in its press.\n\n\"Tennis club. Asking if I want to join on a permanent basis.\"\n\n\"That's gratifying, Paul. They must be pleased with your game.\"\n\n\"Sounds like it, doesn't it?\"\n\nI drive to Susan's house.\n\n\"I got one too,\" she says.\n\nHer letter is much the same as mine, except more strongly worded. Instead of her membership being terminated \"due to the circumstances\" it is terminated \"due to the evident circumstances of which you will be fully aware.\" The adjusted wording is for Jezebels, for scarlet women.\n\n\"How long have you been a member?\"\n\n\"Thirty years, I suppose. Give or take.\"\n\n\"I'm sorry. It's my fault.\"\n\nShe shakes her head in disagreement.\n\n\"Shall we protest?\"\n\nNo.\n\n\"I could burn the place down.\"\n\nNo.\n\n\"Do you think we were spotted somewhere?\"\n\n\"Stop asking questions, Paul. I'm thinking.\"\n\nI sit down beside her on the chintz sofa. What I don't like to say, or not immediately, is that part of me finds the news exhilarating. I\u2014we\u2014are a cause of scandal! Love persecuted yet again by small-minded petty officialdom! Our expulsion might not have been an Obstacle on which Passion Thrives, but the moral and social condemnation implicit in the phrase \"due to the circumstances\" act, to my mind, as an authentication of our love. And who does not want their love authenticated?\n\n\"It's not as if we were caught snogging in the long grass behind the roller.\"\n\n\"Oh, do be quiet, Paul.\"\n\nSo I sit there quietly, my thoughts noisy. I try to remember cases of boys expelled from my school. One for pouring sugar into the petrol tank of a master's car. One for getting his girlfriend pregnant. One for being drunk after a cricket match, urinating in a train compartment and then pulling the communication cord. At the time, all this seemed pretty impressive stuff. But my own rule-breaking struck me as thrilling, triumphant, and, most of all, grown-up.\n\n* * *\n\n\u2014\n\n\"Well, now look what the cat's brought in\" was Joan's greeting as she answered the door a few afternoons later. I hadn't warned her of my visit. \"Just give me a moment to shut up the yappers.\"\n\nThe door closed again, and I stood by an elderly boot scraper thinking about the distance that had grown between Susan and me since the tennis club's dismissal of us. I had let my exhilaration show too clearly, which displeased her. She said that she was still \"thinking.\" I couldn't see what there was to think about. She told me there were complications I didn't understand. She told me not to come round until the weekend. I felt downcast, like one awaiting judgement even though no crime that I could see had been committed.\n\n\"Sit yourself down,\" Joan instructed as we reached the fag-fogged, gin-scented den that was nominally her sitting room. \"You'll have something to put a few hairs on your chest?\"\n\n\"Yes, please.\" I didn't drink gin\u2014I hated the smell of it, and it made me feel even worse than wine or beer did. But I didn't want to come across as a prig.\n\n\"Good man.\" She handed me a tumblerful. There was a smear of lipstick at its rim.\n\n\"That's an awful lot,\" I said.\n\n\"We don't pour fucking pub measures in this establishment,\" she replied.\n\nI sipped at the thick, oily, lukewarm substance which didn't smell at all like the juniper berries on the bottle.\n\nJoan lit a cigarette and blew the smoke in my direction as if to nudge me.\n\n\"So?\"\n\n\"So. Well. Perhaps you've heard about the tennis club.\"\n\n\"The Village tom-tom speaks of nothing else. The drumheads have been taking a real pasting.\"\n\n\"Yes, I thought you\u2014\"\n\n\"Two things, young man. One, I don't want to know any details. Two, how can I help?\"\n\n\"Thank you.\" I was genuinely touched, but also puzzled. How could she help if she didn't know the details? And what counted as a detail? I thought about this.\n\n\"Come on. What are you here to ask me?\"\n\nThat was the problem. I didn't know what I'd come to ask. I somehow thought that what I wanted from Joan would become clear to me when I saw her. Or she would know anyway. But it hadn't, and she apparently didn't. I tried to explain this, haltingly. Joan nodded, and let me sip my gin and ponder.\n\nThen she said, \"Try lobbing me the first question that comes into your head.\"\n\nI did so without reflecting. \"Do you think Susan would leave Mr. Macleod?\"\n\n\"My, my,\" she said quietly. \"You are aiming high, young man. That's a pair of balls you've got on you. Talk about one step at a time.\"\n\nI grinned inanely at what I took to be a compliment.\n\n\"So have you asked her?\"\n\n\"Gosh, no.\"\n\n\"And, to start at the beginning, what would you do for money?\"\n\n\"I don't care about money,\" I replied.\n\n\"That's because you've never had to.\"\n\nThis was true; but not in the sense that I was rich. My state education had been free, I received a council grant to attend university, I lived at home in the holidays. But it was also true that I didn't care about money\u2014indeed, in my world view, to care about money meant deliberately to turn your eyes away from the most important things in life.\n\n\"If you're going to be a grown-up,\" said Joan, \"you've got to start thinking about grown-up things. And number one is money.\"\n\nI remembered what I'd been told about Joan's early life\u2014her being a \"kept woman\" or whatever, living no doubt from cash handouts and rent-payings and gifts of clothes and holidays. Is that what she meant by being grown-up?\n\n\"I suppose Susan's got some.\"\n\n\"Have you asked her?\"\n\n\"Gosh, no.\"\n\n\"Well, maybe you should.\"\n\n\"I've got a running-away fund,\" I said defensively, without explaining where it had come from.\n\n\"And how much rattles around in your little piggy bank?\"\n\nIt was odd how I never took offence at anything Joan said. I just assumed that beneath her brusqueness she was kindhearted and on my side. But then lovers always assume that people are on their side.\n\n\"Five hundred pounds,\" I said proudly.\n\n\"Yes, well, you could certainly run away on that. It'll keep you for a few weeks in Le Touquet\u2013Paris\u2013Plage as long as you don't go near the casino. And then you'll come running back to England.\"\n\n\"I suppose so.\" Even if I'd never thought of Le Touquet\u2013Paris\u2013Plage as a destination. Was that where fleeing lovers went?\n\n\"You're going back to college next month, aren't you?\"\n\n\"Yes.\"\n\n\"And you're going to keep her in a kitchen cupboard there? A wardrobe?\"\n\n\"No.\"\n\nI felt stupid and hopeless. No wonder Susan was \"thinking\" about it all. Was I merely entertaining some romantic notion of flight, a ladder with no steps attached?\n\n\"It's a bit more complicated than working out how to save me on the gin and the petrol.\"\n\nI had been brought solidly down to earth, as Joan no doubt intended.\n\n\"Can I ask you something different?\"\n\n\"Off you go.\"\n\n\"Why do you cheat at crosswords?\"\n\nJoan laughed loudly. \"You cheeky bugger. I suppose Susan told you. Well, it's a fair question, and one I can answer.\" She took another pull of her gin. \"You see\u2014I hope you never get there yourself\u2014but some of us get to the point in life where we realise that nothing matters. Nothing fucking matters. And one of the few side benefits of that is you know you're not going to go to hell for filling in the wrong answers in the crossword. Because you've been to hell and back already and you know all too well what it's like.\"\n\n\"But the answers are in the back of the book.\"\n\n\"Ah, but you see, to me that _would_ be cheating.\"\n\nI felt absurdly fond of her. \"Is there anything I can do for you, Joan?\" I found myself asking.\n\n\"Just don't cause Susan any harm.\"\n\n\"I'd rather cut my own throat,\" I replied.\n\n\"Yes, I think you might even mean that.\" She smiled at me. \"Now, off with you, and mind your driving. I can see you're not yet hardened to the gin.\"\n\nI was about to put the car into gear when there was a tap at the window. I hadn't heard her behind me. I wound the window down.\n\n\"Don't ever care what they say about you,\" Joan said, looking at me intently. \"For instance, some kindly neighbours assume I'm just a ghastly old lezzer living alone with my dogs. So, a failed lezzer at that. Water off a duck's back. That's my advice if you want it.\"\n\n\"Thank you for the gin,\" I replied, and released the handbrake.\n\n* * *\n\n\u2014\n\nJoan was demanding that I be grown-up. I was prepared to try if it helped Susan; but I still regarded adulthood with some horror. First, I wasn't sure that it was attainable. Secondly, even if attainable, I wasn't sure it was desirable. Thirdly, even if desirable, then only by comparison with childhood and adolescence. What did I dislike and distrust about adulthood? Well to put it briefly: the sense of entitlement, the sense of superiority, the assumption of knowing better if not best, the vast banality of adult opinions, the way women took out compacts and powdered their noses, the way men sat in armchairs with their legs apart and their privates heavily outlined against their trousers, the way they talked about gardens and gardening, the spectacles they wore and the spectacles they made of themselves, the drinking and the smoking, the terrible phlegmy racket when they coughed, the artificial smells they applied to conceal their animal smells, the way men went bald and women shaped their hair with aerosols of glue, the noxious thought that they might still be having sex, their docile obedience to social norms, their snarky disapproval of anything satirical or questioning, their assumption that their children's success would be measured by how well they imitated their parents, the suffocating noise they made when agreeing with one another, their comments about the food they cooked and the food they ate, their love of stuff I found disgusting (especially olives, pickled onions, chutneys, piccalilli, horseradish sauce, spring onions, sandwich spread, stinky cheese and Marmite), their emotional complacency, their sense of racial superiority, the way they counted their pennies, the way they hunted for food trapped between their teeth, the way they weren't interested enough in me, and the way they were too interested in me when I didn't want them to be. This was just a short list, from which Susan was naturally and entirely exempt.\n\nOh, and another thing. The way, doubtless through some atavistic terror of admitting to real feelings, they ironised the emotional life, turning the relationship between the sexes into a silly running joke. The way men implied that women ran everything really; the way women implied that men didn't really understand what was going on. The way men pretended they were the strong, and women had to be petted and indulged and taken care of; the way women pretended that, regardless of the accumulated sexual folklore, they were the ones who had the common sense and practicality. The way each sex blubbingly admitted that, for all the other's faults, they still needed one another. Can't live with 'em, can't live without 'em. And they lived with 'em in marriage, which, as one wit put it, was an institution in the sense of mental institution. Who first said that, a man or a woman?\n\nUnsurprisingly, I looked forward to none of this. Or rather, hoped it would never apply to me; indeed, believed I could make it not apply to me.\n\nSo, actually, when I said, \"I'm nineteen!\" and my parents triumphantly replied, \"Yes, you're _only_ nineteen!\" the triumph was also mine. Thank God I'm \"only\" nineteen, I thought.\n\n* * *\n\n\u2014\n\nFirst love fixes a life forever: this much I have discovered over the years. It may not outrank subsequent loves, but they will always be affected by its existence. It may serve as model, or as counterexample. It may overshadow subsequent loves; on the other hand, it can make them easier, better. Though sometimes, first love cauterises the heart, and all any searcher will find thereafter is scar tissue.\n\n\"We were chosen by lot.\" I don't believe in destiny, as I may have said. But I do believe now that when two lovers meet, there is already so much prehistory that only certain outcomes are possible. Whereas the lovers themselves imagine that the world is being reset, and that the possibilities are both new and infinite.\n\nAnd first love always happens in the overwhelming first person. How can it not? Also, in the overwhelming present tense. It takes us time to realise that there are other persons, and other tenses.\n\nSo (and this would have happened earlier, but I am only remembering it now): I am visiting her one afternoon. I know that at three o'clock, by which time her thieving daily will have left and there will be three-and-a-half hours before Mr. E.P. returns, she will be waiting in bed for me. I drive to the Village, park, and set off along Duckers Lane. I am not in the least self-conscious. The more disapproval, real or imagined, from \"the neighbours,\" the better. I do not approach the Macleod house via the back gate and the garden. I turn down their driveway, walking openly and crunching the gravel, rather than discreetly, adulterously, on the grass edge alongside. The house is red-brick, symmetrical, with a central porch, above which is Susan's narrow little bedroom. On each side of the porch, as a decorative feature, every fourth course of brick has been laid to jut out half a brick's width. A couple of tempting inches, I now see, of handhold and foothold.\n\nThe lover as cat burglar? Why not? The back door has been left open for me. But as I walk towards the porch, a lover's confidence infuses me, and I decide that if I go at it with enough initial speed, I might be able to scoot up the ten feet or so of wall, which will get me to the flat, leaded roof on top of the porch. I take a run at it, filled with bravado, ardour and decent hand-eye coordination. Easy-peasy\u2014and here I am, suddenly crouched on the leading. I have made enough noise to bring Susan to the window, first in alarm, then in surprised glee. Someone else would have rebuked me for my folly, told me I might have broken my skull, expressed all their fear and protectiveness: in short, made me feel a foolish and guilty boy. All Susan does is yank up the window and pull me in.\n\n\"I could always get out the same way if Trouble Comes,\" I say pantingly.\n\n\"That would be a lark.\"\n\n\"I'll just go down and lock the back door.\"\n\n\"Ever the thoughtful one,\" says Susan, getting back into her single bed.\n\nAnd that's true, too. I _am_ the thoughtful one. That's part of my prehistory, I suppose. But it's also about what I could have said to Joan: that I am prepared to be grown-up if it will help Susan.\n\n* * *\n\n\u2014\n\nI am a boy; she is a married woman of middle years. I have the cynicism, and the purported understanding of life though I am the idealist as well as the cynic, convinced that I have both the will and the power to mend things.\n\nAnd she? She is neither cynical nor idealistic; she lives without the mental clutter of theorising, and takes each circumstance and situation as it comes. She laughs at things, and sometimes that laughter is a way of not thinking, of avoiding obvious, painful truths. But at the same time I feel that she is closer to life than I am.\n\nWe don't talk about our love; we merely know that it is there, unarguably; that it is what it is, and that everything will flow, inevitably and justly, from this fact. Do we constantly repeat \"I love you\" in confirmation? At this distance, I can't be sure. Though I do remember that when, after locking the back door, I climb into bed with her, she whispers,\n\n\"Never forget, the most vulnerable spot is down the middle.\"\n\n* * *\n\n\u2014\n\nThen there's that word Joan dropped into our conversation like a concrete fence post into a fishpool: practicality. Over my life I've seen friends fail to leave their marriages, fail to continue affairs, fail even to start them sometimes, all for the same expressed reason. \"It just isn't practical,\" they say wearily. The distances are too great, the train schedules unfavourable, the work hours mismatched; then there's the mortgage; and the children, and the dog; also, the joint ownership of things. \"I just couldn't face sorting out the record collection,\" a non-leaving wife once told me. In the first thrill of love, the couple had amalgamated their records, throwing away duplicates. How was it feasible to unpick all that? And so she stayed; and after a while the temptation to leave passed, and the record collection breathed a sigh of relief.\n\nWhereas it seemed to me, back then, in the absolutism of my condition, that love had nothing to do with practicality; indeed, was its polar opposite. And the fact that it showed contempt for such banal considerations was part of its glory. Love was by its very nature disruptive, cataclysmic; and if it was not, then it was not love.\n\nYou might ask how deep my understanding of love was at the age of nineteen. A court of law might find it based on a few books and films, conversations with friends, heady dreams, aching fantasies about certain girls on bicycles and a quarter-relationship with the first woman I went to bed with. But my nineteen-year-old self would correct the court: \"understanding\" love is for later, \"understanding\" love verges on practicality, \"understanding\" love is for when the heart has cooled. The lover, in rapture, doesn't want to \"understand\" love, but to experience it, to feel the intensity, the coming-into-focus of things, the acceleration of life, the entirely justifiable egotism, the lustful cockiness, the joyful rant, the calm seriousness, the hot yearning, the certainty, the simplicity, the complexity, the truth, the truth, the truth of love.\n\nTruth and love, that was my credo. I love her, and I see the truth. It must be that simple.\n\n* * *\n\n\u2014\n\nWere we \"any good\" at sex? I've no idea. We didn't think about it. Partly because any sex then seemed by definition good sex. But also because we rarely talked about it, either before, during or after; we did it, believed in it as an expression of our mutual love, even if, physically and mentally, it might have given us different satisfactions. After she had mentioned her supposed frigidity, and I had\u2014from my vast sexual experience\u2014airily dismissed it, the matter was not discussed again. Sometimes, she would murmur, \"Well played, partner,\" afterwards. Sometimes, more seriously, more anxiously, \"Please don't give up on me just yet, Casey Paul.\" I didn't know what to say to that either.\n\nFrom time to time\u2014and not in bed, I must point out\u2014she would say, \"Of course you'll have girlfriends. And that's only right and proper.\" But it didn't seem right, or proper, to me, or even relevant.\n\nOn another occasion, she mentioned a number. I can't remember the context, let alone the number; but I slowly realised that she must be talking about how many times we had made love.\n\n\"You've been counting?\"\n\nShe nodded. Again, I was baffled. Was I meant to have been counting too? And if so, what was I meant to count\u2014the number of times we'd been to bed together, or the number of my orgasms? I wasn't in the least interested, and I wondered why the notion had crossed her mind. There seemed something fatalistic about it\u2014as if she would have something tangible, calculable, to hold on to if I suddenly wasn't there. But I wasn't suddenly going to be not there.\n\n* * *\n\n\u2014\n\nWhen, once again, she made reference to my future girlfriends, I said, very clearly and firmly, that she would always be in my life: whatever happened, there would always be a place for her.\n\n\"But where would you put me, Casey Paul?\"\n\n\"At the very worst, in a well-appointed attic.\"\n\nI meant it metaphorically, of course.\n\n\"Like a piece of old lumber?\"\n\nI was hating this conversation. \"No,\" I repeated, \"you'll always be there.\"\n\n\"In your attic?\"\n\n\"No, in my heart.\"\n\nI meant it, I truly meant it\u2014both the attic and the heart. All my life.\n\nI didn't realise that there was panic inside her. How could I have guessed? I thought it was just inside me. Now, I realise, rather late in the day, that it is in everyone. It's a condition of our mortality. We have codes of manners to allay and minimise it, jokes and routines, and so many forms of diversion and distraction. But there is panic and pandemonium waiting to break out inside all of us, of this I am convinced. I've seen it roar out among the dying, as a last protest against the human condition and its chronic sadness. But it is there in the most balanced and rational of us. You just need the right circumstances, and it will surely appear. And then you are at its mercy. The panic takes some to God, others to despair, some to charitable works, others to drink, some to emotional oblivion, others to a life where they hope that nothing serious will ever trouble them again.\n\n* * *\n\n\u2014\n\nThough we were cast out of the tennis club like Adam and Eve, the expected scandal failed to break. There was no denunciation from the pulpit of St. Michael's, no exposure in the _Advertiser &_ _Gazette._ Mr. Macleod seemed oblivious; Misses G. and N.S. were abroad at the time. My parents never mentioned the matter. So by a very English combination of ignorance, true or feigned, and embarrassment, no one\u2014apart from Joan, and that at my invitation\u2014acknowledged the story's existence. The Village tom-tom might have been beating, but not everyone chose to hear its message. I was both relieved by this and disappointed. Where was the merit, and the joy, in scandalous behaviour if the Village declined to be scandalised except behind closed doors?\n\nBut I was relieved, because it meant Susan brought her \"thinking\" period to an end. In other words, we took a deep breath and started going to bed together again, taking as many risks as before. I stroked her ears and tapped her rabbity teeth. Once, to demonstrate that all was still the same, I sprang up the jutting brickwork onto the porch and through her bedroom window.\n\nAnd, as it turned out, she had a running-away fund too. With more than five hundred pounds in it.\n\n* * *\n\n\u2014\n\nI keep saying that I was nineteen. But sometimes, in what I've told you so far, I was twenty or twenty-one. These events happened over a period of two years and more, usually during my student vacations. In term time, Susan would often come and visit me in Sussex, or I would go up and stay at the Macleods'. Six minutes' drive from my parents, yet I never told them I was there. I would get off the train at a previous station, and Susan would pick me up in the Austin. I slept on the sofa bed, and Mr. Macleod seemed to tolerate my presence. I never went into the Village, though I did occasionally think of burning down the tennis club, just for old times' sake.\n\nSusan got to know my circle of friends at Sussex\u2014Eric, Ian, Barney and Sam\u2014and from time to time one or more of them would also stay at the Macleods'. Perhaps they were another kind of cover story\u2014at this distance, I can't remember. They all considered my relationship with Susan an excellent thing. We were on one another's side when it came to relationships\u2014any relationship, really. They also liked the freewheelingness of Susan's household. She used to cook big meals, and they liked that too. We always seemed to be hungry back then; also, pathetically incapable of making a meal for ourselves.\n\nOne Friday\u2014well, it was probably a Friday\u2014Mr. Macleod was chomping on his spring onions, I was playing with my knife and fork and Susan was bringing in the food, when he asked, with more than the usual edge of sarcasm,\n\n\"And how many fancy boys are you providing yourself with this weekend, if I may make so bold as to ask?\"\n\n\"Let me see,\" Susan replied, holding the stew dish in front of her as she appeared to ponder, \"I think it's just Ian and Eric this weekend. And Paul of course. Unless the others turn up as well.\"\n\nI thought this amazingly cool of her. And then we ate dinner normally.\n\nBut in the car the next day, I asked her, \"Does he always call me that? Us that?\"\n\n\"Yes. You're my fancy boy.\"\n\n\"I'm not _that_ fancy. I'm a bit penny plain at times, I think.\"\n\nBut the word hurt. Hurt me for her, you understand. For myself, I didn't care. No, really: perhaps I was even pleased. To be noticed\u2014even to be insulted\u2014was better than to be ignored. And a young man needed a reputation, after all.\n\n* * *\n\n\u2014\n\nI tried to assemble what I knew about Macleod. I could no longer think of him as Mr. E.P. than I could as Old Adam or the Head of the Table. He was called Gordon, though Susan only used that name when speaking of the distant past. He looked a few years older than her, so must have been in his mid-fifties. He worked as a civil servant, though I had no idea in which department, nor was I interested. He hadn't had sex with his wife for many years, though in the old days, when he was Gordon, he had done so, and two daughters were the proof of this. He had declared his wife frigid. He might, or might not, fancy the front half of a pantomime elephant. He believed that rioting mobs of Communists should be shot by the police or army. His wife hadn't seen his eyes, or not properly, for many years. He played golf, and hit the ball as if he hated it. He liked Gilbert and Sullivan. He was good at disguising himself as a shabby but efficient gardener; though according to his own father he could be a difficult row to hoe. He didn't like or take holidays. He liked to drink. He didn't like going to concerts. He was good at crosswords and had pedantic handwriting. He didn't have any friends in the Village, except, presumably, at the golf club, a place I had never entered, and had no intention of doing so. He didn't go to church. He read _The Times_ and _The Telegraph._ He had been friendly and polite with me, but also sarcastic and rude; mainly, I would say, indifferent. He seemed to be cross with life. And was part of what may or may not have been a played-out generation.\n\nBut there was another thing about him, which I felt rather than observed. It seemed to me\u2014I'm sure Macleod wasn't conscious of it, hadn't given it a thought\u2014but it felt to me as if he\u2014he in particular\u2014was somehow standing in the way of me growing up. He wasn't at all like my parents or their friends, but he represented even more than they did the adulthood I regarded with some horror.\n\n* * *\n\n\u2014\n\nA few stray thoughts and memories:\n\n\u2014Shortly after the Sharpeville Incident, Susan reported that Macleod had called me \"a very acceptable young man.\" Desperate for praise, like anyone else of my age, I took it at face value. Perhaps more than that: because he had first shouted at me, then later come to sober judgement, I considered the comment all the more valuable.\n\n\u2014I realise that I had absolutely no notion how the Macleods behaved with one another when I was not there. I was probably too absolutist to give it a thought.\n\n\u2014I also realise that, in comparing the two households, I might have made it sound as if at home we ate peas off a knife while scratching our bottoms. No, we were well brought-up. Our standard of table behaviour was on the whole better than that on display at the Macleods'.\n\n\u2014Also, not all my parents' friends were as passively disapproving of my generation as I may have portrayed them. Some were actively so. One holiday weekend, we all went over to Sutton for dinner with the Spencers. The wife had known my mother since training college days; the husband was a small, aggressive mining engineer, of Belgian origin, who specialised in locating and appropriating the mineral wealth of Africa on behalf of some international company. It must have been a sunny day (though not necessarily) because, peeking from my top pocket, was a recently acquired pair of mirrored sunglasses. I had bought them from Barney, who specialised in the bulk purchase and import of exotic items for resale to those wishing to quietly demonstrate their essential hipsterdom. He had sourced the glasses from somewhere behind the Iron Curtain\u2014Hungary, I think. Anyway, we had scarcely got out of the car when Mine Tiny Host approached me and, ignoring my outstretched hand, ripped the sunglasses from my pocket with the words, \"These are a piece of shit.\" Unlike, say, his own cable-knit sweater, corduroy trousers, signet ring and deaf aid.\n\n\u2014She makes a big cake for the Fancy Boys. Big in the sense of wide and long. When the mixture is poured into the tin, it is three-quarters of an inch deep. By the time it comes out of the oven, it has risen slightly to a height of about an inch. There is mixed fruit inside, all of which has sunk to the bottom.\n\nEven I, back then, can recognise that it is not, by average baking standards, a success. But she has a way of making it so.\n\n\"What sort of cake is that, Mrs. Macleod?\" asks one of the FBs.\n\n\"It's an upside-down cake,\" she replies, flipping it over on its wire rack. \"Look how the fruit has all risen to the top.\"\n\nThen she cuts us big slices, which we scoff.\n\nShe can probably turn base metal into gold, I think.\n\n\u2014I said how my credo was love and truth; I loved her, and I saw the truth. But I must also admit that this coincided with the period when I lied to my parents more often than before or since. And, to a lesser degree, to almost everybody else I knew. Though not to Joan.\n\n\u2014While I do not analyse my love\u2014the whence, why, whither of it\u2014I do sometimes try, when alone, to think about it lucidly. This is difficult; I have no previous experience, and am quite unprepared for the full engagement of heart and soul and body that being with Susan involves\u2014the intensity of the present, the thrill of the unknown future, the discarding of all the mingy preoccupations of the past.\n\nI lie in bed at home, trying to put feelings into words. On the one hand\u2014and this is the part to do with the past\u2014Jove feels like the vast and sudden easing of a lifelong frown. But simultaneously\u2014this is the part to do with the present and the future\u2014it feels as if the lungs of my soul have been inflated with pure oxygen. I only think like this when alone, of course. When I am with Susan, I'm not thinking what it's like to love her; I'm just being with her. And maybe that \"being with her\" is impossible to put into any other words.\n\n* * *\n\n\u2014\n\nSusan never minded my solo visits to Joan; she wasn't possessive about one of the few friends her marriage seemed to permit. I came to enjoy the mugfuls of cut-price gin; after a while, Joan allowed the yappers in, and I got used to the distraction of Yorkshire terriers grazing on my shoelaces.\n\n\"We're leaving,\" I told her one July afternoon.\n\n\"We? You and I? Where are we going, young Master Paul? Do you have your belongings tied up in a red-spotted handkerchief on a stick?\"\n\nI should have known she wouldn't let me get away with earnestness.\n\n\"Susan and I. We're off.\"\n\n\"Off where? For how long? A cruise, is it? Send me a postcard.\"\n\n\"There'll be lots of postcards,\" I promised.\n\nIt was odd, my relationship with Joan was a kind of flirting. Whereas my relationship with Susan barely had any flirting in it at all. We must have gone through all that preliminary stuff without noticing\u2014smack into love\u2014and so had no need for it. We had our jokes and our teases and our private phrases, of course. But I suppose it all felt\u2014was\u2014too serious for flirting.\n\n\"No,\" I said. \"You know what I mean.\"\n\n\"Yes, I know what you mean. I've been wondering about it for some time. Given the circumstances. Half wanting it to happen, half not. But you've got guts, the pair of you, I'll say that.\"\n\nI didn't think of it in terms of guts. I thought of it in terms of inevitability. Also, doing what we both deeply wanted.\n\n\"And how is Gordon taking it all?\"\n\n\"He calls me her fancy boy.\"\n\n\"I'm surprised he doesn't call you her fucking fancy boy.\"\n\nYes, well, probably that too.\n\n\"I shan't say I hope you know what you're doing because it's perfectly obvious neither of you has any idea what you're doing. Now, don't pull that face at me, Master Paul. No one ever does, not in your position. And I'm not going to say, Look after her, and all that stuff. I'm just going to keep my thumbs bloody hard crossed for you.\"\n\nShe came out to the car with me. Before getting in, I moved towards her. She raised a palm.\n\n\"No, none of that fucking huggy-huggy stuff. There's too much of it around, everyone suddenly behaving like foreigners. Be off with you before I shed a tear.\"\n\nLater, I went over what she had and hadn't said to me, and wondered if she'd been spotting parallels I'd missed. No one ever knows what they're doing, not in your position. Off up to London, eh? Fancy boy, kept woman. And who's got the money? Yes, Joan was ahead of me.\n\nExcept that it wasn't going to be like that. I could hardly imagine Susan back on the Macleod doorstep in three years' time, tongue-tied, emotionally blasted, begging silently to be taken in, her life essentially over. I was confident _that_ wasn't going to happen.\n\n* * *\n\n\u2014\n\nThere was no exact Moment of Leaving, neither a surreptitious midnight skedaddle, nor some formal departure with luggage and waving handkerchiefs. (Who would have waved?) It was a long-drawn-out detaching, so that the moment of rupture was never clearly marked. Which didn't stop me trying to mark it, with a brief letter to my parents:\n\n> Dear Mum and Dad,\n> \n> I am moving up to London. I shall be living with Mrs. Macleod. I shall send you an address as and when.\n> \n> Yours, Paul\n\nThat seemed to cover it. I thought the \"as and when\" sounded properly grown-up. Well, so I was. Twenty-one. And ready to fully indulge, fully express, fully live my life. \"I'm alive! I'm living!\"\n\n* * *\n\n\u2014\n\nWe were together\u2014under the same roof, that is\u2014for ten or more years. Afterwards, I continued to see her regularly. In later years, less often. When she died, a few years ago, I acknowledged that the most vital part of my life had finally come to a close. I shall always think of her well, I promised myself.\n\n* * *\n\n\u2014\n\nAnd this is how I would remember it all, if I could. But I can't.\n\n# TWO\nSusan's running-away fund contained enough to buy a small house in Henry Road, SE15. The price was low\u2014gentrification, and juice bars, lay far in the future. The place had been in multi-occupation: a euphemism for locks on every door, asbestos panelling, a squalid kitchenette on a half-landing, personal gas meters and personal stains in every room. Through that late summer and early autumn we stripped it all back, joyfully, the dandruff of distemper in our hair. We threw out most of the old furniture, and slept on a double mattress on the floor. We had a toaster, a kettle, and dined off takeaways from the Cypriot taverna at the end of the road.\n\nWe needed a plumber, electrician and gas man, but did the rest ourselves. I was good at rough carpentry. I made myself a desk from two broken-up chests of drawers topped with cut-down wardrobe doors; then sanded, filled and painted it until it stood, immovably heavy, at one end of my study. I cut and laid coconut matting, and tacked carpet up the stairs. Together we ripped off the parchmenty wallpaper, back to the leprous plaster, then roller-painted it in cheery, non-bourgeois colours: turquoise, daffodil, cerise. I painted my study a sombre dark green, after Barney told me that the labour wards of hospitals were that colour, to calm expectant mothers. I hoped it might have the same effect on my own laborious hours.\n\nI had taken to heart Joan's sceptical \"And, to start at the beginning, what would you do for money?\" Given that I didn't care about the stuff, I could have lived off Susan; but, given that our relationship was going to last a lifetime, I acknowledged that at some point I would have to support her rather than the other way round. Not that I knew how much money she had. I never asked about the finances of the Macleod household, nor whether Susan had a traditional Auntie Maud who would conveniently leave her all she had.\n\nSo I decided to become a solicitor. I had no exaggerated ambitions for myself; my exaggerated ambitions were all for love. But I thought of the law because I had an orderly mind, and a capacity to apply myself; and every society needs lawyers, doesn't it? I remember a woman friend once telling me her theory of marriage: that it was something you should \"dip into and out of as required.\" This may sound dismayingly practical, even cynical, but it wasn't. She loved her husband, and \"dipping out\" of marriage didn't mean adultery. Rather, it was a recognition of how marriage worked for her: as a reliable ground bass to life, as something you jogged along with until such time as you needed to \"dip into\" it, for succour, expressions of love and the rest. I could understand this approach: there is no point demanding more than your temperament requires or provides. But as far as I understood my life at this time, I required the opposite equation. Work would be something I jogged along with; love would be my life.\n\n* * *\n\n\u2014\n\nI began my studies. Each morning, Susan cooked me breakfast; each evening, supper\u2014unless I fetched us a kebab or sheftalia. Sometimes, when I arrived back, she would sing at me, \"Little man, you've had a busy day.\" She also took my washing to the launderette and brought it home for ironing. We still went to concerts and art exhibitions. The mattress on the floor became a double bed, in which we slept together night after night, and where some of my cinematic assumptions about love and sex became subject to adjustment. For instance, the notion of lovers falling blissfully asleep in one another's arms resolved itself into the actuality of one lover falling asleep half on top of the other, and the latter, after a certain amount of cramp and interrupted circulation, gently shifting out from beneath while trying not to wake her. I also discovered that it wasn't only men who snored.\n\nMy parents didn't reply to my change-of-address letter; nor did I invite them to visit the house in Henry Road. One day I returned from college to find Susan in agitated mood. Martha Macleod, Miss Grumpy herself, had descended without warning for a tour of inspection. She was bound to have noted that whereas in the Village her mother had slept in a single bed, now she had a double one. Fortunately, in my dark green study, the sofa bed had been pulled out, and left unmade by me that morning. But then, as Susan remarked, two doubles hardly make a single. My own attitude to Martha Macleod's likely disapproval of our sleeping arrangements was\u2014would have been\u2014one of pride and defiance. Susan's was more complicated, though I admit I didn't spend much time on its nuances. After all, were we living together or were we not?\n\nWhen she reached the two undecorated attic rooms at the top of the house, Martha had apparently said,\n\n\"You should have lodgers.\"\n\nWhen Susan had demurred, her daughter's reply, delivered either as argument or instruction, was:\n\n\"It would be good for you.\"\n\nQuite what she meant by this we debated that evening. True, there was an economic argument for lodgers: they would make the house more or less self-sufficient. But what was the moral argument? Perhaps that lodgers would give Susan something more to do than wait for the return of her shameless lover. Martha might also have intended that lodgers would somehow dilute my noxious presence, and camouflage the reality of number 23 Henry Road\u2014of Fancy Boy Number One living brazenly with an adulteress still more than twice his age.\n\nIf Martha's visit had troubled Susan, it also, on further thought, troubled me. I had failed to consider her future relations with her daughters. My focus had all been on Macleod, on getting Susan away from him, and now, from a safe distance, divorcing him. For our joint sake, but mainly for hers. She had to scrub this mistake from her life and give herself the legal as well as the moral freedom to be happy. And being happy consisted of living with me, alone and unfettered.\n\n* * *\n\n\u2014\n\nIt was a quiet neighbourhood, and we received few visitors. I remember one Saturday morning being stirred from the law of tort by the front doorbell. I heard Susan invite someone\u2014two someones, a man and a woman\u2014into the kitchen. About twenty minutes later, I heard her say, as she shut the front door,\n\n\"I'm sure you feel a whole lot better now.\"\n\n\"Who was that?\" I asked as she passed my door. She looked in to see me.\n\n\"Missionaries,\" she replied. \"God damn and blast them, missionaries. I let them get it all off their chests and then sent them on their way. Better to waste their puff on me than someone they might convert.\"\n\n\"Not _actual_ missionaries?\"\n\n\"It's a general term. Actual missionaries are the worst, of course.\"\n\n\"You mean, these were Jehovah's Witnesses, or Plymouth Brethren, or Baptists, or something?\"\n\n\"Or something. They asked me if I was worried about the state of the world. It's an obvious catch question. Then they bored on about the Bible as if I'd never heard of it. I nearly told them I knew all about it and that I was a flaming Jezebel.\"\n\nAnd with that she left me to my studies. But instead I mused on these sudden bursts of fierce opinion, which so endeared her to me. I had been educated by books, she by life, I thought again.\n\n* * *\n\n\u2014\n\nOne evening, the phone went. I picked it up and gave the number.\n\n\"Who is that?\" said a voice I immediately recognised as Macleod's.\n\n\"Well, who's _that_?\" I replied, with fake casualness.\n\n\"Gor-don Mac-leod,\" he said with extended heaviness. \"And whom might I be having the honour of speaking to?\"\n\n\"Paul Roberts.\"\n\nAs he banged the receiver down, I found myself wishing I'd said Mickey Mouse, or Yuri Gagarin, or the chairman of the BBC.\n\nI didn't tell Susan about this. I didn't see the point.\n\n* * *\n\n\u2014\n\nBut a few weeks later we received a visit from a man called Maurice. Susan had met him before, once or twice. He might have had a connection to Macleod's office. There must have been some arrangement made. It seemed he had picked a time when I would be there too. I'm not sure about it all, at this distance\u2014maybe it was just luck on his part.\n\nI failed to ask any of the obvious questions at the time. And if I had, perhaps Susan would have had the answers, perhaps not.\n\nHe was a man of fiftyish, I suppose. In my memory I have given him\u2014or he has acquired over the years\u2014a trench coat, and perhaps a broad-brimmed hat, underneath which he wore a suit and tie. He was perfectly cordial in behaviour. He shook my hand. He accepted a cup of coffee, he used the lavatory, he asked for an ashtray, and he talked about the bland, general topics adults went in for. Susan was in her hostess mode, which involved tamping down some of the things I most loved her for: her irreverence, her free-spirited laughter at the world.\n\nAll I can remember is that at one point the conversation turned to the closure of R _eynolds News._ This was a paper\u2014R _eynolds News and Sunday Citizen,_ to give its full title\u2014which had fallen on hard times, relaunched itself as a tabloid Sunday, and then finally closed\u2014presumably not long before this conversation.\n\n\"I don't think it matters much,\" I said. I didn't really have any view on the matter. I might have seen a copy or two of R _eynolds News,_ but was mainly just reacting to Maurice's tone of deep concern.\n\n\"You don't?\" he asked civilly.\n\n\"No, not really.\"\n\n\"What about the diversity of the press? Isn't that something to be valued?\"\n\n\"All the papers seem much the same to me, so I don't see that one fewer of them matters much.\"\n\n\"Are you by any chance part of the Revolutionary Left?\"\n\nI laughed at him. Not at his words, but at him. What the fuck did he take me for? Or perhaps, Who the fuck? He might as well have been a member of the tennis club committee, back at the Village.\n\n\"No, I despise politics,\" I said.\n\n\"You despise politics? Do you think that's an entirely healthy attitude? Do you find cynicism a comfortable position? What would you replace them with? You'd close down newspapers, you'd close down our way of doing politics? You'd close down democracy? That sounds like a Revolutionary Left position to me.\"\n\nNow the fellow was really annoying me. I wasn't out of my area of competence so much as my area of interest.\n\n\"I'm sorry,\" I said. \"It's really not that at all. But you see,\" I added, looking at him with melancholy seriousness, \"it's just that I'm a member of a played-out generation. You may think we're a bit young for it, but even so, we're played out.\"\n\nHe left shortly afterwards.\n\n\"Oh, Casey Paul, you are one wicked person.\"\n\n\"Me?\"\n\n\"You. Didn't you hear him say he'd worked for R _eynolds_ _News_?\"\n\n\"No, I thought he was a spy.\"\n\n\"You mean, a Russki?\"\n\n\"No, I just mean he was sent along to check up on us and report back.\"\n\n\"Probably.\"\n\n\"Do you think we should worry about that?\"\n\n\"Not for a couple of days at least, I'd say.\"\n\n* * *\n\n\u2014\n\nYou decide that, since you are a student, and all your fellow students, apart from those who live at home, pay rent, then you should do so too. You ask a couple of friends how much they pay. You take the midpoint: four pounds a week. You can afford this out of your state grant.\n\nOne Monday evening, you hand Susan four pound notes.\n\n\"What's that?\" she asks.\n\n\"I've decided I should pay you rent,\" you reply, perhaps a little stiffly. \"That's about what others pay.\"\n\nShe throws the notes back at you. They don't hit your face, as they might do in a film. They just lie on the floor between you. Awkward silences follow, and you sleep on your sofa bed that night. You feel guilty about not having introduced the subject of rent with more subtlety; it was like when you gave her that parsnip. The four green pound notes lie on the floor all night. The next morning you pick them up and put them back in your wallet. The subject is never mentioned again.\n\n* * *\n\n\u2014\n\nAs a result of Martha's visit, two things happened. The attic rooms were let out to lodgers, and Susan went back to the Village for the first time since we ran away together. She said it would be necessary and practical to return from time to time. Half the house belonged to her, and she could hardly rely on Macleod to pay the bills or remember to get the boiler serviced. (I didn't see why not, but still.) Mrs. Dyer would continue to serve and thieve on a daily basis, and would alert Susan to anything that needed her attention. She promised that she would only go back when Macleod wasn't there. Grudgingly, I agreed.\n\n* * *\n\n\u2014\n\nI said a bit ago that \"This is how I would remember it all, if I could. But I can't.\" There's some stuff I left out, stuff I can't put off any longer. Where to start? In the \"book room,\" as they called it, downstairs at the Macleods'. It was late, and I was unwilling to go home. Susan might already have been in bed; I don't remember. Nor do I remember what book I was reading. Something I'd picked off the shelves at random, no doubt. I was still trying to get my head round the Macleod collection. There were leather-bound sets of the classics, old enough to have been handed down through maybe two generations; art monographs, poetry, a lot of history, some biography, novels, thrillers. I came from the sort of household where books, as if to confirm that they should be respected, were put in order: by subject, author, even size. Here, there was a different system\u2014or rather, as far as I could see, no system at all. Herodotus was next to _The Bab Ballads,_ a three-volume history of the Crusades next to Jane Austen, T. E. Lawrence sandwiched between Hemingway and a Charles Atlas manual of bodybuilding. Was it all an elaborate joke? Mere bohemian muddle? Or a way of saying: we control the books, they don't control us.\n\nI was still musing when the door banged back against the bookcase, then rebounded far enough to be kicked again. Macleod stood there in his dressing gown, which\u2014and this I do remember\u2014was plaid, with a maroon cord tied and dangling. Below were his elephant pyjamas and leather slippers.\n\n\"What are you doing here?\" he asked, in a tone of voice normally attached to the words \"Fuck off.\"\n\nMy default position of insolence kicked in.\n\n\"Reading,\" I replied, waving the book in his direction.\n\nHe stomped across and ripped it from my hand, briefly inspected it, then threw it like a Frisbee across the room.\n\nI couldn't help grinning. He thought he was chucking my book away, when it was one of his own. Hilarious!\n\nThat was when he hit me. Or rather, aimed a succession of blows\u2014three, I'm pretty sure\u2014one of which landed as a wrist slapping the side of my head. The other two flailed past.\n\nI got up and tried to hit him back. I think I aimed one blow, which skidded off his shoulder. Neither of us was doing any snappy defensive work; we were just equally incompetent attackers. Well, I'd never hit anyone before. He, I assume, had, or had at least tried to.\n\nWhile he was concentrating on what to say, or where to hit, next, I squirmed past him, ran to the back door, and escaped. I was relieved to get back to a house where I hadn't been assaulted since a few doubtless-merited spankings a decade and more previously.\n\n* * *\n\n\u2014\n\nNo, that wasn't quite true\u2014about never having hit someone. In my first year at school, the gym master had encouraged us all to enter the annual boxing competition, which was organised by weight and age. I had absolutely no desire to inflict or receive pain. But I noticed that, with only a few hours to go, there were no entrants listed under my category. So I gave my name in, expecting to win by walkover.\n\nUnfortunately for me\u2014for both of us\u2014another boy, Bates, had the same idea at almost the same time. So we found ourselves in the ring together, two skinny, scared things in plimsolls, vests and house shorts, with these big bobbly gloves suddenly at the end of our arms. For a couple of minutes we each did a reasonably good job of feinting attacks and then backpedalling at great speed, until the gym master pointed out that neither of us had yet landed a blow.\n\n\"Box!\" he had commanded.\n\nWhereupon I leaped at the unprepared Bates, whose gloves were down near his knees, and punched him on the nose. He squealed, looked at the sudden blood on his clean white vest and burst into tears.\n\nAnd so I became school boxing champion in the under-12, under-6-stone category. Naturally, I never fought again.\n\n* * *\n\n\u2014\n\nThe next time I went to the Macleod house, Susan's husband couldn't have been friendlier. Perhaps that was when he showed me how to do the crossword, making it some kind of exclusive male preserve. Or at any rate, a Susan-excluding one. So I put the book-room incident down as an aberration. And anyway, it might have been partly my fault. Perhaps I should have engaged him about which version of the Dewey system his library was organised under. No, I can see that might have been equally provoking.\n\nHow much time then went by? Let's call it six months. Again, it was lateish. At the Macleod house, unlike my own, there was a main staircase near the front door, and a narrower one near the kitchen, presumably for those mob-capped servants now replaced by machines. Often, when I visited Susan during term time, I would sleep in a small attic room which could be reached from either direction. Susan and I had been listening to the gramophone\u2014preparing for a concert\u2014and the music was still in my head when I reached the top of the back stairs. All of a sudden there came a kind of roar, and something which might have been a kick or a trip, accompanied by a thump on the shoulder, and I found myself falling back down the stairs. I managed somehow to grab the banister, wrenching my shoulder but just about keeping my balance.\n\n\"You fucking bastard!\" I said automatically.\n\n\"Whatski?\" came an answering bellow from above. \"Whatski, my fine and feathered friend?\"\n\nI looked up at the squat bully glaring down at me from the semi-darkness. I thought that Macleod must be absolutely, certifiably mad. We stared at one another for a few seconds, then the dressing-gowned figure stomped away, and I heard a distant door close.\n\nIt wasn't Macleod's fists I was afraid of\u2014not principally. It was his anger. We didn't do anger in my family. We did ironic comment, snappy rejoinder, satirical elaboration; we did exact words forbidding a certain action, and more severe ones condemning what had already taken place. But for anything beyond this, we did the thing enjoined upon the English middle classes for generations. We internalised our rage, our anger, our contempt. We spoke words under our breath. We might have written some of those words down in private diaries if we kept them. But we also thought that we were the only ones reacting like this, and it was a little shameful, and so we internalised it all even further.\n\nWhen I got to my room that night, I placed a chair at an angle, wedged under the doorknob, as I'd seen done in films. I lay in bed thinking: Is this what the adult world is really like? Underneath it all? And how close beneath the surface does it\u2014will it\u2014lie?\n\nI had no answers.\n\n* * *\n\n\u2014\n\nI didn't tell Susan about either of these incidents. I internalised my anger and shame\u2014well, I would, wouldn't I?\n\nAnd you'll have to imagine long spells of happiness, of delight, of laughter. I've described them already. That's the thing about memory, it's...well, let me put it like this. Have you ever seen an electric log-splitter in action? They're very impressive. You cue the log to a certain length, lay it on the bed of the machine, press the button with your foot, and the log is pushed onto a blade shaped like an ax-head. Whereupon the log splits pure and straight down the grain. That's the point I'm trying to make. Life is a cross section, memory is a split down the grain, and memory follows it all the way to the end.\n\nSo I can't not continue. Even if this is the hardest part to remember. No, not to remember\u2014to describe. It was the moment when I lost some of my innocence. That may sound like a good thing. Isn't growing up a necessary process of losing one's innocence? Maybe, maybe not. But the trouble with life is, you rarely know when that loss is going to happen, do you? And how it will be, afterwards.\n\nMy parents were away on holiday, and my granny\u2014my mother's mother\u2014had been drafted in to look after me. I was, of course, twenty\u2014 _only_ twenty\u2014so obviously couldn't be left in the house by myself. What might I get up to, whom might I import, what might I organise\u2014a bacchanalia of middle-aged women, perhaps\u2014what might the neighbours think, and who might subsequently refuse to come for sherry? Grandma, widowed some five years, didn't have anything better to do. I had naturally\u2014innocently\u2014loved her as a child. Now I was growing up and she seemed boring. But that was a loss of innocence I could handle.\n\nAt this time, I used to sleep quite late during the holidays. It could have been mere idleness, or a belated reaction to the stress of the university term; or, perhaps, some instinctive unwillingness to reenter this world I still called home. I would sleep on until eleven without compunction. And my parents\u2014to their credit\u2014never came in and sat on my bed and complained that I was treating the place like a hotel; while Grandma was happy to cook me breakfast at lunchtime if that's what I wanted.\n\nSo it was probably closer to eleven than ten when I stumbled downstairs.\n\n\"There's a very rude woman asking for you,\" said Grandma. \"She's rung three times. She told me to wake you up. Actually, the last time to 'B' wake you up. I said I'm not interfering with his beauty sleep.\"\n\n\"Good for you, Grandma. Thanks.\"\n\nA very rude woman. But I didn't know any. Someone from the tennis club, persecuting me further? The bank about my overdraft? Maybe Grandma was beginning to lose her marbles. At which point, the phone went again.\n\n\"Joan,\" said the very rude voice of Joan. \"It's Susan. Get over there. She wants you, not me. _You, now._ \" And she put the phone down.\n\n\"Aren't you having your breakfast?\" asked Grandma as I rushed out.\n\nAt the Macleods', the front door was open, and I walked around until I found her fully dressed, handbag beside her, on the sofa in the sitting room. She didn't look up when I greeted her. I could only see the top of her head, or rather, the curve of her headscarf. I sat down beside her, but she immediately turned her face away.\n\n\"I need you to drive me up to town.\"\n\n\"Of course, darling.\"\n\n\"And I need you not to ask me any questions. And absolutely not to look at me.\"\n\n\"Whatever you say. But you'll need to tell me roughly where we're going.\"\n\n\"Head for Selfridges.\"\n\n\"Are we in a hurry?\" I allowed myself that question.\n\n\"Just drive safely, Paul, just drive safely.\"\n\nWe got to near Selfridges and she directed me down Wigmore Street, then left up one of those streets where private doctors practise.\n\n\"Park here.\"\n\n\"Do you want me to come with you?\"\n\n\"I'd rather not. Get yourself some lunch. This won't be quick. Do you need some money?\"\n\nI had indeed come without my wallet. She gave me a ten-shilling note.\n\nAs I turned back into Wigmore Street, I saw ahead of me John Bell & Croyden, where she had gone for her Dutch cap. A terrible realization came upon me. That the system had failed, that she'd found herself pregnant, and was even now dealing with the consequences. The Abortion Law was still going through Parliament, but everyone knew there were doctors\u2014and not just at the backstreet end\u2014who would perform \"procedures\" more or less on demand. I imagined the conversation: Susan explaining how she had got herself pregnant by her young lover, hadn't had sex with her husband for two decades, and how a child would destroy her marriage and endanger her own mental health. That would be enough for any doctor, who would agree to what went down euphemistically in medical records as a D&C: dilatation and curettage. Just a little scraping away at the lining of the womb\u2014which would also scrape away the embryo attached to its wall.\n\nI was working all this out as I sat in an Italian caf\u00e9 having my lunch. I didn't know what I thought\u2014or rather, I thought several incompatible things. The notion of being a father while still a student struck me as terrifying and crazy. But it also struck me as, well, kind of heroic. Subversive yet honourable, annoying yet life-affirming: noble. I didn't think it would get me into the _Guinness Book of_ R _ecords_ \u2014no doubt there were twelve-year-olds hard at work getting their grannies' best friends pregnant, but it would certainly make me exceptional. And irritate the hell out of the Village.\n\nExcept that now it wasn't going to happen. Because Susan was getting rid of our child at this very moment, just around the corner. I felt sudden rage. A woman's right to choose\u2014yes, I believed in that, theoretically and actually. Though I also believed in a man's right to be consulted.\n\nI went back to the car and waited. After an hour or so she turned the corner and came towards me, head lowered, scarf pulled around her cheeks. She averted her face from me as she got into the car.\n\n\"Right,\" she said. \"That's that for the moment.\" There was something slurry about her articulation. The anaesthetic, presumably\u2014if they used any. \"Home, James, and don't spare the horses.\"\n\nNormally I was charmed by her turns of phrase. Not this time.\n\n\"First tell me where you've been.\"\n\n\"The dentist.\"\n\n\"The dentist?\" So much for my imaginings. Unless this was just another euphemism among women of Susan's class.\n\n\"I'll tell you when I can, Casey Paul. I can't tell you now. Don't ask.\"\n\nOf course not. I drove her home, as carefully as I could.\n\nOver the next days, she told me bit by bit what had happened. She had been sitting up late, listening to the gramophone. Macleod had gone to bed an hour previously. She kept playing over and over again the slow movement of Prokofiev's third piano concerto, which we'd heard a few days before at the Festival Hall. Then she put the record back in its sleeve and went upstairs. She was just reaching for the handle of her bedroom door when her hair was seized from behind, and with the words, \"How's your fucking musical education coming along?,\" her husband smashed her face into the closed door. Then he had gone back to bed.\n\nThe dentist's examination showed that her two front teeth were broken beyond repair. The two teeth on either side of them would probably have to go as well. There was a crack in her upper jaw which would, over time, heal itself. The dentist would make her a plate. He asked if she wanted to talk about how it had happened, but didn't press her when she said she would rather not.\n\nAs the bruising came up in all its furious colours, and she powdered over it as well as she could; as I drove her up to town and back for appointment after appointment; as I wasn't able to get her to look at me for days, or kiss me for weeks; as I realised I would never again be able to tap her \"rabbit teeth,\" long discarded in some Wimpole Street waste bin; as I understood that I now had greater responsibilities than before; as I found myself wondering, and not idly, how I might kill Gordon Macleod; as first my grandma and then my returning parents drove me mad with their careful, safe, banal views of life; as Susan's bravery and lack of self-pity nearly broke my heart; as I absented myself from her house a good hour before Macleod's daily return; as I accepted her word\u2014or was it his word?\u2014that nothing like this would ever happen again; as anger and pity and horror washed through me; as I realised that Susan would have to leave the bastard somehow, with me or without me, but obviously with me; as at the same time a kind of impotence overcame me; as all this was happening, I learned a little more about the Macleod marriage.\n\nOf course, that bruise on her upper arm had not just been the size of a thumbprint, it was the imprint of an actual thumb as he forced her to sit in a chair and listen to his denunciations. There had been grabbings and slappings, and more than a punch or two. He would put a glass of sherry down in front of her and order her to \"join in the fun.\" When she declined, he would grasp her by the hair, pull her head back and hold the glass to her lips. Either she drank, or he poured it down her chin, and throat and dress. It was all verbal and physical, never sexual; though whether there was anything sexual behind it...well, that is beyond my competence, or, indeed, interest. Yes, it was usually connected to his drinking, but not necessarily; yes, she was frightened of him, except that mostly she wasn't. She had learned to manage him over the years. Yes, every time he attacked her, it was of course her fault\u2014according to him; she drove him to it with her airy bloody insolence\u2014that had been one of his phrases. Also, her irresponsibility; also, her stupidity. At some point after he had smashed her face against the door, he had gone downstairs and bent Prokofiev's third piano concerto until the record broke.\n\n* * *\n\n\u2014\n\nIt was, I suppose, ignorance and snobbery on my part which had hitherto made me assume that domestic violence was confined to the lower classes, where things were done differently, where\u2014as I understood from my reading rather than from a close familiarity with backstreet life\u2014women would rather their husbands hit them than be unfaithful to them. If he beats you, it shows he loves you, and all that crap. The idea of violence being inflicted by husbands with a Cambridge degree seemed to me incomprehensible. Of course, it was not a matter I'd had reason to think about before. But if I had, I would probably have guessed that violence among working-class husbands was connected to inarticulacy: they fell back on their fists whereas middle-class husbands fell back on words. Both these myths took some years to dispel, despite the present evidence.\n\nSusan's dental plate caused her constant trouble; there were many drives up to town for adjustments. The dentist had also made the four new prosthetic teeth better aligned than the original ones, and shortened the central pair by a millimetre or two. A subtle change, but one always manifest to me. Those teeth I used to tap so lovingly were gone forever; and I had no desire to touch their replacements.\n\nOne thing I never swerved from was the certainty that Gordon Macleod's behaviour was a crime of absolute liability. And his responsibility was also absolute. A man hits a woman; a husband hits a wife; a drunkard hits a sober spouse. There was no defence, and no possible mitigation. The fact that it would never come to court, that middle-class England had a thousand ways of avoiding the truth, that respectability was no more shed in public than clothes, the fact that Susan would never accuse him to any authority, not even a dentist\u2014all this had no relevance to me, except sociologically. The man was as guilty as hell, and I would hate him until the end of his days. This much I knew.\n\nIt was about a year after this that I went to see Joan and announced our intention of moving up to London.\n\n* * *\n\n\u2014\n\nYou are an absolutist for love, and therefore an absolutist against marriage. You have given the matter much thought, and come up with many fanciful comparisons. Marriage is a dog kennel in which complacency lives and is never chained up. Marriage is a jewellery box which, by some mysterious opposite of alchemy, turns gold, silver and diamonds back into base metal, paste and quartz. Marriage is a disused boathouse containing an old, two-person canoe, no longer water-worthy, with holes in the bottom and one missing paddle. Marriage is...oh, you have dozens of such comparisons to hand.\n\nYou remember your parents, and your parents' friends. They were, on the whole, and without giving them too much credit, decent people: honest, hardworking, polite with one another, no more than averagely controlling of their children. Family life meant for them much what it had meant for their parents' generation, though with just enough extra social freedom to let them imagine themselves pioneers. But where was love in all of this, you asked. And you didn't even mean sex\u2014because you preferred not to think about that.\n\nAnd so, when you had come into the Macleod household, and inspected a different way of living, you thought first about how circumscribed your own home seemed to be, how lacking in life and emotion. Then, gradually, you realised that the marriage of Gordon and Susan Macleod was, in fact, in far worse shape than any marriage among your parents' circle, and you became all the more absolutist. That Susan should live with you in a state of love was obvious; that she should leave Macleod was equally obvious; that she should divorce him\u2014especially after what he had done to her\u2014seemed not just an acknowledgement of the truth of things, not just a romantic obligation, but a necessary first step towards her becoming an authentic person once more. No, not \"once more\": really, it would be for the first time. And how exciting must that be for her?\n\nYou persuade her to see a solicitor. No, she doesn't want you to come with her. Part of you\u2014the part that imagines a free, and freestanding, Susan in the near future\u2014approves.\n\n\"How did it go?\"\n\n\"He said that I was in a bit of a muddle.\"\n\n\"He said _that_?\"\n\n\"No. Not exactly. But I explained things to him. Most things. Not you, obviously. And, well, I suppose he thought I'd just bolted. Done a bunk. Maybe he thought it was all to do with the Dreaded.\"\n\n\"But...didn't you explain what had happened...what _he_ did to you?\"\n\n\"I didn't go into detail, no. I kept it general.\"\n\n\"But you can't get a divorce on _general_ grounds. You can only get a divorce on particular grounds.\"\n\n\"Now don't get shirty with me, Paul, I'm doing my best.\"\n\n\"Yes, but...\"\n\n\"He told me that, for a starting point, I should go away and write it all down. Because he could see I found it hard to tell him about it directly.\"\n\n\"That sounds very sensible.\" Suddenly, you approve of this solicitor.\n\n\"So that's what I shall try to do.\"\n\nWhen, a couple of weeks later, you ask how her statement is coming along, she shakes her head without reply.\n\n\"But you've got to do it,\" you say.\n\n\"You don't know how hard it is for me.\"\n\n\"Would you like me to help you?\"\n\n\"No, I have to do this by myself.\"\n\nYou approve. This will be the start, the making, of the new Susan. You try some gentle advice.\n\n\"I think what they need are specifics.\" You know a bit about divorce law by now. \"Exactly what happened, and roughly when.\"\n\nAnother two weeks later, you ask how she's doing.\n\n\"Don't give up on me just yet, Casey Paul\" is her reply. And whenever she says this to you\u2014and you never think it is calculated, because she is not a calculating person\u2014it tears at your heart. Of course you won't give up on her.\n\nAnd then, some weeks later, she gives you a few sheets of paper.\n\n\"Don't read it in front of me.\"\n\nYou take it away, and from the first sentence, your optimism disperses. She has turned her life, and her marriage, into a comic short story, which sounds to you like something by James Thurber. Perhaps it was. It is about a man in a three-piece suit, called Mr. Elephant Pants, who every evening goes to the pub\u2014or the bar at Grand Central Station\u2014and comes home in a state which alarms his wife and children. He knocks over the hatstand, kicks the flowerpots, shouts at the dog, so that there is a spreading of Great Alarm and Despondency, and he rackets away until he falls asleep on the sofa and snores so loudly that tiles fall off the roof.\n\nYou don't know what to say. You say nothing. You pretend you are still considering this document. You know you have to be very gentle and very patient with her. You explain again about them needing to know specifics, the where and the when and, most importantly, the what. She looks at you and nods.\n\nSlowly, over the next weeks and months, you begin to understand that it is not going to happen, not ever. She is strong enough to love you, strong enough to run off with you, but not strong enough to enter a court of law and give evidence against her husband about the decades of sexless tyranny, alcoholism and physical attack. She will not be able\u2014even via her solicitor\u2014to ask the dentist to describe her injuries. She cannot attest in public to what she is able to admit in private.\n\nYou realise that, even if she is the free spirit you imagined her to be, she is also a damaged free spirit. You understand that there is a question of shame at the bottom of it. Personal shame; and social shame. She may not mind being thrown out of the tennis club for being a Scarlet Woman, but she cannot admit to the true nature of her marriage. You remember old cases in which criminals\u2014even murderers\u2014would marry their female accomplices because a wife could not be compelled to give evidence against a husband. But nowadays, far away from the world of criminality, in the respectable Village and many, many similar, silent places across the land, there are wives who have been conditioned, by social and marital convention, not to give evidence against their husbands.\n\nAnd there is another factor, of which, strangely, you have not thought. One calm evening\u2014calm because you have officially given up on the project, and all false hope and annoyance have drained from you\u2014she says to you quietly,\n\n\"And anyway, if I did do it, he'd bring up the matter of you.\"\n\nYou are astounded. You feel you had nothing to do with the breakup of the Macleod marriage; you were just the outsider who pointed out what would have been obvious to anyone. Yes, you fell in love with her; yes, you ran away with her; but that was consequence, not cause.\n\nEven so, perhaps you are lucky that the old law of enticement is no longer on the statute book. You imagine being called as a witness and asked to explain yourself. Part of you thinks this would be wonderful, heroic; you play through the courtroom exchange, in which you are dazzling. Until the final question. Oh, and by the way, young enticer, young seducer, may I ask what you do by way of a job? Of course, you reply, I am studying to be a solicitor. You realise that you might just have to change profession.\n\n* * *\n\n\u2014\n\nYou know that sometimes, after checking on the house she owns half of, she goes to visit Joan. This is a good idea, even if on her return her hair smells of cigarette smoke. Once, you catch sherry on her breath.\n\n\"Did you have a drink with Joan?\"\n\n\"Did I? Let me think...Quite possibly.\"\n\n\"Well, you shouldn't. Drink and drive. It's crazy.\"\n\n\"Yes, sir,\" she agrees satirically.\n\nAnother time, she has smoke in her hair and Polos on her breath. You think, This is silly.\n\n\"Look, if you're going to have a drink with Joan, don't insult my intelligence by chewing a few Polos afterwards.\"\n\n\"The thing is, Paul, there are parts of the drive I don't like. They give me the jitters. Blind corners. I find that a little nip of sherry with Joan calms my nerves. And the Polos aren't for you, darling, they're in case I get stopped by a policeman.\"\n\n\"I'm sure policemen are just as suspicious of drivers smelling of Polos as when they smell of alcohol.\"\n\n\"Don't _you_ turn into a policeman, Paul. Or a lawyer, even if you are going to be one. I'm doing my best. That's all I can do.\"\n\n\"Of course.\"\n\nYou kiss her. You have no more taste for confrontation than she does. Of course you trust her, of course you love her, of course you are far too young to be a policeman or a lawyer.\n\nAnd so you both laugh your way through several uncomplicated months.\n\nBut one February afternoon, she is late back from the Village. You know she doesn't like driving in the dark. You imagine the car off the road, in a ditch, her bloodied head against the dashboard, Polos spilling from her handbag.\n\nYou ring Joan.\n\n\"I'm a bit worried about Susan.\"\n\n\"Why?\"\n\n\"Well, what time did she leave you?\"\n\n\"When?\"\n\n\"Today.\"\n\n\"I haven't seen Susan today.\" Joan's voice is steady. \"I wasn't expecting her either.\"\n\n\"Oh fuck,\" you say.\n\n\"Let me know when she's back safely.\"\n\n\"Sure,\" you say, your mind only half there.\n\n\"And Paul.\"\n\n\"Yes?\"\n\n\"If she comes back safely, that's the main thing.\"\n\n\"Yes.\"\n\nIt is the main thing. And she does come back safely. And her hair is clean, and there is nothing on her breath.\n\n\"Sorry I'm late, darling,\" she says, putting down her handbag.\n\n\"Yes, I was worrying.\"\n\n\"No need to worry.\"\n\n\"Well I do.\"\n\nYou leave it at that. After supper, you pick up the plates, and, making sure your back is to her, ask,\n\n\"How's old Joan?\"\n\n\"Joan? Same as ever. Joan doesn't change. That's what's nice about her.\"\n\nYou rinse off the plates and leave it at that. You are a lover, not a lawyer, you remind yourself. Except that you are going to become a lawyer, because you need to be solid and stable, the better to look after her.\n\n* * *\n\n\u2014\n\nThe log of memory splits down the grain. So you can't remember the quiet times, the outings, the jollity, the running jokes, even the legal studies, which fill the gap between that last exchange and the day when, worried by a succession of late returns from the Village, you say to her, quietly and unchallengingly,\n\n\"I know you don't always go and see Joan when you say you do.\"\n\nShe looks away.\n\n\"Have you been checking up on me, Casey Paul? It's a terrible unloving thing to do, check up on people.\"\n\n\"Yes, but I can't stop worrying, and I can't bear to think of you alone in the house with...him.\"\n\n\"Oh, I'm quite safe,\" she says. There is a silence for a while. \"Look, Paul, I don't tell you about it because I don't want the two parts of my life overlapping. I want to build a wall around us here.\"\n\n\"But?\"\n\n\"But there are practical matters to discuss with him.\"\n\n\"Like divorce?\"\n\nImmediately, you feel ashamed of your sarcasm.\n\n\"Don't badger me like that, Mr. Badger. I've got to do things in my own time. It's all more complicated than you think.\"\n\n\"OK.\"\n\n\"We\u2014he and I\u2014have two children together, don't forget that.\"\n\n\"I don't.\" Though of course, you do. Often.\n\n\"There's money to discuss. The car. The house. I think the place needs repainting this summer.\"\n\n\"You discuss painting the house?\"\n\n\"That's enough from you, Mr. Badger.\"\n\n\"OK,\" you say. \"But you love me and you don't love him.\"\n\n\"You know that's how it is, Casey Paul. I wouldn't be here if it wasn't.\"\n\n\"And I suppose _he_ would like you to return.\"\n\n\"What I _hate,_ \" she says, \"is when he gets down on his knees.\"\n\n\"He gets down on his knees?\" In his elephant pants, I think.\n\n\"Yes, it's awful, it's embarrassing, it's undignified.\"\n\n\"And, what, begs you to stay with him?\"\n\n\"Yes. You see why I don't tell you about it?\"\n\n* * *\n\n\u2014\n\nThe Fancy Boys used to turn up at Henry Road and sleep on the floor, dossing like dogs on piles of cushions. The more of them there were, the more busily relaxed Susan became. So this was all good. Sometimes they brought their girlfriends, whose reactions to Henry Road used to intrigue me. I became expert in sensing covert disapproval. I wasn't being defensive or paranoid, merely observant. Also, I was amused by the orthodoxy of their sexual outlook. You might have thought\u2014mightn't you?\u2014that a girl or young woman in her early twenties would be rather encouraged by the notion that something exciting might happen to her nearly three decades on: that her heart and body would still be excitable, and that her future didn't necessarily have to be a matter of rising social acceptance combined with slow emotional diminution. I was surprised that some of them didn't find my relationship with Susan a cause for cheer. Instead, they reacted much as their parents would have done: alarmed, threatened, moralistic. Perhaps they were looking forward to being mothers themselves, and imagining their precious sons being cradle-snatched. Anyone would have thought Susan was a witch who had entranced me, fit only for the ducking stool. Well, she had entranced me. And to feel the disapproval from women of my own age merely increased my pleasure at Susan's and my originality, and my own determination to continue offending the prim and the unimaginative. Well, we all have to have a purpose in life, don't we? Just as a young man needs a reputation.\n\nAround this time, one of the lodgers moved out, and Eric, having broken up with his (moralistic, marriage-demanding) girlfriend, took over the free room on the top floor. This brought a new dynamic to the house, perhaps even a better one. Eric thoroughly approved of our relationship, and would be able to keep an eye on Susan when I couldn't. He was allowed to pay rent, which made it seem the more illogical that Susan wouldn't take any from me. But I knew how she would react if I renewed my offer.\n\nA few months passed. One evening, after Susan had gone to bed, Eric said,\n\n\"Don't like to mention this...\"\n\n\"Yes?\"\n\nHe looked embarrassed, which was unlike Eric.\n\n\"...but the thing is, Susan's been nicking my whisky.\"\n\n\"Your whisky? She doesn't even drink whisky.\"\n\n\"Well, it's her, or you, or the poltergeist.\"\n\n\"You're sure?\"\n\n\"I put a mark on the bottle.\"\n\n\"How long's this been going on?\"\n\n\"A few weeks. Maybe months?\"\n\n\" _Months?_ Why didn't you tell me?\"\n\n\"Wanted to make sure. And she changed her tactics.\"\n\n\"How do you mean?\"\n\n\"Well, at some point she must have noticed that there was a mark on the bottle. She'd have her nip or glug or however much it was, and then fill the bottle back up to the mark with water.\"\n\n\"That's clever.\"\n\n\"No, it's standard. Banal, even. My dad used to do that when my mum was trying to get him to stop.\"\n\n\"Oh.\" I was disappointed. I wanted Susan always to be as entirely original as she still appeared to me.\n\n\"So I did the logical thing. I stopped drinking from the bottle myself. She'd come up, have a swig, fill up to the pencil mark with water. I let it run and run, until I could see the colour of the whisky fading. Eventually, to confirm it, I had a glass myself. One part whisky to about fifteen of water would be my guess.\"\n\n\"Fuck.\"\n\n\"Yes, fuck.\"\n\n\"I'll have a word with her,\" I promised.\n\nBut I didn't. Was it cowardice, the hope that some alternative explanation might present itself, or a weary refusal to admit my own suspicions?\n\n\"And in the meantime, I'll keep my booze on top of the wardrobe.\"\n\n\"Good plan.\"\n\nIt was a good plan, until the day when Eric said quietly,\n\n\"She's learned to climb up to the top of the wardrobe.\"\n\nHe made it sound like a kind of monkey trick rather than a normal piece of behaviour involving a chair. But that's how it felt to me too.\n\n* * *\n\n\u2014\n\nYou notice there are times when she seems, not squiffy, but out of focus. Not bleary of face, but bleary of mind. Then, by chance, you notice her swallowing a pill.\n\n\"Headache?\"\n\n\"No,\" she replies. She is in one of those moods\u2014lucid, unselfpitying, yet somehow beaten-down\u2014which bend your heart painfully. She comes and sits on the edge of the bed.\n\n\"I went to the doctor. I explained what had happened. I explained that I'd been feeling depressed. He gave me some cheering-up pills.\"\n\n\"I'm sorry you need them. I must be letting you down.\"\n\n\"It's not you, Paul. And it's not fair on you either. But I think if I can get through the...adjustment, then it'll get better.\"\n\n\"Did you tell him you were drinking a bit too much?\"\n\n\"He didn't ask about that.\"\n\n\"That doesn't mean you shouldn't have told him.\"\n\n\"We're not going to quarrel about this, are we?\"\n\n\"No. We're not going to quarrel. Ever.\"\n\n\"Then it'll all come out right. You'll see.\"\n\nThinking about this conversation later, you begin to understand\u2014for the first time, really\u2014that she has more to lose than you. Much more. You are leaving behind a past, much of which you are happy to let go. You believed, and still believe as deeply, that love is the only thing that counts; that it makes up for everything; that if you and she get it right, everything will fall into place. You realise that what she has left behind\u2014even her relationship with Gordon Macleod\u2014is more complicated than you had assumed. You thought chunks could be cleanly amputated from a life without pain or complication. You realise that, if she had seemed isolated in the Village when you first met her, you have made her more isolated by taking her away.\n\nAll this means that you must redouble your commitment to her. You must get through this tricky patch, and then things will become clearer, better. She believes that, and so you must believe it too.\n\n* * *\n\n\u2014\n\nYou take the back route as you approach the Village, to avoid passing your parents' house.\n\n\"Where's Susan?\" are Joan's first words as she opens the door.\n\n\"I've come by myself.\"\n\n\"Does she know?\"\n\nYou like the way Joan always gets straight to the point. You quite enjoy having cold water dashed in your face before sitting down with a streaky tumbler full of room-temperature gin.\n\n\"No.\"\n\n\"Then it must be serious. I'll shut the little yappers up.\"\n\nYou sink into a dog-scented armchair and a drink is put next to you. As you are gathering your thoughts, Joan gets in first.\n\n\"Point One. I'm not a go-between. Whatever you say stays in this room and it doesn't get leaked back. Point Two. I'm not a shrink, I'm not some kind of advice centre, I don't even much like listening to other people's woes. I tend to think they should get on with it, stop moaning, roll up their sleeves and all of that. Point Three. I'm just an old soak whose life hasn't worked out and who lives alone with her dogs. So I'm not an authority on anything. Not even crosswords, as you once pointed out.\"\n\n\"But you love Susan.\"\n\n\"Course I do. How is the dear girl?\"\n\n\"She's drinking too much.\"\n\n\"How much is 'too much'?\"\n\n\"In her case, anything at all.\"\n\n\"You may be right.\"\n\n\"And she's on antidepressants.\"\n\n\"Well, we've all been _there,_ \" says Joan. \"Doctors hand them out like Smarties. Especially to women of a certain age. Do they do any good?\"\n\n\"I can't tell. They just make her woozy. But a different kind of woozy from what the drink does.\"\n\n\"Yes, I remember that too.\"\n\n\"So?\"\n\n\"So what?\"\n\n\"So what should I do?\"\n\n\"Paul, dear, I've just told you I don't give advice. I took my own advice for so many years and look where it got me. So I don't do that anymore.\"\n\nYou nod. You aren't too surprised either.\n\n\"The only advice I'd give you...\"\n\n\"Yes?\"\n\n\"...is have a swig of what's at your elbow.\"\n\nYou obey.\n\n\"OK,\" you say. \"No advice. But...I don't know, is there something that I ought to know and don't? Something you can tell me about Susan, or about Susan and me, that would help?\"\n\n\"All I can say is that if everything goes belly-up and pear-shaped, you'll probably get over it and she probably won't.\"\n\nYou are shocked.\n\n\"That's not a very kind thing to say.\"\n\n\"I don't do kind, Paul. Truth isn't kind. You'll find that out soon enough as life kicks in.\"\n\n\"It feels as if it's kicked in pretty hard already.\"\n\n\"That may be all to the fucking good.\" Your face must look as if it's just taken a slap. \"Come on, Paul, you didn't come all the way down here so that I'd give you a hug and tell you there are fairies at the bottom of the garden.\"\n\n\"True. Just tell me your thoughts on this. Susan goes back to see Macleod every so often. Probably more than she says.\"\n\n\"Does that trouble you?\"\n\n\"Mainly in the sense that if he ever lays a finger on her again, I'm going to have to kill him.\"\n\nShe laughs. \"Oh, I do so miss the melodrama of being young.\"\n\n\"Don't patronise me, Joan.\"\n\n\"I'm not patronising you, Paul. Of course you'd do no such thing. But I admire you for the thought.\"\n\nYou wonder if she is being satirical. But Joan doesn't do satire.\n\n\"Why don't you think I would?\"\n\n\"Because the last murder in the Village was probably committed by someone wearing woad.\"\n\nYou laugh, and take another sip of gin. \"I'm worried,\" you say. \"I'm worried that I shan't be able to save her.\"\n\nShe doesn't reply, and this annoys you.\n\n\"So what do you think about _that_?\" you demand.\n\n\"I told you I'm not a fucking oracle. You might as well read your horoscope in the _Advertiser &_ _Gazette._ I said when you ran away together, you've got guts, the pair of you. You've got guts, and you've got love. If that isn't good enough for life, then life isn't good enough for you.\"\n\n\"Now you are sounding like an oracle!\"\n\n\"Then I'd better go and wash my mouth out with soap.\"\n\n* * *\n\n\u2014\n\nOne day, you return to find her with cuts and bruises to her face, and her arms held defensively against her.\n\n\"I fell over that step in the garden,\" she says, as if it were a known hazard you had previously discussed. \"I'm getting very trippy, I'm afraid.\"\n\nShe is indeed getting \"trippy.\" Nowadays, as a reflex, you take her arm as you walk with her and keep watch for uneven pavements. But she also has a giveaway flush to her face. You call the doctor\u2014not the private one she went to for her cheering-up pills.\n\nDr. Kenny is a fussy, inquisitive middle-aged man, but the right sort of GP\u2014one who believes that house calls provide useful background when it comes to diagnosis. You take him upstairs to Susan's bedroom; her bruises are coming into full colour.\n\nDownstairs again, he asks for a few words.\n\n\"Of course.\"\n\n\"It's rather puzzling,\" he begins. \"It's unusual for a woman of her age to take a fall.\"\n\n\"She's been getting very trippy lately.\"\n\n\"Yes, that's the word she used to me. And, if I may ask, you are...?\"\n\n\"I'm her lodger...no, more than that, kind of godson, I suppose.\"\n\n\"Hmm. And it's just the two of you here?\"\n\n\"There are two more lodgers in the attic rooms.\" You decide not to promote Eric to the status of second godson.\n\n\"Does she have family?\"\n\n\"Yes, but she's kind of...estranged from them at the moment.\"\n\n\"So she has no support? Except for you, that is?\"\n\n\"I suppose not.\"\n\n\"As I say, it's rather puzzling. Do you think there was drink involved?\"\n\n\"Oh no,\" you say swiftly, \"she doesn't drink. She hates the stuff. That's one of the reasons she left her husband. He's a drinker. Flagons and gallons,\" you add, without being able to stop yourself.\n\nYou realise two things. First, that you lie automatically to protect Susan\u2014even if the truth might have helped her more. You also begin to see how your relationship, or rather, your cohabitation, might appear to an outsider.\n\n\"So, if I may ask, what does she do all day?\"\n\n\"She...does some volunteer work for the Samaritans.\" This isn't true either. Susan has mentioned the idea; though you are against it. You think she shouldn't try to start helping others when she is the one needing help.\n\n\"That's not much, is it?\"\n\n\"Well, I suppose she...keeps house.\"\n\nHe looks around. The place is clearly in a mess. You realise that he is finding your answers inadequate. And why shouldn't he?\n\n\"If it happens again, we'll be obliged to investigate,\" he says. Then picks up his bag and leaves.\n\nInvestigate? you think. Investigate? He can tell you've been lying. But investigate what? Perhaps he guesses you are her lover, and suspects you might have been beating her up. Christ to that, you think: in your desire to protect her from being thought a drinker, you seem to be opening yourself up to a charge of assault. Perhaps he was giving you a final warning.\n\nNot that the police would necessarily be interested. You remember an incident from a year or two before. You are in the car with Susan and have scarcely gone a quarter of a mile when you notice a couple rowing on the pavement. As you see the man bearing down on the woman you have flashbacks to the Macleod household. He is not exactly hitting her, but looks about to do so. Maybe they are drunk, you can't tell. You wind down the window and the woman yells, \"Call the police!\" Now he is holding her. \"Call the police!\" You speed home, dial 999, and are picked up by a patrol car which takes you to the scene of the reported possible crime. The couple have moved on, but you soon track them down a couple of streets away. They are ten yards apart, bellowing obscenities at one another.\n\n\"Oh, we know them,\" says the young constable. \"It's just a domestic.\"\n\n\"Aren't you going to arrest him?\"\n\nThe two of you are probably about the same age, but he knows he has seen more of life than you have.\n\n\"Well, sir, it's not our policy to interfere in domestics. I mean, not unless it really kicks off. They're just having a bit of a barney by the looks of it. Friday night, after all.\"\n\nAnd then he drives the two of you home.\n\nYou realise that you want official interference into other people's lives but not into your own. You also realise that your truthfulness has become dangerously flexible. And you wonder if you should have got out of the car and tried to pull the man away from the woman.\n\n* * *\n\n\u2014\n\nOne of your problems is this: for a long time it remains inconceivable to you that she is a drinker. How could she be, given that her husband is a drinker, and drink disgusts her? She hates even the smell of it, as she hates the bogus emotions it sets off in people. It makes Macleod coarser, angrier, more crudely sentimental; when he grabbed her hair and forced a glass to her lips, she would rather the sherry went down her dress than her throat. Nor has anyone in her life ever offered a credible counterexample: alcohol as glamorous, as usefully disinhibiting, as fun, as something you can control, knowing when to give the stuff its hour and when to refuse it.\n\nYou believe her. You never query her increasing lapses and latenesses. When you come in to find her blank-faced and bleary, you tell yourself that she has mistakenly swallowed an extra cheering-up pill\u2014which is sometimes the case. And because you inevitably believe that one of the reasons she is on antidepressants is because you are failing to make her so happy that she doesn't need them, you feel guilty, and this guilt forbids you from questioning her. So when, out of her bleariness, she looks up, pats the sofa beside her, and asks,\n\n\"Where've you been all my life?\"\n\nyou feel a ripping and a tearing inside you, and there is nothing you want more in the world than to make everything all right for her, and on her own terms, not yours. So you sit down next to her and take her wrists.\n\nJust as you believe your love to be unique, you believe your problems\u2014her problems\u2014to be unique. You are too young to understand that all human behaviour falls into patterns and categories and that her\u2014your\u2014case is far from unique. You want her to be some kind of exception, rather than any kind of rule. If anyone had ventured such a word as codependency to you back then\u2014assuming the term had even been invented\u2014you would have laughed it off as American jargon. However, you might have been more impressed by a statistical linkage of which you were then unaware: that the partners of alcoholics, far from being repulsed by the habit\u2014or rather, despite being repulsed by the habit\u2014frequently succumb to it themselves.\n\nBut the next stage for you is to accept a percentage of the evidence in front of your eyes. You understand that in certain, very limited circumstances, she needs the small lift of a small drink\u2014as she now occasionally admits. Obviously, she has to keep Joan company when she goes to the Village; obviously, she's sometimes frightened by the increasing traffic on the roads, and by that sudden twisty climb over the hills, so a little nip helps her; obviously, she is sometimes very lonely when you're away at college for most of the day. She also has \"my bad time,\" as she calls it\u2014usually between five and six in the evening, though as the days draw in and dusk falls earlier, so her bad time accordingly starts earlier, and obviously, extends just as late as it did before.\n\nYou believe what she says. You believe that the bottle she keeps beneath the sink, behind the bleach and washing-up liquid and silver polish, is the only bottle she drinks from. When she suggests that you put a pencil mark on the bottle so you can both monitor how much she drinks, you are heartened, and think these pencil marks are quite different from the ones on Eric's whisky bottle. Nor do you imagine there are other bottles elsewhere. When friends try to tip you off\u2014\"I'm a bit worried about Susan's drinking,\" says one, and \"Boy, you could smell the booze from the other end of the phone,\" says another\u2014you react in various ways. You protect her by denying it; you admit there are occasional lapses; you say the two of you have talked about it and she has promised \"to see someone.\" You may even say all three things in the course of a single conversation. But you will also be offended by your friends' attempted helpfulness. Because you do not need help: the two of you, since you love one another, will be able to sort the matter out, thank you very much. And this slightly alienates your friends, and also alienates them from her. Increasingly, you find yourself saying, \"She was just having a bad day,\" and you believe it yourself by dint of repetition.\n\nBecause there are still many good hours, and good days, when sobriety and cheerfulness fill the house, and her eyes and smile are just as they were when you first met, and you do something simple like drive for a walk in the woods, or go to the cinema and hold hands, and a sudden rush of feeling tells you it is all very easy and straightforward, and then your love is reaffirmed, yours for her, hers for you. And you wish you could display her to your friends at times like this: look, she is still herself, not just \"underneath,\" but here, now, on the surface too. You never suspect that one reason your friends tend to see her half-cut might be because she has persuaded herself, by some tortuous argument, that she needs a little Dutch courage before facing them.\n\nEach stage rolls seamlessly into the next. And here comes a paradoxical one that you initially struggle with. If you love her, as you unwaveringly do, and if loving her means understanding her, then understanding her must include understanding why she is a drinker. You run through all her prehistory, and recent history, and current situation, and possible future. You understand all this, and before you know where you are, you have passed somehow from total denial of the fact that she drinks to total comprehension of why she might do so.\n\nBut with this comes a brute chronological fact. As far as you know, Susan only drank occasionally in all her years with Macleod. But now that she is living with you, she is\u2014has become, is still becoming\u2014an alcoholic. There is too much in this for you to entirely acknowledge, let alone bear.\n\n* * *\n\n\u2014\n\nShe is sitting up in her quilted bedjacket, the newspapers around her, at her elbow a mug of coffee long gone cold. She has a frown on her, and her chin is pushed forward, as if she has been ruminating all day. It is now six in the evening, and you are in your last year of law studies. You sit on the side of her bed.\n\n\"Casey Paul,\" she begins, in an affectionate, puzzled tone, \"I've decided that there's something seriously wrong.\"\n\n\"I think you may be right,\" you answer quietly. At last, you think, perhaps this is the moment of breakthrough. That's what's meant to happen, isn't it? Everything comes to a moment of crisis, and then the fever breaks, and all becomes clear and rational and happy again.\n\n\"But I've been searching my wits all day and can't get a handle on it.\"\n\nNow where do you go? Do you start straight in again with the drinking? Suggest seeing the doctor again, a specialist, a psychiatrist? You are twenty-five, and quite untrained for this kind of situation. There are no articles in the newspaper headed, HOW TO COPE WITH YOUR MIDDLE-AGED FEMALE ALCOHOLIC LOVER. You are on your own. You have no theories of life yet, you only know some of its pleasures and pains. You still believe, however, in love, and in what love can do, how it can transform a life, indeed the lives of two people. You believe in its invulnerability, its tenacity, its ability to outrun any opponent. This, in fact, is your only theory of life so far.\n\nSo you do the best you can. You take one of her wrists, and talk about how you met and fell in love, how you were chosen by lot and then threw in your lot together, how you had run away in the finest tradition of lovers, and you continue like this, meaning and believing every word, and then you gently suggest that she's been drinking a little too much lately.\n\n\"Oh, you're always going on about that,\" she replies, as if this were some tedious and pedantic obsession of yours, nothing really to do with her. \"But if you want me to say so, then I will. Maybe I occasionally take a drop or two more than is good for me.\"\n\nYou quell the prompting inner voice which says: No, not a drop or two, a whole bottle or two more than is good for you.\n\nShe goes on, \"I'm talking about something much bigger than _that._ I think there's something seriously wrong.\"\n\n\"You mean, something that causes your drinking? Something I don't know about?\" Your mind heads towards some terrible, defining event in her childhood, much worse than a \"party kiss\" from Uncle Humph.\n\n\"Oh, you really can be a Great Bore at times,\" she says mockingly. \"No, much more important than that. What's behind it all.\"\n\nYou are already losing a little patience. \"And what do you think might be behind it all?\"\n\n\"Maybe it's the Russkis.\"\n\n\"The R _usskis_?\" You\u2014well, yes\u2014you yelp.\n\n\"Oh Paul, do try and keep up. I don't mean the _actual_ Russkis. They're just a figure of speech.\"\n\nLike, say, the Ku Klux Klan or the KGB or the CIA or Che Guevara. You suspect that this one brief chance is slipping away, and you don't know if it is your fault, her fault, or nobody's fault.\n\n\"OK,\" you say. \"The Russians are a figure of speech.\"\n\nBut she takes this only as sly impertinence.\n\n\"It's no good if you can't follow. There's something behind it all, just out of sight. Something which holds it all together. Something that, if we put it back together, would mend it all, would mend us all, don't you see?\"\n\nYou give it your best shot. \"You mean, like Buddhism?\"\n\n\"Oh don't be absurd. You know what I think about religion.\"\n\n\"Well, it was just an idea,\" you say jokingly.\n\n\"And not a very good one.\"\n\nHow quickly it has gone from something tentative and gentle and hopeful to something irascible and mocking. And how far away from what you consider to be the problem, not just behind it all but on the surface and at all points in between: the bottles under the sink, under the bed, behind the bookshelves, in her stomach, in her head, in her heart. It may be true that you don't know the cause, if indeed there is a single, identifiable cause, but it seems to you that you can only work with\u2014against\u2014the manifestations that erupt every day.\n\nYou know what she means about religion, of course. There is her adamantine disapproval of missionaries, whether they seek to convert in distant lands or on suburban doorsteps. And there is also the Malta story, which she has told you more than once. When the girls were small, Gordon Macleod was posted to Malta for a couple of years. She went out and lived there for some of the time. And her abiding memory was of the priest's bicycle. Yes, she would explain, it's terribly Catholic out there. The church is all-powerful, and everyone's very obedient. And the church keeps them down by making the women have as many children as possible: it's absolutely impossible to obtain birth control on the island. They're very backward in that regard\u2014John Bell & Croyden would be run out of town\u2014so you have to take the equipment out with you.\n\nAnyway, she goes on, it sometimes happens that a young bride doesn't get pregnant immediately after marriage, say for a year or two, despite all her prayers. Or maybe there's a woman who has two children and desperately wants a third but it isn't happening. And in such cases, the priest will come round and prop his bicycle outside the front door, so everyone\u2014especially the husband\u2014knows not to interfere until the bicycle has gone. And when, nine months later\u2014though of course it may take several goes\u2014the family is blessed, that blessing is known as \"the priest's child,\" and thought of as a gift from God. And sometimes there is more than one priest's child in the family. Can you imagine that, Paul? Don't you think it's barbaric?\n\nYou do think it's barbaric\u2014you say so every time. And now part of you\u2014the doomful, despairing, sarcastic part of you\u2014wonders whether, if it isn't the Russkis who are behind it all, then it might be the Vatican.\n\n* * *\n\n\u2014\n\nYou still share a bed, but haven't made love for a long time now. You don't ask yourself how long in calendar terms, because what counts is how it feels in terms of the heart. You discover more about sex than you want to\u2014or more than you should be allowed to discover while still young. Certain discoveries should be kept for later in life, when they might hurt less.\n\nYou know already that there is good sex and bad sex. Naturally, you prefer good sex to bad sex. But also, being young, you think that even so, all things considered, taking the rough with the smooth, bad sex is better than no sex at all. And sometimes better than masturbation; though sometimes not.\n\nBut if you think these are the only categories of sex that exist, you find you are mistaken. Because there is a category which you had not known to exist, something which isn't, as you might have guessed had you heard about it before, merely a subcategory of bad sex; and that is sad sex. Sad sex is the saddest sex of all.\n\nSad sex is when, the toothpaste in her mouth not fully disguising the smell of sweet sherry, she whispers, \"Cheer me up, Casey Paul.\" And you oblige. Though cheering her up also involves cheering yourself down.\n\nSad sex is when she is already doped by a cheering-up pill, but you think that if you fuck her, it might cheer her up a bit more.\n\nSad sex is when you are yourself in such despair, the situation so insoluble, the prehistory so oppressive, the very balance of your soul in doubt from day to day, moment to moment, that you think you may as well forget yourself for a few minutes, for half an hour, in sex. But you don't forget yourself, or your state of soul, not for even a nanosecond.\n\nSad sex is when you feel you are losing all touch with her, and she with you, but this is a way of telling one another that the connection is still there, somehow; that neither of you is giving up on the other, even if part of you fears that you should. Then you discover that insisting on the connection is the same as prolonging the pain.\n\nSad sex is when you are making love to a woman while thinking about how to kill her husband, even if this is something you would never be able to do, because you are not that sort of person. But as your body continues, so does your mind: you find yourself thinking, Yes, if you discovered him in the process of strangling her, you can imagine hitting him on the back of the head with a spade, or maybe stabbing him with a kitchen knife, though you realise that, given your hopelessness at fisticuffs, you might end up with the spade or the knife skidding off him and striking her instead. Then this parallel narrative in your head gets even madder, proposing that if you were to miss him and hit her instead, then it might be that you secretly wanted to harm her, because she\u2014this woman now naked beneath you\u2014has got you into this insoluble morass so early in your life.\n\nSad sex is when she is sober, you both desire one another, you know that you will always love her regardless, just as she will always love you regardless, but you\u2014both of you, perhaps\u2014now realise that loving one another does not necessarily lead to happiness. And so your lovemaking has become less a search for consolation than a hopeless attempt to deny your mutual unhappiness.\n\nGood sex is better than bad sex. Bad sex is better than no sex, except when no sex is better than bad sex. Self-sex is better than no sex, except when no sex is better than self-sex. Sad sex is always far worse than good sex, bad sex, self-sex and no sex. Sad sex is the saddest sex of all.\n\n* * *\n\n\u2014\n\nAt college you meet Paula\u2014blond, friendly, direct\u2014who has switched to law after a short-service commission in the Army. You like her handwriting when she shows you a case summary from a lecture you missed. You invite her for coffee one morning, then start having sandwich lunches in the nearby public gardens. One evening you take her to the cinema and kiss her good night. You exchange phone numbers.\n\nA few days later, she asks, \"Who's that madwoman who lives in your house?\"\n\n\"I'm sorry?\" Already there is a chill spreading through you.\n\n\"I rang you up last night. A woman answered the phone.\"\n\n\"That would have been my landlady.\"\n\n\"She sounded as mad as a hatter.\"\n\nYou take a breath. \"She's a little eccentric,\" you say. You want this conversation to stop, immediately. You wish it had never started. You wish Paula had never phoned the number you gave her. You very much don't want her to be specific, but you know she is going to be.\n\n\"I asked when you'd be back, and she said, 'Oh, he's very much the dirty stop-out, that young man, you can't rely on him from one moment to the next.' And then she came over all genteel and said something like, 'If you will excuse me while I fetch a pencil, I shall pass on any message you may choose to leave.' Well, I put the phone down before she came back.\"\n\nShe is looking at you expectantly, sure that you will provide her with an explanation that will satisfy her. It doesn't have to be much; a joke might even do it. Various extravagant lies cross your mind until, preferring the quarter-truth to the self-interested obfuscation\u2014and also feeling stubborn and defensive about Susan\u2014you repeat,\n\n\"She's a little eccentric.\"\n\nAnd that, unsurprisingly, is the end of your relationship with Paula. And you realise that such a pattern is likely to repeat itself with other friendly and direct girls whose handwriting you admire.\n\n* * *\n\n\u2014\n\nAround this time, you stop thinking of her family by their nicknames. All that Mr. Elephant Pants and Miss Grumpy stuff was fine and funny at the time, part of the first silliness and proprietoriness of love. But it was also a facetious minimising of their presence in her life. And if you are beginning to think of yourself as grown-up\u2014however forcedly and prematurely\u2014then they should be allowed their own maturity as well.\n\nAnother thing you notice is that you no longer fall easily into the private, teasing love language that used to pass between you. Perhaps the weight of what you have taken on has temporarily crushed out love's decorativeness. Of course, you still love her, and tell her so, but in plainer terms nowadays. Perhaps, when you have solved her, or she has solved herself, there will be room again for such playfulness. You can't be sure.\n\nSusan, however, continues using all the little phrases from her side of the relationship. It is her way of maintaining that nothing has changed, that she is fine, you are fine, all is fine. But she, you and it aren't, and those familiar words sometimes cause a prickle of embarrassment, more often lurching pain. You let yourself into the house, deliberately making enough noise to alert her, and as you come down the short flight of stairs into the kitchen, you find her in a familiar pose: red-faced by the gas fire, wrinkling her brow at a newspaper as if the world really does need to sort itself out. Then she looks up brightly and says, \"Where've you been all my life?\" or \"Here's the dirty stop-out,\" and your cheerfulness\u2014even if briefly assumed\u2014drains like bathwater. You look around and take stock of the situation. You open the store cupboards to see if there is something you can make into something. And she lets you get on with it, while offering occasional remarks designed to convey that she is still well capable of understanding a newspaper.\n\n\"Things seem to be in a frightful mess, don't you agree, Casey Paul?\"\n\nAnd you ask, \"Where exactly are we talking about?\"\n\nAnd she replies, \"Oh, just about everywhere.\"\n\nAt which point you might throw the emptied tin of plum tomatoes into the bin with some force, and she will chide you,\n\n\"Temper, temper, Casey Paul!\"\n\n* * *\n\n\u2014\n\nBy months of manoeuvring, you get her first to a GP and then to a consultant psychiatrist at the local hospital. She doesn't want you to come with her, but you insist, knowing what will probably happen otherwise. You turn up at a quarter to three for a three o'clock slot. The waiting area already contains a dozen other patients, and you realise it is the hospital's policy to book everyone in for the same time, which is when the consultant's session begins. You can see their point: mad people\u2014and at your age you use the term pretty broadly\u2014are presumably not among the world's most punctilious timekeepers: so it's best to summon them all _en bloc._\n\nShe makes what might be an attempt to escape, heading off to the ladies. You let her go with a fifty-fifty expectation that she won't return. But she does, and you find yourself reflecting cynically that she probably went to the hospital shop to check if they stocked booze, or maybe asked a few nurses where the bar was, only to receive the annoying news that the hospital doesn't have one.\n\nYou realise how sympathy and antagonism can coexist. You are discovering how many seemingly incompatible emotions can thrive, side by side, in the same human heart. You are angry with the books you have read, none of which have prepared you for this. No doubt you were reading the wrong books. Or reading them in the wrong way.\n\nYou feel, even at this late, desperate stage, that your emotional situation is still more interesting than that of your friends. They (mostly) have girlfriends and (mostly) have peer sex; some have been inspected by their girlfriends' parents, receiving approval, disapproval or judgement suspended. Most have a plan for their future life which includes this girlfriend\u2014or, if not, one very similar. A plan to become furrow-dwellers. But for the moment, they have only the traditional clear-skinned joys, sane dreams and inchoate frustrations of young men in their mid-twenties with girlfriends of the same age. Yet here you are, in a hospital waiting area, surrounded by mad people, in love with a woman who is being characterised as potentially mad.\n\nAnd the strange thing is, part of you feels exhilarated by it. You think: not only do you love Susan more than they love their girlfriends\u2014you must do, otherwise you wouldn't be sitting here among all the nutters\u2014but you are having a more interesting life. They may measure their girlfriends' brains and breasts, and their future parent-in-laws' deposit accounts, and imagine they have won; but you are still ahead of them because your relationship is more fascinating, more complicated and more insoluble. And the proof of this is that you are sitting here on a metal stacking chair, half-reading some discarded magazine, while your beloved dreams of\u2014what? Escape, no doubt: escape from here, escape from you, escape from life? She too is staggering beneath the weight of extreme, unbearable and incompatible emotions. You are both in deep pain. And yet, aware as you are of the stupid, bolloxy world of male competitiveness, you tell yourself that you are still a winner. And when you get to this point in your thoughts, the next logical stage is: you're a nutter as well. You are obviously one stark staring, complete and utter nutter. On the other hand, you are the youngest fucking nutter in the whole waiting area. So you have won again! Former under-12, under-6-stone school boxing champion becomes hospital's under-26 nutter champion!\n\nAt this moment a round, bald, suited man opens the door of the consulting room.\n\n\"Mr. Ellis,\" he calls quietly.\n\nThere is no reply. Familiar with the inattention, selective deafness and other failings of his patients, the consultant raises his voice:\n\n\"Mr. ELLIS!\"\n\nSome old fool wearing three sweaters and an anorak gets to his feet; a towelling headband restrains the ten or so wisps of white hair that sprawl from his crown. He stands looking round for a moment, as if perhaps expecting applause for having recognised his own name, then follows the consultant into his office.\n\nYou are not prepared for what happens next. You hear the psychiatrist's voice, quite clearly, say,\n\n\"And how are we today, Mr. Ellis?\"\n\nYou look at the closed door. You see that there is a three-inch gap between the foot of it and the floor. You guess that the consultant must be facing the door. You do not hear a reply from the deaf old fool, but perhaps there hasn't been one, because next, loud enough to rouse the other nodding nutters, come the words,\n\n\"SO HOW'S THE DEPRESSION, MR. ELLIS?\"\n\nYou are not sure if Susan has been paying attention. For yourself, you think this is unlikely to work.\n\n* * *\n\n\u2014\n\nThere is her shame, which is ever present. And then there is your shame, which sometimes presents itself as pride, sometimes as a kind of noble realism; but also, mostly, as what it is\u2014just shame.\n\n* * *\n\n\u2014\n\nYou come back one evening to find her pie-eyed in a chair, the water glass by her side still containing a good inch or so of non-water. You decide to behave as if all this is completely normal\u2014indeed, what domestic life is all about. You go into the kitchen and start looking around for something to turn into something. You find some eggs: you ask if she would like an omelette.\n\n\"It's easy for you,\" she answers belligerently.\n\n\"What's easy for me?\"\n\n\"That's a clever lawyer's answer,\" she replies, taking a swig right in front of you, which she rarely does. You are about to go back to cracking eggs, when she adds,\n\n\"Gerald died today.\"\n\n\"Which Gerald?\" You can't immediately think of a Gerald among your mutual friends.\n\n\"Which Gerald indeed? Mr. Clever. _My_ Gerald. My Gerald that I told you about. The one I was engaged to. This was the day he died.\"\n\nYou feel terrible. Not because you have forgotten the date\u2014she has never told you it before\u2014but because, unlike you, she has her dead to remember. Her fianc\u00e9, the brother who disappeared out over the Atlantic, Gordon's father\u2014whose name you no longer remember\u2014who had been soft on her. You have no such figures in your life, no griefs, no holes, no losses. So you don't know what it's like. Everyone should remember their dead, you believe, and everyone else should respect this need and desire. You are in fact rather envious, and wish you had a few dead of your own.\n\nLater, you become more suspicious. She has never mentioned the day of Gerald's death before. And there is no way for you to check. Just as, in happier days, there was no way for you to check when she told you how many times the two of you had made love. Perhaps, when she heard your key turn in the lock, and she was unable to get up and also unwilling to hide the glass at her elbow, she decided\u2014no, this is perhaps too deliberate a verb to describe her mental processes that evening\u2014she \"realised,\" yes, she suddenly realised that it was the day of Gerald's death. Though it could equally have been Alec's, or that of Gordon's father. Who could tell? Who knew? And who, in the end, cared?\n\n* * *\n\n\u2014\n\nI said I never kept a diary. This isn't strictly true. There was a point, in my isolation and turmoil, when I thought writing things down might help. I used a hardback notebook, black ink, one side of the paper. I tried to be objective. There was no point, I thought, in merely venting my feelings of hurt and betrayal. I remember that the first line I wrote down was:\n\n> _All alcoholics are liars._\n\nThis was, obviously, not based on a huge sample or broad research. But I believed it at the time, and now, decades later, with more field experience, I believe it to be an essential truth about the condition. I went on:\n\n> _All lovers are truth-tellers._\n\nAgain, the sample was small, consisting mainly of myself. It seemed to me evident that love and truth were connected; indeed, as I may have said, that to live in love is to live in truth.\n\nAnd then the conclusion to this quasi-syllogism:\n\n> _Therefore, the alcoholic is the opposite of the lover._\n\nThis seemed not just logical, but also consistent with my observations.\n\nNowadays, a lifetime later, the second of these propositions seems the weakest. I have seen too many examples of lovers who, far from living in truth, dwelt in some fantasy land where self-delusion and self-aggrandizement reigned, with reality nowhere to be found.\n\nYet, even while I was compiling my notebook, searching to be objective, the subjective kept undermining me. For instance, I realised, looking back at our time in the Village, that whereas I thought of myself as both lover and truth-teller, the truths I had told were only to myself and Susan. I told lies to my parents, to Susan's family, to my own close friends; I even dissembled at the tennis club. I protected the zone of truth with a rampart of lies. Just as she was now lying to me all the time about her drinking. As well as lying to herself. And yet she would still affirm that she loved me.\n\nSo I began to suspect that I was wrong in considering alcoholism as the opposite of love. Perhaps they were much closer than I imagined. Alcoholism is certainly just as obsessive\u2014as absolutist\u2014as love; and maybe to the drinker the hit of booze is as powerful as the hit of sex is to the lover. So could the alcoholic be merely a lover who has shifted the object and focus of his or her\u2014no, her\u2014love?\n\nMy observations and reflections had filled a few dozen pages when I came home one evening to find Susan in a state I knew all too well: red-faced, semi-coherent, quick to take offence, yet at the same time genteelly pretending that all was for the best in this best possible of worlds. I went to my room and discovered that my desk had been inexpertly rifled. I had, even then, a habit of orderliness, and knew what lived where. Since the desk contained my Notes on Alcoholism, I assumed, wearily, that she had probably read them. Still, I thought, perhaps in the long run the shock might have a useful effect on her. In the short run, evidently not.\n\nThe next time I went to my notebook to make an addition, I saw that Susan had done more than just read it. She had left an annotation beneath my last entry, using the same black ink from the same pen. In an unsteady hand, she had written:\n\n> _With your inky pen to make you hate me._\n\nI didn't accuse her of rifling my desk, reading my notebook, writing in it. I could imagine her saying, in a tone of polite protest, \"No, I don't think so.\" I was weary of constant confrontation. But then, I was equally weary of a constant pretence that all was well, a constant evasion of the truth. I also realised that it would be impossible for me to write anything down in the future without picturing her at my desk studying my latest denunciations. This would be intolerable for both of us: the annotation of pain on my part, the dim yet irate acknowledgement of pain caused on her part. So I threw away the notebook.\n\nBut that half-formed sentence of hers, written by a wonky hand with an unfamiliar pen, remained with me, and always has. Not least because of its ambiguity. Did she mean, \"You use your inky pen to write down things which will then make you hate me\"? Or did she mean, \"I have left my mark with your inky pen because I want to make you hate me\"? Critical and aggressive, or masochistic and self-pitying? Maybe she knew what she meant when she wrote the words, but there was no subsequent clarity to be found. You may judge the second interpretation oversubtle, and designed to let me off the hook. But\u2014and this formed the basis of another of my long-lost notes\u2014the alcoholic, in my experience, wants to provoke, to push away help, to justify her own isolation. So if she managed to convince herself that I hated her, all the more reason to turn for comfort to the bottle.\n\n* * *\n\n\u2014\n\nYou are taking her somewhere in the car. There is no need for her to fear the journey, and you will pick her up later and drive her home. But there are the usual delays before you can get her into the car. And as you are about to release the handbrake, she rushes back into the house and returns with a large, bright yellow plastic laundry bag, which she puts between her feet. She does not explain. You do not ask. This is where things have got to.\n\nAnd then you think, Oh Fuckit.\n\n\"What's that for?\" you ask.\n\n\"The thing is,\" she replies, \"I'm not feeling entirely well, and it's just possible I might be sick. What with the car and all that.\"\n\nNo, you think, what with being drunk and all that. A doctor friend has told you that alcoholics sometimes throw up so violently that they can perforate their own oesophagus. As it happens, she doesn't need to vomit, but she might as well have done. Because she has already filled your head with an image of her throwing up into this yellow bag, and you cannot stop seeing it. You might as well have listened to her dry retching and then wet retching, and can hear the vomit trickling into the bright yellow plastic. The smell, too, of course, in your small car. The excuses, the lies. Her lies, your lies.\n\nBecause it is no longer just a question of her lying to you. When she does so, you have two choices: call her out on it, or accept what she says. Usually, out of weariness and a desire for peace\u2014and yes, out of love\u2014you accept what she says. You condone the lie. And so become a liar by proxy. And it is a very short step from accepting her lies to lying yourself\u2014out of weariness, and a desire for peace, and also out of love\u2014yes, that too.\n\nWhat a long way you have come. Years ago, when you started off lying to your parents, you did so with a kind of relish, reckless of consequence; it almost felt character-building. Later, you began to tell lies in all directions: to protect her, and to protect your love. Later still, she starts lying to you, to keep you from knowing her secret; and now she lies with a kind of relish, reckless of consequence. Then, finally, you begin lying to her. Why? Something to do with the need to create some internal space which you could keep intact\u2014and where you could yourself remain intact. And this is how it is for you now. Love and truth\u2014where have they gone?\n\nYou ask yourself: Is staying with her an act of courage on your part, or an act of cowardice? Perhaps both? Or is it just an inevitability?\n\n* * *\n\n\u2014\n\nShe has taken to going to the Village by train. You approve: you think this comes from a recognition of her unfitness to drive. You take her to the station, she tells you the time of her return train, though, as often as not, she doesn't turn up until the next one, or the one after. And when she says, \"Don't bother meeting me,\" she is protecting her inner world. And when you reply, \"Fine\u2014sure you'll be all right?\" you are protecting yours.\n\nThe phone goes one evening.\n\n\"Is that Henry?\"\n\n\"No, sorry, wrong number.\"\n\nYou are about to put the phone down when the man reads your number to you.\n\n\"Yes, that's right.\"\n\n\"Well, good evening, sir. This is the transport police at Waterloo Station. We have a lady here in a...slightly distressed condition. We found her asleep in the train and, well, her handbag was open and there was a sum of money in it, so you see...\"\n\n\"Yes I see.\"\n\n\"She showed us this number and asked us to call Henry.\"\n\nIn the background you hear her voice. \"Call Henry, call Henry.\"\n\nAh, her shorthand for Henry Road.\n\nSo you drive to Waterloo, find the office of the railway police and there she is, sitting up, bright-eyed, waiting to be collected, knowing that it would happen. The two policemen are courteous and concerned. They are doubtless used to helping drunk old ladies found snoring in empty carriages. Not that she is old, just that when she is drunk, you think of her, suddenly, as a drunk old lady.\n\n\"Well, thank you very much for looking after her.\"\n\n\"Oh, she was no trouble at all, sir. Quiet as a mouse. Look after yourself, Madam.\"\n\nShe gives a rather stately acknowledging nod. You take the arm of this piece of left luggage, and off you go. Your annoyance and despair, however, are cut by a certain pride in her having been \"no trouble.\" Though what if she had been?\n\n* * *\n\n\u2014\n\nEventually, more out of despair than hope, you try tough love, or at least what your understanding of that concept is. You don't let her get away with anything. You call her out on her lies. You pour away whatever bottles you find, some in obvious places, some in such strange locations that she must have hidden them there while drunk, and then forgotten where she had put them. You get her banned from the three local shops which sell alcohol. You give them each a photo to keep behind the till. You do not tell her this; you think the humiliation of being refused service will jolt her. You never find out, and she merely gets round the obstruction by travelling further afield.\n\nYou hear reports. Some people are shy about mentioning things to you, others not. A friend, on a bus a mile or so away from Henry Road, has spotted her down an alleyway next to an off-licence, raising a newly bought bottle to her lips. This image burns deep, and transforms itself from another's account into your own private memory. A neighbour tells you that your auntie was in the Cap and Bells last Saturday night, downing five sherries in succession until they stopped serving her. \"It's not the kind of pub someone like her should be in,\" the neighbour adds concernedly. \"They get all sorts in there.\" You picture the scene, from her ashamed first order at the bar to her unsteady walk home, and this too becomes part of your memory bank.\n\nYou tell her that her behaviour is destroying your love for her. You do not mention hers for you.\n\n\"Then you must leave me,\" she says. She is flushed, dignified and logical.\n\nYou know that you are not going to do this. The question is, whether or not she knows it too.\n\nYou write her a letter. If spoken words of rebuke fly unhindered straight out of her head, perhaps written ones will stick. You tell her that the way she is going on, she will almost certainly die of a wet brain, that there is nothing more you can do for her, except come to her funeral, whenever that might be. You leave the letter on the kitchen table, in an envelope with her name on it. She never mentions receiving it, opening it, reading it. _With your inky pen to make you hate me._\n\nYou realise that tough love is also tough on the lover.\n\n* * *\n\n\u2014\n\nYou are taking her to Gatwick. Martha has invited her out to Brussels, where she is working as a Eurocrat. To your surprise, Susan agrees. You promise to make it as easy as possible for her. You will drive her to the airport and see her through check-in. She nods, then says straightforwardly,\n\n\"You might have to let me have a drink before getting on the plane. Belgian courage.\"\n\nYou are more than relieved: almost encouraged.\n\nThe night before she is half-packed and half-drunk. You go to bed. She continues packing and drinking. The next morning she comes to you with a cupped hand over her mouth.\n\n\"I'm afraid I don't think I shall be able to go.\"\n\nYou look at her without speaking.\n\n\"I've lost my teeth. I can't find them anywhere. I think I may have thrown them into the garden.\"\n\nYou don't say anything except, \"We have to leave by two.\" You decide to let her go on destroying her life.\n\nBut perhaps your failure to respond\u2014to offer neither help nor rebuke\u2014is, for once, the correct approach. An hour or two later, she is walking around with her teeth in, never alluding to either having lost them, or found them.\n\nAt two o'clock you put her case in the back of the car, double-check her ticket and passport and set off. There have been no last-minute diversions, no scurrying for a bright yellow laundry bag. She sits beside you quiet as a mouse, in the railway policeman's words.\n\nAs you are approaching Redhill, she turns and says in a demurely puzzled way, as if you were more her chauffeur than her lover,\n\n\"Would you mind very much telling me where we are going?\"\n\n\"You're going to Brussels. To visit Martha.\"\n\n\"Oh, I don't think so. There must be some mistake.\"\n\n\"That's why you've got your ticket and your passport in your handbag.\" Though they are actually in your pocket, as you don't want them going the way of her teeth.\n\n\"But I don't know where she lives.\"\n\n\"She's meeting you at the airport.\"\n\nThere is a pause.\n\n\"Yes,\" she says, nodding, \"I seem to remember about this now.\"\n\nThere is no further resistance. Part of you thinks she should have a large label round her neck with her name and destination written on it, like a wartime refugee. With perhaps her gas mask in a box as well.\n\nAt the bar you buy her a double schooner of sherry, which she sips with inattentive gentility. You think: it could be worse. This is how you react to situations nowadays. You have the lowest of expectations.\n\nThe trip turns out to be a success. She has been shown the city, and brings you some postcards. Miss Grumpy, she announces, is nowadays Much Less So, perhaps influenced by a charming Belgian boyfriend who was in attendance. Her memories are clearer than usual, a sign that she has been temperate. You feel happy for her, if slightly resentful that she can clean up her act for others more easily than for you. Or so it seems.\n\nBut then, she tells you that on the last morning, the real reason why her daughter invited her out became clear. She, Miss Grumpy, is of the opinion that her mother ought to go back to Mr. Gordon Macleod. Who is now very contrite and promises to be on his best behaviour if she returns. According to Susan, according to her daughter.\n\n* * *\n\n\u2014\n\nTo save time, and to save emotion, you address her, straightforwardly, as a drinker. No longer, There seems to be a problem, Do you know what it might be, Perhaps I can suggest...none of that. So one day you suggest Alcoholics Anonymous, not knowing if there is a branch near you.\n\n\"Not going to the God-botherers,\" she replies firmly.\n\nGiven her dislike of priests, and extreme disapproval of missionaries, this response is understandable. No doubt she thinks of AA as yet another bunch of American missionaries interfering in other countries' belief systems, bringing the foreign halt and lame into the radiant presence of their God. You do not blame her.\n\nMostly, you can only deal with the day-to-day crises. Occasionally, you look to the future, and find one outcome which has a terrifying logic. It goes like this. She doesn't drink all the time. Not every day. She can go a day or two without the comfort of a bottle. But her memory, as a result of the drink, is getting poorer. So the logic runs: if she carries on destroying her memory at the present rate, maybe she will reach the stage when she has actually forgotten she is an alcoholic! Might that happen? It would be one way to cure her. But you also think: you might as well simply blast her with ECT and be done with it.\n\nHere is one of the problems. You don't, at bottom, think of alcoholism as a physical disease. You might have heard that it is, but you aren't really convinced. You can't help thinking of it as many people\u2014some of whom you might not want to be associated with\u2014have thought about it for centuries: as a moral disease. And one of the reasons you do so is because she does too. When she is at her most lucid, her most rational, her most gentle, and as much tormented by what is happening to her as you are, she tells you\u2014as she always has\u2014that she hates the fact that she is a drinker, and feels deep shame and guilt about it: so you must leave her, because she is \"no good.\" She has a moral disease, which is why hospitals and doctors cannot cure her. They cannot fix a flawed personality from a played-out generation. She urges you again to leave her.\n\nBut you cannot leave Susan. How could you bear to withdraw your love from her? If you didn't love her, who would? And maybe it is worse than this. It is not just that you love her, but that you are addicted to her. How ironic would that be?\n\n* * *\n\n\u2014\n\nAn image comes into your head one day, an image of your relationship to one another. You are at an upstairs window of the house on Henry Road. She has somehow climbed out, and you are hanging on to her. By the wrists, of course. And her weight makes it impossible for you to pull her back inside. It is all you can do to stop yourself being pulled out with her, by her. At one point she opens her mouth to scream, but no sound emerges. Instead, her dental plate comes loose; you hear it hit the ground with a plasticky clatter. You are stuck there, the two of you, locked together, and will remain so until your strength gives out, and she falls.\n\nIt is only a metaphor\u2014or the worst of dreams; yet there are metaphors which sit more powerfully in the brain than remembered events.\n\n* * *\n\n\u2014\n\nAnother image, based on a remembered event, comes into your mind. You are back at the Village, the two of you, in the flush of love, quietly but entirely intent on one another. She is wearing a print dress and, knowing that you are watching her\u2014because you are always watching her\u2014she goes over to the chintzy sofa, plumps herself down, and says,\n\n\"Look, Casey Paul, I'm disappearing! I'm doing my disappearing act!\"\n\nAnd, for a moment, as you look, you can see only her face and the stockinged part of her legs.\n\nNow she is doing another disappearing act. Her body is still there, but what lies inside\u2014her mind, her memory, her heart\u2014is slipping away. Her memory is obscured by darkness and untruth, and persuades itself towards coherence only by fabulation. Her mind oscillates between stunned inertia and hysterical volatility. But it is the disappearing act which her heart is doing, oh, that is the hardest part to bear. It is as if, in her thrashing about, she has stirred up the mud that lies at the bottom of us all. And what is now coming to the surface is unfocussed anger, and fear, and frustration, and harshness, and selfishness and mistrust. When she tells you solemnly that in her considered opinion your behaviour towards her has been not just beastly but actively criminal, she really thinks it is true. And all the sweetness of her nature, the laughingness and trustingness central to the woman you fell in love with, can no longer be seen.\n\nYou used to say\u2014when putting off friends who wanted to visit\u2014\"Oh, she's having a bad day. She's not herself.\" And when they saw her drunk, you'd say, \"But she's still the same underneath. She's still the same underneath.\" How many times did you tell this to others, when the person you were actually addressing was yourself?\n\nAnd then comes the day when you no longer believe such words. You no longer believe that she is still the same underneath. You believe that being \"not herself\" is her new self. You fear that she is, finally and utterly, doing her disappearing act.\n\n* * *\n\n\u2014\n\nBut you make one final effort, and she does too. You get her admitted to hospital. Not the National Temperance, as you had hoped, but a general, all-female ward. You sit her down on a bench while they are admitting her, and explain gently, once more, how it has come to this, and what they will do for her, and how it will help.\n\n\"I'll give it my very best shot, Casey Paul,\" she says sweetly. You kiss her on the temple, and promise to visit her every day. Which you do.\n\nAt first they put her to sleep for three days, hoping to peaceably flush the alcohol out of her system, while also calming her disturbed brain. You sit by her lightly sleeping form and think that this time, surely, it will work. This time, she is under proper medical supervision, the problem has been stated clearly\u2014even she isn't ducking it\u2014and at last Something Will Be Done. You look at her calm face and think of the best years of your time together, and imagine that everything you had back then will now return.\n\nOn the fourth day she is still asleep when you go in. You ask to see a doctor and some twenty-year-old houseman with a clipboard presents himself. You ask why she is still sedated.\n\n\"We woke her up this morning, but she immediately became disruptive.\"\n\n\"Disruptive?\"\n\n\"Yes, she attacked the nurses.\"\n\nYou don't believe this. You ask him to repeat it. He does.\n\n\"So we put her under again. Don't worry, it's a very light sedative. I'll show you.\"\n\nHe adjusts the drip slightly. Almost instantly, she begins to stir. \"You see?\" Then he adjusts the drip again and sends her back to sleep. You find this deeply sinister. You have yielded her care to the authority of some youthful technocrat who has never met her.\n\n\"You're her...?\"\n\n\"Godson,\" you reply automatically. Or maybe you say \"Nephew,\" or possibly \"Lodger,\" which at least contains four correct letters in it.\n\n\"Well, if we wake her up and she's that disruptive again, we'll have to section her.\"\n\n\" _Section_ her?\" You are horrified. \"But she's not mad. She's an alcoholic, she needs treatment.\"\n\n\"So do all the other patients. And they need the nurses' attention. We can't have nurses being attacked.\" You still don't believe his initial allegation.\n\n\"But...you can't just section her by yourself.\"\n\n\"You're right, there have to be two signatures. But it's just a formality in cases like this.\"\n\nYou realise you have not brought her to a place of safety after all. You have delivered her over to the kind of zealot who in the old days would have prescribed a straitjacket plus a course of electroconvulsive therapy. Susan would have called him a \"little Hitler.\" Who knows, perhaps she did. You partly hope so.\n\nYou say, \"I would like to be there when you next wake her up. I think it would help.\"\n\n\"Very well,\" says the curt young man whom you have already come to hate deeply.\n\nBut\u2014such is the way of hospitals\u2014this arrogant little shit isn't there when you next come, and you never see him again. Instead, a female doctor operates the drip. Slowly, Susan wakes. She looks up, sees you and smiles.\n\n\"Where've you been all my life?\" she asks. \"You dirty stop-out.\"\n\nThe doctor reacts with slight surprise, but you kiss Susan on the forehead, and the two of you are left alone together.\n\n\"So you've come to take me home?\"\n\n\"Not just yet, darling,\" you say. \"You've got to stay here for a while. Until you're cured.\"\n\n\"But there's nothing wrong with me. I'm perfectly well and insist on being taken home at once. Take me to Henry.\"\n\nYou grasp both her wrists. You squeeze very hard. You explain that the doctors won't release her until she is cured. You remind her of the promise she made when you brought her here. You say that the last time they brought her round, she attacked the nurses.\n\n\"No, I don't think so,\" she says, in her most distant, genteel manner, as if you are some ill-informed peasant.\n\nYou talk at length to her, asking for her promise to behave until you come back tomorrow. At least until then. She doesn't respond. You press her. Then she promises, but with a stubbornness of tone you are all too familiar with.\n\nThe next day, you approach the ward expecting the worst: that she's been sedated again, or even sectioned. But she is looking alert, and her colour is good. She greets you rather as if you were her guest. A nurse walks by.\n\n\"The maids here are frightfully good,\" she says, giving a wave to the passing figure.\n\nYou think: What's the right tactic? Go along with it? Challenge it? You decide that you mustn't indulge her dream world.\n\n\"They're not maids, Susan, they're nurses.\" You think she might have confused \"hospital\" with \"hotel,\" which after all would not be much of a verbal slippage.\n\n\"Some of them are,\" she agrees. Then, disappointed with your lack of perspicacity, adds, \"But most of them are maids.\"\n\nYou let it go.\n\n\"I've told them all about you,\" she says.\n\nYour heart sinks, but you let that go as well.\n\nThe next day, you find her agitated again. She is out of bed, sitting up in a chair. On the tray in front of her are five pairs of spectacles and a copy of a P. G. Wodehouse novel she has mysteriously acquired.\n\n\"Where did you get all those glasses?\"\n\n\"Oh,\" she replies casually, \"I don't know where they come from. I expect people have been giving them to me.\"\n\nShe puts on a pair which are evidently not hers and opens the book at random. \"He's frightfully funny, isn't he?\"\n\nYou agree. She has always enjoyed Wodehouse, and you take this as a good, if slightly confused, sign. You tell her what's in the newspapers. You mention a postcard you've had from Eric. You say that all is well at Henry Road. She listens idly then seizes a different pair of glasses\u2014though still not her own\u2014opens the book at random again and, probably seeing it no more in focus than the previous time, announces,\n\n\"It's frightful rubbish, this, isn't it?\"\n\nYou think your heart will break, now, here, immediately.\n\nThe following day she is again under sedation. The woman in the next bed chats to you and asks what's wrong with Your Nan. You are so weary of it all that you answer,\n\n\"She's an alcoholic.\"\n\nThe woman turns away in distaste. You know exactly what she is thinking. Why give a good hospital bed to a drinker? Furthermore, a female drinker? One thing you have discovered is that male alcoholics are allowed to be amusing, even poignant. Young drinkers of either sex, when out of control, are indulged. But female alcoholics, old enough to know better, old enough to be mothers, even grandmothers\u2014these are the lowest of the low.\n\nThe next day she is awake again and refusing to look at you. So you just sit there for a while. You glance at the tray in front of her. This time, her nocturnal ward-rambling has netted only two pairs of other patients' glasses, plus a tabloid newspaper she would never have in the house.\n\n\"I do think,\" she announces finally, \"that you will be remembered as one of the greatest criminals in the history of the world.\"\n\nYou are tempted to agree. Why not?\n\nThey do not threaten to section her\u2014that little Hitler is off practising his black arts on other, less disruptive patients. But they tell you that they cannot treat her further, that the rest may have done her some good, that this is not the appropriate place for her and they need to free up the bed. You see their point of view entirely, but ask yourself: Then what is the appropriate place for her? Which stands in for a wider question: What is her place in the world?\n\nAs the two of you leave, the woman in the next bed pointedly ignores you both.\n\n* * *\n\n\u2014\n\nIt has taken some years for you to realise how much, beneath her laughing irreverence, there lies panic and pandemonium. Which is why she needs you there, fixed and steadfast. You have assumed this role willingly, lovingly. It makes you feel grown-up to be a guarantor. It has meant, of course, that for most of your twenties you were obliged to forgo what others of your generation routinely enjoyed: the mad fucking around, the hippie travelling, the drugs, the going off the rails, even the stonking idleness. You were also obliged to forgo the drinking; but then, you were hardly living with a good advertisement for the stuff. You didn't hold any of this against her (except perhaps the lack of drinking); nor did you treat it as some unfair burden you were assuming. It was just the given of your relationship. And it has made you age, or mature, if not by the route normally taken.\n\nBut as things begin to fray between you, and all your attempts to rescue her fail, you acknowledge something you haven't exactly been hiding from, just didn't have time to notice: that the particular dynamic of your relationship is triggering your own version of panic and pandemonium. While you probably strike your friends at law college as affable and sane, if a little withheld, what roils beneath your own surface is a mixture of groundless optimism and searing anxiety. Your inner moods ebb and flow in response to hers: except that her cheerfulness, even when misplaced, strikes you as authentic, your own as conditional. How long will this present little stretch of happiness last, you are continually asking yourself. A month, a week, another twenty minutes? You can't, of course, tell, because it doesn't depend on you. And however calming your presence is on her, the trick doesn't work the other way round.\n\nYou never think of her as a child, not even in her most selfish delinquencies. But when you watch an anxious parent tracking its offspring\u2014the alarm at each bandy-legged footstep, the fear of each \"trippy\" moment, the wider fear of the child simply wandering off and getting lost\u2014you know that you have been there yourself. Not to mention the child's sudden switches of mood, from blissful exaltation and absolute trust to rage and tears and a sense of abandonment. This too is familiar. Except that this wild, shifting weather of the soul is now passing through the brain and body of a mature woman.\n\nIt is this, finally, which breaks you, and tells you to move out. Not far, just a dozen streets, into a cheap one-room flat. She urges you to go, for reasons good and bad: because she senses that she must let you go a little if she is to keep you and because she wants you out of the house so that she can drink whenever the mood takes her. But in fact, little changes: you are still living just as closely. She doesn't want you to remove a single book from your study, or any knick-knack you have bought together, or any clothes from your wardrobe: such actions will throw her into a fit of grieving. Sometimes you sneak back into the house to remove a book, shuffling others along the shelf to cover the theft; occasionally, you stuff in a couple of cheap paperbacks from Oxfam to disguise the betrayal.\n\nAnd so you live an oscillating life. You continue to have breakfast with her, and also supper\u2014which you mostly cook; you go on expeditions together; and you get reports from Eric on her drinking. Eric, being merely fond and concerned, rather than in love with her, is a more reliable witness than you ever were yourself. Susan continues to do your laundry, and some of your best shirts come back lovingly scorched. Drunken ironing: that is one of the lesser, but still painful, things life has surprised you with.\n\n* * *\n\n\u2014\n\nThen, almost without your noticing it, what is close to the final stage kicks in. You may still desperately want to save her, but at some level of instinct or pride or self-protection, her devotion to drink now strikes you more sharply, and more personally: as a rejection of you, of your help, of your love. And since few can bear to have their love rejected, resentment builds, then curdles into aggression, and you find yourself saying\u2014not aloud, of course, because you find it hard to be overtly cruel, especially to her\u2014\"Go on, then, destroy yourself, if that's what you want.\" And you are shocked to discover yourself thinking this.\n\nBut what you don't realise\u2014not now, in the heat and dark of it all, only much later\u2014is that, even without hearing you, she will agree. Because what she is leaving unspoken is this reply: \"Yes, that's exactly what I want. And I _am_ going to destroy myself, because I am a worthless person. So stop bothering me with your well-intentioned meddling. Just let me get on with the job.\"\n\n* * *\n\n\u2014\n\nYou are working for a South London practice which specialises in legal aid. You enjoy the range of cases you handle; you enjoy the fact that in the majority of them you can solve things. You can get people the justice they deserve, and thereby make them happy. You are aware of the paradox of this. Also, of another, longer-term paradox: that in order to support Susan, you need to work, and the more you work, the more you are away from her, and the less able to support her.\n\nYou have also, as Susan predicted, found yourself a girlfriend. And not one who will run off at the first phone call. Anna is, perhaps inevitably, also a lawyer. You have told her some of Susan's history. You have not tried to get away with merely saying she is \"eccentric.\" You introduce the two of them, and they seem to get on. Susan says nothing to embarrass you, Anna is brightly practical. She doesn't think Susan looks after her diet well enough, so once a week takes round a loaf of proper bread, a bag of tomatoes, a pound of French butter. Sometimes the door remains unanswered, so she leaves her offering on the step.\n\nYou are home one evening when the phone goes. It is one of the lodgers.\n\n\"I think you'd better come round. We've had the police. With guns.\"\n\nYou repeat the words to Anna, then run for your car. In Henry Road there is an ambulance outside the house, its blue light revolving, its doors open. You park, walk across, and there she is, in a wheelchair facing out towards the street, with a broad bandage around her forehead which has pushed her hair up into a _Struwwelpeter_ shock. Her expression, as often when a sudden crisis has worked itself out, is one of slightly amused calm. She surveys the street, the ambulance men fixing the wheelchair in place, and your own arrival, as if from a throne. The blue light revolves against the steadier sodium orange. It is real and unreal at the same time; filmic, phantasmagoric.\n\nThen the chair slowly rises on its hoist, and as the ambulance doors are about to be closed, she lifts her hand in a pontifical blessing. You ask the ambulance men where they are taking her and follow in your car. When you get to the A&E department, they are already taking preliminary details.\n\n\"I'm her next of kin,\" you say.\n\n\"Son?\" they ask. You nearly agree, for speed, but they might query the difference of surname. So, once again, you are her nephew.\n\n\"He's not really my nephew,\" she says. \"I could tell you a thing or two about this young man.\"\n\nYou look at the doctor, lying to him with a slight frown and a tiny movement of the head. You collude in the notion that Susan is temporarily off among the nutters.\n\n\"Ask him about the tennis club,\" she says.\n\n\"We'll come to that, Mrs. Macleod. But first...\"\n\nAnd so the process continues. They will keep her in overnight, perhaps run a test or two. It may just be shock. They will call you when they are ready to release her. The ambulance men have said it was just a cut, but as it was on the forehead there was a lot of blood. It may need a stitch or two, maybe not.\n\nThe next day, they release her, still in full dispossession of her faculties.\n\n\"About time too,\" she says, as you walk her to the car park. \"It really has all been frightfully interesting.\"\n\nYou know this mood only too well. Something has been observed, or experienced, or discovered, which has little to do with anything, yet is of extreme, overwhelming interest, and must be reported.\n\n\"Let's wait until we get you home first.\" You have slipped into the language of the hospital, where everything is done or asked for in the name of \"us.\"\n\n\"All right, Mr. Spoilsport.\"\n\nAt Henry Road, you take her to the kitchen, sit her down, make her a cup of tea with extra sugar and give her a biscuit. She ignores them.\n\n\"Well,\" she begins, \"it was all so fascinating. Such fun. You see, these two men with guns got into the house last night.\"\n\n\"With _guns_?\"\n\n\"That's what I said. With guns. Do stop interrupting before I've barely even started. So yes, two men with guns. And they were going round looking for something. I don't know what.\"\n\n\"Were they robbers?\" You feel you are allowed to ask questions which don't challenge the essential veracity of her fantasy.\n\n\"Well, that's what I thought might be the case. So I said to them, 'The gold bullion is under the bed.' \"\n\n\"Wasn't that a bit rash?\"\n\n\"No, I thought it would put them off the scent. Not that I knew what the scent was, of course. They were both quite polite and well mannered. For gunmen, that is. They didn't want to bother me, they would just go about their business if I didn't mind.\"\n\n\"But didn't they shoot at you?\" You indicate her forehead, now decorated with a large gauze patch.\n\n\"Lord, no, they were much too polite for that. But it was rather an interruption to the evening, so I felt obliged to call the police.\"\n\n\"Didn't they try and stop you?\"\n\n\"Oh no, they were all in favour. They agreed with me that the police might help them find what they were looking for.\"\n\n\"But they didn't tell you what that was?\"\n\nShe ignores you and continues.\n\n\"But the thing I really wanted to tell you was that they had these feathers everywhere.\"\n\n\"Gosh.\"\n\n\"Feathers sticking out of their bottoms. Feathers in their hair. Feathers everywhere.\"\n\n\"What sort of guns did they have?\"\n\n\"Oh, who knows about guns?\" she says dismissively. \"But then the police came, and I answered the door to them, and they sorted everything out.\"\n\n\"Was there a gunfight?\"\n\n\"A gunfight? Don't be ridiculous. The British police are far too professional for that.\"\n\n\"But they arrested them?\"\n\n\"Naturally. Why else do you think I called them?\"\n\n\"So how did you cut your head?\"\n\n\"Well, of course I can't remember that. It's the least interesting part of the story in my view.\"\n\n\"I'm glad it all worked out in the end.\"\n\n\"You know, Paul,\" she says, \"sometimes I'm really disappointed in you. It was so enjoyable and so fascinating, but you keep coming up with these banal comments and banal questions. Of course it all worked out in the end. Everything always does, doesn't it?\"\n\nYou don't answer. After all, you have your pride. And in your opinion, the notion that everything works out in the end, and the counternotion that nothing ever does, are both equally banal.\n\n\"Now don't sulk. It's been one of the most interesting twenty-four hours of my life. And everyone\u2014 _everyone_ \u2014was very nice to me indeed.\"\n\nThe gunmen. The police. The ambulance men. The hospital. The Russkis. The Vatican. And all's right with the world, then.\n\n* * *\n\n\u2014\n\nThat evening, over takeaway pizzas, I recounted the whole lurid episode to Anna. I told it fondly, concernedly, almost amusedly, if not quite. The fantasy gunmen, the real policemen, the gold bullion, the feathers, the ambulance men, the hospital. I omitted some of Susan's strictures on my character. I was also aware, however, that Anna was not reacting as I had expected.\n\nEventually, she said, \"That all sounds a great waste of public money.\"\n\n\"That's an odd way to look at it.\"\n\n\"Is it? Police, firearms squad\u2014Special Branch\u2014ambulance, hospital. All of them dashing around making a fuss of her, just because she's gone on a bender. And that includes you too.\"\n\n\"Me? What do you expect me to do when the lodger calls and says there are armed police in the house?\"\n\n\"I didn't _expect_ you to do anything different.\"\n\n\"Well then\u2014\"\n\n\"Just as I wouldn't _expect_ you to do anything different if we were going out for a meal, or a film, or leaving for a holiday and already running late for our flight.\"\n\nI thought about this. \"No, I don't expect I would. Behave differently.\"\n\nWe were reaching a stand-off, I realised. One of the reasons I'd gone for Anna in the first place was that she always spoke her mind. This had a downside to it as well as an upside. I suppose all character traits do.\n\n\"Look,\" I said. \"We talked about...all this when we first got together.\" Somehow, I couldn't say Susan's name at that moment.\n\n\"You talked. I listened. I didn't necessarily agree.\"\n\n\"Then you misled me.\"\n\n\"No, Paul, you didn't explain the full extent of it to me. Maybe in future when I get out my diary to write in a dinner date or a play or a weekend away, I should always add a note saying: subject to the extent of Susan Macleod's alcoholic intake.\"\n\n\"That's very unfair.\"\n\n\"It may be unfair but it also happens to be true.\"\n\nWe paused. It was a question of whether either of us wanted to take it further. Anna did.\n\n\"And while we're about it, Paul, I may as well say that Susan Macleod...is not really my kind of woman.\"\n\n\"I see.\"\n\n\"I mean, I shall always try to be kind to her for your sake.\"\n\n\"Yes, well, that's very generous of you. And while we're about it, I may as well say that I once promised her there would always be room in my life for her, even if it was just an attic.\"\n\n\"Paul, I don't want an attic in _my_ life.\" And then she said it. \"Especially not with a madwoman in it.\"\n\nI let that last remark fill the silence that was growing between us. Eventually, no doubt sounding prim, I said, \"I'm sorry you think she's mad.\"\n\nShe didn't withdraw her assertion. I realised that I was the only person in the world who understood Susan. And even if I'd moved out, how could I abandon her?\n\nAnna and I continued for a few more weeks, each of us half-concealing our thoughts from the other. But I wasn't surprised when she bailed on the relationship. Nor, by then, did I blame her.\n\n* * *\n\n\u2014\n\nAnd so, by the end, you have tried soft love and tough love, feelings and reason, truth and lies, promises and threats, hope and stoicism. But you are not a machine, switching easily from one approach to another. Each strategy involves as much emotional strain on you as on her; perhaps more. Sometimes, when, lightly drunk, she is in one of her airy, exasperating moods, denying both reality and your concern for her, you find yourself thinking: she may be destroying herself in the long term, but in the short term, she's doing more damage to you. Helpless, frustrated anger overwhelms you; and, worst of all, righteous anger. You hate your own righteousness.\n\nYou remember the running-away fund she gave you when you were at university. You have never thought to make use of it before. Now, you take it all out, in cash. You go to a small, anonymous hotel towards the bottom of the Edgware Road, just up from Marble Arch. This is not a fashionable or expensive part of town. Next door is a small Lebanese restaurant. In the five days you are there, you do not drink. You want your mind to be lucid; you do not want either your anger or your self-pity to be exaggerated or distorted. You want your emotions to be whatever they are.\n\nYou remove a bunch of prostitutes' business cards from a nearby telephone box. They have been attached with Blu-Tack, and before laying them out on the small desk in your hotel room, you roll off the sticky little balls of adhesive and drop them in the wastepaper basket. You do this in a deliberate way. Then you lay the cards out like a game of patience and decide which of these glamorous women who do \"hotel calls\" you wish to fuck. You make your first phone call. The woman, naturally, looks nothing like the photo on the card. You note this, without caring, let alone protesting: on the scale of disappointment, this is nothing. The location and the transaction are the exact opposite of all you have previously imagined love and sex to be. Still, it is fine for what it is. Efficient, pleasurable, emotion-free; fine.\n\nOn the wall is a cheap print of a Van Gogh cornfield with crows. You enjoy looking at it: again, an efficient, second-rate, counterfeit pleasure. You think there is something to be said for the second-rate. Perhaps it is more reliable than the first-rate. For instance, if you were in front of the real Van Gogh, you might get nervous, be full of jacked-up expectations about whether or not you were reacting properly. Whereas no one\u2014you, least of all\u2014cares how you respond to a cheap print on a hotel wall. Perhaps that is how you should live your life. You remember, when you were a student, someone maintaining that if you lowered your expectations in life, then you would never be disappointed. You wonder if there is any truth in this.\n\nWhen desire returns, you order up another prostitute. Later, you have a Lebanese dinner. You watch television. You lie on your bed, deliberately not thinking about Susan or anything to do with her. You do not care how anyone might judge you if they could see where you are and what you are doing. Doggedly, and almost without actual pleasure, you continue to spend your running-away fund until all that remains is enough for your bus fare back to SE15. You do not reproach yourself; nor do you experience guilt, now or later. You never tell anyone about this episode. But you begin to wonder\u2014not for the first time in your life\u2014if there is something to be said for feeling less.\n\n# THREE\nHe sometimes asked himself a question about life. Which are truer, the happy memories, or the unhappy ones? He decided, eventually, that the question was unanswerable.\n\n* * *\n\n\u2014\n\nHe had kept a little notebook for decades now. In it he wrote down what people said about love. Great novelists, television sages, self-help gurus, people he met in his years of travelling. He assembled the evidence. And then, every couple of years or so, he went through and crossed out all the quotations he no longer believed to be true. Usually, this left him with only two or three temporary truths. Temporary, because the next time round, he would probably cross those out as well, leaving a different two or three now standing.\n\n* * *\n\n\u2014\n\nHe had found himself on a train to Bristol the other day. Across the aisle was a woman with the _Daily Mail_ spread out in front of her. He saw the bright headline, accompanied by a large photo. HEADMISTRESS, 49, SANK 8 GLASSES OF WINE, DROPPED CRISPS DOWN HER TOP, AND SAID TO PUPIL, \"COME AND GET 'EM.\" After such a headline, what need to read the story? And what chance of the reader finding a different moral to the one so fiercely implied? Any more than would have been the case, half a century previously, had the newspaper's hot moralism been applied to a story which, at the time, hadn't even made the local _Advertiser &_ _Gazette._ For the next ten minutes and more he worked on the headline his own case might have elicited. He finally came up with: NEW BALLS, ANYONE? TENNIS CLUB SCANDAL AS HOUSEWIFE, 48, AND LONG-HAIRED STUDENT, 19, EXPELLED OVER RUMPY-PUMPY. As for the text below, it would write itself: \"There were shock waves behind the lace curtains and laurel hedges of leafy Surrey last week as steamy allegations emerged of...\"\n\n* * *\n\n\u2014\n\nSome people, when they grow old, decide to live by the sea. They watch the tides approach and recede, foam bubbling on the beach, further out the breakers, and perhaps, beyond all this, they hear the oceanic waves of time, and in such hinted outer vastness find some consolation for their own minor lives and impending mortality. He preferred a different liquid, with its own movements and its own destination. But he saw nothing eternal in it: just milk turning into cheese. He was suspicious of the grander view of things, and wary of indefinable yearnings. He preferred the daily dealings of reality. And he also admitted that his world, and his life, had slowly shrunk. But he was content with this.\n\n* * *\n\n\u2014\n\nFor instance, he thought he probably wouldn't have sex again before he died. Probably. Possibly. Unless. But on balance, he thought not. Sex involved two people. Two persons, first person and second person: you and I, you and me. But nowadays, the raucousness of the first person within him was stilled. It was as if he viewed, and lived, his life in the third person. Which allowed him to assess it more accurately, he believed.\n\n* * *\n\n\u2014\n\nSo, that familiar question of memory. He recognised that memory was unreliable and biased, but in which direction? Towards optimism? That made initial sense. You remembered your past in cheerful terms because this validated your existence. You didn't have to see your life as any kind of triumph\u2014his own had hardly been that\u2014but you did need to tell yourself that it had been interesting, enjoyable, purposeful. Purposeful? That would be pitching it a bit high. Still, an optimistic memory might make it easier to part from life, might soften the pain of extinction.\n\nBut you could equally argue the opposite. If memory is biased towards pessimism, if, retrospectively, all appears blacker and bleaker than it actually was, then this might make life easier to leave behind. If, like dear old Joan, dead now these thirty years and more, you had already been to hell and back in your lifetime, then what fear of actual hell, or, more probably, eternal nonexistence? There drifted into his mind words caught on the headcam of a British soldier in Afghanistan\u2014words spoken by another soldier as he executed a wounded prisoner. \"There you are. Shuffle off this mortal coil, you cunt,\" the man had said before pulling the trigger. Impressive to have Shakespeare half-quoted on the modern battlefield, he had thought at the time. Why had that come into his head? Perhaps Joan's swearing had been the connection. So he considered the upside to feeling that life was just a fucking coil to be shuffled off. And men were just cunts; not women, men. There might also be an evolutionary advantage to a pessimistic memory. You wouldn't mind making room for others in the food queue; you could see it as a social duty to wander off into the wilderness, or allow yourself to be staked out on some hillside for the greater good.\n\n* * *\n\n\u2014\n\nBut that was theory; and here was practicality. As he saw it, one of the last tasks of his life was to remember her correctly. By which he didn't mean: accurately, day by day, year by year, from beginning to middle to end. The end had been terrible, and far too much middle had overhung the beginning. No, what he meant was this: it was his final duty, to both of them, to remember and hold her as she had been when they were first together. To remember her back to what he still thought of as her innocence: an innocence of soul. Before such innocence became defaced. Yes, that was the word for it: a scribbling-over with the wild graffiti of booze. Also, a losing of the face, and his subsequent inability to see her. To see, to recall what she had been like before he lost her, lost sight of her, before she disappeared into that chintz sofa\u2014\"Look, Casey Paul, I'm doing my disappearing act!\" Lost sight of the first person\u2014the only person\u2014he had loved.\n\nHe had photographs, of course, and they helped. Smiling at him while leaning back against the trunk of a tree in some long-forgotten wood. Windswept on a broad empty beach with a row of shuttered huts in the distance behind her. There was even a picture of her in that tennis dress with the green trim. Photographs were useful, but somehow always confirmed the memory rather than liberating it.\n\nHe tried to get his mind to catch her on the wing. To remember her gaiety, her laughter, her subversiveness and her love for him, before everything became occluded. Her dashingness, and her gallant attempt to make happiness when the odds were always against her, always against them. Yes, this was what he was after: Susan happy, Susan optimistic, despite not having much of a clue what the future held. That was a talent, a lucky slice of her character. He himself tended to look at the future and decide from an assessment of probabilities whether optimism or pessimism was the appropriate outlook. He brought life to his temperament; she brought her temperament to life. It was more risky, of course; it brought more joy, but it left you no safety net. Still, he thought, at least they hadn't been defeated by practicality.\n\nThere was all this; and there was also the way she accepted him simply as he was. No, better: she enjoyed him as he was. And she had confidence in him: she looked at him and didn't doubt him; she thought he would make something of himself, and something of his life. Which in a sense he had done, though not as either of them would have foreseen.\n\nShe would say, \"Let's pile all the Fancy Boys into the Austin and drive to the sea.\" Or to Chichester Cathedral, or Stonehenge, or a secondhand bookshop, or a wood with a thousand-year-old tree at the centre of it. Or to a horror flick, however much they scared the daylights out of her. Or to a funfair, where they would hurtle round the dodgems, stuff themselves with candyfloss, fail to dislodge coconuts from their holders and be whirled into the air by various devices until all the puff had gone out of them. He didn't know if he'd done all these things back then, with her; some perhaps later, some even with other people. But it was the kind of remembering he needed, and which brought her back even if she hadn't actually been there.\n\n* * *\n\n\u2014\n\nNo safety net. One image would always recur, whenever he thought of her. He was holding her out of the window by her wrists, unable to pull her in or let her drop, both their lives in agonising stasis until something happened. And what _had_ happened? Well, he had tried to organise people to pile mattresses high enough to break her fall; or, he had got the fire brigade to hold a jumping sheet; or...But they were locked together at the wrist like trapeze artists: he wasn't just holding her, she was holding him. And in the end his strength gave way, and he let her go. And though her fall was cushioned, it was still very grievous because, as she had once told him, she had heavy bones.\n\n* * *\n\n\u2014\n\nOne entry in his notebook was, of course: \"It is better to have loved and lost than never to have loved at all.\" That was there for a few years; then he crossed it out. Then he wrote it in again; then he crossed it out again. Now he had both entries side by side, one clear and true, the other crossed out and false.\n\n* * *\n\n\u2014\n\nWhen he thought back to life in the Village, he remembered it as being based on a simple system. For each ailment, there was a single remedy. TCP for a sore throat; Dettol for a cut; Disprin for a headache; Vicks for chestiness. And beyond that lay greater matters but still with unitary solutions. The cure for sex is marriage; the cure for love is marriage; the cure for infidelity is divorce; the cure for unhappiness is work; the cure for extreme unhappiness is drink; the cure for death is a frail belief in the afterlife.\n\nAs an adolescent, he had longed for more complication. And life had let him discover it. At times, he felt he had had enough of life's complications.\n\n* * *\n\n\u2014\n\nA few weeks after his row with Anna he gave up his rented room and moved back to Henry Road. Somewhere, in some novel he subsequently read, he had come across the sentence: \"He fell in love like a man committing suicide.\" It wasn't quite like that, but there was a sense in which he had no choice. He couldn't live with Susan; he couldn't establish a separate life away from her; therefore he went back to live with her. Courage or cowardice? Or mere inevitability?\n\nAt least by now he was familar with the patterned patternlessness of the life he was submitting to again. His reappearance was greeted not with happiness or relief, but with a blithe lack of surprise. Because such a return was always going to happen. Because young men must be allowed their delinquencies, but shouldn't be congratulated when they returned to a place they should never have left. He noted this discrepant reaction but didn't resent it; on the scale of things to be resented, it didn't really signify.\n\nAnd so\u2014for how long? another four, five years?\u2014they continued under the same roof, with good days and bad weeks, swallowed rage, occasional outbursts and increasing social isolation. All this no longer made him feel interesting; instead, he felt a failure and an outcast. He never got close to another woman in this time. After a year or two, Eric could no longer stand the atmosphere, and moved out. The top two rooms were rented to nurses. Well, he couldn't get policemen.\n\nBut there was one discovery made during these years which surprised him, and which made his future life, when it came, easier. The office manager announced herself pregnant; they advertised for a stand-in, but could find no one suitable; he suggested himself for the job. It scarcely occupied the whole day, and he continued handling some legal aid cases. But he found the routine of admin, diary-keeping, mail, billings\u2014even the banalities of maintaining the coffee machine and water cooler\u2014gave him quiet satisfaction. In part, no doubt, because he often arrived from Henry Road in a state unfit for much more than low-level administration. But he also took unanticipated pleasure in running things. And his colleagues were straightforwardly grateful to him for making their lives easier. The contrast with Henry Road was blatant. When had Susan last thanked him for making her life less arduous than it would have been?\n\nThe office manager, with many explanations about the thrilling surprise of maternal love, announced that she wouldn't be returning. He took the job full-time; and, years later, this practical ability proved his means of escape. He managed offices for law firms, for charities, for NGOs, and so was able to travel, and move on when he needed to. He worked in Africa, and in North and South America. The routine satisfied a part of him he didn't know existed. He remembered how, back at the Village tennis club, he'd been shocked at the way some of the older members played. They were certainly competent, but inexpressive and uninventive, as if merely following the instructions of some long-dead coach. Well, that had been them, then. Now he could run an office\u2014wherever, whenever\u2014like any grooved old hacker. He kept his satisfactions to himself. And over the years he had also learned to see the point of money: what it could\u2014and couldn't\u2014do.\n\nThere was another thing. It was a job below his qualifications. Not that he didn't take it seriously; he did. But since, professionally, he had now lowered his expectations, he found that he was rarely disappointed.\n\n* * *\n\n\u2014\n\nHe had a duty to see back to how she had been, and to rescue her. But this wasn't just about her. He had a duty to himself. To see back and...rescue himself? From what? From \"the subsequent wreckage of his life\"? No, that was stupidly melodramatic. His life had not been wrecked. His heart, yes, his heart had been cauterised. But he had found a way to live, and continued with that life, which had brought him to here. And from here, he had a duty to see himself as he had once been. Strange how, when you are young, you owe no duty to the future; but when you are old, you owe a duty to the past. To the one thing you can't change.\n\n* * *\n\n\u2014\n\nHe remembered, at school, being guided by masters through books and plays in which there was often a Conflict between Love and Duty. In those old stories, innocent but passionate love would run up against the duty owed to family, church, king, state. Some protagonists won, some lost, some did both at the same time; usually, tragedy ensued. No doubt in religious, patriarchal, hierarchical societies, such conflicts continued and still gave themes to writers. But in the Village? No church-going for his family. Not much of a hierarchical social structure, unless you counted the tennis and golf club committees, with their power to expel. Not much patriarchy, either\u2014not with his mother around. As for family duty: he had felt no obligation to placate his parents. Indeed, nowadays the onus had shifted, and it was the parents' job to accept whatever \"life choices\" their child might make. Like running off to a Greek island with Pedro the hairdresser, or bringing home that gymslip-mother-to-be.\n\nYet this liberation from the old dogmas brought its own complexities. The sense of obligation became internalised. Love was a Duty in and of itself. You had a Duty to Love, the more so now that it was your central belief system. And Love brought many Duties with it. So, even when apparently weightless, Love could weigh heavily, and bind heavily, and its Duties could cause disasters as great as in the old days.\n\n* * *\n\n\u2014\n\nAnother thing he had come to understand. He had imagined that, in the modern world, time and place were no longer relevant to stories of love. Looking back, he saw that they had played a greater part in his story than he ever realised. He had given in to the old, continuing, ineradicable delusion: that lovers somehow stand outside of time.\n\n* * *\n\n\u2014\n\nNow he was getting off the point. Susan and himself, all those years ago. There was her shame to deal with. But there was also, he knew, his shame.\n\n* * *\n\n\u2014\n\nAn entry from his notebook which had survived several inspections: \"In love, everything is both true and false; it's the one subject on which it's impossible to say anything absurd.\" He had liked this remark since first discovering it. Because for him it opened out into a wider thought: that love itself is never absurd, and neither are any of its participants. Despite all the stern orthodoxies of feeling and behaviour that a society may seek to impose, love slips past them. You sometimes saw, in the farmyard, improbable forms of attachment\u2014the goose in love with the donkey, the kitten playing safely between the paws of the chained-up mastiff. And in the human farmyard, there existed forms of attachment which were just as unlikely; and yet never, to their participants, absurd.\n\n* * *\n\n\u2014\n\nOne permanent effect of his exposure to the Macleod household had been a distaste for angry men. No, not distaste, disgust. Anger as an expression of authority, an expression of masculinity, anger as a prelude to physical violence: he hated it all. There was a hideous false virtue to anger: look at me, angry, look how I boil over because I am so filled with emotion, look how I am really alive (unlike all those cold fish over there), look how I am going to prove it by grabbing your hair and smashing your face into a door. And now look what you made me do! I'm angry about that too!\n\nIt seemed to him that anger was never just anger. Love was, usually, in itself, just love, even if it impelled some to behave in ways which made you suspect there was no love present anymore, and perhaps never had been. But anger, especially the sort which coated itself in self-righteousness (and perhaps all anger did) was so often an expression of something else: boredom, contempt, superiority, failure, hatred. Or even something apparently trivial, like a chafing dependence on female practicality.\n\n* * *\n\n\u2014\n\nEven so, and to his considerable surprise, he had finally stopped hating Macleod. True, the man was long dead\u2014though it was perfectly possible, indeed reasonable, to hate the dead; and at one stage he imagined he would live with his hatred until the day he himself died. But it hadn't worked out that way.\n\nHe wasn't sure about the chronology of it all. At some point, Macleod had retired, but continued to live on in that large house, attended by a cook-housekeeper to whom he behaved with elaborate, antiquated politeness. Once a week he would go to the golf club and hit a stationary ball as if it were a personal enemy. He would garden furiously, smoke furiously, turn on the goggle-box and drink along to it until he could still just get himself to bed. Often the thieving Mrs. Dyer would find the blank-screened set still buzzing when she arrived.\n\nThen, one winter morning, while he was planting out cabbages, Macleod had fallen to the hard ground and wasn't discovered for hours; the stroke had done its worst. Half-paralysed, but fully silenced, he now depended on regular visits from a nurse, monthly ones from his daughters, more erratic ones from Susan. Maurice, his old pal from R _eynolds News,_ would drop by from time to time, and, in knowing contravention of medical advice, would pull out a half-bottle of whisky and pour some of it down Macleod's throat while the familiar eyes blinked back at him. By the time the housekeeper found him dead on the floor with the bedsheets wrapped round him, Susan had long since handed power of attorney to Martha and Clara. The house, with many unwanted contents, was sold to a dubious local who might have been fronting for a property developer.\n\nSomewhere in this sequence, he had stopped hating Macleod. He didn't forgive him\u2014he didn't consider forgiveness the opposite of hatred\u2014but he acknowledged that his seething antipathy and nighttime rages had become somehow irrelevant. On the other hand, he didn't feel pity for Macleod, despite all the humiliations and infirmities visited upon him. These he regarded as inevitabilities; indeed, he nowadays regarded most things that happened as inevitabilities.\n\nThe question of responsibility? That seemed a matter for outsiders: only those with a sufficient lack of evidence and knowledge could confidently apportion blame. He was, even at this distance, still far too involved to do so himself. And he had also reached a stage in life where he had started pursuing counterfactuals. What if this had happened rather than that? It was idle but involving (and perhaps held the question of responsibility at bay). For instance, what if he hadn't been nineteen, with time on his hands and\u2014while hardly aware of it\u2014desperate for love when he had arrived at the tennis club? What if Susan, from religious or moral scruple, had discouraged his interest, and taught him nothing more than tactical astuteness when playing mixed doubles? What if Macleod had continued to hold a sexual interest in his wife? None of this might have happened. But given that it had, then if you wanted to attribute fault, you were straight away into prehistory, which now, in two of their three cases, had become inaccessible.\n\nThose charged first few months had reordered his present and determined his future, even up to now. But what if, for instance, he and Susan hadn't been attracted to one another? What if one of their many cover stories had been true? He was a young man who drove her because she needed new glasses. He was a friend of one or both of the daughters. He was a kind of prot\u00e9g\u00e9 of Gordon's. Now, in his state of slowly acquired calmness, he found he could easily imagine things other than they had been; the facts and feelings quite different.\n\nCurious, he pursued this untaken path. For instance, he started helping Old Man Macleod with the gardening. As well as playing tennis with Susan, he took up golf, had lessons at the club and would often partner Gordon\u2014as he'd been asked to call him\u2014round the local eighteen holes when the dew still sparkled on the fairways. There was something about his presence which relaxed Old Man Macleod: that gruffness was only a mask, and Paul was able to help him relax a bit more on the tee; he even taught him (after flipping through an American golf manual) how to love that little dimpled ball rather than hate it. He\u2014Casey Paul, as more than Susan now called him\u2014discovered that he rather liked a drink: gin with Joan, beer with Gordon, an occasional glass of sherry with Susan; though all agreed that at a certain point enough was enough and one more was too many. And then\u2014why not pursue this alternative life to, if not a logical, at least a conventional conclusion\u2014what if he and one of the Macleod daughters became (as their parents would have put it) \"sweet on one another\"? Martha or Clara? Obviously Clara, who took more character traits from Susan. But this was counterfactual, and so he chose Martha.\n\nThe immediate consequence was that the Macleods did indeed come round to have sherry with his parents\u2014an occasion he and Martha had been dreading, but which actually passed off quite well. The two couples were never going to make a harmonious bridge four, but there was nothing like fixing a date with the vicar of St. Michael's for everyone to overlook incompatibilities. And\u2014since this counterfactual had now got way out of hand\u2014he decided to decorate the wedding day with the most extravagantly beautiful weather, even unto a double rainbow. Then, on a whim, he chose to award himself the sister he never had. To stir things up a bit for his parents, he made her a lesbian. Oh, and she brought her baby along to the ceremony. The only baby in the Western world who didn't cry at an inappropriate moment during a wedding. Why not?\n\nHe shook his head to clear this strange vision that had come upon him. There were two ways of looking at life; or two extremes of viewpoint, anyway, with a continuum between them. One proposed that every human action necessarily carried with it the obliteration of every other action which might have been performed instead; life therefore consisted of a succession of small and large choices, expressions of free will, so that the individual was like the captain of some paddle steamer chugging down the mighty Mississippi of life. The other proposed that it was all inevitability, that prehistory ruled, that a human life was no more than a bump on a log which was itself being propelled down the mighty Mississippi, tugged and bullied, smacked and wheedled, by currents and eddies and hazards over which no control was possible. Paul thought it did not have to be one or the other. He thought a life\u2014his own, of course\u2014could be lived first under the dispensation of inevitability, and later under the dispensation of free will. But he also realised that retrospective reorderings of life are always likely to be self-serving.\n\nOn further thought, he decided that the unlikeliest part of his counterfactual was that Martha would ever have considered him a potential husband.\n\n* * *\n\n\u2014\n\nDid he feel regret at what he always thought of as his \"handing back\" of Susan? No: the proper word for that might be guilt; or its sharper colleague, remorse. But there was also an inevitability to it, which lent the action a different moral colouring. He found that he simply couldn't go on. He couldn't save her, and so he had to save himself. It was as simple as that.\n\nNo, of course it wasn't; it was much more complicated. He could have gone on, both fooling and torturing himself. He could have gone on, calming her down and reassuring her even when her mind and memory ran in three-minute loops, from fresh surprise at his presence, even though he'd been sitting in the same chair for two hours, via rebuke for his nonexistent absence, through to alarm and panic, which he would quieten with soft talk and gentle memories that she would pretend to agree with even though she'd long ago drunk those memories clean out of her head. No, he could have gone on, acting as an emotional home help, watching over her progressive disintegration. But he would have had to be a masochist. And by that time he had made the most terrifying discovery of his life, one which probably cast a shadow over all his subsequent relationships: the realization that love, even the most ardent and the most sincere, can, given the correct assault, curdle into a mixture of pity and anger. His love had gone, had been driven out, month by month, year by year. But what shocked him was that the emotions which replaced it were just as violent as the love which had previously stood in his heart. And so his life and his heart were just as agitated as before, except that she was no longer able to assuage his heart. And that, finally, was when he had to hand her back.\n\nHe wrote a joint letter to Martha and Clara. He didn't go into emotional detail. He merely explained that he was obliged to travel on business for an extended period\u2014perhaps several years\u2014and would obviously not be able to take Susan with him. He would be leaving in three months, which he hoped would be enough time for them to make the appropriate arrangements. If, at some future point, it became necessary to put her into some kind of residential care, he would do what he could to help; though at present he was not in a position to contribute.\n\nAnd most of this was true.\n\n* * *\n\n\u2014\n\nThere was one visit he was obliged to make before going abroad. Was he dreading it or looking forward to it? Both, probably. It was five o'clock by the time he rang the bell, answered this time not by a counterpoint of yapping but by a single, distant bark. When Joan opened the door, there was a placid, elderly golden retriever beside her. She looked so foggy-eyed that it might as well have been a guide dog, he thought.\n\nIt was winter; Joan wore a tracksuit with a few cigarette burns on the bosom, and a pair of Russian house socks in which she padded along as softly as her dog. The sitting room mixed woodsmoke with cigarette smoke. The chairs were the same, but older; their occupants were the same, but older. The retriever, which answered to the name of Sibyl, panted from the journey to the front door and back.\n\n\"The yappers all died on me,\" Joan said. \"Don't ever have dogs, Paul. They die on you, and then there comes a point when you don't know whether to get one last one or not. One for the road. So here we are, Sibyl and me. Either I'll die and break her heart or she'll die and break mine. Not much of a choice, is it? The gin's over there. Help yourself.\"\n\nHe did so, choosing the least filthy of the tumblers.\n\n\"So how are you keeping, Joan?\"\n\n\"As you can see. Pretty much the same, except older, drunker, lonelier. How about you?\"\n\n\"I'm thirty. I'm going abroad for a few years. Work. I've handed Susan back.\"\n\n\"Like a parcel? It's a bit fucking late, isn't it? Taking her back to the shop and asking for a refund?\"\n\n\"It's not like that.\" He realised he might have some difficulty explaining to one drunk woman why he was leaving another.\n\n\"So how exactly is it?\"\n\n\"It's like this. I tried to save her, I failed. I tried to stop her drinking, I failed. I don't blame her, it's way beyond that. And I remember what you told me back then\u2014that she was more likely to get hurt than me. But I can't take it anymore. I can't face another ten days of it, let alone another ten years. So Martha's going to look after her. Clara refused, which surprised me. I said that...if they needed to put her into a home at some point, I might be able to help. In the future. If I do well and make some money.\"\n\n\"You've certainly got it all worked out.\"\n\n\"It's self-protection, Joan. I couldn't take anymore.\"\n\n\"Girlfriend?\" she asked, lighting another cigarette.\n\n\"I'm not that heartless.\"\n\n\"Well, finding another woman can bring an exceptional clarity of mind to a man all of a sudden. Remembering my own distant experiences of cock and cunt.\"\n\n\"I'm sorry it didn't work out for you, Joan.\"\n\n\"Your sympathy is about half a century too late, young man.\"\n\n\"I mean it,\" he said.\n\n\"And how do you think Martha will cope? Better than you? Worse? About the same?\"\n\n\"I've no idea. And in a way I don't care. I don't care, otherwise I'll be dragged back into it all.\"\n\n\"It's not a question of getting dragged back. You're still in it.\"\n\n\"How do you mean?\"\n\n\"You're still in it. You'll always be in it. No, not literally. But in your heart. Nothing ever ends, not if it's gone that deep. You'll always be walking wounded. That's the only choice, after a while. Walking wounded, or dead. Don't you agree?\"\n\nHe looked across, but she wasn't addressing him. She was addressing Sibyl, and patting her soft head. He didn't know what to say, because he didn't know if he believed her or not.\n\n\"Do you still cheat at the crossword?\"\n\n\"You cheeky little bugger. But that's nothing new, is it?\" He smiled at her.\n\nHe'd always liked Joan.\n\n\"And shut the door on your way out. I don't like to get up too many times in the course of a day.\"\n\nHe knew not to do anything like embrace her, so merely nodded, smiled and started to leave.\n\n\"Send a wreath when the time comes,\" she called after him.\n\nHe didn't know if she meant for her, or for Susan. Maybe even for Sibyl. Did dogs get wreaths? Another thing he didn't know.\n\n* * *\n\n\u2014\n\nWhat he didn't\u2014or couldn't\u2014tell Joan was his terrifying discovery that love, by some ruthless, almost chemical process, could resolve itself into pity and anger. The anger wasn't at Susan, but at whatever it was that had obliterated her. But even so, anger. And anger in a man caused him disgust. So now, along with pity and anger, he had self-disgust to deal with as well. And this was part of his shame.\n\n* * *\n\n\u2014\n\nHe worked in a number of countries. He was in his thirties, then forties, perfectly presentable (as his mother would have put it), as well as solvent and not obviously mad. This was enough for him to find the sexual companionship, the social life, the daily warmth he needed\u2014until he moved on to the next job, the next country, the next social circle, the next few years of being agreeable to and with new people, some of whom he might see in later years, some not. It was what he wanted; more to the point, it was all he felt able to sustain.\n\nTo some, his way of life might have sounded selfish, even parasitical. But he also took thought for others. He tried not to mislead, to exaggerate what was emotionally available. He didn't linger by jewellers' windows or go simperingly silent at photos of babies; nor did he claim he was looking to settle down, either with this person or indeed in this country. And\u2014though it was a trait he didn't immediately identify\u2014he was generally attracted to women who were...how to put it? Sturdy, independent and not obviously fucked-up. Women who had their own lives, who might enjoy his solid but passing presence as much as he did theirs. Women who wouldn't get too hurt when he moved on, and who wouldn't inflict too much pain if they were the first to jump.\n\nHe thought of this psychological pattern, this emotional strategy, as being honest and considerate, as well as necessary. He neither pretended nor offered more than he could deliver. Though of course, when he laid it all out like this, he saw that some might regard it as pure egotism. He also couldn't decide if his policy of moving on\u2014from place to place, woman to woman\u2014was courageous in admitting his own limitations, or cowardly in accepting them.\n\nNor did his new theory of living always work. Some women gave him thoughtful presents\u2014and that scared him. Others, over the years, had called him a typical Englishman, a tightass, a cold fish; also, heartless and manipulative\u2014though he believed his was the least manipulative approach to relationships that he knew. Still, it made some women cross with him. And on the rare occasions when he had tried to explain his life, his prehistory and the long-term state of his heart, the accusations sometimes became more pointed, and he was treated as if he had some infectious disease to which he should have admitted between the first and second dates.\n\nBut that was the nature of relationships: there always seemed to be an imbalance of one sort or another. And it was fine to plan an emotional strategy, but another thing when the ground opened up in front of you, and your defending troops toppled into a ravine which hadn't been marked on the map until a few seconds previously. And so there had been Maria, that gentle, calm Spanish woman who suddenly began making suicide threats, who wanted this, who wanted that. But he hadn't offered to be the father of her children\u2014or anybody else's; nor did he intend to convert to Catholicism, even if that would have pleased her supposedly dying mother.\n\nAnd then\u2014since misunderstanding is democratically distributed\u2014there had been Kimberly, from Nashville, who had so instantly fulfilled all his unwritten requirements\u2014from laughing him into bed on the second date to embodying the very spirit of freehearted independence\u2014that instead of quietly congratulating himself on his luck he had as near as dammit fallen smack into love with her. And at first she had rebuffed him with references to personal space and to \"keeping things light.\" Yet this only made him the more desperate for her to move into his house that very afternoon, and he'd done stuff with flowers way beyond what he normally did, and found himself gazing at racks of diamond rings, and even dreaming of that perfect hideaway\u2014perhaps an old trapper's shack (with full modern comforts, of course) up some tree-shadowed lane. He had offered marriage, and she had replied, \"Paul, it doesn't work like that.\" When, in his delirium, she had patted his arm and said the kind of stuff he'd said to Maria, he heard himself accusing her of being selfish and manipulative and a cold fish and a typical American woman\u2014whatever he meant by that, as she was the first American woman he'd dated. So she ditched him by fax, and he got punitively drunk to the point of sudden rationality, when he fell into silly laughter, and a sense of the absurdity of all human dealings, and felt a sudden call for the monastic life, while also entertaining fantasies of Kimberly dressed as a nun and them having joyfully blasphemous sex, whereupon he booked two tickets for an early-morning flight to Mexico, but naturally overslept and the message on his answering machine when he woke was not from Kimberly but from the airline company telling him of his missed flight. Somehow he had got into work that day and gave a comic account of his misfortunes, which made his colleagues laugh, and made himself laugh, so that this lighter, distorted fiction swiftly took over from what had actually happened. And in later years he had silently thanked Kimberly for being smarter than he was\u2014emotionally smarter. He had imagined that he'd learned a lot of emotional lessons from being with Susan. But maybe they were only emotional lessons about being with her.\n\n* * *\n\n\u2014\n\nHe kept up with his men-friends when home on leave, or between jobs, over drinks or dinners which felt like sudden jerks of fast-forwarding. Some of them had turned into unremitting furrow-dwellers, and these were the ones who reminisced most sentimentally about the old days. Some were now on to second wives and stepchildren. One had turned gay, after all these years, having suddenly started noticing the napes of young men's necks. For a few, time brought no alteration. Bernard, red-faced and white-bearded, would give him a nudge, a head-toss and an overloud \"Look at the arse on _that,_ \" as a woman walked past their restaurant table. Bernard had been saying the same at twenty-five, though back then with an inaccurate American accent. Perhaps it was useful still to be reminded that some men mistook boorishness for honesty. Just as others mistook primness for virtue.\n\nThese intermittent friends were of different vintages: of the Fancy Boys, only Eric remained in his life. They were companionable for the necessary hours, and alcohol dissolved any distance between them. But in the way of things\u2014or rather, in his way of things\u2014he tended to remember mainly the phrases that either presumed or grated.\n\n\"Still in the game, eh, Paul?\"\n\n\"Footloose and fancy-free?\"\n\n\"Not found Miss Right yet? Or should I say Se\u00f1orita Rita?\"\n\n\"Do you think you'll ever settle down?\"\n\n\"A pity you haven't had kids. You'd have made a good father.\"\n\n\"Never too late. Never say die, old chum.\"\n\n\"Yes, but don't forget: sperm degrades as we knock on.\"\n\n\"Don't you long for that little cottage with a blazing log fire and grandchildren on the knee?\"\n\n\"He can't have grandchildren without having children first.\"\n\n\"You'd be amazed what medical science can do nowadays.\"\n\nHis occasional reappearances made some pleased with how their lives had turned out, and others, if not envious, a little restless. Then, in his fifties, he came home, moved to Somerset, and invested some of his savings.\n\n\"What gave you the idea of cheese?\"\n\n\"Bad dreams for the rest of your life, old chum.\"\n\n\"Maybe there's a little dairymaid involved?\"\n\n\"And look at the arse on _that._ \"\n\n\"Well, at least we'll be seeing more of you now.\"\n\nBut there was no dairymaid involved; and strangely, he didn't end up seeing more of his intermittent friends. Somerset could turn out to be as distant as Valparaiso or Tennessee, if you wanted things that way. And perhaps he chose to remember their heavy joshing because it helped keep them at bay just as he had kept his women friends at bay. Though now some were keeping themselves at bay, having reached the age when illness arrives. There were emails about prostate cancer, and back operations, and that little bit of heart trouble which maybe wasn't such good news. Vitamin pills and statins were consumed, while the World Service kept them company in their sleeplessness. And soon, no doubt, the funeral years would begin.\n\n* * *\n\n\u2014\n\nHe remembered a friend he'd had, a lifetime back, at law college. Alan something. They hadn't kept up, for one reason or another. Alan had spent seven years training to be a vet, but on qualifying had immediately switched to the law.\n\nOne day, he'd asked his friend why he'd thrown up his first career so abruptly. Had he suddenly decided he didn't like animals? Was it the prospective hours? No, said Alan, none of that. He'd always thought it would be a good, purposeful job, helping to cure sick farmstock, bringing them either to safe birth or pain-free death, working outdoors, meeting all sorts of people. And it would have been all that, he knew. But what had finally put him off was a kind of squeamishness. He explained that if you spent several hours of the day with your arm up the backside of a cow, you couldn't help breathing in the animal's noxious exhalations. And that once they were inside you, they would inevitably seek to come back out again.\n\nThat was as far as Alan had gone. But he had naturally imagined Alan in bed with a girlfriend, and all going well between them, until some catastrophic buildup of cow gas hurtles from him, and the girl jumps from the bed, rushes for her clothes and is never seen again. Or perhaps this hadn't happened, but Alan couldn't bear to think of how it might be, if he was with someone he loved.\n\nWhat had become of Alan? He had no idea. But Alan's story had stayed with him ever since. Because once you had been through certain things, their presence inside you never really disappeared. The cow gas would out, in one direction or another. Then you just had to live with the consequences until it dispersed. And yes, it had caused more than one girlfriend to run for her clothes, not just Anna. And no, at those times, he had not been much of a stoic.\n\n* * *\n\n\u2014\n\nIn his youth, hot with pride at his love for Susan, he had been competitive, as all young men are. My cock is bigger than yours; my heart is bigger than yours. Young bucks boasting of their girlfriends' attributes. Whereas his boast had been: look how much more transgressive my relationship is than yours. And then, as well: look at the strength of my feelings for her, and hers for me. Which was what counted, obviously, because the strength of feeling governed the degree of happiness, didn't it? That had seemed blindingly logical to him at the time.\n\nIt used to be said that the Bhutanese were the happiest people on earth. In Bhutan there was little materialism, but a strong sense of kinship, society and religion. Whereas he lived in the materialistic West, where there was little religion and a weaker sense of both society and family. Did this give him an advantage, or a disadvantage?\n\nMore recently, the happiest people on earth were said to be the Danes. Not because of their supposed hedonism, but because of the modesty of their expressed hopes. Instead of aiming for the stars and the moon, their ambition was only to reach the next streetlamp and, being pleased when they did so, were the happier for it. He remembered again that woman, somebody's girlfriend, who said that she had lowered her expectations because this made you less likely to be disappointed. And therefore more happy? Was this what it was like to be Danish?\n\nAs for whether strength of feeling correlated to degree of happiness, his own experience now led him to doubt it. You might as well say, the more you ate, the better your digestion; or, the faster you drove, the quicker you got there. Not if you drove into a brick wall. He remembered that time, out in his Morris Minor with Susan, when the accelerator cable had broken, or jammed, or whatever. They were certainly roaring away up that hill, until he had the wit to disengage the clutch. He'd been doing two things at the same time: panicking and thinking clearly. That's how his life had been, back then. Nowadays, he always thought clearly; but occasionally, he found he missed the panic.\n\nAnd here was another factor, whether you were Bhutanese, Danish or British. If the statistics of happiness depend on personal reporting, how can we be sure that anyone is as happy as they claim to be? What if they aren't telling the truth? No, we have to assume that they are, or at least that the testing system allows for lying. So the real question lay beneath: assuming that those canvassed by anthropologists and sociologists are reliable witnesses, then surely \"being happy\" is the same as \"reporting yourself happy\"? Whereupon any subsequent objective analysis\u2014of brain activity, for instance\u2014becomes irrelevant. To say sincerely that you are happy is to be happy. At which point, the question disappears.\n\nAnd if that was so, then perhaps the argument could be extended. For example, to say that you had once been happy, and to believe what you were saying, was the same as actually to have been happy. Could that be true? No, that was surely specious. On the other hand, the emotional record was not like a history book; its truths were constantly changing, and true even when incompatible.\n\nFor instance, he had noticed during his life one difference between the sexes in the reporting of relationships. When a couple broke up, the woman was more likely to say, \"It was all fine until _x_ happened.\" The _x_ being a change of circumstances or location, the arrival of an extra child, or, all too often, some routine\u2014or not so routine\u2014infidelity. Whereas the man was more likely to say, \"I'm afraid it was all wrong from the start.\" And he would be referring to a mutual incompatibility, or a marriage made under duress, or an unrevealed secret on one or both sides, which had later emerged. So she was saying, \"We were happy until,\" while he was saying, \"We were never really happy.\" And when he had first noticed this discrepancy, he had tried to work out which of them was more likely to be telling the truth; but now, at the other end of his life, he accepted that both were doing so. \"In love, everything is both true and false; it's the one subject on which it's impossible to say anything absurd.\"\n\n* * *\n\n\u2014\n\nWhen he bought a half-share in the Frogworth Valley Artisanal Cheese Company, he had imagined himself as a kind of owner-manager. Co-owner\u2013co-manager. He had a desk and a chair and a rather decrepit computer terminal; he also had his own white coat, though was rarely required to put it on. Hillary ran the office. He had imagined himself running Hillary; but she didn't need running. He offered to help out and muck in; though mainly he watched things happen around him, and smiled. When Hillary went on holiday, he was allowed to take over her desk.\n\nWhere he proved most useful to the company (which only consisted of five people) was stall-holding at farmers' markets. It wasn't easy to find someone regular, and Barry, who'd done it for years, was growing unreliable. He was happy to stand in when required. Driving to one of the nearby towns, setting up the stall, laying out the cheeses, their captions, the tasting plates, the plastic cup of toothpicks. He wore a tweed cap and a leather apron, but knew he hardly passed for Somerset born-and-bred. Behind him was a plastic backdrop bearing a colour photograph of happy goats. The other stall-holders were friendly; he would swap two fivers for a tenner, two tenners for a twenty. He explained to customers the age of the cheeses and their characteristics: this one rolled in ash, this in chives, this in crushed chillis. He enjoyed doing all this. It gave him the level of social interaction he required nowadays: cheerful, mutually sustaining, with no question of intimacy\u2014even if he did sometimes flirt lightly with Betty of Betty's Best Home-Made Pies. It passed the time. Ah, that phrase. A sudden memory of Susan talking about Joan. \"We're all just looking for a place of safety. And if you don't find one, then you have to learn how to pass the time.\" Back then, it had sounded like a counsel of despair; now, it struck him as normal, and emotionally practical.\n\n* * *\n\n\u2014\n\nDespite having no expectation of, nor desire for, some final relationship\u2014or perhaps because of this\u2014he often found himself drawn to all those public displays of wantingness. The personal ads, the \"soulmates\" columns, the TV dating shows, and those newspaper features where couples go for a meal, mark one another out of ten, report on or confess to inept chopstick behaviour, and then answer (or not) the question of whether they had kissed. \"A quick hug\" or \"Only on the cheek\" were frequent responses. Some blokes would answer smugly, \"A gentleman never tells.\" It was meant to sound sophisticated, but showed far too much class deference: \"gentlemen,\" in his experience, were as boastful as any other males. Still, he followed all these brave, tentative forays of the heart with a mixture of tenderness and scepticism. He hoped it might work out for them, even as he doubted that it would.\n\n* * *\n\n\u2014\n\n\"A gentleman never tells.\" Well, it might occasionally be true. For instance, Uncle Humphrey, stinking of booze and cigars, coming into Susan's bedroom to demonstrate \"a party kiss,\" and then demanding one (or more) on an annual basis. He doubted Uncle Humph had \"told.\" But this hardly made him a \"gentleman\"\u2014quite the opposite. Uncle Humph, whose lechery had resulted in Susan not believing in the afterlife. Had his behaviour affected her in other ways? Impossible to tell, at this distance. And so he dismissed that long-dead uncle from his mind.\n\n* * *\n\n\u2014\n\nHe preferred to remember Joan. He wished he'd known her as a bounding tennis champion, then as a girl who went off the rails, then as a kept woman. Was the man who kept her, and then dismissed her, a \"gentleman\"? Susan had withheld his name, and there was no finding it now.\n\nHe smiled at the thought of Joan. He remembered the yappers, and Sibyl, the elderly golden retriever. Which of them had died first, Joan or Sibyl? She'd asked him to send flowers. Though for whom was never made clear. Whenever he'd been tempted to get a dog, he heard Joan's warning voice about them dying on you. So he never got a dog. Nor was he ever tempted to do crosswords or drink gin.\n\n* * *\n\n\u2014\n\n\"Little man, you've had a busy day.\"\n\nThis is the greeting she often sings at you, when you visit her on home leaves.\n\nExcept when it is:\n\n> \"Clap hands, here comes Charlie,\n> \n> Clap hands, good time Charlie,\n> \n> Clap hands, here comes Charlie now.\"\n\nMartha, to your continuing surprise, never objects to your visits, and never asks you for money. She looks after her mother herself, with an occasional nurse in attendance. You get the impression that Martha's husband is doing well in...whatever he does. She told you once, which means you can no longer ask.\n\nSusan's mind has slipped a little more each time you see her. Short-term memory disappeared a while ago, and long-term memory is a shifting, blurry palimpsest from which clear but unconnected phrases will occasionally be picked out by her fading brain. What often rises to the surface are songs and catchphrases from decades previously.\n\n> \"High o'er the fence leaps Sunny Jim,\n> \n> Force is the food that raises him.\"\n\nSome advertising jingle for a breakfast cereal\u2014from her own childhood? from that of her children? In your house, you had Weetabix.\n\nShe has long ago ceased to drink; indeed, she has forgotten that she was ever a drinker. She seems to know that you are, or were, something in her life, but not that she once loved you, and you loved her in return. Her brain is ragged, but her mood is strangely stable. The panic and pandemonium have drained out of her. She is alarmed by neither your arrival nor your departure. Her manner is satirical at times, disapproving at others, but always a little superior, as if you aren't a person of much consequence. You find all this agonising, and try to resist the temptation to believe that you deserve what you are getting.\n\n\"He's a dirty stop-out, that one,\" she will confide to the nurse in a stage whisper. \"I could tell you things about him that would make your hair stand on end.\"\n\nThe nurse looks at you, so you shrug and smile, as if to say, \"What can you do, it's so sad, isn't it?\" while realizing that even now you are betraying her, even in this new and last extremity of hers. Because she could, of course, tell the nurse a thing or two about you, and the nurse's hair might well stand on end.\n\nYou remember her saying that she wasn't afraid of death, and that her only regret would be over not knowing what happened afterwards. But now she has very little past and\u2014literally\u2014no thought for the future. She has only a ghostplay on some frayed screen of memory, which she takes to be the present.\n\n\" _You're_ a played-out generation.\"\n\n\"Got to eat a peck of dirt before you die.\"\n\n\"Clap hands, here comes...Sunny Jim.\"\n\n\"One of the worst criminals in the world.\"\n\n\"Where've you been all my life?\"\n\nAt least, you think, there is something of her still left among these shreds and patches.\n\n> \"Oh dear, what can the matter be?\n> \n> Three old ladies got locked in the la-va-tree,\n> \n> They were there from Monday to Saturday,\n> \n> Reading the R _adio Times._ \"\n\nYes, you remember teaching her that one. So at least she hasn't turned into an entirely different person. You've heard about that happening: pillars of the church screaming obscenities, sweet old ladies turning into Nazis, and so on. But this is faint comfort. Perhaps, if she became unrecognizable and slipped completely out of character, it would all be less painful to deal with.\n\nOnce\u2014and naturally in front of the nurse\u2014she dredges up a football song which can only have come from you:\n\n> \"If I had the wings of a sparrow,\n> \n> If I had the arse of a crow,\n> \n> I'd fly over Tottenham tomorrow,\n> \n> And shit on the bastards below.\"\n\nBut the nurse has, of course, heard far worse in her years of caring for the elderly and demented, so she merely cocks an eyebrow at you and asks,\n\n\"Chelsea supporter?\"\n\nWhat makes it unbearable, what makes you so exhausted and depressed after twenty minutes in her presence that you want to run outside and howl, is this: though she can't name you, never asks you any questions or answers any of yours, she still, at one level, registers your presence and responds to it. She doesn't know who the fuck you are, or what you do, or even your fucking name, but at the same time, she recognises you, and judges you morally and finds you wanting. It is this which urges you to run out of the house and howl; and this which makes you realise that, perhaps at some similar unconscious level, in some remote part of your brain, you still love her. And because this awareness is unwelcome, it makes you want to howl the more.\n\n* * *\n\n\u2014\n\nAnd while he was tormenting himself, here was a question he would often arrive at when his mind followed a particular trail of memory. Handing back Susan had been an act of self-protection on his part. There was no doubt about that; and no doubt in his mind that he had to do it. But beyond this, was it an act of courage, or of cowardice?\n\nAnd if he couldn't decide, perhaps the answer was: both.\n\n* * *\n\n\u2014\n\nBut she had marked his life in so many ways, some for the better, some the worse. She had made him more generous and open to others though also more suspicious and enclosed. She had taught him the virtue of impulsiveness; but also its dangers. So he had ended up with a cautious generosity and a careful impulsiveness. His pattern of life for twenty years and more had been a demonstration of how to be impulsive and careful at the same time. And his generosity to others also came, like a pack of bacon, with a \"use by\" date on it.\n\nHe always remembered what she had said to him after they left Joan's house that day. Like most young men, especially those first in love, he had viewed life\u2014and love\u2014in terms of winners and losers. He, obviously, was a winner; Joan, he assumed, had been a loser, or, more likely, not even a competitor. Susan had put him right. Susan had pointed out that everyone has their love story. Even if it was a fiasco, even if it fizzled out, never got going, had all been in the mind to begin with: that didn't make it any the less real. And it was the only story.\n\nAt the time, he had been sobered by her words, and Joan's story had made him think of her quite differently. Then, over the years, as his life developed, as caution and carefulness began to predominate, he realised that he, no less than Joan, had had his love story, and perhaps there wasn't another one to come. So now he better understood how couples clung to their own story\u2014each, often, to a separate part of it\u2014long after it had gone cold on them, even to the point where they were not sure they could bear one another. Bad love still contained the remnant, the memory, of good love\u2014somewhere, deep down, where neither of them any longer wanted to dig.\n\nHe found himself often wondering about other people's love stories; and sometimes, because he was a calm and unintimidating presence, they would confide in him. Mostly, it was women who did so, but that was unsurprising; men\u2014himself a prime example\u2014were both more covert and less eloquent. And even when he guessed that the love stories of the misled and the forsaken had become a little less authentic with each retelling\u2014that such tales were the equivalent of Winston Churchill in an Aylesbury backstreet, all rouged and made up for the Path\u00e9 News camera\u2014even if this was the case, he was still moved. Indeed, he was more moved by the lives of the bereft and the unchosen than he was by stories of success in love.\n\n* * *\n\n\u2014\n\nOn the one hand, there were the furrow-dwellers, tunnelling deeper into the earth, and who, understandably, were not communicative about their inner selves. And at the other extreme were those who would tell you their entire lives, their only story, either in a series of outpourings, or in a single episode. Where had he been that time? He could see the beachfront bar with its silly cocktails, feel the warm night breeze, hear the thudding backbeat from tinny loudspeakers. He was at ease with the world, watching other people's lives develop. No, that was too grand a way of putting it: he was observing the young get cheerfully drunk and turn their minds to sex, romance and something more. But though he was indulgent\u2014even sentimental\u2014about the young, and protective of their hopes, there was one scene he was superstitious about, and preferred not to witness: the moment when they flung away their lives because it just felt so right\u2014when, for instance, a smiling waiter delivered a mound of mango sorbet with an engagement ring glittering in its domed apex, and a bright-eyed proposer fell to bended knee in the sand...The fear of such a scene would often lead him to an early night.\n\nSo he was sitting at the bar, halfway through his third and theoretically final cigarette of the evening, when a man in beach shorts and flip-flops climbed onto the stool beside him.\n\n\"Mind if I bum one?\"\n\n\"Be my guest.\" He passed across the pack, then some hotel book-matches with a palm tree on the cover.\n\n\"Smokers, we're a dying breed, right?\"\n\nThe fellow was probably in his forties, as lightly drunk as he was, English, genial, unpushy. None of that fake bonhomie you sometimes encountered, the assumption that you must have more in common than you did. And so they sat there quietly, smoking away, and maybe the lack of false small talk encouraged the man to turn and announce in a level, meditative tone,\n\n\"She said she wanted to rest on my shoulder as lightly as a bird. I thought that sounded poetic. Also, bloody brilliant, just what a fellow needs. Never went for clingy.\"\n\nThe man paused. Paul was always happy to supply a prompt.\n\n\"But it didn't work out?\"\n\n\"Two problems.\" The fellow inhaled, then blew the smoke into the scented air. \"Number one, birds fly away, don't they? That's in their nature, as a bird, isn't it? And number two, before they do, they always shit on your shoulder.\"\n\nAnd with that he stubbed out his cigarette, nodded and walked off down the beach towards the gentle tide.\n\n* * *\n\n\u2014\n\nIt came into his head, in one of those whimsical, sentimental moods he always sought to guard against, to try and make one of Susan's famous upside-down cakes. Over the years, he had become a competent baker, and so imagined that he could work out what had gone wrong. Too much fruit, too little baking powder, too much flour\u2014that was his best guess.\n\nThe mixture certainly looked surly and unpromising in the tin. But when he opened the oven door, it had surprisingly risen to its correct height, the fruit looked evenly distributed, and it smelt like...cake. He let it cool, then cut himself a small slice. It tasted fine. Eating it failed to set off any specific memories, for which he was grateful. He was also grateful that he wasn't able to repeat someone else's mistakes, only his own.\n\nHe cut himself another slice and then, suddenly suspicious of his own motives, threw the rest in the bin. He turned on Wimbledon and watched as two tall, baseball-capped men hit aces past one another for game after game. He chewed his cake and wondered idly what might happen if he went back to the Village and presented himself at the tennis club. Applied for membership. Asked to play in, even at his advanced age. The bad boy returned: the Village's own John McEnroe. No, that was another sentimentality. Doubtless there would be no one left who remembered him. Or, more likely, all he would find would be a neat little housing estate. No, he would never go back. He was deeply incurious about whether his parents' house, or the Macleods', or Joan's, were still standing. Those places would hold no emotions for him at this distance. That's what he told himself, anyway.\n\nTowards the end of Wimbledon fortnight, the broadcasters showed more doubles matches: men's, women's, mixed. Naturally, he was most interested in the mixed. \"The most vulnerable spot is always down the middle, Casey Paul.\" Not anymore: the players were so fit, so quick and solid on the volley, and their rackets had sweet spots the size of their heads. Another change was the lack of chivalry, certainly at this level. As he remembered it, back in the day, male players would hit as hard as possible against the opposing man, but when rallying with a woman would hold back on the power, and rely more on a change of angle or depth; maybe throw in a slice or a drop shot. It was a bit more than chivalry, in fact: it was simply boring to watch a man outhitting and overpowering a woman.\n\nHe hadn't played tennis for years; decades indeed. When he lived in the States, a temporary friend had introduced him to golf. At first this felt an ironic surprise; but it was absurd to hold a prejudice against a game just because Gordon Macleod had once played it. He came to know the joy of a perfect contact between club and ball, the shame of a shank; and to appreciate the strategic intricacies of tee to green. Nevertheless, as he took aim down a fairway, his head properly filled with the coach's advice about taking the club back, use of the hips and legs, and the importance of the follow-through, he did occasionally hear, as if in a whisper, the sweet, laughing opinion of Susan Macleod that it was plain unsporting to hit a stationary ball.\n\n* * *\n\n\u2014\n\nGordon Macleod: whom he had once wanted to kill, even if Joan had told him there hadn't been a local murder since the Villagers wore woad. An exemplar of the kind of Englishman he most loathed. Patronising, patriarchal, manneredly precise. Not to mention violent and controlling. He remembered how it had seemed to him that Macleod was somehow preventing him from growing up: not by doing anything, simply by existing. \"And how many Fancy Boys are you providing yourself with this weekend?\" Bravely, Susan had responded, \"I think it's just Ian and Eric this weekend. Unless the others turn up as well.\" Gordon Macleod's words had been like fire; he'd laughed at them, as Susan had done, but they had scorched his skin.\n\nAnd then there was that other occasion when words were spoken which had echoed down his life. That furious, squat man in his dressing gown, his eyes invisible in the gloom, bullying down on him as he gripped the banister in panic.\n\n\"Whatski? Whatski, my fine and feathered friend?\"\n\nAt the time, he had blushed, feeling his skin burn. But beyond this, he thought the fellow must simply be mad. That's to say, mad enough to have somehow listened in to his and Susan's private conversations. Unless he'd hidden a tape recorder beneath his wife's bed. And the thought of that had made him blush all over again.\n\nIt had taken him years to realise that this had not been crazy malevolence, but something quite unintended, which nevertheless held a powerful and destructive resonance. Gordon Macleod, roused from his bed by the sound of his wife's lover, had merely, in that moment, and probably with no ulterior motive, fallen back on the private language he had shared with Susan. Shared? More than that\u2014created. And which Susan had then brought into her relationship with him. Unthinkingly. You say \"darling,\" you say \"my love,\" you say \"kiss me hardly,\" you say \"whatski?,\" you say \"my fine and feathered friend,\" because those are the words which come to you at that moment. With no ulterior motive on her part either. And now he wondered if any of her turns of phrase, which had so beguiled him, had been her own. Perhaps only \"We're a played-out generation,\" because it seemed unlikely that Gordon Macleod, in all his self-importance, believed that he and men of his age were played out.\n\nHe remembered a public service advertisement from the time when the government, grudgingly, had acknowledged the existence of AIDS. There were two versions of the ad, he seemed to remember: a photo of a woman in bed with about half a dozen men, and one of a man in bed with about half a dozen women, all side by side like sardines. The text pointed out that every time you went to bed with someone new, you also went to bed with everyone he or she had previously gone to bed with. The government had been talking about sexually transmitted disease. But it was the same with words: they too could be sexually transmitted.\n\nAnd actions as well, for that matter. Except that\u2014strangely, fortunately\u2014actions had never caused a problem. He had never found himself thinking, Oh, when you did that with your hand or arm or leg or tongue, you must also have done it with _x_ and _y_ and _z._ Such thoughts and images had never bothered him, and he was grateful, because he could easily imagine how ghostly antecedents in your head could drive you mad. But ever since Gordon Macleod's sneer had first made sense to him, he had become conscious\u2014at times, absurdly so\u2014of what must have been going on, verbally, since the day Adam or Eve or whoever it might have been first took another lover.\n\nOnce, he had mentioned this discovery to a girlfriend: lightly, almost frivolously, as if it were natural and inevitable and therefore _interesting._ A day or two later, in bed, she had teasingly called him \"my fine and feathered friend.\"\n\n\"No!\" he had shouted, instantly retreating to his side of the mattress, \"You're not allowed to call me that!\"\n\nShe had been shocked by his vehemence. And he had shocked himself. But he was protecting a phrase which had always been uniquely between Susan and himself. Except that, earlier, it had been a phrase uniquely between the newly married Mr. Gordon Macleod and his hopeful, puzzled wife.\n\nSo, for a while\u2014say, twenty years or more\u2014he had found himself morbidly sensitive to lovers' language. This was ridiculous, of course. He saw rationally that there was only a limited vocabulary available, and it shouldn't matter when the same words were recycled, when nightly, across the globe, billions asserted the uniqueness of their love with secondhand phrases. Except that sometimes it did. Which meant that here, as elsewhere, prehistory ruled.\n\n* * *\n\n\u2014\n\nHe imagined the Village tennis courts replaced by a spread of the finest modern boxes, or perhaps a more lucrative clump of low-rise flats. He wondered if anyone, anywhere, had ever looked at a housing development and thought: Why don't we knock them all down and build a nice tennis club, one with the latest all-weather courts? Or maybe\u2014yes, why don't we go further and lay some proper old-fashioned grass courts, for tennis as it once used to be? But no one would ever do, or even think, that, would they? Things, once gone, can't be put back; he knew that now. A punch, once delivered, can't be withdrawn. Words, once spoken, can't be unsaid. We may go on as if nothing has been lost, nothing done, nothing said; we may claim to forget it all; but our innermost core doesn't forget, because we have been changed forever.\n\n* * *\n\n\u2014\n\nHere was a paradox. When he had been with Susan, they had scarcely discussed their love, analysed it, sought to understand its shape, its colour, its weight and its boundaries. It was simply there, an inevitable fact, an unshakeable given. But it was also the case that neither of them had the words, the experience, the mental equipment to discuss it. Later, in his thirties and forties, he had gradually acquired emotional lucidity. But in these later relationships of his, he had felt less deeply, and there was less to discuss, so his potential articulacy was rarely required.\n\n* * *\n\n\u2014\n\nHe had read, some years before, that a common psychological trope in men's attitude to women was the \"rescue fantasy.\" Perhaps it stirred in them memories of fairy tales in which valiant knights came across pretty maidens locked in towers by wicked guardians. Or those classical myths in which other maidens\u2014usually naked\u2014were chained to rocks for the sole purpose of being rescued by dauntless warriors. Who usually discovered a convenient sea serpent or dragon which had to be eliminated first. In modern, less mythical times, it appeared that the woman about whom men most had rescue fantasies was Marilyn Monroe. He had viewed this sociological datum with a degree of scepticism. Odd how rescuing her seemed inevitably to involve sleeping with her. Some rescue that would prove. Whereas in fact, as it seemed to him, the most effective way to rescue Marilyn Monroe would have been _not_ to sleep with her.\n\nHe didn't think that, as a nineteen-year-old, he had been suffering from a rescue fantasy with Susan. On the contrary, he suffered from a rescue reality. And unlike maidens in towers or chained to rocks, who attracted a whole swirl of knights looking for chivalric action, and unlike Marilyn Monroe, whom every Western man dreamed of liberating (if only to lock her up in a tower of his own making), in the case of Susan Macleod, there was not a great queue of knights, cinemagoers and Fancy Boys squabbling for the right to rescue her from her husband. He had believed he could save her; further, that _only_ he could save her. That was no fantasy; it was practicality and brute necessity.\n\n* * *\n\n\u2014\n\nAt this distance, he realised, he no longer had a memory of Susan's body. Of course, he remembered her face, and her eyes and her mouth and her elegant ears, and what she looked like in her tennis dress; there were photographs to confirm all this. But a sexual memory of her body: that had gone. He couldn't remember her breasts, their shape, their fall, their firmness or otherwise. He couldn't remember her legs, what form they took, and how she parted them and what she did with them when they made love. He couldn't remember her undressing. It was as if she'd undressed as women did on the beach, with lots of prim ingenuity beneath a capacious towel, but emerging in a nightdress rather than a swimming costume. Had they always made love with the lights out? He couldn't remember. Perhaps he'd closed his eyes a lot.\n\nShe had a corset, that he remembered; well, doubtless several. Which had\u2014whatever they were called\u2014straps for holding up her stockings. Suspenders, that was it. How many per leg? Two, three? But he knew she only ever attached the front one. This private eccentricity came back to him now. As for what her bras had been like...At nineteen, he didn't have the slightest underwear fetishism, any more than she took an erotic interest in his vests and pants. He couldn't even remember what his pants had looked like at that age. He'd had a period of wearing string vests, which for some reason he had imagined to be cool.\n\nShe had no coquetry about her, that was certain. No flirty bits of flesh showing. How did they kiss? He couldn't even remember that: Whereas, with later, lesser attachments, there were astonishing moments of sexual freeze-frame still in his head. Maybe, as you got better at sex, the sex became more memorable. Or maybe, the deeper your feelings, the less the particulars of sex mattered. No, neither of these were true. He was just trying to find a theory to explain an oddity.\n\nHe remembered when she had told him, just like that, how many times they had made love. A hundred and fifty-three, or some such number. Back then, it had thrown him into all sorts of pondering. Should he have been counting too? Was it a lapse in love that he wasn't, or hadn't? And so on. Now, he thought: a hundred and fifty-three, the number of times he had come up to that point. But what about her? How many orgasms had she had? Indeed, did she ever have one? There was pleasure and intimacy, surely; but orgasm? At the time he couldn't tell, nor did he ask; nor know how to ask. To put it more truthfully, he had never thought of asking. And now it was too late.\n\nHe tried to imagine why she might have decided to count. To begin with, as a matter of pride and attentiveness, in bed with only the second lover of her life, and that after a long drought. But then he remembered the anguished whisper of her question, \"Don't give up on me just yet, Casey Paul?\" So maybe counting had turned from a matter of acclaim to one of anxiety and dismay: the fear that he might leave her, the fear that she might never have another lover. Was that it? He gave up. He stopped examining the past, chasing down what Joan had memorably called \"my own distant experiences of cock and cunt.\"\n\n* * *\n\n\u2014\n\nOne evening, glass in hand, he was idly following the televised highlights of the Brazilian Grand Prix. He wasn't much interested in the bland plutocracy of Formula One; but he did like to watch young men taking risks. In that respect, the race was gratifying. Heavy rain had made the track dangerous; pools of standing water sent even former world champions aquaplaning smack into the barriers; the race was stopped twice, and frequently brought under the control of the safety car. Everyone drove cautiously, except for nineteen-year-old Max Verstappen of the Red Bull team. He overtook his way from almost last place to third, making moves his elders and supposed betters declined to dare. The commentators, astonished by this display of skill and guts, sought explanation. And one of them provided it: \"They say your risk profile doesn't stabilise until you're about twenty-five.\"\n\nThis statement made him attend even more closely. Yes, he thought: a treacherous circuit, visibility reduced by spray to almost zero, others trepidatious while you felt invulnerable, going flat out thanks to a risk profile as yet unstabilised. Yes, he remembered that all too well. It was called being nineteen. And some would crash and some wouldn't. Verstappen hadn't. So far, anyway: though he had another six years to go before neurophysiology rendered him entirely sensible.\n\n* * *\n\n\u2014\n\nBut if Verstappen was showing youthful fearlessness rather than true courage, did the same age disclaimer apply in reverse: to cowardice? He'd certainly been under twenty-five when he committed an act of cowardice which had haunted him all his life. He and Eric were staying at the Macleods', and had gone off to a funfair in a hilly park. They were walking down from the top, side by side, chatting, and failed to notice a group of youths coming up towards them. As they drew level, one leaned into Eric with his shoulder, spinning him round; another tripped him, and a third went in with his boot. He took all this in with a kind of heightened peripheral vision\u2014how long before Eric was on the ground? a second? two?\u2014and had instantly, instinctively run away. He kept saying to himself, \"Find a policeman, find a policeman,\" but even as he did so, he knew that wasn't the reason he was running. He was afraid of getting beaten up himself. The rational part of him knew that policemen were a rare sight at funfairs. So he ran to the bottom of the hill on this futile, pretend quest, without actually asking anyone where he might find help. Then he walked back up, nauseous at what he might find. Eric was on his feet, blood on his face, feeling his ribs. He could no longer remember what had been said\u2014whether he offered his fake excuse\u2014and they drove back to the Macleod house. Susan bandaged Eric, with liberal use of Dettol, and insisted he stay until the bruising had gone down and the cuts mended. Which Eric had done. Neither then nor later had Eric rebuked him for cowardice, or asked why he'd disappeared.\n\nYou could get through a life, if you were careful, and lucky, without having your courage much tested\u2014or rather, your cowardice revealed. That time Macleod had attacked him in the book room he'd certainly run for it, after throwing one ineffective punch in reply to Macleod's three. And when he'd scuttled out of the back door, he hadn't been trying to find a policeman, either. He had made the probably correct calculation that Macleod was drunk enough and angry enough to go on attempting to hit him until he succeeded. Despite being younger, and reasonably fit, he hadn't fancied his chances at close quarters. It wasn't like facing an under-12, under-6-stone schoolboy of equal timorousness.\n\n* * *\n\n\u2014\n\nAnd then again, more recently. \"Recently\" in the sense of \"fifteen or twenty years ago.\" That was how the mind, and time, worked nowadays. He'd been back in England for only a few years. He'd visited Susan a couple of times, bringing no visible pleasure or benefit to either of them. One evening, the phone rang. It was Martha Macleod, now\u2014for a long time\u2014Mrs. Something-or-Other.\n\n\"My mother has been temporarily sectioned\" was her opening line.\n\n\"I'm very sorry to hear that.\"\n\n\"She's in...\" and cited the mental health department of a local hospital. He knew its reputation. A doctor friend, with professional dryness, had once told him, \"You have to be _really_ mad to get in there.\"\n\n\"Yes.\"\n\n\"It's a terrible place. It's like Bedlam. Lots of people screaming. Either that or they're zombified with tranquillizers.\"\n\n\"Yes.\" He didn't ask which category Susan was in.\n\n\"Would you go and see her? And see the place?\"\n\nHe thought: this is the first time in a quarter of a century that Martha has asked anything of me. Disapproval at first; quiet superiority thereafter; though she had always been polite to him. She must be at the end of her tether, he thought. Well, he had been there in his time too; and knew how elastic the end of a tether could be. So he considered her request.\n\n\"Maybe.\" He was going up to town in a couple of days, as it happened. But he wasn't going to tell her that.\n\n\"I think it would do her good to see you. In the place she's in.\"\n\n\"Yes.\"\n\nHe'd left it like that. After he put the phone down, he thought: I looked after her for years. I tried my best. I failed. I handed her over to you. So it's your turn now.\n\nBut he didn't believe his own bitter logic. It was like saying, \"Find a policeman, find a policeman.\" The truth was, he couldn't face it: he couldn't face seeing her, the remnant of her, whether screaming or zombified, among the screaming and zombified. He tried to think of his decision as an act of necessary self-protection; also, protection of the picture he had of her in his head. But he knew the truth. He was scared of going there.\n\n* * *\n\n\u2014\n\nAs he grew older, his life turned into an agreeable routine, with enough human contact to sustain and divert, but not disturb, him. He knew the contentment of feeling less. His emotional life was recast as a social life. He was on nodding and smiling terms with many, as he stood in his leather apron and tweed cap in front of a colour photograph of happy goats. He prized stoicism and calm, which he had achieved less through some exercise of philosophy, more from a slow growth within him; a growth like coral, which in most weathers was strong enough to keep out the ocean breakers. Except when it wasn't.\n\nSo his life consisted mainly of observation and memory. It was not a bad mix. He viewed with distaste those men in their sixties and seventies who carried on behaving as if they were in their thirties: a whirl of younger women, exotic travel and dangerous sports. Fat tycoons on yachts with hairy arms round thin models. Not to mention respectable husbands who, in a turmoil of existential anguish and Viagra, left their wives of several decades. There was a German expression for this fear, one of those concertina words the language specialised in, which translated as \"the panic at the shutting of the doors.\" He himself felt untroubled by that shutting; though he saw no reason to hurry it up.\n\nHe knew what they said of him locally: Oh, he likes to keep himself to himself. The phrase was descriptive, not judgemental. It was a principle of life the English still respected. And it wasn't just about privacy, about an Englishman's home\u2014even a pebbledash semi\u2014being his castle. It was about something more: about the self, and where you kept it, and who, if anyone, was allowed to fully see it.\n\nHe knew that no one can truly hold their life in balance, not even when in calm contemplation of it. He knew there was always a pull, sometimes amounting to an oscillation, between complacency on one side and regret on the other. He tried to favour regret, as being the less damaging.\n\nBut he certainly never regretted his love for Susan. What he did regret was that he had been too young, too ignorant, too absolutist, too confident of what he imagined love's nature and workings to be. Would it have been better\u2014in the sense of less catastrophic\u2014for him, for her, for them both, if they had indeed had some \"French\" relationship? The older woman teaching the younger man the arts of love, and then, concealing an elegant tear, passing him out into the world\u2014the world of younger, more marriageable women? Perhaps. But neither he nor Susan had been sophisticated enough for that. He had never known sophistication of the emotional life: anyway, to him it sounded like a contradiction in terms. So he didn't regret that either.\n\nHe remembered his own early attempts to define love, back in the Village, alone in his bed. Love, he had ventured, was like the vast and sudden uncreasing of a lifelong frown. Hmm: love as the end of a migraine. No, worse: love as Botox. His other comparison: love feeling as if the lungs of the soul had suddenly been inflated with pure oxygen. Love as barely legal drug use? Did he have any idea what he'd been talking about? Some years later, as it happened, he'd been with a group of friends when an excited junior doctor joined them, having just \"liberated\" a cylinder of nitrous oxide from the hospital where he worked. They were each given a balloon, which they inflated from the cylinder then held tightly by the neck. Emptying their lungs as much as they could, they put the balloon to their lips, and released into themselves the roar and lift of a sudden, rushing, eye-blinking high. But no, it hadn't reminded him at all of love.\n\n* * *\n\n\u2014\n\nStill, were the professionals any better? He took his little notebook from the desk drawer. He hadn't written anything new in it for a long time. At one point, frustrated by how few good definitions of love he could find, he started copying down at the back all the bad definitions. Love is this, love is that, love means this, love means that. Even quite well-known formulations said little more than, in effect: it's a soft toy, it's a puppy dog, it's a whoopee cushion. Love means never having to say you're sorry (on the contrary, it frequently means doing just precisely that). Then there were all those love lines from all those love songs, with the swooning delusions of lyricist, singer, band. Even the bittersweet ones and the cynical ones\u2014always true to you, darling, in my fashion\u2014struck him as the mere counterfactuals of sentimentality. Yes, it was this bad for us, buddy, but it needn't be this bad for you: such was the song's implicit promise. So you can listen with sympathetic complacency.\n\nHere was an entry\u2014a serious one\u2014which he hadn't crossed out in years. He couldn't remember where it came from. He never recorded the writer or the source: he didn't want to be bullied by reputation; truth should stand by itself, clear and unsupported. This one went: \"In my opinion, every love, happy or unhappy, is a real disaster once you give yourself over to it entirely.\" Yes, that deserved to stay. He liked the proper inclusivity of \"happy or unhappy.\" But the key was: \"Once you give yourself over to it entirely.\" Despite appearances, this wasn't pessimistic, nor was it bittersweet. This was a truth about love spoken by someone in the full vortex of it, and which seemed to enclose all of life's sadness. He remembered again the friend who, long ago, had told him that the secret of marriage was \"to dip in and out of it.\" Yes, he could see that this might keep you safe. But safety had nothing to do with love.\n\nThe sadness of life. That was another conundrum he would occasionally ponder. Which was the correct\u2014or the more correct\u2014formulation: \"Life is beautiful but sad\" or \"Life is sad but beautiful\"? One or the other was obviously true; but he could never decide which.\n\nYes, love had been a complete disaster for him. And for Susan. And for Joan. And\u2014back before his time\u2014it might well have been so for Macleod as well.\n\nHe skimmed through a few crossed-out entries, then slid the notebook back in the drawer. Perhaps he had always been wasting his time. Perhaps love could never be captured in a definition; it could only ever be captured in a story.\n\n* * *\n\n\u2014\n\nThen there was the case of Eric. Of all his friends, Eric had truly been a man of good intentions, and therefore had always ascribed good intentions to others. Hence the lack of rebuke after he'd received a kicking at the fair. In his early thirties, working in a local planning department, and with a decent little house in Perivale, Eric had become involved with a younger American woman. Ashley said she loved him; a love which expressed itself as wanting to be with him all the time and never wanting to meet his friends. And Ashley wouldn't sleep with him, no, not now anyway, but certainly later. Ashley had her faith, you see, and Eric, having been religious himself in his youth, could understand and appreciate that. Ashley wasn't a member of an established church, because look at all the harm established churches had caused; Eric could see that too. Ashley said that if he loved her, and agreed with her contempt for worldly possessions, then he would surely join her in such beliefs. And so Eric, temporarily cut off from his friends, put his little house up for sale, planning to give the proceeds to some cockamamie sect in Baltimore, after which the couple would move there and be married by some cockamamie religious theorist, or shaman, or sham, whereupon Eric, in exchange for his Perivale house, would be granted squatter's rights in perpetuity in his new wife's body. Fortunately, almost at the last minute, some survival instinct asserted itself, and he had cancelled his instructions to the estate agent, whereupon Ashley vanished from his life forever.\n\nIt had been a real disaster for Eric. He had lost his belief in the good intentions of others, and with it the ability fully to give himself over to love. If only he'd been inoculated with Susan's suspicion of missionaries. But that hadn't been part of Eric's prehistory.\n\n* * *\n\n\u2014\n\nIt was odd how the long-dead Gordon Macleod still nagged at him. More than Susan did, in truth. She was now resolved in his mind, and would remain so, even if she would also continue to cause him pain. Whereas Macleod was unresolved. So he would find himself imagining what it was like in Macleod's head during those last, mute years, goggling at the wife who had left him, at the housekeeper and nurse whose presence irritated him, at his old pal Maurice, who said, \"Down the hatch, chum,\" then poured whisky straight from the bottle until it soaked his pyjamas.\n\nSo, there was Macleod lying on his back, day after day, knowing that this was not going to end well. Macleod was thinking back over his life. Macleod was remembering when he had first set eyes on Susan, at some dance or tea party, peopled with girls who on the whole wanted to have fun, and men who on the whole were not in respectably reserved occupations. And she was dancing with these spivs and black-marketeers\u2014that's what his envy turned them all into. Even the honest ones were just fancy boys and fancy men. But she went for none of them. Instead she chose that twerp with the goofy grin who could really dance\u2014about the only thing he could do\u2014yet whose flat feet or heart flutter had kept him out of uniform. What was his bloody name? Gerald. Gerald. And then the two of them had danced while he, Gordon, looked on. Then the twerp had died of leukaemia\u2014they'd have done better to send him up in a bomber and let him do a hand's turn before he pegged it, in Gordon's view.\n\nSusan was of course upset\u2014inconsolable, they said\u2014but he, Gordon, had stepped in and declared that he was the sort of chap she could rely on, both during the war and after. She had struck him as not exactly flighty, but a bit\u2014what? irresponsible? No, that wasn't quite right. She eluded him, and she laughed at some of the things he said\u2014and not just the jokes, either\u2014and such improbable, indeed impertinent reactions had made him fall smack in love with her. He told her that it didn't matter how she felt now, because he was confident that she would come to love him in time, and she had replied, \"I'll do my very best.\" Then they'd just thrown themselves into it, as many did during the war. At the altar, he had turned to her and asked, \"Where've you been all my life?\" But it hadn't worked. The being together hadn't worked, the sex hadn't worked, except for successful impregnation; but otherwise, it led to no intimacy. So, their love was a disaster. But that of course was no reason not to stay married, back in those days. Because one could still be fond, couldn't one? And there were the girls. He had long craved a son, but Susan hadn't wanted any more after Martha and Clara. So that was the end of that part of their life. Separate beds to begin with; then, as she complained about his snoring, separate rooms. But one continued to be fond; if increasingly exasperated.\n\nSo he ventriloquised Gordon Macleod, in a way he could never have done while he still hated him. Was he getting anywhere nearer the truth?\n\n* * *\n\n\u2014\n\nHe remembered another angry man: the furious driver with red, hairy ears, hooting and shouting at him on the Village's zebra crossing. And in reply he had sneered, \"You'll be dead before I will.\" At the time he believed that the function of the old was to envy the young. So, now that his turn had come, did he envy the young? He didn't think so. Did he disapprove of them, was he shocked by them? Sometimes, but that was only fair: what they deserved, what he deserved. He had shocked his mother with the cover of _Private Eye._ Now he was himself shocked when some YouTube thread took him to a woman singing of love gone wrong: her title, and refrain, was \"Bloody Mother-Fucking Asshole.\" He had shocked his parents with his sexual behaviour. Now he was shocked when sex was so often portrayed as mindless, heartless, thoughtless shagging. But why the surprise? Each generation assumes that it has got sex just about right; each patronises the previous generation but feels queasy about the succeeding one. This was normal.\n\nAs for the wider question of age, and mortality: no, he didn't think he felt a panic at the shutting of the doors. But maybe he hadn't yet heard their hinges creak loudly enough.\n\n* * *\n\n\u2014\n\nOccasionally, people would ask him, either slyly or sympathetically, why he had never married; others assumed or implied that he must have been, back there, back then. He would deploy an English reticence and an array of demurrals, so the enquiries rarely came to anything. Susan had predicted that one day he would have an act of his own, and she had been proved right. His act, which had developed without his really noticing, was that of someone who had never\u2014not really, not truly\u2014ever been in love.\n\nThere was nothing between a very long answer and a very short one: this was the problem. The long answer\u2014in an abbreviated form\u2014would involve, of course, his own prehistory. His parents, their characters and interaction; his view of other marriages; the damage he'd seen families do; his escape from his own into the Macleod household, and the brief illusion that he'd fallen into some magical world; then a second disillusionment. Once bitten, twice shy; twice bitten, forever shy. So he had come to believe that such a way of life was not for him; and had never subsequently found anyone to change his mind. Although it was true that he had proposed to Susan in the cafeteria of the Royal Festival Hall, and later to Kimberly in Nashville. This would require a parenthesis or two of explication.\n\nThe long answer was too time-consuming to give. The short answer was too painful. It went like this. It was a question of what heartbreak is, and how exactly the heart breaks, and what is left of it afterwards.\n\n* * *\n\n\u2014\n\nWhen he remembered his parents, he often visualised them in some old television play from the black-and-white days. Sitting in high-backed armchairs on either side of an open fire. His father with a pipe in one hand, flattening a newspaper with the other; his mother with a dangerous inch of ash at the end of her cigarette, but always finding the ashtray a few seconds before it would drop onto her knitting. Then his memory would cut to her in that pink dressing gown, picking him up late at night, and flicking her lit cigarette disdainfully out onto the Macleods' driveway. And then the suppressed resentment on both sides, as they made their silent way home.\n\nHe imagined his parents discussing their only child. Did they wonder \"where they had gone wrong\"? Or merely \"where he had gone wrong\"? Or how \"he'd been led astray\"? He imagined his mother saying, \"I could throttle that woman.\" He imagined his father being more philosophical and forgiving. \"There was nothing wrong with The Lad, or how we brought him up. It's just that his risk profile hadn't stabilised yet. That's what David Coulthard would say.\" Of course, his parents had died long before Max Verstappen's exploits at the Brazilian Grand Prix; and his father took no interest in motor sport. But perhaps he might have found some parallel form of exoneration.\n\nAnd he, in turn, now felt retrospective gratitude for the very safety and dullness he had been railing against when he first met Susan. His experience of life had left him with the belief that getting through the first sixteen years or so was fundamentally a question of damage limitation. And they had helped him do that. So there was a kind of posthumous reconciliation, even if one based on a certain rewriting of his parents; more understanding, and with it, belated grief.\n\n* * *\n\n\u2014\n\nDamage limitation. He found himself wondering if he had always misconstrued that indelible image which had pursued him down his life: of being at an upstairs window, holding on to Susan by the wrists. Perhaps what had happened was not that he had lost strength and let her fall. Perhaps the truth was that _she_ had pulled _him_ out with her weight. And he had fallen too. And been grievously damaged in the process.\n\n* * *\n\n\u2014\n\nI went to see her before she died. This was not long ago\u2014at least, as time goes in a life. She didn't know that anyone was there, let alone that it might be me. I sat in the chair provided. Playing through the scene beforehand, I had hoped that in some way she might recognise me, and that she would look peaceful. These hopes were as much for me as for her; I realised that.\n\nFaces don't change much, not even in extremity. But she didn't look peaceful, even though she was asleep, or unconscious, whichever. Her forehead was pulled into a frown, and her bottom jaw pushed out a little. I knew these ways her face worked; I'd seen them many times, when she was in obstinate denial of something, denying it to herself even more than to me. She was breathing through her nose, occasionally giving a small snore. Her mouth was clamped tight. I found myself wondering if she still had the same dental plate all these decades on.\n\nA nurse had brushed her hair, which fell straight down on both sides of her face. Almost instinctively, I reached out a hand, planning to uncover for the last time one of her elegant ears. But my hand stopped, seemingly of its own volition. I withdrew it, not knowing if my motive was concern for her privacy, or fastidiousness; fear of sentimentality, or fear of sudden pain. Probably the last.\n\n\"Susan,\" I said quietly.\n\nShe didn't react, except to continue with her frown, and the obstinate jut of her jaw. Well, that was fair enough. I hadn't come with, or for, any message, let alone for any forgiveness. From love's absolutism to love's absolution? No: I don't believe in the cosy narratives of life some find necessary, just as I choke on comforting words like redemption and closure. Death is the only closure I believe in; and the wound will stay open until that final shutting of the doors. As for redemption, it's far too neat, a moviemaker's bromide; and beyond that, it feels like something grand, which human beings are too imperfect to deserve, much less bestow upon themselves.\n\nI wondered if I should kiss her goodbye. Another moviemaker's bromide. And, no doubt, in that film, she would stir slightly in response, her frown lines uncrease, and her jaw relax. And then I would indeed lift back her hair, and whisper into her delicately helixed ear a final \"Goodbye, Susan.\" At which she would stir slightly, and offer the trace of a smile. Then, with the tears unwiped from my cheeks, I would rise slowly and leave her.\n\nNone of this happened. I looked at her profile, and thought back to some moments from my own private cinema. Susan in her green-piped tennis dress, feeding her racket into its press; Susan smiling on an empty beach; Susan crashing the gears of the Austin and laughing. But after a few minutes of this, my mind began to wander. I couldn't keep it on love and loss, on fun and grief. I found myself wondering how much petrol was left in the car, and how soon I would have to find a garage; then about how sales of cheese rolled in ash were suffering a dip; and then about what was on television that evening. I didn't feel guilty about any of this; indeed, I think I am now probably done with guilt. But the rest of my life, such as it was, and subsequently would be, was calling me back. So I stood up and looked at Susan one last time; no tear came to my eye. On my way out I stopped at reception and asked where the nearest petrol station might be. The man was very helpful.\n\n# A NOTE ABOUT THE AUTHOR\n\nJulian Barnes is the author of twenty-two previous books, most recently _The Noise of Time._ He received the Man Booker Prize for _The Sense of an Ending,_ and has also received the Somerset Maugham Award, the Geoffrey Faber Memorial Prize, the David Cohen Prize for Literature, and the E. M. Forster Award from the American Academy of Arts and Letters; the French Prix Medicis and Prix Femina; and the Austrian State Prize for European Literature. In 2017 he was awarded the L\u00e9gion d'Honneur by the French government. His work has been translated into more than forty languages. He lives in London.\n\n# An A. A. Knopf Reading Group Guide\n\n# The Only Story by Julian Barnes\n\nThe questions, discussion topics, and reading list that follow are intended to enhance your reading group's discussion of _The Only Story_ , the newest novel by Julian Barnes.\n\n# QUESTIONS FOR DISCUSSION\n\n1. The opening line reads, \"Would you rather love the more, and suffer the more; or love the less, and suffer the less?\" Which would you pick? Do you agree with Paul that this isn't a fair questions because \"we don't have the choice\"?\n\n2. Susan and Paul have a quarter-century age difference, yet he repeatedly insists throughout the novel that neither one of them was taking advantage of the other. Do you agree, or do you think there is an inherent power imbalance between them due to that gap?\n\n3. Games and sports feature prominently throughout the story, whether tennis, golf, or crossword puzzles. How do each of these activities, and the attitudes the characters have toward them, illuminate and illustrate the nature of love as they interpret it?\n\n4. Discuss the character of Joan and her role as Paul's only true confidant when it comes to his relationship with Susan.\n\n5. Point of view consistently changes throughout the novel, with part one being in first person, part two in second person, and part three in third, second, and first. Why do you think Barnes chose to do this? How did the different perspectives impact the reading experience and influence how you understand Paul?\n\n6. On this page, Paul presents his theory that memory is like a \"log-splitter.\" How is the nature of memory demonstrated throughout the novel, and do you agree with Paul when he says, \"Life is a cross section, memory is a split down the grain, and memory follows it all the way to the end\"?\n\n7. As Susan's alcoholism progresses, she tells Paul she has \"a moral disease\" caused by her being from \"a played-out generation\" (this page). What do you think is the impetus for her drinking, and how do you interpret her repeated insistence that her generation is \"played out\"?\n\n8. A subsequent girlfriend of Paul's calls Susan a \"madwoman\" in an attic (this page), a reference to not only Charlotte Bronte's _Jane Eyre_ but also the groundbreaking 1979 work of feminist literary criticism of that title by Sandra Gilbert and Susan Gubar. How does Susan fit into the broader tradition of literary housewives? Is she a transgressive feminist, a beleaguered relic of pre\u2013sexual revolution England, or something else entirely?\n\n9. Do you think Paul was right to \"hand back\" Susan to her daughters, or do you think he abandoned her? How did his decision color your opinion of him?\n\n10. As we see throughout the novel, and as is explicitly discussed in part three, Paul is obsessed with defining love. Discuss what it means when, on this page, he posits, \"Perhaps love could never be captured in a definition; it could only ever be captured in a story.\"\n\n11. How is marriage represented in the novel, and how important is it that Paul himself never marries?\n\n12. Gordon Macleod is an extremely complex man\u2014something Paul comes to realize only later in life. Discuss the evolution of their relationship, and Gordon's significance as a man who subscribes to traditional British masculinity.\n\n13. Paul and Susan's final encounter is, on the surface, anticlimactic, but at its core imbued with deep significance. How did you interpret it?\n\n14. After their first match, when Paul apologizes for causing them to lose, Susan says, \"The most vulnerable spot in doubles is always down the middle\" (this page). How does this idea reemerge throughout the novel\u2014that our weakest spot is the space between us and someone else?\n\n15. What is your only story?\n\n# SUGGESTED READING\n\n_The Sense of an Ending_ by Julian Barnes\n\n_The Remains of the Day_ by Kazuo Ishiguro\n\n_The Sparsholt Affair_ by Alan Hollinghurst\n\n_Nutshell_ by Ian McEwan\n\n_Mothering Sunday_ by Graham Swift\n\n# _What's next on  \nyour reading list?_\n\n[Discover your next  \ngreat read!](http:\/\/links.penguinrandomhouse.com\/type\/prhebooklanding\/isbn\/9780525521297\/display\/1)\n\n* * *\n\nGet personalized book picks and up-to-date news about this author.\n\nSign up now.\n  1. Cover\n  2. Other Titles\n  3. Title Page\n  4. Copyright\n  5. Contents\n  6. Dedication\n  7. Epigraph\n  8. One\n  9. Two\n  10. Three\n  11. A Note About the Author\n  12. Reading Group Guide\n\n  1. Cover\n  2. Cover\n  3. Title Page\n  4. Contents\n  5. Start\n\n  1. b\n  2. iii\n  3. iv\n  4. v\n  5. vii\n  6. \n  7. \n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n## Gunship: Omnibus Edition\n\nJohn M. Davis\n\nSmashwords Edition\n\nCopyright 2014 Serenity Valley Publishing\n\nSmashwords Edition, License Notes\n\nThis ebook is licensed for your personal enjoyment only. This ebook may not be re-sold or given away to other people. If you would like to share this book with another person, please purchase an additional copy for each recipient. If you're reading this book and did not purchase it, or it was not purchased for your use only, then please return to Smashwords.com and purchase your own copy. Thank you for respecting the hard work of this author.\n\n### Table of Contents\n\nBook 1: The Flight\n\nBook 2: Glimmeria\n\nBook 3: Reflections\n\nBook 4: Gears And Spears\n\nBook 5: Legendary\n\nAbout the Author\n\nBook 1\n\nThe Flight\n\nAnd so begins the greatest story ever told.\n\nStrangest weather. Adam thought as the high winds continued to push across the barren, almost lifeless ground outside. The border planets always seemed to have the toughest climate, no matter the season.\n\nWhen you were on the fringe of charted space, you could be sure of a few things. Rough weather, rougher living and the roughest damn drinking establishments known to mankind. Those were the facts.\n\n\"Another hand, or should we just count you out Michaels?\" the dealer asked, a thick beard of black covering most of his experienced face as he continued shuffling the deck of cards, only slightly moving his thumbs as the cards crisply fell into place.\n\n\"Let's get something straight right here and now. You don't ever count Captain Adam Michaels out.\" he said as he raised his glass for a long drink of the roughest ale he'd ever fallen victim to.\n\n\"The pay in for this round is seventy credits.\" The dealer said as he surveyed the table, giving his best effort in an attempt to block out the dozens of scoundrels in the background drinking ale and most likely telling lies.\n\n\"I'm out.\" Captain Michaels said quietly as he stood up and walked away from the table, doing his best to ignore the taunts and insults directed his way as if to keep some of his dignity.\n\n\"Watching the news huh?\" he said as he sat down on a raggedy wooden stool beside Dalton at the bar.\n\nWearing a long, brown duster and a stubbly dirt colored beard hiding the scars of his past scuffles, Dalton had been a part of Adam's crew for nearly five years. Old military friends, fighting in the Glimmerian War had brought them close together, each earning the trust and respect of the other.\n\n\"Hell no,\" Dalton replied with a slyly grinned face.\n\nAdam laughed as he motioned the bartender over to them, ordering a tall glass of water as covertly as possible in order to maintain a decent reputation among this night's patrons.\n\n\"All I'm hearing is some pencil necked jackass talking about impending civil war between the colonies. You think I give a damn about politics?\" Dalton asked.\n\nObviously having already consumed his weight in alcohol, or even more if it was a good night, he continued puffing on a withered cigar that sat limply in the corner of his mouth.\n\n\"Tastes like watered down piss if you ask me, or even weaker than that,\" Dalton said loudly, with his full intention of the words hitting the bartender's ears. \"Not that I would know what watered down piss tastes like, or any piss for that matter,\" he quickly added as he turned to convince Michaels of the fact. \"Of course, if I have, I wouldn't know about it.\" Dalton added, drawing a strange look from the Captain.\n\nAdam had drifted away from the conversation a bit, the loud events of the bar in full swing becoming nothing more than white noise as he noticed a man enter the front door of Paulie's Bar. Not the run of the mill scum that usually frequented the place, this man carried himself with a certain swagger, there was a military feel about him.\n\nHe looked the part well enough too, with the dark green cargo pants, solid black t-shirt and shaved head which had a look of shimmering marble to it.\n\nMichaels had needed another crew member for some time now, a hired gun if nothing else. He and Dalton could both handle themselves well in a firefight, but an extra set of arms slinging shells would always come in handy in his line of work. He was a cargo transporter. Not the kind you contact through a government office, more like the type that considered a checklist of felonies to be an honest day's work. The black market kind. Transporting things that needed to be moved, all while skirting the law and dealing with only the worst of the worst. That was his job, and he did it well.\n\n\"Be back in a couple of minutes.\" he said to Dalton as he stood up from the sad excuse for a bar, which was nothing more than aged wooden boards poorly assembled by hand.\n\nAdam made his way toward the stranger sitting at one of the corner tables as the smell of must, smoke and public perspiration hit him in the face like the heel of the heaviest of boots.\n\n\"Mind if I sit?\" Michaels asked as the stranger glanced up at him for a moment, quickly going back to eating his plate of bread without so much as a word. \"Every single time I come here, the bread seems to get worse,\" Adam said as the man continued to eat without missing a beat. \"Got work here or just passing through?\" Adam added, hoping for some kind of response.\n\n\"You Legion?\" the stranger said, looking up from his plate with stern intentions.\n\n\"Legion? No. Hell no,\" Michaels quickly replied. \"Far from it friend. Legion patrols and my ship might be on a first name basis, but it's for all the wrong reasons.\" he added.\n\n\"In that case, no I don't have work yet. Not sure about the just passing through part, I'll get back to you on that one.\" the stranger replied.\n\n\"I see. I'm former military myself, if you don't mind me asking, where did you serve your time?\" Adam asked.\n\n\"Gali Special Forces.\" the stranger replied.\n\n\"Fresh out I take it?\" Adam asked.\n\n\"It's complicated.\" the stranger replied.\n\n\"It usually is. Interested in a job?\" Michaels quickly asked, almost expecting the answer to be an even quicker no.\n\n\"Who's asking?\" the stranger calmly answered.\n\n\"Captain Adam Michaels. Rather not get into the details of the type of work we do, at least not until we make it official. For now, I guess you could consider the type of work we do to be, well, off the grid.\" Michaels replied.\n\n\"Off the grid is where I live.\" the stranger said, as he continued to eat without pause.\n\n\"I see..and you would be?\" the Captain asked.\n\n\"Roman Raines,\" the stranger responded without hesitation.\n\nFinishing his meal, Roman pushed the empty plate to the side, its thick green glass dragging across the table's coarse wooden surface as he looked at the Captain. \"What kind of job?\" he asked.\n\n\"Well, we do pretty much anything that needs to be done. From medical supplies to weapons, we transport anything that pays well enough, under the radar of course.\" Michaels replied cautiously.\n\nRoman gave no reply, just an ice cold stare as a welcome for Adam to continue laying out the details.\n\n\"Truth is, sometimes what we take people aren't willing to let go of. Other times, when we deliver the goods, people start second guessing the whole payment process and, well, gunfights tend to ensue. Would be your job to help protect me, my crew and our ship. In return, you get ten percent of the take and free room and board. I decide to let you go, you go. You decide to leave, you are free to do so.\" Michaels explained in a stern tone of his own.\"Sounds fair enough. I'll think it over.\" Roman replied in a humble tone, his desolate blue eyes focused on the Captain.\n\n\"Well enough then, I'll look forward to it. You can find me at the bar next to the drunk in the brown coat, the one yelling about the alcohol.\" Michaels said as he stared in the direction of the bar, watching Dalton bitch the bartender out over the house brew as only a drunkard could do.\n\nSlowly standing to his feet, Adam took a deep breath as he began heading back toward the bar, his thick leather boots striking hard against the creaking wooden floorboards as he switched himself into babysitting mode.\n\n\"What'd he say?\" Dalton asked, wiping a bit of ale from his lips with the sleeve of his duster.\n\n\"He's thinking it over.\" Michaels replied as he sat back down and motioned for a glass of whatever it was they were passing off as the house special here at Paulie's.\n\n\"Well, that's damn thoughtful of him,\" Dalton said with a sarcastic look on his face as he yelled to the bartender. \"Hey! Do you have anything stronger than this shit?\" he yelled, the chubby man decked out in what looked to be borderline rags for clothes quickly nodded yes and motioned that he would be over in a few moments.\n\n\"Well, let's hope he decides soon, otherwise this trip has been a disaster. Jones fucking us on a deal we came through with, it's been raining since we landed, and who only knows what this shit is that I'm drinking.\" Dalton said under his breath as he turned the glass up, never flinching as every drop of it punished his throat on the way down.\n\nThe night continued on its normal course of action in an establishment such as Paulie's. Stories being conjured and told by those who consumed the most drink. Poker games filled with lies, heavy smoke and even heavier drinking. Pickup line after pickup line used on one of the very few women brave enough to set foot inside the run down dwelling. Not even several hours of drinking what could have doubled as Gunship fuel had Dalton the least bit sideways, although Captain Michaels kept telling himself when it came time to stand up and walk away, he would be hard pressed to.\n\nYep, everything seemed normal on this rainy night, that is until Michaels caught glimpse of a group of Legion guards passing a nearby window; moments from entering the bar.\n\n\"Ah shit.\" Michaels said in a low and discouraged voice.\n\n\"What is it?\" Dalton said, turning to the door as it flew open.\n\nThe Legion soldiers were unmistakable, wearing the dense red outfits that were finely stitched and trimmed in black leather. Solid black riot style helmets with the standard tinted face shields.\n\n\"Ah shit.\" Dalton said moments later, agreeing with the Captain's previous assessment.\n\nHe slowly began to reach for his short barrel shotgun, which looked antique. But the gun, which was covered in a dingy haze, had gotten him out of more than one close encounter.\n\nMichaels quickly grabbed his arm and gave a slight nod. \"Not yet, too many of them. Chances are they aren't even here for us.\" he said as the four guardsmen quickly approached them with battle rifles aimed and ready.\n\n\"Captain Adam Michaels!\" one of them shouted in a loud and administrative voice.\n\n\"Really? Is it too much to ask for one lucky break now and then? Just ONE?\" Michaels said under his breath, almost as if he were talking to some higher power.\n\nAnyone who had known Adam long enough; knew all too well the fact that he had the worst luck of anyone in the Skyla System, or possibly even all of charted space. Women, authorities, games of chance, it didn't matter. He was the unluckiest man alive, and he felt abandoned, if not shunned by the Gods because of it.\n\n\"You are under arrest and subject to prosecution for possession of illegal materials per item seventy-one of the Legion Articles.\" one of the officers pronounced aloud as two more of them disarmed both Michaels and Dalton.\n\nThe fourth guard yelled \"Back, otherwise you will be getting the same!\" as Roman approached the conflict.\n\n\"That's kind of the idea,\" he replied as he looked into the direction of Captain Michaels. \"Twenty percent sounds about right, don't you think?\" Roman asked Michaels as the men remained under gunpoint.\n\n\"What do I think? I think they should be arresting you for attempted robbery. Hell, I would rather rot in a Legion prison cell for the next decade of my life. Twenty percent?\" Michaels asked in disbelief.\n\n\"Have it your way friend.\" Roman said as he turned to start walking out of the bar.\n\n\"On the other hand, fifteen percent sounds about right.\" Michaels replied.\n\nRoman turned back to face them for a moment, his blank stare was an obvious sign of mental calculations at work, not to mention the thought of Adam negotiating a deal while being arrested.\n\n\"I can do fifteen.\" he said as he walked toward the Legion guards and their drunken prisoners.\n\n\"STAND DOWN!\" one of the soldiers yelled as a last warning; not even remotely phasing Roman.\n\nIn what seemed like the most basic of motions, Roman thrust his arm under one of the Legion guards rifle hand, sending a shot bursting into the air. In that same moment, he buried a hard fist into the ribcage of the defenseless man of the law, immediately grabbing the vacated rifle from the air and aiming down the iron sights with precision. Surprised, the remaining three Legion troops fumbled with their rifles for a moment before laying them on the ground and reaching skyward.\n\n\"Well, you can damn sure pick 'em, I'll give you that much.\" Dalton said to Adam as he collected their weapons from one of the hostage soldiers. \"Yea, I would have to agree,\" Michaels said, looking at Roman with amazement.\n\n\"Best be getting out of here, but before we do.\" Michaels said calmly as he returned to the card table, pulling an empty cloth sack from one of the large pockets of his faded brown coat; quickly filling it as he raked the card games winnings of every man into it.\n\n\"Like I said folks, don't ever count Captain Adam Michaels out.\" he said as he grinned widely, obviously proud of robbing the men at gunpoint.\n\nAs Dalton grabbed an almost full bottle of whiskey and raced toward the door, keeping his piece pointed in the direction of the Legion soldiers, Adam glanced back at Roman for a moment.\n\n\"You got the job.\" he said as he dashed out of the door.\n\nIt wasn't the first time Roman had disarmed four men at once, although to his knowledge, it was the first time he had done so and then followed it up with chasing two heavily intoxicated fugitives down one of the busiest streets on the small mining planet of Antillia.\n\nHe began to wonder if he had made the right choice, trying his best to figure out what fifteen percent actually meant in terms of money. At this point, it didn't matter. He couldn't stay here, he was wanted by the Gali government for unspeakable crimes; as well as the Legion because of the incident only moments ago. So at least for the moment he was part of the Gunship crew.\n\n\"Kelly, get ready to get us the hell out of here!\" Dalton yelled into a com unit as he sprinted down the filthy streets of the impoverished town, both Adam and Roman right behind as the rain poured down; a staggering stench of cheap whiskey all over Dalton like a thrift store blanket.\n\n\"Copy that. Anything I should know about?\" a female voice responded.\n\n\"The Usual. Just have us ready to lift when we get there, gonna have a whole lot of the wrong kind of company on our asses.\" Dalton replied.\n\n\"Not again.\" Kelly responded in a puzzled voice.\n\nWithin minutes, they could see the ship. It was easy to spot to the naked eye, strikingly antique; with the only difference being antique gave the impression of value. The Gunship that Adam had poured so much of his heart and soul into, had more of a junkyard feel to it. Different shades of gray on its exterior gave away the fact that it had been pieced together from other ships.\n\n\"I may just hang out here and take my chances.\" Roman said jokingly, earning an immediate response from the Captain.\n\n\"It doesn't look like much, but it's pulled me through a hell of a lot of scrapes.\"\n\nAs they approached the faded blue metal grating of the ship's ramp, a thin Asian man with long brown hair and a white tank top that was littered with grease stains slowly walked down, holding Roman at the end of a single barrel shotgun.\n\n\"Who's the trailer?\" he asked.\n\n\"Relax Kato, he's with us.\" Adam said as he waved everyone inside.\n\n\"May want to put the toy away tan man, before someone gets hurt.\" Roman replied with heavy sarcasm as he walked right by Kato unwaveringly.\n\n\"Don't judge a book by its cover I guess,\" Roman said as he looked around once aboard. \"I'm impressed.\" he added, his eyes skimming the cargo area of the ship.\n\nSeveral wooden crates were stacked nearby, as well as a couple of very large bins made of reinforced sheet metal. The outside of the ship may have looked like a train wreck on its best day, but Roman immediately recognized the interior to be that of a Gunship.\n\nOnce a very popular model, the Glimmerian forces leaned on them heavily during their loss to the Legion in the first Glimmerian War. Once the war had come to an end, they were considered obsolete and quickly phased out in favor of more high tech vessels. The more advanced things became on a ship, the more likely they were to fall apart when you needed them the most, which is why Adam preferred the older model ship. It was solid, built to take a pounding and so basic that it reminded the Captain of himself; his core values simple, yet unwavering.\n\nKato followed the men inside aggressively hitting a red button by the door, followed by a hard turn of a switch to seal it airtight before they took off for orbit.\n\n\"Punch it Kelly, I'm on my way up,\" Michaels said as he pulled the com unit from his pocket and headed up a narrow steel ladder that would eventually lead him to the bridge of the ship. \"Dalton, put Roman in a room and give him the grand tour. I'll catch up with everyone shortly.\" the Captain added as he disappeared up the ladder.\n\n\"You're gonna love the armory.\" Dalton said proudly as he motioned Roman to walk with him.\n\nKato stayed in the cargo bay, shotgun over his shoulder as he held onto a stainless steel handle bolted to the wall until they broke orbit from this sad excuse of a planet, the vibrations of liftoff jolting throughout the ship.\n\nShortly after the Gunship had fully entered the system, Michaels met with Roman on the observation deck of the ship. It was nothing more than a small balcony with two large windows made from hardened glass looking out across the stars, but it was considered to be the perfect place to talk business amongst the crew.\n\nFor the next hour or so, the Captain heard all about Roman's past; or so he thought.\n\nEverything from his many years of service with the Gali special forces, to the recent events during the past couple of years that had him moving planet to planet, doing his best to elude the Legion while making a living for himself.\n\n\"Sounds like it'd make a good book.\" Adam said jokingly, trying to make the newest member of his crew feel a little more comfortable.\n\n\"I suppose it would.\" Roman replied with a slight grin.\n\nAdam went on to do his best in explaining each member of his crew's story to the newly acquired gun. How he had fought alongside Dalton during the Glimmerian war less than a decade ago, and how he was still the weapon obsessed alcoholic which he had become famous for then.\n\nHe explained how Kelly came to them fresh out of flight school, having graduated in the top of her class, she had decided that she would rather work for herself. It presented her a much better chance to see the star system, even the planets on the fringe of uncharted space. It was that, or start making a living taking orders from some suit and tie who would have been signing her paychecks if she would have accepted a job flying a standard commercial transport ship, and that just wouldn't be her style.\n\nKato had been introduced to Michaels during a two week stint in the prison system of Anon. They had formed a bond that only terrible food and forced manual labor can provide, which led to Michaels helping Kato escape a short time after the Captain himself had been released. Since then, Kato had served as the ship's mechanic as well as an extra gun when needed.\n\nAt last, there was Luck. The rest of the crew referred to him as Adam's pet, but the Captain saw him as much more than that. Sure, he was a reminder of the best hand of poker that Michaels had ever played, winning the android in a high stakes game back on Montague.\n\nStill, Adam had grown fond of of him and thought of Luck as being as much a part of this crew as anyone else. Luck had his purpose on the ship, he was a decent mechanic and also knew enough about flying to get the ship to its destination under emergency circumstances.\n\nStill, everyone knew that he was more of a trophy than anything else, giving Adam the opportunity to bring up his rendition of that game whenever he saw fit. Which, painfully for the crew; was quite often.\n\nLong after the conversation was over and Roman had made his way to the crew quarters for the night, the Captain remained sitting in the same chair made of solid stainless steel, finding himself drifting off a bit, thinking of everything important in the here and now.\n\nThe ship's maintenance, his crew's safety and value to the overall scheme of things. He even wondered, as he had done from time to time, what kind of life he would be living if he were settled down on a decent planet somewhere; living a normal life.\n\nAt times, he found himself missing the feeling of settling down with a wife and children, even though it wasn't a feeling that he had been fortunate enough to experience. The thoughts that raced through his head all too often seemed to end up with the same conclusion time and time again. He loved what he was doing, from the near fatal gunfights to the deep space travel. Especially the deep space travel.\n\nIt's hard to explain what space travel is like until you've actually been there. The near frigid temperatures of the ship's interior that you are forced into getting used to. The constant silence of a vast nothingness as you learn to block out the sound of the ship's engines, all while surviving on freeze dried rations and what few supplies you were able to load before leaving the last settled world.\n\nIt sounded horrible, that is until you learned to appreciate the beauty of the stars slowly passing you by; almost as if memories were being revived in slow motion.\n\nMost people will say it all looks the same, but the moment you begin to appreciate the simple things, suddenly you realize how beautifully different every trip becomes. As Adam sat on the observation deck, he had almost as many thoughts racing through his head as stars spread out in front of him.\n\nIt didn't matter though, all of the problems within the system, the challenges his crew faced every single day and the friends he had lost during the Glimmerian war; all of that was simply put on hold every time his ship became interlocked with the stars. It was the one place that nothing else mattered.\n\n\"Into the steel I see.\" Roman said as Dalton continued meticulously cleaning nearly a dozen weapons.\n\nEverything from the basic combat pistol to the punishing double barrel shotgun, it was all right here, laid out on a coarse brown blanket in front of them.\n\n\"Into the steel, hard liquor and the steel that produces the hard liquor,\" Dalton said jokingly. \"I'm also infamous with the ladies, you might have heard?\" he added.\n\n\"Nope.\" Roman replied with a sarcastically fluctuation of his eyes.\n\n\"What about you? I'm sure you've had your hands on plenty of steel yourself in Gali?\" Dalton asked.\n\n\"As well as liquor and women,\" Roman replied as both men laughed quietly. \"Yea, I've been trained with quite a bit of everything, still prefer the tactical blade,\" Roman said, pulling a combat knife from a black leather band that wrapped around his leg. \"Never jams up on you and it's just as accurate.\" he added, throwing the blade end over end throughout the room with blazing speed, hitting the wooden post of Dalton's makeshift gun rack, the bulk of the blade burying deep within its unfortunate wooden fibers.\n\n\"Nice throw, but too much damn unnecessary work for me.\" Dalton replied, quickly shouldering a short barrel shotgun and cutting two shots loose, hitting both over and under the blade; chopping the piece of post down to the thick steel floor.\n\n\"Nice accuracy.\" Roman said.\n\n\"You should see me when I'm sober.\" Dalton replied.\n\nA few moments after he had laid the shotgun back in line with the other weaponry, Kato burst into the room, a semi-automatic rifle in his arms and at the ready, responding to the gunfire.\n\n\"You know, I normally consider it a rule to kill anyone who draws down on me twice.\" Roman said calmly as he looked into the barrel of Kato's rifle.\n\n\"Is that a fact?\" Kato asked, tightening his grip on the weapon as he continued a deadlocking stare with Roman.\n\n\"Calm down jackass, I fired the shots; not him.\" Dalton said, making his way to Kato.\n\n\"I don't care who fired the shots, I'm thinking more about him just threatening to kill me.\" Kato replied.\n\n\"Think you're fast enough, then I'm game.\" Roman said, bringing a tense urgency to the room.\n\n\"RELAX! Nobody's killing anyone,\" Adam said as he entered the room. \"Put the piece away Kato,\" Michaels added. \"NOW!\" he yelled, finally convincing Kato to slowly lower his weapon.\n\n\"You muthafu... \" Roman began to yell, charging into the direction of Kato as the Captain drew his revolver with blistering speed, holding Roman at gunpoint.\n\n\"This ends right now. You both understand?\" he asked with neither man responding.\n\n\"It either ends or I'll personally airlock the both of you and hire two more guns when we land on Tameca,\" Adam added. Seconds after Kato had walked away, Roman held his hands up and backed off slowly. \"Good, after seeing what you did to those damn Legion soldiers, I'd have to hire two men to replace you. And I don't want to spend that kind of money.\" Michaels said in an attempt to lighten the mood of the room, quickly holstering his revolver and walking away.\n\n\"See why I drink?\" Dalton said, turning up a shot glass full of spirits before catching his breath and tossing a spare glass to Roman.\n\n\"I'm starting to paint the picture.\" Roman replied, pouring himself some of the light red bubbly.\n\n\"You know, you may want to be a little more friendly when it comes to Roman, he may end up saving your life one day.\" Adam said, catching up to Kato.\n\n\"That will be the day.\" Kato replied, turning to enter the confined engine room as the Captain stood for a moment; watching him disappear into the steam filled cave made of faded gray steel.\n\n\"System checks out fine.\" Luck said, his upper body under the control panel of the pilot's gauges.\n\n\"Alright, everything looks good up here as well, switching over to self pilot.\" Kelly replied, clicking several buttons as she slowly stood from the plush leather seat to see the Captain board the small bridge area.\n\n\"How's the grid?\" Michaels asked.\n\n\"We didn't pick up anything on our way out other than a little com traffic, looks like a clean getaway.\" Kelly replied.\n\n\"Good. That's good.\" the Captain replied, taking a seat in one of the two chairs that were stationed behind the pilot.\n\n\"We got into a little bit of local trouble with the Legion back there, shouldn't exactly be combing the system looking for us,\" Adam added. \"I need you to plot a new course out for us, we need to make a stop in Tameca before hitting the outer plains.\" Michaels said as both Kelly and Luck looked on.\n\n\"Yes sir Captain. We got a job in Tameca?\" Kelly replied curiously.\n\n\"Not sure just yet, maybe; will be a big payday for all of us if things work out,\" Adam replied, the thought of the Hunters cemented into his mind throughout the entire conversation. \"I'll pull everybody together when we get there, go over all of the details. Try to bring us to the edge of Tameca City somewhere, without drawing a lot of attention. Good job back there.\" the Captain replied with a nod as he left the bridge to head to his rack for a few hours of much needed slumber.\n\nThe hazy gray rain began hammering the Gunship the moment it made its approach into Tameca's atmosphere. It had been a very short flight, having left one of its farthest moons before flying into one of the larger planets in the system. A semi-sweet sight to the Captain as he awaken to see the distant city skyline from a small circular window near his bunk. He was glad to be landing, still he couldn't help but think cautiously of the hidden dangers that always came attached to such a setting. A gigantic planet, which was predominately covered with water, held one of the most highly populated cities throughout the system. Tameca City, known for its massive number of ports; it was a smuggler's paradise.\n\nSure, there was a Legion presence here, however the run of docking ports was so immense that they simply couldn't check them all, relying on the help from its citizens. The same citizens who feared the most ruthless clan of criminals on Tameca, or any of its moons for that matter. The Hunters.\n\nAn almost vampiric group of criminals, they were feared, even by the Legion soldiers themselves; which served almost as a free pass to conduct underhanded business and wreak havoc abroad. It was the cannibalistic nature of the Hunters, combined with the stark white skin tone that caused many people to label them as vampires. Human enough, they bled and occasionally were even killed.\n\nStill, the fact that they moved with exceptional speed and were so proficient with hand to hand combat only added to their legend. It was the Hunters who had brought the Gunship to Tameca, though the rest of his crew remained unaware of the fact.\n\nAdam didn't like the idea of dealing with clientele such as the Hunters; but they had been looking for someone to move goods and were offering a lot of credits to anyone who was brave enough, or stupid enough, to accept the job. A very risky job, one the Captain knew he had to take in order to make financial ends meet; although selling the idea to the crew wasn't going to be the easiest of tasks.\n\nWith the crew gathered in the loading bay area of the ship and the vessel snugly nestled between two large hills of high grass, or as smugglers like to call it, a rural landing pad, the Captain knew that it was now or never. He had to sell the idea of working for a group with a name that was more ruthless than the luck that he had been burdened with his entire life. They would be lucky to finish the interview, much less the job unscathed, and it would take every one of them to pull it off.\n\n\"The Hunters! Have you lost you're damn mind Adam?\" Dalton yelled as the rest of the crew looked on, his loud voice echoing against the hollow walls of the nearly empty cargo bay.\n\n\"I don't care how much it pays! If I can't kill 'em, then I don't want to go into business with the bastards!\" he added.\n\n\"Relax, will ya? We do the job, collect the money. That's it. It is that simple.\" Michaels replied.\n\n\"I'm with you Captain, the Hunters don't intimidate me.\" Kato added.\n\n\"Easy for you to say, sitting back here at the fort while my dick's out there on the chopping block!\" Dalton replied angrily.\n\n\"Captain, what does he mean you can't kill them?\" Kelly asked. Michaels frowned slightly, looking at the floor for a moment before looking in her direction. Before he could say anything, Roman calmly replied.\n\n\"They die. Not easily done, but possible. I've put a few in the grave myself.\"\n\nAdam spun around to see Roman still sitting on an empty crate behind him, quickly asking \"You can? You have?\". Roman replied with a slow nod.\n\n\"Define a few?\" Michaels asked. The look he received from the former Gali commando let him know that it was a good damn many, which Adam wasn't sure should put his mind at ease or throw a red flag.\n\n\"I've put an entire rifle clip into one of the beasts myself! It did little to slow it down; much less kill it!\" Dalton said with disbelief, as the rest of the crew looked on.\n\n\"What do you mean beasts? Captain?\" Kelly asked, glancing hard at Michaels.\n\n\"Go on Adam, tell her.\" Dalton said with heavy sarcasm, earning a long glance from the Captain. \"Everybody just calm down for a minute. Relax. Feels good to be alive doesn't it?\" Michaels said, trying unsuccessfully to lighten the spirit of the crew.\n\n\"Alright, they aren't human; at least not entirely Kelly.\" the Captain said, referring to the Gothic appearance of the Hunters. Their hair and skin void of pigment, solid black leather attire and above all things, their uncanny desire to feast on the flesh of human beings.\n\n\"Any person that eats another person has some damn issues,\" Dalton said with heartfelt emotion. \"Unless she's hot and it aint' for keeps.\"\n\nEven though they kept to themselves, almost like a secret society, they were infamous among the citizens of the star system. Almost demonic in nature; relying on sleek weaponry and a code of honor that rivaled even the Mafia families of the small planet of Benza, which were also notorious for their hardline methods of getting the job done.\n\n\"Doesn't matter. By the time you and Luck get back with the supplies and the ship checks out flight ready, we should be back from our meeting with the Hunters and then it's business as usual.\" Michaels said, trying to calm the crew down before turning to get a list of needed supplies from Kato.\n\nAs Adam turned to the exit of the ship, he could see Dalton and Roman gearing up for the much anticipated meeting; both men looking more than ready. Roman was equipped with nothing more than a large combat knife and a military grade pistol, each strapped to a separate leg with thick leather bands.\n\nDalton, on the other hand, was going heavy. And for him that spoke volumes. He had an arm sized buoy knife strapped tightly to one of his legs, with two older Knocker pistols strapped to the other, earning their unique nickname from the ability to either cut a man in half at point blank range; or bash in the unlucky victim's skull quite easily with the brass shielding at the bottom of the weapon's handle. All of this, however, was overshadowed by his black flack jacket which held several grenades as well as two short barrel shotguns. Both guns over his shoulders and crossed behind his head.\n\n\"A little much don't you think?\" Michaels said as he looked on in disbelief. \"That's what she said.\" Dalton replied, laughing for a moment as he hesitated; finally taking the flack jacket off and tossing it onto a nearby table.\n\n\"Alright, but that's it. The knife stays. The sons of bitches 'aint eating me,\" he replied. \"Hold the fort down now, ya' hear?\" Dalton asked Kato sarcastically as the three men headed for the exit ramp.\n\n\"Kelly, you and Luck grab everything we need as far as supplies go and stay in contact with Kato back here on the ship. Steer clear of any Legion eyes on the street, we don't need that on top of what's already on our plate. Kato, we'll be in touch.\" said the Captain, as he glanced back long enough to see Luck and Kelly prepping the Rover.\n\nIt was nothing more than a large, mechanical vehicle used for carrying cargo, but he sure wished it were him riding inside of it at the moment. They both looked very comfortable to him, meanwhile he started out of the cargo bay door with the others, immediately hammered by a cuttingly cold rainfall.\n\nDalton glanced back for a second, the water falling around them like fireflies as he glanced in Adam's direction, his jacket already thoroughly soaked.\n\n\"If we end up marching in this shit only to die, I'm gonna be seriously pissed off.\" he said as Roman started laughing quietly.\n\nEven Captain Michaels chuckled for a moment. Dalton was right, nearly a mile walk in front of them in some of the worst rainfall he'd ever seen.\n\nThey had walked nearly two hundred yards when the rover passed by, throwing water their way almost as if to mock them. Michaels tried to block out Dalton's cursing and focus on the city that sat on the horizon, but as usual it was easier said than done. He had been here a few times before, but could never quite get past the size of it. To his estimation, millions of people had come from every corner of the system to call Tameca City home. A lot of good citizens lived here, raised families and worked hard everyday to earn their place.\n\nThat said, a city of this magnitude was easy to get lost in; which had a certain luster to it from a criminal's point of view. Although Adam wouldn't admit it, everyone he had ties with in the city was on the wrong side of the legal system. It seemed like everywhere he went in Tameca, he found trouble. Sometimes he had a hand in it. Well, usually he had a hand in it. Still, he tried to convince himself that he had been incorrectly branded because of the few times when he was truly in the wrong place at the wrong time.\n\nIf ever there was a wrong place inside of Tameca City, it was Dusk Tavern. That was one place that never had a right time, full of the scum of the star system from open until close, at which time the more prominent criminals conducted business.\n\nIt was bad enough to be going to Dusk Tavern in the first place, but to march like this; cold liquid piercing their body from head to toe the entire trip? He couldn't think of a single reason that anyone would put themselves through such a drenching march only to wind up at the worst possible destination imaginable.\n\n\"At least they got good drink at the Dusk.\" Dalton said to the group as he grinned from ear to ear.\n\nMichaels just shook his head for a moment before glancing up into the sky, wondering if there truly was a higher power that made a full time job out of torturing him.\n\nAfter walking for what seemed like an eternity in clothes that were clinging to their skin from the rain, they had arrived at the edge of the city. The bright lights and hustle of thousands of bodies were a welcome sight to anyone who had just been on a trip through the system, though the Captain had a knack for attracting the wrong type of attention here.\n\nIt only took a few moments of holding up a fistful of credits before a transport shuttle came to a screeching halt; adding a few more dings to the already wretched yellow paint job. It was like a roll of the dice whenever you used a city shuttle in Tameca City.\n\nSometimes you would land an android at the controls, and they weren't exactly famous for their conversational skills; providing a very serene trip to your destination. Other times, you were presented with a former convict or drunken lowlife who would give you the entire laundry list of rumors throughout the city in five blocks or less.\n\nAfter a couple of minutes of convincing the driver; who Dalton swore did jail time with him several years back on the small moon planet of Jocom, that they truly did want to go to the Dusk Tavern He put down his small flask of booze and began the trip.\n\n\"Everyone is talking about the impending civil war between the planets,\" the driver said as he passed the first intersection on his way to the bad side of town; traffic so thick it was almost unbearable. \"The council should be voting on it one way or another this week, but everyone in Tameca knows that it's just a tactic to stall for time while they raise a large enough army. We will be fighting alongside the Colonial army, or at least that's what every newspaper here has been saying.\" he added.\n\nThe Captain, Dalton and Roman all tried to look and sound interested, though the only person with interest in the war to come was the whiskey laden driver.\n\n\"So, what business do you guys have at the Dusk?\" the driver asked.\n\n\"No business of yours.\" Roman sharply replied, earning him a glance from Michaels.\n\n\"What my friend means is, we are meeting some old friends there. The kind of friends who wouldn't really like to be the subject of conversation.\" Adam replied in a much friendlier tone than Roman had just finished with.\n\n\"Understandable.\" the driver said politely to the Captain as he gave the Roman a look filled with ill intention.\n\n\"I simply meant that it's a rough place. Wasn't sure why anybody would go there willingly is all.\" the driver said in a calm but probing voice as Dalton broke his stare from the window to answer.\n\n\"They got damn good liquor there. Some of the finest I've ever drank, and I've been around.\" The driver looked puzzled for a moment before answering with deep sarcasm.\n\n\"Yes, I bet you have.\"\n\n\"What the hell is that supposed...\" Dalton began to ask, turning his entire body to the driver in the process. But before he could finish, he was cut off by Michaels.\n\n\"Can we stop with the drinking already! Don't you think about anything else?\" he asked in a puzzled voice.\n\nDalton was obviously in a deep, concentrated thought for a few moments.\n\n\"Well yea. Guns. Women. You know, the essentials.\" he responded as Roman broke out into a loud laugh for several seconds.\n\nThe remainder of the ride lasted only a few minutes and was funeral quiet, the shuttle arriving at the Dusk as the driver looked around with paranoia; engine running loudly as the clanging of steel rods could be heard from its engine compartment.\n\nAs Michaels handed the driver a small fist full of credits and thanked him, both Dalton and Roman were already waiting outside, almost as if the drink would run dry if they waited a moment longer.\n\n\"Well, the place hasn't changed a bit.\" Adam said, referring to the borderline condemned look of the Dusk.\n\nWooden boards made up its exterior, the faded red color of the wood washed away with time giving the place a fragile look. The roof of course was still the same shiny aluminum material that had been the topic of discussion for citizens throughout the neighborhood.\n\nIt had been long rumored that the aluminum was put into place to block the overhead scans from the city's security choppers, not that the security force was brave enough to enter the Dusk either way. It was a welcome sight to all three men as they cautiously approached the reinforced steel door that was guarded by two of the largest men any of them had ever seen.\n\nBoth wearing solid black shirts, cargo pants and boots, they fit the profile of higher end mercenaries. Armed with light machine guns, they were the type of security that if you weren't on the list; you weren't getting into anything short of a casket. That is, unless you had a fist full of credits.\n\nAs the three men approached a waist tall podium style computer, Adam placed his hand on the tinted glass top; trying his best to look casual as every fingerprint was thoroughly scanned. After a few moments, the tinted glass illuminated bright green, as the security detail slowly opened the door.\n\n\"Welcome Captain, they're expecting you.\" one of the mercenaries said.\n\nThe men slowly entered, taking extra time to survey the surroundings. It was nothing more that a small wooden bar that was polished to a very fine grain, along with a handful of stools and several thick wooden tables near the entrance. They were also quick to notice a door behind the bar at the rear. Walking into its direction, Dalton would get in position to cover it just in case things went sour; or at least Adam thought.\n\n\"Well, here's the liquor.\" Dalton proclaimed proudly, completely disregarding the rear door while cradling a bottle of molasses black bourbon, smiling as if he had just become a father for the first time.\n\nSo this is what the Dusk looks like after hours. Roman thought as he glanced around the room.\n\nVery dimly lit, there were only a few white track lights on the ceiling and a handful of red lights, giving a subdued crimson glow at the foot of the bar. It was empty for the moment, however that didn't stop Dalton from helping himself to a shot glass, standing behind the counter preparing his drink as if he were working for tips before joining the others at a small wooden table nearest to the center of the room. Michaels kept his eyes locked on the rear door, meanwhile Roman kept watch on the only other door in the place, the entrance.\n\n\"Damn good bourbon, I'll say that much. Of course, I've never really had bad bourbon. Odd, don't ya' think?\" Dalton asked his two comrades in a scratchy voice as he turned his glass up for a second hard shot.\n\n\"I don't like waiting around like this.\" Roman said softly, a statement which Captain Michaels was quick to agree with.\n\n\"So what do you think they are needing moved so damn badly?\" Dalton asked.\n\n\"I don't know, but it looks like we are about to find out.\" Michaels replied as two figures appeared from the door directly behind the bar.\n\nAs they slowly approached, it quickly became obvious that they were in fact Hunters. One of them much larger than the other, they both were darkly dressed in all solid black leather, even down to the thick laced boots.\n\n\"Captain Adam Michaels?\" the smaller of the two asked in a curious voice. \"My name is Anwick. There was never any mention of escorts coming with you to the meeting.\" he said, visibly upset by the intrusion of both Dalton and Roman.\n\nHe motioned toward them slightly with one of his hands, prompting the larger of the two Hunters to approach the men. At least a foot taller than anyone else in the room, his arms were swollen from use; muscles rippling throughout the large limbs.\n\nThe same moment that the large Hunter firmly reached for their weapons in an attempt to disarm the unwanted strangers, Roman spun around and onto his feet; grabbing the Hunter's wrist with one hand and using the other to slam its head straight through the table, throwing wooden shards throughout the room. Anwick stood to his feet swiftly.\n\n\"What is the meaning of this?\" he shouted at Michaels as he began to aid his friend.\n\nAlmost instantly, Anwick found himself looking at the bad end of two shotgun barrels, both in the grasp of a slightly tipsy Dalton. \"You realize that at my signal, I could have this room full of soldiers who would kill and dismember you inside of thirty seconds!\" Anwick stated firmly, glancing hard at Adam.\n\n\"Think so?\" Roman asked calmly, applying pressure to the arm of the larger of the two Hunters, earning a painful screech.\n\n\"Relax! Everyone just relax,\" Adam said loudly as he remained seated. \"You honestly didn't expect me to just walk in here alone did you? We both know what kind of reputation your kind has, and I would hope that you would consider me to be a bit smarter than that. You and I are here to do business, so let's do it.\" Michaels added as he motioned toward the chair that Anwick had left only moments ago.\n\nAs Anwick sat down slowly, he immediately glanced at Roman, who still had a commanding hold on his much larger opponent.\n\n\"Roman, let him go.\" Michaels said.\n\nResponding with a slight nod, he pushed the Hunter several feet back, releasing his arm with the same motion.\n\n\"Lexion! Enough!\" Anwick yelled as the larger Hunter began to walk directly toward Roman a second time.\n\nAn awkward moment of silence fell across the entire room as it quickly became evident to the Captain that should the Hunters have their way, Roman was unlikely to walk out of the Dusk alive.\n\n\"Anwick, I apologize. My crew is very protective when it comes to me as well as the cargo we transport. I hope you view this as an advantage when considering us for the job.\" Michaels said in an attempt to settle the dispute.\n\nAnwick gazed through Roman for a few more seconds before breaking his stare and looking directly at Adam\n\n\"Very well. We have business to discuss,\" Anwick replied. \"It is because you come so highly recommended that we are meeting here today,\" Anwick said, referring to all of the friends Michaels had made in the past that were connected to the wrong side of the Legion. \"Simply put, we have a package that we need to you take and deliver in exactly one week to Gastonia. You will earn fifteen thousand today and another thirty thousand at the time of delivery.\" Anwick added, stating the terms with great clarity, his long and brittle white hair moving slightly as he spoke.\n\nMichaels remained quiet for a few moments, finally giving his answer to the business proposal. \"It's going to cost you twenty five thousand up front and another twenty five thousand upon delivery.\"\n\nAnwick took several long, heavy breaths before replying slowly. \"I do not negotiate Captain Michaels. I offer the terms, you take them. That's how I conduct business.\"\n\nAdam glanced over in the direction of Dalton, trying to figure out how many shots that had made for him already. \"I'm not negotiating. We both know it must be a complicated delivery, otherwise you would have done it yourself. A complicated job pays what its worth in risk. Otherwise, I'm not the right man for the job. That's how I do business.\" Michaels replied.\n\nAnwick stood straight to his feet, prompting Adam to do the same a few moments later. \"Very well,\" Anwick answered as both men stood only inches from one another. \"I'll pay what you ask, however let me tell you this. You have one week. ONE. If you are even a second behind schedule on delivery, you should pray to the Gods that the authorities find you before we do. Otherwise death will be the least of your concerns.\"\n\nAnwick gave a slight hand motion, prompting Lexion to walk to the door behind the bar, giving three hard thumps on it with his bone crushing fist. Seconds later, the door opened as four more men dressed similar to the ones outside of the Dusk entered, however, the difference became quickly obvious to Michaels and his crew.\n\nThey were escorting a female prisoner, her hands in tight rope bonding and her head looking toward the floor. The Captain wasn't sure the reason, but as well dressed as she was, he suspected someone in a powerful position would be trying everything they could to get her back.\n\n\"What's with the girl?\" Michaels asked, dreading he already knew the answer.\n\n\"You mean the package,\" Anwick said smiling, the dim light bouncing off of his shard like teeth. \"One week.\" he added, holding a single finger up closely in front of Adam's face before walking through the door and exiting the room.\n\nOne of the armed men dropped a black leather bag to the ground at Adam's feet; several Legion credits spilling out of the bag a moment after hitting the floor. He looked at the men for a moment, eventually reaching down and grabbing the bag.\n\nLexion held the door open, the polluted air of the city sending a glimmer of light from the sunset as well as a push of civilized air into the Dusk.\n\n\"May we meet again.\" Lexion said smiling.\n\nRoman quickly rose to his feet and replied \"I'll look forward to it.\" as he quietly tapped his fingers across the handle of his combat knife.\n\nDalton secured the half-empty bottle of bourbon into one of the cargo pockets of his pants, finishing off the rest of the shot glass in one fast motion; alcohol burning a familiar path all the way down to the pit of his stomach.\n\n\"My compliments to the chef.\" he said as he handed the empty shot glass to the nearest mercenary.\n\nAs they began to leave, Lexion yelled loudly, prompting one of the well armed soldiers for hire to lead a young boy dressed in tattered blue clothing into the room at gunpoint.\n\n\"Wait, they have a child in there!\" the woman yelled loudly, her soft and soothing voice of sincerity catching Adam's immediate attention.\n\n\"Not our fight,\" Dalton said calmly to Michaels. \"We are talking about Hunters here Adam. Two of them to be exact, not to mention a pile of tin soldiers.\" he added as he nervously awaited a response from his Captain.\n\n\"Please! They are going to kill him!\" she shouted desperately with tears flooding her vivid blue eyes, obviously concerned for the boy's life. \"Please!\" she said once more, grabbing Michaels by the arm; their eyes locking together as they shared a moment of fate.\n\nAdam felt the urge to do the right thing beginning to overtake his better judgment, glancing at Dalton for a moment almost in apologetic fashion.\n\n\"Ahh shit!\" Dalton was able to get out softly before the stinging feeling of worth that came attached to helping someone in need hit Michaels as he had turned back into the direction of the door. Kicking it solidly, the reinforced obstacle quickly flew open, smacking abruptly against the wall behind it and gaining the attention of Lexion as well as the four heavily armed men.\n\nAdam's pistol threw two shots from the chamber almost instantly, the first hitting one of the armed men in the forehead and dropping him like a stone. The second shot pierced the chest of another soldier, throwing him against the wall before hitting the floor with an unforgiving thud; a trail of blood painting the wall bright red above him.\n\nMichaels knew well enough to seek cover before attempting a third shot, diving behind a table in the corner of the room just in time as dozens of slugs began to chew at the wooden barricade that separated them. As the two soldiers began walking slowly in the direction of the Captain, escorting themselves with a blanket of lead from their automatic weapons, they were both knocked to the ground forcefully by shells from the shotgun Dalton held in his right hand.\n\nAn expert of his surroundings, he immediately dropped the empty weapon and spun around into the direction of the front door, putting a two handed grip on his other peacemaker. Anticipating the guards out front rushing them, he blindly fired two shots into the direction of the entrance. The first shell hit the frame of the door, sending up a wooden cloud of splinters, while the second shot found a target on one of the soldiers who fell to the ground in pain for a few moments before checking out of this life. Dropping his second empty weapon, he took shelter beside of the door, drawing both of his pistols at once; the dim light of the Dusk's interior shimmering across the brass inlays.\n\nMichaels had made it to the boy and began to free his hands of the tightly bound rope when Lexion threw a backhand into his direction, sending the Captain several feet across the room. Smashing into the stockpile of intoxicating beverages, Adam fell limply onto the floor behind the bar under a storm of broken glass.\n\n\"Son of a bitch.\" Dalton said of the wasted liquor as he continued to exchange gunfire with the remaining soldier outside.\n\nAs Lexion followed the path of destruction which led to the Captain, Anwick once again entered the room, taken back for a moment by the devastation, he quickly focused his anger on Adam; who remained limp on the floor behind the bar. Anwick's hands clamored for revenge as both Hunters approached the Captain.\n\n\"Hey, bitches,\" Roman yelled across the room, successfully gaining their undivided attention. \"Get some.\" he added, daring the walking beasts to a fight.\n\nAnwick quickly made his way into the direction of the marked man, Lexion following suit and releasing the grip he had on the front of Michaels' shirt allowing him to fall back to the ground. Roman pulled his combat pistol long enough to throw it across the room, sliding roughly on the aged floor as he confidently drew his tactical blade and squeezed his hands for a moment; several of his knuckles busting back to life.\n\nThe boy was able to free himself in the following moments, running to the safety of stunningly beautiful lady as she stood behind Dalton. Firing several shots from the two pistols, he finally clipped the soldier outside on the shoulder, putting him down and rendering him defenseless.\n\n\"Kato, I'm activating the beacon. Come get us now!\" Dalton said out of breath as he pushed a sequence of buttons on the com unit, throwing it out into the street; the small globe on the front beginning to emit a soft white glow. \"Now we hold tight and pray that our ride gets here before they finish us off.\" Dalton said to the woman and child, putting them behind a small table near the door and checking the magazines of his pistols; the unmistakable odor of hooch on his breath.\n\nWhen Captain Michaels began to get his bearings in order, he realized nothing on him was broken and began to slowly crawl in the direction of where his pistol had hit the floor; pieces of glass crunching softly under his bare palms. He didn't see his sidearm right away, instead he caught sight of a large piece of mirror laying in the floor at the corner of the bar. He was confident that he had never seen a fight like the one that was reflecting back to him through the reciprocating glass. Glimpses of Roman's body moving with perfect coordination as he carved flesh with both blade and boot.\n\nHe landed several fast punches into the face of Lexion, momentarily stunning him while his kick found its home in the chest of Anwick, staggering him back a few feet. More surprised than hurt, Anwick quickly unleashed a flurry of lightning fast punches and elbows onto Roman, most of them finding their mark as they sent Adam's newest crew member to one of his knees.\n\nFurious, Lexion wasted no time putting a choke hold on Roman in an attempt to snap his neck and be done with the pesky former Gali commando. His grip hold of Roman's neck was released as a shot from the Captain's pistol embedded into his shoulder, causing Lexion to snarl his razor teeth for a moment; eventually casting a stare in Michaels' direction. That is how his life would end. Roman thrust his combat steel into the face of Lexion, following the stab with a swift elbow that broke the blade from its handle as Lexion fell to the ground, blood spilling from the edges of the eternally lodged instrument.\n\nShocked at the death of one of his best and feeling the sudden sense of morality, Anwick quickly began to exit at the rear of the bar, the room now flooding with soldiers armed to the hilt. Roman had reached the point of no return, rage flowing through his veins as if it were the very blood in his body. He let go of an unrelenting assault with his hands, grabbing the first soldier by the neck and using his free hand to pummel the man with the bottom of his fist. He then grabbed a second soldier with a choke hold that eerily resembled that of the now departed Lexion, this time more effective however as he snapped the man's neck in several places; dropping his lifeless body to the ground.\n\nHe was a killer, a damn good one at that. But even someone of his life ending skill was outnumbered as at least a dozen more armed men filled the room in response to the distress call of the soldier who had pleaded for help earlier during his firefight with Dalton. Michaels had made it to his feet and nearly reached Roman when they both realized the circumstances and were ready to accept defeat. As both men began to put their hands into the sky, Roman heard an unfamiliar sound, one that Adam knew all too well. The deep, throaty sound of the Mauler. It was a name the crew had decided on for the largest flash shotgun any of them had ever seen. Kato stood in the doorway holding the mauler, which gained its technical name not from a flash of light, but the flash drum under the barrel which held thirty rounds and required only an instant to reload.\n\nThe room began to thin out quickly as every burst that fired from the chamber of the massive gun seemed to add a larger variety of body parts to the wall. Kato had barely used half of the drum's capacity when the room was clear, other than a few enduring screams and realizations of lives coming to an end. Michaels and Roman helped one another out of the Dusk as Kato walked behind them, Mauler perfectly positioned to slay anything with a heartbeat that followed the crew outside.\n\n\"Damn I love that gun.\" Dalton said from the cozy confines of the Gunship, which was parked in the middle of the street and attracting a lot of local attention.\n\n\"Get us the hell out of here!\" Michaels said in a weak voice as they passed Kelly on their way up the ramp.\n\nThe roar of the Gunship's engines combined with what seemed like an endless cloud of dust as the vessel began to liftoff. Several soldiers accompanied Anwick to the front of the establishment, firing streams of metal piercing shots into the direction of the ship. Only moments later, following a loud burst of air meeting motion, the ship was gone. The soldiers began to check the dead and wounded as Anwick stood there for several minutes, his eyes still locked onto the sky.\n\n\"All Clear.\" Dalton proclaimed as he made his way down the ladder from the gunner's roost and back onto the bridge of the ship.\n\n\"We need to talk.\" Adam said to to the woman as she continued to harbor the boy they had saved not even an hour ago from the clutches of the closest thing to the devil that he had ever known. She nodded and began to stand up, her long champagne colored hair giving the shimmering illusion of velvet perfection as she made her way toward the Captain and the door of the ship's bridge.\n\n\"Hey big man. First time in space?\" Roman asked as he took a seat beside the boy, who was obviously afraid of the situation he had been cast into.\n\n\"Yes. Yes sir.\" the boy answered with much reserve.\n\n\"You can relax, you are safe and among friends here.\" Roman replied, before asking the child his name.\n\n\"Troy.\" the boy responded a couple of seconds later, sounding much less timid than before.\n\n\"Care to explain to me what in the hell just happened back there?\" Michaels asked in a puzzled and desperate tone. Before she could answer his question, he interjected with a bit of sarcasm. \"Feel free to start with who you are and what the Hunters want with you so badly?\"\n\nIt was a legitimate question from where he stood. Adam had a lot of things racing through his mind at the moment. How Roman had handled himself so well against an opponent who appeared far more capable, how many drinks Dalton had actually consumed back at the Dusk and still been accurate with a weapon or how one of the most beautiful women he had ever laid eyes on, stood in front of him about to answer a question that would most likely affect the remainder of of his life.\n\n\"My name is Sarah Blaine,\" she answered in a soft voice, something Adam hadn't heard a lot of in his line of work.\"My father is the Admiral of the newly formed Colonial Army; they want to use me as leverage against him, at least that would be my guess.\" Sarah added, waiting on a response from Michaels.\n\nHe didn't reply, a combination of letting her finish her story and the angelic sound of her voice were to blame. After several tense moments of silence, she began telling Adam how the Hunters had been working with Legion troops for some time now, how her security detail was ambushed back on Tameca and over a dozen soldiers had given their lives to try and protect her.\n\n\"The boy?\" Michaels asked, almost dreading the answer, and praying that it didn't complicate things any further.\n\n\"I have no idea. I had never laid eyes on him before they drug him into that room.\" Sarah replied, watching the stress overtake the Captain as he stood there speechless.\n\n\"You mean to tell me that my crew just put their lives on the line against the Hunters for a kid that you don't even know?\" he asked furiously.\n\n\"Relax Captain, the boy is still alive because of you and your crew's heroic efforts.\" she replied, standing closely to Adam as they both began to feel a nervous interest in one another.\n\nAdam wanted so badly to be furious with Sarah, to lash out at her with anger. Instead, he found himself thinking of how perfect she was in every aspect. \"For what it's worth, you did the right thing back there.\" Sarah said with a soft and thankful voice.\n\nHe looked at her for a moment, his eyes communicating a tale of great sorrow to come. \"Fantastic. Carve it on my headstone, because once the Hunters catch up to me I'm a dead man.\" he finally replied as he walked away, making his course back onto the bridge.\n\n\"Kelly, Luck, set course for deep space. We need to get as far away from the system as possible for a bit; avoid any Legion or Hunter tracking ships that may be looking for us.\" Michaels said as he sat down in an empty chair, physically exhausted and mentally broken down from the day's events.\n\n\"Yes sir Captain.\" Kelly replied as she and Luck both pressed several buttons on the control panel illuminated in front of them.\n\n\"Thanks for saving our asses back there.\" Dalton said as Kato continued checking the navigational system with a small, hand held data machine.\n\n\"Just doing my job.\" Kato replied, maintaining his attention on the work in front of him.\n\n\"You do your job well.\" Roman added as he stood in the doorway of the engine room, Troy right at his side.\n\n\"Thank you.\" he added, giving a long stare of appreciation to the man he had gotten off on the wrong foot with.\n\n\"You're welcome.\" Kato replied, glancing up for a moment acknowledging the sincere thank you.\n\nAs Sarah walked across the steel catwalk above the cargo area, she stopped for a moment to watch Roman and Troy. The highly trained former soldier was walking the young boy through a set of moves with a combat knife. It was a basic set of motions, still it was impressive to see a child handling a blade built for men of combat as well as he was. Lunging forward quickly, then swiping the blade swiftly as he brought it back in to a neutral position. She hadn't known any of these people for long, but she was glad to see Troy finding someone to begin looking up to. At least the time he spent with Roman was time he wasn't thinking about home, if he even had one.\n\n\"Could be worse, he could be learning Dalton's craft.\" Adam said as he approached Sarah, who was still standing there, admiring the young boy's courage.\n\nEventually she broke her stare and glanced at the Captain for a moment, joining him in placing attention on Dalton. The hardened space cowboy was laying on the deck below them drinking from a large green jar. Its contents were unknown, but from the way he was cradling the thick glass container, it was something of significant importance to him. He glanced up at the catwalk with a huge grin on his brushy face.\n\n\"Nice view huh?\" he said, implying he was well worth the effort.\n\n\"So, if you don't mind my asking, what's the plan? I would very much like to see my father again.\" Sarah asked.\n\nThe Captain tightened his grip on the guard rail of pale blue steel, continuing his stare below for a few moments. \"Well, we are painted into a corner at the moment,\" he replied, while finally looking into Sarah's direction. \"We can't go back to the Skyla System, at least not for a while. Sure to be plenty of hunting parties out there looking for us at this very moment. On the other hand, the supplies aren't going to last forever; eventually we're going to have to make our move.\" he added.\n\n\"We are on the verge of war, a war of a scale that we have never seen I'm afraid,\" Sarah said. \"I fear that not so long from now, the entire system will be engulfed in death and destruction. At some point Adam, you will have to decide which side you're on.\" she added.\n\n\"Doesn't matter. I've got a sheet of outstanding warrants as long as your leg, and now I have the Legion as well as the Hunters out for my blood. I'll have you back with your father soon enough, you have my word. In the meantime, we need to fall off of the map so I can figure out our next move.\"\n\n\"Well, whatever you decide, I'm confident that it will be the right move.\" Sarah said in a thankful tone as she began to walk away, reaching the other side of the catwalk before Adam finally responded.\n\n\"Sarah,\" he said as she turned to face him, her comforting smile causing the Captain to fall silent for a second. \"I know we may not look like much, but I won't let any harm come to you or the boy, I promise.\" Michaels added in an attempt to reassure her of safety from the Hunters.\n\n\"I know that Captain.\" Sarah replied, smiling brightly as she she gave an appreciative nod and slowly turned to walk away.\n\nAdam stood there for a couple of minutes after she was gone, trying to convince himself that Sarah wasn't someone he should be falling for. They were from two different worlds, both literally and socially. Still, he began to feel his heart drifting away from logic as he heard Dalton's voice yell from below.\n\n\"She's a keeper Capt'n.\" Adam glanced down for a moment, giving a sarcastic look, even though he couldn't agree more.\n\n\"No sir Captain, I checked three times. The readouts aren't normal.\" Kato said of the navigational information he had recently pulled.\n\n\"Meaning?\" Adam replied, eagerly awaiting a positive answer.\n\n\"Not sure. It's not time to panic; just thought I would bring it to your attention.\" Kato answered.\n\n\"Look into it.\" Michaels said, giving Kato a quiet slap on the shoulder as an unofficial thank you.  \n\"This area looks good.\" Adam said as he leaned between Luck and Kelly, pointing out a small spot on the pilot's grid.\n\nThe bright neon green lights and numbers of the panel illuminating an otherwise dark bridge. \"Sir that area is uncharted.\" Luck responded after glancing down at the grid for a split second.\n\n\"Uncharted is what we need right now.\" Adam replied as Kelly began punching the coordinates onto a touch screen beside her.\n\n\"Should take us a better part of twenty six hours to get there Captain.\" she said as she double checked the numbers before finalizing them into the ship's navigational system.\n\n\"I can do twenty six hours. Set course, I'll let everyone else know.\" Michaels said as he began to leave the bridge while Kelly pressed a sequence of keys that pushed the ship's thrusters even harder; nearly doubling the torque.\n\nMichaels motioned for Roman as Dalton continued, slightly intoxicated, to show Troy how to aim down the sights of an unloaded combat pistol.\n\n\"Hitting an uncharted area in a few days. Gonna sit tight there for while, wait until this thing blows over. Hate to keep the kid off world that long; no choice though, we go back now and we won't last an hour.\" Adam said in a low voice.\n\n\"No hurry. His entire family was executed by the Hunters for aiding the Colonials with supplies. Poor kid, I feel bad for him.\" Roman responded sympathetically.\n\nAdam felt deep sorrow for Troy; glad he had made the decision to rescue the boy from the clutches of the Hunters, while knowing exactly what it felt like to grow up alone.\n\n\"Well, just try and keep him as busy as possible, he seems to cling to you and our resident drunk.\" the Captain replied. Roman agreed as he turned to join Dalton and Troy in learning to do something he had known how to do for decades, handle a sidearm. Of course, hearing the instructional advice from a man with slurred speech was like learning all over again.\n\n\"Give us a couple of weeks Captain, this boy gonna be dropping Hunters left and right!\" Dalton yelled as Troy smiled wide.\n\nMichaels could only try and imagine the grief Troy must be feeling, having those monsters slaughter his entire family, but at least he was safe now. No matter how feared the Hunters were across the system, he knew that his crew would fight to the death for its two newest passengers.\n\nUsually during deep space travel, Adam would sleep very little and spent most of his time admiring the stars and enjoying true freedom. Not this night however. He had been beaten, nearly killed by gunfire, quite possibly met the woman of his dreams and taken on the responsibility of a child with no home to return to. There would be a lot of sleep in the Captain's future on this night, and he wasted no time; hitting his bunk and escaping into a dreamworld of infinite possibilities only minutes later.\n\nOpening his eyes, Adam was first struck by the absolute silence of the ship's thrusters. A person is taken back by the loud roar and torque of things during their first few trips into the blackness of deep space. Adam had been through it hundreds of times however, and he had grown used to both of them. It was the lack of noise and vibration throughout the ship that raised sudden alarm with the Captain. Jumping to his feet, he grabbed his trusted gray revolver from a solid steel bedside table and used his free hand to rub his eyes for a moment, trying his best to wake up and be clear headed as he expected the worst. Slowly making his way down the narrow corridor, his right hand gripping the sidearm as he checked all of the crew's rooms, every one of them coming up empty.\n\nHad the Hunters somehow found them? Legion soldiers maybe? Many scenarios played out in his mind as he continued his slow but determined walk, trying his best not to make any sound in doing so. As his feet hit the catwalk, he slowed his pace down considerably; knowing full well that its grated steel flooring was notoriously loud to begin with. Glancing over the rail as he slowly made his way to the center, he spotted his entire crew down the iron sights of his combat sidearm.\n\n\"Raise it by two credits.\" Dalton said as he threw a handful of poker chips into a pile.\n\nAdam's initial reaction was that of relief, his crew was safe and sound. As that feeling began to fade, he began to wonder why they hadn't invited him to play in a game that he considered himself to be such an expert in. The Captain walked down the metal steps in a much louder fashion, immediately gaining the attention of the entire crew.\n\n\"You would start a card game without me?\" Michaels said as he approached the table.\n\n\"Hell, you been asleep for nearly two days.\" Dalton replied, a primitive looking cigar that looked as though it had been hand rolled by a blind man hanging from the corner of his mouth. Adam shockingly gazed at his green and brown military style titanium watch, astounded that he had slept for such an extended period of time.\n\n\"We're sitting in uncharted space Captain, thought it best to kill the main engines, save fuel and not gain any unwanted attention.\" Kelly said, waiting for the Captain's approval.\n\n\"Good job,\" Adam said as he pulled up an empty chair, sitting beside Troy to help him play his hand with the best chance of winning. \"Where's Luck?\" Michaels asked, doubting anyone even noticed that he wasn't present.\n\n\"On the bridge running more schematics on the ship's vitals.\" Kato replied as he laid his spread on the table, smiling and raking the winnings in his direction.\n\nUnder normal conditions, Adam would remain focused on the hand of cards in front of him, however these weren't normal conditions. They were on the run, sitting in uncharted space and Sarah Blaine sat across the table from him, her beauty almost impossible to put into words. She had a look of such magnificent innocence, every feature of her face glowing with undeniable passion. He tried to keep it at nothing more than a passive glance in her direction every few minutes, until one of his glances had been answered by one of her own. How could he have let it happen? He dared not glance back for several minutes, going out of his way to ignore her while showing Troy the finer points of becoming a world class card shark in only a matter of minutes.\n\nFinally gaining enough courage, he quickly joked with the crew and threw a glance her way; only to find that she had already been watching his every move. Sarah smiled softly as she continued to stare at him, causing him to begin playing nervously. He couldn't understand what was happening. Since when did he ever get nervous around women? In his line of work he met plenty of them, a lot of them tougher than the average girl; most of them could handle a weapon as well as any man and could hang drink for drink with Dalton himself.\n\nNobody like Sarah however, not even close. She was stunningly beautiful, very well spoken and above all, political. Political? His idea of politics involved small arms fire or even a good old fashioned bare-fisted brawl. They had nothing in common aside from being here at the same moment, the same spot together; something he knew deep down would be long forgotten by Sarah the moment she arrived safely back home. Still, the heart wants what the heart wants, and his was becoming increasingly dependent on seeing her as much as humanly possible.\n\nMaybe he was reading too much into her smile addressed to him, he had to be. A love story between two worlds as different as their own is destined to fail.\n\n\"I think Troy has the hang of it,\" Michaels said as the boy played the winning hand and collected the pot with a smile that was larger than Dalton's tolerance for alcohol. \"I'm going to go check on Luck, you guys enjoy the rest of the game.\" Adam added, being much more polite than usual.\n\nAs he stood up to make his way to the bridge, he glanced at Sarah once more, earning another inviting smile as he nodded his head slightly before walking away; trying to concentrate on not walking nervously as every single step landed abruptly onto the cold steel flooring.\n\nBefore the Captain could make it to the bridge, all hell started to break loose. The last thing you want to see during flight as a passenger, much less a Captain is pitch black. Several seconds of darkness stretched throughout the ship as Michaels reached around and found one of the corridor walls to hold onto. There was a flutter of light throughout the ship for a couple of seconds before going back to complete emptiness. The emergency lights, which were nothing more than small red bulbs mounted into the ceiling throughout the ship powered on, coinciding with a very primitive sounding siren that echoed through the entire vessel. Providing very little light, the red glow produced just enough for the Captain to begin sprinting in the direction of the bridge.\n\n\"What the hell is going on?\" Adam shouted as he entered the bridge. Luck, sitting in one of the two pilot chairs spun around to face him.\n\n\"My guess is we took gunfire when leaving Tameca, must have hit our navigational system. Even worse, it wasn't until we went dark that I tried to fire everything back up. One of the thrusters is not functional and the life support system is substantially strained.\" Luck replied, emotionless as only an android could be in this situation.\n\n\"Can it be fixed?\" the Captain asked frantically, taking a seat in one of the vacant leather chairs.\n\n\"No time sir. With limited life support and only temporary use of our second thruster, we are going to have to land somewhere close and soon. Going to be a hard landing at that.\" Luck replied.\n\nAdam buried his face for several long moments in the palm of his right hand before standing to his feet.\n\n\"Find a suitable planet, I'm on my way to break the fantastic news to the crew.\" Adam replied as he walked quickly out of the door leading to still dimly illuminated hallway. Fucking deep space. he thought as Luck began taking scans of nearby planets.\n\n\"So, secure as many of your personal belongings as possible in your rooms and meet back in the armory. It's the safest spot on the ship, and from my understanding the landing won't be pleasant.\" Adam said as the crew looked on, dazed and facing reality.\n\n\"Are, are we going to be OK Captain?\" a scared Troy asked, having already been through a lifetime of trauma in the past few days.\n\nCaught completely off guard by the young boy, Michaels grew very quiet, unsure of how he should answer. Sarah bent over, placing her hand on the Troy's shoulder.\n\n\"You just stay right beside Roman, everything will work out just fine.\" she said in a very calming voice, glancing at Adam for a moment. He returned a nod of thank you as he quickly made his way back to the bridge with Kelly.\n\n\"He's right Captain, we either land to make repairs or we run out of air to breathe.\" Kelly said after looking over the calculations Luck had made.\n\n\"Fine. Let's do this.\" Adam said as he looked at a map of recent scans of nearby planets.\n\n\"This looks to be the spot sir, all of the planet's vital check out and there's a decent chance we could survive the landing.\" Luck added to the discussion, pointing to a spot on one of the black and white scans.\n\n\"Expecting trouble?\" Sarah said as Dalton filled a field bag full of weapons to sit right beside a slightly smaller wooden crate that he had stocked with his finest drink.\n\n\"I like to stay prepared for anything.\" he replied with a slight grin.\n\nAs they made their way to the armory, the rest of the crew had already strapped in. Thick nylon harnesses were bolted to the wall, surrounding the weapons cache in the middle of the room. Kato held his restraints, nervous, but doing his best to hide it from the rest of the crew. Roman appeared to be expecting a lot less trouble, with only a combat blade strapped to his right thigh; he looked as though he was on the verge of falling asleep, Troy strapped in snugly right beside him. As Sarah strapped herself in with help from Dalton, he turned a small flask up and finished its contents before slinging it onto the floor and climbing into position.\n\n\"Well. It's go time.\" he said with confidence, almost as if he had done this dozens of times.\n\nTwo spots left. Sarah thought as she imagined the Captain would pilot the ship down to the surface. What a brave man. A man's man. she imagined as the red glow of the emergency lights did little to brighten the armory.\n\n\"Luck is going to pilot the ship down.\" Michaels said as he and Kelly entered the room and began strapping in. Smiling slightly in Sarah's direction, completely unaware of the faith she had placed in his courage just moments before.\n\nAnyone who has has been fortunate enough to survive a ship to surface crash can attest to the same thing. The fall. Feeling absolutely helpless as gravity pulls you down faster than any human being should ever go, having nothing more to cling to than a safety harness bolted to the steel plated wall as you pray for forgiveness and intervention.\n\nObjects began to fall throughout the armory as the Gunship dove into a deep descent that slowly led to a nosedive into the unknown, uncharted confines of the planet that the Captain had picked under pressure. It was the first time in a long while that Adam was afraid of dying.\n\nSure, the Hunters could have easily killed him back on Tameca. Still, he had a weapon in hand and at least some control of the situation. Falling from the sky at breakneck speeds tended to make him feel just a bit more helpless. Although he feared for his own life, he feared for his crew and the condition of his ship just as deeply.\n\nGlancing around the room, everyone's individual feelings became crystal clear as their faces revealed all. Sarah cried softly, trying to comfort Troy while doing so. Kato stood solid, grasping very tightly to his safety straps, fingers red from the pressure of his clinched fists. Roman showed no fear, holding his crossed arms securely against his chest while his eyes remained closed. Dalton mumbled under his breath, leading the Captain to question if he was praying or cursing. Either way, he held a field supply bag full of weapons and a wooden crate of drink firmly against the wall with his large black boots.\n\nThe ship jolted roughly, causing a massive pull of gravity to one side of the vessel, nearly pulling the crew out of their confines and sending the wooden crate of alcohol smashing against the weapons cache in the center of the armory; shattering its contents.\n\n\"Oh no,\" Dalton yelled angrily as the crew looked on for several moments, waiting to see if he was brave enough to free himself in an attempt to salvage anything he could from the crate. The overhead lights fluttered for a moment before coming back on permanently, making visible what looked like a stream of ale and other toxic refreshments running out of the armory and falling to waste over the side of the catwalk. \"Oh hell no!\" Dalton added, overwhelmed with discouragement before clinching tightly as the ship dipped into a sudden spinning dive.\n\nAdam heard the ship's thrusters trying to engage, which let him know they were inside the planet's atmosphere and it was almost over, for better or for worse. He could tell because of his extensive time aboard the vessel that the sound he heard was only one of the two thrusters, meaning the other was still malfunctioning.\n\nHe could only hope that Luck put the ship down in a spot that would do the least damage possible. If any of the crew survived, they needed enough of the ship left to attempt a repair job, otherwise it could be a very long time before they would leave this rock. If ever.\n\nThe entire crew could hear a loud alarm throughout the ship, those among them who had military training knew that it was a proximity alert, the time had come. Adam glanced at his crew one last time before settling his eyes onto Sarah Blaine.\n\nHe hoped that some higher power had not purposely placed the woman of his dreams into his life, only to end it a couple of days later. I'm sorry for any wrong I've done. Please, just give me a second chance to make things right, a chance to do something positive for a change. Adam thought.\n\nA big part of him wanted to yell to her loudly that he had started to feel himself falling in love with her, no matter how forbidden it was in the eyes of society. What did he have to lose? Adam had convinced himself that if it truly was the last thing he would ever do, he needed to tell her. But there was no time.\n\nThe impact immediately caused the Gunship to go completely dark, sounds of metal tearing from its hinges all around them. Several of the bolts had torn from his harness upon impact, still enough of them remained intact to hold Michaels in place, he had hoped that the rest of his crew had been so lucky. A couple of minutes after the initial impact, Adam realized that he had survived. Good chance his back had been bruised from beating against the thick plated wall, but he was alive. Sparks flew as the main light above them shorted in and out, the live current hitting exposed wires that now hung above their heads.\n\n\"Everyone OK?\" Michaels asked with true concern, although most of his focus was in Sarah's direction.\n\nThe Captain feared the worst as not a single person responded. Maybe he hadn't spoken loud enough for his voice to overpower the sounds of electric showers and hemorrhaging pipes.\n\n\"IS EVERYONE OK?\" Adam repeated loudly, strong enough to be heard several rooms over.\n\n\"Hell no I'm not! You saw the crate shatter to pieces, all my scratch gone in a matter of seconds!\" Dalton replied, prompting laughter and relief from the rest of the crew, putting the Captain's worries to ease.\n\nAs the crew slowly found its way through the complete darkness to the cargo hold doors, Michaels and Kelly made their way through the mangled corridors which were littered with scattered debris, heading for the bridge to congratulate Luck on a job well done.\n\nThe first one out was Dalton, solid black riot style shotgun in front of him, he slowly made his way down the ramp, which was slightly bent into an angle. Dalton, of course, still steaming over his forced alcoholism intervention.\n\nWhile it wasn't worth a plummet to the surface at blazing speeds in the face of impending death, the scenery wasn't hard on the eyes in the least. Vibrant green grass swaying with a slow wind as they sat in a huge field surrounded by small hills and a distant tree line. It looked almost like a destination for someone in search of a getaway, rather than a spot picked at the last minute for a crash landing.\n\nDalton stepped from the ramp onto the welcoming soil, which gave way under his boots, moist from a recent rain as he began panning his weapon around slowly. The next person down was Roman, walking a bit more calmly with a tight grip on a large combat rifle, he immediately made way to the rear of the ship to try and assess the damage. Seeing a large trench dug behind the ship, dragging out for several hundred feet from the recent impact of the crash.\n\nSarah and Troy, obviously shaken up quite a bit, stepped onto the ground and remained close behind Dalton as he continued to thoroughly evaluate the landscape. Kato exited the ship slowly, holding a small pistol by his side and walking slowly toward the group as the Captain was right behind him.\n\nAdam's first thought was of the sun hammering its rays down onto the ship, the slight wind blowing just enough to keep the otherwise intense heat at bay. They could have landed in much worse conditions, leaving him not only thankful for their new lease on life, but a second chance to tell Sarah exactly what was on his heart. All of it would have to wait for the time, however, as he faced the daunting task of giving the shell shocked crew even more bad news; walking down the ramp with a heaviness of heart to face his crew.\n\n\"Luck?\" Kato asked as Michaels approached them solemnly.\n\nAnswering with only a regretful head shake, it quickly hit everyone that the android pilot, the hero who had saved them from a certain death, had himself perished in the crash. Moments later Kelly walked down the ramp, crying just a bit; not really knowing how to react to the demise of a synthetic human being.\n\n\"Now what?\" Dalton asked as he strategically placed a large cigar in his mouth, nearly the length of a human hand as he sealed the deal with what could have passed as a miniature torch.\n\nGiving him a scowl, Kelly walked to the rear of the ship with Roman, Dalton looked into the direction of Michaels with a puzzled look on his face before shrugging it off.\n\n\"Good question.\" Adam replied a couple of minutes later, looking at the nearby surroundings in the vicinity of the heavily damaged ship.\n\nHe honestly didn't know what move to make his next. Even if they could somehow find a way off of this rock, they were wanted by pretty much anything in the Skyla System that was attached to a heartbeat. The Legion wanted them locked up, especially after the assault that Roman had played such a vital role in. If they were lucky enough to avoid the legitimate authorities, the important people of the underground world surely wanted their heads on platters, and Adam was pretty sure there was a decent sized bounty attesting to the fact; meaning they would eventually have to deal with contract killers and mercenaries along the way. None of it mattered right now though, because from the looks of it, the ship was in no condition to fly; even if they somehow managed to repair it.\n\n\"I'll put Luck to rest, he may have been artificial but he deserves at least that much for saving everyone's life. You get with Roman, gear up heavy and scout the area, Kato and Kelly can look the ship over and assess the damage, see how long we are going to be stuck here.\" Michaels said, finally giving Dalton a definitive answer.\n\n\"What about Sarah and the boy?\" Dalton asked as he started checking the weapons he had with him.\n\n\"They'll stay close to the ship, should be safest for them that way.\" Adam replied as he began to head inside to collect the body of someone very important to him.\n\nNo matter how ridiculous it sounded, Luck had become what the Captain considered to be part of the family, he would truly be missed.\n\nDalton and Roman were geared up and ready to move, waiting for the sunset to begin before heading out to survey the area. It was Roman's suggestion to travel at night, it would be a bit harder to move around, but the shroud of darkness would provide additional cover for the two man team in the event that there was life on the planet. Even worse, if they had been tracked by Hunters, they would need every advantage possible in order to try and even the playing field a bit.\n\nDalton was going heavy of course, with a large assault rifle, standard combat pistol and one of the oldest pump shotguns that Roman had ever seen. The majority of the gun was comprised of a faded blue steel, showing signs of significant use throughout the years. Mounted to a thick wooden stock that was marred with scars of abuse and age; it reminded Roman of something seen in a museum. Of course, he could respect the fact that Dalton knew his iron, so he had no doubt that it would follow through with the task of keeping the peace when called into question.\n\nAlthough no bottle or flask was visible, Roman suspected Dalton had one somewhere in those faded brown pants of his. Most of his stash had just been disposed of in the most unthinkable of ways, so it was very possible that he simply had began to ration what remained of the alcohol.\n\nAs for Roman, he carried a standard issue combat pistol as well as a rugged military grade knife. Most of his gear consisted of rations, binoculars, a chart machine used for surveying and graphing a map of the area and a large com unit that could reach the ship from several clicks away. Michaels had insisted that they stay in contact, just in case they had indeed been followed. It was very unlikely, however, as the Captain made his living by walking with caution and today was certainly no different.\n\nThe two men headed out, wading through the tall grass as Dalton took intensely deep puffs from a cigar, the red embers of the burning end glowing in the darkened surroundings as Dalton glanced back to smile at everyone for a moment as if a parade had been assembled in his honor. Roman had left Troy with a snub nosed pistol and the very important task of protecting the crew. The weapon wasn't hot of course, Roman had removed all of the ammunition from it long before turning it over to the boy, still it gave the young man something positive to focus on and that's how Roman had gotten himself through the toughest of times.\n\nAs Michaels finished covering the grave of his high stakes poker winning turned close friend, Sarah and Troy approached him slowly, trying not to disturb him during a tough time in his life.\n\n\"I'm sorry about your friend.\" Sarah said apologetically as they sat on a decent sized rock near the Captain.\n\n\"Thanks. Truth is, I cheated my ass off the night I won him you know?\" Adam replied, trying to lighten the spirits of the boy, while impressing the girl that he had so quickly fallen for.\n\nThey continued to talk for a while, Michaels showing Troy his method of holding a pistol as Dalton and Roman disappeared from sight, beginning to hit rocky terrain littered with thick brush and small patches of lush trees.\n\nThe two men moved quietly as they made their way to a high peak not far from the crash site, the only noise was an occasional curse word ejected from the lips of a painfully sober Dalton. Obviously high ground would save them a long road ahead if they saw anything of interest. Dalton sat down out of breath on a fallen tree trunk as Roman glared through the his set of binoculars, panning carefully as not to miss any details as the digital readouts of the eyepiece showed distance and altitude.\n\n\"Anything interesting?\" Dalton asked, as he complained through body movements of the short walk they had already taken in, obvious after effects of his motivated smoking habit.\n\n\"Nothing,\" Roman replied as he attached the set of lenses to the sketch unit, allowing it to print out the land they had already covered. \"Not a damn thing.\" he added.\n\nAs the men continued on, down the slope and toward the next decent sized tree line, a slight rain began to fall that quickly turned into an unrelenting downpour.\n\n\"Damn, should have just let them arrest us back at Paulie's and been done with it.\" Dalton said as both me started laughing out loud, cold rain hitting them briskly in the face as they pushed forward.\n\nFinally, they found a thick enough section of trees to shield them from a good bit of the rain, still it had taken its toll on the men; clothes soaked and the feel of filth all over them. Dalton kept his stash of cigars tightly wrapped in a thick brown cloth and tucked away in his supply sack along with God knows what else.\n\nTaking shelter under a huge overhanging slate rock, they decided it was time to make camp for a bit and start back at first light. Dalton offered to take watch first, looking above and quickly painting the stars across what sky could be seen through the thick clouds overhead. Roman had heard enough complaining as it was, however, and didn't want to add legitimacy to it; taking first watch so Dalton could get himself a little shut eye.\n\nQuickly taking up the offer, Dalton climbed into the stained green sleeping bag and rolled onto his side, shutting his eyes and within seconds drifting off into a dreamworld that was sure to include alcohol, women and an endless supply of guns and ammunition. Meanwhile, Roman sat solidly with his back against the rock and a battle rifle casually in the elbow sections of his arms, resting himself while indeed ready for anything.\n\n\"Captain.\" Kato said as he approached Michaels, who himself was pulling watch back at the crash site. Sitting at the top of the damaged cargo ramp, Adam was armed only with a solid black military issue pistol and a miniature set of binoculars, keeping watch on the tall grass around them while most of the crew slept.\n\n\"What is it?\" Michaels asked quietly as Kato walked over, standing next to the Captain's position.\n\n\"One of the thrusters is completely gone, from the looks of it, we took gunfire on the way out of Tameca.\" Kato said regretfully.\n\n\"Fixable?\" the Captain asked with glaring hope.\n\n\"It gets worse sir,\" Kato replied, pausing before sharing the bad news with the man in charge. \"The gunfire hit our main fuel line as well, been leaking it out since we took off, good thing we landed when we did, otherwise we'd be stuck in orbit with no fuel to speak of.\" he added.\n\n\"So we look for some kind of fuel source, patch it enough to get into orbit and then send a distress signal. On the run or not, sounds like our only option at this point.\" Michaels answered, prepared for the worst.\n\n\"It gets worse.\" Kato said apologetically as the Captain shook his head for a moment, replying \"How much worse can it possibly get?\"\n\n\"Even if we found a fuel source, which is unlikely at best, four of the seven battery rods were damaged beyond repair. We got enough juice for systems check, possibly heat or emergency lighting if we needed it, but nowhere near enough to launch into orbit,\" Kato added as Michaels continued to shake his head. \"Truth is Adam, we look to be here for a very long while.\" Kato said as he slowly turned and walked off to make his way back into the Gunship, trying his best not to wake the rest of the crew in the process.\n\nMichaels continued to sit on watch, skimming the area with his eyes in the darkness and cursing the moment that Dalton's stash had crashed into a flood of waste. He could damn sure use a drink himself right about now. Stuck on an uncharted planet, next to no chance of them fixing the ship well enough to get off of the ground and being marked men throughout the entire Skyla System. Adam was definitely in the mood for some hard drink, something to to alleviate his mind of problems, if only temporarily.\n\nRoman had settled in and slept for nearly two hours, when Dalton once again dozed off while pulling his shift of watch. It was almost daybreak, the idea was to get a few extra winks in before they were up and at it again, a full day's walk in front of them. Normally, he was a deep sleeper, nobody had ever come close to accusing him otherwise.\n\nIt was the sound of a small branch being pushed into the ground softly that did the trick, Dalton immediately spinning around with his shotgun, eyes wide open and looking into a thin, dark complected man who was obviously scared and appeared to be unarmed. Throwing his hands into the air, they had almost extended fully by the time Roman was on his feet and quickly approaching Dalton.\n\n\"Enough. Can't you see he's scared shitless?\" Roman said of the trembling man as he pushed the barrel of Dalton's shotgun to the ground, making sure it didn't accidentally discharge into the direction of the native stranger.\n\n\"We mean you no harm stranger. My friend and I, we are having some difficulties with our ship; just looking for anything or anyone to possibly help us get back into space.\" Roman said as the stranger continued to look at the barrel of Dalton's so called peacemaker.\n\nRoman grabbed the weapon from Dalton's clutches and threw it onto the ground, holding his hands up slightly to show the strange local that they truly meant him no harm.\n\n\"The name's Roman, this is my friend Dalton.\" After several moments of silence, the stranger finally spoke.\n\n\"My name is Aira.\" he said, his language broken at best as he remained very skeptical of the two men's intentions.\n\n\"Alright Aira. No way you could survive out here dressed like that,\" Roman said, referring to his thin white shirt that had been poorly stitched and makeshift pants of sack cloth. \"Means you must have a camp nearby, can you take us there?\" Roman added in a friendly tone.\n\n\"My village is not that far from here. Yes, I can take you there.\" Aira replied a bit more calmly.\n\n\"We would appreciate that.\" Roman answered as Dalton had picked his shotgun up and was wiping the dirt from its stock.\n\n\"Actually, I would appreciate not having such a fine weapon thrown to the damn ground like a piece of trash,\" Dalton added as he wiped the dirt from its well worn handle and holstered his weapon once again under his thick leather belt.\"Besides, from the looks of him, I doubt they will be of any use.\" he added.\n\n\"Maybe not, but even the most primitive of races stockpile what they consider to the be the best of drink.\" Roman said convincingly.\n\n\"Lead the way friend.\" Dalton replied, speaking to Aira as if they were long time friends.\n\nThe two men quickly disposed of what had been a campsite through the night and began following Aira to the place he considered home.\n\nAs Captain Michaels started to awaken, he was hit with the overwhelming smell of a cooked meal. Easy to distinguish when you become used to eating rations from airtight silver packages three times a day. He wasn't sure what was being cooked at the moment, but it smelled like heaven and that was easily good enough for him. Standing to his feet, he took a moment to try and work a few kinks out of his body from the past several hours spent sleeping on the very unforgiving ground, and then glanced over in Kato's direction.\n\n\"What's going on? What am I smelling?\" he asked, knowing Kato would have the answers having taken over watch in the middle of the night.\n\n\"Sarah's cooking. Found some rations that she said would taste a bit better cooked over a flame, so she's been working on it for the past hour now.\" Kato replied as Michaels walked into his direction, trying his best to fan the wrinkles out of his clothes by hand.\n\n\"Sure in the hell smells a bit better.\" Adam said, both men chuckling for a moment before he glanced around the corner of the ship to see Sarah doing her best to prepare a decent meal for the crew.\n\nAs the crew ate the thinly sliced and perfectly cooked meat, Adam continued to make strong eye contact with Sarah, the woman who had in may ways given him new reason to continue living.\n\n\"Captain, it's Roman.\" Kato said, handing Adam a portable com unit with one hand while holding a fork full of the tender meat in the other.\n\n\"This is Michaels, go ahead.\" the Captain said; trying his best to cover up the joy of the current meal in his voice.\n\n\"We have made contact with some natives here, several hours from your position. Will be speaking with more of them shortly, how are things on your end?\" Roman voice asked through the crackle of the com speaker.\n\n\"Well, we are making it alright. No home cooked meals, but we'll manage,\" Adam replied, prompting the entire crew to laugh out loud. \"Any chance the natives have replacement parts or fuel to trade?\" Adam asked.\n\n\"Doubtful, the scouts are using sharpened sticks as weapons.\" Roman replied.\n\n\"Outstanding,\" Michaels replied in a sarcastic tone, taking a few moments to glance around at the rest of the crew. \"Find out everything you can. Kato and Kelly have been working with the Rover, it's a little beat up but we should be able to use it if need be. At least until the fuel runs dry.\" Adam added as he laid the com unit down on a nearby piece of scrap metal.\n\n\"Will do.\" Roman replied as the steady crackling of the com went to silence.\n\nAs Aira led Roman and Dalton into the front of the camp, several of the villagers ran indoors, watching the group pass by as they made their way to one of the larger huts. Made of scrap wood, bales of grass and mud, it didn't like a place that anyone should even consider living in.\n\n\"Who knows, The Dusk looks like shit too but they had great liquor.\" Dalton said grinning as Roman just responded with a slight shaking of his head.\n\nMeeting someone in charge was never a pleasant experience, that is unless you were the one with the weapons, in which case it couldn't be matched. As Roman and Dalton entered the hut with Aira, they were face to face with an older gentleman, he wore many different necklaces made of shiny rocks and beads which gave them the indication of him being a very important man among the tribe.\n\n\"Welcome friends, I am Ceria,\" the older man said warmly, welcoming them to sit down on mounds of grass that had been formed to serve as chairs. \"Please forgive my people, they are no doubt afraid to see strange faces among them.\" he added, as he too sat with Roman and Dalton.\n\n\"Where are all of the men?\" Roman asked directly.\n\n\"And alcohol. Where is the drink?\" Dalton asked, stringing his question onto the rear of Roman's.\n\n\"Of course,\" Ceria said as he began pouring a green liquid into rough looking cups that were no more that hollowed out stones. \"The men are gone, most of them at least.\" Ceria said as he handed Dalton a cup of the local favorite.\n\n\"Out in hunting parties? I saw only women and children as we walked through the village.\" Roman responded, wanting to know every detail that he possibly could.\n\n\"No hunting parties,\" Ceria said, lowering his head in silence. After a few moments, he looked at the men and explained a ship coming every thirty moons or so, a handful of well armed creatures taking the men aboard, never to be heard from again.\n\n\"That is what my scout had first mistaken you for, but your friend's gun didn't look anything like that of the beast men.\" Aria added to the conversation.\n\n\"What the hell is wrong with the way my gun looks?\" Dalton asked as he began turning his cup up and stopping abruptly. \"This is some strong shit! What do you mean beast men?\" he added, gasping for air as a side effect of the hardened home brew.\n\n\"They never say anything, they only force men onto the ship, knowing well that we have no means of protection against them. Those who have tried to fight back have been slaughtered on the spot,\" Ceria replied. \"They walk as men do, but they are not men. Nearly two feet taller than any of us; brown tinted skin stretched across them as that of an onion, dark hair growing from them in thickets.\" Ceria added with remorse, thinking of all of the men who had departed against their will.\n\n\"This ship, how large is it? Are there any markings on it that may identify the creatures.\" Roman asked sternly.\n\n\"Twice the size of this very hut maybe.\" Ceria answered, as he drew a marking into the dirt floor of the hut, a large teardrop shape with a smaller shape inside of it.\n\n\"Don't like anything I've ever ran across.\" Dalton said, sipping his drink like a baby from its bottle.\n\n\"Husks.\" Roman replied in a grave tone.\n\n\"Husks? No worse than the Hunters that are no doubt combing the star system for us at this very moment.\" Dalton replied.\n\n\"Much worse. When Hunters have nightmares, they are usually dreaming of Husks. I've only ran into two of them in this lifetime, during my years with the Gali military.\" Roman replied, his eyes locked on Ceria.\n\n\"Well, you're here with us. You obviously killed both of the sons of bitches, so the crew should be able to handle three or four.\" Dalton said as he sat his cup down, not daring to ask for a refill of the strongest drink he had ever consumed.\n\n\"I ran,\" Roman replied, gaining an immediate look of disbelief from Dalton. \"I put five slugs of a pistol directly into the chest of one, it only brought him to a knee. No way in hell I could have dusted two of them,\" Roman added, before switching his attention toward Ceria. \"They usually have a handler with them, a human?\" he asked.\n\"Yes, that is correct. Usually two of the beast men and a human bearing a rifle.\" Ceria replied.\n\n\"The difference between Hunters and Husks, other than the obvious mismatch, is that Hunters kill because they enjoy watching humans die. Husks do everything for the money. Mercenaries you could say, the handler is usually nothing more than the highest bidder at the time. My guess is they are pulling men from this village for off world slave trade.\" Roman said as Dalton remained sitting with a puzzled look on his face.\n\n\"You ran?\" Dalton asked in disbelief.\n\n\"It's the one and only time I've ever ran from a fight. Believe me, I've regretted it every day since then.\" Roman replied as he stood to his feet and approached the door of the hut.\n\n\"Care for more?\" Ceria asked of Dalton, ready to refill his cup. \"No, I'm good.\" Dalton replied, wondering if he had ever muttered those words before.\n\nAs the next few minutes passed, Roman explained through the com system to Captain Michaels about the Husk and it was decided that Adam, Sarah and Troy would load the rover with weapons and supplies and meet up with the others, leaving Kelly and Kato to get the ship in the best condition possible. The villagers weren't expecting a visit from the beast menace for at least another ten days, giving Adam plenty of time to come up with something, anything that got them off of this piece of rock and back into the star system.\n\n\"If we could just send a hail through the com to my father, he wouldn't hesitate to send a rescue ship to our position.\" Sarah said as they traveled to the village, the bumpy terrain of the road that was no more than a path beaten by feet, throwing them back and forth inside of the industrial grade cockpit.\n\n\"The Gunship's com is barely functional, short range for the moment but no way is it capable of reaching off world. Besides, even if we could, as soon as the hail went out everyone with an open com would pick it up. Hunters, Legion, Mercenaries, hell even some of the higher end card clubs.\"\n\n\"What about the com unit of the ship that has the village so afraid?\" Sarah asked.\n\n\"Doubtful. We'd have to somehow overpower the Husk to use their ship,\" Michaels replied, glancing in her direction for a moment as Troy seemed nothing short of amazed by all of the controls inside of the rover. \"Not that I would ever back down from a fight, I just don't see the logic in starting one that we have no chance of winning.\" he added in an attempt to save face in front of the one woman that made his heart thump dramatically faster.\n\n\"We take them out, hijack their ship and use its com to hail Sarah's father, see if he could lend us assistance.\" Captain Michaels said as he sat inside one of the huts, huddled in the floor beside Dalton, Roman, Sarah and Ceria.\n\n\"I thought that wouldn't work?\" Sarah asked, blindsided by Adam's change of plans.\n\n\"Yea. You know I love a fight Captain, but if they are anything close to what Roman says, we'd be in for one hell of a thrashing.\" Dalton added as he regretfully sipped on more of the local drink.\n\n\"It's the only option we have, I've put my head around it a hundred times during the trip from the cash site. Take out the Husk or die on this planet, and personally I'd rather die fighting.\" Adam responded confidently.\n\n\"Even if we did manage to take down the Husk crew, wouldn't a hail bring the authorities right down on us? Not to mention the Hunters, or hell for that matter more Husk?\" Dalton asked.\n\n\"Yea, it would. We'd have to pray that Sarah's father got to us first, and be prepared to hold like hell if he didn't.\" Adam replied, looking in Sarah's direction long enough to see a her smile, every inch of her face glowing with radiance as she realized the Captain had placed his faith in her.\n\n\"Sarah, you'd have to be damn sure that your father sends enough of an escort to get us out of here safely, good chance they would hit some stiff enemies on the way back off planet.\" Adam said very seriously.\n\n\"Trust me, the Legion may have us outnumbered in the ranks, but knowing my father, he'll come with more than enough soldiers to get us back safely.\" Sarah replied with confidence.\n\n\"Husk wouldn't expect to find anyone here other than the usual villagers, the advantage of surprise would definitely be in our favor,\" Roman added to the conversation. \"Not to mention, I've got a score to settle with 'em, it's personal for me.\" he added.\n\n\"Well, we got plenty of weapons, the element of surprise and about nine or ten days to prepare for a fight. That should even the odds just a bit.\" Michaels added, making the plan official.\n\n\"Well, I still don't like it, but if there's going to be a fight then you can damn sure count me in.\" Dalton added, already in a fight against the cup of local drink that he had been so eager to get his hands on.\n\nAs everyone started to exit the hut, Sarah stopped Adam, holding him behind long enough to ensure they would be the last people to leave.\n\n\"Captain, why the change of heart?\" she asked cautiously.\n\nAs he stood there, possible reasons raced through mind. What should he tell her? What had truly changed his mind about meeting certain death head on? Was it his growing love for her, had it clouded his judgment? Finally, breaking several seconds of silence he decided to answer her from deep within his heart.\n\n\"It would be easy to lay low, but the truth is I've been doing just that for too long now, taking the easiest roads along the way. These people can't fight them off, hell I don't even know if we can fight them off, but if it's my time to go then I'd at least like to die for something I think is right, and for those I love,\" he said, staring right into her eyes as the words landed on both her heavy heart and confused mind. \"I guess I'm choosing my side.\" he added, earning a huge smile from her radiant face.\n\n\"I'm not afraid of dying, I'm afraid of living past this. Getting back to the system, only to have to sit and watch you walk out of my life. Having to see you walk away for good, that's what I'm afraid of.\" Michaels said, waiting on some kind of response from Sarah, one that would never come as she emotionally turned and walked outside to rejoin the crew and villagers without so much as a word.\n\nAlthough he wasn't ashamed of his feelings for her, Adam waited several minutes before walking outside himself, trying to give her plenty of time to leave the immediate area. He wasn't sure how she had taken his confession and didn't want to make things any worse than they possibly were at the moment. How could he have been so stupid!\n\nHe had overplayed his hand? Much like he had a habit of doing in a game of cards, except this was far more important to him than any game of chance. This was life, his life, and a life that he couldn't bear to even think of living without Sarah.\n\nOnce outside, the Captain was met by Dalton and Roman as they began pointing out the best spots for an ambush and putting together the intricate details of the plan. Adam heard their words, but every one spoken fell on deaf ears as his mind was preoccupied with his love for Sarah as he continued glancing over his shoulder; trying his best to locate his one true love and wondering how she had taken his earlier comments.\n\nThat evening as the sunset began to take form, everyone in the village would gather to eat a large meal, although basic, it appealed greatly to a crew that had eaten only rations for the last several days. Of course, some of them had been fortunate enough to enjoy Sarah's fine cooking, though they had been sworn to secrecy by the Captain as if the incident had never taken place. They had a solid plan when it came to fighting the Husk, which seemed strange to Adam. They could organize an assault on one of the deadliest species known to man, yet he couldn't explain his heart to Sarah without falling to pieces.\n\nHe only hoped that she would join them for dinner, it would give him some kind of idea as to the extent of damage his words had done to her. He couldn't control his feelings, his heart wanted what his heart wanted; but deep down he knew that a relationship between them could never survive.\n\nHe was nothing more than a smuggler who had no real future to speak of, even if Sarah felt the same way about him society would never allow them to be together. Her father, one of the top military figures for the newly formed Colonial Army would never allow his daughter to fall into the arms of a man with such a history of criminal activity.\n\nAs they sat and enjoyed dinner with the rest of the village, Adam remained focused on the empty chair that should have been filled by Sarah's overwhelming beauty. The biggest benefit of the crew spending time with the people of the village was to Troy, making friends with many of the natives who were not much younger than himself, giving Troy a sense of childhood for the moment without the worries or stress that came along with being alone.\n\nRoman continued to spend time with the young man, however most of the crew had noticed a different side of Adam's newest hired gun, almost a sense of urgency on his part as if to think that he would surely fall to the Husk.\n\nAfter dinner and a formal thank you to the villagers who had prepared it, Adam casually left the group, doing everything he could to find Sarah. If he was lucky enough to speak with her, honestly, he had no idea what he even planned to say. Apologize for the way he felt, maybe, although it was nothing that he had planned or could have avoided. On the other hand, maybe he would just reinforce his feelings to her as well as apologize for the inconvenient position that she had been unwillingly thrust into. One way or another, he had to do something. His heart and mind were too clouded to possibly stave off an assault by the Husk, he had to make things right.\n\nThere she was. As he entered one of the smaller huts, Sarah sat at a small table, the room barely lit by the flicker of a makeshift candle. Obviously deep in thought, Adam apologized for the intrusion and began to leave as Sarah stood to her feet, approaching him slowly; her warm hand soothing his tense shoulder with its touch.\n\nThey kissed deeply, with true passion and undeniable love. Even though it would surely have major repercussions if they were lucky enough to make it back to the system, at least for the moment Adam felt completely at peace. He tried his best to collect himself before responding to a few seconds in time that had truly changed his life forever.\n\n\"I know in my heart that when we return to charted space, the standards of society will eventually pull you back out of my life. All I ask for is this moment, to finally know true love, no matter how short it may last.\" he managed to say, forcing every word out as his heart continued to tremble.\n\n\"Shhhh.\" Sarah said softly as she held a couple of her fingers to his lips. \"Neither of us have any idea what the future holds. All we can control is right now, and right now I want to be with you. My heart has been with you since the moment our eyes first met.\" she added as he pushed her fingers away, grabbed her into the clutches of his arms and kissed her as if his heart must be connected to hers in order to survive.\n\nThe rest of of the night, they held one another tightly, their souls embracing one another as the rest of the crew and village celebrated what everyone had hoped would be the eventual downfall of the reign of terror that the Husk had forced upon them.\n\nThe next morning saw most everyone sleeping in late, after effects of the previous night's festivities. Adam was already awake, having been up for some time, he sat comfortably inside the Rover; watching the thick moisture fall from the sky, the water droplets bursting onto the windshield of the cargo craft as he sipped from a small cup of makeshift coffee.\n\nMost of his morning had been spent in deep thought, what a new life with Sarah at his side would mean for everyone in the unlikely event that his plan actually worked for a change. There had also been a few Dalton sightings throughout the night, staggering around drinking from a large glass bottle, relieving his bladder under the night sky and even one instance where both events took place simultaneously.\n\nThe calm feeling throughout the village came crashing down quickly as a loud noise popped nearby in the air, causing the Captain to spill what remained of his coffee; startling him and no doubt pulling everyone else from a deep and restful sleep.\n\nAdam immediately recognized the sound as he jumped from the confines of the Rover and raced to the bulk of the village in the pouring gray rain, looking straight into the sky, confirming the arrival of the Husk.\n\n\"Shit! How long?\" Adam yelled into the direction of Ceria as Roman and Dalton feverishly placed loaded weapons onto themselves as they tried to awaken.\n\n\"They will land in a field not far past the tree line. Five minutes, maybe less before they are here.\" Ceria replied sleepily as Adam double checked his pistol, ensuring that it was full of ammunition and combat ready.\n\n\"Sarah, stay inside! Keep Troy with you, if things don't go well, stay out of sight until they are gone and rejoin Kelly and Kato at the crash site.\" Adam said loudly, glancing through the shower of water into a nearby hut. Without responding, she began to cry fluidly as she grabbed Troy into her arms, holding him tightly in the corner of the primitive dwelling. \"Sarah!\" Adam said once more in a stern tone, making sure she understood her responsibility in case the worst were to happen.\n\n\"Yes.\" she managed to mutter, crying heavily which prompted Troy to become upset, sharing a few tears of his own.\n\n\"I will do everything I can to keep you both safe. Everything.\" Adam said convincingly as he stared into her eyes for a few moments; their hearts sharing a moment of passion through nothing more than a glance of love, before once again being overtaken by the stinging drops of rain. The plan had been cut short, no time for them to bring the mauler to assist them. It was simply a matter of Adam, Roman and Dalton facing two of the Husk monsters and possibly a human handler. Not very good odds for them by any means, but when a man has nothing left to fight for but the beating heart inside of his chest, it makes for a dangerous man indeed. Michaels was gladly prepared to die if it meant keeping Sarah safe.\n\nDalton remained prone at the far end of the village, only his head and arms were outside of the camouflage surroundings of the forest that surrounded them, his hands gripped a large caliber sniper rifle and his right eye pressed against the scope of the weapon. Roman had disappeared from sight, they weren't sure what he was doing for the moment, but everyone knew that when the killing began he would be in the middle of it, one way or the other. The Captain had nothing more than the long barrel revolver currently at his side and two shotguns tucked into the back of his worn brown pants, giving him a second option if needed. He was surgical with the revolver and knew that if he were accurate enough with the weapon, he would give Sarah and Troy a chance of survival as he positioned himself across from them and watching over the love of his life as if he were a guardian angel.\n\n\"If we don't kill these bastards, good chance they will be pissed enough to murder everyone in the village.\" Dalton said through his com unit, broadcasting his thoughts to both Adam and Roman.\n\n\"Then we kill these bastards.\" Adam replied, tightening his grip on the butt of his pistol and waiting patiently for the beasts to appear on the horizon. Minutes that seemed like days passed; waiting for the inevitable as nerves started to overtake logic.\n\n\"We got a problem.\" Dalton said. Michaels started to ask, when he saw the three figures approaching the village through the curtain of rain.\n\n\"Three of them.\" Adam said frantically under his breath, spotting a three Husk group with no handler. The Captain heard more chatter over the com, although he couldn't make the words out; his body numb from the circumstances almost as if he were in a dream.\n\n\"Captain!\" Dalton's voice finally rang out, forcing Adam back to reality and out of the helpless daze that he had succumb to. \"Do we proceed with the plan?\" Dalton added, waiting for a response from the man in charge.\n\n\"We have no choice.\" Adam forced painfully from his lungs, knowing almost with certainty that they had no chance of winning, the appearance of the walking beasts reassuring that with every step closer toward the village. The sheer size of them had Adam stunned, he had never seen such a build on anything living. Add the proficiency with weaponry and the faster than average movements of the relentless killers, he saw no way of winning the battle ahead.\n\nAs Dalton regained his composure, he began placing the scope of his rifle onto the head of one of the Husk. Waiting for an official OK from Adam, he continued to follow the beast with the telescopic lens of the high caliber weapon. \"Say when Captain.\" Dalton said, anxious to begin the blood letting as he noticed one of the villagers stepping out into the direct path of the Husk soldiers, his head covered with the hood of the sackcloth shirt which was quickly darkening with color at the end of the soaking rain.\n\n\"What the hell,\" Dalton mumbled to himself, wondering which of the locals had a death wish; the Husks merely slowing down, also in disbelief that anyone would dare stand against them. Taking his eye from the scope momentarily, he turned his head slightly to get a good angle on the com unit. \"Adam, are you seeing this?\" Dalton asked.\n\nBefore an answer could follow, Roman removed the sackcloth hood, standing in the hammering rain as if to taunt the three much larger opponents, a rugged combat blade in each hand as the wind blew ripples throughout his shirt.\n\nTwo of the three Husks began to slowly run towards him, each with a leg length machete tightly gripped by the superhuman strength of the nearly tree sized arms of the beasts.\n\nAs they were on him, Adam stepped from the doorway he had been well concealed inside of, holding his revolver forward and strategically firing all six rounds from the rolling chamber; hitting the same Husk with all of the piercing steel slugs, four in his upper chest and two punishing the neck of the beast. Obviously crippling the soldier with what would have killed six normal men, it fell to the ground screaming in pain as it bled openly onto the climate drenched ground.\n\nThe second Husk quickly shifted its attention into the direction of the Captain, who tossed the sidearm to the ground, reaching behind his back and pulling two short barrel shotguns from the back of his belt. Without thinking, his instinct pushed him to fire both weapons, one slug hitting the Husk in the chest and the second nearly missing, only grazing its arm slightly; sending a small cloud of skin and patched hair into the rain. Running through the pain of the recent chest wound, the beast leaped towards Captain Michaels in an attempt to kill him on impact.\n\nIts jawbone and a majority of its chin were wiped clean from its face by a single shot of Dalton's long range killing machine. The bolt action gun spitting an empty shell from its top, and falling hollowly to the ground. Dalton quickly began trying to scope the third Husk up as it went into full sprint in the direction of the two downed warriors. He anxiously fired a round that normally would have hit, however the Husk moved much faster than a normal man as the shot missed, firing wide right of the creature which had reached the other members of the crew.\n\nStriking Michaels with maddening rage, the beast sent the Captain flying onto the ground several feet away from him, sliding across the puddled ground. Adam immediately tried getting back to his feet only to fall back to the ground, simply dazed from the impact only moments ago. Roman thrust one of his blades into the direction of the Husk, who's raw strength knocked the combat steel from his hand, grabbing Roman with his free hand into a choking position.\n\n\"Fuck!\" Dalton said loudly as he had another shell in the chamber, but couldn't get a decent shot in the scope because of the choke, which was slowly bleeding the life from Roman's limp body.\n\nAs he began to stop trying to fight back and had accepted his demise, Roman dropped the other blade to the ground as his arms hung straight down by his sides, the Husk yelling loudly as he applied even more anger into the hold, trying to punish his victim brutally. Roman slid to the ground, falling face first into the mud as the monster suddenly let go of his grip of death. Spinning around, he threw his hand out and placed it around the fragile throat of Troy, who had escaped Sarah's clutches long enough to try and save his mentor by stabbing a blade into the back of the Husk.\n\nAs he squeezed firmly around the child's neck feeling small bones breaking as Troy grasped for air, a shot from Dalton's weapon zipped through the air and pierced the chest area of the beast, who immediately threw the young man who was fighting for his life to the ground. Adam stumbled to his feet, his first thought switched from killing the Husk to running to the aid of Troy, who was obviously in life or death condition. As the Husk looked down, realizing the shot had struck no vital organs, it yelled in rage as both of its friends lay dead on the crimson soaked ground beneath him.\n\nDuring the enraged scream, several of its teeth were jolted from their sockets from a hard right hand thrown by Roman, who had lost his usual ice cold swagger and let emotion take control of his moves as he followed with several more punishing hands to the face of the beast. Not used to the defensive side of battle, the Husk could do very little to block the flurry of commanding fists flying in his direction. Letting a loud and tearful yell of his own out, Roman threw a punch that quickly turned into a cupped hand as it penetrated the throat of the monster, ripping several throat organs out in the process. The monster fell to its knees, slowly dying from the wounds as Roman kicked it in the chest firmly, pushing it onto its back in a large puddle of water. Roman fell to a knee, obviously in pain, slowly making his way back to his feet in an effort to check on Troy as did Dalton, who was sprinting to the child's location.\n\n\"Impressive. Stupid, but impressive.\" the crew heard from behind them as the Husks handler stood with a large automatic pistol pointed in the direction of the group.\n\nThree sharp shots later, he lay face down in the rain, Sarah Blaine standing behind him holding an assault rifle that had been kept in the Rover.\n\n\"No sir, that's impressive.\" Adam said, giving a long look of thanks to the woman who had indeed saved his life at this very moment.\n\n\"How bad off is he?\" Sarah asked, sprinting over to check on the seriously injured youth.\n\nDalton looked up at her with the most serious look she had ever seen him carry. \"It's bad.\" he replied, immediately looking back down to help aid in any way possible. The next few minutes played out as if they were but a faint memory that had been faded with time, Michaels rushing Sarah to the spacecraft of the Husk; boosting the signal as much as possible before sending a message to her father of their position and condition. It was a long shot that he would even receive the distress call, however there was no time to properly prep the ship and hit orbit. They had to hope and pray that the long range com unit would do it.\n\nAs they returned to the nearby village, Dalton and Roman had assisted some of the locals in making Troy the best possible recovering setting possible inside one of the huts, placing him on a soft stack of grass and trying everything to keep fluids in the young man.\n\n\"We have to go. Now. Otherwise, the boy isn't going to make it.\" Roman said in a grave tone, passing the Captain and his newly found love.\n\n\"Can't do it. Help is one the way and we have to be here when they arrive.\" Michaels replied as several feet of heavy rainfall distanced the men.\n\n\"I wasn't asking permission, I was letting you know that I'm going to prep the ship for takeoff.\" Roman replied.\n\n\"That's not going to happen Roman, don't press the issue.\" Adam said, hoping the former special forces soldier would come to his senses.\n\n\"I won't let him die Adam, don't try and stop me.\" Roman said firmly, turning to walk in the direction of the vessel.\n\n\"ROMAN! We WILL wait on help to arrive, that is a direct order!\" the Captain said, pulling his revolver and aiming it at the hardened warrior. Stopping in his tracks, Roman turned to face Adam; both men giving impenetrable stares.\n\n\"I'm through taking orders from you. What? You plan to shoot me?\" Roman said, walking closer to Michaels until reaching the end of the revolver barrel.\n\n\"I don't want to, but I will and I think we both know it,\" Adam replied, pulling the hammer back on the weapon, setting it ready to fire with just the slightest pressure. \"You are a member of this crew until we're back in the system. That said, you WILL obey my direct order.\" the Captain added, calling Roman's bluff.\n\n\"The boy dies, and I'll kill you and anyone else who tries to stop me.\" Roman finally replied, walking past both the Captain and Sarah who continued to stand there for several minutes, finally returning to Troy's side.\n\n\"He ain't fucking around Adam.\" Dalton said, staring out of the small hut window as Roman sat on a large stone directly in the path of the relentless rainfall, a strong grip on his combat sidearm.\n\n\"Neither am I,\" Adam replied, giving Dalton a look to reassure him of that fact. \"I don't want anything to happen to Troy any more than he does, but I'll be damned if I put the rest of my crew in danger to fly back into the system for a single person.\" Adam added.\n\nThe next few days were a mixed bag as Troy continued to fight for his mortality, Roman kept to himself, only leaving the confines of his hut to sit by the boy's side; Kelly and Kato made a list of everything they needed to get the Gunship back into the sky and the rest of the crew stayed right by the side of Troy the moment Roman left it.\n\nThe crew, they all were heroes to the people who lived in this village on what could be considered one of the more primitive planets in the star system, or even uncharted space for that matter. They all knew that eventually Adam and his crew would be leaving by sky, to visit other planets that were much more civilized than this one; still, they made every effort to show their thanks to the heroes of the Gunship.\n\nEvery night filled with immaculate dinners and only the best lodging available, which wasn't much, but it was a gesture of gratitude that the crew wouldn't soon forget.\n\n\"Quick Captain, come quick!\" one of the village natives yelled as he burst into Adam's sleeping quarters, which was nothing more than a back area of one of the larger huts.\n\nAs he jumped to his feet, revolver swiftly in his right hand, he could see the native pointing to the sky. \"Quick, come!\" he restated as he rushed back outdoors. As Adam stepped outside, he immediately recognized the sound of the sonic booms caused by several ships hitting the planet's upper atmosphere.\n\nWithin seconds, the sky was littered with military jets flying overhead, no doubt doing scans of the ground below as Dalton broke a red flare across his upper thigh and threw it to the ground. The markings on the ships as they flew overhead were unmistakable. They were the Allied Colonial Forces, Sarah's father had arrived.\n\nAs the ships broke into formation and began doing defensive flyovers, several heavily armored space to surface choppers began a descent, followed by a solid chrome shuttle. The choppers landed with a compelling thump as dozens of tactical soldiers, much like that of Roman's former employer, began doing security sweeps of the village and surrounding areas. As the shuttle touched down,\n\nSarah sprinted to reunite with her father, who had exited and was waiting with open arms. He was a tall, well dressed man who obviously had the respect of his men. His white hair and peppered beard were indications that he had plenty of experience, and to the crew's delight, he had indeed brought a lot of firepower with him.\n\nRather than introduce himself to her father, Adam quickly explained Troy's condition and need of medical treatment, prompting a team to attend to the young man. Seeing the severity of his wounds, they escorted Troy to a nearby chopper, who was carried in the arms of Roman Raines.\n\n\"We picked up a small Legion fleet of ships approaching as we hit orbit. No doubt locked onto your distress call as well, may be a bit of a bumpy ride out.\" Commander Blaine said quickly to the crew as he hurried Sarah to his private shuttle. The remaining members of the crew, including the Captain, hopped aboard one of the choppers.\n\n\"Thank you for the drink my friend, we will return one day soon enough.\" Dalton said as he boarded, arms full of the local tonic of choice, doing his best to wave for a moment before the chopper door came to a close.\n\nThey had only been airborne a couple of minutes when everyone heard the first explosion. Moments later, it was followed by several more thundering booms as dozens of ships began firing tracer rounds. Adam was in a position to see out of one of the small windows of the crew area, and was simply in awe of the firepower that he lay witness to. Legion ships, three capital ships from what he counted, escorted by several smaller cruiser style ships. He only counted one capital ship bearing the Colonial marking, although there were wave after wave of ship to ship fighters that had launched from it.\n\nHe had known for a while that the star system was on the verge of a Civil War, however he had no intentions of being caught in the middle. He quickly realized, however, with the throaty sound of the ship's pounding cannons; that he was indeed witnessing the first acts of this inevitable clashing of the system's two powers.\n\nAs their chopper began docking to the Colonial capital ship, the entire crew could see that it had taken a brutal beating in an effort to rescue them. The docking area looked more like a make shift hospital as dozens of wounded were being pulled from Spartan star fighters as dozens more were suiting up to launch and try to replace the fallen. Soon after their craft had successfully landed and the rest were accounted for, a sequence of loud beeps rang throughout the ship. None of Adam's crew knew Colonial code, although it obviously seemed to be some type of general retreat order for the fleet, who began landing in the docking bay by the dozens.\n\nIt was only a few minutes after the loud screeches of military code that the capital ship began doing a full burn, all four of the huge thrust engines firing as the huge space fortress had soon put a great deal of distance between the fight and themselves.\n\n\"Father, this is Captain Adam Michaels. Adam, my father, Trevor Blaine.\" Sarah said nervously as she entered the heavily guarded bridge area of the ship along side her new found love.\n\n\"Captain, welcome aboard the Colonial Star Five,\" Commander Blaine said, firmly shaking Adam's hand for a moment before hugging his daughter with great joy. \"We were able to bring your ship up from the surface before jumping away, my engineers are working on it for you as we speak.\" Blaine added, bringing a smile to Adam's face.\n\n\"That's great news. Thank you very much, for the rescue as well as the hospitality.\" Adam replied, much politer than usual.\n\n\"I'm afraid I do have some bad news. Our scout teams have reported that Legion forces have combed the uncharted planet. There is a good possibility that the native camp you were holding up in has been wiped out.\" Blaine said, reading from a paper report he held in his hands at the moment.\n\nMichaels was taken back by the news, almost putting him in tears. \"I don't understand that. They were of no threat to anyone. They had nothing to do with any kind of war, they were just innocent people; good people trying to survive.\" Michaels replied dramatically.\n\n\"I understand your frustration Captain. It's that very same reasoning that led to my taking a position with the Colonial Army. I will not fight under a flag who's military slaughters innocent human beings. The sole purpose of the Colonial Alliance is to put an end to the tyranny and death that the Legion uses to maintain its power,\" Blaine replied. \"I must attend to repairs throughout the ship Captain, I apologize. However I would like to talk to you about this in much greater detail a bit later?\" the Commander asked.\n\n\"Of course. Again, thank you.\" Adam replied, ending the tense conversation with a quick handshake.\n\n\"What in the fuck is that?\" Dalton said to Kelly as he lowered a shiny metal cup from his mouth in awe.\n\nNeither of them had ever seen anything like it before. Standing nearly seven and a half feet tall, it was a soldier of some type, pure titanium exoskeleton with mechanical features intertwined with human bone underneath; its solid steel skull moving from side to side as the soldier patrolled that area.\n\n\"First time seeing a Goliath unit huh?\" one of the Colonial engineers said, raising his head from the thruster of the Gunship that was undergoing repairs by himself as well as Kato. \"They're a spectacle, no doubt about it. If you look closely at its right arm, you'll see that it's nothing more than a mini gun built around a titanium rod, damn things can fire up to five hundred armor piercing rounds per minute. That's enough to take down a small aircraft or a whole lot of anything on foot.\" the engineer added before turning back to continue working on repairs.\n\n\"Yea, a shit load.\" Dalton said in amazement as he took another long drink of his brew, unsuccessfully offering a sip to Kelly.\n\nRoman sat in Troy's room patiently waiting for a miracle to happen.\n\nHe never really considered himself to be a religious person, although he had to wonder why any God would allow such a young child to lay here like this. Troy should be out doing things that normal children do, like playing games of chance or pulling pranks on one another. Yet here he was, no family, no home and very little chance of surviving. Maybe he felt guilty, the boy having help save his life only to find himself fighting for his own.\n\nMaybe it was the need to be there for a young man who was doomed to the same kind of childhood he had thrown on him. Being homeless with no family and no means of survival was the biggest reason he had joined the Gali Army to begin with, or at least what he had claimed to be the Gali Army. When a person is faced with insurmountable odds like that, one of two things happens. Either you succumb to the odds that have been stacked against you, or you say fuck the odds and learn to take what you need. That was his attitude, and that's what made him such a dangerous person. Roman quickly stood to his feet as Troy's doctor entered the room.\n\n\"The plan is a simple one,\" Commander Blaine said as he looked throughout the crowd, which included Adam, Sarah and Dalton among many high ranking military officers and several dozen of the best soldiers that the Colonies had to offer. \"A small team inserts near the heavily fortified compound currently being used as a staging area for the Legion, uses timed explosives to take out its major surface to air defense systems. We then hit them in force, crippling their ability to defend themselves, much less continue killing innocent civilians.\" he added, getting nods of encouragement from most of the room as well as a few loud chants of Colonial patriotism.\n\n\"Captain Adam Michaels will lead the first team, which will be comprised of highly trained Spec Ops soldiers as well as a handful of Husk loyalist warriors.\" the Commander said, immediately taking the breath of his daughter Sarah.\n\n\"You can't!\" she said in a low but frantic voice, having fallen in love with a man that would be walking into a hornet's nest of murderers.\n\n\"Someone has to show these bastards that they aren't the highest on the food chain, not anymore.\" Adam responded proudly.\n\n\"That someone would be me,\" Roman said as he entered the room, turning the head of everyone as he did so. \"I'll go in his place.\" Roman added, approaching the Commander.\n\n\"I appreciate the offer friend, however, this mission is for only the highest qualified of soldi, .\" Commander Blaine had said, trying to reply as two of his personal escort soldiers approached Roman, holding him at the gunpoint of their battle rifles.\n\nPausing for a moment, clearly thinking of his next move, Roman struck one of the soldiers in the neck area, grabbing the rifle from his clutches as he fell to the ground. Before the second soldier had time to respond, Roman pulled two vital pins from the gun, disassembling it into three large pieces and using the stock that was no longer attached to the weapon, slicing the soldier across the face and rendering him defenseless.\n\nSeveral more soldiers rushed into the room, weapons drawn and ready to kill if it meant protecting the Commander.\n\n\"I'm the most qualified son of a bitch in this room. All I know is killing, and I'm particularly fond of killing Hunters, which also makes me the most dangerous,\" Roman said, throwing the cluster of loose combat rifle parts onto the ground; showing Blaine that he had no intentions of harming him. \"You just give me a weapon and tell everyone else to follow close behind and pick up as many of the body parts as they can; I'll kill every one of the bastards myself, and the weapon is optional.\" Roman added, having gotten the news that Troy would eventually recover to full health and feeling in the mood for revenge.\n\n\"I can see you are indeed qualified. Captain?\" the Commander said, asking permission before allowing Roman to go in his stead as armed soldiers continued to hold Roman at gunpoint.\n\nLooking at Sarah for several tense moments, Adam once again focused his attention on the Commander. \"I owe this man a great deal. My life, an apology,\" Adam said, turning to Roman with an apologetic glance. \"If it's his wish to go in my place, then all I ask is he kill as many Hunters as possible.\" Michaels added, smiling at the natural born killer.\n\nIndeed, the plan was a solid one. From the personnel who would infiltrate the stronghold, to the weapons and tactics that would be used once they arrived and now the addition of Roman, whose hatred for Hunters nearly matched his skill in combat. A solid plan indeed. It would have to be, the Legion and its Hunter faithful were no pushovers. One of the more feared races in the charted star system, now with the backing of the Legion army as well as the news that Commander Blaine was about to unveil, the Hunters were even more of a solid opponent.\n\n\"We have confirmed through spies within the Legion Army that the Hunters have genetically developed a new breed of killer,\" Blaine said as the room quickly became deathly silent. \"A super soldier if you will, matching many of the dimensions of our own Goliath units, the Hunters are calling this unit the Fang.\" the Commander added as he pressed a couple of buttons on the podium that stood in front of him, displaying a series of photos onto the wall behind him with vivid detail.\n\n\"We are not exactly sure how many Fangs exist, only that they are currently being used as more of a tank, in terms of being accompanied by a small squad of Hunters. We believe there are not many of them yet, which is why it is imperative that we strike them now, before they have a chance to manufacture any more.\" Blaine added as he waited for a response from anyone in the crowd of gathered military brass and trained warriors.\n\n\"Hell, they don't look that tough to me!\" Dalton proclaimed loudly, as the rest of the room grew loud in support of his statement. He had effectively bolstered the moral of the entire room, sadly, he would remember almost none of it as a side effect of his current drinking binge.\n\nHelping both fallen soldiers to their feet, Roman glanced into the Commander's direction with a look as serious as death itself. \"Just give me a weapon, point me in the right direction and watch what happens.\" he said, slowly leaving the room to check on Troy before the upcoming battle of savages was to take place.\n\n\"He's recovering much faster than we anticipated. In due time the boy will be in perfect health.\" one of the Colonial doctors told Roman as they both stood outside of the ship's enormous infirmary, glancing through a reinforced window made of thick and frothy glass.\n\nHe had to recover quickly, it was a sign of his toughness. He would need plenty of that in the years to come, as life beat him down he would have to be tough enough to withstand the punishment and keep going. That was the way of things, the true mark of a warrior.\n\nAs Roman entered the room to see the young boy, he was amazed at how far he had come in the healing process, several of his neck vertebrae crushed, yet he sat up in his bed unassisted.\n\n\"Did I do good Roman?\" Troy asked, wondering if his knife thrust into the Husk had met his teacher's approval. It was the first time in many years that Roman found himself fighting back tears. Eyes watering as he tried to dissipate his extreme anger against the Husk and his overwhelming feeling of joy, knowing Troy would survive something that he should have never even been involved with.\n\n\"Yes, you did good. Without your help, I wouldn't have made it. You saved all of us.\" Roman replied as he pulled up a chair beside the young man's bed.\n\nHe could see the happiness on Troy's face as he began to think of himself as an important part of the crew.\n\n\"Listen, I have to leave in a few hours to take care of some business. Some other people need that same kind of help. While I'm gone, I need you to look after the crew, can you do that?\" Roman said, handing the boy a brand new combat knife he had collected from the ship's armory.\n\nSeeing Troy's face light up, smiling from ear to ear as he eagerly snatched the blade; nearly was more than Roman could handle emotionally. To watch a young child go through losing his parents, nearly dying at the hands of some piece of shit mercenary and then be so enthusiastic about something as small as a blade. Roman held back, choosing instead to funnel his emotion into ending as many lives of the bastards who force children like Troy down this road, bastards like the Hunters.\n\nDeath, imprisonment, torture; none of these things concerned Roman any longer. He had become a monster, one that was hellbent on making the star system a better place to live, one slaying at a time.\n\nAfter Troy had fallen asleep, Roman waited a few minutes to make sure the boy was comfortable before leaving the infirmary, passing Captain Michaels as made his way into the main hall of the ship.\n\n\"Troy alright?\" Adam asked. Roman nodded his head, letting the Captain know that he was fine.\n\n\"Dunno if it was the alcohol talking or he has revenge on his mind, either way I just wanted you to know that Dalton pitched a fit on me, demanding to go with you and the rest of the first strike team.\" Michaels said. A few moments of silence passed, obviously due largely to the Captain having pulled a gun on Roman not so long ago.\n\n\"He's a hell of a shot, I'm sure there will be a few more Hunters laying dead if he's with us.\" Roman finally replied confidently.\n\n\"Listen,\" Michaels said, stopping Roman in his tracks. \"You get in, plant the explosives and survive long enough for us to get there. You got my word, I'll get you back here in one piece.\"\n\nStanding silent for a few seconds, the former Gali warrior finally looked up at Adam with his fiery but determined eyes. \"Just look after Troy while I'm gone.\" Roman responded.\n\n\"You got it.\" Michaels answered as he watched Roman walk to the end of the narrow wide hallway before turning to make his way back to the armory.\n\nIt was a position that he had been in many times over. Roman sat in the back of a Chopper, waiting patiently as the rest of the crew for the operation slowly climbed aboard. Dalton eventually made his way onto the skiff, strangely enough appearing as sober as Roman had ever seen him.\n\n\"Stay calm. About to be something come through here that you don't want to see. Just remember that we are all on the same team.\" Dalton said softly, as the words had just finished escaping his lips when two heavily armored Husk soldiers boarded.\n\n\"Just stay calm man.\" Dalton added, holding his hand near the chest of his friend as he saw the fury of hell in his eyes.\n\n\"Sorry about the young boy.\" the first of the two Husk said, stopping in front of Roman and speaking with true sincerity.\n\nSitting calmly, or at least it appeared, Roman remained silent until the Husk gave up hope for a response and made his way past the men in the direction of an empty seat. Seizing the moment, Roman lunged towards the Husk, pushing him to the ground while holding a combat blade to his throat.\n\nThe second Husk tried to intervene, only to have his face come inches from the deathly hollow barrel of Dalton's pump action shotgun.\n\n\"No offense friend, but if you take another step towards 'em, not even you are fast enough to escape the shell in this chamber.\" Dalton said, wondering if the Husk was indeed fast enough.\n\n\"You ever so much as speak of the boy again, I'll make sure it's the last words that fall from your mouth.\" Roman said, daring the monster to call his bluff.\n\n\"ENOUGH!\" one of the Colonial soldiers said, walking back from the cockpit area into the passenger section of the chopper. Pulling his sidearm out and pointing it daringly at both Roman and the Husk.\n\n\"This ends now. I'm Lieutenant Avery, I'm in charge of this operation and I swear to whatever God it is that you believe in, if this doesn't end here and now, I will personally shoot the both of you and dump you out of the side of the Chopper. Now, let him up.\" the Lieutenant said in a commanding voice, his body decked out in the Colonial blue and gray coloring.\n\nRoman slowly let the Husk up, easing the edge of his blade from the monster's throat.\n\n\"Wanna be careful with that blade, next time you might not be lucky enough to have him save your ass.\" the Husk said with a smile as he slowly pushed himself to his feet.\n\n\"I make my own luck.\" Roman said confidently as he once again took his seat beside Dalton, who removed his shotgun as the second Husk was quick to be seated. As he started to follow Roman to where he sat, the Husk reached for his long blade, which was attached to the long of his leg.\n\n\"Steiner,\" the Lieutenant said loudly, sliding the top of his pistol back into the ready position. \"Sit your fucking ass down, or make no mistake, they will be scrubbing your brain fragments from the inside of this ship before we launch.\"\n\n\"Now look who's getting lucky.\" Roman said sarcastically as Steiner slowly made his way to be seated, upset that a fight would not ensue between the two, at least not at the moment.\n\n\"You think they're gonna try and kill us the first chance they get?\" Dalton asked his friend.\n\n\"You think it matters if they do?\" Roman replied, letting Dalton know that he planned to kill anyone who stood in his way of reaping revenge for Troy's trauma.\n\n\"Either one of them makes a move on the other, shoot 'em.\" the Lieutenant said to another one of the operation's soldiers, this one wearing solid blue with heavy gray leather markings, signifying the equivalent of Colonial Special Forces.\n\nAs the skiff began to lift off, turning before exiting the launch area of the Capital Ship; Captain Michaels and Sarah were among the crowd that had gathered to watch the departure.\"Well, I had Roman figured out wrong, looks like. I thought for sure he'd try and kill a damn Husk,\" Adam said quietly to Sarah, laughing under his breath as the blast of the chopper hitting launch speed quickly drowned out any noise thereafter. \"Yep, I'm proud of him for behaving so civilized.\" Adam added.\n\nThough a small exterior window inside the infirmary, Troy watched as the skiff went into full burnout, disappearing seconds after it had launched from the huge military carrier. He wondered what Roman and his group of soldiers might encounter, wished that he was along with them for the ride and most of all, hoped that man he quickly began to look up to would make it back safely.\n\n\"Bumpy ride, damn you would think with all of this money they are throwing around they would have designed a more comfortable seat.\" Dalton said aloud as he glanced around at the other soldiers in the passenger area, the strong glimmer of red light filling the cabin.\n\n\"Built for durability, not comfort.\" one of the soldiers responded, his solid blue helmet and visor giving away the fact that he was the only sniper among the group. If that wasn't obvious enough, he held a single barrel weapon that stood nearly three feet high, the butt of the weapon resting on the floor of the chopper.\n\n\"Damn fine piece you got there.\" Dalton replied, using the line that he had reserved solely for weapons and women.\n\n\"Thanks. Thermal EM scope let's me fire from nearly two clicks out and cut a man in half.\" the sniper replied, obviously good at his trade.\n\n\"I prefer to kill them up close myself.\" Steiner added, joining the conversation unexpectedly; staring directly at Roman the entire length of the comment. A long pause between the elite fighters, Dalton finally nodded his approval on the fighting tactics of the Husk.\n\n\"Makes two of us. Give me a shotgun any day.\" Dalton responded, if nothing else, to try an built a rapport with Steiner before turning his back on him once the fighting started.\n\n\"I have to admit Commander Blaine, I'm impressed.\" Captain Michaels said \"I didn't expect to see this many soldiers.\" as he looked over a short metal guardrail from above, watching thousands of Colonial soldiers begin to mobilize.\n\n\"The fate of most of these men, if not all of them, rest on the shoulders of the men aboard that chopper,\" Blaine said. \"If they don't get rid of the surface to air defenses, most of these men will be killed long before they reach the planet's surface.\" he added.\n\nAdam didn't doubt for one second the ability of either Roman or Dalton, rather, he worried about what they were flying into. So little was known of this new strain of Hunter, and he had a hunch that no matter how tough the Fang proved to be, neither man would even consider backing down.\n\n\"Adam, I've seen your warrant file,\" the Commander said. Although Michaels didn't reply in words, his expression was one of shame. \"The truth is I looked them over thoroughly, and to be honest, I didn't see a single charge listed that I feel I should be worried about. I think maybe your heart is in the right place and you've just been tangled up with the wrong kind of crowd.\" Blaine added, having no idea how true those words rang to Adam's ears.\n\n\"I've also read your military file, you fought in the Glimmerian Wars.\" the Commander said.\n\n\"That was a long time ago.\" Adam replied.\n\n\"Once a soldier, always a soldier. At least that's my belief.\" Blaine replied.\n\n\"What I'm getting at Adam, is I believe you would be a good fit here in the Colonial Army. You have the right kind of experience, a heart for the innocent and the love of my daughter.\" he added.\n\nCompletely surprised by the statement, Adam soon realized that their love for one another could no longer be kept a secret, no matter how hard they had tried.\n\n\"A clean start. That's what I'm offering you Adam. A chance to wipe your criminal record clean and become a part of the solution instead of the problem, you and the rest of your crew as well.\" Blaine said. Deep in thought over the possibility of starting fresh, the Captain finally looked at the Commander long enough to reply.\n\n\"I'll think it over and speak with my crew.\" Commander Blaine nodded his approval.\n\n\"Of course Adam, of course.\" the Commander replied as he walked down the metal steps nearby to meet with the troops below in an attempt to keep moral up.\n\n\"Two minutes to touchdown,\" Lieutenant Avery yelled to the soldiers in the passenger area of the chopper. \"Strike team goes first, Sweeper team goes second and sniper is with me, understood?\" the Lieutenant added, trying his best to talk loud enough to be heard through the roar of the ship's engines on decent.\n\nThe soldiers all nodded in compliance as the chopper began a solid nose dive, the overhead light changing to yellow as the entire cabin grew bright enough to see the look on everyone's faces. Dalton counted a total of ten souls aboard, including himself and the Lieutenant.\n\n\"I know it's a bad time, but which team are we again?\" Dalton asked Roman in a calm voice. Gaining an immediate stare from his friend, who was glassy eyed with adrenaline.\n\n\"Just follow me and shoot any son of a bitch who shoots at you.\" Roman replied calmly.\n\nThe chopper spun around, doing nearly a two hundred degree turn as it abruptly hit the ground, the cabin light switching from yellow to green.\n\n\"Go, go!\" Lieutenant Avery yelled as the first four soldiers jumped off, all wearing Colonial Special Ops clothing; the standard issue dark blue with brown leather trim covering them from the neck to boot line. As they exited the craft, panning around the darkness with heartbeat sensors attached to their battle rifles, the second group of four hit the ground.\n\nRoman, Dalton, Steiner and his Husk ally all four began searching the immediate area around the chopper, before finally grouping up with the first team. The last off were the sniper and the Lieutenant. Sprinting for the highest ground in the area, the sniper took position on top of a grassy hill, the wavy green blades at least two foot tall. He quickly pulled a thermal blanket over everything but his head and arms, the metal interior of the blanket setup to deflect any type of heat signatures that his body may give off, while the exterior of the blanket was a green camouflage that was designed to blend in with any surrounding. As he quickly set up his long range rifle, snapping it into a portable tripod; he zoomed the scope around to pick up any hostiles who may have seen the ship's approach.\n\n\"I got nothing in the scope, you're all clear.\" the sniper said as the Lieutenant waved the ship off, taking only seconds before it had disappeared into the night sky.\n\n\"Form up,\" Avery said into his com unit, as the remaining eight men bent to a knee close together, waiting on their commanding officer. \"Strike team, move ahead and take out any patrols on the perimeter of the compound. Sweeper team on me.\" the Lieutenant said as the four Colonial Spec Ops soldiers instantly began a defensive sprint into the direction of the compound, which sat in the distance about a quarter of a mile.\n\n\"Sweeper team, we wait for them to clear the perimeter, then we move up and plant the explosives. Should be a couple of three hundred inch rail guns and a control box near the rear of the largest dwelling that controls the surface to air missiles. We need to take out all three in order to give the Colonial Army a clean landing. Understood?\" Avery said as everyone agreed and waited for a signal from the strike team ahead.\n\nThe signal never came. \"Eyes, got anything in the scope?\" the Lieutenant asked.\n\n\"Negative, I got nothing.\" the sniper replied.\n\n\"Something is wrong. They've been gone way too long.\" Roman said quietly in the direction of Avery.\n\n\"Strike team, come in,\" the Lieutenant said into the com unit, waiting an extended period of time for a response that never came. \"Fuck,\" Lieutenant Avery yelled in a low voice as he contemplated his options. \"Eyes, we are moving up. Keep your fucking eye on that scope, any marks show up, don't hesitate to fire at will!\" Avery said as he looked around at the four men.\n\n\"Fire your weapons only as a last resort. Now move out.\" the Lieutenant said as they began slowly making their way to the compound, taking extra precaution with every step.\n\nThe two Husk soldiers were in lead of the group, with Roman and Dalton following loosely behind them and Avery right on their footsteps. Stopping at the security fencing, the group saw a hole that had been made only minutes ago by the Spec Ops team.\n\n\"Slowly.\" the Lieutenant said as he motioned for the soldiers to continue through the break in the fencing. The first two Husk made their way through and threw their backs to the wall to cover the entrance of the other three men. Once everyone was inside, the first rail gun was in sight, only feet away from their current position out in the open. Waiting several minutes to ensure that they had not been detected, Avery finally gave the go ahead to the group; the explosives had to be planted.\n\nAs the first Husk slowly emerged from the corner into the open, he was immediately hit with a high caliber round from a rifle in the distance, the shell piercing his neck, sending him flailing to the ground. The shot hadn't killed him, but left him defenseless on the ground, reeling in pain as he bit down on his arm to prevent making noise that would give the group's position away. Two more suppressed shots followed, one striking his chest and the other piercing the back of his skull and killing him the very moment. A third shot was fired, this time from behind the group. \"Got him, that last shot gave me his position.\" the Colonial sniper said after eliminating he Legion's sniper.\n\n\"Scanning the area, looks all clear but there is a damn good chance that he wasn't working alone.\" the sniper said as Steiner sprinted to plant the satchel of explosives, glancing only for a moment at his friend who lay in a pool of his own blood. With the charges in place, Steiner grabbed the body of his fallen clansman as he returned to the group, trying to remain out of sight.\n\n\"Surface to air missiles are the most likely to cause trouble for the Colonial Fleet, so they need to go first,\" Lieutenant Avery said, checking a map scan of the Hunter compound. According to the map, the STA Control Room should be in this building.\" he added, pointing to a small building on the computerized blueprint.\n\n\"Eyes open, Steiner take point, move out.\" the Lieutenant said quietly, putting the map away and once again firming his grip on the combat pistol.\n\nAs the approached the marked building, Avery checked his watch before informing the rest of the group that the Colonial force would be striking in less than twenty-two hours. \"Still plenty of time, go in quietly and don't get sloppy. Now move!\" the Lieutenant said quietly as Steiner pushed the large red door, which was made of reinforced red steel, opening it slowly with a small creaking sound.\n\nThe room was pitch dark, as Steiner moved inside slowly, his large shotgun in front and at the ready. Following closely behind, Roman and Dalton moved inside of the military target, Roman with a standard issue rifle while Dalton insisted on using his not so standard pump shotgun.\n\nSuddenly, the interior lights came on, giving the soldiers an immediate picture of the building's interior, which was flooded with Legion soldiers who had all four of the strike team members kneeling at gunpoint.\n\n\"You're late.\" one of the ranking Legion officers said calmly as Lieutenant Avery pointed his pistol directly at the skull of Roman.\n\n\"I'm late because your damn sniper pinned us down.\" Avery replied sternly.\n\n\"Anwick requested you hold these three until he arrives.\" Avery said with a more relaxed tone as the Legion officer nodded.\n\n\"You should know, I make it a point to kill anybody who pulls a gun on me twice.\" Roman said calmly, his hands behind his head.\n\n\"I'll make a note of it.\" Avery replied, motioning a group of the Legion soldiers to lead Steiner, Roman and Dalton to a holding cell.\n\nAs they were leaving the large storage area, they were unfortunate witnesses to Legion soldiers firing a single into the head of each of the strike team members, killing them on the spot.\n\nAs they arrived to the holding area, the Colonial sniper lay in the corner of the room, badly beaten and unable to stand.\n\n\"Get inside you Colonial pieces of shit!\" one of the Legion soldiers said, pushing Steiner into the cell with the backside of his rifle.\n\n\"Looks like I don't want to kill you so badly after all.\" Steiner said to Roman as Dalton tried helping the sniper to his feet.\n\n\"We got to warn the Fleet, they, there is an entire regiment of Legion soldiers here, Commander Blaine is walking right into a trap.\" the sniper managed to say as he continued to fight for a deep breath.\n\n\"First thing's first. We gotta get out of this holding cell.\" Roman said with purpose as he extended his hand to the much larger, former foe Steiner.\n\nAfter the two combat juggernauts shook hands, which sent a shiver up the spine of Dalton, who imagined the possibilities of both men fighting together; they began pacing the cell and looking for anything that may aid in their escape.\n\n\"So, you're thinking my father's offer over seriously then?\" Sarah asked as Michaels read through a stack of papers in his room aboard the Colonial Star Five.\n\n\"Of course, just the idea of starting over scares the hell out of me.\" Michaels replied, continuing to glance at unclassified military reports as he tried to get himself up to speed on the war between the Colonial and Legion armies.\n\n\"Well, you know, starting over with me at your side might not be such a bad thing.\" Sarah said convincingly as she slowly walked over to him, putting her arms around his neck passionately.\n\n\"Wait.\" Adam said, jumping to his feet.\n\n\"Wait? I thought that's what you wanted? What we both wanted?\" Sarah said, very confused by the Captain's sudden change of heart.\n\n\"No, wait.\" Michaels said, holding a single piece of paper in his hand, then grabbing several more as he read them thoroughly.\n\n\"Who was in charge of your security detail back on Tameca?\" Michaels asked impatiently.\n\n\"Um, Lieutenant Avery, I think, why?\" she asked curiously.\n\nAdam bolted out of his room without explanation, several papers tightly in his grasp as he sprinted through the halls of the huge ship on his way to the Commander's quarters. Everyone he passed taking a second to watch his movements, wondering about his sense of urgency.\n\n\"Commander, Adam Michaels here to see you sir?\" a Colonial soldier said softly, as he held the door of Blaine's cabin open.\n\n\"By all means, send him in.\" the Commander said politely as he sat in a very lush leather chair behind a huge wooden desk that looked as sturdy as the ship it rested on.\n\n\"Adam, I take it that you have thought on my offer?\" Commander Blaine said hopefully.\n\nThrowing a small stack of documents onto the desk, Adam tried for a few seconds to catch his breath. \"It's a trap.\" the Captain finally managed to squeeze from his lungs.\n\n\"Excuse me?\" Blaine said, standing out of the chair in search of an answer.\n\n\"Lieutenant Avery is working with the Legion. Right here in the documents are at least a dozen cases where his group came under assault from either Legion or Hunter forces, every single time he was the only one to survive. The Hunters do not leave survivors, trust me on that one.\" Michaels said confidently.\n\n\"Adam, that's outrageous. Avery is one of my most trusted officers.\" Blaine replied skeptically.\n\n\"Commander, your daughter's security detail back on Tameca, Avery survived that assault as well. Believe me, I met the monsters behind that, we damn near didn't make it out of there ourselves.\" Adam said in an attempt to force the Commander to see reason.\n\n\"He's right father, Lieutenant Avery was acting strange that entire day. Our group was hit by four Hunters as well as a full squad of Legion soldiers. When I was taken into custody, I thought for sure that we would surely die, the rest of my detail already lay slain.\" Sarah said, arriving at her father's quarters and nodding in the direction of Captain Michaels to let him know that he had her support.\n\nCommander Blaine stood silent for several very long moments, trying to connect the dots inside his mind.\n\n\"Commander, two of my crew are with the strike team. All I ask is the use of a ship; give me a chance to rescue them if I'm right about this.\" Michaels said.\n\n\"If you're wrong?\" Blaine asked quietly.\n\n\"If I'm wrong, then I will apologize to Avery myself. You told me that my heart was in the right place, well my heart tells me that our men are in trouble.\" Michaels responded.\n\n\"Alright. I'll give you a chopper and a four man strike team, on one condition,\" the Commander said as he slid a small blue box across the fine wood grain of the desk into the direction of Adam. As Michaels slowly opened the box, he was nearly taken over with emotion as he saw the insignia medallion of a Colonial Lieutenant. \"The only way any of the soldiers on this ship will execute an order from you, is if you are a commanding officer. Besides, I know I can trust you with my daughter's life.\" Blaine said firmly.\n\n\"Sarah's life?\" Adam asked, confused.\n\n\"That's right. You don't honestly think I plan on letting the man I'm in love with go it alone do you?\" Sarah relied.\n\n\"But I don't thin... \" Adam began to respond as he was quickly cut off by Commander Blaine.\n\n\"Don't worry Lieutenant, she's plenty capable. Who do you think trained many of the Special Ops soldiers aboard this ship?\" Adam gave a long, blank stare in her direction.\n\n\"I know, I know. Us girls and our secrets.\" Sarah said as she smiled wide.\n\n\"You have a go Lieutenant.\" Commander Blaine said as Michaels was still in shock over the events of the past several minutes.\n\n\"Yes sir.\" Adam finally replied, turning to leave the Commander's quarters.\n\n\"So we wait for Anwick to arrive, more than likely escorted by a couple more Hunters, and then we overpower them as soon as the cell door opens? Seriously? That's the plan?\" Dalton asked in a joking manner.\n\n\"Do you see any other way out?\" Roman replied, confirming the plan he had helped Steiner design.\n\n\"Look, I'm one for a fight, don't get me wrong. Hell, I'm even one for fighting against the odds; but damn, we are talking about a small group of heavily armed HUNTERS HERE,\" Dalton replied loudly. \"We are talking suicide and everyone in here knows it!\" he added.\n\n\"They will kill us regardless. We were only spared so the Hunters could slay us with their own hands.\" Steiner replied.\n\n\"So let's think of something that doesn't involve everyone in this fucking jail cell dying!\" Dalton responded animatedly.\n\n\"All systems are checking out fine.\" Kelly said as she flipped several switches at the helm of the reconstructed Gunship. It still had plenty of wrench time ahead, but the craft was starting to slowly look like a spaceship as a small crew had joined Kato in working on it around the clock.\n\n\"Getting good readings on life support, navigational and cabin pressure.\" she added, yelling the information out of a small hole in the side of the bridge that was being reinforced with plated steel.\n\nThe mechanical crew continued working with engineers as several Spartan model fighter ships also were receiving repairs in the same area of the Colonial vessel. Kato had paused for a few moments, watching the welding torches piecing the small ship to ship fighters back together; wondering what it must be like to fly something that was incredibly fast. The speed was offset by the disadvantage of having only a dual gun mounted to each wing that was capable of firing quick bursts of tracer rounds. It sounded good in theory, however, against a larger ship the tracers were only potent when the Spartans attacked in great numbers.\n\n\"I'm heading down ahead of the fleet.\" Adam said. Kato pulled himself the rest of the way out of the confines of one of the ship's thrusters long enough to see Adam decked out in the Colonial blue and gray, complete with his Lieutenant insignia pinned to the collar of his tightly pressed shirt.\n\n\"I like what you guys have done with the ship. Good job. I'll be back soon enough.\" Michaels said, nodding his approval before walking further down the maintenance deck.\n\n\"What was that all about?\" Kelly asked, glaring through the temporary hole of the Bridge.\n\n\"I dunno. Did he shave or something?\" Kato replied, clueless as to Adam's life altering decision to join the fleet. Grease smeared across his face as he held a massive chrome wrench in his right hand, once again pushing the creeper he rested on back under the work at hand.\n\nAdam stood there for several tense moments, looking at the four man team of soldiers that had been assigned to him. Already seated aboard the Colonial chopper, they were ready to launch at the order of the newly pinned Lieutenant.\n\n\"Alright, listen up. \"I'm sure you all have ready the mission brief and I'm confident that you know what you are doing. Simply put, we get in, get our soldiers and get out. Safely. All weapons stay suppressed unless ordered otherwise. If the intelligence on Lieutenant Avery is incorrect, we will then fall under his command and assist the first team with operations until the Colonial Fleet arrives. Any questions?\" Lieutenant Michaels asked as the soldiers simply nodded firmly in compliance before looking into the direction of the loading door on the craft.\n\nSarah had joined the group, wearing an elegant red colored dress, she had accented her already perfect face with vibrant makeup and had her hair arranged as if she were about to attend a social dinner. Adam stood there speechless, wondering how many more tricks the woman of his dreams had up her sleeve.\n\n\"What? A girl's gotta look good.\" Sarah said comically as she finished boarding the chopper, hoisting a large black combat rifle in her right arm.\n\n\"Yes, you do.\" Adam replied as he sat down in one of the rough black leather seats near the cockpit area, nothing short of speechless.\n\n\"We've been cleared by the bridge sir. On your order.\" the pilot said in a disciplined tone as Lieutenant Michaels gave the go ahead, looking one last time at the safe confines of Colonial Star Five. With his order, the chopper slowly pulled from the thick steel flooring of the shipping bay, turning slowly as several of the crew aboard the capital ship looked on. Seconds later, the small craft executed a full burn, disappearing from sight and showing up on the Star Five's radar system.\n\n\"I'm telling you, it'll work,\" Dalton said loudly to the others, referring to his plan to stage a fight and try to lure one of the less prominent Legion guards to the cell. \"I saw Adam do it once in the public jail on Star City.\" he added.\n\n\"A little difference between a local jail on a space station and a heavily fortified compound led by Hunters.\" Steiner replied grimly.\n\n\"It's the best plan so far.\" the Colonial sniper said, speaking up for the first time since being reunited with the group.\n\n\"Thank you,\" Dalton replied sincerely. \"What's your name again chief?\" Dalton asked. \"Corporal Lassiter.\" the sniper replied, obviously still reeling from the injuries he had suffered during his interrogation time.\n\n\"See, I figure we stage a fight. Hell, Roman and the Husk population aren't exactly on the best of terms anyway. The guards come in to shake down the cell, we turn on 'em and get the fuck out of here.\" Dalton said, as if he were throwing a sales pitch for a new home, even closing things with his own variation of a realtor's smile.\n\n\"Sounds like a solid enough plan.\" Lieutenant Avery said as he approached the cell. Roman cut his eyes at Dalton for a moment, as if to mentally slap him across the face, before turning his attention back to Avery. \"Now, let me tell you a really solid plan. The Colonial Fleet arrives, we purposely hold back our rail guns and surface to air missiles, convincing them that you idiots were successful. Then, once they are far enough in, we launch everything we have from the far side of the planet. They will be outnumbered at least three times over, crushing the backbone of the Colonial Army with one fail swoop.\"\n\nThe men stood there for a moment before Roman finally approached the front of the cell; thick steel bars the only thing separating him from Avery.\n\n\"So how much do they offer a gutless piece of shit like you to turn on his own kind?\" Roman asked. Laughing for a moment, Avery returned the stone glare of Roman.\n\n\"Enough. Ask your friend Lassiter back there, they made us both the same offer, he was just too proud to take it.\" Avery replied.\n\nRoman glanced back at the sniper for a moment, nodding his appreciation.\n\n\"He'll live to see the end of this war. You won't.\" Roman replied, walking away from the front of the cell.\n\n\"So sure are you about that? I plan to kill you myself when the Hunters arrive.\" Avery said confidently.\n\n\"Yea, I haven't heard that line before.\" Roman replied, sitting down beside Lassiter in the corner of the cell.\n\n\"Soon.\" Avery replied, walking away to regroup with his security detail of two Legion soldiers.\n\nKato slid out from his workspace for a moment, wiping his face of excess grease and obviously in deep thought. He quickly grabbed a piece of scrap paper from his workbench and began writing several items down in a concentrated manner.\n\n\"What are you doing?\" Kelly asked as she peeked through one of the cockpit windows.\n\n\"Hopefully, modifications.\" Kato replied, immediately flagging down a deck hand to see if the listed supplies were available.\n\n\"Less than one minute out! Check your gear!\" Lieutenant Michaels said loudly as the crew area of the chopper clicked over from red to yellow. The soldiers began checking their battle rifles and combat pistols, all of which had been previously double checked. The sun had started to break just a bit as the chopper hit ground, the cabin light turning green as everyone dashed out of the passenger door, their solid black boots crunching onto the brittle ground beneath. The chopper had to land farther out, obviously, because of the daylight that was upcoming; leaving the team with a good hike in front of them.\n\n\"Last chance?\" Adam asked as Sarah just looked at him sarcastically for a moment before climbing down to ground level.\n\nEveryone remained low as the chopper turned sharply, rising from the ground before hitting a full burnout. Shortly after they had made it to a small tree line nearby, two Legion jets flew past as a very high rate of speed. \"Daggers. Moving fast, looks like the chopper didn't make it out of here off of radar.\" Sarah said regretfully. They listened quietly as chatter came across the Colonial com units of two fighter planes intercepting the chopper, soon after, the chatter was cut silent.\n\n\"Get in the supply kit and get the reflector blanket out, sure to be scanning the area soon.\" Sarah said.\n\nSure enough, only minutes later the Dagger ships passed over their area several times, scanning the ground for heat signatures. The crew lay quietly under the reflective blanket, as its exterior digitally changed to adapt to the terrain, camouflaging them from anything overhead as they kept eyes open for anything or anyone on foot. Nearly an hour after the Daggers had executed their last flyover, Lieutenant Michaels deemed it safe to move as the squad made their way to the compound.\n\n\"Well, I guess we're back to plan B huh?\" Dalton asked as the rest of the prisoners sat quietly.\n\n\"First person through that cell door is dead by my hands, and I pray that it's Avery.\" Roman said softly, meaning every word of it.\n\n\"They will come at us heavy, knowing what we might be capable of,\" Steiner said. \"Best chance would be to play it down until we are all out of the cell and under gunpoint. More room to move around.\" he added.\n\n\"For what it's worth, there's nobody I'd rather be busting out of captivity and more than likely dying with,\" Dalton replied, trying to lighten the mood of the men, who had the gloom of a death sentence hanging over their heads. \"Of course, I'd like to be doing it with a bottle of whiskey in my hand, but beggars can't be choosers.\" he added, prompting the men to laugh for a moment.\n\nCommander Blaine stood on an overhead catwalk as he watched regiments of men get into formation before boarding troop carriers. They would have to fly down under heavy Spartan escort, having no ship to ship weapons of their own. As impressive as the Goliath units were, there were still far too few of them to launch as an entire regiment, instead they were packaging two of them per troop ship in order to assist the ground forces. Little was known about the new strain of genetically enhanced Hunter, but he couldn't imagine anything strong enough to take down a Goliath unit, not even a Fang.\n\nEveryone among the Colonial Fleet was prepared to fight and even die if needed be, still the Commander placed most of his respect on the Spartan pilots. Having been one himself before moving up through the ranks, he understood that the survival rate for them were the lowest, seeing the most action in a battle such as the one that was to come. Dog fighting Daggers, escorting the ground force to a safe landing and avoiding rail gun fire; these were only a few of the responsibilities that had been cast upon the Spartan pilots. He was damn proud to have them in the Colonial fleet.\n\nAt first glance, the size of the Colonial attack force looked tremendous. However, it paled in comparison to the troops the Legion had at its disposal, which is why Commander Blaine could only hope that they were successful in catching the superior enemy off guard. Otherwise, it was inevitable that the Colonial Army would face a crushing defeat.\n\nAvery waited outside in a slight rain, accompanied by two Legion troops as a small shuttle began descent, making its approach to the landing pad. The strange exterior, which had a ribbed texture and several pointed overhangs, let them know right away that it was indeed a Hunter vessel. At it touched down on the saturated gray pavement, dust blew out from the bottom of the ship loudly, forcing the men to look away momentarily. Soon after, a metal ramp began to lower, offering a glimpse inside; illuminated buttons blinking against the wall in several different color sequences.\n\nA very heavily armed Hunter was the first down the ramp, surveying the immediate area for any possible threats. Seconds later, Anwick followed the armed Hunter down the ramp, followed by two more of the larger killers, all four dressed in black leather from head to toe, Anwick wearing a white medallion while the three soldiers were blanketed in a much heavier armor.\n\n\"Lieutenant Avery I presume?\" Anwick asked, moving close to him, his skin pigmented with lifelessness as he teeth appeared filed to a point.\n\n\"That is correct.\" Avery replied formally.\n\n\"Job well done. I would much like to see the prisoners at once, especially the one called Roman.\" Anwick replied.\n\n\"Then it will be arranged at once.\" Avery replied, extending his arm as to welcome the Hunters into the compound.\n\n\"Two Hunter Carriers will land here for a short period, deploying troops as well as a handful of Fangs, at which time the Carriers will group with your Legion ships and prepare for an air assault.\" Anwick said as the group walked down a well lit hallway of solid white walls and ceiling.\n\n\"Indeed,\" Avery responded. \"The force that we have assembled should be more than enough to shred the Colonial Fleet within minutes.\" he added.\n\n\"Good. Very good.\" Anwick replied.\n\nThe lights in and around the cell switched from dim to full bright as the prisoners jumped to their feet, Lassiter much slower, pulling himself up through the pain of his injuries.\n\n\"Roman Raines,\" Anwick said proudly as he approached the cell. \"Much has changed since my friend Lexion lay dead by your hands.\" he added.\n\n\"If you are referring to my ability to kill Hunters, actually, not much has changed.\" Roman said insultingly.\n\n\"Yes. I was not aware of your past until your crew slipped from my grasp on Tameca. The deeper I dug into your files, the more impressed I became. So, do you mind if I ask you exactly how many of my kind you killed during your time with the Gali Special Forces?\" Anwick asked, knowing Roman had never officially been a Gali soldier.\n\n\"I stopped counting after the first hundred or so.\" Roman replied, gaining instant stares from the other prisoners.\n\n\"Hundred? You said a couple?\" Dalton added in amazement.\n\n\"You see, it seems that your crew has been carrying a cold blooded killer for some time now. Isn't that right Roman?\" Anwick replied, looking heavily at the hardened soldier.\n\n\"In fact, the only reason he is with you is because he is being hunted by his own government,\" Anwick said as Roman looked up at him brashly. \"Seems he turned on his own kind, killed a handful of Gali Special Ops soldiers in the process. As much as I want to kill you with my own hands, even I couldn't turn down the bounty they have placed on your head. I wonder what your precious Captain Michaels would say to this?\" he added.\n\n\"I'd say ten percent sounds about right, don't you think Roman?\" Michaels said from behind the group, standing beside Sarah and two of the Colonial soldiers, the remaining two on the opposite side of the group.\n\n\"I'd rather rot in this Legion prison cell, ten percent is chump change,\" Roman replied smiling as Anwick and his group stood in shock. \"On the other hand, fifteen sounds about right.\" Roman added as Dalton grinned widely, his beard nearly falling from his face in the process.\n\n\"I can do fifteen,\" Michaels replied, looking over to Anwick, Avery and the escort soldiers. \"Now open the door.\"\n\n\"Captain Michaels. You must know that there is no place you can hide that will be safe?\" Anwick asked.\n\n\"Lieutenant Michaels, actually, and I'm done running.\" Adam said, pulling a long black lever on the stock of his rifle; arming it to fire. \"Now. Open the damn door.\" he added.\n\nAs the group stood there waiting for Anwick's reply, Adam fired a shot, hitting one of the Legion soldiers in the head and deadening him on the spot. Pulling the pin back again, Avery stepped to the front, yelling, \"Open the damn door!\"\n\nAs Roman and Dalton exited the cell, Steiner followed close behind, helping Lassiter. As Michaels began to hand the men sidearms, Sarah walked toward the group, bluntly slapping Avery across the cheek of his face.\n\n\"That's for handing me over to them!\" she yelled as Anwick, the three escort Hunters and a Legion soldier looked on.\n\n\"You bitch! I should have killed you myself and been done with it!\" Avery managed to yell before having nearly an entire side of his face blown off by Roman's combat pistol.\n\n\"That's me keeping a promise.\" the Gali warrior remarked as they bolted the cell door shut, getting into position to move out.\n\n\"Ya'll hang tight now.\" Dalton said tauntingly.\n\n\"This isn't over Adam, I promise you that.\" Anwick said calmly with his face pressed against the bars of the cell.\n\n\"If not, this puts you down two to nothing.\" Michaels smiled, infuriating the Hunter, who began shrieking loudly.\n\n\"Damn glad to see you guys.\" Dalton said, getting a slight pat on his shoulder and nod from Lieutenant Michaels.\n\nCommander Blaine stood in the command center of the Colonial Star Five, which was rendezvousing with three more of the Colonial Stars before launching the assault against the Legion. It was nothing more than a staging ground, ships falling into formation, giving the operation the best chance for success while limiting damage as much as possible. Surrounded by neon green grids on all of the walls and a huge grid which was mounted to the floor in the center of the room, crewmen used special graphing markers to update the grids with the position of the ships, while Colonial chatter broadcast throughout the room on several com units.\n\nEven Blaine, who was the ranking officer on Colonial Star Five, didn't know what each panel in the room was used for. All he knew is that they were individually manned by members of his crew, who monitored everything from communication between Spartan pilots to damage control and assessment of the ship. There was no window in the Command Room that overlooked the stars; instead, he had to rely on the information of the grids to be correct. As he watched, a crew member listened to the chatter over his headset for a moment, before aligning the three ships on the central grid that represented the Colonial Star Ships.\n\n\"Everyone is in position Commander, awaiting your go.\" the crewman said as Blaine took one last look around the Command Center, taking in the sights of a ship untouched, the looks on every soul's face as they awaited his command to officially send them into war with the Legion army.\n\n\"I wanted to take a moment to personally thank each and every one of you for your service to the Allied Colonies. To say that it is an honor to serve with you would be an understatement. Even as we get set to embark on an act of war that will forever change the complexion of this star system, I want you all to hold to the belief that we are doing so in the most innocent of fashions. To preserve our freedom and the freedom of those citizens who cannot defend themselves. Good luck and God speed,\" the Commander said proudly through a hand held com unit, his message ringing throughout every ship, every headset and every Spartan pilot's helmet. \"You have a go.\" he calmly said to the crew member as he mounted his com unit back onto the wall.\n\n\"So where's our ride?\" Dalton asked.\n\n\"There is no ride, at least not yet. We find a place and dig down until the Colonial force arrives.\" Adam answered.\n\n\"Nice, but there will be thousands of people shooting, and that's only here on the ground. How the hell do they expect to find us?\" Dalton replied, looking at Adam dumbfounded. Rather than replying, Lieutenant Michaels simply grabbed the medallion signifying his rank and flipped it over, exposing a tracking beacon.\n\n\"Oh, well, I guess that will work, but I would have done it differently.\" Dalton replied in an attempt to save face.\n\n\"We need to make it to the outer perimeter of the compound, much better chance of hanging out unnoticed.\" Adam said to the group as he strategically scouted the horizon.\n\nSeveral rifles shots rang out as two of the Colonial soldiers fell to the ground abruptly. The rest of the group dove for cover, finding refuge behind a stack of metal shipping crates nearby. A small squad of Legion troops had noticed the intruders, sounding a screeching alarm that rang throughout the compound, prompting several sets of soldiers to follow in the pursuit of the Colonial group.\n\n\"Well, there's our dick flapping in the wind,\" Dalton said, pulling one of the seriously injured soldiers to cover, his body dragging roughly across the solid dirt surface. \"Well, not all of us.\" he added after a punishing stare from Sarah. It was a solid spot of cover, but it wouldn't last long with the large number of enemies closing in on them, the metal shipping crates taking a pounding from the relentless gunfire.\n\n\"Well, looks like this may be it.\" Lassiter said.\n\n\"Just hold as long as we possibly can.\" Lieutenant Michaels replied as Roman and Steiner both grabbed rifles to join up with Dalton while Sarah helped tend to the fallen troops.\n\nPressing a makeshift bandage down firmly onto one of the gunshot wounds of a Colonial solider, Sarah stopped suddenly as she heard a thunderous scream nearby.\n\n\"What... the fuck...was that?\" Dalton asked cautiously, joining the other men in staring at one of the Fang soldiers. Hulking tall, it was at least ten feet in height and the largest creature any of them had ever seen walking on two legs. A huge fang extending from the bottom of its mouth on each side, at least as wide as a human hand, solid white hair thickly braided and falling down the upper portion of its back. Heavily armored with a thick chest plate of black plate mail, it quickly became apparent to the Colonial soldiers that the weapons they were firing were having little to no effect as it continued a full sprint toward them unhindered.\n\nThe commanding officer of the group simply stood in amazement, having never seen anything like a Fang before.\n\n\"Adam! We need to move, NOW!\" Sarah yelled, bringing him back into reality for a moment.\n\nHe nodded, frantically checking his surroundings in hopes of finding a way out of the impending slaughter of his team. He needed a miracle of some kind, they were surrounded and pinned down in a spot that provided no exit.\n\nThe bloodcurdling screams of the beast grew louder, before suddenly being silenced by the overpowering booms of hundreds, if not thousands of ships hitting the planet's atmosphere at full burn. The Colonial fleet had arrived with not a second to spare, the attack force descending onto the Legion fortress quickly and without retaliation. Adam noticed the Legion troops, who had been moving in on their position, had began scrambling to nearby buildings to prepare for the inevitable battle at hand. He prayed that the Fang who was bearing down on them would follow suit.\n\nNo such luck. Michaels quickly checked, only to see the monster continuing his rush towards them. Trying his best to hold back his emotion; wanting only to begin firing his rifle into the sky and curse the Gods who had returned the favor his entire life, he simply threw the rifle to the ground in favor of his two revolver style pistols. Adam closed his eyes, tightly gripping the wooden handles of the weapons as he knew the Fang would be on them in only moments.\n\n\"Adam get them out of here, now.\" Roman replied, as he stood there holding a combat pistol and hand length blade.\n\nSteiner quickly joined the other hero, unsheathing his machete and pulling a shotgun to the ready with his free hand. Adam realized at that moment, Roman was prepared to die in a much different fashion than the one he had lived it, as a hero.\n\n\"GO!\" Roman yelled as the Fang was close enough to hear the warrior's command. Adam pointed into the direction of the nearest building he could find as Sarah, Dalton and the remaining Colonial soldiers who were able, helped the wounded quickly to the building's position.\n\nLieutenant Michaels slowed his run down long enough to turn and look back at his friend, who alongside Steiner, was fighting valiantly in the face of death.\n\nHaving been shot several times, at least two rounds hitting it from point blank range directly in the face, the Fang fiercely struck Roman in the area of his neck and face; dropping the former Gali warrior. Before Steiner could intervene with a swing of his machete, the Fang fired two shots from a Hunter rifle, both finding their mark into the chest of Roman who lay limp in a quickly formed pool of his own blood.\n\nTime seemed to stop. Adam considered going back for his friends and trying to help them take out the Fang. He also knew in his heart that Roman was either dead or committed to the act of dying. He glanced the other way, seeing Sarah waving him to her in slow motion, Dalton yelling to him with no sound reaching his ears. This was truly a moment that would stick with him for the rest of his life. He had to make a decision between what he knew was the right and running to the woman that he loved. He was tired of running.\n\nAs he loosened the grip on the handles of his pistols only slightly, Adam turned his attention back to the Fang, who was fighting fiercely with Steiner as Roman continued to lay lifeless in the same spot; blood soaking into the surrounding ground. It was time to fight back or die trying. Adam started a path in the direction of the fight when he was thrown several feet, before hitting the unforgiving dirt with tremendous force by a nearby blast. Explosions rang out around him as the Spartans had made it to near ground level, firing their weapons onto the compound in unison.\n\nIt took only moments for Michaels to recognize the desperation of Sarah's voice, yelling loudly enough to get his attention; his ears still ringing from the deafening explosion. He slowly stumbled toward them, shell shocked and heavily disoriented.\n\n\"What were you thinking?\" Sarah yelled as she and Dalton helped the Lieutenant into the door of the small building. He didn't reply, instead, looking back to his friends only to see nothing but empty ground.\n\nThe thunderous booms of the explosions began to multiply as Legion forces sprang the trap, sending hundreds of air defense missiles into the sky followed by squadrons of Strikewings burning at full speed. Tracer rounds finished lighting the breaking dawn sky as hundreds of Colonial ships engaged the Legion attack planes in dog fighting tactics. Meanwhile, the Colonial Stars moved into position, firing their multiple ship to ship cannons at the approaching Legion Capital ships and Hunter Carriers.\n\n\"Have you located Lieutenant Michaels' tracking beacon?\" Commander Blaine said, standing firmly at the central grid inside of the Command Center.\n\n\"Yes sir, he's indoors and the area is too hot for extraction. Will have to wait for the area to secure or they will have to come to us.\" one of his crew replied, charting several positions on a nearby wall mounted grid.\n\n\"Very well. Concentrate guns onto Legion ships, they will be much more populated than the Carriers.\" Blaine instructed as his crew members acknowledged the order and began relaying the instructions to the ship's weapons room.\n\n\"Our Spartans are taking a pounding up there.\" Lassiter said as the strike team continued to look outside.\n\n\"The entire Fleet is taking a pounding,\" Sarah replied. She was right, the Colonial Fleet was heavily outnumbered and taking great losses across the board. \"Listen,\" Sarah said, quieting the rest of the crew. \"Mini-guns!\" she added as a Colonial sweeper team moved into sight, escorted by two of the very capable Goliath units.\n\nThe heavily armored soldiers of steel slowly made their way through a small portion of the compound, mowing Legion forces down in their wake. The lead continued to pour out of their automatic weapons, chewing into the flesh of the outmatched foes in red and black leather as the Colonial forces had almost reached Lieutenant Michaels and his team; when a group of two Hunters escorted a single Fang into the area. Aiming down the sights of one of his pistols, Adam's arm was quickly pushed away by the swift hand of Sarah.\n\n\"We are no match, not without Roman and Steiner. Gunfire would only give our position away.\" she said, pleading with Adam as he looked back at his team for a moment.\n\nLassiter, as well as two of the Colonial soldiers were wounded, one of them life threatening. She was right, they would be of very little help to the Colonial force outside, all they could do is remain sheltered and wait for the skirmish to end, hoping the Goliaths could end the life of the Fang.\n\nHowever, hope was soon cut short. As the half dozen Colonial soldiers exchanged fire with the two Hunters, the Goliath units began to circle the Fang. The small-arms fire of the others tapered off quite a bit, which was an indication that everyone was watching the battle between the Goliaths and Fang ensue. It was the first of its kind, and a very good bookmark as to which side had the advantage throughout the star system.\n\nFor several minutes, the Goliaths maintained a combination of gunfire and punishing punches by way of their solid titanium arms. As it seemed the mechanical menaces were gaining the upper hand of the fight, one of the Goliath units made the mistake of overreaching during a wide angled punch, the Fang thrusting up to tear the mechanical arm from the shoulder of the super soldier. As the Goliath staggered several feet back, the Fang then pummeled the unit to the grown using its closed fists, turning its attention to the second Goliath. Falling back, the remaining Goliath began to lay down cover fire as the Colonial soldiers retreated.\n\nIt had quickly become obvious to Adam and his crew that the Colonials would fall in defeat at the end of this battle. Their only hope now was to find a way back to the Colonial Star Five before time ran out. The courtyard of the compound was littered with fallen Legion soldiers, ship wreckage from the thick sky above, Colonial men and women who had died for a cause they believed in and even the occasional Hunter.\n\nThey covertly moved from position to position, careful not to alert any of the units in the open that were engaged in the mass bloodshed. The plan was to move back to the rear of the compound, leaving the same way they had arrived and hope that Colonial ships spotted their beacon before Legion forces caught up to them. It wasn't the greatest of plans, but Adam had thrown it together under fire and it was all they had to go with.\n\n\"Lieutenant Michaels.\" Commander Blaine's voice announced through Adam's ear mounted com unit.\n\n\"Go ahead!\" Michaels replied, shocked to hear the Blaine's voice in the midst of he loud fighting around him.\n\n\"We have picked you up on the grid. If there is any chance you can move two clicks to the Southeast, there is a large Colonial unit dug in there to assist you.\" Blaine replied.\n\n\"Southeast. Move out now!\" Michaels yelled to his group, responding to the Commander's request. \"Yes sir.\"\n\n\"Is Sarah alright?\" Blaine asked cautiously, the few seconds of silence seeming like eternity.\n\n\"Yes sir. I have three wounded, four confirmed dead and two missing, presumed dead.\" Adam replied gravely.\n\n\"Understood Lieutenant. Get your people Southeast.\" Commander Blaine replied as the Colonial Star Five shook fiercely, several missiles hitting the large starship at once.\n\n\"Report!\" Blaine yelled as the ship's interior lights continued their fluctuation.\n\n\"Structural damage to the starboard section of the ship, shields have absorbed substantial damage but are holding.\" one of his crew members said lifelessly.\n\n\"Take us down close to the surface. All cannons redirect fire to surface, make my daughter a safe path to walk through.\" Commander Blaine said before slowly taking a wall mounted com unit into his hand.\n\n\"This is the Commander. I am issuing a general order to abandon ship. All remaining Spartans are cleared to launch, remaining soldiers are instructed to rally on the surface and support existing units. It has been my honor to serve as your Commander.\" Blaine said as he slowly returned the com unit to its cradle as the ship was hit with several more breaking explosions.\n\nThe Legion and Hunter ships knew the Colonial Star Five was in serious trouble and had begun concentrating their fire onto the dying masterpiece, which was starting to tear apart at the welded seams. Dozens of Spartan ships launched simultaneously in an effort to protect the sinking ship, most of which were nothing more than small explosions shortly after launching directly into the concentrated gunfire from the Legion ships.\n\n\"Lieutenant, take care of Sarah. That's an order.\" Commander Blaine said across the com unit as the Colonial Star Five exploded, causing a rippling wave of fire throughout the sky.\n\n\"FATHER!\" Sarah screamed as she began weeping uncontrollably, quickly grabbed by Adam as he stood speechless.\n\n\"Lieutenant Michaels, this is Spartan Nine One.\" Adam heard in an unfamiliar voice through his com.\n\n\"Go ahead.\" Michaels said, trying to hold himself together as Sarah continuing to cry loudly in the background.\n\n\"We have your position and are ready to assist you to the rally point.\" the pilot said as eight Spartans flew overhead at incredible speeds.\n\n\"Copy.\" Adam said softly, numb from everything he had lost in only a few short hours.\n\nAs the group looked ahead, they could see a flashing blue light, which every Colonial soldier would immediately recognize as a signal for friendly ground.\n\nThe group began to move as quickly as possible, Adam carrying Sarah as Dalton escorted them by shotgun. Lassiter was slowly moving unassisted, while the remaining Colonial soldiers were responsible for moving the wounded to safety. As they had almost reached the Colonial camp, empty brass shells rained from overhead as the Spartans passed, firing their tracer rounds and killing nearly a dozen nearby Legion soldiers. Moments later the Spartans were heavily engaged by a squadron of Strikewings, however they had been successful in getting Michaels and the crew to safety.\n\nMet by several Colonial troops, the wounded were immediately taken by medics who were decked out in solid blue with white trim.\n\n\"Lieutenant Michaels, you're needed in Outpost Command one of the soldiers said, pointing to a large blue tent which was surrounded by sandbags and three large rail guns pointed into the sky. Sarah accompanied Adam as the rest of the crew got a quick look over by one of the field medics.\n\n\"I'm fine, other than being in serious need of a damn drink,\" Dalton said loudly, wiping a blanket of dirt from his face. \"And a hell of a lot of therapy.\" he added.\n\n\"Sarah, I'm sorry about your father.\" one of the officers said sincerely, placing his hand on her arm for a few moments. Although she had obviously been crying heavily, she did her best to put on a front of strength in front of the soldiers inside of the tent. She had to. They looked at hear as a beacon of strength. They had all lost during this short lived war, she had to press on, at least publicly.\n\n\"Lieutenant Michaels. We've lost the Colonial Star Five, seven of our larger cruise ships and the Colonial Star Three is in danger of being lost to us as well. We have sent a relay back to Colonial Command for reinforcements, it was swiftly denied.\" the officer said.\n\n\"Denied!\" Sarah yelled as Adam try to hold her back while keeping her calm.\n\n\"Yes ma'am. There is a second battle happening this very moment on Tameca. Command relayed back to us that Tameca City is currently controlled by Legion forces, while Colonial forces are mounting to attempt a takeover of the city.\" he replied.\n\n\"What's the plan?\" Michaels asked.\n\n\"Well Lieutenant, two Colonial Stars have been dispatched to help us evacuate everyone from the surface. The plan is holding onto as much rock as we possibly can until they arrive.\" the officer replied.\n\n\"Evacuate! We have lost so many men here, for nothing?\" Sarah asked animately.\n\n\"There is no hope of winning this battle, we were outmatched minutes after hitting the atmosphere. Command wants us to group up with the remaining forces on Glimmeria, become the last line of defense in case Tameca City is a loss.\" the officer added as mortar explosion burst in the background.\n\nGlimmeria. Now there was a word Adam hadn't heard in quite some time, but was far too familiar with. Having fought alongside Dalton in Glimmeria several years back, he was one of the few lucky ones to survive. It was primarily a wasteland, steep canyons of rock and scorching sands marked with the occasional small city. From what Adam had heard over the past few years, the larger of the cities had rebuilt after the grinding war, however the outskirts of the planet remained in ruin, controlled by smaller organized crime families.\n\nWhen the war ended, he had promised himself to never return. Too many of his good friends had fallen, most of them right in front of his very eyes. Now, he was hoping to last long enough for a rescue, only to set a course straight for a planet that had haunted his dreams many nights over. As terrified as a reunion to the soil that had taken part of his soul, strategically it made sense. The Glimmerian government, as well as every crime family with Glimmerian ties he had done business with, had two things in common.\n\nThey were among the toughest people he had ever known, and they had a mutual hatred for the Legion.\n\nSure, their army was miniscule compared to the Legion's forces, but traveling across the system to fight on Glimmeria would put them at a huge disadvantage by stretching them thin and placing a huge strain on their resources. It would give the Colonial army, at least what was left of it, more than enough time to regroup, recruit new planets to their cause and prepare for an assault by the Legion.\n\n\"Well, then let's hold our ground. What do we have to work with?\" Michaels asked.\n\n\"Sarah's father sent us everything he had on Colonial Star Five. He could of turned and ran, but I think he was more concerned with getting his daughter off of this rock safely.\" the officer replied, bringing Sarah to a emotional state.\n\n\"Colonial Star Three is sending us a bulk of its ground force before leaving the battle, should be here and ready within the hour. Looking at about forty battle ready soldiers, a dozen or so Goliath units and by the looks of it; just a handful of Spartans.\" the officer replied.\n\nThe meeting was cut short as the rail guns outside of the tent began thrusting pounding shots of lead into the air, Strikewing units flying overhead firing tracers into the encampment.\n\n\"Good bet if they didn't know we were here, they do now!\" Lieutenant Michaels yelled to the officer as the sound of the gunfire pierced their ears loudly.\n\n\"Sir! Small group approaching on foot!\" one of the privates yelled through the entrance to the tent, prompting the officers to flood outside to direct their squads.\n\n\"Hold this damn ground!\" Adam yelled loudly, walking outside to regroup with Dalton, who had taken position beside Lassiter.\n\n\"Do you think they made it off of the Colonial Star Five before it...?\" Sarah asked before being cut short by Adam.\n\n\"I wouldn't worry, I'm sure they did.\"\n\nIt was a lie of course, Michaels himself had been worried about Kelly, Kato and Troy from the moment he saw the ship go thermal in the explosion. It was a justified lie though, knowing Sarah was still hurting from the death of her father.\n\n\"Holy shit,\" Dalton yelled ecstatically. \"It's Roman and Steiner!\" he added.\n\n\"Give them cover fire!\" Adam yelled, seeing the two friends being closely pursued by a small group of Hunters.\n\nAs the they continued to sprint for the Colonial encampment, Dalton stood up, waving his friends to their direction. Four Goliath units were quickly dispatched to intercept the Hunter party, and intercept they did; the titanium peace keepers sprinted past Roman and Steiner as their mini guns began unleashing a flurry of grave digging lead.\n\nThe Colonial soldiers counted five Hunters, who outnumbered the Goliaths, but were solidly outmatched from the start. Within the span of one minute, all but one of the Hunters lay dead, the fifth fleeing back the way he came only to regroup with a much larger Legion group of attackers. Obeying orders, the Goliaths began sprinting back, two of them not lucky enough to make it, taking a beating from the Legion gunfire until they finally could function no more.\n\nJumping over the sandbag barrier, Roman quickly fell face down, letting the man made shield absorb several shots that were fully intended to end his life. Steiner dove over just seconds behind, not as lucky as he took a piercing shot into his left biceps muscle and taking a good portion of it when it exited. In obvious pain, he remained on the ground for a few seconds as two of the Colonial field medics began numbing the pain while bandaging it the best they could.\n\nDirt began flying as bullets zipped through the Colonial outpost, soldiers diving for cover any place they could find refuge.\n\n\"The more of these bastards I kill, the better I feel about myself and life in general.\" Dalton said to Lasstier as they both fired long range rifles at the approaching group.\n\n\"Pretty accurate with that rifle, huh?\" Dalton asked as Lassiter glanced down at the sniper insignia sewn onto his Colonial uniform.\n\n\"Yea, yea. Well I still say fightin' up close is the best way to handle things.\" Dalton added snidely.\n\nFor hours, the Colonial forces and Legion troops exchanged gunfire from only a short distance; the occasional flyover of aircraft, usually Legion Strikewings spraying stinging clouds of ammunition to the ground below. Roman had joined the fight, gaining a combat rifle from a fallen soldier and using it with precision. Steiner on the other hand, rested near the back of the encampment, a long barrel shotgun in his better arm as the few officers that remained met with Adam.\n\n\"We're being cut to pieces!\" Lieutenant Michaels told a couple of the officers loud enough to overpower the sound of gunshots, who were quick to agree as huge chunks of rocks and debris flew around them while mortar shells hit throughout the camp.\n\n\"We have to fall back.\" Adam said as a rare Spartan squadron flew overhead.\n\n\"There's no place to fall back to. A couple hundred feet behind this camp, nothing but cliffs. Seemed like a strategic advantage at the time, no way of being flanked by a ground assault.\" one of the few remaining officers replied, immediately thrusting Adam into deep thought.\n\nSure, he was Lieutenant Michaels of the Colonial Fleet, but underneath all of that he was still the same guy who specialized in getting himself out of a pinch.\n\nAs he glanced around the Colonial controlled soil, he realized that over half of them were wounded, many who were doomed to perish before day's end. The Legion continued pounding them with fire, although it had slacked up just a bit; which was a good indication that they were organizing a mass assault to overrun the Colonial outpost. Adam saw no way out except for surrender. He knew a lot of the men wouldn't simply lay down arms knowing the Legion and Hunters intended to kill them regardless. Still, the blood wouldn't be on Adam's hands. Rather than leave it in the hands of the remaining officers, he decided to take it upon himself to tell the soldiers on the front line himself.\n\nAs he walked into the direction of Roman, Dalton and Lassiter who would be the first to hear it, three squadrons of Spartans flew past at dazzling speeds. Quickly after, several more squadrons flew past, causing everyone in a Colonial uniform to turn away from the Legion force and stare into the sky behind them. Two of the huge Colonial vessels had entered the fight, pounding the Legion's forces with their cannons without regard.\n\n\"Lieutenant Michaels, this is Commander Douglas of the Colonial Star One Seven, prepare your men for immediate extraction.\" a voice rang out inside of Adam's com unit.\n\nFinally. Finally the Gods above had given him a break, and he wasted no time replying to the Commander emotionally. One of the Colonial Stars moved in directly over the Legion compound, as it began firing relentlessly, causing multiple explosions throughout. Shortly after, two Hunter Carrier ships had come to the aid of the Legion; engaging the Colonial Star as the three ships returned earthquake equivalent shots onto each other.\n\nThe Colonial troops who had fought so bravely to hold their ground, had little time to celebrate as choppers began landing to pick them up literally moments after Michaels had received the message. The Colonial wounded were the first off of the surface, as Michaels opted to stay with his team until the final pickup to provide cover for the other soldiers evacuation.\n\n\"Sarah. Please.\" Adam asked, praying that she would be on one of the first choppers up and out of harm's way. Instead, she refused a reply, grabbing her combat rifle and joining Roman, Dalton and Lassiter on the line. Even Steiner scoured the ground, eventually finding a long range weapon of his own and slowly joining the group in providing cover.\n\n\"We'll be on the next one up! Tell them to save a place for us!\" Lieutenant Michaels said loudly to one of the officers as the chopper's thrust engines nearly consumed the conversation. \"Will do Lieutenant! Good luck!\" the officer replied as he motioned the rest of the survivors onto the skiff, leaving a handful of the Goliath units behind to assist the group of heroes.\n\nAs the chopper left ground, a huge explosion brightened the sky as if it were a sun on the brink of extinction. The Hunter Carriers had eliminated one of the Colonial Stars and every soul aboard it. Adam and his crew simply stood in amazement as thousands of lives came to an end at the hand of those who had murdered so many before.\n\nWith the explosion of the Colonial Star, the Legion ground force that had massed began an all out charge to the crew's location in an effort to eliminate any survivors. Lieutenant Michaels countered the assault the only way possible, unleashing the remaining Goliath units directly into the charging enemies. They stood no chance of course, but they would at least thin them out a bit and maybe buy enough time for the next evacuation chopper to land. As they cut loose the mini gun fire, dozens of Legion soldiers fell instantly, many others joining only seconds behind. Returning fire, and aided by a small group of Fangs; within minutes the Goliaths were nothing more than metal to be recycled long after the battle had come to an end.\n\n\"Just hold them! Hold them long enough for... \" Adam began to yell to his crew before the second Colonial Star started backing away as well, no match for the Hunter Carriers.\n\nContacting them by com was of no use, the chatter through his ear piece was condemning evidence that they were being left to die. A full retreat ordered by the Commander, as those lucky enough to have gotten off this rock would be headed to Glimmeria to fight another day, while Adam and his crew remained.\n\nHe could see it in all of their eyes. The look of defeat, the submission to the inevitable. Sure, they had weapons and would take several of the bastards with them, but in the end there was no way out. Adam threw his combat rifle to the ground and pulled the two revolvers into position. He could think of worse deaths than fighting beside the woman you love, friends who would die for you and above all, freedom. They simply took a few moments, glancing at one another; nodding their appreciation and respect without saying so much as a single word.\n\n\"Let's take as many of the sumbitches to the grave with us as we can.\" Dalton said, readying his favorite toy, the snub nosed shotgun.\n\n\"It's been my honor everyone.\" Michaels said with respect as they heard a Strikewing approaching, coming in at blazing speeds from the sound of it.\n\nHe was wrong, it wasn't pushing high speeds, and it wasn't a Strikewing, it was his ship, the Gunship! He smiled as wide as he ever had before, his eyes full of tears as the titanium skiff lifted up beside them from the cliff below. Even Roman caught himself tearing up a bit as Troy looked out of the passenger window of the cockpit, sitting beside Kelly as Kato manned a large mini gun that had been mounted to the outside of the crew area. It was vastly smaller ship than the day of its crash landing, still Adam had never seen a more beautiful sight in all of his life.\n\nThe mini gun began to ring out, the tearing flesh of Legion soldiers falling from the bone as they screamed in morbid agony. The first in was Sarah Blaine, who immediately turned to begin firing her combat weapon into the direction of the attacking force. Next, Lassiter climbed in before going prone and using his long range rifle to end a few lives of his own. Adam stood there firing one round at a time, dropping a few Legion faithful before Roman could interject.\n\n\"It's your ship Adam, GO!\" he yelled, his chest covered with bloody bandages proving the capable handy work of the Colonial medics. Michaels hesitated for a moment before sprinting to the Gunship to reunite with everyone.\n\nAs Roman and Steiner began to make a run for it, Roman's leg buckled as a bullet passed through, sending him to a knee. Stopping to help him to his feet, Steiner was immediately overtaken by a Fang, jumping on top of him as it tried to rip meat from his skeletal frame. It took four shots from Roman's rifle to get the attention of the beast, who pounced from Steiner to Roman, slicing its claws across his already wounded stomach, as Roman did his best to try and fight back. The gunfire from the ship was enough to hold the Legion guards at bay, but it wasn't possible to get a clean shot on the hulking beast without risking friendly fire.\n\n\"Fuck this!\" Kato said, grabbing the Mauler that he had become so comfortable with over the years and jumping down to assist the members of his crew, while Dalton quickly took his place on the minigun. He sprinted to the fight as the Fang thrust his claws into Roman a second time, tearing vital organs and opening a river of blood onto the ground. With Roman defenseless, Steiner swung his machete with every ounce of power in his body, nearly severing the head of the Fang, which combined with several shots from the Mauler, was enough to send the monster to the ground screaming in pain. Steiner quickly hoisted Roman onto his shoulder and sprinted as best he could to the Gunship, Kato clearing a path behind them with the Mauler.\n\nReaching the ship, Steiner lifted his fallen friend carefully into the arms of Lieutenant Michaels, who was covered in blood within seconds of the transaction, laying Roman onto the floor of the ship as he frantically tried to stop the bleeding. Steiner looked back as he boarded long enough to see Kato fall to his death, several gunshot wounds leading him to it. A small group of Hunters sealed the deal as they sent nearly a dozen more shells from a pistol into the chest of the fallen hero, who lay dead with Mauler in hand.\n\nIn shock from the events, Kelly trembled in the pilot's seat, thinking of nothing but the death of Kato and the imminent demise of Roman Raines.\n\n\"KELLY, GO!\" Adam yelled with no result. \"Damn, we gotta go now!\" Dalton yelled inside the ship as gunfire ricocheted around him; prompting Sarah to quickly make her way to the pilot's chair, tossing Kelly, who was obviously in shock onto the floor.\n\nSarah quickly assumed control of the ship's flight stick, bringing the vessel to a full burn as Dalton shut the exterior door and helped Adam tend to the dying warrior.\n\n\"We've got to catch up to that Colonial Star, otherwise he isn't going to make it!\" Adam yelled desperately, trying to stop the internal bleeding of the former Gali commando.\n\n\"He isn't going to make it either way Adam, it would take a miracle.\" Dalton said solemnly as the Gunship blazed a trail of neon colored fumes across the sky, hitting orbit as it chased its only hope for a miracle to Glimmeria.\n\nBook 2\n\nGlimmeria\n\nThe wind blew steadily through the lush leaves that gave such a buxom appearance to the thick trees behind as Roman stood there, facing the pastel shimmer of the river before him. It was almost a surreal sight, the sky above filled with cotton white clouds moving slowly as he knelt to rub his hand in the dirt of the riverbed, small rocks slipping through his fingers and falling from his battle hardened grasp.\n\nIt had been such a long time since the warrior's eyes had last seen serenity such as was laid out before him at this very moment. Every second that he stood hypnotized by the perfected body of water, he made use of his senses, pulling every small detail into reality. Everything from the birds singing in their melodic language overhead to the sound of the small waves crashing against the front of his boots on the shore. Everything seemed so perfect.\n\nHis attention was immediately broken by movement to his left, a nearing boat full of passengers who stood grouped together on the front deck as it slowly made its way to the edge of the river. Roman's first reaction was to reach for his combat blade, he was a warrior well trained, and any soldier who had seen what his eyes had witnessed through decades of gruesome killing and tasteless wars would be quick to draw a weapon from suspicion. He realized quickly, however, that he was unarmed.\n\nUnusual for the former Gali commando, but that feeling soon dissipated as Roman began to recognize the faces on the boat that was docking with the river bank in front of him. Some were fallen warriors who had passed away beside him during one of many conflicts through the years, while others were familiar faces of family and close friends long deceased.\n\nWhat kind of madness was this? A multitude of things began to run rampant through his mind as he cautiously watched a man step off of the boat, his boots digging firmly into the water drenched sand as he slowly walked directly for Roman. Dressed in official blue, gold buttons holding the jacket closed, the man fit the part of someone who seemed in be in charge of a ferry such as this. As he approached Roman, the commando began calculating self defense tactics in his mind, just in case. He had never laid eyes on the stranger before and wasn't about to let his training go to waste. But there was no time.\n\n\"They are waiting for you Roman.\" the calculated man said in a somber voice, obviously referring to the faces aboard the ferry.\n\nHaving decided he had seen enough, Roman began to slowly back away before turning to quicken his pace into the direction of a heavily wooded area nearby.\n\n\"IT'S YOUR TIME!\" the ferryboat operator yelled in a demonic voice, grabbing Roman by the upper portion of his arm in an attempt to force him onto the vessel of souls.\n\n\"FUCK YOU!\" Roman yelled, pushing the man down onto the ground and quickly turning to run for the thick trees that were only feet away.\n\nThe perfect world of vibrantly painted surroundings soon blended with bright white lights as Roman tried to make sense of things, the birds quickly becoming sounds of advanced medical equipment as he lay on an operating table under nearly a dozen doctors and nurses, the flood of overpowering white lights blinding the man who had cheated death.\n\n\"I've got a pulse!\" one of the doctors yelled loudly, as several machines remained attached to the hardened warrior, giving a readout of every vital sign in his body, both the human and the mechanical side.\n\nRoman let loose a loud shriek of pain that was terrifying for everyone close by, a damn stern reminder of this being the first lifesaving surgical attempt using Goliath parts on a living human being.\n\nSeveral hours passed as Adam, Sarah, Dalton and Steiner waited patiently for some official word on their friend. Kelly and Lassiter had offered to watch over Troy, who was still recovering a bit from his injuries at the hands of the savage Husk and was trying his best to adjust to the desert planet of Glimmeria.\n\n\"Lieutenant Michaels,\" a doctor dressed in solid white said as he entered the waiting room, Adam immediately jumping to his feet, a standard issue Colonial combat pistol having replaced his trusted revolver in the holster on his side. \"Your friend should eventually make a full recovery,\" the doctor said, bringing immediate celebration among the crew who sat behind Adam. \"We lost him for a couple of minutes, but he came back to us on his own. It's unlike anything that I or any of my colleagues have ever seen. He is one tough man. It will take some time for him to get used to the idea of being half mechanical, however I hope he eventually understands that without the Goliath parts, he wouldn't be alive.\" the doctor added.\n\n\"Thank you.\" Adam said, shaking the doctor's hand before turning to his crew.\n\n\"This calls for a damn celebration!\" Dalton said, hinting to everyone that he needed a drink.\n\nGlimmeria may have been a desolate place for the most part, but the Colonial forces had sat down in and around Kamira, which was the planet's largest city. Dalton translated that into having drinking locations nearby, and now that he knew Roman was going to pull through, it was all about keeping his self proclaimed reputation intact.\n\n\"I'm going to stay here, sit with Roman and try to get Troy settled in with a local family.\" Adam said. The Colonials had recently started to place homeless children with families on Glimmeria, which was a much better fate than they would have had if taken in by Legion forces, which would have amounted to nothing short of forced slavery. After the crushing defeat at the hands of the Legion in Tameca City, there were plenty of children without homes, and even on a large planet such as Glimmeria, placing the children would be a daunting task at best.\n\n\"I'm game.\" Steiner said, volunteering to go with Dalton and rage a bit at any of the drinking establishments nearby.\n\n\"Just keep your asses out of trouble, understand?\" Adam asked as Sarah slowly placed her arm around his lower back in order to hold him with affection.\n\n\"Yes sir!\" Dalton said sarcastically as the idea of playing hide and go seek with full bottles of whiskey danced in his head.\n\nHaving caught one of the Colonial Star ships in a narrow escape was the biggest factor in saving Roman's life. As impressive as it all was, it paled in comparison to the sight of things when Dalton and Steiner left the Colonial hospital and made their way onto the sun scorched streets of Glimmeria's largest city. Kamira looked almost as if it were one huge military base now, the outskirts of the city protected by huge underground Mack guns, which were nearly two hundred yards in diameter and could fire bursts of lead from the surface into space, essentially punching holes through the largest of ships.\n\nThey were protected by Razor turrets, nothing more than a clever term for huge steel towers, each manned by two soldiers and fired a twenty inch rail gun in the direction of anything that approached the city unwelcome on foot.\n\nThe new Goliath Model Two soldiers patrolled the city streets, having replaced the original units with much thicker armor plating as well as the addition of carrying surface to air missiles. Last but not least, the Colonial Marines. They weren't as feared as the larger Goliath units or weaponry, but in force, they were still the backbone of the Colonial war effort.\n\nThis was literally the last place that the Colonial forces had to go, and they had thrown all of their eggs into a single basket in order to assure the safety of Glimmeria's citizens. With everyone in Adam's crew having officially joined the Colonial ranks, they all had sworn the allegiance and wore the uniform. Of course, that didn't stop Dalton from covering his with the brown duster that he had become so used to wearing. As much as the rest of the crew dreaded the mere sight of that raggedy ass leather duster that had followed his back so long that it was starting to fall apart at the seams, it was his security blanket, especially when preparing to hammer back as many drinks as he possibly could.\n\n\"Where's the closest pub?\" Steiner asked as they stood in the street, the torturing rays of the blistering sun reddening them every single moment of it.\n\nTwo Swordfish fighter jets quickly blew by overhead, drowning out Dalton's response. They were the design of the Glimmarian military, the extra long nose of the ship as well as the unique ability to hug the ground within a few feet during flight making them virtually undetectable on radar.\n\n\"I dunno, but when we find one, if we see two Swordfish parked out front, somebody's taking an ass whipping.\" Dalton said loudly, upset over the intrusion as Steiner mockingly laughed.\n\n\"I think we can all agree that the ones fighting for control of the Skyla System are not the ones who should be in power.\" Anwick said as he sat at a large wooden table that was polished to a gleaming shine.\n\nHis teeth filed to a razor sharp point, the dead white pupils of his eyes fixated themselves on his business partner. Two of his toughest escort Hunters sat with him, stark white hair also flowing from their scalps as they remained stone faced and heavily armed. Across the table, three well dressed members of the Benzan Mafia sat, finely pressed suits and ties accented by sunglasses of solid silver.\n\n\"The weak do not concern us. Our only wish is to be left alone, so that we may continue to live in true freedom.\" the man sitting in the middle of the Mafia members replied, his lenses reflecting back nothing more than the look of the Hunters to Anwick.\n\n\"True freedom always comes with a price. Always,\" Anwick replied calmly. \"It's time to weed out the weak and useless from our midst, and that begins with Adam Michaels,\" Anwick added, throwing down several sheets of paper that included Adam's photo. \"Prove your capability to me by getting rid of him and in return, I will ensure that the Legion leaves your kind well enough alone once they have crushed the Colonials.\" Anwick said, eagerly awaiting a response from Cyrus.\n\nYelling a handful of words loudly in Benzan, a large man entered the room, very well dressed in a gray pinstripe suit with a face that was without emotion, as if it were cut from marble slate.\n\n\"I will leave the choice to Draco. Should he choose to enter into an arrangement with you, it will remain between the two of you. I have no desire to kill by way of contract, nor do I wish to be directly involved. That said, should he choose to decline, then we are done here.\" Cyrus replied.\n\n\"I'm sure if the money is right, Draco won't mind doing the dirty work. Am I correct Benzan?\" Anwick asked, throwing a sackful of Legion credits onto the table, easily ten thousand or more. Walking over to pick up one of the photos of Adam, Draco smiled widely.\n\n\"Consider him a dead man. Officially. We fought together in the first Glimmerian war. I know how he thinks, how he moves.\" Draco replied in a deep tone of voice, leaving little doubt of his ability to end lives.\n\n\"Very well then. I look forward to hearing from you my brother,\" Anwick said, standing to his feet while smiling wide, the low light of the room reflecting from the large razor's edge of his teeth.\n\n\"Now if you will excuse me, I must meet with the Legion regarding our next major assault.\" Anwick added as Cyrus slowly stood and extended his arm as the two men in charge sealed the deal with a firm handshake, which was a legally binding contract in the world of terrorism.\n\n\"Anyone ever tell you that you are magnificent with children?\" Lassiter asked as Kelly answered his question with only a glowing smile. They had spent the last several hours helping Troy get what few belongings he had together, which amounted to nothing more than the few clothes he had gotten since arriving and a combat blade given to him by the Gali warrior.\n\n\"Kelly. Will I ever have a chance to say goodbye to Roman?\" the young boy asked, his question taking her by complete surprise.\n\n\"No need for a goodbye. As soon as he is done recovering, I'm sure he will visit. The family you have been placed with is right here in Kamira, so you will be seeing plenty of all of us.\" she replied, bringing a brilliant smile to Troy's face.\n\n\"You need a hand getting Troy to his new home?\" Lassiter asked, a bit nervous, but trying to hide it behind the mask of calmness that he wore to disguise his feelings for her.\n\n\"I can handle it,\" Kelly replied, pausing for a moment. \"When I get back though, maybe you'd be interesting in going into the city together? I haven't had a decent sit down meal in a long time.\" she asked, her shoulder length blonde hair keeping his undivided attention.\n\n\"I would love to, as long as it doesn't include Dalton and an open bar.\" he replied, laughing a bit to calm the mood in the room, their unexplored love having grown every single moment since arriving on Glimmeria.\n\n\"Alright, I'll see you soon then.\" Kelly replied smiling happily as Troy gave Lassiter a hug before grabbing his bag and heading for the front door of the small apartment.\n\n\"Not soon enough.\" Lassiter said to Kelly with a smile before turning to Troy, bending down so they could speak at eye level.\n\n\"Don't worry, I'll personally make sure Roman comes to visit soon, alright?\" Lassiter said as Troy grinned ear to ear, shaking his head in approval.\n\nIt was a huge bar, one of the largest Dalton had found himself at in many years. He had long considered himself not only an avid connoisseur of the full spectrum of alcoholic drinks, but a critic of drinking establishments both large and small. Sitting on a wooden stool beside Steiner, he was amazed at the number of tables, nearly a hundred of them, full of people playing cards, drinking and exchanging exaggerated truths.\n\n\"I'll say one thing, these umbrellas are damn classy.\" Dalton stated, referring to the wooden umbrella in his latest mixed drink.\n\n\"Agreed, but some of us are more about quantity and less about quality.\" Steiner replied loudly, trying to talk over the background noise as he held an unmarked brew in each hand.\n\n\"I'm about both, I just call a spade a spade. And these umbrellas are all class, right down to the neon pink.\" Dalton replied, turning up his drink and downing in one lengthy swig what would have taken most men an hour to nurse into their bloodstreams.\n\nMaybe it was the alcohol casting illusions in front of his eyes as it had done so many times before, but Dalton had convinced himself that a brunette sitting at a nearby table, the kind that became more attractive as the drinks became more available, had been throwing stares in his direction. Never mind the fact that she sat at a table with another female and two men wearing the Colonial uniform, or that it wasn't true love. In Dalton's mind a casual glance from the end of an alcoholic beverage was true enough.\n\n\"Sit tight buddy, and watch the master work.\" Dalton told Steiner as he slowly got up from his stool, patting his much larger Husk friend on the back for a brief moment before attempting to walk into the woman's direction.\n\nHis lack of speed was a direct effect of the workload he had recently placed into his bloodstream, still he managed to stagger with swagger as he eventually made it to his destination. Preparing for the show, Steiner turned to watch the events unfold, a frothy glass of house brew in his his oversize hand. Quickly, he realized how he and Dalton viewed the world through different lenses. How the young lady looked very happy with the group she sat with, not to mention the fact that she was far from being even decently attractive.\n\n\"Excuse me miss. Care to join me for some fine wine and casual sex?\" Dalton asked blatantly, causing an immediate look of disgust to fall across the woman's face as he reeked of anything but fine. \"What? You don't like wine?\" Dalton added, confused at being shot down on one of his best pickup lines.\n\nIt had worked so many times before, granted most of them were not that easy on the eyes, in fact they were slumming it at best according to his standards. Still, at his age, getting turned down using his best line made him begin to question if he still had charm.\n\nBoth Colonial soldiers stood to their feet abruptly, one of them calling Dalton a string of words that would scar a child's ears for life, prompting Steiner to stand to his feet as well, the over abundance of muscular tone that was all too common with the Husk race was ever intimidating.\n\n\"Sit down big man, I got this,\" Dalton said loudly, turning to Steiner as he motioned him to sit and let things unfold without his intervention. \"While you're at it, order me another drink. One with the pretty umbrellas.\" he added.\n\nAs Dalton began to turn back, the Colonial soldier who had cursed him only moments ago, struck him across the face with a hooking punch, knocking him onto a nearby table and slinging several drinks onto the floor beneath it.\n\n\"You son of a bi..\" Dalton began to reply, before the second Colonial soldier kicked him in the ribs solidly, knocking the air from the drunken warrior's lungs.\n\nDalton tried to yell for help from Steiner, extending his hand in a begging fashion as he fought to catch his breath. Steiner remained calm, drinking slowly from his glass as he watched the master at work. Scooped up and thrown several feet, Dalton landed on one of the nearby tables, clearing everything from the top as it followed him to the floor, crashing all around him like a glass filled grenade.\n\nSlowly standing to his feet, he finally responded with a thrashing punch of his own to the face of one of the attackers, followed by a punishing kick to the stomach of the other.\n\nAt least that was his intention, but with things blurred extensively because of the night's binge drinking and beating, he wasn't convinced he had even hit the right two men. Standing there for a moment, trying to get his bearings straight, he randomly sucker punched one of the customers who had been sitting at the now flattened table.\n\nFirst on the scene, two Glimmerian warrant officers raced through the door of the large bar, quickly flattened by a sweeping elbow from Steiner, who had watched his friend endure enough. Dalton staggered behind the bar, looking deviously at the coward bartender for a moment as he began pouring a large glass of whiskey, the tall glass nearly emptying the entire bottle before it topped off.\n\n\"And I want a fuckin' umbrella.\" Dalton slurred loudly, pulling one of the neon umbrellas from another patron's drink and placing it in his glass of whiskey.\n\nMoments later, he and Steiner were standing at the end of Colonial rifles held by warrant officers of their own military.\n\n\"Well, I guess we're cut off.\" Dalton remarked softly as both men were escorted out in hand restraints to be processed at a local holding facility.\n\nAdam and Sarah sat in illustrious surroundings as they ate one of the finest dinners he had ever seen, the type of establishment that was usually reserved for those in high political position and the absolute wealthy. Tonight was different however, tonight was the turning point in his life, the defining moment of his soul. With nearly eighty people eating in the large ballroom of red velvet and stained wood finish, Adam stood to his feet for a moment, drawing Sarah's immediate attention as he softly placed his thick red linen napkin onto the table before taking a place beside her on one of his knees.\n\n\"Sarah,\" Adam said with truth and integrity of the heart. Her eyes exploded with emotion, filled with both tears and emotional connection as she stared at Adam, hanging on his every word. \"The moment I met you, I became a different person. What you've taught me is that I have always been this person deep down, I have lived my life, destined to get to this very moment. You bring out all of the best in me, without you I would be a broken man, both heart and soul. I am asking you from the bottom of this thing inside my chest that beats uncontrollably when you are near if you would do me the honor of being my wife?\" Adam asked as the entire ballroom full of citizens looked on.\n\n\"Yes, of course!\" Sarah replied without hesitation.\n\nEveryone started clapping quietly as Adam rose to his feet slowly and embraced Sarah into his arms, kissing her as deeply as the concentration of stars that filled the night sky. After several minutes of affection, the waiter slowly walked over to give Adam the news of his friends checking into lockup. Not the first time Adam had bailed Dalton out of jail, but to have to put such a magical evening on hold to do it should be a felony within itself.\n\n\"It's alright Adam, let's go get our boys out of lockup and spend the rest of the night planning the first day of the rest of our lives together.\" Sarah said softly.\n\nHe nodded, although he planned on giving both Dalton and Steiner a damn good ass chewing when and if he could actually talk the Colonials into releasing them. As they began leaving the confines of the lavish restaurant, everyone once again clapped softly to let them know they had been touched to be a part of such a magical moment.\n\n\"What the hell is your problem?\" Dalton asked Steiner as both men stood in a small holding cell, Dalton pacing a path in front of the door while Steiner stood in a rear corner.\n\n\"Just watching the master at work.\" Steiner replied with heavy sarcasm. Before Dalton could answer with a small piece of his self proclaimed infinite wisdom, they both recognized footsteps approaching.\n\n\"You have a visitor.\" a well dressed and lightly armed Colonial soldier said to Dalton.\n\n\"About damn time, hell Adam's usually here to pick me up long before now.\" Dalton said with relief as Steiner slowly approached the front of the cell. Rather than seeing his longtime friend walking up to the cell, the young woman from the bar cautiously approached.\n\n\"The master son. Don't forget it.\" Dalton said softly as if to pour salt into the wounded pride of Steiner before pausing to grab his brown coat and fluff it a bit.\n\nSeveral hours had passed, which were filled with lies, bedroom eyes and the aftermath of barroom lust between Dalton and the young woman. Never asking her name, it was his belief that a woman's name just complicated things unnecessarily, especially through the steel bars of lockup.\n\nIt was a bittersweet feeling for Kelly. Troy had been lucky enough to find a stable home on Glimmeria with a family that would love and care for him deeply. Still, like the rest of the Gunship crew, she had grown to love and care for him deeply as well. He would be safer here however, there was a raging war that would soon enough reignite only a few miles away.\n\nShe was quickly reminded of that as she stepped outside of the housing unit onto the crowded urban street as a full squadron of Swordfish jets flew overhead. She couldn't wait to catch a transport and get back to the arms of Lassiter, having come to feel a comfort and peace around him that she had never known, a feeling that only true love can bring.\n\nShe never made it back. Standing respectfully as only a Colonial soldier could, she felt a sharp piercing band of lead cut through her chest. Her first reaction was to draw her sidearm, maybe the Legion invasion had began? As she slowly turned, her sidearm shaking roughly while her body fought to remain alive, a second shot hit her chest only inches below the first, the muzzle flash of Draco's sniper rifle alerting her to his position in a second story window. Moments later she died, falling soundly onto the streets below.\n\nDraco tossed a large golden coin out of the window, landing near her lifeless body. A Glimmerian coin that had been out of mint since the day they had lost the first war to the Legion, his calling card to Adam Michaels.\n\nSeveral hours later Adam and Sarah were the first to join the Colonial investigators at the scene of the shooting. Visibly shaken by the body of someone who he had looked at as a little sister, someone he felt compelled to look after, Adam trembled uncontrollably as Sarah held his hand and tried to convince him that there was nothing he could have done to help her. It wasn't his fault. Or was it?\n\nAdam spotted the golden coin laying near her body, immediately recognizing it and deciding there was a huge problem. The coins were only issued to a limited number of Glimmerians nearly a decade ago, a special group, the group he fought beside. His unit. He quickly realized that Kelly had died because of him, as would everyone else around him unless he could somehow find a way to protect them, like he should have been there to protect her.\n\n\"Kelly,\" Lassiter screamed as he sprinted across the street and fell on his knees trying to revive the lifeless body of his lover. \"Get off of me!\" he added as Adam grabbed him, both arms clinched around Lassiter as he drug him away from her body unwillingly.\n\n\"She's gone.\" Adam said softly as Lassiter cried uncontrollably.\n\nThe Colonial jailer escorted Adam down the corridor, the striking of their boots against the stone tiled floor making their presence immediately known.\n\n\"About fucking time. You forget about us or something?\" Dalton asked arrogantly as the cell door opened.\n\nWithout a word, Adam simply turned and grabbed Dalton firmly by the front of his brown coat, picking him up a few inches from the ground and holding him against the cold brick wall. Exchanging nothing more than deep stares, it quickly became obvious to Dalton that his friend was a different person than he was only hours before.\n\n\"Get your fucking hands off of me!\" Dalton yelled as Adam dropped him and loosened his grip. Steiner quickly intervened, easily holding Dalton at bay as Michaels looked onto both of them for several moments with the eyes of a man hellbent on revenge.\n\nA few painstaking days had passed before Adam found himself standing under the large canopy surrounded by friends as a Colonial priest spoke words before committing Kelly's body to the ground. The only sounds that kept the entire funeral from seeming like a dream were the shots from Colonial rifles as the twelve soldiers fired them into the air in unison. Adam watched Lassiter deal with the loss of someone he couldn't manage to live without, he felt for the young man as he tried to imagine losing Sarah Blaine the same way. The thought of it was unbearable.\n\nAfter the service and departure of a majority of those in attendance, Adam gathered his crew.\n\n\"We have a problem. The person who shot Kelly, left this.\" Michaels said, holding the gold coin up as the sunlight shimmered across it.\n\n\"Bullshit, ain't no way Adam. Those coins were specific to our platoon back in the first war. I should know, I dropped hundreds of them at every bar on this fuckin' rock,\" Dalton replied.\n\nAdam answered only with a dedicated stare. \"I'd say we do got a problem then, a big fucking problem.\" Dalton added, feeling Adam's sense of urgency.\n\n\"Whoever did this is coming for Dalton or myself, maybe both of us. The marksmanship and weapon they used to kill Kelly lets me know that none of you are safe. From this moment forward, nobody goes anywhere alone. Nobody.\" Adam added to the conversation.\n\n\"Steiner. You and Lassiter will stay with Dalton at all times. Pack plenty of firepower and stay indoors as much as possible. If they used a sniper rifle the first time, chances are they plan on striking again from a distance,\" Michaels said. \"Roman, you will be with Sarah and I. Lassiter, I realize this is a very tough time, but I have to ask you to stay focused. Until we catch the person who did this, none of us are safe.\" Adam added.\n\nLassiter agreed with a nod, still visibly shaken.\n\n\"I know what Kelly meant to you, she was like a sister to me. I will go through the rest of my life blaming myself for not being able to protect her from this, but you have to shake it off for now. Put it on a shelf, the only way we can help her now is by catching the son of a bitch who did this. Understand?\" Adam said.\n\n\"Yea, I understand.\" Lassiter responded after a long pause.\n\nAs the group split up and walked toward two different Colonial transports, Adam turned to face the others.\n\n\"Dalton, keep your ass out of trouble.\" Dalton stopped walking for a moment.\n\n\"Who me? Sure thing, scout's honor.\" he replied with a smile before once again joining his group.\n\n\"Dalton used to be a scout?\" Sarah asked surprised.\n\n\"No, he didn't. That's what worries me.\" Adam replied, drawing a bit of laughter from Roman.\n\n\"Based on the information our spies have given us, the Colonial base is heavily defended. Our forces will take heavy losses on their way to the surface.\" a Legion advisor said as Flag Officer Andrews stood for a few silent moments watching his forces staging on a wall mounted grid, his crimson red officer's uniform lined with thick black to match his shoes.\n\n\"We will send the invasion in two waves. Send the mercenaries and newly trained soldiers down in the first wave, let them take the blunt of it. Send the officers, battle tested soldiers and heavy equipment down with the second wave. Have them set up a forward base of operations so we can begin to funnel supplies and reinforcements to the front lines.\" Andrews replied.\n\n\"Yes sir. And what of the Hunters?\" the advisor asked.\n\n\"Have them wait in orbit with Legion High Command. If the Colonials attempt an early retreat, the Hunters should be able to finish them off easily. If the Colonials dig in, we'll have the Hunters drop down to the surface to reinforce our lines.\" Flag Officer Andrews replied.\n\n\"Yes sir!\" the advisor responded, immediately ordering his communications officer to relay the message throughout the fleet.\n\nAndrews stood there for several minutes, admiring the view from the other side of the thick plated glass of the ship's bridge. Dozens of Legion capital ships and cruisers along with three Hunter carriers, all working together as squadrons of Legion fighter jets flew past, running drills in preparation of the upcoming battle.\n\n\"Sir. We've picked up a large fleet broadcasting Legion code in outer orbit of the planet!\" one of the grid operators yelled as Commander Edwards walked swiftly toward his station.\n\n\"Set our condition to one, recall all Colonial soldiers and make them fully aware that this is not a drill,\" Edwards replied. \"They are staging from the looks of it, we still have some time. Order all Mack stations fully loaded and Swordfish and Spartans fueled and standing by. Bring every Goliath unit that we have online.\" he added.\n\n\"Yes sir!\" the grid operator announced loudly, punching the orders into a computer that would mirror them to every outpost on Glimmeria.\n\n\"Better not be a fucking drill because this is the first time I've ever left a pitcher of beer sitting,\" Dalton said angrily as Steiner and Lassiter both had their thoughts on the impending battle rather than Dalton's lust for man drink. \"And another thing, they need to find a way around this sweltering fucking heat, I'm already sick of it.\" Dalton added sharply as the group quickly left the Colonial bar along with everyone else, the exterior neon sign easily overpowered by the strength of sunlight as they began their walk back to the Command Center.\n\nHaving almost made it back, the men continued their slow march as they absorbed Dalton's constant barrage of bitching and moaning. Steiner had intended to let him know that his ranting was beginning to resemble that of a married woman, when the right thigh of the massive Husk warrior was struck with a shotgun blast, immediately putting him down to the ground yelling in agony.\n\nDraco walked from an alley nearby, firing the second shotgun blast of the double barrel in the direction of Lassiter, barely grazing his back with the hot lead spray as he dropped the emptied weapon and calmly pulled two pistols. Lassiter found refuge behind a parked transport, drawing his combat sidearm and looking several feet down the road, watching a suddenly sober Dalton pull Steiner to the rear of another one of the transports which lined the sides of the city's streets.\n\n\"Adam! Where the fuck are you! We got the shooter a couple of blocks from HQ, we're pinned down and taking fire!\" Dalton yelled into his com, throwing it to the ground as he stood to fire a blast from his own short barrel shotgun before dropping back down quickly behind cover.\n\nDalton's blast was returned with three ringing pistol shots, shattering the window above the head of the roughly bearded man smothered in his favorite brown coat as he sat with his back firmly against the thin steel of the makeshift barricade. As Draco approached the transport, he slowly rounded the front end, Dalton and Steiner easily in his sights. Before he could finish the task, however, his ribs felt the penetrating plunge of a shot from Lassiter's combat pistol.\n\nThe Colonial sniper had recovered enough to go on the offensive just in time to save the life of his friends.\n\nDalton aimed his shotgun in the direction of the assassin, but it was quickly brushed to the side as the shot made its way deep into the transport, the steel pellets scattering in a circular design as they embedded themselves into the thin steel and aluminum. Draco used his second arm to throw a hooking punch into the face of Dalton who crumpled at the bottom of the transport and laid motionless as Lassiter had began closing in on the Benzan contract killer.\n\nSteiner, still reeling from his flesh wound, realized that if the assassin was to be caught or killed, he was the only one capable of doing so.\n\nDalton was incapacitated, while such an experienced killer would have made short work of a Colonial sniper. Grabbing Draco in a tightly clinched hold, the Husk used all of his strength to tighten the grip even further trying to snap the bones of the assassin. Draco finally was able to loosen himself just enough to swing his right elbow around and bury it into the face of Steiner, who was stunned but still on his feet. A hard right kick from the assassin changed that, sending Steiner to his back with a thud as Draco spun quickly, doing a back fist with his clinched knuckles wrapped in black leather glove, knocked the pistol from Lassiter's hand.\n\nForcefully pushing Lasstier to the ground, the much more qualified Draco jumped onto him, his arms pursuing the neck of the Colonial sniper in an attempt to snap it and be done with the pesky commoner. It was his own neck, however, that felt a sudden stiffness as Dalton wrapped both of his arms around it, burying his elbows deep into the shoulders of Draco. After several moments of struggle between the two, Draco clinched the back of Dalton's brown jacket and using most of the strength he had left, launched Dalton over his shoulder through the air as the wiley soldier of fortune crashed into one of the nearby parked vehicles, knocking him virtually unconscious.\n\nAs with any assassin, Draco had grown tired of the hands on fighting and picked up Lassier's pistol, which was the closest weapon to him, laying just a few feet away.\n\nHe hesitated for a moment, unsure of who to kill first, deciding finally to send Dalton to the afterlife as he aimed down the crisp iron sights at the man he had once fought beside. Meat tore from his shoulder first, followed by skin and muscle from his ribs as\n\nAdam had arrived just in time to fire two shots from his pistol, both hurting the assassin a great deal. Draco quickly made it to his feet just in time to be blinded by several reeling punches and a solid elbow, all thrown into the Benzan's face by Roman Raines.\n\nDraco tried to fight back, throwing several punches with deep intention that would have seriously wounded a normal man, but Roman was no normal man, the farthest thing from it would be the best assessment.\n\nBlocking the punches with the thick aluminum on the backside of one of his partially mechanical arms, the other was thrown to the body of Draco, the sound of ribs breaking under pressure echoed loudly as Roman threw Draco face first into one of the transports, putting him out of commission.\n\n\"Enough,\" Adam said, doing everything he could to pull Roman off of the Benzan assassin. \"We need to find out why.\" he added, Roman stepping away as Sarah continued to tend to Lassiter, Dalton and Steiner.\n\nFlag Officer Andrews watched from a tall glass balcony as thousands of Legion soldiers had gathered in formation under him, awaiting his final words before boarding their drop ships which would be descending in only minutes to the surface of Glimmeria to begin the largest battle ever recorded. The ships were solidly constructed with steel plating thick enough to deflect the paltry shots of small arms fire, however they could hardly withstand the pounding of the surface based Mack cannons.\n\nEvery Legion soldier knew this, understood the significant chance that once they entered the drop ships, they could be entering nothing more than a death sentence. Still they stood loyal as their Flag Officer placed his hands tightly around the bannister and prepared to speak.\n\n\"It is my belief that only the strong should lead. Survive. Take a moment to look around, look into the eyes of those which you fight beside on this very day. You are the strong, and because of this you will survive. The Colonials have tried to separate themselves from the rest of the Skyla System, and in doing so have made a mockery of our very way of life. Today, it is our turn. The Colonials will either submit to our way of life, or they will perish to the will of the strong. May the gods be with all of you!\" Flag Officer Andrews said boldly as every soldier began to cheer before chanting the Legion name.\n\nShortly after, each soldier began boarding their ships with great reserve as Andrews returned to the bridge which would serve as the control center of the Legion High Command for the upcoming battle.\n\nDraco sat on a wooden stool, his hands bound with thick rope and his face bloodied like a scene from a horror film. He knew he was still alive, yet it all seemed surreal, almost as if he were placed snugly into a dream. Steel plated walls surrounding him, the serene silence of emptiness penetrated the air throughout the room as he sat there with no idea of what was to come next. Thick steel began to grind as the door slowly opened, Adam Michaels entering the room and pulling up a wooden chair to sit within arm's reach of the man who had taken the life of Kelly and nearly several others.\n\n\"Funny. Nearly ten years ago, you, I, Dalton and a couple dozen other lucky souls made it off of this rock in one piece. We would have given our lives for each other back then, and when we left we were on good terms Draco, so my only question is why?\" Adam asked. Draco continued to stare at the solid steel door without so much as a blink, no response and no acknowledgement of Adam's presence, well versed in the tactics of interrogation due to his extensive Benzan training.\n\n\"Draco, I am the carrot. I'm in here to try and get the answers I need and maybe let you live to see another day. The rest of my crew is waiting right outside that door, including the man who was deeply in love with the young girl you killed. They are the stick, and trust me, none of them have a problem with cutting the answers out of you and leaving you here to die. So again I ask y..\" Adam said, cut off by Draco.\n\n\"I did it to save my family.\"\n\n\"Who?\" Adam asked in a demanding tone.\n\n\"You're in way over your head here Adam. If you plan to kill me, do it and get it over with. If I say anything, my family is dead, and I promise you I'm not going to let that happen.\" Draco replied.\n\n\"About three years ago, I was in a bad spot Adam. I needed money...a lot of money. I met a few people and before I knew it I was in with the Benzan Mafia. Yea, I know what you must think of me. I kill people, usually in cold blood and I do it for a paycheck.\" Draco said, turning to look into the eyes of his former brother in arms.\n\n\"Adam, you have to believe me when I tell you I didn't have a choice. Usually the people I kill, hell they all deserve it. I do society a favor by erasing some of the worst people you can imagine. This was different. Yea, it paid a lot of money and I know she was innocent Adam, but my family? What was I supposed to do,\" Draco added as Adam turned away. \"WHAT WAS I SUPPOSED TO DO?\" Draco shouted.\n\n\"So the Benzan Mafia is responsible for the death of one of my crew?\" Adam asked bluntly.\n\n\"No. In fact, they wanted no part of it. They just allowed me the meeting, that's it.\" Draco replied.\n\n\"I need you to arrange a meeting, put me in the same room as the head of the Benzan Mafia.\" Adam said boldly.\n\n\"You know I can't do that! Adam, I'm low level with the family, I can't just call 'em up and start demanding that kind of shit!\" Draco replied.\n\n\"I'll do it,\" Michaels said bluntly. \"Who do I need to contact and how do I go about it?\" Adam asked as Draco directed his attention on the other side of the room, reluctant to answer the question. \"WHO GODDAMNIT!\" Adam yelled, pulling his sidearm and pressing it onto the forehead of Draco.\n\n\"This is Commander James Edwards, the superior ranking officer of Colonial Command. Earlier today, our grid systems picked up a large Legion presence entering the orbit of this planet, leading us to only one conclusion, war is imminent. We have solid defenses set up on the planet's surface, the most capable soldiers in the Skyla System, but above all else we have reason. While the Legion looks to impose its will on a free people, we fight for the reason of freedom,\" he said, pausing briefly as everyone near a com unit on Glimmeria's surface continued to stop whatever they were doing to listen to the breaking news.\n\n\"A society in which innocent people may live in peace rather than be slaughtered because of a political agenda. I ask that you all fight with reason, vigilance and the memory of those fallen Colonials who have allowed our freedom up until now. Good luck and God speed.\" Commander Edwards announced throughout every Colonial radio and com unit on Glimmeria.\n\nTurmoil set in as Glimmeria's citizens faced the facts, a war with the Legion was at their doorstep, one that would forever change the face of their planet and those who inhabited it. Most of them were quick to evacuate indoors, trying their best to safeguard their families.\n\nGroups of the mechanical masterwork Goliaths accompanied squads of Colonial soldiers who were armed to the teeth, requesting the citizens stay inside of their homes as Colonial Spartan fighters and Glimmerian Swordfish screamed overhead, their thrusters at full burn and ready for the upcoming fight. The Mack stations were already online, however there became a sense of urgency with the staff as they hustled to run full diagnostics and double check the ammunition storage for the surface to space cannons. They were the primary line of defense for the Colonials and the backbone of their strategic war effort.\n\nMeanwhile, Commander Edwards strengthened the security forces inside of the Colonial Command Center. The large, atrium style room, which was made of plated steel and shatter proof glass, was designed to survive a nuclear strike. Therefore, all of the high ranking officials collapsed into the room with equipment enough to run the entire Colonial war effort behind the closed doors of what everyone had began to refer to as the \"vault\".\n\n\"Where are Lieutenant Michaels and Roman Raines?\" Commander Edwards demanded to know as Dalton, Steiner and Lassiter stood front and center.\n\n\"I'm sorry sir, who?\" Dalton reluctantly answered.\n\n\"Do not play with me at a time like this Dalton!\" Edwards shouted, smelling the hint of soured lager on his breath.\n\n\"Sir. They left to meet with the Benzan Mafia in reference to Kelly's kille...\" Lassiter finally admitted, interrupted by the Commander.\n\n\"THEY DID WHAT? AT A TIME LIKE THIS!\" Commander Edwards added in hysterical disbelief. Steiner and Lassiter both sunk their heads toward the floor as Dalton began to recite his soldier identification number as if being interrogated by the enemy.\n\n\"ENOUGH!\" the Commander shouted, bringing his face within inches of Dalton's own, the sight of his stubbled brown beard furthering the blurred vision of a furious commander.\n\n\"I should have your ass thrown in the brig for such blatant insubordination,\" Commander Edwards yelled as Dalton stared back planning his exit strategy, just in case. \"However, with Lieutenant Michaels off doing business that is completely unauthorized, you are next in line in the chain of command Sergeant. I have no choice other than to put you in charge of the security detail of this building.\" Commander Edwards said.\n\n\"I'm sorry?\" Dalton asked.\n\n\"I have to put you in charge. You can't follow orders worth a damn, but if the Legion makes it this far, I have no doubt in my mind of your ability to give them hell to protect the lives of everyone here.\" the Commander answered.\n\n\"In charge? You mean like, as in giving orders of my own?\" Dalton asked.\n\n\"Yes, in charge. Unless you are too drunk to do so?\" Edwards asked.\n\nDalton stood motionless for a moment, cursing his chewing gum for failing to mask the stench of the morning's alcoholic adventure.\n\n\"No sir, I'm ready and capable.\" Dalton finally replied, stretching his eyes for a moment in an attempt to pull himself from an intoxicated daze.\n\n\"Good. You three hit the ammunition reserve down the street, grab plenty of small arms weaponry as well as the nearest Colonial squad, and then double-time your asses back. Set the squad up strategically and you three collapse inside and protect the vault as a last defensive resort. Sarah Blaine will remain inside of the vault with us, it will be a huge boost to everyone's moral hearing her voice through the com system,\" Commander Edwards ordered, turning to enter and seal the vault.\n\n\"Oh, and if you see Lieutenant Michaels before I do, send him directly to me. Understood?\" Edwards asked.\n\n\"Aye Captain.\" Dalton answered.\n\n\"It's COMMANDER!\" Edwards shouted. \"Huh?\" Dalton confusingly asked.\n\n\"JUST GO!\" Commander Edwards yelled before turning to seal the vault.\n\n\"I wonder what's crawled up his ass?\" Dalton mumbled as the three men turned to walk down the narrow hallway and onto the street.\n\n\"Now what?\" Adam asked as he stood with Roman and Draco, who's hands were tied snugly with thick rope.\n\n\"Now we wait.\" Draco replied.\n\n\"Just to be clear, even though he's given me his word that we will not be harmed, if he does try anything, your ass will be the first one dusted.\" Michaels said of Draco.\n\n\"Relax. The Benzans are always good to their word, especially Cyrus. Of all of the Benzans I have met, he considers a man's word to be as important as the soul in his chest.\" Draco replied as they remained standing beside three of the mangiest horses Roman had ever seen.\n\n\"Looks like you guys have a friend coming.\" Draco said sarcastically.\n\n\"Fuck.\" Michaels replied, catching a glimpse of a Glimmerian lawman approaching them on horseback.\n\n\"Good afternoon.\" Adam said charmingly, unsuccessfully trying to change the scowl on the Sheriff's face as the dust settled around the hooves of the solid brown steed.\n\n\"Adam Michaels, Sheriff Barker. I have a warrant to bring you in to the Colonials.\" Adam hung his head for a moment before looking back at the lawman, who was still mounted on his horse, a large rifle laying across his inner elbows.\n\n\"Can't do it, at least not at the moment. I'm out here on business, when I'm done you can take me wherever you like.\" Adam replied. Sheriff Barker pulled the rifle up, aiming directly at Adam.\n\n\"I'm not in the business of making suggestions, a warrant hits my desk and I go out and fetch 'em. That's how it works.\" Barker said, his rifle's aim still true to Adam's upper body. Putting his hands in front of him to try and keep the peace, Adam replied. \"May want to put the gun away, my friend doesn't take kindly to having them pointed in his direction.\" Barker adjusted the gun barrel slightly to aim it at Roman.\n\n\"Can your friend dodge bullets?\" the Sheriff asked.\n\n\"Well actually.\" Michaels replied, knowing damn good and well that if Roman was pissed off, Barker would have been dead already.\n\nThe Sheriff started to reply, but was quickly drown out by the thrusters of a ship breaking orbit, falling quickly from the sky and hastily approaching to land near them. They all watched as the elongated blade design of the ship's front gave away the fact that it was Benzan, huge clouds of dust rising from the ground as the ship rocked back and forth slightly, finally touching ground.\n\n\"Sheriff, you need to leave now. Please.\" Michaels pleaded, not knowing what the reaction of the Benzans would be.\n\nAs the shuttle door began to open, the Sheriff glanced in its direction for a split second, which was all the time Roman needed. Pulling his combat blade and throwing it end over end with blazing speed and unparalleled accuracy, the knife digging into the flesh of Sheriff Barker's shooting hand and throwing him from the horse.\n\n\"Just stay down until the ship leaves, otherwise they may kill you.\" Roman said to the Sheriff as he walked over to him, grabbing his rifle and tossing it far enough away from him to be irrelevant.\n\nSheriff Barker remained on the ground, doing his best to stop the bleeding as he held the wound moments after Roman recovered his blade with a tug. Obviously enduring a tremendous amount of pain, the Sheriff did his best to remain silent as two Benzans exited the shuttle.\n\nThe first was a tall, slender man with dirty blonde hair braided down his back. He wore a pair of reflective silver sunglasses, a white long sleeve shirt with buttons and carried a solid black pump shotgun. He was everything the men had pictured a member of such a notorious crime family looking like. The second Benzan out, not so much.\n\nBrunette hair with a slight but lustrous curl, she moved with perfect symmetry, deep blue eyes, a white v-neck shirt and gray cargo pants, she had a small machine gun pistol hanging from a black leather strap that draped over her shoulder. Unlike anything Adam or Roman expected, she was simply too damn beautiful to be in this line of work. Not society's definition of beautiful, which amounted to makeup, empty talk and a glamor filled wardrobe. She was a pure beautiful. The kind of woman who could make men do almost anything with a simple request. Of course, if that didn't work, she still had the machine gun pistol as leverage.\n\n\"The agreement was three passengers, not four.\" the tall Benzan man said as the attractive female kept a finger on the trigger of the rapid fire pistol, its barrel pointed to the ground.\n\n\"Relax Oz, the one laying down is a Sheriff who decided to show his damn face only minutes ago.\" Draco replied. With a slight nod of Oz, the female began to walk toward the lawman with intentions of ending his life.\n\n\"No! Please, he is of no harm. Simply came to serve a warrant on me is all.\" Adam said, stepping in front of the Sheriff in an attempt to plead for the man's life. It would be the very first time Adam and the young lady of such immaculate beauty locked stares.\n\n\"It's alright Sasha, leave him be.\" Oz said, standing by the ramp of the ship as he waited for everyone but Barker to board.\n\n\"Weapons.\" Sasha said as she began collecting everyone's killing devices on their way up the steel ramp which was outlined with glowing red bulbs. Taking it upon herself, she quickly reached out and grabbed the combat blade of Roman, attempting to pull it from its casing which was strapped around his leg. Meeting her grasp with one of his own, he gripped her wrist strongly as a warning.\n\n\"Careful.\" Sasha warned, breaking his hold on her and removing the blade as Adam nodded slightly to keep Roman calm.\n\nHe was calm enough, in fact he was impressed. A woman who knew her weaponry, could hold her own with words and had a undeniable beauty throughout her entire body. Not the textbook definition, Roman never had a need for that type of woman, they were much too weak for his taste.\n\nSure, she could have easily worn the clothes of a princess and blended well with the title. However, he thought she had a more gritty perfection about her, she could handle herself well in a fight, this much Roman knew the moment she walked from the ship for the first time. She was the first woman in a very long time who had gained his respect.\n\n\"If they so much as move the wrong way, shoot them.\" Oz said to Sasha as the two men began to buckle their flight harnesses.\n\n\"Get this fucking rope off of me.\" Draco said demandingly as Sasha quickly used Roman's blade to do so, sliding the steel between his wrists and the thick rope, which fell into pieces like a child's toy as she gave a simple upward thrust of the knife. Both Adam and Roman sat there, amazed at a woman who knew her steel well.\n\nAt the moment his bonds were no more, Draco's hand lunged to Sasha's leg, pulling her combat sidearm made of solid black steel and pointing it at the two men. As swiftly as he aimed the weapon, the barrel of Oz's shotgun touched the back of his neck, his tiny hairs standing to attention as the circular design of the barrel commanded respect.\n\n\"No harm is to come to them while they are under safe harbor from Cyrus.\" Oz said, holding the shotgun at point blank range.\n\n\"You would shoot me and watch them go free?\" Draco asked loudly, surprised by the thought. Oz simply answered his question by pushing on Draco's neck with the shotgun, the unforgiving steel digging into his skin and quickly convincing him to lower Sasha's sidearm.\n\n\"It won't be forgotten my brother.\" Draco said sarcastically as Sasha forcefully grabbed her pistol, placing it back into her leg holster.\n\n\"The memories of a man who gets himself caught on a simple hit are of no concern to me.\" Oz replied with a touch of sarcasm himself as he strapped into the pilot's seat.\n\nOnly minutes later, Sheriff Barker stood to his feet, his injured hand wrapped as he collected his rifle and watched the Benzan shuttle once again hit Glimmeria's upper atmosphere.\n\nAlthough Adam was preoccupied with hoping he and Roman would live to see another sunrise, at the same time he found himself intrigued with Sasha's intoxicating beauty and skills with a weapon. Moments after hitting the thin layer of the planet's atmosphere, his attention was quickly drawn to a small window of the shuttle as they entered space. Legion ships as thick as the sand on Glimmeria's surface bunching together for the invasion to come, as he and Roman both sat silently in awe of the force they would soon be fighting against. Within seconds they both knew that the Colonials were outnumbered at least three to one, and there was no doubt that the invasion was being carefully planned before execution.\n\nAdam suddenly found himself wanting to be back on the planet's surface, helping his friends prepare for a fight which would surely lead to the death of them all. Meeting with the Benzans to find the person who initiated Kelly's death was deeply important to him, however, she was gone and nothing would ever bring her back. But he still had a chance to inform his friends of the fight to come, maybe talk them out of a battle in which they had no chance of winning and in doing so, save their lives.\n\n\"Hey, check it out.\" Lassiter said to both Dalton and Steiner as they began the short journey on foot to the ammunition storage building as ordered, the Husk still moving rather slowly as the soreness of his bandaged right thigh had started to catch up to him.\n\n\"What is it Private Lassiter, I'm a busy man.\" Dalton replied, flexing his newly assigned command.\n\n\"Private?\" Lassiter replied, amazed at Dalton's newly found dedication to the uniform, not to mention his ability to use the term outside of the bedroom.\n\n\"Didn't see this coming.\" Steiner said laughing as Dalton returned the comment with a stern look before noticing what Lassiter had seen only moments ago.\n\nGlimmeria's citizens had began crowding the streets, cheering the many soldiers who were preparing for the upcoming battle against the Legion. Many of them clapped loudly, while a few of the women had thrown flowers as a sign of respect. In only a few hours they would no doubt be instructed to remain in their homes, preparing as if a disaster was inevitable while forced to huddle around small radios and listen to the progress of the war. Still they welcomed it with open arms, realizing that these soldiers represented a free tomorrow for each and every one of them.\n\nIt was a very emotional moment for all three of the men, leveled by the sight of so many people who were counting on their help to remain free, rather than slaves who would spend the remainder of their days under horrible conditions while doing the Legion's manual labor. Both Lassiter and Steiner secretly wondered if Dalton's stare of concentration was one of gratitude or lust, as he continued glancing for several moments at the flower bearing women with saliva ridden lips as they slowly continued their course to collect ammunition.\n\n\"Looks like this is the place,\" Dalton said, several armored skiffs exiting through a huge set of steel bay doors which carried the painted insignia of the Colonials. \"Automatic rifles, grenades and as much damn ammunition as we can carry, got it?\" Dalton ordered, flashing his Colonial badge to an officer behind a thick steel desk as Lassiter and Steiner both began to join dozens of other Colonials in rummaging through huge wooden crates full of precision weaponry.\n\n\"What the hell is this,\" Lassiter asked as he began to laugh, Steiner looking for a moment before shrugging without an answer. \"Looks like it belongs in a museum.\" he added, tossing the gun back into the direction of the storage bin.\n\nThe beat up wood grain stock of the single shot rifle, instead found its way to the palm of Dalton's hand as he snatched it from the air just moments before it would have been discarded.\n\n\"This, greenhorn, is a Glimmerian Thumper.\" Dalton said insultingly as he stroked the weapon with vain intention.\n\n\"And they wonder why the first Glimmerian War was lost.\" Lassiter replied with heavy sarcasm of his own.\n\n\"Don't look like much at first glance...\" Dalton said as he was interrupted.\n\n\"Actually, I glanced at it three times, and it still looks like a raggedy piece of shit.\" Steiner said, bursting into laughter as Lassiter quickly joined him.\n\n\"Laugh it up.\" Dalton replied, sliding the thin metal chamber cover open for a second to make sure it was loaded, before sliding it shut forcefully and aiming it to a corner of the warehouse, a quickly created smirk on his face as he fired the weapon.\n\nEveryone's attention was instantly abducted by the loud pop of the shell jolting from the gun's chamber and slamming into the thick stone corner, rock shrapnel flying several feet as the round exploded with force.\n\nStill holding his shit eating grin, he turned to the other Colonial soldiers, who had stopped collecting gear to stand and watch. \"Say something,\" Dalton dared of them as he turned back to his friends. \"I don't give a damn what uniform a man is wearing, one of these exploding rounds hits him and it'll ruin his weekend.\" he added, smiling so widely that his whiskey scarred teeth made a rare appearance.\n\n\"Report!\" Commander Edwards demanded as he leaned over to view the large monitor of one of the many Colonial workstations that was tucked snugly inside of the vault as its alarm rang loudly.\n\n\"Sir,\" a lower ranking officer said, standing to his feet to face the commander. \"Our grids show the Legion staging area has reached the fringe area of our Mack Cannons.\" he added.\n\n\"Good. You have my permission to make them pay for their first mistake of the day. Start pounding the son of a bitches out of the sky.\" Commander Edwards said sharply, ordering the first shot of the battle as he glanced around the room for a moment to admire such a group of loyal Colonials.\n\nOnly a few moments later, everyone on Glimmeria's surface heard the first shot fire. The large, pounding hollow burst of one of the Mack cannons illuminated the sunset filled sky for a moment, sending dedicated lead screaming into outer orbit. Everyone on the street stopped to watch as the city's fallout siren began blaring with painstaking volume, a mask of silence falling onto the city as everyone began to get indoors as quickly as possible. It had begun.\n\nThree more thumping blasts fired in unison, this time panic and screaming overtook the city as everyone both Glimmerian and Colonial began to quicken the pace of whatever they were doing, sprinting for either safety or the nearest military outpost.\n\n\"Ahh shit,\" Dalton said in a dedicated tone of voice, upset that they had not had the adequate time required to fetch a full array of weaponry. \"Grab the whole damn crate, we 'aint got time for bullshittn'.\" he added, Lassiter and Steiner looking at each other and both quickly coming to the solid conclusion that Dalton had been the only one bullshitting.\n\nSteiner easily pulled his end of the huge crate from the floor, his hulking arms barely straining as he hoisted his half to waist level, strength in full supply.\n\nMeanwhile, Lassiter struggled with the other end, repositioning himself several times and trying to find the easiest spot to grab hold of. Finally managing to pull his end off of the floor a bit, doing everything he could to keep it raised at the height of his knees and punishing his back in the process while the rest of his body shook slightly from the immense weight.\n\n\"Let's go private, cowboy the fuck up.\" Dalton ordered, walking quickly outside carrying nothing more than the badly scarred Thumper and leaving the men to follow slowly. Lassiter doing his absolute best to convince himself not to drop the crate of weapons and begin pistol whipping the arrogant man in command.\n\nA few transports zoomed by them before Dalton saw an opportunity present itself, walking in front of the large flatbed truck and holding his newfound rifle in the face of its driver.\n\n\"Need this vehicle, official Colonial business.\" he said brashly.\n\n\"But...but I'm not Colonial.\" the Glimmerian stated, obviously shaken. Rather than give a reply, Dalton simply pulled the ready pin back on the weapon which led to the driver jumping out of the transport as if his clothes were on fire.\n\n\"Aight' boys, throw it in the back,\" Dalton ordered, Lassiter barely able to continue walking, much less throw it anywhere with the additional weight of the crate pulling his frame within a foot of the ground. \"Hey, where the hell do you think you're going?\" Dalton asked, pointing the gun back at the Glimmerian driver and then flicking the barrel in the direction of his struggling Colonial mate.\n\nOutraged at the idea of not only having his transport stolen red handed, but having to help the thieves load their belongings was unheard of in the man's opinion. Of course, opinions don't mean a damn thing when you have a long barrel rifle pointed in your face.\n\nMoments later the man helped Lassiter with his side of the crate, finally lifting it onto the flat bed well enough to appease their ranking officer.\n\n\"Thanks bud.\" Dalton said with gratitude as the man simply told him to fuck off in native Glimmerian and walked away. \"When this war is over, you're hitting the gym little man.\" Dalton said to Lassiter, which would almost have certainly started a fight if the Colonial sniper could have walked upright.\n\nDriving the vehicle only a couple of traffic congested blocks, Dalton leaned out as he spotted a small Colonial assault team setting up a defensive barricade on the corner.\n\n\"Hey you. Take this group to the Colonial Command Center on orders from Commander Edwards, get inside and await further instruction.\" Dalton said loudly, trying to vocally overpower the blasts from the Mack stations which had now become nonstop.\n\n\"Yes sir!\" the squad leader yelled, sprinting for the command center, his eight man squad tucked in closely behind him.\n\n\"Get the lead out, it's about to be on!\" Dalton added.\n\n\"Double time it!\" an impatient Dalton yelled to Steiner and a nearly crippled Lassiter as they had resorted to dragging the crate from the parked vehicle's location into the command center. Opting to pull a small flask from his brown coat and consume a great deal of the contents, Dalton gritted his teeth from the burn before looking into the sky to see the beauty of distant death. The Mack shots streaming as well as the thruster burns of hundreds of Colonial and Glimmerian fighter ships lit up the early night sky as if it were one big fireworks exhibition.\n\n\"Sir, early reports indicate several direct hits on medium size Legion ships in orbit. Still no indication of return fire.\" one of the Colonial officers stated as Commander Edwards stood with Sarah Blaine, dozens of high ranking officers checked data from all of the monitors throughout the room.\n\n\"There won't be any return fire. We are hitting troop ships, they intend to take heavy losses in exchange for getting Legion boots on the surface.\" Sarah said.\n\n\"She's right. Contact all Colonial outposts within the city, tell them to be prepared to cut off the power grid and have every rail gun they have go to standby.\" Commander Edwards ordered as the officer began to relay the message to every military building in the city.\n\n\"I'm not getting a great feeling about this.\" Roman said as he sat with Adam at a huge table made of solid glass inside of a heavily wood trimmed room on board the Benzan base ship. It was a smaller, cruiser size ship, but they had seen plenty of effective cannons on its exterior when the shuttle was boarding. They both had little doubt that the Benzan ship, although a small one, could hold its own against any Colonial or Legion base ship if needed be.\n\n\"Relax. If they would have wanted to kill us, they could have done it as soon as we boarded the shuttle.\" Michaels replied as the security coded door quickly opened.\n\n\"Adam Michaels I presume,\" Cyrus said, entering the room with Oz closely by his side. \"I must admit, I'm surprised that you would have the testicular fortitude to ask for a meeting with someone in charge of a family that has such, well, questionable notoriety,\" Cyrus said curiously. \"My question is why?\" he added, sitting down at the table in front of both men while Oz remained standing.\n\n\"Your assassin killed a young lady that was a member of my crew. Not only that, but she was like a sister to me.\" Adam replied.\n\n\"Yes, Draco informed us of the kill, for the record, I am sorry for your loss. She was collateral damage from what I understand.\" Cyrus replied.\n\n\"Under normal circumstances, I would have buried your knuckle dragger in the desert and been done with it, but these are not normal circumstances.\" Adam replied.\n\n\"No. No, indeed they are not. You are a guest on my ship, at my mercy and alive right now only because I have allowed it to be so.\" Cyrus replied, letting Adam know that he was in control of the meeting.\n\n\"I understand, and I respect the fact that you are a man of your word. My quarrel is not with you, or anyone aboard this ship. Just the person who ordered my execution and in turn led to Kelly being assassinated.\" Michaels replied.\n\n\"I see.\" Cyrus said.\n\n\"I want to know who it was, so I can cut out the middle man and either end his life or my own in trying to do so.\" Adam replied.\n\nCyrus stood to his feet quickly, momentarily startling both Adam and Roman as they remained seated.\n\n\"Adam Michaels, walk with me for a few minutes,\" Cyrus asked as Roman looked at his friend as if to talk him out of it. \"I am a man of my word, do not worry,\" Cyrus replied. Adam slowly stood to his feet, skeptical of the Benzan's intentions but seeing no other choice. \"Hold him here until I get back.\" Cyrus said as Oz nodded, his gun remaining in the direction of Roman Raines.\n\n\"I like you Adam Michaels, it seems as though your heart is in the right place, and on top of that you have zeal,\" Cyrus said as they exited the room, slowly walking side by side. \"It's because of this that I am about to share some privileged information with you, information that only a handful aboard this ship themselves know.\" Cyrus said as he stopped to look over the edge of a very long catwalk of glass which was positioned directly in front of a huge shatterproof window looking out into space. \"I can give you this information, along with the name of your killer, but before I do, you must agree to remain on this ship until the battle on Glimmeria's surface has begun.\" Cyrus said, turning to look at Adam as he awaited a reply.\n\n\"What? My friends are down there this very minute preparing to be invaded. They are counting on me to join them.\" Adam replied.\n\n\"Join them in what, Death?\" Cyrus responded. \"Take a look at the armada that the Legion is staging outside Adam. You and I both know that the Colonials have absolutely no chance of victory here. Your friends will be wiped out inside of a week.\" he added.\n\n\"Maybe, but they are the closest thing to a family that I have,\" Adam replied, gaining the full attention of Cyrus with those very words. \"You have to try and understand, I would rather risk my life to try and save them than to live and watch them die from a distance.\" he added.\n\n\"Interesting.\" Cyrus said, turning back to the thick shatterproof glass which separated them from the stars. \"Alright Adam Michaels. You agree to remain on this ship until the Legion launches their forces. In return, I will personally send you with a crew of my best down in a shuttle to rescue those you speak so highly of. You have my word.\" Cyrus added.\n\nAdam stood there for several moments, trying his best to think the decision through clearly before turning to the head of the Benzan family. \"Deal.\"\n\n\"The man you seek is not a man at all, he's a Hunter named Anwick.\" Cyrus said before pausing.\n\n\"FUCK,\" Adam shouted, putting his fist against his forehead slowly as his knuckles began to turn red from the squeezing of his clinched fist. \"I had that son of a bitch in the sights of my gun and let him go. Kelly would still be here,\" he added as his eyes began to tear a bit with both sadness and anger. \"He's fucking dead!\" Adam yelled, throwing his fist down as rage began to consume him.\n\n\"Indeed. At least he will be if our upcoming meeting ends poorly. It was recently decided among the Benzan family that we will sever all ties with the Hunter tribes. It was an agreement that we were forced into, and it's one that my people simply have had enough of. We are not the murdering mafia that we have been labeled, and any kind of arrangement with the Hunters only strengthens that stereotype of our people. I have asked Anwick to meet with me so I can deliver the news face to face,\" Cyrus said, pausing to look through the large window that held back deep space. \"There is a good chance it will not end well for one of us. It was bound to happen eventually though, the two toughest kids in the school yard fighting,\" Cyrus replied. \"Which is why I need you to remain on this ship.\" he added.\n\n\"But I want to kill Anwick with my own...\" Adam said, Cyrus immediately cutting him off.\n\n\"Have no fear, he may hunt men, but the men aboard this ship he wants no part of. He will be dead within minutes by my own hands if the discussions end as poorly as I believe they will, I can assure you of that.\" Anwick replied.\n\nAdam looked out across the sharp contrast of the stars as he simply replied with a nod of approval and wishing he could go back in time long enough to take care of Anwick when he had the chance.\n\n\"You know Adam, the way you speak of your friends as family, it begins to remind me of myself at a younger age,\" Cyrus said calmly, Michaels giving him full attention. \"I think when all of this is over, you and I should have another talk.\" he added.\n\nThe city streets were littered with Colonial choke points, its military establishing sandbag bunkers on almost every corner which usually protected a large rail gun. Sweeper teams of two Goliaths and four Colonial soldiers patrolled the streets to soon be used as backup firepower against the Legion as both Swordfish and Spartans flew overhead in attack patterns, waiting for the first sign of contact. And the Mack cannons continued to fire. The thumps of life ending lead being launched into the air sounded almost harmonic, their neon blaze trailing behind them momentarily illuminating the sky with every shot.\n\n\"Set up a perimeter here and do not fall back unless I give the order.\" Dalton said, ordering the squad of Colonial soldiers to position themselves in the front lobby of the command center.\n\n\"Yes sir!\" the squad leader quickly replied, motioning his men to take cover with all eyes on the front door as Dalton, Steiner and Lassiter continued to the center of the building to position themselves outside of the vault.\n\n\"We're in place.\" Dalton yelled loudly, his voice penetrating the thick glass of the vault, Commander Edwards nodding his appreciation as he turned to face Sarah Blaine, who's mind was obviously on other things.\n\n\"I wouldn't worry, I'm sure Adam is alive and well.\" Edwards remarked.\n\n\"I just can't understand him willingly going to the Benzan Mafia, they are not a group to be pushed.\" she replied.\n\n\"Based on what little time I have spent around your husband to be, neither is he.\" Edwards replied.\n\n\"I just wish he would contact us, only long enough to let us know that he's alright.\" Sarah added.\n\n\"I'm sure he will soon enough, however, I think the bigger issue is when he returns.\" Commander Edwards said regretfully as Sarah turned to face him, dreading the words she knew would follow next.\n\n\"Sarah, you know that once he has met with the Benzan people, he will be considered a fugitive himself and no longer a part of the Colonial effort.\" Commander Edwards said.\n\n\"I realize that's usually the case, but with everything he has done for us, I thought that maybe...\" she replied, Edwards cutting her off sharply.\n\n\"No Sarah. He cannot be an exception to the rule, you know that. If your own father were here this very day he would tell you the same. You are going to have to put your personal feelings aside if and when he returns. I don't expect we will go as far as to incarcerate him, but I do expect you to stay clear so that I can enforce Colonial law when I dismiss him from duty.\" Commander Edwards said.\n\n\"I understand.\" Sarah replied.\n\n\"I will allow him to stay here on Glimmeria, ensuring that you two may still be married and have a life together.\" Edwards added.\n\n\"Sir!\" one of the Colonial officers yelled across the vault as the grid began picking up contacts. Walking quickly to the station, it soon became evident to everyone inside of the glass chamber that the Legion had launched its invasion.\n\n\"Darken the city and sound the alarms, now we meet them face to face.\" Edwards ordered as every officer who sat at a computer began relaying the message as well as monitor all of the movement from both sides of the battle.\n\n\"My god, there are so many of them,\" Sarah said softly as her eyes locked onto one of the Colonial grid monitors, watching waves of dozens of drop ships begin hitting Glimmeria's atmosphere. \"All squadrons, this is Sarah Blaine. Multiple incoming contacts, cargo unknown, you are authorized to go weapons free. Repeat, weapons free.\" she said into her headset sending the orders to all Colonial fighters in the air.\n\nMoments later, the entire city fell under a shroud of darkness, its electricity cut by Colonial Command, which was using the strategy of forcing the Legion to attempt to land blindly while masking any potential casualties on the ground. Several minutes later, dozens of large blue flares were vaulted hundreds of feet into the air, casting enough light onto the Legion's ships to present them as targets to the Colonial aircraft and rail guns, which had began to stream bursts of piercing metal into the air.\n\n\"Sir, early reports indicate at least a dozen Legion cruisers, each dropping dozens of Spider Pods from the sky.\" one of the Colonial officers said.\n\nThe Commander had encountered the Spider Pods only once before, nothing more than a small round ship that carried a dozen soldiers and once landed, transformed into a mobile rail gun that would escort its cargo.\n\n\"Tell our air units to continue to fire on the small targets, and have the Mack stations redirect their fire from the cruisers to the small targets as well.\" Commander Edwards ordered, knowing the cruisers had no intention of landing.\n\nThe Mack cannons began angling their shots, ripping each Spider Pod it hit into thousands of pieces, the light armor no match for a weapon designed to destroy the largest of ships. Their pounding bursts of lead, combined with the array of rail guns and attack fighters had eliminated nearly half of the small, troop filled ships when the first one landed on Glimmerian soil, marking the first time a Legion military ship had touched the planet's surface since the original Glimmerian war nearly a decade before.\n\nDistant explosions continued to ring out as both Adam and Roman watched Glimmeria through a small glass window inside of their sleeping quarters.\"We've got to get back down there and help them.\" Roman said with desperation as Adam continued to watch the planet, which looked like it had fallen victim to a large thunderstorm, fully engulfed in brilliant flashes and explosions.\n\n\"Not much we could do to help, only two of us Roman,\" Adam replied. \"Besides, they should be able to hold them off long enough for us to get down there with the Benzans.\" he added.\n\n\"Do you think we can trust Cyrus?\" Roman asked, also considering the possibility that the Benzan leader was simply holding them there until Anwick arrived in order to be turned over to the Hunters for reward.\n\n\"I think so, at least that's what gut tells me.\" Adam replied, both men startled suddenly by a light knock at the door of the sleeping quarters.\n\nRoman slowly walked to the small steel door, spinning the large circular handle to unbolt the door to be opened. Sasha stood there trembling, an emotional wreck as she looked at both men with her eyes fluid with tears before falling into the arms of Adam.\n\n\"What's wrong?\" he asked, glancing at Roman curiously. It took Sasha a few moments to collect herself, the woman who had broadcast such a hard appearance when they first met had become nothing short of hysterical.\n\n\"They plan to kill him.\" she managed to finally push from her lungs as she sobbed uncontrollably. Adam continued to look puzzled as Roman spoke up to ask.\n\n\"Kill who?\" Sasha began to pull herself away from Adam just a bit in an attempt to stand on her own as he found himself not wanting her to leave his grasp, which scared the hell out of him.\n\n\"Cyrus. I overheard Draco in his room speaking with the Hunters through a secure com link, he plans to kill Cyrus when the meeting takes place and then turn you both over to Anwick.\" Sasha said a bit more calmly as if to try and suddenly mask her emotions.\n\n\"Why are you bringing this to us? Shouldn't you be having this conversation with Cyrus?\" Adam asked.\n\n\"I tried, but he is in a meeting with Oz and to be honest, I don't know how far this goes. I'm not sure who I can trust with this information, other than Cyrus himself and both of you.\" she replied.\n\n\"What makes you believe you can trust us?\" Roman asked suspectingly.\n\n\"When you first contacted our ship, Cyrus had me run a complete background check on both of you. Adam, you showed up on the sheet twenty seven times for various offenses,\" she said as Michaels smiled wide, proud of his criminal achievements.\n\n\"It was you however, Roman, who threw us for a curve. You showed up but once, almost as if you had never existed. The Gali Special Forces had a condition one on you. Such a high priority to them combined with the fact that the Hunters want you dead so badly could only mean one thing. Greyspine. We know what you've done.\" she said tactfully.\n\n\"Greyspine? What the hell is she talking about Roman?\" Adam asked, never having heard of the small moon planet before.\n\n\"No time for that now Adam. Can you get us any weapons?\" Roman asked of the girl, eager to avoid talking about his past.\n\n\"Small ones. Two pistols would be about all, otherwise they would pick up on them quickly.\" Sasha responded.\n\n\"Pistol works for me.\" Adam replied confidently.\n\n\"I'll pass on the pistol if you can just get me that blade back.\" Roman said speaking of his tactical knife.\n\n\"I'll take care of them both at once. Cyrus has already arranged for the Hunters to dock a shuttle within the hour, so if I get the chance to speak with Cyrus in private...\" Sasha said before being cut short.\n\n\"No. Let us handle this. We just need those weapons and to be put close to the initial meeting, if that's possible?\" Roman asked.\n\n\"I'll make sure you are close.\" she replied as she turned to Adam. \"Thank you.\" she said softly, having regained her composure as she quickly left their quarters to retrieve the two weapons.\n\n\"What's this about Greyspine?\" Adam asked as Roman had sealed the door back tightly with a firm turn of the handle.\n\n\"It was a long time ago Adam. Besides, the real question is what is this shit about soon to be married, but falling for Sasha?\" Roman replied with a question of his own, desperate to avoid the subject for as long as possible.\n\n\"What are you talking about?\" Adam asked, although he knew Roman spoke the truth. He never was very good at masking his feelings for others.\n\n\"You know what I'm talking about. The air in here got thick real quick from the tension you and Sasha were sending back and forth,\" Roman replied. \"Right now though, we both need to concentrate on getting off of this damn ship alive. The rest will work itself out in the end.\" he added.\n\n\"Redirect all Mack and ship to ship fire onto the larger Legion vessels on their way in.\" Commander Edwards ordered.\n\n\"But sir, in doing so we will allow thousands of Legion boots to hit our soil.\" a Colonial petty officer replied.\n\n\"Let them land, but we WILL control the skies of Glimmeria. Now execute my order!\" Edwards replied bluntly.\n\n\"Right away sir!\" the petty officer replied, quickly making his way to a nearby com station to relay his commander's wishes.\n\nAfter taking nearly a minute to readjust the trajectory of the huge subterranean guns, their gleaming shots began tearing into the flag ships of the Legion which had all began to descend behind a cloud of smaller ships filled with grunts. The first few minutes saw a handful of the very large spacecraft ripped to shreds, the rocketing lead punching holes through their most vital areas as they soon became nothing more than large masses of fire free falling to the planet's surface.\n\nThe Legion tried countering the shots with the deck guns built onto the ships, however they were of little to no effect as the overpowering torque of the Mack shots continued reaming not only their most decorated officers, but much needed supplies as well.\n\n\"The Colonials are completely ignoring the smaller craft! Our command ships are being cut into pieces! Call the Hunters from orbit and get their asses down onto the surface!\" one of the Legion Captains commanded, screaming the order across the bridge of the ship as pounding shots steadily hit its reinforced exterior.\n\n\"Yes sir!\" a seated crew member replied, attempting to make contact with the Hunter vessels in orbit as his own ship's power began to flutter from the substantial damage of the Colonial shots it had absorbed.\n\n\"Tell them to wait,\" Anwick replied through the com system aboard his private shuttle as it made its way to the docking area of the Benzan warship. \"Do not follow them into battle without a direct order from me. Do you understand?\" the vampiric monster added.\n\n\"Yes my lord.\" his second in command responded with absolute obedience.\n\n\"Thank you Garrison, I can always count on you my friend. This will not take long. Tell them to stand on their own two feet for a change.\" Anwick said, smiling of the insult and also the excitement of learning that Adam Michaels was being held by the Benzans.\n\n\"Believe me my friends, I have no intentions of turning you over to them. Anwick will arrive and try to force my hand, at which time I will gladly discontinue the treaty he has seemed so willing to force upon us.\" Cyrus said to Adam as he stood with Roman, their hands in front of them locked in thick chains.  \n\"The restraints are necessary only that Anwick would not suspect anything as he boards our ship.\" he added. Adam nodded to agree, a small pistol tucked snugly under his belt and hidden by a lightweight brown jacket. He glanced at Roman, wondering where his friend had hidden the tactical blade delivered to him by Sasha as well as to ensure he was ready to fight, which he was indeed.\n\nOnly two of the Legion's flagships survived the Mack onslaught which was accompanied by both Spartan and Swordfish gunfire. Though intact, both were seriously damaged on their way to landing several miles from Glimmeria's capital city, immediately setting up a forward operating base for the massive group of soldiers on foot who had hit Colonial soil less than an hour before.\n\n\"Any word on the estimated time of the Hunters arrival?\" the highest ranking surviving Legion officer, Sky Admiral Cook asked.\n\n\"They refuse to engage without Anwick, who is tied up in a meeting at the moment.\" one of the deck officers replied.\n\n\"MEETING,\" Sky Admiral Cook yelled loudly as he looked around the badly beaten bridge area of his ship, loose wires sparking as they hung from overhead like snakes from a vine.\n\n\"Have all units fall back and establish a perimeter with the forward operating base. The last thing we need is to be scattered throughout the wastelands of Glimmeria to be picked apart!\" he added.\n\n\"Yes sir, right away.\" the deck officer replied firmly, sending Cook's orders out through the ship's on-board com.\n\n\"Commander, they have set up a few clicks from here.\" Sarah said, pointing to a map in the vault that was made of thick glass and colored marker.\n\n\"No doubt trying to regroup,\" Commander Edwards said as he approached Sarah steadily. \"Have our birds run patterns over that entire area and give them orders to fire on anything that moves. In the meantime, order a large ground assault team to assemble, we need to hit them before they have a chance to dig in too deeply.\" he added.\n\n\"Right away sir.\" Sarah replied, speaking clearly into her headset so all of the Colonial soldiers would understand his orders without question.\n\nWhen Anwick began to exit the Hunter shuttle slowly and board the Benzan warship, an exuberant smile painted to his face as his narrow eyes of emptiness locked onto both Adam and Roman.\n\n\"Cyrus, I must admit that you have done the job well. I did not expect to see both men standing before me so soon, their hands wrapped in chains.\" Anwick said as two very large and heavily armed Hunters followed him down, their semi-automatic rifles looking as though they could have easily punched a hole through the most battle tested of metals.\n\n\"It was nothing of my doing. In fact, they came to me.\" Cyrus replied, as Adam wondered how many more of the Vampiric monsters waited on the shuttle of pointed steel which gave the illusion of fangs.\n\n\"Is that so?\" Anwick said, puzzled but excited nonetheless.\n\n\"It is, as is the fact that I have no intentions of turning them over to you,\" Cyrus said, his words freezing the Hunter in his tracks as he slowly turned his white complexion of death to the Benzan leader. \"Furthermore, this treaty of peace that seems to exist between our two people is no more. We want no part of this political war and refuse to be included in the same sentence as those who mercilessly slay men for pleasure.\" Cyrus added.\n\n\"You should consider your last words wisely Cyrus, as you are no longer the leading member of the Benzan Mafia. I have promised that position to Draco in exchange for his unwaivering loyalty.\"\n\nAnwick said with gleaming confidence as he glanced in the direction of Draco. Pulling his long barrel shotgun up to the face of Cyrus, Draco slowly cocked the hammer of the weapon and solidified his position as the new leader.\n\n\"Tell your people to throw their guns to the floor and I might spare your life.\" Anwick said as both Sasha and Oz had began lifting their guns into the direction of Draco and the Hunters.\n\n\"Draco? I pray that you understand the consequences of your actions.\" Cyrus said as he turned to plead with a man he had placed full trust in.\n\n\"Shut the fuck up old man! Better do as he says and have 'em drop their pieces before all of you die here standing.\" Draco yelled, breaking the heart of Cyrus in the process. \"Do as he says.\" Cyrus said as he nodded uneasily to the remaining Benzans.\n\n\"I'm surprised that you have made it this far Adam.\" Anwick said bluntly as he began to approach both of the prisoners.\n\n\"No choice. Had to find out how deep this treason went.\" Adam replied as within but a single moment's breath, he pulled the small pistol and cemented a hole into the forehead of Draco with unmatched speed. Having taken the entire group by surprise, Adam quickly adjusted his aim, which was still hindered by chains and fired a second shot at the Hunter's head man.\n\nThe uncanny ability to move with urgency was the only thing that saved Anwick's life, a shot which would have penetrated his heart instead passing quickly in and out of the flesh of his shoulder as he turned away. Both of Anwick's escorts turned their weapons in Adam's direction, the first firing wide as Roman clinched his hands together and thrust them upward, knocking the rifle from the grip of the beast.\n\nAs the second escort soldier tried redirecting his aim into the direction of Roman Raines, the blade which had been snuggly tucked into the length of his boot was now in hand, the serrated edge cleaving into the chest of the soldier. Both Sasha and Oz scrambled for a moment to regain hands on their weapons as Cyrus just stood silent, deeply impressed with Adam's pistol skills, Roman's ferocity and above all, the fact that they had went to battle for him. It was in the waiting moment of that very breath that his heart decided both men had earned his respect, which is something the cold body of Draco lacked.\n\nWith Anwick fleeing to his shuttle and both escort soldiers locked into a crimson filled fight to the death with Roman, two final Hunters emerged from the Hunter craft in reaction to the gunfire. One was immediately cut to his grave by the punishing shots of both Sasha's machine gun pistol and Adam's weapon. His one final shot before reload desperately seeking the back of Anwick, but instead striking the back of his leg, crippling the beast momentarily as the remaining escort soldier dragged him onto the shuttle.\n\nRoman had given his best effort, but a mortal man wrapped in chains, no matter how skilled he may have been, was simply no match for two of the toughest soldiers that the Hunter clan had to offer.\n\nHis blade knocked from his hand, the only thing that had kept him in the fight and alive up to this point was his mechanical side, which had began to surrender as well. One of the Hunters dove at him with numbing rage, its teeth fully exposed in an attempt to kill the human menace once and for all. Never making it to him, the Hunter's face and shoulder region became nothing more than a cloud of soiled fluid at the end of the shotgun held by Oz. The second close enough to attempt the same, was faster than a reload by Oz, but not by Adam. He had reloaded his pistol with more speed than any of the Benzans present had ever witnessed, aiming and firing all six shots directly at the beast.\n\nRiddled with holes and screaming in life ending agony, the Hunter was short work for Roman after he reclaimed his tactical knife, making it slow as he carved an apology out of the dying beast. The Hunter's shuttle quickly disengaged, pulling from the deck in a last ditch effort to return to it's base ship. Everyone with a weapon fired into the air, but it was Sasha's gunfire that hit the fuel tanks which burst open and began to bleed a trail of fire and smoke. Anwick had made it back into open space, but with no hope of returning to his ship, they simply began a steep dive into the atmosphere of Glimmeria.\n\nAdam walked with purpose toward the Benzan shuttle which had brought him here.\n\n\"Adam. What are you doing?\" Sasha asked.\n\n\"I think it's time the Hunters became the hunted,\" Adam replied as he made it to the entrance of the craft. \"That son of a bitch will not live to see tomorrow's sunrise, I promise you that.\" he added as he turned and disappeared into the shuttle.\n\nAdam had begun pressing the necessary sequence of buttons when he turned to see Oz carrying Roman on board as Sasha followed closely behind. \"We're going with you, it's the least we can do.\" Sasha said softly as she shut the hatch of the shuttle and sat down to strap in for the decent.\n\n\"Sir! Picking up a warship with Benzan identification firing on the Hunter ships in orbit!\" Sarah exclaimed loudly, knowing in her heart that Adam Michaels had played a role in the unfolding events.\n\n\"The Benzans are still wanted fugitives, stay focused on the battle here on the ground.\" Commander Edwards replied as Sarah patiently watched her com screen hoping the Benzan warship survived the fight.\n\n\"Continue firing all main batteries onto the Hunter Carriers. Those blood sucking bastards have just started a fight they can't win.\" Cyrus said from the elegance of the Benzan ship's bridge. Taken by surprise, the Hunter ships fell victim to several devastating shots before turning their own weapons into the direction of the Benzan Warship.\n\nSeveral minutes passed as they exchanged knockout punches of lead, the surviving Hunters abandoning their sinking Carriers by the dozens in small escape pods, all dropping hastily into the atmosphere of Glimmeria on their way to the surface.\n\n\"Commander, all three Hunter ships have dropped from our grid system but the Benzans remain.\" Sarah said with relief filled joy.\n\n\"Alert the Mack stations, have them fire on the Benzan vessel until it is destroyed.\" Commander Edwards replied sharply.\n\n\"But Adam?\" Sarah asked with desperation, turning to face her commanding officer.\n\n\"Adam isn't up there. We are picking up his Colonial beacon several clicks away from here,\" he replied. Sarah sat motionless, unsure if that was good or bad news as she feared for the man she loved so dearly. \"Don't worry Sarah, I'm assembling a strike team to reach the beacon as soon as possible. In the meantime, have the Mack stations fire on the Benzans, as well as anyone else who shows up in orbit that isn't an official ally.\" Commander Edwards said.\n\n\"Yes sir.\" Sarah replied, slowly turning to direct his orders to the large weapons stations.\n\n\"What the fuck are ya'll cheering about?\" Dalton yelled to the Colonial soldiers in and around Colonial Command who were celebrating the demise of the Hunter Carriers. \"They still outnumber us three to one, get your guns out of the air and point them at Legion troops and shoot!\" he added, his voice ringing through the mortar shocked streets of the city.\n\n\"Dalton, Steiner,\" Commander Edwards said, making a rare appearance outside of Colonial Command. \"Get your asses over here, double time!\" he added, surrounded by a dozen of the finest trained soldiers stationed on Glimmeria.\n\n\"Picked up Adam's beacon a few clicks from here,\" Commander Edwards said, pointing to a spot on a small map as shots rang out around them, the violence moving closer to their position by the minute. \"I'm putting Lassiter in charge of things here, you two take this Colonial strike team to them, see if Adam is still alive. If he is, you bring him back here in one piece, got it?\" Edwards asked. Dalton nodded as clouds of broken concrete mixed with sand began falling around them, Legion troops making their way to the end of the street and meeting stiff Colonial defenses in the process.\n\n\"Aight, listen up! We are gonna backtrack, walk around the gunfire instead of straight into it.\" Dalton said loudly as a much larger Steiner stood beside him, the Colonial soldiers listening to his every word.\n\n\"Sir, we are being pushed back!\" one of the Colonial grunts yelled in Lassiter's direction as a small group of Legion faithful had broken the Colonial lines.\n\n\"Hold the line at all costs!\" Lassiter responded frantically.\n\n\"Ahh hell.\" Dalton said as though he was being hindered, turning to face the Legion threat as he fired six devastating rounds from the Thumper in succession, all but leveling the entire city corner that the Legion troops had occupied just moments before.\n\n\"Keep these fuck stains in line!\" Dalton yelled in the direction of Lassiter, winking at him as he ordered his strike team to deploy in a flanking move that would lead them to the edge of the city and eventually in the direction of Adam's beacon. Lassiter simply gave a glassy eyed look, not from the combat around them or even the wake of death caused by the Glimmerian Thumper, but the term Dalton had just thrown his way. To his estimation it was a new low, even for Dalton.\n\n\"Adam are you alright?\" Sasha asked, her portrait perfect face merely inches from his as he tried to regain his senses.\n\nThe rough landing had thrown him from his seat and against a thick steel plate near the deck of the small ship's interior. He wanted to respond to her, but before he could, his attention was pulled from her angelic face to the rear of the shuttle as Roman and Oz both fired their weapons out of the ship's only exit.\n\n\"Yea, I'm alright.\" he responded slowly. Sasha crouched over him, her lips but inches from his as she smiled softly.\n\n\"Better get up then, I would hate to think your reputation was going to be affected.\" she said softly as her eyes locked with his almost as if they were destined to.\n\nA few seconds later Sasha slowly stood up, her aerodynamic curves no longer straddling him as he was slow to his feet as well. Rather than speak of the fireworks exploding inside every limb of his body which would he feared would eventually force him to deal with conflicting feelings, he instead turned to the two men at the shuttle entrance. \"What's going on?\"\n\n\"We got clipped by rifle fire as we made our approach, had to bring the shuttle in hot. Before we knew it, Hunters were falling out of the sky all around us.\" Oz replied, shouting his response before turning back to fire his shotgun out of the door. Roman steadily dispersed rounds from a battle rifle, but it was obvious through his body movements that the shots were having little effect.\n\nAdam turned to Sasha, taken back for a moment by the look of true infatuation as he slowly took her rapid fire pistol and turned it onto the windshield of the shuttle, spending nearly an entire clip of shells as the thick glass began to crack. He turned to hand her the weapon back, their eyes locking onto one another once again as he found it unnervingly tough to stop himself from continuing the kiss which had played out in his mind only moments ago.\n\nTurning back swiftly, Adam began thrusting his foot forward, the thick sole of his boot cracking the glass in multiple places with each strike.\n\nFinally the entire windshield was knocked from the shuttle, Adam extending his hand to the soft grip of Sasha as he helped her up and out of the newly constructed crawl space. Adam was right behind her, crawling out onto the scorching sand as they motioned for both Roman and Oz. Oz sprinted across the small confines of the shuttle, quickly pulling himself out of the wreckage to join the others. Roman was a bit slower, firing all of the ammunition from the rifle before dropping it and pulling himself through as well.\n\nThey counted six Hunters scattered in the immediate area, even more falling slowly from the sky as the attached parachutes drifted the escape pods to the surface in a motion that resembled that of a tumbling feather. Other than the small town which lay over the horizon, there was literally no cover as the group of four sprinted for the cluster of wooden structures, rifle shots from the distant Hunters clipping the sand behind their heels all the while.\n\n\"Cyrus, we are dead in the water until we are able to repair the ship's engines.\" one of the Benzan members said grimly.\n\n\"Have someone get right,\" Cyrus replied, his words cut short by a massive explosion on the starboard side of the ship. \"Report!\" Cyrus shouted, the remaining Benzans running to both what little working equipment that remained as well as the large glass window of the warship.\n\n\"The Colonials are firing on us!\" one of the men yelled animately as he looked through the thickened glass to see a glowing shot from the surface based Mack cannons scream by them.\n\nIt was a damn tough ship, solid from the treated steel exterior to its heart and very soul. Still, Cyrus knew that it had already taken a beating and would soon succumb to the staggering shots of the Colonials.\n\n\"Give the order to abandon ship. Take any weapons and gear you have time to collect and regroup on Tirious.\" Cyrus said somberly, knowing full well that he didn't intend to make the trip with them to the small snow filled moon.\n\n\"And you sir?\" one of the Benzans asked as the rest scrambled to escape skiffs, the warship's alarm sounding loudly.\n\n\"You know the Benzan way. A commanding officer always goes down with the ship, not that I would have it any other way.\" Cyrus replied as he placed his hands on the cold steel of the guard rail that surrounded the bridge.\n\nHe thought of every single battle that the ship had pulled him through, every deep space run that he himself had made while aboard.\n\n\"As my final order, make sure our people on the ground know the rendezvous point. Adam and his friend saved my life without question, so I wish for them both to be invited into the Benzan family. Now go Helon, carry out my orders and present yourself proudly as a Benzan.\" Cyrus added. Helon nodded, accepting the job of personally delivering the final command of Cyrus.\n\nHelon turned to one of the remaining shuttles, his closely shaved hair of blonde highlights quickly grabbing another one of the scrambling Benzans, this one much taller with very dark skin and stark white dreadlocks as the two men boarded the craft, glancing a final time at Cyrus with a look of complete gratitude.\n\nAs the escape shuttles all bust into flight on their way to Tirious, a single ship broke from the pattern, diving steeply as it hit the atmosphere of Glimmeria to find Adam, Roman and the remaining two Benzans. Cyrus slowly turned his head, looking around the empty Benzan warship without regret as two shots from the surface cannons hit, ripping the large craft into several pieces, all of which burst into explosion only moments later.\n\nSeveral intense minutes of fighting filled everyone's ears with explosions, ringing lead as it zipped through the air and screams of agony as both Colonial and Legion troops died for their factions. Eventually, the shelling died down and Colonial troops outside of the Command Center relayed the message of hope.\n\n\"Legion forces are falling back.\" Commander Edwards walked within inches of the thick glass which separated Lassiter and himself, picking up a mobile com to speak directly with with the loyal soldier in command.\n\n\"Good job. Have your scouts move forward, clear the area and then advance fire teams along with Goliaths.\" the commander said.\n\n\"Yes sir.\" Lassiter responded, turning to direct the men to carry out the orders with haste.\n\n\"Alright, pony up,\" Dalton said, flexing his thin veil of command. \"Lieutenant Michaels is several clicks from here and needs our help. Instead of marching the entire group into a damn Legion ambush, I'm gonna need a scout to get ahead of us and find a clear path.\" he added as a Colonial sniper stood up holding a large bolt-action rifle in his hand.\n\nThe huge scope on the weapon immediately caught Dalton's attention, almost in a romantic sense, as he began to daydream of holding such a fine piece.\n\n\"You'll do.\" he said as Steiner tossed him a long barrel shotgun, then picking up a standard issue combat rifle for himself.\n\n\"Damn you Michaels!\" Anwick yelled loudly as he remained strapped into the wreckage of his shuttle, his escort soldier having died on impact, its head bloodied from the instrument console. Anwick slowly began to crawl from the wreckage as he was met by a small group of Hunters, his eyes fixating on the sky above as dozens more of the escape pods drifted to the surface in a poetic manner.\n\n\"Are you alright my lord?\" one of the larger Hunters asked, all of them wearing heavy black leather clothing and armed to the teeth.\n\n\"I will survive. Regroup with the others and on my command we will sweep through the surrounding area and slaughter anything with a heartbeat.\" Anwick replied, holding his shoulder which he believed had become dislocated during the crash, his two gunshot wounds nothing more than an afterthought.\n\n\"Yes my lord!\" the Hunter replied soundly as the group turned to organize the survivors. \"God damn you Adam Michaels. God damn you!\" Anwick said under his breath as his eyes sliced into the town ahead.\n\n\"Sarah, we got a problem. The numbers you gave me fall about three miles behind Legion lines. Even if we were lucky enough to get to them, there is no way in hell we could get them back. And if we picked up the ships, it's a good bet the Legion did too.\" Dalton said into his com unit, the entire strike team laying low on the top of a sand dune which overlooked the coming road.\n\n\"I'm organizing a large tactical team, just do your best to get to them and dig in until we can get the team to you.\" Sarah replied with desperation.\n\n\"Will do,\" Dalton replied, placing the com unit back on his belt. \"Well fuck. Isn't this some shit? Just a small strike team out here, dicks flappin' in the wind with who knows how many Legion troops ahead.\" Dalton said loudly to the Colonial team.\n\n\"A lot. Looks like they are staging a large ground assault up ahead.\" the Colonial scout said as he kept his eye firmly pressed to the scope of his weapon. Dalton slowly climbed to the scout's position on the sand dune, glancing through a small set of binoculars.\n\n\"Looks like we are going around. These sumbitches are setting up a permanent base!\" Dalton said as Steiner remained under a brown tarp, his eyes locked onto the horizon making sure the Legion had not found them. Minutes later, the group slowly made their way through the sandy cliffs of the area, keeping a safe distance from the Legion base as they worked their way around to get to Adam's location.\n\nShards of seasoned wood flew through the stagnant air of the room as Roman was the first person to walk inside of the abandoned building, kicking the door open as a thick coating of dust lay on top of the tables that once served a purpose for the townsfolk. As he slowly swept his gun from side to side to assure the building was safe, Adam entered followed by Sasha and finally Oz, who spent a couple of minutes trying to secure the door back that Roman had nearly jolted from its hinges.\n\n\"This is as good of a place as any, the Hunters are sure to be on top of us soon and I really hope Anwick is still alive, because I want to kill that son of a bitch.\" Adam said firmly as Oz began to make his way up a narrow ladder to the loft of the building, the creaking floorboards covered with ample dirt.\n\n\"I'll keep my eyes on the door,\" Roman said as he rested behind a stack of wooden crates at the entrance. \"If they come through, we'll be right here to back you up.\" Adam said as he and Sasha found a small stack of their own at the rear of the room, glancing up for a moment to see Oz in position at the only second story window that overlooked the rear of the building.\n\n\"Our scouts report that Legion troops have fallen back and are forming a forward operating base.\" Lassiter said through his hand held com as sporadic gunfire continued in the distance.\n\n\"Copy that. We are going to continuously air strike their position until we are able to mount a ground offensive. Your orders are to further solidify the defenses outside of the Colonial Command Center, understood?\" Commander Edwards asked from the comfort of the vault.\n\n\"Yes sir, copy that.\" Lassiter replied.\n\n\"Gonna air strike the shit out of them! In the meantime, pull your forces up and dig in solid. We are the last line of defense until further notice!\" Lassiter shouted to his troops through a thick succession of explosions. The heavily equipped Goliath units remained back in close vicinity of the command center, along with the mounted rail guns and a handful of Colonial troops as the bulk of the grunts moved forward a couple of city blocks.\n\nOz made a low sound as he held a clinched fist in the direction of the others, Roman immediately placing his back against the wall at the foot of the door with a blade in one hand and a combat pistol in the other. Holding up the number four and moving his hand in a circling motion, the others knew quickly that a group of four were patrolling the area.\n\n\"Hunters?\" Adam asked quietly as he looked to the high loft area. His question was simply answered with a nod. They were Hunters indeed.\n\nAdam buried his face into one of his hands for a mere second before lifting it and firming the grip on his standard issue rifle. Sasha watched him for a few moments to make sure he was alright, before also turning her attention as well as her rapid fire pistol into the direction of the building's entrance.\n\nRoman extended his arm slowly to the others, hand flat to let them know that someone was about to enter the door which was shut but not latched because of the blunt force trauma that his boot had been guilty of. The creaking of wood could easily be heard through the blanket of silence in the room as the door began to open slowly, the light of Glimmeria's moon saturating the otherwise pitch black surroundings of their hideout.\n\nTightening his grip on the blade's handle, Roman counted steps in his head to try and determine how many were gaining entrance, ready to blood let on as many as need be. \"Roman, wait.\" Adam shouted in a whispered voice as everyone watched a dog enter cautiously. The amber colored animal looked half starved and was mange covered, but came right to Michaels as he knelt and extended his hand, quickly grabbing a piece of stringy meat jerky from a supply bag.\n\n\"You are just full of surprises aren't you?\" Sasha asked, impressed with Adam's good hearted nature as she smiled softly. He was true to Sarah Blaine, in his heart he knew that to be a fact. Still, he found it was becoming increasingly tough to deflect the overpowering beauty and compassionate nature of Sasha.\n\nAs Roman slowly began creaking the door back to a closed position, it was thrust open abruptly by a large Hunter clad in thick black armor, knocking the Colonial sworn warrior to the floor. Sasha was the quickest to respond to the surprise guest, her pistol ringing out over a dozen shots that bit into both the armor and flesh of the beast. As it fell to the ground in tremendous pain, a second quickly followed behind, met with equal haste by the tip of Roman's blade as it plunged into the demon's ribcage.\n\nYelling with angered pain, the Hunter grabbed Roman tightly with both hands as the mechanized warrior began throwing a barrage of tightly placed elbows into the head of the beast. Although it didn't break the Hunter's hold, it did loosen it enough for Roman to see two more sprinting behind the monster right outside of the building.\n\nThe grip was let go when the demon dropped Roman to grab the back of its skull, a rifle shot finely placed by Adam which had plunged into the thin bone wall and struck its brain.\n\nThe first Hunter began to rise slowly as the second fell quickly to its death. Oz fired a loud and throaty round from his shotgun which nearly cut the beast's torso in two, dismembering the instantly dead creature as it tumbled back. Without his blade, Roman dashed outside to face the last two on his own, a brave but reckless idea for someone who at his best could only match one of the demons in battle.\n\n\"Roman!\" Adam yelled loudly as he began to dash out to the aid of his friend, very quickly grabbed by the hand of Sasha.\\\n\n\"Adam. There are too many of them! He gave us the opportunity to flee, we must go now!\" she said loudly, Oz opening the window for them to climb out of as their only exit.\n\nAdam turned to her before quickly spinning back to the direction of the front door. Several shots rang out, a few of them were from Roman's pistol while a majority were not. A good indication that their friend lay dead outside on the still hot sands of Glimmeria.\n\nBefore any of them had a chance to respond, the flimsy front door once again flew open, this time slamming against the wall behind it as two Benzans stood in the doorway, the first surveying the room with his rifle drawn as the second, dark skinned warrior drug Roman's bloodied body indoors.\n\n\"Helon, Zavious!\" Sasha shouted loudly as she sprinted to them, relieved to see friendly faces.\n\n\"Appreciate you saving my friend out there.\" Adam said appreciatively as he knelt to check on Roman Raines.\n\n\"Wasn't a problem, we only had to slay one of them. Your friend had already killed the other.\" Zavious replied. Roman slowly stood to his feet unassisted and nodded his gratitude.\n\n\"I'm fine Adam, a little blood never hurt anybody.\" Roman said, wiping his face clean of it with a dirty rag from the building's floor.\n\n\"Best be hauling ass out of here, I'm sure everyone in town heard the gunshots.\" Oz said, quickly stepping down the ladder to greet his old friends.\n\n\"I agree, but before we do, I have to carry out a final order from Cyrus.\" Helon said.\n\n\"Cyrus. How is he?\" Sasha asked impatiently. The shaking of Helon's head was all the answer she needed, the leader of their family was dead.\n\n\"Adam Michaels,\" Helon said with intention as he pulled two Benzan amulets from his light grey jacket of silk and pinstripe. \"Cyrus wanted to extend an invitation to the both of you, wanted you to become official Benzans as reward for saving his life.\" he added. Adam glanced at Roman for a few tense moments before turning back to Helon.\n\n\"I...I don't know. Of course I want to, it's just, it's just I am supposed to be married soon to one of the highest ranking Colonial officers.\" he replied.\n\n\"Yea. You are full of surprises.\" Sasha said, obviously hurt by the news of Adam's previous engagement as she walked outside quickly.\n\n\"At least think it over?\" Helon asked. \"When this is over and we are safe, I'll give you an answer.\" Adam replied.\n\n\"Good enough. And for the record, your Colonials shot down our ship and directly led to the death of Cyrus.\" Helon replied, tossing the amulets to the two men before turning to meet with Sasha outside as Adam stood in disbelief.\n\n\"Fucking politicians.\" Roman replied as the rest of the group slowly made their way outside as well, heavier now by one pitiful sight of a mutt.\n\n\"This fucking sand gets down in your boots and really pisses ya' off.\" Dalton said in a bitchy tone, slowly making his way to the top of a large sand dune to lay beside the Colonial sniper who had his rifle and scope mounted into a tripod.\n\n\"The good news is Lieutenant Michael's beacon seems to be coming from that small town in the distance.\" the sniper said as Dalton squinted his eyes before staring it down through his small set of military goggles.\n\n\"The bad news?\" he asked, almost dreading the answer.\n\n\"Pan your eyes over about fifteen degrees to the right.\" the sniper responded. Dalton slowly moved the binoculars with his eyes firmly pressed against them, stopping and squinting deeply before looking once more.\n\n\"Well I'll be goddamn.\" he said in frustration as he put his lenses down and contemplated the group's next move.\n\n\"That's not the only group of Hunters, I spotted several more throughout the town. Looks like they beat us here and are looking for Adam as well.\" the sniper said.\n\n\"I don't believe this shit.\" Dalton said in a coarse but whining voice. \"Man I hadn't had a hot shower in three days. Aint' had a fucking drink in so long that I'm an ace from putting my tongue in rehab. I Got a promotion through the ranks of a military that hadn't even cut me the first paycheck yet and now here I am, balls deep in the sand out here trying to rescue a couple of friends who are surrounded by a buncha' flesh eating bastards who want me dead.\" he added, trying in his mind to count the days since his last drink.\n\n\"Alright,\" Dalton said as he began back down the dune, pausing to turn to the sniper. \"Keep your ass locked in this spot. Guess me and the boys are going down there. Need you up here relaying the Hunter's movements and ready to bust a cap.\" Dalton added, sliding down the dune slowly before the sniper could even respond.\n\n\"What's the situation?\" Steiner asked.\n\n\"This damn sand is the situation.\" Dalton replied, dumping his boot out while holding his foot off of the ground slightly, his eyes teared with frustration.\n\n\"You alright?\" Steiner asked.\n\n\"We are out here in this shit getting ready to walk into a hornet's nest of Hunters, and Commander Edwards is back at the fort with his ass on plush!\" Dalton replied, forcing his thick brown leather boot back on in a single motion.\n\n\"Hunters? How many?\" Steiner asked. After looking in the direction of the six Colonial soldiers close by, all of them wearing the look of a rookie in combat, Dalton turned back to Steiner. \"Does it really matter how many?\" he replied.\n\n\"Listen up,\" Dalton said, walking to the group slowly, swagger filling his officer's strut with Steiner by his side. \"Any of you rooks seen combat other than boot camp?\" he asked. Two soldiers raised their hands as the other four stood there awaiting his orders.\n\n\"Well then, you two are with us,\" Dalton said. \"You two as well.\" he added pointing to the largest of the two remaining. \"If nothing else, these big sumbitches might slow the Hunters down long enough.\" Dalton said to Steiner as he quietly laughed.\n\n\"Soldier, you stay here and cover the sniper's six. Anything comes up here, you do your best to dust its jimmy off. Got it?\" Dalton asked.\n\n\"Ye...Yes sir!\" the new recruit answered, fumbling with his weapon as he crawled to the sniper's position.\n\n\"What about me sir?\" the last remaining soldier asked impatiently. \"Can you run fast?\" Dalton asked.\n\n\"I...I guess so, why sir?\" the soldier asked.\n\n\"When we get down there, I want you to break off from the group and scavenge for ammunition and alcohol.\" Dalton replied.\n\n\"Alcohol sir?\" the soldier asked.\n\n\"Yea, for the love of God, especially alcohol soldier. Peel them damn eyelids back and find what you can. Grab it, and elbow to asshole your way back to the group. Got it?\" Dalton responded.\n\n\"Yes sir.\" the soldier replied, confused more than just a little bit.\n\n\"Do you think now is really the time to be worrying about whiskey?\" Steiner asked as the men began slowly wondering into the direction of the town under the cover of night.\n\n\"Perfect time. He looks like could outrun me, and besides, we get into it with the Hunters, you really think that scrawny sumbitch is gonna make a difference?\" Dalton replied.\n\n\"Hadn't thought of it like that.\" Steiner replied, a wide grin plastered onto his face.\n\n\"Of course not, that's why they left me in charge.\" Dalton responded proudly.\n\n\"Alright, best ease our way out slowly and hope we don't hit any Hunter patrols.\" Adam said to the group as they slowly moved a few feet from the building in which they had been hiding, using any form of cover they could.\n\n\"Shh.\" Roman said quietly as he clinched a fist to the others, moments later gunshots began to ring out as if they were in the center of an independence parade.\n\n\"Back inside, quick!\" Adam yelled as the entire group sprinted for the entrance once more. \"Are they fighting one another?\" Sasha asked Adam.\n\n\"Not sure.\" he replied, the door cracked as he watched and listened closely.\n\n\"Eat it you whiskey hoarding bastard!\" Dalton yelled loudly.\n\n\"Or, maybe I am,\" Adam said. \"They are with us, actually,\" he added, grinning to the group with embarrassment. \"Let me see your light.\" Adam asked of Helon, who quickly handed him a small flashlight. Adam held it in the crack of the door for a few moments, blinking it repeatedly to gain Dalton's attention, eventually successful, though the mangy looking dog with Adam had caught on much faster.\n\n\"Open it!\" Adam said firmly as Zavious pulled the door open instantly, the rest of the group with weapons drawn as Dalton, Steiner the Colonial sniper and the fleet footed scrawny solider quickly rushed in, bottles clanging inside of a sack that was draped across the shoulder of the weakling greenhorn.\n\n\"It's good to see a friendly face.\" Adam said as he smiled.\n\n\"We'll you are about to see a lot more faces, although I don't suspect they are gonna be looking too friendly.\" Dalton replied as he tried to refill his lungs with the warm Glimmerian air.\n\n\"I thought I told you to keep your ass up on the dune?\" he asked of the sniper, still struggling with exhausted lungs.\n\n\"We were flanked sir, no way I could have held them off on my own.\" the sniper replied.\n\n\"Sir?\" Adam asked as the Benzans looked on, trying to make what they could of both Dalton and the oversize Husk warrior standing at his side.\n\n\"Yea, they put him in charge.\" Steiner said sarcastically with his heavy Husk accent.\n\n\"Seriously?\" Adam asked amazed.\n\n\"Hell yea seriously, they trust me to make the smart calls out here.\" Dalton replied.\n\n\"Yea, like sending your boy into a saloon full of Hunters for whiskey.\" Steiner added.\n\n\"How the hell did I know them certifiably dead son of a bitches would be packed in there drinking the town dry?\" Dalton angrily asked.\n\n\"Alright,\" Sasha interjected. \"We need to worry about how we are supposed to get out of town without getting killed.\" she added.\n\n\"We ain't leaving.\" Dalton replied.\n\n\"We're not?\" Adam asked, still amazed that Dalton had been placed in charge of actual human lives.\n\n\"Nope. Sarah said to hold up here. Said they are forming a large ground team to come in and get us,\" Dalton replied as he bent down with fresh air in his lungs. \"Where did ya'll pick up this cute pup at?\" he asked as everyone in the room looked at him, wondering if he had overlooked the chronic mange of the abused dog. Then it began to make sense to a few of them, viewing both Dalton and the dog as both being a little bit on the trashy side.\n\n\"Poor sumbitch looks like he 'aint ate or drank in weeks. Private, what were you able to grab before they started shooting?\" Dalton asked as the Benzans began picking their spots inside of the building, digging in for any fighting to come.\n\n\"Not much sir, I didn't have time to...\" the Colonial private replied, Dalton snatching a bottle of whiskey from the sack and cutting his explanation off in the process.\n\n\"Double brown whiskey. This shit is so strong it would cure the blind, good job private.\" Dalton said, the young soldier standing a bit taller and proudly answering. \"Yes sir!\"\n\n\"Be a damn good name for you mutt, Whiskey it is.\" Dalton said as he poured a large amount onto the floor, the dog lapping it up before it had time to spread.\n\n\"Unbelievable.\" Sasha said, pausing when she heard Dalton bless the animal with the name Whiskey.\n\n\"Wait until he has a few drinks in him. May want to keep that gun close by your side.\" Adam replied, chuckling a bit.\n\nThe layout was simple. Dalton, Whiskey and the Colonial sniper were in the loft. Meanwhile, Helon, Zavious and Oz covered one side of the room, the three of them behind what looked to be an old wooden sales desk. Adam, Sasha and the scrawny Colonial soldier remained at the rear of the room behind empty crates while Roman and Steiner covered the door.\n\n\"I'd hate to be the unlucky one who barges in first.\" Sasha said, referring to the giant like size of Steiner, who held a short barrel shotgun, and the blade skills of the partly mechanical Roman Raines.\n\n\"Yea, or the first half dozen even. Those two can fight like no other two I have ever seen.\" Adam replied softly.\n\n\"So. Not to pry or anything, but this Sarah that Dalton spoke of...\" Sasha asked as Michaels nodded before she could complete her sentence.\n\n\"Yea, that's her. When we met I was on the run, no real direction in my life. She pulled me into this war I guess, but I couldn't have walked away from her. Love her too much, I guess it was a package deal.\" he replied.\n\n\"Well I hope this isn't over the line or anything, but I just think she is a really lucky woman. I just never understood how you could love someone and try to change them at the same time,\" Sasha said. \"Of course, that's probably just an issue with me. One of the many issues.\" she added, laughing softly. Adam thought deeply about what she had just said. Sarah had changed him, be it for better or worse. Once upon a time he had no direction, lived by the seat of his pants and enjoyed true freedom. And here he was. Trapped inside of a war he truly wanted no part of, decked out in an officer's uniform and under the heel and command of a politician somewhere. All because of his love for Sarah. When, to his knowledge, she hadn't changed anything for him. \"No, you make a good point. And thanks for the compliment.\" he finally replied as he returned her smile.\n\n\"Can I ask you a question Adam?\" Sasha asked in a heartfelt tone.\n\n\"Sure you can.\" he responded.\n\n\"What do you really think will become of myself and my people if and when we do make it back? Do you think the same Colonials who blew our warship out of orbit will let us just walk away when this is over, free and clear?\" Sasha asked.\n\n\"Sure I do,\" Adam responded, although he hadn't even thought about it up until that moment and knew in his heart that she was probably right. \"I give you my word right now, that no matter what I choose to do when this is over, you will walk away free and clear.\" he said.\n\n\"Can you really make such a commitment though Lieutenant?\" Sasha asked.\n\n\"I'm not making that commitment as a Lieutenant, I am making it as Adam Michaels, and my word is good.\" he replied.\n\n\"Thank you. I believe you.\" she responded with a smile. \"The world paints our people to be criminals, but I assure you, we are not.\" she quietly added.\n\nAdam sat quietly, filled with inner emotion as his grief of the loss of Kelly battled the emotional distress he saw on Sasha's face. He had always been a great judge of character, and he knew in his heart that her words were saturated in nothing but the truth.\n\n\"Sasha, you have my word. When we make it back, you will walk out of there a free woman.\" Adam finally replied as he stood to his feet.\n\nOnce the mightiest city on Glimmeria, the Colonial spectacle of a military base looked more like an urban wasteland. Only seconds separated explosions throughout the city as rockets, grenades and anti-aircraft artillery had been going since the Legion had landed on solid ground. Their far superior numbers on the ground had managed to take out all but one of the Mack stations, the last one heavily fortified by the Colonials who still feared assisting armies coming to the aid of their counterparts. What seemed like such a great accomplishment for the Colonials as the Legion base ships in orbit were strategically destroyed by Benzan gunfire, quickly became only a side note. The Legion had already sent its entire force, intending to dig in and fight a war of attrition. The base ships were nothing more than empty warehouses in the sky operated by the smallest of skeleton crews when they exploded. That said, a majority of the Legion's supplies had been destroyed with them, meaning victory would have to come swift if it were to come at all.\n\n\"Sir, Colonial Command remains under heavy guard, and we have the streets locked down tight.\" Lassiter reported to Edwards as the Colonial leaders scrambled to update their war maps, communications coming in from all corners of the city.\n\n\"Good. We have reason to believe that the Legion is staging a very large assault force with plans to hit this area. We are trying to pull back the pockets of soldiers throughout the city and organize a defensive perimeter,\" Commander Edwards replied. \"Get your men resupplied and do your best to assign soldiers to new platoons as they make their way to Command.\" Edwards added.\n\n\"Yes sir!\" Lassiter replied swiftly as he left the vault and returned to the streets outside.\n\n\"Any word on Adam and his crew?\" Commander Edwards asked, slowly approaching Sarah Blaine, who had been organizing all of the communications for the Colonials.\n\n\"No sir. I'm sure they are alright, probably just inside of Valencia and unable to send out a transmit without having their position compromised.\" Sarah answered, trying to convince herself as she spoke the words.\n\n\"I see.\" the Commander replied.\n\n\"Sarah, I want to make sure that you are alright with what has to be done when they do make it back,\" he added. \"Leaving Glimmeria without authorization, conspiring with wanted criminals and direct insubordination.\" the Commander added as Sarah sat in her chair, numb to the fact that if her soon to be husband could beat the odds and make it back alive, she would have to stand by and watch him be discharged while his crew was arrested.\n\n\"Yes sir. I know it has to be done.\" she reluctantly replied.\n\n\"Good Sarah. You are the ideal soldier, putting your Colonial duty above personal feelings. Your father would be so proud of you. We have already been contacted by the Gali government about Roman, seems he is very high on their list of wanted fugitives. Turning him over to them would go a long way toward helping our cause.\" Edwards remarked as he placed his hand on her shoulder for a moment before walking to the rest of the stations in the Vault for updates.\n\n\"Yep, she's locked down tighter than a tick's ass.\" Dalton remarked as he looked through the town streets, the morning sun illuminating the scope of the sniper rifle.\n\n\"Nice.\" Sasha remarked, already disgusted with Dalton's demeanor.\n\n\"He's right, their defenses are solid.\" the Colonial sniper added. Adam took the lenses and looked for himself, noticing the pockets of heavily armed Hunters patrolling the streets.\n\n\"I have an idea.\" Adam said, grabbing Steiner's field pack full of supplies and sifting through until he found a blue flare.\n\n\"Sure you can make the shot?\" the Colonial sniper said after having his rifle commandeered by Dalton, who laid prone with his eye pressed to the scope.\n\n\"Don't worry son, at this distance I could split the hair on a tick's a...\" Dalton responded before being interrupted.\n\n\"What is it with you and your fixation on the ass end of a tick?\" Sasha demanded to know.\n\n\"Adam, unless you got a stout drink to offer or my first paycheck in hand, I don't gotta put up with this shit!\" Dalton responded angrily.\n\n\"Just shoot the damn flare Dalton, everybody else be ready to get down to the back of the stables if they take the bait.\" Adam said as Dalton fired the weapon, its muzzle flash hidden within the silencing tube that was attached to the barrel.\n\nWithin a second, blue smoke began filling the landscape on the opposite side of Valencia. Only moments later, several Hunters began yelling loudly as they braced for a Colonial assault. The few Fang units among them made their way to the far end of the main street, the barbarically large creatures facing the smoke and ready to let their menacing teeth slaughter any approaching Colonials.\n\nA majority of the Hunters dashed to the smoke, weapons at the ready, which was cue enough for Adam and his crew to sprint to the back of the large, wooden stables. A much larger and more strategically placed building, it had already been cleared by the Hunters which made the probability of them checking it again slim to none.\n\nRoman sunk his blade into the thick wood only feet from the ground, pulling the steel slowly across in a straight line, his bulging arms forcefully splitting wood slowly as he followed it with a solid kick that broke open a crawl space about three feet high. One by one they crawled on their stomachs across the desert floor and into the long abandoned building, the last in was Adam, who did the best job he could of putting the fallen boards back into place to appear untouched.\n\n\"Dalton, you're with our sniper up in the loft. Keep eyes on everything and let us know of any movement. Roman, you and Steiner set up near the stable doors. Good chance when they figure out the smoke was a decoy, they'll do a half-ass sweep of the town. They come in here, take care of 'em quick.\" Adam said as everyone took position throughout the large interior and Dalton bitched under his breath about loss of command. Michaels and Sasha dug in behind a stack of musty hay near the newly made entrance at the rear of the building.\n\n\"Plan is, we'll hang here to back up Roman and Steiner if need be, try to get our com system linked up in the meantime.\" Adam said as he pulled the long range com system from a small suitcase in Steiner's field pack, which was a small metal square with several knobs, and hooked it to a table top satellite dish, which he sat on the empty crate in an effort to pull signal in.\n\nThe Hunter force inside of Valencia prepared to do a sweep of the town, just as Adam had predicted, when two Swordfish screamed overhead, dropping hundreds of rounds onto the town while taking return fire from assault rifles. It quickly took the attention off of them and onto the skies as the soon to be sweep was halted. \"Well, that was a lucky break.\" Sasha said.\n\n\"It wasn't luck. My guess is the Colonials were doing a quick flyby to scout the town so they could put together the right group of soldiers to come get us.\" Adam replied.\n\n\"What's coming across the com unit?\" Dalton asked softly, laying on the loft with his head hung over the side smiling, the mange ridden whiskey laying at his side.\n\n\"We can't send anything, it'll give our position away.\" Adam said. \"But from the sound of things, we're getting beat up on the ground.\" he added. \"What about Colonial Command?\" Dalton asked, thinking of the post he held until being sent to save Adam's neck.\n\n\"It's still there, but from what I can make out there is a large Legion force trying to take the ground, sounds like that entire area has become the front.\" Adam replied.\n\nThe sniper whistled quietly, getting Dalton's instant attention. Moments later, Dalton turned back to everyone on the bottom floor of the stable and began giving hand motions to let them know what was outside of the door.\n\n\"Two Hunters coming.\" Sasha relayed to Adam, who had put a rifle to his shoulder and the sights on the door. \"Armed with standard rifles.\" she added as she continued to watch the loft. \"And I have no idea what that means?\" she said, confused of the signal. Adam broke his attention from the door momentarily to glance at Dalton, who was turning his hand upside down next to his mouth.\n\n\"That means that they have liquor.\" Adam said, turning back to the door as Sasha stood there nothing short of speechless.\n\nBoth Hunters pushed the stable doors open slowly, the wood creaking as light from the desert heat permeated the entire building and illuminated thousands of dust particles that had yet to settle. Their thunderous laughter was cut short as Roman moved with exceptional speed, the titanium plating on his elbow crushing the lower jaw of one of the demons. Before either man had a chance to respond, Steiner grabbed the second man and wasted no time snapping his neck under the pressure of his sheer strength as Roman dug the serrated edge of his blade into the intestines of the first.\n\nAs if the stables were engulfed in flames, Dalton scaled the wooden ladder and reached the ground with unmatched haste. It took only moments of searching the bodies before he found a small bottle of rock whiskey.\n\n\"Fucking cheap ass bastards. Could have at least had something decent on 'em.\" Dalton said in disgust as Roman and Steiner dragged the bodies out of sight and quickly closed the front door. \"Oh well, with the day I've had, it'll do.\" he added, climbing the ladder much slower than he had descended, almost as if he had become an instant cripple.\n\n\"Adam.\" Roman said, tossing back a small radio he had found on one of the bodies.\n\n\"Some crew you have here.\" Sasha commented.\n\n\"I know what you must be thinking, but they're loyal and get the job done when it's on the line,\" Adam replied. \"Also, I'm sorry for ever misjudging you.\" he added.\n\n\"I'm not sure what you mean?\" Sasha replied curiously. \"When we arranged to meet with the Benzans, honestly, I did think you were criminals. It's all I had ever heard, ya know? I had always been told that the Benzans were cold-blooded killers, so I guess I had just started to believe it myself.\" he answered.\n\nSasha sat there for a minute, full of emotion and empty on words. Thinking back to her family being slaughtered only a few years back when she was a teen, and how Cyrus had taken her in, trained her and most importantly, been like a father to her. \"Don't worry Adam, I understand.\" Sasha finally replied.\n\n\"No Sasha, I don't think you do,\" Adam said as she smiled slightly.\n\n\"What you said back there made a lot of sense to me. We are more like one another than you will ever know.\" he added.\n\n\"I used to fight the system too. I can remember the deep space runs, just looking across the stars and truly feeling freedom.\" Adam said.\n\n\"I know that feeling. Almost like you are witnessing something magical about the stars that you weren't intended to.\" Sasha replied.\n\n\"Exactly.\" Adam responded. \"Does she share your sense of freedom?\" Sasha asked.\n\n\"No, unfortunately not. I guess the time comes in everyone's life when they have to make the decision though, who they want to be and what they want to live for. I chose love.\" Adam replied.\n\n\"You guys want some of this to take the edge off?\" Dalton said, his head falling in front of them as he hung down from the loft with the rock whiskey bottle in his hand.\n\n\"No thanks.\" Adam replied as he began to remember what life was like only a year ago. His original crew that included Kato and Luck, both victims of the war that Adam had drug them into.\n\n\"Sure, why not.\" Sasha replied, surprising both Adam and Dalton as she turned the bottle up and took a healthy swig.\n\n\"Alright, alright...save a little for me, damn.\" Dalton said nearly prying the bottle from her hands in order to salvage what he could.\n\nAdam hadn't truly seen the kind of smile that Dalton currently had on his face since before they had gotten drug into this damn war. Their lives had consisted of card games, saloon brawls, black market jobs and a type of family atmosphere aboard the Gunship that was hard to explain, unless you were a Benzan.\n\nWhether it was the effects of whiskey or the traumatic events of the day, she spent the next hour explaining to Adam what Benzan life was like. He found himself relating it to the life he used to live when he was free. Before he was Lieutenant Michaels, he was Captain of the Gunship. It didn't have a Colonial logo painted on the side of its thick plated steel, in fact it didn't have anything more than a film of rust coating. He started to realize that even though he was indeed in love with Sarah Blaine, his life had been changed so dramatically for the worse.\n\nAs night began to fall on the town of Valencia, Adam started to do something he had not done in a while. Search his soul. Maybe convince himself that he knew the definition of true happiness, and if by some miracle they made it off of Glimmeria alive, what changes he would have to make in order to become truly happy again. When it came to love, he had followed his heart. However, when it came to being a free and happy man, he realized that he had done everything but follow his heart.\n\nSure, his heart belonged to Sarah, it had since they very first moment he had laid eyes on her. Still, he thought a lot about Sasha as he lay there during the night. Not just because she had a pure beauty about her, but because he envied the life that she had chosen to live. Missing it very much. Living for the moment was the only real definition of happiness in his opinion, and Sasha lay there doing just that, sleeping like an angel and taking life one day at a time.\n\nSuddenly, explosions rang out all around them as the Hunter controlled town had come under siege by Colonial forces. Dalton and Roman had already been wide awake, pulling watch as everyone else quickly jumped to their feet and tried to assess the situation. Realizing it was the Colonials by the sound of their weapons, Adam immediately got on the com unit and sent a communication to let them know they were hold up in the stables. Moments later, the group could hear the explosions backing off of their area a bit, the Colonial cannons redirecting their fire in order to avoid hitting them.\n\n\"I got another idea.\" Adam said, pulling Roman and Steiner to the side. A few minutes later, Adam and Sasha had taken guard at the door as both Roman and Steiner had put on the uniforms they had removed from the dead Hunters.\n\n\"This looks ridiculous!\" the hulking Steiner said, his pants legs a good foot higher than the tops of his boots and so tight you could make out the patterns of Husk hair underneath.\n\n\"A couple more drinks and you might be looking mighty fine to me,\" Dalton said jokingly as everyone took a moment to giggle about the underhanded joke. \"Or not.\" Dalton added in a serious tone as Steiner sent him a blood curdling scowl.\n\n\"The quicker we can get out there and kill some of these damn flesh eaters, the sooner the Colonials can come fetch us.\" Adam said.\n\n\"Shouldn't take us too long.\" Roman replied as he and his horribly outfitted comrade equipped themselves with Hunter rifles and waited at the door for an all clear from the loft before cracking it open and sprinting across the street.\n\n\"You two keep your eyes wide up there, any Hunters get after them, you do anything you can to clear a path.\" Adam said shouting up to the loft as the roar of explosions in the distance intensified.\n\n\"You got it Adam.\" Dalton replied, pulling the slide loader of the sniper rifle back and putting his eye to the scope.\n\n\"Hey that's my gun!\" the Colonial sniper contested loudly. \"Steiner is a big sumbitch...you 'aint. I'll listen to him, but your ass I'll whip.\" Dalton replied snidely, taking no chances in defending his two friends.\n\nIt had only been a handful of minutes when Dalton spotted a small group of Goliath soldiers at the edge of town making minced meat out of the heavily out gunned Hunter soldiers. Several tense moments later as the three mechanical masterpieces began to enter Valiance, a whole squadron of Colonial soldiers behind them, two Fangs entered the street from a Hunter choke point. They were much more intimidating than even the model two Goliath. Sure, the alloy steel and chain guns helped the mechanical backbone of the Colonial army mow through normal foes, but the Fang units were anything but.\n\nSeveral feet taller than even Steiner, they looked more like a monster than a man, the face of of demon with two arm sized fangs hanging from their top jawbone.\n\nThey spent a moment exchanging gunfire before realizing that tactic was useless, both the Goliaths and Fangs had thick enough plating, whether alloy or skin, to deflect any small arms ammunition. Instead, they began to pick up the pace as they geared up for a fist to fist showdown with one another. Passing a large metal dumpster, one of the Fangs was stunned as Roman quickly stood up and swung the stock of his rifle, smashing it to bits across the face of the biological killing machine. The second Fang stopped suddenly, turning into Roman's direction, when the three Colonial Goliaths began letting a blanket of lead unleash onto it, if nothing else to blind and confuse it. Immediately, Steiner jumped onto the face and shoulders of the much more powerful foe, doing his best to hold on long enough to put the barrel of his weapon into the neck area of the oversize flesh eating monster.\n\nHe emptied his entire clip as the Hunter's prized unit began to die slowly of internal bleeding, standing in the same spot as it screeched in agony. Meanwhile, Roman and the second Fang were engulfed in an all out brawl of clashing steel and determined flesh when Roman saw his friend fall.\n\nA Hunter sniper hit Steiner in the back with a high caliber weapon as he stood there to place a full clip into his battle rifle. Right away, Roman sensed the severity of the wound as meat and vital tissue sprung from Steiner's chest, the round making its way completely through the hulking beast as he fell to both knees. A couple more shots followed closely after, knocking Steiner face first into the sands of Glimmeria, dead.\n\nFury took control of Roman as he seized the second Fang and began tearing flesh from any seam of its skeletal frame that he could possibly put his hands on. Two of the Colonial Goliath units had fallen at the end of a hailstorm of Hunter gunfire when Roman found himself beside the remaining Goliath and a small group of Colonial soldiers who had thrown down arms in surrender. There were simply too many Hunters.\n\nAdam had seen the loyal Husk fall dead in the street and made his way up to the loft with extreme haste, grabbing an emotional Dalton and pulling the rifle from his hands before he could get a true shot.\n\n\"Dalton, there are too many!\" Adam said sternly.\n\n\"I don't give a shit anymore Adam! I've had enough of watching my fucking friends die.\" Dalton shouted back as he made his way down from the loft to the front door of the stable.\n\n\"Step back. I don't want to shoot you, but if you leave now you will sign the death warrant of every one of us.\" Sasha said as she aimed her silencer equipped machine gun pistol at the man who had nearly reached the end.\n\n\"Bitch, you better move. None of us would even be in this position if your kind could stop killing for five fucking minutes. Now get out of my way, otherwise you 'aint gonna like how this ends.\" Dalton said angrily.\n\nSasha pulled the load clasp back, assuring him that the weapon was ready to discharge at the slightest bit of pressure.\n\n\"Dalton! You need to calm down before you do something you will regret for the rest of your life.\" Adam said, grabbing the left arm of his pissed off friend.\n\n\"Nope. I aint' gonna regret this Adam.\" he said as he planted his fist firmly into the face of Michaels, knocking him back into the thick wooden wall.\n\n\"Down on your knees, hands behind your head soldier!\" Sasha demanded of Dalton as the Colonial sniper continued watching the Hunters round up any survivors outside, which included a badly outnumbered Roman Raines.\n\n\"No Sasha. If he wants a damn fight, he's gonna get one.\" Adam said boldly as he stood back to his feet and threw the large, Colonial issued pistol to the ground.\n\n\"Sounds good to me, I'm tired of watching you march my friends to the grave over a war that some whore dragged you int...\" Dalton replied as Adam landed a left jab and a harrowing right hook onto the man, stumbling him back as he remained standing.\n\nObviously reeling just a bit, Dalton placed his hand across his beard for a minute, surprised if nothing else that Adam struck him back.\n\n\"Well, you got balls after all.\" Dalton managed to spit out before Adam landed three more punches, followed by a crisp right elbow which he had angled skyward, knocking his belligerent friend onto his back. Adam immediately got on top of him and placed his forearm across the throat of Dalton, choking the life from his body.\n\n\"Yes, I know that Luck, Kato, Kelly and now Steiner have died because of decisions I'VE made. I have to live with it every fucking day, and I'll be damned if another one of my crew's names is going to be on that list. So either you help me free Roman, or you carry your ass back to Colonial Command and wait for me to bring him back,\" Adam said with grave calm before letting go of his choke hold and picking his pistol up, pointing it directly at the face of a broken down Dalton. \"Now what's it gonna be?\" Adam asked.\n\n\"Just give me a fucking gun and let's find a way outta here already.\" Dalton said somberly as if to admit defeat.\n\n\"You aren't seriously gonna give this guy a gun are you?\" the Colonial sniper said, shocked at the notion.\n\n\"This isn't the first time we've had a physical disagreement, I'm just glad I won this round,\" Adam said, smiling a bit as he extended his hand to help Dalton to his feet. \"We go way back. If I step out of line, he puts me back in it, and vice versa.\" Adam added.\n\n\"That elbow got me, you dirty fightin' fuck.\" Dalton replied, using the arm of Michaels to help himself up.\n\n\"You sure you're not Benzan? You act so much like everyone that I know, I'm not even homesick anymore.\" Sasha said to Adam.\n\n\"Our rescue party got their asses kicked, and as thin as the Colonial forces are spread, I seriously doubt they will be sending another,\" Adam said to the group. \"Means we are gonna have to go it on our own.\" he added.\n\n\"The Hunters took the survivors into a building up the street, probably not more than two hundred yards from here. Roman was with the group they took.\" the Colonial sniper said.\n\n\"I've never known Hunters to take prisoners?\" Sasha said in a puzzled tone of voice.\n\n\"They don't. Plan on getting as much information as they can from them and putting a bullet in their head would be my guess, which means we gotta work fast.\" Adam replied. They would have the perfect opportunity soon, as the sun had started to set, painting faded blue and shimmering light red across the sky. The town would soon be engulfed with complete darkness.\n\n\"Sir, reports are indicating that our lines are being pushed back.\" a Colonial officer said as he stood before Commander Edwards.\n\n\"Give a general order to fall back here to Colonial Command. If the Legion tries to take this complex, they will pay dearly in lives.\" Edwards replied.\n\n\"Yes sir!\" the officer replied as he turned to execute the order. \"What is the status of Adam's rescue?\" the Commander asked as he turned to a very distraught Sarah Blaine. With her eyes full of tears, she simply nodded no.\n\n\"Once our remaining forces fall back and we regroup, I will try to put together a second tactical team to go get them, don't worry.\" Edwards replied to calm her before walking to the other stations inside of the Vault.\n\n\"This is a general order of retreat. Repeat, fall back. Remaining Colonial forces are instructed to fall back to Colonial Command and await further orders.\" Sarah said calmly into her com unit, holding the tears back long enough to broadcast with a hope filled voice.\n\n\"In the event that the Legion would begin taking control of this complex, we have strategically placed nuclear warheads on every level of the building. Our engineers have been able to bring Colonial Star Twenty Two to flight ready status. Most of the weapons and defense systems aboard are not functional as of yet, however as a last resort, it will evacuate us and the small group of Colonial soldiers off of Glimmeria to the small moon of Thisia, where we have been assured safe harbor by the Thisian government.\" Commander Edwards reluctantly announced to everyone inside of the Vault.\n\n\"What of the rest of the Colonials?\" one of the officers asked, bringing a harsh expression to the Commander's face.\n\n\"There is simply very limited room on the Colonial Star. The remainder of our forces will be left here to stave off the Legion until we can come back for them.\" Edwards replied. He expected to have to explain the plan in great detail to the most important members of the Colonial government. However, the conversation was cut short as the soldiers outside of the Vault under the command of Lassiter, began firing their weapons in the direction of the building's main entrance. The Legion had made it to their front door and all hell had broken loose.\n\n\"Lassiter! You must hold them until the Colonial Star arrives to evacuate us!\" Commander Edwards proclaimed through the com unit in the Vault as they eyed one another through the shatterproof glass.\n\n\"Will do sir. Highly recommend pulling any available air units to the area to assist.\" Lassiter replied, his voice nearly impossible to make out over the com unit as gunshots rang out all around him.\n\n\"Negative. Remaining air units are escorting Colonial Star in behind command, you must hold the Legion troops back at all costs.\" Edwards pleaded.\n\n\"Understood sir,\" Lassiter replied into the com unit before throwing it to the ground. \"Dig in and earn your independence!\" he yelled loudly to the troops that barricaded the halls of the building, mostly flesh with a few mechanical Goliaths assisting.\n\n\"Sarah. Begin the process of packing up all vital equipment and Colonial documentation. You, remain with Sarah at all times from this point forward.\" Edwards demanded of one of the Vault's very few and heavily experienced soldiers. As Sarah began to explain to Commander Edwards that an armed escort wasn't necessary, she saw several Colonial soldiers gunned down on the other side of the glass as they were in the thickest of firefights, which prompted her to stay quiet and do as he asked.\n\nRoman sat in a small room with six Colonial soldiers, all tied to chairs with thick rope and waiting to join the last two soldiers who were taken out for questioning, along with the remaining Goliath, to the land of the dead. As the door opened up slowly, the same Hunter who had come for the others stood in deep thought, trying to decide who they would torture for information next as two very large and well armed guards stood on the inside of the door. He was dressed in battle tested black leather armor from head to toe with a slight trimming of silver, much of it saturated with dried blood and a dusting of sand.\n\n\"Him.\" he said, pointing to Roman as the two Vampiric guards walked slowly in the direction of the doomed soldier. \"Make it quick, I grow tired of dealing with these pathetic humans.\" he added as Roman suddenly noticed a strange look about his face. As his eyes drifted down from the Hunter's face to his large torso, he could see the tip of a large blade sticking out of his stomach, a waterfall of blood crashing onto the floor. Dalton stood behind him, the Hunter's body falling to the ground shortly after as Dalton reclaimed the arm length machete in Roman like fashion, throwing it end over end at one of the two remaining demons, and in non-Roman like fashion, hitting the Hunter with the handle of the knife.\n\n\"Oh shit.\" Dalton said, stunned as the knife hit the floor making a small clanking sound. Before the flesh eating guards could get the drop on him however, Sasha stepped in and fired her palm sized machine gun pistol, droplets of red mortality finding a home on the wall behind them, quickly bringing a stinging demise to one of the Hunter guards.\n\nThe remaining demon quickly aimed his rifle in Sasha's direction, firing a shot that ricocheted off of the floor, a chair smashing into the side of its head and busting into dozens of splintered pieces. Roman had not only thrown the object in which he had been tied to only moments ago, but had lunged on top of the reeling Hunter, using the thick rope which bound his hands to choke much of the life from the wounded beast.\n\nAfter collecting the large blade, Dalton caught sight of two more of the heavily armed beasts as they dashed into the small dwelling. Once again, he threw it end over end in the direction of the door with the same results as it clanged against the wall and fell pitifully to the ground.\n\n\"Fuck it.\" Dalton said, swiftly pulling his pistol and pecking away much of the flesh of the first demon inside, the screaming Hunter falling against the wall and clutching its face with his claw ridden hands. The second Hunter in was immediately dismissed by a group of shots from one of the Hunter rifles which Roman now held.\n\n\"Dead or alive, bring them to me!\" Anwick yelled, his pale white skin flush with pink tint as revenge overtook him. Several more Hunters, nearly a dozen, quickly ran out of the saloon up the street, cautiously panning their large rifles around as they tried to assess the recent screams. The horrified townsfolk continued glancing from their windows, most having never seen a Hunter in their lifetime, much less watching one fall.\n\n\"Cut their bonds and let's get the fuck out of here!\" Dalton yelled in Roman's direction, having given up hope on ever obtaining any type of skill with a blade. It only took seconds for the former Gali commando to slice his thick bonds, as well as those of the Colonial prisoners as they collected what weapons lay on the floor and hastily proceeded outside.\n\n\"There!\" one of the Hunters exclaimed, pointing to the group of escaping Colonials as they fled the building and hopefully captivity. The Hunters, however, were much quicker than normal men, steadily closing the gap between the fleeing prisoners and those who wished them dead. Anwick remained back at the edge of town as he watched his minions of murder pursue a group of people who he wanted to watch die, slowly and painfully. He had killed thousands of men, women and children during his time, but could not remember a single one that he wanted to perish more than Adam Michaels.\n\nHe would never get that chance. As he watched his men fade into the far reaches of the distance, Adam Michaels stepped from the shadows of the building which marked the beginning of town.\n\n\"Reap it you son of a bitch!\" he said as he stepped from behind two wooden barrels, pointing his standard issue Colonial combat pistol into point blank range of Anwick and firing all six shots in succession. Anwick's inhuman speed was enough to dodge a majority of the shots, although two of them hit him deep in the chest, greatly reducing the flesh demon.\n\nHaving expended his ammunition, Adam struck Anwick across the face with the heel of the pistol before throwing it to the ground. The two wounds, coupled with gunshots suffered earlier had reduced Anwick to nothing more than a normal person. Adam threw several punches into the face of the kneeling Hunter, who eventually threw a drastic backhand of his own, putting Adam onto his back.\n\n\"I will snap your neck and watch you die a slow death!\" Anwick screamed, standing to his feet only to be met by Adam's lust for the revenge of his fallen friends. A devastating uppercut, followed by a thrashing right elbow to the face of Anwick send the monster back to the ground abruptly.\n\nAdam's quest for retribution was cut short however, as he approached Anwick heavy footed and was quickly grabbed by the throat, the Hunter's spiny white fingers wrapping around quickly and their accompanying claws digging in. With maddening rage setting in, Anwick used the grip to pick Adam up off of the ground a couple of feet and hurl him into the wooden fence in front of the nearby building, smashing several feet of it into pieces.\n\n\"You have cost me time, extreme amounts of stress and many faithful lives,\" Anwick yelled loudly, starting to approach Adam before stopping abruptly. \"And now here you are,\" Anwick said, looking at Adam as he lay on splintered wood and scorching sand, broken by the injuries newly gained. \"The worst part of it is if your loyalties would have remained with me, I have no doubt that we could have worked well together. As hard as it was for me to end you, says a lot about the high level of smuggler you must have been.\" Anwick added.\n\n\"You have no idea.\" Adam said, his face bloodied as he sat there prepared to die.\n\n\"Nor will I.\" Anwick replied. \"Does such an outstanding smuggler have any last words?\" he added, slowly approaching Adam for the kill.\n\n\"Yea. I sure could use some Whiskey right about now.\" Adam replied, Anwick stopping for a moment to laugh at the beaten man.\n\n\"I am afraid the saloon is much to far to fill that request.\" Anwick answered.\n\n\"No. I said I sure could use some WHISKEY right about now.\" Adam replied, placing a great deal of emphasis on the word Whiskey as the eye sore of a mutt sprung from one of the barrels and barked wildly.\n\n\"What the?\" Anwick said in a low voice, surprised by the invasion as he turned to the sound of Whiskey's barking. Adam quickly pulled a compact pistol from behind his back, having been tucked away behind his belt throughout the entire fight. Moments later, Anwick lay dead, his blood drenching the sand as Adam continued holding the pistol which smoked from all six shots fired.\n\nThe Colonial force had nearly been caught, when they stopped suddenly and spun to begin firing on the group of Hunters. In unison, sand covered tarps flew into the air as Helon, Zavious and Oz stood entrenched in them, firing heavy weaponry at the suddenly matched foes.\n\nThe Hunters were outnumbered by a group filled with equally tactful killers, several who were pissed off to say the least. The next few minutes were engulfed in a firefight from the worst imaginable hell as lead, blades, blood and cursing all flew freely into the air and onto the sands of that small patch of Glimmerian soil. Adam and Whiskey had sprinted to help his friends after confirming the death of Aniwick, arriving when the fight had freshly come to a close.\n\nOne of the Colonial soldiers was alive, but in bad shape, the rest, including the sniper, lay slain on the sand. Helon and Oz had both fallen, each with grimacing claw marks all over them and multiple gunshots to the chest. Sasha was nearly unscathed, but was grieving loudly at the death of the two Benzans as Zavious lived, effectively making him the highest ranking member of the Benzan family. Roman stood tall, covered in blood, most of which belonged to the fallen Hunters as he did his best to survey the damage around him. Dalton lay with the dead Hunters, his shirt ripped to pieces from both claw and blade as he had tried unsuccessfully to fend off two of the near immortals himself.\n\nIt wasn't until Whiskey went to him, licking his face and leaving tongue trails of cleanliness that Dalton started slowly moving his head around, at first not understanding if he were dead or alive. A few more times of raw tongue dragging unkempt whiskers was all it took to remind him that he was still alive, and in need of a damn drink.\n\n\"Please tell me that's all of the bastards. If not, shoot my ass now and leave my belongings to Whiskey.\" he said, painfully pulling himself to a sitting position as the surviving chuckled a bit.\n\n\"That's all of them. The remaining Hunters and Fangs left a short time ago to aid the Legion's siege on Colonial Command.\" Adam said.\n\n\"And Anwick?\" Roman asked.\n\n\"He won't be waking up to the tongue of a mangy dog. I dusted that son of a bitch like crops.\" Adam replied proudly.\n\n\"He ain't mangy. He's just like me, in need of a damn good woman and a damn good drink.\" Dalton added, Whiskey sitting down right beside him and looking at the rest of the group as if to second the notion.\n\n\"They're inside the building!\" one of the soldiers yelled, relaying the message to Lassiter.\n\n\"Hold your ground!\" he replied, sending what Goliath units that remained up closer to the fight. Legion troops by the dozen had breached the entrance to the building and set up a makeshift barrier using large sandbags as even more troops piled in. The plan was to bring in the heavy Shock Troops next, however they were met with swift gunfire on the outside by several units of Colonial soldiers who had returned to command as instructed.\n\nThat's when everyone outside first saw it. The gigantic shadow cast by the last operational Colonial Star as it made its approach, carefully planning its landing behind the complex. It was met with streaming Legion gunfire, however the small arms weaponry had no more effect on the huge Colonial ship as would rain or direct sunlight. Its hardened plated steel exterior was made for deep space travel and could withstand multiple nuclear strikes, combat rifles couldn't begin to dream of doing any type of significant damage, not even in unison.\n\n\"Sir, Colonial Star Twenty Two is on approach.\" one of the officers announced as a majority of the staff continued helping Sarah prep the equipment and supplies. This was literally the last remaining group of political and military command for the Colonial government, and if they fell here in the battle of Glimmeria, the Legion would win the war and once again rule the Skyla System unopposed.\n\n\"Have them land and load the supplies and equipment, however we have not lost the battle yet. I will remain here with Sarah as long as there is hope of winning, just have the Colonial Star prepared to launch if needed. Instruct all remaining Swordfish and Spartans to do defensive flyovers until ordered otherwise.\" Commander Edwards said.\n\n\"Yes sir!\" the officer responded, quickly turning to carry out the order.\n\nLassiter's men continued the full on gunfight with the few Legion soldiers who had breached the entrance, eventually taking a majority of them out while suffering heavy losses themselves in the process. Four remaining Legion soldiers who realized they were cut off from the central attacking force and surrounded by Colonials, threw their weapons down onto the ground and placed their hands behind their necks in surrender. They were swiftly taken into custody by a returning group of Colonial soldiers who entered the Command Center to rejoin their ranks.\n\n\"Take them to the holding cell on level two.\" Lassiter said loudly as his few remaining soldiers checked the wounded before re-securing the entrance of the building.\n\n\"What's the situation out there?\" he asked.\n\n\"Sir. The streets are torn up pretty bad. Pockets of Colonials held up throughout the city, most of it controlled by Legion forces who are getting one hell of a fight from Glimmerian locals.\" one of the Colonial soldiers answered, his face blackened by the fog of war.\n\n\"Do they have air?\" Lassiter asked.\n\n\"No sir, the only air we have seen has been our own, and they were few and far between. What they do have is a lot of Shock Troops and Spiders on the ground.\" the officer answered as Lassiter forwarded the strategical information into the Vault via com unit.\n\n\"How many troops do we have outside of the building?\" Commander Edwards asked as he stood on the other side of the thick protective glass.\n\n\"Hard to say sir. I'd estimate two dozen who are battle ready. A lot of injured and dying alongside them.\" he responded.\n\n\"Legion troops?\" the Commander asked. \"A lot sir. Hundreds. We were able to get here because we hit them out of the blue and they fell back before they realized we were so few. No way we could hold off an all out assault.\" the soldier replied.\n\n\"Lassiter, post every Goliath you have left at the entrance and bring your men into the Vault. Bring as much ammunition as you can carry, I'm ordering a full evacuation.\" Commander Edwards said.\n\n\"Yes sir.\" Lassiter said hesitantly, thinking of the faithful Colonial soldiers who were holding their ground throughout the city, waiting for reinforcements that would never come.\n\n\"Command One, this is Spartan One-Five-One, your area of evacuation is clear. You have an all green.\" the voice of a pilot passing overhead announced through the com unit in the Vault. Commander Edwards stood beside Sarah Blaine, each one pressing a sequence of keys on the wall which opened a door from the Vault to the area outside the rear of the building, their entire field of vision engulfed by the sight of the massive Colonial Star. As Commander Edwards and the remainder of the soldiers and personnel exited the building to board the base ship, Sarah turned to the Vault door which led to the internal portion of the building, entering her key code and unsealing the door to allow Lassiter's team access to the ship.\n\nIt was only a matter of minutes before the Colonial Command Center would be overrun with Legion forces, the remaining Goliaths would buy them a few more minutes, but they had once again fallen in crushing defeat.\n\n\"What about Adam?\" Lassiter asked as he carried satchels full of ammunition. \"All we can do is wait for him in high orbit. The entire city is overrun, and it will only be a matter of days before the Legion has more ships here. Our window of escape is closing, but we'll give him every last second of it.\" Sarah said, obviously upset at the possibility of her soon to be husband's capture.\n\n\"Colonial Command, this is Sky Commander Tess Weston of the United Gali Army. Please identify the coordinates of your friendlies so that we may assist you.\" a female voice rang out across the Vault's com.\n\n\"What?\" Commander Edwards said under his breath as he turned back to the com system. Visible from outdoors, a huge armada of warships and troop filled landers bearing the seal of Gali had arrived, beginning to darken the sky with such heavy numbers. Without the equipment hooked up and operational, there was no way for Colonial Command to know that Roman's home planet of Gali had indeed sided with the Colonials in the struggle of the Skyla System and had sent the bulk of its armed forces.\n\n\"Commander Weston, this is Commander Edwards of Colonial Command. I will instruct all remaining forces to engage their rescue beacons, everything else is fair game. Thank you.\" he said frantically.\n\n\"You're welcome.\" her voice answered as the roar of the landing shuttles full of Gali troops could be heard throughout Colonial Command. The warships were moved low to the ground, their cannons pounding pockets of Legion troops and occasionally even Hunters as it took only hours for Legion forces to issue a formal order of surrender. It was only one battle, the war was far from over. But at least now the Legion knew it no longer had control of the entire system. Gali, Glimmeria and several other planets who would soon join the Colonial side of the war had taken control of nearly a third of the Skyla System, which would give the people of those planets hope of freedom for their children.\n\n\"Adam continued to try and reach Colonial Command with the portable com unit, his message finally picked up with heavy static by one of the several operators who was collecting and dispatching information to their troops across Glimmeria.\n\n\"Sarah!\" the operator yelled quickly. She had been talking with Commander Edwards as well as a handful of Gali top military, but all of that ceased to matter when she realized it was a message from her soon to be husband, and that he was safe.\n\n\"It's still not safe enough Sarah. Groups of Legion holdouts planet wide. I'm sending a group of Colonial Spec-Ops to bring them home.\" Commander Edwards said, Sarah disappointed but quickly agreeing to speed up the process of bringing them back safely.\n\n\"While they take care of it, you and I need to talk.\" he added, Sarah looking at him suspiciously as he pulled her off into a corner for privacy.\n\n\"Is it nearby?\" Adam asked anxiously, everyone else waiting for Dalton's verdict, a map of the surrounding area in hand as they were all cramped inside of a sand crawler and heading for the extraction point. It was nothing more than a utility vehicle the town had used for repairs, and after seeing what the group had done to the bloodthirsty Hunters, they gladly gave it away with no strings attached.\n\n\"Well fuck me raw.\" Dalton finally proclaimed with disgust, balling the map up and tossing it onto the floor of the Sand Tracker as Whiskey rode shotgun.\n\n\"Want to dumb it down for us maybe?\" Sasha asked sarcastically.\n\n\"We gotta turn this molasses moving bitch around and head the fuck back, the extraction point, I shit you not, is right outside of Valencia. The other side of Valencia. That dumb enough for you?\" Dalton replied snidely before cursing Colonial Command under his breath.\n\n\"You have got to be kidding?\" Adam said, beside himself in disbelief.\n\n\"Do I look like a stand up comedian to you?\" Dalton replied as the Sand Tracker slowly began turning to follow its own tracks back into town.\n\n\"This is ridiculous, if any of the Hunters return we all know they will pick us up the minute we show back up.\" Zavious said.\n\n\"No shit, we might as well hang streamers on this bitch and radio them ahead of time, maybe they can organize a homecoming parade in our honor.\" Dalton replied as everyone inside of the overcrowded antique of a vehicle broke out into laughter for the next few moments.\n\nA little over an hour later, the Sand Tracker had come to a complete stop, everyone huddled outside around the small crumpled map in front of them as the vehicles headlights provided enough illumination for them to see it in an otherwise completely dark environment.\n\n\"Alright, the plan is simple enough. We move everyone to the far side of town on foot with as much ammunition as we can carry, lay low in the hills and wait it out.\n\n\"What about Legion patrols? We got no cover out here.\" Dalton asked.\n\n\"All we can do is hope. \"I can handle first watch,\" Sasha said, holding her suppressed pistol machine gun up. \"Besides, I'm well rested.\" she added.\n\n\"Not by yourself you're not.\" Adam replied. \"I'll cover first watch with you. The group loaded back up, moving cautiously to an area that was perfect for digging in and holding on until the Colonials could save the day.\n\n\"I can't believe we are going to rescue them, and then watch everyone but the man I love be imprisoned by the very government that they have sworn to protect. I mean Dalton and Steiner are there on our orders!\" Sarah yelled.\n\n\"It is out of my hands I'm afraid. I will make it a priority to assure them all a fair trial as well as a very lenient sentence, but when the Colonial Parliament hears of their conspiring with the Benzans, well you have been with us long enough to know the answer to that.\" Commander Edwards replied.\n\n\"They will want them executed!\" Sarah said loudly.\n\n\"I'm afraid you are right. Your father was a lifelong friend of mine, it is that I hold your family in such high regard that I will personally see to it that their lives are spared, Dalton, Steiner and Roman at least. The Benzans will receive no special treatment.\" Edwards replied.\n\n\"Nor would I expect them to.\" she replied, considering them to be cold blooded killers.\n\n\"Sarah, I think it would be best if you were the one who orders them into custody. They will listen to your reasoning before that of an old man in an officer's uniform.\" Edwards added.\n\nSarah looked across the sand of Glimmeria with such mixed emotion. Her entire life had been dedicated to the thought of a star system without corruption. Her father had raised her to value law and righteousness above all things, but was that supposed to include compassion and love? She knew in her heart that the right thing to do would be to rescue the man she loved and bring the rest of them to justice. But she loved Adam, and she knew that in doing so, she may lose him forever. She loved him more than any man that had ever been a part of her life, short of her own father. \"I'll take care of it for you.\" she finally replied, knowing that her struggling heart was no excuse for making the wrong decision.\n\n\"Thank you Sarah. I specifically plotted the extraction point close to Valencia to give them all the best chance of survival. For your sake, I want them alive and well when we arrive.\" Commander Edwards said before slowly taking his leave to return to the bridge of the ship.\n\n\"The son of a bitch wants us dead, I know he does!\" Dalton commented loudly as he and Roman remained awake, the other survivors sleeping solidly.\n\n\"Who wants us dead?\" Roman asked.\n\n\"Commander Edwards, that's who, the cocksucker,\" Dalton replied. \"He sent us all out here while holding Sarah and Lassiter there. She's like a daughter to him and Lassiter is his idea of a perfect soldier. The rest of us he could give two shits about, knowing good and well we were out gunned. Then has us backtrack to the same place we almost died only hours ago.\" he added as everything remained silent.\n\n\"Ya hearing me man?\" Dalton asked as Roman nodded to let him know he felt the same way.\n\n\"My loyalty is to Adam to the end. As much as I like Sarah, I'm starting to realize that since the day we met her, bodies have been dropping faster than panties in my presence.\" Dalton added as the two men laughed quietly, Whiskey sticking his head up for a moment almost as if he was fluent with lingerie.\n\n\"Sounds like your friends are still awake.\" Sasha said, sitting next to Adam as they both watched the sand filled horizon.\n\n\"Yea, I doubt Roman does a lot of sleeping as of late. With his newly fitted mechanical parts, I'm not even sure he needs to sleep anymore. And Dalton, well he rarely gets a good night's sleep unless there's an empty bottle of whiskey close by. I'm sure watching Steiner fall isn't helping much either.\" Adam answered as he and Sasha both remained silent for a moment.\n\n\"So, we got a few hours. Catch me up on the Benzan lifestyle. Seems like pretty much everything that I've been told of them was untrue.\" Adam said.\n\n\"I'm afraid I am not the kind of girl who likes talking a lot about herself.\" Sasha replied. That was a first for Adam, who gave her a huge but faux smile as he couldn't make the connection between a woman and selflessness. Finally, deciding to break what had become a very awkward silence between the two, he spoke up. \"Well, I guess in that case I can ask Dalton to come out and entertain us.\"\n\n\"Where would you like me to begin?\" Sasha asked, bringing a laugh to both of them quickly.\n\n\"Yea, he has that effect on people. He's really not as bad as he comes across when you get to know him. He's loyal and a damn good friend.\" Adam replied.\n\n\"I understand.\" Sasha answered.\n\n\"What I still can't figure out, is how you and the crew ended up fighting a politician's war?\" she asked, changing the mood of the conversation to one of seriousness.\n\n\"Good question,\" Adam answered. \"It wasn't long ago that I was just like you. I thought I had everything figured out. Then, of course, I met Sarah, and it wasn't long after that everything got clouded I guess. Most of the people that the Colonials are trying to protect are honestly good people. I figured it was time to change who I was and start doing something positive.\" he added.\n\n\"If you don't mind my saying so, I don't believe you have changed. This person you seem to think you once were, that's the only person I have seen since you arrived on my ship. You seem to have things figured out to me, loyalty and friendship above all else.\" Sasha said, a very long silence falling after her last word as Adam began wondering if she was right. Had he really changed at all?\n\n\"So, what about you,\" Michaels asked. \"I'm still curious as to how you ended up with the Benzans and what day to day life is like?\" he added.\n\n\"It's a pretty simple story I'm afraid. If the person chooses to wear the amulet, they are from that point forward recognized across all Benzan families as one of their own, taken in and taught the lifestyle.\" she added.\n\n\"And what lifestyle would that be?\" Adam asked.\n\n\"Everything from how to handle a weapon and defend yourself when there is no weapon, to helping the poor and defending the otherwise defenseless,\" Sasha replied. \"In my case, a close friend of the family who was more like an uncle to me, decided after my family was slain that my only hope for survival would be with the Benzans.\" she added.\n\n\"I'm sorry to hear about your family,\" Adam replied somberly, looking at the ground for a few moments. \"So Cyrus took you in?\" Adam added.\n\n\"They all took me in as their own. Cyrus was like a father to me, but there are many other families, just as they are many other clans of Hunters,\" she replied. \"Adam, you should consider the offer to become one of us. I know it will complicate things for a bit, but personally, I would really love the opportunity to spend every day with you.\" Sasha added, taking Adam by complete surprise.\n\n\"Well, I am certainly flattered by the offer. It's only fair to tell you though, when we return, my place is by Sarah's side until death. We had planned a wedding until the Legion arrived.\" Adam replied.\n\n\"Very well, all I can do is extend the offer, but it is your choice as to where life will take you.\" Sasha replied.\n\nFor the next few hours, Adam told Sasha the stories of the first Glimmerian War, his long career as a criminal smuggler and of course, the friends who had fallen that she would never have the opportunity to meet. Sasha continued telling Adam of the Benzan lifestyle as well as of the friends she had made, and sadly lost along the way. They laughed together, showed concern and even let go of some emotional feelings, venting to one another in a strict matter of trust. Personal accounts that only good friends could possibly accomplish through the bonding of a friendship.\n\n\"Adam! Picking up all kinds of Colonial traffic over the com unit.\" Dalton yelled loudly, running out to Michaels and Sasha as the rest of the group slowly followed.\n\n\"Gear up, they're coming.\" Adam said to the group after listening to the chatter for a couple of minutes. A small group of Legion faithful, not even a click away from their position, began firing tracer rounds into the sky. It was a tearful sight for nearly everyone when they saw the tracer rounds begin to be answered by Colonial rail gun fire, which could only mean one thing, their rescue chopper had arrived!\n\nWithin the span of thirty seconds, the rail guns had not only taken out the machine gun nest on top of a large sand dune, but a majority of ground around the Legion troops as the fast moving lead continued chewing into the sand and its underlying layer of clay. The survivors began flocking to the chopper in a single group, the com units continuing to broadcasting a message of the arrival of Gali forces, but this time around it was Sarah's voice.\n\nAdam grabbed the com unit with both hands shaking, pressing the communication button and pausing for a moment to think his words carefully.\n\n\"I've waited a long time to hear that voice.\" Adam said eagerly into the transmitter, nervously awaiting a response.\n\n\"Adam! Where are you, is everyone safe?\" she asked. \"We are on our way home...Steiner didn't make it.\" Adam replied, his lips shaking from the emotion of hearing Sarah's voice again and the loss of his loyal crew member.\n\nA long silence followed before Sarah replied with a heavy heart. \"We see you on our grid now and will be eagerly awaiting your return.\"\n\nThe transmitter was flooded with com chatter as Glimmeria was now overflowing with military help from several planets who had followed Gali's lead. As the shuttle continued its path to the landlocked Colonial Star, Adam took one last look outside of the windshield, promising himself yet again that he would never return to Glimmeria as long as he lived.\n\n\"None of us would have made it out of there alive if it wasn't for everyone's hard work. Thank you.\" Adam said, nodding his appreciation to Roman, Dalton, Sasha and the rest of the exhausted survivors, including Whiskey.\n\n\"He grows on ya' don't he.\" Dalton said as everyone laughed and looked forward to the idea of living to fight another day.\n\nGlimmeria had been lost to the Legion, a crushing defeat under the heels of the far superior force that was assembled by worlds filled with people in search of freedom, though much of its largest cities lay as nothing more than a pile of rubble.\n\nWhat few soldiers remained would join the Colonial Parliament in devising a new strategy based around the newly gained support. Right now none of that mattered to any of them, especially not to Adam, who waited patiently as the shuttle docked onto the floor of the launch bay. He would finally see his future wife's face once more, which is something he had started to doubt would ever be afforded him again by the gods.\n\nAs the shuttle door opened to allow the crew's exit onto the interior of the Colonial Star, they were met by a squad of soldiers aiming combat rifles in their direction.\n\n\"Adam Michaels, step aside. Your crew is being placed under arrest for abandonment of military post as well as aiding known fugitives.\" an older gentleman with a lengthy white beard proclaimed.\n\n\"Well, it looks like Sarah has let herself go all to hell.\" Dalton said, sarcastically comparing her to the old man. The soldiers glanced at the sad excuse for a dog in Whiskey before they began placing hand restraints on the Zavious, Sasha and Dalton, while Roman resisted the attempt.\n\n\"Try putting those on me, see what happens.\"\n\nWith a bit of convincing on Adam's part, Roman soon placed a set of restraints on himself, all while continuing to dare the soldiers to lay hands on him.\n\n\"Don't worry, I'm sure it won't take long to clear up this misunderstanding. Where is Sarah Blaine?\" Adam asked the older gentleman.\n\n\"I'm right here Adam.\" she said, stepping out of the shadows of an unlit room nearby.\n\n\"Adam, I'm sorry. I'm under direct direct orders from Commander Edwards himself to take them into custody.\" she said, her heart pleading with his own.\n\n\"Sarah this is my crew. My family.\" Adam replied somberly, still in disbelief of everything that was happening.\n\n\"I have to admit Adam, she doesn't seem like your type.\" Sasha said, drawing a scowl from Sarah as she tried to figure out the relationship between the Benzan beauty and her husband to be.\n\n\"You realize they haven't built a prison cell yet that can hold us, right?\" Dalton asked Sarah, his question completely disregarded.\n\n\"We'll see.\" the Master of Arms finally responded as the group started to move slowly into the direction of the prison block.\n\n\"I have arranged for the Gunship to be fueled and waiting in hangar bay two when we get there.\" Sarah said to Adam as the group continued its slow pace to lockup.\n\nTaking several minutes, they passed groups of Colonial soldiers who stopped to watch, never questioning her actions because of their deep rooted respect for Sarah and what she had given for the cause of freedom, while at the same time dumbfounded by seeing the crew arrested.\n\nJust as promised, when they arrived, the Gunship sat in one of the launching bays, its reflective chrome exterior immediately catching the eye of a crew who had given up hope of ever seeing it again.\n\n\"It's fueled and supplied, just as you ordered my lady.\" one of the Colonial deck hands said proudly.\n\n\"Thank you.\" Sarah replied before turning to Adam. \"We need to get out of here for a couple of days, just you and I. Sort through all of this.\" she said.\n\n\"Not without my crew Sarah, you know that.\" Adam replied as everyone stopped to watch the heated discussion.\n\n\"Commander Edwards pulled a lot of strings to keep you a free man Adam, your friends chose their own paths.\" Sarah replied.\n\n\"Free? Leaving here in a ship without my crew isn't freedom?\" Adam replied angrily.\n\n\"This war is over for you and your crew, if you stay, the Colonial Parliament will see to your release from the military and not even I can stand in the way of that.\" she replied.\n\n\"Sarah, let's just go. The battle is over, your father is gone. Let's all just go right now, start a new life beyond all of this.\" Adam said, pleading with his lover.\n\n\"I wanted to go alone, just the two of us and have this conversation. But if you insist on doing it in front of everyone else, fine.\" she replied as her eyes watered with emotion. \"We can't start a life together Adam, I'm sorry. My place is here with the Colonials, with a military that longer wants you involved.\" she added.\n\n\"So this is it? Just like that we are finished?\" Adam asked. \"Maybe not forever, who knows, when the war ends you...\" Sarah replied as Michaels interjected.\n\n\"No Sarah. You once told me to make a choice, now I'm asking you to do the same. Either you leave with me and we do things the way we need to, or you stay behind and choose your war and grudge against the Legion over the love we have for each other. Period.\" Adam said sternly. \"Adam, I have to stay, you know that.\" Sarah said emotionally. \"I thought what we had together was so much stronger than all of this, at least until now. Now I don't know what to think Sarah.\" Michaels said, trying his best to hide the pain consuming his every extremity.\n\nAs the group looked on, Adam turned back to look at Lassiter. \"You with us, or staying behind to fight this pointless war to the end?\" he asked, almost already knowing the answer.\n\n\"Adam, you are a true friend to me, but my place is by Sarah's side fighting for freedom. Kelly would want it that way.\" Lassiter replied.\n\nMichaels simply answered with a head shake of pity before turning to the ship's ramp and suddenly stopping in his tracks. He glanced back at Sasha for several tense moments, remembering the promise he had made her not so long ago.\n\n\"I can't let you do this Sarah. I made her a promise.\" Adam replied, turning his attention to his former lover.\n\n\"Adam, it's going to be hard enough watching you leave like this, you know I have to keep them here.\" Sarah replied as she turned to Lassiter to give him the go ahead.\n\nLassiter readied his combat rifle, as he walked toward the prisoners with the full intention of ushering them forward.\n\n\"Belay that order Lassiter! Don't do this Sarah, don't push me into something that we will both live to regret for the rest of our lives.\" Adam said pleadingly.\n\n\"I'm sorry Adam.\" Sarah replied uninterested.\n\n\"Then so am I.\" Adam replied, pulling his pistol and aiming it directly at Sarah Blaine.\n\n\"What the hell are you doing Adam?\" Sarah asked, shocked that the man she had come so close to marrying would go so far as to threaten her life.\n\n\"You leave me no choice.\" Adam replied, holding the weapon surgically steady.\n\nLassiter turned his rifle, aiming it directly at Adam, moments later feeling cold steel held by Roman Raines, who was a master of escaping loose bonds and blade smuggling.\n\n\"Think you're quicker with a combat blade than I am with this trigger?\" Lassiter said jokingly, almost to mock the warrior.\n\n\"Wanna find out?\" Roman replied, digging the edge into Lassiter's neck just enough to draw a trace amount of blood.\n\n\"Adam, there's no going back from this.\" Sarah said sternly, realizing that he was fully prepared to die trying to keep his promise.\n\n\"We outnumber you two to one!\" a Colonial soldier said loudly, all of them taking aim on Adam and his crew.\n\n\"After watching you get your asses kicked twice by the Legion, I'd say that puts the odds in our favor.\" Dalton replied as he laughed, now holding two short barrel shotguns that he had recollected from one of the soldiers, his hands still tied with chains as Whiskey sat close to his feet.\n\n\"What's the logic Adam,\" Sarah asked. \"You have nowhere else to go, we gave you the last chance you'll ever have at a decent life?\" she added.\n\nHe seemed to mentally check out for a few moments, thinking long and hard about his decision to come because of his strong belief in staying true to his word. He looked at his crew, then directly at Zavious.\n\n\"Extend the offer to Dalton and you have yourself a deal.\" Adam said.\n\n\"Done.\" Zavious replied with truth in his voice. Adam pulled the medallion from his pocket and placed it around his neck with a single free hand, his pistol never faltering.\n\n\"Is this what it has come to Adam? You are planning to live a life committed to a family of murderers and thieves? The trash of society.\" Sarah asked.\n\n\"Did this redheaded whore just call me trash?\" Sasha asked Dalton, ready to throw fists with Sarah.\n\n\"I believe she did, and I believe Whiskey needs an official Benzan invitation as well.\" Dalton replied, glancing to Zavious who nodded simply to keep the peace.\n\n\"Relax Sasha, she's not worth the effort,\" Adam replied, holding Sasha back with his free hand, while looking straight through the woman he had once thought to be so perfect. \"Of course, it would have been nice if I would have known this the day we met.\" he added.\n\n\"Go ahead then. Adam, you do realize that once you lift off, I will order our allies to blow the Gunship into nothing more than wreckage inside of a minute?\" Sarah asked, the look on his face was a dead giveaway that he hadn't considered that angle yet.\n\nShe was right, even if he somehow managed to get away, the sky was so thick with Gali warships that they wouldn't have a chance in hell of getting out of Glimmeria's orbit.\n\n\"Give me your guns.\" Roman said, taking the shotguns from Dalton's grasp, putting them on both Sarah and Lassiter at point blank range.\n\n\"You can't be serious?\" Sarah asked, surprised.\n\n\"Dead serious you bitch. Adam may be in love with you, but after hearing you call Sasha's people trash, I got no problem putting both you and your wet nurse on ice.\" Roman replied of Lassiter.\n\n\"Our people trash, actually...now.\" Adam added, Whiskey barking loudly.\n\n\"You've seen my poker face before, if you think I'm bluffing, try me.\" he added.\n\n\"Roman, I'm not gonna leave you behind!\" Adam said loudly. \"What are they going to do? Imprison me? Kill me? I'm already fucking dead Adam, look at me,\" Roman yelled, referring to his synthetic laced body. \"What in the hell else can they do to me that hasn't already been done?\" Roman asked, not expecting an answer. \"Get everyone else the hell out of here, I'll give you the time you need for a clean getaway.\" Roman added.\n\n\"Adam looked into Roman's eyes for a moment with appreciation. \"I will be back for you my friend, and I'm saving a medallion for when the day comes.\" Adam said as he nodded one final time.\n\n\"If you come back Adam, my love for you will not play favoritism again. I will see you imprisoned.\" Sarah said.\n\n\"Hadn't heard that line before,\" Adam replied, not even glancing in her direction. \"Soon my friend.\" Adam said to Roman as he turned to enter the Gunship.\n\n\"This isn't over!\" Sarah yelled, as he turned to face her one last time.\n\n\"Yea. It truly is.\" Adam replied, holding Sasha's hand as they followed Dalton and Zavious up the ramp and into the Gunship, Whiskey stopping momentarily as he pulled a mangy leg up and let urine fly onto the deck of the Colonial Star.\n\n\"Damn I love that dog.\" Dalton said smiling as he slowly sealed the ship's door air tight.\n\nMoments later, he and Dalton made the preparations to lift off, the shiny vessel leaving the Colonial Star in full burn.\n\n\"You've started a war you can't win Roman.\" Sarah said as the heat from the Gunship's thrusters whipped across their skin.\n\n\"Says the bitch who's tucking tail and hiding behind everyone else's army on her last remaining ship.\" Roman replied stone faced.\n\nSeveral minutes later, one of the deck officers on the bridge of the Colonial Star turned from his grid screen.\n\n\"Sir, we just picked up an unidentified craft heading away from us and moving fast!\" Commander Edwards quickly walked to his station to see for himself, the Gunship almost out of striking distance.\n\n\"Set gun coordinates and get the firing solution, quickly!\" he said as he turned to face the entrance to the ship's bridge.\n\n\"Belay that order private.\" Roman said, holding Sarah, Lassiter and several other Colonials at gunpoint as they entered the bridge.\n\n\"This is outrageous!\" Edwards said, motioning for the bridge's security force to take control of the situation. They stopped abruptly as one of the shotguns swayed into the direction of Commander Edwards, the other remaining firmly on Sarah.\n\nWhen it had become clear to him that his own security detail would not be able to intervene until Roman decided he was damn good and ready to let them, Edwards ordered his soldiers to drop their weapons slowly to the ground and place their hands behind their heads.\n\nThe Gunship had gotten away, broken orbit and disappeared into the drop cloth of stars which illuminated the skies of every planet in the system.\n\n\"Officer Lassiter, escort Roman to the prison level.\" Sarah said seconds after he decided Adam was long gone and threw down his weapons.\n\n\"If he even begins to attempt escape, kill him.\" she added.\n\n\"Yes my lady!\" Lassiter agreed with unwaivering loyalty, Commander Edwards doing his best to regain the look of someone with authority.\n\n\"Mr. Raines, I hope you do realize that our willingness to hand you over to the Gali government was a key factor in them coming to our aid.\" Commander Edwards said as Roman turned to face him, his hands wrapped in chain and realizing that he would now have to pay for his role in Greyspine.\n\n\"That's right Roman, you will be on a Gali warship within the hour. May god have mercy on you.\" Commander Edwards added as Roman was led away by Lassiter and the small group of Colonial soldiers.\n\nAdam hurt deeply, trying his best not to show it as they set course for Tirious. He knew that with the companionship of Sasha and a new home among the Benzan family, that with time his heart would heal into something that was once again capable of love. Watching Dalton comb the thick and tangled hair of Whiskey down into a smooth shine, Adam knew that even the worst looking of situations could turn out fantastic in the end. And above all else, he began contemplating an escape plan for Roman as the feeling of freedom began to set in, true freedom. Plenty of time for reflections of the past to run through his mind.\n\nBook 3\n\nReflections\n\n\"I'm surprised you would have the balls to show your face again.\" Walter Jones said with sarcasm.\n\nThe small time crime boss stood in a less than fine suit, two armed men at his side. They weren't killers, that much was obvious by the deer in headlights look upon their faces. They anxiously held Dalton at gunpoint using rifles, and that always had a way of rubbing him wrong.\n\n\"Aw now, no need to be so rude about it. How about you get these boys to lay down arms so we can have ourselves some honest dealings?\" Dalton asked casually.\n\nWalter Jones laughed, knowing Dalton and crew were smugglers and that was about as far from honest as it could get. Not to mention he owed Dalton James and his friend Adam Michaels money from a job previously completed.\n\n\"Now why in the hell would I do that?\" Walter asked.\n\nA split second later, one of his armed men fell to the ground in screaming pain; victim of a sniper's shot from the far distant.\n\n\"Well sir, cause you 'aint got a choice for starters. I got a sniper up above with you all scoped at this very moment. That's the biggest reason I got the balls to show my handsome face again,\" Dalton said, smiling wide. \"Now where's the money?\" he asked.\n\nWalter Jones had planned to screw him a second time of course, but didn't want to risk the possibility of his own demise in the process. Throwing a sackful of credits to Dalton's feet, Walter scowled heavily. \"It's all there.\"\n\n\"You may think I'm somewhat of a stickler, but after you gave Adam and me a sackful of blank paper on the last go round, I think I might be counting it this time.\" Dalton said, bending down to unzip the bag.\n\nIt was filled with credits alright, and he was due two thousand for the job recently completed. \"Looks like about two thousand.\" Dalton said as his fingers quickly fanned through the money.\n\n\"I told you it was all there.\" Walter replied. \"Yea, but see,\" Dalton said walking a bit closer as he pulled a tightly rolled cigar from the pocket of his good as new brown coat, blazing up the tobacco stick and adding heavy smoke to the mix. \"You still owe me three thousand for a job already done. And I can't figure out the exact number without an adding machine, but I'm thinking you need to be handing over the rest of it if you wanna walk out of here alive.\" Dalton replied.\n\n\"Are you insane? You know I don't have three thousand more credits on me,\" Walter Jones said loudly. \"I'd be a fool to carry that kind of money.\" he added.\n\n\"Yes sir, I'd be inclined to agree,\" Dalton replied, stroking rough fingers through the course patch of his beard. \"Well, how about you boys empty your pockets into the bag, including your watches and such. Then we'll just call it even.\" Dalton demanded, kicking the sackful of credits a bit as it moved closer to them in the lifeless dirt.\n\n\"Are you kidding,\" Walter asked snidely. \"Why don't you get on the ground and take the man's gold tooth too for God's sake?\" he added loudly as his gun struck man still lay on the ground, a blanket of pain and agony doing little to quiet his screeching.\n\nSeveral minutes later Dalton stood there with a smile painted to his face and a loaded bag in his hand. \"God damn you Dalton James, you'll pay for this!\" Walter yelled, his man now one tooth shy and rolling on the ground in pure oral pain.\n\nNever one to pass up a suggestion to make money, Dalton had pulled a pocket knife from his brown coat; using its small set of pliers to jerk the golden tooth from the man's skull. It would only fetch thirty or so more credits, but that was money owed to him by Walter Jones and he wasn't about to leave it laying. Of course the man would have to do without tough meat in his diet for the next few days, but that was of no concern to Dalton.\n\n\"Well boys, it's been a blast. I guess this is it until next time.\" Dalton said in his usual wise ass tone, turning to walk away from the deal gone sour.\n\n\"Dalton, if you ever show your face here again I swear I'll cut that damn smile from your skull,\" Walter yelled. \"You tell Adam Michaels I said the same!\" he added.\n\n\"That 'aint gonna happen.\" Dalton replied, his cigar burned down to nothing more than a saliva ridden stub.\n\nShortly after, Dalton boarded the ship, walking up the steel grating of the ramp as he was greeted by Whiskey. His pooch had been with him for a while now, a loyal friend who even sported his own custom made brown leather coat. It wasn't as thick as the one Dalton wore on his back of course, but the couturier had thrown it in for free. Together they looked almost like twins, the fur on Whiskey's face just a tad thicker of course.\n\n\"Where's the Capt'n?\" Dalton asked, petting Whiskey for a moment before standing with the bag of credits, jewelry and that single loose tooth.\n\n\"Right here.\" Cambria said, slowly moving down the spiral stairs that led from the cargo hold to the crew's quarters. She was perfect in every sense of the word, her lush curves tightly wrapped in form fitting cargo pants and tight t-shirt that did wonders in showing off her upper body. The upper body that interested men, of course. Her skin had a glow of white satin about it, which only brought more attention to her vibrant blue hair.\n\nShe was from the Drifts, a series of smaller planets on the fringes of charted space. Some of the planets lacked modern technology, while others simply shunned it altogether. Everyone from the Drifts had a unique look about them, and hers just happened to be a look of insatiable sex and electric innocence.\n\nCambria Sims was still fairly new to smuggling, which was the biggest reason she laid down the kind of money she did for Dalton; which amounted to nothing more than drinking money. He had experience, was wise to the way things worked in this type of life and when things went wrong he was plenty capable of taking care of things with his own two hands.\n\n\"Damn you are a welcome sight for sore eyes.\" Dalton said, watching such a beautiful woman head into his direction.\n\nKneeling down to retrieve the bag full of credits while looking up at him with a smile, her pouty lips only inches away from the most vital area of Dalton's body, the part wrapped in a zipper; Cambria smiled slightly.\n\n\"I still say we should take this to the next level. I could make an honest woman out of you.\" Dalton said as Cambria slowly stood to her feet, purposely keeping herself only inches from his body so their lips could be nearly touching when finished.\n\n\"Maybe one day cowboy. For now though, good job on today's catch.\" she replied softly as though she was ready to kiss him, instead turning to head back to the crew's quarters.\n\n\"One day you are gonna be courtin' me exclusive. You watch and see baby, I'm gonna break you down.\" Dalton said, grinning ear to ear as Cambria walked away slowly, her ass moving with only the slightest of bounces; a perfect testament to her capable curves. Rather than answer, she turned slightly and smiled at the experienced smuggler.\n\n\"Now Whiskey, there goes a real damn woman. I know my away around the bedroom as good as any man, but my gut tells me she'd be able to show me a thing or two,\" Dalton said under his breath with his trusted pooch by his side. \"I'd almost give up drinking for fifteen minutes with...\" Dalton added, interrupted by the shuttle pulling from the planet's surface.\n\nHe quickly made his way to the ship's entrance, spinning the wheel which served as a handle, the metal door sliding shut as he bolted it into place with three locks.\n\n\"Goddamn steam engine, I still hadn't got used to it.\" Dalton said with ill intent.\n\nCambria was Captain of the Outer Heaven. It could house only a small crew but was proudly made in the Drifts, needing nothing more than constant steam to operate. It had its advantages and disadvantages of course, but made almost no sound which was ideal for smuggling. It was a deep space capable ship, though it looked more like an airship or elongated balloon. A mixture of solid steel and thickened glass, the Outer Heaven was a marvel of Victorian technology.\n\n\"Good shootin,\" Dalton said as he turned to nod his appreciation to Skulls. His God given name wasn't Skulls of course, it was Trevor Lagrange. But he had a very odd hobby. He enjoyed collecting skin, bones, teeth and even the occasional shrunken head. A hobby that quickly led to his nickname. \"This is for you pal.\" Dalton said, pulling the still bloody golden tooth from his pocket and flicking it to the strange man.\n\nSkulls was a very tall human, nearly seven feet. He was far from large though, a majority of his frame nothing more than pale white skin and sturdy bones. He wore black leather from his boots to collar, though it was very loose hanging. A black top hat sat firmly on his head as the stringy haired man simply nodded his appreciation.\n\nHis Salvation model sniper rifle hung by a nylon strap down the middle of his back. The Salvation rifle was a much older model and being bolt action made it less popular because of the accuracy needed to make a kill. Skulls loved the weapon because he was accurate. Damn accurate. Anytime he pressed his eye to the telescopic lens mounted onto the rifle, death would surely ensue.\n\n\"Best head up and get your cut.\" Dalton said, turning to make his way up the spiral stairs. They were narrow, made of all steel and noisy as hell; having taken a verbal lashing by Dalton more than once during the routine hangover.\n\nCambria stood near the crew's table with Tank as they emptied the contents of Dalton's bag, credits piling high. Tank also answered to his real name, Greg Shelling, but Tank fit more appropriately. The dark skinned man was huge, at least six and a half feet tall with a muscular frame to go along with it. He stood there in a sleeveless white t-shirt, green pants and boots of black leather. His usual attire, day in and day out.\n\n\"I'm keeping this watch if that's cool?\" Tank asked.\n\n\"Be my guest, too much flash for me anyhow.\" Dalton replied as Tank held up a watch of rock solid silver.\n\n\"Here's your cut, plus a bonus for job well done.\" Cambria said, laying a stack of credits out in front of Dalton, accompanied by a wind resistant lighter that had been salvaged from the pocket of Walter Jones himself.\n\n\"May want to quit giving me gifts like this, people are gonna start talking.\" Dalton replied, winking at the flirtatious Cambria Sims.\n\n\"I'll leave Trevor's cut on the table.\" Cambria said.\n\n\"'Aight. Me and Whiskey are beat, I'm dragging my sorry ass to my rack. Room for two if you change your mind.\" Dalton said, looking heavily at Cambria with a smile.\n\n\"Never know, tonight might be the night.\" she replied with a smile. Of course in the back of his mind he knew it wasn't going to happen, but flirting with a girl who was so perfectly sculpted with genetics seemed to make the trips through space more manageable.\n\nWhiskey was the first one in, immediately jumping onto the foot of the military style bunk.\n\nDalton sat down several moments later, handing Whiskey a long string of jerky before leaning over to take his boots off. \"I'm getting too old for this shit.\" he said under his breath, unlacing his boots a bit before forcing them off. Leaning over to a night table, Dalton picked up a photo taken with Adam Michaels during their first war on Glimmeria.\n\nDalton was decked out in an old brown duster with shotgun in hand, his arm wrapped around the man who had been like a little brother to him. He smiled a bit, remembering the day it was taken and the great night of drinking that followed. \"I miss you old buddy.\" Dalton said in a low voice, finally placing the photo back onto the bedside table as he turned the lamp out and tucked in for bed.\n\nNine months earlier.\n\n\"Next.\" the prison cook said solemnly as a single line of the most incarcerated men in the Skyla System waited patiently.\n\nRoman stood there, gazing out of the small window behind the buffet line that wasn't suited for an animal to enjoy, much less a human being. The stars looked as beautiful as ever, even if only from a window that was less than ten inches wide.\n\n\"Prisoner Raines,\" a heavily armed guard shouted loudly, using the wooden stock of his shotgun to push Roman forward forcefully. \"He said next!\" the guard added as Roman turned to face him for only a moment.\n\nThere was a time when pushing Roman Raines in such a way would have been considered suicide, however this was no such time. His hands and feet were both tightly bound in heavy chain as deep scarring was visible all across his body; a place that once was home to nearly indestructible Goliath shielding. Less than a month after he had been aboard, the warden thought it best that Roman's metal exterior be removed with surgery in order to better protect his guards.\n\nIt was a procedure that had been given only a slight chance of success, and if Roman would have died on the operating table then life aboard the prison ship would have continued without him. And die he did, for nearly two minutes he had escaped this life of caged horror only to be brought back with electrified paddles. The next few months were spent under heavy guard in the infirmary, his gaping wounds healing slowly on their own without the assistance of pain killing treatment. He wished that he would have been left dead, rather than being brought back to this nightmare of bad food and torture.\n\n\"Still looking for your friends I see,\" Zane said, almost in a joking fashion as the badly scarred body and fully bearded face of Roman Raines slowly sat down in front of him in the prison mess hall. \"It's been nine months now. They 'aint comin'.\" Zane added.\n\n\"You let me worry about that.\" Roman said with stern intent, his eyes reflecting a hollow rage.\n\n\"All I'm saying is we need to start working on our own exit strategy. Just in case.\" Zane replied. He was a large man, there was no doubt about that. A bit over seven feet tall, the former soldier had a rock like complexion across his face with a roughly shaven head which left a thick patch of brown hair in mowhawk fashion. Roman took a few moments to let Zane's words set in as he glanced around.\n\nA single file line of once mighty warriors now left humbled, begging for a spoonful of slop as though they were less than human. Then there was the window. That damn window. Every single meal since Roman had been locked up he glanced out of that ten inch window into the cluster of thick stars hoping to see the Gunship arriving to save the day. Adam Michaels had given him his word that he would return, and Roman knew Adam was a man of his word. That said, the possibility that Adam had tried and failed started to become the only good explanation.\n\n\"You're right. We need to get to work on something of our own. I'd rather die trying to get the fuck off of this ship than live like this.\" Roman said, glancing up at the gun rack.\n\nIt was the name of the cage that overlooked the prisoner's mess area and standing inside was a prison guard with a high caliber rifle and the authority to shoot to kill.\n\n\"It's about time my brother. It's about time.\" Zane replied, smiling slightly before once again becoming stone faced as a patrolling guard walked by their table.\n\nIt was Corporal Raykes, a guard that Roman knew all too well.\n\n\"What the fuck are you staring at?\" Raykes asked, walking by the table slowly.\n\nRather than respond, Roman continued his stare, eyes cutting through the man who hid behind the authority of a badge. \"That will be one motherfucker I'll enjoy killing. Just hope I get the chance.\" Roman said in a low voice as Raykes had moved down a few tables.\n\n\"Not your favorite huh?\" Zane asked.\n\nRoman remembered his arrival to the prison ship. How Raykes had spit in his face and dared him to retaliate. How he had heard the faint laughter of Raykes during his forced surgery, steel being pulled from flesh without remorse.\n\n\"Son of a bitches like that put on a good front, but when it's killing time they cower down. I'll either kill the bastard or bring him to the point that he wishes for death.\" Roman replied as the guards began ordering them to stand and return to their cells.\n\n\"They are waiting for you my lady,\" Lieutenant Lassiter said calmly as he approached Sarah Blaine. \"Sarah. Are you alright?\" he asked as she burst into tears.\n\n\"I can't do this anymore,\" she said crying heavily. \"I can't go on without Adam next to me. It's killing me inside.\" she added.\n\n\"My lady, everyone is outside waiting for you to deliver your acceptance speech.\" Lassiter said with panic.\n\n\"Tell them to find someone else. How can I possibly lead the Colonial Army if I can't even sleep at night? I miss him!\" Sarah replied as she continued to cry heavily.\n\nOnce lovers on the verge of marriage, Sarah had chosen her Colonial duty over the man she loved. Truly loved. He was a man of virtue and truth, something she hadn't seen a lot of in the military. She had regretted her decision only minutes after watching him fly away, their storybook love shattered because of a mistake that had haunted her every since.\n\n\"Sarah. Just go out there and tell these people what they need to hear. Let it come from your heart. As soon as you are finished we will get to work on finding Adam.\" Lassiter replied.\n\n\"Really?\" Sarah asked, calming just a bit.\n\n\"Yes my lady, I will personally see to it. Now please, take a moment and then lead these people to the freedom they so desperately need.\" Lassister replied. Sarah simply responded with a nod of gratitude as she began to wipe away the aftermath of tears and poised herself to deliver a speech.\n\nSeveral moments after Lassiter had left, Sarah reached into a drawer on her thick wooden desk. She took several pills, chasing them with a glass of pure water as her eyes fixated onto a photo of Adam Michaels which had remained on her desk. She had no intentions of taking her own life, but rather medicating herself to the point of making life bearable. A habit that had become increasingly dangerous, but made her numb to the pain.\n\nSarah sat for a moment, overwhelmed by the loss of her true love as the medication began to mask the hurt inside of her. She had all but stopped crying, looking out of a small window behind her desk. Hundreds of Colonial brass were outside waiting, each of them sure that Sarah Blaine was excited about taking over the military side of their government.\n\nOf course she would show them the mask of happiness, though she was slowly dying inside. After convincing herself to push through this technicality of taking command; looking forward only to the possibility of once again seeing the man who held both her heart and soul captive, Sarah left to deliver her speech.\n\n\"What's on your mind?\" Dalton asked, a look of whiskey laden concern on his face. His smile was covered in the usual scruff, unkempt hairs flaring wildly.\n\n\"Just thinking that maybe it's time to move on. Starting to give up the idea of somehow finding a way to work things out with Sarah.\" Adam replied as he took a drink from the frothy mug of ale before looking around the lodge. It was one large room built of shaven tree trunks and mortar. The perfect combination for a dwelling that was torch lit and heated by two gigantic fireplaces. It was filled with Benzans, all of which had come in to escape the unrelenting snowfall while grabbing some brew.\n\nWhile they were highly trained killers, the Benzan Mafia did its best to stay out of sight and out of mind. The small moon of Tirious provided perfect cover for them, a refuge of thick trees and constant snowfall. Bitter conditions that kept even the toughest law officers far away. In fact, aside from the Benzan settlement there was no other life on Tirious. Giving them a huge area to train, live and feel the embrace of true freedom.\n\n\"About damn time, we should have been trying to rescue Roman a week after we left.\" Dalton replied, lighting a hand rolled cigar and biting the end off, spitting it onto the floor.\n\n\"I couldn't agree more,\" Adam replied. \"What were we supposed to do though? It's pretty obvious that the Benzans have their own pecking order and we are at the very bottom of it.\" Adam added.\n\n\"They are a strange group, I'll give you that much. But the 'sumbitches treat Whiskey like he's royalty so I can live with strange.\" Dalton said before taking a lung jarring puff from his cigar and turning up a bottle of rough scotch.\n\n\"Yea, Whiskey is a hit. That much is a fact.\" Adam replied, turning to look at their four legged pet for a moment. The once mangy dog was doing a lot better now, his thick fur gleaming as he was outfitted with a small leather saddle which held several bottles of rock whiskey.\n\nHe would move from table to table, a Benzan calling for him and then trading a nice leg of meat or savory cut of bread for some of the hard alcohol. He was considered to be the one waiter that everyone loved and tipped with perfectly cooked beef.\n\n\"Little 'sumbitch eats better than I do.\" Dalton said with near envy.\n\n\"If you want I can saddle you up and let you move from table to table.\" Adam said, smiling wide.\n\n\"I hear ya. Hell, a few more stiff drinks and I might just take you up on that.\" Dalton replied as he noticed Adam deep in thought.\n\n\"My advice is just to forget about Sarah. I mean, hell, you have any idea how many women I've bed down and left sleeping? The number is staggering.\" Dalton said.\n\n\"Yes, sadly I do, we've been running together for many years. And I wasn't bedding Sarah down. Well I was, but it was because I love her.\" Adam replied.\n\nNearly a decade. That's the amount of time since Adam and Dalton first met, fighting side by side against the Legion. Although the first Glimmerian War was lost, they bonded well under fire and had been smuggling together every since. Brothers in every sense of the word outside of birth. Just as Dalton had a passion for booze, Adam had a passion for women.\n\nHe was no Romeo, though he could have easily been. Adam was a man too true to his values for that kind of lifestyle, instead having the habit of throwing his heart out there far too often. He was quick to fall in love.\n\nSarah was different though, and Adam knew it the moment they first locked eyes. He saw through her high profile, lavish clothes and impeccable vocabulary. He saw the true Sarah Blaine, the goodhearted woman beneath that had so long needed someone to see the true her. She had fallen for him because of that very reason. Although he was on the wrong side of things when they met, running from the law and God only knows what else, he looked at her the way she had only wished everyone saw her. They had fallen so madly, deeply in love.\n\n\"Love. Now there's a four letter word for you right there,\" Dalton said, sharply swigging several ounces of scotch. \"Love is an unnecessary emotion. It's for the weak minded. Do you have any idea how many women have gotten the boot because they fell in love with ole' Dalton James?\" he asked, drinking a second helping of the not so smooth scotch.\n\n\"I'm thinking none?\" Adam replied with humor, holding his hand up as if a contestant on some sad excuse for a game show.\n\n\"Shit son, you must not know me well. There have been plenty,\" Dalton replied brashly, puffing deeply on his ill constructed cigar. \"But I live by the creed.\" he added.\n\n\"The creed?\" Adam asked, scared by the notion of his friend standing for anything.\n\n\"The creed. You can give a woman your liquor but you don't ever give her your heart. Ever. That's where you messed up with Sarah.\" Dalton replied.\n\n\"I didn't give her anything, she took it,\" Adam responded. \"That's what love is. You meet someone and they unexpectedly change your life. I loved her, hell I still love her with everything I have.\" he added with conviction.\n\n\"I agree, this scotch is easy to fall in love with.\" Dalton said, drawing a strange look from Adam. Less than a second later Sasha approached their table. Her perfectly rounded hips and fur sheathed breasts were the immediate attention magnets of several men in the room, including Adam.\n\n\"What are you guys deep in discussion about over here?\" Sasha asked, smiling a bit as she sat down with the two men; her slightly curled brunette hair bouncing a bit.\n\n\"I was just telling Adam that he needs to love. Find a good looking woman like yourself and give her his heart.\" Dalton said flirtatiously as he grinned wide.\n\n\"Your friend is right Adam, you should consider his advice.\" Sasha replied. Adam continued to stare at Dalton with a ghostly look unbelief, wanting so badly to reach across the table and smack the hell out of his so called friend.\n\n\"I mean, take Sasha for instance. She is the definition of what a woman should look like. Plus she's impressive with a weapon and mighty good to you. Better scoop her up and make her an honest woman before someone else does.\" Dalton said.\n\n\"Thank you,\" Sasha said with true gratitude before turning to Dalton and noticing scotch born lust painted all over his face.\n\n\"Well then.\" she said, turning back to Adam who was still burning a hole through Dalton with a wide eyed stare.\n\n\"Hey.\" Sasha said, waiving her hands a bit to gain his attention. Adam turned to her, and though Dalton had said what he did out of lust, he was right. Sasha was woman perfected. She was not only beautiful, but treated Adam with such a level of respect. Plus the thought of her and Dalton ending up together was almost unbearable.\n\n\"Well I need a couple more stiff drinks,\" Dalton said, rising to his feet slowly. \"How about you Sasha, need a couple of stiff ones?\" Dalton asked, smiling wide as Adam immediately cast a warning stare to his friend.\n\n\"No thanks, I'm fine.\" Sasha said to Dalton as she slowly pulled her chair closer to Adam, her backside gracefully becoming one with the wooden seating beneath her.\n\n\"I'd be inclined to agree.\" Dalton replied.\n\n\"Barkeep is liable to send you packing if you ask for any more drinks. Says he has never seen someone down so much in a single sitting.\" Sasha added with a smile.\n\n\"Is that so?\" Dalton asked, turning his attention to the clean shaven bartender. \"Well in that case I'm gonna go order a few more just to piss him off,\" he added with a smile of his own, the large cigar saturated with both ash and saliva as Dalton chewed it slightly. \"Ya'll be good.\" he added, starting for the bar and turning for a moment, hidden behind Sasha's back as he did his best to mimic the two of them making love. Adam tried hard not to smile back, knowing it would give his friend's plot away.\n\n\"So.\" Sasha said, her face glowing behind a perfect smile.\n\n\"So.\" Adam replied playfully.\n\n\"We haven't really talked about us since arriving.\" Sasha said.\n\n\"Us?\" Adam asked.\n\n\"Yea, as in you and I together in more than just a friendship fashion.\" Sasha replied, still glowing but with a look of seriousness in her eyes.\n\n\"I'm not trying to avoid the whole conversation. It's just that I don't know if I'm ready to be in another relationship. It's only been...\" Adam replied before being cut off.\n\n\"It's been nine months. Nine months Adam. At some point you have to let Sarah go,\" Sasha said. \"She's gone. You have a different life now and chances are you will never even see her again.\" Sasha added, trying to talk some sense into Adam's reasoning.\n\n\"Hey Adam, come check this shit out!\" Dalton yelled from the bar, both of his hands holding large mugs fulled with ale.\n\n\"I'll be back in a minute.\" Adam said, eager to join Dalton and dodge the current conversation at any cost.\n\n\"Oh you have got to be kidding me.\" Sasha said under her breath as she glanced at the bar to find out why Dalton had stolen Adam.\n\n\"Still a looker, even if she did try to lock me up.\" Dalton said as Adam sat beside him and noticed Sarah Blaine on the large television behind the bar. Dalton had started to explain that she was being promoted to Colonial Commander, but his words were simply mumbled as Adam blocked everything else out and focused his attention onto the woman he had so passionately loved. The more she spoke, the more Adam was reminded of his strong love for her to this very day. Even if it was a one sided feeling, it was a feeling that he couldn't control and his heart pushed his hands to begin trembling slightly.\n\nAdam broke his attention from the television for a moment to casually glance across his shoulder at Sasha. His intentions were to see if he felt the same when looking at her, but his eyes were met with her own as she stared at him with discouragement. Adam of course smiled and tried to act as though his very pulse wasn't directly connected to Sarah's speech, however his head quickly turned back to watch it.\n\nEverything about her was as beautiful as he had remembered. Her hair sparkled with red purity and her eyes told the truth about the beauty of her very soul. It was a forbidden love now, but for Adam it was a love that he continued to find himself lost in.\n\n\"I'm telling you man, between you and me, she was a catch.\" Dalton said in a hushed voice.\n\n\"A catch,\" Adam replied sharply. \"You think I don't know that man? I would do anything to go back to life with her, even if it did mean the Hunters on our asses,\" Adam added, looking back at the television for a moment. \"Besides, what happened to not ever falling in love?\" Adam asked.\n\n\"Yea, well yea,\" Dalton replied, firming up on the bar stool a bit. \"I'm just saying if someone was stupid enough to fall in love, which we both know they shouldn't, it should happen with a woman like that.\" he added.\n\n\"Yea, tell me something I don't know.\" Adam said, once again turning to the television and watching Sarah Blaine's every movement.\n\nHer features were so perfect, her movements soft and her intentions true.\n\n\"Sasha.\" Dalton said a bit under his breath.\n\n\"No, I said tell me something I...\" Adam said, cutting his words short as Sasha took a seat right beside him. Adam glanced over at his sorry excuse of a wingman for a moment, nodding his head as if to ask Dalton to leave them be.\n\n\"Well, I best be going to check on Whiskey.\" Dalton said standing up and tipping his head slightly to the Benzan beauty, only to moments later mock a session of intimacy behind her back. Again Adam gazed at him, stone faced and tight lipped, turning his attention to Sasha.\n\nShe sat there, smiling back although it was obvious that she was truly worlds away in thought.\n\n\"Listen. I don't want you to think I'm avoiding the conversation that we both know needs to happen. It's just that I feel something very strong for you and it scares the hell out of me.\" Adam said.\n\n\"You do? So just to clarify, this feeling isn't one sided?\" Sasha asked.\n\n\"Of course not,\" Adam said, smiling. \"For God's sake I pulled a gun on my wife to be just to get you to safety.\" he added.\n\nIt was the truth. Adam did in fact have feelings for Sasha. Strong feelings. But they were cloudy at best, and he was doing everything he could to figure them out; in need of more time to do so.\n\n\"So where do we go from here Adam Michaels?\" Sasha asked.\n\n\"That's the big question,\" he replied. \"I just need some time to think things over and clear my head. Ya know? Spend time getting to know you without bullets flying at me.\" he added, bringing instant giggles to Sasha's face.\n\n\"You truly are amazing Adam. Take all of the time you need then and just know that as a Benzan I can't guarantee it will be done without bullets flying at you.\" she replied, causing Adam to ease up and laugh a bit.\n\n\"Things will work out for us in the end. Right now I just want to live free and do it with you in my life.\" Adam said, smiling at her as he held her hand a bit.\n\nWas he starting to fall for Sasha? The thought had entered his mind more than once. He knew without a doubt that she truly was in love with him and he didn't have to become someone else to earn it. But what about Sarah Blaine? He simply couldn't ignore the feeling he had been instantly overcome with as he watched her on television only moments before. He couldn't breathe, it felt as though he would die if he couldn't hold her in his arms once again and that wasn't fair to Sasha. She deserved the best man he could be, and that wouldn't be something he could offer until his heart was completely free.\n\n\"Well, looks like we're needed.\" Dalton said, regretful that he would have to leave perfectly good drink behind and turning up his mug to finish as much as possible before approaching their table once again.\n\nAdam turned to see Kraid enter the building, he was the man in charge and it showed. Every single Benzan falling silent as he entered under the guard of two heavily armed warriors, not that he needed it. His legend of ending lives was one of great lore, the very broadsword sheathed to his back had sent a hundred or more Hunters back to the hell that they had been spawned from. He casually glanced across at Sasha, nodding slightly as his curly black hair shifted with ease. As he turned to exit back into the unforgiving peace of the falling snow, he pointed to a table near the entrance and motioned them along as well. Both men stood up slowly, their posture firm as fur covered their otherwise striped dress suits of black silk.\n\n\"Time to put together a rescue plan.\" Adam said under his breath as they all stood to their feet slowly before following Kraid.\n\n\"About fucking time.\" Dalton replied with the smell of must tainted whiskey on his breath as both men waited for Sasha to lead the way.\n\n\"Let's go meet the man.\" she said with a calm resolve as the three walked out into the serene blanket of white. They had been among the Benzans nearly nine months now, and although Adam and Dalton had both caught sight of Kraid in the past, neither man had actually spoken to him.\n\nDalton stopped for a moment, slapping his leg and giving a curl of the lip with a whistle quickly following. Seconds later the clanging of bottles could be heard as Whiskey sprinted out into the snow, running just a bit to the side as the weight of empty bottles in the saddle pushed onto him.\n\n\"Poor 'sumbitch done got vertigo from carrying all this alcohol,\" Dalton said jokingly as he laughed alone and knelt down to pet his most loyal of friends. \"Oh hell no, is this mustard?\" he added, rubbing his fingers harshly across the mutt's back to remove a small mustard stain. \"Oh hell fucking no.\" Dalton added, glancing gratingly back into the door of the lodge and secretly wishing he knew the identity of the condiment perpetrator.\n\n\"Come forth!\" Victoria shouted as she sat in the large throne chair sculpted of rock and gold.\n\nThe Hunter Queen had been awakened following the killing of so many of her species at the hands of the Benzan Mafia. Garrison, the highest ranking of the surviving Hunters from the second battle of Glimmeria approached nervously, kneeling as low as his trembling body would allow.\n\n\"RISE! Tell me of your failures!\" Victoria shouted, standing quickly to walk down and face the unfortunate survivor.\n\n\"My...my Queen,\" the Hunter said shakily. \"The Benzans ambushed our ships and hunted us on the surface...\" he added before his words were cut short.\n\n\"You survive to tell of this story while Anwick lays on a stone slab in the next room,\" Victoria said angrily. \"We are the most feared race in the Skyla System. Do you know why that is?\" she asked with focus.\n\n\"No my lady.\" the Hunter replied out of necessity.\n\n\"Because we always get retribution against those who have betrayed us. Always. You and the other survivors have betrayed me by not fighting to the death to save the life of Anwick.\" Victoria said before turning to walk back to her throne chair and once again be seated.\n\n\"Take him!\" Victoria commanded as two Hunter Elites walked into the throne room. They were a rare sight, even for the fearless members of the Hunter race. The Elites were more like Knights, outfitted with thick black armor and the pupil free eyes of a demon.\n\nThey were a rare sight because there were so few of them, only the finest mortal warriors earning the right to immortality. It didn't take many. One Hunter Elite had the abilities of an entire room full of his standard kin, and they were heartless. Much more heartless. They would slaughter an entire room full of their own kind in order to kill someone who posed a threat to their queen. It was their sole purpose to protect the Queen, a job they did well. And at this very moment they followed a direct order from the one who commanded them, dragging the doomed Hunter into a room filled with the rest of what few survivors had returned from Glimmeria.\n\n\"Please my Queen, I beg of you!\" Garrison yelled as he tried unsuccessfully to escape the grasp of an Elite before resorting to grasping for what little texture the stone wall provided. His large vampiric fingernails dug into the wall, but the sheer strength of the Elites continued forward; splintering the doomed Hunter's nails as one of the Elites pulled a longsword of grey and black. As the door of thick rock and worthy steel shut tightly, screams of her own race began to filter throughout the throne room, a slaughter of cowards the source.\n\nNot so long from now, his flesh along with the flesh of others slain would become feast for the Queen and her most loyal. Tissue tasted the same once applied to the burning of flame, no matter human or otherwise. Cindered meat was the preferred dish of Hunters, and if their hunger grew enough, the source of meat was of no concern. They had their fair share of enslaved humans for the very reason of food, though the occasional failures of a Hunter would bring the Queen to the point of no mercy. They indeed feasted on their own kind.\n\nAfter only a couple of minutes the screams halted, the sound of razor edged blades chopping flesh were all that remained. The door opened once more, creaking heavily as the two Hunter Elites emerged to bow before their queen; covered in both blood and loyalty.\n\n\"Rise my warriors. It is retribution against the Benzans that I now seek. Gather our most battle tested soldiers and see to it personally that not a single Benzan heartbeat sees another sunrise.\" Victoria said confidently.\n\n\"Yes my Queen.\" one of the Elites responded, his voice deep and mighty as if he were a God of battle.\n\nBoth Knights stood firmly and walked away, blood dripping from their armor and forming the path taken on the harsh stone floor of the Hunter fortress.\n\nAs the large monsters of flesh walked slowly through the halls of the Hunter compound, the Elites soon slowed their pace as they approached a large metal door. It was thick steel, both gothic designs and Hunter text forged into the face of it. As the door made a low grinding noise against the rock mounting, both Elites approached their military leader. Vladris. He was one of the few Hunters to claim stake on a hell hound, a dog born again into the ranks of the undead.\n\nThey mutated a bit differently during the turning process, of course, and their skin became thick. Almost leather-like with rising patched across their backs. It gave the look of thorns for eyes that knew no better. The hell hound slowly turned as the Hunters approached, recognizing them a bit while showing his teeth in protection of his master. Its teeth were thick, although needle-like and nearly alive with hunger. Its eyes subtly glowing of crimson red as it made a strange noise of warning. Hell hounds were not to be trifled with. An ability to end lives with ease behind their shard filled smile and eyes of fury.\n\nThousands would be the best estimate when it came to lives ended by the blade of Vladris, the same blade that was tightly strapped to his back as he looked across the bannister of the balcony. Built into the side of a stone mountain, the Hunters had fortified their base to protect and serve their Queen. Though they never slept, each Hunter had a period of downtime in which it would rejuvinate itself.\n\nVladris had been turned by Hunter Elites many years ago, giving such a valiant fight in protecting his King. He had been a Ronical Knight, at least throughout his human life and up until the Hunters felt the need to turn him into an undead warrior. The Ronical Kingdom fell shortly after, succumbing to the rule of Hunters. However, the fact that Vladris in human form had slain two of the mighty Hunter Elites earned him a bridge to immortality when he finally was killed in battle. The hand of a surviving Elite turning him in only minutes. His former kingdom was one of nearly endless rainfall, and the sight of it brought a calm to his soul. At least what remained of it, as Vladris looked across the bannister and into the thick jungle as rain poured heavily, leaving with it a serene melody.\n\n\"Our Queen has commanded us to assemble a group of warriors. One that will prove the demise of the Benzans.\" one of the Elites said as Vladris continued to stand, his hands wrapped around the steel bannister as he looked into the heavy downpour.\n\nFor him it was soothing, and many nights he would spend his downtime simply gazing into the rainfall and enjoying the gift of immortality.\n\n\"Then it shall be so,\" Vladris replied, slowly turning to face the others, reaching down for a moment to calm his hound. \"Handpick our finest and outfit them with the basic tools of slaying. I will personally see to it that our Queen's wishes are carried out.\" he added in a rough but dedicated voice as both Elites nodded and turned to execute his order.\n\nWhile they were all Elite Knights for the Queen, the rest knew Vladris was by far the strongest. A few even suspected the Queen had secretly feared him, something Vladris had also picked up on. His loyalty ran deep, but the fact that he was unstoppable led to a bit of envy by the Queen. The Hunter race followed Victoria out of fear, but on the battlefield they followed Vladris out of respect.\n\n\"You shall soon feast on an endless buffet of man flesh.\" Vladris said, his muscular build kneeling to show affection to the most loyal of pets.\n\nThe hell hound, which stood nearly three feet high, seemed soothed by the voice of his master, turning his head slightly as if to show off his pale white hide.\n\n\"The speech was great my lady. Exactly what our people needed to hear.\" Lieutenant Lassiter said calmly as Sarah sat in a chair of velvety red plush.\n\n\"Thank you. I'm not sure how I got through it, Adam is the only thing on my mind,\" Sarah replied. \"I should be focused on the upcoming confrontation with the Legion, but I just can't. I feel so helpless.\" she added, turning her head a bit to avoid tears.\n\n\"About Adam my lady, I have a plan,\" Lassiter said as he approached Sarah to sit in a chair beside her. \"We have solid information on the Benzan's favorite hideout.\" Lassiter said as Sarah looked at him with unconvinced eyes.\n\n\"They are not a group to be negotiated with, so the hard part will be trying to convince the Benzans to allow us to speak with him,\" Lassiter added. \"I will assemble a strike team, and then plead with Adam to at least return here to meet with you,\" Lassiter said.\n\n\"I have known Adam for some time now. I feel confident he will at least meet with you Sarah, as long as I promise him he's not walking into a trap.\" Lassiter added.\n\n\"No. I will plead my own heart, it's the only way he will listen. I'm coming with you.\" Sarah said.\n\n\"Sarah, listen. We are to begin a major assault on the Legion in less than one day, not to mention the Benzans are not to be taken lightly. There is no guarantee that I will even walk out of there unscathed, the last thing our people need right now is to have its leader in harm's way. I would advise you to stay here and...\" Lassiter replied, sharply cut off.\n\n\"I'm coming Lieutenant Lassiter, that's an order. I ruined a good thing with Adam. It's my fault and now I have to try and make things right and hope he can forgive me. Just make sure the crew we take with us is loyal and capable. I will not let fear of the Benzan people stop me from seeing the man I love again. I'll make sure our war effort is in strong enough hands.\" Sarah said.\n\n\"At once.\" Lassiter replied, turning to leave the room and assemble the best he could. He still wasn't sold on the idea of Sarah coming, but he was the ideal soldier.\n\nShe had given him a direct order and he wasn't the kind to disobey them. All he could do now is place as much of a safety net around her as he could, and give his life for her if needed be. Especially with his growing feelings for her, though they remained hidden. Sarah continued to stand there, looking across the stars through a huge window of reinforced glass and Colonial writing.\n\nShe wondered if Adam ever thought about their near marriage, or if he even bothered himself with thoughts of her. These past nine months had seemed like decades to her. Things were different now, and that was something that scared the hell out of her.\n\nTameca City was once considered one of the best locations to live and prosper. Of course that was before war had torn throughout the Skyla System and pushed the reeling Legion into Tameca City seeking refuge. They were all but beaten, having lost the second war of Glimmeria, but more importantly the support of several planets following the loss. Planets who had shifted to the Colonial side, and now what remained of the Legion consisted of worn down troops, mercenaries and forced volunteers from planets once controlled by the ailing military.\n\n\"They will be on us within days! Our own scouts have confirmed it!\" Lieutenant Rommel shouted across the crude steel table as three more Legion officers remained seated.\n\n\"It takes only one victory, one show of force to turn the tables in any fight.\" a highly decorated officer replied calmly as he remained sitting.\n\n\"My Lord, there can be no victory here! We must issue a formal order of surrender!\" Lieutenant Rommel answered, shouting his reply furiously.\n\nLord Riven stood to his feet slowly, as the highest remaining officer left among Legion forces, he embraced the fight to come and expected no less from those who served beneath him. Standing to his feet, dark red uniform of cotton stitch and black trim; a look of disappointment pasted onto his face as he held a Legion sidearm out and blistered a shot into the forehead of Lieutenant Rommel. Standing for a moment, Lord Riven casually wiped spotted blood from his officer's jacket as the lifeless shell of a body that had been Rommel abruptly fell to the floor.\n\n\"Sometimes, it's simply a matter of rebroadcasting fear into those who look upon us,\" Lord Riven said, slowly sitting back down and laying the glock style pistol onto the cold steel table in front of him. \"There will indeed be victory here, and I will not tolerate anything less. Are you with me?\" Riven asked sternly as both remaining men looked on.\n\n\"Yes my Lord, to the end!\" one of the men said loudly, reaffirming his position in the regime.\n\n\"We'll see.\" Lord Riven replied, standing slowly to walk out of the room and deliver his next order.\n\nIt would be an order that would change the complexion of the war forever. Every epic tale of war had to include a good guy and a bad guy, it was a story as old as time. Lord Riven didn't just accept the role of bad guy, he embraced it. In order to be bad, truly bad, one had to be feared.\n\nFear was earned through acts of violence so savage that onlookers dared not question. Only fear would turn the tide of this war now, and Lord Riven was the perfect man for the job. As he slowly picked up a com unit that would relay his orders throughout every Legion post on Tameca, he smiled a bit. Almost as if pleasure would be forthcoming.\n\n\"I will be away briefly as the rescue mission begins.\" Sarah said as a filled room of both Colonial brass and high ranking soldiers listened.\n\n\"Rescue mission?\" one of the soldiers sitting near the front of the room asked. Sarah looked across to the man who would be handling the Colonial side of the upcoming battle, General Ortega.\n\n\"Yes,\" she replied, turning to answer the soldier directly. \"We received word only minutes ago that Legion forces in Tameca City have started opening fire on civilians. Women, children,\" Sarah said, taken back by the freshly received report. \"Anyone who is refusing to join their war effort is being executed on the spot,\" she added. \"General Ortega will lead our troops into battle, and it appears that we will be doing a bulk of fighting on foot. The Legion appears to have little to no air support left, which means more casualties on our end as well. We will have to meet them on their turf.\" Sarah said.\n\n\"I have asked General Ortega to begin the strike sooner than expected in order to save as many civilian lives as we possibly can. However, your lives will be on the line as well, so I thought it only appropriate to ask all of you face to face.\" Sarah added. One of the Colonial Spec Ops soldiers near the back of the room stood to his feet, looking around for a moment before refocusing his attention onto Sarah Blaine.\n\n\"We'll go. And when we arrive, the Legion will fall.\" the soldier stated proudly as the room began to cheer.\n\n\"Good. That means you are all out of gear, get your asses in check and double time it to the launch bay!\" General Ortega shouted as the room of cheers turned to a full eruption of motivated yelling.\n\nAs the group of soldiers began to file out of the room with courage and conviction, they immediately halted and joined the crowded lobby who stood quietly, stunned by the video footage being broadcast across the mounted video monitors on the adjacent wall. News coverage sending both still images and video feeds out of Tameca City, which was mostly engulfed in flames as explosions rocked throughout the city at random. Legion firing squads lining up and gunning down helpless civilians as they attempted to flee their homes. Sarah began to tear as did many of the Colonial soldiers, still images of bodies piled several feet high being shown, filled with both women and children who were still cindering from a fiery death.\n\nThey watched as Lord Riven gave an interview to the news team, bragging of the Legion's hold on Tameca as he promised to continue the killing while daring the Colonials to intervene. General Ortega looked onto the gutless murderer as he continued his interview for the news crew, smiling as though he were proud of killing so many innocent.\n\n\"The Colonials would be wise to stay clear of Tameca or any other Legion controlled ground.\" Lord Riven stated in a threatening manner as the Colonial soldiers stood silent, watching in horror.\n\n\"Let's go goddammit, these people need us!\" General Ortega shouted, doing his best to break the shocking quiet of his best soldiers while showing them such threats did nothing to intimidate him. And they didn't, in fact General Ortega now thought of the upcoming battle as personal and would have gladly fought it for free.\n\nOther planets were sure to send help as well, but most were days out even by full burn. The Colonials were the closest force, stationed in Glimmeria and stocked with enough firepower to possibly put an end to the massacre at hand.\n\n\"General Ortega, I am more than willing to come...\" Sarah said as the stocky man turned for a moment to cut her short, scars across his face from past battle experience.\n\n\"Thank you for the offer my lady, but that is not necessary. I will not stand for war crimes such as these and give you my solemn promise that when the black of our boots hit Temecan soil, the only massacre will be against those who have committed such butchering. You just work on bringing Adam back to us.\" he said, smiling a bit before turning steadfast and following his soldiers to the three Colonial Star battleships which would be under heavy escort by several well armed ships filled with battle hungry troops.\n\nThe city of Rockheed began to tremble as though a minor earthquake had set in, moments later the large outlines of Colonial ships both massive and commanding respect made their way into the pale green sky of Glimmeria.\n\nCitizens stopped to witness such a fantastic force of freedom seekers heading for war, and leaving behind just enough soldiers to hold the fort down. Ninety percent of Colonial forces were propelling into the sky, their ships in full burn and plotting course for Tameca City, and as the fully equipped soldiers glanced out of the windows lining each ship, the onlooking citizens quickly vanished and were replaced by scattered stars as they hit deep space in full burn.\n\n\"So Adam, Sasha tells me that rescuing this prisoner Roman Raines is a priority for you?\" Kraid asked as he sat calmly in his thick leather chair, a long desk of polished wood separating them.\n\n\"That's right. He stayed behind and gave us an out, otherwise we'd all be locked up.\" Adam replied.\n\n\"You've been with us now, what six months?\" Kraid asked. \"Nine months actually.\" Adam replied.\n\n\"You could say six though because I've been drunk for at least three of 'em.\" Dalton added as both Adam and Sasha turned to stare a hole through him as Whiskey barked loudly to testify to the truth of it.\n\nIt was the moment Adam began to turn his attention back to Kraid that he first noticed it. A photo behind the desk on a small bookcase, Kraid holding Sasha with sheer love on the faces of both. They were ex-lovers, and that complicated things even more for an already overwhelmed heart that slowly beat inside the chest of such an honest man.\n\n\"It's very important we get our friend back,\" Adam said to Kraid, before slowly turning to Sasha. \"He's always been honest with me and that means everything.\" he added, letting Sasha know right away that he had become wise to her past relationship with Kraid.\n\nAfter listening to Adam and staring at Dalton for a moment, wondering what kind of first impression to make, Kraid reached into a door on his desk and pulled out a handgun of gold. Fully functional, it was an semi-automatic pistol although a bit larger than the standard. The bright light from the snowy day outside gleamed into the room and refracted off of the gold plating of the piece as Kraid laid it onto the desk, pushing it into Adam's direction.\n\n\"Every ship's Captain carries one of these, about time you did the same.\" Kraid said with a slight smile, glancing at Sasha for a moment himself.\n\nIt was Sasha who had ended things between them after life became too complicated. Kraid had never gotten over her smile, the touch of her hand. He still held very strong feelings for her even though he dared not show it. As he and Sasha exchanged glances, Adam felt that there was a bunch of history between them, history that couldn't have blindsided him at a worse time.\n\n\"The intel I have on the Gali ship is that it is maximum security, which is why I'm sending two of my very best along with you,\" Kraid said as the two men from the lodge slowly entered. \"Captain Michaels, meet Primal and Stage.\" Kraid added.\n\nIt was easy to see where Primal got his nickname. He looked more beast than man as both Adam and Dalton stared at him to determine if he was actually human or another race entirely. Indeed human, he was badly scarred with bushy brown hair on both his scalp and face. A very tall and stocky man, Dalton found himself instantly becoming more attractive standing beside Primal.\n\nStage seemed to be the opposite. Well dressed with a finely trimmed pinstripe beard of black that perfectly meshed with his crop cut hair. He wore a set of reflective sunglasses and carried a very large pistol on his side that the Benzans referred to as a hammer. It was the most common firearm among their people, its deep and throaty firing sound resembling the strike of a hammer. \"If your friend is still aboard that ship, we'll get him.\" Stage said with calculation.\n\nAdam sat there in deep thought, he didn't like the idea of two men who he knew nothing about coming along for the ride. Especially having just pieced together a past relationship between Kraid and Sasha. He wanted to believe if it came down to it, they would put their lives on the line to save his, but he just wasn't convinced. In fact, he wasn't even sure he knew Sasha at this point.\n\nIt's not the the fact that she had a past, everyone has a history. It was that he had to find out about it by surprise when he had already started to favor her. This whole rescue plan was starting to worry Adam, but it was his only option at this point, and Roman had been locked up far too long. Adam owed his own freedom to the man and the least he could do was return the favor.\n\n\"I appreciate the help.\" Adam said, standing slowly to his feet and extending his hand. Kraid, who wanted to hate Adam so badly for capturing the affections of Sasha, extended his arm to shake hands. It nearly killed him, shaking hands with a man who would be walking back outside with the woman he loved so much. But she had made it clear that her love life wouldn't include Kraid, leaving him helpless to do anything more.\n\n\"I'll have my men fuel a ship for you.\" Kraid said, smiling a bit. Is this son of a bitch planning on killing me? Adam thought as he nodded while broadcasting a fake smile. He didn't want any trouble, but just like his life up until now, doing the right thing always seemed to surround him with it.\n\nBoth Roman and Zane stood there with a small group of prisoners, watching the new arrivals slowly exit the Gali shuttle which brought them in once a week. While the rest of the group meticulously watched fresh meat exit onto the steel deck of the prison grinder, wondering who they could intimidate and control; Roman and Zane continued to look the shuttle over.\n\n\"It's a Zion 400. Gonna take at least three men to pilot it out of here, and no less.\" Roman said as he studied the ship from its rounded silver nose to the dual thrusters which were mounted onto the rear.\n\n\"Going to have to try it with two, I don't know anyone else we can trust enough to include.\" Zane said under his breath.\n\n\"I do.\" Roman replied with confidence as he watched a large man exit the shuttle under heavy guard. Slightly larger than Roman, the man's physique looked as though it had been etched with the sharpest of knives as his closely cropped black hair seemed as confident as his strut.\n\n\"Yea, and who's that?\" Zane asked with interest.\n\n\"My brother.\" Roman added as he continued to stare at a man whom he had close ties with. Quinton.\n\n\"So that's when they caught me,\" Quinton said as he sat at the steel table, its legs bolted to the floor of the recreational area as both Roman and Zane listened. \"I had an entire shipment of weapons, warrants for my arrest on a dozen or so planets and the blood of a Hunter on my hands.\" Quinton added.\n\n\"A Hunter? Can't honestly say I have ever met someone who has fought one and lived to tell about it.\" Zane replied.\n\n\"Really,\" Quinton asked, turning to look Roman in the eyes. \"I take it he doesn't know?\" Quinton said.\n\n\"Know what?\" Zane asked as Roman began to reflect on a past that he had so long tried to leave behind.\n\n\"I'd say you have met someone that has fought a Hunter and lived to tell about it,\" Quinton said with a bit of laughter. \"Roman here has killed hundreds of the bastards, including one of their Queens.\" he added.\n\n\"Bullshit.\" Zane said sharply.\n\n\"No, it's true. I can't believe you've been locked up with him for this long and it hasn't come up yet,\" Quinton replied, gaining a stern look from Roman. \"Ah hell, I'd say it's alright to talk about it now. You've been caught. We both have. Might as well lay it all out there for him.\" Quinton added.\n\nAfter a few moments of looking at Quinton, Roman turned to Zane, ready to reveal at least part of his past.\n\n\"Several years back, my unit was dispatched to work security for the signing of a treaty,\" Roman said reluctantly. \"Seemed like just another security detail, I figured I would be back out drinking a few hours after the politicians did their thing,\" he added as his eyes began to tear just slightly. \"When I was eight I watched both of my parents get murdered in cold blood. Goddamn Hunters cut them down in the middle of the street like they were garbage. I tried to fight them off but I wasn't anything more than a boney ass runt.\" Roman added before taking a long pause.\n\n\"When I walked in to secure the conference room for the treaty signing and found out my own government was signing a treaty with the Hunters, I lost it,\" he added. \"I waited for their Queen to arrive, and even with a large security detail of her own I struck. I buried a blade flush into that bitch's skull and twisted it hard enough to know for a fact she wasn't walking back out of there.\" Roman added.\n\n\"I remember hearing about that. The shit was all over the news. The Greyspine Massacre.\" Zane said.\n\n\"Yea. When I struck Queen Lethra, I guess the Hunters thought it was a coordinated assassination and all hell broke loose in that fucking room. We were the absolute best our government had to offer,\" Roman said, pausing for a moment to glance at Quinton. \"But they had their Elites and eventually I realized there was no winning the fight, it became a matter of survival. Two of my own damn men held me at gunpoint when I tried to take a shuttle. Those are the only two men I have ever killed that I regret day and night.\" Roman replied.\n\n\"No wonder the Gali paid such a high price to lock your ass up.\" Zane said.\n\n\"They have been tracking me down every since, trying to bury me under a prison. The Hunters have been tracking me to cut me into small pieces and I knew deep down that eventually one of them would catch up to me. Can't run forever.\" Roman said.\n\n\"Don't have to run anymore little brother, I got your back.\" Quinton said.\n\n\"After my parents were struck down for nothing more than sport by the Hunters, Quinton and his family took me in. Raised me the best they could,\" Roman said. \"I've dedicated my life from that point on considering them my family and trying to kill as many of the Hunters as I fucking can.\" Roman added.\n\n\"So we can trust him as our third man?\" Zane asked, already knowing the answer.\n\n\"You can trust him with your life, Quinton hasn't ever let me down.\" Roman replied.\n\n\"And I'm not about to start now.\" Quinton added, turning to Zane as a prison guard slowly walked behind them. \"Good. Very good.\" Zane said in a low voice.\n\nNormally dressed for only the most glamorous of events, Sarah was outfitted in a snug fitting pair of blue military pants and a long sleeve shirt which was stitched from thermal material.\n\n\"And you trust everyone coming along?\" Sarah asked with hope filled intent.\n\n\"Yes my lady, they are absolutely your most loyal followers.\" Lassiter said as they boarded through the small entrance which was reinforced with thick bolts and blast shielding.\n\nSarah stopped, glancing across the crew area of the cabin into the faces of the men in which she placed her life into the hands of. There were three of the very large Husk race, barbaric as well as beast like, they bore a heavy resemblance to the mythical minotaurs of old. Two Goliath V2 soldiers were seated at the rear of the shuttle, both programed to use their robotic frames and small mini-guns to defend Sarah at all costs. Finally, Lassiter had hand picked three Colonial soldiers. Human, as well as the most able and trusted he knew. Each had lost so much throughout the war and looked to Sarah with the utmost respect for leading them into the direction of freedom.\n\n\"Umm, Lassiter. Maybe we should discuss the group you have selected.\" Sarah said, unconvinced of this being a group she could trust. One of the large Husk warriors stood to his feet, his near eight foot frame only inches from touching the ceiling of the shuttle. Walking into the direction of Sarah, the thick hair which covered his bulging arms gave them a grizzly appearance as he stood before her. Kneeling down to the floor, the Husk stared low with loyalty.\n\n\"My lady, it is truly an honor to be standing before you this very moment. Please know that I will use every fiber of my being to defend you to my very own death.\" Sarah looked taken back by such loyalty coming from the heart of a warrior who could easily have broken her into pieces.\n\n\"Never mind, they'll do.\" Sarah said calmly as she took a seat with Lassiter near the front of the shuttle.\n\n\"Tigon Twelve to tower. Requesting permission to launch.\" the shuttle's pilot said into his helmet mounted com unit.\n\n\"Affirmative Tigon twelve, safe voyage.\" a voice replied loudly over the console mounted speakers which surrounded the pilot. \"Only the brass know you are onboard, I told everyone else that I was taking out a strike team to bring in a Colonial fugitive,\" Lassiter said. \"I figured the less people in the loop, the better.\" he added.\n\n\"Good thinking.\" Sarah said, continuing to stare out of the window and hoping with every piece of her soul that Adam Michaels would forgive her.\n\nAs the shuttle pulled slowly from the deck of the ship and made its way into the void black of space, Sarah continued looking through the small window positioned near her seat. How could the survival of my heart depend on a single man when all of this exists? Sarah thought as she imagined so many planets around them filled with a variety of people. She started to think about the times shared with Adam.\n\nHow, only minutes after meeting, Adam had risked his life for hers and that of a young boy; going toe to toe with the Hunters in the process. In fact, he had went to bat for her so many times, only to have it thrown back into his face when she asked him to leave.\n\n\"Are you alright my lady?\" Lassiter asked quietly, the shuttle traveling quickly through the curtain of twilight.\n\n\"Yes,\" Sarah replied, her stare into space never faltering. \"Just thinking about everything Adam did for me. I was such a fool.\" she added with truth.\n\n\"Adam is a smart man when it comes to the character of a person. I know you are sincere, and I'm sure he'll see it too.\" Lassiter said.\n\n\"I truly hope so,\" Sarah replied, turning to Lassiter with trembling eyes. \"I miss him so much.\" she added.\n\n\"Attention,\" Kraid yelled, instantly silencing the crowd of Benzans who had been drinking and socializing in the lodge. \"I have just been informed by some very credible sources that the Hunters are assembling soldiers to invade us. They would come to our lands in an effort to spill our blood,\" Kraid added, many loud shouts following his words from Benzans who were ready for a fight. \"I say let them come! For they will not find trembling women and children here, they will not find the usual intimidated cowards they face; only battle tested slayers of the undead!\" Kraid yelled holding his broadsword high into the air, its handle wrapped in leather bonding as the entire room filled with nearly two hundred warriors shouted uncontrollably. The noise pierced not only the smoke filled lodge, but the valley behind the Benzan settlement, crisping the leaves and snow with echos of madness.\n\nThe loud shouting of insanity fueled by rage continued, as did the billowy grey stacks of smoke which climbed from their fires and into the heavens. And the snow continued to fall relentlessly. This was their home, their fields of killing and though the Hunters were damn formidable in combat, the Benzans welcomed the challenge with open arms.\n\nFrom the time a Benzan child learned to walk, they began training. For years they would learn hand to hand combat, everything from breaking bones to the finer points of strangulation. When they reached their teenage years, only the best of Benzan women continued training with men. Wielding anything from a long blade to a compact firearm, they were taught everything about killing. How each of their more notorious foes were slain easily and how the Benzan code demanded of them that no fight should ever be avoided. While the lesser Benzan women began learning survival tactics, the men and stronger women simply learned how to slay without regard.\n\nThey knew all too well that the Hunters were damn hard to kill, almost monster-like in their way of reaving humans. They did not care. When it came to battle, no man, human or otherwise, would ever witness a Benzan back down. Ever. They would fight to the death for the most simple of reasons, and when it came to defending their homes and protecting their families, they would slay any man or beast who stood in front of them.\n\n\"I say come God damn you! Come!\" Kraid yelled loudly, once again holding his sword into the air, its pale blue complexion in need of salty warm blood.\n\nThe sword had been with the Benzan people for centuries, passed down from leader to leader as a token of authority. It had slain so many. As the gigantic arms of Kraid held the blade high into the infusion of smoke and snow, the Benzans yelled wildly. Waiting for the arrival of the immortals so they could be immortal no longer.\n\n\"Adam,\" Sasha said, quickly following him out of the lodge and into the heavy fall of cloud born snow. \"Adam wait, we need to talk about this.\" she added, stopping him in his tracks.\n\n\"No Sasha, we don't.\" he said, turning slowly to face her with the serenity of nature flooding around them.\n\n\"I'm sorry Adam, I didn't think it would matter. It was a long time ago.\" Sasha said, visibly upset.\n\n\"It doesn't. The fact that he obviously still has feelings for you combined with the fact that I was starting to fall for you myself. That matters.\" Adam replied sharply.\n\n\"What are you saying?\" Sasha asked, moving in a bit closer as the snow fell poetically around them.\n\n\"I don't know Sasha, every single time my life begins to make sense it gets turned upside down. I never thought I would say this, but I need calm. I need something solid that I can count on for the rest of my life. Lately I haven't been able to find that.\" Adam replied.\n\n\"I can be that Adam Michaels.\" Sasha said softly, gently putting her arms around him and kissing his lips slowly as luster filled flakes fell from the sky around them.\n\n\"My Lord, do you wish us to stay and fight?\" Stage asked nearly in a yell to be heard over the chants of a war to come.\n\nKraid had seen the kiss between Adam and Sasha, his stare fixated onto them through a small window in the lodge was crushing to say the least.\n\n\"No. I promised Adam we would help free his friend and I am a man of my word. Go, and when you return you can help us count the slaughtered beasts,\" Kraid said confidently. The group of Benzans yelled with hellbent fury as Dalton found himself yelling if for no other reason than to give himself an excuse to raise a little hell without recourse, Whiskey even joining along with vicious barking. \"And when the job is complete, Adam and his friends are not to return. Am I clear?\" Kraid asked in a whisper.\n\n\"Yes my Lord. Crystal.\" Stage replied.\n\nThe Hunters sat in war equipped shuttles inside the face of a large mountain as gripping rainfall moistened the surrounding area. Five shuttles, each holding nearly twenty of the most battle tested Hunters, along with two of the Elite variety. While they would all carry the standard Hunter designed rifle, which was a semi-automatic weapon capable of punching holes through the thickest of men, they were on a different mission. One fueled on revenge and retribution, and for those very reasons it was their intention to cut their foes into pieces. Slicing flesh from bone with perfectly edged swords which they each carried, harnessed onto their backs and nearly alive with the hunger for blood.\n\nAs the five shuttles slowly began to pull from the rocky terrain, many of the undead warriors gazed through the small windows into a curtain of rain and wondered if it would be the last time their eyes would see the gothic beauty of the Queen's cavern. A series of several large doors cut into the face of a mountain with outlines of tribal style writing surrounding them. As feared as the Hunters were, they knew well the ferocity of Benzans in battle. In fact, the Benzans were famous for it. That said, Hunters still considered themselves at the top of the food chain and this was a perfect example of an unstoppable force clashing with an immovable object. Many of each species would perish in the battle to come, if not all of them, which made the presence of Hunter Elites among them bring comfort.\n\nLegend tells of the rise of Hunter Elites. A time when scattered nations of vampiric beasts clashed with one another, the Elites were born to defend their Queen. They were the epitome of horror, easily slaying the most dominant foes in battle, and that was reason enough for the Hunters heading to the upcoming slaughter between nations to feel a bit more confident as they set course for the well known location of the Benzan hideout.\n\n\"The plan is pretty damn simple,\" Zane said as both Roman and Quinton looked on. \"When the next transport shuttle arrives in a week to deliver new prisoners, we take the ship.\" he added.\n\n\"How the hell do you plan on doing that? We were under heavy guard the entire time, at least six riot ready soldiers aboard the shuttle.\" Quinton said.\n\n\"Our ship only places two guards in the landing bay. We take them out and meet the shuttle as normal. Wait for the guards aboard to exit with the prisoners and then use the element of surprise to our advantage,\" Zane replied. \"Third person waits up on the steel catwalk near the entrance of the landing bay. They'll be responsible for keeping everyone on this orbiting hell out of the landing bay until the other two take the shuttle and are ready to make an exit.\" he added.\n\n\"Who's responsible for what?\" Roman asked with concentration. \"Look, I trust you, but I'm not stupid enough to hang out up on the catwalk. I got no intentions of you two taking the shuttle and leaving my ass high and dry.\" Zane said.\n\n\"I'll take the catwalk,\" Quinton said. \"I will only have been here for a week anyway, so there would be a good chance one of the guards aboard the shuttle would recognize me.\" he added.\n\n\"Then it's settled.\" Zane said.\n\n\"One thing,\" Roman responded, looking abruptly into the dead gaze of Zane. \"We will not leave without Quinton.\" he added.\n\n\"Agreed my friend.\" Zane said with a touch of nervousness in his tone.\n\n\"Only question now is how to get to the landing bay?\" Quinton asked.\n\n\"You will ask to be treated in the infirmary,\" Zane responded, his stare directly on Quinton. \"Roman and I will stage a fight. We have to make sure it's severe enough to get thrown into the hole, and we have to time it just right. Otherwise we will miss the chance to jump that shuttle,\" he added. \"When we pass in the hall under guard, they will force you to face the wall,\" Zane said to Quinton. \"The very moment we pass by, Roman and I will both spring on the guards and do our best to overtake them. As we do, you will turn and help us. Grab as many loose weapons as you can, going to need them in order to hold that catwalk.\" Zane added.\n\n\"I like it, it's a good plan.\" Quinton replied as Roman continued his stare on Zane, wondering if he could be trusted. Not that it would matter. He was easily willing to kill Zane if need be in order to make sure he and his terrorist brother Quinton left safely.\n\nPrimal pressed a sequence of several buttons in front of them as Stage turned from the cockpit area of the shuttle to face the rest of the group.\n\n\"Gonna be about a six hour flight to Arch City. Try and get some rest.\" Stage said loudly as the ship's thrusters began to do a slow burn as the shuttle remained grounded in the knee deep snow. He turned to face the instrument panel and assist Primal as both men pressed several buttons and logged their course.\n\n\"Arch City?\" Adam asked, bending over into Sasha's direction slightly while doing so.\n\n\"One of Gali's larger cities. Plan is to overtake the crew of the shuttle that transports prisoners to the ship that houses Roman. Land as though it's a normal drop off, then grab him and get out of there before the guards aboard realize what is going on.\" Sasha replied.\n\n\"Arch City, random moon, I don't really give a damn. Anywhere but here. I'm freezing my ass off.\" Dalton said loudly as his lower jawbone trembled. He sat in the rear of the shuttle huddled into a chair with his thin brown coat pulled around him as though it were a straightjacket; Whiskey sitting pitifully by his side.\n\n\"You know, if you would just have replaced that raggedy ass brown piece of cloth with an actual climate jacket when they offered, you'd be nice and toasty right about now.\" Adam said smiling, his thick green coat with fur lined hood a perfect escape from the cold conditions.\n\n\"Fuck you buddy, the brown coat stays!\" Dalton said with pride filled anger.\n\n\"Mmm...toasty.\" Adam replied in mocking fashion, unzipping his thick coat a bit as if to purposely let cold air in.\n\n\"Everybody wants to be a comedian all of the sudden.\" Dalton said bitterly as he took a quick shot of rock whiskey from a small metal flask, petting Whiskey who gazed at the rest of the crew with the saddest of looks in his eyes.\n\nAlthough Adam continued to chuckle a bit, deep down he knew the humor was a mere cover for the strangling emotions inside of him. His heart, his very soul missed Sarah Blaine so badly. The same heart and soul that knew deep down Sarah was gone, and had started to fall for Sasha. As the Benzan shuttle slowly began to lift from the ground and climb into the heavens, Adam's eyes remained locked onto the cloud of snow that the thrusters had formed below them. He wondered if Sarah ever bothered her own thoughts with him, or had she moved on, making Adam nothing more than a disposable afterthought?\n\nSpace travel had a way of forcing you to think about things that otherwise stayed buried deep inside. It was for this very reason that Adam dreaded their upcoming journey to Arch City, though it was long overdue. Roman had stayed behind so that Adam, Sasha and Dalton could escape. Adam Michaels was a man of his word, and now it was time to make good on a promise to free his good friend. Adam tried to focus his attention to that, although the memory of Sarah remained in the shadows of his every waking moment.\n\n\"What's on your mind?\" Sasha asked nearly an hour after their shuttle hit orbit.\n\n\"Nothing much,\" Adam replied, breaking from his concentrated thought for a moment. \"Just thinking about everything, it's been a crazy year for me.\" he added.\n\n\"Believe it or not, I understand. There was a point in time in my own life when I had given up on the idea of true happiness.\" Sasha said.\n\n\"Really? What happened?\" Adam asked.\n\n\"You came along.\" Sasha replied. Though it would have sounded like the perfect line coming from a smooth talker, Adam knew she was sincere. He could see it all over her face.\n\n\"Does he always do that?\" Sasha asked, breaking the awkward silence between them.\n\n\"Always do what? Snore?\" Adam asked, turning to Dalton.\n\n\"No, I've heard people snore before. That's not what he's doing. In fact, I'm not sure what he's doing.\" Sasha replied as they watched the scruff painted Dalton sleep at the rear of the shuttle. It was a mixture of snoring and mumbling, a bit of laughing thrown in as well.\n\n\"Wondering what he's dreaming about?\" Adam asked.\n\n\"Oh my God!\" Sasha said loudly, but in a whispered voice as Dalton reached down to adjust his man tool while still sleeping, grabbing his crotch area roughly.\n\n\"Congrats.\" Adam said, patting Sasha on the back softly.\n\n\"Congrats for what?\" she replied.\n\n\"You are pretty much the only attractive woman he has seen for some time, so my guess is he's dreaming about you.\" Adam replied.\n\n\"Oh my God!\" Sasha said again as Dalton mumbled in his sleep and readjusted his crotch a second time.\n\n\"Wait. So you're saying I'm attractive?\" Sasha asked with a smile on her face.\n\n\"I'm sure you know the answer to that. Must have hundreds of guys telling you that you're attractive each day.\" Adam replied.\n\n\"I'm not worry about hundreds of other guys, just Adam Michaels,\" Sasha said playfully. \"What do you think?\" she asked.\n\n\"Well ma'am, I think you are very attractive,\" Adam said. \"In fact, I can't see a single feature on you that I would even think of changing.\" he added.\n\n\"Oh really?\" Sasha replied.\n\n\"Yes really,\" Adam said. \"In fact, I would lay you down right now and make passionate love to you if Dalton wasn't awake and smiling at us.\" Adam whispered into her ear. Sasha, who longed for Adam's touch, quickly turned to see Dalton sitting upright and smiling back at her.\n\n\"Oh my God!\" she said as Adam began to laugh.\n\n\"Well a damn fine hello to you too.\" Dalton said snidely.\n\nThey were heading for Arch City, and that was a good thing. It meant they were heading away from the Hunters who were arriving to their own destination with a single purpose.\n\nEliminate the Benzan race once and for all. As the five Hunter shuttles broke from deep space and into low orbit, the Vampiric beasts prepared themselves, double checking both weapon loadouts and armor fittings. Normally, the Hunters feared no man. However, the battle to come was different. Meeting the Benzans on their own home soil meant fighting a race of men who would defend their families with barbaric passion.\n\nTheir assurance of victory was the accompanying Hunter Elite soldiers. The Knights in solid black armor calmly sat at the rear of each spacecraft, almost as if the flight itself were the only burden to them. In just a very short time they would be involved in a small scale battle of flesh shedding and soul reaping. Still they sat there. So calm that they almost looked lethargic. But when the dying began, every Hunter knew the Elites would in fact move and cleave with unnatural speed.\n\nThey had expected to hold the element of surprise, however, as the Hunters descended from the clouds they began to make out figures in the thick of the falling snow. The Benzans were waiting, poised by the heat of large bonfires as they waited in a spread open field that was flat and ripe for the planting of severed limbs.\n\n\"Land there,\" Vladris said to his shuttle's pilot, pointing to a large area less than a half mile from the Benzan's current position. \"We will meet them on their own fields and soak them with the blood of cowards.\" the Elite added, his strangely deep, almost demonic voice enough to rattle any normal man.\n\n\"Yes Vladris, at once.\" the pilot replied as the convoy of Hunter shuttles diverted its path into the direction of its new destination.\n\n\"Prepare yourselves, for tonight we feast of the blood of beasts!\" Kraid yelled as nearly two hundred Benzans began to shout loudly, every single on of them well armed. With the blistering cold of snow consuming most of their vision, the Benzans remained near the bonfires for warmth and in close quarter as they awaited their adversaries.\n\nAs they looked through the thick snowfall at one another, it was a very sobering moment for most. Many would fall in the battle soon to be, and every man knew it to be the truth deep down. Though they believed they were more skilled with both gun and blade, the Hunters were not to be taken lightly. Many of the men had wives, children even. Kraid had sent them into the deep terrain of the mountains, their chosen fall back spot in the event of a battle so dangerous. Nearly fourty wives and children under the escort of only five Benzan warriors, it was simply all Kraid could spare. The women possessed every needed survival skill and the five warriors sent as escort were his absolute finest. While it pained him to not include the five skilled warriors in the battle to come, he knew the others would fight harder knowing their families were well protected. Kraid was a reaver of both man and demon, having sent more Hunters to the grave than any other Benzan still breathing. Even so, his heart thumped with both adrenaline and nerves, knowing damn good and well the vampiric bastards would sent their best, and that was sure to include Elites.\n\nThe first sign of the fight to come was the glimpse of what appeared to be two hell hounds that could be seen through the curtain of white snow. Moments later, the Benzans realized the Hunters were in full charge as they rapidly approached, swords drawn and eyes locked in.\n\n\"They come!\" one of the Benzans yelled frantically as they all prepared for the stinging of steel, cold and yells of death. The exceptional speed of the Hunters allowed them to match the hell hounds in full sprint, stride for stride.\n\n\"Wait. Wait,\" Kraid yelled to calm his men, assuring they remained close together. \"We must fight as one group. Do not let them intimidate you, for today they face the most skilled killers in the Skyla System!\" Kraid added, pulling his broadsword, its edge gleaming a bit from the reflection of the diamond like snow that consumed them.\n\nThe thunder of feet and yells of barbaric and demonic fashion seemed to quiet for a moment, at least in the mind of Kraid who held his broadsword as he drew his gold plated pistol, firing a single shot which hit a Hunter directly between the eyes; his horned helmet had been split in two as blood poured from the skull of the filthy beast, quickly saturating the spongy white snow that would soon become a crimson river.\n\nUsing his free hand, the Benzan leader immediately thrust his sword down, striking one of the hell hounds at the top of its skull and driving the instantly dead beast into the frozen tundra. The second hell hound stopped in mid stride, realizing the Benzans were not the usual buffet of ease.\n\n\"Fight!\" Kraid yelled as within a single instant, the field was filled with hundreds of Benzans and nearly a hundred Hunters, blades exchanging viciousness as the screams and gunshots could be heard for miles in the otherwise calm of falling snow.\n\n\"You alright?\" Lassiter asked as Tigon Twelve continued its full burn, the twin thrusters providing a trailing path of flame.\n\n\"I wouldn't wish this on anyone,\" Sarah said, sitting in her seat and looking out across the landscape of stars. \"The worst part of feeling this way is knowing that I did it to myself. I just want to go back and leave with him, do things right.\" she added.\n\n\"You can't go back,\" Lassiter said, slowly taking a seat beside her. \"So many times I think the same way, wishing I could go back to the day Kelly was murdered, go with her and perhaps prevent it all from happening. Deep down I know it wouldn't matter. It was her time to go.\" Lassiter said.\n\n\"Strangely, this isn't helping much.\" Sarah said, laughing a bit. \"Oh, I'm not saying anything has happened to Adam. My gut tells me otherwise. If there is anything we know about him, it's that he's a survivor. I'm sure he's fine.\" Lassier replied.\n\n\"I hope so,\" Sarah replied in a much more serious tone. \"If anything has happened to him...anything.\" she added, turning to look out of the window of the shuttle once more. \"I wouldn't be able to live with myself.\" Sarah added. Lassiter was at a loss for words, instead placing his hand on Sarah's shoulder for a moment before standing to his feet and checking on the Colonial soldiers who had made the trip with them.\n\nThey were a grit filled bunch to say the least. The much larger Husk sitting to the front of the group, outfitted in thick armor plating they remained solemn.\n\n\"Everyone alright?\" Lassiter asked as he made his way slowly by each seat to the rear of the shuttle. The Husk simply answered with slight nods, quickly returning their attention to the windows which overlooked the stars.\n\n\"Hoping the Benzans are reasonable so we may return to the fight quickly.\" one of the Colonial outfitted soldiers remarked.\n\n\"I understand, but the Legion will fall in due time. As for the Benzans, I wouldn't expect too much reasoning from them.\" Lassiter replied softly, Sarah overhearing his words.\n\n\"And if you're right sir? If the Benzans are not willing to allow us an audience with Adam?\" the soldier replied.\n\n\"We're not going for the opportunity to speak with him. We're going to get him. Sarah has suffered enough, and we will either be allowed to speak with Adam or we will use force to complete the mission.\" Lassiter said, leaning in a bit toward his men. \"When we arrive, you men will remain with Sarah while the Husk and Goliath units accompany me. I will state our demands and be reasonable in doing so. If they are unwilling, we will use force until they become willing.\" Lassiter added, turning to Sarah who was staring out of the window.\n\nI care too much for her to see her suffer like this. Lassiter thought as he watched Sarah for several long moments. His heart still longed for the soothing touch of Kelly, and having held her lifeless body in his arms had been devastating.\n\nThrough his own suffering, Sarah had been there. She saw to it that his heart healed enough to move forward. Now it was time to return the favor. She was a good woman, amazing actually. Everyone made mistakes, Lassiter knew that. It was Sarah's open admission of guilt for letting Adam walk away that made Lassiter understand how amazing she truly was.\n\nSeveral times Lassiter had avoided her because he felt something more, feelings he shouldn't have. They scared the hell out of him, and rather than put Sarah through even more, he wanted to remain her friend. At least for now. Sarah turned to see her best friend watching her as she smiled softly back to him for caring.\n\n\"Face the wall prisoner Raines!\" one of the two prison guards said sternly as Roman slowly stood and faced the wall of his cell.\n\nHe was the only resident, his former cellmate having been put into protective lockup following their one sided melee. Roman was new to the prison life, at least aboard this ship, and his cellmate saw that as an opportunity to prey on the former terrorist. Simple plans of taking Roman's share of food, but stupid plans nonetheless. His cellmate quickly discovered that Roman Raines is intimidated by no man, nearly crippling him before the guards could rush in and stop the carnage. Roman was slapped an even tougher sentence after beating two of the rescuing guards profusely as well, prompting them to treat him with extreme caution from that day forward.\n\nIt was also the first day Zane began to notice him, his effectiveness during the scuffle was exactly what was needed to put together a decent plan of escape. While every prisoner respected Zane and answered to him in some fashion, none could be trusted enough to maim and possibly even kill interfering guards when the time came. Zane saw it in Roman's eyes the day of the melee, and knew from that moment forward that not only was he capable of killing if needed, but there was a good possibility he had done so in his past.\n\n\"You move an inch and I'll paint this fucking wall with your brain matter!\" one of the guards said loudly, holding a riot style shotgun directly to the back of Roman's skull.\n\nMeanwhile, two more guards tossed his cell, throwing what little possessions were inside onto the steel deck of the ship and kicking them out of the eight foot by six foot space. The cell was down to nothing more than the steel sleeping rack which was tightly bolted to the wall. They were given no mattress, and there was no toilet to speak of. They were led down the hall at gunpoint for five minutes every day to the latrine, and that was their one and only time to dispose of human waste outside of their cell floor.\n\n\"Still hadn't figured out what you guys look for?\" Roman said laughingly, his face nearly touching the wall as the guards continued their toss of the cell.\n\n\"What the fuck do you think we're looking for? Weapons and contraband.\" the guard holding him at gunpoint replied.\n\n\"Come on, you've read my rap sheet, do you think I need a weapon?\" Roman said. \"You can read, right?\" he added. \"Shut up!\" the guard yelled.\n\n\"Yea, I can read. Got to be honest, for such a high profile catch you sure don't look like much to me!\" the guard added loudly as his friends finished tossing the cell, laughing at the guard's remark.\n\n\"Don't let the bad shave and dirty clothes fool you. If I wanted you guys dead you'd be dead already,\" Roman replied with a grin. \"Don't worry though. I like you. You remind me of someone I once knew.\" he added before feeling the barrel of a weapon pressing hard against his neck.\n\n\"I told you to shut up.\" the guard replied sternly, reminding Roman of the pecking order while opening a small flask of rock whiskey with his free hand and sneaking a small swig. Roman continued to face the wall, although the smell of the rock whiskey soon brought a smile to his face. It was almost like being back home with the Gunship crew.\n\n\"Alright move!\" the guard said firmly, holding his gun on Roman as the three guards led him down the hall slowly, making their way into the mess hall to join a handful of previously escorted prisoners.\n\nIt had been one of the most violent battles in recent history for both races as body upon body fell lifelessly to the blood drenched ground, a crimson river making its way through the deep drifts of snow as if to become a permanent fixture. The Benzans had all but fallen, only a handful remaining and retreating back into the direction of the lodge, trying to move defensively while avoiding the tripping hazard of dead flesh. Two dozen Hunters remained, including three of the mighty Elites. Their accompanying hell hound feasted on severed flesh, still warm as the snow fell while the remaining Hunters collected themselves. The rest had fallen in battle as a result of the Benzan's concentration of force onto them, taking heavy losses from the standard Hunters in doing so.\n\nKraid, who now stood covered in a mixture of blood, both freshly wet and dried, ordered the remaining warriors to seal the doors of the lodge if only to give them a moment to breathe.\n\nThe Benzans had been defeated, over confidence their achilles heel as they each prepared to die with honor while taking as many of the cannibalistic Hunters down with them as possible. Rather than rush in to slaughter the remaining five Benzans, the Hunters backed away a bit, giving Vladris a moment. The mighty Hunter Elite stood tall, red plasma dripping heavily from his blade as he ran his fingers across the bitter cold steel, placing them into his mouth to taste the spoils of victory.\n\nBending down onto a knee, Vladris ran his hand calmly across the back of his hell hound, which had taken the lives of a couple of Benzans on its own. He slowly rolled his neck, the sound of moving tendons and small bones popping as he prepared for the final showdown.\n\n\"My brothers. If we fall today, know that your families are safely hidden in the mountains and that you died as heroes.\" Kraid said, doing his best to catch a deep breath as the chilled air worked against him. The remaining Benzans were covered in shades of blood and exhausted from swinging iron to flesh, yet they dug deep for the final fight to come. Thousands of wooden splinters filled the air of the room as the Hunters finally burst the lodge door open by force, knocking one side from its hinges.\n\nHolding his mythical sword of silver and severed flesh out in front of him, Vladris pointed it into the direction of Kraid, blood oozing down the blade and dripping steadily onto the grain finish of the wooden floor. Though he realized he was going to die, win or lose, Kraid stepped forward with his broadsword tightly clasped into one hand and his gold plated pistol gripped and positioned near his side. They exchanged a deep stare, the fluid red pupils of Vladris locking with the determined brown eyes of Kraid. Though they fought under different banners, they were both warriors and each knew that one of them would fall today. They were prepared to go with honor and this was their way of once and for all proving which bloodline was the more dominant.\n\nVladris thrust his hulking sword down with thunderous power, Kraid barely quick enough to evade the attack as a foot or more of the sword buried into the thick wooden slats beneath them. Kraid immediately swung his sword with a sweeping motion in an attempt to decapitate the Hunter Elite. Vladris ducked with intent, swiftly pulling himself back upright; grasping the gun hand of Kraid and squeezing until the Benzan dropped the golden piece. Powerfully uprooting his sword, Vladris arched it into a swing which ricocheted off of the Benzan's broadsword.\n\nStaggered for a moment, Kraid quickly thrust his sword forward, clipping the solid black armor of Vladris as thick leather and trace amounts of blood dispersed onto the polished wooden wall behind him.\n\nGlancing at his minor wound for a moment, Vladris was overcome with a beast like rage as he refocused his attention onto Kraid. Inhumanly fast and without warning, Vladris struck the Benzan across the face with a thunderous backhand, sending him reeling. The staggered Benzan leader was merely able to lift his sword just in time to block the downswing of the Hunter Elite's sword, though it did him little good. The trunk of Kraid's sword clanged as the powerful swing knocked it from his hand, the Elite's blade biting deeply into the flesh of his shoulder. It was at that moment the remaining Benzans knew defeat was at hand and quickly lashed out at the Hunters with swords arcing wildly. Moments later, each Benzan had fallen lifelessly to the floor leaving only Kraid.\n\n\"Do you intend to turn him my Lord?\" one of the Hunters asked as Vladris stared at his beaten opponent for several long and silent minutes. Kraid would either be sent into the afterlife for eternity or he would join the ranks with the Hunters. A choice that was now in the hands of Vladris. Hunter Elites possessed the unique ability to turn their defeated foes into Hunters, but reserved the privilege to only a select few.\n\n\"Our work here is done then my Lord?\" one of the Hunters asked of Vladris as the Elite soldier skimmed the area through a large window of the lodge. The sight of butchered bodies lay on the ground around them, slowly becoming hidden by the ongoing fall of heavy snow mixed with the pasty grey smoke of the bonfire. Several Benzan buildings stood in the distance, catching the Elite's attention. \"No. Check the rest of the buildings for any survivors. Anything that may speak of stray cowards away from the nest.\" Vladris commanded.\n\n\"At once!\" the Hunter replied with obedience before turning to lead a small group of the undead warriors into the direction of the buildings as Vladris slowly stepped back outside, focusing his attention into the sky and its blanketing flakes of snow that fell directly down.\n\n\"My Lord, the Colonials are approaching Tameca.\" one of the Legion soldiers said loudly as he entered Lord Riven's quarters. It was a reinforced room, snugly fit into an underground bunker and equipped with everything needed to operate in the worst of conditions.\n\n\"As expected,\" Riven said. \"Wait for them to hit atmosphere and fire all of our ballistics at them.\" he added.\n\n\"All of them my Lord? Every missile?\" the soldier asked. \"Yes.\" Lord Riven said sharply, turning his eyes to the questioning soldier.\n\n\"Right away my Lord!\" the soldier replied loudly with obedience, turning to execute Riven's order.\n\nTameca City was once a proud place, flourishing with opportunity and the citizens who lived there in chase of it. As the soldier exited the bunker to execute Lord Riven's order, the sight of present day Tameca City was gut wrenching. A majority of it on fire, intentionally set by Legion forces as both retaliation against those who opposed Legion rule and a warning to anyone who may approach to give aid. Many of the same citizens who once shared laughter and memories on its streets, now lay dead on the very same paths of asphalt, bodies piled high as they burned without mercy.\n\nIt had become a large scale holocaust, what little remained of the city now part of a Legion controlled warzone. They had been bested during the battle for Glimmeria, losing a majority of their high end weaponry in the process. What remained was a stockpile of surface to air missiles, armored assault vehicles, plenty of small arms weaponry and soldiers who had been intimidated into seeing the war to its end. They had no help coming, no reinforcements. Just a severely outnumbered force that was entrenched into the city and slaying at will.\n\n\"Alright. You've been trained for this and have the superior firepower and numbers. These civilians are being murdered because they want a Colonial government, so as Colonial soldiers it's our job to protect our soon to be people at all costs.\" General Ortega said, scanning the launch bay of a Colonial Star, which was filled with thousands of battle ready troops. \"Most of our convoy is staying in orbit around Tameca, cutting off all escape routes for the Legion in hopes of the war ending here. It's our job to end it, and by God we will end it!\" he added, thrusting the crowd of warriors into a loud cheer. Moments after, the synchronized cheering turned to panic as the Colonial Star began getting pummeled by an array of Legion fired missiles.\n\nIt was a tough ship, built with the very purpose of taking blunt force trauma for extended periods of time. Hundreds of potent warheads gleamed into the sky however, most of them hitting the ship's hull and rocking the already battle weary frame and engulfing it into a curtain of flame.\n\nWhen the general order to abandon ship rang out across the com system of the ship, soldiers began sprinting frantically throughout the halls, trying to make their way into the launch bay where General Ortega had already set up an evacuation detail. \"We stay until the ship loses power!\" Ortega shouted, explosions ringing in all corners of the flying fortress.\n\n\"Make sure everyone is armed to the teeth and well supplied, when the power fails we have to go, at that point it's only a matter of time.\" General Ortega added loudly, the troops near him handing both weapons and survival packs to the evacuating personnel.\n\n\"Mayday, Mayday, this is the Colonial Star Thirty-Seven. We are going down, repeat we are going down. Ship is evacuating to the surface of Tameca. Repeat, ship is...ohh God, ohh God!\" the ship's pilot could be heard yelling up until the second that the Colonial Star was no more, the tomb of steel and circuitry exploding, leaving behind only faint screams and fiery particles of debris as the night sky of Tameca illuminated as though an eclipse were taking form.\n\n\"We barely made it sir.\" one of the Colonial soldiers said frantically as General Ortega looked through the plated glass window of the military shuttle, watching the ripples of destruction where the Colonial Star had been only moments before. Listening to the unorganized chatter being broadcast over the shuttle's com, Ortega finally punched in a security code before picking up the com.\n\n\"This is General Ortega. We have lost our ride but not our resolve. Sky Command, please organize a rally point on Tameca's surface based on my location beacon, we will regroup and complete the mission as planned.\" he said calmly into the hand held com before placing it firmly back onto the communication board near the shuttle's pilot. \"One way or the other this battle will prove the end of the Legion's reign of terror.\" he added, speaking the words to the armed soldiers of his shuttle.\n\nMinutes later, their military issued shuttle began hitting waves of turbulence, a direct effect of the fighting that was taking place below them. The ship's diamond shaped legs slowly bit into the hard crust that covered Tamera's ground, frozen dirt which was typical of the coldest season of the year. A massive Legion force was pushing to their direction, a well organized attempt to end the Colonial force before it could regroup.\n\n\"Laze their positions!\" General Ortega shouted as he exited the shuttle steadfast. One of the Colonial snipers quickly shouldered a large device, similar to a weapon, but with no firepower of its own. \"Keep it locked in!\" Ortega ordered loudly as he contacted the remaining Colonial Star ships with a mobile com. There is a vast quietness of space that is nearly indescribable, the luminous littering of stars everywhere as pure silence engulfs them. This very silence was broken abruptly as ten thunderous booms broke out, all high potent missile strikes firing from the launch tubes of an orbiting Colonial Star and directed to the location of the laze.\n\nThe Colonial outpost, while under a heavy bombardment of Legion fire, had already started to take in Tameca's citizens. Refugees on their own planet, all of them shell-shocked and grieving for those lost as they staggered into the grasp of armed Colonial soldiers. There was a cluster of zinging sounds, followed by rocking explosions throughout the Legion camp nearby. The Colonial warheads struck earth with such velocity, such authority, that only craters remained that filled with ash and flesh. \"General, do we attack?\" one of the soldiers asked as the silence of missile strikes no longer floated through the sky. \"Just wait, hold the line and wait until I give the order.\" General Ortega demanded.\n\n\"Yes sir.\" the soldier responded. Less than a full minute after the missile strikes, one of the remaining Colonial Star ships hovered down slowly from orbit, moving over what remained of the Legion encampment as its deck cannons began chewing into anything that moved. Dispatching dozens of the Glimmerian designed Swordfish fighters, the Colonial Star began a slow ascent back into the heavens and the quiet space which orbited above.\n\n\"Now you may go.\" General Ortega said proudly, a majority of his soldiers running to the sad remains of the large Legion camp as they shouted loudly with pride. \"Lieutenant!\" Ortega shouted. \"Yes sir!\" the man replied with discipline. \"Continue finding our new citizens food and make them as comfortable as possible. I'm taking a strike team in.\" Ortega commanded. \"Yes sir!\" the Lieutenant said loudly, saluting for a moment before turning to continue organizing relief to the citizens lucky enough to have made it to them.\n\n\"Be landing in less than ten minutes, time to wake up.\" Stage said loudly as Dalton slowly left his dreamworld of fast guns and even faster women and began to get his bearings in order. Primal and Stage were both at the controls of the shuttle, gliding it to the planet's surface for the rescue plan ahead. Meanwhile Adam and Sasha sat closely together on a cushioned bench seat a few feet away. Dalton's first thought was that of Sasha's perfect body as he tried his best to convince himself that she hadn't followed him back from the world of dreams. He realized Adam was no longer wearing a climate jacket about the same time he glanced down to see himself snugly underneath it.\n\n\"Man, get this fuckin' shit off of me!\" Dalton said, throwing the Benzan jacket several feet as both Adam and Sasha began to laugh loudly.\n\n\"Aww, but you looked so cute laying there huddled up underneath it.\" Sasha said as Dalton stared her down will ill intent.\n\n\"I 'aint going for cute. I'm going for casual.\" Dalton replied sternly, proudly pulling his faded brown coat together in the middle as if it were a tuxedo. Several small rips littered the jacket, which reeked of cigar smoke, whiskey and God only knows what else; accented by a few stains of dirt and lipstick. Still, Dalton smiled with pride. It was his resume of sorts, a visual reference to his experienced past.\n\n\"There it is.\" Primal said as everyone took a moment to look out of the shuttle's windshield. The entire planet of Gali looked as if it were swampland, thick green brush covering nearly every portion of it. Arch City, however, was a different story entirely. It wasn't the largest city Adam or Dalton had ever seen, but it was large enough. Two huge silver arches cascaded over the city, while several thousand buildings were positioned below, a few skyscrapers and the rest mid sized buildings or less.\n\n\"How are the watering holes here?\" Dalton asked, his tongue growing weak of alcohol starvation. \"They got anything you want here. Drink, women and weapons. You name it, they got it. As long as you got the credits to pay for it that is.\" Primal replied, turning for a moment as he answered.\n\n\"Shit boys, Dalton James hadn't ever had to pay for a woman. I pride myself on that.\" he proudly replied as Whiskey was quick to second the notion with a loud bark.\n\n\"Must be the coat.\" Sasha replied as Adam broke out into an uncontrollable laugh.\n\n\"I hear 'ya. I'm surprised you 'aint sittin' back here with a real man now that you know what one looks like.\" Dalton replied\n\n\"You tell her brother.\" Primal said, chuckling just a bit. \"Alright already, we need to get focused on getting Roman out of lockup.\" Adam said.\n\n\"Agreed,\" Stage said. \"We're going to have to be on top of our game to pull this rescue off.\" he added.\n\n\"Oh, I'll be on my game alright. Hell, Whiskey and me are both liable to walk out of the nearest bar with a nice piece of strange. Shit I'll be on top of something for sure.\" Dalton said, licking his lips a bit as Whiskey began barking loudly.\n\n\"I think I'm going to be sick.\" Sasha said in a low voice as she watched Dalton practice his best pickup lines on Whiskey at the rear of the shuttle while it docked onto the landing port of the city.\n\nWhile the Legion was fiercely outnumbered, they had dug themselves in deep, pockets of soldiers doing what would have been a valiant job if not for the worst of reasons. Equally as resilient, Colonial soldiers fought back. Holding their ground while slowly gaining more. Surviving citizens began to pour in, staggered and starving as the Colonials soon found their outposts overwhelmed. Not with attacking soldiers clad in red and black trim, but common folk, mostly women and children clad in dehydration and the blood stains of their loved ones.\n\n\"Keep them away from windows and potential entry points.\" Lieutenant Scott said, holding a large metal door open as citizens poured in, many falling from gunfire close by as Legion troops ended as many lives as possible.\n\nWhen the senseless murdering first began, over half of the Legion ranks surrendered, their core values not involving the slaughter of unarmed people. However, those who remained were with skill and without soul, gunning down even toddlers as if they were armed soldiers.\n\n\"Seal the door and get me a link to the sky!\" Lieutenant Scott yelled, doing his best to get the survivors into the large rooms of the vacant factory.\n\n\"Here you go sir.\" one of the Colonial soldiers said, handing Scott a wired com unit. \"This is Bravo Forty Two Blue, repeat, this is Bravo Forty Two Blue. Requesting evacuation of Tamecan citizens as well as reinforcements on ground. Repeat...\" Lieutenant Scott said, his plea abruptly cut short.\n\n\"Negative Bravo Forty Two Blue. Be advised we are under fire from hostile forces. I say again, the Republic of Theron has joined the battle and its army has engaged us in war. Advise hold ground and make due until Colonial allies arrive.\" a voice responded, cracking heavily due to the distance between coms.\n\n\"Estimated time of arrival?\" Lieutenant Scott asked, his hopes of a swift victory crushed.\n\n\"Reinforcements are approximately two days out. I say again, two days out.\" the voice replied, distinct sounds of pounding cannons in the background. \"Copy.\" Scott replied.\n\n\"Actual, this is Bravo Forty Two Blue. Did you copy the last?\" Scott asked, his battle ripened hand holding the com only inches away from the smooth shave of his face.\n\n\"Copy that. Do your best to secure our location. I'm ordering several platoons back to your area to assist.\" General Ortega replied.\n\n\"Copy that sir, Bravo Forty Two Blue out.\" Lieutenant Scott replied, handing the com back to the soldiers with him, which were less than fifty and new to the acts of war.\n\n\"Listen up,\" Lieutenant Scott said loudly, gaining the attention of each soldier. \"It is imperative that we fortify our location until help arrives. I want four men at each entrance and heavily armed. As the refugees come to us, two soldiers exit to assist while the other two cover. Every remaining soldier, including myself, will set up a staggered perimeter in the hallways. Protect the rooms of refugees at all costs. Understood?\" he asked loudly.\n\n\"Yes sir!\" they replied, checking their ammunition levels. \"Sergeant Ramon, assign the soldiers to their posts.\" Scott ordered. \"Yes sir.\" the Sergeant replied.\n\nThe Republic of Theron. Bastards. Lieutenant Scott thought as Colonial troops ran around him to secure the building as quickly as possible. Nobody saw it coming, although they should have. The Republic of Theron has been murdering its own citizens for decades now, so why would the Tamecan people be any different. Hindsight is twenty-twenty they say, but the fact that Theron was a major trading partner with the Legion, equipped its army with Legion weaponry and had so much to lose if their trading partners fell made sense. Someone missed the signs and because of the oversight, Colonial ships and Theron cruisers were engulfed in one hell of a fight as they drifted in orbit.\n\nTwo Colonial Star ships remained, their escort cruisers having been sent down to Tameca filled with troops and supplies. Both ships had taken a hell of a beating, but remained intact as they orbited side by side and were exchanging cannon shots with six Theron cruisers. The Theron ships were much smaller than the mighty Colonial Stars, but were designed by Legion engineers and bred for one purpose. War. They were heavily armed with deck cannons, motorized chain guns and only the thickest of steel plating.\n\nStill, the Colonial battleships held their own. Much like two larger and more capable animals defending themselves against a pack of hungry wolves, they continued exchanging shots of bone shaking lead. One of them began to falter, its hull damaged beyond repair and quickly splitting, eventually leading to the order of abandon ship. The Colonials were swarmed, and though they had done significant damage themselves, it was endgame. The soon to be demolished Colonial Star redirected its fire strictly to the surface of Tameca, striking all known Legion locations as its crew abandoned ship in route for the surface.\n\nMeanwhile, the only Colonial Star still intact pulled out of the fighting, setting course for deep space in wait for reinforcements that would arrive from all over the Skyla System. The Theron cruisers didn't attempt chase, knowing full well the injured Colonial ship could still deal extraordinary amounts of damage. The truth was, as the chatter hit the waves of Colonial coms on the surface of Tameca, its soldiers shared one feeling. They were alone. There was no guarantee that help was on the way, and if indeed it was, there was no exact time table for its arrival. They simply had to dig in and do their best not only to survive, but protect the citizens of this once great planet while doing so.\n\n\"My Lord,\" a Hunter said as he approached Vladris, who remained standing in the fields of gore and wrapped in his own blood covered armor of thick leather. \"We went through the personal affections of their leader,\" the Hunter said. \"It seems there are a few Benzans living abroad, as well as a shuttle with five passengers which left only hours ago.\" the Hunter added, handing the flight log to Vladris.\n\nThe demon in charge of the group of Hunters assigned to eliminating the Benzans stood there. Silent, the tribal markings on his smooth scalp visible as the wind blew roughly, hell hound by his side.\n\n\"The remaining Elites will accompany me to Arch City, I will personally see to their executions myself. You are to return to our Queen with the list of Benzans living throughout the Skyla System. Tell her of our great victory here and inform her that as soon as my task in Arch City is finished I will return to resupply and hunt the remaining.\" Vladris said with authority.\n\n\"Yes my Lord, at once.\" the Hunter replied, turning to quickly execute the wishes of the Elite.\n\nAs the other two surviving Elites joined Vladris in standing and looking into the falling snow which surrounded them, he finally pulled his attention away from the sky.\n\n\"Now it is certain that we are the most dominant race in this galaxy. Our brothers fought with bravery against mortal men who fought in fear, which is why their people lay slain on the ground today and we leave to tell of the battle,\" Vladris said in his deep and demonic tone. \"Now we go to Arch City in order to finish our task.\" he added as the three Elites walked to a nearby Hunter shuttle in order to begin their journey.\n\nQuinton sat down at the cafeteria table next to Zane and Roman as all three men began to eat whatever it was scooped onto their tray. \"Not gonna miss this food, that's for damn sure.\" Zane said as he dissected it with his plastic fork.\n\nIt was nothing more than mush formed by a large spoon and stood on its own once it hit the plate. The most common prison feast was algae, although that was a bit too luxurious for men of their standing. They usually got more along the lines of the algae leftovers mixed with thistle weed, which was the fancy name for the stringy weed that excavators removed to get to algae deposits.\n\n\"They treat us like fucking animals.\" Roman replied, eating his food without reserve which included a brown water. It was dubbed \"meat juice\" by the prisoners simply because it gained its brown color from softening good meat before the guards ate it.\n\n\"They've got their shots in on me since I've been here. Slapped me around a bit, spit on me and talked down to me. When the time comes, I plan on gutting as many of these low down bastards as I can.\" Roman added.\n\n\"I don't have much of a story to tell yet, but I do know that last week I was eating buttery Tamecan crab and now this is staring me in the face.\" Quinton said, pushing his plate to the side.\n\n\"Eat my brother, we're going to need every bit of strength we can get.\" Roman said, pushing the plate back in front of Quinton.\n\n\"You got a point.\" Quinton said as he slowly began to choke down each bite of the pathetic bliss of chow.\n\n\"So, when we do get out I guess your first order of business will be with your friends and their broken promise huh?\" Zane asked. Roman sat there for a few moments, his body overtaken by the numb feeling as he thought about the Gunship crew who had been saved because of his staying behind.\n\n\"No. If Adam could have gotten me out of here he would have. It was my choice to stay behind, not his. I don't hold him at fault for this.\" Roman finally replied.\n\n\"Hell, for the right price I'll take care of him for you.\" Zane said jokingly. Quickly grasping the table's edge and slinging it to the floor, Roman grabbed Zane by the front of his shirt and began pummeling him in the face with a closed fist.\n\n\"If you ever threaten another friend of mine motherfucker I'll kill you!\" Roman yelled as the prisoners began to shout in riot fashion.\n\nThe ship's alert siren sounded loudly as shuffling feet thundered their way into the cafeteria. Meanwhile the guard in the gun cage above fired a warning shot, a stern reminder to each prisoner that any man standing when the second shot rang out would be punching his ticket to the grave. Every prisoner, including Quinton, hit the floor with their hands behind their heads as Roman and Zane continued to go at one another.\n\n\"You're a dead man!\" Zane yelled with maddened rage as he threw a hooking punch which found its mark on Roman's chin.\n\nRather than stagger and fall however, Roman turned his head back to Zane and landed a rising elbow under chin and into the throat of his adversary, causing Zane to flip over a nearby table.\n\nBy this time the ship's guards had arrived, six total were first on the scene. Closing in on Roman first, he quickly fed a back fist as he spun around, following it by a numbing elbow that nearly knocked the first guard out of his boots. Jumping onto the second guard that entered, Roman began beating the life out of the man clad in riot gear. Noticing the guard who stood in the gun cage taking aim on Roman, Quinton quickly reached for the first fallen guard's rifle, aiming with precision and hitting the chest of the cage guard which put him down abruptly.\n\nAs the other four guards rushed into the crowded eating hall, the remainder of the prisoners quickly stood and began to overpower the seriously outnumbered officers, besting them within a few seconds. Taking both the baton and rifle of the guard he had beaten to death, Roman quickly glanced at his brother before sprinting out of the room, Quinton following with rifle in hand.\n\n\"You're fucking dead Roman Raines, do you hear me,\" Zane yelled through the crowd. \"Somebody climb up and take the gun cage. You three take a rifle and secure this room, the rest of you follow me. We're taking the ship.\" Zane announced while pointing his finger, the rest of the prisoners following out of his much earned respect.\n\nIt took the Gali prison guards several minutes to truly assess what was really unfolding. A full scale riot. When they finally came to terms with prisoners running loose, well armed and programmed to kill, they sealed off as many doors as possible. Solid steel blast doors closed, their latching mechanisms locking remotely as the guards still had control of the security station.\n\nHow long that control would last, however, remained to be seen. The security station was housed in a circular room, only accessible by a narrow catwalk, and currently it was filled with armed prisoners who were engaging the guards in a gunfight. During which, the large antenna used to communicate both throughout the ship as well as to approaching ships was destroyed. Deliberately, of course, the prisoners hoping they had done so before a distress signal could be sent.\n\nAs she heard the news come across the com, Sarah sat in her seat, stunned just as the rest of the shuttle's crew was. \"Reports indicate several thousand Colonial soldiers perished in the explosion. General Ortega has formed an operating base on Tameca's surface and is asking any civilian nearby in need of rescue, to please relocate to the large set of warehouses near the Tameca City bridge. Again, all civilians and crash survivors are being asked to make their way to the industrial section of Tameca City, the Colonials have a secure forward operating base in the warehouse section of the district. More news as we receive it.\" the man's voice broadcast in a looped message across the Colonial com.\n\n\"Oh God, I should have stayed behind.\" Sarah said, her words filled with regret.\n\n\"Sarah, there is nothing you could have done to help. You may have even been on the ship when it exploded, or fallen into the hands of Legion soldiers. General Ortega is very capable, you need to focus on the idea of possibly seeing Adam again.\" Lassiter replied.\n\nSarah nodded, leaning back in the plush of her chair a bit and turning her attention to the window on her left, casting stares out onto the landscape of what would soon become Arch City.\n\n\"Don't concern yourself with the ongoing war Sarah, just focus on Adam. I want to see you happy again.\" Lassier said, placing his hand on her own for a moment as she looked at him with appreciation.\n\n\"I just hope he hasn't completely forgotten me.\" she admitted, her stare never breaking from the window.\n\nLassiter quickly removed his hand from Sarah's, careful not to show his growing feelings for her.\n\n\"That's very unlikely to ever happen my lady.\" he replied, at a loss as to what else he could possibly say or do to comfort her.\n\nSarah knew Adam wasn't likely to forget her anytime soon, instead, she feared he had possibly grown fond of Sasha's company. She couldn't blame Sasha for wanting to be with Adam if it were the case, but that didn't make it any easier for her to think about. The fact that he was an amazing man was one thing. The fact that she may have lost the affections of such an amazing man forever was something else altogether, and the mere thought of it made her feel helpless.\n\nMeanwhile, the quiet continued at the rear of the shuttle, although everyone checked their weapons for readiness. Arriving at the hideout of Benzans unannounced was not wise for anyone, especially anyone bearing Colonial markings on the side of their ship. They had sought Benzans as criminals, jailed them and fired weapons on them at pretty much every opportunity. All of the soldiers aboard Tigon Twelve knew they wouldn't arrive to open arms, and killing would quite possibly ensue. They had to be prepared for anything.\n\n\"Sir, our allies have arrived and driven the Colonial armada from our skies,\" a Legion soldier said, walking quietly into the room that Lord Riven had declared his. \"However, reports indicate they have help from both Gali and Sion coming.\" the Legion soldier said somberly as Lord Riven stood in his chambers, staring at a battle map posted on the thick walls of the fallout shelter.\n\n\"Good. We must welcome our allies to the fight and coordinate a strike against the Colonials who were left behind to die like the worthless cowards that they are,\" Lord Riven replied. \"We must do this quickly and then ready ourselves for the arrival of more invading armies. In time, enough slaying of heroes will serve as warning to anyone else that dares invade.\" he added.\n\n\"Yes sir. I will ask the Theron armada to land at once.\" the soldier replied with respect.\n\n\"Thank you my loyal friend.\" Lord Riven replied.\n\nThe bunker protecting Riven was still in good shape, repelling the attack from a dying Colonial Star less than an hour before. Other Legion strong points had not been so lucky, many of them laying in shambles, still smoking from the display of desperate measures by Colonial forces. Only a few hundred Legion soldiers remained, at least functional soldiers. Acting on Riven's orders, they began to execute their own, doing away with the injured by way of bullet rather than care for them. It was a sickening display of barbarism, but one Lord Riven felt was necessary in order to continue at full speed.\n\nAs the Theron armada began to land its warships, flooding thousands of fresh bodies to the fight, Admiral Sweed remained on his ship under heavy escort as he awaited a meeting with Lord Riven. His warriors, with skin of deep tan and eyes of greyish tint, were human in every other sense of the word. A heart beat inside of their chest, and they walked upright on two legs with two arms holding weaponry. Still, the eerie sight of their gaze and complexion was not a welcome sight for most. Lord Riven and his Legion faithful were the exception, greeting their longtime allies as if they were brothers.\n\n\"My lady, we are only a few minutes out.\" the Colonial pilot said said as Tigon Twelve began to descend from the stars into the thick snow clouds of the Benzan's last known hideout.\n\n\"Would be best if you stay aboard until we meet the Benzans face to face in order to be sure of peaceful talks,\" Lassister said. \"Sarah, I insist.\" he added, cutting her off before she could even respond.\n\nSarah simply agreed with a nod of the head as the Colonial soldiers and Husk all double checked their weapons and inspected every inch of body armor.\n\n\"Sir, something's not right!\" the shuttle's captain shouted back into the passenger area as the loud thrusters of the ship nearly overwhelmed his words.\n\nSarah's heart began to plunge as she saw huge stacks of smoke rising from what remained of the hideout, most of the buildings burned to the ground with only the lodge remaining. Moments later, mountains of slain bodies became visible to the crew. The crisp from cold corpses left laying in the snow, immediately bringing Sarah to tears. Her sudden burst of emotion became a heavy cry as the shuttle touched down onto the blanket of while, Husks exiting first with large rifles at the ready.\n\nEven for the Husk, the sight was grim. Severed limbs, mortal wounds of gunshot entry points and burning flesh stood visible to them, a small river of blood flowing through the snow beyond their ship a bit. The Hunters had been here, that much was evident by their fallen warriors laying in the deepening drifts of snow. The surviving vampires had taken their time before leaving, that much was also evident by the gruesome cutting and maiming of the already fallen Benzans. They had taken their time with them, filleting the flesh from bone whenever possible. After looking around the immediate area thoroughly, the Husk warriors motioned the rest of the Colonials outside.\n\nThe remainder of the crew exited slowly, Sarah under the protection and escort of the Goliath V2 units.\n\n\"You men search the lodge, find any records or information that may lead to Adam.\" Lassiter said.\n\n\"Yes sir!\" the Colonial soldiers replied, quickly turning to execute his order. Meanwhile the Husk soldiers searched the dead, at least what was left of them, in hopes of some indication of Adam's whereabouts.\n\n\"Don't worry Sarah, if Adam is here we will find him.\" Lassiter said, attempting to comfort the hysterical woman who was deeply in love.\n\n\"The battle is still fresh, not more than a day ago.\" one of the Husk announced, walking toward Sarah and Lassiter. \"No Hunter ships remain, which would lead us to believe that they walked away when the battle was over.\" he added.\n\n\"I can't live without him Eric...I just can't.\" Sarah said to Lieutenant Lassiter as the small Colonial crew continued searching for clues of both battle, and the whereabouts of Adam Michaels.\n\n\"I understand Sarah, believe me, not a day goes by that I wonder what a life with Kelly would be like.\" Lassiter replied.\n\nRather than respond, Sarah stood silently in shock, the heavy snowfall bringing large flakes quietly to the ground. Lassiter took her with his free arm, holding her close to comfort her while his remaining hand gripped a black combat pistol. As the frozen water constructed with such beauty fell onto her, covering her hair and shoulders a bit, Sarah prayed. She simply couldn't survive without the one man who would have loved her unconditionally. Thinking of him made her breathe heavily, thinking of his demise made her not want to breathe at all.\n\nHer life had been mounted onto the wrong beliefs for so long now, she wanted to change, wanted so badly to be truly happy. And that simply wasn't possible without Adam. So she prayed as hard as she could ever remember praying.\n\n\"Sir,\" one of the Colonial soldiers yelled, sprinting over to Lassiter's location. \"It looks like a Benzan shuttle was logged as leaving shortly before the battle. Heading for Arch City.\" the soldier added. Lassiter turned to face Sarah, who was sobbing uncontrollably.\n\n\"My lady, Arch City is where the Gali prison transports are based. Could very well be Adam attempting to rescue Roman.\" Lassiter said, loosening his grip of her, though he didn't want to.\n\nStopping in mid sob out of shock, Sarah continued to look at Lassiter as she began to wonder if it was a possibility. If maybe God had answered her prayers in only a matter of seconds.\n\nWas there a single moment waiting, a moment that she would be held by Adam Michaels once more? Sarah would have gladly traded the remainder of her life for one such moment.\n\n\"If we are going, we need to go now,\" one of the Husk said, holding up a gothic style medallion. \"Hunter Elites fought in this battle, meaning they were here to wipe the Benzans out. Good bet they know about Adam's shuttle and are tracking it as well.\" he added.\n\n\"Hunter Elites?\" one of the Coloniel soldiers asked.\n\n\"Yes,\" the Husk warrior replied. \"Not something we want to run into. Trust me. And if we do, best be damn good and prepared for one hell of a fight. A fight like you've never seen before.\" the Husk replied.\n\nHe went on to explain the superior skills of a Hunter Elite, how they were born to end lives, direct offspring of the Queen as well as only the most tenacious fighters turned undead.\n\n\"It has to be Adam's shuttle, he's nowhere to be found here.\" a second Husk said.\n\n\"My lady, we need to go now if there is any chance of saving him,\" Lassiter said. Sarah agreed with a shaking of her head as Lassiter began to recall everyone back to the ship. \"Don't worry Sarah, we have enough muscle here to take out a Hunter group, Elites or otherwise.\" Lassiter said as they boarded, although the Husk soldiers knew better.\n\n\"Well, that went well.\" Quinton said as both he and Roman took a moment to rest behind a large system of piping.\n\n\"I can do complicated as long as we get the fuck off of this ship.\" Roman replied.\n\n\"Good. From the sound of things it's going to be real complicated,\" Quinton said as heavy gunfire was being exchanged throughout the ship. \"Think he is really capable of taking the ship?\" Quinton asked.\n\n\"Not sure and don't care. As long as we can take the landing bay when the time comes, he can have the ship and all of the bad memories that go along with it.\" Roman replied.\n\n\"How are you on ammo brother?\" Quinton asked.\n\n\"About three fourths full.\" Roman replied after glancing down at the neon green charge count on the rifle's side.\n\n\"A little under halfway for me.\" Quinton said.\n\n\"As long as we do things smart and conserve what we can we should be fine,\" Roman replied, tucking the officer's baton into the back of his prison issued pants. \"We'll lay low and let Zane fight it out with the guards. All we need to do is worry about whoever is left standing when the transport ship arrives.\" Roman added.\n\nAfter nearly an hour of gunfire being exchanged, the armed prisoners finally took the Security room, allowing them to unseal the ship's locked down hatches and move around at will. The Gali prison guards were highly trained, most of them former commandos themselves. However, they were outnumbered, if not overwhelmed by the volume of prisoners on the overpopulated ship. Before they were given warning, a third or more of the ship had already been lost to them.\n\nFor hours the sound of heavy gunfire throughout the ship continued, Gali trained guards fighting the massive riot of former Gali soldiers turned prisoner. Both Quinton and Roman waited patiently as they remained out of sight, positioned behind a large set of steam filled pipes. For the first time in many years Roman began thinking about his past, something he usually avoided. Looking down to his scarred body, he remembered the Goliath parts that had been forcefully removed and events that led him to having them in the first place.\n\nHe thought about everything, even the lies both he and Quinton had told Zane. Glancing at Zane for a moment, he nodded his appreciation. They were indeed like brothers, but for very different reasons than those Zane had been given. Greyspine. Yea, Roman had been there, and he had indeed killed the Queen of Hunters on that very day. But he was not a Gali commando, not even a soldier in the least. He wasn't even supposed to be there. He was a terrorist, at least he had been for the majority of his life. His family had been murdered at the hands of Hunters, that much was true. Quinton and family had taken him in, though the family was a terrorist group named Black Cell.\n\nWatching his family cut down in cold blood had done something to Roman. Opened his eyes, if nothing else, to the reality of how the world worked. The strong survived. And he had promised himself he would survive at any cost while slaying as many Hunters as he possibly could. After Black Cell fell to the Hunters, Roman and a handful of its members went on the run.\n\nKilling the Hunter Queen, Black Cell had been dressed as though they were Gali soldiers. Rather than watching his government sign a treaty of peace with the murdering bastards who had slain his family, he watched the Hunters blame Gali. They called it a setup, and soon that led to war. The war between Gali and the Hunter tribes lasted nearly two years with neither side declaring victory. The Hunters had been killed to the point of near extinction, while Gali fell from a major power throughout the Skyla System to merely an afterthought. Their ranks had depleted to crisis levels.\n\nThey never signed a treaty of peace. Instead, there was a unspoken understanding between the Gali and Hunter tribes. They kept their distance. Live and let live. So many had died on both sides, all because of Black Cell. If they were found in Hunter occupied space, the members of the former terrorist group were painfully executed. As they were rounded up in Gali occupied space, they weren't given the pleasure of death, instead sent to live out a life unfit for a dog aboard this prison ship of horrors. As Roman looked himself from toe to hand slowly, he knew it in his heart. He would never stop killing. Hunters and sympathisers of Hunters. They were and would always be his targets.\n\nMeeting Adam and his crew, Roman's agenda had first been to kill the crew and take the ship. Planning to find as many members of Black Cell as he possibly could and continue the fight. What he quickly discovered, however, was the feeling of family aboard the Gunship. He began to like Adam and crew, feeling accepted among them. They treated him with respect, he felt as though they were what his brothers and sisters would have been like if not slain by the Hunters.\n\nSoon Roman's motives changed and he just wanted to remain with the Gunship crew, hoping to outrun his past forever. As he stared across to Quinton, Roman began to wonder who he was. Was he still the terrorist linked to Black Cell, the killer who had sworn his life to ending the Hunter race? Or had his time with Adam and crew changed him? He knew that in not so many days he would be faced with the very decision, and he wasn't sure where his heart would lead him.\n\n\"There you are, we have been wondering where you took off to.\" Adam said as he slowly approached Sasha. She had made her way back to the shuttle, sitting on its steel ramp and watching the night life of Arch City unfold.\n\n\"I'm fine. Just spend so much time in space anymore that when I get a chance to sit and watch the world in motion, I like to take it.\" Sasha replied.\n\n\"Mind if I sit with you?\" Adam asked, his curious affections for her growing stronger by the minute.\n\n\"As long as you don't pull out any of those horrible pickup lines.\" she replied, both of them laughing as they thought of Dalton's poorly choreographed one liners, many of which had already been used tonight.\n\nThough he had seen no payoff on the empty compliments as of the moment, Dalton began to see a trend. His furry friend Whiskey attracted women, a lot of them, which made him man's best friend indeed.\n\n\"Listen,\" Sasha said, continuing to stare across the highway to the hotel where they would all spend the night, though Dalton's would more than likely be spent in the lobby bar. \"I hope I haven't put any pressure on you. It wasn't my intention at all.\" Sasha said softly, turning to Adam as they continued to sit on the shuttle's ramp, their arms wrapped around peaking knees in front of them.\n\n\"You haven't,\" Adam said, pulling his left arm up and placing it around the shoulders of Sasha to pull her close. \"Besides, I'm the great Captain Adam Michaels. I am wired to deal with pressure.\" he said mockingly as they both laughed softly.\n\n\"Yea, you are quite the living legend.\" Sasha replied, smacking him on the arm playfully. Moments later, they had engulfed one another with a shared kiss, a deep linking of lips which drown in life altering passion.\n\nThe kiss was fantastic, and they both silently agreed on the fact. Looking into each others' eyes afterward and sharing a moment that no amount of time could ever take away.\n\n\"So, I'm not completely over her. Just so you know.\" Adam said, continuing to hold Sasha with his left arm as they turned their attention back to the hotel.\n\nFinally he felt comfortable enough to admit his lingering feelings for Sarah Blaine.\n\n\"I know. I'm alright with that as long as what we do have is something real.\" Sasha replied.\n\n\"It's real,\" Adam said. \"Just wanted to be up front about my feelings is all. Us smugglers and our damn passion for telling the truth.\" he added as Sasha turned to look at him with sarcasm.\n\n\"What about Kraid, are you over him?\" Adam asked.\n\n\"Um, yea. Only about five years ago.\" Sasha replied, huffing a bit.\n\n\"Relax, I was just asking.\" Adam replied in his own defense.\n\n\"In the spirit of honesty though, I don't think he's over me.\" Sasha said.\n\n\"In the spirit of honesty,\" Adam agreed, both of them giggling once more. \"Means he either plans to kill me or sent his boys to make sure I stay away from you.\" Adam said.\n\n\"Why do you say that?\" Sasha asked.\n\n\"Call it a smuggler's intuition. That and the fact that they are heading this way.\" Adam replied, Sasha quickly turning to see both Primal and Stage making their way across the busy street.\n\n\"Damn.\" Sasha said out of frustration.\n\n\"It's alright. That kiss will be lasting me several hours yet.\" Adam replied, smiling as he turned to his sparking love interest.\n\n\"We were able to knock down the transmission tower. Looks like we control the mess hall, security room, cell blocks one through seven and about a third of the administrative wing. The man still controls the rest.\" one of the prisoners said as he stood holding a riot style shotgun.\n\n\"Concentrate as much manpower as you can on the administrative wing. We take that, we take the ship's armory.\" Zane replied. He was certainly in command, several loyal friends at his side as they worked on a strategy to take the remainder of the ship.\n\n\"We outnumber them by a long shot, but they're well trained,\" Zane said as the group looked on. \"We need to take the armory and keep the landing bay. If we lose the landing bay then we lose our ticket off of this ship.\" he added.\n\n\"What about Roman?\" one of the prisoners asked.\n\n\"You let me worry about Roman Raines. You just keep that landing bay secure and we'll leave his ass here high and dry to sort it all out after the fact, and that's only if I don't find him first,\" Zane replied as he shouldered a shotgun and motioned two of his loyal friends to accompany him. \"Time to go hunting boys. Time to go hunting.\" he added.\n\n\"Look, all I'm saying is that I should be getting paid fifteen percent from you, combat pay from the Colonials and 'aint seen a damn dime of nothing,\" Dalton said as he sat with Adam at the long bar of wood and silver polish. \"Which is why drinks should be on you.\" he added.\n\n\"I'd be glad to pay for drinks if you didn't bang 'em down like you had one day left to live. Not to mention you've bought every woman in here a drink, and most of them more than one.\" Adam said, a wide smile on his face.\n\n\"I may only have a day left, hell, with some of the shit you drag us into it's anybody's guess.\" Dalton replied.\n\n\"Tell you what. I'll pay up if you get rid of that raggedy brown coat once and for all.\" Adam said, stoking the already emotionally intoxicated Dalton James.\n\n\"Fuck you and the horse you rode in on buddy,\" Dalton said loudly, drawing unwanted attention from several of the patrons. \"The brown coat stays!\" he added.\n\n\"Here,\" Sasha said, slapping enough credits onto the bar to pay for Dalton's venture. \"Now shut up and quick bringing stares our way, we're here to pull off a heist for God's sake.\" she added sternly.\n\n\"What about Whiskey?\" Dalton demanded to know, his trusted pooch to be treated like royalty now that he understood a woman's attraction to such a cute dog.\n\n\"Sure, order any drink you want.\" Sasha quickly responded, staring at Adam for a moment in disbelief.\n\n\"I meant the dog.\" Dalton said loudly, his demands ignored. \"Damn she's got an ass on her.\" he mumbled, watching the Benzan beauty walk away, her heart shaped backside bouncing inside form fitting pants.\n\nOne of the bar's video feeds switched to the news coming out of Tameca, causing the entire crew to once again join Dalton at the bar in silence as they watched on. It was one of those sobering moments in life, an event takes place and you know that it will remain written in history forever. You never forget where you are when the news arrives, and for everyone here, they remember the bartender turning the volume up loud enough for everyone to hear clearly.\n\nThe hardest hitting of the video segment was a montage of photos showing the level of savagery in which civilians were being murdered. It was the news of a downed Colonial Star, however, that hit Adam hardest as he tried to catch his breath; feeling as though he had been kicked directly in the stomach. He waited, hanging his hopes on every word of the news journalist as he awaited the fate of Sarah Blaine. Instead, they didn't mention her name a single time. She is the elected Colonial leader for God's sake! Adam thought, stunned of no news on the woman his heart still hurt for.\n\n\"Bet them 'sumbitches are getting' paid,\" Dalton said loudly, several patrons of the bar, including his own crew, turning to look in disbelief. \"That's right folks, my name is Dalton James and I work for a deadbeat employer.\" he added, bringing chuckling in the rear of the gathered group.\n\n\"Dalton, people are dying.\" Sasha said snidely.\n\n\"People are always dying. That's politics for you. That's why, unlike pretty women and whiskey, I just can't do 'em,\" he replied, changing his facial expression a bit. \"I mean whiskey as in the drink.\" he added.\n\nSasha, who began to reflect on her own family being murdered as she watched the events unfold on the video feed, grew angry and wanted to lash out at the drunken man.\n\n\"Hey, just let it go,\" Adam said, grabbing her softly and immediately noticing the hurt in her eyes. \"Want to walk outside for a few minutes, just the two of us?\" Adam asked, turning to gaze at Stage and Primal as if to dare them to join.\n\nRather than respond, Sasha got up slowly to accompany him and stared through Dalton with anger as she walked away.\n\n\"Guess she's a voter.\" Dalton remarked as Primal began to explain the fate of her family.\n\n\"What's going on?\" Adam asked with sincerity as both he and Sasha walked away from the hotel a bit, approaching a metal bench located nearby as the hustle and bustle of citizens continued walking around them.\n\n\"I don't know. Seeing those video feeds just made me think about my family I guess. It's tough to walk through the day as though nothing is bothering me.\" Sasha replied.\n\nHer words struck home with Adam, who found himself pushing through each day as though it were an uphill battle as well, wanting to hold Sarah Blaine one last time.\n\n\"Try to focus on the positive things. We're together, we're out in space enjoying a freedom which many will never know. The past can't be changed, no matter how hard we wish it,\" Adam said. \"What you need to understand is, my heart still isn't completely free. Soon, there will come a time when it is and when that happens I'll be knocking down your door begging for a chance to be with you. Hell, any sane guy would. You're beautiful, intelligent and I gotta admit you have a knack with handguns.\" he added, bringing a brilliantly sweet smile to her face as he placed a finger on her cheek to wipe away the most visible tear.\n\n\"And Dalton...\" Adam began to say.\n\n\"Ohh GOD, Dalton James.\" Sasha replied with exhausted patience. \"I know. Believe me, I know. Please understand though, he's a good friend and I would do anything for him. Just like I would you.\" Adam replied, slowly placing his arm around Sasha to hug her a bit in the chilled night.\n\n\"Adam Michaels, you are a good man.\" Sasha said, enjoying the moment as she sat there, held by a man who had her heart and genuinely cared for her.\n\n\"Yea, well there are at least a dozen outstanding warrants that say otherwise.\" Adam replied, both of them starting to laugh quietly as a light rain began to fall.\n\nFirst as nothing more than a mist, the water quickly became heavier. Rather than run for the hotel, Adam pulled his climate jacket up, placing it above both of their heads as Sasha's face remained pinned to his chest.\n\nAs the thick cold of rain fell, Adam protected Sasha with his jacket, using his other arm to hold her tightly as they kissed. Their bodies, even souls, connected at that very moment. Lips passionately feeling each other out as Adam's hand moved up, his fingers quickly flowing through her satin black hair. The rain fell so hard that it was almost impossible to see ahead. Almost. Unfortunately for them both, Stage saw the blossoming love unfolding as he stood at the front entrance of the hotel, watching his leader's love falling for another.\n\n\"They're hitting us with everything they have sir.\" Lieutenant Scott said loudly, the pounding of lead hitting weakening concrete. He stood in one of the building's interior hallways with several Colonial soldiers, including General Ortega.\n\n\"I don't care how much they hit us with Lieutenant, these civilians are to be protected at all costs. Is that understood.\" Ortega asked.\n\n\"Yes sir. I only fear that soon we will have no walls protecting them.\" Lieutenant Scott said.\n\n\"Then we will throw our own bodies in front of bullets to protect them until help arrives. Is that understood?\" General Ortega said sternly.\n\n\"Yes sir, of course.\" Scott replied.\n\nOutside the warehouse, dozens of armored Theron vehicles continued their barrage of gunfire onto the crumbling outer walls. Grey slate turning to rubble and sliding down, both Theron and Legion soldiers by the hundreds waiting a safe distance back, ready to enter and kill everyone when the time was right.\n\nAs General Ortega entered one of the larger rooms on the hall, his two posted Colonial soldiers moved aside to allow his entrance. Nearly fifty refugees, most of them women and children, sat in the floor.\n\nThey had been battered by the stench of war, most of them bloodied and holding their children's ears as the building rocked from intense gunfire.\n\n\"Excuse me. Sir. Can you please talk to my son, explain to him we will be alright.\" a woman said, slowly approaching Ortega.\n\nShe wore what once was a very upscale business outfit, dress pants and button up shirt; covered in the fog of war of course. Dirt and blood was all over the woman who was obviously someone of importance, at least before Tameca City had been destroyed. The General skimmed the room and saw it on all of their faces, even in the eyes of his posted soldiers. They saw no hope. They needed reassurance that victory was still possible.\n\nGeneral Ortega approached the young boy as the entire room fell silent, eyes fixed onto the fearless leader of the Colonial effort.\n\n\"Have you ever seen a Husk son?\" General Ortega asked, kneeling down to the young boy of no more than six years of age.\n\n\"No sir.\" the boy replied timidly.\n\n\"Husk are some of the most battle-tested warriors the Skyla System has to offer. They are strong, savage and much more capable than the cowards who are outside right now firing their weapons at us,\" Ortega said, standing a bit so he may address the rest of the people in the room as well. \"All we have to do is hold out. Gali has sent a large force of soldiers to help all of us, and with them, thousands of Husk will also arrive. All we need to do is hold on until they get here,\" General Ortega said. \"And son, I promise you this,\" he added, bending over once more to place his right hand onto the shoulder of the young boy. \"My soldiers will hold this factory and keep you safe until that time comes.\" General Ortega said, bringing a huge smile to the face of the young man and reassurance to the rest of the refugees who looked on.\n\nThe fire that burned to help these people strengthened inside of the posted Colonial soldiers as well, their demeanor once again becoming that of an unstoppable force.\n\nRoman held his hand out, silently warning his Black Cell brother of footsteps approaching. Quinton was a bit larger than Roman, though both were built sturdily. Quinton gripped the stock of his rifle tight as Roman held both firearm and riot stick, the blunt weapon down at his side as they listened. There were three Gali guards approaching, all slowly and as quietly as possible, weapons in front of them and at the ready.\n\nRoman glanced back to his brother for a moment, his back firmly against the steel of an interior wall just inside the doorway. Outside, the guards approached with caution in the narrow confines of the hallway. Roman held up three fingers, passing the information along to Quinton. Next, he held a clinched fist to let his brother know a fight was unavoidable and without pause he swung the clinched fist around, hitting the guard in front and jarring several teeth loose from the man's skull.\n\nImmediately pulling the helpless guard back to his feet, Roman used the flesh of his unconscious victim to absorb several gunshots sent his way by the remaining two lawmen. Roman then threw the dead flesh to the flooring below, jumping onto the second guard with no remorse. They had beaten him, spit through the bars of his holding cell on many occasions and now that the tables were turned, Roman saw no reason to be gentle. The last guard quickly took aim at the former Black Cell terrorist, but was nearly cut in half with a string of shots fired from the weapon shouldered by Quinton.\n\nKnowing well that more would come to the shots, be they guards of prisoners in hunt of the two men, Quinton did his best to pull Roman from the second guard. It took a few moments, Roman resisting as he continued to punch heavily on the guard he had already beaten to death and beyond.\n\n\"We have to go now! Won't be long before we hear the shuffling of feet. Zane's or more of the man's.\" Quinton said, finally able to pull Roman from the beaten guard.\n\nRoman stood on his own, spitting onto the lifeless body as Quinton let go of his bear hugging grip. \"They got what was coming to them my brother, now we gotta move! Got to get to that landing bay and find a place to hold up.\" Quinton added.\n\n\"Yea they got what was coming to 'em, and I got a lot more to give.\" Roman said abruptly, having found the hate he harbored against the people who had treated him less than human. Checking the guards for weapons, they found a Gali prison radio that would come in handy for the rest of their escape. Quinton placed it into the back of his pants, the two men quickly leaving the scene and heading into the direction of the landing bay.\n\n\"Hit me.\" Dalton said, a stringy brown cigar hanging from his lips. He was the only man in the hotel bar enjoying a cigar that had the undeniable look of poverty.\n\nEveryone else dressed with class and smoked only the finest of rolled pleasure. Not Dalton. Never the type to care about anyone's impression of him, he smiled wide, a bottle of rock whiskey and shot glass in front of of him.\n\n\"He wins again!\" the dealer exclaimed as the small crowd that had gathered at the card table cheered on. Dalton was on a winning streak, and though two beautiful woman had found their way to him primarily because of the credits he had laying on the table, Dalton was convinced his charm was the hook.\n\nRock whiskey was by far the cheapest alcohol in the entire Skyla System, aside from home brew of course. In fact, the only reason the hotel kept it in stock was for the occasional homeless citizen that collected enough money to enjoy a drink of the stagnant brew. For Dalton, it didn't matter. He could be rolling in money and never forget the rough patches he'd been through. For nearly eleven days during the first war of Glimmeria, he and Adam survived on nothing but rust water and vermin. Left behind by their own soldiers, abandoned and given up for dead. What the rest of the hotel lobby didn't understand was compared to rust water, rock whiskey was fine wine. Dalton understood this, and was a survivor because of it.\n\n\"Does he know what inconspicuous even means?\" Stage asked, puzzled with the thought of Dalton drawing so much attention his way before such an illegal heist.\n\n\"I don't think his mind is on the heist.\" Sasha replied as they all watched Dalton slide a free hand down onto the ass of each of the two women.\n\nBoth had bleach blonde hair to go with the look of well used, to say the least. Well used was an afrodisiac to Dalton, however. To him a woman who had been around was a woman worth chasing. She knew the score, understood the game and was less likely to fall in love with the brown coat booty chaser.\n\nHis four legged friend of thick fur was present as well, gaining just as much attention from the women who rubbed on him as though he were their own. Yep, Dalton was hot. Winning hands and hearts, even if the hearts chased his money.\n\n\"Come on, we gotta get going.\" Primal said, tapping Dalton on the shoulder.\n\n\"I'll be up in a few.\" Dalton replied, a handful of cards in his left hand, while his right maintained a firm grip on the heart shaped ass of one of the women.\n\n\"I said now.\" Primal demanded sternly, hushing the crowd. Dalton gained a strange look on his face, one that normally wasn't attached to cheap women and equally as cheap whiskey.\n\n\"Be right back.\" he said, kissing one of the woman deeply and petting Whiskey slightly.\n\n\"What the fuck did you just say,\" Dalton yelled, standing to his feet swiftly and pushing Primal back several feet in the same motion. \"Don't nobody tell me what the fuck to do primate.\" Dalton added loudly, enraging Primal, who dashed at him with ill intent.\n\nA drunk is pretty easy to take out, but unfortunately for Primal, Dalton could hold rock whiskey like a camel of sort. Easily evading his charge, Dalton pushed Primal as he passed, using the Benzan's own momentum to send him crashing into the table and smashing the fine wooden grain into shards on the floor.\n\n\"I was knocking boots when you was wearing diapers you big hairy 'sumbitch.\" Dalton said loudly, jumping on top of Primal and throwing fists into his direction. The Benzan blocked most of the punches, which turned to slaps, letting Primal know he'd been bested.\n\n\"Enough!\" Stage said, starting for the men and his weapon.\n\n\"It's a fair fight,\" Adam said, pulling his sidearm with surgical haste and holding it on Stage who turned back to look at him with bitterness. \"Let them decide when it's enough.\" Adam added.\n\n\"You dare pull a gun on me?\" Stage asked.\n\n\"I've dared to do a lot of things, and you are pretty low down on that list friend.\" Adam replied.\n\n\"Sasha.\" Stage said, waiting for the Benzan beauty to intervene.\n\n\"It is a fair fight Stage.\" she replied, making the man furious. \"First you betray Kraid for this common thief, and now you insult me!\" Stage said, pissed to the highest point that fury could take him.\n\nSasha pulled her automatic pistol out, pointing it at Stage for a moment as she leaned in to kiss Adam deeply. \"Kraid doesn't own me and neither do you.\" Sasha replied, letting the man know right away that she was her own woman.\n\n\"You will be dealt with in due time,\" Stage said to her, glancing at Adam and Dalton for a moment. \"And you two, you're both dead men!\" he added.\n\n\"Now there's a line that's been used more than my damn pickup lines,\" Dalton said, turning to Sasha as if to convince her that his pickup lines were still semi-fresh. \"Plus, if you two clowns really are the best Kraid has to offer then you might want to consider another line of work. Something like ranch work or waiting tables. Hell, apply here, they may hire you on. Cause you sure in the hell 'aint no killers.\" Dalton said, letting Primal back to his feet slowly. Dalton grabbed a fistful of credits from the pile of rubble near the spot that used to be home to a table, pulling one of the women close to him.\n\nA mixture of kissing and talking to the woman as he groped her heavily, Dalton told her to split the remainder of the credits with the other woman. Their company tonight had been well worth the money, which was several thousand credits. Both women hugged Whiskey, a perfect nightcap to their ass kissing of the man with the winning hand. Meanwhile, Primal walked away slowly, his stare never breaking from Dalton while doing so.\n\n\"I guess this means the rescue is off.\" Stage said sarcastically.\n\n\"Nope. Just means you two outlaw wannabees are finding your own ride home,\" Dalton replied. \"Now go on, get!\" he added, shooing them away like unwanted strays. Stage looked at him strangely, quickly realizing the shuttle they had actually arrived in was now Adam's property.\n\n\"Best if you both hit the road.\" Adam said, at gunpoint of course. Sasha wasn't crazy about the idea of turning her back on the two Benzans, but now that she knew what Kraid thought of her; she wasn't crazy about returning to the Benzans either. She had just needed someone to love her, since her parents were executed it had been the Benzans. Now, she had started to look to Adam for that sense of love, and she trusted him with everything inside of her.\n\n\"Simple. We just hang out by the shuttle, when they come out we dust 'em.\" Stage said as the two men furiously walked across the busy street.\n\n\"And Sasha?\" Primal asked.\n\n\"No harm is to come to her. She's just confused, that's all. Kraid will straighten her out.\" Stage replied.\n\n\"Find the ship.\" Vladris said as the other two Hunter Elites carried heavy rifles in hand.\n\n\"There. The numbers match those of the Benzan shuttle.\" one of the Elites said, the pointed claw of his thick white hand showing the way.\n\nVladris led the group, a long blade sheathed to his back as his leg held a pistol and his massive hands clinched the stock of a rifle. A few citizens stared hard. Some had never seen men with such a look, while others knew all too well they were Hunters and were taken by surprise as the demonic clan of murderers rarely appeared in such a public place. Vladris and his two accompanying Hunter Elites slowly approached the Benzan shuttle, the moon reflecting lightly from the shaven head of Vladris. Meanwhile, his accompanying Elites looked equally as savage, their braided locks of shimmer white and teeth of luminous death only two of many features that led the citizens nearby to steer clear. The normal Hunter simply had a lifeless look in its eye, the pupil void of any pigment. Hunter Elites, however, had a slight red color which seemed to glow a bit. Much like the embers of a freshly discarded fire.\n\nIt wasn't clear which group of warriors first caught glance of the other, but to every citizen unlucky enough to be in the area, it was damn clear that lead began to fly without remorse. Stage hit one of the Hunter Elites with two solid shots, both slamming into the trunk of his chest, knocking the undead beast back a bit. Primal was too close for gunfire, his first reaction instead was to begin swinging fists at the demon closest to him. Quickly bested however, Primal fell to his back, sliding quickly out of the way to once again gain footing. The hair covered Benzan drew a long blade from a sheath on his right thigh, his full intention to attack. Quickly however, he found himself on the defense; the large blade barely able to withstand a downward strike from the sword of a Hunter Elite.\n\n\"What is it?\" Adam asked as Sasha looked through the drapes of the hotel room following the distinct sounds of gunshots.\n\n\"Oh God, it's Stage and Primal.\" she replied in shock, Adam joining her at the window seconds later.\n\nHe had never encountered a Hunter Elite before, but was pretty damn sure that's what stood on the ground below. Three of them to be exact.\n\n\"Dalton, pack our shit right now.\" Adam said.\n\n\"Just a second, let me finish this...\" Dalton replied, cut short as he and Whiskey lay on one of the large beds.\n\n\"Now,\" Adam yelled sternly. \"We got Hunters down below and they 'aint the garden variety!\" he added.\n\nSpilling what remained of his tall whiskey glass, Dalton jumped to his feet with haste, quickly sitting to try and put his boots and coat back on with frantic pace.\n\n\"I'm sick and tired of this fucking bullshit,\" Dalton muttered, pulling his brown coat on quickly, though it wrapped his body a bit crooked. \"Going from winning card games, surrounded by fine looking strange and endless streams of whiskey to running from these dead 'sumbitches. It's always these dead 'sumbitches. I told you not to take the fucking job in the first place. Told you not to do business with Anwick. I told you. Didn't I tell you?\" Dalton mumbled loudly as he continued to fight his feet into the thick leather of his worn boots.\n\n\"Shh.\" Adam said quietly.\n\n\"Well I did...\" Dalton began to reply.\n\n\"Shh,\" both Adam and Sasha insisted. \"They're gone.\" Adam said quietly, glancing out of the room's window for a moment. About the time he finished uttering the words, a solid knock came at the door.\n\n\"Shit.\" Dalton motioned with his lips, no actual sound escaping. Sasha motioned Dalton to answer the knock, stalling them long enough for Adam to open the window that led down by fire escape.\n\nDalton looked back at her puzzled, having no idea who was even at the door. Sasha motioned him once more as the knocking grew heavier.\n\n\"Um, if this is about child support you got the wrong room.\" Dalton said cheerfully, drawing a strange glare from both Adam and Sasha. Seconds later, the shots of Hunter rifles blew gaping holes through the solid wooden barrier on hinges. As the door swung open, smacking roughly against the wall behind it, Dalton held his shotgun out and discharged two shots. Blind as to what his target even was, but following the creed of a smuggler. He who gets shot at shoots back.\n\nVladris and his two battle hardened knights of undead walked in, the first Elite hit with both shotgun blasts at close range, putting him down onto the ground in pain. The same Elite had been struck with two shots from Stage's weapon only moments before, and the four combined wounds had seriously injured the Knight of immortality. Dalton was too close to the Hunters and too far away from the window to jump out, opting instead to stand on the other side of the bed. Out of desperation, Dalton began throwing liquor glasses, pillows, magazines and anything else his hands could grab into the direction of the two standing Hunters.\n\nWhiskey pinned his ears back, letting loose a growl proving he was ready to make the ultimate sacrifice for his proud human owner as he caught sight of the Hunter's accompanying hell hound. As the second Hunter Elite stood there, staring at the pathetic halfway house for fleas, he was struck with several shots from Adam's pistol.\n\n\"Dalton, bring your ass!\" Adam yelled, moving as Sasha then peeked in, firing her automatic pistol into the direction of Vladris.\n\nThe Hunter Elites had no choice but to stay hidden behind a wall as the bullets chewed into the wood while a drunk and his dog made their escape. The hell hound growled loudly, its voice a mixture of undead anger and warning.\n\n\"Told you not to take that fucking job,\" Dalton muttered under his breath a bit as his body shook with adrenaline; Adam and Sasha sliding down the large corner rail of the fire escape. \"Don't let him hit the ground!\" Dalton demanded, dropping his four legged friend as though he were a fur covered feather into the waiting arms of Adam Michaels.\n\nHe was safely caught, though the velocity of his fall knocked Adam to the pavement. Dalton slid down right behind, knowing well enough that Hunters would not be far behind.\n\nThe three smugglers turned fugitives ran hard, a brown dog at their side throughout as Hunter Elites trailed them on foot by less than two hundred yards. Dalton stopped abruptly.\n\n\"Go to the shuttle and get that bitch ready, just don't leave me!\" he yelled, Adam pausing slightly before turning to resume his getaway.\n\nSasha was in first, hitting the necessary switches to fire up the engines by the time Adam ran up the steel ramp, his boots striking loudly on the grated steel.\n\n\"Hey boys.\" Dalton yelled, the Hunters slowing their run to a walk. \"Go. Get the fuck out of here!\" Primal yelled under his breath, doing his best to hide below a small ship docked near the Benzan shuttle.\n\n\"Got one hiding under here.\" Dalton said, smiling a bit and pointing to Primal before turning to haul ass and join Adam and Sasha.\n\n\"You bastard!\" Primal yelled loudly, Whiskey barking at the Benzan for a moment before leaving to catch up to Dalton.\n\n\"Go!\" Dalton shouted, he and Whiskey both sprinting up the shuttle's ramp and diving into the passenger area. One of the Hunter Elites had given chase, his arms clinching the edge of the passenger area as the shuttle began to lift off.\n\n\"Told you. Told you not to take that fucking job Adam!\" Dalton said angrily, forcefully kicking the Elite flush in the face several times and eventually causing him to release his grip, dropping back to the ground. Dalton fell back onto the deck of the shuttle for a moment, door still open as he gave a sigh of both frustration and relief.\n\n\"Dalton, close the door so we can break atmosphere!\" Adam said loudly.\n\n\"Wait. I'm trying to decide between shutting the door or jumping out my damn self and putting an end to this shit.\" Dalton replied with sarcasm, finally moving his body over enough to kick the large button which drew the ship's door closed. Primal watched their shuttle ascend into the clouds, glancing for a moment at Stage who lay dead on the ground, his vision filled shortly after with that of the near grin upon a hell hound's face. His screams lasted only seconds as the immortal pet's teeth shaved through the flesh of a life short lived.\n\nTigon Twelve was a solid Colonial shuttle, but its current crew had pushed it to the limit as the silver ship ate fuel without remorse during its full burn.\n\n\"Not long before we get to Arch City my lady,\" Lassiter said loudly as the loyal soldiers all patiently waited with weapons at the ready. \"When we get there, you are to remain onboard with the Goliaths. Is that clear?\" he added.\n\nIt seemed strange, Lassiter giving orders to the leader of the Colonials. However, he was concerned for Sarah's safety and she was aware of the fact, though she had no clue of his mounting feelings for her.\n\n\"Just find Adam before they do.\" Sarah replied.\n\n\"Don't worry my lady, if he's there we'll find him.\" Lassiter responded as he sat directly in front of the mighty Husk warriors who were prepared to fight to the death for Sarah Blaine.\n\nArch City was located on the small but highly populated planet of Gali, known for its upscale living conditions, high priced fashion and modern technology. None of that appealed to Sarah Blaine, instead she only prayed that Adam Michaels was here and that he was safe. Of course, she wanted so badly for him to forgive her. Possibly give her a second chance at the storybook romance they once shared, but her love for him was the unconditional kind. Even if he refused to speak to her again, she just wanted him safe, wanted him happy.\n\nSarah had always been taught that she was in control of her own life, her own destiny. Which is why this was so hard to deal with. She couldn't control her heart's longing for Adam Michaels, it simply wasn't an emotion that could be turned on and off with the push of a button. In fact, nothing about the events which were about to unfold gave her any control. She was helpless. Was Adam alright? Would he forgive her and possibly give her another chance? It was all taking a huge toll on her emotionally, and none of it was to be controlled or decided by her. Instead, she could only pray for his safety and plead for his forgiveness.\n\n\"Sir. You better let us take the lead on this. If the Hunters have an Elite in their ranks it may take all of us to bring him down.\" one of the Husk said with reserve as the human Colonial soldiers looked at one another. It was a staggering thought, anything taking all of the mighty Husk warriors to kill.\n\n\"Well enough, but I will be right behind you with my Colonial team. The Hunters do not intimidate me.\" Lassiter replied proudly.\n\n\"Ever see a Hunter Elite?\" the Husk asked.\n\n\"Not up close, why?\" Lassiter asked. The Husk remained silent as they began checking their weaponry for several tense moments.\n\n\"Just let us take the lead. They are not to be taken lightly.\" the Husk finally replied.\n\n\"We are hitting the upper atmosphere now sir, should be on top of Arch City within minutes.\" the Colonial pilot said as he turned to Lieutenant Lassiter for a moment.\n\n\"Circle the city low and try to locate the Benzan shuttle. We'll have eyes on the ground looking for any signs of Hunters.\" Lassiter replied.\n\n\"Yes sir.\" the pilot responded, turning to bring Tigon Twelve down low and begin looking for the shuttle's identification number and description.\n\nZane slowly walked down the corridor of narrow steel and aged plating as gunshots rang out in the distance. \"Just keep your eyes peeled open and if we find them leave Roman for me.\" he said in a low voice which was answered by silent nods as all three men held riot style shotguns out in front of them.\n\nHearing footsteps, the men stopped and crouched to a knee in wait of what enemies might be ahead. Moments later, two Gali prison guards turned the corner, each with a repeater gun in hand. The Gali repeater was a semi-automatic rifle that fired hollow point shells, though the stock of the weapon was short. Best suited for close quarter combat, the well trained guards preferred them in the event of a prison uprising or riot.\n\nZane and his men stood up, quickly firing bursts from their shotguns as one prison guard immediately fell to his death. The remaining was hit in the leg, falling as he returned fire. The shots fired from his repeater easily cut through both flesh and bone, dropping one of Zane's men and chewing into the plated steel wall behind him. The injured guard tried to crawl back around the corner, blind firing his repeater while doing so. Seconds later Zane stood over him, smiling slightly before burying a shotgun blast into the guard's face at point blank range.\n\n\"Grab their weapons and let's head back. We go much farther and we are liable to run into a nest of the bastards.\" Zane said as they collected the weapons and key cards from the fallen guards, both Roman and Quinton watching from the rafters nearby.\n\nTheir plan was to climb down from the steel beams which held the ship intact shortly after Zane and his remaining henchman had turned back, however additional Gali prison guards arrived before they were able to put the plan into action. Checking their dead, they soon gave chase themselves, a total of eight soldiers carrying the deadly repeater rifles.\n\n\"What now?\" Quinton asked quietly as both men waited a few minutes before slowly climbing down onto the steel deck of the ship.\n\n\"Now we work on making our way to the landing bay.\" Roman said steadfast as he checked both corners of the short and narrow hallway.\n\n\"Gonna be hell taking it, Zane is likely to be expecting us to try.\" Quinton remarked.\n\n\"Zane's got it all figured out. Except for the fact that my hands have killed more men than his eyes have seen. He has no idea what I'm capable of when I'm pissed off.\" Roman replied.\n\n\"And let me guess my brother, you're pissed off right about now?\" Quinton asked. His question was simply answered with a stare that he had only seen Roman give a few times before. Indeed...he was pissed off.\n\n\"They left nothing behind.\" one of the Hunter Elites said as Vladris helped him search through everything. The third Elite had suffered several gunshots and was sitting in the floor near the bed of the room, grimacing with pain.\n\n\"I know where they are heading. At least I will.\" Vladris said, holding the receiver of a tracking module in his hand, the glow of pulsing red running through it. \"I just want to know who were are dealing with,\" Vladris added, turning to his wounded brother who sat in the floor. \"Will you make it?\" he asked. \"In time. The gunshots were dead on but I will heal.\" the Hunter Elite replied, doing his best to stop the bleeding of a nearly foot wide hole that gaped into his chest.\n\nThe Elite Knights possessed the ability to heal quickly, tissue regeneration only one of many Hunter traits. They were damn hard to kill, and the Elite sitting in the floor with a grin of pain had four gunshots attesting to the fact.\n\nTheir voices were easily heard by the approaching Colonial strike team, which walked with light steps into the direction of the room. Its door stood wide open, bloody footsteps leading them from the carved bodies of both Stage and Primal to their current location. The Husk gently pulled the blades from their backs, slowly bringing them to a ready and hoping not to alarm the Hunter Elites. They had no such luck. Vladris heard the silent but distinguishable sound of steel sliding against leather as the Husk unsheathed their blades.\n\n\"At arms!\" Vladris yelled, though one of his Elites was in no condition to fight. Both Vladris and his battle ready soldier pulled their blades with just enough time to meet the Husk on even terms, steel clanging against steel as every one of the warriors fought for survival.\n\nHunter Elites were the top of the food chain. That said, Husk were damn worthy opponents and in numbers they were slayers of Hunters, no matter the breed. The Elite who continued battling previous gunshot wounds slowly made his way to the same fire escape Adam's group had used to flee less than an hour before. A Hunter Elite fleeing was practically unheard of, but catching the sight of a full strike team led by Goliaths in the hallway, the demons knew there was no victory to be had.\n\n\"Get him to the shuttle!\" Vladris yelled. \"We will track them from the air!\" he added loudly, motioning his able Elite to pull the injured out of the window and onto the fire escape.\n\nMeanwhile, Vladris held his Vampiric fashioned blade above his head at a slant, daring the three Husk who circled him to make a move. They knew the dangers of fighting an Elite, even with three to one odds the Husk were over-matched. The equivalent of three finely trained attack dogs circling a Lion. What they didn't know, however, was that they stood before the top Hunter Elite. A fact they would soon learn the hard way.\n\nThe first Husk dashed in, his blade thrusting in front of him. A solid move against most, but not Vladris, who easily slipped past and cleaved his own blade into the shoulder of the beast. The Husk painfully screamed as the small bones of the shoulder area and attached tendons snapped unmercifully. Vladris quickly pulled his warm, salty steel from the Husk flesh, swinging it around and plunging it into the stomach of a second Husk. The beast like warrior immediately dropped his blade, a look of death slowly pasting into his eyes. Only Samuel remained, and he had promised to give his life for Sarah Blaine if need be. He was ready to do just that, though it wouldn't be on this day. Golaiths stormed the room, letting loose a wave of death worthy lead into the direction of a fleeing Vladris.\n\nSarah entered with the remainder of the strike team. She wanted to desperately search for Adam, but her first concern was that of a dying warrior. The Husk who had been cut from shoulder to chest sat in the floor, slowly bleeding out as his life flashed before him. Sarah knelt with him, doing her best to comfort the dying beast, an act that only grew stronger the loyalty of the remaining Husk. Never before had he seen such compassion for his kind by a human being.\n\nNormally the Husk fought beside humans who paid them well and discarded them from thought afterward. Yet Sarah's face told a different story. She cared for the dying Husk as though they were her own children. She was a leader worth dying for, the Colonial cause a noble one and that was reason enough to convince his people to follow her.\n\n\"Rest easy my brother, this is an honorable death. Your attacker will soon join you by my own hands.\" the remaining Husk said.\n\n\"Thank you my lady, I will sit with him. You find the one you love.\" the Husk said with great respect.\n\n\"Is he?\" Sarah asked.\n\n\"No, he's not here. They were though.\" Lassiter replied.\n\n\"How do you know?\" Sarah asked with hesitation. Lassiter held up a half empty bottle of rock whiskey, the preferred drink of Dalton James.\n\nSarah began to laugh a bit, relieved at least that they were alive and well.\n\n\"How will we find them now?\" she asked, looking to Lassiter for some kind of comforting answer.\n\n\"I overheard one of the Hunters say they would track them using a location device. The same kind of location device I had our men plant on the Hunter's shuttle before we entered the hotel.\" Lassiter replied with a grin, holding a small receiver with a luminous blue glow.\n\nSarah breathed a bit easier, knowing eventually she may once again lay eyes on Adam Michaels and simply nodding to Lassiter for his quick thinking.\n\n\"The bad news is when we do catch up to them, we will be seeing more of the killers who caused such quick death of two Husk warriors. I just hope we have enough firepower to fight them off again.\" Lassiter said.\n\n\"You let me worry about that.\" the remaining Husk said, approaching them with the blood of his own race on his hands and a look of revenge in his eyes.\n\nAs Lassiter and the Colonial soldiers cleared the hotel room, Samuel helped Sarah to the safety of the doorway.\n\n\"My lady, I am sorry that we failed you.\" the Husk warrior said regretfully. Rather than reply, Sarah simply broke down into tears.\n\n\"Sarah are you alright?\" Lassiter asked as he hurried back to the room's entrance to find the source of her sadness.\n\n\"These warriors died for me. Died because of my love for Adam, I can't live with that. People are dying because of my mistakes!\" Sarah said regretfully as she sobbed loudly.\n\n\"My lady, the Husk and Hunters live to slay one another. If these warriors would not have fallen here, eventually they would have fallen to Hunters somewhere. We are bred for this, to die in battle is the greatest of honors.\" Samuel said.\n\n\"At least we know Adam made it out alright,\" Lassiter said. Seeing the thought bring comfort to her as she smiled a bit, Lassiter walked closer. \"We know they had to leave in a hurry too, not like Dalton to leave good liquor behind,\" he said as Sarah began to laugh with emotional relief. \"Don't worry, we'll find them soon enough. At least you know Adam's still alive.\" Lassiter added. In Sarah's heart she already knew he was alright.\n\nThis was Adam's world, he made a living escaping with his life. He was a smuggler, and a damn good one at that. Her fear would be his unwillingness to even listen to her pleas as she laid her heart out in front of him. She had to be able to convince him of her true sincerity, otherwise Sarah would have to live out the remainder of her life without him. She wasn't sure that was even possible.\n\n\"It's gonna be tough,\" Quinton said as both he and Roman stood in the shadows of piping near the prison ship's landing bay. \"I'm counting fourteen of 'em, including Zane,\" Quinton added. \"All abundantly armed and waiting for us to try and take the landing bay.\" he added.\n\n\"In other words it's a fair fight.\" Roman replied.\n\n\"Yea, something like that my brother.\" Quinton replied, smiling a bit as he turned to his fellow terrorist. Before they could put any type of plan into effect, the ship began to shift abruptly.\n\n\"What the fuck?\" Quinton asked with shock. \"Looks like Zane's men have taken the control room,\" Roman replied. \"So what's the plan now?\" Quinton asked.\n\n\"Same plan. When that prison transport docks, we gotta be sure that we have control of the landing bay. Otherwise we are going to be stuck on this floating jail a lot longer than expected.\" Roman replied.\n\nThe two men moved a bit farther back into the shadows as a six man group of Gali prison guards rushed by and immediately opened fire into the direction of Zane and his men. Heavily outgunned, the six guards soon began to fall back, not that it would matter. More of Zane's men flanked their position and cut them to shreds with a hail of gunfire within only minutes. Knowing well enough that there was no chance of winning the fight in front of them, Roman and Quinton both remained in the shadows. The sight of dead prison guards being looted reflected back to the cutting pupils of Roman Raines, who did his best to memorize the face of every murdering bastard before him. Eventually their time would come, and when it did Roman would do his best to make sure he was a part of it.\n\n\"This shit 'aint gonna work Adam.\" Dalton said sarcastically as he stood there, dressed in a stolen guard's uniform. It was a snug fit, but manageable. His concern was the fact that they had decked Whiskey out in a guard's shirt as well.\n\n\"Relax.\" Adam said as he and Sasha both wore the same uniforms.\n\n\"Relax?\" Dalton said in a stunned tone. \"How the fuck are we supposed to explain a dog wearing a prison guard's shirt? Hell, the badge on the front is dragging the ground!\" Dalton said loudly.\n\n\"Just tell them it's a K-9 unit or something to that effect.\" Sasha said, turning to Adam as they both began laughing.\n\n\"I 'aint telling 'em shit. If it looks like they 'aint buyin' it then I'm gonna start shooting.\" Dalton replied angrily.\n\n\"Trust me. They are going to smell whiskey on your breath long before they catch a glimpse of the four legged variety.\" Adam added as Sasha burst into full blown laughter.\n\n\"Damn smart ass.\" Dalton replied, covering his mouth and nose for a moment before blowing to test Adam's theory.\n\n\"Relax, the hardest part will be passing as a prison transport shuttle with Benzan markings all over the outside of our ship,\" Sasha said as she began the landing sequence. \"I'm shocked they haven't hailed us by now, the landing bay is sitting wide open. May not buy any of this.\" she added.\n\n\"They don't have to buy anything for long, we don't have a single prisoner. Just land, look official when you exit and then start letting lead fly. Hopefully catchin' their asses off guard will be enough.\" Dalton replied as Sasha guided the ship onto the path of flashing red lights which lay on the prison ship's deck.\n\nRoman's attention was immediately taken by the sight of the transport shuttle to the left as it began entering the ship's landing bay.\n\n\"When it touches down, wait for the guards to exit the shuttle and then we make a run for it. Kill anything with a heartbeat, but do not stop running.\" Roman said.\n\n\"Got it.\" Quinton replied.\n\n\"Wait for them to exit and then spring the trap.\" Zane said as he stood with two of his men who were decked out in Gali prison guard outfits, while almost two dozen armed prisoners waited patiently inside of the room which controlled the landing bay doors.\n\n\"Here we go.\" Adam said as the shuttle touched down and a mixture of grey steam and smoke blew underneath its frame. Sasha followed Adam out of the shuttle door, walking slowly down the ramp as they skimmed the landing bay with an observant eye.\n\n\"How many did you bring us today?\" Zane asked as he approached.\n\n\"Actually we come bearing room for one more.\" Adam said quickly, pulling his gold plated pistol of Benzan craftsmanship and firing the first shot with such speed that Zane and his men still had no idea what was happening.\n\nHis shot surgically pierced the shoulder of the man to the left of Zane, rendering him defenseless as lead passed through the backside of the wound. As Zane and his remaining follower pulled their weapons to the ready, Sasha instantly cut Zane's other loyal friend down, her machine gun pistol tearing both cloth and meat as the man fell down in gripping pain. Outnumbered, Zane caught a glimpse of both Roman and Quinton sprinting for the ship from the corner of his anger ridden eye.\n\n\"Now!\" he yelled as armed prisoners flooded from the landing bay control room in all directions.\n\nLooking up to the shuttle entrance, Adam quickly extended his free hand, catching a fully loaded shotgun which had been tossed by Dalton. Nodding his gratitude, Adam turned and began firing shot after shot into the crowd of armed prisoners. Meanwhile, Dalton remained crouched at the entrance of the shuttle with a Benzan bolt action sniper rifle in hand.\n\n\"Take that ya' little bastard.\" Dalton mumbled as he fired a steadfast shot which instantly dropped one of the armed men. Whiskey remained near the shuttle's cockpit, waiting until the gunshots ceased and it was clear to leave.\n\nRoman was a reaver of men. It didn't matter their aranment or skill, he was simply unmatched when it came to blood letting and moments like these only added to his mounting legend. The combined movements of his Gali rifle and guard's baton allowed him to easily make a path through those who remained standing. Quinton did most of his work by way of rifle, zipping hot lead into the wall of guilty flesh before him.\n\nIt wasn't long before Zane's ranks had thinned and the thought of mortality began to set in. Dalton and Whiskey both exited the ship slowly as they joined their group during the last seconds of the gunfight.\n\n\"No, that's Roman!\" Adam yelled as Sasha pulled her pistol up and into the face of the former terrorist, not recognizing the man who had endured months of hell.\n\n\"Apologies my friend.\" Sasha said, slowly lifting the pistol away from Roman Raines, while continuing to use caution around the warrior's Black Cell brother.\n\n\"I told you I'd be back.\" Adam said proudly, answering the look of gratitude cast by Roman. Through all of the turmoil, Zane managed to gather himself from the floor, sprinting quickly out of the landing bay as his heavy feet clanged against the grated steel deck of the hallway.\n\nRoman thought long and hard about everything he had endured on the Gali prison ship. The food, the guard induced beatings, the inhumane living conditions and the fact that he had started to lose hope. He had truly started to believe that Adam and the Gunship crew had left him for dead.\n\nNone of that mattered now as Roman stood there, continuing to stare at the Captain of the most loyal group he'd ever been a part of.\n\n\"You're a man of your word.\" Roman finally said, extending his arm to shake the hand of Adam Michaels.\n\n\"I told you I would be back.\" Adam said again, proud of the fact that he was able to pull it off.\n\n\"Didn't say you would have Hunters right behind you though.\" Roman replied as everyone turned to see a Hunter distinct shuttle entering the landing bay.\n\n\"Ahh fuck,\" Dalton yelled, smacking his leg frantically and whistling until Whiskey began to run back to their position. \"Get them little legs running!\" Dalton added, crouching for a moment to coax Whiskey into moving quickly.\n\n\"Shit! Must have placed a beacon on our ship back at Arch City.\" Adam said with disgust.\n\n\"We better move now.\" Roman said sternly, motioning the group to follow him down the narrow hallway and into what would be no less than a hornet's nest of guards and rioting prisoners exchanging gunfire.\n\n\"See you lost a little weight and picked up a new friend.\" Dalton said jokingly as the entire group jogged slowly into a storage room. Filled with large wooden crates, it would be the perfect place to dig in and make a last stand if it came to that.\n\n\"This is my brother Quinton. As far as the weight, well I'm sure you all have noticed my lack of chrome.\" Roman said calmly, though his mind had already drifted back to the prison doctor barbarically pulling the meshed steel from his flesh.\n\n\"Your brother? You mean there are two of you?\" Dalton asked, stunned by Roman's statement.\n\n\"Many more than just two.\" Quinton said, a bright smile painted onto his face.\n\n\"I hate to break up the honeymoon here, but it would be wise to knock off the small talk and find cover. We don't know how many Hunters we are dealing with.\" Sasha said, the thought of Hunter Elites still fresh in her memory.\n\n\"She's right. Not to mention the entire prison ship is in a full scale riot. Guards are scattered and the prisoners are armed and on the hunt for us both,\" Roman replied. \"I'll take the front with Quinton.\" Roman said as both of the Black Cell members positioned themselves crouching behind a desk at the entrance to the room. Sasha waited quietly, hoping in her heart that Adam would position himself with her.\n\nIt would give them the perfect chance to be alone, and even under such stressful circumstances her heart felt so many different emotions for the true to his word smuggler. Their kiss in the rain still haunted her every waking moment, she wanted to live in that kiss forever.\n\n\"Dalton, you and Whiskey hide behind something. If we are going to die, I want to do it with my crew.\" Adam said, quickly joining Sasha as they took up residence in the corner behind a forklift type vehicle.\n\nAs Vladris stepped off of his shuttle slowly, gothic black boots striking the ground, the void pupils of his eyes skimmed the perimeter of the room. Quickly joined by his hell hound, who's ears skimmed the area for sound. Their keen senses instantly heard multiple gunshots being exchanged throughout the ship, but a prison riot was of no concern to him. Seeing out the execution of the last remaining Benzans were his only priority. Having left Arch City to regroup with more of the undead murderers, he turned to motion the rest of the group. Two additional Hunter Elites slowly exited the craft, followed by three of the less dangerous Hunter strain.\n\n\"Stay here and hold the landing bay until we return.\" Vladris commanded of the three Hunters as he turned and walked into the narrow hallway under escort of two more Hunter Elites.\n\nSoon the three ultra killers had disappeared from sight, taking with them only precision pistols and articulate swords of destruction.\n\nNearing the storage room, the Hunter Elites were taken off guard by a group of equally surprised prisoners. Heavily armed, the startled prisoners began to open fire without regard, clouds of bullets floating through the narrow confines of the hallway. In a normal scenario, six heavily armed men might stand a chance against a single Hunter Elite. But this was far from a normal scenario.\n\nThree Elites in close quarters led to a speedy slaying of the six prisoners, their flesh gouged to the bone and spines severed from the body. The screams and pleas of the prisoners being mutilated could easily be heard by Adam's group, which was positioned nearby in the storage room. They tried to remain focused, though both Adam and Dalton were horribly reminded of so many friends lost during the first Glimmerian War.\n\nThe screams of a dying man who is unready to go sound the same, no matter the killer or pleading man. The screams always sound the same. And that was a sound that both men had witnessed so many times first hand, a sound that made the skin of both men crawl. Told him not to that that fuckin' job. Dalton thought as they nervously waited.\n\nWith their jet black armor coated with the crimson splatter of blood and boots drenched in it, the Hunter Elites kept moving forward, stopping to glance in each room. Roman Raines was prepared to die to get his friends to safety as he clinched a combat blade in one hand and riot style shotgun in the other. They had came back for Roman and shown a type of loyalty that he had never known, and he wasn't about to let them die in front of him. He glanced at his Black Cell brother for a moment, turning to nod his appreciation one last time to Adam before moving his attention to the door. Vladris and his two skilled slayers stopped for a moment, looking into the room with malicious intent before taking fire from the end of the hallway. A three man group of prison guards held a riot shield and fired piercing rounds from an automatic sniper rifle. Hitting one of the Elites with multiple shots, the monster fell to the deck in agonizing pain. Sprinting to their position quickly, Vladris and the other Elite made short work of the guards, butchering the men into afterlife.\n\n\"It's important to me that these people make it to safety my brother,\" Roman said with a grave tone as Quinton looked at him solidly. \"They are good people. They don't know about our cause, but they are like family to me as well.\" Roman added as Quinton nodded slowly.\n\nRoman stood quickly, motioning the group to the door as the Hunters turned to exit the hallway.\n\n\"We gotta go now and pray they didn't leave many behind to guard the shuttles.\" Roman said.\n\n\"I'll take point, Quinton will watch the flank.\" he added. The group quickly ran out of the storage room and backtracked into the direction of the ship's landing bay.\n\n\"Well, there goes that idea.\" Dalton said as the group sprinted into the landing bay, quickly gaining the attention of the three Hunters left behind. As the Hunters contacted Vladris and his group by radio, Roman dropped his pistol to the ground, reaffirming his grip on the combat blade.\n\nHunters considered themselves to be at the top of the food chain, and when they were challenged by blade wielding cattle, they were quick to answer the challenge. The Hunters quickly drew their swords, impressions of both skulls and demons engraved into the pewter handles of the weapons.\n\n\"We got this.\" Roman said with thirst on his breath. Quinton was quick to join as Dalton tossed him a well built Glimmerian combat blade.\n\nAdam and the remainder of the group stepped back to watch the fight begin, ready to move to the shuttle quickly in the rare case of victory. They could have easily opened fire onto the Hunters, but there was a code of honor among all warriors in the Skyla System, no matter the flag.\n\n\"Be ready to bury some Hunter ass just in case.\" Adam said as the group quickly readied their weapons.\n\n\"All this shit and I still gotta buy my own damn drinks.\" Dalton said jokingly, though Adam knew him well enough to understand Dalton was trying his best to disguise fear. There was a good chance that every single one of them would die right here, the map of their lives drawn to a conclusion in this very landing bay.\n\n\"I've killed a Hunter Queen, you grunts aren't shit to me.\" Roman said insultingly, bringing anger to the faces of the immortal soldiers of death.\n\nThey were some of the best warriors ancient civilizations had produced, and now there was a feeling among everyone here. The best warriors of days gone by facing the most skilled warriors still mortal. While Adam and his crew were stunned at the former Black Cell member's statement, Adam was the only one to think it through. That's why he had been running. That's what led them to meet for the first time, and why the Hunters wanted him dead so badly. Sasha was the first to notice what the rest of the group quickly caught sight of. Three Hunter Elites. Walking into the landing bay, one seemed to be reeling from gunshot wounds slightly. Vladris looked at the group, especially the Benzans as he paced back and forth, dragging the tip of his blade across the floor. They would wait until the current fight was finished before blood letting, but wanted the group to see the sword which would end their lives.\n\n\"You're up next and you motherfuckers aren't walking out of here alive.\" Roman said bluntly, pointing his blade into the direction of Vladris, who quickly shifted his attention to the Black Cell terrorist.\n\n\"Your terrorist group has caused much grief for our people. Acts that you will soon be apologizing for by the edge of our blades.\" one of the Hunters said.\n\nTerrorist group? Adam thought as he watched Roman's movements, perfectly choreographed with those of Quinton.\n\nHe wondered how much he actually knew about Roman Raines. What kind of criminal had he been harboring aboard his ship since day one? It didn't matter, Roman had only been a good person since that day. That's what counted in his book. Especially since all of their lives depended on the skill of Roman and Quinton at this very moment.\n\nThe first Hunter made its move, lunging at the men with life ending intent. His sword was quickly parried away by the boot of Quinton, who threw a bone rattling elbow into the jaw of the beast. As he began to lunge to the wounded Hunter, Roman surprised all by spinning and burying his blade into the face of another Hunter, catching the beast off guard and ending its life in the same motion. Firmly planting his boot onto the chest of the lifeless Hunter, Roman pushed its carcass from his blade, bringing fragments of cartilage and bone with it.\n\nShrieking from anger, Vladris began to pace with purpose, his eyes focused onto the mortal man who had made such short work of the perfect species. While the injured Hunter staggered to gain his senses back, the untouched Hunter dashed for Roman with its blade extended. Side stepping the monster, Roman mockingly kicked it in the back, using its own momentum to push it to the ground. He glanced at Vladris for a moment before turning to slash the reeling Hunter who had made an attempt to attack him from behind.\n\nSlashing the stomach of the Hunter, Roman quickly gave a second slash that filleted the beast from waist to shoulder and then arched a stab which plunged into its back. The dying Hunter screamed before Quinton kicked it directly in the face, jarring several teeth loose from the dead Hunter's mouth. As if Vladris wasn't enraged enough, Roman spit onto the body of the dead Hunter to insult their race.\n\nScreaming with fury, Vladris unbuckled his armor piece by piece, throwing it to the ground with conviction. His pale white skin was fully exposed, throwing every bit of armor from the waist up onto the ship's deck. He wanted to end the life of Roman Raines, and do it without the aid of armor. Stiffening his body up for a moment, Vladris displayed muscle tone matching his skill with a blade. Both in plentiful supply.\n\nRoman and Quinton circled the remaining Hunter slowly, swagger filled steps from both men who waited for the perfect opportunity to strike. It was a feign from Roman that grabbed the Hunter's attention, but the blade of Quinton that ended the Hunter's life. Roman playing decoy just long enough for his Black Cell brother to thrust the shimmering blade deep into the spinal area of the Hunter, twisting the steel bluntly before pulling it upward and spilling the intestines of the creature. The two brothers had made short work of three Hunters and now stood covered in blood as well as anticipation. Vladris was obviously calling the shots, as well as very important to his people; which made him a marked target to the Black cell brothers. \"He's mine.\" Vladris said commandingly as his curved fingernail directed his finger into Roman's direction.\n\nAdam's entire group remained speechless after watching the two brothers make such short work of three Hunters. Finding no suitable words to describe what his eyes had just witnessed, Adam simply gave a look of awe as Quinton made his way over to stand with the group. It was well known that dogs had a dislike for the Hunter race, a fact that had come to fruition because of the Hunter Elite's uncanny ability to turn both humans and animals into undead themselves. Whiskey hid behind the ramp of the crew's shuttle, watching the battle unfold in vivid black and white detail. Meanwhile, Roman and Vladris walked a slow circle, neither warrior removing eyes from the other.\n\nVladris wielded a longsword, an advantage that Roman had quickly countered by wielding a second combat blade. The shimmering silver blade his brother had used only moments ago, now sitting in Roman's second hand. The idea of Roman fighting with a combat blade was testament enough, but when he held two blades the Gunship's crew could only try and imagine the possible destruction which could soon follow.\n\n\"When I cut the soul from your body, I will show your organs to your friends as they too lay dying.\" Vladris said calmly, dragging the tip of his sword a bit as both men continued to circle.\n\n\"Got it all wrong fang man. When this is over, my friends and I will be leaving on that shuttle, and you will know that your sense of immortality was false, that you were nothing more than a mortal masquerading as an elite warrior.\" Roman said, stopping to face the Elite.\n\nJust like the instant of sunlight filling a landscape, Roman dove at Vladris with both blades extended. Stabbing at the Hunter, Roman continued to thrust the blades down as he landed beside the Elite. Using his massive forearm to block the forearms of Roman, Vladris used his sword bearing arm to lay both fist and iron across the face of the Black Cell warrior, knocking him onto his back several feet away.\n\nThough quick to his feet, the nagging sensation of pain shooting throughout his face remained. His sight blurred by throbbing red as trace amounts of blood mixed with sweat and fell helplessly into the whites of his eyes. Such a punishing measure of blunt force trauma usually rendered his foes helpless, and Vladris was impressed with the zeal of Roman Raines, even if he was a man marked for death. Quinton started to intervene, but Adam quickly grabbed him, shedding light on the fact that two move Hunter Elites stood at the ready, their rifles drawn.\n\n\"Come on Roman, dust that 'sumbitch!\" Dalton yelled, knowing deep down that in hand to hand combat Roman was the best shot they had at walking out alive. His motivation was quickly prolonged by the barking of Whiskey, who remained hidden near their shuttle.\n\nRoman began to reflect on everything in his life up to this very moment. His family murdered at the hands of the Hunters, Black Cell being wiped out by the blood sucking bastards and how he felt alone in the Skyla System. Alone, other than Adam and the crew of the Gunship, a group that took him in and treated him like part of the family. He would be damned if he let them fall dead here at the hands of three Hunter Elites.\n\nAs Vladris jumped in for the kill, his blade struck metal, bouncing off of the grated deck of the ship as Roman spun quickly and split the ribs of the Hunter Elite wide open. His blade easily filleted the dead flesh of the mighty Elite, shocking the immortal if nothing else. He was only wounded, still Vladris was more taken back by the thought of a normal man giving such fight to a Hunter Elite. He had not been given a wound of this magnitude in several decades, if not longer.\n\nVladris staggered a bit, sword hanging down by his feet in one hand while the other covered the newly carved flesh wound. \"He always did prefer the blade!\" Dalton said, the idea of him possibly living to see another day exciting him. Wasting no time, Roman dashed in with incredible speed. Though not incredible enough. Vladris instantly used the hand covering his wound to grab the neck of Roman Raines and pull him into the air, looking into the eyes of such a dedicated warrior as he searched his soul. Seconds later, Vladris plunged his massive sword into the chest of Roman, lifting him into the air a bit higher by the length of his blade as he shook the life from the departed warrior.\n\nRoman Raines was dead. After cheating the afterlife so many times before, Roman was finally ended. He lay there on the blade of Vladris, blood gushing down to its handle as Roman glanced at Adam a final time as if to apologize.\n\n\"NO!\" Quinton yelled, running to the aid of his dying brother in arms before being hit with the unforgiving fire of a Hunter rifle. The bullet made a clean pass through the shoulder of Quinton, who lay on the floor suffering from both flesh wound and grief of the soul. Adam was fast to pull his pistol coated in gold and fire six shots, providing enough cover until he could grab Quinton and drag him behind the control booth of the landing bay.\n\nSasha opened fire, letting a swarm of angry lead loose with intention to kill. The Hunter Elites were faster than both normal man and normal Hunter however, dodging the gunfire long enough to seek cover of their own.\n\n\"Hold them back while I turn him!\" Vladris said, laying Roman's lifeless body onto the deck in front of him and pressing his hand firmly to the dead warrior's chest.\n\nSeveral minutes passed as both groups continued to exchange gunfire, when finally Roman's body began to show signs of the transformation. Opening his eyes slowly, the most noticeable trait was the milky white pupil of his eyes, a clear sign that the Hunter Elite's venom had done its job.\n\nRoman had become one of the undead monsters that he had spent a lifetime hunting down. He was now the enemy, a fact that Adam and his crew had yet to discover. As fanged teeth began to slowly push their course into Roman's mouth, his previous wounds reversed; healing themselves in only a matter of moments.\n\nAdam was the first to notice, quickly joined by the others. A Colonial marked shuttle making its approach into the landing bay. They were outgunned and not long removed from a painful death at the hands of Hunter Elites. Still, Adam's world seemed to stop as the Colonial shuttle slowly touched down onto the deck, steel connecting with steel. Standing there without breath or words, the grip of Adam's revolver loosened as he held the pistol at waist level by his side. His eyes had not caught sight of Sarah Blaine, but his heart already knew. His own death was no longer of any concern to him, just seeing her one last time.\n\nBullets continued to zip strategically through the air as Dalton and Sasha remained focused, exchanging gunfire with the Hunters. The Colonial strike team was first to exit the ship, closely followed by Lassiter. Sarah Blaine walked slowly down the ramp under the heavy escort of two Goliath soldiers and her loyal Husk warrior Samuel. Her own eyes immediately locking onto those of Adam Michaels. Sasha slowed her own gunfire, glancing deep into the heart of Adam as Dalton's shotgun continued spending large shells, its shooter cursing in the process.\n\nSarah nervously ordered her own Goliath escorts to the fight in an attempt to save Adam and what remained of his crew. The mechanical warriors quickly sprinted across the long deck of the landing bay, passing the Colonial strike team just before arriving to the fight. A fight began to ensue of epic proportions, the Hunter Elites holding their own against such an incredible force.\n\nWhen Roman rose to his feet, it wasn't obvious to most that he was a Hunter. Or that it was even Roman Raines. So many bodies were slaying, firing weapons and yelling that the confusion distracted almost everyone from the fact. Almost everyone.\n\n\"Roman's a goddamn Hunter!\" Dalton yelled loudly, nearly stretching the beard from his face in the process.\n\nWhiskey began to bark without pause, Dalton holding him back with a free hand by his collar while blindly firing his shotgun with the other.\n\nAdam and Sarah continued a stare of fate. An unspoken communication of sorrow, apology and above all, forgiveness. The stare was finally broken by Adam as Dalton's words rang true. Adam turned abruptly, looking over his shoulder as Dalton sprinted past. Already having thrown his weapon to the ground, he was now carrying Whiskey with both arms and running as fast as he had since boot camp more than a decade earlier.\n\n\"Roman's a 'mutha fuckin' Hunter!\" Dalton added wildly as he ran past.\n\nAdam caught sight of him. Roman stood there at the side of Vladris, the warrior's skin already beginning to turn chalk white. While his eyes milked over to merely void of any color, Roman could still see unhindered. His first sight being that of his brother in arms laying dead on the deck of the ship. His heart continued to feel. And his hatred remained. His hatred for the Hunter race. He reflected back to a time when he was a young child, the Hunters slaying his family as he stood helpless. As he watched Dalton sprint away, Adam and Sasha preparing for their own deaths and the Hunters pushing to them, Roman remembered two important things. His new family was still alive and at the mercy of Hunters. And he was no longer a helpless young boy.\n\nRoman turned to glance at Vladris momentarily, quickly following with a witching backhand that put the leader of the Hunter group to his back. Adam had drawn his pistol, but didn't fire. Instead his body fell into shock as he held his hand out, leading Sasha to quit firing her weapon as well. The remaining Hunter Elites and Goliath units both turned their attention to the newly born super killer. Roman's freshly clawed handed swiped with intent, gouging the throat of an Elite, ripping with it vital organs for survival as the Hunter fell face down. One of the Goliath units had aim on him, its rules of engagement clear. Leave no Hunter standing. However its mechanical arm was violently jerked from place, sparking wires flashing as the swinging arm now held by Roman planted flush into the face of another Hunter Elite. As the Elite stumbled back, his head was aggressively pulled to its side, snapping the neck of the so called immortal.\n\nRoman's flesh was hit several times by the stout slugs of the remaining Goliath's chain gun, only hurting enough to enrage the newly turned monster. Roman jumped onto the mechanical work of art, butchering him quickly by ripping away the parts needed for operation. Both Adam and Sasha stood silent, not knowing where their friend's heart was at by this point. They were both prepared to shoot if need be, though it would have little effect.\n\n\"Roman's a goddamn hunter!\" Dalton screamed, finally making his way to Sarah, Lassiter and her large Husk protector.\n\nSamuel had drawn his sword, standing in front of Sarah Blaine in case his destiny ended here.\n\n\"Did you hear me? I said Roman's a...\" Dalton yelled almost like a screaming child before turning to see the wake of destruction in the near distance.\n\nVladris stood to his feet, his hell hound having backed off a bit and watching from the shadows. He had taken the time to turn Roman Raines, given the man a gift of immortality and now he would have to end the man only recently turned. Vladris drew his sword slowly, walking in a circle as Roman turned to face him. Both had been great warriors in days gone by, in a previous life led by mortal standards. Both had life ending skills now that were unmatched by anyone, alive or otherwise. It was unheard of for a Hunter to turn on his own kind by free will. And though Vladris was shocked at the mortal will of Roman Raines, he knew only one Hunter Elite could walk away from what would be a barbaric display of slaying in the moments to come.\n\n\"Go my brother. It is not your day to die,\" Roman said in a low but stern voice. \"Tell your Queen I will be coming for her soon enough.\" he added.\n\n\"You do not dictate my fate or that of our Queen.\" Vladris replied, his sword still at the ready.\n\n\"Search your soul, it will tell you otherwise. Your Queen is weak, I have seen it. She relies on the true warriors to do her work. Go back and tell her I am coming. There will be a place for you in the future.\" Roman replied, acknowledging that Vladris had given him the gift of life.\n\nVladris knew Roman's words were true. He had known for a while that his Queen had grown arrogant, weak even. As his mind stuck to the fact, he soon found himself more concerned about dealing with a substandard Queen than slaying Roman Raines.\n\n\"Our day will come soon enough,\" Vladris said, pointing his sword directly at Roman before backing away slowly. \"Scucca...come!\" Vladris added, motioning the hell hound to him.\n\nInstead, Scucca slowly approached Roman Raines, sitting down at his feet and making public his new owner. Scowling at his recently lost pet, Vladris glanced back to Roman. \"Soon.\" he said, his stare never faltering. Moments later Vladris turned to slowly jog away, into the shadows which led to the heart of the riot.\n\nAdam and Sasha both stood there silently, trying to make sense of Roman having become one of the immortal Hunters; not to mention Sarah's arrival. \"So are you still Roman enough that...\" Adam had started to ask as Roman turned to them both.\n\n\"Don't worry. I'm not going to kill you.\" They both breathed a bit easier, approaching him with a great deal of caution.\n\n\"Good. Didn't much feel like getting killed. At least not today,\" Adam said with a grin, his words never phasing Roman. \"With just a little time I'm sure the Colonials can fix this. Have you back to normal.\" Adam added. Roman turned directly to him, the newly born Hunter Elite only feet away from both Adam and Sasha.\n\n\"I'm not going back. This is the end of our journey together.\" Roman said. \"There has to be something we can do to help you.\" Adam replied.\n\n\"I don't want to be helped. I am what the Hunters fear now, and I will bring their race to its knees. Don't try and stop me, it's taking everything I have this very moment to suppress the Hunter inside of me.\" Roman said abruptly, his voice altered a bit with demonic dialect.\n\n\"I'm not trying to stop you Roman. You're my friend, hell you're like family to me. I just want what's best for you. That's all.\" Adam replied.\n\n\"Then leave me be. You once told me I was free to leave, and now I am. Adam, it's taking everything I have left inside of me to not continue killing here. The Hunter side of me is strong. It's not safe to be around me now.\" Roman replied, a man torn between immortal and mortal in his eyes as they began to tear slightly.\n\nThe biggest battle to come for him would be the battle within.\n\nAdam understood. His friend wasn't leaving because of a lack of friendship, he was leaving because of his deep respect for the crew. It was for their own good.\n\n\"Well then, It's been an honor.\" Adam finally said calmly, extending his arm. Roman fought it.\n\nThe inner urge to kill for the sport of it, an urge forced onto him by the Hunter DNA. His soul was still strong, at least strong enough to extend his own arm and shake hands with the Captain of the Gunship.\n\n\"Goodbye old friend.\" Adam said as Dalton yelled loudly, clapping his friendship and gratitude.\n\n\"Until we meet again.\" Roman replied, quickly breaking his grip and turning into the direction of the Hunter shuttle that Vladris had arrived in, hell hound following closely behind.\n\nAs Adam watched Roman pull the Hunter shuttle from the ship's deck and slowly exit into the backdrop of stars, it suddenly hit him. Sarah Blaine was waiting. Adam quickly spun around, his eyes locked onto the stare of Sarah as she stood by the Colonial vessel. Dalton cautiously began making his way back to Adam and Sasha, this time forcing Whiskey to follow on foot. After several deep moments of exchanging apologies through their stare, Adam turned slowly to Sasha.\n\n\"Go Adam.\" Sasha said with regret.\n\n\"But I don't know what I'm supposed...\" Adam replied, unsure of which woman held his future in their hands.\n\n\"The look between you both just answered every question I had. I love you Adam Michaels, and I want you to be happy no matter what. So go, please.\" Sasha replied.\n\nAdam wanted to plead with her, maybe ask for time to think things through. Deep down he knew that his time was up. He had to choose between his love for Sarah Blaine or his possible future with Sasha Riley.\n\nSarah stood there, her beautiful red hair swaying slightly as tears filled her eyes. She had prayed for a second chance with Adam, her one mistake having forced her into nearly a year of regret.\n\n\"Just tell me what we had was real Adam? That's enough for me.\" Sasha said. A lie of course, she loved Adam from the pit of her soul. But she wanted him to be truly happy and had seen a look between Adam and Sarah that was undeniable.\n\n\"More real than you will ever know.\" Adam replied, nodding as he slowly turned to begin walking to the woman who had waited so long for the return of his embrace.\n\nSarah began crying a bit from happiness, her hands shaking as she prepared to once again hold the man of her dreams. Meanwhile, Sasha started to cry just slightly, her Benzan toughness standing in the way of her true feelings.\n\nAdam had walked nearly mid-way when he stopped for a moment, realizing that these last fifty or so feet would lead to the rest of his life. Everything he was, everything he would ever be was tied into this very decision. It was in that single moment that he felt it. He thought about a life without Sasha Riley and his breath evaporated, body numb from the thoughts of leaving her behind. He had felt this way about Sarah Blaine as well, but not anymore. As he looked into Sarah's eyes, he knew she would be best loved through the memories they had shared.\n\n\"I'm sorry.\" Adam said in a very faint voice as he stared at Sarah for a moment, turning back to Sasha and moving with a touch of haste.\n\n\"I forgot one thing,\" Adam said as he walked close to Sasha. \"I can't live without you,\" he added, holding a hand out as he waited for her own. Sasha immediately burst into heavy tears as her heart tried to make sense of things. \"My life changed the moment we kissed in the rain. I care for Sarah, and I always will. But I love you. I've loved you every since the moment we first kissed, and I can't go back from that. I just can't.\" Adam added, grabbing Sasha tenderly into his arms.\n\n\"I'm not the leader of a mighty military force Adam. Honestly, I don't even know where my life will go from this point forward.\" Sasha said, her body trembling as Adam continued to hold her with a comforting calm.\n\n\"I don't care. I just want us to go there together.\" Adam replied.\n\nSarah Blaine stood there, devastated. Her life had just been ripped from her from only fifty yards away.\n\n\"Get her onto the shuttle. Fire up and wait for me.\" Lassiter said commandingly, the Colonial Husk nodding and slowly helping Sara Blaine up the ship's ramp.\n\n\"Roman's a fucking Hunter.\" Dalton said as he approached Adam and Sasha locked into an embrace.\n\n\"Yea, I'm well aware.\" Adam replied, both he and his lover giggling a bit.\n\n\"I told you not to take that job Adam. I told you.\" Dalton added as the two lovers laughed a bit harder, locked into a poetic glance of destiny.\n\n\"Enjoying yourself?\" Lassiter asked with a wise tone as he approached Adam.\n\n\"Roman's a fucking Hunter,\" Dalton replied, Lassiter answering with a stone carved gaze. \"Well he is.\" Dalton added, walking away a bit to allow Lassiter time to speak.\n\n\"No hard feelings here Eric.\" Adam replied, finally pulling his arms from holding Sasha in order to face Lassiter.\n\n\"Sarah is back there right now crying her very soul out because of you.\" Lassiter replied harshly.\n\n\"No sir, she's crying because of the mistakes in our past. I care about Sarah a lot, but I can't live in the past. I have to move on.\" Adam replied.\n\n\"Yes. You do. All of you need to move on.\" Lassiter said.\n\n\"Now what the fuck is that supposed to mean?\" Dalton asked with ill manners.\n\n\"It means you all better move on, far away from Glimmeria and as far away from Sarah as you can possibly get. I won't watch her go through this kind of pain for someone hardly worth the effort twice.\" Lassiter replied.\n\n\"Well now,\" Adam said, stepping within inches of Lassiter's face in order to look him dead in the eyes. \"May want to word things with a bit more caution. We aren't in your usual Colonial comfort zone at the moment.\" Adam added. The Colonial shuttle fired its engines, loud torque nearly overwhelming the shouting conversation taking place.\n\n\"No, you're not in Colonial space at the moment. But if you ever venture into it again, I won't put you in prison. That would bring too much pain to Sarah. I'll just have you executed without her ever knowing. Are we clear?\" Lassiter said loudly.\n\n\"Are you threatening...\" Adam said, his question sharply cut off by Lassiter drawing a pistol and pointing it into their direction.\n\n\"Crystal clear,\" Adam replied, motioning Sasha, Dalton and Whiskey to turn and head for the Benzan shuttle. \"Oh and just so we are clear, it's a two way street,\" Adam said, turning to face Lassiter one last time. \"If I were you, I'd make sure this was the last time you ventured outside of your Colonial comfort zone.\" he added, turning to accompany his friends to the shuttle.\n\nIt would be the last time Sarah Blaine would see Adam Michaels through human eyes, watching him enter the Benzan shuttle doors through a window on her own ship. Her Husk protector lay dead as Vladris approached her slowly, his hand extended with clawed fingers gripped her chin.\n\n\"You have fought the Legion valiantly, an effort that is worthy of immortality.\" Vladris said, his voice both calm and demonic.\n\nSarah cried softly, her mortal life about to come to an end as Adam's ship began to pull from the deck; losing both her love and mortal life in the very same instant.\n\nMoments later as Lassiter watched the Benzan shuttle leave distant site, he turned to the Colonial ship, only to watch it pull swiftly from the deck. He wanted to yell at the top of his lungs, wanted his ride to hold long enough for him to get aboard. However the shuttle quickly left, a trail of orange fumes scattering behind by way of full burning thrusters. Seconds later he fell to his death, multiple rifle shots plunging into his thin frame as guards had once again taken control of the prison ship.\n\nRoman's shuttle slowly fell from the clouds on the planet Cres Nine, along with a heavy rain as the Hunter designed shuttle quickly approached for landing. It was a well known gangster hideout, one that Roman had deep ties with. And it was on the moon planet that Roman would begin building his new family. The Roman Empire. A family that would become large enough to stand against the Hunters and their Queen, at least if Roman had his way. The newly born Hunter Elite smiled a bit as his shuttle slowly approached through a curtain of rain, his dagger-like teeth showing crisply as Roman extended his arm down to pet his loyal hell hound.\n\nAs the Benzan shuttle pulled down for its landing on the small mining planet of Antillia, Dalton looked from a window, secretly hoping their faces weren't recognized for events that took place nearly a year before. They had met Roman, beaten several officers of the law and robbed a table full of patrons at gunpoint. Whiskey sat beside Dalton, his head of tangled fur laying on the man's lap as Dalton continued his train of thought.\n\nNormally he was excited when landing here, even if only to refuel. Paulie's was here, and piss poor drink aside, they at least had a stockpile of it available.\n\nThis time was different though. Dalton's mind wasn't on the alcohol. He watched Sasha sit on the lap of Adam Michaels, their overwhelming love for one another very obvious. Dalton was sick and tired of the Hunters. He was tired of running, tired of the Colonial war effort and had started to wonder if there was a Sasha out there waiting for him. How would he ever know if he was constantly on the run from a nation of undead. And now Roman was a fucking Hunter too.\n\n\"Guess now is as good as any time to tell you Adam. I plan on staying here when you two pull out.\" Dalton said with a serious tone, one he rarely used. Sasha turned first, quickly standing to her feet and offering to take the controls so Adam could talk to his friend.\n\n\"What are you talking about?\" Adam asked quickly as he walked to the rear of the shuttle to sit with his friend. \"I know the loss of so many friends has taken its toll.\" he added.\n\n\"That's what has me thinking,\" Dalton said with a touch of sorrow. \"Running all the time, the war. It's not my fate man.\" he added.\n\n\"Come on man, this will pass. You and I are like brothers.\" Adam replied.\n\n\"We are brothers, in every sense of the word,\" Dalton said as he turned back to Adam. \"Which is why I need you to understand, something else is out there waiting for me. I can feel it.\" he added. A moment of silence fell throughout the shuttle as both men dealt with the inevitable. Dalton was leaving Adam's crew to go his own way.\n\n\"I gotta find my Sasha. She's out there somewhere and between you and me I hope she has just as nice of an ass. She doesn't have a sister does she?\" Dalton asked in a hushed tone as both men laughed a bit.\n\n\"I'm not ever going to find her running from the Hunters or fighting this damn war. Hell, I never wanted any of this. I did it because I think so much of you.\" Dalton added with much more seriousness attached to his words.\n\n\"I understand,\" Adam replied with a solid head shake.\"At least let me buy this round of drinks at Paulie's as a thank you.\" Adam added.\n\n\"Damn straight. Hell, if I knew it took leaving to get my drinks paid for I woulda' done it a long time ago.\" Dalton replied with a wide grin. Adam just laughed a bit, putting his hand onto the shoulder of Dalton out of gratitude, petting Whiskey for a moment as he turned to make his way back to the pilot's area.\n\n\"I'm sorry.\" Sasha said gently as Adam sat down beside her, the shuttle on its final moment of landing.\n\n\"No need for that,\" Adam said, reaching over and holding her hand. \"I make it a rule not to feel bad for the end of things, but rather happiness that I ever had a chance to experience them,\" he added. \"Besides, this is hardly the end. Just the end of a single chapter in what is going to end up being a long story.\" Adam said, turning to face the love of his life.\n\nAdam, Sasha and Dalton sat at their usual table in Paulie's bar the entire night. Talking about friends lost and experience gained. They laughed for the most part, though a little crying followed when talking about the life and death of Kelly, Steiner and Kato. Adam also found the time to tell the story of Luck one last time, which prompted Dalton to order another stout drink. The laughs continued as though they were family, Whiskey sitting directly at Dalton's side with the best display of loyalty imaginable.\n\nIt seemed blurred into a single night, though several nights later Dalton sat at the same table. Drinking and laughing by himself as Whiskey stood in the same spot by his feet.\n\n\"That guy? The homeless one at the rear of the bar?\" Cambria asked of the bartender as they both stood there, staring at the broken man tightly tucked into his brown coat.\n\n\"That's the one. Knows more about smuggling than everyone else in here combined. Be a big help to you all.\" the bartender replied.\n\n\"Well he looks down on his luck. May be able to hire him at a fraction of the going rate.\" Cambria said.\n\n\"Down on his luck? This son of a bitch is buried under the casino.\" Tank replied.\n\n\"Guess if we plan on smuggling, this is what the future holds for us one day.\" Skulls added, his loose fitting leather clothing moving just a bit.\n\n\"The hell it does. Not me, I got too much game for that.\" Tank added as they all watched Dalton pour a bit of ale onto the floor for his loyal dog.\n\n\"Wish me luck.\" Cambria said, taking the rest of her liquor in a swift shot before slamming the small glass to the bar and standing to make her way to Dalton's table.\n\n\"Mind if I sit?\" Cambria asked politely.\n\n\"Darlin' with hips like yours, you can do whatever the hell you want to.\" Dalton replied with a grin.\n\n\"Alrighty.\" Cambria said with a smile, slowly placing her perfectly sculpted body onto the small wooden chair. Barkeep says you are the man to talk to about hiring onto my crew. If I can buy you a drink and...\" Cambria said confidently, cut off abruptly by Dalton.\n\n\"I'm in.\" he replied.\n\n\"You're huh?\" Cambria said, taken back by his willingness to work a job he knew nothing about.\n\n\"I just hit on you and instead of the usual slap to the face, you smiled. After which you offered to buy me a drink. I'm in. Consider me hired.\" Dalton said.\n\nCambria smiled a bit, still trying to read the man wrapped in a lipstick stained coat of brown armor.\n\n\"Care to know what my crew plans to do? What your job entails?\" Cambria asked.\n\n\"Don't matter much either way, I've done it all. And twice if it was fun the first go 'round.\" Dalton replied with a grin.\n\n\"Alright then. Order whatever you want and when you are done here, just head outside. I Captain the \"Outer Heaven\" and her crew,\" Cambria replied softly. \"I'll tell the barkeep to put it on my tab.\" she added, standing slowly and noticing Dalton's eyes shift from her own to her perky breasts.\n\n\"Will do Captain,\" Dalton replied as she smiled a bit and turned to walk away. \"Now there goes a real damn woman right there Whiskey. She let me flirt with her, looks good as hell and even bought the booze. This could be love.\" Dalton said in a low voice as he watched her ass bounce slightly, making her way back to be seated at the bar.\n\nWhiskey barked loudly, gaining the attention of Dalton.\n\n\"Bullshit. I done laid claim to her, find your own.\" he added, looking into the sad eyes of his lust filled pooch.\n\n\"What'd he say?\" Tank asked as Skulls looked on.\n\n\"He's in.\" Cambria replied, motioning for another shot of Paulie's finest.\n\n\"Nice. So he knows we plan to smuggle?\" Tank asked.\n\n\"Nope.\" she replied.\n\n\"Well he at least knows we plan to work in The Drifts, right?\" Tank asked.\n\n\"Nope.\" Cambria replied.\n\n\"Well what in the hell did you tell him?\" Tank asked in disbelief. \"Nothing, I just bought him a drink.\" Cambria said, glancing back at Dalton who seemed to be in a deep discussion with his loyal dog. She saw a playful charm about him, almost child-like. She also knew seconds after meeting him that when it was on the line he was the type that would get things done. Exactly what their crew needed. She turned her attention to the window behind the bar, watching the rain fall elegantly onto the ships outside, including her own. The adventure of the Outer Heaven was about to begin, and though she had no idea where smuggling would take them, she looked forward to every moment of it.\n\n\"So I don't understand why we are coming back here if I pulled a gun on two of Kraid's most trusted?\" Adam asked as Sasha piloted the shuttle down onto the snowy surface of the Benzan home world.\n\n\"Benzans have disagreements all the time, you are still one of us and...\" Sasha said, catching sight of bodies laying all over the ground and partially covered in snow.\n\nAdam saw her grief instantly, taking over the controls as Sasha began to cry loudly. These were her people, and from the looks of it most had been slain by the Hunters. The shuttle touched down onto the crisp snow of a nearby field as Sasha hurried to the exit of the ship.\n\n\"Wait.\" Adam said, unable to stop her from making her way outside. He slowly exited to see Sasha standing there, crying as the snow fell from above. Hundreds of bodies lay slain on the ground; nothing short of a massacre.\n\nAdam walked up behind Sasha slowly, placing his climate jacket onto her and wrapping both arms around her waist tightly.\n\n\"Let's resupply, refuel and get out of here. I'll take care of everything, you need to get back on the shuttle. No need to see this.\" Adam said.\n\nAs his words hit the cold of wide open air, Adam and Sasha both caught glance of someone opening the lodge door, prompting Sasha to begin running into its direction. Adam had started to ask her to wait, but instead pulled his sidearm and sprinted behind her; knowing full well she wouldn't stop either way. It was a woman in the doorway of the lodge, a Benzan woman.\n\n\"It's Sasha and Adam!\" the woman said loudly.\n\nThe Benzans inside, which included the women, children and five heavily armed escorts, lowered their weapons as Sasha and Adam entered slowly.\n\n\"What happened? Where is Kraid?\" Sasha asked as Adam tried to catch his breath, not used to the cold planet's air. A few of the Benzan women began crying as one of the escorts shook his head, letting Sasha know Kraid had been slain by the Hunters. It took several moments to sink in. Her former lover and the leader of her people was dead. A vast majority of her people lay slain in the bloody fields of snow right outside and it was only a matter of time before the Hunters returned.\n\n\"We need to go. Now. The Hunters will send a return party to scout the planet, and if we are here they will end us.\" Sasha said, trying to appear calm.\n\n\"Go where? We have nowhere left to run.\" one of the soldiers replied, feeling bad for his slain brothers while a bit lucky that Kraid had selected him as one of the few to protect their women and children.\n\n\"There is something. I need the escorts to come with me, the rest of you pack any kind of supplies you can. Weapons, food and anything vital to survival,\" Sasha said, the survivors taking a moment before dispersing to collect any needed supplies. \"Come.\" Sasha said as Adam and the five Benzan soldiers followed quickly behind.\n\nTheir shuttle had flown for less than ten minutes when Sasha descended it back onto the frozen tundra of their homeworld. \"It's here. Something Kraid had been working on with a few of his closest,\" Sasha said. \"In case of an emergency, and I think this qualifies.\" she added.\n\nAs they all exited the shuttle slowly, Sasha led them into what looked like a large cavern. The first thing that stood out to everyone was the cavern had an open view to the snow filled sky above. Then, their attention quickly shifted to the reason Sasha had brought them here. A ship. A very large ship, which seemed to be outfitted with capable weapons. It was a pale shade of black, the words North Star painted onto the front.\n\n\"Kraid said it would hold a few hundred people if need be. Its navigational unit has information on another existing Benzan colony,\" Sasha said, turning to the group. \"But it's a travel into uncharted space. A long trip from what little Kraid spoke of it.\" she added.\n\n\"Why have we not heard of this before?\" one of the soldiers asked of the large ship.\n\n\"Because he planned to leave you, as well as most of the other Benzans behind in the event of emergency. It's the reason he and I parted ways,\" Sasha said, glancing at Adam for a moment. \"I could not be with a man who thought so little of our people. Every single one of you is important to me, and you will be important to the Captain of the North Star. Adam Michaels.\" she added.\n\n\"Captain?\" Adam asked, stunned.\n\n\"Yes. You are the only surviving Benzan that bears a golden weapon, signifying Captain. And if any of you take issue with the fact, let him speak now and remain behind to fight the Hunters on his own.\" Sasha said, turning to the soldiers.\n\n\"No. We have no issue with Adam as the ship's Captain.\" one of the soldiers said, all five showing a loyal face to the smuggler.\n\nAs the group of soldiers boarded the Benzan shuttle to return to the survivors, Adam and Sasha stayed behind to prep the ship; as well as make love in the most romantic fashion imaginable. Their souls became one for the next several hours aboard a ship that soon would embark on its quest to find other Benzans.\n\nA perilous quest, but one that not only had to be taken in order for their kind to survive, but one that both Adam and Sasha looked forward to. They would begin a life together without a war pulling them down or the thoughts of ex-lovers hindering them. Completely free to love and live their lives. Adam held Sasha softly in his arms as the Benzan shuttle arrived once more, full of survivors on its first of many trips to come; the lovers helping the surviving Benzan women and children aboard their new home. At least their new home for a long time to come. They would all escape the clutches of a war torn solar system...though they would soon fall into the clutches of yet another. One much more brutal.\n\n\"Speak to me of your failures! Who is this you dare bring before your Queen?\" Victoria shouted as Vladris and Sarah Blaine approached with purpose.\n\n\"Your successor.\" Vladris replied sharply, pulling his blade with speed and slashing upward; Victoria's severed head rolling onto the stone floor of the throne room. Victoria's two stationed guards rushed to her aid, seconds later joining her in the afterlife as the blade of Vladris bit sharply into the core of their flesh.\n\nHis unmatched skill with a blade would soon enough be put to the test by Roman Raines. However, there would be no test today. Vladris knew as well as every Hunter in the palace that he was the Elite of all present.\n\n\"My Queen,\" Vladris said, kneeling as Sarah Blaine slowly sat down in the throne chair, her skin having quickly flushed to little pigment as the Hunter venom was doing its job. She was as beautiful as ever, even if undead. \"I will protect you with my life from this moment forward.\" Vladris added.\n\n\"Good. Rise warrior, we have work to do. Together.\" Sarah said, her voice having changed drastically from serene to unnerving.\n\nAs Vladris walked through the palace, he did so with resolve. He knew that Sarah Blaine would be a strong Queen. Someone finally worthy of his unwavering loyalty. Entering the living area of the Hunters, dozens of them catching sight of him, Vladris threw Victoria's severed head to their feet. \"Let any of you who would stand against me do so now,\" Vladris said, drawing his still pulsating sword, a crimson fluid running down the edge of it. \"We have a new Queen. A much stronger Queen. Either you agree to bow before her, or you stand against me and we settle the matter right now, in true Hunter fashion.\" he added, daring any of them who were brave enough.\n\nSeveral minutes later he entered the throne room once more, nearly fifty Hunters behind him as they all knelt to their new Queen. \"The Hunters before you are the very strongest. We are loyal to you my Queen. The others will follow without question.\" Vladris said.\n\n\"Rise warriors. For now we begin our rise to power.\" Sarah said calmly, bringing a grimacing smile to the face of Vladris and the rest of the Elites, including Kraid.\n\nThe union of Legion and Theron forces had pushed General Ortega's soldiers to their breaking point. Tameca City was nothing more than smoldering wastelands as Lord Riven forced his army forward, his attempt to end the assault against their regime. He was nearly successful and knew it. He felt it in his bones as they had formed a massive group outside of the large warehouse that General Ortega had secured the wounded refugees. Lord Riven's feeling of victory instantly turned to nothing more than the black smoke that rose into a sky full of Gali war ships. Help had arrived for what few Colonial soldiers that remained. And the Gali had spared no expense, throwing every able bodied soldier into their effort as the sky quickly blackened out, ships blocking every speck of sunlight and landing with haste.\n\n\"Sir, help has arrived!\" Lieutenant Scott yelled loudly, prompting General Ortega to join the refugees as everyone looked through the dingy glass windows and places where windows once sat.\n\nThe Theron began pounding their weapons into the sky, though their best effort did little to slow the Gali as ships began landing, full of troops with hearts full of payback. They too had seen the horrific video feeds broadcast from Tameca and arrived bearing no offer of quarters. They were there to execute every single person involved with the Legion, Theron descent or otherwise.\n\nAs the young man who had listened to General Ortega stepped up to a small window of his own, he saw them. The huge Husk warriors exiting their shuttles by the hundreds, firing chain style machine guns as they cut through the Theron soldiers; exposing them for the cowards they were. General Ortega was given his first communication since being left on Tameca's surface. Paul Lassiter was dead and Sarah Blaine was missing. Effectively leaving Ortega in charge as the ranking Colonial officer.\n\n\"Sir, we have captured the Legion's leader.\" Lieutenant Scott said following several hours of widespread gunfire throughout Tameca city. Ortega nodded, though his thoughts remained with Adam Michaels. General Ortega knew in his heart that Adam had killed Lassiter and taken Sarah Blaine. Soon enough the search for Adam Michaels would begin. However, the battle at hand was his first priority. Ortega glanced around the room, looking into the faces of the refugees who had lost so much. Family, homes and lives. They had been crushed.\n\n\"Sir, where should we take him for processing?\" Lieutenant Scott asked as two Colonial soldiers held Lord Riven in restraints. General Ortega looked into the eyes of the soulless man, the person directly responsible for such a massacre.\n\nColonial code would call for a trial by jury, a lengthy legal fight to convict a person who had been responsible for such war crimes. But Colonial code was of little concern to General Ortega, who would soon become Commander. As everyone in the room looked on, General Ortega drew his sidearm and fired a single shot, hitting Riven between the eyes and removing a great portion of the back of his skull in the process. \"The shallowest grave you can find.\" Ortega replied, his soldiers dragging the corpse outside. It was soon strung up for all to see, a reminder to everyone that justice had prevailed and Tameca was, and would always be, a world where free people lived.\n\nAnd that's how the story of the original Gunship crew would end. At least a story of how their time spent together would end. For each of them it was just the beginning. Fate had placed them onto their own paths to destiny. Roman had become immortal, and his agenda now included ending as many Hunters as possible. Adam chose the love of Sasha, and together they would begin a journey of their own. One that would introduce them to something horrific and devastating. Sarah took her place as the rightful Queen of the undead, soon to launch a full scale war against every mortal being in the Skyla System. And Dalton found another job. One that included a visually breathtaking Captain and free drinks. It would present him and his faithful pooch with plenty of opportunities to tell the stories of his life so far. Including his favorite, the one where Roman was a fucking Hunter.\n\nBook 4\n\nGears And Spears\n\nGeartown\n\nAnd there he sat. Dalton James. A veteran of several wars and several bars, both the drinking and incarceration type. He wore a heavy brown duster as testament, if nothing else, to his storied life up until this point.\n\nHe was a smuggler, and moving illegal merchandise through space while skirting the authorities had both an upside and a downside.\n\nFor the past decade he had worked for his good friend and former military comrade Adam Michaels.\n\nAfter the first Glimmerian War they decided that no man in political position would ever force them to live out another day, choosing instead, to fetch a ship and begin the black market work of great risk and great reward.\n\nAnything they could do to earn a living, they did it. Every cargo hold of illegal merchandise moved, led to connections on the wrong side of the law. And with each underhanded deal they pulled off, their reputation grew. By the time the older model Gunship, one that the Glimmerian government leaned heavily on during the first war, was fully staffed with crew; they had become notorious.\n\nSuch notoriety led to a job offer from the Hunter Clan, a sadistic group of Vampires who were both wanted and feared. Offering a huge payday, Captain Adam Michaels took the job, double-crossing the Hunters a short time later. He had fallen in love with his cargo, Sarah Blaine. And the direction his heart led him, also led the crew through violent times.\n\nEventually they went their separate ways. Various reasons, of course, but Dalton's reasoning was simple. He was sick of running from the undead. A near death experience has a way of making a man feel alive, even if it includes bullets flying into his direction. But so many near death experiences, sometimes on a daily basis, has a way of wearing down even the best of men.\n\nDalton had reached the point of wanting something more. A peaceful calm, if nothing else. And so he left the Gunship, and her crew, setting off onto his own path in search of his own destiny.\n\nA destiny that was sure to include his running mate Whiskey. The four legged bucket of fleas was Dalton's best friend. Whiskey was a good ear to talk to, had a stomach for alcohol and even a way with the ladies. They wore matching brown coats, and together would take on the Skyla System, one shot of hard liquor at a time.\n\nThe former soldier turned smuggler quickly found himself hired by Cambria Sims, Captain of the Outer Heaven. She was brilliantly beautiful from head to toe, shocking blue hair and cream white complexion only further accenting her perfectly sculpted curves.\n\nShe had offered him a paycheck, even paid for both Dalton and Whiskey to get fitted for custom brown coats, rather than the bargain rack faux leather that currently covered their backs.\n\nIn exchange, Dalton would bring experience to a crew of faces that was fresh to the black market world of smuggling. Though none of that mattered to him.\n\nAll that mattered was the fact that they would no longer be running from the undead. Dalton had made damn sure of it, or at least he had thought so up until the moment his eyes caught sight of a poster that plunged into his heart like a chilled dagger.\n\nThe Outer Heaven had landed in the Drifts. A very primitive, yet extremely elegant string of planets on the fringe of uncharted space. A mixture of Victorian influence and steam powered engineering, its citizens had shunned the modern lifestyle of computers and thrust engines for wind-born airships and a luxurious, yet simplistic, style of living.\n\nDalton had considered it a paid vacation of sorts, going to stay a while in a much calmer enviroment while throwing a few brews back. But all of that changed as he slowly read the header of a poster hanging on the wall of the airship transport terminal. Do not provoke the undead.\n\nAfraid to ask, Dalton simply sat there. Stunned. As they waited for a transport from the terminal to Geartown, he continued to stare at the poster which had shattered everything he thought he knew about their upcoming trip.\n\nThe terminal was the one centralized location on the planet, and each Drift planet had one. Simply put, you landed your ship here and then boarded an airship. Hundreds of airships came through daily, each one stopping in even the most remote locations.\n\nSome of the destinations were large cities, Victorian styled skyscrapers peaking to the heavens with their clockwork shaped tops and brass accented artwork. Others consisted of dusty towns populated by colorful characters who had a curiously playful charm about them.\n\n\"Is he OK?\" Tank asked.\n\nHis God given name was Greg Shelling, but was soon handed the name Tank based on his unbelievable size. A bit taller than anyone currently sitting in the transport station, Tank was packed out in terms of muscular composition. His skin resembled a thin coat of dark paint as it stretched across his physique, barely able to contain the bulging muscles that were easily seen as he wore a solid green shirt with short sleeves.\n\nSkulls simply shook his head.\n\nTrevor Lagrange by birth, Skulls earned his name branding through the odd hobby he took so seriously. Collecting teeth, bones and severed fingers from the dead. Easy enough for him, because when the Salvation model sniper rifle that currently hung down by his side was in hand, people had a tendency of dying.\n\nSkulls resembled an undertaker at first glance. That is until you got close enough to see the solider in him, the wrinkles on his face merely a map of battles seen and horrors lived; at which point you wished he were merely an undertaker.\n\nHis rifle was a very unpopular model, the bolt action considered outdated. But Skulls preferred the weapon because it had pinpoint accuracy when looking through the large telescopic lens mounted on top. It was a very elegant weapon in his mind, and a well respected rifle among snipers.\n\nHis stringy hair flowed from beneath a dark top hat, mushrooming out a bit in the back and falling wildly down between his shoulders. Loose-hanging black leather clothes covered the body of such a tall and thinly boned man. And for such a strange Human, vanity was important, regularly slicking his black pants with grease to create such an obvious luster.\n\n\"Dalton. Are you crying?\" Cambria asked with shock.\n\nDalton didn't reply, though his eyes remained crisped with tears. He simply continued his stare onto the poster warning those entering the Drifts of the undead. Zombies you could call them, though citizens knew them as Drifters.\n\n\"It's alright. They aren't a common sight, more like cattle if nothing else. They are mindless and without intention.\" Cambria said in an attempt to calm Dalton a bit.\n\n\"They're undead,\" he replied, fighting back tears of rage as he bit into his lower lip. \"I'm tired of the undead. So fucking tired of people that should be taking a dirt nap trying to put my ass six feet under.\" he grumbled.\n\nWhiskey gave a long and deeply pitched whine. Even the charismatic dog had seen his share of immortals.\n\n\"Best bite your lip because our ride is here.\" Tank said with a bit of chuckling mixed in as he stood up and began watching a large airship swoop down to them.\n\nIt was the typical transport airship, nothing more than a large and elongated hot air balloon; cabin area below constructed of metal with luxurious wooden trim.\n\nDalton gave a look of ill intent as he also stood to his feet, his stomach turning into knots as he glanced one last time at the poster warning of Drifters.\n\nEverything about the Drifts came across to him as being outdated. Even the very poster which currently had his attention, reminded him of an old military poster. Bold words at the top with a poorly colored sketch below.\n\nBoth Skulls and Cambria slowly stood, the Captain putting her arm around the experienced smuggler for a moment.\n\nShe was from the Drifts, and to her, Drifters were just a common thing. A background detail, like snakes in a sand-filled desert or deer in woodlands. Even so, she tried to empathize with Dalton.\n\n\"We'll be fine. Trust me.\" she said with a poetic tone, her undefinable beauty helping to comfort Dalton.\n\nAs the airship slowly elevated back into the sky, heavier by a couple dozen passengers, Dalton found himself staring out of the thickened glass windows surrounding them and wondering exactly what he had signed up for.\n\nHe had known about the Drifts for most of his life, and honestly, up until now, hadn't cared about them one way or the other. In his mind, anyone who shunned technology deserved to live in huts made of dirt and grass. He had just thought them to be basic and written them off.\n\nThe impression he had gotten since arriving was different. Much different. Sure, they lived without the modern technology that the rest of the Skyla System coveted. But they did it in a very artistic way. Even the very balloon they traveled in now, was a helium filled canvas of linen. The fabric was almost a portrait of style, dark browns accented with gold flakes. And then it was pulled together and held into place by brass links of chain. As it wrapped around the balloon, the links locked together with a large brass medallion; a lion's head designed and pressed into the coin-style lock.\n\nThey were headed for Geartown and from what Dalton could gather, it was full of opportunities for a young smuggling crew. All kinds of people who had goods to move off world, and were willing to pay a smuggling fee in order to avoid having their goods so heavily taxed by local government.\n\nWith his frustration of the walking dead soon turning to anger, Dalton sat in the wooden booth-style seat and continued to look out across the clouds and thriving green pastures below. He quietly cursed the Drifts and their damn regulations on modernized weaponry.\n\nTwenty seven. That's the number of capable weapons he had to leave back on the Outer Heaven. If a shotgun fired too wide of a spread, it was against regulations. A digital counter on the side of a battle rifle, against regulations. Needless to say his grenades had been left behind as well, adding to an already pissed off demeanor.\n\nHe carried only two weapons at the moment, which was as close to naked in front of clothed women as he had been in a very long time. At least in public. A Magnum style revolver that held six rounds inside of a rolling chamber and would damn near cut a man in half, as well as a large buoy blade strapped to his leg that would complete the cut if his revolver failed.\n\n\"I wouldn't sweat it. Hell, I hear they hunt Drifters down here like big game man. We may throw a few beers back and go on the hunt ourselves.\" Tank said in a low voice.\n\n\"I 'aint huntin' shit,\" Dalton said loudly, gaining the attention of every passenger aboard the airship. \"Anything comes at me and can't recite the alphabet is getting shot up.\" he added, turning for a moment to glance across the isle.\n\n\"The fuck you lookin' at?\" Dalton asked belligerently as an older man with literate glasses and a finely pressed suit looked on.\n\n\"Calm down Dalton, you're scaring people.\" Cambria said, quickly sitting beside him.\n\nDalton wanted so badly to mouth off in response, but after catching sight of her beautiful face he started to realize that his soul began to ameliorate every single time she was near. So calm down he did. For the next several hours Dalton was silent, staring out of the window by his seat as the airship coasted passionately through the clouds.\n\nAs they made their approach, Dalton's first reaction was one of curious suspicion. When he had first met Cambria Sims, she stood out. Her loudly colored hair and choice of clothing style was refreshing, but out of place. Looking across the streets of Geartown as the airship landed softly, Dalton realized that he and Whiskey would now be the ones out of place. Every citizen he caught sight of looked unique. Women with blue, watermelon green and even neon purple hair walking abroad. Outfitted in corsets and carrying small umbrellas that were stitched of glamor.\n\nIt 'aint even fucking raining. Dalton thought as he watched the women, all who seemed overwhelmingly attractive to him, twirling their parasols a bit as they walked in Victorian-style dresses. The men he caught sight of, appeared to be the opposite for the most part. Tophats, aviator style caps of leather and even a few gas style masks. Most wore either Victorian influenced shirts filled with ruffle or sharp suits, complete with a pinstriped vest.\n\nHe knew deep down he was about to step off of the airship and into a world he knew nothing about. Usually comfortable in his brown coat, this was the first time he began to feel that he would have to shell inside of it a bit; do his best just to try and fit in.\n\nAnd he felt sorry for Whiskey as well, having to endure the same type of out of place awkwardness. That is until he glanced down at his flea-bitten friend only to discover Whiskey wearing a pair of oversized goggles. Tank and Skulls had placed them on the pooch, and the goggles seemed to have the perfect fit as Whiskey stared back at Dalton. Sad eyes now protected by clear lenses and rounded brass as he stood a bit more firmly, proud of both his brown coat and his Victorian specs.\n\n\"What the?\" Dalton managed to mutter as everyone stood to their feet ready to exit the airship.\n\n\"It's showtime.\" Cambria said playfully as she cast a warm smile into Dalton's direction.\n\n\"You call it showtime, I prefer go time.\" he replied in a low voice, glancing down to make sure his revolver was still holstered before breathing deeply and following the crowd off of the airship.\n\nGeartown wasn't nearly as large as Dalton had envisioned. It was in fact...a town, and a small one at that. The fact that so many people wanted goods smuggled off world held true. It's just that a majority of the citizens in and around Geartown favored the around part. Houses scattered throughout the croplands and wooded terrain that surrounded such a Victorian-style town. Still, Geartown had everything it needed; including a watering hole for those who preferred adult drinks.\n\n\"Trading Post?\" Dalton asked.\n\n\"Yep. That's the name of Geartown's busiest building. Serves as a general store, mail dispatch, surplus shop and saloon.\" Cambria replied.\n\n\"I've never heard of a mail dispatch and saloon in the same building.\" Tank added as the group walked from the recently landed airship into the heart of Geartown.\n\nThey continued to skim the town with their eyes, each wrapping their thoughts around the same idea. If it weren't for the beauty, the damn near artistic perfection of the town around them, it would otherwise be a dusty town on the edge of nowhere. But the Victorian influence around them was obvious, as the gold flaked trim and brass accents of daily life in Geartown were simply marvelous to anyone who visited.\n\n\"At this point who gives a damn. She's buying and I'm drinking, don't really matter what sign is hanging from the front door.\" Dalton replied, a grin of long-overdue plastered onto his face.\n\n\"Well said.\" Skulls added, holding his bolt action rifle behind his neck.\n\nTank and Skulls broke from the group, heading into the direction of Geartown's finest, not to mention, only hotel; The Stage Inn. Meanwhile, Cambria, Dalton and a slightly promiscuous Whiskey made their way to the Trading Post. Dalton quick to notice that Whiskey was walking with a bit more strut.\n\nMust be the goggles. Dalton thought as the three entered the large building of wooden shingles and thick brown logs.\n\nDalton just wanted to fit in, maybe slip into the building unnoticed and hang out until they found a job and got their asses back into the familiar territory of space.\n\nHowever, as the three entered through a heavily creaking door, nearly forty people suddenly turned to see who had arrived. A shroud of unnerving quiet draped across the room as only small sounds of glasses connecting with wooden tables could be heard. Maybe it was Cambria's look of angelic sexuality. Possibly Dalton's rugged look of a poverty-stricken ranch hand. Of course, there was always a dog standing close, outfitted in a thick brown coat with large brass goggles to accent the look.\n\nBut the truth was it had nothing to do with any of the above. They had recognized Cambria Sims, and knew all too well her badly ended romance with Johnny Edmonds. The same Johnny Edmonds who currently sat at the bar looking into her tantalizing eyes, and the same Johnny Edmonds who had earned his nickname the hard way. The Revolver. He was by all accounts the fastest gun in or around Geartown and everyone knew it to be the truth.\n\n\"Welcome back.\" Johnny said as he rose to his feet, clapping loudly in the process.\n\nHe had everyone's attention, except for Dalton, who quickly walked past him and sat down by the bar.\n\n\"Double shot of your strongest.\" Dalton said quietly to the barkeep as he too turned back to watch the former lovers speak.\n\n\"When you told me you were leaving Geartown to live out in the black, leaving to find a ship and crew,\" Johnny said as he stood close to Cambria, their eyes interlocked. \"I had no idea you'd come limping back with a single buster and his homely looking bag of fleas.\"\n\n\"Let it go Johnny, we've been over for a long time now.\" Cambria said, noticing Dalton finishing the large shot of rum before standing to his feet.\n\nShe tried to motion Dalton to sit back down, but it was of no use.\n\n\"Yea Johnny. She went out and fetched her a real man. Aint' got no use for make believe cowboys anymore.\" Dalton said provokingly.\n\n\"Careful outlander. Best sit back down and put those lips on the rim of a glass before they get you killed.\" Johnny said as he continued to stare at Cambria.\n\n\"Only thing these lips are going to be on, boy, is that pretty little woman standing in front of you.\" Dalton said, earning a very strange look from Cambria in the process.\n\n\"Alright. You've had your chance, and now you gotta die.\" Johnny said as he turned slowly.\n\nDalton was the first to go for his revolver, barely raising it from his holster before Johnny's barrel was aiming down at him.\n\n\"Oh shit.\" Dalton said, stunned by the gunslinger's speed. He hadn't seen anyone that fast with a pistol. His good friend and former Captain Adam Michaels maybe, but even that was a stretch.\n\n\"Law says I'm within my right to cut you down right here where you stand.\" Johnny said as Dalton felt a sober panic flow through his rum tainted blood. \"But I'm not going to, I like your demeanor outlander.\" Johnny added, pulling his pistol down, holstering it once more with blazing speed and slapping Dalton on the arm a bit.\n\n\"Damn straight you're not.\" Tank said, hoisting his large shotgun up into the direction of Johnny as he and Skulls entered the building.\n\nImmediately, fourteen men stood to their feet, each pulling a sidearm and taking aim on Tank. Skulls was quick to pull his rifle as well, determined to take a few with him if need be. Reluctant to do so, Dalton finally pulled his revolver and held it to the face of Johnny.\n\n\"Everyone!\" Cambria shouted. \"Calm down! We just came in for a drink and a little down time for the evening.\" she added. \"Johnny, call them off!\" she said, her voice of soothing persuasion doing the trick.\n\n\"Do what she says boys.\" Johnny said, the large group of men slowly putting revolvers back into their side mounted holsters. Johnny then turned back to Dalton.\n\n\"I let you live because you are new here, won't be extending the courtesy twice.\"\n\nBoth Tank and Skulls kept their weapons raised as Johnny and his group slowly left.\n\n\"Gotta go anyway beautiful, taking a Drifter hunting party out tonight.\" Johnny said with a smile before turning to exit the large room.\n\n\"Well that was fun.\" Dalton said as he slowly sat back down and ordered another stout shot of rum.\n\n\"Point of interest,\" Tank said as he slowly sat down. \"The next time someone has that many armed friends, it would be helpful to know BEFORE I draw down on him.\" he added with emotion.\n\n\"Sorry, it all happened so fast.\" Cambria said with apology.\n\n\"Don't sweat it baby, I still love you.\" Dalton added as he slammed the shotglass down and gritted his teeth from the burn.\n\n\"Speaking of which,\" Cambria said as she slapped Dalton across the top portion of his arm. \"What is this about your lips being on me?\" she asked.\n\n\"Did I say that? I never said that?\" Dalton replied in an attempt to back out of his ill chosen words. \"Stay out of it!\" he added angrily as Whiskey barked loudly as if to turns state's evidence on him.\n\n\"Been a long time since we've seen anyone stand up to The Revolver.\" a woman with soft blonde hair said as she slowly approached Dalton, pink accents glimmering across her soothing flow of locks.\n\n\"Mind if I buy you a drink?\" she asked, Dalton turning to his crew with a hard look before turning back to accept her offer.\n\n\"Well, um, I gotta go find some more ammunition anyway.\" Tank said, standing to his feet slowly.\n\n\"And I should look into finding us work.\" Cambria said, slowly standing to her feet as well. Her lushly curved bottom reason enough for Whiskey to stand quickly, though his most important part was already standing to attention, watching her every move through the thick of his goggles.\n\nAs they waited for Skulls to follow suit, the sniper sat there, skimming the interior of the building. Cambria cleared her throat slightly as a suggestive hint, one that never struck home with the skilled killer. Shortly after, his chair was kicked hard by Tank, who motioned him away with a tilt of the head.\n\nSkulls looked at Dalton for a moment and shrugged before standing and following the group.\n\nAbout fucking time ya squatter. Dalton thought, before turning to the blonde with a manufactured smile painted onto his face. \"My name's Selina, and you are?\" she asked playfully.\n\nIn most cases a name meant something, stood for beliefs or heritage. Not to Dalton. In his mind a name was merely words strung together and tied snugly around curves and parts capable of sexual loving. Like a wool blanket. And just like a wool blanket, when it got wet it got clingy.\n\n\"I'm Dalton. Dalton James.\" he said with a grin on one side of his face, though he had indeed contemplated using an alias.\n\nAs Cambria and group exited the Trading Post, once again the Geartown normal seemed anything but to both Tank and Skulls. Glancing through the busy street of such a small town, Skulls noticed a majority of the townfolk glancing back.\n\n\"What's that about?\" Tank asked as he glanced up into the air, a tall wooden tower standing above the entrance to Geartown.\n\n\"Warning system of sorts,\" Cambria said after a quick glance, her explaination falling from such tender lips.\n\n\"If you hear the bell on that tower ring, means a Drifter is nearby. If you are unarmed they ask you to get indoors as a precaution while the sniper up there scopes and shoots.\" she added.\n\nSkulls glanced at that moment, uninterested up until the word sniper was uttered. He was damn good with a scope and knew it. Rightfully so, he was thinking of the art of sniping most times and when another skilled shooter was nearby, Skulls found himself feeling almost competitive.\n\n\"Is he any good?\" Skulls asked.\n\n\"Hasn't been a Drifter reach town before,\" Cambria said softly. \"But I'd say you're a bit better with a rifle.\" she added to calm the artifact of death collector.\n\n\"And here I was trying to calm Dalton down. How bad are these Drifters?\" Tank asked.\n\n\"They wander in close to town sometimes, but you were right to calm him. They are mindless and roam the badlands mostly.\" Cambria responded.\n\n\"Are you crying?\" Selina asked as Dalton indeed teared up a bit, quickly blaming it on the house liquor through hand motions.\n\n\"It's just that Roman was a good friend and now he's a Hunter. One of the walking dead and it's a hard pill to swallow.\" Dalton finally replied.\n\nWell this Roman sounds like a good enough guy. But the way you beat him in a blade fight and turned the Hunters away single-handed, that's amazing.\" Selina remarked.\n\n\"Thanks babe. It wasn't easy. All I had was a blade and a six-shot revolver, but eventually I sent about twenty of the bastards to the grave or running. Got bored with it all to tell you the truth, and that's when I found myself here.\" Dalton said, lying without reserve.\n\n\"In fact, I drew slow on Johnny on purpose. I wanted to see his hand to holster motion, so the next time we meet I'll be well prepared.\" Dalton said, further piling onto the heap of cattle shit verbiage.\n\n\"That's amazing.\" Selina said as she wrapped both of her near glowing arms around the waist of the brown coat wearing weaver of lies.\n\nAnd the day continued, falling slowing into the clutches of night as a fully starred sky draped above Geartown.\n\nThe Trading Post continued to see newcomers to its establishment as the daily airships and roar of a steam powered train brought more outlanders to Geartown. For such a small town, it was booming around the clock with brand new faces. Some in search of smuggling work, just as Cambria and her crew were. Others arriving for the thrill of the hunt, or even just to visit such a beautifully crafted society of simplistic living.\n\nYou could always pick the outlanders from a crowd as they stopped to see what the loud noise of the incoming train actually was. Geartown citizens had grown so used to the iron passenger train screeching into the heart of town that they continued with their routine without pause. For everyone else, the commanding sound of iron sliding recklessly on steel rails was piercing.\n\nThe train came complete with a shotgun toting soldier in the front compartment and a heavily armed rear platform. A gatling style mini-gun was mounted to the rear with four soldiers who stood heavily armed. The train, and others like it, made their way between Geartown and several other towns. Each separated by The Badlands.\n\nThe rolling hills of high grass and thick trees had earned its name for a reason. Drifters roamed, sometimes in groups of four or five through The Badlands, not to mention criminals. Gangs led by Johnny Edmonds and his type, and sometimes they felt the need to take a train by force; robbing its passengers, while stripping the steam engine of any valuable armor and weaponry.\n\nThe Royal Army, which was the military backbone of the Drifts, sometimes routed supplies through The Badlands as well. Of course they knew of outlaw activity in the area, but such an underfunded army was already stretched to its limit. And so the trains served as both a transportation system and armored supply vehichle.\n\nA couple hours after arriving, Cambria and her group had returned to the Trading Post, sitting at the far side this time in order to afford Dalton some personal space.\n\n\"I'll never understand what he sees in women like that.\" Cambria said as she watched Dalton continue his conversation with Selina.\n\nIt was at that very moment that Dalton casually slid his hand down to Selina's ass, prompting Cambria to adopt a look of disgust across her face.\n\n\"Well, if you don't understand, I'd be glad to explain it to you in detail.\" Tank said as Skulls chuckled a bit while Whiskey barked.\n\nEven the flea-induced warrior of a four-legged variety knew the game. And though Dalton was the one about to score, Whiskey knew that when he did, it put the smuggler into a great mood. Which meant a healthy leg of finely-cooked meat and possibly even some smooth hootch. A good complimentary prize to a long night spent swimming under cheap hotel linen.\n\n\"No, that's quite alright. Spare me the details.\" Cambria said with sarcasm.\n\nHer look of sarcasm quickly vanished, turning ghostly white as everyone inside of the Trading Post heard it. The warning tower bell rang once. Quiet fell through the building as everyone listened closely, a single shot firing from the sniper's rifle inside of the tower.\n\n\"Tessa 112.\" Skulls said. \"Accurate but weak, they should be using better.\" he added.\n\nNormally the bell would toll once more, letting the people of Geartown know that the Drifter was down and they were safe. But not this time. The bell began to ring continuously as several shots rang out from both the sniper's rifle and a shotgun held by the tower's other stationed soldier.\n\nPanic of the unknown quickly set in as everyone inside of the Trading Post stood up and began running to the door in search of answers. And answers they got, watching a group of Drifters climb the wooden tower while a larger and more coordinated group made its way into town.\n\nNearly two hundred of them total. Some of the citizens and outlanders began firing their weapons into the crowd of Drifters, while most simply fled as quickly as possible, sprinting for any building that was still located on the side of town the Drifters had yet to reach.\n\nAs the creaking of wood led to the large warning tower falling quickly to the ground below, Dalton grabbed Selina's arm.\n\n\"Stay here!\" he said with compassion. \"Everyone, stay here!\" he yelled, catching the attention of his crew and a handful of citizens who were ready to flee.\n\n\"Close those goddamn doors! Now! Get 'em locked up and start piling anything in front of the windows you can!\" Dalton yelled, knowing deep down if there was one place in town designed to keep people out, it was the Trading Post.\n\nIt didn't matter the planet, the town or the situation. Through his years of experience, Dalton had learned that society went out of its way to protect both money and hooch, putting them smack dab in the best location for survival.\n\n\"Hate the fucking undead.\" Dalton mumbled as he shoved a large table closer to the door.\n\n\"He's right,\" Cambria replied. \"This is the one building in town with all of the valuables. Thicker walls, doors and very few points of entry.\" she added, talking loudly as screams accompanied gunshots outside.\n\n\"Nothing to worry about my rosy red ass!\" Dalton added with anger, upset over another showdown with those who knew life after death.\n\nAs the majority of those few lucky citizens inside began pulling the furniture to the door, doing their absolute best to blockade the entrance, a heavy knocking came.\n\n\"Help me...please!\" a man cried out, continuing to relentlessly beat his fist on the door.\n\n\"Go on! Get! Gonna bring them 'sumbitches over here in hordes!\" Dalton responded with anger.\n\n\"Please, I have a child!\" the man replied.\n\n\"Ah shit.\" Dalton mumbled as he opened the door just slightly, catching sight of the man holding an infant.\n\n\"Move this clutter out of the way!\" Dalton yelled to those inside as he pulled his revolver to the ready.\n\nTank joined him at the door, shotgun in hand as they both waited for the citizens to once again move the heavy furniture. Dalton expected to see the undead as the door opened once again. He had studied the poster at the transport station, even heard the bell tower ringing.\n\nBut as the door opened to allow the man and his child a safe place to hide, Dalton's arm dropped down to his side; still holding the revolver in hand.\n\n\"Oh my God.\" he managed to push from his lungs as Whiskey stood at his feet, both of them witness to hundreds of Drifters killing citizens in the streets. Mutilating their bodies, climbing the exterior walls of buildings in order to reach windows and many of them pulled to the light that now shined outside through the open Trading Post door.\n\nAs a large pack began sprinting to the open door with unnatural speed, Dalton shakily pulled his revolver to the ready. Firing all six shots in succession, five Drifters fell to their deaths. Five headshots and one miss, not bad work in the pitch black of night while under the influence of alcohol.\n\nTwo Drifters remained as they continued their sprint of immortal fury. Dalton had nearly made it back inside when they were on him, infected teeth prepared to dig into his duster smothered flesh.\n\nThe first Drifter quickly became nothing more than dead flesh as Tank fired his shotgun, the Zombie's head dissipating into a fine red mist. The last remaining Drifter of the group lunged, though its entire body was nearly cut in half by a single shell fired by Skulls. The sniper had mounted his bolt-action Salvation rifle to a table at the far end of the Trading Post, using its front end tripod to balance the heavy weapon.\n\nThe gunshots had done away with the small pack, but gained the attention of every remaining Drifter; at least two hundred turning to sprint into the direction of the Trading Post as more of the undead arrived to Geartown. Yet Dalton stood for a moment, simply in shock at how many Zombies were closing in on them.\n\n\"Better,\" Dalton said, still holding his empty revolver. \"Better be getting inside and putting as much between us and them as possible.\" he added calmly, shock having numbed his usual panic.\n\nAs Dalton staggered back inside, Tank quickly secured the door and began helping the citizens pull the heaviest furniture back in an attempt to barricade themselves in. Dalton staggered to the bar, empty revolver in hand and hanging by his side.\n\n\"Are you alright?\" Cambria asked.\n\nUsually full of life and sarcasm, Dalton simply sat on a bar stool, grabbing a bottle of whiskey without even a hint of reply.\n\n\"Dalton. Are you alright?\" Cambria asked again, a bit more firmly as she sat down beside him on a stool of her own.\n\n\"Just fine.\" he replied calmly.\n\nCambria looked down at Whiskey long enough for the pooch to respond with a concerned bark, wet whiskers flaring wildly.\n\n\"Dalton. I need you to be alright. We all do. You are the only one here with enough military experience to see us through this.\" Cambria said.\n\n\"We're going to die.\" Dalton replied in a calm voice, throwing back a shot of the hellacious brown hootch.\n\n\"Hey,\" Cambria said, moving a bit closer while lowering her voice. \"I need you to be positive.\" she added.\n\n\"Oh I am,\" Dalton said as he took another shot. Screams of the dead began to blare throughout the Trading Post as dozens of Drifters beat and clawed the wooden door of the large building. \"I'm positive we are going to die.\" Dalton added.\n\nCambria stood up slowly, taken back by the submissive attitude of such a seasoned warrior. Maybe it was meant for them to die, no hope of surviving to speak of. Still, she knew someone had to take charge of the situation. At least give everyone the faint possibility of hope.\n\n\"Talk to Dalton. See if you can get his head back into the game.\" Cambria said to Tank in a low voice as the dead continued their quest to find a way indoors.\n\nCambria then turned her attention to Skulls, who's eye remained pressed to the telescopic lens of his rifle's scope.\n\n\"Good job staying focused Trevor. Just keep your eye on that door, you are literally our last line of defense if something goes wrong.\" Cambria said softly.\n\nThough Skulls acknowledged her statement, his eye remained on the scope. A few others stood near the door, their pistols at the ready. But even Skulls knew they would delay the Zombies, at best, if they somehow got inside.\n\n\"What are we supposed to do?\" Selina asked loudly, her words frantic as the noisy dead continued their job of trying to rip through the wood. Cambria turned to answer her, though it was a question in the minds of all sixteen surviving citizens.\n\n\"I...I don't...\" Cambria began to reply, sharply cut off by Dalton James.\n\n\"We need to collect as much damn firepower as we can find. Guns, ammunition and if it comes down to it we can even make a few Molotov Cocktails,\" he said, motioning to the large collection of man drink inside of the Trading Post. \"Need to put a bulk of our firepower at the door. It's the only way they can funnel in, and it's a strategic choke point in our favor.\" Dalton added as Cambria began to smile a bit.\n\nShe found her own mind easing as Dalton's extensive military background began to show. He was a lot of things, but the soldier of many wars is who she needed right now. They all did. And as Whiskey rose to his feet, standing proudly beside his best friend, Dalton smiled a bit.\n\n\"But the cocktails are a last resort. Got it?\" he asked as everyone near him nodded. \"Waste not want not.\" Dalton said as he glanced to the alcohol, and quickly back at Cambria who was smiling wide.\n\n\"Also be a good idea if we can get a small group to the roof of the building. Be able to assess how many there are and see if we can find a way out for ourselves.\" Dalton said.\n\n\"The train,\" Cambria replied, gaining everyone's attention. \"The airships will see the Drifters long before landing, and that's sure to turn them away. But the train is heavily armed and couldn't stop in time even if it wanted to. We just need to figure out a way to get on it when it comes through.\" she added.\n\n\"Is there a way to the roof?\" Dalton demanded to know as he clinched the shirt of the bartender in the process.\n\n\"Yes...this way,\" the bartender replied, the chubby man then quickly leading them to a small area in the rear of the Trading Post. \"Ain't many windows to speak of, but this ladder will take you all the way up to the roof.\" the scraggly looking man said as he led them to a narrow red ladder that was firmly bolted to the wall.\n\nCambria was the first one up, her perfected backside serving as a man lure of sorts. Both Tank and Dalton nearly began fighting for the chance to climb up right behind her, Tank finally backing off after reading into the scowl on Dalton's face. As the smuggler started up next, he heard Whiskey, who was whining in a begging fashion.\n\n\"Ah shit.\" Dalton said with frustration, wanting so badly to enjoy the view of Cambria as he climbed but not daring to leave his faithful pooch behind.\n\n\"Don't worry, I'll enjoy the view for you.\" Tank said with a chuckle as he mounted the ladder to climb.\n\n\"Are you guys coming?\" Cambria asked, looking down for a moment.\n\n\"Don't I wish.\" Tank replied under his breath as he climbed steadily.\n\n\"Damned smart ass.\" Dalton said with frustration. As they climbed what seemed to be at least two stories by way of ladder, Dalton clinched Whiskey tightly, all while listening to Tank's lightly spoken sexual innuendos.\n\nFinally making it to the top, Cambria forced open a thick door of steel and anchor bolts. The thickness of the door was such that she couldn't force it completely open, Tank placing his arm around her to help the steel loose. Dalton scowled at Tank making a play on such a beautiful woman, then turning his scowl to the true guilty party. Whiskey glared back as if to apologize, wide eyes staring through the thickened lenses of his goggles.\n\nAs Dalton and Whiskey made it topside, the brown coat booty chaser slammed the door as if to break up a romance. Tank turned to grin as Cambria remained standing, her eyes locked onto the receiving end of a set of binoculars.\n\nDalton approached slowly while the screams of both undead and citizens who would soon join them rang out below, though they had calmed a bit. A very good indication that few living folks remained.\n\n\"Is Skulls on the door?\" Cambria asked, her eyes still skimming the area below.\n\n\"Yep, she's locked down tighter than a tick's ass.\" Dalton said proudly. Cambria turned to look at him, a daze upon her face.\n\n\"That's pretty good.\" she said, smiling a bit.\n\n\"Thanks.\" Dalton replied, his thick beard doing little to cover the grin beneath it.\n\nFinally getting his own hands on the binoculars, Dalton began to skim the surrounding area. \"Has to be a way out of this 'sumbitch,\" he mumbled as his eyes continued to focus. \"What the...\" he said under his breath, locking his sight onto a fixed position.\n\n\"What is it?\" Tank asked, moving closer to the roof's ledge to stand beside Dalton and his trusty sidekick.\n\nDalton stood there, staring out onto the group as he tried to put two and two together. \"The Revolver,\" Dalton replied in awe. \"Johnny fucking Edmonds.\" he added, watching the gunslinger tear his way into town by way of pistol.\n\nSuddenly, the thick steel door sprang open, Selina quickly pulling herself to the rooftop.\n\n\"Woman, what the hell are you doing up here?\" Dalton asked sternly as Tank walked with him, both approaching the bar girl slowly.\n\n\"They're in.\" she replied frantically.\n\n\"Huh?\" Tank asked.\n\n\"They're in!\" Selina yelled to confirm their worst fears.\n\n\"What the fuck do you mean they're in?\" Dalton asked with emotion. His question was immediately answered by the sound of a sniper's rifle firing below.\n\n\"Oh shit.\" Dalton finally said as Tank began helping others up onto the roof.\n\n\"Help them up!\" Cambria yelled with panic as screams flooded beneath them, accompanied by seemingly non-stop gunshots.\n\nThe barkeep was up, as well as Selina and two other men who seemed dazed at best.\n\n\"Where's the kid?\" Dalton asked with no answer following.\n\n\"Where the fuck is the kid?\" he asked again, this time with a commanding voice as his arm grabbed the shirt of the barkeep.\n\n\"We couldn't get anyone else to safety. There are too many.\" the man replied in self-disgust as he tried catching his breath. They got in through the front door, pushed it open and before we knew what was going on, they had flooded the place and separated us in the process.\"\n\n\"Fucking undead.\" Dalton mumbled as he checked the rolling chamber of his capable revolver, reloading it as quickly as he could.\n\n\"Dalton, what the hell do you think you are doing?\" Cambria asked.\n\n\"That kid is down there surrounded by a horde of these non-hygienic fucks.\" he replied as he headed to the steel trapdoor. \"Might be other people down there trapped, we don't know.\"\n\n\"What about Skulls?\" Tank asked.\n\n\"Skulls is a trained killer with a weapon! This kid 'aint got nobody but us. I'm a lot of things that most people would be ashamed of, but I ain't a coward.\" Dalton replied with truth as he readied himself to go downstairs. And at that very moment, Cambria Sims began to feel an emotion she hadn't felt in a very long time. Love.\n\n\"Don't lock this door until I'm back, you got me?\" Dalton asked of Tank, answered with a nod of the head.\n\n\"Cambria, don't you let him...\" Dalton added, turning to his Captain. Whiskey quickly let loose a firm bark to assure his best friend that if needed, he'd protect the door himself.\n\n\"Don't worry, I won't. You just go get that kid.\" she replied, a smile on her face that no crew member had ever seen before.\n\nAs Dalton began to quickly leave the rooftop on his way back down, the group watched him as though he were nothing short of a hero. Something this town needed at the very moment he had arrived, even if their hero cursed the gods on his way downstairs and left a blanket of Rum stench behind.\n\n\"Stay on this door, I'm going to try and get Johnny over here to help.\" Cambria said, moving quickly to the roof's ledge as the walking dead continued to terrorize the streets below.\n\n\"Doesn't he want us dead?\" Tank asked.\n\n\"Well, that was before all of this. Besides, Johnny is easy to forgive. Just keep your eyes on that door.\" Cambria answered.\n\n\"Don't worry, it's locked down tighter than a tick's ass.\" Tank said with a smile.\n\n\"Not really that clever now that I've heard it again.\" Cambria said, turning to begin finding Johnny Edmonds once more.\n\n\"Well you thought it was funny when Dalton said it.\" Tank replied with childish envy.\n\n\"Yea. Yea I did, didn't I?\" Cambria said under her breath.\n\nAs she gazed through her binoculars in an attempt to locate and coordinate with Johnny Edmonds, she continued to think about Dalton James. Of course he wasn't the kind of guy she could see herself ever falling for. But every single time she learned something new about the hardened smuggler, it led her to believe he was a completely different man on the inside. A kind, compassionate man with an exterior that was nothing more than a mask to the world.\n\nBesides, he did have a damn cute dog. As Cambria broke her stare for a moment to glance down, the lustful eyes of Whiskey glared back at her through thickened goggles. Well, maybe not that cute. she thought as the brown coat sporting dog licked his tongue across his lips.\n\nDalton knelt slowly on the floor below the ladder, the tail end of his duster laying slightly on the wood grain. A Colonial soldier is trained to always assess the situation and then react. And as Dalton James remained knelt close to the floor, he first glanced up.\n\nIt was dark, but he saw a handful of stars above that let him know his friends were indeed keeping the door open for his return. As he glanced around the back of the Trading Post, he saw a few bodies, bloody trails and expended shells.\n\nWhat stood out as bothersome to Dalton was the lack of noise. Only moments before, gunshots rang out in near parade fashion. Now things had become deathly quiet and that scared the hell out of the former soldier.\n\nPulling his large revolver to the ready, a trickle of light refracted from the chrome of the barrel as Dalton slowly made his way to the end of a hallway which led to several smaller storage rooms and eventually the main drinking area. It was the first time Dalton had dreaded entering a bar in, well, forever.\n\nGlancing out of the doorway momentarily, Dalton began to enter the main room when he suddenly stopped dead in his tracks. A natural reaction to the sound of steel coming to a ready as a gun stood prepared to fire. He slowly turned to see Skulls kneeling in a small room to his right, large sniper rifle at the ready.\n\n\"It's me damnit!\" Dalton said under his breath.\n\n\"Get down.\" Skulls replied.\n\nTaking a moment to fully register, Dalton finally knelt to the floor. A mere second later, dedicated lead burst from the rifle of Skulls, slicing through the air and into the formerly dead flesh of a Drifter who was approaching.\n\n\"We gotta move, now.\" Skulls said calmly as he pulled his rifle from the ready in order to exit to the roof.\n\n\"I came down for the kid, we gotta find...\" Dalton replied, cutting himself short as he spotted both the man and accompanying child, sitting snugly behind Skulls.\n\n\"Good work,\" Dalton said. \"I'll be right back, don't let them shut me out.\" he added.\n\nSkulls nodded firmly as the quiet killer led the old man and child to the safe confines of the ladder.\n\n\"Where's Dalton?\" Tank asked as he helped Skulls to the roof.\n\n\"Don't know. Didn't say.\" Skulls replied as he ushered the older man and confused child to the rest of the survivors.\n\n\"Didn't say?\" Tank asked in disbelief.\n\n\"Well fuck that, we need to at least shut the door until we know something.\" he added, reaching for the trapdoor.\n\nThe well muscled warrior stopped short, small hairs standing tall all over his body as the sound of a pistol readying halted him.\n\n\"You would dare pull a gun on me?\" Tank asked, furious that Skulls had the guts to hold a piece in his direction.\n\n\"You scrawny little bitch, I'll beat the fucking sense back into you.\" Tank added, walking slowly to Skulls.\n\n\"Tank! Enough!\" Cambria yelled, demanding her crew member stop at once.\n\n\"He pulled a fucking gun on me!\" Tank yelled in reply.\n\n\"He's waiting for Dalton. We all are!\" Cambria shouted back.\n\n\"We'd be doing the same for you if you would have had the guts to go yourself.\" she added.\n\nTank scowled for a moment at Skulls to let him know the issue wasn't over.\n\n\"They're here, they're safe,\" Tank yelled as he pointed to the older man and child. \"My question is where the fuck is Dalton?\" he added.\n\n\"Right here, why did you miss me?\" Dalton replied as he slowly climbed the ladder, gunshots ringing out behind him.\n\n\"Tank wanted to leave you down there.\" Cambria said in disgust.\n\n\"Is that a fact?\" Dalton asked as he stood to his feet. \"Well sir, no liquor for you.\" he added as he pulled three of the tallest bottles of rum from his coat that any of them had ever seen.\n\n\"You risked our lives for liquor? You son of a bitch!\" Tank yelled, storming into Dalton's direction.\n\nThe much larger man stopped abruptly, his face merely inches from the barrel of Dalton's revolver.\n\n\"Best think this through. You got the physique and clothes to fit the part, but I'm a true soldier boy. I'm not playing commando; I am one.\" Dalton said with wily sarcasm, though his words held nothing short of the truth.\n\n\"Damn. I'm falling in love with this guy more and more every minute.\" Johnny Edmonds said as he climbed up behind the experience-worn smuggler, a couple more bottles of whiskey in hand.\n\n\"Now Skulls, you can shut the door.\" Dalton said, his eyes still locked onto those of Tank.\n\n\"I won't forget this. Next time maybe I got your back, maybe I don't.\" Tank said as Skulls secured the lock that bolted the thickened trapdoor into place.\n\n\"I could give a damn young man, just remember that kind of respect is a two-way street.\" Dalton replied as Whiskey growled heavily.\n\n\"Young man. That's clever.\" Cambria said in a low voice, intrigued by the man known as Dalton James.\n\n\"What's it like out there beyond town?\" Cambria asked as Johnny approached her.\n\n\"It's bad girl. Real bad.\" he replied, spitting a bit of tobacco drenched saliva onto the rooftop.\n\n\"What?\" he asked as she stared at him in disgust.\n\n\"Over the ledge maybe?\" Cambria replied.\n\n\"Oh uh, yea.\" Johnny said, spitting a second wad of saliva over into the crowd of undead below.\n\n\"Best we've come up with is getting to the train somehow.\" Cambria said as her eyes skimmed the horizon.\n\n\"That's a bad plan. Really bad plan.\" the gunslinger replied.\n\n\"Huh?\" Cambria replied.\n\n\"Me and the boys,\" he said, taking a moment to pause. \"We hit the train a few hours ago. Left it sitting out there beyond the hills. At least a six or seven mile walk.\" Johnny said.\n\n\"You robbed the train and left our only escape vehicle out in the middle of nowhere? Really?\" Cambria asked furiously.\n\n\"Well hell, I didn't know these son of a bitches was gonna get coordinated today. Hell, we was thinking about the money.\" Johnny replied in his own defense.\n\n\"And the boys?\" Dalton asked as he approached loudly.\n\n\"Some of 'em got dusted during the robbery, the rest are laying nearby. Dead fucks killed 'em on the way in.\" Johnny replied.\n\n\"You know I hated your guts until about thirty minutes ago, right?\" Johnny asked as he turned to Dalton.\n\n\"Yea, I figured as much. You don't have much of a rack on you though darling, so I wasn't going to lose a lot of sleep over it tonight.\" Dalton replied with a grin.\n\nCambria watched the two men converse for a moment, all while deep in thought herself. Johnny was the typical cowboy, strong on the outside and empty on the inside. Even though Dalton appeared the same at first glance, Cambria now saw through all of that.\n\nIf nothing else, the broken man used his brown coat to cover the true man inside. He had a heart, and Cambria suspected it was a big one. Her suspicions were confirmed when Dalton gave himself away with a soft glance into her direction while conversing with Johnny.\n\n\"Cambria,\" Skulls said as the entire group has stopped to stare her way. \"You alright?\" he added.\n\n\"Yes, fine,\" she replied as she quickly broke from her daydream. \"Just trying to think of a way out of here, that's all.\" she added.\n\n\"Aint' no way outta here.\" Dalton replied as a group of Drifters had begun beating on the steel door which separated them from the small group of survivors that had found refuge on the roof.\n\n\"And even if there was, where the hell would we go?\" Johnny asked. \"Me and the boys came back to Geartown because we figured it would be safer. The hills are crawling with the bastards.\" he added.\n\n\"Well we can't stay here, that fucking door 'aint gonna hold forever!\" Tank yelled as he remained sitting.\n\n\"Relax, would ya?\" Dalton said as he glanced at Tank sharply for a moment. \"The door will hold, it's a choke point. Hundreds of the fuckers down there, but they gotta come up the ladder single file. And 'aint one of them gonna be strong enough to bust through,\" Dalton said calmly. \"But eventually we do gotta find a way off this roof. Gonna be needing food and such.\" he added.\n\n\"And you're sure we can't get to the Cliffs?\" Cambria asked of Johnny.\n\n\"Trust me, if it was doable I'd be there and not here.\" Johnny replied.\n\n\"Well we sure in the fuck can't stay here!\" Tank yelled with frustration.\n\n\"Well hop on down there and clear the way for us big boy.\" Dalton replied, a slight grin on his face. Tank's attempt to get up and confront the smuggler was easily stalled by a warning growl on Whiskey's part.\n\n\"Alright, guys,\" Cambria said loudly, trying to overpower the loud beating on the steel trapdoor by the undead. \"We need to work together. Figure out an exit strategy.\" she added in a more convincing tone.\n\n\"Well the hideout up at Mulden Cliffs, like you said, would be the perfect spot to hold up if we could make it. Got plenty of supplies and and weapons just waiting to be dusted off. It's well hidden and only one way in. Even if the fuckers followed us up, we'd be able to thin them out once they started funneling up the narrow pass to us.\" Johnny said as he remained deep in thought.\n\n\"Say they was pretty thick outside of town too?\"Dalton asked.\n\n\"Thick as thieves,\" Johnny replied. \"And coming from a thief that's saying something. They are organized now though, don't make a peck of sense. Just yesterday they were brainless and random.\" he added.\n\n\"Best bet would be for us to sleep on it. Get some rest and then maybe sort things out come sunup.\" Cambria added.\n\n\"Skulls, you take the door and I'll watch the ledge. We'll pull first watch,\" Dalton said. \"In a few hours we'll rotate out with Johnny and Tank. Go on down the line until sunup.\" he added.\n\n\"Sounds like a good plan.\" Cambria said as she looked for a place to rest.\n\nThe roof on its own was pretty large, to her estimation it was about thirty feet wide by thirty feet long. Aside from the steel trapdoor that stood between the survivors and flesh eating dead below, there were a handful of steel pipes along with a flat surface of concrete.\n\nThey had plenty of room. The problem seemed to be the lack of supplies. No type of shelter, little food and nothing to put between themselves and the cold concrete they would be resting on.\n\nShortly after, the group became quiet. It was the perfect opportunity to rest, though little sleep would be had. Dead flesh pounding roughly onto the thickened steel of the roof's trapdoor.\n\nQuiet times were always the worst for Dalton. He was a man with a laundry list of life events, most of them not so great when it came to re-living them. And as he sat on the ledge of the Trading Post, with legs dangling and cigar ashes falling down onto the crowd of Drifters, he began to think about his old crew.\n\n\"Damn woman, you got a death wish or something?\" Dalton said, quickly pulling his revolver to the ready as Selina approached softly.\n\n\"You are alive, right?\" he asked as the barrel remained on her.\n\n\"What does this tell you?\" she replied, wrapping her arms around his neck and kissing him deeply.\n\n\"Tells me this trip 'aint been a total loss.\" Dalton replied with a grin.\n\n\"May I sit?\" Selina asked. Dalton simply pointed to the ledge beside him as Selina sat down gracefully.\n\n\"Do you think we'll be alright?\" she asked in a somber tone.\n\n\"Hard to say,\" Dalton replied as he took a deep inhalation of Rum treated cigar. \"I expect that if anyone makes it out of Geartown alive, it'll be us.\" he added.\n\n\"I hate that all of this happened, but I'm really glad you're here.\" Selina said as she looked deeply into the smuggler's wide eyes.\n\n\"I'm sorta' glad I'm here too,\" Dalton replied, drawing another lung debilitating puff. \"Truth is, this has me feeling more alive than I have in a long time. Don't get me wrong, I could do without the crowd of undead. But otherwise, it's kinda peaceful out here in the Drifts.\"\n\n\"It can be,\" Selina said with a grin. \"At least before all of this happened anyway.\" she added.\n\nDalton agreed with a nod, flicking his cigar stump onto the crowd of animated flesh below.\n\n\"Technology has a way of complicating things. Has a way of pulling people apart and taking the luster away from life's greatest moments.\" Selina said.\n\n\"That's deep,\" Dalton said, his eyes widening just a bit. \"I get it though. \"I've been a lot of places and seen a lot of unhappy people. Most of them were living on the cutting edge of technology.\" he added as he pulled a small flask from his coat.\n\n\"Crash landed once; well, more than once,\" Dalton said prior to taking a healthy swig of whatever his flask was readily offering. \"The planet was off the map, nothing to speak of really. Just a village full of the most primative people you could ever meet. But they were good people...happy people.\" he added as Selina squeezed him a bit in hugging fashion.\n\n\"Really? What happened to them?\" Selina asked, intrigued by his story.\n\n\"Oh they all died,\" he said, starting to take another swig as he realized the damage. \"Not that we are gonna die or anything.\" he added quickly.\n\n\"Well this is a very encouraging conversation.\" Selina replied softly.\n\n\"Ah, don't fret it none babe, I'll come up with something.\" Dalton said.\n\n\"I hope so,\" Selina said with a concerned look. \"Everyone is looking to you.\" she added as she slowly stood up to find a nice spot to rest for the night.\n\nThey were looking to him, especially Cambria at that very moment. Watching his every move. Wondering why such a good hearted man would hide away from the world beneath the shield of his brown coat. As she lay there, she continued to stare at him. The way he sat calmly on the ledge let her know of his experience in battle, but the way he gently placed his hand on the pup beside him let her know of his good spirit.\n\nShe was becoming confused under the worst possible conditions. But still she thought only of Dalton James as she slowly drifted into a world of dreams.\n\n\"All we can do is sleep it off,\" Dalton mumbled as he sat on the rooftop and prepared his duster in a way as to present the illusion of a fine mattress. \"Nothing else we can do.\" he added as he lay down and rolled away from the group of survivors.\n\nIt would be hard for them to sleep. Damn near impossible, actually, as dead flesh continued pounding against the thick trapdoor of steel which separated them. Eventually, seemingly hours later, the beating subsided. Those on watch noticed a lack of interest by many of the Drifters, who began finding their way back out into the streets in search of less difficult victims.\n\nThe interior of the ship was different from anything he had ever seen before. That's what had first caught Dalton's attention. Gauges that looked alien, yet eerily of death.\n\nAs Dalton first sat up, his reaction was to check for Whiskey. But the faithful running mate was nowhere to be found. As was the same fate for his sidearm, in fact, as Dalton felt around for it his holster was no longer a part of the uniform. His eyes told a tale of unanswered questions as he glanced down to see himself clad in solid white. It seemed surreal.\n\nJumping swiftly to his feet, Dalton began to visually search the room. Solid steel walls, though not a single seam could be seen with his naked eyes. He wondered how such a ship could have even been constructed, seemingly flawless in every respect. There did appear to be a door, and as the wily smuggler began to approach it slowly in search of answers, three colored triangles illuminated the wall directly above it. All of them brilliant red and further adding to the mystery.\n\nMoments later, the door opened quickly enough to startle Dalton back against the wall. His reasoning led him to only two possibilites. Either he had been drugged or the Drifters had indeed found him. Maybe his dream was merely a part of the conversion process, leaving the mortal world and joining a very feared nation of walking dead.\n\nNormally, Dalton would have played out each scenario in his head, but his train of though was interrupted as two figures entered the room. Though visible, they both seemed a bit blurred, almost as if his eyes could not focus them. Both were clad in solid white, from boot to riot style helmet. Even the shielding of their helmets glimmered a satin white, while three red triangles formed a larger triangle as the only marking on their helmets.\n\nDalton felt the need to ask questions, and had started to approach them before stopping dead in his tracks. A tail of some sort was visible as the soldiers turned to the side in order to watch the door. It extended slightly from the rear of their helmets and seemed to penetrate the top of their spinal cords. Though the soldiers remained perfectly still, the brown tail wiggled a bit, its tissue both knobby and covered in scales.\n\nIt was unlike anything the smuggler had ever seen before, the sight truly scaring Dalton for the first time in many years. He felt a chill run through his skin, raising bumps all along his arms as his hands shook a bit, preparing to fight for life if needed be.\n\nAll of that changed with the entrance of a third. Solid white cloak, hood pulled up to protect both emotion and identity. Dalton's heart nearly stopped beating when it became evident what was about to occur. The hooded figure carried a glass cylinder in one arm; inside the cylinder a small creature. Its tail matched that of the ones draping from the helmets of the posted guards, though the rest of the creature was also visible.\n\nIt was almost flat, though it had six legs clad in scales of dark brown and the eery tail that appeared to form a large needle-like shape at its tip. The creature began striking toward Dalton, almost as if it were a venomous snake waiting to be loosened from its cage. As the cloaked figure slowly pulled the hood from its head, Dalton began to cry aloud.\n\n\"Help us.\" Adam Michaels said, a monster similar to the one encased in glass attached to his skull with tail running down into Dalton's former Captain's spine. Just as Sasha's overpowering screams could suddenly be heard, Adam threw the encased monster toward Dalton, its glass confines quickly disappearing as the freed monster lunged for him.\n\n\"Dalton!\" Cambria said in a loud whisper, trying her best to force the smuggler to awaken.\n\nHe had started to come to a bit, though he remained, for the moment, between the world of dream and that which required oxygen for survival.\n\n\"Dalton, are you alright?\" Cambria asked again, this time her words filled with concern.\n\n\"Yea,\" he replied, hand gripping his head a bit. \"Splitting headache and,\" he added, seemingly holding back for a moment. \"Really bad dream.\"\n\n\"It's alright, we all have them.\" she replied.\n\n\"Not like this one. God, it seemed so real.\" Dalton said.\n\nCambria gripped his shoulder for a moment, a gesture to let the smuggler know she was by his side.\n\n\"I woke you up because day is breaking,\" Cambria said in a soft voice. \"Be a good idea to assess our situation.\"\n\n\"Off 'yer asses!\" Dalton yelled loud enough to wake everyone who still slept. \"It's time to put our heads together, find out what we got and what we don't got.\" he added, nodding his appreciation to the Captain of the Outer Heaven.\n\n\"It's official. What we got is eleven survivors, five bottles of hooch, three bottles of water and two cans of pork and beans.\" Tank said with a touch of disgust.\n\n\"Twelve survivors.\" Dalton added as Whiskey barked loudly in order to be counted, his angle, simply put, to be considered for the pork and beans.\n\n\"We also have guns.\" Skulls said.\n\n\"Yep, that we do got. May just be our biggest asset at the moment,\" Dalton replied, turning to the remainder of the survivors. \"I know my crew and Johnny are damn capable with a weapon,\" he said with pause. \"Any of the rest of you a decent shot?\"\n\nThe barkeep, older man who held a child in his arms and one of the younger men stepped back a bit. Experience with a weapon was surely not on their list of skills.\n\n\"No Dalton, I don't. I'm sorry.\" Selina said.\n\n\"It all good,\" he replied with a cowboy wink. The kind that was accompanied by fatigue, bad hair and whiskey laced breath. \"What about you?\" he asked of the remaining man.\n\nThe man had medium length hair, dirty blonde, though his clothes were anything but. The light-skinned man wore dress slacks and a pinstripe vest that rested over a white shirt.\n\n\"Yea,\" the man replied. \"I served with the Legion during the first Glimmerian War. I put a right many Colonials into the grave, so I guess that qualifies.\"\n\nDalton bit his tongue as he thought back to the first Glimmerian War. He had been involved as well, on the Colonial side, and had watched dozens of his friends butchered in a losing effort. There was even a good chance that the man who stood in front of him fired a shot or two that may have ended the life of a friend.\n\nBut these were different times. Different circumstances. Once bitter enemies, they now stood together with a common goal. Survival.\n\n\"I fought there as well. Fought for the Colonials,\" Dalton replied as the air that surrounded the group seemingly got very thick. \"And if we were in any other situation, the words you just spoke to me would have been paid for with teeth,\" Dalton added, looking the man from head to toe. \"But the war is long over, and we need every gun we can get.\"\n\n\"Fair enough,\" the man replied, extending his hand. \"The name's Christopher.\"\n\nHe was testing Dalton, and the brown coat smothered man of ill-repute was sure of it. Purposely crossing the line to see if Dalton meant his words. His gut told him to slug the son of a bitchn' redcoat, or better yet, shoot his ass and be done with it. But Dalton soon realized his gut hungered for food too, and the sooner they smoothed things over when it came to Glimmeria, the better.\n\n\"Dalton James.\" he finally replied, shaking hands with a man he would have been under orders to kill back on Glimmeria.\n\n\"We need to figure out if this is the spot we call home, or if we go looking for something else.\" Johnny said.\n\n\"This is the spot. It's a safe spot, otherwise they would have gotten through during the night.\" Cambria replied.\n\n\"Agreed,\" Dalton said. \"So we need to form a small group. Go scavenge what we can and get back before nightfall.\"\n\n\"What are we supposed to do for food all day?\" Selina asked.\n\n\"What we got goes to the kid,\" Dalton said with seriousness. \"We can make it days without eating if need be, but won't have to. Just need to make it through one.\"\n\n\"Well who's going?\" Tank asked.\n\n\"You and I got things to work out, so we're both in.\" Dalton said.\n\n\"Huh?\" Cambria asked with amazement.\n\n\"One of the things you need to understand about smuggling is there will be arguments,\" Dalton replied. \"Lots of arguments,\" he added. \"Hell, my previous Captain and I used to fistfight when it came down to it.\"\n\n\"What the hell does this have to do with scavenging?\" Tank asked.\n\n\"You need to learn that no matter what we got going on between the two of us, we always get each others backs.\" Dalton said.\n\n\"Yea right.\" Tank added, blowing the advice off.\n\n\"Which is why you aren't taking a weapon.\" Dalton said.\n\n\"The fuck I 'aint!\" Tank replied with anger.\n\n\"You'll be carrying all of the luggage. You a big 'sumbitch, we'll be able to carry a lot more this way. And I'll be there to watch you back, just like Johnny will.\"\n\n\"You a crazy motherfucker! No way in hell that I'm going out there unarmed, especially with you and Johnny. I'll be dead within five minutes!\" Tank said.\n\n\"No you won't, cause what you will learn really quick is the smuggler's creed.\" Dalton replied.\n\n\"Oh yea, what's that?\" Tank asked, his words filled with both mystery and sarcasm.\n\n\"That no matter how bad you piss me off, I got your back.\" Dalton said, nodding his head to show everyone the truth of it.\n\n\"Or as I like to say, trust not, eat not.\" Johnny added.\n\n\"Skulls will stay behind,\" Dalton said. \"Not that you're no good in combat my man. But in the event that we come hauling ass back with a crowd on us, you're the only one skilled enough to thin them out.\"\n\n\"I have no problem providing overwatch. It's what I do best.\" Skulls replied with a nod of approval.\n\n\"Ever have to use a slugger to bring peace back to the bar?\" Dalton asked of the bartender.\n\n\"A bat? More times than you can count.\" the barkeep replied.\n\n\"Same principle,\" Dalton said as he un-sheethed his large blade and handed it to the man. \"Drifter comes up, you swing for the hills. This blade will cut the sumbitch' in half.\"\n\n\"I'm coming too. Right?\" Cambria asked, hoping to impose her authority.\n\n\"Not this time out. If things go to shit we need someone capable of putting together a rescue plan.\" Dalton replied.\n\n\"What am I supposed to do? Just hang out here?\" she asked.\n\n\"Hand Christopher a piece and keep these people safe. Stay on that trapdoor and pray we make it back in one piece.\" Dalton said.\n\nSeveral minutes later, the survivors had laid out everything they would need for the scavenging group's trip. Several empty sacks, a couple of throw sacks using thick shirts and a few bare-chested men.\n\n\"Dalton, I cannot believe you had the nerve to ask me for my shirt!\" Cambria said, furious that he had used the excuse of Drifters as reasoning.\n\n\"Hey, just trying to do my part.\" Dalton said, chuckling as Johnny, even Tank, joined in the laughter a bit.\n\nSkulls, Christopher, the barkeep, old man and another survivor all lent their shirts. It would allow the three man scavenging party a chance to scale down the rope, rather than attempt their way down a stairwell which led to sure Drifters.\n\nThey geared up, weapons in hand and belt, while Tank's firm grip held a dozen or so makeshift sacks.\n\n\"We'll drop down and hit the Eastern side of Geartown. Remember to keep sights on us.\" johnny said as a very pale-chested Skulls nodded.\n\nAnd while Dalton embraced Selina one last time, harboring his body in the warmth of her own, even he noticed the dedicated stare of Cambria Sims. He had seen it many times, and knew that it was one of love unexplored.\n\nHe tested his theory, purposely walking past her without saying a single word.\n\n\"Dalton,\" Cambria said, approaching him slowly. \"Please be careful.\"\n\n\"Will do,\" he said with a grin. \"And when I get back, I think it's time we had a little talk.\"\n\n\"Talk. About what?\" Cambria asked, her question answered with a simple glance.\n\nOh God! Cambria thought. How could he know? How could he possibly know my feelings? Are they that obvious? Did keeping them a seret even matter to her anymore? She had fallen so suddenly, so head over heels with a man who might be marching to his own death just to feed the survivors.\n\nKnowing it could be their final moment together, she wanted to reach out and kiss the man so desperately. Plead with him, beg him if needed be, to stay and let someone else go in his place. But she knew why had had thrown himself in the situation. Because otherwise, the scavenging group wouldn't survive.\n\n\"Just take care of my damn dog and quit with the last moment together ever look.\" Dalton said with a smile.\n\nCambria wanted to curse his arrogance, maybe even curse herself for exposing such feelings. Instead she just smiled wide and nodded her head, doing her best to hold back tears that could have easily fallen if she would have let them.\n\nThe plan was a simple one. Repel down the rope made of less than expensive shirts, which they had knotted together to form a rope. Johnny would lead the group, his familiarity of the town was key.\n\nTank would follow next, his large size allowing him to carry the bulk of what they had found. Even though his job for the upcoming trip was technically that of a pack mule, it was also the most important. He would be the one who brought back needed supplies for an extended stay.\n\nThe flank of the three man group was Dalton, who carried his revolver in hand; three more handguns tucked into the leather belt around his waist. The pockets of his pants were shoved full with ammunition clips and spare bullets. He was responsible for protecting himself, as well the pack mule.\n\nAnd so the three assigned to loot Geatown eased themselves over the side of the roof, one at a time, each praying the shirts held as well as they had hoped.\n\nSkulls had spotted some Drifters roaming several blocks away, as well as a group directly in front of the Trading Post. And the sniper's hand motions let the group know where the Zombies were located, directing them almost like some sort of gothic GPS system.\n\n\"We gonna need a lot of food, tents, blankets and weapons if we can find em.\" Dalton whispered, just loud enough for the two men in front of him to hear.\n\nJohnny acknowledged his words with a nod, holding their position at a standstill while peeking from the corner of a nearby building.\n\nAnd though they had to remain hidden for nearly thirty minutes, eventually the three man scavenging group made their way, cautiously slow, out of the sight of Skulls and deeper into Geartown.\n\nCambria felt a chill come across her at the very moment they disappeared from sight. It wasn't the temperature of the low-passing winds, but rather one that made her, for whatever reason, truly believe that she would never see Dalton again.\n\nAnd even though they were no longer visible in the bright rays of sun that fell down onto the Drifts, Cambria continued her stare for several minutes. Praying, if nothing else, that her gut feeling was wrong.\n\n\"Keep on your scope,\" she said, a bit of panic in her voice of deep concern. \"Please.\"\n\n\"Don't worry, I believe they will be fine.\" Skulls replied.\n\nThose who remained back on the Trading Post rooftop seemed to finally grasp the devastation. The previous night had been filled with screams, terror and non-stop fighting for their lives. They had been given no time to sincerely think about the events which had unfolded.\n\nBodies laying out in the streets surrounding their building. Most, if not all, mutilated and dead by horrific measures. There was no doubt that the very building beneath them was filled with more of the same.\n\nDrifters walked among them. And they knew but one purpose. Killing. And as the survivors looked from their rooftop of safety, they saw the bodies, smoking buildings and walking dead throughout the streets.\n\n\"It will be alright.\" the barkeep said, trying his best to comfort Selina as she began to sob heavily.\n\n\"How could this happen?\" she asked, her words falling between two outburst of tears.\n\n\"Let's just hope it's contained to the Drifts,\" Christopher said. \"So we can find a skiff and get the fuck out of here.\"\n\nCambria didn't respond. As she turned to Christopher, she wanted to. So badly, she wanted to respond to the man she knew was less than half the man Dalton James was.\n\nShe didn't know the complete history of Glimmeria, but she did know of both wars there. She knew that Dalton had fought for the very side she would have, if it would have been her. The side that just wanted to be allowed to live. The side that fought, only as a last resort, to protect their families.\n\nAnd as she watched Christopher, his every move giving clue to the character of the man, she knew he was a coward. He had more than likely killed several Colonial soldiers during the war. Soldiers, just like Dalton, who were forced into protecting everything they lived for. But he had done it from a distance, maybe with the odds heavily in his favor. At least that was Cambria's opinion, as she saw a man who only wished he could measure up to a Dalton James.\n\nWhiskey had been quiet since the group of scavengers departed, choosing instead to take his place near the elderly man and child who sat far from the trapdoor. The faithful pooch watched over the pair as though he were a ill-bathed guardian angel. And, while his eyes skimmed the group, his head remained flat to the ground.\n\n\"What's your name old man?\" Christopher asked.\n\n\"Carlos.\" the elderly man replied softly, doing his best to keep the babe in his arms calm.\n\n\"A little old for a baby aren't you?\" Christopher asked.\n\n\"Are you watching the trapdoor or writing for a newspaper?\" Cambria asked sharply.\n\n\"Just trying to get to know everyone,\" he said without even turning into her direction. \"That's all.\"\n\n\"He's not mine,\" Carlos said. \"I found him in the street about two blocks down. Didn't seem right to just leave a infant boy laying.\"\n\n\"What about you, what's your story?\" Christopher asked of the other surviving man.\n\n\"Name's Kieth,\" the man replied. \"I sell prospecting licenses,\" he added. \"Or at least I used to.\"\n\n\"Wow, a licensing official. How exciting.\" Christopher said with sarcasm.\n\n\"True. Not all of us can be cold-blooded killers fighting under the Legion flag,\" Kieth replied. \"If I was the killing kind, I would have killed your kind with a brown coat on my back.\"\n\n\"What the hell did you just say?\" Christopher asked agrily.\n\n\"That's enough,\" Cambria added, stepping between the two men. \"Been enough fighting down below, we don't need it up here too.\"\n\n\"Fair enough, but don't get too comfortable being in charge.\" Christopher said with a grin.\n\nTank held the large sack open as Johnny filled it with haste, pushing his arm onto the shelf and then raking the groceries inside.\n\n\"Seems like you are picking all of the heavy shit.\" Tank said.\n\n\"Getting all of the protein I can find. It will keep us a bit fuller for a bit longer.\" Johnny replied.\n\n\"Shh.\" Dalton added, using one of his arms to calm them to silence as sever Drifters walked by the front door of the small general store.\n\n\"We need to speed this trip up, they seem to be getting thicker outside.\" Dalton said in a whisper.\n\n\"Alright,\" Johnny replied, his voice just as low pitched. \"Tank, you grab as many more groceries as you can carry. I'll look around for anything we can use for an extended stay. Survival gear.\"\n\nTank simply nodded, still steaming a bit over being assigned the group's pack mule. He froze in place, however, as the ringing of a small bell attached to the front door rang out.\n\nA single Drifter had entered the building, leaving the front door open in the process.\n\nJohnny quickly made his way back, a single bag filled with loot and three brand new tents, still folded and sealed, draped across his shoulder.\n\nOnce the Drifter had gotten far enough inside, Dalton sprang from behind to grab the monster around its throat. It tried to bite, tried to fight back, but a quick plunge into the forehead, a knife blade held by Tank, changed the struggle as the Drifter fell limply to the floor.\n\n\"We got to move!\" Dalton said, this time loudly as several Drifters passing by had spotted the killing of one of their own.\n\n\"Might as well use the front door, and don't stop running until we make it back!\" Johnny yelled as the three men began sprinting. Tank, who had never been accused of lacking in the muscle department, hoisted four large sacks filled with groceries with ease.\n\nHis muscles flexed hard, proven by the sudden outline around the bulging in his arms. Yet he was able to keep up with a pistol-wielding Dalton and Johnny as the three men stayed only a dozen or so feet in front of a large crowd of animated flesh.\n\nDalton opened fire, staying to the rear of the group as he executed nearly a half-dozen Zombies at close range.\n\n\"I found an old radio back there,\" Johnny said loudly as the group continued its brisk run. \"It's an older, crank-type model, but it was brand new. Ran across a flare gun too.\"\n\n\"We need every bit of it, good thinking.\" Dalton replied.\n\n\"Should have enough food for weeks if we ration it right.\" Tank added, as the group slowed down, quickly finding every street blocked by a few Drifters.\n\n\"Johnny,\" Dalton said as the three men began to back up into one another. \"What you got in your bag is damn important. You get it back there, ya' hear?\"\n\n\"What are you rambling about?\" Johnny asked.\n\n\"The street to our left runs right back into the Trading Post. I remember the street light,\" Dalton said. \"You and I thin out the bastards blocking that direction, then you run for it. Tank and I will hold up the remaining.\"\n\n\"I 'aint going back without 'ya.\" Johnny said.\n\n\"Yea you are. You have them get a damn rope of some sort ready and waiting for us.\" Dalton replied.\n\n\"Don't have to worry about that, I found a rope back there as well. Two to be exact.\" Johnny replied.\n\n\"Well you get your ass up to that roof and have ropes waiting. We'll be a few steps behind, and coming with a whole lot of bastards on our heels.\" Dalton said.\n\n\"Make it back.\" Johnny said, patting Dalton on the shoulder for a moment before joining the brown coat wearing smuggler in clearing a path. As the last Drifter fell, Johny Edmonds sprinted away, bag of vital equipment over his shoulder.\n\n\"Alright brother, let's dust these 'sumbitches,\" Dalton said, pulling a second pistol out from his belt. \"Just stay behind me.\" he added, shots ringing out a mere moment later.\n\nDropping eight Drifters with his first eight shots, Dalton dropped the empty piece and pulled another from his belt, letting more shots loose in the process.\n\nAnd so continued the killing, the cleansing of life after death by one Dalton James, until his last revolver, a shiny Magnum, clicked empty.\n\nOne remained. A single Drifter remained, coming to the men, slowing only to make its way across the heaping pile of bodies which lay truly dead at the hands of Dalton James.\n\n\"Where are the others?\" Cambria said frantically as the men helped Johnny over the ledge and onto the rooftop. \"Johnny.\" she yelled.\n\n\"They're coming,\" he replied, heavily winded at best. \"They stayed back a ways to cover my flank,\" he added. \"Two ropes in my bag. Get 'em and tie them to the steel pipes up here. Be ready to throw them down in a hurry.\"\n\nCambria stood to her feet, in shock and praying, as the group quickly sifted through Johnny's bag to find the rope. Placing them around the piping system, and re-enforcing the knots to hold, they all patiently waited for sight of Tank and Dalton, as well as whatever horrors may be behind them.\n\n\"Only one of them,\" Dalton said as the Drifter approached, its teeth snarling gruesomely for fresh flesh. \"\"I'll draw him to me, and when I do, you stab that 'sumbitch right in the back of his skull.\"\n\nAs his words closed out the sentence, Tank fired a gunshot that flew by, striking the Drifter in its forehead and killing it stone cold dead.\n\nAs Dalton James turned to find out where the shot came from, he saw the barrel of a solid black pistol bearing down on him.\n\n\"Now let me tell you my version of the smuggler's creed,\" Tank said with a bit of a grin. \"You really think I would come out here without a fucking gun on me? I'm a smuggler!\" he yelled.\n\n\"Yea, I know you are man. Just calm down.\" Dalton replied.\n\n\"Don't tell me to calm down man,\" Tank said. \"You don't know the half of it. I am calm,\" he added. \"The minute I found out you and Christopher fought on different sides of the Glimmerian War, me and him sat down. Had ourselves a real long talk.\"\n\n\"What the fuck are you rambling on about?\" Dalton asked, the sound of Drifters closing in on them becoming louder.\n\n\"You won't get away with this!\" Cambria yelled as Christopher held her at gunpoint. Her own gun, the same one he had, only moments before, used to shoot Whiskey, now reflecting back to her.\n\n\"Shut up you bitch!\" Christopher yelled, a second gun pointed into Johnny's direction. \"I know you're fast with a pistol, but I already got the bead on you. And the sniper over there, just keep your damn eye to the scope. You turn around, just the slightest bit, and I'll end both of them.\" Christopher added.\n\n\"See Dalton,\" Tank said somberly. \"This group of survivors needs to be led by the strong. And while Christopher is up there handling business, it's my job to dust your ass and leave you laying for the Drifters.\"\n\n\"You 'aint got it in you.\" Dalton replied, walking very slowly toward the gunman.\n\n\"Sure I do,\" Tank replied, pulling the slide lock back on his pistol. \"And right about now, Christopher is killing that sack of fleas you travel with. And when I get back, plan on having my way with Cambria.\"\n\n\"I'll fucking kill you!\" Dalton James yelled, Drifters closing in on them quickly.\n\n\"Only one problem with that statement. You'll be dead.\" Tank replied.\n\nAnd with a single shot, the life of a smuggler officially ended.\n\nAs Dalton's eyes fluttered, the cold of concrete drawing warmth from wounded body, the smuggler lifted his head slightly.\n\n\"What happened?\"\n\n\"Luckily, Skulls caught the end of your conversation with Tank,\" Cambria said, her nerves uncontrolled. \"As Tank fired his weapon, he was hit with a shot from our resident sniper. The shot Skulls fired was true, and luckily for you, Tank's swayed off its mark just a bit.\"\n\n\"Don't feel like it was off the mark.\" Dalton replied as pain shot through his extremities.\n\n\"It his your shoulder and passed through,\" Skulls added. \"No doubt it hurts like hell, but you should recover.\"\n\n\"Thanks.\" Dalton replied with sincerity.\n\n\"May want to thank Johnny as well,\" Cambria replied. \"If he hadn't went back over the wall after you, the Drifters would have had you for sure.\" Cambria replied, tears crisp in her eyes.\n\n\"Well, if I am alive, what the hell are 'yall so damn sad about?\" Dalton asked, doing his best to cheer their spirits.\n\n\"Whiskey,\" Skulls replied, filling in as the messenger for Cambria, who had succumb to her emotiions. \"He didn't make it.\"\n\n\"What?\" Dalton demanded to know, instantly forgettting the pain of a flesh wound as he rose to his feet quickly. \"What the fuck are you saying man?\"\n\n\"Christopher shot him,\" Cambria said, trying her best to regain some much-needed composure. \"Planned on shooting us all I think.\"\n\nHer words were received, though no response followed. Just a smuggler who knew loss, kneeling over the dead body of such a faithful dog..\n\nIt was the first time any of the crew had seen Dalton James cry, stroking such rough hands through the long brown fur of the dog's face. Whiskey had been his running mate, and their understanding for each other was hard to explain. Many nights, in the privacy of Dalton's bunk, the smuggler had vented his feelings to the dog. A mutt who seemed to listen with genuine concern.\n\n\"Where is he?\" Dalton asked of Christopher, replacing his overwhelming grief with uncontrollable rage.\n\n\"Dead,\" Cambria replied, doing her best to calm the smuggler. \"When Skulls fired his shot, Christopher had us at gunpoint. As his eyes drifted, Kieth,\" she added, turning to the man. \"Kieth grabbed his arms while Johnny finished him off.\"\n\nNormally, Dalton would have thanked the men. But his grief remained. The loss of such a true friend haunting his every breath of oxygen-ripened air.\n\nFinally, the smuggler knelt one last time, rubbing his hand through the fur of his running mate before removing the dog's goggles. Dalton slowly placed them onto his own head, pulling the banded-leather strapping under his chin.\n\n\"We,\" Dalton said, doing his best to pull things together for the sake of those who survived. \"We need to get our shelters up, figure out what we have. Need to make sure Whiskey didn't die for nothing.\"\n\nCambria simply placed her hand onto his upper back, trying to comfort him while finding comfort for herself in his survival.\n\n\"What should we do with Christopher's body?\" Kieth asked.\n\nDalton swiftly walked to the man, reaching down with a single arm, grabbing Christopher's body and tossing it over the ledge. A motion that forced the smuggler to cringe momentarily for the pain of his gunshot wound.\n\nAs darkness crept in, the survivors huddled around for a well-rounded meal and some plans for the immediate future. Including the child, there were ten of them left, and hundreds of dead below who wanted them badly.\n\n\"Seems like our friends down below are getting more aggressive.\" Johnny remarked, pulling a piece of stringy meat from the bone, and chewing the tender hunk.\n\nThey had built a small fire, using collected lumber from down below and a propane torch Johnny had collected. Each person understood the coming weeks, possibly even longer, would be a much rougher way of living. Still, they all agreed that a single night of luxury was much needed after the previous events. And, if nothing else, they considered it a celebration of the life Whiskey lived.\n\n\"Sumbitches are getting hungrier,\" Dalton replied. \"Fuck 'em\"\n\n\"Still, we do need to put together some sort of plan. A safety net around us while we are forced to remain here.\" Cambria added.\n\nThere was something about her voice. Even though Dalton James sat with Selina by the warmth of their campfire, his eyes remained on the woman who continued to tug at the restraints of his heart. Cambria Sims.\n\n\"Be back girl, I need a little bit of air.\" Dalton said, standing slowly to his feet in order to walk to the opposite end of the building's rooftop.\n\nSlowly he sat down, easing because of the gunshot wound which remained bandaged on his right shoulder. Of course, there was always the fact that a slight stumble would have thrown him to the horde of Zombies below.\n\nDalton glanced up to see Johnny approaching, the light of fire flickering across the seated smuggler's face.\n\n\"So, you really love her huh?\" Johnny asked, sitting down beside him.\n\n\"What you talking about?\" Dalton replied.\n\n\"Come on man. I may be from a planet that's on the low end of technology, but it doesn't make me stupid. Or blind. You and Cambria are both throwing off stares of chemistry that paints a pretty obvious picture.\n\n\"I don't know. And that is the truth,\" Dalton said, turning away for a moment. \"Hard to think about something like that with Zombies screaming below and my running mate dead to the world.\"\n\n\"Well, for what it's worth,\" Johnny said. \"If she won't be with me then I at least want her to be with someone who will take care of her. I'm pretty sure you fit the bill.\"\n\nDalton nodded, his way of showing a mutual respect for the outlaw.\n\n\"Not to get too far ahead or anything, but if you guys need a crew member?\" Johnny asked.\n\n\"Not for me to decide, besides, we got a long wait with the Drifters on our heels.\" Dalton replied.\n\n\"Fair enough. Just thought I'd mention it,\" Johnny said, standing to his feet slowly. \"Cause that train we robbed,\" he added. \"Talking a whole lot of money.\"\n\n\"How much money is a whole lot?\" Dalton asked, feeling as thought the conversation had turned into a job interview of sorts.\n\n\"Enough,\" Johnny replied with a grin. \"Enough to go around and be life changing for everyone.\"\n\nGreat. Dalton thought as he watched Johnny walk away. As if I didn't have enough shit to think about.\n\nMoments later Dalton stood to his feet, turning to rejoin the group and begin a solid plan of safe survival.\n\n\"We rotate in shifts. During the day, Skulls is at the corner with rifle in hand. At night, I'll take his place so he can rest. Daytime, Kieth watches the trapdoor coming up. Nighttime, Johnny takes his post.\" Dalton said.\n\n\"I don't know if I'm the guy to be asking.\" Kieth replied.\n\n\"Come here.\" Dalton said, waving Kieth into his direction. As they walked to the roof's edge, Dalton turned to the man.\n\n\"I appreciate what you did, grabbing Christopher and helping put that sorry 'sumbitch down.\" Dalton said.\n\n\"Anybody would have done the same.\" Kieth replied.\n\n\"No,\" Dalton said. \"No, they wouldn't have.\" he added. \"It takes guts to put it on the line like you did. Guts that a lot of people don't have,\" Dalton said with a pause. \"Cambria told me about you and Christopher having it out. Told me that you said you would have fought against the Legion during the War of Glimmeria.\"\n\n\"That's right.\" Kieth replied.\n\n\"The Legion had us outnumbered fifty to one in that war, but we still gave 'em hell. Wanna know why?\" Dalton asked.\n\nKieth answered with a nod, hanging on the words of the smuggler.\n\n\"Because we were protecting our families and our homes. When you are pushed into that, you tend to fight like hell. Do things you never thought you were capable of. Things like wrestle an armed man to the ground.\" Dalton said with a grin.\n\n\"I guess so.\" Kieth replied with a wide grin of his own.\n\n\"I want you to have this,\" Dalton said, reaching deep into the pit of a pocket on his brown coat. \"It's a Glimmerian coin, specially minted for my unit during the war. Everytime one of us dusted a redcoat, we got one of these,\" Dalton added. \"Truth be told, I pissed all of mine away. Been holding this one as a reminder. But tradition is tradition,\" he said, holding the coin out for Kieth. \"That 'sumbitch more than likely killed plenty of good men. And you did them all a favor when you helped dust his ass.\"\n\n\"Thanks, I don't know what to say?\" Kieth replied, slowly grabbing such a storied piece of war history.\n\n\"Don't say anything. Just know we are fighting a different war now, but the circumstances are similar. This is your family and this patch of roof is home. These fucks down below are trying to take what's yours, and we need you to help us give them hell. Can you do that?\" Dalton replied.\n\n\"I sure can.\" Kieth replied.\n\n\"Good,\" Dalton said, turning to the rest of the survivors. \"Johnny, give this man a piece. Something that can't miss.\"\n\nSeconds later, Dalton's hand snatched hold of a pump-action shotgun, tossed to him by the gunslinger.\n\n\"Pump once, aim in the general direction and get a good look at it. Cause once you fire this bad boy, the sumbitch in your sights is gonna be nothing but a memory.\" Dalton said with a grin, handing the man his weapon.\n\nIt was the simple gestures, things like boosting morale of the common man which really left an imprint on Cambria Sims. And as she watched Dalton joke with Kieth for a few moments, she was even further impressed by his good nature.\n\nSadly, she was not alone. Cambria glanced to Selina only to find the barroom girl also mesmerized by the known as Dalton James.\n\nThey had been on the roof for nearly six days as the sun began to rise. Dalton sat on the roof's corner, Salvation rifle by his side and cigar in mouth. The group still had needed supplies, such as food and water. What they were starting to run out of was hope. Hope that anyone else remained alive in the Drifts. Hope that the Colonial Army would send help, or, for that matter, was even aware of their situation. And Dalton found himself wondering if a growing love for Cambria Sims even mattered. Questioning if they would ever find themselves free of the rooftop to even explore the possibilities.\n\n\"Mind if I sit?\" Cambria asked, shaking the smuggler's calm as he quickly begin to lose his train of thought.\n\n\"No,\" Dalton said, trying to appear calm. \"By all means.\"\n\n\"I was just thinking that we can't win a war of attrition. Seems the Drifters aren't leaving like we had hoped, and another supply run would stretch us thin.\" she said.\n\n\"I agree,\" Dalton replied. \"Just not sure what to do about it.\"\n\n\"Is making a run for it out of the question?\" Cambria asked.\n\n\"Yes, for two reasons,\" he replied. \"We've no idea where to run to and would be carrying a child with us. I couldn't risk that knowing I couldn't guarantee his safety.\"\n\n\"Well, Johnny has said the Bluffs are well supplied and would be easy to defend. Aside from that, it would have to be the terminal and hope our ship is still intact.\" she added.\n\n\"Not a good plan.\" Dalton responded.\n\n\"You, I, Johnny and Skulls could make the trip. Bring our ship back for these people.\" she said\n\"No I can't risk it.\" Dalton replied.\n\n\"Risk what?\" Cambria asked, her tone changing to one of questionably soft.\n\n\"I can't risk anything happening to you out here,\" he replied, turning to look the tender-skinned woman eye to eye. \"I'm willing to go, willing to come back for you. But I can't see you put into harm's way.\" he said.\n\n\"And I want to see you go out there? Is that what you think?\" Cambria asked.\n\n\"It's different,\" he replied. \"My feelings for you are growing by the minute. I can't have a concern for your safety in the back of my mind.\"\n\n\"What about my feelings? You think you're the only one that sees the other and loses an ability to breathe? Do you know how many prayers I said while you were gone on the supply run?\" Cambria asked.\n\n\"Oh really?\" he replied with a smile, as if to brag of his importance.\n\n\"Yes really. I can't even remember what life before you was even like. And now I have to watch you sit with Selina, smile at her jokes and comfort her. And it kills me.\" Cambria said.\n\n\"Well, life before me included a lot less empty bottles, I'm thinking.\" he said with a grin, closing in slowly to kiss the grin of her lips.\n\n\"Cambria, Dalton, come quick!\" Skulls announced loudly as the rest of the group made their way to him as well.\n\nThis guy is forever blowing my game! Dalton thought, grumbling a bit as a kiss with Cambria was cut short.\n\n\"What is it?\" Cambria asked.\n\n\"Picking up a radio transmission from the Royal Army.\" Skulls replied as everyone listened closely to the male voice which transmitted by way of their portable radio.\n\nConstruction continues on Sky City, as we continue to bring in survivors. If you remain alive and well, please do not lose hope. We will get to you with time. The Royal Army is advising against travel as a majority of the countryside is now overrun with Drifters.\n\nThe outbreak of V2 is now thought to be intentional, the strain of virus now being traced back to the Hunter Clans. A Colonial Armada remains in orbit over our planets, strictly prohibiting us to leave until the viral strain is under control. The Colonials, however, do continue to drop supplies to our people. Each silver crate contains weapons, freeze-dried rations, medical kits and a blue flare.\n\nThe following locations are due a Colonial airdrop within the next twenty-four hours: Baltic, Cansoon, Mile View and Geartown. Again, our airships will be looking for blue smoke as a sign of survival.\n\nRemain vigilant, for you are a key part in the survival of our race. I look forward to meeting you soon in our city in the sky, current population, thirteen-thousand and fifty-nine souls.\n\nAnd, as each survivor finished listening to the message, they not only realized that hope existed...but that life as they knew it had officially changed. Forever.\n\nRoman\n\nAnd They were on him. Tu'nak knew well that the Hunters were closing in quickly. He knew not the number, nor the weapons in which they carried; he only knew of their intentions.\n\nThe Hunters were drinkers of mens' souls. Vampires by every definition. They wore Gothic black, carried the most elegant of weaponry, and, above all, thirsted for the salty sting of Human blood.\n\nTu'nak was Human enough, though his structure nearly spoke a different tale. A shade above seven feet in height, the warrior had packed muscle onto his gigantic frame throughout a lifetime. Such meaty flesh was considered a top catch by the Hunters, and that was all the more reason for them to track him down. Slay him and feast on the seasoned frame of the warrior.\n\nYet, they needed no more reasoning aside from the one they carried. Their queen had ordered it. She had demanded Tu'nak, and the remaining warriors who rose against the Vampiric race be hunted down and slaughtered. A group of warriors who answered but to one.\n\nRoman Raines.\n\nRoman had been one of the mightiest warriors ever recorded into history by Human hand, spending an entire lifetime slaying the very Vampires who hunted him.\n\nWhen he met demise, the flicker of a hero's journey coming to a close, the Hunters turned him. He would be a great addition to their cause, becoming one of their mightiest slayers.\n\nHowever, the Hunters underestimated Roman Raines; or rather his hatred for their kind. Indeed, his body had transformed from a dying Human into a warrior of the afterlife. But his objectives remained true. Kill every last Vampire, cleansing Humanity of the bastards forever.\n\nAnd so Tu'nak ran, swiftly and with the guarantee of safety if he were able to make it back. Because the fact was, it didn't matter how many Hunters were on him. Nor did it matter the weapons they carried with them. None would be near the strength in battle as Roman Raines was.\n\nAs the warrior's feet crunched the freshly-fallen snow with haste, his legs continuing a stride of panic; his lungs working hard against the air of frigidness as he finally made it back to their base of operations.\n\nNothing more than a large cabin surrounded by the foliage of trees, blankets of snow covering the round hills which also held the personal shuttle of Roman Raines.\n\n\"Were you followed?\" Roman asked as Tu'nak quickly approached the steps which led to the cabin's entrance, lungs throbbing from such bone-stabbing cold.\n\nRoman wore a coat of fur, one that reached the length of his body and was white with a peppering of black. A single blade hung by his side, one of rugged craftsmanship, but elegant nonetheless.\n\n\"Yes,\" Tu'nak replied, his chest throbbing with the pain of cold air forced in. \"A scouting group.\"\n\n\"Get inside and inform the others. I will join you shortly.\" Roman replied, his voice ringing with a touch of demonic tongue as gray smoke floated up to the clouds, a single campfire to blame. His long hair of white and black nearly blended with the icy surroundings, as did his chalk-white skin.\n\nAnd there he stood, sword in hand as he walked a small circle, slowly facing the thickly-wooded area near camp.\n\n\"You can show yourselves. I smell you. Take arms and fight with honor, our code requires it.\" Roman said loudly, the chill of his voice overpowering that of the air.\n\nWithout warning, a Hunter jumped into sight from the wooded area, the demon holding sword in hand, fully extended and determined to end the life of Roman Raines.\n\nInstead, however, the soldier of immortality quickly became otherwise, the trunk of his body severed in half by a chopping swipe of Roman's blade.\n\nThe two bulks of flesh, both of them completely lifeless, fell to the snow before much of the gushing blood which followed, soaking the ground at the mighty slayers' feet.\n\n\"Go. Run back home with fear and tell your queen of the legend known as Roman Raines!\" he yelled.\n\n\"We will speak to our queen of no such warrior,\" a Hunter replied loudly, a total of two making their way into the clearing. \"My mouth is capable of no such lie.\"\n\n\"Then your lifeless body will speak of it for you.\" Roman said as the two demons began to circle him.\n\nSwordplay was indeed important to the Hunters, looked upon as honorable warfare. But no training they could have completed, no mentor among them, could have prepared the two Vampires for the skill with blade which ran through Roman's veins every moment of every day.\n\nWith just the slightest of feigns, only enough to force one of the beasts into a parry, Roman quickly turned to plunge the bulk of his blade into the chest of the other, unsuspecting demon. Then, knowing well that his enemy was on the door of death, Roman jerked loose his sword, crushing it down onto his other opponent.\n\nThe Vampire, who was recovering from a parry proven unnecessary, only found time to raise his blade to deflect the downswing of such a warrior. And, in most cases, it was a fine move. However, he was fighting Roman Raines on this day.\n\nThe punishing power of the Vampiric slayer of Vampires continued his swing, forcing it through the defending steel and shattering the victim's sword; finding a home in the upper skull of his outmatched foe.\n\nAs Roman forced his blade back out single-handed, bringing with it large fragments of bone, he slowly sheathed it once more while watching the doomed Vampire's lifeless body fall to the ground.\n\n\"We must find another home. The attacks grow more common.\" Gore said as Roman's group of warriors approached at arms.\n\nThe Husk warrior, a race that looked both of Orc and demon descent, had long hair falling from a majority of his large frame. The brown locks that were so common among his race did little to cover such a specimen of muscle, though it was covered in scars. He had seen many battles, and survived to tell of the horrors that had accompanied them.\n\nFor his people hated the Hunters, and their only goal in life was to end as many Vampires as possible; making him a perfect fit for Roman's Empire.\n\n\"I agree,\" Roman replied. \"The scouts they send pose little threat, but may one day be replaced with Hunter Elites.\"\n\nEveryone stood silently at the notion. Hunter Elites were the equivalent of Knights, though they were capable of acts unspeakable. Each had been a slayer of men while alive. So much, in fact, that they had been given the gift of immortality. Roman was one such Knight, or at least it had been planned so. He had been the first to ever resist their influence, and though he seemed unmatched in battle, he would now face the best warriors because of it.\n\nThe best warriors that centuries had spawned. Some kings, others but common men whose legend had grown in battle. Each Hunter Elite was, in his own right, the best warrior of his day. Once turned, they added the Vampiric traits to their own skill in battle. And towering above them all, Vladris.\n\nNo Hunter feared, their genetics would not allow such an emotion. But all Hunters, even Hunter Elites, knew of Vladris.\n\nThey knew of his legend, recognized him to be the greatest of their ranks and respected him because of it. And though he had seen many times of war, ending thousands of lives as both Human and Vampire, he now remained by the side of their queen; Sarah Blaine. Praying for the day that brought a battle which he and Roman Raines both knew was imminent. Their own.\n\n\"Where are we to flee? Draden asked.\n\nThe warrior was of Dragonborn descent. His people were clad in scales rather than skin, his a dark orange. Draden carried a long-blade on his side, and was capable with it. However, his gift was of a defensive nature. His exterior of Dragon hide made him nearly impossible to slay, all but the truest of swings merely deflected.\n\n\"I say we stay and fight the bastards!\" Pica yelled.\n\nThe Human archer was small by definition, his frame only about five and a half feet in height. The longbow that he currently held was about the same. And together, they were mighty. It took nearly everything Pica had to force the bow into a ready position, but the arrow that flew from his weapon of choice had decimating power.\n\n\"I'm glad you feel that way my brother. I have no intentions of fleeing,\" Roman replied, turning his gaze to Draden in the process. \"But we need to leave this cabin and strengthen our ranks before moving forward.\"\n\n\"Moving forward?\" Gore asked.\n\n\"There will come a day not long from now that our fight is taken to the Hunters. To their very doorstep. Their queen will know of my legend,\" Roman replied. \"Come Scucca!\" he added, a Vampiric hound of hell joining his side as the group entered the cabin to work on a thorough strategy. Each warrior taking heed as Scucca could not be trusted, often times lunging after any and all, aside from Roman of course.\n\nSarah Blaine knew of Roman's legend. Not only had word of his slaying made it back to her, but there was a time when both were mortals. Friends even, that had traveled together on the same Gunship crew. A smuggling crew that encountered many dangers along the way, and one in which Roman had defied the odds to save their lives many times over.\n\nSarah knew of Roman's abilities well, and it was the reason behind Vladris remaining at her side. Not that she was afraid of death, not by any means. But because she wanted nothing more than to witness the fall of such a warrior, one that had promised to end her.\n\nAnd though Sarah knew of Roman's legend, she had also started the process of reading the Hunter archives. And with each story she read, came a story of Vladris ending the legend of what Humanity had considered its best. Sarah believed in her heart that Vladris would surely end him.\n\n\"My queen,\" Vladris said, bending low to the floor of her throne room.\n\n\"Rise warrior, for only you are decorated in battle enough to speak to me on the same level.\" Sarah replied.\n\nAs Vladris slowly stood to his feet, his level of respect for her grew. She had replaced a queen who earned nothing of the such from him. She had been a coward, a butcher of her own people. The very reasoning behind Vladris ending her by his own blade.\n\nSarah had done the opposite, giving her race hope while treating them all with a level of respect that none had ever known.\n\nShe wanted to end Humanity, at least the part of it that hunted Vampires, this much was true. But not for the same reasons as those who ruled the Vampire nation before her. She didn't want them slaughtered because of sport; she considered the Hunters to be the most dominant race.\n\nAnd for that very reason, society would either live by their laws of die by their hands. The Hunters were to become the ruling government under her reign, or so she planned.\n\nAs Vladris stood there, a slick-shaven head that was home to several tribal tattoos, he gazed with respect. His pearl white pupils cutting into her with appreciation as a two-handed sword of reaving remained strapped to his back with leather bonding.\n\n\"Thank you my queen,\" he replied, his voice soft and polite by Vampiric standards. \"I have received word that our scouts have been slain.\"\n\n\"As expected.\" Sarah replied.\n\nNormally, during the conversion process, a Human grew worse in appearance. Not Sarah Blaine. She had grown exceptionally more beautiful, if that was even possible. It was almost as if she had been born to become a Vampire, not fully blossoming until her conversion.\n\nBoth her lips and eyes nearly glowed a crimson red, the rest of her skin whitening in a way that brought with it purified beauty. She looked as though she had been hand-carved from ivory, and then accented with the perfect coloring in all of the important places.\n\n\"Should I form a group of Elites?\" Vladris asked.\n\n\"No,\" she said quickly, turning to nod her answer as well. \"Roman will expect such a move and doing so will only bring the loss of mighty warriors to our ranks.\"\n\n\"Shall I track him down myself?\" Vladris asked.\n\n\"No.\" Sarah replied, turning to look out of the large opening which served as a window.\n\nThe Hunters had built what was the equivalent of a castle, inside the confines of a hollowed mountain. For centuries, each queen had lived here, overlooking their homeworld of Ronica through the large opening in the side of their mountain.\n\n\"You do not believe I would prevail against him my queen?\" Vladris asked.\n\n\"I believe nothing less than your victory against him,\" Sarah replied, quickly making her confidence in his abilities known. \"We will need to defeat Roman Raines by out-thinking him. I have a plan, and it involves you doing something completely different.\"\n\n\"Thank you my queen. Speak it and it will be so.\" Vladris replied.\n\nAnd so began the plan of Sarah Blaine, queen of Vampires; a plan that would surely bring an end to Roman Raines.\n\nAs Vladris stood in his own room hours later, built of stone block and vaulted ceilings, he remembered pieces of his mortal life. There was a large door which led to a balcony, and as Vladris walked onto the tiled floor of stone which overlooked a gushing river below the mountain, he remained deep in thought; a curtain of rain falling across their kingdom.\n\nRonica had always been such a beautiful planet. Its warfare being fought with steel, be it sword, arrow tips or axe. Its method of transportation one of horseback, and its climate filled with rain. At least ninety percent of its days filled with the hammering of rain, falling with intention from the heavens above to the soil which its people called home.\n\nThe Ronical Kingdom had been one of the last to fall under the rule of Vampires, staving off the sky-born demons time and time again.\n\nVladris had been one of the mightiest of Knights, slaying hundreds of Vampires over a period of time which spanned nearly a decade. And, as several Hunter Elites finally pinned him down in battle, Vladris remained true to his legend; slaying a few of them as well, before falling himself.\n\nShortly after, The Ronical Kingdom fell. Ronica's people had fought with such courage, such desire for life, that after the war had ended the Hunters allowed them to live. Under Vampire laws and rule, of course, but none were to be harmed without good cause.\n\nVladris had helped earn his people, to some degree, freedom. The Hunters even thought so highly of their champion that they defended the Humans on Ronica as well; placing their castle here and treating the Humans who remained as though they were better than simple cattle.\n\nEvery hard rain that lay before the eyes of Vladris reminded him of his mortal life on Ronica. Reminded him of the King he once served; the woman he once loved.\n\nIn fact, when fighting the Hunters many years ago, it was the thought of his true love which led to such feats on the battlefield. And, even to this very day, Vladris thought of Amelia when locked in battle. Her memory and loss of life fueled Vladris, tapping into an emotional rage with sword in hand that had brought him many victories.\n\nMost across the Skyla System had thought the Hunters to be without emotion. None had ever seen tears flow from the eyes of such a mighty warrior, as they did this very moment. His pain a true love lost forever; forever his death sentence. For he was immortal, cursed to live the remainder of a life never-ending without the warmth of his lover.\n\nAnd so the tears flowed heavily from the eyes of such a mighty soldier, a warrior who gripped the banister of the balcony and prayed for a time when someone could best him in battle, ending such a gripping emotional pain.\n\nRoman and his group of warriors rode hard, their mounted horses pushing forward in the blind of snow. They concerned themselves not with thoughts of bandits, highwaymen or even Hunter groups. For they rode with Roman Raines, arguably the greatest Vampiric warrior in history. Arguably.\n\nBetween that title of legacy and Roman stood Vladris, and in his very soul, Roman knew the day of confrontation was drawing near. Soon the warriors would meet in battle, it was destiny. And though he knew not the outcome, Roman believed on that day Ronica would know with certainty who deserved the title of greatest.\n\n\"Soon the snow will turn to rain. Let us make camp for the night.\" Roman said as the riders began to slow, finally spotting a small patch of trees which would provide cover enough to burn a campfire.\n\nMost notably, Pica was the first to welcome the mention of camp. Though he only carried a longbow, it was one of both exceptional quality and craftsmanship. It remained in such condition because of Pica's caring hands, conditioning the weapon regularly, just as he did on this very night by the light of a campfire.\n\nUsing a small piece of grit paper, the Human slowly brushed across the tough wood of the bow, doing away with any dings it may have recently acquired through use. Then, he brought the longbow back to a shine with a rag which he dipped into a small jar of oil. Clear, almost as if it were alcohol fit for drinking, Pica continued to care for every inch of his weapon.\n\n\"The way you stroke your weapon grows me to wonder of your intentions on this night.\" Gore said as the group burst into laughter.\n\n\"Worry not for me,\" Pica replied with a snarl. \"But for that rust covered blade which rides your back as poorly as you ride your mount.\"\n\nAs the laughter of the group grew, even Roman was forced to break his concentration a bit. He regularly meditated, losing himself into thought and mind as Scucca, his hellhound, stood guard over him.\n\nWith a loud bark which chilled hot tempers, Scucca glanced at the group as if to warn them of their growing noise.\n\n\"This rusty blade has killed many warriors,\" Gore replied, though his eyes remained on Scucca. \"A slew of them holding the very bow that you covet so greatly. The rust will fall from my blade when it finds a new home in bloody flesh, which may be that of a dog should Roman turn a blind eye.\" he added, staring as if to dare Scucca to move on him.\n\n\"And if the blade strike doesn't kill him, the gangrene of rust surely will!\" Draden replied, causing the entire group to burst out loudly.\n\n\"I hear you my brother,\" Gore replied, turning his attention to the Dragonborn warrior turned campfire jester. \"And many of the cowards wielding bows which my blade has slain had scaled-hides as you do.\"\n\nRather than reply, Draden scowled at Gore, defending his people with a stern look. Quickly after, however, the stern look turned to Scucca as well.\n\n\"That damn dog is the devil himself.\" Tunak said, the muscle-bound warrior also eying the guard dog of Roman Raines.\n\n\"Aye,\" Gore said. \"And should the the devil snarl his lips to me one more time, he will lay testament to what my rusted blade can do.\"\n\n\"Save it my brother,\" Tunak said, gently sipping from a metal cup filled with coffee as the light of fire illuminated their faces. \"Not so long from now we will be knee-deep in Vampires. Save what little life your blade has left for them.\"\n\nAnd with that joking insult, the entire group began laughing once more, followed by Draden tossing Gore his oily rag.\n\n\"We'll see whose blade chews more flesh when the dying begins my friends.\" Gore lashed out, though moments later he began shining his blade and ridding the rust.\n\nRoman heard the chatter of his group, and normally would have warned them to be more silenced. But he understood the situation. The Hunters knew they were coming, they must have.\n\nNormally Roman and his group would have encountered several Vampire scouts during their ride. However, they had not encountered a single one. Nothing. Which led him to believe all of the blood-sucking bastards had been called back to the castle to lie in wait.\n\nStrangely, since being turned to the ranks of the undead, Roman not only found the art of meditation relaxing, but felt as though it provided some sort of link to the mind of Vladris.\n\nHe knew not of the location of Vladris, or what thoughts crept into the warrior's mind; but rather a feeling of connection. Roman had often wondered if the same feeling was had by all Hunters, or just himself. He had promised himself that should he ever take a Vampire alive, he'd get the answer to his question. Unfortunately, no day had yet come. He simply enjoyed killing them too much.\n\nRoman also wondered of Scucca. What had spawned such loyalty from the dog, at first sight no less. The Vampiric hound of hell had partnered itself with Roman from the very first encounter, and he was still unsure why. As badly as Roman Raines wanted to watch the Hunters' castle burn to the ground, he felt an overwhelming need to visit their archives first. He wanted to learn as much about the race which he was forced into, because, like it or not; he was indeed a Vampire.\n\nThe battle of Callian saw thousands fall, hundreds of thousands even. The Husk allied themselves with what remained of the Ronical Kingdom, brining their massive blades into battle.\n\nAnd though the damned green-skinned Husk were born to slay us, still we pushed them back. Our Elites were killing Husk Tribals, their warrior of warriors, at a ratio of three-to-one.\n\nAs our great army began to crush what few Husk which remained, an unknown warrior arrived with but twelve knights. And though our enemy's force remained below a hundred warriors, standing against thousands of our own, the warrior led them to victory.\n\nOur own people are beginning to call them the twelve angels. They are whispering that the human warrior is a God among men. Could it be so? Could a God walk among them and protect such a beautiful kingdom?\n\nOur queen has announced such talk is to be considered treason, and that anyone guilty will be led to the gallows. Still, secretive chatter continues among our own ranks.\n\nAs Sarah continued to read the Hunter archives, she grew to understand the power in battle which Vladris possessed. Having him nearby made her feel completely safe, as if the Gods did not have power to harm her. For such a warrior to have been turned, giving him the strength and speed of an immortal, Sarah truly believed it was the Hunters' destiny to rule civilization.\n\n\"We approach Marlock,\" Roman said, his chilled voice only adding to the vivid mist which escaped his lips on such a cold morning. Adding a bit more eeriness to the sight of a hound spawned of hell running beside them.\n\nThe snow had subsided, leaving behind drifts deep with powder and a cold that hurt to the bone. Roman had known of Marlock throughout his lifetime of hatred for the Hunters. And as legend had spoken of it, he had imagined a great city, its buildings filled with warriors who remained true to their fight against the Vampires. A great city marked by the hanging of red skulls from limbs of surrounding trees. A warning to any Hunters who might approach.\n\nIt soon became evident to Roman Raines that the legends spoke of a lie, at least in the description of Marlock. It was no city, not in the least. Merely a village of only a few dozen dwellings, each made of thick logs and cemented mud.\n\nHe hoped that the rest of the Marlock legend held true. It spoke of an army so mighty, so in love with the bloodlust of Vampires, that the Hunters could not remove them from their city. Or in this case, small village.\n\n\"Let us hope the warriors put up a better fight than the beauty of their home.\" Gore said loudly as the group slowed to a trot.\n\n\"Or that their women are easier on the eyes.\" Tunak added, prompting laughter from most.\n\n\"Silence,\" Roman demanded, turning to his group for a moment. \"A warrior can never be judged by his surroundings. To do so is unwise. The mightiest I have ever had the honor of doing battle with, or against, came from such conditions,\" Roman added. \"For these men know of no comfort, they know of no luxury. They only know of pain, which makes for a dangerous warrior indeed. A grizzly among men.\"\n\n\"A grizzly growls in my stomach,\" Pica said, his bow strung to the back of his thick coat. \"Let's hope these dangerous warriors know of food.\" he added, the entire group beginning to laugh once more.\n\n\"Halt!\" a man cried, walking from behind a nearby tree with crossbow in hand.\n\nThe stranger looked impoverished, as though he had not seen the basic Human comforts in many days. Still he stood there, his crossbow aimed directly into the direction of Roman Raines.\n\n\"What business do you have here?\" the man asked, as nearly a dozen more men walked from surrounding trees, each carrying either a large sword or man-slaying axe.\n\n\"We slay Vampires, that is our business. Have we arrived among friends of the same purpose, or do the legends of such a city merely speak a lie.\" Roman responded, turning to face the crossbow wielding man.\n\n\"Liar! You are a Hunter yourself!\" the man yelled.\n\n\"Indeed,\" Roman said calmly, turning his head a bit to view the remaining men as the snow steadily began to fall once more. \"But I am of no threat to you. If it were the case, you and your friends would already lay in a pool of your own fluids.\"\n\nThe man continued to look at Roman, into the eyes of a warrior who knew of life after death, yet traveled with men.\n\n\"You know it to be the truth,\" Roman said. \"We wish to eat, rest and then talk of an assault against the Hunters.\"\n\n\"You lie! To speak of such is to speak of suicide.\" one of the other men cried out, his grip holding a one-handed axe.\n\n\"To speak of suicide would be for another man to call me out as a liar.\" Roman warned.\n\nAs they continued to look Roman over, thick white hair braided down to the warrior's lower back, eyes nearly glowing as hot coals would, the man who held a crossbow nodded a bit.\n\n\"You are the one everyone speaks of. The demon which hunts demons, are you not?\" the man asked.\n\n\"I am the demon who hunts Vladris. I seek to end his reign of terror, slaying with him the legend which follows. The rest of the dead Vampires laying in my wake have simply gotten in the way.\" Roman said sternly, moments later grinning a bit.\n\n\"Then yes, the tales of Marlock are true,\" the man said, returning the grin. \"Take them in at once! They are heroes among us!\"\n\nAnd with that short, but fate-altering conversation, the warriors led Roman and his group into Marlock. Food they would have on this night, only the finest Marlock had to offer. For Roman's legend was growing across Roncia, having already reached the ears of such a secluded village and its warriors.\n\n\"Bring him!\" Sarah yelled, her voice anything but demonic as the Vampire queen sat in a luxurious throne of jewels.\n\nAs Vladris entered her throne room, carrying with him a young man with bound hands, he tossed the baggage down abruptly.\n\n\"Enough! No harm is to come to this prisoner,\" Sarah said loudly, warning her finest Hunter. \"If one of our own but splits a single hair on his head, you had better slay them in my stead.\"\n\n\"Yes my queen.\" Vladris replied, not quite understanding her fascination with the young man, but daring not question her.\n\n\"You will not be harmed. You have my word.\" Sarah replied with a smile.\n\nAnd though the prisoner's face, which remained under the veil of a sackcloth bag, couldn't see the queen of Hunters; her voice sounded familiar to him. Almost soothing.\n\n\"And that's when Roman sprung from the trees and together, we slew nearly a dozen of the bastards!\" Gore said loudly, slamming a steel mug to the table.\n\nIt was filled with a warmed brew. A drink similar to what humans considered ale, though it contained several herbal extracts as well. The people of Marlock considered it to have both healing and relaxation qualities; not to mention it would warm the bones of even the mightiest of warriors.\n\n\"Your numbers are far less than I had expected.\" Roman said with seriousness as the entire group, nearly thirty strong, looked back at him.\n\n\"We were once very formidable, just as your legend speaks. But years of war against an enemy who outnumbers us so greatly has taken its toll.\" Anthony replied. He had been the first to confront Roman's group hours before, crossbow in hand and served as leader of their people.\n\n\"And this other group of warriors you speak of?\" Roman asked.\n\n\"Indeed,\" Anthony replied. \"They are out on a supply raid and are not expected to return for the duration of four more days.\"\n\n\"Four cold days.\" Pica said, causing a chuckling-stir throughout the group.\n\n\"We realized the Hunters hate cold. They will enter snowy fields to do battle as long as they understand it is temporary. So we dug in here,\" a large warrior replied. His hair was a bit short, and fiery red at first glance. His faced was marred with scars of war, his eyes further speaking of their woes. \"They know we are here. Yet they have been unable to break us in a single battle...and unwilling to stay for more.\"\n\n\"This is my first in command, Bral,\" Anthony said. \"One of the finest men you will ever meet, and even more so when it comes to combat.\"\n\n\"Then it is indeed an honor.\" Roman replied, nodding to such a highly of spoken warrior.\n\n\"I do not understand your travels with the Vampires' dog if you are indeed a slayer of the undead?\" a man asked, speaking loudly as a two-handed sword lay snug to his back, strapped with leather bonding.\n\n\"Four!\" Anthony said loudly, scolding his warrior.\n\n\"It is alright,\" Roman said, staring hard at the sword-strapped man. \"I myself do not understand it. From what I have learned along the way, the hellhounds marry themselves to the strongest of warriors. Scucca was by the side of Vladris when we first met, yet he left with me.\"\n\n\"Then you truly are a greater warrior than Vladris? If it is true, we may very well turn the tides of war!\" Four replied.\n\n\"Roman, I apologize. Four speaks out of turn.\" Anthony added.\n\n\"I understand his questions, and have one of my own. Why Four?\" Roman asked.\n\n\"Because, no matter who we face on the battlefield, he fights as well as their four best warriors.\" Anthony replied.\n\n\"Interesting.\" Roman said.\n\n\"So our plan?\" Bral asked, always the one for strategy when it came to battle.\n\n\"Spread the rumor of a great assault against the town of Bainson,\" Roman said. \"The Hunters will deplete their ranks to seek such a battle and defend the town. Then, we strike their castle with our best men. It will be defended, but they should not be able to hold us. We will reach their queen long before the Hunters dispatched to Bainson return.\"\n\n\"It is a solid plan. There are many humans who worship Vampires in Bainson.\" Bral replied.\n\n\"First we must rest for the night and finish filling our bellies.\" Roman said.\n\n\"Agreed, my friend.\" Anthony replied, extending his hand to the Vampire which stalked his own kind. And with their handshake came a treaty of death to come.\n\nSarah's favorite place throughout the castle wasn't the room that held her throne, which had been the favorite of so many before her. Instead, she found herself drawn to the room of archives. She felt compelled to find out as much about the Hunters as possible.\n\nAnd on this great day, the day of our Lord 4072, our army of undead faced its toughest. A small army on the planet of Ronica, led by their King and a warrior among their finest. A warrior which fights like no other our species has ever encountered. A warrior who has slain many of our kind, including four Elite Knights dispatched by our very queen.\n\nPerhaps we have underestimated the will of humankind. The resolve of such an outmatched opponent. They must know of their inevitable demise. They must see the death and destruction around them. Yet this warrior of light gives them hope. Strength. Whatever the cost, whatever measures must be taken, this warrior must be ended if Ronica is to fall under our rule.\n\nSarah continued to read through books which were inked by the hand of Vampires hundreds, possibly even thousands of years before her eyes skimmed the pages of the moment.\n\nToday our species did the unthinkable. Retreat from battle. With our numbers strong and our greatest Elite, Graddon, on the battlefield, swift victory was sure to follow.\n\nYet we were bested, Graddon falling gruesomely to this knight of light which wields a longsword. He is the one our own people have spoken of as an angel. Our queen has called him a glorified farmer, yet he has, by his own hands, killed hundreds of our kin. Members of our own species have even begun speaking of him as the devil, if not the angel he appears to be. The very devil! They say he is un-slayable, though speaking of a human in such a way is considered treason.\n\nSarah stopped ubruptly.\n\n\"My queen.\" Vladris said.\n\n\"What is it my champion?\" the pale white display of beauty replied, a corset snugly fit around curves of perfection.\n\n\"The prisoner is under watch and being treated kind, just as you instructed.\" Vladris replied.\n\n\"Thank you my champion,\" Sarah said with pause. \"I will soon explain to you my reasoning, as well as our plan.\"\n\n\"You need not explain a thing to me, my queen.\" Vladris responded.\n\n\"A warrior of such valor deserves an explanation, even from his queen.\" Sarah said with both admiration and intrigue.\n\n\"Thank you my queen.\" he replied, nodding slightly before turning to leave the doorway.\n\nSarah wanted to know more. More about their race, and just as importantly, more about her champion warrior. Skimming through the pages of the Hunter Archives, she saw talk of a split in their ranks, the coming of a second queen and a war that would see the two Hunter factions fighting each other. Yet, none of it concerned her.\n\nAs she reached the next entry dedicated to the knight of light, her eyes began to focus with concentration.\n\nToday our queen executed dozens of our own species. Each of them guilty for speaking of this knight of light, Vladris, in such a God-like fashion. He slew Graddon, our very best, and did so with such ease. Our next move was to trap his caravan away from the confines of such a beautiful castle, slay him by the roadside. Queen Vivian dispatched her three finest Elites, a sure death sentence for any Human warrior. Yet, Vladris lives.\n\nOur three Elites fell in battle, along with a dozen Hunters as only two arrived back to speak of the battle. They brought with them a prisoner. A woman named Amelia. She seems to have the affections of Vladris, which our queen feels could be the achilles heel of such a mighty warrior.\n\nOnly time will tell of what is to become of her. But, as of this very moment, they are doing things to her. Unspeakable things. Many of our own warriors have reported her screams from behind the wooden doors in which she is being held.\n\nSarah stopped at that moment, having tried to imagine a warrior such as Vladris dealing with the loss of love. She too had lost her true love in Adam Michaels, feeling condemned, even cursed, with immortality. Forever to think about a love never realized; to grieve for the one her heart longed for.\n\nVladris had dealt with this very pain for many years now. And it was a trait that made him even more honorable in her eyes. The strongest warrior among her people growing even stronger in the affections of his queen.\n\n\"Non-sense! There can be no such victory, and to speak of it is insanity!\" the warrior cried out.\n\n\"Silence Ranthra, you speak out of turn!\" Anthony replied loudly, trying to lull the large warrior.\n\nHe was outfitted in brown fur, beneath which, rested a hide of leather that was finely stitched. The hulking warrior's head shaved clean with a scar protruding from the top of his skull to the outer edges of his top lip.\n\n\"I speak the truth,\" Ranthra replied with emotion. \"To march us onto the doorstep of the Hunters is a death sentence to all who would be foolish enough to follow you.\"\n\n\"Careful with your words Ranthra.\" Anthony said in a cautioning fashion.\n\n\"Your courage I do not question, for I have seen it many times over,\" Ranthra replied, turning to Roman. \"It is the courage and abilities of this man. The man who claims himself better than Vladris.\"\n\n\"Not a believer?\" Roman asked calmly.\n\n\"I have seen Vladris fight, even clashed my own sword against his. I have the very scar upon my face to show for it. Luckily, my life as well. Vladris cannot be beaten!\" Ranthra exclaimed.\n\n\"Not by a warrior who walks the path of a coward.\" Roman replied.\n\n\"I walk no such path, and you are in no position to question that statement.\" Ranthra said, drawing his sword and placing its tip to the Hunter Elite's throat.\n\n\"Enough Ranthra!\" Anthony yelled, standing to his feet.\n\n\"No, I want to see what this slayer of Vampires is truly capable of! Vladris went through my group of warriors, six of us in total! If he is as mighty as he speaks, let him prove it right here!\"\n\n\"Only six?\" Roman asked, a bit of sarcasm on his words as he remained sitting for a moment.\n\nThen, without a hint of warning, Roman stood to his feet and spun with inhuman speed, smashing Ranthra's head through the table before them. With the powder of wood filling the air, Roman maintained his grip on the back of Ranthra's neck, pulling him back to his feet abruptly before tossing the bewildered warrior against a nearby wall.\n\nAs Ranthra slid to a sitting position, nearly unconscious, Roman met his next challenge head on. The Vampiric warrior clutched the throat of a second warrior who rushed, applying enough pressure for the man to drop his axe, at which time Roman kicked him square to the chest, knocking him back against the opposite wall.\n\nA third warrior joined the fray, dagger extended fully in an attempt to end Roman's growing legend. Instead, his journey was cut short as Roman grabbed the warrior's wrist, his firm grip snapping the man's wrist as though it were but a dry twig. A backhand sent the reeling warrior to the floor with a thud, Roman having secured his dagger in the process.\n\nThrowing it end over end with surgical precision, the dagger dug into the wall behind a fourth warrior, merely inches from the man's face.\n\nIt was a warning, and Roman made damn sure it was a warning well received as he all but dared the warrior to continue with the gaze of his eyes.\n\n\"Six warriors, six-hundred warriors. It does not matter. I will have my fight with Vladris before whatever God you pray to. Any man who thinks me unworthy of ending the demon of demons, let him speak now.\" Roman said, a harsh truth blanketing his words.\n\nNo man dared step forward, the Hunter Elite having bested many of them in a matter of seconds. At that moment each one of them believed Roman would be the warrior to end Vladris. Every one except for Ranthra, who had been the only warrior who could claim an exchange with both legends of battle. In his mind, the battle could go either way. He was unsure of the outcome, but damn sure that he was officially in on the plan to reunite Roman and Vladris in battle.\n\nVladris stood on the balcony which overlooked the bursting water of a river below. Many things raced through the mighty warrior's head as he continued his stare onto the horizon at distant. Rolling hills littered with trees, all seemingly so full of life.\n\n\"His name is Troy.\" Sarah said, slowly entering the quarters of Vladris.\n\n\"My queen.\" Vladris replied with a slight bow.\n\n\"In my travels with Roman Raines, my human travels, Troy was taken in by our crew,\" she said. \"I am very fond of the boy, but my fondness is minascule compared to Roman's.\"\n\n\"You plan to use the prisoner as leverage?\" Vladris asked.\n\n\"I plan to turn Roman to our cause. I intend to force his hand by making him choose between total submission to our will or the death of Troy, because I know he will not allow the boy to die.\" Sarah replied.\n\n\"But he is a Vampire now. His affections for the boy may not remain.\" Vladris said.\n\n\"They will remain, as does the hatred for our own species. He will be forced to choose to accept his place among us or the death of a young man he thinks of as a son. And he will choose the safety of Troy.\" Sarah responded.\n\n\"And if he does not?\" Vladris asked.\n\n\"Then you will have your chance to prove the better warrior.\" Sarah replied.\n\nVladris longed for the battle. He knew that some of his own race had begun chattering of a demon who possibly could best him. Every chance Vladris had to solidify his own legend, was a chance he yearned for.\n\n\"Speak to me of Amelia.\" Sarah asked.\n\nVladris turned away from her, once again facing the balcony and view of the rolling hills curtained with rain. It was thought to be an insult to turn away from their queen, though he thought the outcome worse had his tears been discovered.\n\n\"My queen?\" Vladris finally responded, masking his overwhelming emotion. The very mention of her name sending shock waves throughout his entire body.\n\n\"My champion, your show of affection for her is not a weakness. It is a strength,\" she said, approaching him easily. \"Turn to face me. Tell me of your method in dealing with love lost, for I am cursed with the same fate.\"\n\n\"Killing,\" Vladris said bluntly as Sarah's fingers comforted the face of such an agonized soldier. \"When I am on the battlefield, sinking blade to flesh and cleaving apologies from enemies, my mind is not concerned with Amelia.\"\n\n\"Tell me of her. It speaks of Amelia in the archives, but briefly. Tell me of the type of woman she was.\" Sarah said.\n\n\"She was perfect. I have never met a woman who brought me more happiness with her mere presence. Everything about her made me better, and I do apologize for my weakness with emotion my queen, but think not that I am weak on the battlefield.\" Vladris said.\n\n\"I know only the opposite. The archives speak in great detail of the warriors who have faced you, all of them falling swiftly.\" Sarah said as they stood only a few feet apart.\n\n\"It also speaks of greater days. A time when the Hunters were a single society, led by both queen and king.\" Sarah said, the look of intentional seduction in her eyes.\n\n\"My queen?\" Vladris said.\n\n\"Become our king. Rule at my side, and let us unify every Hunter to our cause. I am no Amelia, but I may provide a comfort from her constant memory, and you from the constant memory of my own love lost.\" Sarah said.\n\nVladris stood there. At first puzzled, though he soon began to realize the potential behind such a marriage. Maybe Sarah could help him forget, at least to a degree, his Amelia. Together they could rule without their own personal struggles, and do so in striking fashion.\n\n\"My mind must be clear when facing Roman Raines, be it in battle or brotherhood. Once he is either with us, or in an eternal grave, then I will gladly become your king.\" Vladris replied.\n\n\"Good,\" Sarah said, smiling as wide as her powder-white face would allow. \"For these people already look to you as their king. And I look to you as my champion.\" she added, leaning in to kiss the warrior who would help elevate their species back into one of total dominance.\n\n\"I have not seen snowfall this thick in many years.\" Tunak said, several of the warriors sitting inside the cabin, watching the frigid moisture fall to the ground.\n\n\"It is a sign,\" Four said. \"Roman was born to end the life of Vladris!\"\n\n\"Perhaps,\" Ranthra said loudly, doing his best to overcome their loud chants. \"Or perhaps Roman has severely underestimated his enemy.\"\n\nNo chanting followed the statement, only silence as everyone stopped to hear his reasoning.\n\n\"I mean no disrespect to Roman,\" he said, both Tunak and Gore acknowledging his honesty. \"But I may claim to be the only warrior among us to have faced both men in battle, albeit shortly. Vladris moves unlike any man, or monster, that I have ever faced. No legend, besides his very own, tells a tale of such ability to slay.\" Ranthra said.\n\n\"You have yet to see Roman's true ability to slay. For I was with him when he had but a single blade; yet five Hunters still fell. Five!\" Tunak shouted.\n\n\"I do not doubt your claim warrior,\" Ranthra said, easing the tense of their exchange. \"But I ask you this. If Roman is a Vampire, it means he did indeed fall in battle. Who then? Who was the warrior to slay him?\" Ranthra asked.\n\n\"Vladris.\" Roman replied, shutting the door abruptly, his body-length coat of fur covered with snow.\n\n\"Vladris?\" Tunak asked, the entire group stunned with the information.\n\n\"Indeed,\" Roman calmly replied, his stare cutting into Ranthra. \"A better question may be who introduced Vladris to his death?\"\n\n\"I do not know for sure, and think no man does. You would have to dig deep into recorded history to find the answer you seek.\" Ranthra replied.\n\n\"Or dig my blade deep into his own flesh, for Vladris knows the answer. Vladris saw the face of his slayer, much like he will see my own face.\" Roman responded.\n\n\"I think it not important at the moment either way,\" Anthony said, following Roman inside and seeking a source of heat to defeat the chill of his bones. \"This snow will hinder our travels greatly. We would be wise to remain here until the sky breaks for our long journey ahead.\"\n\n\"No,\" Roman said, his voice unwaivering. \"I leave as scheduled. Any man who wishes to remain behind is free to do so,\" he added, the blaze red of his eyes brightening a bit. \"But he who seeks to become legend will accompany me, for Vladris will fall by my sword. I swear of it.\"\n\nAnd the group of warriors remained silent, each man contemplating his own destiny. Wondering what the outcome would be when the two legends finally collided. Anyone accompanying Roman would surely be slain if Roman fell, but if he was successful in bringing down the mighty Vladris? If indeed he possessed the skill to slay the demon of demons? Roman, as well as any warrior who stood with him, would forever be considered legend across Ronica. The dream of every true warrior.\n\n\"You would be wise to hold your tongue to Roman.\" Anthony said, walking quickly to catch up with Ranthra. A few warriors had left the cozy confines of the small lodge to seek good-burning wood for its fireplace.\n\n\"And you would be wise to remain here, as I intend to,\" Ranthra replied, turning to face their leader by committee. \"I do not doubt the abilities of Roman in combat, as everyone suspects. But I do not doubt the abilities of Vladris either. The legends hold true, for I have seen it with my own eyes. To follow Roman into a battle with the demon of demons is to throw yourself on your own sword. All while saving yourself the journey of days by horseback. I'll not ride hard to a rematch of swords which saw Roman fall during the first battle.\"\n\n\"So your choice has already been made?\" Anthony asked, bitter cold masking his voice with droplets of fog.\n\n\"My mind was made the moment I saw Vladris first swing on the battlefield. The Gods spared my life on that very day, and I will not spit in their face by marching into the same scenario again.\" Ranthra replied.\n\nAnthony shook his head for a moment, silently, before turning to begin a walk toward his small home in Marlock.\n\n\"So now you think me a coward?\" Ranthra asked.\n\n\"No my friend,\" Anthony replied, turning to face him once more. \"I have seen your bravery in battle, time and time again. No, I begin to question the definition of courage at this moment, and when it becomes ignorance.\"\n\n\"Go my friend, for you have much to think about on this night.\" Ranthra replied.\n\n\"Are they treating you well?\" Sarah asked, slowly entering the room in which Troy was being held.\n\nTheir castle was complete, right down to a full-size dungeon. Yet Troy had been placed in a room only yards away from Sarah's very own. One normally reserved for important guests among their kind.\n\n\"Yes.\" Troy replied, wanting to hate Sarah for being a Vampire, while also remembering the respect he held for her human memory.\n\n\"I have no intentions of harming you,\" Sarah said, walking closer to him. \"Leave us.\" she added, turning to the two Hunters who had been assigned to Troy's door.\n\n\"I have no intentions of harming Roman either.\" she said.\n\n\"Roman? He's here?\" Troy asked with enthusiasm.\n\n\"Not yet,\" Sarah replied. \"But we expect to see him soon enough. I intend to ask him to remain here,\" she added. \"And there is a place for you as well.\"\n\n\"Roman Raines will never join you! He despises your kind, as do I! Hunters killed my family in front of me, and you ask me to remain here to live among them?\" Troy asked, his voice raising in the process.\n\nIt was the first time Sarah had seen the boy's temper. She remembered holding him as Hunters closed in, remembered calming his fears as Adam Michaels and the Gunship crew protected them both against certain death.\n\n\"I think with time Roman will come around.\" Sarah said with a smile, brilliant white teeth cresting across such an angelic face.\n\n\"I think you're full of shit!\" Troy yelled, prompting the guards to come back into the room. \"I think you will soon find out how mortal you truly are, at the tip of Roman's blade...you bitch!\"\n\nAs the two Hunters approached, Troy decided it time to show his hard-earned skill. Kicking swiftly toward the floor, his foot caught one of the Hunters above its knee, instantly breaking the femur bone and sending the soldier to the ground in pain.\n\nThe second guard was met head on, Troy grabbing the monster's wrist. Of course, a young man barely eighteen years of age, he hadn't the ability to overpower the beast. Mythra. The art of stick-fighting, based around momentum-shifting. Troy had studied the art since losing his family, at first to occupy himself. However, he had quickly mastered the art and it displayed at this very moment.\n\nLetting loose the Vampire's wrist at the perfect moment, Troy used the monster's own momentum, allowing it to slam onto the ground. And as the guard began to get back to its feet, Troy's swift kicks broke the beast's forearm in several places.\n\nThe same foot kicked the base of a nearby table, loosing a wooden leg which was quickly kicked into one of his free hands.\n\n\"Time for me to take my leave you Vampiric bitch!\" Troy yelled, pointing the table leg into her face.\n\nShocked, Sarah simply walked back a bit. She had never seen Troy as anything other than a feeble child. Yet here he stood, two of her mighty warriors laying defenseless and her own safety in jepoardy.\n\nVladris slowly walked into the room, Troy's attention shifting to the large warrior.\n\n\"Back my queen, I will handle this.\"\n\nAnd step back Sarah did, knowing well that she had underestimated Troy this entire time. She also began to wonder if Vladris facing Roman was wise. Had she become too comfy in her new role? Had she underestimated Roman Raines as well?\n\nTroy's art of fighting allowed him to take advantage of larger opponents, and his mind knew that could prove disaster for Vladris, for he was a much larger warrior.\n\n\"Well come on.\" Troy said provokingly, continuing to hold the shattered wooden leg of a table in the Hunter's direction. He grew impatient, tired of waiting for Vladris to enter the fight with a mistake.\n\nWith the speed of the very rays of a sun, Vladris thrust his arm forward, wrapping Troy's attacking hand and squeezing until the wooden leg fell to the room's floor. The demon's freehand then plunged in, clawed fingers gripping Troy by the throat and forcing him to the wall, his feet leaving ground in the process.\n\n\"Vladris!\" Sarah yelled, continuing to want no harm to come to the young man.\n\n\"I have left very few lives to walk away,\" Vladris said harshly, his large eyes brimming with both red and black pigment. \"Know now that if it were not for a queen's command and a lover's wish, your lungs would plead for air as I ripped your head from the scrawny shoulders which held it.\"\n\nAnd with that statement of caution, Vladris loosened his grip of tenacity, letting Troy stand on his own feet once more.\n\n\"Are you alright my queen?\" Vladris asked.\n\n\"I am fine my champion.\" Sarah replied.\n\n\"What happened to you?\" Troy demandingly asked as he began to breathe easier. \"You used to hate the Hunters. I was there when your own father, the same father who fought against the Hunters, died in an explosion of flame!\"\n\n\"Yes, that was unfortunate,\" Sarah replied. \"But those were different times,\" she added. \"Different times indeed.\"\n\nSarah turned to Vladris for a moment while her two Hunters slowly gathered themselves from the floor. This was her species now, and it was her job to protect them at all costs.\n\n\"Change of plans,\" Sarah said firmly. \"If Roman denies his place by our side, Troy will indeed die. In front of the warrior.\"\n\n\"Understood.\" Vladris said with a grin.\n\n\"And double the security detail on Troy's door.\" she added.\n\n\"At once.\" Vladris said.\n\nThe morning light of Marlock brought with it more snow, falling powdery-white onto the ground without reserve. Roman sat on a fine horse, a steed which had been outfitted with the toughest armor available, at least by such a secluded group.\n\nTunak and Gore quickly joined him, as did the rest of his group, slowly galloping their own steeds to his piece of ground.\n\n\"Draden, Pica. You are to remain here. Should I fall, these people need someone to continue the hunt.\" Roman replied.\n\n\"Roman,\" Pica said, his large bow once again strapped to the upper of his back. \"We want to fight.\"\n\n\"A fact I both know and respect,\" Roman replied. \"But you know the location of my ship. You will be able to care for my dog,\" he added, glancing down at Scucca. \"You will succeed me should I fall.\" he added.\n\n\"We understand,\" Draden said, the brilliant-white light of snow refracting from his Dragon-scaled hide. \"And we wish you luck warrior.\"\n\n\"I make my own luck.\" Roman replied, thinking back to a time when his human flesh leaned on the spoken line heavily.\n\n\"And what say you?\" Tunak said loudly, his voice echoing toward the group of warriors led by Anthony. \"Who is ready to accompany us to the fall of the mighty Vladris?\"\n\nFour climbed to his mount quickly, the peppered-gray horse then trotting to Roman's group.\n\n\"That is all? One warrior?\" Tunak asked loudly.\n\n\"Relax my brother, it takes but one to slay Vladris and end his legend of lies.\" Roman said, nodding his respect to Four.\n\n\"Four is as capable as any that I have. He will serve you well in battle, and I will personally see to the proper care of your hound and warriors.\" Anthony replied.\n\n\"Gratitude my friend,\" Roman said, demonic eyes skimming the group a final time. \"I will return in one week's time with the head of a legend so easily spoken,\" he added. \"If not, continue my legacy and bring their race to its knees.\"\n\n\"Either way, it will be done. You have my word.\" Draden replied.\n\nAnd with that final exchange of words, Roman, Tunak, Gore and Four would turn to head away. Disappearing quickly into the thick of snowfall, well-supplied and well-intentioned.\n\n\"Sir, we have just received word of Roman Raines departing Marlock for us.\" a Hunter said, entering the lavish quarters of Vladris.\n\n\"Good. Let him come.\" Vladris replied.\n\n\"Should I double the guard?\" the Hunter asked.\n\n\"You have made me aware, which is as good as doubling the guard. So no, it is not necessary.\"\n\n\"And the queen, sir? Should I attach a personal escort team with her?\" the Hunter asked.\n\n\"I am her personal escort team!\" Vladris yelled, approaching the intimidated warrior of death. \"Do you not think me capable of slaying Roman Raines?\" Vladris asked.\n\n\"Of course sir,\" the warrior replied. \"There are those among our ranks that whisper of his ability in combat, but I am not one such warrior.\"\n\n\"Whispering is the work of a coward. Any warrior who believes in something so strongly, would proudly announce his belief. To whisper is to be weak.\" Vladris responded sharply.\n\n\"Yes sir.\" the Hunter said.\n\n\"You may go. Inform the others I will be speaking shortly.\" Vladris commanded.\n\n\"Yes sir.\" the Hunter warrior replied, turning to execute an order as the black leather of his boots struck loudly onto the stone-tiled floor\n\n\"Forgive them. They do not see you as I do, for they have not seen your history of battle.\" Sarah said, approaching her new love slowly.\n\n\"My queen.\" Vladris said, kneeling slightly in order to show his respect.\n\n\"Many among our ranks have heard of your legend, but have not seen it with their own eyes.\" Sarah said, wrapping her arms around the neck of the capable warrior.\n\n\"Their eyes will learn of it soon enough.\" Vladris said.\n\n\"I will offer him a place at our side, and I am a woman of my word. Please know that if he accepts it, he is not to be harmed.\" Sarah said.\n\n\"I would expect no less from you my queen. But I have seen into this warrior's eyes. He will not submit to our will, and you should expect no less from him.\" Vladris replied.\n\n\"Either way, he will be given a choice. And be it an end to his reign of terror against our race, or an end to the breath in his lungs, this will soon be behind us my champion.\" Sarah said.\n\n\"Yes my queen.\" Vladris replied.\n\n\"I have no idea? He has gone mad!\" one of the Hunters who stood in a watch house on the castle wall said, speaking to another.\n\n\"Surely he knows of Roman's ability in battle? Surely he would rather be enjoying a fine meal this night?\" the other guard remarked as they watched him. Their hero among heroes, sitting outside of the castle gates.\n\nA single chair of banded leather and wood beneath him, and a single sword of massive-scale strapped onto his back. Vladris sat there, patiently awaiting the arrival of Roman Raines.\n\nHe understood, of course, that Roman's arrival was many hours, if not days away. And that the warrior traveled with others. Still, Vladris intended to do things in such a way that his legend would never again be questioned.\n\n\"Sir, is there anything you need?\" a Hunter asked, walking through the gates slowly, daring not disturb the hero any more than necessary.\n\n\"Yes. Assemble our finest and bring them here, before me.\" Vladris demanded.\n\n\"Yes. At once.\" the Hunter replied.\n\nNearly an hour after Vladris had ordered it so, twenty of the finest soldiers the Vampires had to offer slowly made their way outside of the gates. Forming into a single group in front of their hero and awaiting his next command.\n\n\"Most of you have learned of my abilities in battle through words. Legends told, passed down through generations,\" Vladris said, standing to his feet slowly. \"And I am aware that a few have even started to whisper of this man-demon. Roman Raines. Even going so far as to whisper of his ability to best me in combat.\" he added, cutting his words to approach the faces of each warrior.\n\n\"There is a reason that legend of my ability to end lives continues to be passed down. Reasoning that each of you will now know holds true with your very own eyes,\" Vladris said in a sharp voice. \"And while the rest of you whisper in fear, I await the warrior who comes. I long for a fight with the man which so many of you fear. Because, when this man who rides to our gates to end my legend arrives, he will soon beg for grace moments before the very blade strapped to my back slays him. And each of you will then know who to fear. Who to make legend of. Now go, cowards, and do not disturb me further!\" Vladris yelled as if to dare any of them to speak.\n\nAs the group of Vampiric warriors turned to once again enter the castle, Vladris returned to his chair. Turning his back to their castle and facing the rolling hills of Ronica before him as the rain began to fall down.\n\n\"We make good time,\" Tunak yelled loudly as their horses drudged through the thick of mud, finally having left the clutches of snow-filled mountains. \"It is unusual to see no patrols this close to their castle.\"\n\n\"They have cleared a path for us. Their hero believes in the ability to slay me. An ability which he does not possess, but makes for an easier ride indeed.\" Roman replied with confidence.\n\n\"We will be on them by nightfall.\" Four said.\n\n\"Then we will stop to make camp. I wish to face him in the light of day.\" Roman replied, bringing his horse to an abrupt halt.\n\n\"I agree. If we are to slay the demon of demons, the daylight will give us an advantage.\" Gore added.\n\n\"I wish to face him in the daylight, not for an advantage,\" Roman said sternly as his body dropped from horse to ground below. \"I simply wish to slay him during a moment in which many eyes can see.\"\n\nFor the next few minutes, Roman and Tunak made camp with a small tent of white cloth, while Four and Gore collected any dry wood they could find. In such a rain-filled environment the task proved daunting, but eventually they had collected enough to produce heat throughout the night.\n\nVladris knew they were close. He had made the journey several times himself, and knew that they had to be just beyond the tree-line at horizon's end. It also warranted the assumption that Roman wanted his fight during the day, and that sat well with Vladris. He too wanted a thick of witnesses for their battle. And though he knew Sarah would attempt to bring Roman to their cause, the hero among Vampires had a feeling throughout his gut that it wasn't to be.\n\nSo Vladris remained in his chair, dedicating such a quiet night of rainfall and breeze to his thoughts. Though he had no intentions of falling in battle, he knew that if it were to be, he had lived a life worth speaking of. He had known of both lust and love. Known of greed and loss. He stared across the very hills which once worked with Ronical farmers to produce crops to feed their mighty army.\n\nHe remembered the king he served during his mortal life. An honorable man, one that was cut down by the Vampires shortly after Ronica began to fall.\n\nHe remembered, while still in the realm of a mortal, defeating the Hunter champion in battle. His own abilities besting those of the Vampiric legend. Vladris began to wonder if Roman Raines was one such man. Was it meant for Roman to cut Vladris down in battle, ending a life, and along with it the memories of a Ronica that once flourished?\n\nBut soon his thoughts began to turn to Amelia. His love for her had been one of such truthful purity. From the tip of his sword to the bones in his chest, he loved her. Everything he was or would ever be was tied into the woman who would never return.\n\nAnd that very emotion, one of horrific loss that would forever remain, was the emotion which allowed him to fight like a lion. As if he were a hundred lions in battle. And as the chills of a loss never-ending began to consume him once more, he knew it to be the truth. Roman Raines could not best him.\n\nAnd so he spent the remainder of the night thinking of his beloved Amelia, a life before the rule of Vampires and, of course, the rain. The soothing rain which hit the trees of a distant horizon, bringing with it a sound which calmed the warrior and allowed him to control his anger.\n\nVladris was not alone in thought on such a night of pouring rain, however, as Roman also reflected back on his life up until this point.\n\nHis entire family slaughtered for sport at the hands of Hunters. His involvement in killing one of their queens and a life sworn to ending the bloodline of the race which he considered to be the cancer of man.\n\nRoman did not fear death. A direct effect of having nothing to live for. His only purpose was to slay those who deserved death, and he did so well. By most accounts, Roman Raines was the deadliest warrior alive, but not all of them.\n\nMany still spoke of Vladris, hero of the Vampires. If he were able to slay the demon of demons, it would not only send ripples of fear through the Vampiric race, but solidify his legend as the greatest warrior to ever live.\n\nHe believed he could do it, and planned to do it alone. Pulling a dagger from his waist, he planted it firmly into the ground of their campsite while the others slept. Then, pulling a necklace from beneath his robe, a key to his shuttle dangling from it, Roman placed it around the dagger's handle. And with that, the Vampire who hunted his own kind walked away from camp and into the direction of a castle not far away. Alone.\n\nVladris was first pulled back from his memories with Amelia as a bell began to toll throughout the castle, one the Hunters reserved for the unusual event of an army marching to their doorstep.\n\nAs Vladris stood to his feet, he wondered how fitting it seemed. Roman walking to their castle by himself and to the toll of a bell that signified an entire army. One warrior looking to capture the legacy of another, almost as if two complete armies were nearing battle. Though only two champions stood, less than a hundred yards apart.\n\nAs Hunters began to make their way to the castle's entrance, they soon partitioned off a bit to allow their queen a quick passage to the two warriors who had become locked in a stare.\n\n\"I have long awaited this very moment, even seen it in my dreams.\" Roman said, squeezing his fists with crushing power.\n\n\"I respect your courage warrior. Just as I have respected each warrior before you, all of them dead by my hands.\" Vladris replied.\n\n\"Enough!\" Sarah yelled, approaching the two soldiers of destiny with two soldiers of her own. Escort soldiers who held Troy in chains.\n\n\"It's been a long time Roman Raines.\" Sarah said, her demonic tone very soothing by Vampiric standards.\n\n\"Not long enough you crazy bitch.\" he replied.\n\n\"Still the feisty warrior. Such a trait will be of great value to me when you bow your loyalties.\" Sarah responded.\n\n\"A long time indeed, then, as you've forgotten that I bow to no one.\" Roman said.\n\n\"Perhaps,\" she replied with a grin, motioning her escort soldiers to remove a hood from Troy's head. \"Or perhaps you'd like to rethink your loyalty to me.\"\n\nAs Roman laid eyes on Troy, his first reaction was the yearning to pummel those responsible to death. Troy had been like a son to him, at least up until the point of becoming infected with Vampiric DNA. He had purposely avoided contact with the boy from that point forward, hoping to protect Troy from the horrors which now followed him.\n\nHowever, Roman's rage soon turned to desperation, feeling as though he had already been beaten in battle.\n\n\"He has no place in all of this. Free him.\" Roman demanded.\n\n\"Well that depends on you,\" Sarah replied with a heavy tone. \"If you bow before me, before all of those which look on at this very moment; in return I will allow the boy to go free.\"\n\nHe would be bowing to a bloodline which had stolen everything from him. But if he refused, they would take the one thing he had left. Troy was as much like a son as any boy could have ever been; and worse, he was innocent of any crimes against the Hunters.\n\nRoman approached the group slowly, taking his massive blade and burying nearly a foot of steel into the moist ground as he continued his walk.\n\n\"That's far enough.\" Vladris said, holding his own blade out in order to keep Roman and Sarah distanced.\n\n\"All I know is killing,\" Roman said, holding his hands up slightly as a gesture of good faith. \"But Troy has a chance to live a life of value. Become something more than all of this,\" he added, staring directly into Sarah's eyes. \"I need your word that he will be allowed to leave, never to be followed. Never to be brought into this again.\"\n\n\"You have it, as long as I have your loyalty.\" Sarah quickly replied, a smile beginning to brim across her face.\n\n\"And I need a moment to say farewell to the boy. Given our history, I'm sure you understand that I want to make peace with him?\"\n\n\"Make it quick,\" Sarah replied. \"Cut the boy loose.\" she added, turning to her personal escort soldiers, each of them clad in banded steel of black.\n\n\"Don't talk, just listen,\" Roman said as Troy approached him. \"Take the set of footprints my boots have made and follow them as fast as you can run. Just beyond those trees, not even thirty minutes away, a group of warriors loyal to me have made camp. When you arrive, tell them I've sent you. Tell them I've left a dagger, necklace and dog for you. Have them wait until midday for me, and should I fail to return, they are to lead you to my shuttle.\"\n\n\"Roman, I don't want to leave you here. Maybe together...\"\n\n\"No!\" Roman replied fiercely. \"I've always looked upon you as my own son. I can't protect those I've lost, but I can still protect you. Go.\" he said, holding his battle-hardened hand out for Troy.\n\nAs the two men locked hands for a moment, showing both respect and truth, Troy turned to begin a sprint for the trees which were only a few hundred yards away. Roman watched the young man run to safety, finally turning back to Sarah and her group.\n\n\"Alright Roman, I've delivered my end of the bargain. Now, bow before me and help us deliver our bloodline to a future of absolute dominance.\"\n\nHe had never before bowed. Not to the Gods above, nor to anyone with blood in their veins. And though it hurt as much, if not more, than a defeat on the battlefield, Roman lowered himself to the ground. His knee had never touched ground for the cause of obedience, yet his pants began to saturate just a bit from the rain-drenched soil.\n\n\"Good,\" Sarah said with a feeling of accomplishment. \"Now rise warrior and join your brothers.\"\n\nAs he rose slowly, waiting for the perfect opportunity to strike, his eyes remained locked onto the queen of Vampires. And with the slightest flicker of his eyes, came a large knife which had been tucked into the back of his pants.\n\nSometimes it's unbelievable how a single moment changes the world around us. One second, frozen in time, that has the ability to alter who we are and how we believe.\n\nAs Roman turned his torso, ensuring every ounce of power went into the flight of the knife which left his hands, even the smallest muscles of his body popped to life under his pale skin.\n\nThough he had bowed, the Hunters still had thought of him a bit cautiously, and that very caution saved Sarah's life. Though the blade struck home, it missed its mark as she slipped to her right. The knife which was meant for the base of her skull, instead drove into the meaty flesh of her shoulder. A wound that would not come close to slaying Sarah, though she screeched as if it would.\n\n\"Protect our queen!\" Vladris yelled, turning to face Roman in what was now an inevitable showdown.\n\nRoman had reclaimed his large blade, pulling the massive handle skyward, its gleaming edge covered with a bit of damp soil.\n\n\"Now you will know of the origin behind my legend.\" Vladris said, holding his sword out provokingly.\n\n\"A legend which dies today.\" Roman replied.\n\n\"We'll see.\" Vladris responded, quickly swooping his blade forward.\n\nRoman was able to parry the shot easily, though he seemed a bit surprised by the sheer amount of force behind the bite of Vladris' steel.\n\nRoman then lunged forward, his sword leading the way as he quickly turned the forward strike into a circular motion, chopping toward Vladris.\n\nHowever, Vladris quickly stepped back to easily avoid it, stepping back a bit more.\n\n\"You disappoint me Roman Raines. You carry the traits of a Vampire, and still I could have beaten you in the days of my own mortality.\" Vladris said in mocking fashion.\n\nThe one thing Roman had learned through his years of fighting was the defining moment. A single breath in which your mind catches up to the body which follows your emotions. A moment when a warrior knows who's better, and as Roman's mind caught up to his extremities, he understood it to be true. Vladris was better.\n\nHe would never admit it, or even show evidence of it on his face. But as Roman's muscular frame fought with all of its might to deflect steady shots by Vladris, he began to grow tired. His hatred for Vampires had fueled his fight so far by rage. However, the rage had began to die out, losing ground to a body that longed for a deep breath and a moment's rest.\n\nVladris continued to strike with unrelenting fury, each angling of his blade similar to that of a serpent's strike. And though silver flashes of his blade were visible to everyone else, Vladris only saw one thing. The face of his love lost, Amelia.\n\nHis rage on the battlefield seemed to slow everything down around him, as it did each time he went to war. Literally a lion with sword in hand, the figure of a ghost nearby, her magnificent body shrouded within a satin-blue dress.\n\nRoman quickly came to understand the warrior he fought against as tears began falling from the eyes of Vladris. Roman was good. Damn good. Yet Vladris struck as though he were a God, piercing shots sapping the life from Roman, not from wounds, but from the simple defense against them.\n\nThe trunk of Vladris' blade hit like a perfectly-forged hammer, taking with it the will of Roman Raines, as he began to understand. Vladris longed to die.\n\nHe had prayed that Roman was the warrior that growing legend spoke of. Perhaps, just maybe, the warrior who was destined to end him; and end his personal suffering in doing so.\n\nHe is not the one. Vladris, you must continue your journey among mortals. For our day will come.\n\nWords which escaped the lips of Amelia's ghost, standing only feet away from the dying battle before his eyes.\n\n\"Yes my love.\" Vladris replied, speaking to a figure which only his eyes saw, leading Roman to think him mad. Not that it would matter as he lay on the ground, sword raised, with only the ability to deflect continuing shots.\n\nRoman Raines had been bested. And while he lay there, sword lifted and shaking roughly from the trauma of steel exchanged, he considered throwing his weapon and allowing Vladris a chance to plunge bitter-cold steel into him. It would be the perfect punishment for the Vampiric warrior who had so easily defeated him. But it was not to be.\n\nHe was falling close to the end, evident to both Tunak and Four as they rode swiftly to the fight. Their horses covered the open ground at alarming speeds, rushing to aid Roman and his plans to slay Vladris.\n\n\"Riders approach!\" a Hunter proclaimed loudly as a dozen or more archers pulled to take aim, the rest of their ranks quickly making way to ground level and, with any luck, the castle gates.\n\nBoth of the warriors who approached on horseback held shields high, arrows meant to end them instead biting into banded leather.\n\nVladris had placed himself outside of the castle's gates in order to prove his abilities, purposely keeping any help out of reach. Both Tunak and Four had depended on his confidence as the tool to slay him. And as they rode within feet of the Vampire Elite, both warriors quickly left mount in order to stand on the same sacred ground shared by Vladris and Roman.\n\nTunak came in, charging with full-rage as Four quickly hoisted Roman to one of the steeds. A quick slap to the animal sent it sprinting for the nearby wooded area.\n\nThe plan was, Four and Tunak would fight Vladris with the man advantage. Overpower the demon, or at least die trying.\n\nHowever, their plan was swiftly cut short as, without the slightest of warning, Vladris thrust his sword forward with incredible speed. Its bite found a mark on the vitals of Tunak, digging into the warrior's chest and pulling his soul out as Vladris jerked his blade loose.\n\nFour was a huge warrior, and his blade matched quite well. But even he saw no victory. Even if he somehow bested Vladris, dozens of Hunters had now made it to them.\n\n\"You have cost me the greatest moment of my finest victory. For that, your death will not come swiftly.\" Vladris said as he walked a small circle around Four, eventually plunging his sword into the ground.\n\nFour had began to ask if Vladris intended a fight without blade. However, the Hunter Elite was quickly on him, grabbing wrist first, and then shoulder, on his way to the kill. And as Vladris sunk huge fangs into the face and upper-skull of Four, the warrior dropped his massive blade, body jerking uncontrollably from pain redefined.\n\n\"Run them down! Kill them...all of them!\" Sarah yelled from her spot nearby as she tended wounds, demanding her champion seek justice.\n\nMoments later, well over a dozen Hunter Elites thundered past the castle gates on steeds of nightmarish appearance, Vladris removing his sword from the damp ground and quickly grabbing the mount of an empty steed reserved for him.\n\n\"Go my champion, for this is your finest day. The Hunter Archives will long speak of your wrath against our enemies.\" Sarah said to herself, proudly watching Vladris and his finest chew up ground as their steeds rode hard into the forest.\n\n\"Help him from his mount, quickly!\" Troy said as both he and Gore gently pulled Roman from the steed.\n\n\"Is Vladris defeated?\" Gore asked, his Husk tone chilling to Troy's ears.\n\n\"Vladris cannot be defeated, this much I am sure of now. We must ride back at once. Inform the others that the Hunters are coming.\" Roman replied.\n\n\"But Four, Tunak?\" Gore asked.\n\n\"I saw Tunak cut down with my own eyes. Four could not have made it out of there with his life.\" Roman said with regret.\n\n\"But he fights like...\" Gore began to reply.\n\n\"Vladris stood before him! As did dozens of Hunters. He is not coming back, but the demons are. We must go now!\" Roman yelled.\n\nAnd with his words, Troy helped Roman back to his mount before joining Gore on their own steeds.\n\nMoments later they were gone. Riding hard toward the city which Hunters could not take. Marlock. Or so the legends claimed. A claim that would soon be put to the test as Vladris and his group of soul-reavers were not far behind.\n\nAnd as the rain slowly turned to snow, an ailing Roman, Troy and Gore raced toward Marlock as though their very lives depended on it, which they did.\n\nTroy wondered what the future held for him against such an immortal army. How a life of simplicity had stood before him until the early morning hours in which Vladris had snatched him, literally getting thrown into a world of warfare.\n\nGore thought of the same thing, every ride to and from battle. Death. He concentrated on what needed to be done in order for him to end up on the right side of the very word of death. On this day, he wondered how he could possibly escape his soul departing a world of mortals, glancing down to a slightly-rusted blade hanging by his side. Quickly remembering the same warriors who mocked it, would give their own lives for him.\n\nAnd as the snowfall began to intensify, large flakes flying into the face and peppered hair of Roman Raines, he thought of a second chance. His first attempt to slay Vladris had been a campaign of legacy building.\n\nAnd though Roman was a warrior of almost supernatural abilities, that only afforded him the equivalent of a puncher's chance in the grandest fight of all-time.\n\nHe was outmatched and knew it. But perhaps, just perhaps, he would be stronger this time. Fighting for the fallen and those who could not fight for themselves, rather than his own legacy. For he too was a lion, and he was wounded and desperate. A hellhound sprinting by his side all the while.\n\n\"Someone approaches!\" Bral yelled as the others came rushing.\n\n\"It's Roman, he has defeated Vladris!\" Anthony said loudly.\n\n\"They do not have the glow of victory upon their faces, and ride as though death follows closely behind.\" Ranthra replied.\n\n\"Quickly, help Roman from his mount!\" Gore yelled as both he and Troy joined the others on foot.\n\n\"What of Vladris? The others...Four?\" Anthony asked.\n\n\"I faced Vladris sword to sword, yet he still lives,\" Roman said with pause. \"I would not be alive now if not for Tunak and Four coming to my aid.\"\n\n\"And what of them?\" Ranthra asked blatantly.\n\nRoman answered his question with a stare, one of both remorse for the fallen and ill-intent toward Ranthra.\n\n\"Roman believes they are coming here to end us all, which means we need to dig in if we are to survive.\" Gore said.\n\n\"No,\" Roman replied, shaking his head with regret. \"You are to take Troy to the ship as planned.\"\n\n\"But Roman, I...\" Gore began.\n\n\"Troy is my son!\" Roman replied, stopping to look at the young man once more. \"If I am to face Vladris again, I cannot do so with the worry of Troy.\"\n\n\"But you said Vladris cannot be beaten.\" Troy replied, catching the attention of the entire group.\n\n\"He cannot. At least not by the hands of a single man. Together, however, we may be able to finally rid our world of the mightiest of demons.\" Roman replied.\n\n\"When will they come?\" Draden asked, the Dragonborn warrior's eyes skimming the tree line.\n\n\"I'm not sure, but it will be soon,\" Roman replied. \"And I cannot ask any of you to stay for this fight, not against such odds.\"\n\n\"We are with you to the end.\" Pica replied, pulling his freshly oiled bow to the ready.\n\n\"As am I,\" Ranthra said, glancing to the warrior in charge as he wielded a massive axe of razor-sharp iron. \"There is strength in numbers, and I will defend my home.\"\n\n\"Indeed,\" Roman replied with a grin, nodding his respect to Ranthra. \"Gore. A word.\"\n\n\"Yes?\" Gore asked, approaching Roman so the two could talk without hinderance.\n\n\"I am asking you to take Troy because I believe we will fall. Perhaps we will be able to thin their ranks, or even slay Vladris. But I do feel as though my time has come.\"\n\n\"Then why stay behind? We can flee to my homeworld, regroup, and then arrive back in force?\" Gore asked.\n\n\"Because I don't flee. If it were not for the longing of seeing Troy one last time, I would have stayed behind with Tunak and Gore to meet my own demise.\"\n\n\"I am not used to you speaking such words of defeat.\" Gore said.\n\n\"Nor am I,\" Roman replied somberly. \"But I have seen into the eyes, the very soul of Vladris. He speaks with ghosts and fights as though he were already dead. He has nothing to live for, and because of that he fights as though he were a lion. I have a concern for Troy now. It has weakened me.\"\n\n\"I will take Troy to my homeworld, you need not worry for his safety. The Husk there, my own family, we will care for him as our own should you fall in battle. Only concern yourself with the fight that is coming and a moment of victory.\" Gore replied.\n\n\"Thank you my brother.\" Roman replied, shaking hands with his loyal friend before turning to Troy.\n\n\"Gore has promised to take you to safety. Await word of my battle, and always do by him as you would by me. Understand?\" Roman asked.\n\n\"Yes,\" Troy said with tear-filled eyes, reaching into a long pocket on his leg. \"This blade belongs to you. You gave it to me when I was but a child. Take it, and you bury it into that big-fanged son of a bitch. If you care anything for me, you'll do that.\"\n\n\"Take care...son. No matter what happens from this moment forward, you are destined to live a warrior's path. Always.\" Roman replied, looking at the young man for a moment before hugging him.\n\nAs Gore and Troy walked away, the young man turned to look upon Roman one last time.\n\n\"They killed my family, your family, our friends, and most of all, tried forcing both you and Sarah into a life of servitude. You make them pay, do you hear me? YOU MAKE THEM PAY!\" Troy yelled, showing for the first time a warrior's rage in doing so.\n\nRoman gave no reply verbally, though his heart had already committed itself to Troy's words. And as Roman watched his son ride off with Gore, headed to the location of a small ship that would pull them from Ronica and place them into the safety of Huven, the large Husk homeworld, he felt as if the Hunters had taken from him once more.\n\n\"We approach my lord, should I have our warriors pull back and scout the area?\" one of the Hunter Elites asked, near-blizzard snow falling around them.\n\n\"No, ride harder. I grow sick of this cat and mouse game which holds my destiny in the balance. Ride hard and slay quickly!\" Vladris demanded as the small, but well-experienced band of undead warriors stormed toward Roman, who stood by himself at the edge of village.\n\nTurning from the entrance of a small building, Pica loosed an arrow that flew as quickly as it killed, hitting an elite in the forehead and fragmenting skull as the demon cried in misery.\n\n\"Run them down like cattle and slaughter them as such!\" Vladris commanded, jumping down from his mount as the horse remained in full-gallop.\n\n\"Now, coward, we finish your journey to the world of ghosts.\" Vladris said, his sword pointing at Roman Raines as warriors, both undead and living, fought in the background. Steel clashing against steel with a hellish-fury.\n\n\"One would ask which world you live in?\" Roman replied, pulling his own blade to the ready. \"I know why you fight; what makes you mighty in battle. For I fight for the same reason.\"\n\n\"All you know of me is what my legend speaks of. A legend which will grow volumes this very day as I sever your skull from its body.\" Vladris replied angrily, cleaving a blade strike down that narrowly missed its target.\n\n\"Perhaps. Or perhaps you fight for a slain family as I do. A woman maybe?\" Roman said, noticing the look in the eyes of Vladris as he spoke the words. \"So it is a woman.\"\n\nRoman's words were cut short as Vladris came in with a soul-damning flurry of sword strikes, each barely blocked by Roman's own blade, and taking with them respect.\n\n\"We are no different Vladris. The Hunters have also taken from me loved ones. It's that very emotional rage which has allowed me to fight the Hunter DNA which runs through my veins.\" Roman said, surprising everyone as he then sunk his sword into the ground beside him. \"Join us my brother.\"\n\nVladris watched with curiosity as Roman Raines stood unarmed, his hand extended to offer the demon of demons a chance to become brothers in arms.\n\nHe also watched the same figure appear, as it did every battle before. The ghost of Amelia walking among them, looking at her truest of champions.\n\n\"This warrior is one of truth. He fights for his loved ones, and protects those who cannot protect themselves. He is a very admirable man. Join him.\" Amelia's ghost whispered softly.\n\nVladris stood there, clashing steel becoming a slight background noise as the mighty warrior remained deep in thought. Both the past and the present weighed heavily as he contimplated the future.\n\n\"What does your gut tell you to do? Your lover?\" Roman asked.\n\n\"My lover speaks of you as an admirable warrior. Thinks I should fight beside you in battle,\" Vladris replied as Roman began to smile a bit. \"Unfortunately for the both of you, my heart has started to love another.\" he added.\n\nJust as quickly as Roman realized Sarah Blaine had the affections of Vladris, he pulled the dagger Troy had gifted him and spun low to the ground, a move which allowed him to escape Vladris' blade while plunging the dagger's steel into the demon's chest.\n\nVladris staggered back, dropping his sword and trying to pull the small blade from his chest. The shock upon his face told the tale of a demon who had never been struck in battle before. At least not until the day he was turned to the ranks of the immortal.\n\n\"You've had your chance and now you must fall, an act which your own legend will speak of.\" Roman said, approaching the wounded lion slowly, carrying his sword with a stone-hardened grip.\n\nAnd as Roman Raines hoisted his blade high, preparing to slay the mighty Vampire of legend, his senses alarmed him with pain. An arrow loosed from the bow of Pica digging through his thick armor and into flesh.\n\nAs Roman turned to speak of the misfire, Pica loosed another and another, both shots deliberately eating into the stomach of Roman Raines, who dropped his blade.\n\n\"Traitor!\" Draden yelled, sprinting over bodies of the dead with a long blade in hand.\n\nThe Dragonborn warrior fell quickly, an arrow of cold steel biting into his face, loosed from Pica's bow only feet away.\n\nAnd so the remaining warriors fell swiftly, leaving only Roman, who had fallen to a knee as life slowly left his body.\n\n\"As your soul departs, take with you this. You are by far the toughest warrior I have ever faced,\" Vladris said, finally ripping the dagger from his chest, then raising his own blade. \"The Hunter archives will speak of you as a legend. This I promise to you my brother.\"\n\nRoman nodded, accepting his ticket to the afterlife. He was done, knowing a life of only misery and killing, except for the brief period he spent with his friends, his family, aboard the Gunship crew.\n\nIt would be the very faces of that crew, the very personalities of those he had grown to love as family, which he would remain in thought about during his eternal rest.\n\nAnd the life of Roman Raines would end with a single strike, an arcing sword which severed his head from its body; just as Vladris had promised.\n\nAt the close of battle, Vladris placed the head of Roman Raines into a large bag of leather stitching, tightening the knot of rope as he glared to his Hunter soldiers, a single contact of eyes which would grow his legend as immortal.\n\n\"I have held my end of the bargain. Now I hope your intentions to hold your end remain true?\" Pica asked as he approached Vladris and the remaining Hunters, bow in hand.\n\n\"Indeed. You will soon be turned to the ranks of the immortal to fight alongside us as a brother. You have proven your loyalty.\"\n\n\"Take care warrior, and enjoy your reign of terror. Because it ends soon at the hands of another.\" Amelia's ghost said, her words blatently ignored by Vladris as he mounted his steed of hell once more.\n\n\"Now we return to our queen. There is a wedding to take place; a wedding which will unify our people.\" Vladris said with a stern voice, as if to dare any Hunter to speak.\n\n\"I don't want to leave him behind.\" Troy said as Gore began firing up Roman's shuttle, pressing dozens of necessary buttons which illuminated red.\n\n\"Roman did not survive. Otherwise, he would have made it back by now. I'm sorry,\" Gore said with regret. \"But you will be among friends on my homeworld, all of which hate the Hunters.\"\n\nTroy began to respond, but as the shuttle slowly lifted from the moist ground, Roman's ghost became visible to the boy.\n\n\"Avenge me son.\"\n\nAs Troy saw those words loose from the lips of such an honorable man, be it from the afterlife, he understood his destiny. Continue building Roman's Empire, and eventually strike against those who have been responsible for so much loss.\n\n\"It's alright. Even now, I feel as though Roman is with us.\" Troy replied with a smile.\n\nScucca remained by the side of Troy since their first encounter, just as he had done with Roman Raines. A sight that was rare among hounds of hell. Much more so than the snarling of teeth currently directed toward Gore.\n\n\"As do I young warrior. As do I.\" Gore replied, casting a stare of dare to the hound before pulling the flight stick harshly as the shuttle went into full-burn and made orbit.\n\nThe trees were crisped with rain, as were the blades of grass which led to such a beautiful place. And though their castle was one of murder, torture and devine obedience for the undead, on this very morning it looked perfect.\n\nSarah Blaine was to wed Vladris, and bring the idea of unity and strength to their people. Hundreds of years had passed since the Hunters were ruled by both king and queen, times spoken of in the archives as the very best among their race.\n\nBut the wedding had grown well beyond a simple gesture to unite their people. They both had grown fond of the other in such a short period of time.\n\nSarah was a compassionate queen, a trait which Vladris respected greatly. She was also both beautiful and comforting to him, bringing with her a soothing calm.\n\nVladris had become Sarah's champion. She had joined those who truly believed he could not be bested in combat, bringing to her the feeling of safety. She felt at peace around him, which made her transition from human to Vampire much more bearable.\n\nAnd as they locked eyes at the alter, hundreds of the best warriors the Hunters had looking on, Sarah grinned softly to her champion.\n\n\"And should any man or woman object to these two being locked in matrimony, let them speak now or forever hold...\" the minister said, stopping short as he joined the rest of the crowd, turning to look to the rear of the chapel.\n\nQuickly, both Sarah and Vladris turned as well, determined to end whoever had intruded on such a perfect ceremony.\n\nHowever, as a man stood to his feet, hooded robe of brown and while; Sarah quickly felt faint. Adam Michaels. Sarah's first true love and former Gunship Captain, removed the hood and smiled.\n\nIt was almost as if relative time stopped, the chatter around them non-existent. The only thing that seemed real to her was his glowing smile, the chisel of his chin; and to him, her undeniable beauty.\n\nAnd though Sarah Blaine was now queen of Vampires, her guards rushing to apprehend her first love, she knew it was only a matter of protocol. She had no intentions of harming him, but rather hearing him out. Why had he arrived, why now? And she wondered if his reasoning even mattered? Because, in the end, Sarah would have another chance to hear the strength of his voice. The ability to look upon him again, which is more than she could have ever dreamed of.\n\nShe cared not how he had managed to find his way into such a heavily protected castle, but rather the fact that her life of clarity had just become murky beyond explaination.\n\nAs Vladris tried to make sense of the man covered in brown satin, his attention quickly shifted to the ghost of Amelia, who stood at the rear of the chapel. She had never before appeared to him beyond the fields of battle.\n\nBut as her lips made out a single word, Vladris felt a chill run across all extremities.\n\nSoon.\n\nBook 5\n\nLegendary\n\n\"Adam,\" Sarah said with gentleness, approaching the smuggler as he stood, watching the rolling hills before them and feeling as content as ever. \"They're ready.\"\n\nFor seconds which seemed to bleed into eternity, Adam stood silent, pulling every crisp moment into his mind. The world of Ronica simply breathtaking.\n\n\"Have you ever just looked out onto the hills? Just to soak them into your memory?\" he asked, remaining mesmerized.\n\n\"More times than you could ever know. Many nights I have stood on my balcony, listening to the calm of rainfall.\" she replied with a hushed tone.\n\nThere was a period of several moments, each filled with the vivid of color and rush of cool breeze. As if the smuggler had never slowed to enjoy such things.\n\n\"I guess it is starting to sink in,\" he said, adjusting his head a bit to watch a small flock of birds soar overhead. \"This may be one of the last times I'm able to see such a view.\"\n\n\"Perhaps,\" she replied, studying the worry in his eyes. \"They're waiting for you.\"\n\n\"Of course.\" Adam replied, slowly turning.\n\nHe did so, of course, as nearly a dozen colonial soldiers held guns onto the queen of vampires. Sarah. Though her personal escort was also nearby, their own weapons at the ready. Their conversation having taken place in the midst of so much nervous firepower.\n\n\"I want you to know that I've done all I can do as their queen,\" she said, the pair walking slowly into the direction of a large castle which the Hunters had called home for many years. \"I've managed to get you an audience with the eldest of vampires, but they will not easily be swayed.\" Sarah added.\n\nHe dreaded it. A meeting with the race of vampires whose acts were usually unspeakable. The Hunters were vampires in every respect, blood sucking monsters that enjoy feasting on human flesh. Though they often seemed to keep to themselves, it wasn't unknown for an occasional member of the race to stray from Ronica to begin his or her own agenda elsewhere, resulting in bloodshed among the human population, which only furthered their stereotype as a horrible race.\n\n\"I know, thank you,\" Adam replied, speaking from the heart to a woman he once loved. \"And I want you to know that I'm sorry for putting you into this position. Truly.\" he added.\n\n\"Just speak to them from the heart Adam, and fate will take care of the rest.\" Sarah replied, a warm smile upon the face of such a beautiful woman.\n\nThough she'd become a vampire, Sarah's beauty had remained intact. Even more so, depending on who you asked. She had the same elegant curves and angelic voice as before, though her skin appeared much whiter.\n\nShe had once been widely considered the most compassionate person alive. Daughter of the acting commander of the colonies, she was a true advocate of humanity.\n\nThe wrong type of Hunter had strayed from Ronica and soon their paths met, dragging Adam into things during the process. Stumbling onto true love during the worst of circumstances. Their love for one another had simply been storybook, though several events tested them.\n\nWhen he faced the decision of a lifetime, Adam walked away from the woman his heart loved so truly. Unknowingly handing her to the Hunters with his decision.\n\nThere were many regrets throughout his lifetime, but none of them greater. A bad call which spawned from pride and bitterness, leading to the death of a woman he truly loved. At least partial death.\n\nHe knew she was a vampire because of his inability to forgive. Still, he saw her just as beautifully as ever. Feeling butterflies while in her presence.\n\nAs Adam was led into the castle, a large security detail following behind, Sarah watched him walk away. Searching her own feelings for a man she once was prepared to marry.\n\nShe had loved him with everything not so long ago. From the pit of her stomach; her very soul. She loved the man who walked away once more \u2013 this time on business.\n\n\"Good luck Adam Michaels.\" she said with a dedicated, yet hushed, voice.\n\nAboard Colonial Star Triumph\n\n\"I am having trouble accepting this.\" Commander Ortega admitted, his emotions torn in half as he sat on a long couch of plush red, several advisers standing nearby.\n\n\"Sir, this decision is not on you.\" one of the advisers remarked.\n\n\"In fact, it is the only choice. We're left with no options.\" another adviser added.\n\n\"Sir, we have been crushed,\" a second adviser added. \"Our mightiest of bases, Glimmeria, fell as though it were a dying leaf in the fall of year. It is in my opinion that nothing can be gained from our situation outside of the greater good.\"\n\n\"Yes,\" Ortega replied, standing to his feet. \"I understand the logic. My heart bleeds for every person we leave behind. Truly, it does.\" he added with sincerity.\n\n\"Sir, if we had time to gather more ships, perhaps we could save them all. We simply do not have the time. The realization of defeat came too late.\"\n\n\"Doctor Arness,\" Ortega said, turning to the high-ranking adviser. \"I wonder if you know what a burden this is, having to decide the fate of thousands by the stroke of a pen? I wonder if you feel for the same for these innocent people as I do?\"\n\n\"Sir, I understand you convictions,\" the doctor said, though his primary role throughout the fleet was political. \"But I also understand that we may not even have time to form the fleet, much less expand it. The horde cares not for our guilt or convictions. They simply come.\" Doctor Arness said.\n\n\"Yes. I'm painfully aware,\" Ortega replied. \"You have often been the voice of reason for me when my heart is worn too openly on my sleeve.\"\n\n\"Just doing my job sir,\" Doctor Arness commented. \"Someone has to be here to keep things in perspective. Sadly, innocent children will have to die in order for us to survive. The fleet needs us.\"\n\nWith the delivery of his statement, the two marines stationed by the door of their private quarters exchanged glances. Obviously shaken by Arness' ease when it came to loss of innocent life, though they understood the chain of command.\n\n\"In just a few moments, you'll be broadcast to every monitor in the Skyla System. Everyone needs to hear strength in your voice and indecision in your heart. Even if it is not entirely there, our citizens need to hear it.\" one of his advisers said.\n\n\"Now more than ever, they need to hear it.\" Ortega replied, agreeing with the logic of his advisers.\n\n\"As soon as we deliver the speech, what remains of our government can begin the process of supplying for the journey to come.\" Doctor Arness said.\n\n\"We have enough ships to make this type of journey?\" Ortega questioned, turning to the men.\n\n\"Yes sir,\" one of them responded. \"Some of them are a bit older, but certainly capable of deep space travel. We can make any necessary repairs once we're on our way.\"\n\n\"Good,\" Ortega nodded. \"Supplies?\"\n\n\"That is our main concern at the moment,\" the adviser admitted. \"Fuel, water, food and ammunition will all be in short supply. Especially after being sacked so many times by the Priests.\" he added, stitching truth into his statement.\n\n\"As will hope. I cannot leave people behind without a fighting chance,\" Ortega said. \"I will not leave them here without at least a small chance of survival.\"\n\n\"Sir, we've selected the location for several reasons. It is the furthest away from the Priest's main force, but it is also one of the resource-richest planets of our system. Full of minerals to fuel a civilization and plenty of land to grow crops. Our survivors would have a good chance to rebuild, should they somehow find a way to turn the Priests away.\"\n\n\"I wonder if your confidence would be as firm if you were to be left behind?\" Ortega asked.\n\n\"Sir?\" the adviser replied, both men with a look of shock.\n\n\"That is the way of a politician, is it not? Save his own skin by sacrificing the skin of others?\"\n\n\"Sir, I assure you. We've done the best job possible given our circumstances. For both sides involved.\" the adviser replied.\n\n\"Indeed,\" Ortega commented, seeming to take a moment to breathe and focus on the task at hand. \"Let's just get it over with,\" he added, scooping up several papers and heading for the door to his quarters, surrounded by the bustling of suits and ties. \"Let's doom nearly half of our race to certain death. For the sake of politics.\"\n\nDrift Planet - Sandila\n\n\"Your card skills are as bad as your Priest-hunting skills!\" a man proclaimed, erupting laughter throughout the small bar setting.\n\nThe boasting man was very much on the chubby side, an unshaven beard masking most of his face as he began reaching for the pot of gold. Or, in this case, three packs of fresh cigars and a near-full bottle of whiskey.\n\n\"Not so fast you cocksucker,\" Dalton replied sharply, slamming his own cards onto the table. \"Three gems and two zips \u2013 now give me my damn winnings.\"\n\n\"That's impossible!\" the large man claimed, grabbing Dalton by the wrist in an attempt to stop him from collecting his winnings.\n\n\"As will be your attempt to eat solid food with a wired-up jaw,\" Dalton warned. \"Which it will be, if you don't unhand me. Friend.\"\n\n\"Let it be Daniel.\" one of the other players commented, the small table holding four players total.\n\n\"Well, alright then.\" the chubby man calmly replied, releasing his grip of the smuggler's wrist.\n\n\"Besides boys, hell, plenty enough to go around.\" Dalton said with a grin, hinting of pouring everyone a shot; a suggestion that immediately brought cheers throughout the sunlight-poor room.\n\nAs his shaky hand began to pour the first shot of whiskey, Dalton's attention followed everyone else to the only door of the sad dwelling. Embers of filth highlighting such a dank pub as rays of light poured in. Its walls held together with nails, while cracks of sunlight bled in through mismatched wooden boards.\n\nDalton had a good idea who was entering. She had a way of showing up each time he was preparing to embark into the land of no good, and it really would have pissed him off \u2013 if not for the fact that she was so incredibly easy on the eyes.\n\nHis captain, Cambria Sims.\n\nCambria slowly entered, her vivid blue hair standing out among the pack of thieves and self-admitted outlaws. And though she eased the door closed once more as if to enter the room unnoticed, the bar had become silent enough to hear the aging of wood. Nearly.\n\n\"Better hide the whiskey mate, your boss lady is here.\" one of the players taunted, chuckling a bit.\n\n\"Dalton James don't have to hide a damn thing.\" he commented proudly, though he casually corked the bottle of whiskey and eased it into the inner of his brown coat.\n\n\"Yea,\" another player mocked with laughter. \"Sure he don't.\"\n\n\"She says jump and the man asks how high.\" the chubby card game loser said.\n\n\"How about I jump across this table and pistol-whip the shit out of you? How about that?\" Dalton asked with a scowl, lips cutting though a beard of brown.\n\n\"Relax boys, she is sitting up there with the monitor.\" one of the men commented, noticing Cambria sitting to the front of the building; her seat at the bar directly in front of the news monitor.\n\n\"Another round?\" a man asked, having scooped up the deck of cards in an attempt to shuffle.\n\n\"Nope,\" Dalton said, though he wanted nothing more than to blister the men once more at a man's game. \"Gotta go check in with my captain.\"\n\nThough laughter ensued, the smuggler shrugged it off, locking eyes onto a man who had eased his way onto a stool beside Cambria.\n\n_So much for honor among thieves._ Dalton thought, casting a hard glance to the man. \"Cocksucker.\"\n\nPausing for a moment, Dalton turned his attention to the pub's door. Believing he'd heard a distant scream, though no others among them heard it. Either they were all too drunk to have heard the bone-chilling cry for help, or Dalton was drunk enough to imagine it.\n\nFinally shrugging it off, he once again focused on his walk to the bar.\n\n\"Hey Cambria,\" Dalton said with a grin, one that was quickly returned by the young lady of impeccable beauty. \"Hey guy with no chance of getting into Cambria's pants.\"\n\nBoth the blue-haired beauty and local patron developed strange looks across their faces, for different reasons, of course.\n\n\"I wasn't going to...\" the man began to reply.\n\n\"Save your shit Mahone, I've known you long enough. Probably using the same damn lines as always on this young specimen of insatiable beauty.\"\n\nIt was the first time he'd openly hit on a woman while warning off another man without at least a pause for breath. A fact that made the smuggler grin a bit.\n\n\"Dalton, really. He was just being nice. That's all.\" Cambria proclaimed.\n\n_Yea right, and I'm the fucking commander of the colonies. That silver-tongued son of a bitch is a bigger booty-chaser than I am, and that's saying something._ Dalton thought.\n\n\"What takes up 12 docking spaces...six women pilots,\" Dalton spouted off, striking home with Cambria, who had heard the joke only moments before. \"Or maybe he hit you with the old...What do you call a woman who can't make sandwiches?\" Dalton asked, knowing he'd exposed the panty-snatcher. \"Single.\"\n\n\"Well, I'll take my leave.\" Mahone stated, knowing he'd been given away by the man who would have been his wingman on any other occasion. This was different though. This was love. Though neither Dalton nor Cambria would admit it.\n\nDalton was careful to watch the man walk away too, as though he were a mighty dog protecting its master.\n\n\"Wow, I feel like an idiot.\" Cambria confessed.\n\n\"Don't,\" Dalton replied, though his eyes cautiously watched Mahone walk away. \"Most of the skanks that come through here would have been in bed with him after the first line,\" he added, quickly realizing the damage of his own words. \"Not to say you're a skank.\"\n\nCambria answered his confession with a laugh.\n\n\"Wow, and I thought his lines were bad.\"\n\nA short moment of silence fell between them, making each a bit uncomfortable.\n\n\"I don't know, I just feel a need to watch out for you. That's all. Places like this, they're in my blood. I know what's going through most of these people's heads before they do. It's like I'm a psychic or something.\"\n\n\"A psychic with whiskey on his breath.\" Cambria replied.\n\n_Ah fuck!_ Dalton thought, though he broadcast a look of confusion.\n\n\"It's alright, this is a bar. I expect it. Besides, it kind of feels good knowing I can come into a place like this and be safe.\" Cambria admitted.\n\n\"Yea,\" Dalton said, his mind wandering to distant places as he thought of her statement. \"I understand that.\"\n\n\"Can I ask you a question? A serious one?\" Cambria asked.\n\n\"Technically you just did,\" Dalton replied with a smile. \"Sure you can, you know that.\"\n\n\"How come you never let anyone close? I see through the whole comedic front you put up, I just don't understand why? Why can't you just be open about what's on your mind? In your heart?\" Cambria asked.\n\nIt was a damn good question, and Dalton knew it. His silent reaction for several moments only confirmed it to her.\n\n\"Not sure really. I reckon I'm just afraid of losing people,\" he admitted, becoming as serious as Cambria had ever seen him before. \"Each time I open up to someone, I lose 'em. One way or another, I lose 'em. Be it in a gunfight or to the undead,\" he added. \"I hate the fucking undead.\"\n\n\"Fair enough.\" Cambria replied, accepting his answer at face value.\n\n\"Now let me ask you something.\" Dalton said.\n\n\"Shoot,\" Cambria replied with a grin. \"Though not literally.\"\n\n\"How come we talk about anything and everything but the obvious? Each time we get close to talking about us, you shy away from it. I don't know what to think?\" Dalton asked.\n\n\"Us?\" she asked.\n\n\"Ah, never mind then. I suppose my idiot brain just conjured that up. Won't happen again.\" Dalton replied, standing to his feet, upset by her lack of feelings for him.\n\nCambria offered no reply, simply turning her head away from him. Though she did so in order to mask her emotions.\n\nA part of her wanted to grab the smuggler and unload her own feelings onto him. Be with him. But her reservations prevented her from it, not seeing how a relationship with him would ever work out for the best.\n\nCambria had a way of over-thinking things, usually for the worse. She was confused, and the tears which began to swell in her eyes were proof of it, eventually beading softly down her cheeks a bit.\n\n\"Dalton, wait.\" she asked, turning to him.\n\n\"What?\" he demanded to know. \"What is it?\"\n\n\"Commander Ortega is on the monitor.\" Cambria said, deflating all of Dalton's hopes for love with her reply.\n\nThe smuggler offered no reply, instead walking to the bar as he joined dozens of others, all preparing themselves for a rare speech by the commander.\n\n\"First off, I would like to confirm the rumors of Glimmeria falling. They are unfortunate, but true. Our army has fought valiantly in the face of this plague of man, but we are steadily losing ground.\n\nMy fair people, this is not a decision that has come easily for me. Though I fear it is our very last option and the window of opportunity is quickly closing.\n\nI have signed an order that will fund an exodus fleet. A fleet of ships that will take us away from this threat once and for all, as we journey into uncharted space to search for a new home.\n\nAgain, I would like to reiterate that my decision has not come lightly, and will forever burden my dreams as I know we simply do not have room for every last survivor.\"\n\nCommander Ortega stopped for a moment as the sound of cameras snapping throughout his conference room seemed to echo the fact that it was an historic moment.\n\n\"My advisory panel confirms that we will have room to house less than half of our survivors. I have personally overseen a lottery system in which we will select those who will accompany us into the stars.\n\nOne-third of the exodus from this plague will be comprised of those with extensive military experience. Their service and zeal will lend in the survival of our race. One-third of our exodus will be comprised of those who are deemed too important to leave behind. Among these people, doctors, scientists and teachers. Our race will not survive without the above mentioned. I truly believe that.\n\nFinally, the remaining one-third will be randomly selected through an exodus lottery. This will provide everyone with a glimmer of hope for the future, and I assure you, the lottery will be random.\n\nIn fact,\" Ortega said, pausing as he carefully planned his next words. \"I have personally decided to stay behind, as I could not live my life knowing I left a single person behind.\"\n\nPausing once more, the commander listened to the deafening sound of camera shutters.\n\n\"Any politician or military person who wishes to remain behind with me, may do so, and in turn free up a spot for a defenseless woman or child. That is a personal decision that I leave to each person listening.\n\nAs for myself, I will deem a commander for the exodus flight and then remain behind to coordinate one final stand against the enemies who approach our doorstep.\n\nI would ask that everyone pray. Not only for your own fate, but for the fate of those around you. Live every single day for the moments contained within it, and prepare for the next chapter in the history of our race. Thank you, and may God find it within his heart to grant us mercy in such troubled times.\"\n\nRather than take questions, Commander Ortega quickly exited from the podium, obviously shaken by his own announcement.\n\n\"Sir, you must reconsider your decision to stay.\" one of Ortega's advisers pleaded, while the other quickly raced out to the group of reporters who seemed as though they were a mob. Readying himself to field questions and assure the people of hope.\n\n\"I've no intention of staying behind,\" Ortega replied, still visibly shaken. \"I simply told the people what they needed to hear. They need hope at a time like this, and I gave it to them. When the plague arrives, none of us will be any wiser as to who is staying and who is leaving. I owe these people hope, but I also owe the exodus fleet stability in leadership.\" he replied.\n\n\"Yes sir, of course.\" his adviser replied.\n\nMoments later, Ortega found himself inside the sanctity of his personal quarters. Deafening silence leading him into even deeper thought.\n\nHe wondered of Adam Michaels' progress, though he also wondered if their exodus flight was the solution to the plague of man, or a prolonging of their own deaths. Knowing such large numbers could easily lead to a lack of fuel, starvation or struggle for power.\n\n_Survivors?_ He thought, feeling pity for them as well. _Survivors for what?_\n\nRonica\n\nAs Adam walked slowly into the Hunters' castle, leather boots striking easily against the stone flooring, his eyes skimmed the immediate surroundings.\n\nCall it a smuggler's intuition.\n\nA large room, vaulted ceiling of multicolored glass, though most of it seemed to refract shades of dark red; surrounded by dozens of vampires.\n\nEach of them wore attire of seemingly the utmost of importance. Most draped in a hooded coat of black, while a handful stood in the decorated armor of a warrior. Among them, Vladris.\n\nHe was the Hunters' champion; having survived hundreds of years while slaying hero after hero, as well as anyone else who opposed the rule of vampires.\n\nA great-sword laced to his back with leather straps, its trunk as wide as a small tree, though its bite was far more deadly. A sword in which only the mightiest could wield; Vladris did so with a single hand, at times swinging it as though it were the weight of a feather. A testament to his brute strength.\n\n\"Speak clearly and choose your words wisely, as our elders have granted you the gift of an audience; one they do not normally grant.\" Vladris said with authority.\n\nHe was right. It was rare for a human to enter the chamber of elders, and unheard of while wearing the Benzan uniform. The mark of a clan of warriors which had waged slaughter against the vampires for centuries, and the vampires likewise slaughtered the Benzans.\n\nNevertheless, Adam Michaels was a Benzan, and having found the lost tribe of Benzans had given him something he was never in short supply of. Confidence.\n\nMost importantly, he understood his job. He was there by the personal request of Commander Ortega. Ronica was the ideal planet to stage their exodus fleet, in fact, it was the only planet. Anything else was too close to the plague of man and would have been too quickly overrun.\n\nAs Adam entered the large chamber, reminding him of a very Gothic-styled judge's chamber, five vampires sat atop their own thrones, each one made of stone and trimmed in the sparkle of jewels.\n\nHe could tell from the look of them that each was hundreds, perhaps even thousands of years old, though he dared not question it.\n\nBowing low to the floor, a sight never before seen by someone bearing the mark of a Benzan, Adam stood slowly, preparing himself for such important negotiations.\n\n\"I thank you all for the opportunity to speak,\" Adam said, pausing for a moment. \"And I have come before you to ask for help. To build a bridge between our people.\"\n\n\"Build a bridge? Your PEOPLE burned that bridge when they began genocide against our race!\" Vladris yelled in response.\n\n\"You said I had been given the gift of an audience with elders, did you not?\" Adam asked with confidence. \"Then let me speak.\"\n\nThough no reply followed, Vladris cut eyes onto the Benzan smuggler that would have intimidated most among mortals.\n\n\"The Benzan family in which I am allied; we have no quarrel with you. With any of you. Just as you do, these Benzans have lived on their own in peace.\" Adam said.\n\n\"Why is it that you would come here during one of our most ceremonious days, as an intruder, and expect help?\" one of the elders asked.\n\n\"I had no idea I was arriving during such a time,\" Adam replied, turning to return a stare of dare onto Vladris. \"For that I apologize,\" he added. \"But I do so out of desperation. We face a new menace now. One that is much greater than any before this day.\"\n\n\"Let them wipe you out! What business is it of ours?\" Vladris demanded to know.\n\n\"Vladris \u2013 let him speak!\" one of the elders warned.\n\nThe Hunters' champion warrior simply bowed with obedience.\n\n\"The human race is only the beginning. Our new threat, the Priests, are after the entire system. Everything we know, everything we love. Including the precious rock we currently stand on; no matter how hidden you believe it to be.\" Adam said.\n\n\"We do not believe it hidden, because we do not hide. We do not cower from an enemy. Any enemy.\" one of the elders proudly stated.\n\n\"And I commend you for that. It is admirable. And though your hatred for the Benzans runs deep, we can all agree that they are a formidable opponent,\" Adam replied, watching the reactions of those in the room. One that spoke of the Benzans being a formidable adversary indeed. \"I saw with my own eyes, hundreds of our best warriors fall within an hour to the Priests,\" Adam stated, doing his best to chide his tears. \"Among them my wife.\"\n\nIt was an admission that struck home with Vladris, having lost the love of his life as well, though the wars were of a different manner. Loss of love feels one in the same.\n\n\"Now I'm asking you,\" Adam said, taking a moment to clear the emotional-cobwebs from his speech. \"I'm begging you,\" he added, kneeling to the elders; something that had never been heard of prior to Adam's gesture, at least by a Benzan. \"You have long been the mightiest warriors of the Skyla System; now I'm asking that you defend those who cannot defend themselves.\"\n\n\"I must admit Benzan, your gesture and statement seem true enough. How are we to know that you indeed speak the truth?\" one of the elders asked, this time an elder of military background.\n\n\"He speaks the truth,\" Sarah Blaine replied, walking into the chamber, her mere presence instilling faith into everyone who lay witness. \"I have known this man for a long time,\" she added, her voice turning to a softer tone. \"Even loved him. He speaks from the heart. I can personally validate that.\"\n\nIt was a statement that struck Vladris by surprise \u2013 though it made sense. He had picked up on Adam and Sarah's uneasiness around him.\n\n\"Very well. Speak to us of your plans Benzan.\" one of the elders insisted.\n\n\"Have you all gone mad? Can you not see that this is a colonial trap?\" Vladris yelled.\n\n\"Silence champion!\" Sarah demanded, showing her status as queen for all to see. \"Your caution as a warrior is welcome but your insubordination to your elders is not!\"\n\n\"Apologies my queen.\" Vladris replied, kneeling a bit though he did so reluctantly.\n\n\"The colonials plan an exodus voyage into the stars. This star system is lost,\" Adam said, thinking of a war they had no chance of winning. \"Of the few remaining planets that are uninfected, Ronica is the only one large enough to use as a staging ground for our journey. We will need time to prep such massive ships and pull resources together.\"\n\n\"You are running away?\" one of the elders asked, the entire chamber taken by surprise.\n\n\"We are beaten. Though it may be honorable to fight to the death, many of those who survive are simply women and children. There is no honor in that.\" Adam replied.\n\n\"Has it really come to this Adam?\" Sarah asked, a concern for those in the Skyla System deeply embedded into her.\n\n\"Yes, it has,\" Adam replied. \"We are overrun by a species that acts as though it is a virus. One for which we do not have the cure.\"\n\n\"And tell me Benzan. What would be the benefit of an alliance between us? What would my people have to gain?\" an elder asked, impatiently waiting for an answer.\n\nAs he thought the coming moments through carefully, Adam reached around his neck slowly, removing his Benzan amulet and throwing it to the floor.\n\n\"The benefit of becoming a single people \u2013 rather than nations divided by race.\" the Benzan replied.\n\nA gesture that sent even the eldest of vampires into deep thought for a moment.\n\n\"Please, I am asking you,\" Adam pleaded. \"I'm asking you to view me as a person in need, not a Benzan. Not anymore. We are all simply people who have been overwhelmed with grief and loss for loved ones,\" he added. \"It is my understanding that you protect the humans of Ronica because they could not protect themselves from war.\"\n\n\"We protect the humans on this planet because they fought us with courage. Our champion was once theirs. His courage earned their freedom.\" one of the elders stated.\n\n\"Our soldiers have died by the thousands, each of them with courage. Though I'm no champion, I am willing to risk my life against your own champion, if need be. My own race at stake.\" Adam replied.\n\n\"It would be a foolish move human. For your own sake.\" Vladris commented with a grin.\n\n\"I know,\" Adam admitted, turning to the large vampire. \"But I'm willing to sacrifice my own life to save those who cannot fight. Women...children. Are you? Are your warriors?\" Adam asked.\n\n\"Silence,\" one of the elders demanded. \"There is no need for our champion to prove himself. Vladris has done so before, countless times. Though we do respect your courage. Tell us of your intentions?\"\n\n\"Our fleet will be comprised of as many ships as possible, but every world in our system will be represented by a capital ship. That ship will receive a vote in whatever our future holds, and if you extend help to us, I can promise your people a vote to call your own.\"\n\n\"Permission to speak?\" Vladris asked, though he did so with great displeasure.\n\n\"Granted champion.\" and elder replied.\n\n\"We have seen threats like this before. When they arrive, we will cut them down, just as we always have.\" Vladris said.\n\n\"I guarantee you \u2013 you've seen no threats like this before.\" Adam stated.\n\n\"Speak to us of this threat.\" an elder insisted.\n\n\"The ability to infect a body, living or dead, by attaching to the spinal cord. Once they have infected, they can use the body just as the warrior was always of their race, bringing with it the battle-prowess of the warrior.\" Adam stated.\n\n\"In other words...\" one of the elders began.\n\n\"In other words, every time a vampire falls, he will join the fight against you. Your own abilities in battle will be your undoing.\" Adam quickly replied.\n\n\"Can the enemy be killed?\" Vladris asked.\n\n\"Yes,\" Adam replied. \"But we've found that the only sure way is by burning the small creature. Or by finding enough luck to strike it in half with bullet or blade.\"\n\n\"Then the trunk of my blade will feast indeed.\" the Hunters' champion said with a smile.\n\n\"Any final words before we begin our own deliberations Benzan?\" an elder asked.\n\n\"Yes,\" Adam replied. \"I would ask that you not consider the history of our two races, but the future. For every warrior here that is willing to die with the honor of battle, there must be a group of people that love him. A family worth saving. I ask that you consider preserving the proud history of your race by becoming part of a larger society. One that is prepared to accept you with open arms. Thank you.\" Adam added, bowing a final time as he began to exit the chamber slowly.\n\n\"Benzan,\" one of the elders announced. \"Your amulet.\"\n\n\"I've no further need for it,\" Adam replied, though his back remained turned. \"I'm just a survivor now.\"\n\nAs his body began to hit the light of day, which draped down from the peaceful Ronican skies above, a group of nearly twenty soldiers awaited him.\n\n\"Did it go well?\" a colonial soldier asked.\n\n\"I don't know,\" Adam replied. \"I don't know.\"\n\nDrift Planet \u2013 Sandila\n\n\"What do you make of it?\" Cambria asked, easing herself down beside Dalton as his concentrated stare into the distant continued.\n\n\"I think if I would have bluffed a bit more I'd have won every hand.\" he replied, using the quick wit of a smuggler.\n\n\"I mean the lottery you meathead!\" she said with a grin, slapping him across the arm playfully.\n\n\"Honestly, hell, I don't know what to think,\" Dalton said as he took a pause for the cause; the swig of whiskey burning its way to his stomach. \"Guess they are gonna do whatever they want to anyway. It's the political way. No sense in getting torn up over something you have no control over. I've never trusted the jack-jawed sumbitches and ain't about to start now.\"\n\n\"Dalton, I don't want you to hate me. I don't want that kind of awkwardness between us.\" Cambria said, her words falling as soft as drizzle.\n\n\"Hate you?\" he asked. \"We live in a world where the dead stalk the living and the living still can't be honest about their feelings. That's what I hate. Not you.\"\n\n\"I know there's something there. Something between us. I just don't know what, I guess I need time. That's all.\" she replied, her statement very candid and honest. \"There, I said it. Aren't you proud?\" she added with a soft smile.\n\n\"Oh. Good. I thought you were about to tell me I wasn't going to get paid or something.\" he remarked as they both began to laugh without hesitation.\n\n\"Well about that, it may be a little late. The whole zombie thing and all.\" Cambria added, causing them to laugh a bit louder.\n\n\"Darling, I hadn't been paid in so damn long that it doesn't even concern me anymore. I figure if I'm alive, got a pillow to lay my head on at night, a bottle of hootch in my hand and people around me that I care about,\" he said, pausing on that statement to glance to Cambria with strong intent. \"I got everything I need.\"\n\nA candid statement as well. One that led to Cambria leaning to the smuggler for a kiss.\n\nShe had thought he'd be completely happy with it, their lips finally having a chance to meet on good terms. However, as she leaned in, feeling a bit awkward, Cambria quickly pulled away and wondered what his hangup was.\n\n\"Dalton, I'm sorry. I shouldn't have.\" Cambria admitted as the smuggler sat there, eyes focused on a small window that was caked with the residue of cigar smoke aplenty.\n\n\"Dalton? Are you OK?\"\n\n\"Zombies! Get outside, quick, otherwise they'll swarm us and trap us in this fucking rat shack!\" he yelled, quickly turning to usher his love interest to the safest possible spot.\n\nRobbing me of my kiss \u2013 'ya undead fucks!\n\nGunshots began to snap out, dozens of survivors firing into a small group that sprinted toward their location.\n\nThey had banded together with others, nearly fifty total, and taken up residence in a very small village on the remote planet. It had been long-abandoned, which served the group just fine.\n\nEach of them knew that eventually their hideaway would be found by the horde they feared, but each had secretly hoped that somehow, by some grand miracle, they would be left alone once and for all.\n\n\"Dalton, what are you doing?\" Cambria yelled, slowing her run to watch the smuggler do a complete turnaround as he began heading into the direction of the vacated bar.\n\n\"Son of a bitches done took everything else from me,\" he responded with a yell of his own. \"I'll be damned if they are getting my whiskey too!\" he added, snatching up a half-empty bottle of rock whiskey before turning to sprint from the door once more.\n\n_Are you kidding me!_ She thought, ready to shoot him herself if he didn't realize the urgency of zombies approaching.\n\n\"He's got balls.\" one of the taunting bar patrons yelled, each of them slowing their sprint of desperation long enough to watch the slightly-tipsy smuggler risk his own neck for a bottle of bargain bin man-drink.\n\n\"Bunch of nasty bastards!\" he yelled, sending two rounds from his magnum-style pistol before turning and beginning the quickest sprint of his life.\n\nThe survivors had dug into the small group of dwellings which was built within a thick nestle of countryside, though they knew that eventually it would come to this. Still, it had been their home for weeks. Each of them growing to appreciate it for what it was. Quiet.\n\nIron Grove \u2013 at least the name of it prior to the plague of man. Most of its citizens had long abandoned the sad excuse for a village, heading out on any available transport they could.\n\nSome stayed. Choosing instead to defend their homes to the death, which looked to be beating down their door like an unwelcome salesman. One that peddled agonizing mortality.\n\nWhat they didn't expect, however, was literally hundreds of Priest-controlled zombies that began filling the nearby hillside, as if it were certified death pouring into a tub of survivors.\n\n_Oh shit._ Dalton thought, his mouth hanging down a bit as even he couldn't fathom an escape from such an ill-fate.\n\nAs quickly as his alcoholic buzz seemed to fade, Dalton realized that the survivors were backed into a corner. A bad thing considering their opponents. He understood the vile tendencies of a zombie, priest, or as he called them, nasty fucks.\n\nDalton understood the survivors were outnumbered ten to one, and had no strategic advantage. His military wisdom finally coming into play.\n\nThey had weapons, sure, but they were about to be swarmed and had the low ground. A snowball's chance in hell. Or, even worse, a rock whiskey bottle's chance in a Glimmerian bar. Zero.\n\n\"Keep running,\" Dalton yelled to Cambria. \"Past our outpost. Get our boy and tell him to elbow to asshole up into the hillside as fast as he can!\"\n\n\"But we would be safer indoors? Nightfall is coming. We don't even have supplies ready?\" she replied.\n\n\"Cambria,\" he shouted, grabbing the woman firmly by both of her arms as gunshots rang throughout the encampment. \"These people are already dead. In a couple of minutes, that horde will be here. A couple of minutes later, everything in this settlement will be dead. Just ain't no way these sad-ass wooden walls are going to hold them back. Get Skulls and head to that brushy part of the hill. Trust me.\"\n\n\"OK...I do.\" she replied with shock.\n\n\"Go.\" he added, the blue-haired bombshell turning to relay the message to their friend.\n\nThere are things in a man's life that he can never shake. Events that plague his dreams, night after night, for the remainder of his life \u2013 and this was one such moment.\n\nAs Dalton readied himself for the forced run ahead, his eyes were witness to survivors, many of them women and children, heading inside to escape the coming horde. He knew they would not last inside of a handful of minutes, and though he wanted to help them escape, it simply wasn't possible.\n\nIt was the first time in many years that Dalton James began to cry aloud. His tough image being overcome by his heartfelt sadness for those who would soon fall victim to the plague of man. The Priests.\n\n\"Dalton,\" Cambria yelled loudly, pausing her wording as she began to watch him, tears falling steady. \"Dalton, we need to go. You said so yourself.\"\n\nSkulls had joined her, the sniper also understanding the need for a quick escape. The settlement was beginning to flood with zombies and would soon be nothing but.\n\n\"Dalton please.\" Cambria said, grabbing his arm and captivating his thoughts for a single moment. Long enough to break him from sadness and throw him back into the military frame of mind.\n\n\"Yea,\" he admitted, quickly drying up his tears. \"Haul ass, I'm right behind you.\"\n\nThe small group of three quickly slipped out through the side of Iron Grove which had not been slammed by zombies yet. Using the cover of a casting nightfall to mask their escape, though they would need to find cover only a few hundred yards from the deafening sound of painful murder.\n\n\"Are you alright?\" Cambria whispered.\n\n\"Yea, I,\" Dalton began to reply, all three survivors nestled snugly into the thick of brush near enough by to witness the genocide unfold. \"I'm sorry you had to see me like that. I don't know what came over me.\"\n\n\"I do.\" Cambria replied, easing her hand onto his, their fingers clasping together as she tried to comfort him.\n\n\"It's one thing when my own ass is on the line, or even soldiers around me,\" he admitted, trying to block the last remaining screams from his mind. \"This is something different altogether. These people ain't equipped for this...they ain't soldiers.\" he added, thinking of the women and children he'd witnessed only minutes before. Each of them butchered by such a vile species.\n\n\"No,\" Cambria replied. \"They don't deserve a fate like this.\"\n\n\"We just have to make sure they did not die in vain. Avenge each of them by the smooth grain of a bullet.\" Skulls added, his right eye pressed firm to the large scope which mounted onto his bolt-action rifle.\n\n\"You can bet your ass on that,\" Dalton responded, nodding his head with purpose. \"Each one of these rank sumbitches are gonna pay.\"\n\nFor nearly an hour, Dalton, Cambria and Skulls had to remain patient. Hiding from plain sight as the walking dead scavenged over those freshly-slain. Eventually raising them from the dead and increasing their army's numbers.\n\nDalton had heard rumors of it, but never seen the turning with his own eyes. The Priests were a small creature, hard shell around it as though it were a sea crab, though it also had eight stringy legs and a rather long tail.\n\nFor the first time, the three survivors watched with horror \u2013 the way of the Priests.\n\nThey slowly attached to the back of a skull of the dead, wrapping clingy legs around to the victim's mouth. Then, as the creature gripped the mouth of the dead, it drove the long tail into the upper spine of the lifeless body, snaking the tale down. Finally covering the entire length of the spine from its inner most spot.\n\nThen, seconds later, the fallen host began to stand. Becoming a soldier for their cause, completely under the control of the Priest which embedded into it.\n\nThe perfect army. The Priests would lose soldiers in battle, but as they fell the small creature would simply detach and then assume control of another body. The entire process taking less than sixty-seconds, depending on how close the body of a fallen host was.\n\nFrom the perspective of a man who put in countless years of military experience, Dalton quickly understood why the populated worlds had fallen so swiftly. Why the colonials were getting their asses kicked. You could kill the Priests, but they would continue to come, using your own fallen soldiers against you.\n\n\"I've never seen anything like it.\" Cambria admitted, though she did so in a whisper as they watched.\n\n\"No,\" Dalton replied, pausing to swig deeply from his bottle of whiskey. \"I've seen a lot of shit before, but I ain't never seen anything like that.\"\n\nRonica\n\n\"He looks just like you.\" Sarah admitted.\n\nA true statement \u2013 the small child looking just as Adam did at the age. The queen of vampires even began to, for a moment, imagine the infant with a duster to its back and a revolver on his hip. A smuggler in training.\n\n\"Thank you,\" Adam replied warmly. \"Many nights, Avery has pushed me forward when I didn't think it was possible. My son has been a godsend when dealing with emotions.\"\n\n\"The loss of your lover?\" Sarah asked.\n\n\"Yes, among other things.\" he admitted.\n\n\"Other things?\" the queen asked.\n\n\"Guilt,\" Adam said, his tone of voice changing to one of true sorrow. \"My group first discovered the Priests. They came back to the Skyla System by way of our ship.\"\n\n\"The Benzans?\" she asked.\n\n\"Yes. Though we didn't realize it at the time.\"\n\n\"Does anyone else know this?\"\n\n\"No,\" he replied. \"And I would really appreciate if you kept it that way. I'm sure the blame of the lost lives would be thrown into my lap.\"\n\n\"Your secret is safe with me Adam.\"\n\n\"I've come to the point where I feel I should let them know. Let them throw the blame for all of this on me. I already harbor the guilt for it.\" Adam responded.\n\n\"Yes, but you did not mean to bring the Priests back? Correct?\" Sarah asked.\n\n\"Of course not,\" he replied. \"We set out in search of a Benzan colony. One that was logged into their history \u2013 its location secret to all but the highest-ranking.\"\n\n\"And you found them?\"\n\n\"Yes,\" he replied. \"Yes we did. It explains the ships and the weapons. The robe even. As we pleaded for their help, finally getting them to agree, we began stocking resources for the return trip.\"\n\n\"Their help in what?\" she asked.\n\n\"Destroying the Hunters.\"\n\n\"You meant to destroy us?\" Sarah asked, taken back a bit by the Benzan's confessions.\n\n\"Well to be fair, I had no idea you were here. When I last saw you Sarah, you were the pinnacle of hope for the colonials,\" he pleaded. \"We were coming back to destroy a race that had taken so much from the Benzans. A race that has taken so much from you.\" he added.\n\n\"They've taken my father, this much is true,\" Sarah countered. \"But they've given me so much more.\"\n\n\"What? What could they possibly have given you that could replace the loss of your father?\"\n\n\"A true understanding of things around me. A true understanding of life.\" she replied.\n\n\"I'm not going to pretend to understand that. I'm just going to say that you seemed to have a good understanding of life before all of this. You had a big heart and a love for your people.\" Adam responded, patting his small son on the back a bit.\n\n\"I'm not the woman I once was Adam.\" she replied.\n\n\"Yea, I'm getting that,\" he responded, looking to her with a great deal of emotion. \"And I harbor the blame for that as well.\"\n\n\"This is not your doing Adam, and it's certainly not a bad thing.\" the queen said with a smile.\n\n\"Yes it is. It is directly my fault. Had I just forgiven you, we would have walked away together and been married. I know that to the very pit of my soul.\" Adam replied.\n\n\"I can still remember you flying away. I remember watching your ship hit full-burn through the glass window of my colonial shuttle.\"\n\n\"Sarah, I'm sorry.\"\n\n\"Don't be. Vladris was too strong and remains so. Had you stayed behind he would have killed you and I would still be here as I am now.\" Sarah remarked.\n\n\"Perhaps. I guess we'll see,\" Adam replied. \"Sarah, I will save you from all of this.\"\n\n\"Adam, it's alright. I'm content with my life now.\"\n\nCutting his eyes to her to reaffirm his statement, the Benzan smuggler told her once more.\n\n\"Sarah, I WILL save you.\"\n\nAboard Colonial Star Triumph\n\nCommander Ortega remained standing, hands behind his back and clasped together while his eyes soaked in the view. The Drift planets seemed so peaceful from above. So relatively unscathed by the zombie plague, though only on the surface.\n\nAs his ship orbited the heavens above, coasting through the stars with a manner of technological beauty, the commander knew that the calm of dusty brown and pale red below was anything but unscathed.\n\nHe'd gotten reports for the past few months of zombies overrunning the planets. The local armies immediately falling to the mercy of such a threat while its citizens were no doubt in hiding and praying for a ship such as the colonial vessel Ortega stood aboard.\n\nHe had never quite understood their desire for such a primitive lifestyle. Blatantly shunning readily-available technology at every opportunity.\n\nThe Drifts had earned a reputation for being primitive, yet beautiful. Spaceships replaced with airships of glass and steel, crafted in a Victorian-style that had long been the signature of the planets below.\n\nA vacation spot, if nothing else, for the people of the system that wished to escape the harsh work schedules which seemed to drag most down. Busy flight schedules by cramped shuttles, flying them to the surface of mundane planets and even worse, to unexciting routine jobs behind desks.\n\nThe Drift planets were of a different sort. They allowed only the essential technology, which meant two things. If you didn't work hard, you didn't eat. Their food came by way of farms across the lands of each planet, trading among themselves and priding themselves on being self-sufficient. The other, of course, is that it was the place to be rowdy. If you had it in you.\n\nThroughout the Skyla System, fighting was heavily frowned upon and usually punishable with jail time. The more advanced society became, the more restrictions governments seemed to place on 'free' people.\n\nThe Drifts settled their arguments in the same manner. Fist to cuff. If a disagreement was had, fighting tended to ensue. Of course when the parties involved were done, they'd usually patch things up over a cold pitcher of beer.\n\nTheir shunning of technology, however, had placed them at a huge disadvantage when it came to the zombie outbreak. They received no warning, nor could they radio for help. Not that they would have regardless.\n\nCommander Ortega knew the rules. He was forced under colonial code, even under such trying times, to make every attempt possible to contact their governments first.\n\nThe Drifts fell under colonial rule, but did not adhere to the same rules. That had been agreed upon many years before Ortega came to power, and though he did not agree with their primitive way of thinking, the commander did respect it.\n\n\"Ready and awaiting your orders sir.\" a high-ranking soldier said, firmly standing in the doorway of command center aboard the Colonial Star Triumph.\n\n\"You've attempted to hail the governments?\" Commander Ortega asked.\n\n\"Yes sir,\" the soldier replied. \"We've made several efforts to reach each of them. Our grid shows no airborne traffic and we cannot reach them through our com systems,\" the soldier added. \"However, we have picked up droves of heat signatures on the surface. All of them consistent with what we've seen across the Skyla System. The Priests are indeed here.\"\n\n\"Very well. Begin taking the fight to them, and advise our soldiers to fire only if they can confirm their targets. These people may live simple lives, but they are still people. Our people. Let's protect the innocent and exterminate the guilty.\" Ortega said.\n\n\"Yes sir, at once.\" the soldier replied, turning to execute his direct orders.\n\nCommander Ortega glanced from a large window that looked across the stars once more. He had always been taught to think of the Drifts as a sanctuary, though he had never personally visited. Many times he'd found himself understanding what must drive people to live here, among the fringe planets of deep space. Family, love, a man's word and the benefit of hard work.\n\nThings the rest of the Skyla System had long forgotten.\n\nThe commander had taken a personal interest in protecting the most innocent civilians he could think of, and though the colonials were losing a war on all fronts, he'd be damned if he would allow such a peaceful people to fall victim to the Priests.\n\nHis thoughts were backed with the remaining might of their fleet, evidenced by dozens of smaller ships, each full of soldiers who were armed to the teeth, descending to the planets below.\n\nThey would confirm their targets and then begin to string napalm in the areas affected. A second wave would provide colonial boots to the ground in an attempt to break the spine of such a large zombie force and finally, his diminishing fleet would deliver supplies to those who were in need. Providing hope to those who wished to remain with their homes, while offering to bring any survivors aboard his ship for protection.\n\nAssuming they had arrived in time and his men were swift when taking the fight to the undead army.\n\nGodspeed.\n\nRonica\n\n\"They must think us fools!\" one of the largest Husk stated firmly.\n\n\"Relax Rylak. We must bide our time and strike when it is meant to be.\" Troy responded.\n\n\"If we take out the head of the snake, the rest of it will fall with swiftness.\" Rylak replied.\n\n\"I agree,\" Troy stated. \"But I have seen firsthand the abilities of the head of the snake. Vladris is not to be taken lightly. We must strike when it is least-expected. That is our only chance of winning.\"\n\n\"Then we will bide our time.\" Rylak agreed, each of the men bowing their heads slightly. A common sight among the might Orc-like Husk.\n\nAs Rylak and several of the muscle-heavy warriors walked away, each of them keeping their plans of deceit closely guarded, Troy began to think of a lot of times gone by.\n\nA man who was like a father to him, not to mention the greatest warrior he'd ever known. Roman Raines. Cut down by Vladris following one of the most epic battles recorded.\n\nTroy began to wonder if Roman would approve of his plan. If Roman would encourage the act of revenge or warn Troy off for his own safety. The young man had been trained in the arts of Husk war, and had excelled during each and every turn.\n\nHe had become one of the best warriors among his people, even if he was human at the core. What he lacked genetically, Troy made up for with speed and wit, becoming one of the most respected warriors in Husk armor.\n\nStill, he knew of Vladris in battle. For many years the mighty Husk had tried to end the champion vampire. By all accounts, Vladris was responsible for hundreds of slain Husk, which provided extra incentive for the young man. Not that he needed an extra push.\n\nWatching a man who had been like a father to him, a man that had first trained him with a combat blade \u2013 watching him fall in battle was push enough.\n\nTroy knew that their time for battle was coming. He planned to strike the lion by surprise and then earn his right among Husk royalty by ending the reign of terror that was Vladris.\n\nAs Vladris approached Adam and his escort which was comprised of colonial soldiers and Benzan warriors, the man stood to his feet.\n\nA former ship's captain, colonial lieutenant and admitted smuggler, Adam had walked many paths in life. Though he understood the most important path was the one which lay beneath Vladris' feet as the vampiric warrior approached.\n\n\"Our elders place great faith in you Adam Michaels. They will allow your fleet to land on Ronica and use our far side as a staging ground under the terms that you proposed,\" the vampire said, his words firm. \"However.\"\n\n\"I'm listening.\" Adam replied.\n\n\"Should I even suspect this is a ploy; a trap to overtake our fine planet,\" Vladris remarked, leaning in closely. \"I won't hesitate to have my finest run you and your precious survivors down like fleeing cattle.\"\n\n\"That's not going to happen.\" Adam said clearly, though it remained unclear as to which part of Vladris' statement he was referring.\n\nThe champion vampire started at him for several moments, both of them unrelenting in eye contact; making it feel as though it were an eternity.\n\n\"Should I find out you're here for Sarah,\" the vampire warned. \"You'll suffer a far-worse fate than death.\"\n\n\"I'm just here to help my people.\" Adam replied.\n\n\"We'll see.\" Vladris warned, turning to begin a long journey back into the castle.\n\n\"A bit of a prick, yes?\" one of the colonial soldiers asked.\n\n\"Yea,\" Adam replied. \"I would be too if this were my home.\" he added.\n\n\"Vladris!\" Adam yelled aloud, jogging slowly as he caught back up to the seasoned warrior. \"Thank you, truly.\"\n\nAs he finished his words, Adam extended his hand to the warrior of vampires, a gesture that was unheard of.\n\n\"Are you sure you want to shake the hand of the warrior who cut your friend Roman Raines to his grave?\" Vladris taunted.\n\nIt was stunning news to Adam Michaels, a confession that mustered up a lot of hatred for Vladris as the smuggler considered drawing his revolver.\n\n\"Yes,\" he finally admitted, hand still extended. \"The fighting has to stop somewhere for trust to begin.\"\n\n\"Interesting,\" Vladris replied, studying Adam a bit. \"Prove your integrity to me Adam Michaels.\" the vampire added, shunning a handshake as he turned to head back into the castle, passing his queen on the way.\n\n\"My lady, you should be under guard at such times.\" Vladris said gently with a bow.\n\n\"I trust Adam and his group. Perhaps you would be wise to do the same?\"\n\n\"My queen, I mean no disrespect,\" Vladris replied, biting a tongue that was growing ever-sharper for Adam Michaels. \"It is in my nature to question the intentions of a race which has hunted us down for centuries.\"\n\n\"I understand your reservations,\" Sarah responded. \"And you are a better champion for it. I have known Adam for a long time. I could easily see through his story if it were not indeed the truth.\" she added.\n\n\"Of course my queen.\"\n\n\"Gather some men and begin to clear enough area for what I understand to be a large grid of ships landing,\" she ordered. \"And Vladris,\" she added, keening the champion's focus on their conversation a bit. \"Keep our warriors posted within the staging area at all times. As a precaution.\"\n\n\"Of course my lady.\" Vladris replied with a grin, knowing Sarah's human side trusted Adam thoroughly, though her vampiric side trusted no one.\n\n\"May we speak Adam?\" Sarah asked, approaching the embattled smuggler as he motioned his small group away.\n\nThere was a time when they communicated with the simplicity of a glance, the emotion of a kiss. But those times had become politically incorrect, which saddened the man throughout his heart.\n\n\"Of course.\"\n\nHe listened, though he wanted to talk. Scoop her up and take her away, just to talk for hours. His heart, his very soul, having missed her so badly that it physically hurt.\n\n\"I wanted to say I'm sorry,\" she admitted. \"About Sasha I mean. I had no idea.\"\n\n\"I'm sorry, is that sarcasm?\" he asked, truly questioning her display of emotions for a woman he'd left with.\n\n\"No, no. I am truly sorry. Becoming a Hunter has heightened my senses greatly and I can see the sadness in your heart. For that I am very sorry.\" Sarah responded.\n\nBreaking down just a bit, Adam refrained from tears, though it seemed nearly impossible.\n\n\"Part of that is indeed for Sasha. Her loss has been unbearable. But much of the sadness you pick up on is for you. Seeing you here, knowing it's because of a choice I made.\"\n\n\"Don't be silly, I'm fine with being here. They have accepted me and their race is truly remarkable.\" Sarah replied.\n\n\"When I look at you, I don't see sadness. I just have the eyes of a smuggler.\" he said with a grin, bringing a grin to her face as well.\n\n\"You always had wit. I'll give you that much.\" Sarah replied, her face warming with an extended grin.\n\n\"I see a woman who I was ready to marry and call my wife. I see war after war, each of them having pulled apart what was once my sole reason for living. I see what should have been.\"\n\nHis confession hit like a blindside of truth, stirring up many emotions that Sarah had dealt with as a human.\n\n\"Now if that complicates matters, I'm sorry. I thought the loss of Sasha was unbearable, and at times it seems that way. But looking at you and not admitting my continuing feelings for you, that's unbearable. I'd rather be cut down by Vladris' blade than pretend I don't still care for you.\" he added.\n\nSarah was shocked by his statement, turning to retreat from such truth into her castle, stopping long enough to ask one final question.\n\n\"Do you mind if I ask how Sasha died?\"\n\nIt was a very blunt question, though such an attitude was not uncommon among the Hunters.\n\n\"She died protecting our son,\" Adam replied, choking back tears. \"The Priests took our ship and there was a single lifeboat left. Two man vessel,\" he added. \"I insisted on staying behind to provide them both time to escape, but Sasha pleaded with me, said I would be Avery's best chance of survival. That I would be able to protect him in the long run,\" Adam said, a single tear beginning to trickle from his eye. \"Sarah, I have to protect my son.\"\n\nTo avoid tears on her own end, Sarah nodded her understanding before turning to enter the castle. Adam's confession of a child and his helplessness in defending the young boy leading her to a single conclusion.\n\nAdam's son would be protected, even if it meant the death of every soul within the Hunter race.\n\nDrift Planet \u2013 Sandilia\n\n\"I don't understand why we don't stay put now that they're gone. We have walls around us and a roof over our heads?\" Cambria asked.\n\n\"Best I can tell, these things are like stray dogs. They seem to move all over the place with no real direction, but I've learned enough about the son of a bitches to know that they do move with direction. They jump from place to place, each one a familiar spot,\" Dalton replied. \"I just think it's best if we scavenge what we can from the village and then hike our asses back up a bit. Find some well-hidden high ground.\"\n\n\"I don't know if I can,\" Cambria admitted. \"Not after getting to know these people.\"\n\n\"I understand,\" Dalton replied. \"You stay put here and cover our asses. Besides, there's a good chance the fuckers dropped ailing bodies for fresh ones.\"\n\nCambria nodded, letting Dalton know she understood. Though she appeared shaken.\n\n\"We'll be done in fifteen, just keep your eyes open,\" Dalton said, comforting the woman with a meaningful hold of her upper-arm. \"Come on worker bee, we got shit to do.\" he added, turning to Skulls with a smile.\n\n\"What if I don't want to be worker bee? What if I want to be something else?\" the sniper replied with sarcasm.\n\n\"Carry your ass.\" Dalton replied, cutting through the chase as the two men walked slowly from the cover of brush. Entering a freshly-fallen camp under the cover of darkness.\n\n\"Looking for anything we might need for survival. Guns and grub, obviously, but anything that can make it more comfortable too. Blankets...\" Dalton whispered, the two men having made it back to the deathly-quiet camp.\n\n\"Whiskey?\" Skulls asked with a sarcastic eye roll.\n\n\"Well if the shoe fits.\"\n\n\"I figured as much.\" the sniper replied.\n\n\"Hey, don't knock my way of getting through things,\" Dalton scolded, though he continued to whisper in doing so. \"We all deal with shit in different ways, and I've been through enough for the both of us.\"\n\n\"It's cool, I get it.\" Skulls admitted.\n\n\"How about you then? How do you deal with shit like this?\" Dalton asked.\n\n\"Me? I just block it out. It plays out in front of my eyes, but I don't store it to memory.\" Skulls replied, rolling the body of a grown man over in search of weapons, instead finding a slain toddler in which the grown man had tried to protect from the horde of undead.\"\n\n\"Yea. Good luck blocking that shit out.\" Dalton said.\n\nAboard Colonial Star Triumph\n\n\"Sir,\" Doctor Arness said as he slowly entered the private quarters of Commander Ortega. \"Preliminary reports show mostly wasteland down below. We've found a few pockets of survivors, but mostly just rolling hordes of Priests.\"\n\n\"Keep the search efforts going.\" Ortega replied.\n\n\"Sir, if I may,\" Arness replied. \"These search choppers are using fuel that will eventually need. The tradeoff for only a handful of lives may not be worth it in the end. It may be in our own best interest to suspend the searches and focus on supplying our exodus fleet.\"\n\n\"I'm not in the business of doing what's in my own person interest. I'm in the business of doing what's best for my people. We don't know how many survivors are down there, but we know they are down there and in need. I won't leave them behind. You have my answer doctor. Continue the searches.\" Ortega said.\n\n\"As you wish sir.\" Doctor Arness replied with a nod.\n\nWalking from the room, the doctor slowly closed the thick door of the commander's quarters.\n\n\"Well?\" a second adviser asked as they both walked from the commander's quarters and into the bustling interior of the large colonial ship.\n\n\"Our commander has lost his way,\" Arness said without hesitation. \"He puts the survival of our fleet in great danger by continuing this pointless search for survivors.\"\n\n\"Perhaps now is our time to strike?\" the adviser asked. \"I have the support of several, many within the commander's own military ranks.\"\n\n\"No,\" Arness replied. \"At least not yet. Our move to power hinges on timing and execution,\" he added. \"If we jump too soon, Ortega will look as though he is a victim, and will have the sympathy of the fleet. However, if we wait for opportunity, eventually one will present itself. At the first overstep of his power, we can strike and look as though we are heroes in removing him from power.\"\n\n\"I understand,\" the adviser replied with a smile. \"It is a solid plan indeed.\"\n\n\"It hinges on your ability to gain support for our cause,\" Arness said, staring onto the slim man of fine clothing. \"We must have a good portion of military with us, otherwise those who back our great commander will stall our efforts quickly.\"\n\n\"Understood,\" the adviser replied. \"I'll get right on that.\"\n\n\"Cautiously,\" Arness warned, pausing his conspiracy partner's walk. \"Be weary of including someone who may not share our vision of the fleet.\"\n\n\"Consider it done.\"\n\nDrift Planet \u2013 Sandila\n\n\"Best be getting some sleep.\" Dalton remarked, turning to welcome Cambria, who eased into a sitting position beside him; both of them sponging in the serenity of a cold \u2013 but clear night.\n\n\"Hard for me to sleep, considering.\" she replied.\n\n\"Yea. I get that way with these stank-undead fucks roaming around here too.\" he said with a grin.\n\n\"Actually, I meant that,\" she replied with a smile of her own. \"Stank-undead fucks aside; it's a beautiful night. A bit chilled with a slight breeze. Quiet. Almost seems like a normal moment. At least what I can remember of normality.\"\n\n\"Yea I,\" Dalton said with pause, eventually removing his brown duster and offering it to the welcoming hands of Cambria. \"I can't get away from leaving those people behind. If there would have been any other way...\"\n\n\"I know,\" the beautiful woman replied, snuggling into the oversized coat and feeling more secure than she had in a long time. \"You didn't do anything wrong back there. If you hadn't of acted so quickly, Skulls and I wouldn't be here.\"\n\n\"I reckon.\" Dalton said, turning for a moment to watch the sniper rest, a deep sleep to be envied.\n\n\"I also have to admit,\" Cambria said, her smile turning a bit more serious. \"There's something to this brown coat. When I first met you, I was skeptical, but now I get it. You're a complex man Dalton James.\"\n\n\"Not by choice, I'll guarantee you that,\" he replied with truth. \"I've just been on the move so damn much I figure I'll bring the comforts of home with me. That usually means whiskey and a coat that doubles as a blanket.\"\n\n\"Sorry I've dragged you into all of this. Truly.\" she admitted.\n\n\"Ah, no need for that. I've been running with the undead at my heels for a long time now. Got nothing to do with you pretty lady.\" he responded.\n\n\"Also sorry for not seizing the moment. With you, I mean. I have this stupid way of not letting anyone get close to me. Even those I love.\" Cambria confessed. Her eyes seeming to gaze onto him with a sky-blue truth.\n\nAs Dalton began to lean in for a kiss, one that both had longed for; the smuggler froze. Staring into her eyes, though seemingly distracted.\n\n\"I'm sorry. I shouldn't have...\" Cambria mumbled with regret.\n\n\"Shh.\" Dalton replied, holding the rough feel of his finger to her soft lips.\n\nThough it seemed an awkward situation, Dalton remained silent, his finger pressed to her lip. Partly because of his attention focused elsewhere, and partly because of her lip feeling so tender. So desirable.\n\nAt that moment, loud blasts streaked across the sky. Each of them trailing from colonial ships traveling at high speeds.\n\n\"Well I'll be a son of a bitch!\" Dalton yelled, understanding their chances of survival had just increased drastically.\n\n\"Does this mean we're safe?\" Cambria asked, turning to Skulls who had snapped immediately from his dream world, battle-hardened grasp onto rifle.\n\n\"Only if we figure out a way to flag 'em down.\" Dalton replied, scrambling around and thinking the situation through.\n\nFinally, he began pouring what remained of his whiskey onto the ground. The dense soil soaking in both alcohol and curse words as Dalton felt bittersweet over the sacrifice. Striking his favorite cigar lighter, he then lit the alcohol and began grabbing any patch of desolate timber he could find.\n\n\"I knew I did the right thing by bringing you in. I knew your experience would eventually save our lives!\" Cambria said with a yell, vibrantly smiling as she thanked the Gods above.\n\n\"Yea, well, you haven't paid me yet.\" Dalton replied, his joking a sure sign of relief as the sounds of thrusters neared their makeshift bonfire, while others streaked across the night sky and slapped burning napalm to the ground.\n\nA moment later, Dalton was paid in full as her lips locked onto his; her statement one of truth. She was committed to the man who had earned her trust, and stolen her heart. Even if he reeked of cheap tobacco and tainted whiskey.\n\nRonica\n\nIt was a moment in which history would never forget. The colonials had begun staging their fleet on the surface of Ronica; large ships undergoing upgrades as they were stocked with supplies and munitions.\n\nHowever, the historic moment before them was much more. As the Husk began to land, topped off eventually with a glimmer-green shuttle that was carrying their leaders, Hunters awaited.\n\nFor many years, thousands upon thousands, the Hunters and Husk had been at war. They had slain one another at every chance for as long as time could remember. The mighty Orc-like Husk clashing with the Hunters, a vampiric community which had remained shrouded in mystery.\n\nAs the shuttle began to open, however, their mightiest warrior was quick to walk onto Ronican soil. A sign that caution had been thrown into the wind during such perilous times.\n\nGore, the musclebound Husk, quickly approached the Hunters' welcoming party; which included several of vampiric society, among them Sarah and Vladris.\n\nSuch an historic event turned heads, and rightfully so, as everyone seemed to pause and watch the exchange. Both curiosity and caution to blame.\n\n\"My people appreciate your help. If I, or any of my kind can be of assistance, please let me know.\" Gore said, extending his arm for a handshake.\n\n\"What would our ancestors think if they saw their two finest shaking hands?\" Vladris questioned, staring directly into the eyes of such a mighty Husk.\n\n\"They would know that we are at the end of times. Yes, I truly believe they would want their finest to band together and write history against such an unbeatable foe.\" Gore replied, his hand remaining extended.\n\n\"Perhaps,\" Vladris replied with authority. \"And if you fight them with as much tenacity as you have fought us, together perhaps,\" Vladris added. \"We can prove them beatable.\" the mighty vampire replied, finally offering his hand.\n\nIt was merely a handshake, though both warriors knew it would be talked about for centuries to come. Perhaps longer, if anyone survived to speak of it.\n\nThough each warrior had fought for a different cause, they knew of the others ability in battle. Each becoming a legend to his own race while remaining a menace to the other. For them, the handshake symbolized a show of respect among warriors, nothing more.\n\nThough it was a sight that most remained locked onto, Sarah quickly pulled her eyes away from the warriors.\n\nHer attention fell to the shuttle as a familiar face exited, accompanied by one of the Hunter's own hellhounds.\n\n\"Troy?\" the queen said in disbelief, walking from her own group's protection in order to greet the boy who had turned into a young man, wrapped tightly in the leather of a Husk warrior.\n\n\"Cookie!\" the young man yelled, scolding the hell hound for baring its teeth as she approached.\n\n\"Cookie?\" she asked with a grin.\n\n\"Couldn't come up with anything better at the time.\" he replied, still possessing a child-like innocence.\n\n\"I feel as though I owe you an apology.\" Sarah admitted, sincerity in her voice.\n\n\"You owe nothing. What is in the past, remains in the past. I just thank you for providing us safety as we try to save as many as we can.\" Troy replied, the large sword of a Husk tied to his back with leather strapping.\n\n\"Of course,\" Sarah replied, looking to him with so many memories. \"You've grown into a strong young man, and, from the looks of it, a mighty warrior. This makes me proud.\"\n\n\"The Husk have shown me the path of a warrior. They have accepted me as one of their own, and for that I am truly grateful,\" Troy admitted. \"Though I continue to wish the best for my old friends.\"\n\nShe answered his statement with a smile, though his mention of Adam sent her attention elsewhere; eyes scanning the background as they finally locked onto Adam Michaels.\n\nOne of the system's most notorious smugglers, not to mention a ranking member of the Benzan Mafia \u2013 at least what remained of it. Yet there he stood. Off to himself a bit and holding a child, no more than a year old, in an attempt to calm his son.\n\nSo many ships around, their massive cargo holds being filled, and colonials briefing others on the flight to come. Hulking vessels being welded for the trip to come. Firming their exterior for a trip that may indeed prove to be one way.\n\nOthers tested thrusters, short bursts of flame firing from engines as pilots gauged their controls; watching as swordfish fighters were loaded into the landing bays.\n\nFor all of the commotion, as though a small city had quickly formed by way of connected ships, there seemed to be a calmness in Adam Michaels. As if he held a son and thought nothing of the turmoil around him.\n\n\"You still love him, don't you?\" Troy asked.\n\n\"Of course not, things are different now. Complicated.\" Sarah responded. Taken off-guard a bit by the young man's direct question.\n\n\"The world complicates things, but the heart doesn't,\" Troy said, nodding his head a bit. \"Just some advice from a Husk-raised human with a dog I can't get rid of.\" he added with a smile.\n\nSarah laughed a bit, though her mind did begin to think of any possibility of a future with Adam.\n\n\"All joking aside, you should tell him. Especially now that all face the certainty of death. No telling how many days we have left to make things right.\" Troy remarked.\n\nShe began to wonder what he had meant about certain death, knowing the exodus fleet was supposed to be their solution for survival.\n\nAboard Colonial Star Triumph\n\n\"I must admit Dalton James,\" Commander Ortega remarked. \"Your ability to survive time after time amazes me.\"\n\nThe commander sat behind a large military-grade desk in his personal chamber, reviewing a file of the brown coat laden survivor who sat directly in front of him.\n\n\"Timing really. I can't take all of the credit.\" Dalton replied, feeling a bit out of place.\n\n\"Well, be that as it may, good timing seems to follow you,\" the commander said. \"Two major wars on Glimmeria, skirmishes with the Hunters and now this? Some would consider you to be a beacon of good luck.\"\n\nDalton thought of that statement. All of the near death encounters with Hunters. The hordes of zombies giving chase. He couldn't understand being a beacon of good luck, having always considered himself the exact opposite.\n\n\"How would you feel about swearing in as a colonial soldier once more?\" the commander asked.\n\nIt was a right hook to Dalton's expectations, stumbling through a list of responses. He had expected to be given a steak dinner, slapped on the ass and dumped off at the next available planet. Not even remotely considering the possibility of enlisting once more.\n\n\"Well sir, I haven't even given it thought to be honest. I haven't showered in a week, and just yesterday saw a lot of innocent people butchered by the undead. I just,\" he replied, taking a moment to emotionally collect himself. \"I just need a few moments to wind down and get past what I've seen. What I've lived through.\"\n\n\"I understand, and your concern for the innocent is both obvious and admirable. The exodus fleet will need people just like you if it is to survive. So you think on it Dalton James. I've signed off on fast-tracking you into the fleet,\" Ortega said. \"It's now a matter of paper that needs your signature.\"\n\n\"I don't understand?\" Dalton asked.\n\n\"It means your record speaks for itself. I'm deeming you too important to leave behind,\" the commander said, grinning a bit. \"And too lucky.\" he added. \"Now, go get some rest and sort your priorities out a bit. Then come back and let me know what you decide on becoming a colonial officer.\"\n\n\"Thank you.\" Dalton replied.\n\n_Officer? Just yesterday I was drinking the cheapest of hand me down whiskey?_ The good-fortune patterned smuggler thought.\n\n\"May I ask you one question sir?\" Dalton asked, standing to his feet.\n\n\"Of course.\" Ortega replied willingly.\n\n\"I saw you on the com stating you were staying behind when the exodus leaves. Just want to know why?\" he asked.\n\n\"Well,\" the commander began to reply, softly laying the Dalton's folder onto his desk. \"I believe that if humanity is going to survive, it will need faith in its government,\" he added. \"It's in my own personal opinion that people will restore their faith in government if its leader sacrifices first.\"\n\n\"But there is no chance of survival if you stay behind. You must know that?\" Dalton asked. \"I've seen these things up close for far too long. There's no winning this one.\"\n\n\"Perhaps. But throughout history, people have fought against insurmountable odds a lot more willingly when doing so behind a leader who showed no fear. If I accompanied the exodus flight into the stars, those left behind would have no hope. Though it may be true that we have no chance of winning, if I remain here to lead a final stand, those who remain behind with me will at least have hope.\"\n\n\"Forgive me commander, but I don't understand the reasoning. It seems as though you would be of more service leaving and keeping the fleet in order.\" Dalton admitted.\n\n\"Think of those innocent faces you saw yesterday Dalton. Think of how they must have felt with no hope. Thinking of certain death before it came. By choosing to sacrifice my own life in staying behind, I save thousands of innocent faces that same fate. They may perish, but will do so with the hope of survival,\" the commander said. \"The sacrifice of self for the better of those in need is the mark of a true leader. Remember those words Dalton James.\" Ortega added.\n\n\"Thank you sir.\" Dalton replied, nodding as he turned to exit the commander's quarters and return to the loving arms of a woman he was meant to be with.\n\nRonica\n\n\"I must admit Adam Michaels,\" Sarah remarked as she approached the man under a perfect Ronican nightfall. \"I never thought I'd see the system's most notorious smuggler comforting a child.\n\n\"Yea,\" he replied with a smile, holding his son close and walking through a small area of tall grass in order to sooth the infant. \"Honestly though, it's what I have always wanted.\"\n\n\"A child?\" Sarah asked, seeming a bit taken back by his confession.\n\n\"A child. The calm a child brings. The innocence of the moment.\" he replied.\n\nHis words were true, and that was obvious to the queen of vampires. Many times she had looked into his eyes as a lover, but this moment was different. Sarah saw a spark of importance, as though he lived for a higher purpose.\n\n\"When nothing else in life seems to be going right, I just hold my son. It's a temporary fix, I know, but when I have him in my arms the world just feels right.\" Adam stated.\n\n\"May I hold him?\" Sarah asked.\n\nFor anyone unfamiliar with their history, it would have seemed odd. Unheard of, even, for the queen of vampires to hold Adam's son.\n\nHowever their history was deeply intertwined. They had loved one another, and thought of the each other as their sole reason for living. Each had made mistakes along the way, and each had harbored both guilt and anger toward the other.\n\nBut times were different. Adam had left Sarah standing alone, choosing another lover during a moment of extremes, and it had led her to the walk of a vampire. Likewise, Adam had suffered the loss of his lover, and both felt it was God's way of punishing him for his mistake.\n\nBoth Adam and Sarah had matured beyond resentment. They had agreed, though unspoken, that the heart-wrenching mistakes of their past would remain in the past, while the good memories would remain.\n\n\"Of course.\" Adam replied, easing the baby over to the shoulders of such a mighty queen.\n\nImmediately, Sarah felt the calm. She understood how something as innocent and precious as a child could eliminate all else. The war, the hatred; all of it.\n\nThough she was now queen of vampires, there was still a piece of her that remained human. A small part of her that remembered her love for Adam and their dream of having a child together. Though Sarah realized it was a life that was no longer possible to her, the Hunter DNA to blame, still she felt comfort in knowing that Adam had found his peace.\n\n\"He seems to take to you.\" Adam remarked, surprised at Avery's snuggling to the queen of vampires.\n\n\"And his father?\" Sarah asked.\n\n\"Well,\" Adam began to reply, finding himself thrown into a very uneasy moment of conversation. \"You must know I still have feelings for you Sarah?\"\n\n\"I do.\" she replied with a smile.\n\n\"I have only two things in this world that push me from the bed each morning. The need to keep my son safe and the belief that I can save you from all of this. I know there must be a way, I just haven't thought it through hard enough. There has to be something I'm missing, because my mind and heart are in so many different places right now.\" Adam said.\n\n\"I'm sorry, I did not mean to overstep.\" she admitted.\n\n\"No, please don't be,\" Adam replied. \"It's just,\" he added, seeming to prepare for the worst. \"I'm still not over Sasha. I'm truly not. Her death has taken its toll on me.\"\n\n\"I understand.\" she replied.\n\n\"No, you truly don't,\" Adam said, his tone changing to one of desperation. \"I love you.\"\n\nHis confession turned both lovers as quiet as the babe whom snuggled to her.\n\n\"Sarah, I love you. I always have. When I left with Sasha, I did it because I was pissed off. I held a grudge against you for holding us at gunpoint, and I just couldn't let that go. You have to understand, I thought a few weeks would pass and you and I would have another showdown. I didn't know this was going to happen. Sasha was a good woman, she truly was. But you and I were meant to be together. We were supposed to end up together, and because of my stupidity, my bad judgment...we're not.\"\n\n\"I,\" Sarah began to say, her emotions getting the best of her. \"I'm not sure what to say?\"\n\n\"There's nothing you can say. Every single night I lay down and think of three things Sarah, three. My son, the mother who died to protect him and the woman I should have married. I love you to the pit of my soul and feel guilty because of it. I hate myself for leaving you there, but I love my son. I live every day in a prison that I created. This, all of this, is on me.\"\n\n\"It's not all on you Adam. We've both had a hand in this. I cannot help but think that perhaps fate has crossed our paths once more for a reason.\" Sarah commented, handing the small child back to the arms that used to hold her so lovingly.\n\n\"If we somehow make it through this...if we somehow survive,\" Adam said. \"I will make everything up to you. I will love you again.\"\n\nSarah responded to his promise with a warm smile, though no words were to follow. She walked away as silently as Vladris had stood, watching their conversation from a distance.\n\nThough the champion vampire felt sudden anger, Vladris also felt the truth in Adam's confession. He too had once loved, only to lose his lover in times of war. Vladris wanted to hate Adam Michaels, but also found a bit of common ground with the smuggler as his stare of concentration continued.\n\nAboard Colonial Star Triumph\n\n\"What's wrong?\" Cambria asked, walking from her rack.\n\n\"Just can't sleep, that's all.\" Dalton replied, sitting at a small table near the cabin door, though he did so in the dark.\n\n\"You should try. We land on Ronica within hours, you'll need your sleep,\" she commented softly, placing a hand on Dalton's back in comforting fashion. \"Our government was unorganized before all of this. You can imagine the hell we'll have to go through now.\" she added, smiling wide.\n\n\"You know, under any other circumstances, I'd be trying to bed you down right about now.\" he said with a bit of a grin.\n\n\"Well you wouldn't have to try all that hard I'm afraid,\" Cambria replied, changing her voice to that of a pure-western girl. \"It seems I've taken to this mysterious cowboy draped in brown and masked in whiskers.\" she added, holding her hand to her forehead.\n\nAs Dalton sat silently, Cambria's own demeanor changed to a more serious one.\n\n\"Well, if my damsel in distress routine didn't get you it must be serious. What's going on?\" she asked.\n\n\"Been thinking a lot about those people we left behind yesterday. Their faces.\"\n\n\"Dalton, it is not your fault. Worlds full of people just like that have fallen all across the Skyla System. The Priests are to blame, not you. We had no choice.\"\n\n\"I was the one who had the experience among them. I should have stayed. Should have stayed behind to give those people hope \u2013 not left 'em to die with fear.\"\n\n\"You wouldn't have made it out alive. None of us would have.\" she replied.\n\n\"Maybe I wasn't supposed to.\" Dalton said.\n\n\"Don't talk like that. Not after I've fallen for you,\" Cambria said, pausing to look deep into his eyes. \"I've fallen for you Dalton James.\"\n\nWith her statement came a kiss. Followed soon after by a bonding of souls' only feet from the door that separated them from their resident sniper. All without the use of whiskey, which, perhaps, was a first for the man known as Dalton James.\n\n\"Wake up my friend,\" Skulls pleaded, doing his best to usher the brown coated smuggler from a foggy world of dreams. \"We're landing on Ronica.\"\n\n\"Damn,\" Dalton was slow to acknowledge. \"My head is splitting. Feel like I've been hit by a train. Almost like a hangover?\"\n\n\"Something like that.\" Cambria replied, covertly eluding to their session of lovemaking.\n\nOf course, Skulls never caught on. He was quick to sway his eye to the scope and fetch a guaranteed kill, that much was a fact. However, his ability to catch onto an inside joke, or, in this case, an inside conversation \u2013 was lacking.\n\n\"Hadn't felt this rough since I got hold of some tainted rock whiskey back on Phinamore.\" Dalton said tauntingly.\n\n\"Oh really?\" Cambria said, playing along. \"You sort of have the look of a long night of binge-drinking alright. Just seems like you've been laid up with the finest of liquor.\"\n\n\"Liquor? You mean hootch?\" Dalton asked, his question changing her facial complexion drastically.\n\n\"No, I mean liquor. As in fine wine. The kind that's been on the shelf for a mighty long time collecting dust. They say it gets better with age.\" Cambria insisted.\n\n\"Well, I'd say as a long time connoisseur...and I mean long...I'd have to admit that it seems like the wine that's been sitting so long is most likely the best I've ever had.\" Dalton replied.\n\n\"Most likely?\" she asked.\n\n\"What about you? Walking a bit rough this morning?\" Dalton asked.\n\n\"So am I, this space travel will do it.\" Skulls added.\n\n\"Wow, you too Skulls? I'm impressed. There are only so many hours in a single night?\" Cambria asked playfully.\n\n\"Oh hell no,\" Dalton insisted. \"There is a select list of consumable wines in my book. That's it, if you know what I'm saying?\"\n\n\"I do,\" she replied, checking her pistol before snugging the oversized revolver into its hip-holster. \"I just feel a little stiff, that's all. I need to get out and stretch my legs a bit. I need some exercise.\"\n\n_If Skulls wasn't here, I'd stretch your legs and give you some damn exercise._ Dalton thought, though he did his best not to moisten his lips with the tip of his tongue. A habit he found hard to break.\n\nAs Skulls prepared his own rifle, Dalton and Cambria continued to stare onto each other. It had been a long time since she'd felt this way about someone. In fact, it had been years. She'd once planned to marry the most notorious gunslinger in the Drifts, but it simply wasn't meant to be.\n\nCambria had a thirst for the stars. A life filled with adventure beyond compare. Her then lover, Johnny, wanted a simple life. He'd always made the claim that he'd both live and die in the Drifts, and the gunslinger wasn't bluffing.\n\nHe had chosen to stay behind. Chosen to do his best in waiting out the plague of men from the rocky cliffs of his home planet, doing so with a small gang of friends and co-criminals.\n\nCambria was saddened by his decision, of course, but it was his to make. She would always have feelings for the man that remained behind, but love is what she had come to feel for Dalton. True love.\n\nShe'd greatly misjudged the smuggler during their first meeting. He seemed as though he was old and broken down. Cambria soon learned, however, that he was quite the opposite. He was alive, and full of the kind of life she thirsted for. Adventure.\n\nHe'd been a part of every major event during his lifetime. Every war, every skirmish. He'd ran with convicts, fought vampires, ran from zombies and even co-existed with the mighty Husk race for a short time.\n\nMost importantly, however, he had a good heart. She could see the hurt in him, a sign of compassion. Her time in the adventurous line of work, smuggling, had been short. Still, she felt safe around the man who reeked of cheap whiskey and cigar smoke.\n\nHe was the closest to a cowboy as Cambria could have imagined. Tough, rugged, experienced, and behind closed doors, sensitive. There was a good man behind that rough patch of whiskers; an honorable man beneath that tattered brown coat.\n\n\"Fuck,\" the honorable man shouted, seeming to break Cambria from her imaginative spell. \"These fucking landings. Every single time, these landings.\" he added loudly, grabbing a steel shelf which was bolted to the ship's wall, as the large vessel descended on Ronica.\n\n\"Such language.\" Cambria replied with a smile, bracing herself as well.\n\n\"Hey, I got liquor or I got language. Right now, I'm fresh out of the good stuff.\"\n\nAs the ship began its harsh descent into the atmosphere of Ronica, forcing everyone aboard to brace themselves while hushing to a quiet calm, Dalton began to think.\n\nHe'd been in so many ships; so many descents. Still, he remembered every single one. His early days during the first Glimmerian War, the smuggler counted nineteen drops altogether. Most had come after a Legion ass-whipping and colonial retreat.\n\nHe also remembered the countless drops from space that he and his former crew had undergone. Most of them successful, while a few were...not so much.\n\nDalton had successfully survived two crash landings under such conditions, and remembered the day he walked away from his first. Promising himself on that very day that his boots would never again touch the inside of a spaceship.\n\nYet here he stood, leaned over and holding onto the frame of a door as if his life depended upon it.\n\nStupid! You are one stupid son of a bitch!\n\n\"Are you alright?\" Cambria yelled with a smile, her lover having grown as quiet as a church mouse, though his thoughts would have scarred said church mouse for life.\n\n\"Fine,\" Dalton said, lying through his teeth in the process. \"About to fall asleep actually.\"\n\n_As long as I live, my damn boots will never touch the inside of another ship. Not for this shit._ He thought.\n\nRoncia\n\nShips had been coming in for days on end, the sound of thrusters seemingly commonplace to those already assigned to Ronica. However, the larger colonial star ships were a different story. Earthquake loud and nearly large enough to blacken the sky, two traits that demanded everyone's full attention.\n\nEspecially Adam, who did his best to calm his son as the shiny armada of the sky slowly touched ground.\n\nThe Hunters called Ronica home. The Husk had arrived, as had the Benzans and every smaller, yet organized, force throughout the Skyla System. Finally the colonials had come. Their commander just moments away from unifying the people who had arrived under his direction.\n\n\"Now my queen,\" Vladris said with a sharp tongue. \"Now we will see with our own eyes. Now we discover the truth of our place among this fleet.\"\n\n\"Indeed,\" she replied, her attention on the recently-landed ship. \"Though I believe they will welcome us. I have a long history with Commander Ortega.\"\n\n\"Does your history include his eyes catching sight of you as a Hunter?\" Vladris asked.\n\nIt was a legitimate question, at least from Sarah's perspective as they joined everyone in watching the large group of soldiers exit, escorting Ortega.\n\n\"People have a way of viewing our race negatively based on our looks. Though our intentions may be true.\" Vladris added.\n\nNear the rear of the group, which consisted of hundreds of lives, Dalton, Skulls and Cambria exited the large ship. Their lungs welcoming the fresh air, though Dalton's craved the smoke of a cigar.\n\nThe fact was, Sarah first caught sight of the small group. Dalton's brown coat a beacon of smuggling among the colonial blue. That said, Dalton immediately caught sight of Adam Michaels a mere moment later.\n\n\"Well fuck me raw.\" Dalton exclaimed loudly.\n\n\"Here? In front of hundreds, even thousands? Well it sounds crazy, but we can try it.\" Cambria replied, motioning for her belt buckle as a gesture, though she had no intentions of going through with it.\n\nWithout an answer, Dalton blew off her advance, immediately walking toward a man who was like his own brother. A hasty strut quickly closing the gap of several hundred yards.\n\n\"Well who'd you go and knock up?\" Dalton asked, the crisp of his voice immediately grabbing Adam's attention.\n\nTurning, Adam's face told of relief. A welcomed reunion among old friends.\n\n\"Dalton James.\" Adam said, doing so after finding himself without words. Happiness in no short supply.\n\n\"Why is it every damn time I run from the undead, you show up? I'm starting to put two and two together.\" the smuggler replied, hugging his old friend firmly, their reunion conjuring up a lot of great memories.\n\n\"I wondered about you. Hoped you had made it out, but had started to worry.\" Adam admitted.\n\n\"Nearly didn't, bunch of half-dead fucks. Didn't know they was up against the old smuggler extraordinaire.\"\n\n\"And who are these good people?\" Adam asked, both Cambria and Skulls approaching.\n\n\"This here is Skulls, the finest sniper around,\" Dalton said, turning to Cambria and quickly finding himself at a loss as to how he should introduce her. Cambria slowly locking her hand into his and squeezing tight. \"This here is my girl. Cambria.\"\n\n\"Your girl?\" Adam said with a grin. \"I thought love was for suckers? What happened to the three women on...\"\n\n\"Hey man.\" Dalton warned.\n\n\"I'm only kidding, glad to meet you,\" Adam said, shaking hands with both the sniper and Dalton's eye candy. \"Any friend of Dalton's is a friend of mine.\"\n\n\"So man, you never gave me an answer. Who'd you knock up?\" Dalton asked.\n\nHesitant to answer, Adam simply began to walk away as Sarah approached. Her champion by her side.\n\n\"What's his problem?\" Dalton asked.\n\n\"Adam is still grieving with the loss of Sasha. His son reminds him of her very much, I think.\" Sarah replied.\n\nBoth Skulls and Cambria seemed intimidated by the idea of Hunters standing before them. Particularly Vladris, who was both hulking in appearance and scarred from battle.\n\n\"You need not fear us,\" Sarah admitted, extending her hand to the blue-haired beauty. \"Dalton is an old friend of mine, and Vladris is a great warrior no doubt, but he fights for your cause. For the defense of the innocent.\"\n\nThough she seemed to be shaken a bit, Cambria indeed shook hands with Sarah, a queen among the race of fear.\n\n\"Ah shit.\" Dalton said.\n\nSaying nothing, Sarah turned to listen to the smuggler.\n\n\"Didn't know he had lost someone important. Put my foot in my mouth, as usual.\"\n\n\"I see,\" Sarah replied. \"Should you need anything, any of you...don't hesitate to ask.\"\n\n\"Thanks Sarah. I appreciate you taking this old hound dog in.\" Dalton replied with a wily grin.\n\nWith hundreds of bodies bustling around in preparation for the exodus launch, Dalton's group watched the queen and her champion walk away. Heading back to the sanctuary of castle walls.\n\n\"She seems a little strange.\" Cambria confessed.\n\n\"The bitch has always been crazy, no doubt about it,\" Dalton replied. \"But the good kind of crazy I guess.\"\n\n\"Wait, were you and her a thing?\" Cambria asked, feeling a bit awkward.\n\n\"Oh hell no!\" Dalton insisted. \"Though I'm sure she's wanted to jump on my Glimmerian stick a time or two. Who hasn't?\"\n\nSmacking the man across his arm as a stern warning, their sniper finally began to make a clear picture of the situation.\n\n\"So you guys are together?\" Skulls asked, the large bolt-action rifle hanging from his shoulder by a thick nylon strap.\n\n\"Um yea, hello?\" Cambria replied.\n\n\"Oh dear God,\" Skulls admitted. \"I thought I dreamed those noises last night. They were so vivid. Some things cannot be undone.\" the sniper confessed with shame, head hanging a bit in doing so.\n\n\"Just start worrying less about my sex life and more on keeping an eye on that hulking sumbitch,\" Dalton said, motioning to Vladris as the champion and his queen made their way through the settlement of survivors. \"He's tried to kill me more than once.\"\n\n\"He didn't? Wow,\" Cambria replied. \"I'm impressed. He's like twice your size.\"\n\n\"Fuck 'em,\" Dalton barked. \"I guess I showed him that size doesn't matter. It's the amount of fight inside.\"\n\n\"Oh, he looks like he has plenty of fight inside of him.\" Cambria said, taunting her lover a bit.\n\n\"What the fuck?\" Dalton questioned, turning to the woman who openly joked about wanting to bed the champion vampire.\n\n\"Oh Dalton James,\" Cambria said, portraying her damsel in distress act once more. \"You know you're the only one for me.\" she added with a snicker.\n\n\"Yea, yea. I hear 'ya,\" Dalton replied, continuing to stare down the vampire's back. \"Bet I could drink that sumbitch under the table.\"\n\n\"Sarah Blaine, it has been a long time. Too long,\" Commander Ortega remarked, surrounded by colonial soldiers as he approached the queen. \"I humbly thank you and your people for providing us a safe haven.\"\n\n\"It was not an easy sell. My people agreed to it with the belief that they will be treated equally.\" Sarah replied, shaking hands with the commander.\n\nShe was surrounded by a group of guards. Though numbering less, their armor spoke of true ability in battle.\n\n\"That will be the case, I assure you. Your people have my word.\"\n\n\"Thank you commander. We should talk in detail on this as soon as possible.\" Sarah pushed.\n\n\"Sooner than I had hoped, I'm afraid. My scouts have confirmed a large armada making course for Ronica. Presumably the Priests. We will need to call a meeting quickly to discuss the future of our people as one, and, unfortunately, devise a plan to defend the invaders long enough to stage our exodus into the stars.\" Ortega responded.\n\n\"We need no plan when it comes to defense. My people are masters when it comes to defending their homes against those who would take it from us.\" Vladris replied swiftly.\n\n\"I do not doubt your ability in combat warrior. For I have heard stories of your ability on the battlefield, you are truly a hero among your people,\" Ortega replied, soothing Vladris a bit. \"But the threat which comes is too large. They have the bodies to simply overrun this planet, growing their numbers as they shorten ours.\"\n\n\"We understand your urgency commander.\" Sarah replied, while casting a stare of scold onto her champion.\n\n\"Sarah, I do think it best if your hero were to lead the defensive efforts. He knows the layout of Ronica as well as any. That is if he is willing?\" Ortega asked.\n\n\"I am willing to go wherever my queen commands \u2013 defending her, along with my home, to the death.\"\n\n\"I agree commander. Vladris should lead our spearhead of defense.\" Sarah replied.\n\n\"Fine. Vladris, you will command your finest, as well as the finest of the colonials, Benzans and Husk. Place them strategically and stagger them in a way that makes it possible to fall back. Eventually, when your force falls back to the fleet ships, you'll board and we'll begin our flight into the stars.\" Ortega said.\n\n\"Understood. If the Benzans or Husk are unwilling to follow the lead of a vampire?\" Vladris asked.\n\n\"Then they will be left behind as our fleet sails into the stars of night. I will brief them myself. You and your people are no longer the enemy,\" Ortega said, his eyes turning to the champion vampire. \"You are a part of this fleet now, I am a man of my word, and you will be treated as such. I'll see to it personally.\"\n\n\"You are an honorable leader. Thank you.\" Vladris replied with a nod.\n\nIt was a scene for the archives of man. A large meeting hall nestled inside the stone of castle walls, heavily guarded by dozens of soldiers of all nations and creed.\n\n\"We must first decide the government that will continue as our exodus begins.\" Commander Ortega said.\n\nHe sat at a large table of polished wood, as did nineteen other leaders, all of them in agreement. So many powerful leaders confined to the meeting hall.\n\n\"I'm not exactly sure how we are supposed to do that during such times of chaos?\" Cherlon, the Theron leader replied in question.\n\n\"We must remain a democracy. If the people have no voice, they have no hope.\" Gore added, proudly representing the Husk.\n\n\"I agree,\" Commander Ortega replied. \"At final count we have forty-seven ships that can make such a voyage. Four of them colonial Stars. It is in my opinion that they should remain to the corners of our fleet. Military buffers against any and all would-be threats.\"\n\nThe room seemed to easily agree with his statement, essentially turning the colonial stars into the backbone of their military.\n\n\"The ships we assign for government should remain to the center and must remain a mixture of our races. Everyone is to be represented. It will make my job a lot easier up there.\"\n\n\"So you are not staying behind?\" Sarah asked.\n\nIndeed she was right, her question followed by an unnerving silence throughout the room.\n\n\"Of course not,\" he replied, chatter begin to spread throughout the room, even among the posted guards. \"Though I do feel it important to have two leaders in place. A civilian leader and a military leader, both equal in terms of power over our future.\" he added.\n\n\"I disagree!\" Gore said with conviction. \"I feel as though a single leader of strong military background is needed if we are to survive.\"\n\nGore's statement caused the whispers to grow louder, finally pushing Commander Ortega to acknowledge them.\n\n\"I do not,\" Ortega confessed. \"Our military should be strong enough to keep our people safe, but our ultimate goal is to find another home. To start over. Rebuild our civilization elsewhere. It is therefore important to retain a leader who will handle everything non-military.\"\n\nWith that, the whispers seemed to calm a bit. Sarah finding herself proud of Ortega, who she'd personally selected as her own successor once upon a time. When she was a Colonial leader, though it seemed so long ago.\n\n\"I think,\" Adam Michaels said, representing the Benzans at the meeting, though very few remained. \"I think it is important to integrate those aboard our ships as well. Force them to respect others from the start so that we may put our past hatred behind us.\" he added.\n\n\"Absolutely not!\" Gore refused. \"My people will not allow Hunters to board their ships and consider them home.\"\n\n\"Yet your people run here to our home, only to be welcomed by the Hunters?\" Sarah replied.\n\n\"Enough!\" General Ortega demanded, silencing the bickering.\n\nA very eerie quiet fell across the room, as if decades of fighting were about to resume.\n\n\"We are a single people now,\" Ortega said, this time a bit calmer. \"The Hunters are to be treated the same as Husk or even my very own race,\" Ortega added. \"I will oversee the exodus lottery in less than one hour. After which, I will personally see to it that each ship's manifest is integrated. Starting with yours,\" he said, looking directly to Gore. \"You will captain a Hunter-built ship as the exodus pulls out, and a Hunter will captain your own. If you, or any among your people disagree, you are free to go your own way. Perhaps leave the luxury of your ships to journey into the nearby hills to defend us, as so many Hunters do this very moment.\"\n\nGore offered no reply, though he cutting stare could be counted as such.\n\n\"And of your mighty colonial star ships? Will they too be integrated, or are your people above your own laws?\" the Theron representative lashed out.\n\n\"Every ship, including my own, will be staffed accordingly. Captains, their executive officers and even the soldiers aboard will be assigned based on their military experience. Fairly and equally.\" Ortega defended.\n\n\"Very well then. You have the blessing of the Therons on this, as well as everyone else I believe.\"\n\n\"Gore? Do the Husk agree to these terms?\" Ortega asked with authority.\n\n\"Yes,\" the mighty Orc-like Husk replied. \"Though I wish to personally see the manifests before they go into effect.\" adding suspicion to his arsenal.\n\n\"Very well.\" Commander Ortega responded, proving to the room he had nothing to hide.\n\n\"The exodus lottery seems to be all that remains?\" A leader of the Drifts commented; his rough beard testament to a life lived simply.\n\n\"Indeed. Now it is time for our computers to decide who is allowed to join us on our exodus into the stars.\" Ortega replied. His statement effectively ending the meeting of leaders among what remained of the Skyla System.\n\n\"How will the moment of our launch play out?\" a leader asked, representing the small moon of Novak, which neighbored Glimmeria.\n\n\"That is a very good question, I'm afraid,\" Commander Ortega replied. \"I don't think anyone has the answer. I do know that those who are in the hills to defend us have been promised a spot on the fleet, should they make it back alive. As they become overwhelmed, I've instructed them to fall back. Several lines of soldiers lay in wait, though they do so in a staggered position.\" Ortega added.\n\n\"You mean to slow them down?\" Gore asked.\n\n\"Precisely,\" Ortega replied. \"Rather than having the horde land and begin a sprint to our fleet, they will be forced to attack the soldiers at each staggered position, which has been strategically placed as far apart as possible. It will force the horde to zig-zag to us, rather than giving them a straight line of advancement.\"\n\n\"That's a good plan.\" Adam commented.\n\n\"It's a good start,\" Ortega replied. \"But I fear as they bear down on us, even the best of plans will begin to fall apart,\" he added. \"Our people will begin to panic. We must do our best to maintain order and provide those who have earned their passage a way through, while keeping those who are to be left behind at bay.\"\n\n\"How do we do that? Begin shooting our own people?\" the Theron representative demanded to know.\n\n\"Of course not,\" Ortega replied. \"If it came to that, we'd be no better than the horde that approaches,\" he added, doing his best to calm the fears of other leaders who carried the same concern. \"We will have to launch our fleet at a different time than announced.\"\n\n\"We are going to do what?\" Adam Michaels asked.\n\n\"We need to give our people a set time in which we will launch. Allow them to drill it into their minds. Then, when the time is right, launch before they have time to respond.\" Ortega replied.\n\n\"You mean lie to them?\" Adam asked.\n\n\"I mean do what's necessary in order to give our people a chance to thrive once more. If that means lie to them, so be it.\" Ortega admitted.\n\nAdam was a lot of things. Smuggler, solider...and even compassionate at times. At this moment, however, his skills at the card table came into play. An ability to read the faces.\n\nHe's bluffing.\n\nSomething told Adam that Commander Ortega was not being honest. At least not completely.\n\n\"Sarah, may we speak?\" Adam asked, catching up to her and her escort team as they left the meeting chamber. As did the other leaders who survived.\n\n\"Of course.\" she replied, turning to face her former lover.\n\n\"My intuition,\" he said, pausing on those words as he carefully mulled his next. \"Something is going on.\"\n\n\"I agree Adam Michaels,\" Sarah replied, having pulled the same feeling away from the meeting. \"Though I'm not sure what.\"\n\n\"Sarah, you need to consider pulling your warriors back.\" Adam suggested.\n\n\"Pull them back?\" Sarah asked.\n\n\"I can't put my finger on it, but I came away with bad vibes about our man in charge. I feel as though he may have suggested Vladris to lead the defenses in order to thin your ranks,\" Adam said, pausing for a moment. \"I think he means to kill you.\"\n\nAs the words left Adam's lips, Sarah's guards firmed up; their muscles tightening as if they were preparing for battle.\n\n\"No,\" Sarah warned them. \"Remain calm as if nothing is suspected. Wait for them to clear out, and then we will all decide on this in private.\"\n\n\"Sarah,\" Adam replied, turning to her with true concern. \"Where is the safest place inside of these walls?\"\n\n\"The keep, I suppose.\"\n\n\"Please,\" Adam replied, turning to her armed escort. \"I need you to escort Sarah there and keep her there. Do not let anyone in, no matter their request.\"\n\nThe soldiers looked onto him for a moment, before turning to their queen.\n\n\"Sarah please.\" Adam said.\n\n\"Alright,\" Sarah replied. \"Only if you agree to join us.\"\n\n\"Huh?\" he asked.\n\n\"If something truly is going on, I want people around me that I can trust,\" she replied. \"I trust you Adam.\"\n\nSeveral moments of silence were strung together by a look of two past lovers.\n\n\"I need to get my son.\" Adam finally replied.\n\n\"Azan,\" Sarah commanded, turning to one of her six escort warriors. \"Go with Adam and make sure he is able to do so unharmed. Protect him as though you were protecting your queen.\" she added.\n\n\"Yes my lady.\" Azan replied, bowing low to the ground.\n\n\"Someone please recall Vladris and his warriors. It would seem as though the enemy is not at the gates \u2013 but already among us.\" Sarah commanded.\n\n\"Please be careful.\" Sarah said.\n\n\"Scout's honor.\" Adam replied with a grin.\n\n\"Gonna be a bittersweet thing,\" Adam remarked, walking through the large hallway of the Colonial Star Destiny. One of hundreds which connected the most vital parts of the large ship. \"Leaving the Skyla System I mean. So many memories.\"\n\n\"Yes sir, I'd have to agree,\" Dalton replied. \"Been locked up on half of the colonized planets. Now we gotta leave 'em behind.\" he added with a chuckle.\n\n\"It's strange. I'm even going to miss that part of it. The shitty food, iron bars and hard labor.\" Adam said.\n\n\"Gonna be different up there. Months, maybe even years before we find something in need of colonization.\" Dalton hinted.\n\n\"Take plenty of booze.\" Adam replied as both men began to laugh aloud.\n\n\"Already covered,\" Dalton replied. \"Didn't mean that earlier today. About knocking someone up,\" the smuggler added, both of the men watching Adam's son at peace in Cambria's arms. A large pane of shatterproof glass separating them as the blue-haired beauty stood outside. \"I honestly didn't know.\"\n\n\"I know that,\" Adam replied with respect, patting his longtime friend on the arm. \"Really hurt me to lose her,\" he added. \"We've lost so many.\"\n\n\"Gonna lose more before this shit is over too.\" Dalton proclaimed.\n\n\"I'm afraid of that.\" Adam admitted.\n\n\"Sarah?\" Dalton asked, though he did so in a casual manner to avoid prying.\n\n\"Seeing her again brought up a swirl of emotions. Things I thought were dead inside of me,\" Adam admitted, turning to his friend. \"I think I still love her.\"\n\n\"Shit son, love is for idiots.\" Dalton boasted.\n\n\"And Cambria?\"\n\n\"Well that's a little different. I mean, the girl is fine looking and has a good heart, but I don't know if we've made it to love just yet.\"\n\n\"Better work quickly my friend,\" Adam replied with a smile. \"Because she is damn fine looking and appears to be real good with my son. I won't back-off forever.\"\n\n\"Like hell!\" Dalton replied, both men laughing.\n\n\"In all seriousness though Dalton, if you feel for the girl at all, you should tell her. Don't wait on it too long. I think of all the time I wasted with Sarah. I let my stubbornness stand between my true feelings and what could have been.\" Adam remarked with seriousness.\n\n\"And the bitch held us at gunpoint. Don't forget that small detail.\" Dalton lashed out.\n\n\"You get what I'm saying?\" Adam asked, staying the course of serious.\n\n\"Unfortunately, yes. It's killing my whiskey buzz actually.\" the smuggler replied.\n\nAs they both laughed aloud, Adam turned to his friend for a heartfelt nod.\n\n\"Dalton, we need to talk.\" Adam said.\n\n\"I thought that's what we were doing. You know, I open my mouth and words begin to come out?\" Dalton replied.\n\n\"I don't think we're safe here.\" Adam suggested.\n\n\"What the fuck are you talking about? We're aboard a military ship.\" Dalton replied.\n\n\"I can't get into details, at least not yet,\" Adam replied. \"I just need you to trust me.\"\n\n\"I do trust you Adam, you know that.\"\n\n\"I just have a really bad hunch that this arrangement between races is going to take a nosedive south very soon,\" Adam said, continuing to speak in a whisper. \"Just be ready to move if that happens.\"\n\n\"Who are you talking to?\" Dalton asked, boasting a bit. \"I'm always ready to move.\"\n\n_Or drink. Or, well, you know._ Dalton thought.\n\nAdam answered his longtime friend with a firm pat on the shoulder.\n\n\"Well, gonna go collect my son from the hottest girl you've ever had a chance with.\"\n\n\"Shit son, I've been with the cr\u00e8me de la cr\u00e8me when it comes to women,\" Dalton boasted. \"She's in my top five though.\"\n\n\"She's in MY top five,\" Adam replied, though he continued a walk away from his friend. \"Which definitely makes her your number one.\"\n\n_Yea, yea. You cheeky bastard._ Dalton thought. Feeling at peace as he was surrounded by his friends once more. Though his mind began to wonder about the alliance of races.\n\nAs the day slowly turned to night, a shadow of moonlight crisping onto the shiny hulls of so many ships, a sense of urgency set in.\n\nUrgency for a fleet which worked night and day in an attempt to stock their ships for a one-way flight into the tapestry of stars, and urgency for Dalton James. The word love having played heavily onto his mind since hearing it earlier.\n\n\"Tell her,\" he said, pouring a shot of rock whiskey and downing it with haste. \"Tell her not.\" he added, pouring yet another.\n\nAs he sat alone, off from the bustling crowd of so many who remained hard at work, Dalton wrestled with the obvious. There was only enough for one more shot.\n\nHe would have to tell Cambria he loved her. The smuggler had feared nothing as much as he feared admitting that to her. Terrified of rejection.\n\n_Fuck it._ He thought, though his mind remained locked onto every word of his coming confession. That he had truly fallen for the woman who was nothing like him. Yet made him complete.\n\nSo as the smuggler draped in his security blanket of brown leather marched on, as if he were marching into the teeth of an execution, he thought of those around him.\n\nWomen and children, who greatly resembled those he'd been forced to leave behind in the Drifts. He saw the faces of desperation and worry. Faces of survivors who seemed bewildered as they worried about their immediate futures.\n\nSuch a sight firmed Dalton up a bit. Realizing the admission of love wasn't the worst fate. Still he stopped in his tracks for a moment. Not for the thought of his admission of love that would soon follow; but for those around him. A group of survivors who had been collected from all parts of the Skyla System. A group that had lost so much, yet continued with hope. He thought of the sacrifice Commander Ortega had made for those very people, and it made him both sad, and proud as hell to be a colonial.\n\n\"Cambria Sims, I have something I...\" Dalton announced proudly as he burst through the small door of their two-room shelter.\n\nStopping his statement however, Dalton saw a girl broken, her crying one of true sadness.\n\n\"What's wrong?\" he asked, immediately grabbing her into his cowboy-strong arms and consoling her.\n\nWith her crying so heavy, no response would follow, just the gut-wrenching somber of a deep cry. Though he would soon discover the reason behind her sadness as he picked up an official colonial memo which lay near.\n\nThe colonial government regrets to inform you that you have been selected to remain behind. Our exodus launch is scheduled in forty-eight hours at the stroke of midnight. You have been assigned to the ground party, led by Commander Ortega. You will report tomorrow at sunrise to your assigned unit, led by Lieutenant James Locke of the colonial army.\n\nPlease know that your denial of flight does not ensure your fate. We are planning a very strategic defense against coming forces and remain confident that we will be victorious.\n\nThank you.\n\nBenton Sanders.\n\nAs Dalton finished reading the death sentence of Cambria Sims, he crunched the official papers into a ball, as if it were designed for ass-wiping, his other arm still firmly wrapped around the woman of his dreams.\n\n\"Where are you going?\" Cambria asked as Dalton pulled away, heading back to the door that had brought him to her.\n\n\"To fix this,\" he replied, angered by the development. \"First though,\" he added, walking back to her arms. \"I love you Cambria Sims.\"\n\nHis statement brought more tears, though it was a mixture of happiness and grief, the woman trembling as she had never trembled before.\n\n\"I love you too.\"\n\n\"I will fix this, I promise you.\" Dalton replied, leaving her grasp as he walked for the door once more.\n\n\"I made the flight!\" Skulls announced, entering the small shelter as the sniper smiled wide.\n\n\"The fuck out of my way.\" Dalton demanded, shoving the sniper and good friend to the side, readying for a showdown with the horde of pencil-necks that would surely follow.\n\n\"Have you recalled our warriors?\" Sarah asked.\n\nThe queen of vampires sat firmly on her throne, which shimmered with both jewels and gothic inscriptions formed into bronze plating.\n\n\"We have sent word, but it will take time,\" one of her soldiers replied. \"Depending on their circumstances, I'm sure we can expect the return of Vladris soon enough.\"\n\n\"Very well,\" Sarah replied, looking onto the group which had gathered. Nearly thirty total. \"We must begin to think of the survival of our own race, nothing more.\"\n\n\"Is it true? Do those who seek out protection plan to eliminate us?\" one of the elders asked.\n\n\"We,\" Adam Michaels replied. \"We don't know. Right now, we're just being cautious.\"\n\n\"You brought this onto us!\" a second elder replied.\n\n\"I didn't know, I swear,\" Adam replied, raising his hand to face level in an attempt to reason with those who had gathered. \"Even now, the commander lies to his own people. He's lost his way, and I fear that he lies to your people as well.\"\n\n\"Adam had no part in this,\" the queen warned. \"In fact, he was the one who warned me.\"\n\n\"What proof do you have?\" one of the elders asked.\n\n\"At the moment, none,\" Adam replied. \"Just a gut feeling.\" he added with regret.\n\n\"A gut feeling?\" the elder replied in mocking fashion. \"You wish us to recall our soldiers and risk annihilation based on a gut feeling?\"\n\n\"Send every soldier you have to face the coming horde, that's not my concern. Nor will it matter. They are coming, and nothing you have can stop that,\" Adam replied, this time with a bit of zeal on his words. \"My only concern is for the queen. YOUR queen.\" he added.\n\n\"He believes they requested our champion lead the defense in order to separate us. Divide our ranks.\" Sarah added.\n\n\"If Commander Ortega has no intention of leaving as scheduled; no intention of remaining behind as promised, why would he send a defense in the first place? Would he not just raise the ships as soon as the horde got close?\" Adam asked.\n\nThe room seemed to be in deep thought, each of them pondering the smuggler's questions.\n\n\"I've been a military man long enough to know that there is only one reason behind his decision,\" Adam said, pausing to look to Sarah. \"He means to thin you out. Either to exterminate your race, or leave you behind to die.\"\n\n\"Was it not you that brought this to our doorstep?\" one of the elders asked.\n\n\"I have no part in this. In fact, if I find out he is indeed trying to kill your queen,\" Adam said, pausing for just a moment. \"I'll gun him down myself.\"\n\n\"He is with us!\" Sarah yelled, commanding her elders back in line. \"He fights with us and is committed to dying with us, if need be. We must begin a plan to counter such colonial treachery.\"\n\n\"Your plans my queen?\" one of the higher ranking warriors asked.\n\n\"They believe they hold the advantage through surprise. Their downfall will be their arrogance and stupidity,\" Sarah replied, grinning a bit. \"We will act as though we are none the wiser. All while slowly putting our own plan into place. When this fleet launches, we will be in control of it.\"\n\n\"A bit late to be out and around isn't it soldier?\" a man asked, decked out in a colonial uniform as he sat behind the desk of an otherwise empty room.\n\n\"Ain't no solider. I'm here to see Commander Ortega.\" Dalton replied, already eying a single door which most-likely led to the commander.\n\n\"Who isn't? Had nearly a hundred of requests today. You'll have to wait your turn. Come back tomorrow and try your luck.\" the colonial soldier replied.\n\n\"I'm sorry, what was that?\" Dalton asked politely, approaching the seated man.\n\n\"I said you'll have to...\" he began to reply, stopping short as he was pulled from his seat by the front of his shirt; clean-jerked directly into a grasp which held him inches from the floor.\n\n\"Now you listen to me, you little puny sumbitch,\" Dalton scolded. \"I was fighting wars and dodging bullets when you were still shitting baby green,\" he added. \"And I've killed men twice your size over nothing more than sales rack whiskey. So believe me when I say it. You're gonna be on that com to inform Ortega that I'm here.\"\n\n\"They'll throw you in the brig for this!\" the smaller man yelled, his only defense.\n\n\"Nowhere I haven't been before. Countless times. Now be smart about it.\" Dalton replied, turning his eyes back to a heavy revolver which hung by his side.\n\n\"Sir,\" the colonial soldier timidly said, \"Hate to disturb you at such an hour,\" the man added, pressing a button on the com in front of him. \"Someone here to see you. A Dalton?\" he said, feeling a firming grip on the front of his shirt. \"He's pretty persistent.\"\n\n\"Of course, send him in.\" Ortega replied through the digital com system.\n\n\"Yes sir, at once.\" the man said, finally removing his finger from the com.\n\n\"See there buddy, nobody even had to die.\" Dalton proclaimed, lowering the man to the ground and slapping his shoulder in good sport.\n\nImmediately, the smaller man pulled away with anger.\n\n\"Of course, I do have to come back through here when I leave.\" Dalton reminded, taunting the man who cast a look of intimidation back.\n\n\"Dalton James,\" Commander Ortega said, welcoming the smuggler into his personal quarters. \"I had hoped to see you again soon. Though I didn't think it would be after hours.\"\n\n\"Something came up.\" Dalton replied.\n\n\"Of course, of course,\" the commander said, motioning to his desk. \"Have a seat.\"\n\n\"Been thinking a lot about what you said before. Most importantly, about the sacrifice you've made for your love of these people.\" Dalton said.\n\n\"You've decided to take up my offer on joining the fleet?\" Ortega asked.\n\n\"No sir, as a matter of fact I'm going to have to respectfully decline,\" Dalton said, his reply shocking the man in charge. \"Though I am going to ask for a favor.\"\n\n\"I see,\" Ortega replied with a hint of disappointment. \"Go on.\" he added.\n\n\"You chose to stay behind and give hope to these people out of love. What I feel is the mark of a true leader; a true man. I also love someone. A girl that I couldn't imagine life without living. I refuse to leave into the stars and leave her behind, knowing it would be a death sentence.\"\n\n\"Perhaps not.\" Commander Ortega injected.\n\n\"Perhaps, but the odds of surviving are the slimmest imaginable,\" Dalton quickly replied, setting the tone of the meeting. \"I wish to give her my spot on the exodus fleet, and I would like to remain behind with you, answering to your command.\"\n\n\"Dalton, I understand your passion. Truly I do. But you could serve these people best by remaining with them in the future. Your experience would be a great asset...\"\n\n\"No,\" Dalton stated firmly. \"My experience means nothing if I fight with an empty heart. I'm as good as dead if I know in my heart that the woman I love is doomed.\"\n\n\"Alright,\" Ortega replied. \"I'll see that it's done.\"\n\n\"I also have one final request.\" Dalton stated.\n\n\"Yes?\"\n\n\"The woman is very, very important to me. I simply ask that she be allowed onto the ship of my own choosing, surrounded by our friends.\" Dalton said.\n\n\"Very well. You have my word.\" Ortega replied.\n\n\"Sir, thank you,' Dalton replied with gratitude. \"I feel at peace with this, and trust me, I'll give you everything I have when it comes time for slinging shells.\"\n\n\"I know you will son.\" Ortega replied, acknowledging the truth of his statement, though his own was riddled with lies.\n\n\"Well,\" Dalton said, standing to his feet and saluting his new superior. \"I'll let you get back to it. Sorry for bothering you at such an hour.\"\n\n\"Dalton, I want you to know something,\" Ortega said, causing the sworn smuggler to pause his walk. \"If our cause had more soldiers like you, we may just be winning this damn war.\"\n\n\"Not a big deal,\" Dalton replied calmly. \"All of these years I've ran from death. But I've ran my course. Now I plan to run to it, guns a blazing.\"\n\n\"You're a model soldier Dalton James.\"\n\n\"Apparently you haven't read my file jacket hard enough.\" the smuggler replied with a grin as Ortega handed him a signed paper. One that would allow Cambria's future to exist on the exodus flight.\n\n\"I'm sorry for the breakdown earlier,\" Cambria admitted, approaching the man of her dreams as he entered their shelter once more.\n\n\"No worries,\" Dalton replied. \"Told you I would fix it.\" he said, handing her the signed paper.\n\n\"What? How?\" Cambria asked with a mountain of emotions running through her.\n\n\"I just did. Got a long history with the man in charge.\" he boasted, cut short by a clinging hug from the woman he loved to the bones inside of his body.\n\n\"This means we'll be together when the flight launches?\" Cambria asked probingly.\n\n\"Of course,\" the smuggler replied, thinking himself a bastard for lying to a woman he loved. \"Just focus on packing enough for the two of us.\"\n\nAs they held one another for several moments, Cambria's arms squeezing the man of her dreams tight, he felt sadness. Not for his coming death, but for the pain which he knew Cambria would eventually feel.\n\nHe also knew that a lie was necessary to get her aboard a ship. If she knew of his deal with Ortega, she'd never agree to it. He only hoped she would recover in time and live a long, happy life in the stars above.\n\n\"Skulls,\" Dalton said, his body still engulfed by the grasp of a love-struck woman. \"Sorry about earlier, I wasn't thinking straight.\" he added.\n\n\"Think nothing of it brother,\" the sniper replied. \"I understand what you must have been going through.\n\nNo, you don't. You truly don't.\n\nWhen the first explosion rocked the surface of Ronica, everyone with a heartbeat knew what was to follow. An invasion. Panic set in throughout the large encampment as daybreak had only begun.\n\nSitting up as though he'd been awake for hours, Dalton immediately grabbed his revolver and readied for a fight.\n\n\"It's alright,\" Cambria said. \"Far off, but it means we're on very limited time now.\"\n\nBolting to the door of their shelter, Dalton stood with revolver in hand and shirtless, watching the sprint of many colonial soldiers and civilians alike. Each of them expediting the process of their plans for the given day.\n\n\"I need to go find Adam. Have to talk to him before they get closer. You finish packing all of the essentials,\" Dalton said, throwing a green shirt on with haste and looking back to his lover. \"Remember, it's a one-way flight.\"\n\n\"Be careful, please.\" Cambria replied, kissing Dalton passionately before turning to finish packing the needed supplies.\n\n\"Hey,\" Dalton insisted, staring at their sniper. \"Keep her close and safe until I get back.\"\n\n\"You got it brother.\" Skulls replied.\n\nAs the smuggler began his own sprint from their shelter, the ground seemed to tremble beneath him, nearly a dozen more explosions landing within a few miles of their basecamp.\n\nThe Priests had arrived.\n\nEvery time they launched an invasion, it began with explosions from warheads that rained down from the stars. Immediately followed by ships filled tight with soldiers of the undead. Infected. Zombies. Priests. Or, as Dalton usually called them, half-dead fuckers.\n\nEither way, Dalton knew he was on borrowed time as he continued a sprint to the Benzan ship where Adam Michaels had been staying.\n\nJust before arriving, his longtime friend exited the large ship, just as so many other Benzans had done only moments before.\n\n\"Adam, we need to talk!\" Dalton yelled, doing what he could to become louder than the increasing explosions.\n\n\"What is it?\" Adam replied, his voice also forced into a very loud tone.\n\n\"Cambria. I volunteered to give up my spot for her.\" Dalton said, though he did so with a heavy heart.\n\n\"What?\" Adam asked, stunned by the admission. \"You did what?\"\n\n\"I'm staying behind so she can live through this. Ortega signed off on it. She's to be put on your ship, and I need your word that you will watch over her?\" Dalton asked with near-desperation. \"Please.\"\n\n\"Dalton, there has to be some other way?\" Adam questioned.\n\nAt the moment he did so, a large explosion rocked through the settlement, destroying a large ship that had been assigned to Glimmerian survivors. One of two. Though it wasn't filled with bodies at the moment, he understood that many people who had been promised a ride into the stars would now be without one.\n\n\"There is no other way Adam, trust me. I don't want to die, I just don't want her to die a little more.\"\n\nAdam was pressed into a tough spot, having been sworn to silence by the Hunters, though failure to speak may have cost his best friend his life.\n\n\"Dalton, listen,\" Adam said, smaller explosions tearing up the countryside around their base camp. \"The Hunters and colonials are about to come to a head. There's going to be war, and you need to be on this side of the fence.\"\n\n\"What?\" Dalton asked.\n\n\"Commander Ortega is lying about a lot of things. He has no intention of staying behind, and he's lying to his people. I also suspect he plans to strike at the Hunters when the time is right.\" Adam replied.\n\n\"What the hell are you talking about? I just talked to Ortega hours ago, he assured me.\"\n\n\"Dalton, he is lying to you.\" Adam replied.\n\n\"Is he? Or have you lost your way?\" Dalton questioned.\n\n\"What?\" Adam asked, shock in his voice.\n\n\"You and Sarah are in love, you always have been, and I get it. I truly do. But she's not one of us anymore.\" Dalton stated.\n\n\"A lot has changed, sure...\" Adam began to reply.\n\n\"A lot has changed, starting with the fact that she's a fucking vampire.\"\n\n\"Dalton, you said you trusted me. I need you to trust me.\" Adam replied.\n\n\"I need you to trust me when I say there's no way I'm putting my life into the hands of a race that's been trying to end it for years now!\" Dalton yelled in response.\n\n\"The colonials will leave you behind the first chance they get Dalton.\"\n\n\"The Hunters won't? You'll be lucky to survive a month with those blood-sucking stiffs!\" Dalton replied.\n\n\"Well,\" Adam replied, nodding his head. \"The answer to your question is yes. Sasha has a place on my ship, as do you. So long as you know it's going to be a Hunter ship.\"\n\n\"Pass.\" Dalton replied with spite.\n\n\"Well, that's your choice to make. Either way old friend, I hope you make it off this rock in one piece.\" Adam replied, turning to walk away from a man he once considered a brother.\n\n\"Hold your positions!\" Vladris yelled. \"The cowards fight with explosives for now, but soon they will test us on the battlefield. A test in which we will pass with the edge of blades!\"\n\nThe champion vampire sat tall, a solid black steed beneath him as nearly two-hundred soldiers looked on. Many of them on horseback as well, though most would fight the good fight. On foot with a single weapon.\n\n\"Those of you who I consider brothers know of our ability on the battlefield,\" Vladris yelled, the thunder of explosions crackling around him as he looked onto the crowd of warriors. Many vampiric, though the sleek armor of Benzan, solid blue of colonials and green tint of Husk were in no short supply. \"The rest of you will learn of it with your own eyes on this day, after which we will become immortal brothers, be it through life or death.\"\n\nThough they awaited the worst enemy the system had ever known, preparing to face odds that were landslide in the favor of their invaders, the warriors cheered loudly. Readying themselves to follow the most-notorious warrior in recorded history into battle.\n\nThough Troy readied for a different battle altogether.\n\nThe colonials and Husk had indeed been conspiring from day one. A plan to rid their departing fleet of anyone deemed a threat to future survival. Which included the Hunters.\n\nTroy could still remember the day Roman was cut down by Vladris' blade. The man, who for so long had been like a father-figure to him, butchered before his very eyes.\n\nAt the time, Troy was too young of a man to strike back. Though he had begun the path of a warrior, he was weak.\n\nThat was long ago.\n\nAfter escaping into the welcoming embrace of the Husk, Troy had learned the art of combat. He had perfected the art, in fact, as though he was meant to swing a sword.\n\nHis own sword not nearly as massive as those of his Husk family, but he had countered the absence of Husk-might with cunning. Troy had learned that timing was just as important as ability when it came to battle. And as the timing became right; Troy would have his revenge.\n\nAdam Michaels glanced throughout the colonial star, standing on a large catwalk above the landing bay as he watched machines load several more Swordfish fighters inside.\n\nHe knew in the pit of his heart that it could be his last look inside of a colonial star, or any colonial ship for that matter.\n\nThe soldier turned smuggler began to think of so many many battles before. Many of which, he wore the colonial patch on his chest.\n\nAdam seemed a million miles away, grabbing hold of several large tubes that would be responsible for pushing steam throughout the ship once it began flight. The primary heating system.\n\nThe former soldier tugged a bit, as if he were testing the integrity of such a mighty ship. Though he broke away to stare to those around him, Adam began looking to their background, focusing his attention to the ship's setting behind them.\n\nHe understood it would be the people's home for the foreseeable future, if not for the remainder of their lives. Their entire lives spent between the beautiful landscapes of planets, each with their own rich history and scenery. Traded now, for an eternity of looking to steel draping around them.\n\nWhat choice did they have? Adam understood that no victory could be had in a war that had become a fight for their own survival. Their only chance of continuing the walk of humanity would be to do so by launching into uncharted space. Planets that had never before been explored.\n\nHe also knew in his heart that no peace would be had until either Commander Ortega and those who sided with him were taken from power, or the Hunters were wiped clean from their massive exodus fleet.\n\nAdam Michaels understood that before their voyage was over, be it by natural death or the discovery of inhabitable soil, these people would come to know every nook, every cranny, of this massive armada of the stars. Perhaps even their children.\n\nA fleet that was near fifty ships, though many of the larger ships included an array of small ones. Each in search of a new home, though politics would play as heavily as before. Every ship with its own agenda.\n\nEach had its own responsibility in the grand scheme of things. The Florentine was responsible for keeping their food in storage, which could prove to be the most important asset to all, depending on how long their journey played out.\n\nThe fleet's mechanics were scattered throughout, though their resources were aboard the Iron Maiden. For lack of better terms, an airborne tugboat full of spare parts, along with every tool imaginable.\n\nFour military ships were assigned, including the one Adam currently stood aboard. All colonial star vessels. The Triumph, God of War, Silver Hammer and Stalwart; though the God of War was designated as a troop ship and housed no smaller aircraft outside of dropships. The remaining three were to be used in protection of the fleet while in flight, relying on the skills of their pilots and housing many ship-to-ship fighters. Glimmerian designed Swordfish, though their design had been altered a bit to better fit deep space.\n\nThey were massive in comparison to the remaining ships of the fleet, even to their former models of colonial star ships. While the original ships stretched for thousands of yards and contained plenty of the Goliath model soldiers within its hull, the newly-designed colonial stars were much larger.\n\nThey were in a sense, small cities with thick armor around them and thrusters to their back. Capable of quartering tens of thousands, though supplies to furnish them would be limited.\n\nThe current model of colonial star ships were also stocked with hundreds of Goliath model soldiers, which was fancy talk for ass-kicking robots. The Goliath model aboard, however, were the recently upgraded V2 versions, which lent them a tougher exoskeleton of steel, along with the standard chain guns and anti-aircraft missiles.\n\nThe colonials had no idea what may lay waiting for them once the fleet made deep space. Be it fruitful planets that were ripe for the picking, or militarized civilizations that would swing at them with might; the colonials had to be prepared for anything.\n\nWhich they were.\n\nAfter catching sight of a four-man colonial marine team, Adam began to think of their job among the fleet. Protecting those in need, while keeping order and carrying out the instructions of the politicians who would lead them. A tall order considering the Magellan.\n\nIt was their political ship, not to mention the one ship that was sure to be the highest priority. Every important political figure surviving, every recognized planet, was to be housed aboard the Magellan. That included the Husk, among others; which was sure to include trouble.\n\nThe decisions that would affect everyone in the fleet would have to be made aboard the Magellan \u2013 a ship that mixed shatterproof glass and steel as if it were a work of art. Several protruding spheres of glass giving it the look of importance.\n\nThe Magellan also had the distinct honor of being the only ship in the fleet, outside of military cruisers, to remain under constant escort. Two Glimmerian-designed swordfish fighters, though more could be scrambled within minutes if needed be.\n\nThe Glimmerians had originally designed the swordfish fighter as a ship-to-ship fighter, which had the ability to hug low to the ground and fly at incredible speeds. In working with the colonial design teams, a newer model was eventually released, making it deep space worthy while retaining all of its original signatures of perfection.\n\nAll of the military marvel housed within the mighty ships of the fleet. It was impressive. Still, Adam found his mind wrapped around a single thought.\n\nSo many races under the same roof. Races that had been slaughtering one another for as long as time could remember, now expected to shake hands and play nice.\n\nAdam wondered how long such a mighty race would bow to the commands of a human leader. He knew it would only be a matter of time before the Orc-like race began its own plans to assume control of the fleet, if it had not already done so.\n\nOf course, the Hunters were now in the mix as well. Vampires in every sense of the word, though they were unhindered by daylight, garlic or religious symbols. That's just a silly assumption to begin with. They are, however, hindered by the burst of a gun and the blade of a sword, when in capable hands. That much had been proven throughout the years. A very important fact that shouldn't soon be forgotten.\n\nAdam wondered if the vampires could be trusted.\n\nThe fleet would make its important decisions aboard the Magellan, shaping humanity's future in the process. That said, their leaders would eventually disagree, and that could be disastrous considering the bloody past between their races.\n\nAdam Michaels realized as he stood firm, watching the bustle of soldiers around him prepping their ship, the most vital part of their fleet. Those around him. Adam promised himself at that very moment that he would never forget that fact. Watching such a wounded people push forward, finding courage in the face of the inevitable loss of their homes. It made Adam feel honored to be a part of such a special group, though his allegiance to the vampires meant his time among the colonials was limited at best.\n\n\"It's amazing to see people continue on. To survive during such times.\" Sarah remarked, approaching Adam slowly.\n\n\"I agree,\" Adam replied, turning to the woman he once loved. \"I was just thinking the same thing. Not to mention how daunting of a task it is going to be when it comes to keeping the peace between such storied races. What are you doing here? You should be in the keep under guard.\"\n\n\"I now know it is our race that will be persecuted at some point. If the others among us have any common ground, it is the hate for my people.\" Sarah replied.\n\n\"Yea, I know,\" Adam replied, turning to the bustle of his ship for a moment and noticing several among his crew staring to a Sarah a bit roughly. \"It isn't going to make my job any easier. You know I'm with you until the end Sarah, as long as my son is safe.\"\n\n\"He shall be.\" Sarah replied, also picking up on the animosity of the crew's glances.\n\nMoments later, several gunshots rang throughout the ship, causing an immediate reaction of panic from everyone. Including Adam.\n\n\"Remain calm Adam. Those are our shots.\" Sarah replied.\n\nBefore questioning her, Adam saw dozens of highly-armed Hunters entering the colonial star.\n\n\"What is this?\" he asked.\n\n\"These people have taken from our race for so long. You didn't expect us to just shake hands and play nice, did you?\"\n\n\"Actually...\" he began to respond as the gunfire intensified.\n\n\"Adam,\" the queen said, pausing to stare onto him. \"You have also taken from us,\" she added with a glare. \"From me.\"\n\n\"Sarah, that's not fair. You know I feel guilt for this.\" Adam replied.\n\n\"As well you should,\" the queen replied with a snap. \"But it does nothing to ease my pain.\"\n\n\"Sarah, I had no idea they would do this to you.\" he replied with a bit of zest on his own part.\n\n\"You left me standing alone Adam, what difference did it make to you where I ended up!\" Sarah shouted.\n\n\"It makes a lot of difference. We've all made mistakes here.\" Adam admitted.\n\n\"Your biggest mistake was trusting our people with your safety,\" she replied. \"The safety of your son.\"\n\n\"Sarah, you better not harm...\" Adam began to reply with desperation, though his words cut short by the end of a rifle. Its stock held by one of several Hunter Elites who arrived to protect their queen.\n\n\"He's safe and will remain that way Adam, as long as you cooperate.\"\n\n\"Cooperate? With what?\" the smuggler demanded to know.\n\n\"My men will escort you to our ship, at which time I will explain your punishment in further detail.\"\n\n_My punishment? You bitch!_ He thought, doing everything in his power not to strike at the woman \u2013 thinking for the safety of his son.\n\n\"I don't understand?\" Adam admitted.\n\n\"Adam, when you first showed up here my heart exploded,\" Sarah replied. \"I wanted to relive our love together,\" she added. \"But as I continued to think about it, I remembered the moment you left me for dead.\"\n\n\"I didn't know, Sarah.\"\n\n\"SILENCE! I've had enough of your excuses! I see the love you still have for Sasha on your face. I know that you left me behind because you loved her, and your love cost me everything.\"\n\nAdam didn't reply. Simply offering a look of disappointment and regret.\n\n\"Soon enough you will know my pain.\" the queen sternly promised, turning to walk away under heavy escort. Adam following slowly behind as well, surrounded by armed soldiers among the vampire race.\n\n\"The explosions seem to be getting closer,\" Skulls claimed. \"A good sign we are losing the fight.\"\n\n\"And you're surprised?\" Dalton asked, pausing to drink a healthy swig from an even healthier bottle of whiskey. \"Been getting our asses kicked up one side and down another for the better part of a year now.\"\n\n\"I just meant we should consider readying ourselves quicker than we had planned.\" Skulls admitted.\n\n\"I was born ready son,\" Dalton confessed. \"I once had a watch with no hands.\"\n\n\"Huh?\" Skulls asked, confused by the smuggler's statement.\n\n\"In other words, it's always go time.\"\n\n\"I don't get it?\" Skulls confessed.\n\n\"Ah fuck, this is killing my buzz,\" Dalton replied as his lover entered the shelter. \"A watch, no hands...always go time. I'm always ready, no matter what time it is.\"\n\n\"That's hilarious.\" Cambria admitted, laughing aloud as Dalton explained himself as though he were on trial.\n\n\"Thank you. I worked hard on that.\" the smuggler replied with a grin.\n\n\"I still don't get it?\" Skulls replied.\n\n\"Fuck sakes man, you need to get out more often.\" Dalton scolded, the taint of whiskey on his breath.\n\n\"Now that, friend, I get.\" Skulls replied, openly acknowledging the truth of Dalton's statement.\n\nIt was at that moment, words rolling from his tongue, that a large explosion took place merely feet from their shelter; slamming into the thick crust of ground near the center of the fleet staging ground.\n\n\"Holy shit!\" Dalton yelled, rushing to the window as alarms sounded only dozens of feet from their shelter's door.\n\n\"Told you they were getting closer,\" Skulls said, turning to make sure both of his friends were unharmed. \"Now what's that watch saying?\"\n\n\"Saying we need to get the hell out of here.\" Dalton replied, forcing their door open as the scramble of desperation had already led hundreds of people outdoors.\n\nIt was the sound of small arms fire, however, that drew Dalton's attention.\n\n\"That's gunfire within the camp!\"\n\n\"You mean the horde has made it in?\" Cambria asked with panic.\n\n\"Not unless the dead sumbitches dyed their hair white and took a liking to leather trench coats.\" he replied.\n\n\"I don't understand?\" Skulls replied.\n\n\"Ah shit man.\" Dalton nodded, preparing himself to explain in detail.\n\n\"He means the Hunters are firing on colonials,\" Cambria interjected, as if to bust up the smuggler's punchline. \"Question is, which side are we on?\"\n\n\"The side that ain't notorious for feasting on human flesh,\" Dalton replied. \"Not to mention being a thorn in my fucking side.\" he added, pushing his brown coat back a bit as his hand slowly pulled the heavy revolver to the ready.\n\n\"I cannot believe this shit,\" Skulls complained. \"I get a winning lottery ticket and now I have to try and find a ship that's safe. People are falling dead all over a camp that's now overrun with vampires \u2013 and zombies are on the way. I do not feel like a fucking winner!\"\n\n\"Winners don't whine,\" Dalton said with a grin, glancing back to the sniper. \"They win.\"\n\n\"How are we supposed to do that?\" Cambria asked, all three of them looking from the shelter door which remained cracked.\n\n\"Just stay close,\" Dalton responded, turning to smile at his lover. \"In case I get the urge for some fine wine.\"\n\n\"I don't get it?\" Skulls admitted.\n\n\"I can still remember seeing you burn away \u2013 your ship screaming into the ocean of stars. I felt helpless. I've lived with the loss every single night, knowing you'd never know my pain,\" the queen said. \"Now you will.\"\n\n\"Sarah, I don't know what you intend to do here, but Avery has nothing to do with this. You need to let him go and deal with me how you see fit.\"\n\n\"I am!\" the queen shouted.\n\nThe group had boarded a large hunter carrier, one of several that was being stocked for the fleet's voyage. At least that's what had been thought, though Adam knew immediately upon entering that it was being stocked for another reason altogether.\n\nWar.\n\n\"You've no intention of joining the fleet.\" Adam stated.\n\n\"We'll join them in the sky, if that's what you mean,\" she replied as their group walked deeper into the heart of the large battleship. \"Of course when we arrive my people will rip the colonial ships to shreds.\"\n\n\"Sarah, these people have done nothing to you!\" Adam argued.\n\n\"Nothing? These people have done nothing to me? For centuries the colonials have hunted my people down and systematically slaughtered them!\"\n\n\"The soldiers, maybe, but the people aboard those ships include a lot of innocent faces. Women and children that have no part in that!\" Adam replied.\n\n\"\"Victims of war,\" she replied without emotion. \"Just as my father was.\"\n\n\"Your father was a good man,\" Adam stated. \"He would not have any part in this,\" he added. \"You're lost to him. To the world that matters. Do what you will to me, I'm done playing games. The Sarah I fell in love with is obviously gone and the, thing, that stands in front of me \u2013 I have no use for.\"\n\nHis words stung her to the bone. Sarah had promised to forget the pain he'd caused her once upon a time, and now he had gashed the wound back open completely.\n\n\"You will remain in this room until I return. If you try to escape, my soldiers will gun you down. If I decide to take your words to heart, my soldiers will gun you down,\" she replied. \"If you value the sight of your son EVER again \u2013 you should pray I continue to feel for you long enough.\"\n\nAdam gave no words; no plea. He simply walked into a small room by gunpoint \u2013 its door slamming shut and bolting from the outside.\n\nThe walls seamless, the sheen of aluminum glaring around him with only a single chair inside. No windows. Nothing. Though it was evident that the ship's thrusters had begun to fire up to a burn.\n\n\"Dalton, we must keep moving.\" Cambria pleaded, helping the distraught smuggler from the colonial star.\n\n\"I cannot keep going. These people need help,\" Dalton responded, his eyes having dried but his heart still gaping wide. \"I can't live with leaving more innocent people behind to die.\"\n\n\"I understand,\" Cambria agreed. \"But one of those innocent people is Adam's son. We need to go get Avery and get him to safety.\"\n\n\"I know that,\" Dalton admitted. \"I just don't know if I can.\" he added.\n\n\"Dalton James, listen to me,\" Cambria shouted, grabbing the man by his coat as gunfire rang out throughout their encampment. \"I need you!\"\n\nHis soldiering began to kick in a bit, knowing deep down that if he could push such great loss aside, at least for the moment, it could save the lives of many. Including Sasha and Avery.\n\n\"Alright,\" he finally responded. \"You've got Skulls' ticket. Now we have to get Adam's son and get to that ship before it leaves us on this damn rock. The rest we can figure out from the sky.\"\n\n\"And you?\" she asked.\n\n\"I've been a soldier most of my natural life. I know a lot of these guys, and they respect what I've done for the old colonial blue. I'll be fine.\"\n\n\"Alright, well, we need to get this done quickly. The horde will be at our front door in only minutes, and that's if our own don't kill us first.\" Cambria said.\n\nIt was a scene of total chaos. Soldiers clad in blue gunning down the vampire nation, and vice versa. Brother against brother, race against race, as everyone seemed to be staking their claim for a spot in the exodus fleet.\n\n\"Samuel. Dirsin. Twylan. All great great warriors who have fallen by your sword.\" Troy said,\n\n\"What is this you speak of human?\" Vladris asked with a demanding voice, though his answer came swiftly.\n\nThe thrust of sword, its tip biting into the lower half of Vladris' torso and sending the mighty champion to a knee.\n\n\"All of them great friends of mine,\" Troy said with a grin, their small group of surviving soldiers feeling as though the end was approaching by way of chaos in their leadership \"But none of them respected by me as was Roman Raines.\"\n\nThe remaining Hunters immediately went for their blades, though the firepower of colonial rifles soon tipped the scales. The humans and Orc-like warriors executing a plan of deception, and a well-executed plan at that.\n\n\"They say you fight as though you are a wounded lion. So wounded lion, what do you think of your chances now?\" Troy mocked.\n\nReaching down with a grimace of pain across his face, lips curling from hurt around jagged teeth, Vladris slowly wrapped a hand around the gushing wound as it drained him of blood.\n\n\"I would say you will fall just as your pathetic army did but moments ago.\" Vladris replied, squeezing pale flesh together in an attempt to slow the bleeding. Extreme duration of pain his biggest hurdle.\n\n\"Then you would be wrong,\" Troy said with a chuckle. \"Do you not see only a handful of vampires, each of them surrounded by Husk steel and colonial weaponry?\"\n\n\"I see only the coward spines of sheep. Such is the way of a lion.\" Vladris replied, looking onto the Husk-raised man with eyes fueled of revenge.\n\n\"Of course Vladris,\" Troy said, snatching an ax from the air, tossed to him by a Husk ally and awaiting the wounded champion to pull his own sword. \"I am up for the task of ending your legend. I will be the one who strikes you down.\"\n\n\"Perhaps,\" Vladris replied, his face speaking of pain as he reached overhead to unsheathe his large sword, pulling flesh away from such a deep wound all the while. \"Or perhaps you will fall just like the list of cowards you so easily name. All who thought they too were up to the task.\"\n\nThe Hunters had never seen their champion injured as badly, blood gushing from wound as though he were a stuck pig. Yet he circled a waiting human as though he were stalking.\n\n\"Your overconfidence will be your demise champion.\" Troy taunted, beginning a circle as well, readying himself for combat against the most legendary of foes.\n\n\"My confidence is earned through battle, while yours is earned through the shouting of retreat.\" Vladris stated, his words striking a nerve with the young man.\n\nWith a commanding leap, Troy slammed down to the ground, his ax biting into the crust of planet only inches from Vladris. Though a bit slower than normal, the champion Hunter arced a glancing shot of sword toward the skilled warrior, who easily parried it away with the thick armor plating of his forearm.\n\n\"Even wounded I expected better.\" Troy said, chuckling a bit as he recovered his ax and readied it.\n\n\"Yet you thrust steel into me from behind. Walking the path of a coward.\" Vladris replied, beginning to feel the effects of a sword shot which would spell his demise.\n\nWasting no additional time, Troy began hammering away by way of his ax, releasing swipe after swipe with the weapon which aimed for the Hunter champion with unrelenting power. At least by human standards.\n\nVladris felt himself weakening, as he continued to parry the ax with defensive blade strikes, finally dropping to a knee once more. His only defense becoming one of desperation as the mighty champion held his blade above him in an attempt to absorb the striking ax.\n\nHe had seen it before. Countless times. When he had an opponent beaten, their last defense before falling was the same.\n\nVladris had no intentions of dying like a wailing pup, opting instead to throw his sword to the ground and let his body absorb the next swing of ax. Something that was worthy of his legend on the battlefield.\n\nFor even the mightiest fall.\n\n\"A final word before I end you?\" Troy asked with a smile pasted to his face.\n\n\"Yes,\" Vladris nodded with defeat. \"Almost.\"\n\n_Almost?_ Troy thought.\n\nWithin the instant, Vladris had pulled a large dagger from his belt, carving into the boy's chest as the entire length of the foot-sized blade dug in. Troy immediately dropping his ax.\n\n\"You almost ended the reign of Vladris,\" the vampire said, slowly rising to his feet. \"But not quite. That feat will one day be reserved for a true hero.\"\n\n\"Are you sure you can make the shot?\" one of the colonials asked. Their ride bumpy to say the least.\n\nDalton James didn't offer a verbal reply. Just a single shot from his revolver which hummed through the thick-air of several hundred feet before striking directly to the forehead of Vladris, shattering the contents inside.\n\nThat's for Roman you son of a bitch.\n\n\"That was one hell of a shot!\" one of the soldiers announced, several more agreeing.\n\n\"I have my days.\" Dalton boasted proudly, though his revolver remained in-hand.\n\nHe knew the very moment he fired his shot, which would become legendary by many eye-witness accounts; the love of his life was boarding a colonial star with Adam's babe in-hand, or so he thought.\n\nDalton also knew that his shot, the single bullet which had turned the tide of civil war, had cemented his seat on the exodus fleet.\n\nHe had Adam Michaels to thank. So many times before, Adam had forced him into practicing with a heavy revolver. Dalton could remember fighting it; even bitching about it more times than not. Still, he remembered Adam lecturing him on a man's sidearm being the most powerful piece of weaponry.\n\nElegance among smugglers.\n\nAll of these years he'd bitched. Growled about his shotguns and coveted the grenades. Yet his best friend, a man who had taught the smuggler so many things over the years, had preached the art of the revolver as if it were a religion.\n\nDalton knew with certainty that Adam had played a big part in slaying the Hunter. A thought which brought a large smile to the brushy-faced man.\n\n\"We'll be there shortly, but it's going to be a bumpy ride from here on out.\" the pilot warned, turning to deliver the warning to Dalton and crew.\n\nBumpy?\n\nBefore he could ask, however, Dalton began to see the horde. So many dead that they literally blacked out the ground below the shuttle with their bodies.\n\n\"Holy shit.\" Dalton admitted, laying witness to tens of thousands of warriors turned from the dead. All of them sprinting toward the colonial encampment.\n\nThe shuttle rocked heavily \u2013 caused by one of many mortar strikes. The colonials were leaning on the explosive rounds; their attempt to thin out the coming horde as the exodus fleet began to launch.\n\nJust as the ground was blackened by the bodies of infected, the sky began to blacken with the shadows of ships lifting into the heavens.\n\nIt had become a non-stop ride of turbulence, with mortar being strung together with desperation. Yet Dalton could gather his bearings well enough to see nearly a dozen colonial ships yet to launch.\n\n\"The Legend of Stars is reporting engine malfunction!\" the pilot yelled, taking a moment to point to the craft.\n\nIt was mid-sized and reserved as the home for survivors of Alowin. A large planet of peaceful, yet tech dependent citizens.\n\n\"At least four-thousand souls aboard that rig!\" one of the soldiers yelled in response.\n\n\"What of the God of War?\" Dalton asked.\n\n\"The rest of 'em are reporting in with green lights. They're good to go. Just waiting for their shuttles to reel back in.\" the pilot replied.\n\nLiterally hundreds of shuttles, small choppers and merchant freighters made their way toward the large craft that were grounded and laying in wait.\n\nThe Legend of Stars seemed the focal point, however, as dozens of engineers worked on its exterior. Each checking the guts of the ship in an attempt to fix it.\n\n\"God of War this is Red Hound Fourteen, requesting landing grid position.\" the pilot said, speaking into his helmet's mic.\n\n\"We have you Red Hound 14, proceed to landing bay Seventy-One.\"\n\n\"Copy that.\"\n\nUpon landing, Dalton's first reaction was to head to the observation deck. He knew his lover was aboard, and that she no doubt cared for Adam's son. His concern was for those innocent lives aboard the Legend of Stars.\n\nIt would seem the case for many others as well, Dalton discovered, as nearly a hundred had gathered on the observation deck to watch colonial engineers race against the coming horde.\n\n\"Dalton,\" Cambria said, her eyes filled with the truth of pain. \"Dalton, they took Adam's son.\"\n\n\"Who?\" he questioned, growling with anger. \"What the fuck?\" he demanded to know.\n\n\"Sarah.\"\n\n\"I don't understand?\" he questioned.\n\n\"Sarah showed up with a group of armed soldiers. Took Avery right from my arms at gunpoint.\"\n\nStanding for a moment \u2013 fighting the numbness that shivered through his extremities, Dalton searched for the reasoning. Searched for the motive.\n\nOh hell no.\n\n\"Someone get me on the damn horn with the commander \u2013 right now!\" he yelled.\n\nWith several of the survivors turning to stare at him with question, Dalton grabbed a nearby soldier by the front of his blue uniform.\n\n\"I said get me on the horn with the commander. Otherwise, I'm gonna start cracking skulls.\" he growled.\n\n\"I can get him,\" a soldier shouted across the large room filled with people. \"What should I tell him?\"\n\n\"Tell him the vampires fucked us. You tell him to be ready for war by the time we top the clouds.\"\n\nSensing the urgency, the soldier began fumbling with his com system \u2013 finally reaching one of the soldiers who answered directly to Ortega.\n\n\"Launch squadron A.\" a voice announced over the com system, each person taking note.\n\nMoments later, the onlookers could see dozens of Swordfish fighters launching from their own ship. Each blazing for the coming horde.\n\nThough it looked impressive, Dalton knew it would not be enough. He had seen the horde with his own eyes. There was simply no stopping it.\n\nEven as the Swordfish began to stream down napalm, bringing death to anything it fell onto, there were just too many Priests coming. Having busted their way through the mighty colonial staggered defenses in the process.\n\n\"Legend of Stars is operational.\" the com announced, bringing cheers to everyone on the observation deck. As their ships began to lift into the heavens, however, their excitement turned to horror as several engineering boxes on the Legend of Stars began sparking with fury.\n\nMoments later, it had lost its altitude of just a dozen feet. Falling back to the surface in the process.\n\nThe survivors on the observation deck watched in horror as the horde began to overrun the fallen ship. Ripping metal sheeting from its hull while others smashed their way through otherwise-shatterproof windows.\n\nDalton just stood in place, watching the horror of mass murder as the colonial star finally broke into the clouds and welcomed its survivors to the black of space.\n\n\"Do not cry for them,\" Dalton scolded, several people having broken down from such a horrific sight. \"Unless you also grieve for the millions we've lost up till now. Their death was no less tragic.\"\n\nWith his words, the smuggler began descending the catwalk steps which led back to C level, thrown a bit to the side as the massive ship shifted its bearing.\n\nAlways with these fucking ships!\n\nFinal Act\n\nDalton's gut had been right, just as it had been so many times before. The colonial stars had made the correct move by lifting to the sky \u2013 guns blazing.\n\nThe vampires awaited them in orbit doing the same.\n\nDalton clinched his lover tightly as she cried aloud, grieving both the young man gone missing and her fear of demise. The brown coated smuggler did what he could to console her, though he had also began to wonder the outcome.\n\nHe was military trained enough to know their ship was being hammered with gunfire. Dalton could hear explosions throughout the lower decks of his ship, which was consistent with approaching the end of the line. At least in a gunfight between sky galleons.\n\n\"You see Adam Michaels,\" Sarah gestured, having led him to the bridge of a vampire-loyal ship under escort. \"The mighty fleet is falling. So many innocent lives.\"\n\n\"What do you want Sarah?\" Adam demanded.\n\n\"For you to make a choice,\" she grinned, waving her soldiers to the ready. Moments later, a woman of Sarah's race carried Adam's son out to him \u2013 sound asleep. \"Tens of thousands of lives aboard the dying fleet. Many children just like this, you said so yourself.\"\n\n\"What Sarah, what?\" he angrily questioned.\n\n\"Either you leave with us as a vampire as you watch the fleet destroyed \u2013 or you watch your son leave with us, from the observation deck of a colonial ship which you helped save. Your choice.\"\n\n\"Sarah, I can't.\" he admitted.\n\n\"Either you make your choice, or I WILL have my soldiers execute you \u2013 at which time I will destroy the fleet anyway, keeping your son to boot.\" she replied.\n\nA broken mess, Adam began crying heavily \u2013 his lips trembling too hard to reply.\n\n\"Your friends are on those ships Adam Michaels. All of them. Are you ready to show your forgiveness to me by joining the nation of vampires and caring for your son? Or are you as committed as you claim to save the innocent? What will it be Adam Michaels, self or selfless?\"\n\nYet the smuggler continued to cry heavily, offering no response to his former lover.\n\nWith the incentive of a gun barrel pressing to his forehead, Adam continued to cry, though holding his hands up and slowing enough to eventually speak.\n\n\"How do I know you'll keep your word? How do I know you'll look after my son \u2013 allow the fleet to go safely?\"\n\n\"Adam Michaels, I hate you,\" she replied, edging closer to him. \"Even so, I loved you once. Because of that I promise you. You have my word as a former lover. As a queen to my people.\"\n\n\"Alright,\" he replied, calming his tears just a bit. \"I'll go.\"\n\n\"Very well,\" she said, ordering her guards to remove Avery from his broken father's sight. \"Now you will watch as your entire world pulls away in full-burn from the thick of glass windows.\" she replied without emotion.\n\nLeading Adam onto a small shuttle, the queen turned to those operating her capital ship.\n\n\"Hail the colonial fleet and let them know we intend to stand down. Tell them we are sending a colonial package to them by way of shuttle,\" the queen said. \"A broken one.\" Sarah added, turning to look onto Adam with disgust.\n\n\"Sarah Blaine,\" Adam said, doing his best to block the loss of his son from memory. \"I will be back for my son,\" he added with truth. \"When I do. I will end you.\"\n\nOffering no reply aside from a look of annoyance, Sarah pressed a sequence of buttons which ejected a thick slate of shatterproof glass, jolting between them.\n\nAs his shuttle began to depart from the vampire's largest battleship, Adam stared at his former lover, signaling her death as he used hand motions to aim at her as if it were a gun.\n\n\"Jump our people to the designated coordinates.\" the queen demanded.\n\n\"Right away.\" a vampire clad in solid black replied, beginning to enter information into their computer systems.\n\nAdam Michaels had never cried as hard as he did the moment the vampires began to fire their thrusters, leaving him helpless as his eyes watched them take his son away \u2013 breaking his heart in the process.\n\nAccepting the lone shuttle, though they suspected a trap, the soldiers aboard the badly-damaged colonial star approached the shuttle with caution.\n\nAs they opened the door, each of the trained protectors of peace expected the worst. Instead, they found Adam laying in the corner \u2013 sobbing without reserve.\n\n\"Everybody out of the way,\" Dalton growled. \"I know this man.\"\n\n\"Who is it?\" a soldier asked, gun still locked onto the broken man.\n\n\"My brother.\" Dalton replied, kneeling down to scoop his longtime friend from the shuttle.\n\nFitting, as Adam had did the same for him during the first Glimmerian war many years before.\n\n\"Cambria, we need to get him to the rack and let him rest for a bit.\"\n\n\"Absolutely.\" she replied.\n\nStanding to his feet moments later, Adam felt weak \u2013 as if his body was just learning to function as an adult.\n\n\"Adam,\" Cambria said, tearing up a bit.\" I'm so, so sorry.\" she added, feeling guilty of the entire event.\n\nStaring at her for a moment, Adam wrapped his arms around her and began crying once more.\n\n\"Nothing you could have done. Nothing I could have done.\" he responded.\n\n\"Alright, alright \u2013 damn. Break it up. My playground brother.\" Dalton yelled with sarcasm.\n\nCausing the broken man to grin just slightly, Adam finally turned to face his longtime friend, offering him the same embrace of a thick hug.\n\n\"Well this is uncomfortable.\" Cambria jested.\n\n\"Yea, no kidding,\" Dalton added. \"Don't start dry-humping my leg or anything.\"\n\nLaughing through the pain, Adam finally broke away and smiled. Though everyone knew what was on his mind. Avery. Revenge.\n\nTwenty-Three\n\nAs the primitive alarm sounded, waking Dalton from a world of dreams, he quietly cursed the Gods for not allowing a longer sleep.\n\n\"Already?\" Cambria asked, turning to her lover as they both lay in bed.\n\n\"Fuck yea.\" Dalton replied, his words loaded with regret.\n\n\"Speaking of.\" she replied with a smile.\n\n\"If I had time,\" Dalton said, grinning ear to ear. \"I'd certainly do the crime.\"\n\n\"Crime? Really?\"\n\n\"Well, you know,\" Dalton replied. \"One of those upscale crimes. You know...high class.\"\n\n\"Well in that case Dalton James,\" Cambria replied, her best damsel in distress now in full swing. \"I suppose I'll save myself for the next time.\"\n\nHe chuckled a bit, but seemed to be distant from the conversation.\n\n\"What is it? Adam?\" she asked.\n\n\"Well it is now!\" he replied with a bark.\n\n\"Oh, yea, it's day twenty-three. Big day, huh?\" Cambria said.\n\n\"Just another day,\" Dalton replied as he stood from their bed and began to drape himself in colonial blue. \"Except I'm without a bottle of scratch.\"\n\n\"I think you can function without it.\" Cambria said with a large grin.\n\nHe simply replied with a tough stare. The one-thousand yard variety.\n\n\"At least I guess you can.\"\n\n\"We'll know soon enough me suppose.\" he replied, though he did so with a bitching tone.\n\nTaking a moment to brush his hands through his whiskers, Dalton stared at himself for what seemed like an eternity. Glaring into the mirror which hung above the steel sink of their quarters.\n\n\"You look nice.\" Cambria remarked.\n\n\"That sucks,\" he replied. \"I was going for mean.\"\n\n\"Well you're one mean hunk of cowboy. That work?\" she playfully replied. Her body nude under the sheets of the large, overstuffed bed.\n\n\"You be sure you have your ass here when I get back. We have some things to,\" he said, glancing to her lower body. \"Work out.\" he added.\n\n\"Oh Dalton James, you simply take my breath away.\" Cambria replied in distress.\n\nPunching several numbers into a keypad by the door, reinforced steel quickly opened to expose a large hallway, well-lit with the white of halogen.\n\n\"Commander.\" one of the two stationed marines said, both men saluting.\n\n\"At ease boys, I ain't officially the commander until this afternoon,\" Dalton replied. \"As of now, I'm still just a guy they suckered into the job,\" he added. \"Take care of my girl.\"\n\n\"Yes sir.\" the marine replied, both men standing guard once more.\n\n_Just another sucker._ He thought as everyone seemed to turn to him, his walk throughout the busy ship one of history. _And a sober one at that._\n\nDuring the final assault between the Hunters and colonials, Ortega had been gunned down.\n\nSomething that did not fade, however, was his commander's log. Mandatory entries, though the last was anything but. He has spoken of both the grit and compassion of Dalton James; even going far enough to name him the man in charge should he ever perish.\n\nHis entry was logged less than twenty-minutes before the Hunters began their final push, and those who survived looked to Ortega as an honorable man. His wishes for succession to go unchallenged.\n\nAs with colonial law, when a commander departed by resignation or death, the government would swear in a successor exactly twenty-three days following. Each day representing a major civilization under the colonial banner.\n\nThough Dalton secretly admitted he knew nothing about leading such a proud people, he also believed that many commanders before him knew less than that, and they were able to pull it off. In fact, he was guilty of slugging one and bedding down the daughter of a second, though her looks fell way-short of her political standing.\n\nBlame the whiskey.\n\nEither way, he was about to lead the colonial people. At least from a military perspective. A civilian was to be sworn in as well, the wishes of Commander Ortega, and they would share power while working together.\n\nSo it was to be.\n\nDalton expected a pretty big ceremony \u2013 though he knew there would be a shortage of man-drink, but he was not prepared for the coming inauguration.\n\nAt least two-thousand people, all seated and awaiting the ceremony. A large podium, with several ranking officers sitting close by, both civilian and military. Everyone awaiting a man who was minutes late for his own swearing in.\n\n_Ah shit._ He thought, knowing it wasn't possible to slip in unseen to the witness of thousands who sat quietly.\n\nThe sound of his boots seemed to echo throughout the landing bay as they clicked to the cement floor. He would have cursed the floor and the people for listening so damn hard, if the military branch of their government hadn't stood up in salute.\n\nIt brought a smile to the man's face, continuing his walk proudly. Though he noticed something odd. A harsh look from the politician in which he would share power. Even a whisper by the man to another high-ranking politician.\n\n\"Doctor Arness has delivered his speech,\" a man said, his role of organizing the event pretty obvious. \"Everyone awaits yours.\"\n\nSpeech? You've got to be shitting me! Nobody mentioned a speech!\n\nStepping to the podium, Dalton cleared his throat a bit, looking onto the crowd and trying to imagine them naked. Quickly realizing a majority of them were male, however, he began to think of his lover naked instead.\n\n\"The truth is,\" Dalton said, adjusting the microphone a bit as the crowd remained silent. \"I don't have a prepared speech.\" he admitted.\n\nThe crowd began to whisper a bit, in disbelief of his statement.\n\n\"Anyone who knows me, knows it wouldn't be my style,\" he said. \"I speak from the heart unless cheap whiskey is involved.\"\n\nHis statement brought laughs and cheers from most of the military branch, while the civilian branch seemed mixed in their reaction.\n\n\"Please, I appreciate it, but don't cheer me. I'm just a man. No different than the millions who perished during this war,\" Dalton said, hushing the crowd in an instant. \"I'm being honest when I say I've worked beneath a variety of men in power. Some that I respected, some that I scratched my head in wondering how they even got into power to begin with,\" he added. \"What I've learned is that I want, more than anything else, the respect of those who work for me. I plan to do anything I ask someone else to do and lead by example.\"\n\nHeavy cheers began to roll in from the crowd, bringing the new commander a moment unlike any he'd ever experienced.\n\n\"So with that, let me just say this,\" Dalton said, giving the crowd a moment to settle down. \"I'm not a commander that believes in delegating my job. You ever have a problem, any of you,\" he added, speaking to them with truth. \"You come and see me personally.\"\n\nThe entire group began to erupt with cheering, although his political equals wanted none of it.\n\n\"Just because you were Ortega's lapdog, doesn't mean you belong here. Just play your part and stay out of my way.\" Doctor Arness whispered as the roar of cheers continued.\n\nAs Dalton looked onto the man, fresh-cut hair of solid black and the face of a young buck, Dalton raised his hand to acknowledge the crowd's growing cheer.\n\n_Lead by example. Fuck it._ Commander James thought.\n\nMoments later, he decked Doctor Arness in front of thousands of eyes which turned cheers into shock. With it, the military began to scuffle with the political \u2013 on stage and in front of those who represented the entire fleet.\n\n\"In fact,\" Dalton said loudly into the microphone. \"As you can see, anyone who speaks for his own benefit and not that of the innocent voices of this fleet,\" he added. \"Really pisses me off.\"\n\nKnowing someone would most likely be killed if he didn't, Dalton exited the podium area and began a walk from the ceremony. This time under guard for his own protection.\n\n\"Sir, that was one hell of a shot.\" one of the marines boasted.\n\n\"Think so?\" Dalton asked. \"I think I knocked two of the son of a bitch's teeth out.\"\n\n\"He has plenty more.\" the marine replied.\n\n\"Hated to put you guys in this position son, I just...\" Dalton began to explain.\n\n\"It's alright sir, several of us heard his words toward you. I've been wanting to deck the bastard for nearly a year now.\"\n\n\"Either way, it will most-likely cause tension within the fleet.\" Dalton admitted.\n\n\"Within the ranks of the political parties' maybe, but not to those who matter. I think that shot to his chin earned you the respect of a lot of civilians, sir.\"\n\n\"I'm hoping with the military as well. If it comes down to a battle of books and guns, I'd love to know you guys have my back.\" Dalton said with a grin.\n\n\"Sir, you had that before you decked him. You've proven yourself time and time again on the battlefield.\"\n\nAt that moment, Dalton James \u2013 in this case Commander James, realized that the marine's statement rang true. All of the shit he'd been through as a military grunt; a glorified fetch boy, had finally paid off.\n\nAs Commander James entered the ship's CIC for the very first time, its staff stood to their feet, clapping loudly and even whistling. They had heard his speech over the com system, as well as the report of him loosening the teeth of a certain politician. Making him a hero in their eyes.\n\n\"Thank you, thank you,\" Dalton said with a grin, calming the CIC crowd down a bit. \"Now, can someone please walk me through what the hell I'm supposed to be doing exactly?\"\n\nHis words erupted laughter as most went back to their workstations.\n\n\"I can help you with that, if you promise not to slug me?\" a well-dressed soldier replied.\n\n\"Alright son, you have my word.\" Dalton said with a grin.\n\n\"Com Officer Paul Anthony, sir,\" the man stated. \"A privilege to have someone like you leading our CIC.\"\n\nDalton nodded his appreciation to the man, who was clean shaven and not a day over twenty-five years of age. Several medals pinned snugly to his chest.\n\n\"Sir, we've mapped out several locations that our fleet may able to survive a journey. I've marked a couple of routes into deep space that our scouts have confirmed may be rich in resources.\" the com officer said.\n\n\"Don't need a map. I know where we're going.\" Dalton replied.\n\n\"Um,\" Anthony replied with confusion. \"Alright sir. Where would that be?\"\n\n\"To get my best friend's son back.\"\n\n\"Sir, with all due respect, we don't even know where the Hunters are?\"\n\n\"No resource ships in their fleet. Means they have to be somewhere in the Skyla System, and I can promise you,\" Dalton said with stern truth. \"There's not a rock in this entire system that the bastards are going to be able to hide under when I'm finished.\" he promised.\n\n\"Alright sir.\" Anthony replied, though he seemed reluctant to do so.\n\n\"So you send our scout ships back out and you have 'em comb the system. When they find the Hunters' fleet, you have 'em float over a message in a bottle,\" Dalton said, choosing his words proudly. \"That we're coming \u2013 and I'm bringing motherfucking revenge with me.\"\n\n\"Will do sir.\"\n\n\"Who in here served on Glimmeria the first time around?\" Dalton asked loudly.\n\nThree hands were raised. Two of them deck officers, while a third was a colonial soldier stationed on the door of the CIC.\n\n\"How many friends did each of you lose?\" Dalton asked.\n\nA strange question, though each respected his command enough to offer their answer. Seven. Four. Eighteen.\n\n\"Congratulations, you're my new XO. Now get your ass up here.\" Dalton demanded of the soldier stationed on the door.\n\n\"Sir?\" the soldier asked.\n\n\"Commander, you should consider this decision wisely.\" Anthony pleaded in a soft voice.\n\n\"Who's a better candidate to lead a group of people that have lost so much - then a man who has also lost, and done so while fighting for the right side since day one?\"\n\n\"Sir, I just think...\" Anthony began to plead.\n\n\"I don't want you to think,\" Dalton replied sharply, staring the man down. \"I want you to execute my damn order. Every minute the vampires are out there is a minute that filthy bitch has her hands on my godson.\"\n\n\"Yes sir.\"\n\n\"Come on private, carry your ass up here.\" Dalton insisted of the guard by the door.\n\n\"Yes sir, I'm just taken back by the promotion. I think very highly of you and appreciate you thinking so much of me. Hope I don't let you down when I admit I know nothing about these computer systems?\" the soldier asked.\n\n\"Well,\" Dalton replied, reaching over to read the soldier's name tag. \"Sergeant Cohen, it can't be too damn hard,\" he added, leaning over to whisper into his new XO's ear. \"Everyone in here has soft hands. How hard can it be?\"\n\nSmiling to his new XO, Cohen eventually smiled back \u2013 understanding that he and Dalton were cut from the same cloth. Hell-raisers for the same cause. Defense of the defenseless.\n\n\"Maybe you're thinking on it a little too much.\" Cambria suggested.\n\n\"Maybe,\" Dalton replied, sitting at a large desk of polished wood as he studied the papers carefully.\n\n\"You should rest,\" she said with concern. \"Particularly your slugging hand.\" Cambria added with a chuckle.\n\n\"That sumbitch had it coming.\" Dalton replied in his own defense.\n\n\"Alright, if you say so,\" Cambria replied, placing her hands in the air. \"Just don't hit me.\" she added with a grin.\n\n\"Oh, you're a wise-ass.\" he commented, offering her a wide grin in return.\n\n\"Speaking of,\" Cambria said, placing her palm to her head as though she were suffering. \"You owe me cowboy.\"\n\n\"I reckon I do.\" Dalton replied, cutting his full-attention to the woman who awaited his embrace.\n\n\"I would hate to think that our mighty leader was no good on his word.\" she said playfully.\n\n\"I'm good for my word alright,\" he answered, standing to his feet as Cambria now lay there, half-nude and teasing him. \"Or, as so many women have called it,\" he added, drawing a disapproving look from his lover. \"Just plain good.\"\n\n\"Really?\" she asked.\n\n\"Just get over here and love on this old hound dog. That's an order.\" Dalton said \u2013 technically his first command as the man in charge.\n\n\"What?\" one of the two stationed guards asked as the other stared to him.\n\nAnswering his question with a continued stare, it was evident that the thickened-steel walls weren't thick enough.\n\n\"Block it out.\" the guard added, Cambria and Dalton growing louder \u2013 as if wild animals were mating on the other side of the door.\n\nA smug look came across the face of the quiet marine. His way of saying impossible.\n\n\"Try.\" the first marine added, escaping into deep thought for a moment.\n\nIt's gonna be a long damn night.\n\nCommander's Log\n\nThe first of many, I suspect, as life seems to have thrown a curveball right into my lap. Still, as commander of this fleet, I'm tasked with providing hope. A task I welcome.\n\nSeeing the pain in the eyes of my best friend has been indescribably hard. He is a brother. The closest thing to family I've known since I was a child. So, as I look into the eyes of those around me among the fleet, I understand their loss.\n\nThe good news is that we've confirmed across the fleet that no infected made the exodus. The bad news, of course, is that we have no real idea of direction. We simply do not know what perils or wealth of resources await us beyond the line we consider uncharted space.\n\nWe've deployed probes that have relayed information back to us, but only the basics. We've no idea if life exists out there. No real number of days we can actually survive based on our food storage and fuel counts.\n\nWe have literally become a race that is chained to our ships. Bringing with us only what we could stock in rush, which doesn't amount to much in terms of comfort.\n\nSome among the fleet think we need to revisit the planets throughout the Skyla System once more. Perhaps salvage more of what we need before setting sail into the stars beyond.\n\nNot revisiting the planets among our former system seems to be the only thing Doctor Arness and myself agree on. The sumbitch. I can already see that he plans to be a constant thorn in my side, though he will do so with a few teeth missing.\n\nThey say that assuming power changes a man. I don't know about that. I've changed, that's for sure, but I do not credit my new position of power for it. I credit loss.\n\nAs I look back on all of the memories gone by. The bars. The bar-fights. The jail time. The bedding of women. Well, to be frank, this list could go on for quite some time. Being an official entry and all, I'll cut to the chase.\n\nThe loss of so many who I cared for has changed me. Many of my good friends. My crew...hell, even my dog. They've all been taken from me, each one taking a piece of me with them to the grave.\n\nThe moment I held true love in my arms - I became a man. So I accept the challenge of leading these people to a new home. I'm ready.\n\nI'll lead these people as the man they need me to be. The man that life has molded me to be.\n\nThe whiskey though, I'm not giving that up. I've grown into a man, but I ain't dead.\n\n-Commander Dalton James\n\nAbout the Author\n\nJohn M. Davis is the author of several books, including three which have grabbed the #2 overall spot in their respective genres. He currently resides in the beautiful mountains of Virginia with his wife and two spoiled children. The journey doesn't end here! Be sure to visit the official blog for information on The Fleet, featuring Commander Dalton James.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTHE  \nIRRATIONALIST\n\nSUZANNE BUFFAM\n\nPOEMS\n\nCopyright \u00a9 2010 Suzanne Buffam\n\nAll rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.\n\nDistribution of this electronic edition via the Internet or any other means without the permission of the publisher is illegal. Please do not participate in electronic piracy of copyrighted material; purchase only authorized electronic editions. We appreciate your support of the author's rights.\n\nThis edition published in 2012 by\n\nHouse of Anansi Press Inc.\n\n110 Spadina Avenue, Suite 801\n\nToronto, ON, M5V 2K4\n\nTel 416-363-4343\n\nFax 416-363-1017\n\nwww.houseofanansi.com\n\nLIBRARY AND ARCHIVES CANADA CATALOGUING IN PUBLICATION DATA\n\nBuffam, Suzanne, 1972\u2013\n\nThe irrationalist \/ Suzanne Buffam.\n\nPoems.\n\nISBN 978-0-88784-307-5\n\nI. Title.\n\nPS8603.U52I77 2010 C811'.6 C2009-906400-6\n\nCover design: Bill Douglas at The Bang\n\nCover image: Old Modern Handicrafts\n\nShip in a bottle, featured on the cover, is manufactured in Vietnam by Old Modern Handicrafts and distributed by Capt. Jim at capt.jimscar-go.\n\nWe acknowledge for their financial support of our publishing program the Canada Council for the Arts, the Ontario Arts Council, and the Government of Canada through the Canada Book Fund.\nFor Mira\nThere is another world, but it is inside this one.\n\nPaul Eluard\nI\nRUINED INTERIOR\n\nIn the beginning was the world.\n\nThen the new world.\n\nThen the new world order\n\nWhich resembles the old one,\n\nDoesn't it? Its crumbling\n\nAqueducts. Its trinkets and shingles.\n\nIts pathways lacquered in fog.\n\nIf all we've done is blink a bit\n\nAnd touch things,\n\nNotice how dust describes\n\nA tin can by not falling\n\nWhere it sits, or how a red sleeve\n\nGlimpsed through curtains\n\nMimics the tip of a flickering\n\nWing, was the whole day a waste\n\nOr can worth be conferred\n\nOn a less than epic urge? Bow-wow\n\nSays the doggie on page two.\n\nAhoy says the sailor.\n\nArise says the tired queen\n\nAnd face the highway\n\nThe donut shops, and the boardwalk.\n\nIt rained today. You can see\n\nPerfect inversions of streetlights\n\nSuspended in drops on the window.\n\nYou can see the skyline\n\nTrying to hold up the sky.\n\nDon't tell me there's another,\n\nBetter place. Don't tell me\n\nThere's a sea\n\nAbove our dreaming sea\n\nAnd through the windows of heaven\n\nThe rains come down.\nIF YOU SEE IT WHAT IS IT YOU SEE\n\nI didn't look at the fire.\n\nI looked into it.\n\nI saw a wall of books\n\nCrash down and bury me\n\nCenturies deep in red leather.\n\nI saw a statue in a park\n\nShake dust from its fist\n\nAnd a ship called Everything\n\nSink down on rusted wings.\n\nTen thousand triangles collapsed\n\nInto a point\n\nAnd the point was this.\n\nI cannot tell you what I saw.\n\nMy catastrophe was sweet\n\nAnd nothing like yours\n\nAlthough we may sip\n\nFrom the same\n\nBroken cup all afternoon.\nAMOR FATI\n\nAny idiot can become a genius if she wants it badly enough.\n\nOne must study how the crow flies.\n\nOne must say to oneself as the crow flies so fly I.\n\nIn the dream I am an empty tree. One by one my branches fill with silent crows that have travelled great distances to reach me.\n\nEach crow contains a golden seed of knowledge locked in its craw and by containing them all in my lofty crown I contain all knowledge of the kingdom.\n\nMy attempts to remember are proof in themselves.\n\nAt times one must accompany a shadow like the moon above a field of bitter greens.\n\nIn this wretched spirit the pilgrim applies herself and is rewarded.\n\n\"I only felt in the midst of my suffering the presence of a love,\" she explains, \"like that which one can read in the smile on a beloved face.\"\n\nI can't help what I want.\n\nThere is no such thing as a dream that comes true.\n\nEvery dream is already true the moment it is dreamed.\nTHE NEW EXPERIENCE\n\nI was ready for a new experience.\n\nAll the old ones had burned out.\n\nThey lay in little ashy heaps along the roadside\n\nAnd blew in drifts across the fairgrounds and fields.\n\nFrom a distance some appeared to be smouldering\n\nBut when I approached with my hat in my hands\n\nThey let out small puffs of smoke and expired.\n\nThrough the windows of houses I saw lives lit up\n\nWith the otherworldly glow of tv\n\nAnd these were smoking a little bit too.\n\nI flew to Rome. I flew to Greece.\n\nI sat on a rock in the shade of the Acropolis\n\nAnd conjured dusky columns in the clouds.\n\nI watched waves lap the crumbling coast.\n\nI heard wind strip the woods.\n\nI saw the last living snow leopard\n\nPacing in the dirt. Experience taught me\n\nThat nothing worth doing is worth doing\n\nFor the sake of experience alone.\n\nI bit into an apple that tasted sweetly of time.\n\nThe sun came out. It was the old sun\n\nWith only a few billion years left to shine.\nHAPPY HOUR\n\nI'll have an Icecap.\n\nMake it a double.\n\nBring me a Fog on the River,\n\nA Niagara Falls on the rocks,\n\nAnd a Tempest with a chaser of Hail.\n\nI don't want to be rescued.\n\nI want to crawl through a honeycomb\n\nOf subglacial passageways,\n\nShove my head under God's faucet\n\nAnd keep chugging until I pass out.\n\nI want thirst to drink me.\n\nI want to come back as a bucket of blood.\nOCCASIONAL POEM\n\nToday is the thirteen thousand, one hundred and forty-first day of the rest of my life.\n\nThere is no way to know how many beans are in the jar without removing them one by one.\n\nIf I find it harder to learn the future tense than the younger students in my Spanish class do, it is because so much more of my life resides in the past.\n\nStill I try to live in the moment, where everything is endlessly happening at once.\n\nThe earth spins, the curtain lifts, clouds appear to be floating, and yet they are, in fact, constantly falling.\n\nTo be ahead of one's time may be the same as being very far behind it.\n\nWhen he saw the bison leaping off the walls at Lascaux, Picasso turned to his guide and lamented the achievements of modern art.\n\n\"We have discovered nothing,\" he is reported to have said.\n\nAnd yet today is different from yesterday.\n\nYesterday only contained itself and the days leading up to it, while today contains itself, yesterday, plus all the endless days before that.\n\nLet us celebrate.\n\nLet us separate the movement from the moving thing.\nDEATH TOLL RISES IN BLACK SEA SINKING\n\nFrom where I stand\n\nThe world is a warm blue bath.\n\nI can wash my feet in it.\n\nI can pick a cloud\n\nAny cloud\n\nAnd watch it nudge a mast\n\nAcross a harbour of light.\n\nI have the potential\n\nFor offspring inside me\n\nAnd if it is desire that I lack\n\nI can sit down on a rock\n\nAnd wait for my lack\n\nTo dissolve.\n\nI can do so\n\nAnd so I do so\n\nWaiting with my feet\n\nAt the edge of the bath\n\nUntil at length it is time\n\nTo lug my body home.\n\nThere I read about a storm\n\nIn the east\n\nIn a story from the west\n\nAbout a boatload of sailors\n\nWho sank that the water would rise.\nTHE SOLITARY ANGLER\n\nOne day I woke up\n\nAnd did not fear the old gods.\n\nI called the number on my fridge\n\nAnd when the movers arrived\n\nI gave them everything.\n\nOn my way out of town\n\nI spat into the wind\n\nAnd did not linger to see where it landed.\n\nWho can say for sure\n\nIf the dream has ended or begun?\n\nA frail dimness rims my craft.\n\nStars swim to the surface of a bottomless well\n\nAnd sink when I take my eyes off them.\n\nThere is no greater calamity\n\nThan to underestimate the strength of your enemy.\n\nThe ancients saw the stars\n\nAnd called them angels.\n\nThey turned everything else into a clock.\n\nI say wear a watch if you must\n\nBut don't count on it. \nTHE ARENA\n\nThe judges invite me to enter a contest of wits.\n\nIf I win, I will be ashamed of myself for having stooped to enter such an unworthy contest in the first place.\n\nIf I lose, I will be ashamed of my not being the one to throw down her laurels in disgust.\n\nThe only solution seems to be to avoid the arena entirely.\n\nAnd yet, not to venture at all seems a concession to anticipatory shame \u2014 a shameful position in its own right.\n\nTo compete or not to compete: this question has dogged mankind since the first hollow sceptre was whittled from a bone.\n\nI will have to turn my shame into a point of pride, I see.\n\nI go home and take off my pants, hang a funhouse mirror from my bathroom door, and force myself to eat soup with a spork.\n\nI call myself every bad word I know.\n\nI heap scorn upon myself by the shovelful until I grow so heavy with shame I can barely lift my body into bed.\n\nBy the end of the week I feel ready to submit.\n\nBut when I show up to take my place at the podium, I discover the contest has ended.\n\nThere is cheering in the stands. I am sentenced to reign in a kingdom of sand.\nPLACEBO\n\nIt is possible to die of fright after being bitten by a non-venomous snake.\n\nAlso, in some cultures, to kill a man by pointing a bone at his heart.\n\nThese are two severe examples of the nocebo effect \u2014 from the Latin for \"I will harm.\" A third is when I forget to wear my lucky orange scarf and lose at craps.\n\nWilliam James writes that often enough our faith beforehand in an uncertified result is the only thing that makes the result come true.\n\nHow often is often enough?\n\nWill faith alone finish this poem?\n\nWhen he ran out of morphine on the Tunisian front, Henry Beecher injected dying soldiers with saline instead. At the time he was relying on the strength of faith alone, but what he could not have known back in 1943 was that this solution in fact released a pain-relieving substance in the patient's aching brain.\n\nWhen asked to discuss his religious beliefs, Luis Bu\u00f1uel replied: I'm an atheist, thank God.\n\nDespair comes from failing to believe new things are possible.\n\nScience: one bold leap after another, suspended by a theory of invisible strings.\n\nObecalp: what the doctor scrawls on the prescription pad when the pharmacy runs out of science.\n\nPlato believed that to release another person from ignorance despite their initial reluctance was a great and noble thing \u2014 and yet every day he woke up and lived in the world as if it were real.\n\nAsleep beneath the sirens and the clashing of arms, who hears the angels gently plucking their inaudible tune?\n\nWho sees the black dog running through the night?\nTO LIVE HERE\n\nPaul Eluard\n\nI built a fire, the blue sky having abandoned me.\n\nA fire to befriend.\n\nA fire to introduce me to the winter night.\n\nA fire to live better.\n\nI fed it what the day had fed to me.\n\nForests, foliage, wheat fields, vines.\n\nNests and their birds, houses and their keys.\n\nInsects, flowers, furs, festivals.\n\nI lived with the solitary sound of crackling flames.\n\nWith the solitary perfume of their heat.\n\nI was like a boat coursing in closed water.\n\nLike the dead I had but one element.\nINFINITIVE INTERIOR\n\nTo be small among voices.\n\nTo wear the black hat.\n\nTo kneel in the shavings.\n\nTo speak of the nameless blue flowers.\n\nTo eat them.\n\nTo retreat to the torn red interior.\n\nTo hear the low engines approaching.\n\nTo button.\n\nTo hammer.\n\nTo have.\n\nTo have not.\n\nTo have sat by the sea and been rewarded with a pair of glinting wings.\n\nTo have held out for more.\n\nTo have had it.\n\nTo have held out for less.\nIDEAL WORLD\n\nNothing matters in an ideal world.\n\nNot the stones you skip,\n\nNot the fat birds overhead.\n\nRun your fingers through the sand all day.\n\nLie still as a ship at the bottom of the sea.\n\nStick out your tongue\n\nAnd taste the wrecked century\n\nIn a melting snowcone purchased for a peso.\n\nAll you taste is the taste of it.\n\nLight plucks the coins from your eyes,\n\nThe heart spends its store\n\nOn a few everlastings\n\nJutting from cracks in the boardwalk.\n\nCall it a kingdom nevertheless.\n\nWatch the light black canopy\n\nLower from the west\n\nWhere a rust-stained tanker spins\n\nA slow pirouette en route to oblivion\n\nVia Sudan.\n\nIf you feel lonely\n\nIt's because you were borne this way.\n\nIf there are clouds here\n\nThey must be ideal clouds.\n\nClouds you can see through.\nII\nLITTLE COMMENTARIES\n\nThere is no one centre of the universe.\n\nNICOLAUS COPERNICUS, COMMENTARIOLUS\nOn Possibility\n\nHere comes the train from Tehuantepec\n\nBringing lovers and rice.\n\nBringing last night's dream of floating ponies\n\nTo today's rusty gate.\n\nSee the faces at the station look up.\n\nSee the mystery deepen.\n\nThere is no train from Tehuantepec.\n\nHere it comes!\n\nOn Necessity\n\nAs a young man Galileo\n\nUnderstood very well\n\nThe workings of the pendulum\n\nBut not until he was an old man\n\nApproaching\n\nThe hour\n\nOf his death\n\nDid he devise\n\nThe pendulum clock.\nOn Shortcuts\n\nI know a painter\n\nWho fills every canvas with sky.\n\nThis makes his landscapes look lonely\n\nAnd his figures bereft.\n\nIt's a shortcut, he concedes.\n\nLike adding wings to men\n\nOr putting birds in poems.\n\nOn Attachment\n\nA house burns all night\n\nIn the middle of a field.\n\nA beautiful sight\n\nEven if the burning house\n\nDoes happen to be mine.\n\nSooner or later\n\nAll burning houses will be mine.\nOn the Fire Sermon\n\nBurning burning burning burning\n\nQuotes T. S. Eliot\n\nA man so famed\n\nFor his chilly demeanour\n\nNot even his pipe\n\nSeemed to give off smoke.\n\nOn Valleys\n\nTo be a valley\n\nFind a hill\n\nAnd lie down at its feet.\n\nOn Clouds\n\nTo be a cloud\n\nFind a hill\n\nAnd swallow it.\nOn the Enlightenment\n\nTake the thing you love most and cut it up.\n\nArrange the parts carefully\n\nAccording to the picture in your head.\n\nNow shine your mind at them.\n\nIf their laws come striding boldly forth at you\n\nYou will soon become a great man of your time.\n\nIf instead they lie there on the table bleeding beauty\n\nYou are probably a poet or a child.\n\nOn Moonlight\n\nMoonlight fills the bathroom sink.\n\nIf a person could drink from it\n\nShe would be her own ghost.\n\nOn Ghosts v. Zombies\n\nSoul without a body or body without a soul?\n\nLike choosing between an empty lake\n\nAnd the same empty lake.\nOn Abstract Expressionism\n\nAd Reinhardt\n\nFilled his black canvases\n\nWith everything he knew\n\nAbout Modern Art.\n\nRobert Rauschenberg\n\nEmptied his white canvases\n\nOf everything he knew\n\nAbout Modern Art.\n\nTogether they express\n\nEverything and Nothing\n\nThere is to know about art.\n\nOn Negative Capability\n\nA man and a woman\n\nSide by side\n\nNot speaking\n\nNot not speaking\n\nCross a moonlit plaza\n\nWithout plans.\nOn Love Poems\n\nA friend says relationships\n\nAre only good for two poems:\n\nOne at the beginning\n\nAnd one at the end.\n\nStevens says better to peddle\n\nPineapples than write love poems\n\nUnless you happen to be\n\nIn love, that is.\n\nWhen your lover shows up\n\nWith a basket of fruit\n\nThank him in advance\n\nFor the poem you will one day receive.\n\nOn First Lines\n\nThe first line should pry up\n\nA little corner of the soul\n\nAs the first ray of daylight\n\nPries open the sleeper's lids.\nOn Quandariness\n\nI do not know which to prefer\n\nThe beauty of Nova Scotia\n\nOr the beauty of France.\n\nDucks landing on the saltmarsh\n\nOr poached in their fat on my plate.\n\nOn Dining in Paris\n\nTake small bites.\n\nChew your food before swallowing.\n\nDo not expect the waiter to congratulate you.\n\nOn Drinking in Paris\n\nGreet the new Beaujolais\n\nWith a mixture of fondness and disdain\n\nLike a childhood friend from the provinces\n\nOr yet another Oscar-winning picture\n\nStarring G\u00e9rard Depardieu.\nOn Parakeets\n\nPractice concentrating on an empty stomach.\n\nPractice making love with a terrible sunburn.\n\nPractice walking with little dried peas in your shoes.\n\nSprinkle sand in your food.\n\nSprinkle salt in your tea.\n\nPitch your tent in a howling gale.\n\nSoon you will be ready to live in the house on the hill\n\nNext door to the house full of parakeets.\n\nOn Could\n\nThere is no cake in the oven, alas\n\nBut a small bit of effort\n\nCould put one there.\n\nOn Travel\n\nNot past my Father's Gate\n\nShe Vowed \u2014\n\nAs she Closed her Eyes\n\nAnd lit out for Zanzibar.\nOn Borges\n\nPut one dream\n\nInside another.\n\nOn Oblivion\n\nThe most harrowing thing\n\nAbout outer space\n\nClaim the astronauts\n\nWho have set foot there\n\nIs not so much the darkness\n\nAs the silence.\n\nBeethoven heard it\n\nAnd composed his final symphony.\n\nOn opening night\n\nHe kept his back to the audience\n\nAnd when the orchestra reached\n\nThe music's echoing close\n\nHe continued to conduct for several bars.\n\nEmbarrassment\n\nRippled through the room\n\nLike cosmic wind.\n\nBut not through Beethoven\n\nBy then light years away.\nOn Romanticism\n\nOut of nowhere\n\nBlows a baleful wind\n\nTo explain your heart to you.\n\nOn Flags\n\nFew things are more stirring\n\nThan a flag in the wind.\n\nA problem of aesthetics vs. ethics.\n\nAll morning I study\n\nA tea towel drying on the line.\n\nA flag without a country\n\nIs like desire without an end.\nOn Metaphor\n\nWhen Job laments that the Lord\n\nHath put my brethren far from me\n\nIt is up to you to decide\n\nIf he is speaking metaphorically.\n\nOn Invective\n\nFuck you and the horse you rode in on\n\nIs often just another way of saying come back.\n\nOn Ad Campaigns in the Underworld\n\nBet you can't eat just one\n\nSmiled Pluto\n\nAs he held out his handful of pain.\nOn Impossibility\n\nI try to write \"automatically\"\n\nBut keep stopping to look at the sky.\n\nWords are in it\n\nAnd a great blue silence\n\nThat fills the distance between.\n\nOn Irrational Numbers\n\nAhoy cries the sunrise\n\nTo the sea's flagging captains\n\nAmong whom you number\n\nInfinity plus one.\n\nOn Seville\n\nEvery tree is an orange tree.\n\nEvery pigeon a dove.\nOn Geological Time\n\nEnjoy the view while you can,\n\nMt. Everest.\n\nOn Middle Names\n\nVictor\n\nSelwyn\n\nHamish\n\nAdeline\n\nAurora\n\nSomerset\n\nLoomis\n\nAnd Stearns\n\nSet out for a picnic\n\nBeneath the green\n\nNever trees.\nOn Normandy\n\nFate piles up\n\nOn the bloody Norman shore.\n\nIf you must swim there\n\nSwim on your back.\n\nOn Last Lines\n\nThe last line should strike like a lover's complaint.\n\nYou should never see it coming.\n\nAnd you should never hear the end of it.\n\nOn Vanishing Acts\n\nThe magician says watch closely.\n\nThe lover says close your eyes.\n\nOn Duration\n\nTo cross an ocean\n\nYou must love the ocean\n\nBefore you love the far shore.\nOn Pi\u00f1atas\n\nNo point in swinging the bat\n\nUnless the blindfold's secure.\n\nOn Fireworks\n\nRadiant pitchforks thunder in the now\n\nTrailing spectres of is.\n\nOn Ambition\n\nLike love\n\nIt grows impatient\n\nOf both rivals and delays.\n\nOn Gold\n\nUnlike love\n\nIts value increases\n\nWhen it sits in the vault.\nOn Yesterday\n\nOf all the bright fruits in the market\n\nI bought only one.\n\nNow my peach\n\nIs a pit.\n\nPlant it deeper.\n\nOn Winter\n\nIt is a good rule of thumb\n\nTo leave your house at least once.\n\nIf you have a little dog\n\nYou can take him for a little walk.\n\nIf a big dog, a big walk.\n\nBut you cannot cross an open field\n\nWhere snow is melting\n\nWithout growing a little bit\n\nOlder with every step.\nOn Fountains\n\nJoy alone upholds the moment\n\nWhen the sparkling ascent\n\nIs overwhelmed.\n\nOn Joy\n\nJoy unmixed with sorrow\n\nIs like a fountain turned off at night.\n\nOn Inverse Relations\n\nThe pleasure I feel\n\nWhen I say the word \"trousers\"\n\nIs equal, exactly\n\nTo the discomfort I feel\n\nWhen I say the word \"slacks.\"\nOn Moving Day\n\nTwo houses stand aloof\n\nIn their emptiness.\n\nThe same dusty sunlight\n\nLicks the floorboards\n\nOf your future and your past.\n\nIt is good\n\nTo be homeless\n\nFor an hour.\n\nOn Exile\n\nOne of the richest men in Rome\n\nSeneca knew many pleasures in life.\n\nIn particular, it is said\n\nHe loved to dine on quail\n\nAt a citrus-wood table\n\nWith ivory legs.\n\nBut he also knew\n\nThe fickle winds of Empire.\n\nFar from home\n\nWithout a friend\n\nOr lover near\n\nWith but a thin gruel\n\nBefore you\n\nAnd plank for a pillow\n\nRecall his recollection\n\nOf these pleasures in a letter\n\nTo his mother from the tower:\n\nBetween them and me\n\nI have kept a wide gap.\n\nOn Evolution\n\nPlace your face\n\nInto your hands.\n\nA perfect fit!\nOn Relativity\n\nA pygmy hippopotamus\n\nIs roughly the size\n\nOf a large family dog.\n\nA capybara\n\nIs an enormous guinea pig\n\nRoughly the size\n\nOf a very small horse.\n\nThe smallest horses\n\nWear very small shoes\n\nRoughly the size\n\nA fairly large person would wear\n\nDuring infancy\n\nA brief phase\n\nIf you are human\n\nA lifetime\n\nIf you are a fly.\n\nOn the New Darkness\n\nWhat's wrong\n\nWith the Old Darkness?\nOn Satisfaction\n\nMake lists\n\nOf things you have already done.\n\nPleasure exists\n\nIn crossing them off one by one.\n\nOn La Gioconda\n\nCrowds press in to glimpse her terrible smile.\n\nWhat diet of secrets sustains it?\n\nThe crowds soon tire\n\nAnd retreat to the buzzing caf\u00e9.\n\nAloof behind her varnish and her bulletproof veil\n\nShe casts her gaze on nothing now \u2014\n\nThe greatest, said Da Vinci\n\nAmong all great things found here among us.\nOn Impediments\n\nChildren play ball\n\nIn the crowded plaza\n\nNot in spite of the crowd\n\nBut because of it.\n\nOn Riding Backwards on Trains\n\nThrough the red hills and over green dells\n\nThe shock of it shakes from you\n\nEndless farewells.\n\nThere goes a fountain. There goes a goat.\n\nBack to the future\n\nHeart in your throat.\n\nOn the Verb Esperar\n\nIn Madrid you may wait your whole life for salvation\n\nWhile hoping all day for your turn at the bank.\nOn Forgiveness\n\nIt is best to forgive all sins in advance\n\nBecause afterwards it can be hard.\n\nOn Advice\n\nDo not offer advice\n\nAnd do not solicit it.\n\nSeek wisdom within.\n\nBut do be sure to bring\n\nA pair of long underwear\n\nFor those long cold hours\n\nIn the desert at night.\n\nOn Space Travel\n\nNot to see the frozen heavens up close\n\nBut to see our leaky planet from afar.\nOn Common Sense\n\nAristarchus of Samos\n\nSealed his fate as a footnote\n\nBy pointing out the movement of the Earth around the Sun\n\nTo Aristotle of Stagira\n\nWho pointed out birds in the sky\n\nKeeping pace with it.\n\nOn St. Augustine\n\nLove and do what you will\n\nIs a dangerous slogan\n\nTo plaster on the walls\n\nOf a freshman dorm in spring.\n\nOn Madame Bovary\n\nMisanthropy starts at home.\n\nSo great was Flaubert's\n\nGrasp of pitiless fact\n\nHe cast himself as a woman\n\nAnd punished her with all\n\nThe love in his heart.\nOn Beauty\n\nNoon comes hammering down on the harbour\n\nRinging all its bells.\n\nMasts tip\n\nThis way and that\n\nScratching God's name on the vault.\n\nOn the Seine\n\nBeggar sailor soldier thief\n\nAnd the dregs of Saint Joan\n\nSlip black as God's mirror\n\nBeneath your bright bateaux mouches.\n\nOn Suicide\n\nPeople who commit suicide don't fail to believe in life.\n\nThey fail to believe in death.\nOn the Logic of Dreams\n\nWhile in dreams it is true\n\nAnything can happen\n\nDreams often seem bound\n\nBy inexplicable rules.\n\nIn this way they resemble Poetry\n\nAstronomy, Geometry, Love . . .\n\nOn Church Bells\n\nEvery\n\nHour\n\nOn\n\nThe\n\nHour\n\nWhat\n\nStirs\n\nThis\n\nSleepy\n\nNowhere\n\nIs\n\nOld\n\nNews\nOn Hummingbirds\n\nThe smaller the heart the swifter the wings.\n\nOn White Flowers\n\nBy moonlight the lily dominates the field.\n\nOn Clear Nights\n\nAt most two thousand stars\n\nCan be seen with the naked eye from earth.\n\nA difficult number to grapple with.\n\nToo large, and, on the other hand, too small.\n\nA simple mathematical equation\n\nMay throw the problem into relief.\n\nConsider a battlefield.\n\nThe fighting has ended\n\nAnd the bodies lie still in the grass.\n\nHow many dead soldiers\n\nEqual the sky overhead?\nOn Antigone\n\nLaw spoke\n\nAnd the land bit its lip.\n\nWhy spit in the wind?\n\nLove too is a law.\n\nOn Where You Live\n\nSomewhere I have never been\n\nThat you\n\nBent over your books\n\nIn snug bronze lamplight\n\nBeyond\n\nNight's silvered reminiscences\n\nWill never see.\nOn Paradise\n\nNo Then.\n\nNo My.\n\nNo Soul.\n\nNo Like.\n\nNo To.\n\nNo The.\n\nNo Lark.\n\nNo At.\n\nNo Break.\n\nNo Of.\n\nNo Day.\n\nNo Arising.\n\nNo From.\n\nNo Sullen.\n\nNo Earth.\n\nNo Sings.\n\nNo Hymns.\n\nNo At.\n\nNo Heavens.\n\nNo Gate.\nOn Everyone\n\nEveryone loves a balloon in the sky.\n\nIt is head-shaped and floating away.\n\nIt knows not why it goes\n\nNor where it will be when it gets there.\n\nIt wags its little tail and disappears.\nIII\nTHE WISE MAN\n\nI am not a wise man. This makes my life difficult in certain ways. But in other ways it simplifies things. I find it hard to sit still very long before I get up and wander the halls in my hat for example. On the other hand I stay warm and keep moving. Could these ways be the same way? A wise man could tell you. A wise man would look out his window and see not a row of low clouds rolling east like a trainload of coal through a crossroads, but a lit glimpse of the infinite, the wise man's only home. A wise man might think of his childhood and smile. Often in a quandary I ask myself what would a wise man do? A fool sees not the same tree that a wise man sees, said a wise man, and when I look out at the spruce I wonder what a wise man sees. A wise man might laugh at such questions. As for me I laugh often, but I don't get the joke. \nTHE TRUE BELIEVER\n\nThere are only two ways of reacting to life. Every scripture in the world will tell you this. Choose the wrong one and no amount of digging will save you. Before me an old woman plucks a stubborn hair from her cheek. Behind me a child weeps into her spinach. I'm the one on her knees with a shovel, wiping starbursts of spit from her chin. Each day rises with the promise of news. Sometimes it isn't time for so long I forget what I'm hoping it's time for.\nTRYING\n\nFor a long time we looked at the world and thought not. Then we looked at our lives and thought maybe. Now we are trying. We bought a new set of sheets for the bed. We bought a thermometer and a book.\n\n*\n\nI find the book on the whole reassuring. It gives lots of examples using real-life names like Gail, Audrey, and Dr. Smith, as well as comic touches here and there like Dr. Rhea Sure, who keeps popping up on Gail's chart. But Appendix K does contain some trying phrases. What, for example, is a Hamster Egg Penetration Assay? What is Nonstimulated Oocyte Retrieval In ( Office ) Fertilization? What is the Male Factor? What is Within Normal Limits?\n\n*\n\nIf procreation were a matter to be decided purely on the basis of rational thought, would the human race still exist? Schopenhauer thought not. Every evening, rain or shine, he would walk his poodle, Atma, for two solid hours through the streets of pre-war Frankfurt, trying to imagine a world as sterile and crystalline as the moon.\n\n*\n\nI try to find it reassuring that Gail is thirty-eight.\n\n*\n\nAfter swallowing a tiny vial of poison that had weakened over time around his neck, Napoleon escaped his island exile on Elba in the autumn of 1812, having built up a tiny island empire there, replete with tiny army and navy, tiny copper mines, and fields of tiny cabbages and beets. When his ship touched shore he rode bravely out to meet his former soldiers alone and dismounted to address them. The entire Grande Arm\u00e9e turned around on the spot and escorted Le Petit Corporal back to Paris, where he renewed his ancient vow to die trying.\n\n*\n\nThis morning I tried on my bikini while my husband walked the dog. I turned around and used a compact to study my backside in the mirror above the sink. The mirror, perhaps mercifully, was dusty, and I did not get a good look.\n\n*\n\nThe majority of husbands remind me of an orangutan trying to play the violin, said Balzac.\n\n*\n\nMy husband, thank heavens, is cooperative. He has even read a few pages of the book. And yet sometimes I worry that it is his fault. That either he is not trying hard enough, or that he is trying too hard, or that no matter how hard he tries or does not try it is his Male Factor that is the problem.\n\n*\n\nWhile searching for advice on overcoming adversity, I find a book called Tom Cruise: Overcoming Adversity. I learn that the public only sees the glamour \u2014 but Phelan Powell shows the significant obstacles Tom Cruise has overcome in order to live his life of pampered opulence. In Cruise's case dyslexia was one obstacle \u2014 it nearly cost him the part of the barman in Cocktail ( he thought it was a film about cockatiels and told his agent he \"didn't do parrots\" ) and he bought his own wildebeest to research the part of Lieutenant Maverick Mitchell in Top Gnu. This description is a direct cut-and-paste from an Amazon.com review by a reader named Henry Raddick in London, UK, who gives the book five stars and titles his review: \"Pig-ignorance no bar to fame and fortune.\" I try to picture what Henry Raddick looks like, and does he live with his mother.\n\n*\n\nThe book lives on the back of the toilet and I try to visit it from time to time throughout the day. Sometimes I just look at it. Sometimes I go in there and sit down and open it at random and get lost looking at the pictures. The follicle develops. The egg begins its journey down the tube . . .\n\n*\n\nThe human soul, wrote Aristotle in his treatise on ethics, has an irrational element which is shared with the animals, and a rational element which is distinctly human. In order to live a virtuous life one must try to achieve some sort of balance between them. In his Poetics, however, Aristotle points out that it is exclusively the irrational upon which the wonderful depends for its chief effects.\n\n*\n\nEvery morning, before getting up to make breakfast, I record my temperature down to the tenth of a degree on a chart, and draw a line between yesterday's and today's. I compare my progress with the sample in the book and try not to worry.\n\n*\n\nPeople who believe in God will tell you that \"trying\" to believe will not work. And yet some believers insist that simply wanting to believe is enough. I keep flipping back and forth between trying and wanting to try.\n\n*\n\nIn his early twenties it was said that Jean Cocteau could bring himself to orgasm without touching himself, purely by the power of his imagination. Here I am trying to live, he wrote, or rather, I am trying to teach the death within me to live.\n\n*\n\nWhen my two oldest friends write to tell me their good news, I try not to let my jealousy show, and sprinkle exclamation marks liberally throughout my replies.\n\n*\n\nOften, I have read, the very act of \"trying\" can undermine one's prospects of success. This makes trying difficult. The trick, they say, is to try without actually \"trying.\" Having finally decided to start trying we must keep on trying while trying not to feel like we are \"trying\" at all. We must above all try not to worry. Sometimes I worry that I am not trying not to try hard enough.\n\n*\n\nIt is one thing to marvel at the miracle of life, but quite another to try to explain it. Almost every freshman biology textbook printed in the last fifty years contains the famous Miller-Urey experiment of 1953, in which Harold Urey and Stanley Miller tried to simulate early atmospheric conditions on Earth, in order to see what they could generate by adding an electrical spark. What they discovered were amino acids, the basic building blocks of life. From there, most books lead straight into a discussion of evolution, prompting the student to conclude that scientists have thus proven life can be created from a few nonliving chemicals. We tell this story to beginning students of biology, admitted Nobel laureate George Wald in his 1954 article, The Origin of Life, as though it represents a triumph of reason over mysticism. In fact, he points out, it is very nearly the opposite. \nTRANS-NEPTUNIAN OBJECT\n\nThe time and place and manner of my death are three facts that don't exist yet.\n\nFacts exist for whole centuries and then suddenly cease.\n\nPluto used to be a planet and now it is a chunk of debris, number 134340.\n\nMy grandmother's house stands on the hill above the sea where she left it.\n\nWhen I come back to visit I discover a crater in its place.\n\nThis room is full of facts.\n\nAll day I let the cat out, let it in, then let it back out again.\n\nI mean this metaphorically.\n\nSome facts never exist.\n\nIt is winter.\n\nIt is summer.\n\nAll night the branches tap at the glass.\nTELESCOPIC INTERIOR\n\nSolar wind singing in a bottlecap.\n\nDistant drone of stone\n\nDrilled through for more stone\n\nLess prone to collapse.\n\nAdd fire. Add feast days and photons\n\nAnd glue the whole mess\n\nTogether in a nerve net\n\nSwinging through the cosmos on a peg.\n\nIf you can sleep through this\n\nSend me a lullaby\n\nFrom your crib of green dreams.\n\nDown here the weather's red\n\nAnd the century's turning\n\nEvery storm back to port.\n\nAt the last resort\n\nThey're selling sand as souvenirs.\n\nThe roses have never looked lonelier,\n\nLess photogenic, but get this \u2014\n\nThey're going ahead with the festival.\n\nThey're addressing the peacock dilemma.\n\nThey're dredging the harbour.\n\nThey're shooting pitchforks at the moon.\n\nThrough a cracked telescope\n\nI watch the late show unfold\n\nIts milky arabesque across the deep.\n\nHow could I sleep? The brightest star\n\nIn the sky tonight is a planet\n\nCalled Tomorrow.\n\nI used to live there.\n\nI should know.\nTHE LAST THING\n\nIn one version heaven is a tidy German town.\n\nIn another it's a shining patch of glory at the centre of the earth.\n\nRead the various accounts and you will learn that happiness consists in greater happiness ahead.\n\nYet every story about the future becomes, in time, a story about the past.\n\nAnother story about the past: you live in a town on the surface of the earth.\n\nThings happen in this town.\n\nA dog barks. A bell rings. A cart overturns on the sidewalk and spills a winter's worth of oranges in your path.\n\nMeanwhile astronomers predict a \"Big Rip\" in the cosmos resulting in a cold, dark, never-ending end.\n\nWhat kind of happiness is this?\n\nDrink your coffee.\n\nUnplug your phone.\n\nDon't believe the stories about life outside your town.\nTHE SOLITARY ANGLER (II)\n\nMa Yuan of the Song Dynasty was famous for painting \"just one corner\" of his canvases. His solitary angler sits small as a stone above a dark sea dissolving into mist. The point is, there is much we can't see.\n\nMan should be a windowpane through which God's light can shine. In his dark night, St. John saw this.\n\nSo long had it been since he'd seen a woman's foot, when he met Joan Baez, Thomas Merton asked her to remove a shoe. Paradise is still ours but we do not know it, he wrote.\n\nSnow falls on birch trees and the branches bend to earth.\n\nIf only I had the tiniest grain of wisdom, wrote Lao Tzu, I should walk in the Great Way, and my only fear would be to stray from it.\n\nMy little way, wrote Th\u00e9r\u00e8se de Lisieux, stroking death's sleepy head nestled in the damp folds of her sheets.\n\nThe brain is a small grey tissue afloat on a wave. Everything we know and can ever know about existence is there.\n\nMan stands in his own shadow and wonders why it is dark.\n\nOne way to know is unknowing, yet another way reports, which must be different than not knowing.\n\nHow else would we know not.\nABSTRACT FIRES\n\n(Mixed media. 1972\u2013present)\n\n#1. Candy canes, tinfoil, flamenco guitar.\n\n#2. Fork, butterfly, dog hair, dust.\n\n#3. Trampoline, harpsichord, rust.\n\n#4. Thumbtacks, chewing gum, forklift, car.\n\n#5. Light bulb, lipstick, ceiling fan, string.\n\n#6. Bathtub, shaving cream, flag.\n\n#7. Disco ball, flashlight, dental floss, diamond ring.\n\n#8. Sand, tin cans, oilcloth rag.\n\n#9. Termites, teacup, artist, microscope.\n\n#10. Sequins, hairdryer, liquid smoke.\n\n#11. Work boots, aquarium, encyclopedia, peach.\n\n#12. Cigarette, rope.\n\n#13. Paper bag, rose.\n\n#14. Glass jug, snow.\nDIM-LIT INTERIOR\n\nI'm done crying in my beer about love.\n\nMy days of riding the shiny brass school bus are behind me as well.\n\nThe changes come slowly but suddenly.\n\nOne day the sun will burn so brightly it will turn all our seas into vast boiling vats.\n\nFreedom comes from understanding our inability to change things.\n\nSo lead me O Destiny whither is ordained by your decree.\n\nJust please don't force me to vacuum the stairs.\n\nThe quiet that follows the storm may be the same as the quiet before it.\n\nLet the wind blow.\n\nLet it blow down each tree on the bright boulevard.\n\nThe things I would most like to change are the things that make me believe change is possible.\nENOUGH\n\nI am wearing dark glasses inside the house\n\nTo match my dark mood.\n\nI have left all the sugar out of the pie.\n\nMy rage is a kind of domestic rage.\n\nI learned it from my mother\n\nWho learned it from her mother before her\n\nAnd so on.\n\nSurely the Greeks had a word for this.\n\nNow surely the Germans do.\n\nThe more words a person knows\n\nTo describe her private sufferings\n\nThe more distantly she can perceive them.\n\nI repeat the names of all the cities I've known\n\nAnd watch an ant drag its crooked shadow home.\n\nWhat does it mean to love the life we've been given?\n\nTo act well the part that's been cast for us?\n\nWind. Light. Fire. Time.\n\nA train whistles through the far hills.\n\nOne day I plan to be riding it.\nVANISHING INTERIOR\n\nLittle patches of grass disappear\n\nIn the jaws of lusty squirrels\n\nWho slip into the spruce.\n\nCars collapse into parts.\n\nSpring dissolves into summer,\n\nThe kitten into the cat.\n\nA tray of drinks departs from the buffet\n\nAnd voila! the party's over.\n\nAll that's left are some pickles\n\nAnd a sprig of wilting parsley on the rug.\n\nWhen I think of all those\n\nGong-tormented Mesozoic seas\n\nI feel a ripple of extinction\n\nAnd blow a smoke ring through the trees.\n\nSoon there will be nothing left here but sky.\n\nWhen I think about the fact\n\nI am not thinking about you\n\nIt is a new way of thinking about you.\nPLAIN GREEK\n\nFate's wind can be cold it is true.\n\nWhat is the wind to you\n\nBut an impression of wind\n\nA phantasia\n\nAs Epictetus puts it\n\nIn his Handbook\n\nA fact you must weather\n\nLike any other fact\n\nSuch as daylight adultery taxes\n\nAnd naturally death.\n\nFace the facts.\n\nThey do not matter.\n\nWhat matters is the use\n\nYou put them to.\n\nThe Iliad consists of nothing but facts.\n\nEpictetus wipes his nose\n\nAnd explains this\n\nTo the students growing restless at his feet.\n\nFact prompted Paris.\n\nFact prompted Helen to follow.\n\nIf fact had prompted Menelaus\n\nTo count his blessings\n\nIn the face of Helen's absence\n\nNot only The Iliad\n\nWould have been lost to us\n\nBut The Odyssey too.\n\nWhen the wind blows\n\nDo not long for warmer climes.\n\nEpictetus puts it\n\nIn plain Greek.\n\nWipe your nose\n\nAnd do not accuse God.\n\nIf all is fire\n\nYou may warm your hands\n\nBy thrusting them here\n\nInto this burning book.\nIDEAL TREE\n\nGod is in the forest counting trees.\n\nYou are in the city writing poems.\n\nYou put a tree in a poem.\n\nA tree without roots or branches\n\nOr squirrels or sap\n\nWithout even a shadow\n\nIn its crown, for it grows\n\nWithout even a crown.\n\nYou are so pleased with your poem\n\nAnd with the sound it makes\n\nWhen you read it out loud\n\nAnd when you whisper it\n\nInto your pillow at night\n\nYou call your poem\n\n\"The Tree of Everlasting Love\"\n\nAnd plant it lovingly\n\nBetween the waiting pages\n\nOf an unwritten book.\n\nThere it dwells for many years\n\nUntainted by moss or regard.\n\nAnd when you finally publish\n\nYour book of sad poems\n\nNo one even notices the tree.\n\nNo one sees it burning coldly\n\nThrough all the foggy mornings\n\nOf your misinterpretable world.\nA PERFECT EMERGENCY\n\nIt was already aflame when I spotted it there in the parking lot.\n\nKids were standing around throwing sticks at it, kicking dirt in its face.\n\nAll I could do was look on in pity as it thrashed at the air like a tiny, vengeful sun.\n\nBut like a tiny, vengeful sun, the burning bush didn't want pity. When I approached with my hands in my pockets, it shook out its golden locks and sang in a language I could see.\n\nI am the Unburnt Bush! it cried. I am Burning but Flourishing! I am Swallowed but I am not Consumed!\n\nIn my head was a page from a musty old book with its useless list of Latin verbs. Before me I could see all the lives I might have lived, lined up and leaping through the same burning gate.\n\nIt was a perfect emergency. The only thing worth saving was the blaze.\nROMANTIC INTERIOR\n\nWind rips splendour from the trees\n\nAnd lays it at our feet.\n\nSome of us hungry,\n\nSome of us lucky to be upright at all.\n\nSeason past sweetness.\n\nStuck in the throat with a fork.\n\nA speck in the spectrum\n\nSpins into a wet little planet\n\nStudded with heartlust,\n\nFlooded with pamphlets\n\nFor classes on how to forget.\n\nWhere Keats sees a reaper\n\nAsleep on the granary floor,\n\nHer scythe set by quietly,\n\nWind playing games\n\nWith the husk of her hair,\n\nI see a dead squirrel.\n\nIt's the end of October\n\nAnd I don't have a costume.\n\nPast lives clutter my closet\n\nA long way from home.\n\nThere's a hole in the ground\n\nWhere my house used to be.\n\nA hole in my head\n\nWhere my heart used to be.\n\nI'm climbing a hillside.\n\nA green patch of laughter. \nEXIT\n\nLow cirrocumulus clouds in the west.\n\nWar in the east.\n\nLift teabag from cup.\n\nAdd milk.\n\nAsk if it is happiness\n\nOr pleasure you prefer.\n\nWatch the storm churn to the surface.\n\nShadows gather in the valley below.\n\nTo count them is to know their many shapes\n\nCannot be counted.\n\nThey must be numbered among.\nNOTES\n\n\"Amor Fati\": The passage in quotations is Simone Weil's, from a letter to Father Jean-Marie Perrin in which she recounts her spiritual autobiography. It appears in the collection Waiting for God.\n\n\"Occasional Poem\": The line about learning Spanish owes a debt to my father. This poem is for him.\n\n\"The Solitary Angler\": The lines \"There is no greater calamity \/ Than to underestimate the strength of your enemy\" are adapted from a Zen proverb.\n\n\"On White Flowers\": This poem owes a debt to my mother, and is for her.\n\n\"Dim-Lit Interior\": The line \"So lead me O Destiny whither is ordained by your decree\" is adapted from a translation of Cleanthes of Assos's \"Hymn to Zeus.\"\n\n\"Enough\": The lines \"What does it mean to love the life we've been given? \/ To act well the part that's been cast for us?\" owe a debt to Epictetus.\n\n\"A Perfect Emergency\": The italicized passage uses language adapted from the Eastern Orthodox Church, and from the mottos for the Presbyterian Church of Ireland and the Church of Scotland. \nACKNOWLEDGEMENTS\n\nGrateful acknowledgement is made to the editors of the following journals who kindly published poems from this manuscript: Alaska Quarterly Review, Boston Review, Columbia Poetry Review, Commonline, Court Green, Crazyhorse, Denver Quarterly, Fou, jubilat, MiPoesias, Pleiades, A Public Space, Ryga, SmallStations, and Washington Square.\n\nAdditionally, some of this work was represented in anthologies, and I am grateful to the editors of these as well: The Canarium Anthology, Vol. 1. ( Canarium Books, 2008 ), VERSschumggel\/reVERSible ( Wonderhorn \/ Editions du Noro\u00eet \/ Literaturwekstatt ).\n\nSeveral poems from the manuscript were published in a chapbook, Interiors, with art by Shawn Kurunero ( Montreal: Delirium Press, 2006 ).\n\n\"Dim-Lit Interior\" was featured in the Poem of the Week section in February 2008 on the website of the Parliamentary Poet Laureate.\n\nThanks to my family, friends, teachers, and colleagues for their love and guidance throughout the writing of this book.\n\nThank you, Chicu. Ideal reader.\n\nAuthor Photograph: Anna Knott\n\nSUZANNE BUFFAM'S first collection of poetry, Past Imperfect, won the Gerald Lampert Memorial Award and was named a Globe and Mail \"Top 100\" Book of the Year. She won the 1998 cbc Literary Award for Poetry and has twice been shortlisted for a Pushcart Prize. Born and raised in Canada, she currently teaches Creative Writing at the University of Chicago.\nABOUT THE PUBLISHER\n\nHouse of Anansi Press was founded in 1967 with a mandate to publish Canadian-authored books, a mandate that continues to this day even as the list has branched out to include internationally acclaimed thinkers and writers. The press immediately gained attention for significant titles by notable writers such as Margaret Atwood, Michael Ondaatje, George Grant, and Northrop Frye. Since then, Anansi's commitment to finding, publishing and promoting challenging, excellent writing has won it tremendous acclaim and solid staying power. Today Anansi is Canada's pre-eminent independent press, and home to nationally and internationally bestselling and acclaimed authors such as Gil Adamson, Margaret Atwood, Ken Babstock, Peter Behrens, Rawi Hage, Misha Glenny, Jim Harrison, A. L. Kennedy, Pasha Malla, Lisa Moore, A. F. Moritz, Eric Siblin, Karen Solie, and Ronald Wright. Anansi is also proud to publish the award-winning nonfiction series The CBC Massey Lectures. In 2007, 2009, 2010, and 2011 Anansi was honoured by the Canadian Booksellers Association as \"Publisher of the Year.\" \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nMore than Matter?\n\nWhat Humans Really Are\n\n## Keith Ward\n\nCopyright \u00a9 2010 Keith Ward  \nThis edition copyright \u00a9 2010 Lion Hudson\n\nThe author asserts the moral right to be identified as the author of this work\n\nA Lion Book  \nan imprint of  \nLion Hudson plc  \nWilkinson House, Jordan Hill Road,  \nOxford OX2 8DR, England  \nwww.lionhudson.com  \nISBN 978 0 7459 6247 4 (print)  \nISBN 978 0 7459 5858 3 (epub)  \nISBN 978 0 7459 5857 6 (Kindle)  \nISBN 978 0 7459 5859 0 (pdf)\n\nDistributed by:  \nUK: Marston Book Services, PO Box 269, Abingdon, Oxon, OX14 4YN  \nUSA: Trafalgar Square Publishing, 814 N. Franklin Street, Chicago, IL 60610  \nUSA Christian Market: Kregel Publications, PO Box 2607, Grand Rapids, MI 49501\n\nFirst edition 2010  \n10 9 8 7 6 5 4 3 2 1 0  \nFirst electronic edition 2011\n\nAll rights reserved\n\nFor more information on the author, Keith Ward, please log on to  \nwww.keithward.org.uk.\n\nCover design: Victoria Dokas  \nCover image: Andy Taylor Smith\/Corbis\n\nA catalogue record for this book is available from the British Library\n\n## Contents\n\nCover\n\nTitle Page\n\nCopyright\n\nIntroduction\n\nChapter 1 Dualism, minds, and bodies: the problem stated\n\nChapter 2 A range of philosophical views about what is really real\n\nChapter 3 The limits of knowledge\n\nChapter 4 Putting minds first\n\nChapter 5 Questions of personal identity\n\nChapter 6 The place of human minds in the cosmos\n\nChapter 7 Dual-aspect idealism\n\nChapter 8 Metaphysics and common-sense philosophy\n\nChapter 9 In defence of dualism\n\nChapter 10 Consciousness, value, and purpose\n\nChapter 11 Thoughts and perceptions\n\nChapter 12 Minds and moral values\n\nChapter 13 Acting for the sake of good alone\n\nChapter 14 The idealist view of life\n\nChapter 15 Can we still speak of the soul?\n\nNotes\n\nShort bibliography\n\nGlossary\n\n## Introduction\n\n\"You, your joys and sorrows, your memories and your ambitions, your sense of personal identity and free will, are in fact no more than the behaviour of a vast assembly of nerve-cells and their associated molecules.\" (Francis Crick)\n\nThis book exists to put a different view. The success of the physical sciences has led to a quite widely held view among the scientifically literate that all that exists is matter or some sort of physical stuff. Human beings are often presented as the accidental results of millions of genetic copying-mistakes and freak accidents of nature. Their cherished ideas of value, freedom, and purpose are illusions, since humans are nothing but the puppets of blind and mechanical forces of nature, and their consciousness is doomed to inevitable extinction, having never been more than a by-product of cosmic processes to which they are completely unimportant.\n\nI believe that this picture of human life is both scientifically questionable and philosophically naive. Moreover, it undermines the belief that human beings, with their thoughts, feelings, ambitions, and moral challenges and ideals, have intrinsic worth, and that worth lies in their mental lives, not in the behaviour of their nerve-cells, however complicated. It is this view, justifying a commitment to the distinctive value of human consciousness and responsible action, to which the deepest reflection on the nature of our cosmos points. That is what I hope to show.\n\nThe view that prioritizes human thought, feeling, and action over the behaviour of physical particles is often called humanism, embodying a commitment to human welfare, interpreted as the fulfilment of the uniquely personal experiences and creative capacities of humans. In that sense I am a humanist; but I think humanism requires ontological backing. That is, there must be good reasons for seeing human experiences and actions as important and human persons as of intrinsic value in a universe like this.\n\nI intend to produce philosophical reasons for such ontological backing, and I find these reasons intellectually compelling. I make little or no reference to religious considerations, based on revelation or religious authority. But I think there is little doubt that many religious believers will find that the arguments I propound have a natural affinity with some religious beliefs about the human soul, and I myself think that some convergence of humanist and religious beliefs will provide the most adequate view of human persons that is available to us.\n\nBut I have confined myself to sustained reflection on the nature of being human, trying to point to phenomena that are available to anyone, whatever their religious beliefs or lack of them. The motto of the book is in fact the reversal of the opening quotation from Crick, and it goes like this: you, your joys and sorrows, your memories and your ambitions, your sense of personal identity and free will, are much more than the behaviour of a vast assembly of nerve-cells, and that \"more\" gives each personal life a significance and value that expresses and points to the ultimate meaning of the universe itself. Human persons are not accidental mistakes in a pointless perambulation of fundamental particles. They are a window into the inner reality, value, and purpose of the cosmos.\n\n## Chapter One\n\n## Dualism, minds, and bodies: the problem stated\n\nGilbert Ryle was an important twentieth-century British philosopher who is famous for attacking what he called \"the myth of Cartesian dualism\", the myth of the \"ghost in the machine\". This is the \"myth\" that minds are different from and more than matter. But was Ryle right? And does his attack also undermine (without meaning to) belief in the unique dignity and value of human persons, which is centred on the nature of their inner experiences and responsible actions? Those are the central questions of this book.\n\nOnce upon a time, when I was studying philosophy at Oxford, my supervisor was Gilbert Ryle. He was one of Britain's outstanding philosophers in the 1950s and 60s, and I was one of his last pupils. He was also my moral tutor, though he said, \"I do not know what a moral tutor is, and I hope I never have to find out.\" And, as far as I know, he never did.\n\nAlthough not as well known as Wittgenstein these days, Ryle was extremely influential and was a kind of Oxford equivalent of Wittgenstein, holding views about philosophy that were very similar to those of the Cambridge philosopher. In a sense, this is a book about Wittgenstein as much as it is about Ryle \u2013 except that people get so emotional about different interpretations of Wittgenstein's gnomic philosophy that I have thought it better only to make rather muted claims in that area. When I was teaching philosophy in Cambridge, Professor Anscombe, who translated and edited Wittgenstein's later work and often discussed it with me, terrified me so much that I have decided that it is safer to leave discussion of Wittgenstein to acknowledged experts like the Oxford philosopher P. M. Hacker. Nevertheless, I believe that much of what I say about Ryle would apply to Wittgenstein, or at least to many popular interpretations of Wittgenstein, as well.\n\nIt is worth talking about Ryle because he was a very good and significant twentieth-century philosopher, and because he wrote the classic critique of Descartes' dualism (usually called Cartesian dualism), the view that the mind and the body are two distinct substances. Ryle originated the expression \"the ghost in the machine\" to describe Descartes' view, implying that the mind is a ghost (an illusion, really) wandering in the machine of the body. I strongly object to this description, though it has been very influential, and want to try to rehabilitate Descartes, at least to some extent. I want to suggest that mind and consciousness are different from, something over and above, molecules and matter, and that they are not at all ghostly. And I will argue that having such a belief is important if you put a great value on individual human experience and responsible moral action.\n\nRyle is particularly interesting because he rejected dualism, but he was still a humanist. He thought that you could defend human uniqueness, freedom, and responsibility without having a belief that minds are something more than matter. I do not think this is true, but examining Ryle's arguments is a good way of finding out what you really think about this issue. In fact, it will take me some time to get around to considering Ryle in detail, as there are problems that have to be cleared up first about the nature of philosophy and about differing philosophical views of the nature of reality. I will use the word \"metaphysics\" to mean a general view about what kinds of things are real (whether, for instance, there are minds in addition to bodies, and whether everything is determined by laws of nature or whether the will is free). Ryle had one such philosophical view, but, oddly enough, he did not think that he had.\n\nHe was certainly not a materialist, a person who thinks that everything that exists, including minds, is purely physical, so that minds are \"nothing but matter\". Nevertheless, his rejection of mind as a distinct substance suggests that, if you are going to have a grand metaphysical theory about what kinds of things exist, you will probably end up as some sort of materialist. And his rejection of materialism has come to seem rather implausible to philosophers like Daniel Dennett, who, as a pupil of Ryle roughly contemporary with me, is a more overt materialist. That is partly because Ryle objected to having any metaphysical theory at all and argued that such grand theories are the results of \"logical howlers\" or grammatical mistakes. His aim in his best-known book, The Concept of Mind, was to correct these mistakes, and rid us of the temptation to think that we could sit in an armchair, philosophize, and thereby discover the nature of reality.\n\nIn my case, he completely failed. Despite the worst (or best) he could do, I went on having metaphysical thoughts. I am still having them. I think that there is a major intellectual battle going on, especially in the West, between those who adopt a purely materialist view of human persons and those who believe that there is a distinctive reality and value about human minds, and that such minds far transcend their physical embodiments both in their nature and in their moral worth. This battle is not about a set of grammatical mistakes. It is about what it means to be human and about the distinctive importance of human personhood in our physical universe. It is a metaphysical battle, a battle about what sorts of things exist and about whether persons are distinctive sorts of things that are different from purely material things. This metaphysical battle is real.\n\nRyle did not hold the fact that I thought this against me. He published my first philosophical article and helped to get me my first philosophical job, in the Logic department at the University of Glasgow. I think he regarded me as a somewhat retarded example of his own early philosophical errors, and expected that I would grow out of them in time. But things have only got worse. I am more than ever convinced that the question of what it is to be a human person is the biggest intellectual question of our day.\n\nGilbert Ryle and common sense\n\nPhilosophy has a job to do. That job is not to provide universally agreed and incontestable answers. It is to examine questions like that of the nature of personhood as deeply as possible, taking account of many different disciplines and points of view. It may help individuals to come to an informed opinion of their own. More likely, it will help to confirm the opinion they already have \u2013 but their opinion will be more informed, more aware of its own limitations and weaknesses, and more appreciative of why other opinions exist, and of why it is so difficult to find any agreed position.\n\nGilbert Ryle made a major contribution to the examination of this question of what it is to be a human person, and whatever he said about metaphysics, he believed that philosophical enquiry was essential for clearing away confusions and needless obscurities. For that reason, as well as out of a sense of relief that he never had to find out what a moral tutor was, I have always valued my discussions with Ryle and have always seen him as one of my philosophical mentors. One of my prized possessions is a signed copy of The Concept of Mind. It was not a book I ever agreed with. My whole being rebelled against it from the first. But it was beautifully written. It had the peculiar property that while I was reading it I believed it. Only when I stopped reading did I know that it was wrong, but I could never quite formulate just what was wrong with it.\n\nThat, I suppose is the mark of good philosophers. They can make you believe that something is supremely reasonable for the space of half an hour, as they take you into their thought world. But, as David Hume used to say, if you put their books down and go and play backgammon, you quickly recover the ordinary beliefs you had before you read their books and wonder what came over you to make you think what they said was so reasonable.\n\nThis is a philosophy book (though I have avoided the lengthy technical discussions that are the mark and pride of the best modern philosophical work), so if it is any good it may have the same effect \u2013 it will seem convincing until you put it down, but perhaps not for much longer. I feel just the same about my own books, so it follows that I am pretty sceptical about my own philosophical conclusions. But I have to say that I am not as sceptical about my conclusions as I am about other people's conclusions, and especially about some of Gilbert Ryle's conclusions.\n\nThat is because Ryle thinks his conclusions are just ordinary common-sense conclusions, as opposed to the incoherent myth of Cartesian dualism. But I think Ryle's conclusions are not at all commonsensical. In fact, they contradict many common-sense beliefs about human persons, their minds, and their bodies.\n\nWhat are these common-sense beliefs about human beings? That we are physical objects in space and time, animals that are born and die, quite quickly wearing out and decaying like most physical objects. Unlike many physical objects, we perceive, think, feel, and act to achieve goals. We have a mental life.\n\nNobody else, not even our closest friends, we often think, really appreciates how we feel. It often seems to us that we are constantly misunderstood and under-appreciated, and nobody else ever really knows what it is like to be us.\n\nFurther, by dint of heroic personal struggles, we make major contributions to the world. These, too, are rarely appreciated by others, who are all too ready to accuse us of dishonesty or selfishness, when we have only been trying to make the best of a very complex situation and improve life for everybody. Or that is what we all tend to think.\n\nWe also have very complicated relations with other people. They teach us our language and skills, but they often try to use or abuse us, and we have to choose our friends very carefully so that we can make our way in a largely hostile and untrustworthy world. Many secret plots and subterfuges are required if we are to acquire a position of status and authority, and if possible \u2013 it seems these days \u2013 become a celebrity whom everyone admires and imitates.\n\nIf we put all these rather gloomy and absurdly self-regarding but common-sense thoughts into philosophical language, we can say that humans have unique memories, thoughts, and feelings to which no one else has complete access. They have the power to act intentionally to achieve various purposes. And they interact with other persons who are also intentional agents with hidden feelings of their own, who are parts of a social culture which shapes the thoughts, feelings, and skills of each individual member of that culture.\n\nI think that Ryle's picture of human persons in The Concept of Mind fails to give an adequate account of this mental life of human beings, and gives an unduly vague and incomplete account of responsible human action. However, it gives a rather good account of the social reality of persons. So it only partly fits our common-sense beliefs about human persons. And, regrettably, it has helped to demonize Descartes in recent philosophy by putting Descartes' views in a very distorted way. I still love Ryle's book and am immensely indebted to him as a teacher and philosophical role model. For a while I even smoked a pipe, as he did, thinking that this was part of being a proper philosopher. But as I grew older I gave up smoking and decided that I really preferred the views that Ryle himself had before he complicated his early common sense with anti-Cartesian philosophy and excessive pipe-smoking.\n\nThe Cartesian myth: the ghost in the machine\n\nWhat about \"Descartes' Myth\", as Ryle terms it in chapter one of his book? It states, according to Ryle, that \"minds are not in space\". Further, \"only I can take direct cognisance of the states... of my own mind\". And \"of at least some of these episodes he [the person whose mental states they are] has direct and unchallengeable cognisance\".\n\nThe thing is that I regard these as plain facts, not myths. Thoughts, feelings, sense-perceptions, mental images, dreams, recalled memories, and intentions, cannot be observed at any location in public space. The temptation these days is to say that they are in the brain. It is of course true that if the brain is not working properly, thoughts or images will not occur. But even if my brain is scanned, observers of the scan can only register the occurrence of electrical activity or enhanced blood flow in a region of my brain. They cannot observe the thought or image I am having. They cannot be directly aware of it in the way that I am.\n\nThe obvious thing to say is that thoughts can only occur in humans if brain activity occurs, and that thoughts are correlated in some way with specific sorts of brain activity. But the content of a thought \u2013 what it is about \u2013 cannot be detected by observation of physical activity in my brain. It can, however, be detected by me, though obviously not by observing my own brain and not by observing any portion of physical space.\n\nEach of us can detect the content of a thought without observing it in space, and in a way in which no one else can detect it. It does not follow that my detection is \"unchallengeable\", though usually we would give an introspective report of what someone is thinking a higher authority than an observer's report of what that person is thinking. But I guess that introspective reports are as fallible as most reports. That is, they are likely to be correct in general, but not always, especially under very unusual or complex conditions (like being in a brain-scanner).\n\nSo far, as far as I am concerned, the myth is not a myth at all, but a statement of what is apprehended to be the case. Did Ryle really think that other people could observe what he was thinking when he sat in his chair smoking his pipe and frowning slightly? I at least had to wait until he spoke, and even then I wasn't always sure what he had just thought. Perhaps he wasn't sure either, but he was surely in a better position than I was to say what he was thinking. He had privileged access by introspection to his own thoughts, even when he was thinking that there was no such thing as introspection or privileged access. Yet that, it seemed to me, was a self-refuting thought to think!\n\nLogical positivism and other people\n\nWhen, after talking to Ryle for an hour, I began to wonder whether anyone had any private experiences, I would take a short walk from Magdalen College to New College to visit Professor Ayer, also in Oxford at that time, and be reassured that he at least had private experiences (which he called \"sense-data\"). In fact, he had hardly anything but sense-data.\n\nAyer was a logical positivist, which meant that he thought all factual assertions had to be verifiable by some sense-experience. Ayer had a very rigorous definition of what a sense-experience (a sense-datum) was. It had to be an immediate datum of some sense (sight, hearing, smell, taste or touch), without including any inferences or theories. So a sense-datum is something like a patch of red or a sound or a sudden smell or taste. Positivists thought that all meaningful words must refer to such data in the end, and you could verify the truth of statements by just having the appropriate sense-data. If you could not do that, words and sentences were actually meaningless. This was positivism, because it insisted that all knowledge is analysable into confrontation with bare sense-data (it is \"positive\", as opposed to speculative or theoretical). It was logical, because it told you what words were meaningful, and what words were not.\n\nA funny thing about sense-data is that they seem to be essentially private. Nobody else can have my sense-data, and I cannot have theirs. In fact, persons are nothing but chains of sense-data, and these chains can never overlap or meet. So when Ayer met other people, what was really happening was that his chain of sense-data included sounds, smells, and sights that looked like the bodies of other people. But they were really just sets of sense-data, and the theory that they were other people was just a theory, a sort of shorthand to make things more convenient (so it is not really so \"positive\" or non-theoretical after all!). The existence of an objective physical world with other people in it was, Ayer thought, a construction out of sense-data.\n\nAs a good logical positivist, Ayer believed that he constructed the whole world, including physical objects and other people, out of his own private experiences. Following this thought through to its logical conclusion, it followed that all the students in his seminars were actually constructed out of parts of Professor Ayer. This was even more disconcerting, however, than Ryle's denial that Ayer was having any private experiences, so I felt the truth must lie somewhere in between these two completely conflicting views of two of the most eminent philosophers in Britain at the time.\n\nWhat did Ryle have against private experiences? I think that he feared that once he got into a world of private experiences, he would never get out again. Each person would be locked into their own little private world, with no way of communicating with, or even of knowing what was going on in, any other person's little private world. \"Absolute solitude is on this showing the ineluctable destiny of the soul. Only our bodies can meet,\" he said.\n\nHe felt that, in a dualist view, minds and bodies would be two completely separate sorts of thing. Since each mind is essentially private, the best we could do would be to infer unobservable mental events from people's bodily movements. But we could never check whether our inferences were correct. We could never be sure anything was going on in the minds of other people at all. And we could never say what sort of transaction was taking place between minds and bodies, since such transactions between two such different realms would belong neither to bodies nor to minds. They would be forever mysterious and beyond reasonable explanation.\n\nOn a Cartesian view, the gulf between mind and body is so wide, Ryle thought, that when we make a remark about someone's thoughts or intentions, \"the onlooker... can never assure himself that his comments have any vestige of truth\". Bodily behaviour might be wholly disconnected from the inner mental events which are forever hidden from others: \"External observers could never know how the overt behaviour of others is correlated with their mental powers.\"\n\nThis is indeed a puzzle that philosophy teachers like to place before first-year undergraduates. If all our knowledge begins from our own experience, how can we ever be sure that anything exists beyond our experience? How do we know that our best friends are not robots, without real minds? How do we know that we are not being totally deceived in thinking there really are any other people with thoughts and feelings?\n\nThis could be called the \"Matrix\" problem (from the film of that name). Might we not all be in some gigantic factory, wired up to machines that produce in our brains the illusion that we are talking to other people, while the truth is that some machine is producing sets of false beliefs in us? There are actually no other people there, just bodies that act and speak as if they were people. And in that case, even the bodies are illusions, so we really are in trouble.\n\nThe irony is that this is exactly the sort of problem Descartes was trying to deal with, as he asked himself what he could be absolutely certain of, without any possibility of doubt. But whereas Descartes thought the thing we could be most certain of was our minds, Ryle claimed that we did not have any minds (in this sense) at all. If we did have minds, we would have all the insuperable problems just outlined. What I want to ask is whether this is really true. I am pretty sure that it is not.\n\nThe Dead Philosophers' Club\n\nIn some obscure corner of limbo we might imagine the Dead Philosophers' Club meeting regularly to discuss whether they had ever really existed. (Or we could have imagined this before Pope Benedict XVI officially abolished limbo.) Now the members of the club have even more reason to suspect that they do not really exist, as their accustomed meeting place has disappeared, by papal command.\n\nNevertheless, we might imagine Professor Ayer meeting up with Gilbert Ryle to discuss deep problems of philosophy, including the major problem of whether they were talking to each other at all.\n\n\"Ryle, I wish you would stop smoking that pipe,\" Ayer might say. \"Don't you know that smoking is very bad for your health?\"\n\nRyle could reply, \"Being dead is not very good for your health either, but I have to put up with it. In any case your cigarettes are much worse than my pipe.\"\n\n\"That,\" Ayer might say, \"is just your subjective opinion. Since subjective expressions of opinion have no truth-value, it follows that it is not true.\"\n\n\"Is it not just your subjective opinion that I am sitting here smoking my pipe?\"\n\n\"Not at all. That is a fact. Statements about facts have a truth-value, and my experience can verify their truth, if they are true. I am quite certain that you are smoking your pipe or at least that someone very like you is smoking something very like a pipe. Admittedly my evidence is not as conclusive as I would like, and you could, after all, be just a set of visual sense-data that give me a strong impression that it is you who is sitting there. I suppose what I am really certain of is that a set of coloured shapes exists in my visual field; but they look remarkably like Gilbert Ryle to me.\"\n\n\"That is a very odd thing to be certain of. I, on the other hand, am quite certain that there are no such things as sense-data, so you can hardly be certain that you are having some of them. That would be to fall into a gross Cartesian error.\"\n\n\"We both agree, Ryle, that whatever else is the case, Descartes fell into a gross error. But we seem to disagree about what the error was. You seem to think he was mistaken in thinking there were any private thoughts. I am sure that there are private thoughts, but I think he was mistaken in thinking that anybody was thinking them. Perhaps, though, we can both agree that he was mistaken in thinking that the mind is an immaterial thing which is churning out thoughts, whereas we know that there is no such thing, no ghost of Descartes hidden in the machine which is his body.\"\n\nAnd there we might leave the Dead Philosophers' Club, with two of its leading members being absolutely certain of totally different things, and being equally certain that the other is wrong. Furthermore, there does not seem to be any way of deciding who is correct, which means that they can go on discussing the issue forever, without ever being in danger of coming to an agreed conclusion. I rather think that this should at least make philosophers relatively modest in their claims and open to the possibility of reasonable disagreement. It may even be quite an intellectual advance to recognize the tentative nature of human guesses about the ultimate nature of reality, and the impossibility of arriving at complete agreement about it. At the very least, we will not be able to regard all those who disagree with us as fools and knaves \u2013 though no doubt some of them are (I'm not saying who).\n\n## Chapter Two\n\n## A range of philosophical views about what is really real\n\nDualism \u2013 the view that mind or spirit is different from body and brain \u2013 is widely derided in much modern thought. But actually it is just one of a number of widely held philosophical views about what is really real, and about what human persons are. In this chapter I outline five of them, all held by reputable philosophers: phenomenalism (the view that all human knowledge is built up from, and is basically limited to, sense-experience); naive realism (the view that the world exists very much as we see it, even when we are not observing it); materialism (the view that only material objects, perhaps publicly observable objects in space-time, are ultimately real); dualism; and epiphenomenalism (the view that minds exist and are different from matter, but that they are wholly dependent upon brains, and play no causal role in anything that happens, as the material brain does all the work). Materialism, though currently very popular, is just one of these views, and I argue that on purely philosophical grounds materialism is less plausible than dualism.\n\nWhat is really real? Are there sense-data, as Ayer supposed? Is there an immaterial thinker, as Descartes supposed? Or are (almost) all statements about minds analysable into dispositional, \"if-then\" statements about the behaviour of publicly observable bodies, as Ryle supposed? It seems as if what these philosophers are doing is to start from a basic set of axioms, trace out what can be inferred or deduced from them, and then see how these consequences fit with their experience of reality. They are each able to see the weaknesses in their opponents' theories, and propose accounts that try to remedy them. Ayer accepts Descartes' claim that mental occurrences like thoughts and images could exist without physical bodies, but rejects the claim that there is an immaterial subject who has such thoughts and images. Ryle accepts Ayer's claim that all factual statements must be verifiable in principle, but rejects the claim that such verification consists in having private sense-experiences which no one else could ever check. I am accepting Ryle's claim that many (not all) statements about mental properties are dispositional statements about what a human animal is capable of doing or is liable to do in various circumstances. But I am querying his claim that there are no private and non-spatially locatable mental occurrences.\n\nDoes this mean there is a ghost in the machine after all? Ghosts are usually visual appearances of people, but they lack solidity and they tend to moan and wail rather a lot. I don't suppose anyone thinks that inside their head there is a wailing gaseous hallucination which pulls the levers that make the brain work. That is plainly ridiculous, and it is meant to be. The metaphor of a ghost in the machine has worked well as a rhetorical device to make people think that we all know the brain (the machine) is real, whereas talk of a mind other than the brain or of mental events in addition to brain-events is talk about something peculiar, not quite real, and probably illusory (a ghost).\n\nIt seems to me that the situation is quite the reverse. Talk of mental events is the most real thing we humans know. We know we have sense-experiences, bodily sensations, thoughts, feelings, and images. We know we experience things in ways that are unique to us and never wholly communicable to other people. We know that all our knowledge of the world has to begin with such experiences. Mental events are real, and to deny them would deprive us of all knowledge. They are not ghosts or hallucinations at all.\n\nBut do we know that brains are real? Well, yes, because we can see and feel them, at least if we are surgeons or pathologists. But may the brains we observe only be appearances to us of a reality which is rather different from what we see? This is another standard problem for first-year philosophers: what would brains look like if we were not looking at them?\n\nWe can never find out by observation, because every time we look at something we only see the way it looks to us. It is no use trying to cheat by closing our eyes and then opening them very quickly, as if to take the world by surprise. However quickly and surprisingly we manage to take a peek, we still only see what things look like to us. We can never manage to take them unawares, and find out what they look like when no one is looking at them.\n\nThis is very frustrating. We are tempted to say, \"Well, obviously, they go on being the same whether we are looking at them or not.\" But there are good reasons for thinking this is not true. One of the most important features of objects in our experience is colour. When we admire a beautiful view, we usually admire the colours objects have. But physicists tell us that colour is a product of the brain. External objects emit electromagnetic waves. Some of these impinge on the cones in human eyes, and cause electrochemical impulses that land up in the visual cortex. Only after that long causal journey do colours appear to us. The original wavelengths have no colour. They cause sensations of colour when they affect human sense-organs and the human brain. Colours are precisely some of those private mental states that Ryle wanted to get rid of. Different individuals may see colours in rather different ways. Colour-blind people certainly do. And some animals do not seem to see colours at all (they have no cones in their eyes). What we see depends on our cognitive apparatus.\n\nBut where are the colours themselves? They are not literally in the brain, as physical objects. The brain does not change colour when we see coloured objects. They are not on the objects we see, which have no colour. Colours are, as John Locke said, following Galileo, secondary qualities. They do not belong to external objects. They are contents of the mind, when stimulated by the brain, which in turn has been stimulated by wavelengths of light. Colours are caused by physical events, but they are not themselves physical events. They are how consciousness perceives physical events. There is a causal basis of conscious events, but it does not exist as we see it.\n\nBrains are rather boring, where colour is concerned. They are usually greyish. But are brains grey when we are not looking at them? Apparently not; they then have no colour at all. In fact, we could go much further than this and say that brains are not the solid porridge-looking objects they appear to be. Any physicist will say that brains are mostly empty space, in which molecules, atoms, electrons, quarks, and other strange particles buzz about in complicated ways.\n\nIt seems as though physical objects, when not being observed, have no colours, and no sounds, smells or tastes either. Sounds, like colours, are not physical events. Neither are smells, tastes or sensations. Things do not smell like, taste like or feel like anything, when nobody is smelling, tasting or feeling them. The physical world, it seems, is totally vacuous. No colours, sounds, smells, tastes or sensations. What on earth is left?\n\nWet philosophers\n\nFor Locke and Galileo, what is left is a world of \"primary qualities\" \u2013 qualities which, they thought (wrongly, as modern physics shows) physical objects cannot fail to have, but which are quite distinct from the \"secondary qualities\" that we see, hear, and feel, but which do not belong to the physical world. The world of primary qualities, the \"real world\", is basically a world of colourless, intangible, inaudible particles located in space, moving around and continually bumping into one another (though of course not realizing that they are doing so).\n\nScience tells us that all objects are made of atoms and other very small particles, and that atoms have no colour, smell or taste. They just have properties like mass, charge, position, and momentum. So they are not exactly how they look to a human being.\n\nWhen oxygen and hydrogen atoms are put together in a particular way, they feel wet to human observers \u2013 they form molecules of water. But if there were no observers, they would not be wet. There would not be any wetness. There would only be what John Stuart Mill called \"a permanent possibility\" of wetness, which would only become actually wet when somebody went for a swim.\n\nThus there is no point in asking Mill whether the sea is warm enough for swimming. He would have to reply, \"Not yet, but it might be if you jump in.\" Anyone who thinks that we see things as they actually are, however, would say that the sea is wet and warm, even if there is nobody about. But there is a problem. How do we know? There is absolutely no way of checking, without jumping in \u2013 and then of course you have not proved that it was wet and warm before you jumped in. This is why, on the philosophers' annual outing to the seaside, you may observe groups of philosophers rapidly jumping in and out of the sea, to try to see whether it remains wet or not. But they can never be sure it is going to be wet before they jump in, and they can never prove that it is still wet after they jump out.\n\nWe might say that it is just common sense to say that the sea continues to be wet when there is no one in it. But it must be admitted that it cannot be proved by sense-experience or observation. It is just much simpler to say that if the sea is wet every time you jump in, it is wet in between jumps. Of course that hypothesis will not work if you only jump in the sea once a year. But if you jump in and out as quickly as possible, it is the simplest explanatory hypothesis to account for why we can more or less predict that the sea is going to be wet.\n\nSo we can assume that the sea continues to be wet even when there is nobody in it. Nevertheless, it is probably safer, even if a little more complicated, to say that the sea is such that, if we jump into it, we will find it wet. There is some continuing causal basis for our feelings of wetness, though there is no actual unobserved wetness in the sea. That is what Galileo, John Locke, and John Stuart Mill, all supposed.\n\nWeird science\n\nBishop Berkeley famously objected to any distinction between primary and secondary qualities. He held, with some justification, that talk about tiny colourless atoms bouncing about is a mere abstraction. Why should we think that such abstractions are real, when in fact this world we observe, with all its smells, tastes, and colours, is the most real thing we know? There is little reason to suppose that science tells us the truth about reality, whereas the senses only provide appearances. It would make more sense to suppose that both science and common sense are concerned with appearances produced by the interaction of a hidden reality with human minds.\n\nThough Bishop Berkeley has become a by-word for silliness among many philosophers, there are things in modern physics that give him some support. The so-called \"real world\" posited by classical Newtonian science is not only an abstraction. It has passed its sell-by date, and it is nowhere near abstract enough. For quantum science, the world has dissolved into sets of probability waves in Hilbert space, of entangled and superposed wave-particles, and of ten-or twenty-six-dimensional curved space-time manifolds, which probably arise by quantum fluctuations in a vacuum. If you want something abstract, try that on for size.\n\nThis scientific view of the real world is about as far from what most people believe, or could even imagine, as it is possible to get. Common-sense consciousness, which is the sort most common people have, experiences things as colourful, tasty, smelly, and wet. Why should we think that the scientific world of probability waves and so on is the real world, as it is in itself when it is not being observed, whereas the common-sense world is just that world as it appears to us?\n\nThere is a great deal of controversy about the interpretation of fundamental physics. Some theorists, like Michio Kaku of the City College of New York, one of the founders of string field theory, boldly claim that \"the matter in the universe and the forces that hold it together... may be nothing but different vibrations of hyperspace\". Superstrings require a ten-or twenty-six-dimensional universe, and the basic forces of electromagnetism, gravity, and the nuclear forces may be caused by the \"crumpling\" of a universe that exists in dimensions far beyond the four-dimensional space-time of our common-sense observations (not that common sense usually gets even as far as four dimensions).\n\nOther physicists, like Niels Bohr, prefer to think that the world of observation is fundamentally real, and that all talk of such things as hyperspace uses mathematical constructs that cannot be mapped onto any objectively existing reality. In other words, quantum field theory does not tell us what the objective world is \"really\" like. As Bishop Berkeley suggested, we cannot get beyond a world of appearances, of things-as-they-appear-to-us, whether they appear as sense-perceptions or as mathematical intelligible mental constructions.\n\nAfter all, probability waves are not actual waves. They are mathematical devices for assigning probabilities to discovering the location of an electron under specific constraining conditions of measurement. You might say that electrons are probably in a number of places at the same time, but they are not actually at any precise place. That is certainly not common sense, and it is very hard to believe. Quantum physics is undoubtedly correct. Its predictions have been verified many times. Yet equally undoubtedly it does not give an adequate description of how things really are. It provides a set of sophisticated mathematical operations for understanding the behaviour of very small energy interactions under controlled and relatively isolated conditions. But one could not say that the variables of the formulae correspond to so-called \"real entities\". They are constructions of the mind.\n\nThe conclusion, for a number of quantum physicists, is not that there is really nothing there at all. It is that reality is hidden from us in its inner nature. All we can do is construct hypotheses that explain the structure of its interactions with our minds. But if we consider that our minds play a fundamental role in constructing both the sensory world that the senses reveal and the mathematical world that the mind explores, we may be led to the opinion that mind or consciousness is a foundational element of reality. As the eminent quantum theorist John von Neumann put it, \"All real things are contents of consciousness.\" Bishop Berkeley lives again, as the solid Newtonian world of classical physics dissolves into the abstract mathematical world of quantum theory, and leaves the mind as the most \"solid\" and basic constituent of the real world.\n\nIt is time to pause and take stock. We have already come across three main philosophical opinions about what the world we experience is really like.\n\nFirst, there is the opinion that the things we perceive are just what they seem to be \u2013 warm, wet, fuzzy, and colourful (they might also be beautiful or ugly, ordered or chaotic, pleasant or unpleasant, though that is disputed). The world of sense-experience is the real world.\n\nSecond, there is the opinion that our senses disclose only appearances or things as they appear to us. In itself reality is very different, and science and intellectual reflection can tell us what it is like. The world of science is the real world.\n\nThird, there is the opinion that appearance and reality are indeed different. But fundamental science deals in abstract mathematical constructions, and such abstractions do not tell us what reality in itself is like. We do not know what the real world is like. But both sense-experience and science are constructions of the mind, so maybe hidden reality is something like consciousness or mind.\n\nThese three philosophical positions, which are basic metaphysical theories (theories about what sorts of things really exist, and what sorts of things are actually real, not just apparent), can be, have been, and still are, ably defended by philosophers. My own preference is for some form of the third view, and I shall try to defend it. The discomfort I feel with Ryle's philosophy is that it seems to espouse some form of the first view, and this basic (and in my view mistaken) commitment is presupposed in all his specific arguments about the concept of mind. If the third view is right, things will look very different, as I shall show.\n\nWhat you see is what you get\n\nOn closer inspection, the three main views I have outlined split into many further into subdivisions. I will restrain myself and mention only a total of eight philosophical possibilities. No doubt we could think of even more divisions and different kinds of overlap between views \u2013 there is probably one sub-division for every philosopher who has ever lived \u2013 but eight is enough to be going on with.\n\nThe reason why it is important to examine these metaphysical views is that the problem of what persons really are cannot be resolved unless we have first come to a decision about what sorts of things fundamentally exist. Many discussions about the nature of persons fail to tackle this more fundamental question or even fail to see that it is a really difficult problem. Many people just assume some sort of common-sense view, that humans are evolved animals in space and time, and that this three-dimensional world that we experience is what is actually real. Other people, more scientifically inclined perhaps, admit that the real world is very different from the one we see and feel, but it is still definitely and completely material. There are no \"spooky\" or \"supernatural\" entities, and Cartesian dualism, in particular, is just a relic of an outmoded superstition.\n\nIf we are going to approach the issue rationally, however, we need to see that these are just two out of a pretty large field of possible metaphysical options. In fact, they are both very controversial and highly contested from a philosophical point of view. It is only when we see the wider range of metaphysical options that we realize how tenuous the arguments for common sense and for materialism are.\n\nI am going to argue that the most adequate view of human persons will fall under what I have called the third group of metaphysical theories, which says that there is a reality underlying our everyday experience whose basic character is consciousness or mind. So I need to establish that this is a coherent and plausible metaphysical view. That means setting it out alongside some other major metaphysical views, and showing their strengths and weaknesses. That is what I am now going to do.\n\nThe first group of metaphysical theories is the \"things are basically what they seem to the senses to be\" group. The most radical version of this is the view that when nobody is observing things, when they are not contents of consciousness, there is nothing there. There is only what David Hume calls a succession of impressions and ideas. The idea of a continuing unobserved world is a postulate of the imagination, and, if we were trying to get the simplest possible account of the world, such a postulate would be regarded as superfluous. Things are what they seem to be, but there is no objective continuing world to support the things we see, hear, and touch. If we attend closely to what we experience, we will realize that we never experience unobserved continuing physical objects. So there aren't any such things.\n\nThis extreme position \u2013 a sort of radical empiricism, sometimes called phenomenalism (I will call it Theory 1) \u2013 is too much for most people, but it makes the point that the common-sense belief that there is a world of physical objects depends upon rational postulates that, strictly speaking, cannot be substantiated simply by appeal to experience. Most people do make such postulates and accept a philosophy of common sense. They assume that things continue to be roughly what they are observed to be even when nobody is observing them. We live in a real world of objects in three-dimensional space, and we observe it more or less as it is (this is Theory 2, sometimes called naive realism, because many philosophers think it really is rather naive).\n\nI think the consideration of properties like colours is sufficient to render this view improbable, especially since we can carry out the same type of analysis for smells, tastes, sensations, and sounds. Nevertheless, I suspect Ryle holds some version of this opinion. At least he speaks as though he does, though to put it like that possibly sounds too \"metaphysical\" for his taste. Like Wittgenstein, he would probably think that \"everything is in order as it is\", that metaphysics raises a host of pseudo-problems that arise from linguistic confusions, and that philosophy can provide no new information about the world, least of all about \"ultimate reality\". Philosophy has no theories.\n\nThe problem is, that is a theory. And Ryle does, as I have suggested, have some beliefs that fly in the face of common sense \u2013 for instance, that there are no objects that are not in publicly observable space (the contents of minds, those \"ghosts in the machine\"). Common sense says that there are dreams, images, sensations, feelings, and thoughts that are not observable in space. At least my common sense does, and most people I meet down the pub think it does, and if we are not common, who is? Common sense says that things are what they seem to most people to be, but we do not explore too deeply what happens to things when we are not looking at them.\n\nThe people who do explore that sort of thing are physicists. Common sense seems to conflict in a marked way with modern physics. This fact leads into the second group of theories about the nature of the world we experience, that science tells us the truth about reality. Modern physicists sometimes say that there are other space-times (\"many worlds\" or a multiverse of universes), and if so those space-times are certainly not publicly observable in this space. Space and time are only four of many possible dimensions, and physicists like Stephen Hawking envisage realms of being in which space or time does not exist at all or can become interchangeable. It seems to be an open question whether there are non-spatial entities, and philosophers are not going to avoid it by saying that it arises from a logical mistake or category error.\n\nThere's more to life than meets the eye\n\nSo we are led beyond common sense toward the currently fashionable position that science tells us about a world that is very uncommonsensical indeed. Because physicists do not like to think of themselves as naive, this is sometimes called critical realism as opposed to naive realism (the unkind term for common sense). The world as it is in itself is not as we observe it in experience. But it has nameable properties such as mass, position, and velocity, and science \u2013 preferably fundamental physics \u2013 can tell us what they are.\n\nSince the 1930s it has become much more questionable what nameable properties fundamental physics does tell us about. If there is five times as much dark matter in the world as there is matter that we can see, if the fundamental forces of our space-time are in fact crumplings of ten-dimensional hyperspace, and if fields and tensors have replaced particles and waves as the fundamental stuff of the universe, we might well begin to wonder what may be coming next. It sometimes seems that physicists are telling us that something is most definitely real, but we are no longer sure just what it is. This realism is so critical that it seems to have become pretty unrealistic.\n\nThis is what I mean by saying that materialism, at least in the sense of saying that everything that exists must have a location and extension in space-time, seems to be scientifically questionable. Many modern physicists have left ordinary space and time well behind. Nevertheless, advances in other scientific areas like neuroscience and artificial intelligence have encouraged a view that all mental events must be identical with spatio-temporally locatable events in the brain. This has led to the adoption of varieties of materialism (Theory 3) which presuppose that, whatever some physicists say, the scientific world is the only one that exists, and the scientific world contains only material entities.\n\nThere are physicists who would support materialism in a more sophisticated form (sometimes called physicalism or naturalism), extending the idea of matter (or energy) considerably, but insisting that all that exists must be material in that extended sense. For instance, if our subjective sense of time passing is the result of a crumpling of hyperspace, or even if time is a fourth dimension in some realistic sense, then our consciousness of passing through time may not be a good clue to what reality is like. Time may be there \"all at once\", existing from beginning to end as a fourth dimension, and we just seem to be passing through it. For such views our sense of personal development, of having interesting and unique experiences one after the other, is an illusion, as is consciousness itself. Only the material, as defined by the latest scientific theory, is real.\n\nIt is not quite clear, however, who would be having such an illusion. Anyway, illusions seem to have some sort of existence. If I seem to see something, there must be something that I seem to see; there must be a \"seeming\" as well as a \"reality\". Or must there?\n\nHeadless women\n\nSuppose a conjurer makes it appear that he saws off a woman's head, and in consequence I seem to see a headless woman. That is an illusion \u2013 the head is there, but I fail to see it. I do not in fact see a woman without a head. I fail to see the head of the woman. I do not see what I think I see. So it is not true that if I seem to see X, there must be an X that I seem to see. However, there is something that I see. I see a woman whose head is concealed, but I misdescribe what I see. That is easily done. Conjuring tricks work because they are easily done. We often make mistakes when we describe what we see. Yet there is something I see, even though I have the description wrong.\n\nCompare this with \"having the illusion\" that I am passing through time, one second after another (whereas, in fact, past, present, and future all exist in one four-(or more) dimensional continuum, and I am not moving through time at all). What is it that I seem to see? One thing happening after another. But could I be describing this wrongly? This would be like saying that time is actually all there \"at once\", but I (wrongly) see it as one thing after another. Even if I am misdescribing it, however, I am seeing one thing after another. Someone (me) is having the experience of a succession of times. So a succession of times does exist \u2013 in my experience.\n\nThat is enough to establish that my experience has properties that objective time does not \u2013 properties of temporal succession. Even if time is an illusion, we must distinguish personal experience from objective physical reality. Therefore it cannot be true that personal experience does not exist. Illusions, too, exist, and have properties that do not belong to objective reality.\n\nThe extreme materialist view that consciousness is an illusion can only be consistently held by philosophers who are not conscious. Therefore most conscious philosophers who are dissatisfied with common sense distinguish personal experience from objective reality, and maintain that while subjective conscious experiences do exist, objective (unobserved) reality has whatever properties scientists ascribe to it. This is actually what Cartesian dualism (Theory 4), Ryle's main target, maintains. Conscious experiences are distinct from material objects, and you cannot get rid of them simply by saying that they are illusions.\n\nYou could, however, say that consciousness somehow \"emerges from\" complex physical structures like the brain, even if you have no idea how it does so. This has been termed epiphenomenalism (Theory 5), since consciousness is regarded as a by-product of the physical brain that does not influence behaviour. Descartes did not think consciousness could emerge from matter, since they are so different from one another, and causes, he thought, can only produce effects like themselves. But you could just say that this need not be the case. If one sort of thing (matter) produces another type of thing (mind), there is no point in complaining that we cannot see how it is done. We just have to live with it.\n\nIt should be noted, however, that if you say this, you will also just have to accept that matter, or changes in matter, might be produced by minds. So minds could produce physical changes or even possibly physical entities, and again it would be useless for philosophers to complain that they cannot understand how it happens. They will just have to get over it. Mind\u2013matter interaction might be real.\n\nNevertheless, scientifically minded philosophers often assert that all genuine causes are physical, whereas the personal or mental is a sort of by-product that plays no effective role in governing what happens in the world. This seems to be just a basic dogma \u2013 all causes must be physical, because I say so. (I should add that I am not against having basic dogmas. We all have them, but at least we should acknowledge that they are dogmas, and are by no means obvious to everybody.) It is certainly not an opinion that is confirmed by observation or by any natural science. Of course the dogma is not produced by some arbitrary whim. It is produced by the adoption of an elegant and comprehensive explanatory hypothesis \u2013 that the unobserved world has the \"primary properties\" noted by physical science, but no \"secondary properties\" which belong only to subjective experience. The hypothesis can become very tightly constraining, when it is postulated that the physical world is governed by a simple set of absolute and inflexible laws which wholly determine everything that happens. This is the \"machine world\" against which Ryle also inveighs. And in that machine personal experiences or the subjects of such experiences become \"ghosts\", illusory apparitions with no causal part to play in the workings of the cosmic machine.\n\nThe big options\n\nI have now collected five of the eight philosophical views I wanted to outline. Phenomenalism (Theory 1) and naive realism (Theory 2) attempt to maintain that reality is basically what it seems to be to the human senses. The advance of the natural sciences introduces critical realism, and this in turn subdivides into materialist views that mind is not a thing at all (Theory 3), dualistic views that mind and matter are distinct (Theory 4), and epiphenomenalist views that mind does exist, but is a product of and wholly dependent upon the physical world that science correctly describes (Theory 5).\n\nOne reason I have listed these views is to make the point that there are many defensible philosophical opinions about the nature of the world we observe and know, and about our own natures as observers and agents in that world. There is not just one obviously true opinion. In fact my suggestion is that the third group of metaphysical theories, which I am just about to embark on, is more plausible than the first two groups. The three members of the third group could be called forms of idealism, which is, in its most general form, the belief that mind or consciousness is more real than matter or provides a better clue to the nature of reality as a whole. If I could establish that, we would be well on the way to showing that human persons are not just complex bundles of matter and molecules. Their moral importance and value lies crucially in their mental lives and acts.\n\nOf course I do not expect to establish it to everyone's satisfaction. No philosopher ever manages to do such a thing. But I do hope to show that idealism is a coherent and plausible view, fully consistent with the best modern scientific knowledge. And I hope to show that the philosophy of materialism, which is assumed by many to be obviously true, is far from obvious and contains major weaknesses from which idealism is free.\n\nUnderlying all this discussion is the belief that metaphysical differences are real and important. Ryle's claim that he is only rectifying the logic of our language about minds by distinguishing logical categories, and not making any statements about the nature of the world, is unconvincing. Ordinary language carries philosophical presuppositions. It presupposes \u2013 contrary to what Ryle claims \u2013 that minds are causes on occasion, that humans are at least sometimes free agents, and that private states of consciousness exist. If we ceased to believe these things, our language might well change.\n\nPhilosophical reflection can make a difference to what we say, to how we speak, and to what we claim to know. The rise of the natural sciences has raised questions about human personhood in a sharp way. It has given rise to radical forms of dualism, which separate minds and bodies completely, and to materialism, which eliminates minds (in this sense) altogether. So another goal I have in mind is to place radical dualism and materialism in historical context as extreme responses to new scientific knowledge, and to point forward to more adequate interpretations of contemporary knowledge.\n\nThis of course is what Ryle thought he was doing. He rejected both radical dualism and materialism as inadequate philosophical theories, and tried to replace them with a more rounded view of humans as essentially social animals. In doing this, he was importantly right. His mistake was to think that he had no philosophical theory, but was only stating the obvious (once people got their logic right). But that, ironically, was his philosophical theory.\n\n## Chapter Three\n\n## The limits of knowledge\n\nAnother major philosophical view is idealism: the theory that reality is mind or mind-like, and that material things are appearances of this reality to consciousness. This is the view I aim to defend. Most classical European philosophers have accepted some form of it. It was most fully formulated by German philosophers like Kant and Hegel, and it dominated British philosophy in the early twentieth century. Immanuel Kant was especially influential, and he is still taught in most philosophy courses today. Kant thought that reality-in-itself is completely unknowable, but that human thought constructs reality-as-appearance (as we see it), and that human reason compels us to think of persons as free moral agents, not determined by the material world. Materialism, Kant thought, confuses appearance with reality. So even though we cannot know reality-in-itself, we must think of it as personal.\n\nAs I approach the topic of philosophical idealism, I will begin with some major philosophical questions raised by the natural sciences: does the objective world really consist only of the basic properties identified by physics? Is it governed by universal and unbreakable laws? Do those laws determine everything that happens, so that no alternatives are even possible?\n\nAffirmative answers to these questions are very ambitious dogmas or postulates indeed; and they could not be confirmed by observation. We cannot observe the unobserved; we cannot be sure that laws of nature are never broken; and we can never know that nothing could have happened except what did happen. The postulates that physics can accurately and adequately describe what the unobserved world is like, that absolute laws of nature in some mysterious sense exist, and that there is one and only one possible effect of every cause, seem to be oddly arbitrary. Why should it be so?\n\nPerhaps these are the basic postulates which helped modern science to get going. Only if we think that the human mind is capable of understanding the structure of nature, only if we think that there are mathematically describable laws or general principles of causality in nature, and only if we think that causality is universal, so that there is always a reason why anything happens, that nothing happens that cannot be accounted for by reference to some general rational principle, is modern science possible. These are, as Immanuel Kant put it, necessary conditions of the possibility of natural science (at least of a Newtonian sort).\n\nBut there is a paradox about these postulates. They give human thought and imagination a central role in comprehending the nature of reality. Human thinking, especially in inventing systems of mathematics and in devising cunning experiments to see how nature behaves, is assumed to be adequate for an understanding of nature.\n\nBut can we trust human reason? \"Reason is the slave of the passions,\" said David Hume. How can we have the arrogance to think that the petty human mind \u2013 the product, according to some evolutionary views, of millions of random genetic copying-mistakes and accidents \u2013 is capable of understanding the origin of worlds or the ultimate nature of things? Perhaps, said Hume, we should accept that habit or custom controls human conduct. We just think the way our brains are set up to think, and they are set up that way, because brains with those thoughts in them have enabled organisms to survive and reproduce better than brains without such thoughts. Hume did not know about evolution. But he certainly would have approved of it, since he believed that enlightened Scottish intellectuals were more evolved than the common herd of humanity.\n\nBrains that thought that you could never predict what was going to happen next died out, because they never learned from experience. Only brains that assumed the future would be like the past had the sense to flee from predators, and only those brains survived. Over many centuries a stock of common-sense beliefs builds up, simply because brains without those beliefs get wiped out. Such beliefs would include the belief that there are predators out there even when you don't observe them (the principle of objective existence), that predators are going to behave in regular and more or less predictable ways (the principle of induction), and that the same predator always produces the same effect \u2013 namely, once you are caught you will be eaten (the principle of efficient causality).\n\nOf course this is just a way of saying that true beliefs tend to be more useful than false beliefs. The reason these sorts of beliefs have survival-value is that they are true; they say what reality is like. You could say that the beliefs are based on long and repeated observations by many people, some but not all of whom got eaten. They are not just abstract rational principles, thought up in some primeval cave.\n\nI was once asked to contribute the section on philosophy in a Reader's Digest book, The Last Two Million Years. I agreed to write the first half and, before the commissioning editors had quite realized what was going on, collected quite a nice fee for saying, correctly, that as far as we know nothing happened in philosophy for the first million years. They nevertheless produced some pretty pictures from the Lascaux Caves, which perhaps could, by a large stretch of the imagination, be called philosophical (that is to say, nobody is quite sure what they are about). Regrettably I missed the opportunity to write a chapter on Neanderthal philosophy, which did not, as it turned out, have a very good survival-value.\n\nToward the unknown\n\nReturning to the question of basic evolved beliefs, the point remains that these beliefs go well beyond simple observation. They begin from observations, but what they do is organize our sense-perceptions so that we interpret these perceptions as perceptions of entities which have objective existence, which threaten, entertain or eat us, and which behave in more or less predictable ways. Such organizing principles are not arbitrary, since they can help us to eat other things before they eat us \u2013 a skill at which humans are remarkably adept.\n\nThese principles no doubt form the basis of the more science-based principles of objectivity, causal law, and determinism. They are nonetheless very different. Common sense sees the world as consisting of predators and prey, not atoms and molecules. We see that predators behave in more or less regular ways, without thinking that they obey universal laws of nature. We think that we can sometimes escape the predator's leap, not that there is no alternative to being eaten. The idea of a Newtonian reality of deterministic and universal law is a leap of imagination that was first clearly formulated by Newton himself, even though it was presaged by some late medieval philosophers, as a combination of ideas of predestination and the creation of the universe by a rational all-determining being.\n\nMoreover, the Newtonian picture omits entirely some other human basic evolved beliefs. Early humans developed a \"theory of mind\", that other animals have purposes and intentions partly hidden from us, but helping us to predict what they will do next. They developed a principle of reciprocity, that you can never quite be sure how other animals will react to you, but your attitude to them will be an important factor in how they subsequently behave. They developed a principle of responsibility, so that on many occasions you can choose between alternative actions, and you are responsible for that choice.\n\nAll these natural principles are hard to accommodate in a Newtonian scheme of nature. Minds, personal relationships, and free actions, are all alien to a wholly mechanistic view of objective reality. Yet it is precisely in scientific investigations that we must assume the importance of understanding and imagination, the cumulative cooperation of many meaningfully communicating minds, and the free selection of experimental conditions and mathematical axioms that will enable us to understand the physical world better.\n\nIs it plausible to consider all this as a higher-order by-product of the unconscious, mechanically determined, exclusively rule-driven behaviour of huge numbers of fundamental particles, each of which is identical in nature to every other of the same sort? Locke baulked at the thought, as had Descartes before him. Thus they were driven to introduce minds into a Newtonian mechanistic scheme that left no room for them. Having constructed the machine of nature, they then had to introduce ghosts to pull the levers of the machine. But how insubstantial ghosts could pull solid material levers, or even where exactly the levers were to be found, remained a mystery.\n\nPerhaps the solution is not to get rid of the ghosts, but to dismantle the machine and start again. That is precisely what happened in the early twentieth century with the advent of quantum mechanics. The machine itself began to disappear. Ryle and Wittgenstein had that thought, too. But they preferred to bypass the findings of the new physics, and to resort to the sort of common-sense beliefs that had preceded physics, and that left questions of the nature of reality as pseudo-problems that could be ignored.\n\nSome physicists agree and regard philosophy as a waste of time. We can do the maths, make the predictions, construct the nano-devices. But we do not need to give any theoretical interpretation to what we are doing. Others, however, feel the need to relate our mathematical computations to our understanding of the world, and to answer the question of what the world is objectively like. But maybe what quantum physics suggests is that the question is unanswerable. For quantum physicists like Bernard d'Espagnat objective reality is forever \"veiled\" from human knowledge, and all we can know is how things appear to us when we observe them (when we collapse wave-functions) in laboratory conditions. Perhaps there is no way of knowing what reality is like in itself, apart from human observation of it. We know how the world appears to us, but its inner reality remains forever veiled. That possibility introduces my sixth philosophical theory about what the world is really like, the theory that human knowledge is confined to how the objective world appears to our minds and senses, but that the world in itself is wholly unknowable. Nevertheless, in some sense the world as we know it is a product of mind. In the history of philosophy, that is the view of Immanuel Kant, who called it \"transcendental idealism\", but in his later writings said that he preferred the phrase critical idealism (Theory 6).\n\nImmanuel Kant and unknowable reality\n\nMost classical philosophers have been idealists \u2013 they have thought that the ultimate character of reality is mind-like. But in modern philosophy the most influential name is that of Immanuel Kant. If we want to trace the roots of modern forms of idealism, we have to start with Kant. There is quite a lot of misunderstanding of his views \u2013 he is thought to have undermined the possibility of metaphysics and to have destroyed all arguments for God, for example. On the contrary, his aim was to place metaphysics on a firm footing, and to defend belief in God as rationally necessary (though not founded on theoretical arguments about causality).\n\nKant's views are very complex, but it is worth exploring them, as they provide the basis for taking modern idealism seriously. He held that space and time themselves, which seem so objective, are in fact forms of our intuition. That is, they are a framework the mind constructs to build a map in which our sense-perceptions can be located. The map is mind-constructed, and if we take away mind, we have no way of saying what is left. This may seem an extremely odd view, but it receives some corroboration from modern physics. In quantum physics our view of space as a three-dimensional Euclidean container, and of time as a kind of cosmic clock which ticks at an absolute rate, adding second to second as we pass through it, is a purely subjective condition of our perception, which does not copy what objectively exists. Quantum physicists tend to say that space-time is non-Euclidean, and is part of a multi-dimensional reality in which space and time may become, under certain conditions, interchangeable. Superstring field theorists say that space and time are perceptually selected fragments of a ten-dimensional hyperworld, and do not exist objectively as the Euclidean space and flowing time that we perceive them to be. Such things can be mathematically expressed, but if we try to imagine what such a strange world is like, imagination fails. Things in themselves are totally beyond our categories of thought.\n\nIt seems, then, that the whole of experienced space-time, and everything in it, is a construction of the mind, and would not exist without the mind. Some philosophers talk about \"constructions of the brain\" rather than constructions of the mind. But that is a fairly crude mistake, since the brain is a material thing in space-time, and therefore cannot be what constructs the reality of experienced space-time. The brain, like all material things, is a construct of the mind and would not exist as it appears to us without that constructive activity of the mind. So the world as we see it is not a construction of the brain, but the world as we see it (including the brain) is a construction of the mind.\n\nThat is enough to refute the simple materialist view that nothing exists, or even can exist, outside of space and time. Beyond observed space-time, there are at least two things. One is the unknowable world of things in themselves and the other is the mind that constructs the world of appearances, by its creative interaction with things in themselves.\n\nBut if the mind is not an appearance (since it constructs the world of appearances), it must be a thing in itself. Or, to be more precise, what appears to us as the constructive activity of the mind in producing the world of appearances, is also an expression of an entity in the world of things in themselves.\n\nEven if the mind as we experience its activity is an appearance, it is as real an appearance as the world of physical objects we sense. It is not the case, as Gilbert Ryle seems to say, that physical objects (bodies) are obviously real, whereas minds (creative intellectual acts that constitute the world of sense-perceptions and thoughts) are ghosts. On the contrary, both bodies and minds are appearances of an objective reality. They have an equally real (or unreal) status.\n\nThe personal world\n\nRyle would of course object that this is even worse than Cartesian dualism. His view is that in the Cartesian world we never know what is going on in other minds. Minds are locked in permanent isolation and never meet. But at least bodies meet. That is enough. We do not need these private islands where minds spend their lives in solitary confinement.\n\nThe Kantian world is much worse. Not only are minds in isolation. The whole of the experienced world is put into isolation, minds, bodies, and all. People cannot ever get out of their own private worlds or know what is going on in anybody else's world. Bodies never meet, for bodies are just parts of essentially private worlds. We are condemned forever to never meet anyone else, and never to know what is really going on. All of us are condemned to permanent ignorance and illusion, from which there is no escape. Is that better than Descartes?\n\nRyle is right. We cannot have a world in which knowledge depends upon making untestable inferences to hidden processes. But Kant does not think that we make inferences to the external existence of objects. Kant proposes that from the very first we interpret what we experience as experience of a world of objects. Our interpretations of the world are not just passive receptions of inert sensations. We are not faced with a lot of private sensations from which we subsequently have to infer the untestable assumption of an external world and other people. Experience is in a sense not shared with anyone else. But that is a realization that comes only from the much later sophisticated reflection that not everyone sees the world as I do.\n\nFrom the very first, Kant argues, human thinking is an active power that necessarily interprets experience as experience of a world of continuing substances in causal interaction. That is, we interpret the coloured shapes we see as appearances of external objects which we encounter through these perceptions. This is a basic interpretative activity of the mind. We do not see sensations. We see objects, presented to us via sensations. Experience comes to us already interpreted by thought. We are aware both that something beyond our control is given to us in experience, and also that thinking actively interprets that \"something\" as a world of causally interacting substances. We are never isolated and alone. On the contrary, we are always encountering other objects and actively responding to them. We are active agents in a world of active agents. That is not an inference; it is a necessary condition of the possibility of knowledge, both in science and in everyday experience of personal relationships.\n\nTo know anything you need both perceptions \u2013 sensory data \u2013 and concepts \u2013 thoughts. So of the world beyond our perceptions, of reality in itself, there can be no theoretical knowledge. This is the proper reply, Kant thinks, both to common sense (naive) realists and to reductive materialists. They both think they know what there is, but both are mistaken. Kant does not know what there is (not theoretically, anyway). But, whatever it is, it must produce the appearances we see and the active minds that interpret them. There is a hidden reality, and the mind plays a positive creative role in interpreting its appearances. This provides the key to understanding what Kant always regarded as the most important, and most misunderstood, part of his philosophy: that which reason forbids me to know, reason compels me to believe.\n\nThe limits of reason\n\nKant argues that if reason claims to tell the truth about ultimate reality it leads to contradictions (he calls these \"antinomies\"). But we do know that if there are appearances, then there must be a reality that appears. In the world of appearances, we seek determining causes for everything. We actually insist that nature conforms to our demand for causes for all events. It is a condition of the possibility of scientific knowledge of the world that we think of the world as consisting of continuing substances in a succession of regular and predictable causal relationships. That is no mere whim, as if it were no better than being inclined to think of the world as moved about by the arbitrary acts of millions of fairies. These inclinations are rational, because they are the very conditions of the possibility of our understanding of an intelligible world. And they are confirmed by constant experience, as science continually finds out more about the natural world.\n\nJust as reason (or understanding, in Kant's terminology) lays down the conditions of the possibility of scientific knowledge, so reason lays down the conditions of a complete rational explanation of the world, including the non-scientific facts of subjective consciousness, freedom, value, and purpose. These conditions cannot be completely confirmed by any specific experience, so they remain postulates \u2013 not irrational leaps of faith, but rational postulates that cannot be fully confirmed by experience.\n\nThe postulates of reason, according to Kant, have both a negative and a positive sense. In their negative sense, they show that materialism, naturalism, and fatalism are indefensible, according to Kant's Prolegomena to any Future Metaphysic. Materialism is the view that the mind is nothing but a material thing, whereas Kant shows, he thinks, that it is the part-creator of reality-as-appearance. Naturalism is the view that nature is self-sufficient, whereas Kant shows that nature is the appearance of an underlying and unknowable reality. And fatalism is the view that all human acts are products of blind necessity, whereas Kant shows that they may indeed be free in reality, even though we can never prove that they are.\n\nTo many philosophers, it sounds impressive to say that the experienced world is only an appearance of a very different underlying reality. But it is not quite so impressive when it is added that we know nothing at all about such a reality. Kant's position of complete theoretical agnosticism about the world of things-in-themselves does not quite ring true.\n\nThe leap of reason\n\nIt is at this point that \"faith\" comes into the picture, and gives a positive sense to the postulates of reason. Faith, for Kant, has absolutely nothing to do with revelation or religious authority, both of which he hated so much that he never went to church and regarded kneeling down to pray as an affront to human dignity. Kant was a great defender of autonomy, of deciding things for yourself, and he tried to get all his disciples to take his word for it that they should decide things for themselves. He added of course that if they decided correctly, they would all agree with him, since he always made the most rational decisions. Insofar as they disagreed, they were not being fully rational.\n\nKantian faith is the positive acceptance of the unprovable but fully rational postulates of reason on practical or moral (not religious) grounds. Kant wrote three \"critiques of reason\", in which he set out to show the limits of reason and its positive role in human thought. At this point I am only concerned with the first two critiques. His first critique, the Critique of Pure Reason, was concerned with examining the role of theoretical reason in helping us to achieve knowledge in science. He argued that it is a condition of the possibility of doing science that you accept some necessary postulates of understanding (most importantly, the objective existence of substances and of causality).\n\nHis second critique, the Critique of Practical Reason, went on to discuss the role of reason in practical action and especially in morality. He argued that it is a condition of the possibility of morality that you accept some postulates of reason (most importantly, the objective existence of free agents who are able to act on rationally chosen principles). Having shown that science does not give a theoretical account of things-in-themselves, he now suggests that morality gives a special sort of insight into that veiled reality. At least it compels us to think of it in a particular way (as a world of free agents), even though we cannot theoretically establish our beliefs.\n\nReason thus plays a positive role in human knowledge. In theoretical matters it requires sense-experience to confirm its postulates. But humans not only think and understand. They also act and set and pursue goals that they believe to be good. Is this a function of reason also? Can reason set goals of action, and are we free to pursue them? For Kant, reason does set two general goals of action \u2013 the happiness of others and one's own perfection, \"the fullest use of one's free powers\".\n\nBut if reason sets such goals, then reason must be assumed to possess a causal role in the world, setting goals of action which I can then pursue. This entails two beliefs \u2013 that I am free to set and pursue rational goals and that it is possible to achieve them (otherwise it would not be a rational pursuit). Thereby results one of Kant's major conclusions: I can never theoretically prove the human will is free. But for rational action and moral commitment to be possible, I must presuppose freedom in practice. Whatever my theoretical indecisions, I must commit myself in practice, and I know that I should commit myself to the wholly good.\n\nMoreover, if I really think free commitment to realizing goodness is reasonable, I must believe that the good \u2013 which consists of both happiness and that form of self-cultivation which is virtue \u2013 is achievable, even in a corrupted world. Kant's point is that if moral action is to be fully reasonable, and not just a matter of arbitrary decision, I must assume that what I am trying to achieve is possible. So fully rational moral action must commit itself to the hope that a world in which happiness is realized in accordance with the practice of virtue can and will exist. That is a presupposition of the rationality of the real world to which the human will commits itself in genuine moral action. For if reality is rationally structured, it will exist in order to realize an envisaged good, and the realization of that good is assured by the rational order of reality.\n\nKant is not, after all, a total agnostic. He believes there can be no knowledge or proof of the nature of the world beyond the senses. But he knows there is and must be such a world. The world of appearance is a world constituted by both an unknown objective reality and the constitutive activity of the mind, which in itself is also unknown.\n\nBut where theory may hesitate, persons must act. This is not as far from Hume as it may seem. Hume also is sceptical about transcendent metaphysical truths obtainable by reason. Hume also thinks we must act on the basis of our human sentiments and habitual inclinations to believe. But whereas Hume thinks that we are driven by passions and habits, which we cannot help, Kant insists that there are rational goals of action, that human persons are free, autonomous (capable of direction by their own free decisions), and that we must assume (without theoretical proof) whatever is necessary to embrace that freedom.\n\nIn this way Kant qualifies his view that we can know nothing of reality in itself. Though that is true of theoretical or testable knowledge, in moral action we commit ourselves to thinking of objective reality in a specific way, as a realm of autonomous rational agents. The limitation of theoretical knowledge to appearances means that such a commitment cannot either be established or disproved theoretically. In that situation, practice takes priority over theory, and, as Kierkegaard was later to put it, we may passionately commit ourselves to what is theoretically uncertain. For Kant, such passionate commitment is not irrational or non-rational. It is supremely rational and a condition of accepting the reasonableness of human moral freedom. It is a leap of reason, not a leap of faith.\n\n## Chapter Four\n\n## Putting minds first\n\nKant's denial of all theoretical knowledge of reality was judged by many philosophers who followed him to be implausible. Other forms of idealism, the best known of which is that of Hegel, more boldly affirm that we can know something about reality-in-itself, namely, that it really is mind-like. There are other forms of idealism too. Many of them can be found in Indian philosophy. In the West, one of them is process philosophy, formulated by the British philosopher and mathematician A. N. Whitehead, which claims that there are many mental realities, rather than just one dominating Absolute Mind. In all its forms, however, idealism is obviously opposed to materialism, and gives mind or consciousness a primary place in our idea of reality.\n\nKant says that the conceptual categories we use only have theoretical meaning within the realm of sense-experience. Yet he also holds that there is a reality beyond sense-experience, of which the sensory world is an appearance. But how can he even say that there is a realm beyond sense-experience, which is the hidden cause of our sense-experiences? He is using the categories of substance (things-in-themselves are described as things, after all), existence, and causality, which should have no meaning. He is even describing the things-in-themselves as noumenal (which means \"mind-like\" or \"only apprehensible by mind\"), intelligible or conceptual as opposed to sensory.\n\nAlthough the freedom of the self cannot be proved by induction or by empirical methods, it must, he says, be postulated as a condition of moral action. The ideas of reason do not have theoretical meaning, but we must act as if they are true of the noumenal world, and the justification for this is that they must be used to achieve unity in our knowledge and to underpin moral action. It is far from satisfactory, however, to hold that we must act as if something is true, when we know it is not, and when we have no idea, theoretically speaking, of what is true. To hold that this is a totally rational procedure stretches the meaning of rationality beyond any reasonable limits.\n\nThe trouble is that Kant provides a wholly mechanistic and deterministic view of phenomena, while free action and the judgments of understanding and reason are allocated to a nonmechanistic and non-deterministic (but also non-temporal and non-spatial) realm. This means that I must regard my moral actions as free and undetermined \u2013 but only in a noumenal realm beyond space and time.\n\nFor most of us, however, free acts take place in time. I am free when I perform a specific action. It is not really much help to say that all my specific actions in time are determined, but that there is some sort of non-temporal freedom as well. The attempt to make sense of this even leads Kant at one point to say that perhaps the place where I am born is freely chosen by me in a non-temporal sense, and that is dangerously near to saying that I am poor and oppressed because I choose to be. I do not think we want to go there. This is worse than a ghost in a machine; it is something completely invisible inside the appearance of a machine. And that threatens to undermine the very morality and rationality the Kantian system was designed to protect.\n\nThe personal, the moral, and the scientific have to be integrated in a better way than this. It is actually Kant and not Descartes who is the chief proponent of the myth against which Ryle inveighs. Descartes doubted only for a while, and in a purely hypothetical way. But Kant advocates an utter scepticism about what reality-in-itself is like. His doubt is total and irrevocable. Of course Kant protests that his philosophy leaves everything as it is, in this phenomenal world. That is why he protests (too much?) that he is not an \"idealist\", in the sense of being a person who thinks physical objects are only ideas in the mind or that the whole of reality is in the mind of a perceiver.\n\nBut the fact is that Kant does think that the whole of reality as it is perceived is a product of the mind of the perceiver. The term he preferred for this philosophical outlook was \"critical idealism\". There is an unknowable reality underlying the world of appearances, but it adds nothing at all to theoretical knowledge. It could simply be ignored, except that it gives the mind a constitutive and active and essential role in experience. Thus it leads us to think of reality-in-itself as mind-like, even though we have no theoretical knowledge of it. Reason does in fact have something to say about the world of things-in-themselves. But it says it only from a practical (Kant says, a \"regulative\") point of view, for the sake of grounding and motivating our moral action and our sense of the intrinsic moral worth of human persons.\n\nMany modern philosophers would recommend avoiding this conclusion by getting rid of things-in-themselves, and reverting to some sort of realism. I think Ryle did that. The trouble is that once you recognize the dependence of human knowledge of the world upon the specific structure of human senses and brains, and once you give thought a constructive part in building up human knowledge, you do seem to be stuck with the conclusion that the world we perceive is largely a product of our perceptual and conceptual apparatus. Even our brains are parts of the perceived world of appearances. Kant is not saying that you only have the appearance of a brain. His assertion is that your brain is only an appearance \u2013 so it cannot be what constructs the world of appearances.\n\nStill, Kant could not get rid of the suspicion that there must be some reality underlying the appearances. Neither could many philosophers after Kant, the best known of whom is probably Hegel, who said that if you can posit that there are things-in-themselves, you must also posit that the mind has some access to what those things really are. Kant really believed this anyway, deep down, but his philosophical system forced him to say that we could have no theoretical knowledge of reality-in-itself. We are moving toward a more fully-fledged statement of philosophical idealism.\n\nThe priority of mind\n\nIn order to get to a more robust form of idealism, we only have to discard the idea, central to Kant's theoretical philosophy, that concepts without \"intuitions\" or perceptions are empty. There are some concepts (roughly, the ones Kant called \"ideas\") that are not conclusively confirmable by sense-observation, but that may be basic to our metaphysical scheme. These will include concepts suggested by reflection upon the causal activity of the mind in both theoretical knowledge and moral action.\n\nKant is convinced that his critical philosophy shows materialism to be false. Minds, since they play a constructive part in creating the objects of knowledge, cannot be just objects of empirical knowledge (brains). Minds are active in two main ways, theoretically and practically. Theoretically they interpret sense-experience in terms of a world of enduring objects having causal relations and often forming integrated wholes. They \"see\" the immediate and transitory data of the senses as mediating a reality beyond themselves. Thus they see the present as mediating the past, in remembering past data and identifying present data as integral parts of what has gone before. This is perhaps clearest in music, where we hear notes as parts of a tune, not as isolated sounds.\n\nMinds also see the present as mediating the future, seeing possibilities for action over stretches of time. Minds see the present as mediating objectivity, a world of objects continuing to exist unobserved. Minds also see the present as mediating a sense of self, a continuing agent which integrates experiences into one unique totality of \"my experience\". And minds see the present as often mediating other agent selves, with goals that may block or help their own.\n\nIn all these ways, I have the idea of myself as a continuing cognitive agent among others in a common world \u2013 something that perception without thought could not generate. My inner mental life of perceptions and thoughts does not, as Ryle suggests, lock me into a private world that could never encounter or be known by others. On the contrary, it is because minds have private thoughts and experiences that they are of interest to others, and that they always remain to some extent mysterious and intriguing. But it is because minds interpret their sense-perceptions as mediating a world of objects and persons (intentional agents) that they know there are others to find intriguing. Persons meet through the mutual interpretative mediation of their thoughts, through language, by which marks or sounds are taken as mediating meaning beyond their sensory appearances.\n\nMinds are also active practically. They consider future possibilities and evaluate them. They choose courses of action and initiate them. They make moral decisions and to some extent shape their own characters by the habitual patterns of action that they choose over time. So persons are praised or blamed for their actions, assuming that they could have chosen otherwise, and that they knew the moral quality of what they were doing.\n\nWe may not know what the veiled reality underlying physical objects is. But there is a good case for saying that we do know what the reality underlying our own mental activity is. That reality is known, not through perception, but in the cognitive and moral activity of the self. At this point reality is known in and through its activity, and that activity is the activity of mind, of free and conscious interpretation and choice.\n\nKant said that we do not know what the mind is in itself; we only know its phenomenal acts. But the mind is known in itself precisely in its activity. Mental activity is not something that appears to us in one way, but exists in itself in another way. Mind is known in its actions, which are not appearances of objects \u2013 that was Hume's mistake when he tried to find the self as an object. The self is not an object that can be perceived like a block of wood or a tree. The self is the unobservable agent that makes all experience of the sensory world possible and is the source of all responsible actions in that world.\n\nIt is thus the activity of the self that gives access to reality, and which creates for us the world of appearances. The primary sense of reality-in-itself that we have is that it has the character of mind, of active mediating knowledge, and responsible choice. To make such a blunt assertion as this is to move beyond Kant's agnosticism to a more fully-fledged idealism, affirming that we are capable of knowing that the basic character of reality is mind or consciousness.\n\nAbsolute idealism\n\nIdealists are aware that there are many human (and perhaps other finite) minds. They claim that the material world (which is the phenomenal world, the world as we know it through perception) exists as an environment created by a primordial mind in which finite minds can exist in mutual self-expression and interaction.\n\nThis is about as far from materialism and as far from commonsense realism as it is possible to get. It totally reverses the modern myth that minds are by-products of a purely material evolutionary process, completely determined by physical events in their bodies and brains. What it says is that physical events as we perceive and know them are appearances constructed by minds. What the reality underlying those appearances may be in detail we do not know. But since minds are the only sorts of reality we know to belong to the world of things-in-themselves, it is reasonable to think that reality does not exist without mind and consciousness, evaluation and intention, understanding and action. Minds are irreducible elements of reality and they play a constructive role in the existence of the phenomenal world that we perceive and know. Minds are not illusory ghosts in real machines. On the contrary, machines are spectral, transitory phenomena appearing to an intelligible world of minds.\n\nWhat idealists maintain is that the ultimate nature of reality itself is mind-like, and that human and other finite minds are the best clues we have to what objective reality is like. The cosmos is not a mindless, unconscious, valueless, purposeless, yet somehow strangely intelligible, mechanism. Such a view is the result of extrapolating a machine-model, very useful in many scientific contexts, to provide the most comprehensive and adequate picture of the real cosmos.\n\nIdealists propose that the human mind provides a better model from which to extrapolate to the cosmos as a whole. That is not because the cosmos looks like a very large human person or because there is some large person hovering just beyond the cosmos. It is because human minds play a creative and constructive role in producing the phenomenal world. They seem to point to a level of reality that is not merely phenomenal or an appearance to consciousness. Human minds generate an idea of reality as mind-like in a way that far transcends human mentality, yet that does include something like consciousness, value, and purpose as essential parts of its nature.\n\nThere are many forms of idealism. Kant's critical idealism is just a little too agnostic for most idealists. Some philosophers, like McTaggart, have proposed that a society of minds underlies and gives rise to our physical cosmos. Perhaps the best-known European idealist, Hegel, hypothesized that there is just one Absolute Mind or Spirit which progressively realizes its nature in the history of the cosmos. This form of idealism, often called absolute idealism (Theory 7), is what I am calling the seventh philosophical view.\n\nIt is found as a major strand in the Indian philosophical traditions, too. Sankara, an eighth-century sage and saint, founded the school of Advaita\u2013 non-dualism \u2013 which holds that the whole of the cosmos and everything in it is an appearance of one Supreme Spirit. Though that Spirit \u2013 Brahman \u2013 is unknowable in its inner nature, it makes itself known as sat-cit-ananda, an infinite reality of intelligence and bliss. Finite minds are appearances of that one Absolute, and each human person can come to know itself as a part of that Absolute Mind, not a separate entity.\n\nThere is an interpretation of absolute idealism to which I should like to draw attention, because it gives a much more positive role to the material world than the rather negative assertion of Advaita that matter and individual minds are in the end illusions. It is best approached through the work of another Indian philosopher, Ramanuja, probably a twelfth-century thinker, who founded the philosophical school of Vishist-advaita Vedanta or qualified non-dualism. This school is non-dualist, because it stresses that all things and all persons are parts or expressions of one supreme reality of Spirit. But it places strong emphasis on personal autonomy and relationship. So Spirit realizes itself by \"becoming many\", by relating to partly autonomous finite spirits which together express the nature of Mind. On this interpretation, it is perhaps clearer that the cosmos of finite persons is not just some sort of illusion, which should ideally be transcended. The cosmos is more like the positive \"play\" (lila) of Spirit, with a positive function of allowing relational properties like cooperation and friendship to exist as proper parts of the self-expression of Absolute Spirit. The ideal is a sort of diversity-in-unity, in which differences remain, but are united in a wider whole. I think that this is rather close to what Hegel meant to convey, though Hegel seemed to suffer from a congenital inability to convey anything very clearly.\n\nProcess philosophy\n\nA rather different form of idealism is more radically pluralistic, and gives causal priority not to one Absolute Spirit, but to a huge, possibly infinite, number of entities, which make up the physical universe. Each of these entities has a psychic or mental aspect, as well as an outer or physical aspect. Bertrand Russell believed such a view for at least two days, calling it \"neutral monism\". I prefer to call it pluralistic idealism (Theory 8), because it gives causal priority to the inner or mind-like aspects of events.\n\nThis may seem very unrealistic if it entails that even rocks and trees are conscious and have feelings, feeling hurt when they get moved around or chopped down. We may, however, confine consciousness as we experience it to higher organisms with brains, and yet say that all physical entities also have an \"inner\" nature that is a sort of embryonic consciousness, an unseen centre of capacities and powers that forms the heart of their being. Such a view \u2013 the eighth and mercifully, you may think, the last view that I will consider \u2013 has received an extensive exposition in A. N. Whitehead's \"Process and Reality\" \u2013 a series of Gifford lectures that is immensely complex and difficult in detail, but contains some basically simple ideas that reframe traditional metaphysical views in the light of modern science.\n\nOne of these simple ideas is that all events have an inner nature, hidden from external observation, which is the source of the creativity that drives the cosmos onwards in time. This philosophy, influenced in part by new developments in mathematical physics, can fairly be seen as a form of radically pluralistic idealism.\n\nLike most really original philosophers, Whitehead begins his philosophy by denying what most other philosophers take to be absolutely certain. Kant, for example, held that we necessarily interpret the world as a collection of substances which continue to exist even when all their non-essential properties change. Whitehead, however, denied that there are any substances at all. There are only processes, chains of events, not held together by any underlying substance. Whitehead's view is that events are discrete existents, and that the chains they form are complexes in which the individual creativity of huge, perhaps infinite, numbers of basic events is the true cause of change in the phenomenal world.\n\nWhitehead, like many philosophers, devises a technical vocabulary that is at times almost impenetrable. But with a bit of over-simplification, his basic scheme is this: the world is made up of huge numbers of \"actual occasions\" or events, which are tiny blobs of \"feeling\". Each one lasts only for a split second, coming into being and perishing in the blink of an eye. Each one \"prehends\" or internally registers influences from a set of other events immediately preceding it. And each one then creatively projects into the future a new integration of these prehensions, which is passed on to its immediately succeeding actual occasions. These occasions form the process of actual being. Even though they \"feel\", they are not conscious, so feeling is a word for their internal structure, not for their emotional state.\n\nActual occasions form complex unities, in which a dominant event presides over a large array of smaller events in an organic whole. The human sense of self is a product of a vast array of actual occasions, organized into a complex organic unity, the human body, under the partial control of a dominant actual occasion which is fully conscious. Yet this dominant occasion, like all others, is in fact a series of evanescent events succeeding each other in a continuous sequence, maintaining the process of prehension and creativity that constitutes our actual world.\n\nTo many people this will seem a strange scheme, and it need not be taken too strictly. As with many philosophical systems, it may be wise to accept some general insights while resisting the intricacies of detail that seem to go with devising a grand philosophical system. Whitehead himself is alleged to have said, as he was dying, \"I want no disciples\" \u2013 to which of course those around him said, \"Yes, Master.\" Process thought is not a final dogmatic system. It is more a spur to thinking in new ways about our complex universe.\n\nWe might pick out its main elements as being these: the world consists of a constant flow of transient moments, a \"process\". There is not a sharp dualistic disjunction between mind and matter. Matter is formed of \"low-grade\" events, and mind of \"high-grade\" events, but there is a continuity of being and a development of new characteristics as organic forms get more complex. What drives the process is not a set of mechanical and blind repetitive routines, but continual creativity and novelty, aiming overall at greater harmony and beauty and some sort of realization of the possibilities inherent in each new situation \u2013 but also realizing many conflicting and negative (narrowly self-satisfying) possibilities as an inevitable corollary of the system.\n\nAbsolute idealists would give a more positive causal role to Absolute Spirit, and might also feel that there is an important distinction of kind between conscious selves that can know and respond responsibly to their situation and the \"lower-grade\" events that have no consciousness, knowledge or freedom of action. Yet Whitehead's form of pluralistic idealism presents a philosophical hypothesis that takes account of the continuity and emergent development of the natural world and stresses the value of creativity, individuality, and freedom in nature that can be overlooked in some forms of absolute idealism.\n\nBoth forms of idealism, absolute and pluralistic, if taken to extremes, may put the concept of persons as autonomous, free, rational agents in question. The seventh theory, the idea that persons are all parts of one cosmic mind, may subordinate individuals completely to one all-determining reality, so that the cosmos becomes nothing but the progressive realization of \"Absolute Spirit\". Hegel has, probably unfairly, often been accused of this.\n\nThe eighth theory, that persons are chains of actual occasions, may undermine the idea of persons as continuing intelligent substances, seeing them as transitory parts of a much wider complex flow of events. Thus a major problem for idealism is to formulate an adequate concept of human personhood.\n\nRyle does not explicitly deal with the question of what constitutes a human person. I suspect he would say that the way we use language shows that we all know what a person is \u2013 not a rock, not an animal, and not an angel. I am not so sanguine. I think there is no more important question in the whole of philosophy and in the whole of life than that of what a person really is. Is a person a by-product of purposeless and blind physical forces, as materialists suppose? Or a single continuous autonomous rational agent, as Descartes thought? A particular finite expression of one universal mind, as Hegel claimed? Or part of an endless stream of sensations, thoughts, and feelings, of actual occasions, as Whitehead asserted?\n\nTo answer these questions, we have to have some theory about what is really real, since persons are certainly parts of reality. That is why I have set out eight main theories of the \"really real\": phenomenalism, common-sense or naive realism, materialism, dualism, epiphenomenalism, critical idealism, absolute idealism, and pluralistic idealism.\n\nI have argued that there are good reasons for rejecting phenomenalism, naive realism, and materialism, and I do not much care for epiphenomenalism and critical idealism. But I have yet to deal with Ryle's objections to Cartesian dualism, which I need to do because I am aiming to develop a view of human persons that lies somewhere between Descartes, as traditionally understood, and a form of idealism.\n\nI believe that persons are continuing centres of consciousness and responsible moral agency. I have said that a process view of persons may threaten to undermine this position. However, I actually think that process thought need not conflict with a view of persons as autonomous continuing selves \u2013 in Descartes' terms, as \"thinking substances\". In the next chapter I will try to show that this is so.\n\n## Chapter Five\n\n## Questions of personal identity\n\nWhat is the nature of human persons? Are they aspects of one Absolute Mind (as in Hegel) or complex combinations of a huge number of mind-like events (as in Whitehead) or something else? I argue that human persons are primarily chains of experiences \u2013 thoughts, perceptions, feelings, and sensations \u2013 and actions (as in Process and some Buddhist forms of pluralistic idealism). But these chains are \"owned\" by continuing subjects (\"mental substances\") who give them their unique identity. It is thus possible to speak, as Descartes did, of continuing mental substances, and to emphasize that those substances (\"minds\" or \"souls\") are primarily made up of inner experiences and intentions.\n\nSo far I have tried to show that there are good philosophical arguments for idealism, and I have hinted that I intend to defend some form of Cartesian dualism, though not the most radical form (which I think is often wrongly ascribed to Descartes). The Cartesian view maintains that human persons are continuing mental substances, \"thinking things\". Idealist views may hold that persons are aspects of one Absolute Mind or that they are composed of streams of \"actual occasions\", as in Whitehead. In defending a form of Cartesian dualism, I want to suggest that there are varieties of idealism that can accommodate that form of dualism, and that such a combination of views may provide the most adequate account of what human persons are.\n\nI will begin the discussion with a consideration of pluralistic idealism, because I think it is a good starting point for seeing what might be meant by considering persons as continuing substances. One reason for thinking of persons as substances has been that this idea enables us to think of a person as \"the same\" person through many changes of personality or in physical and mental properties. But is the idea of \"substance\" really helpful here, and what role does it play? Reflection on a view like process philosophy that explicitly denies the idea of \"substance\" might help to clarify the issue.\n\nAccording to process philosophy, there are no continuing substances, whose properties may change while the underlying substance remains the same. There is just a continual flow of properties, some of which we group together when we speak of things continuing to exist as \"the same things\". Instead of saying, \"The tree (a substance) has green leaves (properties), but they will turn red and fall off in winter (the properties change while the tree remains the same),\" we should say, \"This collection of leaf-like, branch-like, and trunk-like properties will remain the same in some respects, but it will change in others.\" Whether we call it \"the same tree\" or not is a matter of convention. We could say that it is not the same tree, because some of its properties are different. But we normally say that it is the same tree, because its trunk and branches are similar, and the leaves have changed in a generally regular and predictable way.\n\nSome people think that an object can be called \"the same\" only if it is a continuous physical entity in space and time. A tree is the same tree only if it continues to occupy the same space (or a continuously connected series of spaces) throughout one uninterrupted time. But what if one day the tree disappeared and an exactly similar tree instantaneously appeared three feet to the right? Would it be the same tree? I guess that if trees consistently jumped three feet to the right on Tuesdays, we would get into the habit of calling them the same trees.\n\nWould we say the same sort of thing about persons? What if my wife kept disappearing, but then reappeared after a short interval, three feet to the right? I guess I would get used to it in time. I would not say that I had lots of different wives in quick succession, all of them just like the last one. This seems to show that what matters is the process, the way in which properties succeed one another, and we do not need any invisible literally continuing substance to hold the properties together. My wife, and what I value about my wife, is not an invisible unchanging substance. It is the constantly changing, often surprising, always inventive yet reliable and dependable, chain of memories, feelings, likes and dislikes, habits, and goals, that I have lived with for forty-odd years. So far, the process view seems right.\n\nHowever, lots of funny things can happen to processes. In Kafka's novel, The Metamorphosis, a man goes to bed one night and when he wakes up the next morning, he has changed into a large beetle. Whitehead would just say that lots of properties have changed, though what the man's wife would say is another question altogether. She would probably say that her husband was not as reliable as she had thought.\n\nIt is important for finding our way round in the world that people (collections of people-properties) do not suddenly change into collections of beetle-properties. We like to have regularities of change. Without them we could not do science, and it is doubtful whether we could have any long-term human relationships. Could you, for example, marry a man who might change into a beetle? Would you have to say, \"Well, I married him for better or for worse. It just looks like things have suddenly got much worse\"? Or should you have a new marriage vow: \"I marry you for better or worse, unless you turn into a beetle (or some other large and offensive object)\"? We like our processes to be fairly regular and predictable. Fortunately, they usually are.\n\nSo we might be tempted to say that persons are not permanent unchanging substances. They are complex chains of memories, perceptions, imaginings, evaluations, feelings, choices, and thoughts. They are processes. Buddhists manage to live with that thought without much trouble. Nevertheless, Buddhists notwithstanding, these processes are very special. They form unique kinds of unities, and in two main ways. It may turn out that these unities are an important part of what Descartes had in mind when he talked about \"mental substances\".\n\nFirst, there is the unity of many memories, thoughts, perceptions, and feelings in one experience, a unity of experience. All the memories I have are my memories. Even if by mistake I have someone else's memories, when I have them they enter into one present consciousness which includes many thoughts and feelings and perceptions which are all mine. There is a unity of co-presence in consciousness.\n\nSecond, each chain of thoughts and perceptions is separated from every other chain in such a way that it is not possible for anyone to get from one chain to another. Your experiences remain yours and my experiences remain mine, and, it seems, never the twain shall meet. There is a unity of succession in consciousness.\n\nMoreover, conscious events are not just passive experiences. They include active events like attending to or focussing on specific experiences, choosing to seek or avoid specific experiences, and forming intentions to bring future events about. If Kant is anywhere near right, the whole of conscious experience is the result of active acts of attention and interpretation, as well as of passive reception of data.\n\nGiven these facts, we can say that one and the same person continues to exist if there is a unitary consciousness of co-present elements that flows smoothly and continuously through time without gaps, in ways that have at least partly been consciously and intentionally envisaged, evaluated, and chosen. That will do for a first shot at saying what a person is, and what it is for a person to continue as the same person over time and through change. It will need a little amendment to take account of some \"gaps\" in consciousness that obviously exist \u2013 when we go to sleep, for example \u2013 and that will come later in this chapter.\n\nTechnically, for process philosophers, each moment of choice could be seen as a new act, performed by a new, never-before-existing agent. But such choices are made in view of all the knowledge and memories in one consciousness at that time, and as part of a future-directed process whose constituent events are parts of one unique chain of such unitary conscious complexes. Assuming that there are no breakdowns or delusions in such knowledge and memory, and that the pursuit of longer-term goals proceeds by a series of overlapping intermediate steps, we call the series of events that forms one discrete \"chain of events\" or process the acts and states of one and the same person.\n\nThere is in fact no discernible difference between there being a succession of agent-causes, linked together by being members of one unique chain of unitary conscious states, and there being one agent that continues as the \"same\" throughout the existence of such a chain. As long as each agent remembers one and only one immediately past consciousness and intends to modify one and only one immediately future consciousness, this is just what it means to say that the same agent continues through that chain of experiences. So it turns out that a process view of persons as successions of acts and experiences of a unique kind does not differ, when properly examined, from a Cartesian view that there is one continuing mental substance that thinks, feels, imagines, and perceives.\n\nYou cannot peel the substance off from its thinkings and feelings as though it could continue intact without them. That which thinks and feels is the succession of active causes that are normally linked by a continuous chain of conscious events in a way that no other causes are. As David Hume said, I cannot \"see\" a causal agent in addition to all the causal agencies of attending, choosing, and intending, that are the contents of my consciousness. But I do not need to do so, since I am the collection of all my responsible actions \u2013 a collection that is still being formed \u2013 and there is nothing else that I need or should want to be.\n\nOf course the normal process that leads us to speak of personal identity can go wrong. Bodies can radically change, by serious accidents or diseases, for example. The unity of co-presence in consciousness can fragment, as with so-called multiple personality, when different consciousnesses seem to connect with the same body. Memories and thoughts can become imagined or illogical, as when I seem to remember doing things I never did \u2013 which seems increasingly to happen as I get older. Actions can become wrongly directed toward their goals, or those goals may change and disappear from consciousness, so that I may obsessively perform actions over and over again, without any idea why.\n\nPersonal identity is a fragile process and often a matter of degree. There are no hard-edged boundaries that delineate when a person who has changed radically in some ways is or is not \"the same person\". The sort of body we have, the memories that haunt or delight us, the personality that disposes us to be good-humoured or acerbic, our basic evaluations and goals, are all important to being a human person. A major change in any of these aspects might make us wonder if we are the same person. A major change in most of them would probably incline us to say we are not. Nevertheless, there are some clear central, and it is to be hoped normal, cases where \"being the same person\" is not in doubt. We do not have to say that there is some invisible and undetectable substance that continues underneath all the changing states and properties. But in central cases, consciousness and responsible choice are important factors in determining identity of personhood. Mental factors, in other words, are crucially important in determining what persons are, in a way that material, bodily, factors are not, even if we do not want to be forced into the position of denying that humans who lack those factors are persons. Perhaps that may seem obvious. I hope it does. Yet it is something that tough-minded, eliminative materialists deny.\n\nMemories and persons\n\nSo far we have a rough idea of how persons can continue to exist at different times and places. Philosophers, however, will not be satisfied with such a rough idea. They will try to test it to destruction in order to undermine the claim that persons are constituted by chains of memories, thoughts, and feelings. They do this by imagining how they can mix such chains of conscious experiences up in various ways, until we no longer know whether we are talking about the same person or not. If we can do that, they say, this attempt to say that persons are primarily defined by their conscious mental lives will have collapsed.\n\nWhat would happen, for instance, if you took the memories of one person and put them into the mind of another person (perhaps by means of a brain transplant)? This may not seem impossible. It is quite possible that I may remember doing things that somebody else did (I may clearly remember winning Wimbledon, though Pete Sampras did so). The line between imagination, fantasy, and memory is very thin.\n\nIf I remember doing something I never did, have I become the person who did them (John Locke is often accused of thinking this)? Or if I remember doing something nobody ever did, have I become nobody?\n\nThe film Total Recall, starring Arnold Schwarzenegger and based on a story by Philip K. Dick, starts with the idea that future travel shops can save on their carbon footprint by not actually sending you anywhere. Instead, they implant memories that you have already been there in your brain. You pay for the memory of a holiday and, since you usually only remember the good bits, that is in many ways preferable to the hassle of actually travelling and possibly being disappointed.\n\nThe point of the film is that the hero changes his mind and escapes from the memory-implant machine. He then undergoes a series of increasingly alarming experiences, ending in him saving the planet from destruction. But this is exactly the adventure he had in fact paid for in the first place. So did he really escape or did he just falsely remember that he had escaped? We never find out. It's a pretty good story, even though my wife thinks I liked it because of the extreme violence in it. However, I know that it was the philosophical puzzles I liked (at least I think I know that, but I can't quite remember).\n\nSuppose I remember killing someone, though I actually never did so (though no one knows that). I might confess to murder and sincerely regret what I think I did. Should I be punished for it? In the state of California, a philosophically minded defence lawyer has argued that his client, who actually did murder somebody, should not be punished, because he was not a continuing self, and the agent who committed the murder had ceased to exist long ago. Since then, he had been replaced many times by a number of other agents, and the present agent standing in court had only just begun to exist. So he could not have murdered anyone. The defence did not succeed. But why not?\n\nIt is not in question that we often have false memories and that we lose many memories. Do we have to remember something accurately before we can be punished or rewarded for it? Or could we receive a Nobel prize for a discovery we cannot remember making and that perhaps we have forgotten all about? The Oxford philosopher Derek Parfit imagines replacing half of his brain with half the brain of Greta Garbo. Would he then receive half of a lifetime achievement award for what she did? Or would it be he, or half of him, who had done it? Or would she never receive the award, because she had turned into someone else, namely Greta Parfit or possibly Derek Garbo?\n\nThe mind boggles. Normally, if I work hard for a long time I may deserve a reward, and if I have done many things that are harmful to others, I may deserve some sort of penalty. We do think about the future, and about the consequences of our actions. And we think about our future, not just somebody's future. If I save hard for my retirement, I will not be happy if somebody else gets my pension. The point is: I do not do things that will cause some good states later, whoever experiences those states. I do things, at least when I am being prudent rather than altruistic, that will cause good for an agent who is continuous with me now. The experience in future will include memories of me now and will be able to connect a whole series of actions with one continuing set of overlapping experiences.\n\nIn fact, no one could ever perform an action for their own future well-being if they did not believe they would exist in the future. But when we analyse it, what distinguishes me in the future is that there is a temporal continuity between successive sets of private experiences, to which no one else has direct access. \"I\" am constituted by a present unique set of experiences and actions, which continues into the future. So I realize that what happens to this present member of the chain of experiences is a consequence of what some past members of the chain have done. Conversely, what I do now will have consequences for some future agent\/experient who is uniquely connected in time to this chain.\n\nMost of the curious cases that philosophers have thought up are cases in which there is some pathological malfunction in the chain. It is imagined that some members of one mental chain are spliced into another mental chain. Those members will not have been caused by previous free and responsible acts in this chain, but they may have been caused by prior free acts in another chain from which they came.\n\nThe full story must of course include the fact that the chains have been tampered with, and that most instances of tampering will be criminal acts of psychological damage. A normally healthy person will remember events in his mental sequence and will be able to formulate long-term plans for future events in the sequence and act to realize them. Amnesia is an illness, which should be treated if possible, and which should not affect the treatment of present persons as (to the appropriate extent) a product of their previous free acts. Having false memories is also an illness, and since they do not represent genuine past causes, they have no implications for the present treatment of persons.\n\nIf, by some mischance, you have someone else's memories, these should usually be treated as false memories. But if you can imagine two memory-chains being forcibly merged, like Parfit and Garbo, you would have one person with two distinct parallel sets of memories. If, unlike Parfit and Garbo, one was a serial killer and the other a saint, we would have a problem. Maybe we should think that both past persons have died, and we should just start again. We can only hope such a situation would never arise, since it would amount to penalizing the saint's past acts and overlooking those of the serial killer. But who said human life was perfectly fair anyway? We just have to muddle through doing the best we can. Human justice may work in normal cases, but break down in borderline, extremely improbable, situations.\n\nMy conclusion is that fantastic scenarios of memory-change do not undermine the claim that persons are largely constituted by continuous chains of memories, thoughts, and feelings, and that these are much more important than considerations of bodily continuity. This is what underlies our normal notions of moral responsibility, and the idea that we can develop our characters by attention and effort (or of course destroy them by distraction and indolence). It underpins our normal belief that human persons have a distinctive, perhaps unique, moral dignity and value.\n\nPersons and bodies\n\nI have stressed the importance of temporal continuity of experiences to our normal idea of continuing personhood. We also think that spatial continuity is important \u2013 there is normally a continuous spatial track between my body now and my body as it used to be, even though those bodies may look and feel very different. I have suggested, however, that strict spatial continuity may not be essential to my continuing to be the same person throughout a stretch of time. My body may disappear for periods of time, or it may move three feet to the right instantaneously, and I would still be the same person, as long as there is not more than one body that claims to be me.\n\nCould we also dispense with strict temporal continuity? Could there be temporal gaps, so that I could altogether cease to exist for half an hour and then come back into existence again as the same person? In a sense, this already happens, because when I go to sleep there is a gap in the stream of conscious experiences, but that does not usually lead me to say that I die every night and somebody else wakes up in the morning. Of course my body has been there during the night. But I do not have to check that I have the same body before I can be sure exactly who I am when I wake up. I do not feel the need to employ people to keep watching me all night in case my body disappears. As far as I am concerned, my body might have disappeared during the night, and I would never notice. So my sense of being the same person cannot depend just on my having the same body all through the night (or a continuous series of closely connected bodies either). So it seems I can live with temporal gaps.\n\nIf that could happen for an hour or a night, what about a year? Or a million years? Just to make it worse, could I cease to exist for a million years, and reappear as the same person in quite a different body? Or would that be pushing the notion of personal identity too far?\n\nBefore we dismiss such an idea on the grounds that it is not normal, we need to remember that there is a very widespread human belief in reincarnation. Reincarnation offers a solution to the Parfit\u2013Garbo problem. In some other life the merged persons, Parfit and Garbo, would demerge again, and both would get their due deserts in different bodies. But does it make sense to say that the same person might live again in a different body? Is reincarnation possible?\n\nThe philosopher Bernard Williams has argued that reincarnation is not even logically possible. Neither is the rather similar idea of resurrection \u2013 the idea that we might be resurrected, perhaps in a rather different or at least (we hope) reconstructed and improved body, after a huge temporal gap. It would need a very strong argument to show that reincarnation or resurrection could not possibly occur. But Williams does bring out some problems with the idea.\n\nFor instance, there are at this moment about three hundred people who claim that they used to be Napoleon Bonaparte. When, if ever, would we believe them? One vaguely alarming possibility is that they could all be Napoleon. The General has split into three hundred copies, rather like a human stem cell dividing into thousands of copies.\n\nSome quantum physicists believe this actually happens, since every time one possible future is realized in one universe, an alternative possible future is realized in another universe. So Napoleon divided every time he did anything, and there are now millions of Napoleons in existence. But at least they are in different universes, and in any case they are all now dead (unless in some far away universe one Napoleon discovered the secret of eternal life).\n\nBernard Williams argued that none of these would be Napoleon, because you cannot have two identical things existing at the same time. Yet all these people would claim to be identical with the original Napoleon, and therefore they would all be identical with each other.\n\nBut what if one of them was the real Napoleon and all the others were fakes? That, says Williams, is impossible. Because of the lack of bodily continuity, they must all be fakes. And, he says, following this up with remorseless logic, if they are all fakes, then anyone who claims to be Napoleon but exists at a different time from Napoleon must be a fake too. The trouble is, that includes Napoleon himself, who, when he was forty, existed at a different time from Napoleon when he was thirteen. So Napoleon is a fake copy of himself. To be more precise, he is a long series of fake copies of himself. Not only is he a fake, he is not even a very good fake, because he looked quite different at forty than he had done at thirteen. He would not fool anybody.\n\nPeople, as they get older, are never very good copies of their earlier selves. After a certain age, parts begin to drop off, and wrinkles begin to appear where there were no wrinkles in the original (or, if you live in America, parts are added on where there were no parts before). Of course reincarnated people are not physical copies at all, and the three hundred reborn Napoleons look nothing like Napoleon. Quite a large number of them are women, as it happens. So it looks as though I can continue to be the same person even if my body changes considerably. Adding a few temporal gaps would not seem to make such a crucial difference. So reincarnation and resurrection, the change of bodies with physical and temporal gaps, seem to be at least logically possible.\n\nBut how can you know that one person is really the same as a person who used to exist some time ago, perhaps in a different body? Buddhists are the greatest experts at detecting reincarnations, as they have to find reincarnated lamas every few years to run their monasteries. They test candidates by seeing whether they recognize objects or places that dead lamas had known well. And they look out for especially wise words, good behaviour, and mental calm and mindfulness, which suggest an advanced spiritual state suitable for a dead lama.\n\nTheir appeal is to memories and to unusual mental dispositions suggesting wisdom, compassion, and controlled mindfulness. The interesting thing is that Buddhists specifically deny that there is any permanent self which endures through successive incarnations. What they are interested in is a succession of mental acts and events which, they believe, is the result of long and strenuous moral striving and mental training on the part of some past successions of mental events.\n\nDoes it really matter whose mental strivings caused the present set of mental events to exist? The idea is that the present succession does not just come into existence for no reason. It has a cause or a long series of causes. Those causes are not primarily physical causes. They are successive acts of attention and effort, of attending and intending, which have gradually built up a tendency to generate mental states of deep understanding and compassion for all beings. Such states do not just occur. They are brought about by intentional actions. Other chains of mental action have brought about states of ignorance and indifference to or even hatred of other sentient beings.\n\nAll chains of intentional action bring about states of increased or decreased understanding and empathy. To comprehend the nature of such states we need to comprehend how and why they have come into being, for they carry within themselves the causal history of their origin. My present understanding of the world is essentially an understanding that has been developed through a long succession of past acts of insight or obtuseness, and I do not fully understand what I am (my co-present consciousness) unless I understand how this consciousness has come to be.\n\nAs Proust, or someone very like Proust, might say, the reason I like the taste of little cakes like madeleines may be found in experiences in the past which lie in some deep recess of my memory. A full self-understanding would unlock that past succession of memories \u2013 which is no doubt why the Enlightened One, the Buddha, is said to have full recollection of his many lives.\n\nSo I do not just say, \"Someone (it does not matter who) in the past meditated for forty years, and I am now reaping the benefits of that by being calm and collected.\" I say, \"I am continuing a particular causal succession of mental acts and experiences, even though I have a different body and there has been a long temporal interval between the past succession and its present continuation.\" Someone can be Napoleon now, if their dispositions and mental states have in fact been caused by and continue in a different context the succession of mental acts which made up the inner mental life of Napoleon.\n\nThere could be three hundred incarnated Napoleons. But if there were, something would have gone wrong with the causal process. The reason we reject all these putative Napoleons is that they do not have the memories, thoughts, and intentions of the original Napoleon, and in addition they suffer from other behavioural and mental problems that render them subject to delusions. But if they were perfectly capable of ordering their own lives sensibly, and if they did remember things that only Napoleon could have known, we might begin to be impressed. It is because the evidence is not good enough that many people reject reincarnation or resurrection of the dead, not because it is a logical impossibility.\n\nIf, however, there were three hundred reincarnated Napoleons, we would have good reason to doubt the intelligibility of the causal processes of the universe. Whereas the life of Napoleon, insofar as it was the result of free and intentional mental acts, should have generated just one megalomaniac of militaristic and domineering disposition, it would instead have generated three hundred such unfortunate individuals. While that is logically possible, that would be morally unfair both to all the three hundred people who had to suffer because of the acts of just the one previous Napoleon and to everybody who had to put up with these incorrectly processed Napoleons.\n\nThe priority of the mental\n\nThings can go wrong with the universe, and if they do go wrong, we will just have to put up with it. But if things do not go radically wrong, it makes sense to say that I continue to be the same person if I have the same knowledge, memories, thoughts, dispositions, and intentions, even if there were spatial and temporal gaps between the person I used to be and the person I now am, and as long as there exists no other person now who is identical with me (which would complicate, but not completely undermine, the story).\n\nSuch a possibility gives priority to the mental over the physical or bodily. It denies the necessity of having some continuing identical substance to join two sets of conscious events together. And it places enormous importance on the conscious free intentions and efforts of sentient beings, which will affect the future in fundamental ways. It allows for the possibility of reincarnation or of other forms of life after bodily death. But of course its cogency depends upon there really being the sort of mental causality that the theory posits, and that is not at all obvious.\n\nWhether or not we accept the possibility of reincarnation or resurrection (though it is difficult to dismiss them as absolutely impossible), we can hold that at least in this life chains of mental causality are normally (and perhaps always) mediated through temporally continuous physical bodies. This takes account of the fact that fully free intentional acts are probably much rarer in human life than we think. Most of our acts may well be the result of physical hard-wiring and habits established over millennia of evolutionary trial and error. Most of our characters and dispositions may well be due to the adventitious mixing of our parents' DNA, rather than to purely mental causes in the life of some person in the past. So while mental effort does have real effects on our mental life, and while that is of the utmost moral importance, humans are also socially and physically embodied and are subject to physical chains of causality that make any direct causal link between some past life and a present life very difficult to establish. Conscious intention is only one part, though morally by far the most important, of that complex of elements which goes to form a complete human person.\n\nPart of the moral lesson here is that we should never say of some poor or relatively disabled person that it is \"their fault\" they are that way, because it must be due to something wrong they did in some past life. If that were so, we would expect that all millionaires would be past saints and all paupers would be past serial killers. Without trying to be too judgmental, that does not look quite right.\n\nThe conclusion of this Napoleonic episode is that we might agree that the world is made up of successions of events and acts, of processes, ordered in accordance with complex causal laws. Yet in the case of human persons these successions fall into distinct groups of cumulative subjective experiences. And a fairly clear way of distinguishing one such group is to say that, since its parts are subjective experiences, they are the experiences of one common subject of experiences, the \"subject\" being constituted precisely by that unique unity which binds these experiences together in one succession.\n\nWe might also agree that mental acts are very important parts of such processes, and that mental acts are different from passive objects or states and from purely physical or material events. Mental acts, unlike physical acts, normally form successions of causally related private experiences and intentions which work out progressively over time. Constantly renewed mental discipline and effort, attention and intention, result in states of mind and in actions that otherwise would not have existed.\n\nJust as there is one \"subject\" of experiences, which binds them together in a uniquely subjective way, so we can say that there is one creative causal agent that continues to act over time through a succession of experiences. As long as we do not say that this subject and agent of experiences and actions exists separately and quite apart from the flow of events that constitute the life of the mind, it looks as though a \"Cartesian\" view of mental substance (as the subject of experiences and acts) is not in opposition to a process view of personal life. In fact, it is the concept of a continuing mental substance, rightly understood, that enables us to distinguish different sets of mental events from each other, and to see mental events as (at least in part) cumulatively and intentionally built up over time.\n\nThis is the very opposite of Ryle's concept of mind, because it privileges private states over publicly observable ones, yet it seems to follow from a consideration of what is morally important about personal identity.\n\n## Chapter Six\n\n## The place of human minds in the cosmos\n\nIt is important to stress (as Descartes mostly did) that human minds are fully integrated into an evolving material universe and do not inhabit some separate mental world. This very naturally leads to seeing the universe as a purposive process aimed at the progressive realization of intrinsic and objective values in and by finite minds within the universe. Human persons have a positive and responsible role to play within this process, so they are integrated into a material, bodily context and that is their proper environment.\n\nI have argued that the mental is of primary importance in human personhood, but I am in no doubt that in humans the mental and the physical are mixed together in a closely integrated way, and both are implicated in the causal network that drives the world into the future. The problem is to say what this integration is. Whitehead's solution, shared by many philosophically minded scientists, is to say that the mental is the inner life of the physical. All physical phenomena have an inner or \"private\" aspect, in addition to what can be observed by public perception.\n\nThere are process thinkers, like Charles Hartshorne, who have argued that even electrons have free will, and they decide when to jump from one orbit to another within the atom. But it is hard to see why they should jump at one time rather than another or what the point of jumping is anyway. Intelligent consciousness as we know it does seem to depend on the very complex structure of brains, and on the interaction of millions of neurons. Electrons have no complex structure, and they seem to have rather limited opportunities for personal development or for really interesting and adventurous choices.\n\nNevertheless, it can seem plausible to think that consciousness cannot just arise out of nowhere, and be joined onto a brain in a completely accidental and unpredictable way. For many scientists it makes more sense to see consciousness as a natural development out of simpler elements, as an unfolding of potentialities inherent in matter from the first. Consciousness is the phenomenal appearing of things, together with thoughts that interpret those appearances. So maybe even the simplest sorts of things have some sort of phenomenal appearing (some way in which objects are represented), and some sort of interpretative or at least subjective reaction to such appearing. That is certainly what Whitehead thinks.\n\nWe do not have to be committed to the rather elaborate edifice of process philosophy to be attracted to this way of seeing consciousness as a natural development of simpler properties inherent in all material things. Philosophers like Rom Harre of Oxford and scientists like John Polkinghorne of Cambridge accept a very similar view. That is to say, as well as their publicly observable appearance, the basic elements of matter have an inner structure that is not publicly observable, but that drives their causal route through space-time. Materialists (Theory 3) might even say as much, at least insofar as they accept that many forms of material reality (at the subatomic level, for instance) are inaccessible to sense-observation. But materialists wish to remove any even embryonic hint of consciousness, value or purpose from matter. Epiphenomenalists (Theory 5) accept that consciousness emerges from matter, but they typically deny that minds could possibly exist without matter, and that minds have any real causal role to play in the cosmos. So again minds have to emerge by accident, or unintentionally, from a material universe without consciousness or purpose.\n\nIdealist philosophers do not usually think that there are little fully conscious minds driving atoms around. But they do think that even \"material\" processes are not totally random, directionless, mechanistic or wholly determined by absolute and arbitrary laws. Purpose or teleology may be built into nature from the first. The material universe is perhaps more like an organism than like a repetitive machine. Whereas an older generation of scientists and philosophers thought the universe was rather like a watch, many now regard the universe as more like a large organism. It grows and develops, and its first stages can only be properly understood when its completely developed state is perceived.\n\nA human embryo does not unexpectedly and accidentally become an adult person, and it can only be properly understood as a potential adult. So we might think that the primitive elements \u2013 be they quarks or superstrings or something as yet undiscovered \u2013 of the universe do not unexpectedly clump into atoms, which surprisingly form molecules, which accidentally generate proteins, which unforeseeably build organisms, which by pure chance produce brains and societies of organic beings, so that the whole present universe is a totally unexpected accident. On the organic view, this trajectory of development, of increasingly integrated complexity, producing new sorts of properties, and eventually the ability to comprehend and consciously shape the future of the universe, is implicit in the universe at the moment of the Big Bang or in whatever gives rise to that primordial explosion.\n\nFrom this point of view, it is a basic mistake of reductive materialism to try to explain everything in terms of its simplest elements \u2013 as though a large enough group of such simple elements just had to be mixed up at random for a long time, and would then produce brains, thoughts, and the theory of relativity. The main alternative to such reductionism is holistic explanation. Simple elements are explained in terms of the wholes of which they are constituent parts, and of the fullest realization of all the possibilities of a dynamic and developing system.\n\nIt remains a deep mystery how a fully realized society of intelligent and purposive agents (which is what we actually have in this corner of the universe) can be potential in the simple state of infinite density and mass which was the Big Bang. But it may be helpful to think of the space-time universe, not just as a set of separate parts, but as a completed whole.\n\nThe universe does not consist of discrete temporal slices, all isolated in their own little bubbles of time. Causal tracks and connections extend back and forward through time, and a present moment of consciousness can contain echoes of the past and premonitions or anticipations of the future. So we might see the universe not as a set of atomistic time-slices accidentally stuck together, but as one interconnected or entangled space-time whole. We do not see what objects are by seeing just one temporal slice of their existence. That would be like trying to understand a person by looking hard at them when they are asleep. We need to see them from beginning to end of their temporal existence and within the whole context in which they exist.\n\nAn \"object\" is what physicists call a world-line in space-time. The early stages of its temporal existence are properly seen as parts of a whole world-line which adequately describe the object only when it is seen as a whole. Albert Einstein thought that the whole of time, with all its constituent world-lines, actually exists, from the first moment to the last. We only seem to move through it one moment at a time. In reality both our past and our future are timelessly existent and never come into being or pass away. He apparently found great consolation in the fact that his dead friends were not really non-existent. They all exist timelessly, and they only seem to be dead to the rest of us.\n\nI suspect that you would have to be a very advanced mathematical physicist to be consoled by the thought that dead people all really exist, because time is an illusion. Most of us will continue to be more impressed by the thought that we will never meet them again in time. Even if in some strange way we could meet them outside time, we would still not be able to talk to them, since talking takes time, and if there was no time, there would be no time for a good conversation. We would just have to stare at each other, frozen in changeless immobility. That might be all right for Einstein, but it would be very frustrating for most people, especially for the talkative ones.\n\nA \"supernatural\" origin for the cosmos\n\nWhether the passing of time is real or not, it might be just wrong to think that at the moment of the Big Bang nothing existed except a very simple physical state of infinite density. Most cosmologists suppose that there would also have been a whole set of quantum laws, and perhaps those very complex and precise balances of energy that would constitute what is called a \"quantum vacuum\". In modern cosmology, there is something outside space-time, from which space-time originates, which has a rich mathematical structure. There is, in other words, a \"supernatural\" reality; a more fundamental layer of reality beyond space-time.\n\nMaterialists are really shocked at this turn of events. Some of them simply ignore it and say that it is not really science. David Hume held that it was absurd to think that the puny human mind could speculate about the \"origin of worlds\", when nobody had ever observed such things. Nevertheless, there are chairs of this \"absurdity\" in most major Western universities.\n\nWhat is that supernatural reality like? Some think that at least the basic laws and principles that existed at the beginning of the universe must have been very simple \u2013 largely because, they think, if we are going to explain the origin of the universe, the explanation will have to be simpler than what it explains or it will not be an explanation.\n\nThis, however, is plainly false. It is doubtful if it even makes sense. Quantum field theory explains the behaviour of events in a particle accelerator. But in what sense is quantum theory \"simpler\" than very small things smashing into one another at immense speeds in a large hollow tube? Of course it is nice to have an elegant and simple theory if it does the same work as an ugly and complicated one. But the important thing about a scientific theory is that it produces general equations from which the behaviour of physical particles can be predicted. If those equations turn out to be complex \u2013 and they are already so complex that very few people can even understand them \u2013 that is life!\n\nWe have no right to expect that there will be stateable equations that govern all physical behaviour or that they will be simple and elegant \u2013 though we have been very lucky to find there are many such equations (like Newton's laws of motion). We are in no position to lay it down as a necessary truth that there will be simple elegant laws to explain the Big Bang. What we need are laws and principles that will explain the whole space-time structure of the universe, not just some initial state of the universe \u2013 which may turn out to be the least interesting thing in the universe, except insofar as it is the starting-point for the whole cosmic story.\n\nThere might, then, be laws that help to explain how and why this universe was generated 13.7 billion years ago, in terms of the much later realization of complex states that were only implicit in the first state of the universe. Such laws would set out a direction of development, culminating in a completion of initial potencies, a set of possible goals that can be seen as fulfilments of tendencies inherent in the universe itself.\n\nA goal is a future state that is rationally choosable by an intelligent conscious being. A fully intelligent conscious being is one that would choose states that are intrinsically worthwhile \u2013 good just for their own sake and not for any other reason. If anything is ever worth choosing, then there must be some states that are intrinsically worthwhile. If we try to think of the sorts of goals for a universe that could be rationally choosable, we might think of such things as the ability of a material universe to come to understand and shape its own being and to generate communities of organisms that can themselves, as Aristotle put it, find happiness in the pursuit of states and processes that are worthwhile for their own sake.\n\nWe might think of a universe generating beings with conscious capacities such as those for creativity, understanding, appreciation, and free relationship. If this might indeed be the fullest flourishing of the cosmic organism, then the initial laws of the universe will render it inevitable that this, or some very similar, cosmic organism will exist and will come into full bloom before, like all organisms, it decays and dies. The initial laws will, in that case, not simply exist without any point or purpose. They will have a good reason for existing \u2013 the reason being precisely the generation of intrinsically worthwhile states that can be known, appreciated, and produced by intelligent agents. The initial laws will be, in other words, purposive or goal-directed. Purpose has been ejected from scientific explanations for almost four hundred years. Perhaps, in modern cosmology and in idealist philosophy, it is making a comeback.\n\nWhat I have tried to do in this chapter is to suggest how conscious personal life and the material structure of the universe fit together in a coherent way if we suppose that the physical universe has the purpose of producing personal consciousness as the natural realization of its inherent and original capacities. Consciousness is not just an alien substance injected into the material universe at an arbitrary point \u2013 a picture which Cartesian dualism, if interpreted unkindly, may suggest. Rather, consciousness results from the natural generation of capacities inherent in the structure of matter itself, as it develops forms of organized complexity over time. A picture of cosmic evolution that portrays responsible and intelligent minds as a natural, possibly inevitable, outcome of the growth of an organic material universe could be the key to understanding how spirit and flesh, mind and matter, soul and body, can be integrally intertwined, and yet how the primacy of spirit, as the ultimate purposive driving force of an evolving universe, can be maintained.\n\nCould the universe have a goal?\n\nAll this could be seen of course as a piece of human arrogance. Those values of creativity, understanding, and so on \u2013 are they not just the preferences of a small number of rather prissy human beings, projected improbably onto the universe? If you walk into your local bar and say, \"What I'm looking for is creativity, understanding, appreciation, and free relationship,\" the odds are that everybody will ostentatiously look the other way. With luck, somebody might say, \"Well we have darts, and we can probably find some dominoes if you like. How about some pickled eggs?\"\n\nIs it really plausible to think that the universe, so vast in extent, exists just to produce beings like us? Did it really take 13.7 billion years just to produce Big Brother and the Wimbledon Tennis Championships? There aren't, it may be said, any universal or cosmic values. There are just things that people in bars, bus queues, shopping malls, and offices happen to like at the time. And that hardly seems grand enough to merit the title of the goal of the cosmic process, the ultimate reason for the Big Bang. Or would you really want to say that the answer to the question \"Why does the universe exist?\" is \"So that I can enjoy a pint of beer\"?\n\nActually, you might. Some physicists have estimated that, starting from the initial Big Bang, and allowing the known laws of nature to take their course and gradually build up heavy atoms, complex molecules, and central nervous systems, the emergence of intelligent carbon-based life-forms would have taken about 13.7 billion years. Since the Big Bang, the universe has been expanding as fast as it can, so that by now it extends for billions of light-years. Therefore, the reason there is so much empty space around is that it has to exist in order to produce me (assuming that I am a pretty good example of an intelligent life-form. I am certainly a good example of a beer-drinking life-form).\n\nIt seems that I am much more important to the universe than you might think. Or if you think I am a rather disappointing specimen, hardly worth 13.7 billion years of effort, you could say that at least intelligent life is worth having, and if it takes billions of years to produce it, does that really matter? The process itself is not pointless; it is not just a means to an end. It has immense beauty and elegance, and it is worthwhile for its own sake. But that beauty and elegance can only be appreciated when conscious life-forms come into existence (unless there were consciousnesses existing outside the universe, which could appreciate its development).\n\nWe should not think that it took the birth and death of millions of star-systems and the extermination of millions of now extinct organisms, just to produce one man drinking beer in a pub. However, in one sense that man is more valuable than all those galaxies, because there was no one to appreciate their beauty, whereas at least he enjoys his pint. The greatest beauty that passes unknown and unappreciated is in one sense of less value than the enjoyment of a pint of beer, just because there is no one to put a value on that beauty. There is no one who values it, and so in a real sense it has no value \u2013 until it is known and appreciated for what it is. There are no actual values unless someone values them. So consciousness is necessary for there to be any actual values in the universe.\n\nStill, I do not suppose anyone but Homer Simpson would think that drinking beer is the greatest value in the universe. Maybe humans are just one small primitive part of cosmic history. What would be really impressive as a goal for the universe would be the genesis of a consciousness, or perhaps a society of consciousnesses, that understood and could control the cosmos itself, which could devise endless creative purposes and experience unlimited types and intensities of feeling. That such an existence would be of value is not just a projection of a few arbitrary human likes and dislikes onto the universe. It is a goal, a desirable state, which any conscious intelligent being would choose, whether it drank beer or not. It might prefer beer in its present state, but it could at least see, when sober, that a really intelligent being would choose the goal of a cosmic consciousness that could be endlessly creative and that could experience infinite states of enjoyment and beauty \u2013 and then it could have endless kinds of beer whenever it wanted, without any of the ill-effects that beer usually has.\n\nIf we can see humans as only the beginning of the development of intelligent consciousness in the universe, consciousness that will exist for untold billions of years, then it makes more sense to see the long development of cosmic evolution as oriented toward a worthwhile goal. We may think we exist on the last half-page of the many-volume book that is the history of the universe. But if there are even more volumes still to come, that changes the picture entirely. Admittedly this is speculative. It is a sort of ideal utopianism, constructed by thinking what things would be like if there really were ultimately worthwhile goals for the universe that all fully informed intelligent beings would desire. Things might not be like that, though I rather hope they are.\n\nIdealists need not be committed to such a utopian and ultra-optimistic idea. In the Indian traditions there is an infinite series of universes, each of which eventually ends with an age of increasing chaos and evil (a Kali Yuga). The goals of the cosmos are realized only beyond the cosmos, either in absorption into the Supreme Self or in some spiritual realm rather like Western ideas of paradise. Even Hegel, though he remained typically ambiguous, speculated that the final goal of \"reconciliation with Absolute Spirit\" might not exist in this physical cosmos. And Whitehead explicitly denies that there is any realizable final goal of the universe.\n\nWhitehead thinks of the basic directional laws of the cosmos and the states to which they lead as primordial \"aims\": a timeless set of possible goals and of possible tracks to realizing them in many ways and in many forms. He does not think of there being just one final consummating and unsurpassable state. He certainly does not think that human life is it. He rather thinks of the process of creativity as being unending. It is also always compromised by conflict between different organic forms that pursue different aims, and by the fact that some aims will be so limited as to be destructive rather than creative.\n\nWhichever of these varieties of idealism we prefer, for all of them chance, impersonal laws of nature, and creative freedom play an important part in the complex processes of the universe. There may be no final perfection for the universe. Its tendency toward truth, beauty, and goodness is real, but may be always qualified by the existence of elements which resist that tendency, though they too are parts of the cosmic structure. This may be thought of as a modern version of the world of Platonic \"forms\" being imperfectly realized in the material world, though it differs by stressing the importance of time and of evolutionary development, thereby giving the universe a history of which Plato could scarcely have conceived.\n\n## Chapter Seven\n\n## Dual\u2013aspect idealism\n\nIf there is purposive causality in the universe, it will obviously make an enormous difference to what happens. Events will not occur just by chance or accident, but the whole universe will be directed toward the existence of persons and the realization of personal values. Since the natural sciences normally set aside questions of value, they cannot as such establish whether the universe is directed toward the existence of specific values or not. But the findings of the natural sciences are certainly relevant to the question of whether there is direction or progress of any sort in the observed universe. It seems to me that it is a reasonable hypothesis that there is. If so, this helps to confirm the form of idealism I am defending, which could be called dual-aspect idealism, because it stresses the importance of the material aspect as a means of allowing the potentialities of mind to be expressed.\n\nMany modern evolutionary theorists are so opposed to the notion of direction and purpose in cosmic history that they insist on stressing the totally fortuitous and random nature of biological evolution. This is rather odd, because they mostly also think that the basic laws of physics are not at all fortuitous and random. Indeed, many of them are physical determinists and think that at a physical level things must obey the laws of nature and could not be other than they actually are. This is the opposite of randomness!\n\nThere seem to be four main reasons for this resistance to directionality. First, there is too much waste and suffering in evolution for it to be planned. Second, many organisms like bacteria do not evolve, but stay happily as they are, so there does not seem to be a \"universal striving\" in evolution. Third, the evolution of humans depends on a number of freak accidents, like the meteor impact that may have destroyed the dinosaurs and other disasters that turned out well for humans but rather badly for every other form of life. And fourth, the mechanisms of Darwinian evolution, random mutation and natural selection, do not seem to allow for any sort of intelligent or purposive selection. Ideas of direction or purpose smack too much of \"vitalism\", the exploded view that there is some sort of \"life force\" at work in addition to natural mechanisms in evolution.\n\nThis is precisely where basic philosophical beliefs influence what are supposed to be strictly scientific theories. I am not against such influence. On the contrary I am all for it. But in my opinion this is the wrong influence from a false philosophical theory \u2013 or at the very least, from a highly disputed one.\n\nThese arguments of evolutionary naturalism can be very easily dismissed. In response to the first argument, we can say that talk of waste and suffering is completely irrelevant to the question of whether there is direction in evolution. If there is a direction, it is from simplicity to complexity, from unconsciousness to consciousness, from lack of any appreciation of value to the understanding and appreciation of many moral, intellectual, and aesthetic values, and from chance and necessity to intelligent purpose. We can observe this progress in the evolution of human beings from unconscious and unintelligent stardust.\n\nThere is no reason why such a progress should be without blind alleys and eddies which do not lead in the requisite direction, as long as the process as a whole inevitably leads to the \"higher\" states. In fact, one rational structure for such progress is a system which generates a number of alternative possible tracks, where all possibilities are tried and only some lead to cumulative progressive tracks. That may lead to waste and suffering, but that does not make the structure less rational.\n\nIn response to the second argument, it is not at all necessary that every item in the structure should be seeking to mutate in a positive direction. Indeed, it is necessary that most items should not mutate, but should preserve their structure, to provide a solid base on which new mutations can be built. The structure must provide, therefore, for the repetition of dependable elements, with mutations that do not lead to great structural changes. But some, perhaps a small number, relatively speaking, of mutations must lead to structural changes \u2013 and that is what we see.\n\nWhat about the third argument, stressing the role of genuine accident in the evolutionary process? Quantum physics is usually taken to undermine a wholly deterministic interpretation of the laws of nature. But of course it does not undermine the fact that there are laws of nature. What it suggests is that the laws will in general produce predictable results, but at the subatomic level we will have to work with probabilities where not all details are predictable \u2013 and sometimes this will result in larger scale unpredictabilities. In other words, there are elements of genuine randomness, but even they are governed by laws of probability, and most probabilities cancel out at higher levels to leave the inevitability of the general processes of nature intact. Only occasionally, and usually at times not directly observable by humans, will genuine alternatives at the macro level appear. But some of those occasions might have decisive and dramatic consequences.\n\nIt is also a generally accepted rule of quantum physics that physical phenomena are entangled, so that basic physical elements affect one another in non-local ways, and no phenomenon can be considered totally in isolation from other phenomena in the universe. This fact also places constraints on what possibilities are open within the system. As Michio Kaku says, \"Einstein often asked himself whether God had any choice in creating the universe. According to superstring theorists, once we demand a unification of quantum theory and general relativity, God had no choice.\" This is a rather picturesque way of saying that if there are going to be intelligent carbon-based organisms in the universe, then the general structure of the universe, including some of its unpleasant features, could not have been other than it is.\n\nThe limits of human observation mean that we are unlikely ever to know in detail what the necessary constraints on the universe are. We cannot observe the whole universe every time we want to assess the probability of an event occurring, and so we will never know all the laws and constraints that govern the occurrence of any event, much less the structure of the whole universe.\n\nSuppose, then, as a general hypothesis, that there is a basic teleology in nature: a dynamic process, involving elements of chance and also a large element of repetitive law-like behaviour, nevertheless ensures that specific \"developed\" states will inevitably be realized in the cosmic process. This is not the idea of some sort of person interfering in the process from outside. It is the idea that the process itself has inherent direction and goal, set by the fundamental and timeless laws of the cosmos. The existence of such a teleology will make a difference to specific events that occur, for it is part of the laws of nature, not an interference with them.\n\nThe demise of the dinosaurs\n\nWhen an astronomer says that the formation of planet earth and the genesis of organic life on its surface was the result of a fantastic series of hugely improbable events, this has to be seen in the light of the fact that the cosmos may be set up inevitably to realize organic life forms. In that light it is not at all improbable that events conducive to the genesis of life should occur, though the presence of chance elements does mean that some events destructive of life and purpose will also occur.\n\nThe choice, in other words, is not between perfect design and complete chance. It seems more likely that the cosmos moves inevitably toward intelligent consciousness through a partly free (and because free, partly undetermined and therefore partly random) and partly determined (because otherwise there would be no reliable structure) creative process of trial and error.\n\nThere was nobody who specifically planned that a meteor should be dispatched to exterminate the dinosaurs 65 million years ago (assuming that it was a meteor that wiped out the dinosaurs) and open the way for humans to evolve. But it may not have been just a by-product, unfortunate for dinosaurs, of totally blind forces of nature. Dinosaurs, after all, flourished for many millions of years. They had their day in the sun, and perhaps they existed for much longer than humans may. But they did not seem to be going anywhere, and they probably ate anything that looked as if it might be going somewhere. They were blocking the evolutionary development of the universe. They were, to put it bluntly, a creative experiment that had run out of steam. To put it more kindly, the process had succeeded in producing impressively large reptiles, but they had reached a dead end. If they had got any larger, their brains would no longer have been able to communicate with their legs in time to run after their prey.\n\nWe may think of the destruction of the dinosaurs as a state that presented itself as possible but not inevitable on one of those relatively rare occasions when decisive selections between options are likely. What is required to make sense of this is to think of present situations as containing a number of alternative futures. There will be many constraints on which alternatives can be selected, and on which of them might have decisive consequences. But one causal factor in these complex situations, idealists think, will be the general teleological tendency of the cosmos. This of course will be too much for materialists, who have a paranoid fear of any causal factors in addition to the \"blind\" laws of nature.\n\nEven materialists have to admit, however, that according to quantum theory there can be alternative futures, and that we do not know the causal principles that select between them. Materialists can always say that these alternatives cancel out on the molecular scale, that purely Darwinian principles are quite sufficient to explain all evolutionary changes, and that anyway there are no \"mystical\" non-Darwinian principles, so there! This is the fourth argument that sceptics about purpose often produce.\n\nAn appropriate reply is that quantum indeterminacies do not always cancel out, and they may in the right conditions amplify into major changes (as with the \"butterfly effect\", which keeps weather forecasters in a job even when they get everything wrong). Indeterminacy may not be confined to the subatomic world. Quantum theory has shown that physics does not have to adopt a wholly deterministic view of nature. So there may be indeterminacies \u2013 that is, alternative futures without one of them being wholly determined by physical events in the past \u2013 at many points in the physical world. Unless we adopt the dogma that the laws of nature have to determine the future in only one way, an element of indeterminism in nature seems very likely. Why should universal laws completely determine everything that happens?\n\nSpecifically, the intentions of conscious beings may help to decide what happens in the future of the universe, and such intentions, being mental and not physical, may be in principle unpredictable and not wholly determined by previous physical events. Any scientist who says we know all the causal principles there are, and that those principles are sufficient to determine everything that happens, is not being wholly honest. So there is plenty of room in our universe for events that are influenced not just by general laws of nature, but by conscious goals and intentions. There may be general teleological principles built into nature which will help to explain the apparent direction of evolutionary change toward intelligent consciousness and the fully purposive direction of events.\n\nWhen someone resorts to saying, \"This event was just an extremely improbable occurrence,\" we need to remember that science is in the business of making the apparently improbable less improbable. If we cannot do so, that is a defeat for Darwinian theory, not a success! Teleological explanation would render the extinction of the dinosaurs more likely than it would otherwise have been \u2013 always assuming that small mammals were more likely to evolve intelligence than T. Rex was. So while dinosaurs may not like the thought that they were holding up the progress of cosmic evolution, it could be the case that their extinction, while not specifically planned by a dinosaur-hating God, was not wholly an unpredictable accident. Sooner or later they would have been replaced, and the proximity of a meteor provided one way in which this could occur, naturally though not inevitably.\n\nI conclude that modern evolutionary theory, when it is not infected by materialist philosophy, does allow for direction or purpose in the cosmic process. It compels us to include a place for chance in the process, but when combined with physics it sets the operation of chance within limits that are imposed by the fundamental laws and constants of the physical world. That is just what seems to be required for the generation of stable and yet limitedly free intelligent persons. To that extent, evolutionary science, especially when set in the context of a general evolutionary cosmology, is wholly consonant with the hypothesis that there is a goal for the material cosmos.\n\nPurposive explanation and idealist thought\n\nBut is teleological explanation scientific? Final causality was ruled out of science in the seventeenth century, as appealing but fruitless and obstructive \u2013 \"like vestal virgins\", Francis Bacon said. Contemporary science does not deal with purposes in nature. That does not mean there are none. Contemporary science does not deal with the taste of tomato soup either, but there is one. So perhaps purposive explanation exists, but is not part of modern science.\n\nThere is a good reason why not. If you are going to talk about purpose, you have to talk about the goal at which a purpose aims. Then you have to evaluate that goal as good \u2013 nobody aims at something because they regard it as bad. Even masochists, who repeatedly harm themselves, are doing so because (to take just one possibility) they think they deserve it, and it is good to get one's just deserts \u2013 and probably also because they get a sense of sexual pleasure out of it. Some people have very unusual desires. But I am not going to go there.\n\nThe point is that science tries to discount personal evaluations. It tries not to say whether something is good or bad, but to give a neutral description of what happens. It is therefore not the job of science to talk about worthwhile goals, and that stops it from talking about purposes. Teleological explanation does talk about worthwhile goals, and supposes that cosmic processes may be directed to achieving them. But purely descriptive science cannot say what those goals are or use the fact that processes seem to be directed to attain them in order to construct mathematical equations that give predictions that can be publicly tested. Aiming at goals is too infected by creativity and freedom to be captured in mathematically exact predictions. So it is not part of modern natural science.\n\nTeleological explanation, in short, belongs to philosophy, not to science. Ah, materialists may say, so it is just a matter of personal opinion? Indeed it is, just like materialism. None of us can avoid taking up a philosophical position, if we think hard enough. That is very irritating, if you dislike philosophy and think it a waste of time.\n\nThe basic motivation for positing teleological explanation starts from an analysis of human experience. We have purposes and we act intentionally. Then it proceeds to probe the nature of the world as it is experienced by us and finds (perhaps!) that it is an appearance of a veiled and mysterious reality, not accessible to sense-observation. It discovers the positive activity of thought in constructing the world of appearances and so posits that constructive thinking is part of reality as it is in itself. Constructive thinking is intentional; it is freely aimed at the goal of understanding the world. So teleological (intentional) explanation is part of the real world. It is at that point that we may be led to ask whether we can find traces of it in the observed world of appearances and in the general structure of that world rather than just in human experience.\n\nSince materialism has been abandoned at the very beginning of this process, there is no reason to exclude teleology in principle from the general structure of nature. It is a viable option. When we ask how consciousness originates very late in the history of a universe that begins with a Bang, it can become a compelling thought that it is not a total surprise, but a natural realization of the initial potency of matter itself. So now we look for some more primitive states that can naturally give rise to finite consciousnesses through a long emergent process, and idealist philosophy begins.\n\nDual-aspect idealism \u2013 the truth at last?\n\nThe basic laws that structure the universe do not have to be merely mechanical principles with no inkling of the consequences of these mechanical motions. They may include possible goals and processes of value and the future genesis of consciousness as the fruition of matter itself. One way to think about this is to imagine, as some mathematicians do, an abstract space of possibilities, laying down future tracks to future goals and determining, with much room for individual creativity, how the universe will proceed. The actual goals and tracks that are \"chosen\" may depend on many factors, but evolutionary views of the cosmos suggest that there will be an emergent and developmental process in which finite parts of the cosmos become more and more capable of envisaging and pursuing their own creative paths.\n\nIt is consistent with modern quantum theory to regard the whole cosmos as a web of interacting energies, of spatially and temporally located powers. Each part is not, like Leibniz's unfortunate monads, isolated and closed in on itself. Each part is essentially open to the totality of the space-time nexus. Each receives stimuli from all the others that surround it, integrates those stimuli into a unity of being, and actively responds in accordance with its own specific powers. At the simplest level, for instance that of subatomic wave-particles, both stimuli and responses are more or less algorithmic \u2013 they behave in accordance with regular and largely predictable routines, described by the basic forces of nature like electromagnetism, gravity, and nuclear forces. Only in this way can they form stable atoms upon which more complex unities can come to exist.\n\nAs atoms form into molecules and they in turn form long chains of RNA and DNA, patterns of stimulus, integration, and response grow more complex. Primitive sentience is a function of complex organic forms, which increasingly act as individuals, though they essentially function as parts of a larger integrated whole.\n\nProbably at the point when brains begin to exist, there is a radically new form of complexity, for which stimuli are registered with intensities of feeling and responded to with some creative agency. In human beings, the most complex form known to us, the conceptual or interpretative element predominates, and responsive actions become subject to causality by envisaged outcomes (intention). The sense of a continuing and active self emerges, interpreting the stimuli received from \"outside\" as appearances of a world of objects, and intending to modify those objects in accordance with consciously formed purposes.\n\nHumans have private perspectives on, feelings for, and thoughts about, phenomena interpreted as expressions or mediations of external objects (including other persons). They express such feelings and thoughts in external ways, like language. But humans know that language or physical gesture may conceal inner thoughts or fail to state them adequately or be interpreted in many ways, some of them quite mistaken, by those who perceive only the observed expressions. Thus each thought or feeling is known in two ways \u2013 as expressed physically and as experienced internally.\n\nIt is important to note that the physical expression is itself in a sense an \"internal perception\" of some observer, a subjective appearance of something taken to be objective, though it does not exist objectively as it appears to the observer. In that sense, the conscious event has logical priority and primary importance, though it is natural to see it as emerging from a long process of physical development.\n\nIn the first four chapters of this book I argued for a general philosophical position somewhere between dualism and idealism, and suggested that some form of dualism and some form of idealism probably converged on an acceptable view. That would mean that matter would not merely be an illusion or a content of some mind or minds, but would have its own form of reality. Yet matter would ultimately depend for its existence on a mind-like reality.\n\nA pressing problem for modern philosophy is how to relate matter and mind in a satisfactory way. An evolutionary view would find much to resonate with in Whitehead's view of the world as a succession of transitory events, each of which has an \"inner\" aspect as well as an outward physical appearance. For idealists, that inner aspect is the causal driving force of a cumulative and creative process of increasing organized complexity, generating richer forms of consciousness and purposive causality.\n\nI call this a form of idealism because the engine of the process is not the mechanical movements of non-purposive physical entities, but the potentially mind-like reality or realities that are creatively and progressively expressed in the physical cosmos, and in the gradual unfolding of values that were implicit from the first in that cosmos. It is sometimes, however, called monism, because it insists that \"feelings\" (the inner aspect) and matter are bound together as aspects of a unitary reality. I argued that this is actually identical with a form of dualism that sees mind as the individuating agent of the unity and cumulative development of different streams of individual consciousness and purposive agency, but which allows that such streams of consciousness are firmly embodied in (bound together with) a physical cosmos.\n\nIt may seem confusing to mix idealism, monism, and dualism in this way. But perhaps what that shows is that we should not pour our theories too strictly into neatly labelled jars. What I am suggesting is a form of qualified absolute idealism. It could be called dual-aspect idealism. I do not mind the name, as long as it is clear that I mean to argue in favour of two hypotheses. First, that the most important feature of human persons is that they are streams or chains of mental acts and events, streams that are distinguished from one another by each containing experiences of one and only one uniting and cumulatively shaping subject. Second, that these streams of consciousness are the inner aspects of complex organized physical systems, with a long evolutionary history and an inherent potentiality for generating and realizing consciously created and appreciated values.\n\nThis provides a general picture of what human persons are that I think is philosophically plausible and consistent with the best modern science. Yet it is undoubtedly the case that many people feel that any sort of dualism or idealism, even or perhaps especially a dual-aspect idealism, is somehow incompatible with contemporary science. This is my cue to return to the work of Gilbert Ryle and re-examine why he thought Cartesian dualism was a myth. It may be that Ryle actually helped to invent the myth, and that Ryle's own account is capable of a much more idealistic interpretation than he ever dreamed. Of course Professor Ryle never actually had any dreams, since such things are private events, which do not, he thought, exist. So I will have to dream for him.\n\n## Chapter Eight\n\n## Metaphysics and common-sense philosophy\n\nGilbert Ryle's arguments aim to undermine both dualism and idealism. In the rest of this book, I seek to assess just how strong his arguments really are. It is necessary first of all, however, to examine Ryle's general view of philosophy. For he thought that metaphysics, in the sense of a general theory of the nature of reality, is a fairly useless or even impossible undertaking. That would make my project superfluous. I try to show that no thinking person can really escape some sort of metaphysics (some theory of what kinds of things really exist), and that Ryle himself actually had one, which was some sort of common-sense (or ordinary language) philosophy. So the question of metaphysics is important and unavoidable \u2013 and I think the reason why many people are materialists is that they do not take the metaphysical question seriously enough.\n\nI may seem to have come a long way from Ryle's The Concept of Mind. But the point has been to see how Ryle's denial of inner processes is almost the exact opposite of the idealist claim that it is just such inner processes, culminating in intelligent consciousness, that form the core and the inherent teleological aim of reality.\n\nMost of our ideas and beliefs are pretty much determined by the very first steps of thought we take. Ryle starts The Concept of Mind with a set of philosophical assumptions or premises that are going to guide everything else he says. What I have been trying to do is to show that those premises are far from obviously true. For lots of philosophers, they are obviously false. Only if you accept them as true will the rest of Ryle's book seem convincing \u2013 in fact, it will come to seem pretty obvious, and we might wonder why nobody thought of it before.\n\nBut you can get out of Ryle's system by rejecting his first assumptions. What I have done is to set out some alternative assumptions that other philosophers make. There are living philosophers who still follow each of the alternatives that I have outlined. I myself am convinced that some version of idealism is true, and I am attracted to some key ideas of process philosophy. I am pretty sure that materialism is false.\n\nPhilosophy, it must be said, is not good for decisiveness. If you have a reasonably open mind, you will probably be persuaded by every new philosophy book you read \u2013 at least for a day or two. But as you get to know more and more theories, and you see all the good arguments philosophers can find for them, you get less and less able to decide which one is true. I once asked a very well known philosopher of religion whether he believed in God. \"Well,\" he said, \"as a Catholic I do, but as a Buddhist I don't.\" \"But are you a Catholic or a Buddhist?\" I rather foolishly asked. To which his reply was, \"Sometimes.\" I suppose that in his Buddhist moods he could have quoted the Buddha and said, \"I am and I am not, and it is not the case that either I am or that I am not.\" But that would not have been much better.\n\nI actually find the indecisiveness of philosophy rather helpful. Sometimes you just have to make decisions and stick by them. But it can be helpful to remember that we are rarely as certain as we think we are, and our own basic philosophical beliefs are rarely as obviously true as we pretend they are. Maybe the best we can do is to say which beliefs seem to us most obviously false and which seem most appealing, and admit that not everyone is going to agree with our decisions.\n\nSo I suppose the minimum thing I would say about Ryle is that his view is not as obviously true as he thinks it is \u2013 although it is not quite clear that he always thought it was obviously true. He did confess to me that on one occasion he thought he might have had a mental image. But then he recovered his composure and decided that he had only formed the belief that he had nearly had a mental image, without actually having nearly had one.\n\nGilbert Ryle was a common-sense philosopher \u2013 except that, because he was after all an Oxford man, he probably thought that only the members of Oxford Senior Common Rooms really possessed common sense. Most other people are superstitious and deluded most of the time. And that is the trouble with common sense. It is just not very common. What most common people think is mostly nonsense. But there is no system of philosophy that calls itself \"common-nonsense\" philosophy. At least the nonsense talked by philosophers is not common. And the common sense that philosophers share is very unlike what common people think.\n\nRyle was a leading member of a common-sense school that is sometimes called \"ordinary-language philosophy\". We must remember that this is Oxford ordinary language, and it uses words that most people have never heard of. Philosophers of this school do not try to find out exactly what common-sense beliefs might be. Instead they talk about how people actually use language. They do not of course actually go out onto the streets and listen to ordinary people talking. The unwritten union rules of British universities state that only sociologists are allowed to do that. What philosophers must do is to sit in armchairs and think about people talking.\n\nIt turns out that as long as these imaginary people keep talking, everything goes reasonably smoothly. But when they start to think about what they are saying, things go horribly wrong. Then they start to ask questions like, \"What did I mean by what I just said?\" But in order to answer this question they first have to know what they mean by meaning. As they cannot work this out, because they are not quite sure what the question means, what they say quickly becomes meaningless. And philosophy is born.\n\nIt is a bit like riding a bicycle. If you just keep peddling, you will get along nicely. But if you ask yourself exactly how you keep your balance, you will probably fall off while you are trying to work it out. Most philosophers are like cyclists who fall off their bicycles, because they cannot work out exactly how they manage to stay on them.\n\nSo the job of ordinary-language philosophers is to stop people asking philosophical questions. Wittgenstein, a leading exponent of this type of philosophy, though regrettably he was based at Cambridge, thought that it was ridiculous to claim to be a professional philosopher \u2013 it was equivalent to claiming to be a professor of nonsense and grammatical mistakes. Unfortunately he was a professional philosopher and so were many of his pupils. They remained sane, when they did, by claiming that all other philosophers were talking nonsense, but that they had seen through the nonsense, and their job was simply to stop people being deluded by philosophers. They largely succeeded and in consequence they have largely become extinct. There was nothing left for philosophers to do.\n\nOn trying not to have a philosophical theory\n\nRyle and Wittgenstein both wrote so beautifully and were such commanding personalities that this became a dominant form of common-sense philosophy in mid-twentieth century Britain. It is common-sense because it refuses to invent grand theories about the universe just as a result of sitting in an armchair and thinking, and it insists on the diversity and flexibility of language, not as a clue to the ultimate nature of reality, but as a natural expression of human social behaviour.\n\nThe killer question, however, is this: it is said that philosophical theories arise out of misinterpretations of language use. But isn't that a philosophical theory? It is, after all, an important insight that language has many different functions, that we learn it from others in a particular social context, and that a large part of its use is practical or action-guiding, not theoretical or descriptive. That is a view about what language is and about how it relates to the real world. It is a pragmatic theory, implying that the relation is one of usefulness, that there are many sorts of usefulness (depending on what we want it to be useful for), and that preferences and interests may well differ from one society, time, and place, to another.\n\nThis theory also claims that most traditional philosophical problems, like that of materialism versus idealism, determinism versus free will, passion versus reason or the subjectivity versus the objectivity of values, result from misinterpretations of language. I guess most philosophers who take one of these positions would be annoyed at being told they had merely made some kind of grammatical mistake. They think they are disagreeing about what is the case, even though with these sorts of question it does not seem possible to decide with certainty what the case is.\n\nPhilosophy deals with undecidable yet apparently factual questions, which sometimes have great practical consequences. One of those, which I spent some time on in earlier chapters, is whether observed objects remain the same when they are not observed. Ryle might say that is an unreal question, since it only arises in philosophy seminars, which can seem pretty unreal occasions. Professor Ayer, however, believed that the answer to that question was very important. Ayer agreed with the realists that what we see is what there is, but added that what there is is nothing except what we see. The idea of a world of unobserved physical objects is a logical construct, invented for pragmatic reasons \u2013 it helps us to find our way around the world if we pretend that it is really there. But if we realize that it is not really there, we will see that there is no point in talking about it or in discussing its hidden nature. All our language will be concerned with what we see, hear, touch or smell, and not with supposed hidden or \"'supernatural\" realities.\n\nAyer summed this up with what he called the \"verification principle\" \u2013 the meaning of a statement is the method of its verification. To put it another way, if you cannot see or smell something, it makes no sense to talk about it. This is no doubt why Ayer never paid much attention to what other people thought. He couldn't smell their thoughts, so it was meaningless to talk about them. This philosophy got rid of most traditional philosophical questions by showing that they were not just grammatical mistakes; they were actually meaningless.\n\nIt follows that both Ryle and Ayer thought that each other's beliefs were mostly meaningless. Ayer was sure that it was nonsense to talk about unverifiable objects (like Ryle's unvoiced thoughts), and Ryle was sure that it was nonsense to talk about \"sense-data\" (like Ayer's smells). It is not surprising that they could not understand each other. They each thought the other's beliefs were not just mistaken. They were literally nonsensical.\n\nThis situation provides a clue to the real nature of philosophy. Philosophers do not deal with particular factual questions, where everybody agrees what a \"fact\" is, and how to decide whether something is a fact or not (for example, whether light moves in a straight line or not). Philosophers deal with questions about what general scheme of concepts most adequately makes sense of the world. They deal with general conceptual frameworks for understanding and interpreting the world. Anybody who has a different conceptual framework will seem to them to be talking nonsense.\n\nSuch frameworks differ considerably from one another. Some think that in the end we should rely on common-sense beliefs and not be led astray by weird theories. Others (sometimes the same people on different days) think that many common-sense beliefs are actually based on linguistic mistakes and that we need a bit of linguistic hygiene to eliminate such mistakes. Some think that, whatever common sense says, we should only trust our sense-perceptions, which tell us how things really exist. Others think that only sense-perceptions really exist and there is nothing else to tell. And yet others, influenced by modern physics, think that we should not rely on either common sense or on sense-perception, since the real world is very much stranger than we think and wholly unrepresentable by the senses.\n\nWhat is the evidence for the truth of such views? There is nothing that would conclusively settle a dispute between them. However hard Ayer tried to get Ryle to have some sense-data, Ryle would refuse to have them. Not only that, Ryle would deny that there were any such things, even though Ayer was having them all the time. This is clearly not a matter of evidence. It is a matter of which most basic or general concepts we are going to use to interpret our experience. Of course arguments can be presented for and against such basic interpretations, and those arguments seek to show that one favoured view presents an interpretation of the data of human knowledge that is adequate, comprehensive, consistent, fruitful, elegant, and appealing, while all other views are nonsensical \u2013 or, it might be more tolerant to say, less adequate interpretations of reality.\n\nMy dispute with Ryle, which I probably only have any hope of winning because he is safely dead, is a philosophical debate in this sense. It is not about the ordinary usage of words or about how best to tidy up the informal grammar of our language. It is a dispute about what human persons really are. It is about whether what is of unique importance about human persons is their possession of a rich inner mental life, over which they have some degree of responsible control, and which is quite different in character from and irreducible to the law-governed motions of physical particles in space. It is, to put it in more traditional (though almost universally misunderstood) terms, about whether it is helpful to speak of the human soul as what makes humans of distinctive value and significance.\n\nRyle and dualism\n\nThe Concept of Mind is largely devoted to arguing that the \"Cartesian myth\" of humans as \"ghosts in mechanical machines\" is nonsensical, and to proposing a different model of human beings as social animals exhibiting distinctive kinds of intelligent behaviour. The stress is on the primary importance of publicly observable behaviour. But it is not empiricist in the narrow sense that confines all knowledge to immediate data of the senses. Indeed it regards talk of \"immediate data of the senses\" as artificial and misleading jargon invented by philosophers, when we should just say that sometimes we hear and see things without using telescopes or microscopes.\n\nIf Ryle's book convinces, it is not because it provides hitherto unconsidered evidence. It is because it gives a convincing model for understanding human beings as evolved social animals. This fits well with evolutionary biology, with an interest in social psychology and anthropology, and with a general loss of belief in a \"soul\" which is only possessed by human beings, marking them out as totally different in kind from all other animals. It is also a very non-mechanistic and non-reductionist view, and so makes it possible to retain a form of humanism, stressing the distinctiveness and importance of human capacities and excellences.\n\nRyle's view, largely shared by Wittgenstein, is proposed as a perspective from which to perceive human nature, and it has been influenced by many converging strands of new factual knowledge, new capabilities, and novel evaluations in rapidly developing social systems. Philosophy changes as previous basic interpretations are felt to be inadequate in some way. It is the fate of each philosophical insight to become the nonsense of its succeeding generation. Descartes celebrated the birth of the new science in the seventeenth century, but Ryle set out to show that the Cartesian philosophy was nonsensical. Perhaps Ryle knew that the same thing was bound to happen to him.\n\n## Chapter Nine\n\n## In defence of dualism\n\nI begin with the question of whether there are inner experiences to which persons have privileged access and which may not be material objects with spatial location. These would be experiences like sights, sounds, smells and so on (they have been called \"sense-data\" or \"qualia\"). Against Ryle, I argue that sensible dualists do not think the mind is a separate hidden world, connected arbitrarily to the body. It is the inner aspect of the material person, but it is a realm of partly unverifiable privately accessed data, and its rich, value-filled complex of feelings, thoughts, and intentions (its \"inner life\") is a key element of human personhood. Moreover, it is logically possible that this inner aspect could, very unusually, exist without the body. But its proper and normal place is precisely as the inner aspect of a material body and brain, situated in a shared social environment.\n\nDualism, the original sin of Descartes, is not yet dead. Dualists can be found hiding in the philosophical undergrowth, slightly cowed perhaps but still defiant. The heart of dualism, in the sense relevant to this discussion, is that mind and matter are two distinct sorts of thing. Minds do not exist in space, whereas matter is defined in terms of its location and extent in space. Minds think, feel, and perceive, and matter does not. I have suggested that this is just as much common sense as is Ryle's view, though it is quite possible that common sense may change its mind, if it has one.\n\nIt has to be admitted that there is almost a consensus among many modern writers that common sense should give up belief in dualism. Malcolm Jeeves and Warren Brown, both distinguished neuropsychologists who write about the philosophical implications of their work, say, \"We believe it is no longer helpful or reasonable to consider mind a nonmaterial entity that can be decoupled from the body.\" We should no longer consider the \"I\" to be a separate inner agent, but we must accept that the mind is \"a functional property of our brain and body\".\n\nI can see what they mean, but I think that what they say is not quite right. In fact, the rest of their excellent book shows just why it is not quite right. It is, to put it bluntly, a bit of undigested anti-Cartesian prejudice. They say that the mind is not a \"nonmaterial entity\". Yet they also say that consciousness, intelligent thought, and moral decision-making are emergent properties of a complex material system. \"The basis of consciousness,\" they write, \"is a dynamically self-organising complex system within the cerebral cortex.\" But they are careful not to say that consciousness is a system in the cerebral cortex and nothing more.\n\nThat consciousness is nothing more than physical brain-behaviour they call \"reductive physicalism\", and they distance themselves from it. They espouse what they call \"non-reductive physicalism\". Complex systems generate new properties that are not just combinations of properties possessed by the simpler parts of such systems. Nevertheless, they hold that such emergent properties do not introduce any new \"stuff\" into what the universe is made of. The only stuff out of which things are made is material stuff \u2013 fundamental particles or waves or vibrations in a fifth dimension or whatever the favourite basic material stuff may be. (It is not insignificant that physicists are not agreed about what it is. It is not after all totally convincing to be told that everything in existence is definitely made of something, but we are not quite sure just what that something is.)\n\nAn example of an emergent property might be the sound of an orchestral chord in a Beethoven symphony, which is an emergent property from the arrangement of fundamental particles which makes up an electromagnetic set of wavelengths. The only physical stuff around is electromagnetic waves at a specific frequency whose physical properties can be specified accurately. Those properties do not include any reference to what a chord sounds like or to its beauty. But when those waves hit the ear and get transmitted to the appropriate area of the brain, hey presto, a beautiful sound appears!\n\nBut is the sound of a chord not a kind of stuff? If it exists at all, it is stuff. Could it be just mental stuff? Conscious minds hear specific sound waves as having a specific timbre, pitch, and emotional tone, though none of those properties exist when the sound waves are unheard.\n\nHeard sounds do not appear to be objective physical properties. They are what humans experience sound waves to be, when such waves cause stimulation of the brain. We say that some sounds are beautiful. But what we mean is that we experience them as beautiful. This new property of heard sound, with a pleasing or displeasing character, is not some new behavioural principle that applies to complex arrangements of fundamental particles, whether or not they are being perceived. It is an actual occurrent feeling of something being experienced as emotionally resonant. If that heard sound is not a physical property (one that exists in the absence of conscious beings), then it follows that there must be at least one non-physical property. So this is not a form of \"physicalism\" at all, either reductive or non-reductive. It has introduced at least one piece of non-physical stuff.\n\nIt is true that complex organized physical systems behave in different ways from simple, relatively isolated physical particles. The laws governing the behaviour of such systems cannot be deduced from a study of fundamental particles alone. We could say that the laws of complex organized systems are not reducible to laws governing the behaviour of fundamental particles considered in isolation. Yet that would not license anyone to talk of \"new occurrent properties\". The properties would remain the same; they would just behave in different ways.\n\nFeelings of wetness, perceptions of colour, and sounds of sonority are occurrences that do not exist in the absence of minds. These are what Ayer used to call \"sense-data\", though many modern philosophers call them \"qualia\", and recognize that they are logically distinct from physical properties. They do not just emerge from complex physical systems. They emerge when such complex systems give rise to perceptions. For this reason, Professor John Polkinghorne, from whom I have learned almost all that I know about physics, nevertheless does not seem to me to be quite right when he says, \"We can accept a structural reductionism... the units out of which all the entities of the physical world are constructed are just the elementary particles studied by fundamental physics.\"\n\nHe is right when he says that in the physical (unobserved) world there are only elementary particles (or whatever), in various more or less complex arrays. But he is wrong if he means that all that exists is part of the physical world. Perceived sound is not a bit of \"meaning\" or \"information\" stuck onto physical particles. It is something we perceive and feel as what the physical world is like when it is apprehended by us.\n\nThe obvious conclusion is that perceived properties are caused by physical processes, including brain processes, and those processes must be functioning correctly for us to perceive correctly. But the perceived properties are not out there in the physical objects. They are additional properties caused by complex organized physical systems, and they exist only for and in consciousness.\n\nWe assume, most of the time, that effects are different from their causes, and that cause and effect could, logically, exist apart. They could be decoupled. So the \"non-reductive physicalists\" are really saying that mental properties are caused by physical properties, that when a certain sort of complex organized physical system comes into existence, it causally generates new sorts of conscious stuff. Minds or mental properties emerge from matter. That is a very important thing to say. But it implies that minds are different from matter. They therefore could in principle be decoupled from matter, as any effect could in principle be decoupled from its cause. Causal connections are, after all, contingent. They may hold universally in our universe. But they could have been otherwise. Therefore if the brain\u2013mental state relation is causal, the brain state and the mental state are not strictly identical. Whether Jeeves and Brown like it or not, mental states could in principle be decoupled from brain states, and minds are more than just functional properties of our brains and bodies. We may just have to wait and see.\n\nDo zombies exist?\n\nSome philosophers, who obviously have a liking for horror films, address this point by asking whether there could possibly be zombies. A zombie, for philosophical purposes, is a human body that acts and talks exactly like a human, but has no consciousness, no mental states, at all. It would obviously pass the Turing test \u2013 that is, after talking to it for as long as you like, you would still not be able to tell that it was a zombie. You would probably think that it had thoughts and feelings and sensations, just like you. But it might not have.\n\nIs such a thing possible? I have to say that I think it is \u2013 and not just because I really suspect that some of the people I meet are zombies. I prefer to give everyone the benefit of the doubt. Yet I can think of borderline cases. For instance, do ants possess consciousness? Do they feel anything or like and dislike things? I tend to think ants operate solely by reacting to chemical stimuli \u2013 by stigmergy or programmed behaviour caused by previous behaviour that has changed the immediate environment, and that looks as though it is intelligent (like termites building a rather complicated nest).\n\nIf someone built a robot that was more or less indistinguishable from a human being, but the creator knew that his robot consisted just of integrated circuits and programmed routines, she would have good reasons for thinking she had built a zombie. Descartes infamously thought that animals were all zombies. How do we know that they are not? Well, their physiology is very like ours, they have central nervous systems and brains, and their behaviour seems to suggest awareness of pleasure and pain.\n\nA number of experiments by the physiologist Michel Cabanac claim to show that animals and humans both act, at least in laboratory conditions, not to maximize their biological fitness, but to maximize their sensory pleasure. The feeling of pleasure, he holds, is the cause of many behavioural and physiological changes. This suggests that animals are similar to us in being conscious \u2013 they feel pleasure. Of course you could still maintain that the feeling is a by-product of the release of chemicals in the brain, which is the real cause of behavioural changes, thereby preserving a materialist view.\n\nThere is no experimental way of resolving the issue. It is natural to think that the more like us something is, the more it will share the same sorts of property, including conscious properties. I conclude that zombies are possible, but it is a good thing to extend a personal set of interpretations of and reactions to the world as widely as possible. We take a particular moral stance toward something when we regard it as a conscious agent. Perhaps, as Kant suggested, this is a basic attitude we must adopt to the world if we are to be fully moral agents ourselves, even though it cannot be based on conclusive evidence.\n\nThough we cannot refute Descartes' view of animals, most of us probably now feel that our greatly increased knowledge of the genetic, biological, and psychological properties of animals leads us to adopt a more morally concerned attitude to them. Zombies are possible, but it is a moral failure to assume that they are actual without a very good reason. This shows that moral failures can be failures to interpret the facts, so values and facts are not as distinct as some philosophers have claimed. It also shows that some of our most basic philosophical attitudes are not based on evidence, in the sense of publicly accessible data that can be publicly observed and agreed upon by all competent observers.\n\nThere are some widely, though not universally accepted, beliefs that Alvin Plantinga has called \"basic beliefs\". I will adopt this term for beliefs that are not based on agreed evidence (since we all agree on the available evidence), but for which reasons can be given, reasons that argue for particular beliefs as conditions of a general conceptual scheme that we (some of us) think it right to adopt. They include the belief that people (and animals) are not zombies, that they have an inner consciousness; the belief that the laws of nature will continue to operate, and that there is no event without some sort of cause; the belief that I am a continuing agent of experience and action; that there is mental causality or freedom of action; that I can remember my dreams; that human embryos have a right to life; and, for people like me, the idealist belief that reality is ultimately mind-like.\n\nCommon-sense philosophers are right to think that some of our fundamental beliefs are like this. But they do not always see that such beliefs are not all universally agreed, and therefore need not be as common as all that. The reasons given for such beliefs may depend on deep underlying ways of seeing and responding to human experience. But there is no way of gaining universal agreement on such basic ways of seeing and believing. One of these beliefs \u2013 the one I am concerned with at present \u2013 is that other people are not zombies, but that zombies are possible, and one might in future be constructed in a laboratory.\n\nSeeing faces\n\nMalcolm Jeeves and Warren Brown say they have a rather different basic belief, a belief in the truth of physicalism. They would not like saying that zombies are possible; it entails, after all, a form of dualism, that mental properties are different and decouplable from physical properties. Yet they do say something that strongly implies their possibility. For, they say, their work in neuroscience suggests that subjective experiences often exert a causal influence (a \"top-down\" influence) on the physical system and its parts.\n\nThey stress, for instance, that the cerebral cortex of young humans is plastic or partly unformed and continues to organize and reorganize functional networks of neurons \"through experiences, learning, imagination, and thought\". They quote Nancy Kanwisher of MIT as saying that \"evidence indicates important roles for both genetic factors and specific early experience, in the construction of the Fusiform Face Area\". That is, the ability of the adult human to recognize faces depends in part upon the subjective experience of interacting with other people's faces in the first few months of life. Such experience actually modifies the face-perception neurons in the brain to enable them to function more efficiently. Subjective experiences or acts (thoughts) cause changes in the physical structure of the brain. Emergent mental properties and experiences \"have a real influence on behaviour\". How, then, can they continue to say that \"humans are taken to be entirely physical\"?\n\nPerhaps they mean that humans are physical entities that possess nonmaterial properties, but that there are no nonmaterial entities. Unfortunately, that would require a clear distinction between entities and properties that would be very hard to defend. Properties just are entities that coexist with or depend upon other sets of properties.\n\nThey could still say that nonmaterial properties (like conscious experiences) could not exist without the brain properties which are their basis. But what is the force of such a \"could not\"? We might say that there is a strong causal connection between them \u2013 though, as I have noted, this causality runs both ways, from brain states to subjective experiences and from thoughts to neuronal connections in the brain. Where there is a causal connection in fact, there need not have been or there could have been a different one. We could have brains without consciousness or consciousness without brains. Those who believe there is a God who knows what is going on in the world actually believe that there is a consciousness without a brain \u2013 God is conscious, but has no brain. Admittedly God is very different from human minds, but the point is that many people at least think they can conceive of a consciousness without a body. If they are right, we cannot rule out the possibility of a human consciousness existing without its normal body by saying that such a thing is inconceivable.\n\nThe world would be decidedly odd if causal connections between brains and experiences kept breaking down. This is another basic, unevidenced attitude we take to the world \u2013 we trust that causal laws will not break down for no reason. Nevertheless, in extreme situations, like the death of the body for example, such connections may well break down. We cannot refute the hypothesis of continued consciousness after death, without a brain, just by saying it is impossible. Philosophers sometimes do say that, admittedly. But even philosophers should not deny possibilities a priori, just because they seem peculiar to them.\n\nThe embodied mind\n\nI conclude that minds, subjects of experiences and intentional acts, are after all nonmaterial and can be decoupled from specific brains. Moreover, that is entailed by any genuinely non-reductive physicalism, which is not, when analysed more closely, really physicalism (materialism) at all. Yet what Jeeves and Brown are saying is important. Minds need an environment to provide experiential data, and also to provide an arena in which to perform actions and to interact with other minds. Our physical world is not the only possible form of environment for minds. There may be other universes different from ours, with different kinds of minds in them. But human minds need to be genuinely embodied and embedded in some sort of environment. Minds are directed upon a world of objects, both by attention and by intention. Our minds may be causally limited to this physical world and they have certainly emerged within it by a long process of evolutionary development. They are not self-contained and only externally related to the physical world.\n\nWe can thus say that minds read or interpret the configuration of neurons which store information that the brain has received from its environment. They may then influence this configuration by thought and further experience. The configuration is stored in the brain (as a symphony is stored on a CD) and is ready to be read again (remembered) at a later time.\n\nThis way of putting the matter requires that the brain functions properly. Configurations and interpretations can go wrong, and if the physical basis is blocked or damaged, they will go wrong. So it is not at all surprising that brain damage will cause predictable mental defects or that particular mental processes are often found to be localized in specific areas of the brain. Such facts are to be expected by sensible dualists, who of course believe that minds think about objects in the world and try to perform actions in the world, and have never believed that minds only contemplate special mental objects and perform special mental actions in a special mental world. There are mental objects (like images) and there are mental actions (like thoughts). But these are derivative from the physical objects toward which minds are naturally and properly directed. That is just what poor maligned Descartes actually thought.\n\nWhat is important about dualism is that it retains belief in the moral primacy of personal experience and morally responsible action. But it does not deny that human experience is primarily of a physical world and that free acts are performed in a physical and public world.\n\nLiving in two worlds\n\nIt is possible that our subjective mental life may not in fact be separable from the activity of our bodies. I think that is a matter that argument alone cannot resolve. It may be that the interweaving of mind and body is so close and complex that it would not make sense, if we fully analysed it, to speak of a mental life anything like ours continuing without the very same physical world in which it has developed and lived. Then what Jeeves and Brown say would be correct. But I do not think we can be sure of this, and I think it is wiser to remain open to the possibility of a continuation of mental life after bodily death.\n\nIn dreams we experience the possibility of conscious knowledge and action which does not take place in an actual physical world. Admittedly dream images seem to be constructed from past experiences of a physical world. But it is fairly easy to conceive of images that occur in the mind without any physical cause and which might be shared between different minds. We can conceive of image-worlds that some groups of minds share, but which do not form the sort of objective causal nexus that exists in our physical universe.\n\nThe philosopher H. H. Price has sketched such an image-world in his little book, Lectures in the Philosophy of Religion. In his image-world, persons meet in rather flexible image-bodies (rather like avatars in a video game) insofar as they are psychically attracted to one another, and the causal laws are such that desires directly cause desired \"physical\" states to exist. This may sound very attractive, until we realize that always getting what we want may turn out to be a disadvantage, as King Midas could tell us. And also the people we would be psychically attracted to might turn out to be as horrible as we are, which could be very uncomfortable. Indeed, it might be a sort of hell.\n\nAnother idea of an image-world is presented by the Oxford philosopher Anthony Quinton, who sketches the slightly alarming possibility that every night when we go to sleep, we may have such coherent dreams that we actually seem to live in a quite different world. After a hard night's work in the other world, we go to sleep, and immediately dream of being back in this world. That seems a coherent idea, though it also threatens to be a very tiring one. When, if ever, would we get a good night's sleep? Anyway, if this happened which would be the real world? I suppose they would both be equally real, and we would be the same person, not just in two different bodies, but in two different worlds.\n\nSuch possibilities require some mechanism by which images could be generated or by which minds could continue to exist with new forms of experience and capacities for intentional action. In the absence of such a mechanism, it will probably be true that even if minds are immaterial (or contain important sets of non-physical properties), they will not in fact be separable from the bodies and brains that form their physical substrata.\n\nThere may, however, be various mechanisms that could ensure the decoupling of minds from bodies. The most obvious one is a supreme cosmic mind of some sort that could arrange for appropriate experiences to occur to \"decoupled minds\", perhaps as moral consequences of what they have done in their physical bodies. Or, as in some Buddhist philosophies, there could be an impersonal moral law (karma) which brings into existence image-worlds (detailed in The Tibetan Book of the Dead, for example) in which the moral consequences of human life-choices can be worked out. Yet another possibility is that, if there are many alternative universes which have sprung from the primal quantum vacuum, we might not need to travel through a black hole to get into another universe. We may just have to go to sleep \u2013 a very economical form of travel.\n\nThe philosopher Professor Ayer, one of the most vociferous philosophical atheists of his day, was extremely surprised to find that he continued to have experiences after he had died. He came back to life after being clinically dead, and his experiences, though they were odd in the extreme, were momentous enough for him to accept the possibility of life after death. This he found very disappointing personally, as it conflicted with most of his previous philosophy. Yet for anyone who is not a materialist, some kind of existence after death remains a possibility, disappointing or not.\n\nThat possibility requires that either dualism or idealism is true. So I do not think we should give up the idea that there is an immaterial self, in the sense outlined in chapter five, a subject of experience and action that is not bound to the conditions of this earthly body and that could be decoupled from this body and brain. But we should certainly accept the overwhelming evidence from neurological research that our minds are emergent realities from physical processes and are strongly embedded in physical environments through brain, body, and networks of social-historical relationships. Such evidence is quite compatible with sophisticated forms of both dualism and idealism, and it even points to the need for a more holistic view of minds as embedded in networks of social relationships, rather than as isolated units confined to individual skulls. That is where the work of Ryle and Wittgenstein provides a healthy antidote to materialism.\n\nIn this chapter I have argued that there is a real question of whether minds and contents of consciousness are special kinds of entities (\"stuff\") that are not reducible to material properties. This is not a pseudo-problem or one that could be resolved simply by appeal to how we ordinarily use words. Yet it is also not a question that can be resolved by appeal to evidence, in the sense of publicly accessible and decisively verifiable data. It is more a question of interpreting the data we have in the most adequate way \u2013 and what is most adequate is itself not universally agreed or conclusively decidable.\n\nI suggested that it is partly a moral stance to adopt the basic belief that other persons are genuine moral agents, not zombies. But the existence of qualia or \"private\" experiences, and the seeming existence of \"top-down\" causation in neuropsychology can lead to adopting the belief that minds are more than brains. It seems to follow that the mind\u2013brain connection is causal and contingent, and that minds could logically exist without brains. However, this may be only a bare logical possibility, and it certainly seems that human minds emerge from and are closely integrated with brain-events. It is not essential to Cartesian dualism that minds actually do exist in a disembodied state. What matters is that, even if human minds are essentially embodied, what is morally important about human persons depends upon their uniquely appropriated experiences and responsibly free actions; that is, upon distinctively mental properties.\n\n## Chapter Ten\n\n## Consciousness, value, and purpose\n\nA crucial question is whether human persons are morally free and responsible. This is a question which there seems to be no possibility of resolving empirically, yet to which arguments, like the ones discussed in this book, are relevant, and which it is important to decide in practice. The psychologist Benjamin Libet has tried to put the issue to experimental test, but the tests do not really seem to touch on the issue of moral freedom, and they do not appear to resolve the issue. I accept Immanuel Kant's view that we need to decide the issue on practical grounds \u2013 it is morally important to treat persons as free and responsible, even if we cannot theoretically prove they are.\n\nIn addition, we experience ourselves as envisaging possible futures, evaluating them, and choosing between them. It is reasonable to take these experiences as telling us what is true of reality \u2013 we really are free and able to cause physical changes by thought and intention. No relatively abstract theory should cause us to renounce these deliverances of experience.\n\nThe real threat to Ryle has not been a revival of dualism. It has been the rise of materialism. The development of neuroscience and of artificial intelligence has suggested to some philosophers that consciousness is not as important as we used to think. Computers can be built to perform intelligent operations. They (or at least one of them) can even beat the best human chess-players in the world, and they can think millions of times more quickly than humans. But they do not need to be conscious.\n\nStudies of the brain have shown that humans do not need to be conscious either. Experiments with patients with brain damage have shown that they can tell the location or movements of objects in their vicinity with an accuracy well above average even when they are not aware of any visual stimuli \u2013 that is, when they cannot see them. This remarkable phenomenon, known as blindsight, seems to show that the brain can respond to stimuli without being conscious of them. Some experiments seem to show that the brain can \"make decisions\" before any conscious decision has been made. These experiments have been hailed by materialists as showing that awareness and free will are by-products of physical brain processes, and that the belief that I can make decisions freely is an illusion.\n\nThe best-known experiments in this respect are those by Benjamin Libet. In his most famous experiment, he put human subjects in a laboratory, attached EEG electrodes to their skulls to measure electrical activity in their brains, and attached an electromyograph (EMG) to their arm muscles. He then asked them to press a button at any time selected by them within a short period of time. The EEG measured when a particular sort of cortical activity occurred (called the \"readiness potential\", which generally occurs as pre-motor planning of volitional movements). The EMG reading measured when the button was actually pressed. And the subjects were asked to note the moment (as measured by a dot moving around a huge clock) when they made the decision to press the button.\n\nThe results, confirmed over many experiments, were that the readiness potential occurred in the brain about 300 milliseconds before the stated decision to act, which in turn occurred about 200 milliseconds before the button was actually pressed. Libet's conclusion was that the brain initiates movement, and the apparently \"free\" decision comes after movement has been initiated. So it seems all causation happened at the unconscious level of the brain, and the conscious decision to act played no part in the proceedings at all. Free will has been disproved!\n\nIf this were true, it would be very worrying indeed for anyone, like an idealist, who thinks that the mind can initiate actions in a non-physical way, and that such decisions have physical effects. It seems to show that all real causes are physical, and that the sense of \"acting freely\" is totally superfluous and, in fact, illusory.\n\nAs we might expect, however, this sort of experiment is very controversial, and what it really shows is not very clear. One major consideration is the very artificial situation of the person being asked to press the button. When we are interested in free decision-making, we are mostly concerned with important moral decisions, when someone has to choose between right and wrong and can be held responsible for their choice. Or we might be concerned with the nature of creative choices in painting or music or with reasoning activities in mathematics or philosophy. These are real-life issues where we want to know if mental states can make a causal difference, which might have been otherwise, to what happens in the physical world.\n\nAll such factors are missing in the Libet experiments. The subject is asked to press a button at a completely arbitrary time, and say when he or she was \"aware of the urge to act\". In fact, the primary intention has been completed before the experiment begins. That intention is to execute a slight and pointless physical movement at some arbitrary time in the near future. There is no moral or rational or artistic significance in the action; it really does not matter when or how often it takes place.\n\nSince we know that a great many bodily actions \u2013 like breathing or eating \u2013 are virtually automatic, it is hardly surprising that the mind (whose job it is to find reasons for acting) hands over the task of pressing a button to the automatic brain, because there is no reason for choosing one time over any other time. The intention is to press the button at an arbitrary time. There is no reason to press it or not press it at any particular time. The brain has been instructed to do that, so it does.\n\nWhen the conscious mind \"becomes aware of the urge to act\", which has been arbitrarily selected by the unconscious motor system of the brain, there is no reason to suppress that urge. The mind assents and records the time at which it does so. It may seem as if the mind is merely following the dictates of electrical activity in the brain. But the fact is that the mind had already agreed to do precisely that when it formed the intention to take part in the experiment and to press the button arbitrarily (that is, for no reason). The only point at which a significantly free decision was made was at the point of deciding to participate in the experiment. So all the experiment shows is that some fairly arbitrary \"decisions to act\" are parts of behavioural routines that proceed automatically, but which may themselves be initiated by real mental decisions.\n\nLibet himself was a believer in freedom of the will, and argued that the mind could always veto the \"urge to act\" if it had reason to do so. But my point is rather wider than that. It is that experimental situations are very special. They abstract from real-life situations, and introduce artificial factors like \"moving a finger for no reason\" to substitute for really significant factors like \"deciding whether to kill someone for personal gain\". Naturally we do not want to construct experiments that will put such real-life questions to the test. But that means the results of the experiments we can do, fascinating though they are, do not address the problem of human freedom in any really illuminating way.\n\nExperimental tests on human persons mimic real-life situations in a distorted and artificially restricted way. They do not show what \"really happens\" in real-life situations. They show how some abstracted aspects of real-life situations can be mimicked in a laboratory. Much useful information about the brain can be obtained from such experiments. But it should not be thought that laboratory results show us what really happens in real life, where a huge variety of complex, interconnected, and unmeasurable factors contribute to our conscious experience of acting and intending to act.\n\nIs consciousness useless?\n\nThere is good reason to think that conscious states are real and causally efficacious, but it could be argued that consciousness is a disadvantage, an evolutionary mistake. If we were not conscious, we would not feel pain, we would not get depressed, and we would not waste time by asking silly questions like \"Why am I here?\" or \"What's it all about?\" Consciousness helps animals to see and try to escape their predators. But it would be more efficient to have an automatic stimulus-response-escape mechanism. Possibly ants have. They hardly have big enough brains to be conscious, yet they manage to fight, reproduce, and communicate very well. So we might have survived very well as communities of unfeeling robots. Information could have been processed, long molecules of DNA could have reproduced, and vast banks of knowledge could have been built up \u2013 but no one would have understood such knowledge or would even have been aware of it. It would just have been there, coded in complex systems of condensed algorithms. It would have been a universe in which CD-ROMs of amazing complexity would have been assembled, encapsulating every possible word and melody and thought that ever could exist, but without those thoughts and notes ever actually occurring to any consciousness. All the great literature of the world would have existed, but would never have been read or understood.\n\nIf you ask, \"What would be the point of that?\", the point is that there is not, and never has been, any point. It just is. Of course there may be no point in the existence of consciousness either. But the existence of consciousness introduces evaluation into the discussion. There may be no ultimate point in anything, but some things are more pointless than others. Counting the number of hairs on my legs is more pointless than trying to alleviate a headache. Since consciousness involves pleasure and pain, there is always a point in ending pain and obtaining pleasure.\n\nPleasure can be pointless, too, but it is always preferable to pain \u2013 unless the pain is a warning of greater pain to come. The greatest pleasures, according to philosophers as diverse as Aristotle and John Stuart Mill, are those which engage the human faculties in active and creative ways. People are happiest when they do things they are good at and interested in, with a reasonable degree of success, and when others appreciate what they are doing. Consciousness is essentially bound up with intentional action, because conscious beings seek to avoid pain and pursue pleasure. The very complex physical structures of the brain that make awareness possible also enable intentions to guide the physical movements of the body.\n\nComputers do not feel pain or seek to avoid it. They are designed (programmed) by humans to do specific tasks, information is inputted by humans, and outputted information is then interpreted by humans. Computers, like telescopes, are instruments designed to enable conscious beings to achieve new knowledge and to modify the world in new ways. Computers are tools and take no pleasure in winning chess games. They will win if they are programmed well, and if they are instructed to, and that is it. Computers are very bad models for human beings, as they lack awareness, evaluation, and purpose, the very qualities that are distinctively personal.\n\nThis is the enormous difference that consciousness makes. Consciousness may not be necessary for complex organisms to develop, but it does introduce a new element to reality, the element of understanding and appreciating a state of affairs. This element is quite different from any purely physical state, and it cannot be reduced to or described in terms of any purely physical state. It also introduces a new purposive form of causality, based on the envisaging and evaluation of future conscious states and activities. In having a purpose, minds have to envisage a possible state as a future state. This representation of something as future is a distinctively mental property. The occurrent state, which of course exists now in some mind, represents a state that does not yet exist. Knowing that some state is about the future, or even about the non-existent, is not itself a physical property. It is the \"aboutness\" that makes the difference. It is what has been called an \"intensional state\", a state that symbolizes something other than its own physical properties.\n\nThe evaluation of a possible future as good or bad, desirable or undesirable, is similarly a mental property. Machines can be programmed to reach specific physical states and then stop. But there is no sense in saying that they think about such states and are pleased when they reach them. \"Goodness\" is not a physical property of anything. To say that something is good is to say that it is the object of rational choice, choice directed by the thought that something would be pleasant or otherwise desirable. There is no place in any physical science for such concepts.\n\nThe formation of an intention to do something in future is also not a physical property, as it signifies that a future state will be brought about by intelligent action. The idea of final causality, of acting in order to obtain a desired future, as a cause of action through ideas, is a distinctively mental idea of causality. Bacon was right; there are no final causes in a purely material world.\n\nThe natural sciences presuppose the basic postulate that there are measurable and intelligible law-like regularities in nature. This postulate is confirmed by experience and is the basis of scientific explanation and prediction. In a similar way the experiences of practical reasoning, creative imagination, and moral commitment presuppose the basic postulate that there exists free and reason-based action. This postulate too is confirmed by experience and is the basis of personal relationship and understanding. Any adequate account of our world must include both postulates in a coherently related framework.\n\nThe idea of purely physical causality is a very sophisticated concept that remains almost wholly mysterious in nature. It could be said that our primary sense of causality, the one that we most directly and commonly experience, is that of bringing something about for the sake of a desired goal. Personal causality is something we experience and have to assume if we are to live as persons. Physical causality is, by comparison, a stripped-down abstract idea of regular law-like connections between physical events considered in artificially isolated conditions. In physical causality, one thing follows another, but nothing is actually brought about by any active power, and there are no goals set by any being. It seems strange, then, that anyone should think that personal causality may be an illusion, whereas only physical causality is objectively real. Such a counter-intuitive belief can only be accounted for by a strong prior commitment to a strictly materialist point of view that ignores the all-pervading evidence of personal experience.\n\nWe can make a meaningful distinction between behaviour that blindly executes a routine without any goal being actually envisaged, and conscious pursuit of an envisaged goal. Even those who think that the idea of intelligent design in nature is an illusion have to know what an intelligent, purposive action would be or they would not be able to say what it is that does not actually exist. Once that possibility is allowed, it is a matter of experience to determine whether it is actual. It is our own experience that we envisage, evaluate, and intend, and it would take an overwhelmingly strong theory to overturn that experience. I do not believe there is any such theory.\n\nWhy consciousness is essential\n\nIt is only with consciousness that the concepts of value and purpose, which are central to our social life, to living as persons among other persons, make sense. I doubt if these concepts make any sense for a ruthlessly materialist view of reality. They would, it may be said, be purely subjective. But even in that case purely subjective realities would exist, and that is just what a materialist denies.\n\nThe philosopher G. E. Moore asked people to envisage a very beautiful and elegant universe that was not consciously perceived by any being and to compare it with a very ugly and chaotic universe, similarly unperceived. He thought the beautiful universe was obviously better. But only philosophers can seriously claim that objects that no one has ever seen or will ever see are beautiful. It is rather like saying that ice cream that has never been eaten tastes wonderful. It might taste wonderful if someone ate it. Otherwise it has no taste at all. Most of us know that conscious experience adds properties \u2013 like beauty and pleasure \u2013 to the universe that otherwise would never exist at all.\n\nI have discussed the fact that many philosophers, beginning with Galileo, make a sharp distinction between \"secondary qualities\" like colour, taste, smell, pleasure, and pain, that only exist in consciousness, and \"primary qualities\" like mass, position, and momentum, that exist in non-conscious reality. Yet such primary qualities in turn seem to dissolve into spatially located or extended fields of force or even into the completely \"veiled reality\" that Kant and d'Espagnat postulate.\n\nThis is a long way from materialism, which claims that primary qualities are pretty well known and are the only constituent elements of reality. For some quantum physicists, like Niels Bohr, one of the fathers of quantum physics, the primary qualities of the physical world are almost wholly mysterious and seem more like mathematical abstractions from the directly experienced world of colours, sounds, smells \u2013 and of beauty, elegance, pleasure, and pain.\n\nIt may well seem that the \"real\" world is the experienced world, the world that exists in conscious experience. This is the view taken by idealist philosophers, who completely reverse the materialist hypothesis. For materialism, the emergence of awareness, evaluation, and purpose from unconscious, unfeeling, and purposeless physical processes is a puzzle, and a pretty pointless one at that.\n\nIdealists hypothesize that fundamental reality includes awareness, evaluation, and purpose \u2013 that mind is more basic, more causally efficacious, than matter. For an idealist, the existence of a material world is not such a great puzzle, because finite minds need physical expression if they are to relate to one another; if they are to have objects of knowledge that can please or displease them; and if they are to have goals that they can try to achieve in a partly pliable but partly resistant environment. Moreover, minds give an additional sort of explanation for why physical laws exist as they do \u2013 namely, they exist in order to allow a wide variety of worthwhile states to be realized and appreciated.\n\nMost idealists think that since the universe existed long before any finite minds came to exist within it, there is something mind-like at the basis of the physical universe itself. There is a cosmic awareness that envisages all possible states, ranks them evaluatively, and has the purpose of realizing its nature by generating societies of minds in an emergent universe \u2013 minds of many grades of awareness and intelligence, which can act in many creative ways to generate new forms of awareness, of beauty and intelligibility. This cosmic awareness is often called God, at least by philosophers.\n\nIt sounds a little too personal or anthropomorphic to some idealists, and they prefer to speak in a more Platonic way of an eternal realm of possibilities, beyond even mind or consciousness as we understand it, which necessarily realizes itself through the emergence of conscious and purposive beings, by processes of complex integration between simpler spatio-temporal elements. This is generally known in the German and early twentieth-century British idealist tradition as \"the Absolute\", and in the Indian Advaita tradition as Brahman. Consciousness as we know it then is emergent rather than primordial, but it is still an irreducible and causally efficacious element of reality, and it is the inherent goal or purpose of the cosmic process.\n\nThat seems to me the most satisfactory picture of the cosmos that can be obtained both from philosophy and from modern science. It may seem that Ryle would disagree completely with this picture, as he seems to deny that inner private conscious events exist at all. But it is not quite so simple. A different, more sympathetic way of interpreting Ryle is that he is saying (and this is even more so in the case of Wittgenstein) that inner conscious events need to have public and physical expressions, and could not exist as they do without being parts of networks of social relationships. Persons are only persons in community, not in isolation. That is why we might want a \"dual-aspect\" view rather than an outright dualist one.\n\nThe inner reality exists, but it could not exist fully and properly as it does without the outer expression. Materialism and idealism both err if they deny any existence to mind or to matter. Both must go together, but for an idealist the driving force of the whole process is in the end the mind-like, the conscious and intentional, with its values and purposes. It is to that existence that the material cosmos points, and in which it finds the fulfilment of its inherent potentialities.\n\n## Chapter Eleven\n\n## Thoughts and perceptions\n\nThoughts are an integral part of human consciousness. Without thoughts we would be unable to interpret or reflect upon our sense-experiences. Of course we learn the language in which we usually think from others. But what we think, the way in which we think, and the specific content of our thoughts, is often hidden from others. This does not mean, as Ryle says, that our thoughts take place in a hidden parallel universe that can never be discovered at all. It means that there are important elements of our inner lives that remain unique to us and that build up, if we are fortunate, an integrated and structured perspective on the world that imparts to our lives a unique value and significance. It is our inner lives of thoughts and feelings that make us what we are.\n\nFor Ryle, who thinks that the Cartesian myth of substantial minds imagines ghosts hiding inside bodily machines, idealists sound as if they are saying that there is a huge super-ghost (a cosmic substantial mind) in the cosmic machine or perhaps that lots of ghosts flit around its more complicated parts. In one sense Ryle is right. There is no cosmic machine and there are no ghosts hiding in its works. Idealists think that the universe includes an important dimension of awareness and purpose or perhaps many sorts of awareness and many diverse purposes. Perhaps these are present as real potentialities in a realm beyond space-time, and space-time is the expressive medium in which they develop to their fullest realization. This might very well suit Ryle's vision of humans as neither angels nor machines, but as intelligent beings whose behaviour expresses their natures in social and creative interaction.\n\nYet Ryle is also importantly wrong. There are few real dualists who have believed in ghosts. They do not usually believe that there are two completely parallel worlds, one visible and one invisible. What Ryle attacks are often straw men, or straw ghosts, after all. His own language discloses that he is, paradoxically, nearer to the Descartes of reality, not of myth, than he suspects.\n\nConsider what Ryle says about the intellect, in chapter two of The Concept of Mind. He opposes the claim that if we speak of an \"intelligent\" action we are referring to occult episodes in which intelligent minds contemplate a set of true propositions. On such a view, intelligent overt acts are the effects of such unobservable acts of inner contemplation. Ryle wishes to maintain that assertions about intelligent action are explanatory-cum-predictive assertions that can be publicly observed and tested.\n\nBy an occult episode he means some invisible action, like assenting to a theoretical truth, which is complete in itself, but which then contingently happens to cause some physical movement of the body. So we could give a complete account of occult actions without mentioning the body at all. It is an almost accidental truth that I think, \"If I move my arm I will catch this ball. I want to catch the ball, so I will need to move my arm,\" and then my arm moves. Ryle believes that on such an account, the intellectual act and the physical act just coincide \u2013 though if we are not well coordinated they may not coincide very often. Every physical act is preceded by an invisible piece of syllogistic reasoning, and the link between that reasoning and my bodily movements is wholly mysterious. Life is a perpetual surprise, as my thoughts are often followed by bodily movements, though I have no idea how or why. Most actions also take rather a long time, as I have to go through a series of internal arguments before I do anything. Dualists of this kind are likely to be very slow-moving creatures, who will probably be eaten before they have gone through the arguments in favour of running away.\n\nDoes anyone seriously think this, however? Humans are organisms, constantly receiving stimuli from their environment and actively responding to them. Catching a ball is usually pretty automatic and takes no thought at all. But I can pay attention to my technique and improve it through practice. Is that \"paying attention\" an occult episode? Of course calling it occult is meant to make it seem weird. The occult is the secret and slightly crazy. And my paying attention is not occult in the sense of being completely unknowable by anyone else. I know when you are paying attention. You stare fixedly, you do not just look out of the window, your body is tensed. That is because attending is not a purely private act. It guides bodily processes, modifying their natural responses by targeted mental effort. The mind directly affects the body, to increase the amount and quality of information the senses provide.\n\nNevertheless, attending is not just bodily behaviour. If attending \"consisted in\" the presence of testable predictive assertions about behaviour, then it would not matter if I was conscious or not. Yet, as Ryle says, \"There are some things which I can find out about you only... through being told of them by you.\" They include facts about exactly what you see when you attend, what it looks like, and how clearly you see it.\n\nSo Ryle amends the account that attending consists in specific sorts of bodily behaviour, and retreats to saying that attending is normally expressed in bodily behaviour. But he holds that our private internal imaginings are only a small part of human lives, and that only for the mentally ill do they play more than a very minor role. If that is so, there are many more mentally ill people around than Ryle supposed. The richness, complexity, and variety of what I see when I attend is known to me, but cannot be conveyed with any adequacy either in words or in behaviour. Poets and novelists are better at it than philosophers, and they are the first to say that no words can capture the sheer exuberant profusion of sensory experience.\n\nWe can observe people attending, but we cannot observe what they see when they attend. Similarly, we observe people intending, exerting efforts, and trying to perfect their actions. But we cannot observe just how much their efforts contribute to their behaviour, how difficult it is for them or what degree of freedom they have to modify their actions by strength of will.\n\nPersons have hidden lives, which others know nothing about and could never learn from their behaviour, which is often ambiguous or even dissimulative. Where Ryle is right is in seeing that the hidden is capable of being disclosed, and that it is essential that it should normally be disclosed. Our mental acts do not take place in a different parallel world. They take place in this shared and physical world, as experiences of it and actions within it. The unspoken thought and the uttered word are two aspects of the same reality, and though they may exist apart, it is natural for them to exist together, and for each aspect to be modified by the other. I learn a language from others, but I can then modify the way language is used by my own creative deployment of words. This, I think, is what Descartes actually thought, though it seems that his thoughts did not get expressed in words that conveyed his thoughts clearly enough for some other philosophers.\n\nThinking of coffee\n\nThe statement that I am now thinking about a cup of coffee is, despite Ryle's denials, an untestable categorical proposition, and is not just an improbably large set of testable, hypothetical \"if-then\" statements about how I am liable to behave in proximity to coffee cups. No one can know whether I am lying about my thought or not. You have to take my word for it. It is a prime example of a factual claim that is not capable of public verification.\n\nI could say, as Ayer did, that I can privately verify it. But is that really so? After all, my thought passes very quickly, and my memory is very bad. By the time I get round to verifying whether I am thinking about coffee, I have stopped thinking about it, and instead I am thinking about verification. So I have to depend on memory, and I have no way of checking that my memory is correct. The truth is that I cannot even conclusively verify my own experiences. I can only say that it seems to me that I had them. I am certainly the best judge of whether or not I did have them, but I am not wholly reliable. There is no reason why introspection should be infallible, but that does not mean introspection does not exist. I just have to put up with saying that it seems to me that I thought about coffee \u2013 and if I did not believe myself most of the time, I would be in deep trouble.\n\nMaybe we have to drop the verification principle, in the sense of proving the truth of a statement by being confronted with an undeniable sense-experience. We could replace it with something like a fiduciary principle. I have some thoughts or perceptions, and I just have to trust that I have identified them and remember them correctly.\n\nThe amazing thing about natural science is that many people have very similar perceptions, and they can produce descriptions that predict some future perceptions pretty well, because perceptions happen to recur in regular ways and in partly controllable conditions. But when we come to our own thoughts, some of us have highly unusual ones. They are not widely shared by others, they do not occur in accordance with some general law, and they do not recur in experimentally controlled conditions.\n\nConsider Einstein thinking of the theory of relativity. This was a totally new and unique thought at the time. There is no law that will make it occur to people in regular ways. And there is little you can do to make it occur to others at all. You can teach them of course but that is conveying the thought to them, not causing it to occur in their brains by creating the right conditions. Nobody doubts that Einstein thought of the theory of relativity. Yet nobody, not even Einstein himself, can verify that he did. All we can do is to claim that he was the person who first wrote it down, and trust all those who have told us that, and that they got it right.\n\nSo there are millions of facts the occurrence of which is not strictly verifiable, either privately or publicly. They include all factual claims about our own thoughts, perceptions, and feelings. We just make a claim on the basis of personal experience and trust that we are probably right. If our behaviour conflicts with such a claim \u2013 if, for instance, we say that we hate smoking, while at the same time inhaling a cigar \u2013 we have a problem. But it is not obvious that the behaviour should always overrule our personal report. We may indeed hate smoking, but have decided, just once, to be bloody-minded and do something completely unreasonable, just to show that life is absurd. Thoughts and perceptions are different from and not reducible to behaviour.\n\nIn a similar way, my thoughts and perceptions are normally, and perhaps always, accompanied by or \"expressed in\" electrochemical activity in my brain. Yet no amount of electrical or chemical terminology will tell anyone the contents of my thoughts. It may be possible to say what kind of experience I am having when a particular part of my brain displays enhanced blood-flow or electrical activity. I can tell whether it is a visual experience or an aural experience, for example. But to find this out I have to ask the brain's owner what sort of experience is occurring, and then assume that similar brain-events will correlate with similar experiences in future. I may be able to tell that Einstein is thinking in abstract mathematical terms about space-time. But I will not be able to state the theory of relativity just by inspecting Einstein's brain.\n\nEinstein's brain\n\nEinstein's brain was for many years kept in a jar by Dr Harvey of Princeton Hospital, perhaps in the hope that it might one day be stimulated to produce a new and revolutionary physical theory. But if it did, how would we understand it? If we had a neurological dictionary correlating each brain-event with some word or letter, we could write down all the words the brain uttered, like reading a morse-code message. As long as the brain spoke English (or some other known language), we would understand Einstein's brain.\n\nThis assumes that there is a one-to-one correlation between phonemes and identifiable and discrete brain states (which is not at all obvious and almost certainly not the case), and that thoughts are always formulated in some grammatically correct language (which is even less obvious). Still, it may be possible to \"read thoughts\", as long as people are happy to stay connected to brain scanners while they are thinking. That would not be very different from telepathy, an alleged direct knowledge of what someone else is thinking. It would certainly not show that thoughts and brain states were identical. We would just have to assume that each brain state correlated with a specific thought or part of a thought. Whether or not this is so is a factual question, the answer to which cannot be decided by a mere verbal definition. Yet we could never check that or verify it. The best we could do would be to use the fiduciary principle and believe what the brain said when we asked it.\n\nI suspect it would say, \"Get me out of here!\" or \"Leave my brain alone.\" There might be problems in getting it to concentrate on problems in physics, if it realized its rather unusual situation. But the stubborn fact is that while we can observe neurons firing in the brain, we cannot observe what the brain is conscious of when its neurons are firing.\n\nThere is an additional problem. Presumably brains function in accordance with the laws of physics. But Einstein's brain also functions in accordance with the laws of logic and of mathematical deduction. Could we, just using the laws of physics, predict what Einstein's brain would say next? Or would we need to know how to do differential equations before we could say what state his brain would be in when he had finished solving such an equation?\n\nComputers, functioning entirely in accordance with the laws of physics, can solve differential equations \u2013 but only if the rules of deduction have been put into them by intelligent minds. Physical pathways can be constructed to produce results in accordance with stateable rules, of logic or of anything else, such as chess. But computers will not know that they have reached a result. They simply follow the rules and then stop, without any idea of what they were trying to do, and without any knowledge that they have solved anything.\n\nThis shows that \"trying to achieve a goal\" and \"succeeding\" are not ideas that occur in physics, though they can be expressed in purely physical events, if those events are organized in a specific way.\n\nA Brahms symphony can be fully expressed in the physical structure of a compact disc. But a recital of the string of binary digits that make up the compact disc would not sound as attractive as hearing the symphony. Brahms was not trying to write strings of binary digits. He was trying to write beautiful music. That could be put into binary strings, but then some device is needed (a CD-player) to turn those strings back into sounds, which can be heard as violin, not just binary, strings.\n\nIf I suggest that the binary strings just organize themselves without even having any conception of what music is, and by chance they happen to play a Brahms symphony, it would be hard to take me seriously. It is the same with the brain. People do not try to make various sets of neurons fire. Most people do not even know what neurons are exactly, much less how to start firing them. Someone may try to solve an equation. The mental acts they perform in the course of doing this can be translated into physical brain states, but it is the purposefully directed acts that decide the order in which brain states occur. They do not just put themselves into a certain order, which miraculously makes me argue in a correct deductive way. The brain states then have to be translated back into mathematical symbols, understood by a human mind to be an argument with premises and a conclusion.\n\nOn this account, the brain is a highly organized mechanism for performing mathematical operations, for storing them, and for translating them into occurrent thoughts \u2013 axioms, operations, and conclusions. The brain's operations are all purely physical, but its structure, the ordering of its successive states in a logical argument, and the understanding that what has gone on is an argument, and not just a succession of physical states, are all non-physical.\n\nThe brain from the inside\n\nRyle insists that it would be a mistake to think that there is some inner replica of the brain which structures the brain, and which orders and understands its physical states. He is right; the mind is not an inner replica of the brain. It does not repeat what the brain does on an invisible inner stage. Yet the brain has an inner aspect. Not only does it appear to others as an electrochemically active lump of porridge, but it also is known to itself as a set of thoughts, feelings, and intentions.\n\nThe structure of the brain has evolved from simpler stimulus-response structures, as organisms come to have more complex and sensitive responses to their environment. Rocks do not have a rich and vibrant inner life. They do not even have the complex organization that would make such a life possible. It is not until central nervous systems develop that anything like consciousness appears in some sort of feeling-life, the nature of which we can hardly imagine. Intelligent consciousness is the inner aspect of the complex organized physical systems we call brains.\n\nAt that stage, the organization itself becomes a new phenomenal and causal factor in the behaviour of organisms. At simple inorganic levels the laws of nature are fairly simple, routine, and repetitive in their operation. The laws of mechanics are almost sufficient to describe what is going on. But at complex organic levels new laws come into operation which are more subtle, and exhibit novelty and creativity. The inner life of organisms is not captured by simple physical laws \u2013 that is the mistake of reductionism. More complex laws of psychology, of social relationship, and of personal creativity, must be added to the simpler physical bases of behaviour in order to understand human life. Some behaviour will not fall under general laws at all, because of its unique complexity.\n\nWe need not think of mind as a complete substantial entity, which mirrors the activity of the brain or body. We can think of it as developing various degrees of awareness and creative responsiveness, as the inner aspect of what appears to us as the physical organism. We might say that there are two modes of access to the same reality, which is active and emergent. There is no threat to this view in saying that all mental operations are strongly correlated with brain processes. But we might want to stress the importance of the inner mental aspect of such processes. That aspect is what contributes value to being, what adds understanding to existence, what adds individual uniqueness to what otherwise might be universal processes, and what adds purposive direction to quasi-mechanical routines.\n\nThis is what after all Ryle himself believes. He says that when people concentrate on what they are doing, they do not perform two operations, but what they do has to be described \"in terms of semi-dispositional, semi-episodic epithets\". He is not saying that human behaviour is just a series of \"if-then\" statements. There are inner episodes of attending, being careful, and so on. I think that it is slightly misleading, and that it actually did mislead Ryle on occasion, to say that these episodes \"do not signify the concomitant occurrence of extra but internal operations\". For there are extra internal operations, which make a real causal difference to what happens in the external world \u2013 intelligent action is observably different from inattentive action. The point is that these internal operations and episodes are precisely the inner aspects of external operations, not intrusions or interferences from some other quite separate spirit-world.\n\nVolitions and the will\n\nDoes the inner aspect of attending, concentrating, and intending make any causal difference to what happens in the world? The whole of human life assumes a positive answer. We are commanded to pay attention. We have to concentrate in order to improve our tennis strokes or our violin playing. We are blamed if we do not try our best. And we are usually praised if we try really hard, even if we fail. My school reports used to say: \"He has done everything that we could have expected from him.\" Fortunately my parents always thought that was good, as they assumed that my teachers' expectations were high. I knew better, but I said nothing. The point is that my teachers did expect me to try, knew that what I did was limited by my abilities, but assumed that I could improve them by exerting a little more effort.\n\nEducation would be very different if we knew that everything people did was controlled by their brains and so could stick them into a brain scanner for half an hour and modify their brain states in the desired way, without them having to revise for days on end. Some schools seem to work on a principle rather like this, but we still think we can tell the difference between people whose brains have been efficiently modified by rote learning so that they know all the answers to standard examination questions and people who show originality and ingenuity, even though they have never even heard of the standard questions.\n\nMore importantly, perhaps, when it comes to moral issues, we assume that people generally know the difference between right and wrong and can stop doing wrong if they choose. There are exceptions of course. In France \"crime passionnel\" used to allow you to shoot your wife's lover if you found him in her bed. In England, however, this has never been allowed, since the English are expected to be able to control their passions or, more probably, it is assumed that they do not have any passions to control.\n\nRyle is adamant that there are no such things as \"volitions\", special mental acts that precede and cause physical acts. Yet of course he recognizes the difference between acting purposefully and acting inattentively or without thought. The difficulty is to capture that difference.\n\nWe should begin by recognizing that it is not brains who act, but persons. Persons have brains, just as they have fingernails, but neither fingernails nor brains act. A person is a very complex structure, including a brain, a body, a context of social relationships, and a place in history. Ryle stresses that we cannot count volitions. We do not know how many volitions we made this morning and whether they were hard or easy, painful or pleasant. But we cannot count how many actions we did this morning either. The mistake is not in positing mental acts. The mistake lies in thinking of mental acts as countable, discrete episodes.\n\nThe answer to the question \"How many volitions did you have when you wrote this book?\" is that that I was voliting continuously. There are no quanta of voliting. There is just a stream of volit, which I could split up in various ways. I can certainly decide to do something, but even then my decision may not be one identifiable action. It is a sort of redirection of attention, and it may occur over a long period of time.\n\nIt is a bit like asking, \"When you were looking at that painting, how many looks did you have, and were some of them longer and more painful than others?\" I probably looked at the painting for some time, more or less attentively. I certainly intended to look at it, which is to say that I directed my attention in a particular way. I am, when I am awake, continuously attending and intending. These are continuous mental processes. They are inner, if you like \"occult\", acts, though they normally have behavioural symptoms and perhaps always have correlative brain states.\n\nThe deepest argument for mental action is that without it there would be no knowledge of the world at all. Indeed, the world as a phenomenon would not exist. For I construct the phenomenal world as a sort of inner simulation of external reality. Some say that it is the brain that constructs such a reality-simulation. But they forget that the brain is part of the simulation. What appears to us as the brain is part of the phenomenal world that \"we\" construct. So the \"we\" cannot be the brain as it appears to any of us. It can, however, be the hidden reality of which the brain is an appearance, a reality not accessible to the senses, to which my conscious awareness of perceiving and thinking and feeling gives a distinctive mode of access.\n\nThat hidden reality, however, does not just consist of whatever gives rise to the physical brain. It is expressed in the brain as it exists in a specific body and in a wider social context and in a still wider set of historical and cultural occurrences. All these relationships go to make up that complex structure that gives rise to a simulation of reality, as it perceives and responds in feeling, evaluation, and action to the world from its own unique perspective.\n\nWhen I say that persons act, I mean that there are centres of perception and innovation which exist within a many-layered web of physical, organic, personal, social, and historical relationships. They reflect that web from their own point of view, and they initiate changes in that web in response to their evaluations of what they perceive.\n\nThe brain has a central focal position in the web, but to simply concentrate on the correlations between brain and mental state as if they existed in isolation is to omit most of the rich complexity and diverse forms of relationship which go to form the content of mental states and the principles of causal change that are initiated within such complexes.\n\nPerception itself is already a principle of causal change. It selects and integrates simulated sensory data from a surrounding environment. The fact that it occurs changes the causal, integrated, relational structure of which it is part. My perception of the world changes my response to the world. This happens continuously while consciousness exists.\n\nIs this not just what Ryle thinks? In a way, yes. But not in the way he thinks. For he concentrates on the public, on the phenomenal, to the virtual exclusion of the private, which exists beyond the reach of sensory observation. Yet what is really important about persons, what we value about them, is their inner unique reflection of and response to the wider reality of which they are part. Persons have a unique series of experiences, which they integrate into their mental life in a unique way. That integration governs how new experiences are interpreted and builds up patterns of reactive action that become established as exercises of attention, practice, and achievement. The process is a continuous one of reception, integration, and response.\n\nPersons are essentially responsive to a wider reality, not isolated in private asylums. They are essentially active causal agents in modifying that wider reality in response to their own perceptions and feelings. The inner life of a person is what drives the behaviour of a person. While it is correct to stress the vital importance of behaviour, of social context, and of active relationships within that context, it is radically misleading to deny that any unique inner perspective, evaluative feeling, and agency exist.\n\nThe inner life\n\nPsychiatric counselling is the activity that most clearly reveals the importance of such an inner mental life. A good psychiatrist will try to empathize with patients, to be sensitive to what it is like to think and feel as they do. This requires a non-objective view of patients, an attempt to understand their inner lives, though without either approving of or condemning those lives. It takes years of training and skill to do this, and to make sense of the weird-sounding beliefs of some psychiatric patients demands a high degree of imagination and sensitivity. Yet the attempt can be made to understand how they see the world and interpret their experience, and why they see it as they do.\n\nSometimes, at least, patients can be encouraged to deal with their obsessions or neuroses by using specific techniques. They may be invited to take active control of their thoughts and habits, and it is assumed that they may be able to control such things. Psychiatrists know that there may be severe limits to such self-control, and sometimes there are physical pathologies of brain function that need to be dealt with medically. Nevertheless, an important area of self-analysis and reflection may remain, and to acknowledge that is part of respecting the proper personhood, the real if limited responsibility patients have for much of their mental lives.\n\nPatients may be asked to consider their goals and the means to achieving them, and in that process they are invited to guide their thoughts (and thus their brain states) and actions by conscious focussing and effort. If such psychiatric counselling is valued at all, it presupposes that reflective thought can have a causal role in influencing future behaviour, whereas it would be considered invasive to simply modify the brain directly in a physical way, without attending to the wishes and desires of the patient.\n\nThere is no question that sometimes physical intervention with the brain is necessary. The brain can malfunction in many ways, and this may produce irrational thoughts and feelings. But we speak of behaviour being determined physically only when patients cannot exercise rational control, when there is something abnormal with their behaviour. In normal cases we assume that conscious thought can modify brain states, though we have no idea of the mechanics of this process.\n\nBut this is not unusual. We have no idea of the mechanics of how causes bring about effects in the publicly observable world either or of how it happens that causes are regularly followed by specific effects. That is just the way things are, and we have to accept it. The slightly different thing about thoughts causing brain states is that there is no public access to a person's thoughts, and there is no way of measuring thoughts quantitively. Thus we cannot set up equations describing the interactions of thought and brain, equations that could be checked experimentally. We cannot set up equations describing how brains give rise to thoughts either, but it is obvious to most of us that they do.\n\nOne way to think of this is to think of mental and physical events as two aspects of one composite reality, which taken together form a single whole. It is that single whole which is the causal complex that gives rise to the whole that succeeds it (the effect). It seems natural that the causal properties of such a composite whole will be significantly different from the causal properties of non-sentient matter, not organized in such a way as to generate mental events.\n\nAt the simple inorganic level there are very regular and repetitive laws, which can be plotted, at least in relatively isolated situations, with a high degree of simplicity and predictive accuracy. This simple scheme breaks down at the subatomic level, where indeterminacy and entanglement introduce much more complicated factors. But it operates at the level of everyday observations, where wave-functions collapse into observable events. At the level of highly structured brain states, operating in organic bodies and rapidly changing social contexts, the operation of laws is much looser and permits novel, unique, and unpredictable occurrences. Subjective evaluations and a relatively free choice of goals, social constraints, and learned and modifiable behaviour patterns become factors that economists and social scientists have to take into account in explaining human conduct.\n\nIt is not that ethereal minds follow one set of laws (where evaluations, reasons, and goals rule), and brains follow another set of laws (where only charge, mass, and spin rule), and each must in some mysterious way mesh with the other. Rather, when complex brains come to exist and operate properly, the sorts of laws that govern their behaviour expand to include new factors of subjective evaluation, creative response, and empathetic relationship. Rocks and planets do not evaluate things, adjust their behaviour accordingly, and react to the perceived feelings and purposes of other rocks. Rocks have no feelings or purposes. But people, complex organized physical organisms, do. There are not two separate worlds. There is one world, existing at various levels of emergent complexity. At the higher levels, the behaviour of objects is modified by the presence of feelings and purposes, which add new causal influences that shape the immediate future states of that part of the world.\n\nRyle puts this by speaking of the \"bogy of mechanism\". The ghost is not in the machine; it is in the thought that the universe is a machine. Mechanical laws only apply to a certain sub-class of physical events \u2013 namely, mechanical events. But there are lots of non-mechanical events, including, most obviously, people, who are not machines. So \"the discoveries of the physical sciences no more rule out life, sentience, purpose or intelligence from presence in the world than do the rules of grammar extrude style or logic from prose\". Exactly so. Unfortunately, Ryle goes on to say that sentience and purpose do not imply extra entities which have any causal properties \u2013 that would, he thinks, be dualism, a very horrid thing.\n\nYet an intention is often an occurrent thought that causes physical changes to occur, and sentience is almost always a set of occurrent feelings and perceptions that affect the way observers behave. Thoughts and feelings are parts of the complex whole from which an immediately succeeding state is generated. They make a difference to what happens next.\n\nRyle argues that it is not true that I first of all do something (have a thought) which causes my body to act. For then my doing something mental would have to be preceded by an earlier act that caused me to do that thing, and so on ad infinitum. But that is not so. Not all my doings have to be preceded by prior doings. I can just do some things straight away (Ryle obviously agrees, because he says so). Very often, though not always, what I do straight away is form a thought about what I might do next. Not all my acts need to be preceded by theorizing explicitly. But some do, when I wonder what I am going to do next. Such wondering often does take place, and it involves some occurrent yet occult acts, which can issue in a later physical action (though it may not, since I may change my mind). There is no infinite regress here, just the fact that sometimes I think before I act, though quite often, unfortunately, I do not.\n\nRyle and I agree that it is misleading to speak of minds as knowing propositions and producing hidden volitions in a quite separate mental world, which is then mysteriously causally connected to this physical world, in which the body resides. We agree also that it is very important to see human beings as intelligent agents with purposes and values, living in a social world, developing or failing to develop their abilities and dispositions, and interacting with others, whom they know more or less well. We disagree about whether our ordinary human actions in the world have a vitally important aspect which is private, capable of being completely hidden from others, and which has an important causal influence on some purely physical movements. And we disagree about whether philosophical reflection can cast any light on this mental aspect by bringing out its metaphysical foundations and implications or whether philosophy simply asks us to attend closely to how we actually apply words to observable human behaviour. Behind these differences lies a different view of what human persons are, and of what the place of consciousness, value, and purpose is in the world. I think Ryle's rejection of materialism and mechanism, and his affirmation of distinctive human values, is entirely correct. My only complaint is that he does not bring out the philosophical underpinning of the humanist understanding he wishes to defend. In the end, an appeal to common sense is too arbitrary and insubstantial to support a totally humanist view.\n\n## Chapter Twelve\n\n## Minds and moral values\n\nFeelings are not just, as Ryle seems to say, tickles, urges, and tinglings. They are our deepest forms of response to the world in which we find ourselves, and they are closely connected with our most basic evaluations and active responses. As Aristotle said, what is distinctive of human personhood is the ability to pursue human excellence in relation to what is good \u2013 that is, the pursuit of virtue. The distinctive virtues of a human life are intelligent freedom and creativity, conscious appreciation and understanding, and the fostering of social cooperation and compassion. These are the objects of intelligent and informed moral choice, and they involve the cultivation of feelings that can more fully realize the possibilities of a good human life.\n\nIn this chapter and the next I want to explore further just what it is about the inner lives of human persons that gives them unique moral value; what enables them to live a genuinely human life, a life that is good for human persons as such. I want to argue that it is very largely their capacity to have a rich and complex feeling-life, in which states and objects are felt and evaluated, and become the basis for intentionally aiming at future goals. Moreover, this is not just a matter of purely subjective feelings, which can vary indefinitely between different people. There are some objective goals that are rationally worthy of choice, and the moral uniqueness of human persons lies in the ability to choose or to ignore such goals. Without the capacity to feel, to evaluate, and to choose future goals, humans would not be the morally responsible agents, worthy of special respect and compassion, that most of us take them to be. Human moral dignity does not lie, as many materialists suppose, in pursuing actions that have proved conducive to survival in the long evolutionary past of humanity. It lies in the capacity to choose goals of action that are (whatever their past or present relevance to survival) objectively good, and that are known to be good by rational reflection upon inner human experience. That is why an affirmation of dualism (or dual-aspect idealism, as I have called it) is of the greatest practical importance.\n\nI believe that almost everything that we value about human beings belongs to our inner lives. Of course we like having bodies, and we are very sorry when bits of them do not work. But even then what is important about bodies is that they give us pleasure or pain \u2013 conscious feelings \u2013 and they enable us to perform some interesting activities and pursue some of our purposes and desires.\n\nEverybody would prefer to be happy than to be in intense pain \u2013 well, almost everybody. For most of the time, and for most people, pain is bad. So straight away we find that good reasons for action involve feelings of pleasure and pain. It is reasonable to avoid pain and seek pleasure; pain is bad and pleasure is good \u2013 that, for most people, is just obvious.\n\nI do not think that seeking pleasure and avoiding pain are the only good reasons for action, by any means. Such a view fails to distinguish personal, human life from that of other animals. But the notion of reasons for action only begins to make sense when some form of consciousness \u2013 minimally, sentience, or the capacity for feeling pleasure and pain \u2013 exists.\n\nSome philosophers think that is all there is to morality. Jeremy Bentham, who had himself stuffed and put on show at University College, London, once said that the only good reason for acting was to seek pleasure and avoid pain. Presumably he thought that seeing him stuffed would make a lot of people very happy.\n\nThe best hope of personal happiness may lie in fulfilling our natures as human beings, as essentially related and interdependent and social agents. We are not machines that accumulate as many units of pleasure as possible, and who see other human machines as competing pleasure-units who are primarily useful to us as providers of our own pleasure.\n\nWe like things, states or activities that make us happy. But, as Ryle says, there is not one thing called happiness that we can seek just for its own sake. When I enjoy gardening, I do not experience two distinct things: gardening and a feeling of pleasure. I feel pleasure in gardening. But of course pleasure does exist in addition to digging behaviour. It is the subjective feeling I have when I dig. I certainly know the difference between having that feeling and not having it when digging on a cold and rainy day.\n\nPeople feel pleased or disgusted by different things. Bentham hoped that we could measure pleasure and so calculate just how much pleasure any action might bring to everybody concerned. But, he said, \"Pushpin (pool or snooker) is as good as poetry.\" Five units of pleasure from lying in a jacuzzi is five times as pleasant as one unit of pleasure from reading Dostoevsky. This finding may seem to be corroborated by the unfortunate (or perhaps very happy) monkeys who had their brains electrically stimulated to produce orgasms whenever they pressed a red button on a desk in front of them. What happened was that they pressed the red button repeatedly all day until they were completely exhausted. But maybe that was because they couldn't read Dostoevsky.\n\nOrgasms may be very pleasant. But even Jeremy Bentham might hesitate at the thought that we ought to aim at the greatest number of orgasms for the greatest number of people. Most philosophers tend to agree with that other great utilitarian philosopher, John Stuart Mill, that the \"higher pleasures\", pleasures of the mind, are worth more than sensual pleasures. They engage the mind; they require learning and practice; they require concentration and skill; they extend understanding and creativity; and they seem to be worthwhile in themselves, not to be just ways of filling in the time. Even orgasms, for many of us, are enriched when they occur in the context of a relationship of genuine love and personal commitment. They are not just bodily sensations.\n\nWhatever arguments may be used to speak of distinctively human pleasures, they essentially involve types of human consciousness and understanding, in which feelings of many different sorts are involved. Feelings are, as Ryle rightly said, specified by their objects, by the sorts of activities that evoke them. Such activities are often chosen because of the feelings associated with them. To the extent this is true, the feelings that only an individual can experience are real elements of reality, with real causal influence. They add a vital element to observable behaviour, and they belong to the inner life of socially engaged human animals.\n\nGilbert Ryle's peculiar feelings\n\nGilbert Ryle was basically an Aristotelian, who agreed that true happiness is to be found in the exercise of distinctively human skills and excellences (in the pursuit of virtue, as Aristotle put it). Strangely, however, when Ryle came to say what feelings were, he provided the following list: \"thrills, twinges, pangs, throbs, wrenches, itches, prickings, chills, glows, loads, qualms, hankerings, curdlings, sinkings, tensions, gnawing and shocks\". One can only assume that his inner life was a succession of more or less alarming episodes of chronic indigestion.\n\n\"How are you feeling?\" we can imagine his colleagues at Magdalen College asking. \"This morning,\" he might reply, \"I had three twinges, a small curdling, and two throbs.\" If he did, we might imagine them backing away with a slightly alarmed expression. This is certainly a very odd list of feelings. When listening to Mozart, one may be overcome by feeling, without being able to identify any pangs or itches.\n\nThe \"myth of feeling\" that Ryle opposes is the myth that feelings occur in some hidden world, and that they cause actions in the physical world to occur. Whereas, he says, feelings or emotions are primarily dispositions to behave in certain ways. Dispositions are not causes. To be vain, for example, is to be liable to say vain things and to parade around in front of people when the opportunity arises.\n\nIt is true that there is not such a thing as a feeling of vanity that may unexpectedly descend upon you when you are walking down the street and that suddenly causes you to say things like, \"What a handsome person I am.\" Vanity is a personality trait, a disposition to say vain things, to preen yourself in the mirror, and so on. But part of this behavioural disposition is that when you look at yourself in the mirror, you do have a feeling of pleasure in your own handsomeness. Those feelings may not be causes of your behaviour, but they are important parts of being vain. Vanity is not in itself a feeling, but there are feelings of vanity, which may be characterized as the sort of feelings you are apt to have when you think unduly well of yourself in relation to other people.\n\nRyle is right; feelings are not usually causes. Yet he is wrong to imply that feelings are not occurrences that are quite different from physical behaviour, and that are very important to a person's life. Such feelings are part of a very complex integration of many factors \u2013 the brain is in specific states. The body exhibits physical symptoms and behaves in characteristic ways. That body exists in a specific social and historical context, which makes evaluative comparisons with other people possible \u2013 the vain fop must have the belief that he is better than others, he must perceive himself in a specific way, and then there must be that sort of \"puffed up\" feeling that consists in the felt awareness that one is better than others. The cause of vanity, insofar as there is one, is a specific sort of awareness in a social context, which gives rise to evaluative beliefs that have a felt emotional quality, a distinctive sort of pleasure that is naturally expressed in bodily sensations and physical behaviour of a peculiarly obnoxious kind (to others).\n\nIt is the evaluative element that is important to feelings and emotions. We feel pity if we feel pain at the suffering of others, and evaluate such suffering as bad and to be avoided where possible. Pity is a very natural feeling. It involves the perception of another, the belief that they are suffering, and the evaluation that it would be better for them if they were not. The feeling of pity is the felt awareness of another's suffering, coupled with some desire to alleviate it if possible. There is good reason to seek states that bring pleasure and avoid states that bring pain. But one kind of pain is the pain you may feel at the perception of another's suffering. Humans are so bound together as social animals that good humour is infectious, and so is sorrow. We are well aware of how one person can cheer up a whole gathering, while another person might spoil any part by their depressed utterances. It would be very hard to have a great party while one of the guests was noisily expiring in a corner. There may be sadomasochistic parties where that would be regarded as adding to the fun. But they are not the norm. Because most of us are taught to laugh by our mothers and soothed by them when we feel pain, it is entirely natural to respond to the happiness or gloom of others by mirroring their state in ourselves.\n\nIn our emotional inner lives we naturally appreciate beauty of various sorts and are repelled by certain smells and sensations. When we perceive the mental states of others \u2013 and Ryle is entirely correct to say that we do so normally by interpreting their behaviour \u2013 we appreciate their happiness or intellectual acumen, and we are repelled by their pain or vanity. There is a social dimension to most feelings, as we evaluate the evaluations that are made by others (we see that they dislike pain, and so we dislike their pain too).\n\nThe emotional lives of humans are governed by the evaluations we make of the complex world we encounter in awareness (perception interpreted by thought). That world includes the dispositions and intentions and feelings of other persons and of ourselves. Our evaluations are not arbitrarily subjective \u2013 that is, they are not correlated contingently with the objects of awareness. We do not see the aurora borealis and say, \"What a horrible green mess.\" Of course we might. It is reported of Dr Johnson that when he first saw a hillside of gorgeous heather fully in bloom in Scotland, he simply said, \"What horrid blotchy purple hills.\" There is no accounting for taste. But we can learn to appreciate purple and many other things too. Or at least generations of musicians and artists and literary theorists think and hope we can.\n\nTo dislike the music of Wagner is one thing. But to say that it is trivial or superficial or devoid of merit is another. You can learn Wagner's place in musical history, how harmonic traditions grew more complex, and how Wagner developed one operatic tradition to express it in a richer way, which was outstandingly excellent of its kind \u2013 and you can still not bear to listen to The Ring. But you would have to say that it displays outstanding excellence, and that it develops a tradition in a profoundly creative and distinctive way. You may not like that tradition \u2013 but you will have to admit that others do and are deeply moved by the music, and that it meets superbly certain criteria of excellence, which you may not like very much.\n\nThere are seemingly irreducible differences of taste. For decades no one liked the music of J. S. Bach, and then he became, for many, the greatest of all musicians. Who is right and who is wrong? Or is everyone's opinion as good as everyone else's? Maybe these are the wrong questions. The fact is that some people will never like Bach \u2013 too austere, too intellectual, too tuneless, some would say (except for one or two favourite pieces). Yet there is much to appreciate in Bach, and you can be taught to appreciate it better if you have the initial \"feeling for it\".\n\nIt is a bit like wine-tasting. Some people will never drink wine. Some do not like it. Some like it, but cannot tell which wines are good and which bad. But we might all admit that some people have a more discriminating palate, and can distinguish between different vintages of the same wine with no difficulty. They are the people we would consult if we had an initial interest in wine. They are the experts, even if they are experts in areas that nobody else is interested in (like philosophy).\n\nSo maybe the bottom line is this: you cannot help your initial feelings, your basic likes and inclinations. But you can cultivate them and have a \"refined palate\" or a \"refined musical taste\" \u2013 and then you might get a job as a wine or music critic. Of course different critics will still disagree very strongly about many things. But they will, however grudgingly, have to accept that there are degrees of expertise, of knowledge, experience, and skill in appreciating wine or music or art.\n\nIn these areas, strong disagreement and generally accepted degrees of skill or competence go together. I have to admit \u2013 I gladly admit \u2013 that Gilbert Ryle was a learned, wise, skilful, and innovative philosopher. But I think he was blind to the rich evaluative and emotional life of human beings, and so I disagree strongly with some of his conclusions about such things.\n\nWhat makes human life worthwhile is our rich emotional inner life. Our perceptions are integrated and interpreted by thought, evaluated by feeling, and responded to by intention and creative action, within a social community of other beings with thoughts, feelings, and intentions. It is interpretative thought, evaluative feeling, creative intention, and cooperative empathy that distinguish human persons as autonomous moral agents and give to human life a special sort of value. Such a thought may, thank goodness, be implicit in much common-sense thought. But it can be undermined by a critical reason that overlooks or even denies the importance of the subjective personal life of feeling and intention. To that extent, Ryle's objections to Cartesian dualism may, however unintentionally, undermine a reasonable commitment to the moral value of human personhood. For such commitment is founded on a perception of the moral priority of the embodied mind.\n\nThe objectivity of values\n\nRyle says, \"A person can usually... tell without research whether he enjoys something... but so can his associates.\" What is right about this is that we naturally express our thoughts and feelings in our behaviour and in the things we say. Yet our behaviour may be misleading, and what we say often does not express what we mean. There may often be no behaviour at all \u2013 which is why, when I am thinking deeply and profoundly, my wife thinks I have gone to sleep. Also, our feelings vary in tone and intensity. They can inhibit or enhance our actions, but no one else can know just how intense my feelings are or what their complexity and quality is.\n\nUndeterred, Ryle goes on to say that if we want to find out about our feelings when they are concealed, other people are more likely to succeed than we are. It is true that we often deceive ourselves and do not admit that we are vain or ambitious or aggressive. But these are, as Ryle says, not so much feelings as dispositions or character traits, and we often conceal our characters from ourselves \u2013 probably because we do not like ourselves very much. We can, however, practice self-knowledge and explore what our feelings are like in ways that others cannot. Admittedly we do not have many linguistic resources for doing so. If I look into myself and try to describe my feeling-state at this moment, I may say something rather feeble, like \"There is a pang of contentment, with a few twinges of lassitude around the edges, a dollop of resolution, and two throbs of anticipation.\"\n\nThat is why introspective psychology came to a halt in the early twentieth century \u2013 people just didn't have the words to describe what they felt. And when they did describe, it was of no scientific use \u2013 it did not help to predict or explain causally. It just reported, in very boring ways, what was going on in someone's mind. Some philosophers concluded that there was nothing to describe or that \"what went on inside\" was irrelevant. But poets and novelists, musicians and artists, manage better by finding ways to express feelings that communicate in a special way to others or at least to those who are initially sympathetic. Even the most peculiar examples of modern art express a personal vision of the world that communicates a certain feeling toward it. That feeling could be one of disgust or alienation or sadomasochism, which is why art is so powerful. It communicates a feeling, evaluative response that is part of the artist's awareness and personal interpretation of the world and that arouses a responsive (not identical) feeling in those who are able to resonate with it.\n\nFeelings differ enormously in different people and are not separable from individual awareness, interpretation, and responsive actions. We cannot sensibly aim at having feelings as discrete items isolated from the states or activities they are feelings about \u2013 that is the utilitarian mistake. Yet without feelings we would be automata, and it would be hard to give any value to human awareness or any purpose to human actions. Human life is largely concerned with the kinds of feelings we seek or avoid, with the activities that evoke or sustain them, and with the social relationships that enrich or destroy them.\n\nAre feelings of this sort just matters of purely personal taste? Or are there some feelings that would be accepted by all reasonable people as bad and others that are good?\n\nIn the arts it is difficult to claim that there are absolute standards of good and bad \u2013 though we can distinguish different levels of skill, originality of vision, creativity, and expertise in a specific tradition or style. Within classical harmonic music, Mozart stands out as particularly skilful, original, and expressive of profound feeling. Anyone who appreciates that sort of classical music should be able to recognize Mozart as an outstanding composer. Within a particular tradition of creative art, there are standards of excellence that it would be insensitive not to appreciate. But we are perhaps prepared to accept that some people simply do not like classical music. So objective evaluations do exist in the arts, but they are tradition-specific, and it does not make much sense to say that classical music is better than romantic music or jazz. That is why it does not make sense to say that anyone, even Mozart, is \"the best composer ever\". He is the master of a specific style, but we are not required or expected to appreciate every style of music there ever was.\n\nThere are some things, however, which every reasonable person has a good reason to avoid. It would be hard to understand someone who said, \"I am really looking forward to an agonizing toothache.\" If some religious fanatic said that, we might understand that he thinks if he endures the toothache he will get some great heavenly reward. Even then he would not look forward to the toothache for its own sake, and if he did not think it was a necessary means to some greater good, he would soon change his mind.\n\nReligious fanatics apart, we would expect everyone to think that agonizing pains are to be avoided wherever possible. That is to say that pain is bad. It is objectively bad \u2013 every being that can feel pain has a good reason to avoid it, and if we care about other beings at all, we have reason to help them avoid it. Why should we care about other beings? We don't always. Little boys routinely torture insects or small animals. But because we are social animals, we do care about some other beings, if only our own children or parents. We find happiness in making them happy, on birthdays and holidays. Even if we do not care about people we have never met, we can see that they have a reason to avoid pain, and we have no reason to cause them pain, unless it will produce some good for us (which, alas, it often does).\n\nOther things than pain are objectively bad. There are very good reasons to avoid ignorance, since ignorance can lead us to make disastrous mistakes. There are good reasons to avoid powerlessness, which can cause us to fall prey to the whims of the powerful. There are good reasons to avoid hating other people, since that will usually cause them to hate and harm us if they can.\n\nThese are not absolutes. Sometimes we may prefer to be ignorant \u2013 about the time of our death or about our chances of getting diabetes, for instance. We may prefer to be slaves, if we can get a fairly easy life with a lax slave-owner. We may successfully hate other people, if we can terrify them into doing what we want. Nevertheless, in general and for most of us pain, ignorance, slavery, and hatred are things to be avoided. They are prima facie evils \u2013 things which in general there are good reasons to avoid.\n\nWhere there are prima facie evils, there must be prima facie goods. Pleasure, knowledge, freedom, and sympathetic feeling are objective goods. We will train children to seek them, and will reckon that most people will have a happy life if they achieve them or a reasonable degree of them.\n\nThese are very general values, and in human lives they will be filled out in a huge variety of ways. As I have noted, pleasure is not one thing that everyone will pursue in the same way. Happiness is to be found in many mental states and in many activities, and such states will be very diverse. But it is not an accident that knowledge, freedom, and sympathetic feeling are objective values, things that humans have a good reason to pursue and sustain. For these values point to the distinctive character of human consciousness. They are states or acts that make us fully and distinctively human. If we lack them, our distinctive humanity is impaired. A suffering, ignorant slave who hates everyone is a diminished human being, not exercising the capacities that make humans what they are. It may not be the slave's fault, but it is still a life that is objectively frustrated in many ways. Conversely, a happy, wise, free, and compassionate person is living a life which is fully human, because it exercises the capacities that are distinctive of human beings.\n\nHuman animals, supremely among all species on earth, are capable of abstract intellectual understanding and of delight in beauty and in artistic artefacts. They are able to be creative in their thoughts and actions, transforming their environment in new ways. They are able to empathize with the feelings of others, and to cooperate in learning and in action, so that they can share in the experiences of others and work jointly with them to devise new cultural projects.\n\nThese capacities are not discrete and isolated from one another. They are intertwined, so that understanding and appreciation of beauty are creative acts that involve cooperation with others. They suggest an ideal for human personhood that consists, as Aristotle put it, in the unimpeded exercise of distinctively human capacities. To understand and appreciate fully, to act creatively, to be compassionate, and to cooperate with others \u2013 these are the virtues of a genuinely human life. Each human being is born into a unique situation and faces problems and possibilities never shared in detail with anyone else. Nevertheless, there are goals for human life as such, and in every historically particular situation the mind can be disciplined to love those excellences that are distinctive to human personhood.\n\n## Chapter Thirteen\n\n## Acting for the sake of good alone\n\nIn the end, the essential capacity of human persons is the capacity to choose the good for its own sake (in Christian terms, to love God with mind and heart and strength). The privacy of inner experience and the importance of introspection and growth in self-knowledge is of crucial moral importance for human personhood. That is why Ryle's attack on dualism, implicitly on idealism, and on the inner life as primarily distinctive of human personhood, despite its many insights, should be resisted.\n\nIf we are disposed to think of mind as having ontological priority, it will be natural to think of the fullest development of mind as the innate goal of human history or even of the cosmic process as a whole. Human beings could well be only one of many forms that personhood or mind takes in the cosmos. But the development of personal powers embodied in a particular species \u2013 homo sapiens \u2013 provides an objective goal for human life and feeling.\n\nThe goal is founded upon the distinctive capacities of human nature, which characterize what is rationally good and desirable for a being with such capacities, and what is bad or diminishing and is to be avoided. This moral ideal is concerned with feelings, but feelings considered as integrated with social and personal activities, with knowledge and understanding, and with rational evaluations of envisaged states and acts. Feelings are not, in other words, twinges, tingles, and throbs. They are affective and reactive evaluations based on inner experience of the complex world with which we consciously interact. They give to human lives a sense of value and a sense of purpose (to realize objective values), which is perhaps the most important and distinctive feature of being human.\n\nMental discipline is not easy. Knowledge may be a good thing, but anyone who has taught in school knows that it is quite hard to get some pupils to learn. Interests and abilities vary enormously, and some lack the interest to learn very much. They may also lack the motivation and application that is necessary to learn. Moreover, humans are very prone to exaggeration, gossip, and rumour. Lies and misinformation spread easily, and very few of us have the time and patience to gain full information even about topics we are interested in. The ideal might be truth, but the reality is often a mixture of hearsay, deceit, and ignorance. Even those who sincerely seek for truth can have a very limited view of where truth is to be found and of the best way of finding it.\n\nSome religious movements, for instance, think that believers must be prevented from reading anything that would disturb their beliefs. That attitude is not confined to religion. Ruling political parties may \"massage the truth\" so that it presents a one-sided and distorted message, and they may censor opposing opinions. In fact, it is entirely typical of human communication in general that it distorts truth and propagates prejudice and hatred.\n\nWhy should people lie? That is simple. Lies are usually more interesting than the truth. The imaginary adventures of Baron Munchausen are much more interesting than the daily drudge of an assembly-line worker. People like good storytellers. And if the storyteller can claim to be the hero of his own stories, this will add considerably to his reputation. Good liars can increase their social status enormously, as long as they are not found out \u2013 which usually requires a constant stream of even bigger lies to conceal the much less interesting truth.\n\nReally big liars will do almost anything to prevent being found out. In such a world the pursuit of truth may be difficult, even dangerous. Partly for that reason, some people who wish to pursue truth try to leave the world behind and live in relatively closed societies where truth of a certain sort is valued. Unfortunately, this policy does not really work, for most truths have to be discovered precisely in that ambiguous social world where truth and deceit are inextricably mixed up. The renunciation of secular knowledge is not, after all, a very reliable path to knowing as much about reality as possible. It is possible that cloistered contemplation does disclose very important facts about the inner life of the mind, about mindfulness, equanimity, and universal compassion, for instance. But it does not disclose much about quantum physics or genetic mutation. If we are really to comprehend the world in its fullness, we must know both about our inner lives and about the nature of the world, including the evil that is in it. The life of meditation may be one way of discovering truth, but it cannot be the only way. The pursuit of the human ideal of comprehension of truth involves many different kinds of knowing. We may only be able to pursue some, but we should not deny the existence of the others, and we should encourage their pursuit as much as we can.\n\nThe quest for truth, then, is difficult and often requires prolonged struggle against strong opposition. The same is true of the search for the other basic personal values \u2013 of appreciation, creativity, compassion, and cooperation. They require effort and sometimes sacrifice in a world filled with indifference, the will to power, hatred, and conflict. It is not surprising that a fully human life is rare and attained only by few. Is it reasonable to demand it of any human being?\n\nMany philosophers have held that there is an ultimate moral choice in every human life. We can choose what is good for its own sake or we can choose what is good for us, regardless of others. We can put goodness first or we can put ourselves first. Reason cannot decide between these choices. Neither can desire, for we must decide which of our desires to follow. Nor can the choice be seen as an arbitrary decision of will, without cause or reason, for such a choice would be absurd. No, we decide whether to subordinate ourselves to the good or to subordinate the good to ourselves. That choice determines the sort of person we are, and so is of fundamental importance to our lives, yet no further reason can be given for it than that one ought to choose the good (or alternatively that one ought to choose one's own long-term good). This is the truth behind Kant's dictum that we must do our duty for its own sake.\n\nDuty and desire\n\nKant is often misunderstood on this point. He is supposed to have said that you only know you are doing your duty if you act in opposition to your own desires. Otherwise you might be acting in accordance with desire, not because it is your duty. So you are only really good if you are doing what you do not want to do. It is as if you are really desperate for a drink, yet you know you ought not to drink because you are going to drive. Being English, you pass a pub every few yards. Every time you come to a pub, you have a huge craving for a drink, and you have a huge struggle to overcome that craving. After a few heart-stopping minutes, the sense of duty wins, though the struggle leaves you emotionally exhausted. Heaving a sigh of relief, you walk on \u2013 only to come to another pub, where the same process takes place again. By the time you get to your car, nine pubs later, you are a quivering wreck. Having done this every day for a week, you might decide it is better never to go out at all. Except that it might be your duty to go out. So every day you have a nerve-racking struggle about whether to leave the house or not.\n\nTo be fair to Kant, he did warn against \"moral fanaticism\", which is the mistake of letting your life be filled with crucial moral decisions. As Aristotle said, it is better just to become the sort of person who is habitually good. Then you can pass all the pubs with equanimity, because, once you have made a firm decision not to drink and drive, you are not even tempted to have a drink.\n\nKant said that it was just about impossible to tell if you were acting from a purely moral motive (because it is right) or from long-term selfish desire. Maybe we should not agonize about that, but just get on with trying to do things because it is right to do so, without worrying about what our deepest motives are. There are people who will say that all human actions are really self-interested. Even if we give all our money to the poor, we really do so because we gain some inner satisfaction or some social approval from doing so. You can never refute that argument. But you could equally well say that all our selfish actions are really altruistic, because they are aimed at the happiness of someone who does not yet exist. I exist now, but people who exist in the future are not identical to me, and so even my selfish actions are aimed at someone other than me (some future person who happens to share some memories and character traits with me and who happens to have a body very like mine), which is a form of altruism. I don't suppose that argument seems very convincing, but it is no worse than the argument that the more altruistic I try to be, the more selfish I actually am.\n\nThe upshot seems to be that reasoning can tell me what things are good, in general. But the decision to do things because they are good is not made either by more reasonable people or by people who just happen to have the most altruistic desires. Moral action is not unreasonable, and it is not necessarily opposed to desire. But it is quite distinct from both reason and desire. It is a basic option that determines what sort of person you are.\n\nIt was for this reason that Kant felt it would actually be harmful if you knew there was a life after death, where you would be judged for what you had done during this life. That, he thought, would turn moral action into long-term prudence, and so it would undermine the whole point of morality. You have to do your duty without thought of personal reward. In fact, if you do things out of self-interest, you will go to hell. So everybody who does things because they want to go to heaven will land up in hell. And only people who do not care whether they go to heaven or not will get there. It seems that hell will be full of people who have spent their lives trying to get to heaven, while heaven will be full of people who do not particularly want to be there. There must be something wrong somewhere.\n\nKant's suggestion actually is that you should do things because they are right. But it is all right to hope that it will make you happy in the long run, as long as that is not your main motive. He even, notoriously perhaps, argued that the hope for \"happiness-in-accordance-with-virtue\" was a condition of the possibility of rational moral action. Perhaps that is a way of saying that it is pointless to aim at a just society if such a thing will always remain impossible. A reasonable action will only aim at something that is at least possible, even if very difficult.\n\nBut the reply to that is that moral action can always make things better than they would have been, even if it will never achieve a perfect society (whatever that would be). If the world is like that, it is tragic \u2013 good people will often die in vain, unknown, unrecognized, and unappreciated. Naturally, we would wish the world not to be tragic. That is probably one major motivation for some religious beliefs. It does not make such beliefs true, but it may at least ground them in reasonable and humane and comprehensible human desires, rather than in irrational, vindictive or self-centred motivations.\n\nIt may be said that Kant makes an unduly subtle distinction between hoping for heaven and believing that there is a heaven. If you can hope for heaven without undermining your moral motivation, then surely you can believe there is a heaven without making that your main motivation for acting. After all, even believing that there is a heaven is not absolute theoretical certainty. It is a belief that may easily be mistaken. And that may be Kant's main point \u2013 though even then I am not sure that absolute theoretical certainty about rewards and punishments would make it impossible to act purely for the sake of goodness. It would just make it more difficult to distinguish that from long-term prudence. It is a relief to find that religious people can be really good and also to find that non-religious people can be genuinely good too, just because they feel goodness is worth aiming at for its own sake, and that a genuinely human life is one that acts for the sake of good alone.\n\nThat is the main conclusion of this chapter, which has been chiefly concerned to affirm that the capacity to act for the sake of goodness, and for no other reason, is one of the most important capacities of human persons. That capacity cannot be reduced to, or exhaustively described in terms of, any genetic account of how moral awareness originated through evolution or any purely material description of what human beings are. The commandingness of morality requires an affirmation of the uniquely individual appropriation of inner experience and the responsible freedom of moral agents. That requires that human persons are more than matter, and in that \"more\" lies the significance and meaning of their lives.\n\nInner experience as evidence for idealism\n\nPart of Ryle's goal is to escape from what he thinks of as the Cartesian myth that each person has a totally private and hidden mind which has to be inferred from bodily movements in a completely uncheckable way. Ryle suggests that humans are not either ghosts or machines, much less ghosts in machines. They are social animals. Their minds as well as their bodies have a physical, genetic, evolutionary history, which is an important component of what they are. And humans are essentially social, finding themselves in a particular society with a unique history and culture, and defining themselves in relation to other people with whom they continually and directly interact.\n\nSo far so good (except that Descartes did not himself believe the \"Cartesian myth\"). But humans are also essentially morally responsible agents, capable of pursuing goodness for its own sake and of shaping both the world and their own characters in the light of their moral ideals \u2013 or, more often, failing to do so. They have a rich inner life, in which understanding, feeling, evaluation, and intention play a major role, a life and quality of experience which is not open to others to inspect.\n\nThis inner life is known by introspection, by a form of self-examination which inspects a person's feelings, beliefs, evaluations, and motives without the use of the senses. Introspection is non-sensory knowledge of one's own states of mind, and it is an important part of coming to know if and to what extent and in what ways one is pursuing a life which is wholly good.\n\nIt is quite difficult to work out exactly what Ryle thinks of introspection. He did not really like being called a behaviourist \u2013 someone who denies that there are any causally relevant mental states \u2013 though he thought it was a fairly harmless appellation. And he speaks of \"semi-episodic\" dispositional states, which include talking to oneself or knowing that one has a toothache. But he leaves a strong impression that such inner states are parasitic upon observable bodily states, like talking to others or crying out in pain, and that they are not of primary importance in human life.\n\nThis impression is strongest when he speaks of mental images, the existence of which he seems flatly to deny. I would sit in his study and say, \"I am now having a mental image of Big Ben.\" \"No you are not,\" he would reply. \"You are just imagining that you are seeing Big Ben. You are pretending to see it. There is no image there. There cannot be any image, because there is nowhere for it to be. It is not inside your skull, and it is certainly not in front of your eyes, because I can see what is in front of your eyes, and there is nothing there. This supposed image is nowhere. And that is not surprising, because you are just imagining that it is there.\"\n\nIt was very hard to think of a satisfactory reply, since I was sitting there having the mental image, and Ryle simply couldn't see it. Of course he couldn't. It was a mental image after all, and he wasn't supposed to see it. But how could he deny that I was seeing it? He was quite right that I was just imagining that I was seeing Big Ben. That is what having a mental image of Big Ben is! I did not think I was magically transported to London and was standing in front of the clock tower. I pretended that I was in London, looking at Big Ben. But I was not looking at the actual thing. I was constructing a picture of Big Ben, without actually drawing and without leaving my armchair. And there was something there \u2013 a mentally constructed picture of Big Ben, in full colour. I was imagining it. But there was an object that I imagined, and it was an object that only I could observe, not with my physical eyes, and not in any publicly observable space.\n\nRyle was so opposed to the idea of a private inner space that he had the confidence to tell me that I was not really having a mental image when I quite clearly was. As a matter of fact, I could also hear music \"in my head\", but I did not tell him that, as he might have given me an aspirin and called a doctor.\n\nI am quite convinced that I have non-sensory knowledge of mental images both visual and auditory, including dreams, and that I also have thoughts and feelings that I am careful never to tell anyone about. Ryle seemed to think that I could tell anyone about them if I chose, and that I could not know anything about them if I had not first learned from others what my thoughts and feelings were. In addition, other people are better at knowing what my thoughts and feelings are than I am. All these claims seem to me to be mistaken.\n\nHidden knowledge\n\nThere is, as usual, something important in what Ryle says. One main purpose of self-examination is to review your behaviour in the recent past and ask if it was motivated by selfish interests or if it was genuinely altruistic and compassionate. It is true that others may see aspects of our behaviour that we might miss. We are mostly prejudiced in our own favour and tend to put our behaviour in the best possible light. It is true that it is not easy to see ourselves honestly, and that we repress many feelings and motives that may be obvious to others, but that we disguise from ourselves. Psychiatrists exist partly in order to make this very clear.\n\nNevertheless not all our behaviour is overt. The \"smiling depressive\", for example, is someone who shows no outward signs of depression at all, but who may unexpectedly try to take their own lives, to the surprise and consternation of their friends. We might be well-advised to keep many of our thoughts and daydreams, many of our hopes and ambitions, to ourselves. While we may often disguise them, some of them are obvious enough to us, but to no one else. As Machiavelli pointed out, the wish to be rich and famous is best hidden from others, while we publicly express only the desire to help others and to bring about world peace. Successful hypocrisy is never discovered; it is only that most of us are not very good at it. But practice makes perfect, and if we try really hard, we can probably manage it.\n\nIt is not that we have perfect knowledge of our own feelings, while others have no access to them at all. We normally do express our feelings in observable ways, and we often do not recognize our own feelings for what they really are. We learn to formulate our thoughts about such feelings in language we have learned from others, and that may be felt inadequate for what we want to convey. Nevertheless, we do have a mode of access to our feelings and thoughts that no one else has. That mode of access is non-sensory, and it is required for full knowledge of a person. Without it, we would not be able to discipline ourselves in the love of virtue, and our lives could not be completely centred on the good and beautiful, as they could and ought to be.\n\nWe would not believe someone who said, \"I know the answer to this mathematical problem; I just cannot express it very well.\" The proof of a thought lies in its written expression, and if the expression is nonsense, then probably the thought was nonsense too.\n\nDoes that show, however, that there are no non-expressed thoughts? Surely not. Fermat's last theorem, despite having now been proved by Andrew Wiles, remained without a proof for decades. Did Fermat actually have the proof, though he never wrote it down? We shall never know. But he might have done.\n\nMany of us know that we enjoyed a performance at a theatre even if we never told anyone we were even there. We could have said so. But if we did not, and if no friends were there to see us grinning inanely from time to time, no one might ever know. Even if our friends had been with us, they might have thought we had indigestion. We might even have cried with happiness. Unless we explain ourselves, no one will never know we are happy.\n\nAnd would it make sense for us to say, \"I think I am really enjoying this play. But I could be wrong. Perhaps I dislike it intensely\"? As Ryle says, this is not a matter of watching the play, and then watching ourselves intently on some inner stage to see if there is some enjoyment there. It is a matter of watching the play in an enjoying sort of way. But that enjoyment, though often expressed in behaviour or in language, may remain wholly unexpressed and unobserved and yet make all the difference to our view of the play.\n\nSo there are forms of knowledge that may have no linguistic or behavioural expression. The test of whether it is knowledge will be some public expression \u2013 obviously, for that is what a \"test\" is, publicly observable evidence. But the introspectionist claim is precisely that there is some untestable knowledge. And it seems that the only test of that claim is to have some yourself. Such a test will itself fail the test of being publicly observable, since no one else can test whether you have had some untestable knowledge. So it is a paradoxical sort of test; it is a \"suck-it-and-see\" test, which depends on assuming what is to be proved \u2013 that there is untestable knowledge.\n\nThe importance of memories\n\nThat is not such a strange idea as it may at first seem. Most of our memories, particularly of things that only we have seen, are in practice untestable. We simply have to believe that they are trustworthy. Although Ryle does not believe in introspection, he does believe in retrospection. We may not have special inner access to what we are doing now, but we may know what we did a few moments ago. Ryle says, \"The fact that retrospection is autobiographical does not imply that it gives us a Privileged Access to facts of a special status.\" We know what we did five minutes ago in the same way that anyone else does or could do, Ryle claims. He says that our knowledge of what we did in the past is \"of the same kind\" as the knowledge anyone else may have of what we did five minutes ago.\n\nIs this so? For it to be true, we have to make the extraordinary assumption that if I had spoken my thoughts out loud, and if there had been anyone there to hear me, and if they remembered what I said, then \u2013 and only then \u2013 their knowledge of what I did would be of the same kind as my knowledge of it. In fact I did not speak out loud, and there was no one there to hear or remember what I actually did not say, and yet I remember what I thought. I can remember unspoken thoughts when no one else is there. Nobody else can ever do that with my thoughts. Here, then, is a \"kind of knowledge\" that nobody else could ever have, in principle.\n\nAnyway, what is the point of a test that can never in practice be carried out? It is like saying, \"If I were as small as an electron, then I could observe electrons buzzing busily about.\" In a similar way, if I had been at the Battle of Waterloo, then I could have observed Napoleon looking worried. But I was not there, and I never could be there. Does a purely hypothetical and indeed impossible hypothesis make an actual memory claim more reliable? Not at all.\n\nMy memory claims about the past are of the same kind as other people's memory claims about the past \u2013 they are both often untestable. Having two or more untestable claims does not strengthen the evidence for anything. For if I cannot trust one person's claims, the situation cannot be helped by trusting the claims of lots of people, none of whose claims we can trust! It would be rather like asking a twelve-year-old student to state the theory of relativity. Just to check what he says, we ask all his classmates to give their version of the theory. That is only going to make things worse. So if we distrust one person's memory claims, it will not help to ask lots of other people what they remember, because we have no good reason for preferring their memory claims. Adding many untrustworthy memory claims together, so that we get them to agree with each other, is not a guarantee of reliability.\n\nThe fact is that we just have to trust uncheckable memory claims. Then, if many people make the same sorts of claims, that does increase our general confidence in their claims. If many fallible people agree, that does not make them infallible. But it normally increases our confidence in our own claims if others make very similar claims (even though we know crowds are more subject to illusions than single individuals).\n\nSo we just have to accept, for the sake of our own sanity, that memories are in general trustworthy \u2013 which means that we have good reason to accept the memory claims of others. Far from other people's memories serving as necessary confirming evidence of our own memories, it is in fact our belief in the reliability of our own memories that gives rise to the thought that other people's memories are in general reliable too. That thought is then confirmed by our agreeing about the same things, when those things have been publicly observable. But if we have memories about things that nobody else observed, we are still justified in thinking them reliable, other things being equal. Ryle's preference for retrospection over introspection does not work, because many retrospections are about introspections, and both rely on the untestable but wholly reasonable claim that our memories are reliable in general.\n\nWhat goes for memories goes for inner experiences in general. Some non-sensory untestable knowledge turns out to be essential for human knowledge in general, and such knowledge can and does influence behaviour. Self-knowledge, obtained through introspection, can cause major changes in behaviour and can be of decisive moral importance for a human life. We do well to take the advice of others when assessing our own characters. But we would do very badly to take their advice as the only or last word on what we are and know. Our behaviour is a good clue to what kind of people we are. Yet our unspoken thoughts and feelings should often modify judgments based purely on behaviour in important ways. In a perfect world, we might perhaps express our thoughts and feelings in an uninhibited and open way. In the world as it is, only a very silly or a very saintly (or possibly a very rich) person would do so.\n\nWhat is most important about a human life is the unique quality of experience a person has and the unique moral choices that a person makes. A person's life is normally the result of a whole series of unique perspectives and choices about what to attend to, what to do, and how to do it, and of a complex of evaluations and feelings in regard to the varied situations that person has encountered and responded to. Those responses are unknowable in many respects by anyone else, and even if they sometimes remain unrecognized to the agent concerned, that agent has an access to them that no one else could have.\n\nThe ability to know oneself fully and to direct one's own actions freely to a personally chosen goal in positive cooperation with others is what defines a fully human life. That ability is rarely exercised to any great extent, and it remains for most of us a distant ideal rather than an actual achievement. Ryle is importantly right in drawing attention to the importance of behaviour and of conscious personal interactions with others in human life. He is right to oppose the thought \u2013 whether or not Descartes believed it \u2013 that human minds are isolated private and ethereal worlds where invisible levers are pulled to make bodies move. But he tends to minimize or, at his worst, to deny that which makes human behaviour and relationship personal \u2013 the inner experience and feeling that is what is expressed and shared in personal behaviour and social relationship, and the inner choices and motives that are only partly manifest in outward actions. Human life carries with it the possibility of a fundamental option for the good. That option is often deeply hidden by the ambiguities of our lives. But it may be what defines us as fully human beings.\n\n## Chapter Fourteen\n\n## The idealist view of life\n\nSo I propose philosophical idealism as the most adequate, consistent, and plausible metaphysical view of reality. It carries with it a theory of human persons as experientially unique, morally free, and fully embodied subjects of experience and action, living in an interpersonal world of similar beings \u2013 a community of social and self-realizing conscious agents. Idealism is not to be accepted because it is comforting or wish-fulfilling. It is to be accepted because it makes a reasoned claim to be the most intellectually adequate view of reality and of human personhood that human thought has devised.\n\nOf course any philosophical view will remain contested and less than overwhelmingly convincing. This is, after all, philosophy and not chemistry. But we can scarcely escape having some such view, and idealism will always continue to be one of the most intellectually impressive high-points among human attempts to achieve real insight into the nature of the complex and mysterious reality of which we humans are part.\n\nThe point of this discussion has been to emphasize that we all have privileged, though not infallible or complete, access to our own inner lives, our thoughts, memories, feelings, and intentions. This fact gives us very important information about the world \u2013 namely, that conscious experiences and intentional actions are real, not reducible to material and publicly observable facts, and morally crucial for the way we live.\n\nMaterialists, however, have a very different view. They would in general say that conscious experiences, if they exist, are byproducts of material processes, so thinking and feeling does not give any special or privileged access to reality. Reality is accessed through scientific (and materialist) theories. Yet for materialists such theories are a by-product, and an unforeseen one at that, of blind laws of nature. So, on a materialist theory, we would not expect that our theories (including the theory of materialism) are particularly reliable or informative vehicles of information about the world.\n\nMaterialism gives abstract theory priority over concrete experience, while at the same time it undermines the reliability of the reason and understanding that provides us with our abstract theories. That is like saying that we should trust reason even when what it tells us is that reason is untrustworthy. Something seems to be wrong somewhere.\n\nThe idealist resolution of this paradoxical situation is to say that our thoughts, feelings, and perceptions do give us access to reality. They suggest that reality itself may be founded not on blind chance or unconscious necessity, but on some form of purposive consciousness.\n\nMaterialists will probably protest that consciousness is a very late evolutionary development, requiring complex brains for its existence. But we need to distinguish the embodied consciousness of beings that exist in a common space-time manifold from a possible non-embodied consciousness that expresses itself in, or forms the causal basis of, the whole space-time manifold itself. The former types of consciousness do come into being as the result of a long process of evolutionary development, and they have a strong causal and epistemic dependence upon their physical environment. The latter type of consciousness does not come into being at all. The physical world depends for its very existence upon the primordial consciousness and perhaps expresses the inner nature of that consciousness, as it brings into being dependent consciousnesses which can share relationships with it.\n\nWhy should we think that there is such a primordial consciousness? The strongest form of idealism, that of Bishop Berkeley for instance, would say that only minds are fully real, and there cannot be material things that are not contents of some consciousness. Also the only sort of causal power we directly experience is our own intentional action, which brings things about for a reason. The idea of \"material causality\" or, more exactly, regularity of succession, is by contrast a relatively abstract notion. On this strong idealist view, the ultimate reality has to be mind, and ultimate causality has to be intentional agency.\n\nWeaker versions of idealism would insist that value (the positive evaluation of personal experiences) and purpose (the effort to bring about future states) are elements of reality that are not analysable in purely material terms. Therefore, mind must at least be a fundamental component of what there is, and if there is some ultimate explanation for the existence of the universe, it must include a mind-like element. Personal explanation (explaining something by saying that it is chosen for a good reason) must be part of the explanation of why the cosmos exists as it does. And it implies that alleged experience of a non-embodied mind, known in and through material appearances, is a real possibility that may be veridical and could provide confirming evidence of the existence of such a mind.\n\nIf we give great importance and value to the inner personal lives of human beings, and if we find the distinctiveness of human life to lie largely in the ability to choose goodness for its own sake, then we are committing ourselves to a view of reality that affirms the reality and value of conscious experience and moral goodness. This is opposed to any hard-line materialism that denies the existence or importance of consciousness, and it is opposed to any relativist view of ethics that maintains that the good is whatever certain people happen to prefer. There are real values and they lie in conscious states or actions. Since such values are objective, they suggest the existence of an objective non-human consciousness in which they can be found.\n\nWe may also feel that two very different sorts of reality, the material cosmos and the existence of free conscious and intelligent agents, need to be integrated within a coherent and plausible metaphysical framework. The materialist thesis that inner experience and purpose just happen to originate by chance and have no lasting significance does not seem very plausible. Consciousness and purpose simply become inexplicable add-ons to the physical process, and for a tough-minded materialist they may even be quite illusory.\n\nAn idealist framework will explain the physical cosmos as existing in order to permit beings who have personal experiences and purposes to share a common environment in which they can learn, develop, and act. But this \"in order to\" introduces purpose into the basic structure of the physical cosmos, and that introduction suggests that the basic structure of the physical cosmos is mind-like in important respects.\n\nFurther, the apparent intelligibility of the world suggests that there are good reasons for why things are as they are; that events do not happen solely by chance. Even the laws of nature exist for a reason, and the best reason is that they exist for the sake of desirable goals which the universe may realize. We are then thinking of a primordial mind that can envisage and evaluate possible goals and bring them about intentionally. That is the heart of the idealist case.\n\nMany idealists would call this primordial consciousness \"God\". But others find it better to avoid the word \"God\", because of the many different, and often silly, things that it might mean to them. Idealists do not mean to speak of a person, probably male, who lives outside the universe and interferes in it from time to time, in completely unpredictable ways. They believe that the ultimate nature of reality is more like consciousness than either blind necessity or pure chance. It is a consciousness very unlike human embodied consciousness. It is an independent and generative consciousness, not a dependent and largely receptive one. It contains as part of its substance something like a set of ideal possibilities, which perhaps necessarily express themselves in varied forms of finite material being. Its nature is known by human beings, if it is, as all other minds are known, by seeing the temporal processes of the phenomenal world as expressions of a largely hidden mental content.\n\nFor idealists like Hegel, primordial mind is not some being apart from the physical universe. It is the inner nature of the universe itself. It does not conjure up the universe out of nothing by some arbitrary act of will. It realizes its own necessary nature in a progressive process of temporal unfolding. It is not a person who chooses to create suffering and tragedy just because it wants to, when it could easily have chosen otherwise. Suffering and tragedy are parts of its own being, possibilities necessarily inherent in its self-objectification. Nevertheless, it could be said to have a moral goal or purpose \u2013 as a rational consciousness, it \"aims at\" the realization of many sorts of valuable states for their own sake. And it may actualize freedom and mutuality as well as necessity and a monopoly of causal power. That is, the cosmos it objectifies may generate many personal agencies capable of moral choice and social relationship, with a degree of autonomous causal power. Indeed in this cosmos it seems to do precisely that.\n\nPresumably such a mind will have knowledge of whatever becomes actual in the cosmos. Most idealists suppose that while it will retain complete knowledge of all that has ever existed, within its own cosmic experience it will mitigate destructive and painful experiences by subordinating them to and placing them in a wider context of creative and valuable experiences. Within that context they may be seen as inevitable, or at least as non-removable, parts of an emergent cosmos in which societies of creatively and morally free finite agents exist. In this way it will fulfil the goal of its own self-realization.\n\nIdealism is certainly a \"grand metaphysical theory\", and as such it is the main philosophical competitor with materialism. And both idealism and materialism are competitors with a Rylean or Wittgensteinian disinterestedness in such grand metaphysical systems.\n\nPhilosophy and metaphysics\n\nI suspect that Ryle would say, in response to all this, that if we are thinking about the concept of mind, and we get too far away from the ordinary language that we use about human minds and the everyday contexts in which we use that language, our concepts will no longer have any purchase on reality. They will be like cog wheels spinning energetically on their own without connecting to any useful mechanism. We should look at the informal logic of mental-conduct concepts as they are actually used, and not try to invent purely hypothetical and untestable theories about how they might be used in totally different, probably inconceivable, circumstances. Talk of \"ultimate reality\" is like talk of \"ultimate Platonic trousers\": all very well in theory but not very good for covering your legs.\n\nUnfortunately, once philosophers had eliminated the great classical philosophical systems as based on grammatical mistakes, and had demonstrated that the language that ordinary people speak is quite in order as it is, as long as ordinary people do not ask silly questions about it, there was nothing left for philosophers to do. It is rather ironic that Ryle, as a great philosophy teacher, was able to place most of his pupils (including me) in university jobs as professional philosophers, where they were paid to proclaim that there was no such profession as philosophy.\n\nI had to teach a course in moral philosophy in the 1960s (I will not say where), and I recall the widespread philosophical view that there was no point in asking moral philosophers about difficult ethical issues, since their opinion was no better than anyone else's. Moral philosophers could write long monographs about how people use words like \"good\" or \"right\", but they were not qualified to express any moral opinions. Looking around at some of my colleagues, I thought that perhaps this was a very good thing.\n\nIn a similar way, metaphysics was a form of armchair palaeontology \u2013 the study of fossilized philosophical systems that were now all extinct. If asked what the nature of reality was, philosophers would reply, \"It all depends on what you mean by real,\" and then patiently explain that real hair consisted in not wearing a wig or that real antiques were not new things that had been shot full of holes to look like ancient woodworm. If unwise undergraduates persisted in saying that they were seeking for what things were made of, they would be told that some things were made of wood, some of plastic, and some had not been made at all. It is not surprising that philosophy classes became progressively less well attended. Ancient philosophers like David Hume had played backgammon because philosophy was too difficult. Modern philosophers played backgammon because philosophy was too easy and consisted mostly in getting people to stop asking philosophical questions \u2013 which they could do best by not going to philosophy lectures.\n\nThat, however, is no longer the case. Partly because physicists have stepped in where philosophers refused to tread, and partly because medical practice and new technology generated a new range of ethical problems that do need some expertise to address, philosophy has started to get interested in the large traditional metaphysical and ethical issues again. \"What is the real?\" and \"What is the good?\" are no longer naive questions. People really want to know if the natural sciences are the only ways of finding out the truth, and if there is any way of reasonably resolving the ethical dilemmas that modern medicine puts before us.\n\nImagining minds\n\nMaterialism battles with idealism \u2013 and scientists line up on both sides. Utilitarianism battles with moral absolutism \u2013 and both politicians and doctors find themselves on opposing sides here too. Any objective observer would say that we are in a situation where many diverse and competing views can be reasonably defended. I am not persuaded that ordinary linguistic usage should be the final test of whether we are talking sense or making sense of human existence. If anything, I tend to think that ordinary linguistic usages are liable to be quite misleading, to express prejudices and half-digested philosophical theories whose origin has been forgotten.\n\nI certainly think that if we are considering the concept of mind, we should not limit ourselves to the sorts of minds we know best \u2013 human minds. We should explore the possibility of other kinds of minds, by imaginative extrapolation. The idea of a cosmic mind is a particularly interesting one, because it takes the notions of knowledge, feeling, and power to the highest degree we can think of. If we grant the existence of consciousness and objective value, we can try to conceive a consciousness of supreme value, knowledge, feeling, and power, and try to say what in general it would be like. That is indeed what many writers of science fiction do \u2013 and they often describe possible realities that some philosophers cannot imagine. Perhaps, as Tom Stoppard once said, \"There are more things in my philosophy than are dreamed of in reality.\" Or perhaps such dreams give a hint of what reality might really be like.\n\nThe cosmic mind may not have all these properties to the highest possible degree, and, if not, it should perhaps try harder. Anyhow, idealist views are not committed to the \"highest degree of mind\" view, pleasing though that might be. What they are committed to is that the reality underlying sensory appearances is more like a conscious and purposive cause of the phenomenal cosmos than it is like unconscious globs of unintelligent stuff (or maybe one super-glob of super-stupidity) from which the cosmos emerges without purpose or design. Or if purpose and design remind you too much of God, we might say that an idealist cosmos would at least exist for a reason, and it would exhibit an intelligible order.\n\nIf an idealist philosophy is adopted, this will have implications for the way human persons see themselves in relation to the cosmos of which they are parts. Most basically, they will find in their own inner lives of apprehension, feeling, and intention, a resonance or unity with the inner reality of the cosmos. Human consciousness will not be a freak, transient accident in a basically lifeless and indifferent universe. It will have a place in the self-realization of cosmic mind. The world revealed to us in sensory experience will be seen as an expression of cosmic mind \u2013 sometimes terrifying and dangerous, sometimes beautiful and awe-inspiring, but always expressive of a deep consciousness beyond the veil of the senses. The world may be \"read\" as an interaction with cosmic mind, somewhat as other human bodies are read as expressions of conscious feelings and aims. Humans will be fundamentally \"at home\" in the universe; not alien intrusions into a realm of blind laws, but integral parts of a self-expressive process oriented toward the realization of objective personal values.\n\nWithin such a world view, the arts can be seen as participations in the creativity of the cosmos, in a power beyond the finite self that yet works through and can heighten the insights and skills of artistic endeavour. Great works of art, music, and literature will be disclosive of what George Steiner calls \"real presence\", communications of transcendent mind as perceived by the immanent and embodied minds of human beings.\n\nScience will be seen as the discovery of a real intelligible order in the natural world, an order with a beauty and rationality that can be partly captured by mathematical exploration. The ancient Greeks were right \u2013 mathematics is the key to the order of the universe. For some peculiar reason, they did not devote much time to testing their mathematical theories by experimental observation, insisting that their maths was right without looking to see. But they did see what modern science has often failed to see, that a rational universe will be one in which the laws of nature will themselves have a reason. The only plausible candidate for such a reason is not the existence of a further set of hyper-laws. It is that for the sake of which all laws exist, the efficient realization of intrinsic values.\n\nMorality will be a response to an objective moral ideal and goal for the cosmos itself, a goal which humans can play a part in realizing. It will not be the construction of compromise rules which can mediate between opposing social interests. The good life will be a life lived for the sake of good, both in oneself and in society. It will be part of the realization of distinctive values which forms the ultimate reason for the existence of the cosmos. Devotion to the good for its own sake will thus also be devotion to that ultimate mind by which the good is conceived and in which it is progressively realized. This makes possible that \"intellectual love of God\", of the source and completion of the good, which was the centre of Spinoza's philosophy.\n\nHuman history will not only be a story of the interactions of human persons as they formulate cultural ideals and seek to make a distinctive contribution to the human world. It will also be an expression of the self-realization of transcendent mind, as it generates many diverse embodied minds, allows them creative and moral freedom, interacts with them as they impede or realize its purposes, and leads them through objectification and alienation toward a wider goal of a global society of societies. Hegel developed most fully this view of what he saw as the planetary history of Absolute Spirit. We may view his specific analyses and predictions with some scepticism. Karl Marx even found it necessary to stand Hegel on his head and get rid of a higher consciousness altogether, though he did not do any better himself. But that should not put us off the thought that in some way human relationships may also be the vehicles of the expression of the inner nature of the cosmos as it moves toward a moral goal that is inherent in its own inner being.\n\nCosmic optimism\n\nIdealists tend to be optimistic about the universe, thinking that value will be progressively realized in it. Even then, however, they usually accept that the universe will come to an end in a few billion years or so. Their optimism is not unqualified. Indeed some of them go around looking rather gloomy, and saying, \"It may seem good now, but mark my words, in a few billion years this will all be gone.\" The super-optimists, however, refuse to accept this depressing thought. Bolstered by some of the weirder extremes of cosmology and artificial intelligence research, they think that human beings will manage to download themselves into supercomputers and move off into inter-galactic space, as our sun runs out of steam (more precisely, out of heat).\n\nBy the time our universe runs down, they will then have become even cleverer, and will be able to travel through a black hole into a brand new universe. So they can keep going forever, moving from one universe to another, intelligences of immense knowledge and power, living in the huge seeming emptiness between the galaxies, hidden from all the relatively minute intelligences slowly emerging in those universes.\n\nIf this is so, then in our universe today there are probably huge numbers of super-intelligences living in outer space, and having super-committee-meetings about how they might guide the affairs of the little primitive carbon-based life forms emerging in their local universe. Religions may sometimes seem weird, but they have nothing on some modern cosmological speculations. A favourite among such speculations as I write is the many-worlds or multiverse theory. It holds that every possible universe exists, the whole set of universes forming a super-universe or multiverse.\n\nIt is not often realized that on this theory, every possible array of gods, goddesses, devils, and angels will exist somewhere in some universe. So every possible religion and philosophy will be true, but all in different universes. We can correctly believe in God in one universe and correctly be an atheist in another. We can even be parts of God in some universes and parts of Lucifer in others. The possibilities are endless. Even materialist atheism can be true in some universes, where no super-intelligences ever visit, and only the illusion of consciousness exists (I am not sure what that would feel like or who would be having the illusion, but now we've got the story going, let's run with it as far as possible).\n\nSpeculative science has thus far outstripped idealism in its classical forms. But the surprising thing is that these possibilities, though they are indeed fantastic, seem to be not only logically possible, but physically possible. Common-sense philosophy can be seen to be merely a restriction on human imagination. Perhaps common-sense philosophers should stay in more and read more science fiction. If they do that, they will soon find themselves asking, \"Which universe do I actually live in?\" And I do not think the answer is at all clear.\n\nOf course that is the main reason for being a common-sense philosopher. We cannot make our minds up between all the possible universes we might be living in, and so we drop the subject and just respond to particular problems as and when they come up.\n\nOn making ultimate metaphysical decisions\n\nIdealists, like everyone else, can distinguish between speculation and real existential decisions \u2013 decisions which make a real, practical, important difference to the way we live. The question of whether our descendants might one day decant themselves into a supercomputer and disappear down a black hole is purely speculative. It opens the mind to the possibility of the continued existence of intelligent life even after this universe has run down and given up the ghost (or all its ghosts). But it is not something we should count on or look forward to in the next decade or so.\n\nHowever, our answer to the question of whether this cosmos has an objective moral goal which we can play a part in actualizing might make a difference to how we see our lives and to how we live. If we see human lives as lived in a dialectical or ambiguous relationship to the goals of cosmic mind, as alienated by hatred, greed, and ambition and yet haunted by the ideal of a more creative and compassionate global community, that might make us see ourselves and other persons in a new light. We might see persons as \"called\" to a distinctive way of life, which is to be achieved only by overcoming the attractions of pleasure, power, and indifference that would frustrate such a life. Each person would be both a potential vehicle of expressing a higher consciousness and also a partial obstacle to its expression. In that dialectic of expression and obstruction each person would be part of a continuing interplay between transcendent mind and many socially embodied minds, an interplay which weaves history into the complex patterns it displays.\n\nWhether the pattern of history moves inevitably toward a society on this planet in which values are freely created and shared without restriction is perhaps not as clear as Hegel and Marx both thought. It might be that our world will, despite T. S. Eliot, end with a bang and not with a whimper. Humanity could be a cosmic experiment that is about to fail, and, as The Hitchhiker's Guide to the Galaxy predicts, our planet might be removed to make way for a cosmic motorway or for some more successful form of life. Since we cannot be sure, maybe the best we can do is to hope and work for the best and yet prepare for the worst. That way we might at least find, as tourists do when they contemplate the weather while on holiday in England, that we will be very pleased that things did not turn out to be quite as bad as they could have been. And that could be part of the recipe for a happy life.\n\nIdealism need not be religious, in the sense of leading to membership of some religious institution. Many idealists regard most religions as a mass of superstitions and legends, with unduly submissive attitudes toward various holy books, all of which contradict one another. But idealism may be sympathetic to some kinds of religion or spirituality, as attempted human responses to transcendent mind, intended to achieve liberation from self-centred egoism and to evoke a sense of union with or positive relation to a reality of wisdom, compassion, and bliss.\n\nFor some people, idealism is best seen as an abstract intellectual attempt by weak and partly corrupted human reason to approach a truth that has been independently revealed in a more emotionally accessible way by God \u2013 that is, I think, what most Christians think about Hegel. Even if that is so, idealism will remain an intriguing testimony to a residual sense of the ontological primacy of mind, of the reality of the soul and of God. For others, the reverse will be true, and various religions may be symbolic and pictorial ways of approaching the philosophical truth of idealism \u2013 that is what Hegel thought about most Christians.\n\nWhatever view you take about that, idealism seems to me the most coherent, comprehensive, integrative, and plausible conceptual scheme for understanding the world of which we are parts. As a basic metaphysical scheme, it does not need and cannot have \"evidence\", in the sense of publicly observable experimental demonstration. Like any ambitiously large-scale philosophy, it is based on reasons drawn from a wide range of data. Different philosophies stress different aspects of these data as of primary importance, and idealism stresses the irreducibility and importance of consciousness, reason, and morality.\n\nThere are alternative metaphysical schemes. For some, the conceptual disentangling of mind from the context of its human embodiment in brain, body, and society is a step too far, and may seem like the invention of a fantastic fiction. They will not be able to see the life of the human mind as one of response to transcendent mind through art, science, morality, and quotidian experience. It will still be possible, however, to see the life of the mind as one of creative and empathetic interaction with other embodied minds, and to find the best kind of human life to be the common pursuit of the good and the beautiful, even in a world plagued by violence and injustice and destined to end in universal cosmic death. Some would hold that human life becomes more precious by recognition of its sheer fortuitousness, its brevity, and its inevitable end.\n\nWhile not sharing that view myself, I recognize in it a tragic nobility, and I feel that a life committed to such a pursuit would be a life well worth living. Maybe it is really all one can have. Yet it would not be unreasonable to hope for more. Some of us think we have more, not particularly in visions or peculiar and overwhelming \"religious experiences\", but in a general confirmation throughout many forms of personal experience of a sense of transcendent presence that speaks of a mind other and greater than ours or of what Matthew Arnold called \"an enduring Power, not ourselves, which makes for righteousness\". I doubt if this is a matter that philosophy alone can resolve, although philosophical reflection may help to make clearer the different presuppositions it is reasonable to adopt and some of the logical consequences of adopting them.\n\n## Chapter Fifteen\n\n## Can we still speak of the soul?\n\nI conclude the book with some references to the views of Plato, Aristotle, Descartes, and the Bible. I draw attention to particular affinities that exist between their views and those expressed here. Idealism is a philosophy, not a religion, but it has an elective affinity with some religious views insofar as they are genuinely humanistic, and it shares with them an aversion to materialism, especially in its reductive or eliminative forms. Insofar as religions have a philosophical basis, it will usually be a form of idealism. Insofar as idealists seek ritual, moral, and communal expression for their views, they may find it in some form of religion.\n\nHowever that may be, I will conclude with a final homage to and divergence from Gilbert Ryle. I should say that he would agree with the first of the following sentences, and probably remain baffled by the second: there is no ghost, and there is no machine. There is only the reality of mind, and its expression and appearance in the dynamic and developing forms of an open and emergent material universe.\n\nI have mentioned the soul once or twice in previous chapters, though I have generally preferred to talk about the human person or the mind. Yet of course I have been speaking throughout of what has been traditionally thought of as the human soul, and in this last chapter I will connect what I have said with some traditional philosophical and religious views.\n\nWe do not speak much of the soul these days, though the ancient Greek word for soul \u2013 psyche \u2013 can be translated as \"mind\". Perhaps psyche is rather broader in reference, since it can be used to refer to all living things rather than just to mind. But it does not have to refer to something higher than minds and detachable from them. The soul is nevertheless often thought of as a distinct spiritual substance which is added to the body at some early stage of life, and which departs at death to continue its existence in its own immaterial way. Plato did seem to speak of the soul in that way, saying in some dialogues that all souls were destined to return to their true homes in the stars, once they had escaped their bondage to physical bodies. One problem with this view is that it is hard to see what souls will do all day if they have no bodies. Perhaps they just twinkle or spend their time doing arithmetic in their heads \u2013 except of course they have no heads. That is not something that most of us would look forward to with keen anticipation.\n\nBut being Plato, he did not always think that, and he had a much more positive view of physical bodies in the Timaeus, where he even spoke of the physical cosmos as a \"visible god\", and of souls as finding visible and sometimes beautiful expression within the cosmos.\n\nPlato divided the soul into three parts, but he did so largely in order to make a neat and tidy comparison between the individual and society. In The Republic he divides the classes of society into three \u2013 the rulers, the guardians, and the rest. The rulers, after long training, could tell what the best kind of society would be. The guardians were the executive power, an armed guard who could defend the state and enforce the decisions of the rulers. The rest carried on trade and did other sorts of menial and unworthy but necessary things. Plato wished to argue that justice ruled when all these parts of society were content with their proper place in the social order.\n\nWhen he came to speak of justice in the individual soul, he similarly tried to speak of it as a sort of internal order. Reason should rule. There should be a part for enforcing rational decisions. We might call this \"the will\", which puts decisions into effect. But Plato called it thumos, a \"spirited\" element, whose defining virtue is courage and a sense of honour or righteous indignation. The third element is desire. In the well-ordered soul, honour would execute the decisions of reason, and control desire. In a badly ordered soul, desire, and passionate action might overwhelm reason.\n\nThis threefold Platonic division is not quite that of Ryle's pet hate \u2013 the division of mind into thought, will, and feeling, three separate faculties each doing their own job. But it is pretty close. There is a contemplative faculty, an executive faculty, and a faculty closely bound up with bodily needs, wants, and feelings. This, Ryle says, is \"such a welter of confusions and false inferences that it is best to give up any attempt to re-fashion it\".\n\nRyle's point is that there are so many different sorts of mental activity that it is pointless to try to divide them up into a specific number of ways of thinking, as though there was a little committee inside the brain which divides mental jobs up rather neatly between its members. It is only fair to recall, however, that Plato is not seeking to give a systematic analysis of mind. What he basically wants to say is that in the just soul the passions and desires should be ordered to good ends by reason, and that to order often unruly desires some fairly aggressive and resolute discipline is called for. The inner mental disciplinarian is thumos, which we might call resolution or determination, the guardian and overseer of desire. It is not unlike Nietzsche's \"will to power\". If it is not controlled by reason, it will seek vitality and ambition for their own sakes. But when controlled by reason, it will order desires toward the cultivation of personal excellence and the realization of that which is good and beautiful.\n\nWe need not speak of \"three parts of the soul\". We could simply say that we have bodily appetites, but we are capable of reflecting on what is fully good and worthwhile, and we can discipline our appetites to aim habitually and by nature at such things. A good human life requires desires to be controlled effectively by dispassionate understanding and an iron will. Plato's metaphorical talk of the faculties of the soul makes this point very well. It is after all true that human minds think abstractly and understand, feel pleasure and pain and many diverse subjective responses to what impacts on human bodies, and are able (or at least think they are able) to act to control their bodies and environments to some extent. These are the main elements appealed to in forms of personal explanation. So even if Plato's terminology differs somewhat from ours, it alludes to the main basic elements that characterize human mentality, and it clearly places an emphasis on the importance of moral action (action for the sake of good) in human life. It also brings out the importance of having a view of human nature that can make sense of moral action in a universe like this. To have a picture of the nature of morality is already to have a picture of human nature. What philosophy does is try to make that picture explicit.\n\nAristotle on the soul\n\nDespite the immense importance of Plato in the history of Western philosophy, it was Aristotle who formulated the most influential Western definition of the soul, when he said that it was the \"form\" (eidos) of the body. A form is the essential nature of something, that which makes it what it is and not another thing. All things, for Aristotle as for Plato, incarnate forms; they have essential or proper natures. But organic forms, the natures of organisms, have a special role as defining what it is a developing organism is \"aiming at\" or tending by nature to be. Thus the form of a human embryo is to be a specific human adult, with the capacities and dispositions of that fully grown organism. Or the form of an acorn is the fully grown oak tree.\n\nEach organic form has a set of specific capacities that are proper to it. For Aristotle, vegetative forms have the capacity for growth and reproduction. More complex animal forms have the capacity for movement and sentience. Human, intelligent forms have the capacity for abstract thought and responsible action. Thus in speaking of forms we are speaking of sets of capacities proper to organisms of a specific kind. When we say that the soul, more properly the intellectual soul, or the mind, is the form of a human body, we mean these sets of capacities are proper to human beings as such.\n\nAristotle's doctrine of the soul or mind is thus that some intellectual and moral capacities and propensities are distinctive of and proper to organisms that are members of the human species. To say what the human soul is is to say what the distinctive, proper capacities of human beings are. I believe that Gilbert Ryle would accept this view, with one important and decisive proviso. All essentialist and teleological elements must be removed from the idea of a \"form\".\n\nThe rise of modern science involved a rejection of the idea that substances have essential natures, which they are \"meant\" to embody. Rather, the natural world is a continuum of many different properties, capacities, and forces. One shades into another without any clear demarcation line, and there are many blurred edges and overlaps. As Wittgenstein put it, there are many family resemblances between things and processes, but there need be no central defining characteristic that makes a thing what it is.\n\nEvolution is often interpreted in this way, leading to views that humans are not sharply distinguished from higher primates, and that, as a class, they have no necessary and sufficient defining characteristics. Some humans have no intelligence, and some intelligences are not human. Humans are beings that resemble each other (and also other animals) in varying ways and to varying degrees, like different members of a family. Therefore (it is sometimes said) humans have, as such, no peculiar moral value or worth.\n\nIn the same way, the idea of final causality was widely rejected in the seventeenth century. Organisms do not \"aim at\" some ideal state, for the sake of which they exist. They usually pass through a cycle of growth and decay, without aim or purpose, just happening to reproduce on the way. Therefore (it is sometimes said) there is no particular sort of thing that humans ought to be or no sort of life that humans are meant by nature to live.\n\nI think Aquinas was right in suggesting that the Aristotelian view only made sense based on the supposition that there is some kind of cosmic mind which contains the ideal forms that define what things properly are in their fullest development or what the innate goals of their existence are. Idealists will naturally have no problem with such a view, and it fits well with the belief that intellectual understanding, creative freedom, and interpersonal empathy are objective goals of the cosmic process. It is not the human species, as such, that is of special value. Any being capable of understanding, freedom, and empathy has special moral worth \u2013 just the \"faculties\" of thought, will, and feeling so derided by Ryle. For this reason, \"humanism\" is not the most appropriate term for that concentration on personal flourishing that was one of the theoretical marks of the European Enlightenment. It would be better to speak of personalism, the postulation of intelligible thought, creative will, and empathetic feeling as intrinsic moral goods, to be protected and maximized wherever they occur.\n\nI think it makes good sense to say that any beings with such capacities are of distinctive worth and should be respected as such. It also makes sense to say that human beings are such beings or are parts or offspring of such societies of beings, and so incarnate a specific character or form which is morally important. To sustain, enhance, and cherish such capacities, and to oppose all that frustrates them, both in oneself and in others, as far as is possible, is a distinctive moral goal for human lives.\n\nWhat modern science opposes is answering questions about the natures of things with merely verbal definitions. What is required is experimentation, observation, and testable predictions. Natural science is not concerned with the aims natural things might have, but with the regular processes by which things change. But that does not mean that there are no goals in nature or that those goals cannot be defined in terms of distinctive capacities. Values and purposes have not been the concern of the natural sciences for some centuries. But the exclusion of the personal from the realm of natural science does not exclude it from reality. It is for philosophical reflection to define what worthwhile purposes may be, using all the available data provided by the natural sciences, but adding data from the rich historical records of human personal and moral experience, and making the most comprehensive personal evaluation of all this data.\n\nCartesian dualism and beyond\n\nWhen the Aristotelian philosophy was replaced by the more mechanistic approach of classical science, it became difficult for philosophers to integrate personal values and purposes into the increasingly influential world view of natural science. Cartesian dualism was one symptom of this difficulty, separating mental substance from material substance in such a way that it was difficult to see how one could interact with the other. As I have emphasized, Descartes believed in such integration, but did not find a plausible way of formulating it. It was to take the discovery of evolution to do so, with the picture it opened up of a gradually more complex and emergent process leading to the development of mind as an increasingly autonomous inner aspect of matter.\n\nA Cartesian hangover from Aristotle was the notion of substance, an enduring substratum that could contain various changing properties. For many philosophers, this was slowly replaced by the idea of process, of a flowing succession of properties, located in space and time. The idea of substance could be retained as the idea of a core bundle of properties that could change more or less gradually over time. The idea of strict numerical identity was replaced by the idea of a succession of properties, \"identity\" being largely a matter of degree and convenience. A tree generated from the pruned stump of a prior tree could be called the same tree or not, almost at will.\n\nIn modern science, questions of identity are often treated as matters for conventional decision, upon which nothing much turns. With personal and mental lives, however, the idea of identity becomes morally important. It matters to me whether tomorrow I will remember the plans I made yesterday and be able to continue them or whether I will take over the plans of someone else and pass them on to another person in turn. \"This is just what I planned to do\" is very different from \"This is what he (someone otherwise just like me) would have wanted me to do.\"\n\nRyle notwithstanding, it is the chain of privately accessible experiences and actions, thoughts and intentions, that makes the difference. When one and only one person can have such private access, then I could reasonably say that the same personal subject of experience and action continues to exist. The sense of being one and the same continuing subject of many experiences and acts is important to a person. That, I think, is what Descartes meant by a mental substance. For that reason, there is little or no substantial difference between being a mental substance and being a process of privately accessible, temporally flowing events and acts. That is just what being a mental substance is.\n\nDescartes held that the mental substance could exist without the physical substance of the body. But he did not hold that it should, that it did or that it would be a full and proper person if it did. In this, too, he is surely correct. There can, logically, be a succession of thoughts and feelings without any physical body or brain or universe. Perhaps they would be thoughts and feelings about a universe in which we used to have a body and a brain. Perhaps disembodied minds would be condemned to repeat their past embodied experiences over and over again.\n\nOn the other hand, perhaps chains of mental acts could find new forms of embodiment in different environments. When Plato speaks of the soul's journey of a thousand years between earthly births, he is speaking as if there are bodies of some sort, receiving experiences from some form of environment, good or bad. And his story is relatable without obvious contradiction. Perhaps it is in some such world, unfathomably far in our future and from this space-time complex, that minds once embodied here could find a fulfilment rare and strange, where fully shared understanding, cooperative creativity, and mutually reinforcing happiness could ameliorate and transfigure all the conflicts and sufferings of our previous lives. And yet those past lives would have been the way of bringing us to our final goal, a way that we were partly thrown into and partly chose, but which was always destined for final good, and which, if we saw it whole, we would embrace with repentance and with joy.\n\nThat is the dream of idealism. But is it true? It must be plainly said that there is no proof. Yet it is more than an idle wish. It is rooted in the firm belief that mind is the ultimate nature of being, and that intelligent mind aims, as far as it can, at goodness, at what is worthwhile for its own sake. This dream is what mind would realize if it could, if it were at all possible. It does seem possible, since it contains no contradiction. Dare we then hope for it?\n\nWhat we should commit ourselves to is the importance of freedom, of moral commitment, of nobility of character, of intellectual and moral virtue, of personal fulfilment and flourishing for all. With this goes a sense of the importance of mind or personhood, of a moral intelligence embedded in an emergent physical world. One might be agnostic about questions concerning an objective spiritual basis of reality. I fully understand that. But moral action requires passionate commitment, not based on evidence (in the sense of demonstration beyond reasonable doubt or even probabilistic judgment), but consonant with one's most basic world view.\n\nThat world view may remain largely implicit and obscure, even to oneself. Yet if one commits to the importance of morality, this is an implicit commitment to the importance of mind, if morality is not to be fundamentally irrational. It is entirely reasonable to root these commitments in a basic ontological option for the rationality and goodness (moral orderedness) of being. Then we can root our morality in a response to what is most fully real, what is possible of fuller realization, and what will satisfy personal striving for fulfilment in many diverse and unique historically situated vocations. This is faith \u2013 trust that reality is what it seems to us in our highest moments of insight to be. It is validated by practice and by the experience of a life sustained by love.\n\nThe soul and immortality\n\nSome people speak of belief in immortality as based on a desire to find some supernatural sanction for moral conduct. If there is a supernatural power that will reward goodness and punish badness, this adds force to moral motivation. But that is to my mind an unduly negative way of putting things. Belief in immortality is primarily based on belief that the ultimate character of reality is mind. This builds consciousness, value, and purpose into the universe in a fundamental way, and prevents one from seeing existence as pure chance, accident or unforeseen happenstance. It roots moral action in a perception of objective purposes of value which it is a human responsibility to further. It gives rise to a presumption that such goals will be achieved if it is possible to achieve them. The point is not that I will be punished or rewarded for my acts. The point is that moral action will not be in vain. My turning away from justice and compassion will lead to my inhabiting a world of injustice, violence, conflict, and hatred. My growth toward greater justice and love will lead to my living in a world of friendship, cooperation, and universal empathy.\n\nIn the world as we see it, it is in very general terms true that heroic moral acts lead to a happier and more flourishing society, and that violence breeds its own destruction. But this is only true in general. In millions of particular cases, the innocent perish and the evil flourish. Moral action seems ineffective, and personal sacrifice is in vain. It is not a selfish desire to continue to exist that leads to belief in immortality. It is the impact of the thought that goodness will not triumph, and that the noblest moral sacrifices will fade into insignificance in an indifferent universe.\n\nIn a universe in which mind is primary and aims at good, it would contradict the rational structure of the universe if that were the last word. If value and purpose are primary in the universe, then there must be some possibility of rectification of the ills of this life, space for fuller development of our feeble moral efforts, and the possibility of a fuller realization of the value of finite being than seems to be possible in this life. Thus it is deeply rational to hope for a life of the mind beyond the death of this body.\n\nThat does not mean that this life becomes less important, while we simply wait for better times hereafter. This life is part of the human pilgrimage, and what is done in it determines what shall become of human lives. In a world corrupted by evil, conflict, and injustice, we have a unique personal part to play in an unfolding scenario of trials encountered and endured, of values envisaged and realized, of tasks undertaken and achieved. Our life is not sound and fury, signifying nothing. It has been like a journey into a far country, from which, chastened and taught by experience, we must at last return. As Plato puts it in Orphic mythology, after our \"journey of a thousand years\" through the earth, we may shine as stars in the sky. Or, in a Christian version of the same hope, after we have passed through the fires of purification (Mark 9:49 and 1 Corinthians 3:13\u201315), we may at last shine like the sun in the kingdom of the Father (Matthew 13:43).\n\nThis mention of Orphic and Christian symbols may be thought to be introducing religion into a philosophical work. And it may indeed seem that if there is a supreme cosmic mind, it would be rather strange if it disclosed nothing of its nature and purpose in some form of revelation. Nevertheless, this book belongs to philosophy, because it makes no reference to any revealed truths, and its arguments do not depend upon any specific religious teachings. What it does, though, is give many religious interpretations of the human soul a strong rational foundation, suggest the wisdom of at least attending to religious claims for revelation, and support the importance of an understanding of human persons as moral agents whose lives have unique value and a moral purpose that gives them inalienable significance and meaning.\n\nBiblical views of the soul\n\nIn a book about the nature of human persons, it would be absurd to overlook religious views entirely. So I will end by saying something about the biblical view of the soul that has been so influential, especially in Western culture. One major problem here is that Plato and Aristotle, or a mixture of the two, have often been more important than what the Bible says. For this reason, there is frequently much confusion about what the Christian or Jewish view of the soul is. The strange truth is that there is, strictly speaking, no biblical, Hebrew or Greek word for \"soul\", even though the word \"soul\" is constantly used in English translations of the Bible.\n\nIn Hebrew there are three words \u2013 nephesh, neshamah, and ruach \u2013 which can each be translated as breath, soul or spirit. None of them seems to denote a thing, substance or entity. They are more like activities or energies. In Genesis 2:7, God formed man \"from the dust of the ground\", and breathed into his nostrils \"the breath of life\". Later, in Genesis 7:22, it is reported that, during the great flood everything on dry land in whose nostrils was \"the breath of life\" died.\n\nThe breath of life is common to all breathing animals, so in this sense \"the soul\" might be taken as that which keeps animals alive and breathing. It is not particularly mental or conscious, but it is some sort of vitality or energy, perhaps something like what the philosopher Henri Bergson called the \u00e9lan vital, life force.\n\nHumans are made from dust and filled by God with vital energy. The first creation story in Genesis 1:26 adds that humans are made in the \"image and likeness\" of God, having dominion over other life forms. There is much discussion about what exactly this means, but one main interpretation is that humans are entrusted with stewardship of the earth, so that they can act as God does with justice and mercy toward other lives, and stand in the place of God upon this planet.\n\nSome writers have protested that this gives humans too much power, licensing them to do whatever they want with the earth and with other animals. This seems to be against the spirit of the biblical account, however, since all things belong to God, and the role of humans can only be to do God's will, though to do it with a sense of personal responsibility.\n\nFor the Genesis stories, humans are special not because they alone have souls, but because, being material and breathing beings, they have a responsibility to order the living things in the world in a just and compassionate way, thereby helping to realize the divine purpose for creation and sharing in the divine work of bringing finite things to the fulfilment for which they were created.\n\nThe Hebrew Bible does not show any great interest in a separate spiritual realm apart from this material world. Indeed, for most of the Hebrew Bible, there is no positive view of any life beyond death. The Bible is concerned with realizing justice and compassion in this material world, and humans, as fully material beings \u2013 \"dust\" \u2013 have the responsibility to do that.\n\nIt is for that reason that when belief in an afterlife became more important, at the time when the very latest passages of the Hebrew Bible were being written, it was thought of in a rather material way. There might be a \"resurrection of the body\", not just a continuance of disembodied souls. There might be a \"new heaven and earth\", not a non-physical existence of some sort. We can conclude, then, that the Hebrew (Old Testament) view of the soul is that it is not something that could or should float free of the body. A piece of matter has a soul if it has a sense of responsibility and all that implies \u2013 knowledge, freedom, and moral sensibility. Any material entity with those capacities would have a soul. So the possession of a soul is the possession of a set of capacities that entail consciousness, evaluation, and purposive causality (the existence of mind). But those capacities are possessed by a material entity, by dust enlivened by the divine life, not by an immaterial entity that may or may not be embodied in matter.\n\nNevertheless, there was some idea of an afterlife in early biblical thought. Sheol is the place of the dead, like Hades in the Latin tradition. It is not much to be looked forward to, but past persons do seem to reside there \u2013 Samuel was called up from Sheol by the witch of Endor (1 Samuel 28). There were also angels of various sorts, and some Old Testament characters were said to have ascended to heaven \u2013 Enoch, Moses, and Elijah were popular candidates. Apparently there were other realms apart from this earthly one, where past persons could exist in some sense.\n\nSince ordinary physical bodies do not go to Sheol, it must be possible for persons to transfer to other sorts of bodies. The soul may not be an immaterial entity, but persons can be embodied in different ways. In later Jewish thought, ideas of a place of punishment (probably temporary) and of paradise developed, presumably as intermediate states before the general resurrection.\n\nIn later Christian tradition, this became the doctrine of purgatory or, in the Eastern Orthodox traditions, of the intermediate state, where experiences of various kinds occur to persons who neither exist in their earthly flesh-and-blood bodies, nor in their perfected and glorious resurrection bodies, which will exist only when the whole of heaven and earth and all living things have been renewed.\n\nThe whole biblical tradition of course is opposed to materialism. It is idealist, in that it is centred on a knowing, intending, and creating God who is not embodied in any sense. Ruach, the breath or spirit of God, is the energizing power of God that shapes creation out of primal chaos (Genesis 1:1\u20132), and that inspires prophets and heroes. Part of being created \"in the image\" of God is being able consciously to share in this creative power of God, and so participate in a spiritual and non-material power that has causal influence in the material world.\n\nIn the New Testament, nephesh is rendered into Greek as psyche, and ruach as pneuma. Again, psyche is usually translated into English as \"soul\", but again it primarily refers to any embodied living animal, anything that \"has breath\". As Christian thought developed, humans were distinguished as a sub-class of animals that have an \"intellectual soul\", a specific set of intellectual capacities, especially knowledge, freedom, and moral sense. These capacities were seen as capable of being embodied in different non-flesh-and-blood forms (purgatory or paradise), though it was important that they had been generated and cultured in earthly life. And their final destiny was thought of as a transfigured yet fulfilled form, still individual, still social, still personal, of their first earthly embodiment.\n\nIn the key New Testament passage on the resurrection of the dead, Paul distinguishes between the soma psychikon and the soma pneumatikon (1 Corinthians 15:44). We die, he says, in the former, in bodies filled with the breath of physical living things. But we are resurrected in the latter, in bodies filled with the divine breath or Spirit. They are in some sense bodies, yet they are not the same in kind as ordinary material bodies. They have been transfigured by the divine Spirit, to become knowers and mediators of Spirit, freed from ignorance and selfish desire, free to cooperate with the creative purposes of Spirit. The environment in which they live will be different from this world, free from the laws of transience and decay, and yet the later prophets foresee that this world will be transfigured too, that there will be \"a new heaven and earth\", a new cosmos.\n\nThe biblical idea of the soul is complex and many-stranded. Some of these strands, however, are widely shared. They emphasize that some form of embodiment is proper to such conscious, feeling, willing beings as human persons are. But they refuse to limit embodiment to the precise physical bodies that exist now on this planet. They envisage some form of continuance in which human moral choices for good or evil can be seen and felt with all their implications, and can be worked out to their ultimate conclusion. They give free and responsible moral choice a decisive causal role in the formation of the human future. And they look forward to a final transfiguration of material embodiment that will enable it to become a fully conscious cooperation with and a fully conscious expression of the self-realization of Supreme Spirit in a world of many embodied, unique, and freely acting souls.\n\nIdealist philosophy did not develop independently of such a religious tradition. Both the biblical tradition and the analogous Indian tradition of Brahman \u2013 one Supreme Self of the cosmos \u2013 have been closely interwoven with the growth of philosophical ideas about the nature of reality. Alleged personal experiences of God or of the Supreme Self have for many confirmed the thought that the heart of reality is conscious mind.\n\nNevertheless, even apart from that, rigorous philosophical thought about the nature of human persons, and about the place of mind in the cosmos, does, in my view, point toward idealism. The argument for idealism stands on its own, and it offers a view of human life that stands in stark opposition to the materialism that characterizes many popularizations of modern scientific thought. It gives human life a value, significance, and purpose of enduring worth. Not only that. It might actually be true.\n\n## Notes\n\nIntroduction\n\n1. Francis Crick, The Astonishing Hypothesis, New York: Simon and Schuster, 1994, p. 3\n\nChapter One\n\n1. Gilbert Ryle, The Concept of Mind, London: Hutchinson, 1949, p. 11\n\n2. Ibid, p. 12\n\n3. Ibid, p. 15\n\n4. Ibid, p. 15\n\n5. Ibid, p. 21\n\nChapter Two\n\n1. Michio Kaku, Hyperspace, Oxford: OUP, 1994, p. 15\n\n2. Gilbert Ryle, op. cit., p. 16\n\nChapter Seven\n\n1. Michio Kaku, op. cit., p. 177\n\nChapter Nine\n\n1. Malcolm Jeeves and Warren Brown, Neuroscience, Psychology and Religion, West Conshohocken, PA: Templeton Press, 2009, p. 52\n\n2. Ibid, p. 117\n\n3. John Polkinghorne, One World, Oxford: Templeton Foundation Press, 2007, p. 102\n\n4. Malcolm Jeeves and Warren Brown, op. cit., p. 49\n\n5. Ibid, p. 111\n\nChapter Eleven\n\n1. Gilbert Ryle, op. cit., p. 61\n\n2. Ibid, p. 48\n\n3. Ibid p. 47\n\n4. Ibid, p. 79\n\nChapter Twelve\n\n1. Ibid, p. 84\n\n2. Ibid, p. 110\n\nChapter Thirteen\n\n1. Ibid, p. 167\n\nChapter Fifteen\n\n1. Ibid, p. 62\n\n## Short bibliography\n\nI have not given classical texts a publisher, since there are many translations and editions. For them I have given the date of original publication.\n\nIntroduction\n\nFrancis Crick, The Astonishing Hypothesis, New York: Simon and Schuster, 1994.\n\nGilbert Ryle, The Concept of Mind, London: Hutchinson, 1949.\n\nChapter One\n\nA. J. Ayer, Language, Truth and Logic, Harmondsworth: Penguin, 1936.\n\nAyer's later rejection of logical positivism is expressed in his Gifford lectures, published as The Central Questions of Philosophy, London: Weidenfeld and Nicolson, 1973, especially chapter 2.\n\nRene Descartes, Meditations on First Philosophy, especially Meditation 6, second edition, 1642.\n\nKeith Ward, The God Conclusion, London: Darton, Longman and Todd, 2009, chapter 3.\n\nLudwig Wittgenstein, Philosophical Investigations, trans. G. E. M. Anscombe, Oxford: Blackwell, 1953.\n\nChapter Two\n\nGeorge Berkeley, A Treatise Concerning the Principles of Human Knowledge, second edition, 1734.\n\nMichio Kaku, Hyperspace, Oxford: OUP, 1994.\n\nJohn Locke, An Essay Concerning Human Understanding, book 2, 1690.\n\nA readable, popular account of the interpretation of quantum theory is: Nick Herbert, Quantum Reality, New York: Anchor Books, 1985.\n\nFor a much more difficult technical account, Paul Dirac, The Principles of Quantum Mechanics, Oxford: OUP, 1958.\n\nAnd for a selection of \"weird\" views (where \"weird\" may well mean \"true\"), see J. A. Wheeler and W. H. Zureck, Quantum Theory and Measurement, Princeton: Princeton University Press, 1983.\n\nChapter Two\n\nDavid Hume, A Treatise on Human Nature, volume 1, 1739.\n\nChapter Three\n\nBernard d'Espagnat, Veiled Reality, Reading, MA: Addison-Wesley, 1995.\n\nImmanuel Kant, Prolegomena to any Future Metaphysics, 1783. Kant wrote this to explain the \"Critique of Pure Reason\", and it is in some ways a simpler exposition of his thought.\n\nKeith Ward, The Development of Kant's View of Ethics, Oxford: Blackwell, 1972.\n\nChapter Four\n\nOn Hegel, see Stephen Houlgate, An Introduction to Hegel, Oxford: Blackwell, 2004.\n\nKeith Ward, Concepts of God, Oxford: Oneworld, 1998.\n\nA. N. Whitehead, Process and Reality, ed. David Griffin and Donald Sherborne, New York: Macmillan, 1979.\n\nA simpler exposition of Whitehead's thought: John Cobb and David Griffin, Process Theology; an Introductory Exposition, Philadelphia: Westminster 1976. Although called theology, there is a lot of philosophy in it.\n\nChapter Five\n\nDerek Parfit, Reasons and Persons, Oxford: Clarendon Press, 1984.\n\nKeith Ward, Religion and Human Nature, Oxford: Clarendon Press, 1998.\n\nBernard Williams, \"Personal Identity and Individuation\", in Problems of the Self, Cambridge: CUP, 1973.\n\nChapter Seven\n\nSimon Conway-Morris, Life's Solution, Cambridge: CUP, 2003.\n\nChapter Nine\n\nMalcolm Jeeves and Warren Brown, Neuroscience, Psychology and Religion, West Conshohocken, PA: Templeton Press, 2009.\n\nMalcolm Jeeves, ed., From Cells to Souls \u2013 and Beyond, Grand Rapids: Eerdmans, 2004.\n\nFor a good account of non-reductive physicalism, see Warren Brown, Nancey Murphy and Newton Malony, Whatever Happened to the Soul? Minneapolis: Fortress, 1998.\n\nAnd for a modern fully dualist account see Richard Swinburne, The Evolution of the Soul, Oxford: Clarendon Press, 1986).\n\nAlso, Karl Popper and John Eccles, The Self and its Brain, London: Routledge and Kegan Paul, 1977.\n\nJohn Polkinghorne, One World, Oxford: Templeton Foundation Press, 2007.\n\nZombies, stigmergy, and Michel Cabanac can all be found in David McFarland, Guilty Robots, Happy Dogs, Oxford: OUP, 2008.\n\nAlvin Plantinga, \"Reason and Belief in God\" in Faith and Rationality, ed. Plantinga and Wolterstorff, Notre Dame, Indiana: University of Notre Dame, 1983.\n\nH. H. Price, Essays in the Philosophy of Religion, Oxford: OUP, 1972, especially chapter 6.\n\nA. Quinton, \"Spaces and Times\", Philosophy, volume 37, 1962.\n\nChapter Ten\n\nDavid Armstrong, A Materialist Theory of Mind, London: Routledge, 1969.\n\nDan Dennett, Consciousness Explained, Harmondsworth: Penguin, 1991.\n\nBenjamin Libet, \"Do We Have Free Will?\", in The Volitional Brain, ed. Libet, Freeman and Sutherland, Thorverton, UK: Imprint Academic, 1999.\n\nPeter Strawson, \"Freedom and Resentment\" in Freedom and Resentment and other essays, London: Methuen, 1974.\n\nChapter Fourteen\n\nSpinoza, Ethics, 1675.\n\nFrank Tipler, The Physics of Immortality, New York: Doubleday, 1994.\n\nChapter Fifteen\n\nThomas Aquinas, Summa Theologiae, Part 1a, question 76.\n\nAristotle, De Anima.\n\nPlato, Timaeus and the Republic.\n\n## Glossary\n\nAbsolute idealism The theory that there is one Absolute Mind of which all finite minds and all matter are parts, or which that Mind generates by inner necessity.\n\nAdvaita One sort of Vedanta (see below), claiming that there is in reality only one non-dual Reality of consciousness and bliss, and that the appearance of many souls and things is an illusion.\n\nAntinomies Kant's word for the contradictions into which, he held, human reason falls when it tries to talk about reality-as-it-is-in-itself.\n\nBasic beliefs Beliefs that are not based directly on evidence \u2013 like the belief that all beliefs must be based on evidence (which is false if there are basic beliefs).\n\nCartesian dualism Descartes' view that mind and matter are different substances, and can exist apart. However, as I argue, this view is often misunderstood, since Descartes did not think that they should exist apart.\n\nCommon-sense philosophy The philosophical theory that the world really is just as most people believe it to be, and common-sense beliefs need no rational justification.\n\nCritical idealism (also called transcendental idealism) Kant's theory that we must assume that reality is ultimately mind-like, but we cannot theoretically demonstrate it, and the material world is not just in our (human) minds.\n\nCritical realism The theory that objects really do exist apart from us, and we have true knowledge of them. But the way we see them depends upon our special sense-organs and mental apparatus.\n\nDeterminism The theory that every event is wholly determined by some initial state and a set of general laws. So no event could be other than it is.\n\nDual-aspect idealism The theory that mind is the ultimate reality, but in the case of human beings, mental events are correlated strongly with physical brain-events, so that a mind can even be called the \"brain seen from inside\".\n\nDualism The belief that humans are made up of two parts, body and mind (or soul). They normally exist together, but can in principle exist apart.\n\n\"Emergent\" universe Universe in which new properties, like consciousness, emerge as matter becomes more complex and organized through time. Such a universe is not just a recycling of properties that have always existed and always will exist.\n\nEpiphenomenalism The theory that minds, consciousness, and mental properties exist, but are wholly caused by brain-events, and have no causal role in the universe.\n\nEvolutionary naturalism The belief that evolution develops by wholly physical laws, without any supernatural or mental influence.\n\nFatalism The theory that, because determinism is true, there is nothing we can do to change the future. Fate rules!\n\nHolistic explanation An explanation of the parts in terms of the whole of which they are part. For example, you may explain why a cell of the body develops as it does by referring to the whole body of which it is part.\n\nHumanism The view that it is more important to care for human welfare than to practise religious rituals. Historically, many Christians have been humanists, but in modern usage humanism often implies the rejection of belief in God.\n\nIdealism The theory that material objects would not exist without mind or consciousness, so that mind is the primary form of reality, and causes material things to exist.\n\nIntensional state A mental state that refers to something beyond itself. That \"something\" could be imaginary or fictional, like a character in a novel.\n\nKarma A Sanskrit term referrering to a law of moral reward and retribution, which determines what happens to souls in their different reincarnations.\n\nLogical positivism A philosophical system that is \"positive\", in that it believes all true assertions must refer to scientifically verifiable facts, and that is \"logical\", because this is a doctrine about meaning \u2013 that factual statements only have meaning if they do refer to such facts.\n\nMaterialism The belief that nothing exists except matter \u2013 publicly observable things with location in space and time.\n\nMetaphysics A systematic attempt to say what sorts of things exist (minds, bodies, numbers, moral truths, God, etc.), and how they relate to each other. Sometimes called a \"world view\".\n\nMultiverse The theory that our space\u2013time universe is only one of many such universe, possibly even an infinite number of them.\n\nNaive realism The theory that objects continue to exist just as they appear to us, even when we are not observing them.\n\nNaturalism Usually, the theory that all true statements must refer to \"natural\" objects \u2013 objects subject to physical laws in space and time.\n\nNon-reductive physicalism The belief that materialism (physicalism) is true, but that matter in very complex states (in brains) produces consciousness, which may even have causal power. So minds cannot exist without brains, but they have new, \"emergent\" properties which simpler forms of matter do not have.\n\n\"Open\" universe A universe which is not deterministic. The future is open to many possible alternative tracks.\n\nPhenomenalism The theory that nothing really exists except sense-data. Physical objects are logical constructs out of sense-data.\n\nPluralistic idealism The theory that mind, not matter, is the ultimate reality. But there are many minds, not just one (not just the mind of God, for example).\n\nPrimary qualities For Galileo, the qualities that any physical object must possess, whether or not it is observed \u2013 properties like position, velocity, and mass, for example.\n\nProcess philosophy The metaphysics of A. N. Whitehead, which denies there are enduring substances, but asserts that there is a continuing process of distinct events (\"actual occasions\") which make up the natural world.\n\nQualia Immediate objects of the senses, just like sense-data. But there might also be feelings, sensations, and mental images.\n\nSecondary qualities Qualities that only belong to objects when they are observed \u2013 like colour and smell. Such qualities are often said to be \"in the mind\".\n\nSense-data The things we are immediately aware of by means of our senses \u2013 patches of colour, smells, touches, and tastes.\n\nSubstance From Aristotelian philosophy, an enduring \"thing\" which is the bearer of properties (like a tree that remains the same while its leaves grow and die). One continuing substance can have many changing properties.\n\nTeleological explanation Explanation in terms of a goal or purpose, not in terms of general laws that bring some state about. For instance, \"I run in order to lose weight\" is a teleological explanation.\n\nVedanta An Indian philosophical system which assumes the truth of the Hindu scriptures (the Vedas and Upanishads), especially that \"all is Brahman\" [Absolute Mind], and builds various metaphysical systems on that.\n\nVerification principle The rule that all meaningful statements must in principle be capable of being shown to be true by some sense-experiences.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nFirst Vintage Books Edition, August 1984\n\nCopyright \u00a9 1983 by Robert Farris Thompson\n\nAll rights reserved under International and Pan-American Copyright Conventions. Published in the United States by Random House, Inc., New York, and simultaneously in Canada by Random House of Canada Limited, Toronto. Originally published by Random House, Inc. in 1983.\n\nLibrary of Congress Cataloging in Publication Data  \nThompson, Robert Farris.  \nFlash of the spirit.  \n1. Blacks\u2014America\u2014History. 2. Afro-Americans\u2014History.  \n3. Arts, Black\u2014America.  \n4. Afro-American arts.  \nI. Title.  \nE29.N3T48 1984 973\u2032.0496 83-40298  \neISBN: 978-0-307-87433-7\n\nv3.1\n**For Dr. and Mrs. R. F. Thompson,  \nthe late Mrs. E. B. Hood, and  \nHenriette Wyeth Hurd**\n\n# **Acknowledgments**\n\nAn artist in Nigeria once taught me a proverb: One tree cannot make a forest _(Igi kan k\u00ec s'igbo)_. In other words, a person who does not work together with his colleagues and his friends will not accomplish very much. The recognition of this truth lies behind all acknowledgments.\n\nThus with profound pleasure and gratitude I record, first of all, the names of the women and men without whose fundamental contributions in the field of African or Afro-American aesthetics I doubt this book would have been conceived at all\u2014W. E. B. DuBois, Melville J. and Frances S. Herskovits, Zora Neal Hurston, Maya Deren, Harold Courlander, Alan Lomax, Alan Merriam, Roy Sieber, Richard Alan Waterman, Marshall W. Stearns, Nat Hentoff, William Bascom, Jean Price Mars, Alejo Carpentier, Argeliers Le\u00f3n, Lydia Cabrera, Arthur Ramos, Ralph Ellison, William Fagg, Kenneth Murray, Frank Willett, Roger Bastide, Pierre Verger, Ulli Beier, Janheinz Jahn, Kwabena Nketia, and Don Fernando Ortiz. Their work encouraged my lifetime interest in African\/Afro-American musical and visual contributions.\n\nSeveral friends and colleagues generously read my manuscript, in part or in its entirety, and made many useful suggestions. Nancy Gaylord Thompson, William Rout, Richard N. Henderson, Richard and Sally Price, George Kubler, Henry Drewall, Sidney Mintz, Paul Newman, Roger Abrahams, John Szwed, Perkins Foss, John Janzen, Jan Vansina, Wyatt MacGaffey, Fu-Kiau Bunseki, Daniel Biebuyck, Nehemiah Levtzion, Charles Bird, Keith Nicklin, Araba Eko, Clarence Robins, Lydia Cabrera, Dennis Warren, Roy Sieber, Claude Savary, Harold Courlander, L. Prussin, and K. Campbelle.\n\nThe following persons helped in various ways: Vincent Scully, Nelson Ikon Wu, Charles Seymour, Sheldon Nodelman, Robert and Fi Herbert, Leonard Doob, Leon Lipson, Sumner Crosby, Arthur Wright, Anne Hanson, John Blassingame, Richard Morse, Henry Louis Gates, Jr., Margaret Webster Plass, Bernard Fagg, Jacqueline Delange, Charles Ratton, J\u00fcrgen Zwernemann, Ekpo Eyo, Kenneth Murray, John Picton, Oginnin Ajirotitu, A. Onayemi Williams, Alaperu Iperu, Adisa Fagbemi, Wole Soyinka, Tahonzagbe Hountondji, C. Malcolm Watkins, Clarence Robins, Ted Holiday, John Amira, Sunta, John Mason, Babalosha Adefunmi, Akinleye Awolowo, B. C. de Groot, J. Douglas, Valdelice Gir\u00e3o, Katherine White Reswick, Morton D. May, Huguette van Geluwe, Vinigi Grottanelli, Albert Maesen, Ezio Bassani, Aldo Tagliaferri, Carlo Monzoni, Jos\u00e9 Adario dos Santos, Deoscoredes e Juana dos Santos, Vivaldo da Costa Lima, Clodimir Menezes da Silva, Renato Ferraz, Christopher Healey, Janina Rubinowitz, Gaama Gazon, Leo Fonseca e Silva, John Dwyer, Arlete Soares, Balbino, Arnaldo e Helma Marques, Oresto Mannarino, Alain Chevalier, Ragnar Widman, Bertil S\u00f6derberg, Torben Lundbaek and Paul Mork, Sylvia de Groot, Andr\u00e9 K\u00f6bben, Ambassador Franklin Williams, Ambassador and Mrs. Oakley, Lloyd and Sarah Garrison, Agogohene of Agogo, Thomas Akyea, Mr. and Mrs. Michael Hoyt, Rick Lawler, Peter Eno, Charles Miller, Chiefs of Ngbe and Obasijum at Itu and Defang (Cameroon), Obon Ndidem, Big Qua Town (Calabar), Gerry Amede, Ludovique Elien, Nathaniel Spear III, Mrs. Edward Webb, Albert Sanders, Judith Wragg Chase, Harris Lewis, Andrew Lewis, Muriel and Malcolm Bell, Mary Granger, Andr\u00e9 Phanord, Andr\u00e9 Pierre, Fr\u00e8re Cornet, Nestor Seuss, Gweta Lema, Shaji, Mrs. William R. Eve, Mr. and Mrs. Clemens de Baillou, Barbara Rivette, Elisabeth Coffman, Earle S. Teagarden, Patrick Pleskunas, Junellen Sullivan, J. Carter Brown, George Ellis, Regan Kerney, Robert L. Bernstein, and P. Lawford.\n\nI salute the strong support of my editor at Random House, Erroll McDonald, who helped to shape and protect this book; the inspiring presence of my children, Alicia and Clark; and, above all, the priestesses and priests of traditional Africa, and of Afro-America, who guard the coming of divine aspiration from heaven to this world.\n\nR.F.T.\n\n_New Haven_\n\n_October 1982_\n\n# **Contents**\n\n_Cover_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\n_Acknowledgments_\n\nIntroduction\n\n[ONE Black Saints Go Marching In:  \nYoruba Art and Culture in the Americas](Thom_9780307874337_epub_c01_r1.htm)\n\n[TWO The Sign of the Four Moments of the Sun:  \nKongo Art and Religion in the Americas](Thom_9780307874337_epub_c02_r1.htm)\n\n[THREE The Rara of the Universe:  \n _Vodun_ Religion and Art in Haiti](Thom_9780307874337_epub_c03_r1.htm)\n\n[FOUR Round Houses and Rhythmized Textiles:  \nMande-Related Art and Architecture in the Americas](Thom_9780307874337_epub_c04_r1.htm)\n\n[FIVE Emblems of Prowess:  \nEjagham Art and Writing in Two Worlds](Thom_9780307874337_epub_c05_r1.htm)\n\nNotes\n\n_About the Author_\n\n# **Introduction**\n\n# **The Rise of the Black  \nAtlantic Visual Tradition**\n\nListening to rock, jazz, blues, reggae, salsa, samba, bossa nova, juju, highlife, and mambo, one might conclude that much of the popular music of the world is informed by the flash of the spirit of a certain people specially armed with improvisatory drive and brilliance.\n\nSince the Atlantic slave trade, ancient African organizing principles of song and dance have crossed the seas from the Old World to the New. There they took on new momentum, intermingling with each other and with New World or European styles of singing and dance. Among those principles are the _dominance of a percussive performance style_ (attack and vital aliveness in sound and motion); _a propensity for multiple meter_ (competing meters sounding all at once); _overlapping call and response_ in singing (solo\/chorus, voice\/instrument\u2014\"interlock systems\" of performance); _inner pulse control_ (a \"metronome sense,\" keeping a beat indelibly in mind as a rhythmic common denominator in a welter of different meters); _suspended accentuation patterning_ (offbeat phrasing of melodic and choreographic accents); and, at a slightly different but equally recurrent level of exposition, _songs and dances of social allusion_ (music which, however danceable and \"swinging,\" remorselessly contrasts social imperfections against implied criteria for perfect living).\n\n_Flash of the Spirit_ is about _visual_ and _philosophic_ streams of creativity and imagination, running parallel to the massive musical and choreographic modalities that connect black persons of the western hemisphere, as well as the millions of European and Asian people attracted to and performing their styles, to Mother Africa. Aspects of the art and philosophy of the Yoruba of Nigeria and the Republic of Benin; the Bakongo of Bas-Zaire and neighboring Cabinda, Congo-Brazzaville, and Angola; the Fon and Ewe of the Republic of Benin and Togo; the Mande of Mali and neighboring territory; and the Ejagham of the Cross River in southeastern Nigeria and southwestern Cameroon, have come from sub-Saharan Africa to the western hemisphere.\n\nAll of these traditions are ancient and charged with nobility of blood and purpose. And all of them, with the exception of the villager Ejagham, are urban. Since the Middle Ages or earlier, the ancestors of the Yoruba, the Bakongo, the Fon, and the Mande peoples have lived in commanding towns, centers of visual richness and creativity. Even the Ejagham, with their widely imitated important men and women associations, founded respectively under the sign of the leopard and the crocodile\u2014emblematic of intimidating powers of moral vengeance and strong government\u2014live surrounded by miniature versions of the trappings of the courts of ancient urban queens and kings.\n\nThese civilizations not only were impressive for their urban density, refinement, and complexity, but were empowered with an inner momentum of conviction and poise that sent them spiraling out into the world, overcoming accidents of class, status, and political oppression. The rise, development, and achievement of Yoruba, Kongo, Fon, Mande, and Ejagham art and philosophy fused with new elements overseas, shaping and defining the black Atlantic visual tradition. To portray not only the originating impulses of these different black civilizations but some sense of the special inner drive and confidence that has kept them going\u2014that showered the northeast of Brazil with famous beads and emblems and gowns of the Yoruba and Dahomey; that fundamentally enriched the culture of North America with profound and sophisticated Kongo- and Angola-influenced herbalism, mental healing and funereal traditions among black people of the Old Deep American South and so on\u2014this is the scope and sweep and purpose of this book. For several decades, scholarly concentration upon shared main organizing principles of dance and music have shown generalized African cultural unities linking the women and men of West and Central Africa to black people in the New World. This book begins the project of identifying _specifically Yoruba, Kongo Dahomean, Mande_ , and _Ejagham_ influences on the art and philosophies of black people throughout the Americas.\n\n_Flash of the Spirit_ opens with a discussion of the art and ideals of the Yoruba, black Africa's largest population, creators of one of the premier cultures of the world. The Yoruba believe themselves descended from goddesses and gods, from an ancient spiritual capital, Ile-Ile. They show their special concern for the proprieties of right living through their worship of major goddesses and gods, each essentially a unique manifestation of _\u00e0she_ , the power-to-make-things-happen, a key to futurity and self-realization in Yoruba terms. There are thousands of deities in Yoruba territory, western Nigeria and eastern Benin Republic, but only the most widely worshipped and important survived the vicissitudes of the Atlantic Trade. These deities include Eshu, spirit of individuality and change; If\u00e1, god of divination; Og\u00fan, lord of iron; Yemoja, goddess of the seas; Oshun, goddess of sweet water, love, and giving; Oshoosi, god of hunting; Obaluaiye, dread spirit of disease and earth; Nana Bukuu, his mother; Sh\u00e0ng\u00f3, the fiery thunder god, who has inspired thousands of Afro-Americans (two Afro-American religions\u2014Sh\u00e0ng\u00f3 in Trinidad and Xango in Recife in Brazil\u2014bear his name). There are many other spirits of the Yoruba, alive and brilliantly worshipped by thousands in the New World, such as Obat\u00e1l\u00e1, deity of creativity; Orisha Oko, a celestial judge and restorer of fertility in life and cultivation of the earth; Eg\u00fang\u00fan, ancestral spirits, virtual moral inquisitors, who form the world of the dead. In _Flash of the Spirit_ , I offer portraits of ten _orisha_ , deities of the Yoruba, to represent the impact of the mind and spirit of millions of Yoruba in West Africa on key black urban populations in the Americas, most notably in Havana, Salvador, Brazil, and the heavily Hispanic barrios of certain cities of the United States, especially Miami and New York. In other words, the richness of detail, moral elaboration, and emblematic power that characterize the sacred art of the Yoruba in transition to Brazil, Cuba, and the United States, as sampled in this volume, is but an introduction to a wider universe of interlocking forms that will require future books fully to explore and explain.\n\nAlso widespread across the black Atlantic world are the signs and insights of the great Kongo people of Za\u00efre, Angola, Cabinda, and Congo-Brazzaville. There is a clear connection between the cosmographic signs of spiritual renaissance in the classical religion of the Bakongo and similarly chalked signs of initiation among blacks of Cuba, Haiti, the island of St. Vincent, the United States, and Brazil where numerous Kongo slaves arrived. The Kongo sign of the four moments of the sun\u2014dawn, noon, sunset, and the mirrored noon of the dead that we call midnight\u2014the master icon of their religious and philosophic world, informs the rituals of heavily Kongo-influenced parts of the New World. Kongo, for instance, largely sparked the rise of a beautiful ideographic tradition among the Haitian people, ground-signatures of deities called _v\u00e8v\u00e8_ , and an equally famous and pervasive ground-sign tradition among the blacks and African-influenced people of Rio de Janeiro\u2014signs that are called _pontos riscados_. They can be compared with yet more signs, richly reflecting Christian and other Western influences, on the isle of St. Vincent in the black Caribbean and with others found among Kongo-Cubans and U.S. mainland blacks from Memphis to the Carolina coast. These drawings and their accompanying rituals of healing and\/or initiation reflect the confluence of originating Kongo impulses plus other African and European influences.\n\nIn Haiti occurred a deep synthesis of the main forms and tenets of the classical religions of the Yoruba, the Dahomeans, and the Bakongo that was partly informed by the saints of the Roman Catholic Church and by their attributes. The result was _vodun_ : formally speaking, one of the richest and most misunderstood religions of the planet. In the chapter on Haitian _vodun_ I will not only review those elements of Yoruba and Kongo art and thought informing the main forms of vodun, but also focus in particular on the Dahomean influence. I will describe the greatness of _vodun_ art by way of an examination of perhaps the religion's two most beautiful expressions: the ground-paintings called _v\u00e8v\u00e8_ and sequin-studded flags of mediation, for the goddesses and gods, called _drapeau de vodun_.\n\nTwo mainstreams of visual lore and creativity came from the ancient Mande to the Americas. The cone-on-cylinder building traditions of the Mande and their neighbors\u2014their special spatial patterns and proportions\u2014were models for Afro-Mexican cone-on-cylinder structures along a stretch of the southwestern littoral of Mexico locally called the \"little coast\" (La Costa Chica), a few hours south of the well-known city of Acapulco.\n\nThe specificity of design of the rhythmized textiles of the Mande and their neighbors and of the far-flung replications of this mode, mediated through the Dyula and other Mande traders, has been handsomely reexpressed in the western hemisphere. Four main Afro-American textile traditions\u2014Barbadian aprons, Afro-Suriname Djuka _as\u00e9esenti_ , and Saamaka _as\u00e9esente_ , some U.S. black quilt-tops, and Afro-Bahian _panos da costa_ \u2014show traits deriving in part from Mande and Mande-related textile traditions. Chief among these features is the sewing together of narrow strips to form multistrip textile compositions in which the accents of one strip are staggered in relation to the accents of the strip immediately adjoining it but matched with the accents of other strips in the composition. This striking style of suspended patterning is one of the most dramatic elements of African influence not only in the visual art of the Afro-American world but also in its dance and music.\n\n_Flash of the Spirit_ ends with a discussion of the ideographic writing of the Ejagham of eastern Nigeria and western Cameroon. Through the accident of their proximity to Calabar, one of the notorious slaving ports of Africa, their art and ideology of accomplishment and prowess, specially symbolized among men by the sign of the leopard, indelibly influenced the art history of western Cuba. There men of Calabar-area descent founded Cuban chapters of the male \"leopard\" associations of Calabar and the Ejagham to the north with many aspects of their art and ideographic writing recalling the sumptuousness of the feathered art of the noblewomen of Ejagham. Ejagham artistic influence in the Americas extends a deeply rooted and significant African ideographic writing system, surprising only to those who still believe that Africa alone among the continents was without letters before the arrival of whites, without a means for recording and transmitting moral and folk lore. The transformation of Ejagham _nsibidi_ writing into the creolized offshoot in Cuba known as _anaforuana_ is one of the signal achievements of the black New World.\n\n_Flash of the Spirit_ , then, illumines art and philosophy connecting black Atlantic worlds. I hope, in opening these lines of inquiry, that the identification and explanation of some of these mainlines, intellectually perceived and sensuously appreciated, will provide a measure of the achievement of African civilizations in transition to the West, for theirs is one of the great migration styles in the history of the planet.\n\n# **One**\n\n# **BLACK SAINTS  \nGO MARCHING IN**\n\n# **Yoruba Art and Culture  \nin the Americas**\n\nOne bright morning in the middle of the nineteenth century, a young American missionary, R. H. Stone, ascended a lofty granite boulder and looked down upon the Yoruba city of Abeokuta. He wrote:\n\nWhat I saw disabused my mind of many errors in regard to... Africa. The city extends along the bank of the Ogun for nearly six miles and has a population approximately 200,000... instead of being lazy, naked savages, living on the spontaneous productions of the earth, they were dressed and were industrious... [providing] everything that their physical comfort required. The men are builders, blacksmiths, iron-smelters, carpenters, calabash-carvers, weavers, basket-makers, hat-makers, mat-makers, traders, barbers, tanners, tailors, farmers, and workers in leather and morocco... they make razors, swords, knives, hoes, billhooks, axes, arrow-heads, stirrups.... women... most diligently follow the pursuits which custom has allotted to them. They spin, weave, trade, cook, and dye cotton fabrics. They also make soap, dyes, palm oil, nut-oil, all the native earthenware, and many other things used in the country.\n\nThe city of Abeokuta seethed with creative activity, belying the condescending Western image of \"primitive Africa.\"\n\nThe Yoruba are one of the most urban of the traditional civilizations of black Africa. Yoruba urbanism is ancient, dating to the Middle Ages, when their holy city, Ile-Ife, where the Yoruba believe the world began, was flourishing with an artistic force that later provoked the astonishment of the West. At a time, between the tenth and twelfth centuries, when nothing of comparable quality was being produced in Europe, the master sculptors of Ile-Ife were shaping slendid art, as exemplified by a Berlin Museum terra-cotta head (Plate 1). In the elegant conception of the head, perhaps representing a person of status or a most important spirit, can be seen the signs of spiritual alertness (the searching gaze) and self-discipline and discretion (the sealed lips), which suggest, in Yoruba symbolic terms, the confidence of the people's monarchic traditions, and the complexity and poise of their urban way of life.\n\nPLATE 1\n\nLike ancient Greece, Yorubaland consisted of self-sufficient city-states characterized by artistic and poetic richness. The Yoruba themselves cherish the creators of their aesthetic world, as one of their hunters' ballads states:\n\nnot the brave alone, they also praise those who know how to shape images in wood or compose a song.\n\nWhen the first missionaries penetrated the city of Abeokuta in the 1840's, they found that the talents of the local master carver Kashi \"had procured for him the headship of the artisans in Abeokuta, and he had great influence among the people.\"\n\nThe Yoruba assess everything aesthetically\u2014from the taste and color of a yam to the qualities of a dye, to the dress and deportment of a woman or a man. An entry in one of the earliest dictionaries of their language, published in 1858, was _amewa_ , literally \"knower-of-beauty,\" \"connoisseur,\" one who looks for the manifestation of pure artistry. Beauty is seen in the mean _(iwont\u00fanwonsi)_ \u2014in something not too tall or too short, not too beautiful (overhandsome people turn out to be skeletons in disguise in many folktales) or too ugly. Moreover, the Yoruba appreciate freshness and improvisation per se in the arts. These preoccupations are especially evident in the rich and vast body of art works celebrating Yoruba religion.\n\nThe Yoruba religion, the worship of various spirits under God, presents a limitless horizon of vivid moral beings, generous yet intimidating. They are messengers and embodiments of _\u00e0she_ , spiritual command, the power-to-make-things-happen, God's own enabling light rendered accessible to men and women. The supreme deity, God Almighty, is called in Yoruba Olorun, master of the skies. Olorun is neither male nor female but a vital force. In other words, Olorun is the supreme quintessence of _\u00e0she_.\n\nWhen God came down to give the world _\u00e0she_ , God appeared in the form of certain animals. _\u00c0she_ descended in the form of the royal python _(ere)_ , the gaboon viper _(oka olushere)_ , the earthworm _(ekolo)_ , the white snail _(lakoshe)_ , and the woodpecker _(akoko)_. God, within these animals, had, according to Yoruba belief, bestowed upon us the power-to-make-things-happen, morally neutral power, power to give, and to take away, to kill and to give life, according to the purpose and the nature of its bearer. The messengers of _\u00e0she_ reflect this complex of powers. Some are essentially dangerous, with curved venomous fangs. Others are patient and slow-moving, teaching deliberation in their careful motion. Even the earthworm has its power, \"ventilating and cooling earth without the use of teeth.\"\n\nWe find the avatars of _\u00e0she_ dramatically suggested on the body of a superb ritual ceramic bowl for the thunder god and other deities _(ikoko shango)_. Made in Oyo Yoruba territory, perhaps in the late nineteenth century or the early decades of the present century, it was found in a market in the city of Ibadan in 1964. The object (Plate 2) now forms part of the exhibition collection of the Yale University Art Gallery. Deferring for the moment a discussion of the central motif\u2014the square representing the four corners of the earth along with the three concentric circles within the square that are the triple sign of the Yoruba goddess Earth\u2014we note strong zigzag patterns in relief that suggest the coming down of _\u00e0she_ in the form of lightning. These patterns also signify the embodiment of _\u00e0she_ within the python, gaboon viper, and many other serpent messengers of the deities. Y-shaped representations bespeak the balancing of this fiery enabling power upon thunderstaffs, i.e., double meteorites upon a royal scepter. Above these Y-shaped thunderstaffs appear smooth rectangular emblems representing thunderstones themselves come from heaven. Time and again the story of the descent of God's _\u00e0she_ , in multiple forms, in multiple avatars, is suggested ideographically upon this important vessel. To the right and left of the central square emblem appear chiefly scepters, underscoring the essential nobility of the persons who embody and comprehend the power-to-make-things-happen. The three concentric circles suggest three stones, the kind Yoruba women use to support their cooking vessels, meaning that adherence to the moral sanctions of Earth supports us all, safeguarding the equilibrium of the country and its people.\n\nSome trees are also thought to be avatars of _\u00e0she_ , sentinels guarding the Yoruba universe. The mighty teaklike _iroko (Chlorophora excelsa)_ is often so honored in traditional towns by the tying of a white cloth to its trunk as an offering. Iron, as time-resistant as the towering _iroko_ when specially consecrated, is also believed to contain _\u00e0she_. _\u00c0she_ may also be present in a drop of semen or a drop of blood\u2014for many Yoruba, red, \"supreme presence of color,\" signals _\u00e0she_ and potentiality.\n\n\u00c0she has other formal representations, including iron staffs, iron sculptures of serpents and long-beaked birds, and even the fluid of snails poured over altar emblems.\n\nPLATE 2\n\nA thing or a work of art that has _\u00e0she_ transcends ordinary questions about its makeup and confinements: it is divine force incarnate. This richest of all privileges is merited by the highest women and men of the land: the master priestesses, the diviners, the kings, the most important chiefs. All have _\u00e0she_. And even their words are susceptible to transposition into spirit-invoking and predictive experiences, for _\u00e0she_ literally means \"So be it,\" \"May it happen.\"\n\nYoruba kings provide the highest link between the people, the ancestors, and the gods. Their relation to the Creator is given in the praise poem _Oba alashe ekeji orisha_ , \"The king, as master of _\u00e0she_ , becomes the second of the gods.\" Birds, especially those connoting the _\u00e0she_ of \"the mothers,\" those most powerful elderly women with a force capable of mystically annihilating the arrogant, the selfishly rich, or other targets deserving of punishment, are often depicted in bead embroidery clustered at the top of the special crowns worn by Yoruba kings (Plate 3) signifying that the king rules by mastering and participating in the divine command personified by them. These feathered avatars brilliantly rendered in shining beads protect the head of the supreme leader. The veil that hangs across the wearer's face protects ordinary men and women from the searing gaze of the king in a state of ritual unity with his forebears.\n\nPLATE 3\n\nRitual contact with divinity underscores the religious aspirations of the Yoruba. To become possessed by the spirit of a Yoruba deity, which is a formal goal of the religion, is to \"make the god,\" to capture numinous flowing force within one's body. When this happens, the face of the devotee usually freezes into a mask, a mask often (but not always) held during the entire time of possession by the spirit. _\u00e0she_ is untranslatable. But it is clearly manifest in prophecy and predictive grace; hence persons possessed by the spirit of a Yoruba deity are believed to speak of things yet to come. They attract large crowds wherever they appear. They look about grandly with fixed expressions, with eyes sometimes wide and protuberant. The radiance of the eyes, the magnification of the gaze, reflects _\u00e0she_ , the brightness of the spirit. According to the Yoruba:\n\nThe gods have \"inner\" or \"spiritual\" _eyes (oj\u00fa in\u00fan)_ with which to see the world of heaven and \"outside eyes\" _(oj\u00fa ode)_ with which to view the world of men and women. When a person comes under the influence of a spirit, his ordinary eyes swell to accommodate the inner eyes, the eyes of the god. He will then look very broadly across the whole of all the devotees, he will open his eyes abnormally.\n\nIn addition to _\u00e0she, iwa_ (character) is another crucially important consideration in Yoruba religion and art. The Yoruba impart to many figurations a sense of ideal noble character by details of attitude and gesture as well as the use sometimes of white-colored media. Character is a force infusing physical beauty with everlastingness. \"I want to deliberate on this,\" an elder of Ipokia, capital of the Anago Yoruba, once told me, \"beauty is a part of coolness but beauty does not have the force that character has. Beauty comes to an end. Character is forever.\"\n\nThe importance of good character _(iwa rere)_ , which is virtually synonymous with coolness, with gentle generosity of character _(iwa pele)_ , is poetically rendered by the Yoruba:\n\nA man may be very, very handsome\n\nHandsome as a fish within the water\n\nBut if he has no character\n\nHe is no more than a wooden doll.\n\nPLATE 4\n\nPLATE 5\n\nPLATE 6\n\nAccording to the Yoruba art historian Babatunde Lawal, the Yoruba see the force of inner character operating as a smoky flame _(eefin n'iwa)_ easy to detect, for outward beauty can be burned through by inner ugliness or selfishness.\n\nLike _\u00e0she_ , good character originates in God. God is praised as Lord of Character _(olu iwa)_. Hence, that which attains proximity to the divine generally is progressively imbued with fine character. Artistic signs communicating this noble quality, _iwa_ , are often white. Immaculate white cloths may be honorifically draped over sculpture honoring gods or ancestors famed for spotlessness of reputation, as in the case of the cult of Obatala, the god of creativity. Purity of sculptural presentation; symmetry; balance: these qualities can memorably imply _iwa. Iwa_ also means custom, the traditional ways of life. An image portraying a person fulfilling the canons of the land in terms of fine posture or careful hands-to-the-sides gestures of spiritual alertness (Plate 4), or giving to an elder in the correct and prescribed manner\u2014with _both_ hands\u2014(Plates 5, 6) suggests submission to moral authority or to higher forces.\n\nA main focus of the presentation of ideal character in Yoruba art is the human head, magnified and carefully enhanced by detailed coiffure or headgear. This tendency, in combination with the use of the symbolic color of good character, white, is strikingly present in the Yoruba shrine of the head _(il\u00e9 or\u00ed)_. It is often a pointed, crownlike box, lavishly covered with a sheath of cowrie shells (Plate 7) to represent the riches a good head\u2014good character\u2014will bring, for cowries were the traditional currency of Yorubaland.\n\nAccording to traditional authority, shrines of the head also conceal, in the covering of the shining white shells, an allusion to a certain perching bird, whose white feathers are suggested by the overlapping cowries. This is the \"bird of the head\" _(eiye ororo)_ , enshrined in whiteness, the color of _iwa_ , and in purity. It is the bird which, according to the Yoruba, God places in the head of man or woman at birth as the emblem of the mind. The image of the descent of the bird of mind fuses with the image of the coming down of God's _\u00e0she_ in feathered form.\n\nThe sense of certainty, which character and _\u00e0she_ confer, is enriched by mystic coolness _(itutu)_ , whose emblematic color is often blue or indigo or green. Lawal introduces us to this sovereign concept, which cuts across virtually the whole of Yoruba figuration:\n\nTo tame or pacify is to \"cool the face\" _(tu l'oju)_. Thus, providing the non-figurative symbol of an orisha with sculptured face facilitates the pacification of that orisha, for what has a face is controllable.\n\nMuch Yoruba art is informed by _itutu_. To carve a calm face upon a represented thunderstone, or upon an abstract divination tray, or to incise it upon the swelling curves of a calabash for sacred things is to provide critical focus for acts of sacrifice and devotion.\n\nFurther manifestations of aesthetic coolness in Yoruba art include representations of idealized action. We must take care not to stress character and coolness as separate semantic structures because they shade into each other and also blur into the existential definition of _\u00e0she_. The interrelation of the concepts is explicitly given in vernacular testimony from the capital of the Anago:\n\nPLATE 7\n\nCoolness or gentleness of character is so important in our lives. _Coolness is the correct way you represent yourself to a human being_. When I saw you, I opened my cap. It is _itutu_ , answering past _itutu_ you made to me.\n\nThe phrasing is significant\u2014 _opening_ the cap, instead of doffing or removing it, in Western parlance, is a sign of generosity of response, of coolness in life and art. Thus Yoruba art abounds with images of men or women proffering a vessel. Witness a thunder god scepter-image of a naked kneeling woman supporting her breasts, with both hands (Plate 8). This is a sign of giving\u2014\"This milk shall be the sustenance of my children.\" It relates to another frequent image, the man or woman who presents an open, empty kola bowl (Plate 9), held with both hands, again as a sign of honor and respect. In the case of women, the location of the bowl at the level of the womb deliberately hints at the giving of children to the world.\n\nGenerosity, the highest form of morality in Yoruba traditional terms, is suggested yet another way: by the symbolized offering of something by a person to a higher force through the act of kneeling. Thus a senior priest: \"If you wish to talk to an elder, you do not stand, you kneel. When presenting a plate of food to someone important, _kneel_ as you make the presentation. _Kneel_ and give with _both_ hands, the left with the right, the 'mother hand' and the 'father hand,' the hand-which-keeps and the hand-which-acts\" (Plate 5). Giving with both hands, in a gesture of submission, emphasizes in traditional terms the act of giving as an embodiment of character and perfect composure, a point given further focus, in both art and life, by the firmness of the facial expression that accompanies the noble act.\n\n\"Constant smiling is not a Yoruba characteristic,\" a village elder once told me. Sealed lips, frequent in Yoruba statuary, are a \"sign of seriousness.\" They, too, imply the coolness of the image, as in an idiom that refers to discretion in ordinary discourse: \"His mouth is cool\" _(\u1eb9nu \u00e8 t\u00fat\u00f9), which is one of the ways the Yoruba would say, \"He fell silent.\"_ 22 _Like character, coolness ought to be internalized as a governing principle for a person to merit the high praise \"His heart is cool\" _(okan \u00e8 tutu). In becoming sophisticated, a Yoruba adept learns to differentiate between forms of spiritual coolness: (1) direct sacrifice _(ebo)_ , the cooling of the gods by the giving of cherished object\u2014such as the proffering of a ram to the thunder god; and (2) propitiation _(irele)_ , the utterance of conciliatory words or acts to hardened or angered deities, entreating them to become generous and concerned at times of crisis, such as birth, death, or initiation._ _A three-figure image for the deity Earth (ere_ Ogboni_), carved in Abeokuta in the twentieth century (Plate 10), splendidly illustrates cooling by propitiation. Here, where a male helper holds a fan, a literal sign of coolness and rank, a woman cools _(tu)_ her spiritual superior, the towering Ogboni cult leader with his staff of office, by prostrating herself on her right side. Through this gesture, called _yinrinka_ , a female member of a compound traditionally paid her respects every morning to the head man and head woman of the compound. (The corresponding male gesture of submission, _idobale_ , involved a complete prostration of the body on the ground. It was used also in saluting superiors.) The notion of coolness in Yoruba art extends beyond representations of the act of sacrifice and acts or gestures of propitiation. So heavily charged is this concept with ideas of beauty and correctness that a fine carnelian bead or a passage of exciting drumming may be praised as \"cool.\"\n\nPLATE 8\n\nPLATE 9\n\nPLATE 10\n\nCoolness, then, is a part of character, and character objectifies proper custom. To the degree that we live generously and discreetly, exhibiting grace under pressure, our appearance and our acts gradually assume virtual royal power. As we become noble, fully realizing the spark of creative goodness God endowed us with\u2014the shining _ororo_ bird of thought and aspiration\u2014we find the confidence to cope with all kinds of situations. This is _\u00e0she_. This is character. This is mystic coolness. All one. Paradise is regained, for Yoruba art returns the idea of heaven to mankind wherever the ancient ideal attitudes are genuinely manifested.\n\nThe Yoruba remain the Yoruba precisely because their culture provides them with ample philosophic means for comprehending, and ultimately transcending, the powers that periodically threaten to dissolve them. That their religion and their art withstood the horrors of the Middle Passage and firmly established themselves in the Americas (New York City, Miami, Havana, Matanzas, Recife, Bahia, Rio de Janeiro) as the slave trade effected a Yoruba diaspora\u2014reflects the triumph of an inexorable communal will.\n\nYoruba traditional culture and religious art had seemed destined for total obliteration in the wake of the slave trade, civil wars, and modernization. By the early decades of the nineteenth century the stability of western Yorubaland had been destroyed. The protective military arm of the old empire of the Oyo Yoruba, centered in their capital city of Oyo-Ile near the Niger River, collapsed during severe political dislocations following the end of the reign of King Awole in 1796. The western and northwestern groups of Yorubaland were consequently harassed by Dahomean slave-hunting warriors and Fulani horsemen, and later even by other Yoruba. Refugees fled to the south, where they banded together in new settlements, such as Ibadan and Abeokuta, for protection. These camps rapidly became cities, only to come in conflict with ancient established kingdoms, such as the Ijebu.\n\nWith the restraining hand of Oyo weakened, inevitably civil wars flared, starting with conflict at the ancient city of Owu in 1821. Ijebu began selling Egba and Ibadan captives; Egba and Ibadan retaliated. By 1845 Ibadan, successor to Oyo's power, had evolved into a military state and marched east to capture and sell Ijesha Yoruba, for the latter's capital, Ilesha, was the strongest northeastern kingdom and hence an obvious candidate for conquest in Ibadan's search for expansion into the area. While all this was happening, the Yoruba were beset by the Atlantic slave trade.\n\nNew World Yoruba emerged from all this strife. Ketu Yoruba men and women captured by the Dahomeans turned up in Haiti and Brazil, where to this day they are called by the Dahomean word _nago_. Oyo and other captives of the Fulani were brought to Cuba, Brazil, and the Caribbean, notably Trinidad. The Yoruba of Cuba were called Lucum\u00ed, probably after an ancient Yoruba phrase meaning \"my friend\" _(oluku mi)_. Thus the deities\u2014the orisha\u2014of the city-states of Oyo and Ketu were introduced to Cuba and Brazil, where the names of particular goddesses and gods of abiding fame\u2014Sh\u00e0ng\u00f3, Yemoja (often called Yemay\u00e1 in the New World), Oshoosi, Orisha Oko, Eyinle\u2014derived from the Oyo region. The cults of Oyo and Ketu deities (the latter including especially Obaluaiye, Omoolu, and Nana Bukuu) were reinforced by their encounter with most of the principal Yoruba deities: Eshu, Ifa, Osanyin, Ogun, and Obatala. Yoruba-influenced Ewe, Popo, and Fon slaves from territory directly to the west of Yorubaland brought their own cults and also influenced the syncretism of deities. A remarkable fusion of _orisha_ , long separated by civil war and intra-Yoruba migrations, took place in the New World.\n\nWhat is more, especially in Cuba and Brazil, New World Yoruba were introduced to the cult of Roman Catholic saints, learned their attributes, and worked out a series of parallelisms linking Christian figures and powers to the forces of their ancient deities. Thus the smallpox deity was equated, in some places, with Saint Lazarus because of the latter's wounds illustrated in chromolithographs. Thus the Virgin Mary was sometimes equated with the sweet and gentle aspect of the multifaceted goddess of the river, Oshun. Thus Sh\u00e0ng\u00f3, the Yoruba thunder god, in Cuba was frequently equated with Saint Barbara, whose killers were struck dead by God with lightning.\n\nYoruba-Americans, outwardly abiding by the religious proprieties of the Catholics who surrounded them, covertly practiced a system of thought that was a creative reorganization of their own traditional religion. Luminously intact in the memories of black elders from Africa, the goddesses and the gods of the Yoruba entered the modern world of the Americas. They came with their praises _(or\u00edk\u00ec)_ , extraordinary poems of prowess that defined the moral and aesthetic reverberation of their presence.\n\n## **Portraits of Major _Orisha_**\n\n### Eshu-Elegba\n\nAccording to legend, at a crossroads in the history of the Yoruba gods, when each wished to find out who, under God, was supreme, all the deities made their way to heaven, each bearing a rich sacrificial offering on his or her head. All save one. Eshu-Elegbara, wisely honoring beforehand the deity of divination with a sacrifice, had been told by him what to bring to heaven\u2014a single crimson parrot feather _(ekodide)_ , positioned upright upon his forehead, to signify that he was not to carry burdens on his head. Responding to the fiery flashing of the parrot feather, the very seal of supernatural force and _\u00e0she_ , God granted Eshu the force to make all things happen and multipy _(\u00e0she)_. _Outward_ signs of submission and material bounty were no match for wisdom and humility. Once granted his powers of dominion, Eshu, instead of arrogantly subordinating everyone to himself, did the \"cool\" (generously appropriate) thing: he gave a vast commemorative feast to share his newfound prestige, and to honor God for the priceless treasures of _\u00e0she_. And he warned those who did not recognize his status that he would bend them, \"like a string upon a bow,\" or pound them \"like a shell.\"\n\nThe sign of the crimson feather worn upon the brow, the seat of mind and judgment, was clearly interpreted as a symbol of _\u00e0she_ and the techniques of ritual assuagement _(etutu)_ that lead to the attainment of _\u00e0she_. Representing both the means and the end, the red parrot feather is seen today in initiatory contexts in the Yoruba religion ranging from the Benin Republic to Bahia in Brazil. _\u00c0she_ is a privilege of righteous living, not a right, and it can be seriously diminished when someone has slighted a deity or an important person. This means that one must cultivate the art of recognizing significant communications, knowing what is truth and what is falsehood, or else the lessons of the crossroads\u2014the point where doors open or close, where persons have to make decisions that may forever after affect their lives\u2014will be lost.\n\nEshu consequently came to be regarded as the very embodiment of the crossroads. Eshu-Elegbara is also the messenger of the gods, not only carrying sacrifices, deposited at crucial points of intersection, to the goddesses and to the gods, but sometimes bearing the crossroads to us in verbal form, in messages that test our wisdom and compassion (\"Is this true; shall I help him; what larger purpose opens up beyond this message?\"). He sometimes even \"wears\" the crossroads as a cap, colored black on one side, red on the other, provoking in his wake foolish arguments about whether his cap is black or red, wittily insisting by implication that we view a person or a thing from all sides before we form a general judgment.\n\nBecause of his provocative nature, Eshu has been characterized by missionaries and Western-minded Yoruba alike as \"the Devil.\" Outwardly mischievous but inwardly full of overflowing creative grace, Eshu-Elegbara eludes the coarse nets of characterization. Even his names compound his mystery. Some call him Eshu, \"the childless wanderer, alone, moving only as a spirit.\" Others call him Elegbara (or Elegba), \"owner-of-the-power\" (the ever-multiplying power communicated by the crimson feather that he bore to heaven), a royal child, a prince, a monarch. He is, of course, all these beings and more\u2014the ultimate master of potentiality. Eshu becomes the imperative companion-messenger of each deity, the imperative messenger-companion of the devotee. The cult of Eshu-Elegbara thus transcends the limits of ordinary affiliation and turns up wherever traditionally minded Yoruba may be.\n\nSo it was that Eshu-Elegbara became one of the most important images in the black Atlantic world. Blacks honor him in Cuba, for example, where \"men or women of African descent pour cool water at crossroads\u2014unobtrusively if white strangers be about\u2014in honor of Eshu.\" Cuban blacks associate Eshu with change: \"favorable, he modifies the worst of fates; hostile, he darkens the most brilliant of happenings.\" And thousands honor him today in Rio de Janeiro, where candles begging Eshu's favor may be lit in the gutters at intersections, in the very shadows of the skyscrapers that line the beaches of Ipanema or Copacabana. Aware of the complexity of his nature, whose different sides are symbolized by different names, Afro-Cubans share with Nigerians the belief that Eshu is a homeless wanderer, \"whose mood is perpetual evil,\" while Elegba is demonstrably less difficult.\n\nWitness against this shared exegetic backdrop the richness of art concerning Eshu-Elegbara on both shores of the Atlantic. The most important icons of this spirit in Africa are figures in lateritic earth and clay. These forms took root deeply in Cuba, Brazil, Miami, and Spanish Harlem. Wood sculpture, which is regarded within the cult as less ancient and consequently less essential, is divided in Nigeria and the Benin Republic into several categories: tiny supplicant figurines; paired male and female images united by a richly cowrie-studded strand; large ceremonial, figurated dance-hooks, which rest in ceremonial context upon the performer's shoulders; large votive images for decorating shrines and gates to the compounds of the illustrious and the great. Of these, only paired male and female icons sometimes reappeared on New World altars, and even they had attributes\u2014such as an upright knife upon the head, as an emblem of Eshu's wonder-working powers, the limitlessness of his _\u00e0she_ \u2014that were more generally shared and communicated by images in clay in the black Americas.\n\nThe descendants of the Yoruba in Cuba tell a myth relating to Eshu's clay imagery:\n\nOnce upon a time there was a child named Eshu who was always telling lies. One day young Eshu met a pair of terrifying eyes, shining in the shadows of a shell of a cocoanut, lying by a crossroads.\n\nHe told his parents about this marvel (his parents were king and queen) but no one would believe him, such was his reputation for mendacity. And so Eshu was left alone, soon to die a mysterious death, caused by unknown influences emanating from the eyes beside the crossroads. He had not honored these disembodied eyes [which may have been his own!] and that is why he died.\n\nNor did the members of the court see fit to offer sacrifice beside these eyes, still burning with a sinister light. Suddenly death and disaster struck the world with annihilating force. Divination priests were summoned. Ultimately they found the crossroads where the eyes once gleamed. But by this time the shell had weathered into nothing; the eyes were gone. Whereupon the priests then selected a certain stone, soothed it with assuaging fluid. By this rite they caused the spirit of the god Eshu to come from the forest to live within this stone, there to receive their profferings of honor. And so Eshu was properly honored, by sacrificial signs of honor and respect, and order returned to the world.\n\nIn Yorubaland itself, the \"stone\" of atonement was actually a piece of lateritic earth and believed to be the oldest icon of Eshu. Consider a letter, written in 1919 by a Christian convert at Ibadan, relating to Eshu sculpture in wood that once belonged to the writer's father and was probably carved toward the end of the nineteenth century:\n\nMy late father was once the chief worshipper of the gods of mischief, Eshu or Elegbara. Formerly these gods were not represented in human form [i.e., figurated wood] but were adored in the shape of a stone\u2014a kind of laterite or sandstone\u2014but the number of their worshippers increasing, wood statues of these gods began to be made.\n\nLaterite is said to be the oldest and most important medium for representing Eshu, Eshu-Yangi, father of all Eshu. I assume that the custom of marking the presence of Eshu with lateritic cones of hard red clay extends at least as far back as 1659; surviving exemplars of relief sculpture representing Eshu at that time indicate that other iconographic particulars of his image were complete.\n\nThe cone of laterite _(yangi)_ appears in Yoruba markets over which cult officials pour daily offerings of palm oil to maintain Eshu's problematic coolness. Laterite-cone altars to Eshu recall a myth whereby Eshu devoured enormous quantities of fish and fowl offered to him by his mother, and finally devoured his mother, too. Whereupon his father, the god of divination, alarmed, himself consulted a divination expert and was told to sacrifice a sword, a male goat, and fourteen thousand cowries. The god of divination did as he was told. Consequently, when Eshu threatened to devour him, too, the god took a sword and hacked him to pieces, and the pieces became individual _yangi_ , lateritic shards. Orunmila pursued Eshu through nine heavens until finally, in the last heaven, Eshu was pacified by Orunmila and said that all the particles of his spirit, the _yangi_ stones and shards, would become his representatives. All Orunmila had to do was to consult them (make sacrifice upon them and ask a blessing) whenever he wanted to send them on a mystic mission. Eshu then returned his mother, alive, to the world. His terrifying gluttony had therefore concealed an abundant generosity, the many pieces of laterite, the myriad Eshu. This represents the fact that he can take anything away\u2014or give it back\u2014according to whether his surrogates in clay are worshipped with sacrifice and devotion.\n\nJuana Elbein dos Santos strikes to the core of this legend:\n\neach individual is constituted and accompanied by his personal Eshu, the element which permits his birth, ultimate development, and progeny.... in order that [Eshu] can fulfill harmoniously a person's cycle of existence, the person must without fail restore, through sacrifices, the _\u00e0she_ devoured, in a real or metaphoric way, by his principle of individualized existence.\n\nEshu the prince devoured the truth by lying, never sacrificing, heedless of the damage done, and paid for his arrogance, when he finally told the truth, by dying. Thus in the Afro-Cuban myth Eshu devours himself but once again returns when proper sacrifice, centered upon a piece of stone, is made. The story of Eshu is an intricate retelling of the Yoruba belief that the highest form of morality is sharing and generosity\u2014the strongest talisman to hold against jealousy.\n\nAt some time the lateritic cones and pillars, which stand to remind the world of Eshu's power to disintegrate or multiply all happening, became figurated. Adding a note of literal sacrifice, cowrie shells, the ancient Yoruba coinage, mark the eyes and mouth of these figures of Eshu. Such images, like _yangi_ before them, could be fashioned within a shrine or at a crossroads or upon a threshold. They appear very widely in Yoruba and Yoruba-influenced lands.\n\nFrom western Togo comes an arresting exemplar (Plate 11). It was collected at Misah\u00f6he, Togo, in 1912 and is now in the Linden Museum, in Stuttgart. Here Legba (Elegbara) is a tiny rounded head of whitened clay, set on a rounded neck and shoulders also made of clay, the whole set, with feathers, within an earthenware dish similarly coated with whitish clay. This silent, staring little sprite, like the head and shoulders of a human embryo suddenly exposed, projects an image that is at once spectral and unfinished-looking. The gazing eyes recall the Afro-Cuban myth's burning orbs within the shell beside the crossroads. Diminutive dimensions and the use of clay as the main material recall the original animate fragments of laterite into which Eshu was divided.\n\nPLATE 11\n\nThe custom of making small votive images for Eshu was strongly reinstated in western Cuba. Many images were made of clay in Cuba, while some, in stone, were given eyes and mouths either by incisions or with paint, including a sui generis image, shown in a work published by Lydia Cabrera in 1954, carved in the form of a profile of the little spirit's head. Clay heads positioned in an earthenware pan ( _cazuela de barro_ ), analogous to the Ewe mode, emerged also in western Cuba. This is a hint that the rising style acquired its force through fusion of both Ewe\/Dahomean (Arar\u00e1) and Yoruba (Lucum\u00ed) manners of formal exposition. Moreover, clay and small stone sculpture were portable artifacts easy to hide from strangers, even under conditions of far-reaching oppression.\n\nThe vitality of reemergence is evinced by the numerous modes of Afro-Cuban clay statuary for Elegba. An elegantly innovative example of a style that apparently flourished in Havana in the nineteenth century was given as a gift to Lydia Cabrera by Asikpa, El Moro, a follower of the Yoruba gods. The artist took the round earthenware bowl and centered in it the image, as an inspiration for creative repetition of its circular shape, in a ring of gleaming cowries marking the shoulders of the figure, and, in a secondary ring, the neck.\n\nPLATE 12\n\nThe ways of rendering Elegba in stone and clay were countered in the twentieth century by yet another Afro-Cuban mode, Elegba rendered in concrete. By 1954 the mode was well established in Havana, where Sidney W. Mintz collected an example at a market in the summer of 1956 (Plate 12). Such images, essentially conical in shape, were creole transformations, as it were, of the ancient market cone of laterite. Here the tip of the cone is perforated, to receive a single nail, which is meant to suggest the wonder-working knife of Eshu Odara, who worked miracles with a knife erect upon his head.\n\nThe original lateritic shards became two hundred Eshu, and then two hundred more, in the telling of the origin of Eshu-Yangi (Eshu-the-Mound-of-Laterite). And, similarly, Afro-Cuban Eshu images multiplied wondrously. As the guardian of the threshold and keeper of the gate, not unlike the earthen Legba outside the compound entrances of Dahomey, they were placed in containers and concealed behind the main entrance to a person's house. \"Fed, there is no danger, but forgotten, something begins to happen,\" a Cuban woman told me in 1979.\n\nThe tradition of guarding homes with images of Eshu came with black Hispanic people from the Caribbean to New York City and Miami in the decades after World War II. Today clay or concrete images for Elegba in the United States number in the hundreds.\n\nEshu statuary is strongly rooted in Spanish-speaking New York City. Some fine examples\u2014from the collection of Christopher Oliana, himself one of the founding fathers of the Yoruba religion among mainland blacks in New York City\u2014grace the permanent exhibition of Afro-Americana at the Museum of Natural History.\n\nIn the northeast of Brazil at the end of the nineteenth century Eshu-Bara was represented by anthills. Given the incontestable impact of Kongo and Angola cultures upon black Brazil, as well as the importance of the termite mound as a sign of the dead in those portions of Central Africa, arriving Fon, Ewe, and Yoruba slaves may have merged an already well-established mystic usage of earthen mounds with the concept of the lateritic cones of Eshu.\n\nThe Yoruba of Bahia and their cultural allies, the Brazilian descendants of the Ewe (Gege) and the Fon, reintroduced the ancient household altar of Eshu\u2014small head and shoulders set within a dish or bowl. From a turn-of-the-century account: \"a ball of clay congealed with the blood of a bird, palm oil, and an infusion of sacred herbs, reproduces a human face, the eyes and mouth of which are represented by three small shells or cowries, inserted in the mass before it dries.\"\n\nAt some point a nineteenth-century continuity was complicated by a quest for a novel form, a building up of the image within a bowl in such a manner that it would rise up out of its container and begin to gesture. The latter trend was apparent by 1945 in Recife (though it may well have occurred before that time), where an Eshu-Bara was photographed.\n\nRio conceals its share of clay Eshu mounted in bowls. But this former Brazilian capital, like Havana, is heir to a modern mode in which concrete is used in the making of such images. Consider a smoothly finished \"Eshu Boi\" now in the Museu de Pol\u00edcia (Plate 13) that was probably made before 1941. In Dahomean Yorubaland, around Ouidah, I have seen large freestanding images for Elegba with mystic signs of the divination deity marked in inserted cowries on the chest of the image that are precisely one of the distinguishing characteristics of the Rio icon. The solid oval head of this image is hinged to an equally solid mass of truncal concrete, with the latter's starkness relieved by rounded suggestions of shoulders, arms, and trunk. Affection for Eshu softens the hardness of the medium, not unlike a stone transmuted into a child's stuffed animal, and this is appropriate to Eshu's fusion of valences that are both childlike (insatiable eating) and mature (restitution of what is right to those who sacrifice).\n\nPLATE 13\n\nClay and concrete Eshu have symbolic resonance\u2014sacrificial shells, where embedded, bring to life the spirit who, though reduced to shards, has nonetheless retained the energies upon which the development of our individuality depends.\n\nThe powers of Elegba are similarly retold in Nigerian Yoruba wood sculpture, e.g., indigo-painted images of a man and a woman joined by a cowrie-studded leather strap of the kind worn by women devotees, which is deliberately displayed upsidedown. An early European document of the genre appeared in an engraving (Plate 14) in a book about missionary life in the village of Oshielle, eight miles east of the city of Abeokuta. The book was published in 1857. In fact the engraving was based upon a drawing made in Oshielle of an actual specimen of paired male and female Eshu sculptures, which Reverend Townsend, a missionary, took back to Exeter, England, in 1868 (Plate 15). The engraving and the original both show a cylindrical base from which hang strands of cowries and a calabash-of-power ( _ado iran_ ), communicating Eshu's ability to endlessly multiply his force.\n\nPLATE 14\n\nSome Egbado Yoruba villagers say such images represent Elegba and his wife. This complements a deeper interpretation of Eshu as the principle of life and individuality who combines male and female valences. Here both the male and female figures have bulging eyes, which for Yoruba embody the power-to-make-things-happen, the gift Eshu received from God in heaven. This hint of awesome potentiality is softened by the generosity of the woman's gesture, a giving of her breasts, but sharpened by the male's presentation of arms both real (a club or sword) and mystic (a calabash containing power). Their protruding eyes and the male's calabash foretell a miracle that unfolds upon their heads, from which springs up a bladelike element structurally equivalent to the knife-atop-the-head that identifies Eshu in some of his clay and concrete avatars. Here the knives have been transformed into serpent heads, recalling a praise poem for Eshu, who \"makes a whistle from the head of a serpent\" (Plate 16). As if to emphasize the limitlessness of Eshu's wonder-working, calabash containers of self-multiplying power surmount the serpents' heads.\n\nWhen a knifelike element rises out of Elegba's head, it is a sign that the display of his powers has begun, the illustration of the wonder _(ara)_ from which his special name, Eshu Odara, \"the Wonder-Worker,\" derives. Songs for Odara in Ilodo and Ouidah in West Africa mention this wondrous knife as a reference to the fact that the pointed head of Eshu cannot shoulder ordinary burdens. Slightly modified, these songs reemerged among the blacks of Bahia and Havana and, later, in Hispanic New York City and Miami. Their lyrics conceal a visual pun on the single crimson feather that Eshu wore erect upon his head in the presence of the Almighty. Both feather and knife are described as preventing Eshu's head from being used to support an ordinary burden.\n\nIn the wake of this continuing and systematic lore, it was inevitable that nails and other objects would be used to simulate the knife atop the head of Eshu in the clay and concrete sculptures of Havana and Harlem. The sharply pointed knife _(shonsho abe)_ that crowns the head of Eshu is the striking element of a wooden image of him that was collected (probably in 1927) by Arthur Ramos in Bahia (Plate 17). The Ramos piece compares nicely with the Townsend images in Exeter (Plate 15). The expressive play of mass that enlivens the Nigerian images is, however, muted in Brazil. There the simple renderings of line and silhouette are indicative of the maker's apparent wish for leanness of expression. The Brazilian Elegba image holds the calabash-of-power, flaunting the power to make things multiply. This charming Afro-Brazilian figure is in the strongest tradition of visually rendering the Eshu knife.\n\nPLATE 15\n\nPLATE 16\n\nPLATE 17\n\nPLATE 18\n\nFurther transatlantic reinstatements extend the attributes of Elegba. His dance-hook or club becomes a simple hook in Cuba, sometimes painted red and black. The Ulm If\u00e1 divination tray, one of the oldest known pieces of Yoruba wood sculpture and carved before 1659 (Plate 19), includes, at \"seven o'clock\" in the interior circle, an image of Eshu with tailed headdress, sucking his thumb. In the upper right-hand quadrant of the outer square there is another image, perhaps Elegba, smoking a pipe. These are ancient representations of the flagrant orality of Elegba, hints of a propensity to absorb and to devour. His pipe remains an important emblem and so he appears in Cuba, shaped in clay and set in a vessel in the Ewe manner, with a pipe beside him. The icons of Elegba seemingly are infinite. They are figures representing the supreme importance of attaining spiritual coolness through direct sacrifice _(ebo)_ and ritual reconciliation _(irele)_ , acts that protect the mirroring imperative of the actualization of individuality\u2014points intuited in a final example of Eshu sculpture, an image in wood that once graced the compound entrance of Ogabunna, chief of Ikija quarter, Abeokuta, during the first half of the nineteenth century (Plate 18). Here kneeling suggests submission, sacrifice, and propitiation, while the carefully positioned clublike element and the towering headdress are intimations of _\u00e0she_ , and multiplication of the self.\n\n### If\u00e1\n\nWhenever traditional Yoruba encounter change or challenge in the world of Eshu, the limitations of individual calm and wisdom become acute. In such a case a person relies on the accumulated insights of the poetic chants of the Yoruba divination system called If\u00e1 to place his or her individual problem in perspective. Thus Abimbola: \"If\u00e1 divination is performed by the Yoruba during all their important rites of passage such as naming and marriage ceremonies, funeral rites and the installation of kings. In traditional Yoruba society, the authority of If\u00e1 permeated every aspect of life because the Yoruba regard If\u00e1 as the voice of the divinities and the wisdom of the ancestors.\"\n\nPLATE 19\n\nThere are many forms of countering uncertainty with divination in Yoruba culture, including the throwing of kola nuts _(obi)_ on the ground, the casting of cowrie shells _(owo)_ , water-gazing, mirror-gazing, and receiving ecstatic prophecy from a deity speaking through an initiated priest or priestess. Over these systems If\u00e1 rules supreme. The lifetime study demanded to maintain If\u00e1 and the respect traditional Yoruba have for the analytic reach of its verses can be indicated by a single proverb: \"Stargazing is no substitute for If\u00e1's knowledge.\"\n\nThe literature of If\u00e1 divination divides into sixteen main parts called _odu_. Each _odu_ bears the name of an ancient prince. To hear the verses is to come into the presence of a royal voice imparting insight and infinite experience. The priest of divination himself is called the father of the secrets _(babalawo)_ , and he is allowed to possess, like the Yoruba divine kings, beaded treasure and regalia.\n\n_Ikin_ , \"The Sixteen Sacred Palm-Nuts,\" are held the most ancient and important of the instruments of divination. They come from the sacred palm tree of If\u00e1 _(ope ifa)_. Myth tells us how and why these natural elements became an august sign.\n\nOnce If\u00e1 reigned upon the earth and there dispensed his precious wisdom. But one day a son of If\u00e1 arrogantly refused to bow down before his father, whereupon Ifa withdrew to heaven.\n\nWhen the light in the eyes of Eshu dimmed by the road and disappeared, the world began to die. Similarly, the translation of If\u00e1 to heaven occasioned such a terrifying blast of sterility and drought that starving animals attempted to devour sharp razors scattered on the ground, and river basins were covered with dead leaves. The world again was dying. The children of If\u00e1 climbed the sacred palm tree to beg their father to return, and he gave each of them sixteen palm nuts as a concentrated essence of his healing wisdom, replacing himself on earth with the sixteen sacred palm nuts, the _ikin_. Order and life were again restored.\n\nThus, like the division of Eshu into myriad fragments of raw laterite, the division of If\u00e1 into sixteen kernels taken direct from nature restores life. These powerful beliefs give sanction and authority to the most important form of If\u00e1 divination, which involves the use of the _ikin_ and is usually reserved for crucial moments, such as the investiture of a king.\n\nIn such cases, the diviner strews fine divination powder _(iyerosun)_ obtained from the irosun tree (or powdered dry bamboo) on an If\u00e1 divination tray, here illustrated by perhaps the oldest known example, carved before 1659 (Plate 19), and smooths it over. Then, with his middle finger, he traces either a single line or two lines in the powder, depending on the number of _ikin-if\u00e1_ that remain in his hand when he attempts to pick up sixteen _ikin_ in his right hand. If two palm nuts remain, he marks parallel lines on the tray, or, if one _ikin_ remains, one vertical stroke, and so on, until by divining four times with the _ikin_ in this way, one out of a possible set of sixteen patterns completes itself upon the divination tray.\n\nThe sign of a particular section _(odu)_ of the verbal literature of If\u00e1 has now appeared. The diviner recites the verses associated with this sign in three parts: (1) an exemplary myth from the lore of the goddesses and the gods; (2) what happens to the deities within each myth, i.e., cautionary tales on the consequences of failure to make sacrifices indicated by If\u00e1; and (3) application of these themes to the client's problem. The last part usually opens with the phrase \"If\u00e1 says,\" for the client has now placed himself within range of the very voice of the god of divination, speaking through the ritual moves and counters:\n\nIf\u00e1 says that a man should not be covetous and that he should not strive to win a position from the person who holds the position by right. [If\u00e1 also says] that he should make appeasement lest the people of the world take his role from him and give it to another.\n\nThe client himself interprets such verses, for the _babalawo_ is not allowed to know the precise nature of the client's problem. In the verse just mentioned, he might find evidence that an excess of ambitious maneuvering on his part may have angered certain elders who had secretly caused a swift and inexplicable decline in his fortune. Proper sacrifices would be indicated, and he would offer them gladly, knowing that he was making amends suggested by the gods themselves.\n\nThere is another If\u00e1 divining method, involving the use of a divining chain _(opele)_ made of string or metal with four half-nuts of the _opele_ fruit attached to each half of the chain. Each _opele_ nut has a smooth outside surface and a rough inside surface. When the divining chain is thrown, away from the diviner, it falls upon the ground in a rough U shape, with four _opele_ seeds on one side, four on the other, each possibly falling a different way. There are thus sixteen possible forms of presentation, the mystic number of If\u00e1, and each of these presentation forms of the divining chain stands for an _odu_. If the _odu_ that emerges is the one called _Eji Ogbe_ , the diviner writes its sign upon the divining tray:\n\nIf the _odu_ , on the other hand, turns up as _Oyeku Meji_ , the appropriate signature to mark in the _iyerosun_ dust takes another form. And so on. In an example of divining-chain divination in Dahomean Yorubaland at Takon in Benin (Plate 20), all eight seeds have fallen \"open,\" the sign of the richly favorable _odu Eji Ogbe_. This sign has also been permanently rendered in cowrie-shell mosaic (from which one shell is missing) in the floor at the threshold of the owner of the divining chain, to charge the gateway to the house of a chief with good luck and well-being. Thus chain-divining cuts through to _odu_ in a manner similar to the use of _ikin_. The \"readings\" occur faster in the case of the chain, which is more commonly used.\n\nPLATE 20\n\nBoth systems expose Yoruba to the principal sixteen verses of If\u00e1 divination and thus lead to treasures of African verbal poetry and wisdom. It is believed that If\u00e1 encompasses the whole of the wisdom of the ancestors, the whole of the wisdom of the deities, and thus safeguards \"everything that is considered memorable in Yoruba culture throughout the ages.\" Hence the splendor of the image of If\u00e1:\n\nThe life of If\u00e1 surpasses water's coolness\n\nThe life of If\u00e1 surpasses water's coolness,\n\nThe speaker-of-all-languages married a woman\n\nwho herself bathed only in water that is cold\n\nThe life of If\u00e1 surpasses water in its coolness.\n\nWande Abimbola points out that the survival of Yoruba tradition through its turbulent history depended in large measure on a cadre of wise and disciplined diviners steeped in the secrets of the ancestors and of the gods. Thus one can well imagine how cultural treasures were brought across the Atlantic by men who remembered the lore of the _odu_ , the divination verses.\n\nIn Cuba the amazing continuity of Nigerian _odu_ was established in a landmark publication by William Bascom, in 1952, in which he proved that the _babalao_ (creole equivalent to the original Nigerian diviners) had reinstated the _odu_ , with names and explanatory tales virtually intact. In the process, Yoruba divining trays, divining chains (Plate 21), and sacred sixteen _ikin_ were introduced to Cuba.\n\nOf the Cuban divination trays, most are round, like an example from northern Yorubaland (Plate 22), or rectangular, with staring Eshu heads facing each other, and some are called _atefa_ , a Ketu word and thus a hint of the Ketu Yoruba origins of the mode. The divination trays of Bahia are also called _atefa_ , with the same implications. The Museu de Arte Popular, in Bahia, included in its 1968 Afro-Brazilian exhibition a magnificent _atefa_ , attributed to an unknown nineteenth-century carver (Plate 23). The purity of his handling of the ancient themes is manifest in his use of the circle and the head of Eshu (Plate 24), which interrupts the flow of the elaborately carved border, as in the case of the ancient Ulm tray (Plate 19). In the Brazilian example, two heads of Eshu stare at each other across the cosmic circle. There are similarly paired segments of ornamental carving filled with depictions of honorific cowrie shells and V-form elements which may relate cryptically to thunderstones or the Y-like prongs of the thunder ax.\n\nPLATE 21\n\nPLATE 22\n\nPLATE 23\n\nPLATE 24\n\nPLATE 25\n\nYoruba sculptors also make caryatid cups _(agere ifa)_ as vessels for the sacred _ikin-if\u00e1_ or the divining chain.\n\nA Nigerian caryatid figure by Labintan of Otta (Plate 25) supports the If\u00e1 bowl with his head and the soles of his feet. He rests on his elbows and bends up his legs. This marvelously realized gesture conceals, in apparent playfulness, serious submission to predictive grace.\n\nIt is important to grasp the style of Labintan because one of his If\u00e1 divination carvings, an _agere_ emboldened with strongly postured human figures supporting the bowl on their heads, washed up on the shores of Brazil (Plate 26). It was found on the beach at Cal\u00e7ado do Bomfim, Bahia, wrapped in linen, an indication that it had been used by an Afro-Bahian priest or priestess. Apparently upon the death of the owner, the cup had been tossed into the sea to return with the soul of its departed owner to Yorubaland.\n\nThis piece establishes that some Afro-Bahian works of art of the turn of the century were actually carved in Yorubaland, in this case Otta. Attribution to Labintan can be demonstrated by shared traits, including deeply drilled human pupils, hooked noses, hairpin ears, sharply pointed chins, identical hinges, and identically shaped shallow bowls.\n\nPLATE 26\n\nThe ties that bind the \"Rome of the Africans,\" Bahia, to the Yoruba of Nigeria were never more directly instanced. But most of the New World forms of If\u00e1 divination art were indigenously continued and elaborated, giving rise to the _atefa_ of Bahia and Havana, the _opele_ of western Cuba, and the use and knowledge of the green and yellow beads of If\u00e1 throughout the Atlantic world.\n\n### Osanyin\n\nThe art and lore of the cult of Osanyin, god of herbalistic medicine, embody a richness of positive assertion comparable to that of If\u00e1. In Osanyin's name the Yoruba undertook a vast study of the leaves and herbs and roots of the forests, classifying them with regard to their therapeutic properties, and combining them to make the master medicines of initiation and _\u00e0she_.\n\nOsanyin initiates master this taxonomic knowledge, learning what species of forest herbs to collect, mix, and pound to make medicine to soothe a feverish body or to calm an agitated mind. Leaves and roots as elements of healing are to Osanyin what the sixteen _ikin_ and the art divination are to If\u00e1. As we shall see, rich traditions of verbal art and literature also traveled with his worship to the New World.\n\nIn Cuba, for example, traditional segments of the black population honor Osanyin and link him-to a deeper belief in the spirituality of the forest. There the faithful honor the forest, _el monte_ (literally, \"the mountain\") as a source of healing power. They realize that this standing cathedral of shade and moistness belongs to Osanyin and to God, and so they leave small sacrifices in payment for herbs and roots subtracted from his realm, \"for every tree, every shrub and herb has its master, and its protocol.\" Each _orisha_ is served by its own sacralizing herbs, taken with permission from Osanyin, just as each deity is accompanied by his or her Eshu, source of individualizing power and vitality.\n\nAnd yet for all his glory, Osanyin is physically bizarre, having only one eye, one arm, one leg, which are the stigmata of a quondam selfish life, when Osanyin tried to keep all the medicines and leaves to himself. According to legend:\n\nDiviner said he was suffering because he could find no food nor sustenance because all of the work which he might be doing with the leaves was being done by Osanyin. Eshu said that he would help. Eshu caused the stones of the house of Osanyin to fall and maim the deity. Lacking a leg and an arm and an eye, he now needed diviner, urgently, to collect his leaves and continue with the curing of people. And since that time diviners and Osanyin have been working hand in hand.\n\nIt is said that Osanyin also failed to make a proper sacrifice commanded by If\u00e1 and, consequently, lost his voice. Thereafter whenever he opened his mouth to speak, only a comically squeaky voice was heard. Diminished, even vocally, Osanyin is a warning as to what happens to persons who are callously unsharing.\n\nHis is an image now indelibly Atlantic. Both in western Nigeria and eastern Benin, as well as in western Cuba and northeastern Brazil, most of his followers speak of his one-legged, one-armed, one-eyed appearance and his tiny, high-pitched voice. This basic image has been complicated by imaginative additions in Cuba and Hispanic New York. Some Afro-Cuban followers of Osanyin say that one of his ears is of monstrous size but hears absolutely nothing, while the other ear is diminutive but picks up the noise of butterflies in flight, a balance of elements coarse and fine that is reminiscent of Eshu's idiom of extremity. As a matter of fact, both Eshu and Osanyin share the attribute of one-leggedness, and like Eshu, Osanyin was once a prince.\n\nOsanyin, so it is believed in Ijebu Yoruba country, was born with beads shining about his body. Moreover, beads are important to Osanyin because he associates their colors with the hues and qualities of the forest herbs. Thus some of the sylvan fronds are said to be bright green, while other herbs are yellow, black, red, or even white, and each hue denotes a special kind of curing power. In some parts of Yorubaland the beads of Osanyin, in their colors and cool glitter, equal the work of multiple, differently colored, healing herbs. I recall an incident at Edunabon in the south of Oyo country in the winter of 1963\u201364, when a man regarded an ancient, polished blood-red carnelian bead and pronounced it \"cool\" _(tutu)_. But the beads of Osanyin are cool not only in the general sense of giving aesthetic pleasure but in a literal sense of referral to leaf like qualities of refreshing taste or smell, and\u2014most important\u2014powers of restoration.\n\nHeralded by gongs, Osanyin is the crippled king who, crushed to half his size, gained insight into the human condition. He comes not in the body of a possession devotee but in that of a tiny doll, given voice and motion by trained ventriloquists who are also Osanyin priests and healers. Ventriloquism, in fact, is one of the marvels of the cult of the lord of leaves not only in Nigeria but also in Cuba. His tiny voice was once heard also in Brazil, where priests of ancestral spirits attached to their bodies small store-bought rubber dolls of the kind that squeaked when pinched, to suggest the tiny voice of Osanyin\u2014an ingenious \"bending\" of a modern object in the direction of tradition.\n\nOsanyin ventriloquists appeared in the 1950's in and around certain villages of the province of Havana\u2014Perico, Alacranes, Mantilla:\n\nThe Osanyin image of old Federico was a doll. The old man used to sit behind his door, half ajar, to smoke his pipe. One day I happened to be there, and heard a doll say, \"Federico, here comes a woman all dressed in white, looking for a remedy for her husband.\"\n\nHealing, art, and ventriloquism were thus combined in Osanyin worship to provide a striking, theatrical form of consultation to the blacks of Ijebu, Abeokuta, the Orozco sugar mill in Cuba, and other places where the full tradition has been reported or witnessed. I saw the Osanyin puppet in action in southern Ijebu on various occasions in the early 1960's; wearing a miniature beaded veil and beaded gown, it was made to speak in the tiniest of squeaks by the priest-ventriloquist. The reinstatement of this custom in Cuba may not be the work only of the Yoruba and their descendants: Henry and Margaret Drewall found in a study of the Age (hunting deity) cult in Togo in the summer of 1975 that Age is also \"a forest sprite who heals with leaves, has one eye and one leg and whose priest practices ventriloquism.\"\n\nCertain groups in Nigerian Yorubaland allege that the sound of Osanyin's voice relates \"to a little bird that represents him.\" According to this tradition, this bird not only speaks when the deity is consulted, but also lives in the sacred calabash of Osanyin kept upon his altar.\n\nVoice-throwing and bird imagery are integral to the cult of Osanyin, and they explain an important province of Osanyin art: myriad forms of a wrought-iron staff surmounted by one or more birds in iron. If\u00e1 says there are sixteen styles of the Osanyin wrought-iron staff. One kind carries a single \"head\" (a single bird poised at the summit of the staff), while another carries two heads and still another displays three heads and so on until the highest number is achieved, sixteen birds in iron. The last is especially prestigious, \"for the highest people calculate their power by sixteen.\" The divination literature tells us that proliferating bird motifs allude to an ancient time, when Osanyin was magician of the gods, working miracles with one, then two, then three, then four, and, finally, sixteen heads, or birds. The persistent equation of bird with head, as the seat of power and personal destiny, is of the essence in comprehending elaborations of this fundamental metaphor, including staffs (1) showing a single iron bird set upon a single disk of iron surmounting several bells of iron, the _osun_ staff or _orere_ , (2) a bird set over a radiating display of miniature iron implements for the iron god, Og\u00fan, sometimes interspersed with miniature emblems of other \"hard\" deities, (3) a superbly fashioned bird in a commanding position over a circle of smaller, less elaborately decorated birds. The three staff types appear in various New World cities, notably Rio, Bahia, Gona\u00efves (Haiti), Havana, and New York, but rarely simultaneously.\n\nThe myth of the maiming of Osanyin tells us why he needs If\u00e1 and warns us that the knowledge of the leaves must be shared. But the obverse is equally true: If\u00e1 needs Osanyin. Without the lord of leaves and his many medicines, If\u00e1's effectiveness would be seriously diminished. And so the diviner-herbalist and the herbalist come to share the _osun_ staff, an extraordinary \"text\" wrought in iron on the power to comprehend and check disease.\n\nOdeleogun, a master blacksmith of Efon-Alaiye who flourished toward the end of the last century, is believed to have made our Nigerian example of the genre (Plate 27). It is a work that gracefully illustrates the _osun_ structure: a single iron bird over a single iron disk that covers and conceals parts of the four inverted iron bells radiating out from a single point along the staff below the disk and formally \"answered\" by four more bells, hanging right side up, below this point.\n\n_Osun_ , transmuted into _ase_ by the Popo and the Fon of what is now Benin, have been enriched there by generations of expressive elaboration since 1659 and earlier. There the single bird often rests upon the canonical disk, but the latter element can be fashioned to seal completely the mouth of a single inverted conical container, under which appears a small sphere or sometimes (as in this instance) a hard-shelled seed, which is pierced by the axis of the staff (Plate 28).\n\nFrom Ekiti (northeastern Yorubaland) to Dahomey the association of the presiding bird figure with war on witchcraft is more or less consistent. An informant at Igogo-Ekiti in the summer of 1965 alleged that a bird atop a healer's staff shows \"the mothers\" how powerful the healer is with his herbs. Dahomean chiefs in the city of Abomey told me in 1968 that the bird atop the _ase_ staff (Plate 28) represented the power of King Glele in overcoming not only the armies of Mahi but the \"bird\" (i.e., sorcerous potentiality) of the leader of the enemy forces as well. In addition, a healer at Ipokia, capital of the Anago Yoruba, revealed in December 1962 that the bells on such staffs are sometimes interpreted as metaphoric leaves; the disk is meant to cover the leaflike bells so that uninitiated eyes cannot see what is inside them.\n\nPLATE 27\n\nPLATE 28\n\nA seed as an occasional element on the _osun_ or other Osanyin-related staffs, and the use of bells, express the lore of the guards of If\u00e1. It is told that If\u00e1 brandished a certain seed before the witches of feathered form, saying, \"Witch is not fierce, she cannot eat the hard seed.\" It is also told that If\u00e1, informed by Eshu of the secret vulnerabilities of \"the mothers,\" brandished, among many effective objects, the leaf called _agogo igun_ (bell of the vulture's beak) \"so that everything he asks for, by means of agogo bells will be obtained.\"\n\nThus the metaphor of a text in iron is not to be taken lightly; the bird is both the mind of the healer and a warning to the mothers that he knows their forms and the powers that they inform; the disk covers his secret antidotes to their deadly propensities; and the bells and the seed are guards or extensions of the neutralizing impact of herbalistic medicine.\n\nThe association of a bird with the head or mind of a person is revealed during the initiation of a person into the service of the Yoruba gods in Cuba and Cuban-influenced portions of Miami and New York City. The full ceremony includes one of the most impressive reinstatements of the literature of Osanyin on New World soil\u2014a chanting of some sixteen to twenty-one songs, many based on the same melody. These songs are among the most ancient and precious testaments of Yoruba oral literature that we have in the Americas; they go back at least to the late eighteenth century, and one includes a reference to the immortality of God, unmovable stone under water _(oyigiyigi ota l'omi o oyigiyigi ota l'omi)_. These songs accompany the preparation of the leaves of Osanyin, to complete \"the water of the calm,\" the \"water of the cool\" _(omi ero)_ needed for sacralizing the postulant and the paraphernalia he or she will receive.\n\nAnd then comes the moment when the _\u00e0she_ received by the initiate is sealed in a small incision cut at the uppermost portion of his shaved head and within the container of his personal _osun_ bird-staff with disk and bells:\n\nto make osun is an extremely secret, sacred matter.... Once the person serving as a barber has shaved the initiate's head entirely, he proceeds to paint it white, indigo blue, red and yellow [each color forming a concentric band around the center of his crown].... when the initiator or initiatrix has finished the application of the colors, he or she cuts small incisions at the uppermost portion of the crown and there inserts four important materials, _obi kola, eru, tushe_ , and _osun_ [\"the indispensable seeds of the consagration, seeds imported from Africa\"].\n\nThe four basic seeds\u2014 _osun, eru, tushe_ , and _obi kola_ \u2014placed within the private bird-staff _(osun)_ of the initiate seal the same protective forces that went into his head inside the inverted cone or container beneath the bird of the _osun_ staff. This establishes the bird in iron or metal as the eternal companion and guardian of the initiate. Elements of the ancient \"text\" are reintegrated here\u2014bird, head, seeds, and bells.\n\nMany Cuban _osun_ are strongly African in style, as attested by the older examples of _osun_ staffs in the National Museum in Havana. The latter are remarkable not only for the retention of inverted cones or a single cone in metal under the disk that supports the bird, but also for a shaping of the bird more or less as a flattened, iconic element. Lydia Cabrera published in 1954 a photograph documenting this older style\u2014well-nigh concealed behind a welter of cult detail\u2014at the top of a many-tiered altar to Eshu. Careful study of this area reveals the unmistakable silhouette of an Africanizing _osun_ staff, with its flat, gracefully curving bird at the summit over the disk and inverted cones.\n\nBy 1954 creole transformations had already occurred. The new forms had absorbed Western industrial or cultural fragments\u2014the hubcap of an automobile, a metal rooster from a weather vane or discarded garden furniture, store-bought jingle bells\u2014and invested them with new meaning. The rooster replaced the flattened bird of the elders, the hubcap sometimes became the base, and the jingle bells recalled the _agogo_ gongs. Most important, the single inverted cone underneath the disk became a metal cup (into which the spirit-protecting seeds were placed). It is this modernist form that is most frequently seen today in the Cuban-influenced _bot\u00e1nicas_ of Miami, New Jersey, and New York (Plate 29). In these stores, where herbal charms and medicines frequently are sold, the sign of Osanyin appropriately reappears.\n\nPLATE 29\n\nThere is a further reinstatement of Osanyin iron across the Atlantic that involves staffs with a senior bird of mind or healing positioned at the summit of the staff, above a round of minor birds (Plate 30). Odeleogun of Efon-Alaiye made our Nigerian example, probably in the last quarter of the nineteenth century. This strong and elegant bird-staff carries within its forms a universal thought: the triumph of the mind over the annihilating circle of destruction and disease. The double, spiraled plume of the bird at the summit suggests double power held at once, and displays the ever-multiplying presence of Osanyin, or other healing spirits like Erinle.\n\nPLATE 30\n\nAn echo of this form in Bahia in Brazil is lean and simplified, and was made by an Afro-Brazilian blacksmith, Jos\u00e9 Adario dos Santos, at his atelier on the Ladeira da Concei\u00e7\u00e3o above the port of Salvador in the spring of 1968 (Plate 31). A single stylized bird surmounts six raking bars of pointed iron, suggesting, in one version, Osanyin above the crossroads of Eshu and iron, or, in another, the bird of Osanyin above the sharpened points of Ogun's iron, or, in still another, the bird of Osanyin in the branches of a tree. These ideas receive a full embodiment in Rio de Janeiro, where a bird-staff in iron for Osanyin was collected, perhaps around 1941 (Plate 32). Here the bars below the senior bird suggest again the branches of a tree, each displaying a single minor bird beneath the commanding spirit at the summit. There seems to be a lesson in the cherishing of this ancient image: the senior bird of mind and healing, \"the bird that shows the 'mothers' how powerful the healer is about his herbs,\" teaches that the woods and their medicines are grander than any document, and that one who assumes this privileged forest is hers or his alone will be answerable to God.\n\nPLATE 31\n\nPLATE 32\n\n### Og\u00fan\n\nThe art of Og\u00fan reflects his nature as a \"hard god,\" a deity of war and iron. He lives in the flames of the blacksmith's forge, on the battlefield, and more particularly on the cutting edge of iron. He addresses the forest with a sharpened machete; his spirit moves in the clearing of the bush, in the hoes and knives of cultivators. His worship is a means of thanksgiving for the ambivalent civilizing force of iron and iron-made implements. Og\u00fan served the very creator of the world, so it is believed, by clearing the primordial forests with his iron, making the first sixteen roads that radiate from the ancient holy capital, the roads upon which the original sixteen sons of the first king traveled forth to found the sixteen originating kingships.\n\nPraise-chants for Og\u00fan, those collected in ancient towns like Ire and Ketu and Ilesha, illustrate his ambivalent nature: his power to destroy as well as to construct:\n\n[from the towns of Aramoko and Ilesha]\n\nOg\u00fan, master of the world, support of the newborn child\n\nOg\u00fan is virile\n\nOg\u00fan, master of the yam I cut...\n\nOg\u00fan, with coronet of blood\n\nBurns the forest, burns the bush\n\nLeaves the forest screaming in the sound of flames\n\n[from Ire town]\n\nOg\u00fan cuts, in large or small fragments\n\nHe kills the husband on the face of fire\n\nHe kills the wife on the hearth\n\nHe kills the little people who flee outside\n\nEven with water present in the house,\n\nhe washes himself with blood.\n\nSudden as lightning, he terrifies the lazy.\n\n[from the town of Ilesha]\n\nOg\u00fan promenades, serpent poised about his neck\n\nOg\u00fan, King of Ire, lord, great sovereign of iron.\n\nWith stripes about his body,\n\nSuch as one sees only on the skin of the wild doe\n\nUnless it be Akisale, born of the Gaboon viper\n\nUnless it be Akisale, born of the python.\n\n[from the town of Ketu]\n\nOg\u00fan, allied to the man with a quick hand\n\nOg\u00fan, owner of high fringes of palm fronds\n\nOg\u00fan ties on his cutlass with a belt of cotton\n\nOg\u00fan of the sharp black cutlass\n\nHoe is the child of Og\u00fan\n\nAxe is the child of Og\u00fan\n\nGun is the child of Og\u00fan\n\nOg\u00fan, salute of iron on stone\n\nThe blacksmith of all heaven.\n\nOg\u00fan therefore lives in the piercing or slashing action of all iron. Lord of the cutting edge, he is present even in the speeding bullet or railway locomotive.\n\nHe is honored by liturgical jewelry, iron or brass. For example, in the region of Ilesha, whence came many captives to western Cuba and northeastern Brazil, important priests, Aworo and Owari, wear each year during the Obanifun festival a splendid brass pendant of miniature metal emblems, symbolizing iron's self-multiplying powers and potentiality (Plate 33). This fine display of metal emblems is the pendant of Og\u00fan _(amula ogun)_. Our example comes from the old Ilesha forge of Oginnin Ajirotitu.\n\nAccording to the recollection of the late blacksmith's surviving son, Oginnin completed this pendant for Og\u00fan at some point between 1900 and 1925. Richly patinated, the signs of the _\u00e0she_ of Og\u00fan are here specially made in brass, the customary medium used by Oginnin.\n\nPLATE 33\n\nThe _amula_ bristles with twenty miniaturized Og\u00fan emblems: three curved swords of the kind special to the Og\u00fan cult; an iron hairpin called _ikoti;_ an implement given as _oyiya_ (obscured in the photograph); a flaring instrument used for tapping palm wine; a smaller version of the same implement; a strip of chain symbolizing Og\u00fan's uniting force; the curved clapper of an iron bell; a miniature _agogo_ bell for Osanyin; pincers for Osanyin and for Og\u00fan; a snake \"fighting for Og\u00fan\"; the penis of Og\u00fan, partially obscuring a further curved clapper or, alternatively, a curved stick for drumming; a needle; two iron arrows; a large ceremonial iron bell; a knife for the patron deity, Owari; a sword for Owari. The emblems, counting the heavy chain from which they hang, make up twenty-one separate pieces, a multiple of the characteristic number of Og\u00fan, seven.\n\nMajor and minor versions of the same basic implements sometimes appear together here, and a constellation of further objects united by a kind of visual pun\u2014i.e., penis-stick-needle\u2014further deepens our appreciation of the infinity of Og\u00fan's power, for implements in iron are to his cult as pieces of lateritic stone are to Elegba, or palm nuts to If\u00e1.\n\nThe imaginative force of this visual tradition swept across the Yoruba New World. For example, evidence of Yoruba emblematic iron appeared in Cuba no later than 1868: \"in Guanabacoa an iron bar was found, one third sunk into the ground, with a crescent-form ornament at the top, from the sides of which hung four hooks with blood-sprinkled trinkets in the form of shovels, hoes, knives, and hammers\".\n\nPLATE 34\n\nPLATE 35\n\nOg\u00fan art in Cuba today includes the bucket-shaped iron cauldron (caldero de og\u00fan). Such objects (Plate 34) are full of various expressions of ironwork, such as nails, iron bows and arrows, horseshoes, and fetters, thus fusing token pieces of his medium within the programmatic arrangements of the _amula_ with an iron cooking vessel, as if to prepare a mighty broth of iron. Note that the illustrated cauldron for Og\u00fan is tightly wrapped with chains of iron, echoing a major element of the _amula_. The Cuban migration to North America has resulted in the establishment of the _caldero de og\u00fan_ tradition in Miami and New York, with fanciful additions, such as a shrine in the New York area in 1979 that has a _caldero de og\u00fan_ with an actual pistol (Plate 35).\n\nOg\u00fan is one of the most popular of the _orisha_ among Brazilian blacks. Again, as in Nigeria and the Benin Republic, Brazilians honor Og\u00fan by placing upon his altar separate pieces of iron as well as special miniaturized examples of his implements, called in Bahia _ferramentas de og\u00fan_ (Og\u00fan irons), which, like giant beads or pendants, are often pierced and threaded through a length of wire or piece of chain or thread. Recife is a rich source of such objects, but Salvador is perhaps the Brazilian city most famous for Yoruba symbolic iron. There, in the summer of 1968, I found Jos\u00e9 Adario dos Santos making _ferramentas de Og\u00fan_ at his forge on the Ladeira da Concei\u00e7\u00e3o above the harbor.\n\nPLATE 36\n\nEarlier, in the spring of 1965, Richard and Emerante Morse collected one of his works in the Mercado Modelo (Plate 36), illustrating how Dos Santos follows firm Yoruba canons of expression while at the same time designating modern tools of iron as extensions of Og\u00fan's realm. Here Dos Santos suspends seven iron implements\u2014knife, sword, hoe, spear, knifelike object, shovel, pick\u2014from the iron bow of the hunter-god, Oshoosi, Og\u00fan's mythic brother and companion. The number of the pendants is significant, for in the _candombl\u00e9s_ of Bahia the deity is praised as \"Seven Og\u00fan\" (Og\u00fan Meje), which alludes to the multiplication of his \u00e1she in tools of iron, and this reflects the influence of Nigerian Yoruba praise poetry and lore, as in an Oyo ballad of the hunters where the deity of war and smithing is saluted with the phrase \"seven iron signs of the god of iron.\" But there are also emblems with twenty-one miniature pieces of iron in Bahia, as at Ilesha and in Dahomey.\n\nOg\u00fan's absorption of industrially made iron implements in his arsenal of ritual attributes is paralleled by the advance of the same elements into the realm of Afro-Brazilian myths:\n\n[Og\u00fan]... wished to find his brother [Oshoosi]. He entered his father's forge and there made seven instruments\u2014pick, pick-axe, axe, scythe, spear, cutlass, and shovel. He carried them over his shoulder and entered the forest.... [By means of these implements] he forced his way into the forest until he found his brother... put him on his shoulders, and returned home.\n\nThus, across the Atlantic, iron instruments are all, in the end, the children of Og\u00fan, carried on his broad and mighty shoulders. He directs their energies to benefit those who earn his love through ties of kinship and those who make sacrifices and festivals in his name. The icons of Og\u00fan are not, however, for the lazy or the irreverent. Og\u00fan marches only with the spiritually vital and the quick of hand.\n\n### Oshoosi\n\nThe brother of Og\u00fan, Oshoosi, himself quick and strong, ultimately emerged as the deity of the hunters, the fabled archer of the gods.\n\nThe power of this deity is manifest in the speed and accuracy of his arrow, in prideful assertion of mind and muscle that have been wonderfully honed by the disciplines of forest hunting:\n\nHe is all alone and very handsome\n\nHandsome even in quality of voice.\n\nVital, he arises in the morning,\n\nBow and arrow already about his neck.\n\nSmall or hugely-built, the hunter is stronger than most men\n\nOshoosi quickly unleashes his arrow.\n\nWe see him only to embrace a shadow.\n\nWhen this man of determined muscle becomes a shadow, all that remains is his hunter's fly whisk ( _irukere_ ), his arrow ( _ofa_ ), and his bow ( _orun_ ). His arrow and his bow have been specially rendered in honorific iron or brass since time immemorial. These metal representations of a single arrow heraldically crossing a single bow at its center (Plate 37) form, in Nigerian Yorubaland, part of the traditional Oshoosi sacrificial altar ( _ojubo oshoosi_ ). In the winter of 1964 I visited such a shrine, under the famous Olumo Rock of the Egba and Egbado metropolis of Abeokuta. An early European notice of a shrine for Oshoosi is dated 1857: \"a huge iron bow, heavy with the weight of hundreds of strings of cowries hanging from it, and from small iron cylinders in which are miniature arrows.\"\n\nThe relationship between the cult of Oshoosi and the actual practice of archery in traditional Yorubaland can only be surmised, given present evidence, but it is worth mentioning that in the nineteenth century the English explorer Lander claimed that the Oyo Yoruba \"have the reputation of being the best bowmen in Africa,\" and that Yoruba arrows carried iron heads. This provides another, practical reason for the close association between Oshoosi and the lord of iron.\n\nIn Bahia, Oshoosi represents the pantheon of the hunters of the Yoruba. He is one of the _orisha_ most worshipped by persons of Ketu Yoruba descent. Therefore, the search for antecedents to aspects of Bahian worship of Oshoosi logically begins in Ketu in the People's Republic of Benin. There Deoscoredes dos Santos found that the fly whisk for Oshoosi and its associated lore were similar to Bahian expressions. More important, he found in Ketu a strange, bramblelike shrine, the _ojubo oshoosi_ , for sacrifice to Oshoosi that is similar to certain altars in Bahia for the same deity. If the hunter's fly whisk illustrates the medicine that Oshoosi carries with him into the forest, the bramblelike sacrificial shrine\u2014dry, leafless branches placed in a careful pile on the earth\u2014mysteriously brings the forest to the village.\n\nPLATE 37\n\nMetal bows and arrows for Oshoosi, fashioned in the Nigerian ideographic pattern, took root in Bahia. Among the blacks of Bahia the emblem was sometimes called _damata oshoosi_ , based on a praise-verse that describes the archer as a man who combines the strength of three hunters ( _ode meta_ ) within a single person. It is a praise-verse that came to western Cuba as well as to northeastern Brazil. The interconnections between Oshoosi and Og\u00fan fit the fact that the sign of the lord of the hunters is frequently shown together with the pendant iron implements of Og\u00fan both in Brazil and Cuba. Afro-Bahian versions are straightforwardly additive; they have a metal bow and arrow, the bow of which serves as a bar on which to suspend the seven or twenty-one pendant miniature implements of Og\u00fan (Plate 36). The arrow is the vertical accent and is soldered to a stand or is sharpened at both ends so that it can be driven into the earthen floor of a traditional shrine or altar.\n\nPLATE 38\n\nIn Cuba combined signs of Og\u00fan and Oshoosi have flourished since the nineteenth century, and most of these are simplified statements. But occasionally craftsmen have made elaborate fusions of art for Oshoosi and Og\u00fan, as in an apparently twentieth-century example now in the Institute of Ethnography in Havana (Plate 38). Here the sign of Oshoosi, like a flag of Nimrod, presides over the emblems of Og\u00fan Alagbede (Og\u00fan the Blacksmith). The metal sign of Oshoosi is set atop a kind of carousel of rotating avatars of iron\u2014lengths of chain, a horseshoe, a pick, an ax, a hammer, an arrow, a sledgehammer, a hook, a scythe, and many other objects. Particularly affective is the garlanding of the whole with ornamental pieces of chain, the ancient Yoruba ideogram for the unifying power of the deities. Since World War II similar objects, usually reduced in scale and degree of elaboration, have appeared in the Hispanic barrios of some cities in New England, New Jersey, and New York.\n\nWherever Oshoosi is honored, the canon of his icon has remained intact. He seems to have lifted, invisibly, his bow and arrow, and taken aim, as if to protect the panoply of Og\u00fa's iron, as if to return the debt established when Og\u00fan marched with these implements to find his brother in the forest.\n\n### Obaluaiye\n\nObaluaiye's power to heal or conjure smallpox (or other dread disease) makes his cult feared and respected in Yorubaland. He is an earth deity who strikes down the arrogant and the immoral alike with \"spears\" of pestilence and fever. Informants have spoken of the danger of walking alone when the sun is hot at noon, when Obaluaiye and his followers, all dressed in scarlet, are believed to haunt the earth, and of the danger of wearing red or loud patterns for fear the deity, enraged at the appropriation of his prerogatives, might harm them.\n\nBritish colonial authorities banned the cult in Nigeria in 1917, when Obaluaiye priests were accused of deliberately spreading smallpox. But members of the cult, confident of their insights and their moral worth, refused to be intimidated. They took their worship underground. They worshipped Obaluaiye under different names, e.g., the Lord (Oluwa). The strength of his lore in modern Nigeria is illustrated by the continuity of the old belief that it was dangerous to call him by his name, for one would thereby spread his dread disease, _shoponnon_ (smallpox). Today even some English-speaking Yoruba use the circumlocution \"S.P.\" when alluding to smallpox.\n\nThe terror can be harnessed, however, to shock the thoughtless into social awareness and concern:\n\nIf you are rich, you do not laugh at the poor\u2014\n\nLittle people can become grand\n\nWhen you come into the world\n\nYou own neither a wife, nor a car, nor a bike\n\nNothing you have brought and\n\nNo one knows the future.\n\nThese verses suggest that epidemics\u2014the fiery, annihilating hand of Obaluaiye\u2014can bring about social conscience because they could topple the rich and the powerful. Once aroused by arrogant behavior, the spirit of Obaluaiye is dangerously difficult to appease: \"The glowing embers are difficult to stamp out.\" Obaluaiye is thus a complex god, who inspires a complex set of verses in which the themes of terror and moral retribution are interwoven in a thousand different ways. The following conveys some sense of the peculiar beauty of the poetry chanted in his name:\n\nWild animal, with whom we have been entrusted\n\nYour bird cannot strike my bird.\n\nKing of precious beads, death who flees at dawn.\n\nLeaf poised upon the surface of the water.\n\nBlack hunter, physique covered by a gown of raffia\n\nFalls mightily, blocks the road, as a thorn.\n\nThorn penetrates, man enters town with limp.\n\nNo one should walk alone at noon.\n\nThe images caught within these verses are almost filmic, so swiftly realized are the shifting nightmare changes. Obaluaiye emerges, dressed in broomlike strands of raffia straw, then is suddenly transformed into a mighty force\u2014falling like a tree that blocks the road, and then becomes a thorn.\n\nThe thorn is a metaphor for the pestilential needle of his morally aroused vengeance. Hence a hapless person\u2014foolish enough to ignore warnings not to walk alone at noon when the sun is at its strongest and the sands, domain of the deity, literally heat up between his toes\u2014is pricked by the thorn. We recoil in horror, knowing that with his limp he brings doom to the village.\n\nThorn imagery suggests the specially planted species of cactus dedicated to Obaluaiye and found on public shrines in the Ketu Yoruba area of eastern Benin. It also leads to two of the mystic weapons of his arsenal of disease: an arrowlike object ( _esin_ ) wrapped in scarlet cloth and kept on his shrine; and a lance ( _oko_ ), sometimes plain, sometimes decorated with cowrie shells and special tassels. Both are extensions of the idea of something that pricks the flesh and inflames with lethal pain. Obaluaiye's full complement of attributes also includes the club, one of the most ancient and rudimentary of the weapons of the Yoruba\u2014in this case, the rather generously sized specimen known as the _kumo_ , covered with rust-colored camwood paste.\n\nThere is a line of poetry praising the deity as a man covered with raffia fiber, as if he were a walking broom\u2014a Yoruba traditional whisk broom with a short handle and long fiber. As a matter of fact, extensive broom imagery characterizes the cult of Obaluaiye.\n\nThe Yoruba whisk broom, sacralized by the addition of medicines and camwood paste sprinkled on the straw, is one of the more formidable and famous of Obaluaiye's emblems. If\u00e1 tells us that when he is enraged, Obaluaiye takes this special broom and spreads sesame seeds ( _yamoti_ ) on the earth before him, then sweeps the seeds before him, in ever-widening circles. As the broom begins to touch the dust and the dust begins to rise, the seeds, like miniature pockmarks, ride the wind with their annihilating powers: the force of a smallpox epidemic is thereby unleashed. According to Dos Santos, the cult name for this horrific broom, _shashara_ , is said to represent a fusion of two roots in Yoruba, \"pitting with smallpox\" ( _shasha_ ) and \"human body\" ( _ara_ ).\n\nThe worship of the deity is widely known in the Benin Republic under the Fon term _Sakpata_. Some of the more beautiful of the verses in his honor have been elaborated at his festival in Abomey, Benin. The Dahomean elaboration of his cult traveled, with the Yoruba version, to Cuba and to Brazil, and there they reinforced themselves. For example, the deity is known by a creole Yoruba term in Cuba, \"Babal\u00fa Ay\u00e9,\" the title of a famous Afro-Cuban caf\u00e9 song of the 1940's, but his broom is called by the Dahomean term _ha_ (spelled j\u00e1 in Spanish), and is richly beaded and decked with cowries (Plate 39), as in Dahomey, here illustrated with royal beaded Yoruba-Cuban fly whisks for the gods.\n\nDahomean style is also reflected in the shaping of the smallpox broom in Brazil. The original _ha_ of the region of Abomey has a special medicated handle covered with crimson cloth decked with rich, ornamental bands of cowrie-shell embroidery. Emerging from the handle are the long, bound center fibers of the leaf of the _palma vinifera_. There is a hint of the \"bush\" in the manner in which the fibers of the _ha_ explode this way and that in a twisting of strands at the working end of the instrument, whereas the fibers at the base are tightly bound together. The smallpox priest who carries this mystic broom is dressed in special garments, including a bonnet and armbands, all embellished with cowrie-shell embroidery.\n\nPLATE 39\n\nThe latter elements of Dahomean ritual dress, brilliantly miniaturized and recombined in the making of the creole smallpox brooms of Bahia, give rise to one of the most beautiful objects of the black Atlantic world (Plate 40). The _ja_ of Dahomey in the process of transformation into the _shashara_ of Bahia acquires the noble stillness of a column, so neatly cut and tightly bound are the fibers, which are enclosed by bands of ornamental leather often sited near the bottom and at the middle of the broom fibers as well as at the handle. These bands are richly appliqu\u00e9d, in the Dahomean manner, with cowrie shells. They dress the object with the logic of a Dahomean priest's attire. In the end a whisk broom, glittering with crimson, white, black, and other colors, becomes a kind of abstract jewelry, made of shells and straw and leather.\n\nThese changes parallel an apparent shift in function. When Obaluaiye appears, dressed in his appropriate raiment, in the dances of the temples of Bahian blacks, he rarely carries, so it is reported, his club or arrow. Instead, he bears a lance or broom with motions underscoring a gentler mood. The broom is \"danced\" with gestures that are smooth and orchestrated, as if to suggest a sweeping away of terror, light-years removed from the scattering of sesame seeds amid dark clouds of dust. In short, the Afro-Bahian _shashara_ is more a royal scepter than an object of use\u2014hence the almost speculative manner in which its ancient functions are recalled.\n\nPLATE 40\n\nPLATE 41\n\nBut there is nothing muted about the intimation of African forest energies and fierceness projected in the impressive all-raffia crown ( _ade iko_ ) (Plate 41) and all-raffia gown ( _ewu iko_ ) that are prepared by specialists in Bahia for possession-devotees to wear when the spirit of Obaluaiye fills them. This is a form of dress that directly recalls not only the Nigerian verse about \"black hunter, physique covered by a gown of raffia\" but also the conical shape and veil of both Dahomean smallpox deity costuming and one of the main forms of Yoruba royal headgear (Plates 42 and 43).\n\nPLATE 42\n\nThe roots of the Afro-Bahian raffia gown are therefore various. North of the Fon in the Benin Republic there is a famous shrine to the mother of Obaluaiye, Nana Buk\u00fau, on a platform by a baobab tree, overlooking the Yoruba settlement of Dassa-Zoum\u00e9 (Plate 44). Here there is a standing cylinder of clay over which has been fitted a raffia garment, communicating the presence of a spirit and bringing to mind the image of a whisk broom given animate solidity and form. Obaluaiye is an earth deity and here the body of his mother, as it were, like Elegba, acquires a flesh of laterite, a flesh covered over with raw fiber, the dress of spirits across West Africa. The summit of this strange image is surmounted by a raffia crown with a stemmed finial, also in raffia\u2014details matched by the style of the Afro-Bahian crown for Obaluaiye (cf. Plate 44 with Plate 41).\n\nPLATE 43\n\nAccording to the Lagosian cult of Ejiwa, the earth deity gave Eshu, when the latter was scarred with smallpox marks, a garment made of raffia, of a thickness sufficient to keep flies from swarming about his wounds. Hence, If\u00e1 tells us, the Ejiwa (Eshu) masquerader to this day appears completely shrouded in raffia. Lagos, of course, was intimately linked to Bahia in the Atlantic trade, as were territories to the north of the Fon. The primary image of the Ejiwa cult is a most suggestive clue in the search for the origins of the idea of concealment by raffia of the signs of smallpox. In fact, Afro-Bahian informants explicitly claim that the raffia veil of Obaluaiye hides a face ravaged by the disease.\n\n### Nana Buk\u00fau\n\nNana Buk\u00fau is the mother of Obaluaiye, and such is her importance in Dahomey that there she has come to be considered the grand ancestress of all the Yoruba-derived (Anagonu) deities of the pantheon of the Fon. The name of Nana is famous from Ife in Nigeria to Siade and Tchari in Ghana. It is believed that none other than the kings of Asante and Dagomba sent gifts to her shrines in time of war. These kings sought her blessing because of her fabled powers to bring a city victory, to render in ruins the cities of one's enemies. She herself was a superlative warrior, utterly fearless, who razed the mythic city of Teju-ade.\n\nNana Buk\u00fau is the courage and accomplishment of women, sublimed to the form of an _orisha_. She knows terrifying secrets. She shares with her son, Obaluaiye, the smallpox god, master symbols: raffia-made regalia. Nana and her son are both represented by special staffs made of palm-frond fibers tied together, which collectively symbolize the ancestors and cryptically communicate awesome levels of initiation.\n\nPLATE 44\n\nHer primary icon is the _ileeshin_ (a Ketu term). It is essentially from Ketu, this ancient settlement, its cultural provinces, and neighboring territory that the cult of Nana was brought to Bahia, where she is spectacularly honored with the _ileeshin_. Made from the same materials as the broom of Obaluaiye, the _ileeshin_ is, however, an entirely original instrument with its own logic of form and function. It is not loosely structured at its summit like the _ha_ of the Fon. Instead, its strawlike elements almost disappear beneath rich coatings of camwood paste and leather-covered sections often dyed the darkest blue (Plate 45). The summit of the staff is striking\u2014it curls back upon itself, like the tail of a leopard, leaving a strange oval space outlined at the top. This noose-shaped tip is believed to embody Nana's _\u00e0she_ , Nana's intimidating powers:\n\nPLATE 45\n\nThe _ileeshin_ cannot touch a man or boy. Its tip is dangerous. And if a woman holds it, all during the time that it rests in her hand she must not speak any evil of an animal or a man. But, if a cruel and horrible person stands before her, she can take the _ileeshin gogo_ , thrust it out horizontally before her and strike its looped tip against the belly of the man.\n\nInstantly the man would fall back, arms outspread in a gesture of pain and shock, stomach instantly bloated. Then he would die. Attacking an evil person in the region of the belly, causing it to swell even as the person dies, suggests that Nana presides over the giving\u2014pregnancy\u2014and the taking away of life itself. She is equally capable of bringing children into the world or causing doom, depending on a person's goodness or lack of character.\n\nThe nooselike tip of Nana's staff is glossed by a legend recorded by Dos Santos:\n\nNana has possessed a certain staff from the beginning of her life on earth. The name of the staff was _\u00ccb\u00edr\u00ecr\u00ed_. She was born with this staff; it was not given to her by anyone.... when she was born the staff was embedded in the placenta. Ornamental cowries decked this curled staff, and fine ornaments. After it was born it curled into a noose ( _\u00f3 s\u00ec k\u00e1 k\u00f3r\u00f3b\u00f3j\u00f3_ ). Then they cut it from the placenta and they put it inside the earth. But surprisingly, as the infant grew, the staff grew, too. It was the very staff that Nana used when she went to war against the Teju-ade and it was her son who dug it out of the earth. That is why they call it _\u00ccb\u00edr\u00ecr\u00ed_ , my son-found-it-and brought-back-to-me. She used this staff as a medicine of victory. Many modes of mystic potentiality open up for the person who holds this staff. If the people of a town know to use this staff they will be able to prevent war....\n\nPLATE 46\n\nThe Yoruba staff was creatively transformed in northeastern Brazil, where the _ileeshin_ is called _ibiri_. Appropriately, this emblem of the mother of the smallpox deity is fashioned in Bahia by the Asogba, the supreme head of the devotees of Obaluaiye. The Asogba consecrates, makes, and entrusts to the priestesses mounted by Nana the _ibiri_ of Bahia, and conceals powerful medicines in the base of the object. Our example (Plate 46), photographed in the Museu de Arte Popular in Bahia in 1968, captures the degree of artistic independence of the Brazilian form: there is at once a diminishing of the Ketu tradition of heavily coating the fibers of the object with red camwood and a heightening of the Ketu manner of encircling the straw with strips of leather. A war medicine becomes a work of art.\n\n### Yemoja (Yemay\u00e1)\n\nYoruba riverain goddesses are represented by round fans ( _abebe_ ), crowns ( _ade_ )\u2014some with beaded fringes\u2014and earthenware vessels ( _awota_ ) filled with water collected from the river or from the sea, rounded stones, and sand. Those who worship these powerful underwater women long ago devised an artistic strategy, the use of the round fan as an emblem embodying the coolness and command of these spirits of the water.\n\nMyths explain the meaning of the roundness of the fans. According to the verses of If\u00e1, the round fan was a fashion of the goddesses of ancient Ife. And it came to pass that If\u00e1, god of divination, after a quarrel with the river goddesses, departed from the holy city, whereupon a terrifying famine struck the world. If\u00e1 hid in the forest, in a round house made of leaves and saplings (oddly recalling Pygmy structures of Central Africa). As the crisis worsened people consulted diviners. Speaking through the sacred divination instruments, If\u00e1 said that only the river goddesses themselves, including Yemoja, noted for her use of a round fan, could persuade him to return. Each female spirit took a fan, like the round one of Yemoja, and together they all went into the forest seeking the spirit of If\u00e1, throwing stones at anything solid that they saw. When they threw stones at the round house of If\u00e1, he shouted, \"Who is throwing stones at my house!\" _(Tani s\u00f3 k\u00f2 l\u00f2 run!)_. They replied, \"We are, and we wish to see you.\" When they saw him standing before his round house, they immediately started to fan him, then begged him as they fanned him, gave him a special feast and fanned him again. And If\u00e1 was persuaded to return and the famine ended.\n\nThis is a retelling of a myth with a thousand voices, the myth of conflict that threatens to destroy the world and of the discovery of antidotes in mystic coolness. In another version two round droplets of water remaining in the mouth of a single fish reconstitute the peace that brought back order into the world. Here, in the legend of Yemoja and If\u00e1, a special rounded fan\u2014like a giant soothing drop of water\u2014restores peace and calm, associating itself with the image of If\u00e1's cool round house made of herbs and leaves. An indelible current of association links the roundness of habitations to the roundness of things pertaining to riverain goddesses.\n\nBut the coolness of the riverain goddesses is problematic. Vengeance, doom, and danger also lurk within the holy depths ( _ibu_ ) of the rivers where the goddesses are believed to dwell. In the nineteenth-century Yoruba civil wars, warriors from Ibadan and Ijaiye, after killing the king of Owu, were terrified by what the aroused spirit of the murdered king might do. So they temporarily dammed the river Oshun, buried the king's body in the river basin, and then released the waters. The logic: to bury the king deep beneath the rippling surfaces of the river would muffle the roar of his spirit and neutralize and appease his awesome powers of _\u00e0she_. Similarly, the cool, dark depths of the river may shield mankind from the full blast of the fiery powers of witchcraft harnessed by the river goddesses.\n\nWhat is more, a fan can be dangerous if brandished by a person steeped in moral and medicinal lore. For Yemoja and all the other riverain goddesses, especially Oshun and Oya, who presides over the river Niger, are famed for their \"witchcraft.\" They are supreme in the arts of mystic retribution and protection against all evil.\n\nMany riverain goddesses are visualized as women with swords. The sword, together with the negative uses of the fan, may be said to form in part an image of what Judith Hoch-Smith calls \"radical Yoruba female sexuality\":\n\nWithout the concept of witchcraft, power would have flowed naturally through society, lodging only in socially structured positions, most of which were held by men in the traditional Yoruba patrilineage. However, the concept of witchcraft permitted great quantities of power to become lodged in women, who in turn were thought to use that power against the institutions of society. In this sense, witchcraft symbolizes the eternal struggle of the sexes in Yoruba society over control of the life-force.\n\nWitchcraft, in fact, militates against not only total male dominance but the threat of class formation and drastically unequal distribution of wealth. At the core of the all-powerful council of male elders, the Ogboni Society, lies the awesome image of their deity, Earth, all-devouring, all-seeing. It is believed that man as such and woman as such asked themselves, \"Who is supreme?\" And out of the earth came the white-hot message: I AM SUPREME  And Earth was, in a manner beyond sex or class or any consideration contaminated by singleness of expectation. Just as we arrive among the western Yoruba in the presence of God to discover that \"he\" is not he, nor she, but a pure force\u2014 _\u00e0she_ of _\u00e0she_ \u2014so the impartial laserlike precision of the punishing vision of Earth, who was here before the goddesses and before the gods, implies a power beyond \"her\" sex.\n\nImperially presiding in the palaces beneath the sands at the bottom of the river, the riverain goddesses are peculiarly close to Earth. In the positive breeze of their fans, the ripple of their water, there is coolness. In the darkness of their depths and in the flashing of their swords, there is witchcraft. And within the shell-strewn floor of their underwater province there is bounteous wealth. Yoruba riverain goddesses are therefore not only the arbiters of the happiness of their people but militant witches. Their negative power is alluded to in oral literature:\n\nPLATE 47\n\nYemoja, the wind that whirls with force into the land.\n\nYemoja, angered water that smashes down the metal bridge.\n\nBut this is power used against the human arrogance of Western technocratic structures. The natural goodness within the evil of these water spirits is invoked by righteous devotion. Thus, in Cuba, when a divination priest warned women that a certain river goddess had become dangerously angry, senior priestesses, like Yemoja and her circle before If\u00e1, would face the altar of the offended spirit and fan her image with strong concerted motions.\n\nLike an Ipokia shrine of the river goddess Yewa and the forest dwelling of Orunmila, and like the ancient fans of Ile-Ife, the Afro-Cuban ritual fans are round. They are decorated with a richness of material\u2014beads, cowrie shells, jingle bells, even peacock feathers\u2014surpassing that of the originals. For example, a Ketu Yoruba round fan for the goddess Are, wife of the founding deity Ondo, tutelary spirit for many groups of the western Yoruba, is luxuriously embroidered with cowrie shells (Plate 47). The structure and allusiveness of the round, shell-embroidered fan were complicated in Cuba by metallic or feathered embellishments. Thus a fan for the goddess of the sea, Yemay\u00e1, made in Cuba before 1952 but probably in this century, displays a brilliant surface entirely covered with beads in geometric sections (Plate 48). The design includes a lunette on the right, a lunette on the left, both separated by an ascending column of beaded surface that continues the axis of the handle to the very tip of this pointed fan. Seven cowrie shells in a smooth curve decorate the bottom portion of the fan. Nine peacock feathers, with seven jingle bells, add both pleasing sound and swaying movement. (Nine is one of the numbers of Yemay\u00e1 in Africa, as in the Abeokuta praise verse, \"open river, divided into nine parts.\") The feathers are seductively beautiful, but they all carry a charge associated with some birds, namely, witchcraft. The shells suggest the goddesses' money\u2014all underwater creatures, who are close to the sources of the traditional cowrie currency and to the corals from which, in the ancient imagination, much wealth derived, are believed to be quite rich. The bells provide a gentle watery tinkle recalling a poem praising the goddess of the river Oshun that mentions copper bracelets chiming against each other underwater.\n\nPLATE 48\n\nThe religious history of the blacks of Rio is a palimpsest marked by Kongo, Yoruba, and Roman Catholic infusions, as will be elaborated on in the following chapter. Suffice it to say here that in the fusion of these elements\u2014in the rise of the macumba religious groups of Rio\u2014the arrow or the dagger and the heart of the Most Holy Virgin have combined with the imagery of Yemay\u00e1 in the belief that both the Virgin and the goddess of the sea share qualities of sacred love, faith, and purity. In the creole fusion-image of Yemay\u00e1 and the Virgin Mary, \"sung points\" strike through in the macumba shrines of Rio to an oceanic presence behind a Roman Catholic mask:\n\nO Virgin Mary\n\nLike a flower superb\n\nCelestially harmonious...\n\nReigning monarch of the seas.\n\nIn a superb white metal fan for Yemay\u00e1, probably made in the present century (but no later than August 1939, at which point it was accessioned by the National Historical Museum in Rio), the heart of Mary is returned to the sea in a petaled flower image (Plate 49) of Yemay\u00e1, with reflections of the latter's underwater wealth and celestial love suggested by pendants of miniature hearts and nineteenth-century Brazilian coinage. There is the crucial detail of the touching of the bottom of the heart by a sharpened, arrowheadlike point at the top of the handle of the fan, as in the \"drawn point\" that accompanies Rio sacred songs for Yemay\u00e1\/the Virgin Mary. This is, of course, the seal of Nossa Senhora das D\u00f4res, blending with the Yoruba tradition of honoring powerful underwater denizens with honorific fans. Finally, the scalloped edge circumscribes the entire composition with a sign of water.\n\nThe petals surrounding the inner heart suggest a floral image within water, which is implied by the wavy edge of the fan. Each New Year's Eve thousands upon thousands of black and white worshippers descend from the hills above and around the city of Rio and crowd the famous beaches of Copacabana and Ipanema in one of the most celebrated of the festivals for Yemay\u00e1 in the western hemisphere. And one of the acts rendered, then, in her honor, involves the tossing of sacrificial flowers into the breakers of the sea. Pierre Verger has photographed flowers in a container to be sunk beneath the waves in honor of Yemay\u00e1 in Bahia. The image of flowers within the depths is another recollection of tradition in Yoruba New World worship. It recalls the ending of the festival of the riverain goddess, Oshun, with the throwing of flowers into her stream at Oshogbo, in Nigeria.\n\nPLATE 49\n\n### Oshun\n\nDivination literature tells us that Oshun was once married to If\u00e1 but fell into a more passionate involvement with the fiery thunder god, who carried her into his vast brass palace, where she ruled with him; she bore him twins and accumulated, as mothers of twins in Yorubaland are wont to do, money and splendid things galore. She owned a wealth of bracelets, staffs, needles, earrings\u2014all in brass, the metal the Yoruba regard as most precious. When she died, she took these things to the bottom of the river. There she reigns in glory, within the sacred depths, fully aware that so much treasure means that she must counter inevitable waves of jealous witchcraft by constant giving, constant acts of intricate generosity. Even so, she is sometimes seen, crowned, in images of warlock capacity and power, brandishing a lethal sword, ready to burn and destroy immoral persons who incur her wrath, qualities vividly contrasting with her sweetness, love, and calm:\n\nShe greets the most important word within the water\n\nShe is the orisha you see healing by means of water that is cool\n\nIworo bird with brilliant plume upon her head\n\nTitled woman who heals the children...\n\nWitness of a person's ecstasy renewed\n\nShe says: bad head\u2014become good!\n\nMistress of \u00e0she, of full predictive power,\n\nShe greets the most important matter in the water.\n\nStrong woman who burns a person.\n\nShe cures without fee; gives honey to children\n\nHas lots of money, speaks sweetly to the multitude.\n\nNowhere is the kingly power of Oshun not renowned\n\nOshun has a mortar made of brass\n\nSweet is the touch of an infant's hand\n\nKare, King of the holy depths\n\nShe dances, she takes the crown...\n\nThe chiming bracelets of her dance.\n\nShe smites the belly of the liar with her bell.\n\nMother, O Mother of cool water,\n\nYou, who sired the soothing osun herb.\n\nEcstatic praise literature. It is easy in the context of these verses and Oshun's reputation for great beauty to appreciate why she was romantically transmuted into the \"love goddess\" of many Yoruba-influenced blacks in the western hemisphere. But there are dark aspects to her love and the singers of her praises in Nigeria do not hesitate to mention them: her masculine prowess in war; her skill in the art of mixing deadly potions, of using knives as she flies through the night:\n\nWoman wearing manly crown, oh so rare\n\nOwner of a piercing knife, I take my haven by your side\n\nYou own the inner court, where witch-owl lays her eggs.\n\nYou kill this owl, and make of her a strange cuisine...\n\nArriving, trouble vanishes in coolness.\n\nYeye Kare...\n\nYou hollow out sands beneath the sea,\n\nYou're putting money in there\n\nThe water sounds, _wanran-wanran-wanran_ , like the bracelets of Oshun.\n\nBut Oshun's darker side is ultimately protective of her people. This accounts for the famous material expression of her positive witchcraft among Cuban and New York Hispanic blacks, the _oshun kole_ ornament that is hung from the ceiling of the house of its owner: \"Oshun Kole lives in a calabash adorned with five turkey buzzard feathers, a calabash which is suspended from the ceiling.\" The spray of \"witch feathers\" renders this suspended vessel into a kind of elevated devouring force. Like certain charms suspended from the rafters of reception porches used by some northern Yoruba chiefs and kings, the _oshun kole_ is believed to protect the habitation from all evil. Our example, dated 1968 (Plate 50), was made in Hispanic New York and indicates a change in continuity from Nigeria via Cuba: the buzzard feathers have been replaced by store-bought plumes; the calabash is missing; and a single strand of brilliant copper-colored _oshun_ beads with cowrie shell has been added.\n\nElderly informants of Lydia Cabrera remembered the use, at the turn of the century, of crowns with beaded fringes for Oshun in Havana. The praise-verse \"Crowned Woman, O so rare\" is brought to life by a proliferation of crowns for priests and priestesses of Oshun and riverain spirits in Brazil, many with veils that fall across the wearer's face in the ancient Yoruba manner.\n\nPLATE 50\n\nOshun's positive witchcraft gave rise to feathered charms in Cuba and New York. Her sovereign powers are celebrated by the crowns for Oshun in Brazil. But perhaps the most significant of her New World emblems are her round metallic fans. The idea of Oshun among the Ijesha Yoruba of Nigeria is brilliantly realized in circular fans of polished brass with chased and hammered figuration. One of the finest examples of the genre comes from Ilesha, the capital of the Ijesha; it was made in sheet brass around the turn of the century by the master brassmith Oginnin Ajirotitu Arode Onishona (Plate 51).\n\nThe symbols on the Oginnin fan are remarkable for their richness of allusion. First there is a link between the riverain goddess and the ultimate powers of Earth made explicit in praise literature. Here the closeness of the goddess of the water to the Supreme Being is manifested by the handle. From the bottom of the handle hang three chains, the mystic number of Earth. There are also two pairs of three small hearts chased within the surface of the handle. Altogether there are three statements of the power of Earth, who witnesses two-party vows and covenants.\n\nThe son of Oginnin explained the many meanings embedded in this fan as his father had explained them. The summit of the program of decorative detail is dominated by the figure of Oshun herself, shown with hands held high above her head, \"to greet a person with the sign of gladness, not by song.\" She unifies the world by holding a length of chain about her head, an action called I-tie-all-my-people-together ( _mo so awon enia mi po_ ). Fish, in which her spirit moves within the river, appear as motifs, left and lower right. Alluding to the feathered avatars in which she and her circle fly, there is a \"bird of the river\" ( _karo_ ) standing beneath the fish on the right. Serpents also appear; they are messengers of Og\u00f9n's iron\u2014\"fighting evil men.\"\n\nPLATE 51\n\nFor all the signs of witchcraft and the clash of iron, lessons of the cool and antidotes to evil and to envy are richly given, too: a giant rat ( _okete_ ), \"not for the gods but for the cooling of the witches\"; a tortoise ( _ajapa_ ), \"sacrifice to Osanyin, master of the medicines by which we live\"; a cock; a hen; a Muslim writing-tablet (\"carved with Og\u00f9n's iron\"); and the image of a young woman with a bridal hairdo who brings a calabash filled with kola \"for the greeting of the gods.\" These all connote giving and assuagement, acts that still the wind and calm the surface of the water. And all these images float timelessly about the central icon of the paired heads of Owari and Obanifun, spirits of war and creativity. The tip of the cartouche in which these male valences are contained touches the vagina of Oshun. The implication of the fan's program of symbols is precisely this: without constant rhythms of giving and responsibility, the fruitful union of Owari\/Obanifun and Oshun would be no more and the withering famine described in many myths would return.\n\nThe buzz of mythic voices was silenced, as it were, when the brass fans were brought to Bahia. The chasing of symbols upon the surface in the Ijesha manner was apparently lost. All that remains of the originating tradition is the characteristic shaping of the fan in the round manner of the ancients and the use of polished metal as the medium. But metal fans for the river goddesses in Bahia have a Creole richness of their own, as attested by a fan for Oshun made by Clodimir Menezes da Silva in August 1968 (Plate 52). They fuse basic Yoruba style (round outline, metal medium) with an element of Dahomeanizing design (the use of what might be termed \"metal appliqu\u00e9,\" i.e., the bolting together of various levels of sheet metal to form a composition, as in the great metal standing figures of Abomey). In addition, some artists actually sometimes seal sacred liquid within the hollow, syringelike handle of the fan. In other words, the handle loses its flattened textural quality in the manner of Ilesha and becomes a polished cylinder, a vial. Thus the artists of Bahia have extended the image of the fan of Oshun and the riverain goddesses with an object that mimes, as it were, the balancing of a single, multipetaled sacrificial flower on the rolling, scalloped, chiming waters of Oshun.\n\nPLATE 52\n\n### Sh\u00e0ng\u00f3\n\nThe tempestuous mythic third king of the Yoruba, Sh\u00e0ng\u00f3 is an Oyo deity. He is the thunder god, and his consort is the whirlwind, the goddess Oya, who is also the goddess of the Niger River. As an alleged medieval monarch, he has come to symbolize the powers of the Oyo kings. His royal cult plays an integral role in the installation of each king of the Oyo. In the palace of the Oyo kings there is a special priestess, Iya Naso, who is charged with palace worship of the thunder god, and her disciple Iya Aaafin Iku is responsible for Sh\u00e0ng\u00f3's sacred ram (the motion of whose horns are poetically compared to the thrust and parry of the lightning bolt). These palace customs reach a climax at the annual Beere festival when a masked priest, said to represent Sh\u00e0ng\u00f3's own ancestral spirit, Alakoro, perambulates the palace walls while gesticulating and, in his robe of blazing red and shining mask of polished brass, looking like a crimson ghost. Before each of the main gates to the palace he gestures to heaven and then to earth, to heaven and earth again, and moves on to the next point of blessing.\n\nOnce upon a time, as myth would have it, Sh\u00e0ng\u00f3 was recklessly experimenting with a leaf that had the power to bring down lightning from the skies and inadvertently caused the roof of the palace of Oyo to be set afire by lightning. In the blaze his wife and children were killed. Half crazed with grief and guilt, Sh\u00e0ng\u00f3 went to a spot outside his royal capital and hanged himself from the branches of an ayan tree. He thus suffered the consequences of playing arrogantly with God's fire, and became lightning itself. Like Eshu at the Cuban crossroads, in the lightning bolt Sh\u00e0ng\u00f3 met himself. He became an eternal moral presence, rumbling in the clouds, outraged by impure human acts, targeting the homes of adulterers, liars, and thieves for destruction. \"He dances in the courtyard of the impertinent,\" as composers of his praise-poetry envision his transformation of lightning into moral action. Praise singers among the western Yoruba realize a vision of his spirit in poetry charged with flashing images:\n\nWater by the side of fire at the center of the sky\n\nA strange thing, on the road to Teji Oku\n\nHe strikes a stone in the forest, stone bleeds blood\n\nHe carries a heavy stone upon his head without a cushion.\n\nSh\u00e0ng\u00f3 splits the wall with his falling thunderbolt.\n\nHe makes a detour in telegraphic wire\n\nLeopard of the flaming eyes\n\nLord who wears the sawtooth-bordered cloth of returning ancestors ( _egun_ )\n\nStorm on the edge of a knife.\n\nEarthworm, despite no eyes, plunges deep into the earth\n\nHe dances savagely in the courtyard of the impertinent\n\nHe sets the liar's roof on fire\n\nHe carries fire as burden on his head\n\nThe gaze of this leopard sets the roof on fire\n\nFather, grant us the intelligence to avoid saying stupid things.\n\nAgainst the unforeseen, let us do things together.\n\nSwift king, appearing like the evening moon.\n\nHis very gaze exalts a person.\n\nI have an assassin as a lover.\n\nBeads of wealth blaze upon his frame.\n\nWho opens wide his eyes\n\nLeopard of the flaming eyes\n\nFire, friend of hearth.\n\nLeopard, of the copper-flashing eyes\n\nFire, friend of hearth.\n\nLord with flashing, metallic, eyes,\n\nWith which he terrifies all thieves.\n\nThe power of Sh\u00e0ng\u00f3 streaks down in meteorites and thunderstones, stones both symbolic and real. The _\u00e0she_ of Sh\u00e0ng\u00f3 is found within a stone, the flaming stone that only he and his brave followers know how to balance unsupported on their heads. Flaming stones have become a metaphoric burden: the possession priest or priestess often balances a vessel containing burning oil upon his or her head. The earthworm, one of the first of God's messengers of _\u00e0she_ , plunges like lightning into the soil and, ventilating the earth, creates sustenance for plants. Likewise, Sh\u00e0ng\u00f3, as pestle, crushes the fritter that it may properly be prepared, just as Sh\u00e0ng\u00f3's favorite yams and maize must first be pounded. Only by being broken down do substances yield nourishment. Only through chastisement does the liar ideally change and grow.\n\nSh\u00e0ng\u00f3 sits upon a sacred inverted mortar ( _odo_ ) when he fills the body of his possession-devotee. _\u00c0she_ within his meteoric stones also flashes within his eyes, the eyes of the possessed, whose gaze is believed capable of igniting houses. Sh\u00e0ng\u00f3's _\u00e0she_ flashes dangerously in the amazingly wide-open eyes of thunder priests and in the gaze of the royal leopard, who will kill all felons and enemies of the state.\n\nThe metaphoric and moral range of this poetry is matched by that of the images of the altars of Sh\u00e0ng\u00f3. The thunder ax ( _oshe_ ) is one of his most pervasive attributes. The character of Sh\u00e0ng\u00f3's moral retribution and the control of this power by his followers are represented in one of the earliest _oshe_ brought to Europe, a fine specimen taken to England before 1853 by Lt. W. R. Bent of the Royal Navy (Plate 8). It is now in the Royal Albert Memorial Museum in Exeter. Twinned thunderstones appear above the head of a female follower of Sh\u00e0ng\u00f3. There they are balanced, like the meteorites that Sh\u00e0ng\u00f3 carries upon his head. Character and propitiating coolness ( _irele_ ) are exemplified in her poise; she kneels to show honor, holding her breasts as an offertory. She wears the characteristic red and white beads of the thunder god that refer to, among other images, a line of the praisepoem, \"water by the side of fire at the center of the sky.\" The two colors also suggest the friendship of fiery \"red\" Sh\u00e0ng\u00f3 and Obatala, deity of creativity, whose honorific color is white.\n\nMetaphoric fire balanced on the head of the thunder-god follower is an image that traveled to Bahia, in Brazil, where actual dancing with loads of fire has been reported:\n\nIn the cycle of festivals for Sh\u00e0ng\u00f3 in the shrine of S\u00e3o Gon\u00e7alothere is an impressive ceremony, only realized there, wherein thedaughters of Sh\u00e0ng\u00f3, possessed by their orisha, dance with avessel that contains material in flames, upon their heads. The firedoes not harm them, nor does it burn the hands with which they secure the burning vessel. Later, while still moving in the dance, they eat flaming balls of cotton dipped in oil.\n\nThese miracles indicate to believers that Sh\u00e0ng\u00f3's possession of his followers is actual, not a sham. The balancing of twin bolts of meteoric fire on the head of the devotee is also meant to convey a promise of moral vengeance. This powerful dual metaphor spread to the far corners of the Atlantic Yoruba world. It appears with particular strength in Bahia (Plate 53), where in the late nineteenth century the butterflylike shape of the thunderstones balanced on the represented worshipper's head revealed influence from Ketu, where thunder axes frequently are shaped this way.\n\nSome Western Yoruba believe that when Sh\u00e0ng\u00f3 blinks, a lightning bolt will crash down at that instant. And when a devotee is infused, in spirit-possession, with Sh\u00e0ng\u00f3's _\u00e0she_ , the devotee's eyes will bulge. Notice the cause and effect. This fine piece of nineteenth-century Yoruba sculpture in Brazil also shows the servitor with hands on her stomach, making of her midriff an offertory vessel of life and continuity.\n\nPLATE 53\n\nPLATE 54\n\nAfro-Cuban association of the thunder god with Saint Barbara sometimes gave rise on the island to superbly complex renderings. Our example (Plate 54) dates perhaps from the end of the nineteenth century, and apparently was collected in the region of Havana. The artist affirmed the Catholic aspect but transcended it. He dressed a Sh\u00e0ng\u00f3 follower in a red and white chasuble, deliberately creating raking angles in the phrasing of the garment, so as to harmonize it with and mirror the higher angulation of the thunderstones. The calm, pert face of the devotee has slight indications of the markings of Yoruba patrilineal descent groups. This compact little sculpture is a fusion of images from two cultures and two visions under God.\n\nMany Brazilian and Cuban _oshe_ reflect trade ties and strong ethnicity. But there is another kind of thunder ax made in the Americas, among United States blacks who dropped their Anglo-Saxon names in favor of If\u00e1-indicated Yoruba titles or self-assigned names. Adefunmi, the founder of the Yoruba Temple in Harlem in 1960, a Detroit-born black, learned Yoruba initiatory art and lore in Matanzas, Cuba, and brought an originally Afro-Cubanizing form of Yoruba _orisha_ worship to Harlem. But Adefunmi's imagination was restless, and he ceaselessly studied the Yoruba language, handbooks with illustrations of the classical art of the Yoruba, lectures by New York-area Africanists, and many other sources. In the process, the art of New York Yoruba became a willed phenomenon, a self-conscious renaissance that was distinguishable from modern Afro-Cuban work. In this body of art works are precise, detailed representations of the ideas of the _orisha_. For example, Adefunmi's _oshe_ , carved in Harlem in 1965-68, follows the canon in its essentials\u2014the twin thunderstones carried on the head as a mystic burden, the bulging eyes of _\u00e0she_ , the frozen mouth\u2014but the figure gestures asymetrically, with a miniature _oshe_ in the left hand and a thunder rattle in the right (Plate 55). The figure is not frozen in a pose of giving but in a stance of combativeness.\n\nThis figural wariness is appropriate to the condition of blacks in the crisis decade of the 1960's. Awolowo, another prominent member of the Harlem Yoruba Temple, shaped in c. 1968 a remarkable plaster image of Sh\u00e0ng\u00f3, the thunder king (Plate 56) wearing a kind of _oshe_ -coronet and holding before his body a royal Yoruba fly whisk and a miniature _oshe_. His face is traditionally impassive even as he stares. The sweep and velocity of the lines tracing the folds of the royal garment are unprecedented in the classical statuary, and represent the application of a rich inventive mind to the problems of artistic identity.\n\nSh\u00e0ng\u00f3's complex artistic embodiments\u2014warrior and lover, \"water by the side of fire at the center of the sky,\" \"I have an assassin as a lover\"\u2014extend back to antiquity. As early as 1659 Sh\u00e0ng\u00f3's upsurge into the world had been stylized by particularities of sculpture\u2014the fertilizing thrust of the thunderstone into the earth indicated by an image carved in wood, pointing to his penis with one hand while indicating the source of that energy by pointing to the sky with the other hand (Plate 57). Eva Meyerowitz examined an old brass armlet decorated with Sh\u00e0ng\u00f3 emblems in Dahomey (now Benin), where the traditional owners told her that the object had originally come from Oyo-Ile. On this probably ancient piece, possibly from the city of Sh\u00e0ng\u00f3, the sky-and-penis gesture of the thunder god reappears:\n\nPLATE 55\n\nPLATE 56\n\nPLATE 57\n\nPLATE 58\n\nOne figure presses both his hands against his cheeks, showing his tongue which is of enormous size, while the other carries an object in his one hand which suggests a meteorite or thunderbolt, while his other hand points to his sexual organs.\n\nThis strong phrasing came to Cuba:\n\nThere is no more vehement nor energetic spirit [than Sh\u00e0ng\u00f3]. When a devotee is mounted by the spirit of Sh\u00e0ng\u00f3, he charges three times, head leading, spinning like a ram, towards the drums. Then he opens his eyes to abnormal width and sticks out his tongue, to symbolize a fiery belch of flames, and raises his thunder-axe on high and clamps his other hand upon his scrotum.\n\nThe ancient dual gesture is so persistent that not only does Alakoro perform a modified version, pointing to the sky and to the earth twice, before the gates of the Oyo palace, but it is emblazoned in relief upon a thunder pedestal ( _odo sh\u00e0ng\u00f3_ ) found in Bahia, dating, probably, from the second half of the last century (Plate 58). Here the figure of a priestess makes the dual gesture, rattle to sky, hand to waist.\n\nThe coming of the icons of the Yoruba to the black New World accompanied an affirmation of philosophical continuities. Selected fragments of the liturgies remained alive in songs about the wonder-working blade atop, the head of Eshu-Odara, the irons of Ogun flashing moral vengeance in hearth and home, and the murmur of the chiming bracelets of Oshun beneath the undulating surface of the river. These verses lived in chants, and in icons of the goddesses and gods on New World domestic altars.\n\nThere is, in all of this, the vision of the Yoruba Atlantic world, a metaphoric capture of the moral potentiality inherent in certain powers of the natural world\u2014thunder, oceans, herbs, and stones\u2014and a demonstration that creative persons have shaped certain images, pillars of lateritic clay, implements of iron, metal fans, brooms decorated with leather and cowrie-shell embroidery, so that they illumine the world with intuitions of the power to make right things come to pass.\n\nTime and again, in Yoruba Atlantic art and dance, in the image of the servitor shaped in wood or directly embodied in ritual participation, the ancient praise-poems and divinatory insights were materialized, and made visible. One final example:\n\nA magnificent beaded (probably nineteenth century) Nigerian Yoruba tunic for a thunder-god possession-priest ( _ewu sh\u00e0ng\u00f3_ ) displays the lord of thunder with characteristic axe and rattle, surrounded by four birds of nocturnal power [Plate 59]. Here is Sh\u00e0ng\u00f3, beads of wealth blazing on his frame. Here is Sh\u00e0ng\u00f3, with triangularly represented elements of lightning, each with eyes to seek and find the rooves of the ungodly.\n\nPLATE 59\n\nThere is a telling opposition to be noted here\u2014the clash between the hard-edged, sharply pointed thunderstones and the smooth curves of richly elaborated interlace which almost certainly stands for serpents intertwined, ancient Yoruba image of coolness, peace, and power. These interwoven forces come down like lightning, like the zig-zags glittering on the moving body of the Gaboon viper, harbinger of God's _\u00e0she_ , like the earthworm, another founding form of God's _\u00e0she_ , ventilating earth, opening up the soil as with myriad lightning strikes.\n\nFire (thunderstones, flashing with pinpoint metallic eyes) and water (domain of becalmed serpents, superbly coiled about themselves) meet at the center of the sky. There Sh\u00e0ng\u00f3, axe and rattle handsomely attendant, faces the present and the future on behalf of his numerous followers. On the back of the tunic [Plate 60], facing the past, he whirls as an ancestor-inquisitor ( _eg\u00f9ngng\u00f9n_ ), lappets flaring in the wind, a virtual dervish in striped strips of narrow-loom 'country cloth.' There he spins, a righteous whirlwind, a storm on the edge of the knife dividing this world from the next.\n\nPLATE 60\n\nThis spectacularly allusive tradition, the Yoruba beaded royal tunic, was renewed in Cuba in new form. It reached a peak of decorative quintessence in the making, perhaps in the last quarter of the nineteenth century, of a sumptuously beaded garment ( _bandel_ ), for the mother-drum of a set of three Cuban-Yoruba _bat\u00e1_ drums [Plate 61]. Striking changes are apparent. Instead of Sh\u00e0ng\u00f3 in a field of fiery and acquatic forces, there is a vivid creole fusion of distinct goddesses and gods. Olokun, god of the sea [Plate 62], appears at the summit of the composition, eyes fashioned out of flashing mirror shards. The ax of the thunder-god reappears at left, here centered with a shell-studded jewel of copper color, symbolizing Oshun and her world of wealth. The ancient eyes of Eshu-Elegbara, cowrie shells, stare forward from an implicit cross-roads at the bottom of the tunic. Left, there appears a bird of Yemay\u00e1, intuiting her powers; right, another thunder ax; and handsome segments of leopard pelt allude to _orisha_ of the forest. And there is a segment of green and yellow beads for the deity of divination. The attributes of the deities of the Yoruba, one against the other, protect in unison the body of a drum. The drum itself brings back the rhythmized voices of the ancestors of the Yoruba in Cuba. Like a Trajanic frieze, unfolding history, this beaded drum tunic brings back sacred forces from the Old World to the New.\n\nPLATE 61\n\nPLATE 62\n\nIn conclusion, the grand message of Yoruba Atlantic art, wherever it is found, but especially in Nigeria, Togo, and Benin, the United States, Cuba, and Brazil, handsomely counterpointed, in music, by the rising worldwide popularity of Sunny Ad\u00e9 and other players of Yoruba juju music, and especially evident in the Yoruba Renaissance art and architecture (Plates 63, 64, 65) of Oyotunji, South Carolina, in the 1970's is this: sheer artlessness may bring a culture down but a civilization like that of the Yoruba, and the Yoruba-Americans, pulsing with ceaseless creativity richly stabilized by precision and control, will safeguard the passage of its people through the storms of time.\n\nPLATE 63\n\nPLATE 64\n\nPLATE 65\n\n# **Two**\n\n# **THE SIGN OF  \nTHE FOUR MOMENTS  \nOF THE SUN**\n\n# **Kongo Art and Religion  \nin the Americas**\n\nSpelling Kongo with a _K_ instead of a _C_ , Africanists distinguish Kongo civilization and the Bakongo people from the colonial entity called the Belgian Congo (now Za\u00efre) and the present-day People's Republic of Congo-Brazzaville, both of which include numerous non-Kongo peoples. Traditional Kongo civilization encompasses modern Bas-Za\u00efre and neighboring territories in modern Cabinda, Congo-Brazzaville, Gabon, and northern Angola. The Punu people of Gabon, the Teke of Congo-Brazzaville, the Suku and the Yaka of the Kwango River area east of Kongo in Za\u00efre, and some of the ethnic groups of northern Angola share key cultural and religious concepts with the Bakongo and also suffered, with them, the ordeals of the transatlantic slave trade.\n\nThe slavers of the early 1500's first applied the name \"Kongo\" solely to the Bakongo people. Then gradually they used the name to designate any person brought from the west coast of Central Africa to America. Similarly, the meaning of \"Angola\" broadened over the centuries. \"Ngola\" once referred only to the ruler of the Ndongo part of the Kimbundu culture in what is now the northern part of Angola. According to historian Philip Curtin, \"[Angola's] first European meaning referred quite precisely to the immediate hinterland of Luanda.\" Then the term became the name of not only modern Angola but sometimes the whole west coast of Central Africa, from Cape Lopez in northwestern Gabon to Benguela on the coast of Angola proper.\n\nThe broadening of the meanings of \"Kongo\" and \"Angola\" over the span of the Atlantic trade reflects the expansion of European slave trafficking into the heart of Kongo and Kongo-related societies during the seventeenth, eighteenth, and nineteenth centuries. Thousands of persons were abducted from this culturally rich area. And as opposed to a prevalent view of Africans\u2014as belonging to different \"tribes,\" speaking different \"dialects,\" thrown together in the holds of slave ships, and hopelessly alienated, one from the other\u2014Africans from Kongo and Angola shared fundamental beliefs and languages. When they met on the plantations and in the cities of the western hemisphere, they fostered their heritage. Kongo civilization and art were not obliterated in the New World: they resurfaced in the coming together, here and there, of numerous slaves from Kongo and Angola.\n\nKongo presence unexpectedly emerges in the Americas in many places and in many ways. Take, for example, vernacular English and singing. In the South of the United States, important Ki-Kongo words and concepts influenced black English, especially the lexicons of jazz and the blues, as well as of lovemaking and herbalism. Many a Ki-Kongo-derived word has been described by etymologists as \"origin unknown.\" The word \"jazz\" is probably creolized Ki-Kongo: it is similar in sound and original meaning to \"jizz,\" the American vernacular for semen. And \"jizz,\" suggestive of vitality, appears to derive from the Ki-Kongo verb _dinza_ , \"to discharge one's semen, to come.\" _Dinza_ was creolized in New Orleans and elsewhere in black United States into \"jizz\" and \"jism.\"\n\nThe slang term \"funky\" in black communities originally referred to strong body odor, and not to \"funk,\" meaning fear or panic. The black nuance seems to derive from the Ki-Kongo _lu-fuki_ , \"bad body odor,\" and is perhaps reinforced by contact with _fumet_ , \"aroma of food and wine,\" in French Louisiana. But the Ki-Kongo word is closer to the jazz word \"funky\" in form and meaning, as both jazzmen and Bakongo use \"funky\" and _lu-fuki_ to praise persons for the integrity of their art, for having \"worked out\" to achieve their aims. In Kongo today it is possible to hear an elder lauded in this way: \"like, there is a really funky person!\u2014my soul advances toward him to receive his blessing\" ( _yati, nkwa lu-fuki! Ve miela miami ikwenda baki_ ). Fu-Kiau Bunseki, a leading native authority on Kongo culture, explains: \"Someone who is very old, I go to sit with him, in order to feel his _lu-fuki_ , meaning, I would like to be blessed by him.\" For in Kongo the smell of a hardworking elder carries luck. This Kongo sign of exertion is identified with the positive energy of a person. Hence \"funk\" in black American jazz parlance can mean earthiness, a return to fundamentals.\n\nIn the black American metaphysical traditions of conjuring and healing with herbs and roots, there are words and phrases that cannot readily be traced to European origins. One of the most important words in black United States conjure-work, \"goofer,\" refers to grave dirt, often inserted in a charm. In Kongo territory, earth from a grave is considered at one with the spirit of the buried person. \"Goofer dust\" harks back to the Ki-Kongo verb _kufwa_ (\"to die\"). Another important word in the lexicon of the charm makers is toby. A toby is a good-luck charm. In form and function it almost certainly derives from the _tobe_ charms of Kongo. The original charm was \"made up of a mixture of earth from a grave plus palm wine, and is believed to bring good luck.\"\n\nMore important than the impact of Ki-Kongo on the languages of blacks throughout the Americas, though, is the influence of Kongo civilization on their philosophic and visual traditions. Aspects of Kongo culture retained by blacks enslaved in the Americas inspired these men and women to come together as brothers and sisters in Kongo-American societies virtually devoted to the very idea of being Kongo. At the turn of the century in Montevideo, Uruguay, the outwardly festive look of a Kongo dance society could conceal, as Vicente Rossi observed in a study of the origins of the tango, _Cosas de Negros_ (1926), a serious reaffirmation of autonomy:\n\nThere is something there, in the middle of the circle of black men, something that they alone see, feel, and comprehend... the voice of native soil, a flag unfurled in harmonic syllables. There is something there, in the middle of the dancing ring of black men and it is the motherland! Fleeting seconds of liberty have evoked it, and, once brought into being, it fortifies their broken spirits.... they have, forgetting themselves, relived the [Kongo] nation in one of its typical expressions... in sudden homage, with an expanded power of observation... they dance around the vision.\n\nSo powerful an act of cultural recollection would have inevitably reminded some black Montevidean elders born in Kongo of the grandeur of Mbanza Kongo, the ancient ideal capital, heart of the ancient kingdom, where \"all were born, each clan still has its street and each one has relatives to receive him.\" For the Bakongo envisioned their capital as an ideal, noble place where \"a strong chief, supernaturally inspired, assures every citizen his due.\" Mbanza Kongo was sited at the top of a hill, and its location reflects a vision of the cosmos deeply ingrained in the Kongo mind:\n\nThe N'Kongo [i.e., an inhabitant of the capital of Kongo] thought of the earth as a mountain over a body of water which is the land of the dead, called Mpemba. In Mpemba the sun rises and sets just as it does in the land of the living... the water is both a passage and a great barrier. The world, in Kongo thought, is like two mountains opposed at their bases and separated by the ocean.\n\nAt the rising and setting of the sun the living and the dead exchange day and night. The setting of the sun signifies man's death and its rising, his rebirth, or the continuity of his life. Bakongo believe and hold it true that man's life has no end, that it constitutes a cycle, and death is merely a transition in the process of change.\n\nThe Bakongo symbolized the daily journey of the sun around the mirrored worlds of the living and the dead by means of the spiral shape of the _kodya_ shell. They also expressed the same idea, architecturally, with the concentric spaces of the royal enclosure _(luumbu)_. A newly elected king would make a circular tour of his domain, symbolically passing through the worlds of the living and the dead, thereby acquiring mystic insights proper only to a Kongo king.\n\nIn 1960, while most Western journalists were writing copy about the \"Congo crisis,\" Bakongo women were styling their hair in splendid concentric, spiral patterns to celebrate the restoration of black rule and the circularity of the royal enclosure of the ancient ideal capital. Thus the women of Kongo had both invoked the greatness of Mbanza Kongo and communicated their hope that Kasa-Vubu's tour of the country had given him mystic power.\n\nThe Bakongo viewed their capital as an ideal realm in which images of centering prevailed. It was a world profoundly informed by a cosmogram\u2014an ideal balancing of the vitality of the world of the living with the visionariness of the world of the dead. Charms and medicines were constantly produced in the search for the realization of a perfect vision in the less than perfect world of the living. It is true that, unlike the Yoruba, the Bakongo lack a complex pantheon of deities, but they have, instead, a complex system of _minkisi_ (\"sacred medicines\"), which they believe were given to mankind by God. The religion of Kongo presupposes God Almighty _(Nzambi Mpungu)_ , whose illuminating spirit and healing powers are carefully controlled by the king _(mfumu)_ , the ritual expert or authority _(nganga)_ , and the sorcerer _(ndoki)_.\n\nThe king had the power to execute convicted felons and enemies of the state. As supreme authority, he was expected to live decorously, to embody impartiality. The ritual experts, or _banganga_ (plural of _nganga_ ), were various. Some healed with charms\u2014these were _banganga nkisi_. Others healed with roots and herbs and were called _banganga mbuki_. Still others, _banganga ngombo_ , ministered to the needs of clients by means of divination. Some, the _banganga simbi_ , worshipped powerful and mysterious spirits _(bisimbi)_ , \"the highest class of the dead,\" immortal beings who, because of their good works, are believed to be blessed with the power to resist the organic process. Others who were \"chosen\" were transformed, during initiation, from recovered patients to practicing herbalists, and became members of the healing society of Lemba, a band of healers in existence since the 1660's.\n\nNow the king and the ritual experts controlled mystic powers for the common good. By contrast, the _ndoki_ , or sorcerer, was a caricature of the individualist who, lacking social conscience, bases his security upon the insecurity of others. The belief in the existence of evil, in witchcraft\u2014in utterly selfish action devoid of mercy\u2014remains alive in Kongo. Dialectically, it strengthens the need for the wisdom of the ancestors and the elders who know the necessary antidotes. Not all bad luck or deliberate evil, however, was effected by witches. Offended ancestors could also intervene powerfully in the affairs of the living. Again, in such instances Bakongo turned to ritual authorities for guidance.\n\nVersions of some of the ritual authorities responsible for Kongo herbalistic healing and divination appeared in the Americas and served as avatars of Kongo and Angola lore in the New World. Kongo ritual experts in Cuba took the appropriate ancient name _banganga_ ; those in the United States were known largely as \"conjurors\" and \"root-persons\"; and others in Brazil were called _pae de santo_ and _mae de santo_ , names apparently originating in Yoruba worship\u2014 _babalorisha_ and _iyalorisha_ , \"father of the saint\" and \"mother of the saint.\" These ritual authorities were largely responsible for the dissemination of Kongo religious and artistic culture throughout the New World.\n\nThe influences of and improvisations upon Kongo art and religion in the western hemisphere are most readily discernible in four major forms of expression: cosmograms marked on the ground for purposes of initiation and mediation of spiritual power between worlds; the sacred Kongo medicines, or _minkisi_ ; the use of graves of the recently deceased as charms of ancestral vigilance and spiritual return; and the related supernatural uses of trees, staffs, branches, and roots.\n\n## **Tendwa Nz\u00e0 Kongo: The Kongo Cosmogram**\n\nWyatt MacGaffey, a scholar of Kongo civilization and religion, has summarized the form and meaning of the essential Kongo cosmogram as follows:\n\nThe simplest ritual space is a Greek cross [ ] marked on the ground, as for oath-taking. One line represents the boundary; the other is ambivalently both the path leading across the boundary, as to the cemetery; _and_ the vertical path of power linking \"the above\" with \"the below.\" This relationship, in turn, is polyvalent, since it refers to God and man, God and the dead, and the living and the dead. The person taking the oath stands upon the cross, situating himself between life and death, and invokes the judgement of God and the dead upon himself.\n\nThis is the simplest manifestation of the Kongo cruciform, a sacred \"point\" on which a person stands to make an oath, on the ground of the dead and under all-seeing God. This Kongo \"sign of the cross\" has nothing to do with the crucifixion of the Son of God, yet its meaning overlaps the Christian vision. Traditional Bakongo believed in a Supreme Deity, Nzambi Mpungu, and they had their own notions of the indestructibility of the soul: \"Bakongo believe and hold it true that man's life has no end, that it constitutes a cycle. The sun, in its rising and setting, is a sign of this cycle, and death is merely a transition in the process of change.\" The Kongo _yowa_ cross does not signify the crucifixion of Jesus for the salvation of mankind; it signifies the equally compelling vision of the circular motion of human souls about the circumference of its intersecting lines. The Kongo cross refers therefore to the everlasting continuity of _all_ righteous men and women:\n\n_Nzungi! n'zungi-nzila_ | Man turns in the path  \n---|---  \n_N'zungi! n'zungi-nzila_ | He merely turns in the path;  \n_Banganga ban'e E ee!_ | The priests, the same.\n\nA fork in the road (or even a forked branch) can allude to this crucially important symbol of passage and communication between worlds. The \"turn in the path,\" i.e., the crossroads, remains an indelible concept in the Kongo-Atlantic world, as the point of intersection between the ancestors and the living.\n\nHere is one precise version of the _yowa_ cross:\n\n_Yowa_ : the Kongo sign of cosmos and the continuity of human life\n\nThe horizontal line divides the mountain of the living world from its mirrored counterpart in the kingdom of the dead. The mountain of the living is described as \"earth\" ( _ntoto_ ). The mountain of the dead is called \"white clay\" ( _mpemba_ ). The bottom half of the Kongo cosmogram was also called _kalunga_ , referring, literally, to the world of the dead as complete ( _lunga_ ) within itself and to the wholeness that comes to a person who understands the ways and powers of both worlds.\n\nInitiates read the cosmogram correctly, respecting its allusiveness. God is imagined at the top, the dead at the bottom, and water in between. The four disks at the points of the cross stand for the four moments of the sun, and the circumference of the cross the certainty of reincarnation: the especially righteous Kongo person will never be destroyed but will come back in the name or body of progeny, or in the form of an everlasting pool, waterfall, stone, or mountain.\n\nThe summit of the pattern symbolizes not only noon but also maleness, north, and the peak of a person's strength on earth. Correspondingly, the bottom equals midnight, femaleness, south, the highest point of a person's otherworldly strength.\n\nMembers of the Lemba society of healers had initiates stand on a cross chalked on the ground, a variant of the cosmogram. \"To stand upon this sign,\" Fu-Kiau Bunseki tells us, \"meant that a person was fully capable of governing people, that he knew the nature of the world, that he had mastered the meaning of life and death.\" He thenceforth could move about with the confidence of a seer, empowered with insights from both worlds, both halves of the cosmogram.\n\nDrawing a \"point,\" invoking God and the ancestors, formed only a part of this most important Kongo ritual of mediation. The ritual also included \" _singing_ the point.\" In fact, the Bakongo summarize the full context of mediation with the phrase \"singing and drawing [a point],\" _yimbila ye sona_. They believe that the combined force of singing Ki-Kongo words and tracing in appropriate media the ritually designated \"point\" or \"mark\" of contact between the worlds will result in the descent of God's power upon that very point:\n\n_sikulu dya nene_ | A mighty noise  \n---|---  \n_dyakulumukina_ | causing to descend  \n_Na Nzambi a Mpungu_ | Lord God Almighty\n\nThe cosmogram of Kongo emerged in the Americas precisely as _singing and drawing points of contact between worlds_. On the island of Cuba, when Kongo ritual leaders wished to make the important _Zarabanda_ charm (Ki-Kongo: _nsala-banda_ , a charm-making kind of cloth), they began by tracing, in white chalk, a cruciform pattern at the bottom of an iron kettle. This was the \"signature\" ( _firma_ ) of the spirit invoked by the charm. It clearly derived from the Kongo sign except that the sun disks were replaced by arrows, standing for the four winds of the universe.\n\nSignature of _Nkisi Sarabanda_ , western Cuba, twentieth century.\n\nOne of the major functions of the cosmogram of Kongo, to validate a space on which to stand a person or a charm, remains in force in certain Afro-Cuban religious circles. Kongo-Cuban priests have said, \"All spirits seat themselves on the center of the sign as the source of firmness.\" Songs ( _mambos_ ) are chanted, as in Kongo, to persuade this concentration of power upon the designated point. Kongo-Cuban priests activated old, important charms by singing-and-drawing a sacred point. They chanted sacred texts in Spanish and creolized Ki-Kongo while lowering a charm from the ceiling of a shrine to a chalked sign drawn upon the floor:\n\nTo bring down the bundle was a tedious and delicate operation. \"[the bag] was attached to the ceiling, was enormous, and weighed a hell of a lot.\" Juan O'Farril clearly remembers all the songs that accompanied the descent of the sacred bundle. The latter, with his steward and godmother, swept the earth, singing,... until the ground was immaculate and ready to receive the sacred bundle. The priest asked for chalk ( _mpemba_ ), and traced the cross-within-the-circle \"signature\" at the precise point where the bag was to touch the ground.\n\nAnd all the while that this bag of medicine was coming down the priest was singing the _mambo_ , the song to bring it down:\n\n_Mpati! mpati! mpati!_ | Power, power, power  \n---|---  \n_Mpemba, simbi ko?_ | Kaolin is a simbi spirit, is it not?  \n_Mpemba, simbi ko?_ | Kaolin is a simbi spirit, is it not?\n\nThe song was a warning. It suggested that the charm was like lightning, descending in slow motion, a spirit, a _simbi_ , a force that could not be touched or seen until the nganga had completely lowered it to its appropriate point of relevation.\n\nThere are numerous charm-connected cosmograms chalked on earth in Cuba, including one with pieces of silver placed at the ends of the axes of the cross to complete the ancient Kongo pattern:\n\nThe development of mystic ground-drawings in Cuba parallels the development of Afro-Christian crosses, chalked on the floor as critical \"points,\" among the Trinidad Shouters. Related ground-drawings for a Shaker \"mourning\" ritual on the island of St. Vincent, north of Trinidad in the Windward Islands, receive their basic structure from the cruciform, with disks at the end of each line and each design circumscribed. Within each St. Vincentian ground-drawing largely alphabetically derived ideographic signs float like sidereal dust\u2014signs such as   for travel, or   for baptism, or   for Jericho. A crisscross of spiritual signs of power and mediation is remarkable in this Afro-Caribbean calligraphic art, as instanced by a chalked design made for a St. Vincentian mourning ritual, published by Jeannette Henney in 1968:\n\nSt. Vincentian \"mourning\" ritual ground-drawing\n\nBlack ritual leaders among the Shouters of Trinidad have also elaborated an art of spiritual mediation using ground-signs: geometric patterns filled with myriad mystic ciphers and notations. Consider a drawing documented in 1960 by George Simpson. It breaks the circular seal of Kongo-influenced ways of reaching spirit with graphic signs. Here the sequence of the shapes is virtually architectural, building to the climactic pattern of a star. Every element is filled with spiritual notations, miniature cosmograms, the numbers of certain Psalms, and ciphers received in trance:\n\nTrinidad ground-drawing.\n\nThe result is visual glossolalia, a galaxy of points, indicating spiritual encounter and enlightenment. As Jeannette Henney reveals, \"each pointer [sign-maker] has a different alphabet.\" A woman working with the spirit on St. Vincent can \"point souls,\" by means of chalking mystic signs upon the floor in a state of spiritual enlightenment. Her meanings will not be grasped by all. In these rituals involving ground-writing for the \"pointing\" of souls, on St. Vincent and among Trinidad Shouters, there is overlap with the Kongo ritual of \"singing and marking a mystic point\" ( _iyimbila ye sona_ ), which is suggestive of a trace of Kongo influence, however mixed with Christian signs and concepts.\n\nIn Rio de Janeiro, where there was a heavy importation of Kongo and Angola slaves, we meet simple cruciforms, chalked on the floor of shrines and altars, that have become complex signs fusing diverse Kongo, Yoruba, Roman Catholic, and other references. These signs comprise the blazons of the spirits honored in Rio de Janeiro macumba, a mixture of Kongo, Yoruba, Dahomean, Roman Catholic, Native American, and Spiritualist allusions.\n\nIn Rio at the beginning of this century there were two main African \"nations\" with their characteristic worship: followers of the deities of the Yoruba in rites called _candombl\u00e9_ and followers of largely Kongo and Angola medicines and spirits, called originally _cabula_ but latterly macumba after absorption of further influences: \"All that was needed to produce _macumba_ as it exists today was the adoption, along with... Yoruba elements, of the _caboclo_ [inland Native American] spirits, the Catholic saints, and the spiritists' dead.\" The French spiritualism of Allen Kardec\u2014whose _Book of Mediums_ was widely read in Brazil\u2014especially influenced macumba.\n\nMacumba priests, in the beginning, invoked spirits through simple chalked designs drawn on the ground and called \"marked points\" ( _pontos riscados_ ). Many reflected dual Kongo and Roman Catholic influences, and were essentially cosmograms drawn in the form of a Latin cross. Nevertheless, some are used in a Kongo manner to \"center\" consecrated water and other important liquids in vessels for spirits (Plate 66). The Afro-Brazilian term for these visual invocations of spirit, _pontos cantados e pontos riscados_ , or \"sung points and marked points,\" recalls the Kongo custom of simultaneously singing and marking the centering of spirit ( _iyimbila ye sona_ ).\n\nAs time passes, the ancient cruciforms are complicated by aspects borrowed from the iconography of the Yoruba and enriched by the attributes of Roman Catholic saints identified with Yoruba deities. Even Satan was identified with a Yoruba deity, Eshu, to challenge goodness creatively. Thus the \"signatures\" of Eshu, associated with the crossroads, sudden changes of fortune, and \"devilish,\" that is, unpredictable, behavior, are circular blazons in which Satan's pitchfork, a pinwheel sign of sudden change and motion, a crossroadslike sign, and additional mystic points are recombined:\n\nPLATE 66\n\n_Ponto Eshu Vira-Mundo_\n\n_Ponto Eshu Vira-Mundo_\n\n_Ponto Eshu Tranca-Gira_\n\nAnd then interpreted imaginatively: \"When Eshu's trident is up, the work is for the good; upside down, the work is for evil.\"\n\nKongo \"points\" can at once invoke Almighty God and call upon the noble dead. But Rio macumba \"points\" summon a bewildering multitude of Yoruba goddesses and gods and their matching saints, plus Amerindian spirits, plus departed black elders _(paes velhos)_ , who return in the voice and body of their devotees. More than a hundred spirits are invoked today in Rio through the use of such signs.\n\nMost _pontos riscados_ , traced in chalk on shrine floors or in sand on the beaches at Ipanema, Copacabana, and elsewhere in Rio, became fleeting signs of spiritual invocation and encounter. But some were permanently rendered. The collections of the Rio Museu de Pol\u00edcia include a calabash drinking cup (Plate 67), richly lacquered black, which is incised with the _ponto riscado_ of Pai Velho, a black Kongo ancestor of special power and insight. His _ponto_ warns the world that no one except a person in his spirit, or an appropriate officiant, can use this cup. The _ponto_ , essentially a Latin cross within a Star of David within a circle decorated with six minor stars, compares interestingly with the \"sung point\" for the same spirit:\n\nHe is swung in a circle (is initiated)  \nAncient Father, sovereign one  \nDark stone coming down  \nHe turns, Great God Almighty-Oduduwa  \nAncient Father, sovereign one  \nDark stone, falling, coming down.\n\nThe descent of a meteorite, the spirit of Pai Velho, recalls the lore of Kongo where shooting stars have been interpreted as spirits darting across the sky. The song provides a context for the stars that surround and decorate the cross or cosmogram of Pai Velho.\n\nPLATE 67\n\nPLATE 68\n\nThere is a superb _ponto riscado_ for Og\u00f9n\/St. George, also in the Rio Police Museum, rendered in scarlet thread on a lime green sash of silk (Plate 68) worn with a mirroring sash on which appears the Roman Catholic signature of the same fusion-deity. Representing Og\u00f9n's roads, lines radiate from the center of this cosmogram. They are also crossroads, the element of Eshu\u2014hence, perhaps, the cryptic pitchforks at the termini of three indicated axes. This rich sign on silk captures a complex history of cultural contact and experience in a form of geometric thought. The blending has carried us far indeed from Kongo and from the kingdoms of the Yoruba, but without their encounter in the richest minds of Rio this sign might never have been invented.\n\n## _Minkisi_\n\nThe cosmograms of Kongo also reappeared in the Americas in the form of the _nkisi_ -charm. An _nkisi_ (plural: _minkisi_ ) is a strategic object in black Atlantic art, said to effect healing and other phenomena. Nsemi Isaki, himself a Mu-Kongo, wrote c. 1900:\n\nThe first _nkisi_ , called Funza, originated in God, and Funza came with a great number of _minkisi_ which he distributed throughout the country, each with its respective powers, governing over its particular domain.\n\nHe defined _nkisi_ as\n\nthe name of the thing we use to help a person when that person is sick and from which we obtain health; the name refers to leaves and medicines combined together.... It is also called _nkisi_ because there is one to protect the human soul and guard it against illness for whoever is sick and wishes to be healed. Thus an _nkisi_ is also something which hunts down illness and chases it away from the body.\n\nAn _nkisi_ is also a chosen companion, in whom all people find confidence. It is a hiding place for people's souls, to keep and compose in order to preserve life.\n\nThe belief in an inner spark of divinity or soul within the _nkisi_ leads the Bakongo to suppose that the _nkisi_ is alive:\n\nThe _nkisi_ has life; if it had not, how could it heal and help people? But the life of an _nkisi_ is different from the life in people. It is such that one can damage its flesh ( _koma mbizi_ ), burn it, break it, or throw it away; but it will not bleed or cry out... _nkisi_ has an inextinguishable life coming from a source.\n\n_Minkisi_ containers are various: leaves, shells, packets, sachets, bags, ceramic vessels, wooden images, statuettes, cloth bundles, among other objects. Each _nkisi_ contains medicines ( _bilongo_ ) and a soul ( _mooyo_ ), combined to give it life and power. The medicines themselves are spirit-embodying and spirit-directing.\n\nSpirit- _embodying_ materials include cemetery earth\u2014considered at one with the spirit of a buried person\u2014or equivalents such as white clay ( _mpemba_ ), taken from riverbeds, or powdered camwood, the reddish color of which traditionally signals transition and mediation to Bakongo. Spirit-embodying materials are usually wrapped or concealed in a charm, but such objects as mirrors or pieces of porcelain attached to the exterior of the _nkisi_ may also signify power\u2014the flash and arrest of the spirit.\n\nThe vital spark or soul within the spirit-embodying medicine may be, according to the Bakongo, an ancestor come back from the dead to serve the owner of the charm, or a victim of witchcraft, captured in the charm by its owner and forced to do his bidding for the good of the community (if the owner is generous and responsible) or for selfish ends (if he is not).\n\nSpirit- _directing_ medicines instruct the spirit in the _nkisi_ \u2014by way of puns and symbols\u2014how to hunt down evil or, say, make a person more decisive in daily affairs. Seeds, stones, herbs, or sticks represent action as aspiration. The spiritual commandments of certain medicines are the prerogatives of the head of a clan. Therefore, some significant _minkisi_ bear major Kongo clan names such as Nsumbu and Mbenza.\n\n_Nkisi Nkita Nsumbu_ exemplifies the outward simplicity and inner complexity of Kongo charms. It is a somewhat heavy bag of folded raffia cloth, tied at the neck with a cord. Double _kunda_ wooden bells tied to the neck of the charm suggest its inner powers, as _kunda_ are believed to activate medicines and drive out specters. When _Nkita Nsumbu_ is unwrapped (Plate 69), revealing its medicines, it is like looking through clear water at the pebble-strewn bottom of a river.\n\nThe medicines are embedded in whitish kaolin. The gleaming whiteness of the bottom of the charm, beneath the seeds and pieces of crystal and stones and other things, suggests the lower half of the Kongo cosmogram. _Nkita Nsumbu_ is extraordinary for its large quantity of rock crystals, perhaps because as the Bakongo explained to Karl Laman at the turn of the century: \" _Nkita_ throws stones at the sick person.\" The righteous purpose of the charm\u2014to protect a person from diseases of the skin, which, according to the Bakongo, are caused by supernatural knives and needles thrown at him\u2014may account for the presence of the blade of a knife among the assembled \"medicines.\" Puns on assuagement and healing may lie concealed in the syllables of the the Ki-Kongo words for the beans, plant buds, and various kernels that rest upon the whitish river clay.\n\nThe square-shaped black stone with a concave center near the top of the illustrated center of the charm is used to pulverize kaolin to make a paste that is applied around the eyes of a ritual expert, granting him mystic vision. The foot of a hen and other claws suggest the captivating power of the spirit in the charm. A pearshaped seed called _kiyaala-mooko_ (\"who holds out hands\") is probably included because of its resemblance to the image of two cupped palms joined and extended in a gesture of generosity. According to a Kongo proverb, \"he who holds out his hands does not die\" ( _kiyaala-mooko kufwa ko_ ). Finally, the gleaming white egg near the center of the photograph, according to Fu-Kiau, \"is a symbol of danger, the kind of egg used by Kongo sorcerers to conjure up tornadoes and to bring down thunder.\" Mystically stoning the immoral with diseases of the skin, protecting the righteous, and creating storms and other natural phenomena, Nkisi _Nsumbu_ is believed to represent the world in miniature.\n\nPLATE 69\n\nOther Kongo charms are informed by this metaphor of cosmos miniaturized, such as _Nkisi Mbumba Mbondo_ , which protects persons from swellings of the body or inflicts such illnesses on deserving criminals or enemies. The container of this _nkisi_ is European cotton covered with thick red camwood impasto (Plate 70). White buttons of porcelain and glass are sewn on the charm for glitter. A braided raffia string has been crisscrossed around the bottom and wound tightly at the top to suggest the arrest of spirit. Around the top, which resembles a neck, is a necklace of five strands of green, blue, and white glass beads. This Ter\u00fcren Museum charm is capped by resin, in which feathers have been inserted in the manner of certain feathered headdresses worn by important chiefs and priests in Kongo. Feathers in Kongo connote ceaseless growth as well as plenitude. So if the earth within the charm affirms the presence of a spirit from the dead\u2014from the underworld\u2014feathers capping the charm suggest connection with the upper half of the Kongo cosmogram which represents the world of the living, and the empyrean habitat of God.\n\nPLATE 70\n\nThe mediating function of _minkisi_ is dramatically represented by charms carved in the shape of a dog. In Kongo these are sometimes Janus-like, staring in opposite directions, emphasizing their roles as seers extraordinaire. According to the Bakongo, \"between the village of the living and the village of the dead there is a village of dogs.\" Fu-Kiau extends the metaphor: \"Mirror ( _talatala_ ) and dog ( _mbwa_ ) symbolize the same thing among Bakongo.\" Thus a dog or doglike _nkisi_ is often used by Kongo mystics to see beyond our world.\n\nIn Kongo mythology, Ne Kongo himself, the progenitor of the kingdom, prepared the primordial medicines in an earthenware pot set on three stones above a fire. Clay pots have therefore always been classical containers of _minkisi_. The Royal Museum of Central Africa in Terv\u00fcren, Belgium, has a collection of pottery- _minkisi_ , including one smeared with white clay to represent the other world, and another covered with a mirror to symbolize the water between the realms of the living and the dead. These vessels are filled with spirit-embodying earths and spirit-directing stones and shells.\n\nKongo-inspired _nkisi_ vessels and bundles were profuse in western Cuba in the nineteenth century. Many _minkisi_ produced in Cuba and in Afro-Cuban barrios of the United States today\u2014especially those in Miami and New York\u2014are contained by large three-legged iron cooking pots. Afro-Cubans call such _nkisi_ vessels _prendas_ (pawns), reflecting the ritual obligations shared by the owner of the charm and the spirit within. Here is a Cuban _nganga_ composing his _prenda_ , re-creating the Kongo cosmos in miniature:\n\nFirst one draws a cross, in chalk or white ashes, at the bottom of the kettle. The kettle must be new. One places over this sign five Spanish silver reales, one at the center, the others at each end of the cross... one places to the side a piece of sugarcane filled with sea water, sand and mercury, stoppered with wax, so that the _nkisi_ will always have life, like the flow of quicksilver, so that it will be swift and moving, like the waters of the ocean, so that the spirit in the charm can merge with the sea and travel far away.\n\nThe body of a black male dog may be included to grant the charm the sharp sense of smell associated with that animal... on top of these objects is poured earth from an ant-heap and small pieces of wood. Sticks [from some twenty-seven species of trees] are placed around and leaves and herbs also added... After completion of this level, one throws over it chili, pepper, garlic, ginger, white onion, cinnamon, a piece of rue, pine seed. The work is completed by the addition of the skull of a woodpecker or buzzard.\n\nThis charm is informed by the structure of the sign of the four moments of the sun. And in it are commingled objects associated with flight, through quicksilver, and avatars of earth and sea. Makers of important Afro-Cuban _prendas_ knew exactly what they meant by the ingredients they used:\n\nPLATE 71\n\nPLATE 72\n\nThe _prenda_ is like the entire world in miniature, a means of domination. The ritual expert places in the kettle all manner of spiritualizing forces: there he keeps the cemetery and the forest, there he keeps the river and the sea, the lightning-bolt, the whirlwind, the sun, the moon, the stars\u2014forces in concentration.\n\nOne of the most celebrated and feared of all the _minkisi_ of Cuba was inherited by J. S. Bar\u00f3, a black ritual expert who lived in Marianao, a suburb of Havana. It was an _nkisi_ -kettle (Plate 71), probably originally made by his ancestors in the second half of the nineteenth century and replenished several times in the twentieth. Bar\u00f3's _nkisi_ bears a deliberately long and impressive name: Tree-of-the-Forest-Seven-Bells-Turns-the-World-Round-the-Midnight-Cemetery\u2014a name replete with cosmological allusion. Its ascending structural sequence\u2014round container, encircling sash, luxurious spray of plumes\u2014mirrors that of _minkisi_ forms still seen today in Kongo (Plate 72). Tree-of-the-Forest-Seven-Bells contained bones and earths for the capture of spirit, and beads, and stones, and other matter to tell the spirit cryptically what to do. It was even believed to be enlivened by the \"flash of the spirit,\" the glitter of a falling star, mystically absorbed, as in the case of another famous Cuban _prenda_ , Seven Stars:\n\nThey celebrate this prenda not in the house but in the forest.... the stars come down to this charm. There is an hour in the night when the nkisi is left by itself in the forest, so that the stars may come down, to enter into its power. When you see something brilliantly coming down\u2014it is a star, entering an nkisi.\n\nPLATE 73\n\nPLATE 74\n\nIn addition to _prendas_ , Kongo-Cubans of the nineteenth century made _minkisi_ -figurines to mystically attack slaveholders and other enemies, and for spiritual reconnaissance. Lydia Cabrera has described them: \"magic doll-like figurines about 50 centimeters high, carved in wood... with magic substance inserted in a small cavity.\" Among these figurines figured a so-called _matiabo_ image, documented in 1875 (Plate 73). It was strongly Kongo-influenced, and came equipped with horn _(mpoka). Matiabo_ were runaway slaves who sometimes joined forces with rebel forces in Cuba's nineteenth-century war of independence against Spain. They probably used such charms\u2014even as human figurines mounted on antelope horns were used in Kongo\u2014to expose sorcerers, heal the sick, and locate game (in Cuba the quarry was approaching Spanish soldiers).\n\nAs in many examples of classical Kongo sculpture (Plate 74), the knees of the _matiabo_ image are bent, symbolizing vitality; the eyes sparkle with visionary glass; and medicine has been inserted at the abdomen and in the horn, the latter stoppered with a piece of mirror, again for mystic vision.\n\nIn Haiti, Western Hispaniola, ingenious reformulations of _nkisi_ charms, sometimes called _pacquets-congo_ , rival Afro-Cuban charms in density and allusion. The more elegant _pacquets_ are wrapped in silk (Plate 75) instead of ancestral cotton (Plate 72) or raffia cloth; are tied with broad silk ribbons (secured with pins) instead of cord; are adorned with sequins as well as beads; and sometimes crowned with plumes made of metallic cloth instead of actual feathers. Two _pacquets-congo_ made in 1979\u2014one with a feathered stem called Simbi-of-the-Leaves _(simbi makaya);_ the other, shorter, with arms akimbo called Queen-of-Kongo _(reine kongo)_ \u2014are the work of a _vodun_ priestess and her entourage living on the western outskirts of Port-au-Prince, and represent visual creolization of traditional _nkisi_ forms (Plates 75, 76). Just as the cruciform ground-signs of Kongo expanded to absorb the blazons and iconographic detail of a bewildering number of Yoruba and Dahomean deities in Brazilian _pontos_ , so the persistence of a basic stem-on-globe-shaped-base structure, as evidenced by a comparison of _Nkisi Mbumba Mbondo_ (Plate 70) in Kongo with an important kind of _pacquet-kongo_ in Haiti called _Simbi Macaya_ (Plate 75), cannot hide enormous amounts of creative change. The narrow cords, which bound up the spherical base of the charm in Kongo, have become in Haiti a rich interplay of broad silk ribbons, one of which, horizontal, emphasizes the girth of the charm, while the others, vertical, break up the sphere into armillary-like aesthetic sections. The binding of the charm with these ribbons, carefully secured with pins, emphasizes the capture of forces guarding the households that own such charms.\n\nPLATE 75\n\nPLATE 76  \nCourtesy Terv\u00fcren Museum, Belgium\n\nThe Kongo spirits of the dead, _bisimbi_ , are believed in part to preside over the making of _pacquets-congo_ in Haiti. A priest in _vodun_ told Alfred M\u00e9traux that _pacquets_ were \"guards,\" capable of exciting and heating up the deities in favor of their owners. Deprived of such \"points,\" the deities lose their force. Haitian _pacquets_ , like _prendas_ and _minkisi_ , are therefore sometimes interpreted cosmo-grammatically. They are \"points,\" mediating protective grace between the living and the dead.\n\nThe strong Kongo-Angola influence on the people of Brazil accounts for another _nkisi_ tradition in the New World\u2014Brazilian charms for love and war. The _ponto de segurar_ (\"securing point\") was a small charm in a cloth container that was designed to arrest a spirit or attract a person to its owner. The tight crisscrossing cords\u2014three along one axis, six or more along the other\u2014decisively represent the enclosure of spirit.\n\nCharm from black Rio: _ponto de segurar_ , twentieth century, probably made before 1941.\n\nThe form and function of _pontos de segurar_ resemble those of miniature Haitian charms called _pwe_ (\"points\"), and both types are analogous to the Kongo-Cuban _nkangue_ charms. Cuban _banganga_ make _nkangue_ to \"tie\" a lover for a client, to keep that person from straying from the client's arms. _Nkangue_ literally means \"one who arrests\" in Ki-Kongo, and, in the charm-making process, originally referred to the moment when the _nganga_ \u2014having chanted all incantations and completed filling and binding the charm\u2014says, \"I close the door,\" and the spirit is said to be arrested in the charm as in a tiny habitation (Plate 77). This Afro-Cuban diminutive object\u2014only two inches long\u2014contains a _carga_ (\"charge\") of cemetery earth and powdered bones. Around its carefully folded cloth container, string has been wound tightly. It compares with the symbolically tight binding of Kongo charms (Plate 76). Lydia Cabrera, an authority on the folklore of Cuba, maintains that Afro-Cuban _banganga_ often tie _nkangue_ with four or seven symbolic knots to suggest various sacred meanings. The shape, size, and binding elements of _nkangue_ are similar to those of _muselebende_ love charms of Manianga in northern Kongo today. _Muselebende_ literally means \"to make like a leopard\" _(kala nsele bende)_ \u2014\"for the leopard is a very serious animal; so, when the soul of a lover is put into that charm, it takes the lover by force as does a leopard.\"\n\nUntil very recently the United States was held to be virtually devoid of African-influenced visual traditions. Yet I have discovered that in black communities throughout the States there is a surprisingly strong _minsiki_ -making tradition. Not that the heritage always came direct from Kongo and Angola. The well-known practice during slavery of \"seasoning\" blacks by residence on West Indian islands before bringing them to continental plantations meant that, in some instances at least, the Kongo _nkisi_ tradition was probably already transformed before reaching the United States via Charleston, New Orleans, and other parts.\n\nPLATE 77\n\nThe Kongo heritage is clearly evident in the making of love charms in black American communities\u2014charms whose purpose is to bring back errant lovers. And to the degree that they involve the folding in and tying up of a name, they recall Afro-Brazilian _pontos de segurar_ , Afro-Haitian _pwe_ , and Afro-Cuban _nkangue_. Robert Bryant, a black from New Orleans, describes the making of an Afro-American love charm:\n\nGet a strip of red flannel about a foot long and three inches wide, together with nine new needles. Name the flannel after the absent person.... Fold the flannel three times _towards_ yourself, so that half of it is folded, saying with each fold, \"Come (fold) on (fold) home (fold).... Then turn the other end towards yourself and make three more folds: \"Papa (fold) wants (fold) you (fold).\" Then stick the nine needles in the shape of a cross, working each one toward yourself and sticking each three times through the fabric, saying with each shove such phrases as \"Ma (stick) ry (stick) Smith (stick). Won't (stick) you (stick) come? (stick)\n\nFu-Kiau glosses Robert Bryant's charm as follows:\n\nThe whole action of folding cloth toward my body and inserting pins\u2014this is exciting a girl toward me. This is a kind of _nkisi_ which we use to bless or punish, wrapped up in a small sachet or cloth container called _futu_. It also sounds like _Nkisi Mpungu_ , one of the more powerful _minkisi_ known in Kongo, which you use when you want to make love with a person who doesn't really see you as a lover. You chant three times her name, three times your name. Each time you call her name, and each time call your name, you stick a pin into the charm. This has to be done very carefully, however, for there exists the danger that if one of the needles break, the lover may become deranged. Folding the cloth, to \"tie,\" the lover or person in the charm is called _futika_ (to hem, fold, bind, tie up) _nkisi va taba_ , folding a part of the cloth that wraps up an _nkisi_. The entire action, folding and sticking in a needle while calling a lover's name\u2014all that process is called _siba ye kanga_ , \"calling and tying,\" taking her name, or his name if the charm is for a woman, and putting it in the object with the inserted needles _(zinongo)_ and folding gestures. The _futu_ (sachet) is like a house, and when I have completed the _siba ye kanga_ , I \"close the door.\" He can't go out, he's put inside the charm, he cannot act now by himself.\n\nRobert Bryant's charm is therefore very Kongo; it reaffirms the \"point,\" verbally and materially renders it. Moreover, his practice strongly recalls the making of _n'kondi_ , those famous Kongo statues bristling with half-inserted blades and pins and nails and wedges, each of which may represent a vow or legal agreement.\n\nFrom nineteenth-century Missouri comes another Afro-American _nkisi_ , which was believed to be informed by \"the flash of the spirit,\" and which was supposed to bring its owner good luck:\n\nA more complicated \"luck ball\" was one made for Charley Leland. This contained, briefly, four lengths of white yarn doubled four times (four is a \"luck number\"), four lengths of white sewing silk folded in the same way (to tie your friend to you, while the yarn ties down the devils) and with four knots tied to the whole. Four such knotted strands were used, giving sixteen knots in all. These skeins were made up into a nest, whiskey spit upon them to keep the devils from getting through the knots, and _into it was put tin-foil (representing the brightness of the spirit who was going to be inserted in the ball)_ [emphasis added].\n\nFu-Kiau glosses Charley Leland's charm:\n\nThe \"Charley Leland luck ball\" sounds like the kinds of _minkisi_ we make in Kongo against powers or authorities, to sway these powers in our direction, helping us to find a job. Folding the cloth symbolizes \"tying\" the company or persons to your concern. If they say \"come back and we will discuss your job in three days,\" you tie three knots on the charm, measuring the time, to aim the power of the charm, upon the person or company, with accuracy. If, in another instance, I wish to reply to a prospective employer in six days and I tie only four knots, the _nkisi_ can have no effect. The knots _(makolo)_ have incantations in them; their number records matters of time or space involved in the private affair of the owner of the charm, and they therefore constitute a kind of African ritual mathematical system. The folds, the knots, the insertions in this kind of charm excite the spirit, excite the patron or the lover in your favor.\n\nThe Ki-Kongo verb \"to fold\" _(futika)_ also means \"to bind, to tie up in a packet,\" a double meaning that one may intuit in the Afro-Missourian charm, which was intended to bind the owner to his friends, arrest his enemies in their tracks, and enhance his luck.\n\nBakongo returning from Kinshasa to their villages consult, in many cases, _banganga_ for charms such as Charley Leland's.\n\nAccording to Kongo mythology, the very first _nkisi_ given to man by God was Funza, distributor of all _minkisi_ , himself incarnate in unusual twisted-root formations. So \"when you see a twisted root within a charm,\" Fu-Kiau Bunseki comments, \"you know, like a tornado hidden in an egg, tthat this _nkisi_ is very, very strong\u2014you cannot touch it; only _nganga_ can touch it.\" The association of twisted roots with enormous reserves of power persists in black America, where such charms are called \"high\" or \"big.\"\n\nThe most famous Afro-American charm of this kind is High John the Conqueror, a gnarled and twisted root. If it has a long extension it is considered male; if its salient is short, it is considered female. In _Voodoo in New Orleans_ , Robert Tallant writes: \"Thousands of Negroes carry Johnny the Conqueror roots, not only in New Orleans, but all over the country, for quantities of them are sold in Harlem, in Chicago, in Atlanta and Charleston.\" A man named Abner Thomas told Tallant that \"for love and gambling the _power_ of Johnny is considered supreme.\" He added: \"There is Big John and Little John.... High John is the same as Big John; that is the strongest.\" So, not surprisingly, the blues of the Mississippi delta\u2014an area not far from New Orleans\u2014sometimes make cryptic references to Kongo-influenced charms:\n\nI got a black cat bone\n\nI got a mojo tooth\n\nI got John the Conqueroo\n\nI'm gonna mess with you\n\n\"Hoochie Coochie Blues\"\n\nFinally, there are anti-hex roots galore throughout black America today that are usually wrapped in red flannel, \"so that nobody can put evil on you\u2014if they do it will turn on _them_.\" To the Bakongo these American charms appear to derive from _minkisi wambi_ and _minkisi wa nsisi_ \u2014\"danger _minkisi_ \"\u2014which are contained in crimson cloth.\n\n## **Nzo a Nkisi: The Grave**\n\nNowhere is Kongo-Angola influence on the New World more pronounced, more profound, than in black traditional cemeteries throughout the South of the United States. The nature of the objects that decorate the graves there, as well as in places as diverse as Haiti and Guadeloupe in the West Indies, reveals a strong continuity. That continuity might be characterized as a reinstatement of the Kongo notion of the tomb as a charm for the persistence of the spirit. Recall the container of the Kongo charm or sacred medicine, charged with activating ingredients\u2014a human soul, spirit-embodying and spirit-directing objects.\n\nIn the case of a burial site, the coffin and the mound are the obvious containers; the soul of the deceased is the spark. In addition to the tombstone or headstone, there are decorative objects that, both in Kongo (Plate 78) and the Americas (Plate 79), cryptically honor the spirit in the earth, guide it to the other world, and prevent it from wandering or returning to haunt survivors. In other words, the surface \"decorations\" frequently function as \"medicines\" of admonishment and love, and they mark a persistent cultural link between Kongo and the black New World.\n\nPLATE 78\n\nPLATE 79\n\nBoth Kongo and Kongo-American tombs are frequently covered _with the last objects touched or used by the deceased_. Here is the Kongo rationale for this striking practice:\n\nPlates and cups and drinking-glasses are frequently selected for placement on the surface of a tomb. It is believed that _the last strength of a dead person is still present within that sort of object_... My own mother died while I was away. When I return to my village, and visit her grave, I shall touch her plate and cup. After I touch them, later I will dream... according to the way my mother wanted. By touching these objects automatically I comprehend the _mambu_ (affairs, matters) my mother was willing to transmit to me.\n\nIn black North America the last-used objects of the dead are also believed to be specially charged with emanations, traces of the spirit. One can chart the continuity of this belief from plantation times (1845\u201365)\u2014\"Negro graves were always decorated with the last article used by the departed\"\u2014to St. Helena Island, Georgia, 1919, and Brownsville, Georgia, c. 1939: \"They used to put the things a person _used last_ on the grave. This was supposed to satisfy the spirit and keep it from following you back to the house.\" The Kongo believe that the deposit of such objects safely grounds the spirit, keeping it from harming the living remains. The arrest of the spirit in last-used objects ( _kanga mfunya_ , literally, \"tying up the emanations or effluvia of a person,\" or in another translation, \"tying up the anger of the dead\") directs the spirit in the tomb to rest in peace and honors its powers on earth.\n\nA similarly spirit-directing \"medicine\" surmounts many tombs in Kongo and Kongo-influenced North America: the image of a white chicken _(nsusu mpembe)_. In Kongo it is thought by some that the powers of \"the white realm,\" the kaolin-tinted world of the dead, are released by the sacrifice of a white chicken. In fact, the placement of images of white chickens on Kongo graves symbolizes the presence of the dead within the honorific whiteness of their realm. Again an image both honors the dead and situates their spirit properly. Thus, a nineteenth-century drawing shows a hen carved in wood placed atop a funeral carriage in Kongo. (It was later laid to rest upon the surface of a tomb.) No later than 1816 the custom had been noted in the Caribbean: \"the white washing of tombs is repeated carefully every Christmas morning and formerly it was customary on these occasions to kill a white cock and sprinkle his blood over the graves of the family.\" And on Sapelo Island, off the coast of Georgia, in 1939, a source reported: \"They kill a white chicken when they have set-ups to keep the spirits away.\" Myriad pressed-glass chickens form _nsusu mpembe_ surrogates on twentieth-century Carolina graves, and upon a modern tomb for a child in western South Carolina, dated 1967:\n\nan enormous white rooster guards the tomb, itself sparkling with a careful covering of white driveway gravel and enlivened with further living touches: a pair of miniature shoes in metal, and a small lamp for mystic illumination, like a night light for the bedroom of a child, who will wake up in glory and walk to God, in silver-colored shoes, feet crunching on glittering white gravel.\n\nAnother mediating force on Kongo tombs is the seashell, believed to enclose the soul's immortal presence. Witness a Kongo prayer to the _mbamba_ seashell:\n\nAs strong as your house you shall keep my life for me. When you leave for the sea, take me along, that I may live forever with you.\n\nKongo seashell imagery lingers lyrically in the words of Bessie Jones, a twentieth-century black artist of St. Simons Island, Georgia:\n\nThe shells stand for the sea. The sea brought us, the sea shall take us back. So the shells upon our graves stand for water, the means of glory and the land of demise.\n\nToday the gleam of seashells illumines and identifies Kongo- and Angola-influenced graves from St. Louis, Missouri, through Algiers, across the Mississippi at New Orleans (Plate 80) to Jacksonville, Florida, and from the United States to Haiti (Plate 81) and Guadeloupe. In Guadeloupe the black custom is especially dramatic, with sometimes entire mounds covered over with large conch shells. But nowhere is this Kongo-American custom more beautifully practiced than in the Carolina low country (Plate 82), where the tombs of both early twentieth-century men and women and black soldiers who perished in the tragic conflict in Vietnam are honored by this ancient Kongo emblem of perdurance. The contrast between such burials and Anglo-American tombs (Plate 83) is obvious and striking. In fact, the Kongo-inspired tradition, in further creolization, wherein a cover of gleaming white shells from the sea is replaced by walls of gleaming white bathroom tile\u2014correlated with purity and water\u2014may explain, at least in part, the phenomenon that both modern Kongo and Haitian (Plate 84) cemeteries boast structures built in this striking medium.\n\nPLATE 80\n\nPLATE 81\n\nPLATE 82\n\nPLATE 83\n\nTrees planted on graves also signify the spirit; their roots literally journey to the other world. Hence Kongo elders plant trees on graves (Plate 78), explaining: \"This tree is a sign of spirit, on its way to the other world.\" The mooring of spirit with trees on graves appears in southern Haiti, where the rationale is phrased this way: \"Trees live after us, death is not the end.\" In the continental United States, at Hazelhurst, Mississippi, we learn that \"at the funeral preachers are given a chance with their carefully composed sermons. It is then that the evergreen is planted on the grave. These trees are identified with the departed, and if the tree flourishes, all is well with the soul.\" In other words, the tree stands sentinel above the grave as the immortal presence of the spirit, an image that graces countless Afro-American burials\u2014for example, those in western South Carolina (Plate 79) or Dallas, Texas (Plate 85), where stone and tree represent together the departed person.\n\nPLATE 84\n\nPLATE 85\n\nAmong other Kongo influences on New World black burials are deposits of lamps to light the way to glory (evoking earlier Kongo bonfires lit on graves for this purpose); bottle-enclosed mounds (mirroring similarly represented miniature enclosures on Kongo tombs) to keep evil from the spirit and to prevent the latter's emanations from wandering back to the living; deposits of pipes of all kinds, representing voyage, through smoke or water, from this world to the next; and modern emblems of spiritual mediation, echoing the thought behind the addition of wheels upon a modern Kongo tomb (Plate 86); the rendering of an automobile in concrete, headlights invisibly blazing for eternity (Plate 87); or a toy metallic airplane, such as the one on a Carolina grave (Plate 88), meant to help the spirit to \"get to heaven fast.\"\n\nPLATE 86\n\nPLATE 87\n\nPLATE 88\n\nPLATE 89\n\nA little-noted fact, taken for granted by casual observers, is that numerous Afro-American traditionally decorated graves are adorned with flowerpots, either _turned deliberately upside down_ and embedded carefully in the soil or set right side up but adorned with green floral paper _turned deliberately inside out_ , revealing the gleam of the inner tinfoil (Plate 89) in intimation of the flash of the departed spirit. Both traditions are gestures to the dead. The inversion of pierced white basins and other vessels is common in many Kongo cemeteries. Indeed, the verb \"to be upside down\" in Ki-Kongo also means \"to die.\" Moreover, inversion signifies perdurance, as a visual pun on the superior strength of the ancestors, for the root of _bikinda_ , \"to be upside down, to be in the realm of the ancestors, to die\" is _kinda_ , \"to be strong,\" \"because those who are upside down, who die, are strongest.\"\n\n## **Bottle Trees**\n\nIn addition to the massive reinstatement of Kongo and Angola traditions of grave decorations in the western hemisphere, there is a persistent Kongo-derived tradition of _bottle trees_ \u2014trees garlanded with bottles, vessels, and other objects for protecting the household through invocation of the dead. Our earliest reference to the custom, dated 1776, is from the coast of northern Kongo at Loango:\n\nAll, after having cultivated their field, take care, in order to drive away sterility and the evil spells, to fix in the earth, in a certain manner, certain branches of certain trees, with some pieces of broken pots. They do more or less the same thing before their houses, when they must absent themselves during a considerable time. The most determined thief would not dare to cross their threshold, when he sees it thus protected by these mysterious signs.\n\nJust fifteen years later, in 1791, the coming of this custom to the black Americas had been detected on the island of Dominica in the West Indies. Thus Thomas Atwood marveled at the confidence of the blacks of the island\n\nin the power of the dead, of the sun and the moon... nay, even of sticks, stones, and earth from graves hung in bottles in their gardens.\n\nPLATE 90\n\nAtwood was one of the first to note, without realizing the full cultural and historical implications of what he was observing, the rise of the Creole tradition of the bottle tree. It flourishes today in the Caribbean\u2014the Berbice area of Guyana, in the Djuka (Suriname maroon) _kandu_ tradition of tying bottles and other objects to trees to protect a house or property from thieves (Plate 90), and on the island of Trinidad, where the custom has been enshrined in a short story by Samuel Selvon:\n\nPLATE 91\n\nWhen the mango tree began to flower, Ma Procup tied colored bottles and animal bones to the trunk.... Roaming boys spotted the mangoes, hesitated when they saw the dangling bones and bottles. \"Obeah,\" one of them muttered, \"I not climbing that tree.\"\n\nIt is, however, in the United States that most Kongo-derived bottle trees are to be found\u2014from the Sabine River area in eastern Texas to the coast of South Carolina, with concentrations in Virginia, southern Arkansas, northern Mississippi, and southeastern Alabama. Eudora Welty has described a variety of them:\n\na line of bare crepe-myrtle trees with every branch of them ending in a colored bottle, green or blue [Plate 91] there could be a spell put in the trees, and she was familiar from the time she was born with the way bottle trees keep evil spirits from coming into the house\u2014by luring them inside the colored bottles, where they cannot get out again.... Solomon [a black man] has made the bottle trees with his own hands over the nine years... sometimes the sun in the bottle trees looked prettier than the house did.\n\nToday the function of similar expressions presiding over Kongo graves (Plate 92) is the blocking of the disappearance of the talents of the important dead. Lifting up their plates or bottles on trees or saplings also means \"not the end,\" \"death will not end our fight,\" the renaissance of the talents of the dead that have been stopped, by gleaming glass and elevation, from absorption in the void.\n\nPLATE 92\n\nPLATE 93\n\nIn Mississippi these trees (Plate 93), shorn of life, bearing cold, glittering bottles\u2014visual statements, again, of death and arrest of the spirit\u2014simply block or ward off evil. The custom compares with that in Texas, where \"grave glass will keep the 'evil spirits away' or 'keep away the man's spirit.' \" In that sense Afro-American bottle trees are fugitive specters from a graveyard realm, just as bottle-lined burials are horizontal bottle trees.\n\nPLATE 94  \nCourtesy Smithsonian Institution, Washington, D.C.\n\n## **Coda: Two Afro-American Artists**\n\nThe same cultural forces that sparked the reemergence of the Kongo _nkisi_ tradition in the New World may well account for the medium and the glitter of the late American artist James Hampton's _Throne of the Third Heaven of the Nations Millennium General Assembly_ (Plate 94). _The Throne of the Third Heaven_ represents a unique fusion of Biblical and Afro-American traditional imagery. Lynda Roscoe Hartigan has closely studied this amazing monument, a complex of a hundred and seventy-seven paired architectural pieces set about a central throne crowned with the legend FEAR NOT. She argues inspiration from the Book of Revelation and she is right\u2014therein the Son of Man says, \"Fear not... fear none of those things which thou shalt suffer\"; therein is described a throne set in heaven, with four and twenty seats about the throne; elders clothed in white and crowned in gold, beasts full of eyes and four of these with six wings full of eyes within. It is a vast and intimidating messianic vision. But the genius of James Hampton was to take and abstract these images\u2014seated elders in white and gold, and wings with eyes\u2014and fuse them with the throne itself. The winged furniture of another world therefore appears before our eyes, informed by flash and poetic flight.\n\nRevelation did not tell James Hampton _how_ to impart glitter to his sacred throne. For that he relied on Kongo-American modes of decoration, covering curved or spherical forms, light bulbs and jars of glass, and cast-off furniture with gold and silver foil. The selection of tinfoil, as the covering of the sacred, strikingly recalls the foil-wrapped flowerpots, glass jars, and coffee tins on graves in Louisiana, Carolina, and Florida; the metallic cloth on _pacquet congo_ in Haiti; the insertion of tinfoil to attract spirit into a charm; and many other sources.\n\nHampton was born in 1909 and reared in Elloree, South Carolina, in the heart of an area of strongly decorated Afro-American graves. As he made his sacred furnitures, c. 1950\u201364, in Washington, D.C., apparently for the founding of a religion he never lived to practice, his genius was to shape the images of Revelation as if they were vast receptacles of the spirit, wrapped in gold and silver tinfoil, for siting on a Carolina tomb and perpetually suggesting, in the sparkle, the attainment of God's glory.\n\nWhile James Hampton was piecing together his _Throne of Third Heaven_ , Henry Dorsey, a stonemason of Brownsboro, Kentucky, was transforming the walls and reaches of his property with decorations of celebratory humor (Plate 95). He fashioned a series of interlocking material puns on mediation and transcendence by recombining objects taken from their original industrial functions, thus giving them new meanings. He playfully related one kind of motion to another. His was a consciousness determined in part by knowledge of the structure of traditional Afro-American graves. Dorsey shaped his decorations as a gift, a proffered personal idiom of transcendence, fashioned out of plumbers' pipes and plastic dolls and wheels and doors and axles and pulleys and tires and even an upside-down agitator from a washing machine (Plate 96).\n\nThe pulsating metaphors of this playful universe were gathered in frequently climactic combinations, e.g., a man, under a wheel banner, who is forced to leap onto a horse through a giant hoop under a tiny wheel into a cabin with a doll over a door opening on a phantom booth (Plate 97). Or: a doll on a horse on a stafflike metal stem appears to gallop through the branches of a tree, with a stray tire, hung on other branches, as on a bottle tree. Henry Dorsey may never have heard of Mbanza Kongo or Loango\u2014he did not have to. I suggest that he was their progeny by virtue of the culturally open and responsive spirit of his imagination.\n\nWhen Henry Dorsey hoisted a liquor jug and fan blades (Plate 98), he was not only quoting, indirectly, the image of the bottle tree but adding new dimensions to the form to satisfy his restless mind and to entertain his neighbors. He was vaunting his control of the forces of motion in the modern world. Like the spectral pulleys, wheels, and switches on a modern Kongo drum, he brought together a liquor jug, the blades of an electric fan, and a metal disk to form a material constellation of objects divorced from their functions in the real world and given different meanings by his imagination. With such constructions he begged the questions \"What does it _do?_ \" and \"What does it _run_ on?\"\n\nPLATE 95\n\nPLATE 96\n\nPLATE 97\n\nPLATE 98\n\nThe sculptures ran on combinatory wit and humor, not on electricity (though, to be sure, toward the end of his career, Dorsey took the plunge into electrokinetic art, characteristically mastering fully the intricacies of electrical wiring). The operating currents were, in fact, traces of the theory of correspondences that shaped his mind. What his works did, as in Kongo when seed and claw were commingled in _minkisi_ , was to send to spirits punning messages of positive execution. Dorsey, for example, devised a shining white metal hubcap with images of Jesus, Mary, and Joseph that whirled. The birth of Jesus, the expansion of Christianity throughout the world, were conjured up in the flash of moving chrome. Brilliant gyres in motion were to Henry Dorsey what winged silver thrones were to James Hampton, what mirror-embedded constructions were to black Georgians and the Bakongo.\n\nHenry Dorsey was born the grandson of Maria Prewitt, one of the most famous black women of Oldham County, Kentucky, who lived for a hundred years. She was born a slave in the early 1820's and died in 1924. Her second husband was Louis White, a black man. One of their children was Alice White, who married Clarence Dorsey, a black farmer of Brownsboro. Their son, Henry, was born in 1897.\n\nIn the summer of 1922, while Henry was working in the kitchen of a tearoom on Fourth Street, in Louisville, improper ventilation caused the room to overheat terribly. Dorsey fainted. When he regained consciousness in a hospital, he found himself well-nigh tone-deaf. As if to compensate for the nightmarish loss of one of his senses, Dorsey set off on an odyssey of escape and self-rediscovery. He was twenty-five years old. He rode the rails, as a hobo, as far west as San Francisco. He worked his way south as a roustabout on a steamboat, down the Mississippi to New Orleans. From New Orleans he traveled east, working on the railroads of Mississippi and Alabama. He sometimes saw, in the segregated South of the 1920's, things cruel and horrible:\n\nHenry, he saw a black man have a falling-out with the boss about something, and the boss take this man on the hand-car with him, killed him, and buried him near the railroad in the woods, and come back on the hand-car all alone. I'll tell you, he seen awful tough things in his travels.\n\nHis travels through Louisiana, Alabama, and Mississippi took him through bottle-tree territory, past cemeteries gleaming with the traditional deposits, past black houses with automobile tire sculptures in their yards, past other fleeting images of artistic motion redistilled in the folk propensity to sing of bluesy strains of love and yearning. Such was the grief and joy of being black in the U.S. South in the 1920's. And then one day, just before the Great Depression, Dorsey came back home. He had trained in Brownsboro as a stonemason and, despite the Depression, \"was lucky enough to have as much work as he could do.\" Dorsey married Laura Johnson, a black woman, and lived in the house that his father had inherited from the Dorsey clan. Henry and Laura had four sons and two daughters. By 1956, approaching sixty and feeling his infirmities, he showed his brave and generous nature by countering his physical ailments with images suggesting important cheerful matters. These deliberately entertaining figurated constellations, in iron and plastic and other media from the industrial West, were to occupy him for the rest of his life. He died November 8, 1973. His wife, Laura, died soon thereafter. And the decorated house they lived in remained enigmatic to their neighbors. Few had walked around the house, seen it whole, put Dorsey's labors in cultural perspective.\n\nIt was out of an interaction of house and cemetery, clan and person, that the art of Henry Dorsey emerged. The key to his intent was a sign by his door that said, \"You are welcome as the flowers of May\" (a tag line Joyce himself used in _Ulysses:_ \"Alo! _Bonjour_ , welcome as the flowers of May\").\n\nDorsey taught us to master things rather than to complain about them, to subdue them with artistic weapons of humor and generosity. This is instanced by the care with which he traced the dates of the birth of his own children on a concrete tablet discreetly recessed into a wall on the chimney behind his house. (Plate 99). The tablet shows, for example (Plate 100), that his late son, Zack, was born on 29 May 1930, with the citations of the other children's initials and dates of birth. The _D_ in the Dorsey of Zack's name is decorated with a tiny commemorative white piece of shell; the _D_ in the name of a surviving daughter, Costella A., is similarly embellished. The tablet attests to his historical consciousness. In the beginning he felt these things privately, behind his house, but then, c. 1956, he felt the need to counter his existential void with celebrations of names in a far more assertive and public manner. He began to decorate the front of his house for the entertainment of the world.\n\nPLATE 99\n\nPLATE 100\n\nThe decoration of his house with motifs of industrial detritus can be partially characterized as an idiosyncratic act, and partially as an appreciation of the sensibility implicit in the embellishment of graves with porcelain objects and of trees with bottles.\n\nThe evidence of Dorsey's contact with these traditions exists within his own county, at least with respect to burials; and his travels through black Mississippi and Alabama sharpened his consciousness forever and told him, visually, who he was and where he was coming from.\n\nThe old black families in Oldham County had buried their loved ones with china figurines, last-used cups and saucers, and other material representations of spirit. In 1967\u201368, when the builders of Interstate Highway 71 sliced through one of the oldest black burial sites\u2014the very cemetery in which Alice White Dorsey, mother of Henry, had been laid to rest\u2014the tombs were moved. Guy Dorsey, a surviving brother, remembered: \"When they come, they took up graves, took up dolls, plates, cups, china, all sorts of things. Put 'em in boxes 'n buried them with the bodies at Peewee Valley.\" Eva Dorsey Williams, Henry's sister, added: \"It broke my heart, for Mother to be moved that way.\"\n\nSo Henry Dorsey knew all about black burials. He had buried his own father, Clarence Dorsey, and carved the latter's headstone (Plate 101) and dated it 1937. The design of this stone\u2014sphere over sphere\u2014provides us with a master icon.\n\nThe theme of mirrored circles informs one of his early motion-sculptures, dating from the 1950's (Plate 96). It is a tractor tire that has been transformed into a tondo, framing ice trays clustered about an upside-down washing-machine agitator to form a never-never propeller. The precision and intricacy of these interlocking metaphors of motion reveal a genius for parallel construction.\n\nIn the genial decoration of the ample grounds of Dorsey's house there were multiple hints\u2014the suspension of metal plates on iron trees of pipe (Plate 98) and the making of a garden of hubcaps, suspended like metal zinnias on metal stems\u2014that he had seen and was impressed by the flash of bottle trees in the South.\n\nPLATE 101\n\nHis charm reflected resistance to suffering, through visual contentments and splendidly humorous elaborations. This was made clear by his reaction to the rejection by his brothers and sisters of a headstone that he made in 1963 with his own hands for his late and beloved sister, Bernice. The other members of the family quietly purchased a commercially rendered granite marker and placed it on her grave. Characteristically, Henry did not dwell on the hurt, did not destroy his headstone in a fit of meaningless pique. Instead he took the handcrafted headstone and set it up, in a corner of his yard, atop a boilerlike coffer of iron (Plate 102), a mock tomb anchored on his earth. He flanked this \"grave\" with an abstract silhouette of a locomotive\u2014not unlike the isolated wheeled structure that haunts the funeral processionists of Clementine Hunter's c. 1948 painting, _The Funeral on Cane River_. Dorsey's locomotive is half edifice, half vehicle, for it has only one wheel and is rooted to the ground with miniature supports.\n\nPLATE 102\n\nPLATE 103\n\nA train beside a tomb leads us to a master metaphor: his house as spirit-train. The placing of tractor wheels at certain corners of his house and along the edges of his property implied that the earth itself was wheeled, could be mentally set in motion. Indeed he set his name in motion many times over in the decoration of his house's fa\u00e7ade (Plate 103). He wrote his initials, \"H.D.,\" in pierced green and yellow bleach bottles, threading them, like giant beads, on pipes. He placed this dazzling projection above another act of self-commemoration, a frieze proclaiming:\n\nHenry Dorsey\n\n1956 PRODIGAL SON ALICE'S BOY\n\nAnd then he set this above paired wheels. The point where the frieze is dated \"1956\" is marked with a cluster of seashells (Plate 104). The symbolic shells, lifted out of Afro-American funerary context, become doubly death-defying, a celebration _before_ his death of the permanence of his name. PRODIGAL SON refers to his travels in the South, ALICE'S BOY to coming home. His life is unified and framed with shining shells. As his good-natured vision ripened, Dorsey advanced his art.\n\nHe mastered electrical work. Now he was working with actually moving pulleys, wheels, and wires. Now he was rendering explicit what was once implied. By this time, the late 1960's, writers had come to his door, attracted by the continuous notices of his work in the press of Louisville:\n\nHe seems to have a complete understanding of the maze of electrical cords and fan-belts and bicycle-wheels-made-into-flywheels. He fashioned the thing out of empty space, defining it with cast-off icons. He built it and he certainly knows its music.\n\nHe was painting metaphoric motion, an aspect of the black imagination that led James Hampton to fasten wings on every major portion of _The Throne of the Third Heaven_ , that inspired black musicians to add the sound of disappearing trains to complete an atmosphere of yearning in their blues. Hampton set the rubric FEAR NOT above his composition; Dorsey sited a smiling clown head, a figure of laughter and self-effacement, high within his series of motion-compositions, like a benediction.\n\nPLATE 104\n\nDedicating himself to work that was play, to labor that was festive, displaying his art in a communal round, Dorsey rescued objects thrown away by persons trained to see only single functions in them, recycling them in a deeper sense (Plate 105).\n\nMy appreciation of Henry Dorsey and his art in this chapter on Kongo and Angola influence on New World art is a declaration of marvelous things to come, as black artists everywhere awaken to the implication of this current in Atlantic black art history. The tone of Dorsey's work, his concern with righteous mediation, reflects continuous contact with the _mambu_ (matters) of the graveyard\u2014one block down the street on which he lived. His ancestors had indicated ways of realizing spiritual transcendence through material metaphors. To the degree that he extended their insights he extended the Kongo Atlantic world.\n\nThe house of Henry Dorsey remains a secular _nzo a nkisi_ , a charm for the denial of hurt, for the redirecting of spirit, to greet provocateurs with laughter and generosity, teaching us how to endure, how to bestow honor, even as the ancient Greeks taught their progeny to honor the gods, parents, and strangers.\n\nHenry Dorsey discovered reflections of the Kongo _nkisi_ tradition on his own. The man in touch with his origins, so the Bakongo say, is a man who will never die _(mu kala kintwadi ya tubu i mu zinga)_. Thus the house of Henry Dorsey stands, at the end, as an intimation of the meaning of existence in terms as the Bakongo might have phrased them: If you know where you are going, and where you are coming from, you can decorate the way to other worlds\u2014the road to the ancestors and to God; and your name will merge forever with their glory.\n\nPLATE 105\n\n# **Three**\n\n# **THE RARA  \nOF THE UNIVERSE**\n\n# **_Vodun_ Religion and  \nArt in Haiti**\n\nVoodoo, according to the _Oxford English Dictionary, is_ \"a body of superstitious beliefs and practices including sorcery, serpent worship and sacrificial rites, current among negroes and persons of negro blood in the West Indies and Southern United States, and ultimately of African origin.\" Superficially understood by Westerners since the eighteenth century, voodoo _(vodun)_ has been reviled as abominable primitivism and vulgarized and exploited in countless racist books and films. _Vodun_ , which was first elaborated in Haiti, however, is one of the signal achievements of people of African descent in the western hemisphere: a vibrant, sophisticated synthesis of the traditional religions of Dahomey, Yorubaland, and Kongo with an infusion of Roman Catholicism. What is more, _vodun_ has inspired a remarkable tradition of sacred art.\n\nFrance took formal possession of the western third of the Caribbean island of Hispaniola in 1697. Soon thereafter this part of the island\u2014modern Haiti\u2014developed a vigorous indigo and sugarcane plantation economy. Western Hispaniola, built on the backs of black slaves, became the most profitable French colonial possession in the world, necessitating an enormous increase in the importation of slaves. In 1697 there had been three Europeans to every African in western Hispaniola, but a hundred years later the proportion had radically changed: there were approximately eleven blacks to every white.\n\nThe men and women of African descent who populated Haiti came primarily from Kongo and Angola, but also from Dahomey, Yorubaland, Bamana, and Mande territories in West Africa, with assortments of Igbo, too. As C. L. R. James remarks in _The Black Jacobins:_\n\nTwo-thirds of the population of French San Domingo (at the commencement of the Haitian revolution in 1791) had made the Middle Passage. The whites had emigrated or been exterminated. The Mulattoes who were masters had their eyes fixed on Paris. Left to themselves, the Haitian peasantry resuscitated to a remarkable degree the lives they had lived in Africa. Their method of cultivation, their family relations and social practices, their drums, songs and music, such art as they practiced, and above all their religion which became famous, _Vodun_ \u2014all this was Africa in the West Indies.\n\nActually, _vodun_ was Africa _reblended_. The encounter of the classical religions of Kongo, Dahomey, and Yorubaland gave rise to a creole religion. This religion has two parts: one called _Rada_ , after the slaving designation for persons abducted from Arada, on the coast of Dahomey, itself derived from the name of the holy city of the Dahomeans, Allada; and the other called _Petro-Lemba_ , or simply _Petro_ , after a messianic figure, Don Pedro, from the south peninsula of what is now Haiti, and the northern Kongo trading and healing society, Lemba.\n\nChiefly from Dahomey and western Yorubaland derived the _vodun_ worship of a pantheon of gods and goddesses under one supreme Creator\u2014deities who manifested themselves by possessing (\"mounting\") the bodies of their devotees. This aspect of _vodun_ was reinforced by contact with French services for Roman Catholic saints who were said to work miracles. Chiefly from Kongo and Angola derived _vodun_ beliefs in the transcendental moral powers of the dead and in the effectiveness of figurated charms for healing and righteous intimidation.\n\nBoth _Rada_ and _Petro_ partake of these sources of African influence; neither is traceable to just one source. Both are at once African-inspired and indigenously created. _Rada_ , predominantly Dahomean and Yoruba, is the \"cool\" side of _vodun_ , being associated with the achievement of peace and reconciliation. _Petro_ , predominantly Kongo, is the hot side, being associated with the spiritual fire of charms for healing and for attacking evil forces. The great Haitian painter Andr\u00e9 Pierre, himself a _vodun_ priest, has called _Rada_ \"civilian,\" _Petro_ \"military.\"\n\nIt is important to stress, however, that the two fundamental _vodun_ sections fused similar religious aspects of different African cultures. Thus, the \"hot\" sorcerous potentiality of an otherwise cool Yoruba riverain goddess was reassigned to the _Petro_ side of deities. Correspondingly, the cool, creative Kongo _simbi_ spirits were lifted from the realm of Kongo-inspired \"attack\" charms and reassigned to _Rada_ , where their positive powers were akin to those of the gods and goddesses of Dahomey and Yorubaland.\n\n## **Dahomey: A Distant Paradigm**\n\nThe Dahomean kingdom once flourished in what is now the Benin Republic between Togo and Nigeria on the west coast of Africa. It lacked vast rivers and mountain ranges and therefore was accessible to Yoruba migrations from the east. From Tado, west of modern Abomey, the Adja branch of the ancient Yoruba migration from present-day Nigeria moved south. Around the fifteenth century, according to tradition, Aligbonon, the King of Tado's daughter, while seeking water in a forest, met and was made love to by a leopard spirit; the mystic union resulted in the birth of Prince Agasu, legendary ancestor of all the Fon of Dahomey, whose name is remembered in Haiti.\n\nAgasu's descendants founded the holy Dahomean city of Allada. About 1600 three of his sons contested the throne of the sacred town. The eldest won; the middle son took the throne of Ajase-Ipo (modern Porto Novo); and the youngest, Do-Aklin, trekked north and founded Abomey, capital of Dahomey.\n\nBy 1700 the French had established a permanent slaving base on the Dahomean coast at Ouidah. The link connecting Dahomey and the West Indies, especially western Hispaniola, had been forged. The slave trade intensified the Dahomean warrior way of life. Quickly Abomey emerged from relative obscurity to become a major power in West Africa, with an efficient army, a stable cowrie-shell currency, a strong balance of trade, and firm control of political and social affairs. Defense and economic welfare provided a rationale for military expansion. Abomey struck south, conquered Allada in 1724, and three years later reached the Atlantic Ocean, seizing the trade in European firearms that flourished on the coast. Already Abomey had captured or was raiding Mahi, Savalu, Ketu, and the Anago Yoruba.\n\nPaul Mercier underscores an important aspect of Dahomean cultural history\u2014the transformation of the deities of immigrants and of conquered peoples into Dahomean spirits. The cultures of the conquered\u2014Mahi to the north, Ketu and Anago Yoruba to the east\u2014were fairly close to the Dahomean way of life. Thus their gods and goddesses were assimilated by the Dahomeans.\n\nHowever, the deities of the Yoruba had already made their presence felt in Dahomey over hundreds of years. Yoruba deities were served under different manifestations in Allada before 1659. Therefore, the Abomey conquests brought together Yoruba deities already transformed into Ewe and Fon local spirits, in addition to deities from Ketu and Anago Yoruba. The encounter of Ewe-Dahomean spirits with pure Yoruba _orisha_ in Haiti produced still another synthesis of Yoruba-descended religious practices that, in the course of Dahomean history, had become separated from one another. Fusion and refusion of Yoruba spirits, first in Dahomey and then all over again in Haiti, go a long way toward explaining the phenomenon of multiple avatars of the same Dahomean-Yoruba god. It also helps explain the persistence of the concept of the _orisha_ in the black New World.\n\nThe very Afro-Haitian term for spirit, loa, encapsulates the subtle nature of the syncretions that took place. In Abomey, deities are called _vodun_ (mysteries); in Yoruba, diviner-herbalists are called _babalawo_ (father-of-mysteries), a term creolized by Haitians into _papaloi_ (the name for a _vodun_ priest) through ingenious Afro-Gallic punning. The Haitian words for deity, _loa_ or _myst\u00e9re_ , therefore appear to derive from the Yoruba _l'awo_ for \"mystery.\" The interrelationships binding the gods of Dahomey to the Yoruba pantheon and both to the _loa_ of Haiti are fascinating to behold:\n\nNote that two Ewe-Fon deities merge into a single Haitian spirit, Aizan Velekete. Continuity by combination is a process that throws light on the apparent disappearance of Yemoja, Oya, and Oshoosi in Haiti. Their attributes may have been absorbed in iconographically similar cults.\n\nWith the coming of deities shared by Ewe, Fon, and Yoruba to Haiti, the stage was set for their involvement with the religions of Kongo and Roman Catholicism.\n\n## **Dahomean Influences on Haitian Sacred Art**\n\nThe Dahomean war deity G\u016b was destined to live momentously in an alien land peopled primarily by persons brought from Kongo and Angola. On the Kongo-influenced side of _vodun_ in Haiti, he became the _Petro_ spirit Og\u00fan-Bonfire, or Ogun-of-the-Blazing-Torch. Dahomeans know G\u016b as the personification of iron's cutting edge, which exists in the blade of a razor, in the slicing force of machetes, in the piercing jab of an iron-tipped spear. G\u016b's own sword is represented by one of the monuments of Dahomean iron, the ceremonial blade called _gubasa_ (Plate 106). The image of the openwork disk of the sun relates this weapon to the sky. The master smith who fashioned it in Abomey, probably in the late nineteenth century, attached to its side more than forty miniature iron implements\u2014tiny swords, cutlasses, guns, arrows, hoes, hooks, lances. Arrows thrust out from the cutting edge, like myriad spitting vipers, intensifying the killing aura of the instrument.\n\nPLATE 106\n\nA masterpiece of Dahomean brass-smithing (Plate 107) enlarges our comprehension of this dread power: an image of G\u016b, the iron god himself. This striking figure has been tentatively attributed by Abomey elders to a court brass-smith of Kings Ghezo and Glele, Azidji. When this image arrived in Paris, near the end of the nineteenth century, it was still dressed in its complement of ritual garments (Plate 108), a brass hat for Age (the Dahomean god of the hunt), a cloak in Dahomean-made raffia cloth, a brass pendant and silver charm, and a loincloth of the kind Dahomeans and their Yoruba neighbors associate with forest cultivators, hence suggestive of brute forest energies. The image brandishes with both hands the _gubasa_.\n\nThe bite of iron is matched by the snapping of a lion's jaws in another striking form of Dahomean visual praise (Plate 109). In this example, a lion rampant carved in wood, attributed to the great nineteenth-century court carver, Sosandande Likohin Kankanhau, monumentalizes Glele's famous praise name: \"When the lion shows his pointed teeth, he terrifies the world.\" Fragments of this visual lore of war were brought to Haiti by captives from the Dahomean region and fused in a new cult of the god of war. United with Og\u00fan, the Yoruba god of iron, from whom G\u016b himself originally derived, G\u016b became Papa Og\u00fan. Papa Og\u00fan was, in turn, associated with the warrior saints of the Roman Catholic Church.\n\nIn Haiti, visual representations of the saints of the Roman Catholic Church were viewed with informed sympathy by the blacks. In such imagery they perceived\u2014unbeknownst to the whites\u2014ties to truths they already knew. They noted striking parallels in the lives and attributes of the saints, the _vodun_ of Dahomey, and the _orisha_ of Yorubaland.\n\nThe Church distributed among the slaves, who were forcibly baptized by law, inexpensive woodcuts and lithographs of the saints, demonstrating, didactically, their individual attributes. These were potent images indeed for minds informed by the visual cultures of Dahomean _vodun_ , West Yoruba _orisha_ , and Kongo _minkisi_. In the course of supposed Westernization, Haitians actually transformed the meaning of the Catholic icons by observing their similarities to African spirits. Haitians restructured the identity of the saints of the Catholic Church in terms of their own religious language. Consider Saint James. Michel Leiris remarks:\n\nPLATE 107\n\nPLATE 108\n\nPLATE 109\n\nBecause of a secondary figure, the picture of Saint James the Great [Plate 110]... will receive a double interpretation. The picture shows Saint James on horseback with sword and shield, fighting the Infidels, and escorted by a knight in armor, bearing a red standard with a white cross. All my informants agreed that the principal figure is the blacksmith and warrior god, _Ogun Ferraille_... (the essential attribute of which is a saber and, together with the other Ogun, the color red) but according to some, the figure in the background is _Ogun Badagri_ , the brother of _Ogun Ferraille_ , while others identify him rather as a guede, or graveyard spirit; this is because of the lowered visor of the helmet of the figure in question, which seems to recall the chin-cloth and other attributes of a corpse.\n\nThe vision of _killing by iron_ observed within the lithograph distinctly fits the martial paradigm of Dahomean G\u016b. And just as the _gubasa_ is the central sign of war and smithing in Dahomey, so the _saber_ became the chief icon of Og\u00fan Ferraille in Haiti, in a new and wondrous context, often flanked by honorific banners, symmetrically displayed and inclined, as if nodding in honor of Lord G\u016b.\n\nThe shape of the shafts of the important flags that traditionally flank the sword of the saint in ground-paintings (and the sword of the master-of-ceremonies in _vodun_ dancing) extend a little-noted accent of militaristic assertion. The flags' shafts reverse the S-curved saber's handguard; they are cryptic swords of cloth, following and flanking the lord of the cutting edge, even as a white cross on a field of red in the chromolithograph accompanied the warrior saint.\n\nIn many instances, then, swords and flags are compared and the enlargement of the original meaning of _gubasa_ has been effected by new visual influences of the iconography of Catholicism on the Afro-Haitian aesthetic.\n\nWhen a person becomes possessed by the spirit of Papa Og\u00fan Ferraille, an iron bar, standing in the earth near the altar dedicated to his name, is heated in a fire. The possessed person will then take this staff or some other equally heated bar and dance with it in his or her bare hands to prove that the possession is genuine. Deep mastery of self is the point of the play with fire and heat.\n\nPLATE 110\n\nAn artistic monument of Dahomey also involved with resistance and self-restraint is the circular pendant (Plate 111) symbolizing steadfastness, which the great brass figure of the iron god, now in the Charles Ratton collection in Paris, once wore against his abdomen. It is a disk of brass to which a snarl of brass wire has been attached. Behanzin himself is supposed to have worn this pendant about his person to invoke the spirit of his father, King Glele. The pendant is a praise name of Glele materially rendered: _Togodo_ , The-Circle-of-the-Earth (literally, \"round thing\") a word that connotes a deeper meaning, namely, that Glele, the king (the earth), resists, without moving, the hot, dry, northern winds. The sphere is the earth, and the tangled skein of wire the winds\u2014all of which glosses one of the strongest names for King Glele, \"one knows not how to raise the earth aloft,\" i.e., the king is a force that can never be dislodged. The resistance of earth to wind cosmologizes the king's collectedness of mind and steadfast sense of mission. Glele's intimidating presence promised peace and order. In the realm of zoology the same concept is conveyed by the image of a frog (Plate 112), whose presence correlates with water's coolness, hence with peace imposed by statecraft that is strong and just. These were ancient ideas in Dahomey, long predating their particular association with nineteenth-century kings.\n\nPLATE 111\n\nPLATE 112\n\nPLATE 113\n\nPLATE 114\n\nAnother animal present in Dahomean art\u2014D\u0101 or Dan, the good serpent of the skies\u2014appears not only in Haiti but also in Cuba (Plate 113), and, in mixture with the Yoruba rainbow deity, Oshumare, in Brazil (Plate 114), that is, wherever the Fon and their neighbors arrived as captives.\n\nThe highest deity of the Fon, Mawu-Lisa, combines female (Mawu) and male (Lisa) valences. Mawu is cool and gentle\u2014she is the moon. Lisa is strong, tough, fiery\u2014he is the sun. Their union represents a Fon ideal.\n\nThe good serpent of the sky, D\u0101, is a metaphor for this primary, combinatory sign of order. Like Mawu-Lisa, D\u0101 combines male and female aspects, and is sometimes represented as a pair of twins. Many are his avatars, but principal among them is Da Ayido Hwedo, the rainbow-serpent. Coiling a resplendent bichromatic body about the earth, D\u0101 shaped its globelike form and sustained its balance and existence. Color symbolism in the lore of this rainbow-serpent is potent and direct: \"The male is in the red portion, the female in the blue.\" Aggression and compassion are thus writ large across the skies.\n\nIn one Dahomean myth recorded by Mercier, Da Ayido Hwedo set up four pillars cast in iron at the four cardinal corners of the earth. He did this to hold aloft the sky. And then he twisted around these columns in brilliant spirals of crimson, black, and white to keep the pillars upright in their places. These were colors of night (black), day (white), dawn and twilight (red). Clearly the iridescence of the sacred serpent signified to Dahomeans more than danger. It becomes a perfect metaphor for mind's own ordering motion.\n\nNeedless to say, Dahomean-influenced folk in Haiti, steeped in such lore, were impressed by the chromolithograph of Saint Patrick (Plate 115), dressed in full regalia, and shown at a critical juncture, driving the snakes from Ireland. They saw an elder, white of beard (things white are African attributes of D\u0101), making a gesture of power with his right hand. Instead of a saint treading vermin, banishing all evil\u2014exorcism\u2014they saw multiple embodiments of the serpent of the sky. The proof of this assertion, as will be seen, is how they rephrased this chromolithograph to fit their own religious language (Plate 116).\n\nPLATE 115\n\nIn addition, it can be imagined what an elder from Kongo thought when confronted with such a picture: here was an obvious ritual expert, with a staff of mediatory power, communing with forces incarnate in amphibian reptiles at the watery and grand boundary between the worlds of the living and the dead. Fon titles for this spirit are D\u0101, Dan, and Dan Bada, but the creole name in Haiti is Damballah. This overlaps, in form and meaning, the Ki-Kongo word for flatheaded rainbow-serpent, _ndamba_. MacGaffey elaborates:\n\nPLATE 116\n\nNdamba puns on the Ki-Kongo word for \"to sleep\" in the sense of the ecstatic love-making of two serpents, one male, the other female, who wrap themselves around a palm tree to carnally unite.\n\nThis is remarkably similar to the image summoned by calling the names of Dan and his wife, Ayida Hwedo, in Afro-Haitian _vodun_ and identical with the image used in _vodun_ blazonry to call and designate these spirits.\n\nV\u00e8v\u00e8 ground-painting for Damballah and Ayida Hwedo, Port-au-Prince, twentieth century.\n\nThus, in a commonsensical way, the iconographies of two classical African religions came together: the center post, a ritual site around which _vodun_ devotees dance, is often painted with rainbow stripes or embellished with relief representations of serpents intertwined.\n\n## **Kongo Influences on Haitian Sacred Art**\n\nMoreau de Saint-M\u00e9ry wrote in the eighteenth century regarding a certain _danse \u00e0 Dom P\u00e8dre:_\n\nIn 1768 a black man of Le Petit-Goave [a village on the north shore of the southern peninsula]... taking advantage of the credulity of the blacks with superstitious practices, gave them the idea of a dance analogous to that of the _vaudoux_ , but in which the movements were more sharp and sudden. To give the dance even more of an effect, the blacks mixed well-crushed gunpowder in the cheap rum they drank while dancing... this dance, called Danse \u00e0 Dom P\u00e8dre or simply Dom _P\u00e8dre_ has caused the death of certain blacks....\n\nIn this account, \"superstition\" is mixed with substance. Saint-M\u00e9ry believed that the Dom P\u00e8dre \"dance\" began in 1768, and literally endangered those who danced it. Contesting this image, Melville Herskovits, the American anthropologist, wondered whether _Petro_ spirits derived in fact from a species of African-derived ancestorism, for a Haitian black had told him that he believed _Petro_ was an intimidating _vodun_ priest who had died with the spirit of a god not properly released from his head. Evil and positive divinity therefore lived on within the spirit of this person, perpetually commingled.\n\nActually, lost here was not paradise but understanding. A ritual expert in Kongo today would have little difficulty in explaining plausibly the significance of these hazily remembered legends: Dom P\u00e8dre was a messianic figure or an ancestor come back from the grave to renew his usefulness to the living in the body of a priest or healing charm. In fact, there are figurated healing bundles, or _pacquet congo_ , in Haiti today that bear Dom P\u00e8dre's name. The great ethnologist, Alfred M\u00e9traux, writing in that vein of scientific skepticism tempered with artistic charm that was his alone, could sense\u2014without identifying them\u2014sub-Saharan origins in the legend of Dom P\u00e8dre:\n\nIn the word _petro_ we recognize the Pedro discussed above. Only a very naive person could believe that the complicated (petro) liturgy, which is inseparable from the worship of the petro divinities, could have been introduced by one man, however inspired... In contemporary Voodoo, Domp\u00e8dre is a powerful god who is normally greeted by the detonation of gunpowder. And so it seems certain there must have once been a _hungan_ whose impact was so profound that his name took the place of African \"nations\" who today worship gods bearing his name, _petro_ , and not theirs.\n\nToday additional evidence suggests that those \"African nations\" lie within Ki-Kongo-speaking Central Africa.\n\nIn some parts in the north of Haiti, people sometimes call _Petro_ Lemba, the name of an old and most important Kongo trading society. The doyen of Afro-Haitian studies, Jean Price-Mars, stressed the Lemba- _Petro_ equivalence thirty years ago. If the same cycle of rites is called by a Kongo name\u2014a creole term in the south\u2014in northern Haiti, then _Petro_ must be a rite, not a person; a concept, not a name.\n\nThe fiery militance of the _vodun_ realm was fully evident in 1768 when first contrasted with the idealized coolness of many of the deities of Dahomey. _Petro_ ritualized aggression would seem logically to derive from the spiritualized militancy pervading the world of \"attack medicines\" of Bas-Z\u00e4ire and neighboring territories. The earliest mention of _Petro_ associates the rite with sharp, staccato motions and the tasting of gunpowder\u2014practices sharpened and confirmed by time. Today _Petro_ dancing is defined as \"distinctively intense, almost nervous.... Dancers, instead of riding the beat, as in Rada and Ibo, seem to be running in front of it, as if the beat were whipping them forward.\"\n\nThe gestures of a _Petro_ spirit, filling momentarily the body of a possession priest or priestess, are characteristically stern, hard, fierce:\n\nWhereas Erzulie, the Rada Goddess of Love... is concerned with love, beauty, flowers, jewelry... liking to dance and be dressed in fine clothes... the figure of Erzulie-Red-Eyes, on the _Petro_ side, is awesome in her poignancy. When she possesses a person, every muscle is tense, the knees are drawn up, the fists are clenched.\n\nHere is a spectacle that recalls possession by _bisimbi nganzi_ in northern Kongo, spirits of those who died violently. The Haitian propensity for re-creating and marshaling these incandescent forces is based on a belief in the spirits of _Petro_ 's power to make things burn in a positive healing sense; their flames, their whips, their exploding charges of scattered gunpowder are summoned when cooler _Rada_ cures have failed. _Petro_ altars, in the context of the _houmforts_ (the religious centers of _vodun_ ), powerfully reflect this notion of salvation through extremity and intimidation.\n\n_Houmforts_ betray more than a hint of the spatial arrangement of many compounds in Central and West Africa: the construction of a series of rooms about an open courtyard. The focus of the _houmfort_ is the peristyle, a roofed-over dancing court with sometimes brilliantly embellished posts. The center post ( _poteau mitan_ ), at the center of the dancing area, is a mediating object par excellence, through which the deities are believed to ascend from the watery regions of the dead or come down from the skies. It is the still point of the spinning world of _vodun_ dance: ground-paintings and the dances in honor of the deities symbolized by these paintings are centered on its axis.\n\nThe influences of several African visual traditions are readily discernible in this single column. Firstly, as we have seen, according to a Dahomean belief, the Creator, Dan, \"set up four iron pillars at the four cardinal points to support the sky, and twisted round them in spirals threads of the three primary colours, black, white, and red, to keep the pillars upright in their places.\" There are _poteaus-mitan_ in Haiti showing spiral decoration in the colors of Dan's own rainbow, or serpents entwined around its shaft. Secondly, a cardinal tenet of Kongo culture is that \"a tree stands up tall on the earth, like a chief witness to boundaries.\" Haitians talk about the central column in their songs as a \"planted post\" _(poteau plant\u00e9)_ , as if it were a tree. Thirdly, among the eastern Igbo and the Ejagham who inspired them, there has existed since at least the eighteenth century the tradition of a special central column in the Ngbe (leopard society) house, a column erected with an important stone set before it. The column is planted in a raised, cylindrical dais of shaped clay\u2014details again matched by those of the _poteau-mitan_.\n\nAdjoining the dance court with its all-important center post is usually a building sheltering several shrines that open directly upon the peristyle. Each is a room containing one or more concrete or stonework altars _(pe)_. Each altar, dedicated to particular spirits, displays multiple niches, recessed within its base, in which offering bowls and other objects are placed by the servitors. In _Rada_ shrines, in the area between Port-au-Prince and L\u00e9ogane, one often sees one single, large, dramatic opening for offerings of food to the spirits in the center of the altar's bottom tier. In shape and location this opening resembles the niches or alcoves recessed in Haitian tombs for food offerings on special occasions.\n\nMany _vodun_ altars are built in rising tiers upon which myriad objects, crowded and stained by actual use, impress the beholder with a sense of spiritual aliveness and activity. The power of these altars lies in their representation of complex ideas by specially selected objects:\n\nA vodun altar is a veritable bric-a-brac display of ritual objects: jars and jugs for the deities and for the dead, plates consecrated to twin-spirits, vessels for inititiated priestesses, thunder-stones, swimming in oil, playing-cards, ritual rattles and emblems of the gods, as well as bottles of wine and liquor offered to the deities.... chromolithographs are pinned to the walls. Near the sword for Ogun, driven into the earth, one still finds, in some shrines, _assein_ , those curious supports in iron which are still to be bought in the market of Abomey in Africa.\n\nIt would be appropriate to compare such altars to the store of an apothecary, for herein are also gathered all kinds of bottles and vessels, in different shapes and colors, many meant to lend spiritual confidence, or a healing sense of security, for troubled suppliants brought before the altar. The close gathering of numerous bottles and containers, on various tiers, is a strong organizing principle in the world of _vodun_ altars.\n\nThat unifying concept, binding Haitian _Rada_ altars to Dahomean altars in West Africa, precisely entails a constant elevation of a profusion of pottery upon a dais, an emphasis on simultaneous assuagement (the liquid in vessels) and exaltation (the ascending structure of the tiers). The intermingling statuary and containers on altars in Dahomey and graves in Kongo become the Haitian mingling of chromolithographs of Catholic saints on the wall and offertory and other vessels on the several tiers below.\n\nIn the Port-au-Prince area the _Petro_ room is often one of several chambers adjoining the peristyle. Such rooms are striking for the touches, sometimes of deliberate horror, meant to suggest the moral terror of this fiery side of vodun. Thus a shrine to the _Petro_ spirit Criminel at the sanctuary of Madame Romnus in Bizonton in 1970 included a human skull embedded in a kind of dado near the bottom of the altar dais. The head of a cat, glassy eyes staring eerily ahead, was suspended on a thread above the altar. Elsewhere, in the region between Gressier and L\u00e9ogane, I have seen _Petro_ altars built apart, in a small habitation separated from the main cluster of altars about the peristyle, emphasizing distance and great danger. And there we sometimes find a table, laid for \"work,\" healing, or acts of mystic aggression, instead of a dais with tiers for display of ritual objects.\n\nPLATE 117\n\nI was allowed to photograph a _Petro_ table on the south peninsula of Haiti in the spring of 1975. Deriving from Kongo on this table is the feathered image (Plate 117) of Simbi Macaya, to the right of a skull and in front of a bottle\u2014a promise of the power of _Petro_ to render urgent healing. A cluster of three transparent bottles each with a miniature crucifix within, dramatically displays points of spiritual contact\u2014crossroads in miniature, ensconced in spirit-attracting glass\u2014and deepens the allusion to the dread powers of the _Petro_ spirits. Playing cards have been laid on this table. They are for divination. Renamed in _Petro_ terms, the Queen of Diamonds, for example, has become Erzulie-Red-Eyes, whose poignant possession style we have observed. There is even a human skull, as in Madame Romnus's shrine, here associated, so it was alleged, with the mystic killing of cruel or evil persons through the propitiation of the deity shown seated at this table.\n\nNo sooner do I come into this room than a servitor is instantly possessed with the spirit of Danger-Malheur, Danger-And-Ill-For-tune, a _Petro_ avatar. Face frozen, he wears his traditional crimson hat. Sequinned bottles for assuagement refer in their glitter to the flash of the spirit that has descended here and add beauty to undertones of terror and moral vengeance. To the left of the table the top of an actual cemetery cross in wood rises from a mock grave and in another portion of this room a great black flag guards a carefully deposited piece of meteoritic stone and a long red coffin, lined with silk, has been set upon two chairs, like a presence in a mortuary.\n\nThe art history of the _Petro_ table-altars of southern Haiti remains to be completed, but the presence of a strong Kongo element\u2014fused with ingenious invention and an imaginative local sense of visual structure, pace, and timing\u2014already seems quite clear.\n\n## **_Vodun_ flags**\n\n_Vodun_ flags appear at the beginning of _vodun_ ceremonies and herald the coming of a god or goddess, the possession of devotees. _Vodun_ flags are profoundly liminal. Unfurled and paraded in _vodun_ rites, they stand at the boundary between two worlds. The late _houngan_ ( _vodun_ priest) Loudovique Elien of Gressier told me in April 1970 that _vodun_ flags were used to \"greet the deities\" _(saluer myst\u00e8res)_. Madame Romnus, the mambo of Bizonton mentioned earlier, also in 1970 told me that the meaning of the flags could be summed up in a single word: respect! _(resp\u00e9!)_. When a Haitian approaches another Haitian's yard, making his presence known by shouting _Hono!_ (Honor!), he does not, by the rural rules of etiquette, cross the threshold until he hears from within the corresponding cry of _Resp\u00e9!_ The resonance between this custom and the association of the flags with _resp\u00e9_ underscores the power of these banners as mediating forces. These are creole variations on a fundamental Kongo theme\u2014 _nikusa minpa_ \u2014the ritual agitation or unfurling or \"dancing\" of squares of cloth to open the door to the other world. M\u00e9traux gives us a fine description of _vodun_ flags in ceremonial context:\n\nThey are brought out at the beginning of a ceremony or when a \"great loa\" possesses one of the faithful. Also, important visitors are entitled to the honor of walking beneath two crossed flags. When the moment comes to fetch these flags, the flag party, which consists of two women, goes into the sanctuary escorted by the _la place_ waving his sword. They come out backwards and then literally charge into the peristyle behind their guide who is now twirling his weapon.\n\nThis can inspire a song for Sogbo, lord of thunder and patron of flags, whose thunderous nature, like the sound of cannons on the battlefield, makes an appropriate accompaniment to the use of banners. Then:\n\nThe trio maneuvers and from the four cardinal points salutes the _poteau-mitan_ , the drums, the dignitaries of the society, and finally any distinguished guests, each according to his rank. The la-place and the standard bearers prostrate themselves in turn before them. These show their respect by kissing the guard of the sabre, the staff of the flag, and make the la-place and the standard bearers pirouette. The return of the standards is accomplished in a remarkable rite: the two priestesses, still preceded by the la-place, pointing his sabre before him, run around the _poteau-mitan_ , often making quick changes of direction. This musical ride goes on till the la-place leads them off towards the sanctuary door through which, having first recoiled from it three times, they pass at the double.\n\nThe balancing of a male-handled sword with two female-carried flags suggests a trace of Dahomean influence in the pairing of the sexes, and also perhaps reflects the sharing of power between male and female officiants in _vodun_ in general. Note that two women pair with a single male, as if their banners were co-wives to the spirit. And there is yet one more indication, in formal terms, that the flags themselves embody spirit\u2014the fact that in some styles the silhouettes of deities honored by the banners are carefully described by shining beads on a field of gleaming spangles. In such cases the beads are often reserved for the direct visual rendering of the deity on the banner. In the spring of 1975 the locally famous healer-diviner \u00c1lvares told me in his sanctuary near L\u00e9ogane why he draped shining beads about the tops of swords embedded upright in the earth before his _Petro_ altar: \"These are _kanzo_ beads. They have to be here. We place them on the heads of sabers because there is a spirit in them.\" And, similarly, the bead-embroidered silhouettes of the goddesses and the gods, on those _vodun_ flags where such additional embellishment is found, suggest the presence of spirit within their gathered brilliance, foretelling that mediation is about to happen or confirming its completion.\n\nPLATE 118\n\nTwo masterpieces of the _vodun_ flag tradition\u2014works of the utmost clarity and precision\u2014were fashioned by Adam Leontus, a former dock worker, later celebrated as \"the master decorative artist of Haiti.\" These flags, made c. 1945 in Port-au-Prince for Saint James\/Og\u00fan Ferraille and for Saint Patrick\/Damballah and Ayida Hwedo, incorporate the whole of their tradition (Plate 116). They carry inscriptions identifying the powers for whom they were made. Like the Catholic chromolithographs, they sparkle with allusion to the attributes of the saints. Like the costumed \"majors\" of Haitian _Rara_ street paraders, they achieve maximum glitter with sequin-covered surfaces. And like the head of the sword for G\u016b, the Fon deity of war and iron, observed in the _Petro_ shrine of \u00c1lvares near L\u00e9ogane and elsewhere, the portions of their composition indicating the silhouettes of sacred beings\u2014or their attributes\u2014are covered with bead embellishment.\n\nThe flag for Damballah and his wife, Ayida Hwedo, the serpents of the sky, is cut from a bolt of pale blue silk (Plate 116). It flashes with countless silver sequins. The serpents crushed under Saint Patrick's foot in his chromolithograph (Plate 115) disappear; they reemerge as twin-beaded serpents in the sky, joined in sacramental love about the swordlike palm tree, the sign of Aizan, the wife of the Dahomean trickster Legba. Aizan \"marches\" with the twin serpents. Leontus transformed the color of the snakes into gleaming silver, an affective suggestion of the richness of their powers and their purity. They stand, entwined, as rainbow-serpents, sources of creative coolness. The straightforward opposition of good and evil in the chromo has been transmuted into an extraordinary evocation of divinity and love. The intermingling of Damballah's body with that of Ayida is a sign of union and ecstasy, a sign that echoes throughout the composition, in the pairing of hearts, crosses, and rainbows meeting in a jar. Overmastering sensations of ecstatic union are stated and restated within a field whose starred corners suggest it is the universe itself.\n\nLeontus's other flag bears equal witness to his power to shape a vision of two worlds. Gold sequins sewn on crimson silk outline the passage of the mounted warrior-saint across the composition, as if he were caught in a ray of sunlight (Plate 118). The dissolving of the scene of fallen warriors and abandoned shields of the chromolithograph (Plate 110) releases the saint from a single point in space and time and makes his action heroically continuous and iconic. He, too, is set within four stars, which suggests the translation of the action to the skies.\n\nThe parting of two flags about the standing sword at the bottom of the composition\u2014precisely the act within the peristyle that heralds the coming of the spirit within the flesh of a dancing and initiated servitor\u2014suggests that the hovering mounted figure above this motif is a spirit descending to this world. The miniature banners within the banner not only link this textile to its contexts but also create raking accents that mirror the shaftlike cartouches in which the spirit's Christian name is sited, above the mounted figure. These names descend, like arrows of God, pointing toward the corresponding Dahomean titles. Adam Leontus totally transcended his source of inspiration; he masterfully transformed a single battle against the Saracens into an endless arrest of supernatural power to inform the moral wisdom of this world.\n\n## **_V\u00e8v\u00e8_ : Ritual _Vodun_ Ground-Paintings**\n\n_V\u00e8v\u00e8_ , the celebrated blazons of the vodun goddesses and gods, are traced by priests or priestesses in powdered substances (normally cornmeal) on the earth about the central column of the vodun dancing court. Symmetrically disposed and symmetrically rendered, they praise, summon, and incarnate all at once the _vodun_ deities of Haiti.\n\nThey take their name from an archaic Fon term for palm oil used in the making of simplified squares or rectangles on the ground for certain Dahomean deities. Essentially they take their structure from a reinforcing merger of Fon and Kongo traditions of ritual ground designs, with the cruciform cosmograms of Kongo and neighboring territory the dominating influence not only in terms of design but, critically, in point of context and process, too. Thus John Janzen:\n\nthe tracing of ritual spaces in the Petro phase of the loa service offers some striking resemblances to... Kongo rituals. In the staking out of the cardinal points with candles, the [ritual expert] used a common, perhaps worldwide motif. However, in circling this space in a counter-clockwise direction, and then dividing it into two, one half representing the domestic realm (governed by \"lord of the house\" Mait' Habitation, the other half the realm of the wilds, the deep, of water (governed by \"lord of the deep\" Mait' Source) he was tracing a cosmogram the way it is done in many Kongo contexts.\n\nAnd there are analogous ground-signs, mediatory cruciforms, found among the Tu-Chokwe of northern Angola, the Ndembu of northwestern Zambia, and the Pende of western Za\u00efre, doubtless fragments from a larger, yet to be discovered western Bantu field of visual expression. The Tu-Chokwe have especially developed ground-designs of intricacy and beauty, as illustrated by an ideograph that artfully fuses various motifs:\n\nThe top of this design indicates the realm of God, whom the Tu-Chokwe, like the Bakongo, call Kalunga. There a human figure is shown standing. Twinned serpentlike forces mark the cosmic crisscross, framing the center of the design where nucleated lozenges indicate the sign of the _muyombo_ tree. Myth says that God asked the stars, \"On which side of the cosmos is man found?\" And the stars replied, \"On the side opposite yours.\" God observed: \"Then man will die,\" thus identifying the bottom of the ideogram as representing the realm of the dead, and explaining why a the human figure therein represented lies in a recumbent pose of eternal sleep.\n\nThe symbolizing, with nucleated patterning, of a tree at the center of the cosmos compares closely with Kongo and Angola custom and also with the Haitian tradition of drawing _v\u00e8v\u00e8_ around a treelike post in the center of the peristyle, as shown below. On the other hand, the contrast between the Tu-Chokwe and Haitian illustrated signs of cosmos not only clearly distinguish the creole blazons from their Central African antecedents but also suggests why the explosive variety of _v\u00e8v\u00e8_ emerged. Both the Tu-Chokwe and Haitian signs are centered on cosmic trees, but there the similarity ends. The Central African cosmogram states the same pattern four times. The Haitian illustration combines icons of various traditions and sources\u2014the upper right patterning is for Simbi, the Kongo-descended spirit, with an alphabetic S, and a sacrificial bullock bearing a Masonic emblem, in celebration of Masonic-like mysteries that surround the power of Simbi; at lower right appears a heart for the Dahomean-derived goddess of lovers, Erzulie, with the _M_ of her Catholic counterpart, Mater Dolorosa; at lower left we find the star-nucleated diamonds of Og\u00fan Badagri, framed with volutes relating, by one scholar's reckoning, to French grillwork; and finally, at upper left, the poised and balanced Dahomean serpents of the sky, Damballah and Ayida Hwedo. In other words, this is more than a crisscross of the earth at point of contact with the sky. In effect, this _v\u00e8v\u00e8_ complex provides geometric focus for a constellation of Dahomean, Kongo, and Roman Catholic forces constituting the very fabric of Haitian cultural history. The Kongo and Angola cruciforms invoked God and the collective dead, but the Haitian ground-painting invoked a host of deities and emblems inherited from many lands. In the reduction of historical multiplicity of experience to single New World forms, _v\u00e8v\u00e8_ constitutes the quintessential form of Afro-Haitian art.\n\nPLATE 119\n\nEverywhere in _vodun_ art, one universe abuts another\u2014the gathering of the \"chromos\" of the saints upon the altar walls; the standing of embottled souls upon the altar; the flash of the double _vodun_ flags and swords (cf. Plate 119) about the peristyle; the coming of the deities, responding to this brilliance through the pillar at the center of the dancing court. Luminous force then radiates, so it is believed, from the bottom of this pillar in the form of the blazing chalk-white signatures of the goddesses and gods. These signs, these _v\u00e8v\u00e8_ , are then erased by the dancing feet of devotees, circling around the pillar, even as, in spirit possession, the figures of these deities are redrawn in their flesh. And then the goddesses and gods themselves revolve about this tree, this rainbow, this standing serpent that helped God build the world, creating unities so splendid it is permissible to greet black Haiti as the _coumbite_ of the western hemisphere, the _rara_ of the universe, a school of being for us all.\n\n# **Four**\n\n# **ROUND HOUSES  \nAND RHYTHMIZED  \nTEXTILES**\n\n# **Mande-Related Art and  \nArchitecture in  \nthe Americas**\n\nLimned in one of the masterpieces of Mali oral literature as generous, strong, and brilliant, Sundiata established the Mali Empire in the thirteenth century. He organized men of his age into an army and defeated his principal enemy, Sumaoro, by discovering and exploiting the latter's _tana. Tana_ is a crucial Mande concept. It is an inherited taboo, an animal-familiar that the inheritor should not kill or even touch, for he would thereby lose the concentration of the powers of his ancestors within himself. Sumaoro's _tana_ was the spur of a rooster. He made the fatal error of bragging to Sundiata's sister about his source of power, unaware that she merely pretended to be in love with him in order to uncover precisely this intelligence. She swiftly relayed her finding to her brother. Sundiata is said to have shot an arrow, to which the spur of a rooster had been attached, at Sumaoro, sealing his doom.\n\nAfter this victory, Sundiata went on to unite the twelve towns of Mande, \"the twelve doors of Mali,\" pacifying them and bringing prosperity to the land. About 1240 he absorbed the remnants of the earlier empire of Ghana. After his death the armies of Mali conquered Timbuktu and the Niger town of Gao c. 1325. Mali was then supreme, from the waters of the Gambia and the Senegal to the inland delta of the Niger. Subsequent to these victories, and the establishment of Mali, Mande traders (Dyula) spread Mande civilization to the south, almost to the gates of what would be the emergent civilizations of the Akan in what is now the Ivory Coast and Ghana. For example, the Dyula town of Kong, in what is now the northeast of the Ivory Coast, was founded at the beginning of the eighteenth century, not far from the northwestern marches of the Akan.\n\nThe Mande diaspora did not diminish their cultural identity. Such was the Mande strength of ethnicity that the cultural focus of the civilization was maintained, even among its widely dispersed members: \"the merchants who established trading colonies in Kong and Bonduku in the Ivory Coast, and those who colonized the Gambia, sent their sons back to the Mande to learn the finer points of Mande culture and perhaps to return with a wife.\"\n\nMande partly resolved the tension between tradition and innovation through their rich and extensive oral literature: animal stories; hunting songs; and, especially, the literature of the courts, sung by professional bards. The bards, \"men of words,\" traveled widely, maintaining the integrity of their language, Mande-kan. Thus they themselves contributed to the amazing cohesiveness of Mande culture. Because they were sent to do their apprenticeship at training centers in such places as Kangaba and Keyla in the core area, and because they convened for important national ceremonies, the bards of the Mande\u2014as Charles Bird points out\u2014virtually constituted an academy. They tempered cultural diversification with ancient norms. The Mande prize clarity, strength, and force of character, and they especially admire generosity and adherence to ancient custom.\n\nThe primary Mande aesthetic value is the search for simplicity, for the elegance of science. Thus the Mande-kan word _woron_ \"to get to the kernel\" also means \"to master speech, music, song, any aesthetic endeavor.\" As Charles Bird observes, one masters something in the Mande world by stripping away the superficial covering, by discovering its inner and true nature, as in the poetic concept of _yere-wolo_ (giving birth to yourself), in which a person finds his or her true self, his or her true essence. And Patrick McNaughton has shown that the making of Mande figural sculpture and designs painted in earth on cloth _(bogolanfini)_ is governed by _jayan_ , \"precision and clarity.\" As the cadres of rigorously trained bards kept the language of ancient Mali from splitting into mutually unintelligible languages, so the Mande propensity for the essential apparently preempted the emergence of a tropical baroque in spite of centuries of complex urban life and leisure. The search for the _kolo_ , the kernel, the nucleus, the essential structure, militated against ostentatious display or mindless presentations of one's wealth, which is regarded as being in bad taste.\n\nThe expansion of the Mali Empire had profound cultural effects throughout West Africa that are still being felt today. Two of the most important Mande artistic traditions disseminated by Mande warriors and traders were cone-on-cylinder architecture and the making of multistrip textiles in vibrant colors\u2014with the main accents of one strip staggered in relation to those of an immediately adjoining one but coordinated with those of another and so on. These aspects of Mande visual tradition survived\u2014indeed still thrive\u2014in certain regions of the Americas, where they were first introduced by slaves from Mande or Mande-influenced regions in West Africa. Moreover, they have been blended with local elements and improvised upon for so long that in most cases the practitioners of these traditions have no specific memory of Mande origins.\n\n## **Black Space**\n\nIn the Mande heartland, the basic architectural module is the clay round house (or wattle-and-daub round house) with a conical thatched roof\u2014a mode that has remained unchanged in Mande country since the time of the Mali Empire. According to tradition, the most important cone-on-cylinder building in the land is the famous Mande Bolon, constructed by the workers of Mansa Soulemane as a repository for the sacred texts this leader brought back from Mecca about 1352. This splendid round structure (Plate 120), specially white-washed and written over with Mande ideographs, some of which are alleged to refer to \"Mali\" and to the \"glory of Sundiata,\" is located in the culturally important settlement of Kangaba, where the bards of Mande assemble every seven years to chant the history of the land. And Portuguese explorers observed cone-on-cylinders buildings with straw roofs in the sixteenth century. These documents, material and literary, attest the existence of the cone-on-cylinder round-house tradition in Mande country at the time of the sixteenth-century exportation of Mande and Mande-area slaves to Mexico. For despite its inland position near the headwaters of the Niger and the Senegal, Mande was vulnerable to the reach of European slaving from its earliest period.\n\nPLATE 120\n\nIn the sixteenth century Spain decimated the Native American population on a portion of the west coast of Mexico known locally as the Costa Chica (the little coast), territory 70-odd miles down-coast from the famous port of Acapulco, by bringing smallpox to the land: in 1552 there were 323,00 Native Americans; in 1582, 1,807. As her need for labor intensified, Spain began importing slaves from West Africa. And thus descendants of Sundiata appeared on the Costa Chica in southwest Mexico:\n\nThe great Mande group were, without a doubt, the one who exercised the greatest influence in Mexico during the entire 16th century.... it is easy to prove in colonial archives the role they played in the integration of patterns of culture of the colony, and the persistence of their influx will surely be recognized when ethnographic investigations, motivated by the black groups which still live in Mexico, are undertaken.\n\nSlaves in western Mexico on the Costa Chica were often organized in groups of ten or twelve, and they lived in camps thirty miles or less from one another. In the rugged isolation of the Costa Chica the blacks found themselves in a familiar, tropical environment with unusual opportunites for autonomy: \"What is important to realize is that the Negroes of this coast were no longer slaves... they lived alone with their families... away from the presence of the [absentee] owners of the ranches, free to improvise responsibilities.\" In the process, the blacks intermarried with the surviving Native Americans, as reflected by the physical type prevailing on the Costa Chica. But there were black runaways whose African manner of cultivation and architecture was noted as early as 1591:\n\nI have been informed that in a hilly place called Coyula, two leagues from said town... there are black runaway slaves. And to the present time they are there, _with their houses, maize cultivation, cotton-growing, and other things, as if they were in Guinea_. [Emphasis added.]\n\nPLATE 121\n\nPLATE 122\n\nThe priority of arrival of Wolof and Mande in Mexico and most likely the re-creation of \"their houses\" there by 1591 is a momentous event in the history of African-influenced architecture in the New World.\n\nIn western Mexico, _redondos_ , or rondavels (round habitations with conical thatched roofs), appeared (Plate 121) and are still being built by Afro-Mexicans and their Native American neighbors. A consideration of the building techniques of the African round-house maker, as they have been studied, sharpens our appreciation of the Afro-Mexican rondavels.\n\nThere are four basic techniques of construction in Mande: 1) the superimposition of rows of sun-dried clay bricks sealed by wet clay; 2) the superimposition of string courses of puddled clay left to dry, one after the other; 3) the plastering over of woven wicker walls with a wet clay mix; 4) the arrangement of a ring of posts to support a conical thatched roof. Of the four modes only the last two mentioned\u2014technically the simplest\u2014were mainly practiced in Mexico, evidently because of missing elaborate Mande crafts and because of the exigencies of the new environment.\n\nThe first two techniques enable the making of relatively massive and thick-walled habitations (Plate 122), as evidenced by a photograph of a compound in Kangaba, in the heart of ancient Mande territory, in 1975. But the builders of the black and maroon villages on the Costa Chica availed themselves of the thin-walled wattle-and-daub technique (Plates 121 and 123), and when they built kitchens, where ventilation was a critical issue, they simply drove a ring of posts into the ground, placing over them a conical thatched roof (Plate 124).\n\nThe common housing unit of the central Mande area has four posts emerging from the very summit of the roof (Plate 125), whereas to the north of the core Mande region, the Soninke people complete their conical roofs without the four posts, using instead a woven ring of vegetal matter to hold the roof in place, as Afro-Mexicans do. Among the latter the securing-ring is called the _corona_ (Plate 126). Another hint of a possible Soninke influence on the architectural style of the west coast of Mexico is the fusing of a rectangular dwelling and a cone-on-cylinder kitchen by means of a veranda, a complex strikingly similar to certain constructions in Soninke country.\n\nPLATE 123\n\nPLATE 124\n\nPLATE 125\n\nPLATE 126\n\nCore Mande and perhaps the Soninke contributed builders during the formation of the Afro-Mexican round-house style, and Labelle Prussin, a leading authority in the study of African building traditions, after examining a series of recent photographs of Afro-Mexican _redondos_ observed strong affinities with modes of southern Mandingo\/Malinke construction. \"I am reminded,\" she writes, \"of innumerable 'palaver hosues' in the center of villages in the northern Ivory Coast and further west, i.e., the houses of the male elders of the villages.\" What is more, the spatial arrangement of round houses in Afro-Mexican compounds, in circular plans, evinces further cultural ties to western Africa. Thus, among the Wolof of Senegambia round houses are \"generally grouped in a circle, the house of the compound head being opposite the entrance.\" The reconstellation of round houses in circular compounds was also the prevailing plan on the Costa Chica for which, evidently, there was no clear Iberian nor Native American precedent. The rationale for the grouping of the round houses in circular compounds was given Susan Yecies in the course of fieldwork conducted among a number of villages in the summer of 1970 and the spring of 1971:\n\nCritics constantly told me that my drawings were not complete because the other _redondos_ (round-houses) that form a compound were missing. \"This drawing does not represent a house; know why? Because the other round-houses are missing and without them, there's no balance, between man and man.\"\n\nThe Mande word _lu_ refers not only to a lineage but to the circular compound inhabited by that lineage. Similar usage prevails in Africa wherever Mande influence is strong, such as northern Ghana:\n\nThere is an apparent absence of the concept of \"house\" or \"dwelling\" as a discrete entity among the traditional Western Sudanese cultures. Fortes, in his description of the way in which the Tale of Northern Ghana view their homesteads, notes that no linguistic distinction is made between the name for an extended family unit and the name for the physical architecturally circular compound which such a unit inhabits. Both are _yir_.\n\nTo repeat, as Tod Eddy rightly points out, there is no concept of the solitary habitation among the Mande: \"I refer to the cone-on-cylinder as a multi-habitational unit instead of a house. It is not merely an architectural structure but a representation of the individual within his or her social universe.\" In like fashion descendants of Africans on the west coast of Mexico find the concept of the single round house meaningless or socially askew.\n\nThe strong links between the Mande round houses and circular compounds and the spatial planning and architecture among the blacks and their neighbors on the Costa Chica south of Acapulco indicate the latter's general provenance. But local ingenuity and adaptation to new conditions have considerably transformed the Mande tradition. Old Mande round houses were closely spaced together when villages had to be strongly fortified against attacks by enemy outsiders or slave-raiding parties. The round houses of the Costa Chica are in general spread more apart, because Afro-Mexicans were subject to, chiefly, absentee landlords and because relations with neighboring Native American groups generally were good. Compounds such as those found in the hill-girt town of Amusgo (Plate 127) onspicuously lack the enclosing walls of Mande village compounds in Mali.\n\nPLATE 127\n\nNative Americans on the west coast of Mexico have, over the centuries, adopted the Mande-influenced module as their own. Aguirre Beltr\u00e1n, in his classic study of the black population and culture of the area, writes:\n\nThe Amuzga, Mixtec, and Trique cultural groups also have, as their traditional form of architecture, the round-house form. In this case, we think it is a phenomenon of cultural borrowing, by the Native Americans, from the blacks.\n\nThe adoption of these buildings by the Native Americans occasioned subtle changes in details of construction and plan that distinguish Afro-Amerindian structures from Afro-Mexican. For example, the Amuzga Native Americans use reinforcing crossbeams inside their round structures but this detail is missing from Afro-Mexican rondavels, as in Montecillos village on the Costa Chica.\n\nThe presence of Mandeizing round houses among Native American buildings of the Amuzga, Mixtec, and other indigenous peoples of the Costa Chica in western Mexico raises the possibility that round houses with conical thatched roofs inhabited by Arawak Indians in northeastern Colombia, in South America, might also have reflected certain Mande influences. Colombia was surpassed only by Brazil and the United States in the massiveness of its black population during the Atlantic trade. Colonial Colombia imported slaves from the Mande area as well as from Angola. Among the Arawak of the Sierra Nevada de Santa Marta, in the extreme northeast of the republic, there is a distinct rondavel tradition: cylindrical walls, made of branches coated with clay and topped by a thatched conical roof (Plate 128). These elements are proportioned and scaled in a manner that recalls the world of Mande architecture.\n\nPLATE 128\n\nAlready certain Western architects have experimented with the elegance, simplicity, and the power of social cohesiveness written into round-house forms from other parts of Africa. The Ndebele-inspired Jack Lenor Larsen house in East Hampton, New York, and the Rice House of Christianburg, Virginia, a romantic replication of rondavels from what is now Zimbabwe, are twentieth-century examples. Ideal black architects, availing themselves of increasing knowledge of Mande-influenced round houses in Africa and the Americas, will turn these historical presences into occasions for creative homage and elaboration. The round houses of Mali in the Atlantic world keep families together in socially binding spaces. They will be rediscovered\u2014and maintained.\n\n## **Rhythmized Textiles**\n\nA particular style of Mande and Mande-influenced narrow-strip textile, enlivened by rich and vivid suspensions of the expected placement of the weft-blocks (Plate 129)\u2014thus characterized by designs virtually to be scanned metrically, in visual resonance with the famed off-beat phrasing of melodic accents in African and Afro-American music\u2014introduces to the history of art an extraordinary idiom, unique to the black world.\n\nPLATE 129\n\nThe Mande \"country cloth\" tradition spread west to Senegambia, east to the Djerma, among the Songhai of Niger, and\u2014this is crucial\u2014south by way of Mande Dyula traders to Kong and Bonduku (in what is now the Ivory Coast), then to Gyaman and Akan settlements proper in what is now Ghana, and again down, via the Songhai region, south into Dahomey and among the Ewe of Togo and eastern Ghana. Thus on the west coast of Africa various enslaved peoples, bound for America, converged, sharing Mande-influenced or Mande-related ideas about the composition of textiles. These traditions were reinstated in diverse ways in the Americas\u2014in Brazil, in Suriname, and in the United States, among other places.\n\nBut the creolized cloths of Bahia, the over-one-shoulder capes of the Djuka and Saamaka maroons in Suriname, and the string-quilts of the black South in the United States are not by any means \"pure African textiles.\" Rather, they are works \"carried out in terms of African tradition.\" Variables of Mande and Mande-related cloth-making remain indelibly intact in these Mande, West African-influenced regions of the New World. The recombination of these variables to form novel creole art\u2014also embodying European influences\u2014is an autonomous development in the history of Afro-American visual creativity, especially in Suriname. Nevertheless, the vibrant visual attack and timing of these cloths are unthinkable except in terms of partial descent from Mande cloth, a world of metrically sparkling textiles.\n\nWest African \"country cloth\" was made on the men's narrow horizontal loom in characteristic widths ranging from nine to fifteen centimeters. Neither in Europe, nor in the Middle East, according to Venice Lamb, a ranking authority, do we find such emphasis on narrow-strip weaving, nor such dramatic fusion of such strips in the making of multistrip cloths and textiles. Akan _asadua_ cloth may be made up of as many as twenty-four strips of cloth. Some Nigerian garments call for as many as one hundred narrow strips, each one inch wide. Thus as multiple meter distinguishes the traditional music of black Africa, emphatic multistrip composition distinguishes the cloth of West Africa and culturally related Afro-American sites.\n\nNarrow-strip weaving has been pervasive in West Africa since at least medieval times. Archaeologists, working at the site of the Tellem caves in modern Dogon country, central Mali, have discovered hundreds of old narrow-strip cloths. Some hailed from Berber weaving centers across the Sahara. Others were indigenous. The tradition of the West African multistrip cloth thus dates from the ninth to the twelfth century, a heritage more than seven centuries old. The weight of the tradition has resulted in a related predilection among West Africans for imported striped cloth. Narrow stripes were preferred as equivalent to narrow strips. On this simple substitution, stripe for strip, was to turn a major aspect of the Africanness of certain forms of Afro-American dress.\n\nPLATE 130\n\nThe earliest cloths from Tellem (Plate 130) show a strong use of blue against white, a hint that African cloth has for centuries, as it is today, been distinguishable by deliberate clashing of \"high-affect colors,\" dark blue against bone white, and so forth, in willful, percussively contrastive, bold arrangements.\n\nAnother variable in the making of a number of Mande or Mande-related \"country cloths\" is a vibrant propensity for off-beat phrasing in the unfolding of overall design. As multiple strips are sewn together by their edges, the major accents (weft-blocks) of one strip may be staggered in relation to those of an adjoining strip, with careful alignment of further elements in the same cloth preempting any assumption of accident and indeed confirming a love of aesthetic intensity through this form of special contrast.\n\nOne of the oldest examples of an African textile exhibiting this trait is a robe collected by one Christoph Weichmann \"on the coast of Benin,\" probably in the early seventeenth century (Plate 131). The strips composing the right arm sleeve are staggered in relation to the siting of those of the left arm. Visual aliveness emerges in two related traits: one, a deliberate clash between patterned and unpatterned strips, some white with patterning, some pure dark indigo without a trace of pattern; and, two, the cutting of broadloom cotton (in this case dyed dark indigo) to narrow-loom strip size to complete a multistrip, multidecorative composition. The latter technique, breaking the bulk of received foreign stuff, grants freedom to the maker of the costume: he can achieve a rippling effect by alternating narrow strips in union even without access to strips made from the West African narrow loom\u2014he merely cuts imported cloth down to cultural size, to the frequency modulation, as it were, of the narrow-strip, multistrip style. This was of obvious importance as a precedent for Afro-Americans recapturing the flavor of the past in spite of the loss of the narrow loom.\n\nPLATE 131\n\nPLATE 132\n\nAnd thus emerged strategies for recovering in a special West African way spontaneity in design, without which there can be neither vividness nor strength in aesthetic structure. The prevalence and diffusion of this tendency toward metric play and staggering of accented elements can be demonstrated by a brief consideration of a Fulani _welmare_ 36 cover from Mali and a cloth attributed to the Wolof from Senegal (Plate 132). Both are surcharged with visual syncopation. A detail of the Wolof textile reveals two weft-blocks aligned in the top two rows, three aligned in the next two, the latter design staggered with yet another triple patterning below.\n\nIn the early nineteenth century similar subtleties were achieved in textiles from the Mande trading towns Kong and Bonduku, in what is now the Ivory Coast. Witness a cloth attributed to a weaver who migrated to Bonduku from the Mande settlement of Kong. His work displays multistrip phrasing. There is no symmetry in the top and penultimate strips, but there is in the second and fourth. Red center-striping is continuous, like scarlet drones. There is an equally sustained \"checkerboard\" rhythm (Plate 133).\n\nPLATE 133\n\nThe northeast of Brazil imported slaves from no fewer than three major civilizations with traditions of rhythmic cloths\u2014Mande via the Cacheu slave trade in what is now Guinea-Bissau, Ewe from the slave coast, and Hausa from what is now northern Nigeria. In the pulsating African-inspired ceremonial life of the city of Salvador, multistrip textiles were used by blacks as garments. These textiles were called _pano da costa_ (\"cloths from the coast\"). According to Nina Rodrigues, this originally meant cloths brought by slaves from Cacheu, where Mande influence was strong.\n\nSurprisingly, the narrow-loom complex itself was also reinstated in Salvador. It is alleged that originally there were narrow looms in use among black folk in Cachoeira, across the bay from Salvador, and on the island of Itaparica, in the same harbor. Today, however, only one narrow-strip weaver remains, the master black craftsman Abdias do Sacramento Nobre (Plate 134). About seventy years old in 1980, when I interviewed him, Abdias do Sacramento Nobre inherited his loom from his godfather, Alexandre, who himself had inherited the technique of narrow-strip weaving from his West African forebears, possibly \"Tapa\" (Nupe) in what is now Nigeria. The tradition is said to extend back to the nineteenth century in Bahia.\n\nNevertheless, Abdias's loom is a richly creolized thing. As Beth Jeffe has observed in a recent study, \"The West African Narrow-Strip Weaving Tradition in the New World\": \"Abdias' loom suggests a Portuguese or Portuguese-influenced floor loom later adapted for use as a horizontal narrow-strip loom.... some of the components of his loom also indicate Mande influence, namely the shape and form of the tension bar disks and the nature of the foot pedals.\" Marvelously, inventively, the carrying out of tradition in partially Mande terms is also evident in the composition of resplendent multistrip cloths Abdias has fashioned. There is a handsome collection of his works, evidently dating from the 1970's, in the Museu da Cidade in Salvador.\n\nHere is an example (Plate 135). Though the particular shaping and staggered siting of the narrow, red, rectangular weft-blocks seems Mande, the interpolation of larger squares is an inventive touch by the artist; and he has elaborated his delicate parallel pinstripes in turquoise and dark blue in a manner reminiscent of the dark blue and purple pinstripes of the Fon\/Ewe\/western Yoruba. Moreover, inasmuch as these ritual cloths were meant to be worn in _candombl\u00e9_ ceremonies in Bahia, the colors have symbolic Yoruba-Dahomean meanings. This particular cloth is for Oshoosi, god of the hunt, whom we met in Chapter I; a similar multistrip cloth in red and white honors Sh\u00e0ng\u00f3; a rainbowlike gathering of strips in many delicate hues honors Oshumare, Yoruba serpent-rainbow of the sky, and so forth. In sum, Abdias's creole art fuses off-beat phrasing in the Mande manner with Yoruba-influenced niceties of expression. It could be said that to a heroically singular extent Abdias unfolds and improvises upon the richness of a multiple African heritage of art.\n\nPLATE 134\n\nPLATE 135\n\nAs the special history of Abdias and his Afro-Brazilian forebears indicates, slaves shipped from ports filled with captives from inland Mande-influenced areas certainly must have included weavers who would have remembered their craft in captivity. Indeed, there is another notice of a West African narrow loom having been set up in the Americas, in a black maroon village in French Guiana in 1748. The village was located in forested territory west of Cayenne near the so-called Lead Mountain, where:\n\n... the women spin cotton when the weather is bad and work in the fields in good weather.... Couachy, Augustin, and Bayou weave cotton cloth, which serves to make skirts for the women and loincloths for the men... this cotton material is woven piece by piece and then assembled and decorated with Siamese cotton thread.\n\nThere must have been similar memories of West African multistrip cloth among black runaways of neighboring Suriname, for in 1823 Ferdinand Denis describes and illustrates (Plate 136) an article of dress, given as Carib: \"Men cover what modesty demands with a _camiza_ , or loin-cloth made from a strip of cotton.\" The illustration of this loincloth reveals a telltale multistrip style. Not only that, but two patterned narrow strips are separated by a single band of continuously unpatterned cloth in a manner that points back to early Asante cloths of the nineteenth century (Plate 137), when weavers were working under Mande influences radiating from Kong and from Bonduku, northwest of the Akan and north of Cap Lahou, whence sailed to Suriname 50 percent of a sample of Dutch slaving ships.\n\nPLATE 136\n\nPLATE 137\n\nPLATE 138\n\nBut then there is a mysterious gap, between the multistrip loincloth of 1823 and the splendid multistrip capes, variously called by runaways _as\u00e9esenti_ or _as\u00e9esente_ , that apparently did not emerge among the inland Saamaka maroons until the early decades of this century (Plate 138). Slightly earlier, multistrip cloths called _as\u00e9esenti_ , made by the Djuka maroons of eastern Suriname, brought the tradition closer to its putative early-nineteenth-century roots in whatever area the Carib bought the multistrip loincloth Denis illustrates (for Native Americans of Suriname themselves in general make and prefer solid red loincloths). In addition, the Djuka have elaborated a vernacular terminology for aesthetic assessment of multistrip cloths that betrays full consciousness of its metric play. Thus some Djuka distinguish between _dobi sten_ (\"double voice\") patterned strips and _stiipi_ (literally, \"single stripe\") continuously unpatterned strips in _as\u00e9esenti_ elaboration. \"If one adds one _dobi-sten_ in the middle of two unpatterned strips it's beautiful,\" said Ma Apina of Dii Tabiki, capital of the Djuka, in December 1981. She added:\n\nWhen Djuka paint something, the colors must clash [ _kengi_ , literally, \"argue\"], and where you stop, there must be another color not looking like the one you end with but far away from it [in color].\n\nThis particular prizing of elements in contrastive alternation recalls the consciously clashed elements of the Weichmann robe of the early seventeenth century, as well as early, Mande-influenced Akan cotton multistrip cloths of the nineteenth century (Plate 137).\n\nAn art history of Suriname that considers the visual traditions of the plantation blacks of the coast and those of the inland maroons as an artistic continuum may well find links between Saamaka _as\u00e9esente_ of the twentieth century (Plate 138), Djuka _as\u00e9esenti_ of the late nineteenth century, and coastal black patchwork textiles ( _mamio_ ) of even earlier facture. _Mamio_ , \"different pieces of cloth sewn together\" (Plate 139), is likely to be a link between West Africa and the inland maroons. It crosscuts both sub-Saharan phrasing and Western quilt-making patterns. The latter connection would explain why some examples of Saamaka multistrip cloth look like Afro-American quilt-tops and vice versa without invoking a mysterious collective black consciousness. Both in coastal patchwork, a coastal costume called _a meki sanni_ , \"she makes the moves\" (Plate 140), and in maroon multistrip textiles, we find color-clashing, alternation of patterned and unpatterned strips, and staggered accents. These are abiding traits in the Mande textile world but were recombined differently in Suriname, resulting in the evolution of new styles that reached their culmination in Djuka and Saamaka multistrip expressions of the early twentieth century (Plate 141).\n\nPLATE 139\n\nThe traits that announce an African quotient in Suriname are also present in U.S. black quilt-top making. William Well Brown's My _Southern Home_ documents a period immediately after 1865 at Huntsville, in northern Alabama, where a black-assembled \"queer-looking garment made of pieces of old army blankets\" was observed. Random elements of size, siting, and accent may well have accounted for the strangeness of this multipiece garment to Western eyes. What is certain is that an Africanizing (Mande-izing?) wool blanket was fashioned around 1890 by Luiza Combs of Hazard, Kentucky. Her blanket provides a missing link between the rhythmized cloths of the western Sudan and the similarly vitalized quilt-tops of black North America. Kenneth Combs, the late quilt-maker's grandson, told me in 1980: \"My uncle told me Luiza was born in Guinea... and that she made other such blankets but they are lost.\" The stripes of this two-strip blanket are deliberately staggered in relation to one another (Plate 142). Luiza Combs would have been ten years old in 1863, when presumably she came as a child from \"Guinea.\" There are two-strip cloths among the Tukulor and similarly deliberately skewed striping systems among the Bamana, suggesting the possibility of direct crossover from these western Mande-related areas to Kentucky, but only time and future research can confirm this.\n\nPLATE 140\n\nPLATE 141\n\nDominic Parisi in the summer of 1979 traveled to Hazard, Kentucky, the town where Luiza Combs worked, and interviewed Margaret Adams, a black quilt-maker. What she told him confirmed an abiding taste, in the assembling of cloth strips, for separation by color contrast, a rationale cognate with the Suriname maroon notion of seeking colors that \"fall\" (clash) well in combination, of seeking energy and vitality in juxtaposing _dobi sten_ and _siteepi_ , strips patterned and strips nonpatterned, in the making of Djuka as\u00e9esenti: You don't want no two reds together, no two blues together. Try to get those colors separated.\n\nPLATE 142\n\nPLATE 143\n\nRandomizing sitings of Western-design blocks (\"nine-patch\" and so forth) in the work of black quilt-makers like Lucinda Toomer of Dawson, Georgia, or Amanda Gordon of Vicksburg, Mississippi (Plate 143), suggest a trace of Mande visual influence. But only a trace. Aspects Anglo-Saxon and other African forces have complicated these splendid textiles. They gleam at the center of the traditional Afro-American home (Plate 144), as illustrated by a photograph by Maud Wahlman of the bedroom of Susie Ponds, a quilt-maker of Waverly, Alabama.\n\nOne final question demands an answer: Why the frequent, seemingly imperative suspension of expected patterning? In the British West Indies patchwork dress keeps the _jumbie_ , a spirit, away from a resting place. In Haiti a man procures from a ritual expert, when necessary, a special shirt made of strips of red, white, and blue to break up the power of the evil eye. Nelly Bragg, an old black woman of Warrensville Heights, Ohio, was asked \"Why one red sock and one white sock worn deliberately mismatched?\" to which she replied, \"To keep spirits away.\" For similar reasons, traditional Afro-American cabins once were wall-papered with deliberately jumbled bits of newsprint and crowded squares of magazine illustration. In Senegambia it was important to randomize the flow of paths, since \"evil travels in straight lines.\" And the Mande themselves coded, in discretionary irregularities of design, visual analogues to danger, matters too serious to impart directly. Not all textiles blazing with interrupted patterning communicate such symbolism across the Mande Atlantic world. Nevertheless, those pilgrims of the Mande concept of _fadenya_ (individuality, with all its attendant dangers)\u2014hunters and warriors, heroic wearers of off-beat textiles\u2014continue to venture into disordered regions, mirroring them, deflecting them with their dress, and come back, as Mary Douglas has memorably phrased a parallel accomplishment, \"with a power not available to those who have stayed in the control of... society.\" The double play of Mande influences, _individuality and self-protection_ \u2014suggested by the rhythmized, pattern-breaking textile modes, and the _group affiliation_ mediated by communal rounds of cone-on-cylinder houses\u2014completes a history of resistance to the closures of the Western technocratic way.\n\nPLATE 144\n\n# Five\n\n# **EMBLEMS  \nOF PROWESS**\n\n# **Ejagham Art and  \nWriting in Two  \nWorlds**\n\nThe ideographs of the Ejagham people of southwestern Cameroon and southeastern Nigeria explode the myth of Africa as a continent without a tradition of writing. The Ejagham developed a unique form of ideographic writing, signs representing ideas and called _nsibidi_ , signs embodying many powers, including the essence of all that is valiant, just, and ordered.\n\nNumerous Ejagham women and men in traditional institutions\u2014\"the whole country is honeycombed with secret societies,\" an English explorer observed early in the twentieth century\u2014wear ritual dress and make gestures strongly influenced by the patterns of _nsibidi_ , extending the reach and complexity of Ejagham traditional writing. Titled elders in major institutions (Nnimm, Eja, Ngbe) establish power or prestige by mastering arcane signs of sacred presence and recollection, luminous ciphers of the founding rulers and most important women. The force of _nsibidi_ is a mystery. Some traditional Ejagham continue to regard this mystery as rationale governing their lives and labor.\n\n_Nsibidi_ , in the Ejagham language, means roughly \"cruel letters.\" _Sibidi_ means \"cruel\" in classical Ejagham, according to Peter Eno of Mamfe, Cameroon, an authority on the art and language of his people. P. Amaury Talbot, an English author, discovered while traveling through Ejagham country that _sibi_ meant \"bloodthirsty.\" Consider _nsibidi_ writing, then, as justifiable terror in the service of law and government. As might be expected, the traditional executioner society\u2014men who killed convicted murderers or launched peremptory strikes against outside enemies\u2014was called the Nsibidi Society.\n\nFew see the richest and most awesome of the signs of the _nsibidi_ corpus, the intricate diagrams drawn upon the ground to control a crisis or honor the funeral of a very important person. As Peter Eno remarks: \" _Nsibidi_ signs represent the heart, the very depth of our ancient Ejagham societies, showing the last stage, the final rites, and only the core of the members can come out to view such signs.\" Ejagham ideographic writing both exalts the power of privileged persons and points to a universe of aesthetic and intellectual potentiality. Poetic play and stylized valor, artistic battles of mime and \"action writing\" enliven, with pleasure and improvisation, the dark dimensions of _nsibidi_.\n\n_Nsibidi_ do not derive from Western writing systems. There are no Arabic or Latin letters in the script. It is wholly African. The invocation of divine beginnings occasions the writing of certain of the most important signs, even as does the centering of power upon a seat of final justice. The moral and civilizing impact of _nsibidi_ betrays the ethnocentrism of an ideology that would exclude ideographic forms from consideration in the history of literacy. Educated Western persons continue to assume that black traditional Africa was culturally impoverished because it lacked letters to record its central myths, ideals, and aspirations. Yet the Ejagham and Ejagham-influenced blacks who elaborated a creole offshoot of _nsibidi_ in Cuba have proven otherwise.\n\nEjagham and Ejagham-influenced captives arrived in western Cuba primarily during the first four decades of the nineteenth century, as a result of the immense rise in sugar cultivation in that portion of the island. The slaves included members of the all-important male \"leopard society,\" called Ngbe in Ejagham. These men founded their own \"society,\" promulgating Ngbe values of nobility and government and remembering the master metaphor of masculine accomplishment, the leopard, who moves with perfect elegance and strength. Ngbe in Cuba was known by the creole name, Abaku\u00e1, after Abakpa, a term by which the Ejagham of Calabar are designated.\n\n_Nsibidi_ signs emerged in Cuba no later than 1839, when the archives of the police of Havana received confiscated papers from a black man's raided premises, papers emblazoned with Ejagham-influenced ideographs, signatures of high-ranking Abaku\u00e1 priests. The sacred signs and signatures of Cuban Abaku\u00e1 are chiefly called _anaforuana_ , among other names. More than five hundred signs have emerged among the blacks of the traditional barrios of six cities in western Cuba: Marianao, Havana, Guanabacoa, Regla, C\u00e1rdenas, and Matanzas. Most of these ideographs are hypnotic variants of a leitmotif of mystic vision: four eyes, two worlds, God the Father\u2014the fish, the king\u2014and the Efut princess who in death became his bride. These signs are written and rewritten with mantraic power and pulsation. Mediatory forces, the sacred signs of the _anaforuana_ corpus, indicate a realm beyond ordinary discourse. They are calligraphic gestures of erudition and black grandeur, spiritual presences traced in yellow or white chalk (yellow for life, white for death) on the ground of inner patios or on the floor of sacred rooms, bringing back the spirit of departed ancestors, describing the proprieties of initiation and funereal leave-taking.\n\n## **Nnimm and Sik\u00e1n**\n\nAt the core of the Ejagham artistic phenomenon we find men and women involved in continuous, intensely spirited, brilliant play. The propelling force of the culture\u2014hidden by some of the densest forests in West Africa\u2014is a conscious choice to create objects and sequences of human motion through the playful application of vision and intelligence. In all the arts and rituals the Ejagham play for the sake of formal excellence, transcending, like the Greeks, their material existence through acts of aesthetic intensity and perfection. Their culture naturally made them the envy of neighboring civilizations\u2014the people of Efut, Ibibio, Igbo, and Banyang territory, who eagerly borrowed the artistic fire of the Ejagham.\n\nAlthough traditionally engaged in the clearing of forests for cultivation and other hard work (e.g., digging yam heaps), many Ejagham men left most farm tasks to women or hired laborers. And even their minimal cultivating duties were performed by peer groups, whose members urged one another on with songs and rhythms. The Ejagham male was essentially an artist, a hunter, and a warrior. The beauty of his life was that these roles were for the most part sportive. Wars were few and far between. The Ejagham male could therefore concentrate on what were for him real, important matters: the pursuit of excellence in the making of string-net costumes, body-painting, poetry, mime, and processioneering. He devised new dances or entire dramas and sold them with their appropriate emblems to less talented neighboring civilizations.\n\nBut it was the Ejagham female who was traditionally considered the original bearer of civilizing gifts. Ejagham women also engaged in plays and artistic matters. Their \"fatting-houses\" ( _nkim_ ) were centers for the arts, where women were taught, by tutors of their own sex, body-painting, coiffure, singing, dancing, ordinary and ceremonial cooking, and, especially, the art of _nsibidi_ writing in several media, including pyrogravure and appliqu\u00e9.\n\nIt could be argued that the more Ejagham women, like their men, devoted themselves to creative consciousness and \u00e9lan, the more they liberated themselves from subservience and came into their own. Talbot, in his classic study of early-twentieth-century Ejagham, _In the Shadow of the Bush_ , extolled Ejagham women's \"mastery of outline... far beyond the average to be expected from Europeans.\" In fact, the high quality of Ejagham and Ejagham-influenced women's arts had caught the eye of European observers such as T. J. Hutchinson, who remarked that \"the women of Old Calabar are not only the surgical operators, but are also artists in other matters. Carving hieroglyphs on large dish calabashes and on the seats of stools; painting figures of poeticized animals on the walls of the houses....\"\n\nTheir ultimate triumph was the spreading of the Ejagham \"fatting-house\" tradition among the women of Ibibio and Efik country, where Ejagham arts of feminine elegance and power were taught to women of other Cross River cultures. Efik women, in particular, were fiercely proud of what they borrowed from the Ejagham. Thus Madame Grace Davis, an Efik woman of Calabar, has said:\n\nIt used to be thought that the girls just went into the fatting-house to get fat and idle away their time. This is nonsense. Actually, during the time of their seclusion they were given serious instruction in dancing, comb-making, care of children, embroidery, and many other things, including how to make symbolic appliqu\u00e9d cloths [ _mbufari_.] That is how our Calabar appliqu\u00e9 cloths came into existence.\n\nThe gift of an appliqu\u00e9d _mbufari_ cloth with _nsibidi_ patterns reflected \"cool\" norms, knowing how to make presents judiciously and with style. Artistically underscored acts of generosity in Calabar included acts of honorific submission or service, as in the Efik phrasing \"shows obedience to the person\" _(suk ibuut no enye)_ , which literally means \"cool one's head in relation to a person.\" The women of Calabar continue to give presents on multiple levels of ideographic meaning, for example, by carefully aligning bottles of gin and lager and other refreshments in a brass tray _(akpankpan)_ , itself chased with floral patterning made by women, patterns that in the nineteenth century were richly intermixed with _nsibidi_ signs. The bottles in the tray are covered over with the bannerlike _mbufari_ cloth, bearing appliqu\u00e9d symbols of benison and honor. This orchestrated act of stylized giving\u2013tray, gin, appliqu\u00e9d cloth covering\u2014can be mounted on a traditional stool richly carved with symbolic figuration (Plate 145).\n\nPLATE 145\n\nPLATE 146\n\n_Mbufari nsibidi_ patterning reflects the status of the woman who devises it and the status of her husband. Her proper training and individual wit are both implied in the careful use of symbolic color\u2014with white, for example, standing for \"peace\" or \"stability of relationship\"; red equaling \"success in bringing children into the world,\" a \"sign of life\"; green meaning \"plants,\" \"good harvest\"; gold, \"the light of the sun, the feeling of contentment\"; and blue, \"water... in case your husband hopes to be a fisherman.\" A woman's depth of accomplishment is evidenced by a detail of an _mbufari_ cloth attributed to the grandmother of Mma Eme Abasi Effiom of Calabar, a piece of textile provisionally dated c. 1938 (Plate 146). This cloth communicates a vision of water by means of a chain-motif border in blue. Repeated floral patterns, in red and gold, evoke simultaneous visions of children and contentment.\n\nThe brilliant honorific modes of symbolizing ideas with pieces of cloth (as adapted by Efik and Efut women in Calabar) inform Ejagham wall paintings among the remote forest villages of the Ekwe clan. The wife of Amaury Talbot sketched a mural on an important house at Nfunum, between Lake Ejagham and Ndebiji, just inside what is now western Cameroon:\n\nThe painting is emblazoned with a scalloped border at the top, casting the composition in a watery mode, consonant with important Ejagham myths about the origins of the male and female societies, Ngbe and Nnimm, within the river. Petaled, leafy elements, pulsing like starfish under clear transparent water, add, like _mbufari_ green, notes of power and abundance, thus enhancing and exalting the checked (signifying leopard pelt) and feline symbols of sovereignty and government.\n\nEjagham women with the prestige of membership in the Nnimm Society had even greater occasion to exercise their sharp aesthetic faculties. A woman entered Nnimm as a young girl, before marriage. Partaking of powers mystic and political, she became a leader among women. Nnimm women, in full regalia, were virtual walking charms, laden with material allusions to forces in the skies and in the depths of streams and rivers. Their ritual dress was rich and sumptuous; their body-painting sure and cursive, not unlike the calligraphy of the southern Sung of China.\n\nThe same petaled elegance of line that danced across the walls of the \"treasure house\" at Nfunum here achieves equipoise upon the initiate's brow and cheeks. Glints of forest power, signaled by a feline silhouette and a long-tailed leopard at Nfunum, are materially suggested by the myriad shells and calabash containers making up the bristling initiate's dress. A crown of feathers adorns the head of the Nnimm woman. Monkey bones project starkly and enigmatically from the sides of her impressive coiffure. Concentric necklaces, bound in leather, go about her neck and rest upon her shoulders. Finally a cowrie-fringed wrapper of palm fiber, dyed crimson, completes her costume (Plate 147).\n\nMost important was the single descending bowlike feather emerging from the back of the initiate's head. This was the Nnimm feather. It specially identified the Nnimm woman, marking her as different from ordinary persons. The first Nnimm woman, so it is believed in Ejagham, came down from heaven with this special feather in her hair. Nnimm women moved, as the forest moves, as the sea moves, in a swaying, chiming mass of fibers, bones, and shells. Beyond use, beyond disruption, the image of the Nnimm woman was a multitextured manifestation of potentialities, resisting the pigeonholing of women as purely instruments of labor. Fear of what these women had intuited and stored within their objects provided checks against injustices that might have been committed against them by men or other women.\n\nPLATE 147\n\nPLATE 148\n\nPLATE 149\n\nIn Ejagham country, women maskers wear an impressive plumed headdress called \"calabash-head\" _(echi okpere)_. The flashing twinned light of mirrors can be added to such structures, warding off all evil. Stately plumes, dyed blood-red, rise like feathered sentinels. The plumes can stand for titled women within the Nnimm Society (Plate 149).\n\n_Aban_ dancer with \"hoop-skirt\" _(nkpin)_ , feathered headdress, and beaded panoply. After Kenneth C. Murray, 1939.\n\nLike the bristling \"medicines\" that adorn the Nnimm initiate's body, the calabash-head headdress is characterized by a dazzling recombination of elements: blood-red standing plumes, red and white strands of cotton, diamond-form devices dyed forest- or harvest-green, and miniature versions of the Nnimm feather appearing at the summit of the plumes.\n\nThe structure of these plumes is special to the Ejagham, hence unmistakable when they reemerge in Cuba to attain full and lasting value. Nnimm as an institution did not survive the shock of the Middle Passage. But the whole point of Ngbe ceremonial was to honor, in part, the mother of the sounding leopard, Ebong\u00f3, who in Cuba was replaced by another female figure, Sik\u00e1n. And wherever Ebongo or her Cuban counterpart appeared, we very often find Nnimm-like plumes and decorations. Afro-Cubans, in fact, assume that Sik\u00e1n belonged to some powerful \"matriarchal\" society (i.e., Nnimm) and that women knew the secrets of Ngbe, its rites and writings, before men did.\n\nThus the theme of silence and femininity recurs in the iconographies of Ngbe and Abaku\u00e1. Feather symbolism, connoting silence for instance, informs Ngbe and Abaku\u00e1 sacred musical instrumentation. Ngbe mourners display \"drums of silence,\" short cylindrical, wedge-tuned drums to whose sides plumes are attached. The skin of such drums can be shaded half white, half black to form _nsibidi mboko_ , sign of the messenger of death, or, quartered by two intersecting lines, forming _nsibidi nkanda_ , sign of the \"crown\" of the seven or more Ngbe grades or branches:\n\nThe \"speech\" of Ejagham drums of silence arises from the signs chalked upon their skin and from their special plumes, with the eloquence of prestige and spiritual presence. Such drums, with their plumes, came to Cuba. We meet them in Abaku\u00e1 funerals and initiations, as the _Sese_ drum. \" _Sese_ ,\" said a black Cuban priest of Abaku\u00e1, \"is not a drum as the whites understand such instruments.\" It is an instrument of display, not for use, an instrument of significant silence, not reverberation. It is material writing, to be read, not heard (Plate 150).\n\nPLATE 150\n\nA feather seals a dancer's lips with a sign of silence in Calabar. The addition of wooden shafts wrapped with feathers to Abaku\u00e1 drums similarly transforms them into silent, honorific presences. Abaku\u00e1 consider _Sese_ an incarnation of Sik\u00e1n, a powerful female spirit at the heart of their society. And this belief deepens the continuity of the meaning of the plumes, silence and nobility, and their Nnimm-like bearing. Abaku\u00e1 state explicitly that _Sese_ represents the head of Sik\u00e1n. The drum is an abstract mask, with plumes like _echi okpere_ , representing important personages, and a triangular sign, the signature of the priest called Isu\u00e9, who alone has the right to hold this silently sounding instrument. And thus the plumed and feathered glory of Nnimm-like presence comes alive in Cuba, as documented by a drawing of a _Sese_ drum rendered by an Abaku\u00e1 member and published in 1954 (Plate 150). Tassels and volutes frame this Ejaghamizing vision; they cryptically allude to the first letter of Sik\u00e1n's name, thus complicating one form of writing with another.\n\nPLATE 151\n\nThe standing plumes of Nnimm\/Sik\u00e1n, ciphers of a circle of departed dignitaries, reemerge in a ritual mask representing Sik\u00e1n, great princess of the Efut, a civilization just to the south of the Ejagham (Plate 151). Saibeke, titled priest of Abaku\u00e1, made this mask in the Havana area, according to Lydia Cabrera, between c. 1925 and 1954. It was inadvertently destroyed in 1961. The illustrated photograph, taken by Pierre Verger in 1957, therefore reflects a vanished monument of Afro-Cuban art. The mask is called _Akuaramina_ , creolized Efik for \"Great Spirit.\" The curve of the great tassel-covered calabash with bristling plumes for Nnimm in Ejagham (Plate 149) remains in place but, here, is \"signed\" with ideographs of the priest Isu\u00e9, carefully painted at each cardinal point. Instead of honorific tassels and colors symbolizing richness of achievement and initiation, the curve of the calabash leads directly to an ingeniously constructed Janus mask, one visage staring into the future, the other at the past, symbolizing clairvoyance and presence in two worlds. This mask, honoring a great woman and her power, brings the noble plumes of Nnimm, towering and magnificent, to the world of Afro-Cuban art.\n\nEjagham believe that Nnimm women were privy to ancient secrets first given by\u2014or seized from\u2014beings in the river or the sea. They say that Nnimm is at least as old as Eja, a venerable war society concerned with \"subduing enemies and evil.\" And Eja, we know from a European document, was in existence at Calabar on the coast west of the Efut, south of the heartland of the Ejagham, in 1668. Some Ejagham aver that the Nnimm Society actually derived its powers from the waters of the Ndian River, flowing through northern Efut territory, a little-known Bantu civilization east of the Calabar and south of the Ejagham. The Efut adapted, disseminated and very richly rephrased Ejagham art and culture to the point where Efut lodges of the leopard society carried more prestige and were the costliest to join c. 1877\u201378.\n\nIt was an Efut man from Usaha-Edet, an important cluster of three Efut villages along the creeks of northern Efut, who sold the secrets of the Ngbe Society to the Efik of Calabar c. 1750. The Efik adopted the Ngbe Society and renamed it after their own word for leopard, _ekpe_. The Efik found in Ekpe (Ngbe) a source of secret vengeance and moral circumspection, an ultimate mysterious authority, rumbling \"like the belly of a guilty man\" behind a curtain in the Ngbe hall. In the name of this Voice, laws were made or canceled, debtors forced to pay, and European arrogance defied.\n\nPLATE 152\n\nA new source of authority and moral insight had entered the world of Calabar precisely when the area was heating up under the impact of the growing slave trade in the late eighteenth century. Ekpe (Ngbe) permeated the southern regions of the Cross River partly through the natural consequence of the influence and power and prestige of Calabar. And the Efut of Usaha-Edet, the link to all that was beautiful and morally impressive in Ngbe, was the tutor of Calabar. Usaha-Edet, one of whose citizens sold Ngbe to Calabar, has taken on among some Afro-Cubans the status of a holy city. Usaga\u00e9, as the latter call Usaha-Edet in creole Abaku\u00e1 idiom, is sacred because there, so it is believed in Cuba, the first secrets of God were revealed by the banks of the Ndian creek (Abaku\u00e1: Od\u00e1n, imaginatively further creolized by Christian Abaku\u00e1 as the river Jordan).\n\nAccording to Afro-Cuban informants, the origins of Ngbe are as follows:\n\nTanze was an ancient king of the Ejagham. A long, long time ago, when the king died, his spirit became a fish, which a woman captured, discovering thus in the river the fortune of the kings of Ejagham. And a man came and took this power away from her, killed this woman, and set up the religion [of the leopard society]. Lord Tanze was a departed king who entered the body of a fish and then became the body of female drum ( _Tanze so mismo rey viejo Ekoi. Ne mur\u00ed jaya tiempo, tiempo ante, y p\u00edritu di e bob\u00e9 pecao que mu\u00e9 coge, ne contr\u00e1 lo r\u00edo la suete lo rei ekoi, y var\u00f3n quit\u00e1 neye, mata mu\u00e9, pa pon\u00e9 un religi\u00f3n. Obon Tanze e rey mueto que entr\u00e1 pecao y pas\u00e1 bong\u00f3_ ).\n\nHere is a different version of the same myth:\n\nEfut were the people chosen by God to receive the Secret, there, near Efut Ekond\u00f3, the right-hand side of the Efut monarch's territory, opposite the left-hand territory, Efut Abua, in northern Efut in the region of Usaha-Edet, south of Ibunda.\n\nThere lived a princess named Sik\u00e1n, daughter of the lord of Northern Efut, Mokuire. Every day she went to fetch clear water from the Ndian from a large calabash left overnight within a hollow underneath a towering palm tree standing sentinel upon the riverbank.\n\nOn the fateful morning that she met her destiny, she took water from the river in this manner and placed the calabash-container on her head. She had not gone several paces when she felt something moving in the water in the vessel balanced on her head. A roar like thunder sounded in the water carried on her head. Sik\u00e1n Sina Yantan did not suspect that at that very moment her head had already become sacred by contact with the presence of The Almighty, nor that she, a woman, was carrying salvation to the people of Efut.\n\nThe spirit Sik\u00e1n had captured for her people was a fish who had been king, who some say was God himself. Afro-Cubans know this fish by the name Tanze, which, provisionally at least, seems a creolizing combination of the Efut word for \"lord\" or \"father,\" _Ta_ , with the standard Ejagham for \"fish\" _(nsi)_. As a lord or king, Tanze by definition had his leopard spirit, his leopard double, who leaped when he was in danger, carrying him beyond, making him athletically and milit\u00e1rily invincible. The creolized praise names of Tanze that have come down to us from ancient Efut via western Cuba make these powers plain:\n\nFish of the river\n\nFish of the sea\n\nSacred sounding Voice\n\nExcellent beyond all measure\n\nArriving grandeur,\n\nWho melts into a leopard's body\n\nWho melts into a leopard's body\n\nForce for whom our old diviners yearned\n\nEyes in the water, yellow eyes of life.\n\nAccording to another Afro-Cuban informant:\n\nThe sacred fish lived for a while in its calabash container and then it died. A grand Efut priest, Nasak\u00f3, began the rites to recapture its spirit within a sacred object in which its mighty voice could sound again. Nasak\u00f3 removed the skin from the body of the fish. Later, in his sanctuary, he traced the first symbol of the leopard society upon the skin of the fish... the symbol of renaissance and immortality.\n\nOver the skin of Tanze, Nasak\u00f3 initiated the seven sons of Ngbe, the founding persons. These represented the seven clans of Efut. These seven chiefs, born in the beginning, included titles which have come down to us in the Ngbe hierarchy, such as it was elaborated among Efut: Mokongo, Ekue \u00f3n, Isu\u00e9, Mpeg\u00f3, Iyamba, and Nr\u012bkamo. Their souls are present today in the seven plumes (of the Abaku\u00e1 lodge).\n\n... Nasak\u00f3 then began to build a sacred object for the spirit of Tanze, a base in which his Voice could again resound. He covered, with Tanze's sacred skin, the mouth of the calabash in which the Fish had lived.... But this skin-covered calabash gave back the Voice of Tanze weakly. It was but a shadow of his Voice. Nasak\u00f3's divining-instrument, _ma_  _ongo pabio_ , then spoke, saying it was imperative to bring the instrument to stronger life with Sik\u00e1n's blood. And Nasak\u00f3 then ordered Sik\u00e1n's sacrifice.\n\nBy mere contact with the presence of the Voice, Sik\u00e1n became in essence the bride of God. Hence the death of the fish predestined her demise, and she was sacrificed so that her blood, combined with his, might bring back his spirit to the secret sounding instrument. Nasak\u00f3 summoned her spirit after sacrifice by attaching her skin to the instrument even as he summoned Tanze's spirit by these means. And it came to pass that through this orchestrated union, skin on skin on wood and other media, male and female valences fused within a single object and Tanze's mighty voice\u2014\"the fish that thunders like a bull\"\u2014returned. The accomplished miracle assured the moral continuity of the Ejagham and the Efut and, centuries later, the Abaku\u00e1.\n\nThenceforth the elders could devise laws and sanction them by referring to the living roar of the Voice, the Lord, the Leopard, the Mystic Fish, sounding behind a curtain in the Ngbe hall. The site of the roar became a court of last appeal. The Voice was the moral terror that forced an erring person to mend his ways, that commanded hardened criminals or murderers to be killed, that announced a reported incidence of adultery or some other crime against the well-being of the town. Tanze, within the council hall, became the source of certainty.\n\nBut the crucial role of women in Cross River culture is emphasized by the fact that even today, among the Ejagham of Calabar, when a king dies, his leopard spirit is said to flee into the forest, and cannot be persuaded to return to proper government to start anew, until mystic chains are cast to ensnare him and an elderly woman with a rooster held aloft stands before the leopard-society house and calls out the names of the great ancestors, one by one. Only a woman can call the leopard spirit back because, so it is believed, the leopard society was first a women's society.\n\n## **_Nsibidi and Anaforuana_**\n\nThe late king of Oban in southern Ejagham told me in the summer of 1978 that _nsibidi_ emerged in the dreams of certain men who thus received its secrets and later \"presented it outside.\" That was version one. And then he told me version two: \"How nsibidi started: _mermaids_ showed us how to write nsibidi\" _(nga nsibidi adohe\u2014aku nat\u00f3ngena wud nsibidi)_. A vision of writing emerging from the places where bells and gongs echo beneath the water, when Ngbe is being played upon the land, prepares us for the resonances of this indigenously African form of script. For _nsibidi_ is more to be aesthetically played and displayed, in valiant jousting contests of erudition, than to be written and understood in any strict or linear Western way.\n\nThe power to notate most important happenings with emblems and ideographs and the confidence that comes from contact with and descent from the secret noble source, the lord Tanze within the instrument, inspired the establishment and continuity of Ngbe in Efut and creolized Ngbe (Abaku\u00e1) in Cuba.\n\n_Nsibidi_ symbolize ideas on several levels of discourse. First, there were signs most people knew, regardless of initiation or of rank in the Ngbe Society, signs representing human relationships, communication, and household objects (which themselves were in some instances used as material-ideographs):\n\nlove, unity, comparability:\n\nhatred, disunity, divorce:\n\nword, speech meeting, congress:\n\nmirror, looking-glass:\n\ntable set for drink and meat:\n\ntrek, journey, voyaging, tracks:\n\nAfro-Cuban men, who were able to scale down ceremonial drums to equivalent objects only inches high when occasions for dissembling their religion through miniaturization arose, were indeed the spiritual descendants of men and women who could conjure love within enlacements, separate and drive a wedge between curves to indicate divorce or hate, and let a simple small cross, symbolizing the intersection of two points of view, multiply, in intensity of exposition, until it came to designate a congress or a full-scale meeting. As the signs proliferated in all their variations, they came to resemble a vast musical score.\n\nSecondly, there were serious signs of danger and extremity, the \"dark signs,\" and these were often literally shaded:\n\nShaded signs designated danger, even as drums of silence announcing the death of an important Ngbe member were shaded half black and half white, both to speak of the pain caused by the departure of a family member and to prepare the path to assuagement: black stood for river mud or death, and the white stood for water or freshness and vitality, according to the proverb \"Where there is mud, there must be water.\" Thus the usage of _nsibidi_ musically counterpoints visual and verbal arts.\n\nThe syntax of _nsibidi_ signs also recalls some of the pleasures of music: repetition, call and response, and correspondence. Ritual fans bearing pyrograved _nsibidi_ illustrate the point\u2014fans called _effrigi_ , used in traditional villages in contexts involving initiation. One example, acquired by the Industrial Museum of Scotland in 1859, bears a border filled with patterning said to represent inconstant love. These signs of separated curves, almost always diagnostic of some sort of deeper separation, surround an amazing buzz of different signs, neatly rendered with the heated end of a metal stylus. Another, later example, showing fewer signs, was collected by the Nigerian Museum in Lagos in 1950 (Plate 152).\n\nPLATE 153\n\nThe concepts represented by this fan are not spelled out in treatise form but rendered through repetition, through \"visual music.\" There are twinned signs of social disintegration, harlotry, leg irons or punishment, trouble or speech upon the crossroads; young men gossiping on the council log; high levels of convocation. Emblems of speech are thus countered by ciphers of parlousness and interspersed with signs of trouble and predicament, an implied warning to those who talk idly, to those unaware of the consequences of unprincipled discourse and disputation.\n\nThus far our examples have been purely Ejagham, but the men and women of Usaha-Edet among Efut, and Bende among Igbo, and Ekeya among Okobo Ibibio were renowned for excellence in the arts of Ngbe and the writing of _nsibidi_. Here we consider _nsibidi_ as a women's art, a \"fatting-house\" accomplishment par excellence. The illustrated work is a calabash tray (Plate 153) from Ekeya, made in this century before 1938, on the right bank of the Cross above Oron. Here the _nsibidi_ \u2014stars of love, shaded triangles suggesting power and danger within the leopard society, spirals of motion, and checked patterns restating once again Ngbe grandeur\u2014revolve around the core of the calabash in a playful, vital manner. Leaflike cartouches enclose each sign. These are connected to one another by means of stems or leafy points.\n\nFinally, there were important _nsibidi_ signs of rank and ritual among the higher branches of Ngbe, Nnimm, and other societies. Ngbe houses, for example, pridefully unfurled great cotton cloths with blue and white tie-dyed blazons in _nsibidi_ script. These were the great _ukara ngbe_ cloths, privileged textiles of prestigious houses having the coveted Nkanda branch. In recent years a full spectrum of color has been added to indicate the richness of mind and elaboration represented by the Nkanda branch. A sign circumscribing radial accents could similarly indicate complexity of initiation among the Ngbe members of the Bende Igbo, who call such signs _Ns\u00edbiri_ , meaning \"you all around, you have all the secrets.\"\n\nWomen had their secrets too, knowledge of supernatural underwater transformations, lore about fish that were really leopards, leopards that were really kings, and many other forces. They slyly interposed some of these \"heavy,\" awesome images in the decoration of outwardly secular objects\u2014calabashes, stools, and trays\u2014confident that only the deeply initiated would catch the glint of power veiled by decoration. For example, a chased brass tray _(ekpankpan)_ from Old Calabar now in the Glasgow Museums and Art Galleries (1894.58d) shows two checked \"leopard\" fish beside a \"mermaid\" with a chewing stick. (The \"checkerboard\" motif is an ancient Ejagham symbol of the leopard.)\n\nIn sum, the formal world of _nsibidi_ writing was glorious and vast. It encompassed signs of love and domestic household objects and human communication, signs that the people knew and used. It encompassed shaded signs of moral terror and punishment, shaded signs of danger and extremity. And finally it encompassed complex signs and signatures of the most privileged ranks of the Ejagham societies, particularly Nnimm and Ngbe. To attain high rank was to view at last impressive calligraphs rendered on the floor of inner rooms or upon the earth in secret forest groves. Public signs, shaded emblems of terror and moral intimidation, and complex secret hierarchical signs of deepest initiation\u2014all were _nsibidi_.\n\nIn 1839, nineteen years before the British explorer Hutchinson noted the presence of women's \"hieroglyphics\" in Old Calabar, and nearly a century before a brief flurry of European articles on Nigerian _nsibidi_ appeared in print, a black dock worker named Margarito Blanco attempted to found an Abaku\u00e1 lodge in Havana. Blanco doubtless had been inspired by the success of the first lodge in Regla across the harbor from the Cuban capital some three years earlier. He had planned to take the title of Mokongo, keeper of Abaku\u00e1 justice, a title believed once held by the father of Sik\u00e1n himself, but the police of Havana arrested him before he could set up his lodge and charged him with \"conspiracy.\" They seized his papers. And thus the archives of the police of Havana came into possession of a precious scrap of paper on which Margarito Blanco had traced an early _nsibidi_ -derived sign in Cuba, a circle reported as being identical with the modern Mokongo sign, i.e., quartered and showing two pairs of staring eyes:\n\nAs the many branches of Ngbe were compacted into two in Cuba, the glory and specifications of power that make up Nkanda-branch symbolism in Calabar seem to have infused Abaku\u00e1 writing from its earliest documented instances. Margarito's sign of the Mokongo title strongly resembles the quartered-circle sign of Nkanda as well as the ancient Efut _nsipidi (nsibidi)_ of \"the child of Ngbe;\" _obonekpe_ and cognate Igbo expressions elsewhere in the _nsibidi_ -using universe:\n\nAfro-Cuban _anaforuana_ sign, the signature of the Mokongo title in Abaku\u00e1, Havana, 1839\n\n_Nsibidi_ Nkanda sign, drawn on drum of silence, Oban Ejagham, 1978\n\nAfro-Cuban _anaforuana_ , Mokongo signature, Havana, 1954\n\nAro Igbo (?) brass bangle with _nsibidi_ pattern, nineteenth century (?)\n\nEfut _nsipidi_ , child-of-Expe sign, Calabar area, 1951\n\nEfik _nsibidi_ , \"Nkanda calabash\" Calabar, c. 1925\n\n_Nsibidi_ pattern on skin of a drum collected at Akoa, Anyang area, Cameroon, 1908\n\nAnalogies to the structure of the Mokongo emblem can be found far and wide, and most have to do with the prestige of Nkanda, which a person of the rank of the father of Princess Sik\u00e1n, Mokongo, would certainly have belonged to, and with generalized Ngbe or Ngbe-related symbolism, such as signs collected among the Efut of Calabar and the Anyang of Akoa north of the upper Cross River facing the heartland of the ancient Ejagham.\n\nThe meanings that black Cubans attached to this sign show a richness of imagination and a celebration of the Ejagham sign-making tradition. They say the three plumes atop the circle stand for three settlements in Efut country, Usaha-Edet, Bekura, and Efut-nsun. The circles within the quartered area symbolize the eyes of Tanze and the eyes of Sik\u00e1n, and the fusion of their powers. Other informants in Cuba interpret this sign as symbolic of the primordial calabash in which the Voice was abstracted from the waters, echoing the plumed and quartered \"nkanda calabash\" _nsibidi_ collected in an unpublished study by Jeffreys in 1923. In sum, Margarito Blanco had taken one of the primordial signs of the _nsibidi_ corpus, a sign redolent with associations of prestige and celebration, and adapted it to symbolize a priestly title within the new creole Abaku\u00e1 Society.\n\nEarly _nsibidi_ -derived signs of Cuba had to do not only with priestly signatures but with the identification of entire towns. Thus, in 1882, _La Correspondencia de Cuba_ published the Abaku\u00e1 signature of the town of Regla, where the Abaku\u00e1 rites were born:\n\nThis illustrates the mirror-play of special quartered circles and intersecting arrows with shaded feathers. By this time in Afro-Cuban history, men of Calabar descent not only had devised heraldic devices in their own calligraphic script to signify the sugar towns of western Cuba but had also renamed them, in creolized Efut. Havana became Nunkwe Amanis\u00f3n Yorama. Matanzas was renamed Iti\u00e1 Fondaga. C\u00e1rdenas was called Iti\u00e1 Kanima Usere. There was also in existence in Cuba at this time the usage of the shaded _nsibidi_ of extremity and terror. There was a fabled nineteenth-century sign of \"war and blood\" _(guerra y sangre)_ which Abaku\u00e1 used when one lodge broke off relations with another and was about to fight. This war sign consisted of a quartered triangle with two heavily shaded feathered elements, very likely abstract shafts of war. The triangle also enclosed cross-patterns with small circles at their ends, plus an emphatically overdrawn or shaded cross in the lower right compartment.\n\nThere was another, very expressive, complex sign of \"war and blood\" that has come down to us from the \"days of the Spanish colony,\" i.e., the nineteenth century. The latter sign displays, again, a cross with small circles at each end. This special cross is superimposed on a small circle within a larger circle, the latter figure being shot through with arrows of war. Plumes at the top of the emblem suggest that the circles stand for a drum of honor, resting upon a four-sided stand, with the latter element itself sectioned into areas, many bearing additional signs:\n\nAbaku\u00e1 \"war and blood\" sign, Havana area, 1882.\n\n_Nsibidi_ of argument and poison overlapping Cuban sign\n\n\"two men have an argument every time they meet\"\n\n\"wound\"\n\n\"poisoned bow and arrow\"\n\nAbaku\u00e1 \"war and blood\" sign, northwestern Cuba, attributed to the nineteenth century\n\nShaded feathers recall the \"dark\" signs of war in the _nsibidi_ corpus (adding a note of belligerence and toughness to the nineteenth-century signature of the town of Regla, as well). The cross with circular termini would appear cognate with African _nsibidi_ of poison and danger\u2014shaded black circles.\n\nSigns of initiation were also emergent in Cuba. Talbot described Nkanda initiatory signs chalked upon the back and chest of Nigerian Ngbe postulants, \"five rings made on front and back.\" In 1882 one of the first descriptions of Ngbe-derived Abaku\u00e1 initiation was published, together with three initiatory signs (Plate 154):\n\nPLATE 154\n\nThe sign of the cross is made on his forehead in yellow chalk [yellow is a sign of life] and the design so drawn on him takes the following shape:   Another cross is made on the chest, the form of which is as follows:   This serves as a signature to be marked before the adept leaves the room, so that when he enters into the formal swearing-in those who receive him will see that he is ready to profess the faith of Abaku\u00e1. The third cross is made on his back from the neck to the waist and it takes this form:   This means he had now taken his vows.\n\nMargarito Blanco and the men who came after him concentrated their calligraphic energies on the reformulation of Ngbe signs of initiation, discipline, and funereal leave-taking. In the process, the term _nsibidi_ apparently was lost, and in its place emerged three creole terms: \"signs\" _(anaforuana)_ , \"signatures\" _(gand\u00f3)_ and \"revelations\" _(ereniyo)_. The first-mentioned word is becoming the standard name for Ejagham-Cuban graphic writing. The last term, referring to that which is seen, puts the fundamental concern of _anaforuana_ with mystic vision in perspective.\n\nFor most of the _anaforuana_ signs of Cuba, written like _nsibidi_ in lines of uniform thickness tracing essentially geometric shapes, have to do with mystic vision\u2014the theme of four eyes\u2014just as in Cameroon membership in Ngbe is synonymous with \"seeing Ngbe,\" i.e., demonstrating formal knowledge of its secrets and its constitution. Frequent _anaforuana_ patterns elaborated in western Cuba include quartered circles  , multiple crossed arrows , quartered diamond-forms with mirrored plumes  . A strong preoccupation with crossroads imagery and the mediation of power across worlds, as in Kongo art and writing, is apparent here:  Moreover, a strong liking for the image of the arrow, sign of war, token of aggression and manful self-assertion, reflects more than a trace of the Ejagham male commitment to formalized combats of aesthetic virtuosity and initiatory depth. _Anaforuana_ , a creole concentration upon certain ancient themes, hypnotically repeats and subrepeats a theme of mystic surveillance and completion. _Anaforuana_ writers impart grace and force to sign after sign through the repetition of the theme of the paired eyes of Tanze and Sik\u00e1n at the boundary between worlds. It makes no difference whether a particular sign or emblem is circular, diamond-form, open and curving, a calabash or horn design\u2014the eyes of Tanze and Sik\u00e1n ceaselessly appear:\n\nIt cannot be overemphasized that even as Westerners in the Middle Ages wrote down and memorized what was most precious, most holy to them, the founding priests of Abaku\u00e1 rendered in ideographic writing the basic images and tenets of the society. Thus, at some point after the 1830's, when Canary Islanders began to be brought to Cuba as cheap labor on a plantation where \"there was nothing but Africans and Canary Islanders,\" every Sunday afternoon the Carabal\u00ed (men and women from Calabar slaving areas) came together to relive their culture:\n\nThey used to tell stories, very beautiful stories, and, above all, they remembered their homeland. Whenever it was the saint's day of one or another of them, they would set up an ironing-board and draw with chalk upon this board something like a calabash with seven plumes\u2014\n\nMy uncle used to tell me: \"That be _ereniyo_ \u2014writing\u2014stand for woman they killed, be Sik\u00e1n and the Fish.\"\n\nIndicating the primordial calabash in which Sik\u00e1n captured the sounding fish of God within the waters of the Ndian, this nineteenth-century Abaku\u00e1 design strongly recalls Nkanda plumed calabash themes as well as feathered signs of festivities in Ejagham symbolism:\n\nIn the image of the feathered calabash the creative intelligence of the Ejagham was returned to Cuba. By the time of the present century this particular emblem had been elaborated into a superbly rendered calligraphic work of art: a central, descending arrow, marking the center of the design, led the eye from one position within the founding myth (the finding of the Fish within the calabash) to another (the circle of the instrument wherein the spirits of Sik\u00e1n and Tanze forever were combined). The central arrow, the seriousness and grandeur of which is suggested by its shaded plumes, is flanked by simpler arrows. This image unifies revelations of great power and mystery. And within the curving outline of the calabash of Sik\u00e1n four double pairs of staring eyes challenge us with the persistent vision of the ancients of Ejagham.\n\nThe common aim of _anaforuana_ signs is the unfolding and elucidation of the richness of Abaku\u00e1 art and ritual. As an Abaku\u00e1 informant himself defined them, they are \"reaffirmations of what happened in Africa at the founding of the ancient lodges... evocations, sacred authorization emblems, imparting force to what is done here [in Cuba].\" In addition to the recapture of elements of the past in certain signs such as the one we have just observed above, and the awarding of power to titled priests in the granting to each one his own particular signature, plus the ritual mapping of initiation ceremonies, and ceremonies of leave-taking of this world, _anaforuana_ declare war, maintain discipline, authorize new lodges and brotherhoods, and announce the special Abaku\u00e1 fiestas called _plantes_.\n\n\"Authorization signs\" are drawn to symbolize the swearing-in of a person to a vacant position in the Abaku\u00e1 hierarchy. The sign validates the candidate and indicates he has the permission of the brothers to assume his post. Structurally, such signs combine a representation of the calabash of origin, plus the latter's honorific plume, with a horizontal arrow bearing the emblem of the particular title being filled. In the _gesture_ conjured by this kind of _anaforuana_ sign, one kind of beginning is significantly crisscrossed with another.\n\nEach priest or highly titled member of Abaku\u00e1 also has the privilege of a special blazon, a signature called _gand\u00f3_. In the authorization sign just examined, the crossed-arrows blazon of the Iyamba title is held aloft over the calabash or origin symbol. Iyamba is \"first priest of the leopard-lord,\" and he takes as his signature four crossed arrows, two marking the cardinal points and two marking the points sited in between. The informants of Lydia Cabrera, from whose monumental study of _anaforuana_ (some 512 examples) the following evidence is culled, state that the arrow marking the sign of Iyamba stands for his personal arrow while the perpendicular one indicates his nobility (evidence consistent with the indication of rank along the horizontal axis in authorization signs). The same informants allege that the raking arrow slashing to the left in the Iyamba signature stands for his prowess in sorcery, whereas the corresponding, right-hand raking arrow indicates his prowess in the war. The interpretation here is clearly linked to left-right nuancing of objects and ideas, with the \"left\" being the covert side, where mystic war is waged by secret means, whereas open warfare belongs to the more positive right-hand side.\n\nThe crisscrossed signature of the Iyamba title, shorn of creole flourishes, points, and feathers, very much recalls _nsibidi_ variously interpreted to mean \"speech,\" \"the paw print of a forest feline,\" the sign of leopard society, and even wealth in accumulated copper bars. The latter meanings appropriately modify the image of the feline lord within the sounding instrument, which Iyamba has the honor of guarding. As a matter of fact, Iyamba is the priest in Abaku\u00e1 who actually makes the sacred sounding instrument speak. In the crisscrossing of these arrows, richness of speech and the ferocity of spiritual attack seemingly become visible.\n\nEach of the signature- _anaforuana_ of the Abaku\u00e1 recasts, rediscovers, and exalts, in different ways, essential images of the myth of Tanze and Sik\u00e1n. Sik\u00e1n's executioner, _Ekue_  _\u00f3n_ , takes as his sign an arrow across the line between two worlds whose feather has been half ruined, symbolizing loss, death, execution (Plate 155). The point is well illustrated by one further example of a sign that identifies the possessions and the presence of a high-ranking priest. The signature of Isunekue (Plate 156), guardian of the Voice of the lord within his instrument, combines a diamond with four staring eyes over a slashing arrow with four eyes over a triangle with yet another pair of staring eyes. The masklike, diamonded device at the summit of the sign recalls the signature of the most feared of the messengers of Abaku\u00e1, Nk\u00f3boro. The slashing horizontal arrow represents the martial prowess of this priestly figure and the triangular structure underneath is interpreted, by some Abaku\u00e1 members, to represent the hill at Ibon Nda, in ancient Efut, where the spirit of the lord leopard-fish, Tanze, was first invoked. And, of course, in the eyes within the Voice, the eyes of Tanze and Sik\u00e1n, the power of surveillance and vision, the deepest force of Abaku\u00e1, discloses itself.\n\nPLATE 155\n\nPLATE 156\n\nThe quartered diamond, shared by Isunekue and Nk\u00f3boro as symbolic emblem for stern and fearsome priests, especially Nk\u00f3boro, overlaps the image of dangerous or evil men in _nsibidi_ patterning collected among the Afikpo Igbo of the Cross River.\n\n_Anaforuana:_ the blazon of Isunekue\n\n_Nsibidi:_ \"bad palaver man\"\n\n_Anaforuana:_ the blazon of Nk\u00f3boro\n\n_Nsibidi:_ \"a bad man\"\n\nIf the _gand\u00f3_ or signatures of the ranking priests of Abaku\u00e1 apprehend in compact form both their individual powers and their embodiments of the essential vision and history of the Abaku\u00e1, their complex funereal signs map and indicate passage to the other world. They are structured according to the creolizing genius of the Abaku\u00e1. In other words, such signs represent an assimilation and transformation of _nsibidi_ of death and mediation. _Nsibidi_ emblems of death among Ejagham and the clans of their neighbors often appear in material media of stark and dramatic force. Near the settlement of Nko-Obubra in August 1978 I photographed a shrine to the memory of a dead person\u2014a white enamel basin suspended, upside down, pierced and nailed to the summit of a staff whitened with kaolin, the color of death, and attached to the staff were scraps of cloth fluttering in the breeze. Talbot discovered near Ekuri Eying, between Ikom and Calabar, that the death of a chief inspired the making of a striking material sign: the survivors affixed to the summit of a tall palm staff the dead man's umbrella and his coat, with the garment disposed so that the arms spectrally pointed outwards. In addition, Keith Nicklin found, in Mbubeland, Ogoja, that when a warrior or a hunter dies, the fronds of surrounding palm trees are shot so that the fronds droop earthwards thereafter. This custom is similar to the Ekwe Ejagham tradition whereby members of the Nsibidi Society mourn the passing of a brother by cutting down plantain stalks behind his house.\n\nViewed as a structural whole, these customs represent death's power both to invert all things, hence the reversal of the basin and the bending of the palm fronds, and to set in motion an upward journey to the sky, hence the elevation of the coat and the parasol and the motifs of mourning high among the ruined palm fronds.\n\nAmong the Abaku\u00e1, death was comprehended in similar visual idioms. In Cuba, when the brothers perambulate in a formal procession called _beromo_ (Plate 157), they sometimes hoist on high the goatskin of Sik\u00e1n as a flaglike evocation of her spirit, and this recalls the elevation of the tunic and umbrella of the vanished chiefs of Ekuri Eying. Death's powers of reversal similarly come to the fore in Abaku\u00e1 funereal symbolism. When a brother is sworn into the Abaku\u00e1, arrows are drawn in the yellow chalk of life upon his body, all pointing down to indicate the descent of life and grace into his body from above. When he dies, the direction of such arrows, now written in death's color, white, is reversed. Pointing toward his head, they indicate upward passage to the other world.\n\nPLATE 157\n\nUpward-pointing arrow signs of death are sometimes abstracted from the context of funereal body-painting and drawn upon the earth at funeral ceremonies. In this case they are called _eriku\u00e1_ , the \"arrow of farewell.\" _Eriku\u00e1_ arrows are sometimes rendered in strict verticality and crossed with multiple arrows of Anamangu\u00ed, messenger of death among the Abaku\u00e1, and sometimes they are wavy and cursive in design but always they point up, indicating the return to heaven. Moreover, they indicate the way to the other world at the end of lines of deliberate elongation, recalling the elevation of dress or shattered leaves at the top of shafts or trees in Nigeria. All of these signs suggest the transformation of a person into a spirit, a spirit that has vanished upward, from ruin toward higher worlds.\n\nAfro-Cuban _anaforuana:_ processioneer with goatskin of Sik\u00e1n, hoist on high as evocation of her spirit, 1950's\n\nNigerian material _nsibidi:_ \"the chief is dead,\" Ekuri eying, c. 1910\n\nAfro-Cuban _anaforuana: eriku\u00e1_ , \"arrows of farewell.\" Left arrow is crossed with multiple arrows of the messenger of death, Anamangu\u00ed, 1959\n\nNigerian material _nsibidi_ , funereal motifs, shrine to dead member of a family, Nko-Obubra, 1978\n\nAfro-Cuban _anaforuana:_ emblem of death's messenger, Anamangu\u00ed, 1959. \"Three crescent-form branches with fallen leaves symbolizing grief\"\n\nNigerian _nsibidi:_ a bamboo stick with the leaves turned backwards mourning emblem for a chief, early twentieth century\n\nAn _nsibidi_ sign announcing the death of a distant chief, which is drawn to represent a bamboo stick with the leaves turned backwards, compacts the ritual of the palm tree fired at by the survivors of a hunter or a warrior within a few calligraphic strokes. The design and meaning of this _nsibidi_ almost certainly inspired the rise of a corresponding emblem among the Abaku\u00e1, a signature- _anaforuana (gand\u00f3)_ of the society's masker of death, Anamangu\u00ed. This particular emblem of the death-messenger\u2014he has a variety of emblems\u2014displays a vertical stroke rising from a quartered circle, not unlike the Nkanda calabash sign in Calabar, with the crucial addition of \"three branches with fallen leaves symbolizing the death of the brother.\"\n\nThe signatures of Anamangu\u00ed are various, as if death were too all-encompassing to be concentrated within a single emblem. They form variations on a theme, a feather arrow ruined or rendered incomplete by death:\n\nImages of the masker of death, branches shedding leaves, a skull and crossbones, four parallel arrows, form a vital part of Abaku\u00e1 funerary art. At the death of an important member, the brothers render in white chalk on thick black paper (about 25 by 35 inches) complicated funerary ensigns. These ensigns are mounted on the walls of the inner shrine _(fo-ekue)_ of an Abaku\u00e1 lodge and are called \"altars.\" There is a mihrablike sense of concentration and sacred directionality to their design.\n\nFunereal ensign, or Abaku\u00e1 \"altar\" c. 1959. Note arrows of the masked messenger of death\n\n_Nsibidi:_ seven spears of a murderous father, seven spears used by his son in self-defense, 1911\n\nThe illustrated \"altar\" is set within a pointed frame surmounted by the cross sign of Tanze and Sik\u00e1n. Two \"arms\" bear the multiple parallel arrows of Anamangu\u00ed, phrased in a manner reminiscent of an _nsibidi_ of patricidal strife. Anamangu\u00ed's three branches with drooping leaves again recall the _nsibidi_ of the death of a ruler or a warrior.\n\nThus, Cuban Abaku\u00e1 brought to the New World aspects of the language and ritual costuming traditions of the \u00c9jagham. Together with initiatory and funereal blazons came a tradition of masked messengers of the spirit rumbling in the leopard-drum unseen but fearfully appreciated. Costumes for the messengers of Ngbe\/Tanze form, according to one Cuban folklorist, the \"maximum incarnation of Abaku\u00e1 spirit.\" But that incarnation was brought into being, shaped, and kept in cultural focus by the power of _anaforuana_ to affirm the past. In fact, the whole panoply of Ngbe\/Abaku\u00e1 messengers, with their costumes, characteristically trimmed with fiber at the wrists and ankles, could be accurately characterized as _nsibidi_ or _anaforuana_ in material form. The checked leopard pattern of a Nigerian Ibibio leopard-masker (Plate 158), which tells us where the power of the dancer is coming from, reappears, as it were, on the dress of an Abaku\u00e1 masker of the 1870's (Plate 159). The Cuban masker even crawls, symbolically, to render the feline animality, which is both the glory and means of moral intimidation of his society. Again, the quartered circle with four eyes\u2014spiritual communication and enlightenment, the core emblem among the signs of the _anaforuana_ corpus\u2014becomes, in Abaku\u00e1 masking, a Janus with two eyes seen in front and two implied (or rendered) on a disk at the back of the head (Plate 160). And just as the intersection of two lines in _nsibidi_ communicates the intersection of words of one person with words of another, with the same sign the nineteenth-century Abaku\u00e1 masker visually voices the idea of speech. He crosses broom and wand before his body to indicate a desire to speak of things positive and lasting. The richness of mediation and spiritual embodiment communicated by a nineteenth-century Abaku\u00e1 masker (Janus of mystic vision, checkerboard of feline strength, soft conical headdress characteristic of Ebong\u00f3, in Calabar given as the female dimension to Ngbe, the mother of all leopards) is a marvel of cultural reinstatement, a fusion of sartorial and ideographic means (Plate 161). It provides a measure to assess strong changes occurring now in Calabar (Plate 162), where Ebong\u00f3 is allowed metallic cloth and many mirrors for the glory of her prestige and powers of mystic vision. If Ebong\u00f3 seems purer in Havana, in some senses, than in Calabar, the old Cuban reflection of the special hoopskirted Calabar _Aban_ dancer, a reflection once vibrant and once strong in the 1870's (Plate 163), has disappeared, and only Calabar (Plate 164) continues this visual tradition. We may be sure the hoopskirt was a sign, a piece of ancient writing worn as dress.\n\nPLATE 158\n\nPLATE 159\n\nPLATE 160\n\nPLATE 161\n\nPLATE 162\n\nPLATE 163\n\nPLATE 164\n\nWe turn, at the ending of this chapter, to a climax in the Abaku\u00e1 calligraphy of the dead\u2014the sign of the lifting of the plate _(levantamiento de plato)_ , referring to the custom of taking up the plate of a dead person from his table, \"for no longer will he use it in this world.\" It is a complicated ideograph, which mourns the entire dead of a lodge that has to restore its positions. It is a mapping of a mass keening, a calligraphic machine for honoring myriad ancestors simultaneously. It has a long axis complicated by a cluster of four priestly signatures of high rank about a circular device centered upon two intersecting arrows, often given as the sign of the four winds. The line continues. It traverses a skull and crossbones of Anamangu\u00ed and two extraordinary winglike branches, bending back to form the ancient _nsibidi_ of death, as if pulled down by the weight of phantom priests whose spectral signatures constellate underneath the \"wings\" in sympathy with their curves. The line continues, collides with, and pierces, in an unceasing journey upward, more crossed arrows, a quartered circle of supernatural vision, a calabash of origin, itself crossed with arrows and gleaming with many eyes of Tanze and Sik\u00e1n, a fusillade of arrows from the deadly quiver of Anamangu\u00ed, a final crisscross of two worlds, and then, departure, into the great beyond, on the tip of the arrow of farewell.\n\n# **NOTES**\n\n#### One\n\n 1. Rev. R. H. Stone, _In Afric's Forest and Jungle: Or Six Years Among the Yorubans_ (New York: Fleming H. Revell, 1899), pp. 20\u201321, 23.\n\n 2. See summary comments on Yoruba art and culture in R. F. Thompson, _African Art in Motion_ (Los Angeles: University of California Press, 1974). For Yoruba urbanism, see G. J. Afolabi Ojo, _Yoruba Culture: A Geographical Analysis_ (London: University of London Press, 1966), pp. 104\u201330. A classic article is William Bascom, \"Some Aspects of Yoruba Urbanism,\" _American Anthropologist_ LXIV, 4 (August 1962), 699\u2013709.\n\n 3. Oladipo Yemitan, _Ijala: Are Ode_ (Ibadan: Oxford University Press, 1963), p. 67. Translation from the Yoruba is mine.\n\n 4. Miss Tucker, _Abbeokuta; Or, Sunrise Within the Tropics_ (London: James Nisbet & Co, 1853), p. 165.\n\n 5. T. J. Bowen, _Grammar and Dictionary of the Yoruba Language_ (Washington D.C.: Smithsonian Contributions to Knowledge, 1858). See entries _amewa_ and _mewa_ , and also R. F. Thompson, \"Yoruba Artistic Criticism,\" in Warren d'Azevedo (ed.), _The Traditional Artist in African Societies_ (Bloomington and London: Indiana University Press, 1973), pp. 26, 60, note 10.\n\n 6. Babatunde Lawal, \"Some Aspects of Yoruba Aesthetics,\" _British Journal of Aesthetics_ , Vol. III (1974), p. 239.\n\n 7. Araba Ek\u00f3, conversations with the author, Lagos, Nigeria, 1972\u201373. The Araba, one of the leading priests of divination in Yorubaland, has been enormously kind over the years in sharing his rich knowledge of the literature of divination in relation to the nature of the _orisha_.\n\n 8. Interview with Araba Ek\u00f3, 13 January 1972, Lagos.\n\n 9. Informant: A. O. Williams On\u00e1yemin, Ilesha, 5 August 1965.\n\n 10. _Iroko_ shrines seen in the field, Ketu, Republic of Benin, summer 1972. For a spectacular document of the Afro-Brazilian continuity of this custom, blended with the cognate Dahomean Loko cult, see Caryb\u00e9, _Iconografia dos Deuses Africanos no Candomble da Bahia_ (S\u00e3o Paulo: Raizes, 1980), p. 110.\n\n 11. The definition of red \"as the supreme presence of color\" stems from Claude L\u00e9vi-Strauss, _The Savage Mind (La Pens\u00e9e sauvage)_. (London: Weidenfeld & Nicolson, 1966), p. 65.\n\n 12. Araba Ek\u00f3, Lagos, 13 January 1972.\n\n 13. Informant: the late J. K. Adejumo, Ipokia, capital of the Anago Yoruba, 5 January 1968.\n\n 14. Cf. Denrele Obasa, _Aw\u1ecdon Akewi_ , cited in A. Fajana, \"Some Aspects of Yoruba Traditional Education,\" _Odu_ (July 1966), p. 25.\n\n 15. Babatunde Lawal, _op. cit_., p. 241.\n\n 16. \"Giving with _both_ hands,\" Henry Drewall (personal communication, 4 December 1979) points out, \"signifies as well the union of social and spiritual worlds, for the left is used in greetings by _ori_  _sa_ (i.e., possessed worshippers) to mortals. Thus it is a _sanctified_ gesture of giving.\"\n\n 17. Araba Ek\u00f3, 12 January 1972.\n\n 18. _Ibid_.\n\n 19. Babatunde Lawal, \"The Significance of Yoruba Sculpture\" (mimeographed paper read at the Conference on Yoruba Civilization, University of Ife, Ile-Ife, 26\u201331 July 1976), p. 9.\n\n 20. The late J. K. Adejumo, Ipokia, 5 January 1968. In addition, in the communication cited in Note 13, Henry Drewall further glosses this testimony: \"Opening his cap may also connote peaceful intentions; for, in contrast, the caps of hunters hold medicines for cursing. The gesture of removing one's cap to take out the medicine is an act of war, but 'opening' the cap shows friendship.\"\n\n 21. Araba Ek\u00f3, interview, 17 January 1972.\n\n 22. R. C. Abraham, _Dictionary of Modern Yoruba_ (London: University of London Press, 1958), p. 658.\n\n 23. Araba Ek\u00f3, 12 January 1972.\n\n 24. E. A. Ajosafe Moore, _The Laws and Customs of the Yoruba People_ (Abeokuta: Fola Bookshops, n.d.), p. 7.\n\n 25. For details, see Robert S. Smith, _Kingdoms of the Yoruba_ (London: Methuen, 1969), pp. 133\u201355. See also Rev. Samuel Johnson, _The History of the Yorubas_ (Lagos: C.M.S. Bookshops, 1921).\n\n 26. S. A. Akintoye, _Revolution and Power Politics in Yorubaland 1840\u20131893_ (New York: Humanities Press, 1971), p. 44.\n\n 27. Juana Elbein dos Santos, _Os Nago E A Morte: P\u00e0d\u00e9, \u00c0s\u00e8s\u00e8 e o culto \u00c9gun na Bahia_ (Petr\u00f3polis: Editora Vozes, 1976), p. 13.\n\n 28. William Bascom, _Shango in the New World_ (Austin: African and Afro-American Research Institute, 1972), p. 13.\n\n 29. Focused on the African-influenced life of Port-au-Prince in Haiti, a thoughtful and important study of the punning process in linking attributes of the saints to those of the _vodun_ (Dahomean spirits) and the Yoruba _orisha_ is Michel Leiris's: \"Note sur l'usage de chromolithographies par les vodouisants d'Haiti,\" in _Les Afro-Am\u00e9ricains_ (Dakar: M\u00e9moires de l'Institut Fran\u00e7ais d'Afrique Noire, 1953), pp. 201\u20137.\n\n 30. Bascom, _Shango in the New World_ , pp. 16\u201317: such syncretisms are not consistent, varying from place to place, even from shrine to shrine. Bascom illustrates this point with an excellent table of Sh\u00e0ng\u00f3\/Catholic saint correspondences across the western hemisphere (pp. 16\u201317).\n\n 31. Dos Santos _op. cit_., pp. 171\u201381.\n\n 32. _Ibid_., p. 179.\n\n 33. See Caryb\u00e9, _op. cit_., p. 46, for a watercolor of an Eshu-Elegbara priest in Bahia wearing a headdress that is red on one side, black on the other.\n\n 34. Hence the pitchforks of Satan, horns of the Devil, and other infernal images that cloud his true role as messenger of the deities and principle of individuality, particularly in the Yoruba-influenced religious life of Rio, where a kind of meta-literature has arisen in his name, incredibly creolized and inventive in its references, freely mixing Kongo, Yoruba-Dahomean, Roman Catholic, spiritualist, and other concepts, as illustrated, for example, by N. A. Molina, _Na Gira dos Exu_ (Rio de Janeiro: Editora Espiritualista, n.d.); Antonio de Alva, _O livro dos exus: Kimbas e Eguns_ (Rio de Janeiro: Editora Eco, 1967); and Jos\u00e9 Maria Bitten-court, _No reino dos exus_ (Rio de Janeiro: Editora Eco, 1970).\n\n 35. From a conversation with the Araba Ek\u00f3, Lagos, 18 January 1972.\n\n 36. Juana Elbein dos Santos and Deoscoredes M. dos Santos, _E_  _su Bara Laroye: A Comparative Study_ (mimeographed), Institute of African Studies, University of Ibadan, Nigeria (April 1971), p. 28: \"Esu is the Lord of Power, Elegbara, its controller at the same time as its representation.\" See also their \"Esu Bara, Principle of Individual Life in the Nago System,\" a paper delivered at the conference on _The Notion of the Person in Black Africa_ (Paris: Centre National de la Recherche Scientifique, Colloque Internationaux, No. 544, 1971).\n\n 37. Interview with Lydia Cabrera, Coral Gables, Florida, September 1981.\n\n 38. Lydia Cabrera, _El Monte: Igbo Fina Ewe Orisha, Vititinfinda_ (Havana: Ediciones C.R., 1954), p. 80: \" _Propicio, modifica el peor de los destinos; hostil, ensombrece el m\u00e1s brillante_.\"\n\n 39. Witnessed in Rio, spring 1968.\n\n 40. Cabrera, _op. cit_ , p. 95: \"Eshu is an Elegba ready to do nothing but evil.\"\n\n 41. Dos Santos and Dos Santos, _op. cit_., p. 29.\n\n 42. It may well be, as Henry Drewall suggests (personal communication, 4 December 1979), that \"the iron blade\/tuft\/penis projecting from the top of the head of Esu\/Elegba seems more Fon or more precisely Ewe than Yoruba. I documented a large cement sculpture of a seated Legba figure with iron bursting from his head at the entrance to a compound in an Ewe town north of Lome.\" It is fairly certain that Fon, Ewe, and other Yoruba-influenced groups living to the west of Yorubaland \"came before\" the Yoruba to Brazil, as Drewall adds, \"not with them.\" The complex problem of the merging of Yoruba-influenced Legba imagery, from Ewe\/Fon sources, with Yoruba impulses in Brazil and Cuba will be discussed in a future publication, _The Face of the Gods_. Here it suffices to note the probable reinforcement of Yoruba imagery by already existing Fon\/Ewe customs in the Western Hemisphere.\n\n 43. Argeliers Le\u00f3n, \"Elebwa: una divinidad de la santer\u00eda Cubana,\" _Abhandlungen und Berichte des Staatlichen Museums f\u00fcr Volkerkunde Dresden_ , 21 (1962), pp. 57\u201361.\n\n 44. Letter written by Julius Oyetunde of Ibadan, dated 22 July 1919, and now in the archives of the Museum of Ethnic Art, UCLA, Los Angeles. The letter came with a shrine image carved in wood for Eshu-Elegba, attributed to Agesingbena of Ibadan (Museum of Ethnic Art: X67-600), now at UCLA. I thank George Ellis for a copy of this letter.\n\n 45. Dos Santos and Dos Santos, _E_  _su Bara Laroye_ , p. 29.\n\n 46. Joan Wescott, \"The Sculpture and Myths of Eshu-Elegba,\" _Africa_ XXXII, 4 (October 1962), 338.\n\n 47. Dos Santos, _op. cit_., pp. 135\u201337.\n\n 48. _Ibid_., p. 162.\n\n 49. Cabrera, _El Monte_. The photographs at the back of this important volume are unnumbered; what would be the tenth and eleventh pages of the plates focus on art for Elegba in the late forties and early fifties in western Cuba.\n\n 50. _Arar\u00e1_ (Afro-Cuban for Dahomean\/Ewe worship) is cognate with the Afro-Haitian term _Rada_. Both words stem from the name of an ancient town, Allada, in Dahomey. For a rewarding survey of Ewe art, see Dzagbe Cudjoe, _Tribus_ (August 1969), pp. 49\u201360.\n\n 51. Attribution by Lydia Cabrera, conversation with the author, summer 1979.\n\n 52. I heartily thank Sidney Mintz, of the Anthropology Department, Johns Hopkins University, Baltimore, for bringing this fine piece to my attention, and for details about its collection.\n\n 53. For a broader interpretation of projections from the head throughout Yoruba art, see Margaret Drewall's \"Projections from the Top in Yoruba Art,\" _African Arts_ XI, 1 (Fall 1977), 43\u201349.\n\n 54. Lydia Cabrera, conversation, winter 1979.\n\n 55. Roger Bastide, _Imagens do Nordeste M\u00edstico_ (Rio de Janeiro: Empresa Gr\u00e1fica o Cruzeiro, 1945), p. 129.\n\n 56. _Ibid_., pp. 129\u201330.\n\n 57. _Ibid_., photograph between pp. 160\u201361, caption reads in Portuguese: \"Exu-bara (Photo Recife Department of Police).\"\n\n 58. See Mario Barata, \"Arte negra,\" _Revista da Semana_ (17 May 1941), pp. 16\u201317, 34, where he speaks of \"macumba idols from Rio recently collected by the town police.\" \"Collected\" is a euphemism for police confiscation of objects including an interesting image for Eshu-Elegbara roughly similar to the piece under discussion. Both are now in the Police MuseumofRio. Butthepre-1941 datingoftheobjectunderdiscussionisstill no more than an informed guess, for other than this article no records of accession could be found during several visits to the Police Museum in March and August 1968. See also Mario Barata, \"The Negro in the Plastic Arts of Brazil,\" in _The African Contribution To Brazil_ (Rio de Janeiro: Edigraf, 1966), p. 38, note 21.\n\n 59. M. A. S. Barber, _Oshielle: or Village Life in the Yoruba Country_ (London: James Nisbet, 1857). Illustration faces p. 187.\n\n 60. Susan M. Pearce, _Yoruba Archaeology and Art_ (Exeter: Royal Albert Memorial Museum, 1970) catalog object no. 27 (RAMM no. 1385\/1868).\n\n 61. Pierre Verger, _Notes sur le culture des orisa et vodun_ (Dakar: IFAN, 1957), p. 134: _\u00d3 fi er\u00ed ejo fun fere_ (with the head of a snake he blows a whistle). Verger collected this praise-verse in Ketu, an ancient town over which, according to tradition, Eshu once ruled.\n\n 62. Informant: Araba Ek\u00f3, interviews, Lagos, July 1965.\n\n 63. Verger, _op. cit_., pp. 136, 138.\n\n 64. For Bahia, Verger, _ibid_., pp. 138\u201339; for Cuba, Lydia Cabrera, _M\u00fasica de los cultos Africanos en Cuba_ , Disco 4, Lado 2; for Miami, _Eru Ana: Afro-Cuban Folklore_ (Onix LP: ORLPS-004, Cara A, \"Elegu\u00e1\").\n\n 65. Valdelice Carneiro Gir\u00e3o, \"A Cole\u00e7ao Arthur Ramos,\" _Revista Ciencias Sociais_ II, 1 (1971), 105:\" (1.60.122) _Exu_ , 33 cm. high, wooden statuette, restored... [from the] Candombl\u00e9 da Bahia.\"\n\n 66. As well as in Hispanic New York City. I am indebted to Eshumiwa, a New York priest of Eshu-Elegbara, for the privilege of studying a hook for Eshu, painted red and black, Eshu's colors, made in Harlem in 1965.\n\n 67. Cabrera, _El Monte_ , ninth page of unnumbered photographs.\n\n 68. Wande Abimbola, _If\u00e1 Divination Poetry_ (New York: Nok Publishers Ltd, 1977), p. v.\n\n 69. Cf. William Bascom, _If\u00e1 Divination: Communication Between Gods and Men in West Africa_ (Bloomington and London: Indiana University Press, 1969), p. 11.\n\n 70. Abraham, _op. cit_., p. 274, _If\u00e1: aigbofa la n woke: If\u00e1 kon k\u00f2 si ni para_ (\"owing to being unskilled in divination, we are looking up at the thatch, but it is not a likely place to find him!\").\n\n 71. Wande Abimbola, _Ifa Divination Poetry_ , pp. 2\u20134. For another myth of If\u00e1, see Fela Sowande, _If\u00e1_ (Yaba: Forward Press, n.d., c. 1964). See both Abimbola and Bascom, _op. cit_., for excellent descriptions of the divination process and its instruments.\n\n 72. Fela Sowande, _Ifa. Odu Mimo_ (Lagos: Ancient Religious Societies of African Descendants Association, 1965), p. 28.\n\n 73. _Ibid_.\n\n 74. For a discussion of visual art pertaining to If\u00e1 and its Dahomean equivalent, Fa, see Bernard Maupoil, _La Geomancie a l'ancienne Cote des Esclaves_ (Paris: Institut d'Ethnologie, 1961), reprint of the 1922 edition. There is an excellent survey of If\u00e1 art in relation to the oral literature of Yoruba divination by Rowland Abiodun, \"Ifa Art Objects: An Interpretation Based on Oral Tradition,\" in Wande Abimbola (ed.), _Yoruba Oral Tradition: Poetry in Music, Dance, and Drama_ (Ife: Department of African Languages and Literatures, University of Ife, 1975), pp. 421\u201369.\n\n 75. Abimbola, _Ifa Divination Poetry_ , pp. 150\u201351. I have slightly retranslated the Ifa verse herein cited.\n\n 76. William Bascom, \"Two Forms of Afro-Cuban Divination,\" in Sol Tax (ed.), _Acculturation in the Americas_ (Chicago: University of Chicago Press, 1952), pp. 169\u201379.\n\n 77. Cf. Fernando Ortiz, _Los Instrumentos de la m\u00fasica Afrocubana_ , Vol. I (Havana: Ministerio de Educaci\u00f3n, 1952), p. 193, Fig. 9 (illustration of an Afro-Cuban _irofa_ and _opon ifa_ ). Bascom, \"Two Forms,\" reports (p. 172) that \"in Cuba few diviners know the use of the sixteen nuts, most of them relying on the divining chain. Similarly among the Yoruba the chain is more frequently employed.\"\n\n 78. Abiodun, _op. cit_., makes the important observation (p. 12) that \"Ifa acknowledges the power of \u00c8 s\u00f9 symbolically, placates him and solicits his cooperation through the carved face(s) of E su in the border decoration of _opon_.\" Drewall has examined an _atefa_ in the Deoscoredes dos Santos collection in Bahia which belonged to a servitor of the Yoruba gods who divined. This tray even had a circular indentation on the bottom which in Nigeria, according to my informants, means that the _opon_ is a secret drum with two membranes, one on each side. Peter Morton-Williams (with J. R. O. Ojo), _Museum of the Institute of African Studies, University of Ife: A Short Illustrated Guide_ (Ife: University of Ife Press, 1969), says, on p. 8: \"The back of the tray is hollowed so that it sounds when tapped by the diviner with his ivory rattle to invoke Orumnila.\" Drewall (personal communication, 1979) says it has to do with a covert function of cursing. All three versions dovetail within, and support, the notion of the _opon_ as a most important instrument of communication with the gods.\n\n 79. See Morton-Williams and Ojo, _op. cit_. For a brief discussion of form and meaning in the study of _agere ifa_ , see Abiodun, _op. cit_., pp. 447\u201350.\n\n 80. Roger Bastide, _Le Candombl\u00e9 de Bahia_ (Paris: Mouton, 1958), p. 51, note 77. In Brazil the _agere_ was redesignated \"box for Yemanya,\" goddess of the sea, possibly by analogy with its discovery by the sea.\n\n 81. It is important to note that while the use of the _opele_ in Cuba and Cuban-influenced North America remains strong, our early visual source published in (Fernando Ortiz, _Hampa Afro-Cubana: Los Negros Brujos_ (Madrid: Editorial America, 1906), p. 178, illustrated (Plate 20b) in this book, is actually a drawing, taken from another drawing, published in a late-nineteenth-century Havana newspaper. In translation, \"divining-chain\" became confused with \"necklace,\" and what we have are enough seeds on a chain to make _two opele_ (cf. Plate 20a) with one left over! Nevertheless, the spacing of chain to seed is identical with the Nigerian antecedent, and the same reference gives us an early document of the use of green and yellow beads for If\u00e1 in the western hemisphere (p. 177).\n\n 82. Cabrera, _El Monte_ , p. 15. The citation is a composite of two sentences in which I translate _derecho_ , lit., \"right,\" as protocol.\n\n 83. See R. F. Thompson, \"Icons of the Mind: Yoruba Herbalism Arts in Atlantic Perspective,\" _African Arts_ VIII, 3: 52\u201359, 89\u201390.\n\n 84. _Ibid_.\n\n 85. Cabrera, _El Monte_ , pp. 70\u201371.\n\n 86. For details, see my \"Icons of the Mind,\" noted above.\n\n 87. Cabrera, _El Monte_ , p. 101.\n\n 88. Araba Ek\u00f3, interview, 12 January 1972.\n\n 89. Quoted in my _Black Gods and Kings_ (Los Angeles: Museum of Ethnic Arts, 1971, p. 11\/3.\n\n 90. Informant: Julito Collazo, interviews, Hamden, Connecticut, winter 1970. The melody of an ancient Osanyin song appears in Cuba ( _Cult Music of Cuba_ , Ethnic Folkways LP FE 4410) and also in Brazil ( _Folk Music of Brazil_ , Library of Congress LP AFS L13). Harold Courlander, in the notes to the LP of Cuban black ritual music, was apparently the first to notice the implications of antiquity in the geographic distribution of this song. A full version, mentioning the immovable stone under water, was recorded in the 1950's by Lydia Cabrera, _M\u00fasica de los Cultos Africanos en Cuba_ , Record 3, Side 1 (c. 1958).\n\n 91. Teodoro D\u00edaz Fabelo, _Olorun_ (Havana: Ediciones del Departamento de Folklore del Teatro Nacional de Cuba, 1960) p. 67. The original text interweaves Lucum\u00ed (Cuban creolized Yoruba) and Castilian phrases which I elide together.\n\n 92. _Ibid_., p. 68.\n\n 93. Geraldine Torres Guerra, \"Un elemento ritual: El Osun,\" _Etnolog\u00eda y Folklore_ , Vol. III (1967), pp. 65\u201381.\n\n 94. Cabrera, _El Monte_ , ninth unnumbered page of photographs, \"Altar of a priest, for the Ni\u00f1o de Atocha, Elegba.\"\n\n 95. I am grateful to Owoeye of Efon-Alaiye for an introduction to a corpus of works of art in iron by members of the atelier of Odeleogun and his son, Ajanak\u00fa, in the winter of 1964.\n\n 96. See, for example, Bastide, _op. cit_., pp. 72 (\"tree of life\"), 111 (a bird at summit, surrounded by small branches), which Ramos took to be \"Eshu of the seven roads.\" Cf. Deoscoredes dos Santos, _West African Sacred Art and Rituals in Brazil_ (Ibadan: Institute of African Studies, 1967), p. 80: \"A central bar with six smaller bars and an iron bird on top symbolizes a tree with seven branches and the bird at the top.\"\n\n 97. From oral evidence collected at Igogo-Ekiti, summer 1965, Nigeria.\n\n 98. For an earlier version of this paragraph, see Lewis M. Dabney, _The Indians of Yoknapatawpha_ (Baton Rouge: Louisiana State University Press, 1974), p. 146. Finally, no reference to Osanyin is complete with mention of Pierre Verger's important booklet, _\u00c0won Ew\u00e9 \u00d2sany\u00ecn: Yoruba Medicinal Leaves_ (Ife: Institute of African Studies, University of Ife, 1967).\n\n 99. _Black Gods and Kings_ p. 7\/1. Robert Smith, _Journal of African History_ VIII, 1 (1967), 93, makes an observation that is relevant here: \"Among the Yoruba the sword was regarded essentially as a cutting weapon.\" Finally, for a superb study, see Sandra T. Barnes, _Og\u00fan: An Old God for a New Age_ (Philadelphia: ISHI, 1980).\n\n100. Told me by the Chief of Ipole, near Ilesha, winter 1963.\n\n101. A composite of praise-verses _(\u00e0w\u1ecdon or\u00edki)_ published by Verger, _op. cit_., pp. 178\u201379, verses 1, 3, 9, 8, 9 (in the last verse I expand the idiophonic _rororo_ , the sound that fire makes, into: \"leaves the forest screaming...\", pp. 175\u201377, 180\u201381, 183\u201385, 188.\n\n102. Informant: Onayemi of Ilesha, July 1965.\n\n103. _Ibid_.\n\n104. _Ibid_.\n\n105. Verger, _Notes sur le culture_ , p. 182: \"Seven Ogun in the house of Ire.\"\n\n106. Quoted in Ortiz, _op. cit_., p. 65. The object was in this text misattributed to the cult of Orisha Oko.\n\n107. There may be a trace of Kongo influence here, blended in with Yoruba practice, for some Afro-Cuban informants have told me that one works with an _nfumbi_ (creole Ki-Kongo for \"dead man\") locked in the Og\u00fan cauldron. The cauldron itself resembles a famous pot on a tripod of three stones in which the founder of Kongo primordially \"cooked up\" the first medicines of the land. Cf. the morphology of the famous Afro-Cuban Kongo charm \"Graveyard-Midnight\" (Plate 66), _supra_.\n\n108. Informant: Larry Harlow, New York City, fall 1979. Moreover, a few Afro-Cuban _bot\u00e1nicas_ (herbal stores) in New England sometimes sell protective chains of Og\u00fan to hang at the bottom of front doors, on the inside, for protection.\n\n109. Verger, _Notes sur le culture_ , p. 206.\n\n110. _Ibid_., pp. 208\u20139.\n\n111. The cited verses are a composite, from praise literature for Oshoosi and closely related hunter-deities, in Verger, \"Notes.\" The page numbers, and, in parentheses, the numbers of the verses follow: 221 (4, 5), 225 (65), 215 (village of Save, 1), 216 (4); 222 (13).\n\n112. M. A. S. Barber, _op. cit_., 1857.\n\n113. R. S. Smith, \"Yoruba Warfare and Weapons,\" in S. O. Biobaku (ed.), _Sources of Yoruba History_ (Oxford: Clarendon Press, 1973), pp. 232\u201333.\n\n114. Dos Santos, \"West African Sacred Art,\" Plates 80 and 81.\n\n115. Cf. Babatunde Lawal, _op. cit_., p. 246: \"The person who wears an all-red garment is likely to be mistaken for a priest.\"\n\n116. Abraham, _op. cit_., p. 623.\n\n117. Informant: Adisa Fagbemi, _Ima\u1e63ai_ , northern Egbado, winter 1963\u201364.\n\n118. Verger, _Notes sur le culture_ p. 244. So closely does Obaluaiye relate to his Dahomean derivative, Sakpata, that I follow Verger in combining songs for both deities to illustrate the nature of the redoubtable earth deities. This particular selection is excerpted from _nukoromahan_ , bright, satiric songs of moral allusion that \"invite people to mend their ways or else incur the wrath of Sakpata.\"\n\n119. _Ibid_., p. 244.\n\n120. _Ibid_., pp. 256 (2); 257 (4); 259 (acclamations); 260 (4), an ancient image of nobility and regal poise in Yoruba poetry; 264 (10); 264 (5,6); 265 (7).\n\n121. Dos Santos, \"West African Sacred Art,\" p. 67, Plate 57.\n\n122. Source: Araba Ek\u00f3, 18 January 1972.\n\n123. Juana Elbein Dos Santos, in conversation with the author, January 1980, Bahia.\n\n124. There is a good description of this shrine, which I myself visited on 16 May 1963, in Geoffrey Parrinder, _West African Religion_ (London: Epworth Press, 1961) pp. 28\u201329.\n\n125. The priest of Nana at Dassa-Zoum\u00e9 told me (16 May 1963) that the clay pillar within the crown and sheath of straw was called _ata bukuu_ and added: \"Buk\u00fau lives in there and comes out during spirit possession.\" This spirit can force market prices to drop if the poor are suffering.\n\n126. For notes on the Ejiwa cult in Lagos, see unpublished notes of Kenneth Murray, Nigerian Museum, Lagos. The myth of Ejiwa was explained to me by Ajanaku Araba Ek\u00f3 (13 January 1972): \"Elegba while fishing is known as Ejiwa.\" Obaluaiye gave Ejiwa (Eshu) the gown of raffia because the latter had once been kind to him.\n\n127. Dos Santos, \"West African Sacred Art,\" pp. 50\u201366.\n\n128. Informant: Momuri Orilegbolodo, servitor of Buk\u00fau, Ketu, 8 September 1972.\n\n129. Dramatically pantomimed by Momuri Orilegbolodo in the process of her explanation of the staff's powers. She called it _ileesin gogo_.\n\n130. And her concern with instilling a social conscience in her followers (cf. Dassa-Zoum\u00e9 legends about Nana forcing market prices to fall to favor the poor) matches that of her fiery son.\n\n131. Informant: Babalawo Alawode Ifayemi, apparently recorded in Ketu country. Dos Santos, \"West African Sacred Art,\" pp. 58\u201365. I have retranslated portions of the original Yoruba version of the myth.\n\n132. Interview, Araba Ek\u00f3, 8 December 1975.\n\n133. _Ibid_.\n\n134. Yemitan, _op. cit_., pp. 4\u20136.\n\n135. For example, the altar to Yewa, an important riverain goddess, at Ipokia, capital of the Anago Yoruba, is circular, made of clay, and dyed a splendid indigo blue.\n\n136. Johnson, _op. cit_., p. 243: \"Burying the king in the bed of the river was regarded as an expiation made for his murder.\" The inner power of the riverain goddesses is likewise measured by the immense shield of flowing water that separates them from the world of the living.\n\n137. Judith Hoch-Smith, \"Radical Yoruba Female Sexuality,\" in Judith Hoch-Smith and Anita Spring (eds.), _Women in Ritual and Symbolic Roles_ (New York: Plenum Press, 1978), p. 265. As to the negative dimensions in the mystic use of fans, cf. Araba Ek\u00f3 (17 January 1972): \"There are fans, which the 'mothers' use, concealing poison; while fanning a 'mother' can use it as a means of throwing death or injury or madness upon a person.\"\n\n138. From a myth described to me by a migrant from the ancient city of \u1eccw\u1ecd in southern Ijebu country, winter 1963\u201364.\n\n139. Source: Henry Drewall, who has conducted excellent researches among Ketu, Egbado, and, recently, Ijebu Yoruba.\n\n140. Verger, _Notes sur le culture, p. 297_.\n\n141. Or, cf. Fernando Ortiz, _Los instrumentos de la m\u00fasica Afrocubana_ , Vol. II (Havana: Ministerio de Educaci\u00f3n, 1952), pp. 299\u2013301: \"There are serious cases when one must 'cool' a saint, or rather his stone, with five or seven fans in action all at once, according to what If\u00e1 (the oracle) indicates.\"\n\n142. Verger, _Notes sur le culture_ , p. 298, Abeokuta prayer 2.\n\n143. _Ibid_., p. 426, Ilesha verse 9.\n\n144. Jos\u00e9 Ribeiro de Souza, _400 Pontos Riscados e Cantados na Umbanda e Candombl\u00e9_ (Rio de Janeiro: Editora Eco, 1966), pp. 40\u201341.\n\n145. Informants have also mentioned that a starfish image might be intended here, for which there is good authority in Bahia. Cf. Edison Carneiro, _Candombl\u00e9s da Bahia_ (Rio de Janeiro: Tecnoprint Grafica, 1967), p. 45: \"A large starfish... in homage to Yemanya.\"\n\n146. Dr. Lawrence Longo of Los Angeles has made a film of the Oshun festival at Oshogbo, which was kindly screened for me in the summer of 1970 by George Ellis, then of the Museum of Ethnic Arts, UCLA. The film documents the flower-hurling portion of the Oshun festival.\n\n147. Verger, _Notes sur le culture_ , p. 422 (2,5,8), 423 (10,12,14,15,17), 424 (23,24,29), 425 (1,6) 426 (3,9) 427 (3), 428 (16).\n\n148. _Ibid_., pp. 429 (6,10,12,13,14), 430 (16;4), 431 (8,9), 433 (5).\n\n149. Lydia Cabrera, _Yemay\u00e1 y Och\u00fan: Kariocha, Iyalorichas y Olorichas_ (Madrid: C y R, 1974), pp. 271\u201372.\n\n150. I am extraordinarily grateful to the son of the late Oginnin for taking several days in July 1965 to explain in detail the meanings attached to this fan as his father had explained them.\n\n151. Cf. Abraham, _op. cit_., p. 622.\n\n152. I am grateful to the Alaafin of Oyo for the privilege of witnessing the annual perambulations of Alakoro in January 1964.\n\n153. For an excellent summary of the lore of Sh\u00e0ng\u00f3, see Babatunde Lawal, _Yoruba Sango Sculpture in Historical Retrospect_ (Ann Arbor: University Microfilms, 1970).\n\n154. Verger, _Notes sur le culture_ , pp. 342 (21); 351 (121); 359 (13,14,19;3); 361 (26,30); 362 (31); 362 (48); 363 (49,56); 366 (1); 374 (30); 378 (77,88); 380 (104,112); 381 (12); 390 (12); 392 (4), lit., \"beads, rich, on king of Oyo\"; 393 (11); 395 (4).\n\n155. Morton-Williams and Ojo, _op. cit_., p. 9.\n\n156. Vivaldo da Costa Lima, _Una festa de Xango no Opo Afonja_ (Salvador: IV Coloquio Internacional de Estudos Luso-Brasileiros, Universidade de Bahia, 1959) p. 19.\n\n157. Eva L. R. Meyerowitz, \"A Bronze Armlet from Old Oyo, Nigeria,\" _Man_ XLI, 15\u201337 (March\u2013April 1941), 26.\n\n158. Fernando Ortiz, _Los Bailes y el Teatro de los Negros en el Folklore de Cuba_ (Havana: Ediciones C\u00e1rdenas y Cia, 1951), p. 235.\n\n159. It is possible the modification of the gesture in this Brazilian instance was influenced by the strong presence of the Kongo left-hand-on-hip gesture _(telama lwimbanganga)_ with right hand forward.\n\n160. From a MS. in progress by Robert Farris Thompson, _The Face of the Gods: Art and Altars of the Black Atlantic World_.\n\n161. See Carl M. Hunt, _Oyotunji Village: The Yoruba Movement in America_ (Wahington, D.C.: University Press of America, 1979), p. 63: \"Villagers are skilled in leather crafts, pottery making and cloth dyeing... some of the men are also very good artists and skilled wood carvers.\"\n\n#### Two\n\n 1. Philip Curtin, _The Atlantic Slave Trade: A Census_ (Madison: University of Wisconsin Press, 1969), p. 188.\n\n 2. _Ibid_., p. 105.\n\n 3. Interview with Fu-Kiau Bunseki, winter 1979.\n\n 4. _Ibid_.\n\n 5. For further Kongo and other Bantu sources in black American speech, see Winifred Kellersberger Vass, _The Bantu-Speaking Heritage of the United States_ (Los Angeles: Center for Afro-American Studies, UCLA, 1979). See also J. L. Dillard, _Lexicon of Black English_ (New York: Seabury Press, 1977). This fine text is full of useful observations, such as (p. 61) \"Popular music has established a countertrend in which Black terminology has virtually taken over the domain for American English,\" and (p. 111) \"The terminology of [rootwork and conjure] differs more from mainstream language than does that of any other activity.\" The formula for the original _tobe_ charms is cited in K. E. Laman's _Dictionnaire Ki-Kongo-Fran\u00e7ais_ , Vol. II (Farnborough: Gregg reprint, 1964).\n\n 6. Robert Farris Thompson and Joseph Cornet, _The Four Moments of the Sun: Kongo Art in Two Worlds_ (Washington D.C.: National Gallery of Art, 1981), p. 149.\n\n 7. Vicente Rossi, _Cosas de negros: Los Or\u00edgenes del Tango y Otros Aportes al Folklore Rioplatense_ (C\u00f3rdoba: Imprenta Argentina, 1926), pp. 34\u201335. The source of Rossi's observation would appear to be the French naturalist Alcides d'Orbigny, who witnessed a _candomb\u00e9_ in Montevideo in 1827 which made him recognize that in dance Africans \"regain for a moment their nationality.\" See George Reid Andrews, _The Afro-Argentines of Buenos Aires: 1800\u20131900_ (Madison: University of Wisconsin Press, 1980), pp. 162\u201363.\n\n 8. Georges Balandier, _Sociologie actuelle de l'Afrique noire_ (Paris: Presses Universitaires de France, 1955), p. 39.\n\n 9. Wyatt MacGaffey, _Custom and Government in the Lower Congo_ (Los Angeles: University of California Press 1970), p. 306.\n\n 10. John M. Janzen and Wyatt MacGaffey, _An Anthology of Kongo Religion: Primary Texts from Lower Zaire_ (Lawrence: University of Kansas Press, 1974), p. 34.\n\n 11. A. Fu-Kiau Kia Bunseki-Lumanisa, _N'Kongo Ye Nza Yakun'zungidila: Nza Kongo_ (Kinshasa: Office National de la Recherche et de D\u00e9veloppement, 1969), p. 14.\n\n 12. _Ibid_.\n\n 13. As poetically instanced by the following citation from Karl E. Laman, _The Kongo, III_ (Uppsala: Studia Ethnographica Upsaliensia, 1962), p. 42: \"A chief inheriting the kingdom of Vunda must see the mountain Ludi... before his investiture, to enable him to judge _(ludika)_ with impartiality.\"\n\n 14. For details on Kongo medicine and government, see Thompson and Cornet, _op. cit_., pp. 37\u201342.\n\n 15. A point not lost on W. E. B. Du Bois, who wrote: \"The chief remaining institution was the priest... healer of the sick, the interpreter of the unknown, the comforter of the sorrowing, the supernatural avenger of wrong.\" Cited in John W. Blassingame, _The Slave Community: Plantation Life in the Antebellum South_ (New York: Oxford University Press, 1979), pp. 40\u201341.\n\n 16. From a work in progress kindly shared by Professor MacGaffey with the author.\n\n 17. Janzen and MacGaffey, _op. cit_., p. 34.\n\n 18. _Ibid_.\n\n 19. For further details on the sign of the four moments of the sun, see Thompson and Cornet, _op. cit_., pp. 43\u201352. See also Fu-Kiau Bunseki, _The African Book Without Title_ , privately published MS., 1980.\n\n 20. Fu-Kiau, Bunseki, _The African Book Without Title_ , p. 3\n\n 21. Fu-Kiau Bunseki, _N'Kongo Ye Nza Yakun'zungidila_ , p. 30.\n\n 22. Interview, Fu-Kiau Bunseki, winter 1978.\n\n 23. MacGaffey, _op. cit_., p. 25.\n\n 24. Lydia Cabrera, _El Monte: Igbo Fina Ewe Orisha, Vititinfinda_ (Havana: Edicones C.R., 1954), p. 136.\n\n 25. _Ibid_.\n\n 26. _Ibid_., p. 127.\n\n 27. _Ibid_. Wyatt MacGaffey translated this fragment of Ki-Kongo.\n\n 28. \"Como se prepara una nganga,\" in Cabrera, _op. cit_., pp. 118\u201348.\n\n 29. George Eaton Simpson, \" 'Baptismal,' 'Mourning' and 'Building' Ceremonies of the Shouters in Trinidad,\" _Journal of American Folklore_ , 79 (1966), pp. 537\u201350. Jeannette Hillman Henney, _Spirit Possession Belief and Trance Behavior in a Religious Group in St. Vincent, British West Indies_ (Ann Arbor: University Microfilms, 1968), pp. 202\u20134.\n\n 30. Simpson, _op. cit_., p. 547, Diagram 4. Chalked circles around the eyes of initiates enable them to \"see\" when they travel spiritually, which mirrors closely Kongo belief and custom.\n\n 31. Henney, _op.cit_., p. 203. In modern Kongo, Janzen and MacGaffey report ( _op. cit_., p. 23, Fig. 3), that this is the \"membership card of the Universal Church of the Twelve Apostles. The central device is an example of _glossalalia in print_ [emphasis added]. It was originally seen 'like a billboard in the sky' by the founder of the Church....\"\n\n 32. Roger Bastide, _The African Religions of Brazil_ (Baltimore: Johns Hopkins University Press, 1978), p. 206.\n\n 33. Napole\u00e3o Figueiredo, \"Os caminhos de Exu,\" _Sete Brasileiros e seu universo_ (Brasilia: Departamento de Documenta\u00e7\u00e3o e Divulga\u00e7\u00e3o, 1974), pp. 89, 96.\n\n 34. Jos\u00e9 Ribeiro de Souza, _400 pontos riscados e cantados na umbanda e candombl\u00e9_ (Rio de Janeiro; Editora Eco, 1966), p. 60.\n\n 35. _3000 pontos riscados e cantados na umbanda e candombl\u00e9_ (Rio de Janeiro: Editora Eco, 1975). According to Rainer Flasche, in his _Geschichte und Typologie Afrikanischer Religiositate, Marburger Studien z\u00fcr Afrika\u2013und Asienkunde, Seria A_ (p. 199): \"In the area of Umbanda [as opposed, apparently, to macumba] which is taken to be specially scientific, the _pontos riscados_ are today valued no longer as sacred signs that bring down the spirits but rather only as symbols that elucidate teachings [about such spirits].\"\n\n 36. I wish to thank Senhor Oresto Mannarino, director in 1968 of the Police Museum, for many courtesies involved in the study of these pieces. Both the cup and the sash are probably twentieth-century works, dating perhaps from before 1941.\n\n 37. De Souza, _op. cit_., p. 91. The text of the song combines scraps of Ki-Kongo _(zungi, Nzambi)_ and Yoruba _(Oduduwa)_ with Portuguese.\n\n 38. John H. Weeks, _Among the Primitive Bakongo_ (London: Seeley, Service & Co., 1914), p. 281: \"Shooting stars _(nienie)_ are believed to be spirits... travelling... about in the sky.\"\n\n 39. For a brief discussion of the Eshu\/Og\u00fan liaison in Afro-Brazilian thinking, see Roger Bastide, _Le Candombl\u00e9 de Bahia_ , (Paris: Mouton, 1958), p. 226, 43n. There are also clay images for Elegba in Cuba brandishing iron rods, \"roads of Og\u00fan.\"\n\n 40. Janzen and MacGaffey, _op. cit_., p. 35, 2.4.\n\n 41. _Ibid_., 1.1, 1.3, 1.6.\n\n 42. _Ibid_., 4.2, 4.5.\n\n 43. For details, see Thompson and Cornet, _op. cit_., pp. 37\u201339.\n\n 44. Karl E. Laman, _The Kongo, II_ (Uppsala: Studia Ethnographica Upsaliensia, 1957), Figs. 35A, 38.\n\n 45. Karl E. Laman, _The Kongo, III_ (Uppsala: Studia Ethnographica Upsaliensia, 1962), p. 132.\n\n 46. Interview, Fu-Kiau Bunseki, July 1975.\n\n 47. Interview, Wyatt MacGaffey, winter 1978.\n\n 48. Interview, Fu-Kiau Bunseki, November 1978.\n\n 49. Cabrera, _op. cit_., p. 123. This particular _prenda_ also included holy water from a Roman Catholic church, \"to make the medicine Christian,\" the head of a bat, the head of a chameleon, and many other things.\n\n 50. _Ibid_., p. 131.\n\n 51. Information given me by Lydia Cabrera, telephone interview, winter 1979.\n\n 52. Cabrera, _op. cit_., p. 121.\n\n 53. Lydia Cabrera, _Porqu\u00e9: Cuentos Negros de Cuba_ (Havana: Ediciones C y R, 1948), pp. 248\u201349. Cabrera mentions the names of Afro-Cuban carvers of _minkisi_ \u2014Tata \u00d1unga, Sab\u00e1 Caraballo, and Ta Rafael\u2014and a sugar plantation, Las Canas, where intensive carving of _minkisi_ -figurines was said to have occurred in Cuba in the last century.\n\n 54. _La Ilustraci\u00f3n Espa\u00f1ola_ , Ano XIX, No. XXX (15 August 1875), p. 1.\n\n 55. The _matiabo_ image was picked up from a rebel soldier in Zumarquacam, Cuba, and was said to have been filled with the ashes of Spanish soldiers. On the other hand, Fu-Kiau Bunseki (personal communication, November 1978) says the horn could have been used by an expert in peace-making as well as in war.\n\n 56. Louis Maximilien, _Le Vodou Haitien: Rite Radas-Canzo_ (Port-au-Prince: Imprimerie de l'Etat, 1945), pp. 185\u201391. Cf. also Maya Deren, _Divine Horsemen_ (New York: Chelsea House Publishers, 1970), p. 275: \"Pacquets Congo are bound as magical safe-guards. [their] efficacy depends on the technique of careful wrapping.\"\n\n 57. _Ibid_., photograph facing p. 188.\n\n 58. Informant: Andr\u00e9 Phanord, March 1975, Port-au-Prince.\n\n 59. Source: Lydia Cabrera, November 1978.\n\n 60. Fu-Kiau Bunseki, interview, December 1978.\n\n 61. For example, as early as the eighteenth century in Jamaica there were, among the blacks, notices about charms enclosing cemetery earth.\n\n 62. Newbell Niles Puckett, _Folk Beliefs of the Southern Negro_ (New York: Dover, 1969), p. 273.\n\n 63. Fu-Kiau Bunseki, October 1978.\n\n 64. Puckett, _op. cit_., p. 233.\n\n 65. Fu-Kiau Bunseki, interview, October 1978.\n\n 66. _Ibid_.\n\n 67. Robert Tallant, _Voodoo in New Orleans_ (New York: Collier, 1962), p. 229.\n\n 68. _Ibid_., p. 228.\n\n 69. _Ibid_., p. 229.\n\n 70. Eric Sackheim, _The Blues Line: A Collection of Blues Lyrics_ (New York: Schirmer Books, 1975), p. 432.\n\n 71. Cf. Thompson and Cornet, _op. cit_., p. 200, excerpted from a long interview with Fu-Kiau Bunseki, 30 September 1980. For a brief but brilliant analysis of symbolic usages of tombs in Kongo, see Kimpianga Mahaniah, _La Mort dans la pens\u00e9e kongo_ (Kinsantu: Centre de Vulgarisation Agricole, 1980).\n\n 72. Sarah Hodgson Torian, \"Notes and Documents: Antebellum and War Memories of Mrs. Telfair Hodgson\" ( _Georgia Historical Quarterly_ , 27, 4 (December, 1943), p. 352.\n\n 73. Georgia Writers' Project, _Drums and Shadows_ (Athens: University of Georgia Press, 1940); for further examples of tying the spirit of the dead person to the tomb by means of the last things she or he used, see John Vlach, _The Afro-American Tradition in Decorative Arts_ (Cleveland: Cleveland Museum of Art, 1978), p. 40. Cf. Peter A. Brannon, \"Central Alabama Negro Superstitions,\" _Birmingham News_ (January 18, 1925): \"A Gullah negro on the Santee river explained to me that it was their custom to place the last plate, the last glass and spoon used before death on the grave.\"\n\n 74. R. E. Dennett, _Seven Years among the Fjort_ (London: Sampson Low, Marston, Searle & Rivington, 1887), p. 104. The burial carriage is decorated with other items sometimes appearing on nineteenth-century Kongo tombs\u2014crockery, flags, statuary, vessels, and a wild-cat skin.\n\n 75. Alexander Barclay, _A Practical View of the Present State of Slavery_ (London: 1828), pp. 131\u201333.\n\n 76. _Georgia Writers' Project, op. cit_., p. 167.\n\n 77. Thompson and Cornet, _op. cit_., p. 201.\n\n 78. Laman, _The Kongo, III_ , p. 37.\n\n 79. Telephone interview, fall 1975, also cited in Thompson and Cornet, _op. cit_., p. 198.\n\n 80. Gilbert de Chambertrand, \"La F\u00eate des morts en Guadeloupe,\" _Tropiques_ , Vol. LII (December 1954), pp. 35\u201341. Ramona Austin brought this reference to my attention.\n\n 81. Fu-Kiau Bunseki, interview, 9 October 1977.\n\n 82. Andr\u00e9 Pierre, interview, Port-au-Prince, 22 March 1981.\n\n 83. Ruth Bass, \"The Little Man,\" in Alan Dundes (ed.), _Mother Wit from the Laughing Barrel_ (Englewood Cliffs: Prentice-Hall, 1973), p. 395.\n\n 84. Thompson and Cornet, _op. cit_., pp. 193\u2013210.\n\n 85. Fu-Kiau Bunseki, interview, fall 1977.\n\n 86. L'Abb\u00e9 Proyart, _Histoire de Loango, Kakongo, et autres royaumes d'Afrique_ (Paris: Berton and Crapart, 1776), p. 192\u201393.\n\n 87. Thomas Atwood, _The History of the Island of Dominica_ (London: J. Johnson, 1791), p. 265.\n\n 88. Informant: Edward Kamau Brathwaite, Kingston, Jamaica, September 1981.\n\n 89. I am grateful to Sa. Janina Rubinowitz, who pointed out a _kandu_ tree at Godo Olo, on the Tapanahoni River in Suriname among the Djuka in December 1981.\n\n 90. Samuel Selvon, _Ways of Sunlight_ (London: Longman Group Ltd., 1979), pp. 99, 101.\n\n 91. Eudora Welty, _The Wide Net and Other Stories_ (New York: Harcourt Brace Jovanovich, 1971), p. 156. See also the section on bottle trees in Thompson and Cornet, _op. cit_., pp. 178\u201381.\n\n 92. Thompson and Cornet, _op. cit_., pp. 208, 137n.\n\n 93. _Ibid_., pp. 197, 209, note 167. This section is but a preliminary assessment of the bottle-tree phenomenon in the Kongo New World. John Mack, assistant keeper of the Ethnography Department of the Museum of Mankind, has kindly pointed out (personal communication, 21 December 1981) the Bongo and Moru versions of the bottle tree among East African civilizations. With further evidence we may plot the distribution of intervening Bantu traditions, between East Africa and Kongo, thus revealing myriad possibilities of massive reinforcement of the tradition in the Black New World. Cf. W. and A. Kronenberg, _Die Bongo_ (Wiesbaden: Franz Steiner Verlag, 1981), Figures 54, 55, 73.\n\n 94. Lynda Roscoe Hartigan, _The Throne of the Third Heaven of the Nations Millennium General Assembly_ (Washington, D.C.: National Collection of Fine Arts, 1976); _The Throne of the Third Heaven of the Nations Millennium General Assembly_ (Montgomery, Ala.: Montgomery Museum of Fine Arts, 1977), pp. 1\u201320.\n\n 95. The work might be characterized as monumental visual glossolalia. Hampton devised a secret script in association with his sacred furniture, itself a kind of writing in three dimensions. Study of this work in comparison with \"spiritual writing\" in Rio, Trinidad, and Cuba might unlock unsuspected resonances and intentions, particularly as regards the geometrically enclosed mystic ciphers of the ground-drawings of the holy \"pointers\" of the island of St. Vincent.\n\n 96. Literature on the work of Henry Dorsey includes John Fewtterman, \"Tinkerer Dreams Up a Forest of Things,\" Louisville _Times_ , September 13, 1960; John Christensen, \"Henry Dorsey's House as Art,\" _Scene_ (Louisville _Times_ Sunday magazine), November 18, 1972; Guy Mendes, \"Found People: Some Figures on My Urn\" _Place\/Rogues Gallery_ II, 2 (1973), 44\u201346; Ellsworth Taylor, _Folk Art of Kentucky_ (Kentucky Arts Commission, 1976); Jan Wampler, _All Their Own: People and the Places They Build_ (New York: Oxford University Press, 1977), pp. 150\u201355; Charles McCombs, \"Modern Art in Brownsboro,\" in _Charles McCombs Almanac_ (Brownsboro: Charles McCombs, 1974).\n\n 97. Charles Duvelle, _Musique Kongo: Ba-Bembe, ba-Congo, ba-Kongo-Nseke, ba-Lari_ (Ocora LP OCR 35), accompanying essay, illustration 12: \"friction-drum, a _mukwiti_. This is an open, cylindrical drum to which one skin is nailed; a wooden stick hidden inside the body is firmly attached to one to the center of the membrane.... various metal objects of Western origin (switches, electric-light bulbs, electric cables, etc.) are attached to the body of the instrument and are supposedly connected to an earphone situated inside the drum.... While he plays the drummer operates the switches or turns a small handle....\"\n\n 98. I thank Eva Dorsey Williams, surviving sister of Henry Dorsey, who in an interview in the summer of 1974 let me copy pertinent facts from her family scrapbook, especially as regards the life and times of Maria Prewitt.\n\n 99. _Ibid_.\n\n100. Interview with Guy Dorsey, brother of Henry, February 1974. In a letter dated 21 October 1974 Guy Dorsey added a phrase that in many ways summed up the work of his departed sibling: \"My brother Henry... spent many hours of happy moments getting all his trinkets together and enjoyed showing people that stopped by to see everything in working order.\"\n\n101. Interview, Eva Dorsey Williams, fall 1974.\n\n102. James Joyce, _Ulysses_ (New York: Vintage Books, 1961), p. 49. Joyce apparently refers to a song, which itself refers to a prodigal wanderer, one of the themes of the Dorsey house. But this interesting coincidence cannot detain us here.\n\n103. Interview, Guy Dorsey, March 1974.\n\n104. Interview, Eva Dorsey Williams, July 1974.\n\n105. David C. Driskell, _Two Centuries of Black American Art_ (New York: Alfred A. Knopf, 1976), illustration, p. 89.\n\n106. Guy Mendes, _op. cit_., p. 46.\n\n107. Visible in Jan Wampler, _op. cit_., p. 151.\n\n#### Three\n\n 1. Melville J. Herskovits, _Life in a Haitian Valley_ (Garden City, N.Y.: Anchor Books, 1971), p. 31.\n\n 2. C. L. R. James, _The Black Jacobins_ (New York: Vintage Books, 1963), p. 394. See also Philip Curtin, _The Atlantic Slave Trade: A Census_ (Madison: University of Wisconsin Press, 1969), pp. 75\u201377, 144 (graph), 181\u201383, 191\u201397, 200. Curtin points out (p. 181) that Haiti's population stemmed from slaves who had accounted for more than three-quarters of the entire eighteenth-century French Atlantic slave trade.\n\n 3. See Chapter III, sections on _minkisi_ and _nz\u00f3 a nkisi_.\n\n 4. Interview with Ralph Isham, spring 1976, kindly granted by Mr. Isham.\n\n 5. Alfred M\u00e9traux, _Le Vaudou Haitien_ (Paris: Gallimard, 1958), p. 24.\n\n 6. See Robert Cornevin, _Histoire du Dahomey_ (Paris: \u00c9ditions Berger-Levrault, 1962), pp. 74\u201382.\n\n 7. For details, Karl Polanyi's _Dahomey and the Slave Trade_ (Seattle: University of Washington Press, 1966) provides an excellent analysis.\n\n 8. Paul Mercier, \"The Fon Of Dahomey,\" in _African Worlds: Studies in the Cosmological Ideas and Social Values of African Peoples_ (London: Oxford University Press, 1954), pp. 212\u201314.\n\n 9. This etymology, my own, fits the fact of the intense and creative creolizing pressure brought to bear on many elements of culture from Africa on Haitian soil. The Yoruba term _l'awo_ , \"mystery,\" expands into a broader semantic range when it dovetails with the Dahomean sense of mystery, meaning \"deity.\"\n\n 10. This chart draws on many sources, including Herskovits, _Life in a Haitian Valley_ , pp. 314\u201317; Harold Courlander, _Haiti Singing_ (Chapel Hill: University of North Carolina, 1939); Harold Courlander, _The Drum and the Hoe_ (Berkeley: University of California, 1960); and Maya Deren, liner notes to _Voices of Haiti_ , LP (New York: Elektra Records, c. 1953).\n\n 11. Mercier, _op. cit_., p. 223: \"G\u016b is specially linked with _Lisa_ , strength and the sun.... _Lisa_ comes down to earth holding _Gu_ in his hand in the form of a _gubasa_.\"\n\n 12. I heartily thank Vincent Kinhou\u00e9 Ahokpe of the Mus\u00e9e d'Abomey and his assistant for the privilege of photographing and studying the famous monumental _gubasa_ of Abomey, 10 January 1968. It was given as an \"altar of the spirit of victory.\"\n\n 13. Informants: Sagbaju, son of King Glele and brother of King Gbehanzin, and Akpalosi, son of Sagbaju, Abomey, 10 January 1968.\n\n 14. _Ibid_.\n\n 15. _Ibid_.\n\n 16. Michel Leiris, \"Note sur l'usage de chromolithographies catholiques par les voduisants d'Haiti,\" in _Les Afro-Americains_ (Dakar: Ifan-Dakar, 1953), p. 204.\n\n 17. Alfred M\u00e9traux, _Voodoo in Haiti_ (New York: Oxford University Press, 1959), Plate XV, print of St. James the Elder. For further argument on possible symbolism inherent in the shape of the staffs of the flags of _vodun_ , see my section of _The Four Moments of the Sun_ , pp. 172ff.\n\n 18. Seen on a field trip to Haiti, late August 1961, at Bizonton, in the company of Emmanuel Paul, who kindly led me to a rite for Papa Og\u00fan.\n\n 19. Explication of the iconography by Sagbaju and Akpalosi, Abomey, 10 January 1968. Their testimony jibes with Cornevin, _op.cit_., p. 127: \" _Gle gle ma yon ze_ fields cannot be lifted up [by the wind],\" which he, too, gives as a \"strong name\" of Glele. See also entry for _togodo_ in Julien Alapini, _Le Petit Dahom\u00e9en: Grammaire\u2014vocabulaire_ (Cotonou: \u00c9ditions du Benin, 1969), p. 257.\n\n 20. Informant: Vincent Kinhou\u00e9 Ahokpe, Abomey, 10 January 1968.\n\n 21. Cf. Maximilien Qu\u00e9num, _Au Pays des Fons: Us et coutumes du Dahomey_ (Paris: Larose \u00c9diteurs, 1938), p. 149.\n\n 22. Mercier, _op. cit_., p. 221.\n\n 23. _Ibid_.\n\n 24. I thank Wyatt MacGaffey for bringing this pun to my attention.\n\n 25. Moreau de Saint-M\u00e9ry, _Description topographique... de la partie fran\u00e7aise de l'\u00eele de Saint-Domingue_ , Vol. I (Philadelphia: 1797), pp. 210\u201311. Quoted in M\u00e9traux, _Voodoo in Haiti_ , p. 39. I have reworked the translation.\n\n 26. M\u00e9traux, _ibid_.\n\n 27. Jean Price-Mars, \"Lemba-Petro: Un Culte secret,\" in _Revue de la soci\u00e9t\u00e9 d'histoire et de geographie d'Haiti_ IX, 28 (1938). John Janzen, an anthropologist who has worked for several years in Kongo, has had occasion to analyze this text and finds many strong influences stemming from Kongo in general and the Lemba Society in particular. He discusses these parallels in a recent volume, _Lemba, 1650\u20131930: A Drum of Affliction in Africa and the New World_ (New York: Garland, 1982), pp. 273\u201392.\n\n 28. Jean Price-Mars, \"Lemba-Petro, un culte secret,\" in _Revue de la soc\u00ec\u00e9t\u00e9 d'histoire et de g\u00e9ographie d'Haiti_ , Port-au-Prince, 9, 28 (1938), 12\u201331.\n\n 29. Maya Deren, _Voices of Haiti_ LP.\n\n 30. Maya Deren, _Divine Horseman: Voodoo Gods of Haiti_ (New York: Chelsea House Publishers, 1970).\n\n 31. For discussion, see M\u00e9traux, _Le Vaudou Haitien_ , pp. 66\u201367.\n\n 32. Mercier, _op. cit_., p. 221.\n\n 33. Seen between Bizonton and Port-au-Prince during a survey of peristyles in the area of the capital in March 1975.\n\n 34. P. Amaury Talbot, _In the Shadow of the Bush_ (London: William Heinemann, 1912), p. 25, illustration.\n\n 35. M\u00e9traux, _Le Vaudou Haitien_ p. 68. Translation mine.\n\n 36. These structural observations are inspired by Karen McCarthy Brown's _The Veve of Haitian Vodu: A Structural Analysis of Visual Imagery_ (Ann Arbor, Mich.: University Microfilms, 1975), a landmark study.\n\n 37. Thompson and Cornet, _op. cit_., p. 203, suggesting links between this custom and the rhythmic unfurling and dancing with umbrellas at the traditional jazz funeral in New Orleans.\n\n 38. M\u00e9traux, _Voodoo in Haiti_ , p. 161.\n\n 39. _Ibid_.\n\n 40. And recalling the fact, as Polanyi, _op. cit_., p. 54, points out, that in old Dahomey \"everything went by pairs and even multiple pairs.... every official in the kingdom had his female counterpart.... throughout the army... every male, from the highest ranking officer down to the last soldier, had his female counterpart in the palace.\"\n\n 41. Interview with \u00c1lvares, near L\u00e9ogane, 21 March 1975.\n\n 42. Eleanor Ingalls Christensen, _The Art of Haiti_ (New York: A. S. Barnes and Company, 1975), p. 69.\n\n 43. I am grateful to Pierre Monosiet, curator of the Museum of Haitian Art, for bringing these flags to my attention in the summer of 1961. They have since been stolen from a private collection in New England and their whereabouts are still unknown. Monosiet himself suggested the tentative dating of 1945 for these flags.\n\n 44. John Janzen, Lemba, 1650\u20131930, pp. 284\u201385.\n\n 45. For the Tu-Chokwe design, see Eduardo dos Santos, _Sobre a religi\u00e3o dos quiocos_ (Lisbon: Junta de Investiga\u00e7\u00f5es do Ultramar, 1962) Fig. 1; for the Ndembu ground-sign, Victor Turner, _Chihamba the White Spirit: A Ritual Drama of the Ndembu_ (Manchester: Manchester University Press edition, 1969), Plate 8b; Pende drawings of cosmograms, from an unpublished MS. by R. P. de Sousberghe, were kindly shared with me, in a personal communication (18 August 1977), by Marie-Louise Bastin.\n\n 46. Cover illustration, Milo Rigaud, _La Tradition voudoo et le voudoo haitien_ (Paris: \u00c9ditions Niclaus, 1953).\n\n 47. R. F. Thompson, \"The Flash of the Spirit: Haiti's Africanizing Vodun Art,\" in Ute Stebich, _Haitian Art_ (New York: Abrams, 1978), pp. 34\u201335. _Rara_ refers to Kongo-influenced pre-Lenten street orchestras, _coumbite_ to African-influenced communal work parties fueled by percussion.\n\n#### Four\n\n 1. D. T. Niane, _Sundiata: An Epic of Old Mali_ (London: Longman, 1965).\n\n 2. _Ibid_., pp. 58, 93\u201394, note 62: \"A 'tana' is an hereditary taboo, can also mean a totem... in this case Soumaoro was forbidden to touch ergot, of which a cock's spur is composed, and as long as he observed this he could concentrate in himself the power of his ancestors.\" The presence of a similar belief, in the power of the animal- _tono_ (cf. _tana_ ), among Afro-Mexicans of the Costa Chica south of Acapulco would appear to constitute an important New World Mandeism.\n\n 3. Charles S. Bird, \"The Development of Mandekan (Manding): A Study of the Role of Extra-linguistic Factors in Linguistic Change,\" in David Dalby, ed., _Language and History in Africa_ (New York: Africana Publishing Corp., 1970), pp. 154\u201355.\n\n 4. _Ibid_., p. 156.\n\n 5. Charles S. Bird, personal communication, winter 1976.\n\n 6. _Ibid_.\n\n 7. Personal communication, spring 1980.\n\n 8. Gonzalo Aguirre Beltr\u00e1n, \"Tribal Origins of Slaves in Mexico,\" _Journal of Negro History_ , Vol. XXXI (1945), p. 280.\n\n 9. For details, see Gonzalo Aguirre Beltr\u00e1n, _Cuijla: Esbozo etnogr\u00e1fico de un pueblo negro_ (Mexico City: Fondo de Cultura Econ\u00f3mica, 1958), pp. 52\u201364. As late as 1801, authorities were complaining of the independence of the blacks of the region, \"that they didn't have a fixed residence, that they lived in the fields in scattered huts\" (p. 62).\n\n 10. Lic. Francisco V\u00e1zquez, _Ometepec: Leyenda de un pueblo_ (Puebla: Editorial Cajica, 1964), p. 74.\n\n 11. Aguirre Beltr\u00e1n, _Cuijla_ , p. 60, from a letter by the mayor of Guatulco to the viceroy of New Spain, 1591.\n\n 12. Gerard Brasseur, in his _\u00c9tablissements humans au Mali_ (Dakar: M\u00e9mories de l'Institut Fondamental d'Afrique Noire, 1968), p. 247, reports that the primary characteristic of nuclear Mande architecture is the nearly exclusive position held by the cone-on-cylinder type in traditional villages. In this matrix emerged the four variants of the Mande cone-on-cylinder style. For details, see Tod Eddy, \"Manding Architecture from Africa to the New World: A Study in the History of African and Afro-Mexican Art\" (unpublished paper, spring 1974), pp. 3\u201340. Eddy's paper has the advantage of field study, not only on the Costa Chica but in terms of long residence in Kangaba, the heart of the nuclear Mande. It is an excellent study and I have relied heavily upon it. My own field trips to Kangaba (1975) and to the Costa Chica in Mexico (spring 1977) confirm his findings.\n\n 13. I am grateful to Se\u00f1ora Julia Venedado, of the village of Montecillos, and her colleagues, for the privilege of making a close study of their family kitchen-roundhouses in March 1977.\n\n 14. Eddy, _op. cit_., p. 42, was the first to note this trace of possible (northern Mande) Soninke influuence on the roof types of Costa Chica black architecture. For details on Afro-Mexican roofing, see Beltr\u00e1n, _Cuijla_ , pp. 94\u201395.\n\n 15. Eddy, _op. cit_., p. 52.\n\n 16. Personal communication, 29 July 1977.\n\n 17. David P. Gamble, _The Wolof of Senegambia_ (London: International African Institute, 1967), p. 41.\n\n 18. Susan M. Yecies, \"African Influence in Domestic Mexican Architecture: San Nicol\u00e1s, Costa Chica,\" May 1971, New Haven.\n\n 19. Labelle Prussin, \"Sudanese Architecture and the Manding,\" African Arts III, 4 (1970), 15. Cf. George W. McDaniel, _Hearth and Home: Preserving a People's Culture_ (Philadelphia: Temple University Press, 1982), p. 33: \"Many traditional African societies did not think of a house as a single building; instead, homes consisted of clusters of buildings, the 'rooms' for sleeping, eating, entertaining....\"\n\n 20. Eddy, _op. cit_., p. 27.\n\n 21. Beltr\u00e1n, _Cuijla_ , p. 93.\n\n 22. Eddy, p. 56, was apparently the first to suggest this nuance: \"Two cross beams reinforce the roof from the inside instead of the lower internal ring of [thin and flexible rings of wood].\"\n\n 23. Aquiles Escalante, _El Negro en Colombia_ (Bogot\u00e1: Universidad Nacional de Colombia, Facultad de Sociolog\u00eda, Monograf\u00edas Sociol\u00f3gicas, 18, 1964), p. 5.\n\n 24. _Ibid_., p. 76.\n\n 25. Marquis de Wavrin, _Chez les Indiens de Colombie_ (Paris: Librairie Plon, 1953), p. 218. In Julian H. Steward's _Handbook of South American Indians_ (Washington, D.C.: Smithsonian Institution, 1949), Vol. 5, p. 8, it is stated that \"such houses [with a circular plan and conical roof] are reported for the Arawak of the West Indies, and for the Choco of Colombia.\" These are areas inhabited by blacks as well. The problem of mutual reinforcement of similar ideas in Colombia cannot, in the absence of fieldwork, be resolved. But it is imperative to compare the two traditions in the future on the assumption that outward Amerindian form might conceal a number of traits absorbed from runaway slave architects. Certainly to judge from a published photograph of Colombia Arawak dwellings, Mande influence seems likely.\n\n 26. Cf. _House and Gardens_ 129, 3 (March 1966).\n\n 27. _Virginia Historic Landmarks Commission Survey Form_ , no. 154-10 (June 29, 1973).\n\n 28. The phrase and explication of \"off-beat phrasing of melodic accents\" are to be found in Richard Waterman's classic article, \"African Influence on the Music of the Americas,\" in Sol Tax (ed.), _Acculturation in the Americas_ (Chicago: University of Chicago Press, 1952), pp. 207\u201318.\n\n 29. Venice Lamb, _West African Weaving_ (London: Gerald Duckworth, 1975), pp. 90\u2013100, 180, 190, 198. \"These Ewe blankets resemble strongly the weaving generally found in regions north of the Volta Region\" (p. 212).\n\n 30. _Ibid_., p. 20.\n\n 31. _Ibid_.\n\n 32. _Ibid_., p. 75. See also R. Vedaux, _Tellem_ (Berg en Dal: Afrika Museum, 1977), p. 76.\n\n 33. _Ibid_., pp. 86\u201387.\n\n 34. Cf. Roy Sieber's observation in his _African Textiles and Decorative Arts_ (New York: Museum of Modern Art, 1972), p. 190, apropos of a rhythmized _welmare_ [Fulani men's cover] \"a superb example of seemingly random placement of pattern.... the careful matching of the ends of the cloth dispels the impression of an uncalculated overall design.\"\n\n 35. Lamb, _op. cit_., p. 86.\n\n 36. Pascal James Imperato, \"Blankets and Covers from the Niger Bend,\" _African Arts_ XII, 4 (August 1979), 41. Owen Barfield's _Poetic Diction_ (Middleton: Wesleyan University Press, 1928) is a perfect school of learning for the appreciation of rhythmized textiles like _welmare_. Cf. p. 179: \"Imagination... lights up only when the normal continuum... is interrupted in such a manner that a kind of gap is created, and an earlier impinges directly upon a later\u2014a more living upon a more conscious.\"\n\n 37. Ren\u00e9e Boser-Sarivaxevanis, _Les tissus de l'Afrique occidentale_ (Basel: Pharos-Verlag, 1972), p. 83, Fig. 7.\n\n 38. Beth Ann Jeffe, \"The West African Narrow-Strip Weaving Tradition in the New World\" (senior thesis, Yale University, May 1977), p. 13.\n\n 39. _Ibid_. See also Vania Bezerra de Carvalho, _Mestre Abdias: O Ultimo Artes\u00e3o do pano da Costa_ (Salvador: Studio Domingos, 1982), p. 12\n\n 40. Jeffe, _op. cit_., p. 17.\n\n 41. I am grateful to Juana Elbein dos Santos and to Beth Ann Jeffe for copies of photographs of multistrip cloths by Abdias.\n\n 42. When on 10 January 1980 I asked Abdias, at his atelier in Salvador, why he used offbeat phrasing in his cloths, he replied: \"The design is more beautiful with elements thus separated out\" _(Fica mais linda ainda como separado o disenho)_ , and he added that offbeat phrasing distinguishes his work, as well as his inspiration, from machine-made cloths, \"which cannot make my zigzags.\" He repeated these points in August 1982.\n\n 43. Richard Price (ed.), _Maroon Societies: Rebel Slave Communities in the Americas_ (Baltimore: Johns Hopkins University Press, 1979), p. 317.\n\n 44. M. Ferdinand Denis, _La Guyane: ou histoire, moeurs, usages et costumes des habitants de cette partie de l'Amerique_ (Paris: Nepveu, 1823), p. 130, plus illustration facing p. 131. The word _camiza_ is not Amerindian but maroon black, referring to cotton loincloth. C. Healey brought this reference to my attention.\n\n 45. _Ibid_.\n\n 46. Richard Price, _The Guiana Maroons: A Historical and Bibliographic Introduction_ (Baltimore: Johns Hopkins University Press, 1976), pp. 14\u201315.\n\n 47. From a conversation with the paramount chief of the Djuka at his house in Dii Tabiki, 28 December 1981. Cf. _The Glossary of the Suriname Vernacular_ (Paramaribo: Volklectuur, 1961): p. 15, _dobroesten_ means \"textile type,\" p. 66.\n\n 48. Ma Apina, interview, Dii Tabiki, 29 December 1981: _Te wan Ndjuka sama o felifi sani den felifi mu kengi keele. Te a u go tapu a mu tapu a wan felifi mofu di de faawe fi en_. I thank James Park, an American linguist based in Paramaribo, for checking this translation.\n\n 49. Tony van Omeren of Paramaribo said on 1 January 1982 that _mamio_ , which he defined as \"different pieces of cloth sewn together,\" was more than a hundred years old (i.e., made before 1882) on the coast and that old mid-nineteenth-century examples were still to be found in a few Paramaribo collections. Else Henar-Hewitt of Paramaribo said that when a coastal black woman is expecting, they start to make a _mamio_. She also showed me a coastal costume of a festive type called _a meki sanni_ \"she makes the moves\" (Plate 140), worn by a special hostess whose function is to welcome people. Her costume alternates patterned cloth handkerchiefs set against vertically striped handkerchiefs, not unlike the cross-play between _dobi sten_ and _stiipi_.\n\n 50. John Vlach, _The Afro-American Tradition in Decorative Arts_ (Cleveland: Cleveland Museum of Art, 1978), p. 55.\n\n 51. Telephone interview, spring 1980.\n\n 52. From field notes generously shared with me by Dominic Julian Parisi. See also his excellent \"Needles and Drums: Antebellum Foundations for the Black Quilting Aesthetic\" (M.A. thesis, Yale University, May 1980).\n\n 53. _Yale Course Critique_ , 1973, p. 60.\n\n 54. Melville J. Herskovits, _Life in a Haitian Valley_ (Garden City, N.Y.: Anchor Books, 1971), pp. 254\u201355.\n\n 55. Informant: Ellen Walters, curator of exhibits, Cleveland Museum of Natural History, spring 1977.\n\n 56. Cf. Ruth Bass, \"The Little Man,\" in Alan Dundes (ed.), _Mother Wit from the Laughing Barrel: Readings in the Interpretation of Afro-American Folklore_ (Englewood Cliffs: Prentice-Hall, 1973), p. 393: \"Newspapers pasted on the walls will cause any spirit to stop and read every word on the papers before bothering you.\"\n\n 57. Gamble, _op. cit_., p. 41.\n\n 58. Cf. Sarah Brett-Smith, _Speech Made Visible: The Irregular as a System of Meaning_ , conference paper in press (August 15, 1980), p. 22: \"Bamana methods for transcribing the spoken word... cannot be clear.... [To] fulfill their purpose they must evade linguistic systematization and the socially (perhaps even politically) disruptive possibility of mass communication by introducing aberrant visual symbols which prevent immediate comprehension. Like the spoken, the transcribed word must remain indistinct and allusive; knowledge may thus rest secure in the shadowy realm of the aged or the exceptionally gifted.\"\n\n 59. Mary Douglas, _Purity and Danger: An Analysis of the Concepts of Pollution and Taboo_ (London: Routledge & Kegan Paul, 1966), p. 95.\n\n 60. Charles S. Bird and Martha B. Kendall, \"The Mande Hero: Text and Context,\" in Ivan Karp and Charles S. Bird (eds.), _Explorations in African Systems of Thought_ (Bloomington: Indiana University Press, 1980), p. 17: \"[Power]-laden parts of a hunter's kill\u2014skin, horns, teeth, claws, feather\u2014are incorporated into his... clothing. These serve to control the _nyama_ released by each kill, protecting the hunter from potential destruction; but they also empower him to perform greater deeds.\"\n\n#### Five\n\n 1. A list of writings on _nsibidi_ and _nsibidi_ \\- related material includes: Alfred Mansfeld, _Urwald-Dokumente_ (Berlin: Dietrich Reimer, 1908), Tafeln IV and V; Rev. J. K. MacGregor, \"Some Notes on _Nsibidi,\" Journal of the Royal Anthropological Institute_ , Vol. XXXIX, (1909), pp. 209\u201319; E. Dayrell, \"Some 'Nsibidi' Signs,\" _Man_ , No. 67 (1910), pp. 112\u201314; Elphinstone Dayrell, \"Further Notes on Nsibidi Signs with Their Meanings from the Ikom District, Southern Nigeria,\" _Journal of the Royal Anthropological Institute_ , Vol. XLI (1911), pp. 521\u201340; P. Amaury Talbot, _In the Shadow of the Bush_ (London: William Heinemann, 1912), Appendix G, \"Nsibidi Signs,\" pp. 447\u201361; M. D. W. Jeffreys, \"Corrections,\" _Man_ 67 (Nsibidi Writing), 1910, _Man_ , Nos. 189\u201393 (September\u2013October 1964), p. 155; Kenneth Murray, unpublished MS. fragment on \"nsibidi writing among Ehuman Clan, Enyong, Aro Division,\" Lagos, 1938; M. D. W. Jeffreys, \"Notes on Nsibidi,\" unpublished MS. compiled at Calabar c. 1925, 10 pp.; David Dalby, \"The Indigenous Scripts of West Africa and Surinam: Their Inspiration and Design,\" _African Language Studies_ , Vol. IX (1968), pp. 156\u201397; Robert Farris Thompson, \"Black Ideographic Writing: Calabar to Cuba,\" _Yale Alumni Magazine_ (November 1978), pp. 29\u201333; Kenneth Campbell, _Secrets of Nsibidi_ (unpublished MS. 1978); M.V. Spence, \"Nsibidi and Vai: A Survey of Two Indigenous West African Scripts\" (M.A. thesis, 21 May 1979).\n\n 2. P. Amaury Talbot, _op. cit_., p. 37.\n\n 3. Interview with Peter Eno, Mamfe, Cameroon, 26 May 1969.\n\n 4. Cf. Eno, _ibid_ : \"If you want to execute heavy things, dangerous matters, _nsibidi_ takes the lead.\" Cf. Jacques Derrida, _Of Grammatology_ (Baltimore: Johns Hopkins Press, 1976), pp. 101\u201340, \"the violence of the letter.\"\n\n 5. The late _ndidem_ of Big Qua Town, Nigeria, told me in January 1972 that the reason few Westerners had ever seen such signs was that they were swiftly drawn in chalk on the floor of the deceased person's house, among family members, and then quickly erased before others were allowed into the room.\n\n 6. Peter Eno, 26 May 1969.\n\n 7. Franklin W. Knight, _Slave Society in Cuba During the Nineteenth Century_ (Madison: University of Wisconsin Press, 1970), p. 50.\n\n 8. Daryll Forde (ed.), _Efik Traders of Old Calabar_ (London: International African Institute, 1956), p. 66, note 1 _a_.\n\n 9. Pedro Deschamps Chappeaux, \"Potencias: Secreto Entre Hombres,\" _Cuba_ VII, 72 (1968), 44.\n\n 10. Lydia Cabrera, _La Sociedad Secreta Abaku\u00e1_ (Havana: Ediciones C. R., 1959), includes four terms for Ejagham-related graphic writing in Cuba, including not only the now standard title for the writing, _anaforuana_ , but also _ereniyo_ (\"writing; symbol\"), _gand\u00f3_ (\"chalked sign\"); and _bang\u00f3_ (\"chalked sign\"). See pp. 68, 178.\n\n 11. Lydia Cabrera, _Anaforuana: Ritual y s\u00edmbolos de la iniciaci\u00f3n en la sociedad secreta Abaku\u00e1_ (Madrid: Ediciones R, 1975). I estimate there are approximately 512 discrete signs documented in this important text, a landmark study in the history of black Atlantic writing systems.\n\n 12. For a brief discussion of the Ejagham impact on the art of their neighbors, see R. F. Thompson, _African Art in Motion_ (Berkeley: University of California Press, 1974), p. 173.\n\n 13. Cf. Philip Allison, _Cross River Monoliths_ (Lagos: Department of Antiquities, 1968), p. 17: \"Fighting was a man's normal occupation and on his success... depended his membership of certain important societies.\"\n\n 14. Malcolm Ruel, _Leopard and Leaders: Constitutional Policies Among a Cross River People_ (London: Tavistock Publications, 1969), p. 182. During the fifty years up to 1904 there had been no major warfare between tribes but rather a series of skirmishes between single villages, according to Mansfeld, who administered Ejagham territory in what is now western Cameroon.\n\n 15. For a recent, brief study of the fatting-house institution among the Ejagham's western neighbors, the Ibibio, see Jill Salmons, \"Fat Is Beautiful,\" _Art Links (The Commonwealth Arts Review)_ , September 1981, pp. 23\u201325.\n\n 16. T. J. Hutchinson, _Impressions of Western Africa_ (London, 1858), p. 160.\n\n 17. Interview with Mme. Grace Davis, Calabar, December 1978.\n\n 18. Rev. Hugh Goldie, _Dictionary of the Efik Language_ (Ridgewood: Gregg Press, reprint edition, 1964), p. 279.\n\n 19. Interview: Mme. Grace Davis, Calabar, December 1978.\n\n 20. Talbot, _op. cit_., p. 152 (illustration).\n\n 21. See Ruel, _op. cit_., for data on a cognate women's society, Ndem, among the Banyang; p. 204: \"its primary purpose [was] giving status to women... men seeking community status sought to marry these 'leaders among women.' It was not a matter of chance that both in Tali and Besongabang when I inquired about _Ndem_ , I was sent to the senior wife of the village leader.\"\n\n 22. Cf. Mansfeld, _op. cit_., p. 71: \" _Die ganze Mode soll in strenger Anlehnung an dass historische Vorbild: die erste Mboandemdame, die inalten Zeiten vom Himmel heruntergestiegen kam, verfertigt sein_.\"\n\n 23. I thank the chief and elders of Onun village for the privilege of studying a fine _echi okpere_ , kept in the local Ngbe hall, 27 December 1975. I thank Peter Eno of Mamfe for related data from Ekwe areas.\n\n 24. Informants: Ngbe elders, Oban, August 1978.\n\n 25. Cabrera, _La Sociedad Secreta Abaku\u00e1_ , p. 148.\n\n 26. _Ibid_.\n\n 27. The fact that _Akuaramina_ is an Efik term is given in Cabrera, _La Sociedad Secreta Abaku\u00e1_ , p. 151. I am much indebted to Pierre Verger for his unpublished detail of the _Akuaramina_ mask (Plate 145).\n\n 28. Or, as Ruel, _op.cit_., p. 184, describes it, \"a cult organization by which young men... were organized and ritually strengthened for fighting.\"\n\n 29. Edwin Ardener, \"Documentary and Linguistic Evidence for the Rise of the Trading Polities between Rio del Rey and Cameroons, 1500\u20131650,\" in I. M. Lewis (ed.), _History and Social Anthropology_ (London: Tavistock, 1968), p. 126.\n\n 30. An elder at Oban, for example, told me that Efut Ngbe still carried prestige, and he pointed to songs sung in Efut in Oban Ngbe liturgy as an indication of that fact (August 1978).\n\n 31. A. J. H. Lathan, _Old Calabar: 1600\u20131891_ (Oxford: Clarendon Press, 1973), p. 36.\n\n 32. Cabrera _La Sociedad Secreta Abaku\u00e1_ , p. 88.\n\n 33. _Ibid_., pp. 79, 83\u201384. My translation occasionally modifies, elides, or adds to the renderings of Cabrera's informants with glosses provided by the late King of the Efut at his residence in Calabar in December 1975, e.g., where Afro-Cubans give the creolized term Efokondo, I restore it to Efut, _Efut Ekond\u00f3_ , and suggest its actual location in Efut country.\n\n 34. _Ibid_., p. 87: pez del r\u00edo; pez del mar; _Tanze Uyo_ (lit, Lord Fish of the Voice); _Elori lori; Tanze Kefenbere_ ; Tanze es Ekue (lit., Lord Fish is a leopard); Tanze es Ekue...; _Fuerza...; por quien suspiraban los Adivinos...; oyos amarillos (arema)_.\n\n 35. _Ibid_., pp. 92\u201395.\n\n 36. Ejagham art cryptically refers to a miracle of fusion, the blending of the lordly energy of the leopard within the body of the fish. Talbot's drawings show denizens of the deep, crocodiles, with bodies marked with the checkered symbol of leopard presence, and, as we shall see, the Glasgow Art Museum includes in its study collection a brass tray upon which is clearly represented a leopard-fish. Among the Eastern Ejagham Father Druken of the Mamfe area collected (1969) a myth of a stolen fish. When the fish was restored, order resumed within the world (the very existence of which had been threatened).\n\n 37. Ekwe Ejagham informants recited terse, simple myths of Ngbe origin, most of which coincided with the essential structure of the material published by Cabrera, e.g., \"Our people say woman discovered Ekpe, but could not keep secret... men took over\" (Otu village, Cameroon, June 1969).\n\n 38. See Thompson, \"Black Ideographic Writing.\"\n\n 39. _Ibid_.\n\n 40. Informant: the late Ndidem of Big Qua Town, Nigeria, January 1972.\n\n 41. Pedro Deschamps Chappeaux, _op. cit_., pp. 45\u201346.\n\n 42. Cabrera, _Anaforuana_ , p. 51. In Cuba there is the highest rank, with the title _ntere\u00f1ob\u00f3n_ , and then a generalized rank with the title _ob\u00f3n_. This elides seven or more grades in Nigerian and Cameroonian Ngbe societies.\n\n 43. Informant: Julito Collazo, Hamden, Connecticut, January 1970. Cf. also Cabrera, _Anaforuana_ , p. 183.\n\n 44. Cabrera, _Anaforuana_ , p. 497.\n\n 45. Julito Collazo, interview, January 1970.\n\n 46. Cabrera _Anforuana_ , pp. 496\u201397.\n\n 47. Cabrera, _La Sociedad Secreta Abaku\u00e1_ , p. 40.\n\n 48. D. Jos\u00e9 Trujillo y Monagas, _Los Criminales De Cuba_ (Barcelona: Establecimiento Tipogr\u00e1fico de Fidel Giro, 1882), p. 366.\n\n 49. Cabrera, _La Sociedad Secreta Abaku\u00e1_ , p. 68, 178.\n\n 50. Ruel, _Leopards and Leaders_ , p. 245.\n\n 51. Cabrera, _La Sociedad Secreta Abaku\u00e1: Ese son ereniyo de mu\u00e9 que mat\u00e1; son Sik\u00e1n y pesc\u00e1_.\n\n 52. Cabrera, _El Monte_ , p. 215. See also her _Sociedad Secreta Abaku\u00e1_ , p. 178: \"The graphic signs create and dominate; whatever is not signed has no reality.\"\n\n 53. Cabrera, _Anaforuana_ , pp. 158\u201377.\n\n 54. _Ibid_., p. 181.\n\n 55. Cabrera, _La Sociedad Secreta_ , p. 164.\n\n 56. Cabrera, _Anaforuana_ , p. 187.\n\n 57. Informant: William Tayui, a native of the Obang area, western Cameroon, summer 1969.\n\n 58. Roger Bastide, _African Religions in the New World_ (New York: Harper & Row, 1971), p. 115.\n\n 59. Cabrera, _La Sociedad Secreta Abaku\u00e1_ , p. 257.\n\n 60. _Ibid_., p. 266.\n\n 61. Juan Luis Mart\u00edn, _Vocabularios de \u00d1\u00e1\u00f1igo y Lucum\u00ed_ (Havana: Editorial Atalaya, 1946); Pedro Deschamps Chappeaux, \"El Lenguaje Abaku\u00e1,\" _Etnolog\u00eda y Folklore_ , 4 (July\u2013December 1967), pp. 39\u201347.\n\n 62. Israel Castellanos, \"El Diablito \u00d1\u00e1\u00f1igo,\" _Archivos del Folklore Cubano_ III, 4 (1928), 27\u201337.\n\n 63. _Ibid_., p. 37.\n\n 64. Cabrera, _Anaforuana_ , p. 440. Further glossed by Lydia Cabrera in conversation with the author, winter 1981.\n\n# **A BOUT THE AUTHOR**\n\nROBERT FARRIS THOMPSON was born on December 30, 1932, in El Paso, Texas. He received his Ph.D. in art history from the Yale Graduate School, having attended Yale College as an undergraduate. He is currently professor of African and Afro-American history of art at Yale University, where he is also Master of Timothy Dwight College. Professor Thompson has been a Ford Foundation fellow and has mounted major exhibitions of African art at the National Gallery in Washington, D.C. His previous books, classics in their field, are _Black Gods and Kings_ and _African Art in Motion_.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\nTHE ODYSSEY AND DR. NOVAK\n\nCopyright \u00a9 2018 by Ann Colley\n\nAll rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, digital scanning, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, please address She Writes Press.\n\nPublished May 22, 2018  \nPrinted in the United States of America  \nPrint ISBN: 978-1-63152-343-4  \nE-ISBN: 978-1-63152-344-1  \nLibrary of Congress Control Number: 2017955635\n\nInterior design by Tabitha Lahr\n\nFor information, address:  \nShe Writes Press  \n1563 Solano Ave #546  \nBerkeley, CA 94707\n\nShe Writes Press is a division of SparkPoint Studio, LLC.\n\nNames and identifying characteristics have been changed to protect the privacy of certain individuals.\nTo Alice and Herbert Rowe\n\n## CONTENTS\n\nINTRODUCTION: The Odyssey\n\nCHAPTER 1: Seed Time\n\n**PART ONE: POLAND, 1995-96**\n\nCHAPTER 2: Tangled Threads\n\nCHAPTER 3: The Land of the Poles, That Is Lost, Not Yet Lost\n\nCHAPTER 4: Into the Cold\n\n**PART TWO: UKRAINE, 2000**\n\nCHAPTER 5: A Worried Stillness\n\nCHAPTER 6: In Ruins Laid\n\nCHAPTER 7: Shadows of the Dead\n\nCHAPTER 8: The Leaving\n\nAFTERWORD: The Archives and Dr. Novak\n\n## _INTRODUCTION:_\n\n## THE ODYSSEY\n\n_Sing to me of the person, Muse, the person of twists and turns, who through the touch of a hand is fated to travel from her native land over the boundless earth and across the rush of gusting winds into regions where she knows not the language nor the way of life._\n\n\u2014Adapted from Book I of _The Odyssey_\n\nLooking out of the plane's window at the endlessly starry skies, like Telemachus, I wonder what has possessed me. It is late August 1995. Having abandoned the familiar routines of my life in Buffalo, I am on my way to Poland where for the coming year I shall lecture in English at the University of Warsaw. All night long, wrapped in a fleece, when not surrendering to the enticing oblivion of sleep, I weigh my chosen course and occasionally assuage my anxiety by leafing through a pamphlet on Polish etiquette. Did you know that in polite Polish society, it is thought improper for a woman to walk down the street with her hands in her pockets?\n\nKnowing scarcely any Polish after a prefatory two-week language course that has taught me little beyond how to utter a greeting and how to ask the way, I worry about how I am to make sense of my new surroundings. Condemned to chart my course through either the veil of translation or the hesitating guidance of a nonverbal gesture, such as a telling, subtle turn of a hand, I long for a \"bright-eyed Athena,\" to reassure me that \"some of the words you'll find within yourself, \/ the rest some power will inspire you to say\" ( _The Odyssey_ ). Irving Massey, my multilingual partner, is not to join me for another month.\n\nFive years pass, and again I leave the customary rhythms and language of Buffalo to embark on a year's journey to Ukraine, a country that initially is even less familiar to me than Poland had been. I am to join the Foreign Literature Department at Taras Shevchenko University in Kiev. This time I have received no language instruction at all. To make matters worse, I have no practice in reading the Cyrillic alphabet. As a result, worry and discomfort are once more my traveling companions, but, as before, they do not conspire, like Aeolus, to blow me off course and tear to shreds the sails of my curiosity and wreck my desire to be there. My insecurities might loose the sack and let all the winds burst out, but I grit my teeth and hold firm to my lifelong fascination with that part of the world. Ever since childhood Central and Eastern Europe have intrigued, if not bewitched, me. Not possessing the ability to enter these cultures through the sounds and words of their language, I must tread the land and see things for myself.\n\nWhat follows is a composite portrait of Poland in 1995 and Ukraine in 2000. During these years, both countries are, in their own ways, not only still dominated by the shadows of their pasts (their remembrances of war and the Holocaust) but also suspended between one mode of life and another. Having gained relative autonomy after the collapse of the Soviet Union in 1991, each is trying to chart its course. Caught between its old loyalties to Soviet rule and its increasing devotion to a Westernized economy, Poland is unsettled yet gradually realizing its entrepreneurial dreams. Ukraine, on the other hand, contends with its continuing dependency upon Russia as well as with its own internal quarrels. In spite of these tensions, in 2000, many Ukrainians are optimistically working toward forming more active connections with the West. They have hopes of joining the European Union. Few are yet explicitly imagining the deadly insanity of open civil conflict between Ukrainian nationalists and pro-Russian separatists. Nor are they expecting that, in just over a decade, Russian troops will seize Ukrainian territory and take back Crimea as well as invade eastern parts of the country. For the moment, stillness possesses the nation. All is relatively quiet and stable.\n\nThe resulting collage of my sometimes-solitary life and travels during my tenure in Poland and Ukraine draws upon materials that roughly mimic Wordsworth's \"spots of time,\" for this book assembles those passing, often seemingly insignificant, sounds and sights that remain in substantial lineaments and still invigorate the mind. Sad, absurd, aesthetically pleasing, ugly, painful, humorous, endearing, discordant, frustrating, reassuring, distressing, and often trivial, these spots of time do not fall prey, I trust, to the sterility of hollow generalizations. Rather, through their multitextured, multilayered, and overlapping particularity, these episodes remind one that a place is as intricate and as varied as the people and the landscape that structure its province. Never disowned by memory, these collected moments, like the long dry wind, still blow through my ears and beg to be recorded.\n\n## _CHAPTER 1:_\n\n## SEED TIME\n\n_There are in our existence spots of time, Which with distinct preeminence retain A renovating Virtue . . ._\n\n\u2014Wordsworth, _The Prelude_\n\nBefore I commence the narrative of my experiences in Poland and Ukraine, I must first pause and return to the origins of my journey to these countries, to the seed time of my life. Most particularly, I need to revisit the day in 1946 when, as a child, I met Dr. Novak. It was this indelible moment that was to launch my curiosity about Central and Eastern Europe as well as to send me on my subsequent travels\u2014hence the book's title, _The Odyssey and Dr. Novak_.\n\n### **ENGLAND, 1946-53**\n\nThis is where the odyssey begins. The time is a warm English summer afternoon in 1946. The place is the front garden of the Unitarian parsonage situated in a modest town barely six miles north of Manchester. Holding my six-year-old hand is Dr. Novak, the head of the Unitarian movement in Czechoslovakia. He has come, perhaps (what does a child know?), to talk with my minister father about such matters as the postwar recovery in Prague. Four years earlier, the Nazis had murdered Dr. Novak's predecessor. Behind us, rhododendron bushes bloom; a garden wall half conceals a row of unkempt trees. Then there is nothing. The wreckage has been removed. No longer can one catch a glimpse of the seventeenth-century dissenting chapel peering through the oaks' limbs, for, during the night of December 21, 1940, German incendiary bombs destroyed this historic building.\n\nWhile air raid sirens wailed in the dark, my parents, protecting a ten-month-old child and waiting for their own destruction, crouched in the cupboard under the stairs of the parsonage next door. On that same night, another incendiary bomb fell on the Victorian Sunday school building across the road, but because the device miraculously landed in an open toilet bowl, nothing happened. (Later, during the mid-1950s, when this building finally crumpled\u2014a victim of a sinkhole\u2014remembrances of wartime social gatherings, as well as the sounds of young nursery school children, descended, like Persephone, into the earth.) In the 1940s devastation was everywhere, for this community lay close to factories, mines, and textile mills. Ruins were a normal part of my childhood and played with my imagination.\n\nWhen I glance at the snapshot, my attention, as if drawn by magnetic force (there is no way for me to resist), hastens to the compelling gaze of the figures in the garden. I see and remember a prim child all too willing to bury her hand within Dr. Novak's kindly grasp. Or did I reach for his? The two of us stand before the Kodak or the Brownie in a pose of affectionate understanding. He leans ever so slightly toward me. There's a kindred fondness, a bond that, for me, was to extend miles and years beyond the frame of the moment. I can still see his slicked back hair (it lies as full and as coarse as a badger's back); his eyebrows, which hang like a cliff's promontory over the depth of his eyes; his tailored suit (the buttons seem tight\u2014is the suit a leftover from a younger time?); and, most of all, I remember his elegant walking stick as well as his erect figure and polished shoes. Dear Reader, do I sound too much like Jane Eyre? But I never saw or heard of him again. I do not even recall his first name. What happened before or after remains a mystery. Did he disappear into the maw of the Soviet machine that was to crush a yearning, among many Czechs, to maintain contacts with the West? Was he arrested and imprisoned?\n\nIn spite of all the changes to come, that summer day in 1946 resisted the tyranny of alteration and circumstance. Rather, salvaging itself from the rubble of time, that afternoon lingered long, defying the narrow days of life, and grew incorporate into me. In particular, the kindly pressure of Dr. Novak's hand imprinted itself indelibly on my youthful mind and later led me beyond the garden wall of my childhood to Czechoslovakia and then to the USSR, Poland, Ukraine, and Belarus.\n\nDirectly after World War II, other Czechs came to my father's parsonage. Their visits reinforced my attraction to Eastern Europe and contributed to curiosity's charm. There was Ludik Benes (also from Prague), a young man whom I adored and followed from room to room. I have no photograph of him, but I do possess a slender book, _Little Paul Dombey (Dombey & Son): A Charles Dickens Story Told for Children_, that he presented to me as a parting gift. On the title page he wrote: \"To my little friend Anicka 15th Aug. 1946.\" Ludik was visiting because he was part of an international team of young Unitarians, the International Religious Fellowship (IRF), attempting to promote a better understanding among people of different nations. Fervently, yet delusively, idealistic, the group had rallied its forces and was meeting for a week in Manchester, during which period Ludik boarded at the parsonage. Two years later, when the Soviets seized full power in Czechoslovakia and were purging the so-called \"dissidents\" from all levels of society, Ludik, as well as the other Czech delegates, was denied a passport by the Czech authorities. As a protest, the IRF ceremoniously elected him president in absentia. After 1948, Ludik Benes disappeared.\n\nThere was also Mrs. Kessler, who, to escape persecution and death as a Czech Jew, had come, via an anguished passage, to live in our area. (For several moments when I was writing this, I suddenly forgot her name and sadly realized that I had no way to recall it, no one to help me find it.) She moved to a bungalow a few doors from where Mrs. Rowe, my \"second mother,\" lived.\n\nTo my young eyes, Mrs. Kessler seemed old, but she was probably in her forties. And exactly how she made the passage from Czechoslovakia to where we were, I did not know. All I grasped was that she was displaced. Mrs. Rowe took Mrs. Kessler under her wing. The three of us occasionally drank tea and ate rationed slices of bread and butter together. (Fearing there might be shortages, the Labour government introduced bread rationing in July 1946. Butter had already been restricted since January 8, 1940\u2014the day before I was born. Affected by the rationing, I now consume more butter on my bread than is good for me.)\n\nI liked to listen to Mrs. Kessler talk\u2014I was attracted to her unfamiliar pronunciation of common words. Mrs. Kessler was a gracious, cultured person who spoke of art and music; even at the age of seven, I found pleasure in that. Her house was quiet. There was no one else\u2014no photographs, just furniture. That had not always been the case, but I did not fully understand just what she had lost.\n\nIn 1947, I spent hours with Mrs. Rowe in her kitchen. There we kept a sharp lookout for the postman, who might bring some missive from Czechoslovakia. Like a couple of eagles balanced on an overhanging branch and waiting for a fish to swim through the brook below, we eagerly watched for the postman's bicycle to appear, drift silently down the road, and stop before the gate of 33 Park Lane. Through the open kitchen window, Mrs. Rowe handed the postman letters she had written on behalf of people desperate to escape Soviet rule.\n\nAs a naive seven-year-old, infected by Mrs. Rowe's contagious earnestness, I joined in her endeavors by standing on top of the kitchen table, from which stage I delivered stern homilies to fictitious Stalinist officials. (The script belonged to my imagination.) At the top of my voice I ranted against communism, censorship, and regulation. After a moment or two, the seriousness collapsed and Mrs. Rowe cried with laughter. Reveling in this catharsis, I continued to rage with more passion than ever. My playfulness was strangely raw, for I knew enough to sense how political circumstances had ripped apart the lives of those I had met and those Mrs. Rowe attempted to help.\n\nIn that same year I was sent off from home to a boarding school in Highgate (London N6).\n\n_\"Good-bye, good-bye, to everything!  \nTo house and garden, field and lawn.\"_\n\n\u2014Robert Louis Stevenson\n\nAt school, I lived in a building adjoining another, badly damaged by a parachute mine. The ruins were strictly off limits, but we dared each other to scramble among the piles of rubble and hide among the weeds, which, like a phoenix rising from the ashes, had reclaimed their existence. When I made that move, my memories of Dr. Novak, Ludik Benes, and Mrs. Kessler diminished (but did not disappear), to be replaced, temporarily, by the reality of authoritative teachers who gave out conduct marks and enforced rules (the stale leftovers of a late Victorian era) that governed most minutes of the day - though there were the Sunday walks in the Highgate Cemetery, where, in \"crocodile\" lines, we passed by Karl Marx's grave and saw where his followers left flowers. While my fellow boarders strode on, eager to get back to their afternoon tea, I turned my head and cast one more glance at Marx's larger-than-life dark-granite head. I knew enough to realize that even though Marx had been born in Germany and had lived in London, his writings had inspired the ideology of Eastern Europe. My early curiosity about that area of the world had not departed entirely.\n\nDuring my time in boarding school, when Mrs. Rowe wrote to me, she said not one word about her campaigns to help those fleeing from Soviet oppression; rather, missing the child she had never had, she sent poems that in her Lancashire best she had once recited, at my insistence, over and over again. One was \"That Tempting Apple Pie.\" Her round bright face, her crescent forehead framed by thinning short curls, and her pursed mouth had gleefully shaped each word:\n\n_One day, Mumsy Pig made a fine apple pie,  \nAnd till ready to cook it she put it up high; But Piggy and Wiggy discovered it there,  \nAnd to get it, they simply climbed up on a chair.  \nSoon they cut it and greedily ate the wet paste,  \nFor they said 'twas a shame such a good pie should waste._\n\n_But Mumsy Pig caught them and put them to bed,  \nAnd for dinner they'd physic and water and bread._\n\nIn a very loud voice, she had always added, \"an natty mediton\"\u2014her version of a child's attempt to utter the phrase \"and nasty medicine.\"\n\nLate in 1952, my parents suddenly announced we were emigrating to America. We were to leave in February 1953. One result was that Mrs. Rowe came to visit us in London, to see us for one last time. During her stay, in the rainy autumn, on what seemed to me a sad, long, unlovely street (I did not want to make the move), we met with two Czech refugees with whom Mrs. Rowe had corresponded. They had recently arrived in England. In their presence, remembrances of Dr. Novak, Ludik Benes, and Mrs. Kessler yet again haunted my youthful mind.\n\nAs poignant as this meeting in London was, one other morsel from this disruptive period of my life remained, refusing to be brushed into the litterbin of forgetfulness. Immediately before we were to be interviewed and approved for emigration by an official from the American embassy, my mother took me aside and tearfully begged that I not utter the word \"communism.\" (\"We will not be allowed to enter if you mention the term\u2014and don't tell them about the people who live across the street and display a hammer-and-sickle banner in their window.\") The specter of communism and Russia from my earlier childhood had, a few days before, drawn me to our neighbor's house. Curious as ever about anything to do with Eastern Europe, I had stood staring at their flag. Aware of the hysteria surrounding the McCarthy hearings investigating communist infiltration in the United States, my nervous mother feared I might say something that would cause us to be denied \"alien\" status.\n\n### **PRAGUE 1969**\n\nIn America, years passed, and I was married. My honeymoon was not conventional. In 1969, caught in the curious undertow of my childhood imagination, I spent the first weeks of my married life not only revisiting England but also crossing into a territory that had once been vividly in attendance, yet inaccessible, if not forbidden. After spending several weeks with Mrs. Rowe, my new husband and I traveled, via ship and train, to Prague and Bratislava. I was thrilled that I was actually going to be where Dr. Novak, Ludik Benes, and Mrs. Kessler had once resided. My fantasies were to become more tangible.\n\nThe summer of 1969 was a strange (and, for me, exciting) time to be in Prague, for it was just over a year since the so-called Prague Spring. Commencing in January 1968, broad-based governmental reforms had begun to soften rigid communist doctrines ruling the country. Under Alexander Dubcek's government, Czechs dreamed of democratic elections, greater freedom of speech and religion; they wished to abolish censorship, institute industrial and agricultural reforms, and end restrictions on travel. When we arrived in Prague, however, these hopes had been dashed or at best compromised. Exactly a year before, on August 20, 1968, two hundred thousand Warsaw Pact troops from the Soviet Union, the GDR, Bulgaria, Poland, and Hungary had invaded Prague in what was the largest deployment of military force in Europe since the end of World War II. Alarmed by what appeared to be the imminent collapse of communism in Czechoslovakia, the USSR ordered these troops and their tanks to roll through the streets of Prague. En masse, the city's residents gathered in the streets and protested against the incursion by blocking and climbing over the tanks, tearing down street signs to confuse the soldiers, identifying and following cars belonging to the secret police, setting up underground radio stations (journalists had tried to prevent the troops from taking control of Radio Prague), and sacrificing their lives. More than one hundred protestors were shot. And in January 1969, a student, Jan Palach, protesting the suppression of free speech, set himself on fire in Wenceslas Square.\n\nWhen we arrived in Prague a few months later, in the summer of 1969, the sensation, if not the fact, of a hostile, invading Russian presence lingered on each street corner and hung around the doorways of buildings. Dubcek, the reformer, had become increasingly isolated and had been replaced by a \"realist\" willing to cooperate with the Soviets. Many of Czechoslovakia's intellectuals and business elite had fled, if they could, to the West. The people passing us in the streets were tense, confused, damaged, and distrustful.\n\nI have memories of arriving in the train station and wandering with grudging assistance until we found a rather drab hotel with an available room. My college German was helping a little, but the street signs were still down so when we actually found a place to sleep, we were relieved. Economic difficulties were visible; there was not a sense of plenty, especially when we spent two days in Bratislava and wandered aimlessly around faded, damaged buildings that lined depressed streets like discarded rags. Shopping was difficult too. Goods I took for granted were neither plentiful nor available. I never found out how to buy sanitary napkins (what are now euphemistically called \"feminine products\"); therefore, I was a mess and uncomfortable much of the time (and annoyed with myself for not having anticipated my needs).\n\nFinding a place to eat was always a challenge. Not recognizing the food and not knowing the language, we found a cafeteria in Prague where we pointed to the tray ahead of us and simply motioned \"the same.\" I more often than not ended up with boiled tripe\u2014a dish I despised but that was a favorite of my mother's\u2014a taste developed during poorer times.\n\nUnguided, and always a bit lost (sometimes giddily\u2014we were young), we strolled alone through Prague's streets, gazed at the splendid architecture, and crossed the Charles Bridge (occasionally, whether I imagined it or not, I sensed we were being followed). Eventually George Zidlicky, the brother of one of the Czech refugees whom Mrs. Rowe had sponsored, took charge of us. The arrangement had been made ahead of time.\n\nWith the Zidlicky family, we took pleasure in the warm, relaxed, and lighthearted moments. Within their home, along with the men, we ate generous portions of meat and potatoes while the women sat behind our chairs, watched, and waited for us to finish; we visited George's butcher shop, where he vigorously wielded his sausages, and joined George's wife and her potato-peeling friends in the stone courtyard adjoining their house.\n\nOnly George spoke English, so we communicated with gestures and smiles\u2014and perhaps with a few phrases of my halting German, a tongue uncomfortably familiar to them through Nazi occupation. But there were also sobering, anxious episodes that resurrected thoughts about the circumstances surrounding the refugees and visitors from Czechoslovakia I had met during the immediate postwar period. In particular, I recall George's discomfort when he took us to the historical sites, such as the Prague Castle and Wenceslas Square\u2014places where the militia was conspicuously in attendance. In less populated areas, fearing we were being overheard by an invisible secret police and frightened of their hawk-like surveillance (their \"eyes so sharp that they can even see whose trouser-strap has come undone on the other side of the pavement\" [Gogol]), George periodically whispered that we should keep our voices down. Casting furtive looks, he hastily led us away from spying ears. On the street, he was as wary as a stray cat. He refused to come near our hotel.\n\nWalking the pavements where those I had known had once strolled was exhilarating but paradoxically upsetting, for, strangely, their absence became more definitive than before. So vivid in my childhood, they really were no more. The closer one draws to a memory, the more it dissipates. Being where they had once breathed left me with a better understanding of what had formerly troubled or disrupted their lives. I had a more immediate sense of what it must have been to be subjected to a hostile, authoritative rule. Though the visit reminded me of loss, it also opened doors.\n\n### **USSR, MAY 1985**\n\nYears elapsed, and marriage ruptured. Eventually I met Irving Massey, a professor of comparative literature, whose fluency in Yiddish, Russian, and Polish nurtured my fascination and made it more possible to travel in Eastern Europe. In May 1985, both of us, curious to see with our own eyes the landscape of the Soviet state, spent two weeks on our own in the USSR. Intourist, the official tourist bureau, arranged our accommodations and transportation.\n\nAmong my first memories is the train ride from Helsinki to what was then Leningrad. When we boarded the train in Finland, we were put in a private compartment reserved for foreigners. From time to time, a stout, officious uniformed woman came in to check on us and serve tea. It was when the train stopped at the Soviet border that what had been a moderately comfortable ride turned sour. Thick-booted guards entered our compartment, pulled down the trap door in the ceiling, inspected it for contraband, and then demanded to see our papers and search our luggage. Irving is not the neatest of scholars, so pages and pages of notes and piles of index cards, among which was an address book, spilled out of his cases. Frustrated with his mess, the border guards demanded I leave the compartment. After several minutes, I grew worried and tried to force my way back in. (I had watched too many 1930s spy films; my imagination was creating a dire scenario.) No success, for a guard blocked the door and yet another pushed me up against the corridor window. I gave in and waited. Soon the guards left with Irving's address book in hand. I rejoined Irving, who was none the worse for the interrogation. We waited at least an hour. The book was finally returned. It had been photocopied.\n\nWe were on our way, moving through the Russian countryside, past rows of silver birches, and thinking of Gogol, Pushkin, and Tolstoy. The train started to slow down and\u2014like an old dog that, on its afternoon, walk stubbornly sits down and refuses to move any farther\u2014obstinately came to a full stop. After a few confusing minutes, a passenger poked his head around our compartment door and told us that the train had broken down. Complaining about the heat of the day, he encouraged us to exit through the carriage doors, breathe in the fresh air, and sit for a while on the grassy bank by the tracks. A leap and we were out among the wildflowers, the tall swishing grasses that stretched their fingers toward the cloudless sky, and the Russian passengers, lolling about, enjoying their release. The whole character of the journey had shifted. People were relaxed and singing; one man gallantly handed me a bunch of flowers he had plucked from an adjoining meadow, and even more surprising, the stout matron took off her tight wool uniform jacket, undid the top buttons of her blouse, and while displaying distinct cleavage, flirted and flitted like a pastoral nymph with a willing passenger or two.\n\nA whistle blew. The Arcadian idyll was over. The train had been fixed. It was time to reembark. Another shift. People returned obediently to their cheerless seats; the matron buttoned up her blouse and resumed her bureaucratic posture. Such was the introduction to what, for the passengers on the train, was a normal schizophrenia\u2014a state of mind oscillating between a darkening oppressive authority and the radiant sensuous sentimentality of a nineteenth-century Russian novel.\n\nOnce we were in the Leningrad station, an Intourist guide, immediately recognizing us, swept us away in a long black limousine, down the middle lane of wide thoroughfares normally reserved for officials until we reached our hotel, the Hotel Europeiska, where ladies at desks, placed by each lift and on every floor, watched our comings and goings, night and day. Our room was bugged so we spoke discreetly.\n\nDuring the luminous days and white nights in Leningrad, on foot, we explored the nearby Nevsky Prospect, a grand boulevard designed in the eighteenth century by a French architect. A verse describing the Nevsky Prospect from Pushkin's \"The Bronze Horseman\" came to mind:\n\n_The empty thoroughfares, past number,  \nAre piled, stand clear upon the night;  \nThe Admiralty spire is bright;_\n\n_Nor may the darkness mount, to smother  \nThe golden cloudland of the light,  \nFor soon one dawn succeeds another  \nWith barely half-an-hour of night._\n\nInstead of the packs of wolves that had once roamed this area, sculptured stone lions with one foot in the air guarded eighteenth-century buildings, vaunting their former splendor. Lining both sides of the concourse were palaces, theaters, opulent homes, financial institutions, cathedrals, monasteries, museums, naval headquarters, government offices, and once luxurious shops. At all times of day, streaming past these structures, masses of people flowed up and down the boulevard: pen-pushing civil servants, office workers, financiers, housewives, business men, children going to and from school, bureaucrats, shopkeepers, tradesmen, and, as Gogol once observed, people dashing off from work \"to visit a friend from the office who lives on the third or second floor of a derelict building, in two small rooms with hall and kitchen\" (\"The Overcoat\").\n\nOccasionally, individuals extracted themselves from this torrent of pulsating humanity to pause and look through a shop window. If one wanted to get nearer to see the goods on display, one pushed the person blocking one's view\u2014a behavior that alarmed me until I learned that there was nothing hostile in the gesture. And occasionally someone left the multitude to fill his mug with kvass, a drink made of fermented bread, from a tankard on the back of a wagon parked at a street corner. The striking contrast between the dense movement of the people on the broad pavement and the empty stillness of the road added to my sense of the schizophrenic culture we had entered.\n\nSuch incongruities were to accompany us in Moscow. When we arrived, we stepped off the train into a crowd of unfamiliar faces and waited for our Intourist guide. He was not to be seen. Instead, out of the swarming masses of tired, anxious passengers a petite, bow-legged elderly lady, as if an apparition disembarking from the past, quietly approached. Barely making herself heard above the sound of the hissing steam expelled from the train's massive engine, she asked Irving, in the most polite and elegant Volga Russian, where the toilet was.\n\nThe Intourist machinery had faltered; no one had come to meet us. We felt disoriented. Thanks to Irving's fluent Russian, however, we eventually found a taxi driver willing to take us to our hotel. On subsequent days, we were not left alone\u2014someone was always visibly trailing or covertly listening to us. In the meantime, on the way to the hotel, while Irving chatted with the driver, I, unable to join in, tried to make sense of where I was. Gazing through the vehicle's window, I caught sight of enormous brilliantly colored banners, as long as ten stories high, hanging in multiple rows from Stalinist-style buildings. Displaying massive images of hammers and sickles as well as imposing portraits of political leaders, this drapery impersonated the clich\u00e9s of my imagination and belonged to a fictitious set, designed for a grade B thriller.\n\nIn the days that were to pass, however, I came to realize that such theatrical scenery was more of a reality than not. These authoritative banners surveyed and regulated everything: the people waiting for hours in orderly queues to view Lenin's tomb, even the monotonous, repetitive run-down stalls selling the same fur hats in the once-grandiose Gum Department Store, a building that stretches a quarter of a kilometer along the Kremlin's walls. The ghost of Stalin and his lethal policies now cast a deathly pall over Gum's granite, marble, and limestone gothic arcades. (In 1928, Stalin converted the building into offices, and in 1932, when his wife died, he used the space to display her body.) This opulent structure had deteriorated under the weight of economic difficulties, repressive Soviet regulations, and State control. On the third tier and out of sight, however, a luxurious clothing store was still open but only to the highest echelons of the Communist Party.\n\nBeing Jewish, Irving wanted to visit the synagogue in Moscow. Not knowing the way and having only an address, we took a taxi. When we arrived and stepped onto the pavement, an elderly man, descending from nowhere, hastily warned us in Yiddish, \"Be careful\u2014the rabbi is KGB.\" Like Hermes, god of the golden wand, once the message was delivered, he went on his way and soon was out of sight. Puzzled and slightly shaken, we entered the synagogue and spent a few guarded minutes with the rabbi who seemed superficially pleasant and interested in supporting the aging remnants of a Jewish community. But how to judge? Subject to scrutiny, he too led a double life and had to make compromises.\n\nOn another afternoon, we met with relatives of a Buffalo Russian friend. We had tea in their crowded two-room apartment, where four people lived. Shelves of books and objects, as well as boxes piled from floor to ceiling, took up most of the available space. After tea, we went for a walk. One of the men was extremely nervous. I recall turning a street corner and his urging us to walk quickly and keep quiet. The incident took me back to Prague in 1969.\n\nOur departure from Moscow was fraught with difficulty; we almost missed our plane to New York. Annoying, rule-ridden, niggling customs officers took hours searching our luggage and would not let us through. We were trapped. In the nick of time, an official, who wanted us out of the way, interceded and rushed us to the departure gate. Breathless, we dashed up the rickety steps leading to the plane. The exiles must, for the moment, return home.\n\n## **P ART ONE**\n\n## **P OLAND, 1995-96**\n\n_What had commenced with Dr. Novak's visit to postwar England was fated to carry on, but this time for a more extended interval. Beginning in 1995, I was to reside and work in Poland for an entire year. What follows, I trust, captures the nuances of a life lived in the midst of political and social change. Dreams of economic advancement as well as fantasies of financial success were enticing the public and intermittently intruding upon their remembrance of the country's sad and vicious history._\n\n## _CHAPTER 2:_\n\n## TANGLED THREADS\n\n### **WARSAW, SEPTEMBER 1995\u2014JUNE 1996**\n\n_And I seek the land of the Poles that is lost, that is not yet lost. Some say nearly lost, already lost, lost once more . . . lost to whom, lost too soon, lost by now, Poland's lost, all is lost, Poland is not yet lost._\n\n\u2014 G\u00fcnter Grass. _The Tin Drum_\n\nIn September 1995, strands of communism as well as remnants of Soviet rule are unraveling and clumsily intertwining with the government's increasing commitment to a Westernized economy. Old loyalties are at odds with entrepreneurial dreams of growth and private wealth. Ambition, anxiety, and ownership clash with collectives, unions, and a nostalgia for the security associated with a socialized state.\n\nPoland does not float on an expertly crafted rug that magically rises above the complexities of what is below and ahead so that its rider, in the twinkling of an eye, can be borne thither and land in a place difficult to reach. There is no flying carpet to help Poland find its way. Rather, this Central European country bumps along on a strangely woven fabric through which the warp of the East and the weft of the West incompatibly cross or pass over and under each other.\n\nI am reminded of the time an American friend who, when visiting me in Warsaw, was attracted to a dazzling colored Indian skirt for sale, among other inexpensive clothing, on an outdoor market table reminiscent of an old Soviet style of commerce. She joined a group of women sifting through the garments (like ravens, they foraged by taking turns); selected a skirt; slipped it on over her clothes, and then crossed over to the window of a brand-new Western boutique in order to see, in its distorting, reflective surface, whether the item fit her or not. In the mirror image, a view of her cheaply produced skirt overlaid that of the tailored, luxurious clothes adorning the mannequins standing in the window. These contrasting images partially obscured and blended into each other to create an imperfect collage of a period in which one style of living visibly bleeds into and incompletely superimposed itself on the other. The two merged in the frame of the shop window to fashion a composite portrait of Poland's complexity.\n\nAnother swing of the hips to see if the skirt would do; the bystanders nodded their approval, and my friend, in spite of some doubts, purchased the item and joined the Poland of 1995.\n\nFew escape such a baffling m\u00e9lange of style and orientation. The sight of an elderly woman, shuffling every day to the post office, where she sits and compulsively reads a letter she either wrote or received years ago, is a weirdly disturbing anomaly in a culture that awkwardly and unpredictably mixes the present and the past. Nothing new overlays the news she reads. Clinging to and engaging the fading syntax of a former time, the woman sorrowfully bends over the missive's vanishing ink and mouths each precious lingering word in silence.\n\nFor most who pass by on the street outside, however, no single recurring document either defines their lives or keeps them aloft. Instead, the tangle of the then and the now keeps them alert and in suspense. As they go through their day, they never quite know which thread of ideology or circumstance will snag their progress. At times, an entrenched bureaucracy that stamps everything and won't look at anything unless it has an official seal on it governs, frustrates, and trips them. In the post offices, flush-cheeked women, sitting at desks piled with wobbling packages and heaps of mail, bang rubber stamps three or more times on every letter that comes through the slot. Stamped _POCTA POSKA_ in violet, the letters start their journey or fill laundry baskets marked DEAD LETTERS.\n\nAt other moments, the craving for riches and the desire for the so-called comforts that a Westernized economy promises overtake and propel their lives. Taking advantage of this longing, pyramid businesses prosper. Herablife, an international company that sells a concoction (one is not allowed to call it medicine) that supposedly increases the quality of life, preys on women seeking a magic carpet into the middle classes. Booths, hurriedly erected on the streets of Warsaw, display pamphlets about Herbalife; people dash by sporting \"I Love Herbalife\" buttons, and cars\u2014festooned with Herbalife logos\u2014drive determinedly past icons of poverty. They speed past a ten-year-old beggar-gypsy boy who plays Vivaldi on his violin at the corner of ul. Nowy Swiat.\n\nNot all followers are believers. Some just wear the company's badge so as to work the system cynically. Fully understanding the irony of her actions, Agnieszka, a bright and cunning feminist scholar, decides to take advantage of Herbalife and its exploitation of women's dreams. Needing extra money so she can fly to London and buy books on gender theory, she signs up to be a translator for an Herbalife convention to be held in Warsaw. After a week's work, she earns $3,000. Soon, with the money in her pocket, she disappears on a LOT flight to Heathrow.\n\nFor all these reasons, in 1995 my entrance into Poland is not easy. On several occasions, the tangled threads that complicate life in Warsaw ensnare me. In particular, the sharp, cold-edged remnants of an older Soviet bureaucracy initially snag the tissue of my wellbeing and compromise my commitment to new responsibilities and work.\n\nThe customs office in Warsaw is a setting for trauma, a knuckle-whacking introduction to a system where one must know which strings to pull, what bribes to offer, and what lies to tell if one is to endure.\n\nThe tale begins: Before leaving Buffalo to spend a year teaching at the University of Warsaw, Irving Massey and I, ship, via an agency specializing in mailing possessions to Poland, fifteen boxes packed full of books on American literature (I am told they are impossible to buy or to find in the libraries; I plan to give them to the Institute of English Studies Library), a small portable printer, clothes, shoes, cosmetics, medicines, Kleenex, toilet paper, and even a bathmat. I have been wrongly led to believe that such goods will be hard to find.\n\nInitially, the fifteen boxes are sent to the wrong city (somewhere in eastern Poland), held up in customs, and after a week, forwarded to Warsaw where they are refused clearance. What to do? The liaison from the Fulbright organization, Dorota Rogowska, armed with affidavits of my legitimacy, offers to accompany us and help retrieve our possessions. After a circuitous journey, we reach a dilapidated wooden shed tucked into some godforsaken puddled, grimy train yard almost hidden from view below a concrete bridge. Inside the building, swarms of people, like blind mice, shuffle and scuffle about in a labyrinth of bare corridors and in front of closed wickets or locked doors. No phone is available. We are cut off.\n\nOn the first attempt to pick up my boxes, we are summarily dismissed. Dorota's presence and her affidavits have no effect. For the following four days, we try and try again. Arriving early in the mornings, we wait in line with hundreds of others for our turn to speak to some entity enclosed behind a small barred wicket cut into a concrete wall. The scene is out of Kafka. Each bar in the grating surrealistically crops the official's face so that only a sliced mouth with its metallic teeth peers through it. On the fifth day, after hours of queuing (this time we hope in front of the correct window), we are just one person away from being attended to, but some experienced broker, clutching a fat pile of loose papers, barges in. He leans into the open wicket. His broad leathered rear sticks out into our faces. I hate him. He does not budge. Hours pass while some lengthy official business laboriously takes place. Time to close; the wicket slams shut. Dorota cries; I want to go out in the train yard, scream, and hit something, somebody, anything.\n\nOn the sixth day, the executive director of the Fulbright Commission, Andrzej Dakowski, joins us. Tears of impotency glaze his eyes. Nothing happens except I learn that if I pay $1,000, customs will release the boxes. I refuse. Wishing I could rub Aladdin's lamp so that Baba Yaga might appear and transport me \"beyond thrice-nine lands\" to a destination far from this hell hole, I contact the American embassy. Whoever is in charge refuses to help. He says it is my personal problem. I am stuck.\n\nDesperate, Irving dredges up the name of a Pole he once met at his university in Buffalo. Professor L. plays a prominent role in the Polish government. (It is rumored that he hoped to run for president.) We decide to try. After several inquiries, he is found; we talk on the phone and arrange to meet. Salvation (perhaps), for the politician cum professor is willing to contact the head of customs on our behalf. Before escorting us out of his office, though, he takes us aside (his experienced fingers tug firmly, yet gently, on our sleeves) and deftly coaches us on how to offer a bribe (a good bottle of brandy) and to whom.\n\nTwo days later, after distributing the brandy to certain officials, I have secured an arrangement that I can claim my boxes\u2014that is, _if_ Dorota translates each packaged item into Polish and _if_ I sign an agreement stating that I shall take all my possessions back with me to Buffalo, including the books for the Institute Library. There is no alternative; I concede. After all is signed and sealed, we return, with cash. In the back of the Fulbright van, we cautiously make our way behind the customs shed, pay a hefty storage fee (the system will realize some profit, for my boxes have been sequestered for five weeks), and finally collect my possessions. After living off two sets of skimpy clothes, I can now change into cleaner and warmer attire.\n\nBut not all is over. A week later, the Fulbright office calls to say that our visas are incorrect (they are for only three months). We risk deportation. I also learn that the paper I signed to retrieve my goods from customs will only be legal for three months. It's back to labyrinthine lines and locked doors.\n\nEven though we had paid and applied for a yearlong, multiple-entry visa before we came, we somehow were not granted one. The process and expense begin again. I must get another letter from the rector of the University of Warsaw to say that I am really working at and being paid by the university. Dorota from the Fulbright office loses days getting this document. I also have to produce more photographs, fill out forms, and show an official document to say where I reside in Warsaw. Eventually, after three visits to the visa office\u2014fortunately, not as far away as the customs building\u2014and after standing in line with other anxious clients awaiting their turn to enter through a solidly closed door, I receive my work visa from some entrenched, sardonic, sourjowled, middle-aged man who rules from behind a set of bushy eyebrows and a desk piled with papers and disappointment.\n\nIn the meantime, Irving has his own battle for a proper visa. He brings in evidence that he has already applied for a yearlong visa but is abruptly turned away because he does not have the same last name as I. The official, this time a matronly blond, does not believe we have come together so will not grant Irving an extension unless, she adds, he is legally my husband\u2014and that (she regards us accusingly) we have to prove. We stand as calmly and as patiently as we can before her experienced, distrusting eye. In the end she says she would accept a certified letter from the American embassy to say that we are married.\n\nWe are not. I must lie. Pumped up on adrenaline and propelled by the forces of necessity, I seek out the American cultural attach\u00e9, whom I have already met, and ask him to compose a letter. I make up some cock-and-bull story about _not_ thinking I would need to bring our marriage license to Poland, so it remains among documents at home. Furthermore, I fib, there is not enough time to send for the license because the visa office demands to have proof in two days. He must produce a letter immediately or else. (Would he mind if I suggested some phrasing? I ask.)\n\nThe kindly cultural attach\u00e9 swallows the fiction, avoids eye contact, and promises to have one ready. I should pick it up in a day or so. (Will I be found out? He'll look at my papers and see I am single.) As coolly as possible and armed with a box of chocolates, I later return to the embassy. Within minutes, I pass back through its guarded gates with a typed, signed, and officially stamped letter stating that we are married. Prevarication is my bedfellow. Fearing the Polish official will have either changed her mind or forgotten the arrangement, I rush back to the visa office not only with letter in hand but also clutching the required brandy.\n\nRelief when the officer, after annoyingly perusing the document for far longer than it takes to read it, picks up her stamp, poises it tantalizingly in the air, and both suddenly and forcefully pounds a new visa to life. Following that last, violent motion, her stern, officious eyes disappear into soft furrows plowed by an unexpected cherubic (yet knowing) smile. We pleasantly chat, collect our coats, and leave. As I scamper down the building's marble steps, a nagging reminder of another time shadows the triumph of my lie. I think of the people who lined up anxiously, repeatedly, and daily in a similar Warsaw building in an attempt to flee Nazi persecution. Isolated from help, subject to some barely visible authoritative presence sequestered behind officious wickets, they filled out forms, displayed reams of documentation, and probably offered bribes. Death, not deportation, was their alternative.\n\nI arrived in September. But because of these difficulties, it is not until late November that I can actually, once and for all, feel settled and start to engage fully in what I have come to do.\n\nFrom September 1995 to June 1996, as a senior Fulbright fellow at the University of Warsaw, I teach three graduate classes to exceedingly bright students and, once a week, lecture for forty-five minutes to two hundred undergraduates on the history of American literature. That lecture takes place late Monday afternoons. I enter what was, before the war, a splendid nineteenth-century building (now pockmarked with bullet holes and spoiled by a broken-up and decrepit interior). I climb four flights of marble stairs, enter a large, deep hall lit by the remains of what were once sparkling chandeliers, walk to the front, and mount wobbly steps so as to take my place on a raised wooden stage. The stage looks out, as from a crow's nest on an oceangoing vessel, over a large, distant sea of desks lined up in endless rows. I can hardly make out people's faces. No land in sight except out of the long windows where I am able to spy cars going by as well as the sculptured head of some luminary that stares back at me with its historic stone eyes.\n\nThe head of the department has not prepared me for this task. The two hundred students have no texts, but I at least have a microphone. I muddle through the first lecture and decide that for every subsequent class I shall design a handout for each person. (I live in fear that the few copy machines, usually available in rickety, start-up commercial businesses, will break down, close, or disappear.)\n\nMy other classes are in a building on the elegant ul. Nowy Swiat. Every day, I trudge up six winding flights of stairs to narrow, dull corridors that lead to small shabby classrooms; an unlit foul toilet (lacking paper); and a peeling, bare-walled office, which has one desk, six hard chairs, six professors, and a telephone that from time to time falls off the wall. We take turns sitting at the desk. To get coffee, I descend seven flights to a _klub_ that sells borscht (what my colleagues disparagingly call \"dishrag soup\") and _kanapki_. The one bit of warmth in the building is the porter's cubicle. He hangs up our coats, puts our mail in pigeonholes, listens to opera on the radio, and sells cigarettes, fruit drinks, Kleenex, and candy. Most endearing of all, with pieces of delicate string, he hangs small, lost objects, like tiny notebooks and lipsticks, on his door so that the lost can be found.\n\nThe porter offers an affirming contrast with the divisive atmosphere of my department. My colleagues (all brilliant, well read, and sophisticated) hate, fear, envy, and yet admire each other. Agata does not utter one word either to the head or to Tomek; Agniezka and Barbara do not exchange even a nod. Ambitious, cautious Ewa dashes in and out uttering polite nothings, and Nancy floats blithely above it all in a cloud of poetry. Neurotic, they are on special diets, suffer from stomach ulcers, or are anorexic. For all these reasons, at the end of most days, my colleagues, as if diving under the familiar blankets of their beds in search of protection or oblivion, find relief in leaving the front doors of the institute. There is comfort in plunging into the darkness of the late afternoon and joining the warmth generated by the crowds moving briskly along ul. Nowy Swiat.\n\nThe Institute of English Studies would wreck if it were not for the head, a Marxist, who forcefully and stubbornly, if not sardonically, steers and holds the program on course from her well-fortified office hidden behind a thick cloud of smoke produced by the industry of her chain smoking. Emma is the successor of the Institute's founder, Margaret Schlauch, a medievalist and a leftist who fled the United States in the 1950s to escape the persecution of the McCarthy congressional hearings, devised to rid the country of all those with communist leanings. Not wanting to be subject to the question \"Are you now, or have you ever been, a member of the Communist Party,\" she settled in Poland, where she had family connections and where, during the 1950s, her political ideology would not be associated with subversion or treason. As a member of the University of Warsaw faculty, Schlauch compiled the first anthologies of English literature, and in this way introduced as well as formalized the study of British and American literature into the university curriculum. I am here to add to an American studies component of this structure.\n\nGuarded by her savvy personal secretary as well as by her thick Scots accent, the present chair, Emma, is impervious to criticism. One does not disobey or challenge the department head. I tried once and failed utterly. She has her minion in the form of another Scot who is clever, drinks like a fish, chooses his research libraries according to whether or not they are close to a fine restaurant, and at home in the Scottish highlands keeps a flock of eighty sheep (managed by his wife). He complains to me that his dog nips the sheep too hard. He also advises me on the best mutton for eating.\n\nFriendships are spotty. People are busy, preoccupied. All is work. There are moments of connection but that is the best I can say. Only occasionally do I catch a glimpse of what lies below the surface of people's present lives. And when I do, I am sobered by what emerges. Late afternoon, I accidentally meet a colleague at a nearby caf\u00e9. Before the last bits of cappuccino foam have left the rim of her cup, she tells me about her guilt toward her father. During World War II, he was sent as a political prisoner to Majdanek (one of the largest concentration camps just outside of Lublin). He survived by running through an opening in the barbed wire. When her resourceful mother, a trained pilot who was living in Warsaw, learned of his escape, she managed (through pull and bribes) to get a plane, fly it solo through enemy lines, and reach him. But when he came home, my colleague, as a young child, rejected him. Aged by use and beyond recognition, her father was and continues to be a stranger to her.\n\nWe are all strangers to each other. There seems to be no place to meet people casually; there is no center, except perhaps for once a month, when lines of faculty and staff queue up to be paid (in cash). At a central administration building, I join those who\u2014clutching their identity cards\u2014stand, push, watch, read, and make jokes waiting for their _zloty_ to be pushed by an aged, varicose hand through a window. Once the money is collected, though, the conviviality dissipates and people sprint off in separate directions, unknown to each other. If I were here by myself, I would resemble the forlorn English-language instructor from Glasgow who hangs around the empty lobby of where I live. He does not know what to do with himself so repeatedly asks the person at the desk for the TV room key.\n\nEven the library system is fractured. Anyone interested in researching an American author has to work among four or five libraries that are neither cataloged nor coordinated.\n\nIn addition to settling into my work, I must also become accustomed to my accommodations. We are to live at the Hotel Sokrates, a residence on ul. Smyczkowa, run by the University of Warsaw for international faculty. This apartment building belongs to the bleak gray concrete of a Stalinist era. It sits erect and boringly plain on a scruffy lot next to the last tram stop. Children play with sticks in a nearby, rusted playground.\n\nInside, the place is basic. My apartment has institutional furniture, a washing machine (which leaks), and a microwave (which I succeed in blowing up\u2014toxic yellow smoke fills up the place until I throw the appliance onto a narrow balcony and bang the window shut), but no pots, dishes, towels, knives, spoons, carpets, dishcloths, or reading lamps. Fortunately, there is an outdoor market in an empty, muddy adjoining field, so during our first days, we scout its rows of unfamiliar booths to search for these necessities. Occasionally, there is no water; the management puts up a notice to say there will be none. On these days, a large water truck pulls up outside the hotel and everyone hastens down to fill up containers with water that is actually clearer than the reddish-yellow liquid that regularly drips through our taps. At other times, there is an announcement to say the hot water will be off for half the day.\n\nThe lift usually works, except when a piece of metal between the second and third floors comes loose and trips the mechanism. The toilet cooperates, but the shower floods. And the wiring is unreliable. When I first plug in my computer, yellow flames and blue sparks crackle out of the wall. Black streaks on the plaster register the fact that the Delta computer I have brought with me is cooked, dead. Thanks to the god of fortune, from the tram window, I spot the brand's newly erected corporate headquarters\u2014a shiny, glass-windowed skyscraper that marks the coming of the West. Feeling like David standing before Goliath, I enter with my electrocuted computer and ask for help. Struck by my naivet\u00e9 as well as my suggesting that it might still be under warranty, they look after me and arrange for the hard drive to be replaced without cost\u2014unbelievable.\n\nIn spite of all these problems, however, the Hotel Sokrates is satisfactory, for it is secure and has a frugal bar (with stained tablecloths) that, when open, can be helpful if one has run out of food. The building is far from the caf\u00e9s and restaurants found in the center. A brand-new, bright red-and-yellow McDonald's across the street is the only option. (I start collecting the toys that come with Happy Meals.)\n\nIf something goes wrong, members of the hotel staff fix it. Furthermore, if one needs assistance with the phone or directions, the hotel staff is there to help. Needing to get my hair cut and styled, I ask Anna, who speaks English and works behind the hotel desk, where I might go. Overhearing my request, a group of women lolling about in the lobby immediately gather around with suggestions while others, sitting in front of streaked picture windows partially blocked by tattered brown curtains, join in the game by sliding occasional pieces of wisdom across the lobby floor. They concur that I should go to John Louis David, a French chain, where stylists cut hair with razor blades.\n\nWhat also makes the Hotel Sokrates more palatable is that Agata Preis-Smith, one of my colleagues lives here. Sometimes we go to and from the university together. Dressed sensibly in one of the two outfits she owns, she sprints along while complaining of headaches, liver problems, and a general malaise. The memory of her haunts me. I wonder if she is still alive. When I know her, she is damaged, frightened, and vindictive. Petite, skinny, taut, sharp-featured, and fair, Agata is incredibly brilliant and neurotic. Whenever we walk together, she insists on being on the outside so as to protect me from passersby and traffic. Agata exhibits a sense of decorum that I never fully understand. She is loath to let me see her flat because it is so messy. Indeed, as hard as I try, I never really comprehend her. At times she is silent; at others, uncomfortably confessional. With bitterness, she recalls the difficulties of managing during periods of want and, especially, of holding a young baby while standing for hours in lines with empty milk bottles ready to be filled; yet, when it comes to her turn, the official refuses and says the bottles are not the correct ones. Agata loses her place and must start again. She also speaks, with the rawness of an open sore, about giving birth (cesarean) without an anesthetic during the sparse communist years, when doctors told patients that pain is a sign of health.\n\nMemories of that acute pain mingle with what she knows of her parents' past. Time has stripped their narrative, so what remains is a naked outline that is starving and will die. Agata tells me that as a young woman, her mother worked for the underground during the Warsaw Uprising publishing political leaflets and books. Subsequently arrested, jailed, and eventually put on a train to Auschwitz\u2014not, I believe, the same trains that sent Jews to the camps\u2014she seems to have led a doomed life. She survived, however, for she met a rich Polish baker who befriended her. Having lost his own daughter, he decided to adopt Agata's mother and pay for her release. (Polish political prisoners apparently sometimes had this option.) Saved by the baker, she got off the train at Krak\u00f3w and found a safe haven with the family of her future husband, who, being an avid and young communist himself, had to escape in order to flee from the Germans.\n\nIn the Hotel Sokrates, Agata occupies a tightly packed, three-room apartment with her husband, Phil, a taciturn socialist who has lost his job (he emigrated from England to Poland fifteen years ago); their young son, who exhibits all the bright nervousness of his parents; and Agata's frail, aged, learned father, who sits, his threadbare jacket crumpled over a hand-knitted cardigan, in a corner of the cluttered kitchen. As much as his eyes permit, he reads. He is going blind. The rooms are chock-full of books. Her father rarely leaves the apartment. He suffers from heart trouble for which there is no help. They all live in fear of eviction, since Phil is no longer associated with the university.\n\nThe nightmare of Agata's life comes vividly back to me a few months later in the only letter she sends me after I leave Poland. It seems that one day her father left the apartment and walked out of the Hotel Sokrates to go and see an elderly friend who lived nearby. When Agata returned from the institute, Phil told her that her father had not returned. Agata assumed that he had lingered. When darkness fell and there was still no sign of him, they went out to try and find him. Agata paced back and forth on the pot-holed path that leads to the Hotel. He was not at the friend's place; he was nowhere in sight. On her way back, she saw what she thought was a drunk fallen over in the mud, averted her searching eyes, and walked by disgusted at an all-too-frequent sight. An anxious night followed.\n\nThe next morning the police showed up to say that a body had been found, dead, in the long grasses that waved in the cold winds in the rough empty lots outside the Hotel Sokrates. The \"drunk\" was Agata's father. Realizing that she had seen and walked past him, Agata struggled with the guilt of not having recognized her dying father. Full of self-loathing, she could not deal with the possibility that she might be responsible for his death. (I read her letter and realized she would tell no one at the institute about what had happened. Rather this incident, like so many, would join the strata of trauma and discontent that ran deep below the crust of her being.)\n\nThe tall, unkempt marsh grasses surrounding the Hotel Sokrates hide the dead and the living. I, too, have noticed the drunks who lie and linger with their bottles among their camouflaging stalks. Packs of stray dogs make beds out of the reeds bending in the breeze; their knotted shaggy hair flops carelessly over patches of mange. Squawking pheasants strut and scamper, avoiding the eyes of long-lost cats that dart and pounce through blades and lumps of weeds. When I come home late from the tram, I see that two dogs have treed a helpless kitten. With my umbrella extended, I foolishly, if not ludicrously, storm through the tall grasses and chase away the excited dogs. Later in the evening a fire engine roars up and puts out a blaze that has started in a nearby abandoned house, a shelter for vagrants. The children of Hotel Sokrates call it the Haunted House. The more the smoke rises and the more the firemen smash windows and search the area with weak beams, the more people open their windows and gaze into the troubled night.\n\nThe Hotel Sokrates is twenty minutes by tram number 36 from the center of Warsaw\u2014that is, if all runs according to schedule and if there are no accidents or breakdowns along the route. On my way to get money at the American Express office, the tram suddenly stops. The driver beckons everyone to get off because the electricity on the next section of the tracks has failed. Everything is dark. After much discussion and grumbling, there is a rush to the nearest stop and consultation about what tram and when. On another day, the tram discontinues because a car has just plowed into a group waiting at the next stop. (These stops are narrow wedges in the middle of busy streets.) Seventeen are killed and injured. The police arrive and break up a crowd about to \"lynch\" the drunken driver. A week later, again I must get off. This time it is because someone, too eager to get to the other side of the avenue, dashes across the tracks; tram 36 hits and decapitates its victim.\n\nTo reach the tram from the Hotel Sokrates, one walks on broken concrete slabs past the wild lots and then waits and listens for the whistling of the wires that signal the tram's arrival. One learns that sound prefaces sight. Avoiding the begging gypsy women who park themselves daily at the stop, people cluster by the tracks and stare into the empty space of anticipation.\n\nOne morning, I join a taciturn group facing away from the wind. Our collective attention is caught by the lone figure of a middle-aged woman who, with her shaggy dog, approaches the tram stop across the tracks. Once she reaches the bench, she swiftly picks up her companion, places him on the bench, opens her wicker basket, takes out a brush, and in front of her unsolicited audience, commences her performance. She brushes her dog this way and that, turns him around and around, starts again, and then unceremoniously lifts his tail. While the dog squints, she vigorously scrapes through his whiskers and flattens his forehead with brisk and practiced gestures. When she finishes the grooming, back goes the brush into the wicker basket, down goes the fur into the closest rubbish bin, and off jumps the obedient dog. Satisfied that a morning ritual is complete, I and the other waiting passengers shift our attention and look down the tracks to see that the 36 is on its way. Whatever sense of community crystallized during the previous minutes dissolves when we hear the tram screeching as it rounds the curve and then listen to the sighing and groaning of its doors expelling passengers. Individually, we squeeze past each other up the steps and extend gloved fingers to stamp our tickets. On the tram, competing for space, I avoid the stale alcoholic breath of the rough-skinned man behind me, shield myself from coughs that curl around corners, keep my bag close, and worry unnecessarily about what news I might find in my mail box at the institute.\n\nIt is while waiting, exposed in the middle of the street for a tram, that one becomes acutely sensitive to the bitter cold of winter. The freezing air knows where all the gaps are in one's clothing. Like a cat that finds its way across the rooftops, it knows where the holes and the weaknesses are. It deftly works its way inside and leaves shivering traces of its frozen fur on the skin. Even when one enters what seems at first to be a warm place, the cold follows. Either someone keeps opening the door so that drafts swirl around those seeking refuge within, or a person walks by and the air, like a stone thrown into still waters, responds by dancing in chilling rhythms about the back and down the neck. I have learned the technique of scarves. I wear a soft woolen one inside my coat, then, outside it wrap a long, wide scarf numerous times around my neck and over my mouth. I am not sure when I last did not wear long underwear. I refuse, however, to wear the popular fuzzy, half-mauled berets, sold in the street markets, or purchase the coarse fur caps, some in the shape of baseball caps that look as if the animal has just been pulled out of a trap and is now skulking around a person's ears. I buy, instead, an elegant Italian black felt hat.\n\nFor all my sensitivity to style, however, I am neither willing nor able to walk in high heels over sheets of ice. Rather, necessarily sacrificing aesthetics to survival, I daily pull on ungainly sensible ugly boots to navigate slippery streets. It seems there is no alternative. Imagine my amazement when, one frozen evening, on my way to a performance by the Polish Radio Symphony Orchestra, I notice fashionable, middle-aged Polish women, wrapped in sumptuous perfumed furs fringed with bouncing baubles and perched on high stiletto heels, pecking through the ice and speedily passing my shuffling steps. Clicking along and blithely unconcerned about falling, they link arms with their silk-scarfed husbands, wearing elegant winter coats worthy of a Warsaw tailor, and reach the concert hall ahead of me. How do they magically negotiate the ice as if it were a fashion runway? Left behind, I resemble some ill-shaped, flat-footed Kraken stirring from \"his ancient, dreamless, uninvaded sleep\" (Alfred Tennyson). Once inside, these women and men stand critically before large public mirrors and unself-consciously prune their plumage; they rearrange bits and pieces of their hair, makeup, and attire, as if sculpting statues of themselves. Elsewhere, they would study the windowpane's reflection to make sure their hats were on straight. How different from the quick, furtive looks we take at home in public spaces. What I do not realize at the time is that these people are not necessarily well to do but are subscribing to a culture that respects fashion and encourages a person, no matter what his or her means, to look stylish. They do not own many outfits (perhaps three), but what few dresses, coats, and suits hang in their bedrooms are cherished. The furs, exuding an insect-powder fragrance, are probably rented for the season or for a single afternoon.\n\nIn 1995, Poland is feeling the bite of an extended winter. People find their way in the crisp, dry, windless cold, behind flurries of large snowflakes, amid silent, thick snowfall, or beneath a frosty waxing moon. From time to time, Siberian winds gnaw on Warsaw, and leave the leftovers of their gorging on the ice. Warsaw looks bleak. Its gray concrete structures touch and smear the sky for months. My winter shoes are falling apart and my long underwear is full of runs and holes. Week after week on the tram I see no difference between the gray-whiteness of the sky and fields. Young people on their way to classes do not walk; they make slides out of the ice and glide along where I would fear to fall.\n\nThe snow is still blowing around me when I get off tram 36 this evening. As I walk away from the stop, the tram's overhead wires momentarily burst into electric blue and for a few seconds all of us move visibly toward our destinations. Inside the apartment, I look out of the narrow kitchen window into the frigid darkness and across to a single lit window on the second floor of the nearby army headquarters. (We blame this structure for the bad reception we get on BBC.) All the other apertures are as dark as night. Inside this suspended illuminated space, two officers play table tennis to pass the time. Their bright darting movements in the coldness of the late evening are mesmerizing and reassuring.\n\nLow temperatures are stubborn, omnipresent, and perhaps even omniscient, for they know how to find one's most vulnerable and exposed areas. When I learn that more than one hundred people have frozen to death on the streets of Moscow, I believe it. God, it's cold. Today we hang a pair of Irving's just-laundered woolen socks on the balcony; in five minutes he comes to show me the icicles dangling like stalactites from their heels and toes. The other day I looked out of the window and exclaimed, \"Shadows!\" For the first time in weeks, the sun had come out for a moment and I could see the pattern of our balcony railings imprinted on the balcony's concrete floor. In Poland, shadow is a form of color. But winter is not relaxing its hold.\n\nOutside the Hotel Sokrates, gray-beaked ravens strut about the frozen ground and grope among the dead leaves for whatever they can forage. It is a difficult winter for them too. Each morning, I bring them bread that we have not finished. I throw it over the barbed-wire fence surrounding the nearby military compound. Waiting up in the trees, they fly down, floating, swooping, and dropping like abandoned cloth handkerchiefs conversing with the currents in the air. Some mornings, I forget and guiltily walk by. On the ground, they sit in the snow. In pairs, they lean toward each other, as if supporting themselves and retaining what warmth remains under their ruffled wings. They seem to plead with me:\n\n_\"Build me my tomb,\" the Raven said,  \n\"Within the dark yew-tree,  \nSo in the Autumn yewberries  \nSad lamps may burn for me.  \nSummon the haunted beetle,  \nFrom twilight bud and bloom,  \nTo drone a gloomy dirge for me  \nAt dusk above my tomb.\"_\n\n\u2014Walter de la Mare, \"The Raven's Tomb\"\n\nI also feed a pair of fancy pigeons living near the garbage container. Their feather socks quiver in the wind.\n\nEscape from winter is required. The Eagle Club at the American embassy is pleasant. Occasionally on bleak days, I go there to eat American food, read the _Herald Tribune_ , and watch marine guards flirting with daughters of diplomats. At other times, I push through the shining brass doors of the nineteenth-century Hotel Bristol (once the headquarters of the Nazi Party; Elizabeth Taylor also slept here) and take a seat in the classy Caf\u00e9 Blikle. Under an autographed photograph of Arthur Rubenstein, I savor delicate pastries while looking across the room at a large mirror. Hung at an angle, it frames and reveals the slicked hair of the Warsaw wealthy, the blank faces of customers staring into their fantasies, the triangle of a silk handkerchief peeking forth from a gentleman's breast pocket, and the bowed faces of women confiding in one another. Alone, I eavesdrop and listen to a young woman talk about the breakup of her last affair with a depressed TV sportscaster. (The reporter took her to Florida for a weekend but never left the room. Spending all his time shopping by phone, he bought seventeen shirts. And that was the romance.) And once, desperate to leave the dull grayness of the morning behind, I wander in to the Belvedere Restaurant, which used to be the _orangerie_ on the Lazienki Palace grounds. I sit at a table surrounded by attentive waiters and large tropical plants hanging menacingly over my food. Uninvited sparrows flutter in and out of their leaves. Warm, humid air clouds the greenhouse glass and frames, by contrast, a surrealistic picture of the frost and snow outside.\n\nWhile I am eating, a man passes by and catches my eye as if to say, \"Do you know me?\" I realize I do. How could I mistake the face? It belongs to Jerzy Urban, and I have seen his portrait in the _New Yorker_. He is a Jew who survived the Holocaust by hiding with a Catholic family but is now anti-Catholic. He is the former spokesman for the Communist Party in Poland but has metamorphosed, like so many, into a new capitalist and millionaire. He is somehow the essence of Poland. This genius and opportunist, this irreverent, crude, and sued man appropriately edits Poland's most popular satirical newspaper _Nie_. I look over to where he sits with his entourage and ask the waiter if it is he. The waiter responds in a low, respectful voice, \"Yes, that's Urban.\"\n\nOften to relieve the winter bleakness, Irving and I meet in the large lobby of the once-splendid but still-comfortable Hotel Europeiski. Bits and pieces of the world pass through this lobby. One late afternoon, members of an opera company from Venice that has come to perform at the Wielki Theatre in Warsaw stand either huddled in a corner or melodramatically draped over the settees. They have just learned that their home, the historic opera house in Venice, has burned to the ground. Every last one (and there are about fifty) anxiously puffs a lit cigarette. The lobby fills with smoke. While I wait another afternoon, my eye turns to a pale young man carrying his violin and approaching a person who is to take him over to the symphony hall to perform a Brahms concerto. Even though they are running behind time, the violinist begs the person to wait a moment. The violinist disappears and returns with a large chocolate bar. It is not easy being a performer in winter. The oboist from Tafelmusk slips on the ice and breaks his arm; a pianist friend sprains her ankle and must practice for weeks with her leg elevated.\n\nSometimes the sun comes through and shines for an hour. On those days, rather than hiding in coffee shops, one finds one's way to Lazienki Park near the center, and joins others who have come out to catch what rays they can. Sipping a steaming cup of hot chocolate, topped with whipped cream, I stand close to the ducks in the snow and watch rows of people extending their open, seed-filled hands so that the black-capped chickadees can dart down, perch, and gorge on the offering. One Sunday afternoon in Lazienki Park, I approach a pond in which a mallard swims through ringed currents of melted ice. For a few seconds, the sun moves from behind a cloud and quite suddenly reveals the deep, brilliant oscillating green of the duck's neck. The vividness moves me to tears. It is the first color I have seen in months. A few paces on, I see the sun reappear and illuminate a peacock (introduced by the Polish nobility) perched on a classical garden statue. Its colors are exquisite. Juno would be pleased.\n\nThe Christmas season also offers some brief relief from the dullness. Warsaw is alive with Christmas lights and shoppers who shuffle over pavements of ice. Along the streets are booths where one can buy decorations. I go to a party at the ethnographic museum and buy some traditional decorations made out of straw. There, I meet a Chinese woman who is in Irving's Polish-language class. She tells me she has, just that very day, bought a factory.\n\nBy the beginning of April, the snow has almost gone. In its melting, I watch the rooks roll in the slush and take their first bath in months. Their obedience to the rhythms of their life is impeccable. Even when the cold weather persists, they dutifully build their nests, in clusters easy to see in the bare trees, and sit waiting for rebirth. Dirty patches of ice cling like a cracked, soiled epidermis encasing the earth's body; deep pools of water flood the rutted pores of the road. At street corners, choreographed groups of people jump back in unison to avoid being splattered by passing vehicles. As the thaw continues, from my apartment window I watch the occasional person walking his dog and become distracted by a car splashing in the puddles left over from last night's rain. A child's complaining voice echoes through the courtyard outside Hotel Sokrates and disrupts the reassuring sounds of the songbirds that have returned to the just-blossoming trees that grow behind the army compound.\n\nFinally, the cold weather breaks, and all of a sudden it is hot. The gypsies who beg at our corner tram stop now appear in spotless white short-sleeved blouses. What people call \"a Ukrainian summer\"\u2014a summer that bypasses spring\u2014is upon us. All the windows on the tram are wide open. The cotton fluff from the poplar trees blows in and catches in one's nostrils. Every day for the last two weeks there have been thunderstorms in the late evening that have washed out the sky and left the air a bit fresher. Apple and chestnut blossoms have scattered.\n\nOn Easter Saturday, the blue sky brightens the pale green and yellow buildings in Old Town. People sit and face the sun. In Lazienki Park, among patches of snowdrops, it seems as if the whole of Warsaw is out enjoying the sunlight and queuing for _lody_ (ice cream). While waiting over an hour for my cone, I watch children pick up stones and draw designs in the sand to pass the time; I give money to an elderly woman who does not have enough grosze to buy some popcorn. In the nearby amphitheater, I join many others who sit, heads tilted and eyes closed, to face the sun. Swans and ducks walk on the remaining glistening ice. A layer of water stands on top of the ice, which when it melts releases the trapped leaves of autumn.\n\nThe next day is equally welcoming. Irving and I decide to explore a nature reserve, a few kilometers south of us, that we have seen outlined on a map. We get on a bus and go too far. The bus driver stops especially for us and gives directions about how to walk back, a hike that fortunately takes us by a little cake shop that is open and sells soft drinks. Once we enter the woods, we join hundreds of people who are just as willing as we to get out, feel the warm air, and tramp for hours through a maze of muddied paths.\n\nMay is the time of nightingales. I long to listen to their voices in the passing night and hear what Keats calls their \"plaintive anthem.\"\n\n_Thou wast not born for death, immortal Bird!_\n\n_No hungry generations tread thee down;_\n\n_The voice I hear this passing night was heard_\n\n_In ancient days by emperor and clown._\n\n_Perhaps the self-same song that found a path_\n\n_Through the sad heart of Ruth, when, sick for home,_\n\n_She stood in tears amid the alien corn;_\n\n_The same that oft-times hath_\n\n_Charm'd magic casements, opening on the foam_\n\n_Of perilous seas, in faery lands forlorn._\n\n\u2014John Keats, \"Ode to a Nightingale\"\n\nIn the evenings, poking my head out of the apartment window or stepping in pajamas onto the balcony to brave the cooler evening air, I catch the sounds of the nattering rooks and magpies and pretend that among them is a nightingale:\n\n\"That's one\u2014do you hear it?\u2014over there.\"\n\n\"Yes, maybe.\"\n\n\"Let's call Phil and ask if he has heard them yet.\"\n\nPhil, my colleague Agata's husband, in the middle of the night last year, shouted out: \"I wish those ruddy nightingales would shut up.\"\n\nPhil is not in to help, but over the phone, I persuade Agata to warble like a nightingale so I might better recognize its call. She tries but says she has done it all wrong.\n\nThe quest seems impossible until along comes Barbara, an ornithologist, who for over thirty years has been mapping the bird life in a forest ten miles north of Warsaw. I met her through a friend. Barbara works for a low wage at the Academy of Sciences and lives in a rundown building. The lift smells of urine and her one-room apartment stinks of the two cats she rescued from her basement. She takes me in hand. In a few weeks, I am on my way with her to the woods. We start out at seven in the evening, just at the moment when the murmurous haunt of mosquitoes is at its fullest. Ingesting and inhaling them, we walk a few kilometers, through verdurous glooms and winding mossy ways, then put on boots and sludge through a swamp in the deep-delved earth, up to our knees. \"Darkling I listen.\" My breath is quiet. I finally hear nightingales and the requiem of these \"light-winged Dryads of the trees\" (\"Ode to a Nightingale\"). But there are other sounds: rails resonate like pigs, and the calls of warblers, robins, and cuckoos fill the darkening air with a rippling music that fades away into the leaves, until their voices dissolve \"into the forest dim.\"\n\nBarbara shows me where a wild boar has been digging up the earth and where the badgers have been searching for grubs. She checks the nests (what she calls \"her nests\"). Stooping, she inspects the blackbirds huddled at the base of trees and then approaches a shrike's nest, where she lifts the mother off her fledglings to make sure the young birds are well. After all these years of her solitary daily visits, they let her near and allow her touch. Is it a vision or a waking dream? Unexpectedly, Barbara stops and whispers, \"I hear a moose walking in the swamp.\" Slow, heavy, deliberate squishing noises draw near in the half-light of evening; the moose catches our scent and bounds out of the swamp and into the forest for cover. We too leave. Out of an \"embalmed darkness,\" Barbara and I make our way to the road and reach home by eleven.\n\nNot all is lyrical in spring. A friend visits during the Easter holiday weekend. The only restaurant open is at the five-star Victoria Hotel (the president of Poland works out in its health club). We have ten waiters to ourselves. Because it is late, we decide to take a taxi home. While waiting, we notice a group of well-heeled Poles, wrapped in furs, diamonds, and tailored cashmere, stepping out of their black Mercedes and swaggering into the lobby. Suddenly, distressing screams shred the fabric of their ostentatious wealth. Hearing their car door slam shut, they turn to see their Mercedes disappear into the night\u2014stolen. The doorman, who was going to park their car (keys still in the ignition), momentarily stepped away from the Mercedes in order to obsequiously open the hotel door for these guests. In those seconds, an accomplice hiding in the bushes, jumped out, dashed into the car, and screeched away. By midnight, the vehicle will be across the Polish border and prepared for resale. Like a grouse protecting its family by feigning a broken wing, the doorman drops his extended arm so it droops at his side, simulates anguish, and pleads innocence.\n\nIn the mid-'90s, car theft is rampant. Two Fulbright friends, Denise and her husband Cliff, are robbed of the van they have shipped, at significant expense, from San Marco, Texas, to Krak\u00f3w. Equipped with a lift and a special steering apparatus, this van gives Denise, who has multiple sclerosis, her mobility. One winter night after they return late from a concert, Cliff, exhausted from lifting and maneuvering his wife's wheelchair, decides not to park the van in the secure garage behind their building; instead, he leaves it on the street. The next morning, fighting disbelief, he searches desperately for some trace of the vehicle. Nothing, nothing is there. Calls to the police and posted messages are useless. Until a new van arrives from the States, their university in Krak\u00f3w generously arranges for not only a car but also a driver.\n\n## _CHAPTER 3:_\n\n## THE LAND OF THE POLES THAT IS LOST, NOT YET LOST\n\n### **KRAK\u00d3W AND SURROUNDINGS**\n\nWarsaw and its immediate surroundings are not all I see of Poland. We make several trips to Krak\u00f3w, a city saved from destruction by the fact that, in September 1939, it became the capital of the General Government under the colonial administration of Nazi Germany. During that period, many monuments celebrating the Polish national culture were looted and destroyed. Because the area's splendid medieval and Renaissance structures survive, essentially unscathed, Krak\u00f3w is now an international tourist destination. Visitors meander through its standing historical buildings and wander around one of the largest medieval squares in Europe (the Rynek Gl\u00f3wny). This destination, with its cloth halls (originally designed for the medieval rag trade) converted into restaurants, shops, and entertainment spaces, functions as the center of tourism in the city's old town.\n\nIn the mid-1990s, people are also traveling to Krak\u00f3w to amble through the remains of its Jewish ghetto, search among its tangled, overgrown cemeteries for familiar names, and stroll around Plac Nowy, the Jewish quarter's main square. When I am here, I see the beginnings of development that in the next several decades will bring caf\u00e9s, museums, Jewish-music festivals, bookshops, hotels, and luxury apartments to the area. In the '90s, the legal disputes over these properties are complex, fraught with fear that the former Jewish inhabitants or their descendants will return to reclaim this real estate. Later, visiting Lodz with a Polish-Jewish relative from Montreal, we pass two middle-aged men, who, casting a disapproving eye in our direction, mutter to each other, \"The Jews are back.\" Before World War II, Jews made up one-third of that city's population. By the time the war was over, the Lodz area had lost 300,000 Jews as well as 120,000 other Poles.\n\nAlthough the old city of Krak\u00f3w circulates around its grand central medieval square, I sense that its metropolis is without a core. My response, though, would upset those who adamantly believe that the city, once the capital of Poland, is still central to the country's well-being. Shaped by the length and breadth of its still-visible history, as well as by its aristocratic roots and long-standing contacts with foreign powers, Krak\u00f3w, in their eyes, is not only still the capital of Poland but also eminently superior to a newer, fragmented, and scarred Warsaw. Like the bells of an ancient cathedral regularly marking the time of day, my critics would sound out the enduring rivalry between the two cities. Their preference for Krak\u00f3w would reverberate and swell throughout the land.\n\nIn my mind, however, a nagging disturbing uncertainty, ambiguity, and divisiveness compromise the seemingly solid, enduring stones of Krak\u00f3w's historical castles and towers; deep fissures run through the city's psyche. In the mid-1990s Krak\u00f3w, it seems to me, gathers into itself so many contradictions that there is no place in which to rest my mind or securely settle my feet. Within its boundaries, invasions, massacres, beauty, grace, intellectual curiosity, greed, generosity, prejudice, bigotry, revolution, daring, conventionality, enlightenment, liberation, tolerance, and oppression either trip over each other or commingle in concentrated clusters to create a confusing, unbalanced setting. The city uncannily deviates from itself. I am uncomfortable, for instance, when Krak\u00f3w celebrates the Jewish heritage it once silenced. It is strange being in a culture that listens to Radio Maryja (the voice of anti-Semitism and bigotry) yet simultaneously promotes Holocaust tourism. A devoutly Catholic provost at Krak\u00f3w's prestigious Jagellonian University, founded in 1364, devotes his spare time to studying and reading Yiddish. What does one do with his desire or with a growing impulse on the part of the intellectual classes to study a language that they were once officially taught to despise and silence?\n\nAn express train runs between Warsaw and Krak\u00f3w. In January 1996 after I have turned in final grades, we decide to go. The ride there is lovely with soft, comfortable red seats, a free lunch, and heat. The countryside looks beautiful. A tacit sun lights up the snow glistening on the long, sloping thatched roofs of the older houses and barns, shines on clusters of snow-covered birches and firs, and comes to rest on half-frozen brooks meandering through the fields. The train station in Krak\u00f3w, though, has never seen the sun; dampness seeps through our skin there. The Saski Hotel, where we are to stay, near the main large square, is splendid from the outside. The lobby, featuring an old wrought iron lift with mahogany doors, is promising too. But the rooms are not. Basic and bare, with hardworking radiators that cannot compete with the tall, leaking windows, they offer cold comfort. A wool cap in bed is the ticket.\n\nWhile Irving goes off with one of several librarians, all named Maria, to the French Studies Institute, the National Archives Library, and the University Library, I set out in negative twenty-five degrees Celsius through the streets to find the Ethnographic Museum. After dipping into shops en route to prevent my body from freezing, I reach the museum and find myself alone among exhibits displaying various costumes, masks (many racist and anti-Semitic), and folk sculptures worn or displayed during regional religious ceremonies. A dutiful guard follows me around. She turns on the light as I reach each room and switches it off when I leave. While I rather self-consciously and ignorantly (I cannot read the labels) peer at the exhibits, the guard sits in a folded chair and holds her head in her hands. What, I wonder, are her circumstances?\n\nNot unhappy to leave this marginally awkward setting and longing for warmth, I go in search of a refuge not only from the cold but also from my surroundings' puzzling paradoxes. Near the vast central medieval square, I reach the entrance of the Jama Michalika Caf\u00e9, established in 1895. Inside, the graceful art nouveau furnishings recall a softer time. The flowing stylized patterns of the caf\u00e9's tables, chairs, cabinets, light fixtures, and mirrors temper my recollection of the abrasive, forbidding double-walled fortresses as well as the commanding tower of St. Mary's (Bazylika Mariacka) just beyond the caf\u00e9's doors. According to legend, from the top of St. Mary's tower a trumpeter sounded the alarm before the Tatars invaded the city in the thirteenth century. Supposedly, an enemy's arrow pierced the trumpeter's throat mid warning. As if still alerting those within hearing, a trumpeter now plays at noon every day. To replicate the past, he always cuts short his call.\n\nSecluded from this architecture of conflict, the caf\u00e9's Tiffany-like glass ceilings, windows, and intimate lamps exude warmth and comfort. While sipping hot tea and seated in a mahogany booth, close to a fireplace adjoining a turn-of-the twentieth-century cabinet, I gaze into oversize mirrors suspended, at an angle, high on the opposite wall. Their blemished surfaces gather reflected images of murals and paintings donated by students of the nearby Academy of Fine Art, established in 1818, in exchange for free meals. On other walls, discarded marionettes dangle on fraying strings. Starting in 1905, these puppets were used in cabaret shows protesting bigotry and government censorship. Many satirically depict once-prominent Krak\u00f3wians.\n\nIn January 1996, caf\u00e9s are welcoming spaces and Krak\u00f3w has many. In the Ariel Caf\u00e9 in the old Jewish section, we devour latkes while listening to old recordings of popular 1920s music. This place feeds off Holocaust tourism: a Ukrainian group plays Yiddish songs; waiters dress up as Orthodox Jews, sell Jewish kitsch, and accommodate groups of young Israeli tourists who, led in protective gatherings, troop through the ghetto as if conversing with the forbidden.\n\nOn another late afternoon, we go to a wine cellar, where a draft follows us down the staircase; then one of the Marias, whose family has lived in Krak\u00f3w since the eighteenth century, takes us to a charming coffee shop\u2014the kind one expects in old Vienna, with mirrors, plush seats, intimate booths, small round tables, decorated walls, and chandeliers. Elderly, diminished men and women lean over their espressos and cream cakes while talking about who is there or just sitting and gazing at nothing. When we leave, Irving kisses Maria's hand; she glows and a halo of melted snow circles her feet. We return to Warsaw the next morning. This time, the train is crowded and without heat. We share the compartment with a family suffering from bad colds.\n\nWe are to return to Krak\u00f3w several times in order to explore its surroundings.\n\nAs many feel obliged to do, we travel sixty kilometers south, by bus, so as to visit Auschwitz and Birkenau. I am uncomfortable with myself; I should not be here.\n\nIn the thinly scattered snow of late February, we trudge through the shades of death and past spectacles of suffering into the gas chambers, the crematoria, the cells for political prisoners, the hospital barracks (where the medical experiments were conducted), the holding barracks, the women's barracks (where women slept twelve to a small bunk), the gypsy quarters, the latrine house, the warehouses where the prisoners' possessions were stored after being sorted, the mass graves, the long platform that leads into Birkenau, the commandant's house, and the officers' swimming pool. All are unforgetting witnesses to a pain that is never done.\n\nThis catalog of horrors prefaces another concentration camp, Majdanek, the second largest death camp in Europe, just outside of Lublin, that we later visit. The absence of hundreds of tourists plodding its empty fields\u2014no foot marks their presence\u2014as well as the nonattendance of the all-too-sincere yet transparently practiced dialogue of the Auschwitz guides, allows Majdanek the space in which to make room for the past. In the ensuing silence, death and remembrance reverberate. Alone and undirected, I wander into its still-intact gas chambers and feel what before I saw with \"how blank an eye\" (Coleridge's \"Dejection an Ode\").\n\nLater in February, a Sunday midday meal in Warsaw occasions another vivid view of this unutterable past. Friends from Buffalo have come to see us and want to meet a middle-aged cousin of a Jewish American acquaintance. (Even though they have been warned that the cousin is \"a bit odd,\" our friends are kind and concerned. They have come with money and a thick woolen cardigan for her.) During World War II, Rose was incarcerated in the Warsaw ghetto but miraculously managed to escape and flee to Russia. After the war, she returned to Poland, where she remains. Brilliantly fluent in many languages, she can facilely move from one foreign language to another. Rose now ekes out a living as a translator.\n\nWe all meet for lunch at a modest hotel located in a less-than-picturesque section of Warsaw, close to Rose's rooms. The hotel's restaurant is a dreary smudge of brown except for its crisp white linen tablecloths, which offer some relief to the eye. When Rose enters, removes her coat, and stands before me with her slacks unzippered, I know that not all is right. And when she speaks, I hear her chaos. Rose does not want anyone listening to what we say, so throughout the afternoon, she vigorously shushes our friends and instructs all of us to keep our voices down. Shadows of persecution and loss stalk her distorted life. Like so many Jews in Poland, she fears revealing her Jewish identity.\n\nHowever, when it is time to eat, I more fully sense the damage of her earlier deprivation and maltreatment. At first, claiming she has an upset stomach, she refuses to order food. But as soon as ours arrives, she helps herself to the food on each of our plates, relishes it, and claims that she came in before we arrived to talk to the cook about fixing a special meal for us. By the time we finish, like a starving vulture suffering from a broken wing, she has devoured at least one-half of what we ordered. Then, retrieving a plastic McDonald's goblet from her carrier, she snatches tea bags from our china cups, one by one (her movements are as quick as a frog's tongue catching a fly) and repeatedly (at least ten times) asks the indulgent waiter for hot water. Nine bulging tea bags crowd a single container. To complete the occasion, she dips into her carrier and pulls out jars, which she fills with the salads we have not eaten. These leftovers go back in her bag. Then, after cleaning each plate as if a stern matron or guard were going to inspect it, she clumsily removes the platters to another table and with the side of her worn hand, compulsively, meticulously scrapes morsels of food off the tablecloth. There remains no evidence of anyone's having eaten anything. All is clear. She is safe for the moment. Before we depart, she refuses the money our friends have offered\u2014in fact, throws it back\u2014and exclaims that the cardigan is the wrong color. She does not want it. Her Polish pride refuses the gift. Our friends vow to try to see her again.\n\nBuses from Kark\u00f3w also go to Zakopane, in the Tatra Mountains. In early March, we leave Warsaw in the rain and arrive in a fresh, thick, pristine snow that buries the mountains and swirls in the valley. The morning sky breaks so we join a line of skiers and take a lift to the top of Mt. Kosielec where we can barely see our way so must grope among half-formed images of those who have come for the sport. A few yards from a cluster of practiced skiers, I watch a lone, confused Englishwoman who, without gloves or hat or goggles, is being led by some sort of guide to the head of a trail. Her skis go in all directions at once. Does she survive? I never find out.\n\nAfter a few minutes the clouds roll through the mountains and eradicate the perspective, so we descend to the snow-covered trails below. Water trickles through openings in the frozen streams. I stop to look at a European robin hopping among the waterweeds and pausing behind a yellow marsh flower. The daubs of red upon the bird's chest vibrate among the tints of blossoms edged by snow. Farther on in the brook's rushing water, a dipper, a bird the size of a thrush with large strong feet and a most visible white throat, plunges into the current and walks on the stream's bed, occasionally, with its claws, lifting pebbles to catch insects. The clouds of morning move; we hire a horse and sleigh. Bumping along over places that are beginning to melt, we shield ourselves from the wind and tuck the sheep rug more tightly across our knees. The smell of the horse envelops us; the sound of the sleigh bells beat out our progress. All is tranquil, a pastoral, a picture postcard, until the horse sees the barn and breaks into a gallop. The driver's shouts and sharp tugging motions are impotent. We clutch our seats for dear life until the horse reaches the barn door, halts abruptly, and reaches for his hay. Shaking, we gladly get off and return to the security of shank's mare.\n\nFrom the train window on our way back to Warsaw, in the late afternoon, I glance at the melting corrugated fields; only the indented furrows are white. Just before evening, flocks of returning storks feed in the open areas next to gatherings of trees. Even in the darkling light, I recognize the redness of their beaks. Irving starts a conversation in Polish with our solitary companion, a man reading a book about the House of Potocki, one of the wealthiest and most powerful aristocratic Polish families that originated in Krak\u00f3w. The Potockis have multiple estates and have been known through the centuries for their military prowess and cultural achievements \u2013 not to mention their vodka. Irving asks about the reading, a question that leads to introductions. The man is Jan Potocki himself, a direct descendant, a prince (and also a physicist). We fall silent, and he returns to reading his family's history.\n\nThe Tatras are alluring, so we return in early June to hike into a mountain lake. The path up is long, steep and still covered in ice. Once we arrive, we stare into the deep green lake and skate with our eyes, over its thin, translucent veil of ice. On our return, we meet a group of twenty-five chattering schoolchildren with their teacher on their way up. She instructs them to step aside so we can pass more easily; they do and decorously monitor our slow, clumsy progress on this sheer descent. Once we have maneuvered past them, the teacher instructs them to bow and applaud. (Respect for elders is part of their education.) We are charmed and embarrassed. Soon a fox crosses our path and restores balance. Minutes later we hear the children's distant, excited voices spilling over into the lowering clouds. They have reached the lake.\n\nIt is difficult to leave such beautiful scenery and softness behind, especially when it is time to board the train at Krak\u00f3w to return to Warsaw and its academic responsibilities. The train pulls into the station. Feeling chilled, I decide to get on as soon as the arriving passengers have disembarked. Everyone seems to have gotten off, so wearing my backpack, I climb up the steps and start down the train corridor. Suddenly a group of men, who seem to be in a panic, run out of a compartment and try to push by me. I think, _Oh, they have fallen asleep and think they have missed their stop_. In their rush to get by me in the corridor, they press me up against the wall. Eventually untangling themselves they get off the train and disappear. Confused, I open a compartment door and reach my seat. Irving suggests that they were trying to rob me. I insist that if they were, they did not take anything\u2014that is, until I take off my backpack and find the zipper completely agape, my belongings all shuffled. When I discover that my wallet is missing and my identity gone, Irving dashes off the train to find a policeman. I worry that the train is going to leave without him. The Poles sitting stoic across from me at first ignore my distress, but then, quite suddenly, one of them\u2014a middle-aged woman\u2014rises, leaves the compartment, and, mysteriously, after a few minutes, without one word returns with my wallet, which, she gestures, was tossed onto the platform. Just the money is missing. Lacking the words, I mumble sounds of gratitude. Like Madame Defarge from Dickens's _A Tale of Two Cities_ , she returns to her knitting. Later when I leave the train in Warsaw, I distrust anyone who brushes by me. Two weeks after that, Irving is robbed while boarding a tram\u2014an all-too-common occurrence.\n\nOther surroundings of Krak\u00f3w also attract visitors. In April 1996, when the rain turns the snow into deep slush, we board a local bus and drive from Krak\u00f3w through the fog to visit the historical salt mines (a UNESCO World Heritage Site) in Wieliczka. After descending hundreds of twisting steps, we walk over four and a half kilometers in various levels of the mine, pass through long tunnels and corridors, and enter large rooms with sculptures, chandeliers, and chapels carved out of salt and luminescent crystals. Deeper down, we walk around a museum showing tools, machinery for lowering horses into the mine, treadmills, and clothing of the miners. The mine extends for many miles and is still active. Indeed, children and adults suffering from asthma often go there for the summer. The mines become a sanatorium. The air relieves their symptoms.\n\nA swift and pitch-dark ride in a narrow lift brings us to the top again. The first time I took this restricting lift was a few months earlier when I was with a Fulbright colleague who suffers from claustrophobia. I suggested we sing our way up\u2014I think the strange choice was \"Onward Christian Soldiers.\" It worked.\n\n### **GDANSK**\n\nThroughout the year, in addition to traveling to Krak\u00f3w and its surroundings, we venture elsewhere. In the dead of winter (why, I shall never know), we take a train to Gdansk, on Poland's Baltic coast. (Before the Second Word War the city was a free state.) The seaside in winter beckons me. The snow falls, the winds blow, the ice forms, and we persist. Walking from the train station to our hotel on silent, nearly deserted streets, we occasionally shelter from the wind in the recesses of doorways. We peer into the windows of the amber jewelry shops lining Marjanska Street (amber comes from the Baltic Sea). Few are open at this time of year, but those that are offer what they call a \"winter discount.\" One jeweler gives me a special price because I am the \"first\" customer of the day. It is five in the afternoon.\n\nGdansk (formerly Danzig), a major seagoing and trading port for centuries, has many levels of history (most recently, the beginning of the Second World War and the birth of the Solidarity labor movement in 1980). Before the war treasures, patrician mansions, and handsome buildings displayed a catalog of the city's rich and privileged past, purchased from Teutonic knights and Polish kings. During the war, however, these structures tumbled and crashed into the elegiac, scrap heap of memory. In 1945, Gdansk's old town (with the exception of one fourteenth-century church) was completely leveled, just as was Warsaw's. The restored old section of the city now stands as a tribute to the compulsion to rebuild what was destroyed, to realize the legend of the phoenix rising from the ashes.\n\nAfter the war, what did not survive was reconstructed piece by piece, detail by detail, ornament by ornament. Consulting prewar photographs of the area, the Russians painstakingly rebuilt and, in a sense, revived what had been bombed or burned\u2014if only it were possible to reconstruct lives by transforming snapshots of the departed into living bodies that move and breathe once more. The result is a disturbing beauty, for what one sees is a facade, a front, a mere copy of former splendor. No matter how one tries to pretend otherwise, the past will always be ruined and silenced. A nagging sense that what lies before the eye is not real skulks on the periphery of one's vision.\n\nThe old town borders the city harbor through which large ships pass. Solid and still, these oceangoing vessels stand frozen and waiting. The ice packs in layers upon itself and seems even to freeze the facades of the area's tall, fourteenth-century Hanseatic League dwellings. The wind howls through the old city gates. In an open spot below a footbridge over an inlet, coots, mallards, and swans collect and wait for people to bring them crumbs. The coots stretch their rounded webbed feet, the mallards stand or crouch, and the swans tread clumsily on the ice. Only when they fly and extend their long, graceful necks do they regain their dignity. When they pass directly overhead, I hear their wings rhythmically swishing up and down and listen to the subdued sounds gurgling deep in their throats. Long gone and far from fantasy are the prewar summer days that G\u00fcnter Grass remembers in _The Tin Drum_ when:\n\n_Chimes sent pigeons up in clouds: to cry. But all about was laughter. Women with sunburned children, terrycloth bathrobes, brightly colored beach balls and toy sailboats, descended from trams bearing freshly bathed multitudes from the sea side resorts of Glettkau and Heubude. Drowsy girls licked raspberry ice cream with lithe tongues._\n\nAfter the war, people wear a different look. Men seem demobilized and women smiled while concealing an undertone of the sorrows they've suffered.\n\nNot far from Gdansk is Europe's largest castle, originally erected by the powerful Order of the Teutonic Knights in the thirteenth century. From Gdansk it is an hour's train ride to Malbork Castle. Once inhabited not only by the Teutonic Knights and later by Polish royalty but also, in the 1930s, by the Hitler Youth and the Nazi League of German Girls, this castle was more than half destroyed at the conclusion of the Second World War. A 1945 photograph shows the devastation: shell-damaged walls gaping and pockmarked, arrested avalanches of stones spilt from bombed towers, and brittle charred roofs hanging precariously over piles of rubble. It was not until 1962 that restoration of this site began. We come when that work is well on its way. Now only the main cathedral is still in ruins.\n\nWe arrive on Saturday afternoon, just one hour before the castle closes. We are the only ones there. It is freezing cold. Irving finds a guide who is willing to take us around. She speaks Polish but puts in the occasional German word (sometimes a Russian phrase interrupts her thought). She takes us through one magnificent room after another. The vaulting, frescoes, fireplaces, windows, tapestries, and thirteenth-century doorways (still showing their original paint) are wonderful to behold.\n\nThe guide is loquacious. Irving pretends to follow her patter and does to some extent. She seems pleased to have our company. Taking hold of my hand, she proposes that we walk outside the castle by the River Nogat. People are ice fishing; dogs chase swans, and children slide, with and without sleds, down slick shining hills sloping from the castle to the water. The guide starts to tell Irving about herself. With her ungloved hand she writes in the snow the date of her husband's death; he was run over by a bus in 1974. The lovingly written numbers will dissolve; unlike the refurbished castle, nothing can restore his life.\n\nTo get back to the road, we scramble up an icy hill. Incredulous children show us the way. We return to the train station and eat a snack; then Irving searches for the \"toalety\" but does not know how much to pay for its use. He approaches three skinny young girls clustered like sparrows behind a counter (the only people in the deserted station). As soon as they see him, they start to giggle and chirp. He asks them \"How much?\" They turn away and consult. Eventually one comes back and, wryly cocking her head, inquires, \" _Pissoir_ or _cabinet_?\"\n\nBlushing, Irving replies \" _Pissoir_.\" That will be thirty groszy.\n\nWe go out to the platform. Drunk, black leather-jacketed teenagers lean daringly out of an open train window and jeer menacingly at a group of Polish army men walking just in front of us. The army men wait until the boys are at a safe distance and then coolly raise a finger. Our train arrives and takes us back by Malbork Castle now glowing red in the evening light as if a fiery wreck\u2014a memory of its earlier destruction. We share a compartment with a man on his way to Warsaw to take advanced classes in taxation. Once a farmer (his parents are now alone on the farm), he closes his accounting textbook so he can go to the window and show us the landscape's rich soil in which \"everything grows well.\" He too has not lost a memory. Like so many others rushing to join Poland's rising middle classes, he is studying to become an accountant and is exchanging the countryside for the city. But he still feels the soil beneath his fingernails and treasures that earth.\n\n### **THE PLACES OF EARLY SUMMER**\n\nIn hot late May, we go northwest from Warsaw to Torun. On the five-and-a-half-hour bus ride, people strain to open unwilling windows to let the stifling air circulate and blow through the stuffy interior. Turning toward the window to catch some breath, I notice storks sitting on their nests; their beaks extend beyond the dense elevated mass of branches.\n\nIn Torun, we buy gingerbreads made in ancient molds in the shape of horse-drawn carriages. They are solid; one risks breaking a tooth. Before it is to be declared a UNESCO World Heritage Site in 1997, we walk in the yet-to-be-developed old town where Copernicus was born, stroll downhill through ancient gateways, and reach the river, where hushed barges slowly move their cargo upstream. Farther on, we come to a small zoo where a black bear circles his shadow in an iron cage and where dingo dogs, in a small enclosure, howl and jump, all four feet in the air, whenever the keeper walks by. These images play in the mind like motes in the eye.\n\nIn the weeks remaining before we leave Poland, we take a bus to the village of Kazimierz Dolny, 147 kilometers southeast of Warsaw on the right bank of the Vistula River. There we buy bread in the shape of roosters and crayfish, wander in the old town square, walk among the castle ruins, visit an old Franciscan monastery, and go past a synagogue turned into a movie house. Surrounding the village are walking paths that meander through orchards bordered by blackcurrant bushes. All is sweetened by the early-summer light.\n\nIn the woods, we visit the remains of a Jewish cemetery (1850\u20131930). Like sentries guarding history, each gravestone stands at a distance from its neighbor among the trees. The Nazis once used these _matzevot_ (gravestones) to pave the streets of the town. Long shadows of reflected branches lie across these stones, pied and mottled by the sunlight peering through the leaves. Their wraithlike limbs connect one spent life to another.\n\nThe following weekend we travel three and a half hours by train to Poznan\u2014west of Warsaw, not far from the German border\u2014and accidentally attend the finals of an international violin making competition held every five years. A young man from Italy wins. He is one of 180 entries. Before receiving a free glass of wine and bacon-flavored potato chips, we listen to two sonatas played on the winning violin. Afterward, the evening is still young so we walk up the hill to a Franciscan church and attend a chamber choir concert.\n\nThe next morning, a few kilometers from town, we enter a park dedicated to beekeeping. Hundreds of hives from the fourteenth century on, each more decorative and imaginative than the next, fill the park, as do children dressed as bees, drawing pictures of bees, and dancing a \"bee dance.\"\n\nOn our return to Poznan, we walk by the synagogue, which the Nazis transformed into a swimming pool, and which still stands as such, and come to the old town square, where we find a caf\u00e9, sit, eat, and watch people go by. At the next table are two loud German tourists who have come to Poznan for the weekend. Their presence reminds us that, although the borders between Poland and Germany have often swung back and forth, Poznan is now really a satellite of Germany, which, I believe, finances much of the town's infrastructure and businesses. People are more abrupt and businesslike than those in Warsaw; this city has a harsher edge to it. Yet all is forgiven when I stroll down one of its narrow streets, pass an open window, and overhear a bass voice practicing an aria and a violinist rehearsing a passage from a Scarlatti sonata.\n\nLater, we spend a day in Lodz. I recall little except the streets bordered by deserted textile and cotton mills, six or eight stories high. In the mid-nineteenth century this city was the main textile production center of the Russian empire. The thudding din of the steam-propelled looms pounding inside the factories would have reverberated through openings in the brick walls and joined the clatter of weavers' clogs in the exterior courtyards to create a cacophony of hard work and hard times. Now doors and windows are boarded up, smashed, and scrawled over with graffiti. The mills' whistles that once called people to their shifts no longer blow. All is hollow and silent. Only weeds flourish and push their way through chinks in crumbling bricks. Not a soul is to be seen.\n\nI walk through this wasteland and remind myself that striking workmen once staged protests outside these walls, the largest of which took place in 1892, and that in 1905 Tsarist police viciously fired upon and killed groups of Polish workers carrying placards denouncing the autocracy of the Russian empire. The area reminds me of the Manchester\u2014the industrial North\u2014of my childhood; I cannot help but be drawn subjectively to its landscape. I am back.\n\n_It was a town of red brick, or of brick that would have been red if the smoke and ashes had allowed it; but, as matters stood it was a town of unnatural red and black like the painted face of a savage. It was a town of machinery and tall chimneys, out of which interminable serpents of smoke trailed themselves for ever, and never got uncoiled. It had a black canal in it, and a river that ran purple with ill-smelling dye, and vast piles of building full of windows where there was rattling and a trembling all day long, and where the piston of the steam-engine worked monotonously up and down, like the head of an elephant in a state of melancholy madness. It contained several large streets all very like on another, and many small streets still more like on another, inhabited by people equally like one another, went in and out at the same hours, with the same sound upon the same pavements, to do the same work, and to whom every day was the same as yesterday and tomorrow, and every year the counterpart of the last and the next._\n\n\u2014Charles Dickens, _Hard Times._\n\n## _CHAPTER 4:_\n\n## INTO THE COLD\n\n### **LVIV 1995**\n\nDuring the University of Warsaw's winter break, Irving and I cross the Polish border in order to reach the most westerly region of Ukraine and then travel fifty kilometers to the city of Lviv, now one of Ukraine's main cultural centers. Irving has come to continue his work on philo-Semitism and Leopold von Sacher-Masoch's Jewish stories. Lviv has the bulk of the author's papers.\n\nBefore the outbreak of the Second World War, the city was a vital core of Polish and Jewish culture. Its opulent libraries, palaces, royal residences, early municipal buildings, churches, scientific laboratories, castles, markets, the Ivan Franko University (founded in 1661), the neo-Renaissance-style Lviv Opera House (built in 1901), and even remnants of the so-called Scottish Caf\u00e9 (to be renovated in 2014)\u2014where, in the 1930s, a group of mathematicians collaboratively worked out research problems on the caf\u00e9's white marble table tops\u2014attest to the city's rich cultural history as well as to the intelligentsia that once populated its institutions and overflowed into its streets.\n\nSadly, under both the Soviets and the Nazis, many of the city's intellectuals had been tortured, murdered, deported, or have since passed or moved away. The Soviet repression in the late 1930s, which sent many of Lviv's cultural leaders to their death in forced-labor camps, as well as the Nazi massacre of twenty-five university professors in June 1941, echoes in the memory.\n\nFor the next two weeks, we must find our way through a m\u00e9lange of vociferous Ukrainian nationalists, less audible Russian sympathizers, devoted followers of the former Austro-Hungarian empire, scraps of Jewish culture, diehard German nationalists, and remnants of a once-dominant Polish presence. Subject to the wrath of history, Lviv has multiple identities. The city's variety of names says it all: Lviv (Ukrainian), Lemberg (German and Austro-Hungarian), Lw\u00f3w (Polish), L'vov (Russian), and Lemberg (Yiddish)\u2014and what is its designation among Armenians, who constitute one of the city's minorities?\n\nAs with so many other Eastern European centers, these various allegiances reflect a long (Lviv was founded in 1256), complicated, and often divisive and bloody past that is almost too intricate to map. Shifting borders, battles for control over the city, disputed territorial claims, invasions, dual authorities, multiple languages, mixed allegiances, armed uprisings, genocide, censorship, and deportation have continuously disrupted any possibility of an integrated sense of place and loyalty. The city's unparalleled architectural beauty (from medieval to art deco) is a memorial to this dappled past (the Germans did not bomb Lviv) and leaves the beholder wondering to whom does the city physically or emotionally belong: Austria, Galicia, Germany, Hungary, Lithuania, the Ottoman Empire, Poland, Russia, the Swedes, the Tatars, or Ukraine? All of these nations have, at some time, laid claim to it.\n\nOnly in the last few years has Lviv's primary allegiance been to Ukraine. In the period of liberalization from the Soviet system in the 1980s, the city was the center of political movements advocating Ukrainian independence from the USSR. After the dissolution of the Soviet Union in 1991, Lviv got what it wanted and became a part of independent Ukraine. In the 1990s, during its conflicts with Russia, the city harbors some of the most extreme Ukrainian nationalists, such as the members of Svoboda, a political party that bans atheists and promotes a nationhood built on \"blood and spirit.\" It too is frightening.\n\nWhat I saw and experienced in the frozen residue of 1995 would not be recognizable to people visiting present-day Lviv, nicknamed the \"little Paris of Ukraine,\" where tourists might comfortably enjoy the architectural vestiges of grander times while relaxing in brightly ornamented coffee shops or strolling down manicured boulevards. Money has flowed in since 2012, when Lviv became a UNESCO World Heritage Site and the city turned its attention to attracting foreign visitors.\n\nWhen I arrive in December 1995, tourism is not yet a priority; survival, however, is. Lviv is struggling to pull away from its Soviet past and is economically not at all well. Wrung out from its troubled history of shifting borders, languages, and allegiances, Lviv, like a dirty, wet rag, sits limply waiting to be resurrected and repossess its former glory.\n\nIn the mid-1990s, to those in Warsaw who have never been to these parts, Lviv represents an estranged and threatening world. Ukraine and Poland have often been at war over the possession of it, a city from which the Soviets expelled 100,000 to 140,000 Poles following the Second World War. Though occasional tentative connections, student exchanges, and even familial ties occur between Warsaw and Lviv, Lviv is generally regarded with fear and remains isolated. For most it is an unfamiliar destination, known only through horror stories about thugs in hotel lobbies and roaming Mafiosi. My friends in Warsaw are amazed we are going. They think it dangerous and advise me to hang on to my jewelry, not speak English too loudly, keep a low profile, and avoid crowds. On the train, be sure to lock your couchette doors. One colleague, however, speaks of the city's beauty and suggests that it combines the aesthetic pleasures of Vienna, Paris, Prague, and Krak\u00f3w.\n\n### **THE JOURNEY**\n\nThe December 1995 journey to Lviv is a trip into the unknown and, as such, still rattles down the tracks of my memory.\n\nWith our passports close to our bodies and our money distributed in various pockets (we have to take American dollars), we arrive at the Warsaw-Gdansk Railroad Station at eight in the morning. The snow is blowing; the train station, lit by hollow, harsh fluorescent bulbs, is virtually empty, except for a few people milling around under rafters covered with puffed-up pigeons huddled and sheltering from the cold. Heeding the warnings of Agata, we speak softly, _sotto voce_. We are to leave from Platform 3, so, after crossing the tracks on foot, we join others either lugging massive plastic bags of goods or waiting silently.\n\nThe train looms out of the early darkness. Bulky, black-green, and full of smoke, it emerges from a Soviet past. We find our wagon and step into a dark, cold, and grimy world that will be ours not only on the train for the coming fourteen hours but also for the next fortnight in Lviv. A ghastly dimness makes it hard to grope our way through the train, feel our way past minute buckets of coal, and enter our second-class compartment. Mud, melted snow, and months of memory streak the windows. Ratty stained curtains, stamped in one corner with an official seal, slouch in a corner of the window frame. Bunks with blankets, knotted and inhabited by the germs of humanity's history, occupy both sides of the couchette. The pillows leak feathers; the covers are gray, rough, and used. A small table extends from under the window.\n\nWe are relieved to have the compartment to ourselves, for the travel agent has told us that women and men are segregated on the train and that we, therefore, shall not be able to be together, unless we pay off the conductor. It is ours for an hour, until a university student, who is going home for the semester break, unexpectedly joins us. (Young people from Lviv quite often attend Polish universities.) Communicating with bits and pieces of Polish and Russian, we learn that she was ill after the Chernobyl accident. \"I'm fine now,\" she claims. Once the polite exchange is over, our companion nimbly climbs up to the top bunk and disappears, dead to the world, under the bedding we have rejected.\n\nWhen the train laboriously moves away from the station, we cannot conceive why it will take fourteen hours to go approximately 180 miles to reach our destination. Before too long, however, we begin to comprehend why the journey is to be long and dreadful. The compartment grows colder and colder. Soon we learn that there is no food or drink, so for the next fourteen hours we live off two small boxes of juice and three chocolate wafers we find buried in our pockets. The occupants in the adjoining compartment know better and have brought tins of sausages and bottles of sauerkraut. The putrid smell of our neighbors' repast coasts through the vents and mingles with the smoke from a coal fire the lady conductor stokes and on which she warms water for her tea. The toilets are too awful to use. A dirty rag lies over the seat, which is never cleaned. We both refrain.\n\nFrom time to time the train pulls into stations and stops, but no one, it seems, gets either off or on. When we arrive in Lublin (the largest Polish city east of the Vistula River), there is perhaps a change of engines, and then the train seems to go back the way we came. It is now impossible to see anything out of the window. Though it is growing darker and darker both inside and out, there are no plans to turn on any lights. My feet are like stone.\n\nAfter enduring seven hours of sitting in complete darkness, we come to an abrupt stop. Irving, with the help of a flashlight, has been snatching furtive glances at paragraphs from his novel while I have been listening to books on tape. When the Polish border guards embark, a few fluorescent lights magically switch on. The officials inspect our passports and then instruct us to leave the cabin so they can search for contraband. The lights go off and we are underway for a while, until yet another set of Polish guards enter and look once more at our passports; they carefully compare photographs to faces. Lights off and move. Soon the train stops again; lights on, and we are asked to fill out declaration forms that are written in Ukrainian (and in small print). What to do? The student in our compartment, like a groundhog from its hole, crawls out from the blankets and helps us while repeatedly exclaiming in Russian, \"The trip is a nightmare!\"\n\nLights extinguished; we proceed but not for very long. This time, the Ukrainian guards, wearing their heavy square fur hats, enter and take away our passports and tickets. In the freezing air, their breath trails their commands. After at least half an hour, they return with our documents. The train jerks slowly forward and finally halts at a station full of enormous machinery and lifts. It is like entering some leftover Soviet-style factory or perhaps finding oneself in a frame from a German expressionist film showing massive cogs and wheels laboriously turning, grinding, and producing.\n\nAs a relic of territorial disputes, the train is forced to stop because the tracks in Ukraine are not the same width as they are in Poland. They are narrow-gauge tracks, so in order for the journey to continue, the train's wheels must be taken off and narrower ones put on. (This anomaly was recently corrected; new tracks have been built, and high-speed trains now regularly whiz between Warsaw and Lviv.) For several hours, while still seated on the train, we are shaken, jolted, lifted, and pushed. Groans and clanks accompany the train's moving back and forth. I can stand it no more. I whip out my Sony Walkman and listen to a tape of Helen Reddy singing \"I Am Woman\"\u2014anything to give me strength to endure the ordeal.\n\nFinally, the train pulls into Lviv. Dazed from the long day that has been night, and not yet fully awoken from the nightmare, we step out into a dim station jam-packed with travelers on their way to I know not where. It is 11:00 p.m. Like lost sheep, we follow passengers down stairs and through unlit, filthy, frozen, dripping tunnels, through puzzling corridors, until we emerge into a domed nineteenth-century grand hall, adorned with clumsy imposing friezes displaying communist hammers and sickles.\n\nBrimming with goodwill, our guide, Ihor Markow, greets us and leads us out into the cold night. Across the street a car, stinking of preserved sausages, awaits so it may bump us along to the Hotel Dnister, a place where foreigners often stay. Though our room is dull and modest, the hotel seems bright and clean. Famished and dehydrated, we set out on foot through deserted streets in search of a restaurant. The two places Ihor thinks might be open are shut, so we try the Grand (Mafia) Hotel\u2014the one with armed guards and cell phones. The bar is open; we are thankful when the attendant agrees to make us sandwiches.\n\nOn the way back, a silent snow possesses the earth. Gentle circles of light radiate from the occasional wrought-iron lamppost and illuminate drifting particles of snow. The sight softens my ragged fatigue. It was Lviv that, in 1853, introduced Europe's first street lights.\n\nUnattended, the next morning we walk down to the old celestial city through a sloping park. Children sled and ski, dogs bound through the snow, and parents rush to keep up. Following our eyes (no map is available), we amble from one old square to another, past structures from the sixteenth, seventeenth, eighteenth, and nineteenth centuries. The streets below are thick with poverty, sorrow, sickness, and affliction. Lviv is in pain.\n\nThe trams are crammed because only a few can still run. Because the city lacks the means to build new ones, the older models suffer from inadequate repairs. In some parts of Lviv, water flows only during short periods of the day; the shops lack stock. Inside the historic buildings, one discovers neglect. The city cannot afford lights. We enter the ethnographic museum where nothing is illuminated and where exhibits sleep too long in the dark. At a gift stall we want to buy something but cannot properly see what is for sale. We ask the attendant to turn on a light; she shrugs, \"It does not work.\"\n\n### **LEOPOLD VON SACHER-MASOCH (1836-95): A PILGRIMAGE**\n\nIn the days that follow, I initially accompany Irving on his quest to visit institutes and archives holding the papers of Leopold von Sacher-Masoch, who was born and lived in the city. Interested in the concept of philo-Semitism in nineteenth-century Lemberg, Irving wants to find out more about the composition and circumstances of Sacher-Masoch's Jewish stories as well as to examine the claim\u2014or if you prefer, \"legend\"\u2014that the writer is himself Jewish. The search leads to few definitive answers but reminds one that in Europe there is no such thing as being part Jewish. Any hint of Jewish ancestry and that is enough to label a person as such.\n\nFascinated with the various ethnic cultures that populated the capital of Galicia, Sacher-Masoch, spent much of his life writing stories, in German, about the region's Polish, German, and Galician inhabitants, later translated into Ukrainian, Polish, Russian, and French. In particular, in 1878 and 1881, Sacher-Masoch published two sets of Jewish stories in which he displayed a remarkably thorough knowledge of Jewish customs as well as affection for the Talmudic enterprise. (Irving discovers, however, that many details in these tales are plagiarized from other sources.) Nevertheless, these narratives intrigue Irving, as do Sacher-Masoch's various campaigns to work against local anti-Semitism through adult education and by means of a Leipzig-based monthly progressive literary magazine (1881-85) in which, as editor, he promoted tolerance and integration of Jews in Saxony.\n\nThese stories, however, are not what most people think of when they hear the name Sacher-Masoch. Rather, their attention leaps almost exclusively to his _Venus in Furs_ and its exploration of sexual gratification through physical and emotional abuse. (Indeed until recently this was the only Sacher-Masoch book to be translated into English.) The book endures. Indeed, in one of those bizarre twists of history, a group of entrepreneurs in Lviv have now appropriated it to attract visitors and create a lucrative branding opportunity that, in their view, places the city on the map. Playing upon the popularity of _Venus in Furs_ , these businessmen are promoting masochistic tourism. (And this is progress. The weird distortions of capitalism are depressing.) As a marketing gimmick, one Lviv businessman opened the Masoch Caf\u00e9, decorated with whips, chains, and bras, in which customers, both male and female, can choose to be tied to a chair and, among other painful delights, be lashed, by a waitress, across their bare backs. Outside the caf\u00e9 stands a life-size bronze statue of the writer. All this is surreal, far, far removed from the Lviv of 1995, which revered his work and respected his interest in Galician culture. No intelligence can fathom it.\n\nThe eventual pilgrimage to find out more about Sacher-Masoch's writings and life is fraught and difficult. It has as much uncertainty and complexity as do the questions concerning his Jewish origins. Strangely (because it is a Christian text) and perhaps inappropriately, all I can think of during this quest is a passage from John Bunyan's _Pilgrim's Progress_ , for often _Ignorance_ is our companion through this once-splendid city of Lviv, Lemberg, and Lvov. (Literary parallels sometimes assuage anxiety.) Bunyan aptly captures our dilemma as we try to find our way through the mysteries of ancestry and nationality:\n\n_So they went on, and_ Ignorance _followed. They went then till they came at a place where they saw a way put itself into their way, and seemed withal to lie as straight as the way which they should go: and here they knew not which of the two to take, for both seemed straight before them; therefore here they stood still to consider._\n\n\u2014John Bunyan, _Pilgrim's Progress_\n\nAs the quest continues, _Vain Hope_ joins us. The first stop is an attempt to locate materials in the Austrian Institute, yet another magnificent building gone to seed. After climbing wobbly, creaking backstairs and scuttling past unclean toilets, we enter a splendid receiving hall that proffers a view of the park and the radio towers that Russia erected to interfere with Voice of America broadcasts. (Nobody has bothered to remove them.) Alone sits a smart secretary typing at a desk standing on a fraying Oriental rug that partially disguises a cracked parquet floor that once shone and reflected the brilliance of the Austro-Hungarian Empire. In German, she directs us to an office where a diligent, round-spectacled Austrian suggests that Irving would do better to go across town to the Austrian library. (The Austrians are nostalgic for their historic presence here and maintain what they can of their former presence.) Off we go, back through the old town and ascend a hill. Having reached the Austrian library, we clamber up what was once a splendid flight of stairs, walk through an unmarked door, and enter the library's reading room. Disappointment: the archivist whom Irving needs is away on holiday.\n\nLunch is the only thought, so we return to one of the restaurants that was closed last night. It is Christmas Day in the West, after all. I feel the need for some sort of celebratory \"dinner.\" (I should explain that in the religious calendar of Ukrainian and Russian Orthodoxy, Christmas comes several weeks later.) We enter a dim, cavernous room. The only glow comes from a tinseled Christmas tree attached to which is an electronic gadget that repetitively cranks out familiar seasonal music and traditional carols. A badly dressed doorman, waiters in crumpled tuxes, and a menu we don't understand greet us. We are one of two groups of customers. The restaurant offers little warmth. I take off my shoes and place my feet in Irving's sheepskin gloves; we wrap scarves around us. When the food arrives, it consists of blackened french fries (leftovers cooked innumerable times) and tough, dry meat. Freezing and disappointed, we eat and return to the center of Lviv so we might hunt for the Lviv Archives. At least Irving will know where they are located for another day.\n\nAfter slipping and sliding across the darkening streets of late afternoon, we inquire at some sort of tailor's shop where we might find the door to the archives. An old, wrinkled, arthritic finger that has threaded many a needle wafts vaguely toward an imposing thirteenth-century wooden door that has bolts and locks the size of one's arm.\n\nWith a hefty push, it creaks open and deposits us in a stone hallway. We worry that the door will slam shut and we'll be trapped. Nobody in sight, so we knock on an inner door marked ST. ANN's. It's the wrong one. \"Upstairs,\" we are told. After winding our way up delicate wooden steps, we arrive at another old entrance. We pass through it, open a more modern contrivance that unexpectedly swings, and find ourselves facing a lengthy, dark corridor that resembles illustrations representing the \"vanishing point\" in art. Seventeenth-century porcelain stoves and framed medieval portals line each side of the hallway. Which one should we go through?\n\nThankfully, a door opens and emits some promise of light and heat. We enter. This time it is not Russian, Polish, or German that works for Irving, but French. Pleased to show off her impeccable French, the librarian says that Irving should return tomorrow and she will have materials for him. In my literary frame, she appears as one of the virgins from _Pilgrim's Progress_ who will show the pilgrim the rare ancient documents. I enjoy the walk back to our hotel through the park's quiet darkness. That night we eat in the hotel restaurant, another empty room with little heat and rumpled waiters. The food is not reassuring.\n\nAs they often do, mornings bring promise. For the moment, _Hope_ is our guide. We arise and go down to the chilled hotel restaurant (where a tomcat shrieks), eat smoked chicken and sawdust bread (with the additional delight of rancid butter), and sip on apple juice. Irving takes a taxi back to the archives, which we located the previous afternoon. He has negotiated with the hotel staff for a ride; he has been warned that it is dangerous to catch a cab on the road because he could be robbed. Even though it is bitterly, biting cold, I decide to walk to the archives through the lovely park and take one more look at the historic buildings along the way. When I arrive, I find Irving conversing with yet another librarian, this time in German. She sits in front of an enlarged photograph of herself in army uniform. She is helpful and gives Irving more contacts and phone numbers.\n\nWearing my coat in the reading room, I sidle up as close as I dare to a porcelain stove that produces some semblance of warmth. Once in a while, another reader periodically leaves his seat to poke the stove to tease out more heat from its reluctant and aged orifice. In the poor light, while hunched over a Melville novel I am supposed to teach on my return, turning the pages with gloved hands, I occasionally glance at the rows of seated scholars poring over old maps and newspapers that in this semidarkness are barely visible to the human eye.\n\nIrving keeps getting called away because people, once they learn of his interest, want to meet him. There is an alarmingly avid Ukrainian nationalist, who speaks English, and then a young German, who is researching German settlements in Ukraine. This young man insists on becoming our guide and wants to show us a cheap way to eat lunch. Like _Vain-Confidence,_ he gallops ahead as we struggle to keep up and briskly falls into a deep pit. Much to our horror, he suddenly slips and slides on his back down the icy stairs leading to an underground passage. Concerned, we observe his recovery and marvel at his ability to carry on as if nothing has happened. We cautiously follow him to an outside stall, where he buys some sort of prepared sandwich. All around me people have been losing their footing since the day broke. My back is getting stiff from holding on to myself.\n\nEventually we reach a dingy, barely warm coffee shop that has a few tables. The coffee is delicious, but the cakes and sandwiches are of a doubtful nature. Though difficult to follow when he speaks, our companion soon reveals himself to be an extreme German nationalist, a Beelzebub, who extends the ethos of the 1930s. His aggressive patriotism is alarming, as is his obsessive concern about becoming sick in Lviv.\n\nA beggar child enters. The German instructs us not to give her money. The child, he claims, has been sent in by an adult to find out from which part of our clothing we retrieved our wallets. After we leave, he insists, we will be followed and robbed.\n\nAfter lunch, we shake off our German companion and, progressing on our journey, make our way up another hill of difficulty to yet one more library, the University Library where a Mr. Kutik, the bibliographer, works. We enter and are escorted by a lady who goes to much trouble to locate him. He is found; we go upstairs to a small, chaotic office, where he sits with his hair going in all directions and his unraveling pullover creeping out from beneath his shirt. He is kind and has a wry sense of humor. He promises to help Irving if Irving returns at a certain time tomorrow; he also writes down phone numbers and contacts that might be helpful.\n\nOn the way back to our hotel, I begin to cough but think it caused by some irritant in the air. We stop at yet another restaurant pointed out by our well-meaning guide, Ihor (our _Goodwill_ ). I order pizza and soup. No luck. The soup tastes like a dishcloth and the pizza is an egg salad on top of dense dough. We leave and give the hotel restaurant another try. The meal is refried (I don't know how many times) chips and a cutlet.\n\nThat night it becomes apparent that I am actually sick. The following morning, I have a high fever and a wrenching cough. Ihor calls and promises to send over a doctor. While Irving pursues his research at the University Library, I stay in bed and wait until 3:00 p.m. when a doctor is supposed to arrive with a translator. At that very hour, true to the promise, there is a knock and in comes a Soviet-generation doctor: large, solid, clumsy, nice, and round-faced with instruments as old as they look and not sanitary. For this reason, I don't let her take my temperature. She listens to my chest and bronchial tubes (and takes forever), flips my breasts like pieces of rubber in and out of my bra, and then takes my blood pressure. Before leaving, she gives me a prescription (which the translator refers to as a \"reception\") for an antibiotic, an expectorant (which comes in a horrible-looking bottle I dread going near), a cough suppressant, and aspirin. I ache terribly and am in a bad way. I take some aspirin and wait impatiently for Irving to return. Eventually, in the early evening, he does, but I learn that because there is so much sickness in Lviv, the pharmacies have run out of the antibiotic I require. With the hope of offering me some help, the hotel manager sends a messenger to a pharmacy in another part of the city. He returns with some other antibiotic (labeled in Russian) that might help.\n\nFor over a week, I am sunk in the _Slough of Despond_. Unlike Bunyan's pilgrim, I am not able to drag myself out of the bog. While Irving pursues his search, I am lonely and despondent in this quagmire of fever. I have no means of talking to people. I am stuck with my own ignorance. Each morning, I drag myself to breakfast and leave with extra packets of soft cheese and juice so I shall have something to nourish me until evening when Irving returns.\n\nThe hotel turns off the heat during the day, so in the hope of capturing some warmth, I climb under the blanket, wrap scarves around me, and crawl into a sheepskin jacket. (I am hardly a Venus in furs.) I wait for the maids to come to the room. One housekeeper pities me and wants to make me tea. She brings in a stained cup, takes water from the tap, puts in a coil, fetches tea and sugar, and offers me a hot drink I do not have the heart to refuse. Though we do not share a language, I owe my life to her. I moan, ache, listen to books on tape (they too save my life), and try to watch television (for a few hours in the evenings, there is one static-bitten channel that shows nothing but downhill skiing\u2014and anyway I cannot understand anything being said). I cough, sleep, and waken to hear my lungs whistling. When Irving returns, I listen to him relate his adventures and wait for him to bring me some hot food from the restaurant after he has finished eating.\n\nDuring my sickness, Irving goes to mail a postcard and asks a lady he sees on the street to show him where the postbox is. After they converse in German, she invites him to her apartment so he can help her write Christmas cards to friends in America\u2014her English is imperfect. So immersed am I in the context of this pilgrimage, my mind temptingly and ridiculously entertains thoughts of the model for the masochistic Venus in Furs, a Baroness Bogdanoff, who seduced the writer by asking him to help her improve her writing.\n\n_Her head was wonderful in spite of the dead stony eyes: it was all I could see of her. She had wrapped her marble-like body in a huge fur, and rolled herself up trembling like a cat._\n\n\u2014 _Venus in Furs_ , Part I\n\nBut nothing could be further from the reality. The lady is no Venus in Furs but an elegant, refined, and impoverished remnant of the Austro-Hungarian Empire, a reminder of a softer, more settled, enlightened, and affluent time. When Irving takes her to coffee, she, exhibiting her old-world manners, clings to his arm (\"lest he fall\"), kisses his hand, and invites him, if he should return, to stay with her.\n\nIn the various libraries we have already scouted, Irving meets even more librarians who are more than willing to help. They lead him down dark corridors, through sequestered catalog rooms, and then pause before the place where the Sacher-Masoch drawer is. But once it is opened, the librarian discovers the manuscript is not there. A great scurry. It is, she recalls, in another library, so arranges for it to be sent over. The manuscript contains three unpublished Jewish stories by Sacher-Masoch.\n\nAfter a few days, in the evenings people start coming to the hotel to call upon Irving. There is, unfortunately, not the practice of announcing their presence from the downstairs desk; instead, visitors knock unannounced. At such moments, reluctant and embarrassed, I slip out of bed and try to repair myself in my mortal garments so I am respectable. I often suggest they go elsewhere to talk. On one occasion, the substantial daughter of a well-known Ukrainian writer and her well-fed husband come through the door. She has information for Irving, as well as phone numbers of Sacher-Masoch's relatives. The result is that in a few months Irving meets with the writer's grandson who tells him that Simon Weisenthal tried to reach Sacher-Masoch by holding s\u00e9ances. Moreover, the writer informs Irving that Marianne Faithfull is a descendant. A series of inconclusive emails with the former rock star leads nowhere.\n\nAt another time a Yiddish-speaking, elegiac elderly gentleman who knows nothing about Sacher-Masoch comes sorrowfully to our hotel room to converse with Irving who is fluent in Yiddish. (In Lviv there are few left. If reports are to be trusted, the last Yiddish-speaking resident died in May 2014.) On another night, a man who has just written a history of the Jews in Lviv calls. He tells Irving that he will be emigrating to Los Angeles the next day.\n\nThe invasion goes on and on. One dreadful evening, a crew making a film about Sacher-Masoch knocks, enters, and takes over the room. I crawl into a corner. Another night and another knock\u2014in come Pasha and a young American woman, Katya, who is acting as his interpreter. Pasha is gangly, pale, bad-teethed, Russian-cigaretted, smart, and generous. Katya is voluble, impatient, and manic. Neither of us is sure why they come to the room, for they know little about Sacher-Masoch. Both are, however, familiar with the city and offer to take Irving on a walking tour the next day. In negative-thirty-degree Centigrade temperatures, they tour the old Jewish part of the city. (Before World War II, Jews made up over one-third of the population.) From that day on, the two do not leave us alone. It is exhausting.\n\nAfter many days, I am recovered enough to leave\u2014and we have stayed long enough.\n\nThe night before departure is New Year's Eve. Ihor, our figure of _Goodwill_ , comes to bid us farewell. He keeps saying \"congratulations\" when I tell him I have been sick. His good intentions (incorrectly spoken) remind Irving of a librarian who, concerned about me, repeatedly reassures Irving, \"It will be alright as long as Ann does _not_ get better.\" Still more than happy to stay in bed rather than get up, on television I watch the Ukrainian president give a sober address before midnight (I do not know what he is saying); I listen to people partying in the hotel downstairs. The music continues until 6:00 a.m.; firecrackers go off during the night, and my cough hacks on.\n\nFuzzy-headed and armed with cheese and juice for the return trip, we go down to the hotel lobby and wait for our taxi. Pasha and Katya, who have been up all night, show up and insist on riding with us to the train station. Katya's spectacles are barely holding her eyes together; Pasha's hat and head are more angular than ever. His ever-greasy locks stick out in odd ways from under his hat. It is a good thing the two accompany us, for the taxi driver, misunderstanding our request, is mistakenly taking us to the airport. They correct him.\n\nSilent crowds either walk around the train station's chilled waiting hall or stand watching a television perched high on an ornate balcony overlooking a once-functioning fountain. We make our way back through the darkened tunnels and walk up to an icy platform where monstrous trains spew out coal smoke. Underneath the wagons, the spillage from the cabin toilets hangs, suspended and frozen. Pasha wants to determine whether we are waiting on the correct platform, so instead of going back to the main entrance through the tunnels, he jumps down onto the tracks and crawls under a stationary train to reach the other side. I am incredulous. Katya is becoming more and more voluble; she cannot stop talking. Before long, we are reassured we have the correct train; we climb on and find our compartment.\n\nAnother fourteen hours. This time, Irving asks the conductor for cups of tea and boiled water to fill a hot-water-bottle he has bought in Lviv. Both bring relief to my aching head. Time passes. The train moves. In the darkling of early evening, two large foxes with bushy tails stalk through crisp fields, deer gather close to farms, and a single hare bounds across the meadows' white sheets.\n\nIn contrast with where I have been, Warsaw has never looked so bright and comforting: \"And behold the City shone like the Sun: the Streets also were paved with Gold, and in them walked many men, with Crowns on their heads\" ( _Pilgrim's Progress._ ) All is dreamlike. Warsaw becomes the \"Desired Country.\" I even have warm and joyful thoughts about my flat in the Hotel Sokrates. The food seems heavenly, especially after a diet of smoked chicken, cheese, bread, and burned chips. I am placed under the care of the medical staff at the American embassy. I have pleurisy and pneumonia. More antibiotics, and all will be well. Within days, I return to work at the Institute of English, and life resumes. The past two weeks have been a dream\u2014at times a nightmare.\n\nEventually the year is up and it is time to leave Warsaw and say farewell to Poland. Tearful Eva gives me a painting of the old town, Agata clings, and others promise to stay in touch. During the last week, minutes stretch into sad, protracted hours, yet, paradoxically, the clock speeds ahead. In the end, unable to outrun the swift passage of the remaining days, I must surrender to the inevitable and depart. Flying home, hovering between what was and what is to be, I am nowhere. In months to come, however, my experiences in Poland all too predictably bond with the past. The space between the then and the now expands; remembrances cease to trespass, tumble over, and trip up the present.\n\n## **P ART TWO**\n\n## **U KRAINE, 2000**\n\n_Eastern Europe draws me once more. As difficult and surreal as the visit to Lviv five years earlier had been, my experiences there, nevertheless, had whetted my curiosity. I wanted to return, so, under the auspices of a Fulbright fellowship, I, with Irving, was off again. This time, my curiosity took me, like Alice, down the rabbit hole to other regions of Ukraine, where for a year I unlocked doors and stepped through the looking glass to gaze at the other side of things. And just in time, for later much was to change. Beginning in March 2014, Vladimir Putin destabilized the country by taking back Crimea and then by waging a proxy war, using Ukrainian separatists and \"little green men.\" Lacking a wholehearted commitment from the West as well as adequate financial, political, and military support from its allies, Ukraine, already weakened by its oligarchs and an exploitative civil service, faltered and worried about its future._\n\n## _CHAPTER 5:_\n\n## A WORRIED STILLNESS\n\n### **UKRAINE: FEBRUARY \u2013 MID-DECEMBER 2000**\n\nSweeping clouds of the evening sky chase the frozen fields and soon surrender to the approaching city's sparse grid of flickering lights. The plane from London to Kiev lands. This is the last stage of a journey begun two days ago in Buffalo, New York. It is February 2000.\n\nI have come to a Ukraine caught between its independence from the Soviet Union in 1991 and Russia's bullying return in 2014. Wedged between loyalties to the East and the West, and profoundly sobered by memories of Chernobyl and its consequences, people anxiously hold their breath and get on with life as best they can. There is a tense peace. Frustrated reactions to galloping parliamentary corruption, broken trade agreements, debates about the gas line, and an increasing national debt are audible but kept in check by the harness of political jargon and false promises. A guarded openness typifies and surrounds both the country's governmental agencies and the institutions representing Western interests. (The American Chamber of Commerce in Kiev, promoting American investment in Ukraine, is an armed fortress.)\n\nIn 2000, even though a worried stillness seems to diffuse most everything, people are ripening for the Orange Revolution of 2004, but they are not, as yet, explicitly imagining the deadly insanity of open civil conflict between Ukrainian nationalists and pro-Russian separatists, as well as competition for power among regional oligarchs. Nor is the public expecting that in thirteen years, Russian troops will march into, fight for, and seize Ukrainian territory. I have come at a time between hope and despair.\n\n### **ENTRANCE**\n\nPopping noises of fluorescent bulbs accompany questions: How much money do I have with me? How many rings? Replicating my own crumpled and anxious state of mind, my travel-worn bags move through the X-ray machines and drop untidily onto the floor. Relief. The assistant from the Fulbright office is actually here to meet Irving and me. She is to drive us to a flat we have rented but never seen. On the drive to 36 Bohdan Khmelnitski in the center of Kiev, her singsong English floats like an unheard yet well-rehearsed melody from the front seat of the van to the back, where we sit numbed by both the travel and the realization that I have actually chosen to come here for a year to teach American literature in the Foreign Literature department of Taras Shevchenko University.\n\nWhen the assistant drops us off in the care of Halla, a friend of our absent landlady, I feel alone, particularly since I know neither Ukrainian nor Russian. Evening falls into night. I twist the grudging front-door knob in various illogical directions and, with my booted foot, push the heavy wooden door, which leads into a pitch-black entrance hall where the smell of urine and sounds of dripping water await whoever dares enter. Just enough light from the street lamps leaks through the smeared windowpane above the front door to expose menacing wires hanging from some electric, or perhaps telephone, apparatus attached to a flaking wall. Rusted mailboxes line another wall. Some are locked; others gape and display their neglected mouths. Ahead, my anxious eyes catch the serpentine shadows cast by a chipped marble staircase winding behind a narrow lift suitable for two and a half people.\n\nNo light switch is visible. In the months to come, I learn to carry a small flashlight with me so as to find my way down dark passageways and unfamiliar hallways. Following Halla's shadowy motions, I cautiously walk ahead and pull aside the lift's clanking metal gates, shut them, and as instructed, press button number 3. Complaining, the lift wobbles upward. Later, when visiting a friend's flat, I watch her demurely jump up and down to persuade the lift to move\u2014a technique shared by all who live in her block.\n\nOn this first night, the lift door opens into a hall as dark as a Kiev winter night; Halla finds the light switch. My neighbors are soon quick to tell me that it is essential to turn it off immediately after use\u2014electricity is at a minimum. Even so, I am soon to find out that we are in a privileged section of the city, for we have hot water and electricity most of the day, none of which is metered. In other areas of Kiev and throughout other cities and rural communities in Ukraine, such conveniences are available only a few hours a day (routinely between 4:00 and 6:00 p.m.) and sometimes not accessible at all, depending on the whim of either the Russians, who control the pipeline, or local district officials. Ukraine relies on Russia for half of its gas supplies. Efforts continue to locate alternatives from European providers, such as Germany, the Czech Republic, Slovakia, and Poland.\n\nThe hall light weakly illuminates the door to our apartment. Like so many in the city, this door is solid, seemingly impenetrable, made of ominous steel and padded like a vault. Feeling like Alice after she's tumbled out of her familiar world, through darkness, and landed underground in a hallway where all the doors are locked, I wander from door to door and wonder if the large key Halla has given me will admit us to our flat. Is the lock too large or too small? After I insert my key into what seems to be the right lock, the recalcitrant door to flat number 8 reluctantly gives and we are inside.\n\nAs will happen many times in the months ahead, unlit passageways and unopened doors preface entrances. A few weeks later, for instance, invited to a diplomat's flat, I open the door of his grand building and feel my way through the crumbling plaster of a dim hall. Once in, I step into a diminutive unsavory lift and reach for buttons that have lost their legibility to repetitive fingers. I am reminded of a recurring nightmare in which I attempt to make a phone call to report an emergency and find that all the numbers and letters are the same. After I randomly press a button, the doors close and the lights go out. I hastily push another button: up the elevator groans and disgorges my body onto a landing under construction. To reach the diplomat's residence, I need to go up one more flight, but as I approach the stairs a glimmer of half-light unexpectedly reveals a chasm ready to claim me as its victim (a new staircase is yet to be constructed). Unlike Alice, I am shaken and unable to think \"nothing of tumbling downstairs.\" And never mind considering \"how brave they'll all think of me at home.\"\n\nRather, I am reminded of Robert Louis Stevenson's David Balfour, who, in the blackness of a blustery Scottish night, vigilantly climbs the deteriorating steps in the tower of his murderous uncle's estate ( _Kidnapped_ ). Horrified, David watches as sheets of lightning illuminate a sudden drop where the stairs abruptly end. He, too, knows he could have plunged to his death. I bang on the door of apartment 24 and through a speaker plead for help. A young man, who thankfully speaks English, responds, goes with me to the lift, and pushes another button. Soon I am standing outside apartment 26 where I am meant to be.\n\nThere is life beyond this prefatory darkness, for when the solidly bolted doors in these dimly lit passageways yield, they give way to lighted spaces that exist behind their forbidding opaqueness. Through these entrances, life becomes \"curiouser and curiouser.\" I know \"something _interesting_ is sure to happen\" when, like Alice, I lean up against a padded steel door and step through the looking-glass that \"has got all soft like gauze,\" so I can travel to the other side and see things \"the other way.\" Sometimes the dark squalor of the hallway reverses itself to become a spacious, bright, and even luxurious set of rooms. Behind the diplomat's impermeable door, sunlit spaces with gleaming parquet floors covered in rugs from Kazakhstan dazzle me. I admire a splendid bathroom and a solidly made balcony from which one looks over Kiev and watches the evening sun descend over the city's hills.\n\nOn the first night, I step through my own steel door into a far less luxurious flat, but I do find light, fifteen-foot ceilings, a study with extensive airy windows, a bright kitchen, antique cabinets filled with fine bone china, and in a long, large elegant living room, an out-of-tune concert grand piano the landlady has tucked into a corner. The music of Chopin and Schubert lies scattered and opened on its lid. Our landlady, Luba Zuk, grew up in Ukraine and is a professor of piano at McGill University. Her brother, Ireneus, who is also a musician, often comes to Kiev to perform and teach.\n\nInside, \"cats become bats,\" and \"bats become cats.\" The shower sprays sideways, rather than down, the telephone does not speak, and, worst of all, the bed is not a bed. Halla explains that the better \"bed\" is in the living room\u2014it's a sofa that pushes down to make a single bed. The other option is a narrow sofa, which does not open up, in the study. We try to lie on the one in the living room, but the mattress, consisting of three hard pillows, slopes on one side and, on the other, cuts one's body into some sort of triptych. Months later, we finally walk two miles down Bohdan Khmelnitski to a large department store with creaky wooden floors and thieves. Armed with the right measurements, we find a mattress ($30), wrap string around it, clumsily haul it down three flights of marble stairs, lug it outside to a taxi stand, and go home where we squish ourselves and it into the unwilling lift, step on a resident's foot, and eventually heave our salvation onto the sofa\u2014all in an hour's time.\n\n### **YELENA**\n\nThe morning after our arrival, I hear an insistent ringing of the doorbell. I open the door to find Yelena Abromowich. Her generous body is held together by an apron, smeared with a lifetime of cooking. A maternal smile stretches across a face rutted and left raw by love, death, and care. When she greets me in enthusiastic Russian, her frizzled hair sputters close to her timeworn eyes. Linguistically inept, I am helpless and retreat from the swollen feet moving toward me. I call Irving, who speaks the language. Determined to communicate with me, Yelena persists, even though I can neither respond nor understand. Later, if I manage to articulate one or two Ukrainian or Russian phrases, she assumes that I am miraculously fluent and exuberantly fires away in an increasingly loud voice. I am frustrated with myself.\n\nYelena lives in the adjoining flat. Our landlady told her that we would be there. For the next few months, she repeatedly rings the doorbell (sometimes several times a day and often before we are awake in the morning). She adopts us and supplements her meager income by doing our laundry by hand. It always comes back ironed and folded\u2014even the underwear.\n\nTwo or three times a week, she also cooks for us. Yelena is Jewish and knows Irving is too, so week after week, she brings helpings of kasha, homemade gefilte fish, beetroot salad, borscht, and homemade cherry preserves. For me, she brings cookies. The food is good but I cannot help but notice that it arrives in dishes with bits of the previous meal still on them. We taste the wine made from grapes growing on vines that reach up to her third-floor balcony above the busy street and pavement below. She and her husband planted the vine years ago and waited for it to reach their balcony. Later I will learn that this practice is not uncommon\u2014here, the city is often transformed into a subsistence farm. In poorer areas of Kiev, people grow vegetables, keep chickens, and tether goats on their city apartment balconies.\n\nOne morning, dressed in her gaping slip and a loosely knitted woolen pullover Irving gave her, Yelena rings the bell, plants a big wet kiss on my cheek, and like the Duchess in _Alice_ , affectionately rests her chin on my shoulder. She wants me to see her apartment. It is twice the size of ours. In her kitchen, a pet yellow finch is out of its cage and pecking away at a loaf of bread. It flutters over and perches on a massive sausage sprawled on the kitchen table. With great pride, Yelena leads me into a large bedroom dominated by an equally large, pink satin-covered bed on which she has arranged her dolls for me to admire. Deprived of a real bed, I long to lie on it.\n\nI go into her flat only one more time, in April, for a seder during Passover, a holiday that Irving, though not religious, is eager to observe, particularly in a place where these rituals risk erasure. He is full of expectation. But, of course, we are behind the looking glass and much is the reverse of what it is supposed to be.\n\nWe sit down to dinner with Yelena, her husband, their grown-sorrowful son, and another neighbor, who is a retired biologist and also Jewish. He looks ninety but is only sixty-six years old. He, like Yelena and her husband, survived, somehow, the various persecutions accompanying the Nazi occupation of Kiev. They returned to Kiev after the war and are now among one hundred thousand Jews who are said to live in the city.\n\nIrving has come to the seder with his yarmulke on and a Haggadah tucked under his arm so he might lead the ritual. He tries to read the four questions, but no one knows anything about them or seems to be interested, perhaps with the exception of the retired biologist. During Irving's attempts, the television, which was already on, receives more attention. Yelena and her husband keep twisting their necks to catch a glimpse of the latest crisis taking place on the soap opera, and the son talks with tears in his eyes about how his cat waits for him and how he sings to his cat.\n\nSuddenly the table bursts into a Yiddish wedding song, and, after more vodka fuels that melody, the son, in a lovely tenor voice, warbles a popular tune. Then the biologist speaks about how much TV ruins culture and how hard his life is. Giving up, Irving closes the Haggadah and puts it well aside to make room for the photograph album Yelena places on the table. I look at black-and-white snapshots of her with her first two husbands, the first of whom was killed when a plane crashed on their apartment building. I see a younger Yelena, unembarrassedly spilling out of her bikini, and I see her first husband hugging their crippled daughter. In the midst of all this, the doorbell rings; a neighbor from upstairs anxiously inquires if her missing parrot has flown in through Yelena's window. The dinner collapses.\n\nLike the neighbor's parrot, the rituals and memories of Judaism have taken wing, leaving behind a few tattered feathers of remembrance that congregate in obscure murky corners of the city. With the possible exception of two splendid Moorish-style synagogues, to be renovated after we leave (one has metamorphosed into an ammunition dump, the other into a puppet theater), leftovers of a time when Judaism flourished lie damaged and sequestered waiting to be resuscitated by a gust of fresh wind. A specially organized three-hour bus tour of Jewish Kiev promises to give a glimpse of what remains. With a large group of elderly Jews who have emerged from their underground spaces, I board a crowded bus. Jammed into a seat for what seems an eternity, I see nothing except the huddled shoulders of the people seated in front of me and the thickly coated standing bodies packed into the bus's narrow aisles. My view is blocked, and I am left staring at the crudely woven fabric of the seat cover before me. A static Ukrainian commentary, sputtering through a dysfunctional microphone, completes the obstruction.\n\nDays later, a quest to find the site of Babi Yar (Old Wives Ravine) on the outskirts of Kiev is equally elusive. In 1941, thousands of Jews, who had been rounded up and sent on a forced march from the city center, were lined up, shot, and pitched over into this ravine. None of our acquaintances is sure where it is, not even those who are supposed to know, such as Nellie, a Jewish professor of philology, who attempts to assist us. When we seem close to where she thinks the memorial is, the torn, ragged earth of a construction site for a new metro subway stop foils our futile search. Remembrances of a Jewish past are being driven below levels of visibility and consciousness.\n\nYiddish, which once reverberated through the streets of nineteenth-century Kiev, has also gone underground, muted and almost forgotten. Rummaging among the few remaining voices, Irving and I go in search of the only Jewish center\u2014located, we are told, down a quiet side street not far from where we live. We approach and find neither sign nor nameplate. Hoping we have come to the right place, we walk through a foreboding archway leading downward to a gloomy passage, then out to a dreary courtyard in which a parched, forlorn fountain waits for life-giving water to articulate its thoughts. Shut doors bar our way; we crank one open, go down steps, meet another padded steel entrance, and buzz to be admitted. After descending yet another shadowed stairwell, we emerge into a newly painted basement in which low ceilings almost prevent us from standing up. I am disappointed not to see the White Rabbit rushing through. Instead, I catch sight of a dwindling collection of sedentary elderly people, wrapped in their coats and thick stockings to keep warm. It is a Yiddish group run by a man who is hard of hearing and does not seem to catch anything anyone says. Among those few present, Yiddish, Ukrainian, and Russian, as well as the local hybrid of Ukrainian and Russian idioms battle for attention and stumble over each other. A stout woman, weighed down by both her galoshes and a heavily knitted beret that deflates the bouffant curls of her hair, interrupts the cacophonous confusion and approaches an upright piano. She begins playing (beautifully) a Chopin nocturne. For the moment the piece's melancholic, sweet harmony dissipates the dissonance of the gathering. A few bars in, however, the deaf leader abruptly gets to his feet and loudly proclaims, \"That's not Jewish; play something Jewish!\" (\"Off with her head!\") The next day, the man calls and apologizes for the ragged meeting.\n\nThere are to be many Mad Hatter tea parties. The seder and the Yiddish group preface our own. At the end of May, Irving organizes a party in our flat. The usual arguments ensue about how many and whom to invite. Irving likes to have many people and mix various strata of his life, while I prefer to have fewer. The problem begins the day before when we go through an underground passage where a bedraggled young street musician plays passages from Vivaldi and Bach on an accordion. Irving stands star struck before him. Over my stern but impotent disapproval, he hands the musician our address and asks him to come and play at our party. When he shows up at the door, my Ukrainian friends have a fit, for they fear he is a bandit and will rob us all.\n\nThe relaxed pleasure of the gathering disappears. My friends anxiously retreat to the kitchen and wonder what to do next\u2014it is all a riddle to them. Our neighbors sit nervously in the living room at one end of the table so they can consume most of the food before he takes it all. Unlike the Mad Hatter's tea party, nobody either changes places or moves down a seat. All are frozen. After a while, Irving gives the accordionist a shot of brandy; soon the Vivaldi is \"looped\" or unseasonable, certainly not as mellifluous as it had seemed to be in the underground passage. Upset, people leave earlier than they should have.\n\nThe following day Tamara, Natalia, and Yelena call to see if we are alive. We are, but not without some discomfort, for after they departed, leaving their worried looks hanging on the hallstand, the young man drank more and more brandy, asked to spend the night, slumped into the cushions of our bed-sofa, and, like the dormouse, went off into a doze. Time did not move for him, nor did it for those of us left to endure his unsettling presence. I wished, like Alice, that I could pinch him and he would go. Ellie, an American friend from years ago, was still there. Speaking fluent Ukrainian she came to the rescue and gently, but firmly, maneuvered him out of the door. Before he left, he handed Irving a stolen Rolex watch. Irving, like the Mad Hatter, did not know what to do with it. He looked at it uneasily, shaking it every now and then, and held it to his ear. I went to bed thinking, as did Alice, that it was \"the stupidest tea-party I ever was at in all my life.\"\n\n### **ROOM 73**\n\nIn the morning, the hands of time move again, and, as on most days, I continue to traverse dark passages and open locked doors. The door to room 73 (the Department of Foreign Literature) in the humanities building of Taras Shevchenko University is one that still haunts my waking. This room _is_ the department. Everything takes place here, for no one has an office. To reach it, I pass through a once-splendid, yawning hall where students, carrying their plastic folders, drift in and out of consciousness in the half-light of early risings, and where, in a shadowed corner, a woman sells lukewarm tea and sparkling ruby-jam tarts that tempt the starved eye.\n\nStretching before me, a marble staircase leads to upper floors inhabited by echoing corridors and locked entrances. After coming to room 73, I knock and enter. Inside, the veneer of a long mahogany table, left over from a more affluent time, reflects what light steals through a set of windows. Around the table, mismatched chairs rearrange themselves to suit the needs of the moment. Bookcases and piles of papers line the walls, and off to the side, Luba the secretary, relegated to a worn wooden desk, sits before a manual typewriter that, whenever she uses it, threatens to upset an electric water heater perched close by.\n\nLuba exudes a comforting, matronly presence, but, unfortunately, I cannot communicate with her. When I am alone with her and need something, I dash into the hall to find a student (most of whom know English) who can translate for me. The linguistic abilities of the younger generation are extraordinary; for them, knowing a foreign tongue is a way out, a passport to elsewhere, perhaps to a job with a multinational. In spite of the requirement that courses be taught in Ukrainian, the university is a babble of tongues, a multilingual world. My students are fluent not only in English but also in other languages, and many in the art of simultaneous translation. One of my students, in fact, is periodically absent from lectures because she is the translator for the government's commission to Holland. The dean of the humanities division is President Kushma's personal interpreter. Even the young woman who meticulously cuts my hair spends her free hours poring over what few texts she can afford that give her practice in English. (She hopes to find an American husband.)\n\nMy colleagues are scattered about room 73. One senior professor and former chair, who writes about James Joyce and studies Austrian literature, casts a skeptical, forbidding eye. Wearing close-fitting black leather pants and holding a cigarette between the taut tips of her fingers, Keera leans against an open window in order to expel the smoke and her thoughts into the Kiev sky. Upon my arrival, she turns and nods but does not speak. (Months later, she utters her first words to me when she inquires where my pants are\u2014I have been wearing slacks all winter to protect my legs from the cold, but on this day, I have decided to wear a skirt. Meekly, I cite the coming of warmer weather. Later, she engages me in a mixture of Hungarian, German, and English.)\n\nIn another part of the room, Tamara, who specializes in American literature, tutors a postgraduate. As she bends over the student's paper, her thinning henna hair falls limply over the cascade of her generosity. During this session, Tamara's commitment to her work and her enthusiasm for her subject rejuvenate her face and defy the indelible marks of woe and age that lie beneath the morning's mask of heavily applied powder and smudged eyeliner. Together, they speak Ukrainian.\n\nAt the other end of the table, Victoria and Lydia, under the watchful eye of their senior professor, obediently tutor students. Next to them, one of the many Olgas I am to know underlines sentences in a book of literary criticism sent to her by an acquaintance in America, a corn farmer from Iowa who cares nothing for things literary but remembers her fondly. Later, over coffee at a nearby caf\u00e9, I step into the mire of cultural and linguistic differences and struggle to reach some firmer ground where I might better understand a colleague's thoughts about the meaning of feminism. I am not sure I succeed.\n\nEventually my attention turns to a drab, middle-aged figure in brown. Propped by a bulging, worn briefcase, Natalia stands determinedly erect beneath a prominently displayed framed photograph of her father, who founded the department fifty years ago. (The briefcase was once his.) Beaming through the shadow cast by the paternal portrait, she advances to greet me. From that moment on, she affectionately yet authoritatively directs my life (as she does others'). Every day, she phones to see how I am. In time, I grow dependent on her attention and her will. Only once do I foolishly attempt to resist her iron resolve, when through the Fulbright's generosity, I purchase, from the States, $3,000 of recent critical studies to be put in the bookcases of room 73. The dearth of such books in the various university libraries is conspicuous. As a result, whenever it is financially feasible, my colleagues travel, via grants, to literary conferences in Moscow to acquire what is unavailable in Kiev.\n\nI purchase these books so that the foreign literature students and professors will be able to take one or two from the shelves whenever curiosity or need dictates. However, that is not to be, for as soon as the books arrive and are set in place, Natalia, ready and waiting, with a key held tightly between her thumb and index finger, resolutely locks the bookcases' glass doors. No one, she instructs, may open these without her permission. The books, as well as my hopes, are firmly impounded.\n\nFor days, a battle of wills ensues. Ineffectively, I argue that students and colleagues should be free to look at them, even, heaven forbid, check them out. Frustrated, I try to devise some makeshift system for their circulation. Like Medusa ready to turn me to stone, Natalia glares back, shakes the coiled plaits circling her head, and refuses. I persist and persist, but to no avail. I lose. Since no shield of Perseus materializes to deflect her gaze, the best I can do is convince the secretary to let me sit down at her machine and type a list of the contents so the inhabitants of room 73 might have some idea of what is impounded behind the glass. While I obstinately (perhaps annoyingly) tap the keys, Natalia glares immovable on a chair next to me. That bibliography, I am sure, now lies forgotten and unread in the dark recesses of the secretary's bottom drawer. Is it the fear of theft; is it the wish to maintain a hierarchy that privileges the professor over the student; is it a sense of professional competition that wants to take possession of ideas so that only they have access to them; or is it the value of the books themselves? I never know, but I think it the last. I do know, however, that my colleagues, like dogs with a bone, keep to themselves their books purchased abroad. Once more, cultural difference muddies my understanding, and I fail to reach solid footing.\n\nA paradox of immobility and innovation, Natalia, however, through her sheer willpower, opens many more doors than she closes. Wanting to make use of my \"expertise\" and reform the department's academic program, she requires the students to alter their regular programs and spend time with me. She bends the rigid structures of the traditional curriculum (in the interest of nation building, Russian literature must be read in Ukrainian) so as to erect a more malleable scaffolding to support the courses she wishes me to teach. As a result, while I am there, foreign literature students from each year are required to take multiple eight-week units on African American women writers, minority literature, surveys of modern American literature, and special sessions on individual writers, such as Thoreau. Throughout, I offer supplemental materials through videos, photographs, and tapes I have brought with me. Because the library holdings in American literature and culture are so weak (the American Institute library has more books on business practices than on literature and culture) students have a chance to see materials that are otherwise unavailable, except perhaps through special television programs.\n\nUsed to obediently listening and taking notes, most of these students do not question what I say. Even when encouraged, they prefer, as they have been taught, to absorb information and not, in public at least, ask critical or challenging questions. That is the way. Yet that habit in class does not mean that they are not thinking. I come to realize this when Natalia makes it possible for thirty students from other departments to take a seminar on Ralph Ellison's _Invisible Man_. Having read every word, they come voluntarily every week to discuss two chapters at a time. A faculty member of the Linguistic University occasionally participates. Because we are moving slowly through this complex text, spaces for spoken thought and searching questions open up. I sometimes wonder if my students at times also feel invisible.\n\nBut even Natalia, bolstered by her heritage and her briefcase, cannot always rise above the vindictive and obstreperous world that surrounds her. Occasionally she collapses, and the tightly bound Ukrainian plats that circle her forehead loosen ever so slightly. Once after making a major scheduling error, Natalia fleetingly displays a fragile vulnerability. Painfully embarrassed, she lists what else has gone wrong in the last twenty-four hours: she has smashed her glasses and must now read with an old pair enhanced by a magnifying glass; moreover, she has spilled red-current jam on her blouse and spent hours scrubbing it off. Natalia, as do all of my colleagues, has few clothes and has learned in childhood how to clean wools without the benefit of dry cleaning. Dry-cleaning establishments are costly and are labeled \"American.\"\n\nThe people of room 73 earn the equivalent of a meager $40 per month\u2014that is, if they get paid. There are dry periods, which they must frugally and stoically endure as well as supplement with other work. Olga, Victoria, Lydia, Yelena, Natalia, and Tamara teach courses at other universities, earn extra cash by working at foreign embassies or by translating for corporations headquartered in Kiev\u2014one translates American cartoons and comic strips for Ukrainian media, and another edits a magazine for Ukrainian Airlines. Most of these women live alone and support their grown struggling daughters who, suffering from health problems after the Chernobyl disaster and because of economic hardship, have never left home. The men have absconded and started other lives with unnamed women, never to be heard from again.\n\nUnder Natalia's watchful reign, room 73 is a matriarchy of dedication and determination, except occasionally when an elderly professorial male enters its domain. Then the power of the matriarchy either withers into obsequiousness (such as the time a gentlemanly Hungarian wanders in or when Irving gives a paper and Natalia literally applauds each thought) or deteriorates into vitriolic whisperings (such as the day the university rector, attempting to unseat Natalia, appoints a male favorite to the department faculty). Equilibrium returns only when the culprit blunders back through the door and that door firmly closes to keep at bay the ineradicable dye of patriarchal authority staining the corridors.\n\nCorruption also stalks these halls. In the university system, it is not uncommon for students, particularly an increasing number who are \"private\"\u2014paying their own fees, not admitted through competitive exams, and hence not supported by the government\u2014to pay a bribe for a degree. But not so in room 73 where my colleagues tutor, reexamine, and spend many extra hours with all their charges until their students legitimately pass the exit examination and papers.\n\nClassroom space is limited. As a result, my seminar-style classes must meet in room 73. Frustrating my attempts to fathom the pure waters and depths of Walden Pond, a surfeit of humanity in the room sullies and disturbs the quiet, lyrical passages from Thoreau's _Walden_ that the students read on worn Xeroxed sheets. Around me, the room's usual activities persist unapologetically. Coming and going, colleagues boil water for cups of tea, hang up coats, take off boots, drape wet clothes on the radiator, prepare notes, and gossip. The telephone's repetitive ring, the click of the typewriter keys, the _ch-ch-ch_ overtones of whispering Ukrainian voices, and the opening and closing of the door, as well as the roar of endless traffic rising from the cobbled street below, drown my words and, from time to time, plunge me into despair, in spite of the students' receptiveness to the text. After-wards, I escape and purchase an espresso at a nearby hotel, which also has the benefit of a clean toilet. The bathroom close to room 73 is an old-style hole, smudged and reeking.\n\nThe same struggle to keep my words afloat is true when I move to the British Language Center Library next door. In the midst of a class on _Invisible Man_ , students walk through and chatter as if I too were imperceptible. Sometimes I try to read there but inevitably I am once more interrupted\u2014this time by an overly eager colleague from another department, whom in all honesty I would rather avoid. She dashes in to see me. I am horrified to notice that her face is splotched a pale green and her hair a bright orange. She fell while rushing from the university to another appointment and has just gotten out of hospital. Like a cartoon figure, she now sees stars in her eyes.\n\nYelena, the kind librarian of the British Language Center, tries her best to help, but she too lives in a world of small spaces and noisy clutter. One afternoon I phone her at home but cannot hear a thing she says because of her barking dog. Yelena later explains that she has not only a dog but also a very large puppy whose eager face can rest on a tabletop. Moreover, she owns a cat and a three-month-old kitten. Pandemonium breaks out when the puppy accidentally sits down on the kitten and there is a rush to rescue it. Naively, I ask, \"How many rooms are there in the apartment in which you, your husband, and all these animals live?\"\n\n\"One,\" she replies.\n\n### **THE UNIVERSITY LIBRARY**\n\nSeeking respite from this disorder and in quest of a quiet, spacious setting, I periodically cross the street to the university library, a grand nineteenth-century yellow building that promises silence and space. Oak catalog cabinets stuffed with fading index cards either tilt exhausted in muted corridors or congregate aimlessly in silent side rooms. A curtained space in the gaping, drafty lobby conceals a single Xerox machine (difficult to find elsewhere, particularly functioning ones that do not muddle the paper or print blank pages). A young woman works the machine. I become friendly with her. Wanting an escape from her dead-end life, she conscientiously studies English and practices on me. One day in tears, she tells me that her mother (probably trying to do her best) is pressuring her to break up with her boyfriend and sign up for a marriage bureau that links girls with American men (hopefully rich ones). These men come over for a few days, pick up a bride, and take her home\u2014to who knows what.\n\nLeaning toward her, I urge her to continue to resist her mother's wishes and offer to put her in touch with an organization that might give support. I wonder where she is now. Later, alone with a couple in a funicular on its long way down from Mount Ai-Petri (4,500 feet) in Crimea, I listen to a conversation between a plaid-shirted, potbellied, middle-aged Midwesterner and a thin, blond, desperate Ukrainian woman who replies in broken English. An agency has obviously just linked them up, for there is no real intimacy. Invisible, for they think I am Ukrainian and know no English, I eavesdrop as the man speaks of their future life in suburban America. His words reverberate through the descending compartment and sound out with a controlling timbre that will manage her life if she stays with him\u2014a power probably no longer available to him at home. Her only response is \"I vant a little doggie\"\u2014a request he promptly stomps on and kills by saying \"No, I wouldn't allow that!\" She pleads again. Unhappily, he mutters that he would have to think about it and then quickly shoots back, \"When we have children, you won't need a dog.\" (It is obvious to me that this woman is past childbearing age.)\n\nI feel sick and fantasize that I should interrupt and tell her to get out before it is too late. But dangling in midair and feeling vulnerable myself, I fail. She will find her way to America, but what then?\n\nUpstairs in the university library, I open the door to a large reading area and discover a moderately well-lit place where the daylight peers through a row of tightly shut windows and where flickering fluorescent bulbs articulate their imminent demise with rhythmic popping sounds. Conscientious students wearing coats, gloves, and scarves bend wordlessly over their work at rows of desks. The feeble warmth from the radiators is useless in this vacuous space. As if darkly impersonating a pompous Mr. Bumble in charge of the workhouse in _Oliver Twist_ , a librarian oversees these students from an elevated platform at the head of the room. Afraid to ask for anything, I quietly choose a place and start my work. An imposing portrait of Kushma (the president of Ukraine), replete with medals, lends weight to the librarian's rule. All seems to go well until the librarian rises and with guttural severity gives a strict command, which, of course, I do not understand. Everyone obediently leaves the room; puzzled, I follow their lead.\n\nLike a lost soul, I wait, exiled and wondering, in the dark hall. Eventually, the doors reopen, and with the other expelled readers I start to file back in. The room, however, is far colder than before. The fragile heat, which dared an hour or so ago to try to do battle with a biting cold that intermittingly charged the room's shut windows, is now defeated. Gusts of icy air advance through gaping panes of glass, which have been opened wide during our absence. What has happened? Believing it necessary periodically to rid the room of germs as well as of foul particles, the librarian, following official policy, clears the room twice a day, opens all the windows for a long fifteen minutes (no matter what the temperature outside), and shuts them again before readmitting us. Once a pale, strained, and puzzled face follows me out and asks what on earth is going on. She is an American postgraduate student studying Ukrainian literature. I explain, and she answers that she has had enough culture shock for one day and is going back to her apartment. I never see her again.\n\nIn time, I grow used to this procedure and occasionally even welcome it when yesterday's onions join a lingering troop of body odors and march around the room in search of a billet. I feel comfortable here\u2014that is, except for the time I choose a desk next to a window looking over a concrete courtyard behind the university hospital. Indulging a morbid state of mind, I distractedly watch nurses and white-jacketed attendants come and go between the backdoor of the hospital and a hulking garbage container to the left of the window. They carry bulging white plastic bags (with human parts\u2014the casualties of battle? I wonder).\n\nThis impulse to clear out a room so as to cleanse the air is not only for the benefit of asthmatics (I am told) but also a vestige of the Chernobyl accident. Since April 1986, all public buildings have been annually decontaminated. Nearly fifteen years later, Chernobyl haunts people's minds and bodies (everyone has his or her own narrative), especially when symptoms of some unspecified chronic illness periodically compromise their health or when news of the now-dead village (people also mourn the loss of a culture specifically associated with that village) and pictures of the contaminated area's once-cultivated fields or wild woods periodically make the headlines or fill the television screen.\n\nOne spring afternoon, I, too, am haunted. While I conduct a class in room 73, my attention drifts to the open windows. Time passes. The sky thickens as a yellowish-green fog lowers its heavy head, crawls over tree limbs, obliterates the budding leaves, smothers the street, and swirls uninvited into the room. People dart from their chairs and slam the windows shut. Class comes to an end. Natalia tells me to go home, but says nothing more.\n\nWhen I leave the building, people on the streets, like \"the pestilence-stricken multitudes\" in Shelley's \"Ode to the West Wind,\" seem driven like \"ghosts from an enchanter fleeing, \/ Yellow and black, and pale, and hectic red.\" Something, I worry, has happened at Chernobyl, but the wind this time, rather than blowing north, is heading straight for Kiev. On my way back to 36 Bohdan Khmelnitski, I remember a recurring dream I used to have in the 1960s, in which I was trying to outrun the radioactive dust from a nuclear bomb; now, it has left the darkened eyes of my sleep and become part of my waking. _Is this really happening? Has Chernobyl caught fire again?_ I try to let the thoughts pass.\n\nOur neighbor, Yelena, rushes into the apartment and says that \"because of the strong winds,\" we must shut our windows. This directive puzzles me, for a fortnight earlier she accused me of not opening the windows as much as I should. Believing there is another disaster, she, in her kindly, bumbling way, is attempting to protect us from contamination.\n\nSeveral hours later the fog dissipates; Natalia phones to explain that Kiev has been in a state of panic. Furies from the past have been gusting across the city and its damaged lives. The thickly menacing air, however, was not the result of an explosion at the reactor but the consequence of strong and angry fires blazing in the bogs surrounding Chernobyl. To quell fears, cabinet ministers repeatedly make announcements on the radio and appear on television programs featuring vivid pictures of the conflagration. Moreover, officials at the American embassy issue moderately reassuring statements declaring that after testing, the level of radiation in Kiev has not risen significantly.\n\nWhen I am later in Belarus, a country mapped in swatches of red to mark areas affected by radiation, people nervously ask me about the incident. As always, traumatic memory clings tenaciously to consciousness. In Kiev, there is a museum devoted to the Chernobyl disaster. Natalia takes me there, though she says that she does not need a museum to remind her of that time. In addition to the gruesome, horrific exhibition of fetuses deformed by radiation and preserved in bottles for all to see, two videos in that museum stay with me. One, taken by the KGB, is of the town within the thirty-kilometer circumference surrounding the reactor. The footage is from the very day of the explosion, a few hours after, but before people are told to evacuate (of course, too late). One sees inhabitants going quite normally about their business for the last time in their lives. Another video is of volunteer \"cleanup\" crews, who, with inadequate protection, shortly after the explosion went into the reactor for one and a half minutes each in order to reach what machines could not. Death is their companion.\n\n### **KIEV THROUGH THE LOOKING GLASS**\n\nI am embarrassed to admit that, before I came, I was not properly acquainted with alternative spellings of place names. Months earlier in London, I received the official letter informing me that I would be teaching at Taras Schevchenko University in \"Kyiv.\" Fearing I might have been assigned to some isolated, polluted, ugly industrial city, I wandered the corridors of the Department of Slavic Studies at the University of London and roused a professor, who responding to my knock on his office door, gently assured me that \"Kyiv\" is the Ukrainian-Russian spelling for \"Kiev.\" Relief: I was to spend the next year in one of the most splendid cities in Eastern Europe.\n\nTime continues to pass, and doors to other unfamiliar places continue to open. Kiev is an active cultural city. A few yards from my flat is the magnificent nineteenth-century opera house with its expansive entrance (one imagines the grand carriages that once stood outside), and within easy reach are other theaters and symphony halls. When one goes through these buildings' looking-glass doors, the routines of a hurried, crowded, sometimes-cacophonous life reverse. For the moment, the anxieties inhabiting room 73 or 36 Bohdan Khmelnitski succumb to the fantasies of another time and transport me to another place. Often at the last minute, after surviving the crush of the cloak room (one brings a plastic bag in which to put one's boots, hat, gloves, and scarves) and paying as little as forty cents, at most five dollars, per ticket, I seek consolation from the day in the melodrama of opera (there are frequently three different operas performed per week), in the fluidity of the ballet (dancers move like water over ice), and in the intricate harmonies of orchestral music.\n\nThe reversals in this looking-glass world occasionally do more, however, than offer a pleasant and stimulating reprieve from the business of the working world. Sometimes they quite literally, absurdly, and disturbingly turn upside down and inside out what one is in the habit of knowing and assuming. As if anticipating the terrible reversals of 2013 and 2014, as well as the destabilization of Ukraine's contested borders, realities and boundaries shift to undermine or threaten one's orientation. One is sometimes at war with oneself. The familiar categories of being scatter and leave one unsettled. Alice, as she does at the conclusion of _Through the Looking-Glass_ , sweeps away the tablecloth to create a jumble of things and leaves all in confusion. Order and classification no longer hold.\n\nOne week I attend a serious and moving production of _La Traviata_. The seat, a hard chair in a theater box suspended precariously over stage right, is less than perfect. This location gives me not only sore knees but also an unusual view of the performance. In the midst of a tragic scene, when the beautiful, unjustly wronged heroine is dying, the entire trombone and trumpet sections decide to have a deliriously funny time in the orchestra pit. The trumpet player, while pulling ridiculous faces, tweets the mustache of his colleague, and, in chorus, members of the trombone section stand mischievously and start running in place, pretending they are going to be late for their grand orchestral entrance in the final scene. While the conductor seems blithely to ignore these high jinks, the pudgy oboist, waiting for his entrance, shuffles around the pit in his slippers; in the wings, a tenor picks his nose. Such are the reversals of the sublime. Humpty Dumpty falls off the wall.\n\nThe opera house raises its curtains to other absurdities that topple expectations and tilt perspective so that what seems up is really down. While absorbed in a serious moment in _Rigoletto_ , I become aware of people giggling inappropriately around me. Soon the entire audience collapses in laughter. What is happening? The leading tenor has become ill and has been replaced by an understudy, who knows the libretto _only_ in Ukrainian and _not_ in Italian. Like battering rams, his Ukrainian vocalizations pummel the fluid syllables of the Italian lines and annihilate the tragic, lyrical timbre of the remaining moments.\n\nAnother reversal occurs when I attend what I think is going to be a splendid performance of Bach's _St. Matthew's Passion_ (said to be the first in Kiev for eighty years). Anticipating a splendid performance of a more-than-splendid work, I buy a ticket from a scalper (the event is \"sold out\") and rush into the opera house. I arrive in the dark just as the curtain is going up. As the evening progresses, I sense that something is amiss: the oratorio is staged. The orchestra is not up to par; the soloists are uneven; and, worse, the chorus, dancers, and Roman soldiers are dressed as if they have stepped out of a horrid Sunday-school painting.\n\nThe performance really deteriorates when it comes time for the betrayal and the crucifixion. Judas sports dreadlocks; red and white lights flash; painted clouds move across a rumbling, darkened sky; and a soprano, who nearly forgets her entrance, hurries onto the stage. Moving to the front, an oversized luminous cross precedes dancers propelled by angel wings. Hauling a bible larger than a tabletop, members of the chorus follow. In the grand finale, church members mingle onstage. Thankfully, the house lights go up. I hear snippets of English around me and realize, to my embarrassment, that I am in the midst of evangelical missionaries who have dumped Bach into the wastes of the Bible Belt. Caught in a trap sprung by my nonexistent linguistic skills, I am ensnared by my inability to read the program. (At least Alice could read the labels on the bottles that took her in and out of doors.) It is a relief to exit and get back onto the street.\n\nLater, I learn that the missionary population in Kiev is significantly large. It has money, influence, and, unfortunately, the backing of at least two foreign embassies. At this event, I am disappointed to catch sight of the wife of the US ambassador as well as the British ambassador himself.\n\nThere are to be even more twists and reversals, but not all of them are either absurd or discouraging. Instead, they are curiously wonderful. One evening, I attend a performance presented by the visiting Bolshoi Ballet. Enchanted I follow the lithe, elegant movements of the then-seventy-five-year-old Maya Plisetskaya. Once famous for her revolutionary approach to classical ballet, she continues to dance with the troupe. Her motions flow from the spring of her enduringly muscular body to defy her age. Behind this theater's looking-glass door, age becomes youth.\n\nOn another occasion, behind yet another door, youth metamorphoses into age. One early evening, I walk several miles in order to be present at the finals of the Horowitz Piano Competition for young pianists. The symphony hall is packed. After a performance given by the third- and second-place winners, a child from a godforsaken, depressed part of eastern Ukraine, who has won the competition and is just a few days from her eighth birthday, approaches the piano. Veronica Rogachova's feet barely meet the pedals. Behind her, the Kiev Philharmonic Orchestra sits and waits to accompany her. Dressed in her freshly ironed, girlish frock, a satin bow crowning her hair, she appears young and vulnerable. But when the moment comes and she confidently turns to the orchestra, nods her directive to begin, and renders the opening passages of Bach's F Minor Piano Concerto, all is utterly, utterly changed. The memory still sends shivers down my spine. In that instant, her childish features vanish. Discarding the chubbiness of childhood, her face deepens to display the shadows and authority that can accompany adulthood and that should not yet be hers; some deeper soul rises through the soft, yet-to-be-sculpted tissue of youth to furrow her brow and tighten the skin. She takes command of the music, the piano, and the orchestra (she is conducting from where she sits). Her phrasing and her subtlety are exquisite and moving\u2014beyond the emotional boundaries of her youth. Everyone sits silent and transfixed.\n\nIreneus, the brother of our landlord and one of the judges, says she is a \"genius\" and that there is a tug-of-war over her: Moscow is trying to get her to come and study there, and the music academy in Kiev is making offers to keep her in Ukraine. Still stunned on my way back to 36 Bohdan Khmelnitski, I unexpectedly find myself walking behind her. Close to her mother's side and holding an immense teddy bear given as a prize, she skips nonchalantly along the pavement. Another metamorphosis: she has reverted to being a child again. It is too much for me.\n\nIn other theater spaces in this looking-glass world, gender reverses its supposedly stable text to create puzzling and curious fictions in which masculinity and femininity are as fluid as the gracefully choreographed movements onstage. At the Ukrainian National Theatre, men become women when an all-male ballet company, visiting from St. Petersburgh, performs _Swan Lake_. Wearing tutus and up on pointe, the company's male dancers transform themselves into a delicate corps of cygnets\u2014into a chorus of frail femininity\u2014and perform the principal female roles. Once in a while, a fleeting glimpse or a shadow of a masculine feature, beneath the makeup and whisping feathers, threatens to interfere and sacrifice the troupe's translation to the language of \"camp.\" Perplexed, the audience is not sure whether to titter or not. Respectful and enchanted by the metamorphosis, however, it refuses. Later, when one holds the extraordinary evening up to a mirror, it is difficult to read the performance's text. The more one tries to make sense, the more one becomes entangled in, perhaps attracted to, the puzzle.\n\n_'Twas brillig, and the slithy toves_\n\n_Did gyre and gimble in the wabe:_\n\n_All mimsy were the borogoves,_\n\n_And the mome raths outgrabe._\n\n\u2014\"Jabberwocky\"\n\nA similar enigma follows me when we spend a weekend in Riga (Latvia). I am excited to enter the recently renovated and gilded opera house to attend a classical recital performed by a popular and well-known alto singer. Having walked all day through the old city and filled up on pastries bought at the Lido, Irving and I quickly change into polite clothes and make our way past posters announcing the concert. A bit puzzled, we look at the publicity photograph and see that the soloist has none of the dramatic flair of a prima donna. She wears a turtleneck sweater that struggles to ascend her three double chins. _Well_ , I think, _she must be an awfully good singer if she can get away with looking like this._\n\nWhen she comes out, wearing chunky white heels and dressed in a loose white silk pantsuit that contrasts with her long black hair, I am not reassured. And when she sings arias by Handel, Rossini, Verdi, and Tchaikovsky, I am not terribly sure she is a good singer. Yet the audience is enthusiastic. Her high range seems harsh and strained; her low range, unnaturally full and mellow. From time to time, I also notice that she drops octaves. Nothing fits. I start to look at my watch and hope that time will fly faster than her trills.\n\nIntermission arrives. We go downstairs for refreshments. At a table Irving converses, in Russian, with a lady. \"Curious and curiouser,\" we want to ask about the singer. By this time, we are wondering whether the soloist is having a bad night or maybe has a bad cold\u2014at one point she cracked and had to leave the stage, coughing.\n\nThe woman listens politely to Irving's questions about the singer but annoyingly and frequently corrects his Russian. Every time Irving utters the word for \"singer\" and uses the feminine ending, the lady across the table promptly alters the word to its masculine form. Irving is tempted to follow the example of Humpty Dumpty and tell the lady, \"When _I_ use a word, it means just what I choose it to mean. Who is master here?\"\n\nThen, of course, the truth dawns. The lady is not remarking on Irving's command of the Russian language but is correcting our misunderstanding. The woman onstage is a man. We have been looking and listening backward. The singer is Erik Salim-Meruet from Russian-Central Asia (Kirghizstan), who was once a choreographer but was discovered ten years ago singing the leading female role in _Madame Butterfly_. Ever since then, he has been a sensation and given concerts all over Central Europe. My friends in Kiev have heard of him.\n\nThis general fascination with and acceptance of shifting, uncertain gender boundaries (at least onstage) is what also draws a popular Kiev audience to Boris Moiseev, a gay Russian pop singer, born in jail to a Jewish dissident. In Kiev, after hearing his two most popular songs, \"Charlie Parker\" and \"The Ukrainian Tango\" in street caf\u00e9s, I am curious to see him perform at the Ukrainian Palace.\n\nOn the night of Moiseev's concert, crowds stream toward the entrance of the modern theater. I am there, along with three thousand well-heeled, middle-class Ukrainians, who have also brought their children and paid the equivalent of $20 each to watch his two-and-a-half-hour spectacle of bisexuality. Following a crushing crowd, I squeeze past a demonstration outside the theater. I believe the protestors are ultranationalists who object to his being Russian, never mind his homosexuality. I watch the armed militia arrest a few and clumsily shove them into a paddy wagon.\n\nOnce past a thick column of security guards as well as a set of metal detectors, I find my seat and wait for the show to start. When it does, Moiseev, earringed and sequined, enters with his \"boys.\" To the pounding and sometimes sugared pulse of his music, he dances (he is trained as a ballet dancer), sings, and ingenuously manipulates his costumed body so that for a period he is female (the pant leg becomes a skirt) and then, with a twist, a male (the skirt reverts to a pant leg). Defying classification, his gender seesaws, rotates, and blends. His body and expression are simultaneously as lyrical as the feathers of a swan and as bullying as a swung thick metal pole. His cropped, slick, bleached hair pulls back to expose a powdered, heavily made-up face. Under the glaring, hot lights, sweat trickles through his features to remind one just how vulnerable life is behind the looking glass.\n\n### **UNDERGROUND KIEV**\n\nGetting there and back is an adventure. As yet, I am not comfortably conversant with the metro system. Alien place names composed in an unfamiliar alphabet, as well as the system's web of linking underground passages, intimidate me. Armed with a well-fingered map, I practice traveling to the venue the day before the concert. Like Alice, I don't wish to travel the wrong way. Moreover, I certainly don't want to be told off by a guard who might complain to all around me, \"She ought to know her way to the ticket-office, even if she doesn't know her alphabet.\"\n\nEach metro stop displays a different theme. Many are elegant. Fine works of art, patterned mosaic tiles, and glistening chandeliers grace these platforms. In contrast, the half-lit, damp gray underground walkways leading to these stylish spaces incongruously reverberate with the buzzing bustle of banality and distraction. The place hums as determinedly and as vigorously as if one had mistakenly disturbed a hornet's nest. Underneath Kiev, makeshift, tattered cells of a secular, commercial life flourish. Bartholomew's Fair returns to life through the crowd of tables, stalls, and booths set up by people eking out a living. Musicians, beggars, flower sellers, caf\u00e9s, record shops, toothpaste booths, cigarette stalls (it is possible to purchase one cigarette at a time), butchers selling slabs of lard and ox tongues, mongers marketing stale fish, magazine stands, phone stalls, tables displaying home-made cheeses, baked rolls, sweets, and beer venues crowd the channels. In one particular passage, the young people of Kiev hang out and smoke. In another, a small orchestra plays a tango. Near the entrances, farmwomen cradle kittens and puppies in their arms. Swaddled in innocence and not yet sulking like Alice's kitten or irreverently grinning or disappearing like the Cheshire cat, these balls of fur, like skeins of rare virgin wool, are tempting. \"Come buy, come buy!\" Fearing I shall succumb, I push past.\n\nThe most thrilling moments belong to the metro's escalators. Steeper, longer, and faster than any I have been on before or ever will be again, they take people deep down, down under the city to this hidden stratum of life. One steps onto these moving stairs and wonders if one will ever be released from the descent.\n\nI notice the various postures of those riding up and down. Young people sit; some stand backwards; some have their faces buried in a newspaper or a book; others either pass me in the midst of a quarrel or rise embracing each other, as if they have just resolved their differences; some stare into nothingness; others, gesticulating, chase a fleeing word. A crippled man somehow vigorously maneuvers his wheelchair onto the escalator (there are no lifts) and released from immobility for the moment, glides down effortlessly. A young couple clasps a loose pet rat in their arms. People smile approvingly and utter the \"aws\" of endearment.\n\nBelow Kiev, there are religious as well as secular spaces. Centuries ago, monks lived as hermits underneath the Lavra monastery. Today the devout make pilgrimages to these caves. With a slender, rapidly burning candle, one descends twisting and darkening stone steps, round and round, into a pitch-black passage bordered by coffins in which are wrapped the relics of particular hermits. Small apertures offer obscurely lit views of cells and chapels. Inside, shelves holding bottled skulls, coated in myrrh, catch the sparse light, and tempt the eye. People press by, bend reverently over the coffins, and piously kiss the relics.\n\nBack aboveground and once more in the glow of day, contemporary monks, in flowing black robes, hasten toward the refectory. It is lunchtime. Scurrying to catch up to them, a dwarf in full habit dashes across the courtyard and enters the hall just before the door slams shut. Blown by the gusts of haste, his robes stream after him. Leaving the courtyard and climbing up a bricked hill back to the noise of the streets, one goes by begging women, gypsies, and stray dogs. At the top, chestnut trees are just bursting into leaf.\n\n### **IN THE LIGHT OF DAY**\n\nLike Alice, I often long to get out of so many dark halls or corridors and wander in the light of day. Morning windows are full of promises. Early, before rising in winter, I listen to the sweeping sounds of a woman who spreads sand over the icy sidewalk. She creates a Japanese garden out of her semicircular motions. Then I hear the clanking of the metal poles as people erect their market stalls up and down the street. When I draw back the long, heavy curtains, my attention turns, as it does every morning, to a black-and-white dog leaning up against the window of the third-floor apartment across the street. Adopting the posture of a prying neighbor who regularly appraises the world from her window, the dog carefully considers each pigeon flying by. The sun, when it appears, warms as well as brightens the room with exquisite tints of light. In the late afternoons and evenings, there are other views. Through the window of a second-floor caf\u00e9 facing the opera house, I watch a line of tall narrow cedars sway in the breeze and dip in rhythm with the ballet dancers, practicing inside the rehearsal room. Together, their limbs choreograph a sort of pas de deux. Later that night, before drawing the apartment curtains shut, I gaze at the dazzling red lights of the Budapest Club, kitty-corner from us and cling to illusions of reassuring warmth.\n\nIn the mornings, I step onto the street and join a mass of humanity treading the pavements and streaming up and down steps. When the gray showers gather, my elbows ache from holding an umbrella and my back aches from sitting in unheated rooms. I walk by lines of older women who, exposed to all sorts of weather, stand from nine in the morning until seven at night selling plucked chickens, eggs, bottles of horseradish, worm-eaten apples, lavender sachets, and black woolen leggings arranged on upside-down wooden crates. In season, they offer walnuts and, for two blissful weeks, baskets of freshly picked strawberries. (I gorge.) Attempting to generate some warmth during the biting-cold months, they shuffle and sway in chorus from one foot to the other. I wonder if they, like Oska's grandmother in _The Tin Drum_ , have bricks, heated on wood-charcoal fires, wrapped in newspaper under their skirts to radiate some warmth. After each sale, their swollen hands count the coins. As if transported from Dickens's London, a homeless child glances furtively at their wares. A drunk, dragged by his impatient Airedale terrier, stumbles past.\n\nFarther on, parked on a sidewalk as broad as a road, are tint-glassed limousines and polished Alfa-Romeos that wait indefinitely with their menacing chauffeurs outside expensive restaurants and Mafia bars secured by guards armed with cellular phones and gas guns. Not approving of Irving's less-than-tailored attire, the Dejavu and the Miami twice refuse him admittance.\n\nPedestrians, beware: the nouveau riche thinks nothing of driving on the sidewalk. Nearby, groups of roaming dogs, having developed a road sense, wait until the green crossing light shines. I pass the occasional person rummaging through rubbish. No refusal here, for just outside the covered Rynak, in the city center, sits a ragged woman gnawing a raw, discarded beef bone.\n\nFor all its rough poverty and coarse contrasts, Kiev's beauty is never spent. Corruption, hard lives, crime, and uncertainties do not destroy the visible elegance of another time, which flashes like shook foil to illuminate the city's grandeur. Buildings and structures as well as the lives within them might fall, gall themselves, but their majesty still radiates. Joining the thousands who walk its broad sidewalks, I move with them. At times it is a pleasure to fall into the rhythm of their brisk steps. Past shining white edifices, I find my way up and down hills that look out over the tree-lined Dnieper River, winding through Kiev and navigating the inquisitive eye through a cityscape mapped by monasteries, squares, monuments, offices, universities, museums, galleries, concert halls, boulevards, embassies, palaces, businesses, hotels, parks, apartment buildings, and botanical gardens. Even in the dusk, when one reaches areas trodden and smudged by industry or colored by the gray weariness of areas made desolate by depressing high-rise Stalinist architecture, Kiev shines.\n\nUnder the guidance of Natalia (the head of foreign languages), who thoughtfully devotes her Saturdays to escorting us to various parts of the city, I visit art galleries and museums, where older women, wrapped in coats, scarves, and worn fur hats, loll against the exhibits and serve as guards. (Their body sweat and bad breath make one hesitate to draw close enough to ask questions.) At Taras Shevchenko University, she takes me to an extraordinary cabinet of curiosities that reminds me of the illustration introducing Ferrante Imperato's 1599 _Dell' Historia Naturale_. It displays stuffed mammals, curious shells, small marine specimens, almost lifelike bird skins, covered jars with brains, diseased organs, or malformed fetuses, arranged horns, tusks, minerals, and boxes of unidentified specimens. Hanging from the ceiling are large fish and even a stuffed crocodile. ( _How appropriate,_ I think, _that I, who have found myself in a place that is \"curiouser and curiouser,\" should be taken here._ ) My life is better for these excursions, and so is my understanding of a city that can be simultaneously beautiful and damaged, pastoral and strident, lasting and vulnerable.\n\nTogether we linger before grand homes, such as Gorodetsky House, designed by a nineteenth-century businessman who was passionate about hunting in Africa. Sculpted elephants, tigers, zebras, exotic birds, mermaids, fish, and insects protrude from the corners, doors, windows, and steps of the house and, like gargoyles of empire and authority, gaze down upon the unwitting passersby. We pause while Natalia removes an envelope of paling black-and-white snapshots from her bag so that I might see her (forty or more years ago) as a child, dressed in her buttoned winter coat, standing primly with her parents before the very spot on which we now stand.\n\nShe has spent hours going through albums so she can share these with me. I am moved. But why has she done this? Do these images of time past represent a quest for survival or constancy in a life ruled by revolutions, death, and oppressive policies that continuously overturn and try to make nonsense of what one has known or wishes to be? Or are they some futile gesture intended to reclaim a time of innocence and a feeling of being protected? Perhaps these snapshot are also expressions of identity and belonging: these places, defined by the presence of Natalia's revered father, pictured in the snapshot, seem to comprise the scaffolding of her being.\n\nSometimes Natalia and I meet at 36 Bohdan Khmelnitski, walk up the hill, past the Golden Gate, turn left toward St. Sophia and St. Michael's (in the process of renovation, its gold dome sparkles in the afternoon sun), cast a skeptical glance at the deceptive orderliness emanating from the authoritatively columned exteriors of imposing parliament buildings, and turn to look down over the river, visible to us through spikes of light that dart through openings in the woods. We board a funicular that takes us down among dappled treetops to lower Kiev, where we wander through the peaceful, cloistered grounds of a monastery, stroll by a hat factory, its sewing machines muted for the weekend, and stand outside Kyiv-Mohyla, another large university in the city, where police guards and a busload of militia wait, uncannily silent and at attention, for another attack on the Communist Party headquarters across the road. The day before, radical left-wing demonstrators poured gasoline throughout the building.\n\nWalking back to the funicular, we pause before black ribbons fluttering from Ukrainian flags to commemorate the death (on May 4,1999) of fifty coal miners in the Zasyadko mines, near Donetsk, in the eastern part of the country. These mines are notoriously dangerous, for not only are they unusually deep and produce high levels of methane gas but also they are owned by corrupt politicians who meddle with the hazard-measuring equipment in order to claim that the mine is meeting safety standards. In years to come, three hundred more will fall victim to the black death of the coal mine's blast. In 2015, rescue attempts from yet another disaster at the mine will be hindered by crossfire between Ukrainian government forces and pro-Russian rebels.\n\nGathering our thoughts, Natalia and I enter a compartment. Inch by inch, the funicular rises above the streets below, brushes by round balls of mistletoe attached to the slope's upper branches, and returns us to the bustle of the upper city.\n\nOn another Saturday, we walk through the golden dust of fallen leaves, down a hill, and onto a footbridge that crosses the Dnieper River and leads to a sandy, scruffy island where one can stroll and purchase a drink or a snack. Feeling relaxed in the warming afternoon light, Natalia and I sit at one of the three tables in a caf\u00e9, drink tea, and chat\u2014that is, until a burly, muscular man with slick black hair and the remnants of a broken nose enters with his portable TV, plugs it in, and, after increasing the volume, higher and higher, turns his attention to the boxing match on the screen. Intimidated, annoyed, and momentarily caught in yet another of the brutal contradictions of Kiev, we leave.\n\nWalking through Kiev is a mysterious act, for there is always the unseen and unexpected. Behind the sightless walls of the university's administration building a pleasant park hides in a ravine. Around its uncharted corners, the sweet odors of lilac trees wait for the unsuspecting flaneur. Primroses and violets grow between the tall chestnuts, limes, and birch trees. From a distance, magnolia blossoms glisten like a pond's reflecting surface. On their way to and from work, people find courage in their perfume. In modest restaurants tucked into alleys, caged finches sing while customers sip cups of hot tea; video screens display models parading on fashion-house runways. On back streets, caf\u00e9s buried in basements vibrate to the sounds of a pulsing techno beat that quickens the blood and, like a shot of espresso, keeps the body in motion. After our time together, Irving and I take Natalia to lunch in these hidden spots. In one the owner hands Irving a cigar on a plate, complete with cigar clipper, and brags in a loud whisper, \"Contraband.\"\n\nSometimes I prefer to go silently, without Natalia's direction. One quiet, sunny November morning, I walk by a woman who watches contentedly over her demure hen and indecorously plumed rooster while they scratch around under the fir trees planted outside the opera house. She has carried them out of her crowded flat so they can feel and take nourishment from the earth. Nearby a crow hangs on a grapevine crawling up and clutching the side of a building. The bird delicately but greedily picks with his beak the last two grapes of the season.\n\nFarther on, I rest in one of the many Ukrainian churches so as to be among the colorful religious images painted directly onto every centimeter of speechless walls and columns. Devout elderly people kneel before the altar. Their hands, clutched shut, clasp their thoughts. And then I continue down the hill until I reach Kreshatyk, so I might join yet others who have come to seek release from their crowded living quarters and the niggling cares of life.\n\nOn weekends, this main, expansive boulevard, stretching for several miles, is closed to traffic. The whole of the city comes out stylishly dressed to promenade and talk with friends, no matter what the weather. As if responding to the deep, quietly running currents of the Dnieper River, people gather in groups and move in gentle waves along the boulevard. Observed at a distance from a second-floor caf\u00e9 window, their moving figures resemble hieroglyphic letters spread across an unfurled scroll\u2014a text, or perhaps an illusion, of community. How hard that community is to find now. If only one could superimpose this image over the horror and discord that is to follow in this very spot in 2014.\n\nLoudspeakers lining the streets blast out romantic songs and, for the moment, smooth any disharmony, small or large, that might normally disrupt an afternoon. While promenading down Kreshatyk, those around me give the illusion of stepping away from the confusing linguistic world that can (and will) be divisive and that has the potential to destroy or compromise lives in the years to come. In Kiev, most people speak and read Russian, but public announcements, posters, signs, and university lectures and examinations are, as the government requires, in Ukrainian. The newspapers, however, are in Russian (except for the English-language _Kyiv Post_ ). Most in Kiev speak a mixture of Ukrainian and Russian or speak one language at home and another at work. In the western part of Ukraine, the Russian language is not welcomed. While I am here, ultranationalists murder a musician in Lviv for singing in Russian. In Crimea, however, Russian is preferred. When Russia reclaims Crimea, it bans Ukrainian, as well as the Tartar language from the schools. Supposedly, a fourteen-year-old boy, discovered speaking the Tartar language on a phone, is beaten on March 31, 2014.\n\nIn Odessa I meet an elderly woman, a Ukrainian nationalist, who has vowed never to speak a word of Russian\u2014she forbids it in her house. Tied to the disaster of \"nation building,\" the ammunition of language creates borders and structures its checkpoints. In February 2014, Ukrainian parliament is to ban Russian as the second official language. The vote is later overturned.\n\nOn Kreshatyk, I join the pace of these weekend strollers and enjoy what I understand all too well to be the illusion of becoming part of their world. When one does not speak the language, there is something reassuring, even transformative, about falling into the physical rhythm of the people who surround one. A Strauss waltz clatters imperfectly through the static of the loudspeakers and drowns out the bewildering variety of words and tongues that confuse my daily life. The melody's familiar rhythms silence the bits of modern Greek, Spanish, French, Hungarian, and German darting through the humanities building's corridors. Moreover, the waltz's predictable measure regularizes the syncopated mixture of Ukrainian, Russian, German, Latin, Yiddish, and Byelorussian that I heard the previous evening at an event held at the Actor's Guild (formerly a Persian-style Karaite synagogue), and that generally baffles my understanding.\n\n### **MAIDAN 1999**\n\nThe section of Kreshatyk that leads to the Maidan, or Independence Square, is the heart of the city. At night, too, this area is often closed to traffic so that people can come together and attend outdoor concerts or rallies. One late evening, after attending an opera, a friend from England and I walk down to Kreshatyk to join the crush of thousands who have congregated in the square to listen to a rock concert. Images of the band blast out of a huge television screen. Celia turns to me and says, \"I wonder what it was like being here ten years ago.\"\n\nAs if on cue, bright blue and yellow beams pulsate and send shafts of Ukrainian light across the gathered multitude. Archival footage of celebrations in Independence Square, on the very day (1991) Ukraine separated from Russia, replaces the televised images of the performing musicians. How ironic that almost fourteen years later, in 2014, over eight hundred thousand people are to rally in the same spot to protest the loss of that promise and register their frustration over closer and closer ties with Russia.\n\nIn the Maidan in 1999, Celia and I watch on-screen carefully edited sequences of people fighting for and celebrating their independence. Fireworks burst over our heads. Flaming remnants fall dangerously among us. Dodging rockets, fires, and burning candles, Celia and I, as well as thousands of others, anxiously find our way back along Kreshatyk. Little do they, or we, know that we are benignly enacting what is to be a deadly retreat fourteen years later, when in February, 2014, Independence Square will turn into a battlefield and be the scene of horrific clashes between riot police and thousands of protestors distraught over a repressive, corrupt government that suddenly has turned its back on economic ties with the West.\n\nIn 1999, however, Celia and I are merely worried that the falling fireworks might ignite our coats or our hair, so we rush through streets where fifteen years later the police, armed with guns, and snipers will randomly shoot at protestors, who, barely protected by makeshift shields, will run up a nearby hill for safety. A hotel, close to where we pass, will become a makeshift hospital. The wounded will be rushed in, but the medical supplies will be inadequate and many will die from direct shots to their heart, neck, and lungs. It will be the worst day of violence for nearly seventy years. At least eighty-eight will be killed in the space of forty-eight hours.\n\nFor the time being, though, Kiev holds its breath. Civil war is a distant, rumbling prospect. For the present, at least, congregating in public is a unifying act. Walking especially is a communal experience, particularly on memorial days. Whatever divisiveness might exist seems to surrender to a shared memory and to give way to the collective rhythms of bodies, moving, moving along streets and boulevards.\n\n### **WALKING**\n\nIn March on Women's Day, Kiev becomes a moving garden. Everything shuts down. Through the streets of Kiev, people carry and receive flowers; men (supposedly) cook or take a woman out to dinner, and families meet to stroll and talk together. Main streets, now emptied of moving vehicles, accommodate multitudes of men, women, and children, going to and fro, carrying long-stemmed roses, tulips, or small bunches of lily of the valley and violets.\n\nIn a similar mode, later, on May 9, the entire city comes out to honor \"heroes\" and to pay tribute to those who served in the armed forces. The day begins with a parade of World War II survivors, accompanied by their offspring. Medals pinned to their jackets clang and swing in rhythm with their gait. After-ward, the public walks from monument to monument in order to show their respect and lay flowers at each. Blossoms float in fountains and lie, heaped and fragrant among their stems, on marble steps. Booths selling food, drinks, and small toys edge the street. Millions are walking, walking, walking down and up the hills, by the river, through the parks. After eleven hours on my feet, I reach home.\n\nThe following day, people return to Kreshatyk to see an exhibit of mounted, oversize, archival photographs displaying the destruction that the boulevard suffered in the 1940s. Interspersed among these are images of Babi Yar (for the day, consciousness of this event has moved aboveground), concentration camps, and \"Ukrainian\" military leaders. Memorials have a way of momentarily equalizing loss.\n\n### **THE SEASONS**\n\nWalking is not just for fine, memorialized days; it also part of Kiev's mundane life, in which people diurnally tread and go about their business through the shifting seasons and changing winds. Between February 1, 2000, when I arrive, and my departure, in the middle of December 2000, not only does life fluctuate and reverse itself, but so does the weather, which twists and turns like a petulant child. On foot most of the time, I am acutely sensitive to these forces and variations. The winter I have come to is unusually mild, the warmest since 1895. That, of course, does not mean it is warm. Often, on my way to the university, I feel a damp wind blowing, slicing whatever is in its way. Negotiating slippery hills and skidding on icy streets, I shiver. Irving buys a politically incorrect muskrat hat, and I wear a long, cashmere coat.\n\nIn March, spring drizzles start to replace the icy rain. Buds appear. I actually go out without my hat\u2014a good thing because I am tired of it. By April, it's suddenly warm weather. The central committee or central something-or-other has not yet turned off the heat. In search of relief, we fling the windows wide open and let the flies, yellow jackets, and bees peruse our apartment. The radiators steam away; we wilt. I know, however, that the weather will change and cold wind will yet again blow through Kiev. Outside, crews of scarved women and capped men turn over the soil, flattened by frost. They sow grass seed, tidy the edges of flower beds, and transform the disorder of the winter's freezing into the warmth of order and promise\u2014if only their powers to correct the corrupted earth would extend beyond their purview so as to cultivate a more fruitful governing body.\n\nIn May, lilies of the valley decorate offices, metro stations, and tabletops. Their perfume is refreshing. The tall, old chestnut trees for which Kiev is famous still hang on to their candle blossoms. Only in the last few days do their petals float through the air and settle, crushed on the pavement. Now the maples soar in their greenness. Toward the end of the month, engineers shut off the water for several days so they can clean the pipes.\n\nIn the suns of early autumn, the leaves fall through the smoky light of evening. But in the saturating, dreary rains that follow, this promise of a golden autumn vanishes. Out come the space heater and the hot-water bottles. I even light the oven. Not since childhood have I had to live in an unheated place. Heat will not come on until the end of October. The dampness is debilitating. By November, all the leaves have fallen and now the landscape in the city begins to resemble the way it was when I arrived, last February. Darkness casts broader and heavier shadows over Kiev, so that 5:00 p.m. feels like midnight. In the watery light of day, the trees, brutally pruned of their branches, stand naked like sculpted, limbless torsos, but, as if compensating for their loss, allow uninterrupted views of the city and its architecture.\n\nThroughout the seasons, Natalia and other friends, proud of their national culture, make it possible for me periodically to see other parts of Ukraine. In October we travel, with a hired driver, sixty miles northwest of Kiev to the small town of Pereyaslov (population: thirty thousand), which holds an extraordinary collection of twenty-five museums dedicated to Ukrainian culture. Neither our driver nor we have the slightest idea of how to find or even enter these museums. In Ukraine, there are no pamphlets to guide one.\n\nAs instructed, the driver motors slowly through the town's streets to see if he can spot a certain Michael Ivan Sikorski, who might help. Consulting a faded, crinkled photograph of Sikorski, the driver distractedly steers the car, one eye on the road, the other on the imperfect image of Sikorski. Miraculously, before long, the driver spots the guide cum curator, aged, worn, and weighed down by iron keys, ambling slowly down a side street. He agrees to take us around. The problem is that he speaks only Ukrainian (no Russian is spoken in this town). Irving does his best. As far as we can tell, no other visitors are here. People, curious, stare at us.\n\nOne museum is the house of Skovorada, an eighteenth-century philosopher (perhaps the Ukrainian equivalent of a Thoreau). In this damp, cold, and barely secured dwelling are ten thousand rare books that belonged to Skovorada and his philosophical school. Many lie in disorderly heaps on desks and tables. Most sit compromised and exposed to the elements, as well as to the indiscriminate fingers of the curious. We gasp as the curator carelessly pulls fifteenth-century texts from open shelves, badly in need of repair. After randomly thumbing through ancient atlases, rare Latin texts, and hand-colored illustrations in eighteenth-century natural-history volumes (one partially nibbled and shredded by a mouse that has made a nest of its exotic ancestors), I anxiously inquire if these books are catalogued. The answer is no; there is no funding. While various groups in Kiev rage over where they should go, these forgotten treasures are being left to rot, like forgotten prisoners, behind a locked door.\n\nWe move on to the Museum of Transportation, which houses all shapes and sizes of sledges, once necessary but now obsolete. And then we approach the house of Shalom Alechem, a Yiddish writer of the early twentieth century. When he was alive, 60 percent of Pereyaslov's population was Jewish. Now there are a mere three hundred Jews remaining. As the day continues and we see more of these decaying abandoned places, I feel more and more enveloped by a landscape of loss and deteriorating memory. It is almost a relief to meet our driver so we can go back to the distractions and cosmopolitanism of Kiev.\n\nOn the return journey, we pass through the flat agricultural land that once made Ukraine the breadbasket of the world. Occasionally, fields of sunflowers relieve the sight of dried and gnarled leaves about to fall from the trees bordering the darkening road. Like shaken granules of black pepper, swarms of evening crows spread across the sky. Dropped off at 36 Bohdan Khmelnitski, from the backseat of the car, I gather the apples I have taken from a room in yet another museum, the house of Taras Shevchenko (the nineteenth-century poet, serf, painter, and revolutionary after whom the university is named), so that I can spread them out in flat number 8. Like the ones assembled in tumbling throngs on the tables and shelves in Tschvchenko's dining room, they emit a fragrance that will perhaps help prolong a memory of an enlightened, thoughtful past.\n\nThere are other brief sojourns. One is to Kamianets-Podilskyi, a historic town in western Ukraine. On a cold, dead early-winter morning, a driver arrives. His well-worn car runs alternately and forever off gasoline and propane\u2014a good thing, too, for the reassuring presence of a lighted petrol station is not to be part of our lonely seven-hour journey. Beyond the outskirts of Kiev, we seem to be the only ones who have ventured onto the roads across a landscape still and coated in ice. Only occasionally does a local truck emerge from the frost and come toward us like a phantom of the nameless in search of identity. Frozen crystals coat every branch, leaf, and blade and, from time to time, catch the sun to weave the coarse countryside into an ethereal fabric. The sheer beauty of it helps ease the fear that the car might skid and bring everything to an end. After interminable hours, the driver pulls over and points to a snack bar, an apparition out of nowhere. With the exception of the owner, it is empty. Desperate not only for hot tea and a sandwich but also for a toilet, I ask where the latter might be. The man behind the bar points to the caf\u00e9's entrance and says, \"To the right.\"\n\nI find an outhouse that once had a door. It borders and faces the road. Shivering, I sit, staring into the iced land and slick highway, relieving myself, on view to any vehicle or person who might venture by. I do not care.\n\nIn a few hours, we arrive in the night's darkness and go up to our hotel room. But privacy is to be elusive. No sooner have I swung my overnight case onto the bed than I hear a knock at the door. I open it to find a television crew that wants to come in and interview us. They do. Cameras, interpreter, and interviewer fill up the room. They ask why we are here et cetera. I do my best. The area is starting to promote tourism.\n\nWe are here because Kamianets-Podilskyi is a place that has witnessed one bloody power struggle after another. Ever since the thirteenth century, Mongol Tartars, Poles, Russians, Austro-Hungarians, Armenians, Soviets, Germans, and even Ottomans have claimed it. It was briefly the de facto capital of Ukraine. The city has been partitioned, conquered, divided, oppressed, captured, fortressed, and seized. It has seen horrendous massacres, the most recent of which was the mass murder of 23,600 Hungarian and Ukrainian Jews in the space of just two days, August 27, 28, 1941. Starting in the eleventh century, different peoples and cultures have brought their own culture and architecture to the area. Each has left evidence of its secular and religious life. In the frozen haze, ghosts of old monasteries, churches, cathedrals, schools, town halls, synagogues, Turkish minarets, market squares, and houses congregate like lost and damaged souls among the rubble and cobbled streets of the Old Town. Two hundred buildings, some intact and some in ruins, from the eleventh to the nineteenth centuries, populate the area. The Smotrych River swirls in a deep canyon to create an island that isolates these remains from the central part of the city (a fortressed bridge connects the two). Its rushing waters and dramatic rocks segregate the remembrance of what was once a vibrant, multiethnic, theological-philosophical, though by no means always harmonious, community from its monocultural, industrial, unaesthetic present. The silent, secluded, unpopulated island is a graveyard.\n\nOn the last day, while I walk the streets of the Old Town, my attention wanders to a door, just ajar enough that I can see inside. A group of very young, obedient boys, paintbrushes in hand, bend conscientiously over slabs of wood and pots of paint. An elderly monk instructs them. The boys are learning, from scratch, to design and paint icons. One embodied phantom of the past, at least, haunts the ruins.\n\n## _CHAPTER 6:_\n\n## IN RUINS LAID\n\n### **ODESSA AND CRIMEA**\n\n#### **_Odessa_**\n\nIn April, when spring warms the earth, Natalia arranges for me to go down to the State University of Odessa to offer a series of classes on contemporary American multiculturalism\u2014a generous gesture on the part of the university's Department of Foreign Literature. Because there are no available texts, I bring Xeroxed copies of the literary selections I shall be discussing. The students take notes quietly and attentively. They wait until after each class to ask me questions.\n\nWhile in Odessa, I am also asked to talk about recent trends in American literature to graduate students and professors from Ukraine and, interestingly, also from Russia. (In Odessa, the Russian connection is stronger than it is in Kiev.) Panicked and not sure what to do, I prepare a paper on the electronic poetry I have seen demonstrated at Brown University, as well as in workshops at the University of Buffalo. Though I have no means of showing examples (computers are just not part of this space), I try verbally to illustrate what these works would look like on a computer screen and speculate as well on how these poems not only upset the traditionally static linear form of poetry but also dismiss the single figure of the poet. Rather than standing still on the page, these digitized poems respond to computer programs that allow interactive readers randomly and continuously to rearrange the poems' words, syllables, and lines. On the screen, the text becomes tentative and fluid. To complicate matters, most of the audience is not conversant in English, so I must speak via an interpreter, an experience that I find disorienting, for it slices thought into slim phrases and unnaturally introduces caesura into the running rhythms of consciousness.\n\nLike many occasions, the event makes me anxious, for it places me too much in the position of being an \"expert.\" Throughout my time in Ukraine, one problem for me is that people assume I know anything and everything about American literature. And, of course, I don't. If I am unsure about a particular fact or work, I feel as if I am letting down not only them but also myself. And then I am frustrated because the library holdings are weak; it is more often than not impossible to fill in blanks or dig more deeply into critical responses to a writer. This circumstance increases my admiration for most of my colleagues, who have never been to America or taken many courses in its literature but know more than I. Their commitment to learning, to absorb as much as they can, and incessantly read all they can get their hands on, is incredible.\n\nIn truth, by sending me to Odessa, Natalia is not just putting me to work but also granting me (and Irving) a holiday. She has also allowed me eight days off from teaching to be a \"tourist.\" Even though we are pleased to be going and are packed, the thought of uprooting ourselves from Kiev, making sure that we have the right documents and enough cash feels troublesome. I also dread the fifteen-hour, overnight train journey. I have grown dependent on what has become familiar to me in Kiev, as well as on Natalia's supervision.\n\nAt the train station, I am reminded yet again of my colleagues' selfless generosity. Three have come to see Irving and me off. Like anxious parents, they huddle together on the platform and press their faces up to the windows to make sure we are seated in the correct compartment and that we have brought food and water\u2014no refreshments are available on the train. As they shut the compartment door, they reassure us that people from Odessa will be at the station to meet us. It is awful to admit that at the same time I am moved, as well as reassured, by their attention, part of me looks forward to be going to a place where I am, for a while, to be left to myself.\n\nOnce, Odessa was a splendid city with Eisenstein's Potemkin Steps going down to a harbor, vibrant with bustling naval and military commerce, not to mention the illicit trade of the Mafia. In 2000, it is a city in crisis where buildings are, with a few exceptions, left to crumble; where weeds suffocate once meticulously manicured parks; where projects started ten years ago are abandoned; and where the palaces of the rich stand deserted and lonely. Throughout the city, cracks and smashed windows compromise grand architectural structures. When I walk through the center, it is hard to imagine that fourteen years later, the place will be at war with itself and that on May 2, 2014, forty-two pro-Russian activists will die, trapped in the burning Trade Union Building close to where I stand.\n\nNo tourists come to Odessa in 1999 or early 2000. About ten years ago, there were many visitors from not only Eastern but also Western Europe. We ask why. One answer is that people stopped coming after the fall of the Soviet Union. Pollution is not as regulated as it was during Soviet rule, so now the effluents from the factories and spillage from the sewers get trapped in the circular currents of the Black Sea and cause a \"big stink.\" Some people, at least those with means, have other choices, so they go where they believe the waters are cleaner. Those who cannot move are subjected to the contaminating uncertainties of the present and condemned to a place that is disintegrating around them. No wonder some of the older generation long for another time, whether good or bad, when certainty and predictability seemed to attend Soviet rule.\n\nFor all this, however, in the 1990s Odessa is still a center of culture, though a fading and struggling one. Phantoms of Babel (his house is there) and a young Horowitz (the music academy and concert hall remain) waft in and out of consciousness. At a literary museum, a dogged curator surreptitiously introduces controversial exhibits by hanging items behind doors or up high so that when officials come they don't notice these censored documents. Two art galleries and a dusty archaeological museum continue. And so does the Jewish Library, where, behind a basement door, I discover a small, lighted classroom packed full of young Orthodox Jewish boys, each with a chessboard in front of him. Concentrating over their game, they sit in silent, attentive rows. It is an image to remember.\n\nAfter we leave the train, my first task is to get in touch with the dean of the university in order to know when and where I am to lecture. I am to call her at the prearranged hour of 3:00 p.m. Arriving several hours before then, we decide to eat lunch in the Londonskaya Hotel, the one remaining fashionable hotel in Odessa, situated at the head of the Potemkin Steps. The hotel is a nineteenth-century gothic wonder.\n\nWe enter an ornate dining room and find ourselves utterly alone. During the meal, we suffer through obsequious service. If I move my spoon, the waiter immediately runs to the table and straightens it. We also suffer from eating an oily salad. Later, I joke that it must have been made of surplus ship oil. The result is that in the hotel lobby, from which Irving makes the preliminary call to the dean (I need him to speak Russian to the secretary), he is overcome with an urgency for the toilet, but he cannot get off the phone because this is my only chance to find out what I am supposed to do. While waiting for the dean to get on the line, he squirms and turns to me, saying, \"I can't make it. Ann, you'll have to manage\u2014and where's the toilet?\" Irving throws the phone over to me and rushes down the hall, but as soon as the dean begins talking, the salad oil also turns on me. Thankfully, the conversation is short; then, I, too, shoot down the hall. We emerge, laugh, and recover sitting at the head of the steps, which are disappointingly less dramatic than I hoped they might be.\n\n### **ARCADIA**\n\nThe State University of Odessa has arranged for us to stay a few miles from the city in a sanatorium hotel, the Magnolia, near the sea and just a ten-minute walk from the university\u2014for which I am grateful since one of my lectures is eight in the morning. At that lecture, I discover a swastika chalked on the blackboard. It is erased at my request.\n\nIn spite of this sobering reminder of indelible prejudice, it feels good to be away from the dark, crowded passages of Kiev and out in the open light of the ocean and its skies. From our sparse room, it is possible to look out to sea and cast the eye toward a grassy path that follows the coast and takes the walker eight miles to the center of Odessa. Sometimes, as is the practice, we hitch rides.\n\nLater, I amble along this way, past eight grazing goats and a herder, who, involved in reading his book, commands his dog to keep the animals in order. The goats pay no heed and repeatedly butt the poor animal, who must do his duty until his master lifts his eyes from the page and once more takes control. A few yards along, two middle-aged \"spinsters\" lovingly carry their two pet crows in matching wicker baskets. The ladies talk incessantly to their birds but not to each other. It is the Orthodox Easter weekend, so in the sun of midday the whole of Odessa picnics by the sea. Holidaymakers and dogs peer through gaps among the tall, breezing grasses. Sounds of people sawing branches off trees and of children picking up twigs announce the building of fires. Family and friends cluster around barbecues. Greased bodybuilders flex their muscles before critical and sometimes indifferent eyes. Along the promenades and down on the beaches, children gladly display their nascent acrobatic, karate, and boxing skills.\n\nEventually, snatches of amplified Ukrainian poetry and song drift irregularly toward shore from a platform stretching into the sea. It is time to return. In the early evening, a hedgehog modestly yet deliberately makes its quiet way through the grasses; on a bare branch above him, a species of woodpecker I have never seen before rocks back and forth in search of insects.\n\nWe are staying in an area called Arcadia, a region devoted to sanatoria designed to assuage all manner of troubles and diseases. There are clinics for the digestion, the liver, the eyes, and for alcoholism\u2014paradoxically, this clinic stands next door to a champagne distillery. Enough remains of these grand buildings for one to understand that Arcadia was once utterly blissful, a place of rest for those who were wealthy and luxuriated in their problems. These magnificent places now stand either empty or slightly modernized (but not to a Westerner's eye).\n\nSome sanatoria have been adapted to house patients supported by state funds and a few maintained to accommodate private patients. All the land around them is unkempt and overgrown. When I ramble alone, uninvited, through the grounds of these relics of another time, I watch attendants in white uniforms officiously walk from building to building through tall weeds, past dry fountains, over structurally unsound stone bridges, and then stride by damaged sculptures of discus throwers and Cossack maidens sheltered by old lime, maple, and chestnut trees. Stray dogs live in grottos next to glass conservatories overtaken and burst asunder by the plants, which have grown through and up the sides of these structures. In between stand half-finished modern buildings (abandoned for lack of funds) and all the mess that accompanies incompleteness. Along the paths metal spikes, treacherous holes, and rotten steps lead down to the sea and discarded dreams.\n\nAt the Magnolia, we have full room and board for $20 per day. All the meals are essentially the same. Breakfast consists of gruel, radishes, green onions, sardines, meatballs, kasha, dry bread, and tea with heaping spoonfuls of sugar. Lunch offers soup, dry bread, cabbages, radishes, meatballs, potatoes, and stewed plums. Dinner is composed of radishes, sardines, meatballs, kasha, a roll, and (finally) butter. Thankfully, I am spared the grubby \"healing\" glasses of water served to Ukrainian guests, who have come for their health and after meals recline in grungy-looking mud baths leftover from a previous century or loll in the healing waters piped from some nearby spring into decaying mineral baths. The stray cats and dogs benefit from the meatballs I don't always eat. Ladies wearing frilly aprons speak in diminutives and cast kind but disapproving matronly looks if I leave anything on my plate. Under the long plain wooden table, an orange kitten slides on the linoleum floor and attacks trouser legs. Occupants who obediently eat everything cast curious but polite looks in our direction. We nod back.\n\nTwo men assigned to the dining table next to us move toward us. \"May we have the teapot?\" evolves into a halting conversation in Russian with Irving. Eventually, they ask if we would like to visit a monastery. Ready for anything, though clueless about where and what order it is, we agree.\n\nAt eight that evening (rather late, we think), we climb into their small but well-equipped vehicle. After winding through main streets and alleys for half an hour, and after they stop to buy bottles of liquor, we find ourselves on the grounds of a monastery we can barely see. Abruptly leaving us in the car, the two jump out and obsequiously embrace the monks strolling in the courtyard. After returning to the car, they drive on farther, to where there is a small outer building, which is the dwelling of the most revered of the order. Outside stand long, quiet, shadowed lines of deferential men and women waiting for hours to be blessed. Whenever the door opens and momentarily extends a ray of light, each supplicant excitedly peers over the shoulder of another worshipper to try and catch a fleeting glimpse of his divine presence.\n\nIncreasingly puzzled in the backseat, we watch as our \"friends\" extract the purchased bottles of liquor from the car's boot, break through the rows of patient, devout believers, and like a two-headed Hydra, slither effortlessly and rapidly into the divine's quarters. The door slams behind them. More than curious and a bit impatient, I cautiously get out of the car and peer through a small windowpane proffering a view of the interior. It is a scene out of _The Decameron_ : the divine monk and our two acquaintances are lustily imbibing the purchased liquor, glass after glass. With their feet up on the desk, they laugh raucously and slap their thighs. When a lady attendant catches sight of me, she jumps up and immediately draws the curtains to block my view. I return to the car.\n\nAfter an interminable time, our two companions reappear; the monk follows them out. Oblivious to what has taken place inside, the followers, who have been patiently waiting for an audience, raise their voices in ecstatic religious song and bow down low before the sight of the holy monk.\n\nThis is only the beginning. Our drivers inform us that a friend of theirs (a monk) needs a ride to another monastery. We wait silently in the car. After a number of uncomfortable minutes, to our surprise, both a monk and a nun appear. How are we all to fit in? There are now six of us. I am moved to the front seat, but Irving is relegated to the back, where he sits pressed between one man and a portly monk. The nun, in full habit, places herself on Irving's lap. The ride to the other monastery is wild and reckless. The men stop for yet more liquor. Maneuvering treacherously through alleys and swerving around potholes, gullies, and trenches, the car bounces in the sinister darkness, through a long-deserted military barracks, where it almost crashes into a wrecked, discarded airplane before entering a flooded courtyard that looks more like a lake than like solid ground. Ignoring my complaints as well as the nun's Ave Marias, the driver plows through, scrapes the bottom of the car, and stops suddenly still. In this pitch-black, lonely, gaping yard of a monastery, everything unexpectedly becomes ethereal and dreamlike.\n\nI am not prepared for what happens next. Like a floating apparition, a single Ukrainian Orthodox priest\u2014bearded, tall, and in full regalia\u2014suddenly emerges from the unpeopled darkness into the form-giving brightness of the car's headlights. In measured, hushed, ritual steps, he slowly approaches us. High above his head, he holds the order's most sacred icon. The car's beams catch the sheen of its golden ornamentation. Astonished, we fall quiet and stare. While he and the icon drift silently by before rejoining and disappearing into the black of night, the nun on Irving's lap, overjoyed in the presence of this holy relic, rapturously cries out, \"Christ be with you\" time and time again. That night it is a relief to return to the Magnolia.\n\nAs if needing to return to the underground spaces that are so much a part of our life in Kiev, before leaving Odessa we hire a driver (in a Mercedes\u2014the wife comes along, too) and an English-speaking guide to take us to the catacombs, a series of submerged caves dug into the sides of intricately webbed underground passages, in a village just outside of Odessa. It was in these catacombs that the partisans in World War II hid and conducted their covert operations. Walking through two miles of subterranean walkways, constructed on two levels, we see where these courageous individuals had, in spite of mammoth hardships, slept, cooked, washed, and tapped sources of water.\n\nAfter the tour, we climb back into the Mercedes and bump along ditches to the house of the driver's mother. The driver used to be a chef in the army. He cooks lunch while his dog, knowing he is not allowed in, sulks in the doorway. Next we go to the English guide's flat in the city, where we must listen, for hours, to her daughter's boyfriend play the piano. We have a hard time getting away.\n\nIt is time to leave Odessa. I wait outside the Magnolia on a patio that overlooks a run-down swimming pool, a clinic, and a tacky sculpture depicting a young boy and girl facing each other with adoring looks. When the person from the university arrives to take us to the train, she points to this sculpture and, in her imperfect English, asks me what I think of it. I am about to proffer my opinion, when, fortunately, she interrupts me to say that it is of her daughter and son-in-law. Saved.\n\nBefore we board the train and endure a sleepless, overnight ride back to Kiev (I seem to have bronchitis), our kind faculty escort, attempting to show off her English, pops her head into the compartment and with the best of intentions bids me a hearty \"good riddance!\" I know she means to say \"good-bye\" or something like that, but, once uttered, whether correct or not, the words grate and make me wonder if perhaps there was no error at all. Such is the faulty nature or truth of language.\n\n### **CRIMEA**\n\n_\"Oh, thankless Crimean land! In ruin laid \/  \nAre now the castles that were once your pride.\"_\n\n\u2014Mickiewicz.\n\nMy department chair, Natalia, is eager for me to see as much of Ukraine as possible, so in October 2000 she insists that I take time off from my lecturing and travel to Crimea. \"You cannot be in Ukraine without visiting the area\u2014so beautiful and so restful.\" Memories of that visit to Crimea beg to be recorded. What was possible then is no longer.\n\nIn midmorning, Irving and I board a train for what is going to be an eighteen-hour ride from Kiev to Simferopol, a journey that will become even longer because of an additional four-hour bus ride to Yalta. A uniformed attendant delivers cups of hot tea with lemon and sugar. This tea goes well with the sandwiches I have packed. Waiting for the long night ahead, I look out of the train window. After a week of dreary rains, it is refreshing to see the sparkling sun light up the dewed fields ripe with marrows, pumpkins, beets, sunflowers, and red stubble. It is a season of mellow fruitfulness.\n\nAs the train plods by, the window frames a Constable painting in which a horse attached to a wooden wagon pauses in a stream. Close by domestic geese sway in chorus. A cat sleeps among the zucchini. With her back to the passing train, a woman sits resignedly watching the cows graze from right to left. (I wonder what she thinks about.) As if she belongs to Keats's ode to the season, she watches, with patient looks, hour by hour, the last oozings of autumn. A man riding his bike, loaded with mature acorn squash lashed to its rack, pedals silently through my frame of vision.\n\nAs the afternoon lengthens into evening, long shadows start their trespass through slopes of wheat fields. Their shade will soon subdue the rows of cabbages that, in the late light, resemble turtles lined up on a log to catch the healing warmth of the sun's last rays. Like a moving-picture gallery, the train window makes visible Monet's studies of haystacks that capture changes in their aspect and color between dawn and night.\n\nWhen darkness descends, I pull the blinds, put on the light, open Ralph Ellison's _Invisible Man_ , and then buy a newspaper from a man who knows a little English. I learn that he was once a physicist and a headmaster of a school but lost his job after Perestroika. Now, in his worn uniform, he travels the trains selling newspapers. His eye catches sight of my reading. He tells me he has read George Orwell's _Invisible Man_.\n\nThat night in my bunk, thinking I hear an intruder (it is really Irving opening the door so he can go to the toilet), I sit up screaming. When Russia invades Crimea in 2014, the train I am on will stop running altogether. The border will be closed tight.\n\nAs if I were a worker during that Soviet period, I have been sent for ten days to Crimea for rest and restoration. In 1920, Lenin declared the Crimea the place \"for the medical treatment of the Working People.\" But I am not to go to one of the 138 sanatoria (one for each profession: teachers, doctors, engineers, postal workers, customs officials\u2014even one for actors), where I would be taking three large meals a day, subjecting my body to a complete medical examination, and lowering my blood pressure with the assistance of massages, herbal remedies, and bathing. Neither will I be learning to breathe though the nostrils or the mouth via some medieval-looking contraption that cleans out the lungs, nor enduring the shocks from an electric apparatus in order to restore my balance. And I certainly will not be electing to have leeches applied to my ears so as to improve my hearing. Instead, I am to stay at an Intourist hotel in Yalta\u2014out of season, during a silent, dark, and empty October.\n\nWe decide against the one Natalia has booked, for there is no heat and no hot water. After hearing the continuous running of the toilet, noticing the one dangling lightbulb in our room, and eating leathery dumplings served by a pleasant woman with two surly daughters in a drafty bar, we look for another place. The search introduces us to Yalta, for it leads us from hotel to hotel, private apartment to private apartment, by vine-laden cottages bursting with ripe grapes, and along an old-fashioned seaside promenade, where the surf bursts into spray over the walls and scatters unsuspecting pedestrians who, like startled birds, flit in gatherings out of reach. The places are either too small or too expensive, too dark, up ten flights of stairs, or lacking a view of the sea.\n\nAs a last resort, we climb a hill to the Hotel Yalta. Friends in Kiev have led us to believe it will not be suitable for our taste, but it is and we move in. Our room is modern, with a balcony overlooking the cream, green, and blue bay of Yalta. The window frames a pristine view of the Crimean mountains. In the mornings, Irving watches the gray clouds move from behind these mountains and drift out to sea.\n\nThe Hotel Yalta is monstrous, like some breached, enormous cruise ship. It has 2,300 rooms. Each of the floors has a different character and management. Our floor is run by Intourist and costs $30 per night. Inside are at least five privately owned restaurants, two state-owned restaurants, a post office, a bank, a money exchange, a tourist bureau, shops, a gym, a swimming pool, a casino, a shoe-repair booth, a twenty-four-hour nurse, and slot machines. When it opened in 1973, the hotel employed more than two thousand people and accommodated the rush of tourists during the postwar years from Russia, its neighboring countries, and Germany. It also thrived on cruises originating in Greece and Turkey. (After Russia reclaims Crimea in 2014, tourism drops severely as the region becomes out of bounds for Ukrainians.)\n\nOut of season, the hotel is cavernous and eerie. On the first evening, we sit alone and dwarfed in a vast, unlit restaurant that seats over six hundred people. As if situated in the bowels of some shipwrecked, rolling ocean liner, the restaurant seems to sway in the shadows cast by candles placed on each of the several hundred tables. Adrift at breakfast on the sixteenth floor, we eat in an equally large restaurant, which looks out on all sides at a vast sea disturbed only by its shuddering, dark swells. Eight waiters serve us.\n\nIrving feels unwell and wishes to rest in bed, so I arrange for a driver in a black Mercedes to pick me up early in the morning, tour the area, and return in the autumn evening. On the first day, we travel more than sixty miles, three-quarters of the way up the Crimean limestone mountains. Following the coast, the car winds its way over forested ravines and through exposed hills and small mountain villages to Sebastopol. This is a city like no other, for it accommodates the naval headquarters of both Ukraine and Russia. (The better-equipped Russian sailors live in their own quarters and get paid more than their Ukrainian counterparts.) For the moment, separate and mixed loyalties roam its streets but do not yet indulge the terrible discord to come. (In the early 1990s, the Russian president, Boris Yeltsin, and Ukraine's Leonid Kravchuk agreed to divide the former Soviet Black Sea Fleet between Russia and the newly formed Ukrainian navy.)\n\nFive years earlier, I would have been refused admittance into this armed city on the Black Sea. Now a roadside checkpoint stands vacant and neglected. Even when it was manned, I am told, people regularly bypassed this barrier by walking through the forest. But this alternative is a short-lived reality. It will not be long before such exits and entrances are sealed and the Ukrainian navy is forced to leave. There will be talk of making Sebastopol an exclusively Russian military city to which no unauthorized people can go. Unidentified gunmen will take over the airport (February 28, 2014), the Russian flag will fly over several official buildings, and Vladimir Putin will sign a bill to absorb Crimea into the Russian Federation.\n\nWhen I reach Sebastopol, the long coil of war inevitably commands my attention. Sebastopol has seen some of the bloodiest battles and at times has been entirely destroyed. For instance, the Nazis captured the city on July 4, 1942. After World War II, the Russians felt compelled to rebuild Sebastopol stone by stone. The guide and I, feeling it unwise to linger, drive at a deliberate pace past buildings destroyed in both the Crimean War and the Second World War. We make our way along boulevards bordered by various naval industrial sites, isolated and secured behind barbed-fences, until we reach a relic from the past: a nineteenth-century panorama displaying the 1855 siege of Sebastopol. Excited actually to be entering what I have only seen pictured in Victorian publications, I eagerly get out, pay my money, and, translator in tow, dash up the steps to its entrance.\n\nFrom an elevated circular platform, I look down into a three-dimensional display of the battle, the ships, the canons, the horses, and the fallen men. I read every explanatory plaque. A nineteenth-century engraving published in an 1855 issue of the _Illustrated London News_ comes to life before my eyes. I have a hard time believing that I am at the very spot where Russia and combined British, French, and Ottoman forces vied for territory. As if stepping back in time and becoming Fanny Duberly, the wife of the paymaster for the English troops, I mount a hill to become a spectator at the theater of battle so I might watch the smoking, blazing city and listen to guns and musketry. I look down on the heap of ruins and see the transport carts covered in tarpaulins and carrying the dead. I see stacks of piled-up bodies in every stage of putrid decomposition, flung out into the street. But I do not smell the intolerable stench of death. In her diary entry for September 1855, Duberly writes:\n\n_I think the impression made upon me by the sight of that foul heap of green and black, glazed and shriveled flesh I never shall be able to throw entirely away. To think that each individual portion of that corruption was once perhaps the life and world of some loving woman's heart\u2014that human living hands had touched, and living lips had pressed with clinging and tenderest affection, forms which in a week could become, oh, so loathsome so putrescent!_\n\n\u2014 _Mrs. Duberly's War_\n\nBack in the Mercedes, the journey continues. Time and place shift. Just outside of Sebastopol, tucked into a corner of industrial and military waste, are the remains of an ancient Greek city, Chersonesus (424 BC). The ruins go right down to the sea and, like picked, bleached bones, stretch out under the glimmering sun. \"The Greeks set shining, columned marble here.\" Time has strewn these ancient columns over a heath that, along with its grasses and wildflowers, does battle with its compromised surroundings.\n\nThe guide and I follow a path of cedars, pines, and old oak trees until we are alone with a wind that whips the sky. We stand next to a Greek temple, later transformed into a Christian basilica and turn to look at the clouds racing over a field of sedentary ruins. Suddenly, into the corner of my vision steps a statuesque, young Ukrainian archaeologist, wearing glistening emerald green stiletto heels, who makes her way easily, like some goddess from an ancient pantheon, over shards of remains. As she draws near, her elegant shoes spike the exposed mosaics. The earth that once witnessed long-oared fleets approach its shores accepts her presence. In the far distance, when the sun shines, bathers, like sirens, lie listlessly on adjoining rocks. In the fifth century, as first among a series of invaders and conquerors, the Greeks established colonies along this coast. There is a significant Greek population still in Crimea, even though in June, 1944, the Soviets deported the Greek community, along with Armenians and Bulgarians, as part of an ethnic-cleansing campaign, intent on destroying vestiges of Greek culture. In the 1970s, there are still groups speaking Medieval Greek.\n\nLater, these phantoms of ancient Greece draw me to Balaklava, a long, narrow inlet that for centuries has offered ships protection from the ocean's heaving waves. On one side, a slender street follows this passage to and from the sea. Rows of poplar trees shake like a death rattle in the strong wind and preface a sequence of yellowing, crumbling turn-of-the-century buildings. Above are the remnants of an Italian castle and the foundations of a fourteenth-century church. On the other side are the few remaining husks of nineteenth-century fashionable dachas. Below them, in the harbor, drift the rusted ruins of the Ukrainian navy: like a long-dead walrus, one blackened submarine washes up against the harbor wall. On either side rise steep mountains that stare down upon the harbor. For the moment I share Odysseus's eyes:\n\n_We entered a fine harbor there, all walled around by a great unbroken sweep of sky-scraping cliff and two steep headlands, fronting each other, close around the mouth so the passage in is cramped. Here the rest of my rolling squadron steered, Right into the gaping cove and moored tightly, prow by prow. Never a swell there, big or small; a milk-white calm spreads all around the place._\n\n\u2014Book X, _The Odyssey._\n\nIt is easy to imagine Odysseus suddenly looking up and seeing \"hundreds, not like men, like Giants,\" hurling stones upon his ships from the mountains above.\n\n_Down from the cliffs they flung great rocks a man  \ncould hardly hoist  \nand a ghastly shattering din rose up from all the ships\u2014  \nmen in their death-cries, hulls smashed to splinters\u2014  \nThey speared the crews like fish  \nand whisked them home to make their grisly meal.  \n\"Put your back in the oars . . . now row or die!\"  \nIn terror of death they ripped the swells._\n\nDeath is no stranger to Balaklava, for it was here that the British packed in their war fleets and brought in many of their wounded during the Crimean War.\n\nRoger Fenton's photographs, taken and developed while he traveled in his specially equipped van, not only reveal how dangerously crowded the harbor was but also show images of the nearby exposed hills and valleys looming above this narrow harbor. Spent cannonballs, naval officers, and tent encampments once spread over these plains.\n\nBalaklava catches the imagination, so, when Irving is better and can join me, we return. This time the wind has died down and the sun sees no cloud. We stop for tea at a caf\u00e9 under one of the poplar trees. What some might call the village idiot latches on to us and won't let go. He offers to show Irving the way to a public toilet, as there is none in the caf\u00e9. The two leave; Irving returns with stories of the world's filthiest toilet. The cholera that spread through the troops during the Crimean War seems not that far away.\n\nEventually, a woman at a nearby table rescues us by showing us where we might hire a man to take us in his rowboat to view the harbor. Alone on the water, with our man, we row first to the harbor's entrance, to where the narrow inlet offers passage from the open sea, then to a point from which one can gaze, astonished, at the wild waves dashing against rugged rocks and \"sky-scraping cliffs.\" The boatman then furtively offers to take us into caves burrowed deep into the harbor's surrounding cliffs. Once at their entrance, we duck to avoid becoming entangled in loose strands of barbed wire that once secured these spots from prying eyes. Gliding carefully under sharp, ripping barbs and past skull-and-crossbones signs warning of electric shock and danger, we enter a version of the mouth of hell. Guided by a handheld flashlight, the rowboat moves into a deadly large, dripping, molding interior where the Russians, until 1991, hid their nuclear- submarine fleet from the West's gaze. All is silent except for the occasional splash of the oars. We are in a labyrinth of unfathomable channels under the very ground on which the British troops camped during the mid-nineteenth century. My attention turns to the empty, rusted walkways at the sides of these deep, vaulted caverns; I wonder what feet once trod there and what sounds reverberated through these hidden passages of war. It is almost a relief when the boatman calls a halt to our explorations and pulls his oars back into the sun.\n\nOn the way back to Yalta, the car passes acres of vineyards where women stand by the road, selling sixteen varieties of grapes as well as dried fish, which like last week's washing, hang on lines strung from tree to tree. In twos and threes, people walk into the forests to collect mushrooms; workers holding buckets spilling with grapes (payment for their labor on collective farms) wait for local buses. Evening light replaces the brightness of the day's sun. We take a route thousands of feet above the sea so we can look down upon Gorbachev's dacha, a large, heavily secured white estate, where the Soviet leader went in times of trouble and where his phone line was cut during one of the periods preceding his downfall.\n\nApproaching the Baidarsky Pass, the Mercedes climbs higher and higher, out to the edge of a narrow rock protruding from a cliff that plunges steeply into the surf below. Here perches the Foros Church, four hundred meters above the village of the same name. A Moscow tea trader built this Orthodox church to celebrate Alexander's III's survival in an 1888 train disaster. As if replicating the event, the church's traditional Byzantine domes rise defiantly above the structure's dizzying resting place. This structure, too, is a survivor. It too has outlived the violent wreck of history. In 1921, when Crimea became part of the Soviet Socialist Republic and then subject to Stalin's repressions, officials brutally shut down Foros Church. They exiled its priests to Siberia (to death), and destroyed its frescoes by uncaringly splashing white paint over the paintings' brilliant designs and images. Shattered and looking as if it had toppled off its mountain ledge, what remained was eventually transformed into a snack bar for tourists.\n\nThe destruction continues. After 1969, the caf\u00e9 was closed and the shell of the church sat empty and sacked until the 1980s, when Leonid Kuchma initiated a series of campaigns to finance its restoration. Now, in 2000, alone on the cliff's edge, I gaze at the church as it once was. Its reclaimed white shape, like a paper cutout exhibited on a dark screen, appears before me as an apparition. At its back, the boundless ocean swirls, the night sweeps down from the sky, and gale winds howl. I zip up my leather jacket and tie my scarf more tightly around my neck. Through an open door, the wind whistles and whips down the church's aisles to compete with the solemn, low-pitched, close harmony of the Orthodox priests conducting evening service. Icons and the repaired frescoes, flickering in and out of shadows cast by candlelight, add their voices, too. The priests, the sea, the crags in the darkening night all assemble to remind me that one among them who had raised money for the church's restoration was recently murdered by robbers. Destruction begins again. I wonder now what has happened. Will the whirling gyre of history once more blow its doors shut, exile its priests, and leave the building abandoned?\n\nAnother long excursion takes Irving and me through steep valleys between hills of limestone carved out by the wind, along winding back roads, and over mountain passes to the interior of Crimea, to what was once in the fifteenth century the political and cultural center of the Crimean Tatars. We have come to Bakhchysarai, which surrounds us with a m\u00e9lange of history, conflict, culture, and exile. This central town, now less developed, is a place where live not only Russians and Ukrainians but also Jews, Greeks, Germans, Armenians, Bulgarians, and, most of all, Crimean Tatars. Its architecture helps tell the story of its multiple identities.\n\nWe first visit the Khan Palace, built initially in the sixteenth century and used subsequently by a succession of Crimean khans, vassals of the Ottoman Empire, who once controlled Crimea's coastal areas in the fifteenth to eighteenth centuries, until 1783, when the Russian Empire annexed the region. The palace is a combination of Ottoman, Persian, and Italian architecture. Minarets tower above its walls. Inside are a mosque, built in 1532, a harem, lush gardens, a cemetery, and the khan's living quarters, with a private mosque, tiled floors, carved ceilings, and a \"fountain of tears,\" commissioned by one of the khans to commemorate the death of a young Polish concubine. It was made famous by Pushkin's 1822 \"The Fountain of Bakhchysarai\" and by Adam Mickiewicz's _Crimean Sonnets_. Replicating the khan's grief, the water drops almost silently from one lotus leaf to another, as if a single tear were falling.\n\nAs we leave the palace, the sound of prayers coming from nearby mosques drifts along the dusty streets where descendents of the exiled original Tatars sell deep-fried, honeyed pastries swarming with hundreds of wasps. The muezzin voices vibrating through the town tempt one to imagine camel caravans and traders from Turkey gathering in the square.\n\nWe proceed up a long hill to a walkway overlooking a narrow river valley and come to yet another steep hill that eventually veers to the right, where we continue our ascent on a rocky path beyond the tree line until we reach two thousand feet. The view into the sharply deep fissures and plunge of the river valley is dramatic. One imagines letting go the grip of time and hurling oneself into the abyss.\n\nWe come up to a medieval fortress and the remains of a walled, abandoned community\u2014its dwellings and compartments carved into the mountain limestone\u2014that was home to the Crimean Karaites until the midnineteenth century. Some call it the Jewish Fortress. Because of anti-Semitism elsewhere, these people settled here from the fourteenth century on, especially in the seventeenth and eighteenth centuries. I am reminded of the fact that, from 1923 to 1944, there were efforts to create Jewish settlements in Crimea; concurrently, the Nazis targeted Crimean Jews by recruiting rural Tatar volunteers to search out and murder them.\n\nThe rugged stone remains and husks of synagogues ( _kenesa_ ) testify to a former animated Jewish presence. Hebrew inscriptions from the eighteenth century are still visible, as are burial mounds. It is eerie to walk, sometimes crawl, among these ruins in these deserted, windblown places and think they once teemed with life. Ruts of carriage wheels, as in Pompeii, remain in the rock of roads as silent traces of another time. In the midst of this community are also reminders of a powerful Muslim presence. As I search for the path to take us down, I pass by the delicately carved stone mausoleum protecting the grave of a sixteenth-century khan's daughter. Remains of beliefs and peoples tumble together in the wind.\n\nFascinated, enchanted, we stay up here as long as it is safe. We descend into the sinking sun, walk past local Tatar women selling healing herbs, and come to the eighth-century Uspensky Monastery (Assumption Monastery of the Caves), carved into the mountainside. Just enough light remains in the sky to illuminate the empty, broad, whitewashed steps leading up to its entrance.\n\nSuddenly, as if startled by the sight of a large wading bird taking off unexpectedly across a smooth body of water, I stop in my tracks to watch an Orthodox priest in long black robes, crowned with a flat stovepipe hat, running toward and rapidly ascending all eighty-six steps. The sleeves of his gown, impersonating the bird's wings, spread out to the side and seem to launch him into flight. The billowing blackness of his ritual attire against the blank stillness of the bleached stairs takes my breath away.\n\nThough exhausted, I wait a minute and climb up myself into the damp chapel cave, lit only by candles. Fearful of intruding, I stand at the back and listen to the mysterious, deep harmonies of the evening worship. The priests, made tall and imposing by their elevated headgear as well as by the folds of their robes, move effortlessly, like giant shadows, about the cave. Plato's parable drifts in and out of my consciousness, for it is these shadows that mediate my understanding. When the Soviets took control in 1921, they closed down this monastery. It was only in 1991, when the Ukrainians gained their independence, that it reopened\u2014a history that recalls the fate of Foros Church as well as that of so many institutions victimized by Stalin's repressive measures.\n\nThat night, we arrive late at the Hotel Yalta\u2014too late for the cavernous restaurant. The following day, tired of guides and wanting to see more of the Tatar culture, I alone take the funicular up to the 4,500-foot Mount Ai-Petri. On top are the Tatars, who descend from the Islamic groups that came to Crimea as early as the fifteenth century and eventually grew in numbers to comprise the largest ethnic group in the region. After stepping off the funicular, which climbs far above beech forests and juniper glades, I wander around open wild fields that make a sunken tabletop of the mountain and form a kind of basin that protects one from the wind. Scattered among the grasses of these measureless sea-meadows are paths that lead to places I don't have time to go, horses for riding, and tents in front of which wood fires grill meat and vegetables. I partake of the food and enjoy its freshness. From the sides of tables hang multiple strings of lush-looking, recently harvested red onions that people come from miles to buy, and on top stand rows of bottles containing lethal-looking homemade hard liquor. I am afraid to imbibe. Cooking and selling this food are rural, tough Tatars, darker in skin and hair than their customers. They are the survivors or descendants of a time, in May 1944, when the population of Crimean Tatars was forcibly deported and exiled to Central Asia. (The Soviets accused them of collaborating with the Nazis.) It was not until 1967 that the Crimean Tatars were rehabilitated but banned from legally returning to their homeland until the last days of the Soviet Union.\n\nWhen I am enjoying a pleasant time on top of Mount Ai-Petri and among people enjoying the freshness of a mild day, I have no idea that after the Russian annexation of Crimea in 2014, these Tatars, of which there are over fifty thousand in Crimea, will once more be subject to hardship and violence. Many of their religious schools and homes will be searched under suspicion of possession of literature, which new Russian legislation labels extremist. Several Islamic books that were legal under Ukrainian law will be banned; the Crimean Tatars' religious and administrative buildings, as well as their homes, vandalized; and their language banned from schools.\n\nNor do I imagine, while strolling through the pastoral plains on top of this mountain, that thirty thousand Crimean Tatars will gather in the alpine meadow bowl on the upper plateau of Chatyr-Dah, close to the Simferopol-Alushta highway, to observe the seventieth anniversary of their deportation from Crimea. They will be forced to seek this mountain refuge, for Russian authorities will have forbidden commemorations elsewhere. Moreover, Crimean Tatar university students will be required to apply for a Russian passport and citizenship or else will not receive diplomas. In Bakhchysarai, a Crimean Tatar caf\u00e9 called Musafir suspected of harboring \"illegal\" activity, will be searched by the Russian Federal Security Service.\n\nBefore leaving Crimea, I take a day to myself to wander along the coast near Yalta, just to see how far I can go and what I might find. Like the eponymous lady with the lapdog in Anton Chekhov's story set in Yalta, I, another Anna, here for the first time, promenade by the sea, but there is no Dmitry Dmitrich Gurov to gaze longingly at me.\n\nThe morning mist rises to join the white clouds lying motionless on the mountaintops. Not a leaf stirs on the trees, the cicadas chirp, and the sea, with its monotonous roar, reminds those who listen of the eternal sleep awaiting us all. Minutes and footsteps pass. A warming sun quietly brightens the prospect and with its harmonizing rays composes a pastorale. Along the beaches, sunbathers stretch as if removed from time. Farther along, from a tall cliff, I catch sight of naked bodies strewn blissfully on sands like mermaids waiting for some forbidden pleasure. Only occasionally do undertones of darkness intrude and change the tenor of the lyrical hours. On the surface, all looks peaceful and ideal, but death and war lurk in the moment as well as in the memory. I watch a stray cat stalking two hefty seagulls (the proportions are all wrong) and pause before a bed of petunias where inch-long insects, resembling hummingbirds and yearning for nectar in the death throes of their lives, are vibrating their wings and darting in and out of the flowers' trumpets. Nearby, floating in the sun, is the listless spirit of Chekhov who came to live here to cure his tuberculosis. In the distance, I see his estate and imagine I am there. The curtain rises to reveal the opening scene of _Uncle Vanya_ set in a quiet garden compromised by conflict and thwarted desire.\n\n**Act One:** The garden. Part of the house and terrace can be seen. A table, laid for tea, stands on a path under an old poplar. Benches and chairs. There is a guitar on one of the benches. Near the table is a swing. It is between two and three o'clock in the afternoon.\n\nFarther along, Livadia Palace\u2014where Churchill, Roosevelt, and Stalin sat at a table to divide up Europe in 1945\u2014looks out to a tranquil sea. On the way down through landscaped terraces, I meet a man accompanied by a clipped poodle and an eagle attached to his wrist. He provokes the two to quarrel and offers to place the eagle on my arm. Disgusted, I answer \" _Nie_.\"\n\nContinuing, I arrive at the Alupka Palace, where Churchill stayed and where Rachmaninoff played. The structure strangely combines Edinburgh stone with Moorish ornamentation. On the way back, I bribe a gatekeeper to admit me through locked heavy gates to the Dyulber Sanatorium, built in the nineteenth century to resemble an Arabian palace. A nurse insists on showing me a suite of rooms where members of the Rada had recently stayed. They are luxuriously ornamented, a fantasy, yet oddly uncomfortable and unconvincing.\n\nTime to go home. Back on the overnight train to Kiev, I succumb to the darkness and fall asleep. But it is not to be a peaceful slumber\u2014what I feared on the way down becomes real. At 4:30 a.m., an intruder, who has a key to our private compartment, lets himself in and wakes me. He stands dark and tall at the end of my bunk. I see him because in reaching for the leather jacket I have hung at my feet, he mistakenly catches his arm on the light switch and then stands briefly illuminated in full view.\n\nThis accident saves my life, for earlier the attendant (who probably gave the intruder the key and told him who and where I was) said I should hide my valuables under my pillow. In the dark, the intruder would have come for these and either hit, gagged, or stabbed me. Strangely, this time I do not scream, but emitting some startled gasp, sit up straight. The man flees with my jacket. I wake up Irving, in the opposite bunk; we wonder what to do. The intruder must still be on the train.\n\nStupidly, like characters from Hitchcock's _The Lady Vanishes_ , we furtively glance up and down the empty, narrow, rattling, dimly lit corridors; past windows that show nothing but the smudged raven darkness of early morning; past silent, blocked, locked doors; and naively search for any sight of him. Perhaps he has discarded the clothes that hide his invisibility and has become the invisible man from Orwell's novel, whom the man selling us newspapers on our train journey from Kiev to Crimea remembered and knew well. He has taken advantage of what the protagonist in _The Invisible Man_ understands:\n\n_We have to consider all that invisibility means, all that it does not mean. It means little advantage for eavesdropping and so forth\u2014one makes sounds. It's of little help, a little help perhaps\u2014in housebreaking and so forth. Once you've caught me you could easily imprison me. But on the other hand I am hard to catch. This invisibility, in fact, is only good in two cases: It's useful in getting away, it's useful in approaching. It's particularly useful, therefore, in killing. I can walk round a man, whatever weapon he has, choose my point, strike as I like. Dodge as I like. Escape as I like._\n\nAfter much debate and confusion, Irving decides to look for our train attendant. He has to go through three train carriages before he finally spots an open compartment, complete with samovar, in which sits our attendant, jovially drinking steaming cups of black tea and gossiping with her friends. In Russian, he relates what has happened. Pretending disbelief, she hastily dons her uniform jacket over her corpulent figure and accompanies him back to our compartment where she plops heavily onto my bunk. Crying, wailing, and pretending to be sorry (as well as claiming she will lose her job), she offers to come to our apartment in Kiev to repay us for the lost leather jacket\u2014an unreal proposition because someone on her salary could not possible afford to do this. She asks for our address and phone number. Just in time, I stop a rather-too-willing Irving from giving her our address\u2014I imagine future complications, robberies, or bribes. Once back in Kiev she does phone, but Irving, thankfully, puts her off.\n\nWith this return, the embracing warmth and the illusion of peace in Crimea vanish all too abruptly\u2014not a trace seems left. Such fantasies appear distant, even intangible, when we open the doors to 36 Bohdan Khmelnitski and learn that Yelena, our neighbor who has mothered and protected us since we arrived, has, during our absence, suddenly dropped dead. The morning is bleak and the day blank. The clouds that Irving, from our hotel room in Yalta, watched drift out to sea linger in anticipation of what is to come.\n\n## _CHAPTER 7:_\n\n## SHADOWS OF THE DEAD\n\n### **BELARUS**\n\nBelarus is too close to Ukraine to resist, especially for Irving, whose mother and father spent the first twelve and fifteen years, respectively, of their lives in small, rural villages south of Minsk. Fleeing the threat of pogroms, as well as the dire economic conditions of the early 1900s, his parents, like many others, went to America where they later met and married.\n\nDuring our tenure in Ukraine, we make two journeys to Belarus\u2014the first at the end of May, the second toward the end of October\u2014each for less than a week. These not only introduce us to a contemporary Belarus but also allow Irving to search for the landscape of his parents' childhoods. For this quest, he has only the names of villages that once drifted through his father's memory, descriptions of places from his deceased mother's autobiography (written in Yiddish), and a tattered photograph of a village funeral. These shadows of the dead are his imperfect guides. They do not promise much.\n\nAn invitation to lecture on contemporary American poetry at the European Humanities University in Minsk occasions the first trip in May. We take an overnight train to Minsk. The platform in Kiev is crowded with throngs of impoverished people who haul the hard trials of their lives in overstuffed, red-and-blue-striped plastic square bags that resemble bursting bales of hay. I have no idea where these passengers will put them or even how they have carried them. The plastic handles must cut, even into their calloused hands. After much scrambling, we reach our platform. The train arrives, and we are supposed to get into coach zero. Puzzled by the number that really is not a number, we wonder, _Is this real? Where does one find zero?_ Running up and down the platform from one end of the train to the other, we discover that carriage zero is neither at the end of the train nor at its beginning. In this surreal world we have entered, zero does not precede one. Panic. Minutes before the train pulls out, we discover that zero comes between seven and eight. We jump on.\n\nThe journey lasts for thirteen arduous hours. We attempt to sleep but find it impossible, for in the process of crossing the border that separates Ukraine from Belarus, two sets of police (the Ukrainian and the Belarusian) board the train at midnight and again at 1:30 a.m., rudely wake us up by turning on the compartment light, and demand to examine our Belarusian visa and passport. Tense moments. The guards' shining, well-polished tall boots as well as their glossy pistol cases, slung across tight-fitting uniform jackets, evoke terrifying memories. I see some passengers being removed.\n\nBleary-eyed from the long journey from Kiev, we step off the train and walk into Minsk's monumental railroad station that, in the fog of breaking day, looms like some prehistoric monster out of a neoclassical swamp. Fog is everywhere, creeping, hovering, lying, and drooping. The mist and the pea-green industrial smog of the early morning curl through the openings between the station's tall Corinthian columns and into its cavernous, domed interior where hundreds and hundreds of bodies noiselessly swarm, stand, wander, or squat clustered in groups. The fog softens the harsh edges of their lives. As if already condemned to one of the circles of hell or anxiously anticipating some authoritative verdict, they silently await their fate. Large plastic containers, packed with most of their belongings, stand piled, waiting for someone to undo them. The last line of Matthew Arnold's \"The Scholar Gypsy\" comes to mind: \"the dark Iberians come; \/ And on the beach undid his corded bales.\"\n\nLost among this multitude of tortured, dead souls and confused by the labyrinthine corridors that are conduits for the swirling haze, we stand stunned. A single lightbulb hangs from the ceiling and does little to aid us. Think of the whole of Euston Station or Grand Central lit by one dim bulb. We are supposed to meet a guide, but how? No exit. No place. Irving decides to go on a pilgrimage in search of the man (whose identity is unknown). Fortune is nearby, for after about fifteen minutes he returns with a fur-capped driver-guide in tow. It is a miracle. Soon, in the clearer light of midmorning, we are driving through Minsk, a city that lacks the intimate, ornamental, and extensive beauty of Kiev.\n\nDuring World War II, Minsk was bombed, burned, and flattened, so there is little that is old. Minsk is now a horizontal, new city, spread out and deliberately divided into commanding, charted lines. Gaping spaces between buildings and heavy Russian-style monuments oppress the viewer and, for a moment, remind one of a history of destruction and continuing occupation. Traveling from the Minsk Railroad Station to the Hotel Jubliana is particularly surreal, for the disparity between the visible present and an eradicated past is extreme and unbridgeable.\n\nPersuaded by the bustling, eager present, one perhaps too easily forgets that supporters of Stalin's repression, Soviet partisans, members of the resistance movements, mass murderers, marauding auxiliary police, SS officers, and displaced minorities once actually trod these streets and were the perpetrators or victims of terror. New buildings obliterate memories of utter devastating destruction. And the sight of present-day crowds walking, shopping, taking the buses, driving their cars, and generally going about their ordinary business contradicts the horrendous fact that not so many years ago, Belarus lost a quarter of its prewar inhabitants, including 90 percent of its Jewish population. After that, writers and intellectuals disappeared under Stalinism. (In the Old Town of Minsk, there is a museum dedicated to these writers and thinkers.)\n\nTo negotiate these realities is difficult, uncomfortable, if not impossible. In many ways, it is easier simply to lose oneself in the present. If I am to be honest, for this reason in the days ahead I am sometimes impatient during our journey to locate the landscape of the past. In fairness, however, I should acknowledge that as much as Irving is intent upon finding the villages, he is just as committed as I to engaging the present. In particular, for both of us, the time spent with the university students is memorable. Their dogged, lively curiosity, which seems to be a continuation of the intellectual flavor of Minsk before the Second World War \"cleansed\" the city of most of its scholarly and creative elite, is moving and reassuring.\n\nIn May 2000, we enter at a time when an increasingly isolated Belarus is more and more subject to Russia's demands. Russian has officially been declared to be on equal footing with the Belarusian language and is now the dominant tongue. Russia controls the economy. (Belarus is integrated into the Russian Federation.) Belarus's president, Alexander Lukashenko, a supporter of the Soviets, has been in power for six years. His rule is popularly referred to as \"the last dictatorship\" in Europe. He heads a government (though technically a democracy since 1999) that exerts tight control over institutions and the national press. Among thousands of repercussions, the curriculum in higher education suffers. Soon the American Studies program at the European Humanities University, a private university founded in 1992, where I am to lecture in Minsk, will disappear. Students eager to engage Western philosophical and literary thought will be pressured to shift their attention to Eastern European culture, science, business, and technology. Just three years later, the university's president will be called before the Belarusian education minister and told to resign. And then, in a few months, the university itself will be expelled from Belarus and will relocate across the border in Lithuania.\n\n### **UGL\u00cd, UGLOO, UGLOV**\n\nWhen the conference in Minsk at the European Humanities University comes to an end, we turn our attention to finding the villages where Irving's parents were born. After innumerable phone calls and trips to and from a well-meaning travel agency (Irving's request to find his Jewish family's roots is the agency's first attempt to do anything like this), we sit and talk with those who are willing to help. At first neither of us has confidence that the search is realistic. We consult maps (some of which display \"red zones\" for high nuclear deposits and many of which reveal that there are seventeen places called Ugl\u00ed\u2014a name that simply means \"corners\"). After confronting the reality that names of streets and villages are not only repetitive but also variable (for instance, if one is going to Ugl\u00ed, one is going to Ugloo, and if one is coming from Ugl\u00ed, one is coming from Uglov), all seems speculative and a little fantastic. The quest evokes memories of our almost futile attempt to find the correct train carriage to transport us from Kiev to Minsk. Names as well as numbers shift and, like Proteus, as soon as one tries to take hold of them, morph into other shapes or sequences to frustrate the desire to arrive and grasp what one reaches for.\n\nEncouraged by our guide, we start out anyway\u2014toward an area near one of the many Ugl\u00eds. Traveling south into the countryside from Minsk is an immediate experience. There is no transition: no suburb, no industry, just suddenly fields. The road out of the city is straight. At one point, the police block our way. We pay a bribe and continue. Unlike the thick, cold fog of yesterday's arrival, the weather is now cloudless, hot, and sticky, so the driver halts at a little shop, buys a bottled drink, and agrees to give a woman a ride to her village. The bus goes through only once a day.\n\nThe first village we come to is, if the map is to be believed, probably Doktoworich, the village where Irving's father was born. No sign announces its identity, but when the driver inquires, a man walking down the road replies, \" _Da_ , Doktoworich.\" When Irving hears this local person pronounce the name (the speaker puts the emphasis on the first syllable rather than on the second), he realizes he is listening to the name just as his father and his maternal grandfather pronounced it\u2014a first clue, perhaps, that this is the right area. In old Doktoworich, we manage to find the Soviet, the regional office, where the secretary (already contacted by the travel agency) joins us and tries her best to help. Irving mentions two other villages (Ugl\u00ed and Stanki) that his parents used to talk about. We consult a map we have brought along but do not find either Ugl\u00ed or Stanki. Determined to continue, we get in the car and, within a few miles, believe it or not, notice a sign for Stanki, and then in a few kilometers, one for Ugl\u00ed.\n\nThe close proximity of Doktoworich, Stanki, and Ugl\u00ed suddenly makes this quest real. The burned site of a tavern also adds to the probability. Irving's grandfather ran a tavern. We convince ourselves that we are in the right locale and have arrived in yet another place where the invisible past almost metamorphoses into visibility. Spontaneously, my mind returns to the huddled displaced bodies at the Minsk train station. Their dim silhouettes merge with the ghostly landscape in front of us and remind me that German troops once burned and murdered their way through this area.\n\nThe secretary takes Irving to meet elderly people, hoping that they might remember the Jewish population in these villages. (Few young remain\u2014they leave for the cities. A motorbike parked outside a house is a sign that a young man lives there.) Irving realizes, of course, that there is no chance that anybody might know or have heard of descendents of his family who remained and survived after his parents' departure in the early 1900s. He is introduced to a ninety-two-year-old lady and an aged man (age either unknown or forgotten). Both are deaf (except when the man's cow gets loose, he hears it perfectly and runs out to fetch it). The secretary gently shouts questions, for which there are few answers, if any. The two recall seeing a young Jewish girl being dragged from a house across the street and murdered by the Nazi troops. But that is all.\n\nThe history Irving is curious about has either, at least in the text of their lives, never existed or been erased. The more recent atrocities are remembered with shame and therefore censored. Reluctant to acknowledge what remains, these villagers have repressed as much as they can. Irving decides not to pursue. Indeed, it is a relief to turn away from the past, for Irving is having difficulty imagining that this village was once where his mother lived. As it stands now, its buildings, its layout, its people bear little resemblance to the region she describes in her autobiography. Nothing mirrors her memory. The traditional log cabins (shown in Irving's one photograph) that once circled a village green are gone (the one remaining example is decaying and falling down) and are replaced by ordinary wooden bungalows. The culture has disappeared, Yiddish is silent, and the inhabitants changed.\n\nFor Irving, the villages are disappointing. To him they look barren, drab, and flat. I, on the other hand, unescorted by filial remembrance and, for the moment, repressing the region's history, cast my eyes on dwellings fronted by lush vegetable and flower gardens, blossoming fruit trees, and colorful picket fences that seem to create an idyllic picture. Just beyond these fences are wooden benches where the old people sit and where the barefooted rest. In front of every two houses are deep wells full of spring water, cold and sweet, which comes up in buckets attached to a long wooden pole that one swings or presses down to dip into the well. Two or three times a day, cows stroll down the dirt road to and from the feeding pasture. A woman or a man with a twig follows and occasionally encourages a recalcitrant animal to move on. On their return, the cows know which house or yard to enter. They turn slowly, deliberately, dutifully into their stalls. Around each dwelling, chickens and goslings rest in the shade or, like bleached leaves blown in autumn, scatter across the road. Attached to the houses are small wooden pens out of which the snouts of pigs catch the scent. I stop to say a few words to the pigs. A blurry-eyed man, wiry and humorous, wobbles over and jests, \"You came all this way and spent a lot of money just to see pigs!\" (He has a point.) Ducks perch on top of fences, and tied dogs bark, wag their tails, and pose for their photograph.\n\nAfter the awkwardness of the references to the Holocaust, once the conversation turns more willingly to the immediate surroundings, the villagers (and I, too) are more relaxed. Neighbors hand me flowers and invite us inside their homes. Just before we depart, however, we accidentally unearth a raw vein running through the topography of their seemingly pastoral lives. And that is when Irving turns his attention to a ragged workhorse attached to a farmer's wooden cart.\n\nCurious about what seems to be an unusually arched yoke above the horse's head, he appreciatively points and asks our guide about its design. Startled, the nag's owner, who has been jealously monitoring Irving's interest, angrily rushes up, fists ready to strike. In his eyes, Irving, as outsider, is a gypsy and means to steal his horse. After reassuring words from people who have gathered to witness and mediate the altercation, the owner departs, but unconvinced. Paradoxically, when we are on the way back to Minsk, a gypsy stops our car. He pretends he has lost his way and tries to sell us a \"gold ring.\" Tit for tat.\n\nExhausted at the end of this day, we cross the checkpoint into Minsk. Afterimages of the countryside and farmlands quarrel with the cityscape's avenues of concrete multistory buildings and illuminated pavements visible through the van's window. The two prospects are as incompatible as the past and the present. Thinking that there might be more to learn, we consider returning another time.\n\n### **BELARUS, 2000**\n\nThe second trip to Belarus, in the cold month of October, 2000, is more efficient, systematic, and \"scientific.\" The freezing weather now tempers the heat of novelty. This time a Jewish agency (composed of a young, ambitious Jewish couple) that specializes in helping people locate their roots is in charge. They have arranged for Irving to meet with archivists and to visit old Jewish cemeteries, where possibly his mother's family is buried.\n\nThis time we fly rather than endure the thirteen-hour train trip. And with us is the elderly, vain, and rather lopsided, bent-backed widow of Irving's cousin. (I wait for hours every morning while she arranges her thin wisps of hair into a style she wore when she was beautiful.) She has come over from Canada especially to learn more about her deceased husband's roots. We fly to Minsk via Belavion, a Belarusian airline that specializes in acquiring discarded planes from the Russians. The flight departs from the Kiev national airport (close to the center), which resembles a run-down 1950s Trailways Station I once knew well in Charlottesville, Virginia. We are in the wrong place! Is this actually a terminal? But it is.\n\nA van drawn by an old yellow truck transports passengers to the planes, which some call \"flying tractors.\" Each holds about fifty passengers, is solid, and flies at extremely low altitudes (not far to fall, I think). The interior is straight out of the '50s: aquamarine decorations, wide leather seats, broad aisles, and vanity mirrors (Irving's relative tidies her hair again). We rumble off on tires that seem to have been stolen from a Tonka truck and, strange to say, begin one of the most comfortable flights I have ever experienced.\n\nKnowing the flight route will take us over the Chernobyl nuclear reactors, I keep my eyes glued to the window. Suddenly, 1986 comes into view. The brilliantly clear day and the plane's low altitude make it possible, unbelievably, to see the silent tall abandoned chimneys extending their concrete fingers, as if wanting to stroke the plane's belly and reestablish a contact that is now forbidden. I look at the wrecked buildings, the smoke still spewing from a barely functioning reactor, the polluted Dnieper River (or Soz River) gurgling inappropriately next to the ruins, the adjoining scarred landscape, and the deserted town. How disturbingly appealing the site is, and how odd to know that people continue to enter these poisoned areas not just to try to contain the leaking radiation but also to poach and to plunder. The area around Chernobyl is supposed now to be a good hunting ground.\n\nAn hour and twenty minutes later, the plane lands in Minsk. Waiting in line for passport control is a former dean of the library school in Boston. He is soon whisked away by the US embassy officials. There is also a German professor of business, who admits to me that he is a missionary disguised as an academic. He tells me that because missionaries are forbidden in Belarus, they must come in on other pretexts. He, consequently, teaches business from a Christian perspective. Evangelicals like him are a thorn in the side of Eastern Europe.\n\nThe guide for this second visit is where she says she will be. She takes us to the Hotel Minsk. One look at its drab, dangling lamps, its dull brown walls, towels, carpets, cigarette-holed bed covers, natty curtains, and at its excessively narrow beds convinces me that we should move to the Hotel Jubliana, where we stayed before. Two leather-jacketed thugs trashing the room next door clinch my determination to move.\n\nAt the Hotel Jubliana, Irving's relative is, thankfully, assigned the only remaining \"Western\" single room. She will be comfortable. We climb up stairs to another floor where there are \"Soviet-style rooms,\" which, although still a bit on the brown, frumpy side, are superior to what the Hotel Minsk offers. At first, no hot water runs through our taps. We call a willing but drunk maintenance man, who says we should let the water run for twenty minutes to allow the hot water to reach our room. We try but find only a lukewarm offering. Another ten minutes pass, and the water reverts to its icy origins. The heat is not on, either. I consider calling for a hot-water bottle to warm the damp sheets. Strangely, what runs through my mind is the fact that on September 22, 1943, an SS officer, Wilhelm Kube, made a similar request. The hotel maid promptly came to his room and put the hot water bottle in his bed; however, when the officer crawled under the blankets, the container blew up and mangled him. The maid, either a partisan or an enemy, had placed a bomb inside the bottle. (In Minsk, this officer had thrown sweets to Jewish children who had been tossed into a deep pit of sand. His absurd act was his strange attempt to silence their terrified and failing cries.) Tomorrow, things will be better, we trust, and, for the moment, they are. There is not only hot water but also heat.\n\nIn the morning we drive in a van supplied by our guide to Kapyl ( _Kapulye_ in Yiddish), a town with a Jewish cemetery and a regional archive, which the Soviets established after World War II, when they required each region to compile records of its history and its people. One result is that there are now meticulously collected records of all inhabitants, dead and alive. Alone in the back of the van and hiding from the ceaseless anticipatory chatter among the guide, Irving, and his relative, I watch the morning sun illuminate and melt the previous night's hoar frost. Its golden light settles on tall silver birches, glazes the streams, gilds the poplar leaves, which cluster and shake like precious coins waiting to be spent, and warms the white soil. Some think the name Belarus comes from the pale color of its soil.\n\nIn Kapyl, we stumble through a lumpy, exposed field among the jumbled and overgrown remains of the Jewish graves. Some Hebraic writing is still visible, but most is washed away by the wind, rain, and time. I learn that in old cemeteries within forests the inscriptions, protected by the trees, remain longer. A cow chews the grass and a child, put in charge of it, occasionally pulls on its tether. Realizing the impossibility of this exercise, we halfheartedly pull weeds off the few intact gravestones and with a whisk brush attempt to clear away the algae and dust from the fading letters. Irving was hoping to find his mother's family's name, but all remains as elusive and unknowable as ever. From the field, we walk over to the regional museum\u2014which is, like all public buildings, under _remont_ (renovation). It is locked. After a few despondent minutes, I catch sight of a young woman, carrying her Saturday shopping and heading toward us. She is the curator returning from her lunch break. Her genuine interest in the Jewish past is absolutely astonishing.\n\nFor the next two interminable hours, the guide, Irving, and his relative\u2014engrossed in their task and in the grip of memory\u2014huddle together in a small, unheated office, examining papers from the Soviet archives. Hunched in an attempt to keep what body heat remains under my coat and scarf (even the curator wears a thick vest and lined boots), I sit apart in a corner, next to an ancient typewriter, and watch them painstakingly go over lists of names and through handwritten documents I don't understand and that the eager curator keeps bringing in from other rooms and out of packed files. She finds many references and papers having to do with Mendele Mokher Sforim, a relative of Irving's mother. Some consider him the founder of Yid-dish fiction. Finally, there is some actual, concrete evidence that people Irving's mother mentions in her autobiography lived in this region.\n\nThe curator continues to be helpful. She finds the telephone number of a surviving Jew in Kapyl. She calls; a person answers only to tell her that Mr. Abromowich (meaning \"book peddler\") has left for the market in another town and will not be back until the next morning. A meeting will not be feasible. As if to compensate for the disappointment, the curator tells Irving about a memorial book, compiled from government documents, published in Minsk. The publication exhaustively lists the names of all the people murdered by the Nazis in the Kapyl region. Years later, the tome arrives in Buffalo, courtesy of a friend from Kiev who has traveled to Minsk. Up to that point, purchasing a copy has been impossible. An isolated Minsk does not take checks, credit cards, postal orders, or anything we can think of.\n\nThe morning has been fruitful. Satisfaction, however, strangely and disappointingly does not, in my present, slightly ungenerous state of mind, compensate for the hunger and the biting cold that gnaw on every part of my body. Now that names are found, I simply want to leave and seek relief from the freezing air that contributes to my weariness. Finally, we cross the street to a depressing cafeteria where I devour an aged cutlet and raw rice.\n\nBack in the van, we return to the villages Ugl\u00ed and Stanki. I am, however, eager to see them again. I think of the first time when, in May, the sight of these villages stimulated my sentimental imagination and evoked idyllic images of nineteenth-century rural communities. Returning in the winter breaks this illusion. Now the gilt frame of the picturesque scene is shattered and the canvas damaged. Instead of seeing elderly people sitting in the serenity of the sun, I notice people tramping through mud and muck in galoshes, thick stockings, and bulky coats. The roads that arrange each village don't meander; they are straight and wide\u2014regulated. The houses are badly constructed. The population that works the land, as well as the collective farms, toils hard. People's hands are the large, muscular appendages of laborers; their bodies are burly; their faces round with brilliant blue eyes; and their skin red-cheeked and scarred from the drudge of their lives.\n\nThey remember us. Talkative groups of men and women gather to escort us from house to house. One particularly bent woman, leaning on her stick, leads the way. With her gold-capped teeth, she chews a raw onion and takes us to meet an eighty-nine-year-old woman who stands as straight as a broom. I step into the warmth of kitchens where old-fashioned tiled stoves offer shelves for sleeping (lovely and warm) and ovens for baking. One woman fills my pockets with roasted pumpkin seeds she has removed from the oven with a long wooden paddle. On top of her stove, a stew simmers in an iron pot. I venture into a homemade sauna that a young man is eager to show off. In other dwellings are cabbages piled and stored in cold spaces; apples laid out to dry (their sweet and healing fragrance permeating the room); photographs of lost sons, daughters, and husbands; folk icons propped in a corner; ceremoniously embroidered hanging cloths; celebratory posters of Lenin taped to the wall (the elderly admire the old Soviet system); and platters of late-season bruised pears.\n\nOne lady fills my backpack with several dozen. In a cottage where we drink steaming tea, an old woman opens a trunk and carefully pulls out a tablecloth. It belonged to her wedding trousseau. She despairs, \"What can I do with this now?\" Does she want us to buy it? Does it mean her family has gone? As in so many other instances, we are not sure, so simply admire its exquisite stitching. We listen to stories of their lives, some sad. One woman's husband has just suffered a heart attack, and now she is left with an alcoholic son. I also overhear declarations that I wish were not uttered: \"The Jews were nice, like the Russians.\" It is impossible to understand or respond to these.\n\nIt is time to leave. The horse wagons with their arched yokes pass us by. I gaze at the stork's nest above the grounds of the collective farm. In the evening, the stork perches on an electric pole and looks down upon the blushing haze of the setting sun. Farther down the dirt road is a wide-open, damp, shadowed field of muck in which groups of young people are spread and bent, sorting potatoes with their chapped, raw fingers.\n\nOne evening, the guides from the Jewish agency come over to our hotel in Minsk. We meet in the lobby\u2014an empty brown lounge\u2014and for the next four hours listen to stories of survival they have heard various customers relate. One narrative is about a man who, when he was six, ran away from the ghetto in Minsk. After hiding in the woods, he was recaptured. On the eve of his execution, Latvian officers adopted him as a mascot and errand boy. For months, these officers took him with them to the killing fields. After the war, the young boy found his way to Australia, where he was placed alone in the wilderness. His job was to man a railway crossing. Every few months, the train dropped off food and supplies for his survival. Not being able to take this solitude anymore, he jumped on a circus train that happened to be coming along the tracks. For years, he worked in the circus. Like so many narratives of survival, the story of his nomadic life continues and at one point brings him back to Minsk.\n\nWhile Irving, his relative, and the guides, seated in a closed circle, chat away for hours, absorbed by these stories, my attention starts to wander toward what was an empty bar that is now filling up with prostitutes and German businessmen. It is beyond midnight. The stories continue and those who listen don't see the gamblers going to and from the hotel's casino. Oblivious to their immediate surroundings and the dimly lit present, neither Irving nor his cousin nor the guides notice that two leather-jacketed young men are trying to race into the hotel through automatic sliding doors that move too slowly for their pace. The men run right smash into the doors and fall to the ground; muscular security guards and padded armed militia (who seem to emerge from nowhere) race commandingly toward them. Outside the hotel entrance, they and their violence vanish into the darkness and silence of night and forgetfulness. I never see any of them again. The compelling past, though poorly lit, remains inside, anecdotally recalled and partially visible.\n\nThe national airport in Minsk mimics a Greek temple on the outside but within resembles a deserted marble tomb, cracked, empty, and drafty. The flight is delayed three hours, so we sit on plastic chairs. The toilet is under _remont_ so is not available; the caf\u00e9 is open. Cockroaches run among the open-faced sandwiches. Amusingly some of the customers play with them. Irving's cousin (she escaped from a village in Poland just before the Nazi occupation) becomes anxious as if all the memories and ghosts are assembled here. She cannot find her visa. And when a customs official tries to help her with her heavy suitcase, she cannot believe that he is not inspecting, judging, condemning, or even taking her away. There ensues a tug-of-war that, if it were not tragic, would be absurdly comic. Bent over, her long, full dress almost reaching the floor and her jewelry swinging defiantly, she clings determinedly to the suitcase handles, as if holding on to the last breaths of life. She pulls away from the uniformed official, who politely yet firmly attempts to take it from her resolute fingers so he might place the case on the conveyer belt. Only when we rush forward and put a stop to the struggle, like referees in a boxing match, does she give up. For her, at this moment, the past overwhelms the present and blinds her to what is really going on.\n\nUpon our return, Kiev, in contrast to Minsk, looks even more splendid, bright, and stylish than it did before we left.\n\n## _CHAPTER 8:_\n\n## THE LEAVING\n\nNovember arrives, the days grow shorter, and so does my time in Ukraine. I shall be leaving mid-December. The moment of departure draws closer and closer. We must return to Buffalo. Speeches, presents, group photographs, and promises fill our last week. At the airport, Natalia and Tamara see us off. Leaning over the railing dividing us, our friends watch us place our bulging bags on the conveyer belt. Our eyes touch. In a few minutes, after one last turn of the head, we must move on, knowing that we will never see each other again.\n\nEndings bring with them a certain madness. Mine takes the form of an overwhelming impulse to collect nineteenth-century Ukrainian icons and smuggle them home. (For this reason, I never buy one that has metal on it for the airport X-ray machines will find me out.) Even though I am not religious, the sacred icon held high above the priest's head in Odessa strangely haunts my consciousness, as do others that have become phantoms in my memory. These images carry with them the feelings and habits of those who created them. Rendered by a living, imperfect hand, each contains a still moment of desire for rest and salvation in a long history of conflict, conquest, and difficulty.\n\nKnowing that displaced, stolen, and damaged icons are on the market, my quest leads to antique stalls lining the roughly cobbled Andriyivsky Descent (Kiev's Montmartre), which winds to the lower city, past Mikhail Bulgakov's house where he wrote _The White Guards_ \u2014a novel, set in late 1918, about a family caught up in the various factions (the Whites, the Reds, the Imperial German Army, and the Ukrainian nationalists) that pummel, divide, and destroy their world. As I write, I cannot help but think of the eastern Ukrainians in cities such as Luhansk and Donetsk, split between loyalties and survival, divided among themselves. Those who have not been able to leave and live with relatives elsewhere are caught in basements, afraid to go out, for fear of being bombed or rounded up. Their pensions cut off, they wait for bread lines and food kitchens. On the border between Ukraine and Russia during \"cease fires\" of a never-ending war, explosives fall between already-ruined buildings.\n\nContinuing my search for the displaced, I rummage through makeshift booths set up for a day (at one I find a heavily varnished St. Michael propped on a suitcase fashioned into a table). In the courtyard behind my apartment, I enter a jewelry shop to consider buying a folk icon hanging behind the counter. (Some family has discarded it.) Farther down, I come to a set of broad steps, leading to a higher street, where thick-coated and gloved people, in an attempt to earn a few pennies, sell \"treasures\" from their home. I find nothing but do meet a man who promises to restore damaged icons for a few pennies.\n\nOne day I read about a gathering of antique collectors, who meet every Saturday morning in a cavernous nightclub, appropriately named the New York Club. We are determined to find it, rain or shine. The day we decide to go, it is raining, a heavy, penetrating, deep-puddled downpour. Having no address but the New York Club, we are not sure where to go, but the shopkeeper in our neighborhood jewelry shop does. Off we go via the metro underground across the river. Half an hour later, in an alien residential district, we ask anyone who passes by where the club might be. Eventually, a saturated drunk with a swollen eye leads us in zigzags through nasty concrete apartment courtyards, around corners, and across a muddy park to our destination. Outside the club, like crows standing around fresh roadkill, leather-jacketed, short-cropped male figures consider their next moves. Smoke from their cigarettes swirls cannily through their dealings and the wet. Occasionally a male figure with a painting tucked under his arm or an ancient light fixture in a plastic carrier bag hastily leaves the building like a slanted shadow blown across an open square.\n\nWhen we enter, the darkness in the club is blinding. The only light comes from either a buffet bar or a tiny wicket, where we buy an entrance ticket from a fur-hatted, stout matron whose chubby fingers sport amber rings. With difficulty, we grope our way to the silhouette of a disco floor around which dealers' tables, arranged in three immense circles, create a maze of treasures. Even though an occasional lamp exposes a detail or two, it is too dark to see anything properly. An abandoned, unlit, revolving sparkling light dangles from the center of the ceiling and fails to expose the Minotaur that surely waits patiently for its prey in the center of this labyrinth. Behind each table men display, exchange, and sell their chaotic heaps of wares: collections of medals, photographs, paintings, icons, china pieces, and jewelry. The atmosphere is serious, hushed, and subdued. I am one of only a few women here. This is a man's world. For two hours, until closing time, we roam this den of thieves and work our way through this Bosch-like cavity. I leave with two icons (neither of which I can really see). One is a fine icon from western Ukraine; the other, a lively St. George crawling with wormholes\u2014the man on the steps will fix those in a week's time.\n\nDamp and tired, we duck into a warm caf\u00e9 and then spend two more hours, this time in an open market, looking for hand-knit wool socks. To make sure they are the correct size, the merchant has Irving make his hand into a fist and see if it will get easily into the sock. Ingenuity. I am reminded of when I buy black woolen tights from a stand on the street outside the apartment. The lady takes one look at me (even in a coat) and immediately knows the right size. She is always accurate.\n\nTime passes; I am at the airport to fly back to the States. The socks and the sequestered icons (I have given away most of my clothes to make room for these) glide unnoticed through the X-ray machine and will return with me. The matriarchal society of room 73 leans over a banister separating visitors and departing passengers. We throw kisses, promises, and longing looks. I don't want to let go of this past year, but as the plane moves closer to the West and the chatter up and down the aisles surreptitiously and cautiously shifts to English, all starts to be jumbled. (Where am I?) Like Alice at the conclusion of _Alice Through the Looking Glass_ , I cannot help but pull out the tablecloth from beneath all that has been set upon it during the previous months so that everything tumbles and scatters: people, places, incidents, images, and words begin to fall away and change shape. What once so clearly and definitively classified my life in Ukraine disappointingly starts slipping into some dreamlike state. The taxonomy of my surroundings shifts back to a previous schema. Everything around me begins to revert to the familiar, yet foreign, profile of home. The Red and White Queens become ordinary domestic cats. I am grateful to be met in London by Celia who, because she visited me in Kiev, can affirm that where I have been and what I have seen are real. Standing in line at passport control, I find it momentarily reassuring to overhear, as well as partially understand, two Ukrainians converse behind me. Though their talk is commonplace, their presence is not.\n\n## _AFTERWORD:_\n\n## THE ARCHIVES AND DR. NOVAK\n\nTwo decades after my return from Ukraine, the figures of Dr. Novak, Ludik Benes, and Mrs. Kessler from my childhood continue to haunt my mind. In spite of the tender feelings that grace my remembrance of them, their increasingly fragile presence is disquieting, for there is still much I do not know about their lives during the time I was living in the parsonage a few miles from Manchester. I keep wondering, what exactly did Dr. Novak see in Prague in 1945 and 1946, before he visited my family? What happened to Ludik Benes once he left the 1946 IRA conference in Manchester and returned home? And what did Mrs. Kessler endure before she landed in England in 1947?\n\nIn an attempt to answer these unsolvable questions, I recently embarked on yet another journey. This time I was not to board a plane; rather, I was to travel virtually to the digital archives of the Unitarian Service Committee, housed in the Harvard Divinity School library. Because Dr. Novak and Ludik Benes belonged to the Unitarian movement, I trusted that the postwar documents in these archives, especially those having to do with Czechoslovakia, might reveal something about the circumstances of their lives and perhaps might even mention their names. Alas, and perhaps predictably, I found no reference to their names, but I did discover raw materials that bluntly portray what an individual living in Prague might have known or seen right after the end of the war. Even though I did not learn anything specifically about my childhood acquaintances, I did come away with a more definitive sense of the distressing, and often threatening, situations that challenged their lives from 1945 to 1950. Reading through these papers, I came to understand that bombed-out Manchester, in comparison with the human suffering and condition of their home country, must have looked like paradise. I also gained a more vivid sense than I already possessed of just how horrific the environment Mrs. Kessler survived had been.\n\nWhat follow are a few of these documents.\n\nAmong the most poignant letters in the Unitarian Service Committee files is a letter sent from Prague, dated May 29, 1945, to \"friends\" in the West just twenty days after the liberation by the Soviets of the Theresienstadt ghetto, a concentration camp in northwest Czechoslovakia. Composed after \"years of compelled silence,\" this elegant missive proffers a devastating account of some of the chaotic and dire difficulties in Czechoslovakia right after the war. The letter bears no signature but appears to have been written by a well-educated European, residing in Prague, who was dedicated to and in a position to seek assistance. (The few unidiomatic expressions demonstrate that the writer is not a native English speaker.) Whether or not he was hired by the Unitarian Service Committee, I do not know.\n\nThough the letter's details offer an outline and no more of the larger grief, they catch the essence of the horrific state of affairs that must have been all too familiar to those whom I met in my early childhood and who up to this point must have also lived under the weight of what the correspondent calls \"a compelled silence.\" This anonymous May, 1945, letter begins:\n\n_Dear Friends, I am so glad to be able to send you a message after years of compelled silence. Our thoughts have often been with you all and our hearts beat high when we heard in your radio a few days before the revolution that you were present at a reception of the new Czechoslovak minister at Z\u00fcrich._\n\nThe correspondent continues by isolating some of the dire problems following the release of three hundred thousand workers and prisoners from concentration camps in Czechoslovakia as well as from other parts of Europe, who are now attempting to return to their home country. In particular, he mentions Theresienstadt, which the Nazis ran as a showpiece and which housed many children. In reality, Theresienstadt was used as both a labor and a transit camp for Czech Jews who were deported either to killing centers or to forced-labor camps in German-occupied Poland, Belarus, and the Baltic States.\n\n_Our social work has not been interrupted and now it has to grow enormously. P. has initiated and is leading the organization of four recovery homes for children from Theresienstadt in confiscated castles, in collaboration with the ministries of health and repatriation. There are very many obstacles and even the best organized services would never be able to master the enormous problems that the postwar conditions heap upon this country. Among the worstis the fact that three hundred thousand exported workers and prisoners are flooding back from all sides in a very poor physical, mental and moral state into their home country with the hope of finding paradise after years of privation, in fact finding a robbed-out country disorganized after a long occupation and the excesses of revolution._\n\nIn particular, the writer draws attention to the dire circumstances facing foreign civil prisoners, who are still on Czech soil, especially the foreign Jews at Theresienstadt. Perhaps Mrs. Kessler was among them:\n\n_This old little \"ville de garrison\" [Theresienstadt], which used to have a population of seven thousand, including the soldiers was all turned into one ghetto by the Nazis and is still lodging thirty thousand Jews of many nationalities. Not long ago it had sixty thousand inhabitants, most of which have been killed. The sanitary and food conditions are being improved now, but lodging is still very bad._\n\nWhat the writer of this letter finds especially upsetting are the children who have just been released from Theresienstadt. Many are in terrible condition, physically and emotionally:\n\n_I was there a week ago, when we brought out the first group of orphans to the castle that was first quickly \"adapted\" in a few days. I saw a room of about thirty square meters in which twenty-seven boys of twelve to sixteen live day and night in a state of general apathy. They have been transported here worse than cattle during the last weeks before capitulation from concentration campsthat the Germans evacuated before the enemy._  \n _These boys most look like skeletons covered with yellow skin, many of them with faces of old men. There are still about five hundred such boys at Th. [Theresienstadt], mostly Polish. Very many have died on the way and some are still dying. The children that have been only at Th., not in other concentration camps, look much better, but badly need mental recovery. We hope, in spite of many psychological difficulties, to improve the near future for most of them, but the authorities are interested in sending them as soon as possible back into their countries of origin._\n\nThe correspondent's anguish is prompted not just by the children's condition but also because they have nothing to return to except, as he acknowledges, the reality of a still-active anti-Semitism. As he astutely remarks: \"anti-Semitism has not died with Hitler\":\n\n_Now this is the saddest part of their fate: to be sent back into a country absolutely ruined where they have neither relatives nor friends left and where no one will be interested to help them because anti-Semitism has not died with Hitler. It is especially the case with these shadows of Polish boys, but in lesser degree also with many Germans. There are still five thousand Jews from Germany in Th. Over a fifth of them have no one left in Germany and dread the return to such an inhospitable place._\n\nConcerned because of a lack of organization and communication among agencies supposed to be offering relief, and sensitive as well to the fact that \"the population is longing more for rest than for work,\" he worries that \"very little is done for the future of all foreign Jews.\" He despairs:\n\n_Please, do something about it! Give a copy of this with necessary recommendation to the I.R.C. [International Relief Committee] at Geneva and any organizations likely to be able to do something about offers of hospitality. Of course help of any kind must be offered or mediated by some organization that enjoys a certain consideration with the present chief ally of this country. Mere financial help is of no avail as private initiative here has no possibilities in this field._\n\n\u2014Harvard Divinity School library. bms 16035\/1 (17).\n\nThrough further research, I discovered that \"P.,\" the man referred to in the second quotation from the above letter, is probably Policer, a Hungarian Jew who in 1945 was appointed by Noel Field, the European director of the rescue mission of the Unitarian Service Committee, to direct the Czechoslovak chapter. The two met in Marseilles, where Field had collaborated with a French Jewish humanitarian organization to provide relief for endangered Jewish refugees. When he accepted this position, Policer changed his name to Dr. Gejza Pavlik. (He altered his surname in order to disguise his ethnicity and sound more Slavic.)\n\nPavlik's job lasted only until mid-1946. During that period, he and his wife, Charlotte, who had worked with refugees in Geneva, succeeded, with the assistance of the Unitarian Service Committee, in getting homes for orphaned or displaced children and organizing medical treatment centers. During that period, Pavlik wrote four reports for the USC in which he described the economic and social problems facing postwar Czechoslovakia. A selection from one of these reports, describing the general conditions in the country, is quoted in a letter prepared in December 1945 and sent from the Hotel Alcron in Prague by an unnamed correspondent. (Again, this document is unsigned, but I believe it was submitted by Field, who hired Pavlik.) The paragraph from Pavlik's report, inserted into this letter, describes the appalling circumstances of most living in Czechoslovakia.\n\n_The situation is frequently sadder than one could imagine. The occupation and the war have left the most sinister consequences. The feeding of the urban population in meat, milk, fats is an especially serious problem. The Germans have killed practically all the cattle. This year's harvest was bad. Wherever the fighting has passed the homes have been largely destroyed. In the eastern portion of Slovakia there are 5,000 (?) people without any homes whatever. The public authorities attempt to bring help, but as transportation is lacking, all efforts move too slowly. To understand the situation it is only necessary to remember that the Germans have removed or destroyed 50 percent of the locomotives, they have blown up all of the railway bridges, they have even removed hundreds of kilometers of rails, repair shops have been completely destroyed, etc., etc. So in spite of the efforts of the authorities, a large part of the war victims will remain without shelter during the winter._\n\n\u2014Harvard Divinity School library.  \nbms 16035 (9) seq. 10.\n\nBy 1946, when Dr. Novak and Ludik Benes were visiting my father, they must have been fully aware of not only these dire circumstances and obstructions but also the threats and difficulties arising from the Soviet sphere of influence that was gaining hold in Czechoslovakia. (Recall that Ludik was to be denied a passport.) The Soviets were beginning to dominate plans and strategies for reconstruction. Russia was gaining control over key ministries and thus was able to suppress non-communist activity. Thoughts of a more equitable distribution of power were quickly disappearing, and President Edward Benes's dream that the country could become a bridge between the East and the West were being dashed. By February 1948, the communists had seized full power in a coup d'\u00e9tat. And that is when all visits to our house came to an end.\n\nDuring the period (1953) when my family and I were being benignly questioned by some career bureaucrat in order that we might enter the United States, the fate of those who had worked for the USC in Czechoslovakia was tragic, horrible, and torturously protracted. Gejza Pavlik and Noel Field, among many others, were being subjected to brutal investigations, house arrest, prison, and Cold War show trials, held in Eastern Europe, in which they were charged with serving the interests of Western imperialism. Field was also accused, by the FBI, of feeding information to the Soviets. For this reason, in October, 1947, he was fired from his position with the USC, and his papers in the USC files \"sanitized.\" Field was never able to return to America and was eventually granted political asylum in Budapest. Throughout, he never let go of his belief in pacifism, egalitarianism, and human service. To the last, he clung to an idealism that the historian Arthur Schlesinger, Jr. called both \"simple-minded and indestructible.\"\n\nPavlik, though not as well known, suffered from brutal investigation methods in Hungary. Labeling him a Trotskyite, an enemy, a spy, the Soviets poured strong salt solutions down his throat, beat him, struck the soles of his feet with rubber hoses, deprived him of sleep, and threatened him with \"suicide\" or \"an escape attempt,\" during which he would be shot. At a trial in Budapest, he admitted that he had consciously decided to spy for the Americans, but he later retracted his confession. Finally, on June 29, 1950, Pavlik was sentenced to fifteen years in prison, his wife to ten. And so goes the world.\n\n* * *\n\n_. Field was appointed European Director of the Rescue Mission of the USC in late 1942 or early 1943._\n\n## ACKNOWLEDGMENTS\n\nWith many thanks to those who have chosen to spend time reading through the manuscript. In particular, I would like to acknowledge Carrie Bramen, Regina Grol, and Carolyn Korsmeyer. I am also grateful for the encouragement I received from Kim Chinquee, Joyce Gleason, Ephraim Massey, and Phil Brodrock.\n\n## ABOUT THE AUTHOR\n\n_author photo \u00a9 Bruce Fox_\n\nAnn C. Colley spent the first thirteen years of her life in England during World War II and the postwar years. In 1953, she moved with her parents to the United States. After receiving her PhD in English from the University of Chicago, she taught at Fisk University. In 1980, she moved to the State University College of New York in Buffalo where she is a SUNY Distinguished Professor.\n\nThe author has written extensively on nineteenth-century British literature and culture. She has published with such presses as Harcourt, Brace, Jovanovich, the University of Georgia Press, Macmillan, Ashgate, Palgrave, Routledge, and the Cam-bridge University Press. She has taught on Fulbright Fellowships in Poland and Ukraine. With Irving Massey, she has traveled throughout South America, Central America, Nepal, Turkey, Morocco, Africa, New Zealand, Armenia, Belarus, Hungary, the Soviet Union, Czechoslovakia, Poland, and Ukraine. Always longing for the landscape of home, she often returns to England. In the summers, she lives in the wilderness of Nova Scotia.\n\n## **SELECTED TITLES FROM SHE WRITES PRESS**\n\nShe Writes Press is an independent publishing company founded to serve women writers everywhere. Visit us at www.shewritespress.com.\n\n**_Dearest Ones at Home: Clara Taylor's Letters from Russia, 1917-1919_** edited by Katrina Maloney and Patricia Maloney. Clara Taylor's detailed, delightful letters documenting her two years in Russia teaching factory girls self-sufficiency skills\u2014right in the middle of World War I.\n\n**_Motherlines: Letters of Love, Longing, and Liberation by Patricia Reis._** $16.95, 978-1-63152-121-8. In her midlife search for meaning, and longing for maternal connection, Patricia Reis encounters uncommon women who inspire her journey and discovers an unlikely confidante in her aunt, a free-spirited Franciscan nun.\n\n**_Gap Year Girl_** by Marianne Bohr. $16.95, 978-1-63152-820-0. Thirty-plus years after first backpacking through Europe, Marianne Bohr and her husband leave their lives behind and take off on a yearlong quest for adventure.\n\n**_The Beauty of What Remains: Family Lost, Family Found_** by Susan Johnson Hadler. $16.95, 978-1-63152-007-5. Susan Johnson Hadler goes on a quest to find out who the missing people in her family were\u2014and what happened to them\u2014and succeeds in reuniting a family shattered for four generations.\n\n**_The Butterfly Groove: A Mother's Mystery, A Daughter's Journey_** by Jessica Barraco. $16.95, 978-1-63152-800-2. In an attempt to solve the mystery of her deceased mother's life, Jessica Barraco retraces the older woman's steps nearly forty years earlier\u2014and finds herself along the way.\n\n**_Godmother: An Unexpected Journey, Perfect Timing, and Small Miracles_** by Odile Atthalin. $16.95, 978-1-63152-172-0. After thirty years of traveling the world, Odile Atthalin\u2014a French intellectual from a well-to-do family in Paris\u2014ends up in Berkeley, CA, where synchronicities abound and ultimately give her everything she has been looking for, including the gift of becoming a godmother.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nMAGPIES, SQUIRRELS _&_ THIEVES\n\n# MAGPIES, SQUIRRELS _&_ THIEVES\n\nHOW THE VICTORIANS COLLECTED THE WORLD\n\nJACQUELINE YALLOP\n\nFirst published in Great Britain in 2011 by Atlantic Books, an imprint of Atlantic Books Ltd.\n\nCopyright \u00a9 2011 by Jacqueline Yallop\n\nThe moral right of Jacqueline Yallop to be identified as the author of this work has been asserted by her in accordance with the Copyright, Designs and Patents Act of 1988.\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of both the copyright owner and the above publisher of this book.\n\nEvery effort has been made to trace or contact all copyright holders. The publishers will be pleased to make good any omissions or rectify any mistakes brought to their attention at the earliest opportunity.\n\n10 9 8 7 6 5 4 3 2 1\n\nA CIP catalogue record for this book is available from the British Library.\n\nISBN: 978 1 84354 750 1\n\neBook ISBN: 978 0 85789 561 5\n\nPrinted in Great Britain\n\nAtlantic Books  \nAn imprint of Atlantic Books Ltd  \nOrmond House  \n26\u201327 Boswell Street  \nLondon  \nWC1N 3JZ  \nwww.atlantic-books.co.uk\n'Things' were, of course, the sum of the world.\n\nHenry James, _The Spoils of Poynton_ , 1897\n\n## _Contents_\n\n**List of Illustrations**\n\n**Preface**\n\n**Catching the Collecting Bug**\n\n1 Exhibition Road, London, 1862\n\n2 The Useful and the Beautiful\n\n3 A Public Duty and a Private Preoccupation\n\n[**Making Museums: Collecting as a Career**  \nJOHN CHARLES ROBINSON](9780857895615_13_pt2.html#pt2)\n\n4 On the Banks of the Seine\n\n5 The Battle of South Kensington\n\n6 The Tricks of the Trade\n\n7 Changing Times\n\n[**Ransacking and Revolution: The European Crusade**  \nCHARLOTTE SCHREIBER](9780857895615_18_pt3.html#pt3)\n\n8 Mrs Schreiber's Big Red Bag\n\n9 Pushing and Panting and Pinching their Way\n\n10 The Gourd-shaped Bottle Gourd-shaped\n\n[**Pride, Passion and Loss: Collecting for Love**  \nJOSEPH MAYER](9780857895615_22_pt4.html#pt4)\n\n11 Waiting for the Rain to Stop\n\n12 Mummies, Crocodiles and Shoes for a Queen\n\n13 The Treasures of the North\n\n14 A Larger World\n\n[**Fashion, Fine Dining and Forgeries: Dealing in Society**  \nMURRAY MARKS](9780857895615_27_pt5.html#pt5)\n\n15 Rossetti's Peacock\n\n16 A Notorious Squabble\n\n17 The Fake _Flora_\n\n[**Collecting the Empire: In Pursuit of the Exotic**  \nSTEPHEN WOOTTON BUSHELL](9780857895615_31_pt6.html#pt6)\n\n18 The Route to Peking\n\n19 The Promise of the East\n\n20 Collecting Without Boundaries\n\n**Epilogue**\n\n**Acknowledgments**\n\n**Notes**\n\n**Select Bibliography**\n\n**Index**\n\n## _List of Illustrations_\n\n1. The South Court at the South Kensington Museum. _Illustrated London News_ (6 December 1862). \u00a9 Illustrated London News Ltd\/Mary Evans.\n\n2. Portrait of John Charles Robinson by J. J. Napier. \u00a9 National Portrait Gallery, London.\n\n3. Robinson's collection at Newton Manor. \u00a9 Ashmolean Museum, University of Oxford.\n\n4. Portrait of Charlotte Schreiber by George Frederic Watts. From Charlotte Schreiber's _Journals: confidences of a collector of ceramics and antiques throughout Britain, France, Holland, Belgium, Spain, Portugal, Turkey, Austria and Germany from the year 1869\u20131885_ (1911).\n\n5. Portrait of Charles Schreiber by George Frederic Watts. From Charlotte Schreiber's _Journals: confidences of a collector of ceramics and antiques throughout Britain, France, Holland, Belgium, Spain, Portugal, Turkey, Austria and Germany from the year 1869\u20131885_ (1911).\n\n6. Gourd-shaped bottle. Courtesy of Sotheby's Picture Library.\n\n7. Portrait of Dante Gabriel Rossetti and Theodore Watts-Dunton by Henry Treffry Dunn. \u00a9 National Portrait Gallery, London.\n\n8. The Peacock Room. Freer Gallery of Art, Smithsonian Institution, Washington, D.C. Gift of Charles Lang Freer, F1904.61.\n\n9. Murray Marks. From George Charles Williamson's _Murray Marks and his Friends: A Tribute of Regard_ (1919).\n\n10. Murray Marks' trade card. From George Charles Williamson's _Murray Marks and his Friends: A Tribute of Regard_ (1919).\n\n11. Portrait of Joseph Mayer by William Daniels. Courtesy National Museum Liverpool.\n\n12. The 'Mummy Room' in Mayer's Egyptian Museum. Courtesy of Liverpool Record Office, Liverpool Libraries.\n\n13. Stephen Wootton Bushell. Courtesy of the Peabody Essex Museum, PH118-80.\n\n14. The Bronze Pavilion at the Imperial Summer Palace. Reproduced by permission of Durham University Museums.\n\n15. Satirical engravings by George du Maurier. _Punch Almanack_ (December 1875). Courtesy of the author.\n\n16. Aston Webb's South Kensington complex. Getty Images.\n\n17. The glass palace of the Art Treasures Exhibition, Manchester. _Art-Treasures Examiner: A Pictorial, Critical and Historical Record of the Art-Treasures Exhibition_ (1857). Reproduced by courtesy of the University Librarian and Director, The John Rylands University Library, The University of Manchester.\nMAGPIES, SQUIRRELS _&_ THIEVES\n\n## _Preface_\n\n_T_ oday we are accustomed to the mechanisms that allow collectors to build a collection: auctions and antique dealers, car-boot sales and internet trading sites. We expect to have public collections in town and city museums, even if we rarely visit them. The motivations that drive collectors have frequently been examined by psychologists and psychoanalysts, and this has given us some understanding of why collecting is such a popular activity and how it can become so obsessive. What I want to do with this book is to take a step back, to a period during the nineteenth century when many of the aspects of collecting which we now take for granted were being newly explored, when collectors were emerging into the public eye and when the hunt for objects was at its most inventive and eccentric. The thrills and perils of Victorian collecting will, in some respects, appear very familiar; it was the collectors of the nineteenth century who laid the foundations for later collectors and many of their networks remain. In other ways, however, we will discover outlooks and experiences very different from those which collectors might expect today. Victorian collecting had a character of its own, and it is this robust and intrepid spirit of adventure that I hope to convey.\n\nThere was no single archetypal Victorian collector: individual tastes meant that one collection was very different from another. Some collections were ordered, scholarly or scientific; others were quirky, highly personal narratives. Changes in fashions, attitudes and economic climate also influenced the objects people chose to collect. This book presents the stories of five collectors to give some sense of this diversity and of the ways in which collecting evolved through the nineteenth century. It also uses these individual stories to explore more general issues about collecting. What is a 'collection'? What kind of cultural, social and political factors influence the life of a collection? What drives the collector? What is the relationship between the private collector and the public museum?\n\nEach of my five collectors left lively archives, letters or journals which document their collecting and each has a fascinating story to tell. Between them, they turned their attention to all kinds of art and historical objects. John Charles Robinson was an influential curator at the South Kensington Museum, which would be renamed the Victoria and Albert Museum in 1899, becoming one of the most famous public collections in the world. He collected both for the museum and for himself, indulging a taste for significant, and often expensive, pieces of art and becoming a specialist in the Italian Renaissance. Lady Charlotte Schreiber, the widow of a steel magnate and an intrepid traveller, sought out china and playing cards, fans and glass in showrooms and junkshops across Europe. Murray Marks, a dealer as well as a collector, used his connections in Holland to exploit the fashion for blue-and-white ceramics and his friendship with the Pre-Raphaelites to create elaborate domestic interiors. Liverpool jeweller Joseph Mayer had a taste for Roman remains, Egyptian antiquities, coins, Anglo-Saxon archaeology and quirky objects from history. Stephen Wootton Bushell was sent to China as doctor to the British delegation and ended up becoming a pioneering expert on Chinese art.\n\nThe five individual stories offer a portrait of the collector, from the eccentric and obsessive to the scholarly and the professional. Together, they also show us how art-collecting changed during the Victorian period, moving away from the great pictures and sculpture that had fascinated the wealthy in the eighteenth century towards smaller, more varied decorative objects. Science and natural history collections, which were equally popular and active at this time, could not be covered within the scope of the book, but it is worth bearing in mind that this created yet another growing community of collectors, working alongside \u2013 and occasionally overlapping \u2013 with art collectors. For those whose tastes lay towards stuffed animals, geological specimens or mechanical instruments, there was a correspondingly influential and flourishing network of organizations, museums and collectors to feed their enthusiasm.\n\nNone of my five collectors is well known. Even John Charles Robinson has been little studied; in comparison to his colleague at the South Kensington Museum, Henry Cole, his work has been overlooked. Only some of the objects from their collections were important and valuable; some were just as much a demonstration of individual taste or little more than a frivolous purchase. None of the collections survives intact to be seen now. But each one was significant in its own time; some were even famed. And, taken together, these five histories give an insight into a Victorian phenomenon that went much further than just a handful of extraordinary lives, revealing a passion for collecting that helped shape how we see the nineteenth century and the legacy it has bequeathed to us today.\n\n## _Catching the Collecting Bug_\n### CHAPTER ONE\n\n### _Exhibition Road, London, 1862_\n\n_I_ t was a hot June day in 1862. South Kensington was bustle and dust and noise, as it had been for months. Horses, carriages and omnibuses moved slowly in the packed streets. On Exhibition Road, on a site that would, in twenty years' time, become home to the terracotta columns of the Natural History Museum, two great glass domes shone in the sun. They formed the extravagant centrepiece of the London International Exhibition of Industry and Art, an enormous fair of artworks and manufactures from across the world in the tradition of the 1851 Great Exhibition. Visitors queued at the entrances, eager to see what was on show, crowds inside pushed their way through the glittering displays and Victorian London was, once again, in thrall to the excitement of spectacle\n\nSponsored by the Society for the Encouragement of Arts, Manufactures and Commerce, the new exhibition boasted 28,000 exhibitors from thirty-six countries. The specially erected building, designed by government engineer and architect Captain Francis Fowke, covered 23 acres of land, with two huge wings set aside for large-scale agricultural equipment and machinery, including a cotton mill and maritime engines, and a fa\u00e7ade almost 1,200 feet long, of high arched windows, corner pavilions, columns and flags. At a cost of almost half a million pounds, the exhibition was intended to dazzle and impress even those who had visited the Crystal Palace extravaganza a decade earlier, and the two crystal domes of Fowke's design, each 260 feet high and 160 feet in diameter, were then the largest domes in the world, wider than (although not quite as high as) both St Paul's in London and St Peter's in Rome.\n\nNot everyone liked the building. The popular press seemed to think the vast domes resembled nothing so much as colossal overturned soup bowls; _Building News_ joined other national papers in describing it as 'one of the ugliest public buildings that was ever raised'. But the exhibition inside was very much an attraction. In the six months of its life, between 1 May and 1 November 1862, over 6 million visitors paid between a shilling and a pound, depending on the day, to see new inventions, industrial machinery, home wares, works of art and the occasional splendid folly, such as the huge pillar of gold sent from the Australian gold rush. Not to be put off by a touch of architectural vulgarity, the London crowds flocked to the display galleries where bootmakers rubbed shoulders with baronets; young designers sketched ideas; whiskered manufacturers confided trade secrets; pickpockets, no doubt, flourished, and everyone had a good time.\n\nThe taste for this kind of event was by now well established; most visitors already knew how to negotiate the vast spaces and overwhelming displays. They liked the bewildering array of exhibits, the crowded corridors and the chance to see and be seen. Not only the 1851 Great Exhibition, but other international exhibitions such as the 1855 Paris Exhibition and, closer to home, the 1857 Art Treasures Exhibition in Manchester, had accustomed the Victorians to the idea of huge and eclectic displays, sparkling showcases for the most beautiful, the most efficient and the most innovative. Even those who had never before visited such an exhibition would have read about them in the newspapers, and seen all kinds of prints and photographs of what they had been missing. This new event filled the front pages of the press: the _Illustrated London News_ was just one of the papers to issue a special event supplement, and to use its editorials to update readers with news of the exhibition's progress. Visitors came to the International Exhibition at South Kensington expecting to be amazed and entertained by the practical, the pioneering and the extraordinary.\n\nThe objects brought together under Fowke's clumsy crystal domes included everything from a working forge to sewing machines, from slates and rock salt to the Brazilian 'Star of the South' diamond. The exhibition was a symbol of mid-Victorian aspiration, manufacturing success and consumer confidence. It was a message to the world about the ambition of Britain and its Empire. But to the millions of visitors that pushed through the crowded galleries, it was also, quite simply, a chance to admire and desire beautiful and unusual things. It evoked, on an enormous scale, the Victorian idea of the collection and what it might mean to bring things together, to compare them, to own and value them, to present them in public with pride.\n\nIt was not only the brightly painted and lavishly ornamented arcades of the International Exhibition that hummed with activity. On the other side of Exhibition Road, more objects and more collections were attracting public interest. The new South Kensington Museum, which had been emerging since 1857 in a mish-mash of temporary buildings, was also noisy and crowded. The recently opened permanent galleries of the North and South Courts \u2013 also designed by Francis Fowke \u2013 more than rivalled the glamour and glitter across the road. Here, too, there was a glass roof to let in light and air. Boldly painted walls in deep blood-red and purple grey prepared the eye for corridors packed with elaborate highly coloured arrays of patterned wallpapers, mosaics, friezes and stained glass. Here, too, visitors clustered close together around cases of fascinating and attractive things, preening, laughing and flirting, enjoying the buzz. Voices, and the chink of fine china, drifted from the mock-Tudor refreshment rooms where colourful flocks of women took afternoon tea; in the new display spaces collectors, connoisseurs and the simply curious marvelled at some of the world's finest art objects brought together for the delight of the public.\n\nThis was the 'Special Exhibition of Works of Art of the Medieval, Renaissance, and more Recent Periods, on loan at the South Kensington Museum', a complementary \u2013 or perhaps rival \u2013 attraction to the more manufacturing-based displays across the road. The catalogue played down the ambition of the exhibition, describing it simply as 'fine works of bygone periods', which had been made possible with 'the assistance of noblemen and gentlemen, eminent for their knowledge of art'. In fact, the displays featured some of the rarest and most beautiful objects ever set before the British general public, over 9,000 works of art from over 500 of the country's richest and most influential owners, from the Queen, aristocracy and the Church, to London livery companies, municipal corporations and public schools. These were private treasures, not normally on open display, brought together for the first and only time \u2013 a triumph of negotiation and diplomacy.\n\nThe exhibition covered over 500 years of art production, and included all kinds of media, from painting, ceramic and glass, to metal, ivory and textiles. Objects were organized in the catalogue into forty categories, which more or less corresponded to the layout of the galleries. There was no chronological arrangement to the displays, which were bewildering in their haphazard glamour and sheer abundance. Beginning with sculptures in marble and terracotta, the exhibition moved on to nearly 300 carvings in ivory, bronzes, furniture, Anglo-Saxon metalwork, enamels, jewellery and glass, textiles and illuminated manuscripts. There was a S\u00e8vres porcelain vase in the form of a ship painted with cupids and flowers; a Limoges enamel casket, one of the many enamels on display from the collection of the Duke of Hamilton; a silver-gilt handbell which 'doubtless' came from the chamber of Mary Queen of Scots; and a modest pair of Plymouth porcelain salt cellars, in the form of shells, belonging to the Right Honourable William Gladstone MP.\n\nContributors had loaned not only individual objects but sometimes entire collections, often with ancient family roots. This was particularly true in the case of bookbindings, portrait miniatures, jewellery and cameos, the popular mainstays of country-house collections: 'The Devonshire Gems', for example, assembled in the eighteenth century by William Cavendish, 3rd Duke of Devonshire, was displayed in its entirety, apart from eighty-eight of the cameos which had already been committed to the International Exhibition. So extensive and unique was the collection, and so difficult to assess in the short period of time that had been given over to organizing the exhibition, that the catalogue was forced to confine itself to listing just a few 'of the finest gems'.\n\nThe show was designed to delight and amaze, without reference to the grind of manufacture and industrial production that was the backbone of the displays at the International Exhibition. Its emphasis was on the romantic and the historic, and it focused public attention on the beauty of medieval and Renaissance objects for the first time, introducing unknown styles and techniques. The press marvelled at a display that was 'unequalled in the world' and 'rich beyond all compute and precedent', and the chance to see rare works of art drew visitors from across the country. But this was not just a splendid art exhibition: with objects from over 500 collectors, each of whom was named on labels and in the catalogue, the resplendent galleries were specifically designed to bear witness to the richness and variety of British collecting. The packed cases clearly showed how deeply embedded was the idea of the collection, both in national institutions like the monarchy and in the private lives of many of the country's wealthiest and most influential men and women. And over a million visitors \u2013 more than a third of the population of London \u2013 came to the new museum to see the displays, suggesting how great was the public appetite for the beautiful and quirky, and how widespread the curiosity about, and commitment to, collecting.\n\nMaking all this possible was one man \u2013 John Charles Robinson. Robinson was a collector himself, a connoisseur, but he was also Librarian-Curator at the South Kensington Museum where he had responsibility for buying objects, researching the collections and, as the exhibition triumphantly demonstrated, creating public displays. Crucial to all of this activity, for Robinson, was nurturing a community of collectors, creating an intimate mutual relationship between individuals and the museum that would benefit everyone. And at the heart of this community, he put himself. He knew all the most active and influential collectors, dealers and scholars; he made some of the most glamorous and talked-about transactions. It was Robinson's influential network of contacts that turned the Special Exhibition from a modest display of private loans to the stunning art show that greeted the South Kensington visitors.\n\nRobinson was a founder member of the Fine Arts Club, a London-based club of wealthy collectors. In 1858, he wrote to his superiors at South Kensington to suggest that he and his collecting colleagues should organize a small and select display of historical art, as an adjunct to the forthcoming International Exhibition. It would provide something extra, he suggested, for those whose tastes inclined towards older and more refined pieces than would be included in the trade-based displays at the International Exhibition; it would recognize that the public's interest in all kinds of art had been stimulated by events like the Manchester Art Treasures Exhibition in the previous year, when over a million visitors, from royalty to cotton-mill workers, had clamoured to see everything from sculpture and armour to photographic montages and Limoges enamels. Permission for the loan show was given, but there were plenty of sceptics who were quick to dismiss the proposal, and members of the press warned that Robinson's enterprise would be 'overshadowed by its imposing and all-absorbing neighbour, and. . . be recognized only by the connoisseur'.\n\nRobinson, however, had other ideas. Through the Fine Arts Club, he formed a committee of seventy of the nation's most influential collectors, including the aristocracy and clergy, eight Members of Parliament and Sir Charles Eastlake, President of the Royal Academy and Director of the National Gallery. Between them, these men knew almost every collector in the land. Robinson's exhibition became a rallying point, a prestigious and glamorous event that celebrated collecting. 'Private cabinets were thrown open with an alacrity which showed that the only offence to be feared was not rifling them enough. . .' marvelled the _Quarterly Review_. 'The great floodgate was opened.' Over the coming months, more and more treasures were pledged to the exhibition until the organizers were swamped by both the sheer volume of objects they were being offered and the practical demands of getting everything safely to the London galleries.\n\nWhen the exhibition finally opened its doors, it was something of a triumph, a tribute to Robinson's unrivalled network and what the international press called his 'incessant activity'. Probably no one else could have made it happen. It clearly showed the value of bringing works of art together for comparison, and it began to inspire a new taste for the medieval and the Renaissance, then obscure periods that had received little critical attention. Robinson's scholarly catalogue, too, was recognized by specialists as 'a work of the highest importance'. But perhaps more than anything else, the exhibition signalled to the world the extent of Victorian collecting. For the first time, it offered a glimpse of just how many collectors there might be. It gave a sense of the shared and irrepressible enthusiasm that brought together so many of the country's men and women, and it revealed the range and magnificence of the objects they collected. It brought into the open a fascination with things that was shaping people's homes, forging municipal and national identities, and putting the collector at the heart of an increasingly commodified world.\n\n### CHAPTER TWO\n\n### _The Useful and the Beautiful_\n\n_I_ n all kinds of ways, the buzz of activity at South Kensington during 1862 was a legacy of the enormously successful 1851 Great Exhibition, a monumental enterprise that had as much influence on political, economic and social life as it did on art, science and technology. Like the 1862 International Exhibition, the earlier event was organized by the Society for the Encouragement of Arts, Manufactures and Commerce, which had been established in 1754 to support activities as diverse as spinning, carpet manufacture, tree-planting and painting. It soon became known informally as the Society of Arts (finally to become the Royal Society of Arts in 1908) and by the 1760s it was holding regular exhibitions, both of contemporary art and industrial innovations. But these were relatively modest affairs and there was little British precedent for such a huge and ambitious undertaking as the Great Exhibition. What is now commonly recognized as a landmark of Victorian achievement was a courageous experiment, and almost didn't take place at all.\n\nThe exhibition was intended to restore the fortunes of British trade and manufacturing, by showing off British endeavour, bringing together the best of the world's products as an inspiration to business and providing models for the future. A 'Select Committee on Arts and their connection with Manufactures' reported to the government as early as 1835 that both manufacturers and the public needed educating in matters of art and design. Pandering to a public appetite for novelty, and with an eye on keeping production costs to a minimum, British design standards were falling. Skills were dwindling and foreign competitors were flourishing. In the shops and department stores increasingly powerful middle-class consumers were flexing their spending muscles in unexpected directions, choosing goods from rival countries \u2013 tableware and furniture from France, Italian ornaments, and even glassware or jewellery imported from America. They were displaying what the government regarded as dubious taste; turning up their noses at traditional British wares and instead filling their drawing rooms with showy conversation-pieces that more obviously displayed their wealth and status. In the face of such competition, British manufacturers were suffering and the economy was faltering. What was needed was a showcase where examples of the best design could be displayed; where diligent manufacturers and aspiring young designers could come and study; where they could learn new skills and be inspired by new patterns.\n\nIn the course of the planning process leading up to 1851, however, and in response to the need to raise money and support for the project, the organizers attempted to head off public criticism by accommodating, wherever they could, all kinds of aspirations for the event. The apparently simple original plan \u2013 to raise the standards of British industrial design \u2013 was reshaped to take account of a range of other preoccupations and priorities. Local committees were established to help source objects and raise awareness and funds, but these often had their own agendas, from Chartism to women's rights. While on a national level it became clear that, to survive, the exhibition would need to reflect, among other things, the enthusiasm of the Prime Minister, Lord John Russell, for celebrating commercial liberalism and free trade; the desire of the influential East India Company to show off the Empire's raw materials; the Church's concern with illustrating divine benevolence; and the conviction of the liberal middle classes that the show should assert the British political and social model to the rest of the world. Not surprisingly, these pressures took their toll. The original intentions faded in the face of so many demands and the clear purpose of the exhibition faltered. The final display, while spectacular, was confused and often discordant, and the 'meaning' of the exhibition was further addled by the enormous variety of interpretations offered by the press and visiting public. No one could quite agree, it seemed, exactly what they were looking at.\n\nYet, despite these problems, the idea of the exhibition caught the popular imagination, both at home and abroad. Enough of the original focus on manufacture and trade remained to inspire generations of industrial spectacles across the world over the course of the century, while in the immediate aftermath it spurred the French into organizing a World Exhibition of their own, with an understandable emphasis on restoring French manufacturing and design to pre-eminence. The French, in fact, had already hosted an Industrial Exhibition in 1844 and had a tradition of official exhibitions of art and industry stretching back to the late eighteenth century. The London show, however, had been on a different scale, forcing organizers across the English Channel to redouble their efforts in response. The subsequent 1855 Paris Exhibition was even larger and more elaborate than the Great Exhibition had been, although it was significantly less successful financially (leaving a crippling shortfall of over 8 million francs) and visitor numbers were comparatively low. It succeeded in inflaming old Gallo-British rivalries even further, however, and driving home the idea that British design was still lagging behind its European counterpart. In London, there seemed nothing for it but to stage yet another event, bigger and better than any that had preceded it, demonstrating beyond doubt the technological progress that the British were making across the world. So work began on the 1862 International Exhibition and, despite financial setbacks, conflict in Italy and America which threatened to derail foreign loans, and the untimely death of Prince Albert in December 1861, just months before the scheduled opening, the show managed, in scale at least, to eclipse all others.\n\nBut the International Exhibition was not the only legacy of 1851 at South Kensington. Surplus funds from the Great Exhibition (around \u00a3180,000) were set aside to improve standards of science and art education, and parcels of land were bought around the district for a number of educational and cultural projects. One of these was the new museum, which had been in a temporary home at Marlborough House on Pall Mall, and which also inherited large numbers of objects from the Great Exhibition. Just as significantly, the museum's first Director, Henry Cole, had been a key organizer of both the 1851 and 1862 extravaganzas, and brought with him a commitment to the original Great Exhibition principles of improving standards in manufacturing, raising the quality of industrial design and educating public taste through the display of objects. Perhaps the greatest legacy of the 1851 exhibition, however, was the least tangible one: a growing sense that collecting and display was a serious, public affair. Coverage in the press, political debate and drawing-room gossip began to discuss, appraise and compare works of art, and in the wake of this enormous attention there was an increasing appreciation that objects could have profound personal and social meanings.\n\nThe Great Exhibition offered the most visible, spectacular evidence of a fascination with collecting that had been growing since the beginning of the century. It showed just how many issues might be inherent in the apparently simple idea of making a collection. And, as a result of this, there was increasingly open and varied debate about what kind of role art might have in public life, about the purpose of design, and about thorny questions of taste, patronage, ownership and education. This was a period when even many of the defining terms we take for granted today were new and the distinctions between them ambiguous: what, exactly, was a 'public' collection? What was a work of 'decorative' or 'applied' art? How could you define the difference between an amateur, a connoisseur and a professional? What was a curator's role?\n\nMany of these questions would take decades to resolve; some are still being debated today. But what was immediately clear was that there was more interest than ever in what collecting might mean, and how collections might best be achieved. By the middle of the century, debates that had been rumbling for years, and which were implicit in the collecting activities of Robinson's influential colleagues and the wealthy members of the Fine Arts Club, were being aired ever more frequently in public. It was increasingly evident that collecting could not be seen as a private interest detached from events going on around it; it was part of a wider conversation inextricably linked to political ideology and social change. Robinson's 1862 loan exhibition came, in many ways, at the perfect time. It made collecting visible, topical and desirable at the moment when the discussions inspired by the Great Exhibition were reaching maturity.\n\nAppointing Henry Cole as Director of the South Kensington Museum signalled the government's desire to continue with the work they had begun at the Great Exhibition. Intimately involved with both the 1851 event and the 1862 International Exhibition, and publicly espousing the values behind them, Cole was an enthusiastic advocate for the campaign to improve standards and educate public taste. He would use the objects on display at the museum to show people what they could and could not like; what they did and did not want to buy. He would change habits and restore manufacturing fortunes. He would, he boasted, 'make the public hunger after the objects. . . then they [will] go to the shops and say, \"We do not like this or that; we have seen something prettier at the Museum\"; and the shop-keeper, who knows his own interest, repeats that to the manufacturer and the manufacturer, instigated by that demand, produces the article.'\n\nEven by Victorian standards, Cole was fiercely energetic, an apparently unstoppable force of opinions and activity. By the end of his life, in 1882, he was an institution, known as 'Old King Cole', but both his stubborn, temperamental approach and his reforming zeal tended to arouse admiration and antipathy in equal measure: 'His action has often been harshly criticized,' began his obituary in _The Times_. 'His untiring energy and perseverance have frequently made him enemies.' He was a career civil servant; having joined the Public Record Office in 1823 at the age of fifteen, he was the author of a series of pamphlets advocating reform of the public record system. He wrote for a range of newspapers and journals and in 1840 won a prize for developing a workable Penny Postage plan. By the time he turned his attention to art and manufacturing, around 1845, he was already an experienced and confident administrator. He immediately began to assist the Society of Arts on a series of exhibitions to stimulate industry and invention, finally securing his place, according to his obituary, as 'the leading spirit and prime mover' of the reform impulses which led to the Great Exhibition.\n\nFollowing this success, Cole was invited in 1852 to reorganize the government Schools of Design. These had been established in twenty cities from 1837, one of the earliest indications of the government's commitment to design reform. Created by the Board of Trade, they were supposed to promote an establishment consensus on art and good taste, improving education and training new designers in approved styles and techniques. In practice, however, many of the Schools had concentrated as much on academic art as on industrial design and during the 1840s a series of internal battles raged over exactly what should be taught. Cole aimed to give the system a new sense of direction and control, and put the focus firmly back on to 'useful' skills. As the first step, he put the Schools under the control of a new Department of Practical Art, soon renamed the Department of Science and Art, a subdivision of the Board of Trade. He appointed himself Superintendent, and made clear his intention to direct things with a firm hand: the new department was to be managed from South Kensington, with his prot\u00e9g\u00e9s slotted into key roles.\n\nThe new museum was part of the same package. It was seen as another weapon in the battle for design reform, working alongside the Schools but with a more public emphasis. It fell under the same umbrella at the Department of Science and Art, which meant there was no independent board of Trustees to distract or hinder Cole. It could build on the impetus that had been created by the two industrial exhibitions, giving a more permanent face to government policy and taking its place without restrictions as the flagship of a wider movement for improving practical education nationwide. The galleries were intended to be just as instructional as any school, driving home messages about art and its role in national life. Cole glowed with the righteousness of educational principles: in a rallying speech, he laid out his aim to 'woo the ignorant' and ensure objects at the museum were 'talked about and lectured upon' until no one could be left in any doubt about what constituted good taste. No longer would the manufacturing and shopping public make bad choices: they had someone to guide them. The South Kensington Museum would be useful, practical and modern, displaying things that could be copied, redesigned and mass-produced, providing clear information about process and construction, and lucid explanations of design principles. It could show reproductions, if that's what it took to make a point. It would show art with a purpose; art which could, with Victorian skill and confidence, be made to shape the future.\n\nAs the museum's leading curator, Robinson worked closely with Cole in acquiring objects and arranging them for display. He supported the general principles of education at the museum but he did not fully share Cole's views about the importance of manufacture and the best ways to instruct and influence the public. His real interest was in beautiful and historical things, and he was convinced that he knew better than Henry Cole how to find, treasure and display them. His relationship with his employer was edgy and barbed, liable to erupt in furious explosions. Both men were obstinate, confident and opinionated, and their ambitions collided too frequently for comfortable coexistence. Robinson was not above claiming, in public, that Cole did not know very much about art; Cole struggled to assert his authority over his excitable curator. Inevitably, in time, the friction between them would rub bare uncontainable feelings of anger, disappointment and betrayal. Within a few years, working relations would sour beyond the point of no return. But for now Robinson bided his time. It was with the two contrasting displays either side of Exhibition Road that he made his case.\n\nIn many ways, the two men and their exhibitions represented opposing sides in the debate about the value of art that had been raging for over a century, but into which the Victorians had entered with particular gusto. Did art need to have a function or purpose, or could it exist for its own sake? How did one attach value to an object \u2013 because it was useful or because it was beautiful? Did art have to be linked to social and moral responsibility, or was it autonomous, following its own rules? Such questions had been exercising educated minds since Edmund Burke's influential _Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful_ , published in 1757, and the emergence of the Romantic movement's preoccupation with personal creativity and emotional and artistic truth at the end of the eighteenth century. Proposals to create a national gallery of art at the beginning of the nineteenth century had reinvigorated the debate, in and out of Parliament, setting the case for historic art against modern examples, and looking at ways of organizing displays to best make them meaningful. An exhibition of 'Ancient and Medieval Art' held by the Society of Arts in London in 1850 and the Art Treasures Exhibition in Manchester seven years later both linked the viewing of works of art to a useful purpose: 'To the Artist and Manufacturer, an opportunity is afforded of comparing the handiwork of ancient times with the productions of our own skill and ingenuity,' explained the 1850 catalogue, 'to the Amateur, means are supplied of correcting the taste and refining the judgement'.\n\nBy the middle of the nineteenth century, the discussion was by no means confined to the intellectual elite. Most of those in the public eye, as well as those writing in the popular press, had an opinion. Nor was this a distinctly British trend: across Europe, museums, galleries and governments were discussing what kind of art to collect, and how to show it. Many successful mid-century Victorians considered the fine arts to be some kind of adjunct to the real business of mechanical invention and production, where value was attached to accuracy and originality. Others maintained that art could not be conflated with commerce and that its value was as much emotional as practical. Writers like Tennyson, the poet laureate, and painters like William Holman Hunt preferred their art with a moral message; advocates of the Arts and Crafts movement emphasized the utility and social community of craftsmanship; on the other side, critics like Walter Pater argued vehemently that formal, aesthetic qualities should take precedence over moral or narrative content. In 1857, the critic and writer John Ruskin launched a scathing attack on the South Kensington Museum because, in his opinion, 'fragments of really true and precious art are buried and polluted amidst a mass of loathsome modern mechanisms, fineries and fatuity'; ten years later, Matthew Arnold's influential series of articles, 'Culture and Anarchy', appeared in the _Cornhill Magazine_ , exploring 'what culture really is, what good it can do, what is our own special need of it' and discussing the role of art in a rapidly changing world. It was not a debate that was going to be resolved in the galleries of the South Kensington Museum: at the end of the nineteenth century, it was still a matter of intense argument as the Aesthetic movement and Oscar Wilde made an ostentatious case for 'Art for Art's Sake'. But many of the museum visitors would have been well versed in the discussion, and would have recognized in Robinson's loan exhibition an unexpected contrast to Henry Cole's usual principles of display. Robinson's emphasis on beauty and on collecting for pleasure set the display well apart from the paean to commercial progress that was Henry Cole's International Exhibition.\n\nRobinson was clear on the point: for all Cole's hectoring, what visitors really wanted to see \u2013 what a museum was about \u2013 was the exhibition of the most exquisite and refined objects. 'What', he asked an audience at one of his lectures, 'is there to trust to but the silent refining influence of the monuments of Art themselves?' Not lectures and labels, not modern mass-produced pieces straight from the production line, but the 'monuments' most able to assert their 'silent refining influence' \u2013 the medieval silverware, the exotic oriental screens, the jewelled Renaissance chalices. Robinson was a connoisseur: his enthusiasm for collecting went hand-in-hand with an appreciation of the beautiful. And, as the Victorians continued to debate the ways in which art might influence their lives, more and more people were making their allegiances evident by also choosing to collect, making space for objects in their homes because they liked them there and because the desire for lovely things seemed irrepressible.\n\n### CHAPTER THREE\n\n### _A Public Duty and a Private Preoccupation_\n\n_T_ he mid-Victorian period saw a boom in collecting. Explorers and adventurers to new and distant lands brought back plants, seeds and shells, pinned butterflies, bright beetles and stuffed animals to fill the gardens and display cases of the natural history collector. Enthusiasts dug for fossils, dinosaur bones and geological specimens, while archaeologists unearthed beads, weapons and brooches to fill shelves and cabinets. Art collecting moved away from the country homes of the rich into the fashionable townhouses of the increasingly confident and wealthy middle classes, and in turn attention moved from Old Master paintings and sculptures to more domestic works in materials such as ceramic or glass. A trend for highly decorated parlours, stuffed with all kinds of pictures, statues and objects, became endemic: showing off a collection of curios sent out messages about social status, and helped the owner appear educated and cultured.\n\nCollecting had long been part of the upper-class way of British life. Most of the country's leading families boasted a collection of art and sculpture, jewellery or elite porcelain such as Meissen or S\u00e8vres. The tradition of the Grand Tour had been flourishing since the middle of the seventeenth century, serving not only as a cultural education and a rite of passage but also as an opportunity to add to the family's collection by buying up works of art to ship home. Collecting was considered a fit occupation for a gentleman, drawing on the habits of European royalty and the erudite tradition of the _Wunderkammer_ , or cabinet of curiosities, which had emerged among the wealthy during the Renaissance. The custom for collecting was inextricably linked to power, prestige and riches: during the sixteenth century, the Holy Roman Emperor Rudolph II, for example, was an obsessive collector of paintings, and the seventeenth-century Archduke Leopold Wilhelm of Austria employed the Flemish painter David Teniers the Younger to care for a hugely expensive collection which boasted everything from works by Italian masters to Cromwellian souvenirs of the English Civil War. The fortunes of the royal and aristocratic houses of Europe were clearly reflected in the ebb and flow of their collections, which were regularly sold, moved or broken up as the result of marriage and war, gambling debts and revolution. The Orl\u00e9ans Collection of over 500 paintings was assembled at the beginning of the eighteenth century by the French prince Philippe II, Duke of Orl\u00e9ans and nephew of Louis XIV. Its core, however, already dated back to the 1600s when Swedish troops sacked Munich and Prague during the Thirty Years War, looting masses of objects including the Habsburg collections of Rudolph II and Emperor Charles V from Prague Castle. These were added to by Queen Christina of Sweden, with a particular focus on Renaissance and Roman art, before being sold off at her death in 1689 to cover her debts. When he bought the collection, Philippe II added to it extensively, investing in a range of prestigious works from all over Europe including many of the key pieces accumulated by Charles I of England, another avid collector. The collection was celebrated as one of the gems of European culture, housed in the magnificent Palais Royal in Paris, and asserting the power and splendour of the French monarchy.\n\nWhen Philippe died in 1723, his pious son Louis inherited the collection. But his tastes were not the same as his father's, and he was alarmed by the subject matter of some of the paintings. In a fit of religious zeal, he attacked several of the most famous works with a knife, slashing Correggio's _Leda and the Swan_ which depicted an erotic scene from Greek mythology, and ordering three more of Correggio's works to be cut up in the presence of his personal chaplain. The collection survived, but neither Louis nor his son, Louis Philippe, had the will or money to be active in their collecting, and at the end of the eighteenth century the pieces were put up for sale in a desperate attempt to pay off enormous debts. It was the complex political pressures and the financial uncertainty of the French Revolution which finally decided the fate of the collection, however. Extravagant displays of personal wealth and royal power were out of fashion, a liability, and in 1793, just as Louis Philippe was arrested at the height of the Reign of Terror, almost 200 paintings were shipped over to London's salerooms. In the months prior to his execution at the guillotine in November, negotiations were finalized for selling the rest of the collection, mainly Italian and French paintings, to a Brussels banker.\n\nDuring the mid-seventeenth century and in the period following the French Revolution, political upheaval and changes in European social order saw many collections from the Continent, such as the Orl\u00e9ans collection, dispersed, relocated or re-formed. By contrast, in England the nineteenth century started with considerable security among the landed classes, with the aristocracy generally thriving and their collections benefiting from the distress of their European counterparts. But this gradually changed. In August 1848, _The Times_ reported on the sale of the collection at Stowe, the home of the 2nd Duke of Buckingham and the first of numerous country-house sales that would have been unimaginable a couple of decades earlier. It was, the report explained, 'a spectacle of painfully interesting and gravely historical import . . . an ancient family ruined, their palace marked for destruction, and its contents scattered to the four winds of heaven'. The auction was the first symptom of a lingering malaise among country estates that was to last throughout the century, and to bring more and more art treasures on to the open market. A changing economic climate favoured industry rather than land as a way of making money, and old families feeling the pinch often turned to the sale of at least part of their collections as a way of boosting funds.\n\nIn 1882, the Settled Land Act brought things to a head. The old system of entailment had prevented the sale of land and collections, keeping them intact. Property passed from generation to generation through the eldest son (or, in the event of there being no son, via another male relative). Under the rights of primogeniture, he held a lifetime's interest in the land, house and contents: he was not free to sell or divide anything. With the new Act, this age-old arrangement was overturned. Restrictions on the sale of property were lifted and the tenant for life was given much stronger powers for dealing with his inheritance. Suddenly, family collections could be viewed as marketable commodities, and so converted into cash \u2013 and treasures from the most venerable and celebrated collections soon came under the hammer. In 1882, immediately after the Act was passed, the Duke of Hamilton sold most of the contents of his South Lanarkshire palace for almost \u00a3400,000 (the equivalent today of over \u00a330 million); two years later, at Blenheim Palace, the Duke of Marlborough was given leave to dispose of some of his family's collections in a series of sales. The 18,000 volumes of 'The Sunderland Library' raised \u00a360,000, and over the following years a collection of enamels and some of the palace's major works of art, including Raphael's _Madonna degli Ansidei_ , were also sold.\n\nThe collectors who took advantage of these misfortunes were often members of the newly influential middle classes. After a series of complicated financial manoeuvres, much of the Orl\u00e9ans collection finally found a home in Britain when it was bought in 1798 by a consortium consisting of Francis Egerton, 3rd Duke of Bridgewater, a coal and canal magnate; his nephew Earl Gower; and the Duke of Carlisle. But, in a reflection of changing fortunes and tastes, it was to be broken up yet further. Only 94 of the 305 paintings were retained for the family galleries. The rest were sold on again, raising over \u00a342,000 to cover expenses, finding their way into the hands of a range of buyers that included four painters, four MPs, six dealers, two bankers and six gentleman amateurs. While the sale of the collection into the hands of the Duke of Bridgewater seemed at first to reinforce the impression of ownership by the aristocracy, it was the Duke's new wealth from manufacturing and trade, rather than ancient land rights, that made the deal possible. And the sales which followed showed even more clearly how the finest objects were moving out of the hands of the European elite into the homes of the middle classes \u2013 bankers and MPs, as well as those dealers who were making a profitable profession out of what had once been a royal diversion. Collections that had been as much a display of power and influence as cultural appreciation were being broken up and abridged. The very nature of collecting itself was beginning to be transformed.\n\nOn the death of the Duke of Bridgewater in 1803, his collection was put on semi-public exhibition in Westminster, open on Wednesday afternoons to those who wrote requesting a ticket, or to artists recommended by the Royal Academy. This was a further demonstration that the idea of the inaccessible, aristocratic treasure house was beginning to disappear. Increasingly, private collections were getting a public airing, some being bequeathed to national or regional galleries but many more occupying a grey area by allowing restricted access to a vetted group of visitors. It was a long way from full public access: the Bridgewater collection was exhibited grandly in Cleveland House with a staff of twelve luxuriously liveried attendants, giving an impression not unlike a private palace. But it offered the aspiring middle classes, in particular, new opportunities for viewing and it was an indication of changing perceptions of the role of collecting and collectors. Collecting was no longer the preserve of the very rich; objects that had once been in the collections of royalty were now finding their way into much more modest houses and on to sightseeing itineraries. The changing face of European society was being reflected in increasingly widespread opportunities to collect.\n\nPaintings and sculpture, however, still demanded plenty of wall and floor space, and investing in Old Masters required a particular kind of education. In addition, around the middle of the century, there was a dramatic rise in the prices of the kinds of works collectors had traditionally sought. New directions had to be explored. More and more, the emerging middle-class collectors took an interest in different kinds of pieces, particularly those they could pick up at reasonably limited expense, such as old English glass or pewter, decorations for the home \u2013 including mirrors, vases and textiles \u2013 and objects which might prove to be a shrewd speculation, like Staffordshire ware or silver. Although one or two earlier collectors had been interested in these areas \u2013 most notably, perhaps, the eighteenth-century historian and politician Horace Walpole \u2013 they had, until now, been largely neglected. By the middle of the century, however, this was changing and several large-scale, highly public sales helped cement the idea that collecting had moved away from its traditional preoccupation with expensive masterpieces and was becoming accessible, educational and profitable. In particular, in 1855, Christie's in London undertook the sale of the collection of Ralph Bernal, who had died a year earlier. Bernal epitomized the new type of collecting. He was educated and cultured, but as a barrister and MP he was firmly middle class. His apparently mystical ability to hunt out treasure from ordinary bric-a-brac shops earned him the epithet 'lynx-eyed' in _The Connoisseur_ , an illustrated magazine. And the \u00a320,000 he had invested in his collection was turned into nearly \u00a371,000 at the Christie's sale which lasted thirty-two days and was an extravaganza of the metalwork, ceramics, glass and miniatures that were beginning to fascinate collectors.\n\nThe impulses that were evident in Britain were repeated across Europe as the political and social fabric was rethreaded. Shops and dealers sprang up in major urban centres to cater for the increasing number of collectors seeking out treasure. They were not just the aristocratic sons of the ruling European classes, professional artists or dilettantes; they were merchants and bankers, clergy, military officers, wives and mothers. There was a revolution in attitudes, suggesting everything was within reach, for more or less everybody. What made English collectors unique, however, was their taste for showing things off in their homes: 'They like to _live_ surrounded by their pictures and antiquities,' marvelled Gustav Waagen, art historian and director of the Kaiser-Friedrich Museum (later the Bode Museum) in Berlin, something that was 'virtually unknown elsewhere in Europe'.\n\nThis intimate relationship between a collector and his things intrigued the public. As more and more homes began to display the spoils of private collecting, so the press began to carry detailed coverage of major sales and to record the growing numbers of people seduced by the idea of the collection. After the Bernal auction, _Punch_ noted that the country had been gripped by 'Collection Mania', an 'enormous quantity of money' changing hands and stimulating 'the ambition of great numbers of \"Collectors\" all over the country'. In 1856, Henry Cole came to the conclusion that 'the taste for collecting' was now 'almost universal', and, by the end of the 1860s, _The Graphic_ , an illustrated political journal, was able to go a step further by claiming that 'this is the collecting age'. During the 1870s, _Punch_ embellished the simple reporting of sales, featuring a number of cartoons of dishevelled and obsessive collectors which developed the idea of 'mania'. From enthusiastic shoppers to expert connoisseurs, the collecting habit was becoming entrenched in the homes of the Victorian middle classes.\n\nCollecting was by no means confined to the home, however. Alongside the growing number of private collectors was a developing habit of public collecting. The new galleries at South Kensington were part of an emerging network of museums across the country. Inextricably associated with education and enlightenment, with order and stability and self-improvement, these offered a perfect outlet for both civic pride and visible philanthropy. Many wealthy and influential Victorians saw it as nothing less than a public duty to provide spacious, airy galleries where people could learn about art and science, and to construct imposing and stately buildings that reflected the confidence and affluence of their communities. Public collecting was active and fashionable.\n\nEstablishing a museum sent a clear message about the aspirations of a town, and the apparent generosity and farsightedness of its leading inhabitants. Those with an eye for boosting local dignity and their own reputations found museums an effective way of securing positive publicity and public support: in Sunderland, the Museum and Library, opened in 1879 by US President Ulysses S. Grant, was largely the work of Robert Cameron, a Member of Parliament, Justice of the Peace and temperance campaigner; in Exeter, it was two MPs, Sir Stafford Northcote of Pynes and Richard Sommers Gard, who proposed and developed the plans for the city's Albert Memorial Museum. All over the country, those with an interest in municipal matters and affairs of state began to look at ways of developing and displaying public collections that would entertain and inform, and before long a web of museums started to grow across industrial towns and expanding cities, offering more and more people the opportunity to see precious and fascinating things.\n\nDespite their sudden high profile, the emerging museums did not appear completely out of the blue. Many were an extension of local associations such as Mechanics' Institutes and Literary and Philosophical Societies, established early in the century to encourage and facilitate the research, discussion and display of disciplines as diverse as geology, anatomy and phrenology, shipbuilding, mine engineering, painting and foreign languages. Mechanics' Institutes varied according to local traditions but, although they ostensibly catered for the working classes, many soon became a means for aspirational white-collar workers to advance themselves while also acting as meeting places for the middle class: the _Manchester Guardian_ claimed in 1849 that of the thirty-two Mechanics' Institutes in Lancashire and Cheshire only four had any working-class support. One of the results of this membership profile was an increasing number of exhibitions. No longer confined to providing practical instruction for workers, a number of committees turned their attention to the arts. In 1839, Leeds Mechanics' Institute organized an exhibition of 'arts and manufactures', while in 1848 Huddersfield described its effort as 'a polytechnic exhibition on a small scale . . . of pictures and other works of Art and objects of curiosity and interest'. Perhaps the most actively enthusiastic of the Institutes was Manchester's, which organized five exhibitions, the first from December 1837 to February 1838, and the last finishing in the spring of 1845. These eclectic displays included everything from models of steam engines, ships and public buildings, to paintings, phrenology and geological specimens but the first exhibition was typical in also featuring over 400 examples 'of beautiful manufactures and of Superior workmanship in the Arts'.\n\nBy the 1830s and 1840s, most industrial centres, many smaller towns and even some rural villages had a Mechanics' Institute, generally offering a range of lectures, exhibitions, libraries and specialist classes. The network flourished so rapidly that by 1850 there were over 700 Institutes, boasting more than half a million members. Alongside them, many larger cities also had established Literary and Philosophical Societies, often with roots in the eighteenth century. Combining elements of the academic society and the gentleman's club, they mainly drew their members from among the educated and the influential. But they were not divorced from practical progress and by the 1830s many had evolved into hybrids representing utilitarian, even industrial, interests, while retaining an air of genteel debate. They produced further opportunities for provincial displays, and some even formed collections, often based on a core of natural history objects, with rooms set aside to show everything from prints and paintings to stuffed birds: Newcastle Lit and Phil, for example, opened a special 'Museum Room' in 1826 to display the collection of books, manuscripts, prints and natural history specimens which had originally formed the private museum of Marmaduke Tunstall at his home at Wycliffe Hall near Bernard Castle in County Durham, and which the Society had purchased in 1822.\n\nThe newly developing museums fitted comfortably into this tradition of public learning and debate and increasingly took the next steps in creating and caring for collections. They were part of a general movement towards improving education in general, and art education in particular, with a focus on allowing ordinary people access to extraordinary things. In 1835, when he was asked by the Government Select Committee how to improve the knowledge of art among the general public, Gustav Waagen replied that better public understanding depended on 'accessible collections. . . giving people the opportunity of seeing the most beautiful objects'. By the mid-century, John Ruskin was again making a public case for places where visitors could experience the finest works of art. 'Art', he asserted in his 1859 lecture 'The Two Paths', 'is always helpful and beneficent to mankind, full of comfort, strength and salvation' and his writing indicated a growing and ambitious realization that public access to art could be linked to an improved understanding of everything from social structures to economics, as well as engendering better skills, work habits, personal conduct and even morals. 'A museum,' Ruskin claimed, 'is. . . primarily a place of education. . . teaching people what they do not know. . . teaching them to behave as they do not behave.'\n\nIn 1845, a Museums Act was passed by Parliament, allowing borough councils to raise up to a halfpenny on the rates to go towards funding these emerging museums. This was followed five years later by a Public Libraries and Museums Act. Some towns acted promptly: Warrington opened a rates-funded museum in 1848, and the town council in Colchester agreed in April 1846 to provide a place 'for the deposit of articles of antiquity or curiosity', although it was not until 1860 that a building was finally opened to the public. By the middle of the 1880s, Exeter, Nottingham, Bristol, Liverpool, Wolverhampton, Birmingham, Sheffield, Aberdeen, Leeds, Sunderland and Preston had all taken up the challenge, and so important did a museum become to municipal identity that there was fierce competition to see who would be quickest to create the biggest and best. As museums came to be seen as a symbol of modernity and sophistication, no town of any size wanted to be left behind, and the press fuelled the competition, reporting on efforts nationwide and judging one achievement against another. In 1878, the citizens of Nottingham must have glowed with delight at praise from the _Magazine of Art_ : J. C. Robinson's native town had, it suggested, 'at a single stride, outstripped all the towns in the U-K- in the race to provide themselves with local museums'.\n\nThe new museums were, as we have seen at South Kensington, a way of making concrete philosophical discussions about art and culture. Throughout Europe, there was a drive to develop museums, as politicians, reformers and critics each attempted to prove their point. In Germany, especially, there was a concerted campaign to establish collections in the major cities. For Gustav Waagen, this was 'to advance the spiritual education of the nation through the experience of beauty', but for many of his colleagues the emphasis was on education: 'systematically arranged collections should be for the instruction of the _Volk_ and the advancement of scholarly work', maintained Hermann Grimm, another high-profile German art historian. In France, still rebuilding after the Revolution at the end of the eighteenth century, the focus was also on creating resources for public instruction, and during the 1860s the displays at the Louvre were redesigned to accommodate increased labelling, explanatory materials and guidebooks, further emphasizing its distance from the exclusive habits of its royal past.\n\nThis didactic impulse was part of the ongoing desire to make art generally more accessible, and to include all classes in a richer cultural life. A state museum opened in Berlin, for example, as early as 1830, with the intention of creating an opportunity for 'the general public' to experience art 'without regard to social status or education'. But this kind of rhetoric tended to mask other, more complex, underlying factors. In Britain, the impulse to educate the masses, and to use museums as a tool for education, was, as Ruskin had asserted, also about changing behaviour. As early as 1834, a government select committee established to investigate 'vices of intoxication among the labouring classes' recommended museums as one solution to public drunkenness. And in the wake of the 1832 Representation of the People Act, commonly known as the Reform Act \u2013 with the 1867 Reform Act further extending the franchise to the urban working class \u2013 establishment discomfort about the widening of the right to vote provided another impetus to develop museums. If people were going to be allowed to vote, then they needed to be instructed in the 'right' things, and the choice and arrangement of collections provided one way of moulding attitudes. The emerging museums grew out of a desire for moral authority and universal order as much as from pedagogic fervour or a commitment to beauty. They were a way of organizing people, enforcing norms of behaviour and driving home approved messages.\n\nBy the Victorian mid-century, there was a growing middleclass fear that increasing leisure time would, undirected, encourage the working classes to crime. In 1860, the opening of Manchester's free libraries, for example, was heralded by _Chambers Journal_ as an effective solution for preventing unrest: 'As it is almost certain that the progress of civilization will produce more and more leisure to the human race,' it explained, 'it becomes a subject of the first importance to provide means for occupying that leisure in moral and intellectual progress.' Museums fitted the bill as neatly as libraries. In his 1862 manifesto for a national museum of natural history, Robert Owen, industrialist and reformer, argued that the development of more museums could effectively counter revolutionary impulses, while the opening of the Nottingham Museum in 1872 was lauded for its wholesome and regulatory effect. It was, claimed _The Builder_ , 'an important addition to the educational and refining influences. . . by the means by which those who labour may be lifted upward and at least deprived of their present excuse for merely sensual enjoyments', a respectable alternative to 'the gin-palace and drinking-saloon. . . in which elevation, not degradation, shall result'.\n\nThe philanthropic impulse to provide for 'those who labour' was always complicated. It tended to submerge questions about power and status, and to fudge issues of common access and exclusivity. This was particularly true of the creation of smaller collections by well-meaning individuals which occupied a grey area between the public and private. They were often established specifically with public access in mind. Visitors \u2013 in particular, the working classes \u2013 were given generous admittance and were often allowed a much more intimate relationship to the objects than in the larger museums. But the objects still belonged to the private collector and, harking back to the _Wunderkammer_ tradition of the past, the rooms of marvels were clearly stamped with his character. Although some of these collections in time became the foundation of more genuinely public museums, at the outset they showed how private and public collecting were still in the process of divergence during the nineteenth century, leaving plenty of room for confusion.\n\nIn many cases, these collections existed entirely for their benefit to the visitor: the collector's pleasure was in the wider good the objects bestowed, rather than in the objects themselves. An interesting example of this was John Ruskin's experimental collection which he installed in a small cottage museum in Sheffield in 1875. He chose Sheffield as the archetypal industrial city, with a tradition of craftsmanship but with many poor working-class citizens who might benefit from education and cultural enlightenment. Always vociferous in public debates about art, Ruskin's views often seemed inconsistent: as a collector and critic he espoused the spiritual value of beautiful works, championing the paintings of J. M. W. Turner, for example, but as a social reformer he also firmly believed in the more utilitarian processes of copying, drawing and close observation as a way of improving the life and work of the country's artisans. He intended the Sheffield museum to be the first of a series of similar teaching collections nationwide (the others never materialized), offering carefully chosen works for study, from prints, drawings and paintings to mineral specimens and coins. Ruskin oversaw the way in which things were displayed with a clear view to making the works as approachable as possible. Visitors were allowed to draw from, or handle, artefacts and the curator was requested by Ruskin to 'put everything away but what people can see clearly'. Admission was free and, in an attempt to make the collection of genuine use to the working men of the city, the building was open until 9 at night and, unusually, from 2 until 6 on Sunday afternoons, as well as by appointment.\n\nThe model worked. Entries in the visitor book show that around two-thirds of the visitors were local, representing almost every district in Sheffield including those deep in the heart of the industrial centre and East End where the metal trades and heavy industries were concentrated. Several visitors became regular, suggesting a programme of study, and several identified themselves as 'art metal worker'. There was also a stream of female visitors from a range of social classes, from Lady Cunliffe-Owen and her daughter to the illiterate Mary Newbold of Spital Hill and Mrs Hobson of Pitsmoor, who both signed with crosses alongside the curator's entry of their name.\n\nThe pattern was repeated across the country. The tension between genuinely egalitarian public opening and inherent systemic inequalities may have remained, but museums proved enormously popular on a practical level. Whatever their philosophical basis, a trip to the local museum became fashionable for people from all walks of life. Museums were pleasant places to be, and the collections offered a novel brand of entertainment. The crowds poured in. In Sunderland, loan exhibitions proved so popular that the North Eastern Railway Company laid on special trains, while the original museum building was deemed inadequate for the volume of visitors just fifteen years after its opening in 1879. In Sheffield, a letter to the local paper complained of 'an invasion of Sheffield roughs of both sexes last Sunday pm. . . some hundreds of the worst played at gymnastics over the seats, and afterwards, with girls at their sides, made a promenade of the galleries'. An exhibition in Nottingham in 1872, 'showing the application of fine art to industry', attracted over 2,500 visitors a week, amounting to a total of over 760,000; more than 100,000 visitors admired a special display in Halifax, and the national journals published regular pieces about exhibitions which 'have attracted much attention on the part of the public having been inspected by tens of thousands'.\n\nBehind every public museum project were the private scholars and collectors who made things happen. The money from the Museums Acts could only be used for buying a site or constructing a building \u2013 not for acquiring objects for a collection \u2013 and so inevitably the creation and expansion of the new museums throughout the Victorian period was as much a personal as a municipal achievement. In a few cases, as at Ruskin's Sheffield museum, individual benefactors were to the fore, but more frequently private collectors simply made small cumulative contributions to activity in their towns or cities. They were the lifeblood of the system, lending, giving and bequeathing objects, volunteering their knowledge and expertise, and bringing their experienced eye to bear on acquisitions and displays. Just as Robinson had brought together an army of collectors for his 1862 exhibition at South Kensington, so museums up and down the country were turning to individuals to fill the cases. And just as Robinson was overwhelmed by eager contributors, so most local museums found collectors were only too willing to have their names attached to such a respectable municipal venture.\n\nUnfortunately, this enthusiasm frequently gave an impression of amateurism and muddle. Even though many museums established specialist loan committees to seek out and woo significant local collectors, most found it hard to turn down objects that they were offered on a more ad hoc basis. Not surprisingly, this tended to result in a diverse, and frequently bizarre, mixture of exhibits which often appeared even more confused as a consequence of poor presentation and interpretation. In 1874, a Royal Commission found that 'the only label attached to nine specimens out of ten is \"presented by Mr. or Mrs. So-andso,\" the objects of the presentation having been either to cherish a glow of self-satisfaction in the bosom of the donor, or to get rid under the semblance of doing a good action of rubbish that had once been prized, but latterly had stood in the way'. Bursting with everything from butterflies and minerals to celebrity memorabilia and poorly executed watercolours, these museums were public testimony to the pervasiveness and irresistibility of collecting \u2013 but they also demonstrated the need for studious, focused and devoted collectors who could create something lasting and meaningful.\n\nThis was where the bigger London institutions, the national collections, might have stepped in. Although the provinces were generally new to the idea of museums, public collections in the capital had a longer history. Sir John Soane bequeathed his collection of antiquities, sculpture, furniture and paintings to the public in the early nineteenth century, and in 1838 the National Gallery was opened in Trafalgar Square. By the middle of the nineteenth century, the original collections of books, antiquities and natural history that had been brought together to found the British Museum in 1753 had been expanded to include ethnographic artefacts, Asian treasures, classical sculpture and key pieces such as the Rosetta Stone, engraved with the three scripts of ancient Egypt and brought back from Alexandria by British troops in 1801. In addition, several high-profile private institutions allowed at least some kind of public access: the independent Royal Academy had held influential summer exhibitions since its formation by George III in 1768, while the British Institution, founded by art connoisseurs and private subscribers in 1805, held a series of loan exhibitions at its premises on Pall Mall, a popular society haunt. Throughout the century, the Society for the Encouragement of Arts, Manufactures and Commerce had, as we have seen, been drawing the crowds to its series of exuberant trade shows.\n\nIn many cases, however, the visitor and would-be collector found much less encouragement in London than elsewhere in the country. Exhibitions at the Royal Academy and the British Institution were, in reality, rather closed affairs, partly because there was an entrance fee and partly because they were entangled with the tradition of gentleman's clubs and the artistic elite. They were extremely popular among certain sections of society but they did little to promote the idea of genuinely public access to art, nor did they necessarily suggest that collecting could be an egalitarian activity. The British Institution was governed by the interests of the aristocratic community that had made up its management since it was founded by the Marquess of Stafford; the Royal Academy was as much a stronghold for establishment artists as it had been when it was launched by successful painters like Sir Joshua Reynolds in the 1760s. The membership of these influential art organizations seemed to have evolved little from the days of clubs like the Society of Dilettanti, formed around 1732 with a select membership of the noble and the fashionable: according to Horace Walpole, 'the nominal qualification is having been in Italy, and the real one, being drunk'. Even the growth of private commercial galleries in the 1870s and 1880s did little to alter the situation. With the most valuable markets in mind, most of them concentrated on an upper-class clientele and actively promoted the exclusive feel of the Royal Academy: the Grosvenor Gallery was established in the art marketplace of Bond Street in 1877, for example, but its founder Sir Coutts Lindsay set it apart by nurturing an aristocratic air and a commitment to showing works from private collections. 'The whole has the affect of a private salon, richly and harmoniously furnished,' noted one visitor, while another enthused at how quickly the Grosvenor's series of exhibitions and private viewings had become another stop on the ritualized merry-go-round of elite London social life: 'Sunday afternoon parties at the Grosvenor Gallery, by personal invitation only, were some of the high points of the season.'\n\nThe National Gallery did have a commitment to free opening and its site on Trafalgar Square had been chosen, in principle at least, because it was accessible on foot from the poor streets of the East End. In 1822, Sir Robert Peel talked optimistically about the beneficial reward to society of bringing the rich and poor together in forming a national collection of art. But in general it remained the private preserve of connoisseurs until the 1860s and 1870s, operating on a modest scale, and functioning more or less as a sub-department of the British Museum \u2013 it was only after 1856 that the gallery became independent. Acute lack of space meant that many of the pictures were not displayed at all, or were shuttled around London: the bequest in 1856 of over 1,000 paintings by J. M. W. Turner, for example, had to be moved to South Kensington. Confusion and disagreement between the gallery's trustees also led to periods of almost complete paralysis in terms of developing the collection. Some advocated following the German model being developed in Berlin by Gustav Waagen, buying works to show the progression of art through history, while others, including members of the Royal Academy, argued that acquisitions should be confined to important works in styles derived from the classical and Renaissance periods. The result was that between 1847 and 1850 no acquisitions were made at all, even though London salerooms were awash with affordable masterpieces. In the light of such confusion and missed opportunities, there was an ongoing, very public campaign for the complete reorganization of the collections. The press carried frequent letters about the issue and a public commission in 1857 further aired discontent: in his evidence, John Ruskin complained of the 'continual change of temper and tone of thought' occasioned by the existing arrangements of the works.\n\nThe British Museum was equally disordered and ill-managed. Until the beginning of the nineteenth century, entrance was only permitted after applying to the chief officer of the museum for a signed ticket. It could take months for this to come through, after which the visitor was hurried around the galleries on a rushed guided tour. From 1810, matters improved, but the museum was still only open between 10 a.m. and 4 p.m. on three days a week, to those deemed to be 'of decent appearance'. Daunted by conservative church opposition and suffering from lack of funds and staff, it was the last to offer Sunday opening, while concerns about the hazards of gas lighting meant that evening opening was also severely restricted. As a result, the collections were all but inaccessible to the working classes. In _Little Dorrit_ (1855\u20137), Charles Dickens associates a 'Sunday evening in London, gloomy, close, and stale' with the museum's dark, locked rooms:\n\nEverything was bolted and barred that could by possibility furnish relief to an overworked people. No pictures, no unfamiliar animals, no rare plants or flowers, no natural or artificial wonders of the ancient worlds \u2013 all taboo. . . Nothing to see but streets, streets, streets.\n\nBut it was not only the process of getting into the galleries that frustrated visitors. Once inside, the displays themselves retained an air of muddled academia that made it very difficult to understand or appreciate the collections. In 1860, _Chambers Journal_ suggested that the British Museum was too 'cribbed, cabined and confined' to do any justice to the objects and that it was in danger of degenerating into 'a gigantic curiosity shop'. Two years later, _The London Review_ described the exhibits as 'so crowded together as to be rendered almost useless'. Between 1850 and 1860, a number of parliamentary inquiries and Royal Commissions discussed everything from space in the galleries to conservation requirements, but popular support for the museum was low, and its galleries seemed to present a sad comparison to the lively display spaces of fashionable events like the Great Exhibition.\n\nWith public discontent in the national institutions running high, the relationship between the growing network of regional museums and their larger London counterparts was a complex one. Many of the new museums were fiercely provincial, and unwilling to associate themselves too closely with what they regarded as the failing examples in the capital. Cities and towns outside London were eager to prove they had their own vibrant identities, and events such as the Mechanics' Institutes exhibitions or, on a larger scale, the 1857 Art Treasures Exhibition in Manchester had already provided an important means of doing this. The museums were a novel, potent means of displaying provincial progress and pride. They were an indication of how well the outlying towns and cities were doing, showing off the wealth of key benefactors while also displaying the cultural and educational credentials of the populace more widely. The collections were often local or regional in nature, and the displays were clearly regarded with a certain sense of ownership by visitors. The trades, materials and styles on display were familiar, and the names of local collectors were attached to most of the objects.\n\nThis powerful sense of provincialism was tempered, however, by the importance of loans from the national collections, especially from the South Kensington Museum. These loan exhibitions were seen by the government as complementary to the regional Schools of Design, and as part of the crusade to raise manufacturing standards. They were also crucial in establishing many regional museums, enabling cities to develop audiences for displays and create support for a more permanent local collection. Between 1854 and 1870, loans from South Kensington were made fourteen times to Sheffield, ten times to Birmingham, eight times to Liverpool and Nottingham and seven times to Leeds. At the Mechanics' Institute in Manchester in 1845, the display included 'textile fabrics, bronze, iron and other metal castings, porcelain, and earthenware' and drew visitors from across the city before everything was packed up and moved on to Glasgow.\n\nVictorian advances in transport made such peripatetic exhibitions increasingly possible. As the speed and efficiency of the rail system effectively shrank distance, moving objects around the country seemed as simple as it was desirable. There was no longer any reason for everything to be centred on the national institutions in London, or on London audiences. Displays, and the expertise behind them, could be shared because the railways ran quickly and efficiently, changing perceptions of what was achievable. The magnificence of Robinson's loan exhibition at South Kensington was possible because it had already been demonstrated that substantial collections could be brought together temporarily, and then dispersed, without much fuss. The travelling exhibitions proved just how many visitors could be reached and how popular collecting was becoming nationwide. They also emphasized how modern technologies were altering the very nature of collecting, opening up new possibilities and inextricably linking the collection to travel and discovery. Small local displays were giving way to ambitious exhibitions that drew crowds from a distance; museums aspired to become municipal models for the masses rather than elite palace playgrounds of the rich. And the collector was evolving from the gentleman amateur in his isolated study into the scholar with links across regions and even nations. Times were changing, and the role of the collector was inevitably changing too.\n\n## _Making Museums: Collecting as a Career_\n\nJOHN CHARLES ROBINSON\n\n### CHAPTER FOUR\n\n### _On the Banks of the Seine_\n\n_D_ rawing was John Charles Robinson's escape. In the long days of childhood he would idle around the country lanes near his home in Nottingham, sketching and scribbling, dreaming of what might be. He was not close to his siblings; his father, a printer and auctioneer, was sternly preoccupied with business, and he was largely raised by his grandfather, who was a bookseller. Robinson spent hours by himself. Physically slight and unremarkable, with a sharp mouth and a high, pale forehead, he was little noticed as he sat practising his skills of composition, perspective and shading. But he did not mind this. He preferred his art to anything and was happy to be left alone. He rapidly developed a self-assurance bordering on arrogance and, as his drawing skills developed, so did his confidence. He compared his work to that of others around him, and he became convinced that he had talent. He began to imagine a future for himself as an artist or an architect, leading a life of creativity and inspiration. His ambition began to shine bright and hard.\n\nRobinson knew he could expect little financial support from his family. Moreover, he had few contacts and sparse knowledge of the world of art. The small-scale auctions run by his father were the closest he had come to the international world of the artist. But his early confidence was unshakeable and, soon after leaving school, he began work in an architect's office. Life behind the drawing board was not quite as he had imagined, however. Studying the style of the nineteenth-century Gothic revival, he was, for a while, inspired by the ideals of craftsmanship and the elegance of Gothic decoration, but the day-to-day routine of a small-scale architect seemed largely mechanical and tedious. Robinson was convinced he could do better. And in 1843, at the age of nineteen, he was accepted into a major artists' teaching atelier in Paris. Taking a big step towards creative and personal independence, Robinson turned his back on his life in Nottingham to begin again.\n\nThere was no better place than Paris to learn an artist's trade \u2013 up in Montmartre, the crowded narrow streets were teeming with painters and sculptors and poets. Robinson worked with two of the city's leading ateliers. Under the guidance of Michel Martin Drolling, professor at the Ecole Nationale Sup\u00e9rieure des Beaux-Arts, he studied bright, theatrical scenes from the Bible or from history; with Henri Lehmann, a young German-born draughtsman only ten years older than himself, the emphasis was on graceful portraits and nude studies. Robinson studied hard. He drew from the classics or from copies of masterpieces, worked from plaster casts of sculptures, painted in parks and in the open country and made life drawings. He was robustly confident in his own talents. Every week there was a competition among Drolling's students, who were then arranged around the studio in order of merit. In 1845, Robinson wrote proudly to a friend: 'I am now the \"premier dessinateur\" in the atelier. There are several who have drawn consistently for four or five years whilst I have drawn little more than 8 or 9 months and have already got top of the list.' Robinson may have been exaggerating the speed of his achievements \u2013 he had been in Paris, although not at the same atelier, almost two years \u2013 but it was clear that he had an eye for perspective, form and colour, and was, as with all things he was to undertake, conscientious and determined. He may have looked boyish and angelic, lightweight even, but he fully believed in his own talent to succeed.\n\nRobinson's practical Paris education was indeed a success. He would work as an artist for the rest of his life, exhibiting several times at the Royal Academy and regularly selling his work. Prints of two of his landscapes can still be seen in the collection at the Tate Gallery. But it was another aspect of his time in the French capital that was to have most influence on the rest of his life \u2013 and on the future of British collecting. Robinson's study at the ateliers finished every day at midday, leaving afternoons free. And not far from his lodgings were the high-ceilinged galleries and wide courtyards of the Palais du Louvre. It was there that Robinson discovered for the first time the bold, flourishing masterpieces of the Italian Renaissance, da Vinci's _Mona Lisa_ ; Raphael's angels and demons; Titian's elegiac nudes; bronzes, busts and sculptures by Michelangelo. For a young man from the Midlands, idealistic and lonely, slightly homesick and desperate to succeed, these works of art were both overwhelming and seductive. Robinson developed an interest in the Renaissance that would endure for the rest of his life. The history and beauty of the masterpieces of the past permeated deep into his understanding until he felt the physical tingle of them in the cool breath of the galleries, the magnificent colours and full forms imprinting themselves unforgettably on his mind's eye.\n\nRobinson spent as many hours with the dead artists of the Louvre as with the living ones of Montmartre, and it was there, rather than in his studio, that he metamorphosed from a gauche young man into an erudite connoisseur. With its history as a royal palace, the Louvre was spectacular and ornate. The building had dominated the centre of Paris since the twelfth century, developing from a fortress to a sumptuous expression of imperial power. In 1674, the French king, Louis XIV, moved the permanent court to Versailles, and by the end of the seventeenth century the Louvre buildings were starting a new life as home to a gallery of antique sculpture as well as the Acad\u00e9mie Fran\u00e7aise, the Acad\u00e9mie des Inscriptions et Belles Lettres and the Acad\u00e9mie Royale de Peinture et de Sculpture, the last of which launched a series of popular annual exhibitions, or salons. Visitors came from the French political and academic elite, and the activities within the Louvre were held up as a shining example of the nation's cultural pre-eminence and prestige. But the hierarchical structures of the Acad\u00e9mies, as well as the yearly salons, clearly promoted the established interests of the ruling minority and the Louvre remained closed to all but a handful of visitors.\n\nA century later, however, the French Revolution changed the old systems of power and reinvented its strongholds. The emphasis now at the Louvre was on creating a national collection that belonged to the people and an institution that was open to everyone. Masterpieces 'which were previously visible to only a privileged few. . . will henceforth afford pleasure to all: statues, paintings and books are charged with the sweat of the people: the property of the people will be returned to them,' asserted Abb\u00e9 Gr\u00e9goire, one of the most influential Republican leaders. The Louvre began a long period of development as a genuinely public site, opening in 1793 as the Mus\u00e9e Central des Arts, displaying paintings which had been confiscated from royal and aristocratic families, including the _Mona Lisa_. Admission was free, and large amounts of state money and energy were devoted to building collections that would be esteemed throughout France and beyond.\n\nIn practice, it was not always easy for the newly empowered citizens to view their artistic heritage: at the beginning of its life, the museum was only open to the public at weekends, and artists were always given priority over ordinary visitors. But gradually more of the building was opened as the French collections expanded, and extensions were also added to create more gallery space. The Louvre was as important to the Republic as a symbol of state power as it had been to the monarchy as a royal palace. Its public collections signified national unity and the regeneration of the people, while the display of the nation's new-found artistic wealth was also a means of displaying the political success of the new government; it was hoped that the galleries would forge a visible link between the care and display of respected works of art and the perception of responsible Republican rule. Between 1794 and 1813, treaties with Italy as well as the spoils of Napoleonic conquests swelled the collections: three huge convoys of marbles and paintings from Rome, central Italy and Venice were sent to Paris, with the arrival of the last, in July 1798, prompting a spectacular celebratory festival. Throughout the period, ongoing confiscations from aristocratic families further swelled the Louvre's acquisitions. After Napoleon's defeat at Waterloo in 1815, some of the works looted from Germany, the Netherlands, Austria, Italy and Spain were, grudgingly, returned. For the next couple of decades, until 1830, the unsettled politics around the brief restoration of the monarchy also meant that attention drifted from the Louvre, funding faltered and the galleries were neglected. But this still left plenty to see. By the time of Robinson's visits during the early 1840s, the displays included Egyptian antiquities, ancient bronzes, medieval decorative arts, Etruscan vases, rooms devoted to 'modern' sculpture and a gallery of Spanish art.\n\nRobinson was fortunate to be in Paris at the beginning of a period of active expansion of the Louvre collections. From 1848, in particular, with the birth of the Second Republic, the government invested heavily in the building and the collections, allocating two million francs for repair work. And as the French economy grew during the middle of the century, so too did the number of works on display: over 20,000 works were added between 1852 and 1870. But even in the years before this, it was a unique resource for a young artist. The Revolution and its aftermath had made it possible to view, in one place, a range of artworks from many periods and in many styles arranged specifically to instruct students like Robinson. The emphasis was on works that could be usefully copied, and on hanging pictures chronologically and by national schools of artists (with French painting given pride of place) to explore the historical evolution of style and technique, 'a character of order, instruction and classification', as one of the early museum directors explained. This was a relatively untried way of displaying works, with only a handful of German galleries experimenting with similar principles; the National Gallery, in contrast, modelled its displays on the domestic traditions of private country-house collections.\n\nIn Britain, art was crammed into rooms from floor to ceiling and displays were arranged according to the aesthetic judgements of the organizers, the size and shape of works \u2013 or just to accommodate the space. It was unlikely Robinson could have profited from the type of education he received in France. 'Nothing has so much retarded the advance of art,' suggested John Ruskin, 'as our miserable habit of mixing the works of every master and every century. . . Few minds are strong enough first to abstract and then to generalise the characters of paintings hung at random.' Moreover, British royal and aristocratic collections were still fiercely private, and Robinson's routine of drawing and studying from private collections could not have been easily achieved: 'The country was indeed rich in works of art, richer perhaps than any other,' Robinson admitted, 'but these treasures were the possessions of private individuals, scattered broadcast in a thousand places and town and country houses, for the most part hidden treasures, often unappreciated by their possessors even, and but casually revealed to the world at large.' Even aside from such scattered country-house collections, there was very little for the student. Sweden had had a national gallery of art since 1794; in Amsterdam the Rijksmuseum opened in 1808; and in Madrid the Prado was founded eight years later. London's National Gallery, in contrast, was in its infancy. It did not move into its permanent building until the late 1830s and even then, as we have seen, its development was unsteady; its slowly expanding collections remained a matter of debate and disagreement. In addition, the French Revolution, the turbulence of the Napoleonic Wars and the violence and revolt across Europe in 1848 were fresh in the Victorian imagination. The tendency to public access and universality that was evident in France was in stark contrast to something of a backlash among the English establishment. Haunted by the prospect of unrest and revolution spreading to Britain from the Continent \u2013 and further stoked by the progress of the working-class Chartist movement during the mid-century \u2013 the British ruling classes tended to tighten their grip on power. This showed itself in a variety of ways, from increasing restrictions on women to a greater emphasis on the doctrines of the established Church. It also meant that, while government select committees and initiatives like the Schools of Design voiced the ideals of greater public access to art, in reality those who owned or controlled most of the works were more interested in closing ranks against the perceived threat of the masses. The French model of universal access, born out of the principles of the First Republic, was still some way off across the Channel.\n\nThe only real British alternative lay with a number of exhibition societies that flourished in the second half of the century. While these did not show historic masterpieces, they did offer support to many artists considered too progressive to please the conservative members of the Royal Academy, and they welcomed visitors who wanted to view new work in a sympathetic environment. Many of these societies had open membership and modest subscription rates: the Society for the Encouragement of the Fine Arts founded in 1858 held lectures and debates and organized visits to private galleries for an annual subscription of \u00a31.1s. Several specialized in encouraging artists who worked outside the oil painting tradition favoured by the Royal Academy \u2013 these included the Photographic Salon which met at the Dudley Gallery Art Society in Piccadilly and the Bookplate Society based at the nearby John Baillie's Gallery \u2013 while others provided studios, study rooms and drawing classes, or even allocated grants to young artists. But such opportunities were limited. The exhibition societies were largely confined to the capital and were without real influence or prestige. Perhaps more significantly, their open membership policy and resistance to selection meant that the quality of exhibits could be poor. Displays frequently failed to inspire the serious student, and Robinson would have stood little chance of seeing many significant works of art at first hand.\n\nIn France, it was the Louvre's Italian Renaissance masterpieces that most captivated Robinson's interest. He learned quickly and before long he was an expert. Eventually, he studied other periods and schools of art \u2013 he particularly came to value the workmanship of the medieval craftsman and to admire the detail of eighteenth-century composition \u2013 but his first love was for the extravagance of the Renaissance. It was never to leave him. Inevitably, when his years at the Paris ateliers were over, he brought it with him back to England. In time, this enthusiasm would influence his work at the South Kensington Museum, and have an impact on the type of collections he developed there. But there was more to Paris than the Louvre, and alongside Robinson's education in the public galleries he threw himself into the more informal training of the private collector. The lure of the hidden and the mysterious took him deep into the network of backstreets as well as into the smarter districts of the 7th _arondissement_ where the curiosity shops were clustered. He became known to a number of successful and influential dealers, noting a preference for Monsieur Delange on the Quai Voltaire, opposite the Louvre and the Tuileries Gardens; he regularly bought the more reasonably priced wares of Mister Evans, who owned the next-door shop; and he was a frequent visitor to the small but packed showroom of Monsieur Couvreur, on the rue Notre Dame des Victoires, in the crisscross of streets around the fashionable Op\u00e9ra.\n\nBy 1846, hopeful visits to the most promising dealers had already become an integral part of Robinson's life. A young man of twenty-two, he was collecting in earnest, haggling over the spoils of war and revolution, travelling to Orl\u00e9ans and Tours as well as keeping an eye on bargains to be had in Paris. For the next sixty years, his letters, diaries and published articles, and even his account books and official museum minutes, would show a man completely captivated by the idea of collecting. In Paris, the opportunity to visit several thriving and inspiring collections rooted in private connoisseurship provided a touchstone for what he might go on to achieve, proving what could be done with dedication and commitment. Perhaps most important of these was the collection of Alexandre Du Sommerard in the atmospheric medieval H\u00f4tel de Cluny. Open to the public on Sundays from 1832, the elaborate rooms drew on the influence of the _Wunderkammer_ , evoking the cluttered, treasure-trove effect of the impassioned collector and drawing crowds as large as those at the Louvre.\n\nBy the time of Robinson's visits, the state had taken over the collection, arranging the pieces more systematically and allowing more frequent public access from 1844. The Mus\u00e9e de Cluny was already making news among English politicians and commentators as a resource for educating taste and encouraging the skills of artisans and manufacturers, being held up as a model for British projects like the Schools of Design. Robinson had probably heard of the collection in this light. But it was also a scholarly celebration of collecting the medieval and Renaissance decorative arts. Du Sommerard published a highly illustrated, five-volume catalogue (1838\u201346) with an emphasis on the pieces as 'strange and rare history' instead of as evidence of design standards, and it was this sense of the more idealistic possibilities of collecting that Robinson seems to have found most inspiring. Du Sommerard had a dull day job in the French Audit Office, but his heart was in collecting and for an impressionable young visitor like Robinson this enthusiasm was in itself exhilarating.\n\nRobinson was already finding his way into the circle of collectors. A recommendation from a fellow connoisseur would have granted him access, via three flights of steep stairs, to the tiny apartment of Charles Sauvageot, in rue du Faubourg-Poissoni\u00e8re in an outlying district of northern Paris. Sauvageot lived completely surrounded by his things. 'The objects were so crowded that I tucked in my sleeves from fear,' wrote another English visitor, clearly afraid of accidental breakage. 'It is evident that he buys articles from real love of the beautiful.' Sauvageot, like Du Sommerard, had a 'normal' career, working at the Customs office, and his rooms were small and cramped. He had little money, but he was an indefatigable scavenger, alert to the possibilities for acquiring unusual and valuable works in the long aftermath of the Revolution. His collection showed that with patience and spirit it was possible to achieve a great deal. Robinson took such models to heart. He became acutely aware of the variety and beauty of the objects on offer. He developed a taste for Limoges enamels and an enduring fascination with textiles. He bought sixteenth- and seventeenth-century silks, with their colours still bold, an ornamental carpet and a magnificent altar cover. He became voracious in his collecting, developing a sense of urgency and competitiveness that was to last a lifetime. But late in 1846, after a period of intense collecting, Robinson's money finally ran out. He was forced to return home to Nottingham in the middle of an English winter, complaining in his diary that the city was 'desolate'.\n\nBack in Nottingham, Robinson missed desperately the romance of Paris and the thrill of making contacts there. He continued to paint, but could not find a way of supporting himself as an artist. For a while, his life seemed dull and miserable. His years in France had made an enormous impact on him, inspiring his collecting and offering glimpses of what could be. Short of money and lonely, these possibilities suddenly seemed all too distant. But Robinson was lucky. On 1 June 1847, he was appointed to work at the Nottingham School of Art, before moving just two months later to a post as Assistant Master at the government School of Design in Hanley, where he doubled his salary to one hundred pounds a year. At first glance, teaching drawing skills in the industrial smog of the Midlands potteries was not a glamorous job. But one of Robinson's first tasks was to return to France, to report on the state of pottery and design, and on the teaching of art. He was delighted to be back in Paris, and he was convinced that he had found a quick route to promotion. He was confident that, with his inside knowledge, his report would get him noticed, releasing him from the stagnancy of his provincial backwater and into a more vibrant life in London. In the meantime, he could stroll again by the Seine and rummage in junkshops.\n\nFor the next few years, before starting at the South Kensington Museum in 1853, Robinson travelled as much as his job would allow \u2013 for work, for pleasure and to make himself an expert. In 1851, he discovered the pleasures of Italy, and found for the first time a beauty and romance to challenge anything he had seen in France. Apart from the 'curse' of mosquitoes, which he bemoaned in his letters, there was nothing to upset his enthusiasm. He was overwhelmed by the spectacular sculpture and the magnificent architecture of cities like Verona and Padua; in Florence the glories were so numerous that Robinson was 'too excited to go to bed'; and then there was Venice. 'I have just got in from a moonlight sail through Venice,' he wrote to a friend. 'The moon is at the full \u2013 brilliant \u2013 pouring down floods of light through a deep blue endless sky, such as _you_ have never seen and never will til you come to Italy \u2013 Imagine, but you can't imagine! \u2013 and I am too stupid to describe and no I can't, I can't begin, what an ass I am.'\n\nRobinson was inspired by Italian architecture and art, by the landscape, the language and the culture. He felt he had found a spiritual home, one which was to influence his collecting for the rest of his life. For several months during the summer of 1851, he went from city to city, from Milan to Brescia, Verona to Padua, Ferrara to Bologna. Italy was beginning to occupy a particular place in the Victorian imagination, and Robinson was at the heart of this impulse to celebrate, and romanticize, what could be experienced there. Since the eighteenth century, the British elite had shown a fascination with Italy and its art: half of the paintings sold for more than \u00a340 at London auctions between 1711 and 1760 were by Italian masters, and the Italian towns were well-established highlights of the Grand Tour. Canaletto's idealized paintings of Venice, showing aristocratic palaces alongside pristine canals, were hugely fashionable in eighteenth-century Britain. But travellers like Robinson were beginning to look beyond the well-trodden paths to the Grand Masters. In 1851, the same year as Robinson's journey, John Ruskin published the first volume of _The Stones of Venice_ , an influential book-length essay which held up Italian Gothic architecture, and the communities of craftsmanship from which it emerged, as a model for cultural and social reform. Ruskin's campaign to save neglected and shabby Italian buildings, including St Mark's Cathedral in Venice, took on the vehemence of a crusade, and he arranged for paintings, photographs and plaster casts to be taken of what he considered the most threatened architectural features. Works like _The Stones of Venice_ , and the earlier _Seven Lamps of Architecture_ (1849), inspired the Victorian public to look at Italy in a new light, not just as the fashionable birthplace of Dante or a place of restless politics and retarded modernity, but as a medieval and Renaissance treasure.\n\nThis was something Robinson was delighted to discover first-hand. But collecting in Italy during the middle of the century was not easy. Transport was unreliable, treasure-hunting was notoriously hit-and-miss, and armed scuffles were not at all uncommon. Since the defeat of Napoleon in 1815, the movement towards unification of the Italian states had gathered pace. The years 1848 and 1849, just before Robinson's visit, marked the high point of revolutionary idealism with popular support from all classes: from Sicily in the south to Venice and Lombardy in the north, people took to the streets against both native rulers and the Austrian Habsburgs. Though the insurgency collapsed in the summer of 1849, through a combination of dynastic rivalry, foreign invasion and the withdrawal of key support from papal and Neapolitan forces, the movement rumbled on uncertainly and a series of internecine conflicts broke out, especially in the central Italian states. For travellers this could be disruptive and even dangerous: 'We are expecting a battle every hour, and however exciting and romantic it might be, it would not be pleasant to be rolled amongst the debacle of a beaten army and a panic-stricken population,' Robinson wrote during an eventful journey from Florence.\n\nBut there were advantages to such upheaval. Collecting in Italy had rarely been so rewarding. As the old structures of power were challenged, objects from both the aristocracy and the Church found their way on to the open market and into the hands of collectors. Charles Eastlake, the first Director of the National Gallery (1855\u201365), was just one of those who were drawn to the treasures made newly available by the troubled political climate. He made a series of collecting trips to Italy throughout the 1850s and 1860s; his acquisitions included Fra Angelico's _Adoration of the Magi_ and thirty other paintings from the Lombardi-Baldi collection in Florence in 1857. Whatever the risks, the allure of Italy, it seems, was too great to resist.\n\nDespite his eventful holidays, and his preoccupation with collecting, Robinson's day job continued to go well. He was, as he had foreseen, creating an immediate impression at the School of Design and, by 1852, after just over five years, he had been promoted to Teachers' Training Master. His timing could not have been better. It was the same year that Henry Cole began to reorganize and reinvigorate the government design programme, and Robinson soon came to Cole's notice, partly for his practical work, and partly because he put himself in Cole's way by sending him, with characteristic confidence, a series of suggestions for improving the Schools. Less than a year later, in September 1853, Robinson was in London, at the heart of things, in a new position at the South Kensington Museum. It seemed like the perfect job. He could bring to bear all the art knowledge he had gleaned, he would have public money to spend and he could immerse himself in collecting. What began as a temporary post was confirmed as permanent on 8 July 1854, and Robinson took his place in the ranks of professional collectors.\n\n### CHAPTER FIVE\n\n### _The Battle of South Kensington_\n\n_W_ ithin two years of starting work in London, John Charles Robinson was sitting at his desk, watching the tree tops scraping a bare spring sky. It was early March 1855 at the end of a particularly cold fortnight. Frosts had set hard in the city parks, making them shine white through the smoke-thick air, and the pavement corners were slippery with ice. From the window of his office in Marlborough House, on London's Pall Mall, Robinson could see uniformed doormen stamping their feet to keep the chill from their bones. A discreet notice at the end of the closely raked gravel path tried to entice the public into the warmth to see the displays of 'manufactures', all that yet existed of what would eventually become the splendour of the Victoria and Albert Museum. Visitors made their way steadily into the striking redbrick building and up the marble stairs to peer into the heavy glass cases that lined the rooms inside. In the dignified Georgian townhouse which Sir Christopher Wren had designed for the first Duke of Marlborough and his wife Sarah, porcelain, glass and metalwork were displayed on the first floor; plaster casts were pushed into chimney niches, paintings hung high from the elegant coving and Indian textiles draped over couches. There was plenty to admire.\n\nRobinson watched the visitors arrive. On Mondays, Tuesdays and Saturdays, entrance was free, but the rooms were enormously crowded; on the remaining weekdays, an entrance fee of 6d ensured more space for those who wanted to study quietly. From an average of a thousand visitors a day on free days, attendance on paying days fell to fewer than eighty, but still annual visitor figures for the mid-1850s hovered around the 100,000 mark. During the same period, at the much larger and more established British Museum, figures were sliding rapidly from a peak of over 2 million in the year of the 1851 Great Exhibition to around 300,000. The ideal of public access that had been heralded by the Crystal Palace event seemed to have dissolved and the grand galleries of the British Museum were once again viewed by many as the exclusive haunt of scholars and the wealthy. The six rooms at Marlborough House, in contrast, appealed to the middle classes and the temporary displays were a lively source of entertainment for everyone from serious students to families.\n\nFor Robinson, this popularity was a matter of pride. When the Marlborough House rooms were first opened in 1852, Henry Cole had featured an exhibition of 'False Principles in Design', popularly nicknamed the Chamber of Horrors, a practical demonstration of how not to do things that was supposed to stand as a warning to British buyers and manufacturers: 'a gloomy chamber hung round with frightful objects in curtains, carpets, clothes, lamps and whatnot', wrote one reviewer, a mess of decorative excess and offending aesthetics. By showing people how terrible the worst objects could be, this brash display was supposed to elevate public taste and put the moral case for design reform. It was a clear indication of what Cole thought the museum was about. But in his short time in post, Robinson had set to work rearranging the galleries, reorganizing the showcases, making the exhibitions more varied and, quite deliberately, changing the emphasis of the museum from Cole's focus on educating artisans to his own vision of providing a more general public with the chance to admire the old, rare and beautiful. He was not without sympathy for Cole's general principles of education, but he wanted visitors to use works from history for 'the gradual and progressive cultivation of the judgement, until it assumes almost the readiness and certainty of intuitive conviction'. In his view, the Chamber of Horrors had no part in a museum: 'the object of the Museum is to illustrate the history, theory and practical application of decorative art,' he claimed, making sure that alongside 'objects of utility' there were also 'works avowedly decorative'. He wanted to show the best \u2013 and not the worst \u2013 drawing visitors with the expectation of delight rather than from disgust or curiosity.\n\nIt was the beginning of a battle of philosophies that would divide Cole and Robinson throughout their work at the museum, and would finally highlight the gap between the museum's rhetoric about collecting, and what it collected in practice. Henry Cole displayed occasional enthusiasm for antiquarian objects and pieces of fine art, and he showed characteristic energy in acquiring some of these works for the museum. But, for him, they were always a means to an end. His acquisitions were always made with a view to the larger cause of design reform, and the museum displays were always created with an eye on education. Robinson delighted in beauty and craftsmanship \u2013 at heart he was a connoisseur. He wanted the museum to be more than a training ground; he hoped to create a collection that aimed to foster aesthetic appreciation rather than to deliver design rules.\n\nRobinson argued that the museum he had in mind would serve not only the art student or the general public, but also 'the collector, whose pursuit it is. . . clearly a national duty to countenance and encourage'. For Robinson and his friends, collecting needed to be at the heart of things. There was a sense in which it was much more than an individual preoccupation \u2013 it was 'a national duty'. As we have seen, museums all over the country were emphasizing their public value; there is no doubt that Henry Cole viewed South Kensington as a national crusade. But Robinson was suggesting that it was not only beautiful objects themselves that were important, but also the very 'pursuit' of collecting them. Tracking down objects, studying them, comparing and treasuring them was much more, for Robinson, than an enjoyable habit. It was at the root of the emerging museums network; it was the underlying mechanism that made everything else possible. In statements like this, Robinson was able to give his ransacking of European salerooms a gloss that highlighted its public benefit. But more than that, he was staking a claim for each and every collector, putting their activity into a wider context that gave them influence within the political, economic and social manoeuvring that was striving to make Victorian Britain great.\n\nAs Robinson watched the visitors come to Marlborough House, there was also some sadness. He could feel how things were changing. Just south of Hyde Park on quiet rural land, Henry Cole was working on his site for a grand project \u2013 the building of a permanent museum. He had the support of Prince Albert, who was hopeful that the new building in South Kensington would be the catalyst for the utopian Royal vision of a magnificent 'Albertopolis', a series of striking buildings amid landscaped parkland, drawing together London's learned and artistic societies. So enthusiastic was the Prince, in fact, and so keen to play his part, that he had begun to design the new museum himself. And with his energy and influence, added to Cole's determination, things had progressed quickly. Plans had been drawn up and measurements taken. In just a few more weeks, building would begin.\n\nBut Robinson was fond of the rooms at Marlborough House and did not want to relocate, particularly not to the backwater that was South Kensington. 'Everyone predicts ill luck to the move to Kensington on the score of distance,' he grumbled in a letter to Cole, pointing out what the popular press had already highlighted: that the new site was unfamiliar, isolated, difficult to access and little more than 'wilds and swamps'. What's more, Cole's plans were so vast and disparate that it was not at all clear how it would all fit together. There were proposals for a Patent Museum and for manufacturing displays but also for an exhibition of exotic foods like French snails and Chinese birds' nests, and of animal products, including a case showing silkworms at work. There was to be a huge plaster cast of Michelangelo's _David_ and an extraordinary range of objects that apparently could not find a home elsewhere. There was even to be space on the second floor for fish hatcheries, with the runs of salmon and trout announced in the newspapers. The proposed museum seemed a curious mix of the fairground atmosphere of the International Exhibitions and the old-fashioned jumble of curiosities that had inspired the _Wunderkammer_. It lacked a sense of historical progress and threatened to confuse, and overwhelm, the visitor. Once Cole's Chamber of Horrors had been put away, the rooms at Marlborough House were well focused, with planned collections and organized displays. For a museum of its time, it was remarkably clearsighted. It demonstrated the growing sense of professionalism among curators and rigorous principles of scholarship. Much of this was Robinson's work. He liked the intimacy and coherence of Marlborough House, and was sure that the new museum would be a disaster, nothing better than a 'motley, medley chaos. . . assimilated in an illogical and bewildering manner'.\n\nEntangled with Robinson's discontent about the new museum was a lingering feeling that at the same time his own role was being overlooked and undervalued. He was travelling extensively to seek out objects: in 1854 alone, he had acquired over 1,400 new pieces. He was writing and lecturing vigorously on the collections; he was making contacts. Delighted by the growing number of visitors, he was constantly refining the rapidly expanding displays, and he was bombarding the board with requests for money to make acquisitions. He had taken responsibility for most of the collections, added a couple of titles to his role and had brought some of Europe's finest objects to the rooms in Marlborough House. His work with the Fine Arts Club and networks of private collectors also allowed him to see the wider influence of his activity. His buying on behalf of the museum was changing the market, pushing up prices rapidly. During the museum's five years at Marlborough House, the value of its collections rose so much that Robinson was able to report happily that he had 'speedily doubled or trebled' the original investments. There was also, however, a downside to this, as the competitive Robinson was quick to recognize. As the market boomed, the museum needed to keep pace, he said, with 'a host of wealthy amateurs, who, unfettered by the delays and difficulties impeding all governmental action. . . step in, and, by the power of ready money, triumphantly beat out of the field the unlucky curators of our public collections'. It was clear that the mutual, and complex, relationship between the museum and the private collector, and the markets they shared, was developing quickly. And in this cut-and-thrust environment Robinson felt that his collecting skills were needed more than ever.\n\nSince he was of such benefit to the museum, Robinson could see no reason why his salary, at least, should not be raised in recognition of his contribution. At the beginning of March 1855, he wrote to Henry Cole to ask for more money. Acknowledging his young curator's energy and enthusiasm, Cole agreed promptly to the pay rise. But he just as promptly regretted it because, less than six months later, Robinson was writing again to demand another raise. The collections were continuing to thrive under his hand and his devotion to them was evident. His every thought was about the objects he might bring to Marlborough House and improvements he might make to the displays. He had completed two scholarly catalogues, 'with critical or theoretic illustrations for the information of the student' and more and more visitors were coming to see what was being achieved. He believed he had a convincing case for promotion and, with a successful precedent behind him, Robinson was confident. He could not see how Cole could fail to reward him again.\n\nBut weeks passed, the builders moved on to the site in South Kensington and still there was no official answer to Robinson's request. Slowly the iron frame of the new museum grew into the London landscape, the grass around became muddy with the tramp of workers' boots, and still Robinson waited. He was angry and puzzled. He wondered if Cole and the museum board had somehow overlooked his request in the flurry of activity caused by the building project. But when he tried to raise the matter again, he received only awkward and evasive answers. He was convinced that Cole could do more and he grew increasingly frustrated and cross. Worse still, he could not bring himself to admit the heart of the matter. It was true that he was indispensable to the growing museum. It was clear that it needed his scholarship. He may well have deserved better remuneration for the work he was doing. But more immediately apparent than any of these arguments \u2013 to Robinson and to his disapproving bank manager \u2013 was that he was in desperate need of the extra money for which he was asking.\n\nRobinson had never lived cheaply. He aspired to keep a household that suited his sense of refinement and announced his success. On accepting the job in London, he had taken on both a substantial and elegant townhouse in York Place, Portman Square, along with the requisite number of servants, and a wife. Marian Elizabeth Newton, the daughter of a successful Norwich tradesman, was quiet and undemanding. As far as we know, she did not share Robinson's taste for stalking antiques shops and dealers in search of costly pieces of bric-a-brac, nor his habit of spending money on European travel. But nevertheless, with a household to support, Robinson's costs escalated and, within a few years of marriage, the first of seven children arrived. Robinson's expenses were rising along with his sense of frustration. It was not just that he felt he deserved more money; he _needed_ more money.\n\nFor a young man establishing a family and a professional career, it was not unusual to find daily expenses running over budget. But Robinson's collecting could be seen as something of an extravagance and, in the mid-century, this kind of excess was subject to a raft of moral judgements that, if turned in Robinson's direction, could have threatened his future. Profligacy, bankruptcy, financial disaster and the lack of good moral character they suggested fascinated Victorian writers such as Charles Dickens and Wilkie Collins, who kept the spectre of 'living beyond one's means' firmly in the public eye. The plot of Dickens's _Little Dorrit_ , published between 1855 and 1857, at the same time as Robinson was making his overtures to the museum authorities, highlights the shame, personal decline and poverty that accompany bankruptcy, while almost a decade later, in _Our Mutual Friend_ (1864\u20135), the contrast between the simple living of the impoverished Mortimer Lightwood and the glittering, unpaid-for interiors of the Lammle household has a clear moral message: 'how often have I pointed out to you that it's the moral influence is the important thing?' asks Lightwood's friend Eugene Wrayburn. Too overt a display of possessions was not only an indication of suspect taste, but also of dubious character. The glut of objects on show in the Lammle household acts to conceal both their lamentable financial situation and their sinister intentions:\n\nThe handsome fittings and furnishings of the house in Sackville Street were piled thick and high over the skeleton up-stairs, and if it ever whispered from under its upholstery 'Here I am in the closet!' it was to very few ears. . .\n\nRobinson needed to take care that his collecting did not come to be seen as simply an expression of wanton spending and lack of moral fibre. Nor would he have wanted to get caught up in the Lammle fashion for objects merely as display. To maintain his course towards serious, meaningful collecting, it was imperative that he steady his financial situation.\n\nDisappointed with Cole and obstructed by the board, Robinson continued to wait. If he could not succeed in increasing his regular salary, then the obvious next step was to sell some of his collection to supplement what he earned. But such a move was not to be taken lightly. Souvenirs of his youth in Paris; bargains from French and Italian junkshops; finely framed paintings and mirrors; jewel-coloured tapestries \u2013 they all meant something special to him. In the spring of 1855, Robinson was still, at thirty-one, a young collector. In time, he would become accustomed to the idea of changing his collection, of selling on some pieces to buy others, of investing for profit. His own preferences would evolve as his knowledge increased. At this stage, he found such a prospect difficult to contemplate. It seemed like a betrayal of the idea of the collection that he was still building. Daunted by such a step, nothing much happened.\n\nThe weeks and months passed, and no letter came to signal an upturn in Robinson's fortunes. The final touches were put to the building at South Kensington, and articles quickly appeared in the press to mock it. Reporters marvelled at its ugliness and its spirit of utilitarianism. Prince Albert's giant structure, clad in iron and looking like a cross between a beached ship and an oversized oil drum, was ridiculed as 'a threefold monster boiler'. And the name stuck. Robinson shuttled backwards and forwards between Marlborough House and the new South Kensington site, 'The Brompton Boilers'. He moved objects and arranged displays. With the criticism of the Prince's design ringing in his ears, he battled with the weaknesses of the clumsy architecture: leaking roofs and poor drainage; galleries that baked in summer and were brittle with winter cold. Hastily, he had to move the plaster casts from their planned home upstairs as the floors groaned and buckled under the weight. But in June 1857, despite all the doubts and disturbances, he stood proudly at the opening of the new museum, holding himself tall and straight for the curious eyes of the press and public, stupefied for a moment by the enormity of what had been achieved and feeling his future unfolding before him.\n\nThe opening of the new buildings at last prompted the chequewriting hands of the board into action, and Robinson settled into his new professional home with a slightly improved salary of \u00a3450 a year, the dual responsibility of Librarian-Curator and an assistant. This was a comfortable living: probationary police officers were paid only \u00a310 a year, a chaplain could expect a salary of around \u00a330 and a letter sorter at the General Post Office earned \u00a390 a year, about the same as a senior teacher, although solicitors and barristers could expect to earn as much as \u00a31,800 a year by this time. But in the context of Robinson's professional colleagues, the pay rise was minimal, far from the \u00a3600 a year, plus accommodation, enjoyed by the curators at the British Museum or the \u00a3750 salary granted to the Keeper of the National Gallery. Moreover, it still did not pay the bills. There was no choice: some of Robinson's collection would have to be sold.\n\nI have not found a record of exactly what Robinson chose to part with at this time. Certainly, the core of his collection remained intact, but even so the process was a wrench. Characteristically, he took refuge from the distress of the sale, and the frustrations of life at the museum, in the showrooms of Europe's antique dealers. The museum may have been unwilling to raise his salary substantially, but it was prepared to fund continuing research trips abroad. Robinson undertook a series of yearly expeditions, each lasting several months, and each yielding 'an infinity of treasures. . . at fractional prices'. In the summer of 1857, just after the museum opening, Robinson took off to Dresden and Vienna. By the summer of 1859, he was back in Italy on the trail of 'cartloads of majolica ware, innumerable cassoni, terra-cottas, and bronzes'. He was in the mood for aggressive collecting. The spoils of Italian cultural life 'must be diligently fought for', he asserted, throwing himself wholeheartedly into the crusade to find new objects and writing home enthusiastically that 'Florence has not had such a raking out as this within the memory of man'.\n\nRobinson relished the independence of collecting on the move. His days took on the rhythms common to collectors across the Continent: bursts of enthusiastic buying, long hours of frustrated travelling and the enduring desire to discover more and better things. If there were gaps in his knowledge, he plugged them with scholarly reading and conversation, so that he could hunt out and identify the best objects. Alongside his study of sculpture, he launched into the appreciation of bookbinding and manuscripts, textiles, metalwork, furniture and jewellery. He travelled quickly and bought extensively. He felt liberated again. He could ignore the irritations of the museum, and what he viewed as Henry Cole's attempts to limit him.\n\nBut such freedom was not to last. Robinson, with his nose on the trail of prestigious collections, perhaps misread the mood back at South Kensington. The museum, installed in its new site, was becoming more professional, and Henry Cole, free of the burden of the building project, was turning his attention to improving administration; there was a growing realisation that the museum should provide a model of what could be done, a blueprint for budding municipal projects up and down the country, and, indeed, across the world. In the years after the opening of the South Kensington Museum, a number of institutions were established along similar lines, with an emphasis on educating and developing public taste, raising the quality of manufactures and providing a showcase for craftsmanship. In Vienna, the Museum of Applied Arts opened in 1864, and in Berlin and Hamburg, two large and ambitious museums began assertively collecting the decorative arts during the 1860s with such success that by 1878 _The Athenaeum_ was proud to announce that 'Art and Industrial Museums, humble copies of our own parent establishment in South Kensington, continue to spring up all over Germany.' And it was perhaps in the USA that the specific example provided by South Kensington was most admired and emulated. _Harper's New Monthly Magazine_ claimed that the founding of museums in Cincinnati, Chicago, Philadelphia and Boston was in direct and admiring imitation of the London model and hoped that this new generation of public institutions would be as successful as South Kensington in 'widening waves of taste and love of beauty through the country'. Meanwhile, the _New York Herald_ looked forward to the opening of the Metropolitan Museum of Art in early 1872 by explaining that it would be modelled on 'the splendid South Kensington Gardens London [sic], which is probably the most perfect thing of the kind in the world. . . a great work in behalf of civilization and education'.\n\nIn the light of such praise, Cole was no doubt aware that others were scrutinizing his models of curating and display. With the management of the British Museum and the National Gallery drawing such frequent criticism, he may well have seen a further chance to impress. Public discussion of the National Gallery had singled out the museum at South Kensington \u2013 as well as his leadership \u2013 for commendation: 'We cannot refrain from bearing testimony to the ability, knowledge, and devotion displayed by Mr Cole in the management of the Kensington Museum,' suggested an article in _Quarterly Review_. 'The varied and admirably arranged collections exhibited there now form one of the most useful and interesting exhibitions of the metropolis, and are a convincing proof of what may be accomplished in a short space of time by well-directed and unfettered energy.' With such approval from the critics, and as the museum matured, Cole turned his attention more and more to securing its administration. Creative collecting and active educating needed reinforcing with clear and accountable day-to-day systems. This meant the careful management of staff and finances \u2013 and it meant keeping a close rein on Robinson.\n\nIn Rome in 1859, Robinson sat down to write again to the museum from the cramped cold of his winter lodgings. This letter was swiftly scribbled, the tone urgent and sure. Robinson had his mind firmly on collecting and not on museum administration. He wanted to buy fifteen key pieces from the collection of Giampietro Campana who had spent years leading archaeological expeditions and bringing together Etruscan, Roman and Greek art, as well as Renaissance masterpieces. Campana was well connected \u2013 he was adviser to the Grand Duke of Saxe-Weimar, and his wife's family was related to Napoleon III. He was also responsible for the papal Monte di Piet\u00e0, a system offering financial loans. It worked almost like a pawn shop: wealthy patrons contributed to a central fund, without the expectation that their money would be returned. Those needing a loan could apply to the fund, exchanging an object of value for ready cash. In 1857, however, Campana was arrested for using the Monte di Piet\u00e0 to finance his own collecting. He was stripped of his collection and imprisoned. Almost immediately a scramble began across Europe to take advantage of his disgrace. Collectors poured into Rome, and Robinson was just one of the international connoisseurs negotiating for the best pieces as they came on sale. Campana's magnificent collection finally raised about \u00a3207,000, the largest amount ever realized, to that date, for a single collection.\n\nRobinson also wanted over half of the collection (almost seventy pieces) which had belonged to Campana's agent, who had also found the Monte di Piet\u00e0 funds a useful way of supporting his personal collecting and who had been caught up in Campana's downfall. He believed he had negotiated a fair price. He wrote to the museum and made the case for the purchase plainly, requesting that the money be sent without delay so that he would not have to risk fending off rival collectors. But again, Robinson had to wait and, when an answer finally came, it infuriated him: there would be no funds forthcoming, he was told, and he was to make no immediate purchases. The Campana treasures went elsewhere \u2013 many were bought by Stepan Gedeonov on behalf of the Imperial Hermitage in St Petersburg and by Napoleon III for the French collections \u2013 and Robinson had no choice but to return to London empty-handed.\n\nIt is easy to imagine the gloom of the journey. It was a wet spring and tensions were running high among the various Italian factions vying for power as unification loomed. Roads were frequently blockaded and travellers stopped and searched. Progress was slow. And all Robinson had to think about as he waited at barriers was the humiliation and disappointment of having his advice spurned by the museum. He was not a man to doubt himself and he could see no satisfactory reason why his acquisitions should have been blocked. The museum was clearly mistaken and short-sighted in its refusal to buy the Italian works. He felt that his ambition for the collections was being wasted, and his vision for their future distorted. He was burning with a sense of injustice and grievance that was to smoulder inside him for a very long time. And on his return to South Kensington in the autumn of 1859 there was a furious row.\n\nBoth Robinson and Henry Cole, it seems, were tired of playing cat and mouse with each other. Their rumbling complaints and accusations cracked open in a series of irate exchanges. Robinson bemoaned the uncertainty of his position, his poor salary and the lack of trust the museum appeared to have in his judgement. Belligerent and cantankerous, he stamped and puffed and hollered. In return, Cole, who was more than able to hold his own in an argument, prodded away at Robinson's administrative shortcomings. He demanded to know why Robinson was not keeping any record of his movements and transactions and why his official diary was little better than blank.\n\nRobinson's old-fashioned, self-reliant way of collecting was finally colliding with Cole's administrative priorities and the growing sense of the curator as a public servant instead of a free-wheeling connoisseur. Robinson's habits of collecting for the museum were much the same as the habits of a gentleman collector spending his own money. He believed that the justification for buying the best things was self-evident, and he could not understand the need to constrict or explain his activities by means of record-keeping. There was a sense in which he saw himself as some kind of crusader, travelling Europe to liberate the most beautiful treasures, like those of Campana's collection, and bringing them back to safety. The spheres of private and public collecting were intimately entangled during the Victorian period, more so than ever before or since: the collection being developed at South Kensington was so much a part of Robinson's character and had his identity so clearly written into it that it was difficult to see any distinction between his personal choices and those he was making on behalf of the nation. He took advantage of the ambiguities of a role that was still evolving, letting his personal collecting priorities drive what existed of a public policy.\n\nCole's accusations only acted to entrench Robinson yet further in his ways, and to reignite the continuing battle over the philosophical basis for the collections. The two men seemed to be moving ever further apart. Robinson complained that the office accommodation was cramped and uncomfortable, moaning that 'we are all packed close as pigs' in the square rooms. But alongside his physical discomfort he also started to complain about the 'ceaseless tramp' of visitors to the popular new galleries. It seemed as though the practicalities of open public access were beginning to grate on Robinson's sensibilities. His discontent with Cole's mish-mash of modern collections at South Kensington was becoming identified with a wider disillusion with the way the museum was developing. Robinson's interests remained with Europe's elite collectors, and, as his contacts grew among the members of the Fine Arts Club and his own collecting became increasingly expensive and erudite, he was perhaps beginning to feel superior to the populist approach at South Kensington. His sense of what was right for the museum was constantly being offended: in 1861, funds for Robinson to take another planned trip to Italy to study Renaissance sculpture were diverted instead for Cole to go to the International Exhibition in Florence to buy objects for the less refined museum displays, including food and animal products as well as examples of contemporary design. Cole's new attack about record-keeping only added to Robinson's feeling that policy was heading in the wrong direction. He was desperate to prove that Cole was wrong \u2013 and that he was right.\n\nIt was a relief, in the end, to close the 1862 exhibition and to be able to walk through the galleries again in the heavy quiet, with the visitors gone and the noise of activity in the rest of the museum somehow distant and without meaning. Robinson could take a moment now to breathe deeply in the calm, catching the scent of the past. The exhibition had proved a success, and had reasserted his position as the champion of the rare and the beautiful. It had boosted the collections: many of the lenders had been persuaded to donate their objects permanently to the museum. And it had cemented Robinson's role at the heart of the Fine Arts Club, and at the heart of British collecting.\n\nRobinson had founded the Fine Arts Club just six years earlier in 1856, dedicating it to helping collections grow and improve, and to finding the best objects. The Club, which was later rechristened the Burlington Fine Arts Club, began quietly enough with evening receptions in Marlborough House, 'frequently', as Robinson boasted, 'of the most elaborate and costly character': fine wines and fine food shared among those who enjoyed talking about art. But it soon became extremely popular and membership had to be limited to two hundred, to make it manageable.\n\nThe club's membership was erudite and varied, drawn from all kinds of professional backgrounds and crossing political affiliations. There were practising artists such as John Henderson, a gifted amateur painter and archaeologist, and Baron Carlo Marochetti, a sculptor. There were wealthy establishment figures such as Sir Henry Thompson, surgeon to the Queen. Politicians included Lord Overstone, a celebrated banker and High Sheriff of Warwickshire, and Alexander Beresford Hope, a vehement opponent of the 1867 Reform Bill which aimed to extend the vote to the urban working class. There were also, inevitably, representatives of the English aristocracy, such as the Duke of Hamilton and Sir William Holburne, whose renowned collection was displayed in his fashionable Bath townhouse. And there were members who were very much engaged in the debates about public education and access to art. Meetings were attended by both Charles Eastlake, director of the National Gallery, and John Ruskin. The membership also included the Liberal politician William Gladstone, who was to serve the first of his four terms as Prime Minister from 1868, and who managed to persuade Parliament to pass the Education Act of 1870, to establish public school boards. In a departure from the usual model of the gentleman's club, women, too, were invited and by 1867 there were eight female members.\n\nThe club's mixed membership no doubt led to some lengthy discussions, but the emphasis was firmly on collecting, rather than politics, and members clearly relished the chance to meet, if only to keep an eye on what others were doing. It was an indication of the power of the collecting impulse that the club managed to draw together such a disparate crowd with one common aim. It was also a testament to Robinson's diplomatic skills, and it demonstrated just how confident he was \u2013 as a curator from an ordinary background \u2013 mixing in such circles. Under Robinson's guidance, meetings flourished and when the museum became established in South Kensington, so too did the Fine Arts Club. But in time meetings were also held increasingly in private houses. This clearly allowed members to show off their collections to each other, but it might also have signalled a growing closing of ranks, and a desire to create a more elite atmosphere. The election of members by invitation created a structure for excluding more modest collectors; the membership limit of two hundred could be seen in this light as a matter of policy as well as practicality, ensuring the club was reserved for a comfortable coterie of learned friends. Collecting at this level was still entangled with the traditions of social class, and many members of the Fine Arts Club were no doubt keen to differentiate their activity from populist spectacles like the Great Exhibition. Robinson himself was certainly alert to such distinctions: the 1851 event at Crystal Palace was, he recognized, 'the apotheosis of commerce and the shopkeeper. The aristocratic and cultured classes had comparatively little sympathy with the great exhibition, and had little or nothing to do with it', whereas the Fine Arts Club 'comprised every connoisseur of note in the country'.\n\nDespite some of the club members' more liberal credentials, it was this atmosphere of exclusivity and culture for the privileged few that became an issue in the wake of the 1862 exhibition. Cole found he was having to fend off assaults from the press and Parliament who accused the museum of forgetting its original educational purpose. The splendour and refinement of the 1862 show \u2013 displaying, perhaps too clearly, the obvious wealth of those who organized it \u2013 reinforced the impression that the museum had abandoned its roots in favour of a more glamorous and elegant existence. Despite all of Cole's hard work on the core principles of design education, many visitors were getting the impression that the South Kensington galleries were straying away from sound and useful practice. During one debate, MPs harangued Cole by name and claimed the museum was no longer fulfilling its express original purpose of providing examples for education and training and had become instead 'nothing but a great toyshop for the amusement of the residents in the west-end'.\n\nAlthough Cole vociferously defended the museum against such criticism, there was no disputing the fact that the evidence looked damning. Students themselves, at design schools both in London and the provinces, complained continually that the museum was letting them down and that the collections were inaccessible and of little use to their studies. A written statement from the students at the London School of Design in 1864 declared the displays of 'limited utility' and one Lambeth student claimed that the museum 'might as well be in the moon' for any use it was to him. Cole was under increasing pressure to justify the amounts that had been spent \u2013 mostly by Robinson \u2013 on paintings and other objects which were of only slight relevance to design education: 'What use to the practical workmen of the present day is the reliquary purchased recently. . . in Paris for the enormous sum of \u00a32,142?' demanded _Art Journal_. And indeed, when the accounts were finally examined, it was found that most of the public money given to the museum by the government in the decade since its opening had been used for non-educational purposes: to fund the increasingly complicated administration, to develop and maintain the growing complex of buildings, to finance travel abroad and to buy expensive works of art for the collections, in line with Robinson's private tastes. Visitor numbers continued to be healthy: figures hovered around 40,000 to 50,000 a month, for admission on free days, with as many people coming after work during evening opening as during the day. But the purpose of the museum seemed, especially to those who were funding it, to be increasingly opaque and confused.\n\nCole could not shake off the feeling that much of the criticism was Robinson's fault. During the grey, smoggy days of January and February 1863, the mood at the museum became fierce and gladiatorial. Cole found a host of accusations to level at Robinson, complaining in particular about his unremitting insolence. Once again, he ripped into the state of Robinson's record-keeping. This time, however, he had a new complaint. Cole suspected something other than just administrative incompetence. Robinson was not simply forgetting to record his movements and to write up his sales \u2013 he was apparently being purposely secretive. He was, Cole believed, ensuring that the museum could not trace his movements nor piece together his activities. With the wet sleet falling thickly on to the glass roofs of the new galleries, melting into the courtyards in heavy streams and stifling the rattle of the world beyond, Cole finally voiced the accusation that had been gnawing within him for so long: was Robinson collecting for the museum, or for himself ?\n\nRobinson did not respond. It was clear that the line between his public and private collecting was becoming increasingly blurred, and that his enthusiasm for collecting was apt to override all other concerns, but in reply he simply repeated his own grievances: he complained again about the trouble he had getting his purchases sanctioned, about the heating of the cranky building and about the poor lighting in the galleries. Cole's question remained unanswered and, as the cold days dragged on, so too did the series of bitter exchanges. Cole could not break down Robinson's cantankerous self-confidence; Robinson continued to make it clear that he regarded much of the museum's achievement as his own work. Inevitably, the dispute leeched into the wider workings of the museum, drawing the staff and authorities into the venomous quarrel. During March 1863, Robinson bombarded two of the museum board members, Lord Granville and Lord Lowe, with a series of letters. In reply, the board registered its strong disapproval of Robinson's ragged financial paperwork and bristled at the news that he had made purchases in Italy without waiting for permission, taking the decisions on behalf of the museum 'on his own responsibility'. It noted that Robinson's accounting from the 1862 loan exhibition was 'loose and unsatisfactory' and in desperation withheld the cash gratuity Robinson was due as a reward for his hard work.\n\nThe unending squabbles took their toll on Robinson's health. He suffered from almost permanent indigestion, headaches stopped him working and he found he was getting unusually confused by even quite simple tasks. But he would not give up the fight. His tone was confrontational, contemptuous and bitter. His irresistible ambition for the collections was at the heart of his continuing complaints, but his subject matter, more often than not, was Henry Cole. He continued to send a string of disparaging notes to individual board members, damning Cole's 'administrative talents and practical energy' with faint praise and openly challenging his superior: 'his specific knowledge of Art', he wrote on one occasion, 'is in my opinion not equal to the actual work of forming and technically directing a public Collection of such a varied and comprehensive nature'. As if that was not condemnation enough, more was in store. Cole was, Robinson claimed, unfit to make decisions or negotiate deals on the museum's behalf: 'Unfortunately, Mr Cole's selections were rarely either timely or judicious, made on the spur of the moment by accident or momentary impulse,' he alleged.\n\nRobinson's accusations were not without precedent elsewhere. In November 1853, William Gladstone, as Chancellor of the Exchequer, had bought sixty-four German paintings for the National Gallery. By the summer of 1854, William Dyce, an established artist and previously the head of the government School of Design in London, had looked at the paintings and found that most of them were of low quality. Only seventeen were worth keeping for the national collection. On this occasion, it was the gallery, rather than Gladstone, that attracted criticism: the incident was used as further evidence that the trustees were failing and that public money was being ill-spent. By directing a similar attack at Henry Cole \u2013 perhaps with the Gladstone incident firmly in mind \u2013 Robinson was reminding the board members that they were likely to be held to account if Cole's alleged incompetence came to light. He was clearly hoping to use the board's discomfort for his own ends. His concern for the use of public funds seems ironic, given his habitual free-spending on the museum's behalf, and it was more likely that he was hoping for a fundamental change in leadership that would allow him to take charge, or at least rid him of an irritant.\n\nAs spring arrived in South Kensington, the leaves splitting green on the trees and the rotten mud in the streets beginning to harden, the internecine war at the museum was at its most vicious. Cole was at his wits' end with his subordinate's defiance, and Lord Lowe, on behalf of the board, proclaimed angrily that Robinson's letters must stop. Robinson, however, remained in rebellious mood and, when the museum failed to allocate him one of four new purpose-built staff residences, he took it as the final slight. By Easter, as a flush of warm weather blew into the city, things were at boiling point.\n\nIt was just before the Easter of 1863 that Henry Cole asked to see Robinson. Cole had had enough of his irritating curator and had decided to act: 'never keep a man who is dissatisfied, he is mischievous. . . this would perhaps apply to J.C.R.', he admitted to his diary. He told Robinson that he would have to leave his post as curator. The museum was changing; it needed a new staff structure, and Robinson was being offered the role of salaried consultant or 'Art Referee', without control of the collections but still with a remit to give advice on purchases. Robinson was taken by surprise. At first, he was convinced that he would find a way to overturn the decision against him \u2013 and get rid of Cole in the process. But in the end there was nothing to be done. The board told him firmly that 'he must submit or go' and for a while \u2013 and against his natural instincts \u2013 Robinson chose submission. But he was uncomfortable in his new role and his wounded pride blistered. He soon found there were to be three art referees, and he was not even destined to be the most senior of them: that role was to be filled by an old friend of Cole's from the Schools of Design, Richard Redgrave. Pushed further down the hierarchy, Robinson felt that his vision was being purposely slighted.\n\nBeyond the museum, Robinson's expertise was already being missed and his demotion noted with alarm. His fellow collectors knew all too well that they were determining the shape of the national collection at South Kensington just as much as the utilitarian creed of Cole and the board, and Robinson's network soon began to rumble with the discontent of losing their linchpin. At the Fine Arts Club, influential members were dismayed to find that their interests were being sidelined and the discussion at meetings regularly slipped away from the objects on display to an animated discussion of Robinson's fate. A stinging piece in the influential _Art Journal_ publicly voiced the growing dissatisfaction. In an opinionated article that was read by all the most important collectors, dealers and curators, the magazine took a snipe at Cole's leadership and lauded Robinson. 'It is to his [Robinson's] indefatigable industry, no less than to his sound and matured knowledge, that we are mainly indebted for the value of the Museum,' _Art Journal_ announced. The article made its loyalties plain and recognized, too, where the real power lay \u2013 not with the museum or with its administrators, but with individual collectors up and down the country. Collectors mattered, and Robinson, it claimed, 'has the confidence of every collector in the kingdom'. As the journal warmed to its subject, its attack on Cole became increasingly spiteful. Cole's abilities, it noted scathingly, were 'not at all equal' to Robinson's scholarly connoisseurship and it was with distinct pleasure that it offered a final, damning judgement: 'it is not pleasant for a colonel to know that a corporal he commands is a better soldier than himself '.\n\nWith the backing of his collectors, and in defiance of his real lack of influence, Robinson harried and annoyed the museum staff and directors, and Cole, of course, in particular. For four years, he goaded and bothered and griped. Minutes of the museum board meetings noted, wearily, that Robinson still needed constantly reminding of the limits of his duties, but, even in the face of persistent reprimands, he still could not believe that he would not make progress. He continually tried to conjure up authority for himself, anxious to make it appear that he retained the vestiges of power. Even in the face of a Select Committee of the House of Commons in 1867, he raged against Cole's commitment to purchasing examples of contemporary design for the museum, arguing fearlessly and proudly that the works he had been buying were 'vastly more important'. It became clear to Cole that even at arm's length Robinson was making a nuisance of himself. He decided to act before he was undermined by Robinson's constant sniping and the distressing criticism from his allies. He looked around for something \u2013 anything \u2013 that would give him the excuse he needed to make the final break. And in his neat piles of paperwork and closely written documents, he found it.\n\nIt was Christmas 1867. The windows of the small stores along the Brompton Road, a short stroll from the South Kensington site \u2013 including the premises of the rapidly expanding Harrods \u2013 were bright with gifts and treats and lights. Delivery boys pushed their barrows and carts anxiously through the traffic as the ladies of the fashionable middle classes, bobbing within the frames of their wide crinolines, shopped diligently and, if they could, stylishly. Robinson's seven children were no doubt lightheaded with the excitement, with the promise of change from their strict routine, with the soft smells drifting through from the kitchen and with the thought of puddings and sweets and oranges.\n\nAlone in the museum, Henry Cole was reading a museum staffing report written two years earlier for the Treasury. The language was formal and elaborate, the tone sombre. There was nothing festive about it. But contained within the convoluted sentences was the ammunition he needed to rid himself of Robinson for good. The report proposed saving money by reducing the museum staff to its general administrators. Specialist expertise, it suggested, could be bought in temporarily, as and when it was needed. The idea of art referees drawing a salary for consultancy work was, perhaps unsurprisingly, highlighted as uneconomic. A committee, drafted in to look at the implications of restructuring, agreed. Two days before Christmas, on 23 December 1867, Robinson's post was abolished. 'Request that Mr Robinson's superannuation may date from 1 April 1868. He is to have notice from 1 January,' noted the official memo blandly. Robinson's life was about to change irrevocably. As he celebrated the close of the year at home with his family, the unaccustomed lightness and gaiety of the season lifted, for a moment, his disappointment and frustration. But there was no escaping the bleak realization that his tenure at South Kensington was over. On 10 January 1868, the board wrote to Robinson again, reaffirming their decision, and he was forced to accept that things were really at an end. He would have to leave.\n\nWorking out his notice was frustrating and painful. His fifteen years as a curator had shaped him. He had started as an eager and knowledgeable young man, ambitious to succeed in London, but na\u00efve and obscure. He was leaving as an established connoisseur and scholar, respected at home and abroad, and with a reputation for clever and imaginative collecting. He had become integral to the growth of the South Kensington Museum from a few borrowed rooms in Marlborough House to an important international collection in purpose-built galleries. He had developed the first British collection of medieval and Renaissance decorative art and, in doing so, helped create a whole new field for British collectors. His spiky relationship with Henry Cole and his bitter exchanges with the board had sharpened his determination and clarified his sense of purpose. His self-belief had swelled and his confidence matured. Now everything was crumbling.\n\nSuspicions and accusations that had been grumbling poisonously in the background for months, and even years, came bubbling to the fore now that it was certain that Robinson was leaving. There were plenty of disgruntled colleagues willing to step forward to tell tales. Robinson was not an easy man to work with and he was not afraid to offend those he regarded simply as clerks, bureaucrats and charlatans. Now, with his power draining from him, he found he was prey to the spite of gossip and unconstrained resentment. And the museum authorities, thoroughly weary of Robinson's defiance, seemed eager to make it clear just how right they were in dismissing him. Cole's long-held allegation that Robinson had been using his position at the museum to make money as a dealer \u2013 that he had been buying and selling on his own behalf when he was travelling on museum business \u2013 resurfaced with unpleasant vigour: 'An Art Referee has not considered himself precluded when abroad from purchasing objects of art for friends,' noted the board brusquely in early January. Robinson responded to the allegation with his usual energy. He may have been out of favour, but he remained defiant. As far as he was concerned, his name was being slandered and his work belittled. So furious did he become that the board backtracked, perhaps unwilling to get too embroiled when it was only a matter of weeks until they were free of Robinson's bickering: 'It is not intended to cast any imputation or reflection on your honour and fidelity as a public servant,' the museum reassured Robinson on the 20th. But it was a small victory, and a chilly valediction. By the spring of 1868, Robinson was gone.\n\nCole and Robinson seemed to enshrine two possible futures for the museum at South Kensington, with Cole's breadth of activity and administrative professionalism standing in contrast to Robinson as the scholar and collector. Cole was committed to the principles of public service and was eager to explore all kinds of different projects from inventing commercial Christmas cards to mounting international trade fairs; Robinson could not be shaken from his independent habits and was happiest at the heart of his collection. But the sense of the two philosophies being completely irreconcilable was largely illusory. Across Britain and Europe, at other museums, the same principles were being debated and the same changes were being worked through as the ideas and practice of curating collections evolved. It was largely as a result of the clash of two belligerent and opinionated men that the process at South Kensington was so painful.\n\nA few miles away at the British Museum, another collector, Augustus Wollaston Franks, was making his own personal mark, creating an environment of professionalism and scholarship that evolved a little more peaceably than at South Kensington. He joined the staff at the British Museum in 1851 (two years before Robinson first took up his post at Marlborough House), with responsibility for developing the neglected collections of British objects, and he remained at the museum for almost half a century, until his death in 1897. Determined and strong-willed, and occasionally temperamental, he completely changed the collecting policy and departmental structure. He made significant personal contributions to the collections, and oversaw the growth of the British galleries from 154 feet of wall cases and a handful of table cases to 2,250 feet of wall cases, 90 table cases and 31 upright cases. He is regarded by many as the founder of the modern British Museum.\n\nEducated at Eton and Trinity College, Cambridge (where he spent much of his time collecting and pursuing his own architecture and archaeology projects), Franks was keen from the outset to develop an intellectual environment at the British Museum. His sympathies lay with the scholar: he would countenance general visitors to the museum if he had to, but he outlawed children, who could, he was sure, 'derive no benefit from seeing the collections'. He became an expert on a range of subjects from Japanese flint instruments and Anglo-Saxon ivories to Indian sculpture and English ceramics. He wrote original and learned articles (though he did not publish very much), kept up an energetic correspondence with other collectors and travelled widely, becoming an important member of an international group of scholars who met throughout Europe to discuss anthropology and archaeology. Like Robinson, he combined a commitment to public collections with a private collecting habit. Alongside his curatorial work, he amassed an encyclopaedic personal collection, some of which was left to the British Museum in a vast bequest that included 3,300 finger rings, over 500 pieces of highquality European porcelain, 1,500 netsuke (miniature Japanese sculptures), 30,000 bookplates and other pieces of jewellery. 'Collecting,' he admitted, 'is a hereditary disease, and I fear incurable.'\n\nFranks' drive to collect \u2013 for himself, his friends and colleagues and his museum \u2013 meant that his activities often occupied the grey area of confusion between the public and private. Like Robinson, the strength of his personality seemed to be the cohesive force that often made sense of what and how he collected, although to modern eyes the failure to delineate between his personal and public roles can seem strange. Objects came and went freely between his own collection and the museum displays. In 1885, for example, Franks was cataloguing a 'painted figure of a bald person' brought from Japan for the British Museum, when he decided to swap it for a better one of his own: 'in place of this, removed by me from the collection, has been substituted a much finer wooden figure', he noted. Between 1864 and 1883, when Franks was looking after part of the British Museum's collections in a store in Victoria Street, Westminster, he explained that 'I paid myself the clerk or secretary who worked at it with me', rather than burdening the museum with extra costs. There is no doubt that he saw little difference between the different aspects of his collecting; to him, the priority was to amass the finest things, however that might happen. To Franks' advantage was that he was a more conscientious administrator than Robinson and so his decisions and acquisitions were better documented.\n\nOn his arrival at the British Museum, Franks, like Robinson, soon began to reshape the collections. The British Museum had been dominated by the library departments, with the Director also being Librarian, and it was objects from the classical world that mostly filled the galleries. Almost single-handedly, Franks created collections from other countries and other periods that established whole new areas of interest: instead of classical pottery, he bought Japanese porcelain; he introduced English pre-historic material; and he invested in Chinese bronzes and objects from European archaeological excavations. Just as Robinson was challenging Cole's vision for South Kensington, so Franks was circumventing the traditions and bureaucracy in Bloomsbury to achieve what he wanted.\n\nUnlike Robinson, however, Franks was also a skilled politician. After some initial sparring about the direction in which he seemed to be steering the museum, those above and around him came to respect and support his work. He was friendly with many of the museum's trustees. Coming from a distinguished aristocratic and banking family, the grandson of a baronet and cousin to the Earl of Harewood, and being a man of considerable fortune (although it is unclear exactly where this came from), Franks was also in the enviable position of being able to sweeten his politicking with generous, practical gestures. He gave over 7,000 objects to the museum during the time he was a curator there. With this kind of incentive, it is perhaps no surprise that his work was so widely admired by those above him in the hierarchy, nor that, unlike Robinson's superiors, they never cast doubt on whether public money was being siphoned off to feed Franks' personal collecting habits. No matter how he disguised it, Robinson could not escape the fact of his humble provincial origins. The leeway Franks was given, in contrast, was a reflection of his upper-class background and the consequent, instinctive, assumption that he could be trusted. It appeared to be taken for granted that Franks should be allowed to collect, obsessively, and that in the end the British Museum would benefit; there seemed to be little need to bother Franks with discussions about a conflict of interest.\n\nFranks became a mainstay of Victorian collecting, the epitome of the scholar-collector. And the work he undertook to move the British Museum towards a more scholarly, collections-based approach was not unlike Robinson's crusade at South Kensington. For Robinson, however, it was the nature of the collections at South Kensington \u2013 as well as his clash with Henry Cole \u2013 that meant progress was particularly tortuous. South Kensington embodied a new and largely untried idea of a museum that might collect neither antiquities nor paintings, but something from the vast number of objects that fell somewhere beyond and between such categories. It is little surprise that the politics were complicated and fiery, nor that the identity of the collection was sometimes vague and confusing. The work both Robinson and Franks were doing, however, showed the increasing importance and influence of professional collectors. Working with large sums of public money, among a community that stretched across Europe, these new collector-curators were very much at the heart of things, forging a new vision for collecting that would last for generations.\n\n### CHAPTER SIX\n\n### _The Tricks of the Trade_\n\n_D_ espite Robinson's spirited defence, and the board's retraction, the notion that he had, for some while, been combining his work for the South Kensington Museum with work for private collectors proved hard to shake. There were plenty of people who marvelled at his ability to run his smart household on a curator's salary alone, and the conspicuous blanks in his museum diaries, especially when travelling abroad, were commonly held to conceal all kinds of private transactions. The belief that he had been using his knowledge and contacts as a collector to make money from dealing was widespread \u2013 and the idea stuck because there was truth in it. Travelling for the museum across Europe, Robinson was ideally placed to know where and when pieces were coming on to the market. From his discussions at the Fine Arts Club, he also knew exactly what people were collecting, and which works they were eager to acquire. It was not difficult to put the two together for profit. Robinson was a collector who revelled in the thrill of collecting, the sport of it, the hunting out and tracking down, the negotiations, intrigues and manoeuvrings; he also had a voracious appetite for acquisition. Funding this was desperately expensive. Adding to his museum salary seemed to make sense, either by buying directly for friends or by speculating on pieces that he knew he could sell on himself for a reasonable profit at auction.\n\nRobinson bought and sold furiously. It was not necessarily unusual at this time for the distinction between collector and dealer to appear hazy. Dealers could end up amassing far more stock than they could sell, and so metamorphosed into collectors; the lively and extensive interaction between collectors meant that it was common for prized objects to change hands between them. But Robinson's official status at the South Kensington Museum made his situation sensitive; his foray into dealing, especially while on official trips abroad, was a disquieting complication of his role as a public servant with responsibility for public money. On top of this, there was also a lingering sense among some collectors that collecting and dealing should remain separate.\n\nWhen Augustus Franks applied to join the British Museum in 1851, there was much discussion among his family as to whether a gentleman of his standing should be considering paid work. He had a ready entr\u00e9e into high society and the reputation of being a keen member of his prestigious gentleman's club, the Athenaeum. He was certainly wealthy enough to be dismissive of the salary on offer, regarding it as pocket money. In these circumstances, the idea of a 'job' was distinctly uncomfortable. In the end, however, Franks managed to construct a philosophical basis for his work that satisfied his family's sensitivities, while allowing him to continue professional collecting: he positioned himself very clearly as a public servant, in the same way as other gentlemen of means in the Foreign Office, Parliament or the Church. He accepted a paid role at the museum, but also managed to keep himself apart from the usual administrators and workers. He made frequent purchases from his own money that he then donated to the museum, amounting to a value of around \u00a350,000 (which would correspond to as much as \u00a32 million today). This alone set him apart, reinforcing the impression that he was a gentleman of rank.\n\nRobinson, it was clear, was treading a tricky path. Dealers were men of trade, while collectors were gentlemen. It would not do for the two to be confused. Austen Henry Layard, a diplomat, MP and archaeological pioneer, as well as a trustee of the National Gallery, voiced the concerns of the establishment. Robinson, he snorted, was 'bumptious and odious. . . the more I hear of Mr Robinson the less I like him. He is nothing but a dealer \u2013 up to every trick of the trade.' Layard was, like Robinson, a collector. He was an excellent draughtsman and a scholarly historian and writer, specializing in European art. The two men's interests intersected particularly in their enthusiasm for Italy \u2013 where Layard was born and where he was to retire in the 1880s to write on Italian art and to develop his collection. They should, perhaps, have been friends. But Layard's assessment of Robinson, and his easy dismissal of dealers in general, was a clear indication that in many minds there remained a gulf between collecting and dealing. To be 'nothing but a dealer', Layard implied, was to be nothing much at all.\n\nRobinson may well have harboured similar reservations himself. He was now a man with excellent social connections, a reputation for scholarship and a first-rate collection, and he was surely loath to sully any of these with the suggestion of commercial coarseness. But in 1868, with his tenure at the museum now at an end, he needed funds more than ever. He put fifty-six paintings and seventy-one Old Master drawings into a sale in Paris in May, but still money was tight. He could not afford to ignore the potential for using his knowledge to make a living. Moreover, he was distinguished, energetic and, at forty-three, still young. He was well equipped to deal with the demands of a new challenge, and well placed for forging a new career. If he was obliged to reinvent himself, what could be better than becoming a helping hand for other collectors, an expert friend, a seeker out of bargains and a stalwart at the sales? He would, at least, be free from the restrictions of public administration and accountability.\n\nIn 1871, Robinson was put forward for the directorship of the National Gallery, but when this fell through it seemed certain that his formal connection with British museums was over. His new life trading objects had begun in earnest. Nevertheless, it was likely that Robinson regarded himself as a successful collector, rather than as an outright dealer. He never owned a shop and he only worked with a range of refined, high-quality objects. But in practice the outcome was the same: collecting was making a profit, and Robinson's new career flourished rapidly. In 1873, just five years after leaving the South Kensington Museum, he had made enough money to begin looking around at suitable properties to supplement his London townhouse at York Place in Portman Square. He chose Newton Manor, near Swanage in Dorset, a secluded seventeenth-century house with walls of local Purbeck limestone, nestling among tall elms that creaked in summer winds.\n\nWhen Robinson first discovered the old manor house, it had been used for years as a farm store and was putrid, neglected and shabby. Nonetheless, he was immediately taken with it and quickly set to work clearing the mess of low agricultural buildings and lean-tos from the back, and emptying the rooms of their animal inhabitants. 'Bats, rats and mice occupied the bedrooms, a colony of owls was established in one of the stone chimneys, and a swarm of bees was installed. . . at the other end of the roof,' he noted delightedly in a pamphlet of his reminiscences. Even better, 'we possess a private breed of spiders, fine, big, long-legged creatures, as active as race horses. One of their amiable customs is to drop down on the shoulders of our lady guests at dinner time.' Robinson improved land on the estate, building new roads and planting trees, and discovered an interest in horticulture. He took to spending time outdoors and became fond of the many fine views stretching away across England's south coast.\n\nThere was just one disappointment \u2013 the house lacked a ghost. It certainly looked as though it should be haunted but, no matter how hard Robinson listened out for tell-tale bumps and moans, the ghosts failed to appear. To make matters worse, the little cottage at the edge of the estate, which had traditionally been used as the dower house, was notoriously haunted. Robinson could not suppress his envy: 'my house has not got a ghost whilst the cottage rejoices in the possession of a first-rate one,' he complained. Despite this cause for regret, however, Robinson was in his element at Newton. He would be intense and emotional to the end of his days \u2013 without his 'extreme enthusiasm', noted his wife patiently, he would be a lesser man, a 'Samson with shorn locks' \u2013 but Newton Manor provided a sanctuary from the worst of life's irritations. The angry days at South Kensington seemed to have passed.\n\nNewton Manor was also the perfect location for an intimate, informal showroom. Robinson was keen to maintain the distinction between what he was doing and the tradition of the shopkeeping dealers pressed into Europe's major cities. Here he could create an environment that established him as the country gentleman. He could fill the house with his collection in the tradition of the aristocratic elite and create tasteful, elegant displays that had the illusion of being completely divorced from the financial realities of trade. Over the years, Robinson made a small fortune out of his knowledge of the art markets, but he spent much of it making Newton Manor beautiful. The house and its grounds became an extension of his collection. The garden was filled with Venetian sculpture, columns and fountains, including a life-size statue of Sylvanus, a muscular Roman woodland god, which puzzled and apparently alarmed locals passing by on the road. The dining room was hung with sixteenth-century Spanish panelling, Italian fireplaces and heavy carved wooden doors which gave it the sombre magnificence of an ancient ancestral home. Seventeenth-century suits of armour stood guard in the corridors; Roman busts gazed out across the estate and all kinds of fine paintings lined the walls of the old house, including a full-scale portrait of a nameless _Dutch Gentleman_. At Newton Manor, Robinson displayed his learning and scholarship, his eye for grace and refinement, his collector's instincts and his exquisite taste \u2013 everything his clients wanted in someone buying and selling works on their behalf. But Newton Manor had the added advantage of conveniently obscuring the fact that Robinson was a first-rate dealer and not a very wealthy private collector. In the simple solid walls of an old manor, Robinson set up a complicated illusion, a house of mirrors, the perfect salve for Victorian anxieties about class and the changing world of trade.\n\nThe genteel lifestyle at Newton Manor was as important to the way Robinson saw himself as to how he wanted others to see him. He bemoaned the break-up of the English country-house collections of the past and there is little doubt that he admired the tradition of collecting that had its roots in the aristocracy. As a dealer, he drew on his conservative credentials to emphasize the discretion and sheer class of the service he was offering. He tried to tempt even the most wary of men with his promise of prudent and tactful dealing, and he was able to count society heavy-weights such as William Gladstone, who somehow found time to be a ceramics collector, among his most honoured customers. Gladstone, who was Chancellor of the Exchequer at the time, sent Robinson a speculative \u00a32 in 1869 for the purchase of 'an interesting piece of Italian ware' and, suitably reassured by the return on his investment, continued to patronize Robinson from time to time.\n\nNot all of Robinson's clients were society figures, but they were all influential or wealthy, or both. Nearly six hundred works, including some by renowned artists such as Michelangelo and Raphael, passed through his hands to John Malcolm, the 14th Laird of Poltalloch, in Argyll, Scotland, who inherited three houses in 1857. He traded energetically for Robert Napier from Dunbartonshire, who had made a fortune patenting a new naval engineering method, and for Sir Francis Cook, a textile trader, who was left more than \u00a32 million by his father. Abroad, he worked for influential collectors such as Wilhelm von Bode, the creator of the Kaiser-Friedrich Museum who shared Robinson's taste for the Italian Renaissance, and the art historian Stephan Bourgeois. He even managed to establish himself as a royal dealer, cultivating the business of the Kaiserin Victoria, more familiarly known as Princess Vicky, the Empress of Germany, eldest daughter of Queen Victoria and wife of the German Emperor Frederick III. Shrewdly recognizing the value of such exclusive clients, Robinson seemed content to use high-profile royal transactions as a loss leader: he sold the Princess a still life of _Dead Game and Fruit_ by Frans Snyders in 1877 for nothing more than the price he had paid for it, generously adding a complementary new frame, and during their long association as dealer and client he carefully adhered to a non-profit policy. He even presented a number of valuable works as gifts to celebrate the silver wedding anniversary of Vicky's marriage to the Kaiser in 1883, including a _Sketch of Hampstead Heath_ by Constable and Reynolds' _A Portrait of a Child with Doll_.\n\nPrincess Vicky was a collector, from a family of collectors. Her father, Prince Albert, had thrown himself enthusiastically into raising the quality and reputation of the arts; the Royal Collection expanded in all areas under her mother's rule. Together, Victoria and Albert commissioned artists for all occasions, treated themselves and each other to sculpture for Christmas or birthdays, shopped excitedly for decorative pieces to furnish their houses at Osborne House and Balmoral, and began to encourage scholars to publish about the Royal Collection, which was begun by Charles I. Growing up with this enthusiasm, and with unrivalled access to great works of art, Vicky was knowledgeable, cultured and sophisticated, if rather conventional in her tastes. During years of political machination in Germany, and three bitter wars of German unification during the 1860s and 1870s, the Princess's love of fine things had been a comfort and an inspiration. And the environment Robinson created at Newton Manor clearly offered her a retreat: _The Times_ noted that she visited there with her four daughters.\n\nSuch a royal visit was a great achievement for Robinson. It linked him to collecting in the very highest circles and was a clear indication that his activities as a dealer had been accepted. He seemed to be offering a certain type of wealthy Victorian collector just the kind of refined and discreet personal service they demanded. And, in return, they seemed content to reward him with a great deal of money. Details of Robinson's account books reveal the considerable profit he made on reselling works to his elite network of collectors: he bought a Dutch painting of A _Lady at a Harpsichord_ at a sale in May 1877 for \u00a388, and sold it two months later for \u00a3320. Similarly, a Rubens portrait bought for \u00a390 was sold a few weeks later for \u00a3300; a painting described simply in his notes as 'Venus and Cupid' and acquired for just \u00a35 was marked up to \u00a3375; and a _Rainbow Landscape_ , again by an unknown artist, secured Robinson well over \u00a3500 profit. As his client list grew, and as he counted more and more obsessive collectors among his customers, Robinson was making transactions with impressive ease. He was working in London, travelling to Europe, entertaining at Newton Manor, networking through his cluster of clubs and societies, and making deals at every turn. He never tired of the adventure of finding the right object for the right client and, during a typical three-month period of the 1870s, he made almost 800 sales of paintings alone, as well as the unrecorded business in his other diverse interests such as Old Master drawings, sculptured bronzes, oriental ceramics and glassware.\n\nRobinson's energy and success as a dealer can be seen as a blueprint for other internationally renowned dealing careers at the end of the nineteenth and beginning of the twentieth century. Those that followed adopted Robinson's enthusiasm for mixing in influential circles, and combining erudition with a sharp business sense. Joseph Duveen, for example, one of the most well known and influential of British art dealers, began work in his father's business in 1886 and made a fortune by selling works to wealthy Americans and trading on the idea that owning pieces of art was a way to acquire social standing. Like Robinson, he had an eye for the art of the Renaissance and he too moved easily in the company of royalty and millionaires. He was also, like Robinson, confident, ambitious and single-minded. Before the First World War, he established a virtual monopoly on the trade in Old Masters and his talent for salesmanship allowed him to deal on an unprecedented scale. With his success assured, he donated money and paintings to many British galleries and funded the building of the Duveen gallery at the British Museum to house the Elgin Marbles. He was finally knighted for his philanthropy in 1919.\n\nBut Duveen was not alone. The growing number of collectors throughout the Victorian period meant a corresponding growth in the number of dealers who, in turn, created the kind of active, profitable market that later allowed men like Duveen to make a fortune. By the last decades of the nineteenth century, the advertising pages of the daily papers teemed with colourful advertisements for dealers and their showrooms. Businesses such as Gladwell Brothers offered everything from 'new and choice etchings' to cornices and console tables; the family firm of William Dyer boosted trade as picture restorers by selling a few works on the side, while higher-class dealers such as Agnew's were responding to the growing market for Old Masters and eighteenth-century portraiture. Others were following more closely in Robinson's careful footsteps, negotiating the tricky boundaries between trade and social prestige: in the 1880s, for example, Marcus Huish combined his occupation as a barrister and a role as director of the Fine Arts Society with a profitable career in art dealing. In the late 1860s and early 1870s, Robinson was at the vanguard of this new profession. He was one of the first to take on the demands of dealing alongside his own collecting. Cloaked in the disguise of the gentlemanly amateur, he was forging new possibilities as a commercially minded professional, and in the process altering perceptions both of collectors and the markets in which they moved.\n\nThe contradictions of Robinson's career at the South Kensington Museum were not entirely confined to the past, however. He could not shake off his attachment to the South Kensington collections, nor the idea that he should continue to play a part in shaping them. In 1873, little more than five years after Robinson's ignominious dismissal, Henry Cole retired from his post as Director, worn out by battles with the government and half a century of life as a civil servant. He was replaced by Philip Cunliffe-Owen, who had been primed by Cole as his successor: after a number of posts in the Science and Art Department, he was officially made Cole's deputy in 1860, taking particular responsibility for the British contributions to the International Exhibitions which took place all over Europe during the 1870s. He was efficient and shrewd, and praised by _The Times_ for being 'robust and capable of much hard work. . . the capable man of business'. More significantly, he had no axe to grind with Robinson. His work on the International Exhibitions had kept him largely out of Robinson's way, and he had spent so much time travelling that he had been able to maintain a healthy distance from the worst of the squabbles. Besides, he made no claims to be an expert on objects or collecting and so did not see Robinson as any kind of direct threat to his authority. He was content to be an organizer and, when he succeeded Cole as Director, he was happy to focus on administration. He was a man, noted _The Times_ , of 'little expert knowledge, [who]. . . had little to do with the actual purchases of objects'.\n\nWith Cole out of the way, Robinson launched something of a charm offensive. He kept an eye out at sales for lots that might tempt the museum and began offering some of his own pieces on loan in the hope that they might prove indispensable. He moved quickly and confidently. By 1879, over 300 of Robinson's objects were on display as loans, but with a view to a sale, including a stunning twelfth-century Flemish standing cross with a base inlaid with crystal; an elaborately decorated dress sword; a set of eighteenth-century Italian clerical vestments; strings of jewels, silver chalices, old Venetian glass, blue-and-white china, bronzes, wood carvings, ivories and books. The objects were magnificent and conspicuous and they filled gaps in the existing collections. But Cunliffe-Owen was wary about directing public money towards Robinson without being sure it was to the museum's advantage, and he drew on support and expertise from other national institutions to bolster his position. The pieces were scrupulously inspected by Franks at the British Museum, who found them 'remarkable for the taste displayed in collecting them', and by the painter Edward J. Poynter, a member of the Royal Academy who was, at the time, Principal of the National Art Training School and who was later to become Director of the National Gallery. He too approved, writing to Cunliffe-Owen that 'ALL the specimens appear to me to be admirably chosen for their artistic value.' With the government establishment in accord, Robinson found his objects once again taking pride of place in the South Kensington galleries.\n\nTo consolidate the impression that he was still invaluable, Robinson offered the pieces as part of a deal which, he assured Cunliffe-Owen, would be 'as advantageous as possible to you'. There would be no haggling over price: Robinson was prepared to offer the museum a bargain, selling the whole lot at a rate 'very much smaller' than the market value. This still amounted to the not inconsiderable sum of \u00a36,800, and the museum, perhaps unsure about exactly how generous Robinson was inclined to be towards an institution that had sacked him, called on Poynter and Franks again for advice. When both advisers approved the price as being 'very moderate', however, the deal was made. It was a marked divergence from Cole's desire to concentrate on modern manufactures, a very obvious demonstration that after his retirement in 1873 the museum moved more rapidly to adopt a policy that owed much to Robinson's early collecting. The idea of making contemporary acquisitions was gradually abandoned, so much so that by 1880 the displays of modern manufactures that had been at the heart of the original museum were moved from the main building to an outpost in Bethnal Green. In their place, Cole's successors concentrated their efforts on creating a vibrant visual encyclopedia of connoisseurship. Potential acquisitions were no longer assessed on whether they could be used to improve public taste or as models for students but instead, as Poynter said, 'for their artistic value'.\n\nA decade after he had been dismissed by Cole from the South Kensington Museum, it was as if Robinson had never left. The eagerness and zeal of youth was quickly rekindled, and once again he was undertaking lengthy journeys to Europe to buy on behalf of the museum, sending back extravagant deliveries of packing cases which piled up awkwardly in the corridors 'awaiting instructions' on his return: in January 1882 alone, he sent the museum twenty-nine cases of miscellaneous objects from Italy, as well as seventeen cases of marbles and bronzes. Once again, Robinson was apt to commit sums for acquisitions with little regard for the due forms of process or finance. In January 1881, during a huge snowstorm that brought much of London to a standstill, Robinson, undeterred by the weather, acquired two Florentine inlaid marble tables at a sale at Christie's. They were rare and lovely, but expensive. Robinson bid for them without a second thought, and wrote unrepentantly to Cunliffe-Owen that 'there was no time to tell you about them and I am aware that your funds for purchase for this year are exhausted \u2013 However, I considered them so important that I determined to purchase them.'\n\nBy 1881, Robinson was writing to the amenable Cunliffe-Owen to explain that recent months had 'quite revived the old habits. . . In short almost without intending it, I have found myself doing just the same kind of work as before.' A year later, he was plainly preening himself in front of the museum board, making sure that everyone knew to whom credit was due for years of success. 'May I be allowed to remind your Lordship,' he wrote to Earl Spenser, 'that it is to my having assumed personal responsibility. . . and to having risked my own pecuniary resources on occasions that the Nation owes a large proportion of the great monuments of art which form the pride and glory of the Museum. . . The building of the South Kensington Collection has been mainly my work.' Henry Cole, who would be dead within a few weeks, had little chance to set the record straight. Robinson was displaying a talent for rewriting the past to fit the way he saw things, and he seemed to be reinstating himself with ease into the museum's history books. Better still, as a recognized dealer and an independent man, he was vigorously running a double life, combining public and private business without fear of reprimand, and keeping himself determinedly unfettered. He refused to accept a salary from the museum, maintaining a discreet distance from its committee of art referees, and choosing for himself whether to be in London, at Newton Manor or travelling in Europe. He was making all his own decisions about the buying and selling of objects. He was earning substantial sums of money from private clients, and winning priceless prestige for his work at the museum. It was the best of both worlds.\n\nRobinson was astute enough to recognize the value of his work with museums to his broader activity as a dealer: the National Gallery, Birmingham Museum, Dublin Museum and the Bowes Museum at Barnard Castle all benefited from both his connoisseurship and his commitment to supplying pieces for public display at little or no profit. Occasionally, he even offered works as donations: in April 1877, for example, he bought El Greco's _Christ Driving out the Moneylenders from the Temple_ for \u00a325. 20s. 0d from a sale at Christie's and later presented it to the National Gallery. At South Kensington, this policy was mingled with a lingering sense of responsibility. He often sold objects to the collections there at cost price, and was flexible about payment so that the museum could stagger instalments to suit its budgets. He offered only what he considered the finest pieces, worthy of a place in the galleries, and he was happy to take back works that the staff rejected. When he returned from a trip to Italy in early 1881, he brought with him a wealth of objects wheedled out of churches, palaces and private collections, allowing the museum to make a choice depending on what could be afforded. 'I want it to be thoroughly understood', he said reassuringly to Cunliffe-Owen, 'that I have bought these things entirely on my own account, to please myself, and that I have not the slightest desire to _urge_ them on the Museum.' Nevertheless, in the end, Robinson made a very satisfactory transaction, selling a range of pieces from a 1560 sundial from the Pitti Palace in Florence, priced at \u00a38, to a sixteenth-century wooden coat rail, at 8 shillings.\n\nAs ever, Robinson's relationship with the museum was complicated. In everything he did, there was ambition and pride, but there was also an unshakeable, altruistic belief in the importance of the collection and its value to generations of visitors. Robinson was delighted to be back at the centre of things because he believed so fervently in what he was doing \u2013 and that he was the best man to do it. He trusted his own judgement absolutely and he had faith in his ability to use his talents for the future glory of South Kensington. He boasted an imposing knowledge of European art, and was certain he could find the most important pieces; he understood the markets, and was confident of getting the best deals. He saw himself as a champion for collectors and what they could achieve.\n\nYet Robinson spent the best part of the late 1880s and 1890s in a very public, and damaging, crusade to highlight the faults at South Kensington, complaining that his own heyday of astute and visionary collecting had been replaced by a 'mechanical system, carried out by mere laymen, superabundant clerks, secretaries and shopkeepers'. He could not free himself of the idea that he should have influence over the beloved collection; even as an established dealer, leading a life quite separate from the museum and more than twenty years after he had been sacked, his relationship with the objects at South Kensington remained intimate and highly personal, his desire for control absolute. He typified the ambiguous unresolved distinction of the time between public and private collecting; in turn, this confusion helped fuel a character that was naturally difficult and contradictory. He even stood before a government select committee in July 1893 to reiterate the litany of complaints that had started life in his battles with Henry Cole. His old habits died hard. He did not seem to be able to shake off his discontent with those whose views failed to match his own, and he was openly critical of the museum that had first nurtured his talents.\n\nBut, as a collector, Robinson's energy and expertise, confidence and commitment could not be faulted. His ongoing dedication to collecting \u2013 no matter on whose behalf \u2013 could never be doubted. And, in many ways, his stubborn insistence on publicly raising issues about South Kensington and its collecting policy was good for collectors. His constant letters of complaint to _The Times_ , his frequent articles and his impassioned contributions to public discussions continued to cement the idea that it was, as he had once claimed, a 'national duty' to encourage good collecting and the systems that made it possible. He clearly raised the profile of collecting in general, pushing it into public notice alongside other political and economic matters. His vociferous criticisms helped ensure that the debate about what and how to collect remained lively in the public eye for the rest of the century.\n\n### CHAPTER SEVEN\n\n### _Changing Times_\n\n_R_ obinson's collecting continued to focus on the expensive and the rare. As his wealth grew, so did the amounts he invested. He assembled a collection of nearly 300 rare Renaissance portrait and commemorative medals cast in copper, tin, gold or silver; he developed a substantial collection of paintings by major British artists including Constable, Gainsborough, J. S. Cotman, Girtin, Landseer and Turner; he took to buying Louis XVI clocks, Japanese lacquerware and jewellery. His expenditure was so large that, despite his dealing success and a rash of sales in his final years, Robinson had little over \u00a3500 in his account at the London, County and West-minster Bank at his death in 1913. It was a collection to impress, a lifetime's work and a testament to his knowledge and energy. In recognition, he was acquiring a number of titles: Fellow of the Society of Antiquaries, Curator of the Royal Society of Painters and Etchers, honorary member of the Academies of Art in Rome, Florence, Bologna, Antwerp, Madrid and Lisbon, and Knight Commander of the Order of Isabella in Spain and Portugal. Then, in 1882, at the age of fifty-eight, Robinson received the acknowledgement he had always felt he deserved for his role at the forefront of British collecting: he was appointed Surveyor of the Queen's Pictures.\n\nBy the time Robinson was appointed as Surveyor, the Queen was sixty-three years old, and had been widowed for over twenty years. Art had been something she had shared with her husband and, without him at her side, her enthusiasm for collecting waned. During the 1870s and 1880s, when she finally resumed public and political life long after Albert's death, she found affairs of state sat too heavily to allow for personal pleasures: she was travelling extensively on a tiring succession of engagements in an attempt to suppress a growing republican voice and she had little free time to think about buying or commissioning works, even had she wanted to. Without special attention, the collection became little more than a passive receptacle for randomly acquired pieces. It became increasingly muddled and disordered, separate from the life of the sovereign and no longer an integral part of royal affairs. And for Victoria, there was something more: a lingering sense of loss that was made particularly poignant by the paintings and sculptures she had enjoyed in happier times. The collection acted as a reminder of life as it had once been and as a memorial to Albert's taste. The Surveyor was expected to maintain the status quo and to content himself with the minor administration associated with keeping the collection in reasonable shape. He was expected to make periodic reports and deal with any gifts from grateful subjects. He was not required to create opportunities for either acquisition or public exhibition. There was not a lot to do. It was the prestige of the role that was important, embedding its holder firmly in the ranks of the elite establishment \u2013 and it was this that Robinson cherished. Finally, in 1887, at the Queen's Jubilee, he was knighted, confirmation of the distance he had travelled from his days as an impoverished art student.\n\nAs a collector, however, it was important for Robinson that his objects, too, were given official recognition, and as the century drew to a close he began to make numerous overtures to public collections to secure the future of his pieces. He demonstrated a special fondness for the Whitworth Institute in Manchester, which had been established with the fortune left by Sir Joseph Whitworth, a manufacturer of machine tools. The Institute was in the process of developing an art gallery under the governorship of William Agnew, a fellow London art dealer. Robinson knew Agnew well, and was happy to 'show any assistance in my power' to make the new Whitworth scheme a success. At first this remained a vague commitment, but the Manchester board was determined to keep him to his word and was soon negotiating to buy 1,000 items from Robinson's magnificent collection of textiles, even luring him into 'coming from my own home to personally arrange the specimens'. The pieces had to be teased out of Robinson, who had a collector's reluctance to relinquish them, but the sale was finally settled in 1891. For the modest sum of \u00a33,570, he delivered a collection of European ecclesiastical textiles from the sixteenth, seventeenth and eighteenth centuries, vestments from the celebration of High Mass, rare altar tapestries and a handful of British embroideries of exquisite workmanship and design.\n\nApparently undeterred by the fact that until recently he had been openly and vehemently critical of the work of the South Kensington Museum, Robinson also approached the curators there, confident that he had long ago earned the right to a permanent welcome. Rather remarkably, he was greeted with patient goodwill. The latest in the long line of Henry Cole's successors, Caspar Purdon Clarke, was, perhaps fortunately, known 'for geniality and good fellowship. . . a well-liked man both by his subordinates and the general public'. Perhaps he saw Robinson as little more than an eccentric old duffer, an occasional thorn in the side; or perhaps he was clear-sighted enough to see beyond any personal difficulties with Robinson to the prized pieces the museum was being offered. Whatever the reasons for his tolerance, he listened enthusiastically to Robinson's proposals, and, although he politely turned down the full extent of the collection Robinson had hoped to deposit, he happily agreed to take a smaller selection for display in 1901. In the end, this amounted to seventy-three choice objects, including a miniature Spanish prayer book, set with rubies, which had belonged to Charles V; another Charles V jewel, the 'Bezaar stone', set in gold, and presented to the king by Hernan Cortes, the 'conqueror of Mexico'; a fourteenth-century Hungarian reliquary cross; and the walking staff which had once belonged to Sir Nicholas Bacon, Lord Keeper of the Great Seal to Elizabeth I.\n\nBut even as the pieces were being prepared for display, as a permanent recognition of Robinson's achievements as a collector, it was becoming clear that times were changing. The works were being assessed and arranged by the museum's Assistant Director, Arthur Banks Skinner, who had climbed through the ranks during his twenty years at South Kensington. Robinson had written the display labels to accompany the pieces, presumably because he believed that he was best placed to explain them to the public. Skinner, however, whose 'knowledge of works of art was singularly wide and accurate' and who 'enjoyed the confidence and esteem of many foreign collectors and heads of museums', had his reservations. 'These descriptions were prepared by him, and in my opinion are open to considerable doubt,' he explained to Purdon Clarke, before asking incredulously, 'Do you wish them to go out as they are?' Purdon Clarke, ever the diplomat, proposed a compromise. He suggested that the labels remained unedited, but that it was made abundantly clear that they were Robinson's own idiosyncratic work, and not a reflection of the museum's scholarship.\n\nIt was a small but significant exchange between the new generation of professionals at South Kensington. By the end of the nineteenth century, men like Arthur Banks Skinner were taking the haphazard knowledge of the hosts of Victorian collectors and connoisseurs and beginning to formalize it into the discipline that we now know as art history. This movement had begun in earnest in the 1850s with what was later referred to as the Vienna School of Art History. In 1852, Rudolph Eitelberger was appointed to the world's first chair of the history of art at the University of Vienna, and a succession of scholars there followed in his footsteps, trying to find ways to make the appreciation of art more objective. This meant taking into account historical sources and adopting a more factually based approach to the examination of artistic achievement. To some extent, this academic work was consolidating the more practical efforts of curators like Gustav Waagen. Alongside his publications, Waagen's displays at the Kaiser-Friedrich Museum in the 1830s and 1840s presented objects within their chronological and social contexts, supported by explanatory labels and catalogues. This allowed visitors to get a sense of how one artist's work related to another's, and how practice had evolved across time and in different places. Waagen also advocated re-creating a sense of the original spaces in which the works of art would have been displayed, 'to lessen as much as possible the contrast which must necessarily exist between works of Art in their original site, and in their position in a museum. . . to realise in some degree the impression produced by a temple, a church, a palace or a cabinet'.\n\nBy the mid-nineteenth century, as had been seen at the Louvre, other European curators were also beginning to experiment with presenting art in 'schools', grouping works together to explore similarities in technique, style or subject matter, and to demonstrate ways in which artists influenced each other. During the 1880s and 1890s, the principles of this art historical approach were further developed by Heinrich W\u00f6lfflin, a Swiss scholar and critic. While teaching in Basel, Berlin and Munich, he formulated pairs of opposing ideas \u2013 such as whether a work demonstrated a 'linear' or a 'painterly' approach \u2013 to underpin the study of art.\n\nIt was not until well into the twentieth century that the academic discipline of art history became fully established, but British curators in the nineteenth century were certainly aware of what was being accomplished by their European counterparts. From the 1850s, articles and pamphlets discussed the arrangement of picture galleries in particular, trying to define the ideal display system. The debate was focused around the National Gallery: the shortcomings of the existing gallery had prompted numerous calls for something completely new, on a different site, and this in turn acted as the catalyst for a wider discussion about different ways of displaying and interpreting the works inside the proposed new building. Waagen himself contributed to the debate, outlining his ideas in a lengthy article in the _Art Journal_ in 1853 and including a model classification for presenting paintings within a public museum. Robinson, Ruskin and Eastlake were just some of the others to enter the fray in a process that seemed to typify the Victorian enthusiasm for inquiries, reports, articles and counter-articles, protests, statements and official proceedings.\n\nIn the end, the plans for a new National Gallery fizzled out into a modest extension of seven rooms on the existing site and the display principles remained inconsistent. But the idea that art history was a serious area for study continued to flourish and the new breed of scholars, as Skinner demonstrated, became increasingly confident in its challenge to the accepted wisdom of the past. As Skinner's incredulous note suggested, Robinson was no longer, in other people's eyes, at the vanguard of scholarship; he was simply a respectable veteran collector, part of a generation that was being left behind as knowledge about art continued to evolve. He was a Victorian in the dawning years of a new century and a different age.\n\n* * *\n\nRobinson's collections fared little better in creating a lasting impression for posterity. In the early 1890s, at the same time as he was agreeing terms with the Whitworth, he was putting works into four different sales at Christie's with a view to improving his finances. Three hundred and seventy-five oil paintings, thirty-five prints, forty-one Old Master drawings and a hundred and sixty-eight pieces of decorative art went under the auctioneer's hammer at these sales alone, allowing him to meet the not inconsiderable expenses of his lifestyle. With the death of Queen Victoria in January 1901, however, Robinson lost the position of Surveyor in a sudden change of royal politics that came as 'a most painful surprise'. Suddenly, the pressures on his purse grew even more acute and in April 1902 Robinson parted with a significant portion of his collection of sculpture and ceramics at Christie's, quickly followed in two more sales by the majority of his Old Master drawings. In contrast to his reputation for slapdash administration at the museum, these personal pieces were carefully recorded; Robinson noted the source of each of the works, how and when he had bought them, and what he knew about their earlier provenance. In May, over 450 drawings, attributed to the principal artists of the Italian, Dutch, French, German and Spanish schools, were announced in the auctioneer's catalogue as the 'valuable collection. . . formed by a well-known amateur over the last 40 years', and Robinson watched with mingled pride and regret as other collectors clamoured to acquire what he had taken so long to amass. His collection was being irretrievably scattered. After a lifetime's accumulation came months of determined dispersal. There was plenty left \u2013 Newton Manor was still a treasure trove \u2013 but the sales set a tone of closure, the sad parting of a collector with the objects he prized.\n\nRobinson's collection did not survive him. After his death at the age of eighty-eight in April 1913, the rest of the works were gradually sold on. By September, many of the drawings were being put up for sale, and by February 1914 the Robinson name was appearing regularly at Sotheby's. With an eye perhaps on this kind of insecurity, Robinson had left a number of financial gifts in his will, primarily to organizations that encouraged collectors and artists and preserved their work for the future. The Burlington Fine Arts Club, not surprisingly, was top of the list, but there were also legacies to Dorset County Museum, the National Art Collections Fund, the Royal Society of Painters and Etchers, the Society of Antiquaries and the Society for the Protection of Ancient Buildings. Through the work of these kinds of institutions, Robinson knew, arts scholarship would continue to progress and flourish, collectors would meet like-minded colleagues, and the best objects would be admired, studied and displayed.\n\nThe South Kensington Museum had changed greatly since Robinson's years as a curator, becoming known as the Victoria and Albert Museum from 1899 and unveiling a massive new building and a grand stone fa\u00e7ade ten years later. Nonetheless, Robinson left to the collections there a parcel of his own etchings, which he selected just weeks before his death. He wished to be remembered, it seems, not just as a collector and dealer, a learned middleman, but as a peer of the painters, sculptors and craftsmen whom he so much respected. The etchings, however, were not enough to accord Robinson a lasting presence in the museum. The opening of the new building in June 1909 \u2013 almost fifty years after Robinson had scurried across from Marlborough House to see the first work begin on the site \u2013 was seen by staff as an opportunity for rearranging the collections and asserting their modern relevance. There was even a backlash against what was seen as the excesses of Victorian taste: the art critic of the _Daily Telegraph_ noted that the practical new building was so plain that it gave 'the general impression. . . of some immense, finely-appointed modern hospital' and until the outbreak of the First World War there was a controversial campaign to remove the most highly decorative elements of the original building because they 'belonged to a bygone age'. Robinson was increasingly out of fashion and his etchings out of place in a collection which was once again reinventing itself for new generations.\n\nRobinson's long life straddled the Victorian age and his biography as a collector reflects the changes of his time. In the complicated twists and turns of his career \u2013 as private connoisseur, professional curator and successful dealer \u2013 we begin to see the complexities of the Victorian relationship to the collection, the achievements and disappointments, the shifts and disputes and contradictions. In the fate of Robinson's beloved things, we also glimpse a motif which will be repeated at the heart of these collecting stories \u2013 a collection does not last. Tastes change, financial and family concerns intervene; what seems to one collector like a logical arrangement of lovely pieces often looks to the next generation like a hotchpotch of odd personal trinkets. Individual objects survive through history, often gathering meaning and value; the collection is ephemeral. It acquires life and meaning from the collector who makes it. It is a very personal treasure.\n\n## _Ransacking and Revolution: The European Crusade_\n\nCHARLOTTE SCHREIBER\n\n### CHAPTER EIGHT\n\n### _Mrs Schreiber's Big Red Bag_\n\n_I_ t was a long and tedious journey to Paris. Charlotte Schreiber and her husband Charles were woken in time to join the noisy huddle of travellers at the inn in the centre of Bordeaux, all clamouring in the cold dawn for a place on the omnibus to the capital. They had arrived from Spain the previous day by train, reaching Bordeaux in the middle of the night, but there was no chance of taking a break. Paris was emerging from the Franco-Prussian War and was in the middle of an armed uprising which was to become known as the Paris Commune. Transport to and from the city was unpredictable: railway bridges had been destroyed or cut off, trains halted, roads blocked by barricades, and horses, carriages and carts caught up in the confusion. Places on the omnibuses that were still making the journey to the city were at a premium. In the courtyard of the inn, the mood was hectic and slightly aggressive; the driver was surly and blatantly profiteering. But as a seasoned and levelheaded traveller, Charlotte conducted her negotiations amicably and in impeccable French. She was a wealthy woman, and offered to pay handsomely for places in the coach for herself, her maid and her husband. She was polite and unruffled. She agreed a price for three seats on the fourteen-seat omnibus, and, with her appetite sharpened by the brisk May air, she set her mind to breakfast.\n\nThe problem came with the bag \u2013 not with the usual cases and hat boxes that could be piled on to the roof and tied on to the railings for safety, but with the floppy, soft red velvet bag that the Schreibers insisted on carrying between them. The driver would not take it inside, and they would not relinquish it to the roof. It was a stalemate. And since it was a stalemate of a particularly French kind, quiet and smoky and stubborn, there was little the middle-aged English couple could do to press their point. Just over an hour later, the omnibus pulled slowly out of Bordeaux, the horses steaming in the early-morning air, and Charlotte and Charles Schreiber were left behind \u2013 with their bag \u2013 in the chilly shadows of the city's medieval streets.\n\nSince there were no more omnibuses and no trains, the Schreibers ended up riding to Paris in a farm cart with a horse Charlotte described in her journal as 'sturdy' and a 'loquacious' driver. They rattled slowly through the French countryside, bumping along rutted roads, with no shelter from the wind and occasional spitting rain, and the incessant chatter of the driver drumming in their ears. It was a long and uncomfortable journey, but they had their red bag safe in the cart between them, and that was the main thing.\n\nThe bag itself was not, on the whole, remarkable. It was roomy and practical, and slightly patched. What was special was what was in it. By May 1871, when the Schreibers were making their steady progress towards the outskirts of the French capital, they had been on the road already (not to mention the seas, the railways and the back alleys) for almost a year. Throughout their journey, Charlotte had maintained her lifetime's habit of keeping a daily journal, recording the trials of collecting on the move and giving us a colourful picture of what it took to be a collector. They had trekked through the South of France before attempting the 'wretchedly bad' crossing of the Pyrenees, where the roads were no better than 'ill-ploughed fields' and progress was only made by taking pickaxes to the wheels and whips to the horses.\n\nIn Spain, hardly pausing for breath, they moved down through Gerona and Barcelona to Seville (with Charles suffering from headaches, a sore throat and inflammation of the eyes) and then on to Cadiz before crossing to Gibraltar. When their train ground to a halt in front of a broken railway bridge between Valencia and Cordoba, during a thunderstorm, Charles, along with other passengers, stepped in to help as the carriages were uncoupled and then pulled, one by one, across the ravine. At another unsound and unfinished railway bridge, they were transferred from their train to 'a sort of temporary contrivance' which shunted them across 'very slowly' with the men still at work around and below them. They had endured a twenty-six-hour journey between Granada and Madrid, having stayed up all night in advance to savour Granada's fabled nightlife. They had toiled on foot through Northern Spain and back into France during an unseasonable spring heatwave, carrying their big red bag between them, and stopping at shabby inns whose owners displayed a distrust of such odd English visitors. They had even spent one memorable night in Valladolid in the omnibus on which they had arrived in the city, much to the dismay of the driver who simply unharnessed his team of horses and pushed the carriage into the inn yard. The Schreibers were woken the next morning, early, by the stamping of mules, the neighing of horses, a cacophony of cocks and hens and 'the tinkling of a bell on a very playful, restless goat'.\n\nEvery moment of that year, every bright day in a pale Spanish town, every dusty evening in a Mediterranean port, every rain-sodden afternoon in a small, damp French village, had been filled with the unrelenting, resolute and entirely absorbing hunt for things. And the spoils of the hunt \u2013 a delicate fan of exotic birds' feathers, a perfect silver serving jug, pieces of fine china carefully wrapped in paper, unusual figurines and a bright enamel \u2013 were safely stowed away in Charlotte's big red bag. So intense and energetic was Charlotte's search, so absorbing and dogged, that, when the farm cart finally trundled into the outskirts of Paris, the Schreibers were in for a shock. With her eyes fixed on the details of fine china, larger events in Europe had largely passed Charlotte by. The Franco-Prussian War, which had devastated Northern France and Germany for over a year, had registered simply as an inconvenience, prompting her to note blandly in her journal that Calais 'looked sad and chastened' and Paris was 'impassable'. None of this had really mattered, as long as the search was not interrupted. But now the Schreibers were thrown into a city in turmoil.\n\nAfter the French defeat by the Prussians in the autumn of 1870, Paris had failed to accept the surrender offered by the rest of France. The victorious Prussian army had laid siege to the city with over a quarter of a million troops, and throughout the winter months Paris had defiantly starved. When the national government finally tried to enforce a truce in the spring, the workers and people of Paris, inspired by the ideal of ' _la r\u00e9publique d\u00e9mocratique et sociale_ ', had declared the city a separate commune. From March until May 1871, while the Schreibers had been further south, the revolutionary citizens had built over 600 defensive barricades. This time, however, it was not the Prussian army but the French army which marched on the capital. It arrived just a fortnight ahead of the Schreibers, overwhelming the revolutionaries in a series of violent, hand-to-hand street battles. Charlotte, her husband and her maid finally arrived on 1 June. Three days earlier, the revolutionaries had taken their last stand at the end of a week which saw as many as 30,000 casualties. Riding expectantly into Paris, what Charlotte discovered was not the lively cultural capital she so much admired, with its network of dealers and showrooms and junkshops, but instead what she described forlornly as a 'City of the Dead'.\n\nCharlotte found Paris ravaged, and its people demoralized. No corner was untouched by the ferocious fighting of the past weeks. There was, she wrote in her journal, 'no life or animation; scarce anyone in the streets. . . the Tuileries and other public buildings still smoking; the Vend\u00f4me Column lying in pieces on the ground'. She was shocked and bewildered, moved to tears by the destruction, but still undeterred by smoking ruins and bodies half-hidden in the rubble. She managed to secure a small apartment \u2013 with 'a sort of cupboard' for the maid \u2013 in a building where workmen were bricking up the ground floors against the threat of arson. During the last days of the Commune, the Palais de Tuileries and the H\u00f4tel de Ville had been burned down and rumours were rife that working-class women \u2013 _les p\u00e9troleuses_ \u2013 were stalking Paris with bottles full of petroleum or paraffin, ready to start fires through unprotected cellar windows. In fact, _les p\u00e9troleuses_ turned out to be mythical, a construct of fear and suspicion, but at the time they were considered a formidable threat. The sale of flammable liquids was banned for several months, and as far as Charlotte was concerned, her lodgings could be burned to the ground at any moment in an act of revolutionary spite. There is no doubt that she was frightened and overwhelmed by the situation she and her party had wandered into blindly. Perhaps because of this, she set her mind even more firmly on collecting. Within minutes of arriving in Paris, she went out into the ruins to try to discover what had become of her trusted dealers and whether there were any bargains to be had.\n\nIt took time to establish exactly what had happened. Entire streetscapes had been altered, first by the building of the huge barricades and then by their destruction and the hand-to-hand fighting. Houses and shops were burned out and deserted, and many Parisians were sheltering in cellars or attics, afraid to venture on to the streets for bread, far less to talk about antiques with English travellers. Eventually, though, through perseverance and with the aid of a few substantial banknotes, Charlotte began to get a trickle of information. And the news was not good. Two of her oldest and most trustworthy dealers, Mme Caillot and Mme Oppenheim, had, Charlotte noted sadly, 'both died of fright'. Nothing was left of their stock, which had been burned or looted. For another of Charlotte's contacts, 'poor old Fournier', the fear and confusion had proved too great: he had reportedly gone mad, and could not be found. The only dealer Charlotte could locate was Mme Flaudin, and the next morning she set out early for the small dark shop which she had visited many times before on happier occasions. Remarkably, it was open and largely unscathed by the fighting. Mme Flaudin was waiting with a jug of hot coffee and a table covered in things she hoped would tempt her intrepid customer. But even in the aftermath of revolution, Charlotte would not be rushed. She took her time browsing, and carefully examined any of the objects that caught her eye. She asked questions, and bartered fiercely. She was in her element; she had found what she was looking for, a surviving Paris dealer, and, better still, a dealer with things worth buying. Not even her walk back through the bloody wreckage of the city could blunt her joy at unearthing some decent china, a 'matching old maroon Chelsea set'.\n\nRescued from the rubble of Paris, Charlotte's set of china was tenderly wrapped and given a temporary home in the big red bag. Produced in the eighteenth century by a factory founded around 1740 in the rapidly expanding but still rural town on the outskirts of London, Chelsea china was regarded as the very finest of English porcelain. Its famous claret-red tableware, daintily decorated with gilded borders and golden curlicues, was prized above everything. It had been popular since the day it was made and so demanded high prices, finding its way into the homes of the very wealthy. Charlotte admired the deep tones and fine shapes of the china, and was shrewd enough to recognize that what she was offered in Mme Flaudin's was a bargain. The Chelsea set was to become one of her treasured pieces. Fittingly for such a glamorous refugee of the Paris violence, it was to end up in the refined cabinets at South Kensington, part of the huge collection that Charlotte donated to the museum \u2013 one of the finest, most detailed and most interesting collections of English china ever amassed.\n\nA couple of weeks before arriving in Paris, Charlotte Schreiber celebrated her fifty-ninth birthday. It was a day like any other, spent hunting for collectables with her husband. Good wishes had been sent from some of her children in the letters which straggled into foreign post offices in the days running up to the occasion, but 19 May 1871 was passed quietly in Tangiers. The most special and welcome thing about the day was the quality of the Victoria Hotel, chosen by chance rather than as a birthday treat, but worth noting in Charlotte's journal as 'clean and comfortable, unlike any since leaving England'.\n\nCharlotte looked like any other elderly traveller, taking in the fabled and relaxed culture of the North African port. She was slightly plump and matronly; she was not, and had never been, particularly beautiful, and she dressed sedately and neatly in sombre colours, even in the African heat. Perhaps her one concession to vanity was that her grey hair was dyed. This was not an easy thing to achieve in the midst of so much travelling, and Charlotte may well have resorted to using Condy's Fluid, a mixture of the mineral pyrolusite (largely magnesium dioxide) with potassium hydroxide that had been patented in the mid-nineteenth century and which could be conveniently carried as a disinfectant as well as being used to colour hair. Even with Condy's Fluid, however, the process was not straightforward and would have given Charlotte's maid a great deal of work: each hair had to be dyed individually. In addition, it was something that would have had to be done discreetly. While it became very popular for Victorian women to dye their hair, and magazines were awash with advertisements for products, it was not quite accepted as thoroughly respectable, especially among the middle and upper classes. 'Above all dyes will be renounced', proclaimed Baroness Staffe authoritatively, in a French etiquette manual that was influential enough to be translated into English towards the end of the century. 'The natural colour of the hair will be kept and grey hair itself will not be powdered. . . At this cost the hair will remain abundant and vigorous, even in those of advanced age, and will allow of being prettily and gracefully dressed.'\n\nEven with the fashionable rebelliousness of dyed hair, Charlotte did not look like a woman whose name was becoming known across Europe. She looked ordinary; a woman of her class and time. And even she, as she admitted in her journal, was surprised at the way things were unfolding. She had not expected to be ransacking foreign lands for china. She had not planned to spend her days hurrying from one dealer to the next, excited by rumours of the rare and beautiful, exhausted and exhilarated by the chase. There was little in her earlier life to suggest that she was to become such a dedicated and inventive collector, and she was almost fifty when the obsession overwhelmed her. Her enthusiasm for collecting came late, and apparently out of the blue.\n\nBut, while there may have been few clues in Charlotte's first fifty years to suggest she would become such a significant and respected collector, the energy, determination and curiosity that characterized her collecting were clearly evident in the range of activities to which she dedicated herself as a younger woman. She was born in 1812 with a title, Lady Charlotte Elizabeth Bertie, the daughter of Albemarle Bertie, the 9th Earl of Lindsey, an army general and an MP. She was the first of three children, but her father was already sixty-eight years old when Charlotte was born, and he died just six years later. By the time Charlotte was nine, her mother had married again and her new husband, the Reverend Peter Pegus, had moved into the family home at Uffington House in Lincolnshire; a year later, Charlotte began the journal which she was to keep candidly and faithfully until she was seventy-nine years old and almost completely blind.\n\nCharlotte hated her new stepfather. Pegus was a bully and a drunk who was once so desperate for beer that he downed a mug of lamp oil instead. Unpredictable and violent, he was quick to flare into a rage; passionate about the most trivial of domestic habits, on one occasion he sacked the entire household of servants on the spot. Family life was stormy and miserable, and Charlotte was lonely. The house at Uffington was isolated and there were no suitable friends for Charlotte nearby. Her eldest brother suffered from what would now be regarded as a mental disability; her younger brother was baffled by her bookish tastes; and she was never close to her mother, who sank into decline after her second marriage. Charlotte spent hours alone in the garden, particularly among the avenue of lime trees, and she immersed herself in her studies. She was serious, literary and fiercely bright. She worked with her brothers' tutor whenever she could (falling in love with him at the same time) and studied hard outside the hours officially allocated to her schooling, rising each morning to begin work by four, setting herself strict routines, devouring the books in the library and reciting poetry in her garden walks. She taught herself a variety of languages, including Arabic, Hebrew and Persian; she worked diligently at her mathematics; she practised her drawing skills until she became an accomplished draughtswoman; she played piano and harp, she read Chaucer and Ariosto for pleasure \u2013 and she played a mean game of billiards.\n\nBy the time she was sixteen, Charlotte had a reputation for learning and culture. She followed contemporary affairs closely from the newspapers and her journal was full of her thoughts on politics. In contrast, she found the demands of fashionable society \u2013 the dress fittings and the long afternoons, the predictable parties and the 'gazing at each other in listless indolence' \u2013 tedious and unsatisfying, preferring the company of scholars to the chatter of female acquaintances. This may have done little to impress the pleasure-loving young men of the Regency, but her title, and a not insignificant personal fortune, more than made up for her unfortunately thorough education. She was something of a catch, and society was not slow in conjuring up all kinds of potential matches for her. Terrified of being trapped in a miserable alliance to satisfy her family's ambition, however, she refused a marriage arranged by her mother in 1832 to the sixty-seven-year-old Robert Plumer Ward, a politician and novelist who was already twice widowed, and instead enjoyed a brief, but chaste, flirtation with the future Prime Minister Benjamin Disraeli.\n\nDisraeli was handsome and dashing, described by Charlotte's cousin as 'wearing waistcoats of the most gorgeous colours and the most fantastic patterns. . . velvet pantaloons and shoes adorned with red rosettes. . . his black hair pomatumed and elaborately curled and his person redolent with perfume'. Not yet elected to even the most minor of political offices, he was instead writing novels which Charlotte admired, and he took her to a series of fashionable concerts and bought her flowers. He even wrote enthusiastically to his sister about her fortune. But the relationship came to nothing (Disraeli later married one of Charlotte's closest friends), and in April 1833 Charlotte left Uffington for London.\n\nAlmost immediately upon her arrival in the capital, at the age of twenty-one, she met the forty-eight-year-old Welsh ironmaster and MP Josiah John Guest, known simply as John. Already a widow, John Guest was a handsome, curly-haired, practical man who, like Charlotte, was a rather diffident outsider amidst the glamour of London society. After a whirlwind romance, Guest walked with Charlotte in Kensington Gardens on 12 July 1833 and proposed to her. Charlotte accepted.\n\nMost of Charlotte's family seem to have been pleased to be rid of her to a wealthy man. Although he was in trade, they could console themselves with the knowledge that it was no ordinary trade \u2013 the rail tracks made at Guest's factory in the Welsh town of Merthyr Tydfil were to criss-cross the globe, opening up lands from Russia to India. They celebrated at Uffington with a ball and an ox roast. Her stepfather Pegus, however, was horrified. He objected vociferously to the marriage and continued to detest his new son-in-law, even calling him out for a duel on one occasion, which, Charlotte duly noted, 'is too absurd an idea'. He was habitually rude and dismissive of Charlotte's new husband, and he was not alone. London society in general was offended at the thought of a daughter of the aristocracy marrying a man whose hands were dirtied by the dust of the factory; a man descended from farmers and coal miners and who worshipped in the dissenting chapels that challenged the established hierarchy of the English Church. The new couple were ostracized: they were not invited to the events that mattered, and Charlotte's drawing room was distinctly quiet during the usual visiting hours.\n\nCharlotte was naturally emotional and prone to sentimental tears, but she wasted few of them on the newlyweds' situation. Relishing the thought of a different life, she was untroubled by the opinions of the social elite. She believed in her husband, and in herself, and she believed that marriage to an active, industrious and political man would, in the end, open more interesting and challenging doors than the ones that were being closed to her. In August 1833, after a honeymoon tour of Sussex, Charlotte accompanied her husband for the first time to his ironworks in Dowlais, in South Wales, where it was evident that London sensibilities were not shared by John Guest's friends and workers. Charlotte described with great delight the couple's arrival in Cardiff, which was celebrated by 'a volley canon, fired in grand style'. Furthermore, 'a triumphal procession and an illumination were planned' to greet them, intended on such a grand scale that 'no less than from 15 to 20 thousand people would probably have collected on the occasion'.\n\nThe thought of such a reception horrified Guest, however, who, despite his public role, was a shy man; he had experienced a similar event before, during which a boy had been killed in the press of the crowd. But he did not want to seem ungrateful or heavy-handed in putting an end to the celebrations. In a pattern that was to be often repeated during their twenty years of marriage, John Guest turned to his wife for advice, and they decided to order a prompt dinner at a local inn, allowing them to press on to Dowlais and arrive a day earlier than expected, so that, diplomatically, 'the complimentary machinations for the following day might be eluded without being rejected'. In compensation, John and Charlotte gave out celebratory beer to over 4,000 people at the ironworks a couple of days later and 'they gave us some discharges of canon in return'.\n\nCharlotte's married life was extraordinarily active and successful. She embraced the Victorian ideal of the wife and mother, following her husband's political and industrial interests with pride and giving birth to ten children in the space of thirteen years. She also took her responsibilities for the Dowlais workers' welfare to heart, developing housing and recreational projects and working hard to introduce a fresh water system. She was particularly keen that all the children should have the benefit of a good and useful education, and she founded six schools, raising large amounts of money for their upkeep by private subscription. So important did she become to the success of the ironworks, and the lives of the people who worked there, that a friend noted that in many respects she was more influential than her husband: 'for in all that he was deficient she excelled, and while we credit him with founding the greatest ironworks in the world, and giving sustenance and substantial comfort to twenty thousand souls, it is chiefly to her influence we must look for all that was done in the way of moral and mental elevation'.\n\nIn order to better accommodate their rapidly expanding family, Charlotte also set her mind to improving their home. In August 1835, she gave birth to Ivor Bertie Guest, a son and heir, and with the dynasty secure, she encouraged Guest in 1846 to buy Canford Manor in Dorset, a convenient distance from business both in London and Wales. The manor of Canford Magna had history. It dated back to Saxon times and had been an important base for the Earls of Salisbury. It had also been the home of William Ponsonby, the brother of Byron's lover Lady Caroline Lamb. It boasted extensive rolling parkland, dropping down to the River Stour, shaded woodland paths and neatly kept formal gardens, but the house itself was unremarkable, a muddled conglomeration of earlier buildings and practical extensions to the original medieval manor. Charlotte thought it dull and was immediately determined to make some alterations; to stamp her character on the place.\n\nThe Guests commissioned the architect Sir Charles Barry (who was, at the same time, rebuilding the Palace of West-minster with Augustus Pugin) to improve and expand Canford. Together, they created a fairytale, battlemented neo-Gothic house, complete with towers and turrets and a spectacular garden pavilion that housed twenty-six ancient Assyrian sculptures, including two colossi, one with two human heads and a lion's body, the other in the form of a bull. Charlotte worked closely with Barry, enjoying the challenge of learning about architecture and undertaking extensive discussions with the skilled stonemasons, carpenters and designers on site. John Guest, too, took a close interest in the house but he had less time to spend in Dorset and his role was primarily to pay the bills: the improvements became so costly that he became known locally as 'paying Guest'.\n\nSo much was being achieved in the valleys of South Wales; so spectacular was the Canford house becoming and so influential was John Guest proving in public life that news could not fail to reach fashionable London society of Charlotte's increasing wealth and influence. What had amounted to social ostracism came to be conveniently forgotten, and during the 1840s and 1850s showers of invitations duly arrived from those eager to make the family's acquaintance. Along with her commitments in Wales and Dorset, Charlotte became famed as one of London's most entertaining and gracious hostesses. Before long, there were splendid dinners at the Guests' London house in Spring Gardens, Westminster, with the children playing on the back stairs, the table laden with veal cutlets, woodcock pie and boars' head, and leading statesmen such as Sir Robert Peel and the Duke of Wellington discussing politics in the parlour.\n\nNevertheless, Charlotte's life was not confined to entertaining, society events, philanthropic projects and running a family. Even with the interest of rebuilding Canford, and the challenge of keeping ten children healthy and occupied, she wanted more. She saw her marriage as a professional, as well as a personal, partnership and was eager to take an active role in the development of the ironworks. Just as she had been willing to challenge Victorian convention in marrying Guest, so she was determined not to be confined by the usual nineteenth-century restrictions on a woman's role. The day after she first arrived at Dowlais with her new husband, she insisted on seeing the furnaces and the forges, and after dinner she walked back to the works to watch the iron being cast. She was soon able to take an informed view of the processes and before long she was translating technical documents into French, to better disseminate developments in the industry, and was writing pamphlets on ironworking techniques and improvements. The prospect of entrepreneurial business excited her, not so much for the wealth and prestige which accompanied it, but because it offered a means of reaching and understanding people and cultures across the world. During her husband's frequent absences, she was in sole charge of the expanding Dowlais empire and she entertained streams of influential visitors: iron masters from France, Germany, Holland, Italy and Poland, the Nawab of Oudh from India and Russia's Grand Duke Constantine. Charlotte believed fervently in the value of what was being achieved at Dowlais and, when John's work as a politician and industrialist was recognized in 1838 with a baronetcy, she thought he deserved more: 'I consider it a paltry distinction and was much averse to his taking it. . . I shall not rest till I see something of more value bestowed upon him.'\n\nAlongside her husband's industrial interests, Charlotte also became increasingly involved in the political issues closest to him: free trade, the abolition of slavery and the reform of the Church. The curiosity she had shown for political discussion as a teenager flourished into an active engagement that even took her on to the streets. She canvassed openly on Guest's behalf, and was frequently to be found sneaking into large and unruly public meetings in Merthyr market to hear him speak. She was a practical, informed and unflappable spouse for a politician attempting to push through liberal reforms in what could be stormy times: when the works and house were threatened by Chartist riots during the autumn of 1839, Charlotte made composed plans for 'the case of a siege' and arranged for the children to be sent away to safety, noting in her journal only that since there were 'from fifty to a hundred special constables all in the house. . . the succession of suppers and tea-drinkings that went on amongst all that entered was really a curious thing'.\n\nRemarkably Charlotte also found time \u2013 in the midst of political riots, the children's scarlet fever scares and entertaining high society \u2013 to continue with her studies. She began to teach herself Welsh within weeks of her marriage, and took on what proved to be the eight-year task of translating eleven medieval tales collectively called _The Mabinogion_ , providing extensive scholarly notes to explain the stories and their contexts. The Mabinogion myths were at the heart of Welsh folklore and are some of the greatest tales of Celtic literature, set in a bizarre and magical landscape, part Wales and part mysterious underworld, populated by giants, enchanted horses and magnificent heroes. They introduced figures like King Arthur and Merlin into European storytelling, and they brought together some of the most influential myths of the oral tradition. Welsh scholars knew of the stories and William Owen Pughe, a Welsh teacher, antiquarian and writer of grammars, had left an unfinished translation at his death in 1835; but no English speaker had thought them worth much trouble.\n\nCharlotte took a different view. She valued _The Mabinogion_ as more than an obscure element of Welsh tradition or a personal academic challenge. It was a way of understanding and popularizing the culture which she had embraced so enthusiastically. It was a means of sharing the stories with her own children, and of exploring the magical cultural landscape which was too often obscured by the noise and dirt and fire-blowing furnaces of the Dowlais ironworks. It was a labour of love. But unexpectedly, when it was published in several beautifully produced volumes between 1838 and 1849, it also became something of a publishing sensation. Charlotte's translation, which remained the standard for almost a century, made the tales famous: they became popular and fashionable almost immediately, not just in England and Wales, but also across Europe and America.\n\nWith both industrial wealth and a family background in the English gentry and aristocracy, Charlotte was perfectly placed to become a collector. Her interest in ceramics and smaller decorative objects such as fans and playing cards placed her at the forefront of the changing fashions in collecting, moving on from paintings and sculpture to more portable and domestic pieces. She was clever and studious, and in many ways she inherited her interests from the long line of gentlemanly amateurs who had gone before her. But her collecting was in no way inevitable. She might have continued as an industrialist and a reformer, and been remembered as a mother, a scholar and a pioneering translator.\n\nThere were a few signs in her early life that suggested Charlotte might be destined to become a collector. When she was nineteen years old, her mother gave her a feathered fan which she treasured, not only as a gift, but as a work of art and a cultural artefact, reading more into the fan's construction and design than might most young women of the period. She was also a close friend of her cousin, Austen Henry Layard, the man who would later dismiss J. C. Robinson as 'nothing but a dealer' and who was related to Charlotte on her mother's side. She was a keen supporter of Layard's pioneering archaeology while he, only six years her junior, came to rely on her as a patron. There has been some suggestion that the two were lovers, but any romantic relationship between them was certainly conducted with the utmost discretion and no evidence has been found to support the rumours of the time. There is, however, little doubt that at least some of Layard's enthusiasm for the idea of adventurous collecting had rubbed off on his cousin: he gave Charlotte numerous pieces from his excavations, and in time, as we have seen, she made over a building at Canford, known as 'the Nineveh Porch', to show off his finds. This included a relief taken from the throne room of the Assyrian king Ashurnasirpal II (883\u2013859 BC) and brought back from the ancient city of Nimrud in northern Mesopotamia (now Iraq), which Layard began excavating in 1845. The relief, among the first Assyrian pieces to be seen in Britain, was rediscovered at Canford in 1992 and sold two years later at Christie's for \u00a37.7 million, by far the highest price ever paid for an antiquity and a pleasant windfall for the boarding school that now occupies the Canford buildings.\n\nWithout a change in circumstances, however, the early feathered fan might have remained nothing more than a treasured gift and Charlotte might well have remained a passive bystander to Layard's own collecting. Her future as a collector depended on a combination of events in the early 1850s which altered her personal and professional circumstances, and introduced her to the temptation of rare and beautiful objects. The first of these was the Great Exhibition in Joseph Paxton's magnificent Crystal Palace in 1851. As members of the social elite, the Guests were invited to the official opening of the exhibition on 1 May; they spent the previous day researching the best places to sit so that when the pageant began they would have an unobstructed view of the Queen and her entourage.\n\nBut Charlotte was not just there to see and be seen. When the pomp was over, she made return visits and, along with more than six million other visitors, she was fascinated and delighted by the display of objects from around the world. The examples of manufactures absorbed her; the details of international trade and commerce intrigued her; and the models of technological innovation excited her. Taking her place in the crowds, she enjoyed the sheer exuberant spectacle: the circus, tightrope walkers, dog shows, pigeon shows and flower shows, the life-size reproductions of dinosaurs, the replica lead mine, and the fountains in the park with over 12,000 individual jets of water. But what was to change her life were the packed and dazzling displays of furniture, textiles and china, jewellery and silver, glass and sculpture, paintings, carvings and antiquities from ancient civilizations. It was a treasure trove. Charlotte had never before seen so many beautiful and fascinating things gathered together, the past and the present \u2013 as well as glimpses of what was to come \u2013 so jumbled and interweaved. For a woman devoted to learning, there was much to study. For a woman with a sharp eye for detail and value, there was much to admire. And for a woman with wealth, there were many opportunities to buy. Charlotte did not become a collector overnight as a result of visiting the Great Exhibition, but it gave her a sense of what might be possible.\n\nOne of the reasons Charlotte could not, or would not, indulge any enthusiasm that had been sparked by her visits to the Crystal Palace was that by 1851 John Guest was seriously ill. Her visits to the Great Exhibition were overshadowed by her concern for his deteriorating health, and by November, with her husband often feverish and incoherent, Charlotte was doing everything for the business. He could not travel, nor attend meetings, nor even sign letters or cheques. Doctors tried a variety of treatments but were, on the whole, baffled and pessimistic. There were periods when Guest had the energy to make short journeys and entertain, but these became shorter and fewer. Charlotte could do nothing but take on the burden of the business and hope.\n\nGuest's illness dragged on another year, coming and going, but in November 1852, after a sharp turn for the worse, he finally died. It was the end of a marriage, an unusually intense and equal business partnership, and of Charlotte's direct influence in political circles. It was also an enormous personal loss. Charlotte was, for many weeks, inconsolable. Two months after the funeral, when she visited Canford for the first time since her husband's death, she tried to describe her intense sorrow: 'When we stopped at the door, I got out silently, and leaving them all went straight to the Library, where luckily there was a light. A slight veil had been thrown over his bust, which at once I removed and then I flung my arms around it, and remained clasping it for some minutes, kissing the cold lips \u2013 not colder than his own when I kissed them last \u2013 and shedding torrents of passionate tears. And this cold marble is now all that is left me!'\n\nHowever, it was not John Guest's death which affected Charlotte's future as a collector so much as her remarriage. John's death inevitably involved much change, but as executor to his will, and a trustee of the estate, Charlotte continued to be heavily involved with the day-to-day running of the Dowlais works, much as she had been in the past. She continued to take charge of the business, struggling to make the works profitable in a changing economic and industrial climate, and negotiating a resolution to a strike among the workers in South Wales in 1853 \u2013 a fraught affair that turned her hair white. But she was hardly out of her heavy widow's weeds when she caused another scandal with her blatant disregard for Victorian proprieties. In 1855, at the age of forty-three, Charlotte married again. This time she chose 'Charley' \u2013 Colonel Charles Schreiber \u2013 a handsome soldier from a military German family, not yet thirty years old, a classical scholar and a fellow of Trinity College Cambridge. Respectable enough, perhaps, but he was also the tutor who had first come to Canford to school Charlotte's oldest son Ivor just twenty-four days after John's death. Her friends and acquaintances, and even her daughters, were shocked and horrified. It was not only that Charlotte had been so recently widowed, and that Schreiber was so much her junior, but Charles's role as a family servant put the relationship beyond the pale.\n\nOnce again, Charlotte was challenging Victorian convention, but she did not seem to care if her choice of husband was regarded as unfortunate. She did not worry that many considered that she had demeaned and disgraced herself, nor that her daughters were horrified by the thought of having Charles Schreiber as a stepfather. And she was quite relieved when Pegus disowned her. Her love affair reinvigorated her. She was ready for a new challenge. Under the terms of John Guest's will, a second marriage meant she had to relinquish the running of the ironworks to the other trustees. It also left her substantially less buoyant financially, and she immediately had to forgo luxuries like ponies and a carriage of her own. But she was content to sacrifice material comfort. Charles Schreiber's modest nature, good humour and energetic mind delighted her, and it was with growing excitement that they made plans together.\n\nThe marriage, in April 1855, was a quiet affair. This time, there was no cannon salute or triumphal procession. And the first few years of Charlotte's new life were unsettled and sometimes sorrowful. By 1856, the newlyweds were overspent by \u00a36,000, and money was a worry. Schreiber's personal wealth was negligible \u2013 or he would never have been forced to take on the role of tutor. The youngest of Charlotte's children by Guest was still only eleven years old, and the financial demands of raising her family were not inconsiderable. It was ten years before the Schreibers' affairs settled down and money pressures eased, with three daughters safely and successfully married. In addition, Charlotte was having to labour, for a second time, to edge back into polite society, being seen riding out in Rotten Row and attending Court. But her progress was slow and frustrating, and it was hard not to look back at the glittering social events she had been hosting just a few years earlier. Worst of all, however, were the disappointment and agony of three miscarriages, the last of which marked the end of the Schreibers' hopes to have children together, followed by the death, in 1862, of Charlotte's son Augustus at the age of just twenty-one.\n\nIt was not until all this upheaval and grief passed, almost ten years after Charlotte's second marriage, that she could begin to look around and think what she might like to do with her new life. And it was now that she found herself increasingly attracted to the idea of collecting. Things from her past, like the feathered fan, provided a catalyst for new areas of study. The people around her, particularly her husband and her cousin, tempted her with the idea of travel and adventure. The allure of beautiful and curious objects held out a romantic promise that seemed suddenly irresistible. Charlotte did not fall into collecting; she made a conscious decision that it was how she wanted to spend the rest of her life. By the mid-1860s, she had devoted herself to the idea, one which took her far from Dowlais and its furnaces, and which would beguile and consume her for the next forty years. She confided every high and low of her collecting, every expectant journey and successful transaction, to her numerous volumes of journals, kept in her elegant handwriting. Sometimes these are pithy notes relating little more than travel schedules, objects and prices, but more often the entries run away with colourful descriptions of journeys, people and places, glowing with delight in what she is doing and giving a real taste of what it was like to be a traveller at this time. Her journals take us to the heart of the day-to-day life of the Victorian collector. They reveal Charlotte's dedication, scholarship and courage; they give a lively sense of an all-consuming desire to collect, and they show just what it took to manoeuvre successfully in the cut-throat world of nineteenth-century collecting.\n\n### CHAPTER NINE\n\n### _Pushing and Panting and Pinching their Way_\n\n_I_ t was spring 1873 and the collecting season had begun in earnest. Fine weather and easier travelling conditions had coaxed collectors out of their drawing rooms and into the streets. Dealers, too, were restocking their shelves and refilling their displays. The Schreibers were far from alone as they made their way through the now familiar towns of Holland and Belgium. Speed and stamina were everything: on 8 April, after a half-past-five start on a misty morning, they squeezed in visits to Gouda, Utrecht and Rotterdam, before reaching The Hague in the evening in violent rain and with Charles, perhaps unsurprisingly, feeling 'ill and tired'. Nevertheless, their efforts proved to be in vain. There was nothing in any of the shops worth the trouble of the journey, and the Schreibers' big red bag remained empty. It was not that dealers were struggling to find new objects, nor that the usual supply routes from impoverished gentry and declining European aristocrats were drying up. It was that they were being outmanoeuvred by the competition. Exasperated, disappointed and defeated, Charlotte confided to her journal: 'we find everywhere that Duveen of Hull has been there before us making wonderful purchases.'\n\nRivalry between collectors throughout Europe was fierce. When J. C. Robinson was sitting in Rome in 1859, waiting for the South Kensington Museum's reply to his request to buy works from the Campana collection, it was the manoeuvrings of other collectors that most concerned him. During the 1860s and 1870s, an increasingly sophisticated European travel network was making it possible to cover ground reasonably quickly, cheaply and easily. By 1860, there were nearly 800 miles of railway in Britain for every million people, amounting to almost 10,000 miles of track. In France and Germany, railway development emerged more slowly during the early part of the century, but a boom in construction meant that by 1880 France had caught up with Britain and boasted some 15,000 miles of track, while Germany's political unification in 1871 was facilitated by almost 20,000 miles of railway that joined the major states in a public system of _L\u00e4nderbahnen._ In addition, the increasing number of museums emerging across Europe, and the growing state investment in collections, meant that men such as Robinson, Franks and Eastlake were travelling and collecting with the wealth and power of national institutions behind them. If and when they could disentangle themselves from government bureaucracy, they could often beat private collectors \u2013 who had fewer resources \u2013 to the spoils.\n\nClubs and galleries and organizations brought collectors together across Britain and Europe, but collecting in the mid-nineteenth century was as much about competition as about cooperation. As an expression of state prestige on a national level, museums were anxious to lead the way for their country \u2013 or indeed within their country. J. C. Robinson always asserted that there was no conflict between his collecting at South Kensington and the medieval and Renaissance treasures being accumulated at the British Museum by Augustus Franks, but there was little doubt that the two men were at times competing for the best pieces. In 1855, it was the rivalry between the two institutions which put a stop to the idea of acquiring the Bernal collection in its entirety for one or the other. Instead, the grant had to be shared. Robinson at once recognized what this meant: 'the state virtually committed itself to the formation of two concurrent new undertakings of the same character. . . with little co-operation betwixt the managers of the separate establishments, if not indeed with tacit rivalry betwixt them.' Fortunately, he and Franks came to an understanding for dividing the Bernal spoils, but the rivalry went further than just the two principal collectors. Henry Cole had a distinctly predatory attitude towards the British Museum: in 1852, he proposed bringing 'the overflowings of the Brit Mus' to South Kensington, and after the Bernal sale he was delighted to hear Antonio Panizzi, who was soon to be appointed Principal Librarian of the British Museum, suggest that 'the whole shd be handed over [to South Kensington] & even Franks with them'.\n\nBy 1859, Cole was talking to Panizzi about taking long-term loans from the British Museum, which would in effect strip the Great Russell Street galleries of their entire medieval collections. After Henry Cole's retirement, the rivalry continued but it was the British Museum that appeared now to have the upper hand: a Select Committee investigating the relationship between the two museums in 1873 proposed that the South Kensington Museum should be subsumed into its older rival. The autonomy of South Kensington was, for a while, under real threat, but the move was fiercely opposed by staff there, and by the mid-1870s it had been agreed that the two museums should continue to coexist. As the collections at South Kensington turned their attention even more from manufactures and towards precious objects, however, there were inevitably areas in which interests clashed.\n\n* * *\n\nAmong individual collectors, the instinct to be first was compelling. Despite constant weariness and not infrequent ill-health, the Schreibers took night trains and battled with omnibuses when easier options were available, not just to reduce costs, but also because they needed to pounce quickly on objects. In many cases, they were successful: in March 1874, for example, they 'espied a printed Battersea box, very handsome' in an auction in Brussels, and stepped in to buy it before it was offered to the gathered dealers, 'coming away well pleased with our bargain'. They took the competition seriously, and had the advantage over many collectors in that they travelled almost everywhere together. This meant that, when one of them, as was not unusual, felt too tired or sick to be out of their hotel, the other could continue the constant round of shopping that marked their days. In Duveen, however, as Charlotte's journal suggests, the Schreibers felt they had found their match.\n\nIn 1873, when Charlotte was writing, Joseph Joel Duveen was thirty years old, energetic, astute and beginning to establish a dealership that, in the hands of his successors, would influence the markets for generations to come. It was his son, also Joseph, who would in time become one of the wealthiest and most prominent dealers ever known, shipping paintings across the Atlantic to the American millionaires of the early twentieth century and funding celebrated extensions to the British Museum and the Tate Gallery. In the early 1870s, however, all this was just beginning. Joseph Joel, sent by his Dutch family to launch an import business in Hull specializing in Delft porcelain, was still in the throes of establishing himself as a dealer, drawing on his connections across Europe and familiarizing himself with the machinations of the market.\n\nDuveen had learned his lessons as an apprentice in an ordinary porcelain merchant in Hull. The shop he joined as a young man was not doing well. European suppliers were pulling out of deals, apparently without explanation; objects which were expected from Europe would fail to arrive and then turn up days or weeks later in competitors' showrooms. Somehow, information was leaking out, making the business increasingly vulnerable. And so the young Duveen took himself off early one weekday to the town's busy docks and watched as shipments were loaded and unloaded in the morning half-light. He stood amongst the soot and the salt and the baffling noise, watching as papers were signed and consignments counted; and, against the cranking of the machinery, the dull hooting of the ships and the shouting of dockers on the wharf, he listened. Gradually, things became clear. He heard details of the shop's transactions \u2013 its trade secrets \u2013 flung from one part of the dockside to another. He saw confidential lists sold on. He heard new, rival, deals struck. By the end of the morning, he knew exactly how information was escaping, who was betraying the company, and just how important it was to know exactly what was going on, all the time, everywhere.\n\nDuveen immediately booked himself a berth on a fishing boat to Holland so that he could visit personally the company's old suppliers and win back lost business. He was handsomely rewarded for his initiative: within a year, his salary was raised by his grateful employers from fifteen shillings to fifteen pounds a week, making him the highest-paid employee in the Hull import business. But this was not enough. It was not money that drove Duveen's ambitions \u2013 it was the thrill of competition. He soon began trading for himself, setting up a company with a colleague called Barney Barnett and using what he had learned in the commotion of Hull docks to make it thrive. With a network of friends and family acting as informants across Europe, he would pursue the whisper of a deal quickly and decisively, making the gathering of information as critical to his activity as the amassing of objects. He was, as Charlotte found out, first off the mark and first to the spoils. He had a superb memory for people, places and, above all, things, and, while at home his wife Rosetta was busy raising fifteen children and his brother Henry was establishing family showrooms in Boston and New York, Joseph Joel was slipping quietly from one European town to the next, following his precious tip-offs and outstripping competitors.\n\nIt was his connections in Holland that were most active and effective, and it was here in the spring of 1873 that Duveen first heard rumours of some valuable pieces of china. It was not quite clear what the pieces were, only that they were old and rare and in pristine condition. Duveen did not need to know more: immediately, he set off to buy them. But the journey was far from straightforward. Duveen's informant had been travelling through small, remote villages, making inquiries about anything local families might be willing to sell, and the pieces that had come to light were in an old house in a tiny hamlet deep in the Dutch countryside, miles from the railway or a town of any size. The only means of travel was by heavy carriage on the country tracks that wound their way slowly through woods and farmland. It was a long and tedious journey. Duveen spent hours rattling past isolated settlements, the carriage wheels dragging through the mud of the lanes. There were no inns to provide refreshment and nothing to see but flat fields stretching out bare to the horizon. It was with enormous relief that, finally, he caught sight of the roofs of the houses in the village where he was heading and knew that his journey was nearly over. He would negotiate quickly, clinch the sale and return to civilization as soon as he could.\n\nBut just as the tired horses were pulling the final half-mile, Duveen spotted a fast-moving fly coming towards him from the village. With just two large wheels and a cabin for the passenger, the fly was lighter and quicker than a normal carriage and could cover the uneven ground at greater speed. It was not the kind of vehicle Duveen expected to see in such an out-of-the-way place and he was puzzled. Perhaps he even had an inkling that, for once, he had been outdone. Because as the fly came rumbling towards him, he saw Charlotte Schreiber, bumping along with the quick jolting rhythm, a large red bag safely snuggled in her lap. As he was to tell amused listeners for many years to come, it soon became all too clear that he had been defeated: Charlotte 'had snatched the prize, which she was carrying off with her'; and Duveen was left to face the monotonous return journey in the knowledge that his efforts had been thoroughly wasted.\n\nThere is no exact record of what Charlotte managed to snatch from under Duveen's nose \u2013 he remembered it only, rather vaguely, as being 'objets d'art'. I find it surprising that Charlotte herself does not mention the victory in her journal, but she was probably unaware that Duveen was behind her and, with her new objects to savour, failed to notice him rattling past in his carriage. If so, it seems a shame. Such a triumph would surely have been sweet to her and would have reassured her that she was winning a place among the competitive community of collectors. There can be little doubt that the story added to Charlotte's reputation among her rivals scouring Europe for the best pieces. The very fact that Duveen dined out on the tale, passing it on with admiration as proof of her ability to outwit the opposition, shows just how seriously she was regarded by her colleagues, and how widely her name was becoming known.\n\nFrom the very beginning of her collecting career, however, Charlotte was at a disadvantage: she was a woman pursuing a distinctly male occupation. Most of the collectors and dealers were men, with easy access to learned societies, social clubs and the male hierarchies of the museums. Despite her obvious enthusiasm and knowledge, and even with her husband at her side, Charlotte sometimes found relations with the more misogynistic of European shopkeepers difficult. Among her family and friends, too, her collecting was regarded by many as a bizarre idiosyncrasy that distracted from her role as wife and mother and prevented her from attending to the essential demands of running a household. For the Victorian middle and upper classes, the home was a sanctuary from the uncertain social and economic forces of the world beyond, a still point in the changing, and increasingly commercial, environment of modern life. The royal couple of Queen Victoria and Prince Albert with their obvious devotion to each other and their children, and a public domestic life built round a clear sense of duty, offered an embodiment of respectability to which those below them could aspire. Within this ideal, the woman's place was to anchor the home and provide a safe private space that would reassure and fortify her husband, however buffeted he was by public life.\n\nThis was not Charlotte's way. Her collecting was not confined to Europe \u2013 she spent weeks and months tracking down objects in Britain, too \u2013 but it always required travel, and most of her attention. Her son, Montague Guest, remembered that she would 'come back, after weeks on the Continent to Langham House, where she lived, rich with the fruits of her expeditions', and, once reinstalled, she would spend a great deal of time and energy sorting, cleaning, assessing and cataloguing what she had acquired. In the eyes of conventional upper- and middle-class Victorians, these were not the actions of a sane and proper woman; instead, such behaviour suggested neglect of her family and household, and a disregard for her social position. For nineteenth-century women of the middle and upper classes, some knowledge of art, and the formation of an artistic taste, might be valued as a fashionable accomplishment, but it was expected to remain an amateur hobby. Engaging with dealers while researching and financing a substantial collection was unusual, and, in devoting herself to her collecting, Charlotte was openly flouting convention.\n\nSuch expectations did not mean women were prevented entirely from collecting, but they certainly restricted the aspirations of most female collectors. The growing enthusiasm for natural history and science collections, for example, threw up few successful female role models. The celebrated Mary Anning was a skilled fossil hunter, palaeontologist and dealer in Lyme Regis during the 1820s and 1830s, but not only was she a woman, she was also a poor woman from a working-class, dissenting family. Despite being well known in geological circles, and winning a certain degree of respect from many of her male colleagues, she remained an outsider and was never given full credit for her contributions to a scientific community that was dominated by wealthy Anglican gentlemen. Similarly, with the development of disciplines such as botany and zoology over the course of the nineteenth century, many women found themselves increasingly excluded and marginalized, forced to study, collect and draw informally, as amateurs based at home, rather than being included in the developing hierarchies of professionalized practice.\n\nWomen also often found themselves excluded from areas such as painting, sculpture and antiques, which were all very much embedded in a male tradition that had, for generations, governed the stately homes, scholarly clubs and even schools, where the habit of collecting often began. Women were not admitted to the Society of Antiquaries until 1920, and of the two nineteenth-century archaeological clubs only one, the Royal Archaeological Institute of Great Britain, was open to women. Despite a petition to admit women in 1859, the august schools of the Royal Academy did not allow female members until 1922 (even though two women had been founder members in 1768); women students were admitted from 1861, but only under restrictive rules that insisted on their drawing from draped models. Similarly, the Royal College of Art shunted its women students into a special 'Female School' where life drawing classes featured a series of well-clothed dummies. It was not until the end of the century that many conservative societies began to extend membership to include women: the Royal Society of Painters in Water-colours, for example, did not give full membership rights to women until 1890.\n\nThe Fine Arts Club, as I have already mentioned, allowed women to attend meetings from the mid-century, but, even then, it did so rather grudgingly. For the first year after its foundation, female connoisseurs were only afforded honorary instead of full membership and of the 201 members only eight were female. Charles Schreiber joined the club early in its life in 1858, and was an active contributor, exhibiting his first objects there in 1863 and organizing a display of porcelain, lacquer and jewellery for fellow members at the Schreibers' home in April 1866. Charlotte was encouraging and often attended meetings, as well as hosting those that were held at her house, but she never became a member, perhaps because even she felt uncomfortable in an environment that so closely resembled a gentleman's club. While the example of pioneers like Mary Anning suggests that class was one of the factors determining the degree to which women were allowed to participate in a scholarly community, it was clear that even for Charlotte Schreiber, with an impeccable family background, the right upbringing and a place in elite social circles, it was far from easy to earn the respect of male peers or to take advantage of the opportunities offered to male collectors.\n\nSuch restrictions inevitably influenced the type of objects Victorian women tended to collect. Instead of ancient classical artefacts and Old Masters, their attention often turned to things that did not appear inappropriately scholarly or assertive, or which many of their male counterparts regarded as trivial. This was not necessarily a mark of defeat. Charlotte was typical in exploiting the lack of interest of most male collectors in objects such as playing cards, fans and especially ceramics, not only creating for herself a new and defined area of expertise but also allowing her to take shrewd advantage of less volatile, and less expensive, markets. Many male dealers knew little or nothing about these less traditionally popular types of object: 'ignorance was the prevailing characteristic,' explained Charlotte's son in introducing her journal. 'Nobody wanted Old Sheffield Plate, Pinchbeck, old English jewellery, needlework pictures, old English glass, pewter, Staffordshire ware, excluding Wedgwood, old steel, brass etc. . . my mother was able to pick up the finest specimens of china and other such articles for quite a moderate outlay.'\n\nThe enthusiasm for 'specimens of china', especially porcelain, among female collectors was particularly pronounced. England's Queen Mary II had collected oriental porcelain in the seventeenth century, helping to make the craze for Japanese china fashionable and respectable. In his _History of England_ , Macaulay looked back scathingly at the way the fashion had taken off, dominating many women's lives: 'In a few years almost every great house in the kingdom contained a museum of these grotesque baubles. . . a fine lady valued her mottled green pottery quite as much as she valued her monkey and much more than she valued her husband.' At first, only noblewomen and women of the gentry could afford to follow in Queen Mary's footsteps but despite, or perhaps because of, this the fashion flourished. In the first half of the seventeenth century, imports of Japanese ceramics peaked while cheaper, less ornamental pieces from China began to flood into the European market, making it easier for less wealthy collectors to join the fray.\n\nThere are many reasons why women took to collecting china, and particularly porcelain. Not only were ceramics enormously fashionable, but such a collection was associated with wealth and power and so was also extremely prestigious \u2013 owning even a few pieces could go a long way towards boosting social standing. The growing popularity of domestic rituals around tea-drinking also gave women the opportunity to take a more visible role in polite gatherings. Their tea sets placed them at the heart of a new kind of social event, taking control of the occasion and using their china to orchestrate it. For many, however, the attraction of china was much more straightforward: seventeenth- and eighteenth-century women were seduced by the bright colours, the extravagant decoration and the sheer sense of luxury of their porcelain treasures.\n\nBy the nineteenth century, this tradition meant that there was a great choice of secondhand 'old' china for collectors, alongside new china being produced or imported to meet the demands of the market, and the growing sense that china had a history started to change the way in which it was perceived. It was no longer viewed as simply decorative and fashionable \u2013 it began to acquire the status of art. When public museums like the South Kensington Museum, and popular exhibitions like the 1857 Manchester Art Treasures Exhibition, included ceramics in their displays, this consolidated the idea that ceramics had artistic merit, and might be worth collecting. The 1860s and 1870s witnessed the publication of a variety of books and pamphlets offering histories of English and Continental ware, as well as enticing the collector with hints on which pieces to look out for, which shops and dealers offered the best bargains and which were the most profitable hunting grounds. Charlotte and Charles Schreiber, beginning their collecting around 1865, were at the vanguard of this Victorian chinamania. They made the most of the fact that knowledge was not yet quite widespread nor the market entirely prepared for the onslaught of collectors. They also set an example for those who came after them with their energetic and skilful ceramic collecting, remaining somewhat critical of those who were ignorantly swept along by what Charlotte called the 'ridiculous rage' for china.\n\nIncreasingly, the wealthy and influential housewives of the middle classes began to collect ceramics, even if modestly. Some men, too, developed a taste for ceramic collecting, but this was more unusual. China came to be seen as so much a female speciality that a man's enthusiasm for ceramics was often regarded as evidence of effeminacy. Charles Lamb, the English essayist of the late-eighteenth and early-nineteenth centuries, was one of the small number of men who admitted to a fascination with ceramics, and in an essay on 'Old China' he openly admitted that this was not an entirely masculine predilection: 'I have an almost feminine partiality for old china,' he confessed. In contrast, for women, a small collection of china was considered a proper and reputable outlet for their energies. An enthusiasm for china collecting could be roundly approved by the male establishment: it was useful enough for the tea table and pretty enough to keep a woman amused. It accorded neatly with Victorian ideals about the woman's role in the home, and a few nice pieces of china or a handful of tasteful 'objets d'art' reflected well on the standing of the household. Surveying the various types of possible collections, one commentator noted condescendingly: 'CHINA. This is a hobby that ladies should cultivate. . . Women, as a rule, have little taste for collecting books, prints or pictures, but it is a fact that they evince quite an attachment to their ordinary china service.'\n\nBut it was not simply that china was viewed as an appropriate distraction for women, particularly those of the upper classes; it was also usually regarded as personal \u2013 instead of family \u2013 property. Unlike jewels, classical antiquities or paintings, which would usually pass from father to son, pieces of china were rarely subject to the strict rules of primogeniture inheritance and could often pass quietly from mother to daughter, or from sister to niece. This had the advantage of potentially circumventing the restrictions on female ownership. Unmarried women, though often marooned in a social no man's land, at least had the privilege of owning their possessions. Married women, in contrast, handed over the rights of ownership to their husbands. If they became collectors, by law they could not own any of the objects they collected, nor choose to whom they left them after their deaths. It was not until 1870 that the government passed the Married Women's Property Act, which allowed women to own possessions they had acquired after their marriages, and it was another twelve years before the same right was extended to objects owned before marriage. Until these laws came into force, a woman's possessions, like herself, were considered to be simply an extension of her husband's property. A few pieces of china, however, were frequently considered to be unworthy of much attention; husbands often allowed their wives to dispose of such a collection as they wished. Bequeathed through the generations in this way, the best or favourite pieces could become heirlooms down the female line.\n\nCollecting was what the Schreibers did together, and they worked side by side in every aspect of creating and developing their collection. But other women were not so fortunate as to have a family which sympathized with the all-consuming urge to collect. Many Victorian men believed that their wives, mothers and daughters would be unable to control themselves if they became too involved with the stimulating world of 'things'. They feared that such obsession would lead to personal instability, the breakdown of domestic order and even sexual licentiousness. During the Great Exhibition of 1851, _Punch_ highlighted a variety of dangerous impulses that might be prompted by the glittering displays. In one cartoon, middle-class ladies are shown resisting police eviction from the Crystal Palace at closing time with a defiant, even violent, show of solidarity, opposing 'civil power at the point of a parasol'. In another, 'The Awful Result of Giving A Season Ticket to Your Wife' is a neglected and empty house. Worse still, in another cartoon, flocks of women are depicted relishing the highly physical, sexually charged atmosphere of the Exhibition as they 'push, and pant, and pinch their way amongst each other' to see the famous and fabulous Koh-i-Noor diamond. Over twenty years later, the idea that collecting was somehow at odds with healthy female instincts still lingered. A _Punch_ cartoon of 1874 shows a dishevelled middle-class mother bemoaning the breakage of a valued vase. Her oldest daughter tries to comfort her as five younger siblings look on, but a family is apparently no consolation: 'You child! You're not unique!! There are six of you \u2013 a complete set!' wails the mother disconsolately. The message was clear: left to their own devices, women could not be trusted to bring self-control and dignity to the collecting and display of lovely objects.\n\nThis discomfort with display and a woman's part in it revealed all kinds of Victorian anxieties. During the eighteenth century, the capitalist spirit was frequently seen as a positive attribute which could keep in check other more unruly and immoral inclinations, and act as a counterweight to undesirable behaviour. By the middle of the nineteenth century, however, attitudes towards conspicuous consumption had become much more ambivalent. Despite the growth of a commodity culture, the development of department stores, extravagant shopping habits and spectacular one-off events like the International Exhibitions, the Victorians' response to the objects with which they liked to surround themselves was complex. The activity of creating wealth had acquired overtones of coarseness and taint, and the realities of commercial enterprise needed, at least in public, to be separated from the higher moral priorities of family and social or religious responsibility. The display of too many objects within the home ran the risk of being perceived as exhibitionism, which in turn could be linked to degenerate and even immoral behaviour. Women, in particular, who exhibited too openly a preference for 'things', for shopping and luxury and spending, could be portrayed as artificial and untrustworthy, and even sexually promiscuous.\n\nIn the light of these attitudes, it is perhaps no surprise that female collectors were often reticent about their activity and, officially at least, appeared to be thin on the ground: of the 131 ceramics collectors recorded in 1851 by Joseph Marryat, the first writer in English to address the history of ceramics, only twelve were women. Even though these numbers rose to thirty-three by the late 1860s, female collectors still represented only a fraction of the total. But for all the reasons we have seen, collecting was, in fact, not unusual among women. On the other hand, it _was_ unusual for women to be high-profile collectors like Charlotte Schreiber and it was also rare for them to make large bequests to public museums to cement their place in collecting history. (It is perhaps revealing that, while Charlotte's name is associated with the collection of English ceramics now at the Victoria and Albert Museum, it was Charles Schreiber who first proposed a permanent public home for it.) Nonetheless, collecting was a popular female pastime. It tended, however, to be a more private pleasure than it was among men, and it was often linked to the home education of children, particularly girls. It was often viewed as a welcome means of self-expression, rather than an historic investment, and it tended to be an aesthetic as much as a scholarly pursuit: Charlotte Schreiber's collection is unique among Victorian women's collections in the range, quality and extent of the documentation that she maintained.\n\nWith few opportunities to join the male-dominated art and collectors' clubs, those women who did enjoy collecting had to find more informal ways to meet to discuss their enthusiasm, exchange information and arrange for the buying and selling of objects. Compared to male networks, the female equivalents were small and distinctly less powerful, but nonetheless they brought like-minded collectors together to share ideas. Charlotte's journals are littered with references to helping out her sister collectors. In November 1869, for example, she spent 'two very pleasant hours' with Mrs Haliburton, a widowed china collector who became a regular visitor and correspondent, and in June 1884 she called on Lady Camden, in Eaton Square, to discuss china. By the 1870s, Charlotte was already being recognized as an expert and she was able to use her unusual level of access to the male worlds of curating and dealing to act on behalf of her female friends both at home and abroad. In October 1873, while in Rotterdam, she called at Boor's Bazaar 'to execute a commission for Mrs Haliburton', and a few months later she took a ceramic mould to elicit an opinion from Augustus Franks at the British Museum on behalf of a friend, Dorothy Neville, another active collector.\n\nCharlotte's women friends clearly recognized that she had battled her way into the male bastions of the collecting world. Her collecting was undertaken on a different scale from almost any other woman of her time, and this was what set her apart. She herself often refers to her collecting in her journals as 'work', an indication of just how serious and consuming a matter she considered it. And the male establishment certainly did not view her collecting simply as a genteel lady's hobby. Augustus Franks was a close friend and collecting confidant. The two shared regular discussions about their objects, and frequently bought and sold pieces between themselves. Franks was, according to Charlotte's son Montague, 'a constant visitor at her house'. When she was abroad, Charlotte mixed as an equal with museum professionals: in Tarragona in 1870, the director of the museum took time to show her personally 'all over the antiquities collected there', and in November 1874 she was shown around the Utrecht Museum by its founder, Monsieur Van Huckelm, 'a most well-informed and agreeable man. . . making this collection. . . quite wonderful'. In the 1870s, when the china craze was at its height, the London dealers Mortlocks, based in Oxford Street, even relied on Charlotte as a 'runner', asking her to use her European connections to purchase blue-and-white ceramics on their behalf. Mortlock openly admitted 'that nearly all he knew about China he learned from her', something which, as Charlotte's son acknowledged, was 'a marvellous tribute indeed, from the professional to the amateur'. In spite of being a woman, Charlotte was respected, and even feted, as a collector. So completely did she challenge the conventions of what female collectors could achieve that few of her companions, male or female, could boast such success. Few kept up such an energetic life of travelling, research and bartering; few made so many confident and profitable deals; and few demonstrated such commitment to collecting as a way of life.\n\n### CHAPTER TEN\n\n### _The Gourd-shaped Bottle_\n\n_C_ harlotte and Charles Schreiber made twenty-three foreign trips between 1869 and 1882, spending around half of each year, and sometimes more, travelling in Europe. This had the advantage of being cheaper than maintaining an appropriately large and fashionable household in England, and they were eager to see the sights in the towns and cities they visited. But the driving force by far was their collecting. Prices abroad tended to be more reasonable, even for British wares, and Charlotte's journals reveal how the Schreibers sought out shops and dealers, private collectors and museums, in a constant rush to view, compare, buy and sell objects. Whatever the length and tribulations of a journey, it was usually just a matter of minutes before Charlotte was out into the streets of a new destination, continuing her search: in August 1869, for example, she bemoans arriving so late in Gouda that they were 'barely in time to save daylight to ransack the old dealers' stock'; on 8 March 1872, they disembarked from the night train in Amsterdam and immediately 'took a carriage and went straight to Van Houtum's to see what he might have'. The Schreibers continued their collecting at a similarly frantic pace during the months back in England, but the attention given to the European crusade in Charlotte's journals indicates just how absorbing and exciting it was to be collecting abroad at this time.\n\nAs we have seen, Europe was ready for collectors. Thanks to the French Revolution, the Napoleonic Wars, the Franco-Prussian War, the revolutions and political instability that accompanied nationalist movements in Italy, Germany, Denmark, Hungary and Poland; changes to the old structures of royalty, nobility and the church; growing urbanization; shifting patterns of wealth; fashionable commercialism and economic fluctuation, there had never been a better time to be collecting. In Holland, the Schreibers hoped to pick up the distinctive blue-and-white ceramics of seventeenth- and eighteenth-century delftware, or even 'original' blue-and-white that had been imported from China. Such pieces had been popular as domestic furnishings in the seventeenth century but had since dropped out of fashion and been relegated to back rooms and store cupboards, junkshops, auctions and markets. In Dresden, the Schreibers were on the trail of more ceramics, in particular from the renowned Meissen works, but they were also offered ecclesiastical relics and the entire interior decoration of a local church. In Belgium, they were shown choice examples of oriental china, imported from Japan, Chelsea ware that had made its way from Britain, and an ornamental metal picture frame that had once graced the palace of Louis XIII of France. All over Europe, objects that had been hidden away in homes, castles and churches were coming on to the open market, while new and expanding trade routes were moving objects across the Continent, and even the world.\n\nThe Schreibers travelled from north to south, taking advantage of a changing Europe. But for all the riches to be had elsewhere, it was in France that the network for collectors offered most. As Robinson had discovered as a student in Paris during the 1840s, by the middle of the century, the French had already discovered a widespread interest in collecting. In Germany, in contrast, undisturbed collections of treasures from the Middle Ages and Renaissance continued to languish quietly in the houses of the nobility. But in France, in the aftermath of the Revolution, greater instability created a much more vigorous marketplace in which similarly impressive objects changed hands frequently. 'The French Revolution,' according to writer and critic Jules Janin, 'began to break everything, to destroy books, to cut paintings in pieces. . . to melt gold and silver. . . to sell \u2013 at auction even \u2013 the marbles of tombs.' This, however, was not the end of the story. What was cast out was usually immediately rescued, Janin went on to explain, by 'a whole army of antiquarians. . . whose life and fortune are spent in collecting these scarce remains, in saving from oblivion these precious remembrances'. Medieval and Renaissance art works could be picked up cheaply in junkshops and market stalls \u2013 everything from sixteenth-century furniture to prints by D\u00fcrer or Rembrandt, from armour and ceramics to marriage caskets. Across France, church silver pulled from the altars now appeared in auction rooms, and family collections were broken up and sold. Within a generation, a treasure trove of previously hidden, often completely unknown, objects became available: silver and tableware that had belonged to princes and nobles; paintings, tapestries and jewellery; precious religious objects and even entire walls, altarpieces and massive doorways torn from churches, private chapels and monasteries. Some of Europe's most beautiful and valuable pieces were now appearing for sale to anyone who could afford them.\n\nBy the 1850s, when the fashion for the leftovers of the French _ancien r\u00e9gime_ had really taken off, Paris was the heart of the collectors' circuit. Calling at a renowned dealers in Utrecht in 1869, Charlotte found there was 'absolutely nothing' because 'he had 84 cases packed up for Paris', and visits to the curiosity shops of Alicante in March 1870 were equally unsuccessful because 'the best things' were all kept back to be sold to the French capital. The violence of the Commune in 1871 interrupted Paris's preeminence but the setback was temporary. By May 1872, Charlotte was noting that 'all the great English dealers' were again gathering at the Paris sales, and not long afterwards she was delighted to be finding so many promising shops. 'We enjoyed our _chasse_ very much,' she observed.\n\nThe Schreibers' peripatetic lifestyle was not always so enjoyable, however. Charlotte's journals include plenty of accounts of days when nothing was bought. The couple were sometimes ill, and frequently tired. The stock of worthwhile objects was often exhausted and dealers could be obstructive, rude or even mad. Disappointment was commonplace.\n\nThe story of Charlotte's gourd-shaped bottle at first appeared to be one such disappointment. It was February 1873 and, unusually, Charles was alone on a short trip to dealers in Holland, leaving his wife at home. We know very little about Charles as a partner in the couple's collecting. Charlotte's journals provide her point of view, and, although she sometimes records her husband's opinion \u2013 referring to him always as C. S. \u2013 we learn little of his character. We discover most about his often precarious state of health, the apparently constant headaches, sore throats, inflamed eyes, undefined pains and exhaustion that may have been symptoms of an underlying and untreated condition. He was clearly as enthusiastic about collecting as his wife, however, and it may even have been his encouragement which first inspired Charlotte's interest. Beyond this, it is difficult to be sure of the nature of his contribution.\n\nAs a scholar, it is likely that he shared in the research that made their collecting of such value, but he never published anything. The impressive catalogue of the collection of English ceramics at the Victoria and Albert was written after his death and bears his wife's name. Like Charlotte, he clearly had an interest in politics and was elected MP for Cheltenham in 1865, but he lost the seat three years later and did not stand again until 1880, when he became MP for Poole without distinguishing himself on any particular political issue. Apart from his parliamentary role, he did not work. Charlotte herself suggests that he was often at a loose end: after a visit to the 1857 Art Treasures Exhibition in Manchester, she noted: 'Having no profession, no pursuit at this particular moment, I cannot describe how pleased I was to see how heartily he threw himself into the spirit of all around him.' I could find no other contemporary accounts of him and in his portrait he appears unremarkable, even insipid. It is only from his anecdotal role in incidents from Charlotte's collecting life, such as the story of the gourd-shaped bottle, that we get any sense of his perspicacity, dedication and skill as a collector.\n\nThe trouble with Charles Schreiber's Dutch trip was that he had bought too many things. His boxes and cases were full and heavy. Although he had already made arrangements to send as much as he could back to England, without Charlotte and her big red bag, he found he could not carry all the remaining purchases alone. Worse still, he found himself amongst the plinths and shelves of a cluttered showroom in Rotterdam known as Kryser's, and he wanted to buy more: a large Wedgwood bowl and cover, which had been partly decorated in Amsterdam in 1808, and a strange bottle, also apparently Wedgwood, but shaped like a gourd and decorated with a painted landscape bright with flowers. Charles clearly felt he had to have both pieces, even at the risk of leaving things behind, and so he negotiated with Kryser. But when it came to finding space for them in his already overstrained luggage, he realized that he had overreached himself. All he could do was ask Kryser to put the purchases to one side so that he could collect them on a future visit. It was not the ideal solution, because they were two of the finest pieces he had managed to unearth during the trip, but he knew that he and Charlotte would be back in Rotterdam before the spring was out.\n\nIn the end, it was not until the beginning of April that the Schreibers managed to return. It was a few days before Easter, and in the warmer weather it was a pleasure to stroll together around the busy port and into the labyrinth of lanes behind. The town gardens were beginning to blossom, there was more life in the streets, and it was easy to spend the spring hours idling in front of shop windows. Doors were thrown open to the street, and the alluring treasure in the dark interiors twinkled in the spring sunlight. The Schreibers were not here to dawdle, however; their minds were fixed on retrieving their treasure from Kryser's. But, when they reached the shop, Kryser pretended that he had never seen either of them before, even though they had both shopped there on many previous occasions. Worse still, he denied all knowledge of the sale a few months earlier.\n\nPuzzled and annoyed, the Schreibers clattered up the wooden steps to the top rooms of Kryser's shop and there found the bowl and bottle. Kryser's memory, however, still refused to be jogged. He was adamant that no previous transaction had ever taken place. The negotiation had to begin again. Finally, a sale was agreed. The Schreibers paid a sovereign, again, for the bottle, and carried it off with them at last. Back at the hotel, their purchases safely carried between them, it was the bottle that took pride of place. Charlotte did not know exactly what it was but she was sure it was uncommon and of the best quality. Perhaps it was Delft china, or perhaps something even rarer. Whatever it was, she was certain they had a bargain. Only a few days later, however, she was upset by a very different valuation. She was in The Hague, hosting a tea party for some friends, fellow travellers from England and old Dutch acquaintances. Charlotte proudly showed off her newly acquired bottle and it elicited a great deal of enthusiastic comment. Hardly anyone there had seen anything like it. They admired its unusual pregnant form and the colours of the decoration. Almost everyone thought, like Charlotte, that it had been a good buy. But one of the guests, the painter Christoffel Bisschop, disagreed. He was an expert on fine Dutch china, so he said, and this was not fine Dutch china. It was German, cheap and ordinary. It was attractive, perhaps, but nothing more than a trinket and essentially worthless.\n\nThe Schreibers had made bad purchases before but Bisschop's verdict inevitably came as a disappointment. Something about the bottle still excited them, and when the guests had drifted away, and the tea things had been cleared, they talked for a long time about whether or not to keep the bottle, just in case. Pragmatism, in the end, prevailed. They could not afford to encumber themselves with second-rate pieces. They needed to save space, as well as money, for truly rare things \u2013 for the very best. So they agreed to resell the bottle, and on arriving in Utrecht a few days later, on 8 April, the matter was settled when a dealer there confidently announced that he knew an amateur collector who would give good money for it. The journal volumes record the Schreibers' relief: '\"Then for goodness' sake let him have it,\" said C.S., who had been put thoroughly out of conceit with his purchase, and now only cared to be rid of it.'\n\nThe bottle was left behind, but as the Schreibers passed through Utrecht again six months later, in October 1873, the story begins once more. Relating the tale delightedly in his notes on his mother's collecting, Montague Guest recounts the dealer's verdict: '\"I sold your bottle to the great amateur for 100 florins,\"' the dealer explained. '\"He took it home and was delighted with it, but in a few days he brought it back to me and would have nothing to say for it, for his friends had told him it was only a piece of German ware and of no value whatever.\"' History, it seemed, was repeating itself. The Schreibers had to take their bottle back, and forgo their hundred florins. Frustrated by the entire experience, Charles was determined not to waste energy on carrying the disappointing bottle to England and a few days later he called at a sale in Antwerp to offload it: 'He got a letter to inform him that the troublesome bottle had been sold \u2013 this time sold outright, and had realized eighteen shillings.'\n\nThe sale represented a financial loss, but at least they were rid of the bottle, and as Charlotte noted stoically, 'These things will happen to the best regulated collectors.' Besides, they could be sure to recognize, and avoid, such inconsequential German pottery when they saw it in the future. This might have been some consolation, except that the next time they came across a piece exactly like their bottle was almost a year later, in August 1874, and it was not in a dingy junkshop backroom but in the elegant, beautifully lit cases of the Rouen Museum in Normandy, set on the most conspicuous shelf at the heart of the town's collection. There, carefully arranged 'in the very place of honour', was a group of bottles exactly like the one the Schreibers had dismissed, the same size and strange gourd shape, the same cheerful decoration. Aware that they had made a terrible mistake, they began to investigate further, and as they discovered more about the precious bottles they realized that 'nothing in Europe, or indeed in any other part of the world, [was] as rare as these productions. . . only about a dozen being hitherto known'.\n\nIn fact, the bottles were so rare that the experts could not agree as to exactly what they were: some claimed that they were the work of Denys Dorio, an obscure Italian who had once made a few such things at Rouen en route to setting up a studio in Holland; others claimed they were seventeenth-century Delft china from the renowned workshop of Aelbrecht de Keiser. Whatever they were, experts agreed that there were few known examples. Two were in a private collection; the rest were at the Rouen Museum. Except, of course, for the one that Charlotte and Charles Schreiber had once owned, that was now somewhere in Holland, sold at an obscure sale for a song. It was infuriating: 'And now indeed there was wailing and gnashing of teeth! To have been unconsciously possessed of such a treasure, and to have been at such pains to have deprived oneself of it was almost more than amateur nature could endure.'\n\nThe Schreibers returned despondently to England. The thought of what they had lost preyed heavily on them. They did not have the heart to look over the things they had brought back, and they found no comfort in the familiarity of the collection waiting for them at home. All they could think about was what might have been. The weather in England now seemed chill and uninviting; the house seemed dull. They could not bear the thought of admitting their mistake to friends, and within the week they were back on the boat to Brussels, making the choppy Channel crossing and hoping that something would turn up to distract them from their disappointment.\n\nThe shops in the Belgian capital seemed intent on annoying them further; there was little to tempt them. Their latest journey had all the hallmarks of another wasted trip, but it was on this visit to Brussels, almost two years after Charles Schreiber had first spotted and bought the gourd-shaped bottle in Rotterdam, that something unexpected happened. They were about to leave a little-known dealers, Polonet's, in the newly built Rue Gr\u00e9try by the covered market, a shop, according to Charlotte, 'of no great account'. Like the other dealers, Polonet had had nothing to interest them. But, just as they were leaving, Charlotte happened to cast her eyes 'to a topmost shelf ' and there saw a bright and familiar object. By some extraordinary luck or accident, some tantalizing twist of fate, the Schreibers had a second chance: in a dark corner of the humble Brussels shop, explained Charlotte in triumph, 'I spied the lost and much-lamented Rouen gourd-shaped bottle, whose very history is a romance.'\n\nPolonet had acquired the bottle at the Antwerp sale in the spring and, like many people before him, he guessed that it was nothing more than simple German ware of more decorative than financial value. He had no hope of a great profit. But 'C.S. . . instantly paid whatever he was asked without demur.' The price was forty francs, quite a lot of money for a nondescript piece of German pottery. But next-to-nothing for the rare and beautiful work of art that the Schreibers had been tracking across Europe. With a nod of the head and a shake of the hand, the deal was done. The bottle was swiftly dusted and packed, and the Schreibers carried it 'triumphantly away'.\n\nThe bottle did eventually turn out to be German. It was finally identified as a rare faience ceramic vase by one of a group of seventeenth-century studio painters known as the Nuremberg _Hausmaler_ , artists who specialized in decorating ceramics or faience. The vase itself was probably made in Frankfurt, but it was the decoration which made it so eminently collectable, and it is believed it was painted by _Hausmaler_ Abraham Helmback around 1700. The Schreibers treasured it, and, on Charlotte's death, it was inherited by her daughter Blanche, the Countess of Bessborough, who was herself a keen collector. It remained in the Bessborough family at their home, Stansted Park, on the Hampshire\/West Sussex border, until October 1999, when it was sold by Sotheby's, attracting a guide price of \u00a310,000\u2013\u00a315,000. Charlotte's investment, it seems, had ultimately been proved a good one.\n\nFor the Schreibers, the protracted story of the bottle ended in a sweet victory that bolstered their confidence in their own judgement. The confusing episode had shown how well-meaning friends and eager dealers could be wrong in their appraisals, and could not be relied upon to give the final word on an object, especially when so much collectors' pride was at stake. If the best pieces of European treasure were to be extracted from their hiding places, then Charlotte and her husband needed to trust to their own resources. But, for Charlotte, the open display of too much scholarship was not without its own problems. As we have seen, at a time when the intellectual and social activities of respectable women were constantly being scrutinized and judged, those who seemed too learned risked blurring the boundaries between an amateur hobby, which was acceptable, and a professional undertaking, which was not.\n\nCharlotte was in the potentially awkward position of needing to acquire scholarly knowledge to make her collecting a success, and needing to obscure it to retain her social standing. Like collecting, art criticism remained a staunchly male domain, and there were few women willing to challenge this convention. Those who did had to be prepared to stand out from the crowd and have their names associated with a wide range of feminist causes. One of the best known of these was Emilia, Lady Dilke, who attended the South Kensington Art School in the 1850s before beginning to publish in a range of journals, becoming a contributor to the _Saturday Review_ in 1864 and fine art critic for _Academy._ By the end of the century, she had published several major works of art history, including two volumes on _The Renaissance of Art in France_ (1879), a critique of _Art in the Modern State_ (1888) and a series of works exploring French architecture, sculpture, engraving and furniture.\n\nBut Emilia was a controversial woman, the subject of much discussion in polite Victorian society. During her unhappy first marriage to Mark Pattison, Rector of Lincoln College, Oxford, and a reputable scholar, she was publicly and frequently unfaithful, as a result of which several thinly veiled portrayals of adulterous wives and cuckolded husbands appeared in popular novels of the time. Her second marriage, to Charles Dilke, also carried a hint of scandal: they were married very shortly after Dilke was publicly named in a divorce case which ended his influential political career. In addition, Emilia's close association with the trades unions \u2013 in particular the Women's Trade Union League, of which she was president \u2013 and with feminist causes such as women's enfranchisement meant that she continually courted controversy. Certainly, her behaviour provided plenty of ammunition for those in the establishment who wanted to argue that female scholarship led to a breakdown of social order and the undermining of accepted values.\n\nAnother of the rare female art critics was Anna Jameson, who made her way from the twilight existence of a governess to become a woman of letters. In 1842, she wrote a _Handbook to the Public Galleries of Art in and near London_ , and two years later she published _A Companion to the Most Celebrated Private Galleries of Art in London_. Between 1843 and 1845, she contributed over forty essays on Italian painters to journals. Her work was clever, analytical and original. She gained widespread respect for her knowledge and her ability to convey her ideas accessibly: 'her contemplation of the great works of the great schools was more intelligent than that of men in many ways more learned than she', observed one later commentator in 1879. For her series on _Sacred and Legendary Art_ , which appeared from 1848 until after her death in 1860, she had lengthy discussions with Charles Eastlake, who had been working on a similar theme. Eventually, in a rare act of scholarly magnanimity and as a clear indication of his respect for her work, Eastlake gave her all the material and references he had collected himself. Despite such achievements, however, Anna had to make her writing non-polemical and uncontroversial, weeding out phrases that seemed too explicitly feminist in an effort to get readers, both male and female, to take her views seriously. To sell her books to a wide public, it was critical that she seemed respectable, inoffensive and unchallenging. She had to consider what her writing might imply about other areas of her life, and was always conscious of her sensitive position as a woman in the masculine world of art scholarship.\n\nDeveloping scholarship as an integral part of her collecting, Charlotte would certainly have been aware of the potential hazards of her situation. As far as we know, she was not particularly feminist in her views \u2013 her journals give little indication that she was interested in feminist issues \u2013 nor was she keen to disrupt her family life with controversy. But she was naturally intellectual and ambitious, and, while she may not have been willing to take up a wider feminist cause, her marriages, her close involvement with the Dowlais steel business and her Welsh studies all indicate that she was happy to challenge, on a personal level, Victorian preconceptions about what she could do as a woman. As her collecting developed, so too did this challenge to convention. Privately, without fuss, she was developing a level of erudition to rival any male collector's. Wide reading, learned discussion, an excellent visual memory and a talent for thinking on her feet got Charlotte the pieces she wanted. The history of the gourd-shaped bottle shows how her scholarship was constantly evolving and how important this was to her reputation as a collector.\n\nHer knowledge paid dividends. In one 'very bad' shop in Genoa, 'a dilettante shoemakers', Charlotte was offered a set of cups decorated with red landscapes. They were fine and unexpected, and they immediately attracted her attention. When she turned one of them over to check the maker's mark, she found the name 'Jacques Boselly' painted crudely across the base, linking them to a small eighteenth-century ceramics factory in Savona, near Genoa in Italy. If they were genuinely Boselly cups, they were quite rare and desirable, a real find. But Charlotte continued turning them over. Three displayed the same scrawled signature. But fortunately Charlotte was not content until she had turned each one of the set and peered closely at every scratch and imprint, and that's when she saw something else: the name 'Wedgwood' impressed into the glaze: 'I confess the English name was rather faint so that the ingenious foreigner might be excused from expecting that it would escape ordinary attention,' she explained in her journal. At a time when Wedgwood was out of fashion and distinctly less collectable than Boselly, the dealer had tried to upgrade the English cups into something more sought after and valuable. But those with a good knowledge of the trade were not to be so easily fooled: 'the mark was quite strong enough to be quite clear to the initiated,' Charlotte noted.\n\nIn the competitive world of European collecting, there was no substitute for acquiring better knowledge than other collectors, or acquiring it more quickly. Charlotte's scholarship gave her the opportunity to make profitable purchases and to speculate for the future. At the dealers in Genoa, Charlotte was quite as happy with Wedgwood cups as with Boselly. She had already decided that Wedgwood was quality china, worth collecting, and this was another opportunity to add the English ware to her collection at a time when ceramics from the factory were of little interest to the majority of European collectors and consequently relatively cheap to acquire. And as Charlotte's knowledge grew, so did the opportunities for such bargains. In March 1872, she was scouring the shops in Amsterdam when she noticed a china figure of a youth holding a comb. It was a fine piece, and in perfect condition. Charlotte at once recognized that it was scarce Bristol china, a hard, white eighteenth-century porcelain manufactured from Cornish china-clay and much sought after for the quality of its glaze. She was so astonished at the unexpected discovery 'that I put it down again, hardly believing my eyes at so great a find'. But looking around, she found that the dealer was paying little attention to her, and, when she tentatively asked him about the piece, she realized that he 'had no idea what it was'. Taking advantage of his ignorance, she struck a deal for the figure, along with two sauceboats which caught her eye, for \u00a37. The shrewd speculation, Charlotte noted, 'amply repays the trouble of a seven hour drive'.\n\nA few years later, in early 1880, she instructed one of her French contacts, a dealer named Fournier (possibly the son of the Fournier who had disappeared during the violence of the Paris Commune) to buy on her behalf a small figurine. Charlotte had glimpsed the piece only once, briefly, as it lay unregarded in a display window, but she was immediately sure it was sixteenth-century Spanish. Fournier disagreed. Since he was being paid for his trouble, he continued to act on Charlotte's orders, but he maintained that the piece was nothing more than a modern reproduction. He wrote to her, warning her that she was likely to be disappointed, and finally he brought the figure to her rooms at the Hotel Drouot in Paris. The minute she saw it, Charlotte knew she was right. And her instincts were, once again, impeccable. After some straightforward research, she was able to show beyond doubt that the piece was special, and a bargain: 'It proved to be, as I anticipated, a very fine specimen of Spanish Cinquecento art, and we were wonderfully fortunate to obtain it for about \u00a38. It was worth quite 3 times as much, but the ignorant French dealers did not know it.'\n\nCharlotte's journals are full of small incidents like these, where her scholarly approach to collecting bests even experienced dealers. And it was not just with ceramics that she could show off her skill. She was fascinated by the variety and history of board games and before long had acquired all kinds of counters and draughts and dice. Still cherishing the gift from her mother many years before, she also collected fans which had been a popular, and exclusive, fashion item across Europe since the sixteenth century, and which had the added advantage of folding easily for transport in the big red bag. Individual fan leaves in silk, linen or animal skin, supported on wooden, ivory or mother-of-pearl sticks, were printed with historical, political or social illustrations: Charlotte's collection included a scene from the coronation banquet of George II in 1727, maps, landscapes and pastorals, biblical stories such as Moses in the wilderness and a spangled black gauze fan made to mourn the death of Louis XVI.\n\nIn September 1880, she was in Munich when something else caught her eye \u2013 her first set of playing cards. Hand-painted and early-printed playing cards featured all kinds of suits, from coins, cups and bells to acorns, sceptres and cudgels, as well as a variety of royal households that often extended beyond king, queen and knave to valets and maids. The Germans were the most imaginative and lively of early card-makers, and Charlotte was immediately captivated by the intricate designs. But the cards were expensive, even after some spirited negotiating, and Charlotte was strong enough to refuse them and return to her lodgings for dinner. Her resistance did not last, however. After desert had been served and eaten, she set off back through the quiet streets to the shop and bought three complete packs and two fragmented sets, 'for which I am ashamed to say we paid \u00a310'.\n\nIt seemed inexcusably extravagant. Charlotte was used to paying well for things that she knew were fine pieces and a sensible speculation, but with the cards she did not feel confident that her knowledge was strong enough to justify such expenditure. 'I am so inexperienced as yet in this branch of collecting, that I do not know if they are worth anything,' she admitted to her journal. Yet they were more than just a whim; she bought them because she liked them, but also as part of her ongoing collecting education, as an investment in her future as a collector as much as in the cards themselves. 'I must pay to learn,' she reminded herself.\n\nThis commitment to developing original areas of collecting, and backing it up with thorough scholarship, identified Charlotte with the increasing community of collectors who were moving away from the idea of simple connoisseurship (where collecting was largely a matter of personal taste) towards a more professional approach, based on wide learning. The work of men like J. C. Robinson and Augustus Franks in the public museums had set a standard of collecting \u2013 and an ambition for the collector \u2013 that began to be seen also among private collectors. Just ten years after its foundation, the more ambitious members of the Fine Arts Club, including Charles Schreiber, had become frustrated with its emphasis on socializing and dilettante connoisseurship. They felt its more serious aims were becoming lost, and set about founding a more scholarly offshoot which was established in 1866 as the Burlington Fine Arts Club and which operated alongside the original club as a refuge for the most committed collectors. By the 1880s, when Charlotte was beginning her research on playing cards, this impetus had grown. It was no longer enough simply to be enthusiastic. The influence of the museums and the increasing number of collectors in new fields meant that those who wished to distinguish themselves needed to demonstrate originality and a commitment to extending their knowledge. While Charlotte may have been excluded from professional collecting by her gender, she nonetheless developed a rigorous and independent approach that made her professional in everything except the fact that she was not usually paid for her work.\n\nBefore very long, Charlotte had begun to master her new field of study, and to collect playing cards feverishly. She admitted to her diary that, within days, the thought of finding new patterns and designs 'now occupies me very much' and she was tormented by the urgency of age and lost opportunity, afraid that 'I have begun too late to make my collection a very handsome one'. But her fears were unfounded. With characteristic thoroughness, she soon added a detailed knowledge of playing cards to her lengthening list of specialities, and her international collection of over 1,000 packs, from Poland and Russia, Denmark, India, Java and Japan, as well as Italy, England, France, Germany and Spain \u2013 with designs featuring everything from seventeenth-century Latin grammar to the military science of fortification \u2013 proved good enough in the end for the British Museum.\n\nCharlotte's purchases could be modest: \u00a310 for the playing cards; \u00a37 for the figurine and sauceboats; \u00a38 for a piece of Spanish fifteenth-century art. In these cases at least, her bargains might have cost a lot more, but she could not always be trading on her knowledge to find things that had been overlooked, and most of Charlotte's transactions were at more or less market value. And even though she chose to collect in many areas that were unfashionable or relatively unknown, her collecting still cost her a great deal in comparison with everyday items. In the 1880s, when Charlotte was buying her first playing cards, the annual farm rent on an acre of decent land was between 15 shillings and \u00a31, while, on the railways, excursion and season ticket fares around the home counties were set at under a penny a mile; a gross of quills, packed in a box, cost less than 2d; and a ton of large coal could be had for around 8 shillings. While collecting was gradually becoming more accessible and widespread, it was still not within everyone's reach. Certainly, those hoping to buy as often as Charlotte did needed both a healthy income and an unshakeable commitment to building a collection.\n\nBy 1883, the bouts of illness that Charles had long been suffering had become more serious and protracted. Charlotte was alarmed, and, when a three-month trip to the south coast failed to have much effect, the doctor prescribed a bracing sea voyage. The Schreibers set off on their longest, and last, journey together, taking the slow ocean route to Cape Town where the climate was said to be good for ailing health. This time, there was no opportunity, or desire, to collect. There was not a bowl or fan or plate that could distract Charlotte from the aching fear of losing her husband; the showrooms of Paris and the bustle of Italian auctions seemed a world away. For eleven tense months, Charlotte was too busy acting as nurse to think of much else, but, despite her care, her husband's alarming symptoms refused to be assuaged by South Africa's warm air. And when it became clear that Charles's health was in fact getting worse, she took the decision to return to London, to the support of family and friends.\n\nCharles never made it home. He became steadily weaker as the steamer travelled slowly up Africa's Atlantic coast. Confined to his cabin, he could not eat and hardly seemed to know that Charlotte was by his bed, holding his hand. He died finally at Lisbon on 29 March 1884. Charlotte could comfort herself only with the thought that it was 'within an appreciable distance of home'. She was seventy-two years old, and her eyesight was failing fast. She felt more bewildered and alone than she had ever done in her life. Her second husband's illness and death seemed to drain her characteristic energy and optimism and on her return to England she went to live with her youngest daughter Blanche, in Cavendish Square, welcoming the comfort of family. Looking back wistfully on 'the days that were over; dreams that were done', she did not have the will to continue the European crusade alone: 'So ends my _life_ on earth,' she wrote. 'It has been a very happy one, and I have much to be grateful for.' Even so, she did not neglect the collection. There was still much that could be done without the habitual travelling. After decades of actively pursuing objects across a continent, Charlotte set her mind now to the quieter tasks of sorting, cataloguing and arranging. She recognized that this was just as important as the work that had gone before and she welcomed the more contemplative aspects of life of a collector. Indeed, in her bereavement, her collection became more to her than ever: 'that is the only thing that gives any relief to my sadness,' she noted.\n\nIt was shortly after Charles's death, in May 1884, that Charlotte had her first serious discussion with staff at South Kensington about giving the best part of the collection of English porcelain to the museum as a memorial to her husband. Over the years, she had loaned many pieces for exhibition and display, and had spent many hours in the galleries studying. Charles himself, possibly with an eye on posterity, had raised with his wife the idea that they might donate some of their collection to a public museum. The thought, noted Charlotte, 'was not a new one to me', and the museum, in its turn, was delighted at the prospect of acquiring such a unique collection of china. In preparation, Charlotte cleared the two largest rooms in the house and arranged the pieces on seven long tables. Then staff from South Kensington arrived to take stock of what might be on offer and, with her husband no longer around to advise, Charlotte turned to Augustus Franks to help finalize the details of the bequest.\n\nIn the months that followed, Franks became an even more familiar face at Charlotte's home, calling on an almost daily basis to work with her on cataloguing the English ceramics and to help select the best pieces for South Kensington. When he could not spare the time, Charlotte went to him instead in the offices of the British Museum. It was clear evidence that, in her old age, Charlotte had become a respected collector, one of the most influential of her time. With Franks' help, a catalogue was written and printed, although not before Charlotte succumbed, even at such a late stage, to making the occasional extra purchase as a last-minute addition to the collection. She found it hard to break the habit of looking for treasure: 'Really I must stop these morning rambles into curiosity shops,' she chastised herself after a dark November day spent shopping. 'I shall be ruined \u2013 another \u00a35 \u2013 but then it is all going to the South Kensington.' There was satisfaction for Charlotte in securing her treasured pieces a good home, and there was a certain pleasure in seeing her lifetime's collecting honoured by such a robust institution, but all in all she found the process of transferring the collection unsettling and troublesome. She could not sleep. When the terms of the agreement were finally settled in October 1885, it was a relief: 'now that I have got the load of the settlement of the collection off my mind, all else seems trifling', she wrote, and promptly went to see a workman about dealing with her blocked drains. A month later, on 19 November, the collection was finally packed and sent to its new home at South Kensington. It was a poignant parting. 'I close this sad volume with my adieux to the collection,' she wrote, perhaps even then unaware of what a singular slice of history she had just bequeathed to the nation.\n\nOver the next ten years, Charlotte campaigned vigorously for causes ranging from Turkish refugees to London cabbies, funding a Cabmen's shelter at Langham Place, off Oxford Circus. But her collecting years were behind her. Her journal entries finally stumbled to a halt and her days were filled with the comfortable, unremarkable routines of any wealthy Victorian matriarch. She died, almost totally blind, on 15 January 1895, at the age of eighty-two. Her more ephemeral and quirky pieces \u2013 the fans and the playing cards \u2013 she left to the British Museum. During the last years of her life, she worked with staff there on the fans in particular, publishing books on both English and European designs: the Worshipful Company of London Fan Makers awarded her the Freedom of the Company, one of only two Victorian women to be honoured in this way by livery companies. One piece in particular was displayed as a highlight in the immense labyrinth of the British Museum's galleries: an eighteenth-century English fan, given to the museum by Charlotte in 1891 and celebrating the magnificent and chaotic firework display that marked the temporary end of the War of Austrian Succession in April 1749. With the design etched in paper, coloured by hand and mounted on ivory sticks, the fan showed rockets bursting over a huge makeshift palace erected in St James's Park. Meanwhile, across London, the English ceramics were installed at South Kensington. The rest of Charlotte's collection was handed down to her children, particularly her son Montague and daughter Blanche, who both inherited their mother's interest in collecting.\n\nIn some respects, Charlotte's collecting derived from the genteel habits of the past. Born into a wealthy, aristocratic family, she had the benefits of a decent education and she always mixed in elite circles. She was a rich woman with the time and money to indulge her hobbies. But her collecting could also have been restricted by such a background: she could have been a wealthy connoisseur with a fine aesthetic appreciation, or a fashionable society figure with a collection to catch the eye, or a woman at home, her collecting mediocre in scope and modest in ambition. Charlotte was much more than this. Using money from her marriage into the industrial, and industrious, middle classes, she helped show the way for new generations of collectors whose enthusiasm would be supported by erudition. She challenged the conventions about how female collectors could act and what they could achieve. She took collecting out of the drawing rooms of her aristocratic heritage and on to the streets of a changing Europe.\n\n## _Pride, Passion and Loss: Collecting for Love_\n\nJOSEPH MAYER\n\n### CHAPTER ELEVEN\n\n### _Waiting for the Rain to Stop_\n\n_J_ oseph Mayer was having his portrait painted. It was around 1840 and he was in his late thirties, a successful silversmith and Liverpool businessman, socially secure, reasonably wealthy, a man of fashion \u2013 and an emerging collector. This portrait, the first Mayer had ever commissioned, was a way of marking how far he had come, and where he was going. His business was thriving: he was a partner in his brother-in-law's jewellers in Lord Street, at the heart of the flourishing Victorian town. He lived in considerable ease and some splendour in a townhouse on Clarence Terrace, Everton Road (since demolished), establishing himself as a member of Liverpool society. But this was not just another standard Victorian portrait, reinforcing respectability and professional success. It was a statement about Mayer's passions and ambitions, a portrait of an aspiring connoisseur. Surrounded by vases and busts, sculpture, classical marbles, Greek and Etruscan antiquities, Mayer was showing off the part of him that mattered \u2013 his collection.\n\nWhen it came to choosing an artist for the commission, Mayer needed someone who would depict the story of his life in the objects around him, its bonds with the past and its potential for the future; someone who would pay as much attention to portraying the artefacts crowding the edges of the painting as to the man at its centre. He chose William Daniels. Daniels was ten years younger than Mayer, but was already becoming known in local circles. He was everything a painter should be: proud and eccentric, unreliable, passionate and poor. The town's middle classes delighted to hear how he had been discovered as a boy in a ditch in the brickfields, modelling figures from the clay; how he had been adopted out of the notorious Scotland Road district, Liverpool's poorest and roughest area, and introduced to the Liverpool Academy of Art by one of the masters, Alexander Mosses, and how he had learned so well and so quickly that he had won first prize for a drawing of a dying gladiator. Ashamed to clatter on to the wooden platform to receive his award in his clogs, Daniels had apparently borrowed a pair of boots from a gentleman's son. This poignant tale of talent in adversity further charmed Liverpool society when the penniless Daniels fell in love with a pretty girl called Mary Owen, who was also penniless and whom he painted as a rosy-cheeked gypsy pedlar in an attempt to raise money for a wedding. When the portrait was immediately bought by Sir Joshua Walmsley, MP and Mayor of Liverpool, Daniels had secured a future. He had a patron, and, better still, a romantic reputation.\n\nIt was not necessarily a straightforward commission. Daniels could be tetchy, puffed up by his sudden success and celebrity. He drank heavily, and, if he was in a sensitive mood, he would rage against the humiliation and sterility of being asked to paint to order; 'his inspiration was always fleeting,' explained one of his early Liverpool patrons, 'and when out of temper he would slash a partly completed canvas or daub it with aimless strokes'. But for Mayer, Daniels seemed willing to cooperate. Perhaps Mayer adopted a ruse that had helped some of Daniels' other customers manage the temperamental artist: one explained how he carefully displayed a 'few guineas on a convenient mantelshelf in the hope of attracting him, thus doling out the price as the picture went on'.\n\nHowever it was, the portrait was completed. Daniels painted the dim light idling through heavy curtains, throwing the room into shadow so that it seems something more than a study or even a museum, something like a great ancient tomb, newly opened and packed with objects. He lets the papers twist open under the table, a glove fall to the floor, and Mayer's spaniel peer, half-curious, from under his chair. The atmosphere is homely and relaxed, yet simultaneously strange and distant, as though Daniels is attempting to capture the sense of the past accumulating in the heavy quiet. And at the very centre is Mayer, contemplative and still; smaller than his sculptures and busts and the huge carved chair in which he sits, another piece in the collection, not separate from it but alive within it, a natural part of the things with which he is surrounded.\n\nIt was a portrait of its time, evoking the idea of the chaotic, romantic antiquarian study that had been made popular by writers like Walter Scott and which was the height of fashion when Daniels was completing the commission for Mayer in the early 1840s. In Scott's 1816 novel _The Antiquary_ , he describes the 'sort of den' in which the collector might be found:\n\nIt was a lofty room of middling size, obscurely lighted by high narrow latticed windows. One end was entirely occupied by book-shelves, greatly too limited in space for the number of volumes placed upon them. . . while numberless others littered the floor and the tables amid a chaos of maps, engravings, scraps of parchment, bundles of papers, pieces of old armour, swords, dirks, helmets, and Highland targets. . . The top of this cabinet was covered with busts, and Roman lamps and paterae, intermingled with one or two bronze figures. . . A large old-fashioned oaken table was covered with a profusion of papers, parchments, books and nondescript trinkets and gewgaws, which seemed to have little to recommend them, besides rust and the antiquity which it indicates. . . The floor, as well as the table and chairs, was overflowed by the same _mare magnum_ of miscellaneous trumpery, where it would have been as impossible to find any individual article wanted, as to put it to any use when discovered.\n\nIn Daniels' new portrait, Mayer's study fitted the model perfectly. It would have been immediately clear to anyone seeing the painting, just what kind of man Mayer was (or at least how he wished to present himself); the viewer would have understood that by surrounding himself with objects in this way he was displaying himself as both fashionable and apparently learned. Daniels painted Mayer to look exactly how most Victorians thought a collector should look.\n\nIn one hand Mayer holds a book. This is both serious and fitting, an indication of his connoisseurship. But when it came to selecting something particular for him to hold in his other hand, there was perhaps more of a problem. The random selection of objects in the painting aptly illustrates the haphazard and eclectic nature of Mayer's collecting, which was enthusiastic rather than particularly knowledgeable. Although he no doubt wished to appear studious and erudite in his portrait, he was in fact much more likely to be guided by his instincts than by learning. He was not, as he admitted himself, particularly scholarly. He was the product of an average education at a local grammar school and his classical, historical and scientific understanding was limited and uneven. He could show Daniels an intriguing jumble of things to be painted but not the ordered, comprehensive collection of an intellectual. Some of his pieces were significant and historic; others were merely curiosities, and a few were little more than household junk. Mayer liked all sorts: rare antique objects from the ancient civilizations of Rome, Greece and Egypt, and broken archaeological finds from Anglo-Saxon burial sites; arms and armour, swords, guns and all things military; manuscripts from Burma, scraps from German prayerbooks, illuminated medieval missals, books of hours and fine bindings; Napoleonic memorabilia; gems, ivories and enamels; engravings, cartoons, the occasional oil painting and examples of local Liverpool pottery. There were few things he could resist if they had the scent of history about them.\n\nMayer's table was littered with Roman and Etruscan antiquities, with candlesticks and antique figurines. But what he chose to hold in the portrait, in the end, was a miniature vase, shaped like an ancient Greek urn and set on a pedestal. It was small enough to fit comfortably in one hand, light enough for him to hold up through the long sittings, and pretty enough to hold Daniels' attention. But, in fact, it was a very ordinary piece, an example of English pottery from the Wedgwood works and unfashionable. It was not a status symbol; it did not testify to wealth or learning; it had not been unearthed during a risky foreign voyage. It was just something Mayer liked and apparently it did not disturb him that it was relatively new, or that in the 1840s no one else seemed to share his taste in china. Wedgwood pieces pleased him; he admired their colours and forms and their strong references to the ceramics of the past. It was, he thought, a fitting object to include in his portrait.\n\nWithin a few years, the room which Daniels had so carefully depicted had disappeared. By 1844, Mayer had given up the smart house in Clarence Terrace, and the gentleman's study, and had moved instead into modest and simple accommodation above his business on one of Liverpool's busy shopping streets. He had been apprenticed at the age of nineteen to his brother-in-law James Wordley and had been a successful partner in the business since the early 1830s, but now he was looking for change and greater independence. In 1843, he broke the partnership with Wordley, acquired premises a few doors down, and set up alone. Without ties, he could do as he pleased. He could try his hand at designing and making his own silver, and he could begin to establish his own name in the trade.\n\nMayer's new workshop flourished. He became highly skilled in designing large, often ceremonial pieces of gold and silver plate, spectacular trophies and civic regalia, and he studied innovative techniques, exploring the commercial potential of affordable new processes like electroplating. The business became respected and thriving; the shop imposing and magnificent. Mayer presented himself within the tradition of Renaissance silversmithing, and he designed a new building to invoke a sense of monumental art and of continuum with an impressive past. Either side of the entrance door were two massive frescos, one of the Renaissance smith and sculptor Benvenuto Cellini, and one of the seventeenth-century goldsmith and royal jeweller George Heriot. Two huge busts, one male and one female, gazed down from the balustraded fa\u00e7ade on to the shoppers below. Immense plate-glass windows were piled high with objects to the height of three or four men, and Mayer's name was emblazoned, twice, above the display. Inside, watches and clocks, cameos, precious stones, heraldic engravings and a glittering array of silver and gold were all carefully arranged in specially designed showcases.\n\nWith such a prosperous showroom, it was not enforced economies that persuaded Mayer to give up the ease of the Clarence Terrace house. It was the lure of collecting. All his energies and resources became focused on finding and buying objects: it became his obsession, his life's purpose. Unlike J. C. Robinson, he could not bear to have the distractions of fashionable living divert his attention, and his money. He was fascinated by the past, and the objects it had left behind, and he did not mind giving up the comforts of the present if it meant bringing him closer to history.\n\nWhile Mayer contented himself with simple living in the rooms above his shop, he was anything but reclusive. He was naturally a sociable man, but, more than this, he understood that to be a successful collector he needed to explore new places and meet new people. He travelled whenever he could, sending frequent parcels and crates back to England. He made his first trip, at the age of twenty-five, in 1828, and he spent the next few decades exploring the archaeological sites of Italy and the ruins of the ancient world, steaming down the Rh\u00f4ne to the towns of Southern France, and uncovering objects in Bavaria, Austria, Bohemia and Prussia. Mayer was only nine years older than Charlotte Schreiber, but was setting out on his journeys when she was busy at Dowlais, long before she had begun to collect; twenty years the senior of J. C. Robinson, he was beginning to explore the treasures of the Continent at a time when popular Victorian collectors' routes were still being etched into the European map. In the 1820s and 1830s, collecting off the beaten track of the established Grand Tour had a sense of trailblazing; it seemed exciting, daring and new. Mayer travelled for business, to buy stock, make deals and keep abreast of fashionable trends. But his journeys soon became about much more than the demands of trade. His moment of conversion to collecting, he always maintained, came when he was convalescing from fever during a business trip to Dijon, and had time to wander around the town's museum and picture gallery. The antiquities he saw there captivated him, and suddenly turned his journey from business to pleasure; from commerce to collecting.\n\nAt home, too, Mayer's urge to collect encouraged him to make contact with scholars, connoisseurs and other private collectors in an effort to improve his knowledge and expand his collection. He was an enthusiastic member of the local Arts Association and a founder member of the Historical Society of Lancashire and Cheshire. He made regular visits to the library, meeting rooms and gallery of Liverpool's Royal Institution. Whereas the local clubs, like the Athenaeum and the Lyceum, were gentleman's clubs with restricted membership and predominantly daytime hours, the Royal Institution was open to everyone and was available for use during the evenings for those who, like Mayer, had to work during the day. Beyond the region, he also made an effort to know as many influential collectors as possible and as time went on his collection became a favourite stopping-off point for those travelling north from London: Charlotte Schreiber was among the visitors from the capital who made sure to see what Mayer was collecting. Making the journey in the opposite direction, Mayer frequently combined business in London with visits to museums, in particular the British Museum, and he became a member of a number of respected London organizations, including the British Archaeological Association, the Royal Society and the Royal Geographic Society. By 1850, he had also become a Fellow of the Society of Antiquaries, which put him in touch with some of the most knowledgeable and enthusiastic scholars of his day, including Augustus Franks at the British Museum.\n\nMayer's particular interest in antiquarian objects drew him into different circles of collectors from those of which Robinson and Charlotte Schreiber were a part. The term 'antiquarian' loosely \u2013 and usefully \u2013 covered a wide range of artefacts from the past, including archaeological finds, books, remains of the 'antique' classical civilizations and even scientific instruments, with a particular emphasis on objects as historical evidence of past lives. It had long been a popular area with collectors, not least on account of its variety, and a tradition of antiquarian writing went back as far as the Middle Ages. As early as 1572, a national society had been established to encourage the study and preservation of antiquarian remains and in 1717 this was officially constituted as the Society of Antiquaries, receiving a charter from George II in 1751.\n\nIn the nineteenth century, the enthusiasm flourished. The Victorians' adulation of progress and empire was accompanied by a reverence for history and an urge to preserve ideals, traditions and objects from the past. A taste for historical painting pointed to a nostalgic longing for an idealized lost age, while 'the antiquarian style' became a fashionable architectural genre during the mid-century. The antiquarian look brought together artefacts from a variety of historical styles and periods to create interiors full of everything from Greek sculpture to medieval tapestry and Elizabethan furniture. Robinson's Newton Manor owed much to this kind of eclectic decoration, but it was Sir Walter Scott's carefully designed home at Abbotsford in Melrose, created between 1812 and 1832, that was at the forefront of the trend. The fashion became associated with mystery and romance, and Scott's novel _The Antiquary_ \u2013 featuring an historian, archaeologist and collector, family secrets, hidden treasure, a mysterious aristocrat, a ruined abbey and hopeless love \u2013 only added to a popular fascination with re-creating an imagined past. Even reasonably ordinary objects became increasingly entangled with the idea of memory and nostalgia: 'inanimate and senseless things' took on a new importance as the 'object of recollection', suggested Dickens in his 1841 novel _The Old Curiosity Shop_. 'Every household god becomes a monument.'\n\nThe popularity of all things antiquarian influenced what was being collected both by individuals and institutions. Since its foundation in 1753, the British Museum had largely turned its attention abroad, and had ignored British antiquities in favour of pieces from the Mediterranean and the Near East, such as the famous Elgin marbles or the Assyrian sculptures from Henry Layard's excavations. The increasing expertise and enthusiasm of British collectors by the middle of the nineteenth century, however, brought growing calls to fill the gaps. In 1845, the British Archaeological Association, of which Mayer was a member, met at Winchester to express the hope that a museum of national antiquities would be formed, and three years later, when a hoard of Iron Age bronzes was found at Stanwick Park in Yorkshire on the land of Lord Prudhoe (later Duke of Northumberland), it was offered to the British Museum on the condition that a special room was set aside for these and other British antiquities. Although this was agreed, by 1850, nothing had happened. Politics, finance, the constraints of space and simple procrastination seemed to be in the way. It was not until the appointment of Franks, in March 1851, that any progress was made. His commitment to collecting British antiquities, and his links with collectors and collecting organizations, brought the issue to the forefront of the debate about what kind of role the British Museum should have. With the support of colleagues like Mayer at the British Archaeological Association and the Society of Antiquaries, Franks managed to persuade the museum's trustees and management that this popular area of interest deserved space and attention, and in 1866 he finally won approval for a new and distinct department devoted to British and Medieval Antiquities, with himself at its head.\n\nMeanwhile, a new generation of private collectors began to move the study of the antiquarian on to a more professional footing. Dedicated and discriminating, they started to develop practices for fieldwork, and to create frameworks for shaping and preserving their collections. Some were members of the nobility. Lord Londesborough (1805\u201360) was an enthusiastic antiquary who published his collection of 'ancient, medieval and Renaissance remains' in an illustrated volume in 1857. The 3rd and 4th Dukes of Northumberland began to collect pieces found on their estates in the 1820s and went on to undertake their own excavations at home and abroad. Their collection included a number of objects from Pompeii; its museum in Alnwick Castle still survives today. Most of the enthusiasts, however, were of more modest means. Compared to art collecting, antiquarianism was a relatively cheap interest to indulge. The breadth of antiquarian interest meant there were always neglected objects to be had at a reasonable price: the medieval manuscripts, Anglo-Saxon finds and medieval ivories which Mayer collected were largely ignored at the time, unwanted by the national collections and passed over by wealthier collectors.\n\nAntiquarianism brought collectors together without great emphasis on class, status or wealth, and, although most of the collectors were men, it was otherwise a relatively egalitarian activity. In Yorkshire, John Mortimer, a corn merchant, began a personal programme of research based around the Wolds. Inspired by a visit to the 1851 Great Exhibition, Mortimer collected stone implements, fossils and geological specimens before moving on to excavate prehistoric barrows and Anglo-Saxon cemeteries, eventually displaying his rapidly expanding collection in its own museum in Driffield and publishing his life's work in an illustrated volume. John Evans, a partner in a paper manufacturing firm, undertook scholarly work in periods from pre-history to the post-medieval, as well as in geology and numismatics. His publications, especially _Ancient Stone Implements_ (1872) and _Ancient Bronze Implements, Weapons and Ornaments of Great Britain_ (1881), marked an advance in antiquarian study. They shaped the way in which many antiquarian collectors worked, and are still important today. After Evans's death in 1908, his collection was presented by his son to the Ashmolean Museum in Oxford. A close friend of Franks, and President in turn of the Geological Society, the Numismatic Society, the Anthropological Institute, the Society of Chemical Industry and the Society of Antiquaries, as well as a Trustee of the British Museum and a Fellow and Treasurer of the Royal Society, Evans was held in high regard across Europe and was knighted for his work in 1892. His visionary and committed collecting set the standard for many of those who shared his interests, raising the quality and profile of antiquarian study.\n\nThe status of antiquarianism, the role of antiquarian collectors and the importance of antiquarian objects were fashionable issues, and in the middle decades of the century Mayer was at the heart of these debates. As a member of the key societies, he was able to contribute both locally and nationally, and to meet his peers across the country. While this was important for all collectors, it was especially crucial for Mayer. Unlike Robinson, for example, Mayer had never acquired any particular assurance or expertise. He lacked confidence in his knowledge and scholarship, and much preferred learned company to having to display learning himself. Left to his own devices, he could be muddled and uncertain \u2013 an easy target for swindlers \u2013 and he looked to his more cultured colleagues for something more consequential than the usual social round. He needed them to help him make informed decisions about his collecting.\n\nAmong his many new acquaintances, Mayer came to rely in particular on two useful and loyal friends: Joseph Clarke, a natural historian, enthusiastic amateur archaeologist and Keeper of Saffron Walden Museum, and Charles Roach Smith, a pharmacist and Fellow of the Society of Antiquaries, as well as a leading light in the British Archaeological Association. Both men were collectors in their own right, and Roach Smith in particular was becoming an expert on objects that were exposed as London expanded and trenches were dug on new building sites throughout the city. Roach Smith's collection was accumulated over twenty years by bartering directly with labourers. He had no opportunity to study the building sites or to undertake any kind of contextual research, but he did what he could, rescuing objects from across London and relying on old texts and maps to piece together their history. By dint of his energetic legwork and extensive investigation, his collection eventually amounted to a coherent history of the capital, a 'Museum of London Antiquities', which included 'a very curious collection of swords and spear heads from the Thames' and 'an enamel buckle or brooch similar in workmanship to the Alfred Jewel. . . an object of great rarity'. This is now recognized as an Anglo-Saxon treasure known as the Dowgate Hill brooch, a late-tenth- or early-eleventh-century gold disc brooch decorated with colourful cloisonn\u00e9 enamel and a filigree border set with four large pearls, found in 1831 at the foot of Dowgate Hill on Thames Street. By 1857, Roach Smith's collection was so extensive and so widely admired that it was given a home in the British Museum.\n\nRoach Smith and Clarke appreciated the impulses driving Mayer and were conscientious on his behalf. They were his eyes and ears in London and for many years they were attentive to Mayer's needs, constantly on the lookout for objects he might like. 'If you set me to work I go to it in earnest,' declared Clarke enthusiastically, and on one memorable occasion he discovered an entire shop full of pieces for Mayer: 'The owner told me he had not been in the rooms for seven years and only once for fourteen and if you had seen me when I came out you would have laughed, no chimney sweep would have been blacker, three washings, I am not clean yet.' During the 1850s and 1860s, Clarke and Roach Smith acquired a variety of ancient artefacts to send back to Liverpool: 'I have received for you a fine British urn. . . found. . . at Felixstowe. . . I got it. . . for a little above \u00a32,' enthused Roach Smith in a letter of December 1852. In April 1856, he wrote again, 'I yesterday secured a Roman vessel, from Blackfriars Bridge & two old English vessels from the Fleet Ditch for you.'\n\nMore significantly, it was Roach Smith who was instrumental in helping Mayer acquire a number of complete collections that firmly established his reputation as one of the most influential collectors in the country. The first of these, the Faussett collection, had been assembled by Bryan Faussett of Heppington in Kent, an eighteenth-century churchman and amateur archaeologist. Faussett collected thousands of Roman and English coins, and had so many duplicates that he was able to melt down 150lbs of bronze to be cast into a bell. His real interest, however, was graves, and he spent all his time between writing sermons and visiting parishioners excavating Anglo-Saxon barrows in the woods and downs across Kent. He opened up over 630 graves in an eight-year period during the 1760s and 1770s, amassing a collection of grave goods, from plain hammered-metal bowls to intricate brooches. Unlike many amateur archaeologists of the time, he kept painstaking journals of all his excavations, sketching the sites as he found them, and recording every detail of his discoveries in five substantial volumes which were kept with the objects at his house.\n\nBy the middle of the nineteenth century, however, the collection had been forgotten. Faussett had died and the objects were simply gathering dust in the family home. Strolling in Kent in 1842, Roach Smith found himself close to Heppington and found the name half-familiar. Eventually, he managed to recall the details of an eighteenth-century work, _Nenia Britannica_ : _or, a sepulchral history of Great Britain; from the earliest period to its general conversion to Christianity_. This was, as its title suggests, a wide-ranging survey published in 1793 by another Kent clergyman, James Douglas, who was also an officer in the Corps of Engineers. Douglas used his military expertise to survey barrows and other archaeological sites, and was one of the first antiquaries to meticulously record, draw and publish his findings to a high standard, laying the foundations for the work of the Victorians who followed. In _Nenia Britannica_ , he recounted something of Faussett's collection, being the first to realize that the objects were Anglo-Saxon. Roach Smith recalled the description of Faussett's work, his archaeological explorations, and the diversity of his collection and he decided to pay a visit, unannounced, on whichever surviving relatives he could find.\n\nRoach Smith was taking a chance. He had no idea whether the collection still existed, what state it might be in or what Faussett's descendants might make of his unsolicited inquiries. But he was fortunate enough to receive a warm welcome from Henry Godfrey Faussett, the collector's grandson, who was delighted that someone was, at last, taking an interest in the family treasure. Henry spoke enthusiastically about his grandfather's achievements and spent hours showing Roach Smith boxes and drawers and shelves full of the antiquities that he had collected from local sites. The two men pored over slips of glass and pottery, over amber and amethyst beads, brooches, axes and spearheads, keys and daggers. Henry Faussett was eager that the collection should finally receive official recognition and urged Roach Smith to spread the word about what he had found. Roach Smith needed little encouragement. He was astounded by the quality and variety of the objects packed into the churchman's study and, when he finally left the Faussett house, as dusk closed in over the pale Kent hills, he promised to return with a delegation from the British Archaeological Association.\n\nHenry Godfrey Faussett was never to see the collection receive the attention it deserved, however. The British Archaeological Association did indeed inspect the finds and confirmed Roach Smith's evaluation of the objects as important and valuable, but no one in the Association could find the time or the money to publish the details of the collection and nothing more was done. Other collections became newsworthy instead, and when Henry Godfrey Faussett died, eleven years after Roach Smith had visited, interest in the Faussett finds had largely waned. It looked as though the collection might once more lapse into obscurity. Although the terms of Henry Faussett's will urged that it should be sold complete and intact, no public organization could be persuaded to find the \u00a3700 Faussett's heirs were requesting, despite the actual value of the collection being much greater. In 1853, Franks made four attempts to persuade the British Museum to make the purchase, but without success. The Department of Antiquities had an annual purchase grant of \u00a33,000 but this was stretched by a previous liability of \u00a3550, and, without extra funds from the government, the museum was unwilling to commit to the Faussett sale. Even though the Society of Antiquaries sent a series of protests, and complained that 'there is a strong probability that it will be purchased in France; and that it will be a disgrace to England that English objects thought worthy of being purchased in France, should leave this country', the museum once again declined. A trustees' meeting just a month later confirmed a request to the Treasury for a special grant of \u00a386,000 to build and fit out the Round Reading Room in the Inner Courtyard. They perhaps thought better of asking for yet more money to acquire the Faussett collection, and were no doubt unwilling to jeopardize the ongoing programme of building work. The opportunity was lost.\n\nJoseph Mayer, however, was much taken by the thought of the collection. As early as November 1853, when staff at the British Museum were still, in theory, negotiating for the pieces, he wrote a friendly and enthusiastic note to Franks, forewarning that he was in the market: 'Hurrah! For the Faussett Collection. I hope you will get them \u2013 not hearing from you I wrote to the Revd Faussett to ask about them if you refused them.' If he managed to make the acquisition, he knew he could claim to have saved the treasure for the nation: 'they ought to belong to the national collections and may yet when I have done with them,' he said. In the end, he moved quickly, and with a sensitivity to public relations' opportunities. Three days after the collection was finally and formally rejected by the British Museum, Mayer travelled to Kent to pay a first instalment and, with Roach Smith's help, he had the objects parcelled up and put on a train to Liverpool. He then approached Thomas Wright, an eminent historian, to give a lecture at the town's Philharmonic Hall to the British Archaeological Association about Anglo-Saxon antiquities in general, and the Faussett collection in particular. And he sponsored Roach Smith to begin work on publishing the Faussett archive \u2013 the journal notes, observations and sketches \u2013 in an illustrated volume that placed the collection in its historical context.\n\nMayer had always been competitive. As a youth, he had revelled in all kinds of sports, admitting a particular fondness for bear-baiting. The purchase of the Faussett collection was an opportunity to take on his rivals, and declare victory. Acquiring such a complete set of objects was a substantial triumph in itself, but it was all the more pleasing to Mayer because he had more or less trumped the British Museum, highlighting its deficiencies and putting himself forward in its place as the saviour of a national treasure. By afterwards supporting learned research about the collection, Mayer consolidated the impression of his success, and aligned himself with modern, forward-thinking antiquarian scholars such as John Evans. He emphasized the distance between himself and dilettante, amateur collectors whose studies and drawing rooms were cluttered with random historical objects. He was no longer confined to Walter Scott's model or Daniels' fashionable portrait, but was beginning to cut his own distinct path, creating a unique identity as a collector.\n\nMayer's purchase of the Faussett collection was a strong declaration of his intent to invest everything he had in collecting. The acquisition not only demanded the considerable financial outlay of \u00a3700, but more significantly it demanded that Mayer reveal his ambitions to his fellow, and rival, collectors. He seemed inspired and excited, more than daunted, by the prospect of such high-profile collecting, and for a while he seemed to relish the attention that came with the Faussett purchase. Many of his friends wrote to congratulate him on the success, and Roach Smith was particularly delighted, seeing more clearly than others how the acquisition might secure Mayer's place in history: 'the work will inevitably bring you GREAT returns in honourable fame. . . I am SURE of this:-many will envy you; and many will regret they are not in your position. There are chances, my dear Sir, which occur only once in an age, & the Faussett collection was a CHANCE OF CHANCES.'\n\nBut Roach Smith was wrong. This was not a 'once in an age' opportunity, at least not for Mayer. Only a year after acquiring the Faussett finds, Mayer again swelled his collection by acquiring a large number of objects at once. In 1855, he bought the classical and medieval ivories, some Mexican pottery and a large amount of prehistoric metalwork from the Hungarian Fej\u00e9rv\u00e1ry collection, which belonged to Franz Pulszky, an exile living in London. Again the British Museum declined to buy the pieces, reluctant to spend its limited budget on little-known medieval Byzantine ivories at the expense of such museum mainstays as classical sculpture. Its refusal to buy drew renewed and widespread criticism from antiquarians, archaeologists and the press: the _Art Journal_ accused the trustees of 'incapacity of judgement' in failing to secure the ivories. When Mayer stepped in to purchase everything for \u00a31,500, he was able to present himself, once again, as some kind of national champion, at least to the scholarly community.\n\nWithin a few years, several other important collections came Mayer's way. In 1857, he acquired medieval antiquities and a range of objects from Anglo-Saxon cemeteries belonging to W. H. Rolfe, a friend of Roach Smith's and an antiquarian from Sandwich. And, in 1868, he bought up the collection of his exhausted friend Joseph Clarke who had written plaintively to Mayer, explaining that 'I am utterly tired of collecting anything. . . if you but come and see me. . . and take the pick of my collections nothing would delight me more.' Through such acquisitions, the range, variety and sheer size of Mayer's collection was growing at speed, filling his rooms and establishing him as a figure among the community of collectors.\n\nBy purchasing existing collections wholesale, Mayer was able to build on other collectors' work and expertise, and save on a great deal of bartering and research. He certainly had an eye to the future: his negotiations for both the Faussett and the Rolfe collections made much of his intentions to save them from obscurity and keep them intact, emphasizing the public-spirited nature of his motives. But there was also perhaps something obsessional about Mayer's bulk acquisitions. The cut-and-thrust of the hunt itself clearly excited Mayer as much as the final ownership of the objects; the overwhelming desire to possess was a driving force. Such intense collecting was later to be described \u2013 especially by Freud and other twentieth-century psycho analysts \u2013 as 'fetishistic'. But it can also be seen as typically Victorian. At a time of imperial ambitions and unprecedented industrial expansion, Mayer's wholesale collecting could be viewed as part of a compulsive urge to do things bigger and better, to bring as many elements as possible under one protective umbrella, to control, to regulate and to extend. Those collecting on this scale shared many of the same motives and priorities as those championing rapid industrialization, competitive free trade and mass-production; Mayer's collection was as much an expression of the age as cotton mills or railways.\n\nAs with the Industrial Revolution, such progress was not without its pitfalls, however. Mayer was not an especially wealthy man; he had not inherited any fortune. None of his ledgers or account books survives, and there has been some speculation that he perhaps traded in diamonds, or was fortunate in some kind of financial speculation. But as far as we know he had to rely solely on the income from his jewellery business to fund his collecting. The scale and ambition of what he achieved convinced many that he must have had some other mysterious source of funds, but such wealth was an illusion created by the magnificence of the collection. Mayer lived simply, traded successfully \u2013 and invested everything he had in his objects. When he could not quite afford what he wanted, he turned once again to his friends for help. But large-scale collections necessarily require large-scale expenditure, and, in his overwhelming desire to acquire, Mayer sometimes underestimated the costs of his activity. His collecting had an element of brinkmanship about it and once or twice, perhaps inevitably, he came unstuck.\n\nWhen the Bram Hertz collection of classical sculpture, cameos and over 1,700 gems came on to the market in 1856, priced at \u00a312,000, Mayer put together a consortium of Liverpool businessmen to help him acquire it. He pressed and cajoled them into partnership, tantalizing them, so he thought, with the promise of rare and handsome objects which they could display in their home town. He presented the purchase as a civic duty, part of the growing movement to establish municipal collections. Here was the chance for private individuals to make an act of public generosity. But most Liverpool businessmen were not like Mayer. They were not particularly taken by the idea of collecting. Although the \u00a312,000 was promised, and formal ownership of the collection was transferred into Mayer's hands, the consortium quickly crumbled, leaving him with a treasure trove he could not pay for.\n\nDespite his profession as a jeweller, Mayer was no connoisseur of gems, and the Bram Hertz collection was probably overpriced: 'the British Museum people would have niggled about as many hundreds as he has given thousands for it!' confided a sceptical Joseph Clarke to Roach Smith. Nonetheless, Mayer was devastated when the deal fell through: 'I cannot bear the idea of breaking up and scattering a collection,' he complained. But he could not find the necessary funds to make the purchase alone and he had to part with things. Within a few months, some of the Bram Hertz collection was back on the market, followed a year or two later by a two-week auction at Sotheby's. For Mayer, the sale was heartbreaking. The collection was dispersed, just as he had feared and, to make matters worse, the \u00a310,011. 2s. 6d. achieved by the sale left Mayer with substantial losses, after commission, of almost \u00a33,000, which, as Clarke remarked, 'I am afraid prey on his spirits'. Only the diligence and goodwill of Mayer's loyal friends, and of Roach Smith and Clarke in particular, managed to salvage anything at all from the disaster. When the collection came under the auctioneer's hammer, they bid for a number of gems which, before long, found their way back into Mayer's hands.\n\nAs the Bram Hertz affair made clear, relying on the support of other people to build a collection could be perilous. It was helpful to be able to draw on an extensive network of contacts and friends, but collecting could never just be about the large-scale public purchases, the notices in the press and the ambitious acquisitions. It was a personal and sometimes a painful quest; it demanded sacrifice, dedication and sensitive attention. Mayer was always destined to be a great collector. For all his high-profile purchases, and the gratification of buying intact existing collections, he also took his time over much smaller acquisitions, spending blustery evenings on site at archaeological excavations, waiting to barter for the smallest of Romano-British finds. He walked out on wet Sundays along the banks of the River Dee and across the Wirral, talking to villagers about whatever they might have turned up; he rescued dozens of Roman artefacts when the London, Midland and Scottish railway cut a new line through to Lancaster. He was constantly and intensely on the lookout for objects. Better still, luck was clearly on his side, giving him the opportunity to make rare and inventive acquisitions, and taking his collection into the realms of the spectacular.\n\nThe first moment of luck came when Mayer was on a visit to London in 1845. He was invited to a tea party given by the wife of an old friend, and, as the slices of cake were passed around, he was introduced to a Mr and Mrs Wedderburne, an unexceptional elderly couple. There was no immediate reason for Mayer to find them of any interest, but the grapevine of collectors' gossip suggested that Wedderburne had once worked for Wedgwood, travelling for forty years to promote the company and sell their wares, overseeing their outlets and advising on the ceramic trade. When the opportunity arose, Mayer drew Wedderburne away from the company. Wedderburne was jovial and obliging. As we have seen, very few people at this time thought Wedgwood worthy of much attention and Wedderburne was unused to meeting anyone who cared for his stories. He was happy to entertain Mayer with anecdotes, but the more he went on, the more surprised he became at Mayer's attention. He ventured into increasing detail, emphasizing his technical knowledge of factories and sales figures, as far as he could remember them. He talked about the history of the works, expressing opinions as to which periods had seen the finest ceramics being produced at the Etruria factory in Stoke-on-Trent, and he described his day-to-day experience of selling, the accounting procedures, the tricks of getting Wedgwood into the big stores and to the front of the shelves. The more Wedderburne told, the more attentive and enthusiastic Mayer became. The more of the minutiae he remembered, the more delighted Mayer seemed to be.\n\nWedderburne could not help but be pleased and flattered by such attention, and after a long conversation he was lured into making an admission. He had been chief clerk and adviser to Wedgwood's London warehouse when it was shut down. Losing impetus after the death of its founder, Josiah, trade with Europe had slowed towards the end of the eighteenth century and the breaking up of the London premises had become inevitable. Wedderburne, who had spent so many years developing the business, had watched in dismay as pieces from the company's most successful and productive years were cast aside and sometimes smashed; great packages going to auction to be sold at bargain prices. And so, in a moment of sentimental folly, he had stepped in himself to buy up a vast quantity of the best and oldest examples, rescuing some of Wedgwood's most special objects, so that they could be safely stored and protected for the future. But the cost of his intervention had almost ruined him. He had not been a wealthy man, he explained, and his expenditure had made him considerably and uncomfortably poorer. His savings had dwindled, he was stretched by household expenses, and all he had to show was an attic packed full of old and unfashionable Wedgwood china.\n\nMayer became convinced that he had stumbled on to an undiscovered treasure trove. He asked Wedderburne to bring him a few specimens, to prove his story. He was not disappointed. 'When I saw them my heart beat as I asked him if he had any more, and how much he would want for them,' Mayer admitted. 'He suggested that I should visit him next day and see the remainder.' And so, a day later, Mayer made his way through the quiet residential streets to see the rest of the china for himself. Soon he was surrounded by a vast and distinctive collection of Wedgwood \u2013 by plaques and vases, by jasper ware and basalt and pottery, representing every period and style of the company's history, as well as trial pieces from the factory, held back from normal sale, unique pieces, imperfect and blistered and misfired. Mayer bought everything. The terms of the sale were agreed there and then, the two men sitting in the dust of the garret, the sun filtering pale through the sky-light. The pieces were packed up and put on a train for Liverpool. Mayer's collection of Wedgwood had begun in earnest.\n\nMayer's meeting with Wedderburne had been a rewarding stroke of luck, the kind of once-in-a-lifetime opportunity of which myth is made. But better was yet to come. Within a short time, another fortunate encounter was to bolster Mayer's holdings yet further. He had, through Wedderburne, bought up the wealth of Wedgwood's past; now he was to discover the key to its history. It was a wet afternoon in Birmingham in 1848, three years after Mayer had done the deal with Wedderburne. He was in the Midlands on business, and found himself in Birmingham's back streets as a sudden thunderstorm broke over the low roofs of the workshops in the jewellery quarter. As the rain pummelled the pavements and the sky darkened, there was nothing for it but to take shelter and Mayer scrambled into the nearest shop. In his haste, he did not have time to notice anything about the place in which he was taking cover; he just pushed open the door and flung himself out of the rain.\n\nThe shop turned out to be a scrap dealer's, piled high with waste iron and copper and brass, broken furniture, pipes and cogs and wheels. It was dusty and slightly oily, and the floors were littered with slips of wood and paper, screws and bolts, oddments of cloth, bits of string and a good helping of grit. Everything was grimy. There should have been nothing at all to interest him. But he had to wait for the rain to stop. Looking out through a small window, he could see it still, battering hard on the street outside and throwing gobbets of mud against the close-packed brick buildings. He paced up and down, idling and looking around in the dim light. And in the midst of everything, what caught his eye were a number of heavy old ledgers, piled up at the end of the counter, leather-bound, scruffy, and instantly alluring to Mayer. They were not, he could be sure, scrap. He realized that they had to be a set, more or less complete, and, with the thunder still creaking overhead, he had nothing better to do than to sit down and take a look at them.\n\nThe scrap merchant thought Mayer was quite mad. They were just old papers, he explained, that he sold to butchers and greengrocers who used the broad thick sheets for wrapping bacon and butter. What on earth did Mayer want with them? But Mayer had begun to turn the close-written pages and was already absorbed. He could not be put off by the dealer. He was already experiencing the incomparable excitement of unearthing a find. What Mayer had discovered would have meant nothing to Birmingham's shopkeepers, had they even thought to look. The books were, in many ways, perfectly ordinary, even dull: they were filled with lists of names and calculations, business accounts and transactions, day-to-day dealings; they detailed workmen's wages. What made them special was that they had belonged to the two factories in Stoke-on-Trent, at Burslem and Etruria, that were the core of the Wedgwood operation. Pages that might have seemed tedious and commonplace to almost anyone else in Britain, books that would have been passed over without a second glance, had found their way to the notice of the one man who knew what he was looking at, and who could recognize them as valuable.\n\nMayer naturally asked about the history of the ledgers, and the dealer explained that he had bought them all, as a job lot, when Josiah Wedgwood's son had died five years earlier, in 1843, and the factories were being cleared out. He had got them for almost nothing, and had not thought much about them. There was, he guessed, a little profit to be made if he split them up and sold the paper. Now, however, seeing the gleam in Mayer's eye, he sensed the possibility of something greater than the small return he had envisaged. To Mayer's delight, he explained that there were more packets of papers and piles of documents stored in the loft above the shop, all from the same source, and all available for Mayer to see, should he care to.\n\nThe thunder had moved off and the rain was easing, but neither Mayer nor the scrap dealer noticed the change in the weather. They were now both absorbed by the promise of a deal. In the dingy loft, there were, as the merchant had promised, reams and reams of papers, far too many for Mayer to read there and then, stuffed in boxes, tied, folded and sometimes torn. Mayer needed only a brief glance to tell him that everything related to the Wedgwood factories, as he had hoped, and, crouched in the low loft space, he offered a price that satisfied both men.\n\nMayer spent long months back in Liverpool poring over the Wedgwood archives, deciphering, arranging and cataloguing them. He laboured long and studiously. He may have been lucky to have discovered the papers in an obscure scrap merchant's, but he was more than willing to make the luck work for him; to reinforce good fortune with industry. In the bundles of papers, his collection of ceramics came alive. He could show how each piece slotted into the history of ceramics; he could trace the evolution of the art of English pottery. To those doubters who questioned his taste for Wedgwood, he could better justify his choices. With the documents in front of him, it became clear how the leading factories were making progress in technical and stylistic developments, and he could trace more openly the manufacturing links with the craftsmen of the past.\n\nBy the time Mayer had finished his work on the documents, the reputation of Wedgwood was beginning to change. It was spoken of with growing respect and prices were rising: it was becoming recognized as one of the most desirable names in English ceramics. In the 1830s and 1840s, few other people were interested in Wedgwood and hardly any collectors were making any kind of systematic attempt to acquire the factory's china. But by the 1850s and 1860s, collectors like Charlotte Schreiber were beginning to look in new directions, while at the same time changes in fashion were making the wares once again desirable. During the second half of the nineteenth century, the early Wedgwood pieces that Mayer had found in Wedderburne's attic were to become the most sought-after examples of the company's output, and, by the end of the century, the demand for Wedgwood was so widespread and intense that one commentator noted that 'not only the shops but the private dwellings of France, Germany, Italy, Holland and Belgium have been ransacked by enthusiastic collectors'. As Wedgwood china began to catch the eye of fashionable society, Mayer's collection was able to boast some of the most interesting and superior pieces in private hands, as well as the most comprehensive archives, testament to both his good fortune and his astute opportunism. Thanks to two strokes of luck, he stood at the vanguard of the rediscovery, accidentally leading the followers of fashion, and winning himself a reputation as a collector of exceptional shrewdness and guile.\n\n### CHAPTER TWELVE\n\n### _Mummies, Crocodiles and Shoes for a Queen_\n\n_J_ oseph Mayer presented all his friends with etchings of the Daniels portrait. In an age before the widespread use of photography, this was a standard way to prompt them to remember him. It was a means of keeping in touch and was not an uncommon thing to do: Augustus Franks gave his friends a specially made medallion struck with his name, degree and motto to help them remember him. This was made of gold, and was the mark of a gentleman, the forerunner of a bookplate which Franks later designed, giving his full title and dignities in Latin and displaying coats of arms that linked him to several ancient families. Mayer was not, yet, going this far. But the gift of the engravings seems to suggest that he was anxious not to lose ground to some of his wealthier and more well-connected colleagues; it was an indication of a certain vanity, and of his aspirations to be an important man.\n\nFranks never had his portrait painted. It has been suggested that this was because it would have been in bad taste for a bachelor gentleman to 'advertise' himself so blatantly. Mayer, lacking Franks' upper-class sensitivities, had no such reservations. In 1851, at the age of forty-eight, in his prime and unmarried, he commissioned another portrait, a bolder image, with the dignity of age and professional success. This time, he did not want a furnished study or a clutter of objects to distract and he opted for a simple image of his head and body against a featureless background \u2013 nothing but himself; no references to anything but his essential respectability. It is perhaps no coincidence that this was the year of the Great Exhibition. Mayer had been invited by the organizers to draw on his connections in the silversmithing world to exhibit 'works of ancient and medieval art'. Instead, he used the opportunity to show off his own work, displaying fifteen pieces of silver plate (then a new technique) and eighteen items of his jewellery. The display got him noticed, and several of his pieces were commended by the judges and rewarded with an 'honourable mention'. It was in the light of this success that he commissioned another portrait to further consolidate his rising place in the world.\n\nAs an indication of this growing confidence, Mayer chose George Freizor (G.-A. Freezor), a London man of the art establishment. Freizor painted Mayer half turning to an imagined audience, and gave him weight across the chest, the decorum of middle age, the gravity of learning in his high forehead. In this second portrait, Mayer looks distinguished and active, a man pausing for a moment in the midst of important and pressing duties. It came at an apt moment. With the successful contribution to the Great Exhibition behind him and his collection growing, Mayer was embarking on his most ambitious project to date \u2013 his own museum.\n\nIt was on a visit to the British Museum that the idea came to him. Mayer was a frequent visitor to the huge dark galleries, packed with objects; he savoured the heavy scent of ancient lives, and the way the precious treasures stretched out into the shadows as if forever. And in particular he enjoyed the newly opened Egyptian galleries, pausing among the tombs of the Pharaohs, standing alone and quiet, without the press of the present in his head. It was an inspiration. It prompted him to think about his own collecting, and ways in which he could give it shape and authority and consequence. He decided he would open a museum, somewhere with proper space for his best pieces, where they could be organized and catalogued and flaunted to the public. A museum of his own would be a brilliant triumph and, when the time came, the perfect memorial. It would allow him to share with the world the greatness of his collection.\n\nMayer had no doubt that the museum should be in Liverpool. He was proud of the town and his place within it and he wanted it to grow and prosper; he wanted its citizens to have all the advantages of those of the capital. Liverpool was thriving. It had everything to rival London: the magnificent stone buildings, the busy international port and packed docks, the shops and parks \u2013 as well as the slums and grime and starving poor. Moreover, it had a long tradition of successful private museums. As early as 1800, William Bullock, another jeweller and silversmith, had opened a popular museum of over 30,000 objects in Church Street, having tried out the idea for several years in his native Sheffield. The displays included works of art, armour, natural history specimens and ethnographic objects, many of which had been brought back from Captain Cook's expeditions abroad. The museum flourished, and by 1812 was such a success that it was deemed worthy of a London home and was transferred to its own 'Egyptian Temple' (later the Egyptian Hall), purpose-built in Piccadilly. Then, in 1851, just as Mayer was shaping his own plans, the 14th Earl of Derby opened a natural history museum in Liverpool's Duke Street, preserving some of the unrivalled collection of birds and mammals that his father had kept and bred in his zoo at Knowsley. Visitors flocked to see the stuffed, pinned and desiccated remains of colourful and bizarre animals, proof that the Liverpool public had an appetite for the attractions of museum visiting.\n\nAs yet, Liverpool had no permanent collection of antiquities or decorative art; this Mayer decided to change. In May 1852, he opened his new museum at Number VIII Colquitt Street, a few doors down from the Liverpool Royal Institution, at one of the town's most reputable addresses. Everything was done properly. He sought advice from many of his friends about what to display, and he drew on Franks' expertise to create a cataloguing system. He appointed a full-time curator who lived on the top floor of the building, charged a shilling entrance fee to contribute towards basic costs, and filled the rooms with the most precious and lovely things from his collection.\n\nPerhaps because he could not shake off the memory of the moment in the British Museum galleries when the inspiration came to him, Mayer called his new project the 'Egyptian Museum'. The name usefully brought to mind not only the British Museum but also other successful private ventures, including Bullock's Egyptian Hall in London which had built on the original collection from Liverpool to put on a number of spectacular, and profitable, shows in the early years of the century, including a display in 1816 of Napoleonic relics which attracted over 200,000 visitors. The name also focused attention on at least some of the objects on display, which included ancient Egyptian inscribed tablets, carved figures and bronzes, as well as a designated 'mummy room' on the ground floor. More than anything, however, it was a way of enticing visitors. The name 'Egyptian' exuded mystery and romance; it was the buzzword of its day, a marketing phenomenon. By calling his new project the 'Egyptian Museum', Mayer was associating his collection with everything that was fashionable, stimulating and glamorous.\n\nIn the eighteenth century, and earlier, when travel in the Middle East was difficult and hazardous, only a trickle of Egyptian relics reached Europe, and there were consequently few collectors. In Britain, the Revd Robert Huntington, chaplain to the Levant Company in Aleppo for ten years from 1671 to 1681, had collected manuscripts and some antiquities. In the 1760s, Edmund Wortley Montagu (son of Lady Mary Wortley Montagu, the renowned eighteenth-century lady of letters) sent back a painted wooden coffin and mummy which became one of the earliest Egyptian exhibits at the newly opened British Museum. But, on the whole, few people were willing to brave the dangers of Middle Eastern travel in search of pieces to collect. It was after Napoleon unexpectedly occupied Egypt in 1798 that things changed. Suddenly the country was modernized and opened up to European travellers and entrepreneurs. Collectors poured into the ancient sites, flooding the markets with objects and sparking a fashion for things Egyptian.\n\nThe novelty of travelling in Egypt, and the promise of what could be collected there, drew men from a variety of backgrounds: George Annesley, Viscount Valentia, who amassed an enormous Egyptian collection at Arley Castle in Staffordshire in the early nineteenth century; Joseph Sams, a Darlington bookseller who visited Egypt and Palestine in the 1820s and brought back antiquities which he sold to the British Museum; and Henry Stobart, a clergyman who travelled extensively during the 1850s. The market in Egyptian objects boomed and private collections were soon appearing around the country. But the obsession was not confined to Britain. Expeditions to Egypt were funded by most European monarchs or governments. The most notable, led by Karl Richard Lepsius, was funded by the King of Prussia and sent back 15,000 antiquities and plaster casts for the Royal Museum in Berlin. Across the Continent, mummies were publicly unwrapped in staged spectacles that nodded only in passing to the interests of science, and publications proliferated, from the downright sensationalist to such scholarly works as the encyclopaedic, five-volume _Egypt's Place in Universal History_ , written in German by Christian Bunsen but also translated into English and published between 1848 and 1857. The visit of the Prince of Wales (the future Edward VII) with his new bride, Princess Alexandra, to Egypt in 1868\u20139, and international politicking around the opening of the Suez Canal in November 1869, only consolidated the impression that an interest in Egyptian history and affairs was quite the right thing for a person of fashion, good standing and sound education.\n\nMany of the archaeologists and antiquarians exploring Egypt were respectable men of the gentry, government and the church. But they often employed 'colourful' agents to rummage, cheat and steal on their behalf. One of the most active of these early collectors was the British Consul-General in Cairo, Henry Salt, himself a portrait painter, whose agents Giovanni d'Athanasi and Giovanni Belzoni ransacked ancient sites such as Thebes, the Pyramids and the Sphinx at Giza in a series of 'excavations'. Among the objects they removed was a colossal granite bust of Rameses II, and enough material to form three huge collections: one was acquired by the British Museum, another was sold to King Charles X of France for \u00a310,000, and a third was auctioned at Sotheby's in over 1,000 lots (some of which Mayer later acquired) after Salt's death in 1827. The enormous popularity of Egyptian objects, and the prestige attached to discovering them, meant that many early collectors and their agents would stop at nothing to secure the pieces they wanted. Excavation methods were violently destructive, obliterating as much as they uncovered. Digs, explained Lepsius, often took place 'hurriedly and by night and with bribed assistance'. There was ferocious rivalry, and the competitors could frequently be seen clambering over stones and broken sarcophagi, rooting through rubble and bartering with boys whose pockets were stuffed with ancient remains. There was little attempt at documentation and the contexts for the objects, as well as the opportunities for studying them on site, were almost always lost.\n\nIn his early travels, Mayer himself was not above such instant archaeology. There was undoubtedly a thrill to be had from hands-on intervention, and, at a time when European empires were expanding, there were few who would have questioned the right of British collectors to pieces from more 'primitive' cultures. What is now regarded as looting was seen by the Victorians as, at best, the rescue of historic objects for posterity and, at worst, little more than enthusiastic opportunism. In Britain, many expeditions were funded by the government, national and regional museums, or by respectable establishment organizations, and the pillaged artefacts were proudly displayed to an admiring public. Private and public collectors alike benefited from the free-for-all. As early as 1799, the Rosetta Stone was looted during Napoleon's Egypt campaign and eventually given over to the British, shown first at the Society of Antiquaries and then, from 1802, at the British Museum. Mayer's collection boasted numerous large pieces taken from Egypt, including the great granite sarcophagus of Bakenkhonsu, stolen from his tomb at Thebes, and thirty-three of the stone funerary tablets known as stelae.\n\nBut Egypt was not the only source of new objects. Lord Elgin's expeditions to Greece in the early nineteenth century famously resulted in the Parthenon Marbles finding a home in the British Museum; Austen Henry Layard sent the results of his excavations in Mesopotamia (now Iraq) to private enthusiasts like Charlotte Schreiber and also to the national collections. His looting was regarded as thoroughly respectable and his work was recognized with a trusteeship of the British Museum in 1866. In the 1860s, British troops sent to Abyssinia (now Ethiopia) took sacred wooden tablets, religious manuscripts, ceremonial crosses, a solid gold crown, chalices, textiles and jewellery from the imperial palaces in an operation that is said to have required 15 elephants and 200 mules as transport. Again, the British establishment openly condoned the thievery: the hoard, known as the Magdala Treasure, was largely divided between the Queen, the British Museum, the South Kensington Museum and the British Library. The remaining pieces were auctioned to private owners.\n\nAs the century moved on, however, there were signs that the consequences of such widespread looting were beginning to be acknowledged. In June 1871, less than four years after the arrival of the Magdala Treasure in Britain, Prime Minister William Gladstone conceded to the House of Commons that the whole affair had been 'unsatisfactory. . . from first to last' and that he 'deeply regretted that those articles were ever brought from Abyssinia and could not conceive why they were so brought'. Gladstone was, as we have seen, a collector himself, and his observation points towards a growing change of heart in some collecting circles. Clearly, there would remain unscrupulous and profiteering individuals, buying, selling and collecting objects thoughtlessly. The driving political imperatives of Empire still remained, but the increasing influence of scholarly curators such as Franks and Robinson, and of expert private collectors such as Charlotte Schreiber, meant that the emphasis was gradually shifting from acquiring objects at any cost and ripping them from their ancient sites to valuing and understanding them within their original contexts.\n\nThis was not, of course, as simple a matter as I might have made it sound. Attitudes were complex and progress was erratic. Some of the scholarship of the most learned of Victorian curators and collectors derived from studying looted objects; many early archaeologists like Layard combined a certain gung-ho acquisitiveness with a genuine spirit of inquiry; public and private collectors are, as we still see today, notoriously slow to agree to the repatriation of other people's treasures. Nevertheless, the nineteenth century saw a movement towards more structured, scholarly and responsible collecting, and this was reflected in the changing attitudes of collectors like Joseph Mayer. Through discussions with other, often more learned, collectors, Mayer came to appreciate the importance of documentation, research and publication. Though his own methods were not necessarily scholarly, he grew to appreciate \u2013 and support \u2013 scholarship in those around him. His friendship with Roach Smith in particular taught him to value objects as much for the light they shed on history as for their status as treasure. Roach Smith, Mayer admitted, gave him a 'valuable education', in which he learned how 'all kinds of antiquarian subjects. . . gave an impetus' to study and acted as 'corroborative help to written history'. Over the fifty years that he was collecting, Mayer's attitudes changed. The instant gratification of on-site rummaging which had excited him in his youth gave way to a more considered and respectful approach to objects. This may have had something to do with simply growing up, and becoming older, wealthier and more reputable, but personal circumstances do not explain it all. Joseph Mayer was part of a collecting age when new and intriguing objects, like those from Egypt, created different collecting habits. He was part of a generation that was discovering largely uncharted territory and, in turn, developing new ways of doing things, influencing attitudes and forming opinion.\n\nAs with Wedgwood, Mayer was fortunate to be collecting Egyptian objects at the right time, when interest was at its height and when pieces of unique variety, character and quality were first becoming available on the open market. Such good timing, allied with unerring enthusiasm, meant his lack of expertise hardly mattered: he was still able to amass important Egyptian pieces that had both scholarly and aesthetic significance, and that would be valued in years to come. But callers at the Colquitt Street museum who came expecting no more than a display of Egyptian antiquities were both surprised and fascinated by what they found. There was a stuffed crocodile hung high over the ancient sarcophagi; there was Roman pottery; sculpture from Assyria and Babylonia; Anglo-Saxon metalwork; coins and commemorative medals; Burmese manuscripts; a range of fine Persian leather bindings; and examples of Chinese calligraphy. Visitors could browse an extensive reference library or study books in a dozen languages \u2013 none of which Mayer could read himself \u2013 before moving on to admire his collection of jewellery, or the artefacts of medieval life he had picked up while wandering the Wirral. The ground floor had a room set aside for the history of English pottery, especially local Liverpool ware and Wedgwood, and the first floor boasted a room devoted to Renaissance Italian majolica. On the stairs between the two floors hung a number of illustrations explaining the technical processes of engraving; a cartoon depicting a scene from Dante's _Divine Comedy_ ; and a painting of Noah's Ark breasting the rising waters of the biblical flood.\n\nMost of all, though, Mayer valued objects that connected to famous lives of the past. Earlier generations would have dubbed him a 'curio': the type of collector who chose things not for their intrinsic aesthetic value but for their rarity, novelty or romantic associations. In many ways, he was part of the tradition of collecting for a 'cabinet of curiosities'; these cabinets of wealthy seventeenth-and early-eighteenth-century gentlemen embraced everything from geological specimens to coins. His museum owed much to eclectic forerunners such as John Tradescant, whose collection was described by a visitor in 1638 as including, among other things, 'two ribs of a whale, also a very ingenious little boat of bark. . . foreign plants. . . a salamander, a chameleon, a pelican. . . the hand of a mermaid, the hand of a mummy. . . all kinds of precious stones, coins, a picture wrought in feathers, a small piece of wood from the cross of Christ, pictures in perspective of Henry IV and Louis XIII of France. . . a scourge with which Charles V is said to have scourged himself and a hat band of snake bones'. Tradescant's collection was, according to another admirer, 'where a Man might in one daye behold and collecte into one place more curiosities than hee should see if hee spent all his life in Travell'. It was a marvel to his contemporaries, and an inspiration to those that followed, eventually becoming the core of the Ashmolean Museum in Oxford.\n\nIn this tradition, Mayer's displays of genuinely significant objects at the Egyptian Museum were interspersed with a range of pieces of doubtful aesthetic value, but each of which came with a good story. There was a signet ring and a snuff box which Mayer claimed had belonged to Napoleon, as well as jewellery and another, gold-lined, snuff box which had apparently once been in the possession of the Empress Josephine. There was a 'large wheel-lock Gun' which, the label proudly proclaimed, had been 'used at the execution of Mary Queen of Scots', and, in the same room, 'The Armoury', there was 'a cocoa-nut Cup, set in silver, formerly belonging to Oliver Cromwell', as well as a pair of boots, also said to have been Cromwell's. Mayer also treated his visitors to the sight of a 'Pair of Shoes, worn by her present Majesty, on the night of her marriage to Prince Albert'.\n\nFew of Liverpool's citizens had ever before seen this kind of display; they marvelled at the exotic, the far-away and the historic now all within reach. Mayer's idiosyncratic and unpredictable approach to collecting was amusing and spectacular, a stimulus to the imaginations of adults and children alike. Locals quickly added the museum to their list of entertainments, and many made return visits. Even in his own museum Mayer lacked the confidence to speak directly to visitors or answer queries about the collection, and the curator was kept busy explaining the displays and acting as tour guide for those unfamiliar with the town.\n\nAt a time when the distinctions between private and public collections could be hazy, Mayer's museum straddled the border between the two. Although there was an admission charge of one shilling for adults and sixpence for children (which Mayer regretted having to impose), the museum was genuinely accessible, useful and popular. It was created with the general visitor in mind, as much to entertain as to educate. While many museums \u2013 from the South Kensington Museum to Ruskin's educational experiment in Sheffield \u2013 were forthright in their didactic ambition, Mayer seems simply to have enjoyed the idea that people might like to come to see his things. Certainly, it seems clear that he regarded the Egyptian Museum as largely 'public', as an enterprise for the town of Liverpool that just happened to be funded by his personal wealth and made possible by his personal obsession.\n\nWith his museum open, Mayer felt an increased belief in himself as a collector. He commissioned a third portrait from John Harris, a competent local craftsman without pretensions. It is imposing and substantial. Mayer stands at the heart of his museum and replacing the eccentric clutter of his first portrait are a few choice objects: a colossal statue from the Abu Simbel temples, one of ancient Egypt's richest and most haunting sites; a rare thirteenth-century German prayerbook in an intricate medieval binding; and a beautiful ancient carved ivory, the Asclepius-Hygieia diptych, which had graced a Roman site of worship and been part of the Fej\u00e9rv\u00e1ry collection. Behind him, retreating into the arcaded spaced beyond, can be seen the forms of his cases; under his hand the gilt-edged top of a heavily carved scholar's table and at the front of the picture, bold and large this time, both distinct from his objects and master of them, the figure of Mayer himself, half-smiling at his visitors, welcoming them to the museum he had created with an open, confident sweep of the hand.\n\n### CHAPTER THIRTEEN\n\n### _The Treasures of the North_\n\n_F_ ive years after opening his museum, Joseph Mayer found himself standing in a crowded field under a hot May sun, choking in the dust of passing horses and staring with slightly dismayed wonder at a vast vaulted iron and glass building whose low wings seemed to stretch far into the distance. He was with a group of his Liverpool friends. They had travelled together by train the few miles to Old Trafford, just outside Manchester, where, in the heart of the industrial grime and pragmatism of England's north-west, they were about to join the snaking queue to see an unprecedented event. Larger and more impressive even than the famous Great Exhibition, this was Manchester's Art Treasures Exhibition of 1857. Here, 16,000 works of art were collected under one roof: oils, watercolours, engravings and drawings; the controversial and avant-garde work of the Pre-Raphaelites; Michelangelo's unfinished 'Manchester Madonna'; photographs, textiles, sculpture, armour, furniture \u2013 and Joseph Mayer's choicest and most precious pieces.\n\nLike Robinson's South Kensington exhibition a few years later, the 1857 Manchester display drew on the generosity of, and was an inspiration to, private collectors. 'Art in England may be said to have derived all its encouragement from private persons,' explained the organizers. Articulating the tension between private ownership and public display, they went on: 'The pictures of our leading artists, the work of our best Sculptors, as well as the most select of all other objects coming under the denomination of Fine Arts, are distributed in private houses throughout the kingdom, instead of being found as in Continental Countries in National Collections accessible to the public.' Like the 1862 show designed by Robinson and the Fine Arts Club, the purpose of the Art Treasures Exhibition was to display the finest of these works, to celebrate the achievements of private collectors, and to go some way towards redressing the balance from 'periodic gatherings of the production of industry' to exhibitions of more aesthetic interest: 'There appears no reason why an effort should not be made to collect together in one central locality, and in a suitable building, the Treasures of Art with which Great Britain abounds.'\n\nEveryone, it seemed, was eager to be involved. As for the later London show, the Queen and Prince Albert agreed to lend a number of pieces and, keen to be included in such exalted company, collectors from across the country hurried forward to make their possessions available. Members of the Fine Arts Club were asked to lend, and Charlotte Schreiber offered three paintings by the Welsh watercolourist Penry Williams to the section of 'Modern Masters'. But this was to be a regional showcase as much as a national event. It was a chance to polish provincial pride and highlight what could be done by the townsmen of the North; an opportunity for men like Mayer to prove that, even in the regions, there were collectors with taste and refinement whose objects rivalled the best in the world. Local collectors, in particular, readily committed their support and the organizers emphasized the choice of Manchester as a venue. 'Situated as it is, in the centre of the kingdom, in the midst of a dense surrounding population with railway facilities admirably adapted for bringing and returning visitors within one day', the city boasted both practical advantages 'calculated to ensure the financial success of the scheme' and, just as significantly, a tradition of active collecting and art appreciation: it was, pointed out proud local dignitaries, 'a district where individuals have done so much to encourage art'.\n\nIn just fourteen months, the committee succeeded in building an enormous palace on land at Old Trafford leased from the Manchester Cricket Club and later converted into the city's botanical gardens. They filled the cavernous spaces, hanging paintings three or four high on the walls, installing cases with ceramics and glassware and creating corridors of sculpture in the airy nave. Over the entrance was a quotation from Keats \u2013 'a thing of Beauty is a joy for ever' \u2013 and over the exit a line of Pope's, 'to wake the soul by tender strokes of art'. These set the tone for the exhibition. There were over 6,000 paintings, both by respected English artists such as Hogarth, Gainsborough and Constable and by the European Old Masters \u2013 Rubens, Raphael, Titian and Rembrandt. 'Modern' works by Turner and the Pre-Raphaelites were included, and the emerging art of photography was represented by evocative images of the Crimean War by James Robertson and an ambitious allegoric montage of _The Two Ways of Life_ by the Swedish photographic pioneer Oscar Gustave Rejlander. The hall was subdivided by partitions, creating separate galleries, each shaded with calico to prevent damage to the artworks on sunny days. When things inside got too hot, firemen sprayed water on to the roof in a rudimentary form of air conditioning. Two public refreshment rooms also helped to reinvigorate visitors: in the convivial surroundings of the second-class restaurant, one reviewer noted that 'John Bull and his female may be seen in full gulp and guzzle, swallowing vast quantities of cold boiled beef, thoroughly moistened with porter or bitter'.\n\nPride of place in the Old Trafford palace was given to the impressive collection of medieval and Renaissance decorative arts brought together in the 1830s and 1840s by the Toulouse lawyer Jules Soulages. It was a collection J. C. Robinson had already seen and admired. In 1855, when it had come on to the market after Soulages' death, a photographer had been sent by Henry Cole to France to record the objects, with a view to making a purchase. His photographs showed a Toulouse townhouse bursting with vases and platters, highly decorated chests, carved chairs, tables and mirrors, and elaborate fireplaces. It was clearly a unique and valuable collection, and, with the support of Prince Albert, Cole began arrangements to buy everything, in the meantime bringing the objects to England for temporary display in the Marlborough House galleries. There, Robinson began the meticulous task of cataloguing, producing an illustrated descriptive inventory for publication in December 1856, just as plans for the Art Treasures Exhibition were taking shape.\n\nBut the sale was not completed. The Prime Minister at the time, Viscount Palmerston, and the Treasury could not see what Cole and Robinson wanted with Soulages' rare and historic objects, nor how they could be used to educate designers and artisans. Once again, there was a debate about the purpose of the South Kensington Museum and the types of objects it should be collecting. Finally, emphasizing the principle of utility, the expenditure was vetoed. Instead, it was the organizers of the Manchester spectacle who audaciously agreed to pay the \u00a313,500 needed to secure the collection for Britain. Robinson was forced to repack all the objects and the Soulages pieces left Marlborough House for Old Trafford, where they formed the core of the Art Treasures show. 'After repeated communications with the London managers which failed to secure its loan, the members of the Executive Committee, in their individual capacity and on their personal responsibility, agreed to purchase the Collection. . . and thus secured it,' explained the committee. Manchester had scored a victory over the capital, and let it be fully known in the press. Numerous announcements and illustrations drew the northern crowds to see what it was that had been snatched from London's hands.\n\nLacking a formal organization like the Fine Arts Club to oversee the event, Manchester relied instead on bringing together an ad-hoc general council specifically for the duration of the project. This included the usual selection of gentry, MPs, wealthy merchants and municipal dignitaries. A few were also members of the London-based Fine Arts Club, such as Lord Overstone, but many more were attracted by the opportunity to make a local contribution. James Aspinall Turner, for example, was a Manchester cotton manufacturer, Whig MP and naturalist, who owned a scholarly collection of entomological specimens; Stockport-born Joseph Whitworth was a mechanical engineer who invented an hexagonal, high-performance rifle in 1859 and who left a number of bequests to the city of Manchester, including the core collection of the Whitworth Art Gallery; Thomas Goadsby, mayor of Manchester from 1861 to 1862, presented an imposing memorial of Prince Albert to the city in 'grateful acknowledgement of public and private virtues'; Sir Humphrey de Trafford held nearly 2,000 acres of land in Cheshire and was to become an implacable opponent of the building of the Manchester Ship Canal in the 1880s \u2013 it was his agreement to give over the lease of the Old Trafford land at favourable terms which made the exhibition possible.\n\nWhile it was largely drawn from, and committed to, the area, the organizing committee was in no way parochial. It expressed ambitious hopes for the exhibition, setting it in a national context that went beyond 'the mere gratification of public curiosity, and the giving [of ] an intellectual entertainment to the dense population of a particular locality'. As Prince Albert suggested in a letter lending his support to the project, such an intention might well be 'praiseworthy in itself ' but the show could accomplish so much more: 'National usefulness', enthused the Prince, 'might be found in the educational direction which may be given to the whole scheme. . . If the collection you propose to form were made to illustrate the history of Art in a chronological and systematic arrangement, it would speak powerfully to the public mind, and enable, in a practical way, the most uneducated eye to gather the lessons which ages of thought and scientific research have attempted to abstract.' Such an achievement, Prince Albert went on, would have not just national, but also international impact, showing off Britain to the world and creating 'for the first time a gallery as no other country could produce'.\n\nThe Manchester organizers decided to follow the Prince's advice, and the show was the first large-scale exhibition in Europe to present works chronologically in an attempt to reveal the historical development of art. More usually, art was displayed by theme, or by prominent schools. Here in Manchester, works from northern Europe, for example, were hung opposite artworks of the same period from southern Europe in order to highlight differences in style and technique. The conventional accepted hierarchies were set aside, so that equal emphasis was given to Italian, Spanish, Dutch and German painters. And, as the Prince had suggested, this approach was to prove influential beyond the immediate life of the exhibition. At a time when national museums were struggling to make their displays meaningful, the Art Treasures show became associated with some of the most forward-thinking curatorial practices in Europe. It came to be lauded as a model of 'teaching the mind as well as gratifying the senses', and discussions about the principles of display at the National Gallery, in particular, frequently referred to what was regarded as Manchester's great success. 'Each work of Art appears as a link in a great chain, which receives an influence from the one preceding it, and imparts an influence to the one following,' explained Gustav Waagen admiringly in the exhibition catalogue. 'Each work is thus illustrated and made intelligible, while instruction is combined with enjoyment.' Gradually, the example of the Art Treasures exhibition began to make itself felt in permanent displays across the country. When George Scharf was appointed Secretary of the newly founded National Portrait Gallery in London in the year of the Art Treasures Exhibition, he took with him the experience he had gained as art secretary of the Manchester show and the principles espoused there, and continued to uphold the value of chronological display and clear interpretation. In this way, he set in train the development, on a national scale, of new ways of displaying art inspired by the 1857 event.\n\nMost visitors to the Art Treasures Exhibition, of course, were more concerned with enjoying themselves and seeing a fine spectacle than considering the event's effect on museum policy. And in this too it did not disappoint. On 5 May 1857, Prince Albert opened the show. Season tickets were sold in advance for one guinea; a two-guinea ticket included admission to both the opening and the official visit of Queen Victoria on 29 June. On most days, admission was charged at 1 shilling, although, on Thursdays and for the first ten days of the show, the price was set at half a crown. By the time the doors closed in the middle of October, over 1 million people had passed through the turnstiles. These visitors included European royalty, the politicians Disraeli, Gladstone, Lord Palmerston and the Duke of Wellington, the writers Charles Dickens, Alfred Lord Tennyson, Elizabeth Gaskell and Nathaniel Hawthorne, and critics such as Gustav Waagen and John Ruskin. The Schreibers visited the show on five consecutive days.\n\nMany of the visitors had never before seen any works of art, and certainly had never seen so many gathered together in one place. This was a show to delight ordinary visitors as much as to impress eminent ones. As the organizers had predicted, railways were the key to getting masses of working people to the site: excursions by train were laid on not only from the north-west but from as far afield as Shrewsbury, Leeds, Grimsby and Lincoln. Titus Salt, the Bradford manufacturer and philanthropist, commissioned three trains to take nearly 3,000 of his factory workers to see the exhibition in September, while Thomas Cook, who had arranged trips to both the Great Exhibition in 1851 and the 1855 Paris Exhibition, this time advertised romantic 'moonlight' excursions that arrived in Manchester at dawn.\n\nSuch enthusiasm did not convince everyone that the exhibition was a worthwhile venture, however. For those who knew Manchester as a blackened industrial hub of factories and mills, it seemed an incongruous place to show art. Hawthorne, who visited the displays many times, commented that 'it is singular that the great Art-Exhibition should have come to pass in the rudest great town in England'. More antagonistically, the Duke of Devonshire, whose Chatsworth estates lay on the other side of the Pennines and who refused to lend to the show, was sharply dismissive of Manchester's right to an event of any cultural significance: 'What in the world do you want with art in Manchester?' he demanded. 'Why can't you stick to your cotton spinning?' The Duke was a collector himself, with a particular taste for illustrated botanical books, and he was not without philanthropic impulses: he donated land for public gardens in Buxton and endowed the Cavendish Laboratory at Cambridge, where he was Chancellor of the University for thirty years from 1861. But his apparent belief that art did not belong among the lower classes highlights the continuing reluctance of many of the country's wealthiest and most powerful men to countenance any change to the established systems for showing and viewing art. It reveals the lingering fear that accessible events like the Art Treasures Exhibition, and the public museums they inspired, might in the end encourage unrest and distract workers from the essential business of their 'cotton spinning'. Cavendish recognized that, by taking works out of their more usual context of aristocratic privilege and placing them on open display, the Manchester spectacle was doing something potentially subversive. It was asking private collectors to share the treasures they and their families had amassed over the centuries with the rest of the nation \u2013 and not just the genteel part of it.\n\nThe Art Treasures Exhibition was a particular triumph for Joseph Mayer. His objects were a celebrated part of the display. He had already played a public role on other notable occasions: he designed the silver trowel with which Sir Philip Egerton, MP, laid the foundation stone for Birkenhead docks in October 1844 and the prize plate for the Royal Mersey Yacht Club. As we have seen, his pieces were well received at the Great Exhibition. Now, at this latest spectacle, he found his collection widely honoured and admired. Scholars asked for information and photographs, and even Gustav Waagen spoke 'of the importance of your great collection'. The review volume that accompanied the exhibition carried illustrations of all of Mayer's pieces along with a description of one of his Wedgwood vases as 'an artistic treasure', an indication that Wedgwood was becoming noticed and valued. It also cited eleven of Mayer's pieces as among the finest examples of ceramic art. In the galleries, a selection of his late antique and Byzantine ivories were laid out chronologically in two cabinets to display the history of European sculpture from the classical period through to the Renaissance, and his collection was praised for its exceptional quality: 'There is an excellent opportunity to make a perfect study of sculpture in monuments of small scale,' suggested Waagen, for example, in his commentary on the displays. 'To do so it is advisable to begin with the many fine antique sculptures in bronze and terracotta from the collection of Mr Hertz, now belonging to Mr Joseph Mayer in Liverpool.'\n\nOne piece that attracted particular attention was an almost lifesize bronze statue of the Buddhist goddess of compassion, Guanyin. Another looted treasure, the statue had been taken from the pilgrimage island of Putuo, east of Ningbo near Shanghai in China, by Major William Edie, an officer in the British army during the First Opium War (1839\u201342). Made during the fifteenth century, it showed the contemplative goddess with twenty-two outstretched arms seated on a carved pedestal. Mayer had acquired it, along with four other devotional statues from Putuo, as part of the Bram Hertz collection. While he was waiting for the statues to be sold at Sotheby's, Mayer displayed the pieces in Manchester, no doubt aware that such publicity would do no harm in raising their profile and boosting his standing as a collector.\n\nWith such a showcase, Mayer's reputation continued to grow, within the region and beyond. The museum in Colquitt Street flourished. In 1862, it became a 'Museum of Antiquities', perhaps to emphasize the increasing range of the objects on display, and five years later it was renamed again as the 'Museum of National and Foreign Antiquities', leaving no doubt as to its breadth and ambition. The experiment of an accessible museum had clearly been proved a success, and in 1860, embracing the growing fashion for public museums, Liverpool Town Council opened a splendid purpose-built building with money and land provided by William Brown, a merchant, banker and politician. The neo-classical style of the new building was typical of the emerging Victorian museums. It was grand and imposing, a statement of municipal order, and it also recalled art of the classical period, still regarded as the standard by which all other art was judged. Just as the British Museum concentrated much of its energies and resources on collecting artefacts of the classical civilizations, so the architectural presence of these new museums suggested that they, too, would be displaying equally prestigious objects. Preston, Sheffield, Norwich, Cambridge and Manchester all raised new neo-classical buildings to house their municipal collections and act as temples to the arts. Elsewhere in Europe, neo-classicism was also the style of choice for new museums. In the 1780s, the Prado in Madrid was built to a neoclassical design by Juan de Villanueva; the Altes Museum in Berlin was designed in the same manner by Karl Friedrich Schinkel during the 1820s, using the Stoa in Athens as a model; and the New Hermitage in St Petersburg, designed by the German architect Leo von Klenze, combined elements of the classical, Renaissance and Baroque, all interpreted in a neo-classical style.\n\nAt first, the displays in the imposing new Liverpool building consisted almost entirely of natural history specimens, bequeathed from the collection of the 13th Earl of Derby, but, as elsewhere, private collectors soon began to make an increasing diversity of objects available through loans or legacies. In 1867, Mayer decided that the museum was the proper home for the 14,000 pieces of his collection. Roach Smith tried to persuade him that the town was unworthy of anything so splendid: 'There are certain classes to whom the gift of food is nothing, unless at the same time you chew it for them,' he fretted. 'You will find the Liverpool Citizens. . . of this class. Gratitude nor appreciation can never possibly spring from them. . . There are higher and better regions for you.' But Mayer, for once, did not listen. He had made up his mind. Of course, he knew that the galleries of the British Museum were splendid; he had walked the corridors of South Kensington's Brompton Boilers and had admired the displays there. But he was closer to local collectors, curators and scholars and shared their regional pride. He had no desire for his collection to be scattered across the shelves of a large institution. And so Mayer wrote to the Liverpool authorities, offering his museum to the town, so long as they promised 'that the Collection shall be kept together and be known as and called \"The Mayer Collection\"'.\n\nWhen his offer was accepted, it seemed as though his future in Liverpool was secured, his reputation and his local roots honoured. 'Your museum has been nobly given and has most deservedly earned you a niche in the temple of Fame,' Joseph Clarke assured him. 'Your name will float down the stream of time.' Some of the more idiosyncratic pieces were set aside by the municipal curators but much of Joseph Mayer's jumble of objects was duly catalogued and displayed, while the Egyptian pieces in particular formed the nucleus of the new, genuinely public, collection that made the Liverpool Museum known across the world, in Mayer's lifetime and today.\n\n### CHAPTER FOURTEEN\n\n### _A Larger World_\n\n_J_ oseph Mayer had decided it was time to move. With his ambitious collecting, the rooms above his shop were becoming increasingly cramped, overwhelmed by objects, books and papers. But there was also a new belief that he was worthy to take his place among the scholars he so admired. It was time for something grander than a shopkeeper's existence. So in the late 1850s Mayer took a proper house, one of substance and even grandeur, at Dacre Park on the Rock Ferry escarpment, looking across the Mersey from Cheshire to Liverpool. The guise of the country gentleman suited him \u2013 and his collecting \u2013 so well that just a couple of years later, in 1860, he moved a mile or so south to Bebington, where he established himself in a solid brick-built double-fronted farmhouse with gardens sweeping into open land, the potential for expansion and room for even the most vigorous of collectors and his possessions.\n\nAs with his museum, Mayer wanted a name for the house that would inspire, and he called it Pennant House after Thomas Pennant, the eighteenth-century traveller, antiquary, topographer and naturalist. Pennant was an amiable and progressive country squire, erudite, inquisitive, eloquent and fascinated by history. Born in 1726 at Downing Hall in Flintshire, he wrote on subjects as diverse as earthquakes and the history of quadrupeds, but it was his remarkably popular series of travel books, beginning with _A Tour in Scotland_ in 1769, and including accounts of Wales, Snowdon, and a _Journey from Chester to London_ , which came to be seen as important records of antiquarian relics. His correspondence with Gilbert White formed the basis for White's _The Natural History and Antiquities of Selborne_. In taking his name for the Bebington farmhouse, Mayer was setting out a vision for his own future. Just as the 'Egyptian Museum' had captured the mystery and romance of the past, so Pennant House promised dignity, refinement and far-reaching scholarship.\n\nIn his new home, Mayer surrounded himself with light and bustle and company. He invited his unmarried sister Jane to move in with him as well as their niece Mary Wordley. Two of his nephews, Frederick and Henry Boyle, lived close by; Mayer opened his doors to them, enjoying the commotion of having young men in the house. He took to gardening, and his collecting instincts found a new outlet in rare plants, often fragile and fickle in the northern cold. He held dinners for his friends, entertaining with enthusiasm, and he talked earnestly to local residents, eager to find out what interested them. He was energetic and inventive, boyish even. He should have been happy, but there was something or, more properly, someone missing. There was a place in the house that could not be filled by nephews and nieces, or even by the most sympathetic of friends.\n\nMayer had never married. He had invested his time, energy and money in his business, and in his collection. But he enjoyed female company and, while he had resisted the best efforts of the Liverpool matrons to fix him up, he had been happy enough to chatter and flirt and dance with their daughters. He was approachable and friendly and he had an eye for the beautiful, the quirky and the rare. While he may have escaped the demands of marriage, however, he could not elude the complications of love. There had been Peggy Harrison. When Mayer came to commission from the Italian sculptor Giovanni Fontana a series of marble reliefs and busts of his friends, it was Peggy who was one of the first to sit for him. The sculpture, Mayer said, would stand as a lasting commemoration to the 'elegant manners, cultivated taste and affectionate friendship', which made Peggy 'a delightful companion in my pilgrimages to shrines of art and antiquities in many lands'.\n\nVery little is known about Peggy Harrison, or her exact relationship with Mayer. She certainly travelled abroad with him extensively, and the marble bust is a clear sign of his admiration and respect. Perhaps it was also a way of making amends. In an age with strict rules of propriety, Peggy may well have expected a proposal of marriage. A woman who travelled with a man to whom she was not married, who accompanied him to archaeological sites and along tangled streets of antique dealers, who studied sales catalogues in cold hotel rooms and who entertained, amused and comforted her companion on long journeys, risked a great deal \u2013 presumably for love. There is obvious intimacy in Mayer's evocation of 'a delightful companion' who had travelled so far and for so long with him, and there is clear admiration in his assessment of her manners and character. For a connoisseur like Mayer, too, his judgement of Peggy's 'cultivated taste' signals the highest praise. But perhaps she was too independent to submit to married life with a man like Mayer, or perhaps she was not quite what he was looking for. However it was, their relationship was never formalized, and eventually faltered. By the time Mayer had settled himself into Pennant House, their intimate companionship had somehow come to an end.\n\nAs Mayer's circle of friends grew, there were inevitably other women to whom he was attracted, and who understood his devotion to collecting. Not long after moving into Pennant House, he spent some weeks working with Elizabeth Meteyard, who was writing a biography of Josiah Wedgwood. Elizabeth was bright and appealing in her middle age. She was studious and she shared Mayer's taste in ceramics and his fascination with the past. Mayer was delighted to be in such company and gave Elizabeth free access to all his papers, helped fund her research and offered her the benefit of introductions to his long list of useful acquaintances, including members of the Wedgwood family. Elizabeth also noted demurely, that he 'permitted me to work for a fortnight under his personal guidance' and entertained her with stories. There is no doubt that he found her a charming and knowledgeable companion. He may, for a moment, have thought of more. But Elizabeth had a career and was as committed to her work as Mayer was to his collection. Her biography was soon to be followed by five other books on Wedgwood. Making her way as a professional writer, she contributed articles and short stories to numerous periodicals under the pen-name 'Silverpen' and she also found success as a popular novelist. She could not spend too much time at Pennant House, no matter how pleasant it was, and she was unwilling to be distracted by Mayer. For each of her ceramics books, she drew on the resources of Mayer's collection, and over the years she became a familiar face in Bebington, attending the opening of an art exhibition there in September 1871 and visiting again to research Wedgwood books that were published in 1873 and 1875. Such visits seem to have been no more than an agreeable interlude for them both, and all too quickly she returned to London. Meteyard clearly retained a lifelong admiration for Mayer and his work, however, and at her death in 1879 her will bequeathed all her personal papers to Liverpool Museum; the wish was not respected by her executors.\n\nIn the summer of 1863, Mayer's sixtieth year, he finally became engaged. He wrote delightedly to all his closest friends, announcing his forthcoming marriage with pride, and they wrote back in turn, expressing their surprise and sharing in his pleasure. 'I am very happy to think that you have partially weaned yourself from things antique. . . that you have distracted your attention from coats of mail and bent it upon crinoline,' wrote Joseph Clarke, adding that he was 'in high glee at the pleasure of paying my most ardent but humble respects' to the new Mrs Mayer. The wedding plans, it seems, were quickly set \u2013 there was little reason for delay. At Mayer's age, he could dispense with the reticence of youth.\n\nWe don't know who it was that Mayer was so close to marrying. All we know is that she lived in Kent, some 300 miles away. It is possible that Mayer had known her a long time and I believe he may have met her during the negotiations to buy the Faussett collection ten years earlier when he had spent time in and around Heppington. He may have been wooing her for a decade, or he may have had to shelve his advances until the way finally became clear on the death of a father or a husband. Nor do we know why the marriage never took place. There might have been last-minute nerves or a disagreement. The lady who Mayer had chosen for a wife may have faltered at the thought of uprooting herself and moving so far from familiar things. They may have been cheated, in the end, by death, leaving Mayer to mourn the woman he should have married. All we know for sure is that he was left alone.\n\nThe Victorians were inclined to think of avid collectors as life's losers. Although there was money to be made from collecting, and a certain prestige too for those who, like Mayer, managed to earn a prominent position within the community, it was difficult to resist the idea that a singular devotion to things was the mark of a man who had failed in other ways. Collecting was increasingly being seen as a means of expressing identity \u2013 or of compensating for personal shortfalls. In literature, the stereotype of the collector was associated with the withdrawn, the anti-social and the strange, like Wilkie Collins's Mr Fairlie in _The Woman in White_ , a character portrayed as living in 'profound seclusion' and completely preoccupied with his possessions. In popular journals like _Punch_ , the collector was drawn as weak, obsessive and comical. A cartoon from the _Punch Almanack_ of 1875 shows a 'pale enthusiast' in the grips of 'Chronic Chinamania (incurable)' and so absorbed by his collection that he is apparently oblivious to the charms of the young women encircling him. Similalry, a set of French caricatures features M. de Menussard, an imaginary collector who lives alone, never goes out, has no friends and is 'a little old man, dry, wrinkled, worn down, patched up'. Even among collectors themselves, there could be ambiguity and embarrassment about their activity: _The Connoisseur: A Collector's Journal and Monthly Review_ noted defensively that its readers might well display 'amiable weaknesses', and might even be in the grip of an 'unwise but never despicable passion'.\n\nThe idea that collecting was somehow 'unwise' harked back to a more widespread discomfort with art, and how it might encourage potentially disruptive, or even immoral, pleasures. The behaviour of avant-garde artists such as the Pre-Raphaelites only reinforced this uneasiness. In 1850, John Everett Millais' work, _Christ in the House of His Parents_ , was attacked as blasphemous, while the unconventional lives and loves of the painters attracted continual attention and at times caused outrage: John Millais' relationship with Effie, the wife of John Ruskin, her subsequent divorce and remarriage to Millais was the subject of a public scandal. Those of a religious turn of mind could be suspicious of art's appeal to the senses, and its portrayal of unpredictable emotional and psychological states. The emphasis on the utility of art objects \u2013 Henry Cole's commitment to proving their relevance to industry and manufacturing \u2013 was one way of divorcing them from desire or sensual pleasure, removing them from the domain of aesthetic pleasure to the less unsettling and more pragmatic structure of commerce.\n\nThe anxieties about such unhealthy passions were particularly evident when the Victorians thought about men collecting the decorative arts or antiquarians in their cluttered studies. Science collections were typically regarded as scholarly and progressive; paintings, sculpture and classical antiquities were respectable and had long been valued by the leisured and landed classes. Robinson's fondness for historical works, especially Renaissance masterpieces, was considered properly intellectual and cultured \u2013 but a preoccupation with what became known as the 'lesser' arts was more suspect. Modern manufactures carried the taint of the factory and warehouse; china, glass, silverware and textiles were a matter for household affairs and were associated with excess, decoration and femininity. As we have seen with the collecting of china, men who chose to collect these things risked being seen as odd, emotionally deficient or even effeminate.\n\nFor some Victorians \u2013 and for later psychoanalysts such as Sigmund Freud \u2013 there was something about collecting that suggested abnormality, sexual impotence and personal failure. Writers like Henry James and George Meredith peopled their novels with collectors who turned to their objects as a substitute for love. In James' _The Portrait of a Lady_ , the relationships of one of the central characters, Gilbert Osmond, are described in terms of collecting: 'We knew that he was fond of originals, of rarities, of the superior, the exquisite. . . he perceived a new attraction in the idea of taking to himself a young lady. . . in his collection of choice objects.' Osmond's emotional inadequacy and cold sexuality are directly linked to his collecting, which is seen as an impulse that disengages him from the more normal passions of those around him. During the collapse of his marriage, he turns to his objects, breaking off from an emotional scene with his wife to study some of his favourite pieces: 'He got up, as he spoke, and walked to the chimney, where he stood a moment bending his eye, as if he had seen them for the first time, on the delicate specimens of rare porcelain with which it was covered. He took up a small cup and held it in his hand. . .' Elsewhere in the same novel, another character, Ned Rosier, pursues love in a similarly frigid way, unable to dissociate his collecting from his emotional life: 'She was admirably finished,' he notes, on finding the woman of his dreams. 'She had had the last touch: she was really a consummate piece. He thought of her in amorous meditation a good deal as he might have thought of a Dresden-china shepherdess.' In _The Egoist_ , Meredith explores similar ideas. The wealthy aristocrat at the heart of the novel, Sir Willoughby Patterne, is a collector. His overwhelming urge to collect people as well as things is presented not in terms of sexual desire but as a more insidious and general obsession with control. His collecting impulses become all-embracing \u2013 and dangerous \u2013 reducing him finally to 'a stone man'. With such precedents all around in popular culture, it seems likely that knowing nods would have been exchanged in Liverpool drawing rooms when a man like Mayer, failing to marry, immersed himself instead in his collection.\n\nWith the knowledge that he would never now be a family man, Mayer threw himself so vigorously into collecting that, despite its size, Pennant House was soon full of objects, papers, letters and ledgers. Having already given his Egyptian Museum to the public, he now accumulated a second collection, one that was even more personal. In the quiet garden rooms that might in other circumstances have been used by his wife, he accumulated packets of autographs from famous people whom he admired. In cabinets and on tables were displayed over 200 engraved gems and rings. There were illuminated manuscripts, wood engravings, ivories, enamels and embroideries. There were calendars of plays performed at Covent Garden and the Theatre Royal, Drury Lane, during the early 1740s and the seventeenth-century manuscript of John Crowne's play _Darius_ , _King of Persia_. The greatest solace was all kinds of papers and objects relating to pottery.\n\nThe house began to feel small under the pressure of the expanding collection, and so Mayer was forced to extend. First a two-storey bay was added, but it was almost immediately filled and, just a few months later, in the autumn of 1873, Mayer added a new wing on each side of the house, as well as a high tower. He set himself the monumental task of writing a history of art in England, drawing on his collection for material, and he collaborated with his nephew Frederick Boyle on two books: _Early Exhibitions of Art in Liverpool: With some Notes for a Memoir of George Stubbs R.A_., which he published in 1876, and _Memoirs of Thomas Dodd, William Upcott and George Stubbs R.A._ , three years later. The great art history was never completed and neither of Mayer's published collaborations was particularly remarkable: both were almost certainly ghosted by Boyle on his uncle's behalf. It was a much less ambitious work, a paper on the 'History of the art of pottery in Liverpool', first contributed to the _Transactions of the Historic Society of Lancashire and Cheshire_ in 1855, and revised by Mayer in 1873 during this period of learned writing, that proved that his lifelong lack of confidence in his own scholarly credentials was misplaced. It became the definitive text on the subject, and is still significant today.\n\nAs the years passed and he retired from his business, finally, at the age of seventy-one, as his friends became too old to travel, too tired to collect or too ill to spend time with him \u2013 and as loneliness proved ever harder to keep at bay \u2013 Mayer must have seemed at times like the archetypal collector, shuffling around rooms piled high with the past, obsessed with the minutiae of history. But he was not one to shut himself away. His collecting had always been a sociable activity, a way to make friends and meet colleagues, and even in old age he displayed no desire to become reclusive. His nature was philanthropic and he spent his last years looking beyond the collection at Pennant House. When the evenings closed in, quiet and dark, he relished the hours reading his papers or writing notes, but he was proud, too, that his days were active and generous and useful. The village of Bebington was thriving, and Mayer was, by and large, responsible for its orderly growth.\n\nWith the construction of the railways and new roads, and the introduction of a ferry boat link with Liverpool, the population of Bebington more than trebled between 1841 and 1871. But whereas, in cities, the rapid mid-century rise in the number of inhabitants frequently resulted in overcrowded slums and desperate poverty, Bebington's growth saw an influx of professional people, rather than unskilled labourers, and, thanks to Mayer, there was a sustained programme of investment. 'Without losing its rustic air, the village has gained advantages such as many a town might envy,' boasted a local paper, _The Standard_ , in 1878. 'A very few years since. . . it had neither gas nor pavements, its younger population went barefooted and women fetched water from two miles distance.' Now, however, the paper noted proudly, 'Bebington may be held up as a model village [and] the honour is chiefly due to Mr Joseph Mayer.'\n\nMayer's involvement with the local people started simply enough, and had its roots in his collection. Arms, armour and things military had always fascinated him. At the age of twelve, he had a brief taste of army life, marching from his home in Newcastle-under-Lyme to Macclesfield as a drummer boy with the 34th Regiment of Foot in the weeks before Waterloo. As an adult, flintlocks, swords, daggers, axes, crossbows, suits of armour, powder flasks and spurs were prominent in his museum and he developed a particular interest in the history of local defence. In 1860, he was appointed Commanding Officer of the Liverpool volunteer borough guard. Four years later, he established a volunteer company of the 1st Cheshire Rifles: he became captain of the 4th Bebington Company, which he supported financially and presented with two challenge cups for shooting. Within six months, over a hundred men had enrolled to be part of Mayer's troop and he took satisfaction from serving as their commander. Mayer enjoyed the expeditions: the would-be soldiers camped together at Hooton, further along the Wirral, telling stories in the smoke of camp fires, testing themselves against the cold and trapping animals for food.\n\nThe 4th Bebington Company was one of many. The Victorian volunteer movement was a phenomenon which began in the 1860s, when fears of war with France were at their height. In 1858, reprisals were expected after a failed assassination attempt on Napoleon III was traced back to exiled French conspirators based in England. In March 1860, France annexed Nice and Savoy. Such territorial ambitions on the part of the French raised the spectre of invasion and were regarded as a threat to national security. A programme of coastal fortification led to the construction of defences such as Fort Nelson, near Portsmouth, and many towns began to establish local volunteer Rifle Corps.\n\nThe idea was an eighteenth-century one: volunteers were called upon between 1797 and 1805, before the Battle of Trafalgar brought to an end, for that period at least, the French naval threat. To a new generation, it proved a popular and sociable way to be seen to do one's duty. According to the report of a Royal Commission in 1862, the Volunteer Force consisted of 162,681 men including engineers, light horse and mounted rifles and, by the 1870s, one in twelve men throughout England had joined up. Across the country, drill halls, training grounds and rifle ranges were quickly established. The attraction, as at Bebington, was not just military protection: musical, sporting and recreational activities were commonly organized for members, local teams travelled to take on rival corps, and men from all walks of life came together for target shooting, cricket and football. Each corps had its own character, depending on the area in which it was based and the type of men it attracted. One of the more well known was the 38th Middlesex (Artists') Rifle Volunteers, founded in 1860 by an art student called Edward Sterling. Consisting largely of painters, sculptors, engravers, musicians, architects and actors, the corps' first commanders were the painters Henry Wyndham Phillips and Frederic Leighton, and the regiment became celebrated in time as it attracted members such as William Morris, John Everett Millais, Edward Burne-Jones and Dante Gabriel Rossetti. The battalion was not finally disbanded until 1945: during the First World War, over 15,000 men fought under its banner, including the poets Wilfred Owen and Edward Thomas, the artist Paul Nash and the sculptor Frank Dobson.\n\nAs Mayer got to know his men, discovering how they lived and what it was they valued, he gradually immersed himself in their welfare. It was a small step from the needs of the volunteers of the 1st Cheshire Rifles to the needs of their families. Convinced that physical activity and fresh air were to everyone's benefit, Mayer founded bowling, cricket, quoits and football clubs, as well as starting a horticultural society and setting aside land for allotments. But he also heard from the men in his company how difficult and uncertain working life could be, and how everyday chores were unnecessarily exhausting and demoralizing. So he raised funds for a village hospital and campaigned for the introduction of gas and water. The obvious next move, in the tradition of the great Victorian patrons, was, of course, to open a school, but there was already a schoolmaster in the village doing an adequate, if modest, job, and, in any case, Mayer believed strongly in the value of informal, self-directed education, the kind of learning that had allowed him to take his place alongside scholars. He chose instead to invest in a public library. 'He who reads lives in a larger world, and has a knowledge and grasp of possibilities far wider than he who is without the art, and even prospers when the latter would succumb,' he informed the village, and to make this larger world of literacy available to as many local people as possible, he rented a house, stocked it with 20,000 volumes on travel, poetry, history, natural sciences and art, as well as a good number of novels, appointed a librarian and a team of volunteers, established evening opening hours for study after work and set out the arrangements for borrowing books.\n\nThe library was an instant success, and an extra room had to be provided almost immediately. Three years later, in 1869, Mayer had the opportunity to buy a farmhouse and barn alongside Pennant House and by January 1870 he had created a public garden, with walks of chestnut trees, a grand new reading room brightly lit with gas burners, and a public hall for art exhibitions and lectures. He was clearly joining the philanthropic trend of his age, aligning himself with wealthy industrialists and merchants across the country who were founding public facilities and open spaces. From the ambitious model villages at Saltaire, outside Bradford, and Bournville, near Birmingham, to smaller gifts like the Derby Arboretum, England's first public park, created for the people of the Derby in 1840 by textile manufacturer Joseph Strutt, philanthropic Victorians were reshaping the urban landscape. Mayer's projects were respectable and predictable outlets for his energies, positioning him at the heart of Victorian society. But his generosity was not simply a product of conformism. Mayer was a genuinely kind man; he was not interested in gestures. He envisaged sturdy and practical facilities for the young and the old, for learning and for entertainment, not an empty monument to his wealth.\n\nAll of this, he achieved. 'All classes meet at the library, farmer's boys in corduroy and heiresses in sable, clergy of the church and little girls of their Sunday School, servants and masters,' noted _The Standard_. Many villagers walked four or five miles or more with their families to use the library and 'each night a crowd assembles around the door as if it were a theatre, waiting with patience' until the readers could be admitted to browse the shelves. Mayer was spoken of fondly and with respect. Far from shutting himself off with his things, he reached out energetically to the community, to the benefit of all in Bebington.\n\nThere was satisfaction in seeing the village thrive around him, and not a little pleasure in being greeted so warmly by local people. As the pains of age grew sharper and his breath came shorter, Mayer delighted in sitting by the counter at the library, helping the staff issue readers' tickets or watching from the terrace of Pennant House as local boys gathered up the chestnuts in the autumn. With his museum apparently in the safe hands of the Liverpool town council, Mayer made arrangements that, on his death, the objects in Pennant House should be sold to finance the public projects in Bebington. He left clear instructions detailing the sale, and he established a trust to manage the resulting funds and look after the necessary day-to-day administration. With one collection safely stowed for posterity, he could afford to use the other for more practical purposes: the sale of the Pennant House objects would, he knew, reach out into the future, imprinting his name on the village he had adopted as home. Everything was settled. He would leave the best of both worlds, the Liverpool Museum and the Bebington facilities, which, he believed, would last. His memorial would be secure. There would be no need of further portraits.\n\n## _Fashion, Fine Dining and Forgeries: Dealing in Society_\n\nMURRAY MARKS\n\n### CHAPTER FIFTEEN\n\n### _Rossetti's Peacock_\n\n_T_ he carriage pulled up quietly outside number 16 Cheyne Walk. Everything about this eighteenth-century Chelsea street was hushed and genteel. A light mist drifted up from the Thames and the bells from the medieval Old Church were subdued. Murray Marks paid the driver and stood for a moment on the wide stone pavement looking across the river. It was 1863, and Cheyne Walk was one of London's most desirable addresses. In 1810, Elizabeth Gaskell had been born at number 93 and in the early decades of the century, Isambard Kingdom Brunel had lived for nearly twenty years at number 98, a divided seventeenth-century townhouse. Admiral William Henry Smith, founder of the Royal Geographic Society and Vice-President of the Royal Society had lived at number 3 in the 1840s and 1850s, while next door to him, at number 4, was the Scottish painter William Dyce. Further along the street, J. M. W. Turner had lived quietly since the mid-1840s, before dying in 1851 at number 119. At 101, James McNeill Whistler had recently arrived. It was a street of celebrity, a place of invention, creativity and achievement, and Murray Marks had come to visit another of its famous inhabitants, the poet and artist Dante Gabriel Rossetti. He relit his cigar, settled his hat on his head after the ride and made his way through the high gate to the tall, flat-faced white house.\n\nMarks became aware of a strange squawking, something like the shriek of a banshee. He stopped and looked around. But the citizens of Cheyne Walk, apparently unflustered, remained in their houses, and for a moment all was quiet again. Then the squawking continued; as Marks pulled the doorbell, it was louder and shriller than ever. It was coming from within the house, he could tell that, but he could not place the sound. He had not heard anything like it before; it was harsh and otherworldly; it did not belong. But when the door was opened to him, Marks saw nothing terrifying or supernatural. Nor did he notice, at first, the wide entrance hall with its stuccoed ceiling and sweeping stairway, because the source of the sound was showing off and Marks was confronted with the dazzling blue of a disgruntled peacock, in full display, flashing green and gold in the morning light from the street, and confronting a kangaroo.\n\nMarks was not as surprised as he might have been. He had heard about the menagerie. In this most elegant of streets, Rossetti kept a zoo: among its inhabitants, described by his brother William, were: 'a Pomeranian puppy named Punch, a grand Irish deerhound named Wolf, a barn-owl named Jessie, another owl named Bobby. . . rabbits, dormice, hedgehogs, two successive wombats, a Canadian marmot or woodchuck, an ordinary marmot, armadillos, kangaroos, wallabies, a deer, a white mouse with her brood, a racoon, squirrels, a mole, peacocks, wood-owls, Virginian owls, Chinese horned owls, a jackdaw, laughing jackasses (Australian kingfishers), undulated grass-parakeets, a talking grey parrot, a raven, chameleons, green lizards, and Japanese salamanders'. It was a haphazard arrangement, and the collection of exotic creatures was not easy to control. The tent constructed in the garden to house the animals was by no means secure; William Rossetti wrote in his memoirs of how 'one or other bird would get drowned' or 'the dormice would fight and kill one another' or, worse still, the racoon 'followed his ordinary practice of burrowing, [and] turned up from under the hearthstone of a neighbour's kitchen, to the serious dismay of the cook, who opined that, if he was not the devil, there was no accounting for what he could possibly be'.\n\nUndaunted, Marks tried to take his unusual surroundings in his stride. The obvious eccentricity was unlike his own steady, respectable home, as though he had stepped over the threshold from the ordinary to the avant-garde. But there were also more familiar things to draw his eye; despite the shuffles and smells of strange animals, Marks admired the high ceilings, tall windows and elegant proportions of Rossetti's panelled rooms, strikingly painted in deep colours. Above all, he could not fail to notice Rossetti's idiosyncratic collecting. The house was full of bric-abrac picked up at junkshops, an uncoordinated assemblage of brass and pewter, oriental rugs, velvets and chintzes; a jumble of musical instruments valued not for how they played, but for how they looked; Sheraton furniture and Spanish cabinets set alongside worthless junk; and banks upon banks of mirrors, reflecting the city light into the rooms.\n\nMarks and Rossetti soon became friends. Rossetti had a collector's instincts. He liked to surround himself with the curious and the rare, and to talk excitably and at length about his finest objects. He could forget himself so absolutely in his curiosity that he was capable of ignoring the dictates of etiquette: upending a porcelain dish at a dinner party to check the maker's marks on the bottom, he promptly upset a salmon all over the tablecloth. But his fascination with collecting and display needed a reliable and respectable foil, and this is what drew him to Murray Marks. Marks was modest and quiet, reticent with strangers, and distinctly unflamboyant. He had a calming influence on the long-haired, ostentatious Rossetti. He was reliable and sympathetic, happy to help create the rooms the artist wanted with the minimum of fuss. Theirs was a marriage of opposites: Rossetti was excitable and unconventional; Marks was studious and proper. They were brought together by the desire to discover and acquire lovely objects, to dress Rossetti's home with the finest furniture and ornaments, to create a Pre-Raphaelite interior that would provide the perfect backdrop for the preening peacock.\n\nIt took a long time for the house at Cheyne Walk to be completed. Rossetti was strong-willed and unpredictable, and could be difficult to do business with. His life was full of distractions and often hectic and fractured. The Pre-Raphaelite brotherhood of painters that he had founded with John Everett Millais and Holman Hunt in the late 1840s was becoming increasingly popular for its aesthetic principles, and increasingly notorious for its extravagant lifestyles. By 1863, when Marks called on the Cheyne Walk house, Rossetti was working on a series of jewel-like watercolours and experimenting with the highly decorated, richly coloured oil paintings of beautiful, faintly exotic women that were to make him famous. His wife and muse Elizabeth had recently died after taking an overdose of laudanum, an opium-based, highly addictive painkiller that was commonly prescribed at the time for ailments ranging from headaches to tuberculosis. He was sharing his home, on and off, with his brother William, his mistress and model Fanny Cornforth, the controversial poet Algernon Swinburne, and the writer George Meredith, whose own wife had recently left him for another of Rossetti's artist friends. He was suffering from ever-lengthening bouts of depression, drank heavily and was addicted to chloral hydrate, a sedative and hypnotic drug, marketed for insomnia, that would eventually wreck his health.\n\nNot surprisingly in the circumstances, Rossetti's collecting could be erratic. Marks might send notice of a striking table, or a pretty writing desk, only to have his note returned if his client were lost in a misanthropic haze of drugs and depression. For long weeks, Rossetti would refuse any guests, before a sudden obsession with a particular piece had him hounding Marks for information. His demands were exacting, and he expected his dealer to be thoroughly diligent on his behalf. He would think nothing of giving Marks apparently impossible commissions: 'get me two thimble-shaped lids for two little round pots I have', he demanded on one occasion, sending Murray to scour the length and breadth of Holland for just the right lids, of the exact shape and size he required.\n\nBut Marks was patient and amenable, and prepared to ride the peaks and troughs of his client's moods. He found Rossetti 'the most amusing and at the same time the most intellectual man I ever met', and he was conscientious in his quests on Rossetti's behalf. His friendship with the artist flourished. Marks commissioned portraits of his wife, her face blooming in Rossetti's characteristic red chalk, her long neck and full lips, her flowing hair, typical of Pre-Raphaelite models. Rossetti borrowed objects from Marks to use in his paintings: a mirror from the drawing-room fireplace, a blue jar. And as he mixed his paints, blending and reblending to capture a particular shade, he sent Marks on errands to the markets of Covent Garden to find iris or tulips of a particular colour or a lemon tree in a china pot. When Rossetti's funds dried up, as they were apt to do, Marks would pay some of his bills or negotiate discounts until the crisis passed. He could be found at auctions, bidding up the price of Rossetti's works so that his friend could be sure of a profit. In particularly hard times, he bought parts of Rossetti's collection, at a good price, and kept them safe. Years later, in 1882, Marks was one of the few mourners who made the journey to Rossetti's bleak funeral on the Kent coast, where he had gone to escape both his paralysing addiction and his creditors.\n\n* * *\n\nDespite the evident closeness between the two men, Marks never forgot that the relationship was primarily founded upon the business of collecting and dealing. It was this attention to commercial opportunity, partnered with a sensitivity to the nuances of fashion and the market \u2013 and hard-nosed ambition \u2013 which made Marks in time, perhaps inevitably, a successful man. He was part of a new generation of art dealers that was changing the collectors' world. Born in 1840, he was a near contemporary of both Joseph Joel Duveen, Charlotte Schreiber's rival, who was born just three years later, and of the prominent and respected art dealer Asher Wertheimer, born in 1844. Like many other London dealers, all three came from Jewish \u00e9migr\u00e9 families: Marks and Duveen had roots in Holland, Wertheimer's father was German. They fell outside the social circles that filled museum posts and gentlemen's clubs. They were born into an established community that stretched across all the major cities of Europe, one that viewed art as a business enterprise, that combined connoisseurship with commercialism, and was open about the fact that collecting was a market activity.\n\nMarks, Duveen and Wertheimer were all socially accomplished, knowledgeable and astute. While they had been raised in business families, their love of art was both genuine and inquisitive. All three combined a life dealing in historic works with befriending and supporting emerging young artists. Duveen happily loaned pieces from his showroom to furnish the backdrops to paintings, providing Millais in particular with a number of valuable tapestries; John Singer Sargent dined weekly with the Wertheimers at their house in Connaught Place and painted twelve portraits of the family which were left to the National Gallery after Wertheimer's death in 1918. As social outsiders, they mixed not with the gentry and royalty of Robinson's acquaintance but more commonly with an avant-garde artistic circle that challenged the establishment. But they were also careful to maintain a precious air of respectability that attracted the wealthiest and most influential clients. By the 1860s and 1870s, when all three dealers were at their peak, they had become some of the most powerful men in the collecting world.\n\nMarks had known connoisseurs and collectors since his earliest childhood, helping out in the shop which his father, Emanuel Marks, had opened around 1850 on London's Oxford Street, on the edge of the antiques heartland of Victorian Soho. He was a precocious child: by the age of ten, he was said to be fond of chattering happily to his father's customers, discussing fine points of art history, comparing the masterpieces of the past and arguing the case for forgotten painters. He developed an interest in Chinese art and pestered his father, unsuccessfully, to allow him to undertake a tour of the Far East. A few years later, he attended one of Europe's leading universities in Frankfurt, studying Spinoza and exploring the art of France, Italy, Austria and Germany as he holidayed with his friend, the philosopher Arthur Schopenhauer. He was unusually well educated for a dealer \u2013 erudite, articulate and cultured. As we have seen, Duveen's training in the docks of Hull was more practical, and Wertheimer learned his trade in his father's showroom. Marks, on the other hand, was an intellectual and a connoisseur; he could stand his own in society and, according to Rossetti, his 'taste amounted to genius'.\n\nAfter returning to London from Frankfurt, Marks settled to business, joining his father in the expanding Oxford Street shop that shared a building with Pickford & Co., then just starting out in the removals trade. In the early 1860s, he was conscientious and dutiful, but to a young man with ambition, Emanuel Marks' approach to trade seemed dull and lifeless. There was no sparkle to it. Marks, meanwhile, enjoyed nights at the theatre, spending hours at stage doors waiting for late-night actresses; talking politics and poetry and music with noisy young men; visiting artist's studios with the fashionable crowd for private views. He was warm and witty. He impressed with his learning, but wore his knowledge lightly. He was stylish and modern and smart. And so, despite his father's disapproval, Marks edged his way into the heart of society, to the dinners and dances and parties that mattered. He scoured his father's client list for the names he needed to know; he pressed for invitations; he was seen at all the right occasions. Before long, he had a bohemian circle of influential friends. As well as becoming part of the Pre-Raphaelite coterie, he was close to other men whose names dotted the fashionable journals and pages of _Punch_ : the satirical artist and dandy George Cruikshank, who had once received a bribe of \u00a3100 from George III 'not to caricature His Majesty in any immoral situation'; the son of Charles Dickens, a lively, and often inebriated, socialite; and _The Times_ journalist, and soon-to-be actor and entertainer, George Grossmith.\n\nWith this somewhat outr\u00e9 crowd, Marks kept late hours and attended all the new shows. But he did not forget himself, nor his profession. He was, even as a carousing young man, decorous and slightly reserved. He continued to work diligently, sensing that he was on the verge of making his business one of the most sought after and desirable in London. So, while he might on occasion be fashionable and frivolous, he was also comfortable assuming the character of scholar and intellectual on evenings spent with John Ruskin, or Charles Hercules Read, who worked with Augustus Franks at the British Museum. He took care of his credentials, continuing to read and study widely, and developing links with dealers abroad. And he discovered that even his more ostentatious friends and clients could be useful in sustaining a scholarly network: both Rossetti and James McNeill Whistler belonged to the Fine Arts Club, connecting Marks to the most prominent and active collectors of his day.\n\nIn a campaign to attract the kind of customers he wanted, Marks sought to shed the stereotype of the grimy backstreet dive that Dickens had featured in _The Old Curiosity Shop_ ; the 'receptacles for old and curious things which seem to crouch in odd corners of this town and to hide their musty treasures from the public eye in jealousy and distrust'. He aimed instead for the bright and accessible, the professional but unstuffy, creating a business which would woo the fashionable without alienating serious collectors and the museums. He began by reinventing the Marks name, and with it the character of the art dealer. By 1862, the Marks entry in the London business directory, which for years had simply read 'curiosity dealer', was puffed and polished, and had become instead 'Importer of antique furniture, S\u00e8vres, Dresden, oriental china and curiosities'. It was a statement of Marks' intent and ambition: the business was no longer just about unidentified 'curiosities' nor was he simply a 'dealer', tainted with overtones of the noisier, dirtier London trades. He was an 'Importer', a man who travelled, discerning and learned and slightly exotic; a man with contacts. The new entry associated him with the best and most desirable of European objects and, by association, the leading European dealers. And it kept him ahead of the competition. About the same time, Joseph Joel Duveen was similarly remarketing his own business, printing new stationery as 'Importer of Antique China, Silver and Works of Art' and looking to move to London from Hull, visiting premises on both Oxford Street and Bond Street. In the end, Duveen had to postpone the London move until 1876, but Marks was no doubt aware that other dealers were looking to expand and modernize, and that to succeed he had to find new ways of doing things.\n\nWhile the description of the business was a good place to start, Marks was eager to go further. He tried talking to his father, making suggestions for ways in which they could improve and update the shop, but Emanuel Marks was suspicious of fashion and would not countenance change. So it was after his father set off to Europe for a long trip in 1864 that Marks took his next step, closing a deal on premises in stylish Sloane Street in south-west London, converting them into a showroom and, with a few selected objects carefully displayed, setting himself up alone, as 'Murray Marks, dealer in works of art'. The showroom at 21 Sloane Street was the epitome of fashionable chic: it was everything Emanuel's old shop was not. Its broad windows glistened in the evening gas lights; its heavy door was freshly painted and latticed in gold. The chattering window shoppers, parading between Chelsea and Knightsbridge, could not help but notice the new premises, while the artists and poets and collectors of Marks' crowd were delighted to have somewhere new to be seen. And Marks, sensing what might draw them in, chose for the focus of his display the intense colour and bold forms, the exotic oriental patterns, of blue-and-white china.\n\nRossetti's raucous bird, boldly displaying in the Cheyne Walk entrance hall, was a living, squawking testament to the popularity of the deep blues and glittering eye of the peacock feather. From curtains to carpets, from wallpapers to writing papers, the peacock motif became ubiquitous in fashionable homes as a symbol of the avant-garde and the daring, a motif for those wanting to throw off the mustiness of the mid-century and stand at the vanguard of a changing taste. It was an icon for the poets, writers, artists and designers clustering under the banners of the Pre-Raphaelites, the Aesthetes and the Art Nouveau that, by the end of the century, would be making claims for new ways of representing the world. It was beautiful and ephemeral, detached from establishment arguments about the importance of utility, and so the perfect symbol for those wanting to celebrate and collect art for its own sake. 'Remember that the most beautiful things in the world are the most useless; peacocks and lilies for example,' urged John Ruskin in _The Stones of Venice_.\n\nAlongside the peacock feathers \u2013 on Whistler's mantelpiece and in Rossetti's parlour, in William Morris's country house at Kelmscott Manor and in Oscar Wilde's university rooms \u2013 stood the blue-and-white china that Marks' new shop was showcasing. In his premises on Sloane Street, he was pioneering a trend and promoting one of the biggest collecting crazes of the Victorian age. The fashion for pots and plates with willowy Japanese and Chinese designs would soon create an apparently insatiable market; the homes of the stylish rich and the aspirational middle classes would soon be full of themed rooms, extravagantly decorated in an oriental style with prints, fans, screens and matting and with the distinctive blue-and-white china taking pride of place.\n\nThe techniques of making a pure white ceramic with cobalt-blue decoration beneath the glaze had been introduced to China from the Middle East as early as the ninth century. By the fourteenth and fifteenth centuries, it was already being exported to Europe: during the Yuan dynasty (1271\u20131368) for example, China was part of the Mongol empire that controlled much of Asia, and blue-and-white travelled swiftly to the Mediterranean on its expanding maritime networks. By the end of the seventeenth century, sophisticated pieces from the K'ang-hsi period (1662\u2013 1722) were becoming available to the wealthiest collectors, and the Dutch East India Company was investing heavily in the profitable trading routes that supplied the growing demand in Europe. Blue-and-white became enormously desirable, a way for rich collectors to display their elegance and wealth. The royalty and nobility of Europe created a huge demand and fine examples became so highly prized that they appear in many seventeenth-and eighteenth-century paintings, either as the backdrop to portraits or extravagant allegories, or as subjects in their own right. As early as 1514, Giovanni Bellini's _Feast of the Gods_ showed a satyr and nymph serving fruit to the gods from blue-and-white Chinese bowls. A century later, blue-and-white was even more widely in evidence. The Dutch painter Jan Treck painted a still life in 1649 with two blue-and-white bowls (probably Ming), while in the 1650s Willam Claesz Heda, another Dutchman, evoked the luxury lifestyle of the seventeenth-century merchant with a still life of lobster, crafted gold goblets and blue-and-white; both works are now in the National Gallery in London.\n\nWith a guaranteed market, many pieces were designed and made specifically for export. Thousands of dinner services were ordered in China directly from Europe: drawings of a family's coat of arms would be sent out to China to be copied on to the porcelain which was then shipped back. Traditional oriental subjects were supplemented by those intended to appeal to European taste: Buddhist gods gave way to Tyrolean dancers and pictures of Adam and Eve; cranes and monkeys were replaced by pheasants and dogs. Taking account of the European fashion for tea-drinking, traditional Chinese patterns were used to decorate Western-style wares \u2013 European tea caddies, cups with handles and saucers, and matching milk jugs. European manufacturers, too, were keen to exploit the fashion, and worked hard to perfect techniques similar to those traditionally used in the Far East. By the middle of the eighteenth century, factories in England and the Netherlands were able to produce their own blue-and-white china. Delftware, from Holland, was particularly popular, and most of the major eighteenth-century collections included both imported Chinese ceramics and Delft. But, although the new sources made blue-and-white less exclusive, they did not dent the fashion for oriental-style china, and the demand fuelled some of the earliest mass-production techniques of the Industrial Revolution. By the nineteenth century, blue-and-white was no longer the preserve of the very rich. The more local supply, and the increasing wealth of the middle classes, meant that even relatively modest homes could boast at least a few pieces for display, while the effect of blue-and-white en masse delighted those with a taste for flamboyant decorative schemes.\n\nCollections of blue-and-white varied enormously, from refined and extremely valuable collections of ancient Chinese pieces to mass-market versions produced to cash in on the trend. Some collectors, like Rossetti and Marks, were knowledgeable and scholarly but there were many others who cared less about the provenance of individual objects than about the lavishness of the display they could create at home. For these collectors, fashion was the driving force. The hierarchy of objects and collections tended, of course, to follow class lines: the wealthier, more aristocratic collectors set their sights on fine-quality Chinese and delftware; the enthusiastic collectors of the middle classes often settled for more ordinary, mass-produced examples.\n\nMarks had fallen for the sinewy decoration and pure glazes of blue-and-white during his childhood studies of Chinese art. And with clients like Rossetti he could share his pleasure in ancient pieces, comparing the subtle developments between periods and dynasties, the evolution of forms and the effects of pigments and oxides. Serious collectors came to rely on his expertise and his skilful handling of the import market. But importing blue-and-white from the Far East remained hazardous, slow and expensive, and with his Dutch connections Marks was also well placed to exploit sources closer to hand. Many of his relatives, under the full family name of Marks van Galen, were based in Amsterdam, running shops and showrooms that drew enthusiastic collectors: Charlotte Schreiber was a regular visitor to what she called 'van Galen's'. As a young man, Marks had spent several useful weeks in his grandfather's Amsterdam shop, and as he grew older he used his contacts in Holland to create an unrivalled supply network. Holland was not only the source for delftware, but it also had large quantities of high-quality, original blue-and-white. This had been imported from China to furnish houses in the seventeenth century but had since become unfashionable. No one wanted it. This Chinese blue-and-white was cheap and plentiful and could be easily shipped back to London. Marks knew where to go, and who to talk to. His contacts kept an eye out at Dutch sales and clearances, buying up blue-and-white on his behalf. He had cousins who could stalk the dealers and search the dusty backroom shelves. And he could speak Dutch, so he could hire coaches to take him out into the countryside and knock on the doors of substantial, likely-looking homes himself.\n\nEventually, the Dutch caught on to the English mania and, inevitably, prices rose. In fact, they rose so quickly and so steeply that it actually became cheaper to import directly from the Far East. But for a few years in the 1860s, beautiful and ancient pieces were still to be found on farmhouse dressers in the flat lands along the Rhine or north into Friesland, or in everyday use on kitchen tables. That he was so well placed to manipulate the craze for blue-and-white, that he could respond so quickly and effectively to the booming English market, proved the making of Murray Marks. His artistic and bohemian friends applauded his success; collectors from across the country demanded his services and he expanded from Sloane Street to the bustle of High Holborn. In 1874, when Pickford's moved out of Oxford Street in search of larger premises, Marks seized the opportunity to expand yet again, and to create a showroom to match his crystallizing ambitions.\n\nMarks asked the architect Richard Norman Shaw, one of the most influential of Victorian architects, to extend the Oxford Street premises and make it magnificent. Shaw had been a friend of Marks' for some years, and was a collector himself. In 1866, both men had been involved in creating Glen Aldred, the Surrey country house of the painter E. W. Cooke. Shaw had designed and built the house and helped choose the furnishings \u2013 by ransacking Marks' showroom for its finest pieces. In later years, Shaw would go on to design grand houses with a classical influence \u2013 Bryanston House in Dorset and Chesters in Northumberland in the 1890s, and London's huge Piccadilly Hotel in the early 1900s \u2013 but in the 1870s he was experimenting with a leaner domestic architecture that became known as the Queen Anne style. At 8 Melbury Road, 118 Campden Hill Road and 31 Melbury Road, all in North Kensington, he created three celebrated 'Artists' Houses', each including a light-filled studio space, for the artists Marcus Stone, George Henry Boughton and Luke Fildes. As estate architect at Bedford Park in London, he designed key elements in a pioneering model of suburban living which soon acquired a reputation for attracting 'a particular person, most likely artistic and bohemian': a survey of 168 early residents found that 40 were artists, 16 architects and 9 actors or musicians.\n\nAll these projects were undertaken between 1875 and the early 1880s. Shaw worked with Marks on designs for the new showroom during November 1875. Once again, Marks was in at the beginning, creating rather than simply following fashion. Having seen what could be done at Cooke's house in Glen Aldred, he was convinced that a new kind of artistic architecture could prove popular. Even while Shaw was still experimenting with styles, Marks saw something in his work that fitted perfectly with his own aspirations for a striking aesthetic display space, and he was brave enough to back a new trend, while at the same time raising the profile of his friend's architecture within an influential artistic circle. A visitor to the shop in the 1870s noted admiringly that Marks had achieved 'the first artistic business elevation, in creamy coloured woodwork, which was erected in London in the style of Queen Anne'.\n\nWithin months, nothing remained of Emanuel Marks' old shop. In its place was a ground-floor showroom thirty feet wide, fifty feet long and over twelve feet tall, a splendid airy space, with two more intimate showrooms upstairs. In defiance of the current fashion for blank plate-glass windows, there was a neo-Georgian bow window, with carved wooden frames, to lure customers off the street with a promise of old-fashioned courtesy and hand-picked treasures, and the huge doors were moulded with Renaissance ornament, the name Marks inscribed in large letters above them. And everything was painted in a single colour, a clean cream \u2013 the window frame, the walls, the showcases and doors, the display furniture. This might seem an obvious trick designed to show off the objects, especially the blue-and-white, to best effect, but this kind of attention to the interior of an art dealer's shop was completely new. Some manufacturers had had an eye on the marketing potential of glamorous display rooms since the end of the eighteenth century: the Wedgwood showrooms in London and Bath were carefully designed to appeal to the fashionable, and often female, shopper, and to show off a range of wares at their best. But the art dealers' premises that most collectors knew were cramped and chaotic in the spirit of the bazaar, the goods heaped up, the pictures double-hung, objects, styles and periods all jumbled, and nothing to be had without diligent rummaging. Marks was setting high new standards. He was trying something daring and innovative. And it caused an overnight sensation, attracting would-be buyers and winning Marks widespread publicity and instant celebrity.\n\nWithin two years, Marks' bright, clean, sophisticated method of display was being adopted by any dealer with ambition. In 1877, he was called in to advise on the decoration of the new Grosvenor Gallery, being set up in Bond Street, and the same year the art dealers Agnew's built a new and refined showroom at 43 Old Bond Street, very much with Marks' model in mind. These new shops saw themselves as private galleries. The attention to display was an indication that they were challenging established institutions like the Royal Academy and, perhaps more importantly, it signalled the respect being given to artists and their work. Where the Royal Academy tended to treat its artists as mere producers contributing to a crowded warehouse of paintings, the owners of these new gallery showrooms were developing courteous and creative friendships with their artists and collectors. The harmonious rooms where each piece was allowed generous space were designed to complement the objects, suggesting that works were valued and that those who spent their lives with these works should be valued too. It was a strategy that proved successful. The Grosvenor Gallery had 1,100 customer-visitors a day when it opened in its purpose-built temple to art, and Marks' equally popular showroom became a place to see and be seen.\n\nMarks' new achievements took him back to the quiet pavements of Cheyne Walk, to Rossetti's drawing room and the magnificent peacock. Because he needed one more thing, and for this he called on his friends. And so, on a late spring afternoon, bright with sunshine, three of Europe's leading artists met together with Marks. They drank wine and smoked. There was the flamboyant and controversial James McNeill Whistler, a neighbour of Rossetti's, who was painting moody landscapes and severe and complex portraits. There was Rossetti, of course, in one of his better moods, and there was the rebellious William Morris, craftsman, designer and Socialist agitator. They were men who, between them, were to change the way people thought about art. But first, they had to design a business card for Marks. And what they came up with was gold and blue and maroon, with a heavily decorated border of the type for which Morris is still remembered, Chinese characters emphasizing Marks' connections across the world, and the inscribed names of all the fields in which he was now an expert: carvings, armour, leather, tapestry, furniture, bronzes, oriental and European ceramics and enamels. It was a card that swaggered with commercial confidence and poise, a statement of personal achievement and a flourishing display of Marks' influential friendships. At the centre, framed by the elaborate border and sitting on a highly polished surface, was a gorgeous blue-and-white ginger jar. Tucked inside the jar, and lying nonchalantly alongside, were the twinkling eyes of two large peacock feathers. Perhaps they were picked up from Rossetti's hallway, or out of his garden. Perhaps they came directly from the bird that had greeted Marks so boisterously on his first visit. With their luxuriance, their bold colours and their delicate fine forms, they were a perfect icon of everything Marks had so far accomplished.\n\n### CHAPTER SIXTEEN\n\n### _A Notorious Squabble_\n\n_C_ ollectors, artists, connoisseurs, other dealers \u2013 and simply the followers of fashion \u2013 flocked to see what Murray Marks was offering. Charlotte Schreiber called by at the showroom to see if there was anything to tempt her. Robinson was forced to take note of this energetic competitor. The shop buzzed with activity, commissions poured in and sales flourished. But it was a difficult time for most dealers. During the 1870s and 1880s, rural decline, a dip in world trade and a stuttering British economy led to a corresponding depression in the art market. The greatest success was to be had by those who, like Marks, were prepared to innovate, finding new ways to market their wares and services. As well as updating and improving their premises, many dealers began to specialize. Etching, oil painting and watercolour all began to find distinct homes in particular outlets, while smaller paintings and more portable works of decorative art continued to gain in popularity: 'The exhibition painting has had its day,' suggested Walter Sickert in a magazine article, adding that artists 'will have to learn to work on a small scale'. Like Marks, many dealers were also forging links abroad, scouring Europe for works, bringing together an international client list of collectors and trying to offset a downturn in one nation's market with a potential boom in another. 'Art in England . . . is fast becoming cosmopolitan,' noted _The Times_ in 1879. 'Our picture dealers and buyers lay under contribution the ateliers, not only of Paris, but of Vienna, Munich, and Dusseldorf, Brussels and Antwerp, the Hague and Amsterdam, modern Florence, Milan and Rome.'\n\nOne of Marks' more original marketing ploys was his habit of holding exclusive dinners, with the showroom elaborately decorated and some of the best wares pressed into use on the table where collectors could handle them. The idea was instantly successful. Marks was a genial and accomplished host, and meals were presented with a wealth of exceptional silverware, china, glass and ornament. No hint of Marks' tradesman origins disrupted the carefully constructed atmosphere of refinement and style. He knew his wine, and served only the finest vintages to complement each course. He enjoyed his food, and treated his guests to complex dishes from around the world. For Marks the gourmet, dining was an experience, an adventure, and a chance to show how far he had come. These were dinners to dazzle even the wealthy; chic occasions to flatter the fashionable: like the visiting card, they presented London society with a well-worked image of Marks and his business. This was astute and effective salesmanship.\n\nEncouraged by the success of his new showroom and with interest in blue-and-white china continuing to grow, Marks conceived his pi\u00e8ce de r\u00e9sistance, bigger, more elaborate and more expensive than he had ever before attempted. For this, he required a collaborator, and he turned to one of his best customers, Sir Henry Thompson. Sir Henry was a fashionable man, a member of the Fine Arts Club and the influential Athenaeum Club, founded in 1824 for leading figures in art, literature and science and their patrons. He was surgeon to Queen Victoria and his impressive patient list included several other monarchs: he successfully removed a bladder stone from the King of Belgium, Leopold I, when two other eminent surgeons had failed. He published respected medical works and had the wide-ranging interests of an educated and sociable gentleman: he was an enthusiastic astronomer, his paintings were exhibited twelve times at the Royal Academy, he wrote two successful novels, and he founded the Cremation Society, advocating cremation as an hygienic, convenient and modern alternative to burial. His celebrated 'Octave' dinners, when eight guests were served eight courses, were the talking point of London society. And when the fashion for collecting in general, and collecting blue-and-white in particular, gripped the city, Sir Henry inevitably wanted to be part of it.\n\nSir Henry first turned his attention to blue-and-white in 1870, and Marks soon persuaded the busy surgeon to allow him to act on his behalf. Combining his expertise with Sir Henry's apparently bottomless funds, he was convinced that they could together create the most impressive and important collection of oriental china that had ever been seen. He was under no illusions as to Sir Henry's long-term commitment to collecting china, and guessed that his enthusiasm for blue-and-white might be fleeting. But he recognized another opportunity to set himself apart from other dealers and consolidate his reputation among the most successful and wealthiest collectors. Bringing together the finest pieces was also an intellectual and aesthetic challenge, and Sir Henry's wealth gave him the chance to create a collection he could never afford for himself. He went to work and by 1878 had brought together a wealth of unique pieces on Sir Henry's behalf, sourced from his contacts in Paris, Holland and the Far East. No other man in London could have done so much so quickly. But this was not just a private arrangement with Sir Henry; it was a fashionable enterprise for them both, something of a publicity stunt. The next step was to show the world what they had achieved.\n\nIt was a dim city evening when the preparations were finished for the magnificent exhibition which was to display Sir Henry's collection to a discerning celebrity public. Marks' showroom glittered. Display plinths had been repainted, door handles polished, and extra gas chandeliers fitted down the length of the shop. Marks had gone back to Norman Shaw, now busy designing for his artist clients, and together they had thought through every detail of how the exhibition should look. A witty and elegant invitation was designed, showing many of Marks' friends, including a stern Whistler with his monocle. The window display gleamed, music was played by a liveried quartet and servants flitted to and fro. The immaculate display of blue-and-white was arranged in profusion in every part of the gallery.\n\nGuests flocked out of the damp mist and soon the showroom was bustling with life. There was chatter and laughter, and animated admiration for the pieces on display. To guard against Sir Henry's fickleness, Marks had made sure there was a permanent record of the collection in an illustrated catalogue. It was in itself something to collect, elaborately bound in leather inlaid with a blue-and-white ceramic plaque and with artwork by Whistler; a polite scrum gathered around the table where it was displayed, as guests elbowed their way to the front to inspect it. But the best was yet to come. The highlight of the exhibition was a dinner, described by one of Marks' friends as 'a very recherch\u00e9 supper served. . . on wonderful Blue and White dishes, which formed extraordinary foils to the rich pastries, glowing lobsters and wonderful jellies'. Even by the extravagant standards of wealthy Victorian society, it was a lavish meal: glittering with candles and dressed with flowers and feathers. Long tables ran the length of the gallery, piled high with intricate pastries, decorated sides of fish, piles of seafood, dripping joints of meat and platters of game, sweetmeats, fruits and elegant deserts \u2013 all carefully chosen to complement the shapes and patterns of Sir Henry's china.\n\nBuilding on such landmark occasions, the Marks business continued to prosper, showing the way for a new type of art dealer. Old Emanuel, though impressed by his son's industry and ambition, was bewildered and dazzled. The hybrid of dealership and gallery which his son was inventing was very different to the dark, cramped premises of the original shop. It was a matter of style and celebrity and glamour as well as connoisseurship and collecting. As the last guests drifted away from the Thompson exhibition opening, clustering on the street to hail passing hansom cabs or hurrying through the mist to their clubs for a nightcap, they had been entertained and charmed. They had eaten and drunk well, had met their friends and had been close to the exquisite poise of a very special collection. It had been a memorable night on the fashionable circuit, with an underlying sense of scholarship and good taste lending a satisfying gravity. Marks had, as always, judged well. In the later decades of the nineteenth century, there were almost 500 dealers in London alone, offering collectors everything from prints and drawings to wood carving and fine furniture. It was a cut-throat business \u2013 many dealers struggled, and many never made it from the backstreet premises and jumbled curiosity shops that had been the hallmark of the trade for years. But for those who, like Marks, were inventive in their marketing, knowledgeable and courageous, there were fortunes and reputations to be made.\n\nWhat most clients prized in their dealers was not just the opportunity to socialize and dine well \u2013 although this was clearly a pleasant enough diversion \u2013 but expertise and unerring good judgement. Both Joseph Joel Duveen and his famous son were renowned for what was called 'the Duveen Eye' \u2013 a collector's instincts, a discriminating sagacity, a superb visual memory and the ability to spot hidden treasures. It was something that all successful dealers had to have, or claim to have, but there was no training for it. The only way to educate 'the eye' was through experience. It was, wrote the art critic Bernard Berenson at the end of the century, essential to acquire and understand visual language so that it became as instinctive as spoken language: 'Many see pictures without knowing what to look at. . . We must look and look till we live the painting and for a fleeting moment become identified with it.'\n\nIt was, of course, a quality collectors also wanted for themselves. But even those who boasted an educated eye of their own were often still keen to employ dealers on their behalf. It was convenient to have someone to undertake the hard, often grubby, work \u2013 the travelling and research, the sourcing and bartering. The feeling remained that dealers were at the bottom of the collecting heap, there to do the tasks that collectors preferred not to tackle. In 1870, Charlotte Schreiber recorded a telling moment in her journal, betraying the entrenched attitudes of the elite collector. As she explored the Paris showrooms, she came upon 'a painful scene'. At 59 Rue Bourbon, she called upon Monsieur Bock, who told her his 'sad story': 'to the effect that he was a Russian of private means, had lived many years in England, and, in the course of his travels had made a fair collection; that he lost everything in the failure of a Bank and was obliged to sell it all; and then, having a wife and seven children, turned dealer. He told us of his struggles, but said he never lost courage. . . He had known Mayer, Franks, Panizzi, in fact all our finest collectors. If true, his tale was a very melancholy one,' marvelled Charlotte. Her assumptions are clear: becoming a dealer was the last resort for a man of means and a collector, a source of some shame, requiring the utmost fortitude, a far cry from the honourable society of men like Joseph Mayer and Augustus Franks.\n\nMarks was no doubt aware that he could only mix with the likes of Sir Henry Thompson \u2013 who was at the aristocratic apex of the collecting hierarchy \u2013 as a dealer, an odd hybrid of teacher, designer, banker and servant. But he was fortunate not to have to rely entirely on such high-profile society clients; there were other collectors less judgemental than Charlotte Schreiber. Rossetti and Whistler were both passionate about china, and genuinely fond of the man who supplied them. And alongside the extravagance of this artistic circle, there were also many more secluded collectors up and down the country who dealt with Marks on equal terms. In particular, in the increasingly international world of collecting, there was one set of wealthy clients conveniently detached from the English obsession with class: Marks was beginning to make himself indispensable to a number of Americans.\n\nDuring the second half of the nineteenth century, wealthy travellers from the United States could regularly be seen around the great sites of European history \u2013 sitting in gondolas alongside the Doge's Palace in Venice, loitering in Florentine piazzas, climbing up to the Acropolis, ambling by the Seine and, of course, making their way around the National Gallery and through the museums and parks and squares of London. In a world of changing economic and social power, they were the new Grand Tourists, revisiting and reinventing the routes that had been trodden by eighteenth-century English aristocrats. Crossing the Atlantic was no longer the barrier it had been (although the voyage still took a good fifty days) and first-hand experience of ancient European culture was considered an indispensable way of improving one's education, cultural expertise and social standing (for those who could afford it). It was what Henry James called 'a mild adventure'; for those who preferred to stay at home, writers like James sent back long series of travel articles about Europe which were eagerly bought up by magazines. To educated Americans, Europe appeared more civilized and sophisticated \u2013 though more stultified \u2013 than its transatlantic neighbour, not least because of its long legacy of cultural traditions and artefacts. Among the American middle classes, there was a fashionable enthusiasm for old Europe, and for the objects that represented it. The Americans were becoming collectors.\n\nThe influx of Americans into English drawing rooms caused quite a stir. They tended to disrupt customary ways of doing things, and there was a widespread perception that they were modern and unconventional, less deferential to the rigorously enforced hierarchies of class and more relaxed in their standards of behaviour than the strict English Victorians. The contribution these outsiders made to society was greeted with considerable ambivalence \u2013 and the shrewd awareness that Americans had money to spend. The usual European visit lasted for months, and sometimes years, during which antiques and souvenirs were often required as mementos of place and time. In addition, a growing contingent of expatriate Americans were making their homes in London and other major cities. All in all, the new American collectors were a growing market, and for dealers like Marks it was important to claim a part of it. An outsider himself, he had no difficulty in accepting and working with Americans. His circle of friends included not only the witty and lively Whistler, born in Massachusetts in 1834, but also George Lucas, an American dealer who lived in Paris, and a number of prominent American collectors. The most significant of these was J. Pierpont Morgan, a Connecticut-born financier and philanthropist who was one of the world's richest men and one of its most influential collectors.\n\nPierpont Morgan's taste was eclectic. He became particularly known for the collection of books, manuscripts and seals housed in his private New York library, and for his spectacular collection of gemstones. He also appreciated painting and sculpture, as well as the decorative arts, and he had a prestigious collection of wrist and pocket watches. But his business concerns, in everything from steel and railways to the army, required most of his attention and he often preferred to rely on trusted dealers to put together his collections. His first jewellery collection was largely assembled by Tiffany & Co., and in particular by its chief gemologist George Frederick Kunz, while the New York editor, clergyman and collector William Hayes Ward had brought together the core of his collection of ancient engraved seals. To satisfy his desire for Renaissance treasures, Morgan was drawn to Europe and especially to England, which was the acknowledged centre for the market. In his house on London's Princes Gate, he consulted renowned experts like Robinson, who knew where the finest pieces were located and how to reach them. Before long, he had turned to Murray Marks, relying on him to undertake much of the research on his behalf and to navigate the London trade to arrange the best deals.\n\nWorking for Pierpont Morgan was high-profile and prestigious. It could do no harm to Marks' business to be seen dealing for the world's richest and most influential men. Better still, this kind of prominent collecting was often attended by lectures, articles, catalogues and books that acted as further promotional tools. When Marks helped construct Morgan's collection of Renaissance bronze sculptures and figurines, the result was immortalized in a three-volume publication by the German art historian Wilhelm von Bode, who could not help but mention Marks' contribution: 'I am especially indebted to Mr Murray Marks. . .', he wrote, 'through whose hands so many beautiful specimens have passed.' When Marks undertook the research and investigation that enabled Pierpont Morgan to acquire a famous twelfth-century triptych from the Abbey of Stavelot in Belgium, he was rewarded with an evening at the Society of Antiquaries where the display of the spectacular gold and enamel medieval treasure caused a sensation. His shrewd scholarship and timely involvement were again applauded: 'We have here before us this evening a monument of medieval art workmanship of a kind and importance that is seldom found on the open market. . .' enthused the fellows, before noting that it was dealers like Marks who 'help to make London a market or exchange for works of art and the wealthy buyer is forced to come here to increase his collections'.\n\nMarks clearly had a taste for the sensational and the extravagant. Nowhere was this more evident than in his work to decorate one of Victorian London's most celebrated and controversial interiors, a statement of avant-garde Aesthetic taste which became known as the Peacock Room. Once the prosaic dining room of a house at 49 Princes Gate in Kensington, the Peacock Room belonged to Frederick Richards Leyland, a Liverpool shipping magnate and art collector. In 1867, Leyland took on the tenancy of the gracious timber-framed Speke Hall, on the north bank of the Mersey, but there was little that could be tastefully done to improve an Elizabethan mansion, and so he also bought less distinctive modern London townhouses, first at Queen Anne's Gate and then at Princes Gate \u2013 just a stone's throw from the South Kensington Museum \u2013 with the intention of creating a visually impressive status symbol at the heart of the capital. Leyland was a naturally conservative man and, when he first began collecting art for his homes, he bought traditional landscapes and watercolours. But then he commissioned pieces from the young Rossetti and began to appreciate the work of new, emerging, more innovative artists. He found that he liked being regarded as a trendsetter and, when it came to creating a new interior in the mid-1870s, he turned to two other members of London's thriving modern artistic scene \u2013 James McNeill Whistler and Murray Marks.\n\nWhistler's contribution to the Peacock Room became famous, in no small part because a bitter row broke out between the artist and Leyland before it was even completed. The dining room had been designed by Thomas Jeckyll to showcase Leyland's collection of blue-and-white, and Whistler's initial commission was for a simple decorative piece. Leyland had recently acquired Whistler's painting _La Princesse du pays de la porcelaine_ , but when Jeckyll hung it over the mantelpiece in the new dining room he found that some of the red tones in the antique Spanish leather wall hangings on either side clashed with the colours in the painting. So he asked Whistler to add a few decorative touches to the room to create a more balanced effect, to retouch the leather with traces of yellow and decorate the wainscoting and cornice with a simple wave pattern. And that is how Whistler started out. But he soon began to turn the dining room into a personal evocation of his aesthetic principles, a 'Harmony in Blue and Gold'.\n\nWhile Leyland was in Liverpool, launching a shipping line, Whistler moved in, living on site for months so that he could commit himself entirely to the project. He undertook increasingly ambitious alterations, covering the ceiling with Dutch metal, an imitation gold leaf, over which he painted a vibrant series of lush peacock feathers. He gilded every inch of Jeckyll's walnut shelving and decorated the shutters with four magnificent golden peacocks. He wrote confidently to Leyland, delighted with the changes, explaining that the room was now 'really alive with beauty \u2013 brilliant and gorgeous while at the same time delicate and refined to the last degree. . . there is no room in London like it'. He showed off his work to visitors and called in the press to admire what he had done. But when Leyland returned he was horrified, appalled by the liberties the artist was taking with his home and money. He refused to pay the 2,000 guineas which Whistler was demanding and, on Rossetti's advice, offered half the amount in settlement. But Whistler was a notoriously difficult and passionate man. He took the reduced payment as a personal slight, exacerbated when Leyland wrote out a cheque in pounds instead of guineas. A pound was worth twenty shillings, while a guinea was worth twenty-one, but worse still, while the pound was the currency of trade, the guinea was the currency of gentlemen and artists. Whistler was furious. In revenge, he painted over Leyland's antique leather with blue paint, creating a mural of two enormous gold peacocks, dominating the dining room, strutting and brash. The telling details were highlighted in silver: scattered at the feet of the squabbling birds were the silver coins Leyland refused to pay; the silver feathers on one of the peacock's throats recalled Leyland's taste for ruffled shirts while the silver crest feather on the other resembled Whistler's distinctive lock of white hair. He called the mural _Art and Money; or, The Story of the Room_ and when it was finished, in March 1877, Whistler quit and never set foot in the dining room again.\n\nBehind the bluster and ranting and preening, Marks' contribution to Leyland's distinctive interiors has often been overlooked, but he was an integral part of the process of transforming the Peacock Room, and the rest of the house. He worked closely with Leyland's architect Thomas Jeckyll long before Whistler became involved, putting together the core of Leyland's distinctive collection and integrating it into a decorative scheme on his client's behalf. Marks provided a range of quality blue-and-white pieces that was displayed on the dining room's narrow carved shelves, rising in thin columns up the walls of the room. He was aiming to capture the spirit of the Renaissance, of the _Porzellankammer_ or porcelain rooms which were statements of wealth at the heart of many seventeenth-century collections. But his contribution did not end with the fine china. As well as the blue-and-white, Marks supplied rugs, carvings and tapestries. He provided Botticelli drawings, paintings by his Pre-Raphaelite friends and heavily carved Italian chests. He found the highly coloured stamped leather wall-hangings that Whistler was to paint over in fury, and he set a striking female figure at the foot of Leyland's stairs, taken from the prow of a Renaissance galley.\n\nAfter Whistler's extravagant intervention, much of Marks' work still remained. Despite the controversy, Leyland kept the Peacock Room as Whistler had left it, and it still displayed the blue-and-white, even if the impact was muted in the new design scheme. As the room became famous, the subject of gossip and scandal, so visitors came to see what Whistler had done, and in turn were able to admire the beautiful pieces that Marks had assembled. In the other rooms of the house, Marks' original contribution remained untouched and the rest of the collection he had supplied to Leyland continued to provide a model for a fashionable townhouse. But Leyland's collecting was a fashion statement: he was interested in making a splash, not in making any kind of scholarly, long-term commitment. Twelve years after Whistler completed the Peacock Room, it was removed from Leyland's house in its entirety and exhibited in a London art gallery; the collection of blue-and-white was dispersed.\n\nMarks was accustomed to such dispersal. As he had suspected, Sir Henry Thompson's enthusiasm for china also proved shortlived, and in 1880, just two years after the exhibition opening, his pieces of blue-and-white were put into auction at Christie's (where Marks was well placed to buy back some of the most important lots) and he turned his attention to other fashionable pastimes. Marks was more aware than most that there was no single type of collector. Alongside those who entered into an intense, often obsessive, lifelong relationship with their collections, or collectors who combined fashion with scholarship, were men like Leyland and Thompson whose interest was fleeting and who wanted to buy fine things as status symbols rather than for intellectual pleasure.\n\nMarks was unusual among the crowds of London dealers in advising on how to use collections to make a show, to define and create 'lifestyles' in the way we now expect from interior designers. His willingness to embrace this most ephemeral of collecting habits again set him apart. It was a shrewd way of doing business. Joseph Joel Duveen followed Marks' example and persuaded Arthur Wilson, a Hull shipowner, to let him redesign his house in the early 1880s, noting that 'the decoration and furnishing of Mr Wilson's house was the finest advertisement I could have had, and his rich friends almost fell over each other to get beautiful objects, too'. In the early years of the next century, Duveen's son Joseph went a step further, immersing his collectors in Old World luxury and frequently advising his wealthiest clients, especially Americans, on ways to make their homes impressive: 'This decoration always included enormous sales of precious things,' noted Duveen's nephew James. In the late 1890s, in an echo of the Peacock Room, Joseph Duveen oversaw the decoration of the London house of J. P. Morgan, son of Pierpont Morgan, a few doors down from Leyland at 14 Princes Gate, creating a special room for a series of eighteenth-century painted panels by Jean-Honor\u00e9 Fragonard. In 1915, after Morgan's death, he arranged for the sale of the 'Fragonard Room' to the industrialist Henry Clay Frick, reinstalling it at Frick's mansion in New York. At Whitemarsh Hall in Pennsylvania, he helped the banker Edward Stotesbury decorate his huge mansion with oriental rugs and French sculpture, while the American socialite Eleanor Elkins Widener Rice commissioned Duveen to decorate four houses, paying $2 million for her New York dining room in 1925.\n\nFor many years to come, the public face of such ostentatious international collecting owed much to the quiet industry of Marks' research and his early experiments with fashionable collaborations. In an age of conspicuous wealth, when the richest people in the world wanted to create glittering, gilded mansions to flaunt their success, collections were no longer confined to studies and parlour cabinets. They became an essential element of display; rooms, galleries and even entire houses were built to show them off. The luxurious interiors of seventeenth-and eighteenth-century European royalty and nobility were reinvented for a new generation of high-society aristocrats. And as Marks demonstrated, these new collecting showcases owed as much to the dealers who made them possible as to the owners who financed them.\n\n### CHAPTER SEVENTEEN\n\n### _The Fake Flora_\n\n_T_ he wax bust of Flora, Roman goddess of flowers and of Spring, was delicate and elegant. The close-coiled tresses of Flora's hair were intertwined with roses, her smiling face was smooth and noble, the slip of material draped over her shoulders and breasts hardly imagined by the sculptor. It was a classic piece, a masterpiece, and it seemed to bear all the hallmarks of Leonardo da Vinci's distinctive style: the same alluring half-smile as the famous Mona Lisa, the expressive sculptural lines, the modest female form. Given that only a handful of Leonardo's works survived his experimentation with different materials and techniques, and the procrastination of a perfectionist, Flora was not just beautiful \u2013 she was also both important and rare.\n\nAs a new century dawned, Murray Marks was seduced by Flora. It was 1909, and he was sixty-nine years old. In many ways, he had become a figure of the art establishment. He had worked closely with the giants of late-Victorian collecting from Pierpont Morgan to Wilhelm von Bode. Since the late 1870s, he had been in partnership with Durlacher Brothers, an old and established London dealer's, and in 1885 the new business moved into Bond Street, at the heart of the antique trade. His townhouse was replete with his collections: the dining room alone featured a helmeted head and a fine display of blue-and-white, glass showcases exhibiting jewellery, enamels and ceramics, and a French Renaissance cabinet and table. Another house, a seaside retreat at 75 Marine Parade, Brighton, was furnished with a Regency collection. He worked closely with several museums, especially South Kensington, attending auctions on the museum's behalf, and supplying eye-catching objects such as a marble and alabaster rood screen from the Cathedral of St John at Bois-le-Duc (or 's-Hertogenbosch), south of Amsterdam. This seventeenth-century masterpiece with statues of the saints, pillars and angels, delicate carvings, arches, balustrades and massive ornamental candlesticks, was a huge architectural gem 36 feet high and 32 feet long, and became a mainstay of the new Cast Courts (or Architectural Courts) when they were built at the museum in the early 1870s. To acquire _Flora_ would be a crowning glory, an impressive finale to a great career.\n\nMarks was offered the _Flora_ bust by another dealer, and quickly agreed terms. She promised to be one of his greatest finds, and as usual he made the most of her. He may not have been able to arrange a splendid dinner to show her off as he had his displays of china, but he ensured she was presented in the New Bond Street showroom he now shared with the Durlachers. Raised on a high plinth, smiling provocatively at the customers, _Flora_ charmed all who visited and news of her spread rapidly across Europe and the world, as the discovery of this new and lovely Leonardo da Vinci masterpiece became known. The art journal _Burlington Magazine_ ran a feature extolling the virtues of the bust, and Marks' friends and customers travelled long distances to see the _Flora_ he spoke of so enthusiastically.\n\nBy the end of the year, the attention had attracted a number of high-profile buyers, and Marks agreed to sell _Flora_ to Wilhelm von Bode for the Kaiser-Friedrich Museum in Berlin for around \u00a38,000. Bode was opinionated, politically astute and ambitious, in many ways a German counterpart of J. C. Robinson. Many of his contemporaries found him arrogant and egotistical with 'a tendency to lay down the law with more or less pontifical assurance' but he was devoted to his collecting and was one of the first museum directors to openly court private collectors in the hope of acquiring their objects. He rejoiced in making prestigious purchases. He was also Marks' friend and had acquired a variety of pieces from him over the years; in addition, the two men had corresponded energetically as well as collaborating on scholarly publications. They knew each other's tastes and respected each other's judgement. It was Bode who had authoritatively declared the bust to be fifteenth century, and unquestionably the work of da Vinci, when he had first seen it in Marks' showroom. So it seemed fitting that _Flora_ should go to Berlin to begin a prominent new life. Only the British press objected, bemoaning the loss of such a treasure: 'Our own museum authorities might have bought it,' complained _The Times_ , 'but nothing was done', adding that the inability of museums such as South Kensington to recognize the importance of _Flora_ 'is humiliating to our national connoisseurship'.\n\nThe bust had only been in Berlin a few weeks, however, when the press picked up on a more titillating element of the story: a letter to _The Times_ from Charles Cooksey, a Southampton antiquary and auctioneer, claimed that the bust had nothing to do with da Vinci, and had never seen Italy at all, let alone the Renaissance. It was the work, Cooksey asserted, of Richard Cockle Lucas, a Victorian sculptor, who had been copying from a picture of a scantily clad woman draped in flowers. The picture had been shown to Lucas by an art dealer called Buchanan, and might well have had links to da Vinci, but the bust, Cooksey maintained, was nothing more than a second-rate copy that had spent almost forty years rotting in Lucas' garden. When it had turned up in a Southampton junkshop a few years after Lucas' death in 1883, Cooksey had been given the chance to buy it for less than a sovereign, but did not think it worth the investment: he knew its mundane history and, as he pointed out, 'it was in bad condition owing to the long exposure'.\n\nSoon the story, which seemed to be confirmed by Lucas' son, was in all the newspapers. Popular English dailies like the _Daily Mail_ carried opinions on what was dubbed 'the _Flora_ affair', as did the more serious _Times_. In Germany, too, the press published pictures, claims and counter-claims, and the American papers kept an amused eye on proceedings, noting that 'the man in the street in both countries has become interested. . . if he is a German he is very sure that the bust is a genuine Da Vinci, and if he is English he is quite certain that Lucas made it'. In an increasingly tense Europe, such allegiances mattered. For several years, there had been talk of the possibility of war between Britain and Germany, while German naval expansion during 1908 and 1909 had sounded alarm bells in Britain and prompted a nervy arms race: 'we should not complain of Germany's right to build as many vessels as she pleased, she must not take it amiss if we built the number of ships which we thought necessary for our own protection,' explained Sir Edward Grey, British Foreign Secretary, anxiously. With national pride at stake, Kaiser Wilhelm II, son of Princess Vicky, visited the Berlin Museum, examined the bust for over half an hour, and proclaimed definitively that _Flora_ was indeed by da Vinci. He assumed that the weight of his opinion would put any controversy to rest, and that he had saved German blushes. But, as the _New York Times_ pointed out, there was considerable 'British joy over the discovery that the Germans have been fooled' and, despite the Kaiser's intervention, impassioned spats continued between London and Berlin, capturing the popular imagination more thoroughly than any learned art history discussion.\n\nAs we have seen, Marks enjoyed publicity and even a little controversy, but the thought that he had publicly passed on a forgery to one of the world's leading museums horrified him. Collectors trusted him \u2013 and this trust was the foundation of profitable trading. As soon as the questions over provenance arose, he sent a banker's draft for the complete purchase price to Bode, begging him to send _Flora_ back to London, hoping to clear his conscience and forget the entire matter. But the banker's draft remained uncashed. Unfortunately for Marks, Bode was not ready to accept defeat so easily. He had identified the bust as being by da Vinci, and he meant to stick by his opinion. He was not in the least worried by rumours in the press, and he suggested that _Flora_ should be inspected scientifically, to prove to the world that he was right in his assessment.\n\nMarks could do little but wait. But the reports, when they came, were not good. The chemists who examined the composition of the wax were far from convinced that it was fifteenth century. Worse still, it became clear that tucked inside _Flora_ was a grey canvas of English origin (some newspaper reports claimed it was part of a Victorian bed quilt), while scraps of a modern newspaper were jammed into the pedestal. The evidence looked damning. Marks wanted nothing more than to be out of the limelight. He pleaded with Bode again to accept a refund and let the matter rest, and he dreamed of letting _Flora_ slip quietly out of the international news.\n\nBut Bode himself was part of the problem. He had set himself up as an omniscient expert, 'a sort of Jove in art matters', and it was the appeal of deflating him that encouraged many of the attacks: 'He had exercised his authority in the world of art opinion in the most ruthless manner, bowling the little fellows over with a mere wave of the hand when they got in his way,' explained the _New York Times_ , pointing out that it was therefore inevitable that people would welcome the chance to exact revenge, enjoying 'a keen delight over the fact that Dr Wilhelm Bode, curator of the museum, is the particular individual who has been victimized'. What's more, Bode resolutely refused to change his opinion, or even to consider changing it: ignoring the apparent evidence from the investigation, he published a statement in _Die Woche_ , maintaining his position that 'only Leonardo' could have been responsible for _Flora_ and asserting that the modern material was the result of restoration work undertaken by Lucas.\n\nThe outcome was the worst possible for Marks. It was a deadlock, with the dealer stuck helplessly in the middle of the storm. Evidence and counter-evidence, rumour and counter-rumour, swirled around the art world and the whole ' _Flora_ affair' quickly became muddy and unsure. The debate was to drag on for years. Some art historians claimed that the bust was a product of Leonardo's workshop, though not made by the artist himself; others were of the view that the sculptor was one of Leonardo's contemporaries based elsewhere in Europe, perhaps in France. Well into the twentieth century, the belief that _Flora_ was the work of Richard Cockle Lucas remained widespread. But Bode, too, had his supporters, who maintained that he was correct in his original assessment. As late as 1939, the art historian Kenneth Clark again raised the possibility that _Flora_ might indeed have been a genuine work by Leonardo da Vinci: 'Nothing in Lucas's work suggests that he was capable of the noble movement of the _Flora_ , and the evidence advanced of his authorship only proved that he had subjected the bust to a severe restoration,' Clark contended. 'Bode was right in seeing this piece as a clever indication of Leonardo's later sculpture. . . [she is] another of those mutilated documents through which, alas, so much of Leonardo's art must be reconstructed.' In the mid-1980s, however, further chemical tests on the bust seemed conclusive. They revealed the presence of synthetic stearin, added to wax to help it harden, a substance that was not produced before the nineteenth century. It was finally concluded that _Flora_ was a fake.\n\nFor Marks' contemporaries, without the benefit of any modern dating methods, the matter remained one of bitter dispute. Although Bode was the target of most of the criticism, Marks could not help but feel that his reputation had been damaged. With the limited techniques available, it was impossible to be certain what the truth behind the bust might be, but it was disappointing that the prospect of discovering a lost da Vinci masterpiece had embroiled him in such negative debate, perhaps undoing years of careful professional work.\n\nThe Victorian collecting boom and the irrepressible fashion for exotic objects from overseas, such as blue-and-white china, was a godsend for the unscrupulous, the profiteering and the criminal. Forgers flourished. While the _Flora_ affair took place at the beginning of the twentieth century, it was over the previous fifty years or so \u2013 at the highpoint of Victorian collecting \u2013 that the threat from forgeries had really started to affect the market. For every knowledgeable collector, there were plenty of beginners and amateurs who could easily be seduced by a good story and some convincing brushstrokes, and there were many dealers who were more than happy to pass on dubious stock at a profit \u2013 but there were few cases as high-profile or controversial as _Flora_. Mostly it was a matter of an altered signature, a touch-up of paint, or an attempt to give modern objects the distinctive patina of age. Usually it was an inconspicuous sale in a backstreet dealer's or at a country auctioneer's, and in many cases the fact that the piece was not genuine did not emerge until the collection was sold on or broken up \u2013 if then. Without modern ultraviolet or X-rays, carbon-dating or chemical spectroscopy to test pigments, collectors had to trust to their eyes and their instincts. Many died unaware that some of their objects were products of a dubious trade.\n\nThe lucrative rewards of an increasingly international art market were too great a temptation for many forgers. Political uncertainty in Europe throughout the nineteenth century made it easier for unprincipled dealers to concoct convincing tales about a ransacked castle or an impoverished nobleman to explain the sudden appearance of a piece on the open market. And the apparently limitless demand for works from the Italian Renaissance in particular meant that, no matter how many objects appeared, there was always a ready buyer. Many forgers concentrated on perfecting the most profitable styles, such as those of fifteenth-century Florentine sculptors, so that they could produce a stream of credible copies. Under a carefully organized and structured system, the best forgers were contracted to international dealers, supplying them with imitation masterpieces in return for a comfortable salary. These artists were often highly skilled and the objects they created were beautiful in their own right, even if false. Some forgers even became briefly famous for the quality of their work: Giovanni Bastianini was an accomplished Italian sculptor who forged Renaissance busts and figures that thrilled nineteenth-century art historians. Several were acquired by the South Kensington Museum, and even when their actual provenance was revealed they were considered to be so fine that they were kept on display.\n\nThe growing taste for objects from beyond Europe also created new opportunities for deception. Oriental ceramics and Islamic art, for example, were relatively unstudied compared to the more familiar European genres, leaving the door open for the unscrupulous to dupe the ignorant. Victorian scholars struggled to keep pace with the fashion for foreign pottery, tiles, textiles, architectural and archaeological objects, and it became relatively easy to pass off modern copies as originals. A macabre fascination with quartz crystal skulls in the second half of the nineteenth century led to the sudden appearance of examples in both private and public collections, apparently from ancient sites in Mexico. Some were little over an inch high; a few were larger. Many dealers boasted a skull for sale, two were exhibited at the Exposition Universelle in Paris in 1867, and the life-size example in the British Museum was acquired at the end of the nineteenth century from the reputable New York jewellers Tiffany and Co. But no scientific archaeological excavations had yet been carried out in Mexico, and knowledge of pre-Columbian objects was scarce. This was a perfect environment for forgers. Scientists at the British Museum found that their skull had traces of tool marks which showed that it had been extensively worked with rotary cutting wheels, unknown in Mexico before the arrival of the Spanish in 1519, and analysis of the quartz was damning: it had come from a nineteenth-century mine in Brazil or Madagascar, far beyond ancient Mexican trade links.\n\nForgeries were so commonplace that everyone was deceived at some point, no matter how careful and scholarly they might have been, or might have liked to think themselves. Charlotte Schreiber became an expert on forgeries in order to be able to identify attempts to cheat her. Joseph Mayer's collection displayed a number of objects that turned out to be forged, from antique ivories and Babylonian onyx to French miniatures of Napoleon and Josephine. On one occasion, Mayer bought a large number of medieval pilgrim badges from a London dealer only to find that they had been manufactured by two Thames scavengers known as Billy and Charley, who had shrewdly created 11,000 pieces and sold them on with the tale that they were salvaged from a wrecked ship.\n\nJohn Charles Robinson used the issue of forgeries to emphasize the distance between himself and Henry Cole, accusing Cole of being too easily duped into spurious deals which showed him up as an amateur. In 1864, Robinson was delighted to pronounce that Cole 'fell into every hotbed of falsification and fraud' while on a buying trip to Germany, handing over \u00a3260 to a dealer in Hanover for textiles of the most commonplace kind: 'they ought not to have imposed on anyone possessing even the most rudimentary acquaintance with art,' snorted Robinson dismissively. A few years later, Robinson high-handedly ordered the return of some late-fifteenth-century French playing cards that Cole had bought in Paris and which Robinson believed to be forgeries, going on to discuss in a twenty-three-page letter to the board and colleagues at the British Museum why his own assessment was indubitably more accurate than the Director of South Kensington's.\n\nAfter Cole's retirement, as Robinson made his case to be part of the new museum hierarchy, he was keen to emphasize his record on spotting fakes and forgeries. He felt he had strong justification, particularly when the press weighed in on his side, alleging that the loss of Robinson had opened the door to incompetent and costly mistakes. 'The authorities have been in the habit of buying spurious or counterfeit articles as originals, and. . . they have given a price considerably above what might be deemed the market value,' noted the _Daily Telegraph_ in 1885, taking it for granted that the market was more or less flooded with forgeries. There was a popular and lingering perception that the staff at South Kensington had been all too easily fooled, and Robinson was perfectly happy to encourage the idea that, without him, the museum was floundering. When he was invited to present the prizes at the Birmingham School of Art in 1888, he used the occasion to lambast the 'dull and comparatively stagnant regime' of 'mere administrators and inexperienced, irresponsible amateurs' who had attempted to run South Kensington in his absence, and pointed out their record of 'astonishing blunders' that, he claimed, had become 'notorious throughout Europe'.\n\nBut matters were not quite as clear-cut as Robinson tried to make them appear. Like Murray Marks, he was involved in a complicated and frequently obscure trade. The market for forgeries was so rampant and sophisticated that it was almost impossible for collectors to avoid being taken in on occasion. Most famous of Robinson's errors was a large ornamental dish, which he believed was made by the sixteenth-century Parisian ceramicist Bernard Palissy. Palissy's works were enormously fashionable at the time. Consequently, there were plenty of craftsmen making a living by churning out pieces in the same style, and both Palissy originals and more modern copies had been shown at all the international exhibitions, including London's Great Exhibition. In 1887, on one of his trips to France, Robinson was delighted to be offered the chance to buy a genuine Palissy platter; on his return he sold it to the museum for \u00a350. It was just the kind of prestigious, high-quality piece that he was known for supplying. It appeared to be another of his triumphs.\n\nThe platter looked as though it had been broken at some point and reassembled with strips of canvas cemented to the china, crossing each other on the reverse. At the South Kensington Museum, after a couple of months on display, staff noticed that the damp in the cases had made the canvas sag, and they took the dish to the workshops to be cemented more firmly. It was a job that had to be done delicately, and it took some time, but, as they reached the point where the canvas strips crossed, it became clear that a maker's mark was inscribed on the plate, hidden by the cloth. As the restorer eased away the last of the canvas, the letters PULL were revealed, the mark of Georges Pull, a German-born soldier and naturalist who took to making ceramics in Paris in the 1860s and 1870s. It was a modern mark, identifying the piece as no more than a copy worth around \u00a310. Robinson had been cheated.\n\nPull was not a charlatan. He made precisely crafted ceramics inspired by the Palissy style, and often so closely modelled on the originals that it was difficult to tell the difference. But he signed his work, and did not try to pass it off as anything other than his own. It was unscrupulous dealers who, with a quick fix of sealing wax, tried to persuade unsuspecting buyers that they were holding original sixteenth-century ware in their eager hands. Robinson could not believe that he had fallen for such a simple trick. He was still convinced that the dish was genuine, and when he heard of the discovery at the museum he immediately sent a note arranging to call by and examine it himself. He presumed there was some mistake, that the staff were, once again, just being ignorant and crass. But when he arrived to inspect the platter, he could not ignore Pull's clearly marked signature. Robinson had to accept that he had been swindled, and had to begin the delicate task of vindicating himself to the museum.\n\nRobinson wrote long and detailed notes of justification to Cunliffe-Owen, explaining that an extraordinarily astute French dealer 'with great ingenuity' had been so clever with the dish that it was 'practically indistinguishable from the ancient specimens of the same type'. He assured Cunliffe-Owen that it was no reflection on his competence as a dealer, nor on his ability to continue working for the museum. He offered to repay the \u00a350; the offending dish was destroyed. And, in his own defence, Robinson emphasized the enormity of the forgery problem, and the constant battle collectors faced in their search for genuine pieces: 'It has become almost impossible even for the most learned and experienced connoisseurs and experts,' he pointed out unrepentantly, 'to avoid being from time to time deceived.'\n\nAmong European experts, Augustus Franks at the British Museum became perhaps best known for his prowess in detecting forgeries. Despite acquiring objects at a rapid rate, across a variety of different fields and from a range of sources, Franks was rarely taken in. He, like Duveen, was famed for his 'good eye', 'a marvellously wide knowledge of every kind of antiquity. . . an almost uncanny faculty of recognizing forgeries whenever and wherever he saw them', in the words of a colleague. In 1875, for example, he was granted a week's leave to travel to Switzerland to examine Palaeolithic remains excavated from a cave in Kesslerloch. His report to the museum trustees focused on several objects \u2013 a decorated bird bone, a perforated bone head, and engravings of a fox and a bear on bison bone fragments \u2013 that had been declared as genuine by numerous experts. Franks noted his 'grave doubts', however, as to the artefacts' authenticity. A few months later, Ludwig Lindenschmidt, founder of the R\u00f6misch-Germanisches Zentralmuseum in Mainz, published a paper proving beyond doubt that the pieces were forgeries and that the designs had been copied from the illustrations in a German children's book.\n\nFranks could count numerous such successes: there seemed to be something about the detective work involved with unmasking forgeries that he particularly enjoyed. An anecdote recounted by one of his colleagues at the Society of Antiquaries suggested that he relished the opportunity to expose cheats, and was able to appreciate the humour in what could be tense situations:\n\nDredging operations were going on at St Paul's wharf, and a large number of curious things were found. Medals of considerable size were among the finds. [Franks] drew an obverse and reverse of a medal, took it to the foreman of the dredging operations, and asked him if they had come across that medal. No, they had not, but of course they might come across one; if they did they would let him know. In three or four weeks' time they had the astonishing luck to come upon the very identical thing. The foreman himself brought it to Franks, so unfeignedly delighted was he to have found the desired medal. The thing was correct in every point. Franks asked the man if he knew the meaning of the inscription under the head of the obverse. No, he did not; he understood it was Latin. 'So it is; S. Fabricotus, the forged Saint.' The man fled.\n\nYet, despite making himself an expert in forgeries, even Franks was by no means infallible. Several of the large collections he acquired for the British Museum contained one or two suspect pieces, and a German stoneware vase he accepted from his friend Charlotte Schreiber, moulded with the heads of European kings and dated 1587, is now known to be a forgery. A French porcelain vase painted with flowers and cupids, which he bequeathed to the British Museum from his own collection, is also fake. Even Franks could not afford to be complacent about the scale of the problem facing collectors. Indeed, like Robinson, he was keen to emphasize the pervasive threat of the forgery trade and the unscrupulous who profited from it: 'there is scarcely an object in the range of ancient or medieval art to which the attention of the forger has not been given,' he explained to the Society of Antiquaries. He went on to note that this was a problem not only for the individual, 'the hapless collector', but for collecting as a whole, since the market in general was inevitably distorted by the volume of forgeries being produced. The forger, Franks complained, tended 'to depreciate the value of even genuine remains of the past by his dishonest industry'.\n\nThe ease of making forgeries, getting them into the shops and persuading collectors to buy them was driving prices down. With their livelihoods potentially at risk as a result of such widespread dishonesty, it would perhaps have made sense for dealers to be at the forefront of the battle to control the forgery rackets. But matters were not so simple. One or two Victorian dealers were notoriously crooked: in London, Louis Marcy, for example, was known to trade almost entirely in fakes and forgeries. A volatile character, his real name was Luigi Parmiggiani, and his motive was not so much personal gain as an attack on the capitalist art world and all it represented. He indulged an anarchistic streak by intentionally skewing the market with forgeries in an attempt to disrupt and embarrass those he considered pompous speculators living off an unfair system. His notable victims included J. C. Robinson, who bought a gold-mounted sword from the rogue dealer in the belief that it had belonged to Edward III, alongside two daggers, once allegedly in the possession of Edward III and Edward, the Black Prince. Few of those at the heart of the London collecting market were as blatant as Marcy, but equally few crusaded to shore up the value of genuine objects by stemming the influx of forgeries. It was too difficult a task. Many dealers acted honestly, but some of the forgeries were so skilfully made that even the most experienced and astute buyers were deceived. In addition, the existence of forgeries, while depressing the market as a whole, offered the individual the promise of riches. Many dealers could be tempted to turn a blind eye if there was enough profit to be made.\n\nOften the circumstances in which a forgery came on to the market were so opaque that it was difficult to be sure just how much the dealer knew about the provenance of the object. Murray Marks appeared trustworthy, and his reaction to the _Flora_ debacle emphasized his resolve to keep his good name unsullied. But there were murkier moments. In 1912, towards the end of his career, drawings of 980 designs by German goldsmith and jeweller Reinhold Vasters came \u2013 somehow \u2013 into his hands. Born in Aachen in 1827, Vasters was a consummate craftsman. Like Pull, he did not necessarily set out to make fraudulent sales, at least at first. His early work was clearly marked. But he also found that he was able to produce pieces convincingly like those of the Middle Ages and the Renaissance, and in time he began deliberately making forgeries. Between 1853 and 1890, he created a series of beautiful jewelled pendants, gold and enamel spoons, chalices, cups and bowls which were sold as genuine, with prices to match. When Marks acquired the Vasters drawings, he offered them to the Victoria and Albert Museum, but staff there turned them down and he took them back. They were sold at his death for \u00a337.16s and then presented to the museum by the new owner. What became clear, almost immediately, was that the designs were the blueprints for Vasters' forgeries. They were clearly modern production drawings for pieces that had been sold during the last fifty years as genuine works from the Renaissance. Marks had owned the very documents that proved Vasters had duped the market, and its leading collectors.\n\nIt took time to prove exactly what Vasters had been up to, and it was a hundred years before further study revealed that many fine metalwork objects in collections across Europe and America, thought to be Renaissance, were actually examples of Vasters' work: refined, skilful, very beautiful \u2013 but false. The collections of the Rothschild banking dynasty and of William Waldorf Astor, the American financier and statesmen, both included Vasters' fakes; the Victoria and Albert Museum, the British Museum and New York's Metropolitan Museum are today among the public collections which own Vasters' work, often still on display because of the quality of its craftsmanship. But Marks, who was keenly interested in forged bronzes, was in a good position to identify what Vasters was doing. He had traded one or two of Vasters' pieces to clients; he now had in his hands almost 1,000 modern drawings that were identical to objects he must have seen coming into auction houses, showrooms and collections. He may not have had sophisticated dating techniques to help him, but he would surely have recognized some of the pieces in the Vasters portfolio. Was he suspicious? Did he turn a blind eye in the interests of business, or because he was daunted by the extent to which Vasters' forgeries had penetrated the market? Was there something about the audacity of Vasters' scheme that he admired? It remains a mystery. As with many of the numerous sales of forgeries during the nineteenth century, the exact role of the dealer is difficult to determine.\n\nThe waters were muddied even further by the fact that some Victorian collectors were not at all concerned by the idea that dealers might be supplying them with forgeries. To them, what mattered was the prestige or intrinsic beauty of the object, rather than its provenance. During the later nineteenth century, the Rothschild family employed a Parisian goldsmith and restorer, Alfred Andr\u00e9. After his death in 1919, Andr\u00e9's workshop was found to be full of plaster casts and wax models for making forgeries. Like Vasters, he had been producing pieces in the Renaissance style to satisfy the demand of Victorian collectors. Three of his pieces were found in the Rothschild collection, clearly marked, suggesting that the family was willing to buy beautiful work that looked genuine, even in the knowledge that it was forged.\n\nSimilarly, many of Vasters' forgeries were bought by Fr\u00e9d\u00e9ric Spitzer, 'the greatest genius among nineteenth-century collectors', according to a German art historian in 1902, and also a dealer who made a substantial income from Vasters' work. Spitzer seemed unconcerned by the fact that his renowned collection was based on a complex series of forgeries. Indeed, he sought out forgers other than Vasters who might supplement his collecting, including Alfred Andr\u00e9. In 1910, the year after Vasters' death, another German art historian, Stephen Beissel, noted that 'as is well known', Spitzer happily 'employed for almost fifty years a series of first rate artists in Paris, Cologne and Aachen etc. who made old things'. In some ways, Spitzer was simply defying the experts by filling his collection with forgeries. But his carefree attitude also raised a variety of complex questions that collectors had to address on a more or less daily basis: did authenticity matter? If the experts could not tell the difference between an original and a forgery, was the forgery not just as good? Which was the most important, an object's intrinsic value, based on its beauty, artistry or craftsmanship, or its monetary value, based on its provenance?\n\n* * *\n\nIn January 2010, a painting called _La Belle Ferronni\u00e8re_ was sold at Sotheby's for $1.5 million. Thought to be of Lucrezia Crivelli, the mistress of an eighteenth-century Duke of Milan, the painting is the 'sister' to another version in the Louvre. For more than eighty years, the identity of the artists for both works has been a matter of bitter dispute. In 1920, when the owner of _La Belle Ferroni\u00e8re_ , Harry Hahn, tried to sell it as a work by da Vinci, authenticated by a French expert, a reporter from the _New York World_ rang Joseph Duveen in the middle of the night to get an opinion. A sleepy Duveen instantly dismissed the work as a copy, even though he had never seen it. The only real da Vinci of Lucrezia Crivelli was the one in the Louvre, Duveen maintained. Many of the potential buyers took his word for it, and pulled out of the sale. A furious Hahn took Duveen to court, claiming half a million dollars of damages for the apparent slander, but the 1929 trial concentrated not so much on what Duveen had said, but why. The prosecution argued that Duveen wanted to control what was sold to whom, thus making sure that he remained the dominant force in the art market. Declaring an object to be a forgery was one sure way of deterring buyers and scuppering a deal in which he was not involved.\n\nThere is no evidence that Duveen was acting so manipulatively, but it is certainly true that expert connoisseurs were in a powerful position, with most of the market prepared to accept their judgement on key authentications. This was particularly the case for those like Franks, Robinson and Murray Marks, who were working before the widespread use of scientific techniques to identify forgeries. For most of the time, Victorian collectors had to rely on their 'eye' to decide what it was they were looking at. Only at the beginning of the twentieth century, and in high-profile cases like the _Flora_ bust, did science begin to have much of a part to play. Even then, there was a lingering tendency to believe that such new-fangled techniques could not possibly add much to a connoisseur's verdict. During the last decades of the nineteenth century, art historians such as the Italian Giovanni Morelli and Bernard Berenson working in Europe and America developed a brand of connoisseurship which was identified as 'scientific' \u2013 and considered more than a match for emerging technology. Instead of relying on instinct and taste, vague spiritual qualities, Morelli and Berenson adopted a detective approach to small concrete clues. Their technique relied on the close scrutiny of tiny details that could be used to identify the style of a particular artist, often by looking at things that had been created almost unconsciously, like the folds of an ear, and which were unlikely to be closely imitated by a forger. 'Just as most men, both speakers and writers, make use of habitual modes of expression, favourite words and sayings, that they employ involuntarily, even inappropriately, so too every painter has his own particularities that escape him without his being aware of them,' wrote Morelli, and his books included images of disembodied hands and ears by famous Renaissance artists to make his point.\n\nNot everyone was convinced by these new methods: Wilhelm von Bode was particularly critical of such a schematic approach. But, in many ways, this kind of connoisseurship was a refinement of the expertise that collectors such as Schreiber, Robinson and Franks had been developing through the middle years of the century, continuing principles of detailed and informed observation. It also had the added advantage of retaining the air of gentlemanly scholarship that the Victorians valued. To many, this approach was the ideal 'scientific' compromise, and even well into the twentieth century there were art market stalwarts who did not want to sully their reputations with too great a reliance on less genteel techniques. As late as the 1920s, when Duveen was taken to court by Hahn over the _Belle Ferronni\u00e8re_ dispute, such Victorian sensitivities lingered. The American jury was distinctly unimpressed by Duveen's reliance on vague statements about the quality of his 'eye', and in the absence of factual evidence could not agree on a verdict, forcing Duveen to settle out of court to the tune of $60,000. In fact, Duveen had experts in modern X-ray techniques, as well as archival research, examine _La Belle Ferronni\u00e8re_ , but he never used their evidence, even though it supported his conclusions. He seems to have preferred to maintain his reputation as a connoisseur of impeccable instincts, and pay up, rather than be seen publicly to doubt his own connoisseurship or to relinquish the power of identifying forgeries to a new generation of scientists.\n\nVictorian collectors were caught in this changing world, where untried science was beginning to encroach on long-trusted scholarship, and where the art market was continually confused by accusations of forgery. The more specialized and knowledgeable connoisseurs became, the more forgeries were discovered, and the more collections were undermined and the market disrupted. The thorny question of what was genuine had always mattered to serious collectors: in the seventeenth century, for example, Rubens' patron, Sir Dudley Carleton, quizzed the artist at length to find out which works were by Rubens himself and which were the product of studio collaboration. But it was during the nineteenth century that issues of authenticity became widely discussed, and collectors began to realize the extent of the forgery problem. In such an atmosphere of doubt, it is perhaps not surprising that many of them tried to shore up their authority and retain some control over the market.\n\nIn 1881, Murray Marks suddenly acquired a very beautiful twelfth-century enamelled gold reliquary cross. Like the Vasters drawings, the cross, which had originally belonged to Ludwig I, King of Hungary and Poland, appeared in Marks' showroom apparently from nowhere. It was spectacularly beautiful, and inevitably caused a stir. Marks, taken by the intricate workmanship of the piece, showed it off to customers with pride and eventually sealed a profitable deal. But the cross was never his to sell. It was not a forgery, but it was part of an ingenious plot to make money from forgeries. Twenty years earlier, a Hungarian dealer working in Vienna, Salomon Weininger, had been asked to do some restoration work on the cross and three other pieces which came from the city's Imperial Treasury. While the objects were in his workshop, he copied them, and then returned the forgeries in place of the originals. The genuine pieces he sold on secretly to collectors: one of them, the Holy Thorn Reliquary of Jean, Duc de Berry, eventually ended up in the British Museum. So immaculate were the copies that the art world was confused for many years about which were the authentic objects and which were the forged ones. The case quickly became notorious, and Marks, who openly admitted that the cross came from Vienna, must have suspected that it was one of the pieces at the centre of the controversy. He may not have been sure whether it was an original or not, but that hardly makes his decision to act as dealer for the sale seem less dubious. It appears that he intentionally sold an object that was either a forgery or which he knew had been stolen from one of Europe's greatest treasure houses.\n\nThere was nothing secretive about the way Marks went about selling the cross. He openly invited clients to inspect it, and he used his contacts with other dealers to ensure everyone knew what a fine piece was in his possession. Robinson, in particular, was intimately involved. Like Marks, he was drawn to the quality of the reliquary's workmanship and, like Marks, it seems, he was willing to set aside any niggling doubts as to provenance. As far as we know, Robinson was not involved in the sale, but he was certainly part of the marketing drive to find the reliquary a new home: in March 1891, he borrowed it from Marks to display at the Society of Antiquaries in London where he wrote disingenuously that 'nothing is known of the history of this cross'. No doubt the two men shared quiet conversations, speculating over the origins of this most collectable of objects. Why did they choose not to make their suspicions public? We cannot, of course, be sure, but I doubt that it was simply a case of assuring Marks a profit. Robinson, after all, had no reason to have any interest in that. It is more likely that a complicated combination of factors kept them silent. The sale was prestigious, and turning a blind eye to the fact that the cross might be stolen or forged allowed Robinson and Marks to reflect the glory, asserting their positions at the top of their profession. Moreover, casting doubt on the piece's provenance would have destabilized the market yet again, and perhaps involved them both in an international scandal. In contrast, celebrating the cross as a legitimate masterpiece sustained the Victorian system of connoisseurship that they had both worked so hard to establish. It allowed them \u2013 like Duveen later \u2013 to stake a claim for saying what could and could not be sold. It put the power to decide in their hands.\n\nWhile the _Flora_ affair was to rumble on into the twentieth century, Marks found himself soon absolved for his part in it: 'there never was any ground for impugning in the slightest degree the good faith of the vendor,' reassured the _Times Literary Supplement_. He also received a vote of confidence from staff at the South Kensington Museum, who were no doubt mindful of the number of loans and advantageous deals he was apt to make on their behalf. He provided 'much valuable information respecting the provenance and history of many. . . objects', asserted an official report. With his name cleared and his reputation intact, Marks' association with South Kensington continued, and became one of the longest and most fruitful of the many enduring alliances he made over the course of his life. Having begun in the 1860s by luring Henry Cole with the splendour of the Dutch rood screen, he went on to work with five other Directors and generations of museum staff. His correspondence reflected the technological advances of a changing age: his handwritten letters gave way to typewritten notes and, finally, to telephone calls as he negotiated deals on the museum's behalf. His generosity filled the galleries with lovely things even as a new world order took shape: when he retired from business and moved permanently to Brighton in May 1916, during the darkest hours of the First World War, he offered all the objects from his London house on loan to the museum, brushing aside the staff's concern that the 'Zeppelin danger' was too great to risk such a display.\n\nIn the five years before his death in 1918, at the age of seventy-seven, Marks gave numerous pieces to the collections. He marked his fifty-year friendship with South Kensington with photographs, carvings, portraits, ironwork, sculptures, an oak door and, inevitably, a lovely example of K'ang-hsi period blue-and-white china. In a final flourish of generosity, he also gave the museum a group of bronzes, showing Bacchus and his faun, which he believed to be a rare fifteenth-century work from Florence. It was finely modelled, elegant and yet substantial, and, since bronzes were something of a speciality for Marks, it was a fitting tribute to his lifetime's work. Yet there is now considerable doubt over the origins of the Bacchus bronze. Recently re-evaluated, there is something not entirely right about the modelling of the faun's 'knee', as well as evidence of later techniques in a cast screw; the work is now considered to be relatively modern. As it has not turned out to be as special a piece as Marks believed, it is not on display in the galleries. It seems that, in his final handsome gesture, Marks was himself undone by the rampant Victorian trade in forgeries.\n\n## _Collecting the Empire: In Pursuit of the Exotic_\n\nSTEPHEN WOOTTON BUSHELL\n\n### CHAPTER EIGHTEEN\n\n### _The Route to Peking_\n\n__M__ urray Marks' success with Chinese blue-and-white ceramics showed just how widely Victorian collectors were beginning to spread their net. The idea of collecting from remote and unfamiliar lands was taking hold, part of a fascination with voyage and adventure that characterized an age of discovery. In the early 1850s, the slightly scandalous Captain Richard F. Burton captivated audiences with accounts of how he undertook the haj pilgrimage to Mecca disguised as a Muslim. From 1852 to 1856, news of David Livingstone's conquest of the African wilderness was delighting the public. By the 1870s, the British Empire was at its peak, with Victoria declaring herself Empress of India in 1877 and extending her rule to huge swathes of the globe. The opportunities for travel had never been so great.\n\nIt was no longer only the rich, or the heroic, who had the chance to see new sights, and it was no longer only the familiar, traditional European destinations of the Grand Tour which drew the crowds. Taking the opportunities for employment that lay in building railways, roads, bridges and canals; in ministering to the sick and the religious; and in completing the reams of paperwork that kept the Empire working, the sons of the middle classes soon filled the berths on ships taking them to a new life overseas. The rising numbers of ordinary people who set sail from ports all over the country only added to the fascination with travel, bringing it ever closer to home, while a generation of writers brought tales of exploration into libraries, parlours and nurseries in articles, stories, poems and biographies. The flamboyant Henry Morton Stanley published journal pieces and reports of his African journey throughout the 1870s, including his famous meeting with Livingstone in 1871. In the 1880s and 1890s, Rudyard Kipling's exotic, romanticized tales of Indian life secured him a place as one of the most popular writers of his age. Victorian society relished the frisson of the foreign, and everywhere there were accounts of new lands, astonishing peoples and remarkable journeys.\n\nThe appetite for real-life stories about those who came and went across the Empire was accompanied by an equally voracious desire for the objects they carried with them. The world was brought into the Victorian home in the shape of silks and muslins; extraordinary plants, pinned insects and butterflies and stuffed birds; shells, furs and feathers; spices, teas and spirits \u2013 and in portable art objects. As today, few travelled to foreign lands without picking up something to remind them of their journey, and before long there was a booming international trade in souvenirs. Whereas in previous centuries souvenir-collecting had been largely the domain of aristocratic travellers on the Grand Tour, the Victorian middle classes were now adopting the same habits, in new contexts. The souvenir trade was by no means confined to exotic destinations. By the middle of the century, the main sites of Europe, such as the Alps and the great Italian cities, were swamped with low-grade memorabilia on sale to tourists. In Dickens' _Little Dorrit_ (1855\u20137), Mr Meagles boasts a collection of 'model gondolas from Venice; model villages from Switzerland; morsels of tesselated pavement from Herculaneum and Pompeii. . . Roman cameos, Geneva jewellery, Arab lanterns, rosaries blest all round by the Pope himself, and an infinite variety of lumber'. But this familiar trade in cheap collectables was also being reinvented in far-flung lands as new types of objects caught the traveller's eye.\n\nOn the whole, as in Europe, these foreign souvenirs were manufactured quickly and cheaply to supply the rapidly expanding market and to give even the poorest visitor the opportunity to purchase a memento. Modelled clay figures from India, known colloquially as Poona figures, for example, were shown at all the international exhibitions, including the Great Exhibition, and in turn became extremely popular with European travellers, so much so that the manufacture of affordable souvenir versions helped sustain the economy of Pune, the Indian city where they were produced. Visiting Mexico City in 1884, the American anthropologist and archaeologist William Henry Holmes was amazed to find 'relic shops' on every corner, selling ceramic vessels, whistles and figurines.\n\nMost of these souvenirs tended to be clumsy imitations of genuinely exotic objects, rapidly produced to cash in on current fashions. Quite often they were not even made abroad. While the people of Pune and Mexico City manufactured modern reproductions and fashionable cultural artefacts to sell to tourists, plenty of other objects were made in Europe, with European tastes firmly in mind. In Thomas Hardy's 1873 novel _A Pair of Blue Eyes,_ na\u00efve young architect Stephen Smith tries to impress Elfride Swancourt by sending her some glamorous souvenirs of his life in India, but he manages only to waste his money: 'One day I bought some small native idols to send home to you as curiosities, but afterwards finding they had been cast in England, made to look old, and shipped over, I threw them away in disgust,' he laments. Hand-made objects crafted by local people were supplemented by mass-produced souvenirs churned out in enormous quantities in British manufacturing centres such as Birmingham. With the Victorian commitment to exploration, and rapid improvements in shipping and rail, the volume of travellers across the world had become industrial and demanded souvenir production on an industrial scale. Little bits and pieces for tourists became another cog in the huge wheel of Empire.\n\nFew of these objects were meant for serious collectors. They acted more as evidence of distances covered and lands visited than as objects of aesthetic pleasure or financial investment. Those with a more incisive eye were nonetheless aware that among all the false knick-knacks and tacky keepsakes there were important and handsome objects that encapsulated the mystery and glamour of the places they were made. But you had to know where to look. Just as in Europe, serious collectors had to be both wary and astute. They needed to be energetic enough with their research to keep one step ahead of the mass-market travellers and the traders who profited from them. And, just as in Europe, it was often a great advantage to be first on the scene. As Stephen Smith discovered in _A Pair of Blue Eyes_ , by the middle of the nineteenth century, countries at the heart of the Empire, such as India, were enmeshed in a complicated and profitable souvenir trade: those who wanted to find the best objects had to be prepared to go to more remote, inaccessible, and often more risky, destinations.\n\nOn 29 February 1868, Stephen Wootton Bushell sailed from Southampton on a boat bound for Shanghai. He was twenty-three years old and had graduated as a doctor from Guy's Hospital in London less than two years previously. He was clever, likeable and mildly adventurous, but his experience of life so far had been sheltered and comfortable. He had grown up on his father's substantial farm in Kent, spent happy years at a minor public school and won easy success as a scholar. His natural talent for learning, an excellent memory, a personable manner and money in his pocket had allowed him to glide through childhood and adolescence and to emerge as a thriving and upright young man well equipped to begin a medical career. Now, however, his life was changing. For the first time, he was facing a challenge and striking out from the solid middle-class ways of his upbringing. As he joined the bustle in the Southampton docks, showed his papers and began the steep walk up the wooden gangplank, Bushell was setting foot firmly in the unknown.\n\nIt was a long journey. Back at his father's farm, the spring lambing was over and the summer crops ripening before Bushell reached his destination. At sea, there were wide, alien views and strange stories. There were long days of bright calm, and nervous stormy nights. There were dinners and concerts and games. And there was plenty of time to think. Bushell was not on a pleasure trip. He had his formal letter of appointment as Medical Attendant to the British Legation in Peking (now Beijing), a promise of \u00a3600 a year, and official approval 'to engage in private practice at Pekin' if he so wanted. He was determined to use the months at sea to prepare himself properly for his new job. He began the task of learning Chinese and he brushed up on the medical conditions he might expect to encounter. He read and reread the scant official documentation he had been given. Yet he did not know what to expect. Very few European travellers had spent time in China, or travelled extensively there, and there was limited reference material for him to consult. He had little idea what his life was going to be like and he could not help but suspect that his preparations might all turn out to be useless. Huddled in his berth as he sailed across the world, Bushell had nothing in his experience to compare with where he was going. He would just have to wait and see.\n\nShanghai was noisy, crowded and prosperous. It was a city growing quickly and erratically, a newly affluent port on the mouth of the Yangtze river, ideally placed to trade with the West. It was a hub of commerce, a centre for travellers and a maze of narrow streets leading into still narrower alleys. From the moment he landed, Bushell was captivated by the colours, smells and sheer noise of such foreignness. He found himself welcomed into a thriving cosmopolitan community. But his journey was not over; he was hardly acclimatized to the light and the dust \u2013 or to the unfamiliar cadences of the Chinese language \u2013 before he was travelling onwards to Peking, around 1,300 kilometres to the north, away from the mercantile centre of China to a much more traditional heartland. As he left the modernity of Shanghai, Bushell became more and more a foreigner, a strange man with curious ways.\n\nIf Bushell had been hoping to find himself part of a community comparable to the one in Shanghai, then he was to be disappointed. When he finally arrived in Peking, tired and exhilarated by his journey, he discovered that there were fewer than fifty non-Chinese residents enclosed in a compound within the city, walled in with their families and servants, a fives court and bowling alley, a library, a billiard room and a stage for regular performances of amateur dramatics; an entrenched and unremarkable group of administrators and bureaucrats carefully recreating European society deep in the heart of China. There were luncheons and parties and evenings around the piano. There were lengthy discussions of British politics and imperial ambitions. The national anthem was sung, and the Queen toasted. But the restrictions of life within the legation did not dishearten Bushell. He enjoyed the security and comfort, and he became determined to use the legation as a springboard for exploration instead of a barrier to it, a safe retreat that would allow him to make forays into the bewildering city that lay all around him.\n\nBushell's fascination with China, inspired by his brief stopover in Shanghai, had become only more potent as he had moved north. His ten-day journey by boat up the Beihe river, and then overland on a mule cart, had given him the opportunity to begin a study of the country. Now he was eager to discover more. Not content to shut himself off in an expatriate enclosure, he wanted to learn more of the language, meet more of the people and explore more of the landscapes and the intriguing histories that China seemed to offer. Above all, he wanted to acquire some of the tempting things he had seen. China, he had already observed, was a country of extraordinary objects. On landing at Shanghai, Bushell wrote, 'one sees on the wharf a number of pedlars offering for sale teapots and cups of quaint form' fashioned like a dragon rising from the waves or a mythical phoenix, a gnarled tree trunk or an elaborate flower. In Peking, 'a garden of any pretension must have a large bowl and cistern for goldfish, and street hawkers may be seen with sweetmeats piled up on dishes a yard in diameter, or ladling syrup out of large bowls; and there is hardly a butcher's shop without a cracked Waul-li jar standing on the counter to hold scraps of meat'. Although Bushell had not been a collector before, the extravagance and abundance of Chinese objects quickly seduced him.\n\nDespite his enthusiasm, the contrast with the neatly kept fields of Kent was a culture shock, and settling in took time. Bushell knew he could not simply go out and start bartering for wares. First, he had to become familiar with the arcane workings of the British government abroad and the idiosyncrasies of his patients. He had to learn how to fit in with his colleagues and their families \u2013 and he had to improve his Chinese and begin to understand the habits of the Peking inhabitants, who were frequently hostile. Throughout the East, the British presence was often resented: in Japan, the legation's native interpreter was murdered at the compound gate in 1860, and the following year the building was stormed by angry locals. In China, the Second Opium War of 1855\u201360 pitted French and British troops against the Chinese in a series of bitter battles punctuated by incidents of kidnap, torture and looting. Unsurprisingly, when Bushell arrived in Peking in 1868, relations between the British and their hosts could be awkward, and the city could seem unwelcoming. Mary Crawford Fraser, the wife of the secretary to the legation, noted in her journal that the British 'were detested in the city and never passed outside the Compound without being made to feel it', and many preferred to spend as much time as possible in the Western Hills where they felt more welcome and secure.\n\nBut Bushell was not intimidated. With the determination and optimism of youth, he knuckled down to establishing himself, and soon found that, with such a small British contingent to serve, his medical duties were reasonably light and relaxed. He had a great deal of spare time. So he set his mind to the community beyond the compound, studiously reading about Chinese history, improving his language skills and researching the country's ancient culture. He worked enthusiastically and conscientiously, and he made rapid progress. What might have begun as the na\u00efve experiment of a young man soon developed into something more. His excursions from the compound became increasingly lengthy and ambitious; his conversations with the locals increasingly fluent. He grew in knowledge and confidence and within a couple of years he was moving boldly across the city, a familiar figure among its residents and an object of wonder to Mrs Fraser: 'our good Doctor Bushell,' she noted in amazement, 'could talk to the people of their ailments in excellent vernacular and gave them medicines free of charge. This fact alone set him quite apart from other human beings in their estimation.'\n\nThe more Bushell saw and read of Chinese art, the more it fascinated him. He used his local contacts to find the best shops and market stalls in the Outer City, way beyond the compound, and he became familiar with the monthly markets in the temple courtyards and the lanes clustered around them, where all kinds of food, clothes and trinkets were on display. The market at Wu Men, the main gate into the walled Forbidden City, was, he discovered, a particularly rewarding source of objects from private houses, and he browsed the stalls there regularly. Sometimes, too, the objects came directly to him without his having to leave the compound: small-time dealers in curios would call daily at the legation and on its Western residents. 'The morning was spent in studying and cheapening the wares brought by native merchants and spread all over the floors,' explained an American visitor, apparently dazzled by the display, an alluring array of 'bronzes, porcelain, jasper, jade, amethysts and emeralds' as well as furs from 'sea otter, sable, Tibetan goat, Astrakhan, wolf, white fox, red fox, bear, panther and tiger skins'.\n\nBushell was especially fortunate, however, in not having to rely only on curiosity dealers. 'I have obtained access, in the exercise of the duties of my profession,' he admitted gratefully, 'to several palaces and private houses, and have in this way had many opportunities of seeing the treasures of native collectors, which usually are so rigidly closed to foreigners.' He began to acquire a variety of things: textiles and silks; enamels, ivories and jewellery; carved stone sculptures, bronzes, pictures and architectural specimens. He was increasingly impressed with the skills of both ancient and modern Chinese craftsmen. By 1873, just five years after his arrival in Peking, he felt confident enough to begin corresponding with Augustus Franks at the British Museum, alerting him to the treasures he was unearthing and sending one or two pieces to Franks for his personal collection. He also published his first scholarly article, a study of the 'mountain boulders roughly chiselled into the shape of drums' which stood in the city's Temple of Confucius.\n\nBushell had settled quickly. The sheltered young man had grown into the role of pioneer and explorer. With archaeology yet to uncover the sites around Peking, he was becoming an expert in the ancient literary sources that made sense of Chinese history and its objects, and he was already on his way to becoming the first Westerner to undertake the serious study of Chinese art. In 1872, he set off for nearly 200 miles to the north with Thomas Grosvenor, a secretary at the legation, on a journey beyond the Great Wall, to Inner Mongolia, where he visited the ruins of Shangdu, the summer capital of Kublai Khan's Yuan dynasty. It was a fabled city, a byword for opulence, made famous in Britain as Xanadu in Coleridge's popular poem 'Kubla Khan' (1816). There was little left to see, but there were enough relics to captivate Bushell. The two men were the first Europeans to visit the site since Marco Polo in the thirteenth century, and the feeling of discovery was intense. Bushell used the journey to collect all kinds of pieces, and to begin to understand more fully the culture of ancient China. On his return, he took a period of leave in 1874 to return to Kent to marry, and persuaded his new wife Florence \u2013 herself the daughter of a doctor \u2013 to return with him to China the following year. The long voyage back was very different from the first. It was a journey home. The talk in the Bushells' cabin was about everything Florence could look forward to, the people she would meet, the places her new husband would show to her \u2013 and the abundance of things still waiting to be discovered.\n\nThe objects accumulating in Britain's museums provided tangible evidence of the Victorian fascination with their expanding world. The extension of the Empire, the growth of trade routes and the increasingly far-flung voyages of intrepid travellers such as Bushell were brought to life for museum visitors in the pieces beginning to appear in the display cases. At the South Kensington Museum, for example, alongside the Italian Renaissance sculptures and medieval French treasures that Robinson had tracked down, or the English manufactured wares championed by Henry Cole, there were now increasing numbers of objects from less familiar destinations: carpets, textiles and tiles from Persia and the Islamic world; models, busts, Buddhist reliefs and imperial treasures from India; Japanese porcelain and pottery; arms, ammunition and jewellery from Afghanistan; lace, costumes and embroidery from Mexico and South America. New forms, colours and ideas were beginning to appear in the museum, presenting visitors with alternative ways of looking at the world and its peoples. Collecting was making itself truly international.\n\nThese new kinds of objects required new methods of description. There was a long tradition of European anthropology that had often sat alongside abstract philosophical thought: the eighteenth-century German philosopher Immanuel Kant, for example, gave a series of lectures on anthropology for over twenty years from 1772. But such anthropology was not necessarily specifically concerned with foreign races and cultures (Kant mostly used examples from close around him to make his points) and there was no term at all for the tangible artefacts that collectors were now acquiring. In 1834, the term 'ethnography' was coined, and this was quickly adopted to cover anything that did not fit into the established hierarchies of works of art or that came from some distant, apparently primitive, possibly exotic, land.\n\nHaving a word for these things made it easier to define and categorize them, but it did nothing to make them more familiar. The arrival of objects from all over the Empire and beyond was creating a challenge for museum staff. Asian art, Japanese ceramics and Indian sculpture were not traditional subjects of study; they had not been considered important. A young gentleman's education in the Classics rooted him firmly in European culture; the Grand Tour and its Victorian hybrids prepared him for an appreciation of Western art. At small private endowed schools or grammar schools for the middle classes and respectable working classes, boys and girls had little chance of seeing any art, let alone works from distant cultures. These schools tended to be insular in their philosophies, with an emphasis on discipline and vocational skills, and even at progressive schools run by education reformers the aims were modest: 'to write a letter grammatically, to calculate rapidly without a slate and to keep accounts by single and double entry'. Most ordinary people knew very little, if anything, about countries other than their own, while, even among dealers and connoisseurs, scholarship about non-Western objects was distinctly patchy. The world was simply growing too quickly.\n\nThere were exceptions. J. C. Robinson developed a love for Chinese ceramic art, which he regarded as 'the perfect consonance of material and decoration'. And at the British Museum, Augustus Franks was making himself something of an expert in non-European works. Franks himself did not travel outside Europe. His contemporary and friend in Germany, Adolph Bastian, was a field collector, personally acquiring 2,000 artefacts from his travels to South Asia alone, and creating the Museum f\u00fcr V\u00f6lkerkunde in Berlin in the 1880s with ethnographic collections six or seven times larger even than those at the British Museum. But Franks did not let his relative lack of field experience defeat him; he kept in close contact with numerous travellers, negotiated with a steady stream of collectors who brought objects to the museum on a more or less daily basis, and used his visits to Europe to acquire pieces by exchange, both with private collectors and museums.\n\nWhen Franks joined the British Museum in 1851, there were around 3,700 ethnographic objects on show in a single gallery; by the end of his work there in 1896, the museum boasted an outstanding display of almost 40,000 pieces in a suite of rooms. As well as individual objects, Franks was instrumental in securing a number of substantial complete collections, such as the Christy Collection, which brought more than 1,000 items to the museum. A banker and textile manufacturer, Henry Christy had been a friend of Franks' and had spent much of the early 1860s undertaking excavations of cave sites in southwest France. Alongside this archaeological material, however, he had collected almost 1,000 items from travels in Mexico, North America and Canada. After Christy's death in 1865, Franks worked on cataloguing the objects. He also purchased a further 20,000 pieces with \u00a35,000 bequeathed to the museum by Christy, which Franks had invested into a fund specifically for making acquisitions. The museum's ethnographic collection grew quickly, and Franks' knowledge grew with it. He soon became an acknowledged expert on non-European objects, and was often called on to assist other institutions with their collecting; when the South Kensington Museum acquired a large consignment of Japanese ceramics from the International Exhibition in Philadelphia in 1876, Philip Cunliffe-Owen turned to Franks to research and catalogue it.\n\nNevertheless, there were few other scholars working in Britain on the new material from around the world, so the public collections were forced to look much further afield for help. Staff began to recruit collectors to work alongside their Western European specialists and provide a much-needed link with distant lands. They looked to the colonial service and the army, in particular, to start creating a collecting network that spanned the globe. In the first half of the nineteenth century, the British Museum acquired important objects from officers in the Royal Navy and the East India Company, and from public servants working in Africa, the Arctic and Australia. Captain Marryat, also known as a writer for children, gave a large lacquered Buddha and a colossal stone carving of Buddha's footprint in 1826, and the museum became the home for the stunning collection of Sir Stamford Raffles, Lieutenant-Governor of Java, who founded modern Singapore in 1819. By the mid-century, as the number of British territories grew and increasing numbers of people were involved in administering them, the quantity of objects arriving on ships from around the world rose steadily, and from the 1880s, collections made by professionals abroad were frequently arriving at the museum. Very few of these distant collectors were experts: most of them collected as a diversion from their day jobs and chose objects that caught their eyes as opposed to pursuing a disciplined programme of study. But they were enthusiastic and often acquired a sound knowledge of their local area and customs. And help was at hand to guide them, especially from Franks. He regularly briefed travellers in person about what to collect, and even handed out questionnaires for them to complete on his behalf, demanding details such as the kinds of clay employed in making pottery or the type of objects used for marriage and funeral customs.\n\nMany of these overseas collections, rather like Franks' own, had an odd semi-official status. They were made during the course of official duties but they were usually privately owned. Once they were finished with \u2013 usually on the death of the collector, or on his return to Britain \u2013 they were commonly given, rather than sold, to public museums, as though it were understood that they had been borrowed by the individual for the duration of his time overseas. Collecting on this basis was clearly a popular distraction for those posted far from home, a way of understanding a new culture while marking public service with something tangible. The collecting network was expanding alongside the ambitions of Empire. In the early 1870s, the South Kensington Museum made contact with Major Robert Murdoch Smith, director of the British Telegraph Service in Persia, who agreed to source examples of historic and contemporary Persian art on the museum's behalf, and whose expertise enabled the museum to publish a guidebook to its Persian material. Then, in 1874, two cases of bronze vases, mirrors and bowls and twenty-three small spearheads, daggers and other weapons arrived in South Kensington. All the pieces were Chinese, and they were all on offer for exhibition. The consignment had come direct from China, via the Temple Club on the Strand, and the unfamiliar and occasionally bizarre objects were the property of Stephen Wootton Bushell.\n\nBushell's association with South Kensington was to be a long one, but it started humbly enough. When they were unpacked, the pieces turned out to be small and largely unspectacular. But they were a taste of things to come. And for Bushell, it was a means of establishing professional contact with the museum staff. When his marriage to Florence had been celebrated in 1874, he used his time back home to discuss his initial discoveries with the curators at South Kensington. He tried to describe the variety and wealth of objects waiting to be discovered in China, emphasizing how much more there was than the limited range of Chinese porcelain already familiar in fashionable circles. His enthusiasm was infectious, and the staff were delighted. Here was a learned man at the heart of one of the most mysterious countries on earth, with a collector's eye for the unique and historic, and with a willingness to ship his objects back to England and lend them for display.\n\nBushell turned out to be as good as his word. When he returned to China with his new wife, he found that his conversations at the museum had given him new impetus and direction, and immediately he began assembling a consignment to send back to South Kensington. Over the next few years, he began collecting with the museum in mind, acquiring an assortment of carefully chosen objects that would make a coherent display in the galleries. 'I have selected during my residence at Peking a few more typical specimens. . . to add to my small collection now being exhibited in the Loan Department,' he wrote modestly to Cunliffe-Owen in the summer of 1880. The arrangement seemed to work perfectly. The museum had found a reliable source of expertise and a cost-free way of acquiring the best pieces, and Stephen Bushell could collect to his heart's content, knowing that his efforts were being appreciated back in England and that the South Kensington display cases would ease the pressure on the modest doctor's accommodation he was allocated in the Peking enclosure.\n\n### CHAPTER NINETEEN\n\n### _The Promise of the East_\n\n__T__ he objects Bushell was selecting for South Kensington and the British Museum, and the increasing number of pieces being sent to London by travellers from around the world, were creating something of a dilemma. With their odd symbolism, their unfamiliar materials and their sometimes na\u00efve techniques, they were not easy to classify. Were they really art? Museum staff found it difficult to decide whether such things deserved to be displayed alongside European objects or whether their very difference marked them out as cultural curiosities, of social instead of artistic value, and inferior to the established displays. Bushell's Chinese objects were shown along with other foreign works, mainly on loan from private collectors, in the aptly named 'East Cloister' at South Kensington. It was here, at the edge of the museum, that visitors could see pieces from overseas, set out to dazzle and intrigue. The gallery was colourful and exotic, specially decorated in elaborate mock-oriental designs that emphasized its difference, and hung with objects that even the most diligent student of art would never have seen before. It was deliberately distinctive, distinguished by a kind of alluring otherness that set it apart from the rest of the museum. The damp grey of the London streets receded as visitors were invited to step into another world and to discover the strange skills and odd tastes of the people who lived there.\n\nThe gallery was far more than just a window on other cultures: it was at the same time a way of ordering and controlling the unfamiliar and unknown. Persia, China, India and Japan fascinated the nineteenth-century public. The apparently vast markets for British exports, as well as the promise of imported luxury goods such as teas and silks, excited merchants, shipowners and shopkeepers. Political uncertainty in many Eastern countries held out the prospect of the further expansion of British territory and fired imperial ambitions. But, above all, many Victorians were seduced by the idea of a glamorous Orient, a place of sultry nights, outlandish wildlife and bizarre sensual rituals. They were intrigued by news of each new discovery and were anxious to see for themselves the unexpected and beautiful objects that emerged from the East.\n\nThe new wonders of Japan, for example, were a source of endless fascination to many ordinary Victorians. When Augustus Franks first joined the British Museum in 1851, Japan was still a closed country, governed by a policy called _sakoku_ that had been in place since 1639 and which prevented the Japanese from travelling abroad and foreigners from entering the country. Little information about Japan leaked out, and few things: a small concession of the Dutch East India Company in Nagasaki supplied Europe's palaces with fine porcelain exports, known as 'Old Japan', from the 1660s onwards, but the only chance most people had had of seeing Japanese objects was the tiny handful of items included in the 1851 Great Exhibition, under the auspices of the Chinese display. In the 1850s, however, _sakoku_ ended abruptly after the American naval commander Commodore Perry forced the Japanese to sign a 'Treaty of Peace and Amity' (or Treaty of Kanagawa) in 1854. By the end of the decade, Japan had links with most Western nations, including a further treaty in 1858 which opened three of its ports to British traders.\n\nSuddenly, Japanese art and antiquity were available to collectors \u2013 and they took full advantage. During the 1860s and 1870s, in Britain and France, the fashion for collecting Japanese artefacts was intense. A special term was coined in 1872 to describe enthusiasts \u2013 _Japonistes_ \u2013 and netsuke (miniature sculptures) and tsuba (sword-guards), in particular, were enthusiastically acquired by Westerners. The influence of Japanese objects and aesthetics was felt across fashionable and artistic society: Degas, Renoir, Whistler, Monet and Gauguin were among the painters influenced by Japan, while, in the Victorian home, Japanese-inspired jewellery, ceramics, furnishings and wallpapers became hugely desirable. In 1875, Liberty's opened in London's Regent Street as a specialist supplier of Japanese goods, and in the popular theatre works such as Gilbert and Sullivan's comic opera _The Mikado_ (1885) continued the craze for all things Japanese towards the end of the century.\n\nBut such widespread enthusiasm for new cultures and their objects was not without a darker dimension. There was also a sense of alarm, a niggling fear that disorder and disruption might creep into Britain from these mysterious lands with their weird customs. In many museums, the strange ethnographic objects from beyond the usual Western canon of art were shown together as evidence of the 'backwardness' of other cultures, rather than as beautiful and valuable objects in their own right. They were generally classed as wondrous oddities rather than being afforded the status of art, and were often displayed alongside natural history specimens to make a point about racial progress. African objects, in particular, seemed to emphasize to the Victorians the gulf of racial difference and these, of all pieces, were most commonly kept separate from more familiar works of art. When Henry Townsend, a young missionary, first travelled to Sierra Leone in 1836, and then on to Abeokuta, a Egba kingdom inland from Lagos in 1843, he was bewildered and distinctly unnerved by the vastness of Africa, by the heat, the darkness, the wary tribespeople and the primitive transport arrangements, but as he settled into African life during the 1850s and 1860s it was the objects which occupied his attention. As well as converting suspicious worshippers to his God and energetically spreading the Christian message, Townsend took time to barter with the heathen villagers, and was frequently delighted with the unusual jewellery, sculptures and vessels he discovered. While fellow churchmen at home were collecting Anglo-Saxon archaeology or Greek antiquities to display in their parlours, Townsend demonstrated a more adventurous taste.\n\nRegular shipments from Britain brought Townsend personal supplies and the kinds of things any active missionary might require: Bibles direct from Queen Victoria, money collected for the building of a church, and a steel corn-mill from Prince Albert for the practical improvement of African life. In return, he sent back colourful and curious things. Many of these objects simply confirmed to friends back in Britain just how much the Egba people needed European culture and values: an Eshu fetish figure, given to Townsend by Chief Ogubonna, for example, had been used at his door to protect against witchcraft before his conversion to Christianity. Back in Exeter, Townsend's home town, no one had ever seen anything quite like the strange African objects. Staff at the museum received Townsend's parcels with some consternation and in the end the decision was taken to keep all the African works physically separate from the main museum, carefully displaying them instead in a 'temporary depot' set apart from the displays of Western painting and sculpture.\n\nEven once it was agreed that ethnographic objects did not constitute art, however, it was still not easy to decide on an authoritative way of exhibiting them. During the eighteenth and early nineteenth centuries, collectors tended to adopt one of three different approaches to ethnographic display. There was the eclectic gentleman's cabinet of curiosities which could easily accommodate a random collection of objects of varying significance and needed little philosophical rationale beyond the driving passions of the individual collector. Then there was the model adopted by collectors like Walter Scott or Lieutenant-General Augustus Henry Pitt Rivers, who brought together 20,000 objects from across the world and gave them to the University of Oxford in 1884. This type of display often focused on arms and armour, and removed the works from any social or geographical context. Instead, it emphasized visual similarities and encouraged the visitor to follow the development of each type of object in a linear way, from the simplest to more complex forms: 'Specimens were arranged according to their affinities, the simpler on the left and the successive improvements in line to the right of them,' explained Pitt Rivers in 1874. The displays positioned objects along these evolutionary timelines from primitive to modern, irrespective of the actual date they were made. They tended to enforce strict hierarchies and, in doing so, stressed racial differences. Taking up the work of anthropologists, the layout aimed to reveal the history of humanity and compare the cultures of different peoples. It was implicitly understood that the examples from distant countries were, even if modern, really only of value as evidence of the more primitive stages of Western development rather than as objects of a specific place and culture.\n\nFinally, from the 1780s onwards, as explorers like Captain Cook brought ever more artefacts back to Britain, there was a growth in the third model of display: contextualized shows defined by geography. The enthusiasm for foreign objects by now could mean that sometimes over half the pieces on display in public museums had been shipped to Britain from far away, and plenty of galleries offered a 'South Sea Room', for example, or indeed an 'East Cloister'. Increasingly, merchants, diplomats, sailors, soldiers and churchmen were all taking time off from their trades to send things back to entertain and astound the public. A sledge rescued from William Parry's expedition to find the North-West Passage; a 'pair of garters worn by the Queen of Tahiti'; costume, ornaments, baskets and bowls from South America, Madagascar and the Philippines collected by plant-hunter John Gould Veitch; 'a valuable tortoiseshell comb, of immense size, as worn by the ladies of Mexico'; and a 'twentyfive Rupee note apparently Burmese, printed on native paper, taken from the body of a Rebel Sepoy shot by the donor at the recapture of Dharwalagiri in the Nepaul mountains' are just some of the objects I have seen noted in contemporary accessions registers.\n\nFollowing the third model of display, these various artefacts were sorted into orderly presentations defined by geography, and cases and printed guides were arranged to show material specific to one culture, such as the Canadian Inuit. Again, there was little sense of chronology. Objects were clearly linked to ideas of voyage and adventure rather than to historical scholarship, and were often shown to give visitors an idea of what travel might be like in strange and distant lands instead of examining cultural development in any detail. Over the course of the nineteenth century, this last approach to display became the most common \u2013 at the British Museum, the galleries were arranged geographically for over half a century, from 1808.\n\nYet the controversial question of how to classify and display such objects was far from resolved. In the 1860s, the unspoken principles behind these display choices were increasingly brought out into the open and discussed in public. During a decade of change, popular ideas about overseas cultures were challenged, and curators and collectors began to look again at the way their collections were presented. Previous work on evolution was transformed by the publication of Charles Darwin's _On the Originof Species_ in 1859, followed by Herbert Spencer's work on evolution and natural selection, _Principles of Biology_ , in 1864 and, at the end of the decade, Francis Galton's experiments based on theories of biological inheritance. Traditional religious beliefs seemed to be under threat, scientific opinion was divided and often confusing, and the very nature of humankind was up for reconsideration. As debates raged, ethnographic collections in museums and in private hands were inevitably caught up in the fray. How and what people collected, and how these things might be displayed, became connected to controversies over race, religion, biology and 'the survival of the fittest'.\n\nAs Darwinian ideas spread, the evolution of human societies became the topic of urgent debate, as did the relationship between technologically advanced countries such as Britain and more apparently primitive cultures. Ethnographic collecting and display was at the heart of this discussion, providing for 'the scientific study of manners and customs of particular peoples' and showing, according to the British Museum, 'their development from savagery towards civilization'. Displays of objects were an excellent way of presenting the complex intellectual ideas underpinning evolution. Visitors were encouraged to marvel at simple native objects, and draw comparisons with the 'better', more refined examples of pottery, clothing, jewellery, tools or artworks in their own homes. Foreign pieces were a source of entertainment, even laughter; the displays offered an experience comparable to travelling shows of living 'exotic' people that toured throughout Europe, becoming particularly popular during the 1870s and 1880s, and exhibiting novelties like Laplanders herding reindeer or African bushmen preparing meals. Closely linked to principles of social evolution, the display of ethnographic artefacts reinforced the Victorians' view of themselves as advanced, sophisticated and superior.\n\n* * *\n\nWith the display of travellers' objects apparently fortifying Victorian notions of supremacy, collections became especially entangled with imperial ambitions. Empire building and collecting went hand in hand, and, by the 1870s and 1880s, public collecting could clearly be seen as a tool of state, an expression of how Britain saw itself on the world stage. In 1886, this was fully articulated in a Colonial and Indian Exhibition at South Kensington. Much more concerned with the display of imperial might than earlier extravaganzas like the Great Exhibition, this demonstrated how foreign objects could be used to illustrate and validate the glory of the Empire: 'It meant a proclamation to all and sundry that Victoria rules the Empire,' trumpeted the _Illustrated London News_ , 'a just and equitable, but firm and fearless rule to the uttermost ends of the world, to the extremist limits of human civilization.' It was not simply that Indian textiles were beautiful, for example: the display of such works in the exhibition also helped to highlight the dominance of the Raj in India and remind people of the extent of British power. One of the most popular pieces in South Kensington was the famous Tippoo's Tiger, an animated mechanical organ, made of painted wood and carved to look like a tiger mauling an Englishman. It had originally been commissioned by the Indian ruler Tipu, Sultan of Mysore, a symbol of his opposition to British authority, and in particular of his resistance to the British East India Company. Its presence in the museum not only recorded Britain's victory in India, but also reminded visitors of the need to take the cultivated ways of Empire to a place that was once governed by such a sadistic oriental despot.\n\nBecause of the long British involvement in India, the public collection and display of Indian objects was often more structured and considered than the presentation of art from other Eastern nations. The East India Company, trading since the days of Elizabeth I, had set up an Indian museum in London's Leadenhall Street as early as 1801. Despite being crowded, chaotic and squalid \u2013 it was described by one journal as 'practically inaccessible, and hid away out of sight. . . a bonded warehouse and not a museum' \u2013 it drew crowds of visitors, who were fascinated and horrified in equal measure by the eccentric hotchpotch of ancient tablets, religious sculptures and elephant heads. There was soon a crisis of space and in 1858 a 'New Museum' was opened to the public, but the East India Company was already in decline. The Indian Rebellion of 1857, followed a year later by the Government of India Act, which set up a new government department for the subcontinent, unsettled old ways of doing things and reduced profit. The museum went through several decades of uncertainty and upheaval before finally being disbanded in 1879. The collection was then divided between the British Museum and South Kensington, where treasures like Tippoo's Tiger took pride of place in the Eastern Galleries. With the added attraction of more convenient and salubrious premises, the Indian material once again drew huge audiences: nine rooms and an adjoining landing were given over at South Kensington to the display of architectural fragments from Mughal palaces, models of Indian domestic scenes and religious festivals; fabrics, carpets and homewares. One of the most striking exhibits was a massive plaster cast of the Eastern Gateway from the domed Buddhist monument of the Great Stupa at Sanchi in India, a huge pillared entrance elaborately carved with elephants and scenes from the life of Buddha. So magnificent and popular were the displays that the travel publisher Baedeker even included a guide to the collections in its London itinerary.\n\nBut interest was by no means confined to India, and, as British ambitions expanded, so too did collecting from the length and breadth of the world. In many cases, countries that were not yet under British rule were the priority, rather than potentially more reliable allies like India. Acquiring objects from intransigent states was one way, at least, of announcing supremacy. While the Victorians were trying to gain control of Afghanistan, for example, the spoils of war helped to make British victories visible and tangible, both to those whose treasures were being removed and to an influential public back at home: a wooden panel from the gateway of Ghuzni, taken during the first Afghan war in 1842, and arms from the second Afghan war in 1878\u201380 all found their way into the galleries at South Kensington. Japan and China were similarly important. At a time of fierce rivalry for dominance in the region with other European and world powers, collecting acted to assert territorial claims ever more strongly. In a competitive world, acquiring objects from the Far East assumed unique symbolic and diplomatic value.\n\nAt South Kensington, an active programme of collecting from Japan began in the 1850s with the purchase of some modern Nagasaki lacquer work. It was soon boosted by Queen Victoria who added swords, armour, textiles and ceramics during the 1860s, while Britain's minister to Japan from 1865 to 1883, Sir Harry Parkes, also donated a fourteenth-century sword that he was given by the Meiji Emperor in 1872. At the British Museum, as well as decorative objects and sculpture, Franks persuaded the Trustees in 1881 to buy a collection of over 3,000 Japanese paintings put together by William Anderson, a doctor who lived in Tokyo from 1872 to 1880. In China, however, progress had been slower. The country was so huge, so apparently undeveloped and so completely unknown to all but a few Western travellers \u2013 and trade was so tightly controlled by the Chinese government \u2013 that the objects which emerged tended to be of limited variety, and sometimes low quality, often specifically designed for the Western markets. When Stephen Wootton Bushell sent his first consignment of boxes to South Kensington, there was little with which to compare it. The small bundles of objects surprised the Victorians with new ideas about Chinese art.\n\n* * *\n\nBushell was delighted to be back in Peking. There was the pleasure of sharing everything with his new wife, and of being welcomed so enthusiastically to the compound. There was news to catch up on, patients and friends to visit. And there was the satisfaction of rediscovering his favourite places and immersing himself once again in the sights, sounds and smells of China that he had missed so much. Peking was not an obviously handsome place. It was one of the world's largest cities, a cramped collection of simple, low buildings, their roofs propped on poles, clustered around dusty, unmade streets, rutted by mule carts. The view was frequently monotonous: there was little vegetation, and the rocky ground was open and wind-blown. A huge blank wall, fourteen miles long, broken only by occasional gates and towers, loomed over the close-packed houses and divided the city in two, with a Tartar City to the North and a Chinese City to the south, the ancient and enclosed Forbidden City lying behind yet more walls of its own. There were no parks, and the only gardens were tucked out of sight in private homes. But the temples were beautiful, though often overgrown and weedy, with grass covering the roofs or pushing through the cracks in the stone paving. In the tranquillity of the Confucius temple, gnarled cypress trees sprawled among squat pavilions, and silk and paper prayer offerings fluttered in the breeze. Here Bushell could linger and plan for the future.\n\nIn his early thirties, with a settled family life and his medical career running smoothly, Bushell was in an enviable position. His visits to South Kensington had given him the support he needed to extend his collecting with confidence, and his conversations in London had convinced him that his scholarship would be valued. It was clear to Bushell from his early explorations that the understanding of Chinese art in Britain, even among collectors and so-called experts, was poor. The more he discovered, the more it seemed to him that the most important works were underrated and misunderstood. Even the study of Chinese ceramics, which had become so fashionable in Victorian London, was much more interesting and varied than he had imagined. There was just so much more to the craft than had so far been revealed.\n\nThere was, of course, the K'ang-hsi porcelain which changed hands in European showrooms for large sums of money and which, Bushell claimed, was 'the culminating epoch of the ceramic art in China by common consent of all connoisseurs, eastern and western'. But this was just the beginning. What caught his eye as he sat in the quiet, darkened rooms of his wealthier Chinese patients, or followed a local guide along the narrow and steep paths to secluded shrines, was the fine form and lustre of much earlier Chinese ware, virtually unknown in the West, and certainly not yet at the forefront of fashion. This was historic work that was valued by the Chinese, and which they had kept safely out of the hands of traders and merchants; the ancient pieces of the Sung dynasty (960\u20131279) which Bushell could pick up surprisingly cheaply, and the refined work of the Ming dynasty, dating back to the 1360s. Here was a whole new perspective on the study of blue-and-white, a long and successful tradition waiting to be discovered by a pioneering collector, and a thrill for Bushell as he rummaged through the markets of Peking.\n\nBushell was keen to show what depth and quality there was to traditional Chinese blue-and-white, but it was the unusual, the quirky and the distinctly 'foreign' that really caught his eye. At heart he was an explorer, and simply re-evaluating the wares that could be found in London drawing rooms was not enough to satisfy him. What fascinated him were the pieces of art that helped define the people and explain the ways in which they lived. His acquisitions were more than just pieces for decoration: Bushell understood that they were cultural artefacts, historic markers which gave him an insight into a unique country. Every journey became an adventure, even if it was a brief trip into the markets of Peking to see what was on the stalls. Every conversation was a revelation of the links between the past and the present. Every object, however simple, began to fit into his growing understanding of the culture and its history, adding colour and diversity to the picture. Bushell was eager to find out everything he could about the pieces he saw: how and when they were made; where they came from; who had owned and used them; what they meant. He became a collector of stories.\n\nOne of the things he observed, particularly among his more respectable Chinese friends, was that it was common to wear a plain, flexible band of gold or silver around the neck or arm. At first he did not take much notice. He had never had much of a taste for jewellery, and there was always something else to interest him. Increasingly, he was able to leave his black leather doctor's bag tucked neatly away at home and venture out just for the pleasure of it. He was calling socially on old patients, and in the streets Chinese locals who had once welcomed him only for his medical skills were more than happy to talk about other matters. All the time, he was being shown new things and, with so much to attract his attention, the plain rings and bangles seemed insignificant.\n\nBut, as time went on and Bushell saw more and more people wearing the bands, he began to ask questions. Listening to the stories, he discovered that the necklaces and bracelets were much more interesting than they had first appeared. The bands were far more than simple ornament or fashion \u2013 they were portable property, the family wealth. But unlike the diamond necklaces and elaborate tiaras of the English upper classes, brought out with a flourish on important occasions to signal status and prosperity, this Chinese jewellery was practical and plain. What was important was its solidity: the weight and purity of the metal were far more significant than any craftsmanship or decoration. When times were hard, it was relatively easy to strip off pieces as required, sell them on and melt them down to provide funds. His appetite whetted, Bushell asked more questions and discovered that the custom of wearing bands was so ancient and widespread that it had become carefully organized and systematized: each piece was stamped by the original jeweller who, at any time in the future, was honour bound to buy back strips from his customer, by weight, without questioning the quality. Bushell was now alert to the bracelets: what had seemed insignificant and undistinguished became full of interest. He could not help noticing the bands everywhere he went, amazed at himself for not seeing much sooner that this was more than a random taste for body ornament. In a city full of beautiful and unfamiliar things, Bushell realized that even the smallest, plainest and most intimate objects were important. His European perspective was shifting, and he was beginning to see the details of Chinese life with new eyes.\n\nSuch small discoveries delighted Bushell, but he was not one for publicizing his breakthroughs. He kept his notes diligently but inconspicuously, and he treasured each advance that brought him a step closer to understanding China without feeling a need to broadcast his growing expertise. All over the world, royal, government and diplomatic representatives were making inroads into new lands, and Bushell knew that he was just a tiny part of a vast imperial network. He was, officially, nothing more than a respectable physician, helping the British machine to function and upholding its values. Many other expatriate administrators collected, and many local and national museums were grateful for the objects they sent back for display. Bushell was under no illusions as to his uniqueness or distinction. But even so, in his quiet way, through his systematic note-keeping, careful observation and scholarly collecting, Bushell was making himself special. He was doing more than simply picking up curiosities so that people back in Britain could wonder at the strangeness of the world \u2013 he was changing attitudes.\n\n* * *\n\nWhen it came to thinking about the East, the Victorians tended to view Japan as sophisticated and cultured, producing a range of objects which showcased the exquisite skills of Japanese craftsmen and were worthy of fashionable attention. Japanese objects were being imported in large quantities by the 1870s and popular shops like Liberty's set the tone for stylish middle-class homes. Japan was romantic. Yet, despite its appeal to the popular imagination, a patronizing mindset still lingered, a sense that beneath the refinement lurked a semi-barbarous 'foreignness'. Even Britain's first minister to Japan, Sir Rutherford Alcock, described the country as a place of primitive violence and natural disasters, noting that experiencing Japanese life was like taking a 'step backward some ten centuries to live again the feudal days'.\n\nBut such condescension was nothing like the disdain and suspicion directed towards the Chinese. If the West was the source of rationality, order and progress, then China especially remained resolutely corrupt, chaotic and unsound. A taste for ' _chinoiserie_ ' had been fashionable among the European elite since the eighteenth century, a special exhibition of 'Ten Thousand Chinese Things' had been shown in Hyde Park in 1841, and Chinese objects had been part of London's Great Exhibition, but these events had done little to change perceptions of the country, all too easily associated in Victorian minds with the seediness and corruption of the opium trade. The Chinese were typically viewed as childish and incapable, militarily weak and embroiled within a stagnant culture: 'toddling, little-eyed, little-footed, little-bearded, [and] little-minded', suggested the writer Leigh Hunt dismissively. With the experimental new science of photography, travellers like John Thomson, an Edinburgh photographer, were able to document some of the most remote and spectacular areas of China, bringing to the English middle classes images of unfamiliar landscapes and communities, of pagodas, river villages and the Great Wall. But, even so, opinions remained entrenched: what the British saw in the photographs simply reaffirmed the distinction between the barbarous East and their own more civilized way of life.\n\nSuch attitudes were fully on display in public museums. Most visitors expected the presentation of the collections to confirm the way in which they saw the world; in turn, the museum used the objects as a way of shaping opinion. The mechanics of this complicated mutual relationship were rarely explicit, but were clear enough in the arrangement of the exhibits and the language used to discuss them. The American _Harper's New Monthly Magazine_ thought the displays of Chinese objects at South Kensington were important not only for bringing to light 'the romance of the East' but also, and more significantly, for their ability 'to revise, correct and estimate the traditions of the Oriental world'. Only through the discipline imposed by the museum, it implied, might a respectable and acceptable impression of China emerge. The 1872 _Catalogue of Chinese Objects_ at the South Kensington Museum was blunt about what it called 'the Chinese character', and was anxious to emphasize the distance between the middle-class visitor and the Chinese on display. 'It would hardly be supposed that an effeminate race like the Chinese should have a taste for working in metal,' it pronounced, 'but it must be remembered that they have not always been a degenerate race, softened by luxury and by too great a facility for enjoyment, but that on the contrary, they are still a hardy race.'\n\nStephen Bushell, of course, had been brought up with such attitudes, and would have been familiar with the contempt which the British were apt to show to unfamiliar parts of the world. He worked for several years in Peking under Sir Rutherford Alcock, whose views of both the Chinese and Japanese were uncomplimentary, and life in a government enclosure would have given Bushell a daily dose of imperial pride and prejudice. But, from his own explorations of Peking and the land around, he knew that there was much more to China than most British people could ever imagine. The stories he listened to outside the compound revealed vibrant histories and contemporary achievements that confounded the common view of the Chinese as 'degenerate'. And he understood that perhaps the best way for the Victorian imagination to grasp the complexity, significance and beauty of Chinese culture was through the artefacts that he was able to collect and explain.\n\nBushell found an ally in Augustus Franks. Franks, too, was unwilling to be seduced by unthinking and damning caricatures. His wanted his ethnographic collection to speak for itself, to encourage intelligent minds into a new perspective on the world. He wanted more than just an attractive display of curious objects; he was determined to extend Western knowledge of distant countries and understanding of different cultures. His enthusiastic collecting of Indian pieces, for example, had little to do with asserting imperial supremacy or showing off the bizarre accomplishments of a heathen race. Instead, he wanted to raise understanding of Indian art until it was regarded as important and beautiful in its own right: 'I am ambitious to show the fanatics for Greek and Roman sculpture that the art of India is not to be despised,' he wrote in 1881. Similarly, his energetic pursuit of Islamic pieces helped persuade museum staff and visitors that these objects were worthy of a place alongside the esteemed Western antiquities of the ancient world. Franks drew the line at costumes and examples of modern manufacturing from other countries, but otherwise he was quick to see the value of even the most ordinary objects, and their ability to change the way his fellow Victorians looked at cultures beyond their own: even 'the commonest things of the country', he maintained, had a place in explaining and understanding the world.\n\nBushell soon recognized a kindred spirit in Franks, and, as well as working for the South Kensington Museum, he also joined the international network of collectors serving the British Museum curator. He was not content, however, with just sending back 'the commonest things' and worked hard to find objects that would intrigue, delight and inform back in Britain. He began to establish, and study, a considerable library of books to help him in his search, and he gradually extended his network of local contacts. As time went on, he developed an astounding breadth of knowledgeable interests in his curiosity about all things Chinese. Scrambling up trees, climbing rocky slopes and sliding down riverbanks, he collected plant specimens and seeds for the Royal Botanic Gardens at Kew, thinking nothing of wet feet, muddy knees and the bites of strange insects. He began to study a range of complex extinct Chinese scripts, coming to recognize the difficult characters and in some cases decipher them. He even led a small expedition to the sacred burial grounds of the Qing emperors in the mountains outside Peking so that he could capture rare monkeys that he had seen there, scampering around the ancient stonework. He successfully managed to lure a pair of young animals into his trap, before sending them back on the long journey to the Zoological Society in London.\n\nHis correspondence with Philip Cunliffe-Owen and the staff at South Kensington became increasingly enthusiastic. In 1880, he sent a consignment of bronzes for display, followed a couple of years later by four pieces of carved and perforated stone that were traditionally used as weapons and which were, Bushell added proudly, 'relatively rare in China'. Soon, he was given permission to buy officially on the museum's behalf. Instead of financing everything personally, and loaning his own pieces back for display in London, Bushell was now given \u00a3250, with the understanding that he would act as an authorized agent and use his expertise to acquire 100 objects at around 50 shillings each. He was delighted. The arrangement gave him both extra funds to collect and the museum's formal sanction for his activities. 'The specimens will be collected gradually and sent to England when opportunities occur,' he wrote cautiously in February 1882 when the agreement was first made, but such was his eagerness \u2013 and the richness of what was available \u2013 that by November he had already sent a range of objects to the museum, hoping, modestly, that they 'may fill some gaps on the shelves'. By spring 1883, he had asked for, and was granted, another \u00a3250 to spend. Jars, cups and dishes; figures, bottles, bowls and bronzes; a sacrificial wine vessel in the form of a rhinoceros and an ivory lion, as well as three glorious examples of early Ming porcelain, all found their way from Peking to South Kensington under Bushell's watchful connoisseurship. Perhaps more importantly, so too did at least some of his understanding of the culture from which they came. These were not objects from the dark, backward China of the Victorian imagination; they were the evidence of a precious past and of a country worthy of more positive attention. Bushell's collecting was beginning to reveal the East in a new light.\n\nDespite what must have been frequent and lengthy absences from the compound, and a growing collection of lovely things adorning his home within it, Bushell's enthusiastic activity in Peking was rarely remarked upon by his British contemporaries. Caught up in the day-to-day responsibilities of government administration, it seems that they were unaware of the progress he was making in China, or of his contribution back in London. Quiet and unassuming, looking every inch the respectable and uncontroversial Victorian doctor, with a neatly trimmed beard, erect, even slightly stiff, stance and earnest demeanour, Bushell was content to keep his achievements to himself, perhaps unaware of quite how much he was accomplishing. When people asked him about his research into Chinese art, or about his collection of objects, he was modest and self-effacing, keen, he said, to 'disclaim any pretension to authority'. He had no desire to get caught up in the academic debates that were starting to erupt as the study of China, or sinology, became more widespread in Europe, and he was acutely aware that, however much he learned, there was always more out there waiting for him in such a vast, ancient and varied nation.\n\nBushell also knew that he was not alone in his collecting. As Mary Crawford Fraser admitted, 'Of course everybody collected. Half the time there was nothing else to do.' Whiling away a long morning perusing the artefacts 'artistically spread out for inspection', or indulging in 'everlasting bargaining', was just part of the 'native' entertainment of living abroad. During the Peking winter, from November to March, the British delegation was virtually cut off. All provisions had to be brought from Shanghai before the weather closed in and blocked trade routes, and the highlight of the year was provided by a delivery from London, from the Civil Service Stores or the Army and Navy store, once communications reopened. Collecting was a diversion from the enclosed monotony of life, a common talking point. Perhaps this explains why Bushell's activity went largely unnoticed by his colleagues. But, unlike the dilettante souvenir hunter, he was not simply killing time with a hobby. What set him apart was his knowledge and perseverance, his consistent attention to detail and his willingness to dedicate long hours to the study of the pieces he found. In many ways, he was the archetypal scholar, collecting for his own pleasure and to further his own knowledge, happy to share what he knew through the objects he sent back to London, and utterly devoted to his subject.\n\nIn time, Bushell collected his knowledge into an authoritative publication, _Chinese Art_ , a pioneering study of oriental objects and the culture from which they emerged. He chose to end the first volume with a description of a white agate vase. It was elegant and graceful, a fine example of craftsmanship and a rare piece. And, like so many of Bushell's objects, it was intricately entwined with a story, the fable of the successful scholar. The vase, Bushell noted, was carved with three fishes, animated and lithe, captured in the act of springing into the air and becoming dragons. Only after a fish had managed to overcome the rapids and waterfalls of the Yellow River, after long perseverance and much noble effort, was it worthy of such a transformation. Bushell explained that the fish carved into the vase represented the triumphant scholar who, after arduous study, finally managed to have his name added to what the Chinese called the 'dragon list' \u2013 the first step on the ladder of official rank. Poring over the complexities of the Chinese script and overwhelmed by the variety of objects around him, Bushell may well have felt that he was still swimming hard in the river currents. But his collecting increasingly demonstrated an understanding of the combination of material, technique and symbolism, of tangible object and enigmatic myth; showing just how well he was coming to know China, and how close he was to taking the leap as a dragon.\n\n### CHAPTER TWENTY\n\n### _Collecting Without Boundaries_\n\n__M__ agic Chinese bronze mirrors had first been seen in Europe in the early nineteenth century, when distinguished scientists had tried to explain how they worked. When Bushell saw one for the first time, with the sounds of Peking city life buzzing outside, he was enchanted and mystified. It did not seem possible. The front of the mirror was simply a highly polished bronze disc, slightly convex, its surface acting like an ordinary mirror and reflecting the image of anyone looking into it. The reverse was elaborately moulded with mythological figures, animals and birds, and floral scrolls in strong relief. The magic happened when the polished face of the mirror was held up to catch the sun streaming in through the window. Then, on the wall opposite, Bushell saw an exact image of the raised decoration on the back of the mirror. There was still nothing on the face except the glare of the sun, but it was as if the light was passing straight through, as though the mirror had become transparent, a thing no longer of solid metal but of air and light.\n\nBushell never tired of seeing the 'trick' repeated but it was not long before he found a logical explanation. It was, he concluded, 'an accidental effect', a dramatic but explicable phenomenon caused by 'irregularities on the reflecting surface as a result of uneven pressure in polishing'. But this hardly seemed to spoil the spectacle. It remained special, made even more so by the rarity of the mirrors, and Bushell's knowledge that he was privileged to see in action objects that had been a source of pride to emperors as far back as the Han dynasty (206 BC\u2013AD 24). Memories of the mirrors' translucence stayed with him, and, when he came to write up his experiences of Chinese art many years later, he relived the moment, bringing to life once again the mystery of the magic mirrors, not as a bizarre party trick for the Victorian parlour, but as evidence of Chinese tradition and ingenuity.\n\nBushell was always careful to see beyond the enigma of the mirrors, noting how beautifully they were designed and crafted, each a masterpiece of bronze metalworking and finishing, and evidence of the refinement of ancient techniques. With a collector's appreciation of fine workmanship, Bushell emphasized the skill of the decorative moulding and the quality of the materials, as much as the spectacle of illusion. This was what always drew him back to Chinese art: 'The connoisseur always looks at the intrinsic properties of the medium, and its effects in bringing out the skill of the craftsman which ennobles it.'\n\nClose inspection and detailed study became the hallmarks of Bushell's approach. Every style, skill and material came in for rigorous scrutiny. Ivory appealed to him, particularly Canton ivory, because 'there is no material more satisfying to a delicate and refined taste', and he spent long hours in the hot, close Peking summers learning the feel of the bone in his hand, studying the intricacies of the carving and examining the patina of the vessels in the dusty alleys along the city walls. He became an expert in Chinese glazes, watching modern craftsmen at work as well as studying the techniques of the past, and he discovered that it was the addition of lime to the glaze that gave the familiar lustre to Chinese ceramics, creating 'a characteristic tinge of green or blue' as well as 'a brilliancy of surface and a pellucid depth'. He investigated the methods for working jade, and he familiarized himself with the complex practice of lacquering, prizing in particular the elegance and delicacy of Foochow lacquer. All around him was the evidence of unique and historic skills, luring him away from the aches and pains of the British officials and their wives, and into the bustle of the city markets.\n\nArchitectural detail, too, fascinated Bushell. He observed many of the same materials and techniques employed on a larger scale, to create ingenious buildings and stunning decorative effects. At the ChangLing Tomb, the best preserved of the thirteen Ming Tombs in Peking, he admired 'sunken panels worked in relief and lacquered with dragons' and in the Imperial Summer Palace, seven miles outside Peking, he loitered on the Pavilion Bridge, 'hung with bronze bells which tinkle softly in the breeze', and clambered over the steps of the Pavilion of Precious Clouds, 'piled with bricks and bushes to keep off pilferers', to get close to a building that was made entirely of cast bronze, shimmering blue in the light. As with the smaller objects he took home to study, Bushell investigated details of construction and decoration techniques, exploring the intricacies of large-scale design with the same precision he applied to smaller pieces of art. Walking through the trees to discover the turreted pagoda in the Changchunyuan, or Garden of Everlasting Spring, in the Imperial Summer Palace, for example, he was struck by its distinctive colour and form, and went to great efforts to describe the ways in which these effects were achieved. 'The glazes used in the decoration of this pagoda are five in number; a deep purplish blue derived from a compound of cobalt and manganese silicates, a rich green from copper silicates, a yellow, approaching the tint of a yolk of an egg, from antimony, a _sang de boeuf_ red from copper mixed with deoxidising flux, and a charming turquoise blue derived from copper combined with nitre,' he explained, with the scientific eye of a doctor, before adding a note with the enthusiasm of the collector: 'The fivefold combination is intended to suggest the five jewels of the Buddhist paradise.'\n\nBut, even in the 1870s and 1880s, the objects of Bushell's admiration were not untouched. The ruins of the Imperial Summer Palace were not the result of centuries of natural decay and degradation, but of attack by British and French troops in 1860, after the end of the Second Opium War. In a retaliatory operation under the command of James Bruce, 8th Earl of Elgin, whose father had 'collected' the Elgin marbles, 3,500 British soldiers set the buildings alight in a fire that lasted three days. Only thirteen buildings in the enormous complex of palaces remained intact, mostly in remote areas, and the elaborate gardens were destroyed. There was widespread and chaotic looting. Over subsequent years, a variety of important pieces were sold to eager collectors by profiteering soldiers and local adventurers: embroidered robes, sculptures and carvings, furniture, paintings and porcelain. It is estimated that one and a half million relics eventually found their way into more than 2,000 museums in forty-seven countries, while many thousands more ended up in private collections. At the British Museum, Augustus Franks negotiated directly with the military to acquire a pair of ancient vases and some glazed architectural roof tiles \u2013 similar to the ones Bushell so admired \u2013 which were taken as souvenirs by Captain de Negroni, a French officer. Plenty more ceramic pieces were donated by returning soldiers. All over Britain, the growing fashion for things Chinese was swelled by the looted treasures; collectors were anxious to profit from the sudden arrival of so many interesting objects and the market flourished. Back in Peking, amid the ruins of the Summer Palace, Bushell could only go so far with his studies. Ironically, many of the objects that would have most illuminated his work were now scattered across the world.\n\nLooting was just one more way of getting objects on the move. In an age of international collecting, neither collectors nor objects seemed confined by national boundaries. The aristocratic atmosphere and dignified dining of the Fine Arts Club might have made it seem a quintessentially English organization, busying itself with London affairs and serving the needs of British collectors. In fact, it reflected the expanding world of collecting: truly international in scope and aspiration, it was in contact with collectors and connoisseurs from far and wide. It was not just that some members acquired a taste for things Chinese or Indian, for exotic objects from faraway nations. Nor was it that many members had trade interests that took them beyond Britain, making fortunes abroad that they could then spend at home on their collections. There was also a more far-reaching and fundamental internationalism, a growing recognition that collecting was a way of understanding, appreciating and managing the world, and that shrinking distances were allowing collectors from the furthest reaches of the Empire to meet together in London.\n\nIn 1840, the sickly Charles Fortnum, already bored at the age of twenty with his family's business interests, emigrated to Australia to develop land in New South Wales. He did not stay for long; five years later, he was back in London. But the trip had given him a taste for travel. He had discovered a new world whose natural history absorbed him, and a fascination with collecting that was to last a lifetime. Back in England, he married a wealthy cousin, who could fund his new enthusiasm, and immediately set off on buying trips; he began a long association with the Ashmolean Museum in Oxford, the British and South Kensington Museums, developed an expertise in Early German prints and became a member of the Burlington Fine Arts Club. There he met an Australian, George Salting, who had grown up in Sydney. They had plenty to discuss. They knew the same places in the growing Australian city, the bizarre flora and fauna of its bay, and they shared experiences of the protracted sea voyage. They became friends, and, when, at the age of thirty, Salting was left the fortune his father had made in Australia's sheep stations and sugar plantations, both men sat together exchanging collectors' anecdotes and scouring the catalogues and sales bulletins for the furniture, jewellery, bronzes, medals, enamels, ceramics, ivories, glass, textiles, leatherwork, manuscripts and paintings that were to become part of Salting's magnificent collection. It was a long way from the harsh open terrain of New South Wales and the undiscovered wilderness of nineteenth-century Australia; the Fine Arts Club provided a retreat for collectors, no matter how far they had come.\n\nTrends set by high-profile collectors like Salting were taken up all over the world, and particularly in America. Salting's enthusiasm for Chinese porcelain, for example, was shared by men such as William Thompson Walters, a Baltimore liquor merchant and collector who had opened his house to the public (for a visiting fee of 50 cents) in the mid-1870s. There was nothing limited or introspective about this type of collecting. There were no boundaries, and distance was not an obstacle. If an object had to be shipped around the world from its place of origin, sold in one continent and displayed in another, then there were increasing numbers of people with the resources and dedication to make this happen.\n\nWhen William Thompson Walters developed an enthusiasm for Chinese porcelain and decided to assemble a significant collection, Stephen Wootton Bushell was by now the obvious man to guide such a project. By the 1880s, Walters' fortune was vast. What had begun as a relatively modest grain and then liquor business had expanded to take in banking and railways, from Washington DC to Florida and Missouri. He was a director of every line of steamers between Baltimore and the Southern states; he bred horses, patronized local Maryland artists and had amassed more than 3,000 pieces of Chinese porcelain in forty years. All that remained was to record his achievements for posterity, and to find someone with the expertise to catalogue his pieces of oriental art. His first port of call was the British Museum, but staff there, aware of their own limitations, directed him instead to Peking where Bushell was continuing his studies. Without hesitation, Walters commissioned Bushell to write his catalogue, and in the late 1880s arranged for him to make the journey from China to see the pieces for himself.\n\nAfter long years in the low-lying streets of Peking, arrival in Baltimore's busy port was something of a shock for Bushell. A smoggy industrial hub, the city echoed with the noise of shipbuilding and its skyline was already straining upwards, spiked with church spires and factory chimneys. Bushell was taken through the city to Walters' home in Mount Vernon Place, where the imposing townhouses with their European architecture and the neatly laid park around the Washington memorial column brought home to him again how far he had travelled. This bold American grandeur seemed like a different world. But, before too long, Bushell found himself surrounded by familiar objects, by forms and designs and colours that he had seen so many times before in Peking houses and market stalls, and the alarming size of the globe contracted to a room filled with China.\n\nWe don't know how long Bushell stayed in Baltimore to study Walters' collection. After such a journey, and with so much work to do, it seems safe to assume that he spent several weeks at least, and possibly several months, getting to know the collection, cross-referencing with the books he had brought with him from China and drawing on conversations he had had with Peking merchants and collectors. The two men had little in common beyond their collecting, but this proved more than enough to sustain them. Bushell had found a collector who was as passionate about Chinese culture and objects as he was, and in their spare moments there was also a variety of other distractions in Walters' vast collection: contemporary European paintings, sculptures, French landscapes and antique vases. With Walters' enthusiastic support, Bushell described not only the Chinese objects themselves, but also the intricacies of their construction and decoration techniques, their historical and social contexts, and their relationship to similar objects across the East. He translated ancient Chinese texts and consulted modern ones. The catalogue grew and grew, the pages of Bushell's manuscript mounted, his notes expanding into long and erudite descriptions. The time came for him to return to China, and still he wrote. As life settled back into a round of medical visits, the book continued to take shape, with the routine of writing punctuating the days.\n\nRooted in Walters' magnificent collection, the catalogue soon became much more than just a simple record. After returning from Baltimore as a young man in his late thirties, Bushell worked on the book for fifteen years, marking his forties with chapter after chapter of studious text. The result was to be a major work: the first comprehensive study of Chinese, Japanese and Korean art in the English language, a pioneering undertaking. Entering enthusiastically into its scale and ambition, Walters took on the task of commissioning illustrations, scouring America for someone whom he believed could provide accurate pictures to accompany Bushell's writing. He chose Louis Prang, from Massachusetts, to work on the lithographs, instructing him to supply the highest quality regardless of cost, and approving a final count of 2,000 printing stones to create 116 lithographic plates. He looked even further afield for an artist he thought capable of doing the water-colour illustrations, employing an Englishman, James Callowhill, along with his three sons, converting the second-floor north-facing library of his house into their studio and installing them in their own living quarters for the duration of the work. Money was no object. Walters was reportedly willing to pay $250,000 to see the book produced to the highest standards and he was notoriously exacting in ensuring that everything was of the very best quality.\n\nIt was not until 1896 that the book was finally ready. It was entitled _Oriental Ceramic Art_ and it ran to ten volumes, a monument to Bushell's scholarship and nineteenth-century American publishing. Each volume was lavishly bound in yellow cloth, backed with yellow silk, with silk bands to act as markers, each one tipped with a fine bone handle. Alongside the coloured plates, there were over 400 black and white illustrations, and each book in the limited edition of 500 was carefully numbered and recorded; subscribers' names were printed on the title page in return for their $500 investment. The finished article was exactly what Walters had hoped for. But, while Bushell now had this lavish achievement to his name, marking him out across the world as an intellectual heavyweight, his American employer did not live to see the project complete. Walters died two years before the catalogue's publication, in November 1894, with a reputation for 'judgement that was authoritative on all questions of art' and leaving behind him an incomplete masterpiece.\n\nEven before _Oriental Ceramic Art_ was published, the book was attracting attention. Collectors all over the world became aware of Bushell's work, and watched with interest as the project evolved, not only intrigued to know what was in Walters' collection, but also interested to see what impact the catalogue would have. In America, particularly, others were eager to replicate Walters' ambitious venture and to have their own collections recorded in a similar way. The quiet scholar from Peking was in demand, and it was another wealthy American businessman, Heber Reginald Bishop, who was next to employ Bushell's services for posterity.\n\nBishop, who had made a fortune first in exporting sugar from Cuba and later in gas, iron and the railways, had a taste for jade. He had known Bushell for some time \u2013 probably through his connections to Walters \u2013 when in 1889 he asked him to begin a catalogue of his jade collection. This was not a particular speciality of Bushell's, but he already knew something of the long and complex tradition of jade-working, and recognized that 'there is much to be gathered from Chinese sources' to illuminate Bishop's pieces. He agreed to become involved and Bishop, inspired by _Oriental Ceramic Art_ , made the long journey to China to see what they could unearth together. From his home in New York, he travelled overland to Vancouver, before setting sail for Yokohama in Japan and on to Hong Kong, continuing to Shanghai before finally arriving, many exhausting weeks later, in Peking.\n\nFortunately, the journey proved worthwhile. Bishop and Bushell purchased a wealth of spectacular jade objects, from practical but delicate vessels to pendants and ornaments designed for emperors; from flower holders and head rests to sceptres and screens, many with elaborate pierced and relief decoration. Traditionally, Chinese jade was a special material, invoked by poets to reflect the colours of nature and revered as a 'singing stone' because of its resonant qualities. It was reserved for imperial use. Workers laboured in a special compound of the palace enclosure, under close supervision from officials; the pieces they made were banned from sale to the outside world \u2013 so it was not an easy thing for an American visitor to collect. It took guile and bartering skills, Bishop's apparently bottomless purse and Bushell's long-standing network of loyal contacts to extract some of the pieces from their hiding places and amass a shipment to go back to New York.\n\nIn quieter moments, the two men also found time to agree a way of working on the catalogue and it was settled that Bushell would go to America for three months the following year to study the jade and help Bishop put together a publication. For Bushell, it was quite an undertaking. Still in the middle of writing his catalogue for Walters, it meant another long voyage overseas, new and demanding research and the challenges of working with another opinionated American tycoon. It was an intimidating prospect, and there were no doubt moments when Bushell felt he would have preferred things to have remained as they had been before Bishop's arrival \u2013 unhurried days of calm study in and around Peking's familiar monuments and undisturbed buying from trusted traders. Unfortunately, the process of publication turned out to be just as tortuous as Bushell feared: it was ten more years before he had amassed the information he needed about the collection, finished working with Bishop on the text, and completed 'procuring specimens of interesting ancient pieces' on the American's behalf. The final two-volume catalogue was not published until 1906; in the Preface, Bishop praised his British colleague 'for the great assistance given to me, from first to last, in this work and all my studies of Oriental Art'.\n\nHowever daunting the task might have seemed, Bushell had little choice but to take it on: he needed the money. Bishop agreed to pay \u00a3300 for Bushell's work in New York writing the catalogue entries, an amount which was equivalent to half the doctor's annual salary. It was a financial lifeline. Beyond the day-to-day routine of medical consultations, government paperwork, the social life of the Peking compound and his reading and writing, a stalling world economy and general financial uncertainty had left Bushell broke. Just a few weeks after Bishop's visit to Peking, the collapse of the Oriental Bank took with it the family savings, and any kind of security Bushell and his wife might have hoped for. 'We are well-nigh ruined,' Bushell admitted plaintively to Bishop.\n\nBut even the agreement with Bishop was not enough to do more than ease immediate financial worries. In the longer term, Bushell realized that he would have to find additional income to ensure some reserves for the future and funds for his collecting. He was forced to consider what he could sell: his knowledge and expertise. There was already the work for the extraordinarily wealthy like Bishop, acting as researcher and adviser, but he guessed there was also the possibility of acting as an agent for other, less wealthy, collectors who might be willing to pay him to buy in China on their behalf. With the resourcefulness of the impecunious, Bushell was soon contacting friends, colleagues and friends of friends to tender his services. He knew he had plenty to offer: there were still very few Europeans who had spent any length of time in China, or who knew as much as he did about its culture and there was hardly anyone who was as familiar with the markets, or as friendly with the traders. Indeed there were no other collectors in such a privileged position to acquire the finest Chinese objects.\n\nBushell's persistence paid off, and before long he was supplying a number of clients from a variety of backgrounds and with a range of interests: coins for Sir James Stewart Lockhart, Registrar General and Colonial Secretary in Hong Kong; archaeological specimens for Sir Aurel Stein, a British adventurer and explorer in Central Asia; and porcelain for Alfred Trapnell, a humble metal smelter from Clifton in Bristol. He kept his activities low-key, and was loath to advertise himself as any kind of established dealer, but nevertheless the business grew. As with his own collecting, Bushell was thorough in his research and wise in his buying, and no doubt his customers were delighted with the carefully chosen pieces he packaged up to send to them. But, even with a core of loyal paying clients, money was still tight and, having started to profit from his talents, Bushell began to see that with some astute buying and selling he could put an end to his financial woes for good. Perhaps inspired by the example of the American entrepreneurs with whom he was now working, Bushell abandoned the Victorian suspicion of 'trade' and decided to branch out into perhaps the most lucrative of his activities: playing the London art market.\n\nThe fashion for blue-and-white china, particularly pieces from the K'ang-hsi period, had not passed Bushell by. He may have been thousands of miles away in Peking, but he was aware that large amounts of money were changing hands in London showrooms for things he saw every day on local market stalls. The opportunity seemed too good to miss. Setting himself up to trade was not without awkwardness, however. Bushell collected antique and exclusive ceramics for himself and he selected similar pieces for his private clients, but the market in England for 'Old Nankin Porcelain' was apparently so voracious that just as much money was to be made from contemporary reproductions with none of the cultural history that Bushell so valued. There was nothing intrinsically wrong with such modern Chinese pieces, but, designed to look like antique K'ang-hsi wares and 'well calculated to deceive the unwary', they were part of a dubious trade designed to make quick profits out of enthusiastic but na\u00efve collectors.\n\nIt is clear that Bushell had reservations about what he was doing: during a series of sales at Christie's during the 1880s, he was careful to remain anonymous, appearing in the catalogues simply as 'A Gentleman in China', and never attending the sales himself. He was no doubt conscious of his dignity as a public servant, but the secrecy about his identity also suggests, I think, a certain embarrassment in exploiting a fashion purely for profit, even though there were plenty of other dealers doing just that. As a gentleman doctor and government representative, Bushell would have been wary of being seen as just another dealer, which could well have undermined his social standing. As a scholar and a collector, too, he might have wanted to keep this kind of business separate from his renowned work on high-quality pieces. However, as a financial experiment the venture was a success. One auction alone, in August 1888, brought Bushell over \u00a3600 (more than his annual salary), and it seems likely that over the decade he managed to earn as much as \u00a310,000, a considerable supplement to his doctor's income. But by continuing to feed a market that was already frequently unscrupulous, confused and opaque, the sales can also be seen as something of a lapse on Bushell's part, no doubt driven by necessity, but nonetheless threatening to undo the work of long years spent trying to dissociate the serious collecting of Chinese objects and the study of their contexts from the frivolous dictates of popular fashion.\n\nFrom doctor to scholar to dealer, Bushell's life in Peking had taken a number of unexpected turns. He remained in post there for thirty years and never lost his affection for the city, his fascination with its buildings and delight in its objects. But what was a treasure trove for a learned collector was not always so ideal for family life, and Bushell's wife and children had to suffer the hardships of foreign living without the benefits of his diverting interests. Florence was resourceful, positive and loyal, and remained firmly by her husband's side throughout the three decades, but the infant mortality that was ever present in Victorian England was yet more of a fear in Peking. Even a doctor's family suffered, and in the winter of 1892, Sir Robert Hart, Inspector General of Customs in China, wrote: 'We are once more sorrowing for the Bushells who have just lost their little two-year-old daughter Dorothy, the fifth I think to die out of six charming children'. Thankfully, the Bushell's youngest child, a boy, managed to survive into adulthood, but the years of sadness and loss in the Bushell household were long, and Bushell's exhilarating days in the grounds of ancient palaces were tempered by many agonizing nights at the bedside of a dying child. When his own ill-health finally forced him to retire from government service and return to Britain in November 1899, he knew the voyage home was a final one. He would not be returning to China again. As the ship pulled out of the harbour at Shanghai, there was the sorrow of parting and regret at the things that would remain undiscovered, but also a sense of relief at having come through something of a trial.\n\nBushell anticipated using his retirement to reinvigorate research on his personal collection and to begin work on what would become _Chinese Art_. In preparation, he wrote to South Kensington asking for the return of a group of Chinese bronzes which he had collected during the 1880s and loaned to the museum for display. He was formal and reserved, even curt. Since his early visits to the museum as a young man, the staff had changed and he now felt only a professional courtesy to those dealing with his request. Besides, he was eager to begin installing his collection in the house he had taken in Upper Norwood, south London. At the museum, the request was completely unexpected, and the staff were horrified. Accustomed to loans metamorphosing into donations, they had had no expectation that Bushell might actually want his objects back; it was noted with obvious dismay that 'the return of these bronzes will make a serious gap in our Chinese collection'. George Salting advised the museum to try to buy the pieces instead, and, since this seemed the only option, hurried arrangements were made to organize a sale and thus preserve the displays.\n\nBushell, however, was 'not at all willing to part with the collection'. He had plans for it. He entered into a tetchy correspondence with South Kensington, defending his right to the bronzes. In the end, he only agreed to consider the idea of a sale because it was 'put to him that if purchased by us, his name would be attached to the objects which would form an important collection'. Like many collectors, the idea of having his name publicly linked, for posterity, to an assortment of beautiful pieces was too tempting for him to resist. There was fame in it, beyond the usual expectations of a government administrator, and there would be clear evidence for future generations of his pioneering role in understanding Chinese art. Bushell may have been unpretentious and modest, but he was not without enormous pride in his collections, and the museum's offer to publicly recognize his contribution turned out to be irresistible.\n\nThere were further negotiations. Bushell insisted on keeping some weapons and a large vase 'because it was a present to me', but he agreed to sell the rest of the objects for \u00a3400. This in turn upset the museum, since George Salting had valued the collection at only \u00a3350 and budgets were, as ever, under pressure. It looked like a stalemate, but the museum was only impoverished, not foolishly stubborn. It was acknowledged that 'it is improbable that an equally favourable opportunity will occur of purchasing such a collection' and in the end the sale was agreed before Bushell's return to England; museum staff finally accepted his valuation on the grounds that, since he had not wanted to sell in the first place, 'we cannot well ask him to reduce his price'. Florence, who had gone to London ahead of her husband to prepare their new home, accepted delivery of the vase and weapons as Bushell was leaving Peking; the remaining pieces were left, to the museum's relief, to form the core of the galleries' Chinese display.\n\nAfter his retirement, Bushell spent his remaining years quietly, surrounded by the pieces that evoked for him the spirit of the China he had left. He continued to study and worked hard on _Chinese Art_ , which was published in two volumes by the South Kensington Museum in 1904 and 1906. Although it professed to be simply a guide to the Chinese collection, it was of course much more. It was a unique and lasting achievement, a fine piece of scholarship in praise of the exquisite objects and fascinating traditions that had enthralled Bushell for so long. It is still valued today. Although Bushell remained characteristically modest about his achievements, he knew that the publication would identify him forever as a collector and connoisseur, and he was proud of this. There was also something reassuring and satisfying about seeing his contribution made tangible in museum displays, and when he died, two years after the second volume of _Chinese Art_ appeared, Florence agreed to a series of loans and sales which further consolidated his presence at South Kensington. With prices for Chinese objects already rising sharply, the museum took advantage of a number of bargains. Thanks to Bushell's canny dealing in the ceramics markets, Florence was, staff noted, 'not by any means in straitened circumstances' and she seemed willing to preserve her husband's memory at the museum at the most reasonable of costs, agreeing to accept \u00a3700 in payment for vases valued at \u00a31,054 and selling a number of small items for as little as \u00a35 each.\n\nWhen Bushell sailed from Shanghai for the last time, just before the dawn of the twentieth century, China was changing fast. It was already more familiar to Europeans than it had been when he arrived during the 1860s, and was now a target for international territorial ambitions as well as an important trading partner. Peking, too, was becoming increasingly accessible and within a few years was linked by rail to strategic centres across China and so to the rest of the Continent. But it was also becoming a more turbulent and dangerous place for foreigners as locals rejected what they saw as the corruption of the West. Just a year after Bushell's departure, this spilled over into overt violence: during the Boxer Rebellion of 1900, the foreign legations in Peking were put under siege for several months, businesses that dealt with Western clients were attacked and there was fierce fighting between European and Chinese troops. The opportunities for collecting as Bushell had done \u2013 thoughtfully, sensitively and over long decades \u2013 were disappearing, and the contribution he had made to scholarship was increasingly recognized as bold and unique. Few collectors in the future would have such an opportunity to unravel the mysteries of an unknown nation through its objects, or would make so much of the chance when it came.\n\n## _Epilogue_\n\n'To stay what is fleeting. . . to immortalize things that have no duration.'\n\nJohn Ruskin\n\n__W__ riting in _The Stones of Venice_ , in the middle of the nineteenth century (1851\u20133), John Ruskin urged his fellow Victorians to look for something enduring and meaningful in art, a special quality that would arrest the destructive forces of time and 'stay what is fleeting'. It could have been the motto of the Victorian collector. In an age of bewildering technological, scientific and social change, the collection often appeared as a still point, a place outside of time where all that mattered was the objects and where the desire of the collector removed them from the chaos of the world beyond. Then, as now, some collectors embraced the idea of buying and selling, bartering and exchanging; they viewed their custodianship as a temporary hiatus in the dynamic life of the objects they collected. But many more had at least a core to their collection that they wished to protect. Where they could, they put in place mechanisms for ensuring that the collection would survive without them, defending it from the threat of an uncertain fate.\n\nThe collectors in this book were typical of many in imagining a durable future for their collections. They believed that they had created something lasting and self-contained, buffered from forces over which they had no control. Murray Marks and J. C. Robinson both celebrated a lifetime spent acquiring beautiful objects by donating to a range of public museums, making permanent in the galleries their expertise and achievements. Stephen Wootton Bushell arranged for his objects to become part of the collections at the Victoria and Albert Museum after his death, visibly marking decades of scholarly collecting. With her gift of English ceramics, enamels and glass to the same museum, Charlotte Schreiber hoped to preserve a memorial to her beloved husband, a sense of what they had enjoyed and achieved together. Joseph Mayer left both the collection at the Liverpool Museum and the thriving village of Bebington as a monument to his work, convinced he had secured their future by his lifetime's efforts.\n\nAll of these collectors tried to preserve the objects they thought best captured their dedication and expertise. They attempted to immortalize themselves in the continuing life of the collection. But they had all seen other collections come and go. Inevitably, if regretfully, they must have had a sense of collecting as an intrinsically ephemeral occupation. They had seen numerous collections sold and dismantled; they had indeed profited themselves from the ebb and flow of objects coming on to the market. What made them think, then, that this time it would be different?\n\nThe emergence of a network of public museums appears to have convinced many Victorian collectors that their collections would survive their deaths. Private collections, they had witnessed, were subject to death and war, bankruptcy and changes in taste and fashion. Public galleries, it seemed safe to assume, were immune from such vacillations. Once given over to the protection of the British Museum or the Victoria and Albert, it seemed, the fate of the individual was irrelevant to the survival of the collection. Original intentions \u2013 and the personality of the collector \u2013 would be preserved forever. By the end of the nineteenth century, almost every town in the country had a public museum asserting order and permanence, and many boasted impressive examples of philanthropy \u2013 parks and schools and libraries \u2013 that added to the impression that individual names could be saved from obscurity by being associated with lasting municipal projects. Collecting was perhaps not as fragile an occupation as it once had been. Instead, it seemed to be part of a wider public movement.\n\nOf course, it was not always to turn out that way. It is true that public collections would have languished and faded without donations from private collectors. At the end of the nineteenth century, particularly in the uncertain economic climate of the 1880s and 1890s, the sums available for public purchases dwindled, and became increasingly difficult to tease out of government. By 1887, it was estimated that the value of gifts and bequests from individuals to the Victoria and Albert Museum, for example, had exceeded \u00a31 million, at original prices \u2013 over three times as much as had been spent on acquisitions by the museum itself. Nothing displayed the breadth of Victorian collecting so well as the range of objects bequeathed to the South Kensington galleries: books, Old Master drawings and prints in 1876 on the death of John Forster, the historian and critic who became famous for writing the biography of his friend Charles Dickens; six years later, a superb collection of French eighteenth-century furniture donated by John Jones, who had made his fortune in tailoring uniforms for the military; in 1888, hundreds of sketches from the studio of John Constable, given by Isabel, his last surviving daughter; and, at the turn of the century, the library of Emilia Strong, Lady Dilke, author and art historian. Victorian collections like these form the basis of most British museum displays today.\n\nYet these donations and bequests have largely become dissociated from the collectors who made them. Names have been lost; the objects have been re-presented and re-interpreted, often distancing them from their original contexts to emphasize their apparent relevance to modern visitors. In particular, the preoccupations and intentions of the original collectors have frequently been obscured, and very often collections that were meant to remain intact have become scattered. The objects themselves are no less important, rare or beautiful than they were a hundred and fifty years ago, but they have been pared of their personal associations. The ingenuity and pride of the people who acquired them has been dimmed, and the durability they envisaged has often proved fleeting. The safe haven that was imagined to exist in the galleries of public museums has proved precarious after all. As we have seen, even the most self-contained collection is subject to all kinds of social, economic and political forces. For a collection in a public gallery \u2013 rather than tucked away in a private study \u2013 these forces are, if anything, magnified; they are certainly inescapable. No matter how serene and timeless a public museum might at first seem, nothing about it is permanent.\n\nWhile the publication of Charlotte Schreiber's journals gave readers a taste of her collecting adventures, it was not long before the redesign of the Victoria and Albert galleries broke up the collection itself. The pieces were exhibited alongside other similar objects, arranged by type, design or decoration, rather than as a discrete set, and the Schreiber name became a method of curatorial identification, printed in small type at the bottom of display labels, rather than the public celebration of an energetic collecting crusade. The jewellery, plate, needlework and other items that Charlotte had been unable to resist on her travels were inevitably dispersed on her death. So, too, was the not inconsiderable collection of European ceramics and enamels. When Charlotte left Langham House to move in with her youngest daughter, Blanche, Countess of Bessborough in 1890, a sale at Christie's began the process of dispersal, and 217 lots found their way to new homes. The pieces that remained passed to Charlotte's family on her death, were divided among her children and divided again during later generations, until the original shape of the collection was distorted beyond easy recognition.\n\nWhen Joseph Mayer died in 1886, at the age of eighty-three, his instructions to finance the Bebington projects by selling off the complete Pennant House collection \u2013 all the papers, documents and letters as well as the objects \u2013 were carried out promptly. Within weeks, his old home was cleared, the whole collection was put under the hammer, and the resulting funds went to the Mayer Trust to finance the village library, gardens and lecture hall. But Mayer's belief that he had created something permanent through his collecting was quickly shown to be misplaced. A growing village is not cheap to run, and the money raised by the sale was only sufficient to finance the Mayer facilities until 1894. After that, a crisis of funds forced the district council to step in, principally to maintain an efficient library service, and for a while things were run jointly between the council and the Mayer Trust. By 1930, however, the Mayer legacy had been exhausted, and the local authorities took over full responsibility: Mayer's involvement, already largely forgotten, officially came to an end. In 1971, traces of his name were further erased when a new civic complex replaced the original Mayer Library; during the subsequent decades Mayer Hall became neglected and shabby, its walls greying and its roof leaking. It can still be seen, at the heart of Bebington, offering a venue for local meetings and events, but rather unhappily awaiting repair.\n\nMeanwhile, Mayer's injunction to the Liverpool Museum that 'the Collection shall be kept together' was, in time, ignored, or simply forgotten. Visitors lost any clear sense of what it was Mayer had collected, of the range and personality represented by his pieces \u2013 of who he was. The eccentricities disappeared; order was imposed. The Egyptian material was carefully displayed with other Egyptian pieces; the ceramics shown with decorative arts; many of the casts, cameos, prints and medals were lost or completely disassociated from the Mayer name. In May 1941, German bombing further added to the confusion: the Egyptian gallery was completely burnt out, the entire building was ravaged by fire, and it was not until fifteen years later that the museum could reopen. Although some of Mayer's objects had been moved out of the city for safety, many were destroyed. The Egyptian sarcophagus of Bakenkhonsu, for example, was smashed to pieces (although it has recently been restored in a major conservation project), and particularly hard hit was his collection of European pottery, which was almost completely wiped out. The collection that Mayer believed would long outlive him had turned out to be fragile too; collecting proved a precarious way to arrest the uncertainties of time.\n\nMurray Marks' conscientious support of the Pre-Raphaelite painters has largely been obscured by their subsequent popularity and success; the letters, notes and sketches which he kept as proof of his judgement and foresight have not been enough to prevent him from falling into the shadows. His friends found lasting fame; by contrast, Marks and his influential role in constructing Victorian aesthetics and taste have been more or less forgotten. He was not a diligent letter writer, and was apparently unwilling to enshrine his knowledge and experience in print: 'he does know a great deal but seems not to have the slightest faculty for writing it down,' remarked Rossetti. The objects from his London house were dispersed after his death in several summer sales at Christie's in 1918, and have long since passed into the hands of other collectors. Very few of the pieces he bequeathed to the Victoria and Albert Museum are easy to locate today. A handful of blue-and-white vases displayed together in the British Galleries include two pieces which belonged to Rossetti and one which was part of Whistler's collection. They were quite likely to have passed through Marks' hands, and provide enduring evidence of the popular fashion he helped create. It is also possible to see Rossetti's pastel portrait of Marks' wife in the museum's Print Room. But, on the whole, Marks has simply disappeared. His collecting has been forgotten.\n\nSimilarly, few of Stephen Wootton Bushell's objects remain on public display to tell his story. Those pieces he sent back to London as an agent, working on behalf of the South Kensington Museum, are not at all easy to identify: his name is not attached to the labels, and only a prefix '1883' identifies things which Bushell collected. There are rarely more than a handful of objects on display. I would recommend that anyone wanting a real taste of Bushell's interests and wide-ranging research begins with a copy of _Chinese Art_ , which offers a colourful and comprehensive insight into his activities.\n\nEven John Charles Robinson, the most belligerent and ambitious of collectors, could not prevent a new generation from breaking up the collection he treasured. As we have seen, his pieces were quickly scattered after his death at home at Newton Manor on 10 April 1913. By September, many of the drawings were put up for sale, and, by February 1914, works from Robinson's collection were appearing regularly at Sotheby's. Before long, the only reference to Robinson's passionate years of searching, haggling and manoeuvring was a short catalogue entry describing the provenance of an object, recording simply that he had once owned it but revealing nothing of the time and energy he had invested in acquiring it.\n\n* * *\n\nDespite the best efforts of so many of the leading Victorian collectors, the life of their collections proved uncertain. Some found lasting homes, and are still significant today: the Pitt Rivers collection remains a valued asset of the University of Oxford; the collections of Sir John Soane and Frederick John Horniman are both publicly displayed in London; over 4,000 objects, pictures and books belonging to Sir Thomas William Holburne continue to form the core of the Holburne Museum in Bath. But many more have been subjected to dispersal, disinterest and neglect. Each was the result of a particular preoccupation, a very personal adventure that proved not to long outlive the collector. Each was a moment in a specific time, a window on nineteenth-century concerns and attitudes, a quirky glimpse of a changing age. But, while the collections themselves may have been broken up, lost or discarded, their stories continue to offer snapshots of a Victorian legacy that has endured. In these collections, we see the evolution of scholarship, professionalism and expertise that informs today's collectors, curators and connoisseurs; we can understand the emerging possibilities for public collecting and accessible display which are the foundations of twenty-first-century museums; we sense a growing curiosity about other peoples and a respect for their objects which helped change entrenched colonial attitudes; we can trace a developing fascination with the ways in which we display and construct our identity through the things we own; and we witness the increasing sophistication and manipulation of a market which still drives much of the collecting world. The collections may not have lasted, but many of the cultural changes that they embodied still resonate today.\n\n## _Acknowledgments_\n\nSome of the ideas for this book emerged during work on a Ph.D. thesis at the University of Sheffield, and I am indebted to Professor Sally Shuttleworth, now Head of Humanities at the University of Oxford, for the encouragement, guidance and perceptive criticism which set me on my way. As the book developed, however, it took on a character all of its own, and warm thanks is due to the archive staff and a number of specialist curators at the Victoria and Albert Museum who have helped inform and refine the project, especially Julius Bryant, Stephen Calloway, Peta Motture, Marjorie Trusted, Ming Wilson and Hilary Young. I would also like to thank those at the National Museums Liverpool who have given advice on Joseph Mayer, particularly Dr Ashley Cooke, Head of Antiquities, and David Moffat, assistant curator of Decorative Art. Professor Nick Pearce, Head of the School of Culture and Creative Arts at the University of Glasgow, kindly helped me on the trail of Stephen Wootton Bushell and generously shared some of his research.\n\nParticular thanks is due to Tim Bates, for making _Magpies_ happen, to Ravi Mirchandani for trusting that it would happen well, and to all the staff at Pollinger and Atlantic Books. It is always a pleasure working with them. I would also like to thank my parents for their continued encouragement and my lovely husband for everything, but in particular for sharing a moment of inspiration that created the book's title.\n\n## _Notes_\n\n### Chapter 1\n\n. _The Building News_. (22 April 1864), pp. 297\u20138.\n\n. J. C. Robinson (ed.), _The Catalogue of the Special Exhibition of Works of Art of the Medieval, Renaissance, and more Recent Periods, on loan at the South Kensington Museum_ , revised edition (London: Eyre and Spottiswoode, 1863), p. 2.\n\n. 'The New Court, South Kensington Museum', _The Builder_ , 3 May 1862, p. 305; 'The Loan Collection at South Kensington', _Fine Arts Quarterly Review_ (1864), p. 20.\n\n. 'Art Treasures at South Kensington', _Bentley's Miscellany_ (October 1862), p. 349.\n\n. 'The South Kensington Museum and Loan Exhibition', _Quarterly Review,_ 225 (January 1863), pp. 176\u2013207.\n\n. 'Les sculptures du mus\u00e9e du South Kensington', _Gazette des Beaux Arts_ (14: 1863), p. 458.\n\n. Ibid.\n\n### Chapter 2\n\n. Report from the Select Committee on the South Kensington Museum (1860), National Art Library (NAL), pp. 10\u201311.\n\n. 'Obituary of Sir Henry Cole', _The Times_ , 20 April 1882.\n\n. Ibid.\n\n. Henry Cole, speech on 16 November 1867, published in H. Cole, _Fifty Years of Published Work_ (London: George Bell, 1884), p. 293. A more detailed discussion of Henry Cole's vision for the museum can be found in Anthony Burton, 'The Uses of the South Kensington Art Collections', _The Journal of the History of Collections,_ 14 (2002), pp. 79\u201395.\n\n. Quoted in Marjorie Caygill and John Cherry (eds), _A. W. Franks. Nineteenth-century Collecting and the British Museum_ (London: British Museum Press, 1997), p. 18.\n\n. J. Ruskin, Evidence to the 1857 National Gallery Site Commission; M. Arnold, _Culture and Anarchy_ , edited by Jane Garnett (Oxford: Oxford University Press, 2006), p. 32.\n\n. J. C. Robinson, _An Introductory Lecture on the Museum of Ornamental Art of the Department_ , quoted in Anthony Burton, 'The Uses of the South Kensington Art Collections', p. 85.\n\n### Chapter 3\n\n. _The Times_ ,14 August 1848.\n\n. G. Reitlinger, _The Economics of Taste: The Rise and Fall of Picture Prices, 1760_ \u2013 _1960_ (London: Barrie and Rockliffe, 1961), vol. I. p. 30.\n\n. _New York Times_ , 21 August 1897, p. BR7.\n\n. Gustav Waagen, _Works of Art and Artists in England_ , translated into English by H. E. Lloyd (London: John Murray, 1838), p. 98.\n\n. _Punch_ , 28 (June 1855), p. 129.\n\n. _The Catalogue of the Soulages Collection_ , compiled by J. C. Robinson, with contributions from Henry Cole (London, 1856). Archive of the Victoria and Albert Museum; _The Graphic_ , 1:1 (4 December 1869), p. 2.\n\n. Mabel Tylecote, _The Mechanics' Institutes of Lancashire and Yorkshire before 1851_ (Manchester: Manchester University Press, 1957), pp. 69\u201375.\n\n. Ibid., p. 221.\n\n. It is difficult to be sure of accurate figures since membership, and the number of Institutes, as well as their nomenclature, was constantly changing. In addition to Mechanics' Institutes, there were, for example, Mutual Improvement Societies, Institutes for the Advancement of Knowledge, People's Colleges and Artisans' Institutes. W. A. Munford gives a figure of 610 institutions of the type of Mechanics' Institutes in England as well as twelve in Wales, fifty-five in Scotland and twenty-five in Ireland, and Jeffrey Auerbach suggests there were 600 Institutes with half a million members. See W. A. Munford, 'George Birbeck and the Mechanics' Institutes', in _English Libraries 1800_ \u2013 _1850: Three Lectures delivered at University College, London_ (London: Lewis, 1958) and J. Auerbach, _The Great Exhibition of 1851: A Nation on Display_ (New Haven and London: Yale University Press, 1999).\n\n. Report from the Select Committee on Arts and Manufactures, 1835\u20136.\n\n. John Ruskin, _Fors Clavigera_ , letter 59, and _The Crown of Wild Olive,_ Lecture IV (14 December 1869).\n\n. L. Jessop and N. T. Sinclair, _The Sunderland Museum: The People's Palace in the Park_ (Tyne and Wear Museums, 1996), p. 5.\n\n. _Magazine of Art_ , 1 (1878), p. 154.\n\n. James J. Sheehan, _Museums in the German art world from the end of the old regime to the rise of Modernism_ (Oxford: Oxford University Press, 2000), p. 115.\n\n. Ibid., p. 116.\n\n. _Chambers Journal of Popular Literature, Science and Arts_ , 335 (2 June 1860), p. 342.\n\n. _The Builder_ , 22 June 1872.\n\n. John Ruskin, letter to Henry Swan, 13 May 1885, from transcript at the Ruskin Gallery, Sheffield.\n\n. _Sheffield Independent_ , 11 August 1887, p. 6; _Art Union_ , 8 (January 1846), p. 17. The 1872 Nottingham exhibition featured objects on loan from South Kensington; the Halifax exhibition was organized by the town's Mechanics' Institute.\n\n. Royal Commission on Scientific Instruction and the Advancement of Science (Devonshire Commission), 4th Report, (1874), p. 14.\n\n. For an excellent discussion of the eighteenth-century world of the gentleman's art club, see Peter Clark, _British Clubs and Societies 1580_ \u2013 _1800: The origins of an associational world_ (Oxford: Oxford University Press, 2000). Walpole is quoted on p. 78.\n\n. C. E. Clement and L. Hutton, _Artists of the Nineteenth Century and Their Works_ (Boston: Houghton Mifflin, 1897), quoted in C. Denney, _At the Temple of Art: the Grosvenor Gallery, 1877_ \u2013 _1890_ (London: Associated University Press, 2000), p. 54.\n\n. Evidence to the 1857 National Gallery Site Commission, in _Collected Works,_ 12, pp. 412\u201313.\n\n. Charles Dickens, _Little Dorrit_ , edited by Stephen Wall and Helen Small (Harmondsworth: Penguin, 2003), p. 67.\n\n. 'The Mausoleum Marbles', _Chambers Journal of Popular Literature, Science and Arts_ , 317 (28 January 1860), pp. 49\u201352 (p. 52); 'The British Museum', _The London Review of Politics, Society, Literature, Art and Science,_ 29 (March 1862), pp. 304\u20135 (p. 304).\n\n. Cutting annotated 'Some Manchester News (Times?) 1845', in scrapbook of George Wallis, vol. II, p. 81, the Library of the Victoria and Albert Museum.\n\n### Chapter 4\n\n. J. C. Robinson, letter to William Maw Egley, October 1845, copy of Robinson papers, NAL: copies of much of Robinson's correspondence as well as articles, memos, etc. are deposited at the National Art Library, Victoria and Albert Museum, and were consulted there unless otherwise stated. Egley, a painter of miniatures, was one of Robinson's closest friends.\n\n. Quoted in A. McClellan, _Inventing the Louvre: art, politics and the modern museum in eighteenth-century Paris_ (Berkeley: University of California Press, 1994), p. 98, a book which also gives an excellent account of the impulses behind the development of the Louvre.\n\n. Dominique Vivant-Denon, diplomat, courtier and director of the Louvre from 1802 to 1814, quoted in McClellan, _Inventing the Louvre_ , p. 140.\n\n. John Ruskin, _Letters on Architecture and Painting, delivered at Edinburgh, November 1853_ (London: Smith Elder, 1854), vol. I. p. 50.\n\n. J. C. Robinson, 'Our Public Art Museums. A retrospect', _The Nineteenth Century_ , 42 (1897), p. 941.\n\n. Jules Janin, 'A Summer and A Winter in Paris', in George Newenham Wright, _France Illustrated,_ vol. 4 (London: Peter Jackson, 1845\u20137), p. 170.\n\n. Richard Redgrave, one of Henry Cole's closest friends and allies, visited the apartment with Cole in 1855: see F. M. Redgrave, _Richard Redgrave: A memoir compiled from his diary_ (London: Cassell, 1892), pp. 147\u20138.\n\n. Quoted in H. E. Davies, _The Life and Works of Sir John Charles Robinson, 1824\u20131913_ , p. 14.\n\n. J. C. Robinson, letter to William Egley, August 1851.\n\n. This was during a later trip to Italy, in 1859, on behalf of the South Kensington Museum, at a period when armed battles were increasingly common. J. C. Robinson, letter to Henry Cole, 25 April 1859.\n\n### Chapter 5\n\n. Henry Morley, 'A House Full of Horrors', _Household Words_ , VI (4 December 1852), pp. 265\u201370.\n\n. _Art Journal_ (1855), pp. 150\u201352; J. C. Robinson's introduction to the _Catalogue of Decorative Art_ (1855).\n\n. Quoted in Anthony Burton, _Vision and Accident: The Story of the Victoria and Albert Museum_ (London: Victoria and Albert Museum, 1999), p. 38.\n\n. J. C. Robinson, letter to Henry Cole, 7 November 1855, Cole correspondence, NAL; _Building News_ , 6 March 1857, p. 225.\n\n. J. C. Robinson, 'Our Public Art Museums: A Retrospect', _Nineteenth Century_ (December 1897), p. 955.\n\n. J. C. Robinson, _Introductory Addresses on the Science and Art Department and the South Kensington Museum,_ 5 (London, 1858), p. 23.\n\n. J. C. Robinson, letter to Henry Cole, 25 October 1855, Cole correspondence, NAL.\n\n. J. C. Robinson, _Catalogue of the Museum of Ornamental Art (Part 1)_ (London, 1855), Introduction.\n\n. Charles Dickens, _Our Mutual Friend_ , edited by Michael Cotsell (Oxford: Oxford University Press, 1998), p. 245.\n\n. Ibid., p. 223.\n\n. _The Builder_ (19 April 1856), p. 213.\n\n. General Post Office salary from Hansard, Income Tax Bill, third reading, 12 March 1857 (HC DEB, 12 March 1857, vol. 144, cc2272\u20135); others from R. V. Jackson, 'The Structure of Pay in Nineteenth-Century Britain', _Economic History Review_ 4 (1987), pp. 561\u201370.\n\n. J. C. Robinson, 'On Our National Art Galleries and Museums', _Nineteenth Century_ (December 1892), p. 1022; J. C. Robinson, _Italian Sculpture of the Middle Ages and Period of the Revival of Art_ (London: Chapman and Hall, 1862); J. C. Robinson, letter to Henry Cole, 2 May 1858, Cole correspondence, NAL.\n\n. _Athenaeum_ , 7 September 1878.\n\n. Interview with an unnamed artist by Moncure D. Conway, _Harper's New Monthly Magazine_ , 51 (October 1875), pp. 486\u2013503.\n\n. _New York Herald_ , (11 February 1871), n.p.\n\n. 'National Gallery', _Quarterly Review_ (1859), p. 375.\n\n. J. C. Robinson, letter to Henry Cole, 17 December 1858, Cole correspondence, NAL.\n\n. J. C. Robinson, 'Our Public Art Museums: A Retrospect', _Nineteenth Century_ (December 1897).\n\n. Ibid., p. 942.\n\n. Hansard's Parliamentary Debates, 3rd series, vol. 160, p. 1308.\n\n. _Journal of the Society of Arts_ , 16 (1868), p. 179.\n\n. _Art Journal_ (1865), pp. 281\u20132.\n\n. See Geoffrey Swinney, 'Museums, Audiences and Display Technology', University Museums in Scotland Conference, 2002.\n\n. Pr\u00e9cis of the Minutes of the Science and Art Department, South Kensington Museum, 1852\u20133, p. 425.\n\n. J. C. Robinson, letter, unknown recipient, 18 March 1863.\n\n. Diary entry, 3 February 1863; quoted in Burton, _Vision and Accident_ , p. 70.\n\n. See Burton, _Vision and Accident_ , pp. 70\u201371.\n\n. _Art Journal_ (1863), p. 230.\n\n. Pr\u00e9cis of the Board Minutes of the Science and Art Department, 1863\u20131869, pp. 242\u20135.\n\n. For further details of the Select Committee, see Elizabeth Bonython, _The Great Exhibitor: The Life and Work of Henry Cole_ , p. 235.\n\n. Museum minute note, 23 December 1867, quoted in Burton, _Vision and Accident,_ p. 71.\n\n. Letter to J. C. Robinson, 4 January 1868, quoting Board Minute of 30 December 1867.\n\n. Letter to J. C. Robinson, 20 January 1868.\n\n. A. W. Franks, letter, 8 May 1878, British Museum, Department of Medieval and Later Antiquities.\n\n. 'The Apology of my Life', in Caygill and Cherry (eds), _A. W. Franks_ , p. 318.\n\n. Ibid., p. 320.\n\n### Chapter 6\n\n. Layard made his comments when Robinson was being considered for the vacant post of Director at the National Gallery in 1871. Quoted in Helen Davies, _The Life and Works of Sir John Charles Robinson, 1824_ \u2013 _1913_ (Ph.D., University of Oxford, 1992), p. 67.\n\n. A pamphlet on Newton Manor written by Robinson for the Dorset Field Club in September 1896. Robinson Papers, NAL.\n\n. Ibid.\n\n. Marian Robinson, letter to Henry Cole, no date.\n\n. William Gladstone, letter to J. C. Robinson, 8 November 1869.\n\n. Princess Vicky's visit to Newton Manor is recorded, briefly, in the lengthy _Times_ obituary for Robinson (11 April 1913), but few details are known. (Copies of Robinson Papers, NAL.) The young princesses were Charlotte, Victoria, Sophie and Margaret. There were four other children: the family's eldest son, Wilhelm, born in 1859, succeeded his father as Emperor in 1888.\n\n. No. 4 Account Book, in copies of Robinson papers, NAL; also from Davies (see note 1). The account book was kept (as with much of Robinson's paperwork) irregularly during 1877 and concerns paintings only. The extent of Robinson's dealing in other media can only be guessed at, bearing in mind the high levels of activity noted.\n\n. The University of Glasgow runs a project looking at the culture of exhibitions and art dealing in London between 1878 and 1909. See www.exhibitionculture.arts.gla.ac.uk\n\n. Obituary, _The Times_ , 24 March 1894.\n\n. Ibid.\n\n. Museum minutes, May 1879, Robinson archives, Victoria and Albert Museum.\n\n. J. C. Robinson, letter to Philip Cunliffe-Owen, 15 May 1879, Robinson archives.\n\n. Museum minutes, May 1879, Robinson archives.\n\n. Museum minutes, December 1881; January 1882, Robinson archives.\n\n. J. C. Robinson, letter to Philip Cunliffe-Owen, 20 January 1881, Robinson archives.\n\n. J. C. Robinson, letter to Philip Cunliffe-Owen, 6 January 1881, Robinson archives.\n\n. J. C. Robinson, letter to Earl Spenser, January 1882, Robinson archives.\n\n. Cole died suddenly on 18 April 1882.\n\n. J. C. Robinson, letter to Philip Cunliffe-Owen, 6 January 1881, Robinson archives.\n\n. J. C. Robinson, 'On Our National Museums and Galleries', _Nineteenth Century_ (December 1892), p. 1022.\n\n### Chapter 7\n\n. A note from Robinson's bank manager on 23 October 1912, six months before he died, gave the balance of his account as \u00a3573 2s 7d. Copy of the Robinson Papers, NAL. It was quite possible, of course, that he operated several accounts.\n\n. Robinson 'inherited' the position from Richard Redgrave (1804\u201388), a painter. He was Surveyor from 1856 until his retirement in 1880. He and Robinson knew each other well: Redgrave held a number of posts at the South Kensington Museum and the Science and Art Department, and was a staunch ally of Henry Cole.\n\n. Notes for a speech made by J. C. Robinson at the inauguration of the Whitworth Institute, undated. Copy of Robinson papers, NAL.\n\n. Ibid.\n\n. The negotiations were completed in July 1891, almost two years after the Institute first opened. See Ann Sumner, 'Sir John Charles Robinson: Victorian Collector and Connoisseur', _Apollo_ (October 1989), pp. 226\u201330.\n\n. Obituary, _The Times_ , 30 March 1911.\n\n. Obituary, _The Times_ , 9 March 1911.\n\n. Memo from A. B. Skinner to C. Purdon Clarke, 7 August 1901, Robinson archive. Skinner, who entered South Kensington as a Junior Assistant in 1879, went on to succeed Purdon Clarke as Art Director in 1905, when Purdon Clarke left to become Director of the Metropolitan Museum in New York. He remained at the museum until 1908.\n\n. G. F. Waagen, 'Thoughts on the New Building to be Erected for the National Gallery of England', _Art Journal_ (1853), p. 102.\n\n. Details of Robinson's dismissal as Surveyor can be read in a series of letters in the archives, and in _The Times_ obituary. The Queen died at Osborne House on 21 January 1901, and Robinson was told of his dismissal at the end of February.\n\n. _Catalogue of a valuable collection of Drawings by Old Masters formed by a well-known amateur over the last 40 years,_ Christie, Manson and Woods, 12 May 1902.\n\n. Obituary, _The Times_ , 11 April 1913.\n\n. _History of the Victoria and Albert_ ,  \n<http:\/www.vam.ac.uk\/collections\/periods_styles\/ features\/history\/history_buildling>\n\n### Chapter 8\n\n. Quotations, as well as details of the Schreibers' journeys, are taken from Charlotte's entertaining and comprehensive journals of collecting, published in 1911 in a two-volume edited version by her third son Montague Guest and covering the years 1869\u201385. A later edited version, by her grandson, the Earl of Bessborough, was published in 1950, giving more details of her early life and first marriage. The original journal manuscripts are in the National Library of Wales. References given here are to the 1911 published version, unless otherwise stated: Journal, 1 June 1871.\n\n. Journal, 9 March 1871.\n\n. Journal, 25 April and 12 April 1871.\n\n. Journal, 28 May 1871.\n\n. Journal, 11 April 1871.\n\n. Journal, 3 June 1871.\n\n. Journal, 19 May 1871.\n\n. Baroness Staffe, _The Lady's Dressing Room_ , translated by Lady Colin Campbell (London: Cassell's, 1892), p. 125.\n\n. For a more comprehensive narrative of Charlotte's youth, see Revel Guest and Angela V. John, _Lady Charlotte: A Biography of the Nineteenth Century_ (London: Weidenfeld and Nicolson, 1989).\n\n. See Guest and John, _Lady Charlotte_ , p. 10.\n\n. Ibid., p. 18.\n\n. Journal, 1 February 1834, Bessborough edition.\n\n. Journal, 15 August 1833, Bessborough edition.\n\n. Journal, 15\u201318 August 1833, Bessborough edition.\n\n. Quoted in the Introduction to the Bessborough edition.\n\n. Journal, 11 January and 10 September 1837, Bessborough edition.\n\n. Journal, 3 July 1838, Bessborough edition.\n\n. Journal, 4 November 1839, Bessborough edition.\n\n. Journal, 18 January 1853, Bessborough edition.\n\n### Chapter 9\n\n. Schreiber, Journal, 8 April 1873.\n\n. Figures taken from _International Historical Statistics, Europe 1750\u2013 1988_ (London: Palgrave Macmillan), pp. 665\u20139.\n\n. J. C. Robinson, 'Our Public Art Museums: A Retrospect', in _Nineteenth Century_ (December 1897), p. 956; see also Burton, _Vision and Accident_ , pp. 98\u20139.\n\n. Journal, 23 March 1874.\n\n. Editor's note to Charlotte's Journal, vol. II, p. 334. Details of Duveen's life and business can be found in the biography of his son, Meryle Secrest, _Duveen: A Life in Art_ (New York: Knopf, 2005).\n\n. Journal, Introduction.\n\n. Ibid.\n\n. Thomas Babington Macaulay, _History of England,_ vol. 5 (London: Puttnam, 1898), p. 66.\n\n. C. Lamb, 'Old China', _The Last Essays of Elia_ (Pennsylvania: Carey and Lea, 1823), pp. 194\u2013204.\n\n. Maurice Jonas, _Notes of an Art Collector_ (London: Routledge, 1907).\n\n. _Punch_ , 21 (July 1851), p. 10.\n\n. 'Acute Chinamania', _Punch_ , 'Almanack' 1875, 68 (17 December 1874), n.p. This was part of a series of satires by George du Maurier, who waged a long-running crusade against what he saw as the affectations of the Aesthetic movement and its taste for collecting china.\n\n. Joseph Marryat, _A History of Pottery and Porcelain, Medieval and Modern_ (London: Murray, 1857); see also Clarissa Orr (ed.), _Women in the Victorian Art World_ (Manchester: Manchester University Press, 1995).\n\n. Journal, Introduction, 14 March 1870 and 3 November 1874.\n\n. Journal, Introduction, and see also, for example, 27 May 1876.\n\n### Chapter 10\n\n. Schreiber, Journal, 23 August 1869 and 8 March 1872.\n\n. Jules Janin, 'A Summer and A Winter in Paris', in George Newenham Wright, _France Illustrated_ , vol. 4 (London: Peter Jackson, 1845\u20137), p. 112.\n\n. Journal, 2 October 1869 and 22 March 1870.\n\n. Journal, 10 May 1872 and 8 April 1874.\n\n. Quoted in Orr (ed.), _Women in the Victorian Art World_ , p. 130.\n\n. The story of the gourd-shaped bottle is related by Montague Guest in the concluding notes to Charlotte's journals: 'The Adventures of a Bottle', vol. II, pp. 484\u20138, with additional material taken from Charlotte's daily entries.\n\n. For an excellent discussion of female art critics, see Pamela Gerrish Nunn, 'Critically Speaking', in Orr (ed.), _Women in the Victorian Art World_ (Manchester: Manchester University Press, 1995), pp. 107\u201325. This comment is from Alice Oldcastle, quoted on p. 114.\n\n. Journal, 22 May 1869.\n\n. Journal, 8 March 1872.\n\n. Journal, 19 and 20 March 1880.\n\n. Journal, 23 September 1880.\n\n. Journal, 28 September 1880.\n\n. 'Economic and Social History: Industry and Trade, 1500\u2013 1880', _A History of the County of Warwick: Volume 7: The City of Birmingham_ (1964), pp. 81\u2013139.\n\n. Journal, 29 March 1884.\n\n. Journal, 9 June 1884.\n\n. Journal, 3 May 1884.\n\n. Journal, 18 November 1884.\n\n. Journal, 17 October 1885\n\n. Journal, 13 November 1885.\n\n. The book on English fans was finished in 1888, followed two years later by a publication on European fans. Neither is now in print. The other Freewoman was Baroness Coutts, a banker's daughter and philanthropist.\n\n### Chapter 11\n\n. The portrait, now in the Liverpool Museum, is dated 1843, but a letter from the Royal Academy concerning the work is dated 9 May 1840, suggesting it was already in existence then. Mayer was born on 23 February 1803.\n\n. Letter from H. T. Kemball Cook to G. J. Binns, 19 June 1828, archives of Liverpool Museums. For the biographical information on Joseph Mayer and his collecting, I am indebted to the work of the National Museums and Galleries on Merseyside, and in particular _Joseph Mayer of Liverpool 1803\u20131886_ , edited by Margaret Gibson and Susan M. Wright (London: Society of Antiquaries\/National Museums & Galleries on Merseyside, 1988). The majority of Mayer's papers are deposited with Liverpool City Archives or at Bebington Library.\n\n. Ibid.\n\n. Obituary, _Liverpool Daily Post_ , 20 January 1866.\n\n. A. W. Franks, quoted Caygill and Cherry (eds), _A. W. Franks_ , p. 169. The brooch has the accession number MLA 1856.7\u20131, 1461.\n\n. Joseph Clarke, letter to Joseph Mayer, October 1852.\n\n. Joseph Clarke, letter to Joseph Mayer, 19 October 1854.\n\n. Roach Smith, letter to Joseph Mayer, 12 December 1852; letter to Mayer, 11 April 1856.\n\n. For details of the Faussett collection, see Roger H. White, 'Mayer and British Archaeology', _Joseph Mayer of Liverpool 1803\u2013 1886_ , edited by Margaret Gibson and Susan M. Wright.\n\n. Officers' Reports, 15 November 1853, British Museum Central Archive.\n\n. Joseph Mayer, letter to A.W. Franks, 24 November 1853 and 26 February 1854, BM (MLA) papers.\n\n. Roach Smith, letter to Joseph Mayer, 20 February 1856.\n\n. 'Antique Ivory Carving', _Art Journal_ (1 October 1855), p. 276.\n\n. Joseph Clarke, letter to Joseph Mayer, 28 March 1868.\n\n. Joseph Clarke, letter to Roach Smith, 22 November 1856.\n\n. Joseph Mayer, letter to W. H. Rolfe, 8 September 1857.\n\n. See _Joseph Mayer of Liverpool 1803\u20131886_ , pp. 95\u20136 for further details of the sale. Joseph Clarke, letter to F. W. Fairholt, 14 April 1859.\n\n. C. T. Gatty, _The Mayer Collection in the Liverpool Museum considered as an Educational Possession_ (Liverpool Art Club, 1878), pp. 20\u201321.\n\n. A. H. Church, 'Josiah Wedgwood, Master Potter', _The Portfolio,_ 3 (March 1894), p. 100.\n\n### Chapter 12\n\n. Mayer's contribution to the Great Exhibition was first prompted by a letter from Edward Hawkins, Keeper of Antiquities at the British Museum, 11 February 1850 and the pieces were listed in the official catalogue, pp. 674\u20135 ( _Jury Reports_ , III.520, class xxiii).\n\n. R. Lepsius, _Letters from Egypt, Ethiopia and the Peninsula of Sinai_ , translated by L. and J. B. Horner (London, 1853), p. 41.\n\n. Joseph Mayer, letter to Roach Smith, 27 January 1882.\n\n. For these quotations and further information on John Tradescant, see the Ashmolean Museum's extensive web pages at www.ashmolean.org.\n\n### Chapter 13\n\n. _Exhibition of Art Treasures of the United Kingdom Manchester 1857, Report of the Executive Committee_ (London: Longman, 1859), p. 3.\n\n. Ibid., p. 4.\n\n. Nathaniel Hawthorne, _English Notebooks_ (6 September 1857), p. 332.\n\n. _Exhibition of Art Treasures of the United Kingdom Manchester 1857, Report of the Executive Committee_ , p. 17.\n\n. Ibid., p. 31.\n\n. Quoted in Charles Saumarez Smith, _The National Gallery: A Short History_ (London: Lincoln, 2009), p. 74.\n\n. _Exhibition of Art Treasures of the United Kingdom Manchester 1857, Report of the Executive Committee_ , p. 20.\n\n. Nathaniel Hawthorne, _English Notebooks_ (6 September 1857), p. 332.\n\n. The Duke's criticism may be apocryphal, but it amused Manchester commentators and so was frequently repeated; see, for example, _A Handbook to the Gallery of British Paintings in the Art Treasures Exhibition_ (London,1857), p. 3, and the _Manchester Guardian_ , 5 May 1857.\n\n. Gustav Waagen, letter to Joseph Mayer, 18 August 1856.\n\n. J. B. Waring, 'Ceramic Art', _Art Treasures of the United Kingdom from the Art Treasures Exhibition, Manchester_ (London: Day & Son, 1858), p. 31.\n\n. Gustav Waagen, _A Walk through the Art Treasures exhibition at Manchester under the guidance of Dr Waagen, Companion to the Official Catalogue_ (London: John Murray, 1857), p. 74.\n\n. Roach Smith, letter to Joseph Mayer, 11 April 1856.\n\n. Minutes of the Library, Museum and Education Committee, 14 February 1867; quoted in Gibson and Wright (eds), _Joseph Mayer of Liverpool,_ p. 20.\n\n. Joseph Clarke, letter to Joseph Mayer, 28 March 1868.\n\n### Chapter 14\n\n. The inscription on the bust of Peggy (Margaret) Harrison, now in the Walker Art Gallery (7610, _Foreign Catalogue_ ). Very little is known of her, or of her exact relationship with Mayer.\n\n. Elizabeth Meteyard, _The Life of Josiah Wedgwood_ (London: Hurst and Blackett, 1865), vol. I, p. xiv.\n\n. Joseph Clarke, letter to Joseph Mayer, 8 August 1863.\n\n. Wilkie Collins, _The Woman in White_ , edited by John Sutherland (Oxford: Oxford University Press, 1998), p. 176.\n\n. 'Chronic Chinamania (Incurable)', _Punch_ , 'Almanack' 1875, 68 (17 December 1874), n.p; _Les Fran\u00e7ais Peints par Eux-m\u00eames, Encyclop\u00e9die Morale du Dix-neuvi\u00e8me Siecle_ (Paris: Curmer), vol. I, p. 277.\n\n. 'To our reader', _The Connoisseur: A Collector's Journal and Monthly Review,_ 1:1 (January 1895), pp. 5\u20136.\n\n. Freud, writing mostly at the beginning of the twentieth century, identified, for example, a biological drive which directly links collecting to sexual drive, while also identifying a resemblance to hunting and the display of trophies from the aggressive drive. More recently, Jean Baudrillard described collecting as 'a powerful mechanism of compensation during critical phases in a person's sexual development'. (Jean Baudrillard, 'The System of Collecting', in _The Cultures of Collecting_ , edited by John Elsner and Roger Cardinal (London: Reaktion, 1994), pp. 7\u201324 (p. 9).)\n\n. Henry James, _The Portrait of a Lady_ , edited by Nicola Bradbury (Oxford: Oxford University Press, 1995), pp. 328, 558, 386; George Meredith, _The Egoist_ , edited by George Woodcock (Harmondsworth: Penguin, 1968), p. 138.\n\n. 'A Free Village Library', _The Standard_ (1878), pp. 11\u201312.\n\n. H. Cunningham, _The Volunteer Force: A Social and Political History, 1859\u20131908_ (London: Croon Helm 1975), p. 2.\n\n. Mayer's speech at a celebratory dinner after the opening of the Free Library, reported in the _Staffordshire Weekly Times_ , 5 March 1870.\n\n. 'A Free Village Library', _The Standard_ (1878), pp. 18\u201319.\n\n### Chapter 15\n\n. See the reminiscences of Dante Gabriel Rossetti's brother William, in _Dante Gabriel Rossetti: His Family-Letters Edited witha Memoir by William Michael Rossetti,_ 2 vols (London: Ellis and Elvey, 1895), vol. I. p. 28.\n\n. For details of this and other of Rossetti's demands, see George Charles Williamson, _Murray Marks and His Friends_ (London, 1919).\n\n. Quoted in Williamson, _Murray Marks and His Friends_ , p. 52.\n\n. _Times Literary Supplement_ (12 June 1919).\n\n. T. Affleck Greeves, _A Guide to Bedford Park: the first garden suburb_ (1893) (London: The Bedford Park Society, 2010).\n\n. Williamson, _Murray Marks and His Friends,_ p. 13.\n\n### Chapter 16\n\n. Walter Sickert, 'Small Pictures', _The Speaker_ , 2 January 1897.\n\n. 'The French Gallery', _The Times_ , 8 April 1879, p. 4.\n\n. Williamson, _Murray Marks and His Friends_ , p. 38.\n\n. _Exhibition Culture in London 1878\u20131908,_ database, University of Glasgow, 2006; <<http:\/\/www.exhibitionculture.arts.gla.ac.uk\/>>\n\n. Bernard Berenson, Italian Painters of the Renaissance (London: Phaidon Press, 1956), pp. ix, xii.\n\n. Schreiber, Journal, 26 February 1870.\n\n. Henry James, Preface to _The Portrait of a Lady_ (1881). The fascination was two-way: Charles Dickens, for example, found an eager audience for his _American Notes,_ published shortly after his first visit to the United States in 1842.\n\n. J. P. Morgan appears less than sympathetically in Richard Armour's 1953 satire _It All Started with Columbus_ : 'Morgan, who was a direct sort of person, made his money in money. . . He became immensely wealthy because of his financial interests, most of which were around eight or ten percent. . . This Morgan is usually spoken of as \"J.P.\" to distinguish him from Henry Morgan the pirate.'\n\n. The Pierpont Morgan Library in New York was made a public institution in 1924 by his son, J. P. Morgan; many of the gems were donated to the American Museum of Natural History in New York. Pierpont Morgan also left works from his collection to the Metropolitan Museum of Art and founded the Yale Babylonian collection with over 3,000 cuneiform tablets.\n\n. William Bode, _Die italienische Bronzestatuetten der Renaissance_ , 3 vols (Berlin: B. Cassirer, 1907\u201312), vol. I, p. 3; quoted in Clive Wainwright, 'A Gatherer and Disposer of other men's stuffe: Murray Marks, connoisseur and curiosity dealer', _Journal of the History of Collections_ (2002), p. 171.\n\n. Ibid.\n\n. James McNeill Whistler, letters to F. R. Leyland, 2 and 9 September 1876 (University of Glasgow Transcription 08796; Manuscript division, Pennel-Whistler Collection PWC 6B\/22\/1).\n\n. There has been much writing and discussion about Leyland's Peacock Room and Whistler's contribution. For a detailed and authoritative account, see Linda Merrill, _The Peacock Room: A Cultural Biography_ (New Haven and London: Yale University Press, 1998). The room is now on show, complete, at the Freer Gallery of Art in Washington.\n\n. In 1904, twelve years after Leyland's death, the room was bought intact by the American collector Charles Land Freer, who later founded the Freer Gallery of Art in Washington, and it was taken apart, shipped across the Atlantic and reinstalled at Freer's house in Detroit, where it was used to display his own collection of ceramics.\n\n. Quoted in Meryle Secrest, _Duveen: A Life in Art_ (New York: Knopf, 2005), p. 29.\n\n. James Henry Duveen, _The Rise of the House of Duveen_ (London and New York: Longmans, Green, 1957), p. 84.\n\n### Chapter 17\n\n. There is still much discussion among art historians about the exact number of works that can be accurately ascribed, either in whole or in part, to Leonardo da Vinci, but there are around fifteen paintings on panels, murals and drawings on paper which form the core of the accepted body of his work. The two most famous examples are, of course, the _Mona Lisa_ , displayed in the Louvre, and _The Last Supper,_ in the Convent of Santa Maria della Grazia, Milan.\n\n. We know few details about exactly where or from whom Marks acquired the bust.\n\n. The new firm of Marks, Durlacher Brothers gave up the Oxford Street shop in 1885 and moved to 23A Bond Street. By 1887, they had moved again, to 142 New Bond Street, and the Marks name had been dropped, although he was still a partner in the business.\n\n. 'The Newly-Discovered Leonardo', _Burlington Magazine_ , 15, number 74 (1909), pp. 108\u201313. There was also a follow-up article in the same magazine: 'The wax bust attributed to Leonardo', _Burlington Magazine_ , 16, number 81 (1909), p. 123.\n\n. _New York Times_ , 5 December 1909.\n\n. There are 351 letters from Marks to Bode, just covering the period from 1890 to 1910, all of which are in the Zentralarchiv der Staatlichen Museen in Berlin. The first publication was the two-volume catalogue for the Pierpont Morgan bronzes, already mentioned in Chapter 11; the second was a three-volume book on bronzes in general which Bode published between 1907 and 1912.\n\n. Quoted in 'Did da Vinci or R. C. Lucas create _Flora_ ', _The Times_ , 5 December 1909.\n\n. Ibid.\n\n. _New York Times_ , 5 December 1909.\n\n. See Francis H. Hinsley, _British Foreign Policy under Sir Edward Grey_ (Cambridge: Cambridge University Press, 1977), p. 556.\n\n. Ibid.\n\n. Ibid.\n\n. See Kenneth Clark, _Leonardo da Vinci: An Account of his Development as an Artist_ (Cambridge: Cambridge University Press, 1939). The Kaiser-Friedrich Museum in Berlin reopened in 2007 as the Bode Museum, with _Flora_ prominently displayed in the lecture hall (room 220). Museum guides still refer to the origins of the work as a mystery, and the label hedges its bets, describing the sculpture as 'both attributed to Leonardo da Vinci or one of his apprentices and viewed as a forgery of the 19th century'.\n\n. Art Referees Report, 6 April 1864, Robinson Papers, NAL.\n\n. _Daily Telegraph_ , 2 January 1885; quoted in Burton, _Vision and Accident,_ p. 132.\n\n. Speech at the Birmingham School of Art, 1888, Robinson Papers, NAL.\n\n. Museum minutes, 29 March 1887, Robinson archives.\n\n. Wallis Budge, Keeper of Egyptian and Assyrian Antiquities at the British Museum from 1892 to1924, in _By Nile and Tigris: a Narrative of Journeys in Egypt and Mesopotamia,_ 2 vols (London: John Murray, 1920), p. 73.\n\n. A. W. Franks, letter to the British Museum, 7 October 1879.\n\n. Right Revd G. F. Browne, _The Recollections of a Bishop_ (London, 1915), p. 209; quoted in Caygill and Cherry (eds), _A. W. Franks_ , p. 78.\n\n. A. W. Franks, _Proceedings of the Society of Antiquaries_ , IV (1859), pp. 246\u201350.\n\n. E. Renard, 'Die Kunsthistorische Ausstellung, Dusseldorf, 1902', in _Rheinlande: Monatschrift f\u00fcr deutsche Kunst_ (1902), pp. 41\u20132.\n\n. Christie's London, '19th Century \"Renaissance\" Works of Art: A Question of Supply and Demand', _The Collection of the Late Baroness Batscheva de Rothschild_ (14 December 2000), pp. 102\u2013106.\n\n. Giovanni Morelli, _Italian Painters: Critical Studies of their Works_ (London: John Murray, 1892\u20133).\n\n. The Reliquary came to the British Museum as part of the Waddesdon Bequest, left to the museum in 1898 by Baron Ferdinand de Rothschild. It was only in 1959, when the fake reliquary was brought to London and compared with the original, that the truth was finally established.\n\n. _Proceedings of the Society of Antiquaries of London_ , 42 (1891), p. 254.\n\n. _Times Literary Supplement_ , 12 June 1919.\n\n. Official statement concerning the Salting Bequest; see Williamson, _Murray Marks and His Friends_ , p. 198.\n\n. Murray Marks, letter to Cecil Harcourt-Smith, 12 June 1916, Murray Marks archives, Victoria and Albert Museum. Marks assured the museum that his insurance would cover any losses.\n\n### Chapter 18\n\n. Charles Dickens, _Little Dorrit_ , edited by Stephen Wall and Helen Small (Harmondsworth: Penguin, 2003), pp. 236\u20137.\n\n. Jane MacLaren Walsh, 'Legend of the Crystal Skulls', _Archaeology,_ 61.3 (June 2008).\n\n. Thomas Hardy, _A Pair of Blue Eyes_ , edited by Tom Dolin and Alan Manford (Oxford: Oxford University Press, 2005), p. 186.\n\n. For the biographical information on Stephen Wootton Bushell, I am indebted to the work of Professor Nick Pearce, in particular his lecture 'Collecting, Connoisseurship and Commerce: An Examination of the Life and Career of Stephen Wootton Bushell (1844\u20131908)', _Transactions of the Oriental Ceramic Society_ , vol. 70 (2005\u20132006), pp. 17\u201325. For details of Bushell's appointment, see p. 18. Bushell's archives of correspondence are held at the Victoria and Albert Museum where they were consulted, unless otherwise stated.\n\n. Details of Bushell's discoveries, his study of cultural meanings and his evaluation of technique and craftsmanship are taken from his masterly two-volume work _Chinese Art_ , first published in 1904 and 1906 as a handbook to the Chinese collections at the Victoria and Albert. Despite being written in Bushell's old age, it retains the excitement of his early explorations and demonstrates the breadth and thoroughness of his understanding: S. W. Bushell, _Chinese Art,_ 2 vols (London: Victoria and Albert Museum, 1904 & 1906), p. 10; pp. 240\u20131.\n\n. Mary Crawford Fraser, _A Diplomat's Wife in Many Lands_ (New York: Dodd, Mead, 1910). See also Pearce, 'Collecting, Connoisseurship and Commerce', pp. 18\u201319.\n\n. Ibid., p. 19.\n\n. From an account by William Henry Seward, who went on to become US Secretary of State under Abraham Lincoln and Andrew Johnson. _William H. Seward's Travels Around the World_ (New York, 1873), p. 135.\n\n. Bushell, _Chinese Art,_ Introduction, p. 1.\n\n. Ibid., p. 22.\n\n. Shirley Gordon, 'Demands for the Education of Girls, 1790\u2013 1865' (MA, University of London, 1950), pp. 188\u20139.\n\n. H. E. Davies, _The Life and Works of John Charles Robinson_ (Ph.D., University of Oxford, 1992), p. 46.\n\n. S. W. Bushell, letter to Philip Cunliffe-Owen, 20 August 1880, Bushell archives.\n\n### Chapter 19\n\n. For an account of Townsend's career, see G. Townsend, _Memoir of the Rev. Henry Townsend_ (Exeter: James Townsend, 1887). Details of the early history of Exeter Museum are from G. T. Donisthorpe, _An Account of the Origin and Progress of the Devon and Exeter Albert Memorial Museum_ (Exeter: Exeter and Plymouth Gazette, 1868), p. 24.\n\n. Quoted in George Robertson, _Traveller's Tales: narratives of home and displacement_ (London: Routledge, 1994), p. 168.\n\n. See Kate Hill, _Culture and Class in English Public Museums 1850\u2013 1914_ (Aldershot: Ashgate, 2005), p. 75.\n\n. Many acquisitions were noted with pride in local papers. See L. Jessop and N. T. Sinclair, _Sunderland Museum: The People's Palace in the Park_ (Sunderland: Tyne and Wear Museums, 1996), p. 40; _The Exeter Flying Post_ , 6 April 1870. Details of other objects are taken from unpublished accession notes in local museums.\n\n. _A Guide to the Exhibition Galleries of the British Museum_ (London, 1899), pp. 98\u2013101.\n\n. _Illustrated London News_ , 8 May 1886.\n\n. _The Orientalist_ (1869), quoted in J. Jones, 'Fugitive Pieces', _Guardian_ , 25 September 2003.\n\n. _Handbook for Travellers: Baedeker_ (Leipzig: Baedeker 1885), pp. 278\u201380.\n\n. Bushell, _Chinese Art_ , vol. 2, p. 34.\n\n. Sir Rutherford Alcock, _The Capital of the Tycoon: A Narrative of Three Years' Residence in Japan_ (London: Longman Green, 1863), vol. I, p. xix. Alcock (1809\u201395) was, like Bushell, a physician by training. He had previously worked in China, having been appointed Consul at Foochow in 1844. During his ministry in Japan, he was reputedly the first foreigner to climb Mount Fuji in 1860. From 1865 to 1871, he returned to China, where he was British Minister in Peking, working of course with Bushell.\n\n. Leigh Hunt, 'The Subject of Breakfast Continued.\u2013Tea-drinking', _London Journal_ (9 July 1834), p. 113.\n\n. _Harper's New Monthly Magazine_ , 51 (October 1875), p. 658.\n\n. _Catalogue of Chinese Objects_ (London, 1872), p. 57. See also the discussion of Eastern art in 'The Empire of Things: Engagement with the Orient', _A Grand Design: The Art of the Victoria and Albert Museum_ (London: V&A Publications, 1997).\n\n. A. W. Franks, letter to General Alexander Cunningham, 21 February 1881; quoted in Caygill and Cherry (eds), _A. W. Franks_ , p. 259.\n\n. A. W. Franks, letter to Colonel Sir C. Euan Smith, 26 May 1891.\n\n. S. W. Bushell, letter to Philip Cunliffe-Owen, 9 February 1882, Bushell archives.\n\n. S. W. Bushell, letters to Philip Cunliffe-Owen, 23 February 1882, 13 November 1882, Bushell archives.\n\n. S. W. Bushell, _Chinese Art_ , vol. 1, Preface.\n\n. Fraser, _A Diplomat's Wife in Many Lands_ , p. 111.\n\n### Chapter 20\n\n. S. W. Bushell, _Chinese Art_ , vol. 1, p. 94. There were all kinds of early-nineteenth-century explanations for 'magic' mirrors. Some investigators claimed a copy of the mirror's back design was drawn on the face and then concealed by polishing; others that the phenomenon was caused by variations in the mirror's curvature. More recent research suggests that there may have been several methods of producing the mirrors, and that variations in the bronze caused by punching, stamping and polishing may all have produced an effect.\n\n. Bushell, C _hinese Art_ , vol. 1, p. 115.\n\n. Ibid.\n\n. Bushell, _Chinese Art_ , vol. 2, p. 17.\n\n. Lacquering is an ancient technique for laying down extremely thin layers of a natural varnish made from the sap of the lacquer tree.\n\n. Bushell, _Chinese Art_ , vol. 1, pp. 39, 47, 49.\n\n. Bushell, _Chinese Art_ , vol. 1, pp. 50\u201351.\n\n. Figures given by Chen Mingjie, Director of the Imperial Summer Palace, Bejing, in the state-run _China Daily_ newspaper, 19 January 2010.\n\n. By 1874, Salting's collection had completely outgrown the space in his London home and he lent large numbers of pieces to South Kensington. When he died in 1909, he left a huge bequest to the museum which was displayed in separate Salting galleries.\n\n. Obituary, _New York Times_ (23 November 1894), p. 9.\n\n. Born in Worcester, James Callowhill studied at the Worcester School of Art, before working at the Worcester Royal Porcelain Works from 1853. In New York, during the 1880s, he worked for the Faience Manufacturing Company at Greenport, Brooklyn.\n\n. Obituary, _New York Times_ (23 November 1894), p. 9.\n\n. S. W. Bushell, letter to H. R. Bishop, 23 April 1889. On his death, Bishop left his collection to the Metropolitan Museum of Art in New York, and the letters with Bushell are held there as part of the Jade Correspondence.\n\n. H. R. Bishop, letter to S. W. Bushell, 27 February 1891.\n\n. Preface, _The Bishop Collection: Investigations and Studies in Jade_ (New York: privately printed, 1906).\n\n. S. W. Bushell, letter to H. R. Bishop, 24 July 1892.\n\n. W. G. Gulland, an art historian, observed in 1898 that the Chinese interest in porcelain stemmed largely from 'the object of making large profits' from the fashionable market. See Nick Pearce, _Photographs of Peking, China 1861\u20131908_ (New York: Edwin Mellen, 2005), p. 61.\n\n. See Pearce, _Photographs of Peking_ , pp. 60\u201361.\n\n. _The I.G. in Peking: Letters of Robert Hart, Chinese Maritime Customs, 1868\u20131907_ (Cambridge, Mass., 1975), vol. 2, p. 877, letter 827.\n\n. Note in the file, November 1898, Bushell archives.\n\n. Museum minute, 16 January 1899, Bushell archives.\n\n. Ibid.\n\n. S. W. Bushell, letter to Purdon Clarke, 25 November 1898.\n\n. Minute note, 18 February 1899, Bushell archives.\n\n. Minute note, 8 April 1909, Bushell archives.\n\n### Epilogue\n\n. See Burton, _Vision and Accident,_ p. 122.\n\n. Some of Charlotte's pieces remain on display, however, at the heart of the V&A's Ceramics Galleries, and modern technology has even made it possible to glimpse a shadow of the past by virtually re-creating the original collection. Although the pieces on exhibition are still integrated with objects from all kinds of other sources, the Schreiber name is resolutely attached to everything from the original bequest, both on display and in store, and from online records it is possible to bring them together again. Those who want to see exactly what it was that Charlotte sought out so energetically can use searches and keywords to dip back into the past and assemble details and images of her bequest. Although it is unlikely that we will ever again be able to view all the objects together in a real space, breathing their collective connection to history, this virtual exercise can at least give a glimpse of the original magnificence of the Schreiber collection.\n\n. The fully restored sarcophagus is on display in the World Museum, Liverpool, alongside other of Mayer's pieces, identifiable from the letter M prefixing the acquisition number.\n\n. D. B. Elliott, _Charles Fairfax Murray: The Unknown Pre-Raphaelite_ (London: Book Guild, 2000), p. 89.\n\n. See Ann Sumner, 'Sir John Charles Robinson: Victorian Collector and Connoisseur', _Apollo_ (October 1989), pp. 226\u201330.\n\n## _Select Bibliography_\n\nAlcock, Rutherford, _The Capital of the Tycoon: A Narrative of Three Years' Residence in Japan_ (London: Longman Green, 1863)\n\nAltick, Richard, _The Shows of London_ (Cambridge, Mass. and London: Harvard University Press, 1978)\n\nAppadurai, Arjun, _The Social Life of Things: Commodities in Cultural Perspective_ (Cambridge: Cambridge University Press, 1986)\n\nAuerbach, Jeffrey A., _The Great Exhibition of 1851: A Nation on Display_ (New Haven and London: Yale University Press, 1999)\n\nBaker, Malcolm, and Brenda Richardson (eds), _A Grand Design: the Art of the Victoria and Albert Museum_ (London: V&A Publications, 1997)\n\nBarlow, Paul (ed.), _Governing Cultures: Art Institutions in Victorian London_ (Aldershot: Ashgate, 2000)\n\nBennett, Tony, _The Birth of the Museum: History, Theory, Politics_ (London and New York: Routledge, 1995)\n\nBelk, Russell W., _Collecting in a Consumer Society_ (London and New York: Routledge, 2001)\n\nBelk, Russell W., and Melanie Wallendorf, 'Of Mice and Men: Gender, Identity and Collecting', in _The Material Culture of Gender; the Gender of Material Culture_ , edited by Katherine Martinez and Kenneth L. Ames (Hanover and London: University Press of New England, 1997), pp.7\u201327\n\nBlack, Barbara J., _On Exhibit: Victorians and their Museums_ (Charlottesville and London: University Press of Virginia, 2000)\n\nBonython, Elizabeth, _The Great Exhibitor: The Life and Work of Henry Cole_ (London: Victoria and Albert Museum, 2003)\n\nBowlby, Rachel, _Just Looking: Consumer Culture in Dreiser, Gissing and Zola_ (New York: Methuen, 1985)\n\nBriggs, Asa, _Victorian Things_ (London: Batsford, 1988)\n\nBurton, Anthony, _Vision and Accident. The Story of the Victoria and Albert Museum_ (London: Victoria and Albert Museum, 1999)\n\nBurton, Anthony, 'The Uses of the South Kensington Art Collections', _Journal of the History of Collections,_ 14 (2002), pp. 79\u201395\n\nBushell, Stephen Wootton, _Chinese Art,_ 2 vols (London, 1904 & 1906)\n\nCaygill, Marjorie, 'Rare Birds: Female Collectors and the British Museum', _Creating a Great Museum: Early Collectors and the British Museum_ , Columbia University, The Fathom Archive\n\nCaygill, Marjorie, and John Cherry (eds), _A. W. Franks. Nineteenth-century collecting and the British Museum_ (London: British Museum Press, 1997)\n\nClifford, James, _The Predicament of Culture: Twentieth-century Ethnography, Literature and Art_ (Cambridge Mass. and London, Harvard University Press, 1988)\n\nCole, Henry, _Fifty Years of Published Work_ (London: George Bell, 1884)\n\nColley, Linda, _Britons. Forging the Nation 1707_ \u2013 _1837_ (New Haven and London: Yale University Press, 1992)\n\nCrary, Jonathan, _Techniques of the Observer: On Vision and Modernity in the Nineteenth Century_ (Cambridge Mass. and London: MIT Press, 1990)\n\nCumming, Elizabeth, and Wendy Kaplan, _The Arts and Crafts Movement_ (London: Thames and Hudson, 1991)\n\nDavies, H. E., _The Life and Works of Sir John Charles Robinson, 1824_ \u2013 _1913_ (Ph.D. diss., University of Oxford, 1992)\n\nDonisthorpe, G. T., _An Account of the Origin and Progress of the Devon and Exeter Albert Memorial Museum_ (Exeter: Exeter and Plymouth Gazette, 1868)\n\nEastlake, Charles, _Hints on Household Taste in Furniture, Upholstery and Other Details_ (London: Longmans, 1868)\n\nEatwell, Ann, 'Borrowing from Collectors: the role of the Loan Formation of the Victoria and Albert Museum and its Collection (1852\u20131932)', _Journal of the Decorative Arts Society_ , 24 (2000)\n\nElsner, John, and Roger Cardinal (eds), _The Cultures of Collecting_ (London: Reaktion, 1994)\n\nFlint, Kate, _The Victorians and the Visual Imagination_ (Cambridge: Cambridge University Press, 2000)\n\nForgan, Sophie, 'The Architecture of Display: Museums, Universities and Objects in Nineteenth-century Britain', _History of Science_ 32 (1994), pp. 139\u201362\n\nFraser, Mary Crawford, _A Diplomat's Wife in Many Lands_ (New York: Dodd, Mead, 1910)\n\nGibson, Margaret, and Susan M. Wright (eds), _Joseph Mayer of Liverpool 1803_ \u2013 _1886_ (London: Society of Antiquaries and National Museums and Galleries on Merseyside, 1988)\n\nGloag, John, _Victorian Taste: Some Social Aspects of Architecture and Industrial Design, from 1820_ \u2013 _1900_ (London: A & C Black, 1962)\n\nGreenhalgh, Paul, _Ephemeral Vistas: The Expositions Universelles, Great Exhibitions and World's Fairs, 1851_ \u2013 _1939_ (Manchester: Manchester University Press, 1988)\n\nGreenwood, Thomas, _Museums and Art Galleries_ (London: Simpkin and Marshall, 1888)\n\nGuest, Revel, and Angela V. John, _Lady Charlotte: A Biography of the Nineteenth Century_ (London: Weidenfeld and Nicolson, 1989)\n\nGunn, Simon, _The Public Culture of the Victorian Middle Class, Ritual and Authority and the English Industrial City, 1840_ \u2013 _1914_ (Manchester: Manchester University Press, 2000)\n\nHarrison, J. F. C., _A History of the Working Men's College 1854_ \u2013 _1904_ (London: Routledge & Kegan Paul, 1954)\n\nHaywood, Ian, _The Revolution in Popular Literature: Print, Politics and the People 1790_ \u2013 _1860_ (Cambridge: Cambridge University Press, 2004)\n\nHerrmann, Frank, _The English as Collectors. A Documentary Chrestomathy_ (London: Chatto and Windus, 1972)\n\nHill, Kate, _Culture and Class in English Public Museums 1850_ \u2013 _1914_ (Aldershot: Ashgate, 2005)\n\nHilton, Tim, _The Pre-Raphaelites_ (London: Thames and Hudson, 1970)\n\nImpey, Oliver, and Arthur MacGregor (eds), _The Origins of Museums: The Cabinet of Curiosities in Sixteenth- and Seventeenth-Century Europe_ (Oxford: Clarendon, 1985)\n\nInkster, Ian, 'The Social Context of an Educational Movement: A Revisionist Approach to the English Mechanics' Institutes, 1820\u20131850', _Oxford Review of Education,_ 2 (1976), pp. 277\u2013307\n\nJessop, L., and N. T. Sinclair, _Sunderland Museum: The People's Palace in the Park_ (Sunderland: Tyne and Wear Museums, 1996)\n\nJones, Mark (ed.), _Why Fakes Matter: Essays on Problems of Authenticity_ (London: British Museum Press, 1993)\n\nJones, Maurice, _Notes of an Art Collector_ (London: Routledge, 1907)\n\nJones, Robert W., _Gender and the Formation of Taste in Eighteenth-Century Britain. The Analysis of Beauty_ (Cambridge: Cambridge University Press, 1998)\n\nKarp, Ivan, and Steven Levine (eds), _Exhibiting Cultures: The Poetics and Politics of Museum Display_ (Washington and London: The Smithsonian Institution Press, 1991)\n\nKidd, Alan, and David Nicholls (eds), _Gender, Civic Culture and Consumerism: Middle-class Identity in Britain 1800_ \u2013 _1940_ (Manchester: Manchester University Press, 1999)\n\nKowaleski-Wallace, Elizabeth, _Consuming Subjects: Women, Shopping and Business in the Eighteenth Century_ (New York: Columbia University Press, 1997)\n\nLeapman, Michael, _The World for a Shilling: How the Great Exhibition Shaped a Nation_ (London: Headline, 2001)\n\nLevine, Philippa, _The Amateur and the Professional: Antiquarians, Historians and Archaeologists in Victorian England 1838_ \u2013 _1886_ (Cambridge: Cambridge University Press, 1986)\n\nLogan, Thad, _The Victorian Parlour_ (Cambridge: Cambridge University Press, 2001)\n\nMacCarthy, Fiona, _William Morris: A Life for our Time_ (London: Faber, 1994)\n\nMcClellan, Andrew, _Inventing the Louvre: Art, politics and the modern museum in eighteenth-century Paris_ (Berkeley and Los Angeles: University of California Press, 1994)\n\nMacDonald, Sharon (ed.), _The Politics of Display: Museums, Science, Culture_ (London and New York: Routledge, 1998)\n\nMacleod, D. Sachko, _Art and the Victorian Middle Class_ (Cambridge: Cambridge University Press, 1996)\n\nMerrill, Linda, _The Peacock Room: A Cultural Biography_ (New Haven and London: Yale University Press, 1998)\n\nMeteyard, Elizabeth, _The Life of Josiah Wedgwood_ (London: Hurst and Blackett, 1865)\n\nMiller, Andrew, _Novels behind Glass: Commodity Culture and Victorian Narrative_ (Cambridge: Cambridge University Press, 1995)\n\nMunford, W. A., 'George Birbeck and the Mechanics' Institutes', in _English Libraries 1800_ \u2013 _1850, Three Lectures delivered at University College London_ (London: Lewis, 1958), pp. 33\u201358\n\nOrr, Clarissa, _Women in the Victorian Art World_ (Manchester: Manchester University Press, 1995)\n\nPearce, Nick, 'Collecting, Connoisseurship and Commerce: An Examination of the Life and Career of Stephen Wootton Bushell (1844\u20131908)', _Transactions of the Oriental Ceramic Society_ , vol. 70 (2005\u20132006), pp. 17\u201325\n\nPearce, Nick, _Photographs of Peking, China 1861_ \u2013 _1908: Through Peking with a Camera_ (New York: Edwin Mellen, 2005)\n\nPearce, Susan M., _Interpreting Objects and Collections_ (London and New York: Routledge, 1994)\n\nPearce, Susan M., _On Collecting: An Investigation into Collecting in the European Tradition_ (London and New York: Routledge, 1995)\n\nPomian, Krzystof, _Collectors and Curiosities: Paris and Venice 1500_ \u2013 _1800_ (Cambridge: Polity Press, 1990)\n\nRichards, Thomas, _The Commodity Culture of Victorian England: Advertising and Spectacle 1851_ \u2013 _1914_ (London and New York: Verso, 1991)\n\nRobertson, George, _Travellers' Tales: Narratives of home and displacement_ (London: Routledge, 1994)\n\nRobinson, John Charles, _The Catalogue of the Soulages Collection_ , _compiled by J. C. Robinson, with contributions from Henry Cole_ (London: South Kensington Museum, 1856)\n\nRobinson, John Charles, _Italian Sculpture of the Middle Ages and Period of the Revival of Art_ (London: Chapman and Hall, 1862)\n\nRobinson, John Charles, 'Introductory Address on the Science and Art Department and the South Kensington Museum', _On the Museum of Art_ (London: South Kensington Museum, 1858)\n\nRuskin, John, _The Works of John Ruskin_ , edited by E. T. Cook and Alexander Wedderburn, Library edition, 39 volumes (London: George Allen, 1903\u201312)\n\nSchreiber, Charlotte, _Journals of Lady Charlotte Schreiber by Mr Montague Guest_ (London and New York: John Lane, 1911)\n\nSecrest, Meryle, _Duveen: A Life in Art_ (New York: Knopf, 2005)\n\nSteegman, John, _Victorian Taste: A Study of the Arts and Architecture from 1830_ \u2013 _1870_ (Cambridge, Mass.: MIT Press, 1970)\n\nStewart, Susan, _On Longing: Narratives of the Miniature, the Gigantic, the Souvenir, the Collection_ (Baltimore: The Johns Hopkins University Press, 1984)\n\nSumner, Ann, 'Sir John Charles Robinson: Victorian Collector and Connoisseur', _Apollo_ (October 1989), pp. 226\u201330\n\nTownsend, G., _Memoir of the Rev. Henry Townsend_ (Exeter: James Townsend, 1887)\n\nTylecote, Mabel, _The Mechanics' Institutes of Lancashire and Yorkshire before 1851_ (Manchester: Manchester University Press, 1957)\n\nVergo, Peter, _New Museology_ (New York: Routledge, 1992)\n\nVincentelli, Moira, _Women and Ceramics: Gendered Vessels_ (Manchester: Manchester University Press, 2000)\n\nWainwright, Clive, _The Romantic Interior: The British Collector at Home 1750_ \u2013 _1850_ (New Haven and London: Yale University Press, 1989)\n\nWainwright, Clive, 'A Gatherer and Disposer of other men's stuffe: Murray Marks, connoisseur and curiosity dealer', _Journal of the History of Collections_ , 14, 1 (May 2002)\n\nWaring, J. B., 'Ceramic Art', _Art Treasures of the United Kingdom from the Art Treasures Exhibition, Manchester_ (London: Day & Son, 1858)\n\nWaterfield, Giles, _Palaces of Art: Art Galleries in Britain 1790_ \u2013 _1990_ (London: Lavenham Press, 1991)\n\nWhitehead, Christopher, _The Public Art Museum in Nineteenth-Century Britain_ (Aldershot: Ashgate, 2005)\n\nWilliamson, G. C., _Murray Marks and His Friends_ (London: John Lane, 1919)\n\nWright, Susan M., _The Decorative Arts of the Victorian Period_ (Avon: Bath Press; The Society of Antiquaries, 1989)\n\nYanni, Carla, _Nature's Museums: Victorian Science and the Architecture of Display_ (Baltimore: The Johns Hopkins University Press, 1999)\n\n## _Index_\n\n1st Cheshire Rifles ,\n\n4th Bebington Company\n\n34th Regiment of Foot\n\n38th Middlesex (Artists') Rifle Volunteers\n\nAbbey of Stavelot, Belgium\n\nAbbotsford, Melrose\n\nAbeokuta, Nigeria\n\nAberdeen\n\nAbu Simbel temples, Egypt\n\nAbyssinia (later Ethiopia): Magdala Treasure 222\u201323\n\nAcad\u00e9mie des Inscriptions et Belles Lettres, Paris\n\nAcad\u00e9mie Fran\u00e7aise, Paris\n\nAcad\u00e9mie Royale de Peinture et de Sculpture, Paris\n\n_Academy_\n\nAesthetic movement, Aesthetes , ,\n\nAfrican art 333\u201334\n\nAgnew, William\n\nAgnew's ,\n\nAlbert, Prince Consort , , , 103\u20134, , , , , 231\u201334,\n\nAlbert Memorial Museum, Exeter 33\u201334\n\nAlcock, Sir Rutherford ,\n\nAlexandra, Princess (later Queen)\n\nAlfred Jewel\n\nAlicante, Spain 167\u201368\n\nAlnwick Castle\n\nAltes Museum, Berlin\n\nAmsterdam , , ,\n\nAndr\u00e9, Alfred\n\nAngelico, Fra: _Adoration of the Magi_\n\nAnning, Mary ,\n\nanthropology\n\nantiquarianism 196\u2013200\n\nAntwerp ,\n\nArley Castle, Staffordshire\n\nArnold, Matthew: 'Culture and Anarchy' articles\n\nart critics, female ,\n\nart history 117\u201318\n\n_Art Journal_ 84\u201385, , ,\n\nArt Nouveau\n\nArt Treasures Exhibition (Manchester, 1857) , , , , , , 228\u201337\n\nArthur, King\n\n'Artists' Houses'\n\nArts and Crafts movement 23\u201324\n\nAsclepius-Hygieia diptych\n\nAshmolean Museum, Oxford , ,\n\nAshurnasirpal II, King of Assyria\n\nAstor, William Waldorf\n\n_Athenaeum, The_\n\nAthenaeum Club, Liverpool\n\nAthenaeum Club, London ,\n\nBacon, Sir Nicholas\n\nBaedeker (publisher)\n\nBakenkhonsu, sarcophagus of ,\n\nBalmoral Castle, Scotland\n\nBaltimore ,\n\nBarnett, Barney\n\nBarry, Sir Charles\n\nBastian, Adolph\n\nBastianini, Giovanni\n\nBebington, Merseyside , , , ,\n\nBedford Park, London\n\nBeissel, Stephen\n\n_Belle Ferronni\u00e8re, La_ , 308\u20139\n\nBellini, Giovanni: _Feast of the Gods_ 267\u201368\n\nBelzoni, Giovanni\n\nBerenson, Bernard ,\n\nBerlin , ,\n\nBerlin Museum\n\nBernal, Ralph\n\nBernal collection ,\n\nBessborough, Blanche, Countess of , , ,\n\nBessborough family\n\nBethnal Green, London\n\n'Bezaar stone'\n\nBilly and Charley (Thames scavengers)\n\nBirkenhead docks\n\nBirmingham , ,\n\nBirmingham Museum\n\nBirmingham School of Art\n\nBishop, Heber Reginald 361\u201363\n\nBisschop, Christoffel\n\nBlackfriars Bridge, London\n\nBlenheim Palace, Oxfordshire\n\nBoard of Trade\n\nBock, Monsieur (Russian dealer)\n\nBologna\n\nBookplate Society\n\nBoor's Bazaar, Rotterdam\n\nBordeaux ,\n\nBoselly, Jacques ,\n\nBoston, Massachusetts\n\nBotticelli, Sandro\n\nBoughton, George Henry\n\nBourgeois, Stephan\n\nBournville, near Birmingham\n\nBowes Museum, Barnard Castle\n\nBoxer Rebellion (1900)\n\nBoyle, Frederick ,\n\nBoyle, Henry\n\nBram Hertz collection 208\u20139,\n\nBrescia\n\nBridgewater, Francis Egerton, rd Duke of\n\nBridgewater collection ,\n\nBristol\n\nBristol china\n\nBritish Archaeological Association , , , ,\n\nBritish Empire , , , , , , , ,\n\nBritish Institution, Pall Mall, London\n\nBritish Library\n\nBritish Museum, London , , 45\u201346, , , , , , , ,\n\nBritish and Medieval Antiquities department\n\nBushell's contributions\n\nCharlotte Schreiber consults Franks\n\nand Charlotte Schreiber's playing cards\n\nChristy Collection 326\u201327\n\nCole's predatory attitude\n\ncollects artefacts of classical civilizations\n\nDepartment of Antiquities\n\nDuveen gallery ,\n\nEgyptian objects , , ,\n\nand the Faussett collection ,\n\nfavours antiquities from abroad 197\u201398\n\nforgeries issue , , 302\u20133, ,\n\nFranks's contribution 93\u201396, 148\u201349\n\nIndian material\n\nand Japanese art\n\nMagdala Treasure\n\nMayer in 217\u201318\n\noverseas collections 326\u201329\n\nrefuses to buy from Fej\u00e9rv\u00e1ry collection\n\nrivalry with South Kensington\n\nRoach Smith collection\n\nRound Reading Room\n\nand Walters\n\nBritish Raj\n\nBrown, William\n\nBrunel, Isambard Kingdom\n\nBrussels, Belgium\n\nBryanston House, Dorset\n\nBuchanan (art dealer)\n\nBuckingham, nd Duke of\n\n_Builder, The_\n\n_Building News_\n\nBullock, William\n\nBunsen, Christian: _Egypt's Place in Universal History_\n\nBurke, Edmund: _Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful_\n\nBurlington Fine Arts Club (previously Fine Arts Club) , , ,\n\n_Burlington Magazine_\n\nBurne-Jones, Edward\n\nBurslem factory, Stoke-on-Trent\n\nBurton, Captain Richard F.\n\nBushell, Dorothy\n\nBushell, Florence , , , , , ,\n\nBushell, Stephen Wootton\n\nand Bishop 361\u201362\n\nand Chinese culture , 346\u201347, ,\n\nand Chinese portable family wealth 343\u201344\n\ncollecting interests ,\n\nas a dealer 363\u201365\n\ndeaths of his children 365\u201366\n\ndoctor to the British delegation in China 2\u20133,\n\ndonations to Victoria and Albert Museum\n\nenjoys living in Peking ,\n\nenthusiasm for Chinese objects\n\nfinancial problems 362\u201364\n\nfirst scholarly article\n\nfirst Westerner to make a serious study of Chinese art 323\u201324,\n\nand Franks , 347\u201348\n\nlearns Chinese language\n\nand magic Chinese mirrors 352\u201353\n\nmarries Florence ,\n\npersonality ,\n\nretirement\n\nand South Kensington Museum , , 348\u201349, 366\u201367,\n\nstudy methods , 353\u201355\n\nvisits Inner Mongolia\n\nand Walters , 358\u201359\n\n_Chinese Art_ 350\u201351, 367\u201368,\n\n_Oriental Ceramic Art_ 359\u201361\n\nBuxton, Derbyshire\n\nByron, George Gordon, Lord\n\nCaillot, Mme (dealer in Paris)\n\nCallowhill, James 359\u201360\n\nCambridge\n\nCamden, Lady\n\nCameron, Robert\n\nCampana, Giampietro 78\u201379, ,\n\nCanaletto 62\u201363\n\nCanford Manor, Dorset 137\u201338, 141\u201344\n\nCardiff\n\nCarleton, Sir Dudley\n\nCarlisle, Duke of\n\nCathedral of St John, Bois-le-Duc ('s-Hertogenbosch), Holland\n\nCavendish Laboratory, Cambridge ,\n\nCellini, Benvenuto\n\n_Chambers Journal_ , 45\u201346\n\nChangLing Tomb, Peking\n\nCharles I, King 27\u201328,\n\nCharles V, Emperor , ,\n\nCharles X, King of France\n\nChartist movement , ,\n\nChatsworth, Derbyshire\n\nChelsea china 130\u201331,\n\nChesters, Northumberland\n\nCheyne Walk, Chelsea, London , , , ,\n\nChicago\n\nChinese ceramics 267\u201370\n\n_chinoiserie_\n\nChristie's auctioneers, London , , , , , , , 375\u201376\n\nChristina, Queen of Sweden\n\nChristy, Henry 326\u201327\n\nChristy Collection 326\u201327\n\nCincinnati\n\nClark, Kenneth\n\nClarke, Joseph , , , 208\u20139, ,\n\nCleveland House, St James's, London\n\nColchester, Essex\n\nCole, Henry , , ,\n\nand 1862 International Exhibition , 19\u201320,\n\nand abolition of Robinson's art referee post\n\naims of ,\n\n_Art Journal_ article attacks\n\nand design reform ,\n\neducation focus\n\n'False Principles in Design' exhibition ('Chamber of Horrors'; 1852)\n\nfirst Director of South Kensington Museum , 19\u201320, , ,\n\nforgeries issue 298\u201399\n\nand Great Exhibition , 19\u201320\n\npersonality\n\npredatory attitude towards British Museum\n\nrelationship with Robinson , , , , , 85\u201393, ,\n\nand relocation from Marlborough House to South Kensington\n\nreorganises government Schools of Design 20\u201321,\n\nRobinson's pay rises 71\u201372\n\nand Soulages collection\n\non 'the taste for collecting'\n\nColeridge, Samuel Taylor: 'Kubla Khan'\n\nCollins, Wilkie\n\n_The Woman in White_\n\n_Connoisseur, The: A Collector's Journal and Monthly Review_ ,\n\nConstable, Isabel\n\nConstable, John ,\n\n_Sketch of Hampstead Heath_\n\nConstantine, Grand Duke, of Russia\n\nCook, Sir Francis\n\nCook, Captain James ,\n\nCook, Thomas\n\nCooke, E. W. ,\n\nCooksey, Charles ,\n\nCornforth, Fanny\n\n_Cornhill Magazine_\n\nCorreggio, Antonio da\n\nCortes, Hernan\n\nCotman, J. S.\n\nCouvreur, Monsieur (art dealer in Paris)\n\nCovent Garden, London ,\n\nCremation Society\n\nCrimean War (1853\u20136)\n\nCrivelli, Lucrezia\n\nCromwell, Oliver\n\nCrowne, John: _Darius, King of Persia_\n\nCruikshank, George\n\nCrystal Palace, London , , , ,\n\nCunliffe-Owen, Lady\n\nCunliffe-Owen, Philip 106\u201310, , , ,\n\nDacre Park, Merseyside\n\n_Daily Mail_\n\n_Daily Telegraph_ ,\n\nDaniels, William 190\u201393, ,\n\nDante Alighieri\n\n_Divine Comedy_\n\nDarwin, Charles: _On the Origin of Species_ 336\u201337\n\nd'Athanasi, Giovanni\n\nde Trafford, Sir Humphrey\n\nDegas, Edgar\n\nDelange, Monsieur (art dealer in Paris)\n\ndelftware ,\n\nDepartment of Practical Art (later Department of Science and Art)\n\nDerby, 13th Earl of\n\nDerby, 14th Earl of 218\u201319\n\nDerby Arboretum\n\n'Devonshire Gems, The'\n\nDevonshire, William Cavendish, rd Duke of ,\n\nDickens, Charles , ,\n\n_Little Dorrit_ , , 316\u201317\n\n_The Old Curiosity Shop_ ,\n\n_Our Mutual Friend_ 73\u201374\n\nDijon\n\nDilke, Charles\n\nDilke, Emilia, Lady ,\n\n_Art in the Modern State_\n\n_The Renaissance of Art in France_\n\nDisraeli, Benjamin ,\n\nDobson, Frank\n\nDoge's Palace, Venice\n\nDorio, Denys\n\nDorset County Museum\n\nDouglas, James 202\u20133\n\n_Nenia Britannica_ ,\n\nDowgate Hill brooch\n\nDowlais, South Wales , 138\u201339, , , , ,\n\nDowning Hall, Flintshire\n\nDresden\n\nDriffield museum\n\nDrolling, Michel Martin\n\nDu Sommerard, Alexandre\n\nDublin Museum\n\nDudley Gallery Art Society, Piccadilly, London\n\nD\u00fcrer, Albrecht\n\nDurlacher Brothers ,\n\nDutch East India Company ,\n\nDuveen, Henry\n\nDuveen, James Henry\n\nDuveen, Sir Joseph and Americans\n\n_Belle Ferronniere_ dispute , 308\u20139\n\n'Duveen Eye' 279\u201380, ,\n\nfunding of galleries ,\n\nknighted for his philanthropy\n\nmakes a fortune as a dealer ,\n\npersonality\n\nand Robinson\n\ntalent for salesmanship\n\nDuveen, Joseph Joel , 262\u201363\n\napprenticed to a porcelain merchant in Hull 150\u201351,\n\n'Duveen Eye' 279\u201380,\n\nloans pieces from his showroom\n\nredesigning of Wilson's house 287\u201388\n\nremarkets his business\n\nrivalry with the Schreibers 147\u201348, , 152\u201353,\n\nsets up a company\n\nand the thrill of competition\n\nDuveen, Rosetta 151\u201352\n\nDyce, William ,\n\nDyer, William\n\nEast End, London\n\nEast End, Sheffield\n\nEast India Company , ,\n\nIndian museum, Leadenhall Street, London 338\u201339\n\nEastlake, Sir Charles , , , , ,\n\nEcole Nationale Sup\u00e9rieure des Beaux-Arts, Paris\n\nEdie, Major William\n\nEducation Act (1870)\n\nEdward, the Black Prince\n\nEdward III, King\n\nEdward VII, King (as Prince of Wales)\n\nEgba people\n\nEgerton, Sir Philip\n\nEgyptian relics 219\u201327\n\n'Egyptian Temple' (later Egyptian Hall), Piccadilly, London\n\nEitelberger, Rudolph\n\nEl Greco: _Christ Driving Out the Moneylenders from the Temple_\n\nElgin, James Bruce, th Earl of\n\nElgin, Lord ,\n\nElgin Marbles , , ,\n\nethnography\n\ndisplay of ethnographic objects 331\u201337\n\nEtruria factory, Stoke-on-Trent ,\n\nEvans, John 199\u2013200,\n\n_Ancient Bronze Implements, Weapons and Ornaments of Great Britain_\n\n_Ancient Stone Implements_\n\nEvans, Mister (art dealer in Paris)\n\nExeter 33\u201334, ,\n\nexhibition societies\n\nExposition Universelle (Paris 1867)\n\nfans 179\u201380,\n\nFaussett, Bryan\n\nFaussett, Henry Godfrey ,\n\nFaussett collection , , 204\u20136,\n\nFej\u00e9rv\u00e1ry collection ,\n\nFerrara\n\nFildes, Luke\n\nFine Arts Club (later Burlington Fine Arts Club) , , , , , 82\u201384, 88\u201389, , , , , , , ,\n\nFine Arts Society\n\nFirst Afghan War (1839\u201342)\n\nFirst Republic (France)\n\nFirst World War ,\n\nFlaudin, Mme (dealer in Paris) ,\n\nFleet Ditch\n\n_Flora_ affair 290\u201396, , ,\n\nFlorence , , ,\n\nFontana, Giovanni\n\nforgery issue 290\u2013312\n\nForster, John\n\nFort Nelson, near Portsmouth\n\nFortnum, Charles 356\u201357\n\nFournier (a dealer contact of Charlotte Schreiber)\n\nFournier, M. (dealer in Paris) ,\n\nFowke, Captain Francis ,\n\nFragonard, Jean-Honor\u00e9\n\n'Fragonard Room'\n\nFranco-Prussian War (1870\u201371) , ,\n\nFrankfurt\n\nFranks, Augustus Wollaston 93\u201396, , , , , , , ,\n\nbookplate\n\nBritish and Medieval Antiquities department\n\nand Bushell , 347\u201348\n\nand catalogue of Schreiber collection\n\na close friend of Charlotte Schreiber\n\ncollecting skills 180\u201381\n\ndonates from his own collection , , 98\u201399\n\nexpert on non-European works ,\n\nand Faussett collection\n\nforgeries issue 301\u20133,\n\nas founder of modern British Museum\n\nand Japanese art\n\nand Mayer\n\npaid role at British Museum\n\npersonal collection\n\npersonality ,\n\nprofessionalism and scholarship\n\nreshapes the collections 94\u201395\n\nand Robinson , , , , , 148\u201349\n\na skilled politician\n\nspecially struck medallion\n\nFraser, Mary Crawford ,\n\nFrederick III, Emperor of Germany\n\nFreizor, George (G.-A. Freezor)\n\nFrench Industrial Exhibition (1844)\n\nFrench Revolution , , , , , ,\n\nFreud, Sigmund ,\n\nFrick, Henry Clay\n\nGainsborough, Thomas ,\n\nGalton, Francis\n\nGard, Richard Sommers\n\nGaskell, Elizabeth ,\n\nGauguin, Paul\n\nGedeonov, Stepan\n\nGenoa ,\n\nGeological Society\n\nGeorge II, King\n\nGeorge III, King ,\n\nGilbert, Sir W. S. and Sullivan, Sir Arthur: _The Mikado_\n\nGirtin, Thomas\n\nGladstone, William Ewart , , , , ,\n\nGladwell Brothers 105\u20136\n\nGlasgow\n\nGlen Aldred, Surrey ,\n\nGoadsby, Thomas\n\nGothic revival\n\nGouda ,\n\nGovernment of India Act (1858)\n\nGovernment Select Committee on Arts and their connection with Manufactures , 35\u201336\n\nGower, Earl\n\nGranada, Spain\n\nGrand Tour , , , , , ,\n\nGrant, President Ulysses S.\n\nGranville, Lord\n\n_Graphic, The_\n\nGreat Exhibition (London, 1851) , , 15\u201319, , , , 83\u201384, 142\u201343, , , , , , , , , , ,\n\nGreat Stupa, Sanchi, India\n\nGreat Wall of China\n\nGr\u00e9goire, Abb\u00e9\n\nGrey, Sir Edward\n\nGrimm, Hermann\n\nGrossmith, George\n\nGrosvenor, Thomas\n\nGrosvenor Gallery, Bond Street, London 43\u201344, ,\n\nGuanyin statue\n\nGuest, Augustus\n\nGuest, Ivor Bertie ,\n\nGuest, Sir Josiah John 134\u201339, 143\u201344,\n\nGuest, Montague , ,\n\nGuy's Hospital, London\n\nHabsburgs ,\n\nHague, The ,\n\nHahn, Harry\n\nHaliburton, Mrs (a china collector) ,\n\nHalifax\n\nHamburg\n\nHamilton, Duke of , ,\n\nHan dynasty (206 BCAD )\n\nHanley School of Design, Staffordshire ,\n\nHanover, Germany\n\nHardy, Thomas: _A Pair of Blue Eyes_ ,\n\nHarewood, Earl of\n\n_Harper's New Monthly Magazine_ ,\n\nHarris, John\n\nHarrison, Peggy\n\nHart, Sir Robert\n\nHawthorne, Nathaniel\n\nHeda, Willam Claesz\n\nHelmback, Abraham\n\nHenderson, John\n\nHenry IV, King of France\n\nHeppington, Kent ,\n\nHeriot, George\n\nHertz collection\n\nHistorical Society of Lancashire and Cheshire\n\nHobson, Mrs, of Pitsmoor\n\nHogarth, William\n\nHolburne, Sir Thomas William\n\nHolburne Museum, Bath\n\nHolmes, William Henry\n\nHoly Thorn Reliquary of Jean, Duc de Berry\n\nHooton, Cheshire\n\nHope, Alexander Beresford\n\nHorniman, Frederick John\n\nH\u00f4tel de Cluny, Paris 59\u201360\n\nH\u00f4tel de Ville, Paris\n\nH\u00f4tel Drouot, Paris\n\nHuddersfield Mechanics' Institute\n\nHuish, Marcus\n\nHull, East Yorkshire , , ,\n\nHunt, Leigh\n\nHunt, William Holman ,\n\nHuntington, Revd Robert\n\n_Illustrated London News_\n\nImperial Hermitage, St Petersburg\n\nImperial Summer Palace, near Peking 354\u201355\n\nIndian Rebellion (1857)\n\nIndustrial Revolution ,\n\nInternational Exhibition (Florence, 1861)\n\nInternational Exhibition of Industry and Art (London, 1862) 7\u201311, , , , ,\n\nInternational Exhibition (Philadelphia, 1876)\n\njade ,\n\nJames, Henry ,\n\n_The Portrait of a Lady_ 246\u201347\n\nJameson, Anna\n\n_A Companion to the Most Celebrated Private Galleries of Art in London_\n\n_Handbook to the Public Galleries of Art in and near London_\n\n_Sacred and Legendary Art_\n\nJanin, Jules\n\nJapanese art 332\u201333,\n\nJeckyll, Thomas , ,\n\nJohn Baillie's Gallery, London\n\nJones, John\n\nJosephine, Empress\n\nKaiser-Friedrich Museum (later Bode Museum), Berlin , , ,\n\nK'ang-hsi period (1662\u20131722) , ,\n\nKant, Immanuel\n\nKeats, John\n\nKeiser, Aelbrecht de\n\nKelmscott Manor\n\nKesslerloch, Switzerland\n\nKipling, Rudyard\n\nKlenze, Leo von\n\nKnowsley, Merseyside\n\nKoh-i-Noor diamond\n\nKryser's showroom, Rotterdam ,\n\nKublai Khan\n\nKunz, George Frederick\n\n_Lady at a Harpsichord, A_\n\nLamb, Lady Caroline\n\nLamb, Charles: 'Old China'\n\nLandseer, Sir Edwin\n\nLangham House, Portland Place, London ,\n\nLayard, Austen Henry , , , 197\u201398, ,\n\nLeeds ,\n\nMechanics' Institute\n\nLehmann, Henri\n\nLeighton, Frederick\n\nLeonardo da Vinci\n\nand _Flora_ affair 290\u201395\n\n_Mona Lisa_ , ,\n\nLeopold I, King of Belgium\n\nLeopold Wilhelm, Archduke, of Austria\n\nLepsius, Karl Richard ,\n\nLevant Company\n\nLeyland, Frederick Richards 284\u201388\n\nLiberty's store, Regent Street, London ,\n\nLindenschmidt, Ludwig\n\nLindsay, Sir Coutts\n\nLindsey, Albemarle Bertie, th Earl of\n\nLiterary and Philosophical Societies ,\n\nLiverpool , , , , , , , , 218\u201319, , , , ,\n\nNatural History Museum (Duke Street) 218\u201319\n\nLiverpool Academy of Art\n\nLiverpool Town Council ,\n\nLiverpool volunteer borough guard\n\nLiverpool Museum , , , ,\n\nLivingstone, David ,\n\nLockhart, Sir James Stewart\n\nLombardi-Baldi collection\n\nLondesborough, Lord\n\n_London Review, The_\n\nLondon School of Design ,\n\nLouis XIII, King of France ,\n\nLouis XIV, King of France ,\n\nLouis d'Orl\u00e9ans, Duke of Orl\u00e9ans\n\nLouis Philippe, Duke of Orl\u00e9ans\n\nLouvre, Paris , 53\u201356, , , , ,\n\nLowe, Lord , 87\u201388\n\nLucas, George\n\nLucas, Richard Cockle 292\u201393,\n\nLudwig I, King of Hungary and Poland\n\nLyceum club, Liverpool\n\nLyme Regis, Dorset\n\n_Mabinogion, The_ 140\u201341\n\nMacaulay, Thomas Babington, st Baron: _History of England_\n\n_Magazine of Art_ 36\u201337\n\nMagdala Treasure 222\u201323\n\nMalcolm, John, th Laird of Poltalloch\n\nManchester ,\n\nfree libraries\n\nMechanics' Institute 34\u201335,\n\nManchester Cricket Club\n\n_Manchester Guardian_\n\nManchester Ship Canal\n\nMarks, Emanuel , 265\u201366, ,\n\nMarks, Murray 275\u201389,\n\nAmerican clients , ,\n\nand blue-and-white ceramics\n\nbusiness card 273\u201374,\n\nchildhood and education\n\ncollecting interests\n\ncollections , 375\u201376\n\nconnections in Holland\n\ndealer and collector\n\n_Flora_ affair 290\u201396,\n\nforgeries issue 304\u20135, , 309\u201312\n\nfruitful alliance with South Kensington 311\u201312\n\nand the Peacock Room , ,\n\npersonality 259\u201360, , ,\n\nand Pre-Raphaelites\n\nreinvents character of the art dealer ,\n\nreliquary cross issue 309\u201310\n\nand Rossetti 257\u201361, ,\n\nshowrooms , , 270\u201373, ,\n\nMarks van Galen\n\nMarlborough, st Duke of\n\nMarlborough, th Duke of\n\nMarlborough House, Pall Mall, London , 66\u201367, 69\u201372, , , ,\n\nMarlborough, Sarah, Duchess of\n\nMarochetti, Baron Carlo\n\nMarried Women's Property Act (1870)\n\nMarryat, Captain Frederick\n\nMarryat, Joseph\n\nMary II, Queen\n\nMary Queen of Scots\n\nMayer, Jane\n\nMayer, Joseph\n\nacquires Egyptian objects\n\nappreciates scholarship\n\nArt Treasures Exhibition a triumph for him ,\n\nBram Hertz collection 208\u20139\n\nand Charlotte Schreiber\n\ncollecting interests , 192\u201393, , , , 247\u201348\n\ndeath\n\nEgyptian objects , 224\u201325,\n\nand Faussett collection 204\u20136\n\nflourishing workshop\n\nforgeries in his collection\n\nfunding 207\u20138\n\ninvests in a public library 251\u201353\n\ninvolvement in Bebington 249\u201353\n\na Liverpool jeweller , , ,\n\nlove life 241\u201344\n\nhis museum in Liverpool 217\u201327, , , , , ,\n\npersonality\n\nportraits 189\u201393, , , ,\n\npurchases existing collections wholesale 206\u20137\n\nand Roach Smith , ,\n\nstart of his interest in collecting\n\nin the volunteer movement 249\u201351\n\nand Wedgwood 210\u201315\n\n'History of the art of pottery in Liverpool'\n\nMayer, Joseph and Boyle, Frederick\n\n_Early Exhibitions of Art in Liverpool: With some Notes for a Memoir of George Stubbs, R.A._\n\n_Memoirs of Thomas Dodd, William Upcott and George Stubbs R.A._\n\nMayer Collection 238\u201339\n\nMayer Hall, Bebington\n\nMayer Library\n\nMayer Trust\n\nMechanics' Institutes 34\u201335,\n\nMeissen works\n\nMeredith, George ,\n\n_The Egoist_\n\nMerlin\n\nMerthyr Tydfil ,\n\nMeteyard, Elizabeth\n\nMetropolitan Museum of Art, New York City ,\n\nMexican quartz crystal skull forgeries 297\u201398\n\nMichelangelo ,\n\n_David_\n\n'Manchester Madonna'\n\nMilan\n\nMilan, Duke of\n\nMillais, John Everett , , ,\n\n_Christ in the House of His Parents_\n\nMing dynasty\n\nMonet, Claude\n\nMonte di Pi\u00e8ta 78\u201379\n\nMorelli, Giovanni\n\nMorgan, J.P.\n\nMorris, William , ,\n\nMortimer, John\n\nMortlocks (London dealers)\n\nMosses, Alexander\n\nMount Vernon Place, Baltimore\n\nMunich\n\nMus\u00e9e Central des Arts, Paris\n\n_see also_ Louvre\n\nMuseum f\u00fcr V\u00f6lkerkunde, Berlin\n\nMuseums Acts ,\n\nNagasaki, Japan ,\n\nNapier, Robert\n\nNapoleon I, Emperor , , , ,\n\nNapoleon III, Emperor , ,\n\nNapoleonic Wars (1799\u20131815) ,\n\nNash, Paul\n\nNational Art Collections Fund\n\nNational Art Training School\n\nNational Gallery, Trafalgar Square, London , , , , , , , , , , , , , , , ,\n\nNational Portrait Gallery, London\n\nNatural History Museum, London\n\nNegroni, Captain de\n\nneo-classical style ,\n\nNeville, Dorothy\n\nNew Hermitage, St Petersburg\n\nNew York ,\n\n_New York Herald_\n\n_New York Times_ , 294\u201395\n\n_New York World_\n\nNewbold, Mary\n\nNewcastle Literary and Philosophical Society\n\nNewcastle-under-Lyme\n\nNewton Manor, near Swanage, Dorset 100\u2013102, , , , ,\n\nNimrud, northern Mesopotamia (later Iraq)\n\nNorth Eastern Railway Company\n\nNorthcote, Sir Stafford 33\u201334\n\nNorthumberland, rd Duke of 198\u201399\n\nNorthumberland, th Duke of 198\u201399\n\nNorwich\n\nNottingham , , ,\n\nNottingham Museum 36\u201337, ,\n\nNottingham School of Art\n\nNuremberg _Hausmaler_\n\n'Octave' dinners\n\nOgubonna, Chief\n\n'Old Japan' porcelain\n\n'Old Nankin Porcelain'\n\nOld Trafford, Manchester ,\n\nOppenheim, Mme (dealer in Paris)\n\nOriental Bank 362\u201363\n\nOrl\u00e9ans Collection 27\u201328,\n\nOsborne House, Isle of Wight\n\nOudh, Nawab of\n\nOverstone, Lord ,\n\nOwen, Mary\n\nOwen, Robert\n\nOwen, Wilfred\n\nPadua\n\nPalace of Westminster, London\n\nPalais de Tuileries, Paris\n\nPalais Royal, Paris\n\nPalissy, Bernard\n\nPalmerston, Viscount ,\n\nPanizzi, Antonio ,\n\nParis\n\nat the heart of the collectors' circuit\n\nRobinson in 52\u201356, 58\u201362,\n\nthe Schreibers in (1871) 128\u201330\n\nParis Commune , , ,\n\nParis Exhibition (1855) , 17\u201318,\n\nParkes, Sir Harry\n\nParmiggiani, Luigi (Louis Marcy) 303\u20134\n\nParry, William\n\nPater, Walter\n\nPattison, Mark\n\nPaxton, Joseph\n\nPeacock Room, London 284\u201387,\n\nPeel, Sir Robert ,\n\nPegus, Reverend Peter , ,\n\nPeking (now Beijing) 319\u201323, , , , , , , , , , , , ,\n\nPennant, Thomas 240\u201341\n\n_Journey from Chester to London_\n\n_A Tour of Scotland_\n\nPennant House, Bebington, Wirral, Merseyside 240\u201343, , , , ,\n\nPerry, Commodore 332\u201333\n\nPhiladelphia\n\nPhilharmonic Hall, Liverpool\n\nPhilippe II, Duke of Orl\u00e9ans ,\n\nPhillips, Henry Wyndham\n\nPhotographic Salon\n\nPiccadilly Hotel, London\n\nPickford & Co. ,\n\nPierpont Morgan, J. 282\u201383, ,\n\nPitt Rivers, Lieutenant-General Augustus Henry\n\nPitt Rivers collection ,\n\nPitti Palace, Florence\n\nplaying cards , 181\u201382, ,\n\nPolo, Marco\n\nPolonet (Brussels dealer) 173\u201374\n\nPompeii\n\nPonsonby, William\n\nPoona figures\n\nPoynter, Edward J. 107\u20138\n\nPrado, Madrid ,\n\nPrague Castle\n\nPrang, Louis\n\nPre-Raphaelites , , , , , , , , ,\n\nPreston ,\n\nPrudhoe, Lord (later Duke of Northumberland)\n\nPublic Libraries and Museums Act (1850)\n\nPublic Record Office\n\nPughe, William Owen\n\nPugin, Augustus\n\nPull, Georges 300\u2013301,\n\nPulszky, Franz\n\n_Punch Almanack_\n\n_Punch_ magazine 32\u201333, , , ,\n\nPune, India\n\nPurdon Clarke, Caspar 115\u201316,\n\nPutuo island\n\nPyramids, the\n\n_Quarterly Review_ ,\n\nQueen Anne style\n\nRaffles, Sir Stamford\n\n_Rainbow Landscape_\n\nRameses II, Pharaoh\n\nRaphael , ,\n\n_Madonna degli Ansedei_\n\nRead, Charles Hercules\n\nRedgrave, Richard\n\nReform Act (1867) ,\n\nRejlander, Oscar Gustave: _The Two Ways of Life_\n\nRembrandt van Rijn ,\n\nRenoir, Pierre-Auguste\n\nRepresentation of the People Act (Reform Act; 1832)\n\nReynolds, Sir Joshua\n\n_A Portrait of a Child with Doll_\n\nRice, Eleanor Elkins Widener\n\nRijksmuseum, Amsterdam\n\nRoach Smith, Charles and Bram Hertz collection\n\nand Faussett collection ,\n\nLondon collection 200\u2013201\n\nand Mayer , ,\n\nRobinson, Sir John Charles , , , , , , ,\n\naims to foster aesthetic appreciation\n\nambition , ,\n\nappearance ,\n\nbackground and childhood\n\ncareer as a dealer 99\u2013111, ,\n\non Chinese ceramic art\n\nclients 102\u20133,\n\ncollecting skills 71\u201372, , , 107\u20138, , 180\u201381\n\nhis collection , 113\u201316, 119\u201320,\n\non collectors 68\u201369\n\na connoisseur , , ,\n\nand country-house collections\n\ncriticism of South Kensington , 111\u201312\n\ncurator at South Kensington Museum , , 64\u201365, 67\u201368, , ,\n\ndeath (1913) , ,\n\ndismissal from art referee post 90\u201392, ,\n\nhis estate 119\u201320\n\nfinancial matters 72\u201376, , , , , 97\u2013100\n\nFine Arts Club , , , 82\u201384,\n\nforgeries issue 298\u2013301, 303\u20134, , ,\n\nand Franks , , , 148\u201349\n\nill-health\n\nin Italy 62\u201364, , , , , ,\n\nknighted (1887)\n\nlooks for homes for his pieces 115\u201316\n\nand Marks\n\nand Newton Manor 100\u2013102, , , , ,\n\nin Paris 52\u201356, 58\u201362,\n\npersonality , 52\u201353, , , , ,\n\nand Pierpont Morgan\n\nrelationship with Cole , , , , , 85\u201393, ,\n\nrelationship with the museum after Cole's retirement 107\u201312\n\nand Renaissance , , 58\u201359, ,\n\nsells part of his collection ,\n\nSkinner's attitude to ,\n\nand Special Exhibition 12\u201314, , , , , 81\u201382, 228\u201329\n\nSurveyor of Queen's Pictures 113\u201314,\n\ntalent for rewriting the past to his advantage\n\ntownhouse in York Place ,\n\nRobinson, Marian Elizabeth (n\u00e9e Newton)\n\nRolfe, W. H. 206\u20137\n\nRomantic movement\n\nRome , ,\n\nR\u00f6misch-Germanisches Zentralmuseum, Mainz\n\nRosetta Stone ,\n\nRossetti, Dante Gabriel , 257\u201361, , , , , , , , , , , ,\n\nRossetti, Elizabeth\n\nRossetti, William 258\u201359\n\nRothschild family ,\n\nRotterdam , , , ,\n\nRouen Museum, Normandy\n\nRoyal Academy, London , , , , , , , ,\n\nRoyal Archaeological Institute of Great Britain\n\nRoyal Botanic Gardens, Kew, Surrey\n\nRoyal Collection 103\u20134\n\nRoyal College of Art: 'Female School'\n\nRoyal Commissions ,\n\nRoyal Geographical Society ,\n\nRoyal Institution, Liverpool ,\n\nRoyal Mersey Yacht Club\n\nRoyal Museum, Berlin\n\nRoyal Navy\n\nRoyal Society , ,\n\nRoyal Society of Arts (previously Society for the Encouragement of Arts, Manufactures and Commerce)\n\nRoyal Society of Painters and Etchers\n\nRoyal Society of Painters in Water-colours\n\nRubens, Peter Paul , ,\n\nRudolph II, Holy Roman Emperor\n\nRuskin, Effie\n\nRuskin, John , , , 56\u201357, 82\u201383, , , ,\n\nand the Sheffield museum 39\u201340, ,\n\n_Seven Lamps of Architecture_\n\n_The Stones of Venice_ , ,\n\n'The Two Paths' (lecture)\n\nRussell, Lord John\n\nSaffron Walden Museum\n\nSt Mark's Cathedral, Venice\n\nSalisbury, Earls of\n\nSalt, Henry\n\nSalt, Titus\n\nSaltaire, near Bradford\n\nSalting, George , ,\n\nSams, Joseph\n\nSargent, John Singer\n\n_Saturday Review_\n\nSauvageot, Charles 60\u201361\n\nSaxe-Weimar, Grand Duke of\n\nScharf, George\n\nSchinkel, Karl Friedrich\n\nSchools of Design 20\u201321, 46\u201347, , , ,\n\nSchopenhauer, Arthur\n\nSchreiber, Colonel Charles ('Charley') 125\u201328, 144\u201345, ,\n\nArt Treasures Exhibition\n\nceramic collecting , ,\n\ndeath\n\nfamily background and education\n\nFine Arts Club ,\n\ngourd-shaped bottle story 168\u201374\n\nill-health , , , 182\u201383\n\nmarries Charlotte 144\u201345\n\nparliamentary role\n\npersonality\n\nsolo trip to Holland , 169\u201370\n\nstatistics of trips\n\ntutors Charlotte's son Ivor ,\n\nSchreiber, Lady Charlotte Elizabeth (n\u00e9e Bertie) , , ,\n\nachievements of\n\nappearance 131\u201332\n\nand Art Treasures Exhibition\n\nat 'van Galen's'\n\non Bock's 'sad story'\n\nchildhood\n\na close friend of Franks\n\ncollecting interests , , 130\u201331, , 156\u201357, , 179\u201380, 373\u201374\n\ndeath ,\n\ndonation to South Kensington Museum , 183\u201385,\n\nfamily background 132\u201333\n\nand Fine Arts Club\n\nfirst marriage to John Guest 135\u201340\n\nforgery issue ,\n\ngourd-shaped bottle story 168\u201374,\n\nhelps other women collectors 162\u201363\n\nill-health\n\nan intrepid traveller , 126\u201328\n\nand John's death 143\u201344\n\nand Joseph Joel Duveen 147\u201348, , 152\u201353\n\nlate enthusiasm for collecting , , 153\u201354\n\nand Layard's excavations in Mesopotamia\n\nManchester Art Treasures Exhibition\n\nand Marks\n\nmarries Charles Schreiber 144\u201345\n\nand Mayer\n\nmeets and marries John Guest 134\u201335\n\nrespected as a collector , ,\n\nscholarship 177\u201380\n\nstatistics of trips\n\ntranslates _The Mabinogion_ 140\u201341\n\nvisits the Great Exhibition 142\u201343\n\nScott, Sir Walter , ,\n\n_The Antiquary_ 191\u201392,\n\nSecond Afghan War (1878\u201380)\n\nSecond Opium War (1855\u201360) ,\n\nSecond Republic (France)\n\nSettled Land Act (1882)\n\nShangdu, Inner Mongolia\n\nShanghai 318\u201321, ,\n\nShaw, Richard Norman 270\u201371,\n\nSheffield , , ,\n\nmuseum 39\u201340, ,\n\nSickert, Walter\n\nSierra Leone 333\u201334\n\n'Silverpen' _see_ Meteyard, Elizabeth\n\nSkinner, Arthur Banks ,\n\nSmith, Major Robert Murdoch\n\nSmith, Admiral William Henry\n\nSnyders, Frans: _Dead Game and Fruit_\n\nSoane, Sir John ,\n\nSociety for the Encouragement of Arts, Manufactures and Commerce (known as the Society of Arts; later Royal Society of Arts) , , , ,\n\n'Ancient and Medieval Art' exhibition (1850)\n\nSociety for the Encouragement of the Fine Arts\n\nSociety for the Protection of Ancient Buildings\n\nSociety of Antiquaries , , 196\u2013200, , , ,\n\nSociety of Chemical Industry\n\nSociety of Dilettanti\n\nSotheby's auctioneers , , , , , ,\n\nSoulages, Jules\n\nSouth Kensington, London\n\nSouth Kensington Museum (later Victoria and Albert Museum), London , ,\n\nBrompton Boilers ,\n\nand Bushell , , 348\u201349, 366\u201367\n\nceramics displays\n\nChinese displays ,\n\nCole as first Director , 19\u201320\n\ncollecting Japanese objects\n\nColonial and Indian Exhibition (1886)\n\ndescribed 9\u201310\n\nforgery issue , ,\n\nFranks catalogues Japanese ceramics\n\nidentity of the collection\n\nIndian material\n\ninternational objects 324\u201325, 329\u201330, 331\u201332\n\nloan exhibitions ,\n\nMagdala Treasure\n\nrange of objects bequeathed 372\u201373\n\nrelocation from Marlborough House to South Kensington 69\u201372\n\nrenamed (1899)\n\nrivalry with the British Museum\n\nRobinson as Curator , , 64\u201365, 67\u201368, , ,\n\nRobinson offers pieces from his collection 115\u201316,\n\nrole of , , 84\u201385,\n\nRuskin attacks\n\nSchreiber collection 183\u201385\n\n'Special Exhibition of Works of Art of the Medieval, Renaissance, and more Recent Periods, on loan at the South Kensington Museum' (1862) 10\u201312, , , , , 81\u201382\n\n_Catalogue of Chinese Objects_\n\nSpeke Hall, Liverpool\n\nSpencer, Herbert: _Principles of Biology_\n\nSpenser, Earl\n\nSphinx, Giza\n\nSpitzer, Fr\u00e9d\u00e9ric\n\nStaffe, Baroness\n\nStafford, Marquess of\n\n_Standard, The_ ,\n\nStanley, Henry Morton\n\nStansted Park, West Sussex\n\nStanwick Park, Yorkshire\n\nStein, Sir Aurel\n\nSterling, Edward\n\nStoa, Athens\n\nStobart, Henry\n\nStone, Marcus\n\nStotesbury, Edward\n\nStowe House, Buckingham\n\nStrutt, Joseph\n\nSuez Canal\n\nSullivan, Sir Arthur\n\nSunderland\n\nmuseum 40\u201341\n\n'Sunderland Library, The' 29\u201330\n\nSunderland Museum and Library\n\nSung dynasty (960\u20131279)\n\nSweden: national gallery of art\n\nSwinburne, Algernon\n\nTangiers, Morocco\n\nTarragona\n\nTate Gallery, London ,\n\nTemple Club, Strand, London 328\u201329\n\n'Ten Thousand Chinese Things' exhibition (Hyde Park, London, 1841)\n\nTeniers, David, the Younger\n\nTennyson, Alfred, Lord ,\n\nTheatre Royal, Drury Lane, London\n\nThebes ,\n\nThirty Years War (1614\u201348)\n\nThomas, Edward\n\nThompson, Sir Henry , 276\u201378, ,\n\nexhibition opening ,\n\nThomson, John\n\nTiffany & Co. ,\n\n_Times Literary Supplement_\n\n_Times, The_ , , , , , , , ,\n\nTippoo's Tiger ,\n\nTipu, Sultan of Mysore\n\nTitian ,\n\nTownsend, Henry 333\u201334\n\nTradescant, John 225\u201326\n\nTrafalgar, Battle of (1805)\n\n_Transactions of the Historical Society of Lancashire and Cheshire_\n\nTrapnell, Alfred\n\nTreasury , ,\n\nTreaty of Peace and Amity (Treaty of Kanagawa) (1854) 332\u201333\n\nTreck, Jan\n\nTunstall, Marmaduke\n\nTurner, James Aspinall\n\nTurner, J. M. W. , , , ,\n\nUffington House, Lincolnshire , ,\n\nUniversity of Oxford ,\n\nUniversity of Vienna\n\nUtrecht , ,\n\nMuseum\n\nValentia, George Annesley, Viscount\n\nValladolid, Spain\n\nVan Houtum (dealer in Amsterdam)\n\nVan Huckelm, Monsieur\n\nVasters, Reinhold 304\u20135,\n\nVeitch, John Gould\n\nVenice 62\u201363\n\nVerona\n\nVersailles\n\nVictoria, Kaiserin (Princess Vicky) 103\u20134,\n\nVictoria, Queen , 103\u20134, , , , , , , , , , , ,\n\nVictoria and Albert Museum, London (previously South Kensington Museum) , ,\n\nand Bushell\n\nEnglish ceramics collection , 168\u201369\n\nand Marks' bequest\n\nredesign of the galleries\n\nvalue of gifts and bequests\n\nand Vasters 304\u20135,\n\nVienna Museum of Applied Arts\n\nVienna School of Art History\n\nVillanueva, Juan de\n\nVon Bode, Wilhelm , , , 291\u201392, 294\u201395, ,\n\nWaagen, Gustav , 35\u201336, , , , 233\u201334, 236\u201337\n\nWalmsley, Sir Joshua\n\nWalpole, Horace ,\n\nWalters, William Thompson 357\u201360, ,\n\nWar of Austrian Succession (1740\u201348)\n\nWard, Robert Plumer\n\nWard, William Hayes\n\nWarrington, Cheshire\n\nWedderburne, Mr 210\u201312,\n\nWedderburne, Mrs\n\nWedgwood , 177\u201378, , 210\u201315, , ,\n\nshowrooms\n\nWedgwood, Josiah , ,\n\nWedgwood family\n\nWeininger, Salomon\n\nWellington, Duke of ,\n\nWertheimer, Asher 262\u201363\n\nWhistler, James McNeill , , , , , , , 284\u201387, ,\n\n_Art and Money; or, The Story of the Room_\n\n_La Princesse du pays de la porcelaine_\n\nWhite, Gilbert\n\n_The Natural History and Antiquities of Selborne_\n\nWhitemarsh Hall, Pennsylvania\n\nWhitworth, Sir Joseph ,\n\nWhitworth Art Gallery, Manchester ,\n\nWhitworth Institute, Manchester ,\n\nWilde, Oscar ,\n\nWilhelm II, Kaiser\n\nWilliams, Penry\n\nWilson, Arthur\n\nWinchester\n\n_Woche, Die_\n\nW\u00f6lfflin, Heinrich 117\u201318\n\nWolverhampton\n\nWomen's Trade Union League\n\nWordley, James 193\u201394\n\nWordley, Mary\n\nWorshipful Company of London Fan Makers\n\nWortley Montagu, Edmund\n\nWortley Montagu, Lady Mary\n\nWren, Sir Christopher\n\nWright, Thomas\n\nWu Men market, Peking 322\u201323\n\n_Wunderkammer_ (cabinet of curiosities) , , ,\n\nWycliffe Hall, near Bernard Castle, County Durham\n\nYuan dynasty (1271\u20131368)\n\nZoological Society, London\n\nA view of one half of the South Court at the South Kensington Museum, filled with the packed cases of the 1862 loan exhibition. This wood engraving appeared in the _Illustrated London News_ on 6 December 1862, towards the end of the show.\n\nJohn Charles Robinson in middle age, probably around the time he left the South Kensington Museum. The oil painting is by J. J. Napier and is in the collection of the National Portrait Gallery.\n\nThis photograph of a small corner of Newton Manor shows part of Robinson's extensive and varied collection in the domestic setting.\n\nCharlotte Schreiber, around the time of her second marriage in 1855. The drawing is by George Frederic Watts, one of the most influential and renowned Victorian portrait painters.\n\nCharles Schreiber possibly sat for this portrait on his election as MP for Cheltenham in 1865, ten years after his marriage to Charlotte. It is also by Watts.\n\nThe gourd-shaped bottle that tormented the Schreibers during 1873 and 1874.\n\nDante Gabriel Rossetti's collection in the sitting room of 16 Cheyne Walk in 1882. Rossetti is reading to the poet and critic Theodore Watts-Dunton.\n\nThe Peacock Room in the house of Frederick Richards Leyland, decorated by James Whistler in 1877. The photograph shows Whistler's portrait of 'La princesse du pays de la porcelaine' and some of the blue-and-white collection assembled by Murray Marks.\n\nMurray Marks in his London home. The photograph shows Rossetti's characteristic portrait of Marks' wife, as well as some choice pieces from Marks' collection of blue-and-white on the mantlepiece.\n\nMurray Marks' distinctive trade card, designed as a result of a collaboration between Whistler, Rossetti and William Morris.\n\nJoseph Mayer's first portrait, painted around 1840 by William Daniels, shows Mayer at the heart of his growing collection.\n\nA view of the 'Mummy Room' in Mayer's Egyptian Museum in Liverpool, complete with stuffed crocodile.\n\nStephen Wootton Bushell's visiting card with a portrait photograph taken shortly after his arrival in China in 1868.\n\nThe semi-ruined Bronze Pavilion at the Imperial Summer Palace in Peking. This picture was taken by the pioneering Scottish photographer, John Thomson, on a trip to the city in 1871\u20132, during which he almost certainly worked with Stephen Wootton Bushell to document historical sites. Bushell owned an album of Thomson's photographs.\n\nThis pair of engravings by George du Maurier appeared in the _Punch Almanack_ in December 1875. They present collectors as obsessive and strange, physically frail and emotionally incapable of interacting with family and friends around them.\n\nThe architect Aston Webb was chosen to create a new complex of buildings on the South Kensington site in 1891; it finally opened in June 1909. The intervening years saw Webb modify many of his ideas, moving towards the finished buildings we recognize today as the Victoria and Albert Museum.\n\nAn exterior view of the domed glass palace at Old Trafford, Manchester, that housed the Art Treasures Exhibition in 1857. This picture appeared in an illustrated weekly periodical, the _Art-Treasures Examiner: A Pictorial, Critical and Historical Record of the Art-Treasures Exhibition_ , which was published especially for the duration of the exhibition.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**The Forever War**\n\nJoe Haldeman\n\n_For Ben and, always, for Gay._\n\n# Contents\n\nForeword\n\nAuthor's Note\n\nPrivate Mandella\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nSergeant Mandella 2007\u20142024 AD\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nLieutenant Mandella 2024\u20142389 AD\n\nChapter 1\n\nMajor Mandella 2458\u20143143 AD\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nPreview: _Forever Free_\n\nA Biography of Joe Haldeman\n\n# Hey, Joe, I Read Your Book, or, An Open Letter to Joe Haldeman,  \nCleverly Disguised as a Foreword to _The Forever War_\n\nDear Joe,\n\nTo get this letter to you started, and to set the scene for a theme I'll get back to, I want to remind you (and share with the onlookers reading this letter to you) about the first time I met both you and Gay, which was at the Worldcon in Glasgow in 2005. I forget the specific manner in which we were introduced\u2014I suspect my editor Patrick Nielsen Hayden may have made the introduction, as introducing science fictional people to each other is something he's very good at. I remember saying hello and then being marvelously flattered as Gay told me that she had enjoyed _Old Man's War_ , which was at the time my sole novel, having come out six months earlier. After she was done saying very nice things about it, you said, \"I've heard good things about it, but I'm afraid I haven't read it yet.\"\n\n\"That's all right,\" I said. \"I've heard good things about _The Forever War_ , and I haven't read it, either.\" To which you laughed, and then you and I and Gay went on to have a very nice conversation about other things. So that's how we met.\n\nLet me note two things about our meeting: first, you were _entirely_ gracious to me in the aftermath of my attempted witty banter, because in retrospect (i.e., three seconds later) I could see how the comment might have seemed snarky and dismissive, even if it was not meant that way (fortunately for me, you took it the right way); second, in terms of high science fictional crimes and misdemeanors, mine, in not having read your novel, was a far sight greater than yours in not reading mine. My novel was the work of a newbie writer who only a few people knew existed (thus my pleasure in Gay's having read it at all), whereas your novel was (and remains) a science fiction classic\u2014a winner of the Hugo, Nebula, and Locus awards, widely recognized as one of the two cornerstone works of military science fiction, along with _Starship Troopers_. You could be forgiven for not having gotten around to my book. I, on the other hand, did not get off so easily.\n\nIndeed, it's a measure of the significance of _The Forever War_ in science fiction literature that readers and reviewers simply assumed (a) that I had read it of course, and (b) that my own novel of military science fiction was riffing off of yours to some greater or lesser extent. When I admitted to people that, in fact, I had not read your book, I usually got one of two reactions, depending on whether they liked my book or not.\n\nHere's the one for if they liked the book:\n\nReader: I liked your book, man. I really like how you played the changes off of _The Forever War_ in it.\n\nMe: Well, thank you. But I have to admit I haven't read _The Forever War_ yet.\n\nReader: Really?\n\nMe: Yeah.\n\nReader: Have you been, like, _trapped in a box_ for the past thirty years?\n\nHere's the one for when they didn't like it:\n\nReader: Jeez, Scalzi, I sure hope you're paying Joe Haldeman royalties for how much you ripped off _The Forever War._\n\nMe: Well, actually, I haven't read the book.\n\nReader: Uh-huh. So you're not only a thief, you're also a _liar_.\n\nSo it went, for a few years, until, in fact, I actually _did_ start lying about whether I'd read the book, because I was tired of being _told_ I needed to read it. I _knew_ I needed to read it, you know? But I was _busy_ , writing my own books\u2014and, um, being distracted by shiny bits of foil. Yes, that was it. That was it exactly.\n\nFinally, for various reasons, this last year I came to a time and place where I was ready for _The Forever War_. I took it down from the shelf (where it had been, actually, for a few years\u2014did I mention that I am easily distractible?), closed the door of my office, and settled down for a good read.\n\nWhen I finished it, this was the thought I had about it: _Wow, I'm glad I waited until now to read this._\n\nReally, I was\u2014and am.\n\nThere are two reasons for this. The first is a simple and practical writing matter: If I had known going in about all the plot and character choices you made in your novel, I probably wouldn't have ended up making the same basic choices in mine. Because, you know, as a writer I have an ego, and I wouldn't have wanted to step in your footprints, and walked a path you had, even if it were better for my novel to have done so. I would have been self-conscious of it; I would have danced around certain footfalls, and I suspect my own novel would have not been the better for it. There's a whole other letter I could write, unpacking this statement and what it means, but I won't get into that now; suffice it to say for the moment that I would have felt like I would have to be original, even to the detriment of being good. It's easier on the finished end of the writing process to be compared to _The Forever War_ (flattering, too); on the writing end, it would have been an elephant in my head\u2014too much pressure; thanks, no. I'm happy to have missed that. The second reason is that I believe that _The Forever War_ was a novel of its time, and its time, for better or worse, has come around again.\n\nIt's no secret, to you or me or most of the people peering over our shoulders here, that _The Forever War_ comes out of the crucible of the Vietnam War, in which you served, and which, as I understand, marked you for its own, as it did with many who served in it. Science fiction as a genre has the benefit of being able to act as parable, to set up a story at a remove so you can make a real-world point without people throwing up a wall in front of it. You'd already essayed your experience in Vietnam in the contemporary novel _War Year_ (which I had actually read, and gave to my father-in-law, himself a veteran, as a gift), but _The Forever War_ was another, bigger bite of that apple\u2014your chance to explain to people who hadn't been there the confusion and bureaucracy, the muddled aims and random horror, and the alienation that those who went felt when they came back home to a nation and culture that they no longer quite fit into, because both had changed.\n\nI grew up as part of the fortunate generation between Vietnam and 9\/11, the ones whose cohort didn't have to experience what war was, save for a few short weeks in Grenada and Iraq, in '91. There's another generation, behind mine, that did not get to be so lucky. Hundreds of thousands of them went to the Middle East and a good portion of them are still there. Thousands have come back with flags draped over their coffins. Tens of thousands have come back injured, physically or mentally or both, and some portion of them feel the same disassociation to the land they've returned to that Mandella and Marygay felt with theirs. Whether one feels the war in Iraq or Afghanistan is right and necessary or not, there's no doubt a generation will be marked by it and claimed by it.\n\nTo my mind, there are two things that make a novel a \"classic\"\u2014a genuine classic, as opposed to merely \"old and continuing to sell.\" The first is that it speaks to the time in which the novel first appeared. There is no doubt _The Forever War_ did this; its awards and acclaim are signifiers of that fact. The second thing is tougher, and that is that it keeps speaking to readers outside its time, because what's in the book touches on something that never goes away, or at the very least keeps coming around.\n\nI also think there's no doubt _The Forever War_ is doing this, too, right now, in this time\u2014it's a parable whose lessons need to be learned once more. Like its hero, the book has come through time to be part of something; in this case it's to be a reminder to all those who are looking to come home again\u2014and those who care about them\u2014that there's someone who's been where they are now, and who knows what they feel, and why. Maybe it will help them find their way back. I would have missed the power of that if I had read the novel earlier than this. I'm aware of it now\u2014and glad for it.\n\nAll of which is to say: Hey, Joe, I read your book.\n\nEveryone is right about it.\n\nThank you.\n\nYours,\n\nJohn Scalzi\n\nJuly 2008\n\n# Author's Note\n\nThis is the definitive version of _The Forever War_. There are two other versions, and my publisher has been kind enough to allow me to clarify things here.\n\nThe one you're holding in your hand is the book as it was originally written. But it has a pretty tortuous history.\n\nIt's ironic, since it later won the Hugo and Nebula Awards, and has won 'Best Novel' awards in other countries, but _The Forever War_ was not an easy book to sell back in the early seventies. It was rejected by eighteen publishers before St. Martin's Press decided to take a chance on it. 'Pretty good book,' was the usual reaction, 'but nobody wants to read a science fiction novel about Vietnam'. Twenty-five years later, most young readers don't even see the parallels between _The Forever War_ and the seemingly endless one we were involved in at the time, and that's OK. It's about Vietnam because that's the war the author was in. But it's mainly about war, about soldiers, and about the reasons we think we need them.\n\nWhile the book was being looked at by all those publishers, it was also being serialized piecemeal in _Analog_ magazine. The editor, Ben Bova, was a tremendous help, not only in editing, but also for making the thing exist at all! He gave it a prominent place in the magazine, and it was also his endorsement that brought it to the attention of St. Martin's Press, who took a chance on the hardcover, though they did not publish adult science fiction at that time.\n\nBut Ben rejected the middle section, a novella called 'You Can Never Go Back.' He liked it as a piece of writing, he said, but thought that it was too downbeat for _Analog_ 's audience. So I wrote him a more positive story and put 'You Can Never Go Back' into the drawer; eventually Ted White published it in _Amazing_ magazine, as a coda to _The Forever War_.\n\nAt this late date, I'm not sure why I didn't reinstate the original middle when the book was accepted. Perhaps I didn't trust my own taste, or just didn't want to make life more complicated. But that first book version is essentially the _Analog_ version with 'more adult language and situations', as they say in Hollywood.\n\nThe paperback of that version stayed in print for about sixteen years. Then in 1991 I had the opportunity to reinstate my original version. The dates in the book are now kind of funny; most people realize we didn't get into an interstellar war in 1996. I originally set it in that year so it was barely possible that the officers and NCOs could be veterans of Vietnam, so we decided to leave it that way, in spite of the obvious anachronisms. Think of it as a parallel universe.\n\nBut maybe it's the real one, and we're in a dream.\n\nJoe Haldeman  \nCambridge, Massachusetts\n'Man was born into barbarism, when killing his fellow man was a normal condition of existence. He became endowed with a conscience. And he has now reached the day when violence toward another human being must become as abhorrent as eating another's flesh.'\n\n\u2014 _Martin Luther King, Jr_.\n\n# Private   \nMandella\n\n# 1\n\n'Tonight we're going to show you eight silent ways to kill a man.' The guy who said that was a sergeant who didn't look five years older than me. So if he'd ever killed a man in combat, silently or otherwise, he'd done it as an infant.\n\nI already knew eighty ways to kill people, but most of them were pretty noisy. I sat up straight in my chair and assumed a look of polite attention and fell asleep with my eyes open. So did most everybody else. We'd learned that they never scheduled anything important for these after-chop classes.\n\nThe projector woke me up and I sat through a short tape showing the 'eight silent ways.' Some of the actors must have been brainwipes, since they were actually killed.\n\nAfter the tape a girl in the front row raised her hand. The sergeant nodded at her and she rose to parade rest. Not bad looking, but kind of chunky about the neck and shoulders. Everybody gets that way after carrying a heavy pack around for a couple of months.\n\n'Sir' \u2014 we had to call sergeants 'sir' until graduation \u2014 'most of those methods, really, they looked... kind of silly.'\n\n'For instance?'\n\n'Like killing a man with a blow to the kidneys, from an entrenching tool. I mean, when would you _actually_ have only an entrenching tool, and no gun or knife? And why not just bash him over the head with it?'\n\n'He might have a helmet on,' he said reasonably.\n\n'Besides, Taurans probably don't even _have_ kidneys!'\n\nHe shrugged. 'Probably they don't.' This was 1997, and nobody had ever seen a Tauran; hadn't even found any pieces of Taurans bigger than a scorched chromosome. 'But their body chemistry is similar to ours, and we have to assume they're similarly complex creatures. They _must_ have weaknesses, vulnerable spots. You have to find out where they are.\n\n'That's the important thing.' He stabbed a finger at the screen. 'Those eight convicts got caulked for your benefit because you've got to find out how to kill Taurans, and be able to do it whether you have a megawatt laser or an emery board.'\n\nShe sat back down, not looking too convinced.\n\n'Any more questions?' Nobody raised a hand.\n\n'OK. Tench-hut!' We staggered upright and he looked at us expectantly.\n\n'Fuck you, sir,' came the familiar tired chorus.\n\n'Louder!'\n\n'FUCK YOU, SIR!' One of the army's less-inspired morale devices.\n\n'That's better. Don't forget, pre-dawn maneuvers tomorrow. Chop at 0330, first formation, 0400. Anybody sacked after 0340 owes one stripe. Dismissed.'\n\nI zipped up my coverall and went across the snow to the lounge for a cup of soya and a joint. I'd always been able to get by on five or six hours of sleep, and this was the only time I could be by myself, out of the army for a while. Looked at the newsfax for a few minutes. Another ship got caulked, out by Aldebaran sector. That was four years ago. They were mounting a reprisal fleet, but it'll take four years more for them to get out there. By then, the Taurans would have every portal planet sewed up tight.\n\nBack at the billet, everybody else was sacked and the main lights were out. The whole company'd been dragging ever since we got back from the two-week Lunar training. I dumped my clothes in the locker, checked the roster and found out I was in bunk 31. Goddammit, right under the heater.\n\nI slipped through the curtain as quietly as possible so as not to wake up the person next to me. Couldn't see who it was, but I couldn't have cared less. I slipped under the blanket.\n\n'You're late, Mandella,' a voice yawned. It was Rogers.\n\n'Sorry I woke you up,' I whispered.\n\n''Sallright.' She snuggled over and clasped me spoon-fashion. She was warm and reasonably soft.\n\nI patted her hip in what I hoped was a brotherly fashion. 'Night, Rogers.'\n\n'G'night, Stallion.' She returned the gesture more pointedly.\n\nWhy do you always get the tired ones when you're ready and the randy ones when you're tired? I bowed to the inevitable.\n\n# 2\n\n'Awright, let's get some goddamn _back_ inta that! Stringer team! Move it up \u2014 move your ass up!'\n\nA warm front had come in about midnight and the snow had turned to sleet. The permaplast stringer weighed five hundred pounds and was a bitch to handle, even when it wasn't covered with ice. There were four of us, two at each end, carrying the plastic girder with frozen fingertips. Rogers was my partner.\n\n'Steel!' the guy behind me yelled, meaning that he was losing his hold. It wasn't steel, but it was heavy enough to break your foot. Everybody let go and hopped away. It splashed slush and mud all over us.\n\n'Goddammit, Petrov,' Rogers said, 'why didn't you go out for the Red Cross or something? This fucken thing's not that fucken heavy.' Most of the girls were a little more circumspect in their speech. Rogers was a little butch.\n\n'Awright, get a fucken _move_ on, stringers \u2014 epoxy team! Dog'em! Dog'em!'\n\nOur two epoxy people ran up, swinging their buckets. 'Let's go, Mandella. I'm freezin' my balls off.'\n\n'Me, too,' the girl said with more feeling than logic.\n\n'One \u2014 two \u2014 heave!' We got the thing up again and staggered toward the bridge. It was about three-quarters completed. Looked as if the second platoon was going to beat us. I wouldn't give a damn, but the platoon that got their bridge built first got to fly home. Four miles of muck for the rest of us, and no rest before chop.\n\nWe got the stringer in place, dropped it with a clank, and fitted the static clamps that held it to the rise-beams. The female half of the epoxy team started slopping glue on it before we even had it secured. Her partner was waiting for the stringer on the other side. The floor team was waiting at the foot of the bridge, each one holding a piece of the light, stressed permaplast over his head like an umbrella. They were dry and clean. I wondered aloud what they had done to deserve it, and Rogers suggested a couple of colorful, but unlikely, possibilities.\n\nWe were going back to stand by the next stringer when the field first (name of Dougelstein, but we called him 'Awright') blew a whistle and bellowed, 'Awright, soldier boys and girls, ten minutes. Smoke'em if you got 'em.' He reached into his pocket and turned on the control that heated our coveralls.\n\nRogers and I sat down on our end of the stringer and I took out my weed box. I had lots of joints, but we were ordered not to smoke them until after night-chop. The only tobacco I had was a cigarro butt about three inches long. I lit it on the side of the box; it wasn't too bad after the first couple of puffs. Rogers took a puff, just to be sociable, but made a face and gave it back.\n\n'Were you in school when you got drafted?' she asked.\n\n'Yeah. Just got a degree in physics. Was going after a teacher's certificate.'\n\nShe nodded soberly. 'I was in biology...'\n\n'Figures.' I ducked a handful of slush. 'How far?'\n\n'Six years, bachelor's and technical.' She slid her boot along the ground, turning up a ridge of mud and slush the consistency of freezing ice milk. 'Why the fuck did this have to happen?'\n\nI shrugged. It didn't call for an answer, least of all the answer that the UNEF kept giving us. Intellectual and physical elite of the planet, going out to guard humanity against the Tauran menace. Soyashit. It was all just a big experiment. See whether we could goad the Taurans into ground action.\n\nAwright blew the whistle two minutes early, as expected, but Rogers and I and the other two stringers got to sit for a minute while the epoxy and floor teams finished covering our stringer. It got cold fast, sitting there with our suits turned off, but we remained inactive on principle.\n\nThere really wasn't any sense in having us train in the cold. Typical army half-logic. Sure, it was going to be cold where we were going, but not ice-cold or snow-cold. Almost by definition, a portal planet remained within a degree or two of absolute zero all the time \u2014 since collapsars don't shine \u2014 and the first chill you felt would mean that you were a dead man.\n\nTwelve years before, when I was ten years old, they had discovered the collapsar jump. Just fling an object at a collapsar with sufficient speed, and out it pops in some other part of the galaxy. It didn't take long to figure out the formula that predicted where it would come out: it travels along the same 'line' (actually an Einsteinian geodesic) it would have followed if the collapsar hadn't been in the way \u2014 until it reaches another collapsar field, whereupon it reappears, repelled with the same speed at which it approached the original collapsar. Travel time between the two collapsars... exactly zero.\n\nIt made a lot of work for mathematical physicists, who had to redefine simultaneity, then tear down general relativity and build it back up again. And it made the politicians very happy, because now they could send a shipload of colonists to Fomalhaut for less than it had once cost to put a brace of men on the Moon. There were a lot of people the politicians would love to see on Fomalhaut, implementing a glorious adventure rather than stirring up trouble at home.\n\nThe ships were always accompanied by an automated probe that followed a couple of million miles behind. We knew about the portal planets, little bits of flotsam that whirled around the collapsars; the purpose of the drone was to come back and tell us in the event that a ship had smacked into a portal planet at .999 of the speed of light.\n\nThat particular catastrophe never happened, but one day a drone limped back alone. Its data were analyzed, and it turned out that the colonists' ship had been pursued by another vessel and destroyed. This happened near Aldebaran, in the constellation Taurus, but since 'Aldebaraniam' is a little hard to handle, they named the enemy 'Tauran.'\n\nColonizing vessels thenceforth went out protected by an armed guard. Often the armed guard went out alone, and finally the Colonization Group got shortened to UNEF, United Nations Exploratory Force. Emphasis on the 'force.'\n\nThen some bright lad in the General Assembly decided that we ought to field an army of footsoldiers to guard the portal planets of the nearer collapsars. This led to the Elite Conscription Act of 1996 and the most elitely conscripted army in the history of warfare.\n\nSo here we were, fifty men and fifty women, with IQs over 150 and bodies of unusual health and strength, slogging elitely through the mud and slush of central Missouri, reflecting on the usefulness of our skill in building bridges on worlds where the only fluid is an occasional standing pool of liquid helium. \n\n# 3\n\nAbout a month later, we left for our final training exercise, maneuvers on the planet Charon. Though nearing perihelion, it was still more than twice as far from the sun as Pluto.\n\nThe troopship was a converted 'cattlewagon' made to carry two hundred colonists and assorted bushes and beasts. Don't think it was roomy, though, just because there were half that many of us. Most of the excess space was taken up with extra reaction mass and ordnance.\n\nThe whole trip took three weeks, accelerating at two gees halfway, decelerating the other half. Our top speed, as we roared by the orbit of Pluto, was around one-twentieth of the speed of light \u2014 not quite enough for relativity to rear its complicated head.\n\nThree weeks of carrying around twice as much weight as normal... it's no picnic. We did some cautious exercises three times a day and remained horizontal as much as possible. Still, we got several broken bones and serious dislocations. The men had to wear special supporters to keep from littering the floor with loose organs. It was almost impossible to sleep; nightmares of choking and being crushed, rolling over periodically to prevent blood pooling and bedsores. One girl got so fatigued that she almost slept through the experience of having a rib push out into the open air.\n\nI'd been in space several times before, so when we finally stopped decelerating and went into free fall, it was nothing but relief. But some people had never been out, except for our training on the Moon, and succumbed to the sudden vertigo and disorientation. The rest of us cleaned up after them, floating through the quarters with sponges and inspirators to suck up the globules of partly-digested 'Concentrate, High-protein, Low-residue, Beef Flavor (Soya).'\n\nWe had a good view of Charon, coming down from orbit. There wasn't much to see, though. It was just a dim, off-white sphere with a few smudges on it. We landed about two hundred meters from the base. A pressurized crawler came out and mated with the ferry, so we didn't have to suit up. We clanked and squeaked up to the main building, a featureless box of grayish plastic.\n\nInside, the walls were the same drab color. The rest of the company was sitting at desks, chattering away. There was a seat next to Freeland.\n\n'Jeff \u2014 feeling better?' He still looked a little pale.\n\n'If the gods had meant for man to survive in free fall, they would have given him a cast iron glottis.' He sighed heavily. 'A little better. Dying for a smoke.'\n\n'Yeah.'\n\n' _You_ seemed to take it all right. Went up in school, didn't you?'\n\n'Senior thesis in vacuum welding, yeah. Three weeks in Earth orbit.' I sat back and reached for my weed box for the thousandth time. It still wasn't there. The Life-Support Unit didn't want to handle nicotine and THC.\n\n'Training was bad enough,' Jeff groused, 'but _this_ shit\u2014'\n\n'Tench-hut!' We stood up in a raggedy-ass fashion, by twos and threes. The door opened and a full major came in. I stiffened a little. He was the highest-ranking officer I'd ever seen. He had a row of ribbons stitched into his coveralls, including a purple strip meaning he'd been wounded in combat, fighting in the old American army. Must have been that Indochina thing, but it had fizzled out before I was born. He didn't look that old.\n\n'Sit, sit.' He made a patting motion with his hand. Then he put his hands on his hips and scanned the company, a small smile on his face. 'Welcome to Charon. You picked a lovely day to land, the temperature outside is a summery eight point one five degrees Absolute. We expect little change for the next two centuries or so.' Some of them laughed halfheartedly.\n\n'Best you enjoy the tropical climate here at Miami Base; enjoy it while you came. We're on the center of sunside here, and most of your training will be on darkside. Over there, the temperature stays a chilly two point zero eight.\n\n'You might as well regard all the training you got on Earth and the Moon as just an elementary exercise, designed to give you a fair chance of surviving Charon. You'll have to go through your whole repertory here: tools, weapons, maneuvers. And you'll find that, at these temperatures, tools don't work the way they should; weapons don't want to fire. And people move v-e-r-y cautiously.'\n\nHe studied the clipboard in his hand. 'Right now, you have forty-nine women and forty-eight men. Two deaths on Earth, on psychiatric release. Having read an outline of your training program, I'm frankly surprised that so many of you pulled through.\n\n'But you might as well know that I won't be displeased if as few as fifty of you, half, graduate from this final phase. And the only way not to graduate is to die. Here. The only way anybody gets back to Earth \u2014 including me \u2014 is after a combat tour.\n\n'You will complete your training in one month. From here you will go to Stargate collapsar, half a light year away. You will stay at the settlement on Stargate 1, the largest portal planet, until replacements arrive. Hopefully, that will be no more than a month; another group is due here as soon as you leave.\n\n'When you leave Stargate, you will go to some strategically important collapsar, set up a military base there, and fight the enemy, if attacked. Otherwise, you will maintain the base until further orders.\n\n'The last two weeks of your training will consist of constructing exactly that kind of a base, on darkside. There you will be totally isolated from Miami Base: no communication, no medical evacuation, no resupply. Sometime before the two weeks are up, your defense facilities will be evaluated in an attack by guided drones. They will be armed.'\n\nThey had spent all that money on us just to kill us in training?\n\n'All of the permanent personnel here on Charon are combat veterans. Thus, all of us are forty to fifty years of age. But I think we can keep up with you. Two of us will be with you at all times and will accompany you at least as far as Stargate. They are Captain Sherman Stott, your company commander, and Sergeant Octavio Cortez, your first sergeant. Gentlemen?'\n\nTwo men in the front row stood easily and turned to face us. Captain Stott was a little smaller than the major, but cut from the same mold: face hard and smooth as porcelain, cynical half-smile, a precise centimeter of beard framing a large chin, looking thirty at the most. He wore a large, gunpowder-type pistol on his hip.\n\nSergeant Cortez was another story, a horror story. His head was shaved and the wrong shape, flattened out on one side, where a large piece of skull had obviously been taken out. His face was very dark and seamed with wrinkles and scars. Half his left ear was missing, and his eyes were as expressive as buttons on a machine. He had a moustache-and-beard combination that looked like a skinny white caterpillar taking a lap around his mouth. On anybody else, his schoolboy smile might look pleasant, but he was about the ugliest, meanest-looking creature I'd ever seen. Still, if you didn't look at his head and considered the lower six feet or so, he could have posed as the 'after' advertisement for a body-building spa. Neither Stott nor Cortez wore any ribbons. Cortez had a small pocket-laser suspended in a magnetic rig, sideways, under his left armpit. It had wooden grips that were worn smooth.\n\n'Now, before I turn you over to the tender mercies of these two gentlemen, let me caution you again:\n\n'Two months ago there was not a living soul on this planet, just some leftover equipment from the expedition of 1991. A working force of forty-five men struggled for a month to erect this base. Twenty-four of them, more than half, died in the construction of it. This is the most dangerous planet men have ever tried to live on, but the places you'll be going will be this bad and worse. Your cadre will try to keep you alive for the next month. Listen to them... and follow their example; all of them have survived here much longer than you'll have to. Captain?' The captain stood up as the major went out the door.\n\n'Tench- _hut_!' The last syllable was like an explosion and we all jerked to our feet.\n\n'Now I'm only gonna say this _once_ so you better listen,' he growled. 'We _are_ in a combat situation here, and in a combat situation there is only _one_ penalty for disobedience or insubordination.' He jerked the pistol from his hip and held it by the barrel, like a club. 'This is an Army model 1911 automatic _pistol_ , caliber .45, and it is a primitive but effective weapon. The Sergeant and I are authorized to use our weapons to kill to enforce discipline. Don't make us do it because we will. We _will_.' He put the pistol back. The holster snap made a loud crack in the dead quiet.\n\n'Sergeant Cortez and I between us have killed more people than are sitting in this room. Both of us fought in Vietnam on the American side and both of us joined the United Nations International Guard more than ten years ago. I took a break in grade from major for the privilege of commanding this company, and First Sergeant Cortez took a break from sub-major, because we are both _combat_ soldiers and this is the first _combat_ situation since 1987.\n\n'Keep in mind what I've said while the First Sergeant instructs you more specifically in what your duties will be under this command. Take over, Sergeant.' He turned on his heel and strode out of the room. The expression on his face hadn't changed one millimeter during the whole harangue.\n\nThe First Sergeant moved like a heavy machine with lots of ball bearings. When the door hissed shut, he swiveled ponderously to face us and said, 'At ease, siddown,' in a surprisingly gentle voice. He sat on a table in the front of the room. It creaked, but held.\n\n'Now the captain talks scary and I look scary, but we both mean well. You'll be working pretty closely with me, so you better get used to this thing I've got hanging in front of my brain. You probably won't see the captain much, except on maneuvers.'\n\nHe touched the flat part of his head. 'And speaking of brains, I still have just about all of mine, in spite of Chinese efforts to the contrary. All of us old vets who mustered into UNEF had to pass the same criteria that got you drafted by the Elite Conscription Act. So I suspect all of you are smart and tough \u2014 but just keep in mind that the captain and I are smart and tough _and_ experienced.'\n\nHe flipped through the roster without really looking at it. 'Now, as the captain said, there'll be only one kind of disciplinary action on maneuvers. Capital punishment. But normally _we_ won't have to kill you for disobeying; Charon'll save us the trouble.\n\n'Back in the billeting area, it'll be another story. We don't much care what you do inside. Grab ass all day and fuck all night, makes no difference... But once you suit up and go outside, you've gotta have discipline that would shame a Centurian. There will be situations where one stupid act could kill us all.\n\n'Anyhow, the first thing we've gotta do is get you fitted to your fighting suits. The armorer's waiting at your billet; he'll take you one at a time. Let's go.' \n\n# 4\n\n'Now I know you got lectured back on Earth on what a fighting suit can do.' The armorer was a small man, partially bald, with no insignia of rank on his coveralls. Sergeant Cortez had told us to call him 'sir,' since he was a lieutenant.\n\n'But I'd like to reinforce a couple of points, maybe add some things your instructors Earthside weren't clear about or couldn't know. Your First Sergeant was kind enough to consent to being my visual aid. Sergeant?'\n\nCortez slipped out of his coveralls and came up to the little raised platform where a fighting suit was standing, popped open like a man-shaped clam. He backed into it and slipped his arms into the rigid sleeves. There was a click and the thing swung shut with a sigh. It was bright green with CORTEZ stenciled in white letters on the helmet.\n\n'Camouflage, Sergeant.' The green faded to white, then dirty gray. 'This is good camouflage for Charon and most of your portal planets,' said Cortez, as if from a deep well. 'But there are several other combinations available.' The gray dappled and brightened to a combination of greens and browns: 'Jungle.' Then smoothed out to a hard light ochre: 'Desert.' Dark brown, darker, to a deep flat black: 'Night or space.'\n\n'Very good, Sergeant. To my knowledge, this is the only feature of the suit that was perfected after your training. The control is around your left wrist and is admittedly awkward. But once you find the right combination, it's easy to lock in.\n\n'Now, you didn't get much in-suit training Earthside. We didn't want you to get used to using the thing in a friendly environment. The fighting suit is the deadliest personal weapon ever built, and with no weapon is it easier for the user to kill himself through carelessness. Turn around, Sergeant.'\n\n'Case in point.' He tapped a large square protuberance between the shoulders. 'Exhaust fins. As you know, the suit tries to keep you at a comfortable temperature no matter what the weather's like outside. The material of the suit is as near to a perfect insulator as we could get, consistent with mechanical demands. Therefore, these fins get _hot_ \u2014 especially hot, compared to darkside temperatures \u2014 as they bleed off the body's heat.\n\n'All you have to do is lean up against a boulder of frozen gas; there's lots of it around. The gas will sublime off faster than it can escape from the fins; in escaping, it will push against the surrounding \"ice\" and fracture it... and in about one-hundredth of a second, you have the equivalent of a hand grenade going off right below your neck. You'll never feel a thing.\n\n'Variations on this theme have killed eleven people in the past two months. And they were just building a bunch of huts.\n\n'I assume you know how easily the waldo capabilities can kill you or your companions. Anybody want to shake hands with the sergeant?' He paused, then stepped over and clasped his glove. 'He's had lots of practice. Until _you_ have, be extremely careful. You might scratch an itch and wind up breaking your back. Remember, semi-logarithmic response: two pounds' pressure exerts five pounds' force; three pounds' gives ten; four pounds', twenty-three; five pounds', forty-seven. Most of you can muster up a grip of well over a hundred pounds. Theoretically, you could rip a steel girder in two with that, amplified. Actually, you'd destroy the material of your gloves and, at least on Charon, die very quickly. It'd be a race between decompression and flash-freezing. You'd die no matter which won.\n\n'The leg waldos are also dangerous, even though the amplification is less extreme. Until you're really skilled, don't try to run, or jump. You're likely to trip, and that means you're likely to die.'\n\n'Charon's gravity is three-fourths of Earth normal, so it's not too bad. But on a really small world, like Luna, you could take a running jump and not come down for twenty minutes, just keep sailing over the horizon. Maybe bash into a mountain at eighty meters per second. On a small asteroid, it'd be no trick at all to run up to escape velocity and be off on an informal tour of intergalactic space. It's a slow way to travel.\n\n'Tomorrow morning, we'll start teaching you how to stay alive inside this infernal machine. The rest of the afternoon and evening, I'll call you one at a time to be fitted. That's all, Sergeant.'\n\nCortez went to the door and turned the stopcock that let air into the airlock. A bank of infrared lamps went on to keep air from freezing inside it. When the pressures were equalized, he shut the stopcock, unclamped the door and stepped in, clamping it shut behind him. A pump hummed for about a minute, evacuating the airlock, then he stepped out and sealed the outside door.\n\nIt was pretty much like the ones on Luna.\n\n'First I want Private Omar Almizar. The rest of you can go find your bunks. I'll call you over the squawker.'\n\n'Alphabetical order, sir?'\n\n'Yep. About ten minutes apiece. If your name begins with Z, you might as well get sacked.'\n\nThat was Rogers. She probably was thinking about getting sacked.\n\n# 5\n\nThe sun was a hard white point directly overhead. It was a lot brighter than I had expected it to be; since we were eighty AUs out, it was only one 6400th as bright as it is on Earth. Still, it was putting out about as much light as a powerful streetlamp.\n\n'This is considerably more light than you'll have on a portal planet.' Captain Stott's voice crackled in our collective ear. 'Be glad that you'll be able to watch your step.'\n\nWe were lined up, single-file, on the permaplast sidewalk that connected the billet and the supply hut. We'd practiced walking inside, all morning, and this wasn't any different except for the exotic scenery. Though the light was rather dim, you could see all the way to the horizon quite clearly, with no atmosphere in the way. A black cliff that looked too regular to be natural stretched from one horizon to the other, passing within a kilometer of us. The ground was obsidian-black, mottled with patches of white or bluish ice. Next to the supply hut was a small mountain of snow in a bin marked OXYGEN.\n\nThe suit was fairly comfortable, but it gave you the odd feeling of simultaneously being a marionette and a puppeteer. You apply the impulse to move your leg and the suit picks it up and magnifies it and moves your leg _for_ you.\n\n'Today we're only going to walk around the company area, and nobody will _leave_ the company area.' The captain wasn't wearing his .45 \u2014 unless he carried it as a good luck charm, under his suit \u2014 but he had a laser-finger like the rest of us. And his was probably hooked up.\n\nKeeping an interval of at least two meters between each person, we stepped off the permaplast and followed the captain over smooth rock. We walked carefully for about an hour, spiraling out, and finally stopped at the far edge of the perimeter.\n\n'Now everybody pay close attention. I'm going out to that blue slab of ice' \u2014 it was a big one, about twenty meters away \u2014 'and show you something that you'd better know if you want to stay alive.'\n\nHe walked out in a dozen confident steps. 'First I have to heat up a rock \u2014 filters down.' I squeezed the stud under my armpit and the filter slid into place over my image converter. The captain pointed his finger at a black rock the size of a basketball, and gave it a short burst. The glare rolled a long shadow of the captain over us and beyond. The rock shattered into a pile of hazy splinters.\n\n'It doesn't take long for these to cool down.' He stopped and picked up a piece. 'This one is probably twenty or twenty-five degrees. Watch.' He tossed the 'warm' rock onto the ice slab. It skittered around in a crazy pattern and shot off the side. He tossed another one, and it did the same.\n\n'As you know, you are not quite _perfectly_ insulated. These rocks are about the temperature of the soles of your boots. If you try to stand on a slab of hydrogen, the same thing will happen to you. Except that the rock is _already_ dead.\n\n'The reason for this behavior is that the rock makes a slick interface with the ice \u2014 a little puddle of liquid hydrogen \u2014 and rides a few molecules above the liquid on a cushion of hydrogen vapor. This makes the rock or _you_ a frictionless bearing as far as the ice is concerned, and you _can't_ stand up without any friction under your boots.\n\n'After you have lived in your suit for a month or so you _should_ be able to survive falling down, but right _now_ you just don't know enough. Watch.'\n\nThe captain flexed and hopped up onto the slab. His feet shot out from under him and he twisted around in midair, landing on hands and knees. He slipped off and stood on the ground.\n\n'The idea is to keep your exhaust fins from making contact with the frozen gas. Compared to the ice they are as hot as a blast furnace, and contact with any weight behind it will result in an explosion.'\n\nAfter that demonstration, we walked around for another hour or so and returned to the billet. Once through the airlock, we had to mill around for a while, letting the suits get up to something like room temperature. Somebody came up and touched helmets with me.\n\n'William?' She had MCCOY stenciled above her face-plate.\n\n'Hi, Sean. Anything special?'\n\n'I just wondered if you had anyone to sleep with tonight.'\n\nThat's right; I'd forgotten. There wasn't any sleeping roster here. Everybody chose his own partner. 'Sure, I mean, uh, no... no, I haven't asked anybody. Sure, if you want to...'\n\n'Thanks, William. See you later.' I watched her walk away and thought that if anybody could make a fighting suit look sexy, it'd be Sean. But even she couldn't.\n\nCortez decided we were warm enough and led us to the suit room, where we backed the things into place and hooked them up to the charging plates. (Each suit had a little chunk of plutonium that would power it for several years, but we were supposed to run on fuel cells as much as possible.) After a lot of shuffling around, everybody finally got plugged in and we were allowed to unsuit \u2014 ninety-seven naked chickens squirming out of bright green eggs. It was _cold_ \u2014 the air, the floor and especially the suits \u2014 and we made a pretty disorderly exit toward the lockers.\n\nI slipped on tunic, trousers and sandals and was still cold. I took my cup and joined the line for soya. Everybody was jumping up and down to keep warm.\n\n'How c-cold, do you think, it is, M-Mandella?' That was McCoy.\n\n'I don't, even want, to think, about it.' I stopped jumping and rubbed myself as briskly as possible, while holding a cup in one hand. 'At least as cold as Missouri was.'\n\n'Ung... wish they'd, get some, fucken, heat in, this place.' It always affects the small women more than anybody else. McCoy was the littlest one in the company, a waspwaist doll barely five feet high.\n\n'They've got the airco going. It can't be long now.'\n\n'I wish I, was a big, slab of, meat like, you.'\n\nI was glad she wasn't.\n\n# 6\n\nWe had our first casualty on the third day, learning how to dig holes.\n\nWith such large amounts of energy stored in a soldier's weapons, it wouldn't be practical for him to hack out a hole in the frozen ground with the conventional pick and shovel. Still, you can launch grenades all day and get nothing but shallow depressions \u2014 so the usual method is to bore a hole in the ground with the hand laser, drop a timed charge in after it's cooled down and, ideally, fill the hole with stuff. Of course, there's not much loose rock on Charon, unless you've already blown a hole nearby.\n\nThe only difficult thing about the procedure is in getting away. To be safe, we were told, you've got to either be behind something really solid, or be at least a hundred meters away. You've got about three minutes after setting the charge, but you can't just sprint away. Not safely, not on Charon.\n\nThe accident happened when we were making a really deep hole, the kind you want for a large underground bunker. For this, we had to blow a hole, then climb down to the bottom of the crater and repeat the procedure again and again until the hole was deep enough. Inside the crater we used charges with a five-minute delay, but it hardly seemed enough time \u2014 you really had to go it slow, picking your way up the crater's edge.\n\nJust about everybody had blown a double hole; everybody but me and three others. I guess we were the only ones paying really close attention when Bovanovitch got into trouble. All of us were a good two hundred meters away. With my image converter turned up to about forty power, I watched her disappear over the rim of the crater. After that, I could only listen in on her conversation with Cortez.\n\n'I'm on the bottom, Sergeant.' Normal radio procedure was suspended for maneuvers like this; nobody but the trainee and Cortez was allowed to broadcast.\n\n'OK, move to the center and clear out the rubble. Take your time. No rush until you pull the pin.'\n\n'Sure, Sergeant.' We could hear small echoes of rocks clattering, sound conduction through her boots. She didn't say anything for several minutes.\n\n'Found bottom.' She sounded a little out of breath.\n\n'Ice or rock?'\n\n'Oh, it's rock, Sergeant. The greenish stuff.'\n\n'Use a low setting, then. One point two, dispersion four.'\n\n'God darn it, Sergeant, that'll take forever.'\n\n'Yeah, but that stuff's got hydrated crystals in it \u2014 heat it up too fast and you might make it fracture. And we'd just have to leave you there, girl. Dead and bloody.'\n\n'OK, one point two dee four.' The inside edge of the crater flickered red with reflected laser light.\n\n'When you get about half a meter deep, squeeze it up to dee two.'\n\n'Roger.' It took her exactly seventeen minutes, three of them at dispersion two. I could imagine how tired her shooting arm was.\n\n'Now rest for a few minutes. When the bottom of the hole stops glowing, arm the charge and drop it in. Then _walk_ out, understand? You'll have plenty of time.'\n\n'I understand, Sergeant. Walk out.' She sounded nervous. Well, you don't often have to tiptoe away from a twenty-microton tachyon bomb. We listened to her breathing for a few minutes.\n\n'Here goes.' Faint slithering sounds, the bomb sliding down.\n\n'Slow and easy now. You've got five minutes.'\n\n'Y-yeah. Five.' Her footsteps started out slow and regular. Then, after she started climbing the side, the sounds were less regular, maybe a little frantic. And with four minutes to go\u2014\n\n'Shit!' A loud scraping noise, then clatters and bumps. 'Shit-shit.'\n\n'What's wrong, private?'\n\n'Oh, shit.' Silence. 'Shit!'\n\n'Private, you don't wanna get shot, you _tell me what's wrong!_ '\n\n'I... shit, I'm stuck. Fucken rockslide... shit... DO SOMETHING! I can't move, shit I can't move I, I\u2014'\n\n'Shut up! How deep?'\n\n'Can't move my, shit, my fucken legs. HELP ME\u2014'\n\n'Then goddammit use your arms \u2014 push! You can move a ton with each hand.' Three minutes.\n\nShe stopped cussing and started to mumble, in Russian, I guess, a low monotone. She was panting, and you could hear rocks tumbling away.\n\n'I'm free.' Two minutes.\n\n'Go as fast as you can.' Cortez's voice was flat, emotionless.\n\nAt ninety seconds she appeared, crawling over the rim. 'Run, girl... You better run.' She ran five or six steps and fell, skidded a few meters and got back up, running; fell again, got up again\u2014\n\nIt looked as though she was going pretty fast, but she had only covered about thirty meters when Cortez said, 'All right, Bovanovitch, get down on your stomach and lie still.' Ten seconds, but she didn't hear or she wanted to get just a little more distance, and she kept running, careless leaping strides, and at the high point of one leap there was a flash and a rumble, and something big hit her below the neck, and her headless body spun off end over end through space, trailing a red-black spiral of flash-frozen blood that settled gracefully to the ground, a path of crystal powder that nobody disturbed while we gathered rocks to cover the juiceless thing at the end of it.\n\nThat night Cortez didn't lecture us, didn't even show up for night-chop. We were all very polite to each other and nobody was afraid to talk about it.\n\nI sacked with Rogers \u2014 everybody sacked with a good friend \u2014 but all she wanted to do was cry, and she cried so long and so hard that she got me doing it, too. \n\n# 7\n\n'Fire team _A_ \u2014 move out!' The twelve of us advanced in a ragged line toward the simulated bunker. It was about a kilometer away, across a carefully prepared obstacle course. We could move pretty fast, since all of the ice had been cleared from the field, but even with ten days' experience we weren't ready to do more than an easy jog.\n\nI carried a grenade launcher loaded with tenth-microton practice grenades. Everybody had their laser-fingers set at a point oh eight dee one, not much more than a flashlight. This was a _simulated_ attack \u2014 the bunker and its robot defender cost too much to use once and be thrown away.\n\n'Team _B_ , follow. Team leaders, take over.'\n\nWe approached a clump of boulders at about the halfway mark, and Potter, my team leader, said, 'Stop and cover.' We clustered behind the rocks and waited for Team _B_.\n\nBarely visible in their blackened suits, the dozen men and women whispered by us. As soon as they were clear, they jogged left, out of our line of sight.\n\n'Fire!' Red circles of light danced a half-klick down-range, where the bunker was just visible. Five hundred meters was the limit for these practice grenades; but I might luck out, so I lined the launcher up on the image of the bunker, held it at a forty-five degree angle and popped off a salvo of three.\n\nReturn fire from the bunker started before my grenades even landed. Its automatic lasers were no more powerful than the ones we were using, but a direct hit would deactivate your image converter, leaving you blind. It was setting down a random field of fire, not even coming close to the boulders we were hiding behind.\n\nThree magnesium-bright flashes blinked simultaneously about thirty meters short of the bunker. 'Mandella! I thought you were supposed to be good with that thing.'\n\n'Damn it, Potter \u2014 it only throws half a klick. Once we get closer, I'll lay 'em right on top, every time.'\n\n'Sure you will.' I didn't say anything. She wouldn't be team leader forever. Besides, she hadn't been such a bad girl before the power went to her head.\n\nSince the grenadier is the assistant team leader, I was slaved into Potter's radio and could hear _B_ team talk to her.\n\n'Potter, this is Freeman. Losses?'\n\n'Potter here \u2014 no, looks like they were concentrating on you.'\n\n'Yeah, we lost three. Right now we're in a depression about eighty, a hundred meters down from you. We can give cover whenever you're ready.'\n\n'OK, start.' Soft click: ' _A_ team, follow me.' She slid out from behind the rock and turned on the faint pink beacon beneath her powerpack. I turned on mine and moved out to run alongside of her, and the rest of the team fanned out in a trailing wedge. Nobody fired while B team laid down a cover for us.\n\nAll I could hear was Potter's breathing and the soft _crunch-crunch_ of my boots. Couldn't see much of anything, so I tongued the image converter up to a log two intensification. That made the image kind of blurry but adequately bright. Looked like the bunker had _B_ team pretty well pinned down; they were getting quite a roasting. All of their return fire was laser. They must have lost their grenadier.\n\n'Potter, this is Mandella. Shouldn't we take some of the heat off _B_ team?'\n\n'Soon as I can find us good enough cover. Is that all right with you? Private?' She'd been promoted to corporal for the duration of the exercise.\n\nWe angled to the right and lay down behind a slab of rock. Most of the others found cover nearby, but a few had to hug the ground.\n\n'Freeman, this is Potter.'\n\n'Potter, this is Smithy. Freeman's out; Samuels is out. We only have five men left. Give us some cover so we can get\u2014'\n\n'Roger, Smithy.' _Click_. 'Open up, _A_ team. The _B_ s are really hurtin'.'\n\nI peeked out over the edge of the rock. My rangefinder said that the bunker was about three hundred fifty meters away, still pretty far. I aimed a smidgeon high and popped three, then down a couple of degrees, three more. The first ones overshot by about twenty meters; then the second salvo flared up directly in front of the bunker. I tried to hold on that angle and popped fifteen, the rest of the magazine, in the same direction.\n\nI should have ducked down behind the rock to reload, but I wanted to see where the fifteen would land, so I kept my eyes on the bunker while I reached back to unclip another magazine\u2014\n\nWhen the laser hit my image converter, there was a red glare so intense it seemed to go right through my eyes and bounce off the back of my skull. It must have been only a few milliseconds before the converter overloaded and went blind, but the bright green afterimage hurt my eyes for several minutes.\n\nSince I was officially 'dead,' my radio automatically cut off, and I had to remain where I was until the mock battle was over. With no sensory input besides the feel of my own skin (and it ached where the image converter had shone on it) and the ringing in my ears, it seemed like an awfully long time. Finally, a helmet clanked against mine.\n\n'You OK, Mandella?' Potter's voice.\n\n'Sorry, I died of boredom twenty minutes ago.'\n\n'Stand up and take my hand.' I did so and we shuffled back to the billet. It must have taken over an hour. She didn't say anything more, all the way back \u2014 it's a pretty awkward way to communicate \u2014 but after we'd cycled through the airlock and warmed up, she helped me undo my suit. I got ready for a mild tongue-lashing, but when the suit popped open, before I could even get my eyes adjusted to the light, she grabbed me around the neck and planted a wet kiss on my mouth.\n\n'Nice shooting, Mandella.'\n\n'Huh?'\n\n'Didn't you see? Of course not... The last salvo before you got hit \u2014 four direct hits. The bunker decided it was knocked out, and all we had to do was walk the rest of the way.'\n\n'Great.' I scratched my face under the eyes, and some dry skin flaked off. She giggled.\n\n'You should see yourself. You look like\u2014'\n\n'All personnel, report to the assembly area.' That was the captain's voice. Bad news, usually.\n\nShe handed me a tunic and sandals. 'Let's go.' The assembly area\/chop hall was just down the corridor. There was a row of roll-call buttons at the door; I pressed the one beside my name. Four of the names were covered with black tape. That was good, only four. We hadn't lost anybody during today's maneuvers.\n\nThe captain was sitting on the raised dais, which at least meant we didn't have to go through the tench-hut bullshit. The place filled up in less than a minute; a soft chime indicated the roll was complete.\n\nCaptain Stott didn't stand up. 'You did _fairly_ well today. Nobody killed, and I expected some to be. In that respect you exceeded my expectations but in _every_ other respect you did a poor job.\n\n'I am glad you're taking good care of yourselves, because each of you represents an investment of over a million dollars and one-fourth of a human life.\n\n'But in this simulated battle against a _very_ stupid robot enemy, thirty-seven of you managed to walk into laser fire and be killed in a _simulated_ way, and since dead people require no food _you_ will require no food, for the next three days. Each person who was a casualty in this battle will be allowed only two liters of water and a vitamin ration each day.'\n\nWe knew enough not to groan or anything, but there were some pretty disgusted looks, especially on the faces that had singed eyebrows and a pink rectangle of sunburn framing their eyes.\n\n'Mandella.'\n\n'Sir?'\n\n'You are far and away the worst-burned casualty. Was your image converter set on normal?'\n\nOh, shit. 'No, sir. Log two.'\n\n'I see. Who was your team leader for the exercises?'\n\n'Acting Corporal Potter, sir.'\n\n'Private Potter, did you order him to use image intensification?'\n\n'Sir, I... I don't remember.'\n\n'You don't. Well, as a memory exercise you may join the dead people. Is that satisfactory?'\n\n'Yes, sir.'\n\n'Good. Dead people get one last meal tonight and go on no rations starting tomorrow. Are there any questions?' He must have been kidding. 'All right. Dismissed.'\n\nI selected the meal that looked as if it had the most calories and took my tray over to sit by Potter.\n\n'That was a quixotic damn thing to do. But thanks.'\n\n'Nothing. I've been wanting to lose a few pounds anyway.' I couldn't see where she was carrying any extra.\n\n'I know a good exercise,' I said. She smiled without looking up from her tray. 'Have anybody for tonight?'\n\n'Kind of thought I'd ask Jeff...'\n\n'Better hurry, then. He's lusting after Maejima.' Well, that was mostly true. Everybody did.\n\n'I don't know. Maybe we ought to save our strength. That third day...'\n\n'Come on.' I scratched the back of her hand lightly with a fingernail. 'We haven't sacked since Missouri. Maybe I've learned something new.'\n\n'Maybe you have.' She tilted her head up at me in a sly way. 'OK.'\n\nActually, she was the one with the new trick. The French corkscrew, she called it. She wouldn't tell me who taught it to her though. I'd like to shake his hand. Once I got my strength back. \n\n# 8\n\nThe two weeks' training around Miami Base eventually cost us eleven lives. Twelve, if you count Dahlquist. I guess having to spend the rest of your life on Charon with a hand and both legs missing is close enough to dying.\n\nFoster was crushed in a landslide and Freeland had a suit malfunction that froze him solid before we could carry him inside. Most of the other deaders were people I didn't know all that well. But they all hurt. And they seemed to make us more scared rather than more cautious.\n\nNow darkside. A flyer brought us over in groups of twenty and set us down beside a pile of building materials thoughtfully immersed in a pool of helium II.\n\nWe used grapples to haul the stuff out of the pool. It's not safe to go wading, since the stuff crawls all over you and it's hard to tell what's underneath; you could walk out onto a slab of hydrogen and be out of luck.\n\nI'd suggested that we try to boil away the pool with our lasers, but ten minutes of concentrated fire didn't drop the helium level appreciably. It didn't boil, either; helium II is a 'superfluid,' so what evaporation there was had to take place evenly, all over the surface. No hot spots, so no bubbling.\n\nWe weren't supposed to use lights, to 'avoid detection.' There was plenty of starlight with your image converter cranked up to log three or four, but each stage of amplification meant some loss of detail. By log four the landscape looked like a crude monochrome painting, and you couldn't read the names on people's helmets unless they were right in front of you.\n\nThe landscape wasn't all that interesting, anyhow. There were half a dozen medium-sized meteor craters (all with exactly the same level of helium II in them) and the suggestion of some puny mountains just over the horizon. The uneven ground was the consistency of frozen spiderwebs; every time you put your foot down, you'd sink half an inch with a squeaking crunch. It could get on your nerves.\n\nIt took most of a day to pull all the stuff out of the pool. We took shifts napping, which you could do either standing up, sitting or lying on your stomach. I didn't do well in any of those positions, so I was anxious to get the bunker built and pressurized.\n\nWe couldn't build the thing underground \u2014 it'd just fill up with helium II \u2014 so the first thing to do was to build an insulating platform, a permaplast-vacuum sandwich three layers thick.\n\nI was an acting corporal, with a crew of ten people. We were carrying the permaplast layers to the building site \u2014 two people can carry one easily \u2014 when one of 'my' men slipped and fell on his back.\n\n'Damn it, Singer, watch your step.' We'd had a couple of deaders that way.\n\n'Sorry, Corporal. I'm bushed. Just got my feet tangled up.'\n\n'Yeah, just watch it.' He got back up all right, and he and his partner placed the sheet and went back to get another.\n\nI kept my eye on Singer. In a few minutes he was practically staggering, not easy to do in that suit of cybernetic armor.\n\n'Singer! After you set the plank, I want to see you.'\n\n'OK.' He labored through the task and mooched over.\n\n'Let me check your readout.' I opened the door on his chest to expose the medical monitor. His temperature was two degrees high; blood pressure and heart rate both elevated. Not up to the red line, though.\n\n'You sick or something?'\n\n'Hell, Mandella, I feel OK, just tired. Since I fell I been a little dizzy.'\n\nI chinned the medic's combination. 'Doc, this is Mandella. You wanna come over here for a minute?'\n\n'Sure, where are you?' I waved and he walked over from poolside.\n\n'What's the problem?' I showed him Singer's readout.\n\nHe knew what all the other little dials and things meant, so it took him a while. 'As far as I can tell, Mandella... he's just hot.'\n\n'Hell, I coulda told you that,' said Singer.\n\n'Maybe you better have the armorer take a look at his suit.' We had two people who'd taken a crash course in suit maintenance; they were our 'armorers.'\n\nI chinned Sanchez and asked him to come over with his tool kit.\n\n'Be a couple of minutes, Corporal. Carryin' a plank.'\n\n'Well, put it down and get on over here.' I was getting an uneasy feeling. Waiting for him, the medic and I looked over Singer's suit.\n\n'Uh-oh,' Doc Jones said. 'Look at this.' I went around to the back and looked where he was pointing. Two of the fins on the heat exchanger were bent out of shape.\n\n'What's wrong?' Singer asked.\n\n'You fell on your heat exchanger, right?'\n\n'Sure, Corporal \u2014 that's it. It must not be working right.'\n\n'I don't think it's working at _all_ ,' said Doc.\n\nSanchez came over with his diagnostic kit and we told him what had happened. He looked at the heat exchanger, then plugged a couple of jacks into it and got a digital readout from a little monitor in his kit. I didn't know what it was measuring, but it came out zero to eight decimal places.\n\nHeard a soft click, Sanchez chinning my private frequency. 'Corporal this guy's a deader.'\n\n'What? Can't you fix the goddamn thing?'\n\n'Maybe... maybe I could, if I could take it apart. But there's no way\u2014'\n\n'Hey! Sanchez?' Singer was talking on the general freak. 'Find out what's wrong?' He was panting.\n\nClick. 'Keep your pants on, man, we're working on it.' Click. 'He won't last long enough for us to get the bunker pressurized. And I can't work on the heat exchanger from outside of the suit.'\n\n'You're got a spare suit, haven't you?'\n\n'Two of 'em, the fit-anybody kind. But there's no place... say...'\n\n'Right. Go get one of the suits warmed up.' I chinned the general freak. 'Listen, Singer, we've gotta get you out of that thing. Sanchez has a spare suit, but to make the switch, we're gonna have to build a house around you. Understand?'\n\n'Huh-uh.'\n\n'Look, we'll make a box with you inside, and hook it up to the life-support unit. That way you can breathe while you make the switch.'\n\n'Soun's pretty compis... compil... cated t'me.'\n\n'Look, just come along\u2014'\n\n'I'll be all right, man, jus' lemme res'...'\n\nI grabbed his arm and led him to the building site. He was really weaving. Doc took his other arm, and between us, we kept him from falling over.\n\n'Corporal Ho, this is Corporal Mandella.' Ho was in charge of the life-support unit.\n\n'Go away, Mandella, I'm busy.'\n\n'You're going to be busier.' I outlined the problem to her. While her group hurried to adapt the LSU \u2014 for this purpose, it need only be an air hose and heater \u2014 I got my crew to bring around six slabs of permaplast, so we could build a big box around Singer and the extra suit. It would look like a huge coffin, a meter square and six meters long.\n\nWe set the suit down on the slab that would be the floor of the coffin. 'OK, Singer, let's go.'\n\nNo answer.\n\n'Singer, let's go.'\n\nNo answer.\n\n'Singer!' He was just standing there. Doc Jones checked his readout.\n\n'He's out, man, unconscious.'\n\nMy mind raced. There might just be room for another person in the box. 'Give me a hand here.' I took Singer's shoulders and Doc took his feet, and we carefully laid him out at the feet of the empty suit.\n\nThen I lay down myself, above the suit. 'OK, close'er up.'\n\n'Look Mandella, if anybody goes in there, it oughta be me.'\n\n'Fuck you, Doc. _My_ job. My man.' That sounded all wrong. William Mandella, boy hero.\n\nThey stood a slab up on edge \u2014 it had two openings for the LSU input and exhaust \u2014 and proceeded to weld it to the bottom plank with a narrow laser beam. On Earth, we'd just use glue, but here the only fluid was helium, which has lots of interesting properties, but is definitely not sticky.\n\nAfter about ten minutes we were completely walled up. I could feel the LSU humming. I switched on my suit light \u2014 the first time since we landed on darkside \u2014 and the glare made purple blotches dance in front of my eyes.\n\n'Mandella, this is Ho. Stay in your suit at least two or three minutes. We're putting hot air in, but it's coming back just this side of liquid.' I watched the purple fade for a while.\n\n'OK, it's still cold, but you can make it.' I popped my suit. It wouldn't open all the way, but I didn't have too much trouble getting out. The suit was still cold enough to take some skin off my fingers and butt as I wiggled out.\n\nI had to crawl feet-first down the coffin to get to Singer. It got darker fast, moving away from my light. When I popped his suit a rush of hot stink hit me in the face. In the dim light his skin was dark red and splotchy. His breathing was very shallow and I could see his heart palpitating.\n\nFirst I unhooked the relief tubes \u2014 an unpleasant business \u2014 then the biosensors; and then I had the problem of getting his arms out of their sleeves.\n\nIt's pretty easy to do for yourself. You twist this way and turn that way and the arm pops out. Doing it from the outside is a different matter: I had to twist his arm and then reach under and move the suit's arm to match \u2014 it takes muscle to move a suit around from the outside.\n\nOnce I had one arm out it was pretty easy; I just crawled forward, putting my feet on the suit's shoulders, and pulled on his free arm. He slid out of the suit like an oyster slipping out of its shell.\n\nI popped the spare suit and after a lot of pulling and pushing, managed to get his legs in. Hooked up the biosensors and the front relief tube. He'd have to do the other one himself; it's too complicated. For the _n_ th time I was glad not to have been born female; they have to have two of those damned plumber's friends, instead of just one and a simple hose.\n\nI left his arms out of the sleeves. The suit would be useless for any kind of work, anyhow; waldos have to be tailored to the individual.\n\nHis eyelids fluttered. 'Man... della. Where... the fuck...'\n\nI explained, slowly, and he seemed to get most of it. 'Now I'm gonna close you up and go get into my suit. I'll have the crew cut the end off this thing and I'll haul you out. Got it?'\n\nHe nodded. Strange to see that \u2014 when you nod or shrug inside a suit, it doesn't communicate anything.\n\nI crawled into my suit, hooked up the attachments and chinned the general freak. 'Doc, I think he's gonna be OK. Get us out of here now.'\n\n'Will do.' Ho's voice. The LSU hum was replaced by a chatter, then a throb. Evacuating the box to prevent an explosion.\n\nOne corner of the seam grew red, then white, and a bright crimson beam lanced through, not a foot away from my head. I scrunched back as far as I could. The beam slid up the seam and around three corners, back to where it started. The end of the box fell away slowly, trailing filaments of melted 'plast.\n\n'Wait for the stuff to harden, Mandella.'\n\n'Sanchez, I'm not that stupid.'\n\n'Here you go.' Somebody tossed a line to me. That _would_ be smarter than dragging him out by myself. I threaded a long bight under his arms and tied it behind his neck. Then I scrambled out to help them pull, which was silly \u2014 they had a dozen people already linked up to haul.\n\nSinger got out all right and was actually sitting up while Doc Jones checked his readout. People were asking me about it and congratulating me, when suddenly Ho said 'Look!' and pointed toward the horizon.\n\nIt was a black ship, coming in fast. I just had time to think it wasn't fair, they weren't supposed to attack until the last few days, and then the ship was right on top of us. \n\n# 9\n\nWe all flopped to the ground instinctively, but the ship didn't attack. It blasted braking rockets and dropped to land on skids. Then it skied around to come to a rest beside the building site.\n\nEverybody had it figured out and was standing around sheepishly when the two suited figures stepped out of the ship.\n\nA familiar voice crackled over the general freak. 'Every _one_ of you saw us coming in and not _one_ of you responded with laser fire. It wouldn't have done any good but it would have indicated a certain amount of fighting spirit. You have a week or less before the real thing and since the sergeant and _I_ will be here _I_ will insist that you show a little more will to live. Acting Sergeant Potter.'\n\n'Here, sir.'\n\n'Get me a detail of twelve people to unload cargo. We brought a hundred small robot drones for _target_ practice so that you might have at least a fighting chance when a live target comes over.\n\n'Move _now_. We only have thirty minutes before the ship returns to Miami.'\n\nI checked, and it was actually more like forty minutes.\n\nHaving the captain and sergeant there didn't really make much difference. We were still on our own; they were just observing.\n\nOnce we got the floor down, it only took one day to complete the bunker. It was a gray oblong, featureless except for the airlock blister and four windows. On top was a swivel-mounted gigawatt laser. The operator \u2014 you couldn't call him a 'gunner' \u2014 sat in a chair holding dead-man switches in both hands. The laser wouldn't fire as long as he was holding one of those switches. If he let go, it would automatically aim for any moving aerial object and fire at will. Primary detection and aiming was by means of a kilometer-high antenna mounted beside the bunker.\n\nIt was the only arrangement that could really be expected to work, with the horizon so close and human reflexes so slow. You couldn't have the thing fully automatic, because in theory, friendly ships might also approach.\n\nThe aiming computer could choose among up to twelve targets appearing simultaneously (firing at the largest ones first). And it would get all twelve in the space of half a second.\n\nThe installation was partly protected from enemy fire by an efficient ablative layer that covered everything except the human operator. But then, they _were_ dead-man switches. One man above guarding eighty inside. The army's good at that kind of arithmetic.\n\nOnce the bunker was finished, half of us stayed inside at all times \u2014 feeling very much like targets \u2014 taking turns operating the laser, while the other half went on maneuvers.\n\nAbout four klicks from the base was a large 'lake' of frozen hydrogen; one of our most important maneuvers was to learn how to get around on the treacherous stuff.\n\nIt wasn't too difficult. You couldn't stand up on it, so you had to belly down and sled.\n\nIf you had somebody to push you from the edge, getting started was no problem. Otherwise, you had to scrabble with your hands and feet, pushing down as hard as was practical, until you started moving, in a series of little jumps. Once started, you'd keep going until you ran out of ice. You could steer a little bit by digging in, hand and foot, on the appropriate side, but you couldn't slow to a stop that way. So it was a good idea not to go too fast and wind up positioned in such a way that your helmet didn't absorb the shock of stopping.\n\nWe went through all the things we'd done on the Miami side: weapons practice, demolition, attack patterns. We also launched drones at irregular intervals, toward the bunker. Thus, ten or fifteen times a day, the operators got to demonstrate their skill in letting go of the handles as soon as the proximity light went on.\n\nI had four hours of that, like everybody else. I was nervous until the first 'attack,' when I saw how little there was to it. The light went on, I let go, the gun aimed, and when the drone peeped over the horizon \u2014 _zzt_! Nice touch of color, the molten metal spraying through space. Otherwise not too exciting.\n\nSo none of us were worried about the upcoming 'graduation exercise,' thinking it would be just more of the same.\n\nMiami Base attacked on the thirteenth day with two simultaneous missiles streaking over opposite sides of the horizon at some forty kilometers per second. The laser vaporized the first one with no trouble, but the second got within eight klicks of the bunker before it was hit.\n\nWe were coming back from maneuvers, about a klick away from the bunker. I wouldn't have seen it happen if I hadn't been looking directly at the bunker the moment of the attack.\n\nThe second missile sent a shower of molten debris straight toward the bunker. Eleven pieces hit, and, as we later reconstructed it, this is what happened:\n\nThe first casualty was Maejima, so well-loved Maejima, inside the bunker, who was hit in the back and the head and died instantly. With the drop in pressure, the LSU went into high gear. Friedman was standing in front of the main airco outlet and was blown into the opposite wall hard enough to knock him unconscious; he died of decompression before the others could get him to his suit.\n\nEverybody else managed to stagger through the gale and get into their suits, but Garcia's suit had been holed and didn't do him any good.\n\nBy the time we got there, they had turned off the LSU and were welding up the holes in the wall. One man was trying to scrape up the unrecognizable mess that had been Maejima. I could hear him sobbing and retching. They had already taken Garcia and Friedman outside for burial. The captain took over the repair detail from Potter. Sergeant Cortez led the sobbing man over to a corner and came back to work on cleaning up Maejima's remains, alone. He didn't order anybody to help and nobody volunteered. \n\n# 10\n\nAs a graduation exercise, we were unceremoniously stuffed into a ship \u2014 _Earth's Hope_ , the same one we rode to Charon \u2014 and bundled off to Stargate at a little more than one gee.\n\nThe trip seemed endless, about six months subjective time, and boring, but not as hard on the carcass as going to Charon had been. Captain Stott made us review our training orally, day by day, and we did exercises every day until we were worn to a collective frazzle.\n\nStargate 1 was like Charon's darkside, only more so. The base on Stargate 1 was smaller than Miami Base \u2014 only a little bigger than the one we constructed on darkside \u2014 and we were due to lay over a week to help expand the facilities. The crew there was very glad to see us, especially the two females, who looked a little worn around the edges.\n\nWe all crowded into the small dining hall, where Sub-major Williamson, the man in charge of Stargate 1, gave us some disconcerting news:\n\n'Everybody get comfortable. Get off the tables, though, there's plenty of floor.\n\n'I have some idea of what you just went through, training on Charon. I won't say it's all been wasted. But where you're headed, things will be quite different. Warmer.'\n\nHe paused to let that soak in.\n\n'Aleph Aurigae, the first collapsar ever detected, revolves around the normal star Epsilon Aurigae in a twenty-seven year orbit. The enemy has a base of operations, not on a regular portal planet of Aleph, but on a planet in orbit around Epsilon. We don't know much about the planet, just that it goes around Epsilon once every 745 days, is about three-fourths the size of Earth, and has an albedo of 0.8, meaning it's probably covered with clouds. We can't say precisely how hot it will be, but judging from its distance from Epsilon, it's probably rather hotter than Earth. Of course, we don't know whether you'll be working... fighting on lightside or darkside, equator or poles. It's highly unlikely that the atmosphere will be breathable \u2014 at any rate, you'll stay inside your suits.\n\n'Now you know exactly as much about where you're going as I do. Questions?'\n\n'Sir,' Stein drawled, 'now we know where we're goin'... anybody know what we're goin' to do when we get there?'\n\nWilliamson shrugged. 'That's up to your captain \u2014 and your sergeant, and the captain of _Earth's Hope_ , and _Hope's_ logistic computer. We just don't have enough data yet to project a course of action for you. It may be a long and bloody battle; it may be just a case of walking in to pick up the pieces. Conceivably, the Taurans might want to make a peace offer,' \u2014 Cortez snorted \u2014 'in which case you would simply be part of our muscle, our bargaining power.' He looked at Cortez mildly. 'No one can say for sure.'\n\nThe orgy that night was amusing, but it was like trying to sleep in the middle of a raucous beach party. The only area big enough to sleep all of us was the dining hall; they draped a few bedsheets here and there for privacy, then unleashed Stargate's eighteen sex-starved men on our women, compliant and promiscuous by military custom (and law), but desiring nothing so much as sleep on solid ground.\n\nThe eighteen men acted as if they were compelled to try as many permutations as possible, and their performance was impressive (in a strictly quantitative sense, that is). Those of us who were keeping count led a cheering section for some of the more gifted members. I think that's the right word.\n\nThe next morning \u2014 and every other morning we were on Stargate 1 \u2014 we staggered out of bed and into our suits, to go outside and work on the 'new wing.' Eventually, Stargate would be tactical and logistic headquarters for the war, with thousands of permanent personnel, guarded by half-a-dozen heavy cruisers in _Hope's_ class. When we started, it was two shacks and twenty people; when we left, it was four shacks and twenty people. The work was hardly work at all, compared to darkside, since we had plenty of light and got sixteen hours inside for every eight hours' work. And no drone attack for a final exam.\n\nWhen we shuttled back up to the _Hope_ , nobody was too happy about leaving (though some of the more popular females declared it'd be good to get some rest). Stargate was the last easy, safe assignment we'd have before taking up arms against the Taurans. And as Williamson had pointed out the first day, there was no way of predicting what _that_ would be like.\n\nMost of us didn't feel too enthusiastic about making a collapsar jump, either. We'd been assured that we wouldn't even feel it happen, just free fall all the way.\n\nI wasn't convinced. As a physics student, I'd had the usual courses in general relativity and theories of gravitation. We only had a little direct data at that time \u2014 Stargate was discovered when I was in grade school \u2014 but the mathematical model seemed clear enough.\n\nThe collapsar Stargate was a perfect sphere about three kilometers in radius. It was suspended forever in a state of gravitational collapse that should have meant its surface was dropping toward its center at nearly the speed of light. Relativity propped it up, at least gave it the illusion of being there... the way all reality becomes illusory and observer-oriented when you study general relativity. Or Buddhism. Or get drafted.\n\nAt any rate, there would be a theoretical point in space-time when one end of our ship was just above the surface of the collapsar, and the other end was a kilometer away (in our frame of reference). In any sane universe, this would set up tidal stresses and tear the ship apart, and we would be just another million kilograms of degenerate matter on the theoretical surface, rushing headlong to nowhere for the rest of eternity or dropping to the center in the next trillionth of a second. You pays your money and you takes your frame of reference.\n\nBut they were right. We blasted away from Stargate 1, made a few course corrections and then just dropped, for about an hour.\n\nThen a bell rang and we sank into our cushions under a steady two gravities of deceleration. We were in enemy territory.\n\n# 11\n\nWe'd been decelerating at two gravities for almost nine days when the battle began. Lying on our couches being miserable, all we felt were two soft bumps, missiles being released. Some eight hours later, the squawk-box crackled: 'Attention, all crew. This is the captain.' Quinsana, the pilot, was only a lieutenant, but was allowed to call himself captain aboard the vessel, where he outranked all of us, even Captain Stott. 'You grunts in the cargo hold can listen, too.\n\n'We just engaged the enemy with two fifty-gigaton tachyon missiles and have destroyed both the enemy vessel and another object which it had launched approximately three microseconds before.\n\n'The enemy has been trying to overtake us for the past 179 hours, ship time. At the time of the engagement, the enemy was moving at a little over half the speed of light, relative to Aleph, and was only about thirty AUs from _Earth's Hope_. It was moving at .47c relative to us, and thus we would have been coincident in space-time' \u2014 rammed! \u2014 'in a little more than nine hours. The missiles were launched at 0719 ship's time, and destroyed the enemy at 1540, both tachyon bombs detonating within a thousand klicks of the enemy objects.'\n\nThe two missiles were a type whose propulsion system was itself only a barely-controlled tachyon bomb. They accelerated at a constant rate of 100 gees, and were traveling at a relativistic speed by the time the nearby mass of the enemy ship detonated them.\n\n'We expect no further interference from enemy vessels. Our velocity with respect to Aleph will be zero in another five hours; we will then begin the journey back. The return will take twenty-seven days.' General moans and dejected cussing. Everybody knew all that already, of course; but we didn't care to be reminded of it.\n\nSo after another month of logy calisthenics and drill, at a constant two gravities, we got our first look at the planet we were going to attack. Invaders from outer space, yes sir.\n\nIt was a blinding white crescent waiting for us two AUs out from Epsilon. The captain had pinned down the location of the enemy base from fifty AUs out, and we had jockeyed in on a wide arc, keeping the bulk of the planet between them and us. That didn't mean we were sneaking up on them \u2014 quite the contrary; they launched three abortive attacks \u2014 but it put us in a stronger defensive position. Until we had to go to the surface, that is. Then only the ship and its Star Fleet crew would be reasonably safe.\n\nSince the planet rotated rather slowly \u2014 once every ten and one-half days \u2014 a 'stationary' orbit for the ship had to be 150,000 klicks out. This made the people in the ship feel quite secure, with 6,000 miles of rock and 90,000 miles of space between them and the enemy. But it meant a whole second's time lag in communication between us on the ground and the ship's battle computer. A person could get awful dead while that neutrino pulse crawled up and back.\n\nOur vague orders were to attack the base and gain control, while damaging a minimum of enemy equipment. We were to take at least one enemy alive. We were under no circumstances to allow ourselves to be taken alive, however. And the decision wasn't up to us; one special pulse from the battle computer, and that speck of plutonium in your power plant would fiss with all of .01% efficiency, and you'd be nothing but a rapidly expanding, very hot plasma.\n\nThey strapped us into six scoutships \u2014 one platoon of twelve people in each \u2014 and we blasted away from _Earth's Hope_ at eight gees. Each scoutship was supposed to follow its own carefully random path to our rendezvous point, 108 klicks from the base. Fourteen drone ships were launched at the same time, to confound the enemy's anti-spacecraft system.\n\nThe landing went off almost perfectly. One ship suffered minor damage, a near miss boiling away some of the ablative material on one side of the hull, but it'd still be able to make it and return, keeping its speed down while in the atmosphere.\n\nWe zigged and zagged and wound up first ship at the rendezvous point. There was only one trouble. It was under four kilometers of water.\n\nI could almost hear that machine, 90,000 miles away, grinding its mental gears, adding this new bit of data. We proceeded just as if we were landing on solid ground: braking rockets, falling, skids out, hit the water, skip, hit the water, skip, hit the water, sink.\n\nIt would have made sense to go ahead and land on the bottom \u2014 we were streamlined, after all, and water just another fluid \u2014 but the hull wasn't strong enough to hold up a four kilometer column of water. Sergeant Cortez was in the scoutship with us.\n\n'Sarge, tell that computer to _do_ something! We're gonna get\u2014'\n\n'Oh, shut up, Mandella. Trust in th' lord.' 'Lord' was definitely lower-case when Cortez said it.\n\nThere was a loud bubbly sigh, then another, and a slight increase in pressure on my back that meant the ship was rising. 'Flotation bags?' Cortez didn't deign to answer, or didn't know.\n\nThat was it. We rose to within ten or fifteen meters of the surface and stopped, suspended there. Through the port I could see the surface above, shimmering like a mirror of hammered silver. I wondered what it would be like to be a fish and have a definite roof over your world.\n\nI watched another ship splash in. It made a great cloud of bubbles and turbulence, then fell \u2014 slightly tail-first \u2014 for a short distance before large bags popped out under each delta wing. Then it bobbed up to about our level and stayed.\n\n'This is Captain Stott. Now listen carefully. There is a beach some twenty-eight klicks from your present position, in the direction of the enemy. You will be proceeding to this beach by scoutship and from there will mount your assault on the Tauran position.' That was _some_ improvement; we'd only have to walk eighty klicks.\n\nWe deflated the bags, blasted to the surface and flew in a slow, spread-out formation to the beach. It took several minutes. As the ship scraped to a halt, I could hear pumps humming, making the cabin pressure equal to the air pressure outside. Before it had quite stopped moving, the escape slot beside my couch slid open. I rolled out onto the wing of the craft and jumped to the ground. Ten seconds to find cover \u2014 I sprinted across loose gravel to the 'treeline,' a twisty bramble of tall sparse bluish-green shrubs. I dove into the briar patch and turned to watch the ships leave. The drones that were left rose slowly to about a hundred meters, then took off in all directions with a bone-jarring roar. The real scoutships slid slowly back into the water. Maybe that was a good idea.\n\nIt wasn't a terribly attractive world but certainly would be easier to get around in than the cryogenic nightmare we were trained for. The sky was a uniform dull silver brightness that merged with the mist over the ocean so completely it was impossible to tell where water ended and air began. Small wavelets licked at the black gravel shore, much too slow and graceful in the three-quarters Earth-normal gravity. Even from fifty meters away, the rattle of billions of pebbles rolling with the tide was loud in my ears.\n\nThe air temperature was 79 degrees Centigrade, not quite hot enough for the sea to boil, even though the air pressure was low compared to Earth's. Wisps of steam drifted quickly upward from the line where water met land. I wondered how a lone man would survive exposed here without a suit. Would the heat or the low oxygen (partial pressure one-eighth Earth normal) kill him first? Or was there some deadly microorganism that would beat them both...?\n\n'This is Cortez. Everybody come over and assemble on me.' He was standing on the beach a little to the left of me, waving his hand in a circle over his head. I walked toward him through the shrubs. They were brittle, unsubstantial, seemed paradoxically dried-out in the steamy air. They wouldn't offer much in the way of cover.\n\n'We'll be advancing on a heading .05 radians east of north. I want Platoon One to take point. Two and Three follow about twenty meters behind, to the left and right. Seven, command platoon, is in the middle, twenty meters behind Two and Three. Five and Six, bring up the rear, in a semicircular closed flank. Everybody straight?' Sure, we could do that 'arrowhead' maneuver in our sleep. 'OK, let's move out.'\n\nI was in Platoon Seven, the 'command group.' Captain Stott put me there not because I was expected to give any commands, but because of my training in physics.\n\nThe command group was supposedly the safest place, buffered by six platoons: people were assigned to it because there was some tactical reason for them to survive at least a little longer than the rest. Cortez was there to give orders. Chavez was there to correct suit malfunctions. The senior medic, Doc Wilson (the only medic who actually had an M.D.), was there, and so was Theodopolis, the radio engineer, our link with the captain, who had elected to stay in orbit.\n\nThe rest of us were assigned to the command group by dint of special training or aptitude that wouldn't normally be considered of a 'tactical' nature. Facing a totally unknown enemy, there was no way of telling what might prove important. Thus I was there because I was the closest the company had to a physicist. Rogers was biology. Tate was chemistry. Ho could crank out a perfect score on the Rhine extrasensory perception test, every time. Bohrs was a polyglot, able to speak twenty-one languages fluently, idiomatically. Petrov's talent was that he had tested out to have not one molecule of xenophobia in his psyche. Keating was a skilled acrobat. Debby Hollister \u2014 'Lucky' Hollister \u2014 showed a remarkable aptitude for making money, and also had a consistently high Rhine potential. \n\n# 12\n\nWhen we first set out, we were using the 'jungle' camouflage combination on our suits. But what passed for jungle in these anemic tropics was too sparse; we looked like a band of conspicuous harlequins trooping through the woods. Cortez had us switch to black, but that was just as bad, as the light of Epsilon came evenly from all parts of the sky, and there were no shadows except ours. We finally settled on the dun-colored desert camouflage.\n\nThe nature of the countryside changed slowly as we walked north, away from the sea. The thorned stalks \u2014 I guess you could call them trees \u2014 came in fewer numbers but were bigger around and less brittle; at the base of each was a tangled mass of vine with the same bluegreen color, which spread out in a flattened cone some ten meters in diameter. There was a delicate green flower the size of a man's head near the top of each tree.\n\nGrass began to grow some five klicks from the sea. It seemed to respect the trees' 'property rights,' leaving a strip of bare earth around each cone of vine. At the edge of such a clearing, it would grow as timid bluegreen stubble, then, moving away from the tree, would get thicker and taller until it reached shoulderhigh in some places, where the separation between two trees was unusually large. The grass was a lighter, greener shade than the trees and vines. We changed the color of our suits to the bright green we had used for maximum visibility on Charon. Keeping to the thickest part of the grass, we were fairly inconspicuous.\n\nWe covered over twenty klicks each day, buoyant after months under two gees. Until the second day, the only form of animal life we saw was a kind of black worm, finger-sized, with hundreds of cilium legs like the bristles of a brush. Rogers said that there obviously had to be some larger creature around, or there would be no reason for the trees to have thorns. So we were doubly on guard, expecting trouble both from the Taurans and the unidentified 'large creature.'\n\nPotter's second platoon was on point; the general freak was reserved for her, since her platoon would likely be the first to spot any trouble.\n\n'Sarge, this is Potter,' we all heard. 'Movement ahead.'\n\n'Get down, then!'\n\n'We are. Don't think they see us.'\n\n'First platoon, go up to the right of point. Keep down. Fourth, get up to the left. Tell me when you get in position. Sixth platoon, stay back and guard the rear. Fifth and third, close with the command group.'\n\nTwo dozen people whispered out of the grass to join us. Cortez must have heard from the fourth platoon.\n\n'Good. How about you, first?... OK, fine. How many are there?'\n\n'Eight we can see.' Potter's voice.\n\n'Good. When I give the word, open fire. Shoot to kill.'\n\n'Sarge... they're just animals.'\n\n'Potter \u2014 if you've known all this time what a Tauran looks like, you should've told us. Shoot to kill.'\n\n'But we need...'\n\n'We need a prisoner, but we don't need to escort him forty klicks to his home base and keep an eye on him while we fight. Clear?'\n\n'Yes. Sergeant.'\n\n'OK. Seventh, all you brains and weirds, we're going up and watch. Fifth and third, come along to guard.'\n\nWe crawled through the meter-high grass to where the second platoon had stretched out in a firing line.\n\n'I don't see anything,' Cortez said.\n\n'Ahead and just to the left. Dark green.'\n\nThey were only a shade darker than the grass. But after you saw the first one, you could see them all, moving slowly around some thirty meters ahead.\n\n'Fire!' Cortez fired first; then twelve streaks of crimson leaped out and the grass wilted black, disappeared, and the creatures convulsed and died trying to scatter.\n\n'Hold fire, hold it!' Cortez stood up. 'We want to have something left \u2014 second platoon, follow me.' He strode out toward the smoldering corpses, laser-finger pointed out front, obscene divining rod pulling him toward the carnage... I felt my gorge rising and knew that all the lurid training tapes, all the horrible deaths in training accidents, hadn't prepared me for this sudden reality... that I had a magic wand that I could point at a life and make it a smoking piece of half-raw meat; I wasn't a soldier nor ever wanted to be one nor ever would want\u2014\n\n'OK, seventh, come on up.' While we were walking toward them, one of the creatures moved, a tiny shudder, and Cortez flicked the beam of his laser over it with an almost negligent gesture. It made a hand-deep gash across the creature's middle. It died, like the others, without emitting a sound.\n\nThey were not quite as tall as humans, but wider in girth. They were covered with dark green, almost black, fur \u2014 white curls where the laser had singed. They appeared to have three legs and an arm. The only ornament to their shaggy heads was a mouth, wet black orifice filled with flat black teeth. They were thoroughly repulsive, but their worst feature was not a difference from human beings, but a similarity... Whenever the laser had opened a body cavity, milk-white glistening veined globes and coils of organs spilled out, and their blood was dark clotting red.\n\n'Rogers, take a look. Taurans or not?'\n\nRogers knelt by one of the disemboweled creatures and opened a flat plastic box, filled with glittering dissecting tools. She selected a scalpel. 'One way we might be able to find out.' Doc Wilson watched over her shoulder as she methodically slit the membrane covering several organs.\n\n'Here.' She held up a blackish fibrous mass between two fingers, a parody of daintiness through all that armor.\n\n'So?'\n\n'It's grass, Sergeant. If the Taurans eat the grass and breathe the air, they certainly found a planet remarkably like their home.' She tossed it away. 'They're animals, Sergeant, just fucken animals.'\n\n'I don't know,' Doc Wilson said. 'Just because they walk around on all fours, threes maybe, and eat grass...'\n\n'Well, let's check out the brain.' She found one that had been hit in the head and scraped the superficial black char from the wound. 'Look at that.'\n\nIt was almost solid bone. She tugged and ruffled the hair all over the head of another one. 'What the hell does it use for sensory organs? No eyes, or ears, or...' She stood up.\n\n'Nothing in that fucken head but a mouth and ten centimeters of skull. To protect nothing, not a fucken thing.'\n\n'If I could shrug, I'd shrug,' the doctor said. 'It doesn't prove anything \u2014 a brain doesn't have to look like a mushy walnut and it doesn't have to be in the head. Maybe that skull isn't bone, maybe _that's_ the brain, some crystal lattice...'\n\n'Yeah, but the fucken stomach's in the right place, and if those aren't intestines I'll eat\u2014'\n\n'Look,' Cortez said, 'this is real interesting, but all we need to know is whether that thing's dangerous, then we've gotta move on; we don't have all\u2014'\n\n'They aren't dangerous,' Rogers began. 'They don't\u2014'\n\n'Medic! DOC!' Somebody back at the firing line was waving his arms. Doc sprinted back to him, the rest of us following.\n\n'What's wrong?' He had reached back and unclipped his medical kit on the run.\n\n'It's Ho. She's out.'\n\nDoc swung open the door on Ho's biomedical monitor. He didn't have to look far. 'She's dead.'\n\n'Dead?' Cortez said. 'What the hell\u2014'\n\n'Just a minute.' Doc plugged a jack into the monitor and fiddled with some dials on his kit. 'Everybody's biomed readout is stored for twelve hours. I'm running it backwards, should be able to \u2014 there!'\n\n'What?'\n\n'Four and a half minutes ago \u2014 must have been when you opened fire \u2014 Jesus!'\n\n'Well?'\n\n'Massive cerebral hemorrhage. No...' He watched the dials. 'No... warning, no indication of anything out of the ordinary; blood pressure up, pulse up, but normal under the circumstances... nothing to... indicate\u2014' He reached down and popped her suit. Her fine oriental features were distorted in a horrible grimace, both gums showing. Sticky fluid ran from under her collapsed eyelids, and a trickle of blood still dripped from each ear. Doc Wilson closed the suit back up.\n\n'I've never seen anything like it. It's as if a bomb went off in her skull.'\n\n'Oh fuck,' Rogers said, 'she was Rhine-sensitive, wasn't she.'\n\n'That's right,' Cortez sounded thoughtful. 'All right, everybody listen up. Platoon leaders, check your platoons and see if anybody's missing, or hurt. Anybody else in seventh?'\n\n'I... I've got a splitting headache, Sarge,' Lucky said.\n\nFour others had bad headaches. One of them affirmed that he was slightly Rhine-sensitive. The others didn't know.\n\n'Cortez, I think it's obvious,' Doc Wilson said, 'that we should give these... monsters wide berth, especially shouldn't harm any more of them. Not with five people susceptible to whatever apparently killed Ho.'\n\n'Of course, God damn it, I don't need anybody to tell me that. We'd better get moving. I just filled the captain in on what happened; he agrees that we'd better get as far away from here as we can, before we stop for the night.\n\n'Let's get back in formation and continue on the same bearing. Fifth platoon, take over point; second, come back to the rear. Everybody else, same as before.'\n\n'What about Ho?' Lucky asked.\n\n'She'll be taken care of. From the ship.'\n\nAfter we'd gone half a klick, there was a flash and rolling thunder. Where Ho had been came a wispy luminous mushroom cloud boiling up to disappear against the gray sky. \n\n# 13\n\nWe stopped for the 'night' \u2014 actually, the sun wouldn't set for another seventy hours \u2014 atop a slight rise some ten klicks from where we had killed the aliens. But they weren't aliens, I had to remind myself \u2014 _we_ were.\n\nTwo platoons deployed in a ring around the rest of us, and we flopped down exhausted. Everybody was allowed four hours' sleep and had two hours' guard duty.\n\nPotter came over and sat next to me. I chinned her frequency.\n\n'Hi, Marygay.'\n\n'Oh, William,' her voice over the radio was hoarse and cracking. 'God, it's so horrible.'\n\n'It's over now\u2014'\n\n'I killed one of them, the first instant, I shot it right in the, in the...'\n\nI put my hand on her knee. The contact had a plastic click and I jerked it back, visions of machines embracing, copulating. 'Don't feel singled out, Marygay; whatever guilt there is, is... belongs evenly to all of us... but a triple portion for Cor\u2014'\n\n'You privates quit jawin' and get some sleep. You both pull guard in two hours.'\n\n'OK, Sarge.' Her voice was so sad and tired I couldn't bear it. I felt if I could only touch her, I could drain off the sadness like ground wire draining current, but we were each trapped in our own plastic world\u2014\n\n'G'night, William.'\n\n'Night.' It's almost impossible to get sexually excited inside a suit, with the relief tube and all the silver chloride sensors poking you, but somehow this was my body's response to the emotional impotence, maybe remembering more pleasant sleeps with Marygay, maybe feeling that in the midst of all this death, personal death could be very soon, cranking up the procreative derrick for one last try... lovely thoughts like this. I fell asleep and dreamed that I was a machine, mimicking the functions of life, creaking and clanking my clumsy way through a world, people too polite to say anything but giggling behind my back, and the little man who sat inside my head pulling the levers and clutches and watching the dials, he was hopelessly mad and was storing up hurts for the day\u2014\n\n'Mandella \u2014 wake up, goddammit, your shift!'\n\nI shuffled over to my place on the perimeter to watch for god knows what... but I was so weary I couldn't keep my eyes open. Finally I tongued a stimtab, knowing I'd pay for it later.\n\nFor over an hour I sat there, scanning my sector left, right, near, far, the scene never changing, not even a breath of wind to stir the grass.\n\nThen suddenly the grass parted and one of the three-legged creatures was right in front of me. I raised my finger but didn't squeeze.\n\n'Movement!'\n\n'Movement!'\n\n'Jesus Chri\u2014there's one right\u2014'\n\n'HOLD YOUR FIRE! F' shit's sake don't shoot!'\n\n'Movement.'\n\n'Movement.' I looked left and right, and as far as I could see, every perimeter guard had one of the blind, dumb creatures standing right in front of him.\n\nMaybe the drug I'd taken to stay awake made me more sensitive to whatever they did. My scalp crawled and I felt a formless _thing_ in my mind, the feeling you get when somebody has said something and you didn't quite hear it, want to respond, but the opportunity to ask him to repeat it is gone.\n\nThe creature sat back on its haunches, leaning forward on the one front leg. Big green bear with a withered arm. Its power threaded through my mind, spiderwebs, echo of night terrors, trying to communicate, trying to destroy me, I couldn't know.\n\n'All right, everybody on the perimeter, fall back, slow. Don't make any quick gestures... Anybody got a headache or anything?'\n\n'Sergeant, this is Hollister.' Lucky.\n\n'They're trying to say something... I can almost... no, just...'\n\n'All I can get is that they think we're, think we're... well, _funny_. They're not afraid.'\n\n'You mean the one in front of you isn't\u2014'\n\n'No, the feeling comes from all of them, they're all thinking the same thing. Don't ask me how I know, I just do.'\n\n'Maybe they thought it was funny, what they did to Ho.'\n\n'Maybe. I don't feel they're dangerous. Just curious about us.'\n\n'Sergeant, this is Bohrs.'\n\n'Yeah.'\n\n'The Taurans've been here at least a year \u2014 maybe they've learned how to communicate with these... overgrown teddy bears. They might be spying on us, might be sending back\u2014'\n\n'I don't think they'd show themselves if that were the case,' Lucky said. 'They can obviously hide from us pretty well when they want to.'\n\n'Anyhow,' Cortez said, 'if they're spies, the damage has been done. Don't think it'd be smart to take any action against them. I know you'd all like to see 'em dead for what they did to Ho, so would I, but we'd better be careful.'\n\nI didn't want to see them dead, but I'd just as soon not have seen them in any condition. I was walking backwards slowly, toward the middle of camp. The creature didn't seem disposed to follow. Maybe he just knew we were surrounded. He was pulling up grass with his arm and munching.\n\n'OK, all of you platoon leaders, wake everybody up, get a roll count. Let me know if anybody's been hurt. Tell your people we're moving out in one minute.'\n\nI don't know what Cortez had expected, but of course the creatures followed right along. They didn't keep us surrounded; just had twenty or thirty following us all the time. Not the same ones, either. Individuals would saunter away, and new ones would join the parade. It was pretty obvious that _they_ weren't going to tire out.\n\nWe were each allowed one stimtab. Without it, no one could have marched an hour. A second pill would have been welcome after the edge started to wear off, but the mathematics of the situation forbade it; we were still thirty klicks from the enemy base, fifteen hours' marching at the least. And though you could stay awake and energetic for a hundred hours on the tabs, aberrations of judgment and perception snowballed after the second one, until _in extremis_ the most bizarre hallucinations would be taken at face value, and a person could fidget for hours deciding whether to have breakfast.\n\nUnder artificial stimulation, the company traveled with great energy for the first six hours, was slowing by the seventh, and ground to an exhausted halt after nine hours and nineteen kilometers. The teddy bears had never lost sight of us and, according to Lucky, had never stopped 'broadcasting.' Cortez's decision was that we would stop for seven hours, each platoon taking one hour of perimeter guard. I was never so glad to have been in the seventh platoon, as we stood guard the last shift and thus were able to get six hours of uninterrupted sleep.\n\nIn the few moments I lay awake after finally lying down, the thought came to me that the next time I closed my eyes could well be the last. And partly because of the drug hangover, mostly because of the past day's horrors, I found that I really didn't give a shit. \n\n# 14\n\nOur first contact with the Taurans came during my shift.\n\nThe teddy bears were still there when I woke up and replaced Doc Jones on guard. They'd gone back to their original formation, one in front of each guard position. The one who was waiting for me seemed a little larger than normal, but otherwise looked just like all the others. All the grass had been cropped where he was sitting, so he occasionally made forays to the left or right. But he always returned to sit right in front of me, you would say _staring_ if he had had anything to stare with.\n\nWe had been facing each other for about fifteen minutes when Cortez's voice rumbled:\n\n'Awright everybody, wake up and get hid!'\n\nI followed instinct and flopped to the ground and rolled into a tall stand of grass.\n\n'Enemy vessel overhead.' His voice was almost laconic.\n\nStrictly speaking, it wasn't really overhead, but rather passing somewhat east of us. It was moving slowly, maybe a hundred klicks per hour, and looked like a broomstick surrounded by a dirty soap bubble. The creature riding it was a little more human-looking than the teddy bears, but still no prize. I cranked my image amplifier up to forty log two for a closer look.\n\nHe had two arms and two legs, but his waist was so small you could encompass it with both hands. Under the tiny waist was a large horseshoe-shaped pelvic structure nearly a meter wide, from which dangled two long skinny legs with no apparent knee joint. Above that waist his body swelled out again, to a chest no smaller than the huge pelvis. His arms looked surprisingly human, except that they were too long and undermuscled. There were too many fingers on his hands. Shoulderless, neckless. His head was a nightmarish growth that swelled like a goiter from his massive chest. Two eyes that looked like clusters of fish eggs, a bundle of tassels instead of a nose, and a rigidly open hole that might have been a mouth sitting low down where his adam's apple should have been. Evidently the soap bubble contained an amenable environment, as he was wearing absolutely nothing except his ridged hide, that looked like skin submerged too long in hot water, then dyed a pale orange. 'He' had no external genitalia, but nothing that might hint of mammary glands. So we opted for the male pronoun by default.\n\nObviously, he either didn't see us or thought we were part of the herd of teddy bears. He never looked back at us, but just continued in the same direction we were headed, .05 rad east of north.\n\n'Might as well go back to sleep now, if you can sleep after looking at _that_ thing. We move out at 0435.' Forty minutes.\n\nBecause of the planet's opaque cloud cover, there had been no way to tell, from space, what the enemy base looked like or how big it was. We only knew its position, the same way we knew the position the scout-ships were supposed to land on. So it too could easily have been underwater, or underground.\n\nBut some of the drones were reconnaissance ships as well as decoys: and in their mock attacks on the base, one managed to get close enough to take a picture. Captain Stott beamed down a diagram of the place to Cortez \u2014 the only one with a visor in his suit \u2014 when we were five klicks from the base's 'radio' position. We stopped and he called all the platoon leaders in with the seventh platoon to confer. Two teddy bears loped in, too. We tried to ignore them.\n\n'OK, the captain sent down some pictures of our objective. I'm going to draw a map; you platoon leaders copy.' They took pads and styli out of their leg pockets, while Cortez unrolled a large plastic mat. He gave it a shake to randomize any residual charge, and turned on his stylus.\n\n'Now, we're coming from this direction.' He put an arrow at the bottom of the sheet. 'First thing we'll hit is this row of huts, probably billets or bunkers, but who the hell knows... Our initial objective is to destroy these buildings \u2014 the whole base is on a flat plain; there's no way we could really sneak by them.'\n\n'Potter here. Why can't we jump over them?'\n\n'Yeah, we could do that, and wind up completely surrounded, cut to ribbons. We take the buildings.'\n\n'After we do that... all I can say is that we'll have to think on our feet. From the aerial reconnaissance, we can figure out the function of only a couple of buildings \u2014 and that stinks. We might wind up wasting a lot of time demolishing the equivalent of an enlisted-men's bar, ignoring a huge logistic computer because it looks like... a garbage dump or something.'\n\n'Mandella here,' I said. 'Isn't there a spaceport of some kind \u2014 seems to me we ought to...'\n\n'I'll _get_ to that, damn it. There's a ring of these huts all around the camp, so we've got to break through somewhere. This place'll be closest, less chance of giving away our position before we attack.\n\n'There's nothing in the whole place that actually looks like a weapon. That doesn't mean anything, though; you could hide a gigawatt laser in each of those huts.\n\n'Now, about five hundred meters from the huts, in the middle of the base, we'll come to this big flower-shaped structure.' Cortez drew a large symmetrical shape that looked like the outline of a flower with seven petals. 'What the hell this is, your guess is as good as mine. There's only one of them, though, so we don't damage it any more than we have to. Which means... we blast it to splinters if I think it's dangerous.\n\n'Now, as far as your spaceport, Mandella, is concerned \u2014 there just isn't one. Nothing.\n\n'That cruiser the _Hope_ caulked had probably been left in orbit, like ours has to be. If they have any equivalent of a scoutship, or drone missiles, they're either not kept here or they're well hidden.'\n\n'Bohrs here. Then what did they attack with, while we were coming down from orbit?'\n\n'I wish we knew, Private.\n\n'Obviously, we don't have any way of estimating their numbers, not directly. Recon pictures failed to show a single Tauran on the grounds of the base. Meaning nothing, because it _is_ an alien environment. Indirectly, though... we count the number of broomsticks, those flying things.\n\n'There are fifty-one huts, and each has at most one broomstick. Four don't have any parked outside, but we located three at various other parts of the base. Maybe this indicates that there are fifty-one Taurans, one of whom was outside the base when the picture was taken.'\n\n'Keating here. Or fifty-one officers.'\n\n'That's right \u2014 maybe fifty thousand infantrymen stacked in one of these buildings. No way to tell. Maybe ten Taurans, each with five broomsticks, to use according to his mood.\n\n'We've got one thing in our favor, and that's communications. They evidently use a frequency modulation of megahertz electromagnetic radiation.'\n\n'Radio!'\n\n'That's right, whoever you are. Identify yourself when you speak. So it's quite possible that they can't detect our phased-neutrino communications. Also, just prior to the attack, the _Hope_ is going to deliver a nice dirty fission bomb; detonate it in the upper atmosphere right over the base. That'll restrict them to line-of-sight communications for some time; even those will be full of static.'\n\n'Why don't... Tate here... why don't they just drop the bomb right in their laps. Save us a lot of\u2014'\n\n'That doesn't even deserve an answer, Private. But the answer is, they might. And you better hope they don't. If they caulk the base, it'll be for the safety of the _Hope_. _After_ we've attacked, and probably before we're far enough away for it to make much difference.\n\n'We keep that from happening by doing a good job. We have to reduce the base to where it can no longer function; at the same time, leave as much intact as possible. And take one prisoner.'\n\n'Potter here. You mean, at least one prisoner.'\n\n'I mean what I say. One only. Potter... you're relieved of your platoon. Send Chavez up.'\n\n'All right, Sergeant.' The relief in her voice was unmistakable.\n\nCortez continued with his map and instructions. There was one other building whose function was pretty obvious; it had a large steerable dish antenna on top. We were to destroy it as soon as the grenadiers got in range.\n\nThe attack plan was very loose. Our signal to begin would be the flash of the fission bomb. At the same time, several drones would converge on the base, so we could see what their antispacecraft defenses were. We would try to reduce the effectiveness of those defenses without destroying them completely.\n\nImmediately after the bomb and the drones, the grenadiers would vaporize a line of seven huts. Everybody would break through the hole into the base... and what would happen after that was anybody's guess.\n\nIdeally, we'd sweep from that end of the base to the other, destroying certain targets, caulking all but one Tauran. But that was unlikely to happen, as it depended on the Taurans' offering very little resistance.\n\nOn the other hand, if the Taurans showed obvious superiority from the beginning, Cortez would give the order to scatter. Everybody had a different compass bearing for retreat \u2014 we'd blossom out in all directions, the survivors to rendezvous in a valley some forty klicks east of the base. Then we'd see about a return engagement, after the _Hope_ softened the base up a bit.\n\n'One last thing,' Cortez rasped. 'Maybe some of you feel the way Potter evidently does, maybe some of your men feel that way... that we ought to go easy, not make this so much of a bloodbath. Mercy is a luxury, a weakness we can't afford to indulge in at this stage of the war. _All_ we know about the enemy is that they have killed seven hundred and ninety-eight humans. They haven't shown any restraint in attacking our cruisers, and it'd be foolish to expect any this time, this first ground action.\n\n' _They_ are responsible for the lives of all of your comrades who died in training, and for Ho, and for all the others who are surely going to die today. I can't _understand_ anybody who wants to spare them. But that doesn't make any difference. You have your orders and, what the hell, you might as well know, all of you have a post-hypnotic suggestion that I will trigger by a phrase, just before the battle. It will make your job easier.'\n\n'Sergeant...'\n\n'Shut up. We're short on time; get back to your platoons and brief them. We move out in five minutes.'\n\nThe platoon leaders returned to their men, leaving Cortez and ten of us \u2014 plus three teddy bears, milling around, getting in the way. \n\n# 15\n\nWe took the last five klicks very carefully, sticking to the highest grass, running across occasional clearings. When we were 500 meters from where the base was supposed to be, Cortez took the third platoon forward to scout, while the rest of us laid low.\n\nCortez's voice came over the general freak: 'Looks pretty much like we expected. Advance in a file, crawling. When you get to the third platoon, follow your squad leader to the left or right.'\n\nWe did that and wound up with a string of eighty-three people in a line roughly perpendicular to the direction of attack. We were pretty well hidden, except for the dozen or so teddy bears that mooched along the line, munching grass.\n\nThere was no sign of life inside the base. All of the buildings were windowless and a uniform shiny white. The huts that were our first objective were large featureless half-buried eggs some sixty meters apart. Cortez assigned one to each grenadier.\n\nWe were broken into three fire teams: team _A_ consisted of platoons two, four, and six; team _B_ was one, three, and five; the command platoon was team _C_.\n\n'Less than a minute now \u2014 filters down! \u2014 when I say \"fire,\" grenadiers, take out your targets. God help you if you miss.'\n\nThere was a sound like a giant's belch, and a stream of five or six iridescent bubbles floated up from the flower-shaped building. They rose with increasing speed until they were almost out of sight, then shot off to the south, over our heads. The ground was suddenly bright, and for the first time in a long time, I saw my shadow, a long one pointed north. The bomb had gone off prematurely. I just had time to think that it didn't make too much difference; it'd still make alphabet soup out of their communications\u2014\n\n'Drones!' A ship came screaming in just about tree level, and a bubble was in the air to meet it. When they contacted, the bubble popped and the drone exploded into a million tiny fragments. Another one came from the opposite side and suffered the same fate.\n\n'FIRE!' Seven bright glares of 500-microton grenades and a sustained concussion that surely would have killed an unprotected man.\n\n'Filters up.' Gray haze of smoke and dust. Clods of dirt falling with a sound like heavy raindrops.\n\n'Listen up:\n\n_\"Scots, wha hae wi' Wallace bled;_\n\n_Scots, wham Bruce has aften led_ ,\n\n_Welcome to your gory bed_ ,\n\n_Or to victory!\"'_\n\nI hardly heard him for trying to keep track of what was going on in my skull. I knew it was just post-hypnotic suggestion, even remembered the session in Missouri when they'd implanted it, but that didn't make it any less compelling. My mind reeled under the strong pseudo-memories: shaggy hulks that were Taurans (not at all what we now knew they looked like) boarding a colonists' vessel, eating babies while mothers watched in screaming terror (the colonists never took babies; they wouldn't stand the acceleration), then raping the women to death with huge veined purple members (ridiculous that they would feel desire for humans), holding the men down while they plucked flesh from their living bodies and gobbled it (as if they could assimilate the alien protein)... a hundred grisly details as sharply remembered as the events of a minute ago, ridiculously overdone and logically absurd. But while my conscious mind was rejecting the silliness, somewhere much deeper, down in that sleeping animal where we keep our real motives and morals, something was thirsting for alien blood, secure in the conviction that the noblest thing a man could do would be to die killing one of those horrible monsters...\n\nI knew it was all purest soyashit, and I hated the men who had taken such obscene liberties with my mind, but I could even _hear_ my teeth grinding, feel my cheeks frozen in a spastic grin, blood-lust... A teddy bear walked in front of me, looking dazed. I started to raise my laser-finger, but somebody beat me to it and the creature's head exploded in a cloud of gray splinters and blood.\n\nLucky groaned, half-whining. 'Dirty... filthy fucken bastards.' Lasers flared and crisscrossed, and all of the teddy bears fell dead.\n\n' _Watch_ it, goddammit,' Cortez screamed. ' _Aim_ those fuckin things \u2014 they aren't toys!'\n\n'Team _A_ , move out \u2014 into the craters to cover _B_.'\n\nSomebody was laughing and sobbing. 'What the fuck is wrong with _you_ , Petrov?' Strange to hear Cortez cussing.\n\nI twisted around and saw Petrov, behind and to my left, lying in a shallow hole, digging frantically with both hands, crying and gurgling.\n\n'Fuck,' Cortez said. 'Team _B_! Ten meters past the craters, get down in a line. Team _C_ \u2014 into the craters with _A_.'\n\nI scrambled up and covered the hundred meters in twelve amplified strides. The craters were practically large enough to hide a scoutship, some ten meters in diameter. I jumped to the opposite side of the hole and landed next to a fellow named Chin. He didn't even look around when I landed, just kept scanning the base for signs of life.\n\n'Team _A_ \u2014 ten meters, past team _B_ , down in line.' Just as he finished, the building in front of us burped, and a salvo of the bubbles fanned out toward our lines. Most people saw it coming and got down, but Chin was just getting up to make his rush and stepped right into one.\n\nIt grazed the top of his helmet and disappeared with a faint pop. He took one step backwards and toppled over the edge of the crater, trailing an arc of blood and brains. Lifeless, spreadeagled, he slid halfway to the bottom, shoveling dirt into the perfectly symmetrical hole where the bubble had chewed indiscriminately through plastic, hair, skin, bone, and brain.\n\n'Everybody hold it. Platoon leaders, casualty report... check... check, check... check, check, check... check. We have three deaders. Wouldn't be _any_ if you'd have kept low. So everybody grab dirt when you hear that thing go off. Team _A_ , complete the rush.'\n\nThey completed the maneuver without incident. 'OK. Team _C_ , rush to where _B_... hold it! Down!'\n\nEverybody was already hugging the ground. The bubbles slid by in a smooth arc about two meters off the ground. They went serenely over our heads and, except for one that made toothpicks out of a tree, disappeared in the distance.\n\n' _B_ , rush past _A_ ten meters. _C_ , take over _B_ 's place. You _B_ grenadiers, see if you can reach the Flower.'\n\nTwo grenades tore up the ground thirty or forty meters from the structure. In a good imitation of panic, it started belching out a continuous stream of bubbles \u2014 still, none coming lower than two meters off the ground. We kept hunched down and continued to advance.\n\nSuddenly, a seam appeared in the building and widened to the size of a large door. Taurans came swarming out.\n\n'Grenadiers, hold your fire. _B_ team, laser fire to the left and right \u2014 keep'm bunched up. _A_ and _C_ , rush down the center.'\n\nOne Tauran died trying to run through a laser beam. The others stayed where they were.\n\nIn a suit, it's pretty awkward to run and keep your head down at the same time. You have to go from side to side, like a skater getting started; otherwise you'll be airborne. At least one person, somebody in A team, bounced too high and suffered the same fate as Chin.\n\nI was feeling pretty fenced-in and trapped, with a wall of laser fire on each side and a low ceiling that meant death to touch. But in spite of myself, I felt happy, euphoric, finally getting the chance to kill some of those villainous baby-eaters. Knowing it was soyashit.\n\nThey weren't fighting back, except for the rather ineffective bubbles (obviously not designed as an anti-personnel weapon), and they didn't retreat back into the building, either. They milled around, about a hundred of them, and watched us get closer. A couple of grenades would caulk them all, but I guess Cortez was thinking about the prisoner.\n\n'OK, when I say \"go,\" we're going to flank 'em. _B_ team will hold fire... second and fourth platoons to the right, sixth and seventh to the left. _B_ team will move forward in line to box them in.\n\n'Go!' We peeled off to the left. As soon as the lasers stopped, the Taurans bolted, running in a group on a collision course with our flank.\n\n'A team, down and fire! Don't shoot until you're sure of your aim \u2014 if you miss you might hit a friendly. And fer Chris' sake save me one!'\n\nIt was a horrifying sight, that herd of monsters bearing down on us. They were running in great leaps \u2014 the bubbles avoiding them \u2014 and they all looked like the one we saw earlier, riding the broomstick; naked except for an almost transparent sphere around the whole bodies, that moved along with them. The right flank started firing, picking off individuals in the rear of the pack.\n\nSuddenly a laser flared through the Taurans from the other side, somebody missing his mark. There was a horrible scream, and I looked down the line to see someone \u2014 I think it was Perry \u2014 writhing on the ground, right hand over the smoldering stump of his arm, seared off just below the elbow. Blood sprayed through his fingers, and the suit, its camouflage circuits scrambled, flickered black-white-jungle-desert-green-gray. I don't know how long I stared \u2014 long enough for the medic to run over and start giving aid \u2014 but when I looked up the Taurans were almost on top of me.\n\nMy first shot was wild and high, but it grazed the top of the leading Tauran's protective bubble. The bubble disappeared and the monster stumbled and fell to the ground, jerking spasmodically. Foam gushed out of his mouth-hole, first white, then streaked red. With one last jerk he became rigid and twisted backwards, almost to the shape of a horseshoe. His long scream, a high-pitched whistle, stopped just as his comrades trampled over him. I hated myself for smiling.\n\nIt was slaughter, even though our flank was outnumbered five to one. They kept coming without faltering, even when they had to climb over the drift of bodies and parts of bodies that piled up high, parallel to our flank. The ground between us was slick red with Tauran blood \u2014 all God's children got hemoglobin \u2014 and like the teddy bears, their guts looked pretty much like guts to my untrained eye. My helmet reverberated with hysterical laughter while we slashed them to gory chunks, and I almost didn't hear Cortez:\n\n'Hold your fire \u2014 I said HOLD IT, goddammit! _Catch_ a couple of the bastards, they won't hurt you.'\n\nI stopped shooting and eventually so did everybody else. When the next Tauran jumped over the smoking pile of meat in front of me, I dove to try to tackle him around those spindly legs.\n\nIt was like hugging a big, slippery balloon. When I tried to drag him down, he popped out of my arms and kept running.\n\nWe managed to stop one of them by the simple expedient of piling half-a-dozen people on top of him. By that time the others had run through our line and were headed for the row of large cylindrical tanks that Cortez had said were probably for storage. A little door had opened in the base of each one.\n\n'We've _got_ our prisoner,' Cortez shouted. ' _Kill_!'\n\nThey were fifty meters away and running hard, difficult targets. Lasers slashed around them, bobbing high and low. One fell, sliced in two, but the others, about ten of them, kept going and were almost to the doors when the grenadiers started firing.\n\nThey were still loaded with 500-mike bombs, but a near miss wasn't enough \u2014 the concussion would just send them flying, unhurt in their bubbles.\n\n'The buildings! Get the fucken buildings!' The grenadiers raised their aim and let fly, but the bombs only seemed to scorch the white outside of the structures until, by chance, one landed in a door. That split the building just as if it had a seam; the two halves popped away and a cloud of machinery flew into the air, accompanied by a huge pale flame that rolled up and disappeared in an instant. Then the others all concentrated on the doors, except for potshots at some of the Taurans, not so much to get them as to blow them away before they could get inside. They seemed awfully eager.\n\nAll this time, we were trying to get the Taurans with laser fire, while they weaved and bounced around trying to get into the structures. We moved in as close to them as we could without putting ourselves in danger from the grenade blasts, yet too far away for good aim.\n\nStill, we were getting them one by one and managed to destroy four of the seven buildings. Then, when there were only two aliens left, a nearby grenade blast flung one of them to within a few meters of a door. He dove in and several grenadiers fired salvos after him, but they all fell short or detonated harmlessly on the side. Bombs were falling all around, making an awful racket, but the sound was suddenly drowned out by a great sigh, like a giant's intake of breath, and where the building had been was a thick cylindrical cloud of smoke, solid-looking, dwindling away into the stratosphere, straight as if laid down by a ruler. The other Tauran had been right at the base of the cylinder; I could see pieces of him flying. A second later, a shock wave hit us and I rolled helplessly, pinwheeling, to smash into the pile of Tauran bodies and roll beyond.\n\nI picked myself up and panicked for a second when I saw there was blood all over my suit \u2014 when I realized it was only alien blood, I relaxed but felt unclean.\n\n' _Catch_ the bastard! Catch him!' In the confusion, the Tauran had gotten free and was running for the grass. One platoon was chasing after him, losing ground, but then all of _B_ team ran over and cut him off. I jogged over to join in the fun.\n\nThere were four people on top of him, and a ring around them of about fifty people, watching the struggle.\n\n'Spread out, dammit! There might be a thousand more of them waiting to get us in one place.' We dispersed, grumbling. By unspoken agreement we were all sure that there were no more live Taurans on the face of the planet.\n\nCortez was walking toward the prisoner while I backed away. Suddenly the four men collapsed in a pile on top of the creature... Even from my distance I could see the foam spouting from his mouth-hole. His bubble had popped. Suicide.\n\n'Damn!' Cortez was right there. 'Get off that bastard.' The four men got off and Cortez used his laser to slice the monster into a dozen quivering chunks. Heart-warming sight.\n\n'That's all right, though, we'll find another one \u2014 everybody! Back in the arrowhead formation. Combat assault, on the Flower.'\n\nWell, we assaulted the Flower, which had evidently run out of ammunition (it was still belching, but no bubbles), and it was empty. We scurried up ramps and through corridors, fingers at the ready, like kids playing soldier. There was nobody home.\n\nThe same lack of response at the antenna installation, the 'Salami,' and twenty other major buildings, as well as the forty-four perimeter huts still intact. So we had 'captured' dozens of buildings, mostly of incomprehensible purpose, but failed in our main mission, capturing a Tauran for the xenologists to experiment with. Oh well, they could have all the bits and pieces they'd ever want. That was something.\n\nAfter we'd combed every last square centimeter of the base, a scout-ship came in with the real exploration crew, the scientists. Cortez said, 'All right, snap out of it,' and the hypnotic compulsion fell away.\n\nAt first it was pretty grim. A lot of the people, like Lucky and Marygay, almost went crazy with the memories of bloody murder multiplied a hundred times. Cortez ordered everybody to take a sed-tab, two for the ones most upset. I took two without being specifically ordered to do so.\n\nBecause it _was_ murder, unadorned butchery \u2014 once we had the anti-spacecraft weapon doped out, we hadn't been in any danger. The Taurans hadn't seemed to have any conception of person-to-person fighting. We had just herded them up and slaughtered them, the first encounter between mankind and another intelligent species. Maybe it was the second encounter, counting the teddy bears. What might have happened if we had sat down and tried to communicate? But they got the same treatment.\n\nI spent a long time after that telling myself over and over that it hadn't been _me_ who so gleefully carved up those frightened, stampeding creatures. Back in the twentieth century, they had established to everybody's satisfaction that 'I was just following orders' was an inadequate excuse for inhuman contact... but what can you do when the orders come from deep down in that puppet master of the unconscious?\n\nWorst of all was the feeling that perhaps my actions weren't all that inhuman. Ancestors only a few generations back would have done the same thing, even to their fellow men, without any hypnotic conditioning.\n\nI was disgusted with the human race, disgusted with the army and horrified at the prospect of living with myself for another century or so... Well, there was always brainwipe.\n\nA ship with a lone Tauran survivor had escaped and had gotten away clean, the bulk of the planet shielding it from _Earth's Hope_ while it dropped into Aleph's collapsar field. Escaped home, I guessed, wherever that was, to report what twenty men with hand-weapons could do to a hundred fleeing on foot, unarmed.\n\nI suspected that the next time humans met Taurans in ground combat, we would be more evenly matched. And I was right.\n\n# Sergeant   \nMandella   \n2007\u20142024 AD\n\n# 1\n\nI was scared enough.\n\nSub-major Stott was pacing back and forth behind the small podium in the assembly room\/chop hall\/gymnasium of the _Anniversary_. We had just made our final collapsar jump, from Tet-38 to Yod-4. We were decelerating at 1\u00bd gravities and our velocity relative to that collapsar was a respectable .90c. We were being chased.\n\n'I wish you people would relax for a while and just trust the ship's computer. The Tauran vessel at any rate will not be within strike range for another two weeks. Mandella!'\n\nHe was always very careful to call me 'Sergeant' Mandella in front of the company. But everybody at this particular briefing was either a sergeant or a corporal: squad leaders. 'Yes, sir.'\n\n'You're responsible for the psychological as well as the physical wellbeing of the men and women in your squad. Assuming that you are aware that there is a morale problem aboard this vessel, what have you done about it?'\n\n'As far as my squad is concerned, sir?'\n\n'Of course.'\n\n'We talk it out, sir.'\n\n'And have you arrived at any cogent conclusion?'\n\n'Meaning no disrespect, sir, I think the major problem is obvious. My people have been cooped up in this ship for fourteen\u2014'\n\n'Ridiculous! Every one of us has been adequately conditioned against the pressures of living in close quarters _and_ the enlisted people have the privilege of confraternity.' That was a delicate way of putting it. 'Officers must remain celibate, and yet _we_ have no morale problem.'\n\nIf he thought his officers were celibate, he should sit down and have a long talk with Lieutenant Harmony. Maybe he just meant line officers, though. That would be just him and Cortez. Probably 50 percent right. Cortez was awfully friendly with Corporal Kamehameha.\n\n'Sir, perhaps it was the detoxification back at Stargate; maybe\u2014'\n\n'No. The therapists only worked to erase the hate conditioning \u2014 everybody knows how _I_ feel about that \u2014 and they may be misguided but they are skilled.\n\n'Corporal Potter.' He always called her by her rank to remind her why she hadn't been promoted as high as the rest of us. Too soft. 'Have you \"talked it out\" with your people, too?'\n\n'We've discussed it, sir.'\n\nThe sub-major could 'glare mildly' at people. He glared mildly at Marygay until she elaborated.\n\n'I don't believe it's the fault of the conditioning. My people are impatient, just tired of doing the same thing day after day.'\n\n'They're anxious for combat, then?' No sarcasm in his voice.\n\n'They want to get off the ship, sir.'\n\n'They _will_ get off the ship,' he said, allowing himself a microscopic smile. 'And then they'll probably be just as impatient to get back on.'\n\nIt went back and forth like that for a long while. Nobody wanted to come right out and say that their squad was scared: scared of the Tauran cruiser closing on us, scared of the landing on the portal planet. Sub-major Stott had a bad record of dealing with people who admitted fear.\n\nI fingered the fresh T\/O they had given us. It looked like this:\n\nI knew most of the people from the raid on Aleph, the first face-to-face contact between humans and Taurans. The only new people in my platoon were Luthuli and Heyrovsky. In the company as a whole (excuse me, the 'strike force'), we had twenty replacements for the nineteen people we lost from the Aleph raid: one amputation, four deaders, fourteen psychotics.\n\nI couldn't get over the '20 Mar 2007' at the bottom of the T\/O. I'd been in the army ten years, though it felt like less than two. Time dilation, of course; even with the collapsar jumps, traveling from star to star eats up the calendar.\n\nAfter this raid, I would probably be eligible for retirement, with full pay. If I lived through the raid, and if they didn't change the rules on us. Me a twenty-year man, and only twenty-five years old.\n\nStott was summing up when there was a knock on the door, a single loud rap. 'Enter,' he said.\n\nAn ensign I knew vaguely walked in casually and handed Stott a slip of paper, without saying a word. He stood there while Stott read it, slumping with just the right degree of insolence. Technically, Stott was out of his chain of command; everybody in the navy disliked him anyhow.\n\nStott handed the paper back to the ensign and looked through him.\n\n'You will alert your squads that preliminary evasive maneuvers will commence at 2010, fifty-eight minutes from now.' He hadn't looked at his watch. 'All personnel will be in acceleration shells by 2000. Tench... hut!'\n\nWe rose and, without enthusiasm, chorused, 'Fuck you, sir.' Idiotic custom.\n\nStott strode out of the room and the ensign followed, smirking.\n\nI turned my ring to my assistant squad leader's position and talked into it: 'Tate, this is Mandella.' Everyone else in the room was doing the same.\n\nA tinny voice came out of the ring. 'Tate here. What's up?'\n\n'Get ahold of the men and tell them we have to be in the shells by 2000. Evasive maneuvers.'\n\n'Crap. They told us it would be days.'\n\n'I guess something new came up. Or maybe the Commodore has a bright idea.'\n\n'The Commodore can stuff it. You up in the lounge?'\n\n'Yeah.'\n\n'Bring me back a cup when you come, OK? Little sugar?'\n\n'Roger. Be down in about half an hour.'\n\n'Thanks. I'll get on it.'\n\nThere was a general movement toward the coffee machine. I got in line behind Corporal Potter.\n\n'What do you think, Marygay?'\n\n'Maybe the Commodore just wants us to try out the shells once more.'\n\n'Before the real thing.'\n\n'Maybe.' She picked up a cup and blew into it. She looked worried. 'Or maybe the Taurans had a ship way out, waiting for us. I've wondered why they don't do it. We do, at Stargate.'\n\n'Stargate's a different thing. It takes seven cruisers, moving all the time, to cover all the possible exit angles. We can't afford to do it for more than one collapsar, and neither could they.'\n\nShe didn't say anything while she filled her cup. 'Maybe we've stumbled on their version of Stargate. Or maybe they have more ships than we do by now.'\n\nI filled and sugared two cups, sealed one. 'No way to tell.' We walked back to a table, careful with the cups in the high gravity.\n\n'Maybe Singhe knows something,' she said.\n\n'Maybe he does. But I'd have to get him through Rogers and Cortez. Cortez would jump down my throat if I tried to bother him now.'\n\n'Oh, I can get him directly. We...' She dimpled a little bit. 'We've been friends.'\n\nI sipped some scalding coffee and tried to sound nonchalant. 'So that's where you've been disappearing to.'\n\n'You disapprove?' she said, looking innocent.\n\n'Well... damn it, no, of course not. But \u2014 but he's an officer! A _navy_ officer!'\n\n'He's attached to us and that makes him part army.' She twisted her ring and said, 'Directory.' To me: 'What about you and Little Miss Harmony?'\n\n'That's not the same thing.' She was whispering a directory code into the ring.\n\n'Yes, it is. You just wanted to do it with an officer. Pervert.' The ring bleated twice. Busy. 'How was she?'\n\n'Adequate.' I was recovering.\n\n'Besides, Ensign Singhe is a perfect gentleman. And not the least bit jealous.'\n\n'Neither am I,' I said. 'If he ever hurts you, tell me and I'll break his ass.'\n\nShe looked at me across her cup. 'If Lieutenant Harmony ever hurts you, tell me and I'll break _her_ ass.'\n\n'It's a deal.' We shook on it solemnly.\n\n# 2\n\nThe acceleration shells were something new, installed while we rested and resupplied at Stargate. They enabled us to use the ship at closer to its theoretical efficiency, the tachyon drive boosting it to as much as 25 gravities.\n\nTate was waiting for me in the shell area. The rest of the squad was milling around, talking. I gave him his coffee.\n\n'Thanks. Find out anything?'\n\n'Afraid not. Except the swabbies don't seem to be scared, and it's their show. Probably just another practice run.'\n\nHe slurped some coffee. 'What the hell. It's all the same to us, anyhow. Just sit there and get squeezed half to death. God, I hate those things.'\n\n'Maybe they'll eventually make us obsolete, and we can go home.'\n\n'Sure thing.' The medic came by and gave me my shot.\n\nI waited until 1950 and hollered to the squad. 'Let's go. Strip down and zip up.'\n\nThe shell is like a flexible spacesuit; at least the fittings on the inside are pretty similar. But instead of a life-support package, there's a hose going into the top of the helmet and two coming out of the heels, as well as two relief tubes per suit. They're crammed in shoulder-to-shoulder on light acceleration couches; getting to your shell is like picking your way through a giant plate of olive drab spaghetti.\n\nWhen the lights in my helmet showed that everybody was suited up, I pushed the button that flooded the room. No way to see, of course, but I could imagine the pale blue solution \u2014 ethylene glycol and something else \u2014 foaming up around and over us. The suit material, cool and dry, collapsed in to touch my skin at every point. I knew that my internal body pressure was increasing rapidly to match the increasing fluid pressure outside. That's what the shot was for; keep your cells from getting squished between the devil and the deep blue sea. You could still feel it, though. By the time my meter said '2' (external pressure equivalent to a column of water two nautical miles deep), I felt that I was at the same time being crushed and bloated. By 2005 it was at 2.7 and holding steady. When the maneuvers began at 2010, you couldn't feel the difference. I thought I saw the needle fluctuate a tiny bit, though.\n\nThe major drawback to the system is that, of course, anybody caught outside of his shell when the _Anniversary_ hit 25 GS would be just so much strawberry jam. So the guiding and the fighting have to be done by the ship's tactical computer \u2014 which does most of it anyway, but it's nice to have a human overseer.\n\nAnother small problem is that if the ship gets damaged and the pressure drops, you'll explode like a dropped melon. If it's the internal pressure, you get crushed to death in a microsecond.\n\nAnd it takes ten minutes, more or less, to get depressurized and another two or three to get untangled and dressed. So it's not exactly something you can hop out of and come up fighting.\n\nThe accelerating was over at 2038. A green light went on and I chinned the button to depressurize.\n\nMarygay and I were getting dressed outside.\n\n'How'd that happen?' I pointed to an angry purple welt that ran from the bottom of her right breast to her hipbone.\n\n'That's the second time,' she said, mad. 'The first one was on my back \u2014 I think that shell doesn't fit right, gets creases.'\n\n'Maybe you've lost weight.'\n\n'Wise guy.' Our caloric intake had been rigorously monitored ever since we left Stargate the first time. You can't use a fighting suit unless it fits you like a second skin.\n\nA wall speaker drowned out the rest of her comment. 'Attention all personnel. Attention. All army personnel echelon six and above and all navy personnel echelon four and above will report to the briefing room at 2130.'\n\nIt repeated the message twice. I went off to lie down for a few minutes while Marygay showed her bruise to the medic and the armorer. I didn't feel a bit jealous.\n\nThe Commodore began the briefing. 'There's not much to tell, and what there is is not good news.\n\n'Six days ago, the Tauran vessel that is pursuing us released a drone missile. Its initial acceleration was on the order of 80 gravities.\n\n'After blasting for approximately a day, its acceleration suddenly jumped to 148 gravities.' Collective gasp.\n\n'Yesterday, it jumped to 203 gravities. I shouldn't need to remind anyone here that this is twice the accelerative capability of the enemy's drones in our last encounter.\n\n'We launched a salvo of drones, four of them, intersecting what the computer predicted to be the four most probably future trajectories of the enemy drone. One of them paid off, while we were doing evasive maneuvers. We contacted and destroyed the Tauran weapon about ten million kilometers from here.'\n\nThat was practically next door. 'The only encouraging thing we learned from the encounter was from spectral analysis of the blast. It was no more powerful an explosion than ones we have observed in the past, so at least their progress in propulsion hasn't been matched by progress in explosives.\n\n'This is the first manifestation of a very important effect that has heretofore been of interest only to theorists. Tell me, soldier.' He pointed at Negulesco. 'How long has it been since we first fought the Taurans, at Aleph?'\n\n'That depends on your frame of reference, Commodore,' she answered dutifully. 'To me, it's been about eight months.'\n\n'Exactly. You've lost about nine years, though, to time dilation, while we maneuvered between collapsar jumps. In an engineering sense, as we haven't done any important research and development aboard ship... that enemy vessel comes from our future!' He paused to let that sink in.\n\n'As the war progresses, this can only become more and more pronounced. The Taurans don't have any cure for relativity, of course, so it will be to our benefit as often as to theirs.\n\n'For the present, though, it is _we_ who are operating with a handicap. As the Tauran pursuit vessel draws closer, this handicap will become more severe. They can simply outshoot us.\n\n'We're going to have to do some fancy dodging. When we get within five hundred million kilometers of the enemy ship, everybody gets in his shell and we just have to trust the logistic computer. It will put us through a rapid series of random changes in direction and velocity.\n\n'I'll be blunt. As long as they have one more drone than we, they can finish us off. They haven't launched any more since that first one. Perhaps they are holding their fire... or maybe they only had one. In that case, it's we who have them.\n\n'At any rate, all personnel will be required to be in their shells with no more than ten minutes' notice. When we get within a thousand million kilometers of the enemy, you are to stand _by_ your shells. By the time we are within five hundred million kilometers, you will be in them, and all shell compounds flooded and pressurized. We cannot wait for anyone.\n\n'That's all I have to say. Sub-major?'\n\n'I'll speak to my people later, Commodore. Thank you.'\n\n'Dismissed.' And none of this 'fuck you, sir' nonsense. The navy thought that was just a little beneath their dignity. We stood at attention \u2014 all except Stott \u2014 until he had left the room. Then some other swabbie said 'dismissed' again, and we left.\n\nMy squad had clean-up detail, so I told everybody who was to do what, put Tate in charge, and left. Went up to the NCO room for some company and maybe some information.\n\nThere wasn't much happening but idle speculation, so I took Rogers and went off to bed. Marygay had disappeared again, hopefully trying to wheedle something out of Singhe. \n\n# 3\n\nWe had our promised get-together with the sub-major the next morning, when he more or less repeated what the commodore had said, in infantry terms and in his staccato monotone. He emphasized the fact that all we knew about the Tauran ground forces was that if their naval capability was improved, it was likely they would be able to handle us better than last time.\n\nBut that brings up an interesting point. Eight months or nine years before, we'd had a tremendous advantage: they had seemed not quite to understand what was going on. As belligerent as they had been in space, we'd expected them to be real Huns on the ground. Instead, they practically lined themselves up for slaughter. One escaped and presumably described the idea of old-fashioned in-fighting to his fellows.\n\nBut that, of course, didn't mean that the word had necessarily gotten to this particular bunch, the Taurans guarding Yod-4. The only way we know of to communicate faster than the speed of light is to physically carry a message through successive collapsar jumps. And there was no way of telling how many jumps there were between Yod-4 and the Tauran home base \u2014 so these might be just as passive as the last bunch, or might have been practicing infantry tactics for most of a decade. We would find out when we got there.\n\nThe armorer and I were helping my squad pull maintenance on their fighting suits when we passed the thousand million kilometer mark and had to go up to the shells.\n\nWe had about five hours to kill before we had to get into our cocoons. I played a game of chess with Rabi and lost. Then Rogers led the platoon in some vigorous calisthenics, probably for no other reason than to get their minds off the prospect of having to lie half-crushed in the shells for at least four hours. The longest we'd gone before was half that.\n\nTen minutes before the five hundred million kilometer mark, we squad leaders took over and supervised buttoning everybody up. In eight minutes we were zipped and flooded and at the mercy of \u2014 or safe in the arms of \u2014 the logistic computer.\n\nWhile I was lying there being squeezed, a silly thought took hold of my brain and went round and round like a charge in a superconductor: according to military formalism, the conduct of war divides neatly into two categories, tactics and logistics. Logistics has to do with moving troops and feeding them and just about everything except the actual fighting, which is tactics. And now we're fighting, but we don't have a _tactical_ computer to guide us through attack and defense, just a huge, super-efficient pacifistic cybernetic grocery clerk of a logistic, mark that word, _logistic_ computer.\n\nThe other side of my brain, perhaps not quite as pinched, would argue that it doesn't matter what name you give to a computer, it's a pile of memory crystals, logic banks, nuts and bolts... If you program it to be Genghis Khan, it is a tactical computer, even if its usual function is to monitor the stock market or control sewage conversion.\n\nBut the other voice was obdurate and said by that kind of reasoning, a man is only a hank of hair and a piece of bone and some stringy meat; and no matter what kind of a man he is, if you teach him well, you can take a Zen monk and turn him into a slavering bloodthirsty warrior.\n\nThen what the hell are you, we, am I, answered the other side. A peace-loving, vacuum-welding specialist _cum_ physics teacher snatched up by the Elite Conscription Act and reprogrammed to be a killing machine. You, I have killed and liked it.\n\nBut that was hypnotism, motivational conditioning, I argued back at myself. They don't do that anymore.\n\nAnd the only reason, I said, they don't do it is that they think you'll kill better without it. That's logic.\n\nSpeaking of logic, the original question was, why do they send a logistic computer to do a man's job? Or something like that... and we were off again.\n\nThe light blinked green and I chinned the switch automatically. The pressure was down to 1.3 before I realized that it meant we were alive, we had won the first skirmish.\n\nI was only partly right.\n\n# 4\n\nI was belting on my tunic when my ring tingled and I held it up to listen. It was Rogers.\n\n'Mandella, go check squad bay 3. Something went wrong; Dalton had to depressurize it from Control.'\n\nBay 3 \u2014 that was Marygay's squad! I rushed down the corridor in bare feet and got there just as they opened the door from inside the pressure chamber and began straggling out.\n\nThe first out was Bergman. I grabbed his arm. 'What the hell is going on, Bergman?'\n\n'Huh?' He peered at me, still dazed, as everyone is when they come out of the chamber. 'Oh, s'you, Mandella. I dunno. Whad'ya mean?'\n\nI squinted in through the door, still holding on to him. 'You were late, man, you depressurized late. What happened?'\n\nHe shook his head, trying to clear it. 'Late? Whad' late. Uh, how late?'\n\nI looked at my watch for the first time. 'Not too\u2014' Jesus Christ. 'Uh, we zipped in at 0520, didn't we?'\n\n'Yeah, I think that's it.'\n\nStill no Marygay among the dim figures picking their way through the ranked couches and jumbled tubing. 'Um, you were only a couple of minutes late... but we were only supposed to be under for four hours, maybe less. It's 1050.'\n\n'Um.' He shook his head again. I let go of him and stood back to let Stiller and Demy through the door.\n\n'Everybody's late, then,' Bergman said. 'So we aren't in any trouble.'\n\n'Uh\u2014' Non sequiturs. 'Right, right \u2014 Hey, Stiller! You seen\u2014'\n\nFrom inside: 'Medic! MEDIC!'\n\nSomebody who wasn't Marygay was coming out. I pushed her roughly out of my way and dove through the door, landed on somebody else and clambered over to where Struve, Marygay's assistant, was standing over a pod and talking very loud and fast into his ring.\n\n'\u2014and blood God yes we need\u2014'\n\nIt was Marygay still lying in her suit she was\n\n'\u2014got the word from Dalton\u2014'\n\ncovered every square inch of her with a uniform bright sheen of blood\n\n'\u2014when she didn't come out\u2014'\n\nit started as an angry welt up by her collarbone and was just a welt as it traveled between her breasts until it passed the sternum's support\n\n'\u2014I came over and popped the\u2014'\n\nand opened up into a cut that got deeper as it ran down over her belly and where it stopped\n\n'\u2014yeah, she's still\u2014'\n\na few centimeters above the pubis a membraned loop of gut was protruding\n\n'\u2014OK, left hip. Mandella\u2014'\n\nShe was still alive, her heart palpitating, but her blood-streaked head lolled limply, eyes rolled back to white slits, bubbles of red froth appearing and popping at the corner of her mouth each time she exhaled shallowly.\n\n'\u2014tattooed on her left hip. Mandella! Snap out of it! Reach under her and find out what her blood\u2014'\n\n'TYPE O RH NEGATIVE GOD damn... it. Sorry\u2014Oh negative.' Hadn't I seen that tattoo ten thousand times?\n\nStruve passed this information on and I suddenly remembered the first-aid kit on my belt, snapped it off and fumbled through it.\n\n_Stop the bleeding \u2014 protect the wound \u2014 treat for shock_ , that's what the book said. Forgot one, forgot one... _clear air passages_.\n\nShe was breathing, if that's what they meant. How do you stop the bleeding or protect the wound with one measly pressure bandage when the wound is nearly a meter long? Treat for shock, that I could do. I fished out the green ampoule, laid it against her arm and pushed the button. Then I laid the sterile side of the bandage gently on top of the exposed intestine and passed the elastic strip under the small of her back, adjusted it for nearly zero tension and fastened it.\n\n'Anything else you can do?' Struve asked.\n\nI stood back and felt helpless. 'I don't know. Can you think of anything?'\n\n'I'm no more of a medic than you are.' Looking up at the door, he kneaded a fist, biceps straining. 'Where the hell are they? You have morph-plex in that kit?'\n\n'Yeah, but somebody told me not to use it for internal\u2014'\n\n'William?'\n\nHer eyes were open and she was trying to lift her head. I rushed over and held her. 'It'll be all right, Marygay. The medic's coming.'\n\n'What... all right? I'm thirsty. Water.'\n\n'No, honey, you can't have any water. Not for a while, anyhow.' Not if she was headed for surgery.\n\n'Why is all the blood?' she said in a small voice. Her head rolled back. 'Been a bad girl.'\n\n'It must have been the suit,' I said rapidly. 'Remember earlier, the creases?'\n\nShe shook her head. 'Suit?' She turned suddenly paler and retched weakly. 'Water... William, please.'\n\nAuthoritative voice behind me: 'Get a sponge or a cloth soaked in water.' I looked around the saw Doc Wilson with two stretcher bearers.\n\n'First half-liter femoral,' he said to no one in particular as he carefully peeked under the pressure bandage. 'Follow that relief tube down a couple of meters and pinch it off. Find out if she's passed any blood.'\n\nOne of the medics ran a ten-centimeter needle into Marygay's thigh and started giving her whole blood from a plastic bag.\n\n'Sorry I'm late,' Doc Wilson said tiredly. 'Business is booming. What'd you say about the suit?'\n\n'She had two minor injuries before. Suit doesn't fit quite right, creases up under pressure.'\n\nHe nodded absently, checking her blood pressure. 'You, anybody, give\u2014' Somebody handed him a paper towel dripping water. 'Uh, give her any medication?'\n\n'One ampoule of No-shock.'\n\nHe wadded the paper towel up loosely and put it in Marygay's hand. 'What's her name?' I told him.\n\n'Marygay, we can't give you a drink of water but you can suck on this. Now I'm going to shine a bright light in your eye.' While he was looking through her pupil with a metal tube, he said, 'Temperature?' and one of the medics read a number from a digital readout box and withdrew a probe. 'Passed blood?'\n\n'Yes. Some.'\n\nHe put his hand lightly on the pressure bandage. 'Marygay, can you roll over a little on your right side?'\n\n'Yes,' she said slowly, and put her elbow down for leverage. 'No,' she said and started crying.\n\n'Now, now,' he said absently and pushed up on her hip just enough to be able to see her back. 'Only the one wound,' he muttered. 'Hell of a lot of blood.'\n\nHe pressed the side of his ring twice and shook it by his ear. 'Anybody up in the shop?'\n\n'Harrison, unless he's on a call.'\n\nA woman walked up, and at first I didn't recognize her, pale and disheveled, bloodstained tunic. It was Estelle Harmony.\n\nDoc Wilson looked up. 'Any new customers, Doctor Harmony?'\n\n'No,' she said dully. 'The maintenance man was a double traumatic amputation. Only lived a few minutes. We're keeping him running for transplants.'\n\n'All those others?'\n\n'Explosive decompression.' She sniffed. 'Anything I can do here?'\n\n'Yeah, just a minute.' He tried his ring again. 'God damn it. You don't know where Harrison is?'\n\n'No... well, maybe, he might be in Surgery _B_ if there was trouble with the cadaver maintenance. Think I set it up all right, though.'\n\n'Yeah, well, hell you know how...'\n\n'Mark!' said the medic with the blood bag.\n\n'One more half-liter femoral,' Doc Wilson said. 'Estelle, you mind taking over for one of the medics here, prepare this gal for surgery?'\n\n'No, keep me busy.'\n\n'Good \u2014 Hopkins, go up to the shop and bring down a roller and a liter, uh, two liters isotonic fluorocarb with the primary spectrum. If they're Merck they'll say \"abdominal spectrum.\"' He found a part of his sleeve with no blood on it and wiped his forehead. 'If you find Harrison, send him over to surgery A and have him set up the anesthetic sequence for abdominal.'\n\n'And bring her up to _A_?'\n\n'Right. If you can't find Harrison, get somebody\u2014' he stabbed a finger in my direction, '\u2014this guy, to roll the patient up to _A_ ; you run ahead and start the sequence.'\n\nHe picked up his bag and looked through it. 'We could start the sequence here,' he muttered. 'But hell, not with paramethadone \u2014 Marygay? How do you feel?'\n\nShe was still crying. 'I'm... hurt.'\n\n'I know,' he said gently. He thought for a second and said to Estelle, 'No way to tell really how much blood she lost. She may have been passing it under pressure. Also there's some pooling in the abdominal cavity. Since she's still alive I don't think she could've bled under pressure for very long. Hope no brain damage yet.'\n\nHe touched the digital readout attached to Marygay's arm. 'Monitor the blood pressure, and if you think it's indicated, give her five cc's vasoconstrictor. I've gotta go scrub down.'\n\nHe closed his bag. 'You have any vasoconstrictor besides the pneumatic ampoule?'\n\nEstelle checked her own bag. 'No, just the emergency pneumatic... uh... yes, I've got controlled dosage on the 'dilator, though.'\n\n'OK, if you have to use the 'constrictor and her pressure goes up too fast\u2014'\n\n'I'll give her vasodilator two cc's at a time.'\n\n'Check. Hell of a way to run things, but... well. If you're not too tired, I'd like you to stand by me upstairs.'\n\n'Sure.' Doc Wilson nodded and left.\n\nEstelle began sponging Marygay's belly with isopropyl alcohol. It smelled cold and clean. 'Somebody gave her No-shock?'\n\n'Yes,' I said, 'about ten minutes ago.'\n\n'Ah. That's why the Doc was worried \u2014 no, you did the right thing. But No-shock's got some vasoconstrictor. Five cc's more might run up an overdose.' She continued silently scrubbing, her eyes coming up every few seconds to check the blood pressure monitor.\n\n'William?' It was the first time she'd shown any sign of knowing me. 'This wom\u2014, uh, Marygay, she's your lover? Your regular lover?'\n\n'That's right.'\n\n'She's very pretty.' A remarkable observation, her body torn and caked with crusting blood, her face smeared where I had tried to wipe away the tears. I suppose a doctor or a woman or a lover can look beneath that and see beauty.\n\n'Yes, she is.' She had stopped crying and had her eyes squeezed shut, sucking the last bit of moisture from the paper wad.\n\n'Can she have some more water?'\n\n'OK, same as before. Not too much.'\n\nI went out to the locker alcove and into the head for a paper towel. Now that the fumes from the pressurizing fluid had cleared, I could smell the air. It smelled wrong. Light machine oil and burnt metal, like the smell of a metal-working shop. I wondered whether they had overloaded the airco. That had happened once before, after the first time we'd used the acceleration chambers.\n\nMarygay took the water without opening her eyes.\n\n'Do you plan to stay together when you get back to Earth?'\n\n'Probably,' I said. ' _If_ we get back to Earth. Still one more battle.'\n\n'There won't be any more battles,' she said flatly. 'You mean you haven't heard?'\n\n'What?'\n\n'Don't you know the ship was hit?'\n\n'Hit!' Then how could any of us be alive?\n\n'That's right.' She went back to her scrubbing. 'Four squad bays. Also the armor bay. There isn't a fighting suit left on the ship... and we can't fight in our underwear.'\n\n'What \u2014 squad bays, what happened to the people?'\n\n'No survivors.'\n\nThirty people. 'Who was it?'\n\n'All of the third platoon. First squad of the second platoon.'\n\nAl-Sadat, Busia, Maxwell, Negulesco. 'My God.'\n\n'Thirty deaders, and they don't have the slightest notion of what caused it. Don't know but that it may happen again any minute.'\n\n'It wasn't a drone?'\n\n'No, we got all of their drones. Got the enemy vessel, too. Nothing showed up on any of the sensors, just _blam_! and a third of the ship was torn to hell. We were lucky it wasn't the drive or the life-support system.' I was hardly hearing her. Penworth, LaBatt, Smithers. Christine and Frida. All dead. I was numb.\n\nShe took a blade-type razor and a tube of gel out of her bag. 'Be a gentleman and look the other way,' she said. 'Oh, here.' She soaked a square of gauze in alcohol and handed it to me. 'Be useful. Do her face.'\n\nI started and, without opening her eyes, Marygay said, 'That feels good. What are you doing?'\n\n'Being a gentleman. And useful, too\u2014'\n\n'All personnel, attention, all personnel.' There wasn't a squawk-box in the pressure chamber, but I could hear it clearly through the door to the locker alcove. 'All personnel echelon _6_ and above, unless directly involved in medical or maintenance emergencies, report immediately to the assembly area.'\n\n'I've got to go, Marygay.'\n\nShe didn't say anything. I didn't know whether she had heard the announcement.\n\n'Estelle,' I addressed her directly, gentleman be damned. 'Will you\u2014'\n\n'Yes. I'll let you know as soon as we can tell.'\n\n'Well.'\n\n'It's going to be all right.' But her expression was grim and worried. 'Now get going,' she said, softly.\n\nBy the time I picked my way out into the corridor, the 'box was repeating the message for the fourth time. There was a new smell in the air, that I didn't want to identify. \n\n# 5\n\nHalfway to the assembly area I realized what a mess I was, and ducked into the head by the NCO lounge. Corporal Kamehameha was hurriedly brushing her hair.\n\n'William! What happened to you?'\n\n'Nothing.' I turned on a tap and looked at myself in the mirror. Dried blood smeared all over my face and tunic. 'It was Marygay, Corporal Potter, her suit... well, evidently it got a crease, uh...'\n\n'Dead?'\n\n'No, just badly, uh, she's going into surgery\u2014'\n\n'Don't use hot water. You'll just set the stain.'\n\n'Oh. Right.' I used the hot to wash my face and hand, dabbed at the tunic with cold. 'Your squad's just two bays down from Al's, isn't it?'\n\n'Yes.'\n\n'Did you see what happened?'\n\n'No. Yes. Not _when_ it happened.' For the first time I noticed that she was crying, big tears rolling down her cheeks and off her chin. Her voice was even, controlled. She pulled at her hair savagely. 'It's a mess.'\n\nI stepped over and put my hand on her shoulder. 'DON'T touch me!' she flared and knocked my hand off with the brush. 'Sorry. Let's go.'\n\nAt the door to the head she touched me lightly on the arm. 'William...' She looked at me defiantly. 'I'm just glad it wasn't me. You understand? That's the only way you can look at it.'\n\nI understood, but I didn't know that I believed her.\n\n'I can sum it up very briefly,' the commodore said in a tight voice, 'if only because we know so little.\n\n'Some ten seconds after we destroyed the enemy vessel, two objects, very small objects, struck the _Anniversary_ amidships. By inference, since they were not detected and we know the limits of our detection apparatus, we know that they were moving in excess of nine-tenths of the speed of light. That is to say, more precisely, their velocity vector _normal_ to the axis of the _Anniversary_ was greater than nine-tenths of the speed of light. They slipped in behind the repeller fields.'\n\nWhen the _Anniversary_ is moving at relativistic speeds, it is designed to generate two powerful electromagnetic fields, one centered about five thousand kilometers from the ship and the other about ten thousand klicks away, both in line with the direction of motion of the ship. These fields are maintained by a 'ramjet' effect, energy picked up from interstellar gas as we mosey along.\n\nAnything big enough to worry about hitting (that is, anything big enough to see with a strong magnifying glass) goes through the first field and comes out with a very strong negative charge all over its surface. As it enters the second field, it's repelled away from the path of the ship. If the object is too big to be pushed around this way, we can sense it at a greater distance and maneuver out of its way.\n\n'I shouldn't have to emphasize how formidable a weapon this is. When the _Anniversary_ was struck, our rate of speed with respect to the enemy was such that we traveled our own length every ten-thousandth of a second. Further, we were jerking around erratically with a constantly changing and purely random lateral acceleration. Thus the objects that struck us must have been guided, not aimed. And the guidance system was self-contained, since there were no Taurans alive at the time they struck us. All of this in a package no larger than a small pebble.\n\n'Most of you are too young to remember the term _future shock_. Back in the seventies, some people felt that technological progress was so rapid that people, normal people, couldn't cope with it; that they wouldn't have time to get used to the present before the future was upon them. A man named Toffler coined the term _future shock_ to describe this situation.' The commodore could get pretty academic.\n\n'We're caught up in a physical situation that resembles this scholarly concept. The result has been disaster. Tragedy. And, as we discussed in our last meeting, there is no way to counter it. Relativity traps us in the enemy's past; relativity brings them from our future. We can only hope that next time, the situation will be reversed. And all we can do to help bring that about is try to get back to Stargate, and then to Earth, where specialists may be able to deduce something, some sort of counterweapon, from the nature of the damage.\n\n'Now we could attack the Tauran's portal planet from space and perhaps destroy the base without using you infantry. But I think there would be a very great risk involved. We might be... shot down by whatever hit us today, and never return to Stargate with what I consider to be vital information. We could send a drone with a message detailing our assumptions about this new enemy weapon... but that might be inadequate. And the Force would be that much further behind, technologically.\n\n'Accordingly, we have set a course that will take us around Yod-4, keeping the collapsar as much as possible between us and the Tauran base. We will avoid contact with the enemy and return to Stargate as quickly as possible.'\n\nIncredibly, the commodore sat down and kneaded his temples. 'All of you are at least squad or section leaders. Most of you have good combat records. And I hope that some of you will be rejoining the Force after your two years are up. Those of you who do will probably be made lieutenants, and face your first real command.\n\n'It is to these people I would like to speak for a few moments, not as your... as one of your commanders, but just as a senior officer and advisor.\n\n'One cannot make command decisions simply by assessing the tactical situation and going ahead with whatever course of action will do the most harm to the enemy with a minimum of death and damage to your own men and material. Modern warfare has become very complex, especially during the last century. Wars are won not by a simple series of battles won, but by a complex interrelationship among military victory, economic pressures, logistic maneuvering, access to the enemy's information, political postures \u2014 dozens, literally dozens of factors.'\n\nI was hearing this, but the only thing that was getting through to my brain was that a third of our friends' lives had been snuffed out less than an hour before, and he was sitting up there giving us a lecture on military theory.\n\n'So sometimes you have to throw away a battle in order to help win the war. This is exactly what we are going to do.\n\n'This was not an easy decision. In fact, it was probably the hardest decision of my military career. Because, on the surface at least, it may look like cowardice.\n\n'The logistic computer calculates that we have about a 62 percent chance of success, should we attempt to destroy the enemy base. Unfortunately, we would have only a 30 percent chance of survival \u2014 as some of the scenarios leading to success involve ramming the portal planet with the _Anniversary_ at light speed.' Jesus Christ.\n\n'I hope none of you ever has to face such a decision. When we get back to Stargate, I will in all probability be court-martialed for cowardice under fire. But I honestly believe that the information that may be gained from analysis of the damage to the _Anniversary_ is more important than the destruction of this one Tauran base.' He sat up straight. 'More important than one soldier's career.'\n\nI had to stifle an impulse to laugh. Surely 'cowardice' had nothing to do with his decision. Surely he had nothing so primitive and unmilitary as a will to live.\n\nThe maintenance crew managed to patch up the huge rip in the side of the _Anniversary_ and to repressurize that section. We spent the rest of the day cleaning up the area; without, of course, disturbing any of the precious evidence for which the commodore was willing to sacrifice his career.\n\nThe hardest part was jettisoning the bodies. It wasn't so bad except for the ones whose suits had burst.\n\nI went to Estelle's cabin the next day, as soon as she was off duty.\n\n'It wouldn't serve any good purpose for you to see her now.' Estelle sipped her drink, a mixture of ethyl alcohol, citric acid and water, with a drop of some ester that approximated the aroma of orange rind.\n\n'Is she out of danger?'\n\n'Not for a couple of weeks. Let me explain.' She set down her drink and rested her chin on interlaced fingers. 'This sort of injury would be fairly routine under normal circumstances. Having replaced the lost blood, we'd simply sprinkle some magic powder into her abdominal cavity and paste her back up. Have her hobbling around in a couple of days.\n\n'But there are complications. Nobody's ever been injured in a pressure suit before. So far, nothing really unusual has cropped up. But we want to monitor her innards very closely for the next few days.\n\n'Also, we were very concerned about peritonitis. You know what peritonitis is?'\n\n'Yes.' Well, vaguely.\n\n'Because a part of her intestine had ruptured under pressure. We didn't want to settle for normal prophylaxis because a lot of the, uh, contamination had impacted on the peritoneum under pressure. To play it safe, we completely sterilized the whole shebang, the abdominal cavity and her entire digestive system from the duodenum south. Then of course, we had to replace all of her normal intestinal flora, now dead, with a commercially prepared culture. Still standard procedure, but not normally called for unless the damage is more severe.'\n\n'I see.' And it was making me a little queasy. Doctors don't seem to realize that most of us are perfectly content not having to visualize ourselves as animated bags of skin filled with obscene glop.\n\n'This in itself is enough reason not to see her for a couple of days. The changeover of intestinal flora has a pretty violent effect on the digestive system \u2014 not dangerous, since she's under constant observation. But tiring and, well, embarrassing.\n\n'With all of this, she would be completely out of danger if this were a normal clinical situation. But we're decelerating at a constant 1\u2014\u00bd gees, and her internal organs have gone through a lot of jumbling around. You might as well know that if we do any blasting, anything over about two gees, she's going to die.'\n\n'But... but we're _bound_ to go over two on the final approach! What\u2014'\n\n'I know, I know. But that won't be for a couple of weeks. Hopefully, she will have mended by then.\n\n'William, face it. It's a miracle she survived to get into surgery. So there's a big chance she won't make it back to Earth. It's sad; she's a special person, _the_ special person to you, maybe. But we've had so much death... you ought to be getting used to it, come to terms with it.'\n\nI took a long pull at my drink, identical to hers except for the citric acid. 'You're getting pretty hard-boiled.'\n\n'Maybe... no. Just realistic. I have a feeling we're headed for a lot more death and sorrow.'\n\n'Not me. As soon as we get to Stargate, I'm a civilian.'\n\n'Don't be so sure.' The old familiar argument. 'Those clowns who signed us up for two years can just as easily make it four or\u2014'\n\n'Or six or twenty or the duration. But they won't. It would be mutiny.'\n\n'I don't know. If they could condition us to kill on cue, they can condition us to do almost anything. Re-enlist.'\n\nThat was a chiller.\n\nLater on we tried to make love, but both of us had too much to think about.\n\nI got to see Marygay for the first time about a week later. She was wan, had lost a lot of weight and seemed very confused. Doc Wilson assured me that it was just the medication; they hadn't seen any evidence of brain damage.\n\nShe was still in bed, still being fed through a tube. I began to get very nervous about the calendar. Every day there seemed to be some improvement, but if she was still in bed when we hit that collapsar push, she wouldn't have a chance. I couldn't get any encouragement from Doc Wilson or Estelle; they said it depended on Marygay's resilience.\n\nThe day before the push, they transferred her from bed to Estelle's acceleration couch in the infirmary. She was lucid and was taking food orally, but she still couldn't move under her own power, not at 1\u2014\u00bd gees.\n\nI went to see her. 'Heard about the course change? We have to go through Aleph-9 to get back to Tet-38. Four more months on this damn hulk. But another six years' combat pay when we get back to Earth.'\n\n'That's good.'\n\n'Ah, just think of the great things we'll\u2014'\n\n'William.'\n\nI let it trail off. Never could lie.\n\n'Don't try to jolly me. Tell me about vacuum welding, about your childhood, anything. Just don't bullshit me about getting back to Earth.' She turned her face to the wall.\n\n'I heard the doctors talking out in the corridor, one morning when they thought I was asleep. But it just confirmed what I already knew, the way everybody'd been moping around.\n\n'So tell me, you were born in New Mexico in 1975. What then? Did you stay in New Mexico? Were you bright in school? Have any friends, or were you too bright like me? How old were you when you first got sacked?'\n\nWe talked in this vein for a while, uncomfortable. An idea came to me while we were rambling, and when I left Marygay I went straight to Dr Wilson.\n\n'We're giving her a fifty-fifty chance, but that's pretty arbitrary. None of the published data on this sort of thing really fits.'\n\n'But it is safe to say that her chances of survival are better, the less acceleration she has to endure.'\n\n'Certainly. For what it's worth. The commodore's going to take it as gently as possible, but that'll still be four or five gees. Three might even be too much; we won't know until it's over.'\n\nI nodded impatiently. 'Yes, but I think there's a way to expose her to less acceleration than the rest of us.'\n\n'If you've developed an acceleration shield,' he said smiling, 'you better hurry and file a patent. You could sell it for a considerable\u2014'\n\n'No, Doc, it wouldn't be worth much under normal conditions; our shells work better and they evolved from the same principles.'\n\n'Explain away.'\n\n'We put Marygay into a shell and flood\u2014'\n\n'Wait, wait. Absolutely not. A poorly-fitting shell was what caused this in the first place. And this time, she'd have to use somebody else's.'\n\n'I know, Doc, let me explain. It doesn't have to fit her exactly as long as the life-support hookups can function. The shell won't be pressurized on the inside; it won't have to be because she won't be subjected to those thousands of kilograms-per-square-centimeter pressure from the fluid outside.'\n\n'I'm not sure I follow.'\n\n'It's just an adaptation of \u2014 you've studied physics, haven't you?'\n\n'A little bit, in medical school. My worse courses, after Latin.'\n\n'Do you remember the principle of equivalence?'\n\n'I remember there was something by that name. Something to do with relativity, right?'\n\n'Uh-huh. It means that... there's no difference being in a gravitational field and being in an equivalent accelerated frame of \u2014 it means that when the _Anniversary_ is blasting five gees, the effect on us is the same as if it were sitting on its tail on a big planet, on one with five gees' surface gravity.'\n\n'Seems obvious.'\n\n'Maybe it is. It means that there's no experiment you could perform on the ship that could tell you whether you were blasting or just sitting on a big planet.'\n\n'Sure there is. You could turn off the engines, and if\u2014'\n\n'Or you could look outside, sure; I mean isolated, physics-lab type experiments.'\n\n'All right. I'll accept that. So?'\n\n'You know Archimedes' Law?'\n\n'Sure, the fake crown \u2014 that's what always got me about physics, they make a big to-do about obvious things, and when it gets to the rough parts\u2014'\n\n'Archimedes' Law says that when you immerse something in a fluid, it's buoyed by a force equal to the weight of the fluid it displaces.'\n\n'That's reasonable.'\n\n'And that holds, no matter what kind of gravitation or acceleration you're in\u2014In a ship blasting at five gees, the water displaced, if it's water, weighs five times as much as regular water, at one gee.'\n\n'Sure.'\n\n'So if you float somebody in the middle of a tank of water, so that she's weightless, she'll still be weightless when the ship is doing five gees.'\n\n'Hold on, son. You had me going there, but it won't work.'\n\n'Why not?' I was tempted to tell him to stick to his pills and stethoscopes and let me handle the physics, but it was a good thing I didn't.\n\n'What happens when you drop a wrench in a submarine?'\n\n'Submarine?'\n\n'That's right. They work by Archimedes'\u2014'\n\n'Ouch! You're right. Jesus. Hadn't thought it through.'\n\n'That wrench falls right to the floor just as if the submarine weren't weightless.' He looked off into space, tapping a pencil on the desk. 'What you describe is similar to the way we treat patients with severe skin damage, like burns, on Earth. But it doesn't give any support to the internal organs, the way the acceleration shells do, so it wouldn't do Marygay any good...'\n\nI stood up to go. 'Sorry I wasted\u2014'\n\n'Hold on there, though, just a minute. We might be able to use your idea part-way.'\n\n'How do you mean?'\n\n'I wasn't thinking it through, either. The way we normally use the shells is out of the question for Marygay, of course.' I didn't like to think about it. Takes a lot of hypno-conditioning to lie there and have oxygenated fluorocarbon forced into every natural body orifice and one artificial one. I fingered the valve fitting imbedded above my hipbone.\n\n'Yeah, that's obvious, it'd tear her \u2014 say... you mean, low pressure\u2014'\n\n'That's right. We wouldn't need thousands of atmospheres to protect her against five gees' straight-line acceleration; that's only for all the swerving and dodging \u2014 I'm going to call Maintenance. Get down to your squad bay; that's the one we'll use. Dalton'll meet you there.'\n\nFive minutes before injection into the collapsar field, and I started the flooding sequence. Marygay and I were the only ones in shells; my presence wasn't really vital since the flooding and emptying could be done by Control. But it was safer to have redundancy in the system and besides, I wanted to be there.\n\nIt wasn't nearly as bad as the normal routine; none of the crushing-bloating sensation. You were just suddenly filled with the plastic-smelling stuff (you never perceived the first moments, when it rushed in to replace the air in your lungs), and then there was a slight acceleration, and then you were breathing air again, waiting for the shell to pop; then unplugging and unzipping and climbing out\u2014\n\nMarygay's shell was empty. I walked over to it and saw blood.\n\n'She hemorrhaged.' Doc Wilson's voice echoed sepulchrally. I turned, eyes stinging, and saw him leaning in the door to the locker alcove. He was unaccountably, horribly, smiling.\n\n'Which was expected. Doctor Harmony's taking care of it. She'll be just fine.'\n\n# 6\n\nMarygay was walking in another week. 'Confraternizing' in two, and pronounced completely healed in six.\n\nTen long months in space and it was army, army, army all the way. Calisthenics, meaningless work details, compulsory lectures \u2014 there was even talk that they were going to reinstate the sleeping roster we'd had in basic, but they never did, probably out of fear of mutiny. A random partner every night wouldn't have set too well with those of us who'd established more-or-less permanent pairs.\n\nAll this crap, this insistence on military discipline, bothered me mainly because I was afraid it meant they weren't going to let us out. Marygay said I was being paranoid; they only did it because there was no other way to maintain order for ten months.\n\nMost of the talk, besides the usual bitching about the army, was speculation about how much Earth would have changed and what we would do when we got out. We'd be fairly rich: twenty-six years' salary all at once. Compound interest, too; the $500 we'd been paid for our first month in the army had grown to over $1500.\n\nWe arrived at Stargate in late 2023, Greenwich date.\n\nThe base had grown astonishingly in the nearly seventeen years we had been on the Yod-4 campaign. It was one building the size of Tycho City, housing nearly ten thousand. There were seventy-eight cruisers, the size of _Anniversary_ or larger, involved in raids on Tauran-held portal planets. Another ten guarded Stargate itself, and two were in orbit waiting for their infantry and crew to be outprocessed. One other ship, the _Earth's Hope II_ , had returned from fighting and had been waiting at Stargate for another cruiser to return.\n\nThey had lost two-thirds of their crew, and it was just not economical to send a cruiser back to Earth with only thirty-nine people aboard. Thirty-nine confirmed civilians.\n\nWe went planetside in two scoutships.\n\n# 7\n\nGeneral Botsford (who had only been a full major the first time we met him, when Stargate was two huts and twenty-four graves) received us in an elegantly appointed seminar room. He was pacing back and forth at the end of the room, in front of a huge holographic operations chart.\n\n'You know,' he said, too loud, and then, more conversationally, 'you know that we could disperse you into other strike forces and send you right out again. The Elite Conscription Act has been changed now, five years' subjective in service instead of two.\n\n'And I don't see why some of you don't _want_ to stay in! Another couple of years and compound interest would make you independently wealthy for life. Sure, you took heavy losses \u2014 but that was inevitable, you were the first. Things are going to be easier now. The fighting suits have been improved, we know more about the Taurans' tactics, our weapons are more effective... there's no need to be afraid.'\n\nHe sat down at the head of the table and looked at nobody in particular.\n\n'My own memories of combat are over a half-century old. To me it was exhilarating, strengthening. I must be a different kind of person than all of you.'\n\nOr have a very selective memory, I thought.\n\n'But that's neither here nor there. I have one alternative to offer you, one that doesn't involve direct combat.\n\n'We're very short of qualified instructors. The Force will offer any one of you a lieutenancy if you will accept a training position. It can be on Earth; on the Moon at double pay; on Charon at triple pay; or here at Stargate for quadruple pay. Furthermore, you don't have to make up your mind now. You're all getting a free trip back to Earth \u2014 I envy you, I haven't been back in fifteen years, will probably never go back \u2014 and you can get the feel of being a civilian again. If you don't like it, just walk into any UNEF installation and you'll walk out an officer. Your choice of assignment.\n\n'Some of you are smiling. I think you ought to reserve judgment. Earth is not the same place you left.'\n\nHe pulled a little card out of his tunic and looked at it, smiling. 'Most of you have something on the order of four hundred thousand dollars coming to you, accumulated pay and interest. But Earth is on a war footing and, of course, it is the citizens of Earth who are supporting the war. Your income puts you in a ninety-two-percent income-tax bracket: thirty-two thousand might last you about three years if you're careful.\n\n'Eventually you're going to have to get a job, and this is one job for which you are uniquely trained. There are not that many jobs available. The population of Earth is nearly nine billion, with five or six billion unemployed.\n\n'Also keep in mind that your friends and sweethearts of two years ago are now going to be twenty-one years older than you. Many of your relatives will have passed away. I think you'll find it a very lonely world.\n\n'But to tell you something about this world, I'm going to turn you over to Captain Siri, who just arrived from Earth. Captain.'\n\n'Thank you, General.' It looked as if there was something wrong with his skin, his face; and then I realized he was wearing powder and lipstick. His nails were smooth white almonds.\n\n'I don't know where to begin.' He sucked in his upper lip and looked at us, frowning. 'Things have changed so very much since I was a boy.\n\n'I'm twenty-three, so I was still in diapers when you people left for Aleph... to begin with, how many of you are homosexual?' Nobody. 'That doesn't really surprise me. I am, of course. I guess about a third of everybody in Europe and America is.\n\n'Most governments encourage homosexuality \u2014 the United Nations is neutral, leaves it up to the individual countries \u2014 they encourage homolife mainly because it's the one sure method of birth control.'\n\nThat seemed specious to me. Our method of birth control in the army is pretty foolproof: all men making a deposit in the sperm bank, and then vasectomy.\n\n'As the General said, the population of the world is nine billion. It's more than doubled since you were drafted. And nearly two-thirds of those people get out of school only to go on relief.\n\n'Speaking of school, how many years of public schooling did the government give you?'\n\nHe was looking at me, so I answered. 'Fourteen.'\n\nHe nodded. 'It's eighteen now. More, if you don't pass your examinations. And you're required by law to pass your exams before you're eligible for any job or Class One relief. And brother-boy, anything besides Class One is hard to live on. Yes?' Hofstadter had his hand up.\n\n'Sir, is it eighteen years public school in every country? Where do they find enough schools?'\n\n'Oh, most people take the last five or six years at home or in a community center, via holoscreen. The UN has forty or fifty information channels, giving instruction twenty-four hours a day.\n\n'But most of you won't have to concern yourselves with that. If you're in the Force, you're already too smart by half.'\n\nHe brushed hair from his eyes in a thoroughly feminine gesture, pouting a little. 'Let me do some history to you. I guess the really important thing that happened after you left was the Ration War.\n\n'That was 2007. A lot of things happened at once. Locust plague in North America, rice blight from Burma to the South China Sea, red tides all along the west coast of South America: suddenly there just wasn't enough food to go around. The UN stepped in and took over food distribution. Every man, woman, and child got a ration booklet, allowing them to consume so many calories per month. If tha went over ther monthly allotment, tha just went hungry until the first of the next month.'\n\nSome of the new people we'd picked up after Aleph used 'tha, ther, thim' instead of 'he, his, him,' for the collective pronoun. I wondered whether it had become universal.\n\n'Of course, an illegal market developed, and soon there was great inequality in the amount of food people in various strata of society consumed. A vengeance group in Ecuador, the Imparciales, systematically began to assassinate people who appeared to be well-fed. The idea caught on pretty quickly, and in a few months there was a full-scale, undeclared class war going on all over the world. The United Nations managed to get things back under control in a year or so, by which time the population was down to four billion, crops were more or less recovered, and the food crisis was over. They kept the rationing, but it's never been really severe again.\n\n'Incidentally, the General translated the money coming to you into dollars just for your own convenience. The world has only one currency now, calories. Your thirty-two thousand dollars comes to about three thousand million calories. Or three million k's, kilocalories.\n\n'Ever since the Ration War, the UN has encouraged subsistence farming wherever it's practical. Food you grow yourself, of course, isn't rationed... It got people out of the cities, onto UN farming reservations, which helped alleviate some urban problems. But subsistence farming seems to encourage large families, so the population of the world has more than doubled since the Ration War.\n\n'Also, we no longer have the abundance of electrical power I remember from boyhood... probably a good deal less than you remember. There are only a few places in the world where you can have power all day and night. They keep saying it's a temporary situation, but it's been going on for over a decade.'\n\nHe went on like that for a long time. Well, hell, it wasn't really surprising, much of it. We'd probably spent more time in the past two years talking about what home was going to be like than about anything else. Unfortunately, most of the bad things we'd prognosticated seemed to have come true, and not many of the good things.\n\nThe worst thing for me, I guess, was that they'd taken over most of the good parkland and subdivided it into little farms. If you wanted to find some wilderness, you had to go someplace where they couldn't possibly make a plant grow.\n\nHe said that the relations between people who chose homolife and the ones he called 'breeders' were quite smooth, but I wondered. I never had much trouble accepting homosexuals myself, but then I'd never had to cope with such an abundance of them.\n\nHe also said, in answer to an impolite question, that his powder and paint had nothing to do with his sexual orientation. It was just stylish. I decided I'd be an anachronism and just wear my face.\n\nI don't guess it should have surprised me that language had changed considerably in twenty years. My parents were always saying things were 'cool,' joints were 'grass,' and so on.\n\nWe had to wait several weeks before we could get a ride back to Earth. We'd be going back on the _Anniversary_ , but first she had to be taken apart and put back together again.\n\nMeanwhile, we were put in cozy little two-man billets and released from all military responsibilities. Most of us spent our days down at the library, trying to catch up on twenty-two years of current events. Evenings, we'd get together at the Flowing Bowl, an NCO club. The privates, of course, weren't supposed to be there, but we found that nobody argues with a person who has two of the fluorescent battle ribbons.\n\nI was surprised that they served heroin fixes at the bar. The waiter said that you get a compensating shot to keep you from getting addicted to it. I got really stoned and tried one. Never again.\n\nSub-major Stott stayed at Stargate, where they were assembling a new Strike Force Alpha. The rest of us boarded the _Anniversary_ and had a fairly pleasant six-month journey. Cortez didn't insist on everything being capital-M military, so it was a lot better than the trip from Yod-4. \n\n# 8\n\nI hadn't given it too much thought, but of course we were celebrities on Earth: the first vets home from the war. The Secretary General greeted us at Kennedy and we had a week-long whirl of banquets, receptions, interviews, and all that. It was enjoyable enough, and profitable \u2014 I made a million k's from Time-Life\/Fax \u2014 but we really saw little of Earth until after the novelty wore off and we were more or less allowed to go our own way.\n\nI picked up the Washington monorail at Grand Central Station and headed home. My mother had met me at Kennedy, suddenly and sadly old, and told me my father was dead. Flyer accident. I was going to stay with her until I could get a job.\n\nShe was living in Columbia, a satellite of Washington. She had moved back into the city after the Ration War \u2014 having moved out in 1980 \u2014 and then failing services and rising crime had forced her out again.\n\nShe was waiting for me at the monorail station. Beside her stood a blond giant in a heavy black vinyl uniform, with a big gunpowder pistol on his hip and spiked brass knuckles on his right hand.\n\n'William, this is Carl, my bodyguard and very dear friend.' Carl slipped off the knuckles long enough to shake hands with surprising gentleness. 'Pleasameecha Misser Mandella.'\n\nWe got into a groundcar that had 'Jefferson' written on it in bright orange letters. I thought that was an odd thing to name a car, but then found out that it was the name of the high-rise Mother and Carl lived in. The groundcar was one of several that belonged to the community, and she paid 100k per kilometer for the use of it.\n\nI had to admit that Columbia was rather pretty: formal gardens and lots of trees and grass. Even the high-rises, roughly conical jumbles of granite with trees growing out at odd places, looked more like mountains than buildings. We drove into the base of one of these mountains, down a well-lit corridor to where a number of other cars were parked. Carl carried my solitary bag to the elevator and set it down.\n\n'Miz Mandella, if is awright witcha, I gots to go pick up Miz Freeman in like five. She over West Branch.'\n\n'Sure, Carl, William can take care of me. He's a soldier, you know.' That's right, I remember learning eight silent ways to kill a man. Maybe if things got really tight, I could get a job like Carl's.\n\n'Righty-oh, yeah, you tol' me. Whassit like, man?'\n\n'Mostly boring,' I said automatically. 'When you aren't bored, you're scared.'\n\nHe nodded wisely. 'Thass what I heard. Miz Mandella, I be 'vailable anytime after six. Righty-oh?'\n\n'That's fine, Carl.'\n\nThe elevator came and a tall skinny boy stepped out, an unlit joint dangling from his lips. Carl ran his fingers over the spikes on his knuckles, and the boy walked rapidly away.\n\n'Gots ta watch out fer them riders. T'care a yerself, Miz Mandella.'\n\nWe got on the elevator and Mother punched 47. 'What's a rider?'\n\n'Oh, they're just young toughs who ride up and down the elevators looking for defenseless people without bodyguards. They aren't too much of a problem here.'\n\nThe forty-seventh floor was a huge mall filled with shops and offices. We went to a food store.\n\n'Have you gotten your ration book yet, William?' I told her I hadn't, but the Force had given me travel tickets worth a hundred thousand 'calories' and I'd used up only half of them.\n\nIt was a little confusing, but they'd explained it to us.\n\nWhen the world went on a single currency, they'd tried to coordinate it with the food rationing in some way, hoping to eventually eliminate the ration books, so they'd made the new currency K's, kilocalories, because that's the unit for measuring the energy equivalent of food. But a person who eats 2,000 kilocalories of steak a day obviously has to pay more than a person eating the same amount of bread. So they instituted a sliding 'ration factor,' so complicated that nobody could understand it. After a few weeks they were using the books again, but calling food kilocalories 'calories' in an attempt to make things less confusing. Seemed to me they'd save a lot of trouble all around if they'd just call money dollars again, or rubles or sisterces or whatever... anything but kilocalories.\n\nFood prices were astonishing, except for grains and legumes. I insisted on splurging on some good red meat: 1500 calories' worth of ground beef, costing 1730K. The same amount of fakesteak, made from soy beans, would have cost 80K.\n\nI also got a head of lettuce for 140K and a little bottle of olive oil for 175K. Mother said she had some vinegar. Started to buy some mushrooms but she said she had a neighbor who grew them and could trade something from her balcony garden.\n\nAt her apartment on the ninety-second floor, she apologized for the smallness of the place. It didn't seem so little to me, but then she'd never lived on a spaceship.\n\nEven this high up, there were bars on the windows. The door had four separate locks, one of which didn't work because somebody had used a crowbar on it.\n\nMother went off to turn the ground beef into a meatloaf and I settled down with the evening 'fax. She pulled some carrots from her little garden and called the mushroom lady, whose son came over to make the trade. He had a riot gun slung under his arm.\n\n'Mother, where's the rest of the _Star_?' I called into the kitchen.\n\n'As far as I know, it's all there. What were you looking for?'\n\n'Well... I found the classified section, but no \"Help Wanted.\" '\n\nShe laughed. 'Son, there hasn't been a \"Help Wanted\" ad in ten years. The government takes care of jobs... well, most of them.'\n\n'Everybody works for the government?'\n\n'No, that's not it.' She came in, wiping her hands on a frayed towel. 'The government, they tell us, handles the distribution of all natural resources. And there aren't many resources more valuable than empty jobs.'\n\n'Well, I'll go talk to them tomorrow.'\n\n'Don't bother, son. How much retirement pay you say you're getting from the Force?'\n\n'Twenty thousand K a month. Doesn't look like it'll go far.'\n\n'No, it won't. But your father's pension gave me less than half that, and they wouldn't give me a job. Jobs are assigned on a basis of need. And you've got to be living on rice and water before the Employment Board considers you needy.'\n\n'Well, hell, it's a bureaucracy \u2014 there must be somebody I can pay off, slip me into a good\u2014'\n\n'No. Sorry, that's one part of the UN that's absolutely incorruptible. The whole shebang is cybernetic, untouched by human souls. You can't\u2014'\n\n'But you said you _had_ a job!'\n\n'I was getting to that. If you want a job badly enough, you can go to a dealer and sometimes get a hand-me-down.'\n\n'Hand-me-down? Dealer?'\n\n'Take my job as an example, son. A woman named Hailey Williams has a job in a hospital, running a machine that analyzes blood, a chromatography machine. She works six nights a week, for 12,000k a week. She gets tired of working, so she contacts a dealer and lets him know that her job is available.\n\n'Some time before this, I'd given the dealer his initial fee of 50,000K to get on his list. He comes by and describes the job to me and I say fine, I'll take it. He knew I would and already has fake identification and a uniform. He distributes small bribes to the various supervisors who might know Miss Williams by sight.\n\n'Miss Williams shows me how to run the machine and quits. She still gets the weekly 12,000K credited to her account, but she pays me half. I pay the dealer ten percent and wind up with 5400K per week. This, added to the nine grand I get monthly from your father's pension, makes me quite comfortable.\n\n'Then it gets complicated. Finding myself with plenty of money and too little time, I contact the dealer again, offering to sublet half my job. The next day a girl shows up who also has \"Hailey Williams\" identification. I show her how to run the machine, and she takes over Monday-Wednesday-Friday. Half of my real salary is 2700K, so she gets half that, 1350K, and pays the dealer 135.'\n\nShe got a pad and a stylus and did some figuring. 'So the real Hailey Williams gets 6000K weekly for doing nothing. I work three days a week for 4050K. My assistant works three days for 1115K. The dealer gets 100,000K in fees and 735K per week. Lopsided, isn't it?'\n\n'Hmm... I'll say. Quite illegal, too, I suppose.'\n\n'For the dealer. Everybody else might lose their job and have to start over, if the Employment Board finds out. But the dealer gets brainwiped.'\n\n'Guess I better find a dealer, while I can still afford the fifty-grand bite.' Actually, I still had over three million, but planned to run through most of it in a short time. Hell, I'd earned it.\n\nI was getting ready to go the next morning when Mother came in with a shoebox. Inside, there was a small pistol in a clip-on holster.\n\n'This belonged to your father,' she explained. 'Better wear it if you're planning to go downtown without a bodyguard.'\n\nIt was a gunpowder pistol with ridiculously thin bullets. I hefted it in my hand. 'Did Dad ever use it?'\n\n'Several times... just to scare away riders and hitters, though. He never actually shot anybody.'\n\n'You're probably right that I need a gun,' I said, putting it back. 'But I'd have to have something with more heft to it. Can I buy one legally?'\n\n'Sure, there's a gun store down in the Mall. As long as you don't have a police record, you can buy anything that suits you.' Good; I'd get a little pocket laser. I could hardly hit the wall with a gunpowder pistol.\n\n'But... William, I'd feel a lot better if you'd hire a bodyguard, at least until you know your way around.' We'd gone all around that last night. Being an official Trained Killer, I thought I was tougher than any clown I might hire for the job.\n\n'I'll check into it, Mother. Don't worry \u2014 I'm not even going downtown today, just into Hyattsville.'\n\n'That's just as bad.'\n\nWhen the elevator came, it was already occupied. He looked at me blandly as I got in, a man a little older than me, clean-shaven and well dressed. He stepped back to let me at the row of buttons. I punched 47 and then, realizing his motive might not have been politeness, turned to see him struggling to get at a metal pipe stuck in his waistband. It had been hidden by his cape.\n\n'Come on, fella,' I said, reaching for a nonexistent weapon. 'You wanna get caulked?'\n\nHe had the pipe free but let it hang loosely at his side. 'Caulked?'\n\n'Killed. Army term.' I took one step toward him, trying to remember. Kick just under the knee, then either groin or kidney. I decided on the groin.\n\n'No.' He put the pipe back in his waistband. 'I don't want to get \"caulked.\" ' The door opened at 47 and I backed out.\n\nThe gun shop was all bright white plastic and gleamy black metal. A little bald man bobbed over to wait on me. He had a pistol in a shoulder rig.\n\n'And a fine morning to you, sir,' he said and giggled. 'What will it be today?'\n\n'Lightweight pocket laser,' I said. 'Carbon dioxide.'\n\nHe looked at me quizzically and then brightened. 'Coming right up, sir.' Giggle. 'Special today, I throw in a handful of tachyon grenades.'\n\n'Fine.' They'd be handy.\n\nHe looked at me expectantly. 'So? What's the popper?'\n\n'Huh?'\n\n'The punch, man; you set me up, now knock me down. Laser.' He giggled.\n\nI was beginning to understand. 'You mean I can't buy a laser.'\n\n'Of _course_ not, sweetie,' he said and sobered. 'You didn't know that?'\n\n'I've been out of the country for a long time.'\n\n'The world, you mean. You've been out of the world a long time.' He put his left hand on a chubby hip in a gesture that incidentally made his gun easier to get. He scratched the center of his chest.\n\nI stood very still. 'That's right. I just got out of the Force.'\n\nHis jaw dropped. 'Hey, no bully-bull? You been out shootin' 'em up, out in space?'\n\n'That's right.'\n\n'Hey, all that crap about you not gettin' older, there's nothin' to that, is there?'\n\n'Oh, it's true. I was born in 1975.'\n\n'Well, god... damn. You're almost as old as I am.' He giggled. 'I thought that was just something the government made up.'\n\n'Anyhow... you say I can't buy a laser\u2014'\n\n'Oh, no. No no no. I run a legal shop here.'\n\n'What can I buy?'\n\n'Oh, pistol, rifle, shotgun, knife, body armor... just no lasers or explosives or fully automatic weapons.'\n\n'Let me see a pistol. The biggest you have.'\n\n'Ah, I've got just the thing.' He motioned me over to a display case and opened the back, taking out a huge revolver.\n\n'Four-ten-gauge six-shooter.' He cradled it in both hands. 'Dinosaur-stopper. Authentic Old West styling. Slugs or flechettes.'\n\n'Flechettes?'\n\n'Sure \u2014 uh, they're like a bunch of tiny darts. You shoot and they spread out in a pattern. Hard to miss that way.'\n\nSounded like my speed. 'Anyplace I can try it out?'\n\n''Course, of course, we have a range in back. Let me get my assistant.' He rang a bell and a boy came out to watch the store while we went in back. He picked up a red-and-green box of shotgun shells on the way.\n\nThe range was in two sections, a little anteroom with a plastic transparent door and a long corridor on the other side of the door with a table at one end and targets at the other. Behind the targets was a sheet of metal that evidently deflected the bullets down into a pool of water.\n\nHe loaded the pistol and set it on the table. 'Please don't pick it up until the door's closed.' He went into the anteroom, closed the door, and picked up a microphone. 'OK. First time, you better hold on to it with both hands.' I did so, raising it up in line with the center target, a square of paper looking about the size of your thumbnail at arm's length. Doubted I'd even come near it. I pulled the trigger and it went back easily enough, but nothing happened.\n\n'No, no,' he said over the microphone with a tinny giggle. 'Authentic Old West styling. You've got to pull the hammer back.'\n\nSure, just like in the flicks. I hauled the hammer back, lined it up again, and squeezed the trigger.\n\nThe noise was so loud it made my face sting. The gun bucked up and almost hit me on the forehead. But the three center targets were gone: just tiny tatters of paper drifting in the air.\n\n'I'll take it.'\n\nHe sold me a hip holster, twenty shells, a chest-and-back shield, and a dagger in a boot sheath. I felt more heavily armed than I had in a fighting suit. But no waldos to help me cart it around.\n\nThe monorail had two guards for each car. I was beginning to feel that all my heavy artillery was superfluous, until I got off at the Hyattsville station.\n\nEveryone who got off at Hyattsville was either heavily armed or had a bodyguard. The people loitering around the station were all armed. The police carried lasers.\n\nI pushed a 'cab call' button, and the readout told me mine would be No. 3856. I asked a policeman and he told me to wait for it down on the street; it would cruise around the block twice.\n\nDuring the five minutes I waited, I twice heard staccato arguments of gunfire, both of them rather far away. I was glad I'd bought the shield.\n\nEventually the cab came. It swerved to the curb when I waved at it, the door sliding open as it stopped. Looked as if it worked the same way as the autocabs I remembered. The door stayed open while it checked the thumbprint to verify that I was the one who had called, then slammed shut. It was thick steel. The view through the windows was dim and distorted; probably thick bulletproof plastic. Not quite the same as I remembered.\n\nI had to leaf through a grimy book to find the code for the address of the bar in Hyattsville where I was supposed to meet the dealer. I punched it out and sat back to watch the city go by.\n\nThis part of town was mostly residential: grayed-brick warrens built around the middle of the last century competing for space with more modern modular setups and, occasionally, individual houses behind tall brick or concrete walls with jagged broken glass and barbed wire at the top. A few people seemed to be going somewhere, walking very quickly down the sidewalks, hands on weapons. Most of the people I saw were either sitting in doorways, smoking, or loitering around shopfronts in groups of no fewer than six. Everything was dirty and cluttered. The gutters were clotted with garbage, and shoals of waste paper drifted with the wind of the light traffic.\n\nIt was understandable, though; street-sweeping was probably a very high-risk profession.\n\nThe cab pulled up in front of Tom & Jerry's Bar and Grill and let me out after I paid 430K. I stepped to the sidewalk with my hand on the shotgun-pistol, but there was nobody around. I hustled into the bar.\n\nIt was surprisingly clean on the inside, dimly lit and furnished in fake leather and fake pine. I went to the bar and got some fake bourbon and, presumably, real water for 120K. The water cost 20K. A waitress came over with a tray.\n\n'Pop one, brother-boy?' The tray had a rack of old-fashioned hypodermic needles.\n\n'Not today, thanks.' If I was going to 'pop one,' I'd use an aerosol. The needles looked unsanitary and painful.\n\nShe set the dope down on the bar and eased onto the stool next to me. She sat with her chin cupped in her palm and stared at her reflection in the mirror behind the bar. 'God. Tuesdays.'\n\nI mumbled something.\n\n'You wanna go in back fer a quickie?'\n\nI looked at her with what I hoped was a neutral expression. She was wearing only a short skirt of some gossamer material, and it plunged in a shallow V in the front, exposing her hipbones and a few bleached pubic hairs. I wondered what could possibly keep it up. She wasn't bad looking, could have been anywhere from her late twenties to her early forties. No telling what they could do with cosmetic surgery and makeup nowadays, though. Maybe she was older than my mother.\n\n'Thanks anyhow.'\n\n'Not today?'\n\n'That's right.'\n\n'I can get you a nice boy, if\u2014'\n\n'No. No thanks.' What a world.\n\nShe pouted into the mirror, an expression that was probably older than _Homo sapiens_. 'You don't like me.'\n\n'I like you fine. That's just not what I came here for.'\n\n'Well... different funs for different ones.' She shrugged. 'Hey, Jerry. Get me a short beer.'\n\nHe brought it.\n\n'Oh, damn, my purse is locked up. Mister, can you spare forty calories?' I had enough ration tickets to take care of a whole banquet. Tore off a fifty and gave it to the bartender.\n\n'Jesus.' She stared. 'How'd you get a full book at the end of the month?'\n\nI told her in as few words as possible who I was and how I managed to have so many calories. There had been two months' worth of books waiting in my mail, and I hadn't even used up the ones the Force had given me. She offered to buy a book from me for ten grand, but I didn't want to get involved in more than one illegal enterprise at a time.\n\nTwo men came in, one unarmed and the other with both a pistol and a riot gun. The bodyguard sat by the door and the other came over to me.\n\n'Mr Mandella?'\n\n'That's right.'\n\n'Shall we take a booth?' He didn't offer his name.\n\nHe had a cup of coffee, and I sipped a mug of beer. 'I don't keep any written records, but I have an excellent memory. Tell me what sort of a job you're interested in, what your qualifications are, what salary you'll accept, and so on.'\n\nI told him I'd prefer to wait for a job where I could use my physics \u2014 teaching or research, even engineering. I wouldn't need a job for two or three months, since I planned to travel and spend money for a while. Wanted at least 20,000K monthly, but how much I'd accept would depend on the nature of the job.\n\nHe didn't say a word until I'd finished. 'Righty-oh. Now, I'm afraid... you'd have a hard time, getting a job in physics. Teaching is out; I can't supply jobs where the person is constantly exposed to the public. Research, well, your degree is almost a quarter of a century old. You'd have to go back to school, maybe five or six years.'\n\n'Might do that,' I said.\n\n'The one really marketable feature you have is your combat experience. I could probably place you in a supervisory job at a bodyguard agency for even more than twenty grand. You could make almost that much, being a bodyguard yourself.'\n\n'Thanks, but I wouldn't want to take chances for somebody else's hide.'\n\n'Righty-oh. Can't say I blame you.' He finished his coffee in a long slurp. 'Well, I've got to run, got a thousand things to do. I'll keep you in mind and talk to some people.'\n\n'Good. I'll see you in a few months.'\n\n'Righty-oh. Don't need to make an appointment. I come in here every day at eleven for coffee. Just show up.'\n\nI finished my beer and called a cab to take me home. I wanted to walk around the city, but Mother was right. I'd get a bodyguard first. \n\n# 9\n\nI came home and the phone was blinking pale blue. Didn't know what to do so I punched 'Operator.'\n\nA pretty young girl's head materialized in the cube. 'Jefferson operator,' she said. 'May I help you?'\n\n'Yes... what does it mean when the cube is blinking blue?'\n\n'Huh?'\n\n'What does it mean when the phone\u2014'\n\n'Are you serious?' I was getting a little tired of this kind of thing.\n\n'It's a long story. Honest, I don't know.'\n\n'When it blinks blue you're supposed to call the operator.'\n\n'OK, here I am.'\n\n'No, not me, the _real_ operator. Punch nine. Then punch zero.'\n\nI did that and an old harridan appeared. 'Ob-a-ray-duh.'\n\n'This is William Mandella at 301\u201452\u2014574\u20143975. I was supposed to call you.'\n\n'Juzza segun.' She reached outside the field of view and typed something. 'You godda call from 605\u201419\u2014556\u20142027.'\n\nI scribbled it down on the pad by the phone. 'Where's that?'\n\n'Juzza segun. South Dakota.'\n\n'Thanks.' I didn't know anybody in South Dakota.\n\nA pleasant-looking old woman answered the phone. 'Yes?'\n\n'I had a call from this number... uh... I'm\u2014'\n\n'Oh. Sergeant Mandella! Just a second.'\n\nI watched the diagonal bar of the holding pattern for a second, then fifty or so more. Then a head came into focus.\n\nMarygay. 'William. I had a heck of a time finding you.'\n\n'Darling, me too. What are you doing in South Dakota?'\n\n'My parents live here, in a little commune. That's why it took me so long to get to the phone.' She held up two grimy hands. 'Digging potatoes.'\n\n'But when I checked... the records said \u2014 the records in Tucson said your parents were both dead.'\n\n'No, they're just dropouts \u2014 you know about dropouts? \u2014 new name, new life. I got the word through a cousin.'\n\n'Well \u2014 well, how've you been? Like the country life?'\n\n'That's one reason I've been wanting to get you. Willy, I'm bored. It's all very healthy and nice, but I want to do something dissipated and wicked. Naturally I thought of you.'\n\n'I'm flattered. Pick you up at eight?'\n\nShe checked a clock above the phone. 'No, look, let's get a good night's sleep. Besides, I've got to get in the rest of the potatoes. Meet me at... the Ellis Island jetport at ten tomorrow morning. Mmm... Trans-World information desk.'\n\n'OK. Make reservations for where?'\n\nShe shrugged. 'Pick a place.'\n\n'London used to be pretty wicked.'\n\n'Sounds good. First class?'\n\n'What else? I'll get us a suite on one of the dirigibles.'\n\n'Good. Decadent. How long shall I pack for?'\n\n'We'll buy clothes along the way. Travel light. Just one stuffed wallet apiece.'\n\nShe giggled. 'Wonderful. Tomorrow at ten.'\n\n'Fine \u2014 uh... Marygay, do you have a gun?'\n\n'It's that bad?'\n\n'Here around Washington it is.'\n\n'Well, I'll get one. Dad has a couple over the fireplace. Guess they're left over from Tucson.'\n\n'We'll hope we won't need them.'\n\n'Willy, you know it'll just be for decoration. I couldn't even kill a Tauran.'\n\n'Of course.' We just looked at each other for a second. 'Tomorrow at ten, then.'\n\n'Right. Love you.'\n\n'Uh...'\n\nShe giggled again and hung up.\n\nThat was just too many things to think about all at once.\n\nI got us two round-the-world dirigible tickets; unlimited stops as long as you kept going east. It took me a little over two hours to get to Ellis by autocab and monorail. I was early, but so was Marygay.\n\nShe was talking to the girl at the desk and didn't see me coming. Her outfit was really arresting, a tight coverall of plastic in a pattern of interlocking hands; as your angle of sight changed, various strategic hands became transparent. She had a ruddy sun-glow all over her body. I don't know whether the feeling that rushed over me was simple honest lust or something more complicated. I hurried up behind her.\n\nWhispering: 'What are we going to do for three hours?'\n\nShe turned and gave me a quick hug and thanked the girl at the desk, then grabbed my hand and pulled me along to a slidewalk.\n\n'Um... where are we headed?'\n\n'Don't ask questions, Sergeant. Just follow me.'\n\nWe stepped onto a roundabout and transferred to an eastbound slidewalk.\n\n'Do you want something to eat or drink?' she asked innocently.\n\nI tried to leer. 'Any alternatives?'\n\nShe laughed gaily. Several people stared. 'Just a second... here!' We jumped off. It was a corridor marked 'Roomettes.' She handed me a key.\n\nThat damned plastic coverall was held on by static electricity. Since the roomette was nothing but a big waterbed, I almost broke my neck the first time it shocked me.\n\nI recovered.\n\nWe were lying on our stomachs, looking through the one-way glass wall at the people rushing around down on the concourse. Marygay passed me a joint.\n\n'William, have you used that thing yet?'\n\n'What thing?'\n\n'That hawg-leg. The pistol.'\n\n'Only shot it once, in the store where I bought it.'\n\n'Do you really think you could point it at someone and blow him apart?'\n\nI took a shallow puff and passed it back. 'Hadn't given it much thought, really. Until we talked last night.'\n\n'Well?'\n\n'I... I don't really know. The only time I've killed was on Aleph, under hypnotic compulsion. But I don't think it would... bother me, not that much, not if the person was trying to kill me in the first place. Why should it?'\n\n'Life,' she said plaintively, 'life is...'\n\n'Life is a bunch of cells walking around with a common purpose. If that common purpose is to get my ass\u2014'\n\n'Oh, William. You sound like old Cortez.'\n\n'Cortez kept us alive.'\n\n'Not many of us,' she snapped.\n\nI rolled over and studied the ceiling tiles. She traced little designs on my chest, pushing the sweat around with her fingertip. 'I'm sorry, William. I guess we're both just trying to adjust.'\n\n'That's OK. You're right, anyhow.'\n\nWe talked for a long time. The only urban center Marygay had been to since our publicity rounds (which were very sheltered) was Sioux Falls. She had gone with her parents and the commune bodyguard. It sounded like a scaled-down version of Washington: the same problems, but not as acute.\n\nWe ticked off the things that bothered us: violence, high cost of living, too many people everywhere. I'd have added homolife, but Marygay said I just didn't appreciate the social dynamic that had led to it; it had been inevitable. The only thing she said she had against it was that it took so many of the prettiest men out of circulation.\n\nAnd the main thing that was wrong was that everything seemed to have gotten just a little worse, or at best remained the same. You would have predicted that at least a few facets of everyday life would improve markedly in twenty-two years. Her father contended the War was behind it all: any person who showed a shred of talent was sucked up by UNEF; the very best fell to the Elite Conscription Act and wound up being cannon fodder.\n\nIt was hard not to agree with him. Wars in the past often accelerated social reform, provided technological benefits, even sparked artistic activity. This one, however, seemed tailor-made to provide none of these positive by-products. Such improvements as had been made on late-twentieth-century technology were \u2014 like tachyon bombs and warships two kilometers long \u2014 at best, interesting developments of things that only required the synergy of money and existing engineering techniques. Social reform? The world was technically under martial law. As for art, I'm not sure I know good from bad. But artists to some extent have to reflect the temper of the times. Paintings and sculpture were full of torture and dark brooding; movies seemed static and plotless; music was dominated by nostalgic revivals of earlier forms; architecture was mainly concerned with finding someplace to put everybody; literature was damn near incomprehensible. Most people seemed to spend most of their time trying to find ways to outwit the government, trying to scrounge a few extra K's or ration tickets without putting their lives in too much danger.\n\nAnd in the past, people whose country was at war were constantly in contact with the war. The newspapers would be full of reports, veterans would return from the front; sometimes the front would move right into town, invaders marching down Main Street or bombs whistling through the night air \u2014 but always the sense of either working toward victory or at least delaying defeat. The enemy was a tangible thing, a propagandist's monster whom you could understand, whom you could hate.\n\nBut this war... the enemy was a curious organism only vaguely understood, more often the subject of cartoons than nightmares. The main effect of the war on the home front was economic, unemotional \u2014 more taxes but more jobs as well. After twenty-two years, only twenty-seven returned veterans; not enough to make a decent parade. The most important fact about the war to most people was that if it ended suddenly, Earth's economy would collapse.\n\nYou approached the dirigible by means of a small propeller-driven aircraft that drifted up to match trajectories and docked alongside. A clerk took our baggage and we checked our weapons with the purser, then went outside.\n\nJust about everybody on the flight was standing out on the promenade deck, watching Manhattan creep toward the horizon. It was an eerie sight. The day was very still, so the bottom thirty or forty stories of the buildings were buried in smog. It looked like a city built on a cloud, a thunderhead floating. We watched it for a while and then went inside to eat.\n\nThe meal was elegantly served and simple: filet of beef, two vegetables, wine. Cheese and fruit and more wine for dessert. No fiddling with ration tickets; a loophole in the rationing laws implied that they were not required for meals consumed en route, on intercontinental transport.\n\nWe spent a lazy, comfortable three days crossing the Atlantic. The dirigibles had been a new thing when we first left Earth, and now they had turned out to be one of the few successful new financial ventures of the late twentieth century... the company that built them had bought up a few obsolete nuclear weapons; one bomb-sized hunk of plutonium would keep the whole fleet in the air for years. And, once launched, they never did come down. Floating hotels, supplied and maintained by regular shuttles, they were one last vestige of luxury in a world where nine billion people had something to eat, and almost nobody had enough.\n\nLondon was not as dismal from the air as New York City had been; the air was clean even if the Thames was poison. We packed our handbags, claimed our weapons, and landed on a VTO pad atop the London Hilton. We rented a couple of tricycles at the hotel and, maps in hand, set off for Regent Street, planning on dinner at the venerable Cafe Royal.\n\nThe tricycles were little armored vehicles, stabilized gyroscopically so they couldn't be tipped over. Seemed overly cautious for the part of London we traveled through, but I suppose there were probably sections as rough as Washington.\n\nI got a dish of marinated venison and Marygay got salmon; both very good but astoundingly expensive. At first I was a bit overawed by the huge room, filled with plush and mirrors and faded gilding, very quiet even with a dozen tables occupied, and we talked in whispers until we realized that was foolish.\n\nOver coffee I asked Marygay what the deal was with her parents.\n\n'Oh, it happens often enough,' she said. 'Dad got mixed up in some ration ticket thing. He'd gotten some black market tickets that turned out to be counterfeit. Cost him his job and he probably would have gone to jail, but while he was waiting for trial a bodysnatcher got him.'\n\n'Bodysnatcher?'\n\n'That's right. All the commune organizations have them. They've got to get reliable farm labor, people who aren't eligible for relief... people who can't just lay down their tools and walk off when it gets rough. Almost everybody can get enough assistance to stay alive, though; everyone who isn't on the government's fecal roster.'\n\n'So he skipped out before his trial came up?'\n\nShe nodded. 'It was a case of choosing between commune life, which he knew wasn't easy, and going on the dole after a few years' working on a prison farm; ex-convicts can't get legitimate jobs. They had to forfeit their condominium, which they'd put up for bail, but the government would've gotten that anyhow, once he was in jail.\n\n'So the bodysnatcher offered him and Mother new identities, transportation to the commune, a cottage, and a plot of land. They took it.'\n\n'And what did the bodysnatcher get?'\n\n'He himself probably didn't get anything. The commune got their ration tickets; they were allowed to keep their money, although they didn't have very much\u2014'\n\n'What happens if they get caught?'\n\n'Not a chance.' She laughed. 'The communes provide over half the country's produce \u2014 they're really just an unofficial arm of the government. I'm sure the CBI knows exactly where they are... Dad grumbles that it's just a fancy way of being in jail anyhow.'\n\n'What a weird setup.'\n\n'Well, it keeps the land farmed.' She pushed her empty dessert plate a symbolic centimeter away from her. 'And they're eating better than most people, better than they ever had in the city. Mom knows a hundred ways to fix chicken and potatoes.'\n\nAfter dinner we went to a musical show. The hotel had gotten us tickets to a 'cultural translation' of the old rock opera _Hair_. The program explained that they had taken some liberties with the original choreography because back in those days they didn't allow actual coition on stage. The music was pleasantly old-fashioned, but neither of us was quite old enough to work up any blurry-eyed nostalgia over it. Still, it was much more enjoyable than the movies I'd seen, and some of the physical feats performed were quite inspiring. We slept late the next morning.\n\nWe dutifully watched the changing of the guard at Buckingham Palace, walked through the British Museum, ate fish and chips, ran up to Stratford-on-Avon and caught the Old Vic doing an incomprehensible play about a mad king, and didn't get into any trouble until the day before we were to leave for Lisbon.\n\nIt was about 2 A.M. and we were tooling our tricycles down a nearly deserted thoroughfare. Turned a corner and there was a gang of boys beating the hell out of someone. I screeched to the curb and leaped out of my vehicle, firing the shotgun-pistol over their heads.\n\nIt was a girl they were attacking; it was rape. Most of them scattered, but one pulled a pistol out of his coat and I shot him. I remember trying to aim for his arm. The blast hit his shoulder and ripped off his arm and what seemed to be half of his chest; it flung him two meters to the side of a building and he must have been dead before he hit the ground.\n\nThe others ran, one of them shooting at me with a little pistol as he went. I watched him trying to kill me for the longest time before it occurred to me to shoot back. I sent one blast way high and he dove into an alley and disappeared.\n\nThe girl looked dazedly around, saw the mutilated body of her attacker, and staggered to her feet and ran off screaming, naked from the waist down. I knew I should have tried to stop her, but I couldn't find my voice and my feet seemed nailed to the sidewalk. A tricycle door slammed and Marygay was beside me.\n\n'What hap\u2014' She gasped, seeing the dead man. 'Wh-what was he doing?'\n\nI just stood there stupefied. I'd certainly seen enough death these past two years, but this was a different thing... there was nothing noble in being crushed to death by the failure of some electronic component, or in having your suit fail and freeze you solid; or even dying in a shootout with the incomprehensible enemy... but death seemed natural in that setting. Not on a quaint little street in old-fashioned London, not for trying to steal what most people would give freely.\n\nMarygay was pulling my arm. 'We've got to get out of here. They'll _brainwipe_ you!'\n\nShe was right. I turned and took one step and fell to the concrete. I looked down at the leg that had betrayed me and bright red blood was pulsing out of a small hole in my calf. Marygay tore a strip of cloth from her blouse and started to bind it. I remember thinking it wasn't a big enough wound to go into shock over, but my ears started to ring and I got lightheaded and everything went red and fuzzy. Before I went under, I heard a siren wailing in the distance.\n\nFortunately, the police also picked up the girl, who was wandering down the street a few blocks away. They compared her version of the thing with mine, both of us under hypnosis. They let me go with a stern admonition to leave law enforcement up to professional law enforcers.\n\nI wanted to get out of the cities: just put a pack on my back and wander through the woods for a while, get my mind straightened out. So did Marygay. But we tried to make arrangements and found that the country was worse than the cities. Farms were practically armed camps, the areas between ruled by nomad gangs who survived by making lightning raids into villages and farms, murdering and plundering for a few minutes, and then fading back into the forest, before help could arrive.\n\nStill, Britishers called their island 'the most civilized country in Europe.' From what we'd heard about France and Spain and Germany, especially Germany, they were probably right.\n\nI talked it over with Marygay, and we decided to cut short our tour and go back to the States. We could finish the tour after we'd become acclimated to the twenty-first century. It was just too much foreignness to take in one dose.\n\nThe dirigible line refunded most of our money and we took a conventional suborbital flight back home. The high altitude made my leg throb, though it was nearly healed. They'd made great strides in the treatment of gunshot wounds, in the past twenty years. Lots of practice.\n\nWe split up at Ellis. Her description of commune life appealed to me more than the city; I made arrangements to join her after a week or so, and went back to Washington. \n\n# 10\n\nI rang the bell and a strange woman answered the door, opening it a couple of centimeters and peering through.\n\n'Pardon me,' I said, 'isn't this Mrs Mandella's residence?'\n\n'Oh, you must be William!' She closed the door and unfastened the chains and opened it wide. 'Beth, look who's here!'\n\nMy mother came into the living room from the kitchen, drying her hands on a towel. 'Willy... what are you doing back so soon?'\n\n'Well, it's \u2014 it's a long story.'\n\n'Sit down, sit down,' the other woman said. 'Let me get you a drink, don't start till I get back.'\n\n'Wait,' my mother said. 'I haven't even introduced you two. William, this is Rhonda Wilder. Rhonda, William.'\n\n'I've been so looking forward to meeting you,' she said. 'Beth has told me all about you \u2014 one cold beer, right?'\n\n'Right.' She was likable enough, a trim middle-aged woman. I wondered why I hadn't met her before. I asked my mother whether she was a neighbor.\n\n'Uh... really more than that, William. She's been my roommate for a couple of years. That's why I had an extra room when you came home \u2014 a single person isn't allowed two bedrooms.'\n\n'But why\u2014'\n\n'I didn't tell you because I didn't want you to feel that you were putting her out of her room while you stayed here. And you weren't, actually; she has\u2014'\n\n'That's right,' Rhonda came in with the beer. 'I've got relatives in Pennsylvania, out in the country. I can stay with them any time.'\n\n'Thanks.' I took the beer. 'Actually, I won't be here long. I'm kind of en route to South Dakota. I could find another place to flop.'\n\n'Oh, no,' Rhonda said. 'I can take the couch.' I was too old-fashioned male-chauv to allow that; we discussed it for a minute and I wound up with the couch.\n\nI filled Rhonda in on who Marygay was and told them about our disturbing experiences in England, how we came back to get our bearings. I had expected my mother to be horrified that I had killed a man, but she accepted it without comment. Rhonda clucked a little bit about our being out in a city after midnight, especially without a bodyguard.\n\nWe talked on these and other topics until late at night, when Mother called her bodyguard and went off to work.\n\nSomething had been nagging at me all night, the way Mother and Rhonda acted toward each other. I decided to bring it out into the open, once Mother was gone.\n\n'Rhonda\u2014' I settled down in the chair across from her.\n\nI didn't know exactly how to put it. 'What, uh, what exactly is your relationship with my mother?'\n\nShe took a long drink. 'Good friends.' She stared at me with a mixture of defiance and resignation. 'Very good friends. Sometimes lovers.'\n\nI felt very hollow and lost. My mother?\n\n'Listen,' she continued. 'You had better stop trying to live in the nineties. This may not be the best of all possible worlds, but you're stuck with it.'\n\nShe crossed and took my hand, almost kneeling in front of me. Her voice was softer. 'William... look, I'm only two years older than you are \u2014 that is, I was born two years before \u2014 what I mean is, I can understand how you feel. B \u2014 your mother understands too. It, our... relationship, wouldn't be a secret to anybody else. It's perfectly normal. A lot has changed, these twenty years. You've got to change too.'\n\nI didn't say anything.\n\nShe stood up and said firmly, 'You think, because your mother is sixty, she's outgrown her need for love? She needs it more than you do. Even now. Especially now.'\n\nAccusation in her eyes. 'Especially now with you coming back from the dead past. Reminding her of how old she is. How \u2014 old I am, twenty years younger.' Her voice quavered and cracked, and she ran to her room.\n\nI wrote Mother a note saying that Marygay had called; an emergency had come up and I had to go immediately to South Dakota. I called a bodyguard and left.\n\nA whining, ozone-leaking, battered old bus let me out at the intersection of a bad road and a worse one. It had taken me an hour to go the 2000 kilometers to Sioux Falls, two hours to get a chopper to Geddes, 150 kilometers away, and three hours waiting and jouncing on the dilapidated bus to go the last 12 kilometers to Freehold, an organization of communes where the Potters had their acreage. I wondered if the progression was going to continue and I would be four hours walking down this dirt road to the farm.\n\nIt was a half-hour before I even came to a building. My bag was getting intolerably heavy and the bulky pistol was chafing my hip. I walked up a stone path to the door of a simple plastic dome and pulled a string that caused a bell to tinkle inside. A peephole darkened.\n\n'Who is it?' Voice muffled by thick wood.\n\n'Stranger asking directions.'\n\n'Ask.' I couldn't tell whether it was a woman or a child.\n\n'I'm looking for the Potters' farm.'\n\n'Just a second.' Footsteps went away and came back. 'Down the road one point nine klicks. Lots of potatoes and green beans on your right. You'll probably smell the chickens.'\n\n'Thanks.'\n\n'If you want a drink we got a pump out back. Can't let you in without my husband's at home.'\n\n'I understand. Thank you.' The water was metallic-tasting but wonderfully cool.\n\nI wouldn't know a potato or green bean plant if it stood up and took a bite out of my ankle, but I knew how to walk a half-meter step. So I resolved to count to 3800 and take a deep breath. I supposed I could tell the difference between the smell of chicken manure and the absence thereof.\n\nAt 3650 there was a rutted path leading to a complex of plastic domes and rectangular buildings apparently made of sod. There was a pen enclosing a small population explosion of chickens. They had a smell but it wasn't strong.\n\nHalfway down the path, a door opened and Marygay came running out, wearing one tiny wisp of cloth. After a slippery but gratifying greeting, she asked what I was doing here so early.\n\n'Oh, my mother had friends staying with her. I didn't want to put them out. Suppose I should have called.'\n\n'Indeed you should have... save you a long dusty walk \u2014 but we've got plenty of room, don't worry about that.'\n\nShe took me inside to meet her parents, who greeted me warmly and made me feel definitely overdressed. Their faces showed their age but their bodies had no sag and few wrinkles.\n\nSince dinner was an occasion, they let the chickens live and instead opened a can of beef, steaming it along with a cabbage and some potatoes. To my plain tastes it was equal to most of the gourmet fare we'd had on the dirigible and in London.\n\nOver coffee and goat cheese (they apologized for not having wine; the commune would have a new vintage out in a couple of weeks), I asked what kind of work I could do.\n\n'Will,' Mr Potter said, 'I don't mind telling you that your coming here is a godsend. We've got five acres that are just sitting out there, fallow, because we don't have enough hands to work them. You can take the plow tomorrow and start breaking up an acre at a time.'\n\n'More potatoes, Daddy?' Marygay asked.\n\n'No, no... not this season. Soybeans \u2014 cash crop and good for the soil. And Will, at night we all take turns standing guard. With four of us, we ought to be able to do a lot more sleeping.' He took a big slurp of coffee. 'Now, what else...'\n\n'Richard,' Mrs Potter said, 'tell him about the greenhouse.'\n\n'That's right, yes, the greenhouse. The commune has a two-acre greenhouse down about a click from here, by the recreation center. Mostly grapes and tomatoes. Everybody spends one morning or one afternoon a week there.\n\n'Why don't you children go down there tonight... show Will the night life in fabulous Freehold? Sometimes you can get a real exciting game of checkers going.'\n\n'Oh, Daddy. It's not that bad.'\n\n'Actually, it isn't. They've got a fair library and a coin-op terminal to the Library of Congress. Marygay tells me you're a reader. That's good.'\n\n'Sounds fascinating.' It did. 'But what about guard?'\n\n'No problem. Mrs Potter \u2014 April \u2014 and I'll take the first four hours \u2014 oh,' he said, standing, 'let me show you the setup.'\n\nWe went out back to 'the tower,' a sandbag hut on stilts. Climbed up a rope ladder through a hole in the middle of the hut.\n\n'A little crowded in here, with two,' Richard said. 'Have a seat.' There was an old piano stool beside the hole in the floor. I sat on it. 'It's handy to be able to see all the field without getting a crick in your neck. Just don't keep turning in the same direction all the time.'\n\nHe opened a wooden crate and uncovered a sleek rifle, wrapped in oily rags. 'Recognize this?'\n\n'Sure.' I'd had to sleep with one in basic training. 'Army standard issue T-sixteen. Semi-automatic, twelve-caliber tumblers \u2014 where the hell did you get it?'\n\n'Commune went to a government auction. It's an antique now, son.' He handed it to me and I snapped it apart. Clean, too clean.\n\n'Has it ever been used?'\n\n'Not in almost a year. Ammo costs too much for target practice. Take a couple of practice shots, though, convince yourself that it works.'\n\nI turned on the scope and just got a washed out bright green. Set for nighttime. Clicked it back to log zero, set the magnification at ten, reassembled it.\n\n'Marygay didn't want to try it out. Said she'd had her fill of that. I didn't press her, but a person's got to have confidence in their tools.'\n\nI clicked off the safety and found a clod of dirt that the range-finder said was between 100 and 120 meters away. Set it at 110, rested the barrel of the rifle on the sandbags, centered the clod in the crosshairs, and squeezed. The round hissed out and kicked up dirt about five centimeters low.\n\n'Fine.' I reset it for night use and safetied it and handed it back. 'What happened a year ago?'\n\nHe wrapped it up carefully, keeping the rags away from the eyepiece. 'Had some jumpers come in. Fired a few rounds and scared 'em away.'\n\n'All right, what's a jumper?'\n\n'Yeah, you wouldn't know.' He shook out a tobacco cigarette and passed me the box. 'I don't know why they don't just call 'em thieves, that's what they are. Murderers, too, sometimes.\n\n'They know that a lot of the commune members are pretty well off. If you raise cash crops you get to keep half the cash; besides, a lot of our members were prosperous when they joined.\n\n'Anyhow, the jumpers take advantage of our relative isolation. They come out from the city and try to sneak in, usually hit one place, and run. Most of the time, they don't get this far in, but the farms closer to the road... we hear gunfire every couple of weeks. Usually just scaring off kids. If it keeps up, a siren goes off and the commune goes on alert.'\n\n'Doesn't sound fair to the people living close to the road.'\n\n'There're compensations. They only have to donate half as much of their crop as the rest of us do. And they're issued heavier weapons.'\n\nMarygay and I took the family's two bicycles and peddled down to the recreation center. I only fell off twice, negotiating the bumpy road in the dark.\n\nIt was a little livelier than Richard had described it. A young nude girl was dancing sensuously to an assortment of homemade drums near the far side of the dome. Turned out she was still in school; it was a project for a 'cultural relativity' class.\n\nMost of the people there, in fact, were young and therefore still in school. They considered it a joke, though. After you had learned to read and write and could pass the Class I literacy test, you only had to take one course per year, and some of those you could pass just by signing up. So much for the 'eighteen years' compulsory education' they had startled us with at Stargate.\n\nOther people were playing board games, reading, watching the girl gyrate, or just talking. There was a bar that served soya, coffee, or thin homemade beer. Not a ration ticket to be seen; all made by the commune or purchased outside with commune tickets.\n\nWe got into a discussion about the war, with a bunch of people who knew Marygay and I were veterans. It's hard to describe their attitude, which was pretty uniform. They were angry in an abstract way that it took so much tax money to support; they were convinced that the Taurans would never be any danger to Earth; but they all knew that nearly half the jobs in the world were associated with the war, and if it stopped, everything would fall apart.\n\nI thought everything was in shambles already, but then I hadn't grown up in this world. And they had never known 'peacetime.'\n\nWe went home about midnight and Marygay and I each stood two hours' guard. By the middle of the next morning I was wishing I had gotten a little more sleep.\n\nThe plow was a big blade on wheels with two handles for steering, atomic powered. Not very much power, though; enough to move it forward at a slow crawl if the blade was in soft earth. Needless to say, there was little soft earth in the unused five acres. The plow would go a few centimeters, get stuck, freewheel until I put some back into it, then move a few more centimeters. I finished a tenth of an acre the first day and eventually got it up to a fifth of an acre a day.\n\nIt was hard, hardening work, but pleasant. I had an earclip that piped music to me, old tapes from Richard's collection, and the sun browned me all over. I was beginning to think I could live that way forever, when suddenly it was finished.\n\nMarygay and I were reading up at the recreation center one evening when we heard faint gunfire down by the road. We decided it'd be smart to get back to the house. We were less than halfway there when firing broke out all along our left, on a line that seemed to extend from the road to far past the recreation center: a coordinated attack. We had to abandon the bikes and crawl on hands and knees in the drainage ditch by the side of the road, bullets hissing over our heads. A heavy vehicle rumbled by, shooting left and right. It took a good twenty minutes to crawl home. We passed two farmhouses that were burning brightly. I was glad ours didn't have any wood.\n\nI noticed there was no return fire coming from our tower, but didn't say anything. There were two dead strangers in front of the house as we rushed inside.\n\nApril was lying on the floor, still alive but bleeding from a hundred tiny fragment wounds. The living room was rubble and dust; someone must have thrown a bomb through a door or window. I left Marygay with her mother and ran out back to the tower. The ladder was pulled up, so I had to shinny up one of the stilts.\n\nRichard was sitting slumped over the rifle. In the pale green glow from the scope I could see a perfectly round hole above his left eye. A little blood had trickled down the bridge of his nose and dried.\n\nI laid his body on the floor and covered his head with my shirt. I filled my pockets with clips and took the rifle back to the house.\n\nMarygay had tried to make her mother comfortable. They were talking quietly. She was holding my shotgun-pistol and had another gun on the floor beside her. When I came in she looked up and nodded soberly, not crying.\n\nApril whispered something and Marygay asked, 'Mother wants to know whether... Daddy had a hard time of it. She knows he's dead.'\n\n'No. I'm sure he didn't feel anything.'\n\n'That's good.'\n\n'It's something.' I should keep my mouth shut. 'It is good, yes.'\n\nI checked the doors and windows for an effective vantage point. I couldn't find anyplace that wouldn't allow a whole platoon to sneak up behind me.\n\n'I'm going to go outside and get on top of the house.' Couldn't go back to the tower. 'Don't you shoot unless somebody gets inside... maybe they'll think the place is deserted.'\n\nBy the time I had clambered up to the sod roof, the heavy truck was coming back down the road. Through the scope I could see that there were five men on it, four in the cab and one who was on the open bed, cradling a machine gun, surrounded by loot. He was crouched between two refrigerators, but I had a clear shot at him. Held my fire, not wanting to draw attention. The truck stopped in front of the house, sat for a minute, and turned in. The window was probably bulletproof, but I sighted on the driver's face and squeezed off a round. He jumped as it ricocheted, whining, leaving an opaque star on the plastic, and the man in back opened up. A steady stream of bullets hummed over my head; I could see them thumping into the sandbags of the tower. He didn't see me.\n\nThe truck wasn't ten meters away when the shooting stopped. He was evidently reloading, hidden behind the refrigerator. I took careful aim and when he popped up to fire I shot him in the throat. The bullet being a tumbler, it exited through the top of his skull.\n\nThe driver pulled the truck around in a long arc so that, when it stopped, the door to the cab was flush with the door of the house. This protected them from the tower and also from me, though I doubted they yet knew where I was; a T-16 makes no flash and very little noise. I kicked off my shoes and stepped cautiously onto the top of the cab, hoping the driver would get out on his side. Once the door opened I could fill the cab with ricocheting bullets.\n\nNo good. The far door, hidden from me by the roof's overhang, opened first. I waited for the driver and hoped that Marygay was well hidden. I shouldn't have worried.\n\nThere was a deafening roar, then another and another. The heavy truck rocked with the impact of thousands of tiny flechettes. One short scream that the second shot ended.\n\nI jumped from the truck and ran around to the back door. Marygay had her mother's head on her lap, and someone was crying softly. I went to them and Marygay's cheeks were dry under my palms.\n\n'Good work, dear.'\n\nShe didn't say anything. There was a steady heavy dripping sound from the door and the air was acrid with smoke and the smell of fresh meat. We huddled together until dawn.\n\nI had thought April was sleeping, but in the dim light her eyes were wide open and filmed. Her breath came in shallow rasps. Her skin was gray parchment and dried blood. She didn't answer when we talked to her.\n\nA vehicle was coming up the road, so I took the rifle and went outside. It was a dump truck with a white sheet draped over one side and a man standing in the back with a megaphone repeating, 'Wounded... wounded.' I waved and the truck came in. They took April out on a makeshift litter and told us which hospital they were going to. We wanted to go along but there was simply no room; the bed of the truck was covered with people in various stages of disrepair.\n\nMarygay didn't want to go back inside because it was getting light enough to see the men she had killed so completely. I went back in to get some cigarettes and forced myself to look. It was messy enough, but just didn't disturb me that much. _That_ bothered me, to be confronted with a pile of human hamburger and mainly notice the flies and ants and smell. Death is so much neater in space.\n\nWe buried her father behind the house, and when the truck came back with April's small body wrapped in a shroud, we buried her beside him. The commune's sanitation truck came by a little later, and gas-masked men took care of the jumpers' bodies.\n\nWe sat in the baking sun, and finally Marygay wept, for a long time, silently.\n\n# 11\n\nWe got off the plane at Dulles and found a monorail to Columbia.\n\nIt was a pleasingly diverse jumble of various kinds of buildings, arranged around a lake, surrounded by trees. All of the buildings were connected by slidewalk to the largest place, a fullerdome with stores and schools and offices.\n\nWe could have taken the enclosed slidewalk to Mom's place, but instead walked alongside it in the good cold air that smelled of fallen leaves. People slid by on the other side of the plastic, carefully not staring.\n\nMom didn't answer her door, but she'd given me an entry card. Mom was asleep in the bedroom, so Marygay and I settled in the living room and read for a while.\n\nWe were startled suddenly by a loud fit of coughing from the bedroom. I raced over and knocked on the door.\n\n'William? I didn't\u2014' coughing '\u2014come in, I didn't know you were...'\n\nShe was propped up in bed, the light on, surrounded by various nostrums. She looked ghastly, pale and lined.\n\nShe lit a joint and it seemed to quell the coughing. 'When did you get in? I didn't know...'\n\n'Just a few minutes ago... How long has this... have you been...'\n\n'Oh, it's just a bug I picked up after Rhonda went to see her kids. I'll be fine in a couple of days.' She started coughing again, drank some thick red liquid from a bottle. All of her medicines seemed to be the commercial, patent variety.\n\n'Have you seen a doctor?'\n\n'Doctor? Heavens no, Willy. They don't have... it's not serious... don't\u2014'\n\n'Not serious?' At eighty-four. 'For Chrissake, mother.' I went to the phone in the kitchen and with some difficulty managed to get the hospital.\n\nA plain girl in her twenties formed in the cube. 'Nurse Donalson, general services.' She had a fixed smile, professional sincerity. But then everybody smiled.\n\n'My mother needs to be looked at by a doctor. She has a\u2014'\n\n'Name and number, please.'\n\n'Beth Mandella.' I spelled it. 'What number?'\n\n'Medical services number, of course,' she smiled.\n\nI called into Mom and asked her what her number was. 'She says she can't remember.'\n\n'That's all right, sir, I'm sure I can find her records.' She turned her smile to a keyboard beside her and punched out a code.\n\n'Beth Mandella?' she said, her smile turning quizzical. 'You're her son? She must be in her eighties.'\n\n'Please. It's a long story. She really has to see a doctor.'\n\n'Is this some kind of joke?'\n\n'What do you mean?' Strangled coughing from the other room, the worst yet. 'Really \u2014 this might be very serious, you've got to\u2014'\n\n'But sir, Mrs Mandella got a zero priority rating way back in 2010.'\n\n' _What the hell is that supposed to mean?'_\n\n'S-i-r...' The smile was hardening in place.\n\n'Look. Pretend that I came from another planet. What is a \"zero priority rating\"?'\n\n'Another \u2014 oh! I know you!' She looked off to the left. 'Sonya \u2014 come over here a second. You'd never guess who...' Another face crowded the cube, a vapid blonde girl whose smile was twin to the other nurse's. 'Remember? On the stat this morning?'\n\n'Oh, yeah,' she said. 'One of the soldiers \u2014 hey, that's really max, really max.' The head withdrew.\n\n'Oh, Mr Mandella,' she said, effusive. 'No wonder you're confused. It's really very simple.'\n\n'Well?'\n\n'It's part of the Universal Medical Security System. Everybody gets a rating on their seventieth birthday. It comes in automatically from Geneva.'\n\n'What does it rate? What does it mean?' But the ugly truth was obvious.\n\n'Well, it tells how important a person is and what level of treatment he's allowed. Class three is the same as anybody else's; class two is the same except for certain life-extending\u2014'\n\n'And class zero is no treatment at all.'\n\n'That's correct, Mr Mandella.' And in her smile was not a glimmer of pity or understanding.\n\n'Thank you.' I disconnected. Marygay was standing behind me, crying soundlessly with her mouth wide open.\n\nI found mountaineer's oxygen at a sporting goods store and even managed to get some black-market antibiotics through a character in a bar downtown in Washington. But Mom was beyond being able to respond to amateur treatment. She lived four days. The people from the crematorium had the same fixed smile.\n\nI tried to get through to my brother, Mike, on the Moon, but the phone company wouldn't let me place the call until I had signed a contract and posted a $25,000 bond. I had to get a credit transfer from Geneva. The paperwork took half a day.\n\nI finally got through to him. Without preamble:\n\n'Mother's dead.'\n\nFor a fraction of a second, the radio waves wandered up to the Moon, and in another fraction, came back. He started and then nodded his head slowly. 'No surprise. Every time I've come down to Earth the past ten years, I've wondered whether she'd still be there. Neither of us had enough money to keep in very close touch.' He had told us in Geneva that a letter from Luna to Earth cost $100 postage \u2014 plus $5,000 tax. It discouraged communication with what the UN considered to be a bunch of regrettably necessary anarchists.\n\nWe commiserated for a while and then Mike said, 'Willy, Earth is no place for you and Marygay; you know that by now. Come to Luna. Where you can still be an individual. Where we don't throw people out the airlock on their seventieth birthday.'\n\n'We'd have to rejoin UNEF.'\n\n'True, but you wouldn't have to fight. They say they need you more for training. You could study in your spare time, bring your physics up to date \u2014 maybe wind up eventually in research.'\n\nWe talked some more, a total of three minutes. I got $1000 back.\n\nMarygay and I talked about it through the night. Maybe our decision would have been different if we hadn't been staying there, surrounded by Mother's life and death, but when the dawn came the proud, ambitious, careful beauty of Columbia had turned sinister and foreboding.\n\nWe packed our bags and had our money transferred to the Tycho Credit Union and took a monorail to the Cape.\n\n'In case you're interested, you aren't the first combat veterans to come back.' The recruiting officer was a muscular lieutenant of indeterminate sex. I flipped a coin mentally and it came up tails.\n\n'Last I heard, there had been nine others,' she said in her husky tenor. 'All of them opted for the Moon... maybe you'll find some of your friends there.' She slid two simple forms across the desk. 'Sign these and you're in again. Second lieutenants.'\n\nThe form was a simple request to be assigned to active duty; we had never really gotten out of the Force, since they extended the draft law, but had just been on inactive status. I scrutinized the paper.\n\n'There's nothing on this about the guarantees we were given at Stargate.'\n\n'That won't be necessary, Lieutenant.' I handed back the form. So did Marygay.\n\n'Let me check.' She left the desk and disappeared into an office. After a while we heard a printer rattle.\n\nShe brought back the same two sheets, with an addition typed under our names: GUARANTEED LOCATION OF CHOICE [LUNA] AND ASSIGNMENT OF CHOICE [COMBAT TRAINING SPECIALIST].\n\nWe got a thorough physical checkup and were fitted for new fighting suits, made our financial arrangements, and caught the next morning's shuttle. We laid over at Earth-port, enjoying zero gravity for a few hours, and then caught a ride to Luna, setting down at the Grimaldi base.\n\nOn the door to the Transient Officers' Billet, some wag had scraped 'abandon hope all ye who enter.' We found our two-man cubicle and began changing for chow.\n\nTwo raps on the door. 'Mail call, sirs.'\n\nI opened the door and the sergeant standing there saluted. I just looked at him for a second and then remembered I was an officer and returned the salute. He handed me two identical faxes. I gave one to Marygay and we both gasped at the same time:\n\n* *ORDERS * * ORDERS * * ORDERS\n\nTHE FOLLOWING NAMED PERSONNEL:\n\nMandella, William 2LT [11 575 278] COCOMM D Co\n\nGRITRABN\n\nAND\n\nPotter, Marygay 2LT [17 386 907] COCOMM B Co\n\nGRITRABN ARE HEREBY REASSIGNED TO:\n\nLT Mandella: PLCOMM 2 PL STFTHET STARGATE\n\nLT Potter: PLCOMM 3 PL STFTHET STARGATE.\n\nDESCRIPTION OF DUTIES:\n\nCommand infantry platoon in Tet-2 Campaign.\n\nTHE ABOVE NAMED PERSONNEL WILL REPORT IMMEDIATELY TO GRIMALDI TRANSPORTATION BATTALION TO BE MANIFESTED TO STARGATE.\n\nISSUED STARGATE TACBD\/1298-8684-1450\/20 AUG 2019 SG:\n\nBY AUTHO STFCOM Commander.\n\n* *ORDERS * * ORDERS * * ORDERS\n\n'They didn't waste any time, did they?' Marygay said bitterly.\n\n'Must be a standing order. Strike Force Command's light weeks away; they can't even know we've re-upped yet.'\n\n'What about our...' She let it trail off.\n\n'The guarantee. Well, we were given our assignment of choice. Nobody guaranteed we'd have the assignment for more than an hour.'\n\n'It's so dirty.'\n\nI shrugged. 'It's so army.'\n\nBut I couldn't shake the feeling that we were going home.\n\n# Lieutenant   \nMandella   \n2024\u20142389 AD\n'Quick and dirty.' I was looking at my platoon sergeant, Santesteban, but talking to myself. And anybody else who was listening.\n\n'Yeah,' he said. 'Gotta do it in the first coupla minutes or we're screwed tight.' He was matter-of-fact, laconic. Drugged.\n\nPrivate Collins came up with Halliday. They were holding hands unself-consciously. 'Lieutenant Mandella?' Her voice broke a little. 'Can we have just a minute?'\n\n'One minute,' I said, too abruptly. 'We have to leave in five. I'm sorry.'\n\nHard to watch those two together now. Neither one had any combat experience. But they knew what everybody did; how slim their chances were of ever being together again. They slumped in a corner and mumbled words and traded mechanical caresses, no passion or even comfort. Collins's eyes shone but she wasn't weeping. Halliday just looked grim, numb. She was normally by far the prettier of the two, but the sparkle had gone out of her and left a well-formed dull shell.\n\nI'd gotten used to open female homosex in the months since we'd left Earth. Even stopped resenting the loss of potential partners. The men together still gave me a chill, though.\n\nI stripped and backed into the clamshelled suit. The new ones were a hell of a lot more complicated, with all the new biometrics and trauma maintenance. But well worth the trouble of hooking up, in case you got blown apart just a little bit. Go home to a comfortable pension with heroic prosthesis. They were even talking about the possibility of regeneration, at least for missing arms and legs. Better get it soon, before Heaven filled up with fractional people. Heaven was the new hospital\/rest-and-recreation planet.\n\nI finished the set-up sequence and the suit closed by itself. Gritted my teeth against the pain that never came, when the internal sensors and fluid tubes poked into your body. Conditioned neural bypass, so you felt only a slight puzzling dislocation. Rather than the death of a thousand cuts.\n\nCollins and Halliday were getting into their suits now and the other dozen were almost set, so I stepped over to the third platoon's staging area. Say goodbye again to Marygay.\n\nShe was suited and heading my way. We touched helmets instead of using the radio. Privacy.\n\n'Feeling OK, honey?'\n\n'All right,' she said. 'Took my pill.'\n\n'Yeah, happy times.' I'd taken mine too, supposed to make you feel optimistic without interfering with your sense of judgment. I knew most of us would probably die, but I didn't feel too bad about it. 'Sack with me tonight?'\n\n'If we're both here,' she said neutrally. 'Have to take a pill for that, too.' She tried to laugh. 'Sleep, I mean. How're the new people taking it? You have ten?'\n\n'Ten, yeah, they're OK. Doped up, quarter-dose.'\n\n'I did that, too; try to keep them loose.'\n\nIn fact, Santesteban was the only other combat veteran in my platoon; the four corporals had been in UNEF for a while but hadn't ever fought.\n\nThe speaker in my cheekbone crackled and Commander Cortez said, 'Two minutes. Get your people lined up.'\n\nWe had our goodbye and I went back to check my flock. Everybody seemed to have gotten suited up without any problems, so I put them on line. We waited for what seemed like a long time.\n\n'All right, load 'em up.' With the word 'up,' the bay door in front of me opened \u2014 the staging area having already been bled of air \u2014 and I led my men and women through to the assault ship.\n\nThese new ships were ugly as hell. Just an open framework with clamps to hold you in place, swiveled lasers fore and aft, small tachyon powerplants below the lasers. Everything automated; the machine would land us as quickly as possible and then zip off to harass the enemy. It was a one-use, throwaway drone. The vehicle that would come pick us up if we survived was cradled next to it, much prettier.\n\nWe clamped in and the assault ship cast off from the _Sangre y Victoria_ with twin spurts from the yaw jets. Then the voice of the machine gave us a short countdown and we sped off at four gees' acceleration, straight down.\n\nThe planet, which we hadn't bothered to name, was a chunk of black rock without any normal star close enough to give it heat. At first it was visible only by the absence of stars where its bulk cut off their light, but as we dropped closer we could see subtle variations in the blackness of its surface. We were coming down on the hemisphere opposite the Taurans' outpost.\n\nOur recon had shown that their camp sat in the middle of a flat lava plain several hundred kilometers in diameter. It was pretty primitive compared to other Tauran bases UNEF had encountered, but there wouldn't be any sneaking up on it. We were going to careen over the horizon some fifteen klicks from the place, four ships converging simultaneously from different directions, all of us decelerating like mad, hopefully to drop right in their laps and come up shooting. There would be nothing to hide behind.\n\nI wasn't worried, of course. Abstractedly, I wished I hadn't taken the pill.\n\nWe leveled off about a kilometer from the surface and sped along much faster than the rock's escape velocity, constantly correcting to keep from flying away. The surface rolled below us in a dark gray blur; we shed a little light from the pseudo-cerenkov glow made by our tachyon exhaust, scooting away from our reality into its own.\n\nThe ungainly contraption skimmed and jumped along for some ten minutes; then suddenly the front jet glowed and we were snapped forward inside our suits, eyeballs trying to escape from their sockets in the rapid deceleration.\n\n'Prepare for ejection,' the machine's female-mechanical voice said. 'Five, four...'\n\nThe ship's lasers started firing, millisecond flashes freezing the land below in jerky stroboscopic motion. It was a twisted, pock-marked jumble of fissures and random black rocks, a few meters below our feet. We were dropping, slowing.\n\n'Three\u2014' It never got any farther. There was a too-bright flash and I saw the horizon drop away as the ship's tail pitched down \u2014 then clipped the ground, and we were rolling, horribly, pieces of people and ship scattering. Then we slid pinwheeling to a bumpy halt, and I tried to pull free but my leg was pinned under the ship's bulk: excruciating pain and a dry crunch as the girder crushed my leg; shrill whistle of air escaping my breached suit; then the trauma maintenance turned on _snick_ , more pain, then no pain and I was rolling free, short stump of a leg trailing blood that froze shiny black on the dull black rock. I tasted brass and a red haze closed everything out, then deepened to the brown of river clay, then loam and I passed out, with the pill thinking _this is not so bad_...\n\nThe suit is set up to save as much of your body as possible. If you lose part of an arm or a leg, one of sixteen razor-sharp irises closes around your limb with the force of a hydraulic press, snipping it off neatly and sealing the suit before you can die of explosive decompression. Then 'trauma maintenance' cauterizes the stump, replaces lost blood, and fills you full of happy-juice and No-shock. So you will either die happy or, if your comrades go on to win the battle, eventually be carried back up to the ship's aid station.\n\nWe'd won that round, while I slept swaddled in dark cotton. I woke up in the infirmary. It was crowded. I was in the middle of a long row of cots, each one holding someone who had been three-fourths (or less) saved by his suit's trauma maintenance feature. We were being ignored by the ship's two doctors, who stood in bright light at operating tables, absorbed in blood rituals. I watched them for a long time. Squinting into the bright light, the blood on their green tunics could have been grease, the swathed bodies, odd soft machines that they were fixing. But the machines would cry out in their sleep, and the mechanics muttered reassurances while they plied their greasy tools. I watched and slept and woke up in different places.\n\nFinally I woke up in a regular bay. I was strapped down and being fed through a tube, biosensor electrodes attached here and there, but no medics around. The only other person in the little room was Marygay, sleeping on the bunk next to me. Her right arm was amputated just above the elbow.\n\nI didn't wake her up, just looked at her for a long time and tried to sort out my feelings. Tried to filter out the effect of the mood drugs. Looking at her stump, I could feel neither empathy nor revulsion. I tried to force one reaction, and then the other, but nothing real happened. It was as if she had always been that way. Was it drugs, conditioning, love? Have to wait and see.\n\nHer eyes opened suddenly and I knew she had been awake for some time, had been giving me time to think. 'Hello, broken toy,' she said.\n\n'How \u2014 how do you feel?' Bright question.\n\nShe put a finger to her lips and kissed it, a familiar gesture, reflection. 'Stupid, numb. Glad not to be a soldier anymore.' She smiled. 'Did they tell you? We're going to Heaven.'\n\n'No. I knew it would be either there or Earth.'\n\n'Heaven will be better.' Anything would. 'I wish we were there now.'\n\n'How long?' I asked. 'How long before we get there?'\n\nShe rolled over and looked at the ceiling. 'No telling. You haven't talked to anybody?'\n\n'Just woke up.'\n\n'There's a new directive they didn't bother to tell us about before. The _Sangre y Victoria_ got orders for four missions. We have to keep on fighting until we've done all four. Or until we've sustained so many casualties that it wouldn't be practical to go on.'\n\n'How many is that?'\n\n'I wonder. We lost a good third already. But we're headed for Aleph-7. Panty raid.' New slang term for the type of operation whose main object was to gather Tauran artefacts, and prisoners if possible. I tried to find out where the term came from, but the one explanation I got was really idiotic.\n\nOne knock on the door and Dr Foster barged in. He fluttered his hands. 'Still in separate _beds_? Marygay, I thought you were more recovered than that.' Foster was all right. A flaming mariposa, but he had an amused tolerance for heterosexuality.\n\nHe examined Marygay's stump and then mine. He stuck thermometers in our mouths so we couldn't talk. When he spoke, he was serious and blunt.\n\n'I'm not going to sugarcoat anything for you. You're both on happyjuice up to your ears, and the loss you've sustained isn't going to bother you until I take you off the stuff. For my own convenience I'm keeping you drugged until you get to Heaven. I have twenty-one amputees to take care of. We can't handle twenty-one psychiatric cases.\n\n'Enjoy your peace of mind while you still have it. You two especially, since you'll probably want to stay together. The prosthetics you get on Heaven will work just fine, but every time you look at his mechanical leg or you look at her arm, you're going to think of how lucky the other one is. You're going to constantly trigger memories of pain and loss for each other... You may be at each other's throats in a week. Or you may share a sullen kind of love for the rest of your lives.\n\n'Or you may be able to transcend it. Give each other strength. Just don't kid yourselves if it doesn't work out.'\n\nHe checked the readout on each thermometer and made a notation in his notebook. 'Doctor knows best, even if he is a little weird by your own old-fashioned standards. Keep it in mind.' He took the thermometer out of my mouth and gave me a little pat on the shoulder. Impartially, he did the same to Marygay. At the door, he said, 'We've got collapsar insertion in about six hours. One of the nurses will take you to the tanks.'\n\nWe went into the tanks \u2014 so much more comfortable and safer than the old individual acceleration shells \u2014 and dropped into the Tet-2 collapsar field already starting the crazy fifty-gee evasive maneuvers that would protect us from enemy cruisers when we popped out by Aleph-7, a microsecond later.\n\nPredictably the Aleph-7 campaign was a dismal failure, and we limped away from it with a two-campaign total of fifty-four dead and thirty-nine cripples bound for Heaven. Only twelve soldiers were still able to fight, but they weren't exactly straining at the leash.\n\nIt took three collapsar jumps to get to Heaven. No ship ever went there directly from a battle, even though the delay sometimes cost extra lives. It was the one place besides Earth that the Taurans could not be allowed to find.\n\nHeaven was a lovely, unspoiled Earth-like world; what Earth might have been like if men had treated her with compassion instead of lust. Virgin forests, white beaches, pristine deserts. The few dozen cities there either blended perfectly with the environment (one was totally underground) or were brazen statements of human ingenuity; Oceanus, in a coral reef with six fathoms of water over its transparent roof, Boreas, perched on a sheared-off mountaintop in the polar wasteland; and the fabulous Skye, a huge resort city that floated from continent to continent on the trade winds.\n\nWe landed, as everyone does, at the jungle city. Threshold. Three-fourths hospital, it's by far the planet's largest city, but you couldn't tell that from the air, flying down from orbit. The only sign of civilization was a short runway that suddenly appeared, a small white patch dwarfed to insignificance by the stately rain forest that crowded in from the east and an immense ocean that dominated the other horizon.\n\nOnce under the arboreal cover, the city was very much in evidence. Low buildings of native stone and wood rested among ten-meter-thick tree trunks. They were connected by unobtrusive stone paths, with one wide promenade meandering off to the beach. Sunlight filtered down in patches, and the air held a mixture of forest sweetness and salt tang.\n\nI later learned that the city sprawled out over 200 square kilometers, that you could take a subway to anyplace that was too far to walk. The ecology of Threshold was very carefully balanced and maintained so as to resemble the jungle outside, with all the dangerous and uncomfortable elements eliminated. A powerful pressor field kept out large predators and such insect life as was not necessary for the health of the plants inside.\n\nWe walked, limped and rolled into the nearest building, which was the hospital's reception area. The rest of the hospital was underneath, thirty subterranean stories. Each person was examined and assigned his own room; I tried to get a double with Marygay, but they weren't set up for that.\n\n'Earth-year' was 2189. So I was 215 years old, God, look at that old codger. Somebody pass the hat \u2014 no, not necessary. The doctor who examined me said that my accumulated pay would be transferred from Earth to Heaven. With compound interest, I was just shy of being a billionaire. He remarked that I'd find lots of ways to spend my billion on Heaven.\n\nThey took the most severely wounded first, so it was several days before I went into surgery. Afterwards, I woke up in my room and found that they had grafted a prosthesis onto my stump, an articulated structure of shiny metal that to my untrained eye looked exactly like the skeleton of a leg and foot. It looked creepy as hell, lying there in a transparent bag of fluid, wires running out of it to a machine at the end of the bed.\n\nAn aide came in. 'How you feelin', sir?' I almost told him to forget the 'sir' bullshit, I was out of the army and staying out this time. But it might be nice for the guy to keep feeling that I outranked him.\n\n'I don't know. Hurts a little.'\n\n'Gonna hurt like a sonuvabitch. Wait'll the nerves start to grow.'\n\n'Nerves?'\n\n'Sure.' He was fiddling with the machine, reading dials on the other side. 'How you gonna have a leg without nerves? It'd just sit there.'\n\n'Nerves? Like regular nerves? You mean I can just think \"move\" and the thing moves?'\n\n''Course you can.' He looked at me quizzically, then went back to his adjustments.\n\nWhat a wonder. 'Prosthetics has sure come a long way.'\n\n'Pross-what-ics?'\n\n'You know, artificial\u2014'\n\n' _Oh_ yeah, like in books. Wooden legs, hooks and stuff.'\n\nHow'd he ever get a job? 'Yeah, prosthetics. Like this thing on the end of my stump.'\n\n'Look, sir.' He set down the clipboard he'd been scribbling on. 'You've been away a long time. That's gonna be a leg, just like the other leg except it can't break.'\n\n'They do it with arms, too?'\n\n'Sure, any limb.' He went back to his writing. 'Livers, kidneys, stomachs, all kinds of things. Still working on hearts and lungs, have to use mechanical substitutes.'\n\n'Fantastic.' Marygay would be whole again, too.\n\nHe shrugged. 'Guess so. They've been doing it since before I was born. How old are you, sir?'\n\nI told him, and he whistled. 'God _damn_. You musta been in it from the beginning.' His accent was very strange. All the words were right but all the sounds were wrong.\n\n'Yeah. I was in the Epsilon attack. Aleph-null.' They'd started naming collapsars after letters of the Hebrew alphabet, in order of discovery, then ran out of letters when the damn things started cropping up all over the place. So they added numbers after the letters; last I heard, they were up to Yod-42.\n\n'Wow, ancient history. What was it like back then?'\n\n'I don't know. Less crowded, nicer. Went back to Earth a year ago \u2014 hell, a century ago. Depends on how you look at it. It was so bad I re-enlisted, you know? Bunch of zombies. No offense.'\n\nHe shrugged. 'Never been there, myself. People who come from there seem to miss it. Maybe it got better.'\n\n'What, you were born on another planet? Heaven?' No wonder I couldn't place his accent.\n\n'Born, raised and drafted.' He put the pen back in his pocket and folded the clipboard up to a wallet-sized package. 'Yes, sir. Third-generation angel. Best damned planet in all UNEF.' He spelled it out, didn't say 'youneff' the way I'd always heard it.\n\n'Look, I've gotta run, Lieutenant. Two other monitors to check, this hour.' He backed out the door. 'You need anything, there's a buzzer on the table there.'\n\nThird-generation angel. His grandparents came from Earth, probably when I was a young punk of a hundred. I wondered how many other worlds they'd colonized while my back was turned. Lose an arm, grow a new one?\n\nIt was going to be good to settle down and live a whole year for every year that went by.\n\nThe guy wasn't kidding about the pain. And it wasn't just the new leg, though that hurt like boiling oil. For the new tissues to 'take,' they'd had to subvert my body's resistance to alien cells; cancer broke out in a half-dozen places and had to be treated separately, painfully.\n\nI was feeling pretty used up, but it was still kind of fascinating to watch the leg grow. White threads turned into blood vessels and nerves, first hanging a little slack, then moving into place as the musculature grew up around the metal bone.\n\nI got used to seeing it grow, so the sight never repelled me. But when Marygay came to visit, it was a jolt \u2014 she was ambulatory before the skin on her new arm had started to grow; looked like a walking anatomy demonstration. I got over the shock, though, and she eventually came in for a few hours every day to play games or trade gossip or just sit and read, her arm slowly growing inside the plastic cast.\n\nI'd had skin for a week before they uncased the new leg and trundled the machine away. It was ugly as hell, hairless and dead white, stiff as a metal rod. But it worked, after a fashion. I could stand up and shuffle along.\n\nThey transferred me to orthopedics, for 'range and motion repatterning' \u2014 a fancy name for slow torture. They strap you into a machine that bends both the old and new legs simultaneously. The new one resists.\n\nMarygay was in a nearby section, having her arm twisted methodically. It must have been even worse on her; she looked gray and haggard every afternoon, when we met to go upstairs and sunbathe in the broken shade.\n\nAs the days went by, the therapy became less like torture and more like strenuous exercise. We both began swimming for an hour or so every clear day, in the calm, pressor-guarded water off the beach. I still limped on land, but in the water I could get around pretty well.\n\nThe only real excitement we had on Heaven \u2014 excitement to our combat-blunted sensibilities \u2014 was in that carefully guarded water.\n\nThey have to turn off the pressor field for a split second every time a ship lands; otherwise it would just ricochet off over the ocean. Every now and then an animal slips in, but the dangerous land animals are too slow to get through. Not so in the sea.\n\nThe undisputed master of Heaven's oceans is an ugly customer that the angels, in a fit of originality, named the 'shark.' It could eat a stack of Earth sharks for breakfast, though.\n\nThe one that got in was an average-sized white shark who had been bumping around the edge of the pressor field for days, tormented by all that protein splashing around inside. Fortunately, there's a warning siren two minutes before the pressor is shut down, so nobody was in the water when he came streaking through. And streak through he did, almost beaching himself in the fury of his fruitless attack.\n\nHe was twelve meters of flexible muscle with a razor-sharp tail at one end and a collection of arm-length fangs at the other. His eyes, big yellow globes, were set on stalks more than a meter out from his head. His mouth was so wide that, open, a man could comfortably stand in it. Make an impressive photo for his heirs.\n\nThey couldn't just turn off the pressor field and wait for the thing to swim away. So the Recreation Committee organized a hunting party.\n\nI wasn't too enthusiastic about offering myself up as an hors d'oeuvre to a giant fish, but Marygay had spearfished a lot as a kid growing up in Florida and was really excited by the prospect. I went along with the gag when I found out how they were doing it; seemed safe enough.\n\nThese 'sharks' supposedly never attack people in boats. Two people who had more faith in fishermen's stories than I had gone out to the edge of the pressor field in a rowboat, armed only with a side of beef. They kicked the meat overboard and the shark was there in a flash.\n\nThis was the cue for us to step in and have our fun. There were twenty-three of us fools waiting on the beach with flippers, masks, breathers and one spear each. The spears were pretty formidable, though, jet-propelled and with high-explosive heads.\n\nWe splashed in and swam in phalanx, underwater, toward the feeding creature. When it saw us at first, it didn't attack. It tried to hide its meal, presumably so that some of us wouldn't be able to sneak around and munch on it while the shark was dealing with the others. But every time he tried for the deep water, he'd bump into the pressor field. He was obviously getting pissed off.\n\nFinally, he just let go of the beef, whipped around and charged. Great sport. He was the size of your finger one second, way down there at the other end of the field, then suddenly as big as the guy next to you and closing fast.\n\nMaybe ten of the spears hit him \u2014 mine didn't \u2014 and they tore him to shreds. But even after an expert, or lucky, brain shot that took off the top of his head and one eye, even with half his flesh and entrails scattered in a bloody path behind him, he slammed into our line and clamped his jaws around a woman, grinding off both of her legs before it occurred to him to die.\n\nWe carried her, barely alive, back to the beach, where an ambulance was waiting. They poured her full of blood surrogate and No-shock and rushed her to the hospital, where she survived to eventually go through the agony of growing new legs. I decided that I would leave the hunting of fish to other fish.\n\nMost of our stay at Threshold, once the therapy became bearable, was pleasant enough. No military discipline, lots of reading and things to potter around with. But there was a pall over it, since it was obvious that we weren't out of the army; just pieces of broken equipment that they were fixing up to throw back into the fray. Marygay and I each had another three years to serve in our lieutenancies.\n\nBut we did have six months of rest and recreation coming once our new limbs were pronounced in good working order. Marygay was released two days before I was but waited around for me.\n\nMy back pay came to $892,746,012. Not in the form of bales of currency, fortunately; on Heaven they used an electronic credit exchange, so I carried my fortune around in a little machine with a digital readout. To buy something you punched in the vendor's credit number and the amount of purchase; the sum was automatically shuffled from your account to his. The machine was the size of a slender wallet and coded to your thumbprint.\n\nHeaven's economy was governed by the continual presence of thousands of resting, recreating millionaire soldiers. A modest snack would cost a hundred bucks, a room for a night at least ten times that. Since UNEF built and owned Heaven, this runaway inflation was pretty transparently a simple way of getting our accumulated pay back into the economic mainstream.\n\nWe had fun, desperate fun. We rented a flyer and camping gear and went off for weeks, exploring the planet. There were icy rivers to swim and lush jungles to crawl through; meadows and mountains and polar wastes and deserts.\n\nWe could be totally protected from the environment by adjusting our individual pressor fields \u2014 sleep naked in a blizzard \u2014 or we could take nature straight. At Marygay's suggestion, the last thing we did before coming back to civilization was to climb a pinnacle in the desert, fasting for several days to heighten our sensibilities (or warp our perceptions, I'm still not sure), and sit back-to-back in the searing heat, contemplating the languid flux of life.\n\nThen off to the fleshpots. We toured every city on the planet, and each had its own particular charm, but we finally returned to Skye to spend the rest of our leave time.\n\nThe rest of the planet was bargain-basement compared to Skye. In the four weeks we were using the airborne pleasure dome as our home base, Marygay and I each went through a good half-billion dollars. We gambled \u2014 sometimes losing a million dollars or more in a night \u2014 ate and drank the finest the planet had to offer, and sampled every service and product that wasn't too bizarre for our admittedly archaic tastes. We each had a personal servant whose salary was rather more than that of a major general.\n\nDesperate fun, as I said. Unless the war changed radically, our chances of surviving the next three years were microscopic. We were remarkably healthy victims of a terminal disease, trying to cram a lifetime of sensation into a half of a year.\n\nWe did have the consolation, not small, that however short the remainder of our lives would be, we would at least be together. For some reason it never occurred to me that even that could be taken from us.\n\nWe were enjoying a light lunch in the transparent 'first floor' of Skye, watching the ocean glide by underneath us, when a messenger bustled in and gave us two envelopes: our orders.\n\nMarygay had been bumped to captain, and I to major, on the basis of our military records and tests we had taken at Threshold. I was a company commander and she was a company's executive officer.\n\nBut they weren't the same company.\n\nShe was going to muster with a new company being formed right here on Heaven. I was going back to Stargate for 'indoctrination and education' before taking command.\n\nFor a long time we couldn't say anything. 'I'm going to protest,' I said finally, weakly. 'They can't make me a commander. Into a commander.'\n\nShe was still struck dumb. This was not just a separation. Even if the war was over and we left for Earth only a few minutes apart, in different ships, the geometry of the collapsar jump would pile up years between us. When the second one arrived on Earth, his partner would probably be a half-century older; more probably dead.\n\nWe sat there for some time, not touching the exquisite food, ignoring the beauty around us and beneath us, only conscious of each other and the two sheets of paper that separated us with a gulf as wide and real as death.\n\nWe went back to Threshold. I protested but my arguments were shrugged off. I tried to get Marygay assigned to my company, as my exec. They said my personnel had all been allotted. I pointed out that most of them probably hadn't even been born yet. Nevertheless, allotted, they said. It would be almost a century, I said, before I even get to Stargate. They replied that Strike Force Command _plans_ in terms of centuries.\n\nNot in terms of people.\n\nWe had a day and a night together. The less said about that, the better. It wasn't just losing a lover. Marygay and I were each other's only link to real life, the Earth of the 1980s and 90s. Not the perverse grotesquerie we were supposedly fighting to preserve. When her shuttle took off it was like a casket rattling down into a grave.\n\nI commandeered computer time and found out the orbital elements of her ship and its departure time; found out I could watch her leave from 'our' desert.\n\nI landed on the pinnacle where we had starved together and, a few hours before dawn, watched a new star appear over the western horizon, flare to brilliance and fade as it moved away, becoming just another star, then a dim star, and then nothing. I walked to the edge and looked down the sheer rock face to the dim frozen rippling of dunes half a kilometer below. I sat with my feet dangling over the edge, thinking nothing, until the sun's oblique rays illuminated the dunes in a soft, tempting chiaroscuro of low relief. Twice I shifted my weight as if to jump. When I didn't, it was not for fear of pain or loss. The pain would be only a bright spark and the loss would be only the army's. And it would be their ultimate victory over me \u2014 having ruled my life for so long, to force an end to it.\n\nThat much, I owed to the enemy.\n\n# Major   \nMandella   \n2458\u20143143 AD\n\n# 1\n\nWhat was that old experiment they told us about in high school biology? Take a flatworm and teach it how to swim through a maze. Then mash it up and feed it to a stupid flatworm, and lo! the stupid flatworm would be able to swim the maze, too.\n\nI had a bad taste of major general in my mouth.\n\nActually, I supposed they had refined the techniques since my high school days. With time dilation, that was about 450 years for research and development.\n\nAt Stargate, my orders said, I was to undergo 'indoctrination and education' prior to taking command of my very own Strike Force. Which was what they still called a company.\n\nFor my education on Stargate, they didn't mince up major generals and serve them to me with hollandaise. They didn't feed me _anything_ except glucose for three weeks. Glucose and electricity.\n\nThey shaved every hair off my body, gave me a shot that turned me into a dishrag, attached dozens of electrodes to my head and body, immersed me in a tank of oxygenated fluorocarbon, and hooked me up to an ALSC. That's an 'accelerated life situation computer.' It kept me busy.\n\nI guess it took the machine about ten minutes to review everything I had learned previously about the martial (excuse the expression) arts. Then it started in on the new stuff.\n\nI learned the best way to use every weapon from a rock to a nova bomb. Not just intellectually; that's what all those electrodes were for. Cybernetically-controlled negative feedback kinesthesia; I felt the weapons in my hands and watched my performance with them. And did it over and over until I did it right. The illusion of reality was total. I used a spear-thrower with a band of Masai warriors on a village raid, and when I looked down at my body it was long and black. I relearned epee from a cruel-looking man in foppish clothes, in an eighteenth-century French courtyard. I sat quietly in a tree with a Sharps rifle and sniped at blue-uniformed men as they crawled across a muddy field toward Vicksburg. In three weeks I killed several regiments of electronic ghosts. It seemed more like a year to me, but the ALSC does strange things to your sense of time.\n\nLearning to use useless exotic weapons was only a small part of the training. In fact, it was the relaxing part. Because when I wasn't in kinesthesia, the machine kept my body totally inert and zapped my brain with four millennia's worth of military facts and theories. And I couldn't forget any of it! Not while I was in the tank.\n\nWant to know who Scipio Aemilianus was? I don't. Bright light of the Third Punic War. _War is the province of danger and therefore courage above all things is the first quality of a warrior_ , von Clausewitz maintained. And I'll never forget the poetry of 'the advance party minus normally moves in a column formation with the platoon headquarters leading, followed by a laser squad, the heavy weapons squad, and the remaining laser squad; the column relies on observation for its flank security except when the terrain and visibility dictate the need for small security detachments to the flanks, in which case the advance party commander will detail one platoon sergeant...' and so on. That's from _Strike Force Command Small Unit Leader's Handbook_ , as if you could call something a handbook when it takes up two whole microfiche cards, 2,000 pages.\n\nIf you want to become a thoroughly eclectic expert in a subject that repels you, join UNEF and sign up for officer training.\n\nOne hundred nineteen people, and I was responsible for 118 of them. Counting myself but not counting the Commodore, who could presumably take care of herself.\n\nI hadn't met any of my company during the two weeks of physical rehabilitation that followed the ALSC session. Before our first muster I was supposed to report to the Temporal Orientation Officer. I called for an appointment and his clerk said the Colonel would meet me at the Level Six Officers' Club after dinner.\n\nI went down to Six early, thinking to eat dinner there, but they had nothing but snacks. So I munched on a fungus thing that vaguely resembled escargots and took the rest of my calories in the form of alcohol.\n\n'Major Mandella?' I'd been busily engaged in my seventh beer and hadn't seen the Colonel approach. I started to rise but he motioned for me to stay seated and dropped heavily into the chair opposite me.\n\n'I'm in your debt,' he said. 'You saved me from at least half of a boring evening.' He offered his hand. 'Jack Kynock, at your service.'\n\n'Colonel\u2014'\n\n'Don't Colonel me and I won't Major you. We old fossils have to... keep our perspective. William.'\n\n'All right with me.'\n\nHe ordered a kind of drink I'd never heard of. 'Where to start? Last time you were on Earth was 2007, according to the records.'\n\n'That's right.'\n\n'Didn't like it much, did you?'\n\n'No.' Zombies, happy robots.\n\n'Well, it got better. Then it got worse, thank you.' A private brought his drink, a bubbling concoction that was green at the bottom of the glass and lightened to chartreuse at the top. He sipped. 'Then they got better again, then worse, then... I don't know. Cycles.'\n\n'What's it like now?'\n\n'Well... I'm not really sure. Stacks of reports and such, but it's hard to filter out the propaganda. I haven't been back in almost two hundred years; it was pretty bad then. Depending on what you like.'\n\n'What do you mean?'\n\n'Oh, let me see. There was lots of excitement. Ever hear of the Pacifist movement?'\n\n'I don't think so.'\n\n'Hmm, the name's deceptive. Actually, it was a war, a guerrilla war.'\n\n'I thought I could give you name, rank and serial number of every war from Troy on up.' He smiled. 'They must have missed one.'\n\n'For good reason. It was run by veterans \u2014 survivors of Yod-38 and Aleph-40, I hear; they got discharged together and decided they could take on all of UNEF, Earthside. They got lots of support from the population.'\n\n'But didn't win.'\n\n'We're still here.' He swirled his drink and the colors shifted. 'Actually, all I know is hearsay. Last time I got to Earth, the war was over, except for some sporadic sabotage. And it wasn't exactly a safe topic of conversation.'\n\n'It surprises me a little,' I said, 'well, more than a little. That Earth's population would do anything at all... against the government's wishes.'\n\nHe made a noncommittal sound.\n\n'Least of all, revolution. When we were there, you couldn't get anybody to say a damned thing against the UNEF \u2014 or any of the local governments, for that matter. They were conditioned from ear to ear to accept things as they were.'\n\n'Ah. That's a cyclic thing, too.' He settled back in his chair. 'It's not a matter of technique. If they wanted to, Earth's government could have total control over... every nontrivial thought and action of each citizen, from cradle to grave.\n\n'They don't do it because it would be fatal. Because there's a war on. Take your own case: did you get any motivational conditioning while you were in the can?'\n\nI thought for a moment. 'If I did, I wouldn't necessarily know about it.'\n\n'That's true. Partially true. But take my word for it, they left that part of your brain alone. Any change in your attitude toward UNEF or the war, or war in general, comes only from new knowledge. Nobody's fiddled with your basic motivations. And you should know why.'\n\nNames, dates, figures rattled down through the maze of new knowledge. 'Tet-17, Sed-21, Aleph-14. The Lazlo... \"The Lazlo Emergency Commission Report.\" June, 2106.'\n\n'Right. And by extension, your own experience on Aleph-1. Robots don't make good soldiers.'\n\n'They would,' I said. 'Up to the twenty-first century. Behavioral conditioning would have been the answer to a general's dream. Make up an army with all the best features of the SS, the Praetorian Guard, the Golden Horde. Mosby's Raiders, the Green Berets.'\n\nHe laughed over his glass. 'Then put that army up against a squad of men in modern fighting suits. It'd be over in a couple of minutes.'\n\n'So long as each man in the squad kept his head about him. And just fought like hell to stay alive.' The generation of soldiers that had precipitated the Lazlo Reports had been conditioned from birth to conform to somebody's vision of the ideal fighting man. They worked beautifully as a team, totally bloodthirsty, placing no great importance on personal survival \u2014 and the Taurans cut them to ribbons. The Taurans also fought with no regard for self. But they were better at it, and there were always more of them.\n\nKynock took a drink and watched the colours. 'I've seen your psych profile,' he said. 'Both before you got here and after your session in the can. It's essentially the same, before and after.'\n\n'That's reassuring.' I signaled for another beer.\n\n'Maybe it shouldn't be.'\n\n'What, it says I won't make a good officer? I told them that from the beginning. I'm no leader.'\n\n'Right in a way, wrong in a way. Want to know what that profile says?'\n\nI shrugged. 'Classified, isn't it?'\n\n'Yes,' he said. 'But you're a major now. You can pull the profile of anybody in your command.'\n\n'I don't suppose it has any big surprises.' But I was a little curious. What animal isn't fascinated by a mirror?\n\n'No. It says you're a pacifist. A failed one at that, which gives you a mild neurosis. Which you handle by transferring the burden of guilt to the army.'\n\nThe fresh beer was so cold it hurt my teeth. 'No surprises yet.'\n\n'And as far as being a leader, you do have a certain potential. But it would be along the lines of a teacher or a minister; you would have to lead from empathy, compassion. You have the desire to impose your ideas on other people, but not your will. Which means, you're right, you'll make one hell of a bad officer unless you shape up.'\n\nI had to laugh. 'UNEF must have known all of this when they ordered me to officer training.'\n\n'There are other parameters,' he said. 'For instance, you're adaptable, reasonably intelligent, analytical. And you're one of the eleven people who's lived through the whole war.'\n\n'Surviving is a virtue in a private.' Couldn't resist it. 'But an officer should provide gallant example. Go down with the ship. Stride the parapet as if unafraid.'\n\nHe harrumphed at that. 'Not when you're a thousand light years from your replacement.'\n\n'It doesn't add up, though. Why would they haul me all the way from Heaven to take a chance on my \"shaping up,\" when probably a third of the people here on Stargate are better officer material? God, the military mind!'\n\n'I suspect the bureaucratic mind, at least, had something to do with it. You have an embarrassing amount of seniority to be a footsoldier.'\n\n'That's all time dilation. I've only been in three campaigns.'\n\n'Immaterial. Besides, that's two-and-a-half more than the average soldier survives. The propaganda boys will probably make you into some kind of a folk hero.'\n\n'Folk hero.' I sipped at the beer. 'Where is John Wayne now that we really need him?'\n\n'John Wayne?' He shook his head. 'I never went in the can, you know. I'm no expert at military history.'\n\n'Forget it.'\n\nKynock finished his drink and asked the private to get him \u2014 I swear to God \u2014 a 'rum Antares.'\n\n'Well, I'm supposed to be your Temporal Orientation Officer. What do you want to know about the present? What passes for the present.'\n\nStill on my mind: 'You've never been in the can?'\n\n'No, combat officers only. The computer facilities and energy you go through in three weeks would keep the Earth running for several days. Too expensive for us desk-warmers.'\n\n'Your decorations say you're combat.'\n\n'Honorary. I was.' The rum Antares was a tall slender glass with a little ice floating at the top, filled with pale amber liquid. At the bottom was a bright red globule about the size of a thumbnail; crimson filaments waved up from it.\n\n'What's that red stuff?'\n\n'Cinnamon. Oh, some ester with cinnamon in it. Quite good... want a taste?'\n\n'No, I'll stick to beer, thanks.'\n\n'Down at level one, the library machine has a temporal orientation file, that my staff updates every day. You can go to it for specific questions. Mainly I want to... prepare you for meeting your Strike Force.'\n\n'What, they're all cyborgs? Clones?'\n\nHe laughed. 'No, it's illegal to clone humans. The main problem is with, uh, you're heterosexual.'\n\n'Oh, that's no problem. I'm tolerant.'\n\n'Yes, your profile shows that you... think you're tolerant, but that's not the problem, exactly.'\n\n'Oh.' I knew what he was going to say. Not the details, but the substance.\n\n'Only emotionally stable people are drafted into UNEF. I know this is hard for you to accept, but heterosexuality is considered an emotional dysfunction. Relatively easy to cure.'\n\n'If they think they're going to cure _me_ \u2014'\n\n'Relax, you're too old.' He took a delicate sip. 'It won't be as hard to get along with them as you might\u2014'\n\n'Wait. You mean nobody... everybody in my company is homosexual? But me?'\n\n'William, everybody on Earth is homosexual. Except for a thousand or so; veterans and incurables.'\n\n'Ah.' What could I say? 'Seems like a drastic way to solve the population problem.'\n\n'Perhaps. It does work, though; Earth's population is stable at just under a billion. When one person dies or goes offplanet, another is quickened.'\n\n'Not \"born.\" '\n\n'Born, yes, but not the old-fashioned way. Your old term for it was \"test-tube babies,\" but of course they don't use a test-tube.'\n\n'Well, that's something.'\n\n'Part of every creche is an artificial womb that takes care of a person the first eight or ten months after quickening. What you would call birth takes place over a period of days; it isn't the sudden, drastic event that it used to be.'\n\nO brave new world, I thought. 'No birth trauma. A billion perfectly adjusted homosexuals.'\n\n'Perfectly adjusted by present-day Earth standards. You and I might find them a little odd.'\n\n'That's an understatement.' I drank off the rest of my beer. 'Yourself, you, uh... are you homosexual?'\n\n'Oh, no,' he said. I relaxed. 'Actually, though, I'm not hetero anymore, either.' He slapped his hip and it made an odd sound. 'Got wounded and it turned out that I had a rare disorder of the lymphatic system, can't regenerate. Nothing but metal and plastic from the waist down. To use your word, I'm a cyborg.'\n\nFar out, as my mother used to say. 'Oh, Private,' I called to the waiter, 'bring me one of those Antares things.' Sitting here in a bar with an asexual cyborg who is probably the only other normal person on the whole goddamned planet.\n\n'Make it a double, please.'\n\n# 2\n\nThey looked normal enough, filing into the lecture hall where we held our first muster, the next day. Rather young and a little stiff.\n\nMost of them had only been out of the creche for seven or eight years. The creche was a controlled, isolated environment to which only a few specialists \u2014 pediatricians and teachers, mostly \u2014 had access. When a person leaves the creche at age twelve or thirteen, he chooses a first name (his last name having been taken from the donor-parent with the higher genetic rating) and is legally a probationary adult, with schooling about equivalent to what I had after my first year of college. Most of them go on to more specialized education, but some are assigned a job and go right to work.\n\nThey're observed very closely and anyone who shows any signs of sociopathy, such as heterosexual leanings, is sent away to a correctional facility. He's either cured or kept there for the rest of his life.\n\nEveryone is drafted into UNEF at the age of twenty. Most people work at a desk for five years and are discharged. A few lucky souls, about one in eight thousand, are invited to volunteer for combat training. Refusing is 'sociopathic,' even though it means signing up for an extra five years. And your chance of surviving the ten years is so small as to be negligible; nobody ever had. Your best chance is to have the war end before your ten (subjective) years of service are up. Hope that time dilation puts many years between each of your battles.\n\nSince you can figure on going into battle roughly once every subjective year, and since an average of 34 percent survive each battle, it's easy to compute your chances of being able to fight it out for ten years. It comes to about two one-thousandths of one percent. Or, to put it another way, get an old-fashioned six-shooter and play Russian Roulette with four of the six chambers loaded. If you can do it ten times in a row without decorating the opposite wall, congratulations! You're a civilian.\n\nThere being some sixty thousand combat soldiers in UNEF, you could expect about 1.2 of them to survive for ten years. I didn't seriously plan on being the lucky one, even though I was halfway there.\n\nHow many of these young soldiers filing into the auditorium knew they were doomed? I tried to match faces up with the dossiers I'd been scanning all morning, but it was hard. They'd all been selected through the same battery of stringent parameters, and they looked remarkably alike: tall but not too tall, muscular but not heavy, intelligent but not in a brooding way... and Earth was much more racially homogenous than it had been in my century. Most of them looked vaguely Polynesian. Only two of them, Kayibanda and Lin, seemed pure representatives of racial types. I wondered whether the others gave them a hard time.\n\nMost of the women were achingly handsome, but I was in no position to be critical. I'd been celibate for over a year, ever since saying goodbye to Marygay, back on Heaven.\n\nI wondered if one of them might have a trace of atavism, or might humor her commander's eccentricity. _It is absolutely forbidden for an officer to form sexual liaison with his subordinates_. Such a warm way of putting it. _Violation of this regulation is punishable by attachment of all funds and reduction to the rank of private or, if the relationship interferes with a unit's combat efficiency, summary execution_. If all of UNEF's regulations could be broken so casually and consistently as that one was, it would be a very easygoing army.\n\nBut not one of the boys appealed to me. How they'd look after another year, I wasn't sure.\n\n'Tench- _hut_!' That was Lieutenant Hilleboe. It was a credit to my new reflexes that I didn't jump to my feet. Everybody in the auditorium snapped to.\n\n'My name is Lieutenant Hilleboe and I am your Second Field Officer.' That used to be 'Field First Sergeant.' A good sign that an army has been around too long is that it starts getting top-heavy with officers.\n\nHilleboe came on like a real hard-ass professional soldier. Probably shouted orders at the mirror every morning, while she was shaving. But I'd seen her profile and knew that she'd only been in action once, and only for a couple of minutes at that. Lost an arm and a leg and was commissioned, same as me, as a result of the tests they give at the regeneration clinic.\n\nHell, maybe she had been a very pleasant person before going through that trauma; it was bad enough just having one limb regrown.\n\nShe was giving them the usual first-sergeant peptalk, stern-but-fair: don't waste my time with little things, use the chain of command, most problems can be solved at the fifth echelon.\n\nIt made me wish I'd had more time to talk with her earlier. Strike Force Command had really rushed us into this first muster \u2014 we were scheduled to board ship the next day \u2014 and I'd only had a few words with my officers.\n\nNot enough, because it was becoming clear that Hilleboe and I had rather disparate philosophies about how to run a company. It was true that _running_ it was her job; I only commanded. But she was setting up a potential 'good guy-bad guy' situation, using the chain of command to so isolate herself from the men and women under her. I had planned not to be quite so aloof, setting aside an hour every other day when any soldier could come to me directly with grievances or suggestions, without permission from his superiors.\n\nWe had both been given the same information during our three weeks in the can. It was interesting that we'd arrived at such different conclusions about leadership. This Open Door policy, for instance, had shown good results in 'modern' armies in Australia and America. And it seemed especially appropriate to our situation, in which everybody would be cooped up for months or even years at a time. We'd used the system on the _Sangre y Victoria_ , the last starship to which I'd been attached, and it had seemed to keep tensions down.\n\nShe had them at ease while delivering this organizational harangue; pretty soon she'd call them to attention and introduce me. What would I talk about? I'd planned just to say a few predictable words and explain my Open Door policy, then turn them over to Commodore Antopol, who would say something about the _Masaryk II_. But I'd better put off my explanation until after I'd had a long talk with Hilleboe; in fact, it would be best if she were the one to introduce the policy to the men and women, so it wouldn't look like the two of us were at loggerheads.\n\nMy executive officer, Captain Moore, saved me. He came rushing through a side door \u2014 he was always rushing, a pudgy meteor \u2014 threw a quick salute and handed me an envelope that contained our combat orders. I had a quick whispered conference with the Commodore, and she agreed that it wouldn't do any harm to tell them where we were going, even though the rank and file technically didn't have the 'need to know.'\n\nOne thing we didn't have to worry about in this war was enemy agents. With a good coat of paint, a Tauran might be able to disguise himself as an ambulatory mushroom. Bound to raise suspicions.\n\nHilleboe had called them to attention and was dutifully telling them what a good commander I was going to be; that I'd been in the war from the beginning, and if they intended to survive through their enlistment they had better follow my example. She didn't mention that I was a mediocre soldier with a talent for getting missed. Nor that I'd resigned from the army at the earliest opportunity and only got back in because conditions on Earth were so intolerable.\n\n'Thank you, Lieutenant.' I took her place at the podium. 'At ease.' I unfolded the single sheet that had our orders, and held it up. 'I have some good news and some bad news.' What had been a joke five centuries before was now just a statement of fact.\n\n'These are our combat orders for the Sade-138 campaign. The good news is that we probably won't be fighting, not immediately. The bad news is that we're going to be a target.'\n\nThey stirred a little bit at that, but nobody said anything or took his eyes off me. Good discipline. Or maybe just fatalism; I didn't know how realistic a picture they had of their future. Their lack of a future, that is.\n\n'What we are ordered to do... is to find the largest portal planet orbiting the Sade-138 collapsar and build a base there. Then stay at the base until we are relieved. That will be two or three years, probably.\n\n'During that time we will almost certainly be attacked. As most of you probably know, Strike Force Command has uncovered a pattern in the enemy's movements from collapsar to collapsar. They hope eventually to trace this complex pattern back through time and space and find the Tauran's home planet. For the present, they can only send out intercepting forces, to hamper the enemy's expansion.\n\n'In a large perspective, this is what we're ordered to do. We'll be one of several dozen strike forces employed in these blocking maneuvers, on the enemy's frontier. I won't be able to stress often enough or hard enough how important this mission is \u2014 if UNEF can keep the enemy from expanding, we may be able to envelop him. And win the war.'\n\nPreferably before we're all dead meat. 'One thing I want to be clear: we may be attacked the day we land, or we may simply occupy the planet for ten years and come on home.' Fat chance. 'Whatever happens, every one of us will stay in the best fighting trim all the time. In transit, we will maintain a regular program of callisthenics as well as a review of our training. Especially construction techniques \u2014 we have to set up the base and its defense facilities in the shortest possible time.'\n\nGod, I was beginning to sound like an officer. 'Any questions?' There were none. 'Then I'd like to introduce Commodore Antopol. Commodore?'\n\nThe Commodore didn't try to hide her boredom as she outlined, to this room full of ground-pounders, the characteristics and capabilities of _Masaryk II_. I had learned most of what she was saying through the can's force-feeding, but the last thing she said caught my attention.\n\n'Sade-138 will be the most distant collapsar men have gone to. It isn't even in the galaxy proper, but rather is part of the Large Magellanic Cloud, some 150,000 light years distant.\n\n'Our voyage will require four collapsar jumps and will last some four months, subjective. Maneuvering into collapsar insertion will put us about three hundred years behind Stargate's calendar by the time we reach Sade-138.'\n\nAnd another seven hundred years gone, if I lived to return. Not that it would make that much difference; Marygay was as good as dead and there wasn't another person alive who meant anything to me.\n\n'As the major said, you mustn't let these figures lull you into complacency. The enemy is also headed for Sade-138; we may all get there the same day. The mathematics of the situation is complicated, but take our word for it; it's going to be a close race.\n\n'Major, do you have anything more for them?'\n\nI started to rise. 'Well...'\n\n'Tench- _hut_!' Hilleboe shouted. Had to learn to expect that.\n\n'Only that I'd like to meet with my senior officers, echelon 4 and above, for a few minutes. Platoon sergeants, you're responsible for getting your troops to Staging Area 67 at 0400 tomorrow morning. Your time's your own until then. Dismissed.'\n\nI invited the five officers up to my billet and brought out a bottle of real French brandy. It had cost two months' pay, but what else could I do with the money? Invest it?\n\nI passed around glasses but Alsever, the doctor, demurred. Instead she broke a little capsule under her nose and inhaled deeply. Then tried without too much success to mask her euphoric expression.\n\n'First let's get down to one basic personnel problem,' I said, pouring. 'Do all of you know that I'm not homosexual?'\n\nMixed chorus of yes sirs and no sirs.\n\n'Do you think this is going to... complicate my situation as commander? As far as the rank and file?'\n\n'Sir, I don't\u2014' Moore began.\n\n'No need for honorifics,' I said, 'not in this closed circle; I was a private four years ago, in my own time frame. When there aren't any troops around, I'm just Mandella, or William.' I had a feeling that was a mistake even as I was saying it. 'Go on.'\n\n'Well, William,' he continued, 'it might have been a problem a hundred years ago. You know how people felt then.'\n\n'Actually, I don't. All I know about the period from the twenty-first century to the present is military history.'\n\n'Oh. Well, it was, uh, it was, how to say it?' His hands fluttered.\n\n'It was a crime,' Alsever said laconically. 'That was when the Eugenics Council was first getting people used to the idea of universal homosex.'\n\n'Eugenics Council?'\n\n'Part of UNEF. Only has authority on Earth.' She took a deep sniff at the empty capsule. 'The idea was to keep people from making babies the biological way. Because, _A_ , people showed a regrettable lack of sense in choosing their genetic partner. And _B_ , the Council saw that racial differences had an unnecessarily divisive effect on humanity; with total control over births, they could make everybody the same race in a few generations.'\n\nI didn't know they had gone quite that far. But I suppose it was logical. 'You approve? As a doctor.'\n\n'As a doctor? I'm not sure.' She took another capsule from her pocket and rolled it between thumb and forefinger, staring at nothing. Or something the rest of us couldn't see. 'In a way, it makes my job simpler. A lot of diseases simply no longer exist. But I don't think they know as much about genetics as they think they do. It's not an exact science; they could be doing something very wrong, and the results wouldn't show up for centuries.'\n\nShe cracked the capsule under her nose and took two deep breaths. 'As a woman, though, I'm all in favor of it.' Hilleboe and Rusk nodded vigorously.\n\n'Not having to go through childbirth?'\n\n'That's part of it.' She crossed her eyes comically, looking at the capsule, gave it a final sniff. 'Mostly, though, it's not... having to... have a man. Inside me. You understand. It's disgusting.'\n\nMoore laughed. 'If you haven't tried it, Diana, don't\u2014'\n\n'Oh, shut up.' She threw the empty capsule at him playfully.\n\n'But it's perfectly natural,' I protested.\n\n'So is swinging through trees. Digging for roots with a blunt stick. Progress, my good major; progress.'\n\n'Anyway,' Moore said, 'it was only a crime for a short period. Then it was considered a, oh, curable...'\n\n'Dysfunction,' Alsever said.\n\n'Thank you. And now, well, it's so rare... I doubt that any of the men and women have any strong feelings about it, one way or the other.'\n\n'Just an eccentricity,' Diana said, magnanimously. 'Not as if you ate babies.'\n\n'That's right, Mandella,' Hilleboe said. 'I don't feel any differently toward you because of it.'\n\n'I \u2014 I'm glad.' That was just great. It was dawning on me that I had not the slightest idea of how to conduct myself socially. So much of my 'normal' behavior was based on a complex unspoken code of sexual etiquette. Was I supposed to treat the men like women, and vice versa? Or treat everybody like brothers and sisters? It was all very confusing.\n\nI finished off my glass and set it down. 'Well, thanks for your reassurances. That was mainly what I wanted to ask you about... I'm sure you all have things to do, goodbyes and such. Don't let me hold you prisoner.'\n\nThey all wandered off except for Charlie Moore. He and I decided to go on a monumental binge, trying to hit every bar and officers' club in the sector. We managed twelve and probably could have hit them all, but I decided to get a few hours' sleep before the next day's muster.\n\nThe one time Charlie made a pass at me, he was very polite about it. I hoped my refusal was also polite \u2014 but figured I'd be getting lots of practice. \n\n# 3\n\nUNEF's first starships had been possessed of a kind of spidery, delicate beauty. But with various technological improvements, structural strength became more important than conserving mass (one of the old ships would have folded up like an accordion if you'd tried a twenty-five-gee maneuver), and that was reflected in the design: stolid, heavy, functional-looking. The only decoration was the name MASARYK II, stencilled in dull blue letters across the obsidian hull.\n\nOur shuttle drifted over the name on its way to the loading bay, and there was a crew of tiny men and women doing maintenance on the hull. With them as a reference, we could see that the letters were a good hundred meters tall. The ship was over a kilometer long (1036.5 meters, my latent memory said), and about a third that wide (319.4 meters).\n\nThat didn't mean there was going to be plenty of elbow-room. In its belly, the ship held six large tachyondrive fighters and fifty robot drones. The infantry was tucked off in a corner. _War is the province of friction_ , Chuck von Clausewitz said; I had a feeling we were going to put him to the test.\n\nWe had about six hours before going into the acceleration tank. I dropped my kit in the tiny billet that would be my home for the next twenty months and went off to explore.\n\nCharlie had beaten me to the lounge and to the privilege of being first to evaluate the quality of _Masaryk II's_ coffee.\n\n'Rhinoceros bile,' he said.\n\n'At least it isn't soya,' I said, taking a first cautious sip. Decided I might be longing for soya in a week.\n\nThe officers' lounge was a cubicle about three meters by four, metal floor and walls, with a coffee machine and a library readout. Six hard chairs and a table with a typer on it.\n\n'Jolly place, isn't it?' He idly punched up a general index on the library machine. 'Lots of military theory.'\n\n'That's good. Refresh our memories.'\n\n'Sign up for officer training?'\n\n'Me? No. Orders.'\n\n'At least you have an excuse.' He slapped the on-off button and watched the green spot dwindle. 'I signed up. They didn't tell me it'd feel like this.'\n\n'Yeah.' He wasn't talking about any subtle problem: burden of responsibility or anything. 'They say it wears off, a little at a time.' All of that information they force into you; a constant silent whispering.\n\n'Ah, there you are.' Hilleboe came through the door and exchanged greetings with us. She gave the room a quick survey, and it was obvious that the Spartan arrangements met with her approval. 'Will you be wanting to address the company before we go into the acceleration tanks?'\n\n'No, I don't see why that would be... necessary.' I almost said 'desirable.' The art of chastising subordinates is a delicate art. I could see that I'd have to keep reminding Hilleboe that she wasn't in charge.\n\nOr I could just switch insignia with her. Let her experience the joys of command.\n\n'You could, please, round up all platoon leaders and go over the immersion sequence with them. Eventually we'll be doing speed drills. But for now, I think the troops could use a few hours' rest.' If they were as hungover as their commander.\n\n'Yes, sir.' She turned and left. A little miffed, because what I'd asked her to do should properly have been a job for Riland or Rusk.\n\nCharlie eased his pudgy self into one of the hard chairs and sighed. 'Twenty months on this greasy machine. With her. Shit.'\n\n'Well, if you're nice to me, I won't billet the two of you together.'\n\n'All right. I'm your slave forever. Starting, oh, next Friday.' He peered into his cup and decided against drinking the dregs. 'Seriously, she's going to be a problem. What are you going to do with her?'\n\n'I don't know.' Charlie was being insubordinate, too, of course. But he was my XO and out of the chain of command. Besides, I had to have _one_ friend. 'Maybe she'll mellow, once we're under weigh.'\n\n'Sure.' Technically, we were already under weigh, crawling toward the Stargate collapsar at one gee. But that was only for the convenience of the crew; it's hard to batten down the hatches in free fall. The trip wouldn't really start until we were in the tanks.\n\nThe lounge was too depressing, so Charlie and I used the remaining hours of mobility to explore the ship.\n\nThe bridge looked like any other computer facility; they had dispensed with the luxury of viewscreens. We stood at a respectful distance while Antopol and her officers went through a last series of checks before climbing into the tanks and leaving our destiny to the machines.\n\nActually, there was a porthole, a thick plastic bubble, in the navigation room forward. Lieutenant Williams wasn't busy, the pre-insertion part of his job being fully automated, so he was glad to show us around.\n\nHe tapped the porthole with a fingernail. 'Hope we don't have to use this, this trip.'\n\n'How so?' Charlie said.\n\n'We only use it if we get lost.' If the insertion angle was off by a thousandth of a radian, we were liable to wind up on the other side of the galaxy. 'We can get a rough idea of our position by analyzing the spectra of the brightest stars. Thumbprints. Identify three and we can triangulate.'\n\n'Then find the nearest collapsar and get back on the rack,' I said.\n\n'That's the problem. Sade-138 is the only collapsar we know of in the Magellanic Clouds. We know of it only because of captured enemy data. Even if we could find another collapsar, assuming we got lost in the cloud, we wouldn't know how to insert.'\n\n'That's great.'\n\n'It's not as though we'd be actually lost,' he said with a rather wicked expression. 'We could zip up in the tanks, aim for Earth and blast away at full power. We'd get there in about three months, ship time.'\n\n'Sure,' I said. 'But 150,000 years in the future.' At twenty-five gees, you get to nine-tenths the speed of light in less than a month. From then on, you're in the arms of Saint Albert.\n\n'Well, that is a drawback,' he said. 'But at least we'd find out who'd won the war.'\n\nIt made you wonder how many soldiers had gotten out of the war in just that way. There were forty-two strike forces lost somewhere and unaccounted for. It was possible that all of them were crawling through normal space at near-lightspeed and would show up at Earth or Stargate one-by-one over the centuries.\n\nA convenient way to go AWOL, since once you were out of the chain of collapsar jumps you'd be practically impossible to track down. Unfortunately, your jump sequence was pre-programmed by Strike Force Command; the human navigator only came into the picture if a miscalculation slipped you into the wrong 'wormhole,' and you popped out in some random part of space.\n\nCharlie and I went on to inspect the gym, which was big enough for about a dozen people at a time. I asked him to make up a roster so that everyone could work out for an hour each day when we were out of the tanks.\n\nThe mess area was only a little larger than the gym \u2014 even with four staggered shifts, the meals would be shoulder-to-shoulder affairs \u2014 and the enlisted men and women's lounge was even more depressing than the officers'. I was going to have a real morale problem on my hands long before the twenty months were up.\n\nThe armorer's bay was as large as the gym, mess hall and both lounges put together. It had to be, because of the great variety of infantry weapons that had evolved over the centuries. The basic weapon was still the fighting suit, though it was much more sophisticated than that first model I had been squeezed into, just before the Aleph-Null campaign.\n\nLieutenant Riland, the armory officer, was supervising his four subordinates, one from each platoon, who were doing a last-minute check of weapons storage. Probably the most important job on the whole ship, when you contemplate what could happen to all those tons of explosives and radioactives under twenty-five gees.\n\nI returned his perfunctory salute. 'Everything going all right, Lieutenant?'\n\n'Yessir, except for those damned swords.' For use in the stasis field. 'No way we can orient them that they won't be bent. Just hope they don't break.'\n\nI couldn't begin to understand the principles behind the stasis field; the gap between present-day physics and my master's degree in the same subject was as long as the time that separated Galileo and Einstein. But I knew the effects.\n\nNothing could move at greater than 16.3 meters per second inside the field, which was a hemispherical (in space, spherical) volume about fifty meters in radius. Inside, there was no such thing as electromagnetic radiation; no electricity, no magnetism, no light. From inside your suit, you could see your surroundings in ghostly monochrome \u2014 which phenomenon was glibly explained to me as being due to 'phase transference of quasi-energy leaking through from an adjacent tachyon reality,' so much phlogiston to me.\n\nThe result of it, though, was to make all conventional weapons of warfare useless. Even a nova bomb was just an inert lump inside the field. And any creature, Terran or Tauran, caught inside the field without the proper insulation would die in a fraction of a second.\n\nAt first it looked as though we had come upon the ultimate weapon. There were five engagements where whole Tauran bases were wiped out without any human ground casualties. All you had to do was carry the field to the enemy (four husky soldiers could handle it in Earth-gravity) and watch them die as they slipped in through the field's opaque wall. The people carrying the generator were invulnerable except for the short periods when they might have to turn the thing off to get their bearings.\n\nThe sixth time the field was used, though, the Taurans were ready for it. They wore protective suits and were armed with sharp spears, with which they could breach the suits of the generator-carriers. From then on the carriers were armed.\n\nOnly three other such battles had been reported, although a dozen strike forces had gone out with the stasis field. The others were still fighting, or still en route, or had been totally defeated. There was no way to tell unless they came back. And they weren't encouraged to come back if Taurans were still in control of 'their' real estate \u2014 supposedly that constituted 'desertion under fire,' which meant execution for all officers (although rumor had it that they were simply brainwiped, imprinted and sent back into the fray).\n\n'Will we be using the stasis field, sir?' Riland asked.\n\n'Probably. Not at first, not unless the Taurans are already there. I don't relish the thought of living in a suit, day in and day out.' Neither did I relish the thought of using sword, spear, throwing knife; no matter how many electronic illusions I'd sent to Valhalla with them.\n\nChecked my watch. 'Well, we'd better get on down to the tanks, Captain. Make sure everything's squared away.' We had about two hours before the insertion sequence would start.\n\nThe room the tanks were in resembled a huge chemical factory; the floor was a good hundred meters in diameter and jammed with bulky apparatus painted a uniform, dull gray. The eight tanks were arranged almost symmetrically around the central elevator, the symmetry spoiled by the fact that one of the tanks was twice the size of the others. That would be the command tank, for all the senior officers and supporting specialists.\n\nSergeant Blazynski stepped out from behind one of the tanks and saluted. I didn't return his salute.\n\n'What the hell is that?' In all that universe of gray, there was one spot of color.\n\n'It's a cat, sir.'\n\n'Do tell.' A big one, too, and bright calico. It looked ridiculous, draped over the sergeant's shoulder. 'Let me rephrase the question: what the hell is a cat doing here?'\n\n'It's the maintenance squad's mascot, sir.' The cat raised its head enough to hiss half-heartedly at me, then returned to its flaccid repose.\n\nI looked at Charlie and he shrugged back. 'It seems kind of cruel,' he said. To the sergeant: 'You won't get much use of it. After twenty-five gees, it'll be just so much fur and guts.'\n\n'Oh no, sir! Sirs.' He ruffed back the fur between the creature's shoulders. It had a fluorocarbon fitting imbedded there, just like the one above my hipbone. 'We bought it at a store on Stargate, already modified. Lots of ships have them now, sir. The Commodore signed the forms for us.'\n\nWell, that was her right; maintenance was under both of us equally. And it was her ship. 'You couldn't have gotten a dog?' God, I hated cats. Always sneaking around.\n\n'No, sir, they don't adapt. Can't take free fall.'\n\n'Did you have to make any special adaptations? In the tank?' Charlie asked.\n\n'No sir. We had an extra couch.' Great; that meant I'd be sharing a tank with the animal. 'We only had to shorten the straps.\n\n'It takes a different kind of drug for the cell-wall strengthening, but that was included in the price.'\n\nCharlie scratched it behind an ear. It purred softly but didn't move. 'Seems kind of stupid. The animal, I mean.'\n\n'We drugged him ahead of time.' No wonder it was so inert; the drug slows your metabolism down to a rate barely adequate to sustain life. 'Makes it easier to strap him in.'\n\n'Guess it's all right,' I said. Maybe good for morale. 'But if it starts getting in the way, I'll personally recycle it.'\n\n'Yes, sir!' he said, visibly relieved, thinking that I couldn't really do anything like that to such a cute bundle of fur. Try me, buddy.\n\nSo we had seen it all. The only thing left, this side of the engines, was the huge hold where the fighters and drones waited, clamped in their massive cradles against the coming acceleration. Charlie and I went down to take a look, but there were no windows on our side of the airlock. I knew there'd be one on the inside, but the chamber was evacuated, and it wasn't worth going through the fill-and-warm cycle merely to satisfy our curiosity.\n\nI was starting to feel really supernumerary. Called Hilleboe and she said everything was under control. With an hour to kill, we went back to the lounge and had the computer mediate a game of _Kriegspieler_ which was just starting to get interesting when the ten-minute warning sounded.\n\nThe acceleration tanks had a 'half-life-to-failure' of five weeks; there was a fifty-fifty chance that you could stay immersed for five weeks before some valve or tube popped and you were squashed like a bug underfoot. In practice, it had to be one hell of an emergency to justify using the tanks for more than two weeks' acceleration. We were only going under for ten days, this first leg of our journey.\n\nFive weeks or five hours, though, it was all the same as far as the tankee was concerned. Once the pressure got up to an operational level, you had no sense of the passage of time. Your body and brain were concrete. None of your senses provided any input, and you could amuse yourself for several hours just trying to spell your own name.\n\nSo I wasn't really surprised that no time seemed to have passed when I was suddenly dry, my body tingling with the return of sensation. The place sounded like an asthmatics' convention in the middle of a hay field: thirty-nine people and one cat all coughing and sneezing to get rid of the last residues of fluorocarbon. While I was fumbling with my straps, the side door opened, flooding the tank with painfully bright light. The cat was the first one out, with a general scramble right behind him. For the sake of dignity, I waited until last.\n\nOver a hundred people were milling around outside, stretching and massaging out cramps. Dignity! Surrounded by acres of young female flesh, I stared into their faces and desperately tried to solve a third-order differential equation in my head, to circumvent the gallant reflex. A temporary expedient, but it got me to the elevator.\n\nHilleboe was shouting orders, getting people lined up, and as the doors closed I noticed that all of one platoon had a uniform light bruise, from head to foot. Twenty pairs of black eyes. I'd have to see both Maintenance and Medical about that.\n\nAfter I got dressed.\n\n# 4\n\nWe stayed at one gee for three weeks, with occasional periods of free fall for navigation check, while the _Masaryk II_ made a long, narrow loop away from the collapsar Resh-10, and back again. That period went all right, the people adjusting pretty well to ship routine. I gave them a minimum of busy-work and a maximum of training review and exercise \u2014 for their own good, though I wasn't naive enough to think they'd see it that way.\n\nAfter about a week of one gee, Private Rudkoski (the cook's assistant) had a still, producing some eight liters a day of 95 per cent ethyl alcohol. I didn't want to stop him \u2014 life was cheerless enough; I didn't mind as long as people showed up for duty sober \u2014 but I was damned curious both how he managed to divert the raw materials out of our sealed-tight ecology, and how the people paid for their booze. So I used the chain of command in reverse, asking Alsever to find out. She asked Jarvil, who asked Carreras, who sat down with Orban, the cook. Turned out that Sergeant Orban had set the whole thing up, letting Rudkoski do the dirty work, and was aching to brag about it to a trustworthy person.\n\nIf I had ever taken meals with the enlisted men and women, I might have figured out that something odd was going on. But the scheme didn't extend up to officers' country.\n\nThrough Rudkoski, Orban had juryrigged a ship-wide economy based on alcohol. It went like this:\n\nEach meal was prepared with one very sugary dessert \u2014 jelly, custard or flan \u2014 which you were free to eat if you could stand the cloying taste. But if it was still on your tray when you presented it at the recycling window, Rudkoski would give you a ten-cent chit and scrape the sugary stuff into a fermentation vat. He had two twenty-liter vats, one 'working' while the other was being filled.\n\nThe ten-cent chit was at the bottom of a system that allowed you to buy a half-liter of straight ethyl (with your choice of flavoring) for five dollars. A squad of five people who skipped all of their desserts could buy about a liter a week, enough for a party but not enough to constitute a public health problem.\n\nWhen Diana brought me this information, she also brought a bottle of Rudkoski's Worst \u2014 literally; it was a flavour that just hadn't worked. It came up through the chain of command with only a few centimeters missing.\n\nIts taste was a ghastly combination of strawberry and caraway seed. With a perversity not uncommon to people who rarely drink, Diana loved it. I had some ice water brought up, and she got totally blasted within an hour. For myself, I made one drink and didn't finish it.\n\nWhen she was more than halfway to oblivion, mumbling a reassuring soliloquy to her liver, she suddenly tilted her head up to stare at me with childlike directness.\n\n'You have a real problem, Major William.'\n\n'Not half the problem you'll have in the morning, Lieutenant Doctor Diana.'\n\n'Oh not really.' She waved a drunken hand in front of her face. 'Some vitamins, some glu... cose, an eensy cc of adren... aline if all else fails. You... you... have... a real... problem.'\n\n'Look, Diana, don't you want me to\u2014'\n\n'What you need... is to get an appointment with that nice Corporal Valdez.' Valdez was the male sex counselor. 'He has empathy. Itsiz job. He'd make you\u2014'\n\n'We talked about this before, remember? I want to stay the way I am.'\n\n'Don't we all.' She wiped away a tear that was probably one percent alcohol. 'You know they call you the Old C'reer. No they don't.'\n\nShe looked at the floor and then at the wall. 'The Ol' Queer, that's what.'\n\nI had expected names worse than that. But not so soon. 'I don't care. The commander always gets names.'\n\n'I know but.' She stood up suddenly and wobbled a little bit. 'Too much t' drink. Lie down.' She turned her back to me and stretched so hard that a joint popped. Then a seam whispered open and she shrugged off her tunic, stepped out of it and tiptoed to my bed. She sat down and patted the mattress. 'Come on, William. Only chance.'\n\n'For Christ's sake, Diana. It wouldn't be fair.'\n\n'All's fair,' she giggled. 'And 'sides, I'm a doctor. I can be clin'cal; won't bother me a bit. Help me with this.' After five hundred years, they were still putting brassiere clasps in the back.\n\nOne kind of gentleman would have helped her get undressed and then made a quiet exit. Another kind of gentleman might have bolted for the door. Being neither kind, I closed in for the kill.\n\nPerhaps fortunately, she passed out before we had made any headway. I admired the sight and touch of her for a long time before, feeling like a cad, I managed to gather everything up and dress her.\n\nI lifted her out of the bed, sweet burden, and then realized that if anyone saw me carrying her down to her billet, she'd be the butt of rumors for the rest of the campaign. I called up Charlie, told him we'd had some booze and Diana was rather the worse for it, and asked him whether he'd come up for a drink and help me haul the good doctor home.\n\nBy the time Charlie knocked, she was draped innocently in a chair, snoring softly.\n\nHe smiled at her. 'Physician, heal thyself.' I offered him the bottle, with a warning. He sniffed it and made a face.\n\n'What is this, varnish?'\n\n'Just something the cooks whipped up. Vacuum still.'\n\nHe set it down carefully, as if it might explode if jarred. 'I predict a coming shortage of customers. Epidemic of death by poisoning \u2014 she actually drank that vile stuff?'\n\n'Well, the cooks admitted it was an experiment that didn't pan out; their other flavors are evidently potable. Yeah, she loved it.'\n\n'Well...' He laughed. 'Damn! Wait, you take her legs and I take her arms?'\n\n'No, look, we each take an arm. Maybe we can get her to do part of the walking.'\n\nShe moaned a little when we lifted her out of the chair, opened one eye and said, 'Hello, Charlee.' Then she closed the eye and let us drag her down to the billet. No one saw us on the way, but her bunkmate, Laasonen, was sitting up reading.\n\n'She really drank the stuff, eh?' She regarded her friend with wry affection. 'Here, let me help.'\n\nThe three of us wrestled her into bed. Laasonen smoothed the hair out of her eyes. 'She said it was in the nature of an experiment.'\n\n'More devotion to science than I have,' Charlie said. 'A stronger stomach, too.'\n\nWe all wished he hadn't said that.\n\nDiana sheepishly admitted that she hadn't remembered anything after the first drink, and talking to her, I deduced that she thought Charlie had been there all along. Which was all for the best, of course. But oh! Diana, my lovely latent heterosexual, let me buy you a bottle of good scotch the next time we come into port. Seven hundred years from now.\n\nWe got back into the tanks for the hop from Resh-10 to Kaph-35. That was two weeks at twenty-five gees; then we had another four weeks of routine at one gravity.\n\nI had announced my open door policy, but practically no one ever took advantage of it. I saw very little of the troops and those occasions were almost always negative: testing them on their training review, handing out reprimands, and occasionally lecturing classes. And they rarely spoke intelligibly, except in response to a direct question.\n\nMost of them either had English as their native tongue or as a second language, but it had changed so drastically over 450 years that I could barely understand it, not at all if it was spoken rapidly. Fortunately, they had all been taught early twenty-first century English during their basic training; that language, or dialect, served as a temporal _lingua franca_ through which a twenty-fifth century soldier could communicate with someone who had been a contemporary of his nineteen-times-great-grandparents. If there had still been such a thing as grandparents.\n\nI thought of my first combat commander, Captain Stott \u2014 whom I had hated just as cordially as the rest of the company did \u2014 and tried to imagine how I would have felt if he had been a sexual deviate and I'd been forced to learn a new language for his convenience.\n\nSo we had discipline problems, sure. But the wonder was that we had any discipline at all. Hilleboe was responsible for that; as little as I liked her personally, I had to give her credit for keeping the troops in line.\n\nMost of the shipboard graffiti concerned improbable sexual geometries between the Second Field Officer and her commander.\n\nFrom Kaph-35 we jumped to Samk-78, from there to Ayin-129 and finally to Sade-138. Most of the jumps were no more than a few hundred light years, but the last one was 140,000 \u2014 supposedly the longest collapsar jump ever made by a manned craft.\n\nThe time spent scooting down the wormhole from one collapsar to the next was always the same, independent of the distance. When I'd studied physics, they thought the duration of a collapsar jump was exactly zero. But a couple of centuries later, they did a complicated wave-guide experiment that proved the jump actually lasted some small fraction of a nanosecond. Doesn't seem like much, but they'd had to rebuild physics from the foundation up when the collapsar jump was first discovered; they had to tear the whole damned thing down again when they found out it took time to get from _A_ to _B_. Physicists were still arguing about it.\n\nBut we had more pressing problems as we flashed out of Sade-138's collapsar field at three-quarters of the speed of light. There was no way to tell immediately whether the Taurans had beat us there. We launched a pre-programmed drone that would decelerate at 300 gees and take a preliminary look around. It would warn us if it detected any other ships in the system, or evidence of Tauran activity on any of the collapsar's planets.\n\nThe drone launched, we zipped up in the tanks and the computers put us through a three-week evasive maneuver while the ship slowed down. No problems except that three weeks is a hell of a long time to stay frozen in the tank; for a couple of days afterward everybody crept around like aged cripples.\n\nIf the drone had sent back word that the Taurans were already in the system, we would immediately have stepped down to one gee and started deploying fighters and drones armed with nova bombs. Or we might not have lived that long: sometimes the Taurans could get to a ship only hours after it entered the system. Dying in the tank might not be the most pleasant way to go.\n\nIt took us a month to get back to within a couple of AUs of Sade-138, where the drone had found a planet that met our requirements.\n\nIt was an odd planet, slightly smaller than Earth but more dense. It wasn't quite the cryogenic deepfreeze that most portal planets were, both because of heat from its core and because S Doradus, the brightest star in the cloud, was only a third of a light year away.\n\nThe strangest feature of the planet was its lack of geography. From space it looked like a slightly damaged billiard ball. Our resident physicist, Lieutenant Gim, explained its relatively pristine condition by pointing out that its anomalous, almost cometary orbit probably meant that it had spent most of its life as a 'rogue planet,' drifting alone through interstellar space. The chances were good that it had never been struck by a large meteor until it wandered into Sade-138's bailiwick and was captured \u2014 forced to share space with all the other flotsam the collapsar dragged around with it.\n\nWe left the _Masaryk II_ in orbit (it was capable of landing, but that would restrict its visibility and getaway time) and shuttled building materials down to the surface with the six fighters.\n\nIt was good to get out of the ship, even though the planet wasn't exactly hospitable. The atmosphere was a thin cold wind of hydrogen and helium, it being too cold even at noon for any other substance to exist as a gas.\n\n'Noon' was when S Doradus was overhead, a tiny, painfully bright spark. The temperature slowly dropped at night, going from twenty-five degrees Kelvin down to seventeen degrees \u2014 which caused problems, because just before dawn the hydrogen would start to condense out of the air, making everything so slippery that it was useless to do anything other than sit down and wait it out. At dawn a faint pastel rainbow provided the only relief from the black-and-white monotony of the landscape.\n\nThe ground was treacherous, covered with little granular chunks of frozen gas that shifted slowly, incessantly in the anemic breeze. You had to walk in a slow waddle to stay on your feet; of the four people who would die during the base's construction, three would be the victims of simple falls.\n\nThe troops weren't happy with my decision to construct the anti-spacecraft and perimeter defenses before putting up living quarters. That was by the book, though, and they got two days of shipboard rest for every 'day' planetside \u2014 which wasn't overly generous, I admit, since ship days were 24 hours long, and a day on the planet was 38.5 hours from dawn to dawn.\n\nThe base was completed in just less than four weeks, and it was a formidable structure indeed. The perimeter, a circle one kilometer in diameter, was guarded by twenty-five gigawatt lasers that would automatically aim and fire within a thousandth of a second. They would react to the motion of any significantly large object between the perimeter and the horizon. Sometimes when the wind was right and the ground damp with hydrogen, the little ice granules would stick together into a loose snowball and begin to roll. They wouldn't roll far.\n\nFor early protection, before the enemy came over our horizon, the base was in the center of a huge mine field. The buried mines would detonate upon sufficient distortion of their local gravitational fields: a single Tauran would set one off if he came within twenty meters of it; a small spacecraft a kilometer overhead would also detonate it. There were 2800 of them, mostly 100-microton nuclear bombs. Fifty of them were devastatingly powerful tachyon devices. They were all scattered at random in a ring that extended from the limit of the lasers' effectiveness, out another five kilometers.\n\nInside the base, we relied on individual lasers, microton grenades, and a tachyon-powered repeating rocket launcher that had never been tried in combat, one per platoon. As a last resort, the stasis field was set up beside the living quarters. Inside its opaque gray dome, as well as enough paleolithic weaponry to hold off the Golden Horde, we'd stashed a small cruiser, just in case we managed to lose all our spacecraft in the process of winning a battle. Twelve people would be able to get back to Stargate.\n\nIt didn't do to dwell on the fact that the other survivors would have to sit on their hands until relieved by reinforcements or death.\n\nThe living quarters and administration facilities were all underground, to protect them from line-of-sight weapons. It didn't do too much for morale, though; there were waiting lists for every outside detail, no matter how strenuous or risky. I hadn't wanted the troops to go up to the surface in their free time, both because of the danger involved and the administrative headache of constantly checking equipment in and out and keeping track of who was where.\n\nFinally I had to relent and allow people to go up for a few hours every week. There was nothing to see except the featureless plain and the sky (which was dominated by S Doradus during the day, and the huge dim oval of the galaxy at night), but that was an improvement over staring at the melted-rock walls and ceiling.\n\nA favorite sport was to walk out to the perimeter and throw snowballs in front of the laser; see how small a snowball you could throw and still set the weapon off. It seemed to me that the entertainment value of this pastime was about equal to watching a faucet drip, but there was no real harm in it, since the weapons would only fire outward and we had power to spare.\n\nFor five months things went pretty smoothly. Such administrative problems as we had were similar to those we'd encountered on the _Masaryk II_. And we were in less danger as passive troglodytes than we had been scooting from collapsar to collapsar, at least until the enemy showed up.\n\nI looked the other way when Rudkoski reassembled his still. Anything that broke the monotony of garrison duty was welcome, and the chits not only provided booze for the troops but gave them something to gamble with. I only interfered in two ways: nobody could go outside unless they were totally sober, and nobody could sell sexual favors. Maybe that was the Puritan in me, but it was, again, by the book. The opinion of the supporting specialists was split. Lieutenant Wilber, the psychiatric officer, agreed with me; the sex counselors Kajdi and Valdez didn't. But then, they were probably coining money, being the resident 'professionals.'\n\nFive months of comfortably boring routine, and then along came Private Graubard.\n\nFor obvious reasons, no weapons were allowed in the living quarters. The way these people were trained, even a fistfight could be a duel to the death, and tempers were short. A hundred merely normal people would probably have been at each other's throats after a week in our caves, but these soldiers had been hand-picked for their ability to get along in close confinement.\n\nStill there were fights. Graubard had almost killed his ex-lover Schon when that worthy made a face at him in the chow line. He had a week of solitary detention (so did Schon, for having precipitated it) and then psychiatric counseling and punitive details. Then I transferred him to the fourth platoon, so he wouldn't be seeing Schon every day.\n\nThe first time they passed in the halls, Graubard greeted Schon with a karate kick to the throat. Diana had to build him a new trachea. Graubard got a more intensive round of detention, counseling and details \u2014 hell, I couldn't transfer him to another _company_ \u2014 and then he was a good boy for two weeks. I fiddled their work and chow schedules so the two would never be in the same room together. But they met in a corridor again, and this time it came out more even: Schon got two broken ribs, but Graubard got a ruptured testicle and lost four teeth.\n\nIf it kept up, I was going to have at least one less mouth to feed.\n\nBy the Universal Code of Military Justice I could have ordered Graubard executed, since we were technically in a state of combat. Perhaps I should have, then and there. But Charlie suggested a more humanitarian solution, and I accepted it.\n\nWe didn't have enough room to keep Graubard in solitary detention forever, which seemed to be the only humane yet practical thing to do, but they had plenty of room aboard the _Masaryk II_ , hovering overhead in a stationary orbit. I called Antopol and she agreed to take care of him. I gave her permission to space the bastard if he gave her any trouble.\n\nWe called a general assembly to explain things, so that the lesson of Graubard wouldn't be lost on anybody. I was just starting to talk, standing on the rock dais with the company sitting in front of me, and the officers and Graubard behind me \u2014 when the crazy fool decided to kill me.\n\nLike everybody else, Graubard was assigned five hours per week of training inside the stasis field. Under close supervision, the soldiers would practice using their swords and spears and whatnot on dummy Taurans. Somehow Graubard had managed to smuggle out a weapon, an Indian chakra, which is a circle of metal with a razor-keen outer edge. It's a tricky weapon, but once you know how to use it, it can be much more effective than a regular throwing knife. Graubard was an expert.\n\nAll in a fraction of a second, Graubard disabled the people on either side of him \u2014 hitting Charlie in the temple with an elbow while he broke Hilleboe's kneecap with a kick \u2014 and slid the chakra out of his tunic and spun it toward me in one smooth action. It had covered half the distance to my throat before I reacted.\n\nInstinctively I slapped out to deflect it and came within a centimeter of losing four fingers. The razor edge slashed open the top of my palm, but I succeeded in knocking the thing off course. And Graubard was rushing me, teeth bared in an expression I hope I never see again.\n\nMaybe he didn't realize that the _old queer_ was really only five years older than he; that the _old queer_ had combat reflexes and three weeks of negative feedback kinesthesia training. At any rate, it was so easy I almost felt sorry for him.\n\nHis right toe was turning in; I knew he would take one more step and go into a savate\u00b4 leap. I adjusted the distance between us with a short _ballestra_ and, just as both his feet left the ground, gave him an ungentle side-kick to the solar plexus. He was unconscious before he hit the ground. But not dead.\n\nIf I'd merely killed him in self-defense, my troubles would have been over instead of suddenly being multiplied.\n\nA simple psychotic troublemaker a commander can lock up and forget about. But not a failed assassin. And I didn't have to take a poll to know that executing him was not going to improve my relationship with the troops.\n\nI realized that Diana was on her knees beside me, trying to pry open my fingers. 'Check Hilleboe and Moore,' I mumbled, and to the troops: 'Dismissed.' \n\n# 5\n\n'Don't be an ass,' Charlie said. He was holding a damp rag to the bruise on the side of his head.\n\n'You don't think I have to execute him?'\n\n'Stop twitching!' Diana was trying to get the lips of my wound to line up together so she could paint them shut. From the wrist down, the hand felt like a lump of ice.\n\n'Not by your own hand, you don't. You can detail someone. At random.'\n\n'Charlie's right,' Diana said. 'Have everybody draw a slip of paper out of a bowl.'\n\nI was glad Hilleboe was sound asleep on the other cot. I didn't need her opinion. 'And if the person so chosen refuses?'\n\n'Punish him and get another,' Charlie said. 'Didn't you learn anything in the can? You can't abrogate your authority by publicly doing a job... that obviously should be detailed.'\n\n'Any other job, sure. But for this... nobody in the company has ever killed. It would look like I was getting somebody else to do my moral dirty work.'\n\n'If it's so damned complicated,' Diana said, 'why not just get up in front of the troops and tell them how complicated it is. Then have them draw straws. They aren't children.'\n\nThere had been an army in which that sort of thing was done, a strong quasi-memory told me. The Marxist POUM militia in the Spanish Civil War, early twentieth. You obeyed an order only after it had been explained in detail; you could refuse if it didn't make sense. Officers and men got drunk together and never saluted or used titles. They lost the war. But the other side didn't have any fun.\n\n'Finished.' Diana set the limp hand in my lap. 'Don't try to use it for a half-hour. When it starts to hurt, you can use it.'\n\nI inspected the wound closely. 'The lines don't match up. Not that I'm complaining.'\n\n'You shouldn't. By all rights, you ought to have just a stump. And no regeneration facilities this side of Stargate.'\n\n'Stump ought to be at the top of your neck,' Charlie said. 'I don't see why you have any qualms. You should have killed the bastard outright.'\n\n_'I know that, goddammit!'_ Both Charlie and Diana jumped at my outburst. 'Sorry, shit. Look, just let me do the worrying.'\n\n'Why don't you both talk about something else for a while.' Diana got up and checked the contents of her medical bag. 'I've got another patient to check. Try to keep from exciting each other.'\n\n'Graubard?' Charlie asked.\n\n'That's right. To make sure he can mount the scaffold without assistance.'\n\n'What if Hilleboe\u2014'\n\n'She'll be out for another half-hour. I'll send Jarvil down, just in case.' She hurried out the door.\n\n'The scaffold...' I hadn't given that any thought. 'How the hell are we going to execute him? We can't do it indoors: morale. Firing squad would be pretty grisly.'\n\n'Chuck him out the airlock. You don't owe him any ceremony.'\n\n'You're probably right. I wasn't thinking about him.' I wondered whether Charlie had ever seen the body of a person who'd died that way. 'Maybe we ought to just stuff him into the recycler. He'd wind up there eventually.'\n\nCharlie laughed. 'That's the spirit.'\n\n'We'd have to trim him up a little bit. Door's not very wide.' Charlie had a few suggestions as to how to get around that. Jarvil came in and more-or-less ignored us.\n\nSuddenly the infirmary door banged open. A patient on a cart; Diana rushing alongside pressing on the man's chest, while a private pushed. Two other privates were following, but hung back at the door. 'Over by the wall,' she ordered.\n\nIt was Graubard. 'Tried to kill himself,' Diana said, but that was pretty obvious. 'Heart stopped.' He'd made a noose out of his belt; it was still hanging limply around his neck.\n\nThere were two big electrodes with rubber handles hanging on the wall. Diana snatched them with one hand while she ripped his tunic open with the other. 'Get your hands off the cart!' She held the electrodes apart, kicked a switch, and pressed them down onto his chest. They made a low hum while his body trembled and flopped. Smell of burning flesh.\n\nDiana was shaking her head. 'Get ready to crack him,' she said to Jarvil. 'Get Doris down here.' The body was gurgling, but it was a mechanical sound, like plumbing.\n\nShe kicked off the power and let the electrodes drop, pulled a ring off her finger and crossed to stick her arms in the sterilizer. Jarvil started to rub an evil-smelling fluid over the man's chest.\n\nThere was a small red mark between the two electrode burns. It took me a moment to recognize what it was. Jarvil wiped it away. I stepped closer and checked Graubard's neck.\n\n'Get out of the way, William, you aren't sterile.' Diana felt his collarbone, measured down a little ways and made an incision straight down to the bottom of his breastbone. Blood welled out and Jarvil handed her an instrument that looked like big chrome-plated bolt-cutters. I looked away but couldn't help hearing the thing crunch through his ribs. She asked for retractors and sponges and so on while I wandered back to where I'd been sitting. With the corner of my eye I saw her working away inside his thorax, massaging his heart directly.\n\nCharlie looked the way I felt. He called out weakly, 'Hey, don't knock yourself out, Diana.' She didn't answer. Jarvil had wheeled up the artificial heart and was holding out two tubes. Diana picked up a scalpel and I looked away again.\n\nHe was still dead a half-hour later. They turned off the machine and threw a sheet over him. Diana washed the blood off her arms and said, 'Got to change. Back in a minute.'\n\nI got up and walked to her billet, next door. Had to know. I raised my hand to knock but it was suddenly hurting like there was a line of fire drawn across it. I rapped with my left and she opened the door immediately.\n\n'What \u2014 oh, you want something for your hand.' She was half-dressed, unself-conscious. 'Ask Jarvil.'\n\n'No, that's not it. What happened, Diana?'\n\n'Oh. Well,' she pulled a tunic over her head and her voice was muffled. 'It was my fault, I guess. I left him alone for a minute.'\n\n'And he tried to hang himself.'\n\n'That's right.' She sat on the bed and offered me the chair. 'I went off to the head and he was dead by the time I got back. I'd already sent Jarvil away because I didn't want Hilleboe to be unsupervised for too long.'\n\n'But, Diana... there's no mark on his neck. No bruise, nothing.'\n\nShe shrugged. 'The hanging didn't kill him. He had a heart attack.'\n\n'Somebody gave him a shot. Right over his heart.'\n\nShe looked at me curiously. 'I did that, William. Adrenaline. Standard procedure.'\n\nYou get that red dot of expressed blood if you jerk away from the projector while you're getting a shot. Otherwise the medicine goes right through the pores, doesn't leave a mark. 'He was dead when you gave him the shot?'\n\n'That would be my professional opinion.' Deadpan. 'No heartbeat, pulse, respiration. Very few other disorders show these symptoms.'\n\n'Yeah. I see.'\n\n'Is something... what's the matter, William?'\n\nEither I'd been improbably lucky or Diana was a very good actress. 'Nothing. Yeah, I better get something for this hand.' I opened the door. 'Saved me a lot of trouble.'\n\nShe looked straight into my eyes. 'That's true.'\n\nActually, I'd traded one kind of trouble for another. Despite the fact that there were several disinterested witnesses to Graubard's demise, there was a persistent rumor that I'd had Doc Alsever simply exterminate him \u2014 since I'd botched the job myself and didn't want to go through a troublesome court-martial.\n\nThe fact was that, under the Universal Code of Military 'Justice,' Graubard hadn't deserved any kind of trial at all. All I had to do was say 'You, you and you. Take this man out and kill him, please.' And woe betide the private who refused to carry out the order.\n\nMy relationship with the troops did improve, in a sense. At least outwardly, they showed more deference to me. But I suspected it was at least partly the cheap kind of respect you might offer any ruffian who had proved himself to be dangerous and volatile.\n\nSo _Killer_ was my new name. Just when I'd gotten used to _Old Queer_.\n\nThe base quickly settled back into its routine of training and waiting. I was almost impatient for the Taurans to show up, just to get it over with one way or the other.\n\nThe troops had adjusted to the situation much better than I had, for obvious reasons. They had specific duties to perform and ample free time for the usual soldierly anodynes to boredom. My duties were more varied but offered little satisfaction, since the problems that percolated up to me were of the 'the buck stops here' type; those with pleasing, unambiguous solutions were taken care of in the lower echelons.\n\nI'd never cared much for sports or games, but found myself turning to them more and more as a kind of safety valve. For the first time in my life, in these tense, claustrophobic surroundings, I couldn't escape into reading or study. So I fenced, quarterstaff and saber, with the other officers, worked myself to exhaustion on the exercise machines and even kept a jump-rope in my office. Most of the other officers played chess, but they could usually beat me \u2014 whenever I won it gave me the feeling I was being humored. Word games were difficult because my language was an archaic dialect that they had trouble manipulating. And I lacked the time and talent to master 'modern' English.\n\nFor a while I let Diana feed me mood-altering drugs, but the cumulative effect of them was frightening \u2014 I was getting addicted in a way that was at first too subtle to bother me \u2014 so I stopped short. Then I tried some systematic psychoanalysis with Lieutenant Wilber. It was impossible. Although he knew all about my problem in an academic kind of way, we didn't speak the same cultural language; his counseling me about love and sex was like me telling a fourteenth-century serf how best to get along with his priest and landlord.\n\nAnd that, after all, was the root of my problem. I was sure I could have handled the pressures and frustrations of command; of being cooped up in a cave with these people who at times seemed scarcely less alien than the enemy; even the near-certainty that it could lead only to painful death in a worthless cause \u2014 if only I could have had Marygay with me. And the feeling got more intense as the months crept by.\n\nHe got very stern with me at this point and accused me of romanticizing my position. He knew what love was, he said; he had been in love himself. And the sexual polarity of the couple made no difference \u2014 all right, I could accept that; that idea had been a cliche\u00b4 in my parents' generation (though it had run into some predictable resistance in my own). But love, he said, love was a fragile blossom; love was a delicate crystal; love was an unstable reaction with a half-life of about eight months. Bullshit, I said, and accused him of wearing cultural blinders; thirty centuries of prewar society taught that love was one thing that could last to the grave and even beyond _and if he had been born instead of hatched he would know that without being told!_ Whereupon he would assume a wry, tolerant expression and reiterate that I was merely a victim of self-imposed sexual frustration and romantic delusion.\n\nIn retrospect, I guess we had a good time arguing with each other. Cure me, he didn't.\n\nI did have a new friend who sat in my lap all the time. It was the cat, who had the usual talent for hiding from people who like cats and cleaving unto those who have sinus trouble or just don't like sneaky little animals. We did have something in common, though, since to my knowledge he was the only other heterosexual male mammal within any reasonable distance. He'd been castrated, of course, but that didn't make much difference under the circumstances. \n\n# 6\n\nIt was exactly 400 days since the day we had begun construction. I was sitting at my desk not checking out Hilleboe's new duty roster. The cat was on my lap, purring loudly even though I refused to pet it. Charlie was stretched out in a chair reading something on the viewer. The phone buzzed and it was the Commodore.\n\n'They're here.'\n\n'What?'\n\n'I said they're here. A Tauran ship just exited the collapsar field. Velocity .80c. Deceleration thirty gees. Give or take.'\n\nCharlie was leaning over my desk. 'What?' I dumped the cat.\n\n'How long? Before you can pursue?' I asked.\n\n'Soon as you get off the phone.' I switched off and went over to the logistic computer, which was a twin to the one on _Masaryk II_ and had a direct data link to it. While I tried to get numbers out of the thing, Charlie fiddled with the visual display.\n\nThe display was a hologram about a meter square by half a meter thick and was programmed to show the positions of Sade-138, our planet, and a few other chunks of rock in the system. There were green and red dots to show the positions of our vessels and the Taurans'.\n\nThe computer said that the minimum time it could take the Taurans to decelerate and get back to this planet would be a little over eleven days. Of course, that would be straight maximum acceleration and deceleration all the way; we could pick them off like flies on a wall. So, like us, they'd mix up their direction of flight and degree of acceleration in a random way. Based on several hundred past records of enemy behavior, the computer was able to give us a probability table:\n\nUnless, of course, Antopol and her gang of merry pirates managed to make a kill. The chances of that, I had learned in the can, were slightly less than fifty-fifty.\n\nBut whether it took 28.9554 days or two weeks, those of us on the ground had to just sit on our hands and watch. If Antopol was successful, then we wouldn't have to fight until the regular garrison troops replaced us here and we moved on to the next collapsar.\n\n'Haven't left yet.' Charlie had the display cranked down to minimum scale; the planet was a white ball the size of a large melon and _Masaryk II_ was a green dot off to the right some eight melons away; you couldn't get both on the screen at the same time.\n\nWhile we were watching, a small green dot popped out of the ship's dot and drifted away from it. A ghostly number 2 drifted beside it, and a key projected on the display's lower left-hand corner identified it as 2 \u2014 _Pursuit Drone_. Other numbers in the key identified the _Masaryk II_ , a planetary defense fighter and fourteen planetary defense drones. Those sixteen ships were not yet far enough away from one another to have separate dots.\n\nThe cat was rubbing against my ankle; I picked it up and stroked it. 'Tell Hilleboe to call a general assembly. Might as well break it to everyone at once.'\n\nThe men and women didn't take it very well, and I couldn't blame them. We had all expected the Taurans to attack much sooner \u2014 and when they persisted in not coming, the feeling grew that Strike Force Command had made a mistake and that they'd never show up at all.\n\nI wanted the company to start weapons training in earnest; they hadn't used any high-powered weapons in almost two years. So I activated their laser-fingers and passed out the grenade and rocket launchers. We couldn't practice inside the base for fear of damaging the external sensors and defensive laser ring. So we turned off half the circle of gigawatt lasers and went out about a klick beyond the perimeter, one platoon at a time, accompanied by either me or Charlie. Rusk kept a close watch on the early-warning screens. If anything approached, she would send up a flare, and the platoon would have to get back inside the ring before the unknown came over the horizon, at which time the defensive lasers would come on automatically. Besides knocking out the unknown, they would fry the platoon in less than .02 second.\n\nWe couldn't spare anything from the base to use as a target, but that turned out to be no problem. The first tachyon rocket we fired scooped out a hole twenty meters long by ten wide by five deep; the rubble gave us a multitude of targets from twice-man-sized on down.\n\nThe soldiers were good, a lot better than they had been with the primitive weapons in the stasis field. The best laser practice turned out to be rather like skeetshooting: pair up the people and have one stand behind the other, throwing rocks at random intervals. The one who was shooting had to gauge the rock's trajectory and zap it before it hit the ground. Their eye-hand coordination was impressive (maybe the Eugenics Council had done something right). Shooting at rocks down to pebble-size, most of them could do better than nine out of ten. Old non-bioengineered me could hit maybe seven out of ten, and I'd had a good deal more practice than they had.\n\nThey were equally facile at estimating trajectories with the grenade launcher, which was a more versatile weapon than it had been in the past. Instead of shooting one-microton bombs with a standard propulsive charge, it had four different charges and a choice of one-, two-, three- or four-microton bombs. And for really close in-fighting, where it was dangerous to use the lasers, the barrel of the launcher would unsnap, and you could load it with a magazine of 'shotgun' rounds. Each shot would send out an expanding cloud of a thousand tiny flechettes that were instant death out to five meters and turned to harmless vapor at six.\n\nThe tachyon rocket launcher required no skill whatsoever. All you had to do was to be careful no one was standing behind you when you fired it; the backwash from the rocket was dangerous for several meters behind the launching tube. Otherwise, you just lined your target up in the crosshairs and pushed the button. You didn't have to worry about trajectory; the rocket traveled in a straight line for all practical purposes. It reached escape velocity in less than a second.\n\nIt improved the troops' morale to get out and chew up the landscape with their new toys. But the landscape wasn't fighting back. No matter how physically impressive the weapons were, their effectiveness would depend on what the Taurans could throw back. A Greek phalanx must have looked pretty impressive, but it wouldn't do too well against a single man with a flamethrower.\n\nAnd as with any engagement, because of time dilation, there was no way to tell what sort of weaponry they would have. They might have never heard of the stasis field. Or they might be able to say a magic word and make us disappear.\n\nI was out with the fourth platoon, burning rocks, when Charlie called and asked me to come back in, urgent. I left Heimoff in charge.\n\n'Another one?' The scale of the holograph display was such that our planet was pea-sized, about five centimeters from the _X_ that marked the position of Sade-138. There were forty-one red and green dots scattered around the field; the key identified number _41_ as _Tauran Cruiser_ (2).\n\n'You called Antopol?'\n\n'Yeah.' He anticipated the next question. 'It'll take almost a day for the signal to get there and back.'\n\n'It's never happened before,' but of course Charlie knew that.\n\n'Maybe this collapsar is especially important to them.'\n\n'Likely.' So it was almost certain we'd be fighting on the ground. Even if Antopol managed to get the first cruiser, she wouldn't have a fifty-fifty chance on the second one. Low on drones and fighters. 'I wouldn't like to be Antopol now.'\n\n'She'll just get it earlier.'\n\n'I don't know. We're in pretty good shape.'\n\n'Save it for the troops, William.' He turned down the display's scale to where it showed only two objects: Sade-138 and the new red dot, slowing moving.\n\nWe spent the next two weeks watching dots blink out. And if you knew when and where to look, you could go outside and see the real thing happening, a hard bright speck of white light that faded in about a second.\n\nIn that second, a nova bomb had put out over a million times the power of a gigawatt laser. It made a miniature star half a klick in diameter and as hot as the interior of the sun. Anything it touched it would consume. The radiation from a near miss could botch up a ship's electronics beyond repair \u2014 two fighters, one of ours and one of theirs, had evidently suffered that fate, silently drifting out of the system at a constant velocity, without power.\n\nWe had used more powerful nova bombs earlier in the war, but the degenerate matter used to fuel them was unstable in large quantities. The bombs had a tendency to explode while they were still inside the ship. Evidently the Taurans had the same problem \u2014 or they had copied the process from us in the first place \u2014 because they had also scaled down to nova bombs that used less than a hundred kilograms of degenerate matter. And they deployed them much the same way we did, the warhead separating into dozens of pieces as it approached the target, only one of which was the nova bomb.\n\nThey would probably have a few bombs left over after they finished off _Masaryk II_ and her retinue of fighters and drones. So it was likely that we were wasting time and energy in weapons practice.\n\nThe thought did slip by my conscience that I could gather up eleven people and board the fighter we had hidden safe behind the stasis field. It was pre-programmed to take us back to Stargate.\n\nI even went to the extreme of making a mental list of the eleven, trying to think of eleven people who meant more to me than the rest. Turned out I'd be picking six at random.\n\nI put the thought away, though. We did have a chance, maybe a damned good one, even against a fully-armed cruiser. It wouldn't be easy to get a nova bomb close enough to include us inside its kill-radius.\n\nBesides, they'd space me for desertion. So why bother?\n\nSpirits rose when one of Antopol's drones knocked out the first Tauran cruiser. Not counting the ships left behind for planetary defense, she still had eighteen drones and two fighters. They wheeled around to intercept the second cruiser, by then a few light hours away, still being harassed by fifteen enemy drones.\n\nOne of the Tauran drones got her. Her ancillary crafts continued the attack, but it was a rout. One fighter and three drones fled the battle at maximum acceleration, looping up over the plane of the ecliptic, and were not pursued. We watched them with morbid interest while the enemy cruiser inched back to do battle with us. The fighter was headed back for Sade-138, to escape. Nobody blamed them. In fact, we sent them a farewell\/good luck message; they didn't respond, naturally, being zipped up in the tanks. But it would be recorded.\n\nIt took the enemy five days to get back to the planet and be comfortably ensconced in a stationary orbit on the other side. We settled in for the inevitable first phase of the attack, which would be aerial and totally automated: their drones against our lasers. I put a force of fifty men and women inside the stasis field, in case one of the drones got through. An empty gesture, really; the enemy could just stand by and wait for them to turn off the field, fry them the second it flickered out.\n\nCharlie had a weird idea that I almost went for.\n\n'We could boobytrap the place.'\n\n'What do you mean?' I said. 'This place is booby-trapped, out to twenty-five klicks.'\n\n'No, not the mines and such. I mean the base itself, here, underground.'\n\n'Go on.'\n\n'There are two nova bombs in that fighter.' He pointed at the stasis field through a couple of hundred meters of rock. 'We can roll them down here, boobytrap them, then hide everybody in the stasis field and wait.'\n\nIn a way it was tempting. It would relieve me from any responsibility for decision-making, leave everything up to chance. 'I don't think it would work, Charlie.'\n\nHe seemed hurt. 'Sure it would.'\n\n'No, look. For it to work, you have to get every single Tauran inside the kill-radius before it goes off \u2014 but they wouldn't all come charging in here once they breached our defenses. Least of all if the place seemed deserted. They'd suspect something, send in an advance party. And after the advance party set off the bombs\u2014'\n\n'We'd be back where we started, yeah. Minus the base. Sorry.'\n\nI shrugged. 'It was an idea. Keep thinking, Charlie.' I turned my attention back to the display, where the lopsided space war was in progress. Logically enough, the enemy wanted to knock out that one fighter overhead before he started to work on us. About all we could do was watch the red dots crawl around the planet and try to score. So far the pilot had managed to knock out all the drones; the enemy hadn't sent any fighters after him yet.\n\nI'd given the pilot control over five of the lasers in our defensive ring. They couldn't do much good, though. A gigawatt laser pumps out a billion kilowatts per second at a range of a hundred meters. A thousand klicks up, though, the beam was attenuated to ten kilowatts. Might do some damage if it hit an optical sensor. At least confuse things.\n\n'We could use another fighter. Or six.'\n\n'Use up the drones,' I said. We did have a fighter, of course, and a swabbie attached to us who could pilot it. It might turn out to be our only hope, if they got us cornered in the stasis field.\n\n'How far away is the other guy?' Charlie asked, meaning the fighter pilot who had turned tail. I cranked down the scale, and the green dot appeared at the right of the display. 'About six light hours.' He had two drones left, too near to him to show as separate dots, having expended one in covering his getaway. 'He's not accelerating any more, but he's doing point nine gee.'\n\n'Couldn't do us any good if he wanted to.' Need almost a month to slow down.\n\nAt that low point, the light that stood for our own defensive fighter faded out. 'Shit.'\n\n'Now the fun starts. Should I tell the troops to get ready, stand by to go topside?'\n\n'No... have them suit up, in case we lose air. But I expect it'll be a little while before we have a ground attack.' I turned the scale up again. Four red dots were already creeping around the globe toward us.\n\nI got suited up and came back to Administration to watch the fireworks on the monitors.\n\nThe lasers worked perfectly. All four drones converged on us simultaneously; were targeted and destroyed. All but one of the nova bombs went off below our horizon (the visual horizon was about ten kilometers away, but the lasers were mounted high and could target something at twice that distance). The bomb that detonated on our horizon had melted out a semicircular chunk that glowed brilliantly white for several minutes. An hour later, it was still glowing dull orange, and the ground temperature outside had risen to fifty degrees Absolute, melting most of our snow, exposing an irregular dark gray surface.\n\nThe next attack was also over in a fraction of a second, but this time there had been eight drones, and four of them got within ten klicks. Radiation from the glowing craters raised the temperature to nearly 300 degrees. That was above the melting point of water, and I was starting to get worried. The fighting suits were good to over a thousand degrees, but the automatic lasers depended on low-temperature superconductors for their speed.\n\nI asked the computer what the lasers' temperature limit was, and it printed out _TR 398-734-009-265_ , _'Some Aspects Concerning the Adaptability of Cryogenic Ordnance to Use in Relatively High-Temperature Environments,'_ which had lots of handy advice about how we could insulate the weapons if we had access to a fully-equipped armorer's shop. It did note that the response time of automatic-aiming devices increased as the temperature increased, and that above some 'critical temperature,' the weapons would not aim at all. But there was no way to predict any individual weapon's behavior, other than to note that the highest critical temperature recorded was 790 degrees and the lowest was 420 degrees.\n\nCharlie was watching the display. His voice was flat over the suit's radio. 'Sixteen this time.'\n\n'Surprised?' One of the few things we knew about Tauran psychology was a certain compulsiveness about numbers, especially primes and powers of two.\n\n'Let's just hope they don't have 32 left.' I queried the computer on this; all it could say was that the cruiser had thus far launched a total of 44 drones and that some cruisers had been known to carry as many as 128.\n\nWe had more than a half-hour before the drones would strike. I could evacuate everybody to the stasis field, and they would be temporarily safe if one of the nova bombs got through. Safe, but trapped. How long would it take the crater to cool down, if three or four \u2014 let alone sixteen \u2014 of the bombs made it through? You couldn't live forever in a fighting suit, even though it recycled everything with remorseless efficiency. One week was enough to make you thoroughly miserable. Two weeks, suicidal. Nobody had ever gone three weeks, under field conditions.\n\nBesides, as a defensive position, the stasis field could be a death-trap. The enemy has all the options since the dome is opaque; the only way you can find out what they're up to is to stick your head out. They didn't have to wade in with primitive weapons unless they were impatient. They could keep the dome saturated with laser fire and wait for you to turn off the generator. Meanwhile harassing you by throwing spears, rocks, arrows into the dome \u2014 you could return fire, but it was pretty futile.\n\nOf course, if one man stayed inside the base, the others could wait out the next half-hour in the stasis field. If he didn't come get them, they'd know the outside was hot. I chinned the combination that would give me a frequency available to everybody echelon 5 and above.\n\n'This is Major Mandella.' That still sounded like a bad joke.\n\nI outlined the situation to them and asked them to tell their troops that everyone in the company was free to move into the stasis field. I would stay behind and come retrieve them if things went well \u2014 not out of nobility, of course; I preferred taking the chance of being vaporized in a nano-second, rather than almost certain slow death under the gray dome.\n\nI chinned Charlie's frequency. 'You can go, too. I'll take care of things here.'\n\n'No, thanks,' he said slowly. 'I'd just as soon... Hey, look at this.'\n\nThe cruiser had launched another red dot, a couple of minutes behind the others. The display's key identified it as being another drone. 'That's curious.'\n\n'Superstitious bastards,' he said without feeling.\n\nIt turned out that only eleven people chose to join the fifty who had been ordered into the dome. That shouldn't have surprised me, but it did.\n\nAs the drones approached, Charlie and I stared at the monitors, carefully not looking at the holograph display, tacitly agreeing that it would be better not to know when they were one minute away, thirty seconds... And then, like the other times, it was over before we knew it had started. The screens glared white and there was a yowl of static, and we were still alive.\n\nBut this time there were fifteen new holes on the horizon \u2014 or closer! \u2014 and the temperature was rising so fast that the last digit in the readout was an amorphous blur. The number peaked in the high 800s and began to slide back down.\n\nWe had never seen any of the drones, not during that tiny fraction of a second it took the lasers to aim and fire. But then the seventeenth one flashed over the horizon, zig-zagging crazily, and stopped directly overhead. For an instant it seemed to hover, and then it began to fall. Half the lasers had detected it, and they were firing steadily, but none of them could aim; they were all stuck in their last firing position.\n\nIt glittered as it dropped, the mirror polish of its sleek hull reflecting the white glow from the craters and the eerie flickering of the constant, impotent laser fire. I heard Charlie take one deep breath, and the drone fell so close you could see spidery Tauran numerals etched on the hull and a transparent porthole near the tip \u2014 then its engine flared and it was suddenly gone.\n\n'What the hell?' Charlie said, quietly.\n\nThe porthole. 'Maybe reconnaissance.'\n\n'I guess. So we can't touch them, and they know it.'\n\n'Unless the lasers recover.' Didn't seem likely. 'We better get everybody under the dome. Us, too.'\n\nHe said a word whose vowel had changed over the centuries, but whose meaning was clear. 'No hurry. Let's see what they do.'\n\nWe waited for several hours. The temperature outside stabilized at 690 degrees \u2014 just under the melting point of zinc, I remembered to no purpose \u2014 and I tried the manual controls for the lasers, but they were still frozen.\n\n'Here they come,' Charlie said. 'Eight again.'\n\nI started for the display. 'Guess we'll\u2014'\n\n'Wait! They aren't drones.' The key identified all eight with the legend _Troop Carrier_.\n\n'Guess they want to take the base,' he said. 'Intact.'\n\nThat, and maybe try out new weapons and techniques. 'It's not much of a risk for them. They can always retreat and drop a nova bomb in our laps.'\n\nI called Brill and had her go get everybody who was in the stasis field, set them up with the remainder of her platoon as a defensive line circling around the northeast and northwest quadrants. I'd put the rest of the people in the other half-circle.\n\n'I wonder,' Charlie said. 'Maybe we shouldn't put everyone topside at once. Until we know how many Taurans there are.'\n\nThat was a point. Keep a reserve, let the enemy under-estimate our strength. 'It's an idea... There might be just 64 of them in eight carriers.' Or 128 or 256. I wished our spy satellites had a finer sense of discrimination. But you can only cram so much into a machine the size of a grape.\n\nI decided to let Brill's seventy people be our first line of defense and ordered them into a ring in the ditches we had made outside the base's perimeter. Everybody else would stay downstairs until needed.\n\nIf it turned out that the Taurans, either through numbers or new technology, could field an unstoppable force, I'd order everyone into the stasis field. There was a tunnel from the living quarters to the dome, so the people underground could go straight there in safety. The ones in the ditches would have to fall back under fire. If any of them were still alive when I gave the order.\n\nI called in Hilleboe and had her and Charlie keep watch over the lasers. If they came unstuck, I'd call Brill and her people back. Turn on the automatic aiming system again, then sit back and watch the show. But even stuck, the lasers could be useful. Charlie marked the monitors to show where the rays would go; he and Hilleboe could fire them manually whenever something moved into a weapon's line-of-sight.\n\nWe had about twenty minutes. Brill was walking around the perimeter with her men and women, ordering them into the ditches a squad at a time, setting up overlapping fields of fire. I broke in and asked her to set up the heavy weapons so that they could be used to channel the enemy's advance into the path of the lasers.\n\nThere wasn't much else to do but wait. I asked Charlie to measure the enemy's progress and try to give us an accurate count-down, then sat at my desk and pulled out a pad, to diagram Brill's arrangement and see whether I could improve on it.\n\nThe cat jumped up on my lap, mewling piteously. He'd evidently been unable to tell one person from the other, suited up. But nobody else ever sat at this desk. I reached up to pet him and he jumped away.\n\nThe first line that I drew ripped through four sheets of paper. It had been some time since I'd done any delicate work in a suit. I remembered how in training, they'd made us practice controlling the strength-amplification circuits by passing eggs from person to person, messy business. I wondered if they still had eggs on Earth.\n\nThe diagram completed, I couldn't see any way to add to it. All those reams of theory crammed in my brain; there was plenty of tactical advice about envelopment and encirclement, but from the wrong point of view. If you were the one who was being encircled, you didn't have many options. Sit tight and fight. Respond quickly to enemy concentrations of force, but stay flexible so the enemy can't employ a diversionary force to divert strength from some predictable section of your perimeter. _Make full use of air and space support_ , always good advice. Keep your head down and your chin up and pray for the cavalry. Hold your position and don't contemplate Dienbienphu, the Alamo, the Battle of Hastings.\n\n'Eight more carriers out,' Charlie said. 'Five minutes. Until the first eight get here.'\n\nSo they were going to attack in two waves. At least two. What would I do, in the Tauran commander's position? That wasn't too far-fetched; the Taurans lacked imagination in tactics and tended to copy human patterns.\n\nThe first wave could be a throwaway, a kamikaze attack to soften us up and evaluate our defenses. Then the second would come in more methodically, and finish the job. Or vice versa: the first group would have twenty minutes to get entrenched; then the second could skip over their heads and hit us hard at one spot \u2014 breach the perimeter and over-run the base.\n\nOr maybe they sent out two forces simply because two was a magic number. Or they could launch only eight troop carriers at a time (that would be bad, implying that the carriers were large; in different situations they had used carriers holding as few as 4 troops or as many as 128).\n\n'Three minutes.' I stared at the cluster of monitors that showed various sectors of the mine field. If we were lucky, they'd land out there, out of caution. Or maybe pass over it low enough to detonate mines.\n\nI was feeling vaguely guilty. I was safe in my hole, doodling, ready to start calling out orders. How did those seventy sacrificial lambs feel about their absentee commander?\n\nThen I remembered how I had felt about Captain Stott that first mission, when he'd elected to stay safely in orbit while we fought on the ground. The rush of remembered hate was so strong I had to bite back nausea.\n\n'Hilleboe, can you handle the lasers by yourself?'\n\n'I don't see why not, sir.'\n\nI tossed down the pen and stood up. 'Charlie, you take over the unit coordination; you can do it as well as I could. I'm going topside.'\n\n'I wouldn't advise that, sir.'\n\n'Hell no, William. Don't be an idiot.'\n\n'I'm not taking orders, I'm giv\u2014'\n\n'You wouldn't last ten seconds up there,' Charlie said.\n\n'I'll take the same chance as everybody else.'\n\n'Don't you hear what I'm saying. _They'll_ kill you!'\n\n'The troops? Nonsense. I know they don't like me especially, but\u2014'\n\n'You haven't listened in on the squad frequencies?' No, they didn't speak my brand of English when they talked among themselves. 'They think you put them out on the line for punishment, for cowardice. After you'd told them anyone was free to go into the dome.'\n\n'Didn't you, sir?' Hilleboe said.\n\n'To punish them? No, of course not.' Not consciously. 'They were just up there when I needed... Hasn't Lieutenant Brill said anything to them?'\n\n'Not that I've heard,' Charlie said. 'Maybe she's been too busy to tune in.'\n\nOr she agreed with them. 'I'd better get\u2014'\n\n'There!' Hilleboe shouted. The first empty ship was visible in one of the mine field monitors; the others appeared in the next second. They came in from random directions and weren't evenly distributed around the base. Five in the northeast quadrant and only one in the south-west. I relayed the information to Brill.\n\nBut we had predicted their logic pretty well; all of them were coming down in the ring of mines. One came close enough to one of the tachyon devices to set it off. The blast caught the rear end of the oddly streamlined craft, causing it to make a complete flip and crash nose-first. Side ports opened up and Taurans came crawling out. Twelve of them; probably four left inside. If all the others had sixteen as well, there were only slightly more of them than of us.\n\nIn the first wave.\n\nThe other seven had landed without incident, and yes; there were sixteen each. Brill shuffled a couple of squads to conform to the enemy's troop concentration, and she waited.\n\nThey moved fast across the mine field, striding in unison like bowlegged, top-heavy robots, not even breaking stride when one of them was blown to bits by a mine, which happened eleven times.\n\nWhen they came over the horizon, the reason for their apparently random distribution was obvious: they had analyzed beforehand which approaches would give them the most natural cover, from the rubble that the drones had kicked up. They would be able to get within a couple of kilometers of the base before we got any clear line-of-sight of them. And their suits had augmentation circuits similar to ours, so they could cover a kilometer in less than a minute.\n\nBrill had her troops open fire immediately, probably more for morale than out of any hope of actually hitting the enemy. They probably were getting a few, though it was hard to tell. At least the tachyon rockets did an impressive job of turning boulders into gravel.\n\nThe Taurans returned fire with some weapon similar to the tachyon rocket, maybe exactly the same. They rarely found a mark, though; our people were at and below ground level, and if the rocket didn't hit something, it would keep going on forever, amen. They did score a hit on one of the gigawatt lasers, though, and the concussion that filtered down to us was strong enough to make me wish we had burrowed a little deeper than twenty meters.\n\nThe gigawatts weren't doing us any good. The Taurans must have figured out the lines of sight ahead of time, and gave them wide berth. That turned out to be fortunate, because it caused Charlie to let his attention wander from the laser monitors for a moment.\n\n'What the hell?'\n\n'What's that, Charlie?' I didn't take my eyes off the monitors. Waiting for something to happen.\n\n'The ship, the cruiser \u2014 it's gone.' I looked at the holograph display. He was right; the only red lights were those that stood for the troop carriers.\n\n'Where did it go?' I asked inanely.\n\n'Let's play it back.' He programmed the display to go back a couple of minutes and cranked out the scale to where both planet and collapsar showed on the cube. The cruiser showed up, and with it, three green dots. Our 'coward,' attacking the cruiser with only two drones.\n\nBut he had a little help from the laws of physics.\n\nInstead of going into collapsar insertion, he had skimmed _around_ the collapsar field in a slingshot orbit. He had come out going nine-tenths of the speed of light; the drones were going .99c, headed straight for the enemy cruiser. Our planet was about a thousand light seconds from the collapsar, so the Tauran ship had only ten seconds to detect and stop both drones. And at that speed, it didn't matter whether you'd been hit by a nova-bomb or a spitball.\n\nThe first drone disintegrated the cruiser, and the other one, .01 second behind, glided on down to impact on the planet. The fighter missed the planet by a couple of hundred kilometers and hurtled on into space, decelerating with the maximum twenty-five gees. He'd be back in a couple of months.\n\nBut the Taurans weren't going to wait. They were getting close enough to our lines for both sides to start using lasers, but they were also within easy grenade range. A good-size rock could shield them from laser fire, but the grenades and rockets were slaughtering them.\n\nAt first, Brill's troops had the overwhelming advantage; fighting from ditches, they could only be harmed by an occasional lucky shot or an extremely well-aimed grenade (which the Taurans threw by hand, with a range of several hundred meters). Brill had lost four, but it looked as if the Tauran force was down to less than half its original size.\n\nEventually, the landscape had been torn up enough so that the bulk of the Tauran force was able to fight from holes in the ground. The fighting slowed down to individual laser duels, punctuated occasionally by heavier weapons. But it wasn't smart to use up a tachyon rocket against a single Tauran, not with another force of unknown size only a few minutes away.\n\nSomething had been bothering me about that holographic replay. Now, with the battle's lull, I knew what it was.\n\nWhen that second drone crashed at near-lightspeed, how much damage had it done to the planet? I stepped over to the computer and punched it up; found out how much energy had been released in the collision, and then compared it with geological information in the computer's memory.\n\nTwenty times as much energy as the most powerful earthquake ever recorded. On a planet three-quarters the size of Earth.\n\nOn the general frequency: 'Everybody-topside! Right now!' I palmed the button that would cycle and open the airlock and tunnel that led from Administration to the surface.\n\n'What the hell, Will\u2014'\n\n'Earthquake!' How long? 'Move!'\n\nHilleboe and Charlie were right behind me. The cat was sitting on my desk, licking himself unconcernedly. I had an irrational impulse to put him inside my suit, which was the way he'd been carried from the ship to the base, but knew he wouldn't tolerate more than a few minutes of it. Then I had the more reasonable impulse to simply vaporize him with my laser-finger, but by then the door was closed and we were swarming up the ladder. All the way up, and for some time afterward, I was haunted by the image of that helpless animal, trapped under tons of rubble, dying slowly as the air hissed away.\n\n'Safer in the ditches?' Charlie said.\n\n'I don't know,' I said. 'Never been in an earthquake.' Maybe the walls of the ditch would close up and crush us.\n\nI was surprised at how dark it was on the surface. S Doradus had almost set; the monitors had compensated for the low light level.\n\nAn enemy laser raked across the clearing to our left, making a quick shower of sparks when it flicked by a gigawatt mounting. We hadn't been seen yet. We all decided yes, it would be safer in the ditches, and made it to the nearest one in three strides.\n\nThere were four men and women in the ditch, one of them badly wounded or dead. We scrambled down the ledge and I turned up my image amplifier to log two, to inspect our ditchmates. We were lucky; one was a grenadier and they also had a rocket launcher. I could just make out the names on their helmets. We were in Brill's ditch, but she hadn't noticed us yet. She was at the opposite end, cautiously peering over the edge, directing two squads in a flanking movement. When they were safely in position, she ducked back down. 'Is that you, Major?'\n\n'That's right,' I said cautiously. I wondered whether any of the people in the ditch were among the ones after my scalp.\n\n'What's this about an earthquake?'\n\nShe had been told about the cruiser being destroyed, but not about the other drone. I explained in as few words as possible.\n\n'Nobody's come out of the airlock,' she said. 'Not yet. I guess they all went into the stasis field.'\n\n'Yeah, they were just as close to one as the other.' Maybe some of them were still down below, hadn't taken my warning seriously. I chinned the general frequency to check, and then all hell broke loose.\n\nThe ground dropped away and then flexed back up; slammed us so hard that we were airborne, tumbling out of the ditch. We flew several meters, going high enough to see the pattern of bright orange and yellow ovals, the craters where nova bombs had been stopped. I landed on my feet but the ground was shifting and slithering so much that it was impossible to stay upright.\n\nWith a basso grinding I could feel through my suit, the cleared area above our base crumbled and fell in. Part of the stasis field's underside was exposed when the ground subsided; it settled to its new level with aloof grace.\n\nWell, minus one cat. I hoped everybody else had time and sense enough to get under the dome.\n\nA figure came staggering out of the ditch nearest to me and I realized with a start that it wasn't human. At that range, my laser burned a hole straight through his helmet; he took two steps and fell over backward. Another helmet peered over the edge of the ditch. I sheared the top of it off before he could raise his weapon.\n\nI couldn't get my bearings. The only thing that hadn't changed was the stasis dome, and it looked the same from any angle. The gigawatt lasers were all buried, but one of them had switched on, a brilliant flickering searchlight that illuminated a swirling cloud of vaporized rock.\n\nObviously, though, I was in enemy territory. I started across the trembling ground toward the dome.\n\nI couldn't raise any platoon leaders. All of them but Brill were probably inside the dome. I did get Hilleboe and Charlie; told Hilleboe to go inside the dome and roust everybody out. If the next wave also had 128, we were going to need everybody.\n\nThe tremors died down and I found my way into a 'friendly' ditch \u2014 the cooks' ditch, in fact, since the only people there were Orban and Rudkoski.\n\n'Looks like you'll have to start from scratch again, Private.'\n\n'That's all right, sir. Liver needed a rest.'\n\nI got a beep from Hilleboe and chinned her on. 'Sir... there were only ten people there. The rest didn't make it.'\n\n'They stayed behind?' Seemed like they'd had plenty of time.\n\n'I don't know, sir.'\n\n'Never mind. Get me a count, how many people we have, all totalled.' I tried the platoon leaders' frequency again and it was still silent.\n\nThe three of us watched for enemy laser fire for a couple of minutes, but there was none. Probably waiting for reinforcements.\n\nHilleboe called back. 'I only get fifty-three, sir. Some may be unconscious.'\n\n'All right. Have them sit tight until\u2014' Then the second wave showed up, the troop carriers roaring over the horizon with their jets pointed our way, decelerating. _'Get some rockets on those bastards!'_ Hilleboe yelled to everyone in particular. But nobody had managed to stay attached to a rocket launcher while he was being tossed around. No grenade launchers, either, and the range was too far for the hand lasers to do any damage.\n\nThese carriers were four or five times the size of the ones in the first wave. One of them grounded about a kilometer in front of us, barely stopping long enough to disgorge its troops. Of which there were over 50, probably 64 \u2014 times 8 made 512. No way we could hold them back.\n\n'Everybody listen, this is Major Mandella.' I tried to keep my voice even and quiet. 'We're going to retreat back into the dome, quickly but in an orderly way. I know we're scattered all over hell. If you belong to the second or fourth platoon, stay put for a minute and give covering fire while the first and third platoons, and support, fall back.\n\n'First and third and support, fall back to about half your present distance from the dome, then take cover and defend the second and fourth as they come back. They'll go to the edge of the dome and cover you while you come back the rest of the way.' I shouldn't have said 'retreat'; that word wasn't in the book. Retrograde action.\n\nThere was a lot more retrograde than action. Eight or nine people were firing, and all the rest were in full flight. Rudkoski and Orban had vanished. I took a few carefully aimed shots, to no great effect, then ran down to the other end of the ditch, climbed out and headed for the dome.\n\nThe Taurans started firing rockets, but most of them seemed to be going too high. I saw two of us get blown away before I got to my halfway point; found a nice big rock and hid behind it. I peeked out and decided that only two or three of the Taurans were close enough to be even remotely possible laser targets, and the better part of valor would be in not drawing unnecessary attention to myself. I ran the rest of the way to the edge of the field and stopped to return fire. After a couple of shots, I realized that I was just making myself a target; as far as I could see there was only one other person who was still running toward the dome.\n\nA rocket zipped by, so close I could have touched it. I flexed my knees and kicked, and entered the dome in a rather undignified posture. \n\n# 7\n\nInside, I could see the rocket that had missed me drifting lazily through the gloom, rising slightly as it passed through to the other side of the dome. It would vaporize the instant it came out the other side, since all of the kinetic energy it had lost in abruptly slowing down to 16.3 meters per second would come back in the form of heat.\n\nNine people were lying dead, facedown just inside of the field's edge. It wasn't unexpected, though it wasn't the sort of thing you were supposed to tell the troops.\n\nTheir fighting suits were intact \u2014 otherwise they wouldn't have made it this far \u2014 but sometime during the past few minutes' rough-and-tumble, they had damaged the coating of special insulation that protected them from the stasis field. So as soon as they entered the field, all electrical activity in their bodies ceased, which killed them instantly. Also, since no molecule in their bodies could move faster than 16.3 meters per second, they instantly froze solid, their body temperature stabilized at a cool 0.426 degrees Absolute.\n\nI decided not to turn any of them over to find out their names, not yet. We had to get some sort of defensive position worked out before the Taurans came through the dome. If they decided to slug it out rather than wait.\n\nWith elaborate gestures, I managed to get everybody collected in the center of the field, under the fighter's tail, where the weapons were racked.\n\nThere were plenty of weapons, since we had been prepared to outfit three times this number of people. After giving each person a shield and short-sword, I traced a question in the snow: GOOD ARCHERS? RAISE HANDS. I got five volunteers, then picked out three more so that all the bows would be in use. Twenty arrows per bow. They were the most effective long-range weapons we had; the arrows were almost invisible in their slow flight, heavily weighted and tipped with a deadly sliver of diamond-hard crystal.\n\nI arranged the archers in a circle around the fighter (its landing fins would give them partial protection from missiles coming in from behind) and between each pair of archers put four other people: two spear-throwers, one quarterstaff, and a person armed with battleax and a dozen throwing knives. This arrangement would theoretically take care of the enemy at any range, from the edge of the field to hand-to-hand combat.\n\nActually, at some 600-to-42 odds, they could probably walk in with a rock in each hand, no shields or special weapons, and still beat the shit out of us.\n\nAssuming they knew what the stasis field was. Their technology seemed up to date in all other respects.\n\nFor several hours nothing happened. We got about as bored as anyone could, waiting to die. No one to talk to, nothing to see but the unchanging gray dome, gray snow, gray spaceship and a few identically gray soldiers. Nothing to hear, taste or smell but yourself.\n\nThose of us who still had any interest in the battle were keeping watch on the bottom edge of the dome, waiting for the first Taurans to come through. So it took us a second to realize what was going on when the attack did start. It came from above, a cloud of catapulted darts swarming in through the dome some thirty meters above the ground, headed straight for the center of the hemisphere.\n\nThe shields were big enough that you could hide most of your body behind them by crouching slightly; the people who saw the darts coming could protect themselves easily. The ones who had their backs to the action, or were just asleep at the switch, had to rely on dumb luck for survival; there was no way to shout a warning, and it took only three seconds for a missile to get from the edge of the dome to its center.\n\nWe were lucky, losing only five. One of them was an archer, Shubik. I took over her bow and we waited, expecting a ground attack immediately.\n\nIt didn't come. After a half-hour, I went around the circle and explained with gestures that the first thing you were supposed to do, if anything happened, was to touch the person on your right. He'd do the same, and so on down the line.\n\nThat might have saved my life. The second dart attack, a couple of hours later, came from behind me. I felt the nudge, slapped the person on my right, turned around and saw the cloud descending. I got the shield over my head, and they hit a split-second later.\n\nI set down my bow to pluck three darts from the shield and the ground attack started.\n\nIt was a weird, impressive sight. Some three hundred of them stepped into the field simultaneously, almost shoulder-to-shoulder around the perimeter of the dome. They advanced in step, each one holding a round shield barely large enough to hide his massive chest. They were throwing darts similar to the ones we had been barraged with.\n\nI set up the shield in front of me \u2014 it had little extensions on the bottom to keep it upright \u2014 and with the first arrow I shot, I knew we had a chance. It struck one of them in the center of his shield, went straight through and penetrated his suit.\n\nIt was a one-sided massacre. The darts weren't very effective without the element of surprise \u2014 but when one came sailing over my head from behind, it did give me a crawly feeling between the shoulder blades.\n\nWith twenty arrows I got twenty Taurans. They closed ranks every time one dropped; you didn't even have to aim. After running out of arrows, I tried throwing their darts back at them. But their light shields were quite adequate against the small missiles.\n\nWe'd killed more than half of them with arrows and spears, long before they got into range of the hand-to-hand weapons. I drew my sword and waited. They still outnumbered us by better than three to one.\n\nWhen they got within ten meters, the people with the chakram throwing knives had their own field day. Although the spinning disc was easy enough to see and took more than a half-second to get from thrower to target, most of the Taurans reacted in the same ineffective way, raising up the shield to ward it off. The razor-sharp tempered heavy blade cut through the light shield like a buzz-saw through cardboard.\n\nThe first hand-to-hand contact was with the quarterstaffs, which were metal rods two meters long that tapered at the ends to a double-edged, serrated knife blade. The Taurans had a cold-blooded \u2014 or valiant, if your mind works that way \u2014 method for dealing with them. They would simply grab the blade and die. While the human was trying to extricate his weapon from the frozen death-grip, a Tauran swordsman, with a scimitar over a meter long, would step in and kill him.\n\nBesides the swords, they had a bolo-like thing that was a length of elastic cord that ended with about ten centimeters of something like barbed wire, and a small weight to propel it. It was a dangerous weapon for all concerned; if they missed their target it would come snapping back unpredictably. But they hit their target pretty often, going under the shields and wrapping the thorny wire around ankles.\n\nI stood back-to-back with Private Erikson, and with our swords we managed to stay alive for the next few minutes. When the Taurans were down to a couple of dozen survivors, they just turned around and started marching out. We threw some darts after them, getting three, but we didn't want to chase after them. They might turn around and start hacking again.\n\nThere were only twenty-eight of us left standing. Nearly ten times that number of dead Taurans littered the ground, but there was no satisfaction in it.\n\nThey could do the whole thing over, with a fresh 300. And this time it would work.\n\nWe moved from body to body, pulling out arrows and spears, then took up places around the fighter again. Nobody bothered to retrieve the quarterstaffs. I counted noses: Charlie and Diana were still alive (Hilleboe had been one of the quarterstaff victims), as well as two supporting officers. Wilber and Szydlowska. Rudkoski was still alive but Orban had taken a dart.\n\nAfter a day of waiting, it looked as though the enemy had decided on a war of attrition rather than repeating the ground attack. Darts came in constantly, not in swarms anymore, but in twos and threes and tens. And from all different angles. We couldn't stay alert forever; they'd get somebody every three or four hours.\n\nWe took turns sleeping, two at a time, on top of the stasis field generator. Sitting directly under the bulk of the fighter, it was the safest place in the dome.\n\nEvery now and then, a Tauran would appear at the edge of the field, evidently to see whether any of us were left. Sometimes we'd shoot an arrow at him, for practice.\n\nThe darts stopped falling after a couple of days. I supposed it was possible that they'd simply run out of them. Or maybe they'd decided to stop when we were down to twenty survivors.\n\nThere was a more likely possibility. I took one of the quarterstaffs down to the edge of the field and poked it through, a centimeter or so. When I drew it back, the point was melted off. When I showed it to Charlie, he rocked back and forth (the only way you can nod in a suit); this sort of thing had happened before, one of the first times the stasis field hadn't worked. They simply saturated it with laser fire and waited for us to go stir-crazy and turn off the generator. They were probably sitting in their ships playing the Tauran equivalent of pinochle.\n\nI tried to think. It was hard to keep your mind on something for any length of time in that hostile environment, sense-deprived, looking over your shoulder every few seconds. Something Charlie had said. Only yesterday. I couldn't track it down. It wouldn't have worked then; that was all I could remember. Then finally it came to me.\n\nI called everyone over and wrote in the snow:\n\nGET NOVA BOMBS FROM SHIP.\n\nCARRY TO EDGE OF FIELD.\n\nMOVE FIELD.\n\nSzydlowska knew where the proper tools would be aboard ship. Luckily, we had left all of the entrances open before turning on the stasis field; they were electronic and would have been frozen shut. We got an assortment of wrenches from the engine room and climbed up to the cockpit. He knew how to remove the access plate that exposed a crawl space into the bomb-bay. I followed him in through the meter-wide tube.\n\nNormally, I supposed, it would have been pitch-black. But the stasis field illuminated the bomb-bay with the same dim, shadowless light that prevailed outside. The bomb-bay was too small for both of us, so I stayed at the end of the crawl space and watched.\n\nThe bomb-bay doors had a 'manual override' so they were easy; Szydlowska just turned a hand-crank and we were in business. Freeing the two nova bombs from their cradles was another thing. Finally, he went back down to the engine room and brought back a crowbar. He pried one loose and I got the other, and we rolled them out the bomb-bay.\n\nSergeant Anghelov was already working on them by the time we climbed back down. All you had to do to arm the bomb was to unscrew the fuse on the nose of it and poke something around in the fuse socket to wreck the delay mechanism and safety restraints.\n\nWe carried them quickly to the edge, six people per bomb, and set them down next to each other. Then we waved to the four people who were standing by at the field generator's handles. They picked it up and walked ten paces in the opposite direction. The bombs disappeared as the edge of the field slid over them.\n\nThere was no doubt that the bombs went off. For a couple of seconds it was hot as the interior of a star outside, and even the stasis field took notice of the fact: about a third of the dome glowed a dull pink for a moment, then was gray again. There was a slight acceleration, like you would feel in a slow elevator. That meant we were drifting down to the bottom of the crater. Would there be a solid bottom? Or would we sink down through molten rock to be trapped like a fly in amber \u2014 didn't pay to even think about that. Perhaps if it happened, we could blast our way out with the fighter's gigawatt laser.\n\nTwelve of us, anyhow.\n\nHOW LONG? Charlie scraped in the snow at my feet.\n\nThat was a damned good question. About all I knew was the amount of energy two nova bombs released. I didn't know how big a fireball they would make, which would determine the temperature at detonation and the size of the crater. I didn't know the heat capacity of the surrounding rock, or its boiling point. I wrote: ONE WEEK, SHRUG? HAVE TO THINK.\n\nThe ship's computer could have told me in a thousandth of a second, but it wasn't talking. I started writing equations in the snow, trying to get a maximum and minimum figure for the length of time it would take for the outside to cool down to 500 degrees. Anghelov, whose physics was much more up-to-date, did his own calculations on the other side of the ship.\n\nMy answer said anywhere from six hours to six days (although for six hours, the surrounding rock would have to conduct heat like pure copper), and Anghelov got five hours to 4\u00bd days. I voted for six and nobody else got a vote.\n\nWe slept a lot. Charlie and Diana played chess by scraping symbols in the snow; I was never able to hold the shifting positions of the pieces in my mind. I checked my figures several times and kept coming up with six days. I checked Anghelov's computations, too, and they seemed all right, but I stuck to my guns. It wouldn't hurt us to stay in the suits an extra day and a half. We argued good-naturedly in terse shorthand.\n\nThere had been nineteen of us left the day we tossed the bombs outside. There were still nineteen, six days later, when I paused with my hand over the generator's cutoff switch. What was waiting for us out there? Surely we had killed all the Taurans within several klicks of the explosion. But there might have been a reserve force farther away, now waiting patiently on the crater's lip. At least you could push a quarterstaff through the field and have it come back whole.\n\nI dispersed the people evenly around the area, so they might not get us with a single shot. Then, ready to turn it back on immediately if anything went wrong, I pushed. \n\n# 8\n\nMy radio was still tuned to the general frequency; after more than a week of silence my ears were suddenly assaulted with loud, happy babbling.\n\nWe stood in the center of a crater almost a kilometer wide and deep. Its sides were a shiny black crust shot through with red cracks, hot but no longer dangerous. The hemisphere of earth that we rested on had sunk a good forty meters into the floor of the crater, while it had still been molten, so now we stood on a kind of pedestal.\n\nNot a Tauran in sight.\n\nWe rushed to the ship, sealed it and filled it with cool air and popped our suits. I didn't press seniority for the one shower; just sat back in an acceleration couch and took deep breaths of air that didn't smell like recycled Mandella.\n\nThe ship was designed for a maximum crew of twelve, so we stayed outside in shifts of seven to keep from straining the life-support systems. I sent a repeating message to the other fighter, which was still over six weeks away, that we were in good shape and waiting to be picked up. I was reasonably certain he would have seven free berths, since the normal crew for a combat mission was only three.\n\nIt was good to walk around and talk again. I officially suspended all things military for the duration of our stay on the planet. Some of the people were survivors of Brill's mutinous bunch, but they didn't show any hostility toward me.\n\nWe played a kind of nostalgia game, comparing the various eras we'd experienced on Earth, wondering what it would be like in the 700-years-future we were going back to. Nobody mentioned the fact that we would at best go back to a few months' furlough and then be assigned to another strike force, another turn of the wheel.\n\nWheels. One day Charlie asked me from what country my name originated; it sounded weird to him. I told him it originated from the lack of a dictionary and that if it were spelled right, it would look even weirder.\n\nI got to kill a good half-hour explaining all the peripheral details to that. Basically, though, my parents were 'hippies' (a kind of subculture in the late-twentieth-century America, that rejected materialism and embraced a broad spectrum of odd ideas) who lived with a group of other hippies in a small agricultural community. When my mother got pregnant, they wouldn't be so conventional as to get married: this entailed the woman taking the man's name, and implied that she was his property. But they got all intoxicated and sentimental and decided they would both change their names to be the same. They rode into the nearest town, arguing all the way as to what name would be the best symbol for the love-bond between them \u2014 I narrowly missed having a much shorter name \u2014 and they settled on Mandala.\n\nA mandala is a wheel-like design the hippies had borrowed from a foreign religion, that symbolized the cosmos, the cosmic mind, God, or whatever needed a symbol. Neither my mother nor my father knew how to spell the word, and the magistrate in town wrote it down the way it sounded to him.\n\nThey named me William in honor of a wealthy uncle, who unfortunately died penniless.\n\nThe six weeks passed rather pleasantly: talking, reading, resting. The other ship landed next to ours and did have nine free berths. We shuffled crews so that each ship had someone who could get it out of trouble if the pre-programmed jump sequence malfunctioned. I assigned myself to the other ship, in hopes it would have some new books. It didn't.\n\nWe zipped up in the tanks and took off simultaneously.\n\nWe wound up spending a lot of time in the tanks, just to keep from looking at the same faces all day long in the crowded ship. The added periods of acceleration got us back to Stargate in ten months, subjective. Of course, it was 340 years (minus seven months) to the hypothetical objective observer.\n\nThere were hundreds of cruisers in orbit around Stargate. Bad news: with that kind of backlog we probably wouldn't get any furlough at all.\n\nI supposed I was more likely to get a court-martial than a furlough, anyhow. Losing 88 percent of my company, many of them because they didn't have enough confidence in me to obey the direct earthquake order. And we were back where we'd started on Sade-138; no Taurans there, but no base either.\n\nWe got landing instructions and went straight down, no shuttle. There was another surprise waiting at the spaceport. Dozens of cruisers were standing around on the ground (they'd never done that before for fear that Stargate would be hit) \u2014 and two captured Tauran cruisers as well. We'd never managed to get one intact.\n\nSeven centuries could have brought us a decisive advantage, of course. Maybe we were winning.\n\nWe went through an airlock under a 'returnees' sign. After the air cycled and we'd popped our suits, a beautiful young woman came in with a cartload of tunics and told us, in perfectly-accented English, to get dressed and go to the lecture hall at the end of the corridor to our left.\n\nThe tunic felt odd, light yet warm. It was the first thing I'd worn besides a fighting suit or bare skin in almost a year.\n\nThe lecture hall was about a hundred times too big for the twenty-two of us. The same woman was there and asked us to move down to the front. That was unsettling; I could have sworn she had gone down the corridor the other way \u2014 I _knew_ she had; I'd been captivated by the sight of her clothed behind.\n\nHell, maybe they had matter transmitters. Or teleportation. Wanted to save herself a few steps.\n\nWe sat for a minute and a man, clothed in the same kind of unadorned tunic the woman and we were wearing, walked across the stage with a stack of thick notebooks under each arm.\n\nThe woman followed him on, also carrying notebooks.\n\nI looked behind me and she was still standing in the aisle. To make things even more odd, the man was virtually a twin to both of them.\n\nThe man riffled through one of the notebooks and cleared his throat. 'These books are for your convenience,' he said, also with perfect accent, 'and you don't have to read them if you don't want to. You don't have to do anything you don't want to do, because... you're free men and women. The war is over.'\n\nDisbelieving silence.\n\n'As you will read in this book, the war ended 221 years ago. Accordingly, this is the year 220. Old style, of course, it is 3138 ad\n\n'You are the last group of soldiers to return. When you leave here, I will leave as well. And destroy Stargate. It exists only as a rendezvous point for returnees and as a monument to human stupidity. And shame. As you will read. Destroying it will be a cleansing.'\n\nHe stopped speaking and the woman started without a pause. 'I am sorry for what you've been through and wish I could say that it was for good cause, but as you will read, it was not.\n\n'Even the wealth you have accumulated, back salary and compound interest, is worthless, as I no longer use money or credit. Nor is there such a thing as an economy, in which to use these... things.'\n\n'As you must have guessed by now,' the man took over, 'I am, we are, clones of a single individual. Some two hundred and fifty years ago, my name was Kahn. Now it is Man.\n\n'I had a direct ancestor in your company, a Corporal Larry Kahn. It saddens me that he didn't come back.'\n\n'I am over ten billion individuals but only one consciousness,' she said. 'After you read, I will try to clarify this. I know that it will be difficult to understand.\n\n'No other humans are quickened, since I am the perfect pattern. Individuals who die are replaced.\n\n'There are some planets, however, on which humans are born in the normal, mammalian way. If my society is too alien for you, you may go to one of these planets. If you wish to take part in procreation, I will not discourage it. Many veterans ask me to change their polarity to heterosexual so that they can more easily fit into these other societies. This I can do very easily.'\n\nDon't worry about that, Man, just make out my ticket.\n\n'You will be my guest here at Stargate for ten days, after which you will be taken wherever you want to go,' he said. 'Please read this book in the meantime. Feel free to ask any questions, or request any service.' They both stood and walked off the stage.\n\nCharlie was sitting next to me. 'Incredible,' he said. 'They let... they encourage... men and women to do _that_ again? Together?'\n\nThe female aisle-Man was sitting behind us, and she answered before I could frame a reasonably sympathetic, hypocritical reply. 'It isn't a judgment on your society,' she said, probably not seeing that he took it a little more personally than that. 'I only feel that it's necessary as a eugenic safety device. I have no evidence that there is anything wrong with cloning only one ideal individual, but if it turns out to have been a mistake, there will be a large genetic pool with which to start again.'\n\nShe patted him on the shoulder. 'Of course, you don't have to go to these breeder planets. You can stay on one of my planets. I make no distinction between heterosexual play and homosexual.'\n\nShe went up on the stage to give a long spiel about where we were going to stay and eat and so forth while we were on Stargate, 'Never been seduced by a computer before,' Charlie muttered.\n\nThe 1143-year-long war had been begun on false pretenses and only because the two races were unable to communicate.\n\nOnce they could talk, the first question was 'Why did you start this thing?' and the answer was 'Me?'\n\nThe Taurans hadn't known war for millennia, and toward the beginning of the twenty-first century it looked as though mankind was ready to outgrow the institution as well. But the old soldiers were still around, and many of them were in positions of power. They virtually ran the United Nations Exploratory and Colonization Group, that was taking advantage of the newly-discovered collapsar jump to explore interstellar space.\n\nMany of the early ships met with accidents and disappeared. The ex-military men were suspicious. They armed the colonizing vessels, and the first time they met a Tauran ship, they blasted it.\n\nThey dusted off their medals and the rest was going to be history.\n\nYou couldn't blame it all on the military, though. The evidence they presented for the Taurans' having been responsible for the earlier casualties was laughably thin. The few people who pointed this out were ignored.\n\nThe fact was, Earth's economy needed a war, and this one was ideal. It gave a nice hole to throw buckets of money into, but would unify humanity rather than dividing it.\n\nThe Taurans relearned war, after a fashion. They never got really good at it, and would eventually have lost.\n\nThe Taurans, the book explained, couldn't communicate with humans because they had no concept of the individual; they had been natural clones for millions of years. Eventually, Earth's cruisers were manned by Man, Kahn-clones, and they were for the first time able to get through to each other.\n\nThe book stated this as a bald fact. I asked a Man to explain what it meant, what was special about clone-to-clone communication, and he said that I _a priori_ couldn't understand it. There were no words for it, and my brain wouldn't be able to accommodate the concepts even if there were words.\n\nAll right. It sounded a little fishy, but I was willing to accept it. I'd accept that up was down if it meant the war was over.\n\nMan was a pretty considerate entity. Just for us twenty-two, he went to the trouble of rejuvenating a little restaurant-tavern and staffing it at all hours (I never saw a Man eat or drink \u2014 guess they'd discovered a way around it). I was sitting in there one evening, drinking beer and reading their book, when Charlie came in and sat down next to me.\n\nWithout preamble, he said, 'I'm going to give it a try.'\n\n'Give what a try?'\n\n'Women. Hetero.' He shuddered. 'No offense... it's not really very appealing.' He patted my hand, looking distracted. 'But the alternative... have you tried it?'\n\n'Well... no, I haven't.' Female Man was a visual treat, but only in the same sense as a painting or a piece of sculpture. I just couldn't see them as human beings.\n\n'Don't.' He didn't elaborate. 'Besides, they say \u2014 he says, she says, it says \u2014 that they can change me back just as easily. If I don't like it.'\n\n'You'll like it, Charlie.'\n\n'Sure that's what _they_ say.' He ordered a stiff drink. 'Just seems unnatural. Anyway, since, uh, I'm going to make the switch, do you mind if... why don't we plan on going to the same planet?'\n\n'Sure, Charlie, that'd be great.' I meant it. 'You know where you're going?'\n\n'Hell, I don't care. Just away from here.'\n\n'I wonder if Heaven's still as nice\u2014'\n\n'No.' Charlie jerked a thumb at the bartender. 'He lives there.'\n\n'I don't know. I guess there's a list.'\n\nA man came into the tavern, pushing a cart piled high with folders. 'Major Mandella? Captain Moore?'\n\n'That's us,' Charlie said.\n\n'These are your military records. I hope you find them of interest. They were transferred to paper when your strike force was the only one outstanding, because it would have been impractical to keep the normal data retrieval networks running to preserve so few data.'\n\nThey always anticipated your questions, even when you didn't have any.\n\nMy folder was easily five times as thick as Charlie's. Probably thicker than any other, since I seemed to be the only trooper who'd made it through the whole duration. Poor Marygay. 'Wonder what kind of report old Stott filed about me.' I flipped to the front of the folder.\n\nStapled to the front page was a small square of paper. All the other pages were pristine white, but this one was tan with age and crumbling around the edges.\n\nThe handwriting was familiar, too familiar even after so long. The date was over 250 years old.\n\nI winced and was blinded by sudden tears. I'd had no reason to suspect that she might be alive. But I hadn't really known she was dead, not until I saw that date.\n\n'William? What's\u2014'\n\n'Leave me be, Charlie. Just for a minute.' I wiped my eyes and closed the folder. I shouldn't even read the damned note. Going to a new life, I should leave the old ghosts behind.\n\nBut even a message from the grave was contact of a sort. I opened the folder again.\n\n_11 Oct 2878_\n\n_William_ \u2014\n\n_All this is in your personnel file. But knowing you, you might just chuck it. So I made sure you'd get this note_.\n\n_Obviously, I lived. Maybe you will, too. Join me_.\n\n_I know from the records that you're out at Sade-138 and won't be back for a couple of centuries. No problem_.\n\n_I'm going to a planet they call Middle Finger, the fifth planet out from Mizar. It's two collapsar jumps, ten months subjective. Middle Finger is a kind of Coventry for heterosexuals. They call it a 'eugenic control baseline.'_\n\n_No matter. It took all of my money, and all the money of five other old-timers, but we bought a cruiser from UNEF. And we're using it as a time machine_.\n\n_So I'm on a relativistic shuttle, waiting for you. All it does is go out five light years and come back to Middle Finger, very fast. Every ten years I age about a month. So if you're on schedule and still alive, I'll only be twenty-eight when you get here. Hurry!_\n\n_I never found anybody else and I don't want anybody else. I don't care whether you're ninety years old or thirty. If I can't be your lover, I'll be your nurse_.\n\n_\u2014 Marygay_.\n\n'Say, bartender.'\n\n'Yes, Major?'\n\n'Do you know of a place called Middle Finger? Is it still there?'\n\n'Of course it is. Where else would it be?' Reasonable question. 'A very nice place. Garden planet. Some people don't think it's exciting enough.'\n\n'What's this all about?' Charlie said.\n\nI handed the bartender my empty glass. 'I just found out where we're going.'\n\n# 9   \nEpilogue\n\nFrom _The New Voice_ , Paxton, Middle Finger 24-6\n\n14\/2\/3143\n\nOLD-TIMER HAS FIRST BOY\n\nMarygay Potter-Mandella (24 Post Road, Paxton) gave birth Friday last to a fine baby boy, 3.1 kilos.\n\nMarygay lays claim to being the second 'oldest' resident of Middle Finger, having been born in 1977. She fought through most of the Forever War and then waited for her mate on the time shuttle, 261 years.\n\nThe baby, not yet named, was delivered at home with the help of a friend of the family, Dr Diana Alsever-Moore.\nTurn the page to begin reading from the follow-up to _The Forver War_\n\n# One\n\nWinter is a long time coming on this god-forsaken planet, and it stays too long, too, I watched a sudden gust blow a line of cold foam across the grey lake and thought about Earth, not for the first time that day. The two warm winters in San Diego when I was a boy. Even the bad winters in Nebraska. They were at least short.\n\nMaybe, we were too quick to say no, when the magnanimous zombies offered to share Earth with us, after the war. We didn't really get rid of them, coming here.\n\nCold radiated from the windowpane. Marygay cleared her throat behind me. 'What is it?' she said.\n\n'Looks like weather. I ought to check the trotlines.'\n\n'Kids will be home in an hour.'\n\n'Better I do it now, dry, than all of us stand out in the rain,' I said. 'Snow, whatever.'\n\n'Probably snow.' She hesitated, and didn't offer to help. After twenty years she could tell when I didn't want company. I pulled on wool sweater and cap and left the rain slicker on its peg.\n\nI stepped out into the damp hard wind. It didn't smell like snow coming. I asked my watch and it said 90 percent rain, but a cold front in the evening would bring freezing rain and snow. That would make for a fun meeting. We had to walk a couple of klicks, there and back. Otherwise the zombies could look through transportation records and see that all of us paranoids had converged on one house.\n\nWe had eight trotlines that stretched out ten meters from the end of the dock to posts I'd sunk in the chest-deep water. Two more had been knocked down in a storm; I'd replace them come spring. Two years from now, in real years.\n\nIt was more like harvesting than fishing. The blackfish are so dumb they'll bite anything, and when they're hooked and thrash around, it attracts other blackfish: 'Wonder what's wrong with that guy \u2013 oh, look! Somebody's head on a nice shiny hook!'\n\nWhen I got out on the dock I could see thunderheads building in the east, so I worked pretty fast. Each trotline's a pulley that supports a dozen hooked leaders dangling in the water, held to one-meter depth with plastic floaters. It looked like half the floaters were down, maybe fifty fish. I did a mental calculation and realized I'd probably just finish the last one when Bill got home from school. But the storm was definitely coming.\n\nI took work gloves and apron off a hook by the sink and hauled the end of the first line up to the eye-level pulley wheel. I opened the built-in freezer \u2013 the stasis field inside reflected the angry sky like a pool of mercury \u2013 and wheeled in the first fish. Worked it off the hook, chopped off the head and tail with a cleaver, threw the fish into the freezer, and then rebaited the hook with its head. Then rolled in the next client.\n\nThree of the fish were the useless mutant strain we've been getting for more than a year. They're streaked with pink and have a noxious hydrogen-sulfide taste. The blackfish won't take them for bait and I can't even use them for fertilizer; you might as well scatter your soil with salt.\n\nMaybe an hour a day \u2013 half that, with the kids helping \u2013 and we supplied about a third of the fish for the village. I didn't eat much of it myself. We also bartered corn, beans, and asparagus, in their seasons.\n\nBill got off the bus while I was working on the last line. I waved him inside; no need for both of us to get all covered with fish guts and blood. Then lightning struck on the other side of the lake and I put the line back in anyhow. Hung up the stiff gloves and apron and turned off the stasis field for a second to check the catch level.\n\nJust beat the rain. I stood on the porch for a minute and watched the squall line hiss its way across the lake.\n\nWarm inside; Marygay had started a small fire in the kitchen fireplace. Bill was sitting there with a glass of wine. That was still a novelty to him. 'So how are we doing?' His accent always sounded strange when he first got back from school. He didn't speak English in class or, I suspected, with many of his friends.\n\n'Over the sixty percent mark,' I said, scrubbing my hands and face at the work sink. 'Any better luck and we'll have to eat the damned things ourselves.'\n\n'Think I'll poach a big bunch for dinner,' Marygay said, deadpan. That gave them the flavor and consistency of cotton.\n\n'Come on, Mom,' Bill said. 'Let's just have them raw.' He liked them even less than I did. Chopping off their heads was the high point of his day.\n\nI went to the trio of casks at the other end of the room and tapped a glass of dry red wine, then sat with Bill on the bench by the fire. I poked at it with a stick, a social gesture probably older than this young planet.\n\n'You were going to have the art zombie today?'\n\n'The art history _Man_ ,' he said: 'She's from Centrus. Haven't seen her in a year. We didn't draw or anything; just looked at pictures and statues.'\n\n'From Earth?'\n\n'Mostly.'\n\n'Tauran art is weird.' That was a charitable assessment. It was also ugly and incomprehensible.\n\n'She said we have to come to it gradually. We looked at some architecture.'\n\nTheir architecture, I knew something about. I'd destroyed acres of it, centuries ago. Felt like yesterday sometimes.\n\n'I remember the first time I came across one of their barracks,' I said. 'All the little individual cells. Like a beehive.'\n\nHe made a noncommital noise that I took as a warning. 'So where's your sister?' She was still in high school but had the same bus. 'I can't keep her schedule straight.'\n\n'She's at the library,' Marygay said. 'She'll call if she's going to be late.'\n\nI checked my watch. 'Can't wait dinner too long.' The meeting was at eight and a half.\n\n'I know.' She stepped over the bench and sat down between us, and handed me a plate of breadsticks. 'From Snell, came by this morning.'\n\nThey were salty and hard; broke between the jaws with an interesting concussion. 'I'll thank him tonight.'\n\n'Old folks party?' Bill asked.\n\n'Sixday,' I said. 'We're walking, if you want the floater.'\n\n'\"But don't drink too much wine,\" ' he anticipated, and held up his glass. 'This is it. Volleyball down at the gym.'\n\n'Win one for the Gipper.'\n\n'What?'\n\n'Something my mother used to say. I don't know what a gipper is.'\n\n'Sounds like a position,' he said. 'Server, spiker, gipper.' As if he cared a lot about the game _qua_ game. They played in the nude, mixed, and it was as much a mating ritual as a sport.\n\nA sudden blast of sleet rattled against the window. 'You don't want to walk through that,' he said. 'You could drop me off at the gym.'\n\n'Well, you could drop _us_ off,' Marygay said. The route of the floater wasn't registered; just the parking location, supposedly for call forwarding. 'Charlie and Diana's place. They won't care if we're early.'\n\n'Thanks. I might score.' He didn't mean volleyball. When he used our ancient slang I never knew whether it was affection or derision. I guess when I was twenty-one I could do both at the same time, with my parents.\n\nA bus stopped outside. I heard Sara running up the boardwalk through the weather. The front door opened and shut fast, and she ran upstairs to change.\n\n'Dinner in ten minutes,' Marygay called up the stairs. She made an impatient noise back.\n\n'Starting to bleed tomorrow,' Bill said.\n\n'Since when do brothers keep track of that,' Marygay said. 'Or husbands?'\n\nHe looked at the floor. 'She said something this morning.'\n\nI broke the silence. 'If there are any Men there tonight...'\n\n'They never come. But I won't tell them you're off plotting.'\n\n'It's not _plotting_ ,' Marygay said. 'Planning. We'll tell them eventually. But it's a human thing.' We hadn't discussed it with him or Sara, but we hadn't tried to keep them from overhearing.\n\n'I could come someday.'\n\n'Someday,' I said. Probably not. So far it was all first-generation; all vets, plus their spouses. Only a few of them, spouses, were born on this thing Man had called a 'garden planet,' when they gave us a choice of places to relocate after the war.\n\nWe usually called 'our' planet MF. Most of the people who lived here were dozens of generations away from appreciating what we'd meant by 'middle finger.' Even if they did know, they probably didn't connect the acronym with the primal Oedipal act.\n\nAfter living through an entire winter, though, they probably called the planet their cultures' versions of 'motherfucker.'\n\nMF had been presented to us as a haven and a refuge \u2013 and a place of reunion. We could carve out an existence here as plain humans, without interference from Man, and if you had friends or lovers lost in the relativistic maze of the Forever War, you could wait for them on the _Time Warp_ , a converted battlewagon that shuttled back and forth between Mizar and Alcor fast enough to almost halt aging.\n\nOf course it turned out that Man did want to keep an eye on us, since we comprised a sort of genetic insurance policy. They could use us as a baseline if, after X generations, something bad cropped up in their carbon-copy genetic pattern. (I once used that term with Bill, and started to explain, but he did know what carbon copies were. Like he knew what cave paintings were.)\n\nBut they weren't passive observers. They were zookeepers. And MF did resemble a zoo: an artificial simplified environment. But the zookeepers didn't build it. They just stumbled onto it.\n\nMiddle Finger, like all the Vega-class planets we'd found, was an anomaly and a cartoon. It defied normal models of planetary formation and evolution.\n\nA too-young bright blue star with a single planet, Earth-sized with oxygen-water chemistry. The planet orbits at a distance where life can be sustained, if only just.\n\n(Planet people tell us that there's no way to have an Earth-type planet unless you also have a Jupiter-type giant in the system. But then stars like Vega and Mizar shouldn't have Earths anyhow.)\n\nMiddle Finger has seasons, but they're provided not by inclination toward the sun, but by the long oval of its orbit. We have six seasons spread over three Earth years: spring, summer, fall, first winter, deep winter, and thaw. Of course the planet moves slower, the farther it is from its sun, so the cold seasons are long, and the warm ones, short.\n\nMost of the planet is arctic waste or dry tundra. Here at the equator, lakes and streams ice over in deep winter. Toward the poles, lakes are solid permanent ice from the surface down, with sterile puddles forming on warm summer days. Two-thirds of the planet's surface is lifeless except for airborne spores and micro-organisms.\n\nThe ecology is curiously simple, too \u2013 fewer than a hundred native varieties of plants; about the same number of insects and things that resemble arthropods. No native mammals, but a couple of dozen species of large and small things that are roughly reptiles or amphibians. Only seven kinds of fish, and four aquatic mollusks.\n\nNothing has evolved from anything else. There are no fossils, because there hasn't been enough time \u2013 carbon dating says nothing on or near the surface is more than ten thousand years old. But core samples from less than fifty meters down reveal a planet as old as Earth.\n\nIt's as if somebody had hauled a planet here and parked it, seeded with simple life. But where did they haul it from, and who are they, and who paid the shipping bill? All of the energy expended by the humans and the Taurans during the Forever War wouldn't have moved this planet far.\n\nIt's a mystery to them, too; the Taurans, which I find reassuring.\n\nThere are other mysteries that are not reassuring. Chief among them is that this corner of the universe had been inhabited before, up to about five thousand years ago.\n\nThe nearest Tauran planet, Tsogot, had been discovered and colonized during the Forever War. They found the ruins of a huge city there, larger than New York or London, buried in drifting dunes. The husks of dozens of alien spaceships drifted in orbit, one of them an interstellar vessel.\n\nOf the creatures who had built this powerful civilization, not a clue. They left behind no statues or pictures, which may be explainable in terms of culture. Neither did they leave any bodies, not even a single bone, which is harder to explain.\n\nThe Tauran name for them is Boloor, 'the lost.'\n\nI usually cooked on Sixday, since I didn't teach then, but the Greytons had brought by a couple of rabbits, and that was Marygay's specialty, hassenpfeffer. The kids liked it better than most Earth food. They mostly preferred the bland native stuff, which is all they got at school. Marygay says it's a natural survival trait; even on Earth, children stuck to bland, familiar food. _I_ hadn't, but then my parents were strange, hippies. We ate fiery Indian food. I never tasted meat until I was twelve, when California law made them send me to school.\n\nDinner was amusing, Bill and Sara trading gossip about their friends' dating and mating. Sara's finally gotten over Taylor, who had been her steady for a year, and Bill had welcome news about a social disaster the boy had caused. It had stung her when he declared himself home, but after a few months' fling he turned het again, and asked her to take him back. She told him to stick to boys. Now it turns out he _did_ have a boyfriend over in Hardy, very secret, who got mad at him and came over to the college to make a loud public scene. It involved sexual details that we didn't used to discuss at the dinner table. But times change, and fun is fun. \n\n# Two\n\nThe thing we were plotting actually grew out of an innocent bantering argument I'd had with Charlie and Diana some months before. Diana had been my medical officer during the Sade-138 campaign, our last, out in the Greater Magellanic Cloud; Charlie had served as my XO. Diana had delivered both Bill and Sara. They were our best friends.\n\nMost of the community had taken Sixday off to get together at the Larsons' for a barn-raising. Teresa was an old vet, two campaigns, but her wife Ami was third-generation Paxton. She was our age, biologically, and they had two fusion-clone teenaged daughters. One was off at university, but the other, Sooz, greeted us warmly and was in charge of the coffee and tea.\n\nThe hot drinks were welcome; it was unseasonably cold for late spring. It was also muddy. Middle Finger had weather control that was usually reliable \u2013 or used to be \u2013 but we'd had too much rain the previous couple of weeks, and moving clouds around didn't seem to help. The rain gods were angry. Or happy, or careless; never could tell about gods.\n\nThe first couple to arrive, as usual, were Cat and Aldo Verdeur-Sims. And as usual, Cat and Marygay embraced warmly, but only for an instant, out of consideration for their husbands.\n\nOn her last mission Marygay, like me, was a het throwback in a world otherwise 100 percent home. Unlike me, she overcame her background and managed to fall in love with a woman, Cat. They were together for a few months, but during their last battle, Cat was severely wounded and went straight to the hospital planet Heaven.\n\nMarygay assumed that was it; the physics of relativity and collapsar jump would separate them by years or centuries. So she came here to wait for me \u2013 not for Cat \u2013 on the _Time Warp_. She told me all about Cat soon after we got together, and I didn't think it was a big deal; a reasonable adjustment under the circumstances. I'd always been easier with female homosex than male, anyhow.\n\nSo right after Sara was born, who should appear but Cat. She'd met Aldo on Heaven and heard about Middle Finger, and the two of them switched to het \u2013 something Man could easily do for you and, at that time, was required if you were going to Middle Finger. She knew Marygay was here, from Stargate records, and the space-time geometry worked out all right. She showed up about ten Earth years younger than Marygay and I were. And beautiful.\n\nWe got along well \u2013 Aldo and I played chess and go together \u2013 but you'd have to be blind not to see the occasional wistfulness that passed between Cat and Marygay. We sometimes kidded one another about it, but there was an edge to the joking. Aldo was more nervous about it than me, I think.\n\nSara came along with us, and Bill would come with Charlie and Diana after church let out. We unbelievers got to pay for our intellectual freedom by donning work boots and slogging through the mud, pounding in the reference stakes for the pressor field generator.\n\nWe borrowed the generator from the township, and along with it got the only Man involved in the barn-raising. She would have come anyway, as building inspector, after we had the thing up.\n\nThe generator was worth its weight in bureaucrats, though. It couldn't lift the metal girders; that took a lot of human muscle working together. But once they were in position, it kept them in place and perfectly aligned. Like a petty little god that was annoyed by things that weren't at right angles.\n\nI had gods on the brain. Charlie and Diana had joined this new church, Spiritual Rationalism, and had dragged Bill into it. Actually, they didn't have gods in the old sense, and it all seemed reasonable enough, people trying to put some poetry and numinism into their everyday lives. I think Marygay would have gone along with it, if it weren't for my automatic resistance to religion.\n\nLar Po had surveying tools, including an ancient laser collimator that wasn't much different from the one I'd used in graduate school. We still had to slog through the mud and pound stakes, but at least we knew the stakes were going where they belonged.\n\nThe township also supplied a heavy truck full of fiber mastic, more reliable than cement in this climate, and easier to handle. It stayed liquid until it was exposed to an ultrasonic tone that was two specific frequencies in a silent chord. Then it froze permanently solid. You wanted to make sure you didn't have any on your hands or clothes when they turned on the chime.\n\nThe piles of girders and fasteners were a kit that had come in a big floater from Centrus. Paxton was allotted such things on the basis of a mysterious formula involving population and productivity and the phases of the moons. We actually could have had two barns this spring, but only the Larsons wanted one.\n\nBy the time we had it staked out, about thirty people had showed up. Teresa had a clipboard with job assignments and a timeline for putting the thing up. People took their assignments good-naturedly from 'Sergeant Larson, sir.' Actually, she'd been a major, like me.\n\nCharlie and I worked together on the refrigeration unit. We'd learned the hard way the first years on this planet, that any permanent building bigger than a shed had to sit on ice year-round. If you carve down to the permafrost and lay a regular foundation, the long bitter winters crack it. So we just give in to the climate and build on ice, or frozen mud.\n\nIt was easy work, but sloppy. Another team nailed together a rectangular frame around what would be the footprint of the building, plus a few centimeters every way. Max Weston, one of the few guys big enough to wrestle with it, used an air hammer to pound alloy rods well below the frost line, every meter or so along the perimeter. These would anchor the barn against the hurricane-force winds that made farming such an interesting gamble here. (The weather-control satellites couldn't muster enough power to deflect them.)\n\nCharlie and I slopped around in the mud, connecting long plastic tubes in a winding snake back and forth in what would be the building's sub-foundation. It was just align-glue-drop; align-glue-drop, until we were both half drunk from the glue fumes. Meanwhile, the crew that had nailed up the frame hosed water into the mud, so it would be nice and deep and soupy when we froze it.\n\nWe finished and hooked the loose ends up to a compressor and turned it on. Everybody took a break while we watched the mud turn to slush and harden.\n\nIt was warmer inside, but Charlie and I were too bespattered to feel comfortable in anyone's kitchen, so we just sat on a stack of foamsteel girders and let Sooz bring us tea.\n\nI waved at the rectangle of mud. 'Pretty complex behavior for a bunch of lab rats.'\n\nCharlie was still a little dull from the glue. 'We have rats?'\n\n'A breeding herd of lab rats.'\n\nThen he nodded and sipped some tea. 'You're too pessimistic. We'll outlast them. That's one thing I have faith in.'\n\n'Yeah, faith can move mountains. Planets.' Charlie didn't deny the obvious: that we were animals in a zoo, or a lab. We were allowed to breed freely on Middle Finger in case something went wrong with the grand experiment that was Man: billions of genetically identical non-individuals sharing a single consciousness. Or billions of test-tube twins sharing a mutual data base, if you wanted to be accurate.\n\nWe could clone like them, no law against it, if we wanted a son or daughter identical to us, or fusion-clone like Teresa and Ami, if some biological technicality made normal childbirth impossible.\n\nBut the main idea was to keep churning out offspring with a wild mix of genes. Just in case something went wrong with perfection. We were their insurance policy.\n\nPeople had started coming to Middle Finger as soon as the Forever War was over. Vet immigration, spread out over centuries because of relativity, finally totaled a couple of thousand people, maybe ten percent of the present population. We tended to stick together, in small towns like Paxton. We were used to dealing with each other.\n\nCharlie lit up a stick and offered me one; I declined. 'I think we could outlast them,' I said, 'if they let us survive.'\n\n'They need us. Us lab rats.'\n\n'No, they just need our gametes. Which they can freeze indefinitely in liquid helium.'\n\n'Yeah, I can see that. They line us up for sperm and egg samples and then kill us off. They aren't cruel, William, or stupid, no matter what you think of them.'\n\nThe Man came out to get the manual for her machine, and took it back to the kitchen. They all looked alike, of course, but with considerable variation as they got older. Handsome, tall, swarthy, black-haired, broad of chin and forehead. This one had lost the little finger of her left hand, and for some reason hadn't had it grow back. Probably not worth the time and pain, come to think of it. A lot of us vets remembered the torture of re-growing limbs and members.\n\nWhen she was out of earshot, I continued. 'They wouldn't kill us off, but they wouldn't have to. Once they had sufficient genetic material, they could round us up and sterilize us. Let the experiment run down, one natural death at a time.'\n\n'You're cheerful today.'\n\n'I'm just blowin' smoke.' Charlie nodded slowly. We didn't have the same set of idioms, born six hundred years apart. 'But it could happen, if they saw us as a political threat. They get along fine with the Taurans now, but _we're_ the wild card. No group mind to commune with.'\n\n'So what would you do, fight them? We're not summer chickens anymore.'\n\n'That's \"spring\" chickens.'\n\n'I know, William. We're not even _summer_ chickens.'\n\nI clicked my cup against his. 'Your point. But we're still young enough to fight.'\n\n'With what? Your fishing lines and my tomato stakes?'\n\n'They're not heavily armed, either.' But as I said that, I felt a sudden chill. As Charlie enumerated the weapons we did know them to have, it occurred to me that we were in a critical historical period, the last time in human history that there would be a significant number of Forever War veterans still young enough to fight.\n\nThe group mind of Man had surely made the same observation.\n\nSooz brought us more tea and went back to tell the others that our little mud lake had frozen solid. So there was no more time for paranoia. But the seed had been planted.\n\nWe unrolled two crossed layers of insulation sheet, and then went about the odd business of actually raising the barn.\n\nThe floor was the easy part: slabs of foamsteel rectangles that weighed about eighty kilograms apiece. Two big people or four average ones could move one with ease. They were numbered 1\u201340; we just picked them up and put them down, aligned with the stakes we agnostics had pounded in.\n\nThis part was a little chaotic, since all thirty people wanted to work at once. But we did eventually get them down in proper order.\n\nThen we all sat and watched while the mastic was poured in. The boards that had served as forms for the frozen mud did the same for the mastic. Po and Eloi Casi used long, broom-like things to push the grey mastic around as it oozed out of the truck. It would have settled down into a level surface eventually, but we knew from experience that you could save an hour or so by helping the process along. When it was about a handspan deep, and level, Man flipped a switch and it turned into something like marble.\n\nThat's when the hard work started. It would have been easy with a crane and a front-end loader, but Man was proud of having designed these kits so they could be put up by hand, as a community project. So no big machines came along with them, unless it was an emergency.\n\n(In fact, this was the _opposite_ of an emergency: the Larsons wouldn't have much to put into the barn this year, their grapes almost destroyed by too much rain.)\n\nEvery fourth slab had square boxes on either end, to accept vertical girders. So you fasten three girders together, ceiling and wall supports, put a lot of glue into the square boxes, and haul them into an upright position. With the pressor field on, when they get within a degree or so of being upright, they snap into place.\n\nAfter the first one was set, the rest were a little easier, since you could throw three or four ropes over the rigid uprights and pull the next threesome up.\n\nThen came the part of the job that called for agile young people with no fear of heights. Our Bill and Sara, along with Matt Anderson and Carey Talos, clambered up the girders \u2013 not hard, with the integrated hand- and toe-holds \u2013 and stood on board scaffolds while hauling up the triangular roof trusses. They slapped glue down and jiggled the trusses until the pressor field snapped them into place. When that was done, they had the easier job of gluing and stapling down the roof sheets. Meanwhile, the rest of us glued and stapled the outer walls, and then unrolled thick insulation, and forced it into place with the inner walls. The window modules were a little tricky, but Marygay and Cat figured them out, working in tandem, one inside and one outside.\n\nWe 'finished' the interior in no time, since it was all modular, with holes in the walls, floor, and roof girders that would snap-fit with pre-measured parts. Tables, storage bins and racks, shelves \u2013 I was actually a little jealous; _our_ utility building was a jerry-built shack.\n\nEloi Casi, who loves working with wood, brought a wine rack that would hold a hundred bottles, so the Larsons could put some away each good year. Most of us brought something for the party; I had thirty fish thawed and cleaned. They weren't too bad, grilled with a spicy sauce, and the Bertrams had towed over their outdoor grill, with several armloads of split wood. They fired it up when we started working on the inside, and by the time we were done it was good glowing coals. Besides our fish, there was chicken and rabbit and the large native mushrooms.\n\nI was too tired and dirty to feel much like partying, but there was warm water to scrub with, and Ami produced a few liters of skag she'd distilled, which had been sitting for months with berries, to soften the flavor. It was still fiery, and revived me.\n\nThe usual people had brought musical instruments, and they actually sounded pretty good in the big empty barn. People with some energy left danced on the new marble floor. I tended the fish and mushrooms and broiled onions, and drank almost enough skag to start dancing myself.\n\nMan declined our food, politely, and made a few stress measurements, and declared the barn safe. Then she went home to do whatever it is they do.\n\nCharlie and Diana joined me at the grill, setting out chicken pieces as I removed fish.\n\n'So you'd fight them?' she said quietly. Charlie'd been talking to her. 'To what end? If you killed every one of them, what would it accomplish?'\n\n'Oh, I don't want to kill even one of them. They're people, whatever else they claim to be. But I'm working on something. I'll bring it up at a meeting when we have the bugs ironed out.'\n\n'We? You and Marygay?'\n\n'Sure.' Actually, I hadn't discussed it with her, since the thought had only occurred to me between the mastic and the girders. 'One for one and all for all.'\n\n'You had some strange sayings in the old days.'\n\n'We were strange people.' I carefully loosened the grilled fish and slipped them onto a warm platter. 'But we got things done.'\n\nMarygay and I talked long into the night and early morning. She was almost as fed up as I was, with Man and our one-sided arrangement, breeding stock staked out on this deadend arctic planet. It was survival, but only that. We should do more, while we were still young enough.\n\nShe was wildly enthusiastic about my scheme at first, but then had reservations because of the children. I was pretty sure I could talk them into going along with the plan. _At least Sara_ , I thought privately.\n\nShe agreed that we ought to work out some details before we brought the thing to meeting. Not present it to the kids until after we'd talked it over with the other vets.\n\nI didn't sleep until almost dawn, blood singing with revolution. For several weeks we tried to act normal, stealing an hour here and there to take a notebook out of hiding and jot down thoughts, work on the numbers.\n\nIn retrospect, I think we should have trusted Bill and Sara to be in on it from the first. Our judgment may have been clouded by the thrill of shared secrecy, and the anticipated pleasure of dropping a bombshell. \n\n# Three\n\nBy sundown the rain had gone through sleet to soft sifting snow, so we let Bill go straight to his volleyball game, and walked over to Charlie's. Selena, the larger moon, was full, and gave the clouds a pleasant and handy opalescence. We didn't need the flashlight.\n\nTheir place was about a klick from the lake, in a copse of evergreens that looked disconcertingly like palm trees on Earth. Palm trees heavy with snow sort of summed up Middle Finger.\n\nWe'd called to say we were coming early. I helped Diana set up the samovars and tea stuff while Marygay helped Charlie in the kitchen.\n\n(Diana and I had a secret sexual history that not even she knew about. Conventionally lesbian before she came here, during Sade-138 she had gotten drunk and made a pass at me, just to give it a try the old-fashioned way. But she passed out before either of us could do anything about it, and didn't remember it in the morning.)\n\nI lifted the iron kettle of boiling water and poured it over the leaves in two pots. Tea was one thing that adapted well to this planet. The coffee was no better than army soya. There was no place on the planet warm enough for it to grow naturally.\n\nI put the heavy kettle back down. 'So your arm's better,' Diana observed. She'd given me an elastic thing and some pills, after I pulled a muscle working on the roof.\n\n'Haven't lifted anything heavier than a piece of chalk.'\n\nShe punched a timer for the tea. 'You use _chalk_?'\n\n'When I don't need holo. The kids are kind of fascinated by it.'\n\n'Any geniuses this term?' I taught senior physics at the high school and Introduction to Mathematical Physics at the college.\n\n'One in college, Matthew Anderson. Leona's boy. Of course I didn't have him in high school.' Gifted science students had classes taught by Man. Like my son. 'Most of them, I just try to keep awake.'\n\nCharlie and Marygay brought in trays of cheese and fruit, and Charlie went out to get another couple of logs for the fire.\n\nTheir place was better suited than ours, or most others', for this sort of thing. Downstairs was one large round room, the kitchen in a separate alcove. The building was a metal dome that had been half of a Tauran warship's fuel tank, doors and windows cut in, its industrial origin camouflaged inside with wooden paneling and drapes. A circular staircase led to the bedrooms and library upstairs. Diana had a small office and examination room up there, but she did most of her work in town, at the hospital and the university clinic.\n\nThe fireplace was a raised circle of brick, halfway between the center and wall, with a conical hood. So the fire was sort of like a primitive campfire, a nice locus for a meeting of a council of elders.\n\nWhich is what this was, though the ages of the participants ranged from over a thousand to barely a hundred, depending on when they were drafted into the Forever War. Their physical age went from late thirties to early fifties, in Earth years. The years here were three times as long. I guess people would eventually become used to the idea of starting school at 2, puberty before 4, majority at 6. But not my generation.\n\nI had been physically 32 when I got here, although if you counted from birth date, ignoring time dilation and collapsar jumps, I was 1,168 in Earth years. So I was 54 now \u2013 or '32 plus 6,' as some vets said, trying to reconcile the two systems.\n\nThe vets began to arrive, by ones, twos, and fours. Usually about fifty showed up, about a third of those within walking distance. One was an observer, with a holo recorder, who came from the capital city, Centrus. Our veterans' group had no name, and no real central organization, but it did keep records of these informal meetings in an archive the size of a marble.\n\nOne copy was in a safe place and the other was in the pocket of the woman with the recorder. Either one would scramble itself if touched by Man or Tauran; a film on the outside of the marble sensed DNA.\n\nIt wasn't that a lot of secret or subversive discussion went on here; Man knew how most of the vets felt, and didn't care. What could we do?\n\nFor the same reason, only a minority of the vets ever came to the meetings, and many of them just came to see friends. What was the use of griping? You couldn't change anything. Not everyone even believed things needed changing.\n\nThey didn't mind being part of a 'eugenic baseline.' What I called a human zoo. When one Man died, another was quickened, by cloning. Their genetic makeup never changed \u2013 why mess with perfection? Our function was to go ahead and make babies the old-fashioned way, random mutation and evolution. I suppose if we came up with something better than Man, they'd start using our genetic material instead. Or perhaps see us as dangerous rivals and kill us off.\n\nBut meanwhile we were 'free.' Man had helped us start up a civilization on this planet, and kept us in touch with the other inhabited ones, including Earth. You could even have gone to Earth, when you mustered out, if you were willing to pay the price \u2013 be sterilized and become one of them.\n\nA lot of vets had done it, but Earth didn't sound at all inviting to me. One big city, full of Man and Taurans. I could live with these long winters, for the sake of the company.\n\nMost of the people were reasonably content here. I was hoping to change that tonight. Marygay and I had been hatching a plan, and I was going to throw it out for discussion.\n\nAfter about a half-hour, forty people had shown up, clustered around the fire, and I supposed weather was keeping the rest away. Diana tapped on a glass for attention, and introduced the woman from Centrus.\n\nHer name was Lori. Her English had the flat Man accent of most Centrans. (All of us vets spoke English, which had been the default language during the Forever War, for people born centuries and continents \u2013 or even planets \u2013 apart. Some of us only spoke it at get-togethers like this, and the strain showed.)\n\nShe was small and slender and had an interesting tattoo that peeked out from under her singlet, a snake with an apple in its mouth. 'There's not much to report that hasn't been in the news,' she said. 'A number of Taurans landed and stayed for one day of meetings, evidently some sort of delegation. But they never appeared in public.'\n\n'Good thing,' Max Weston said. 'I don't care if I never see one of those bastards again.'\n\n'Don't come to Centrus, then. I see one or two a day, in their bubbles.'\n\n'That's gutsy,' he admitted. 'Sooner or later somebody'll take a shot at them.'\n\n'That may be their purpose,' I said. 'Decoys, sacrificial lambs. Find out who has the weapons and the anger.'\n\n'Could well be,' Lori said. 'They don't seem to do much but walk around.'\n\n'Tourists,' Mohammed Morabitu said. 'Even Taurans might be tourists.'\n\n'Three are permanent,' Cat said. 'A friend of mine installed a heat pump in their apartment in the Office for Interplanetary Communications.'\n\n'Anyhow,' Lori said, 'these Taurans came in for a day, were put on a blacked-out floater from the Law Building, spent four hours there, and returned to the shuttle and left. A couple of cargo handlers saw them; otherwise they could have been in and out without being noticed by humans.'\n\n'I wonder why bother with secrecy,' I said. 'There've been delegations before.'\n\n'I don't know. And the shortness of the visit was odd, as well as the number four. Why should a group mind send more than one representative?'\n\n'Redundancy,' Charlie said. 'Max might have run into them and killed three with his bare hands.'\n\nAs far as we could tell, the Tauran 'group mind' was no more mysterious than Man's. No telepathy or anything; individuals regularly uploaded and downloaded experiences into a common memory. If an individual dies before tapping into the Memory Tree, new information is lost.\n\nIt did seem uncanny, since they were all physically twins. But we could do the same thing, if we were willing to have holes drilled into our skulls and plugs installed. Thanks, no. I have enough on my mind.\n\n'Otherwise,' Lori continued, 'not much is happening in Centrus. The force field bunch got voted down again, so we'll be shoveling snow another year.'\n\nSome of us laughed at that \u2013 with only ten thousand people, Centrus wasn't big enough to warrant the energy expenditure to maintain a winter-long force field. But it _was_ the planetary capital, and some citizens wanted the field as a status symbol as much as a convenience. Having the only spaceport, and alien visitors, didn't make them special enough.\n\nTo my knowledge, no Taurans had ever been here to Paxton. It might be unsafe; with our large vet population, a lot of people were like Max, unforgiving. I didn't bear them any animus myself. The Forever War had been a colossal misunderstanding, and perhaps we were more at fault than they.\n\nThey were still ugly and smelled weird and had killed a lot of my friends. But it wasn't _Taurans_ who had sentenced us to life imprisonment on this iceberg. That was Man's idea. And Man might be a few billion twins, but they were still biologically human.\n\nA lot of what went on in these meetings was just a more splenetic version of complaints that had already been sent through channels. The power grid was unreliable and _had_ to be fixed before deep winter, or people would die, and the only response from Centrus was a schedule of municipal engineering priorities, where we kept getting shoved back in favor of towns that were closer to the capital. (We were the farthest away \u2013 a sort of Alaska or Siberia, to use examples that would be meaningless to almost everyone.)\n\nOf course, the main reason for these secret meetings was that Centrus did not really reflect our concerns or serve our needs. The government was human, elected representatives whose numbers were based on population and profession. But in actual administration, Man had oversight that amounted to veto power.\n\nAnd Man's priorities were not the same as ours. It was more than just a city\/country thing, even though it sometimes took that appearance. I called it 'deliberate speciation.' About half the population of Men on the planet lived in Centrus, and most of the ones sent out to places like Paxton usually only stayed one long Year before going back. So whatever benefited Centrus benefited Man. And weakened us, out in the provinces, however indirectly.\n\nI'd worked with Man teachers, of course, and a few times dealt with administrators. I'd long gotten over the uncanniness of them all looking and, superficially, acting the same. Always calm and reasonable, serious and gentle. With just a grain of pity for us.\n\nWe talked about the grid problem, the school problems, the phosphate mine that they wanted to build too close to Paxton (which would also bring a freight monorail that we needed), and smaller problems. Then I dropped my bombshell.\n\n'I have a modest proposal.' Marygay looked at me and smiled. 'Marygay and I think we all should help Man and our Tauran brothers out with their noble experiment.'\n\nThere was a moment of absolute silence, except for the crackling fire. The phrase 'modest proposal' meant nothing to most of them, I realized, born a millennium after Swift. 'Okay,' Charlie said. 'What's the punch line?'\n\n'They want to isolate a human population as a genetic baseline. Let's give them isolation with a vengeance.\n\n'What I propose is that we take the _Time Warp_ from them. But we don't just go back and forth between Mizar and Alcor. We take it out as far as it can go, and come back safely.'\n\n'Twenty thousand light-years,' Marygay said. 'Forty thousand, here and back. Give them two thousand generations for their experiment.'\n\n'And leave us alone for two thousand generations,' I said.\n\n'How many of us could you take?' Cat asked.\n\n'The _Time Warp_ 's designed for two hundred, crowded,' Marygay said. 'I spent a few years on it, waiting for William, and it wasn't too bad. We would probably want a hundred fifty, for long-term living.'\n\n'How long?' Charlie said.\n\n'We'd age ten years,' I said. 'Real years.'\n\n'It's an interesting idea,' Diana said, 'but I doubt you'd have to _highjack_ the damned thing. It's a museum piece, empty for a generation. Just ask for it.'\n\n'We shouldn't even have to ask for it. Man's claim to ownership of it is a legal fiction. _I_ paid for one three-hundred-twelfth of it, myself,' Marygay said. There were 312 vets in on the original 'time shuttle' deal.\n\n'With wealth artificially generated by relativity,' Lori said. 'Your salary piling up interest, while you were out soldiering.'\n\n'That's true. It was still money.' Marygay turned to the others. 'Nobody else here bought a piece of the shuttle?'\n\nThere was a general shaking of heads, but Teresa Larson raised her hand. 'They stole it from us, pure and simple,' she said. 'I got billions of Earth dollars, enough to buy a mansion on the Nile. But it won't buy a loaf of bread on Middle Finger.'\n\n'To be devil's advocate here,' I said, 'Man offered to \"assume stewardship\" of it, if the humans were going to abandon it. And most of the humans had no interest in it after it had served its purpose.'\n\n'Including me,' Marygay said. 'And I don't deny having been a willing collaborator in the swindle. They bought back our shares with money we could only spend on Earth. It was amusing at the time, worthless money in exchange for a worthless antique.'\n\n'It _is_ an antique,' I said. 'Marygay took me up there once to show me around. But did it ever occur to you to wonder _why_ they keep it maintained?'\n\n'Tell me,' Diana said. 'You're going to.'\n\n'Not out of sentiment, that's for sure. I suspect they're maintaining it as a kind of lifeboat for themselves, if the situation gets difficult.'\n\n'So let's _make_ it difficult,' Max said. 'Stack 'em in there like cordwood and shoot 'em back to Earth. Or to their Tauran pals.'\n\nI ignored that. 'No matter what their plans are, they won't just let us have it. It may be three Earth centuries old, but it's still by far the largest and most powerful machine in this corner of the universe \u2013 even without weapons, a Class III cruiser is a lot of power and materiel. They don't make anything like them anymore. It probably comprises a tenth of the actual material wealth in the system.'\n\n'It's an interesting thought,' Lori said, 'but how do you plan to get there? Both of the orbital shuttles on the planet are at Centrus. You'll have to highjack at least one of those before you highjack the time shuttle.'\n\n'It will take some planning,' I admitted. 'We have to manufacture a situation where the alternative to letting us take the _Time Warp_ is unacceptable. Suppose we had kidnapped those four Taurans and threatened to kill them?'\n\nShe laughed. 'They'd probably say, \"Go ahead,\" and send for four more.'\n\n'I'm not convinced of that. I suspect they may be no more actually interchangeable than Man is. We only have their word for it \u2013 as you say, if they're all the same, why go to the expense of sending four?'\n\n'You could just ask them for the ship first,' said Ami Larson. 'I mean, they _are_ reasonable. If they said no, then\u2014'\n\nPeople were murmuring, and a couple of them laughed out loud. Ami was third-generation Paxton, not a vet. She was here because she was married to Teresa.\n\n'You grew up with them, Ami.' Diana kept a controlled neutral expression. 'Some of us old folks aren't so trusting.'\n\n'So we go out for ten years, or forty thousand, and come back,' said Lar Po. 'Suppose Man's experiment has been successful. We'll be useless Cro-Magnons.'\n\n'Worse than that,' I said cheerfully. 'They'll probably have directed their evolution into some totally new direction. We might be like house pets. Or jellyfish.\n\n'But part of my point is that you and I and most of us here have done this before. Every time we came back from a campaign, we'd have to start over \u2013 even if only a few dozen years had passed on Earth, most of our friends and relatives had died or aged into totally different people. Customs and laws were alien. We were largely unemployable, except as soldiers.'\n\n'And you want to do it again, voluntarily?' Charlie said. 'Leave behind the life you've built for yourself?'\n\n'Fisherman-teacher. I could tear myself away.'\n\n'William and I are in a better situation than most,' Marygay said. 'Our children are grown, and we're still young enough to strike out in a new direction.'\n\nAmi shook her head. She was our age, biologically, and she and Teresa had teenage daughters. 'You aren't curious about how your kids will turn out? You don't want to see your grandchildren?'\n\n'We're hoping they'll come along,' she said.\n\n'If they don't?'\n\n'Then they don't,' I said. 'A lot of children leave home and start off on their own.'\n\nAmi pressed on. 'But not many parents do. Look at the choice you're giving them. Throw away their own world to join their parents.'\n\n'As time travelers. As pioneers.'\n\nCharlie butted in. 'Forget about that aspect for a minute. Do you actually think you can recruit a hundred, a hundred fifty people without anybody going to Man and pointing the finger at you?'\n\n'That's why we want to keep it among vets.'\n\n'I just don't want to see my oldest friend in jail.'\n\n'We're _in_ jail, Charlie.' I made a gesture that didn't knock anything over. 'We can't see the bars because they're over the horizon.'\n\nBuy _Forever Free_ Now!\n\n# A Biography of Joe Haldeman\n\nJoe Haldeman is a renowned American science fiction author whose works are heavily influenced by his experiences serving in the Vietnam War and his subsequent readjustment to civilian life.\n\nHaldeman was born on June 9, 1943, to Jack and Lorena Haldeman. His older brother was author Jack C. Haldeman II. Though born in Oklahoma City, Oklahoma, Haldeman spent most of his youth in Anchorage, Alaska, and Bethesda, Maryland. He had a contented childhood, with a caring but distant father and a mother who devoted all her time and energy to both sons.\n\nAs a child, Haldeman was what might now be called a geek, happy at home with a pile of books and a jug of lemonade, earning money by telling stories and doing science experiments for the neighborhood kids. By the time he entered his teens, he had worked his way through numerous college books on chemistry and astronomy and had skimmed through the entire encyclopedia. He also owned a small reflecting telescope and spent most clear nights studying the stars and planets.\n\nFascinated by space, the young Haldeman wanted to be a \"spaceman\"\u2014the term _astronaut_ had not yet been coined\u2014and carried this passion with him to the University of Maryland, from which he graduated in 1967 with a bachelor of science degree in physics and astronomy. By this time the United States was in the middle of the Vietnam War, and Haldeman was immediately drafted.\n\nHe spent one year in Vietnam as a combat engineer and earned a Purple Heart for severe wounds. Upon his return to the United States in 1969, during the thirty-day \"compassionate leave\" given to returning soldiers, Haldeman typed up his first two stories, written during a creative writing class in his last year of college, and sent them out to magazines. They both sold within weeks, and the second story was eventually adapted for an episode of _The Twilight Zone_. At this point, though, Haldeman was accepted into a graduate program in computer science at the University of Maryland. He spent one semester in school. He was also invited to attend the Milford Science Fiction Writers' Conference\u2014a rare honor for a novice writer.\n\nIn September of the same year, Haldeman wrote an outline and two chapters of _War Year_ , a novel that would be based on the letters he had sent to his wife, Gay, from Vietnam. Two weeks later he had a major publishing contract. Mathematics was out of the picture for the near future.\n\nHaldeman enrolled in the University of Iowa Writers' Workshop, where he studied with luminary figures such as Vance Bourjaily, Raymond Carver, and Stanley Elkin, graduating in 1975 with a master of fine arts degree in creative writing. His most famous novel, _The Forever War_ (1974), began as his MFA thesis and won him his first Hugo and Nebula Awards, as well as the Locus and Ditmar Awards.\n\nHaldeman was now at his most productive, working on several projects at once. Arguably his largest-scale undertaking was the Worlds trilogy, consisting of _Worlds_ (1981), _Worlds Apart_ (1983), and _Worlds Enough and Time_ (1992). Immediately before releasing the series' last installment, however, Haldeman published his renowned novel _The Hemingway Hoax_ (1990), which dealt with the experiences of combat soldiers in Vietnam. The novella version of the book won both the Hugo and Nebula Awards, a feat that Haldeman repeated with the publication of his next novel, _Forever Peace_ (1997), which also won the John W. Campbell Memorial Award for Best Science Fiction Novel.\n\nIn 1983 Haldeman accepted an adjunct professorship in the writing program at the Massachusetts Institute of Technology. He taught every fall semester, preferring to be a full-time writer for the remainder of the year. While at MIT he wrote _Forever Free_ , the final book in his now-famous Forever War trilogy.\n\nHaldeman has since written or edited more than a half-dozen books, with a second succession of titles being published in the early 2000s, including _The Coming_ (2000), _Guardian_ (2002), _Camouflage_ (2004)\u2014for which he won his fourth Nebula\u2014and _The Old Twentieth_ (2005). He also released the Marsbound trilogy, publishing the namesake title in 2008 and quickly following it with _Starbound_ (2010) and _Earthbound_ (2011).\n\nA lifetime member and past president of the Science Fiction Writers of America, Haldeman was selected as its Damon Knight Memorial Grand Master for 2010. He was inducted into the Science Fiction Hall of Fame in 2012.\n\nAfter publishing his novel _Work Done for Hire_ and retiring from MIT in 2014, Haldeman now lives in Gainesville, Florida, and plans to continue writing a novel every couple of years.\n\n**The author and his brother, Jack, around the year 1948. The image is captioned \"Stick 'em up or I'll shoot. Woy Wogers and the Long Ranger.\"**\n\n**Haldeman in third grade, the year he discovered science fiction.**\n\n**Haldeman's mother, Lorena, and a bear cub in Alaska around the year 1950.**\n\n**Joe and Gay Haldeman on their wedding day, August 21, 1965.**\n\n**The author, with a cigarette, a beer, and a book, waits for a helicopter to arrive on the tarmac in Vietnam, July 1968.**\n\n**A pamphlet with details on how to handle prisoners of war. Haldeman carried this with him in Vietnam.**\n\n**The author in Vietnam, examining bullet holes on a US Army vehicle.**\n\n**Haldeman and the actor Jimmy Stewart in Cam Ranh Bay, Vietnam, 1968.**\n\n**The author in Vietnam with a book and sandbags.**\n\n**Joe and Gay Haldeman with their friend, prominent science fiction personality Rusty Hevelin (at right) in Alaska, 1993.**\n\n**The author with his Questar telescope in 2004.**\n\n**Janis Ian, Joe Haldeman, and Anne McCaffrey at the 2005 Science Fiction & Fantasy Writers of America Nebula Awards weekend.**\n\n**Honoring tradition, Haldeman wears the infamous tiara after winning the James Tiptree, Jr. Award for his novel _Camouflage._**\n\n**Celebrated science fiction author Harry Harrison (at left) and Haldeman dressed as pirates during the 2005 World Fantasy Convention in England.**\n\n**Joe and Gay Haldeman enjoying the Valley of the Kings in Egypt during a trip to see a total solar eclipse in 2006.**\n\n**The author outside St. Augustine, Florida, on the first day of a cross-country bicycle trip with his wife in February 2013.**\n\n**The author's Hugo and Nebula Awards.**\n\n**Joe and Gay Haldeman, 2013.**\nAll rights reserved, including without limitation the right to reproduce this ebook or any portion thereof in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.\n\nThis is a work of fiction. Names, characters, places, events, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.\n\nCopyright \u00a9 1975 by Joe Haldeman\n\nCover design by Michel Vrana\n\n978-1-4976-9235-0\n\nThis edition published in 2014 by Open Road Integrated Media, Inc.\n\n180 Maiden Lane\n\nNew York, NY 10038\n\nwww.openroadmedia.com\n\n# **JOE HALDEMAN**\n\n**FROM OPEN ROAD MEDIA**\n\nFind a full list of our authors and\n\ntitles at www.openroadmedia.com\n\nFOLLOW US\n\n@OpenRoadMedia\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nThe author and publisher have provided this e-book to you for your personal use only. You may not make this e-book publicly available in any way. Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com\/piracy.\nContents\n\nTitle Page\n\nCopyright Notice\n\nDedication\n\nAcknowledgments\n\nPrologue\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nChapter 34\n\nChapter 35\n\nChapter 36\n\nChapter 37\n\nChapter 38\n\nChapter 39\n\nChapter 40\n\nChapter 41\n\nChapter 42\n\nChapter 43\n\nChapter 44\n\nChapter 45\n\nChapter 46\n\nChapter 47\n\nChapter 48\n\nChapter 49\n\nChapter 50\n\nChapter 51\n\nChapter 52\n\nChapter 53\n\nChapter 54\n\nEpilogue\n\nA Cellar Chronology\n\nTitles by Ken Englade\n\nCopyright\nFor my patient and loving wife, Sara, whose support was invaluable.\nAcknowledgments\n\nThis book could not have been written without the aid of Chuck Peruto and his partner, Jack Fognano, who went beyond the legal issues and helped me see the human side of a very complicated story. Likewise, for helping me feel my way around an unfamiliar city, I owe a special thanks to Mildred Keil and Hilda Schoenwetter.\n\nFor sharing the efforts of his reporting I am especially grateful to Kurt Heine of the Philadelphia Daily News. For helping fill in the blank spots, my thanks to Juan Guerra of Channel 29; Doris Zibulka and her father, Warren Hensman; the men of the Sheriff's Department, especially Deputy Jim Lee; and Judge Lynne Abraham's staff and others around City Hall.\n\nFor helping educate me about the intricacies of Gary Heidnik's complex personality, I owe special thanks to Jack Apsche.\n\nI am particularly grateful as well to Todd Moritz for his technical support, and to Diana Clark, Mike Webb and Katherine Krell for sharing their expertise in the insanity defense. Advice and counsel came from many sources, but I am especially indebted to my editor, Charlie Spicer, and Betsy Graham, David Snell and Rick Gove.\nPrologue\n\nFebruary 10, 1987  \n5:05 P.M.\n\nOfficer Julio Aponte shook his head. \"No,\" the rookie cop said emphatically. \"I can't do it. Not without checking with my supervisor.\"\n\nIt was cold. Too cold to stand shivering on the porch of a beat-up house in a gone-to-seed neighborhood in North Philadelphia arguing with a civilian. But Warren Hensman was insistent. A cop himself once in California, where it never got this cold, Hensman wanted only one thing from Aponte. He wanted him to break into this house to see if the man who lived there, a weird duck named Gary Heidnik, was alive and well.\n\nHensman and a few other neighbors were worried. They hadn't seen Heidnik for several days, and there was this bad smell, this really bad smell, emanating from his house. Hensman knew Heidnik was depressed. When his neighbor's Filipino wife had left him a year before, he had choked down a six-week supply of Thorazine\u2014almost forty times the lethal dose\u2014and nearly died. Betty had come back after that, but now she was gone again, this time apparently for good. There was no telling what Heidnik was likely to do. When this stench seeped out of the house and settled over the neighborhood like a malodorous fog, they called the cops.\n\nAponte's shift started an hour before, and already it looked like his day was off to a crummy start. He had been peacefully patrolling his assigned sector in the notorious Twenty-sixth District, Philadelphia's dope capital, when his radio crackled.\n\n\"Check on the well-being of a person and a foul odor at 3520 North Marshall Street,\" the dispatcher ordered.\n\nNow that he was here, what could he do? He didn't have to put a stethoscope to the door to hear the hard rock blasting away. Despite the music, when he listened carefully he thought he could detect the sounds of someone moving around inside. But no one answered when he knocked. He went to a window and tried peeping in, but the shades were drawn skin tight. Even the rips were covered with duct tape. He walked around the left side of the house, where a driveway separated 3520 from Hensman's residence, and banged on the back door. Again, he got no response. The shade was not down all the way on one of the back windows, and when he looked in he could see into the kitchen. It looked normal enough. On the top of the stove was a large aluminum stockpot, which was frothing over.\n\nExcept for the smell being particularly pungent at the back, Aponte didn't notice anything unusual. The smell was worse than anything he had ever before encountered, but in this neighborhood nothing surprised him. On that February evening it was just another annoyance that he could well do without, just as he could do without Hensman hassling him to knock down the door just because his neighbor wanted to be left alone.\n\nIt wasn't just Hensman either. Standing there waiting for him to do something was Hensman's daughter, Doris Zibulka, whose house abutted Heidnik's, and a few others from up and down the street. They demanded that Officer Aponte kick down the door.\n\n\"That smell is so bad it's making my eyes water,\" complained Zibulka, a normally jovial mother of three young girls. \"I can't cook in my house because the smell makes me nauseated.\"\n\n\"Yeah,\" chorused the others. \"Do something. It smells like burning bodies.\"\n\nAponte looked at the door. It was metal. Could he break it down? Fat chance, he thought. He looked at the windows. They were protected by sturdy iron burglar bars. No hope there either.\n\n\"Look,\" he said, trying to be reasonable, \"I'll call in and try to get a supervisor out here.\"\n\nTurning, he walked the thirty yards to his car and was explaining his problem to the dispatcher when the door creaked open and a white man with a beard and piercing pale eyes poked his head out. Everybody jumped.\n\n\"Good God!\" Hensman cried in relief, smiling at Gary Heidnik. \"You're all right.\"\n\n\"Sure, I'm all right,\" Heidnik replied affably, \"Everything's okay.\"\n\n\"Cancel that request,\" Aponte said into the microphone.\n\n\"We thought you were dead,\" said Hensman. \"There's that terrible smell and we thought it was you.\"\n\n\"Oh no, I'm okay,\" Heidnik repeated. \"I just burned my dinner is all.\"\n\nAponte studied Heidnik. He certainly wasn't dead; in fact, he looked pretty healthy. He wasn't ranting or raving, and he didn't talk crazy. As far as the smell went, he had explained that satisfactorily. He himself had seen the boiling pot. You can't arrest a guy for being a lousy cook.\n\nAponte sighed. That turned out to be an easy assignment. Jumping into his blue and white, he drove off.\n\nIf Aponte had gone inside that house that day, he would have gotten the shock of his young life. In the basement he would have found four semi-nude young black women, tethered like livestock with heavy chains. In the oven he would have found a human rib cage burning to a cinder. And if he had looked in the bubbling caldron to see what Heidnik was having for \"dinner,\" his blond hair might have turned gray. Boiling away on the back burner of Gary Heidnik's stove was the head of a young black woman.\n\"She came in, and as we were walking up the steps, she was rambling on, you know, talking real fast about this guy having three girls chained up in the basement of this house and she was held hostage for four months... She said that he was beating them, raping them, had them eating dead people, just like he was a cold-blooded nut. Dogs was in the yard eating people bones. I just thought she was crazy. I really didn't believe it, and I still don't believe this shit.\"\n\nVincent Nelson, Josefina Rivera's boyfriend\nChapter 1\n\nNovember 26, 1986\n\nJosefina Rivera was having a rough night. In another hour it would be Thanksgiving, but so far she didn't have a lot to be thankful for.\n\nShe was still fuming about her argument with her boyfriend, a thirty-year-old black man named Vincent Nelson. She stomped out of his apartment in a huff. Then, when she went back an hour later to apologize, they started fighting all over again. That's when she said to hell with it and left a second time, figuring she might as well go to work.\n\nPacing back and forth at the corner of Third and Girard, she cursed under her breath and kicked at the trash littering the sidewalk: empty coffee cups; bright aluminum cans, which would soon be scooped up like treasure by neighborhood bums on their nightly rounds; and soggy mounds of what had been sheets from that afternoon's Daily News, now reduced to mush by a heavy, early-evening shower.\n\nThe rain had been the forerunner of a Canadian cold front barreling down the Eastern seaboard, bringing a jolting taste of winter to Philadelphia's northside slums. As she performed a little hop-and-skip around the puddles of dirty water, she burrowed deeper into her thin windbreaker, seeking relief from the plunging temperatures.\n\nAll the while, even as she swore against Nelson and the weather, she kept an eye on the street, sensitive to the cars that braked and cruised slowly past while the drivers gave her the once-over. Every time one of them seemed about to stop, she made an effort to look cheerful, flashing a fake, airline-hostess smile.\n\nWhat the drivers saw was a thin, striking-looking woman of medium height clad in sneakers and skin-tight jeans. Josefina Rivera had fine, well-developed features, inherited from her Puerto Rican father rather than her black mother. Her nose was long and straight, her lips pencil thin, and her skin the color of coffee with double cream. Fortunately for her, the dim light hid the hard lines at the corners of her mouth and the flat glint in her eyes; the cool stare that was much older than her twenty-five years. When it looked as though a potential john was really interested, Rivera gave her head a quick jerk, setting the waves in her outsized wig jiggling like a fat man's stomach.\n\nUsually she had no trouble attracting men, but tonight was exceptionally slow. As the evening wore on, she became increasingly desperate. It was cold and wet on the glum street corner, but she couldn't afford to give up yet. She didn't want to quit without at least one quick trick. She needed the money. A rabbit-fast liaison in a seedy motel or the back of a car would make her night, give her enough money for a hot Turkey Day dinner.\n\nAs she reached the edge of her self-defined boundary and reversed her course to make another circuit, a pair of headlights glared, went slightly past her and stopped. Glancing over, she opened her eyes slightly in surprise when she saw that it was a shiny new pewter-over-white Cadillac Coupe De Ville, complete with gleaming continental kit.\n\nAs she stared, the window glided down and a man spoke. It was a white man's voice, soft and low. \"Hi,\" he said, leaning forward. \"You hustling?\"\n\n\"Yeah,\" Rivera responded, straining to see inside. Despite the gloom, she caught a reflection of light off a big, expensive-looking watch on the man's left wrist.\n\n\"How much you want?\" he asked affably.\n\nShe named a figure and he made a counter offer: \"Will you take twenty?\"\n\nIt didn't take her long to decide. In response, she opened the door and slid into the passenger seat, noticing as she did the initials GMH painted on the door in flowing blue script. The smell inside the car was intoxicating, an overwhelming aroma of leather and wax. The Caddy was only nine days off the dealer's floor.\n\n\"My name's Gary,\" the man said.\n\n\"I'm Nicole,\" she answered, using her favorite alias. Nicole was a nice name, she felt, infinitely fancier than Josefina. To her it had class; it went better with her image of herself as a slightly exotic hooker.\n\n\"I want to make a quick stop first,\" the man said as he pulled away. A few minutes later he aimed the big car into the crowded lot of a McDonald's.\n\nWhen he walked inside, she went with him. He bought coffee but offered her nothing. Clutching his steaming cup tightly in his right fist, he strolled to a back table and sat facing the parking lot. She slid into a chair across from him.\n\nIn the restaurant's bright light she could see him clearly, appraising with interest the thick gold chain and gold cross visible through the open neck of his plaid flannel shirt. In counterpoint to the jewelry and the heavy watch, which she now saw carried the Rolex name, the man wore an inexpensive cowhide jacket with leather fringes down the arms, the same kind of garment Jon Voight favored in the movie Midnight Cowboy.\n\nThe jacket was stained, and in spots the suede had rubbed through, leaving irregular shiny patches that looked like moth holes. It smelled, too, of sweat and grease, a fact evident now that it wasn't camouflaged by the new-car scent.\n\nThe man, Rivera noticed, was not the cleanest john she had ever done business with. His dark beard was neatly trimmed and his hair had recently been styled, but now it was unwashed, hanging in greasy ringlets over his ears. His shirt had a slept-in look, and his jeans, although fairly new, were marked with traces of oil and dirt. He had a strong jaw, though, and a straight nose. His most arresting feature was his eyes; they were as expressionless as two blue glass marbles. Looking into them, a shiver ran up her spine.\n\n\"What's your name?\" she asked.\n\n\"I already told you,\" he said. \"Gary.\"\n\n\"Gary what?\" she persisted.\n\n\"Gary Heidnik,\" he said, lapsing into silence, quietly sipping from his steaming cup.\n\n\"Let's go,\" he said after a few minutes.\n\n\"Where are we going?\"\n\n\"My house,\" he answered, already heading for the door.\n\n* * *\n\nHeidnik pulled out of the lot and pointed the car north, deeper into the slum district. Speeding recklessly down the potholed streets, one foot on the brake and one on the gas, Heidnik said nothing as they maneuvered through an expanse of row houses, block after block of dwellings sitting literally on the cracked sidewalks.\n\nIn years past, this section of Philadelphia had been home to hardworking blue-collar immigrants, Germans for the most part, who took pride in their surroundings and kept the streets as spotless as their houses. When these immigrants and their descendants abandoned the neighborhood for the suburbs, the blacks and Hispanics who moved in were not as fastidious.\n\nBy 1986 the neighborhood had earned the nickname \"The OK Corral\" because of a highly publicized, middle-of-the-street shootout between would-be drug lords\u2014an incident that left several bystanders wounded while the participants escaped unscathed. In a demonstration of perverse pride, neighborhood toughs sewed the name \"OK Corral\" on their jackets and strutted the title proudly through the neighborhood, especially when they worked the corners hawking crack, coke, and pot to passing motorists.\n\nWhen they got to North Marshall Street, Heidnik took a sharp left, nearly clipping an abandoned Chevy parked at the curb. The car's windows were broken out and its wheels had been removed. All vital parts had long ago been carted off. At one point someone had attempted to torch the vehicle, and the path of the flames were still visible on the rusting exterior.\n\nJust past the wreck, Heidnik turned left again, swinging through a gap in a waist-high chain-link fence into a trash-littered yard. The number on the post in front was 3520. Gary Heidnik was home.\nChapter 2\n\nNumber 3520 was an anomaly. For blocks around, in all directions, there were nothing but row houses; street after street of grim, deteriorating dwellings lying on their deathbeds with their chins in the street. But Heidnik's house was different. It was not only set back a dozen yards from the sidewalk, but it was unattached on one side, leaving enough space for a small yard and a driveway, which led to the rarest structure of all, considering the neighborhood: a garage.\n\nIn Heidnik's case the garage was a ramshackle building with a definite starboard list; a fragile-looking structure of badly weathered board topped with a row of barbed wire to keep trespassers from climbing over from the alley. He also had lined the inside of the creaky doors with metal after a group of neighborhood punks fired several shots at the building the previous summer. One of the bullets had damaged a Cadillac Heidnik had parked inside, the predecessor of his current De Ville. He had vowed that would never happen again. Heidnik was very attached to his cars.\n\nAs Heidnik eased into the garage, Rivera noticed a dark shape occupying the other half of the building. Barely visible in the dark, it was a 1971 Rolls-Royce\u2014Heidnik's pride and joy. He had paid $17,000 in cash for the vehicle less than a year before, but in the few short months he owned it he had managed to burn out the engine and transmission, which he replaced with Chevrolet parts. Unsatisfied, too, with the finely crafted radio that came with the vehicle, Heidnik had mounted a cheap tape player under the dash. Scattered around on the Rolls' floor were inexpensive cassettes of music he had taped himself.\n\nAs he hurried toward the house, anxious to get out of the rising wind, Heidnik pulled a ring from his pocket and twirled it until he came to a stumpy piece of metal with an unevenly serrated edge. Rivera eyed it curiously.\n\n\"What's that?\" she asked.\n\n\"A key,\" he grunted.\n\n\"I never seen a key like that before.\"\n\n\"I made it,\" Heidnik said. \"I put the regular key halfway in and sawed it off. The front half stays in the lock permanently.\"\n\n\"Why'd you do that?\" she inquired.\n\n\"So no other key except mine will work,\" he replied, swinging open the door and stepping in front of her into a small kitchen, the walls of which were half covered with pennies which had been meticulously glued into place.\n\n\"This way,\" he said, leading her into a sparsely furnished living room.\n\nOn one wall, under a heavily barred window, was a battered orange couch, stained and swaybacked. Opposite the couch was a stand holding a stereo tape deck and turntable, a TV and a VCR. Next to the stand was a cabinet jammed with dozens of videotape cassettes, each with a handwritten label.\n\n\"You want to see a movie?\" he asked.\n\nRivera glanced at the titles; she could see he was heavily into porno flicks, horror films, and sappy comedies. Disinterested, she made a show of looking at her watch.\n\n\"Let's skip the movie,\" she said. \"I'm running a little short on time.\"\n\nCatching the quick flash of anger that jumped across Heidnik's face, she hastily added: \"I have three children at home, and my babysitter leaves at midnight.\"\n\nIt was a lie, but Heidnik didn't know that. She was beginning to feel very uncomfortable, and all she wanted to do was finish the job, collect her money, and go back to see if she could straighten out her disagreement with Vincent Nelson.\n\n\"Okay.\" Heidnik shrugged, apparently unoffended. Turning on his heel, he led the way up a flight of rickety stairs. As they entered a narrow hallway at the head of the steps, Rivera did a double take. The wall there, instead of being covered with pennies, had been partially papered with one- and five-dollar bills.\n\n\"In here,\" Heidnik said, opening a door into a room containing a waterbed, a dresser, and two chairs. \"Here's your money,\" he said, digging in his pocket and extracting a soiled twenty dollar bill, which he handed to Rivera. Without another word, he peeled off his clothes and jumped into bed.\n\nRivera put the bill on top of the dresser, slipped out of her shirt and jeans, and climbed in with Heidnik.\n\n* * *\n\nAfter a few minutes of active but emotionless coupling, Heidnik stood and walked across the room toward his pile of clothes. Rivera got up, too, and reached for her shirt. She had it on and was about to step into her jeans when she felt a pair of strong hands clamp securely around her throat. Twisting, she looked up into Heidnik's eyes, which seemed colder than ever. Without expression, he tightened his grip, slowly squeezing her life away. Just before she passed out, she squeaked her surrender.\n\n\"All right,\" she croaked. \"I'll do anything you say, but don't hurt me.\"\n\nWhen Heidnik relaxed his pressure, she sagged to the floor. As she fell, she realized he had slipped a handcuff onto her right wrist.\n\n\"Stand up and put your hands behind your back,\" Heidnik ordered.\n\nWhen she complied, he slipped the twin cuff on her other wrist. Dragging her over to the dresser, he scooped up the bill he had given her earlier and returned it to his pocket. Then he pushed her out the door, down the stairs, and through the living room and kitchen. He opened another door and she saw a second flight of stairs, these narrower than the first and unprotected by a banister. Pushing her again, he forced her down into a cold, damp, dimly-lit room that smelled strongly of mildew and dust. The chill air reminded Rivera that she was wearing nothing but a shirt. As her feet hit the icy concrete floor, she jumped in surprise. Then she started shivering.\n\n\"Over there,\" Heidnik said, maneuvering her toward a lumpy, bare mattress pushed into one corner of the small room.\n\n\"I can't see out of my right eye,\" she complained.\n\n\"Shut up,\" he told her.\n\n\"My vision is blurred,\" she insisted.\n\nHe grabbed a piece of lumber that was lying among other bits of debris on the dirty floor. \"If you don't shut up, I'm going to hit you,\" he said, waving the club.\n\nShe shut up.\n\nWalking across the room, Heidnik picked up a small cardboard box and extracted from it a metal rod that had been bent in the middle to form a skinny U. Looking closely, Rivera saw that each end of the rod was threaded. Actually, the device was a commercially-made product called a muffler clamp. Mechanics attach them to the underside of cars to cradle and support a vehicle's exhaust pipe.\n\nHeidnik ran one end of the clamp through a heavy chain, which he pulled from another box, and then forced the clamp over Rivera's ankle. A small metal bar fit between the two prongs to seal off the open end of the U. He dug in the box again and came up with two nuts, which he screwed onto the threads after first wetting them down with super glue. From out of nowhere, it seemed, he pulled a hair dryer and aimed that at the glue to make it dry faster. Then he repeated the procedure with a second clamp.\n\nWhile Rivera stood frozen in shock, Heidnik flipped the loose end of the chain over a five-inch-thick pipe that came out of the ceiling and ran across the room into the opposite wall.\n\nStanding back to survey his handiwork, Heidnik nodded in satisfaction. \"Sit down,\" he told her, pointing at the mattress.\n\nWhen she did, he stretched out beside her, put his head in her naked lap and went to sleep.\nChapter 3\n\nNovember 29, 1961\n\nGary Heidnik had a thing for November. Important things happened to him then. He was born in that month, in 1943. He faced his first criminal charges in November 1978. He took Josefina Rivera captive in November eight years later. He would take one more captive before the month rolled over. And in November 1961 he had committed what may well have been his last-ever conventional act. One week after his eighteenth birthday he joined the Army.\n\nEach of these events was significant. Each was traumatic. Each was a starting point or a turning point. Come December, in 1961, 1978, and 1986, Gary Heidnik may as well have been born again, because his life was changed forever.\n\nThousands of teenagers join the Army every year. For most of them it is a bittersweet occasion. They're growing up. They're leaving family and friends, at least temporarily. But there was nothing temporary about it for Gary Heidnik. When he rolled out of Cleveland, Ohio, bound for Fort Leonard Wood, Missouri, he wasn't ever going back. Well, maybe once. But only for a few days. Later, he would tell psychiatrists his father would fight with him. Disown him. His father admitted they didn't speak for twenty-five years. They still don't. His father told reporters he has no use for his son. And Gary has none for him. When Gary Heidnik ended up in jail, charged with all sorts of monstrous crimes, his father had no sympathy. If he did those things, the father said, he hoped they gave him the electric chair. He would even pull the switch.\n\n* * *\n\nFather and son had been fighting all their lives. Gary Michael Heidnik was Michael Heidnik's firstborn. Gary's father was a tool-and-die maker in the Cleveland suburb of Eastlake. His mother Ellen, a not unattractive woman of Creole descent, was a beautician. They lived in a middle-class house in a middle-class subdivision. They probably were not very happy. Not really. Not ever.\n\nA year and a half after Gary was born, Ellen gave birth to another boy. They named him Terry. As they grew, Gary and Terry fought. And they both fought their father. Except Gary and Terry maintained their relationship longer. The Terry\u2013Michael equation was just as unbalanced as the Gary\u2013Michael one. Terry later claimed that his father had broken all relations with him, too, just as he had with Gary.\n\nAccording to the records, Gary and Terry didn't match with Ellen either. Neither did Michael. In 1946, when Gary was two and Terry was a toddler, Michael and Ellen split. In her divorce petition, Ellen accused Michael of \"gross neglect of duty.\" Michael said she was a \"wild woman.\" And \"a boozer\" to boot.\n\nGary and Terry went with their mother, who had remarried. It started a pattern. Before she committed suicide in 1970, she would marry three more times. Her last two husbands would be black.\n\nGary and Terry lived with Ellen until it was about time for Gary to start school. Then they went to live with Michael, who also had remarried. Gary and Terry didn't get along with their stepmother worth a flip. But their relationship with Michael was even worse. They both said their father was a ferocious disciplinarian. When he wet his bed, Gary claimed in later conversations with psychologists, his father made him hang the soiled sheet out the window of his second story bedroom for all the world to see. Gary also claimed that when Michael thought his son was really bad, he'd hang him out the window too. Grab him by his ankles and dangle him headfirst twenty feet or more off the ground. One time, according to Terry, when he was mad at both Gary and Terry, Michael painted bull's-eyes on the seats of their jeans and sent them off to school. In his more gracious moments, Gary would describe his father as an \"Archie Bunker type\" or a \"cold fish.\" The psychologist to whom he told this thought those were extraordinarily benign descriptions. Michael Heidnik, 74, contended his children were raised in a normal household, as he has said since his son was arrested on March 25, 1987. He denied he had beaten them or taught them to be prejudiced. Heidnik described the environment in which his sons were raised as \"average.\"\n\n\"He was an average kid,\" he said of Gary. \"He went to school. He played baseball like the other kids, but, Jesus, he must have lost his buttons.\"\n\n* * *\n\nGary had other troubles besides those with his father. According to Terry, Gary was ridiculed because he had an oddly shaped head. The other kids called him \"football head.\" Terry said it was the result of an accident. When Gary was in an early grade he fell out of a tree, flat on his crown. After that, Terry said, Gary's personality changed. That has the ring of an apocryphal story. As an adult, Terry would also have mental problems. He later told police he had spent considerable time in mental institutions and had tried suicide several times. The interesting thing is, he never fell out of a tree.\n\nGary did what most boys who grew up in suburbia in the 40s and the 50s did. He joined the Boy Scouts and he worked at summer jobs. One summer he spent his vacation painting fire hydrants. He dated girls, but he was shy, so he didn't get around a lot.\n\nHe had two interests in life: business and the military. He devoured the financial sections of newspapers like other kids did comics. \"One day,\" he used to say, \"I'm going to be a millionaire.\" When he was in the eighth grade, he wore oversize military fatigues everywhere he went. His ambition, besides making a lot of money, was to go to West Point.\n\nWhen he was in the ninth grade, Michael scraped up enough money to send him to military school. He picked the Staunton Military Academy in Virginia, a fairly prestigious school at the time. Its alumni include John Dean and Barry Goldwater. The school closed in 1976 but a former superintendent told the Plain Dealer, Gary's hometown newspaper, that during his two years there, Gary Heidnik scored \"exceptionally fine marks\" and was not a disciplinary problem.\n\nIt was roughly during that period, too, Gary later admitted to mental health professionals, that he made his first visit to a shrink. What prompted this has never been disclosed. Neither is it known how long the treatment lasted. Gary has revealed only one thing about that early encounter: He was never given any medication.\n\nMaybe it was because he had simply burned out. Or perhaps the pressures were getting too great. Whatever, he dropped out of Staunton at the end of his junior year. He went back to the Cleveland suburbs, where he lived with Michael and a stepmother he thinks hated him. He enrolled in Eastlake's North High School but didn't remain long. Six weeks later he switched to East High School in Cleveland. That lasted a month. Then he joined the Army.\n\nWhen he walked out the door of his suburban Ohio home on the cusp of winter in 1961, his psychological baggage was considerably heavier than the barracks bag he clutched in his hand. He was leaving his childhood behind, and he was damn glad of it.\nChapter 4\n\nNovember 27, 1986\n\nWhen Josefina Rivera awoke, she was alone in the cellar. She had no idea how long she had been asleep or how long her captor had been gone. Looking around, she saw an unmoving bulk in the far corner. Her heart leaped and she stifled a cry. Then, squinting, she saw that it was a washer-dryer. She sighed in relief.\n\nThere was a window in the room, located at head height a few feet from where she huddled in fear, but it had been boarded up so no light filtered in from outside. The only illumination came from a dim overhead bulb which threw long shadows around the depressing space. Across the room, farther than she could reach when she stretched her tether to its full length, was a white chest-style freezer. It looked like a coffin, she thought. Later\u2014much later\u2014it would prove to be just that. Virtually at her elbow was a battered pool table, its felt surface blotched and torn. She saw nothing to give her hope, nothing to cheer her up.\n\nHugging herself, she realized suddenly that she was still naked from the waist down and that she was cold. The floor and walls were bare concrete, which contributed to the damp chill that seeped into her bones.\n\nOne area caught her eye. At a spot not far away, the concrete on the floor had been broken up and there was a shallow pit in the dirt underneath. For a fleeting moment the word grave popped into her mind, but she dismissed it. The broken spot in the concrete was smaller than a washtub and not even that deep. She had no idea what it was for.\n\nLetting her shoulders droop, she hugged her knees and tried to think about life in the outside world. She wondered if people living in real time were sitting down to a holiday feast. That thought reminded her that she was hungry; hungry enough, in fact, to eat an entire turkey, feathers and all.\n\nAs though in response to her thoughts, she heard the upstairs door open and Heidnik unexpectedly clump down the stairs. In his hand was her holiday meal: an egg sandwich and a glass of orange juice.\n\nShe looked at it suspiciously. She was hungry, but not that hungry. For all she knew, his offering was poisoned or drugged. He held it out to her. She shook her head. \"I think I'll wait.\"\n\n\"Suit yourself,\" he said, turning and carrying it back upstairs. A few minutes later he returned, lugging a pick and shovel. The word grave flashed through her mind again.\n\nTurning his back on her, Heidnik began digging at the earth in the spot where the concrete had been removed. First he deepened the hole, then he used the shovel as a scoop to widen the pit under the concrete that had not been chipped away.\n\nAs he worked, he talked. He told her what he wanted most in the world was to have a big family but that something always happened to ruin his plans. He told her that in his lifetime (he had celebrated his forty-third birthday six days earlier, but he didn't tell her that) he had fathered four children by four different women. But all four were now gone, taken away from him either by the mothers or by the authorities. He was particularly bitter about an experience with a woman named Anjeanette. They had a baby girl, but the state took her away immediately and put her in a foster home. Before he could have more children by her, he got into serious trouble for trying to help her sister escape from an institution. The sister, he said, was thirty-four years old and had willingly had sex with him, but they charged him with rape and sent him away to prison. When he returned four years later, Anjeanette had disappeared.\n\n\"Society owes me a wife and family,\" he told Rivera. \"I want to get ten women and keep them here and get them all pregnant. Then, when they have babies, I want to raise those children here too. We'll be one big happy family.\"\n\nRivera shivered. He didn't have to tell her that she was Number One.\n\nHeidnik put down the shovel and walked over to where she was sitting. Unzipping his pants, he pulled his penis out and ordered her to take it in her mouth. After a few minutes he inserted it in her vagina and pumped away until he climaxed. Then he went back upstairs.\n\nLeft on her own, Rivera started working on the nuts on her shackles. With some effort she freed her left ankle, giving her the mobility to scurry around the room as far as the chain would allow\u2014about twelve feet. She went to the window and started working on the covering until she pried it open enough to see daylight. Desperately searching for a tool, she found a cue stick under the pool table and used that as a lever to force the covering back so she could open the window. Then, boosting herself up, she wiggled through the opening into the backyard. She crawled as far as the chain would stretch and started screaming.\n\nScreams were not unusual in the OK Corral. All the time, day and night, someone was screaming. The residents learned to ignore them; they became as much a part of the background noise as the squeal of tires and the boom of gunshots. Rivera yelled and yelled. She yelled until she was hoarse; until Heidnik heard her.\n\nRed-faced, he raced into the yard and silenced her with a smack. Then he tried to shove her back through the hole she had climbed out of. She went limp and he couldn't get her back in. So he ran back inside, down the stairs. Grabbing the chain, he started hauling her in like a giant fish. Once he got her inside, he shoved her into the pit in the floor and dragged over a piece of plywood to put over the top. However, the hole wasn't deep enough to hold her. Even with her legs folded against her chest, her head stuck up too much for the board to lie flat. She started screaming again.\n\n\"I can't breathe,\" she pleaded. \"I need air.\"\n\nHeidnik shoved the board aside, grabbed her by her hair, and yanked her out of the pit. Then he picked up a stick and started beating her. When she whimpered surrender, he shoved her back in the pit, pulled the board over the opening and forced it down, making her bend her neck until her chin was on her chest. He put several bags of dirt on top to hold it in place, then left her alone again.\n\nHe came back a few minutes later with a radio, which he tuned to a rock station. Setting the volume all the way up, he left again and didn't come back for twenty-seven hours. She knew precisely how long it was because the station's deejays were fanatic about announcing the time.\n\nA wave of frustration and depression swept over her. Never had she felt so frightened and dejected. Now, with the radio blasting away, even if she screamed, no one would ever hear her. For the first time she could remember, she cried with all her heart.\nChapter 5\n\nJanuary 1962\n\nGary Heidnik was a good soldier. He kept his nose clean. He stayed out of trouble. He saved his money. But he probably didn't make many friends.\n\nAfter boot camp he asked to be sent to Military Police school. The Army turned him down. The minimum age for MPs was nineteen; Gary was barely eighteen. So he asked to be a stenographic clerk. The Army again said no. A parts clerk? No. A heavy-equipment operator? No. An electrician? Still no. What the Army said was: Gary, do we have a deal for you. We're going to make you a hospital corpsman. A medic. So Gary Heidnik shipped off to Fort Sam Houston in San Antonio, Texas, for training. When he graduated, his ticket was stamped \"excellent\" in both conduct and efficiency.\n\nFrom what he said later, Gary Heidnik learned more than how to change bandages during medic training. Suddenly he had more money than ever before in his life, so he decided to put it to work. He later told a friend of his, John Francis Cassidy, that he became a loan shark. He hoarded his pay and allowances, and loaned it to other soldiers, charging interest. He was making good money. Then the Army interfered. In May 1962 he was posted to the 46th Army Surgical Hospital in Landstuhl, West Germany. His job title was Orderly.\n\nCassidy said Heidnik got to Germany, realized what had happened, and almost had a heart attack. He had almost $5,000 out in loans, and he had left in such a hurry he didn't have time to collect. Fat chance he had in Germany. He had to kiss that money good-bye.\n\nWithin a few weeks Heidnik took the test for his high school equivalency diploma. He did very well, scoring ninety-six percent. That was twenty-eight points above the standard. Then something really strange happened.\n\nThree months after arriving at Landstuhl, he went on sick call. On August 25, 1962, he asked to see a doctor, saying he had dizzy spells, headaches, and blurred vision in his right eye. He was nauseated much of the time and was vomiting, he added. The doctor noted Heidnik also had a visible tic; at irregular intervals his head would jerk horizontally.\n\nA hospital neurologist examined him and decided he was suffering from gastroenteritis. He also noted that Heidnik seemed to exhibit symptoms of a mental illness: he was either schizoid or schizophrenic.\n\nDr. Jack Apsche, a Philadelphia psychologist who spent months piecing together Heidnik's complicated medical records, said those initial symptoms the soldier Heidnik suffered pointed not so much toward psychosis as to the after-effects of hallucinogens. While schizophrenics are noted for emotional frigidity, aloofness, and an inability to develop close relationships, Heidnik's complaints were subtly different. He admitted he did not respond well to authority, and he also griped that others did not like him because he performed his jobs better than they did. \"I was by far the best,\" he told one doctor, and others \"pulled rank out of jealousy.\"\n\nAlso, when Apsche saw the drugs that were prescribed for Heidnik, an alarm went off. His medication included Stelazine, which particularly attracted Apsche's attention. It is a major tranquilizer. Not a Valium. Stelazine has K-I-C-K. \"If one examines the adverse reaction of these drugs as listed in Physicians Desk Reference, it would be questionable why someone would be placed on these without a diagnosis of severe psychosis or some other psychiatric definition,\" said Apsche. \"It is obvious that they are drugs of choice for someone who is hallucinating.\" Apsche believed the Army must have understood Heidnik's condition was serious.\n\nOn October 25, two months to the day after he reported sick, Heidnik was transferred back to the United States, to the military hospital in Valley Forge, Pennsylvania. There, in addition to his other symptoms, an attending physician said Heidnik also complained of \"seeing things moving.\" The doctor noted this suggested an \"hallucinatory experience.\"\n\nThis had not been recorded in his Landstuhl file although Heidnik later told Apsche he had listed it among his symptoms when he was first examined. Its absence in his initial file suggested to Apsche that it may have thrown the entire diagnosis out of kilter.\n\n\"Either Heidnik was having a psychotic reaction (vis-\u00e0-vis the hallucination) or was responding to an hallucinatory agent. Since he was not a schizoid personality type,\" said Apsche, \"it is understandable that this diagnosis (of schizophrenia) was totally erroneous and that Heidnik was having an hallucinatory experience.\"\n\nThree months later, on January 23, 1963, an Army board reviewed the diagnosis of Heidnik as someone suffering from a schizoid personality disorder and recommended he be given an honorable discharge. Heidnik protested, but it did no good. A week later he was dismissed from the service. Ultimately, he would be judged one hundred percent mentally disabled.\n\nOn January 30, 1963, without being released from the hospital, Heidnik was given an honorable discharge. He served only fourteen months of a thirty-six month enlistment. He also received a pension for a service-connected mental disability. The pension was originally ten percent, but once his papers worked their way through all the channels, it was raised to one hundred percent and made retroactive to his discharge date.\n\n* * *\n\nAfter he left the Army, Gary settled in Philadelphia. He graduated as a Licensed Practical Nurse and got his state certificate. He also enrolled at the University of Pennsylvania and earned credits in chemistry, composition, anthropology, history, sociology, biology, commercial law, and marketing.\n\nWith his ticket as an LPN, he went to work in the University of Pennsylvania hospital (from which he was later fired because of sloppy work) and started training as a psychiatric nurse at the Veterans Administration hospital in Coatesville, near Philadelphia. He never finished that course; he was booted out after about four months because of poor attendance and a bad attitude.\n\nAbout that time, he decided to try to make up with Michael and his stepmother. It didn't take. He went back to Cleveland but left again before the stamp on his ticket was dry. That time it was for good. He never went back.\n\nGary's relationship with his mother was not much better. He said years later that one of the reasons he joined the Army was because his mother was hitting him up for money to buy booze. Over the years, they kept loosely in touch, but were never close. On May 30, 1970, Ellen Heidnik, n\u00e9e Rutledge, riddled with bone cancer and the effects of alcoholism, gulped her last drink: the contents of a bottle of mercuric chloride, a chemical commonly found in beauty shops. Gary had her cremated and then took her ashes to Niagara Falls, where he dumped them over the precipice.\n\nHis own destructive path was written in his stars even then. He already had a long history of admissions to mental hospitals. He would have a lot more.\nChapter 6\n\nNovember 29, 1986\n\nThe next time Gary Heidnik came into the cellar, Josefina Rivera heard him before she saw him.\n\nShe was still squatting in the shallow pit, cramped beyond belief, when Heidnik thumped down the stairs. This time, though, he had someone with him\u2014a woman who was crying and saying she didn't want to go.\n\n\"Shut up,\" she heard Heidnik say. \"I'm not going to hurt you.\"\n\nDespite the radio which was still blaring away, Rivera heard him repeating the shackling procedure with the new arrival. Finally, Heidnik came over and slid the board back. Reaching down, he grabbed her arm and pulled her up. She was so stiff she could hardly stand. She was dizzy, too, from lack of food, and the whole world seemed to be spinning. When it slowed down she looked up, straight into the eyes of a dark-skinned, pleasant-looking woman about her age. The woman's jaw hung slightly open and her eyes looked as big as saucers behind the thick lenses of her pearl-colored plastic glasses. She wasn't comprehending this at all, Rivera realized. She didn't know what was going on.\n\n\"Nicole,\" Heidnik said, using the only name by which he knew Rivera, \"this is Sandy.\" Turning to the new woman: \"Sandy, this is Nicole.\" He might have been introducing two women at an afternoon tea.\n\nWhen Heidnik left, Sandy explained that her name was Sandra Lindsay and she had known Gary Heidnik for four years.\n\n\"I met him when he used to hang around Elwyn,\" she said, explaining painstakingly to Rivera that the Elwyn Institute was an institute for the mentally and physically handicapped which was located on Market Street, across the Schuylkill River, on the other side of town.\n\n\"Gary was good to me,\" she added. \"He used to bring me here all the time and take me to McDonald's. We used to have a lot of fun.\"\n\n\"Did you ever have sex with him?\" Rivera asked.\n\n\"Oh, yeah,\" Lindsay replied. \"Him and Tony both.\"\n\n\"Who is Tony?\" Rivera wanted to know.\n\n\"Tony Brown,\" said Lindsay. \"A black man. A friend of Gary's. He used to drive Gary's car.\"\n\nLindsay confided that she had once been pregnant with Heidnik's baby, but the thought of being a mother frightened her so she had an abortion.\n\n\"That made Gary really mad,\" she said. \"He criticized me at church on Sunday and said abortions were evil. Then he offered me a thousand dollars to have his baby. But I don't want to do it. Now he says I'm going to have his baby whether I want to or not.\"\n\nRivera looked carefully at Lindsay. Her skin was a deep, chocolate brown and as smooth as an infant's. She had rather large, sturdy bones and carried herself well. Basically, she seemed to be a happy person, caught up in a circumstance she could not comprehend. In fact, Lindsay, formally classified as mildly retarded, could not understand why her friend Gary had suddenly turned against her.\n\nAs Rivera watched, Lindsay took off her glasses\u2014which was all she was wearing, except for a wrinkled blouse\u2014and cried. They were big, deep sobs that made her whole body shake.\n\n\"Let me have your glasses,\" Rivera said, reaching for the spectacles. \"You won't need them here.\" Carefully she folded the glasses and stuck them in the only place available, the ball rack on the pool table.\n\nAs Lindsay cried, Rivera thought about her own life and how it contrasted markedly from Lindsay's. Although Lindsay had a high school diploma, she had graduated through the special education program. Rivera, too, was a high school graduate, but she was the product of a tough inner-city girl's school run by Catholic nuns. While Lindsay led a protected home life, watched over as carefully as possible by a solicitous mother, Rivera had been on the streets as long as she could remember. Lindsay had had a single abortion; Rivera had three children. That part of what she had told Heidnik had been the truth. But none of the children lived with her, and she certainly had not hired a babysitter to watch them while she turned tricks.\n\nRivera was twenty-five years old, with the heart and cunning of a fifty-year-old; Lindsay, the same age, had as much savvy and sophistication as a fifteen-year-old.\n\n* * *\n\nBefore Heidnik dashed back upstairs, Rivera had sensed that he was nervous about Lindsay. He should have been; his decision to bring her into the basement was a reckless one. Lindsay's family knew who he was, at least indirectly. They definitely knew Tony Brown, who was a frequent visitor in their home. And they knew of Brown's relationship with the strange white man named Gary.\n\nBrown, despite the fact that he also was mildly retarded, could still put it together if someone from Lindsay's family went to him and asked where Sandy was likely to have disappeared.\n\nIf Heidnik was feeling uneasy, he covered it with rare hospitality. An hour after he left them alone, he returned with a welcome chant. \"It's dinner time,\" he said, producing a handful of crackers and a bottle of water.\n\nRivera looked at the meager offering and cursed herself for refusing the sandwich and juice the day before. If she had it in front of her now, she wouldn't have had any qualms about digging in. Being poisoned was probably better than starving to death. At least it was quicker.\n\nWithout another word Heidnik turned and again retreated up the stairs, leaving the women to get better acquainted. In a way, they were sort of related: two wives in a growing harem.\n\nA few hours later Heidnik returned and resumed digging in the pit, anxious now to make it large enough for two. Again, after some vigorous digging, he stopped and went over to Lindsay. Rivera knew what was coming; she had been there already. She was quickly learning his pattern. First he forced Lindsay to take his penis in her mouth, and then he entered her in the conventional way. Then he forced Rivera to have sex with him too. Afterwards, he turned mildly talkative.\n\n\"Sandy has been promising to have one of my babies,\" Heidnik said, confirming what Lindsay had already said. \"She keeps backing out, but this time she's going to have to go through with it.\"\n\nThe next morning, after preparing and serving the women a breakfast of hot oatmeal, they all were startled by a loud pounding on the front door. Heidnik raced upstairs and peeked through the shades. Outside was Sandra Lindsay's sister, Teresa Lomax, and two cousins.\n\nHe ignored their pounding and went back downstairs. \"That's your sister,\" he told Lindsay. \"She must have gotten my address from Tony.\"\n\nAfter a few minutes the pounding stopped and the visitors went away. Heidnik went to check to make sure they were gone, and when he returned, he was carrying several blank sheets of paper and a pen.\n\n\"Write what I tell you,\" he commanded Lindsay, dictating slowly so she could print it.\n\n\"'Dear Mom,'\" Heidnik ordered her to scribble, \"'do not worry. I will call.'\" Laboriously, Lindsay wrote it down.\n\n\"Now sign it,\" he ordered.\n\nWhen she did, Heidnik ordered her to address an envelope. \"I'm going to mail this from New York,\" he told her, sealing the note inside. \"Then your mother will think you just ran away. She won't be coming here looking for you.\"\n\nWhen he left, the two women looked unhappily at each other. Instinctively they knew darker days were ahead.\nChapter 7\n\nJanuary 1963\u2013September 1972\n\nMost people's lives are measured by traditional milestones: births, deaths, marriages, educational achievements, career moves, and the like. Gary Heidnik's is measured in admissions to psychiatric hospitals.\n\nFrom the time he entered the Army hospital in Landstuhl in August 1962, until he was arrested in March 1987, he had been in and out of different hospitals twenty-one times. Those are documentable admissions. There is a huge six-year gap in his records that shows no hospitalizations. Given his history before and after, that is virtually impossible. He almost had to be somewhere. But no one knows where.\n\nDuring that time he also attempted suicide thirteen times. Documentable. At his trial the prosecutor would scoff. They weren't serious attempts, he said. Not serious? Once he drove his motorcycle head-on into a truck. Another time he tried to hang himself. Once he chopped up a light bulb and gulped down the glass. Several times he tried to O.D. on Stelazine or Thorazine, his two most common medications. It was usually after such attempts that he was hospitalized, more frequently than not, voluntarily. Either he would surrender himself, or his brother Terry would take him in. Sometimes he would return the favor by taking Terry to the hospital when he was suicidal.\n\nHis astounding admissions record began not long after he was discharged from the Army. Within a year he tried suicide by ingesting thirty 50-milligram tablets of Thorazine, a cousin to Stelazine. That didn't work, so he tried rat poison. He had the doctors' attention.\n\nTwo psychiatrists examined him after his second attempt. One said he had an undetermined psychosis and was suffering from acute depression. The other diagnosed him as a schizophrenic.\n\nHeidnik wasn't helping them much, since he decided to go mute. He would do this periodically for the next quarter century. Just quit talking. Sometimes he wrote notes. Sometimes he used sign language. Sometimes he just sat and stared.\n\nThis time, his silent period lasted three weeks. According to the records, when he started talking again his speech was blurred. His thoughts wandered. He was often incoherent.\n\nWhile doctors were trying to decide whether to release him or hold him, he walked away. In mental-health-institution jargon it's called AMA\u2014Against Medical Advice. Over the years he AMA'ed a lot.\n\nA few days after he strolled off, the cops nabbed him for driving a motorcycle at night with a broken headlight. He admitted he was a mental patient and they took him back to the hospital.\n\nTwo weeks later he was released. But two nights after that, on Halloween eve, 1966, he had the head-on collision. He said he wanted to kill himself. He was only slightly injured. Then, while being treated at a regular hospital for the accident injuries, he took some Stelazine capsules he had hoarded during his previous hospitalization. His doctors sent him back to the nut ward.\n\nThis time he was placed on suicide watch and examined more thoroughly. The judgment was by now becoming familiar: Heidnik was diagnosed as suffering a schizophrenic reaction and an anxiety reaction. He also was found to be a catatonic type. When a series of intelligence tests were given him, he scored in the ninety-fifth percentile of the entire adult population. The record noted \"a score at this level is classified as superior, and even when compared with college students, his score surpassed seventy-five percent of them.\"\n\nThere was one other finding that was significant, especially when viewed in light of what would occur later. Another test revealed that Heidnik desperately needed to be in a position of authority.\n\n* * *\n\nAfter that, Heidnik's hospitalizations tumbled one atop the other. By then, too, a pattern was developing. The nurses had to watch him or he wouldn't take his medicine. He'd hoard it for a killer cocktail. Mutism also became a Heidnik trademark. In February 1968 he walked into the VA hospital in Coatesville but refused to speak. A nurse asked him to remove his watch. He ripped it off, threw it on the floor and stomped on it. When he undressed for the physical, doctors found a string tied tightly around one toe. What was that for? Heidnik scribbled a note saying he wanted to induce gangrene, which he hoped would spread throughout his system and kill him.\n\nIn December 1968 he was readmitted again after he hit his brother Terry over the head with a wood plane.\n\nAs Terry lay in his hospital bed recuperating from the blow, Gary walked through the door.\n\nTerry looked up, surprised to see his brother. \"What if you had killed me?\" he asked angrily.\n\nGary looked down at him and waited a long time to answer.\n\n\"I would have put your body in the bathtub and poured acid over it to dissolve the bones,\" he finally said. \"I would have had to be careful while mixing the acid, though, because I wouldn't want to damage the drainpipes. I would leave you there to soak for two or three days, and if there were any big bones left, I'd saw them up and put them in a trash compactor. Then I'd distribute them around the neighborhood in various trash cans.\"\n\nTerry was speechless. For a long time after that, he said, he had nightmares in which Gary would be chasing him with a wood plane.\n\n* * *\n\nFor almost two years Gary was hospital free. Then he was back, complaining of depression over his mother's suicide. Since she killed herself by taking a fatal dose of mercuric poison, Heidnik refused medication containing a mercury derivative.\n\nDoctors decided to give him a little test. They asked him to make up two stories. One of his tales was a symbolic one. It represented the passage from darkness to light, or, psychologically interpreted, from depression to a more optimistic outlook. The second story was more like a brief essay. It was a story about a religious fanatic who jumped out a window. His last words were, \"God, here I come.\"\n\nWhich of those tales more closely represent your views? the doctors asked. \"The abstract one,\" Heidnik replied. How about the other one? \"I'm not very religious,\" he told the doctor. This later became significant.\n\nAlso listed in the record at this time was the fact that the day before he was admitted, his live-in girlfriend walked out on him. At the time, he was twenty-seven years old. His girlfriend was a decade or so older, a mentally retarded black woman who had spent almost one third of her life in mental institutions.\n\n* * *\n\nIn 1971 he added another eccentricity to his growing personal list. It was summer. It was hot. He showed up at a VA hospital in Maryland wearing a black-leather motorcycle jacket. He refused to take it off. And he refused to talk.\n\nSilently, he bent over and rolled up his right pants leg. \"When I do that,\" he wrote, \"it means I don't want to be bothered.\"\n\nBy this time some of his psychiatrists were beginning to wonder if he was going to be around forever. \"The patient seems to be settling in for life,\" one doctor wrote in his chart.\n\nIn August he invented another trademark: a snappy military salute. The interesting thing about these eccentricities is they would follow him over the years. Once adopted, never forgotten. The salute. The mutism. The disregard for hygiene. Heidnikisms all. One doctor with a way with words later inscribed for posterity his opinion of Heidnik's bathing habits. \"He is quite unkempt in appearance and... is still wearing the same leather jacket [which is] creating a social disturbance through its odoriferous emanations.\"\n\nAnother psychiatrist described him as \"a residual schizophrenic\" with lack of motivation or initiative and a \"disregard for the usual customs, both social and cultural.\"\n\nIt was about this time, Apsche feels, that Heidnik began seeing himself in a separate world governed by his own sense of morality. \"This,\" he said, \"was Heidnik reality.\"\n\n* * *\n\nOn September 19, 1972, Gary Heidnik was discharged from another hospital. This is notable only because it marked the beginning of an inexplicable hole in his record. There is nothing else until 1978. To Jack Apsche this does not mean that Heidnik's mental problems were in some sort of remission. He could have gone to a hospital in another state or he could have been in a private hospital. One thing can be assumed with confidence, Apsche said: Heidnik was not getting any better.\n\n\"It is apparent that there were precursors and predictions, maybe even prophecies, of what was going to transpire,\" said Apsche. \"Every sign of extreme psychosis was in operation. Here was a man with no control, screaming for help, but no one listened. He had delusions, hallucinations, both auditory and visual; he had attempted suicide and he was suffering from extreme depression. Here was a man who had lost touch with reality. There were many clear warning signs that this man was heading for a total psychotic suicidal or homicidal experience.\"\n\n* * *\n\nWhen Heidnik came to trial, the defense attorney called Apsche as a witness. However, the bulk of the psychologist's material detailing Heidnik's convoluted hospital history never got before the jury. In an unusual move, the judge ruled that Apsche could not go into detail. He could talk about dates, medicine, and symptoms, but not about diagnoses. This was a boon for the prosecution. Without the defense being able to proffer a pattern of medical opinions, the prosecutor argued loudly and apparently convincingly that Heidnik had been faking mental illness for twenty-five years. His alleged motive: some $2,000 a month in benefits from the Veterans Administration and the Social Security Administration.\nChapter 8\n\nSummer and Fall, 1971\n\nThat $2,000 a month, though, was paltry compared to what he could collect from another sure-fire moneymaker: a church.\n\nIn the spring of 1971 Gary Heidnik, who had never before professed any strong religious leanings, opened a dialogue with God. One morning he walked out of his Philadelphia apartment and climbed into his battered Plymouth. He planned, he said, to drive down the street for coffee and a doughnut. He headed west. When he got to the Pacific Ocean he stopped. Standing in the surf at Malibu, gazing at the sunset, Gary Heidnik said he was visited by the Almighty. He told Heidnik to go back to Philadelphia and start a new church. Someone needed to care for the mentally and physically handicapped. Years later he would say he also conversed from time to time with Jesus. Jesus gave him stock-market tips.\n\nA few months after the California trip, while he was under treatment at one of the mental hospitals, he applied to incorporate the United Church of the Ministers of God. The application was approved on October 12, 1971. Founding members included his brother Terry, a retarded woman he was living with at the time, and a handful of other people. Attached to the application was a raggedly typed five-page \"constitution\" setting forth the church structure and principles.\n\nThe document mandated the selection of a five-member board of directors. It would select a \"bishop\" who would rule \"for life or until the so selected chooses to resign.\" Overnight, ex-private Gary Heidnik became Bishop Gary Heidnik.\n\nThe bishop had \"full control and responsibility for church funds.\" He also had the power to determine how these funds would be raised. Among the suggested avenues were \"loans, stocks, bingo games, business ventures, and other endeavors.\"\n\nIn 1975 Bishop Heidnik plunked down $1,500 and opened an account in the church's name with Merrill Lynch, Pierce, Fenner & Smith. Since he was a child he had been obsessed with Wall Street. Now he could play it for real. Over the next twelve years, no matter which hospital he was in or where he went, Bishop Heidnik always made sure that Merrill Lynch's monthly financial statements got to him. He also kept in close touch with his broker. Over the years that $1,500 mushroomed into a $545,000 portfolio.\n\nHis broker would later testify that Bishop Heidnik's game plan was solid. \"He was,\" he admitted, \"an astute investor.\" Astute but not perfect. He had one weakness. In a child-support hearing years later, Heidnik would appear embarrassed about his one not-so-wise investment. \"I just couldn't resist Crazy Eddie,\" he confessed, a stock whose value plummeted soon after Heidnik invested $158,000.\n\n* * *\n\nThe \"constitution\" also spelled out how the church funds would be dispersed. Assets, it said, would be used \"for furthering the goals of the church, such as purchase of buildings for worship, administration, education, and heat, etc.\" This apparently included motor vehicles, too, since his fleet was church-listed.\n\nOver the following months the genuineness of the church would become a highly debatable issue. One school argued that it was a blatant strategy to evade taxes. Another asserted just as sincerely that it was a humanitarian association.\n\nIf it was a con game, however, it was an elaborate one.\n\nThe \"constitution\" covered religious as well as secular matters and was quite specific on points of ecclesiastical doctrine. It defined the Bible, preferably the New World version, as the \"guiding inspiration.\" However, it also noted, \"it is to be remembered that it is only an interpretation of the original Scriptures.\"\n\n\"The divinity of Christ is questionable,\" the document continued, \"but He is recognized as God's prophet and our savior, hence his claim of Divine Origin is to be played down.\"\n\nAs with secular issues, the \"constitution\" gave virtually unlimited power to the bishop: \"His is the final word on interpretation of the Bible or the settling of religious disputes.\" There was one exception: \"divine intervention.\"\n\n* * *\n\nWhen he wasn't in the hospital, Bishop Heidnik regularly held \"services\" wherever he happened to be living. Early on Sunday, Tony Brown would climb behind the wheel of the most commodious vehicle then owned by the church and make the rounds, picking up congregants. Since many of them couldn't read, Heidnik would painstakingly teach them the words to hymns and crank up his stereo to play recorded gospel music. Usually he would preach a short sermon. Then they would pile back into the car and Heidnik would take them all to lunch at McDonald's or Roy Rogers. Sometimes they would go to a theme park in New Jersey or some other recreational spot.\n\nThose who saw the \"church\" in operation swore to its authenticity. Heidnik's friend, John Francis Cassidy, said he walked into Heidnik's house one Sunday morning and found twenty people standing around, singing hymns. He was impressed. \"Up until then, I always thought it was a dodge,\" he said.\n\nBy 1986, fifteen years after it began, the church was not only thriving, but wealthy. Within two years of that, a long list of people would be fighting over its funds, including the U.S. government's Peace Corps. Heidnik himself inadvertently brought the federal agency into the battle through the will-like last paragraph in the \"constitution.\"\n\n\"If and when dissolution [of the church] should ever become necessary,\" it said, \"the total assets of the church should be given directly or sold and the profits given equally to the federal government's Peace Corps and Veterans Administration.\"\nChapter 9\n\nNovember 30\u2013December 21, 1986\n\nFor Sandra Lindsay and Josefina Rivera, life in Gary Heidnik's dingy cellar settled into a nightmarish routine. They adjusted as best they could, but it wasn't easy, considering the grim surroundings: the bare, littered floor, the empty walls, a constantly blaring radio, and an overhead light that never went off; the cold, the damp\u2014the sheer indignity of it all.\n\nSince the only article of clothing they were allowed to wear was a thin shirt, they often huddled together for warmth and pleaded for blankets and more clothing. Even worse was the lack of contact with the outside world, the constant sexual demands by their captor, the ever-present threat of being beaten, and\u2014always present\u2014the hunger. Heidnik fed them little, and what he did give them was hardly appetizing. Occasionally he served them oatmeal for breakfast, but usually it was Pop-Tarts, crackers, and white bread. Dinner was rice and shriveled hot dogs, and for special treats, takeout fried chicken.\n\nThe two women were totally subject to Heidnik's sexual whims, which were more patterned than bizarre. He preferred the first contact to be oral-genital, but he did not want to climax in their mouths. His aim was not sexual gratification but a single-minded desire to get them pregnant. Every day, at least once a day, he would visit them in the basement and demand sex from one or both of them. They didn't even think of refusing.\n\nHeidnik warned them over and over not to scream, not to yell, not to do anything that would alert anyone to their presence. His threats were not always heeded. When Heidnik took to beating them, Rivera and Lindsay howled and shrieked, cursing and begging for mercy. That usually made Heidnik beat them harder.\n\nHe wasn't particularly worried about the neighbors taking notice, but just in case, he tacked soundproofing material to the ceiling to help deaden any noise they might make. But the best remedy was to convince them not to make any noise at all. If they did, he beat them with a shovel handle. Or he put them in the pit, \"the hole,\" and covered it with a board and put sandbags on top. Then he would feed them only bread and water or nothing at all for a day or two. Or he used the stretch punishment.\n\nSoon after he brought them into the cellar, he screwed a large eye hook into a ceiling beam so it would be about seven feet off the floor. If one of them misbehaved, he would put a handcuff on one wrist and the matching cuff through the eye hook. The woman being punished would then have to stand, for hours on end, with one arm above her head, unable to lie down, sit, or shift positions.\n\nThese punishments and precautions were necessary, he told them, because there was a constant danger they would be discovered and his plan would be wrecked. Lindsay's relatives were still coming by and banging on his door, so the letter he made her write in an attempt to convince them she was not in his house had been only partly successful, even when he followed it up with a Christmas card early in December and enclosed a five dollar bill.\n\nWhile the correspondence may not have done much to discourage Lindsay's family, it apparently did have the desired effect on the police. Lindsay's mother reported her missing the Monday after the Saturday she left home. The case was assigned to Sergeant Julius Armstrong, a black officer with seventeen years in the department. She told him that she suspected her daughter might be a prisoner of a man named Gary who lived in a house at 3520 North Marshall Street. She even gave him a telephone number. The only thing she did not know was Gary's last name.\n\nLike Lindsay's sister, Sergeant Armstrong came by and beat on Heidnik's door. Like her, he got no response. He dialed the number. No one answered. So he went looking for Tony Brown, the contact between Lindsay and Heidnik. He found him at a McDonald's in West Philadelphia, a popular hangout for Heidnik and his retarded friends from the Elwyn Institute.\n\n\"What is Gary's last name?\" Armstrong asked Brown.\n\n\"Heidnik,\" said Brown.\n\n\"How do you spell that?\" Armstrong wanted to know.\n\nBrown thought about it. \"H-E-I-D-A-I-K-E,\" he answered.\n\nArmstrong wrote it down in his pad.\n\nLater, back at his district headquarters, Armstrong punched the name Brown had spelled for him into the computer. The search produced nothing.\n\nBy this time Lindsay's mother had received the letter. She took it to Armstrong. Ah-ha, thought the cop, Lindsay is a twenty-five-year-old runaway, not a kidnap victim. Although he didn't stop working on the case altogether, he admitted in his testimony later that after he was shown the letter and the card, he did not give it a high priority. Under cross-examination by Heidnik's lawyer, Armstrong confessed he made no concerted attempt to identify Gary, not even following through on such basic procedures as checking the tax rolls or utility records to find the surname of the man who lived at 3520 North Marshall. His sole source for the identification had been a mentally deficient man.\n\nIf he had followed through and learned that the Marshall Street Gary was Gary Heidnik, and then if he had punched H-E-I-D-N-I-K into the computer, the result would have sent him running to his patrol car. He could have discovered, by pressing a few keys, that a half-dozen years previously, Gary Heidnik had done time for kidnapping a retarded black woman and hiding her in a basement storage bin. That information might have saved two lives.\nChapter 10\n\nJanuary 1976\u2013May 1978\n\nThe computer, in fact, would have spit out more than one mention of Gary Heidnik. He may have been better known on the mental-health circuit, but he was certainly no stranger to the Philadelphia Police Department. He had an arrest record going back to 1976, when he was charged with aggravated assault, carrying a pistol without a license, and carrying a firearm on the public streets.\n\nThe charges stemmed from an incident involving a black man named Robert Rogers. Rogers' then-girlfriend, later to be his wife, was renting an apartment from Heidnik, who at the time owned a dilapidated three-story house on Cedar Avenue in a run-down neighborhood in West Philadelphia, not far from the University of Pennsylvania.\n\nHeidnik and Rogers' girlfriend had an argument, so Heidnik went into the basement and turned off all the electricity in the building. When Rogers came home, the woman told him what had happened, and he went to the cellar to turn the power back on. When he got to the basement door, however, it was locked. So he went outside, opened a cellar window and climbed in. Inside, sitting on the floor on a baby mattress, was Heidnik. He had a rifle in his hands. When Rogers came in, Heidnik put the rifle down and picked up a pistol.\n\n\"I got you and I'm going to kill you and say you're a burglar,\" Heidnik said, lifting the pistol, aiming it at Rogers' face and pulling the trigger. Rogers turned his head just in time. The bullet grazed his cheek.\n\n\"I kind of talked him out of shooting me a second time,\" Rogers recalled. \"We went outside, and when the police came up, I snatched his gun and struck him.\"\n\nThe charges were dismissed a week later. The record does not say why.\n\nNot long after the incident, Heidnik sold the house to a University of Pennsylvania administrator, who was interested not so much in the property as in getting Heidnik out of the neighborhood. When the man and his wife started cleaning up the building, they found it littered with pornography and garbage.\n\nEven more disturbing, they found a place in the basement where Heidnik had chopped a hole about two-feet square through the concrete, then scooped out the dirt under the slab to create a pit large enough to hold an adult. Did Heidnik dig the hole to hide in? To climb into to protect himself? Or as a cell for a captive?\n\n* * *\n\nIf his behavior in the fall of 1976 failed to elicit much notice from law enforcement officials, Heidnik certainly got their attention twenty months later.\n\nAfter he vacated the Cedar Avenue house, he moved in with Anjeanette Davidson, a retarded black woman who had an apartment at 2331 North Fifty-eighth Street\u2014another poor Philadelphia neighborhood.\n\nWhile not feebleminded enough to require institutionalization, Anjeanette was unable to read or write. Later, a psychiatrist would gauge her IQ at 49. (Heidnik, tested at the same time, scored 130.) Generally, a score of 100 is recognized as average, and 70 is the cutoff point to certify retardation. A score of 130 is considered superior.\n\nAnjeanette got pregnant, but Heidnik, who had a certificate as an LPN, insisted on treating her at home and refused to allow her to receive outside medical attention. A month before the baby was due, Anjeanette's older sister went to the apartment with a police escort and took Anjeanette away. A medical examination disclosed she had a large fibroid tumor that would have prevented a normal delivery. On March 22, 1978, she delivered, by Caesarean section, a seven-pound, eight-ounce girl. Because of Heidnik's strict eating regimen, Anjeanette had gained only five pounds during her pregnancy. As soon as the baby girl was born, she was put in a foster home.\n\nA few weeks later, on May 7, Heidnik and Anjeanette drove to Selinsgrove Center, an institution for the mentally retarded near Harrisburg, in central Pennsylvania, to visit Anjeanette's sister Alberta.\n\nWith an IQ of only 30\u2014roughly the mental capacity of a five-year-old\u2014Alberta could not read, write, or distinguish between coins. She could, however, feed, clothe, and clean herself. Then thirty-four, she had been in the center since she was fourteen.\n\nThe two sisters were delighted to see each other. While they chatted happily in a corner, Heidnik filled out a pass signing her out of the institution for a short-term visit. The time was twelve-thirty, and Heidnik promised to return her to the center by the next morning at the latest. They never came back.\n\nWhen Alberta had not returned by May 16, center officials secured a court order demanding her return. But first they had to find her. The first stop was Heidnik's and Anjeanette's apartment in Philadelphia.\n\nHeidnik answered the door and told the woman from the center that Alberta was not there. If she did not believe him, he added, swinging the door wide, she could come in and look for herself. The woman walked through the apartment and did not find Alberta.\n\n\"Where is she?\" she demanded.\n\n\"I put her on a bus back to Selinsgrove,\" Heidnik replied.\n\nReluctantly, the woman left.\n\nBy the next day, when Alberta still had not returned, center officials were getting frantic. The woman who had been to Heidnik's the day before went back, but on the way she stopped to pick up a police escort.\n\nAgain Heidnik told her Alberta was not there, and again she went through the apartment without finding Alberta. This time, though, they decided to search the entire building.\n\nThey found Alberta cowering in an unused storage room in the basement. When Alberta saw the woman from the center, she ran to her and hugged her tightly. She was trembling violently.\n\n\"Let's go,\" the woman said. \"We're going home.\"\n\n\"Wait a minute,\" interjected Heidnik. \"If you go with them,\" he told Alberta, \"they're going to lock you up and you'll never see your sister again.\" She went anyway.\n\nOnce back at the center, Alberta was given a thorough physical. Doctors discovered a tear on the vestibule of her vagina, which indicated recent intercourse. Even more upsetting to center officials, a chemical test showed traces of sperm in her mouth and gonorrhea in her throat, a condition that could result only from oral-genital sex.\n\nThree weeks later, at the center's urging, police returned to Heidnik's apartment. This time they arrested him and charged him with kidnapping, rape, false imprisonment, unlawful restraint, involuntary deviate sexual intercourse, interfering with the custody of a committed person, and recklessly endangering another person. He pleaded innocent.\n\nHis defense seemed off to a good start when he agreed to submit to a test for venereal disease. It was negative, meaning Heidnik did not have gonorrhea. Not at that time anyway.\nChapter 11\n\nNovember 1978\u2013March 1983\n\nSurprisingly, when the case came to trial the following November, Heidnik took the stand.\n\n\"When we left the center on May seventh,\" he swore, \"we went to a restaurant named Gary's, where we stayed for an hour and a half. From there we went to get ice cream at a Dairy Queen. Then, when we were driving Alberta back to the center, she started crying. She said she wanted to be with her sister, so we brought her back to Philadelphia.\"\n\nThe next day, Heidnik testified, he telephoned the center and asked officials to list Alberta as being on vacation so she could stay longer. He said the request was tentatively approved.\n\nThat night, he said, he went to work, and when he came home the next morning, Anjeanette and Alberta were crying and upset.\n\n\"What happened?\" he asked. \"What's the matter?\"\n\nAnjeanette told him the center had telephoned and said Alberta had to come back.\n\nInstead of returning her to the center, Heidnik said, they took her shopping and bought her two dresses, a purse, an inexpensive watch, and a wig. Over the next few days, he said, he treated calluses on her hands and feet and worked with her to help her learn how to handle money. He hid her, he admitted, because she said she did not want to go back to the center. He thought he was helping her. He denied keeping her captive or forcing her to have sex with him.\n\nUnder cross-examination Heidnik was forced into admissions that damaged his credibility. For one thing, although he had tested negative on a VD test, he admitted that he had access to antibiotics that he could have taken to cure the disease. He also admitted lying to Selinsgrove officials about Alberta's whereabouts.\n\nAs a mitigating circumstance, he said he was being treated for schizophrenia.\n\nSince he had waived a jury trial, the decision on his fate was left to Common Pleas Court Judge Charles P. Mirarchi Jr., an owlish-looking veteran on the criminal circuit who had heard a lot of wild tales in his time. As far as he was concerned, this story of Heidnik's was one of the wildest tales he'd heard yet. The judge's instincts told him Heidnik was big trouble, especially with a psychiatric report in hand that said that even with an IQ of 130 Heidnik was a mental basket case.\n\n\"He appears to be an extremely insecure and confused individual,\" the report said. \"Records indicate he is suffering from a major mental illness, which apparently has been of long standing. He is also psychosexually immature. He appears to be easily threatened by women whom he would consider to be equal to him either intellectually or emotionally. His defense cannot tolerate criticism. Gary needs constant acceptance and self-assurance that he is an intelligent, worthwhile human being.\"\n\nMirarchi found Heidnik guilty, but ordered the standard presentence report done. It was even more foreboding.\n\n\"Heidnik appears to be manipulative, and he is certainly lacking in judgment,\" wrote Joseph A. Tobin, the presentence criminal investigator. \"He impresses me as one who sees himself as superior to others, although apparently he must involve himself with those distinctly inferior to himself to reinforce this.... It is my opinion, based on my extensive investigation, that he is not only a danger to himself, but perhaps a greater danger to others in the community, especially those who he perceives as being weak and dependent. Unfortunately, it seems to me that he will not significantly change his aberrant behavior pattern in the near future.\"\n\nThat was enough for Mirarchi. He gave Heidnik as stiff a sentence as he could, considering that the felony charges\u2014rape, kidnapping, false imprisonment, and involuntary deviate sexual intercourse\u2014had to be dropped because they could not be prosecuted without Alberta's testimony. She was judged too retarded to take the stand. What Mirarchi was left with was a handful of misdemeanors: recklessly endangering another person, interfering with the custody of a committed person, and unlawful restraint. It was enough. He sentenced Heidnik to three to seven years in the state penitentiary.\n\n\"If it had been within my power to give him longer, I would have,\" Mirarchi said ten years later. At the time, long before Heidnik's name would be in headlines, Mirarchi sensed something sinister about Heidnik, \"something evil and dangerous.\"\n\nHeidnik's attorney, however, thought the sentence was severe and pleaded for leniency, arguing that Heidnik had no prior convictions, that he had not been accused of committing a violent crime, and that no weapons were involved. \"It is a crime which would lead a court, in my opinion, to believe that he is not dangerous to society.\"\n\nMirarchi ordered still another report. This one, from court psychiatrist Dr. Wayne C. Blodgett, was the harshest yet, one which, a decade later, would look like a crystal-ball prophecy.\n\nBlodgett predicted there was a \"high probability\" Heidnik would commit similar crimes in the future. \"Of particular concern,\" he wrote, \"is the defendant's potential for engaging in sexually assaultive crimes against females. In order to avoid such a tragedy in the future, it will be necessary for him to be very closely supervised and for him to receive continuing surveillance over a long period of time.\"\n\nAs it turned out, the time was not long enough.\n\n* * *\n\nHeidnik, who already frequently referred to himself as \"bishop,\" now took on a third identity: he became prisoner number F-9748. Despite his new persona, however, he never did a day of hard time. Instead, he spent the next four years, two months, and fourteen days bouncing from one state hospital to the other. He went from the facility at Graterford Prison to Norristown State Hospital to the maximum-security hospital for the criminally insane at Farview and back to start the circle all over again. Over the course of his prison time, Heidnik was transferred from one hospital to another at least a half-dozen times.\n\nHalfway through the sentence, he went mute. He simply quit talking. As he had done in the VA hospital, he rolled up one pants leg and wrote a note saying it was a signal that no one was to talk to him. When asked why he wouldn't talk, Heidnik scribbled a note saying the Devil had shoved a cookie down his throat.\n\nOne doctor told him to open wide and say aaahh. After a quick look with a flashlight, he told Heidnik he couldn't see any cookie. Heidnik scribbled another note: \"Pray for eyes.\"\n\nThe mute period continued, with only brief relief, for the next two-and-a-half years. One of the exceptions occurred during a visit from his old friend, John Francis Cassidy. When the two met, Heidnik started chatting away as though nothing was amiss.\n\nThe guard, who had known only the silent Heidnik, almost fainted. \"Hey! He's talking,\" he exclaimed.\n\nHeidnik flashed him a cutting glance. \"Yeah, it must be a miracle,\" he said drily.\n\n* * *\n\nThe parole board, though, did not see any humor in Heidnik's case. Three times he came up for parole and three times the board decided he needed more psychiatric help. Once he wrote a note to the board and signed it not G. M. Heidnik, but \"G. M. Kill.\"\n\nFinally, on March 24, 1983, near the end of his sentence, he was considered eligible for release. His parole, however, was dependent on his being accepted into the in-patient program at Coatesville, the VA mental unit. Some three weeks later, on April 12, he was freed, but he had to remain under state supervision for another three years. It had no effect on his future behavior.\nChapter 12\n\nDecember 22, 1986\n\nGary Heidnik got his Christmas present early.\n\nHe was cruising the north side in his Cadillac on Monday afternoon, three days before Christmas, when he spotted a saucy-looking young woman bopping down Lehigh Street. It was a pert nineteen-year-old named Lisa Thomas, who was sashaying along in jeans and a blue thigh-length down jacket. Thomas had dropped out of high school in the eleventh grade because she was pregnant. She later had a child, and currently the two of them were living at home with Lisa's mother. They subsisted on welfare payments.\n\nHeidnik let the Cadillac drift to a stop beside Thomas, rolled down the window and asked her a question: \"You want to see my peter?\"\n\nThomas was offended. \"I'm no prostitute,\" she said angrily.\n\nHeidnik apologized and asked if she wanted a ride.\n\n\"No,\" she said, still angry. \"I'm only going to my girlfriend's house.\"\n\n\"Where's that?\"\n\n\"Just around the corner.\"\n\n\"Why don't you get in and I'll take you there. I'm not going to hurt you.\"\n\nLisa looked at the handsome car. She took another look at the driver, who had changed his tune now that she had told him she wasn't a hooker. He didn't look mean. There's no danger in accepting a ride, she thought.\n\nHeidnik took her to her friend's and said he'd wait for her. Lisa went in, retrieved a pair of gloves she had forgotten on an earlier visit, and glanced out the window. Heidnik and his Cadillac were waiting at the curb.\n\n\"Look at that car,\" she bragged to her friend.\n\nLisa was back in a flash.\n\n\"Let's go get something to eat,\" Heidnik suggested.\n\n\"Okay,\" she agreed.\n\nHeidnik favored McDonald's and Roy Rogers, but he apparently was anxious to impress this new woman, a dark-skinned, fresh-looking teenager with a bright smile and an adventurous spirit. He took her to a TGI Friday's.\n\nWhile she was devouring a cheeseburger and fries, Heidnik asked her if she would go with him the next day to Atlantic City.\n\n\"I don't have anything to wear,\" she said.\n\n\"We can fix that,\" Heidnik said, reaching into a stuffed wallet and extracting a fifty dollar bill.\n\nShe looked at it suspiciously.\n\n\"It's for new clothes,\" explained Heidnik. \"When you finish eating we'll go over to Sears and you can use this to buy whatever you want.\"\n\nShe bought two pairs of jeans and two tops. Then Heidnik asked her if he could put them on her.\n\nShe shrugged. \"Okay.\"\n\nHeidnik took her to North Marshall Street, gave her a wine cooler, told her to make herself comfortable. While she pulled at the drink he popped a cassette of \"Splash\" into the VCR.\n\nThe drink did it for Lisa Thomas. At the restaurant she had taken an allergy pill, and it was just starting to kick in. With the wine, she got so drowsy she couldn't hold her head up. In a few minutes she was sound asleep, sprawled in front of the VCR.\n\nWhen she awoke sometime later, she discovered Heidnik had undressed her. She was completely nude. Heidnik carried her upstairs, put her on the waterbed, and they had sex.\n\nLisa got up and reached for her clothes. \"Will you take me back to my girlfriend's house?\" she asked.\n\nIn reply, Heidnik wrapped an arm around her throat and squeezed. She started seeing stars.\n\n\"Wait a minute,\" she cried. \"Wait a minute. Quit choking me and I'll do whatever you want.\" He clapped a pair of cuffs on her wrists and pushed her down the stairs.\n\nWhen she entered the basement, Lisa was still groggy and confused. She looked around and saw an empty room littered with white plastic bags.\n\n\"It's body parts in those bags, isn't it?\" she asked shrilly.\n\n\"No,\" Heidnik answered.\n\n\"You're going to kill me. You want to kill me.\"\n\n\"No, I'm not going to kill you. Trust me. I'm not going to kill you.\"\n\nHeidnik pointed to the board covering the pit in which Rivera and Lindsay were cowering.\n\n\"I'm going to introduce you to my two friends down here,\" he said.\n\n\"They're dead down there, aren't they?\" Thomas screamed. \"Aren't they?\"\n\n\"No,\" Heidnik said. \"But if you don't shut up, I'm going to hurt you.\"\n\nHe moved the board, and a half-nude woman climbed out. As Lisa watched in total amazement, the first woman was followed by another, also nude from the waist down.\n\n\"I'm Nicole,\" said the slimmer, lighter-colored one.\n\n\"I'm Sandy,\" said the other, who seemed a little dim.\n\nHeidnik stood by beaming. Like a happy host, he produced peanut butter and jelly and started to make sandwiches. But before anyone could eat, there was a ritual that had to be completed.\n\nTurning to Lisa, Heidnik gave her a quiet command, intent on establishing his authority. \"Kiss my behind,\" he said.\n\nShe did.\n\n\"Who's the boss?\" Heidnik asked.\n\n\"You are,\" she replied.\n\nHe looked at her again. \"Suck my balls.\"\n\nShe complied.\n\n\"Suck my peter.\"\n\nShe did that too.\n\nThen he forced her to have intercourse with him.\n\nAfter that they ate their sandwiches and Heidnik left Thomas alone with Rivera and Lindsay.\nChapter 13\n\nFor all of his adult life, Gary Heidnik's women were almost exclusively black. This undoubtedly went back to his childhood, when his mother, whose last two husbands were black, told Gary that he was of mixed race. If he was, it certainly didn't show. His skin was milky white and his eyes as pale as ice that had been in the freezer too long. Despite what the mirror told him, he apparently thought otherwise. Later, on two separate occasions when he entered mental hospitals, he listed his race as \"colored.\" He did not like the term black.\n\nAfter he got out of the Army he invited a married black woman to come live with him. She did. They had a baby girl, but not long afterwards the woman went back to her husband and took the child with her. That was in the mid-sixties.\n\nAfter that, another black woman, named Dorothy, came to live with him. She stayed around for a long time, a decade or so, until one day she simply disappeared.\n\nDorothy was almost a dozen years older than Heidnik and, from all accounts, mentally handicapped. She had lived in institutions for fourteen years by the time he met her, and showed signs of deep neglect. Sometime over those years her teeth had disappeared one by one. By the time she moved in with Heidnik she was, as dentists say, edentulous. She had no choppers.\n\nLinda Rogers, the tenant who had married the man Heidnik shot in his basement in 1976, said Heidnik treated Dorothy very badly at times, yelling at her, criticizing her, occasionally beating her. When he really wanted to punish her, though, he refused to give her food. At such times she would knock on Linda's door and Linda would take her in and feed her.\n\nLinda was a nurse, and one day Dorothy came to her and showed her a gash on her shin. It was a gunshot wound. How in the world did that happen? Linda wanted to know. Dorothy said Gary had accidentally shot her. Linda didn't say anything; she knew Heidnik kept several guns in his apartment.\n\nWhen Dorothy disappeared, Heidnik seemed very worried. He said he feared she had wandered off and was unable to find her way back. A few years later, when his friend John Francis Cassidy came to visit him in prison, Heidnik told him he had seen a television documentary on street people and he thought one of the women in the film was Dorothy. He wanted to hire Cassidy to find her. Cassidy declined. \"I'm not a detective,\" he told Heidnik.\n\nAfter an exhaustive search, the police eventually found Dorothy \"alive and well\" but mentally more confused than ever.\n\n* * *\n\nHeidnik's next long-term relationship, after Dorothy, was Anjeanette. When he got out of prison he tried to find her but was unsuccessful. He couldn't figure out where she had gone.\n\nLater, police would wonder too. They were unable to trace Anjeanette. Some investigators think Heidnik killed her and buried her in some grave they have not yet found.\n\nUp until the time he left prison, Heidnik's batting average with long-term relationships was not too good. His first serious lover left him and took their daughter. Dorothy, although she bore him no children, also left. And now he couldn't find Anjeanette, although he knew exactly where his daughter was and occasionally visited her. In the summer of 1986 his visitation rights would be terminated despite his efforts to prevent it.\n\nBetween the spring of 1983, when he got out of prison at age forty, and the fall of 1985 he had no single mate. There was a white woman who was practically always at the Marshall Street house, but she shared Tony Brown's bed as well as Heidnik's. She later had a son which she said was Heidnik's, and indeed was named after Heidnik. He always referred to the boy as \"Little Gary.\"\n\nThe child went into a foster home, too, and she began seeing more of Tony Brown than she did of Heidnik. Eventually she married Brown and they had a child.\n\nBut she was just one of the women having a sexual relationship with Heidnik at the time. Another was Jewel, a black woman of Heidnik's age. He had met Jewel when they were both LPNs at a hospital in Philadelphia in the sixties, and they had been lovers ever since.\n\nThe relationship developed a degree of regularity after Heidnik got out of prison. Every other Saturday for almost two years Heidnik would pick her up, take her to North Marshall Street, and she would spend the night. On Sunday she would attend Heidnik's \"church services,\" then go out to eat with the congregants. After that, Heidnik would take her back home.\n\nWhen she first met Heidnik, she said, when she was younger, he used to tell her what a \"fine specimen\" of a woman she was and how he wanted her to bear him a son. When she made it clear she didn't want to have any more children, he eased off.\n\nBut his sex habits also got kinkier.\n\n\"When I would go to the house, Gary would want to make love with two women at the same time,\" one of his former girlfriends told police. \"I always wanted to be the first because I didn't want him to have sex with another woman and then put his penis in me. So I would have sex with Gary, and he would like to have another woman in the bed and would bite on her breast. Then Gary would have sex with the other woman and bite my breast.\"\n\nOne of the other women who participated in these activities claimed to have had Heidnik's child. Another was a friend of Heidnik's from the Elwyn Institute. Her name was Sandra Lindsay.\n\nEarly in 1987 Heidnik varied his Saturday routine with Jewel and changed the rendezvous day to Wednesday. He would pick her up at noon, take her to his house, and they would make love in the waterbed. About five P.M. they would get dressed, go get something to eat, and Heidnik would take her home.\n\nThis routine continued during almost the entire time Heidnik was holding women captive in the basement and demanding sex from them on a daily basis.\n\nOne Wednesday, said Jewel, Heidnik brought another woman with him when he came to pick her up. To Jewel, the woman looked like a Puerto Rican. She said her name was Nicole.\n\n\"She seemed to be a sweet girl,\" Jewel said.\n\nThe three of them lunched at Wendy's, did some shopping for Nicole in a thrift store, then went to North Marshall Street. When they got there, she and Heidnik went upstairs for their midweek sexual tryst and Nicole stayed downstairs. When they came down later, Nicole was sitting alone in the kitchen, said Jewel.\n\nA white woman a few years younger than Heidnik said she also met Gary in the sixties, at the hospital, although she was a patient, not a member of the staff. Over the years she had a sexual relationship both with Heidnik and with Tony Brown. Once, she said, she had sex with Heidnik in the back of his van when it was parked near the Elwyn Institute as a convenient bedroom-on-wheels for whatever conquests he could make at the institution.\n\nA black hooker told police she lived in a room at North Marshall Street off and on for several weeks in 1986. During that time, she said, the place was like a zoo.\n\nTony Brown and a retarded black woman, who also had a physical handicap, were living there in addition to a parade of other women who came and went. Sandra Lindsay was one. (\"She would spend the night once in a while.\") Also there was a black girl (\"she would have epileptic seizures all the time\"), a \"real skinny woman from Elwyn\" who was severely retarded (\"she had three kids to look after and couldn't count or dial a telephone\"), and an obese white woman who \"looked like a bag lady,\" said the black hooker. \"Gary never messed with her.\"\n\nIn contrast to what a former girlfriend said about Heidnik's penchant for active sex, the black hooker said around her he was entirely different. When they were in bed, she said, \"Gary just wanted me to lay there and not move.\"\n\n* * *\n\nIn 1983 Heidnik decided to expand his sexual-racial repertoire. While he clearly preferred blacks to whites, he decided to look east as well. He went to a matrimonial bureau and directed them to find him an Oriental virgin.\nChapter 14\n\nJanuary 1\u201318, 1987\n\nTen days after adding Lisa Thomas to his horror harem, Gary Heidnik went on the prowl again. This time he came home with another black woman, a twenty-three-year-old named Deborah Johnson Dudley.\n\nHow or where Heidnik picked up Dudley is still a mystery. One thing is known, however: she was a constant thorn in his side. Almost from the time he took her to join the others in the basement, she proved to be a never-ending source of aggravation.\n\nDisharmony was already surfacing among the captives, and more dissension was not what Heidnik wanted, especially not from a woman who challenged his authority at every opportunity.\n\nAs the number of women grew, a pecking order developed. Rivera, the most streetwise of the group, was learning how to manipulate the manipulator. As time went on, Rivera was punished less frequently than the others and worked herself steadily into Heidnik's trust. Lisa Thomas, who had been abducted on December 22\u2014the third member of the group\u2014said she never saw Heidnik beat Rivera, although the others were regularly thrashed.\n\nOne of Heidnik's favorite tactics was to pick one of the group to be in charge when he left them alone, a sort of officer-of-the-day approach. Later he would come back and ask the one who had been responsible who had misbehaved so he could dole out punishment. Discipline usually consisted of being whacked with the shovel handle, but it could also include a restricted diet, time in \"the hole,\" or being handcuffed to the eye hook. If the one in charge said no one had misbehaved, Heidnik punished her.\n\nFrequently, too, he made them beat each other, and if the one administering the punishment was not doing it vigorously enough, he would reverse the roles. Or he would take over himself.\n\nHis sexual appetite showed no sign of declining. It was a rare day when he didn't force at least one of them to have sex with him. Sometimes he would go from one to the other, like a bee pollinating a flower bed, until he finally climaxed or grew tired. Later, too, according to Thomas, he made the women, under threat of death, have sex with each other while he watched.\n\nHygiene was minimal. Heidnik had brought a portajohn in for the women to use as a toilet, and for feminine hygiene he brought them tampons. But in the early days he refused to let them bathe. The only way they had of cleaning themselves was with disposable premoistened towels of the type many parents use on babies when changing their diapers. One day Thomas accidentally pulled two of the towels out of the container, which sent Heidnik into a rage. Accusing her of wasting property, he laid into her with the shovel handle.\n\nLater he relented and every day would take one of them upstairs to wash. They always carried their chains with them, even into the bathtub. After they soaked for a few minutes, Heidnik would push them into bed and have sex with them.\n\nWhile the cleanliness situation improved somewhat, the food situation deteriorated.\n\nOne day he was feeding his two dogs, a huge part-Lab he called Bear and a scruffy mutt of collie descent named Flaky, and he had an idea. The next time one of the women needed to be punished, he grabbed a can of chicken-flavored dog food and ordered them all to eat. They balked. \"Eat it or take a beating,\" he commanded. They ate it. From then on dog food became a regular part of their diet. Later it would take on a much more grisly aspect.\n\nThere were, however, occasions when he bent his rules. On Christmas Day he brought a takeout menu from a Chinese restaurant into the basement and let them pick their meal.\n\n* * *\n\nOn January 18 their number grew by one more with the addition of Jacquelyn Askins, a petite, soft-spoken eighteen-year-old. Despite her childlike qualities, Askins was no na\u00eff. At lunchtime that day she had been working the street in front of a seedy northside hotel, hoping somebody would come along looking for a nooner. That somebody turned out to be Heidnik, who was driving his blue Dodge van with the imitation fur interior, the one he fancifully referred to as \"Bugs Bunny.\"\n\nShe went home with him and, according to the established pattern, was dragged into the cellar. As soon as they got downstairs, Heidnik grabbed a plastic rod and hit her five times across the buttocks.\n\n\"That's what you're going to get if you don't do what I tell you,\" he said.\n\nTo impress that upon her, he hit her five more times.\n\nWhen he went to shackle her in the customary fashion, with muffler clamps, he discovered her ankles were so small she could slip right through the devices. So he had to use a pair of handcuffs.\n\n* * *\n\nThat night he was in an merry mood. He not only had a new recruit, but he thought both Rivera and Lindsay were pregnant. It turned out to be a false alarm; none of the women, despite Heidnik's efforts, ever conceived. A psychiatrist would later testify that Heidnik became extremely depressed when he saw the women at a court hearing \"and all their stomachs were flat.\" Dr. Clancy McKenzie said Heidnik \"took one look at them and told himself, 'All that for nothing.'\"\n\nThere was, however, in addition to his mistaken belief in their pregnancies, another reason for celebrating: the next day was Rivera's twenty-sixth birthday. In honor of the double occasion he brought home more Chinese food and a bottle of champagne, a luxury the women had never expected to see. While they eagerly gulped it down, Heidnik stood by and sipped slowly from a small glass. Drinking was not one of his vices.\nChapter 15\n\nOctober 1985\u2013April 1986\n\nA few weeks after Gary Heidnik's visit to the matrimonial bureau, a starry-eyed Filipina named Betty Disto was thumbing through a pen-pal brochure when a certain paragraph caught her eye. American male, it said. Single. Age: thirty-six. Occupation: nurse. She grabbed a piece of stationery and fired off a note of introduction to the person who placed the listing: Gary Heidnik, Philadelphia, PA.\n\nA friendship-by-correspondence flowered. He sent her a color picture. It showed an intense-looking, dark-haired man with a thin but rather handsome face. She was unaccustomed to close contact with Caucasian males; she could not tell from the snapshot that he was actually five years older than he had claimed in the brochure. On the back of the picture there was a message: \"Dearest Betty, Greetings from the land of ice and snow. Your friend, Gary.\"\n\nShe sent a picture of herself by return mail. Heidnik opened the envelope, stared at the image of a raven-haired young woman with dark eyes, a pouty mouth, and smooth, yellow skin. She said she was in her early twenties, but she looked younger. Absolutely ravishing. He smiled. Here was his Oriental beauty.\n\nIn a lengthy interview with Michael E. Ruane of the Philadelphia Inquirer, before she went into virtual seclusion, Betty spoke fondly of those months of getting acquainted.\n\nFor almost two years, she said, they exchanged letters (Betty wrote every week) and, occasionally, telephone calls. In one transpacific hookup, Heidnik told her he was a minister in a church he had founded. It sounded strange to her, but people in America did strange things. Then he proposed.\n\nThis created something of a crisis in the Disto household, halfway across the world and a thousand years away. She was the youngest of six children. Her father, a former military man and rural policeman, had died when she was twelve. Her mother was not anxious for her to leave the nest, especially when it involved a trip to an unknown land among unknown people. \"Don't go,\" her mother urged, arguing that Betty knew virtually nothing about the man. He might be a witch; a voodoo man. Betty laughed. Her mind was made up.\n\nShe applied for and received a visa. Heidnik sent her a ticket. Then she packed a suitcase, and on September 29, 1985, boarded a plane for a long, wearying trip: Manila to Tokyo to New York to Philadelphia.\n\nHeidnik was waiting for her, wearing a favored black-leather vest, black trousers, and a black shirt. Later she would reflect on that first encounter. \"He looked old,\" she told Ruane. Then shivering slightly, \"He looked like Dracula.\"\n\nThat day, however, she was too excited to be very perceptive. He kissed her chastely on the cheek and welcomed her to the United States. This public display of affection embarrassed her. People didn't do that where she came from.\n\nSince they were not yet married, she said, he wondered if she would be more comfortable at a hotel. \"No,\" she replied. \"Don't worry about it. I trust you.\"\n\nThose were her days of innocence. They didn't last long.\n\nWhen they got to North Marshall Street, Heidnik ushered her into a bedroom that contained a waterbed. She had never seen a waterbed and was a little uncomfortable at the thought of sleeping on one. She became even more uncomfortable when she discovered she was going to have to share it with a retarded black woman who was already snoring away. She had never seen a black person up close before either.\n\n\"Who is this?\" she asked.\n\n\"A tenant,\" Heidnik replied. \"She rents the room for two hundred fifty dollars a month. That isn't bad, is it?\"\n\nBetty didn't know what to say, so she didn't say anything.\n\nIt was a whirlwind romance. On October 3, before she even had a chance to recover from her jet lag, Betty drove with Heidnik to Elkton, Maryland, where they were married. Afterwards they returned to Philadelphia and checked into the Marriott. Then they went back to North Marshall Street.\n\nFor a while he treated her as though she were a princess, calling her \"honey\" and doing whatever he could to please her. They talked about having children, and Heidnik said he really wanted a son. If she had a boy, he said, he wanted to name him Jesse after King David's father in the Old Testament.\n\nThe honeymoon lasted a week.\n\nOne day she came home from shopping and heard a strange noise in the bedroom. Nudging open the door, she peeked inside and had the shock of her life. Her husband was in bed with three black women. All of them were naked and twisted into strange positions. She cried out, turned and ran down the stairs, sobbing hysterically.\n\nHeidnik ran after her. \"Give me a ticket,\" she said. \"I'll go home.\"\n\n\"No,\" Heidnik said. \"You don't understand. That is normal here in America.\"\n\n\"I can't stand it,\" she sobbed.\n\n\"You're going to have to,\" Heidnik said. \"I'm the boss.\"\n\nFrom then on it got worse.\n\nHeidnik spent a lot of time hanging out near the Elwyn Institute. On Sundays he held \"services\" at the North Marshall Street house, jamming the place with a couple of dozen handicapped people\u2014the world's outcasts, who had either physical or mental deformities and were ashamed to go to regular church because people stared at them.\n\nBetty estimates she attended twenty of these \"services,\" and never saw Heidnik accept a penny. It was against the church \"constitution\" to pass a collection plate. In addition, if any of them were having a hard time or were temporarily homeless, Heidnik would let them stay there with him and Betty.\n\nThis did not mean he had given up other women. He would bring home retarded clients from Elwyn or hookers he picked up on the street. Never during the entire time she was there, Betty said, was there not another woman in the house.\n\nWhen she complained, he would give her a withering look with eyes that changed in a flash from calm and cool to flashing and murderous. Then he'd punch her, usually on the arms, and punish her. He'd make her stand in the corner for up to twelve hours at a time. He would refuse her food. Or he'd laugh and make her watch him have sex, then force her to cook for him and his partner or partners.\n\nSunday, January 12, 1986, was a particularly bad day; a day when Heidnik was even more violent than usual. When she complained yet again about his lifestyle, he grabbed her by the hair, slapped her face, and punched her on the arms and legs. Then he forced her to submit to regular and rectal intercourse.\n\n\"If you run away, I'll kill you,\" he told her.\n\nA stranger in a strange land, she didn't know where to turn. Finally, in desperation, she made contact with other members of the Filipino community. They urged her to leave.\n\nFour days later, three and a half months after she arrived, she put her passport and a spare dress in a plastic bag and hid it outside. Saying she was going shopping, she slipped out the door, picked up her bundle and fled. She also reported him to the cops.\n\nOn January 27 the district attorney's office charged Heidnik with spousal rape, involuntary deviate sexual intercourse, simple assault, and indecent assault. He was picked up two days later, the precise day his parole expired on the Alberta Davidson incident.\n\nIn March Betty failed to appear at a preliminary hearing called to determine if there was enough evidence to hold him for trial. Without her testimony there was no case, so the judge had no choice but to dismiss the charges.\n\nHeidnik didn't know it, but Betty was pregnant. On September 15 she delivered Heidnik's child, a boy she named Jesse John, nicknamed J.J. She sent him a postcard telling him about it.\n\nOver the months she steadfastly refused to divorce him. At the time of his trial they were still officially married, but she virtually boycotted the proceedings. But then again, so did Heidnik's father and brother.\nChapter 16\n\nJanuary 14, 1987\n\nSeparated or not, Betty Heidnik expected financial support. When her husband fell dreadfully behind in his $135 a week payments, which didn't even factor in J.J., she hauled him into court. She did not know it, but Gary Heidnik had more pressing things on his mind.\n\nThirteen days after he added Deborah Dudley to his basement group\u2014four days before he would pick up Jacquelyn Askins\u2014he appeared before Judge Stephen E. Levin Jr. in the family court division of Common Pleas Court. If it were not such a serious issue, it would be humorous. A transcript of the proceeding has overtones of a Smothers Brothers skit. What makes it particularly interesting, though, is it is one of the few public recordings of a dialogue with Gary Heidnik. Although he took the stand during his 1978 trial, that was a set performance. With Levin, however, there was give and take. Heidnik, acting as his own lawyer, jousted with the judge. Actually played games with him. The judge didn't appreciate it. He came very close to throwing Heidnik in jail for contempt. If he had, the saga of the women in the cellar might have had a very different ending.\n\nAt the time, Heidnik probably was feeling pretty cocky. He had four women in the basement, two of whom he thought were pregnant. He thought his plan was working. And then along came Judge Levin and his runaway wife to try to screw things up.\n\nJudge Levin opened the proceedings by reminding Heidnik of a decision the previous August ordering him to pay support to Betty. J.J. had not then been born.\n\n\"There are allegations in the file,\" Judge Levin began, \"that defendant [Heidnik] has substantial assets, and, number one, the plaintiff [Betty] would like an increased amount on the arrearages, and, secondly, I believe there is a problem in having the wages attached. The record indicated that the VA will not deduct the ordered child support amount from the check. Is that so, sir?\"\n\n\"I'm,\" Heidnik stuttered, \"I'm still having a problem where you said I had a lot of assets.\"\n\nJudge Levin: \"Well, according to the court record, you have twenty-eight thousand\u2014\"\n\nHeidnik: \"If I got that, I sure would like to know where it's at.\"\n\n\"Well, either we have a mistake here or we have the wrong case. Do you have an account with Merrill Lynch?\"\n\n\"No.\"\n\n\"Do you have a bank account?\"\n\n\"Yes, sir.\"\n\n\"How much is in your bank account?\"\n\n\"About two thousand.\"\n\n\"Okay,\" said Levin.\n\n\"I'd like to know where the rest of that's at,\" Heidnik added.\n\nFor the next several minutes they skirmished. Levin probed. Heidnik played coy. Judge Levin was able to get Heidnik to admit he received about $400 a month in Social Security.\n\n\"Okay. Let me ask you a question,\" said Levin. \"Why are you getting Social Security and disability? What is the problem?\"\n\n\"You mean what's my medical problem?\"\n\n\"You must have a problem. Social Security does not give money to a healthy man... at your age. So something is the problem.\"\n\n\"Yeah. I... I got a nervous disorder.\"\n\n\"All right,\" Judge Levin responded. \"Now how much do you receive from the VA a month?\"\n\nHeidnik reluctantly admitted it was about $1,300.\n\nAfter a long discussion about Heidnik's alleged inability to remove his common-law wife (Dorothy, circa 1971) from his Social Security records, the topic swung back to Heidnik's VA checks.\n\n\"How long have you been on disability?\" Judge Levin asked.\n\n\"About twenty, twenty-one years.\"\n\n\"Oh. Okay. In other words\u2014\"\n\n\"Well, in actuality, since I got out of the service.\"\n\n\"When did you get out of the service?\"\n\n\"I got out of the service... I got out in '63.\"\n\n\"Okay.\"\n\n\"And then\u2014\"\n\n\"So when you got out of the service, at one point, you claimed\u2014\"\n\n\"Then, in '66, they made me a hundred percent. I was, like, ten percent or something in between there.\"\n\nThe discussion took another abrupt turn. Back to Social Security. Heidnik claimed one reason he was behind in his support was because he had been unable to convince the Social Security office to replace Dorothy with Betty on his monthly checks.\n\nJudge Levin prompted: \"Then, when you came with Betty, they said, 'How can you prove she's your wife? You still have the other wife.'\"\n\n\"Yeah,\" Heidnik agreed. \"Something like that.\"\n\n\"Okay. Now I understand.\"\n\n\"But then I tried to tell them I hadn't seen the woman [Dorothy]. They had actually taken her off. My understanding was that they had taken her off my checks. See, I was getting money for both me and her. Not Betty, but the other woman, right? And then something like '78, '79, '80, somewhere around there, they actually took that woman off my checks and I was only getting money for me. So I... I assumed that they had dissolved the marriage, the VA, because, you know, that was the only place, you know, it was registered, so to speak. Then, when I went in to get Betty on the checks, they dropped that one on me.\"\n\nJudge Levin decided to explore Heidnik's mental history.\n\n\"What is your mental problem, sir?\"\n\n\"What is my mental problem?\"\n\n\"Yes.\"\n\n\"You know, the doctors aren't too specific about that when they talk to me about that. They... they have me going to group therapy every month and I... I get medicine. Things like that.\"\n\n\"Mr. Heidnik,\" Levin said, getting back to money, \"didn't you have eleven thousand dollars in Fidelity in 1986?\"\n\n\"That's possible.\"\n\n\"What happened to that money?\"\n\n\"Well, I bought Betty a car, for one thing.\"\n\n\"When did you buy her a car?\"\n\n\"'Eighty-six.\"\n\n\"'Eighty-five, sir,\" interjected Betty Heidnik.\n\n\"Oh. 'Eighty-five?\" Heidnik queried.\n\n\"You had $11,902 in a bank account in 1986,\" Levin pressed. \"What happened to that money?\"\n\n\"I'm... I'm paying a mortgage on a house, for one thing.\"\n\n\"You still have it there, don't you?\" Levin asked accusingly.\n\n\"No, I don't. You're talking about Fidelity?\"\n\n\"Yes... What happened to the Fidelity money?\"\n\n\"It's down to about two thousand now.\"\n\n\"It is down two thousand? There is only nine thousand there?\"\n\n\"No. No. It's down to about two thousand.\"\n\n\"What happened to the nine thousand?\"\n\n\"I spent it.\"\n\n* * *\n\nSeeing he was getting nowhere, Judge Levin switched to the church.\n\n\"What is the United Church of the Ministers of God?\" he asked Heidnik.\n\n\"That's the church I belong to.\"\n\n\"Are you the bishop?\"\n\n\"It's an honorary title.\"\n\n\"Whose money is at Merrill Lynch?\"\n\n\"Merrill Lynch?\"\n\n\"Yes.\"\n\n\"Merrill Lynch's, I guess.\"\n\nThis must have angered Levin. \"You do not have any money at Merrill Lynch?\" he asked.\n\n\"No, I do not.\"\n\n\"Well, do you control money for the United Church of the Ministers of God? Do you personally handle their money?\"\n\n\"Yeah. I guess you could say that.\"\n\n\"Well, how much money do you handle for them?\"\n\n\"I just cost them about eighty thousand. So it went down a little bit.\"\n\n\"You cost them eighty thousand?\" Judge Levin asked in disbelief.\n\n\"See, there was a stock called Crazy Eddie's\u2014\"\n\n\"Yes.\"\n\n\"\u2014and I bought it around sixteen, and it's down around nine now. I'm not too popular after that little move. I couldn't resist Crazy Eddie, though.\"\n\n\"Well, how did you buy it? Who gave you the authority to buy it?\"\n\n\"The board of directors.\"\n\n\"Who is the board of directors?\"\n\n\"Reverend Mosley. My brother. What the heck's her last name?\"\n\n\"You are saying the money in Merrill Lynch is not your money?\"\n\n\"No. It's the church's money.\"\n\n\"It is the church's money. Why did you not tell me about it? Why are you being evasive?\"\n\n\"You asked me if I have any money\u2014\"\n\n\"Sir, you are being evasive.\"\n\n\"No. You asked me\u2014\"\n\n\"Do you know what the word evasive means?\"\n\n\"Yeah, I think so.\"\n\n\"I think you do too. You are jerking my chain. You know all about the Merrill Lynch account. You had three hundred thousand dollars in the Merrill Lynch account. Over three hundred thousand dollars.\"\n\n\"That's not my money.\"\n\n\"Then why didn't you tell me, 'It wasn't my money'?\"\n\n\"I did. You said did I have any money with Merrill Lynch. I said no, I do not. You're talking about my money, right? I do not have any money with Merrill Lynch.\"\n\n\"It is their money?\"\n\n\"The church's money.\"\n\n\"How about the Fidelity bank account?\"\n\n\"That's my money.\"\n\n\"And how much is in the bank account right now?\"\n\n\"Two thousand, roughly.\"\n\n\"And what happened to the other nine?\"\n\n\"Well, I... I didn't itemize it, but I know, like, I paid my gas bill, which they were going to shut off. I put a thousand dollars in my car, which was falling apart, at Pep Boys over there. I've got receipts for that one, like I said.\"\n\nJudge Levin was weary. \"Okay,\" he said.\n\nAfter discussing the possibility of getting money for J.J. with Betty, he promised not to let the matter drop.\n\n\"We are going to have another hearing as soon as I have a mental health evaluation on him,\" he dictated for the record. \"I am going to ask our department to look at him and ask for his VA records, so that I can know who I am dealing with. I do not know who I am dealing with. I am looking at a man who appears to be very intelligent, who appears to be evasive. I do not know whether that is deliberate or he really is answering honestly. I cannot tell, all right? I know that he had control of over three hundred thousand dollars. I do not know whether that is his money or the church money.\"\n\nTurning to Heidnik, Judge Levin offered one piece of advice: \"Hold on to the Crazy Eddie stock. It may go up.\"\n\n* * *\n\nThe tests Judge Levin ordered were performed in March. By that time things were going very badly for Heidnik. In January he was haughty; in March his personality almost certainly was entirely different. When he took the tests, two of his captives had died. It was a time of extreme crisis. The tests were very revealing. But they came too late.\nChapter 17\n\nFebruary 7, 1987\n\nSandra Lindsay looked like hell. For a week she had been dangling by her wrist from an overhead beam in a punishment session that began because she tried to push aside the plywood sheet covering \"the hole.\" The punishment dragged on when she refused to eat and Heidnik, still believing her to be pregnant, tried to force-feed her. He shoved small pieces of bread in her mouth and then held his hand over her lips until she swallowed.\n\nBut she was wearing down. For the last couple of days she had been vomiting and she complained of a fever. Heidnik ignored her and kept insisting that she eat.\n\nJosefina Rivera knew a crisis was coming when she looked up and saw that Lindsay had slumped down and showed no signs of rising. Realizing it would just make Heidnik angrier if he came down and found her like that, Rivera and the other three began yelling encouragement to the semiconscious woman. She didn't move.\n\nAbout that time Heidnik came down the stairs, took one look at Lindsay and ordered her to stand up. She did and he left. However, a few minutes later she collapsed again. This time when Heidnik came down, he did more than talk to her.\n\nHe unconnected her handcuffs and she dropped to the floor in a heap. Heidnik angrily kicked her into the hole. \"She's faking,\" he said.\n\nTurning his back on her, he walked over to a large chest-type freezer on the other side of the room and dished out three bowls of ice cream. He handed one to Rivera, one to Thomas, and kept one for himself. He took his upstairs. When he came back a few minutes later and saw that Lindsay still had not moved, he dragged her out of the hole and tried to find a pulse. There wasn't one. She was dead.\n\nHeidnik looked at her for a minute and then said, \"She choked on a piece of bread.\"\n\nHer death presented a major problem for him. It not only set back his plan to gather a group of human baby machines, but also he had to figure out what to do with the body. If he put it somewhere and it were found, she could be identified. And if she were identified, she could be traced to him. Now and then, her sister or cousins still came by looking for her. He simply couldn't get rid of the body, he reasoned; he had to destroy it.\n\nHoisting Lindsay over his shoulder like a sack of cement, Heidnik carried her up the steps. Sometime later the women in the basement heard what they thought was a power saw. They looked at each other and shivered.\n\n* * *\n\nHeidnik's dogs, Flaky and Bear, had the run of the house. Occasionally, when the door to upstairs was open, they would bring some of their food into the basement and take it under the pool table across the room to eat. Several hours after Heidnik disappeared with Lindsay's body, Bear came down dragging a long white bone with chunks of red meat clinging to it. The women looked at the bone, then they looked at each other, and each thought the same thing: I wish I were close enough to grab the meat.\n\nInvestigators later were unable to find bloodstains on an electric saw found in Heidnik's house, prompting them to believe he got rid of the tool used to carve up Lindsay. A few days after Lindsay died, he bought a food processor, and while no bloodstains or body parts were found on it, investigators are convinced he used it to grind up parts of Lindsay's body. He mixed the processed meat with dog food and fed it, he told Rivera, to Flaky, Bear, and the surviving captives. What he did not immediately grind he put in white plastic bags, which he stacked neatly in the freezer compartment of his upstairs refrigerator.\n\nThe parts he could not grind\u2014the head, hands, feet, and rib cage\u2014he tried to destroy by cooking. That created a terrible stench, though, which almost choked the women in the basement and was so pervasive it had the entire neighborhood up in arms.\n\nAlthough the neighbors yelled for the police, a rookie cop answered the call and went away when Heidnik told him he had simply overcooked the roast he was preparing for dinner. The smell hung around for days: polluting the air; permeating what little clothing they had; and, most noticeably, virtually soaking into Heidnik himself. That night, for the first time since he took his first captive on November 26, Heidnik did not come down to the basement demanding sex. But for many days afterwards, when he resumed his sex-fix visits, he smelled so strongly of burning flesh it was all the women could do to keep from gagging.\n\n* * *\n\nHeidnik didn't know it yet, but his days were numbered; his grandiose plans for creating a basement baby factory were rapidly unraveling. Before they collapsed completely, however, his captives would undergo a lot more suffering and one more of them would die.\nChapter 18\n\nFebruary 8\u2013March 18, 1987\n\nAfter Sandra Lindsay's death, Heidnik became even more paranoid. Convinced now that the women were plotting against him, he encouraged an informer system so he could stop any escape plans before they got started. The reward for snitching would be exemption from the hole, a little better food, and slightly more freedom.\n\nHeidnik's paranoia about escape plans was not totally unjustified. At one time the four women worked out a scheme whereby Deborah Dudley was going to hit Heidnik over the head with a piece of iron pipe they had dug up and the others would grab whatever was handy and stab him. Askins later testified at Heidnik's trial that before they could put the plan into effect, Rivera tipped him off.\n\nHeidnik also had decided that he could keep them even more off balance if they didn't know where he was, whether he was upstairs or out of the house altogether. When he was home they could easily hear him walking, and when he left they could hear the door close and his car drive away. The way to solve that, he figured, was to stop them from hearing him. That led to one of the most cruel tortures he was ever to use against the women.\n\nOne by one he took Deborah Dudley, Lisa Thomas, and Jacquelyn Askins over to the eye hook and cuffed them with one hand above their heads. He also cuffed their feet. Then he stuffed a plastic bag in their mouths as a gag and secured it with duct tape wound around their heads. Finally, looping an arm around their throats to hold them still, he took a screwdriver and gouged in their ears, trying to damage the eardrums.\n\n\"He used three kinds of screwdrivers,\" Thomas swore later, \"small, medium, and large. He twisted them in our ears until pus came out.\"\n\nAgain, Rivera was exempt.\n\n* * *\n\nHeidnik's chief antagonist during this period was Deborah Dudley, who fought him in everything he tried to do. One day, in an attempt to frighten her into submission, he unhooked her and dragged her upstairs. They were gone only a few minutes, and when they returned, Dudley was uncharacteristically quiet. Rivera was particularly inquisitive. \"What happened?\" she asked. \"What did he do?\"\n\nFinally Dudley whispered, \"He showed me Sandra Lindsay's head in a pot. And he had her ribs in a roasting pan and a bunch of her other body parts in the freezer. He told me if I didn't start listening to him, that was going to happen to me too.\"\n\nThe sight didn't chastise her for long; within a day or two she was again pushing him as far as she could.\n\nAbout this time Heidnik introduced still another new torture: electric shock.\n\nHeidnik snipped off the plug end of an ordinary electrical extension cord and stripped the insulation to leave a bare wire. Then he plugged the other end into a socket. With current then flowing through the wire, he touched the bare end to the women's chains and laughed while they jumped and screamed. For extra effect he submerged them in water.\n\nOn Wednesday, March 18, he decided all the women except Rivera needed punishment, so he put them in the hole and stood by while Rivera filled the pit with water using a hose and a nearby faucet. Earlier he had drilled several holes in the plywood covering as air vents. When the women were in the hole they could see a little of what was going on in the basement.\n\nOn this day they crouched in the hole in fear. They knew what was coming. Thomas, physically the largest of the three, was on the floor of the pit with Askins and Dudley in her lap. Through one of the air holes they could see the wire. They watched as it inched toward their chains, which snaked out of the hole around a pipe. When the bare wire touched a metal link, the current shot through their bodies. They screamed.\n\n\"I couldn't see Nicole with the wire, but I could hear her saying she had it,\" Thomas said later.\n\nThe wire went straight to Dudley's chain, so she took the heaviest jolt. \"He's killing me,\" she shrieked.\n\nShe was right. Seconds later she went limp. Thomas and Askins heard a splash and looked over. Dudley was bent over, face down in the muddy pool.\n\n\"She's dead,\" Thomas yelled. \"You've killed her. I don't feel no pulse.\"\n\nHeidnik, thinking the wire was faulty, had run upstairs to get another one. When he returned Rivera told him that Thomas claimed Debbie was dead.\n\n\"No,\" Heidnik replied, \"nothing's wrong with Debbie. I don't want to hear that bullshit.\"\n\nThomas lifted the board. \"Something's wrong with Debbie,\" she said. \"She's dead.\"\n\nHeidnik walked over to the pit and looked in. \"You know, Debbie's face is down in the water,\" he told Rivera.\n\n\"Hold your hand up,\" he told Thomas, unfastening the cuff that linked her to Dudley. He lifted Askins out of the hole and then reached for Thomas, but she looked to be in shock. He grabbed her arm, lifted her out, and pulled her clear. Then he bent down and pulled Dudley's body out.\n\nThe next thing Thomas knew she was stretched out on the concrete basement floor and her handcuffs had been removed. She looked up and saw Heidnik calmly making dog food sandwiches. A few feet away was Dudley's body.\n\n\"Aren't you glad it wasn't one of you,\" Heidnik commented conversationally. Nodding at the dead Dudley, he added, \"That's the asshole that caused Sandy's death.\"\n\nAfter making them sandwiches, Heidnik went upstairs, returning a few minutes later carrying writing implements. Handing a clean sheet of paper and a pen to Rivera, he told her to write.\n\n\"Put the date at the top,\" he said. \"March eighteenth, 1987.\" She did.\n\n\"Now the time.\"\n\nShe scribbled in \"6:30 P.M.\"\n\n\"Now write this: 'I, Nicole Rivera, and Gary Heidnik killed Debbie Johnson (Dudley) by applying electricity to her chain while sitting in a pool of water in a hole in a basement of 3520 Marshall Street.'\"\n\nHe ordered her to sign it, then added his signature below hers. He also commanded Askins and Thomas to sign it as witnesses.\n\n\"Now I've got the letter,\" he told Rivera. \"If you ever go to the cops, I can use this as evidence that you killed Debbie.\"\n\nBending over, he unlocked Rivera's shackles and told her to go upstairs and put on a clean shirt and a pair of pants. It was the first time in almost four months that she had not been chained. It also was the first time she had worn a pair of pants. The others enviously watched her go.\n\n* * *\n\nHeidnik left Dudley's body on the floor overnight. The next morning he wrapped her head and feet in plastic and carried the body over to the freezer. After shuffling some of his cartons of ice cream and other foodstuffs, he laid the body inside, closed the lid, and put several bags of dirt on top. Then he started making plans to dispose of it. Since no one could connect him to Dudley as they could have to Sandra Lindsay, he was not faced with the same constraints. This body he would simply dump as soon as he found a suitable spot. With her safely in the freezer, though, there was no hurry. He went upstairs to find Rivera. That night he would give her a treat. They would go out to eat.\nChapter 19\n\nMarch 18\u2013March 23, 1987\n\nFollowing Deborah Dudley's death, Josefina Rivera's lifestyle improved considerably.\n\nSecure in the belief that the letter he had Rivera sign protected him from treachery, Heidnik treated her more as a girlfriend and less as a captive. One night they went to the McDonald's near the Elwyn Institute where Heidnik used to hang out. Heidnik chose it specifically so his friends could see him with another woman and reinforce his contention that Sandra Lindsay was not with him. While there, they ran into Tony Brown and Gail. They asked if he knew where Sandy was.\n\n\"I think her family has her stashed away somewhere so they can cash her Social Security checks,\" Heidnik replied.\n\nAfter that they went to a home-supply store so Heidnik could get two new cabinets for his kitchen.\n\nOne day they went to an auto-supply store, where Heidnik purchased a can of transmission fluid for his Rolls. They took the Rolls to a garage for repairs.\n\nThey did the fast-food circuit, all the places Heidnik liked to go: McDonald's, Roy Rogers, Denny's, Wendy's. At one restaurant Rivera and Heidnik held hands.\n\nThey went shopping for a new wig for Rivera. They had sex. He took her into his confidence.\n\n\"If I ever get caught,\" he told her, \"I'm going to act crazy. I'm going to go into court and salute everybody.\" He bragged that he could beat the tests. \"I know them inside and out,\" he said. \"I've learned so I can keep getting my government checks.\"\n\nOne night they drove across the Delaware River into New Jersey. When they got to a forested area known as the Pine Barrens, Heidnik stopped, flicked on his brights and looked around. \"This place would be perfect to dump Debbie's body,\" he told Rivera.\n\nOn the way back they stopped at a flea market and a record shop so Heidnik could buy an album and four or five 45s.\n\nJust before midnight on Saturday, March 22, Heidnik said it was time to take care of Dudley. \"Get a blanket,\" he ordered, \"and spread it on the kitchen floor.\"\n\nHe went into the cellar, got Dudley's body out of the freezer, and carried it upstairs, stretching it out on the blanket.\n\n\"I want you to look at her,\" he told Rivera.\n\n\"I don't want to,\" she said.\n\nHe insisted, peeling back the plastic from her face. \"Look,\" he commanded. \"It will make you strong.\"\n\nRivera took a quick peek.\n\nHeidnik carried the body into the backyard and loaded it into the trunk of his old Dodge Dart, which would not be nearly as conspicuous as his Cadillac. They returned to the Pine Barrens and drove down a dirt road. After a few minutes Heidnik stopped. No lights were visible for miles.\n\n\"This looks good enough,\" he said.\n\nRivera huddled in the car while Heidnik retrieved Dudley's body from the trunk and carried it across an open space to a grove of trees. She could hear the twigs crunching as he walked.\n\nIn a few minutes he was back. Throwing his black leather gloves on the dash, he sped away. When they got back to Philadelphia he stopped to buy a newspaper. \"I want to check my stocks,\" he told Rivera.\n\nThe next day he brought up the subject of finding a replacement for Dudley. His harem was now down to two\u2014Thomas and Askins\u2014if you didn't count Rivera. He was a long way from the ten-woman group he envisioned as ideal.\n\n* * *\n\nOn Monday night, March 23, he and Rivera piled into his Cadillac and drove around looking for a candidate. It didn't take them long before they saw a face they both recognized.\n\nStanding on a corner, waiting for a customer, was a small, dark, twenty-four-year-old woman named Agnes Adams. Rivera knew her as \"Vickie\" and she knew Rivera as \"Nicole.\" They knew each other from a strip club called Hearts and Flowers where they both had worked in the past.\n\nHeidnik also knew her, but not by name. He had picked her up one night in January, apparently looking for some sex outside his basement crowd. At that time he had agreed to pay her thirty-five dollars for oral sex and she had climbed into his Cadillac and driven with him to North Marshall Street. When they got there, a car was parked across the entrance to Heidnik's driveway and he was unable to find a parking place anywhere in the vicinity. He paid her ten dollars for her time and took her back downtown.\n\nShe had seen him again in February, when he picked her up at noon from the corner of Fifth and Girard. This time he had better luck getting into his driveway. He took her into the kitchen, gave her a soda, and watched while she played a couple of games on Mr. Do. Then they went upstairs and she performed fellatio on him. He paid her thirty-five dollars, saw her out the door, and locked it behind her. She walked home. He made no attempt to capture her and make her part of his cellar group. He was picky about whom he selected.\n\nThis time, however, the situation was different.\n\nHe negotiated a thirty-dollar price with her, and the three of them\u2014Heidnik, Rivera, and Adams\u2014drove to North Marshall Street. When they got there, Heidnik gave Rivera the key and told her to lock up. He and Adams went upstairs and had sex, followed by what some cynics now refer to as the \"Heidnik maneuver.\" He choked her, cuffed her, took her downstairs and threw her in the hole. Rivera remained in the kitchen, sipping wine coolers.\n\n\"That was easy,\" Heidnik said to Rivera. \"We can do that again tomorrow.\"\n\nRivera just nodded. She had plans of her own.\nChapter 20\n\nMarch 24\u201325, 1987\n\nJosefina Rivera sensed it was time for her to make her move. All day Tuesday, March 24, she worked on Heidnik, convincing him she needed to see her family. After all, she had been gone for four months. If he let her do that, she said, she promised to find another woman for him.\n\nHeidnik finally agreed. He had the letter, didn't he? \"But if you try to run,\" he cautioned, \"I'll kill the others.\"\n\nLate that night they left in Heidnik's Cadillac. Everything was quiet in the cellar. The new arrival, Adams, was in the hole, and Thomas and Askins were curled up on the mattress. After Dudley's death, Heidnik had eased up considerably and provided the cellar dwellers with pillows, blankets, and even a TV.\n\nAccording to the plan, Heidnik would drop Rivera, who would go see her children, pick up a woman she knew, and meet him back at a gas station parking lot at Sixth and Girard at about midnight.\n\nAs soon as Heidnik drove off, Rivera beelined to the apartment of her boyfriend, Vincent Nelson, four blocks away.\n\nNelson was shocked. This woman had walked out on him in the middle of an argument four months ago and he hadn't heard a word from her since. Now here she was, he told police later, banging on his door and ringing his door bell at the same time and ranting about somebody who had been holding her captive.\n\n\"She came in, and as we were walking up the steps, she was rambling on, you know, talking real fast about this guy having three girls chained up in the basement of this house and she was held hostage for four months. She said that two of the girls were dead and he had three more in a hole in the basement floor chained up and he was going to kill them if she doesn't come back with another girl in a certain amount of time.\"\n\nThe more she talked, the less believable her story became.\n\n\"She said that he was beating them, raping them, had them eating dead people, just like he was a cold-blooded nut. Dogs was in the yard eating people bones. I just thought she was crazy. I really didn't believe it, and I still don't believe this shit.\n\n\"I said I was going up there with a hammer and I was going to fuck him up, and she said, 'No, that might scare him and he might go back and kill the girls.'\"\n\nAccording to what Nelson later told Inquirer reporter A. Paolantonio, he and Rivera left his apartment and started walking toward Sixth and Girard. However, when they got about a block away, Nelson reconsidered and decided it would be better to call the cops.\n\nThey stopped at a pay phone. Nelson dialed 911 and handed the receiver to Rivera. At first the cops didn't believe her either. An officer told her to stay where she was and a unit would be there presently.\n\nMinutes later officers David Savidge and John Cannon pulled up.\n\nAs everyone else had been that night, Savidge and Cannon were skeptical. The story Rivera disgorged was just too wild to be true. Then they looked at the abrasions and scars on her ankles where she said the chains had been, and they became believers.\n\nAfter Rivera told them Heidnik was waiting for her three blocks away, Savidge and Cannon wasted no time getting there. They coasted into the lot and pulled up behind a 1987 pewter-over-white Cadillac. The vehicle's engine was running, and there was a man sitting behind the wheel.\n\nCannon, a short, compact man with graying hair, took the passenger side; Savidge, a heavily-built blond a head taller than his partner, approached on the driver's side. They had their weapons drawn. Heidnik saw Savidge coming and he raised his hands.\n\n\"What's this all about, officer?\" he asked in surprise. \"Didn't I make my child support?\"\n\nSavidge studied him closely. \"No,\" he replied quietly, \"it's a little more serious than that.\"\n\nHe cuffed Heidnik and took him to the Sex Crimes Unit building where Rivera was being questioned in more detail.\n\nAgain, Heidnik looked at Savidge.\n\n\"Is it child support?\" he repeated.\n\nSavidge thought he sounded like a broken record.\n\n\"No,\" Savidge told him for the second time. \"It's about kidnapping, rape, and homicide.\"\n\nStill, it was a song Heidnik would sing all night: that the police really wanted him for missing his child-support payments.\nChapter 21\n\nMarch 25, 1987  \n4 :30 A.M.\n\nSergeant Frank McCloskey was ready for some action. He had been on call outside 3520 North Marshall Street for the last four and a half hours, waiting for a command to do something. When he first got to the scene, he did the same thing Officer Julio Aponte had done forty-three days earlier. He knocked on the doors. He tried the handles. He checked out the windows. He got nowhere. When he called in to report his lack of success, the dispatcher told him to sit tight.\n\nAll he knew was that a man named Gary Heidnik had been arrested about midnight after a near-hysterical woman called the Twenty-fifth District and spun some wild tale about how she had been a captive in this house for four months and three women were believed to be alive and still in the cellar. He had to wait until they got a search warrant before he could go busting in.\n\nJust before five, more police showed up. They had the warrant. Among the new arrivals was Officer David Savidge, who had been the first to hear Rivera's story face to face.\n\nWhile Savidge and McCloskey watched, another sergeant tried unsuccessfully to open the doors with keys he had gotten from Heidnik. When he saw the keys weren't going to work, Homicide Lieutenant James Hansen called for a crowbar.\n\nNext door, Doris Zibulka was sleeping. When Hansen went to work with the crowbar, she thought she was being raided. \"They just about knocked me out of bed.\"\n\nOnce the door was down, Hansen dashed for the kitchen; Savidge and McCloskey headed for the basement.\n\nAll around the room, McCloskey noted, were white plastic shopping bags. A mattress was lying in the middle of the room and on it were two black women, covered in blankets and snuggled up against each other for warmth. They were asleep.\n\nWhen they heard the commotion, they jerked awake and started screaming.\n\n\"Police,\" McCloskey yelled. \"We're the police. We're here to help you. No one is going to hurt you.\"\n\nThe women bolted to their feet, letting the blanket drop. Except for socks, they were both nude from the waist down. Both had shackles on their ankles and were connected to heavy chains. One was Lisa Thomas, the other was Jacquelyn Askins.\n\n\"Are there any other women in the house?\" McCloskey asked. \"Is anyone here but you?\"\n\nBoth pointed to a pile of white plastic bags sitting on a board on the other side of the room.\n\n\"She's there,\" Thomas said.\n\nMcCloskey, remembering the woman's tale included a story about dismemberment, picked up one of the plastic bags.\n\n\"Here?\" he asked incredulously.\n\n\"No, under the board,\" Thomas replied. \"She's in the hole.\"\n\nMcCloskey pushed away the bags and slid the board aside. Squatting at the bottom of the shallow pit was Agnes Adams. She tried to stand up, lost her balance and tottered. McCloskey grabbed her arm and lifted her out. She was completely nude. She was shackled like the others, and in addition, her hands were cuffed behind her back.\n\nShe started screaming too.\n\n\"It's all right,\" McCloskey said. \"We're here to help you.\"\n\n\"We're free!\" Thomas and Askins yelled. \"It's the police!\" They grabbed McCloskey's hands and smothered them with kisses. \"We're saved!\" they shrieked.\n\nAdams was still in shock. \"He took my thirty dollars,\" she yelled angrily. \"Get my thirty dollars back.\"\n\n\"Don't worry about it,\" McCloskey said.\n\nHe dug out his handcuff key and tried to unlock the bracelets binding Adams' hands. It didn't fit.\n\n\"Try this one,\" another officer said, proffering his cuff key. It did the trick. They then worked at removing the leg irons.\n\nMcCloskey noted that their ankles were covered with bruises and sores. Some were fresh and some had been scabbed over. They couldn't undo the nuts, and sent word upstairs to fetch some bolt cutters. They also called for an ambulance and some extra hospital shifts for the women to wear.\n\nThey were so skinny, they looked like POWs. They said they were starving.\n\n\"Gary kept ice cream in the freezer,\" Askins said, pointing to the box where Deborah Dudley's body had been kept on hold until Heidnik had found a place to dump it. \"Can we eat that?\"\n\n\"I don't think you ought to have any ice cream yet,\" said McCloskey. \"Wait until the doctors check you over.\"\n\nWhen their ankles had been freed and they were taken to the ambulance, they went through the dining room where Heidnik had left some cookies on the table. They grabbed them and wolfed them down.\n\nWith the women on their way to medical care, the officers fanned out through the house. On the shelf in a closet they found a large stack of porno magazines, all of them featuring black women.\n\nWhile McCloskey headed upstairs, Savidge went into the kitchen. He looked at the stove and noted an aluminum pot. The inside was scorched and covered with a yellowish material. In the open oven he saw a metal roasting pan that was charred on the inside and contained a piece of bone that looked suspiciously like a rib. On the counter was a heavy-duty food processor, obviously used. He opened the freezer and, right there on the front shelf, was a human forearm.\n\nThat was too much for even the veteran Savidge. Feeling the bile rise in his throat, he ran outside and gulped the fresh air, trying mightily to keep from vomiting.\nChapter 22\n\nMarch 26, 1987\n\nWhen the story broke, the news media went wild. There were so many lurid details to cover, reporters weren't sure which way to turn first. The crimes in themselves were gruesome enough: murder, rape, bondage, torture, dismemberment, and cannibalism. Race was involved. Plus, swirling in the background were twin specters guaranteed to get attention anytime they were tied to scandal: money and religion. Gary Heidnik definitely was not a penniless ghetto dweller, not with a stock portfolio at Merrill Lynch valued at some $550,000, even with the Crazy Eddie loss. The stock account was in the name of the United Church of the Ministers of God, but Heidnik also had a personal account, police later discovered, containing more than $16,000. There may be other accounts that police still haven't found.\n\nIt was a case of overload for the media; there were simply too many sensational elements for any newspaper to handle at once. But they tried.\n\nThe timing of Heidnik's arrest gave the newspaper break to Philadelphia's tabloid Daily News. When the paper's morning edition hit the street about eight A.M., there was a relatively small piece on page three headlined, YOUNG WOMEN FOUND CHAINED; HUMAN ARM DISCOVERED IN NORTH PHILADELPHIA REFRIGERATOR. By the time the afternoon edition rolled off the press, though, Heidnik was the front page. HORROR ON MARSHALL STREET was the memo line, and in smaller type, He put chains on my legs; victims tell of beatings, sex, and deaths of 2 women. Inside were a half-dozen stories, including a profile on Heidnik compiled from interviews with neighbors. The main story was bannered, THREE CHAINED WOMEN FREED, HUMAN BODY PARTS FOUND; MAN ARRESTED.\n\nThe more staid Inquirer couldn't publish the information until the next day, but it compensated in length for what it lacked in immediacy. A bold, black banner across the entire eight columns of page one screamed, MAN HELD IN TORTURE KILLING, with 3 found chained in cellar in smaller type. The story ran about halfway down the front page and took up most of an inside page. The paper also carried several ancillary stories.\n\nWhen it was discovered that Heidnik was a self-proclaimed bishop in a church he created, and drove expensive cars, the headline writers went into passing gear.\n\nOn March 26 the New York Post's main story was headlined, MADMAN'S SEX ORGY WITH CHAINED WOMEN, with the subhead, Body parts found in Philly's house of horror. A subordinate story dubbed Heidnik the \"Rolls-Royce 'Reverend'\" and proclaimed he was \"into stocks & bondage.\"\n\nAnother New York tabloid, the Daily News, yelled, Sex slaves & mutilation in Philadelphia, and in larger type, FIND WOMEN CHAINED IN HORROR DUNGEON.\n\nThat same day USA Today said, PHILA. ROW HOUSE SITE OF DEATH, TERROR... 3 women chained; body parts are found.\n\nEven the ultraconservative New York Times got into it, albeit a bit more slowly. On March 27, a day behind most of the others, the Times devoted a quarter of a page to a story headlined, PROSECUTOR TO ASK DEATH PENALTY IN TORTURE CASE. Included in the spread were pictures of Heidnik, Tony Brown, and the North Marshall Street house.\n\nThe Times followed with a second story the next day, captioned, STRANGE PORTRAIT OF TORTURE SUSPECT.\n\nUncounted stories also were broadcast on radio and television. The facts were sent worldwide by the wire services. The man who would later be Heidnik's lawyer received a clipping from a friend in Australia.\n\nCoverage in Philadelphia, of course, was voluminous, with the Daily News and the Inquirer, both owned by Knight-Ridder, devoting a task force of reporters and thousands of inches of print to the incident.\n\nThe intensity and breadth of the media coverage would prove to be an issue when it came time to select a jury to try the case. Heidnik's lawyer had no trouble digging up plenty of evidence to show that the whole world knew about his client's crimes.\nChapter 23\n\nWhen police arrested Heidnik and freed Thomas, Askins, and Adams, their work was just beginning. It would take them days just to sift through the piles of paper and mounds of physical evidence they would need to present a comprehensive report. Whatever crimes Heidnik would be charged with, investigators fervently wished disorderly record keeping and slovenliness could be added to the list. Just the inventory of what he had on him when he was brought in by Savidge and Cannon was enough to cause immediate confusion.\n\nIn his pockets, among other things, was $1,962 in one-, five-, ten-, twenty-, fifty-, and hundred-dollar bills, along with a silver dollar and a half dollar. He had a bank card, a Sears credit card, a telephone card, a VA identification card, a driver's license, an ID card from his church carrying his picture in a clerical collar, and a nurse's identification card from the VA hospital at Coatesville. He had a pocket calculator, a gold Cross pen, a penknife, and a handcuff key. He had papers referring to a '72 Dodge, a '64 Plymouth, a '71 Cadillac, and a '73 Serro Scotty house trailer, some of which were in the name of his church and some of which were in his name. He had a stack of receipts from traffic court and a notice saying his driver's license had been suspended. He also had receipts from several retail stores and a statement from Merrill Lynch showing his account balance as of the previous October 31 was $577,382.52.\n\nBut that was as nothing compared to the list police made of items found in his house. It took them eleven days to compile it. But it was thorough.\n\nThe North Marshall Street inventory was a hefty sheaf of typewritten pages that carefully listed almost everything of note, a grab bag of items ranging from bloodstained clothing and chains to bags of bones and an aluminum pot with \"a tan crusty residue around the rim.\" A footnote: \"Analysis did not determine the identity of this tan material.\"\n\nInvestigators found several saws, including a jigsaw which failed to show traces of blood and a hacksaw that yielded paint samples matching flakes found on Lindsay's dismembered body. There were piles of used cotton swabs, a mayonnaise jar containing \"what appeared to be onions\" and a \"thick slab of moldy, round, light-yellow, cream-colored cheese.\" Screwdrivers aplenty were found, along with a box of no-frills dog biscuits and\u2014off in a corner\u2014\"a dirty, moldy, used condom.\"\n\nPolice spent days digging up Heidnik's yard, looking for more bodies. They never found any, but the search was not without its form of black humor.\n\nWith the curious crowding the streets and cheering encouragement, officers dug so many holes in the tiny yard it looked like an artillery target zone. One digger yelled and held up something that looked suspiciously like a human arm. It turned out to be a tree limb. Soon afterwards searchers uncovered a mass they thought was the remains of a human body. A frantic call went out for a medical examiner. After he examined it, he said it was simply a collection of refuse. Despite days of digging, all the searchers ever found, beside a few bones they immediately identified as belonging to Sandra Lindsay, was the skeleton of a cat.\n\nAll this poking, probing, and excavating was not without reason. When the captives gave statements to police, they said Heidnik had boasted of killing six women whose names he rattled off: Sandy, Debbie, Sallie, Jody, Marcie, and Carol. Two things added strength to this claim. At the time, no one had seen either of Heidnik's two former live-in women, Dorothy or Anjeanette Davidson, in years. Even more damning was a statement from Heidnik's former roommate, Tony Brown.\n\nThe day Heidnik was arrested, the thirty-one-year-old Brown wandered around to North Marshall Street to see what all the fuss was about. When a cop told him a guy named Gary Heidnik had been arrested for killing and torturing women, Brown volunteered that he was Heidnik's best friend and once lived in the house. They quickly whisked him off to headquarters for some detailed conversation. Eighteen hours later, according to the lawyer who later defended Heidnik, Brown was charged as an accomplice. In a statement leaked to the Inquirer, Brown embellished on Heidnik's crimes.\n\nHe said he had been with Heidnik in the summer of 1985 when Heidnik picked up a prostitute at the corner of Broad and Poplar, offering her a hundred dollars to come with him. Brown allegedly drove back to the house and went to bed, leaving Heidnik alone with the woman, whose name he did not know.\n\nThe next morning he went to the basement to put his dirty clothes in the washing machine. The woman was there. She was chained. He said she asked him to set her free. He refused, afraid of what Heidnik would do to him.\n\nAbout a week later, Brown said, he had been out running errands, and when he returned he smelled a terrible odor. He went down into the basement and saw Heidnik cutting up her body. Later, he saw Heidnik bury the parts in the front yard.\n\nBrown also told police he was present in the basement when his girlfriend, Sandra Lindsay, died. The tale he told was similar to that of the victims. He said he saw Heidnik trying to stuff bread in her mouth and she was gagging, but he kept pushing it in. Then she died.\n\nHe and Heidnik then carried Lindsay upstairs. \"I could tell she wasn't breathing,\" he said. \"I knew she was dead.\" He added that he saw Heidnik having intercourse with the corpse. He said he left when Heidnik dismembered the body.\n\nThere was only one thing wrong with the story. None of the victims had ever seen Brown anywhere, much less in the basement the day Lindsay died. There was another minor problem too: Brown's IQ is only 75, barely above the official level for mental deficiency. That does not qualify him as incompetent for trial, however.\n\nDespite these issues, Brown was charged with murdering Sandra Lindsay and with conspiracy, kidnapping, and rape. He was thrown in jail and bail was set at $50,000.\n\nA month later, on May 1, Brown was released on his own signature. The district attorney's office asked for his release, saying he could aid the investigation of Heidnik more if he were free than he could if he were kept in jail. Two weeks after the Heidnik trial, the charges quietly were dropped. Brown was a free man.\n\n* * *\n\nThere was no chance, though, that Heidnik was going to be released on bail. Never in a million years. Not under any circumstances short of divine intervention. He was in jail to stay, and both he and the system were going to have some adjusting to do. It was not going to prove easy on either.\n\nOn March 25, within hours of his arrest, Heidnik was attacked by another prisoner. He suffered a broken nose. Following that incident, he was put in a special section of the Philadelphia Detention Center and was allowed out of his cell only two hours a day. Even then he was not removed until all other inmates in the section had been locked down.\n\nDespite these precautions, he was able within a few days to get alone long enough to try to kill himself. On April 2, his ninth day in jail, he went into the shower. While a guard was occupied trying to adjust the water temperature for Heidnik, he looped one end of his T-shirt over the shower nozzle, the other around his neck. He let his knees sag. When the guard glanced up, he summoned help. Heidnik was already unconscious. But they found him in time.\n\nMany seasoned observers shook their heads. It was only going to be a matter of time before one or both of those episodes was repeated. Either he was going to try to kill himself or someone else was going to kill him. In addition to that, District Attorney Ronald Castille had said he planned to seek the death penalty. At the moment that seemed the least of Heidnik's worries.\n\"Tell me, Gary, what kind of seasoning did you use?\"\u2014Defense Attorney A. Charles Peruto Jr.\n\nHeidnik, staring hard at Peruto: \"Man, you're crazy.\"\nChapter 24\n\nGary Heidnik led a queer life, no doubt about it. But his preoccupation with the women in his basement was not all-consuming. During that four-month period he did a number of things that would not be considered explicitly abnormal. For instance:\n\n\u2022 Early in November he went to one of the city's largest Cadillac dealers, shopping for a new car. His previous one had been damaged in an accident. It didn't look as classy anymore. He knew precisely what he wanted: a De Ville with all the extras. \"He wanted flash,\" sales manager David Pliner testified later. \"He wanted everything you could put on an auto: a custom roof, continental kit, custom grill, wire wheels, the best tires\u2014everything.\" He negotiated a trade-in on his other Cadillac and paid the difference in cash, a $12,240 check drawn on Merrill Lynch.\n\nIntroducing himself as a bishop in the United Church of the Ministers of God, Heidnik struck up an acquaintance with the salesman. Whenever he brought his car in for service, he and Pliner chatted about the stock market. Pliner gave him a stock tip that must have proved fruitful because he called Pliner a month later and expressed his gratitude.\n\n\u2022 In December he returned to a VA psychiatric clinic in Philadelphia after a ten-month lapse in treatment. He was greeted warmly by some of the long-term patients and seemed touched by the response his return elicited. \"Hey,\" he told Dr. Richard W. Hole, \"they're glad to see me.\" He returned for group therapy regularly until the time of his arrest. Every time he left, he had in his pocket a supply of Thorazine which he was supposed to ingest. He apparently never did, because a considerable amount of the drug was found in his bedroom on North Marshall Street by police.\n\n\u2022 In January there was the hearing before Judge Levin in family court. Although the judge was unsure how much Heidnik understood about the proceeding, he did not appear so deranged to cause concern. His wife Betty apparently did not think he acted abnormally either. At least she did not mention it to the judge.\n\nThroughout this period, too, he was bringing other women into his house for sex. One of these was a longtime girlfriend who used to meet him every week. Another was Agnes Adams, who was not taken captive until her second time inside number 3520. Why did he capture her then and not the first time? Only Heidnik knows, and so far he hasn't volunteered that information. Police suspect that there were a number of others as well, streetwalkers and perhaps some of his friends from Elwyn.\n\nThe incidents with the other women, though, are instructive in one respect. They show that Heidnik was using some kind of selection process. He wasn't grabbing women willy-nilly off the street, taking them home and chaining them up. But his criteria are still a mystery. No one except he knows why he picked some and rejected others.\n\n* * *\n\nThese glimpses into Heidnik's \"worldly\" persona are fascinating but not necessarily enlightening. In fact, they only cloud the issue. His lawyer spent long hours thinking about it. If a jury was going to have to question Heidnik's mental process, how could these seeming bouts of sanity be explained? Was mental illness something that could be turned on and off? Was it visible sometimes? Invisible at others? If someone was crazy, did they have to act crazy all the time? Was Gary Heidnik a nut or not? If he was a fruitcake, how could he sometimes act in an understandable way? If he was not, what could explain the perfectly atrocious and revolting things he did? Lawyers call it the \"mad\/bad line,\" the wavy, far-from-clear divider between someone being insane and just being vicious.\n\nInsanity would be one of the two main prongs of Heidnik's case. To get off, to be found not guilty, was never considered a viable possibility. Three women were found in his basement, led there by a fourth. All of them unequivocally identified Heidnik as the man who had held them captive. His freezer was half full of body parts. Human bones were found in his yard. One of the former captives led investigators, like a steel ball to a magnet, to a second body. There was just too much evidence against him; he couldn't very well say \"wrong guy.\"\n\nThe other main prong would be the argument that he did not intend to kill Sandra Lindsay and Deborah Dudley. Although District Attorney Ronald Castille had already decided to go for the death penalty, he could not ask for that, under Pennsylvania law, until he first won a conviction for first-degree murder. A conviction under anything less than first degree maxed out at life in prison. But Castille looked forward to the fight.\n\n* * *\n\nAn outspoken law-and-order man, Castille fought as a Marine platoon commander in Vietnam. He left a leg there, on a dusty battlefield in the area known as I Corps. He had a reputation as a scrapper. Witness the fact that he was holding his office at all\u2014a white Republican in a city dominated by black Democrats. He had beaten a black opponent in the election just a few months before, and this was a good opportunity to show that he could practice what he preached when he promised hard justice.\n\nLike all major American cities, Philadelphia has a high percentage of violent crime. As a rule, there is roughly a murder a day in the city. There are arrests in about eighty percent of them. So every year, somewhere around three hundred murder charges are filed by the D.A.'s office, give or take a dozen. Not all of those go to trial, of course. And not all go to trial with the prosecutor seeking the death penalty. Usually there are twelve to fifteen capital-punishment cases a year. Castille's office scored about a sixty-six percent success rate. Death penalty cases were far from rare, but there wasn't one every week either.\n\nFor a while Castille even flirted with the idea of trying the case himself. He decided against it when he saw how strong the evidence was against Heidnik. In the end, he knew, it would be primarily a battle of psychiatrists: the Commonwealth's shrinks against the defense's shrinks. That didn't sound like his kind of case. But it did sound like a perfect case for one of his top assistants, a dedicated worker down the hall named Charlie Gallagher.\n\n* * *\n\nCharles Gallagher, curiously, was into the case well before Castille, before anyone else in the D.A.'s office, as a matter of fact. He was home in bed, asleep, when his phone rang soon after midnight on March 26, 1987. As acting chief of the D.A.'s homicide division, the forty-year-old Gallagher was accustomed to middle-of-the-night phone calls. It was policy that either the chief or assistant chief had to be notified when a charge of murder was being contemplated, no matter what the hour. And Gallagher had learned that murderers rarely respected a public servant's work schedule. When he groped sleepily for the phone, he suspected it was trouble. It was. Police Lieutenant James Hansen was on the line with some not so cheery news.\n\nAs Gallagher stifled a yawn, Hansen succinctly laid out the basics of the complaint against Gary Heidnik. He said he wanted a search warrant to get into Heidnik's house. He needed it pronto.\n\n\"I think that can be arranged,\" Gallagher said. After getting the warrant moving, the ADA went back to bed. He had an early day staring him in the face. He had a funeral to go to. A popular judge had died and he was being buried that morning.\n\nAs soon as he could break away from the services, Gallagher hurried to the office and called Hansen for a status report. It was about ten o'clock. The homicide detective said things were progressing smoothly, that everything had, indeed, been exactly as the complainant\u2014a woman named Josefina Rivera\u2014had outlined. While they were talking, she was leading another detective on a search of Wharton State Park across the Delaware River in New Jersey to see if she could find the spot where she saw Heidnik discard the second body.\n\nAs Hansen ran through some of the details for him, Gallagher felt his stomach tighten. To do all these things, Gallagher thought, this man Heidnik had to be more than crazy. He had to be evil.\n\nSuch thoughts loom large in Gallagher's mind. He grew up in a strict Catholic home. One of his sisters is a nun. He describes himself as a \"strong\" believer. Added to his religion-inspired perception was the fact that his childhood had not been a sheltered one. His father had been a cop for thirty-one years.\n\nAs more details emerged and the case against Heidnik began taking substance, Gallagher made up his mind that he wanted to try this one. If ever there was a case that screamed for the death penalty, this was it. The Catholic church's position might be different, but Gallagher was also a prosecutor, and in this case his public duty came first. Without a qualm\u2014not even a flicker of hesitancy\u2014he went to Castille and asked for permission to make the charge first-degree murder.\n\n\"The death penalty may not be a deterrent to crime,\" Gallagher said, \"but it sure would deter Gary Heidnik.\"\nChapter 25\n\nAt the time the Heidnik story broke, A. Charles Peruto Jr. was trying a murder case in Lancaster, west of Philadelphia, deep in the heart of Amish country. During a break in the proceedings he routinely called his Philadelphia office and was put through to John Fognano, a cousin and a member of Peruto's small firm. Fognano was bubbling with excitement.\n\n\"You'll never guess who's been trying to reach you,\" he said excitedly. Before Peruto could respond, Fognano broke the news: \"Gary Heidnik.\"\n\n\"Who?\" said Peruto, not sure he had heard correctly.\n\n\"Gary Heidnik,\" Fognano repeated. \"He wants to hire you. He wants you to represent him.\"\n\n\"Is this a joke?\" Peruto asked in surprise. \"How did he find out about me? Why me?\"\n\nWhen Peruto recounted this scenario he was being uncharacteristically modest. He was known throughout Philadelphia and southern New Jersey for two reasons: he was a high profile, shoot-from-the-lip criminal defense attorney who absolutely loved sensational cases, and he was the thirty-two-year-old son of one of the city's most colorful and respected defenders, A. Charles Peruto Sr. The father was Chuck, the son Chuck Jr. Chuck had a reputation for being clever, quick-witted, highly quotable, and a top-notch lawyer. Chuck Jr. had a reputation for being clever, quick-witted, and highly quotable. He was still working on the top-notch lawyer tag. It was not that he was a bad lawyer or even a mediocre one. But he was invariably compared to his father, and his father cast a long shadow.\n\nPeruto\u2014Chuck Jr. (he detests \"Chuckie\")\u2014had read enough in the newspapers in the last few days to know that agreeing to represent Gary Heidnik would be as sensible as trying to juggle vials of nitroglycerine. The word in the legal community was that his father had advised him to keep away. But Chuck Jr. was curious.\n\n\"He says he got your name from the guy in the next cell,\" Fognano explained, mentioning a name to Peruto.\n\nOf course, Peruto thought, so simple. A few weeks before, he had defended the cellmate on a murder charge and won a verdict of not guilty by reason of insanity. That explained how but not why. Anyway, even if Gary Heidnik did want him, he was not sure he wanted Gary Heidnik. He wasn't sure at all.\n\n\"You go see him,\" Peruto told Fognano after thinking it over. \"You see if he's really the one who made the call and if you think he's serious. If that works out, I'll come see him as soon as I get back to Philadelphia.\" He was hooked, but he didn't want to admit it.\n\nTwo days later Peruto met with Heidnik in a common area in the center of a cell block at the Philadelphia Detention Center. When he walked in, Heidnik was waiting for him, clad in a standard orange prison jumpsuit. When he saw Peruto, Heidnik bounded to his feet and flipped a salute. \"Mr. Peruto, I presume,\" he said.\n\nHoly shit, thought Peruto, looking straight into the icy eyes of Gary Heidnik.\n\n\"Talk about weird experiences,\" Peruto recalled later. \"Here we were, sitting in the middle of all these other cells, and the guys in there kept calling out to me, 'Hey, Chuck, come talk to me when you're through.' Apparently that impressed Heidnik. After a few minutes he said quietly, 'Boy, you're pretty popular around here.'\"\n\nThe interview lasted for more than two hours. During that time, Heidnik sat quietly at the table that was bolted to the floor in the middle of the room and, in a soft, matter-of-fact voice, told the lawyer the story of his arrest and the events that led up to it. Peruto couldn't believe what he was hearing; it was just too grotesque. The whole time Heidnik was talking, Peruto grew increasingly disgusted. It was probably the worst story he'd ever heard. Finally, when Heidnik got to the part about how he had cooked Lindsay's body, Peruto couldn't handle it any longer. He responded in the way that is his second nature: he let his wise-guy persona take control.\n\n\"That really got to me,\" Peruto said. \"I couldn't help myself. I had to ask a question.\"\n\nLeaning over, Peruto whispered confidentially, \"Tell me, Gary, what kind of seasoning did you use?\"\n\nThere was a long silence. \"He just stared at me,\" Peruto said. \"Then he answered, very distinctly, 'Man, you're crazy.' The way he said that made the hair on my neck stand up. I thought, 'Oh Chuck, you've gotten yourself into it this time. This is going to be a real experience.'\"\n\n* * *\n\nAn attorney who makes his living defending criminals cannot always be particular about his clients. He figures he is automatically going to get a high percentage of riffraff no matter how picky he tries to be. Referrals\u2014and getting their names in the newspapers\u2014are what keep defense attorneys in business. Given the nature of the profession, the jailhouse grapevine is a crackerjack advertising medium. It doesn't always bring in the cr\u00e8me de la cr\u00e8me, but then again, the percentage of cr\u00e8me de la cr\u00e8me that gets charged with gruesome crimes is relatively low.\n\nDefense attorneys, at least those who do it day in and day out, get a lot of clients of the type most people wouldn't sit next to on a bus. But there are limits, and Peruto was not sure Heidnik didn't exceed the parameters. Defending Heidnik would get his name in the papers for sure, but he was not altogether convinced he needed the publicity that much. The crimes Heidnik was accused of committing were downright repulsive. Just talking about them made most people queasy. But defense attorneys, in that sense, are not ordinary people. Where others see sleaze, a good defense attorney sees challenge. When others think of senseless violence, defense attorneys think of motivation. What made the accused commit that violent act often is more important in the courtroom than the violence itself. There is always a reason, even if the reason is not readily apparent; even if the reason is insanity. The defense attorney's task is to make a jury see the mitigating circumstances, make them understand that their client had a basis for doing what he did.\n\nIf a jury can see logic behind behavior, even twisted logic, it becomes comprehensible. When others envision hopelessness, defense attorneys see hope. They also see dollar signs and headlines. But there are limits. No defense attorney builds a reputation by consistently standing up next to losers. No one is going to rush out to hire you if the last three or four people you defended were presented open-ended reservations at Graterford, Pennsylvania's main penitentiary. On the surface, Heidnik's case looked unwinnable. If ever there had been a born loser, Heidnik was one. If ever there had been a series of repulsive crimes, this was it. If ever there was a case in which the police had caught someone redhanded, it was Heidnik's. At that point neither Peruto nor anyone else would have said that Heidnik had even the slimmest of chances of escaping the death penalty.\n\nOn one hand, Peruto was intrigued; on the other, repelled. There are better ways to make a buck, he figured. Still...\n\nThe way to settle it for sure, he decided, the way to keep from having to make a moral and professional issue out of it, was to let economics be the first hurdle. Peruto figured he would set an outlandish fee, one so high that Heidnik would immediately back off.\n\n\"I'll defend you,\" Peruto told him, \"but it's going to cost you a hundred thousand dollars.\"\n\n\"Okay,\" Heidnik replied without hesitation.\n\n\"Jesus,\" Peruto said to himself.\n\n\"I wanted to say,\" he admitted later in an interview, \"'Gary, I'm not a hundred-thousand-dollar lawyer. I usually charge a flat ten thousand fee for a homicide.'\"\n\nBut then, this wasn't any ordinary homicide.\nChapter 26\n\nApril 23, 1987\n\nChuck Peruto didn't have to wait long to see just how unordinary it was. On this Thursday, as April was sliding into May, he made his first major appearance with his client. It was a traumatic experience.\n\nThe occasion was a preliminary hearing, a procedure fixed by law to formally determine if there is probable cause to hold an accused for trial. In Heidnik's case it was superfluous, but the law said it had to be done. To Peruto's consternation and Gallagher's delight, it was a showcase for the district attorney's office, because it gave the Commonwealth a chance to brag about what a solid case it had against Gary Heidnik. From start to finish it was strictly a prosecutorial event.\n\nThe formal charges against Peruto's client were murder, kidnapping, rape, aggravated assault, involuntary deviate sexual intercourse, plus indecent exposure, false imprisonment, unlawful restraint, simple assault, making terroristic threats, recklessly endangering another person, indecent assault, criminal solicitation, possession and abuse of a corpse. The latter charge was defined as a crime in which Heidnik \"unlawfully did treat a corpse in a way that he knew would outrage ordinary family sensibilities.\"\n\nWaiting in Room 675 in City Hall were seven witnesses: the four former captives, a sister of one of the murder victims, and two medical examiners for the city. Also present were Municipal Court Judge Charles Margiotti Jr.; the prosecutor, Assistant District Attorney Charles Gallagher; Peruto and, of course, Gary Heidnik.\n\nWith a nod from Judge Margiotti, the parade began.\n\nLisa Thomas was the third woman to join Heidnik's harem, but she was first on the stand. In a steady voice she told the silent crowd how she was picked up, how Heidnik enticed her with food and presents, how he had sex with her and then took her captive. Not a sound could be heard when she explained how Heidnik had reinforced his domination over her by forcing her to perform oral-genital sex in front of the others.\n\nThomas' tale was concise and unemotional, accurate yet withdrawn, a perfect warm-up for Josefina Rivera.\n\nIf Thomas was impressive, Rivera was spectacular. In a flat, detached voice which gave her narrative even more power, Rivera bombarded the crowd with a tale so horrible, so detailed, that even veteran court watchers were moved.\n\nIn the beginning her story paralleled Thomas', differing only in the fact that she had been the first captive. She also had been hoodwinked, she said, forcibly captured and horribly mistreated. But when she got to the part of how Lindsay died, the story took a direction of its own. Truth, indeed, had proved stranger than fiction.\n\nWhen Lindsay had collapsed that second time and Heidnik came down the stairs, she said she could feel his anger. \"He lifted her left wrist and let it fall.\" There was no response. He uncuffed her and she fell in a heap. \"Then he kicked her into the hole and felt for her pulse. When there wasn't one, he said she had choked on a piece of bread.\"\n\nFive weeks after Sandra Lindsay's death, Rivera said, it was Deborah Dudley's turn. On March 18 Heidnik was punishing Thomas, Dudley, and Jacquelyn Askins. He decided to give them electric shock.\n\nHeidnik ruled the women with an iron hand, she said, and he expected to be obeyed. When he told her to help punish the other three, she never even considered telling him no.\n\nWhile Heidnik watched and chuckled, Rivera dragged a garden hose over to a hole in which the captives were kept and filled it with water. Thomas, Dudley, and Askins were already in the pit. When the hole was full, Heidnik plugged a white extension cord into an outlet across the room. Then he plugged a second cord into that. At his direction, Rivera touched the wires onto Dudley's chain. All three of them screamed. They then grew silent. Heidnik thought the cord had burned up, so he dashed upstairs to get another one. Askins yelled that Dudley was dead.\n\n\"There's nothing wrong with Debbie,\" said Heidnik. But when he walked over and peered into the pit, he saw Dudley was limp and her face was in the water.\n\nRivera told how Heidnik had forced her to write a confession admitting to Dudley's death. \"'That's so you will be guilty too,'\" she said he told her.\n\n* * *\n\nRivera's testimony that it was she who had held the electrical cords when Dudley died was not lost on Peruto. When his turn for cross-examination came, he probed Thomas about Rivera's participation, not only in Dudley's electrocution but in other torture sessions conducted in the basement.\n\nRivera frequently beat the other captives, Thomas admitted, both when Heidnik was there and when he wasn't. \"She enjoyed it.\" She was the boss of the basement, Thomas said. \"She would beat us on our bottoms and on our legs with a stick, laughing the whole time.\"\n\nAccording to Thomas, it also was Rivera who gave Heidnik the idea of using electric shock.\n\nThomas' assertions, however, were contradicted by Askins, who swore that Rivera never beat them unless Heidnik ordered her to.\n\nNevertheless, Askins added, Rivera had been a snitch. She said that at one point all of them agreed they would try to overpower Heidnik and escape. But before they could execute their plan, Rivera tipped off Heidnik. As a result, all of them except Rivera were punished severely. Rivera, on the other hand, was rewarded. \"[She]and Gary would go out and then come back and tell us what a good time they had. And then they would laugh about it,\" said Askins.\n\nWhen asked what possible motive Heidnik could have had in this incredible escapade, all of the victims agreed: he had his mind set on getting them all pregnant and raising the babies in the cellar.\n\n\"He wanted ten girls and as many babies as possible before he died,\" said Askins.\n\n\"He wanted me to get pregnant because he would like to have a lot of kids running around in the basement,\" confirmed Thomas.\n\nSince the fourth captive, Agnes Adams, was only in the basement for one night, there was nothing substantial she could add to the testimony of the other three.\n\nBut it had been a wrenching day, an emotionally draining experience. By the time the two witnesses from the medical examiner's office took the stand, their appearance was almost anticlimatic.\n\nDr. Robert L. Catherman's testimony was exceedingly brief. Basically, he confirmed that Dudley had died of electrical shock. But his colleague, Dr. Paul Hoyer, was considerably more forthcoming. It was Hoyer who had examined the body parts found in Heidnik's freezer.\n\nPainstakingly, ADA Gallagher led Hoyer through the steps that led to the discovery, beginning with his arrival at the house on North Marshall Street.\n\n\"Did you observe anything peculiar in the kitchen?\" Gallagher asked.\n\n\"Yes, I did,\" responded Hoyer.\n\n\"And what was that, Doctor?\"\n\n\"Two items. First of all, there was a white refrigerator in the kitchen. And located in the freezer compartment of the refrigerator were several white... bagged parcels, as you would perhaps wrap up a piece of meat to keep it in the freezer, double wrapped in a white plastic bag. One of these bags had been previously opened. And I looked inside this bag and I believe I found two human forearms.\"\n\n\"And what did you do? Were there other items wrapped similarly in the refrigerator?\"\n\n\"There were other items wrapped similarly in the refrigerator,\" Hoyer confirmed. \"I removed an additional parcel, opened that parcel, and found what appeared to be a human upper arm.\"\n\n\"And were there any other items found in the refrigerator?\" Gallagher queried.\n\n\"Yes. There were other parcels found in the refrigerator, which were subsequently brought to my office for examination.\"\n\nGallagher and Hoyer interrupted each other, both trying to talk at the same time. Then Hoyer resumed.\n\n\"... in sum, there were two forearms, one upper arm, two knees, and two segments of thigh. Each of these pieces had... the bone end had been cut apparently with a saw. And the skin and muscle, soft tissue, was still on the bone.\" Twenty-four pounds in all, it was determined.\n\nHad he examined the oven? Gallagher wanted to know. Certainly, said Hoyer. \"There were things in the oven that had been cooked and cooked and cooked for hours.\"\n\n* * *\n\nThe room was abuzz. If this crime was not the most gruesome in recent Philadelphia history, no one could think of a worse one. If Gallagher's purpose had been to build public opinion against Heidnik, he had certainly succeeded.\n\nThere was one problem, however, which became more evident in retrospect. It was that Gallagher's presentation may have been too expansive. All the Commonwealth had to do was present enough evidence to justify holding Heidnik for trial, but Gallagher went far beyond this. Some shrewd court watchers mumbled that Gallagher exposed too much of the Commonwealth's case. This gave the defense a good look at what evidence the prosecution had, plus it locked in the four witnesses to testimony they might give at the trial. They could elaborate on their testimony, but if they tried to deviate, it threw them open to impeachment.\n\n\"Josefina Rivera would have been quite sufficient,\" said one lawyer with a close eye on the case.\nChapter 27\n\nWhile most of the focus was on the criminal charges against Heidnik, a different drama was being played out in the city's civil courts. Eventually it would become an issue in the criminal case as well.\n\nAs soon as the captives learned how much money Heidnik had in the bank, they filed suit against their former captor, seeking to dip into the half-million dollars he was reported to have tucked away in an account at Merrill Lynch. But Heidnik contended the money did not belong to him but to his church, the United Church of the Ministers of God.\n\nNot everyone agreed with this distinction. Some claimed the church was just a front, that Heidnik was the church and its money was really his money. Before this issue could be settled, however, it had to be determined which court had jurisdiction, state or federal. That would be the first major battle to determine the legitimacy of Heidnik's church and the distribution of funds.\n\nOn April 6, four days after Heidnik tried to hang himself, attorneys for Lisa Thomas filed a petition in the civil division of Common Pleas Court asking that the church's assets be frozen and a conservator appointed. Judge Samuel M. Lehrer immediately complied with the first part of the request. But he waited fourteen days, until April 20, to name a conservator.\n\nOne hour and forty-five minutes before Lehrer got around to announcing his choice as conservator, Heidnik filed a Chapter Eleven bankruptcy petition in federal court. That document listed three of the surviving captives plus the estates of the two murdered captives as creditors. All told, the document noted, they were seeking several million dollars in damages. An incidental creditor, interestingly enough, was the City of Philadelphia, asking $6,800 in back child-support payments.\n\nSignificantly, the filing of bankruptcy set the stage for the battle over jurisdiction. Judge Lehrer, of the Common Pleas Court, was determined not to surrender. In an April 27 ruling he ordered his newly-named conservator to seize the church's assets, including the half-million-dollar Merrill Lynch account, plus a local bank account containing $16,000, a 1987 Cadillac valued at $34,000, and a 1971 Rolls-Royce.\n\nHe also used that occasion to launch a bitter attack against the organization, labeling it a \"sham\" and a church that \"exists in name only.\" Angrily, Lehrer made it clear he thought the church was a fabrication devised by Heidnik solely to evade taxes. \"Do you have a church program?\" he demanded. \"A schedule of Sunday services or Saturday services? Do you have a membership list? Books? Pamphlets? Prayer guides?\" These were rhetorical questions; he never waited for answers. His indignation was barreling along like a sixteen-wheeler without brakes.\n\n\"I find that it is in the public interest that such an entity be found to be what it is,\" Lehrer raged. \"The mere thought that this is a church is offensive and an insult to all persons in the community. Further, it is also offensive and an insult to the sensibilities of the victims and their families... that such crimes could be perpetrated... in the name and the guise of something called a church.\"\n\nLehrer's position was not unique. At first even Peruto was ready to believe the church was simply a con. \"But after I took the case, people started calling me saying they had been members and they wanted to know if someone was still holding services. I was amazed. The more I found out about it, the more I was convinced it wasn't a tax dodge.\"\n\nDr. Clancy McKenzie, a psychiatrist who spent more than a hundred hours with Heidnik after his arrest, also vehemently denied accusations that it was a mock operation.\n\n* * *\n\nWhile some were terribly uncomfortable in thinking of the church as a church, there also were some who were equally uncomfortable with Lehrer's presumption of jurisdiction. The day following Lehrer's ruling, Tuesday, April 28, U.S. District Judge David A. Scholl held a hearing to determine if the case should be in Lehrer's province at all or in the federal courts. Three weeks later, on May 19, he handed down his decision. He said he\u2014not Judge Lehrer\u2014had control.\n\nIn a twenty-page decision, Scholl said the jurisdiction shifted when the Chapter Eleven petition was filed. At that point the federal bankruptcy court became \"all-powerful and exempt from compromise.\"\n\nAlthough he thought Judge Lehrer's anger at \"what he believed was the sacrilegious hypocrisy of cloaking the financial support of [Heidnik's] activities in the name of a church strictly for the purposes of tax evasion\" was \"totally appropriate,\" he nevertheless slammed Lehrer's conclusion that the church was a sham. In his words, the state court judge's decision was \"legally void.\"\n\nJudge Scholl also appointed two trustees, one for Heidnik and one for the church, to administer the estate.\n\nThose proceedings had one very important immediate effect on the criminal case. When the church's assets were frozen, Peruto saw his promised $100,000 fee go down the tube. To his credit, he refused to submit a bill for time spent and walk away. Soon afterwards he was formally appointed Heidnik's lawyer, so he picked up exactly where he left off. Except this time it was at the rather unsensational fee of sixty dollars an hour. Despite the pay cut, he resolved to stick with his client.\nChapter 28\n\nMay 14, 1987\n\nFive days before Judge Scholl's ruling in federal court, Heidnik was called into Common Pleas Court for still another proceeding, called a status hearing. It would be his final court appearance until jury selection got under way.\n\nHeidnik hobbled into the courtroom and came face to face with Judge Charles L. Durham, a suave black man who also served as the court's calendar judge. With his ankles chained together and his hands cuffed in front of him with a chain running to a thick leather belt, Heidnik looked like Houdini being prepared for an escape trick. As a sheriff's department officer reached over to uncuff Heidnik, the officer was overcome by the drama of the situation; his hands shook so badly he couldn't fit the tiny key into the lock. Heidnik watched the white-shirted officer's performance with interest. As the officer fumbled around, Heidnik turned to Peruto and softly whispered, \"Something's wrong with this man.\" It was the longest sentence he would utter during the entire proceeding, and the only one that indicated he knew what was going on.\n\nPeruto explained to a grim-faced Durham that his client would not be testifying.\n\nThat was all right, Durham replied, all he wanted to do was ask Heidnik a few easy questions to make sure he understood where he was and what was happening to him.\n\nHeidnik, his hands now free, ripped off a military salute to the judge. Standing at a loose parade rest, he cast his eyes down, intently eyeing the floor. Occasionally his head twitched.\n\nJudge Durham looked him up and down. \"Mr. Heidnik!\" he called.\n\nHeidnik popped to attention and saluted again.\n\n\"Mr. Heidnik\u2014\" Durham began.\n\n\"Yes, sir,\" Heidnik snapped.\n\n\"Do you know where you are?\"\n\n\"Yes, sir.\"\n\n\"How old are you?\"\n\n\"Yes, sir.\"\n\n\"Do you speak English?\"\n\n\"Yes, sir.\"\n\n\"Are you going to the moon today?\"\n\n\"Don't answer that,\" Peruto whispered in his ear.\n\nHeidnik remained silent.\n\nWhen asked if he wanted to plead guilty or not guilty, Peruto answered for him: \"Not guilty, your honor.\"\n\nFrustrated, Durham accepted his plea and moved to the next case.\n\nLater, in a brief hallway news conference, Peruto exonerated Heidnik for not speaking, saying he had been under his command to keep quiet. He planned to plead his client insane, and he did not want a prosecution psychiatrist to be able to use anything he might say in court later as evidence to damage his case.\n\n* * *\n\nAfterwards, Peruto got down to more mundane tasks: the blizzard of paperwork that surrounds cases such as this.\n\nOn May 22, eight days after the status hearing\u2014just long enough for all the salacious tidbits to be displayed and debated in the media\u2014Peruto requested a change of venue. Media coverage had been \"highly prejudicial and inflammatory and in deprivation of the defendant's rights to a fair and impartial trial and his rights to due process,\" he claimed. \"The details, events, and facts surrounding the matters relating to this case are now generally so well known to the citizens and inhabitants of this county that a fair and unbiased jury would be impossible to select.\"\n\nHe also filed several other motions, dealing with the search of Heidnik's house and the seizure of his property, but it was pretty standard legal maneuvering that would have been expected in almost any significant criminal case. The most important issue was the request for a change of venue. Wisely, Judge Durham decided not to rule on that until a trial date had been set. No one knew it then, but that was even more distant than the most pessimistic observers believed. After all, Peruto had only started to romp through his repertoire of motions.\nChapter 29\n\nApril 4, 1988\n\nFor everyone, except possibly Gary Heidnik, the previous eleven months had gone by in a blur. The case was out of the headlines and off the six o'clock news. Time and the vagaries of the news business had seen to that.\n\nWhen the details of Heidnik's crimes were made public, hard-nosed cops and toughened murder buffs said they could not remember incidents as grisly as those attributed to the man from North Marshall Street. Then, in less than five months, they had another case that almost overshadowed Heidnik's.\n\nOn August 9, 1987, a steamy Sunday, police got a call from an angry resident of a disreputable northside ghetto who said blood was dripping into his apartment from the cubicle upstairs. Probably nothing less would have prompted such a call. The building involved was so loathsome that its occupants, lacking even running water and indoor plumbing, defecated in buckets and dumped the contents in the trash-filled backyard.\n\nWhen police arrived, they were almost overcome by the smell. Donning gas masks, they forced their way into a back bedroom whose door had been nailed shut. In the room were remains of five women. The body of a sixth woman was found wrapped in sheets inside a closet in the room, which also had been nailed shut. Still one more body was found, part of it on the roof and part in the basement of a neighboring building. A few days later a retarded twenty-nine-year-old black man built like a tight end surrendered. Harrison \"Marty\" Graham said he began killing the women the previous winter, the same time Heidnik was holding his captives less than two miles away. Graham said he lured the women with offers of drugs, then strangled them during intercourse.\n\nHe also confessed to being a necrophiliac. Until he could no longer stand the odor, he said, he would periodically return to the bodies for sex.\n\n* * *\n\nThe details of Graham's revolting escapades made Philadelphians temporarily forget about Heidnik, who had been left to vegetate in a psychiatric ward while Peruto and Gallagher built their files for the trial. Officially, the proceeding was to start on the first Monday in April. Of course, it did not start then. Nor would it for many weeks, as a matter of fact. But everyone had to go through the motions. The entire cast, minus only Heidnik, that had been in Judge Durham's courtroom the previous May, was back in Room 613 on April 4.\n\nPeruto and Gallagher stood shoulder to shoulder in the small chamber, waiting for Durham to formally assign the case to a judge who would preside at the trial. Except for the fact that both men were roughly the same height\u2014about five feet eight\u2014the two lawyers could hardly have been more different.\n\nPeruto was a dandy with a taste for custom-made suits and shirts, gold cuff links, and fancy shoes. He wore a beard, which was short and immaculately groomed. His hair was styled just a tad longer than the current fashion and frozen with hair spray. He had a weightlifter's torso and a subtle arrogance of bearing. In profile he resembled Burt Reynolds. Like the movie star, he was not unaware of his physicality and the image he presented to admirers\u2014the younger and the more shapely, the better\u2014who flocked to the courtroom to see him in action.\n\nGallagher, in contrast, was as unpretentious as a Naugahyde sofa. He dressed in off-the-rack suits in gray, grayer, and black, with pure white shirts and red foulard ties. His shoes were drill-field practical. If he was growing a little thick around the middle, it didn't seem to bother him; he simply substituted suspenders for a confining belt. He wore his hair cut short, in the military style, with his graying sideburns chopped off at mid-ear. He wouldn't have known a can of hair spray if it fell in his lap. His rimless spectacles could have been Navy issue, the frames stamped with a bold USN rather than YSL. He looked like a chubby Ollie North.\n\nDurham, wearing the hat of calendar judge, wasted no time. Trying the Heidnik case, he announced, would be Lynne M. Abraham. It would be up to her, he added, to set a date for jury selection and to rule on all future motions, including Peruto's request for change of venue and a request he had just filed to force the prosecution's psychiatrist, Dr. Robert Sadoff, to provide a list of the cases in which he had testified.\n\nSuddenly, the hearing was finished. It took less than ten minutes. The reporters who crowded the room went away without much to report, but with some research to do. They all scurried to their computers to search their electronic libraries to see what they could find on the new character in the continuing drama: Lynne Abraham. What was the story on this female judge?\nChapter 30\n\nThe story on Lynne Abraham was that she was, by her own admission, one \"tough cookie.\" A heavyset, slightly masculine woman who wore her salt-and-pepper hair cut short and brushed back over her ears, she was a formidable veteran of both the Philadelphia court system and Philadelphia politics.\n\nWhen she graduated from Temple University Law School a generation ago, there were not many lawyer jobs available for women in public-service agencies, so she went to work with the U.S. Department of Housing and Urban Development. Secretly, she was searching for a way to get into the courtroom.\n\nShe devoted her days to HUD and her nights to graduate classes at Temple. Soon she became friendly with one of her instructors, who happened to be District Attorney Arlen Specter's top deputy. The man introduced Abraham to Specter, and he offered her a job. Her main reservation had been that she would be assigned to juvenile court, but she negotiated an agreement with her new employer to work magistrate's court instead. That would do, she figured, until she could find a way to get to where she really wanted to be: the homicide division, prosecuting cases in Common Pleas Court. It didn't take her long.\n\nOnce she got there, she quickly won the respect of the other ADAs and the judges for her knowledge of the law and her aggressiveness. Her acceptance among defense attorneys was more guarded because she was hard to beat. She was a perfectionist, and the job became a passion. Take forensics, for example. She became preoccupied with the basics of the cases she was prosecuting, and decided the best way to get a thorough grounding in her cases was to start at the beginning. Much to the surprise of the city's medical examiners, she began turning up at autopsies. It helped her, she said, when she got into court. Knowing exactly how a victim had died\u2014and why\u2014gave her an edge when she asked a jury to come down heavy on the murderer. She showed no mercy to defense attorneys or their clients. She seldom compromised; \"plea bargain\" was a phrase that made her stomach turn.\n\nIn quick order she became known as an obdurate, relentless prosecutor.\n\nHer name came to the attention of Mayor Frank L. Rizzo, who was having personnel problems in his administration and was looking for someone from the outside to come in and clean up the mess. He offered her a job as head of the Redevelopment Authority. It was Rizzo who hung the \"tough cookie\" nickname on her and bragged that if anybody could erase the corruption in the agency, she could. Five years after taking her first case, Abraham left the district attorney's office. She would never step into a courtroom again except as judge.\n\nThe honeymoon with Rizzo, however, was brief. Within a year the two were disagreeing vehemently. After fifteen months he fired her, claiming developers didn't want to work with her because she was too arrogant. She said he wanted to get rid of her because she wouldn't hire his buddies or agree to turn a blind eye to procedures involving certain contracts.\n\nWiser for her initial foray into politics, she looked for another way back into the courtroom. In 1975 she was elected the city's first female Municipal Court judge. Four years later she ran for a job with more authority, a seat on the bench in the Common Pleas Court. During that campaign she took a page from Mayor Rizzo's book; she won public support by standing on street corners handing out \"tough cookies.\"\n\nAlthough she was back in the courtroom, politics was still pulling her. In 1985, at the age of forty-four, she feinted toward a run for district attorney but backed out when support she thought she was going to get from Mayor W. Wilson Goode dissolved. In that case the Democratic black power structure decided a black man would make a better candidate than a white woman. But the man they put up was trounced by Republican Ronald Castille, a hard-charging former Marine who left a leg in Vietnam.\n\nThe forces that drove Abraham to be an absorbed, unyielding prosecutor and a potential candidate for political office became apparent when she took over the Heidnik trial. Her instincts as an inflexible ADA were still there, coupled with a tendency to be pedantic. During jury selection particularly she drove the defense attorney, Peruto, to distraction, and though he would never admit it, perhaps Gallagher as well, by repeatedly interrupting their questioning of potential jurors, called \"veniremen\" in legal parlance. \"Let me rephrase that,\" was one of her favorite lines, a phrase she repeated whenever she thought one of the attorneys had not been explicit enough in his grilling. \"Let me put that another way,\" was another of her favorites. Or, fixing the venireman with a steely gaze that was sure to intimidate, she would say, just after one of the lawyers finished a query, \"Don't answer that! This is what Mr. Peruto [or Mr. Gallagher] meant.\"\n\nIt wasn't so much that her interpretations were inaccurate, it was just that they seemed to be superfluous, almost always wordy and theatrical, and at times even condescending. In many cases the potential jurors appeared to understand all too well what the attorney was asking but of course, were in no position to point this out to Judge Abraham.\n\nWhen Abraham chose to, she exhibited a robust sense of humor; she could also send everyone running for cover, particularly members of the media to whom she devoted special attention. It often seemed she held reporters in contempt, though at other times she seemed bent on making a good impression. Perhaps she doesn't like the press (although her husband is a local radio personality), but she is certainly shrewd enough to recognize its usefulness, particularly if she has her eye on something higher than a seat on the Common Pleas bench. The word among insiders at the time of the Heidnik trial was that she had written off the possibility of seeking the district attorney's job and was directing her efforts instead to a possible campaign for a spot on the state supreme court. If that were true, what better launch vehicle than a trial guaranteed to take an exceptional share of the headlines? She would not be the first jurist whose career was helped by presiding over a sensational case. Nor would she be the last.\nChapter 31\n\nMay 16, 1988\n\nGary Heidnik was not only out of mind, he was out of sight as well. Except for his lawyer, a few shrinks and prison personnel, no one had seen him for a year. But that quickly changed. On a cool, overcast Monday in the middle of May, twelve months and two days after his last public appearance, he sprang\u2014or rather, he shuffled\u2014back into public view. When the heavily bolted door in Judge Abraham's courtroom swung open, a dozen or so reporters and a handful of spectators strained to catch a glimpse of the man who, despite Marty Graham, still held a lock on grisly crime in Philadelphia.\n\nAs they watched in open-mouthed amazement, a wraithlike Gary Heidnik skated into the room, slowly sliding his feet across the floor. He moved like a cross-country skier who had unexpectedly run out of snow. His scuffed, ankle-high black boots may have been made of lead or attached by a powerful force to the room's stained, green carpet. Left... right... left... right, slowly he lurched along. Reporters looked at one another and raised their eyebrows. \"Is this Gary Heidnik?\"\n\nIn the intervening months Heidnik's reputation as a crazed woman killer had grown. The Madman of Marshall Street had taken on monstrous proportions. He had become the infamous Gary Heidnik. In the myth he was seven feet tall; brawny; defiant; scornful; a walking stick of dynamite. But the reality was something else. What the world was expecting was not what it got.\n\nThe old Gary Heidnik had a reputation in his neighborhood as something of a neat dresser, even if he did have a thing for cowhide. In the past one of his favorite articles of clothing was a sleeveless black-leather vest. And he absolutely loved the fringed buckskin jacket he was wearing when he was carted off to jail. In Philadelphia you saw a lot of Gary in that jacket because the local TV stations liked to show those film clips every time there was a development in the story. The deputies who guarded him, however, hoped they never saw that jacket again. They complained it was indelibly saturated with the stench of burning flesh.\n\nIt was said on North Marshall Street that Heidnik always dressed tidily, even if he was only going into the garage to work on his cars. Apparently, incarceration had influenced him to turn over a new sartorial leaf. When he stumbled out of the holding cell passageway behind the courtroom for the beginning of jury selection\u2014a process known in the trade as voir dire; in Latin, \"to say the truth\"\u2014he certainly gave no hint that he ever took pride in his attire. He was an emaciated wretch in a clown costume.\n\nHis khaki pants were clean and neatly pressed, but they were at least two sizes too large. They bunched at the waist, bagged in the rear, flared in the thigh and dragged the floor. He wore a dark blue, wool, navy watchcoat. It was fully buttoned, but it was not known if that was in deference to the weather or because it was his established psychiatric pattern to overdress. Beneath the coat, his shirt of choice was the same faded-black Hawaiian number with the purple orchids and bright green leaves that he had worn the year before at his arraignment. Maybe it had been washed in the interim. Maybe it hadn't. Later, during the trial, Peruto had to get a court order to make him change his shirt and take a shower.\n\nIn fairness, his wardrobe selection was limited. All of his own garments, including his treasured buckskin jacket, were still being held by police as evidence. The man in charge of the jail's used-clothing room had become his haberdasher. Obviously they weren't good friends.\n\nBut there was more to Heidnik's mildly outlandish garb than that. Peruto wanted him to look scruffy. Like a street person. Like a nut. If you're a lawyer building an insanity defense, you don't want your client coming to court looking like he just stopped off at Brooks Brothers for a trial suit. Peruto helped pick the clothing. He told Heidnik to shuffle. To look vacant and disconnected. To salute him each morning. It was part of the game. There was more to being a defense lawyer, Peruto insisted, than appearing in court and arguing motions.\n\nOf course, Peruto got more than a little help from prison doctors, who kept Heidnik under control with three hundred milligrams a day of Thorazine.\n\nOne thing that was not faked, though, was Heidnik's physical condition. It was deplorable. Anyone watching the film clips of his arrest saw a not unhandsome man. That Heidnik had a short, neatly groomed beard. A barbered coif. A straight nose and a well-filled form. Generally a healthy-looking specimen. This Heidnik was about thirty pounds lighter. His neck had grown long and scrawny. His arms were matchsticks. His handcuff-encircled wrists were bony and had a bluish cast. But the big difference was the hair. This Heidnik had not cut his hair for fourteen months, and it hung in long, greasy tangles to his shoulders. Obviously he had not shampooed it in days, perhaps weeks. His beard, streaked with gray, tumbled untrimmed and uncombed to his chest. His nose was humped and bent off center, a souvenir of an attack by other inmates the day of his arrest. In short, he looked like a wreck. Just like Peruto wanted him to.\n\nOne thing remained unchanged, however: his eyes. Blue-gray aggies as cold as a Philadelphia winter, they were still wild, still angry. \"Creepy eyes,\" his lawyer called them. Charlie Manson eyes. A fanatic's eyes.\n\nHeidnik's appearance had an unreal feel to it. A spooky feel. But it fit the venue. Gary Heidnik might not be your usual, everyday accused murderer, but then the building in which his trial was being held, Philadelphia's City Hall, was not your usual, everyday brick-and-glass county courthouse.\nChapter 32\n\nCity Hall. Style. History. Nobility. Degeneration. Squalor. Take your pick; they all apply.\n\nPatterned after the New Louvre, in the style known as French-Victorian, the building was erected in the 1880s in what was then the city's central square. Today it sits all alone at the center of a huge traffic circle, smack in the middle of the city's business district, rising out of a sea of cars like a man-made Gibraltar. The first seven floors are constructed in a huge block, which is divided into work space. Atop the block is a graceful tower capped by a statue of William Penn, the city's founder and the state's namesake. From the ground to the top of Billy Penn's hat it is 491 feet, forty stories give or take. Until 1987, when a sixty-one story skyscraper known as One Liberty Place was erected nearby, it was the city's tallest building. Although bested in height, it has lost none of its majesty.\n\nIt is a pity that the building's grandeur is only skin deep. Inside, the building is no longer so grand at all. The corridors are as wide as some of the surrounding old streets, and just about as dingy. The linoleum floors are cracked and pocked by cigarette burns; permanently stained by mysterious, corrosive liquids. Discarded food wrappers, crumbled coffee cups, empty soda cans, and old newspapers pile up in the halls, which are home to criminals and police alike.\n\nOccasionally they are used for other things as well. One day just before the trial, when a persistent heat wave discouraged outdoor running, a jogger wearing a blue tank top and gray Reeboks determinedly did his laps around the sixth-floor rectangle. The circuit is about a third of a mile.\n\nThe corridors are lined with tall windows, a delight for those who appreciate good design. Unfortunately, they look almost exclusively inward, opening onto empty air-shafts and undeveloped courtyards strewn with rubbish and broken furniture.\n\nIf the corridors seem bad, the stairways are worse. Much worse. Although designed to angle elegantly around a broad open area to give a feeling of spaciousness, they have deteriorated into dark, evil-smelling passageways. When the early morning cleaning crews come in, the first thing they tackle are the stairwells, where they commonly find puddles of urine and piles of excrement.\n\nCity Hall's deterioration is not entirely the result of a lack of concern. The building simply is overused and underfunded. It is a working building, not a museum. There are courtrooms for eighty judges in the building. The criminal division of the Common Pleas Court, the largest user, has almost fifty judges and hundreds of attendants: law clerks, calendar clerks, minute clerks, recorders, secretaries, receptionists, telephone answerers, bailiffs, criers, stenographers, court reporters, and sergeants-at-arms. Each year some thirteen thousand criminal proceedings are held in the building, including about three hundred fifty murder trials.\n\nEach day some two hundred fifty prisoners pass through City Hall's southwest portal, the traditional depository and pickup point. Their transportation isn't fancy: a large brown or blue school bus with heavy wire over the windows and an interior partitioned into a large cage. But Heidnik got special treatment. As a potential suicide, he almost always traveled alone, either in a van or station wagon. However, that special treatment ended at the portal. Once there, he marched to an elevator like all the prisoners and got an express to the seventh floor, where the holding cells are located.\n\nHeidnik was usually mixed in with other prisoners in one of the fifty-man cells. Commonly, he wiggled his way into a corner and either read from the pocket-sized New Testament he carried in a smudged manila envelope or slept until it was time for him to be taken downstairs to Room 653, Judge Abraham's bailiwick.\n\n* * *\n\nRoom 653 is somewhat infamous around the courthouse; it has its own little spot in the building's history books. What makes it unusual is an event that happened there on August 18, 1986. Judge Lynne Abraham was on the bench, and in the defendant's chair, instead of Gary Heidnik, was a thirty-nine-year-old unemployed construction worker named Robert McPeake, charged, as was Heidnik, with rape. For five days a jury had been listening to testimony in his case. McPeake had an earlier conviction for the same crime and knew if he was found guilty again, he could expect no mercy at sentencing. When the jury came back, the verdict was against him. Exercising his option, McPeake's attorney asked that the jurors be polled. One by one they stood and individually confirmed that they had judged the defendant guilty. McPeake, in the meantime, sat stiffly at the defense table, shooting nervous glances around the room. Suddenly, just as the penultimate juror rose to authenticate the decision, McPeake leaped from his chair and dashed across the room. Raising his hands over his head like a diver on the high board, he leaped out a window, plummeting six floors to his death. After the courtroom had been cleared, a still-shocked Abraham shook her head. \"I've been in court for a lot of years,\" she said sadly, \"but nothing could have prepared me for this.\"\n\n* * *\n\nMcPeake taught Abraham a lesson. When it came time for Gary Heidnik to make his appearance, the judge wasn't taking any chances. When he materialized, his hands were cuffed securely behind him. At each shoulder was a burly deputy. In front of the window through which McPeake exited was an oscillating fan on a stand. Since it was never turned on, it was assumed that its function was to serve as a barrier in case anyone wanted to do a repeat-McPeake.\n\nIt was not outlandish to assume that Heidnik might harbor such thoughts. According to medical records, he was no stranger to suicide attempts. This time, though, there seemed little immediate cause for worry. As he hobbled across the room it was evident he didn't have the strength to sprint the fifteen yards to the windows, even if it had not been for the heavy-duty escort.\n\nWhen he got to his spot at the defense table, the cuffs were removed and he slid quietly into the straight-backed wooden chair on Peruto's left. He stiffened his spine and sat ramrod straight, just as he had been taught in military school decades before. He stared vacantly at the back wall and outwardly showed little concern with what was going on around him. He might have been in a trance; indeed, he had been diagnosed in the past as catatonic. But as the proceedings progressed, he became more animated. By day two he had dropped his pretense of noninvolvement and was carefully following the action. By day three he had developed a little spring in his step, which was quickly corrected by Peruto. \"Shuffle when you walk,\" he reminded him. \"Don't lift your feet.\" By day four he was laughing and joking with his lawyer.\n\nBut he continued to refuse to change his shirt, wash his hair, or bathe. As the days dragged by, Peruto began having second thoughts about the cleverness of his instructions to his client. \"When I told him not to shower, I didn't mean forever,\" Peruto moaned. \"A few days of that and he was getting pretty ripe.\" Eventually, midway through the trial, he begged Judge Abraham to sign a court order forcing Heidnik to take a shower. As his trial moved into its sixth day, Heidnik surprised everyone, even Peruto, by showing up in a rumpled but clean pale-blue, long-sleeved shirt. The next day, though, it was back to the Hawaiian number.\nChapter 33\n\nIn sharp contrast to his client's shabbiness, the thirty-three-year-old Peruto was the peacock of Room 653. On the opening day of jury selection he was decked out in a dark suit, burgundy tie on a red-and-white striped shirt with a white collar. His dark beard was trimmed and neatly brushed; his hair was blow-dried and cemented in place. He could have been a model in GQ. While waiting for Judge Abraham to appear, he paced nervously around the defense table, occasionally shaking his head. Not a single hair quivered.\n\nOn the other side of the narrow aisle was Charles Gallagher, the intent, zipper-lipped prosecutor looking for Heidnik's head on a pike. Since the last time Gallagher was in court on the Heidnik case, he had been promoted from an assistant to a deputy district attorney. If Peruto looked as if he stepped out of a men's fashion magazine, Gallagher wore a slightly rumpled dark suit that looked as though he had dug it out of his seabag a half hour before court and didn't have time to have it properly pressed.\n\nPeruto was flamboyant, a wiseacre, a shoot-from-the-lip lawyer who depended on style as much as substance to make his points with the jury. Gallagher was a plodder, an administrator whose expertise was appellate law. This was his first death-penalty case. He was not in Abraham's courtroom to make a fashion statement or play mind games with jurors. He was there to send a man off to the death house.\n\n* * *\n\nPeruto and Gallagher nervously rustled their papers; two fighters dancing in their corners waiting for the bell. Gary Heidnik had been in jail for some four hundred twenty days, and still his trial had not started. They didn't know it yet, but they were going to have to wait a little longer.\n\nIf it were up to Peruto, he would wait a lot longer. The more successful he could be in delaying the proceedings, the better. If he could postpone the trial another full year, he would be delighted. That would be just that much more time before he faced what all observers considered was the inevitable: Heidnik was almost certainly going to get the stiffest possible sentence. Anything else would be a minor miracle; a victory for Peruto and his client. But the chances of that, he knew, were slim to nonexistent. The momentum had begun. The time of reckoning was at hand.\n\nBy all appearances Heidnik was oblivious to this tension. He stared straight ahead, studiously following his lawyer's advice to look disinterested. He may actually have been disinterested. The former Army medic, former nurse, self-ordained bishop, church founder, satyr, recluse, and stock market manipulator par excellence, perhaps didn't really seem to care if he was shipped off to Rockview and strapped into the electric chair. It may have been some consolation to him, however, to know that although there were about ninety men on death row, Pennsylvania had not executed a criminal since 1962.\n\nThe fourth major player in the forthcoming fight, Judge Abraham, looked down through rimless glasses at the mismatched trio before her\u2014the spooky-looking defendant, the dapper defender, and the intense, solemn prosecutor\u2014and struck the gong.\n\n\"Let's go into my chambers,\" she said brusquely.\n\n* * *\n\nThe invitation did not extend to Heidnik, the press, or the public, only to Gallagher, Peruto, and the court reporter, who silently picked up his stenograph machine and followed the others offstage.\n\nFor the next hour Heidnik, a dozen or so spectators, and an almost equal number of reporters were left to their own devices. It was a huge room with a thirty-foot ceiling and a spectator section configured to seat a hundred fifty on straight-backed chairs set up in neat rows. The occupants were so few and the room so large, it seemed virtually empty. But that would change once the trial started; later the room would be jammed with a waiting line outside. On that day, though, Heidnik stared at the wall, alternately clasping his hands in front of him on the table or linking them lightly behind his back. He wasn't asleep, because his legs pumped slowly, methodically. Up. Down. Left. Right. Hour after hour, seldom changing pace; never stopping. Maybe he was nervous. More likely it was because of his medication. Doctors tell people who are taking Stelazine and Thorazine to \"pump\" their legs before standing to avoid dizziness that comes as a side effect.\n\nWithout preamble Peruto and Gallagher reappeared, followed by the court reporter toting his recorder. Abraham swept up the rear and went directly to the bench.\n\nThe back-room discussion, she explained, dealt with motions filed by Peruto. His request for a change of venue, that is, to have the trial moved to another city\u2014definitely not \"Nome, Alaska,\" as Peruto had mentioned earlier to reporters as his city of choice\u2014had been withdrawn. He had not, however, given up on his second major petition.\n\n\"If I grant Mr. Peruto's request for a change of venire,\" Abraham said, giving it the Latin pronunciation, vee-nigh-ray, \"that means the trial will still be held here but the jury will be selected somewhere else.\"\n\nIt was, she added, a motion she was reluctant to grant, at least not without first trying to pick a jury in Philadelphia. The basis for Peruto's request was that Philadelphians had been deluged with too much Garyiana. They had been told so much about the case for so long, Peruto contended, that they could no longer be objective.\n\nThe primary question, Abraham declared, was not whether prospective jurors had heard about the case, but whether what they had heard had caused them to form unshakable opinions about Heidnik's guilt.\n\nBefore beginning to question jurors, Abraham wanted to get a flavor for what the press, primarily TV, had been saying. With a nod from the judge, a deputy rolled in a dolly supporting a large television set and VCR. Stopping in front of the empty jury box, he swung the equipment so the screen faced Judge Abraham, the two lawyers, and Heidnik. Peruto stepped forward and inserted the first of a series of videotapes into the VCR. As the film reliving Heidnik's arrest and a series of stories that dealt with events that transpired on North Marshall Street was played again and again, a strange thing occurred. Heidnik, who had been studiously ignoring the proceedings up to that point, became more involved. As his image appeared repeatedly on screen\u2014here wearing his favorite leather jacket with the fringes on the sleeves, there being quick-marched into court for a preliminary hearing\u2014he abandoned his aloofness and swiveled his chair to get a better view. His face remained impassive, but his ice-blue eyes darted back and forth as the images flashed in front of him. Occasionally he leaned over and whispered in Peruto's left ear.\n\nWhen the show was over, Abraham ordered everyone home for the day. On the morrow, she promised, the quest would begin to try to find a dozen unbiased citizens. On Tuesday the first order of business would be the examination of a panel of sixty veniremen.\nChapter 34\n\nMay 17\u201318, 1988\n\nThey marched in single file, coming from an anteroom to the judge's right. Abraham had cleared the left side of the spectator section for them, and as they trooped in, they had to travel diagonally across the front of the room, each of them passing within a few feet of Heidnik. As they filed by, each of them gave him a searching glance then quickly looked away. He ignored them, staring unblinkingly at the rear wall. No one had told them what case they might be selected to hear, but by now Heidnik was recognizable to everyone in Philadelphia old enough to watch television, a fact that would be proved within a few minutes.\n\nAfter they were sworn, en masse, Judge Abraham took over.\n\nThis was the case of the Commonwealth versus Gary Heidnik, she explained. No one looked surprised. She paused, examining the group. It was a mixed bag, maybe a few more women than men, but on the whole, pretty evenly matched. They ranged in age from about twenty to about seventy. Roughly half of them were black.\n\n\"That's Gary Heidnik,\" she repeated for emphasis, \"not Marty Graham.\"\n\nGraham's trial ended two weeks earlier in a neighboring courtroom. After waiving a jury, Judge Robert A. Latrone found him guilty of killing seven women. Minutes after Graham was judged guilty, he was smiling, laughing, and signing autographs for sheriff's deputies. Although the Commonwealth had sought Graham's execution, Latrone, apparently in a moment of compassion, ignored the plea. Instead he handed down a complicated sentence that meant the feebleminded murderer, who at one crucial point in his trial had asked the judge to return his favorite hand puppet\u2014which he called his \"Cookie Monster\"\u2014would spend the rest of his life in prison.\n\nLatrone sentenced Graham to life for the first murder and death for the other six, plus seven misdemeanor sentences of one to two years each for abusing the corpses of his victims. The way the sentence read, Graham would first serve the misdemeanor sentences then begin the life term. If he ever sought parole on that sentence, the death order would kick in. It was, legal authorities agreed, a very unusual arrangement which was sure to be challenged by the Commonwealth on grounds that death sentences are supposed to take precedence over those specifying life imprisonment. But that was a battle for the higher courts, definitely not something for Judge Abraham to consider. She wasn't, in fact, the least bit worried about Graham or his legal entanglements. She just wanted to make sure the veniremen could separate the two cases in their own minds.\n\n\"Would those of you who have heard of the Heidnik case please stand,\" Abraham commanded. Not surprisingly, everyone did.\n\nFor the next twenty minutes she quizzed them as a group, winnowing out the ones who felt Heidnik was guilty and did not think they could change their minds. They would not be acceptable for the jury. Neither would those who were opposed to the death penalty, nor those to whom a month or more of sequestration would pose an undue hardship. When she finished, fifty of the sixty-member panel had been eliminated. But that was only the first round. She sent the remaining ten into an anteroom, from which they would be called individually and questioned further in an attempt to determine their suitability. As soon as they left the room, Peruto jumped to his feet.\n\n\"I want to call it to your honor's attention,\" said Peruto, in a pseudo-outraged tone, \"that one of those women was winking at me. She must have winked at me a dozen times.\"\n\nAbraham smothered a grin. \"Which one was it?\" she asked sweetly.\n\n\"I don't know,\" Peruto replied, \"but she was sitting right there on the front row.\" Turning, he counted off the number of seats. \"Right there,\" he pointed, \"one, two, three, four. The fourth seat from the aisle.\"\n\nAbraham suggested postponing a decision on the winking woman until she came up for more thorough interrogation. When she did, Peruto dismissed her almost immediately by using one of his precious peremptory challenges.\n\nUnder Pennsylvania law, the defense and prosecution each have twenty peremptory challenges\u2014opportunities to dismiss twenty potential jurors without having to offer an excuse. Veniremen also can be disqualified for cause, such as prejudice or extenuating circumstances that would hinder their capabilities as jurors\u2014illness, for example\u2014but that decision has to be made by the judge, usually at the request of the defender or the prosecutor. Peremptory challenges are valuable as each side tries to shape a jury.\n\n* * *\n\nAt the end of the day four jurors had been selected, a remarkable feat considering the obstacles. Judge Abraham, who still had not ruled on Peruto's motion for a change of venire, was particularly pleased. \"I've seen it go slower than this in much-less-publicized cases,\" she confessed.\n\nBut one thing bothered her. All four of the jurors who had been picked were white. During the day, Gallagher had used two of his twenty peremptories and Peruto three. But all three of Peruto's challenges had been against blacks. She was getting the strong impression that Peruto wanted an all-white jury and was going to peremptorily dismiss any black who made it through the initial screening. That, she said, would be intolerable.\n\nFixing the defense attorney with a hard glare, Abraham warned him that he was tiptoeing through a mine field. \"I don't want to have to face accusations that the defense is using discrimination to systematically exclude blacks,\" she said. \"I don't want any charges of reverse discrimination.\"\n\nPeruto put on his innocent look. \"There won't be, your honor,\" he promised.\n\n* * *\n\nTuesday's feeling of progress carried over into Wednesday.\n\nAs on the previous day, all of the sixty veniremen in the panel admitted to having heard about Heidnik and his alleged crimes. But that was hardly unexpected. When the culling was complete, only five potential jurors were left, half as many as the day before. But it was the quality that counted.\n\nDuring the morning session two jurors were chosen, both white. But Peruto also used two peremptories to dismiss two blacks. After the second one, Judge Abraham straightened and leaned forward, locking her eyes on him. \"I'm reserving the right to strike the whole jury,\" she told him angrily, \"if I feel there has been systematic exclusion of blacks.\"\n\n\"I assure you we're not trying to do that,\" Peruto swore.\n\nAt the end of the morning session Abraham had promised to call up another panel in the afternoon. Obviously pleased with the way jury selection was going, she wanted to delay ruling on Peruto's change of venire motion.\n\nThose plans were abruptly cancelled, however, when members of the new panel complained that television crews were filming them in the hallway as they filed into the antechamber preparatory to being examined.\n\nThat made her furious. \"They're upset, and I don't blame them. They don't want their pictures on TV if they think they're going to be on the jury.\" Trying to control her temper, she issued a stern warning to the news crews: Either refrain from filming the veniremen, or face orders excluding the cameramen from certain areas of the courthouse.\n\nBefore the admonition could be tested, it became moot. Although half a panel had been chosen in only two days, Heidnik insisted, through his attorney, that Abraham approve a change of venire.\n\nAbraham sighed and granted the motion. Three and a half days work went for nothing. Despite numerous requests, it was only the third such entreaty to be granted in Philadelphia in the previous eight years. Abraham set the new voir dire for June 13 in Pittsburgh, three hundred miles away and just about as far west as one can go from Philadelphia and still be in Pennsylvania. On that date she, Peruto, Gallagher, and Heidnik would again begin trying to select an unbiased panel, this time among the residents of Allegheny County.\nChapter 35\n\nJune 13, 1988\n\nPhiladelphia's City Hall and the Pittsburgh courthouse both date from the same decade, and both are made from granite. That is all they have in common.\n\nIf City Hall looks like a structure from the center of Paris, the Allegheny County Courthouse and Jail, built between 1884 and 1888, resemble buildings that might be found in a medieval countryside; towered and turreted, with faces of rough-cut stone and steeply pitched roofs.\n\nUnlike City Hall, where the interior spaces are littered and dismal, one of the best features of the Pittsburgh facility is its winsome quadrangle, a semi-secluded spot complete with tumbling fountain, leafy trees, comfortable benches, and flowering shrubs. Throughout the day, weather permitting, passersby treat the sun-splashed alcove like a sanctuary, a place to escape the hustle of the city. At noon on Fridays concerts are held there.\n\nBut the Philadelphia visitors were not there to sample the cultural amenities. When Judge Abraham and her entourage arrived well past mid-morning, they went straight to Room 315. Judge Robert E. Dauer, the usual occupant, had abandoned the sunny, cheerful chamber and accommodatingly moved into smaller quarters next door so the Philadelphia crew could be more comfortable.\n\nWhile only about half as large as her usual courtroom back home, Room 315 was much cheerier. It should have put everyone in a happy mood. From the beginning, though, Abraham was out of sorts. Breezing into the building in a dark, pleated skirt and cool-blue silk blouse decorated with large flowers, she was met, not with the anonymity she expected, but chaos. TV lights almost blinded her and still-photographer's strobes winked in her face. \"Philadelphia was a [media] freak show,\" she grumbled. \"As soon as I got off the elevator [in Pittsburgh] I saw it was going to be that situation all over again. All we needed was sawdust and peanuts and we would have had a real circus.\"\n\nIt seemed to put her in a bad mood for the entire day. She certainly came down hard on the media. Calling an impromptu news conference, she laid down the law.\n\n\"There will be no photos of any prospective jurors or of those who may finally be chosen,\" she barked. \"You will not be permitted to follow them down the street or talk to them. We want an atmosphere of calmness and dignity. I will not let you interfere with the trial.\" Disobedience, she warned, would result in immediate banishment. \"I'll keep you out so long your presence here will be totally wasted.\"\n\nTo make her point, she called up a panel of prospective jurors corresponding exactly to the number of available seats: seventy-seven. Since there was, therefore, no room for the press to sit, they would have to go. When Kurt Heine of the Philadelphia Daily News asked if she would allow a pool representative to remain, she turned a deaf ear. When the Inquirer's Fen Montaigne protested, she angrily told him to go \"file a suit.\"\n\nWhen the first prospective juror, a shy middle-aged widow with a cheek full of gum, took the witness stand for detailed quizzing, just about everyone in the room was sizzling. Even though the air conditioner was going full blast and the room was cool enough to serve as a meat locker, steam was rising off just about everyone's brow. Abraham was still fuming at the press, which was mad enough to throw her out in the corridor. Peruto was boiling because he didn't like the look of the Pittsburgh panel and the judge had ignored his request to take the whole show on the road once again. On top of that, no one on the panel looked too excited about the prospect of spending half the summer in sticky Philadelphia when they could be in the mountains or on the shore. The only one not grinding his teeth\u2014at least not at that stage\u2014was Gallagher, who remained unruffled despite the turbulence surrounding him.\n\nIt turned out to be a short-lived appearance for the nervous widow. Within minutes she was dismissed by Peruto, using the first of his peremptory challenges. Since this was a new proceeding, both he and Gallagher started over again with their allotments. Peruto was now down to nineteen. Looking puzzled but relieved, and still chewing at sixty bites to the minute, the woman hastily exited, only to be followed almost as quickly by a painter for the city school system (a Peruto peremptory), a chubby college coed who begged off because of summer-school classes (dismissed for cause), a half-deaf plumbing inspector (cause), a husky-voiced practical nurse who hesitated when asked about her willingness to impose the death penalty (a Gallagher peremptory), and an eighth-grade dropout who said he thought anyone convicted of first-degree murder should automatically get the chair.\n\nIt wasn't until the seventh venireman of the afternoon that both Peruto and Gallagher found a juror they could live with. Somewhat surprisingly, it was an outspoken nurse married to a policeman for twenty-two years. At first this caused Peruto some concern. Tilting back in his chair and stroking his beard with his left hand, he posed a carefully worded question.\n\n\"If a police officer were to say something was red and somebody else said it was blue, would you believe it was red just because a police officer said it?\"\n\n\"No way,\" she fired back. \"I know police officers too well.\"\n\nHer response drew laughter from the press, which had finally been allowed inside the courtroom. That prompted another rebuke from Judge Abraham. \"We're not here to treat this as a laughing matter,\" she pointed out.\n\nFor a while it looked as though the nurse had stepped over the line as an undesirable juror by admitting that her views could be influenced by her friends and the media. But Abraham led her out of the swamp by eliciting her assurances that she could view the situation with an open mind.\n\nAlthough eight more prospective jurors were questioned before Abraham knocked off a little after four o'clock, an hour earlier than normal, only two others were selected. One was a taciturn welder with an Elvis haircut who blushingly admitted a brief brush with the law ten years ago when he was fined thirty-seven dollars for starting a barroom ruckus. The other was a well-spoken elementary school teacher's aide\u2014mother of three preteens, and wife to a marketing manager for Westinghouse.\n\nThe ones who were not selected, however, were as significant as those who were. Among the rejects were a self-possessed young police officer married to a nurse, an indecisive electrical engineer, and two black women\u2014one a data entry clerk and the other a college-educated counselor for a travel agency. Both black women were dismissed by Peruto, who, at the end of the day, had used up seven of his peremptory challenges, one more than one third of his total. And the jury was only one-fourth complete. Gallagher, on the other hand, had used only one peremptory. The others were dismissed by Abraham.\n\nAt the rate he was expending his peremptories, things were not looking up for Peruto. \"I want to go back to Philadelphia,\" he mumbled disgustedly, jabbing forcefully at the down button on the hallway elevator. Despite questioning fifteen prospective jurors, only three had been selected. And this was in a city where the jury pool supposedly had been untainted by media coverage of the case. At the end of the first day the ratio of chosen jurors was well below that of Philadelphia, where four out of seven had been chosen in the opening flurry.\n\nFor just about everyone June 13 had turned into a blue Monday. Everyone, that is, except Heidnik. Of all those present, he was the only one who seemed to be enjoying himself. Clad in a different pair of pants but the same old Hawaiian shirt, he perched alertly at the defense table, his slate-blue eyes sparkling and jumping anxiously. While Gallagher plodded through his usual litany of questions in his best Joe Friday drone\u2014just the facts, ma'am, nothing but the facts\u2014Heidnik rocked back and forth in a padded swivel chair that had replaced the wooden straight-back assigned to him in Philadelphia. Twice he laughed aloud, and several times he got into deep, energetic discussions with Peruto. When it came time for him to return to his cell for the night, he even forgot to shuffle.\nChapter 36\n\nJune 14, 1988\n\nIn Pennsylvania it was Flag Day, a state holiday except in Room 315 of the Allegheny County Courthouse. Judge Lynne Abraham, who was in Pittsburgh to select a jury, not sit around a hotel pool, decreed it would be a working day. It was a spectacularly successful one, as it turned out. In less than four and a half hours, which may have been close to record time, Gallagher and Peruto signed up nine jurors and six alternates, completely filling the panel. Of the original seventy-seven veniremen, two were left when they were through. Even Peruto was happy, which was surprising, considering he'd started off by asking Judge Abraham for another change of venire.\n\n\"You have to be joking,\" she responded when Peruto made his motion. He wasn't, but neither was she. \"I can't believe the Supreme Court of Pennsylvania is going to entertain another motion for a change of venire. This is it,\" she added, ending the discussion.\n\nDespite Peruto's misgivings, the first potential juror called\u2014a frizzy-mopped young hairdresser\u2014was approved in a matter of minutes. That broke the log jam. By lunchtime six more jurors had been picked, including a research chemist who was clearly upset about having to leave his job at Pittsburgh Plate Glass to go to Philadelphia to hear a murder case. His company, he said, would have great difficulty getting along without him. That could be corroborated, he said, by the fact that he had taken only three days vacation at any one time in the last five years.\n\n\"PPG is a pretty big company. What would they do if you became ill?\" Judge Abraham asked.\n\n\"They would not meet their deadlines,\" the chemist replied.\n\n\"That would not be the end of the world,\" Abraham responded.\n\nThe chemist, however, didn't want to give up. He said his absence would result in the loss of \"millions of dollars.\"\n\nJudge Abraham was losing patience.\n\n\"That's not money out of your pocket,\" she said somewhat testily.\n\n\"Only so far as I am a stockholder,\" the chemist shot back.\n\nThat was enough for Abraham.\n\n\"Inconvenience to your employer is not a hardship. I'm sorry,\" Abraham said, not looking or sounding sorry at all, \"but that's what jury duty is all about.\"\n\nGallagher popped to his feet, straightening his coat. \"The juror is acceptable to the Commonwealth, your honor.\"\n\nAbraham cut her eyes to Peruto, who would have the last say on the man's eligibility. \"Acceptable to the defense,\" he blurted, then quickly swiveled so his back was to the bench. His face was as red as a ripe tomato while he struggled to contain his laughter.\n\nThe chemist looked as though he were going to cry out in frustration, and finally left the room.\n\n\"Boy, was he pissed,\" Peruto chuckled later. \"I just hope he blames it on the prosecution.\"\n\nA few minutes later another juror, a man married for thirty-seven years, brightened when told he would have to go to Philadelphia to hear the case.\n\n\"Can I bring my wife?\" he asked innocently, then looked crestfallen when Abraham explained that he would be sequestered and unable to see her.\n\nThe final two jurors were selected back to back shortly after the lunch break. When the last member, a sixtyish caseworker for the state Department of Public Welfare, was picked after minimal questioning, Heidnik flashed Peruto a big grin. It was an all-white jury, and Peruto still had half of his peremptories left.\n\nEven with an all-white jury, the panel was impressively diverse. Picked to decide Heidnik's fate were six males and six females ranging in age from their early twenties to their late sixties. The group included one single woman and one divorc\u00e9e with three grown children, including a son who was a lawyer. There was the hair dresser, a toll collector, a welfare-department caseworker, a derrick-boat operator, the chemist, and a skilled laborer. One man had his own business. One of the jurors was a nurse and another was married to one. Of the six female jurors, all but one worked outside the home at least part time. If there was any common denominator, it appeared to be ancestry. Five had Italian surnames.\n\nA newsman, observing that those of Italian descent might be more conservative and thus inclined to deal more harshly with Heidnik, asked Peruto why he didn't use more of his peremptories to get a more liberal panel.\n\nPeruto was waiting for that one. \"They don't know it yet,\" he quipped, \"but my father is president of the Italian Sons and Daughters of America.\"\n\n* * *\n\nWith the jury selection out of the way, there was only one thing left to do before opening the trial: haggle over what evidence each side would be permitted to introduce and how far afield they would be allowed to roam in eliciting testimony.\n\nJudge Abraham had already made it clear she planned to keep a tight reign. Whenever Peruto mentioned that the trial might take three to five weeks, Abraham jumped in. \"No way,\" she said more than once. \"This trial is not going to take that long. Two to three weeks at most.\"\n\nTo make sure everyone knew the ground rules to which she planned to adhere, Abraham scheduled a hearing in her Philadelphia courtroom on Thursday, June 16, to give Peruto and Gallagher the chance to settle those issues in advance and not tie up valuable trial time.\n\nIt had been Peruto's intention to introduce testimony designed to link Heidnik's treatment at the West German military hospital to Army experiments with LSD and other hallucinogens. What he wanted was to create a circumstantial trail showing that Heidnik may have been a guinea pig for drug tests and that the experiments irreparably fried his brain.\n\nThe problem was, Judge Abraham wasn't buying it. She wanted to know if Peruto actually planned to pursue this approach.\n\n\"Yes, I do,\" Peruto replied. \"If my client were to testify, he would say he went into an Army hospital where LSD experiments had been carried out. He would say he went in for a stomach problem and they gave him something that kept him up for three or four days.\"\n\n\"That's not proof,\" Abraham said.\n\n\"The Army said he would get a one hundred percent disability,\" Peruto persisted, \"but his attorney fought that. He didn't want to leave the Army. He didn't know until published reports in the 1970s that he had been a guinea pig.\"\n\nAbraham said that all Peruto had offered so far was speculation, and that wasn't going to be enough to allow testimony on the subject.\n\nProof is hard to come by, Peruto admitted.\n\n\"Is it your contention that Mr. Heidnik was one hundred percent mentally well when he went into the Army?\" Abraham asked.\n\n\"He did have problems,\" Peruto confessed. \"He was psychotic. He wasn't perfect. But that also made him the perfect guinea pig.\"\n\n\"It's a long way from showing he was a perfect guinea-pig subject to being a guinea pig,\" Abraham commented.\n\nPeruto argued that other things had to be considered as well, such as the Army's desire to get Heidnik out of the service with a full disability. \"Only one in eighteen thousand people who file for mental disability get one hundred percent,\" Peruto said. \"And only one in seventy-nine thousand receive it for life. My client gets it for life. And he didn't want it; he didn't file for it.\"\n\nThe judge was unimpressed.\n\n\"I haven't heard anything yet about competent evidence that shows me your client was subject to LSD experimentation. You can't use that for mitigation without competent evidence. It would be relevant, but you don't have the evidence,\" she said.\n\n\"If you're asking me if I can put someone on the stand who says my client was given LSD, I can't,\" said Peruto.\n\n\"All you've mentioned so far are conclusions, not proof,\" said Abraham. \"If you have anyone to confirm this, I'll allow it, but from what I've heard so far, you don't.\"\n\n\"It's a clear circumstantial picture,\" Peruto persisted, unwilling to let go.\n\n\"I'm not going to allow any speculation,\" Abraham repeated. \"No guesswork. If you have any competent information about LSD, I'll allow it. But from what you've told me, you're just guessing.\"\n\nIn the end Peruto had to abandon that approach. He didn't have the horses.\n\nThere was one more thing Abraham wanted to settle before the trial began: the issue of the all-white jury. Before going to Pittsburgh she vowed that she would not permit an all-white jury to hear the case, but after seeing the racial composition of the Allegheny County venire list, she had softened her stance. \"The panel of potential jurors had far fewer blacks than we are used to seeing in Philadelphia,\" she conceded. \"I don't know if [the racial makeup of the jury] is going to be an issue. The tale will be in the telling. In the end, it is not who is on the jury but how it was selected.\"\n\nShe still was not comfortable with the result, she admitted, but she did not plan to strike the entire panel.\n\n\"It is a racial case,\" Peruto interjected. \"However, please note that I excluded more whites than blacks.\"\n\n\"I'm just noting it,\" the judge responded. \"It may be a nonissue. I'm not taking you to task, I just want you to know that it didn't go unnoticed.\"\n\"Any person who puts dog food and human remains in a food processor and calls it a gourmet meal and feeds it to others is out to lunch.\"\n\n\u2014Peruto\n\n\"Just because someone does bizarre acts, the law doesn't recognize them as insane.\"\n\n\u2014Gallagher\nChapter 37\n\nJune 20, 1988\n\nChuck Peruto considers himself an honest man. He also considers himself a crafty defense lawyer. Not that the two are mutually exclusive, but on opening day he saw an opportunity to be both and he leaped at it.\n\nWhen he walked into Room 653 in Philadelphia's City Hall, his crisply starched, custom-made shirt already wilting in the blanket of heat and humidity hanging over the city and seeping into the courtroom, he had only an ill-formed plan for his opening statement. But as he sat there listening to Charles Gallagher outline how he intended to prove that Heidnik murdered, raped, kidnapped, assaulted, and did any number of other hideous things to six young women between the ages of eighteen and twenty-five, he had an idea. It was risky, he thought, but it might be worth it. What the hell, he told himself, what does my client have to lose?\n\nGallagher, looking cool despite the heat, delivered his opening statement in a dull monotone. He didn't have to resort to dramatics to grab the jury's attention. It was the first time the Pittsburghers had heard any specifics about the case, and their eyes popped open. Gallagher was telling them things that went beyond their wildest nightmares. This was horror-movie material. Gallagher didn't have to lay it on.\n\n\"Gary Heidnik took these women home with him,\" Gallagher began. \"He plied them with food and in some cases sex. He assaulted them. He choked them. He handcuffed them, and he took them to his basement, where he put muffler clamps on their ankles.\"\n\nWhen Gallagher is nervous he fiddles with the flap of his jacket pocket or pulls on his coattail. He displayed both those mannerisms as he paced before the jury.\n\n\"He starved them,\" he continued, his voice steady and matter-of-fact. His left hand dipped briefly into his left jacket pocket. He withdrew it and straightened the flap. \"He tortured them. He repeatedly had sex with them.\" In the process, he added, he killed two of them, one of whose bodies he dismembered, cooked, and fed to the others. Some of the jurors paled. One spectator groaned loudly.\n\nEnough was enough, he thought. He didn't need to overdo it. Once he started calling his witnesses, the testimony would more than speak for itself. But he wasn't quite through. He didn't want to mention the death penalty at this stage, but he did want to get them thinking about first degree. And he felt he had to get them thinking about Heidnik's mental state. He wanted to demolish Peruto's insanity defense before he could make it.\n\n\"The evidence will show that from the eve of Thanksgiving 1986 up through March 25, 1987, the defendant committed repeated sadistic and malicious acts,\" Gallagher said, tugging on the hem of his jacket. \"He did them in a methodical and systematic way, and he concealed them in a methodical and systematic way. He knew exactly what he was doing, and he knew it was wrong. He took advantage of underprivileged people.\"\n\nGallagher gave a final yank to his coattail and walked back to his chair. His statement took only twelve minutes.\n\n* * *\n\nFor several moments Peruto sat motionless\u2014for effect, for theatre.\n\n\"You're not required to make an opening statement,\" Judge Abraham prompted. \"You can postpone it until later or you can waive it all together.\"\n\n\"No,\" said Peruto, springing to his feet. \"I want to go ahead.\" Striding the twenty feet to the jury box, he leaned on the rail, searching for eye contact among the eighteen Allegheny County residents, many of whom still appeared to be in shock by what Gallagher had told them.\n\n\"The judge said something this morning about people being innocent until proven guilty,\" Peruto began, having convinced himself that boldness was the best approach. \"My client is not innocent,\" he said. He paused. \"He is very, very guilty.\" He paused again, a little longer this time.\n\nThat was not what he wanted to say, he admitted later; it just seemed the right thing to say at the time. Before the trial he had offered to plead Heidnik guilty to any charge the district attorney's office wanted except first-degree murder. But there was no compromise in the deputy district attorney's heart; he wanted the death sentence. Peruto was determined he was not going to get it.\n\nEven as he mouthed the words about his client's guilt, he wondered about the wisdom of it. But as he listened to Gallagher he knew he would not be able to refute the charges. He had to be honest, he told himself. His client was guilty. He had done everything the Commonwealth said he had. Except for one thing. He had to convince the jury that Heidnik did not intend to kill the two women who died in his cellar. If he could do that, it would not be first-degree murder. Heidnik would dodge the electric chair. The best he could do, Peruto felt, was plant a seed with the jury, give them some reason to start thinking that a man who did everything Heidnik was accused of doing could not possibly be sane. That was his only hope.\n\n\"There's no mystery here,\" Peruto continued. \"This is not a whodunit. If all we had to decide here was who did it and what was done, it would be easy.\"\n\nGallagher had promised the jury he would construct a trail of evidence leading right to Heidnik's door. Peruto was telling them he didn't want to stop there; he wanted to go one step farther. He intended to open that door and show the man inside.\n\n\"You're not here to determine if Gary Heidnik is going to walk out of here a free man,\" Peruto continued. \"He's never going to see the light of day. He will be behind bars or in some mental institution. Any person who puts dog food and human remains in a food processor and calls it a gourmet meal and feeds it to others is out to lunch.\"\n\nThat was inspired, Peruto thought, congratulating himself. Nobody was going to disagree with that, least of all the jurors.\n\nThen he switched direction. He began insinuating that they should consider viewing Heidnik more as a victim and not so much as a cold-blooded killer. When his turn came, Peruto told them, he planned to present overwhelming psychiatric testimony showing that the signs of Heidnik's disintegration had been there for a long time, but no one took the time to read them.\n\nIt isn't as though he had never seen a psychiatrist before or never been treated, Peruto said. \"He's had twenty-something hospitalizations.\" Any number of psychiatrists who examined him warned that he was dangerous. Some said he was a time bomb waiting to explode. He was so far gone, Peruto said, that he was no longer able to tell right from wrong.\n\n\"You have to keep an open mind,\" he cautioned, laying the groundwork for his insanity plea. \"Dr. McKenzie will testify that Gary Heidnik suffered from such a disease of the mind for so many years that he could not tell right from wrong. Dr. Kool will say the same thing. In fact, you're not going to hear from anybody who came to any other conclusion.\"\n\nPeruto sensed it was time to quit, time to shut up and sit down. One more point to make. \"Understand two things,\" he said. \"One, Gary Heidnik didn't want anybody to die, and two, because of his mental illness, he couldn't tell right from wrong.\"\n\nHis presentation took only nine minutes, three minutes less than Gallagher's.\n\nNot surprisingly, Peruto found he was sweating. So was Gallagher. So was everybody, as a matter of fact. Outside, the thermometer was on its way up to the mid-nineties. Inside, the air-conditioning was broken. It promised to be a long trial.\nChapter 38\n\nIt was hard\u2014maybe the hardest thing Jeanette Perkins had ever done. Sitting there in the witness box talking about her dead daughter and staring at her murderer, she wanted badly to cry or scream or jump up and slap his face. But she couldn't. All she could do was answer the questions and pray that he would get what was coming to him. As if he cared. He wouldn't even look at her. Gary Heidnik may as well have been on another planet, from what she could tell. He just sat there staring straight ahead, seeing nothing, saying nothing, maybe hearing nothing too.\n\nPerkins, Gallagher's first witness, tried to ignore Heidnik, tried to forget he was present. Instead she concentrated on the jury, on telling her story to the eighteen men and women sitting a few feet from her on her left and, unlike Heidnik, watching her intently.\n\nA pleasant-looking, middle-aged woman wearing a white dress with blue stripes, Perkins leaned forward slightly to get closer to the microphone. In a calm, soft voice that gave no hint of her inner turmoil, she told the jury that her slightly retarded daughter, Sandra Lindsay, told her on the Saturday after Thanksgiving, 1986, that she was going to the corner store to buy a package of Midol. She never returned.\n\nBy Monday, Perkins said, she was starting to get frantic. \"It was unusual for her to be gone that long without calling,\" she explained. First she went to the Elwyn Institute, where her daughter was a client. When she couldn't find her there, she went to the police to report her missing. After that she searched out a man who Sandra had been seeing a lot of: Tony Brown.\n\nBrown had not seen Sandra either, but he suggested Perkins try \"Gary's house\" at 3520 North Marshall Street. She did. \"Me and two of my cousins went there and knocked on the door. We could hear the music playing real loud, but nobody answered our knocks. We went back at seven-thirty, but nobody came to the door then either.\"\n\nA week to the day after Sandra disappeared, Perkins said, she received a note in her daughter's hand. It was mailed the previous day from New York. A few days afterwards she received a Christmas card from Sandra, also postmarked New York.\n\nDespite the correspondence, she could not shake the feeling that it was all a hoax; it was too much out of character for her daughter. Another daughter and the cousins made several more trips to the North Marshall Street address, but no one ever came to the door. The next thing she knew, she said, police were knocking on her door to tell her that parts of a body found in a freezer at 3520 North Marshall Street were believed to be Sandra's.\n\nPeruto did not cross-examine. But when Perkins was cornered by reporters in the hallway, she said that the police should have done more. That time when she spoke, the tears started flowing.\n\n* * *\n\nPeruto may have been kind to Perkins by waiving his chance to question her, but he had no reason to be gentle with Gallagher's second witness, police sergeant Julius Armstrong, the slim, black officer who handled Perkins' missing-person report.\n\nDuring direct examination Armstrong, a veteran of seventeen years with the department, explained that he, too, had gone to the North Marshall Street house. He got no response to his knocks, either, and no one answered the phone when he called. He also looked up Tony Brown and got Gary's last name from him. Brown misspelled it, but Armstrong admitted he did not double check. When he fed the name Brown gave him into the police computer, it came up blank. Then, when Perkins told him about the note and the card, he rested much easier. From then on, he confessed, he did not give the search a lot of thought.\n\n\"You knew [Gary's] name and you knew the address?\" Peruto asked.\n\n\"Yes,\" replied Armstrong.\n\n\"Did you ever check the tax rolls or the utility files to see what Gary's last name was?\"\n\n\"No.\"\n\n\"Tony Brown is a little slow, isn't he?\"\n\n\"He seemed all right to me.\"\n\n\"Did you go next door and ask the people if they had seen Sandra?\"\n\n\"No,\" Armstrong said, then quickly added, \"I was satisfied she was safe when we got the letter from New York.\"\n\nPeruto gave him a disgusted look and slumped in his chair. \"No more questions.\"\n\n* * *\n\nGallagher's third witness was his star: Josefina Rivera.\n\nShe walked quickly and proudly to the stand, her head held high. Taking the oath with her hand on the Bible, she looked straight at Heidnik. He glanced at her quickly, then ignored her just as he had Perkins and Armstrong. Just, in fact, as he would almost everyone called to testify.\n\nOf the four surviving captives, Rivera would be the most articulate. In a low-pitched, sedate voice, the thin former prostitute, wearing a stylish sun dress decorated with large blue flowers and a huge, flowing wig, described her capture, her imprisonment, and the indignities she was forced to endure.\n\nThe courtroom was packed, but there was no chatter, no comment. At the more gruesome parts of the testimony some of the spectators moaned softly, but Judge Abraham had sternly warned the audience about displays. \"If anyone interrupts, they will be arrested and jailed,\" she pronounced. \"I won't permit any outbursts, no matter how sad or bizarre the testimony is.\"\n\nOn the first day she was a captive, Rivera said, she listened in fascination as Heidnik told her how he planned to bring other women into the cellar, impregnate them and start a colony.\n\n\"Why did he say he wanted to do this?\" Gallagher asked innocently. It was, in fact, a loaded question. He knew, although the jury did not, that her answer would open a door revealing Heidnik's criminal record. If he had not been able to introduce the issue through Rivera, he may not have been able to bring it up at all because it would, by its very nature, be prejudicial to Heidnik. The fact that he planned to do this was brought out in discussions out of the jury's presence earlier in the day. Peruto opposed it vigorously, but Judge Abraham agreed to allow it if Gallagher could introduce it through Rivera. It was the first of Abraham's rulings that seemed to come down heavily on the side of the prosecution. It would not be the last.\n\nRivera testified that Heidnik was obsessed with having children because he had been sent to prison for trying to help the sister of the woman he had been living with. The woman, Anjeanette Davidson, had a daughter by him. The girl had been placed in a foster home.\n\n\"He said he went away for five years, and when he came out he couldn't find Anjeanette and he felt he had been deprived of his family. He felt as though society owed him a wife and family.\"\n\nSeveral of the jurors looked surprised when Heidnik's prison record was revealed. Abraham interrupted. \"Don't try to interpret that,\" she told the jury, letting the statement hang. Later, from other witnesses, they would hear a lot more about the crime and Heidnik's prison record.\n\nWithout skipping a beat, Rivera continued her tale. She told how the others\u2014Sandra Lindsay, Lisa Thomas, Deborah Dudley, Jacquelyn Askins, and Agnes Adams\u2014were brought in one by one, and about how they, too, were abused and misused.\n\n\"What did he feed you?\"\n\nRivera shrugged. \"Crackers... oatmeal... rice... hot dogs... sometimes waffles... chicken... bread... water... hot chocolate. Usually we would eat early in the morning and late at night.\"\n\nDid he say anything about Lindsay having his baby?\n\n\"He said Sandra had promised to have a baby by him, but she kept backing out, and this time she wouldn't be able to.\"\n\n\"What did you do to pass the time?\" Gallagher wanted to know.\n\n\"Nothing too much,\" she replied nonchalantly, \"outside of just having sex and staying in the hole. Three times we were down in the hole and we ran out of air and we couldn't breathe. We started screaming and hollering, and Gary came down and beat us... We didn't take any baths or wash our hair. We just had baby washcloths... Music was going twenty-four hours a day.\"\n\nThe hygiene situation improved after Christmas, she added. \"Then everybody had a day to go bathe. He would take your chains off the sewer pipe and take you upstairs and put you in the tub with the chains, and afterwards he would take you in a little bedroom and have sex with you.\"\n\nBy the end of January, when there were five captives, they were actively considering plans to overpower Heidnik and escape, said Rivera. \"We had an escape planned for January twenty-ninth but nothing was done. Gary had heard us talking about it.\"\nChapter 39\n\nFor more than three hours Josefina Rivera buried Gary Heidnik with a wealth of detail, mainly about the deaths of Sandra Lindsay and Deborah Dudley.\n\n\"He had Sandra on bread and water and he kept her chained with one arm over her head. She had been throwing up and she said she wasn't feeling good. She passed out once... [and] I asked Lisa to go over and make her stand up. She stood up for twenty or thirty minutes, but then she collapsed... Gary came down and undid her handcuff, and her body just crashed to the floor... then he kicked her into the hole and got in there with her and said she was dead.\"\n\n\"What happened after that?\" Gallagher prompted.\n\n\"He carried her upstairs and we heard an electric saw, and then we smelled a terrible odor. He smelled like it and so did the food he brought us.\"\n\n\"Did he have sex with anyone that night?\"\n\n\"No.\"\n\n\"How about the next day?\"\n\n\"Yes, with Lisa and [Askins].\"\n\nRivera said Heidnik had stopped beating her early in January, apparently because he was beginning to feel he could trust her. Besides, she said, he was having continuing problems with Dudley, and that occupied much of his time.\n\n\"He always had trouble with Debbie... Debbie always fought back.\"\n\nNot long after Lindsay died, Heidnik took Dudley upstairs on a mysterious mission.\n\n\"In about five minutes she came back,\" Rivera said. \"She was very quiet.\"\n\nFinally Dudley told the others what had occurred. Rivera remembered her saying, \"He showed me Sandra's head cooking in a pot, and her ribs were cooking in a roasting pan in the stove, and her legs and arms were in the freezer.\" He told Dudley that if she kept it up, it would happen to her. Six weeks later Dudley, too, was dead.\n\n\"Everybody went on punishment in early March,\" Rivera said. \"We were eating dog food mixed with body parts.\" Over a period of days, the punishment became more severe. Finally, Rivera said, Heidnik filled the hole with water, put Thomas, Askins, and Dudley inside, and began shocking the three with a bare electric cord which he placed in contact with Dudley's chain. When she collapsed, Heidnik thought she was faking. \"He said, 'She can't be dead.' Then he went over and looked at her and said, 'She's dead.'\"\n\nAfter that, Rivera testified, Heidnik began trusting her because he made her sign a letter saying she was responsible for Dudley's death.\n\nOn March 24, after promising to find him another woman, Rivera was able to get away alone. She went to the cops.\n\n\"Until then,\" Gallagher asked, \"did you have any chance to escape?\"\n\n\"No.\"\n\n\"What did he say would happen?\"\n\n\"He said he would kill the other girls.\"\n\n* * *\n\nAlthough it was late in the afternoon, Peruto was anxious to begin his cross-examination. If he could shake Rivera, he felt, he could perhaps insert that seed of doubt in the jurors' minds. At the preliminary hearing in the spring of 1987 he had attempted to prove that Rivera played a much larger role in what happened to the other women than she had admitted. He was going to have a second chance to see if he could get her to crack.\n\n\"Why did he say he was keeping you captive?\" Peruto asked.\n\n\"He wanted us to have children,\" Rivera answered. Her tone had changed from that used with Gallagher. Her voice was still impersonal, but now it carried a heavy overlay of hostility.\n\n\"Why did he pick the cellar?\"\n\n\"He said he didn't want to do it in the conventional way because the city kept taking them away.\"\n\nPeruto was trying to make a point; he wanted to show how unbalanced Heidnik was. \"How many women did he say he wanted and how many babies?\"\n\n\"He wanted ten women to have ten children, all in the basement.\"\n\n\"He walked, talked, and acted the same, but what he was doing may not be what you would do, right?\"\n\n\"Yes.\"\n\nPeruto let a long silence build while he and Rivera glared at each other.\n\n\"Before you went to the police, you went to your boyfriend, is that right?\"\n\n\"Yes.\"\n\n\"What did you tell him?\"\n\n\"I told him I had been held captive for four months.\"\n\n\"Was there anything else to it?\"\n\n\"No.\"\n\n\"Your boyfriend is your pimp, isn't he?\"\n\n\"No,\" Rivera answered angrily. \"I never worked for anybody but myself.\"\n\n\"Are you aware you're being sued by the other victims because of your connection with Heidnik?\"\n\n\"No.\"\n\n\"Are you wearing a wig today?\"\n\nThis flustered Rivera. \"Yes,\" she stammered.\n\n* * *\n\nTry as he might, Peruto was unable to make much headway. At one point he tried to get her to confess that she and her boyfriend, Vincent Nelson, planned to rob Heidnik before calling the police. She denied it.\n\n\"Did you get any of your information from the media?\" Peruto asked, trying to show she used news reports to embellish her tale.\n\n\"I was there,\" she shot back. \"I don't need to get it from the newspapers.\"\n\nAlthough he was able to get her to admit that there were times when Heidnik was not supervising her and she could have escaped, she said if she had, he simply would have taken it out on the others. Her plan all along, she said, was to work her way into Heidnik's trust and bide her time until she could assure the safety of the others as well as herself.\n\nPeruto sat silent, looking for a new line of attack.\n\n\"Is it a fair statement that you would like to see the defendant convicted of first-degree murder?\" he asked.\n\nRivera shot a venomous glance at Heidnik. \"That's right,\" she spat.\n\nJudge Abraham called time. It had been a long day, she said, and the temperature in the courtroom was nearing the boiling point. Peruto could resume his cross-examination on Tuesday. It would give everyone a chance to cool off.\nChapter 40\n\nJune 21, 1988\n\nThe cool-down period didn't take: not for the weather, not for the participants. It was even hotter in the courtroom on Tuesday than it had been the previous day, so hot, in fact, that the first thing Judge Abraham did was take off her robe and invite the lawyers to remove their jackets. Peruto did; Gallagher didn't. The jurors had anticipated this development. When they appeared, most of the men had abandoned their ties for golf shirts and the women gave up their suits for sun dresses. It was time to get down to business.\n\nThe battle between Peruto and Rivera was no less bitter on Tuesday, but it was much shorter. Rivera had swapped her dress for a white blouse and purple slacks and her big wig for a less ostentatious one. It was the only indication that Peruto had gotten to her at all. She had not left any of her hostility toward the defense at home.\n\nAgain Peruto was unable to get her to admit complicity or to trap her into contradicting her testimony at the 1987 hearing. After about ten minutes he gave up and turned her back over to Gallagher for redirect.\n\nThe DDA had three more points he wanted to make before he let her go, three more things to show that Heidnik knew what he was doing during the period he held the women hostage.\n\n\u2022 He asked Rivera what was the first thing Heidnik did when they drove back into Philadelphia after dumping Dudley's body in New Jersey.\n\n\"He stopped to buy an Inquirer so he could check his stocks,\" Rivera said.\n\n\u2022 He asked why Heidnik had stopped to get another woman to go with them when they went to pick up Heidnik's Rolls at the garage, and why he took that woman home with them to play video games.\n\n\"She was Sandy's friend,\" Rivera said, \"and he wanted her to see inside the house so she could go to [Lindsay's] family and tell them she wasn't there.\"\n\n\u2022 Gallagher asked Rivera if she knew where Heidnik got his ideas.\n\n\"Yes,\" she answered brightly. \"He got them from watching movies and TV. He got the idea for feeding us parts of Sandy's body from the movie Eating Raoul, and his ideas on punishment from Mutiny on the Bounty. He also saw [The World of] Susie Wong, and he liked the way Oriental women were. That's why he picked a Filipino wife.\"\n\nGallagher called two more witnesses: one of the policemen who arrested Heidnik, and an attractive black policewoman who questioned Rivera. The latter testified to Rivera's condition when she was first brought in\u2014 \"she was trembling, crying,\" said Officer Denise Turpin.\n\nWith the groundwork laid, Gallagher called the other three captives in quick succession.\n\nLisa Thomas, now a chunky twenty-year-old with her hair in braids, took the stand wearing turquoise slacks, a T-shirt, and long, dangly earrings.\n\nFor the most part the story of her capture paralleled Rivera's, as did those of Jacquelyn Askins and Agnes Adams. But the details were no less horrific.\n\nThe day after attaching her shackles and leaving her to join Rivera and Lindsay, Heidnik seemed to experience a touch of compassion and substituted a longer chain for the one connecting her ankles.\n\n\"Why did he do that?\" Gallagher asked.\n\n\"So I could open my legs wider to have sex,\" she said.\n\n\"Did he beat you too?\" the DDA inquired.\n\n\"Yes,\" she said. \"Almost from the first moment. I said something to make him mad, and he said he was going to hit me. He did. He hit me five times with a thick, brown stick.\"\n\n\"Was there anything else?\"\n\n\"He told me to beat Sandy regularly; he'd get his kicks from seeing us beat each other... He'd get on top of me and make me suck his penis. He said if I didn't, he would beat me.\"\n\nOn cross-examination Thomas at first said she couldn't remember her earlier testimony about Rivera beating the others even when Heidnik wasn't there.\n\n\"Oh, yeah, I remember,\" she said. \"Nicole [Rivera] would laugh when she put water in the hole [prior to electric-shock torture]. The electric shocks were her idea.\"\n\n\"[Rivera] was just feeding a sick mind, wasn't she?\" Peruto asked quickly, seeing an opening.\n\n\"Objection,\" Gallagher screamed, jumping to his feet.\n\n\"Sustained,\" Judge Abraham said quickly, giving Peruto a withering look.\n\n\"What else would she do?\" Peruto continued.\n\n\"She and Gary would go out and then they would come back and brag about what they had done that day.\"\n\n\"Did you ever plan an escape?\"\n\n\"Yeah, but then Nicole told Gary about it.\"\n\nGallagher regained some of the lost ground on redirect, getting Thomas to clarify that Rivera was always in chains until Dudley was electrocuted. While Peruto made much more headway with Thomas than he had with Rivera, his path was unclear. Was he defending Heidnik or prosecuting Rivera?\n\n* * *\n\nJacquelyn Askins looked scared to death. The youngest of the captives, she was still only nineteen when she took the stand. Her eyes, which had a slight Oriental tilt, were brimming with tears and her voice shook. She was hooking on a street corner at lunchtime when Heidnik picked her up, took her home, had sex with her, handcuffed her and took her to the cellar. The first thing he did to her in the basement was whip her with a switch. That night, she added, he decided to have a \"party.\" Her description of the events was the first (and last) testimony to indicate Heidnik indulged in group sex.\n\n\"I'd suck his penis and another girl would suck his balls,\" she said. \"Then he'd have sex with one girl and I'd lie next to him so I could catch his juice.\"\n\nLater Heidnik seemed concerned about Askins getting enough to eat. She was so petite that the muffler clamps Heidnik used on the other women were too large for Askins' ankles. He had to use handcuffs. When it seemed as though she was not eating properly, he brought her instant breakfasts and peanut butter-and-jelly sandwiches\u2014rare treats in Heidnik's cellar.\n\n\"Debbie was so hungry she said she'd eat dog food,\" Askins said. \"After that, we all got dog food.\"\n\nShe said Heidnik was constantly testing them, pushing them to see how they would react, perhaps to give him another excuse to beat them. Even when police rushed in to rescue them, Askins was not sure it was not another Heidnik ruse. \"I thought [the police raid] was him,\" said Jacquelyn Askins. \"I thought he was trying to get us to scream so he could beat us.\"\n\nThe petite teenager was coming apart on the stand, quietly sobbing and dabbing at her eyes with a wadded handkerchief. Peruto studied her quietly for a few seconds and decided to waive cross-examination.\n\nAs Askins left the room on the arm of a burly matron from the district attorney's office, those inside the courtroom were jolted by a loud piercing scream and a husky voice yelling, \"Get away, get away.\" Askins had been waylaid by the camera crews waiting in the hallway.\n\nJudge Abraham's face turned red. \"Bring them all in here,\" she ordered. \"Right now.\"\n\nWith the jury out of the room, Abraham angrily lectured the half-dozen photographers and cameramen lined up before the bench.\n\n\"You don't have any sympathy at all,\" she told them. \"If that were your mother or sister, you'd be the first to punch someone out. That was a cheap, tawdry trick to play on a victim, and you ought to be ashamed. You have no sensitivity, no feeling, no sense of dignity. You have nothing going for you except your cameras.\"\n\nShe took the one course of punishment open to her: she ordered them away from the courtroom. \"You're off the floor,\" she said furiously. \"I'll have the sheriff remove you. You're out. If you want to take pictures, you're going to have to do it from outside.\"\n\nDismissing them, she stalked into her chambers to cool off. As she did, the spectators in the jammed courtroom applauded loudly. It was a demonstration she did not try to control.\n\nTwenty minutes later testimony resumed when Gallagher called the last captive, Agnes Adams, to the stand.\n\n* * *\n\nAdams' tale differed from the others by the fact that she had been with Heidnik twice before the night he took her captive. She met him the first time, she said, around the end of 1986 or early 1987, when he picked her off a street corner and offered her money for sex. They went to the North Marshall Street house but couldn't get in the driveway because a car had it blocked. Neither could Heidnik find a parking place, so he paid her for her time and dropped her off.\n\nThe second time was March 20, 1987. That time, he took her to his bedroom and had sex with her. Afterwards, he let her out the back door. He carefully locked it behind her. The third time, she said, was on the night of March 23, when she was approached by Heidnik and Rivera together. She said she also knew Rivera\u2014although she was then using the alias Vanessa\u2014from the days when they both worked in the same strip club.\n\nShe said she left with them in Heidnik's Cadillac, went to North Marshall Street, and had sex with Heidnik while Rivera waited in the living room downstairs. It was then that Heidnik handcuffed her and dragged her into the basement.\n\nDuring cross-examination Peruto got her to concede that Rivera could have left Heidnik's house at any time while she and Heidnik were upstairs. She also said she had seen Rivera at a northside street corner two weeks before she was captured, which would have been more than a week before Rivera claimed she was released from chains. Afterwards Peruto admitted that Adams had been mistaken about the sighting.\n\n* * *\n\nIn only two days the Commonwealth had presented its major witnesses\u2014a process that Gallagher originally estimated would take at least twice as long. Proceedings were moving much faster than everyone had expected. If all went well, Gallagher promised Judge Abraham, he would finish his case-in-chief by lunchtime on Wednesday. He proved to be right on target.\nChapter 41\n\nJune 22, 1988\n\nBefore Charles Gallagher could rest his case, however, there were still some loose ends to be tied up. With the jury out of the room, Gallagher, Peruto, and Judge Abraham waded through a large pile of exhibits the Commonwealth wanted to enter as evidence. Gallagher was relentless about exhibits. Before the trial was over he had offered almost two hundred items ranging from photographs of the site where Dudley's body was found to a red plastic bucket full of chains. The DDA also proffered a series of slides showing the body parts found in Heidnik's freezer, but Abraham previewed them in her chambers and decided the jury didn't need them. By the time they finished sorting through the items and the jury was summoned, it was well after mid-morning.\n\nWhen the court recessed on Tuesday, police lieutenant James Hansen was on the stand. A veteran homicide detective, Hansen had been in charge of the search party at Heidnik's house and had later headed the investigation. Most of his testimony dealt with the physical evidence Gallagher wanted to introduce.\n\nHansen, however, did add one piece to the growing puzzle. Peruto managed to get from him the fact that there was a second person\u2014Cyril \"Tony\" Brown\u2014charged in connection with the events in Heidnik's cellar.\n\nThe jurors looked puzzled when Brown's name came up, and Judge Abraham quickly ordered the defense attorney to abandon that line of questioning.\n\n\"You can't ask any questions about him,\" Abraham ruled abruptly.\n\n* * *\n\nThe rest of Gallagher's witnesses were scene setters. A detective who headed the search for Deborah Dudley's body testified in excruciating detail about the route he and Rivera followed as she tried to lead him to the site.\n\nDr. Robert Catherman, the deputy medical examiner who performed the autopsy on Dudley, confirmed she had died of electrocution. \"It was an almost classic example of electrothermal injury,\" he said dispassionately.\n\nCatherman also mentioned one other curious fact: Dudley's body showed clear evidence of recent considerable weight loss. \"She had a lot of excess skin,\" he said. \"She had more skin than body.\" That condition occurs, he added, when there is rapid weight loss and the skin has not had time to shrink to adjust to the loss of fat.\n\nOther witnesses came and went quickly. There was Dudley's sister, Diane Dudley, a handsome woman with close-cropped hair, who told the jury the last time she had seen Deborah was on Thanksgiving, six weeks before she disappeared.\n\nOfficer Julio Aponte testified that he responded to neighbors' complaints about a terrible smell on North Marshall Street in February 1987 but did not investigate further after Heidnik told him he had simply burned his dinner.\n\n\"You realize you don't burn your dinner for days, don't you?\" Peruto asked on cross-examination.\n\n\"I don't know that,\" Aponte replied. \"I don't cook.\"\n\n\"Didn't you tell us that you actually observed something cooking in the pot on the stove?\"\n\n\"Yes.\"\n\n\"Did you ask [Heidnik] about that pot?\"\n\n\"No.\"\n\n\"Did you ever smell anything like that in your life?\"\n\n\"No.\"\n\n\"So you were satisfied that everything was all right?\"\n\n\"I had no reason to believe anything was wrong. He was alive.\"\n\n* * *\n\nAnother grisly bit of testimony came from Dr. Paul Hoyer, the assistant medical examiner who studied the parts taken from Heidnik's freezer.\n\nIn addition to the frozen parts, Hoyer said, investigators found in the yard fragments of a bone from an upper left arm that were the same size and shape as a matching body part found in the freezer. \"The fragments had dog hairs on it,\" Hoyer said.\n\nIn the oven there were several rib fragments, a number of tooth fragments, and one whole tooth. \"The intact tooth had a very unusual wear pattern,\" Hoyer testified. \"But when we looked at a photograph [of Sandra Lindsay], we saw that she had a very unusual bite pattern.\" Investigators also found in the stove several hand fragments, skull fragments, part of a shoulder bone, and a piece of vertebra. \"There was nothing there to suggest the remains were from more than one individual,\" he added.\n\nHoyer said identification was made by studying a left wrist that was found in the freezer. Lindsay had injured her wrist a year before and an X ray was on file. That was compared with the wrist found in the freezer. \"The overall shape and size matched,\" said Hoyer, \"plus the pattern matched. There is a pattern in bone, and each pattern is unique. By comparing this we were able to say this was Sandra Lindsay.\"\n\nFrom the parts remaining, however, he said he had been unable to determine how Lindsay died. \"But I can say her death was unnatural, based on the extreme effort to which someone went to conceal the death. We ruled that the manner of death was homicide.\"\n\nPeruto asked Judge Abraham to strike Hoyer's testimony regarding the homicide conclusion.\n\n\"Nope,\" Abraham answered.\n\n\"I move to strike the cause of death,\" Peruto persisted.\n\nAgain Abraham refused. \"I don't think the method of death needs to be proved to determine murder,\" she said. \"The question is, has the Commonwealth established this was an unlawful killing?\"\n\nGallagher rose and straightened his pocket flap. \"The prosecution rests,\" he said.\n\nImmediately Peruto was on his feet asking Judge Abraham to remove the charge of first-degree murder. \"There was no specific intent to kill,\" he argued.\n\n\"Overruled,\" Abraham said.\n\nGlancing at the clock over the door, she ordered a recess. The defense would begin its case after lunch.\nChapter 42\n\nChuck Peruto could hardly have gotten off to a worse start.\n\nHis first witness was Dr. Clancy McKenzie, a plumpish, frizzy-haired, unconventional psychiatrist who looked as much like a shrink as Santa Claus looks like an IBM executive. With his blue suspenders and detached grin, McKenzie had a folksy, relaxed air about him. When he spoke, it was slowly, deliberately, almost shyly. Jurors sized him up quickly and were prepared to like him. They could even be ready to listen to him.\n\nThere was a problem, though. Judge Abraham had no use for him.\n\nIronically, McKenzie almost didn't squeak through the prosecution challenge of his credentials to qualify as an expert witness.\n\nWith his own psychiatric witness, the much sought-after Dr. Robert Sadoff, hanging over his shoulder and feeding him questions, Charles Gallagher put McKenzie through the paces.\n\n\"Do you belong to the American Psychiatric Association?\"\n\n\"I did, but I dropped my membership.\"\n\n\"How long ago?\"\n\n\"About three years ago.\"\n\n\"Do you belong to any professional institutions or organizations?\"\n\n\"No, I do not.\"\n\n\"Do you have any publications?\"\n\n\"Some. I have written about my work.\"\n\n\"Any specific publications?\"\n\n\"No.\"\n\n\"You have not written?\"\n\n\"No, that's not true. I didn't send my work in for publication.\"\n\n\"How much work have you done in forensic psychiatry?\"\n\n\"I've testified maybe half a dozen times. It's not what I choose to do.\"\n\n\"How many criminal cases?\"\n\n\"Two.\"\n\nWith a smirk, Gallagher sat down. If he had chosen to object to McKenzie's qualification, he might well have been backed by Judge Abraham. On the other hand, if he did not object, McKenzie might damage Peruto's case more than help it\u2014at least that's what Gallagher seemed to be banking on.\n\n* * *\n\nWhile Gary Heidnik had shown little interest in the testimony of any of the witnesses called so far, he listened intently as McKenzie spoke. Since shortly after his arrest until the trial started, Heidnik and McKenzie had met thirty times for a total of almost one hundred hours. That was infinitely more time than Heidnik spent with all of the other psychiatric witnesses combined. Obviously he and McKenzie got along.\n\nIf McKenzie could have stuck to what was asked him, his testimony might have been a major benefit for the defense. As it was, it turned into a distinct liability.\n\n\"I can't understand it,\" Peruto complained later. \"During our prep sessions he was right on target; he had everything together. Then when he got on the stand, he just refused to take any direction. He had his own agenda; he wanted to present his theory and he wouldn't take my guidance at all.\"\n\nIndeed, from the beginning McKenzie exhibited a tendency to say what he wanted to say rather than what Peruto wanted him to say. Heidnik's defense would have been better served, by far, if the psychiatrist would have let the lawyer be the quarterback.\n\n\"I'm not here so much to defend Gary Heidnik as to explain him,\" McKenzie began. \"I believe you should never judge a man until you've walked in his moccasins, and it's pretty hard to walk in Gary Heidnik's moccasins.\"\n\nIt was his theory, he said, that every schizophrenic has two minds\u2014that of the adult and that of the troubled infant\u2014in one skull.\n\n\"Is Gary Heidnik schizophrenic?\" Peruto asked.\n\n\"Oh, yes,\" McKenzie replied.\n\n\"Between November 1986 and April 1987 did he know right from wrong?\"\n\n\"During that four to five month period he could not have known right from wrong. But what I'm going to talk about are events that occurred twenty years ago\u2014\"\n\n\"If you're going to start back in history,\" Judge Abraham interrupted, \"you're going to lose the jury long before you get to where you want to go.\"\n\n\"What caused his schizophrenia?\" Peruto queried.\n\n\"Ah,\" said McKenzie. \"This is my forte. There is a survival mechanism in everyone. If you give a puppy a powerful electrical shock and then, as grown dogs, they are shocked again, they retreat to puppy behavior...\"\n\n\"I don't want to give the jury a medical degree,\" said Peruto, losing patience. \"Listen to me very carefully. Specifically, what finding did you come to regarding Gary Heidnik?\"\n\n\"In schizophrenia every piece of behavior matches the behavior of an infant... everything the mentally disturbed person experiences relates in some way to infancy. The most common thing I've found was the birth of a brother or sister in the first eighteen months of life. Gary Heidnik had a brother seventeen months younger\u2014\"\n\nJudge Abraham broke in and tried to bring McKenzie back on track. \"You're killing them with words, Doc.\" Her effort was unsuccessful. He ignored her.\n\n\"Trauma in the present returns an adult to trauma in the past,\" McKenzie continued. \"I explored every one of Gary Heidnik's acts to see if it came from a seventeen-month-old brain or if it came from an adult brain. If it came from the seventeen-month-old brain, Gary Heidnik wouldn't be able to tell right from wrong\u2014\"\n\n\"Just answer everything in one or two words,\" Peruto interjected. \"If it requires elaboration, no more than fifteen. What trauma did Gary Heidnik have when he was seventeen months old?\"\n\nMcKenzie didn't answer. Instead he launched into a long, complicated explanation of schizophrenia.\n\n\"The jury is never asked to define schizophrenia, just to determine if he knew right from wrong,\" said Abraham. \"The jury doesn't have to know what all the ramifications are.\"\n\n\"He had a sibling born when he was seventeen months old,\" McKenzie said. \"His wife left him the first time, and he tried to commit suicide. Then she came back and left him again in April 1986, and that started the whole thing. He decided no woman was ever going to leave him again and no one was going to take his babies away again. This was the fourth time. He went totally berserk. He gets the women, chains them up, then develops this concept that he's going to raise normal babies in his basement. That's the seventeen-month-old brain.\"\n\n\"Why did he like retarded women?\" Peruto asked.\n\n\"He felt rejected, so he sought those who were also rejected.\"\n\nPeruto asked how this fit in with his treatment of the women in the cellar.\n\n\"Like a baby would do with a pet,\" said McKenzie. \"Shove a couple of pieces of bread in its mouth. When he had to dispose of the bodies, it was like a two-year-old trying to hide the candy wrappers. He put body parts in a pot on the stove, in the oven, in the backyard, everywhere. The only thing he didn't do was run them up the flagpole. Do you know why he put body parts in the freezer? He was planning to start the babies\u2014once they were off the breast\u2014on human flesh. Infants suck for six months, and then they have a wish to devour the mother's flesh. This was the infant brain beyond a shadow of a doubt... He punished the way he was punished as a child... He had a beer party for them every Friday night. He had hot buttered popcorn. When he heard the girls were planning to kill him, the punishment got worse... one day some dog food dropped on the floor and one of the girls came along and scooped it up. He couldn't understand why anyone would want to eat dog food\u2014\"\n\n\"How could he make stock investments?\" Peruto interjected.\n\n\"At the time he became very ill he made very poor investments... He lost a hundred thousand dollars in a very short time when he started getting ill...\"\n\nAt two minutes before four o'clock, a half hour before the normal quitting time, a frustrated and exhausted Peruto turned McKenzie over to Gallagher for cross-examination. You could almost hear the DDA licking his chops.\nChapter 43\n\nJune 23, 1988\n\nCharles Gallagher, cross-examiner, was not the same as Charles Gallagher, prosecutor. So far in the trial, Gallagher had been quiet, noticeably polite, restrained, almost deferential. As a cross-examiner he became arrogant, loud, and impatient. It was a new persona, and he picked a good time to reveal it: Clancy McKenzie was still on the stand. Before he could begin really grilling the psychiatrist, though, there were some other matters to be dealt with.\n\nWith the jurors not yet in their seats, Peruto told Judge Abraham that he was considering asking that the judge instruct the jury to consider the possibility that Josefina Rivera was an accomplice to Gary Heidnik.\n\n\"If the defendant is clever enough to enlist the aid of an accomplice, he knows what he's doing,\" Abraham said. \"This could be a problem if you charge Rivera.... If you want an accomplice charge, I'll give you one, but Rivera is no fool. It is no great leap to say if your client got someone to work with him, if he's smart enough to get an accomplice, he's not insane. Why don't you talk it over with your client.\"\n\nIn the courtroom it was hotter than ever. The day before in Philadelphia, for the first time in eleven years, the temperature had hit a hundred degrees. When the jurors filed in, only two of them wore jackets. Peruto hung his over the back of his chair, but Gallagher kept his on. Giving his coattail a quick jerk, the DDA opened up on McKenzie with evident relish.\n\n\"What is the legal definition of insanity?\" Gallagher asked, trying to determine if McKenzie knew the provisions of the \"M'Naghten Rule.\" Adopted in England in 1843 and later imported to this country, M'Naghten (named after an insane Scots woodcutter who tried to assassinate the prime minister) holds that a person shall not be held responsible for criminal acts if he, because of a \"disease of the mind,\" is unable to know the \"nature and quality\" of his acts, or does not know that such acts are wrong. It is a standard used in courtrooms in twenty-five states besides Pennsylvania.\n\n\"A person did not know...\" McKenzie began, then fumbled for words. \"I don't have the textbook definition.\"\n\nGallagher handed him a photocopy of a page from the Pennsylvania statues. McKenzie read: \"... someone who did not know what he was doing or did not know right from wrong.\"\n\n\"Which is it?\" Gallagher demanded.\n\n\"Both.\"\n\n\"Did he know what he was doing between November and April?\"\n\n\"I'd have to go to my diagram...\"\n\n\"Did he pick up a mentally deficient woman to satisfy his sexual desire?\"\n\n\"I don't agree with that statement.\"\n\n\"Did he know when he lured mentally deficient girls to his basement?\"\n\n\"No. But which 'he' are you speaking of? His adult brain or his infant brain?\"\n\n\"Did the adult brain know?\"\n\n\"The adult brain was operating in greatly diminished capacity.\"\n\n\"Yes or no?\"\n\n\"The adult brain had only partial awareness. It knew what he was doing when he was driving a car, but it was not there sufficiently... No, absolutely he did not know right from wrong.\"\n\n\"Did Heidnik know it was wrong when he proceeded to conceal the identity of Sandra Lindsay by dismembering her and destroying the head and fingerprints?\"\n\n\"No. There was not enough adult mind present... it was not until he was thrown into jail and someone beat him up and he was given anti-psychotic medicine that he came out of it. When he was shown their pictures he didn't even recognize them, except for Sandra, who was a good friend of his. That's because he was looking at them with his infant brain. His adult brain hadn't seen them.\"\n\n\"Is there a possibility he was faking it?\"\n\n\"Absolutely not.\"\n\n\"Is it possible he was... telling you lies to get an advantage over you?\"\n\n\"Absolutely not.\"\n\n\"Isn't it true that someone can fake mental illness?\"\n\n\"He can't fake schizophrenia with me,\" McKenzie insisted.\n\nMcKenzie was excused soon afterwards, much to Peruto's relief. Gallagher gave a small grin. He rather enjoyed it.\n\n* * *\n\nAt this point in the trial a strange thing became noticeable: Judge Abraham also appeared to assume a new persona, one that certainly appeared pro-prosecution. Given her background as a highly aggressive prosecutor, it would not have been unusual for her sympathies to be in that camp. Also, she might have been having great difficulty remaining unbiased toward Gary Heidnik, considering the viciousness of his acts. But, discernably, her temper was getting shorter with Peruto. Their personalities might have clashed under the best of conditions, and the tensions within a courtroom were not conducive to improving relations.\n\nWhatever it was, it began emerging when Clancy McKenzie was on the stand and grew steadily more pronounced. Clearly, she had little patience with the psychiatrist, and at times was overtly scornful. There was, as well, another incident relating indirectly to McKenzie.\n\nOne of the first actions Abraham took after being assigned the case was to issue a gag order prohibiting the lawyers from talking to the media. Ostensibly, it was an attempt to keep publicity in check until a jury could be selected. But the order continued in force even after the jury was picked and sequestered. The lawyer it damaged the most was Peruto, because he had much more to gain from carrying on a dialogue with reporters than did Gallagher. As the trial progressed, Peruto chafed more under the restriction. \"There is no reason for the gag order,\" he grumbled out of Abraham's hearing. \"The jury is locked away and can't read the newspapers or watch TV.\"\n\nThe McKenzie fiasco put a definite strain on Peruto's patience; he felt he had to say something. During the lunch break, after McKenzie had finished testifying, he slipped furtively down a set of back stairs and filmed a brief interview with local TV reporter David Henry. During the interview the words \"McKenzie\" and \"flake\" came up in the same sentence. This would later surface to haunt Peruto. Back in the courtroom Peruto also began having second thoughts about the wisdom of allowing himself to be filmed. He asked Judge Abraham about the order.\n\n\"Oh,\" she said, \"I lifted that yesterday. I read it into the record in open court.\"\n\nPeruto looked puzzled. So did the dozen or so reporters who had been covering the proceedings on a daily basis. Covering a trial can be like watching a football game from the bench: you're close to the action but you can't always see what's going on all over the field. Between sidebar conferences, discussions in chambers, and tidbits thrown into the middle of obtuse legal arguments, one reporter can't always be sure of catching everything. In this case, however, not one of the half-dozen reporters asked had heard Judge Abraham rescind the gag order. This was not a major issue, but it did register as a bleep on some of the reporters' highly tuned antennae.\n\nThe gag-order issue was not crucial, but very quickly the subject got more serious. When Peruto called his next witness, Judge Abraham handed down a rigid ruling that seemed to pose a considerable handicap for the defense while immensely benefiting the prosecution.\nChapter 44\n\nAs a youth Jack Apsche was a helicopter door gunner in Vietnam. He survived the furious battles in the infamous A Shau Valley. On that Thursday, June 23, as an adult approaching middle age, he almost crashed and burned on a mission in Judge Lynne Abraham's courtroom.\n\nNow a psychologist who specializes in records research, Apsche spent weeks tracing Gary Heidnik's complicated trail, which wound through hospitals on two continents. He came into court prepared to share his findings, which he had gathered in a large, black box he lugged with him to the witness stand. Abraham was not impressed.\n\nAs Apsche pulled out his stack of index cards and settled in for what he thought was going to be at least two days of testimony, Judge Abraham launched into an exhaustive explanation of hearsay evidence. In legal terminology, hearsay is evidence that is not within the personal knowledge of a witness. Generally, someone is not allowed to testify to hearsay for two very good reasons: it is inherently unreliable, and the parties involved should have the right to confront those who furnished the information. There are certain exceptions to the rule. Medical records are almost invariably excepted. But this was a special case. Judge Abraham was not going to grant an exception on Heidnik's medical records.\n\n\"I can see where his testimony is going to go,\" Abraham said as Peruto asked his first question, \"and I can tell you it isn't going to be acceptable. Even if Dr. Apsche used these records to prepare his diagnosis, they can't be admitted. Opinions, diagnoses, and conclusions will not be admissible.\"\n\nPeruto was stunned. This was a tremendous and perhaps irreparable blow to his case. His entire defense was based on the insanity plea. With Dr. Clancy McKenzie down the tube, Peruto was going to have to hang his case on his two remaining psychiatric witnesses, one of whom was Apsche. The foundation of his insanity plea was in Apsche's black box\u2014the documentation showing how a whole string of psychiatrists, over a period of a quarter of a century, had found Gary Heidnik to be suffering from a mental illness so severe that his reasoning ability was incapacitated. Ninety percent of Apsche's testimony was scheduled to revolve around those diagnoses. But Abraham was not going to let the jurors hear about them. Peruto didn't know what to say. The only thing he could do was file an objection.\n\nThis was an extremely stringent ruling, and a very unusual one. Even more unusual, it appeared to apply only to the defense. Later, when Gallagher wanted to introduce medical records, Abraham permitted it. She herself even read to the jury one record of the type she had prohibited Apsche from discussing. Significantly, that was a report supporting Gallagher's contentions.\n\nPeruto wanted to show that Heidnik's mental problems began when he went on sick call in the U.S. military hospital in Landstuhl. \"He went in for a medical problem and he was treated for a mental one,\" Peruto argued.\n\n\"But the doctor who made that diagnosis is not here,\" said Abraham. It was the last word.\n\nUnder her ruling, Apsche would be able to testify only about Heidnik's admission dates, what symptoms he was complaining about at the time, and what medication he was given. \"He can't say what some other doctor concluded,\" she said. \"He doesn't know if that doctor was drunk or deranged.\"\n\nFurther, Apsche would not be allowed to testify about test results listed in the records unless the raw examination data also were included. Armed with raw data, Apsche would be able to interpret the results on his own and not have to depend on someone else's conclusions. The catch was, none of the records contained all the raw data.\n\nPeruto wanted to wail in frustration. He would have to do the best he could within the restrictions.\n\n\"In your opinion,\" he asked Apsche, \"did Gary Heidnik know right from wrong?\"\n\n\"No,\" replied Apsche, \"in my opinion he did not know right from wrong.\"\n\n\"In your opinion, did he know the nature of his acts?\"\n\n\"He did not know the nature of his acts.\"\n\n\"Was he suffering from a mental disease?\"\n\n\"Yes.\"\n\n\"What disease?\"\n\n\"Severe schizophrenia.\"\n\nApsche spent the rest of the morning cataloging Heidnik's extensive experiences in mental hospitals stretching from the time he was in the Army virtually until he was arrested. There was one large gap, the period between 1972 and 1978, when Apsche was unable to find any admissions into Heidnik's usual circle of hospitals. Also there was a period between 1983 and 1986 when the records were not clear. During that period Heidnik was admitted to one hospital, walked away, and went to another. The record was confused because the first hospital still carried him on its rolls. Even though he was listed as a patient, he was not physically present in the institution. He could have been anywhere.\n\nFinally, testimony was getting to the real core of the case: whether Heidnik was insane or whether he was simply faking insanity. Peruto, of course, contended that Heidnik met the criteria for legal insanity. Gallagher, on the other hand, wanted the jury to believe that Heidnik had been faking his mental illness\u2014malingering\u2014since he was a private in the Army. His motive, Gallagher would further assert, was to collect his disability pay and Social Security. In an attempt to strengthen his own position and weaken Gallagher's, Peruto asked Apsche if any of the doctors who had examined Heidnik felt he had been malingering.\n\n\"Objection,\" interjected Gallagher.\n\n\"Sustained,\" said Abraham.\n\n\"Are there tests for malingering?\" Peruto asked, trying another tack.\n\n\"Yes,\" said Apsche.\n\n\"Was Gary Heidnik given these tests?\"\n\n\"Yes.\"\n\nBefore Peruto could widen this wedge, Judge Abraham cut him off, reminding him about the restriction on test results without the raw data. This touched off another heated discussion, during which Apsche quickly inserted, \"No, the evidence does not support the fact that he was malingering.\" Abraham shot him a forbidding glance.\n\nWith tempers running high, she called a lunch break, hoping that feelings would cool over a sandwich.\n\n* * *\n\nIt worked for a while. For the next hour and a half after the noon recess, the situation was relatively calm while Apsche completed his litany of Heidnik's convoluted paper path.\n\nThings heated up again, however, when Gallagher got the witness on cross-examination.\n\nThe DDA zeroed in on the 1983\u201386 period when Heidnik was difficult to track. When Heidnik was paroled in April 1983, he didn't go on the street. It was a condition of his release that he be admitted to the VA hospital at Coatesville. At some point between then and March 1986, when he was officially discharged from Coatesville, he was not physically at the institution.\n\nGallagher contended that Heidnik, in fact, left the VA hospital in November 1983.\n\nThat wasn't true, Apsche argued. Shuffling through his black box he produced a handful of papers. \"Here are nurse's notes showing he was in the hospital in '84, '85, and '86,\" he said.\n\nGallagher changed direction, attacking Apsche's records as \"sketchy.\"\n\n\"You said Heidnik checked himself into the VA hospital at Perry Point, Maryland, in April 1971, because he was depressed over his mother's suicide five days before?\" Gallagher asked.\n\n\"I didn't say that,\" Apsche answered. \"The record said that.\"\n\nDigging into a stack of papers, Gallagher handed a sheet to Apsche.\n\n\"What was Heidnik's mother's name?\" Gallagher queried.\n\nApsche said he didn't remember.\n\n\"It was Ellen Rutledge,\" said Gallagher. \"And what I've just handed you is a copy of a death certificate from Lake County, Ohio, for Ellen Rutledge. It is dated 1970, not 1971.\" His point was that Heidnik was claiming to suffer depression over his mother's death although it had occurred months previously rather than just a few days before.\n\nApsche argued that was irrelevant. It's not unusual, he said, for someone to become depressed a year or more after a traumatic event. That is called post-traumatic shock disorder. It is common among Vietnam veterans and others who go through awesome events.\n\n\"That makes two errors in your record,\" Gallagher contended, ignoring Apsche's answer.\n\n\"I don't think that's an error,\" said Apsche.\n\n\"I do,\" interrupted Judge Abraham.\n\nDigging again into his stack of papers, Gallagher produced another document and handed it to Apsche as well. It was a discharge summary from one of Heidnik's admissions to the VA hospital at Coatesville.\n\nGallagher asked him to read it.\n\nApsche looked at Abraham and raised his eyebrows. Included in the document was a diagnosis and several specific test scores\u2014two items he had earlier been prohibited from reading.\n\n\"You can read it,\" Abraham said. Apsche did, then tried to read a reentry note that was part of the same record. He wanted to get it before the jury because it indicated Heidnik was readmitted to Coatesville two months later, suffering from delusions and hallucinations. Abraham stopped him.\n\nApsche complained that the discharge summary was \"sketchy.\"\n\n\"Ah,\" said Gallagher. \"'Sketchy.'\"\n\n\"That's your word, not mine,\" Apsche retorted.\n\n\"Would you also say that your opinion that this man didn't know right from wrong is also 'sketchy'?\"\n\n\"No,\" Apsche answered emphatically.\nChapter 45\n\nJune 24, 1988\n\nChuck Peruto had one big gun. Dr. Kenneth Kool was known throughout the legal community as a courtroom psychiatrist. That is, he is one of a relatively small group of psychiatrists who makes his living by testifying. Sometimes he testifies for the defense. Sometimes he testifies for the prosecution. Over the years, he has appeared at more than a thousand trials. But before the jury could hear from Kool, there were some procedural matters to untangle.\n\nPeruto jumped up as soon as Judge Abraham took her seat and said he was considering calling Vincent Nelson, Josefina Rivera's former boyfriend. Nelson, Peruto said, would testify that he and Rivera discussed electric-shock torture a number of times before Rivera was taken captive by Heidnik. He also would say, Peruto promised, that when Rivera came to him on March 25 she wanted Nelson to go with her so they could rob Heidnik. It was Nelson, not Rivera, who insisted they call the cops, Peruto said. Nelson's availability to testify was not a problem, the defense lawyer added. He was in jail for robbery and illegally carrying a gun. He wasn't going anywhere.\n\nGallagher objected. \"Mr. Nelson's testimony would be irrelevant,\" he complained.\n\n\"No,\" Judge Abraham responded. \"Mr. Peruto can attack Ms. Rivera's credibility. He is entitled to impeach the witness.\"\n\nBut she had more weighty things on her mind. \"I'm worried about Dr. Apsche,\" she said. \"Nobody asked him if he was licensed. No one ever verified the authority, the verity of the records. At least, at least, he should have attempted to talk to the doctors. My real problem is whether that evidence is even competent. I think I want you gentlemen to go find some [legal] references for your various positions now that the jury has heard the testimony.\"\n\n\"It wasn't objected to,\" said Peruto. \"The Commonwealth has not made a motion to strike it.\"\n\n\"It's the judge who makes that decision,\" Abraham replied. \"It goes to the admissibility of the evidence that he testified to. The question here is not admissibility but competency. I don't have to wait for one of you to make a motion. Otherwise I might just as well wait in my office.\"\n\nAfter some discussion of previous cases that might serve as precedent, Gallagher launched an attack on a report Apsche filed before taking the stand. It outlined the areas his testimony would cover.\n\n\"His report is filled with errors,\" the DDA said. \"Now we have a report today from Dr. Kool in which he says he met with Dr. Apsche.\" Gallagher's argument was that Kool relied on Apsche's report, and if Apsche's report was wrong, Kool's testimony would be wrong as well.\n\nGallagher also complained that Peruto had not turned in reports from his psychiatric witnesses far enough in advance to allow him to study them. \"I was given Dr. McKenzie's report ten minutes after he took the stand. Now Mr. Peruto hands in a handwritten report from Dr. Kool. I don't think he has complied with the mandates of this court. Mr. Peruto's actions are not in accordance with the rules, and I ask that Dr. Kool not be allowed to testify. I also ask that Dr. Apsche's testimony be stricken.\"\n\n\"You can cross-examine Dr. Kool on Dr. Apsche's report,\" Judge Abraham said. \"But this just points out an infirmity in our criminal procedure... I place that blame on Mr. Peruto. It is a flagrant abuse of my orders.\"\n\nBut that was still not what really worried her. After perusing Kool's report, she said she feared that his testimony was going to infringe on the jury's prerogatives.\n\nKool's handwritten, three-page document, dated the previous day, addressed some touchy issues. Three, to be specific. Insanity. Diminished capacity. Guilty but insane. All different concepts but all dealing with a defendant's mental state at the time a crime was committed. Peruto's defense involved all of these concepts. He was covering all the bases.\n\nDiminished capacity, or diminished responsibility, is recognized as a defense to a criminal charge in many states, including Pennsylvania. When making such a plea, the defense admits that the accused is legally sane but is substantially lacking in mental capacity. As interpreted in the courtroom, it means that the defense may be admitting that the defendant is not insane but neither does he have the mental capacity to premeditate. That is, to plan a murder. Therefore, lacking the ability to form premeditation, he cannot be found guilty of first degree (that is, intentional) murder. In Pennsylvania a defendant can be sentenced to death only if convicted of first-degree murder. Any other murder conviction carries a maximum sentence of life in prison.\n\nIf Peruto could convince the jury that Heidnik did not have the mental capacity to plan a murder, he could only be convicted of a charge lesser than first degree. That would save him from execution.\n\nHowever, he was not going to get any support for diminished capacity from Kool. The psychiatrist's report was unequivocal on the issue. \"Diminished capacity is not to be considered in this case,\" he wrote.\n\nGuilty but mentally ill is a separate concept. Diminished capacity concedes a person is sane. So does guilty but mentally ill, but it has the added acknowledgment that a person is suffering from a severe mental illness. While this illness may not rise to the level of legal insanity, the very fact that he has this illness means that he cannot be held legally responsible for a crime to the same extent as someone who is not mentally ill. Generally, a person found guilty but mentally ill is treated at a hospital instead of being sent to prison. Once his mental illness has been cured, he begins serving his prison term.\n\nThis concept is causing quite a bit of trouble in the legal community. No one, it seems, has been able to get a good handle on it. Just about every guilty-but-mentally-ill law that has been passed has received serious complaints from one quarter or another. Legal scholars agree that the concept is good but the execution is flawed.\n\nJust as Peruto could not look to Kool to support a diminished capacity argument, neither could he ask the psychiatrist to back a guilty-but-mentally-ill one.\n\n\"I am of the opinion that the criteria for guilty but mentally ill are not met,\" Kool wrote. \"That finding requires a capacity Heidnik did not possess at the time. I find no evidence that he could discern he was delusional, no evidence of mental capacity to reflect upon, recognize, or have clear cognition of right and wrong or nature and quality\" (knowing what he was doing).\n\nBoth diminished capacity and guilty-but-mentally-ill findings differ from a straight insanity verdict. If a person is acquitted by reason of insanity, he can be committed to a mental institution but cannot be held responsible for his crime. Once he is declared cured of his mental illness or is no longer thought to be a danger to the public, he can be released. There is no prison sentence attached.\n\nPeruto was going to get support from Kool on Heidnik's alleged insanity. Strong support.\n\n\"It is my opinion,\" the psychiatrist wrote, \"that during that [November\u2013March] period, Mr. Heidnik was suffering from an exacerbation of his long-standing schizophrenia, and it was of such intensity that it denied him the mental capacity to know the nature and quality of his behaviors and denied him the mental capacity to know right from wrong.\n\n\"The basis for this opinion is that he was reacting to systematized delusion and had no conscious awareness of, nor insight into, the fact that he was psychotic at the time and that his distorted perceptions and behaviors were a direct product of that psychosis. His 'reality' was a delusion. Not having the mental capacity to recognize that fact, he reacted to what his psychosis caused him to perceive as real. The psychosis denied him the mental capacity to knowingly reflect and make informed decisions. His lifestyle was bizarre, chaotic, and regressed under the dominance of a profound psychosis... I am of the opinion that this is a case that meets the criteria for [insanity].\"\n\nThat was pretty strong stuff. But it was not what had Abraham concerned. She was worried that Kool's testimony was going to be too authoritative; that it was going to confuse the jury.\n\n\"In looking at Dr. Kool's report,\" she said. \"The problem I have is that he is saying that evidence of guilty but mentally ill is not here. I don't think it's appropriate for him to tell that to the jury. The doctor can say that the defendant was not suffering from a mental illness under the statute, but he cannot say the jury cannot return a verdict [to the contrary]. That usurps the function of the jury... How can he say a man is crazy, but not guilty but mentally ill? That's going to confuse the jury.\"\n\nPeruto bounded to his feet. \"He goes beyond guilty but mentally ill. He does not say Gary Heidnik doesn't suffer from mental illness.\"\n\n\"It causes the jury too much confusion,\" said Abraham. \"I can't let Dr. Kool tell the jury they can't find him guilty but mentally ill. He may not intrude into the fact-finding function.\"\n\nPeruto disagreed. \"He can intrude because he is an expert.\"\n\n\"No,\" Judge Abraham snapped. \"An expert may never intrude. He can give his opinion and that's all.\"\n\n\"He just wants to say it goes beyond guilty but mentally ill,\" Peruto argued.\n\n\"He may not overrule the legislature,\" Abraham said. \"The legislature has put that in the jury's prerogative.\"\n\nThat was the end of the discussion. Peruto sat down.\n\nBut Abraham was not through with the defense lawyer. She had seen the morning Inquirer, which had a story quoting Peruto as calling Dr. Clancy McKenzie a flake. \"I have a grave problem with a lawyer calling his own witness a flake,\" she said, glaring at Peruto. \"I don't want either of you,\" she added, including Gallagher in the order, \"calling McKenzie a flake to Dr. Kool.\"\n\nThe discussion had taken an hour and fifteen minutes. The jury, in the meantime, was kept sitting in an anteroom\u2014wondering, no doubt, what was going on that they were not privy to. When they were called in, though, they looked chipper and ready to work. A cool front had swept through during the night, clearing the blanket of smog. It was still warm in Room 653 because the air-conditioning had not yet caught up, but the inside temperature would drop during the day. That may have been due, in part, to Kool, who was every bit as calm and collected as his name indicated. He was a pro. Gallagher wasn't going to rattle him.\nChapter 46\n\nKenneth Kool is stamped in the mold of Gary Cooper when Gary Cooper was in his prime: tall, slim, laconic, unflappable. As a courtroom veteran, he was unintimidated by his surroundings. He looked more at home behind the bar than either of the lawyers facing him.\n\nIn his radio announcer's voice, speaking distinctly and confidently, he said he spent about seven hours examining Gary Heidnik and another three with Jack Apsche, going over Heidnik's voluminous records.\n\n\"Have you reached an opinion about Mr. Heidnik?\" Peruto asked overanxiously.\n\nSlow down, warned Judge Abraham: \"Lay your foundation first.\"\n\nIn response to a series of questions from Peruto, Kool said he had carefully studied Heidnik's medical history and had reached a conclusion based on the information he had and his experience studying mentally-ill defendants. \"I don't know every element [of his personality], but I have what I assume to be the highlights,\" he said.\n\n\"From November to March, did he appreciate the nature and quality of his acts?\" Peruto queried.\n\n\"No,\" said Kool, adding that Heidnik also did not have the capacity to know right from wrong.\n\n\"He could drive a car,\" said Peruto. \"Did that make any difference?\"\n\n\"No.\"\n\n\"If I told you he placed a body in New Jersey, would that change your opinion?\"\n\n\"No.\"\n\nUnder questioning from Peruto and Judge Abraham, Kool repeated the details he had already outlined in his report: that Heidnik was delusional and did not know it; that he was psychotic; that he had an unbalanced perception of reality; that his lifestyle was bizarre and regressed.\n\n\"What is a delusion?\" Peruto asked.\n\n\"A delusion is an unreality perceived by the victim of it to be a reality,\" answered Kool.\n\n\"Are you aware of his goal?\"\n\n\"I'm aware of his delusional goal,\" Kool began. \"He had evolved a paranoid and grandiose delusion that God wanted him to produce a number of children. He looked upon it as a partnership with God. This was a fixed delusion, and he had it for a long time. It was totally out of context with reality. He believed that if he were detected, he would be prevented from carrying out this delusion.\"\n\nPeruto asked Kool if he thought Heidnik knew what he was doing was unlawful.\n\n\"He had some awareness of man's law, what the laws of the Commonwealth were, but he saw God's law as superior. He did not have the capacity to reflect upon these things.\"\n\n\"Did you see any progression in his medical records?\"\n\n\"Yes, I did. I saw the predictors of what would occur. The fact that he kidnapped a young lady and kept her in a closet in his basement was one of those predictors. His writing the parole board and signing the letter 'G. M. Kill' was another thing I saw as an indicator. Another was his profound reaction to the loss of his known children. The neglect of his wife when she was pregnant... history indicates all the elements were there for a tragedy.\"\n\n\"You would agree,\" interrupted Abraham, \"that two psychiatrists can disagree, and one can be just as qualified as another?\"\n\n\"Yes,\" answered Kool.\n\n\"Would it change your opinion if Gary Heidnik had been mute with you?\" Peruto asked.\n\n\"No,\" said Kool, \"but it would change the basis on which I formed my opinion.\"\n\n\"Would it change your opinion if I told you that the first thing he wanted to do after dumping Deborah Dudley's body was to buy a newspaper to check his stocks?\"\n\n\"No.\"\n\nPeruto ticked off several things that would seem to indicate that Heidnik had periods of apparent normality and asked Kool how this squared with an opinion of insanity.\n\n\"His psychoses are primarily in the areas of reproduction\u2014having babies and completing his pact with God,\" replied Kool. \"He does not have it in areas that are not conflicting.\"\n\n\"You often see malingerers, do you not?\" asked Peruto, changing course.\n\n\"Yes.\"\n\n\"Did you find any history of malingering throughout his history?\"\n\n\"No,\" answered Kool. \"I'm of the opinion that he is not sane and has not been so over the years. There are things that auger against claims of his malingering. Look at how many times he wanted to remain in hospitals when they wanted him to leave. His writing the parole board. His habit of stockpiling medicine. His remaining mute. There was never anything to be gained by malingering over all those years. There was no advantage; no secondary gain.\"\n\n* * *\n\nWhen Charles Gallagher took over on cross-examination, his approach was quite different than it had been with McKenzie and Apsche. Gallagher was almost deferential to Kool.\n\n\"You saw no evidence of malingering, right?\"\n\n\"Right.\"\n\n\"Did you compare his hospital admissions with the dates he filed claims with the Social Security Administration and the VA?\"\n\n\"I don't remember those specifically.\"\n\nGallagher outlined the circumstances of Heidnik's discharge from the Army and how his pension benefits were increased from ten percent to a hundred percent.\n\n* * *\n\nBefore he could get deep into his examination, however, Peruto asked Judge Abraham to excuse the jury so he could discuss something else. The issue was Vincent Nelson. Peruto said he had dropped his plans to call him to the stand.\n\n\"Mr. Peruto was concerned [Nelson] was going to commit perjury in order to aggrandize himself,\" Abraham announced after a brief sidebar conference.\n\n\"He may have been seriously mistaken [about some of his claims],\" Peruto added.\n\n* * *\n\nWith the jury back in, Gallagher resumed his questioning of Kool. First, though, he handed Kool a sheet of paper and asked him if he had ever seen it before.\n\n\"No,\" said Kool. \"I did not see this.\"\n\nUnder Gallagher's prompting, Kool read extensively from the document, which dealt with Heidnik's examinations at military hospitals in Germany and the United States. Included was a diagnosis by an Army doctor that said Heidnik had a schizophrenic personality but was not psychotic.\n\nIt was striking that Judge Abraham allowed Kool to read a previous doctor's diagnosis\u2014at Gallagher's urging\u2014while she would not permit Apsche to read diagnoses at Peruto's request. Asked about this during the lunch break, Abraham said: \"This was information he had not seen, and it might make him change his opinion. This was a different witness. This was a doctor who had spent seven hours examining Heidnik and he testified that he based his opinion in part on records. This was a record he didn't have. This was a whole different kettle of fish.\"\n\nDespite Peruto's objections, all of which were overruled by Judge Abraham, Gallagher presented document after document to Kool and asked him to read from them. One was a report of a security check done on Heidnik in 1965 in response to an application for employment. Gallagher presented this because he wanted to get on the record that Heidnik had trained for several months as a psychiatric nurse with the VA.\n\n\"If someone has an IQ of 148, what does that mean to you?\" Gallagher asked.\n\n\"IQ scores are relative. Ninety to 110 is average; 148 is very high.\"\n\n\"How about 130?\"\n\n\"That's superior to very superior.\"\n\n\"On May 24, 1978, Gary Heidnik was evaluated...\" Gallagher began.\n\n\"Objection,\" yelled Peruto.\n\n\"Overruled,\" said Abraham.\n\n\"... and he had a score of 120 on the verbal section, 127 on performance, and 130 full scale.\"\n\nAgain Judge Abraham had allowed Gallagher to present the same type of evidence she had forbidden Apsche and Peruto from proffering. She told Apsche he could refer to test results only if he had all the raw data and scored them himself. She did not ask Gallagher if he had the raw data. He did not volunteer the information.\n\nChanging direction, Gallagher asked Kool if he agreed with McKenzie's theory of the cause of schizophrenia.\n\n\"Very, very few psychiatrists share the theory that early life experiences are the sole factor in determining schizophrenia,\" Kool said, wording his reply carefully. \"Many feel it is primarily a genetic illness... [but] all the votes are not in yet. Like cancer, it is a very complicated phenomenon and is not curable. Medication complicates it. There is a justifiable place in psychiatry for all these theories... we don't have a real handle on schizophrenia yet.\"\n\nGallagher asked: What percentage of psychiatrists agreed with McKenzie that early-life experiences were paramount?\n\n\"Much, much less than five percent regard it as the exclusive cause,\" Kool said.\n\nGallagher repeated Heidnik's pension-benefit history and asked if he thought that indicated that Heidnik was malingering.\n\n\"No,\" said Kool. \"In my opinion he has a major mental illness.\"\n\n\"Do you think he was telling you the truth when you examined him?\"\n\n\"Yes.\"\n\nGallagher wanted to know if he was certain.\n\n\"I can't read minds,\" Kool replied. \"I'm giving you my best opinion. I can't tell if someone is lying.\"\n\nStill pursuing the issue, Gallagher detailed Heidnik's frequent transfers between mental hospitals when he was serving his prison sentence for his attack on Alberta Davidson.\n\n\"I don't have any knowledge of this,\" said Kool.\n\n\"Doesn't all this indicate to you that he is a malingerer?\"\n\n\"No,\" Kool said forcefully.\n\nWith a nod to Judge Abraham, Gallagher indicated he had finished his cross-examination of Kool. After the trial, on the late-night talk show with Larry King, Gallagher would brag about how he \"demolished\" all three of the defense's psychiatric witnesses. If Kool, for one, had been demolished, he didn't act it.\n\nSince Kool was the last defense witness, Gallagher began his rebuttal after lunch. His first witness was Dr. Robert Sadoff, another courtroom professional on the same level as Kool. Like Kool, Sadoff testifies for either side. This time he was lined up with the prosecution. Like Kool, he was virtually unflappable.\nChapter 47\n\nCharles Gallagher yanked at the back of his jacket and called his witness in a loud, clear voice: \"Robert Sadoff.\" A neatly dressed man with short, brown hair and a relaxed manner walked briskly to the witness stand and raised his right hand. Like Kenneth Kool, Sadoff had been in so many courtrooms, he probably carried his own Bible.\n\n\"I've testified in federal and state courts in twenty states,\" he admitted modestly when Gallagher questioned him to establish his credentials. \"I've testified before a Senate committee to discuss what should be done with people who are found guilty but mentally ill.\" And Gallagher, anticipating an insanity defense by Heidnik's lawyer, signed him up to testify for the prosecution within weeks after Heidnik's arrest.\n\nDespite the long-term commitment, Sadoff wasn't able to get much out of Heidnik. He spent about twenty minutes with him before he gave up in disgust. Heidnik refused to talk. The two never exchanged a word.\n\nIt didn't matter, though, he told Gallagher. He fell back on Heidnik's medical records, his personal business records, police records, and testimony from other expert witnesses.\n\n\"What is the legal definition of insanity?\" was Gallagher's first question.\n\n\"The difference between someone who is insane and someone who is mentally ill is that the insane person doesn't know the nature and quality of his act and he doesn't know right from wrong.\"\n\n\"Did Gary Heidnik know what he was doing when he picked up those girls?\"\n\n\"In my opinion, he did. He wasn't just grabbing people off the street as a person with a seventeen-month-old brain would do.\" He couldn't resist the dig at McKenzie.\n\n\"Did he pick them up to satisfy his sexual desire?\"\n\n\"I'm not so sure that's wrong. Men do it all the time.\"\n\n\"Did he know what he was doing when he choked and cuffed them?\"\n\n\"In my opinion, yes.\"\n\n\"Did he know it was wrong?\"\n\n\"In my opinion, he did know.\"\n\nFor the next ten minutes Gallagher led Sadoff through a series of similar questions covering all of Heidnik's major acts during the four-month period. Sadoff answered each one affirmatively, prefacing each answer with, \"In my opinion...\"\n\n\"Based on your review, do you have an opinion about Gary Heidnik?\"\n\n\"I think he is probably schizophrenic. I would agree with those who call him that.\"\n\n\"Is he a malingerer?\"\n\n\"At times he has shown evidence of exaggerating his mental illness... of being manipulative.\"\n\nGallagher asked if his mutism was a sign of mental illness or a sign of malingering.\n\n\"I think he was exaggerating his mutism, for whatever reasons,\" Sadoff answered. \"He would choose not to talk, to present a psychotic image to the doctors, but behind the scenes he was doing what he always did.\"\n\nThe DDA asked if Sadoff thought Heidnik had been malingering when he presented himself for admission to the Maryland hospital in 1971 soon after he had asked the Social Security Administration to reconsider its action on his request for benefits.\n\n\"It was either incredibly coincidental, which I don't think it was, or this was a very bright man who decided to go to Maryland where he could have a bright new start [with a new Social Security office].\"\n\nHe added that he also thought Heidnik was malingering when he was serving his prison sentence. His motive was to stay in mental hospitals because it was more pleasant than being in the prison's general population. No matter where he was transferred within the system, Sadoff said, Heidnik always managed to let his stock broker know where he was.\n\n\"Is it possible for someone to fake mental illness?\"\n\n\"I think one can fake mental illness easier than insanity. Mental illness is not hard to fake.\"\n\n\"In your opinion did he have a major mental illness that caused him not to know the nature and quality of his acts?\"\n\n\"Maybe he had a major mental illness,\" Sadoff said, \"but the evidence indicates he was not so deprived of his reason that he did not know the nature and quality of his acts.\"\n\nGallagher asked Sadoff if he thought Heidnik should be held responsible for his acts.\n\n\"You have to hold a person responsible or not responsible,\" said Sadoff. \"There is nothing to indicate that Heidnik did not know what he was doing was wrong at the time he was doing it.\"\n\n* * *\n\nJust as Gallagher had been unable to shake Kool, Peruto was unable to shake Sadoff. In response to a question from Peruto if he had ever been wrong in his opinions, Sadoff calmly admitted that in some cases the courts had found differently. \"But I'm not sure it was my mistake or someone else's.\"\n\nAlthough it was only 3:45, forty-five minutes before the usual quitting time, Judge Abraham was so pleased with how rapidly the trial was progressing, she decided to knock off for the weekend. While she initially planned Saturday sessions, things were going so smoothly she cancelled that idea.\n\nBefore she adjourned, however, she picked up a document and read into the record the results of an IQ test given to Heidnik on March 16, 1987, nine days before he was arrested. The test was part of the battery ordered by Judge Levin in family court. She said Heidnik scored 148.\n\nBy doing this, Judge Abraham violated her own prohibition on test results levied when Apsche tried to read some scores to the jury. She also did not say that the test on which Heidnik scored a 148 is generally regarded by testing experts as inferior to the type of test on which Heidnik twice scored a 130. Apsche was trying to get the 130 score before the jury as well as the results of the tests Heidnik took which purported to show he was not faking his symptoms.\n\nJudge Abraham also announced that she had decided not to erase Apsche's testimony. Doing that, she said, \"may strike a blow to the defense that they needn't suffer.\" Peruto didn't say thank you.\nChapter 48\n\nJune 27, 1988\n\nOnce the psychiatrists were through with their main testimony, the tension in the room lifted like a hot-air balloon. The end was in sight now and everyone seemed to be relieved. When the jury was called in, the change was evident. They had more spring in their step and looked almost festive. Six of the men wore ties. Gary Heidnik felt it too. Earlier he showed up wearing a blue, Oxford cloth, long-sleeve shirt carefully buttoned at the wrist. It was too large by several sizes, but it was clean and pressed. It was the first time in fifteen months he had not worn his faded Hawaiian shirt during a public appearance. Even the weather was cooperating. Another cool front moved in over the weekend, and the temperature inside Room 653 was the most comfortable it had been since the proceeding started.\n\nBefore the jury was summoned, however, there were\u2014as usual\u2014several procedural matters to haggle over.\n\nCharles Gallagher told Judge Abraham that among the mop-up rebuttal witnesses he intended to call was someone to testify that Heidnik had bars put on the downstairs windows of his house in 1985. This indicated, he said, that Heidnik was planning his abductions even then.\n\nAbraham shook her head. \"I think you're going way overboard,\" she said. It was one of her rare rebukes to the prosecution.\n\nGallagher also wanted to recall police lieutenant James Hansen to testify about some books police recovered in their search of Heidnik's house.\n\n\"What kind of books?\" Abraham asked.\n\n\"One was Criminal Law in a Nutshell,\" said Gallagher. \"One was on abnormal psychology. One was on how to buy stocks. There was one about the Pinkerton detective agency, and there was a New York phone book.\"\n\nPeruto objected. \"Some of those might be prejudicial to the defendant,\" he argued. \"Plus, there is no indication when he got the books. If the prosecution can be selective, I'd like to show what else was in the house.\"\n\n\"Fine,\" Gallagher responded. \"I'll bring the other dozen boxes or so over here too.\"\n\n\"What is the relevance?\" Abraham wanted to know.\n\n\"To show he is a malingerer,\" said Gallagher.\n\nIn the end Gallagher did not introduce the books. Nor did he call the workman who put the bars on the window.\n\nBut the DDA was not the only one running into roadblocks over witnesses. Peruto told Judge Abraham he would like to call Betty Heidnik, but Abraham balked.\n\n\"There's no benefit to your client to have this woman produced,\" the judge said. \"Your client is very dangerous to women, and this witness could be too prejudicial. I will not let her talk about how dangerous she thought he was.\"\n\n\"As an attorney, I feel it is not prejudicial,\" Peruto argued. \"I think it's helpful.\"\n\n\"It renders your defense useless,\" Abraham countered. \"It shows he was cold and calculating. It doesn't help him at all. All it proves is that he is a cold-blooded killer.\"\n\n\"It is prejudicial not to let it in,\" Peruto persisted.\n\n\"There goes your insanity defense then,\" said Judge Abraham. \"You might as well plead guilty. It shows he is a cold-blooded, calculated killer. It doesn't advance your theory of the case. It shuts it down.\"\n\n\"If the Commonwealth is allowed to introduce doctors and reports... I can't take it in a vacuum,\" Peruto said. \"The jury has to have it all. They have to have the whole thing.\"\n\nAfter all the argument, Betty Heidnik was not called. On reflection, Peruto conceded she could only have damaged his case. Not only could she testify about how he abused her, but her recitation of Heidnik's promiscuity would wreck his argument that he was having sex with the women in his basement only to reproduce.\n\nThat did not mean there was a shortage of last-minute witnesses. In a blur nine people trooped to the stand as prosecution rebuttal witnesses to bolster Gallagher's contention that Gary Heidnik was sane. Among them was Heidnik's stock broker, one of Heidnik's former girlfriends, the sales manager from the Cadillac agency, the psychologist who tested Heidnik in March 1987\u2014just before his arrest\u2014and a psychiatrist from a VA outpatient clinic who saw Heidnik three times during the four-month period he was holding the captives.\n\n* * *\n\nWhen Robert Kirkpatrick took the stand, Heidnik underwent a remarkable transformation. So far during the trial Heidnik seemed to exhibit little interest in the testimony or the arguments. Occasionally he conversed with Peruto, but for the most part he simply stared at the wall behind the judge's bench and rocked back and forth on the back legs of his straight-backed chair. But when his former broker took the stand, Heidnik became animated. When Merrill Lynch's representative spoke, Heidnik listened. Intently.\n\nKirkpatrick was Heidnik's broker because he happened to answer the telephone when Heidnik called one day in December 1974. The broker said the caller introduced himself as \"Bishop Heidnik\" and said he was interested in opening an account. Several weeks later he followed up the call with a letter. Enclosed was a check for $1,500. The account was opened on March 21, 1975, in the name of the United Church of the Ministers of God.\n\nOver the next dozen years, Kirkpatrick said, he saw Heidnik only about four times in person because he normally handled affairs by telephone or through the mail.\n\n\"Did anyone else control this account?\" Gallagher asked.\n\n\"No,\" answered Kirkpatrick.\n\nThe broker produced a sheet showing how the account grew progressively larger. By October 27, 1979, the balance was $21,000. By July 31, 1982, it had increased tenfold, to $256,000. By the following May it had taken another large jump, to $361,000. For the next couple of years it fluctuated slightly. But by November 28, 1986\u2014two days after Heidnik captured Josefina Rivera and the day before he took Sandra Lindsay\u2014the balance was $545,000.\n\nDuring most of this time, Kirkpatrick testified, Heidnik was an enthusiastic player and kept in touch frequently. In May 1983 he wrote Kirkpatrick and reminded him not to forget the church's thirty-five percent discount. At another time he chided Kirkpatrick for not transferring $60,000 from an inactive account to an active one so it would draw interest.\n\n\"Did he ever lose money on Crazy Eddie stock?\" Gallagher asked.\n\n\"No, he did not,\" Kirkpatrick answered.\n\n\"What kind of investor was he?\"\n\nPeruto objected, but Judge Abraham overruled.\n\n\"An astute investor,\" the broker said.\n\nKirkpatrick severely damaged the defense. The picture he painted of Heidnik was of a bright, fervent investor who knew what he was about. The best Peruto could do was try to show that underneath, Heidnik was not as rational as he appeared on the surface.\n\n\"Did he ever tell you he was raising ten babies in his basement?\"\n\nKirkpatrick admitted he had not.\n\n\"If he had, what would you have done?\"\n\n\"I would have reported it to my superiors immediately and recommended that the account be closed.\"\n\n\"Every time you saw him, was he wearing the same clothing?\"\n\n\"You mean the same style?\"\n\n\"No, I mean the same clothing.\"\n\n\"I have no idea,\" Kirkpatrick said, explaining that there was as large a gap as six years between their meetings.\n\nPeruto also wanted to clarify the Crazy Eddie issue.\n\n\"When you said he didn't lose any money on Crazy Eddie, you mean he didn't sell it at a loss?\"\n\n\"Yes.\"\n\n\"But if he had sold it, he would have been down considerably?\"\n\n\"Yes.\"\n\n* * *\n\nKirkpatrick put a big dent in Peruto's insanity plea. So, too, did Gallagher's next witness, a former girlfriend named Shirley Carter.\n\nA forty-year-old black woman suffering from cerebral palsy, Carter testified she had a sexual relationship with Heidnik extending from the period before he went to prison until not long after he got out in 1983. She testified that Heidnik was very intelligent, frequently telling her how to invest her money and what stocks to buy. \"He even told me how he set up a church so he wouldn't have to pay taxes,\" Carter said.\n\n\"Did he ever do anything bizarre?\" Gallagher asked.\n\n\"Not really,\" Carter answered.\n\nCarter's most damning evidence against Heidnik, however, would not be brought out. In a folder on the table in front of him, Gallagher had a copy of a ten-page handwritten letter from Heidnik to Carter. It was dated June 9, 1982. At the time, Heidnik was still serving his prison sentence.\n\nBeginning, \"Dear Sweety bun,\" the letter dealt mostly with financial issues. It read:\n\nSay things go good for us, I get paroled and sent to Coatesville. You then move out there, have the two children we're planning on (Lisa and Gary Jr.)... when we achieve this situation, it involves an increase in our income of about $700 a month... The good part is that this is extra money. I'll still continue getting my $1,800 a month also. That doesn't even count the money from the stock market. That will give us a tax exempt income of over $70,000 a year. WOW! If you figure we're also getting other benefits, like free health care, that brings it up to about $75,000 a year. WOW! Since this is all tax exempt, it's equivalent to someone else making and paying taxes on about $110,000 a year. WOW!! This isn't even taking into consideration other plans I have for increasing our income (like a couple of choice boarders, getting welfare or SSI, who will also be willing to help out around the house with things like cooking and cleaning...). No wonder you don't have any interest in working. In many ways, including financial, it seems our union was made in heaven...\n\nThe rest of the letter dealt with problems Carter was having with her mother, a used car Heidnik wanted to buy for her, how he thought he was given an unjust prison sentence, his scores on the IQ tests (\"I passed with a 130 score, which is pretty high\"), and about how he had applied for a department-store credit card. \"If I get it, you'll get a nice little present and Price [Carter's son] will get a basketball set. It includes a basketball and pump even. So if you get a basketball, I'm sure you'll know it's for Price and not for you (he, he, he).\"\n\nGallagher had wanted to introduce the letter, but Judge Abraham had ruled it was too remote to apply to the current case. While it would have tainted Heidnik's insanity plea in one way, it would have helped in another. Even then Heidnik showed a preoccupation with having children. More than once in the letter he talked about their plans to produce a son and daughter. \"... Lisa and Gary Jr. I like writing their names,\" he penned. He also laid down a plan he would follow once he got out: buy a house, find a couple of boarders drawing Social Security and use their income as well as their manual labor. Conceivably, Apsche or Kool could have argued convincingly that these were still more predictors.\nChapter 49\n\nAt first Dr. Richard Hole didn't want to talk about Gary Heidnik. \"Don't worry about the doctor-patient privilege,\" Judge Abraham told the psychiatrist from a Philadelphia VA outpatient clinic. \"You can feel perfectly free to answer the questions.\"\n\nOn the third Wednesday of December 1986, Hole said nervously, Heidnik unexpectedly came back to the clinic for treatment after a ten-month lapse. When he left, he was participating in group-therapy sessions and taking 200 milligrams a day of Thorazine.\n\nHole said Heidnik was not complaining of any symptoms \"nor did he show manifestations of ongoing or poorly treated schizophrenic illness.\"\n\nWhen asked to define Heidnik's problem, Hole said he was a paranoid schizophrenic, but his illness appeared to be under control.\n\n\"I asked him if he was depressed,\" Hole said, \"and he denied it.\" He also denied he was suicidal, paranoid, hallucinatory, or delusional. \"He essentially denied all psychiatric symptomatology,\" Hole said. Nevertheless, the psychiatrist prescribed more Thorazine.\n\nIn January and February Heidnik returned and his condition appeared not to have changed. That was the last personal contact Hole had with him, although he said Heidnik telephoned in mid-March. He wanted Hole to write a letter to the court asking for a renewal of visiting rights with his eight-year-old daughter.\n\nDuring cross-examination Hole denied that Heidnik exhibited a common symptom of schizophrenia, the numbness of spirit called the \"flat affect.\"\n\n\"If he had, he wouldn't have given a big smile to the other patients he hadn't seen in ten months.\"\n\n\"Did you perform any tests to show he was not taking his medication?\" Peruto asked.\n\n\"No,\" Hole replied, adding that Heidnik did not show any signs of a \"poorly controlled schizophrenic,\" so he assumed he was taking his Thorazine.\n\nIf he had no symptoms, Peruto asked, why did he need medication?\n\n\"To prevent an outbreak or a relapse of the illness,\" Hole replied.\n\nEarlier Gallagher had proffered the theory that Heidnik had not taken his medication (evidenced by the fact that a considerable amount of Thorazine was found in Heidnik's house after the arrest) because it would reduce his sex drive. Peruto asked Hole if that were true.\n\nThe amount of Thorazine he had prescribed was unlikely to impair a healthy young man's sex drive, Hole said.\n\n\"Hold on,\" Judge Abraham interrupted. \"That's not the reason he was called. He was called only to talk about his experiences with Heidnik between December and February.\"\n\n\"Four times a day with four different women?\" Peruto said, plunging on.\n\n\"Objection,\" yelled Gallagher.\n\n\"Sustained,\" snapped Abraham.\n\n\"Do you have any reason to believe he was malingering or faking it?\"\n\n\"Based on the time frame of December, January, and February, I would say no.\"\n\n\"He was not faking his medical symptoms?\"\n\n\"Correct.\"\n\n* * *\n\nThe final major prosecution witness was Eva Wojciechowski, the court psychologist who examined Heidnik on Judge Levin's orders. The examination took place on March 16, 1987, some five weeks after Sandra Lindsay died and two days before Deborah Dudley was electrocuted.\n\nAlthough he scored a 148 on the IQ test, Wojciechowski explained that the type of test given him was not as authoritative as the test that is usually given, called the Wechsler Adult Intelligence Scale, or WAIS. When Heidnik was evaluated in 1978 after being charged with abducting Alberta Davidson, and three other times as well, he was given the WAIS. Twice he had scores of 130.\n\nGallagher asked the psychologist how common a score of 148 was.\n\nOnly one half of one percent of the population scores in that range, she said. Such a score is near genius.\n\nGallagher dwelled on the score at length, implying that Heidnik was too smart to be crazy.\n\nWhen Peruto got the witness, his first question was aimed at undermining the foundation of Gallagher's theory that Heidnik's mental illness was an elaborate hoax.\n\n\"Is there any indication he was faking?\" Peruto queried.\n\n\"No,\" Wojciechowski said decisively.\n\n\"What was the purpose of the tests?\"\n\n\"To see if he was employable.\"\n\n\"Was he?\"\n\n\"I didn't think so,\" replied Wojciechowski. \"[The tests showed] he has very strong, aggressive, violent tendencies, and he is apt to go off if he doesn't get what he wants.\"\n\n\"What do you think he wanted?\"\n\n\"I think he wanted custody of his son.\"\n\n* * *\n\nAlthough Gallagher had called more witnesses than expected, the trial was winding down on schedule. Peruto planned to recall Kenneth Kool on Tuesday to counter Sadoff, and that would be the end of testimony. After that the two lawyers would present their closing arguments. The jury would have the case by Wednesday afternoon.\nChapter 50\n\nJune 28, 1988\n\nChuck Peruto's first question to Kenneth Kool was totally predictable. What in the world, he wanted to know, does IQ have to do with schizophrenia?\n\n\"Nothing at all,\" Kool answered calmly. \"Schizophrenia is a major mental illness that can develop in any human being.\"\n\n\"When Dr. Hole testified yesterday, he said he asked Heidnik if he was delusional. Is that a proper question?\"\n\n\"No,\" Kool said with a fleeting grin. \"Asking a schizophrenic if he is delusional is kind of ludicrous. They will say no because they perceive their delusion to be reality. They will deny it. Asking them that is an exercise in futility.\"\n\n\"What about malingering [when he was being interviewed by Hole]?\"\n\n\"If he wanted to malinger, he would have told the doctor he had all those mental problems.\"\n\nPeruto asked Kool if he placed any significance on the fact that the Thorazine prescribed by Hole was found in Heidnik's house.\n\n\"That makes it even more likely that without his medication his fixed delusional system was operating,\" Kool stated.\n\nHow could Kool explain Heidnik's different behavior? Peruto asked. How could he do some things\u2014negotiate for a car, buy stocks\u2014and act the way he did at other times?\n\n\"His delusions were not involved with buying cars or stocks,\" answered Kool. \"They were in the area of his relationships with people, particularly women and children. There you can see a marked contrast which is explainable in terms of his psychosis.\"\n\nAs soon as he got the witness, Gallagher jumped to his feet. \"So Gary Heidnik had selective delusions?\" he asked.\n\n\"No,\" replied Kool. \"That is not the word. 'Selective' suggests an ability to turn it on and off. Heidnik had no control.\"\n\nGallagher: \"His delusional spigot was on when he was in the house and off when he was out buying cars?\"\n\nKool: \"This man did not reveal his delusional behavior to other people. It fits. He was hiding his behavior the same way he did the bodies.\"\n\nGallagher hammered at the delusional concept, bringing in Heidnik's church, his testimony during the support hearing before Judge Levin, and his treatment at Farview when he was under the prison sentence.\n\n\"Isn't this man pretending?\" he asked pugnaciously.\n\n\"Nooo waaaay,\" Kool shot back.\n\nWhen Kool stepped down, Judge Abraham sent the jury out of the room. She wanted to lay down the ground rules for closing arguments.\n\n\"I don't want counsel to talk about penalty,\" she said. \"That doesn't mean you, Mr. Peruto, can't mention guilty but mentally ill.\"\n\nTurning to Gallagher. \"Here's something to remember. You can say Mr. Heidnik has been released from mental institutions twenty-two times, ergo, he might be released again.\"\n\nTo both of them: \"I don't want any inflammatory language. Don't bring up other cases. Keep to the evidence, logical inferences, and your position in regard to the case.\"\n\nJudge Abraham said Peruto would go first. He objected, but was overruled. In Pennsylvania it is customary for the prosecution to have the last say. The rationale is that it is fair because the prosecution has the burden of proof. However, many legal scholars consider such a situation a tremendous advantage for the prosecution. Each lawyer could talk as long as he wanted, Abraham said.\n\n* * *\n\nPeruto took the floor at 11:06. He was wearing a luxurious light-gray suit and a red-and-gray rep tie. His blue shirt had the initials ACP embroidered in dark thread, running vertically down the front, parallel to the placket.\n\n\"The question is not whether Gary Heidnik did these heinous acts,\" he said softly, \"but whether or not he was insane. We're not contesting that these women were raped, that these women were kidnapped, that these women were killed. What we're here to determine is the level of culpability of the defendant. Even though we have conceded that these acts took place, we are not conceding first-degree murder\u2014the specific intent to kill.\"\n\nWalking briskly back and forth before the jurors, he suggested they consider what the prosecution was asking them to believe.\n\n\"Let's say he's a malingerer. A faker. That he went into the Army with plans to develop a paranoid schizophrenic personality so one day he could make a living that way... Could he fool all those doctors all the time? Could he fake mental illness\u2014schizophrenia\u2014for twenty-five years? You have to believe that. You have to believe he did this for twenty-five years so when he got caught building his family in his basement, he could say he was insane. That's the prosecution's case. Does that make sense?\"\n\nPeruto's voice rose emotionally. The jurors remained stone-faced, but they followed him carefully with their eyes.\n\n\"What was Gary Heidnik's purpose? His purpose was to raise ten kids, not to kill anybody.\" He was punishing the women for disobeying him, he said, not trying to murder them. \"Third-degree murder is reckless disregard for human life. This is a classic case of third degree.\"\n\nOne of the women was retarded, he said. One was illiterate. Three were prostitutes. \"As sick as it is, these were his chosen people. These were the girls he wanted to reproduce with. Is that sane? It's a case of Dr. Jekyll and Mr. Heidnik. Isn't it more likely he's insane than not?\n\n\"What kind of mentality does it take to have human flesh in front of you. A human being. And to cut through that body. To cut through flesh. To cut through bone, and to take some of those body parts and wrap them up and put them in the freezer. And then to cook some and feed it to the others? Who was he trying to impress with that delusion? The body was wrapped, but he wasn't.\"\n\nSandra Lindsay's mother, who was sitting in the front row of the packed courtroom, began sobbing. Another relative took her arm and led her out.\n\nPeruto seethed when he talked about Josefina Rivera. \"She was feeding a sick mind... she knew he was sick and she turned that to her advantage... Listen to Lisa [Thomas]: she said Rivera was laughing about beating the others. That she did it with or without Gary. Did Lisa make that up? What is her motivation? Ladies and gentlemen, I am going to argue to you that Josefina Rivera acted criminally... that she did a couple of things that went too far. She was putting sugar on his plan.\"\n\nSwitching back to Heidnik, he summed up. \"I don't want a verdict based on sympathy,\" he said. \"Nor do I want one based on prejudice... I want you to find him not guilty based on insanity.\"\n\nPeruto sat down. It was 11:51. He had spoken for exactly forty-five minutes.\n\n* * *\n\nJudge Abraham gave the jury a brief recess, and then it was Gallagher's turn. It was four minutes after noon when he began.\n\nStanding solemnly before the jury, Gallagher searched his left jacket pocket, then folded his hands in front of him. \"I want you to rely upon your good old-fashioned common sense,\" he said quietly. \"Rely upon your powers of observation. Go over the evidence with me.\"\n\nTurning quickly, he raised his arm and pointed at Heidnik, who was, as usual, staring at the rear wall, apparently oblivious to the proceedings.\n\n\"This man,\" Gallagher said, his voice rising, \"repeated sadistic and malicious acts upon six victims. He planned it. He did it, and he concealed it. Ladies and gentlemen, I submit to you\u2014make no mistake about it\u2014that this man committed murder in the first degree... It's clear that Sandra Lindsay and Deborah Johnson Dudley were killed as a result of being taken into that basement. It's clear that Gary Heidnik did it... It was premeditated. It was deliberate. It was intentional.\"\n\nAll anyone knew about Heidnik, he said, was what Heidnik had told them. \"None of the experts can read a person's mind. None of them can tell when a person is lying. But I submit that you can tell. You all have common sense because you deal with people day in and day out.\"\n\nGallagher said Heidnik planned each of the abductions, choosing the women, \"plying them with sex or food,\" choking them, handcuffing them, and chaining them in the basement.\n\nHe killed Lindsay and Dudley \"in a cold, premeditated fashion,\" he said, and then disposed of Lindsay's body by dismembering it and cooking the parts that could lead to her identification\u2014the head and the hands.\n\n\"Just because someone does bizarre acts, the law doesn't recognize them as insane... What he did was premeditated, deliberate murder.\"\n\nHe, too, had been pacing. Then he stopped and looked intently at the jurors. \"Reject this defense,\" he said. \"Reject the idea that this man is legally insane. Seek the truth, and I think you will find that this man, Gary M. Heidnik, is guilty beyond reasonable doubt of the specific intent to kill two young girls.\"\n\nWhen he quit it was 12:49. His argument, by remarkable coincidence, had also taken precisely forty-five minutes.\n\n* * *\n\nJudge Abraham looked at the clock and announced that she was giving everyone the afternoon off. \"I need to prepare my charge,\" she said, warning that it would be lengthy.\n\nWhen Peruto finished his argument, he whispered to Heidnik as he slid into his chair, \"How'd I do?\" Heidnik simply grunted.\n\nThen, after Gallagher's argument, when he was being led out of the courtroom back to a cell, Heidnik stopped when he was opposite Gallagher. Turning to Peruto, he said, \"Your speech was better than his.\" The jurors were gone. They did not hear the remark.\nChapter 51\n\nJune 29, 1988\n\nOvernight Room 653 was transformed from a courtroom into a classroom. Behind the judge's bench, where before there had been nothing, was a large green chalkboard and a three-foot-tall paper pad on an easel. Written on the top sheet of the pad, in large black letters, were the names of the captives. The chalkboard was blank.\n\n\"What is this?\" Peruto wanted to know, referring to the equipment as \"props.\"\n\n\"I always use these when I instruct a jury,\" said Judge Abraham. \"You guys use charts and graphs to help you out, why shouldn't I?\"\n\nBefore calling the jurors in, Abraham went through the usual morning ritual of discussing issues they didn't need to know about. The topic that morning was what she had included in her charge, that is, what suggestions she had culled from lists submitted by Gallagher and Peruto and what she had decided upon on her own. One thing she wanted to put before the jury was the issue of Josefina Rivera as a possible accomplice.\n\n\"I object to that,\" Gallagher said.\n\n\"Okay,\" Judge Abraham said amiably. \"But I think it's an issue for the jury. It's not a question of law, but one of fact for the jury to decide.\"\n\nTo help make the jurors' job easier, Abraham added, she had removed the misdemeanor charges from the list. \"They'll merge with the felonies anyway,\" she pointed out.\n\nIt was almost ten-thirty. \"Anyone who wants to leave the courtroom has to go now,\" the court crier announced. \"No one will be allowed in or out during the judge's charge to the jury.\"\n\nAbraham estimated it would take two hours. She was off by only half an hour.\n\n* * *\n\nJudge Abraham was in her glory; a captive audience and a chance to exercise her inherent pedantic tendencies. \"Sometimes I may have to spell words to you that you've never heard before,\" she began condescendingly.\n\nWalking over to the chalkboard, she scratched, PRESUMPTION OF INNOCENCE.\n\n\"The fact of Mr. Heidnik's arrest, the fact that charges have been filed against him, the fact that he is on trial, doesn't mean he's guilty. He is presumed innocent.\"\n\nHer hand moved again, BRD, she wrote. \"That means Beyond Reasonable Doubt\u2014beyond the doubt that any reasonable person would consider. You must find Mr. Heidnik not guilty when the Commonwealth doesn't prove his guilt... you can't hold it against him because he did not testify... you have to assess the credibility of everyone in this case,\" she said, writing the word CREDIBILITY on the chalkboard. \"That means believable.\"\n\nUnder credibility she wrote DEMEANOR.\n\n\"That means how they behave. It is, part of the test of credibility.\"\n\nShe scratched, MOTIVE.\n\n\"Motive is a reason for testifying one way or the other,\" she said, adding an equal sign and the word bias beside motive.\n\n* * *\n\nAt 11:25, an hour into the charge, she took a short recess. She still had not gotten to the crux of the matter. After the break, though, she plunged in.\n\n\"You have several options for verdicts,\" she said. \"One is 'guilty.' Another is 'not guilty.' Another is 'not guilty by reason of insanity.' There is 'guilty but mentally ill.' Finally there is 'diminished capacity,' which is not really a verdict but an evidentiary issue, and it applies only to first-degree murder.\n\n\"A verdict of 'not guilty' means the state has failed to prove its case... To render a verdict of 'not guilty by reason of insanity' you must first conclude that he committed the crimes, but was legally insane... A verdict of 'guilty but mentally ill' means he is guilty beyond a reasonable doubt and he's not insane but mentally ill.\"\n\nShe explained that insane was a legal term, not a medical one. \"Doctors use the term 'mental illness.'\"\n\nTo be found legally insane, she said, Heidnik would have to meet the conditions set down in the M'Naghten Rule. \"If at the time of the crime he is, as a result of mental disease, unable to know the nature and quality of his acts or unable to distinguish right from wrong, he meets the legal test. Otherwise, he is not legally insane.\"\n\nA mental illness alone, she cautioned, cannot qualify someone as legally insane. And, she added, legal insanity must be proved by the defense by a preponderance of the evidence.\n\n\"Did Mr. Heidnik suffer from a mental disease or mental defect? Did he know what he was doing? Was he capable of doing it? If he knew what he was doing would cause injury or death, he knew the quality of his act. Did he know it was wrong?\" These were questions they would have to answer, she said.\n\n\"If you first find him guilty, and the Commonwealth has proved he was sane but mentally ill at the time, he can be found guilty but mentally ill.\"\n\nHow do you know if someone is mentally ill? \"If he lacks substantial capacity either to appreciate the wrongfulness of his conduct or is incapable of doing anything about it.\"\n\nDiminished capacity kicks in, she added, only in first-degree murder. \"That's where the defendant admits he committed the crime of murder but says he is not guilty because he was unable to meditate and couldn't form a specific intent to kill.\" In effect it lowers murder of the first degree to murder of the third degree. \"If he had a fully formed intent to kill and was not laboring under a mental defect, then it's first degree.\"\n\n* * *\n\nFollowing another five-minute recess, she meticulously outlined the degrees of murder.\n\nFirst-degree murder is premeditated. \"The killer had a specific intent to kill. He was able to formulate a plan and carry it out.\n\n\"Second degree murder also is known as felony murder. That is when death occurs in some special felonies, such as arson, burglary, or kidnapping.\" Since Heidnik was charged with multiple counts of kidnapping, she stressed that word. \"The Commonwealth doesn't have to prove beyond a reasonable doubt that there was a specific intent to kill in second-degree murder,\" she said.\n\n\"Third-degree murder is where there is no intent to kill, but there is an intent to inflict grievous bodily harm.\"\n\nShe glanced at the clock again. \"Let's all stretch,\" she said, standing and throwing her arms wide.\n\nShe was nearing the end now. In the next twenty minutes she raced through the other charges: rape, kidnapping, involuntary deviate sexual intercourse, and aggravated assault.\n\n\"Take your time,\" she told the jurors, \"don't rush.\" It was one o'clock. Eat your lunch and then you can start deliberations, she said, sending them out of the room.\n\nPeruto and Gallagher were to come back after lunch, she said, so they could work through the huge pile of exhibits, to determine what would be sent to the jury room as evidence. The state alone had filed almost two hundred items.\n\n* * *\n\nDuring the lunch hour two inmates unhappy with their day in court and looking for a victim jumped Heidnik while the three were alone in a holding cell. They beat him to his knees and were kicking him when the deputies heard the commotion and broke it up. He was not seriously injured. It was the second time since he was arrested that he had been attacked. Earlier, another inmate had smacked him in the face, breaking his nose.\nChapter 52\n\nJune 30\u2013July 1, 1988\n\nWaiting is never easy. While the Pittsburghers were off in a back room\u2014presumably arguing, discussing, and sifting through documents\u2014reporters, members of the captives' families, and a handful of diehard spectators loitered in the dreary hallway outside Room 653. Just about everybody, including Chuck Peruto, expected a quick verdict. The consensus also was that a quick verdict would be very bad for Gary Heidnik. But there was nothing quick about this jury. At five-thirty Judge Abraham told them to quit for the day.\n\nThe next morning, Friday, Abraham took pity on the crowd and opened the courtroom. It was better than the hallway. At least the hangers-on would have chairs and air-conditioning. Just before noon the jurors sent a note asking Judge Abraham for additional instructions. That required a gathering of all the participants: Abraham, Gallagher, Peruto, and, of course, Heidnik.\n\nThey had four questions, Abraham said.\n\n1. What are the various degrees of murder?\n\n2. At what point did Heidnik begin receiving one hundred percent disability?\n\n3. What is legal insanity?\n\n4. What is mental illness?\n\nThey would have to decide number 2 on their own, because that was a matter of fact for the jury to remember, Judge Abraham decided. In order to answer the other questions, she summoned them back into the courtroom. Again the doors were locked while she went through an abbreviated version of the charge she delivered the day before. It took an hour.\n\nAfter the jury was sent back to argue some more, Judge Abraham looked at Peruto and Gallagher and shrugged. \"The diminished-capacity provision is mumbo-jumbo,\" she acknowledged, \"and the guilty-but-mentally-ill statute is not well-written. But that's the law. I didn't write the constitution. What do you want me to do?\"\n\nPeruto, not to be outdone in the quote department, praised the Pittsburgh jurors for their diligence. \"The people in Philadelphia didn't even think he deserved a trial,\" he quipped, \"much less deliberations.\" Pressed for a prediction, he smiled. \"Something less than first degree.\"\n\nSurely, veteran court watchers felt, the verdict would be forthcoming quickly now. Peruto was so confident that things were rolling, he hung around Room 653 for the rest of the afternoon, chatting with reporters and members of the victims' families. Although the family members had been hostile to him initially, they had warmed during the trial. Now they were feeling him out about representing them in possible suits against the hospitals that turned Heidnik loose. Philadelphia is a very litigious community.\n\n* * *\n\nFriday morning came and went. As the hours went by, Peruto's spirits rose. Surely they couldn't be out this long, he thought, if they were going to come back with a hanging verdict.\n\nFinally, shortly after three o'clock, after about sixteen hours of deliberations over two and a half days, the jurors sent word that they were ready.\n\n\"I think it's going to be second degree,\" a grinning Peruto whispered to a reporter as the participants assembled.\n\nBefore the jurors were called in, Judge Abraham issued a stern warning to the spectators. No clapping, shouting, cheering, or crying. \"There will be no expression of pleasure or displeasure,\" she cautioned, \"under threat of contempt and immediate jailing.\" Reporters, too, would have to wait until the proceeding was over to leave the room.\n\nWhen the jurors filed in, they looked grim and tired. Although they passed directly in front of Heidnik and Peruto, not one of them glanced at the defense table. That was a bad sign.\n\nAfter aligning themselves in the jury box, a petite, brown-haired woman with a heart-shaped face and intelligent, dark eyes, identified herself as the foreperson. She was Betty Ann Bennett, a registered nurse married to a suburban Pittsburgh cop for twenty-two years. One by one she went down the list of charges: guilty in the first degree of murdering Deborah Dudley... guilty in the first degree of murdering Sandra Lindsay... guilty, guilty, guilty. Eighteen guilties: guilty on two counts of first-degree murder, five counts of rape, six counts of kidnapping, four counts of aggravated assault, and one count of involuntary deviate sexual intercourse. There was one acquittal\u2014an involuntary deviate sexual intercourse charge involving Josefina Rivera. They gave no explanation for that finding.\n\nWhen Peruto asked that they be polled, they stood individually and repeated the findings. Bennett, Kimberly Higgins\u2014a young, single woman\u2014and Marcella Lenhart, a bright-eyed brunette with three small children, appeared about to break into tears. But one juror, August Manfredo, glanced malevolently at Heidnik every time he said \"guilty.\"\n\nPeruto took it worse than Heidnik. When the first verdict of guilty of murder in the first degree was read, he deflated. He put his head in his hand and stared at the floor. Heidnik, on the other hand, didn't blink. He just sat upright, staring straight ahead, rhythmically pumping his legs.\n\nThe jury had returned at three-thirty. At 3:53, with the verdict read and the jurors polled, Judge Abraham called a brief recess.\n\nAs reporters scrambled out the door and dashed for the telephones, a deputy walked up to Heidnik and asked him if he wanted a glass of water. \"No,\" he said quietly, still staring at the wall. Then he stood and put his hands behind his back to be cuffed. Still showing no emotion, he was led out of the room. When he came back, he pointedly stared at the area of the courtroom where the captives' family members had sat throughout the trial. And as he folded into his chair, he glanced over his shoulder and stared again in the same direction. It was the first indication he knew they were there. His gaze was not friendly.\n\n* * *\n\nUnder Pennsylvania law a jury that returns a guilty verdict for first-degree murder must also set the penalty. They have two choices: life in prison, or death in the electric chair. It is a second, separate verdict requiring another immediate deliberation session. But before they could begin, the lawyers would have a chance to talk to the jury again, to argue aggravating and mitigating circumstances\u2014their reasons for why he should or should not be sentenced to death. This procedure is called the penalty phase. Judge Abraham decreed it would begin at nine the next morning.\n\nAs the jurors marched out of the room, they again ignored Heidnik and Peruto. Some courthouse veterans noted this and commented that it did not bode well; that they may already have made up their minds about the punishment.\n\nIn the hallway the reporters clustered around Gallagher. \"The jury rejected his plea of insanity,\" said the beaming DDA. \"They decided this man had the specific intent to kill, and he did it with malice.\"\n\nPeruto was in a rare morose mood. The jury acted emotionally, not intellectually, he said. \"There is absolutely no evidence of first-degree murder. There's no question in my mind that Gary Heidnik is severely insane.\"\nChapter 53\n\nJuly 2, 1988\n\nThe Saturday morning Inquirer, in three-quarter-inch letters across the top of the front page, screamed: HEIDNIK IS CONVICTED OF MURDER. And, in smaller type, His plea of insanity is rejected by jury. That almost guaranteed a fair crowd at the penalty session, even though it was held early on a Saturday morning in the middle of one of the most popular holiday weekends of the year.\n\nWith the jury still out of the room, Gallagher jumped up to make one of his most forceful objections of the trial. He questioned Judge Abraham's decision that allowed Peruto to make the last argument before the jury began deliberations. \"The position of the Commonwealth is that the Commonwealth always goes last,\" he asserted.\n\n\"It doesn't hurt the Commonwealth if it goes first and the defense goes last in this situation,\" Abraham said. \"I'm going to stick to that.\"\n\nReluctantly, Gallagher sat down.\n\n\"In the penalty phase,\" the judge said, turning to Peruto, \"the defendant has the right to testify and call witnesses, including his wife. Does he want to testify?\"\n\n\"No, your honor,\" said Peruto.\n\n\"Does he want to call any witnesses?\"\n\n\"No,\" Peruto repeated.\n\nPeruto said earlier he was considering calling Betty Heidnik. Indeed, she was in the courtroom that morning for the first time during the trial\u2014an attractive, olive-skinned woman demurely dressed, hiding behind a pair of large, dark sunglasses. She sat in the back, with Dr. Clancy McKenzie, and left before the jury came back with its final verdict. Peruto said he decided not to call her because he was afraid her testimony would hurt him more than help him.\n\nWith the formalities taken care of, the jury was called in. They looked more rested, but the strain of the last few days was still clearly visible. They looked pale and grim, as if dreading the task still facing them. One man walked as though he were treading through a mine field; that the slightest misstep would result in disaster.\n\nGallagher began his final argument at 9:31. \"The law isn't simple, but it's clear,\" he said. \"If you find by unanimous decision that there are aggravating circumstances and no mitigating circumstances, the decision must be death.\"\n\nDressed in a conservative gray suit and solid white shirt, Gallagher paced slowly in front of the jury box.\n\n\"You've already found there was specific intent to kill Sandra Lindsay,\" he said. \"She was chained, cuffed and naked, hung by her wrist for eight to twenty-four hours. She was not only mentally retarded, but she had trouble swallowing. She suffered from starvation and malnutrition. There is evidence to indicate she was beaten while she was hanging there... Gary Heidnik gave the other girls ice cream while Sandra Lindsay was hanging there. He said she was faking\u2014a word he knows too well.\n\n\"In the case of Deborah Dudley,\" he continued, \"she was tortured. She was taken upstairs and shown the remains of Sandra Lindsay's body and told to behave or it would happen to her. She was repeatedly beaten, a tape was put around her head, and screwdrivers were driven in her ears.\" Then, he added, she was killed.\n\n\"Don't rely upon leniency for this man or mercy for this man,\" he urged them. \"Rely upon the evidence.\"\n\nHis speech lasted seventeen minutes. It was Peruto's turn.\n\n\"Before you sentence someone to die, it is my belief that you should know the whole picture,\" he began. \"I can't argue what I know. I can't argue what Dr. Apsche knows. I can only argue the evidence.\n\n\"Look at him,\" he said, swiveling and pointing at Heidnik, who appeared oblivious to the situation. \"He's full of Thorazine. His behavior today is not what it was like between November and March. When he formed, fully formed, his specific intent to kill either one of them\u2014\"\n\nJudge Abraham interrupted. \"You can't argue that at this point. That's already disposed of.\"\n\nThrown off stride, Peruto lowered his head and pinched the bridge of his nose between his thumb and forefinger. He stood that way for four or five seconds, looking defeated. Lifting his head, he turned to the jury and said softly: \"I just hope you do the just thing and do not rely on procedural rules.\" Slowly, he returned to his chair and sat down. He had taken only four minutes.\n\nJudge Abraham spent twenty-three minutes going over the details of deliberations with the jurors before sending them out to consider the options.\n\nFifteen minutes after the jury left the room, a court official whispered that they had ordered lunch. It looked like another long day. Peruto, ever the optimist, predicted they would not reach a verdict by Sunday evening and the judge would dismiss them and impose the mandatory life sentences.\n\n* * *\n\nHe was wrong. At 12:10, only one hour and fifty-five minutes after they began deliberations, they sent word that they had a verdict.\n\nAfter Heidnik was brought back in and took his seat, Peruto turned and flashed him a questioning look. Heidnik just shrugged.\n\nBetty Ann Bennett, looking grimmer than ever, stood and announced the decisions: death for the murder of Sandra Lindsay; death for the murder of Deborah Dudley.\n\n\"I thank you for your service, not for your decision,\" Judge Abraham said quickly.\n\nHeidnik, as on the previous day, showed no emotion. Peruto winced, folded his arms on the table and laid his head down. He stayed that way for almost a minute. Gallagher sagged in relief.\n\nSpectators had been warned again about demonstrations, so there were no overt displays. However, one of Lindsay's sisters sobbed silently.\n\nPeruto stood and asked Judge Abraham to issue an order to prison officials to make sure Heidnik was kept in isolation. \"If he is put in the general population, the jury's wishes will be carried out immediately,\" he said.\n\n\"If your client is going to commit suicide, he's going to do it,\" she said. \"Prison officials don't want that, but neither do they want prisoners killing other prisoners. I don't have the authority to order him held in isolation, but I will suggest they carefully watch him for suicide and carefully house him.\"\n\nTo set the wheels in motion, she immediately sentenced Heidnik to death for Lindsay's murder and set a date three months in the future for sentencing on Dudley's killing.\n\nOutside, Denise Dudley told reporters she was satisfied with the verdict. \"My sister can rest in peace,\" she said. \"We got what we wanted.\"\n\nLindsay's mother, Jeanette Perkins, said she was glad it was over. \"I'm happy for what happened,\" she said, \"because he sentenced my daughter to death.\"\nChapter 54\n\nJuly 1988\n\nGary Heidnik spends most of his time sitting in his cell, staring at the wall. Sometimes, officials said, he reads his Bible. Not much else for him to do. His former recreational activities are a thing of the past. No more video games. No more movies. No more cars. No more women.\n\nSince he is under death sentence\u2014and since he lives constantly under threat of harm from other inmates\u2014authorities at the State Correctional Institution at Pittsburgh keep him isolated. Prisoner number F1398 is confined to a cubicle by himself in a cell block referred to as a restricted housing unit. There are another hundred or so men in the section. Some of them are, like Heidnik, murderers, awaiting execution. Some are informers, put there for their own protection. And some of them are just plain mean. Too mean and generally too unfit for the company of others. For one reason or another they all are segregated.\n\nBut even among such a group Heidnik gets extra attention. When he showers, he does it alone. If he exercises, he does it alone. He eats his meals in his cell. Alone.\n\nIsolation is a distinct form of punishment. But his punishment is also his protection. On his way to the Pittsburgh prison, when guards stopped the bus to transfer other inmates, Heidnik was left alone momentarily with other prisoners on the bus. As soon as the guards' backs were turned, they jumped him, and were still beating him when the guards rushed back to break it up. Such an incident, in prison slang, is called \"a tuning up.\" Although Heidnik was not severely injured, when the other inmates finished with him, he was well tuned.\n\nOn the other hand, no one put him in a hole in the ground. No one beat him with shovel handles. No one dug in his ears with screwdrivers. And no one plugged him in to see if he shined like a floodlight at Three Rivers Stadium, a few minutes' walk away. Not yet anyway. It is ironic that his is one of the rare cases where the ultimate punishment coincidentially fits his crime. If he is electrocuted, he will die the same way his second victim, Deborah Dudley, died.\n\nIt is ironic, too, that he was assigned to a prison in Pittsburgh. Just on the other side of the wall are the men and women who convicted him; the men and women Heidnik came all the way across the state to pick to hear his defense. He was sent initially to the state's main prison at Graterford, near Philadelphia, for two weeks of probing, testing, and processing. What, if anything, the prison people discovered has not been disclosed. Probably nothing that has not already been regurgitated. Heidnik is a master at hiding things.\n\nThere are 1,600 men in SCI-Pittsburgh, and in varying degrees they all miss their freedoms. At no time, probably, is this more evident than on Sunday afternoons in autumn and early winter, when they can hear the crowds cheering the Steelers. Undoubtedly this sets off strong reminders of lost liberty. A mournful sound. The urban man's call of the wild goose. They remember what it was like to cheer a team. To drink a beer. To have a high time with a lady.\n\nNobody ever said anything about Gary Heidnik being a football fan. There was testimony that he drank only rarely and then very temperately. But the women! Gary Heidnik always had at least one woman in the house.\n\nOn the other hand, maybe thoughts of freedom are fleeting for Gary Heidnik. He has been in one institution or another since he was fourteen. It began when he assumed the rigid discipline of military school. Then he went into the Army. His story is, he didn't want to leave. Then he didn't want to leave the hospitals. Perhaps now he's finding a home behind the walls. At least one psychiatrist who treated him during his chain of hospitalizations noted that Heidnik was becoming excessively institutionalized. Now he is institutionalized in spades.\n\nThis time, though, he's not in the military. Not in a hospital where he can check himself in or out. Where he can slip away AMA\u2014Against Medical Advice. He's not even back in Farview, or one of the institutions where his girlfriend used to visit him and they'd get it on in the ladies room; in the elevator; in the bushes. He's in a real prison now. Not exactly hard time, but close enough.\n\nStill one more irony is that, despite his imprisonment, his income continues: $1,212 every month from the VA, and another $708 in Social Security. It goes into a bank account watched over by the courts. He has no place to spend it.\n\nA few reporters wrote him asking for interviews. He turned them down. While he was still at Graterford he had two visitors one day: Betty and his infant son J.J. Whether they will travel across the state to see him has yet to be seen.\n\nGary Heidnik's prospects of ever getting out of prison are zilch. The best he can hope for is to have his death sentences lifted or be transferred to a mental hospital. If he lives that long. Some don't think he will. Chuck Peruto for one.\n\nAfter the closing arguments, but before the case went to the jury last June, a glum Peruto slumped behind the marble pedestal desk in his office and unburdened himself of his melancholy thoughts. Gary Heidnik didn't stand a snowball's chance in hell of getting any mercy, he figured. Even, even, if he was not sentenced to die, Heidnik was at the end of his road, Peruto said.\n\n\"I'll make a prediction,\" he volunteered. \"I predict Gary Heidnik's going to kill himself. I predict he'll be dead before your book hits the stands.\"\n\nThe question no one was able to answer\u2014in fact, which no one even addressed\u2014was why Gary Heidnik was on the street to begin with. Ten years previously, Judge Mirarchi had sensed impending catastrophe and sent Heidnik away for as long as he could. Other psychiatrists subsequently recommended that he not be released. Certainly he had drawn the attention of law enforcement officials and mental health authorities more than once. Yet he was unconfined.\n\nPart of the problem undoubtedly is a result of the tension that exists between the courts and mental health professionals. There was no disagreement during Heidnik's trial that psychiatry is an inexact science. But viewed in the light of what occurred, that term becomes meaningless. To say that the majority of the conclusions about Heidnik were \"inexact\" is a tremendous understatement. To say they were based on science is even more ludicrous. If those diverse opinions are among the best the \"science\" can offer, there is only one conclusion: judging the degree of someone's mental illness is guesswork.\n\nThe demands of the criminal justice system contribute to the confusion. The law wants absolutes. Psychiatry cannot produce them. Unhappily, a solution is not on the horizon. Twenty-six states still use M'Naghten as the formal test for legal insanity. M'Naghten was promulgated 145 years ago. Since then, apparently, there have not been sufficient advancements in psychiatry to devise a more reliable indicator of a person's mental state. We simply don't know what makes a man like Gary Heidnik tick. Even more frightening is the comprehension that no one\u2014not the police, not the courts, not the mental health people\u2014knows how many other Gary Heidniks are out there.\nEpilogue\n\nIn the aftermath:\n\n\u2022 Josefina Rivera, Lisa Thomas, Jacquelyn Askins, and Agnes Adams melted back into their pre-Heidnik existences. All the captives, and Heidnik's wife Betty, as well, are protected from reporters by a screen of attorneys they have hired to get part of the money in the Merrill Lynch account.\n\n\u2022 Philadelphia Daily News reporter Maria Gallagher telephoned Michael Heidnik, Gary's father. She asked him if he wanted to know what the verdict was. \"I'm not interested,\" he said. \"I don't care. It don't bother me a bit. All I want is for you people to leave me alone. I don't care what happens.\"\n\n\u2022 Chuck Peruto went straight from Heidnik's trial to another homicide case. He looked back on the trial with some bitterness, claiming the jury was too overwhelmed by the viciousness of Heidnik's acts to objectively view the evidence of his insanity. He filed notice of appeal, citing alleged errors by Judge Lynne Abraham in, among other things, allowing DDA Gallagher to introduce material through Peruto's witnesses, Jack Apsche and Kenneth Kool, that the defense attorney was prohibited from presenting. Despite his efforts, he was pessimistic about the chances for a second trial. \"Heidnik was lucky to get one trial,\" he commented cynically, \"there's no way he's going to get another.\"\n\n\u2022 Charles Gallagher continued to lambast Heidnik. \"Is Gary Heidnik crazy?\" he echoed in response to a question. \"Crazy like a fox! He has a personality disorder. He has a medical problem. But he's not legally insane. He definitely is not suffering from any major mental illness.\" To what did he attribute his victory? \"I thought I destroyed any evidence the defense presented.\" On several talk shows afterwards, Gallagher made an issue of saying he was the only one who knew the \"real\" Heidnik, but he was so restricted by the system that he could not present the \"whole story.\" He and police lieutenant James Hansen were considering writing a book about the case.\n\n\u2022 Tony Brown, Heidnik's friend and former driver, was pronounced a free man. Two weeks after Heidnik's conviction, at Gallagher's request, Judge Charles L. Durham dismissed all charges against him.\n\n\u2022 The federal bankruptcy judge is not expected to make any decision on the complicated case until well into 1989. In addition to the victims, the Peace Corps is seeking some of the funds, and the IRS is looking to see if the bulk of the money should not go to the government for taxes. These proceedings may delve into the question of the legitimacy of the church. Was Gary Heidnik religiously motivated? Was it a tax scam?\u2014questions that were not addressed during the criminal trial.\n\n\u2022 Judge Lynne Abraham declined to be interviewed about the Heidnik case, citing the danger it might present to the appeals process. Gallagher on the judge: \"Her rulings were right down the middle. She was a good judge in that I wanted to do some things for the Commonwealth, but she wouldn't let me.\" Peruto: \"If I had any other judge, I might have won this case.\"\n\n\u2022 Dr. Jack Apsche began work on another case for Peruto and was making plans to enroll in law school.\n\n\u2022 Dr. Clancy McKenzie was trying to find a lawyer to represent him in a slander action against Peruto for calling him a \"flake.\"\n\n\u2022 An official appraiser hired by the city examined Heidnik's house at 3520 North Marshall Street and evaluated the three-bedroom, one-bath dwelling at $3,000, which was $12,900 less than he paid for it in 1983 and $11,000 less than he still owes on it. Repairs\u2014including a new roof, replacement of broken windows, and filling in holes in the basement floor\u2014would cost an estimated $15,000. In a separate note, the appraiser wryly added: the estimated value \"does not account for possible market resistance due to any stigmatization that may be caused by the notoriety of alleged acts committed on the premises by the current owner.\"\n\nAmong the stacks of documents and pieces of information that did not get an airing at the trial, one of the most revealing was a fifteen-page typewritten manuscript composed by Heidnik under the appropriate pseudonym \"Gary Bishop.\" Entitled \"Fortieth Street Soaps or Life in the Slow Lane,\" it appears to be the beginning of a book. It has a title page, a dedication (to Shirley Carter), a prologue, and twelve pages called \"Chapter One.\" It is about the underground culture that thrives in the neighborhood of Fortieth and Market Streets, around the Elwyn Institute. It talks about the disadvantaged\u2014the mentally and physically handicapped\u2014who rely on each other for support, companionship, and love.\n\nIn the story, the author-protagonist, obviously Heidnik, is reemerging into the world after more than four years in state hospitals for the criminally insane. It tells how he goes to stay with a friend he identifies as John, probably his old friend, John Francis Cassidy. In the story the author spends some time with an old girlfriend (Carter?). He shops for a van. He goes to see his old friends at McDonald's. He becomes attached to his host's strange cat. It is a rambling story, going nowhere really. But it is an upbeat tale, not a depressing one. While it no doubt accurately reflects what was happening in Heidnik's life at the time, it does not go into the past or give any indication what was going to happen in the future.\n\n* * *\n\nAnother revealing tidbit was a brief interview with Albert Leavitt, a psychologist for the Common Pleas Court. Leavitt's colleagues twice examined Heidnik, giving him a full battery of tests.\n\n\"Do you think Heidnik is crazy?\" I asked.\n\nLeavitt paused for a moment. \"Let me put it this way,\" he said. \"If there were an Olympics for craziness, Heidnik would be a gold-medal winner.\"\n\nHe's not just manipulating the system?\n\n\"He's not manipulating anything. He's absolutely stone crazy.\"\n\nIn Leavitt's opinion, Heidnik is on the same level as Charles Manson and Jim Jones, the charismatic leader who set up an ill-fated colony in Jonestown, British Guiana. Heidnik should be put under a microscope, Leavitt said, because he is the rare perfect example of an insane person as defined by M'Naghten.\n\n* * *\n\nTwenty-one times Gary Heidnik entered mental health facilities. Twenty-one times he was discharged. Even when he said he wasn't ready to go. Even when he begged to stay.\n\nThe truth is, Gary Heidnik should never have been free. If some of the people he came into contact with\u2014just the people he met during the four-month period in which he was holding the captives\u2014had been a little more perceptive, a little less cautious, a tad less cynical, a smidgen less bureaucratic, Sandra Lindsay and Deborah Dudley might be alive.\n\nIf Sergeant Armstrong had carried his investigation into Sandra Lindsay's disappearance just one step further... if Officer Aponte had shown a little more curiosity about what Heidnik was cooking for dinner... if Dr. Hole had been a little more searching in his examination and a little less anxious to dash off a prescription... if Judge Levin, during the hearing in family court, had followed his instincts a little more closely... the story may have had a different ending.\nA Cellar Chronology\n\nNovember 26, 1986 | |\n\n\u2014Gary Heidnik offers Josefina Rivera twenty dollars for sex, takes her home and makes her his first captive.\n\n---|---|---\n\nNovember 29, 1986 | |\n\n\u2014Sandra Lindsay, an old friend of Heidnik's, is taken captive and joins Rivera in the basement.\n\nDecember 22, 1986 | |\n\n\u2014Heidnik picks up teenager Lisa Thomas, woos her with a cheeseburger and clothing from Sears, and locks her in the cellar.\n\nJanuary 1, 1987 | |\n\n\u2014Deborah Dudley becomes number four.\n\nJanuary 18, 1987 | |\n\n\u2014Eighteen-year-old Jacquelyn Askins becomes the youngest member of the horror harem.\n\nFebruary 7, 1987 | |\n\n\u2014Sandra Lindsay dies. Heidnik dismembers her body; tries to destroy parts of it by cooking.\n\nMarch 18, 1987 | |\n\n\u2014Deborah Dudley is electrocuted. Heidnik stores her body in his basement freezer, alongside the ice cream.\n\nMarch 22, 1987 | |\n\n\u2014Heidnik disposes of Dudley's body by dumping it in a New Jersey park.\n\nMarch 23, 1987 | |\n\n\u2014Heidnik and Josefina Rivera pick up Agnes Adams and add her to the captives group.\n\nMarch 24-25, 1987 | |\n\n\u2014Rivera escapes and goes to police. Heidnik arrested.\n\nJune 20, 1988 | |\n\n\u2014Heidnik goes on trial for killing Lindsay and Dudley, plus multiple charges of kidnapping, rape, aggravated assault, involuntary deviate sexual intercourse, and others.\nTITLES BY KEN ENGLADE\n\nCellar of Horror\n\nMurder in Boston\n\nBeyond Reason\n\nDeadly Lessons\n\nA Family Business\n\nA Dark and Deadly Love\n\nBlood Sister\n\nHot Blood\n\nEverybody's Best Friend\n\nFROM THE TRUE CRIME LIBRARY OF ST. MARTIN'S PAPERBACKS\nCELLAR OF HORROR\n\nCopyright \u00a9 1988 by Ken Englade.\n\nAll rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles or reviews. For information address St. Martin's Press, 175 Fifth Avenue, New York, NY 10010.\n\nISBN: 0-312-92929-3\n\nEAN: 80312-92929-9\n\nSt. Martin's Paperbacks edition \/ March 1989\n\nSt. Martin's Paperbacks are published by St. Martin's Press, 175 Fifth Avenue, New York, NY 10010.\n\neISBN 9781466853270\n\nFirst eBook edition: August 2013\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n### Table of Contents\n\nCopyright\n\n\"HORIZONS OF DECEIT\n\nPrologue\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nChapter Thirteen\n\nChapter Fourteen\n\nChapter Fifteen\nSpace: 1889 & Beyond\u2014Horizons of Deceit, Book I\n\nBy Jonathan Cooper\n\nCopyright 2013 by Jonathan Cooper\n\nSpace: 1889 \u00a9 & \u2122 Frank Chadwick 1988, 2014\n\nCover Design & Art \u00a9 Tom Webster and\n\nUntreed Reads Publishing, 2014\n\nSpace: 1889 & Beyond developed by Andy Frankham-Allen\n\nThe author is hereby established as the sole holder of the copyright. Either the publisher (Untreed Reads) or author may enforce copyrights to the fullest extent.\n\nThis ebook is licensed for your personal enjoyment only. This ebook may not be resold, reproduced or transmitted by any means in any form or given away to other people without specific permission from the author and\/or publisher. If you would like to share this book with another person, please purchase an additional copy for each person you share it with. If you're reading this book and did not purchase it, or it was not purchased for your use only, then please return to your ebook retailer and purchase your own copy. Thank you for respecting the hard work of this author.\n\nThis is a work of fiction. Any resemblance to the living or dead is entirely coincidental.\n\nOther Titles in the Space: 1889 & Beyond Series\n\nJourney to the Heart of Luna\n\nVandals on Venus\n\nThe Ghosts of Mercury\n\nA Prince of Mars\n\nAbattoir in the Aether\n\nDark Side of Luna\n\nConspiracy of Silence\n\nMundus Cerialis\n\nLeviathans in the Clouds\n\nA Fistful of Dust\n\nThe Forever Journey\n\nhttp:\/\/www.untreedreads.com\n\n\"HORIZONS OF DECEIT\n\nBOOK I\"\n\nBy Jonathan Cooper\nPrologue\n\nThe Bomb and the Dead Men\n\n1.\n\nTHEY WERE THE fingers of an artisan, and they worked quickly\u2014he didn't know how much time they had. The choking smog that descended, night after night, on the streets of Westminster had bought them a few minutes, but no more. They were out in the open, and even with Potsdam on guard he felt vulnerable. The opalescent sheen of soot and chemical residue would mask them only for so long\u2014if the prying eyes of Scotland Yard's finest got within a few feet of the monument even the dimmest of their number would quickly realise that something was terribly amiss.\n\nHe clicked the timing mechanism's ratchet into place. A proud man, he couldn't help but feel satisfied by the elegance of his design, one built to please his eye as much as it was built for murder. He'd often been mocked due to the time and effort he took in producing his little devices\u2014but let the others use jam jars, scrap brass and broken watch springs. To him, his trade was as much about producing works of art as it was about making engines of death.\n\nWith the timer in place, he began to wind the handwheel like a seamstress at her machine. A constant stream of clicks, inexorably increasing in volume, began to emanate from the clockwork mechanism in the bomb's elaborate housing. From somewhere behind him Potsdam growled his disapproval\u2014this was too much noise. Klopstock agreed, sweating despite the cold. If Potsdam got riled the clicking would be irrelevant, drowned by noises far louder and fleshier in nature. He looked up from his kneeling position to the eyes of his watchdog, but as usual, the thick round discs of black glass he wore gave absolutely nothing away.\n\nAs he wound the wheel he tensed, sympatico with the spring he was tightening. He looked up at the monument. Erected in 1886 to commemorate the British killed at the Battle of Bous, the plinth held two soldiers standing back-to-back, one British, one German. At their feet lay a French shako, tipped to one side. Klopstock bristled. He found the whole thing insulting, and would've spat on the ground had propriety and excellent schooling not curbed the instinct. To be relegated to a hat on the ground\u2014that was as much recognition as this ham-fisted sculptor would give the French, lacking the spine and the imagination to go against instructions from his over-privileged, overfed and autocratic paymasters.\n\nWith the timer fully wound, Klopstock reached to the back of the device to ensure the clamps holding the dynamite were sufficiently tight. There was only one last thing to do\u2014sliding the polished wooden lever inset onto the front of the device would release a catch attached to the timing mechanism, dropping a ball bearing into a cradle that sat beside the fuse. If the bearing hit the metallic sides of the cradle, it would complete a circuit, activating a clockwork arm that pulled a pin from the timer, causing it to run down almost instantly and then, Klopstock thought\u2014and he couldn't help but allow himself a wry smile at the vulgarity of it all\u2014boom. His mother would not have been proud.\n\nAs he reached for the lever, the sound of boot-steps from behind him made his spine turn to steel and his head whip round. Potsdam stepped forward, one monstrous clodhopper crunching the debris on the cobbles, his dark grey greatcoat almost imperceptible against the night. Klopstock held out a hand to stop him, and Potsdam relented. But only just. Breathing out steadily, Klopstock brought the lever across and heard the almost imperceptible tap! as the ball bearing fell from its housing. Breathing a sigh of relief, he stood up and turned around, suddenly glad for the cool night air and the dimness of a London lit by the hiss of gaslamps.\n\nAnd found himself looking at an unmistakable silhouette, roughly six feet in front of him. The domed hat, the cape...there was no mistaking a member of the constabulary. If Klopstock could see him he could surely see Klopstock, but Potsdam was standing slightly further behind. The smaller man brushed the grime from his knees and wordlessly urged his hulking companion to step back into the night, and be ready.\n\n\"Bit late in the day for a constitutional, isn't it, sir?\"\n\nHis nose was pockmarked and ruddied by rum. Under his helmet were bulbous cheeks and a moustache that, despite obvious and continual waxing and trimming, still managed to look unkempt. But his eyes...they were sharp and keen, and despite the ruptured capillaries that betrayed a drinker, tonight was one night when this particular constable had abstained.\n\n\"Not at all, officer, not at all!\" beamed Klopstock. \"For I am, you see, a member of that curious breed known as the artist. My mind is as restless as paper scraps in the breeze, sir. As active as a tiger by night, sir. As fanciful as these great ships that take us to the stars and back, sir. I am here\u2014\" and Klopstock breathed in, puffing out his chest and spreading his arms wide, \"\u2014for inspiration.\"\n\n\"Well,\" replied the policeman, raising an eyebrow. \"I doubt you'll find it here, sir.\" He stepped closer to Klopstock, who smiled at him beatifically. \"Have you been drinking at all, sir?\"\n\n\"Not a drop, constable. Not a drop.\"\n\n\"And your place of residence?\"\n\n\"No fixed abode, officer. I am but a dandelion seed on Melpomene's wind.\"\n\n\"Melpomene, eh? I see. If you could just step over here, sir? Into the light. I wouldn't mind asking you a couple more questions.\"\n\n\"Of course, constable! Of course! Although if you'll excuse my impertinence, I must beg you to not delay me too long.\" The policemen stopped and eyed Klopstock wearily.\n\n\"Inspiration suddenly striking, is it, sir?\"\n\n\"Not at all. It's more a matter of patience, you see.\"\n\n\"When it's police business, sir, you'll be as patient as I need you to be.\"\n\nKlopstock made a noise at the back of his throat, something caught between a tut and a snort of laughter.\n\n\"Ah, well, that's it, isn't it? The thing is, it's not my impatience that you need worry about...\"\n\nA figure, fully a head taller than policeman\u2014helmet and all\u2014appeared from the smog. A massive hulking rectangle of a man, dressed in a docker's greatcoat, scarf and an almost comically undersized bowler hat. He seemed to be made almost entirely of black, the only specks of brightness on his person the dim reflections of gaslight in the welding goggles strapped to his face. The policeman whirled round, but too late! Without even time to yell or dash away, the policeman's neck was gripped and he was lifted, eyes bulging, a full foot-and-a-half from the floor. And yet he did not feel the flesh of man tightening around his throat\u2014there was nothing but cold bars of metal pressing the very life from him.\n\nHe began to kick out, booting the monster in the chest and groin, but to no avail. On the contrary, this only seemed to enrage the beast, and he lifted the policeman higher still. For a moment that lasted forever in the unfortunate constable's mind, and yet was no more than a brief moment in the life of the world, he was held up high and granted one last look at the stars above London, dimmed by a chemical smog. And then with brutality the body was brought down and the skull was smashed on the gritty, sewer-soaked cobblestones of Horseguard's Parade.\n\nLike a dog with a rag doll, Potsdam straightened his arm and offered the corpse to his companion for inspection.\n\n\"Ergh, I don't want to see that,\" said Klopstock. \"Throw it in the river.\"\nChapter One\n\nThe Errant Guest\n\n1.\n\nEVERY GIRL HAS nerves upon her wedding day, but before the day was out the nerves of Annabelle Somerset would be decidedly more fraught than that of the average bride. She was making her way through the streets of Greenwich, and to her left the Thames bustled with sludge and merchant ships. The air had cleared since last night, she noticed, and the day might almost have threatened some of that elusive English sunshine later in the afternoon. This cheered her, and she allowed herself a moment of joyous, schoolgirl glee that she was heading to the church in which she would be wed, Our Layde, Star of the Sea.\n\nOh, she knew all that old bosh about the bride arriving last. But Annabelle Somerset was a modern woman\u2014the natural upshot of her life of adventure on other planets, in the piston-oil black between worlds, and the battle scars that lingered in her body and her mind. This was her day, and hers alone, and she demanded distraction. She had waited seven long hard months for it, and so she resolved to rise before any other lady had even had their bodice pressed and head to the church to see how the day was beginning.\n\nPerhaps from a hidden nook in the transept, just to be discreet.\n\nAnd so, having arrived, she ensconced herself thus before the first sparrow had sung. Soon, she heard gentle murmurs begin and the heavy clank of the huge door's cast-iron catch lifted up and down, hard heels treading softly over stone floors in reverence for the silence. Annabelle suddenly realised she had not thought this through. The nook had seemed perfect at first but as the guests and friends began to enter she felt she'd had to shuffle back piecemeal to avoid being seen, and had now shuffled back so far she couldn't see a blasted thing. What's more, more and more people were arriving every moment.\n\nShe shrank back as much as she could and tried to think. What to do? Just go out there, yell ta-daaa! and get the whole sorry shambles over and done with? Maybe there was a way yet that she could sneak herself out. But she found herself bundling up more and biting her lip when she heard the crisp, loud smack of men's fine leather shoes striding straight towards where she was hiding.\n\nBugger, she thought, and tried to think of a line she could use to brazen her way out of this. She took a deep breath and held her head high, the footsteps two metres, a metre away....\n\nAround the corner from the nave came Nathaniel Stone, and when he saw Annabelle standing there with her jaw thrust impudently out he stopped in his tracks and stared.\n\n\"Annabelle,\" he whispered.\n\n\"Now before you defend all this stuffed-shirt British wedding nonsense, which you will, and perhaps that may do me some good, I'd just like to make a few things clear. To whit... Nathaniel, why are you staring?\"\n\n\"Annabelle,\" he said, shaken from his stupor and beaming widely. \"You look radiant.\" Perhaps not for the first time that day, but certainly not the last, Annabelle Somerset felt herself blushing, and the pink tinge to her cheeks seemed all the softer and more delicate when compared to the pristine whiteness of her dress. A wide belt in light brown leather was wound around her waist, held by a large gold double buckle with a watch chain looped from one end. She had worried about her prosthetic leg and the problems it may cause\u2014she had tried numerous dresses at the dressmakers and ended up tearing some of the wretched things. So the front of her skirt had been adjusted accordingly, and a suitably demure slit came up the front (tripped with fine lace just to be on the safe side). One shoulder was bare, the other covered with a delicate ruffled fabric that caught the light. Her hair was in ringlets and adorned with pearls.\n\n\"I'd be happy just to stand here for the rest of the day, dear Annabelle, and admire the beauty that I've seen you become.\" Nathaniel grinned. \"But I believe you've got a wedding to attend.\"\n\n\"Yes, Nathaniel, but the problem is I'm already there.\"\n\n\"Hmm,\" he muttered. \"So you are. Two ticks.\"\n\nNathaniel popped his head back towards the nave and scanned the pews, taking in the faces. He noticed Captain Folkard at the back. He was standing with his hands clasped behind him, upright and resplendent in his full naval regalia; despite his insistence that now their mineral collecting mission was over he wished to resign his commission, the Admiralty had persuaded him otherwise. He was still one of their finest captains, and they had need of him. There was a man next to him, shorter than Folkard but wiry, secretive, and Nathaniel noticed he was dressed more for a funeral than a wedding. Head to toe in black, of all things. He was whispering in Folkard's ear, and when Folkard looked like he might glance Nathaniel's way, Nathaniel bobbed his head back and returned to Miss Somerset.\n\n\"Folkard's arrived,\" he said. \"He's brought his cloak and daggers. Arnaud's there, and I heard Commander Bedford is to arrive any minute... All his family's here, I recognise his father from the squint. Hang on.\" Nathaniel stopped, struck by a thought he feared to voice. \"If you're here, where's your uncle? Where's Cyrus?\"\n\nAnnabelle sighed. It seemed that seven months ago her uncle's mind had settled once again, but after her most recent stop-over at Earth, some two months back, Uncle Cyrus' increasingly erratic behaviour had resurfaced; a deep and debilitating worry, but now, after being back on Earth for a couple of weeks, it had hardened into a sort of anger. Her parents dead, who else but her only living relative was to give her away? She had feared this and tried to forget it, but the inevitability of the disappointment had caught up with her at last.\n\n\"Oh, Nathaniel,\" she breathed, the inflection of it filled with the potential of tears. She held out her hands and he let her come to his shoulder, the soft, velvet safety of a life-long friend. Her mind bobbed to the past, forgotten things, tragedies on the great American plains... She thought of Venus, Luna, of Mars and the Phobos, how she had lost her leg to a gunshot. And all the time, Nathaniel, who had dragged her over extra-terrestrial deserts and guarded her surely through nightmarish times, had been there.\n\n\"Wait a moment,\" she snapped, bolting up, and there was no trace of tears in her voice now. \"You can do it!\"\n\n\"What? Well... Really?\"\n\n\"Of course! Why, now that I come to think of it, I wouldn't want anyone else!\"\n\n\"But... Your uncle, he may just be delayed, and on a day like today, where might haste and imprudence lead us?\"\n\n\"I do not wish to hear my uncle mentioned again today, and I'd be forever grateful if that sentiment could be communicated to the rest of the guests. The shadow of his madness will not linger on my wedding day.\" She looked into Nathaniel's face and placed her palm upon his cheek. \"A brighter and more constant light must lead me down the aisle.\"\n\nNathaniel frowned.\n\n\"The only problem being, Miss Somerset\u2014and I realise that that may be the last occasion I ever get to address you as that\u2014is that you are already down it.\"\n\n\"Well,\" grinned Annabelle, \"I've never been one for convention.\"\n\nThe hubbub in the nave seemed to subside momentarily and the clank of the door sounded finally. There were hushed congratulations and a decorous smattering of applause, and soon the hubbub had returned, this time infused with a current of anticipation. Nathaniel bobbed his head out again and there, inevitably, was the groom. Annabelle peered discretely from behind Nathaniel. George, dressed in the sharp-lined livery of a career naval man. He was shaking hands warmly with Folkard, the two sharing a moment to reminisce. George looked coy.\n\n\"Excellent,\" hissed Nathaniel. \"That's bought us some time.\"\n\nWith all of the attention directed to the groom at the front of the church, Nathaniel tiptoed across to the organist. Annabelle, distracted by the absurdity of it all, idly wondered if the poor fellow had to put up with these sorts of shenanigans often. Nathaniel reached his destination mostly unseen, and had a quick word in the man's ear.\n\nMoments later he was back with Annabelle.\n\n\"He says he'll do it.\"\n\n\"Really? However did you manage that?\"\n\nNathaniel raised an eyebrow. \"I know my way around the men of the cloth.\"\n\nAnd so, with the organist duly instructed and the guests beginning to settle, George Bedford slowly and solemnly made his way towards the altar. Nathaniel, no doubt estimating his speed, the distance...mentally computed the perfect time to go, which was, in fact, right now, nodded across at the organist while taking Annabelle's arm, and started walking.\n\nWhen George saw her step from the transept ahead of him, only a scoundrel would have admitted seeing the groom stumble momentarily in awe at the first sight of his bride. And yet he did not miss a beat, and neither did she, and as the organ resounded with the bridal march the two met at the centre of the alter together, and took each other's hands.\n\n2.\n\nAT THE RECEPTION, which was held in the Admiralty's grand ballroom (and grand it was, belonging as it did to the finest Navy of the inner planets), Nathaniel was enduring some ribbing at his expense. Unversed in the matrimonial ceremony he had mostly had to follow the priest's hissed instructions. His repeated failure to accurately do so had caused Annabelle an unfortunately uncontrollable bout of mirth, and Bedford stepping in to help did not, in fact, help. But when they managed to compose themselves the whole thing went off without a hitch, and the memories of that morning had instantly become cherished. Those same memories were now being regaled, at Nathaniel's expense, notwithstanding the assistance of champagne.\n\n\"And didn't you tread on the poor fellow's foot, at one point?\" asked the newly-married Mister Bedford, leaning back into his chair.\n\n\"I don't think so. It's all a bit of a blur, to be honest.\"\n\n\"Oh, Nathaniel,\" smirked Annabelle. \"It was memorable, at least.\"\n\n\"Quite.\"\n\n\"Don't pout, Nathaniel. It's unattractive.\" She glanced over at Arnaud, another trusted friend, who was lazily badgering the ma\u00eetre d' for more champagne.\n\nThe wedding feast had been and gone. Almost by mutual assent each guest had settled into their chair, perhaps surreptitiously unfastened a waistcoat button or two, taking a moment to recover from the day and revel in plenitude.\n\nThis pleasant malaise, however, did not seem to affect Captain Jacob Folkard. He nimbly wound his way towards the group at the top table.\n\n\"Jacob!\" hollered Bedford. \"Care for some champagne?\"\n\nFolkard might have had a reputation of earned respect and results gained when commanding a vessel, but Nathaniel had come to know him more for his wit and warmth. These endearing qualities were always evident when he gave orders to his crew, though they were hidden\u2014when one came to know him as the wedding party did, these foibles became the key to his character. And so it was that everyone noticed something was amiss when those selfsame virtues were all but absent in Folkard's face and demeanour on this, the happiest of wedding days.\n\n\"Thank you, George, but no. May I congratulate you once more on a triumphant day? Not without its surprises, but I would expect nothing less from you, Mrs Bedford.\"\n\n\"You're too kind, Captain. Won't you at least join us?\"\n\n\"Not right now. Nathaniel, might I show you something?\"\n\nNathaniel, mildly drunk and therefore momentarily nonplussed by being the centre of attention once more, drained the last of his champagne and stood, it had to be said, rather unsteadily. \"Of course.\"\n\nAs Folkard led Nathaniel through the throng of revellers, cutting a swathe between flute chinks and chatter, Nathaniel peered over his shoulder and noticed Bedford, Annabelle and Arnaud exchange an ominous glance.\n\n3.\n\nFOLKARD LED NATHANIEL to the atrium, striding ahead, straight-backed. He suddenly veered to the left to push through a door, and Nathaniel found himself half-jogging to keep up as they traversed a corridor lined with pot plants on plaster columns. Before he could even attempt to protest Nathaniel found himself following through a smaller\u2014though no less ornate\u2014atrium, and out into a well-kept, walled garden where he was finally allowed to catch his breath. Folkard was silent as he did so.\n\n\"You made a fine stand-in, Professor,\" he said eventually. \"It was a joy to see you up there. I feel your brother would have been proud of you.\"\n\n\"Yes, well,\" muttered Nathaniel, appreciating the sentiment, reminded of his and Folkard's previous antagonism over Edwin's death. It was good that both men had finally been able to bury the hatchet over that unfortunate event. \"You weren't the only one to find it amusing.\"\n\n\"I meant that sincerely, Nathaniel. It was just all so...fitting.\"\n\n\"Unlike your friend in black.\"\n\n\"I don't like it any more than you do, Stone. But it was\u2014and remains\u2014necessary.\" Folkard crossed to a small gate beyond the rose garden and placed his hands on the top. He seemed to grimace, and Nathaniel looked down to see his knuckles go white as ice.\n\n\"Captain? What's wrong?\"\n\n\"I'm sorry to have to do this. Today of all days... But you must understand, it had to be you. You're the least involved.\"\n\n\"Captain, please. You're talking in riddles.\"\n\nNathaniel searched for Folkard's eyes, which were darting and restless. The captain, over the course of their adventures, had demonstrated peculiar mental and telepathic talents. These were inexorably tied to Luna, and the brooding alien relic known as the Heart that ran through the moon's very belly\u2014the relic that had caused the madness of Annabelle's uncle, Cyrus Grant, and the grotesque deformation of the insane Russian scientist Vladimir Tereshkov. Madness and the moon, even in this enlightened age, seemed to remain cosy bedfellows.\n\n\"It falls to me, Nathaniel,\" he said slowly, \"to destroy this day.\"\n\n\"Now steady on...\"\n\n\"Listen. Keep your peace and be quiet, man! Just listen.\"\n\nAnd so, stunned, Nathaniel listened.\n\n\"I don't hear anything,\" he said, unconsciously lowering his voice to whisper.\n\nFolkard hissed. \"Precisely...\" Having said that, he stood bolt upright and began to pace, his volume returning to normal and startling Nathaniel. \"And yet,\" he continued. \"It is such a bright September day in a choice and, many say, a pleasant part of town.\"\n\nNathaniel listened again, and his face fell.\n\n\"Where's the sound of lovers?\" asked Folkard. \"Or mothers with their children and fathers with their sons? Commerce. Trade. Noise. Where is it?\" Shocked by the shout, Nathaniel heard it: Nothing. London at a deathly, unnatural standstill. The hairs on the back of his neck stood on end.\n\n\"There has been an attack on sovereign soil and an attempt on the life of Her Majesty,\" Folkard said flatly. \"Perhaps we should go inside.\"\nChapter Two\n\nIntroducing Mister Enderby\n\n1.\n\nTHE SCENT IN the room was one of ageless grace. The soft musk of panel oak, cognac and polished leather mixed together against a backdrop of book-lined walls and pearled gaslights. The very seat of the Empire. It was in rooms such as this that the fates of worlds beyond Earth were dictated.\n\n\"Brandy?\"\n\nNathaniel watched as Folkard crossed to a large mahogany globe, the reach of England's arm marked out in pastel pink across the continents. Folkard split the world in half with a creak, revealing decanters and crystal glasses, and began to pour himself a drink. Nathaniel silently declined Folkard's offer, and the Captain gestured for him to sit. Folkard sat opposite him, brandy cupped in his hands but untouched.\n\n\"This morning,\" he began, \"the body of a police officer, one Constable Jeremiah Graves, was discovered snagged on a girder near Blackfriars Bridge. He hadn't returned from his beat the previous night, but no-one thought much of that. Fellow was apparently a bit of a drinker. More than once the morning watch had picked him up, uniform and all. But it turns out it wasn't the drink that got him.\"\n\nNathaniel swallowed dryly.\n\n\"We originally paid the death no heed. Why would we? Early morning, another corpse. Thought the poor wretch was three sheets to the wind, tumbled into the river and caved his own head in. But when the bomb went off we discovered its location was right on his route, and drowned men tend to have more water in their lungs.\" It was only then that Folkard took a sip of his brandy, and a minute one at that.\n\n\"And you said,\" Nathaniel asked, \"Her Majesty?\"\n\n\"Today is the anniversary of the Sacrifice at Bous, and Her Majesty was scheduled to attend a short service at St Paul's followed by laying a wreath at the memorial. Ecumenical red tape\u2014or just a slow sermon\u2014delayed Her Majesty. But were she on time there is doubt, Stone, but on God's good Earth she would have been killed.\"\n\n\"And Her Majesty?\"\n\n\"Is shaken, but otherwise unharmed.\"\n\nIt was not Folkard who made this declaration. The voice came from behind Nathaniel\u2014precisely behind him\u2014and so he whipped around to engage the man who had uttered it.\n\nHe had seen this man before, whispering in Folkard's ear, and later, chewing on a canap\u00e9 and watching from the ballroom's corners. And yet when he had walked in to this room, there had been no trace. Nathaniel doubted he would have been far from Folkard's side today, and suddenly he knew he was faced with a man who only allowed himself to be seen when he wished it.\n\n\"This,\" said Folkard, \"is Mister Enderby.\"\n\nEnderby was not a tall man. Folkard and Nathaniel would have easily towered over him, and yet he had a moulded, triangular frame\u2014not stocky, but lithe. He was dressed in plain, neat black, his hands clasped in front of him. He had an almond-shaped face with whiskers and shoulder-length hair that kinked around his ears; you could not guess his age. He might have been a haggard twenty-two or a youthful fifty-five, and when Nathaniel looked back he inadvertently met his gaze. He found a still gaze that neither challenged nor invited an inch.\n\n\"Charmed,\" said Nathaniel.\n\n\"Mister Enderby,\" said Folkard, as Enderby crossed to stand behind him, \"is charged with the safety and security of Her Majesty's person, although I do not think he will be overly troubled if I inform you his duties extend rather more broadly than that.\"\n\nEnderby smiled, and nodded.\n\n\"Hence his presence here today. And I want to take this opportunity now, before events drag us further into turmoil, to say that I am truly sorry to have to do this.\"\n\n\"But, Captain, it's...\"\n\n\"It's the wedding of your ward and a very great friend of mine. And yet, once more, I find the perfidious attempts of our enemies and the politics of alien worlds has wounded the memories of this day, and it falls to me to administer the coup de grace.\"\n\nFolkard was losing his faith in this world. He knew it and Nathaniel knew it. It had begun as 1889 has come to a close, and it continued apace whenever he returned to Earth. Folkard sighed and drank his drink, no doubt wishing he was far away... Far away in the caves of Luna, perhaps. It seemed, to Nathaniel, that sometimes the good captain's only affinity lay with the Heart and the memory of his wife and daughter.\n\n\"Because the only people,\" Folkard continued, \"that do not know of this attack are the bride and groom. George and dear Miss Annabelle. And it has fallen upon me to tell them.\"\n\nFolkard put his head in his hands. Enderby, proud as you like, reached forward and took Folkard's glass. He padded over to the drink cabinet, refreshed Folkard's brandy, and returned. Folkard took the glass while Enderby removed and lit a small cheroot from a silver case.\n\n\"We were lucky enough to find fragments,\" he said. \"Enough to go on. Given the right...expertise?\"\n\n\"I'm sorry, Nathaniel,\" muttered Folkard.\n\nIt was such a rare occurrence for Folkard to use Nathaniel's given name. The depth of his guilt over this was obviously huge. But why Folkard should take the blame was beyond Nathaniel. He looked up to Enderby's eyes, intermittently obscured by smoke as the man breathed out. \"Can we not at least give them the rest of the day?\" he asked.\n\n\"We could,\" observed Enderby coolly, \"but your presence is still requested.\"\n\n\"Hell's teeth, man. I won't be shanghaied like this.\"\n\n\"It's a matter of national security, Professor Stone.\"\n\nSo much for being considered dead, Nathaniel thought. Only when it serves the government's agenda.\n\n\"And you are giving me no choice but to choose patriotism over friendship?\"\n\n\"Unfortunately so.\"\n\n\"Which is why,\" Folkard breathed, \"It must be me that tells them.\"\n\nA silence fell upon the room. Enderby removed the cigarillo from his lips and held it loosely between his fingers, unmoving. Folkard made a move to get up before thinking better of it, and Nathaniel simply stared at his clasped hands.\n\n\"No,\" he said, finally.\n\n\"I'm sorry?\" said Enderby.\n\n\"No. I will not abandon my friends on the happiest day of their lives in order to thrust myself once more into a world of danger and death. We have all, all of us, seen too much of both, and if we are not allowed respite on a day such as today, then it is a world I wash my hands of.\"\n\n\"Nathaniel, please...\"\n\n\"No, Captain. No. You know this is the right thing to do.\"\n\nFolkard looked up to Enderby, who stared back impassively. He leaned forward and stepped between the two men, extinguishing his cheroot on a gilt ashtray that sat on the low table before them. It was not even partially smoked.\n\n\"These things, we find, have only a small window of opportunity to pursue.\"\n\n\"You have fragments,\" said Nathaniel, bluntly. \"Are they to dissolve before tomorrow? And if, sir, you are in pursuit, how are we to assist with that? Bodily? Shall we strap on our galoshes and go tramping through the banks of the Thames for clues?\"\n\n\"I'm not without sympathy, Professor Stone. But there are greater things afoot than wedlock and revelry.\"\n\n\"Not today there isn't.\"\n\nSilence.\n\n\"I think you have your answer, Mister Enderby,\" said Folkard.\n\nHaving said this, he drained his brandy and stood, a weight having seemingly lifted. \"In the course of the day I will inform Commander Bedford of events. I see no reason for Mrs Bedford to find out, though doubtless she will before long. We will convene at the Admiralty, say, nine o'clock sharp tomorrow morning?\"\n\n\"Seven,\" said Enderby. \"Sharp. Doctor Arnaud Fontaine must also be there. Commander Bedford may attend, if he wishes. We believe his experiences may be of use.\"\n\n\"Very well. Stone?\"\n\n\"Do I have a choice?\"\n\n\"You have made your choice, Professor, and bought yourself the day. Be grateful,\" Enderby purred.\n\nAnd yet Nathaniel felt anything but grateful.\n\n2.\n\nTHEY RETURNED TO the ballroom to the find the guests reinvigorated. The detritus of fine dining had been cleared and people were milling in the grandiose hall in which entertainment, in the form of a full orchestra and dancing, would be enjoyed. Upon his return to the throng, Nathaniel could suddenly sense that something was afoot. Movements were quick, and panicky\u2014a ringing laugh was stifled unnecessarily quickly, high-ranking elderly man formed in groups to tut and shake their heads.\n\nAnd in the centre of it all were Bedford and Annabelle, standing by the cake and chatting idly to relatives and guests. Blithe and blissfully ignorant, thought Nathaniel. Let them have their day. He nimbly pinched another glass of champagne from the tray of a passing waiter.\n\nFolkard appeared, and drew Bedford to one side.\n\n3.\n\nLATER, IN THE bridal suite, George and Annabelle gazed into each other's eyes.\n\n\"Mrs Bedford,\" twinkled George.\n\n\"Husband?\" replied Annabelle. They smiled.\n\n\"Despite everything,\" she sighed, flopping exhausted onto the bed, \"it really was a rather glorious day.\"\n\n\"It was never going to be normal though, was it?\" said George, flopping beside her. He put his arm around her shoulder and she nuzzled closer to him, rubbing her nose against his neck.\n\n\"I wouldn't have it any other way.\"\n\nThis was their time, their first time, together and at peace as a married couple. Each enjoyed the other's presence, the gentle tingle of a touched fingertip and the closeness and heat of each other.\n\n\"A bomb though,\" said George. \"An attack. I can barely credit it.\"\n\n\"In a world such as ours, dear George, these things happen all the time. It's just that on this occasion, it has happened to us.\"\n\n\"For our wedding day to be remembered as that, though...\"\n\n\"Is but nothing,\" intercut Annabelle. \"Do you not think we will also remember Nathaniel's endearing attempts to give me away? Or the priest who had to limp for the rest of the afternoon? Or Arnaud's champagne fountain, and the mess that that idea caused? This was our day.\" They looked at each other, shoulder to shoulder on the bed.\n\n\"And though the worlds may turn and events run riot, this is still the day that we were wed. My darling George. My husband. Let the power-mongers blow themselves to pieces today. And if the day itself was not perfect, what matter? We have the rest of our lives to make it so.\"\n\nGeorge, though a gentleman, could not restrain himself. He leaned over and cupped Annabelle's cheek with his palm. Though this kiss was gentle it was a kiss more forceful and sexual than either had ever experienced before.\n\nAfter a moment, George stopped.\n\n\"Annabelle,\" he breathed. \"I have never been...\"\n\n\"Shh,\" soothed Annabelle. \"The day was taken, but the night is ours.\"\n\n4.\n\nTHE PARTY HAD continued later than anyone could have reasonably expected. It was almost as if some unconscious drive had pushed the guests to enjoy themselves as much as was possible in the time that remained that night; they all knew, deep down, that with the bomb attack they would wake the following morning in a world that may well have irrevocably changed.\n\nIt had, in fact, gone on so long that as the guests filtered out much decorum had been abandoned. They was loud laughter and flirting, promises of dinner dates made and stolen, hard breathed-kisses. Even Bedford's two maiden aunts, both in their sixties and pillars of the community, had somehow managed to procure a cucumber from the kitchens (nobody dared to ask how) and were waving it around and cackling in the courtyard with an infectious, carefree joie de vivre.\n\nNathaniel was taking solace in this happiness. His clandestine meeting with Folkard had imbued him with such ambivalence, the joy of the day against the sorrow of the world. At present, he was content to stand in the shadows, near the door to the Admiralty courtyard, and take strength in the joy of people other than himself.\n\nThere was a bang behind him, and Nathaniel turned quickly to see Arnaud slam gracelessly into the side of the door. His legs weren't working particularly well, his eyes were unfocussed and he had a lop-sided, goofy grin on his face. He looked up and spotted Nathaniel, and beamed widely.\n\n\"Why Professor Stone.\" He grinned, piling on every ounce of his French charm. \"Fancy seeing you here...\"\n\n\"Arnaud,\" Nathaniel found himself saying, in a voice that wasn't entirely admonishing. \"You're drunk...\"\n\n\"And you, mon toujours, are looking very, very, good...\"\n\nNathaniel looked up, towards the windows of the grand building, but only for a second. It was a movement he knew would not be noticed by Arnaud, who had come with other things on his mind.\n\nWith remarkable rapidity Arnaud seemed to throw off his alcohol-induced cloak and strode up forcefully to Nathaniel, who, almost unconsciously, pressed his back against the cold stone pillar behind him. Nathaniel swallowed loudly. They were in the dark, away from prying eyes. Slowly, hesitantly, Arnaud's strong arms wrapped around Nathaniel's waist. Nathaniel reciprocated, curling his own arms around Arnaud's broad back and running them up to his neck. Gently, firmly, Arnaud pulled Nathaniel towards him and they began to kiss deeply, their lips exchanging exquisite, tingling touches.\n\nArnaud relaxed into the passion, hungry but content, blissful, enjoying the danger and the shame... He let go, began to kiss back, Nathaniel's playing the same delicate game. The two lovers began that immaculate see-saw of predator and prey, chasing each others' desires in the moonlight.\n\nAnd yet, fatefully and disastrously, their passionate encounter did not go unobserved.\n\n5.\n\nFOLKARD, WRACKED WITH guilt over how his duties had so hatefully encroached upon the day, had exiled himself to an upper office in the Admiralty for the remaining hours. He had meant to get some work done, prepare for the meeting the following day, but had in truth spent most of his time gazing from the window in a cloak of self-loathing. It was while he was doing this that he noticed Nathaniel idly watching the revellers, and then Arnaud's appearance, and everything that subsequently transpired.\n\nWas he losing his mind, or did he really see Stone glance up to look him straight in the eye, moments before the whole vile spectacle had unfolded?\n\nAlmost instantly any respect he had for the professor and Fontaine dissolved into disgust, into disbelief and loathing. Previously, Stone's lack of romantic entanglement had seemed natural\u2014these science boys were all the same, more interested in test tubes and tinkering than the fairer sex\u2014and he'd assumed Arnaud was more of a man-about-town, promiscuous perhaps, but not deviant. And yet here it was, laid before him, and the sickness that he felt, the betrayal, the perversion... It permeated his every nerve, sat heavy like lead in his gut. He gritted his teeth, the anger overtaking him, throwing a full tumbler of brandy to shatter into shivers on the opposite wall.\n\n6.\n\nFOLKARD SLEPT FITFULLY that night.\n\nThe blankets around him seemed itchy and hot. He threw them off, and sat dolefully on the edge of the bed to think. He suddenly realised he was grinding his teeth and, only with great mental effort, stopped himself from doing so. He was cold. He reached across to the blankets and draped them around his shoulders where he sat.\n\nArnaud and Nathaniel's secret gnawed at him; a sickening, repulsed sensation that would not give him peace. Yet even this turn of events was not the uppermost in his mind, for a far larger and more cosmic fear was searching for him, soothing and teasing in equal measure, an itch that could not be scratched.\n\nHe could see nothing in the dark. Only feelings. Insomnia or brandy on an empty stomach made erratic, organic shapes flash and pulse before his eyes. Only in those shapes he could feel them, hear their voices reaching out to him over the yawning chasm of space... Charlotte, Felicity, both gone, but whispering from The Heart. The Heart of Luna. Closing his eyes, he could see them, more real than his own breathing or the cold of the room around him. He tried to soothe them with his thoughts, telling them, I will return to you, my heart, my darlings.\nChapter Three\n\nRendezvous at the Admiralty\n\n1.\n\nAS WAS TO BE wholly expected, Mister Enderby was already in the room by the time Folkard and Nathaniel arrived. Laid out on the table in front of him was a calico sheet onto which had been arranged charred and twisted relics of metal and glass.\n\n\"Where is Doctor Fontaine?\"\n\n\"En route, and rather the worse for wear,\" said Folkard bitterly, removing his cap and hanging it on the hat-stand. \"There was talk of a champagne fountain, as well as other...distractions. My apologies.\"\n\n\"Quite. I see no reason not to proceed,\" said Enderby stiffly.\n\nNathaniel looked across to Folkard. The naval man stared back coldly.\n\nNathaniel was nonplussed by Folkard's odd behaviour. It was almost as if Folkard couldn't bear to look him in the eye. He shrugged it off, and when Nathaniel bent down to examine the remnants laid out on the table, any irritation he felt over the spy's brusque attitude or Folkard's rudeness instantly dissipated. If Enderby lacked manners, he could be forgiven for bringing Nathaniel so intriguing a collection. The pieces laid on the sheet ranged in size from an almost complete side of casing roughly five inches long to tiny cogs and brass coils no more than a centimetre in circumference. There were also chunks of thick glass, the inner curve of which was covered in a gritty powder, almost like a pale green sand.\n\n\"Fascinating,\" muttered Nathaniel, removing a pair of ornate, multi-lensed pince-nez which he quickly adjusted and balanced on his nose, peering ever closer at the fragments.\n\n\"Our chaps have already given it the once over, of course,\" sniffed Enderby.\n\n\"Do you have a copy of their findings?\" asked Folkard in a clipped tone, and Enderby removed a sheaf of parchments from a black leather carpet bag and passed them across. Folkard flicked idly though the first couple of pages.\n\n\"All ancient Greek to me, I'm afraid. Stone, you'll have to look.\"\n\n\"Yes yes,\" muttered Nathaniel, distracted. \"One moment.\" He removed a pencil from his pocket and began to prod at a piece of glass. \"This substance,\" he said. \"The silica. What did your chaps make of that?\"\n\n\"Possibly some form of copper residue, they thought. Right kind of colour.\"\n\n\"Hmm.\" Nathaniel rubbed his finger on the inside of one of the glass pieces and rubbed the grit between his fingers before sniffing it experimentally. He recoiled.\n\n\"I'm not sure I agree. May I remove some of it for study?\"\n\n\"So long as some of it remains for our chaps.\"\n\nSatisfied (at least for the moment) Nathaniel folded his glasses up and replaced them in his case. He took the papers from Folkard and began to pace as his eyes flicked through the figures and sketches at a remarkable speed.\n\n\"Your...'chaps',\" he said to Enderby as he paced, \"none of them are horologists, I take it?\"\n\n\"Clockmakers, you mean? What difference would that make?\" asked Enderby shortly, before adding, \"Professor Stone.\"\n\n\"Nothing in terms of function, but a lot in terms of artistry. I notice in these notes that none of your,\" (and here he cleared his throat) \"chaps seemed to notice the single most interesting thing about these remains.\"\n\n\"And that is?\" asked Enderby archly.\n\n\"The design.\"\n\n\"If I could refer you to the third page of the document, Professor Stone, the design is described as operating at a level above the usual backstreet competence, with a higher than usual explosive yield and several mechanisms inserted to prevent tampering, diffusion or lateral transport of said device.\"\n\n\"Very good,\" said Nathaniel, looking up from the papers. \"That was word-for-word. But I'm not talking about the prosaic mechanisms of the device's function, Mister. Enderby. I'm talking about aesthetics.\"\n\n\"Aesthetics?\" asked Folkard, engaging with Nathaniel for the first time. \"Do you mean to say...?\"\n\n\"That this device was not built by someone blind to beauty, yes.\"\n\nAt Nathaniel's beckoning, the three men gathered around the table. He pointed out the largest surviving piece of the bomb's casing, which was etched with an ornate, curling wave pattern from end to end. \"Of course, it could just be that the blackguard that built it used some scrap metal with the design already on it. But then I noticed something more interesting.\" Using a pair of tweezers he had removed from the inside of his frock coat, Nathaniel picked up one of the gears from the cloth\u2014about half an inch round, and remarkably intact\u2014and gestured around the circumference with his little finger.\n\n\"A cog, I'm sure you will agree, that could be found in any broken watch throughout the Empire. Only look here, around the rim. Any industrially produced cog would be rougher in cast, but this is exquisite workmanship\u2014polished, refined. I'll wager it was hand-made. The same goes for many other of these pieces, gentleman. We are dealing with a sociopath who takes an uncommon amount of pride in his work.\"\n\n\"And what about the residue?\" asked Folkard.\n\n\"I'd need to get a sample to my lab. Arnaud may be of no small assistance there.\"\n\nAnd speaking of the French devil, he appeared. Despite the gravity of the situation, Arnaud's eyes were puffy and red and his face seemed to sag around his dewlap. More hirsute than usual and with his collar undone, he walked with a stoop and groaned with every step.\n\n\"Good night, Arnaud?\" commented Folkard, his voice chilled. Without answering, Arnaud flopped into a chair and put his head in his hands. Smirking, Nathaniel crossed to the drinks cabinet and fetched a soda siphon to help alleviate the young geologist's suffering.\n\nAfter Arnaud, entered Bedford and Annabelle. The latter looked radiant, and the former had a spring in his step that had never, Nathaniel noted, been noticeable before.\n\n\"Mrs Bedford,\" smiled Enderby icily. \"I did not think you would be attending us this morning.\"\n\n\"That,\" said Annabelle brightly, \"is because you did not invite me. However, I decided not to let that one small detail prevent me from accompanying my husband. After all, why should you boys have all the fun, eh?\"\n\n\"I can assure you, Mister Enderby,\" said Folkard, watching as Nathaniel passed Arnaud a glass of seltzer, \"that Mrs Bedford is as useful an asset as any here, and probably far more so than Doctor Fontaine, given his current state.\"\n\n\"C'est des conneries,\" muttered Arnaud, sipping and grimacing. \"This isn't brandy...\"\n\n2.\n\nNATHANIEL HAD INITIALLY wished to use his own finely-calibrated equipment to continue the investigation, but Enderby would hear none of it. Expanding, as it was, from a grand and rather stuffy seat of military leadership into a substantial interplanetary power, the Admiralty was forever adapting and adding to its myriad corridors and anterooms. Many of these additions had been scientific in nature, with both the Crown and the Navy keen to maintain their technological superiority over the rest of the world. Therefore, Nathaniel and the rest of the party were led through the labyrinthine building to a wing that smelled of ozone and sulphur. Wooden walls had given way to white tiling, the clack of typewriters replaced by the hum and groan of steam and engines.\n\nHaving begged a brandy, Arnaud was suitably recovered and was being informed of events by Bedford. Enderby strode ahead, with Nathaniel, Folkard and Annabelle maintaining a discreet distance behind him. He reached a door, spun around on his heels, and waited for the rest of the group to catch up with him.\n\n\"Mrs Bedford,\" said Enderby. \"I'm afraid this is where your journey today must end.\"\n\n\"Now look here,\" scowled Annabelle, \"as Captain Folkard so rightfully pointed out, I've done my fair share of adventuring across the inner planets, and if you think I'm about to turn back after everything I've been through...\"\n\nEnderby's face, usually so stoic, seemed to soften as he looked at Annabelle's resolution. It seemed he knew something she did not. \"Mrs Bedford, please understand that your capabilities are not, nor will they ever be, in question. My reticence to allow you further is rather more...delicate in nature.\"\n\n\"And which issue,\" hissed Annabelle, \"is the most pressing this time? That I am a woman, that I am a cripple, or that I am a foreigner?\"\n\n\"Alas, Mrs Bedford, it is none of the above.\"\n\nEnderby's callous and condescending manner pushed Annabelle over the edge. Before Bedford could even dare to reach forward to hook her arm, she had barged past Enderby and swept through into the door beyond with a swish of her skirts.\n\nTo find herself staring into the eyes of her uncle, the volatile genius known as Cyrus Grant. She was taken aback, stopped and stared at her shoes, while her uncle looked down, away from her, gripping the edge of a workbench tightly. The others followed Annabelle in, with Enderby entering last. Faces fell, discomfort and unease were prevalent in the air. Enderby cut through the mire of discontent with typical sang froid.\n\n\"I did say, Mrs Bedford, it was delicate.\" As he picked a mote of fluff from his lapel, he was the only one who moved.\n\nMoments passed.\n\nAnnabelle gritted her teeth.\n\n\"Where were you, uncle?\" she hissed.\n\nGrant couldn't look her in the eye, waved a vague circle with his left hand and turned away. \"The bombing,\" he said, his back to her. \"Matter of national security. Have to show them which side you're on.\"\n\n\"My wedding,\" she begged. And with his back to her he waved her off again, like an idle farmhand would bat away a fly.\n\n\"Doctor Grant has been assisting us,\" purred Enderby. \"He seems to believe there's a link between yesterday's bombing and an Irish...connection we have. And if you follow the fuse the match was lit in Russia.\"\n\n\"Russian dogs,\" muttered Grant, hugging himself.\n\nEnderby caught Nathaniel's eye.\n\n\"We were wondering, Professor Stone,\" he said sadly. \"If you could carry on where the good doctor left off. His notes are in the left-hand pile.\"\n\nNathaniel glanced at the left-hand pile, and saw more the beginnings of a small campfire than scientific rigour. \"It'll... take some time,\" he murmured.\n\n\"Then perhaps it may be advisable for you to start afresh, Professor.\" Both Nathaniel and Enderby looked at Grant, who was pacing in a small circle and muttering to himself. For a second, Nathaniel caught a glimpse of his old friend's face\u2014a face quite unlike the man he had known, with wild, roving eyes, a sallow, jaundiced complexion and tufted, wrangled hair. Grant looked away almost instantly.\n\n3.\n\nWHILE NATHANIEL TRIED to make head from tail of Grant's notes\u2014a mishmash of formulas, esoteric sketches and snatches of wild, improbable conspiracy theories concerning the moon and a Russian incursion\u2014Arnaud busied himself with the mysterious green residue found within the bomb. He had set up a workstation opposite Nathaniel and was gazing into an ornate brass microscope. Next to him whirred a powerful clockwork centrifuge while a test tube rack held various brightly-coloured acids and solvents to his left. Once in a while, the two would look up and catch each other's eye, and more than once Nathaniel found himself gazing at the Frenchman's cheeky grin. Yet Nathaniel could only muster a wan grin in return. The notes over which he pored were not written by a man of sound mind. Often, numbers would transmute into letters, while formulas mutated into nonsensical, paranoid rants and back again, often employing strange, arcane symbols that were not the written language of any earthly tribe. Whatever Grant had gleaned from his studies of the charred remains of the explosive device, no light would be shed by these madcap scribblings. It was like they were written by two different people holding the same pen.\n\nGrant had remained in the lab after Folkard, Enderby and Bedford had left to discuss events in a room of the Admiralty more conducive to such discussions. As far as he could tell, the erstwhile doctor was now sat at a bench on the other side of the room, his back to everybody else. A collection of seemingly unconnected fragments of machinery was sat on the desk before him, and as he arranged these in patterns and piles he muttered to himself and, more worryingly, giggled occasionally.\n\nAnnabelle had declined Bedford's offer to retire to a more comfortable environment, and elected to stay in the laboratory with her uncle. As yet she had not dared to approach him. She merely sat and gazed across at the old man's back, her face flitting uncertainly between disdain and concern. She did not even seem to acknowledge that Nathaniel and Arnaud were there.\n\n\"Nathaniel, if you please.\"\n\nWithout lifting his gaze from the microscope, Arnaud beckoned for Nathaniel to join him. Relieved to abandon the perturbing papers, he crossed to Arnaud's side. \"Have a look,\" Arnaud said.\n\nNathaniel peered into the eyepiece. It was much as he has suspected\u2014definitely crystalline in nature, with some of the larger chunks hexagonal in shape. They were a curiously pale green, the colour of desiccated grassland, and yet flecks of a darker emerald shade pitted the insides of the crystals.\n\n\"Curious impurities,\" he murmured.\n\n\"My thoughts exactly.\"\n\n\"And the composition? Could it be some sort of copper derivative?\"\n\n\"No, I don't think so. As far as I can tell, it seems to be silicate in nature. Silicate and... Organic.\"\n\nNathaniel bolted up. \"Organic?\"\n\n\"Oui. And here,\" Arnaud gently removed the slide and carefully slid another into the stage clips. \"Look at this.\" He twisted the lens around to increase the magnification fivefold.\n\nNathaniel looked again. Another, larger crystal greeted him, and here the impurities were far more defined. While some remained as flakes and grains, others looked like snapped pieces of hollow tubing, running parallel to the crystal's grain.\n\n\"Why,\" whispered Nathaniel, \"I could almost swear they were...\"\n\n\"Parenchyma,\" finished Arnaud. \"Or schlerenchyma.\" He grinned. \"I could never remember which was which.\"\n\n\"Schlerenchyma,\" grinned Nathaniel back. \"The supporting tissues of plants. But that would mean...a plant made of crystals. Of silicon.\"\n\n\"It's only a hypothesis.\"\n\n\"And an intriguing one, at that.\"\n\n\"Have you ever seen anything like it?\"\n\n\"Not in my travels, no. But there remains much of the cosmos to explore. It's hardly worth saying that, if true, this is a plant that has doubtless never grown on this planet before, perhaps not even the inner planets...\" He looked at Arnaud, like an expectant teacher before a pupil, and asked \"What else do you think?\"\n\nSuddenly, Grant swept his arm across the desk and pitched his assorted detritus across the room.\n\n\"No!\" he yelled, bolting up and pacing, tugging on his hair. \"I won't have it, no!\"\n\n\"Doctor Grant...\"\n\n\"Uncle!\" Annabelle, wrenched from her despondent reverie by her uncle's manic outburst, dashed across to him and grabbed him by the arm. Grant roughly pushed her away and, unsteady on her prosthetic limb, she tumbled into a desk. Arnaud, who was closer, swooped to help while Nathaniel dashed to try and calm the crazed scientist.\n\n\"Doctor Grant, really!\"\n\n\"No, Stone, no! I won't have it, I just won't have it! Do you hear me! No!\" He barged past Nathaniel and stormed to the door, ripping it open and slamming it shut in his wake.\n\nArnaud comforted Annabelle, who had settled onto a high stool and was breathing heavily, on the verge of tears.\n\n\"Oh Nathaniel,\" she breathed, \"I try to put him from my mind, I really do. I could do that yesterday, when grander things were happening, but now...to be confronted with his madness... Pain and discord and danger, whenever he appears in my life. To think he is the only member of my family that remains....\"\n\nNathaniel took her hands in his, and smiled gently. \"Well, that's not strictly true now, is it?\" Annabelle sniffed, and Nathaniel continued. \"I believe that yesterday you made a family of your very own.\"\n\nDespite herself, Annabelle looked up and smiled. \"Regardless,\" she said, \"to see Uncle Cyrus so beset by his madness... He's like a different person, Nathaniel. A person I don't know any more.\"\n\nNathaniel nodded, grave. That so disturbing a degeneration could follow such brilliance\u2014was that the inevitable price of genius?\n\nIt was at that point the door creaked open. Three heads whirled around to see the entrants, and when Bedford saw his bride with puffed eyelids and damp cheeks, he rushed across to take her hand. Folkard stepped in calmly behind him.\n\n\"Annabelle,\" fussed Bedford, running his hand through her hair, \"are you all right?\"\n\n\"I'm fine, George, fine,\" and Annabelle composed herself. \"Please stop fussing.\"\n\n\"I take it,\" said Folkard gravely, \"That Mrs Bedford's distress is somehow linked to Doctor Grant's speedy exit?\"\n\n\"We saw your uncle make his way down the corridor away from us,\" said Bedford to Annabelle. \"He was...agitated. We called, but he didn't respond.\"\n\n\"Out of nowhere!\" exclaimed Arnaud, throwing his hands in his air. \"The man just exploded! Pushed poor Annabelle to one side and rushed from the room.\"\n\nFolkard looked around for somewhere to sit but, judging a lab stool inadequate, remained standing. \"Yes,\" he said gravely. \"I have been informed that the good doctor's behaviour of late has been...troubling.\"\n\n\"To put it mildly,\" said Nathaniel. \"How on Earth did he come to be working in the Admiralty?\"\n\n\"Just turned up one day,\" Folkard said. \"Mad as a March hare and twice as jumpy. Demanded to be let in, then pretty much locked himself away and wouldn't speak to anyone. I did try to persuade him to attend yesterday, Annabelle. To no avail. I hope you can forgive me for keeping his presence hidden.\"\n\nAnnabelle nodded her forgiveness.\n\n\"His sudden arrival was sinister, to say the least. As are...other developments.\" Folkard grimaced.\n\nArnaud, Nathaniel and Annabelle stared at Folkard. Folkard looked away, disturbed.\n\n\"We have intercepted certain heliographs. We can't as yet be sure of their source but they seem to indicate that one of Doctor Grant's crazed ramblings is correct. There is, indeed, a connection to the Irish Separatist movement.\"\n\n\"How so?\" asked Nathaniel.\n\n\"We're not sure as yet. All very vague. It will require some digging on ground level, so to speak.\" And here Folkard looked pointedly at Nathaniel, before looking away. He crossed to the bench Arnaud had occupied for most of the afternoon and lifted a test tube, held it to the light and peered into it. \"Fontaine, Stone... How did your investigations fare?\"\n\n\"We've made some progress,\" answered Arnaud, non-committally.\n\n\"And that progress is...?\"\n\n\"We can't be sure,\" said Nathaniel. \"The residue seems to be some sort of plant. A plant not known on this planet.\"\n\n\"And we're not even sure of that,\" added Arnaud.\n\n\"Is there a way to be sure?\" said a voice.\n\nEverybody wheeled around. Standing nonchalantly by the door was Enderby, once more holding a lit cigarillo.\n\n\"Ah, possibly,\" said Arnaud, somewhat stunned. \"An old associate... Teacher, I guess you would say. When I studied at the Sorbonne. But it may be difficult.\"\n\n\"My job,\" purred Enderby, \"is dealing with the difficult. Go on.\"\n\n\"His name is Garrecreux. Fabrice Garrecreux. A genius when it comes to organic crystallography. As an undergraduate both he and Professor Fournier wanted me to assist each of them with their research. I was flattered at the time, no doubt pompous\u2014though now I realise it was more to do with academic rivalry than my own talents, however substantial they may be. In the end, Garrecreux's advances were...unwelcome.\" Here Arnaud looked down, and didn't look up. \"I chose Professor Fournier, thankfully, and Garrecreux never spoke to me again.\"\n\n\"Will he speak to you now?\" asked Enderby.\n\nArnaud shrugged.\n\n\"The last I heard he had fled France, was living in India. Somewhere near Calcutta, researching the crystalline properties of a hemp plant. Cannabis indica. This was all years ago.\"\n\n\"I'll see what we've got on him,\" said Enderby, stubbing out his untouched cigar on a worktop. \"Pack your safari suit, Doctor Fontaine. And a hat for the heat.\"\n\n\"And that's it?\" Nathaniel asked. \"You're packing Arnaud off to India?\"\n\n\"And you to Dublin, Professor Stone. Pack a windbreaker. Possibly galoshes.\"\n\nNathaniel shot a glance at Folkard, his gaze furious.\n\n\"Captain, you can't...\" Folkard looked him straight in the eye.\n\n\"I can, Stone,\" he said. \"And will. There has been an attack on our soil and on her Sovereign Majesty. And there are calls louder and greater than even the Navy can deny. Is there a problem, might I ask, with you and Doctor Fontaine being separated?\"\n\nNathaniel raised an eyebrow, his posture still and uncompromising. He scrutinised Folkard for a moment.\n\n\"And if we are, what of you?\" he asked.\n\n\"Captain Folkard is required elsewhere,\" said Enderby. \"Unfortunate, but necessary. Please believe me, Professor Stone. I am more than aware of the dangers you have faced, and overcome, as a group. But when faced with such a pernicious enemy, our only hope is to spread our net wide, and hope to God the gaps are wide enough to catch a kraken.\"\n\n\"I cannot help but notice,\" said a small, sudden voice, \"that I have not been mentioned.\" The voice came from Annabelle, and it was pure and pointed. \"And neither has my husband.\"\n\n\"Well,\" grinned Enderby, carelessly lighting another cheroot, \"that may be because you have a honeymoon to enjoy. Courtesy of the Crown.\"\nChapter Four\n\nThe Spreading Net\n\n1.\n\nTHEY CALLED IT Dear Old Dirty Dublin Town.\n\nNathaniel, for all his travels between the inner planets, had never actually been to Britain's closest island neighbour. At Enderby's insistence he was to travel lightly and without ostentation, and so had caught the train from King's Cross to Liverpool under the watchful eye of one of Enderby's mute underlings. He had not even been allowed the luxury of first class (an eye opening experience that he instantly resolved to omit from his diary) and was thankful for the small meal of smoked salmon and cheeses he had had his housekeeper prepare. From the Liverpool docks he had boarded the ferry and, after watching England slink into the distance and several minutes staring out to sea, he realised that his sea legs were not so sturdy as his space legs and elected to have a little sit down and drink some weak tea until the whole sorry voyage was over.\n\nEnderby's lapdog had returned on the ferry, but not before breaking his silence to quiz Nathaniel on the minutiae of his mission. Rankled by the man's apparent skepticism concerning both his abilities and his short-term memory, his reply had been terse.\n\n\"I am to meet a man named Charles Cahalleret\u2014known locally as 'Tally'\u2014in a public house known as The Bleeding Horse, located in St Kevin's Port on the city's south side. And yes, before you ask, I have studied the man's daguerreotype so minutely I couldn't fail to describe him to his own mother. He has information pertaining to the gentleman who built the bomb, and I am by no means to trust him. After that, your superior has trusted my instincts as to how the investigation should progress and, by Heaven's eye, man, I'd be grateful if you did the same.\"\n\nWithout even a goodbye, the lapdog had left.\n\nBack on solid ground again, a rather picturesque tram ride brought Nathaniel to Pearse Station, and turning south onto Westland Row he met the city proper.\n\nSectarian squabbles aside, Nathaniel had not been sure how he would react to what was ostensibly the Empire's \"Second City\". He had to admit that, despite having arrived in the chilly air of a September afternoon, he was really rather enjoying himself. Popular belief would have had any visitor imagine the city a destitute hive, inflamed by Times editorials and heated talk of Home Rule. And yet, as Nathaniel paced idly down Nassau Street, the grandeur of Trinity College to his right, he felt curiously at home. Georgian architecture abounded, young lovers and smart-dressed gents took their strolls\u2014the whole thing smacked, he thought wryly, of Putney. There was even a game of cricket being played on the college green, overseen by an imposing and rather matronly pavilion.\n\nOne could not deny, however, the squalor that bristled at the city's edges, in the corners of market squares, taverns and side-streets. Dublin's population had grown, and with that a need for jobs that fell sorrowfully short. London's expansion meant a wealth of factories and workhouses which were sadly absent here, and the guttersnipes and destitutes that flitted between the city's grander occupants\u2014though no stranger to Nathaniel's eye\u2014seemed all the more dissolute as a result. He saw two women pulling at each other over a shawl, and a small boy wrestle away a man's pocket-watch before fleeing. But such were the strains of city life, and he instead decided to concentrate on the brightness and curious energy of the place.\n\nHe had memorised his route, took a left down Dawson Street and crossed through St Stephen's Green. Here, the air seemed even brighter\u2014ethereal, almost\u2014the trees halted in their journey to turn an autumnal brown. Nathaniel drank it in, this unspoilt mirror of a London blackened and choked by Industry's metal-and-mortar maw. Such was the price of progress, he reasoned.\n\nHe headed south, out of the park. Before long he had reached St Kevin's Port, and he looked up at the building in which he was to meet his informant and guide.\n\nThe Bleeding Horse was an ugly building, its extended gable and darkened windows looking somehow like a frown. As Nathaniel reached for the big brass handle the door was pushed open and a large man, pock-marked and unsteady on his feet, lurched out. He looked Nathaniel up and down and smirked before tumbling on his way towards the city's centre. As Nathaniel watched him go, he felt his eyes drawn upwards to the sign that hung on a great steel bar that extended from the building's front like a warning. The picture showed a great white horse, rearing up, its throat pierced. Its eyes were wide and panicked, a jet of crimson arcing from the wound. Nathaniel made his way into the fug of smoke and fumes of porter, and wondered how Arnaud and the newlyweds were faring.\n\n2.\n\nARNAUD WAS NOT a natural smoker of cigars, but in this environment it was somewhat gauche to be otherwise. Despite himself, he was finding the thick stub of tobacco silkily decadent, no doubt made silkier by the cognac. The great glass sheets of the observation deck stretched around him like a halo, sunlight flecking from steel girders and chandeliers alike. He found, much to his surprise, that he could actually ease back further into the depths of his chair, and he groaned momentarily in delight. God bless her Majesty! Libert\u00e9, \u00e9galit\u00e9 and fraternit\u00e9 could all go hang. This was living.\n\nThe pleasant tremble of the airship's engines hummed under his feet, adding to the air of lazy indulgence he was more than happy to encourage, both in himself and the staff on the sidelines. He need only gesture for them to refill his glass, or bring him an amuse-bouche or another fine cigar. He was no longer a simple geologist, and Enderby had insisted that he play the part assigned to him\u2014that of the spoilt son of an exiled marquis who had fled the revolution to the haven of Florence. And Arnaud (or at least, who Arnaud was supposed to be) was in good company. Lords and Ladies took in the day while the landed and the nouveau riche argued over card games, decorated Germans got solidly drunk and relived their past glories with a raucous camaraderie. Arnaud drank it in, all the while drinking cognac.\n\nAs he placed his glass back down, a black-clad arm swooped down and picked it up. He looked up into the face of the man Enderby had sent along with him. What was his name again? Ashford, or Ashforth, something like that. Pleasant enough, somewhat mild and milky-eyed, and as Arnaud had adopted the role of the nobility, so this milquetoast had adopted the role of the serf and servant. It was only keeping in character, Arnaud thought hazily, that he could not remember the fellow's name.\n\n\"Bon, Ashton,\" he said, waving lazily. \"Encore.\" Ever the professional, the servant of her Majesty's secretest of services bowed and left to bring him more booze and nibbles.\n\nArnaud had noticed the gazes of ladies with their d\u00e9butante daughters drawn his way. He would often catch their eyes and smile, smirking as mothers pulled their daughters aside to instruct them in the next steps of snagging old French money. He would not give them any truck, of course. But the game and the surroundings would provide him an amusing distraction, he decided, as the airship chugged its inexorable way towards the Indian subcontinent.\n\nIt was mostly the British on board the dirigible. Factory owners were expanding their businesses and career soldiers sailed to make their names in hot, uncertain climes. The vagaries of Empire business and the flirting of socially mobile schoolgirls didn't trouble him much right now, and besides, he had plenty of time to sober up before the ship reached port. Keeping up appearances, especially in these lavish surroundings, never felt so good. He had two days more to enjoy it.\n\n\"I hear,\" said an excitable matron unsubtly to his left, \"that he's technically a dauphin...\"\n\nArnaud took his drink from his uncomplaining valet and waved him away, smiling. He could get used to this, and wondered idly how his friends were getting on.\n\n3.\n\n\"THIS IS NOT how I imagined my honeymoon progressing, George.\"\n\nShe was furious, and, having been married for the best of two days, the only thing Bedford truly wanted was to make his bride, well...slightly less furious. This was proving somewhat difficult thanks to circumstances that were frustratingly beyond his control. He was becoming used to married life.\n\nFirstly there had been the problems with the luggage. Enderby\u2014or at the very least, somebody in his employ\u2014had been tasked with bringing it to the aerodrome for their departure. But their cases and Annabelle's grand selection of hats had somehow failed to materialise.\n\n\"It's never easy, is it?\" she grumbled, stepping up to board the airship Pall Mall with a swoosh that was, to Bedford at least, dismissive.\n\nIn all fairness, Enderby had been apologetic. He was clearly unused to such hiccoughs and was duly penitent, a welcome crack in his otherwise standoffish demeanour. \"I guess we'll have them sent along,\" he muttered, avoiding Annabelle's poison gaze. \"All we wanted was for you to enjoy a trip that would prove memorable. Speaking of which, may I introduce Mister Boon? Bertrand Boon. He'll be joining you for the journey.\" And Enderby extended his arm to introduce a black-clad man.\n\n\"Really, George,\" Annabelle fumed as they'd closed the door on their cabin. \"It's not too much to ask, is it? A little romance and a change of clothes,\" and here she raised her voice unnaturally, angling her head towards the cabin door, \"and privacy!\"\n\nThere was the soft clearing of a throat from beyond the wooden portal. Having served on so many ships\u2014both on the seas and in the void between worlds\u2014Bedford had to admit that their accommodation was, to put it mildly, somewhat basic. It was spartanly decorated, all straight lines and a chest for their non-existent luggage, maybe five feet by eight. But what had really been the icing on the cake was the bunk beds.\n\n\"I'll talk to the steward,\" he said. \"Mister Boon must have some sway, as do I. After all it's a British ship, and I've more than a few shillings in my pocket. We'll sort something out, my dearest.\"\n\n\"I am not your dearest,\" growled Annabelle. \"I'm your wife.\" Bedford thought it best to back off. He did so, unsure of what to do next. Annabelle sighed.\n\n\"I'm sorry. It's not the surroundings, it's the situation. Do you not think something...untoward in all this? That we're provided with a vacation, all,\" she glanced around the room, frowning, \"expenses seen to, and to add insult to injury they've packed us off to Arizona. Now don't get me wrong, it's not that I'm not looking forward to returning, it's just...\" She sighed.\n\n\"Oh Annabelle, I know...\" Bedford understood the conflict Annabelle must have felt deep within herself. Knowing her pain so intimately, he knew there was little he could offer to truly ease her mind. Arizona was the state of her birth, her home\u2014and yet also the place in which she was captured and incarcerated for two years by the indigenous Indians. It was an experience that had made her the woman she was, and the woman he loved, but also one which had deeply affected her, and left scars and terrible memories far beneath the surface of her skin. Even he was not aware of all the traumatic events that must have transpired\u2014he was not even sure Annabelle recalled the worst of them, and for a woman as strong as she, this must surely indicate some horrific ordeals indeed.\n\nAnnabelle looked him in the eye, took his hand. \"George, we could be sleeping in a back-alley slum, cabbage leafs and cardboard for our bed, and we would still have each other. And I would be happy if you were at my side. But I fear it will never be the case. I fear we will never have the chance to live happily as a married couple. It's not the fault of either of us, of course, but the callings of a world that may well never leave us in peace until we're dead. Had you thought of that?\"\n\nOf course he had. In fact, the very same thought had not been far from his mind since Enderby had proposed their surprise excursion. His allegiance to the Navy was one thing, but this was quite another! Yes, Enderby may well have had the safety of her Majesty (and, logically extending from that, the realm) at heart. Bedford had no qualms about that allegiance, and was proud of the fact he shared it\u2014and yet, and yet...he could not shake the fact that he was being taken advantage of, that his good nature and fortitude was being exploited by people whose motives remained worryingly opaque. He had his loyalties\u2014fealty to the Navy, to Crown and Country\u2014but his love for Annabelle and his marriage vows put her above all other pledges, and he desperately wanted to prove this to her.\n\n\"Mister Boon!\" he called forcefully. The door opened and Mister Boon's face\u2014thin, cautious, with a grossly receding hairline\u2014popped between the gap beside the jamb.\n\n\"Yes, Commander Bedford?\"\n\n\"Ah, so you are aware of my rank, then, sir?\"\n\nThe man furrowed his brows.\n\n\"I am, sir.\"\n\n\"Well then kindly look around you, Mister Boon. Do these look like a commander's quarters? And what's more, do they look like the quarters of a commander on his honeymoon?\"\n\n\"They do not, sir.\"\n\n\"So glad you agree. See to it, eh, Boon?\"\n\nMister Boon smiled wanly, but there was a certain warmth there. \"With pleasure, sir,\" he said.\n\n4.\n\nEYES FOLLOWED NATHANIEL as he traversed the room. Some were hazed by drink, others were suspiciously sharp. Everyone, it seemed, had noticed him. This may not have been due to Nathaniel merely being an outsider. The inside of the pub, which pretty much resembled a large, hollow cube, was crammed with low, skulking tables and darkened nooks along the edges. A solid wooden staircase climbed the left hand side of the room, inviting drinkers up to a mezzanine that looked down on the main floor. Behind the bar, other rooms and corridors branched off into the darkness. Tally could be anywhere, and in order to make sure Nathaniel did not miss his man he found himself scrutinising every single face, and it was clear that his interest was unwelcome. He had already noticed the barman's gaze follow him relentlessly, whether the man was idly polishing glasses or pouring a customer their dark, black pint.\n\nNathaniel suddenly found himself shoulder-barged by a passing drinker, and as he wheeled around, babbling a quick apology, he found the man had already taken offence.\n\n\"Why, there y'are, ya gobshite!\" he growled, and suddenly Nathaniel felt two arms like iron girders wrap around his waist and yank him down. To the general cheers of the crowd the lout grasped Nathaniel in a headlock and yanked the helplessly protesting professor back outside. As the air of the outdoors hit him he struggled, was nudged around by his assailant and brought upright. Nathaniel raised his fists to defend himself, pistol-keen and ready to duck a punch, but instead found himself looking into the face of a rather familiar man, smiling and brushing off Nathaniel's lapels.\n\n\"Now here's a piece of advice for you, Professor Stone. If you find yerself walking into an Irish inn, as a Britisher you do one thing. You look at the walls. If you see our fair Isle's declaration of independence posted up there in English, get out while your legs still can. If you see it written in Irish, well, that just means they just haven't decided how they're gonna kill you yet.\"\n\nNathaniel rubbed his throat and got his breath back. \"And which,\" he asked, gasping, \"is the case in there?\"\n\n\"Neither,\" smiled the man, extending a hand. \"But that's hardly the point now, is it? Tally Cahalleret at your service.\"\n\nHe was much as he had looked in the picture. A hardened man, clearly, albeit a handsome one. In his early thirties, he had jet-black hair, blue eyes, strong jaw line, and was somewhat shorter than Nathaniel had imagined him to be. He wore a waistcoat and open collar and a wide, flat beige cap was balanced diagonally over his head.\n\n\"Pleasure to make your acquaintance, Mister Cahalleret.\"\n\n\"Tally, please. Mister Cahalleret was my arse of a father. Sorry about the roughhousing. It's just I had to get ya out of there before one of them boys heard ya talkin', or worse, tapped ya for a drink.\"\n\n\"It's quite alright.\"\n\n\"Right so. And speaking of drinks, Professor Stone, can I indulge ya? I believe we have some business to discuss.\" He opened the door for Nathaniel. \"After you.\"\n\n\"But...\"\n\n\"Ah, don't be soft, man. I'll tell 'em ya paid what ya owed me and that's what we're drinking on. C'mon now. Don't be shy, but keep your gob shut. At least 'til we're in private.\"\n\nFollowing shortly after Nathaniel, Tally yelled to the crowd \"He's alright, fellers!\" and waved at the barman. \"Stephen,\" he said, \"rack a couple up and bring 'em through. We'll be in the snug.\"\nChapter Five\n\nThe Empire's Second Cities\n\n1.\n\nIT WAS WITH the greatest of sorrows that Arnaud packed his suitcase as the airship prepared to dock. Childish it may have been, but he resented being deprived of the lifestyle into which he had grown so accustomed\u2014over two days, at least. For the rest of the first day he had got solidly blotto on cognac, and then was invited by a certain Lady Hawkston for an evening reception and lunch. In a cramped and second-class cabin she had introduced her daughter Cecily (mentioned many times but not yet seen) as if she were the Duchess of Dubai. Cecily had clearly scrubbed up for the occasion. He had had to concentrate hard in order to not fall over, and in all honesty, had felt more attraction for the Skreelan of Venus.\n\nThen followed lunch in the ballroom, along with wine. Arnaud didn't remember much after that, and awoke feeling like the Devil's own cleaver had been brought down hard between his eyeballs. He had stayed in bed until 4pm, and after bathing for another hour and feeling sorry for himself, he threw on some clothes and staggered off to find something to eat. There was no sign of the fellow, Ashpan, or whatever the devil's name was, the devil take him. An early evening buffet had begun and, using his clout as a potential dauphin, he persuaded the chef to knock up some kedgeree, and sent for a brandy. As he was snaffling the spicy rice, eggs and haddock Lady Hawkston passed by, and turned her nose up. Of poor Cecily there was no sign.\n\nIt was only after he'd stuffed himself with the kedgeree, followed by English muffins and three cups of coffee, that Arnaud had begun to feel even marginally human again. Feeling sedated and sleepy, he elected to spend the rest of the day gazing from a porthole and snoozing, only to find the black-clad man waiting back in his cabin, reading a paper with his legs resting on Arnaud's bed. He uncrossed his legs and put the paper down.\n\n\"Feeling better, are we, dauphin?\" he asked.\n\nTo cut a long story short, it turned out the bugger's name was Coyne and he was dead set against letting Arnaud stay on the ship to Bombay.\n\n\"We'll be docking south of Calcutta within the hour, Arnaud,\" he said, bored, as he passed Arnaud another shirt to pack. \"You've had your fun, and now we've got work to do.\"\n\nArnaud was still mildly hung-over. He packed shirts and prayed for somewhere close to the landing pad where he could sleep. He could already feel the heat of the Indian sun beating at his temples through the window.\n\n2.\n\nTHE HEAT WAS also bothering Annabelle and George. Annabelle felt momentarily flushed, and lifted her fan. But not because of the temperature, no\u2014the pleasant Arizona breeze rushing through the train's open windows kept her cool\u2014but because she was curious to see what would happen if she dropped it.\n\nGeorge was sitting next to her. In order, he maintained, to get into the proper holiday spirit, he would put away his usual histories and biographies and read something a bit different. Unfortunately, this flash of literary derring-do had only occurred to George after they boarded the airship (having thankfully been relocated to a comfortable double room with a nice view. It turned out, incidentally, that the luggage had been in this room all along). He had insisted on scouring the nearest dead-end town and didn't leave until he ended up procuring several penny dreadfuls and Volume II of Don Quixote. Have discovered Don Quixote to be written in the original Spanish, he was now attempting to enjoy some lurid tale, which, inasmuch as Annabelle could work out, was mostly about soldiers shooting things. He was absorbed, and Annabelle was bored.\n\nAnd so she raised her fan. This was a test for him.\n\nAffecting a touch of the vapours she tilted her head back and let out a little sigh, then nonchalantly dropped the paper fan. She kept her head back, waiting for George to acknowledge her feigned distress. He did not, and kept his nose in the book. She cleared her throat. He turned a page. She cleared her throat again.\n\n\"George,\" she hissed.\n\n\"Hmm?\"\n\n\"I've dropped my fan, George.\"\n\nGeorge looked up, then down at Annabelle's feet.\n\nHe beamed. \"Oh, so you have!\" He picked it up and dropped it in her lap, and then got back to his book. \"I'll just finish this chapter.\"\n\nAnnabelle crossed her arms and harrumphed. At least Mister Boon had left them alone for the present, reasoning, she assumed, that he thought there wasn't much she could get up to on a speeding train. Her eyes fell on the seven or so other cheap novellas George had acquired at such great cost. There was one about an adventure on Venus (full of factual errors on even the briefest glance), an historical one about previous generations of soldiers shooting things in Africa, and another historical one that caught her eye.... The Lusty Knight, it was called. She picked it up, and the frontpiece showed a rough illustration of a wench with a cavernous bosom being swooned upon by a shining, softly handsome knight. Medieval drapes and stone walls swam, badly inked, in the background. Flicking through, she noticed Chapter IV (subtitled 'A Passionate Encounter') was by far the longest in a fairly short book, and Annabelle decided she might as well sit and have a read too.\n\n3.\n\nMEANWHILE, AT THE Admiralty, Folkard and Enderby were hardly so pleasantly diverted. They were packing for a trip of their own. Yet they would not experience the pleasure of a first-class cruise, nor the manic squalor of Calcutta's slums and the danger that waited therein. They would not feel the Arizona winds on their faces nor the warm embrace of a Dublin pub, because for them, the destination was far bleaker.\n\nFolkard was calm. As he packed thick clothes, bully beef and ammunition into his rucksack he gripped each item tightly. The Heart was in his head, his lost ones were singing, and tonight would be a full moon in a clear sky. Enderby watched him, nervously.\n\n4.\n\nMISTER BOON HAD returned and, much to Annabelle's delight, had brought a covered tray of breakfast with him. George had dozed off and was snoring while Annabelle had immersed herself in a rather racy tale that (on more than one occasion) had caused her to pause and look up to George. Her radiant knight, drooling onto a paperback. He was still asleep when Boon had lifted the platter to reveal poached eggs, toast, grilled kidneys and bacon, coffee, tea and the morning paper. George became instantly awake.\n\n\"Little bit of home, Commander,\" said Boon. \"My treat. It's your honeymoon after all.\"\n\nIt was easy to see. George was touched. Boon nodded, pleased, and turned on his heels.\n\n\"Bertrand, is it?\" asked George.\n\n\"Well, Mister Boon, professionally. But in all honesty, I prefer Bert.\"\n\n\"Well, Bert, that's... That's incredibly generous of you!\"\n\n\"Thank you, Bert,\" added Annabelle.\n\n\"Thing is, sir. I know how it is.\" It seemed as if he'd relaxed a bit, his natural South London accent coming though. \"This job, all that. They cart you off without a by-your-leave and half the time you don't even know where you're going, let alone what you're supposed to do when you get there. But they told me all about your wedding, so I felt I had to. Missed seeing my first born, see. Thanks to the job. Anyway, not my place. I just wanted to make it feel like a special occasion for you.\"\n\n\"Bert, please,\" insisted George. \"There's more than enough for the carriage, let alone us two. Won't you join us?\"\n\n\"I'd love to, sir. Really would. But I've gotta have a quick recce of the rest of the train. Couple other of our fellows aboard, see. Pays to crack the whip. 'Fraid I couldn't get you The Times, sir. Just a fairly local thing, but it's got a crossword, at least. Enjoy your breakfast.\" And with that, he was gone.\n\n\"He's a boon indeed,\" said George, rubbing his hands together and ogling the food before him. \"I wonder if I should give him a tip?\"\n\n\"Don't be asinine, George,\" said Annabelle and, feeling she had just uttered a line worthy of one of Mister Dickens' heroines, she picked up a piece of buttered toast and nibbled at the edge. George poured himself a cup of tea and tucked into the meat with gusto.\n\n5.\n\nAFTER THEY WERE both sated, George got back to his boy's own novella and left Annabelle to her own devices. She was tired of her own paperback, the passionate encounter of Chapter IV having been and gone, and so instead decided to riffle through The Blackwater Ledger, the newspaper Boon had so thoughtfully provided. She found the news she read there oddly comforting. It seemed the west had changed little\u2014there were the same old announcements of births and deaths, the opening of a new railroad or the closure of a spent silver mine. Murders and their perpetrators, acts of faith and villainy, tiny type cramped between adverts for horses, brand new ventures and snake oil. And then one headline caught her eye.\n\nProspectors Spared Indian Savagery! The Reason: Innocence?\n\nTwo prospectors, to whose request of anonymity we have graciously conceded, have today returned to Blackwater after a frightful ordeal in the desert\u2014involving Indians! The pair, who had been informed of possible deposits of gold to the west of Horton's Pass, set out and rode perhaps ten to fifteen miles beyond where they had intended. Lost, thirsty and disorientated, one prospector claims they were set upon by a starving mountain lion, while the other insists it was nothing more than an ill-tempered fox. Irrespective of these minor details, one of the pioneers found himself with a sorely twisted ankle, unable to walk or even move. Thanks to a spot of squirrel hunting and previously packed supplies, the duo managed to survive the night, with the uninjured of the two intending to set back to the nearest town at first light\u2014perhaps four days' ride away.\n\nAnd yet! His journey was not to be! He awoke in the silky dusk light to find his friend had been rolled away from him, towards the dying embers of the fire, and that in the stygian gloom he could just make out a hulking figure...leant over the prostrate form of his companion!\n\nInstantly, the prospector tells us, he was alert and had drawn his pistol. The figure had simply raised a great hand akin to that of a bear and waved him off. It was then that the prospector noticed a young girl, perhaps eight or nine years old, standing behind the man, holding out a sheaf of herbs. The old man was chewing on the herbs and rubbing the resulting, foul-smelling mixture on his companion's ankle, and binding it with strips of cloth and leather.\n\nIn what was described as \"Excellent English\", the Indian advised the pair to ride to our fair town of Blackwater, far closer than the original destination and, in the opinion of this humble reporter, a far more desirable one. He advised the injured man that he could ride after a day's rest, so long as he rode side-saddle and kept the weight off the bound leg. With that, the Indian and his young accomplice took three tins of beans and two roasted squirrels, and departed into the night. The following day, the prospectors began their journey to our town, where they sought out our fair publication forthwith, in order that their story might be told.\n\nCould these two men have survived an encounter with the famed Apache outlaw Geronimo? This reporter thinks not. If they had encountered that misanthropic beast of the wilds they would surely have been scalped alive. And yet, perhaps it was the presence of that innocent, the young girl of the tribe, that tempered the savage heart. Even though it is not uncommon for a squaw to be married and bear children at such a shockingly young age, perhaps this Wild Man retains some semblance of humanity, and chose to spare the girl the sight of such unconscionable bloodshed.\n\nIt was clear to Annabelle that this was the kind of disreputable journal that sold itself primarily on hyperbole and populist cant; it was all gossip and ill-informed conjecture, nothing more. She tossed the paper angrily aside.\n\n6.\n\nTWO DAYS PREVIOUSLY, in the lounge of The Bleeding Horse, Nathaniel was directed to a booth at the back of the lounge, one of several, all numbered, with doors that could close the occupants snugly in. This room sat at the very rump of the pub, down a short flight of stairs and possibly underground. There were no other tables on the floor and, save the dimmed ruckus from the main bar, the room was silent.\n\n\"They nickname it 'The Confessional',\" smirked Tally, as he closed them in the booth.\n\nOut of politeness more than curiosity, Nathaniel had accepted a pint of Guinness from Tally. It now sat before him, untouched, while Tally's was nearly halfway gone.\n\n\"You not gonna touch your Guinness, Professor?\" asked Tally, genuinely interested.\n\n\"I'm not much of a drinker, let alone of ale.\"\n\n\"Well, there's yer first mistake, as this ain't ale. G'wan. Give it a go. What's the worst that could happen?\"\n\nNathaniel couldn't really argue with his logic. He gripped the glass tightly. It felt slightly heavier than he remembered from carrying it across, and he brought the thick black liquid topped with thick, white foam to his lips, and sipped. A deep, chocolaty goodness passed into his mouth, followed by a rich taste of velvet molasses that coated his throat as he swallowed.\n\n\"Dear Lord,\" he muttered. \"That really is rather pleasant.\"\n\n\"First taste of Ireland. Welcome to Dublin, Professor Stone.\"\n\nNathaniel smacked his mouth appreciatively and took another sip.\n\n\"Careful now,\" said Tally. \"That stuff'll leave you with a fair old head in the morning.\"\n\n\"Yes, quite,\" agreed Nathaniel. \"And I fear I'll need my faculties over the coming days. I'm assuming it's safe to talk?\"\n\n\"Safe as houses.\"\n\n\"Then I'm led to believe you have information pertaining to the identity of the Horseguard's Bomber.\"\n\n\"Well, that's the thing,\" said Tally. \"I do and I don't.\"\n\n\"You do and you don't?\" blurted Nathaniel loudly. \"What's that supposed to mean?\"\n\n\"Whoa there, Professor,\" cautioned Tally. \"These things are private, but they ain't soundproof.\" He lowered his voice to a whisper. \"What I mean by I do and I don't is that I do know something, but I don't know yer man who lit the fuse.\"\n\n\"Well,\" pressed Nathaniel, \"what is it?\"\n\n\"It's me nephew. He's a zookeeper.\"\n\n\"A...zookeeper?\"\n\n\"Aye. Up in Dublin Zoological Gardens. You never heard of Dublin Zoo?\"\n\n\"Does it matter?\"\n\n\"Aye, I guess not. Anyway, young Simon (for that's his name), young Simon notices one day the Earth sorta shiftin' underneet his feet, like a tremor, an earthquake or what-have-you. Pretty soon after that, see, parts of the zoo get closed off, no explanation, and young Simon says he could hear the sound of drilling, an' explosions. All this over the past couple of months. Anyway, they've opened it up again now, but our Simon noticed some shady goings on in that exhibit, the new one, what with stuff from Mars. Visitors after closing time, that sort of thing. When young Simon told the gaffer about it he was told to shut his hole and feed the ferrets.\"\n\n\"But what's that got to do with the bomb?\"\n\n\"Not a lot. But there's been rumblings in certain...organizations. That an expert was in town, just doing his own thing, but he suddenly opened himself up to giving certain organizations lessons in new an' deadly ways to make things go bang, if you get what I mean. Apparently, this expert was seen a lot in Phoenix Park, not far from the Zoo.\"\n\n\"What does he look like?\"\n\n\"Ah, nobody's all that sure. The lads who attended his little masterclass have apparently been sworn to silence. But he's one of your lot, a Britisher. Doesn't seem to care to hide it, neither. Tweed, leather gloves, an upturned nose and a stick up his arse. Only... Only there's always this other feller with him, and he's the type that tends to draw the eye.\"\n\n\"Oh? How so?\"\n\n\"He's built like a side of beef on both sides, is how so. They say he's got brass hands, always wears these damn welding goggles. Never talks.\"\n\n7.\n\nTHEY HAD DISCOVERED their destination in the final briefing: Severnaya, Russia, miles from the middle of nowhere. It was here, they had been informed, that the Russians were studying and perfecting dangerous new technologies that had the potential to eclipse those of the British Empire. Grant had provided some information, either through rambling, enraged rants or spidery notes pushed out from under doors as the doctor wailed in the locked room beyond. The Secret Service Bureau was dug in well in Dublin, and though the intel was vague there was a definite Russian connection. Treading on enemy soil\u2014it was by far the most dangerous of missions. Yet Folkard relished the prospect. Not for the adventure nor the protection of his homeland\u2014it seemed that every passing moment brought him closer to the mission, and that also brought him closer to the Heart, and, ultimately his dear wife.\n\nThere was no choice. They could not have gone by rail or commercial flyer. This clandestine mission required far much more finesse and resources, and so HMAS Sovereign had been commissioned for their flight.\n\nFolkard felt a strange sense of nostalgia being back on the ship, but not once did he ever feel the need to go to the bridge. He mostly stayed in his room. The rumblings of the engines through the walls\u2014that exact pitch\u2014was one he knew so well, like the close embrace of someone long lost. But with those rumblings came an energy, a direction, that was bringing him closer to the Heart. He could feel it.\n\nHe had taken a stroll round, of course. The old girl was looking fit as a fox, he had to admit, and he noted that even some of the improvements he had suggested to the Admiralty following his decommission had been implemented. He felt proud to have made the ship what she was. But he was also aware that many of the men would know his face, catch his eye\u2014he would cause unrest on board if he made his presence obvious. One or two trusted souls he'd like to speak to, of course... But for the most part, he wished his presence to be ghostlike.\n\n8.\n\nARNAUD WAS RATHER surprised at how easy this was all turning out to be\u2014despite this damnable, oppressive heat. He had felt the temperature rising steadily over the second day of his journey east; by the evening the observation deck of the airship had become sweltering due to the great glass windows, and all but the hardiest of passengers had retired to their rooms to snooze and escape the sun.\n\nThat selfsame sun was now beating down on Arnaud like a hammer at a forge. He was wearing a safari suit made of cream-coloured cotton, but even this light fabric was sodden with sweat and clung to his body like a wet bathing costume. After the airship had docked, Arnaud had been one of the first passengers to disembark. The aerodrome (if you could call it that\u2014a more accurate, though less flattering, description would be a raised mound of earth surrounded by a few sorry looking huts, with telegraph poles spidering away from them towards the horizon) was located some miles south of Calcutta, and yet with commendable alacrity Coyne had procured them a sort of rickshaw affair, piloted by a local peasant in a loincloth on a bicycle that had been bolted to the passenger's bench in a worryingly ramshackle fashion. Neither the driver's emaciated appearance or the shoddy design of the vehicle were built for speed, and any hopes for relief from the heat by a cooling breeze were remote. Arnaud found himself constantly fanning his face with his hat or wiping the sweat from his eyes with a damp handkerchief. Coyne, still in black, seemed unperturbed.\n\nThe city of Calcutta crawled up to meet them. It was not like approaching London, he thought. Or even Paris. Calcutta snuck up on you slowly. First came an increase in locals by the roadway, often walking towards the city, mostly alone, often carrying or carting improbable loads\u2014mangoes, scrap metal, bundles of newspaper tied with strips of leaves. Skeletal dogs trotted beside them or slumped, breathing heavily on their sides, by the edges of the road.\n\nThen the slums began. The first of the hovels had shocked Arnaud\u2014it seemed incomprehensible to him that people could even survive\u2014let alone make a living\u2014in such appalling squalor. From a distance, he had thought the first was some kind of rest-stop dropped carelessly onto the barren landscape, little more than a few wooden boards propped together, topped with a waxed fabric roof. It was perhaps five feet square. As they approached Arnaud saw an ancient looking Sikh sitting cross-legged on a mat beside the door, smoking a sort of upright wooden pipe. Through the doorway he could see figures sleeping, cramped together on woven mats, and on the other side of the hut a broad-hipped woman in a colourful sari was pegging out washing on a line. It was then Arnaud realised that this was not a rest-stop but a home, and he thought back guiltily to his two days of sloth and drunken indulgence.\n\nThereafter, similar huts began to pepper the roadside, all equally cramped and ill-constructed. He began not to be shocked by them, and then not to notice them at all, and occupied his time drifting off into hazy, febrile daydreams. Occasionally, children would run alongside the rickshaw with their arms outstretched to beg, only to have the driver curse at them in a rapid, angry staccato and wave his arms at them to clip their ears.\n\n\"This is nothing compared to the slums in the city proper,\" said Coyne. Was Arnaud mistaken, or did he detect a sort of malicious glee in the way the fellow said it? Either way, the Frenchman was too wilted for conversation and grunted non-committally, returned instead to his daydreams and, eventually, thoughts of Garrecreux.\n\nBizarrely, or perhaps because of this feverish heat, one of the only things that Arnaud could recall of his would-be mentor was his eyebrows. Great, big, black bushy things they were\u2014vast to the point of ridicule. More than once an undergraduate night of drinking had ended with a cooled chunk of charcoal being pilfered from the grate to facilitate an impromptu impression, and Arnaud smiled at the memory of long-gone, carefree days. It struck him that they had mocked Garrecreux because they did not respect him\u2014not in the same way they had respected Professor Fournier. Garrecreux had an insincerity about him that Fournier lacked, a shifty, squint-eyed suspicion that would wash across the lecture theatres like the beam from a lighthouse. Where his gaze was keen, his teaching was dull and dry. To him, rocks were no more than dead objects to be studied, minerals ground down to powder to make statistics out of scientific composition. What was there, was there; they had no more stories to tell after they were reduced into dust. Fournier's approach\u2014and consequently Arnaud's, thanks to the former's tutelage\u2014was to find magic, life and mystery in the floor beneath humanity's feet.\n\nAs well as that, there was Garrecreux's instance of one-to-one tuition and an alarming penchant to lean uncomfortably close when checking Arnaud's notes. Arnaud had been little more than a boy then (though at the time, naturally enough, he believed himself to be far more), and even as a man the thought of those weasely eyebrows brushing against his cheek made him shudder. He steeled himself. This was not to be a pleasant reunion of old university acquaintances. His only hope, he reasoned, was to challenge the old man's pride, and force him to prove that he could, in fact, help.\n\nYet whatever happened, Arnaud was certain that he would be remembered.\nChapter Six\n\nIntrigue in the World's Four Corners\n\n1.\n\nTHE TERRIBLE SCREECH of a train's brakes roused George from his idle slumber and punted Annabelle from her seat into the opposing\u2014and thankfully unoccupied\u2014bench. She was relieved to see George instantly awake and tending to her current discomfort, offering his hand to help her rise and asking if, please God, she was unharmed. Troubled, perhaps, but unharmed\u2014she wondered to herself if she might not have taken such a tumble had she possessed two fully functioning legs.\n\nScreams and consternations emanated from other booths along the carriage. George and Annabelle had scarcely got their breath back before Boon pulled open the door to look down upon them.\n\n\"Are you all right?\" he asked quickly.\n\n\"We're fine, Bert,\" replied Annabelle, unfazed. \"What's happening?\"\n\n\"No idea, miss. Legged it here as soon as this bloody thing stopped shuddering.\"\n\n\"Well,\" said George, dusting himself off pointlessly, \"shall we go and find out what we're dealing with, hmm?\"\n\n\"Right you are, Sir,\" beamed Bert. \"Right you are.\"\n\nGeorge turned to Annabelle.\n\n\"Coming, Mrs Bedford?\" he asked, offering the crook of his elbow.\n\n\"I thought you'd never ask, Mister Bedford,\" she said, and took his arm.\n\nThey glanced out of the window but the mound of earth onto which the tracks were built was rather steep. The rails ran round along the left towards the horizon, and it was easily seen that the tracks ran into a plateau near the engine. Already, the driver and a couple of engineers had disembarked. Their bright blue overalls stood out starkly against the burnt orange of the Arizona soil as they approached an ominous black blob that sat, unnervingly still, some twenty metres on the track before the front of the train.\n\nThe trio dashed up the length of the carriages until they reached the partition to the hopper filled with coal. When they reached the door, a surly and overweight railway worker had held up his hand to stop their progress before they were even ten feet away.\n\n\"Summin' on the line, folks,\" he drawled, bored. \"We clean it up, we mosey on. Get back ter ya carriage, drink yer tea. We's got it all under control.\"\n\n\"Excuse me!\" started Annabelle.\n\n\"Now hush hush, little lady! You don't wanna be getting' your drawers in a disturbance, you just leave this to...\"\n\nAnd the worker found his drawl curtailed by Annabelle's knee, which had swiftly and viciously found itself embedded in his groin.\n\n\"And to think I was about to invoke Queen and Country,\" muttered Boon.\n\n\"We're in my country now,\" smiled Annabelle. \"Let's go.\"\n\n2.\n\nANNABELLE, GEORGE AND Bert clambered down the short ladder onto the dusty ground, and within moments were dashing towards where the driver and his cohorts were milling around the blockage on the track, scratching their heads and rubbing their stubble. As they approached, it became clear the problem on the line was animal in nature\u2014not black, but dark brown, lying on its side with two great horns jutting out from the crown of its head. A buffalo! Annabelle recognised it instantly, sped her step towards the corpse of the fallen beast, and as she did so, the train driver extended her arms and waved for her pleadingly to stop.\n\n\"Whoa, whoa, missy!\" he began to plead, \"this ain't no sight for a lady!\"\n\n\"I've seen more of these gutted than you've had hot dinners,\" she admonished, stopping the man in his tracks as she finished her jog towards him. \"And what's more, you're all idiots.\"\n\n\"Well hold your tongue there, missy! This is Injun country, an' caution ain't gonna do us no harm.\"\n\nGeorge, breathless, caught up to them, Boon close on his heels.\n\n\"I'd listen to her if I were you,\" said George, resting his palms on his knees. \"Your colleague nursing himself in the train will testify as much. What's wrong, Annabelle?\"\n\nAnnabelle had approached the fallen buffalo. Three arrows jutted from the hump in the beast's back, and with a twist of her wrist Annabelle yanked one out and studied the tip.\n\n\"These aren't Apache arrows. Can't be. Look at them! The shaft, you couldn't fire this straight if you tried. And what's more, three arrows in a buffalo's hump? It would barely stop the thing from goring you, let alone kill it.\" She tossed the arrow aside and dusted off her hands. \"Push this thing to the side,\" she instructed the two engineers. \"I'll wager we'll find a great shotgun wound there. And if there is...then where's all the blood?\"\n\nThe two engineers, seemingly powerless to resist Annabelle's barked instructions, hefted their shoulders against the weight of the animal and began to push.\n\n\"Come on, boys,\" said Annabelle, \"Put your backs into it!\" Yet her bravado was curtailed by George, who tapped her lightly on the shoulder.\n\n\"Darling,\" he protested gently, \"this all well and good, but I think we may be faced with more pressing concerns.\"\n\nAnnabelle followed his gaze. On the horizon, a dust cloud loomed, the dark silhouettes of riders and the trembling thunder of hooves cloaked inexorably, terrifyingly, within.\n\n\"Injuns!\" yelped the driver.\n\n\"Indians,\" corrected Annabelle. \"But which tribe? Are they friendly?\"\n\nUnder the hoof-beats, the sound of whoops and war-cries began rising steadily in volume. Once or twice came the crack of a carbine being fired into the air.\n\n\"Well, I guess that answers that question...\" She lifted her skirts and, along with George, Bert, and the ashen-faced railway workers, began to pelt back towards the train.\n\n3.\n\nNATHANIEL HAD AWOKEN at some point in the mid-morning, his head not unpleasantly heavy thanks to a couple of pints of stout. He smacked his mouth experimentally, just hoping that, should he once more desire a taste of Ireland's most famous export, he would not have to head into the more salubrious of London's inns in order to satisfy his newly-found craving.\n\nHe was stretched out on an old, threadbare divan in Tally's garret, his feet poking off the edge. He began to recall the previous night. After Nathaniel had quizzed the Irishman on everything he knew about the strange seismic disturbances under Phoenix Park (which turned out to be little more than he had already been told) the two had moved on to other, more diverting topics, including local politics, the state of the Empire and life in Dublin in general. They had ordered a large platter of Dublin Bay prawns from the barman\u2014sweet, succulent, covered in butter and accompanied by heavy and toothsome soda bread. As promised, Nathaniel had picked up the bill and at some point in the early hours the two had stepped out into the warm night air. Nathaniel had enquired about a local hotel he might stay at\u2014even The Bleeding Horse itself\u2014but Tally had insisted on showing Nathaniel the extent of Celtic hospitality. This had turned out to be the aforementioned threadbare divan in a garret, somewhere in the backstreets of the suburb of Rathgar.\n\nHe could, just about, see Tally in his shirtsleeves, his back to Nathaniel by the fire. A gentle sizzling and the sweet, salty smell of bacon wafted temptingly into the room. Nathaniel sat up to greet the heady aroma, and Tally turned around.\n\n\"Ah, top o' the morning to ya,\" he grinned. \"And just so's you know, nobody really says that here.\"\n\n\"Duly noted,\" said Nathaniel.\n\n\"Thought I'd whip us up a bit of fry,\" said Tally, turning back to his pan. \"Bacon and white pudding, the best Dlugacz's Butchers had to offer. You hungry? Kettle's on too.\"\n\n\"Marvellous,\" exclaimed Nathaniel, stretching his arms and rising. \"What's our plan of action for the day?\"\n\n\"Well,\" said Tally slowly, prodding at the crisping meat, \"I've a little business to take care of, but that works well as we don't want to get to the zoo 'til later. No point in us hanging around the park all day anyways, reckon we'll be best making our investigations under night's sultry cover, eh, Professor Stone?\"\n\n\"I couldn't agree more,\" nodded Nathaniel. \"And, ah...if it's not to impertinent a question, Tally, what is it, day-to-day, I mean, that you actually, ah...do?\"\n\n\"I get by,\" said Tally glibly as he tipped their breakfast onto a couple of plates. He turned around and passed Nathaniel his share of the thick, delicious-looking rashers.\n\n\"Tell me,\" he continued, folding up a piece of bacon with his fingers and pushing it into his mouth, \"with all them degrees and book learning and what-have-you, have you ever cracked a safe?\"\n\nNathaniel let his fork drop down with a clink and stared, slack-jawed, at Tally. The Irishman stared back at him with steely, serious eyes.\n\n\"Well, I, I mean, Tally...\"\n\nBut Tally could compose himself no longer, burst into a huge peal of laughter, and doubled up. \"Ah, the look on your face! Priceless. Don't be worrying your head, Professor. It's just a couple of acquaintances I've gotta see. May not be strictly above board, in the truly legal sense, but we ain't harming a hair on no innocent's head. Like I say, just getting by. No need for either of us to run off to confession any time soon.\"\n\n\"Well, that's a relief.\"\n\n\"Ain't it just? Do you have milk with your tea? Only I can't afford milk.\"\n\n4.\n\nTHOUGH FULLY AWAKE, sleep being a commodity Folkard found himself increasingly untroubled by, the dispossessed captain nevertheless ignored the knock on his cabin door. He was lying in bed, wrapped in a rough grey naval blanket, as Sovereign made its way through low orbit to the ice sheets of darkest Russia. The knocks came again, and Folkard drew the blanket around his head tighter still, turning his back to the door.\n\n\"Folkard,\" came Enderby's voice. \"Folkard!\" He rapped again, louder. Like a child in the dark startled by shadows on a window pane, Folkard scrunched his eyes tightly shut and pulled the pillow close over his head. And in the same way this action cannot stop the terrors of the night assailing the imagination, so it did not stop Enderby from opening the door and entering Folkard's cabin.\n\nFolkard feigned being roused from a deep and heavy slumber, raised himself up on his haunches and slowly rubbed his eyes.\n\n\"Mister Enderby,\" he groaned sarcastically. \"To what do I owe the pleasure?\"\n\n\"To your continued absence on board, for one,\" sniped Enderby. His tone was curt. \"We're beginning our descent on Severnaya in just over forty minutes. It's time to get up and get ready.\"\n\n\"Very well.\" Folkard swung his legs over the side of the bed and rubbed his temples. His eyes felt red. His whole body ached. \"Allow me some coffee and I'll be right with you.\"\n\nFolkard had intended this to implicitly suggest that Enderby should leave him be, but the agent merely stood with his hands behind his back. He eyed Folkard suspiciously.\n\n\"Are you enjoying your return to the Sovereign, Captain?\"\n\n\"It's just Sovereign, Enderby, not the. naval tradition. And I'm no longer a captain, at least not of this ship. But yes, I can't deny a certain...nostalgia.\"\n\n\"You like this ship then? Are you proud of it?\"\n\nFolkard bristled. \"What kind of a question is that, Enderby? Barely warrants an answer. Of course I am. In many ways, I still consider this my ship, irrespective of the current situation. We adventured many times, this old girl and I. More of a home than a ship, and more of a friend than a mere means of transport.\"\n\n\"Good. May I sit? There's something we need to discuss.\"\n\nFolkard idly waved him to the seat next to the cabin's desk. Every move careful, Enderby pulled out the chair, twisted it around, and sat to face Folkard directly.\n\n\"It is time,\" he said, \"that you learned of the true nature of the threat we are dealing with, and the extent of the conspiracy that digs its foul roots into our most treasured and distinguished institutions. To their very heart.\"\n\nFolkard winced at the use of the word. Noticing this but not commenting, Enderby continued.\n\n\"As was made clear in our briefings, it appears that the Russians are compiling and experimenting with sets of technologies that may pose the single greatest threat that the Empire has ever seen. What has been kept from you, up until now, is that the basis of this threat is inexorably linked with this very ship, the ship that you hold so dear.\" Folkard snapped his head up to look Enderby straight in the eyes.\n\n\"How so?\" he asked. Enderby returned his gaze, and the glib, slightly mocking tone the agent usually commanded was all but undetectable. This was all business, and serious business at that.\n\n\"Are you aware,\" said Enderby, \"of the mineral deposits we believe the Russians have been searching for and mining, throughout the inner planets?\"\n\n\"Of course,\" replied Folkard. \"I've read the files. Rumours and scuttlebutt, by the sound of it. But it's of little surprise that they got wise to the mission of Esmeralda 2 and her gallant crew....\"\n\n\"Very true, very true,\" observed Enderby. \"And there's the rub. Oh, it pays for us to keep an eye on what occurs in the Siberian climes, and while the Russians are certainly antagonistic, and expanding, their capabilities concerning the art of espionage are, shall we say, somewhat lacking compared to our own. Jackboots and rifle fire is much more their style, though I hardly have to tell you that.\"\n\n\"Quite,\" frowned Folkard.\n\n\"So we are left with a conundrum. It's certain that the Russians lack the kind of intelligence network that helps keep her Majesty's subjects sleeping so soundly. And yet how, should these unsettling reports turn out to be true, could they possibly undertake such a task without us ever getting any real, solid evidence? Like you say, rumour and scuttlebutt. But they're certainly looking for something. The same minerals your own \"gallant crew\" were searching for. From what we have since learned of the situation on Ceres, they got there before you. And almost certainly followed a path similar to your own crew. But why? And why, Jacob Folkard, have we also discovered that the Russians have a full set of blueprints for Sovereign?\"\n\nFolkard was silent, dumbfounded. Enderby's gaze bored into his brow. The answer was obvious, but Folkard could not bring himself to voice it. Enderby had no such problem.\n\n\"They are building their own version of the Empire's flagship, Folkard. And what's more, they're imbuing it with the power of these mysterious minerals, in order to create a vessel far more powerful than the one in which we currently sit. While our own scientists toil away with refining the minerals brought back from your previous mission, it would appear the Russians are ahead of us....\"\n\nHe forced Folkard, through sheer force of will, to look into his eyes. \"There is a spy, sir,\" he said gravely. \"Someone loose with secrets in the heart of the Admiralty. And treason is a crime Her Majesty's Secret Service Bureau takes very, very seriously.\"\n\nThe hum of Sovereign's engines changed momentarily in pitch, a lowering in tone that signalled preparations for descent. Folkard knew it well.\n\n\"A crime,\" Enderby said coldly, \"that they will happily hang you for.\"\n\n5.\n\nRUSHING, GEORGE HELPED Annabelle up the short ladder back into the train's first carriage. He hopped up after her, followed by the driver and the two engineers. The raiders were perhaps half a mile away, and closing fast.\n\n\"You!\" yelled the engineer Annabelle had previously\u2014and so successfully\u2014emasculated. He turned to the driver, who had just clambered up. \"Billy, this hellcat caught me straight in the family jewels!\"\n\n\"Shut yer yap, Ignatius,\" snapped the driver. \"Would you rather have your balls ache or the top of your head cut off? Hmm? Have you looked out the window?\" Ignatius duly shut his trap.\n\n\"Weapons,\" said George. \"I'm assuming you have weapons for this sort of contingency?\"\n\n\"That we do, sir,\" said Billy. \"Maybe a half dozen Winchesters, an' fifty shells to go with 'em.\"\n\n\"Excellent. Bring them. Mister Boon?\" Boon smiled and reached into his jacket. With a flourish he produced an elaborate revolver quite unlike anything either George or Annabelle had seen before. A large clockwork housing was bolted on top of the chamber, with a brass belt that automatically fed fresh shells into the cylinder. As well as that, thin copper pipework ran up the length of the barrel, presumably to facilitate an increased and more deadly velocity to the bullets.\n\n\"I call her Elizabeth,\" said Mister Boon. \"Strictly speaking, wasn't meant to take her from the prototype lab, but well...no-one was looking, and I was a Boy Scout.\"\n\n\"Well I'm glad you came prepared,\" said George with a smile. Billy and one of the engineers quickly returned, awkwardly carrying the seven Winchester Repeaters between them. One was handed to George, another to Ignatius, while Billy held on to his own.\n\n\"But...but...I don't know how to shoot!\" whined Ignatius.\n\n\"I thought you said you were brought up on a farm, son!\" said Billy.\n\n\"Yeah, but I just milked the cows, boss! I never shot 'em!\"\n\n\"Good grief,\" muttered Annabelle, yanking the rifle from his hands. She took a handful of cartridges from the engineer and began loading the rifle with a precision and alacrity that had the poor farmhand stunned. She cocked the rifle and aimed down the barrel. \"It'll do,\" she said.\n\n\"Lord have mercy,\" muttered Ignatius.\n\n\"Right, how do we do this?\" said George.\n\n\"If they're just here for a raid,\" said Annabelle smartly, \"they'll generally surround the train and shoot in until we're subdued. Speaking of which, you, the milk maid.\"\n\n\"Yes, ma'am?\"\n\n\"Head down the train and tell everyone to keep their heads down, then get back here.\"\n\n\"Yes, ma'am.\" Ignatius waddled off as fast as he could go, happy to be at least moderately useful.\n\n\"Fifty shells should be more than enough to send them running,\" said Annabelle. \"These kind of attacks are rarely more than scare tactics, perhaps a little thieving off terrified passengers. It's a protest more than anything else. I doubt they'll even fire at us.\"\n\nAs soon as Annabelle had said this, the glass window closest to them exploded in a shower of shards.\n\n\"Well, bang goes that theory,\" said George, before adding, \"Darling.\" Annabelle frowned at him. They quickly took their places to fend off the attack.\n\n6.\n\nARNAUD'S JOURNEY TO the heart of Calcutta was proceeding apace. The rickshaw containing himself and the taciturn agent had passed through a huge shanty town on the southern side of the banks of the river Hooghly. The wide expanse of the waterway, its flow sluggish with the effluent and detritus of the displaced thousands, stretched to their right. On the other side of the river more of the same sad, decrepit patchwork shacks jutted and grew from the plains almost as far as the eye could see. Studded into these like islands were larger structures, crudely but at least semi-competently built, the municipal buildings and permanently-manned watchtowers at the eastern end of Britannia's reach.\n\n\"Black Town,\" said Coyne. \"The arse end of the armpit of the Empire. After this lot decided to rattle their shackles back in '57 it's been growing day by day. Didn't have a lot of trouble here in Bengal, but that didn't stop 'em piling in and claiming a home wherever they could stick a tent peg. Rest assured, Fontaine, if there's trouble brewing in Calcutta that's where we'll end up.\"\n\nArnaud pondered the possibility. Right now it didn't look too unappealing, the souks and vendors looking like something out of Byron or The Arabian Nights. It would be a very different story, he knew, when night fell.\n\n\"Still,\" smiled Coyne thinly. \"We'll dredge that sewer when we come to it. White Town will, I'm sure, be far less trying on your soft French sensibilities.\"\n\nArnaud frowned. Was it treason to punch him? He tossed the thought aside. As the rickshaw zig-zagged up the haphazard track, the river curved out of view before looming back in again, at some points so as the wheels were almost up to the waterline. The water was fetid, bobbing and thickened with God-knows-what. Arnaud, reluctant to demur to Coyne's stereotyping, gazed into the water unflinching for several minutes. That'd teach this reprobate, he thought stubbornly. Yet when he finally turned around, he found himself confronted by the side of Calcutta that had felt the improvement of a European presence: White Town.\n\nThe streets were spotless. The architecture was a mix of blocky, almost Georgian-style housing with trim and fittings in the local style, lots of white and latticed windows in the shape of minarets. There was a needle-shaped cenotaph in the centre of a plaza, and the sudden clang of the bell of a tram startled the driver, who veered towards the pavement so as not to get them killed. Arnaud wondered why they could not have dispensed with the old man's pedalling and boarded the tram quite some time ago.\n\n\"Where are we headed?\" he asked, trying to hide his annoyance.\n\n\"Where else would one find a world-renowned geologist?\" replied Coyne, with a sneer. \"The university bar, of course.\"\n\n7.\n\nNATHANIEL HAD FELT the need for another two cups of milkless tea and another helping of white pudding, just to make sure he was feeling his best. He would have been quite happy reclining on the divan\u2014with which he had by now formed something of a bond\u2014and pondering this whole affair until Tally had returned from whatever twilight deals he was in the process of completing. But Tally had insisted that the air would do him good, and that a nod was as good as a wink to a blind horse, and by the time he had his shady meetings done with it would be time for them to head west to Phoenix Park anyway. Nathaniel reluctantly agreed, and grabbed his jacket.\n\nIt was another fine day in Dublin, a fact that had been hidden from Nathaniel up until that point by the rough cloth Tally had tied to the top of his curtain rails. Grinning, Tally lit a cigarette and offered Nathaniel one, with the latter declining on the pretext that he could smell the tobacco before Tally had even removed the packet from his pocket. The Irishman shrugged, lit his Woodbine and sucked on it deeply. He pointed towards the city centre and started walking, leaving behind a trail of smoke for Nathaniel to follow in the glorious midday sun.\n\nAnd yet that same warming sun was not the only thing obscured by Tally's makeshift curtains. It had also hidden a black hansom cab, utterly unnoticeable on any street, looking slightly unkempt with the paint peeling at the edges. The driver tended to draw the eye, though. A huge man, he wore a bowler hat and huge, thick black gloves that held the reins steady and tight as telegraph wires. He only ever looked forwards, unfazed, slightly bored through smoked welder's goggles, while in the cab his comrade watched Nathaniel and Tally head towards town. The cab's occupant smoked a cigarette from a long ebony holder. In his head, the seconds ticked down towards the big bang, a glorious, gorgeous expression of his art that had already been set in motion.\nChapter Seven\n\nThe Secret of Phoenix Park\n\n1.\n\nNOT ONLY HAD Annabelle been mistaken about the attacker's use of firearms, she had also been disastrously wrong about their tactics. She had expected them to fan out a few metres from the train, moving around to take the carriages from both sides. Yet they had stayed as a fairly close-knit group, ploughing on relentlessly towards the engine. Something, she thought, was terribly amiss.\n\nOf course, it was dashed generous of them to stay so close together, as it made them a far larger target. Bert had winged one of the horses with his incongruously-named pistol, sending the rider head first into the sun-baked soil, where he lay still. But the riders were sending up so much dust and were approaching at such a pace that finding and keeping a target proved difficult.\n\nAs the attackers yanked back on their horse's reins, the wave of dust they had brought with them hit the carriage and spilled in through the open window, making Annabelle and Billy gasp for breath. They heard muffled shouts from outside, and with an alarming crack the lock was shot from the carriage door.\n\n\"Everybody, quick, backwards,\" said George, glancing to check Annabelle's progress as he made his way to the next carriage down. \"We can set up defences, pick 'em off!\"\n\nThe door was kicked open and a bullet whizzed into the space, making everyone duck instinctively. Bert had already crouched behind a table to cover their retreat, and as the first silhouetted assailant boarded the train he took him down with a deadly shot between the eyes. That would give them pause for thought! George ushered Annabelle and the others through, then took his place in the booth opposite Bert. The two men nodded at each other.\n\n\"The others will get everyone out and make sure we're safe to follow them,\" said George. \"Annabelle knows what she's doing.\"\n\n\"You're a lucky man,\" said Bert, and saying this he narrowed his eyes and raised Elizabeth towards another assailant's face. In a split second he had taken aim and pulled the trigger.\n\nMomentarily deafened, George wheeled around to see where Bert had shot, and caught a glimpse of one the riders falling from his saddle into the dust. A neat bullet hole sat at the centre of the window.\n\n\"I'm doubly blessed,\" said George.\n\n\"I'll say,\" said Bert, winding his gun's clockwork mechanism once more. \"Shall we see about that retreat?\"\n\nThey pulled the sliding door behind them just as the invaders had begun to storm the carriage. Annabelle had already set herself up to take the shot through the window in the door. She steadied her Winchester and put a bullet straight through to the first attacker's forehead. She managed a couple more volleys before the strengthened glass became too criss-crossed with cracks to adequately see through.\n\n\"We need to open the door,\" she yelled.\n\n\"What?\" came a voice.\n\n\"The door, we need to open it to drive them back!\"\n\nIgnatius, not the most graceful of gents at the best of times, lumbered forward as quick as he was able. He threw himself heavily to the ground (George was too polite to mention later that he felt the carriage lurch), bullets thudding into the door from both sides.\n\n\"Get ready!\" he yelled. They readied their weapons.\n\nIgnatius inserted his fingertips into the minuscule gap between the door and the jamb and, using his considerable strength, flung the door wide open before covering his head with his arms. He closed his eyes and whispered a prayer. Already prepared, George, Annabelle, Bert and the armed train workers unleashed a wicked salvo of gunfire through the gap. Three of their assailants dropped almost instantly, the rest diving for cover or fleeing back out into the desert.\n\nThere was a moment of calm after the ferocious volley, the air thick with cordite and the iron smell of blood. The advantage was now on the defender's side as they trained their weapons straight through the door to the corridor beyond. As soon as one of the pinned-down attackers tried to move they were wisely advised not to try by one of Bert's pinpoint bullets ricocheting from a nearby surface. The attackers returned fire occasionally, but blindly, unable to find a target without exposing themselves to a shot to the gullet.\n\n\"Right,\" muttered George, mostly to himself. \"What do we do now?\"\n\n\"Stalemate,\" agreed Bert. He ducked nimbly across the aisle and took a quick peek outside. Some of the riders had already fled, and as they spurred their horses into the distance, one of the raiders\u2014who Bert could only assume was their de facto leader\u2014was bawling at them for their return, yelling curses and promises of retribution in a mid-western drawl. Definitely not Indians then...\n\nHe rode back to where his compatriots milled around the carriage and began banging on the windows, yelling for them to crawl off their yellow bellies and fight like men. Somewhat affronted by this sleight against her sex, Annabelle quickly leaned from her position and took a pot shot at him through the door, taking the man's hat clean from his head. The bandit reeled and ducked down on his horse before turning the beast around and spurring it into a gallop from the crime scene.\n\n\"Your leader has abandoned you!\" called Annabelle into the next carriage. \"Surrender is your only option!\"\n\nThere was a moment of silence, and then came a reply. The reply, strangely enough, was fizzing.\n\nA red cylinder had been tossed through the carriage, landing several inches before the prostate Ignatius and rolling towards his face. The farmhand ceased making peace with his creator and opened one eye experimentally. When he saw the fuse sparking away right in front of him, he genuinely wished he hadn't.\n\n\"Dynamite!\" he yelled.\n\n\"Technically, I suppose that's another option,\" opined Annabelle coolly.\n\nAs soon as the explosive had landed the bandits in the carriage had begun to move, grabbing the bodies of their comrades-in-arms and dragging them back outside.\n\n\"Quickly, Ignatius, grab it!\" said Bert.\n\nIgnatius hoisted himself up and grabbed the stick of dynamite, tossing it from hand to hand like a hot potato.\n\n\"What do I do, what do I do?\"\n\nBert leaned himself back and shot out the window nearest to him. \"Throw it, you lunk!\" he bawled.\n\nIgnatius pulled his arm back and pitched the dynamite out of the window, throwing it some twenty feet from the train. As he did so, one of the attackers took the opportunity from hauling his deceased companion to ready his six-shooter and take a shot at the big man's frame. The bullet struck Ignatius in the shoulder with the sound of splintering wood and the big man fell heavily to the floor. Not wasting a moment, George raised his Winchester and fired in the direction of the gunshot, taking down the cowardly bandit with hot lead to the heart.\n\nSuddenly, the train was rocked by the explosion of the dynamite. All the windows were blown in simultaneously and the carriage rocked perilously for a moment. The defenders were all thrown to their feet, with George landing heavily on the injured Ignatius. The big man did not react to the impact, and George noticed sadly that his face had gone grey and the wound was bleeding profusely into the fine, first-class carpets.\n\n2.\n\nONE UNIVERSITY BAR, discovered Arnaud, was very much like another the world over.\n\nIn fact, Arnaud was becoming increasingly more stunned by the manner in which certain areas of Calcutta resembled Europe. As well as the clean splendour of White Town, the university building itself wouldn't have been out of place in the streets of Marylebone or Munich. It was an ornately uniform building, its front decorated with arches on each of its three levels and a large, cuboidal portico supporting two large pillars at the entranceway. The whole building seemed solid and safe, yet somehow stern\u2014not surprising from a building composed of so many right angles, and hardly inappropriate for a seat of higher learning.\n\nThey had made their way into the interior, which Arnaud found blissfully cool. Impeccably dressed students, mostly Indian but with a good mix of Caucasian faces, filtered silently through the hallways, carrying books or huddling together momentarily to pass the time of day. Signs written in Bengali and English pointed them to the back of the building, where the bar was located.\n\nArnaud felt pleasantly nostalgic to be back on a campus, even if it was on the pretence of finding an academic he had no real desire to see. The nooks and crannies of the bar were filled with students smoking, drinking tea and arguing loudly over their chosen subjects. Arnaud noticed with some pleasure that, more often than not, these arguments were conducted in a volume more conducive to elaborating the intelligence of the speaker, rather than getting the point across. That definitely took him back....\n\n\"Where do we start?\" he asked Coyne.\n\nCoyne shrugged.\n\n\"I'm just child minding, you're the university sort. You sort it out.\"\n\nArnaud could feel his patience running thin.\n\n\"Yes,\" he said, controlling his temper, \"but it was your idea to come here in the first place. I'd have thought you'd have more to offer than just getting me to 'sort it out.'\"\n\n\"Well, you thought wrong. Where would you start?\"\n\n\"The geology department, perhaps?\"\n\n\"Not my fault if you didn't think of that sooner.\"\n\nArnaud gritted his teeth and headed to the bar, where a helpful and smartly-dressed Hindu offered them tea, then gave them directions to the geology department.\n\n3.\n\nNATHANIEL HAD ASSUMED, perhaps naively, that Tally was to conduct his business quickly, maybe shunt Nathaniel off to a nearby park or even leave him waiting outside. He did not expect to be part and parcel of the deception.\n\nTally and he had walked about halfway up to the town centre, with Tally either doffing his cap and waving a cheery hello at his acquaintances or hastily grabbing Nathaniel's arm and pushing him across the street to avoid someone. Whether the person Tally would greet or dodge was arbitrary\u2014it ran from guttersnipes through to businessmen, from decidedly rough-looking characters to elegant ladies in their dotage. Tally was clearly a well-known man about town, and Nathaniel wasn't entirely sure whether this was a good or a bad thing. He was gently leaning towards bad.\n\nThey had walked several feet up George Street before Tally had hooked Nathaniel's arm once more and pulled him into a professional-looking office building set slightly back from the main road.\n\n\"Stand at the back,\" said Tally. \"Don't say anything, and look stern.\" Nathaniel then found himself being winked at.\n\nWithout even a by-your-leave Tally had pushed into the building and headed straight for a door on the ground floor. He didn't knock or present himself, just breezed into the office as if he owned the place and sat down in the chair opposite the room's occupant\u2014a balding, sweaty, harried-looking man in a waistcoat and watch chain, sat before a precarious pile of papers. Mustering all the sternness he was capable of in the circumstances, Nathaniel sidled in and stood at the back of the room, his arms crossed.\n\n\"This here's Mister Walsh,\" said Tally, gesturing behind him. \"From the Galway branch.\"\n\nThe harried-looking man behind the desk sputtered and looked up at Nathaniel, almost knocking over his ink pot in the process.\n\n\"I didn't think,\" he said nervously, shuffling his papers, \"that is, I never imagined, Mister Walsh...\"\n\n\"Shush, shush now, Cormac. Can I call you Cormac, Cormac?\"\n\nThe man looked up nervously to Nathaniel. Not knowing what else to do, Nathaniel nodded at him. Cormac transferred his nod to Tally, who beamed.\n\n\"See, there we are!\" he said. \"All good and friendly. Y'see now, Cormac, I told you I'd have it sorted an' now everything's nice and easy and above board. So there's nothing stopping you sorting me out now, is there?\"\n\nCormac, who had begun sweating profusely and tugging at his collar, nodded. \"Of course, of course. There's still a few papers to get through, stamps to, ah, stamp. That kind of thing. I'll be with you presently.\" Slicking his hair back with the sweat from his brow, Cormac rose clumsily and headed to the door. As he passed Nathaniel he looked at him beseechingly, half in terror, half in awe.\n\n\"I'm sorry, sir,\" he stammered. \"I never meant, I mean...\" Nathaniel eyed him stoically. \"I'll go and get the papers,\" finished Cormac weakly, closing the door gently behind him.\n\nNathaniel waited a moment.\n\n\"And what was all that about?\" he exploded.\n\n\"Ah, relax. You did grand.\"\n\n\"I'm not asking for a critique of my performance, Tally, I'm asking what the hell you've just got me embroiled in!\"\n\n\"Nothing serious, nothing illegal. Technically, anyhow. These pencil-pushers just need a nudge every now and again, just to get 'em to see through the red tape. See? Yer man Cormac'll be back in a jiffy with the papers, then it's a quick stop by the National Library to see a fella who owes me ten bob and we're golden. A trip to the park awaits.\"\n\n4.\n\nIGNATIUS, IT TURNED out, was bleeding fairly badly but only from a flesh wound. A local sawbones had been enlisted from the petrified passengers in the rear carriages. He looked Ignatius up and down, diagnosed the bullet had grazed his shoulder blade, and set to work patching up the farmhand with a torn-up shirt. Pleased that the man who had proved himself so worthy was going to be all right, Annabelle and the others got down to investigating the body of the attacker George had managed to fell at the last moment, along with that of his fellow ne'er-do-well.\n\nIt was like something out of a school play, or a children's book. These men\u2014they were clearly American, or Caucasian, at least\u2014looked like they'd found their Indian costumes in a dressing-up box. Blankets or hessian sacks had been cut up to make ponchos and rough chaps. Moth-eaten bits of leather or shoelaces had been tied around arms or linked together to make tassels. Feathers from seagulls and ravens were tied onto headbands made of old bandage. Dried mud had been used to simulate war paint.\n\n\"It's ridiculous,\" fumed Annabelle. Her relationship with the indigenous American Indians was a complicated one, of that there was no doubt, but this was just plain insulting. \"How on Earth could anyone believe that these were genuine Indians?\"\n\n\"Maybe we were never meant to get such a close look at them,\" said George. \"If they're a bandit gang it makes sense. Attack a wagon from afar, maybe convince whoever you're attacking to drop any valuables and flee... Survivors might report an Indian attack, and any marshals sent out wouldn't be looking for brigands. Just Indians.\"\n\n\"But why attack the train?\" pondered Annabelle.\n\n\"Desperation?\" offered George. It was hardly a convincing argument, but at present nobody had any better ideas.\n\nBert had got down on one knee and was going through the pockets of the two fallen men for clues. He came away with a few rounds of ammunition and a bowie knife, but no real leads.\n\nGeorge turned to Billy.\n\n\"Has there been an increase in Indian attacks recently? Anything like that?\"\n\nBilly frowned. \"Not really, sir. Been much the same as ever. Although...\"\n\n\"Although what?\" asked Annabelle quickly.\n\n\"Though there's been no attacks, there's been...rumours, I guess. Sightings.\"\n\n\"What kind of rumours?\"\n\n\"And what kind of sightings?\" added George.\n\nBilly looked down and fiddled with his thumbs. \"Well, I don't give much credence to ghost stories, sir...\"\n\n\"Well,\" said George, trying to calm the man, who was pale and clearly suffering from some sort of delayed shock after the incursion, \"we do. Trust me, Billy, we've seen it all, and you're not in some tavern you're going to get laughed out of. Come on. Take a seat.\" He gently took Billy by the arm and set him down on the nearest bench. Billy breathed heavily, composing himself, and looked to George.\n\n\"Sorry sir,\" he gasped. \"I always thought that when it really got down it, I'd be a bit more use in a fight. But when it's there, right in front of you, and when some Injun is trying to take your life an' alls I could think of was my darlin' Babette back at home...\"\n\n\"You did fine, Billy,\" soothed Annabelle, kneeling next to him. \"And you'll see Babette soon enough. But this is important. What have you heard?\"\n\n\"Mostly rumblings,\" said Billy after a deep breath. \"Prospectors coming back into town, the odd rancher who's been out on the plains. Say there's been a lot of activity roundabouts Greenore Gulch, though no-one in their right mind would have any business there. Tried to mine for silver there back in '64, only it got abandoned, and there ain't been much talk of it since.\"\n\n\"How so?\" asked George.\n\n\"Well originally, sometime in the spring of '65, they say the miners were chased off by Injuns who reckoned it was a cursed place, warning the white man off. Now normally I'd just put that down to a savage's mumbo-jumbo, only my Uncle Tobias spent a bit of time working there. He says the further they dug down, the more strange things started happenin'. Began, he says, with this itchin' in your head. This crazy feelin' you couldn't scratch. Some of 'em even used to say that the ore in the walls used to vibrate somethin' ungodly, and the seams used to glow bright green in the moonlight. Sent a couple of men stark raving mad, so they say.\"\n\nGeorge and Annabelle shared a glance. This all sounded worryingly familiar. \"And how about now?\" asked George carefully. \"What's happening up there at the moment?\"\n\n\"Well this it,\" said Billy, looking up at him helplessly. \"No-one's quite sure. They say they've seen all sorts up there\u2014Injuns, miners, soldiers, the lot. There's even been talks of skirmishing, gunfire and explosions. One old cowpoke says he even felt the earth move, and green lights blazin' up the sky like ball lightning. Only I don't pay him no heed as he's halfway blind and fully pickled on moonshine.\"\n\n\"Well, any information is useful.\"\n\n\"But there's been no attacks on other trains?\" asked Annabelle.\n\n\"Not that I know of, miss.\"\n\n\"So what's so special about this one?\"\n\nAnnabelle, George and Bert looked at each other.\n\n\"Quite,\" said George.\n\n\"Bert,\" said Annabelle, \"did anybody else know of our plans to come out here? Anyone in the Admiralty, or your Bureau?\"\n\nBert shrugged.\n\n\"I couldn't tell you, ma'am. Certainly Mister Enderby and the other few agents sent off with your friends. We were all briefed on that. Captain Folkard knew, of course, and your uncle. It's highly unlikely many people knew. A half dozen, at most.\"\n\n\"So if we were targeted, then...\" interjected George. He seemed reluctant to finish his sentence.\n\n\"It scarcely bears thinking about,\" agreed Annabelle.\n\nThis sombre and troubling realisation caused the three of them to ponder for a moment. Annabelle, lost in thought, found herself glancing at Billy who, if it were possible, seemed to have gone an even whiter shade of pale. She initially feared he may be having some sort of cardiac seizure and was about to call for the doctor again, but his eyes were wide as cue balls and he pointed to the windows behind Annabelle.\n\n\"I...I...Injuns!\" he stammered.\n\n\"I told you Billy, they weren't...\"\n\nBut Bert and George were following Billy's gaze, and out of the windows, staring through at them into the wrecked carriage, were four very genuine Indian braves dressed in their full regalia.\n\n5.\n\nHAVING TRAMPED UP countless flights of stairs and through corridors ripe with the scent of learning, past lecture halls and professor's office from which drifted the sounds of mathematics and history in languages exotic and familiar, Arnaud and Coyne found themselves in the reception area of the University of Calcutta's Geology Department.\n\nA young British girl, maybe seventeen years of age, sat behind the desk at reception looking resolutely bored. Her face lit up when Arnaud and his companion entered, chiefly because they were giving her something to do that didn't involve a student waxing cock-and-bull on why their essays were late.\n\n\"Hullo!\" she beamed, sitting upright to look more professional. She wore thin-rimmed spectacles, her hair was up in a bun and she had a chipper, upper-class accent that smacked of a childhood in the home counties. \"Welcome to the Geology Department. How may I help you today?\"\n\n\"Well,\" said Arnaud, and the girl's interest noticeably peaked when she caught his French accent, \"we were wondering if we could talk to someone regarding the whereabouts of one of the teachers here.\"\n\n\"Of course, monsieur,\" she said, doubtless feeling very cosmopolitan. \"Which of our esteemed professors can I help you locate?\"\n\n\"We're looking for Doctor Garrecreux,\" said Arnaud cheerfully. \"Doctor Fabrice Garrecreux.\"\n\nThe girl's face fell.\n\n\"Well,\" she said, pursing her lips. \"I can't say I'm surprised. Give me a moment, if you would. I'm afraid this is a matter I've been instructed to bring straight to the Vice-Chancellor.\"\n\n6.\n\nARNAUD AND COYNE had waited patiently in the ante-room adjoining the reception, pointedly not talking to each other and studying portraits on the walls and the pot plant in the corner. About twenty minutes passed, then the young girl popped her head around the corner and cleared her throat.\n\n\"Gentlemen,\" she said, somewhat more formally than before, \"Vice-Chancellor Bannerjee will see you now.\"\n\nShe led them efficiently and silently through the department and up a flight of stairs at the end, heading to the administrative heart of the college. Typewriters clacked behind closed doors, punctuated by the shunt of filing cabinet drawers being pushed into place. The wooden walls and air of quiet decorum could have come from any of the colleges in Oxford or Cambridge, though the bright light through the windows offered more illumination than had been seen in those dusty corridors for many, many years. The girl led them to a large set of double doors and curtseyed gently.\n\n\"He's expecting you,\" she said. \"Go on in.\"\n\nSir Gurudas Banerjee held the distinction of being the first Indian Vice-Chancellor of Calcutta University, and had inauspiciously begun his post by having to deal with a scandal in the geology department that was not of his making, the details of which Arnaud was about to learn.\n\nHis office, like much of the university, was a seamless integration of British institutionalism and Indian exoticism. It had wood-panelled walls that eschewed the rectangles of Imperial design, instead choosing curls and onion-shaped domes. A classically large bookcase stood against one wall, an Indian rug covered the floor and a tea set sat before a green leather Chesterfield to the left of where Arnaud and Coyne entered. Vice-Chancellor Banerjee smiled at them warmly. He was a thick-set man, authoritative but welcoming, dressed in a turban atop a morning suit that had the collar open to accommodate the heat.\n\n\"Gentlemen,\" he smiled, \"please take a seat. Can I offer you any refreshment?\"\n\nArnaud and Coyne politely declined, and Vice-Chancellor Banerjee relaxed back in his chair as his guests sat before him.\n\n\"Let us dispense with any formality,\" he said. \"Miss Warrington informed me you have asked after Doctor Garrecreux.\"\n\nArnaud looked at Coyne, unsure as to whether he or the agent should take the initiative. Coyne's dismissive, feline look told the young Frenchman he was on his own; it was almost as if the spy found the whole mission to be a dull chore.\n\n\"That's true, Vice-Chancellor. We are.\"\n\n\"And why, might I ask, are you looking?\"\n\n\"We were hoping to count on his expertise. Did you hear of the bomb attack in London recently?\"\n\n\"There's very little of British business that does not get reported here, sooner or later,\" sighed the Vice-Chancellor. \"And so we heard of it. A terrible affair.\"\n\n\"Bon. And so we have been sent by Her Majesty's Admiralty to investigate any possible leads that may lead to the perpetrators, and we believe Dr Garrecreux may be able to assist us with the identification of certain...\"\n\n\"Please,\" said Gurudas, \"I am sympathetic to your goals, but am not sure how I can help you achieve them. Doctor Garrecreux is gone, Mister...?\"\n\n\"Fontaine,\" said Arnaud, \"Doctor Arnaud Fontaine.\"\n\n\"Doctor Fontaine,\" said the Vice-Chancellor, leaning forward, \"while we are more than aware of the Empire's business on these shores, perhaps the scandals of Her Majesty's outposts do not reach London. If they did, surely you would have heard of Dr Garrecreux's...in the words of your countrymen, faux-pas.\"\n\nArnaud shifted in his seat. Coyne gazed away, still bored. The Vice-Chancellor waved away their discomfort.\n\n\"Politics is something one must face as the head of a university, Doctor Fontaine. It is nothing new. The pressures of dealing with such issues are manifold in India, on a local and global scale. This,\" he cleared his throat, \"faux-pas of Doctor Garrecreux was something far less pressing than other difficulties my country may be facing, but was nevertheless an issue I could have done without, both personally and professionally.\"\n\n\"Of course,\" nodded Arnaud, feeling like a chided schoolboy.\n\n\"Good.\" Having got this out of his system, Vice-Chancellor Banerjee seemed to relax, satisfied he had made his point. It was almost as if he blamed Arnaud for his ersatz teacher's behaviour on account of their shared heritage.\n\n\"Do not misunderstand me,\" he continued, carefully. \"I have the utmost respect for Her Majesty's Navy. I have heard tales of the ruthlessness with which the Mutiny was suppressed; the last thing I wish is for such a power to be brought to bear on Calcutta. You have seen, I take it, Black town?\"\n\n\"I have,\" answered Arnaud.\n\n\"Then you have seen what the Navy's guns could do to such a settlement\u2014the people there are barely protected from the weather, let alone cannon fire. And yet I fear, Doctor Fontaine, that Black Town is an area you may have to get more familiar with, should you wish to find Garrecreux.\"\n\n\"Told ya,\" muttered Coyne.\n\nBoth Arnaud and the vice-chancellor ignored him.\n\n\"Any information you could give me regarding his whereabouts would leave me sorely in your debt, Vice-Chancellor.\"\n\n\"I'll tell you what I know, of course, along with any other information I have that may help you navigate Black Town safely. But I would ask you, with all due respect, that you keep the scandalous nature of Garrecreux's behaviour to yourself. This is a respected place of learning, and I do not want my tenure at its helm to be tarnished any further by that irascible, weak-willed blackguard.\"\n\n\"Of course not,\" said Arnaud, once more feeling he was somehow being blamed by implication. The vice-chancellor narrowed his eyes and studied Arnaud's face for a moment, and Arnaud knew that any gesture of uncertainty, even the slightest indication that he was not being wholly sincere, would have caused Arnaud and Coyne to leave none the wiser of Garrecreux's current location.\n\nAfter a moment, it seemed Arnaud had passed the test.\n\n\"Very well,\" nodded the vice-chancellor. \"I will tell you. As so often happens, the cause of Fabrice Garrecreux's fall into infamy was a woman. You will not be surprised to learn that she was young, beautiful... And dangerous.\"\n\n7.\n\nPHOENIX PARK COULD be found just west of the centre of Dublin, an expansive green space north of the Liffey that provided Dubliners young and old, rich and poor, with respite from the chaos of the city. Despite having quickly and efficiently reclaimed his debt from his acquaintance at the Library (the details of which Nathaniel was spared, much to his relief), Tally had refused to pay for a cab and insisted on walking the mile or so from Temple Bar. The sky had turned ashen as the afternoon had aged, a cold wind blowing in from the bays on the east. Not having eaten since breakfast, Nathaniel found his mood becoming ever more tetchy, not helped by Tally cheerfully reminding him every few minutes he was ten bob up.\n\n\"I don't care if you're rich beyond the dreams of Solomon,\" grumbled Nathaniel, after the fifth time Tally's suspiciously-gained wealth was mentioned. \"Are we nearly there yet?\"\n\n\"Ah, quit your gabbin', and be grateful the Zoological Gardens are on the side of the park closest to us.\"\n\nNathaniel had endured many perils and unenviable situations during his adventures, but none had yet had the ability to irritate him quite as much as the combination of an empty belly and a chipper Irishman. Eyes peeled for somewhere he could at least grab himself a snack, he trudged on.\n\nWhen they reached the park, they walked up through the people's garden, a delightful evocation of current horticultural fashion, which now seemed mournful due to the cold and the fading light. Dublin's denizens were scampering back to hearths and home-cooked food, almost as if they were primordially aware of the heinous feats of villainy that the area would be subjected to that very night... And, in fact, had already and violently begun.\n\nTally strode up to the entrance lodge, a small cottage building with a thatched roof, as proud as you like and as happy as a clam. He greeted the porter who stood in the building like an old friend, but the man's reaction was such as to make it clear that he didn't know Tally from Adam, and nor did he wish to.\n\n\"Hullo, my good man,\" began Tally's charm offensive. \"I was wondering if we might not take a quick squint round, take in the beasts in the dying light, so to speak.\"\n\n\"Well you can wonder otherwise, and sling yer hook into the bargain. Nobody in tonight. That's what they told me, and that's what's happening.\"\n\n\"Ah, we've still got an hour or two of good light left. Maybe more if the clouds blow over. We're paying customers.\"\n\n\"Are your ears blocked, sonny, or do I need ta box 'em open?\"\n\n\"Now now, no need for that, my man. Truth is I got a nephew works in there, feeding the ferrets and that. Simon, his name is. And young Simon, y'see, his Ma's awful sick with the gout, an'...\"\n\n\"Young Simon, you say?\" said the porter, stiffening up.\n\n\"Aye, that's him. Simon Sheen.\"\n\n\"Well if that's the case his Ma's got more to worry about than her gout. I hate to be the bearer of bad news, but yer nephew had an accident, not more an' an hour ago. Seems as if he went and dropped himself in with the lions. Poor bugger didn't stand a chance.\"\n\n\"Simon? You're telling me Young Simon's dead?\"\n\n\"Well, what's left of him ain't movin' around so much, so I'd guess so.\"\n\n\"Well I gotta get in there man, I'm family!\"\n\n\"That's as maybe, but the constabulary were pretty firm in tellin' us there's no use having all an' sundry gawkin' at bloodstains. I'm sorry for yer loss, fella, I really am. But it ain't gonna happen tonight.\"\n\n8.\n\nTALLY WAS SITTING with his head on his hands on the kerbstone. He and Nathaniel had walked away quietly, Tally clearly shaken, and had taken up residence in a darkened stretch of Chapelizod Road, where the Irishman had stopped and sat to gather his thoughts.\n\n\"I wasn't even really his uncle,\" he said, mostly to himself. \"I was just knocking off his ma....\"\n\nNathaniel wasn't sure what to say. He genuinely felt for Tally, whose mood of gregarious energy had been replaced by one of sorrow and deep despondence. Once or twice he thought to reach forward to comfort the man, but Tally's blank-eyed stare across to the other side of the road seemed to indicate that any solace Nathaniel might have offered would have gone at least unheeded, at worst returned with anger. Nathaniel was left in a state of disturbed contemplation. This turn of events was worrying indeed, and the fact that Tally's contact at the zoo had met such an ignominious end was a coincidence he was sure had grave and dangerous connotations. It was clear that Tally was thinking the same.\n\n\"It's not right, is it?\"\n\n\"I'm sorry?\" said Nathaniel, drawn suddenly from his mulling.\n\n\"The fact that Young Simon noticed all that, with the ground shakin' and him asking questions... That you and I were only talkin' about it last night, and then he meets his end? Just like that?\" Tally's quiet anger had been replaced by resolution, and the fog of mourning had parted to reveal a need for answers in his heart...and retribution in his eyes.\n\n\"I must be honest with you, Tally,\" said Nathaniel, easing himself to sit beside his friend. \"I've found that in these matters there is no such thing as chance, no arbitrary occurrences... These people I was sent here to stand against, to fight... They have no mercy, no remorse, no respect for the lives of young or old, or for families....\" He looked Tally straight in the face, and the Irishman's eyes had turned as hard as old bones. \"The plots that I have encountered, the greed, the violence, it goes far beyond any petty squabbles our respective countries may have. They are but trifling when compared with the evil men may do to harness the powers opened up to humanity when we discovered how to fly between the void. I know this may seem harsh, but there is far more at stake at the moment than the freedom of Ireland, or the life of your poor, departed nephew.\"\n\n\"Oh, I know that well enough,\" said Tally. There was steel in his voice. \"And that's it though, isn't it? Simon, he was a, he was...\" And Tally began to laugh, recalling him. \"He was pretty feckin' useless, truth be told. His ma used to badger me to get him along on a few of me jaunts, y'know? Get him to earn a bit more coin, have him as a bit more than the man who tosses grain in the cages. But I knew he couldn't hack it an' I knew he was happy where he was. Harmless to boot. An' these bastards still think it fit to take his life.\"\n\n\"These are the men,\" said Nathaniel quietly, \"that must be fought.\"\n\n\"Then fight 'em I shall,\" said Tally. \"And a lion's claws will be like God's grace compared to what I'll do when I get my hands on 'em.\"\n\nAt that sentiment Nathaniel found himself smiling.\n\n9.\n\nTALLY SEEMED RENEWED and, rather worryingly, imbued with a vengeance. Nathaniel knew that those at their angriest were prone to make the greatest mistakes, and while he appreciated how Tally's zeal may help them, he was also concerned that it might prove to be their undoing.\n\nThey had returned the way they had come and were skirting back around the walls of the park. As soon as they had found a sufficiently darkened and quiet section of the wall, Tally hoisted himself up and was offering Nathaniel a hand in order to haul him after him. After a short scuffle accompanied by only the mildest of swearing from Tally, they were both over the brink and into the darkened park. The night was still and cold, illuminated by a frozen and unfeeling moon. The grass seemed crisp beneath their feet, the trees still in the black and breezeless air. Tally, it seemed, was in his element. He slinked around, quickly and half-crouched, with an almost instinctive ability to stay unseen in the shadows. Such was his ability to lose himself in the dark he had to backtrack once or twice to show Nathaniel the way after losing him, and quickly the pair made their way towards the silent stillness of the zoo.\n\nAs a child, Nathaniel had been fascinated by exotic beasts, and had spent as much time in London Zoo studying them first-hand as he had poring over them in books and magazines. Nevertheless, he knew his timorous childhood self would have felt somewhat differently having encountered the zoo at night\u2014this was a place of terror. Growls emanated from cages, permeated by the padding of clawed feet more suited to the dust of the African plains. Birds with beaks as long as their bodies would shriek suddenly, the clatter of their black wings filling the air and dying again just as suddenly. Though Tally seemed impervious to these shocks to the senses, Nathaniel felt the need to steel himself against such savage noises. Even a Russian could be reasoned with; the same was not true of a tiger. And yet, if his fears proved true concerning the fate of Tally's nephew, he was genuinely unsure which killer he would choose to face.\n\nHe almost cried out when he bumped into Tally, who had stopped and crouched in the darkness before him.\n\n\"Good grief,\" he hissed. \"You're like a ghost.\"\n\n\"Aye, well, when you've been about as much as I have, Professor, you know when it pays to keep yourself hidden. Quick question.\"\n\n\"Go on.\"\n\n\"Now we're here, what the feck do we do next?\"\n\nIt was a good point. Nathaniel had been too concerned with stealth to give much thought to a plan, but now that they had stopped he devoted his mind to thinking about the situation logically and practically.\n\n\"If we're going on the assumption that your nephew was murdered, we must also assume that his observations were correct\u2014that there is, indeed, something rotten under the grounds of the zoo that out enemies will kill to keep secret. So we need somewhere that is deeper than the rest of the park, but close... Now, you mentioned earlier these two men. The English, the bomb-maker, he seems happy to kill from afar, with fire and cowardice. But the other one, well... I think he's more of the 'hands-on' of the two, regardless of what his hands are actually made of. And if he draws the eye, their base of operation must be close to the lion's enclosure in order that this big fellow is seen as little as possible. Any ideas?\"\n\n\"Not just an idea,\" said Tally, who seemed to have regained some of his previous joie-de-vivre, \"but a good one to boot. C'mon.\"\n\nShortly they found themselves outside a darkened, circular building made of thick, solid stone. Though built in an older style, the freshness of the mortar and the cleanliness of the bricks was apparent to Nathaniel even in the sombre lunar light.\n\n\"Of course!\" he exclaimed, chiding himself. \"You said as much yourself. This must be the area that was closed off before....\"\n\n\"Aye.\" Tally grinned. \"That it was. And you can smell the lions from here.\" He delved into his pocket and brought out a small pen-knife and a lockpick. \"Give me a jiffy,\" he murmured, before getting to work on a lock that Nathaniel could barely see in the gloom, let alone attempt to open.\n\nShortly, and with a muttered swear-word of triumph, the lock clicked heavily and the door squeaked open just an inch on its hinges.\n\n\"I do wonder about your past, Tally,\" observed Nathaniel.\n\n\"Bugger the past,\" said Tally. \"Just appreciate I'm a professional in the present.\" Unable to argue with the logic, Nathaniel followed him inside.\n\n10.\n\nTHEY FOUND THEMSELVES in a large, doughnut-shaped atrium that arced away to the left and right. The air was humid and still. Lining the walls were glass-fronted exhibition cabinets filled with various earthly environments\u2014the fronds of ferns, thick branches leaning diagonally between them, shallow green pools and patches of desert sand. In the dim light that glowed within them, the animal eyes of reptiles and rodents would flash momentarily before fleeing into the shadows.\n\n\"Let's take a look around,\" whispered Nathaniel.\n\nThey began to move along the curving corridor. Nathaniel's eyes scanned each exhibit minutely, ever alert for something out of place. Tally would occasionally cup his hands against the glass to take a curious peek at the exotic beasts rustling around within, but more often than not was unable to see what was inside. Nathaniel frowned. As far as he could tell, everything seemed to be in order\u2014this was just a normal zoo, made marginally spooky by the silence and the dark. Yet he couldn't shake the keenness of his instincts, which told him that this must be where they needed to be.\n\n\"There must be some sort of chamber behind this inner ring,\" he muttered, aping Tally and cupping his fingers against the glass to see if there was a hatch on the back wall. A large bearded lizard blinked lazily back at him. \"Some sort of access point, perhaps to feed the animals and clean the cages. Hello...what's this?\"\n\nAfter stepping back from the dragon's gaze his eyes fell upon an exhibit distinctly different from the rest. It was set inside the inner wall, directly opposite and facing away from the door through which they had entered. He strode up to it purposefully, Tally close behind him.\n\n\"Something caught your eye, Professor?\"\n\n\"Indeed, something has...\"\n\nWhat had piqued Nathaniel's interest was a free-standing exhibit that purported to show the flora and fauna of Mars. It looked as if it belonged more in a museum than a zoo\u2014a crude diorama of the bleak Martian landscape, with two statues of Martians made from resin standing in its centre. The figures, Nathaniel noted, were rough and inaccurate, as if their sculptor had only the most basic sketches from which to craft their likeness. Seemingly based on the more civilised Canal Martians, their long, elfin features were out of proportion, with eyes that were far too wide and too many fingers on each hand. They were also incongruously dressed in a patchwork of skin and fur, more redolent of cavemen than the actual baroque styles the Canal Martians commonly wore.\n\n\"Bit of a rum pair,\" observed Tally.\n\n\"Quite,\" rejoined Nathaniel, \"and they're not the only thing that's rum here....\"\n\nNathaniel quickly unhooked the maroon velvet rope that served as a barrier to curious hands and restless children, and carefully stepped around the resin figures to the back wall. A pale red Martian sky had been painted there, and Nathaniel started running his hands up and down the wall to gauge if there was some sort of break that could act as a doorway. He quickly found a crack and, tracing it with his fingers, discovered the distinct outline of a door. He tried pushing it, pulling it and even sliding it to one side\u2014all to no avail.\n\n\"There must be some sort of switch,\" he said to Tally, exasperated. \"Have a look around, see if you can see anything.\"\n\nTally got to work on the left of the scene while Nathaniel looked down to his right. Clearly, at this point, the model maker either knew something Nathaniel didn't or was just letting his imagination run free, for he had built a couple of plants into the ground that resembled nothing Nathaniel had ever seen on Martian soil. They resembled a kind of thistle with a corkscrew-shaped stem from which vicious looking spikes emanated, each being about four inches long, the plants themselves a couple of feet high. These fantasy plants were a pale green colour, looked almost translucent, and were topped with a large black bloom like a sunflower. Nathaniel knelt down to examine them. The thorns, he noticed, were razor sharp.\n\n\"Curious...,\" he muttered, experimentally prodding and withdrawing his finger quickly. He looked at the base of the plant closest to him, and noticed something odd. Where the bottom of the stem curled into the ground there seemed to be some sort of recess, a break in the otherwise unbroken floor in the exact shape of a circle. It was totally unnoticeable unless you were looking, and was surrounded by a plethora of the angry-looking spikes.\n\nCareful so as not to pierce his skin, Nathaniel edged his hand inside the curling stem and pushed down firmly on the button.\n\nThere was a loud clunk and the sound of groaning, and he looked up to see the outline of the door had swung backwards a fraction.\n\n\"Bingo,\" he breathed. He quickly extricated his hand and stood, and with Tally looking on he gave the door a gentle push. It eased open smoothly on well-oiled hinges. There was nothing but darkness beyond.\n\n\"After you,\" said Tally cheerfully. Nathaniel frowned at him.\n\n11.\n\nTHEY FOUND A couple of lanterns in a cupboard on the wall just inside the door and, having lit them with a couple of matches provided by Tally, took in their surroundings. Nathaniel's previous assumption had been correct\u2014they were in a circular room at the centre of the ring. A spiral staircase built from wood jutted from the walls surrounding a dark, empty hole at the centre. There were no handrails, and the stairs creaked ominously as they began their descent down the well.\n\nTheir journey was a long and careful one, each step tested carefully before Nathaniel, who was leading, would trust his weight to it. They were, he estimated, perhaps two hundred feet down before the stairs stopped at an unevenly-hewed stone floor. A tunnel roughly cut from the bare rock yawned before them.\n\n\"Someone's been busy,\" remarked Nathaniel.\n\n\"What is this?\" asked Tally. \"Some sort of mine?\"\n\n\"I'd guess so. The Russians do seem overly keen on digging, I've found. It's either that or some sort of...escape route? But why?\"\n\n\"Only one way to find out,\" said Tally. He gestured like a valet for Nathaniel to take the lead once more. Ducking his head, Nathaniel entered the tunnel. He had not gone far before the roof began to open out and a soft light from ahead allowed them to abandon their lanterns. They proceeded with the utmost caution, but the closer they got to the source of the light, the more apparent it became that this mine\u2014or whatever it was\u2014was deserted. There was no sound of any sort, not the thrum of machinery or the barked instructions of soldiers, which prompted Nathaniel to wonder why the lights had been left on....\n\nShortly, the tunnel opened out into a larger cavern. Compared to some of the Russian cave systems and mining operations Nathaniel had seen on Luna, this one was decidedly modest. It was only a couple of hundred feet across and roughly dome-shaped, with a rough ledge carved around the circumference half-way up the wall. Wooden props and staircases were dotted around haphazardly, and the mouths of several lesser tunnels opened ominously into darkness on both levels. Hemp ropes and rusted pulleys hung from the ceiling, deathly still. It was the centre of the cave that proved the most intriguing. A mineshaft had been dug into the floor and, now convinced that whoever had dug it had absconded, Nathaniel jogged up to peer into the dark. He had expected the hole to yawn sickeningly into the depths, but found the shaft was not as deep as he had thought\u2014he could easily see the ground, which was littered with abandoned chunks of machinery, pickaxes and cables that fed the spectral lights. A large tarpaulin sheet covered something in the centre of the pit, and was Nathaniel mistaken, or did he detect a faint green pulsing from underneath it...?\n\n\"Whoever they were they must have known the location of what they were digging for to a high degree of accuracy. This is far more efficient than you'd expect....\"\n\n\"Well that's just dandy,\" said Tally. \"Shame the same thing can't be said about us. We're no closer to knowing a damn thing.\"\n\n\"Not the point, Tally. Not the point. What's important is that we're finding out. Otherwise, what's the point of being here?\"\n\n\"Exactly the question I was asking meself....\"\n\nNathaniel noticed a ladder leaned up against the lip of the pit. \"Chin up, Tally!\" he said, clapping the Irishman on the shoulder. \"Onwards and downwards....\"\n\nRelieved that the ladder was at least sturdily constructed, Nathaniel began his descent into the hole. Tally followed him down.\n\nAfter dusting himself off primly, Nathaniel surveyed the scene.\n\n\"It looks like all the answers will be discovered under that tarpaulin,\" he noted, and strode across to the wax-covered cloth. It had been tied down firmly at the corners onto pegs that had been drilled into the floor, but there was enough give at the sides to allow Nathaniel a quick peek underneath. He had not been mistaken\u2014he found himself looking at a sheer green wall, crystalline in nature and glowing faintly.\n\n\"Remarkable! Tally, have you still got that penknife?\"\n\n\"No, I left it behind on account of its terrible weight. Of course I've bloody got it.\"\n\n\"Well then don't be facetious and hand it here.\"\n\nTally did as he was instructed. Nathaniel scratched at the pale, translucent material, hoping to carve off a chunk\u2014but the penknife barely made a scratch.\n\n\"Whatever it is, it's tough,\" he remarked, before using the penknife to cut away at the rope that held one corner of the tarpaulin down. He quickly sawed through, and lifted up the corner to reveal a small section of the crystals more fully. They spiked and branched from a central root, looking almost organic in nature. Nathaniel gripped one of the outcroppings firmly, a hexagonal chunk roughly six inches long that tapered into a deadly-looking point. He gave it a yank, and felt it give; put all his weight against it and his shoe against the side before tugging with all his might. With a delicate snap, the piece came away in his hand and he fell backwards into the grit.\n\n\"Laterally,\" he said, as Tally helped him up, \"it's firm as a diamond, but it's far more brittle on the horizontal plane. As if it's composed of some sort of tubing....\" His mind began to whirr with possibilities. \"Here, Tally, help me undo the rest of these ropes. I need to get a good look at the formation as a whole.\"\n\nTally got to work unfastening the knot at the other corner with his fingers while Nathaniel cut away at another. They both finished their task simultaneously and, catching Tally's eye and nodding, Nathaniel whipped the tarpaulin up and away with a whoosh of air. His heart fell when he saw what he had revealed.\n\n\"Oh dear,\" he muttered, swallowing loudly. \"Oh dear, oh dear, oh dear.\"\n\n\"I'm thinkin' you weren't expecting to find that,\" noted Tally, his mouth dry.\n\n\"If I'd had my wits about me, Tally, I'm afraid that's exactly what I should have expected to find!\"\n\nThe crystal growth had been revealed in all its glory. The main trunk was about two feet high, from which smaller gems branched like fronds of lightning. The top of this trunk had been sheared away like a felled tree. Attached to this plateau was a device whose construction Nathaniel instantly recognised, even though previously he had only seen it in shards and blackened pieces.\n\nWith a sinking feeling deep in his abdomen he realised how much bigger this version looked than the one he had examined in the bowels of the Admiralty....\n\nThe mechanism and construction of the bomb were almost identical to how Nathaniel had imagined them, as he'd carefully pieced the machine together in his mind. The body was built up of copper, beautifully crafted with intricate fractal patterns and polished to a gleaming shine. A clockwork mechanism jutted from the main section, ticking ominously and surrounded with a glass dome, and Nathaniel felt almost breathless when confronted with the sheer ingenuity and elegant perfection of this device of death. He noted the steel tubing strapped to one side, calibrated, he deduced quickly, to prevent the housing from being moved even a fraction of an inch. Yet there was a distinct difference in this far more terrifying version.... Before, where a glass tube had held the crystalline powder, wires ran from the heart of the infernal contraption and were embedded solidly in the depths of the crystal's root.\n\n\"We need to get out of here,\" said Nathaniel. \"We need to get out of here as quickly as possible!\" Any thoughts of immediate escape for Tally and Nathaniel were curtailed by a sound far more ominous than the soft ticking of the bomb. From the ledge above them came the sound of a slow hand clap, and a nasal voice in clipped, upper-class tones purred.\n\n\"Well congratulations, Professor Stone. I knew you'd get there in the end.\"\n\nTally and Nathaniel's gaze shot upwards. The pit was not so deep that they could not fully see an impish figure sitting with his legs dangling into the hole. He wore immaculate green tweed, a shallow-brimmed hat with a small brown feather in the band and was smoking a cigarette from a long ivory holder gripped tightly between his teeth. Next to him stood a giant of a man, perhaps seven feet tall and nearly half as wide, dressed in a huge, double-breasted greatcoat. He wore welding goggles over his eyes and a black bowler hat, and his hands...his hands were surrounded by a dull brass framework, their hinges and sprockets drilled into the very flesh itself.\n\n\"Emphasis on the word end,\" said the smaller man, grinning like Mephistopheles.\nChapter Eight\n\nThe Succubus of Black Town\n\n1.\n\nTHE SHAMEFUL EXTENT of Fabrice Garrecreux's depravity had been revealed. Vice-Chancellor Banerjee had recounted the scandalous tale in a slow monotone, with no inflection of emotion save a vague, weary sorrow. It was almost as if he were giving a lecture on a point of law in which he had no interest, or was discussing the dull minutiae of college finances.\n\nThe tale went as follows: Garrecreux had been a nuisance from the very start. His research was ostensibly into cannabis sativa, a subspecies of hemp that produced a bud infused with a psychoactive crystalline compound. This, Banerjee explained, was a blight on the lower castes who would inhale its vapours to induce an indolent, carefree haze that blunted the realities of their otherwise dismal existence. Garrecreux had procured two dozen hessian sacks of the stuff and had stunk out the geology department for six weeks afterwards. He became less and less dependable in his teaching duties, often missing his scheduled lectures for days on end in order to concentrate on his research into the plant. He attempted, utterly unsuccessfully, to synthesise the active compound and grow larger and larger crystals, analysing its properties and, so it was rumoured, becoming more and more addicted to the thrall of the weed's euphoric effects.\n\nMuch as in Arnaud's time, Garrecreux had become a figure of ridicule in the university, among the faculty as a whole and, in particular, the students\u2014with the single exception of a young Bengali graduate by the name of Esme Moonsinge. She came from a wealthy family and, to all intents and purposes, seemed an introverted, brooding girl with a fierce intelligence and a terrible temper. On account of her privileged stock she insisted she be called Madame Moonsinge by peers and professors alike, and when a classmate humiliated her in the common room on account of her close association with Garrecreux, the unfortunate boy was found the following morning with his face slashed open by a knife. The boy lost an eye, and yet nothing could be proved\u2014such things happened when the young wandered Black Town at night. Yet the whole thing reeked of suspicion.\n\nIf this wasn't troubling enough, further distressing reports began to reach Vice-Chancellor Banerjee's desk. Garrecreux and Moonsinge would lock themselves away in labs to which they (strictly speaking) were not allowed access. They were seen together outside the Campus, often frequenting the riskier corners of Black Town's most notorious slums. When working in the Campus they would cause fires, small explosions and outbreaks of noxious, headache-inducing gasses. Disciplinary procedures were already well underway before talk came of a disturbing shift in the nature of their relationship\u2014Madame Moonsinge had ceased to be the student and had become the mistress, in every sense of the word. Garrecreux, it was said, was now nothing more than a simpering lapdog, so besotted with the young, tempestuous beauty he would wear dark glasses all hours of the day to counter, it was said, the brightness of her radiant countenance.\n\nThere was nothing more the vice-chancellor could offer, he admitted with a sigh. He wanted nothing more than to wash his hands of the whole affair and concentrate on encouraging students and teachers who would bring acclaim and recognition to the fine institution he was tasked with running, as opposed to ruin and ridicule.\n\n\"Where do you think we can find them?\" asked Arnaud, as Coyne\u2014for the hundredth time\u2014inspected the cleanliness of his nails.\n\n\"I can direct you to some of the areas of Black Town where they've been seen. It's not a course of investigation I would recommend, however. Black Town is a dangerous place, Doctor Fontaine, and not one for tourists or outsiders. Even the Queen's Regiments despise being posted there. Yet I can see your resolution is firm, and your mission of no little importance, so allow me to offer a guide, a trusted friend of mine, and a driver to take...\"\n\n\"That won't be necessary,\" interjected Coyne, idly. \"I'm all the protection he needs.\"\n\nWhen the vice-chancellor led them to the door of his office, he shook Arnaud's hand firmly and, his eyes sharp, instructed the young Frenchman to watch his back at all times, and to always carry a loaded pistol.\n\n2.\n\nSOVEREIGN WAS HOVERING forty feet above the windswept tundra, a blinding and barren sheet assailed by ghostly wisps of snow made living by the wind. Hardy trees as cold as stone stood sentinel hundreds of feet apart, forlorn for lack of warmth or company. Crops of rock would occasionally peek above the ice, the colour of gunmetal and the only hint of the true nature of the terrain below. They were several miles west of the Russian base at Severnaya, and while the blizzard's white-out offered them almost perfect concealment, the great airship did not dare venture any closer. Folkard, Enderby and four hand-picked men from the Secret Service Bureau would be continuing on foot.\n\nA cutter was deemed too risky in these conditions, and so weighted rope ladders were thrown down from the sides of Sovereign. Instantly the wind hit them, buffeting them wildly and making the ends whiplash around like a viper in the throes of death.\n\nEven with the weight of the men in full kit, the ladders were bucked and buffeted by the gale on their descent, and more than once Folkard had been forced to cease his climb and grip the ladder close to his body until the tempest had momentarily subsided. Yet eventually, with cheeks already chapped red-raw and facial hair peppered with frost, the party set foot on the enemy's inhospitable turf.\n\nThey regrouped quickly and watched as Sovereign began its mournful ascent, soon lost in the maelstrom of sleet and grey cloud. Folkard watched it until the hull became indistinct, then to the point where he was sure he would not be seeing it again. He almost raised a hand to wish her farewell, but checked himself before he could perform this sentimental gesture. Nevertheless, he felt guilty for not having said goodbye to the ship he loved so dearly, and turned to find the rest of his men with a bitter feeling in his heart.\n\nEnderby consulted a heavy-duty compass and a map wrapped in translucent wax paper, determined which direction was east, and the party set off in silence and single file.\n\nThe trudge across the wasteland was a long and arduous one. With the lack of vegetation, cover was impossible; they were truly trusting to luck that they should not be spotted. At least the weather was on their side in that respect. Enderby took the lead with Folkard second, and even though the men were only a few yards apart Folkard could not see the last in line when he looked behind him. As the snowstorm worsened the cold became almost unbearable, and as much as it aided them with concealment it sapped all the energy from Folkard's already exhausted body. It was only the thought of the Heart, and his certain reunion with Charlotte, that warmed him and spurred him on.\n\nAfter perhaps four hours of hiking, the storm lessened its assault and rough natural tracks began to appear in the terrain. More and more trees began to dot the landscape, lessening the depth of the snow on the ground, and this along with the uneven yet mercifully solid paths allowed them to pick up their pace. The paths soon became paved with large, flat, irregular stones, tessellated as much as was possible. This was the first sign of civilisation the men had seen in hours, and Folkard felt a brief surge of relief\u2014if not amusement\u2014at his joy upon seeing a badly-made road.\n\nThey followed the road for another quarter mile or so, before Enderby ducked to his haunches without warning. Instantly, the rest of the men followed suit and Folkard crawled up to see what had startled the spy.\n\nEnderby pointed ahead of him. Just visible, standing around a dimly-glowing steel brazier, two Russian soldiers huddled against the cold. They slapped their hand against their arms for warmth, their incessant, guttural grumbling carried towards the British on the wind. Enderby and Folkard looked at each other nodded. Enderby reached down to his waist and pulled out a stiletto knife, its blade tempered black to prevent any reflection. Folkard did the same and Enderby quickly gestured: You take the one on the left, I'll take the one on the right. Without prompting, the men behind them readied their rifles should the assassins be detected.\n\nSwiftly and silently, Folkard and Enderby approached the two unsuspecting soldiers until they could feel the heat of the fire on their faces. Enderby counted down on his fingers, three, two, one...and the two men sprang into action, both plunging their stilettos into the base of the soldiers' necks with deadly, instant accuracy. The Russians fell to the ground, dead weights, and before the blood had even started to colour the ground, the four service operatives who had taken up the rear were bundling the bodies into the undergrowth, where they would be covered by snowfall in mere minutes\u2014if the wolves didn't get to them first.\n\nKnowing that the Russian presence was a very real danger from here on in, stealth became paramount. Still in single file, the agents and Folkard eased their way up the road at a crouch, scanning the horizons for any sign of trouble. A larger camp containing a small platoon of Russians was deemed to be too much of a risk, so the party once more took to the snowdrifts and skirted around the danger as if they were never there. The increase in troops surely meant they could not be far from their objective.\n\nBefore long they came across a large stone bridge. The eaves of its impressive arches were decked with foot long icicles, the bridge itself spanning a gorge at the base of which lay a frozen river, unmoving and impassive. The cover offered by the bridge seemed a perfect place to stop, rest and assess the situation, and it was clear the men were grateful for even the briefest of respites.\n\nThey eased their way down a shallow slope to where the buttress met the ground. There was a small stony area, neatly secluded, where the men could break out their tinned rations. Set into the large buttress was a heavy wooden door, locked when they tried it. This didn't stop Enderby from keeping one beady eye on it as he sliced off strips of tinned ham with his knife.\n\nHe threw the empty tin into a bush and looked to Folkard, then the door.\n\n\"Shall we see where it goes?\" he asked.\n\n\"Why not?\" replied Folkard, wiping his hands on his trousers and standing.\n\nThough the door was locked it was rotten with age and obviously disused, and it didn't take much time or force to open. Inside, a dark stone staircase led up, musty, ancient and cracked.\n\n\"Like something out of a ghost story,\" remarked Folkard.\n\n\"Then it's lucky I don't believe in ghosts.\"\n\nFolkard raised an eyebrow at Enderby.\n\n\"Trust me,\" he said. \"If you'd seen everything I have...\"\n\nInstructing the rest of the group to guard the entrance, the two men made their way up the steps. There were four flights, all at right angles, and for a while in the middle the two adventurers were in total darkness. When they reached the top, a thin rectangle of light indicated the presence of another door, and Enderby pressed his ear against it to listen for a moment.\n\n\"Nothing,\" he noted. \"A light step is hardly the Russian army's fort\u00e9, so I think we're safe.\"\n\nOnce more the door proved no great obstacle, and after a brief peek through to check the coast was clear, Enderby and Folkard stepped out onto the top of the bridge.\n\n\"Handy little short cut,\" said Folkard.\n\n\"But somewhat exposed. Look.\"\n\nEnderby pointed across the bridge. A half a mile away, the Russian's secret base at Severnaya sat like some great, squat beetle. The main building was a vast hangar, surely the resting place of the Russian's terrifying new weapon of war, and around it several smaller square buildings were grouped, as if in praise or reverence. Several smaller airships were tethered to the hangar, bobbing contentedly in the sky, with cutters darting quickly between them, flies on the bellies of these great beasts of the air.\n\n\"We've found it,\" said Enderby. \"Welcome to Severnaya, Captain Folkard.\"\n\n3.\n\nBOBBING UP AND down on a horse, Annabelle felt a curious sense of nostalgia that somehow managed to override the apprehension that, by all rights, she should be feeling.\n\nThey had left Billy and Ignatius to tend to the rest of the passengers and Annabelle, George and Bert had moved outside to greet the Indians.\n\n\"Will we be safe, miss?\" Bert had asked.\n\n\"Trust me, Bert. These are Apaches. If they'd wanted us dead we wouldn't have even known they were there.\"\n\n\"I'll defer to your judgement, miss.\" There was no trace of irony or fear in his voice, and Annabelle realised this man trusted her completely. She just wished he'd stop calling her \"miss\"...\n\nIf Annabelle was convinced that the Indians posed no danger, her assumption was buoyed even more when she thought she recognized the face of one of the braves. She squinted at him, remembering the high cheekbones and soft brown eyes of a boy she had once known\u2014a boy that had grown into a man.\n\n\"Wapi!\" she exclaimed.\n\n\"Annabelle,\" replied Wapi from his horse, nodding gently at her. \"It has been a very long time.\"\n\n\"I'll say!\" replied Annabelle. \"Look at you!\"\n\nWapi was dressed in a curious fusion of traditional Apache and Mexican dress. A woven poncho covered his shoulders and bare torso, moccasin-clad feet thrust firmly into his horses stirrups. A bandoleer full of bullets crossed his chest one way, the strap of a Winchester rifle the other. She looked at his face, a face she remembered always being blessed with a gap-toothed smile as they had teased scorpions together or chased each other around the tents and campfires of their ever-nomadic home. Though his youthful looks remained, the smile had gone, replaced by a sombre seriousness that indicated their re-acquaintance, however welcome, was not a matter for celebration just yet.\n\n\"The years have made you more beautiful as you have grown, Annabelle,\" he said, and as she attempted to conceal the reddening of her cheeks she was simultaneously amused that George's bristling behind her was almost palpable, \"but now is not the time to discuss how those years have shaped you. We must ride, and quickly. There is danger in the desert. Evil men are awakening spirits that should be left to sleep.\"\n\n\"Where are we going?\" asked George.\n\n\"Wapi, this my husband George,\" said Annabelle, stepping to one side to introduce George. \"And Bert, our...travelling companion.\"\n\nGeorge stepped forward.\n\n\"Happy to make you acquaintance, Wapi.\" He held out his hand, but Wapi kept his firmly on the reins, looking down his nose at the Englishman whose arm was so comically outstretched.\n\n\"There will be time for talk on the journey. Annabelle, you must ride with me. This husband of yours and the man in black can go with Bly and Gopan.\" He paused before adding; \"Whichever they prefer.\"\n\n4.\n\nTHEY HAD RIDDEN due south at a fast pace for what, Bedford guessed, must have been around forty minutes. After that, when they had reached a seemingly vast expanse of desert, they had slowed to allow the horses some respite from the heat of the high Arizona sun. They were heading, it seemed, to a mesa in the distance, the only landmark visible in a horizon perpendicular to the unbroken blue of the sky. Wapi led, Annabelle swaying lazily as she sat behind him, and Bedford goaded Gopan into trotting up until they were side-by-side. When she noticed his approach, Annabelle looked back and smiled.\n\n\"Enjoying the ride?\" she asked.\n\n\"The scenery's magnificent,\" said Bedford drily. \"But seriously, Annabelle, what's all this about? It can't be coincidence.\"\n\n\"Of course it isn't, George my darling. And don't be coy, you know as well as I that the only way to find out why is to follow the route we've been given. And find, so to speak, the root of the problem.\"\n\n\"But the attack on the train. We've been targeted, Annabelle, and dash it all but you're my wife now, and I won't see you in danger.\"\n\nAnnabelle turned to him sharply.\n\n\"And, as I believe I've said before, you're my husband. Do you think I wish to see you in danger? I trust the Apaches, and Wapi in particular. Though he may represent a time in my life you were no part of, you are a part of my life now, and a significant one at that. I can't erase my past, however much I'd like to, so trust me when I say that we'll both be safe if we stick with Wapi.\"\n\n\"I agree,\" chipped in Bert, trotting some feet behind them. \"It's the best course of action. It seems we share an enemy. Or enemies, at least.\"\n\nWapi, who had been gazing off into the distance, seemed to come to out of some mystic reverie and nodded. \"Our chief, the wise and revered Goyahkla, has had many visions of the deaths that may have to be dealt. On these walks inside the mind he has seen you, Annabelle, travel the sky with the same ease as a horse steps through a shallow river. On some of those occasions he has been with you in spirit, hearing with your ears and seeing through your eyes. He has told us of those worlds in the sky; those that looks similar to this, our land, and the ones that look so strange he could barely describe them, or the beasts that dwell there.\"\n\n\"Who's Goyahkla when he's at home?\" asked Bert, with genuine interest.\n\n\"He is better known to you,\" said Wapi, turning his head once more to the horizon, \"as Geronimo.\"\n\n5.\n\n\"OH, CORKS,\" SAID Tally.\n\n\"Corks indeed,\" muttered Nathaniel.\n\n\"Mister Potsdam, the ladder, if you'd be so kind.\"\n\nFollowing the instructions of his weasely master, the brute thumped across to where the ladder lay against the rim of the pit and lifted it clear as if it were matchwood.\n\n\"Can't have you climbing out and ruining all the fun now, can we, Professor Stone?\" said the well-dressed man, licking the tips of his fingers and using them to extinguish his cigarette with a hiss. \"Although I'm sure you'll feel happier going to you death knowing that I'm truly honoured to finally kill someone who appreciates my art.\"\n\n\"What can I say?\" quipped Nathaniel. \"I'm a fool for aesthetics.\"\n\n\"Now now, Professor. Let's not get catty. You wouldn't want to meet your maker feeling all cross now, would you?\"\n\n\"Bit late for that,\" Nathaniel muttered to himself.\n\n\"I take it you've used your not inconsiderable powers of deduction to work out the purpose of that delicately-hued crystal?\"\n\n\"I'm guessing it's some sort of accelerant. There was a small amount, perhaps a few ounces, in the Horseguard's bomb, and that seemed to imbue the device with a significant lethality.... The extent to which that power is magnified with a block this size is scarcely imaginable....\"\n\n\"Who needs to imagine! You'll be seeing for yourself in but a few minute's time. I would ask you what you thought, but then, what's the point in asking questions of a bloody, vaporised mess, as my dear old mother used to say.\"\n\n\"Look,\" said Nathaniel hotly, \"this witty fluff and bluster is all very good but it's starting to grate on my nerves a bit. Just who the hell are you, anyway? I caught the name of your pet brick wall\u2014Potsdam, wasn't it?\u2014and yet here you are, content to remain the man of mystery.\"\n\n\"It's funny that the doomed have such a fondness for propriety, don't you think, Mr Potsdam?\" Potsdam remained silent and stony-faced. \"Still, I suppose my manners have deserted me. My name is Klopstock. KP Klopstock. Inventor, renaissance man, artiste. Also hired killer.\"\n\n\"He's off his chops,\" said Tally.\n\n\"I'll let that slide, Mister Cahalleret. I'm gracious like that. Nevertheless, Professor Stone, as well as its glorious ability to make things go bang that little bit bigger, this marvellous mineral you see before you has many other curious qualities. It's quite the wonder stuff, and my employers are as keen to keep it to themselves as your government is in keeping gravitar on the QT. Which is, clearly, not working out so well for them. And so naturally, any Tom, Dick or Harry who goes poking their nose into the wrong places is bound to suffer misfortune of a rather fiery and terminal sort, thanks to my delicate art. I do so hope you like my little masterpiece down there, Professor. I put some extra work into it just for you....\"\n\n\"I'm flattered.\"\n\n\"That's marvellous to hear. Anyway, time does fly when you're having fun with gloating, as the saying goes. Mister Potsdam and I really must be off. I believe we have an appointment with your dear friend Arnaud in Calcutta, and he's hardly going to explode himself now, is he?\"\n\nNathaniel's face became set. His eyes narrowed and his teeth clenched, his hands unconsciously balling into fists. \"I swear to you, Klopstock, I'll be coming for you.\"\n\n\"Oh right. And how do you intend to do that, may I inquire? As a ghost? Or do you plan to reconstruct your mangled corpse by sheer force of will and shamble after me like some ungodly revenant? Come, come, Professor Stone. Be reasonable.\"\n\n\"No force on Earth will be able to stop me.\"\n\n\"Handy, seeing as the forces you're dealing with here have no Earthy provenance anyway. Toodle pip, Professor. Do die nicely.\"\n\nWith that, Klopstock hopped up and brushed the grit from the palm of his hands. \"Come along, Mister P. Places to kill, people to mangle.\"\n\n\"Klopstock!\" yelled Nathaniel. \"Klopstock!\"\n\nBut Klopstock ignored him, jauntily sidling to the cave entrance with Potsdam lumbering behind him like some mythical behemoth. As the vast man ducked into the tunnel entrance, he turned around briefly and, using his unnatural and metal-infused strength, yanked at a timber beam that supported the roof. He used the length of thick wood to strike at the top of the tunnel and, ignoring the rocks that fell onto his head and shoulders as if they were no more than snowflakes, had soon totally blocked the only available route of escape.\n\n\"Well that's us well and truly fecked,\" said Tally.\n\n6.\n\nGIVEN THE CHOICE, Arnaud may well have picked the dank and cold of that deadly underground cavern over the heat of the Calcutta sun\u2014bomb or not. Vice-Chancellor Banerjee had sent information regarding the whereabouts of Garrecreux and the mysterious Madame Moonsinge to the hotel that Coyne had managed to procure for himself and Arnaud. It turned out, Arnaud thought bitterly to himself, that the fellow wasn't totally useless after all.\n\nAfter they had washed and dressed and had eaten a light meal (Arnaud had declined the more exotic Indian food on offer, which looked like some sort of pungent stew served with flatbreads\u2014though Coyne had lapped it up like a native) they left the hotel and headed in the direction of Black Town. Arnaud was still impressed by the European elegance of Calcutta's wealthier side. He almost felt at home strolling down paved roads and comparing the similarities (and differences) of the architecture in Paris. Everyone seemed well-dressed, affluent, busy\u2014if only there were some respite from this damnable heat.\n\nArnaud saw a bench under the shade of an awning and beckoned for Coyne to stop. Coyne tutted dismissively.\n\n\"I'm sorry,\" snapped Arnaud, \"but I'm just not used to this...this relentless sunshine!\"\n\n\"Which may have something to do with you sweating out three days' worth of brandy.\"\n\n\"Couldn't we at least hail a hansom? The tram? Even one of those death-trap rickshaw contraptions would do....\"\n\n\"Do you think you often see a fancy-pants hansom cab in Black Town, Fontaine? Don't be soft. They'd see us coming a mile off.\"\n\n\"Oh, and I suppose a sweaty Frenchman and man dressed entirely in black is utterly inconspicuous.\"\n\n\"More than conspicuous wealth, yes. As it happens. Oh, you'll see Europeans in Black Town, all right. None of 'em look you back in the eye and they'll be moving pretty sharpish, as whatever you want in Calcutta that isn't on the level can be found there, if you jangle your rupees in the right darkened corner. Weapons, hashish, girls...oh, this lot have picked up quick how to cater for tastes that you don't talk about in the drawing room. You've got five minutes.\"\n\nArnaud didn't even think he'd been given three before Coyne was chivvying him to get up and get moving again. It was almost as if the agent was more keen to find Garrecreux than the man who had actually been charged with procuring his help.\n\n\"Do you even know where we're meant to be going?\" whinged Arnaud as he dragged himself to his feet. He was already soaked with sweat and feeling miserable. Coyne turned sharply on his feet and jabbed Arnaud in the chest with his index finger, taking Arnaud aback.\n\n\"Listen up,\" said Coyne, his voice clipped. \"I don't like you and you don't like me. I think we've established that. Right? Right. So get this into your Bastille-storming, grape-crushing French head, and get it there quick. I was trained for this. This is what I do, right? I don't tell you how to look at bloody rocks. If I say we walk this way, we walk this way. If I say we go here, we go there. If I say stab this chump in the gut, we stab him. Is that understood?\"\n\n\"Now look here!\"\n\n\"No. I won't. I won't look anywhere because you haven't got the first idea what you're doing. But I do. And that's enough to mean I don't have to take any of your guff. Or at least no more than I've already taken. So wipe your face, shut your mouth and follow me.\"\n\nHe'd turned smartly on his heels and strode off before Arnaud could even begin to reply. The Frenchman was stunned. It was not so much the rudeness of the man\u2014it was true Arnaud didn't like him, and so ultimately his opinion mattered little. But so often on his adventures he had been accompanied by his friends, those he trusted and loved, and now he found himself stuck with a petulant, ill-mannered hothead who clearly could not be relied on when the chips were down. He wondered what on Earth Folkard had been thinking when he chose this degenerate for the job. Arnaud was glad he had taken Banerjee's advice to heart, and though the loaded pistol was a weight on his side, it did do something to lighten his worries.\n\n7.\n\n\"WHERE'S YOUR FIGHTING Irish spirit, Tally?\" Nathaniel chided. \"Come on, give me a hand.\"\n\nNathaniel began to race around the floor of the pit, inspecting the chunks of machinery and discarded mining detritus. The ropes that hung from the ceiling were out of reach, but Nathaniel reasoned that with both his and Tally's height, and a little boost from a bit of abandoned equipment, they just might make it to the lip of the hole.\n\n\"How long have we got?\" asked Tally, as he and Nathaniel hefted their shoulders against an old engine housing.\n\n\"I don't think that madman is prone to leaving much to chance. But we'll certainly know when our time runs out.\"\n\n\"Comforting thought.\"\n\n\"Quite. Come on, man, put your back into it!\"\n\nThe ticking from the bomb strapped to the crystal seemed to increase in intensity. More worryingly, the device seemed to be giving off a low, reptilian hiss. Tally and Nathaniel had managed to manoeuvre the engine block to the wall, and Nathaniel motioned for Tally to climb onto it. Tally began to protest.\n\n\"No time for debates, Tally. You're stronger and I'm lighter, and as soon as I'm up I'll put the ladder down for you. Hurry, man!\"\n\nTally did as he was told and scrambled up. Nathaniel followed\u2014with the two of them standing there barely any room remained and the block rocked precariously whenever either of the men moved. Nathaniel squashed himself as close as he was able to the wall and Tally bent down to interlock his fingers and hook them under the sole of Nathaniel's shoe.\n\n\"You ready, Professor?\"\n\n\"Ready as I'll ever be.\"\n\n\"Three, two... Hup!\"\n\nHis back suddenly wrenched with pain at the awkward weight he was attempting to lift, Tally gritted his teeth and tried to lean his weight forwards, giving Nathaniel as much height as he could. Nathaniel scrambled up the side of the wall, dirtying and blooding his fingernails with every frantic, hard-fought inch of progress. As Nathaniel's foot passed his shoulder the Irishman howled in pain and effort, Nathaniel yelling unheard encouragements. Mustering all his strength, and still holding Nathaniel's shoe, Tally slowly, carefully started to stand, wincing with the terrible effort.\n\n\"Keep going, keep going!\" hollered the Professor. \"I'm nearly there!\"\n\nWith a gasp that seemed to hollow out his lungs and a push from a source of strength unbeknownst in himself, Tally pushed Nathaniel's leg up as far as it would go. The whole of Professor Stone's weight was now on Tally's outstretched arms, the hardness of the heel digging cruelly into his palms. The whole of Tally's body began to shake.\n\n\"I can't hold it!\" he cried. \"I can't hold it!\"\n\n\"A second more!\"\n\nBut Tally could not even muster the energy for that one second more, the second that might have saved both of their lives. He let go and fell away, winding himself as he dropped heavily on his back. He had expected Nathaniel's weight to slam onto him a mere millisecond later, perhaps break a couple of ribs or his wrist, and Tally thought with grim humour that whatever got broke, at least he'd have the pain to concentrate on until the bomb went off and killed him, Nathaniel and goodness knows how many innocent Dubliners.\n\nOnly Nathaniel didn't drop onto him at all.\n\nTally opened his eyes quickly to see, with a relief that bordered on the spiritual, Nathaniel's ankles disappearing over the rim of the pit.\n\n\"Oh, you little British beauty!\" he yelled, wincing as he got up.\n\n\"I had a leg up from a very dependable Irishman,\" came Nathaniel's disembodied voice. Moments later, the ladder was manhandled inexpertly back into the pit and Tally scrambled up it with remarkable speed.\n\n\"Right,\" he said. \"What now?\"\n\n\"Well, it looks like that way is a no-go,\" said Nathaniel, indicating the exit Potsdam had blocked with rubble. He furrowed his brow before exclaiming, \"Lucifers!\"\n\n\"No need for that language,\" said Tally, surprising himself that the gift of the gab hadn't deserted him even in these trying times.\n\n\"Matches, Tally! Do you have any matches?\"\n\n\"Aye, here y'are.\" Tally pulled a box from his pocket and handed them across. \"Time for one last smoke, is it?\"\n\n\"Don't be asinine, man! Light one and watch the flame!\"\n\nThey both took a match from the box and struck it on the sandpapered side. Nathaniel lifted his just above his head and watched the flame intently. Nothing happened\u2014the match just burnt down, agonisingly slowly, half way down, then more... The flame was about to nip Nathaniel's finger when suddenly, almost imperceptibly, it was caught by a secret breeze.\n\nBoth Nathaniel and Tally's eyes darted in the direction opposite the flame's movement. A dark tunnel on the other side of a cave, clearly apart from the rest that dotted the walls of the cavern.\n\n\"There,\" said Nathaniel, already beginning his sprint. \"It has to be!\"\n\n\"I ain't arguing,\" said Tally, hot on his heels.\n\nSending up a rooster-tail of grit and gravel behind them, they dashed towards the darkened cave mouth. Behind them, the bomb began to hiss louder, a kettle about to over-boil.\n\nThe entrance was wide enough to continue their run, the light from the cavern behind them providing a little illumination. It wasn't long before they were forced to grasp the walls in the darkness, pulling themselves along as fast as they dared. The tunnel branched several times, and the duo had to move onwards solely by instinct, choosing the routes that sloped upwards and the larger openings. In the stygian depths behind him, Nathaniel heard Tally cry out as he slipped and fell. He whipped round, but the blackness was so absolute it disorientated him.\n\n\"Tally!\" he cried.\n\n\"I'm here, I'm here, I'm all right! Just a little\u2014\"\n\nBut Tally's sentence was cut short with a blinding flash that seemed to run against the walls of the tunnel like some ethereal beast. The tiniest fraction of time passed, and the sharp, deafening crack and the low, resounding rumble of the blast followed in its fearful wake. The walls of the tunnel began to shake, dislodging dust and pebbles from the ceiling; secondary blasts rocked the very air itself as the deadly chemical reaction made its way down into the roots of the alien crystal. The flash of light had given Nathaniel enough time to reach his arm out to Tally, who grabbed it and hauled himself up. There were the sounds of rocks falling toward them as the tunnel began to collapse. Dragging the dazed Irishman behind him, Nathaniel hurried on\u2014yet it was not mere self-preservation that spurred him, but the thought of protecting Arnaud from the evil plots of Klopstock and his monstrous, mute companion. The ground would not stop shaking. It felt as if the very world was ending, and if it proved that Nathaniel could not escape, the explosion that rocked the city of Dublin that night would be but a fleabite compared to the end that awaited the Earth.\n\n8.\n\nTHE REST OF the walk passed in an uncomfortable silence. It was not long before they reached the banks of the Hooghly River, and while the return of its unutterable stench was unwelcome, the soft breeze that rose up from its waters was not. They needed to cross, and head east into the dark heart of Black Town that spread like a stain into the distance. As one approached it, the gentle chatter and clatter of horses' hooves was replaced by the loud shouts from markets, haggling, threats and wails. They crossed a bridge, Arnaud lagging deliberately behind, mulling silently, and before they'd even crossed its span the poverty and human degradation of Black Town had begun.\n\nIt seemed like every second person you encountered was begging. But Coyne had neglected to mention that even raggedy stallholders and the lower-caste Calcuttans were game for the pleas and grasping hands of the dispossessed. Arnaud saw one emaciated boy beaten viciously for snatching a fly-blown hunk of fish from a tabletop, and as the stall's owner was occupied with his strop other chancers and vagabonds descended to help themselves. British troops watched idly from street corners, smoking, powerless or too desensitized to help. Was this, thought Arnaud, the inevitable price that must be paid for the progress and abundance across the river? Did encouraging culture, life and knowledge always mean suffering for those who lived beyond its reach? And was such a price worth paying?\n\nThe crowds, the noise...it was almost too much for Arnaud to take. Tents and shacks were cobbled roughly together at the sides of what passed for roads. Everything was bleached a dirty white by the sun. The further they got in, the more noise assaulted them, the more people jostled them, the more the tension rose. It seemed to Arnaud that every eye glanced in their direction at least once, with some holding their gaze suspiciously for as long as the duo remained in sight. Disregarding his previous unprofessional outburst, Arnaud was almost glad to find Coyne sidle up beside him in the bustle.\n\n\"Keep your wits about you, Fontaine. And keep your hand on that pistol you think I haven't noticed in your pocket.\"\n\nArnaud did so.\n\n\"What's the plan?\" he asked.\n\n\"We're coming up to a little cluster of buildings. Here on the right. See 'em? They sell a sort of local rot-gut there, Calcutta moonshine. Don't touch it, whatever you do. Not even a drop. I know what you're like. It stings when you get it on your fingers, God alone knows what it does to your insides.\"\n\n\"So essentially, this is another trip to a bar.\"\n\n\"Yeah, that's exactly what is it, Fontaine. They sell roast beef and have the Sunday papers, and the barmaid will call you 'squire' and give you the wink. Just stick with me and don't go aggravating the locals, eh?\"\n\nArnaud was getting increasingly tired of Coyne's attitude, but felt that for the moment is was best to put up with it. He'd certainly have the upper hand when it came to talking to Garrecreux, and bided his time, mentally preparing a complaint to issue to Folkard upon his return.\n\nShortly, they reached the building Coyne had indicated. Pretty much the whole of the front wall had crumbled, and a tattered, flapping awning held up by sticks stretched over a small courtyard that sat before the bar. Arnaud suddenly understood Coyne's sarcasm when it came to the comparison with a pleasant little British inn. This looked more like somebody's house, filled with clutter and second-hand junk. About two dozen Indian men sat on the ground or, if they were lucky, lounged around on pillows. They drank a milky, oily liquid from tin cups or puffed away on stained hookahs, dropping their conversations mid-sentence as Arnaud and Coyne snaked around their prostrate forms. An impromptu bar\u2014more like a plank of wood held up by bricks and optimism\u2014ran across the back wall. The bartender who stood behind it was conspicuous by his corpulence; he was the only denizen of Black Town Arnaud had seen so far with more than an ounce of fat of him.\n\nHe had a large, thick, black beard and restless eyes, and as the two men approached he adopted a stance of wilful defiance and ill-concealed displeasure. In a thick accent, he said;\n\n\"I think you gentleman have the wrong building.\"\n\n\"We're here to ask some questions,\" said Arnaud, matching the man for obstinacy.\n\n\"Then you definitely have the wrong building.\"\n\n\"Listen, chum,\" said Coyne, crossing his arms and leaning in menacingly. \"I happen to know we've got the right place, and you're the right man. So unless you want a visit from some of the more unhinged members of Her Majesty's Colonial Peacekeepers at ooh, let's say, 3am tomorrow, I suggest you start acting a little more helpful.\"\n\nThe bartender sneered. \"British,\" he spat. \"You are all the same. You are not content with taking over our country, you have to keep the right to threaten us as well. It is a land of cowards, your England. Cowards and bullies.\"\n\nWith lightning speed Coyne lurched over the bar and grasped the man by his collarless shirt. He bunched his fists tight and yanked the bartender forward, and the agent's forehead met his nose with not inconsiderable force and a sickening crack. The big man's legs fell from under him, his face streaming with blood, but Coyne kept him upright. There was a commotion behind them, the smashing of a hookah pipe and the sound of the crowd bolting upright, and Arnaud whipped around. The clients in the seedy bar were all on their feet, eyes sharp and dangerous, edging towards where the intruders stood. With a speed that surprised even himself, Arnaud's pistol was in his hand and he jabbed it at the encroaching crowd, who backed off warily.\n\n\"I think we've got his attention,\" said Coyne. \"Ask your questions, Fontaine.\"\n\n\"Whatever happened to not aggravating the locals?\"\n\n\"Well if he'd have kept his manners I'd have kept my temper. Come on, Frenchie. We haven't got all day.\"\n\nStill keeping his gun trained on the mob, Arnaud turned to the bruised and battered bartender, who was whimpering and cursing under his breath. He looked up, dazed, at Arnaud.\n\n\"We're looking for a man,\" said Arnaud in a firm voice. \"A man and a girl. He's called Garrecreux, a lecturer from the university. The girl is called Moonsinge, Madame Esme Moonsinge\u2014one of his old students. We hear they've been seen around here.\"\n\n\"Moonsinge,\" moaned the bartender. \"Yes, I know her. We all know her. When we thought Black Town could not get any more desperate and depraved, she appears like a typhoon and steals out friends and children.\"\n\n\"Where can we find her?\"\n\n\"Nobody knows. She is a phantom, an evil that lingers everywhere but can never be found....\"\n\n\"Stop talking in riddles, man!\" yelled Coyne, hoisting the bartender up roughly. \"Tell us what you know!\"\n\n\"That's all I know, I swear!\" he babbled, before slumping down momentarily and, bizarrely, starting to laugh. It started as a low chuckle, but had soon reached a crescendo of a booming guffaw that unsettled Arnaud far more than the baying drunkards that threatened them from the courtyard. \"But now, now...you fools! You won't find her, she will find you. And when she does, I pray that the cruelties she inflicts on your bodies and your minds are a thousand times worse than those she has inflicted on Black Town.\"\n\nCoyne released his grip and the man fell back against the rear wall, still chuckling in a low, threatening bass.\n\n\"I think we'd better get out of here,\" said Arnaud.\n\n9.\n\nIT HAD BEEN a long and terrifying crawl through a tunnel barely the width of Nathaniel's shoulders. The tremors had subsided and the main shaft had ceased its collapse perhaps twenty feet behind them\u2014Nathaniel could barely stop himself shaking with the adrenalin, and Tally seemed to be in a state of considerable shock. They got their breath back and, having no alternative, continued up the passage with a prayer on their lips that it would lead to some form of escape. As the route began to narrow, Nathaniel noticed the air around them beginning to freshen, and a thin trickle of water on the floor\u2014he even thought he could see the welcoming softness of moonlight ahead of them. The only question was whether the tunnel became too narrow to prevent their escape.... Soon they were on their haunches, then their knees, and finally face down, crawling with barely an inch between their heads and the ceiling of the claustrophobic passage.\n\n\"There's light, Tally! There's definitely light!\"\n\n\"Saints be praised,\" said Tally, without a hint of irony.\n\nThe possibility of finally escaping the dismal tunnel spurred Nathaniel on, despite his almost total exhaustion. There! He could taste it! Fresh air! Even the ice-cold stream of water that froze his fingertips and soaked his shirtfront seemed refreshingly pleasant. He crawled quicker, desperate, excited. Inch by slow inch an opening appeared in front of him, pale with the light of the moon, a small hole that maybe, just maybe, they could force themselves through....\n\nAs if acting as some final, cruel appendix to the terror and discomfort they had endured all night, the opening was not quite large enough. Nathaniel gashed his waist as he struggled out, gasping, desperate, fearful he might get stuck. With an almighty yank he pulled himself free and tumbled from the hole, filthy, bleeding and soaking wet. Every part of him ached, and all of his senses seem unnaturally keen and alert. He turned around and helped tug Tally out, the two men falling together as Tally popped from the hole like a cork from a bottle of champagne. For a moment the two of them lay on their backs, breathing heavily.\n\n\"Dublin air,\" said Tally eventually. \"Best in the feckin' world.\"\nChapter Nine\n\nThe Daughter of Yohana\n\n1.\n\nWHEN THE APACHE camp came in sight, Wapi deftly dismounted and began leading the horse, Annabelle still resting in the saddle. Once more she felt a curious mix of nostalgia and long-suppressed fears. It almost seemed as if nothing had changed for the Apaches\u2014the tents, the smells, even the atmosphere in the settlement, simultaneously so serious and so happy, were all exactly the same. Only, she reasoned, things were not the same. Not for her, not any more. She had seen distant worlds, befriended aliens and battled tyranny and cruelty, suffered great hardship...she had even lost a leg. And yet all that seemed so trifling compared to the biggest change in her life, the one that had happened only days ago, even if it felt like weeks. She was married now, she and George professing their undying love and commitment to each other, the biggest and best change in her life so far.\n\nAs they approached the outermost tepees of the camp, a pair of small figures causing a ruckus became discernible. Two children were fighting behind one of the tents. There was a clear winner, and as the combatants resolved themselves in the shimmering haze that rose from the arid desert floor, it was clear that that winner was a girl. She was perhaps eight years old and had striking ebony hair flowing to her waist, with thick, woven bands of leather wrapped around her chest and waist. She was mercilessly drubbing her opponent, a young boy, with a large stick. Though the blows were not so heavy as to break the skin, the boy had curled up into a foetal position and was wailing at her to stop. She did not seem overly keen on doing so, and it was only when she looked up and noticed the approach of Wapi's party that she did, instantly dropping her bludgeon and pelting across the dusty floor towards them, her attack on the unfortunate boy forgotten.\n\n\"Wapi!\" she squealed. She did not decrease her speed or hesitate as she dashed towards them, and when she got within a foot of Wapi she leapt nimbly up to wrap herself around his chest, nuzzling her face into his poncho. \"I was so worried about you!\"\n\n\"My journey was not without danger, little one, but I returned to you safely, just like I promised. But I see you have not kept your promise to me...\"\n\nThe girl looked up, coy and shame-faced.\n\n\"What promise, Wapi?\"\n\n\"The promise you made to behave, Cactus Flower. Who was that? Nenko? Why were you beating him?\"\n\n\"It was his fault, Wapi! I swear! He said I'd never be a warrior. He said he was going to marry me and then I'd have to cook for him and wash his smelly trousers. And I don't want to wash his smelly trousers so I had to teach him a lesson.\"\n\n\"Well I think he has learnt it, little one. And now you must apologise! Nenko is your friend.\"\n\n\"He's not my friend, he's stupid.\"\n\nWapi laughed and tousled her hair, setting her back down on the ground. It was only then that the girl noticed Annabelle, George and Bert. Beguiled by their strange dress, Annabelle's beauty and George's strong, handsome face, she looked down shyly at her bare feet. She beckoned for Wapi to lean down, and as he did so she reached up on tiptoe and whispered something in his ear. Once more, Wapi laughed warmly.\n\n\"These are friends of the Apache, Kai. They have come to help us restore peace to our land. You will meet them properly soon, when the time is right. But it is far more important now that we speak with Geronimo, and start our fight against the evil men that would awaken spirits to destroy us. Go, run ahead! Tell him we are here.\"\n\nThe girl nodded dutifully and scampered off towards the camp. Wapi turned around to look at Annabelle.\n\nShe was almost catatonic. Her eyes were wide and her face had blanched to a deathly white. She tried to swallow but her mouth was so dry she was incapable even of this simple, reflexive action. Delirious visions on Mars, memories stirred by an assault on Ceres, and the haunting face of a child stalking her cabin on Esmeralda... Her head felt light, her eyelids fluttered and her pupils shot up into her skull. She could just about feel her prosthetic leg unhook itself from the stirrup as she plunged towards the hard, hot ground.\n\n2.\n\nFOLKARD AND ENDERBY had scouted up closer to the Russian base, taking in its perimeter, the number of guards and the best route to get to the hangar undetected. Night had begun to crawl in like a living thing, and with it came the bright searchlights from the airships, swooping over the base like ominous eyes. A stealthy approach would be difficult, a point that Enderby and Folkard did not feel the necessity to share, both already knowing it far too keenly.\n\nThey returned to the bridge to find the men waiting on full alert. Not wasting any time, Enderby briefed them on the situation. The length of fence at the northern side of the base seemed closer to the main hangar than the rest, and so it was decided this was the best point of entry. Speed was of the essence, and though it was unlikely that the base would be less guarded at night, at least the darkness outside the walls would help mask their approach. Once more, the fickle weather had turned its cold shoulder on the mission. The sky had cleared, and in the absence of snow-heavy clouds a plethora of stars stood out starkly against the absolute darkness of the sky.\n\nIt took the men mere seconds to pack their supplies and ready themselves for the next stage of the expedition. They took the stairs inside the buttress, once more checking the horizon both ways to see if any Russian patrols were on the approach. All seemed clear, and keeping their heads down, the group of six skirted along the bridge and headed out into the deeper tundra. There was very little cover from trees and vegetation surrounding the Severnaya installation\u2014there was no need for concealment so far away from civilization, and it made sense that all approaches to the base were clear. Expansionist and antagonistic they may be, but the Russians had lost nothing of their guile. To underestimate them would be a fatal mistake.\n\nThey were a few hundred metres away from the walls, encroaching upon the west side, when an almighty clank reverberated around the landscape. Instantly the six men dropped to the floor, Enderby's agents with their carbines instantly ready and primed. Shouts, rushed activity and an alarm bell sounded out from the base, ringing across the desolate white plains. This was followed momentarily by the long, ominous droning of machinery, vast gears grinding together, the screech of metal on metal. The beetle of the huge black hangar was extending its carapace. It was opening.\n\nEven from this distance, the scale of the engineering was awe-inspiring. The curved roof began to flower, pushed open by monstrous iron joists that hinged on an oil-black wheel halfway down their length. As they opened further, an intricate mechanism of flywheels and cogs the size of millstones was revealed, slowly cranking the roof apart. And yet even the noise they made was not loud enough to drown out another sound\u2014an ungodly thundering that came from above their heads, throbbing like a heartbeat from the chest of a Titan. The men looked up.\n\nIt moved at such speed that it was upon them before they had a chance to even comprehend the magnitude of the thing\u2014and after the initial shock of the scale, there came to Folkard a greater terror...one of recognition.\n\nThe entire body of the beast, from the plates bolted to its hull to the chimney stacks that stretched from bow to stern, was coloured a matt black, and so at first discerning its true shape against the night sky was difficult. And yet as it passed ever closer the light from the stars and the moon's bright glare picked out details, reflecting dully from surfaces and gun turrets...a shape Folkard knew only too well, down to the very last bulkhead.\n\nSovereign!\n\nOnly this was no sleek, majestic British airship. It was more like Sovereign's warped, bastard twin. It was, Folkard estimated, perhaps twenty-five percent larger, and where Sovereign had been built with concessions to beauty and craft, this behemoth eschewed any pretence to grace. It had been built as nothing more than a weapon of war. Extra armour plates covered the entire body of the ship, giving a brutish, unstoppable look. Extra fins, struts and gantries criss-crossed the hull, seemingly grafted on at random, the design of a madman. Bulky protuberances emerged from the black behind the armour plates, their purpose mysterious and alarming, and Folkard counted at least a dozen extra guns at either side of the prow, sticking out like poisoned spines. Then there was the engine. It was easily twice the size of Sovereign's, and was constructed with the same haphazard alchemy that characterised the rest of the ship, covered in exposed gears and a row of pistons beating in and out of the body with a terrifying precision. A faint green glow emanated from within its housing, and in its wake it left a thick cloud of choking black dust that gave the air a scent of sulphur and coal, and something else... Something not of this Earth.\n\nAs it passed overhead the noise was overwhelming, forcing the men to cover their ears. The searchlights from the numerous dirigibles tethered to the hangar whipped up in unison to illuminate the warship and guide it in to dock, further showing the patchwork brutality of the Russian's secret weapon\u2014a truly awe-inspiring and terrifying sight.\n\nIt let out a long, low hoot like a bellowing Kraken and, as it did so, a flash of hope leapt into Folkard's heart. The ship was coming in too fast! He'd steered Sovereign long enough to know when an approach was miscalculated, and miscalculated this one surely was. At its current rate the bow would skew as it turned to descend into the hangar... The Tereshkov-patented propeller governor (Folkard had no doubt that this ship possessed one) would overcompensate, destabilizing the entire weight of the vessel and causing it to list uncontrollably. There was no doubt the ship would suffer a huge, catastrophic crash.\n\nHis breath held, Folkard hoped against hope that this mission would be over before its most dangerous segment began, that this machine that could change the destiny of the world would be written off as just another failed Russian experiment. And yet his hope was all in vain. In an almost breathtaking display of helmsmanship, the Russian warship's prow seemed to hang in the air as it hovered over the open hangar roof. The back end began to slide round with an ease and grace that belied the ship's monstrous appearance, and deftly stopped dead when the length of the vessel hung over the open bay doors. As light as a feather, with its engines purring and cooling, the dreadnought gently sank into its nest, and once more the air was filled with the gigantic doors clanking shut.\n\nAfter that, there was nothing but silence. Even the wind, it seemed, held its breath in fear.\n\n3.\n\nTHE DRIVER OF the cab had taken some persuading to take Nathaniel and Tally as fares. Considering their appearance and the lateness of the hour, this was hardly surprising, but it was nothing the promise of a few extra shillings couldn't solve. When they returned to Rathgar and let themselves into the garret, the sight of his lumpy old mattress almost made Tally weep with gratitude.\n\nWhen Nathaniel awoke, late the following afternoon, every muscle in his body roared with strain. He stretched experimentally, and despite a few bruises and the cut on his side (which turned out to be not as deep as he had feared) everything seemed to be in good working order. Tally, fully clothed, lay face down on his ratty bed, snoring at such a volume that it rattled the windows. Nathaniel got the kettle on and, just as it was coming up to the boil, Tally eased himself up with a groan and rubbed his eyes.\n\n\"Christ,\" he moaned, swinging his legs over the side of the bed. \"When I first opened me eyes I thought, 'Jayze, old Jenny must have put me through me paces last night'. Then you get to thinking it hurts a bit too much for that, and maybe I've gone a few rounds with Slugger O'Toole. It's only when I sets eyes on you it all comes back, and bugger me blind how I wish it hadn't.\"\n\n\"Well it hasn't seemed to affect your blarney, Tally. Here, drink this.\" Nathaniel handed him a steaming cup of tea.\n\n\"Right!\" said Tally eventually, draining the last of his mug and standing. \"Shouldn't take me long to get my stuff together and then we'll be on our way.\"\n\n\"Stuff?\" said Nathaniel, nonplussed. \"We?\"\n\nTally had turned his back to Nathaniel and was sorting through a pile of clothes heaped on the floor in the corner, occasionally tossing a choice article on the bed.\n\n\"Aye, the ferry back to England. I needn't pack much anyway, your boys have said they'll sort me out when we reach London. Digs and togs, all that. Part of the deal, they said.\"\n\n\"Deal?\" said Nathaniel, wondering if he'd somehow managed to doze off despite himself and wake in the middle of a totally different conversation. \"I'm sorry Tally, I have to confess I'm not entirely sure what you're talking about.\"\n\n\"Well what d'ya think all that business yesterday with yer man Cormac was all about? Sorting out me papers.\"\n\n\"Papers? You mean, passports and such?\"\n\n\"Aye, that's it.\"\n\n\"You mean to say you're coming back to London with me?\"\n\nTally stopped his rummaging and turned around slowly.\n\n\"You mean to say they didn't tell you I was coming back to London with you?\"\n\n\"Now that you mention it...no.\"\n\n\"Why those slimy, double-crossing English sons o' whores!\" exclaimed Tally, before adding; \"Present company excepted.\"\n\n\"Believe me, Tally, I'm tempted to echo the sentiment. What did the Bureau offer?\"\n\nTally sat back down on the edge of the bed and looked Nathaniel straight in the eyes.\n\n\"Now you're not foolin' me about here, are you, Professor? All on the level?\"\n\n\"Absolutely.\"\n\n\"I thought as much. You're a stand up chap, sure y'are, and it makes sense they'd mess you about as much as they have me. Okay. So here's the craic.\" He took a deep breath before continuing. \"Yer' a smart man, Professor, so you'll probably picked up that my line of work here in Dear Old Dublin isn't exactly what you'd call 'blessed with legality'. And that sort of employment, if you aren't careful, is wont to ruffle a few feathers here and there. And don't get me wrong, I love my mother country an' I want to see us free as much as the next man, but setting bombs and shooting soldiers ain't the way to go about it, I reckon. That's just gonna make things worse. And so, when I was given the chance, I may have, well...stopped a few bombs going off and a few soldiers getting shot. Needless to say, my countrymen weren't best pleased. I needed a way out before I ended up floating in the Liffey, arse pointed at the sky.\"\n\n\"And that's when the Bureau stepped in.\"\n\n\"Right so. They said they'd get it all sorted no bother, just had to help out one of their men with a job. That'd be you.\"\n\n\"As I gathered.\"\n\n\"Only it's me who has to go chasing around to get me papers, and naturally the Bureau contact over here vanishes when you need him most. Guess they hoped the Fenians would catch up with me or I'd just cark it down in that feckin' mine. God Bless England.\"\n\n\"Tally, I can only apologise...\"\n\n\"Sure, it ain't your fault, Professor. But I've got a few choice words for your employers, and not all of them would please your maiden aunt.\"\n\n\"Rest assured, Tally, they're not my employers. And it seems like I have a few choice words for them myself. I think it'll be much more satisfying if we both air our grievances together, don't you think?\"\n\n\"You mean you'll help me?\"\n\n\"Tally, after everything that happened last night, I feel I'm duty bound to.\"\n\nTally leapt up and pounced on Nathaniel, throwing his long arms around him and slapping him on the back. \"Ah, Jaysus,\" he said. \"Long-live the decent and honourable subjects of Her Majesty!\"\n\n\"And God Bless Ireland,\" said Nathaniel, gently pushing the smiling Irishman away.\n\n4.\n\nCOYNE, HIS OWN pistol drawn, was taking the lead, with Arnaud keeping a skittering eye on the rear. They had slunk into the back streets, every sense alert, moving through the shadows with no real plan, or sense of where they were going. Arnaud was only too aware that Madame Moonsinge posed a serious threat, and the grip she held on the slums meant that a reckoning was due. The barman had seemed terrified of her power and influence, and Arnaud had been adventuring long enough to trust his instincts when it came to the greater dangers. Not only that, Coyne was a loose cannon, and he felt himself keeping as much of an eye on the unstable agent as he did their surroundings.\n\nThey came to the corner of a derelict building and Coyne stopped. They had been lucky so far; with the exception of a few feral dogs and the odd pair of eyes peering suspiciously from the inside of ramshackle huts, the darkened byways had been deserted. Arnaud found little comfort in this, however, and the state of the back streets\u2014littered with human waste and the stench of poverty and decay\u2014made him all the more sickened and uneasy.\n\nHe sidled up to Coyne, who had taken a quick peek around the corner.\n\n\"What can you see?\" he asked.\n\n\"Trouble,\" said Coyne, as he brought the butt of his pistol viciously down onto Arnaud's temple.\n\n5.\n\nTHERE WERE MURMURS, the crackling of wood spitting in a fire...then a soothing coolness on her forehead that seemed to gently coax her senses back to full consciousness. She tried to move her arms up to rub her eyes but they felt light and unresponsive; it was only when she moved her head and heard a groan that her body felt like it belonged to her once more, and she was able to open her eyes. Even this basic task felt an uncommonly strenuous thing to achieve.\n\n\"Easy, Annabelle, easy....\" It was George's voice that filtered into her ears, hazy and indistinct, but it gave her something solid to concentrate on\u2014something to aim for in the fog that clouded her faculties. Gradually his face came into focus, as pale as a ghost and as fretful as she had ever seen him. He smiled in relief.\n\n\"You gave me quite a shock, Mrs Bedford,\" he said soothingly, rubbing her cheek softly with his palm. She tried to raise her torso but her head swam, so George curled his arm around her waist and helped her into a sitting position. \"Here,\" he said, offering her a clay cup. \"Drink this, they said it'd help.\" She curled both her hands around the proffered drink but it was really George who tipped the liquid gently into her mouth. It was mildly sweet, herbal, and as she swallowed it gave to her throat a pleasantly warm sensation. Already feeling rejuvenated, she drank a little more, and the restorative effects of the infusion coursed into her belly and her bloodstream.\n\n\"Thank you,\" she muttered, finding her voice but not her memory. \"What happened?\"\n\n\"You passed out... Dropped off the horse like a brick off a bridge and gave yourself a pretty good thump when you landed. Probably this blasted heat getting to you.\"\n\nAnnabelle didn't respond and looked around her. She was sitting on a mattress made of buffalo hide in a large tepee. A fire crackled gently in the centre, and dozens of totemic constructions of feathers, sticks and bone hung from the walls. The shaman's hut, she was sure. She looked down, and with anger and embarrassment found that her prosthetic leg had been removed and was laying a couple of feet to her left. Stiff and ungainly as it was, she had grown used to wearing it, and the weightlessness and uselessness of her stump irked and shamed her. She quickly grabbed for the leg and, not caring for decorum, hoisted up her skirt and began to strap it back into place.\n\n\"You shouldn't have taken it off,\" she hissed to George, not looking up.\n\n\"I'm sorry, I know, but the fall dislodged it and we were much more concerned in getting you to safety... I'm sorry, Annabelle. Truly.\"\n\nShe had chided him unnecessarily, and felt guilty. Once more he had made her safety paramount, and she needed to remember that this came not from any doubt in Annabelle's strength or ability, but from the love he felt for her so sharply in his heart.\n\nShe locked the leg into place and tightened the buckles until they were comfortable, then replaced her skirts to a more modest position. It was only then that she properly looked around the tent, and noticed that she and George were not the only occupants. Sitting across from them, at the opposite side of the fire, was Geronimo. He was looking at her with intelligent eyes the colour of obsidian that reflected the fire's dance. Dressed far more traditionally than Wapi or the other braves she had hitherto seen, he sat cross-legged on a raised bear pelt surrounded by carved poles of wood, like a throne. His face, framed by two braided lengths of black hair, was a roadmap of wisdom through age. It was wrinkled, pinched and sternly serious, the colour of a roasted nut with a strong jaw-line and forehead. Wapi stood at his right, and on his left the young girl, Kai, sidled up to him as a child would to her grandfather. She looked across to Annabelle with wide, apprehensive eyes.\n\n\"You have travelled far, Yohana,\" said Geronimo. His voice was low, but strong. \"At times I have been with you on your journeys, watching, fearing for your life as you have feared for your own. I was with you when you traversed the red planet, when you dug into the belly of the white lady in the sky, and fed and helped her children. I have gazed through your eyes at the stars, and seen our mother Earth from heights that the eagle can only dream of. I have been with you, Yohana, and I have prayed for your deliverance.\"\n\nWapi stepped forward.\n\n\"The great Geronimo is wise beyond the skill of many tribes,\" he said. \"But he cannot see the future. He simply knew you would come when you were needed most, as the rain comes when the earth is at its most thirsty.\"\n\n\"This need,\" said Annabelle, hoisting herself up. \"You mentioned that before. Spirits that must be left to slumber. What do you mean? What's going on?\"\n\nGeronimo nodded. \"As I had hoped, your spirit is drawn in the wake of your passing, and follows you here. It is this, and the pure hearts of your companions, that will aid us in cleansing our land. Wapi is a brave and noble warrior, and yet what he sees as evil spirits, you and I know, Yohana, in truth come from a far more solid plane. Our lands have been a waypoint for visitors since the lands themselves were young, and as a warrior's steed leaves its tracks in the sand, so too did these travellers from beyond leave their mark. It is these forgotten footprints that have brought evil men here, men who would exploit powers far beyond their understanding.\n\n\"We have passed down secret knowledge over countless generations, acted as the guardians of these forbidden places. Once they were worshipped, but always they are feared. Metal and money are the ways of nations now, not the plants and beasts and land, as our people believe. The balance must be restored, the hunger of this progress must be tempered, not satisfied, a hunger that has no care for the life that blesses the world. If we fail, a great cataclysm of fire will engulf all of mankind.\"\n\n\"I thought you said you couldn't see the future?\" commented George, before realising his interruption may have come across more accusatory than he intended.\n\nGeronimo looked at him stoically.\n\n\"To see the events of futures untold is not necessary, husband of Yohana. Merely to observe the actions of greedy and careless men is enough, for the consequences are always the same. Death. Misery. Slavery. Pain. You know this, or you would not take arms against them.\"\n\n\"Of course,\" mumbled George. \"I'm sorry.\"\n\nAnnabelle put an arm on his shoulder, and turned back to Geronimo.\n\n\"What are we to do?\" she asked.\n\n\"First, we must rest. The days ahead will punish the spirits and bodies of us all. We must focus, become one with ourselves, each other, our purpose. And you, Yohana. You must reconcile with your past, with the crimes that were imposed upon you when you were but a child.\"\n\nFor the first time, Geronimo moved, he looked down sadly into his lap before looking back up and meeting Annabelle's eyes. She could almost swear the fire reflected in them had become misty, dimmed by the tears that welled there...\n\n\"You must reconcile,\" he said quietly, gesturing to his right, \"with your daughter.\"\n\n6.\n\nTALLY AND NATHANIEL packed quickly. There was no point wasting any more time in Dublin, and Nathaniel was keen to return to the Admiralty so he could make his report and begin his investigation on the chunk of crystal he had rescued from the caverns under Phoenix Park. The city itself was in a state of panic. The news of the bombing was on the tongue of every Dubliner, discussed in loud voices in pubs, in hushed tones in market corners and yelled in newsprint in every paper from Howth to Tallaght. There was shock and consternation, fear and guilt, the city boiling over into a maelstrom of accusation and counter-accusation. Many believed it was a British plot to break the spirit of the Fenians, a show of fearful strength and a promise of retribution. Others believed the Fenians themselves were responsible, experimenting with new explosives and destroying a large chunk of the city in the process. There was political turmoil at the highest levels and a tightrope tension in the streets. Now, where a resting place for lovers and a playground for children had once stood, there only remained a vast sinkhole. Charred debris and twisted metal poked evilly out of the ruined land, mixed with the splinters of uprooted trees and mangled hunks of exotic animal flesh.\n\nThe city was swollen with fear and mistrust, and it seemed that if you did not appraise every face in the street with a look of suspicion then you, too, were somehow suspect. Not that it was even that necessary\u2014despite the weather, which was once more balmy and promised glorious sunshine later\u2014the streets were pretty much deserted. This didn't stop curtains from twitching when Tally and Nathaniel walked past them, and neither of the pair dared to look back inside. Tally, demonstrating admirable sense and forward-thinking, lent Nathaniel a shirt, waistcoat and one of his old, grubbier overcoats to replace his battered and conspicuously British suit. Nathaniel was also conscious that his accent might not be particularly well-received, so as they walked, they walked in silence. This seemed to pair well with the sombre mood of the eerily desolate town.\n\nThey boarded the tram at Westland Row and it trundled sluggishly past the famous Martello Towers on the coastline, on towards the port near Ringsend. As Nathaniel had suspected, there was a distinct increase in the number of security measures for those wishing to leave the Emerald Isle. Passports were scrutinised far more closely, eyes were met and held by the squinting, suspicious gaze of officials. Tally did his best not to look shifty; he was sure that the papers he had acquired with Nathaniel's help were legitimate, but he had been betrayed so many times that any underground deal he'd successfully completed always came with that faint whiff of panic. Normally he'd have enjoyed it\u2014but today was a very different day.\n\nThe staunch and humourless official eventually waved them through with an irate flick of the hand, and only then were the two men able to breathe again. They boarded the ferry with a minimum of fuss, and as the great boat's engine's hummed into life Tally sauntered to the stern and leant on the aft railing, watching as he sailed away from the city he called his home.\n\nNathaniel followed him, and leaned next to him in silence. The Irishman was gazing landward, smiling sadly. He took a packet of cigarettes from his pocket, pulled one out and, thanks to the wind, lit it with some difficulty. He looked sideways to Nathaniel, curled the corners of his lips, and looked back to the city with a nod.\n\n\"Dear Old Dirty Dublin,\" he said, raising an imaginary glass. \"Be seein' ya.\"\n\n\"This was,\" said Nathaniel, somewhat unsure, \"your home...\"\n\n\"What kind of feckin' stupid question is that? You worry me sometimes, Nath. Yeah, sure it was my home. Born an' bred. And this might be the last I see of it. Say,\" said Tally, turning to Nathaniel with a smile, \"sure I'll be fine. The streets of London are paved with gold! All I needs do is pull up a cobblestone or two. Right?\" He looked across to Nathaniel, who was staring back at the land with an eerie intensity.\n\n\"Right...?\"\n\n7.\n\nTHEY BOTH MOVED slowly around the tepee, shuffling wolfskin rugs, making the place their own as a new wife would rearrange her living room. The weight of the day, and the news it had brought with it, was upon them. For Annabelle, years of denial had been replaced by acceptance and anger, emotions that tugged her in opposite directions. George was stunned, and moved around their home for the night as if in a daydream, barely able to comprehend the truths that had been told.\n\nDid he really now have a daughter?\n\nBoth were thinking of the sorry tale Geronimo had told, the history that Annabelle had held, hard-pressed against the back of her mind, for eight long years. When the Apaches had murdered Annabelle's parents and kidnapped her, it was not long before the young girl's spirit had begun to show itself. She was feisty, adventurous, never content to be a victim and always keen to be a part of the wild world that had captured her. When told \"no\", she would always respond with \"why?\" When forbidden to do something, she would always go out of her way to try it anyway when elder eyes weren't watching. She'd practise with bows or rifles she'd stolen from the secret cache after dark, or followed the braves out to hunt while the elder women stayed around the fire to weave.\n\nSuch fortitude was noted and respected, even if it was never talked about openly. Even the squaws who would harry and chide her for her precociousness would smile with pride when they thought she wasn't looking.\n\nAs she had grown, and grown beautiful, she was accepted as no other outsider had been\u2014totally and with absolute trust. She was known to the women as feminine and wise, and the men knew her as a warrior with a heart of fire. Yet, in such a close-knit tribal group, there were always those who would take more than they had earned, who would want more than they were offered...\n\nSuch a man was Deerpak. Even as a child he had a darkness about him, spending too much time on his own and rarely talking to his peers. Where Annabelle's rebellious nature had endeared her to the rest of the tribe, both young and old, male and female, Deerpak's natural status of outsider had been sly, quiet and suspicious. Annabelle would befriend young coyotes and tousle with them playfully; when he thought no-one was looking, Deerpak would kick the pups around the dust, or pierce scorpions into the ground with an arrowhead and pull off their legs, one-by-one. He was nearly three years older than Annabelle, and as manhood overtook him, so did jealousy and lust.\n\nAnnabelle did not care to remember that night when he found her, alone, still a girl, scratching pictures in the dirt with a stick. How, far beyond the sight of those who would have stopped it, he forced himself upon her, the smell of his teenage sweat and the pain and the bleeding. How, after he had satisfied himself, he had got up and walked away as if nothing had occurred, leaving her to make the red dust of the desert black with the stain of her tears. How she feared to tell the elders or the women; how she cried at night when her lunar cycle ceased and her belly began to swell with a hated, invading seed.\n\nWhen the women of the tribe knew what was happening, they stood together as one. They took Annabelle, still too young to understand, and quizzed her ceaselessly. The truth was soon discovered.\n\nHe tried to escape. Cunning came easily to him, and when his crime had been discovered he knew his time with the tribe was at an end. At night, he stole a horse and supplies, still young, still stupid, for the noble men knew he would attempt to flee. They were ready, and to spare the noises carrying back to the camp, they let him get far enough away before they caught him.\n\nIt was Wapi, Annabelle's childhood friend and confidant, who had insisted on exacting the whole tribe's retribution. He was fourteen, and it was the first time he had killed a man. Fittingly, after Geronimo had told this sad and sorry tale, it was Wapi who held out to Annabelle a package wrapped in leather; a package containing the scalp of the man who had attacked her. She took it without speaking, and held it loosely in her hands.\n\nThat same package sat beside the fire of their tent, unbidden, unwanted. A sore, sorry point. Annabelle breathed in deeply. She looked at George, still tending the fire. He sensed her unease, and looked back.\n\n\"Annabelle,\" he said.\n\n\"George...,\" she replied.\n\nIt was as much as they could manage. She tried to smile.\n\n\"I'm so sorry,\" he said.\n\nShe sighed, uncomfortable. What could she say? She loved him deeply, and knew that he loved her. Wasn't that enough? She wanted it to be, wished that that was enough for him...but how would he see her now? She had told only one person of the incident before, and even then it had not been the complete truth. But George, he deserved the whole truth.\n\n\"Annabelle,\" he said. \"I love you so much....\"\n\n\"And I you, George....\"\n\n\"No, stop.\" He said. \"Do you think you have to make excuses, to explain yourself?\" His voice was terse with anger, directed, she knew, only towards himself. \"You are the woman I love,\" he continued. \"The woman who I have loved absolutely from the moment I set eyes upon you. To know that that woman I fell for had been through so much, endured so much pain and hardship.... It only makes me love you all the more.\"\n\nHe breathed in, deeply.\n\n\"I love you for your strength, your beauty, for everything that has made you who you are. I love you Annabelle, and am proud to be your husband. I am yours...\" He faltered. Looked about, momentarily frustrated, then up to her.\n\n\"And that's all that I can say.\"\n\n\"Oh, George...\"\n\nShe skipped up, suddenly happy, and trotted across to him. Resolute as he was, he smiled as she wrapped her arms around him and kissed him on the mouth.\n\n\"We're still on our honeymoon,\" she teased, and as she arced herself back she found her waist slipping easily into his welcoming arms. \"And you and I in a tepee? Bearskin rugs, an open fire...\" She looked him in the eye and smiled, teasing and content. He kissed her.\n\n\"Promises promises,\" he said.\n\nShe leaned back, happy as he held her in his arms and letting out a slow groan of pleasure.\n\n\"I need to disrobe,\" she said glibly, pulling away. \"Just do one thing for me.\"\n\n\"Anything.\"\n\nShe looked towards the parcel at the base of the fire.\n\n\"That,\" she said, suddenly cold. \"I want it gone. Promise me, George. Take it out and bury it. Leave it to rot in the desert. Will you do that for me, husband?\"\n\nHe smiled at her. \"Of course, my love.\"\n\nShe smiled back, coquettishly.\n\n\"And I promise that the sooner you get back, the sooner I'll be here waiting for you. And fur does feel so good against bare skin...\"\n\nShe pecked him on the lips, and skipped back happily to their bed. She grinned at him; he grinned at her. He turned. Casually, he bent down in his stride and picked up the leather package is if it were nothing more than a piece of rubbish.\n\n\"Back soon,\" he said, as he flipped the tent's entrance aside.\n\n\"I'll be waiting,\" said Annabelle, beginning to undress.\n\n8.\n\nBEDFORD LOOKED UP. The Arizona night sky was like nothing he had ever seen before. No clouds, no light, and nothing but a thousand stars to dazzle his eyes. He was far from the light of London, and glad for it, too. He breathed in the air, clean and cold. Breathed out. Looked to the package that he held firmly in his left hand.\n\nAnnabelle had not unwrapped it. But he wanted to. He wanted to see the dead skin of the man who had wronged her. He did so, slowly, surprised by how human it felt when his fingers touched human hair, and the thickness of the skin that once hugged so close to a living scalp... He held it, for a moment, and felt pleasure. The woman he loved had been avenged, and no amount of stars in the blackest of skies could bring light to the soul of the man who had hurt her. He would always be damned, that man. And he didn't deserve a burial. Such was the way of their people, Bedford knew. He slipped the scalp into his pocket as a keepsake. Nobody would hurt his Annabelle like that again. He promised.\nChapter Ten\n\nJacob's Ladder\n\n1.\n\nFOLKARD FELT DIZZY, but knew he needed to press on. The sight of the Russian aether battleship had engendered a steely determination in his men\u2014now they knew the true scale of their enemy's plans the importance of success was brought into sharp relief, as were the terrible odds they faced. It was a mutual feeling that linked them, one that did not need to be voiced lest the enormity of their mission overwhelm them. Still, what Folkard wouldn't give for a battalion of heavily armed men, artillery and air support...\n\nHis mind was racing with possibilities as they sneaked up towards the fence. They got as close as they dared and stopped to form a tight-knit circle. Enderby nodded to him to give the men a final briefing and pep talk before proceeding. Folkard nodded back, and looked to each of the men in turn. They were strong, resolute, prepared. None let fear show in their faces.\n\n\"Men,\" he began, \"the display of the Russian's new technology we just witnessed was a shocking one, and only serves to demonstrate just how important our mission is\u2014though I hardly have to say that. In light of this discovery, our primary mission objectives are as follows: Number one, sabotage. This new weapon represents a threat to the security and safety of not only our home nation, but potentially the entire world. If possible it must be destroyed, or put out of commission as much as is possible with the resources we have at hand.\n\n\"Number two: Intelligence gathering. As you have no doubt been made aware, this new ship has been built using the Sovereign's blueprints, and I need not point out the similarities between this new vessel and the pride of Her Majesty's fleet\u2014you have seen that well enough yourselves. It is of paramount importance we discover how the Russians managed to obtain these secret plans, as well as how their version differs from our own in terms of build and capability. In short, keep your eyes peeled and grab any files or folders, however irrelevant they may seem. At present, I'd be happy to acquire the cleaning rota if it gives us even the smallest advantage.\"\n\nThe men smiled ruefully, glad for this burst of irreverence in an otherwise gloomy monologue.\n\n\"In short,\" continued Folkard, \"keep your wits about you and we'll all see England again. An England which, God willing, will continue to be the glorious seat of the Empire and the pride of the seven continents that we know. Our efforts today depend upon it.\"\n\nThe men nodded solemnly. \"Anything to add, Mister Enderby?\"\n\nEnderby shook his head. \"I believe you have covered everything, Captain. Eloquently and poignantly, I might add.\"\n\n\"Thank you. Let's just hope our actions reflect our sentiments.\"\n\nThe men quickly checked their weapons and had a last look through their kit. The plan was to employ stealth as much as possible, get as close to the hanger\u2014if not the ship itself\u2014without being detected, though the likelihood of this was diminished greatly by the scores of patrols that paraded around the base. They were to split into two groups, Folkard leading the first and Enderby the second, and should one team be discovered the other was to follow the sound of gunfire and hopefully get the jump on their assailants.\n\nA large, nondescript building sat close to the nearest section of fence, and would offer them some small protection from the Russian's line of sight. They hurried across, and Enderby quickly and neatly snipped out a hatchway using a heavy pair of pliers. He yanked up the section he had cut free and ushered the men through, Folkard last.\n\nThey pressed their backs firmly to the wall of the closest building, ever vigilant. Folkard's team was to set off first, heading clockwise around the base's perimeter towards the huge hangar, with Enderby and his group waiting five minutes before setting off counter-clockwise.\n\n\"Good luck, Folkard,\" nodded Enderby.\n\nFolkard returned the gesture.\n\n\"You too. See you at the hangar.\"\n\nFolkard ducked down and began to lead his men into the dark and the depths of the base. Cover was minimal. This was a typically sparse Russian set-up; besides the looming shadow of the hangar and the several windowless, redbrick buildings that surrounded it, the ground was mostly bare. Watchtowers peaked at the base's four corners, with two or three others built inside, their searchlights sweeping endlessly across the bare, snowy floor. Folkard feared their presence would not remain undetected for long. The prevalence of the patrols and the fact that each man had to dash between the outhouses separately meant their journey to the airship's nest was taking far longer than Folkard felt comfortable with.\n\nThey had progressed maybe fifty metres in. Folkard and one of his men were flattened against a building, looking back nervously to where the third of their party was crouched at the corner of the previous hut. He was waiting for a chance to make a break for it. Two soldiers had stopped directly opposite where he hid, laughing idly and passing a lit cigarette between them. Folkard watched them and swallowed drily, sweating in spite of the cold. They smoked to the very nub before tossing the end into the snowdrifts; one of the soldiers gazed lazily about and Folkard unconsciously pressed himself harder against the wall. The soldier who had tossed the butt gestured to his companion to wait for a moment. Folkard caught a snatch of his words, as he vulgarly detailed the satisfaction he would feel after relieving himself behind the building... The very building where Folkard's other man was hiding. Folkard caught his eye, gestured quickly that an enemy soldier was approaching. The agent nodded his understanding and readied himself, slowly drawing a knife from the sheath strapped to his thigh.\n\nThe Russian whistled tunelessly as he ambled towards his death without a care in the world, condensation jetting from his pursed lips in time with the noise. Folkard almost felt sorry for him\u2014he couldn't have been more than seventeen, and hadn't even managed to grow the beard that was so necessary for such a desolate outpost. The boy must've been so cold...\n\nBefore he had even rounded the corner the British agent's arm had whipped up quickly to choke him. The Russian didn't even have time to cry out before the black-bladed knife was thrust up viciously into the soft underside of his jaw. Blood began to flow from the wound, soaking the snow and the agent's arm... The young soldier began to spasm, his leg kicking out wildly, before he slumped down, a lifeless rag doll. The agent laid him gently on the ground and peeked out to where the other soldier waited, signs of impatience already apparent.\n\n\"Sergei!\" he called back. \"Ty, chto tak dolgo?\" he emitted an annoyed grunt and tramped across to the building. \"Ya zamerzaiu, perestan\u02b9 tratit\u02b9 moye vremya...\" The agent was ready again. He leapt up and grabbed the man as he rounded the corner, but this was no boy soldier. The Russian was big, fifteen stone if he was an ounce and covered in muscle. He struggled against the agent's grasp, sweating and swearing, and with a strong, deft swipe of his paw he managed to deflect the knife before it connected with his throat. He reached back and twisted the agent round, fumbling with his right hand for the pistol in his holster. This brief movement was all the agent needed, and in a split second he brought the knife to bear once more and plunged it into the Russian soldier's side. The Russian fell back, the hilt of the blade sticking out of him. As he crashed to the ground he yanked his gun up, grimaced in pain, and fired. The shot was like a thunder-crack, the eerie silence of the base shattered in an instant.\n\nThe bullet caught the agent in the right side of his face, and once where there was a brave Englishman's smile, there now only remained a crater of splintered bone and sinew. A livid spray of blood, the shape of a rooster's tail, coated the wall behind him. The Russian groaned and rolled over, trying to yank the knife from his belly, but before he could even compose himself Folkard had dashed back and dispatched the man by plunging his knife into his chest.\n\nThere was no time to hide the bodies\u2014he could already hear yelled Russian commands and the heavy clump of boots running towards his position. Folkard looked back and caught the eye of his remaining compatriot.\n\nThere was nothing to do now but try to get to the hangar as quickly as possible. The need for secrecy was long gone, and death was in the air.\n\n2.\n\nOF COURSE HE was tied up. The blitzing headache he could cope with; numberless mornings after too much brandy had seen to that. What really irritated him was the ponderous predictably of it all. Still, he's never been knocked unconscious in Asia before, so that was something.\n\nHe realised he was blindfolded, too. Of course! Naturally, when the blindfold was removed, he would find himself being threatened with some weapon or other. Plus \u00e7a change, plus c'est la m\u00eame chose. He mentally shrugged to himself.\n\nAnd as for that putin Coyne, well... If Arnaud was lucky he'd be stuck wherever he was long enough to think of some particularly grim way to avenge himself. Something ungentlemanly that, at the very least, would begin with a dashed good thrashing at Arnaud's hands. He might not have been a trained spy but he'd spilled enough acids and corrosive chemicals on his hands during his years as a geologist to put up with a bit of pain. As well as that, the anger he felt over his own stupidity\u2014and how Coyne's backstabbing may well have put his friends' lives in danger\u2014had imbued in him a torrential fury that would only be sated by giving the blackguard the drubbing of his measly life. And Nathaniel... he prayed with a sincerity that came from a deeper place that Nathaniel was safe.\n\nStill, the revenge fantasies could wait. More pressing was deducing the situation he faced, however much that was possible. He mused sardonically on the fact that, even though he had been denied the power of sight by his captors, the cretins has not bothered to stop his ears. As smart a man as Arnaud could glean much just by keeping his mouth shut, and listening well.\n\nThe low clank of machines\u2014no, tools, in the background. Subtle movements and low footsteps, a few metres directly in front of him. Someone padding around, so the floor was soft. Carpeted perhaps? Closer to him came another chink, glass on glass, but not the sound of drinks. It was a sound he knew equally well, test tubes and conical flasks, the mixing of molecules and formulas. The smell pretty much confirmed this, eggy sulphates and bitter acidic tangs, the familiar fug of the lab.\n\nHe also felt warm\u2014not the blazing, oppressive heat of the unyielding Calcutta sun but a pleasant, homely warmth. Interesting. Whosoever had captured him was at least rich enough, intelligent enough or both to keep the temperature at a tolerable level.\n\nThen came another sound\u2014a match being struck, the long slow suck of someone inhaling heavily on what sounded like a pipe. But there was a desperation in the draw inwards, a neediness quite unlike the gent-about-town puffing contentedly on a cheroot. Arnaud's nostrils were suddenly filled with a dense, herbal aroma that caught the back of his throat with a harsh, chemical haze. He held his breath, desperate not to cough.\n\n\"Oh darling, do you have to?\" purred a female voice, sultry and seductive, impossibly cool and controlled. It was tinged with the delicate lilt of an Indian accent.\n\n\"Yes!\" a European male voice barked back. \"Yes! Always! Now he's here... Oh, my angel, my queen, but I didn't mean to shout...\"\n\nIt was the voice of Garrecreux. Older, cracked and wheezing, yes, but the nasal drone and Gascon twang were unmistakable. To hear it after so long, after so many years... It was strange. Arnaud had no respect for the man, even less so after his obsequious fawning to the woman, whose identity Arnaud was sure he knew. But Garrecreux disgusted him. Without thinking, the bitter memories and disgust made him sigh, and he breathed in heavily. The acrid sting of the unusual smoke caught the back of his throat once more and, powerless to stop, he coughed loudly.\n\n\"Oh tish, Arnaud,\" purred the woman's voice, suddenly close to his ear. \"And you managed to last so long...\"\n\nThe blindfold was whipped from his head so quickly that even the gloom of the laboratory made him blink. After a moment, when his eyes adjusted and he was able to focus, he found himself unexpectedly agreeing with Garrecreux.\n\nHe was looking into the face of an angel.\n\nShe was, thought Arnaud, quite simply the most exquisite-looking woman he had ever laid eyes upon.\n\nHer face was the shape of a teardrop-cut jewel, wide at the top and gently tapering down to the rounded tip of her chin. Her eyes were open and welcoming, a deep velvet brown, her nose a button in perfect proportion. She had flawless skin the shade of lightened mahogany and her hair was so dark and heavily lustrous that it seemed not a single strand could be defined... It was only the cruel smirk that twisted her full lips that betrayed the malice infesting her veins.\n\nShe was wearing a traditional Indian sari modified to accentuate her curves. It displayed her status as half-siren, half-Amazon, a golden breastplate studded with lines of rivets, curved from her shoulder around her chest, with another piece sweeping gracefully across her thighs. The bright, exotic oranges and purples of the material shone through the gaps in her armour, stark and beautiful in contrast. Her smile twisted again, her eyes narrowed and she leaned in to kiss Arnaud fully and softly on the lips.\n\n\"I hope you like that smell, my love,\" she whispered breathlessly, \"for you're going to have to get used to it...\"\n\nAs she pulled from the kiss there was an agonised groan from behind her. She turned gracefully, and as her hips sashayed seductively she revealed the pathetic sight of Fabrice Garrecreux cowering behind her.\n\nIf Madame Moonsinge was a paradigm of beauty, Garrecreux's state represented the very basest level of degradation and weakness. He had aged terribly, his skin raggedy and sallow, covered in nicks, bruises and liver spots. Thin, yellowing hair receded from his balding scalp, fronds and snatches of it sticking out piecemeal from behind his ears. His lips were thin with patches of dried spittle in the corners. He stooped like a hunchback, wearing a long, buttoned-up double-breasted lab coat with a high Nehru collar. As Arnaud had heard, a pair of thick, tinted welding goggles were strapped to his face with a thick, studded leather band. Even from here, Arnaud could see the skin around the lenses was sore, weeping and corrupted. It was he that had been smoking the upright wooden pipe that was the source of the noxious odour, and he placed it next to him on the lab and hopped manically across to where Arnaud was tied.\n\n\"Arnaud, Arnaud!\" he jittered excitedly, \"You're here, you're here at last. Here to help me, you bright, bright boy!\"\n\nArnaud spat to his right.\n\n\"Go to hell, Garrecreux. If you had any decency left in your rotten old corpse, you'd be helping me.\"\n\n\"Oh yes, oh yes!\" Here the madman giggled in a shrill, high-pitched waver. \"I heard, I heard about your plan! Oh, I can tell you everything you need to know about our wonderful discovery. Bright boy, bright boy! And what a joyous coincidence it was to discover you were coming for me, when for so long I have dreamed, have dreamed of employing your mind to assist me in my aims! And they brought you to me, they brought you, just as I asked! You know what that means, Arnaud? She's happy with me! Happy! She wants me to have the things I most desire!\"\n\n\"You're deluded, Garrecreux. A drug-addled, degenerate wreck. I'll never help you, you or your vile succubus.\"\n\nMadame Moonsinge spun around in mock surprise.\n\n\"Succubus! I like that, the boy's got spirit. What a pretty little plaything he'll make.\" She gestured to Garrecreux, who bowed and wrung his hands before skipping across to a bench. With his back to Arnaud, he began to unpack a small, wooden box.\n\n\"And as for your assistance, my exquisite little pet, I do not recall offering you a choice. You'll soon see, Arnaud. Beautiful Arnaud. We'll take you to such places that you'll beg to help... You'll beg to have me kick you in the dust.\"\n\nGarrecreux appeared behind her, brandishing a large, blunt-ended glass syringe that was filled with an iridescent, pale green solution that glimmered as it caught the light. Arnaud began to struggle against the ropes, but they were too tight. His right arm was bared to the elbow, the crook and its vein open and exposed... Madame Moonsinge leaned in again, and nuzzled against his ear.\n\n\"Oh my darling,\" she breathed, her hot breath contrasting with the coldness of the needle as it pierced his skin. \"The things you'll see...\"\n\n3.\n\nFOLKARD AND HIS remaining agent were dashing full-pelt towards the hangar. Commotion followed them. Folkard guessed that the discovery of the three bodies they'd left behind was still occupying the majority of the guards, and he was glad that this, at least, had bought them some time. But was it time enough? Slapdash and confused as they appeared to be, the Russians would quickly focus their forces on guarding the hangar...and that's when things would get tricky.\n\nThe hangar was directly before them, and Folkard and his man skidded to a halt behind a building directly opposite the door at the front. Folkard cursed. They were quicker than he anticipated, and the large doors they had to enter to reach the Russian's secret weapon were now surrounded by perhaps a dozen guards, each one scanning their surroundings with rifles held ready at their shoulders. As each did a sweep they would call to the others; \"Nyet... Nyet... Nyet...\"\n\n\"Blast,\" hissed Folkard. \"Outmanoeuvred.\"\n\n\"If I might suggest something, sir?\"\n\nFolkard looked at his companion-in-danger\u2014what was his name again? He had been told both his and the other agent's name during their briefing at the Admiralty, but for the life of him he couldn't remember it. The last emotion Folkard would have thought he'd be experiencing at this time was embarrassment, but there it was, mixed in amongst the dread, adrenaline and, most startling of all, a sudden memory of Charlotte.\n\n\"What's your idea?\" he asked.\n\n\"Distraction, sir. I could pelt it back to that building there. I'm quick enough on my heels and it's not all that far. Should be able to make it before they get too many shots off. If we're lucky it should draw a few of them away.\"\n\nFolkard couldn't argue with the logic, however risky it sounded. Yet he knew that if he agreed, the agent's chances of making it out alive were slim to none. He had ordered men to their deaths before, but before he had always known that there was a chance of survival if they kept their wits about them and had fate on their side. This time, however, the chances of this man seeing another dawn were so slim as to be negligible. Embarrassingly so\u2014there was that alien emotion again.\n\n\"I know what you're thinking, sir. But I've seen that monstrosity the Russians have cobbled together and if anyone can bring the bastard down, it's you. I owe you every chance. Countless lives and the safety of nations depend upon it.\"\n\nFolkard swallowed. The agent's brave and noble words cut him to the quick\u2014yes, the security of the world was at stake, and innocents would surely suffer if the Russian battleship wasn't somehow disabled... But he knew. He knew from the moment he set eyes on it that this ship was the key to him returning to the Heart, and to Charlotte. The two were linked in some strange, ethereal manner that he could sense at the back of his mind, a yawning tug that rose to the forefront of all his senses.\n\n\"What's your name, agent?\" he asked quietly.\n\n\"Fleming, Sir. Lieutenant Robert Fleming. Friends call me Bobby.\"\n\n\"Well, Bobby,\" and Folkard clasped his hand to Lieutenant Fleming's shoulder. \"You're a braver man than I. Godspeed.\"\n\n\"Thank you, sir. Tell 'em I died well.\"\n\nNothing more needed to be said. The two men locked eyes for a moment, and Folkard nodded. Lieutenant Fleming turned to make his way to the other end of the building when a terrible screech and the clanking of gears filled the air around them. The great roof of the hangar was slowly opening again...\n\n\"They're trying to get away before we find out too much! Go, man! Go!\"\n\nFleming nodded and sprinted to the other end of the wall. He paused for a moment, took a breath to ready himself, and ran out into the full sight of the Russian soldiers.\n\nTheir yells intermingled with the deafening roar of machinery, the soldiers began firing as Fleming sprinted away from them, a savage volley of bullets in his wake. Folkard took a quick look back\u2014it seemed as if the brave Lieutenant was going to make it to the next hut! As he disappeared around the corner, eight of the guards took up pursuit\u2014a far better number than Folkard could've hoped for. Taking out the remaining four would be difficult, but not impossible. He unholstered his pistol. Time was of the essence.\n\nWithout thinking he whipped round the corner and brought the gun to bear. His first two shots downed the guard closest to him with ease, the next winged the guard standing next to him. The other two Russians were quick to ready their rifles and returned fire, pinning Folkard to the wall as the shots gouged chunks out of the brickwork inches from his face. The shots stopped and Folkard leaned out again\u2014the soldier he had winged was on the ground, his trouser leg soaked in blood, but he had nevertheless managed to prop himself up and was assisting his comrades with the shooting. Unfortunately for him, his inability to move made him an easy target and Folkard put him down with a single shot to the forehead.\n\nThen came the second volley and Folkard hid again, taking the opportunity to reload. He couldn't keep this up. Slow as the roof of the hangar was to open, it wouldn't be long before the airship ascended and his mission\u2014as well as his striving to reconcile with Charlotte\u2014would be for naught.\n\nHe tried to return fire but to do so in that interminable hail of gunfire would be suicide. He gritted his teeth, frustrated. He would not let it end like this! A wild notion overtook him, the madness sparking in his brain, to just run towards the guards, pistol blazing, screaming out his lungs... He took a deep breath, and yelled.\n\nAll six of his bullets were spent in but a second or two, and all went wide. He screamed at the top of his voice, wide eyed and bestial, knowing he would tear the throats from the Russian dogs with his teeth, even if he had to do so with a slug of lead in his gut.\n\nAnd yet the Russians were otherwise engaged! One of them suddenly jerked back, the top of his head blown clean off, and almost simultaneously the other was knocked off his feet by two fatal shots to the chest. Still dashing towards the hangar door, Folkard looked across to see Enderby and his two agents running in the exact same direction.\n\n\"Folkard!\" he yelled, \"Quickly man, go! It's nearly open!\"\n\nFolkard redoubled his efforts, the muscles in his legs starting to ache. He reached the door and dashed inside, taking in the sight of the ship once more.\n\nUp close it seemed even more brutal, its stark paint work and sheer scale intimidating and awe-inspiring in equal measure. He noticed on the prow, written in blood red, stencilled Cyrillic, the word \u0418\u043c\u043f\u0435\u0440\u0430\u0442\u043e\u0440\u044a. Imperator\u2014the Tsar's self-proclaimed title. Underneath this, wrapped around the claws of a golden depiction of the Russian Imperial Eagle, was the rest of the ship's name\u2014\u0410\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440 II. The full title of this monster was revealed.\n\nImperator Aleksandr II. The rest of the hangar seemed deserted. A huge pit stood in the floor to accommodate the length of the battleship's aether propeller governor, the belly of the ship resting on huge struts drilled at its edges. The walls were taken up with various pieces of bulky machinery, steel lockers and tools for maintaining the ship. Much of the upper reaches of the vast room were taken up with the gigantic engines which cranked the roof open, and even now Folkard could hear the deafening thrum of Imperator's engine cranking up to speed.\n\nHe quickly looked for the easiest way to get aboard. A gangplank still stood on the right-hand side of the vessel\u2014hardly a sensible procedure, but he guessed the Russians were too keen to get Imperator away as quickly as possible\u2014and so Folkard sprinted towards it.\n\nEnderby appeared at the doorway and quickly took stock of the room. He noticed where the captain was headed and, wishing them luck, instructed his two men to stay by the door and fend off the guards pursuing them. The number of men the British had managed to sneak onto the base was as yet unknown to the Russians, and his men were well-trained when it came to resisting the methods the Russians used to force their prisoners to talk. The Russians would eventually believe the mission had been a total failure\u2014with two captured British agents, and two dead\u2014and any incursion on the Russian battleship would go unnoticed until it was too late.\n\nFolkard sprinted up the gangway. A Russian guard stood at the top before an open bulkhead door and, taken completely by surprise as Folkard raced towards him, fumbled inexpertly with his rifle. Folkard reached him and instantly knocked the man unconscious with a swift, heavy punch to the jaw.\n\n\"Folkard!\"\n\nThe airship began to rise! Folkard turned around to see Enderby flailing with the gangplank. The slight buffeting of the ship had dislodged it and it began to slide away, would leave the scrabbling Enderby stranded...\n\n\"God damn it, man! Help me!\"\n\nHe reacted quickly, grabbing the end of the gangplank firmly and resting his weight there to keep it steady. Enderby launched himself up on all fours. He rested his hands on his knees as he got his breath back, the gangplank clattering to the floor as the ship began its ascent.\n\n\"Welcoming committee seen too, I take it?\" said Enderby, nodding at the unconscious Russian.\n\n\"Not quite. Help me get his clothes off. We'll tip him over the side when we're over open territory.\"\n\nThe two men set to work as Imperator drifted upwards into the chilling night sky over Severnaya.\nChapter Eleven\n\nReturns and Expeditions\n\n1.\n\nANNABELLE AWOKE FEELING more rested than she had in as long as she could remember. She almost felt absolved, the memories of her terrible ordeals she had endured now behind her. A freshness and a newness seemed to invigorate her mind and senses. George dozed peacefully beside her, naked from the waist up and wrapped in a bearskin rug. She smiled to herself\u2014he really did look rather adorable like that, bear on the outside, teddy bear on the inside. Careful so as not to wake him, she slipped from under the covers and took a deep gulp of the warm, clean morning air of Arizona that blew gently through the flap of the tepee.\n\nShe pulled on her blouse and skirts and walked barefoot to greet the day, not even caring that the carved wood and brass-work of her false foot was on full display. It was bright, but not overly hot; the wind across the plains mussing her hair and softly brushing her cheeks. The rest of the camp was already awake, activity in every corner accompanied by laughter and bright, cheerful banter. She spotted Wapi talking with one of the braves that had accompanied them from the train, and waved to get his attention. When he saw her, he smiled, and it seemed for a second that her childhood friend had finally returned. He patted his fellow brave on the shoulder and crossed to greet her.\n\n\"Yohana,\" he said, nodding to her reverently.\n\n\"You seem a lot happier this morning.\"\n\n\"That is because I may not see the night again, and am enjoying the sun above and the ground below before I become one with them again. Our mission today is a perilous one, Yohana. We may not all be returning.\"\n\n\"Oh, come now!\" joshed Annabelle. \"Remember that time in the mountains? With the rattlesnake nest and that blind donkey? We'll be fine.\" Wapi simply stood and stared, his face once more stony, his eyes impassive.\n\n\"Where is the man?\" he asked.\n\n\"Man? You mean George?\"\n\n\"Yes. The man.\"\n\nAnnabelle had had enough dealings with men over the years to have a highly-tuned and perspicacious female intuition. She knew jealousy when she heard it, and even with Wapi's striking cheekbones and a physique toned by a life of wilderness living, pouting was never attractive. The little boy, it seemed, was back.\n\n\"Wapi...\" she began.\n\n\"No,\" he said suddenly, looking down at the floor. \"I want you to know, Yohana, that had you not left, there was a day, I hoped...even with everything that occurred...that we...\"\n\nAnnabelle's face twisted into sudden, violent anger. With a totally automatic reaction she pulled her hand back and slapped the Indian viciously across his cheek. His head whipped sideways with the impact.\n\n\"Even from an old friend, I will not be seen as some sort of...object. I am my own person, Wapi, and my experiences\u2014however traumatic\u2014have made me the woman that I am today. I am not the little girl you once knew. I do not need your protection and, what's more, I certainly don't need your pity.\"\n\n\"Yohana, I... I...\"\n\n\"I what, Wapi? What do you have to say for yourself?\"\n\nWapi could barely speak with the embarrassment and humiliation. He kept his eyes locked firmly on the ground, not even daring to raise them up high enough to glimpse the tips of her toes.\n\n\"We ride in two hours,\" he mumbled.\n\n2.\n\nBACK ON BRITISH soil, Tally had no time to appreciate his first visit to England as he and Nathaniel rushed to the Admiralty post-haste. A report had to be made, and lines of enquiry followed before the trail of Potsdam and Klopstock ran cold. There was also the crystal shard that still weighed down the pocket of Nathaniel's borrowed coat to deal with. The professor was keen to see if there had been any news of his friends. If they were facing the same level of danger he and Tally had encountered in Dublin; their safety was not in the least guaranteed. Memories of them all were stark in his mind, clear to see...easy to remember.\n\nThey took the train to Whitehall. Tally gazed around him incessantly with a wide-eyed, almost childish glee. Metropolitan as it was, Dublin was on a far more modest scale than the Empire's beating heart.\n\n\"Jayze,\" he muttered, his face pressed against the window as the train clattered through Mayfair. \"Look at it... The buildings! The landscape! And the people! The buggers are everywhere!\" Nathaniel turned to him and smiled warmly. \"Not so sure on the air, though. Compared to the homeland it's a bit, well...ripe.\"\n\nAs indeed it was. One could hardly blame Tally for his sense of awe and his curiosity towards London. Airships, not an uncommon sight in Ireland, were far more common here. The way buildings had been raised, improved upon\u2014grown almost organically to accommodate the burgeoning population\u2014was a world away of the make-do attitude of his native home. Modern technological wonders were abundant, steam driven contraptions on whose purpose he could only guess, the fashions bedecked with the latest clockwork gadgetry and the rich jewels brought back from the inner planets. And then there was the belt of factories that surrounded the city like an encroaching shadow, their tireless engines steering the course of industry and civilisation, churning out gears, pistons and steel, blackening the sky with the ashen refuse of progress. They threatened from afar, those factories; in a way it was they who controlled, their imposing fa\u00e7ades a reminder to London's people that, in truth the power lay elsewhere, in bodies constructed of stronger stuff than flesh and bone.\n\n\"If only me ma could see me now...\" said Tally as they strode up to the Admiralty. \"And that gobshite Mister Kelly from St Michael's, he always said I'd never amount to anything.\"\n\n\"A little decorum, Tally, please.\"\n\n\"Right y'are.\"\n\nThey entered the Adam screen, strode across the courtyard and entered the Admiralty through the main gate beyond. Time being of the essence, Nathaniel had not had the luxury of wiring ahead to inform them of his return. As such, when the powers-that-be were informed of his arrival there was a flurry of activity and, before long, some sneering high-level minion appeared to take him through to debriefing.\n\n\"Just you, Professor,\" said the minion in a nasal drone. \"Your...friend is no longer required.\"\n\n\"Now look here,\" started Nathaniel hotly. \"This man, who is, as you put it, my friend, has survived an encounter with the very men central to this whole insidious plot. Not only that, he has demonstrated a fortitude and bravery without which I would not be standing here. Even if that were not enough to imbue him with a certain debt of gratitude, the Secret Service Bureau made certain promises to him which I intend to see honoured.\"\n\n\"Aye,\" said Tally, equally impassioned, \"and if sticking with the professor is the best chance I've got for getting my hands on the bastard that killed Young Simon, then by the Holy Mother you'd better believe I'm sticking to him like glue on a stamp.\"\n\nThe minion sighed dismissively, clearly thinking himself far above an argument. \"Very well, Professor, but don't blame me if your...friend is out on his ear before you can say Jack Robinson. Fair warning.\"\n\nNathaniel frowned, and he and Tally followed the uppity, overdressed steward to Folkard's office, nestled in the grandeur of the upper floor of the Ripley Building. He showed them to the door, opening it for them while revelling in his disdain, and strode off without even a farewell. Tally was tempted to punch him in the back of the head. He closed the door behind them as they entered, and watched as Nathaniel slumped heavily into an armchair.\n\n\"So what's the plan now?\" he asked, eyeing up the drinks cabinet.\n\n\"I haven't the first idea, Tally. Doubtless we'll be recounting every last detail of our adventure numerous times before numerous committees, who will then hamstring any action by arguing amongst themselves over the next course of action. Folkard should be here. At least he has the weight and mental fortitude to make difficult decisions.\"\n\n\"Still, you've got your wee bit of crystal there, no? Get the old grey matter onto figuring what that's all about. Sure, you'll have the answers in no time, smart feller like yerself.\"\n\n\"It's more than just an explosive, Tally. Something far more dangerous. Far older, far more...powerful. Its potential for destruction alone is clearly limitless. That the Russians have obtained it is a catastrophe. For the British Empire to exploit it too... I fear that would be apocalyptic.\"\n\n\"Still,\" said Tally, looking around. \"Could be worse, eh? Nice warm digs, fancy surroundings, bet there's a good kitchen, and oh!\" he added nonchalantly. \"Is that a bit of a drinks cabinet I spy in the corner? D'ya mind?\"\n\nNathaniel, too troubled to care, waved his assent. Tally could barely contain his glee.\n\n\"Necking the finest of the Britisher reserves,\" he chuckled to himself as he poured. \"Now that's living.\"\n\n3.\n\nFOLKARD AND ENDERBY had successfully stripped the Russian soldier of his uniform and dumped the body over the side before Imperator had gained too much altitude. Locking the bulkhead door behind them, Folkard quickly began to undress.\n\n\"I take it you're taking the initiative then, Captain?\"\n\n\"Don't be ridiculous, man. This monstrosity is Sovereign in all but a name. I know it like the back of my hand. As well as that, I speak perfect Russian.\"\n\n\"You assume I don't? You also assume I'm not acquainted with the layout of the ship. Knowledge I had, I might mention, before we even started out on our little jaunt to Severnaya. Besides which, any half-informed Russian will know what you look like. No Russian has yet seen my face and lived to tell the tale.\"\n\n\"Which is why they'll pay less heed to the man in uniform, and more to the man dressed as a British spy. Now be a good chap and hand me those pantaloons.\"\n\nEnderby didn't argue. They found a porthole and stuffed Folkard's discarded attire outside, where the high winds whipped them away in an instant.\n\n\"Where first?\" asked Enderby.\n\nFolkard looked around. The effect of being on the ship was profoundly disturbing for him\u2014like visiting your childhood home only to find it wracked by decay, twisted beyond the happy memories of youth. Here there was none of the pride and fine craftsmanship that had gone into Sovereign's construction\u2014though each measurement was the same, each strut identical and each seam aligned, the Russians seemed to have cobbled this together from bits of junk yard waste, without care and at great speed. Where wood and burnished coppers adorned his old ship, here were studded steel plates and the ugly scar-lines of welding. Where Sovereign shone, Imperator brooded, angry and insular and brutally functional.\n\n\"We should hole you up,\" Folkard said to Enderby. \"Somewhere quiet, out of the way. That way I can have a proper look around, assess the true capabilities of this vessel. I may even attempt to get some sort of coded message to the Admiralty...\"\n\n4.\n\nTHE PARTY SADDLED up, preparing to ride. Bedford felt weighted down with the sheer amount of weaponry strapped to his person\u2014two pistols on his hips, a shotgun on his shoulder and two bandoleers of shells across his chest. Mister Boon, who had stayed up much of the previous night enthralling the younger members of the tribe around the campfire with tales of London's mechanical marvels and his own adventures as a spy, was decked out much the same. Thanks to his eloquence (and a surprising knack for silly voices) he had acquired quite a following among the children, who were grouped around him as he strode towards his horse, tugging on his trouser-legs and begging for one last tale.\n\n\"Later, little ones! Later! Old Bert needs to go and find more stories to tell; I've already told you all of mine!\" The babes whined and pouted until a broad-busted and smiling squaw ushered them away from the war party.\n\nThe air was filled with the click of rifles being loaded and the gentle whinny and stamp of impatient hooves. There was Bedford, Annabelle, Boon, Wapi and six other braves. Geronimo, against a unanimous volley of protest, had insisted on accompanying them. The sacred place, now so desecrated by those hungry for power and bloodshed, must be evaluated and cleansed\u2014there was no other Apache with the wisdom or the spiritual strength to do so, and he would not brook any argument.\n\nWapi remained bitterly aloof after his previous altercation with Annabelle. He did not even dare to look her in the eye, and Bedford could sense the tension between them. Annabelle had not told him what had occurred, but the way she had stamped around the tepee and awoke him, he knew for sure that something was up. It was something that niggled him, certainly, but now was not the time for such discussions. His beloved had endured such mental anguish since arriving at the camp; he knew his fussing would neither be welcome nor helpful. To everybody else she would seem the same old Annabelle: frank, spirited and strong. But he noticed in her both a deflation and a new fortitude, as if the revelations of the previous night had drained parts of her resolve, but drastically bolstered others. He put his worries aside, told himself once more that now was not the time, and concentrated instead on helping her up into her saddle.\n\nIngenious as it was, the prosthetic leg Annabelle had acquired after being shot by the sadistic criminal mastermind La Boeuf was not designed with horses in mind. She cussed in a shockingly unladylike manner and Bedford was forced to grasp her leg under her skirts and shove, receiving a face-full of thigh in the process.\n\n\"I'm not a sack of potatoes!\" she squealed.\n\n\"More's the pity,\" muttered Bedford.\n\n\"George Bedford!\"\n\n\"Sorry, dear.\"\n\nHe grunted and gave her one last shove, and she managed to hook her other leg around and settle, after a momentary wobble, into the saddle. Bedford stepped over to his horse, the reins being held by a younger brave of the tribe, and hopped up onto the grand beast's back. Wapi turned around to face the group.\n\n\"If we ride fast we will arrive in good time. Do not spare the horses, they are strong and have been well fed. If there...\"\n\n\"Shiwoye!\" yelled a voice. \"Shiiiiwoye!\"\n\nWith the exception of Wapi, the whole group tugged on their reins to bring their horses to bear on the source of the cry. It was Kai, belting barefoot towards them, looking as mad as all hell.\n\n\"Shiwoye!\" she yelled to Geronimo. \"I'm coming, Shiwoye, I'm coming with you and Wapi and Yohana!\"\n\nGeronimo looked down stoically from on high, adjusting his reins as the horse trotted forward. \"I have told you, Kai,\" he said softly. \"You cannot. It is too dangerous for one as young as you.\"\n\n\"But I've been before! With you, with the other braves, I've been out in the desert lots!\"\n\n\"And each and every time you have come, we were sure there was no danger. Yet this time there is much danger, and this time you will not come.\"\n\n\"But...but...\" she stammered. She looked up to Annabelle. \"Yohana,\" she said, and faltered, before seeming to make a decision that clicked suddenly in her own mind. \"Mama. You're my mama. Shiwoye Geronimo told me all about you, your adventures... I want to hear them all, and you're my mama, so you have to listen to me!\"\n\nAnnabelle was struck dumb. This girl...this beautiful, precocious, stubborn little girl did not feel like her daughter. To Annabelle, she was just another child of the tribe, with no more familial connection than she would have with any of the other wild, fierce little youngsters here. Yet she could see her features in Kai's soft face; even the way the little girl stood with her chin jutting out was a pose Annabelle knew she adopted when she was cross. And it was these tiny reminders that drew out hitherto unknown instincts in her, that knew she wanted to know and love this girl like no-one else could, to protect her. But these were weak stirrings of emotion as yet, and Annabelle, despite all evidence to the contrary, had not become a mother overnight.\n\n\"You must stay, Kai. Your shiwoye is right; it's too dangerous for you. And if I am one day to be your mother, your shi ma, then danger is something we must both avoid. It will happen, Kai, someday. I promise.\"\n\nSensing the finality in her voice, Kai made one last plea. She turned to Wapi, but he merely stared down at her without even saying a word.\n\n\"Go back, Kai,\" said Geronimo gravely. \"Go back and pray that the spirits be with us in battle, and that the wind of the wolves drive us back safely.\"\n\nKai scrunched up her face in anger, turned on her heels and belted back towards the comfort of the camp.\n\n\"We've wasted too much time,\" said Wapi. He yanked on his reins, startling the horse, and viciously spurred it into a ferocious gallop. The other riders, mimicking his speed but not his anger, followed suit.\nChapter Twelve\n\nThe Green Gods\n\n1.\n\nHE WAS ARNAUD Fontaine! He knew he was Arnaud Fontaine. And he was having such a great time... Mon Dieu! Brandy didn't have anything on this. His head was light, full to bursting with euphoria. His arms drifted at his sides like tethered balloons and everything, just everything, was so funny...\n\nThey let him wander about a bit, which was nice... But why would he want to go any further? He had everything he needed right here. Right within his grasp! Well there, there...there was Garrecreux! His old mate Garrecreux, bent over some fusty old science or something, crushing up the lovely sparkly bits and well...being a scientist, because that's what he did, old Garrecreux. They let him help out and he liked it when he helped, because then he felt helpful, and that made him happy. But he was pretty happy anyway. Life was a dream!\n\nAnd Coyne! Coyne was here, too. Arnaud loved Coyne. If it wasn't for Coyne, he'd never be here in the first place. Merci, Coyne! Merci! Coyne didn't seem to care, but Arnaud didn't really care about that. Arnaud knew Coyne would talk to him if only he could! But he can't, not where he is, poor Coyne, but it didn't really matter. Arnaud knew Coyne loved him, too.\n\nAnd love! Speaking of love, here was Madame Moonsinge again, coming in, seeing how everything was getting on. It was very important Arnaud helped to get everything done, he knew that, because if he did it right he'd get cuddles and kisses, which he liked. If he did it wrong Madame Moonsinge would get mad, and if she did Arnaud would get not love but horrid hurts, and Madame would take all the fun away until he got all shivery and sad and horrid... He had to make Madame Moonsinge happy, because Madame Moonsinge said they haven't got long left, and everything that Madame Moonsinge said was fine and dandy by him. It was! Yay!\n\nArnaud was happy. Arnaud was content. Arnaud had been allowed a little rest, and so Arnaud decided he would go and sit outside the lab, in the big cave that\u2014Shh!\u2014is a big old secret from the world. He leaned back against a rock that he knew was a rock but was really very soft and fluffy, like a pillow at home. And look at the cave! It's oh so nice. In the middle of the cave, so Madame Moonsinge told him, there used to be a big crystal tree. Arnaud giggled at the thought of that, and the sound of the giggle made him giggle again. Trees aren't made of crystal! That's silly! Madame Moonsinge is right, though, he told himself in a telling-off voice. Madame Moonsinge is always right.\n\nThe tree isn't there any more. There's a bit of it, a whatchacallit\u2014a stump! Arnaud giggled again. Stump was such a funny word. There were all these little children and old women and old men and they were ever so careful when they chipped bits of stump\u2014stump! Teehee!\u2014away. They carried it so carefully to bring it up to Arnaud from the big, wide, cave. When they brought it he and his Best Friend Garrecreux would mix it up together into magic potions, and with those they will make the world a much better place! Much better! That's what they tell him.\n\nBut ooh, there's angry voices from behind him and he knows he's been naughty, Arnaud knows he must get back to work. Can't disappoint!\n\nThere's no disappointing Madame Moonsinge!\n\nArnaud loves Madame Moonsinge.\n\nMadame Moonsinge gave him kisses and the lovely green stuff.\n\n2.\n\nTHE RIDERS POUNDED on, sending a torrent of red dust in their wake. The horses were not spared, relentlessly spurred onwards to their destination. Even Geronimo kept his head down low over his steed, his skill at riding and strength of purpose belying his seventy years of age. The vast, burnt topography of the Arizona landscape surrounded them, unbroken and daunting, lethal to the unwary despite its beauty. Wapi pointed ahead, indicating a hilly area that was barely a low, shallow hump on the horizon. Compared to the vivid, rusty sienna around them, this formation seemed to be composed of an entirely different mineral\u2014it was a pale, jaundiced yellow, as if the rich iron pigment of the soil had been leached away, the bones of the rock left to bleach in the sun.\n\nWapi's indication was enough. All the riders whipped their reins down on the horses' necks and lowered their bodies against the wind. The day was already half-old; the sun had begun its descent down the other side of the sky with a lazy impatience that reflected its ambivalence towards the affairs of mortal men. It would fall and set without caring if these riders reached their destination, and if the stories were to be believed, then the riders would be wise to arrive while light still remained to illuminate this dark and accursed place.\n\n3.\n\nAND YET AN unwanted rider followed them. A sneak, a thief who had cajoled a boy she had beaten into helping her onto a horse too big for her, a boy she had threatened to beat again should he breathe a word of her transgression. She was a rider who learned quick, who kept her mouth shut and her eyes wide open, had absorbed the skills of tracking and hunting and skinning without ever being given a word of instruction.\n\nThe girl whose father did not deserve life, the daughter of a woman whose spirit had spanned the stars. She had been adopted by Geronimo but knew she did not fit, and in the eyes of Annabelle Bedford she had found that life she had always dreamed of, and so she left her old one in order to pursue it.\n\n4.\n\nNATHANIEL PACED FOLKARD'S office while Tally, who had pointed out that it was the height of rudeness not to enjoy a fine smoke with fine drink, had sat back in a maroon Chesterfield with a thick cigar he had purloined from a humidor on Folkard's desk. Nathaniel had no patience with all this waiting, and had badgered a steward that he be granted an immediate audience with Admiral Sir Richard Vesey Hamilton, the current First Sea Lord and professional head of the Navy. If he had known how, he would have also insisted that some message be directed to the offices of Secret Service Bureau with a demand that whosoever called the shots in that clandestine faction should also attend. The true extent of the dangers they faced\u2014on a personal, political and global scale\u2014was only slowly coming to light, each possible trail seeming to end, literally and figuratively, in a frustrating stump. It was time for a true course of action to be decided.\n\nThe steward returned within minutes to inform Nathaniel and Tally that such a meeting had already been scheduled for later that afternoon but, with the current situation and Nathaniel's insistence holding considerable sway in the right circles, schedules had been cleared and it was due to commence momentarily. The two were led quickly through the corridors of the Admiralty to the State Room that lay at its heart. Here was a room reserved for the highest level meetings, the whispers within which could alter the fate of the world.\n\nAdmiral Hamilton's secretary, a studiously decorous and professional-looking man in a morning suit, nodded as he opened the grand double doors to the State Room. Inside this opulent space was a vast oval oak table with only two of its many seats occupied. Maps affixed to portable boards surrounded the far end of the table and a magnificently bright crystal chandelier hung low in the centre of the room.\n\nAdmiral Hamilton sat forward in his seat, poring over a document. He was wearing a double-breasted naval tunic adorned with braided epaulettes and bright brass buttons, and as the door clicked softly behind Tally and Nathaniel he beckoned them wordlessly to join him. The pomp and ceremony of the admiral's uniform totally overshadowed the appearance of the man who sat to his right, a man Nathaniel had never seen before. He was dressed in the attire of a clerk or accountant. He wore a dark, nondescript suit with an ash-coloured scarf wrapped around his neck. Roughly fifty years of age, his dark hair was thin and cropped close to his skull. His grey eyes were shadowed under his brow, a fact that did not seem to diminish their intelligence and attentiveness. The other features on his face seemed oddly out of proportion\u2014too large a nose, an oddly-rounded chin, but the overall effect was not one of ugliness, but of the mundane.\n\nTally stepped forward to take a seat but Nathaniel caught his arm\u2014it would not do for them to sit with a man of such rank unless invited, and so they remained standing.\n\n\"Good afternoon, Professor Stone. And this, I take it, is Mister Cahalleret?\"\n\n\"Indeed it is, sir.\" said Tally, somewhat bashfully. \"Pleasure to make your acquaintance.\"\n\nThe admiral smiled briefly at Tally's nervousness, but the look was soon lost as he turned to more serious matters. Nathaniel noted that the other man had neither been introduced nor introduced himself; his purpose here as yet a mystery and therefore prosaically obvious. Probably best not to make Tally aware of this just yet, lest the Irishman attempt to box his ears in retribution for broken promises.\n\n\"Gentlemen, the fact of the matter is that we have very little idea of the current state of play. Everything seems to be in confusion, threads dangling and disappearing almost as soon as they appear. I fear we are being played, to put it in frank terms, and are in a position of being permanently on the back foot.\"\n\n\"Which is why it is imperative,\" said Nathaniel, his tone remaining respectful but bordering on a command, \"that Sovereign be commissioned forthwith to take us to Calcutta. It is there, I believe, that the villains who set the Horseguard's bomb are headed.\"\n\n\"And why is that?\" asked the unnamed man quickly, his voice warm, precise and questioning.\n\n\"Because they told me,\" said Nathaniel simply.\n\n\"Tell me exactly what happened in Dublin,\" the man continued. \"You may spare me the explosion and its upshot. I read the papers.\"\n\nNathaniel gritted his teeth momentarily. \"We discovered a cavern under Phoenix Park. It had been mined, if my guess is correct, by the Russians, and was the provenance of the mysterious green residue that imbued the Horseguard's bomb with such deadly power. We discovered a sample there, far larger, and it was this that was used an accelerant to destroy all evidence of the operation. The bomb was set by as distasteful a pair of blackguards as I have ever had the misfortune to meet.\"\n\n\"And who are these... villains?\" asked Hamilton.\n\n\"An inhuman brute and a degenerate narcissist with a penchant for gloating. They call themselves Potsdam and Klopstock. Beyond that I have no idea.\"\n\n\"Ever heard of them?\" the admiral asked the mysterious man. He shook his head blankly and looked back down at the paper before him.\n\n\"Look,\" started Nathaniel, leaning down to quietly address the gentleman with the scarf. \"I know who you're here representing. That much is obvious. You must have some knowledge on the progress of Arnaud or Captain Folkard. And while we're on it, what was the nature of the captain's mysterious mission? I don't appreciate being kept in the dark, you know. Who exactly are you? A figurehead? A messenger? You haven't even told us your name.\"\n\nThe man looked up at Nathaniel and blinked slowly.\n\n\"You can call me Tooler,\" he said. \"For now. As for your other requests, they are not without merit. We have heard nothing from any of your companions and must therefore assume the worst. However, seeing as you have mentioned Sovereign, allow me to offer you some small relief. It is currently en route to London, after which it has been ordered to seek out and assist the Bedfords in their American investigation.\"\n\n\"Investigation?\" sputtered Nathaniel. \"So the honeymoon was all just a smokescreen for your insidious Bureau's ends!\" Tally's ears pricked up, but he remained calm.\n\n\"Regrettably, yes,\" said Tooler smoothly. \"But rest assured; we are doing the best we can, Professor Stone.\"\n\n\"And Jacob?\"\n\n\"Captain Folkard's mission was of the highest priority, and he was chosen due to his unique relationship with Sovereign. I regret that it falls to me to tell you, but we discovered...\"\n\nTooler's revelation was stopped short by a commotion coming from behind the doors at the other end of the room. A heated debate had sprung up, and Admiral Hamilton barely had time to utter \"what the deuce?\" under his breath before the doors were flung open and in strode Thomas St John Curnoble, 28th Lord of Chillingham, his face a dozen shades of red.\n\n\"Hamilton!\" he bellowed. \"By God, I'll have you strung up for this! Do you really think you could convene such a meeting without me finding out? A demonstration not only of your lack of foresight but also of your lack of respect for my position. Needless to say I shall use these defects of character as principal reasons to call for your dismissal.\"\n\nHamilton seemed unfazed. \"Lord Chillingham,\" he said coolly. \"Please take a seat.\" Lord Chillingham! Upon hearing the name, Nathaniel stiffened noticeably. He looked across, and for the briefest of moments the two men's eyes met.\n\nChillingham broke the contact as if nothing had occurred. He huffed over and sat to the admiral's right, dropping his bulk into the chair. He gritted his teeth, looked briefly over the papers on the desk without picking them up before sweeping them to one side with a gesture of impetuous irritation.\n\n\"I should have been in on this d\u00e9b\u00e2cle from the start, Hamilton,\" he said. \"If I were, I shouldn't have let you make such a hash of it all.\"\n\n\"And to which hash are you referring?\" asked Nathaniel coolly. Chillingham barely registered his presence.\n\n\"Yes, Stone,\" he muttered. \"I might have known you'd be here, poking your nose where it doesn't belong.\"\n\n\"You haven't answered my question, Lord Chillingham.\"\n\n\"Well,\" said Chillingham, shifting in his seat. \"All of it. Obviously. The whole sorry mess.\"\n\n\"It appears, Lord Chillingham,\" cooed Tooler, a faint whiff of satisfaction in his voice, \"that you're not as well as informed as you would have us believe.\"\n\n\"And who the devil are you, sir?\"\n\n\"Oh,\" said Tooler offhandedly. \"I'm just chipping in.\" Lord Chillingham opened his mouth to reprehend Tooler but Admiral Hamilton raised his hands in a gesture of pacification.\n\n\"If you wish to be better informed, Lord Chillingham, I suggest you sit back and let our friend here, as he so modestly puts it, chip in.\"\n\n\"Very well. But I do so only under sufferance.\"\n\nTooler waited a moment, then began.\n\n\"As I was saying, Professor, I'm afraid I must inform you that the blueprints of Sovereign have, by means unknown to us, found themselves in the hands of the Russians. It has long been our fear that they have been producing their own version of the ship. As you know, the Russians have been aware for some time of your covert mission on behalf of the Crown, and indeed your own survival was...\"\n\n\"Yes. Whatever did happen to Mister Fenn?\" Nathaniel asked, his mind drifting back to Mars and the prompt arrest of Jack Fenn following his admission of his betrayal of their mission to the Russians.\n\nTooler waved his question aside. \"That is not something I am at liberty to discuss. Needless to say, Folkard was sent with a small team of our men to confirm or deny our fears about the Russians version of Sovereign. Then there is the issue of this mysterious compound. Reports have come to our attention that indicate the two are inexorably linked. The Russians, Lord Chillingham, are making a play for power...but to what end?\"\n\nChillingham's face burnt a deeper red, as if the question were an insult. \"You said it yourself, sir!\" he exploded vehemently. \"Power! The only goal those quarrelsome swine have is to unseat us as the greatest nation on God's Earth.\"\n\n\"Quite,\" said Tooler, pointedly. Chillingham bristled, but said no more. A moment of silence descended on the room. Here were the representatives of three bodies\u2014the Government, the Navy and the Secret Service Bureau\u2014all sharing a roughly common goal but having such disparate approaches and conflicting agendas they were barely able to have a civilised conversation. Tally was out of depth, having nothing to add, and so Nathaniel thrust his hand into his pocket and brought out the weight he had carried there from Ireland, the hunk of green mineral he had chipped from the crystal root. All three representatives of England's foremost powers watched carefully as he set it down neatly on the table before them.\n\n\"This, gentlemen, is a sample of the mineral the Russians have been pursuing so ferociously. To uncover its mysteries is to uncover the true reason behind the Russian plot.\" He turned to Hamilton. \"Admiral, do you know when Sovereign is due to return?\"\n\n\"It is set to arrive at first light tomorrow.\"\n\n\"Excellent,\" said Nathaniel. \"That should hopefully be enough time for me to begin my investigations into the properties of the crystal. I can continue my research on Sovereign, should Doctor Beverly be good enough to let me use the sickbay as an impromptu laboratory. There should be plenty of time during the voyage to Calcutta, via, of course, Arizona to rendezvous with Annabelle.\" He turned around. \"Tally,\" he continued, \"can I rely on your help?\"\n\n\"Ever to the end,\" said Tally, smiling, \"and at least until I've knocked the head of that bastard with the brass hands.\" Nathaniel returned his smile, but the moment was usurped by Chillingham slamming his fists on the table and launching himself to his feet, sending his chair crashing back behind him.\n\n\"Am I to understand,\" he seethed, \"that not only are we now taking orders from a civilian\u2014a civilian whose actions on Peregrine Station remain under close scrutiny, and whose character is in serious doubt\u2014but we are also allowing into the Admiralty an Irishman, whose fellow countrymen deem it fit to rabble-rouse with talk of independence from the Empire?\"\n\n\"Yes,\" said Nathaniel simply. \"That is what you are to understand. Particularly so, seeing as any fears concerning my involvement in the Peregrine tragedy have long since been allayed.\"\n\nChillingham fumed. He understood Nathaniel's emphasis well. Chillingham's own role in Peregrine's destruction and the subsequent cover-up was less known. As a result, it was almost as if he didn't know where to direct his substantial ire first, and was therefore left helpless. Admiral Hamilton stepped in.\n\n\"Lord Chillingham,\" he said, \"despite your notorious reach of influence there is nothing you can do to stop me allowing Professor Stone passage to India on Sovereign. And as he says, that course of action may well be the only way to apprehend the terrorists responsible for such death and devastation. If you have a better idea, you're welcome to share it.\"\n\nChillingham stayed silent, brooding. Nathaniel sensed Chillingham's petulant mind whirring, thinking of how the right words in the right ear, the right men sent out to snoop in the appropriate places, could bring retribution on those in the room who had emasculated him. Chillingham was not a man accustomed to defeat, but tactical retreat was quite another matter.\n\n\"I'll bide my time,\" he said eventually. \"But a report will be made to the House of Lords concerning this matter. I'm sure they'll be interested in how the Navy uses vital resources for holidaymaking, and what's more, assisting a known saboteur...\"\n\n\"Who is, officially, dead and was cleared of such a charge,\" Hamilton pointed out.\n\nChillingham barely paused, showing no indication of hearing Hamilton's well served point. \"...and, above all things, a damn Fenian.\"\n\nTally was about to rise to the bait, but Tooler caught his eye and gave him a warning glance. He spoke to Chillingham without looking at him, directing his attention instead to the papers before him like a true bureaucrat. He squinted at them for a moment.\n\n\"Mister Cahalleret, so I have been informed, has done far more to stem the bloodshed of an Irish uprising than you are ever likely to, Lord Chillingham. Speaking of which, Mister Cahalleret,\" he said, turning in his seat to smile at Tally, \"I would rather like a word with you in private at some point. You've proved yourself very capable, it seems, and certain promises were made. We have much to discuss.\"\n\n\"Right so,\" said Tally, a note of uncertainty in his voice. \"But I was hoping on accompanying the professor, what with having a score to settle, an' that...\"\n\n\"Quite so, quite so. I think that's permissible, is it not, Admiral?\" The admiral nodded.\n\n\"Then it's settled.\"\n\n\"Not in my book,\" said Chillingham.\n\n\"Your protest has been duly noted,\" soothed Tooler, not without a hint of sarcasm.\n\n\"I'll get a telegram off to Sovereign,\" said Hamilton. \"Professor, we'll keep you informed, but in the meantime I suggest you prepare for your journey in what little time you have remaining.\"\n\nNathaniel nodded. Chillingham looked at him with a wholly undisguised contempt.\n\n\"Then I think it wise I talk to Doctor Grant,\" said Nathaniel, ignoring the lord's petty ire. \"If his madness has not subsumed him, he may well be of some assistance.\"\n\nHamilton and Tooler exchanged an uneasy glance. Nathaniel picked up on it instantly, and looked at them with a challenge in his eyes.\n\n\"I'm afraid, Professor Stone, that isn't possible,\" said Admiral Hamilton.\n\n\"Oh? And why is that?\"\n\n\"Because, regrettably, it is my duty to inform you that Doctor Grant has disappeared.\"\n\n\"He seems to be making a habit of that,\" Nathaniel said.\n\nGrant was not the only one\u2014disappearances in the Admiralty now seemed to be quite the done thing. If the knowledge of the unstable doctor's disappearance was not troubling enough, it was only then that Nathaniel noticed the chunk of mysterious crystal he had placed on the table had vanished.\n\nAny of the three men could have taken it.\n\n5.\n\nTHE PALENESS OF the earth was eerie. It was almost as if they had ridden onto another world, a sensation Annabelle and George were not unfamiliar with. Yet here the sense of strangeness and dislocation felt far more acute. Perhaps it was because of the way it was their own planet that suddenly felt so alien. Perhaps it was simply the way the wind seemed to die away as soon as they approached, leaving the area cloaked in a still and sudden silence. Whatever it was, it was abundantly clear why the Apaches thought this a cursed place.\n\nThe low mesa that sat to the right of Greenore Gulch seemed much taller now they were close to it; this effect was enhanced by the basin in which the mine sat. Yet there was no evidence of this ever being a place for prospecting, a place where hopeful Americans had come with avarice in their hearts and silver in their mind's eye. The entrance was nestled at the base of the mesa, hidden by an overhang and obvious only to those who were already aware of its existence. The riders approached cautiously\u2014even the horses were spooked.\n\nThey looked around them. Everything seemed desolate and deserted. There was nothing on the horizon, no enemy on the approach... A blessing, possibly, which gave the already tense atmosphere a further layer of ill-omen. This was a quiet that should not be trusted. The horses scuffed down the steep side of the basin, sending up clouds of sallow, sickly dust. The group arranged themselves in a semi-circle facing the cave opening, and Wapi hopped down from his horse. There was nowhere to tie their steeds, whose wide eyes and ceaseless, shifting trots indicated they were moments away from bolting.\n\nOne-by-one the riders dismounted, with Wapi helping Geronimo down. They grouped together again, each keeping the reins of their horses held tight. Wapi reached into a large saddle-bag and brought out half a dozen wooden torches, the rags tied to the ends infecting the air with the stench of kerosene.\n\n\"Gopan and Iki, you stay here,\" he said. \"See if you can find somewhere to tie up the horses. Bly, Devra\u2014scout the area, you must keep an eye on all approaches. If you see anyone or anything, come to find us, without haste.\"\n\nThe braves nodded and swiftly obeyed Wapi's instructions. Geronimo, wrapped in a tight shawl and seeming somehow frailer than before, stared at the young brave with a slow, sad intensity. Wapi turned away. He still could not look Annabelle in the eye, and avoided George's too. The atmosphere felt as sour as the barren, washed-out landscape around them. Bert, sensing the tension, attempted to inject a little life into the party.\n\n\"So,\" he said, in his chipper London accent. \"Who's up for exploring a mine, then?\"\n\n6.\n\nTHE TORCHES WERE lit and the group, with Wapi at its head, descended into the tunnel. Even here, the only indication of previous human interaction with this strange, accursed place was the odd scrape-mark of a pick on the walls. There were no pit-props, no tools...nothing.\n\n\"This place holds evil close in its heart,\" said Geronimo, as they descended the gentle slope into the tunnels. \"The white men who came did not respect the spirits as we do, but still they felt their presence. Their minds became bloated with fear before they could reach the true depths, the Green God Tree that nestles at the centre, the tree that needs neither sun nor water to thrive. It is said that only those who are true of spirit can come close to the Tree.\"\n\n\"I once kicked a pigeon when I was drunk,\" said Bert. \"Hope that means I qualify.\"\n\nWapi spun around, and suddenly the flame of the torch was close enough to Boon's face to singe his eyebrows.\n\n\"You think this funny, magaanii? You think this evil place is one deserving of laughter, foolish talk of drinking and pigeons?\"\n\n\"No, no,\" said Bert, taken aback. \"I'm sorry, I just...\" He faltered.\n\nWapi stared at him for a moment, before turning around and heading once more into the gloom.\n\n\"...Wanted to lighten the mood,\" Bert finished lamely. Geronimo placed a hand on his shoulder.\n\n\"Excuse him,\" he said. \"The return of Yohana has brought back many memories, many emotions. They boil within him like a pot on the fire.\"\n\n\"That's as maybe,\" said Boon, \"but he doesn't have to be so ruddy rude.\" Geronimo patted him and smiled warmly, passed by Boon's shoulder and weaved after Wapi downwards into the blackness. The forms of Annabelle and Bedford resolved themselves out of the dark to join Boon.\n\n\"Everything okay?\" asked Bedford.\n\n\"Well the wind's got up old Wapi, that's for sure,\" answered Bert. \"What's his problem?\"\n\nAnnabelle and Bedford exchanged a glance.\n\n\"A tale for another time, Bert,\" said Annabelle. Bert seemed to pick up on the insinuation pretty quick.\n\n\"Righty-ho,\" he said. He looked behind him, checking that Wapi and Geronimo were out of earshot.\n\n\"Look,\" he whispered. \"What's going on? Fact is, when I was sent out here by the Bureau they gave me sweet Fanny Adams to work with, to put it bluntly. Normally I'm a bit more informed than this. It was that slimy git Coyne who was meant to prep me, the one who's with your French friend. Only he swans off without a by-your-leave. 'Just look after the lovebirds' is all he said. It was almost as if he was expecting you to make all the headway, which, credit where it's due, is exactly what you've been doing.\"\n\n\"Well,\" said Annabelle, \"we're no more enlightened than you, Bert. The raiders who attacked us might be linked to this place, but they might not. Fact is, we can't ignore the possibility we were targeted, and that this place might hold the key to their identity. It's not much, but it's all we've got to go on.\"\n\n\"And trust me,\" added Bedford. \"I don't like not knowing any more than you do.\"\n\n7.\n\nIT DIDN'T TAKE them long to catch up with Wapi and Geronimo. The latter had overtaken the former as head of the party, leading with a confidence that the others were truly grateful for. Suddenly, his already wrinkled brow wrinkled further in the torchlight, a tight pursing of his lips indicating that everything was not, somehow, quite right.\n\n\"The feeling,\" he muttered.\n\n\"Feeling?\" asked Bedford. \"What feeling?\" Geronimo didn't turn; in fact, it was almost as if Bedford's question has strengthened his resolve, and he moved forward with a renewed vigour.\n\n\"The feeling that always comes,\" he called back quickly, \"that teases the mind in these tunnels, one I have felt each time I have entered here, somehow, now... It is gone. We must hurry.\"\n\nThough the caverns branched and tangled together, Geronimo seemed to know exactly where he was going. The intensity of his purpose seemed to render any further conversation moot. They continued ever downwards, and that keen sense Geronimo had mentioned, the one that allowed him to explore the spaces beyond the corporeal veil, began to tickle the minds of the others in the party. It was faint, indistinct, but unmistakable. It seemed to jar memories of past slights and jealousies in their mind, tug at them, almost mocking them. While at the same time reminding them of moments of pride, passion and success. It was unsettling and alien, the shadow of an intruder in the consciousness.\n\nGeronimo stopped.\n\nThe tunnel ahead of them was illuminated with a faint, pale green glow. He turned back to the rest of the party, looked to each of them in turn for sign of the mental distress that coiled like tentacles from this haunted, foreboding place.\n\n\"We have arrived,\" he said.\n\nWith a sudden caution he began to pick his way across the tunnel floor to the glowing chamber. The others followed suit.\n\nIt was not a particularly large space, and compared to the barren darkness of the tunnels, stepping into the cavern was like entering a nightmare from a long stretch of heavy, empty-minded sleep. Here the walls seemed to have had any trace of life or nutrients bleached from them; they were a crumbling, chalky white. Hung on this alabaster canvas was a myriad of offerings and totems. Skulls adorned with feathers hung from blackened strips of leather, wood and animal horns had been carved into grotesque, twisted parodies of people and the creatures of the desert. It was obvious, even to the untrained eye, that these pagan relics came in a myriad of styles and age; they were not the work of just one civilisation. This had been a place that had shocked the hearts and minds of many tribes over countless thousands of years.\n\nYet their work had been desecrated. Wherever the relics had been reachable by human hand, they had been torn from their moorings and smashed on the floor. All, that is, except for one, which sat at the back of the room undisturbed before a great, green, crystalline stump that rose from the centre of the floor, perhaps three foot high and half as tall, jagged where it had been cut away.\n\nBut the malignant shard was not what caught the attention of Bedford and Annabelle. Their gaze was trapped and held by the skeletal figure that oversaw this ancient place, strapped to a rough wooden seat in a sitting pose, monarch of this desolate kingdom. The seat was surrounded by more elaborate offerings of old feathers and intricate carvings, and had the look of some ancient heathen throne.\n\nIt had the rough shape of the people of Earth but its proportions were warped, almost apelike. The skull was bulbous and enlarged, the eye sockets large and pushed towards the sides of the head. The face seemed almost squished between the large forehead and long, protruding chin.\n\nMemories of their previous excursion to Luna came rushing to the fore, and Annabelle remembered one of the images rendered on the monolith protruding from the surface of Phobos. It seemed these creatures had travelled further than the moons of the Solar System.\n\n\"A Drobate.\" whispered Annabelle.\n\n\"You know of it,\" said Geronimo. It was a statement, not a question. The five of them gathered around the alien, misshapen thing, captivated by its other-worldliness. The very look of it seemed to reach out on a primal level, intrinsically linked to the subconscious pull of this place. It was no wonder that, despite the destruction meted on the rest of the shrine, this reminder of ancient lives had remained untouched.\n\n\"They're a race of creatures that inhabit the deeper reaches of Luna,\" said Annabelle, reaching forward to touch the Drobate skull but just stopping short.\n\n\"From what we can tell,\" continued Bedford, \"the Drobates used to be a flourishing civilisation with an astounding degree of technological and scientific advancement. But something happened. Their society declined and collapsed to the point where they live little better than animals, savage and unruly, territorially aggressive\u2014vicious, telepathic fighters. The presence of this feller here would seem to indicate that, at least at one point in their race's history, they visited Earth.\"\n\n\"It is said,\" intoned Geronimo, \"that in the stories passed down about the great crystal tree, that the Shaltak came. The meaning of Shaltak has been lost as languages grew and intertwined, but my own grandfather used to think it could be translated as 'the spirits of the minds in the sky'.\"\n\n\"Certainly ties in with your theory,\" Bert said to Bedford.\n\n\"I cannot claim ownership of the theory; merely sharing what has been discovered by Professors Stone and Quintana, as told to me by Annabelle.\"\n\nGeronimo nodded at Bedford, and continued. \"Nobody knows if the Shaltak planted the Crystal God Tree, or if they came in search of it.\" He indicated the stump in the middle of the cavern. \"But the legends state that the Shaltak crawled into the minds of our ancestors and pleaded with them that the tree should be protected and revered. It is my greatest shame, and one I shall carry to my grave, that it was in my lifetime this sacred and ancient duty was betrayed. I relied too heavily on the tree's power to ward men off. I underestimated the evil in the minds of those who came and desecrated this place.\"\n\n\"Having faith in the essential goodness of people isn't something you should beat yourself up about,\" said Bedford. He stopped, suddenly realising how trite and unimportant that sounded with the gravity of the situation.\n\n\"Whatever the circumstances,\" stepped in Annabelle quickly, \"the fact remains that we're too late.\"\n\nGeronimo, suddenly leaden with the weight of his years, shuffled over to the stump and bent down to touch it gently.\n\n\"Look at it,\" he said quietly. \"Look at what remains. Beyond the effects it has on the minds of mortals, the Crystal God Tree was a wonder, a thing of great beauty, its branches spanning the whole of this cavern. A thing of nature and awe, Yohana, cut down by the ignorance of evil. It is my failure. No good will come of this.\"\n\nAnnabelle crossed to him and gently lifted him up. \"I understand the pain and disappointment you must feel, Goyahkla. But we have friends who can help, who will work with us to ensure the danger you speak of is fought, curtailed, and brought to justice. Any failure here may only be the start of a greater victory for the entire world. Have faith.\"\n\nGeronimo, who had suddenly seemed so aged and broken, stared at her, the light in his eyes reigniting.\n\n\"You are wise beyond your years, Yohana,\" he said. \"You give faith back to this old man. There is nothing more we can do here.\"\n\nTaking one last look at the strange and disturbing place, the group grabbed their torches and began the long climb back up the tunnel to the surface. They had got maybe half-way before Gopan came pelting down the passageway towards them, a flaming torch in one hand, the other holding his rifle aloft.\n\n\"Riders!\" he cried.\nChapter Thirteen\n\nBetrayals\n\n1.\n\nARNAUD STARED INTO the barrel of the pistol, giggling occasionally and pressing his forehead against the cool tunnel of iron. He was sitting cross-legged on the floor of Garrecreux's lab, having been given an hour's rest thanks to the arrival of Madame Moonsinge's latest visitors\u2014Potsdam and Klopstock.\n\nIt was Klopstock who held the gun. He sat on a chair before the stupefied Frenchman, waving the weapon around and watching contemptuously as Arnaud's eyes followed the barrel around like a mystified toddler.\n\n\"Well this is boring,\" he said aloud. \"Where's the fun, Moonsinge? Where's the challenge? I could shoot him in the face now. Pow! Dead. And while the visual impact of doing so would, I think, be rather a thrill, it seems somehow...inelegant.\"\n\n\"Be quiet, idiot,\" snapped Moonsinge. She eyed Potsdam warily, a hulking threat that took up much of one corner of the room. \"Anyway, you can't kill him yet. We still need him. He's torpid and suggestible, yes, but his skills in helping us prepare the Aleksandrite are no less diminished by its effects. Do you want to answer to our Lord and Master for delays?\"\n\n\"No more than you do, my sweet. Funny you won't say his name.\"\n\n\"Would you?\" she snapped back. Klopstock pouted childishly, and waved her away. Moonsinge scowled at him. \"Well then,\" she said.\n\nGarrecreux, who had been hunched over a lab bench, let out a sudden howl of irritation. \"Will you stop all this pointless bickering!\" he yelled. He took in the look of utter contempt on Madame Moonsinge's face, and in an instant he was on his knees before her, his arms wrapped around her waist.\n\n\"I'm sorry, so sorry, my mistress, forgive me.\"\n\n\"Get up, you worm,\" sneered Moonsinge. \"Prepare the minerals.\"\n\n\"As you wish, as you wish.\" He beckoned for Arnaud to assist him in further refining and producing the various arcane compounds that came from Aleksandrite, which the Russians had named in honour of their Tsar.\n\nMadame Moonsinge nodded at Potsdam, monstrously motionless in the corner, his arms by his side like a mannequin.\n\n\"And him,\" she said. \"The brute. Does he need a refill?\"\n\n\"Couldn't hurt, couldn't hurt,\" said Klopstock blithely. \"Top him up.\"\n\n\"I wasn't offering to help,\" said Moonsinge. \"You know I can't stand the sight of it.\" Despite the curious effect her demeanour had on the men of this world, Klopstock seemed strangely immune. This was hardly an issue for her, however, as she couldn't stand the man and was happy to admit it. A cabinet stood against one of the walls. She removed a key from a chain looped around her neck and unlocked the cupboard door.\n\nInside were rows and rows of phials, each full of a powdery, pale green substance whose minute facets caught the light. Each phial was labelled differently: The true extent of the versatility of Aleksandrite. She pointed down one of the rows, grabbed one of the phials and tossed it casually across to where Klopstock sat. As soon as he saw it flying towards him, he panicked, reached out, desperate to grab the phial before it shattered on the floor. Moonsinge laughed at him.\n\n\"It's no laughing matter,\" said Klopstock darkly. \"This is dangerous stuff.\"\n\n\"And you're nowhere near as dangerous as you make out,\" mocked Moonsinge. \"Feed the demon.\"\n\nKlopstock, gripping the phial of compound tightly, approached Potsdam. The huge man did not even register his approach, and Moonsinge looked over idly from one corner.\n\n\"How are his eyes?\" she asked.\n\n\"Worse,\" said Klopstock. He reached up and unbuckled the goggles from the back of Potsdam's face. They fell away and revealed the true effects of the chemical on this part of his anatomy\u2014his eyes were glowing a bright, burning green, the whole of their surface encrusted with a spiky, crystalline residue. The flesh around them was raw and cracked, crimson capillaries spreading like spider's web.\n\n\"He'll be blind soon,\" said Moonsinge. \"Then we can finally put him down like the rabid dog he is. Garrecreux's the same. How fortunate their usefulness is so soon coming to an end. Maybe we can have them fight to the death?\"\n\n\"Mistress!\" cried Garrecreux from the corner, appalled.\n\n\"I don't remember giving you permission to listen to me, Fabrice,\" she chided.\n\n\"I'm sorry, I'm sorry, mistress!\" he pleaded, before turning to Arnaud and whispering in schoolboy conference. \"Did you hear? She called me Fabrice!\" Arnaud giggled idiotically.\n\n\"Well,\" said Klopstock as he began to unbutton Potsdam's greatcoat. \"I know who my money would be on.\"\n\nHe finished unbuttoning and, pulling the scarf away from the great brute's neck, let the greatcoat slip to the floor. The true monstrosity of Potsdam was revealed.\n\nThe braces that caged his hands extended up his arms and cradled his torso. Tight bronze bands were clamped to his wrists and neck that made his head and hands unnaturally bloated and ruddy with blood\u2014yet the rest of him was not so healthily fattened. What remained of his body, hung within the burnished framework of metal, was a translucent, deathly grey. Veins and bones stood out on parchment-thin skin, his abdomen distended and sagging low over his groin. The metal cage held his trousers up; the body simply dangled into empty space like a chicken strung up in a butcher's window. The cavity smelt like sweat and sour milk.\n\nThe metal poles and joists drilled into his fingers continued up into the joints of his elbows and shoulders, with pistons and gears fed by tubes running their lengths. These tubes were wrapped around his skeletal limbs, all of them snaking and coiling until six of them were fed into the space on his chest directly before his heart. Klopstock unlocked a clasp on the front of the cage and opened a door. On the back of it was strapped a bulky clockwork mechanism, redolent with the ornate swirls and craftsmanship that typified Klopstock's deadly art. He unscrewed a lid on the top of the engine and gently began tapping the Aleksandrite compound inside.\n\n\"I think I'll sort of miss him when he's gone,\" said Klopstock. \"Do you know how many prisoners we killed before we managed to perfect it? Dozens. A hundred, maybe. But he was the only one with the fortitude to survive. Or the stubbornness. They say he killed nearly forty people before he was caught, you know. At least half of them were children,\" he added conversationally.\n\n\"You keep such fashionable company,\" proffered Moonsinge.\n\n\"As do you, dear heart. As do you.\"\n\n\"What of Professor Stone?\"\n\n\"Dead,\" said Klopstock simply, concentrating on his task. \"Blown to bits. Kaput. No more. You've got me to thank for that. We met Imperator on a godforsaken stretch of the West Irish coast and, after a thankfully brief pit-stop in Severnaya, came right here.\"\n\n\"So Stone and Fontaine are accounted for.\"\n\n\"Indeed. Bedford and the woman, well... they're too late to do anything. We exhausted that vein weeks ago, and should they unwisely decide to go sniffing around they'll be dealt with. I hardly think a few savages will be able to withstand the might of the veritable army of bandits our...heh, Lord and Master, as you so delicately put it, has in his pocket.\"\n\nMoonsinge did not seem convinced. \"Is he ready?\" she said, indicating Potsdam. \"I've a job for him.\"\n\n\"Oh, goody!\" beamed Klopstock. He screwed the lid back on the top of the mechanism and wound a clockwork key set into the side of the device. Potsdam seemed to straighten his back slightly as the new drugs entered his system, the gearwheels and pistons whirring and hissing gently. \"There we are. Tip-top shape.\"\n\n\"Excellent,\" said Moonsinge. \"Garrecreux, Fontaine!\" the two men instantly turned at the sound of her voice like obedient dogs. \"Bring him in.\"\n\nThey scuttled out of the room and, moments later, returned with a man dressed in black held between them. He was struggling violently, a burlap sack over his head. Moonsinge crossed and whipped the bag away.\n\n\"Mister Coyne,\" she soothed. \"Why, you were oh so useful to us, oh so very happy to betray your country and your principles for the promise of so great a reward. And here it is. May I introduce you to Mister Potsdam?\"\n\nPotsdam, unheeded, strode towards Coyne's struggling form. The appearance of his body in motion, and the emaciated torso dangled in the cage like a dead thing as the pistons drove the power, was a vile and unholy sight. The metal-encased hands shot out and grabbed either sides of Coyne's head and lifted the traitor out of Garrecreux and Arnaud's hands. Coyne screamed, a horrendous, panicked, animal sound.\n\n\"Ah, ah, ah!\" said Moonsinge gently, as the giant relented slightly. He looked down at her with dumb and glistening green eyes.\n\n\"Take your time with this one,\" she said. \"Enjoy yourself. Make it slow.\"\n\n2.\n\nENDERBY WAS AT a total loss. Never before in all his years of service had he felt so rudderless, so constrained and frustrated. They had been working for months on the mission to discover the nature of the Russian's secret weapon, and now they were here, he was powerless.\n\nThey had managed to ensconce themselves in a supply bay deep in the heart of the ship. At the back, he and Folkard had boxed themselves off with a few crates where they could easily evade detection and snatch enough supplies to survive. There were no portholes, and the only indication of their movement was the malignant, heartbeat thrum of the engines that assailed them through the walls.\n\n\"There's something about this ship,\" said Folkard, pressing his ear against the bulkhead to listen. \"It sounds so different to Sovereign.\"\n\nThere had been something of a breakdown in communications since the men had boarded Imperator. Perhaps, Enderby thought idly, the loss of the men they had commanded had left the two at loggerheads\u2014the two of them were fine having others to lead, but now the balance of power had shifted. It was true what Folkard had said. He did know the layout of the ship far better than the agent, but that did not mean Enderby couldn't be of some use. It was clear that the captain was deeply troubled by the Russian battleship's similarity with the vessel he knew so well, but at the same time he sensed a renewed confidence in Folkard that came, he assumed, from sailing once again.\n\nEnderby was a patriot and a reasonable man. He had accepted the Bureau's offer in order to serve his country, and it frustrated him that that duty of service was being denied to him when it was potentially needed most. It seemed now that Folkard spent much of his time reconnoitring, and it was in those moments that Enderby felt at his most vulnerable. If, by chance, a luckless Russian sailor had entered to grab vittles for the mess Enderby would have snapped his neck and snatched his uniform without a second thought. And yet no such sailors came. He was left in the dark, alone, his head whipping towards each unbidden, unexpected sound, one hand constantly on the hilt of his knife.\n\nHe had even thought, once or twice, of heading off on his own to explore. The calculations in his mind ripped at him. If they stayed concealed, which he was sure they could do, they may well be taken somewhere important to the Russian plot, have a chance to expose and counteract whatever machinations they were planning. He could not risk discovery, not here, not now. The Russians would throw the corpses of himself and Folkard in the British government's face, and war would be inevitable. He sat, biding his time, frustrated and unsure.\n\nThe last time the captain had returned he at least had some solid news, having discovered Imperator's destination\u2014Calcutta, India, where Coyne and Arnaud had been sent to investigate. He bitterly hoped the two were having better luck. Perhaps it was restlessness that made him disingenuous, but surely Folkard should have discovered more than that by now? Then again, perhaps he had, and if he had, why was he keeping the information from Enderby? What purpose did that serve? Did Folkard not trust him? Or did Folkard's allegiances lie elsewhere? There was still the mystery of how Sovereign's plans had found their way into Russian hands... Blueprints that Folkard certainly had access to.\n\nHe'd been gone for hours now, and troubling thoughts were assailing Enderby in the darkness. For now he wondered at Folkard's behaviour, and put such qualms down to an over-active, incarcerated mind given too much time to mull over too many uncertainties. Still the worry stayed with him.\n\nThe heavy door to the bay smacked open and Enderby, ever alert, already had his knife poised before it squeaked to a halt against the frame. He pressed himself back against the wall, and only the keenest observer would have noticed him grip his knife tighter. From his position, squirreled behind a packing crate, he could not see the entrance. There came the sounds of boots on grating, and Enderby strained to count the steps. Definitely more than two, but whether it was three or four he couldn't be sure.\n\nThe Russian squad were keeping together as a group\u2014a solid tactic when searching for a dangerous quarry. They were moving clockwise around the room, and with a little luck and careful movement he could flank them. There was no chance he could take one out and steal a uniform, not now.\n\nAn icy thought struck him, sudden and sharp. Had Folkard been captured...or had he given away Enderby's hiding place?\n\nThe soldiers were approaching, their footsteps slow. Enderby eased around the packing crate and bolted out to hide behind a set of steel shelving lined with tins and boxes stamped with Cyrillic. The soldiers were on the other side. He flexed his fingers across the knife's handle and moved at a crouch down the length of the shelves. The time he'd spent stuck in this godforsaken supply bay had not been wasted, and he knew if he could reach the end a large crate had been strapped down just beside the wall, with enough space to crawl behind until the patrol had moved on. He readied himself to bolt across, held his breath, and moved out.\n\nThere was a click behind him, the solid crack of an oiled bolt-handle nestling home. He froze. Even from the noise he knew the make, one of the newer ones, the .42 Mosin-Nagant. The Russian military, technically, were still testing it. It was hardly a revolutionary design, but then, from this distance it hardly mattered. The barrel was, after all, pressed against the back of Enderby's head.\n\nHe felt sure they must have known exactly where he was.\n\n3.\n\nPELTING OUT INTO the Arizona day, Bedford was blinded by the light. It was as if the tunnel entrance had rushed towards him faster than he approached it. As he suddenly found himself in the air the whole of the world assaulted him with noise and brightness. He took in the situation.\n\nThe Apache warriors that had remained above ground were gathered in a tight semi-circle around the entrance to the cave. Two were hunched forward on their knees while the others stood behind them, rifles held aloft to produce a classic wall of gunfire. To his right, some feet off, two of the horses lay dead. The others were nowhere to be seen, and Bedford suddenly realised he had unconsciously shouldered his shotgun and was already aiming down the barrel. All around the rim of the basin, black figures were darting between outcrops of cover. Bedford tried to count. A dozen? Two? The antlike smatterings of movement were impossible to assess. Once or twice the head of a mounted rider would snake into view before pulling his horse back to safety.\n\n\"Geronimo!\" Bly hollered over his shoulder. \"Protect Geronimo!\" He quickly brought his Winchester up and fired a round.\n\nWapi grabbed Geronimo and yanked him unceremoniously back into the mouth of the cave. The old man protested, but weakly, and seconds later Wapi was back with own weapon readied. Boon was at Bedford's side, cranking the handle of his custom pistol and scanning the overhang with keen, expert eyes.\n\n\"Twenty, at the very least,\" he said. \"And that's only the ones up close.\"\n\n\"They're firing down,\" added Annabelle, who had dropped uncomfortably onto her prosthetic knee and was cocking her rifle. \"Not only are we horribly outnumbered, but they've got the tactical jump on us too.\"\n\n\"If you weren't here, darling, I'd be inclined to worry.\"\n\n\"Oh, George, you say the sweetest things when we're about to die.\"\n\n\"Not today we're not, my sweet. Not today.\" He trailed his shotgun over the lip of the crater and let out a powerful blast. A scampering silhouette fell, the buckshot hitting him on his left side. Not lethal, Bedford could tell, but enough to put the bandit out of action and make his compatriots think twice about sticking their necks up.\n\n\"Custer's Last Stand?\" opined Boon.\n\n\"Rather an insensitive comparison,\" countered Annabelle, not without a smirk, as she let off a round of her own that ricocheted mere inches from one of their shadowy attackers' heads. \"Damn and blast,\" she muttered.\n\nThere was no real cover. To retreat into the cave would be a slow suicide; not only would their line of fire be pitifully diminished it would be all too easy for the bandits to crowd down from the sides of the basin and overwhelm them. No. Their only option was to stand their ground.\n\nYet these were a ragtag, drunken band of marauders. There was no military training, no finesse\u2014they put too much faith in their sheer force of numbers, that enough men with enough ammunition guaranteed victory. As such they were arbitrary, careless. They took random pot-shots that betrayed the fact they cared more about their own individual skins than their group as a whole. Three, four, were picked off in careful succession by the more measured shots of the braves and their companions. But it was not a tactic that could endure. The Apache Annabelle had noted as Iki was struck with a bullet in his right breast; he fell and gurgled helplessly as the blood began to fill his lungs, the others powerless but to listen to his ghastly death-throes as they fought with their all to survive the onslaught.\n\nSlowly, as the defenders started to become conscious of how many bullets remained to them, the attackers advanced. No longer were they antlike and far off; they would skid quickly down dusty hillsides, finding cover in the rocky outcrops on the slopes of the basin. Onward and down they piled, in such a torrent it was impossible to catch them all with rifle-fire. Soon the only option would be retreat into the cave, and after that, certain death.\n\nThe bandits were getting braver now, peeking their heads up for longer, taking more and more shots each time. Bly took a bullet to the knee, keeled over, but momentarily dragged himself back upright and fought on with his teeth so tightly gritted they drew blood. Perhaps seven or eight of the bandits started to crowd toward them across the basin floor. Three were taken out with ease but others, buoyed by their companion's success, were joining the encroaching ranks. More than twenty of the bandits lay dead or dying, perhaps thirty or more remained.\n\nYet all of their attention was directed towards the cave mouth\u2014a fatal mistake for the foremost of the attackers who, with a swoosh of wind like wildfire, stood suddenly bolt upright with the knowledge of death in his eyes and an arrow jutting horizontally through his neck. A split-second later, a second fell the same way. As the bandits turned, stunned by this new threat, three more were dispatched by Boon and Annabelle's fire.\n\n\"Kai!\" wailed Wapi, a terrible trembling in his throat as he saw the girl approach. \"Kai!\"\n\nShe was thundering down the hillside to the left of their position, her body steady and upright on the back of her stolen horse. With a smooth, practised motion her hand whipped behind her head to pull another arrow from her quiver; quick as a blink the bolt was drawn back and shot, only to miss and thud into the dirt. The braves took full advantage of the distraction, firing into the confusion, but as Kai belted across their line of fire they were powerless to shoot for fear of hitting the girl. Her horse rounded, slowed, and as it did so Wapi leapt towards her.\n\nThe great animal Kai had stolen was so large a target the feckless bandits couldn't help but aim towards it. As a multitude of cordite booms resounded from the landscape's bleached and shallow sides Wapi leapt up behind Kai, curled his body around her, and put his back towards the bullets.\n\nThey struck him mercilessly. He fell away. The horse screamed and tumbled, its forelegs buckling into the dirt. Wapi and Kai fell with it, falling cradled in the nest between the dead steed's legs and belly, bullets thudding into the beast's back, while Wapi held Kai tightly in his arms as she struggled to get free.\n\nIt had bought them some time, but not much. One or two of the bandits fell, but as they approached the defenders backed imperceptibly towards the darkened cave, where Geronimo knelt muttering a low, strong Apache prayer. More gunfire came, closer, more accurate, and in the dying times Bedford turned to his new wife and reached out to her hand, his eyes pleading how much he loved her, how he would strip the skin from his own bones for it not to end like this, right now.\n\nAnd then came the boom of the breech guns.\n\nThe opposite wall of the dip exploded into fragments, chunks of unnaturally pale rock sent to spin like pinwheels in the air. The bodies of the bandits flew with them, some of them falling into tattered shreds of limbs and sinew as they crashed back down to earth.\n\nA thundering came from above them, and all eyes shot skyward. Slowly sailing into view as its prow breached the cover of the mesa came Sovereign, the strong Arizona light gleaming in bursts from its metalwork, its shadow crawling across the floor of the basin. Two more retorts sounded from the guns, sending up a shower of gravel and taking more of the bandits to their graves. The rest looked up; a couple even had the wherewithal to shoulder their weapons and fire pointlessly at the pride of the British Fleet. Yet when they turned to discover the rest of their band were fleeing, scrabbling up the slopes over the bodies of their dead, they soon followed. All their horses had bolted when the cannons thundered and they were left to flee on foot, panicking and stumbling across the deadly terrain.\n\nWith the exception of Geronimo, the remaining Apaches were almost as terrified by the aether battleship as the bandits. They had seen them before, of course, lazily drifting through the atmosphere, but the might and majesty of Sovereign up close was an entirely different spectacle. As the belly of Sovereign hovered into view Boon strode forward to calm them. Annabelle threw her rifle to the floor and ran, breathless, to where Kai still struggled in Wapi's fading grip. She skidded to the floor before them, and only when Annabelle's face appeared in his diminishing field of vision did he slowly, and with a palpable relief, let the daughter of Yohana go.\n\nKai struggled out, got up, turned around. When she saw the extent of Wapi's injuries, the strength of this precocious warrior failed her. Her whole attitude snapped, her eye widening and her face softening. Where moments before there were battle cries and blood-rage, there only now remained fear and confusion on the face of a little girl.\n\n\"Wapi,\" cried Annabelle.\n\n4.\n\nHER HANDS QUICKLY felt around his chest and back, assessing the wounds, but they were so disparate and numerous she was at a loss as to where to start. He was slick with blood, pale, his lips trembling as he fought against the weakness overpowering him. Annabelle howled in frustration and rage. Bedford came to her side but she batted him away.\n\nThe brave reached up, his eyes fogged and unfocussed, the pupils wide with death. He could barely keep his arm aloft. Kai crowded in but Annabelle pushed her gently away as she leaned down to Wapi, his blood-soaked arm snaking up to pull her closer.\n\n\"Yohana,\" he whispered. It was only now, Annabelle knew, that he displayed how frightened he was. \"Kai. Be a mother to her. Your husband...a father. More than I could be.\" He choked. \"Promise me, Yohana.\"\n\nBefore she could reply he was gone.\n\nSuddenly there was a battering on her side. Kai, a furious tempest, was pummelling her with bunched up fists, pushing ineffectively for Annabelle to move. Stunned by the fact that the boy she once teased scorpions with was gone, she half-moved, half-fell sideways. Kai shot towards Wapi's lifeless form, his head lolling horribly as the young girl tugged at him with arms that lacked the strength to lift him. Squealing, frustrated, she let his dead weight flop back and screamed his name, battering his chest with a child's ineffectual smacks. Annabelle, her back to the noise, thought she would soon stop. But she did not.\n\nShe was grey with shock; cold all over. Suddenly it all seemed too much, this paradigm shift of her life. In the space of only these last few paltry days it seemed as if everything had changed. Her own fate, and the fate of those who she loved, Nathaniel, Uncle Cyrus, George...Kai. There was Kai. The name, the concept, just seemed so alien to her among everything else that had happened. She had gained a husband, a daughter, found a long-lost friend only to see him snatched from her once more, only this time forever. She began to shiver, barely able to control herself. George rushed to her side once more and once more she angrily pushed him away. What more was there to do?\n\nIn the faint and still Arizona air, heard by her ears but not registered by her consciousness, came the drone of a cutter, the distant putt-putt of its engines stuttering as it wheeled in to land. Faintly she was aware of Boon's black-suited form running up to greet it, but he was hazy, indistinct. Kai still wailed behind her. All the strength that Annabelle had accrued through her hardships, her trying days, seemed to seep from her fingers and leach into the pale dirt. She wanted to press her face to it, feel the grit against her cheeks, go to darkness, let the world sleep without her. Even as she thought this, it disgusted her how she could feel so weak, so overcome.\n\nShe knew George still stood by her, powerless, and hated herself for making him watch her unravel.\n\nBoots, clumping, the stirring up of dust. The flashed blue of naval uniforms and the glint of gun-barrels in the sun. Sand was in her eyes, her unbidden tears failing to clear her vision, and so it was more the movement, the poise that she recognised when she looked to see him run towards her. She screamed his name in her mind, unable to produce the words with the self-loathing that choked her throat.\n\nShe grunted.\n\nHer arms felt like wet rope but she fought to control them, to haul herself up on all fours. She managed to do so, fell back onto her knees, he head hanging back and her arms lolling behind. The thought of Deerpak's rancid hands filled her mind and it was this image she fought against, that made her grit her teeth. She fell forward again, grabbed at the earth and pushed out, struck against it and managed to raise herself onto unsure legs. As she did so, Nathaniel Stone skidded to a stop before her.\nChapter Fourteen\n\nIllusions of Choice\n\n1.\n\nDISARMED, STRIPPED OF equipment, Enderby was led through the hot, dank corridors of Imperator like a prize on parade. He knew the two men that followed, their rifles aimed at his head and his back, would be keeping a steely gaze on his every movement. When he passed open hatchways off-duty Russian sailors would mockingly raise their tin mugs to him, jeering with rotten or knocked out teeth. Those in the corridors would flatten themselves against walls and hiss threats or derision, once or twice yelling feral obscenities an inch from his face, goading him into a reaction that would surely have him shot. As the procession turned corners, Enderby would flick his glance sideways and back, noting what he saw.\n\nThen he saw Folkard.\n\nThe captain, still dressed in his Russian uniform, was standing chatting with a pair of soldiers like they were old chums reminiscing at the bar. All three figures turned to watch Enderby pass, and as they did his eyes locked onto Folkard's. The captain's mouth tightened, his jovial chat interrupted, and for a moment the two men simply stared. Then Folkard nudged his compatriot, nodded towards Enderby and made a low comment in Russian. The three laughed bawdily, and as Enderby was marched out of view the mocking hoots followed him.\n\nFor over an hour he was marched around the ship. Everyone, it seemed, from cabin-boy to captain, had to take a look. Not that he was brought anywhere of importance, of course\u2014they didn't even approach the engine room or the bridge\u2014but surely that was half the point. This was all about humiliation, but no matter. Enderby's mind was still his, and though he knew now was not the time for action, he began to look for the possibilities of a plan.\n\nAfter his forced march through endless grimy corridors, the kitchens and the comm bay, he was at last taken down to the hangar where the cutters sat in rows like coffins. Enderby had learnt much from his tour; far more, he assumed, than the Russians assumed he would. The other side of the coin being, naturally, that he could learn as much as he wished in what little time remained to him. The knowledge, ultimately, would do him little good. He had no illusions that they were to kill him soon, but it wouldn't happen just yet. He knew the way these barbarians worked, and something else was planned\u2014something dictated by a power far greater than the boy soldier with a barrel to his back.\n\nEnderby was shoved roughly into one of the Russian's cramped cutters. There were no windows and the benches were made of tin plate; one had to shuffle to even attempt comfort thanks to the hexagonal rivets that stuck up from the seat. On one side sat the captive, on the other four surly, distrustful Russians that passed a foul smelling cigarette between them. The smoke choked the air and the ceiling was too low; it stank of oil and rough construction. With a jolt that nearly punted Enderby from his pew the cutter took to the skies.\n\nThe soldier directly opposite Enderby looked him straight in the eye. He took one last, deep lungful of his dry, powdery tobacco smoke and threw the nub to the floor without even bothering to extinguish it. Still holding Enderby's bold gaze the Russian\u2014the largest of the quartet by far\u2014reached behind him and produced a hessian sack with a rough hemp rope threaded into the opening. Grinning slavishly, he leaned forward and dumped the sack down heavily over Enderby's head. When he tightened the cord he did it with a heavy-handed sadism until the brute was sure the rope was cutting into Enderby's neck. He could barely breathe.\n\nThe Russian chuckled to his comrades as he sat back down. Enderby listened for a moment. There was no more noise.\n\n2.\n\nSHE WAS BEATING his chest, pleading with him, she remembered that. She remembered him holding her shoulders firmly, trying to look into her eyes as she shook her head to dodge his gaze. How can I leave now, she was screaming. Not now, I can't! George was there at her side. She felt the irritation of the Navy boys, a demeanour she had come to know so well, anxious to set sail. Their work in this wasteland was done. Nathaniel's head bobbed as he tried to look at her, Annabelle resisting, Geronimo looking on sadly from the sidelines.\n\nShi ma, came a voice. It was small but strong. Shi ma. The Apache word for mother.\n\nI'll find them, she remembered hearing, that small voice still strong. Me and Shiwoye, we'll find these men and we'll find each other again.\n\nShe had felt Geronimo's old hands, strong and gnarled as an oak, turn her round from her daughter so George could wrap his arms about her shoulder and lead her towards the cutter, Sovereign hovering placidly above. She had looked back, seen the strong, small girl and her wise old protector, the defiance in the eyes, the hope that her mother would travel to the stars and right a greater wrong; to unseat a more desperate demon. She kissed her hand and waved it to her daughter; the daughter kissed her own and waved back. The old man smiled. All three, as the door to the cutter slammed shut and the naval slang was yelled back from port to starboard, knew she would return.\n\n3.\n\n\"LET HER SLEEP!\" yelled Bedford, his knuckles closed and white.\n\n\"Commander,\" said Stone, without emotion, \"I know...\"\n\n\"You know nothing, Stone!\" yelled Bedford, slamming both fists on the desk. \"Nothing! It's like you're some bloody calculating machine, not a man anymore.\"\n\nThe two of them had retired to an unoccupied cabin on the starboard side. An ensign had brought them a tray of tea that sat cooling on the table by the bed. Bedford, in truth, wanted a cup desperately, but Stone's sudden and uncharacteristic callousness had infuriated him so much he wasn't sure he'd be able to pour without spilling it everywhere\u2014or pelting a china saucer at the professor's head. He just didn't understand it. Why was Stone suddenly so insensitive to Annabelle's health? It was clear she needed rest, even if only for an hour or two, before he quizzed her.\n\n\"By Christ man, don't you care?\"\n\n\"Of course I care, which is why it's so important.\" Bedford stared at him again. There was almost a flicker in Stone's eyes, like a tic, that he remembered the friendship he and Annabelle Somerset had shared, or the countless number of times they had saved each other's lives. Perhaps something had happened in Dublin that had irrevocably changed Stone. He did not know. He had briefly been introduced to Tally Cahalleret, Stone's companion on that trip, a gent who seemed roguish but dependable and who clearly had a lot of respect for Stone. If only, Bedford thought bitterly, I had the same right now.\n\n\"What's so important about it, anyway?\" he asked, finally overcoming his ire and pouring himself a cup of Earl Grey.\n\n\"The stump of the tree you mentioned. Tally and I saw one too, in the cave under Phoenix Park. This mineral, Commander, it's deadly. Its properties are manifold and utterly destructive. I need to find as much information about it as quickly as possible, even the smallest thing you or Annabelle might have noticed may be of the utmost importance. It's vital. Can't you understand that? For all our safety.\"\n\nBedford relented somewhat. At least now the professor seemed to be coming from a place that held their welfare at heart. \"Of course, Stone, of course,\" he sighed. \"But I've told you all I know, which isn't much. A stump, much like the one you yourself described. I fear you may have missed a trick, however.\" Stone arched his eyebrow and looked at across at Bedford, his head still.\n\n\"Trick?\" he snapped. \"What trick?\"\n\n\"Old Geronimo. He seemed to know a fair old bit about it, told us about how it used to be some gigantic glass tree. Said his forefathers had been looking after it for aeons.\" Stone shook his head sadly, made a gesture of impatience as he turned back around.\n\n\"With all due respect, Geronimo's stance on the crystal is unscientific. It's jumbled about with superstition and fables half-heard or badly retold, a ransacked and reconstituted oral history. No. What I need are facts. You and Annabelle have no small experience in these matters and, as you know, some sort of psychic resonance may be the key to a greater insight. Even the smallest observation may unearth a wealth of information, however insignificant it may have appeared at the time.\"\n\n\"Well there's one thing that isn't insignificant, dear chap\u2014the presence of that Drobate skeleton. Could they be behind the whole thing?\"\n\n\"In their current state? Unlikely, especially considering the age of the remains. But there must be a connection. As to whether the Drobates brought the crystal or were in search of it, I haven't the faintest idea. I'm working with assumptions and half-baked theories, Bedford! Is it any wonder I'm keen to get as much information on this whole sorry mess as I can?\"\n\n\"No, Stone, it's no wonder at all.\" There\u2014that terseness had returned to Bedford's voice. He set his cup back down before continuing. \"But Annabelle's health and recuperation must come first. There are...events of which you are unaware.\"\n\nThe professor cocked his eyebrow, and this time the mildly patronising gesture irritated Bedford to his core, making him start to hate the offhand and supercilious attitude Stone was taking to him, and Annabelle's distress in particular.\n\n\"Which are?\" asked Stone testily.\n\n\"It isn't for me to say. And if your emotional intellect was even a fraction of your scientific one, you certainly won't go asking Annabelle about it until she's damn well good and ready. And if you upset her, Professor Stone, by God you'll have me to answer to. It seems to me now that there are many things beyond the comprehension of a genius such as yourself.\" He snorted, sardonic. \"To think you were the man whom we always turned to for answers. I say that less out of spite, Stone, and more out of pity. But I warn you again: Do not upset my wife. She has been through enough.\"\n\nThe low thrumming of the engines was tempered by the gentle rattle of the teacup in its saucer as Sovereign cut through the lower atmosphere. Bedford was unsure as to whether his words had touched Stone, made him rethink his conduct, or that the professor had simply retreated within himself and was ignoring them.\n\nThe door banged open without a knock. The ensign who had brought them their tea was in the doorway, breathless. Bedford and Stone both stared at him.\n\n\"The bridge, gentlemen,\" wheezed the ensign. \"The captain wants to see you. There's been news.\"\n\n4.\n\n\"DECISIONS, DECISIONS.\"\n\nThey had landed in Calcutta some time ago. Shortly, Enderby found himself in a large stone cell, more-or-less a perfect cube twelve feet square. They had removed the bag from his head\u2014not before the Russian sailors had given him a few hard and cowardly knocks and dragged him here\u2014and now he found himself faced by a semi-circle of six malicious figures, arranged like dolls on show. Two stood to his left, three on his right, illuminated by lamps directed straight down. In the middle, directly opposite him, a man sat in shadow, the one who had spoke. There was no light above to show him\u2014here was a total silhouette, with nothing to discern beyond his outline. The voice buzzed in his ears. Russian accent, drawling yet urbane, and was there the hint of something deeply and disturbingly familiar about it? His head was spinning and his vision was fogged. In his current state he could not be sure.\n\n\"There's a choice on offer, Mister Enderby,\" droned the shadow.\n\nEnderby betrayed no emotion, staring ahead at the black form outlined by the other lights. He would not give this torturer the satisfaction of surprise, that his name was known; Enderby was resolute he would not give an inch. The blood from a cut on his eyebrow had dried down his face, and even with the smallest movements of his breathing he could feel this coagulated dribble start to crack and peel away. Still, he stared.\n\n\"Not the talkative type,\" came the voice again. \"Hardly surprising, considering your unenviable position. Poor, poor little British agent. Let me introduce you to the choices on offer. Kindly direct your gaze to the left? Now now, don't be petulant. A man of your training should take such an opportunity to know his enemies. So I implore you. You'll surely learn a lot. Excellent, so, going from your far left inwards...\"\n\nThe man on the far left leaned on a cane, dressed impeccably in green tweed and a brown, shallow-brimmed hat sloped at an angle. His face was ratty, sallow, looked pockmarked and ill, with a neatly trimmed moustache and a flat nothing of a chin. Clearly a degenerate, he smoked from a long ivory cigarette holder clamped between sharp, too-small teeth.\n\n\"This man, Mister Enderby, will kill you with fire and fear. Very quick, very violent, but not, it has to be said, neat. It's the word quick there that's the problem here. Not to mention that to waste his talents on a single victim is somewhat, if you'll excuse the pun, missing the target. So, onwards to the left. His companion.\"\n\nA great hulking thing, towering two heads above everyone else in the room. Wide and unmoving, dressed for winter even in these scorching Asian climes, the goggles that covered his eyes seemed white with the light's reflection. He held his hands dumbly by his sides, one holding a ragged and bloody stump of flesh. The hands were surrounded by some sort of metal brace, catching the lamps at odd angles and distorting their light.\n\n\"Now, Mister Enderby, this is more my style. I mean, really...look at him! Imagine what a giant like that could do to a man.\" He paused, self-consciously dramatic. \"Only we don't have to imagine. If you would, you vile monster, show the man what you can do.\"\n\nThe behemoth stepped forward. It only took him a couple of strides to reach the seat where Enderby was tied, and he leaned down without ever changing the set, thin-lipped expression on his face. He raised the hand holding the gore-soaked thing and slapped it down into Enderby's lap, as if he were tossing some begrudged debt. He raised himself upright again like a machine, stepped back into position. As he did, Enderby looked down.\n\nIn his groin sat a face. Or part of one, at least. It was the right-hand side of a skull, ripped from its other constituent parts, a claggy maroon around the edges and pale where the skin was, tattered at the side with bone-white splinters sticking out. Still damp on the underside, and soft. Even though the eye had been torn from its socket Enderby recognized the features.\n\nIt was half of the face of Edward Coyne.\n\nTrying not to show his disgust, Enderby bristled, and shook his knees sideways to dislodge the lost piece of that man. The brute looked on, impassive.\n\n\"I think that neatly demonstrates what our friend is capable of. Though I'm happy to elaborate. No? Then onwards still. Next in line, you'll notice, is me. Thing is, Mister Enderby, I don't much go in for the dirty work these days. As such, we'll pass on, and move to my right. Now, isn't she a peach?\"\n\nThe woman was clad in traditional Indian dress with studded brass plates stretching across her breasts and hips, fabric tightly hugging the seductive curves between them. She was flawlessly, effortlessly beautiful, with numerous rings and piercings in her ears, a snarl to her lips and cool lust in the vortices of her eyes. She had the poise of a ballet dancer and was ready to strike.\n\n\"This exquisite little example is quite the firebrand, Mister Enderby. She's been known to make men weak at the knees and she's oh so very smart. How will she dispatch you? Well, if those looks of hers don't get you first, she's quite the mistress of the khukuri. If she doesn't break your heart, she'll slice it out.\"\n\nThe woman smiled at Enderby cruelly, twisted her left wrist around twice and tugged. She was holding a black leather leash that arced down to the clasp at the top of a laboratory coat. The ingratiating animal dominated by the Indian siren whimpered, a frail and beaten, old, half-dead thing with yellow skin stretched across a scuffed, bruised face and hair finer than cobwebs stuck with dust. As with the giant, he wore black-tinted goggles, simpering while his hands quivered ceaselessly around his drooling mouth.\n\n\"Now, this one doesn't look like much, I'll be the first to admit. Indeed, if you could reach far enough to kick him you could kill him, and if you did you'd be doing him a favour. Ah, but the things he can do with poisons! He's a master there, not the wreck he appears. Agony for days, weeks, if he chooses. An assortment of chemical brutality or bliss, the doctor with the deadliest medicines. Excuse my foray into the florid.\"\n\nEnderby snorted mockingly. The man made a brief chuckle of satisfaction from his diaphragm, equally mocking in return. He waited for a moment, giving Enderby enough time to reply and break his stubborn silence. When the agent still refused to talk, the shadow of a man continued with the jolly tones of a salesman's patter.\n\n\"And so,\" he said, affecting excitement, \"last but not least, the wild card. The final player in this band. And what do you think of him, Mister Enderby, hmm? What might his speciality be?\"\n\nOf course this sadist would leave him for last. Years of training allowed Enderby to feign indifference in the face of trials and physical pain, but that didn't allay the sickening of his gut and the hot, pure anger that is commonplace to every mortal man. Out of everyone in the room, this last in the line was the only one who did not look at him. He gazed upwards, a bemused child distracted by a passing butterfly, his eyes unfocused and his face a saggy, lop-sided mess. His clothes were covered in filth and he looked malnourished, his arms dangling straight down by his sides.\n\nArnaud, thought Enderby. By God, what have they done to you?\n\n\"If I had to pick,\" the silhouette continued, leaning back in his chair. \"I think I'd go for our plucky fallen hero on your far right. All the others here, well...I've seen their work. Their craftsmanship is a joy to behold, but without putting too fine a point on it, I've seen it all before. This new toy of mine, however... What is he capable of? What would he do when commanded to kill? Ah, the thrill of Chaos, Mister Enderby. It's intoxicating.\n\n\"But maybe it's not for me to pick. And so this is the decision I have for you. If you had to pick, which one would you choose? Poison, dismemberment, the slice of the knife? In all sincerity I'd advise you, at this juncture, how inadvisable it would be to keep to your vow of silence.\"\n\nEnderby stared out, desperately trying to make out a feature\u2014anything\u2014of this man who sat and threatened him. But the darkness of the figure was absolute, like a slab of obsidian that drew in and jealously held any light. Still, he continued to stare. He opened his mouth to speak.\n\n\"Ah ah ah,\" interrupted his torturer. \"Before you decide, I should point out that your friends are on their way, so there may well be an audience. An audacious rescue plan is on their minds, I shouldn't wonder! Hope is such a wonderful thing. Don't you agree?\" He sat forward slightly, but his face was still utterly obscured. \"And with that on your mind, Mister Enderby, have you made your choice?\"\n\nEnderby swallowed and smiled.\n\n\"I'd have the coward do it,\" he said, his voice low and strong. \"You, you craven bastard.\"\n\nA short burst of genuine laughter emanated from the dark shape of a man.\n\n5.\n\nTHEY HURRIED THROUGH Sovereign's corridors, Nathaniel's long legs allowing him to take the lead ahead of Bedford and the ensign. Their haste seemed perfectly in tune with the mood of the rest of the ship. A restless energy buzzed in the air, naval ratings noticing their swift passage and putting it down, as ever, to trouble ahead. The whole crew seemed to sense that this was no ordinary mission; the rumble of the engines at full steam were felt with an uncommon clarity, and even the smallest tasks necessary for the ship's wellbeing were performed with an uncommon speed and precision.\n\nThe door to the bridge was open, and Stone sprinted in before jerking to a stop. Bedford was only a second behind him, and without even pausing to catch his breath he turned and ordered the ensign to fetch Tally, Boon and Annabelle\u2014so long as the latter were sufficiently rested. The ensign saluted and left.\n\nNathaniel always felt a thrill on the bridge; there was something indescribable about it, like looking out through the eye of a leviathan. The sleek wooden panelling and the brass glint of instruments that stretched before the observation window, the chink and hiss of delicately tuned apparatus worked by the nimble fingers of highly-trained men, men who went about their work with a quiet, intense pride. Yet there was a ghost at this feast. Even now it seemed as if Folkard's presence lingered still, a disquieting feeling which still seemed, Nathaniel noted with interest, to affect Commander Bedford. As if to hammer this point home, the captain turned from the bank of instruments at the helm.\n\nIt did not help that Captain Benjamin Theobald was no match for Folkard when it came to commanding Sovereign. His experience previous to his promotion had been wholly Earthbound, his rise to captain more a matter of politics than prowess. He knew this, of course, but was a proud man. He was loathe to admit Bedford's ability greatly surpassed his, yet remained an earnest and solid\u2014if slightly unimaginative\u2014leader. He seemed to have aged slightly; an increase in the grey of the mallen streaks around his temples, a darker shade of black around his eyes. His relief at seeing Bedford, his second-in-command, was almost palpable. It was as if a weight of culpability had been lifted from his shoulders.\n\n\"Gentlemen,\" said Theobald, nodding gravely. \"Let us not waste time. The matter in hand is twofold, and as ever, Commander Bedford, I would appreciate your views.\" The fact he admitted this was worryingly indicative of the seriousness of the situation. \"Firstly,\" he continued, \"we have received reports of a...presence over Calcutta. A battleship, apparently Russian, of a greater magnitude than we have ever witnessed. It is currently stationed above the slums, hovering like some Godforsaken bird of prey. Our forces deployed there dare not take any action against it. While it has not yet exhibited any antagonistic action, descriptions indicate that it is heavily armed and primed for firing. It is a threat, plain and simple.\"\n\n\"Surely Sovereign can match it?\" said Bedford.\n\n\"Commander Bedford, I have as much faith in this ship as any man on board, yourself included. But to put her in the line of fire would be too great a risk. Besides which, another matter has arisen, one which has a far greater bearing on how we should proceed.\"\n\n\"Show me,\" said Nathaniel quickly, stepping forward.\n\nCaptain Theobald picked up a small piece of paper lying next to the aether tiller controls. \"A heliograph,\" he said, \"apparently received from the comm deck of this mystery dreadnought. You'd... You'd better read it yourselves.\"\n\nHe held the paper out and Nathaniel snatched it away. Bedford crowded by his elbow to read.\n\nCyrus Grant is mine. Stop. Folkard is mine. Stop. Come one, come all. Stop. My blackbird will sing in Calcutta. End.\n\nThe note was followed by a set of co-ordinates. Nathaniel swallowed dryly, unconsciously crumpling the missive as he dropped his hands to his sides. \"Worse than I thought,\" he muttered. \"Worse than I thought.\"\n\nSub-lieutenant Barry turned in his seat from his station at the horizontal inclinometer.\n\n\"On the approach to Calcutta, sir,\" he said. \"Naturally enough the captain ordered that we keep our distance for now.\"\n\n\"A wise move,\" said Nathaniel. \"Seeing as it's clearly a trap.\"\n\n\"What choice do we have?\" asked Bedford. Though it was said aggressively, the statement held no malice. It was merely a reflection of powerlessness, the panic of the ensnared. \"We've been chasing shadows, Stone. Putting ourselves and our friends in danger, and for what? To turn around and go home? If there's hope we can fight back, hope that there's the slimmest chance that what we know to be right could prevail? Who are we to turn our backs?\"\n\nSovereign's broad observation window, hitherto misted and specked with the condensation of cloud, began to clear. Suddenly, as the last of the wisps of the nimbus cleared, the city spread before them like a toy thing, a military map on which figures of soldiers were placed and knocked down. The splendour of Calcutta's noble city, defined by solid brick buildings, wide boulevards and white brickwork, spanned to their left in a serene and careful order. Then came the river, a meandering muddy split, and beyond that the smear of the slums, a cancerous mould of smoke and ruin.\n\nAbove the slums hung the great black battleship. Even from such a distance its scale was obscene, its stillness in the air a promise of violence. Like a bruise against the blue of the sky it sat there, angular and squat, the beckoning bully-boy of the horizon, just waiting for the weak to get close enough to punch.\n\n\"The result of the stolen blueprints... There is no choice,\" said Nathaniel. \"All through this... There never was.\"\nChapter Fifteen\n\nThe Horizons of Deceit\n\n1.\n\nTHEY BEGAN THEIR final descent.\n\nThe rumble of the cutter belied the silence. No-one spoke. Stone's cutter was the first down, a small battalion of marines following in the second. Boon and Tally sat side-by-side. They had started chatting soon after being introduced, and even in these tense times a burgeoning friendship was apparent to all. They had spent the trip to Calcutta telling each other stories of times long past and just gone, with Tally quizzing the experienced agent on what Tooler's offer might entail, and all he might have to look forward to in the service of the Bureau.\n\nAnnabelle was rested, but remained edgy. Bedford knew it was pointless telling her to remain aboard Sovereign. For starters there was no guarantee she would be any safer there. Secondly, if she were with him he knew he could protect her better, and thirdly... Well, thirdly he knew if she were with him then he had his own angel of protection. If she were elsewhere he knew he would fret about her safety, become distracted, maybe even make a fatal mistake. With her by his side he felt better in a multitude of ways.\n\nThe co-ordinates turned out to be a small patch of wasteland half a mile from the North-Eastern side of Black town. As the cutter approached, Barry turned back from the cutter's helm.\n\n\"They've got forces waiting, Commander,\" he said to Bedford as he eased their descent. \"Russian marines, a couple of dozen. Tanks too. Two other figures there, I can just about make out... Though one of them ain't hard to miss. Christ alive, he's the size of a shire horse.\"\n\nNathaniel and Tally caught each other's eye.\n\n\"We need to be very, very careful,\" said Stone.\n\n2.\n\nKLOPSTOCK, HIS EVER-PRESENT cigarette holder clamped between his peg-like teeth, puffed contentedly as he brought the binoculars down from his eyes and smiled. The holder jutted up in his mouth like an antenna.\n\n\"Second one!\" he said, in his quick, nasal tones. \"Has to be. If you will, Mister Potsdam.\"\n\nPotsdam lurched forward, his tread heavier than usual. Strapped to his right arm with leather braces three inches wide was a boxy, mechanical contraption the size of a small chest of draws. The corner opposite the straps had been cut away to reveal the mechanism inside; a huge and tense lateral spring poking from the bottom in a semi-circle. The whole device was patterned with Klopstock's trademark flourishes, oak polished to a sheen and gently carved fractal patterns. Parallel to Potsdam's arm was another arm of less organic origin\u2014that of a finely wrought, clipped-in catapult. A small, brass crosshairs flipped up automatically as Potsdam hefted the weight perpendicular to his body. His smallest movements mirrored that of Sovereign's second cutter; he stepped back, clenched his fist, and fired.\n\nThe catapult arm shot forward, releasing a cylinder that drew a faint black-green mist behind it. It arced into the sky with perfect aim. Potsdam reeled from the recoil but quickly righted himself, watching as the projectile flew, the peaceful and soundless seconds as it whipped through the air. It drew a perfect curve, catching the second cutter square on the port midship. The cutter shot horizontally some thirty feet; far more so than would be expected from a conventional explosive. After that, there was no pretence to lift, no hope left in those engines or for the luckless men that burned inside. The cutter dropped from the sky in a graceless plummet, slamming to the ground without fire or fanfare, a mishmash of metal in which no-one could have survived.\n\n\"Y'know,\" said Klopstock, turning around and patting Potsdam on his free arm. \"The nicest thing about that was how it was all so unnecessary.\"\n\n3.\n\n\"YOU SAW WHAT we did,\" called Klopstock from several feet away, as the door to the cutter swung open. \"I assure you we've rewound our little trebuchet again, should you desire to leave this world as a flaming mess. Keep your weapons down, your movements slow, and...\" He squinted at the figures filing out of the surviving cutter. \"Why, dash it. It can't be.\"\n\nHe screamed, loudly and uncontrollably.\n\n\"Stone!\" he screeched. \"Stone, you bastard! Baaaaah-stard!\" He started stamping, then jumping up and down on the spot in an irritated little circle. \"And, and...that bastard Mick you dragged along! Here's here too? Bastards, bastards, buggery cripes and blasphemy! Don't you understand!? That was my masterpiece, the months I spent crafting... And you walked away! Walked away like, like...a pigeon! A pigeon from a... Gah!\" He started to hop around again.\n\nHe finished his tantrum and breathed out heavily. He pinched the bridge of his nose, regaining some semblance of control. Potsdam remained impassive. The bomb-maker turned and addressed the Russian platoon.\n\nNe trogat kater. Okrujite ih. Svyajite ih. Zavyajite im glaza.\n\nThe Russian soldiers began their advance, marching in rows, their bayonets ahead of them.\n\n4.\n\nTHE JOURNEY IN the tanks had been rough and uncomfortable, but otherwise the group had been handled gently. Even Klopstock ignored the possibility of a swift, cheap punch while Nathaniel was prone. When they had reached their destination they were ushered out with the same firm but undamaging insistence, each of the party with a Russian hand on their shoulder, guiding the blind and bound.\n\nWhen the masks were removed, Nathaniel and his party may well have wished their captors had the compassion to keep them in the dark. But compassion was in short supply in the monstrous scene that confronted them. The first thing they saw when the blindfolds were removed was a charnel house of corpses stacked around a green and glistening stump.\n\nThey were in a rough cave, the stump sat in a shallow dip. The bodies of the slum's disaffected and disposable lay in piles, arms and legs entangled, elderly and children alike wrapped in rags. Wide yellow eyes stood out like headlights from emaciated faces, bones jutted from hips, a withered pile of discarded human jackstraws going soft in the omnipresent heat. The buzz of the flies was almost deafening. A semi-circle of Russian rifles was pointed towards Stone and his compatriots, steady, unmoved by the stench and the slaughter. A Russian thug on the right of the line stepped forward, producing a glinting dagger from his waist with a soft, sharp schink. He approached Nathaniel and stood behind him. The barrels of the rifles continued to point.\n\nA thick Russian voice came from behind, made all the more ominous by its mixing of the malevolent and the horribly familiar.\n\n\"Nathaniel Stone,\" it said. \"No, don't turn around.\" As his name was said, Nathaniel felt the rope that bound his hands to his back cut away.\n\n\"Commander George Bedford. Welcome.\" The soldier with the knife stepped to one side, snipped away George's restraints with a flick. He moved along the line as each name was intoned.\n\n\"Mrs Annabelle Bedford, n\u00e9e Somerset.\" When this was said, the voice seemed to crack...Annabelle's maiden name pronounced with an almost American twang. Annabelle stiffened.\n\n\"Charles 'Tally' Cahalleret. And last but not least, Mister Bertrand Boon. Welcome. Welcome all.\"\n\nThe thug with the knife rejoined his line, slung his rifle up and pointed it at the group once more.\n\n\"Now you may turn around. No, wait,\" there was a pause. \"Now I shall allow you to turn around.\"\n\nThe prisoners did so in unison as the voice continued. \"You may think me proud for unburdening you of your bonds. You may think me foolish. But you think that from a lesser position, one of weakness, a place that hopes for the smallest of victories. Let me assure you that no such victory is possible. I allow you total freedom because you pose no threat to me; none at all. No more than a fly can pose a threat to a bear. You are useless. Inadequate. So much so, in fact...\" He spoke a command in Russian. Vy svobodny. Vernites na korab. The guards lowered their weapons, faced to the left as one, and marched out.\n\nWhen they turned, all of them\u2014Nathaniel, Bedford, Annabelle, Boon and Tally\u2014were confronted by a raised stage, behind which lay the grubby windows of a small laboratory. A row of figures stood on the platform. Potsdam and Klopstock, a beautiful Indian woman who held two chains in her arms, one attached to a decrepit walking corpse, the other to Arnaud, now a grinning, insensate fool. Nathaniel almost flew forward at that point, to wrench his friend and lover from that far-off grasp. In the centre of the room, the legs of his chair jammed perilously into another, smaller pile of Calcuttan bodies, was Enderby, his mouth taped up, breath ragged through his nose.\n\nAbove this ghastly scene was the face of a man whose appearance made Nathaniel's blood run cold.\n\nHe sat in a high-backed chair, the comfort of it incongruous compared to the death and desolation over which he presided. He was leaning to one side, one knee across the other in a pose of total relaxation.\n\nIt was the face of Cyrus Grant, his eyes ablaze with madness.\n\nNathaniel glanced at Annabelle. The horror in her eyes was intense, although how she managed to keep the emotions from her face was beyond Nathaniel. Bedford looked at his wife, concern and anger quite clear on his own countenance.\n\nSmiling like he owned the world Grant looked down upon his captives. Since their last encounter in the Admiralty labs his face had grown pale and hard. He had shaved and his hair had been neatly slicked back across his skull; a different man entirely to the one who had acted as Annabelle's guardian, who had worked with intense rigour alongside Nathaniel to perfect the most wondrous scientific advancements of the age. Insanity, finally, had consumed his last shred of morality.\n\n\"Uncle!\" cried Annabelle, her voice faltering with heartbreak.\n\n\"Annabelle, don't,\" warned Bedford.\n\n\"Don't what!?\" spat Grant, and again his voice seemed to crack and once more take on his usual transatlantic inflection. \"Don't reason with him? Don't play his games? Look at me, Annabelle.\" He slammed his fists down onto the arms of his chair. \"Look at me, all of you! Do you really think\u2014and I say this to you in particular, Professor Stone\u2014that I would have ever granted you even the smallest chance of success? All of you, no more than puppets, dancing to my tune across the four corners of the Earth.\n\n\"You saw those stumps of the mineral trees because I wanted you to see them. You cannot imagine how long I have been working against you. Ever since the British government kindly allowed me to set up the research team, while they sent you off on your mission. Guided by the Heart, but working for me without even knowing it! We finished mining weeks ago! All I needed was a little more time to complete my plans, so I got your attention with that little bang on Horseguard's Parade, split you up and sent you toddling off to buy the time I needed. It was insultingly easy.\n\n\"Admittedly, Professor Stone, I would have preferred to have you die in Dublin, but life is full of such little disappointments. More so for you than me, naturally. And yet my victory is only just beginning, and it pleases me that you will witness it\u2014you, above all, will appreciate all I have accomplished. You see, this epic deception of mine was just the start. Your dissolute Empire would never allow a man such as me any real power. You feared my genius, all of you, and so I sought out an ally that would respect and cherish it. I will find Plypolyplon, Stone...\"\n\nAt this Nathaniel smiled; it was subtle, and with everyone's attention focussed on Grant, none noticed.\n\nNot even Grant, who continued in his rant. \"...and from thence the might of Russia will grow to encompass the entire Solar System. And I will not do it without assistance. There is one final deception, my dear old friends, a betrayal so perfect as to make my ascension truly immaculate. Here is your worthless spy, Enderby. But what of his cohort? What of your friend, the gallant captain?\"\n\n\"Folkard,\" breathed Bedford. \"Dear God, no...\"\n\n\"Dear God yes, Commander Bedford. At present Captain Folkard awaits my return aboard Imperator. Together, he and I will traverse the asteroid belt to discover the worlds beyond. You took me for a raving fool, but the brave, stout, loyal Jacob Folkard... How could any of you have ever doubted him?\"\n\nGrant watched with a sadist's smile as the weight of his words crushed his captive's spirits.\n\n\"I have achieved everything I needed to here. Imperator awaits me.\"\n\nHe stood to leave, brushed an imaginary speck of dirt from his lapel, and turned to the room. \"Potsdam, Klopstock\u2014kill them all. Use your imagination. Moonsinge, come with me. I still have need of your talents.\" He started to stride towards the door to the lab, but stopped suddenly and turned, theatrically raising his index finger as if he'd remembered something \"Oh! And, Professor Stone?\"\n\nNathaniel glared at him, his expression impassive.\n\n\"They're going to make sure you die last, old friend. The frantsuzskiy korova will not be happy, but...\" Grant waved it away with a smile. \"Enjoy the show.\"\n\nAnnabelle, so silent during the whole exchange, suddenly yelled. It was a roar of frustration, of hatred, of pure, blind rage. She whipped to her right with remarkable speed and her hand went into Boon's jacket. Quick as a dart she had pulled out Elizabeth, his custom machine-pistol, took aim and fired across the room at Grant, the shot only just missing Potsdam's head as the brute lumbered towards them. A feral grin bisected her face as she watched the bullet fly.\n\nGrant's crowing gesture was his downfall. Had he not raised his arm to mock Nathaniel, he would never have lost it.\n\nThe round hit Cyrus Grant near the crook of his elbow; there was a flowering of flesh with the almighty burst of force. His arm twisted away and dropped in a pathetic arc, its weight snapping tendons as it fell. For a second there was no blood, and then it flowed in a sickening spatter to the floor. Grant stumbled, grimaced\u2014but he did not fall and he did not scream. He leaned back on his back foot and regained his posture.\n\nHe looked Annabelle in the eye as the smoke from the pistol barrel rose and wisped away. His teeth were gritted, his right hand clutching the torrenting stump by instinct. \"Give her pain!\" he half-hissed, half-screamed to Potsdam and Klopstock. \"Give her pain beyond the limits of imagining!\" He turned to Moonsinge and let her cradle him as she led him away into the laboratory and beyond, dropping Garrecreux's leash as she went.\n\nAs Potsdam passed Enderby in the centre of the room he struck out and batted the bound agent sideways with his tree-trunk arm. Enderby grunted as he fell, but still Potsdam advanced, each step as slow and lethal as the turning of the tide.\n\nTally watched carefully as the brute stomped towards them. He turned to Annabelle and grabbed the gun from her unresisting fingers. Potsdam was no more than four feet from him. Tally aimed and screamed as he pulled the trigger time and again, each bullet shredding the thick black greatcoat and bouncing off the frame that held the monster's rotten bones. His flesh inside was revealed; a sour milk stench filled the cavern. The rancid zombie torso was punctured and bleeding, but still the behemoth advanced. Tally tossed the empty gun away. He breathed deep and hard through his nose; he concentrated.\n\nHe reached back into his pocket, pulled out the penknife and with a swiftness honed on dangerous streets flicked the blade out. Potsdam reached towards him with a huge metallic hand but Tally, more lithe, dived out of reach between his arms. There was just enough room, and he had just enough skill, to thrust the blade forward though the gaps in Potsdam's brass ribcage and plunge the point into the beast's modified, ungodly heart. Pale green ooze seeped weakly from the wound; the monstrosity spasmed and twitched, uncomprehending. As the abomination fell back, his face still an unfeeling slate, he pulled Tally with him, the Irishman's arm still caught between the bars of the cage that held Potsdam's rancid and desiccated corpse. Tally screamed as his wrist snapped.\n\nBoon dashed across to free Enderby. Klopstock, his eyes wide, fled suddenly to follow Grant and Moonsinge. Meanwhile, on the platform, another death was unfolding. Garrecreux, seeing his beloved mistress abandon him, had begun to crawl towards the entrance to the lab, but Arnaud, still lost in the vile chemical's thrall, grabbed his ankle and pulled him backwards. The degenerate scientist twisted onto his back as Arnaud pinned him down with his legs, his arms reaching up to the old doctor's throat.\n\n\"Le vert?\" Arnaud hissed through gritted teeth. \"O\u00f9 est le vert majestueux?\"\n\n\"L\u00e2che-moi, laissez-moi!\" Pleaded Garrecreux. \"Je peux vous donner c'est... Je peux vous donner tout cela!\"\n\n\"Pas assez,\" said Arnaud, wrapping his hands around Garrecreux's neck. He gripped tighter, furious, the old man's eyes bulging as he choked. He had soon lost the strength to even fight back.\n\nA hand on Arnaud's shoulder whipped him around. A fist, tight and bunched, connected with his jaw within a second. Arnaud fell unconscious as Nathaniel waved his bruised knuckles in the air. He looked down. It was too late for Garrecreux. This was no great pity.\n\nBoon was helping Enderby to his feet, the senior agent pushing him away as if insulted. Nathaniel caught Boon's eye, called him over. \"Pick him up and bring him,\" he said, indicating Arnaud. \"He's one of us.\" Boon nodded and hefted Arnaud's passive form between his shoulders.\n\n\"We need to get back to the cutter,\" said Bedford. \"And fast.\"\n\n5.\n\nTHEY EMERGED FROM a rocky tunnel into an abandoned shell of a building somewhere in a deserted corner of the slum. Sovereign hovered some way to the East, and they dashed towards it. It did not take them long to reach the plain on which the remaining cutter sat.\n\nArnaud, still unconscious, had been laid between the benches. Bedford chivvied the pilot into getting the cutter airborne as quickly as possible. With all of them inside it was a bit of a squeeze, Boon and Bedford allowing Annabelle and the injured Enderby to sit. Tally nursed his broken wrist but brushed off any attention from the others\u2014he'd had worse. Bedford pulled Nathaniel to one side.\n\n\"Is it really wise to pursue them?\" he asked quietly.\n\n\"What other choice do we have? It may be the only way to discover what Grant's been working towards all this time, what he's been using these crystals for.\"\n\n\"And if it's some sort of aether weapon? What's stopping him from simply turning about and blasting us out of the sky?\"\n\n\"Nothing. Hardly a comforting thought.\"\n\nAs if in acknowledgement, the cutter was suddenly buffered by turbulence.\n\n\"And then,\" muttered Bedford, looking down, \"there's Folkard.\"\n\nNathaniel nodded sombrely. \"Indeed. Though I am far more inclined to believe Grant is a liar than Folkard a traitor.\"\n\n\"Folkard's a traitor alright,\" growled Enderby. The duo turned, surprised, not even aware he'd been listening. \"He'll hang.\"\n\nA tinge of spite crept into Nathaniel's voice. \"Mister Enderby, I assure you I have known Captain Folkard for quite some time now...\"\n\n\"Well clearly you didn't know him well,\" said Enderby simply, his voice still low. \"I saw him on Imperator, laughing and joking with his damn Cossack friends. He's turned, Stone. I'm sure of it. Why else wouldn't he have attempted to rescue me?\"\n\nNathaniel, unsure, stayed silent.\n\nBedford noticed Annabelle staring ahead of her, the faint ghost of a smile slightly curling her lips. He pushed past Nathaniel and knelt in front of her, careful not to tread on Arnaud.\n\n\"Annabelle, are you all right?\" Her reverie broken, she looked up at him suddenly.\n\n\"Of course I am, George.\" She said it matter-of-factly, as if in a conversation in a drawing room. \"Why wouldn't I be?\"\n\n\"After everything we've been through, the last few days... Your uncle...\"\n\nAnnabelle smiled cruelly. \"He's not my uncle, George. Not anymore. He betrayed us. Tried to have us killed. His madness was one thing, but this...?\"\n\nThe hatred in her eyes was palpable.\n\nNathaniel watched this exchange, and tilted his head slightly, considering all that Annabelle had been through. What future would she have? What kind of woman was she becoming? It was clear that it was also something Bedford was wondering, too. Nathaniel looked forward to finding out.\n\n6.\n\nFOLKARD DESPAIRED. HE was unsure how much longer his pretence could keep up, and was feeling a great unease regarding Oleg Olkhovsky and Valentin Utterklo, his two new best friends.\n\nThey had accosted him as he was making his way towards the comm deck. If Imperator was anything like Sovereign it would be staffed by at least six men around the clock. If he was lucky he might be able to get inside on some pretext and discover any changes between Imperator's version and Sovereign's\u2014and how he might exploit them. Either that or somehow lure out, incapacitate and replace one of the heliograph operators there. Folkard didn't like it. It was a plan in which far too many things could go awry, but in his present desperate state it was the best he had.\n\nHe had first spotted Olkhovsky and Utterklo striding down one of the corridors towards him, just as he was making headway towards his objective. He quickly noted their insignias\u2014both the Russian equivalent of a petty officer. In many ways, it had been fortuitous the uniform he had acquired belonged to a mere rating, as the majority of the crew ignored him and it made it easier to avoid the sight of officers who may have been better informed of Folkard's appearance. The two men were burly, seasoned sailors who took up most of the corridor. They muttered unintelligibly to each other, and did not seem to have noticed him yet. Adopting a stance of respect rather than caution, and careful not to move too quickly, he moved to one side in the corridor to let them pass.\n\nAs they approached, one of the men slapped the back of his hand to his companion's broad chest, and pointed directly at Folkard.\n\n\"Oi, you,\" he barked in Russian. \"What're you doing?\"\n\nFolkard stepped out from the shadows with his chin high, recalling the days it was necessary for him to show deference to a superior officer. \"Message for the comm room, sir.\"\n\n\"Message my arse,\" chuckled the other petty officer. \"What are you really doing?\"\n\n\"Skiving, that's what he's doing. Probably got some poor muzhlan on the floor he's supposed to be mopping. What's your name? C'mon, out with it!\"\n\nFolkard's mind raced. He found himself instantly replying \"Kuznetsov, sir.\"\n\n\"Well, Kuznetsov, let me tell you this: I've never liked lazy men, and nobody gets to your age without being promoted unless they're a very, very lazy man. Christ above, even the galley boys get to cook a borscht sometimes.\"\n\n\"What'cha reckon, Oleg?\" said the other sailor, rubbing his chin. \"Reckon we should teach him what hard work feels like?\"\n\n\"Oh, yes indeed, Valentin. Yes indeed.\"\n\nThey had frogmarched him away from the comm room. It didn't exactly feel like he was being press ganged, and Folkard could well imagine this being a tactic these downtrodden, frustrated bully-boys often employed against those they felt they could abuse. He had to hope that was the case now, and not that his deception had been uncovered.\n\nWhen they put him to work, scrubbing the mess floor on his hands and knees, it almost made him feel relieved. If his true identity had been known he'd be straight to the brig and probably beaten to boot\u2014if a little expenditure of elbow grease was the alternative Folkard considered himself fortunate.\n\nOlkhovsky and Utterklo had not left his side. They now both leaned with their backs to a table, crossing their arms and watching him in silence as he scrubbed.\n\nUtterklo called out. \"Where you from then, Kuznetsov? Anywhere nice?\"\n\nThe time spent labouring had given Folkard ample opportunity to fabricate a story. From their accents these two were Moscow born and bred, so he picked somewhere provincial and replied without looking back.\n\n\"Kimra. My father was a cobbler.\"\n\n\"Oh, Kimra. Your sister lives in Kimra, doesn't she, Oleg?\"\n\n\"Used to.\"\n\n\"What made you leave then, Kuznetsov? The shoemaking life a bit too much like hard work, was it? Thought you could get your meal ticket from the Imperial Navy?\"\n\n\"Something like that,\" said Folkard. \"I guess I just wanted to see the world.\" Olkhovsky snorted derisively.\n\nAs he worked they carried on quizzing him. How did he get into the Navy? Why the hell hadn't a man his age ever been given a promotion? And finally, the one that stung sharpest of all; \"You got a sweetheart back home, Kuznetsov?\"\n\nHe answered each of their questions diligently, avoiding unnecessary detail that might trip him up later. After he'd scoured the whole floor\u2014his hands red raw and wrinkled from the tepid, greasy water\u2014it was almost like the three of them were old friends. Utterklo wandered across to one of the tables and hooked three tin mugs with his fingers. As he wandered back he used his free hand to produce a small brown bottle from his back pocket.\n\n\"Here, Kuznetsov, c'mere,\" he said as he poured. It was more an order than an invitation, and he eyed Folkard suspiciously as he handed him the drink. \"Keep it to yourself but I've a friend in the sick bay. Amazing what he can do with a potato, imagination and a few glass bottles.\" Folkard took the mug. Its contents smelled like kerosene. \"To the Tsar!\" Utterklo said with a smile, and held Folkard's eye.\n\n\"To the Tsar!\" parroted Folkard.\n\nAs they brought the mugs to their lips a commotion came from outside. Olkhovsky jumped down from the table he was perched on and glanced out the door before looking back to grab Utterklo and Folkard's attention.\n\n\"Hey, you two,\" he said, beckoning. \"Come and look at this.\"\n\nUtterklo pocketed his moonshine and the trio wandered out into the corridor. What Folkard saw made him glad he had a drink in his hand.\n\nHe had had to force himself to keep absolute control when he saw Enderby paraded by him. The agent's face was steely, unperturbed by the insults and catcalls yelled at him. He looked pale but was otherwise unharmed. Folkard quickly assessed the situation\u2014two soldiers led Enderby and two brought up the rear, but with all the hullabaloo it seemed the entire crew was on deck, watching the captive at the height of his humiliation. Realistically, there was nothing Folkard could do at present. That Enderby had been captured was a blow, but he was still at large and while Imperator still sailed his mission was clear.\n\nOlkhovsky hovered back after a brief chat with one of his shipmates. \"English spy, apparently,\" he noted casually as he knocked back his drink. \"Wouldn't want to be in his shoes, eh, Kuznetsov?\"\n\n\"Not for all the tea in England,\" Folkard quipped, and Olkhovsky and Utterklo roared with laughter.\n\n\"You're all right, Kuznetsov,\" said Utterklo, slapping him on the back. \"Look, we've all been a bit uptight recently, what with everything going on. Big changes happening, too much bloody work, and if clapping some slimy British opezdol in irons isn't reason to celebrate, I don't know what is. Come, drink with us.\"\n\nOnce more, it was not an invitation, but an order.\n\n7.\n\nTHE BRIDGE WAS ablaze with activity. Captain Theobald had, with no small relief, shunted himself to the background to allow Bedford to take control. The only vestige of command that remained was his pompous and upright posture. Bedford yelled commands, directing his men, wringing every nanowatt of power from the engines and ensuring each carefully calculated movement of the ship directed it as swiftly as possible through the aether.\n\nNathaniel suddenly burst in, skidding to a halt. He reached out an arm and steadied himself on the command chair, speaking in breathless bursts.\n\n\"I've tweaked, George...done as much as I can...without totally overloading the propeller governor. Three, four knots perhaps. Anymore and we risk tearing Sovereign apart.\" He breathed in deeply and stood up, his head spinning.\n\n\"Boswell agree?\"\n\nNathaniel nodded.\n\n\"I defer to your judgement, then. Sub-lieutenant Barry, what news of the Russian vessel?\"\n\n\"Every time we accelerate it simply pulls ahead, sir. Outstripping us without even breaking a sweat. Whatever's powering that beast is beyond me.\"\n\n\"Are you sure,\" said Bedford, turning to Nathaniel, \"that this is as fast as we can go?\"\n\n\"Are you?\" replied Nathaniel, not without humour. \"Any more and we'll shake the bolts from their housings.\"\n\nBedford nodded solidly. \"Very well,\" he said. \"We'll chase these bastards into the very mouth of hell if we have to.\"\n\n\"Commander Bedford.\"\n\nBedford turned. Boon was standing before him, bolt upright.\n\n\"If I can help, I will.\"\n\nThe commander slapped him on the arm, grinning. \"Good man!\" he said. \"Not beyond your remit, though?\"\n\nBoon smirked. \"I'm sure we can figure something out.\" Bedford slapped his arm again.\n\n\"As can I,\" said another voice behind Nathaniel.\n\nThe three men turned. Annabelle stood in the doorway to the bridge defiantly, legs apart and her hands on her hips. She brushed off their stunned looks. \"Well, what do you expect me to do? Macram\u00e9?\" she said brusquely, striding into the room. \"If I can help, I will. We're short-handed as it is. You, darling husband, are needed on the bridge. If anyone can make heads or tails of what the Russian ship is all about, it's Nathaniel...even from a distance. Enderby is as battered as a snake in a mongoose den, Tally's wrist is badly broken and Arnaud...\" Suddenly, her vigour abandoned her, and she looked at Nathaniel sadly. \"Poor Arnaud...\"\n\n\"How is he?\"\n\n\"Unconscious. Still. But restless. You should go to him. You may be able to do something.\"\n\nNathaniel frowned.\n\nBedford strode up to the banks of instruments before the wide window through which blind night winked with stars. Ahead of them, the moon was the size of a ship's biscuit, but bright and clear, the Russian ship an ink-blot against its pristine surface. He breathed in. Hands behind his back, he turned to his men. Took in the sight of his wife and his friends.\n\n\"Onward,\" he said, \"into the unknown.\"\n\n8.\n\nHAVING FOUND HIMSELF at rather a loose end on the bridge, Nathaniel decided to take Annabelle's advice and visit Arnaud in the sick bay. When Nathaniel had left him there, what fleeting moments of fevered lucidity the Frenchman had were filled with rambling shouts and spasms. He sweated almost constantly but was cold to the touch. The pernicious drugs he had been forced to take were slowly leaving his system, and as they did so they dragged suffering and anguish with them, displaying an almost organic will to torture the man who dared to abandoned their embrace.\n\nHe had witnessed first-hand the way this insidious chemical could ravage the human body\u2014a cursory glance at the bodies of Potsdam and Garrecreux would communicate that to even the dimmest of intellects. Their usage had been an addiction, taken to the extreme, and Nathaniel only hoped that Arnaud's relatively brief spell under its destructive thrall was reversible. He chastised himself for losing the sample of crystal in the Admiralty. What a fool he had been to take his eyes from it, even for a second! If nothing else he could analyse its structure and composition, perhaps synthesise some sort of antidote or dilution to lessen Arnaud's terrible symptoms...\n\nThese fretful thoughts were suddenly curtailed by an unmistakable noise\u2014gunfire! Two or three shots, a pistol by the sound of it, that came from the direction of the sick bay. With an icy dismay that hit him cold and hard in his gut, Nathaniel broke into a run.\n\n9.\n\nOLKHOVSKY AND UTTERKLO, their arms around each other, were onto the sixth verse of some interminable Russian drinking song. They had manhandled Folkard to their quarters, all the while promising him the best home-made vodka he'd have tasted since leaving, well, home. He protested as much as he could, made up a whole sorry story about an unforgiving, sour-faced skipper who'd been gunning for him from the second he stepped aboard, but to no avail. The two petty officers simply brushed his concerns aside, assuring him they'd have a word and smooth things over as soon as the drinking was done.\n\nWithout taking a breath they launched into the seventh verse. Between them they had managed to consume a bottle and a half of the rancid concoction, which burned the throat like wildfire and had a metallic, greasy taste. This, Folkard thought, was the kind of thing normally reserved for dissolving piston grease, not drinking. In as much as he could he had avoided the stuff, either only pretending to pour himself a shot or slyly disposing of it when the two burly Russians were distracted. More than once, however, when they proposed a toast and their eyes were on him, he found himself having to drink. He could feel the stuff stinging his stomach and making his head light, but he'd be damned if some Siberian rot-gut would jeopardise this mission. Still he played his part, cackling along with some off-colour anecdote about a one-legged prostitute and joining in with what few words he'd picked up of the song's chorus.\n\nUtterklo emptied the rest of the latest bottle into his mug and upended it theatrically. He slurred a curse and rose to his feet, but the movement was too much for his sodden brain to take. He wobbled for a moment, tottered and pitched forward, crashing headlong into the ground. Olkhovsky burst out laughing, a deep and rough sound. Folkard joined in.\n\n\"Looks like we have a winner,\" smirked Olkhovsky, getting down on his hands and knees to crawl across to Utterklo, who had started snoring. Olkhovsky nudged him playfully. \"Hey, hey myshka... Now's not the time for dozing, we've gotta look after Kuznut... Kaznit... Ach, you know who I mean.\"\n\nHis senses dulled by booze, Olkhovsky didn't have even have time to react as Folkard brought one of the empty bottles down sharply onto the back of his head.\n\n\"Sweet of you, Olkhovsky,\" muttered Folkard as he pilfered the unconscious Russian's pistol and checked the chamber. \"But I'm quite capable of looking after myself.\"\n\n10.\n\nNATHANIEL RUSHED HEADLONG toward the sick bay, any thoughts other than those for Arnaud's immediate safety stricken from his mind. He skidded around the door frame to the sick bay, his lungs fit to burst. He took it all in in a microsecond\u2014the crumpled body of a sailor, hunched into a foetal position and deathly still. His blood was mingling with the soup he had been carrying in for Arnaud. Stooping over him was an orderly, who had just finished resting the poor man's head to the ground.\n\n\"You man!\" Nathaniel snapped. \"What the devil's happened?\"\n\n\"It's Doctor Fontaine, sir! He must've got hold of his pistol, shot him there and then!\"\n\n\"Which way did he go? Quickly, man!\"\n\n\"Ah, he must've gone towards the stern, sir. Reckon I'd have seen him otherwise.\"\n\nNathaniel reckoned the same. The orderly's protestations faded quickly as he, eternally grateful for the mobility the gravitar on the ship brought him, pelted towards the engines. Had they been on any other aether ship, such speed would have been impossible.\n\n11.\n\nFOLKARD MOVED, SILENT as a shadow, through Imperator's darkened corridors, his thoughts not too distant from Nathaniel's. Judging by the existence of gravity on Imperator, it appeared the Russians had also been mining gravitar. In every way that mattered, Imperator truly was Sovereign's sister ship.\n\nWhat little he had imbibed in the petty officers' cabin was still having an effect. He was woozy and his step was light; he had to fight to keep his balance and vision steady. He shook his head, trying to ward off the fog that clouded his judgement. Thoughts of the mission, of duty to the Crown and fidelity to his friends and countrymen, spurred him onwards.\n\nHis first idea was sabotage. If he could somehow make his way to the engine room undetected he may have been able to disrupt the engines, perhaps scuttle the warship in the aether and, at least for the moment, lessen the threat Imperator posed to the world. But what then? He could not guarantee Her Majesty's Navy were aware of Imperator's position, let alone its dreadful capabilities. Not only that, he was certain any damage he might do would only be temporary. This ship may not have been elegant, but its construction was sturdy and methodical\u2014and with that came a veritable army of engineers who'd have it back up and running in next to no time.\n\nCould he perhaps set a fire? Or rig up some sort of explosive? Whatever he needed to do, he needed to do it quick. A sudden, panicked coldness gripped his belly as he took in the vastness of Imperator's many decks and passageways. He estimated it to be perhaps a third bigger than Sovereign, or was that guess just the vodka needling at his vulnerabilities, making him second-guess himself and ask what, realistically, a single man could do against such might?\n\nHe shook his head again, clearing these morbid thoughts. It was true that there was not much only one man armed with a single pistol and five bullets could achieve against such odds, even if that man was Captain Jacob Folkard. But the might of Sovereign and a battalion of fully armed British ships was another matter, and not a force to be resisted even by this grotesque behemoth. He had to let Sovereign know where he was and, if need be, tell them to blast this black beast out of the sky with him inside it.\n\nHe had to get to the comm room.\n\n12.\n\nHE WAS DEFINITELY on the right track when he found the marine was still breathing but out cold. Arnaud had clearly got the jump on him, and Nathaniel noticed with dismay that the unconscious man's pistol was also missing\u2014the next crewmember to encounter Arnaud might not be so lucky. If only he could corner him, restrain him, give it time for these destructive chemicals to leach away so the old Arnaud could return. Only time was the last thing he had. Finding the injured sailor had only served to confirm his worst suspicions\u2014that an armed and unbalanced Arnaud was heading straight towards the engine rooms.\n\nIt must be some sort of post-hypnotic suggestion, he reasoned. Nathaniel's body was charging down corridors but his mind was calmly using the time to make connections, to think ahead and plan. Potsdam and Garrecreux had appeared highly susceptible and weak-willed, and creating some sort of ingrained mental link between withdrawal symptoms and a specific set of instructions would not, he presumed, be all that difficult to do. But how strong would the impulse be? He had to assume that the real Arnaud's will was stronger, and that only by coaxing his friend's mind back to full consciousness could the conditioning be shaken off...and that was only if he could find Arnaud before he'd been shot dead.\n\nHe was nearly at the engine room when another pistol shot cracked and reverberated through Sovereign's corridors. It was very close ahead.\n\nNathaniel just caught Arnaud's heels disappearing around the corner. In seconds he would be in the engine room. To his left Nathaniel saw a sailor ducked into an ante-room off the main gangway, cocking his pistol and readying himself to lean out from cover and fire.\n\n\"Don't shoot!\" Nathaniel yelled as he raced past. \"He mustn't be harmed!\" Stunned, the sailor watched Nathaniel blaze by before taking to his own heels in pursuit.\n\nMere moments after Arnaud had rounded the same corner, Nathaniel followed. But the beast was in control of Arnaud, and that beast was wily and cunning\u2014he sprang out from his hiding place as soon as he had seen Nathaniel. For a split second Nathaniel could see his ravaged eyes, bloodshot and wide, and the crusts of spittle that had dried around the corners of his mouth. With a strength Nathaniel was unaware Arnaud possessed the unbalanced doctor whirled him around and barred his forearm across Nathaniel's throat. Nathaniel attempted to struggle but ceased as soon as he felt the gun barrel pressed against his temple, still warm from Arnaud's last crazed shot. He was breathing like a crazed, wounded animal and when the sailor from the ante-room skidded to a halt before them Arnaud closed his arm tighter around Nathaniel's throat.\n\n\"Go!\" He growled, gesturing with the pistol barrel. \"Leave.\"\n\n\"Can't do that, sir,\" said the sailor calmly. He began to reach for his holster.\n\nWithout pause or compunction Arnaud levelled the pistol and shot the man twice in the stomach, the blasts almost deafening Nathaniel. He was still recovering when Arnaud began to drag him towards the engine room's imposing door, the barrel of the gun pressed to his head once more.\n\n\"You,\" he hissed into Nathaniel's ear, \"you try and escape and I swear by God I'll shoot you where you stand.\"\n\n\"Arnaud, this isn't you! It's that perfidious drug, it's got a hold of you, man! You've got to fight it!\"\n\n\"No!\" Shouted Arnaud. \"No Nathaniel, no! You're not going to talk me out of this, they told me all about you, all of you, how you'd use your guile against me!\"\n\nArnaud kicked open the engine room doors and slammed it shut with his back. Such was the noise of the engine room the bang was barely noticed. Nathaniel's previous tweaks were really putting Sovereign through her paces, and he noted with alarm the high and fevered whine of straining heat and energy emanating from every pipe and furiously pumping piston. Had he tried to be too clever again, putting his own pompous scientific largesse before the safety of himself and others? At this juncture, the slightest disruption to the systems could be catastrophic...\n\nAs Arnaud shoved Nathaniel roughly into the room a couple of the engineers had taken note and began backing away. Arnaud whipped the pistol out to point at the nearest one.\n\n\"You,\" he spat, \"lock the door!\" His arms in the air, the engineer complied. Arnaud locked his arm tighter around Nathaniel's throat again, and put his mouth towards Nathaniel's ear. \"You will tell me,\" he whispered, as the professor began to suffocate, \"how to destroy this ship!\"\n\n\"Arnaud,\" Nathaniel gasped, \"think of what you're doing. All the lives you'll destroy...\" A sudden bang on the door. Reinforcements had arrived. Arnaud released his grip slightly, alarmed, and pointed the pistol in a frenzy around the room. \"It's over, Arnaud. Come back, come back to me. We'll make you well again...\"\n\nGentle as they were, the words seemed to strike Arnaud like blows. As the battering at the door was reaching its climax Arnaud roared, a desperate and angry sound that came from some primal and untempered source. He flung Nathaniel aside, raised his pistol, and emptied the chamber into the first bank of instruments he saw.\n\n13.\n\nON THE BRIDGE, the lights dimmed and flickered for a moment. There was an ungodly screech of metal from the stern and Sovereign began to list, losing precious speed as she did so.\n\n\"Lieutenant Sykes!\" yelled Bedford. \"Get onto the engine room, find out what the hell's going on. Tally, Mister Boon,\" he whipped around to face the two men, \"get down there, see if you can help.\"\n\nSub-lieutenant Barry looked up worriedly from a pressure gauge. \"We're losing all power in the engines, Commander. Sovereign... She's dead in the water, sir!\"\n\nAs the stricken ship continued to list, outside the menacing hulk of Imperator sailed effortlessly away over Luna's horizon, the unearthly green of its engines blazing.\n\n14.\n\nSUDDENLY IMPERATOR'S CORRIDORS were alive with activity and excited chatter. Folkard ignored the ruckus, moving with a singular purpose towards the comm deck, grateful the enemy's attention was directed elsewhere. On the way he picked up half-heard chunks of gabbled conversation, something about a great blow to the British and that Imperator was shortly to bring a great glory to the Russian Empire, but it was difficult to separate hard facts from all the scuttlebutt. If anything, it only steeled his resolve to reach the comm room as quickly as possible.\n\nIronically, his pace had quickened tenfold now the corridors were full and lively with activity. Folkard could hop from group to group or stride purposefully along, waving and smiling at his purported crewmates. He reached the door to the comm room in no time, and just as he was approaching a midshipman with a somewhat noble bearing breezed out of it. Folkard stopped dead right in front of him, and the midshipman looked him up and down.\n\n\"It's a great day to be a Russian, eh, sailor?\" he said.\n\nFolkard nodded eagerly. \"Indeed, sir. I never thought I'd see the day...\"\n\n\"Quite, quite. Do you, ah... Do you have any business in the comm room?\"\n\nFolkard stuttered for a second. Through the open door he could see the room was deserted, the power in the lights above the heliograph turned down low.\n\n\"Well sir,\" he began, stalling for time by scratching the back of his head. \"I was just thinking, I've a brother back in Kimra, he's been sick with influenza and this news...\"\n\nThe midshipman smirked wryly. \"Ah, of course. Want to send a message, eh? Cheer up the folks back home. Well, I can't see the harm. Go in, but don't touch anything. I'll send someone by presently to see to your needs.\" He stepped to one side to allow Folkard into the comm room.\n\n\"Thank you sir, that's very kind of you, sir.\"\n\n\"And don't touch anything.\"\n\n\"I won't, sir.\"\n\nFolkard squeezed by and the midshipman softly closed the door behind him. The comm room was unnaturally dark, and here and there he could see shadowed backs murmuring into pipes from the dark hoods of cubicles. Ahead of him, the heliograph was unmanned. He strode across and, without bothering to sit, began to take in the controls. The setup was markedly different from the one on Sovereign, but if could just employ a little Russian logic to it... He reached out and flicked a switch experimentally.\n\nThe first thing he thought of, in those meagre final moments, was that it all must have been happening to somebody else. That the shot had come from outside, that some drunken officer from Kazakhstan had got overexcited and let off some steam with a bang and a whoop. But no; the shot had come closer than that, much closer, and after a lifetime of realisation the warm, wet patch that had been spreading around his left kidney gave way to a sharper, excruciating pain. He pitched forward onto the controls.\n\n\"Ah ah ah,\" came a voice from behind him. \"Weren't you listening? He told you not to touch anything.\" It was a voice he knew, the inflections, the timbre, all the same but the accent was different. Gasping against the pain he managed to turn himself over to face the man who had shot him in the back.\n\n\"I hope you enjoyed your little tour, Folkard. I arranged it specially. Olkhovsky and Utterklo were happy to help. I wanted to show you the very best of Russian hospitality. Hard work and vodka, Jacob. That's what'll make you a man.\"\n\nIt couldn't be... Before him stood a relic of a man he once knew, the same face, the same voice, but all out of proportion like he was being controlled by a bad puppeteer. His face had hardened and his eyes were wide and maniacally focussed. Folkard noticed that his right arm was bandaged in a stump at the elbow.\n\n\"Grant,\" hissed Folkard, clutching at his side. \"It was you. All along, it was you...\"\n\n\"Well, yes and no. An exercise in control, if you will,\" continued Grant, padding across to Folkard with an insouciant air. \"You see, Cyrus Grant isn't here anymore...\" Folkard, overcoming the weakness that had begun to course through him, reached down for the pistol at his side. If he could just get one shot off... Grant casually took aim and shot Folkard in the thigh, and the captain crumpled to the floor.\n\n\"We have met several times, Captain. You thought your man had killed me, but my will is beyond the reach of the mortal now.\" He strode over and slowly grabbed both sides of Folkard's head in his hands, and angled it up so Folkard could look him in the eyes. \"The perfect disguise to destroy you and your weak little band of crusaders, righting your perceived wrongs and daring to presume you could meddle with my destiny. I am Vladimir Tereshkov, killer of Jacob Folkard, the greatest mind Russia has ever produced, reborn and redefining the horizons of humanity!\"\n\nFolkard looked into his eyes, his strength failing. \"Grant,\" he whispered. \"Are you there, Grant?\"\n\nIn the last moments before his vision blurred, Tereshkov's face seemed to shift and sag. The eyes were still wide, but this time were filled with horror and sadness. It was Cyrus Grant's soft Midwestern accent that whimpered \"Captain!\" with unrelenting confusion, and then Folkard felt Tereshkov take control again and throw his head to the floor. He was suddenly and painfully aware that he was not breathing.\n\nHe felt the pressure in his temples ebb, replaced with a soothing coolness and an inky blot that swam on the edges of his vision. He tried to close his eyes but could not, and suddenly there came a blinding whiteness in his mind. Images, information, sounds and senses overcame him, crowding out every other thought and memory, every notion of who Jacob Folkard was, is, or ever would be...\n\nThe Heart called out to him once more. It is time. Your journey's end.\n\nThe blank void blazed for a moment and started to resolve itself, images he remembered, picked out in pastel shades and bathed in light. That sunny meadow, the apple tree with his little girl skipping around it, in the distance and slightly out of focus, but he knew his daughter would be everything he dreamed of. A picnic of fine cheeses, wine and baked bread, ham and slices of summer pudding. It was spring, you could smell it in the air, and she sat on the blanket and patted where she wanted him to sit, right beside her.\n\n\"Charlotte,\" breathed Folkard. He began to smile.\n\n15.\n\nMessage begins.\n\nLarge Russian vessel designated Imperator has breached asteroid field stop attempted blockade resulted in critical damage to HMAS Tartarus stop source of damage believed new Russian weapon stop request immediate assistance stop\n\nMessage ends.\n\n_TO BE CONTINUED..._\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Artist and designer Della Sigrest combines passion with talent in her new business, Bound To Be Great Co., crafting custom journals and sketchbooks. Sigrest took up bookbinding as a hobby on a whim. She found the process \"meditative,\" and a business was born.\n\"Holding the artifact of a finished project is always satisfying, but the added functionality of hand-bound sketchbooks is particularly exciting to me,\" she says. Sketchbooks cost about $30.\nSigrest, who says she uses durable, quality materials, began selling her products in March on Etsy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Mission of the Virginia State Bar.\nBeginning in August 2016, the VSB began hosting a portal to the ABA's online pro bono website that allows low-income Virginians to ask legal questions in a private forum and have the questions answered by volunteer Virginia lawyers.\nLawyers can register directly on the Virginia Free Legal Answers website.\nAll of Virginia's legal aid offices have well-established private attorney involvement programs with pro bono coordinators who will help support your volunteer lawyering work. Contact your local legal aid office for more information.\nPro Bono Clearinghouse \u2013 a referral service linking volunteer attorneys with nonprofit organizations in need of legal representation on a variety of transactional matters.\nPro Bono Promise \u2013 law firms and law departments are invited to join Pro Bono Promise with a pledge to commit a percentage of time to the provision of pro bono legal services to the disadvantaged and\/or to include financial support to legal services organizations annually.\nJusticeServer \u2013 an all-inclusive case management and referral system that enables private attorneys to accept and work on pro bono cases from their own computers. Attorneys are able to create a confidential profile, view pro bono opportunities available in designated practice areas\/jurisdictions, and find the resources to assist with handling the legal matter.\nOther Volunteer Opportunities \u2013 GRBF also provides a menu of volunteer opportunities available in the greater Richmond area.\nProbono.net\/va contains resources for pro bono and legal services lawyers to assist in their representation of low income or disadvantaged clients, including the Pro Bono Opportunities Guide, which allows volunteer lawyers to find legal opportunites tailored to their specific requirements.\nSt. Anthony Legal Clinic, Falls Church, VA \u2013 The St. Anthony Legal Clinic is an informal group of volunteers operating under the auspices of St. Anthony Catholic Church, 3301 Glen Carlyn Rd. in Falls Church. The clinic operates on Saturday mornings from 9:00 to 11:00 AM, working out of the church office and using the church conference rooms. This is a walk-in clinic, no appointments, and completely free. While no screening is done for income, it is apparent that most clients are poor and cannot afford a lawyer at normal rates. Many of the clients are Hispanic\/Latino, and the clinic is staffed with interpreters for non-Spanish speaking lawyers. The majority of clients have problems relating to immigration and family law\/divorce. Other issues that arise include landlord-tenant issues, criminal actions, DMV problems and employment issues. Attorneys are expected to provide advice and guidance, and limited follow-up if needed, but not to represent clients in court. The clinic maintains a list of referral agencies that can provide more long-term representation. For more information, contact Norman Hicks, (703) 203-4484.\nVirginia Bar Association Veterans Issues Task Force. The VBA provides military veterans and service members with greater access to legal assistance. Volunteer assistance opportunities include: initial intake interviews, one-on-one counsel, speaking at Yellow Ribbon or similar events, mentoring other attorneys, providing pro bono or low-cost assistance, and supervising law students in legal clinics assisting veterans.\nHogar Immigrant Services (703) 534-9805 or (571) 208-1572. Hogar Immigrant Services' mission is to welcome the stranger by offering immigration legal assistance at a reduced rate to those who could not afford it otherwise. Hogar also hosts Naturalization Workshops every month. At each workshop, members of the Hogar Legal team, as well as trained attorney and non-attorney volunteers, are on hand to assist eligible legal permanent residents (LPRs) in completing naturalization applications. Please contact volunteer.hogar@ccda.net or call 571-208-1572 x102 for further information.\nJust Neighbors (703) 979-1240. Just Neighbors provides immigration legal services to low income immigrants in the northern Virginia area. Clients must make 200% or less of federal poverty guidelines and must live in northern Virginia to qualify for services. Just Neighbors has a special emphasis on humanitarian-based immigration cases and does not take: removal\/deportation cases, asylum cases, family-based petitions. Clients interested in our services can be screened by calling 703-979-1240, Tuesday - Friday 9 am - 5 pm. If you are an attorney interested in volunteering with Just Neighbors, please call our main number and ask to speak with the Director of Legal Services.\nSee the Calendar for upcoming webinars or visit Webinar Recordings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The i-tec USB 3.0 HDMI 2x 4K Ultra HD Display Adapter. 4k monitor support out of a USB 3.1 connector that can run for hours.\nI like i-tec stuff. You may not have heard of them, but their gear seems well screwed together and survives inhospitable environments \u2013 like the bottom of my bag \u2013 very well. They get stuff that hits highest specifications or newest standards to market quickly as well, which is always nice.\nOne of their little boxes of magic I've had in long term daily use is the i-tec USB 3.0 HDMI 2x 4K Ultra HD Display Adapter. Dual 4k over regular USB 3.1 seems impossible, but with the latest Display Link drivers and a half-way decent laptop it actually works fine. Note this isn't going to give you the framerate you want for gaming, but even video plays acceptably well. It's not just a handy thing for the bottom of the laptop bag, in extended use driving a 4k monitor it will happily run for hours and hours without overheating or in any way being unreliable. I mention this because not all connectors of this kind can manage extended duties without failing (I think, because they overheat). Like most quality gear, this is not the cheapest USB-to-video connector on the market, but if you want one that is going to meet all possible requirements, Just Works, and continues to Just Work after many hours, consider getting one of these.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Among the little known gems in the bay area is the Nike Missile historic site in the Golden Gate National Recreation Area. It's open with docent tours the first Saturday of every month, and it's a fascinating look into what the Cold War really meant.\nOK, so, it's not quite a year ago (yet) that we did this, but, you know, the pictures keep the memories fresh. Here's part 1 and part 2 for those that aren't so fresh.\nIn case this was before your time, as is likely to be the case, 65 years ago today, the Japanese bombed Pearl Harbor. Since I blog about things that are all too easily forgotten, what better a subject to post on.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Marshall Net Impact is a chapter of national Net Impact, an organization that mobilizes new generations to use their skills and careers to drive transformational social and environmental change.\nMarshall Net Impact provides opportunities for USC Marshall students to develop their own capacity to drive transformational societal change as the business leaders of tomorrow. We support Marshall graduate students in building skills and finding their path to make a positive difference in the world through their careers, no matter the industry or function. By partnering with USC, the Los Angeles community, and the nationwide Net Impact network, we give Marshall students the experiences, connections, and community to pursue the intersection of purpose and profit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"When is a Labrador Retriever a \"Pit Bull\"?\nWe were contacted this week by one of our readers in response to the article \"Persecution Ontario Style.\" She was offering support in light of a very similar incident that happened last year to her mother, who wishes to remain anonymous.\nThe reader's mother lives in Thornhill, Ontario, Canada. Thornhill is a well-to-do area just north of Toronto where the houses are large and well-kept and the residents tend to be educated and successful.\nOn a hot summer day in July 2006, out of the blue the doorbell rang.\nThe 64-year-old woman opened the door to find an Animal Control Officer (ACO) standing there. As it was such a hot day, she invited the officer into the house where it was cooler. He asked if he could see her dog. The woman obliged, assuming they were following up on the licensing program in Thornhill.\n\"Lucky,\" a five-year-old Labrador Retriever, was licensed with the city. The woman went into the backyard and brought Lucky, who was wearing her collar and license tag, into the house to meet the ACO.\nThe officer began to take many pictures of Lucky. He didn't ask Lucky's breed, age or anything else about her.\nAfter he had taken the photos, he pulled out some papers that explained DOLA (Dog Owners Liability Act) and the amendments that ban American Pit Bull Terriers, Staffordshire Bull Terriers, American Staffordshire Terriers and any dogs that are substantially similar to those breeds.\nHe then informed the woman that he was taking Lucky with him to be destroyed.\nThe shocked and horrified dog owner didn't understand what was happening. She started to cry for fear of what would happen to her dog. To have an ACO arrive at your house unexpectedly and to invite them in on a hot day and then hear them say that they are taking your dog for destruction is almost incomprehensible.\nAt this point, the woman's husband came into the room. He promptly escorted the officer off the property and told him not to come back unless he had a warrant and court order for the destruction of their dog.\nOne month later, two police officers showed up at this couple's door and charged them under DOLA with owning a prohibited dog.\nLucky is a five-year-old black, purebred, registered Labrador Retriever.\nThe couple took the registration papers to their set court date which proved their dog is not banned in the province of Ontario. The charges were dismissed.\nImagine what would have happened had Lucky not been a registered purebred dog. How many dogs in the province are unregistered? How many are mixed breed dogs that CANNOT be registered? There is NO proof of your dog's breed without registration papers. The Canadian Kennel Club estimates that only 10% of Ontario dogs are purebred and registered. That means that 90% of all dogs in Ontario cannot have their lineage proved. That means that 90% of all dogs in Ontario, when found in this situation could lose their lives if they are not Lucky!\nThe couple feels this whole situation may have started over a neighbor dispute over how to cut the hedge bordering on both properties. They have no proof because Animal Control won't release the names of complainants, but can't think of any other reason why that ACO came knocking on their door on that nightmarish day.\nBetween the time of the visit and subsequent charges being laid and the court appearance the woman had a heart attack and was diagnosed with stomach ulcers. Her doctor attributed both to stress.\nWe hope she is well on her way to recovery!\nIt is sickening to think about what this Ontario Liberal Government has done to responsible citizens of this province.\nIt is now a crime to own somebody's idea of the wrong shape of dog in Ontario.\nIt makes us very angry to know that responsible dog owners in the Province of Ontario were legislated into second-class status and had their constitutional rights stripped because of grandstanding and government irresponsibility!!\nLucky was named as such because the woman felt she was \"lucky\" her husband agreed to let her get a dog.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Compare and buy tickets for California Extreme 2019, Weekend Pass in Santa Clara, CA on 07\/27\/2019: Get your tickets for California Extreme 2019, Weekend Pass now!\nFace-value tickets to California Extreme 2019, Weekend Pass through TicketWeb: Tickets should be available until Jul 27th, 2019 or until they sell out.\nJoin us Now and get more choices for California Extreme 2019, Weekend Pass tickets in Santa Clara sent to your inbox.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The brothers behind a Thai eatery are opening a fried-chicken restaurant tomorrow at the Grand Rapids Downtown Market.\nBlon Hang and Yang Hang, owners of the Downtown Market's Rak Thai Bistro, will open Juju Bird Chicken + Sides at the market, at 435 Ionia Ave. SW.\nJuju Bird will serve a variety of items: fried chicken; boneless and traditional wings; chicken strips; side dishes, such as curly fries, macaroni and cheese, mashed potatoes and gravy and deli salads; and chicken sandwiches, including a classic crispy chicken, Nashville Hot, Juju BLT and chicken salad sandwich.\nIt will offer dine-in, takeout and catering services.\nJuju Bird sources non-GMO, hormone-free, all-natural Amish-raised chicken, according to the restaurant.\nIt is located in the space between Field & Fire bakery and Blue Spoon Pasta.\nFor Friday's grand opening, which coincides with Downtown Market's Fall Flavors day, Juju Bird will feature a special sweet-and-savory menu item called the \"cone-a-copia,\" a handmade cornucopia-shaped waffle cone filled with fried chicken and a choice of sauce.\nIt leases space to emerging and experienced \"food artisans\" and merchants and features a 24-vendor Market Hall and two restaurants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nontoxic Anti Aging Hair Care Can Get You Results!\nCan I get a show of hands of people; Men, Women, and Children of all ethnicities- who have \"something\" they don't like about their hair? That is my point.\nWe eat organic food, have air purifiers in our home, buy natural cleaning supplies, and make sure our skin care products are naturally based and cruelty free. Then, when it comes to our hair- all bets are off. We would put motor oil on our head if it said it regrew hair, or cured split ends, or made our hair smooth and silky. (Okay, a slight exaggeration but, really not that far off, right?) Have you noticed that 99% of the products available out there, that purport to solve these \"issues\", are made up of chemicals, toxic substances, synthetic ingredients, and unnatural fragrances? Science shows that chemicals in our shampoo\/hair products take twenty six seconds to enter our blood stream! We use these same products on our children. Something needs to change. However, when you try the \"organic\", enviro friendly products there is one BIG problem\u2026they typically don't work, or get us the results we want and need.\nEnter Monat. Their name is combination of the words Modern Nature. They are the \"only\" Anti Aging Hair Care that is; naturally based, vegan, gluten free, nontoxic, has no harmful chemicals, no sulphates, no parabens, no phthalates, cruelty free and \"Leaping Bunny Certified\", no DEA\/MEA, no Phenoxyethanol, no ethanol, no Petrochemicals, no Sodium Chloride, no harmful colors, and no harmful fragrances. Their children's line, or \"Junior Line\", is also tear free, ophthalmologist and dermatologist tested and approved for young children.\nMonat works by removing build up and toxins that have clogged the hair follicles. These, along with the active ingredients \u2013 Which include, red clover extract (reduces scalp inflammation, strengthens and thickens hair, and hydrates the scalp which stimulates natural noticeable hair growth), pea extract (reduces the production of DHT, the hormone that contributes to hair loss, combats premature hair thinning, also protects color and shine), and Crodasorb (protects against oxidative stress, UVB damage, and preserves natural pigmentation, keeps strands stronger and locks in moisture) all work together to give you your best hair possible.\nMonat offers a variety of \"Systems\" to cater to every hair type.\nThe \"Volume System\" is for individuals who have fine limp hair, or thinning hair. The Revive Shampoo and Revitalize Conditioner are the \"go to\" products to pump up lifeless limp hair. These products increase hair density, hair and scalp health, provide weightless volume, and more. When you add in the Root Boost Spray that rounds out this System you are in for true volume and bounce without the heavy dirty feeling that a lot of volume products on the market today have.\nThe \"Hydration System\" helps everyone who needs some extra moisture to tame the frizzy fly away hair that can be the bane of our existence. This system is packed with Monat's trademark Rejuvinique Oil (which I will give you a summary of after the Systems). There are several gentle cleaners (which make this a favorite product for color treated hair, curly hair and ethnic hair) and conditioners that hydrate the hair leaving it looking more youthful looking and supple with extraordinary shine. There is also a lightweight leave in conditioner and revitalizing masque which complete the trio.\nThe premier product for hair growth is the \"Let It Grow\" System. If you want fast hair growth, or new hair growth- this is for you. There is twice the amount of red clover extract, or Capixyl, in this Intensive Repair Treatment Shampoo product than in the previously mentioned two Systems. This makes it an incredible combination with the other ingredients to reduce scalp inflammation and DHT build up. In turn, it allows for rapid repair of the follicles and new growth to occur. It comes with a concentrated Intensive Repair Spray that gives you product to apply to particular areas on the scalp that need a bit of a boost in the growth department.\nThe Men's, or \"Black System\" \u2013 has a two in one shampoo, also with the double amount of Capixyl. There is also a shave cream, moisturizer, and grooming clay. This shampoo is also part of the \"Classic Confidence\" System with the Intense Repair Spray \u2013 It is a simple 3 step System wash twice, towel dry and use the Spray. Take a before picture and then one every four weeks. You will be astounded by the difference.\nThe \"Junior Line\" consists of a shampoo, conditioner, and detangler that makes bath time a non issue when it comes to combing out snarls and curls. As mentioned previously these products are tear free, dermatologist and ophthalmologist tested and approved.\nMonat also has various styling products which also have the same qualities of Rejuvinique Oil and naturally based ingredients that allow you to style your hair with products that do what they claim. The hair spray is even non flammable!\nRejuvinique Oil is the flagship product of the Monat line. It is made up of the purest form of 13 rare botanical oils from around the world. A major component is Abyssinian Oil. This Oil won't dry your hair, rather it nourishes your hair and scalp by penetrating them on a cellular level. It fills and expands the hair shaft and soothes the cuticle with the perfect balance of moisture. There are 101 documented uses for this oil- everything from hair growth to psoriasis and skin conditions. Patent Pending.\nRejuvabeads is the Rejuvinique Oil condensed with nano technology into beads that when applied to the hair actually \"heals\" split ends by bonding the hair together. The results are staggering and impressive! Patent Pending.\nWe have addressed the gambit of \"Hair 911\" situations. Not only addressed, but SOLVED the issues \u2013 with naturally based products.\nFor a free consultation and personalized hair prescription for you and your family, with these amazing naturally based products, Contact Me Today while I have space available!\n* Join as a Market Partner \u2013 30% off all products, Free Shipping on flex ship orders, No Inventory requirements, No Parties, No Quotas, Make Commission off of all your sales and Bonus Incentives! Plus a lot more\u2026 Let's talk if this interests you!\nMy Husband I moved to Montana @ 5 yrs ago. Living in a cabin w\/our 2 dogs and 2 cats is a dream come true. Boating & Photography are faves of mine. Most of all I enjoy helping people live the life they have always wanted and motivating them to make their dreams come true.\nWhat Made Me Try Naturally Based Haircare?\nHow To Find Time\u2026 Fitting a Side Hustle Into Nooks and Crannies Successfully.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The airport was built for a mere \u00a335,000, and opened in summer 1935.\nIn the 1950s, package deals resulted in 5,500 passengers travelling through Newcastle per year.\nThe airport was renovated and added to during the 1960s.\nThe number of passengers travelling through Newcastle international continued to grow - 700,000 per year during the 1970s, 1.2 million in the 1980s and 1.7 million in the 1990s. The current figure for the turn of the 21st century is 3.8 million passengers - over twice what it was ten years ago.\nThroughout the 80s and 90s a great deal of development took place, including new terminal buildings and updated facilities inside.\nThe most recent development has been a \u00a327million extension to the building.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"You emphasize and exploit the potential of the special material from which your shield is made.\nBenefit(s): When wielding a shield made of one of the materials listed below, you can use the listed ability twice per day.\nDarkwood: As a free action as part of an attack with your shield, you can drive splinters of your shield into your opponent, dealing an amount of bleed damage equal to 1 + the shield's enhancement bonus.\nDragonhide: As an immediate action when you take energy damage of the type to which the dragonhide shield is immune, you can reduce the amount of damage you take by 10 hit points.\nLiving steel: As a free action as part of an attack with your shield, you can attempt a sunder combat maneuver check against a metal weapon wielded by your target. If you succeed at the attempt, the weapon gains the broken condition; if the weapon already has the broken condition, it is destroyed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So that concludes the module and this course.\nIn the first module, Morgan discussed how to ensure you're making the right product.\nof processes used in the industry to create successful, well designed products.\nto achieve a high quality product.\nIntroduced the purpose of metrics.\nIn the third module, Morgan detailed how to ensure the project is managed right.\ncharts, and a task board to monitor the status of ongoing and completed work.\nIn this final module, Bradley explained two kinds of retrospectives.\nSprint retrospectives and project retrospectives.\ndiscovering lessons learned to improve future work.\nWe covered a lot of approaches in this course.\nIt takes ways to improve the product, process and project.\nTo achieve the goals of the right product, done right and managed right.\nthe next course is the Capstone.\nthe capstone serves as the all important head.\nBuilt upon legs, body and arms.\nthat represent the preceeding courses in the specialization.\nThe capstone brings together a lot of what you've learned so far.\ndeal with change, follow the progress, address risks and so much more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How much is Dan Quayle Worth?\nDan Quayle net worth: Dan Quayle is an American politician who has a net worth of $3 million. In 1988, Quayle estimated his net worth to be $1.8 million. His net worth was primarily derived from real estate and a financial trust set up before he became Vice President. Dan Quayle was born in Indianapolis, Indiana in February 1947. He is a Republican who graduated from DePauw University and earned his J.D. from Indiana University School of Law \u2013 Indianapolis. Quayle is best known for serving as the 44th Vice President of the United States under President George H.W. Bush from January 1989 to January 1993. He served in the United States Army from 1969 to 1975 and attained the rank of Sergeant. Quayle served as a member of the U.S. House of Representatives from Indiana's 4th district from January 1977 to January 1981. He was then a United States Senator from Indiana from January 1981 to January 1989. He earned his J.D. He practiced law after college and as Vice President he served as chairman of the National Space Council. Currently Quale is the chairman of global investments at Cerberus Capital Management. His memoir Standing Firm was published in 1994.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"__________ his brother, he is a _________________.\"\n2. Who were the first children of Adam and Eve?\n3. Why did Cain attack his brother Abel and kill him?\n4. What did Cain say when the Lord asked him \"Where is your brother?\"\n5. Name one of the saints who are called \"wonder-worker\" because of the miracles God performed through him?\nA. St. Nicholas Bishop of Myra, St. Spiridon, etc.\n6. Which of the apostles is seen carrying the Scriptures?\n7. Which of the apostles is seen carrying the Keys of the Kingdom?\n8. Who were the first disciples?\n9. What do we celebrate on Theophany?\nA. The revelation of the Holy Trinity and the manifestation of Christ to the world. The beginning of His public life.\n10. What is the name of the parable in which the rich man's son says \"Give me what is mine.\"\nA. The Parable of the Prodigal Son.\n11. What are the signs of new life we receive in Baptism?\nA. The water, white robe and candle.\n12. What happens in Chrismation?\nA. We receive the gift of the Holy Spirit, who dwells in us and makes us His living temples. The Holy Spirit is the giver of life and we are sealed in our new life with the Holy Spirit.\n13. How do we become members of the people of God?\n14. Why is Holy Communion so important?\nA. Holy Communion is the bread of life. It makes Jesus present to us and gives us nourishment. It will bring us to eternal life.\n15. What are three things that unite us to God?\nA. Prayer, Fasting, and the holy Mysteries.\n16. What is the greatest commandment?\nA. You shall love the Lord your God with all your heart, with all your soul, with all your mind, and with all your strength.\n17. What are the first three Commandments?\nA. I am the Lord your God. You shall not have other Gods before Me. You shall not take the name of the Lord, your God, in vain. Remember to keep holy the Lord's Day.\nA. Honor your Father and mother. You shall not kill. You shall not commit adultery. You shall not steal. You shall not bear false witness against your neighbor. You shall not covet y our neighbor's wife. You shall not covet your neighbor's goods.\n19. Name two reasons why we pray.\nA. When we pray we talk to God, praise Him , thank Him for His many blessings and ask Him for help.\n20. What is our privilege as God's people?\nA. We can call God \"Our Father.\"\nA. We must stay close to God.\n22. What do we remember in the Diving Liturgy?\nA. We remember all that Jesus did for our salvation, particularly The Last Supper of the Lord.\n23. In what two ways do we receive Jesus at the Divine Liturgy?\nA. We receive Jesus through the words of the Gospel. We receive Jesus in Holy Communion.\n24. What does Jesus want us to do when we have sinned?\nA. To ask forgiveness of Jesus. The Priest gives us a special blessing called absolution as a sign of our reconciliation with God.\n25. What is the name of the Mystery in which we ask forgiveness?\n1. What mystery involves the use of water, a lighted candle, a white garment, and oil?\n2. What mystery gives us the seal of the gift of the Holy Spirit?\n3. In what mystery do we become participants in Christ's Mystical Supper?\n4. In what mystery are our sins forgiven by Christ?\n5. In what mystery is the eternal union of a man and a woman blessed?\n6. Who can receive the mystery of Holy Oil?\nA: A Christian desiring to be healed of illness.\n7. What mystery ordains bishops, priests, and deacons for the service of the Church?\n8. What is the collective name given to the first five books of the Bible?\n9. When did St. Nicholas live?\nA: In the fourth century.\n10. To what does St. Paul compare the Christian household?\nA: He compares it to the Church.\n11. How does St. Paul define love?\nA: It is patient, kind, not jealous, never rude, doesn't rejoice in wrong, and never fails.\n12. What is Moses holding in the icon of the Transfiguration?\nA: The tablets of the Law.\n13. Which three apostles appear in the icon of the Transfiguration?\n14. What does Christ's Passion show?\nA: That He took upon Himself every humiliation and sin of mankind.\n15. The Son of God became like us in everything except one thing. What is it?\n16. What Feast celebrates the revelation of the Holy Trinity?\nA: The Feast of Theophany.\n17. Who are the three most famous Fathers of the Eastern Church?\nA: St. Basil the Great, St. Gregory the Theologian, and St. John Chrysostom.\n18. What are the processes called by which we learn self-control and dedication to God?\n19. Recite the Jesus Prayer.\nA: Lord Jesus Christ, Son of God, have mercy on me, a sinner.\n20. What is our response to Jesus' command to \"do this in remembrance of me\"?\n21. What is a vocation?\nA: God's calling of certain men to be part of His plan for the world through Holy Orders, and His calling of certain men and women to dedicate themselves wholly to Him.\n22. What is the sign of truly faithful Christians?\nA: Their love for one another.\n23. What is the basic command Jesus gave us?\nA: Love God and love your neighbor.\n24. What is often used as a model of the love of God for His people?\n25. What does \"Messiah\" mean?\n1. Who is known as the protomartyr or first martyr for Christ?\nA. St. Stephen, one of the seven deacons ordained by the Apostles.\n2. In the Eastern Catholic Church, which mysteries do we receive immediately after Baptism?\nA. Chrismation and Holy Communion.\n3. How does the name \"Gospel\" tell us of its meaning?\nA. It is the Good News (Good Spel) of the coming of Christ our Savior.\n4. What are the antiphons?\nA. An antiphon is a short hymn made up of verses taken from the Old Testament Psalms. They are called antiphons because they are meant to be sung by two groups alternating in answer to one another.\n5. The evening service, consisting of psalms hymns, biblical canticles and litanies is called?\n6. What is the Greek word meaning conversion or repentance and the continual life of inner change and growth?\n7. What are the first five books of the Old Testament called?\n8. What are the names of the first five book of the Old Testament?\n9. What does the word \"Genesis\" mean?\n10. Who was the last prophet?\n12. What do we celebrate on Palm Sunday?\nA. The entrance of our Lord into Jerusalem.\n13. What great miracle is celebrated on the Saturday before Palm Sunday?\nA. The raising of Lazarus \u2014\u00ad the miracle of the triumph over death and the symbol of our own resurrection.\n14. The Church is one because?\nA. As the body of Christ it cannot be divided. Jesus meant it to be one when He prayed of His Apostles: \"I pray not only for them, but also for those who will believe in me through their word, so that they may all be one, as you, Father, are in me and I in you, that they also may be in us, that the world may believe that you sent me. And I have given them the glory you gave me, so that they may be one, as we are one. . .\" (St. John 17:20-22).\n15. The Church is Holy because?\nA. Her Founder is holy. The Church it continues Christ's work of sanctification with the guidance of the Holy Spirit.\n16. The Church is Apostolic because?\nA. The Church continues to proclaim what was revealed by Christ to His apostles, and continues to evangelize in the same spirit that characterized the apostolic era.\n17. Who makes and blesses once a year the Holy Myron for the Sacrament of Holy Chrism?\nA. The Patriarch on Holy Thursday.\n18. Why does the Priest cut the hair of the child after applying the Chrism?\n19. What are the mysteries of the Holy Church?\nA. Baptism, Chrismation, the Holy Eucharist, Repentance, Holy Orders, Matrimony, and Holy Oil.\n20. This Apostle kissed Christ in order to betray Him to those who were about to crucify Him.\n21. What do we call the words which the priest pronounces over the bread and the wine during the Divine Liturgy?\nA. The Words of Institution.\nA. The Sunday of the Pharisee and the Publican.\n23. Who was the first person to ask Christ to remember him when He came into His Kingdom?\nA. The good thief, who was crucified next to Christ.\n24. What is the unwritten teaching transmitted by the Church?\n25. What is another name for the Last Supper?\n26. Who are the spiritual descendants of the Apostles?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you under obligation to dematerialize your public sector (B2G) invoicing in the near future ?\nAs of 1 January 2017, e-invoicing is gradually becoming mandatory for all suppliers invoicing the state, regional and local authorities and public entities.\n1 January 2020: mandatory for very small\/micro enterprises (under 10 employees).\nChorus Pro provides a platform for the deposit, receipt, transmission and follow-up of B2G e-invoices, for use by local authorities and public entities, government departments, and their suppliers.\nTX2 Concept provides you with an automated interface with the CHORUS PRO platform.\nWe help you set up the project, whether based on EDI or PDF files.\nAll your customer invoices will be issued in compliance with legal requirements.\nInvoices are archived for a period of 10 years, the legal retention period essential to satisfy the conditions for approval of tax-compliant e-invoicing.\nYou can monitor all invoice flows to date by means of a user-friendly interface.\nYou will also receive a daily statement of all invoices sent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztskw b/data_all_eng_slimpj/shuffled/split2/finalzztskw
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index 0000000000000000000000000000000000000000..7f2167c7b4f3ef9c2cc6a10b31d8402c7c90106b
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+{"text":"This site is my hero..\nMillion thanks for the developer!!\nThis article describes how to create easily basic and ordered bar plots using ggplot2 based helper functions available in the ggpubr R package. We'll also present some modern alternatives to bar plots, including lollipop charts and cleveland's dot plots.\nYou need to install the R package ggpubr (version >= 0.1.3), to easily create ggplot2-based publication ready plots.\nCreate ordered bar plots. Change the fill color by the grouping variable \"cyl\". Sorting will be done globally, but not by groups.\nThe deviation graph shows the deviation of quantitative values to a reference value. In the R code below, we'll plot the mpg z-score from the mtcars data set.\nI'm greatly enjoying your different package(s), thanks for sharing!\nI'm having a small hiccup trying to plot my data.frame with a lollipop chart.\nEven though I don't use the \"group\" option within the function, it still always sorts within groups.\nAnd the output is still sorted descending per group.\nThis happens even if I copy directly from your tutorial, .\nGreat graphs! Especially the lollipop graph. I made a plot using your instructions but the problem I am having is that for some comparisons (by group) are too close together, distorting the view. Is there a way around this? Maybe something similar to jitter? See attached plot.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Computational Cosmolog Group@NAOC mainly study some of the most of fundental questions in Science: What were the first objects in the Universe? How do matter contends of the Universe distribute? What is the nature of dark matter? How do galaxies form and evolve? High-end Supercomputers are essential for our studies, just like telescopes for observers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In April 2015, Semco Outdoor launched the new website you're now experiencing with the intention of making it easier for people to find the products we have available, learn about our brand and communicate with our sales team. We're always trying to make your experience at Semco better, and if you have any suggestions for improving this site or your Semco experience, please share with us! Thanks for taking the time to help us make your retail experience the best it can be!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have a Asus RT-86U with merling and torguard VPN. The VPN is connected and i have activated Policy rules (strict). I have only one client going through the VPN and this client is added to the routing rules list. The rest of the clients goes on the normal WAN provider. However none of my android mobile phones can connect to the internet. They are showing as connected to the router, but no net access. Mobile phones with apple device works and all computers works. I did also have this problem with my previous TR-66U so its not router specific.\nUpdate: Torguard recommends by default \"Accept DNS configuration\" set to Strict. This had to be changed to exclusive in order to successfully route android devices outside the VPN tunnel. This still prevents DNS leak for the clients routet through the VPN tunnel. Setting the tab to \"relaxed\" however will give a DNS leak. I guess if one wants to route also the android mobile phone devices through the VPN tunnel have to change the DNS settings in the phone (apple devices work ok).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good Leadership is Contagious. So is Bad Leadership.\nIt's flu season. At this time of year, most of us alter our behavior a bit to avoid \"catching\" anything that might be contagious. But how many of us are on the lookout for contagious behavior? Do we \"catch\" certain behaviors from those around us?\nResearch by UC San Diego's James Fowler and Harvard's Nicholas Christakis has shown that many behaviors are, indeed, \"catching.\" Their research showed that if you have overweight friends, you're more likely to be overweight. If you quit smoking, your friends are more likely to quit. Rose McDermott of Brown University even found that divorce is contagious: if you have a close friend who's divorced, you are 33% more likely to split with your spouse.\nWe know that good leadership creates engaged employees and that good leaders have a major impact on retention, productivity, job satisfaction, and so on. But if you're a good leader, do the people around you \"catch\" that behavior and become good leaders, too? And which behaviors are most readily \"caught\"?\nZenger and Folkman found that the best-performing leaders had the best-performing direct reports, and vice versa. To determine the impact on the organization as a who le, they used a 360-degree feedback instrument to question subordinates about their own level of engagement. The results were sobering. \"HL managers whose overall leadership effectiveness was in the top 10% had direct reports (MLs) whose engagement scores were in the 81st percentile, and the subordinates of these ML managers had engagement scores in the 74th percentile. Those whose overall leadership effectiveness was in the bottom 10% had direct reports (MLs) whose engagement scores were in the 15th percentile, and the direct reports of those ML managers had engagement scores in the 24th percentile.\" In other words, managers doing a sub-par job erode not only the engagement of their direct reports, they also drag down the engagement of the people working for them, too.\nAt Executive Forum, we teach people how to lead. Our leadership development pathway offers a clear route for helping your good leaders expand their skills while filling the leadership pipeline and ensuring long-term continuity.\nContact us. Our systematic approach can help your organization develop leaders with the skills needed to enhance performance, motivate teams, discover new opportunities, and support sustainable success.\nRescheduled. Fall session begins October 29.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuken b/data_all_eng_slimpj/shuffled/split2/finalzzuken
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+{"text":"Further, if you really try to understand dataset, please check out CSLA http:\/\/www.lhotka.net\/cslanet\/default.aspx .\nIn the past, I used the data binding book to understand more CSLA; now, I am proposing that you read CSLA to understand the data binding book.\nYou need to think about CSLA custom objects (without the portal part), then, you can really understand and appreciate dataset, and how to use it with simple (\"no framework approach\") custom objects without custom collections (again, \"no framework approach\").","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"L.A.P.E.C.\u2013P.D.L.E.S. does not offer general or technical support via telephone or email without student applying the following procedures, or receiving mandatory support form. Once support form is received, support manager's response time is less than 24 business hours.\nAt L.A.P.E.C. we are always very serious about providing accurate, to-the-point required support to current or past student(s), as soon as possible. A prompt and accurate reply is possible only when we have accurate and to-the-point information from students.\nWe treat all students equally and with respect. Inquiries are handled in sequence as they arrive in our office, and response depends upon volume at any given time.\nPlease note: L.A.P.E.C.staff does not provide or offer any personal computer or software-related technical assistance or training to members. If you still encounter problems in this situation, please try to explore a solution via the following common questions\/issues. If you still face difficulty and cannot find a solution, then submit a support ticket form.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The source of the displayed data is either the property owner or public record provided by non-governmental third parties. It is believed to be reliable but not guaranteed. Listing information provided courtesy of Hampton Properties. Information Copyright 2001, Multiple Listing Service of Long Island, Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"White Rose Wall Slogans | Wei\u00dfe Rose Stiftung e.V.\nThe group of friends experiments with other ways of expressing their opposition as well: Hans Scholl, Alexander Schmorell and Willi Graf paint slogans of freedom on walls across Munich during three nights in February 1943. Alexander Schmorell has fabricated a stencil (25 x 15 cm) for the purpose and the slogans are painted with black tar coal. They write \"Freedom\" and \"Down with Hitler\" in large letters on the entrance to the university several times on 04 and 09 February 1943. The slogan \"Mass murderer Hitler\" and crossed-out swastikas appear on buildings around the central areas of Marienplatz and Viktualienmarkt on 16 February. The slogans are applied to about 30 different facades. The Gestapo has them removed immediately, but not always entirely successfully.\nAlexander Schmorell stated after his arrest that these activities were meant to \"address the masses mainly\", something that was not as easily possible in distributing printed leaflets.\nThe leaflets remain the central activity that the resistance of the White Rose is famous and remembered for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The adenosine antagonist 7-methylxanthine (7-mx) works against myopia in animal models. In a clinical trial, 68 myopic children (mean age 11.3 years) received either placebo or 7-mx tablets for 12 months. All participants subsequently received 7-mx for another 12 months, after which treatment was stopped. Axial length was measured with Zeiss IOL-Master and cycloplegic refraction with Nikon Retinomax at \u22126, 0, 12, 24, and 36 months. Axial growth was reduced among children treated with 7-mx for 24 months compared with those only treated for the last 12 months. Myopia progression and axial eye growth slowed down in periods with 7-mx treatment, but when the treatment was stopped, both myopia progression and axial eye growth continued with invariable speed. The results indicate that 7-mx reduces eye elongation and myopia progression in childhood myopia. The treatment is safe and without side effects and may be continued until 18\u201320 years of age when myopia progression normally stops.\nTrial registration: ClinicalTrial.gov. Registration number NCT00263471. www.clinicaltrials.gov.\nNormally, the axial length of the human eye increases from 17 mm at birth to 24 mm at 14 years of age. From an initial hypermetropia the refraction gradually changes toward emmetropia. It has been hypothesized that a growth-control mechanism exists, which can either accelerate or slow down axial growth in response to defocusing of the retinal image during the period of normal eye growth. Results from experimental myopia in mammals suggest that this mechanism works by regulating the scleral extensibility through complex changes in the synthesis and degradation of extracellular matrix components [1, 2, 3].\nIn myopic children an excessive eye elongation is typically seen from age 6 years until 20 years. The axial length will then be in the range of 24\u201328 mm. Until the age of 14 years, the increase in myopia is approximately linear , after which a slow deceleration is seen . With maturity, an increase in collagen cross-linking occurs , which probably explains the gradual stabilization of myopia. The myopic eye differs in shape from normal eyes, because the increase in length (0.35 mm per diopter) is greater than the increase in height (0.19 mm per diopter) and width (0.1 mm per diopter), resulting in the characteristic \"egg-shape\" of highly myopic eyes .\nWith increasing axial length there is an increased risk of potentially blinding complications later in life, such as retinal detachment, macular degeneration, or glaucoma. About 23% of all cases of retinal detachment occur in eyes with myopia of more than \u22126 dioptres . This underlines the importance of finding an efficient and safe way to prevent excessive axial elongation.\nThere is today no treatment that efficiently and without side effects controls the progression of myopia . Topically applied muscarine antagonists, like the non-selective antagonist atropine, and the M1 selective antagonist pirenzepine have been shown to reduce myopia progression and axial eye growth [10, 11, 12]. The mechanism of action of atropine and pirenzepine has not been established, but it is assumed that it is independent of their action on accommodation, and that it may depend on receptors in the retina or the sclera . Reduced accommodation and dilation of the pupil, however, remain a problem both with atropine and to a lesser degree with pirenzepine.\nThe biomechanical properties of sclera are determined by the contents and organization of proteoglycans and collagen . Collagen accounts for about 80% of the dry weight of sclera . It is known that plastic deformation (creep) of tissues is directly related to the proportion of small diameter fibrils . The diameter of the collagen fibrils in sclera from myopic eyes is reduced compared to normal eyes, as are the contents of collagen and glycosaminoglycans .\nExperimental myopia in mammals is accompanied by a decrease in the scleral content of proteoglycans and collagen , with reverse changes taking place during recovery . In mammalian models of high myopia, a thinning of the posterior sclera and an increased number of small diameter collagen fibrils in posterior sclera is found [20, 21].\nTreatment of young rabbits with a metabolite of caffeine and theobromine, 7-methylxanthine (7-mx), increases the thickness of the posterior sclera, increases the content of collagen-related amino acids in posterior sclera, and at the same time increases the diameter of the collagen fibrils in the posterior sclera . Recently, 7-mx has been shown to reduce the amount of myopia and eye elongation induced by form deprivation in guinea pigs and to counteract the thinning of posterior sclera and of collagen fibrils induced by form deprivation .\nTheobromine and its metabolite, 7-mx, both have very low toxicity [24, 25] and no carcinogenic effects . The aim of this study was to test whether 7-mx is useful against axial elongation and myopia progression in childhood myopia.\nOne hundred seven children, aged 8\u201313 years, with a minimum myopia of \u22120.75 diopters (cycloplegic) in one eye were screened using two subject visits with an average interval of 6 months. Eighty-three were found to have an average axial growth rate of 0.075\u20130.39 mm per 6 months, and were enrolled in the study. The low cut-off was chosen in order to exclude children with a normal or close to normal axial growth rate. The high cut-off was chosen in order to exclude atypical cases with an extremely fast progression. The normal axial growth rate in children of this age-group is around 0.05 mm per 6 months .\nAll participants were Danish. Children using soft contact lenses were accepted. All children used single vision lenses. The degree of under-correction was not assessed, but the parents were encouraged to have the children's correction adjusted at regular intervals.\nThe study followed the tenets of the Declaration of Helsinki. Informed consent was obtained from the participants and their parents. Approval was obtained from the Danish Medicines Agency and the Ethics Committee for the municipality of Copenhagen and Frederiksberg. The trial was registered at www.clinicaltrials.gov, Reg.NCT00263471.\nDuring a 12-months placebo-controlled trial, participants received one tablet every morning. The tablets had a diameter of 12 mm and a thickness of 3 mm. Active tablets contained 400 mg 7-mx, placebo tablets only standard filling and binding substances.\nPeak serum concentration after an oral dose of 400 mg 7-mx in adults is around 20 \u00b5mol\/l with a half-life of 200 min (unpublished results, SM Ribel-Madsen & K Trier, 2004). In the rabbit experiment mentioned above , peak serum level of 7-mx was around 70 \u00b5mol\/l and half-life around 1 h.\nThe 7-mx substance was synthesized by Sigma-Aldrich Co. (Fluka works, Buchs, Switzerland). The tablets were manufactured and packed in a pharmacy certified for pharmaceutical production (Glostrup, Denmark). Participants were asked to bring unused tablets at the end of the follow-up period and the remaining tablets were counted.\nThe primary outcome measure was axial growth rate. Secondary outcome measure was myopia progression.\nThe axial length was monitored with non-contact, partial coherence interferometer (Zeiss IOL-Master) . The refraction was measured with an autorefractor (Retinomax, Nikon) 30 min after one drop of 1% cyclopentolate in each eye. In case of astigmatism, half of the astigmatism was added to the sphere. The same Retinomax autorefractor and IOL-Master was used throughout the trial. Measurement stability was confirmed by means of test eyes. Outcome measures were taken four times and the average calculated.\nWeight was measured at base-line, and weight, height, blood pressure, heart rate at the end of the trial, and participants were questioned specifically for CNS, gastrointestinal or cardiopulmonal symptoms.\nFrom a pilot trial, a standard deviation of 0.063 mm on axial growth per year was found. The smallest difference in mean value of axial growth not to be overlooked was set to 0.050 mm. With an accepted risk of type I error of 5% and of type II error of 20%, it was calculated that the number of participants needed in each group was 24.\nFor the 12-month placebo-controlled trial, participants were randomized in a 1:1 ratio to receive 7-mx or placebo. The allocation sequence was generated in the pharmacy and sealed and numbered containers consecutively handed out to the participants. Outcomes were assessed unaware of group assignment at Trier Research Laboratories. No participants or investigators became unmasked during the first 12 months of the trial.\nThe distribution of demographic data did not deviate substantially from the normal and variances were homogeneous, hence parametric tests were used. For all data on the state and development of myopia was calculated the average of data from each subject's right and left eye, and these average data were subjected to significance tests. The significance of differences between mean values referring to the respective treatment groups was tested with Mann\u2013Whitney test, because of inequality of variances of some of the variables. The significance of differences in axial length or myopia progression rates between periods within the same individual was tested with paired Student's t test. Correlations were analyzed with Pearson's test and proportions with Fisher's exact test.\nAfter completing the 12-month placebo-controlled trial, children from both the placebo and the 7-mx group were given the option of continued treatment with 7-mx 400 mg either once or twice per day for another 12 months.\nAfter completing 24-month follow-up, all participants discontinued treatment with 7-mx, and axial growth and myopia progression was followed for another year by measurement at the end of the total of 36 months' follow-up.\nThe 12-month placebo-controlled trial was completed by 77 children.\nThe reason for not completing was in no case stated as side effects. Compliance among those completing the 12-month placebo-controlled trial was 89% in the 7-mx group versus 92% in the placebo group. No differences were found as to weight, height, blood pressure, and heart rate. No subjective side effects were reported.\nAxial growth rate during the 12-month placebo treatment period was found to be proportional to base-line level (p\u2009<\u20090.01, Fig. 1). To ensure comparability between the 7-mx and the placebo group, data were therefore compared within two stratification layers, one with a base-line axial growth rate of 0.075\u20130.190 mm per 6 months, including 19 treated with 7-mx and 27 treated with placebo, and the other with a base-line axial growth rate of 0.200\u20130.390 mm per 6 months, including 16 treated with 7-mx and 15 treated with placebo.\nTables 1 and 2 show axial growth, myopia progression, and change in corneal curvature in the two stratification layers. In both, axial growth was reduced in the 7-mx group compared with the placebo group. The difference was \u22120.055 mm per year (95% Cl \u22120.114 and +0.005 mm per year) for children with a moderate base-line axial growth rate (p\u2009=\u20090.073), and \u22120.031 mm per year (95% Cl \u22120.150 and +0.087 mm per year) for children with a high base-line growth rate (p\u2009=\u20090.593).\nAlso the myopia progression was lower in the 7-mx group of both stratification layers, with a difference of \u22120.108 diopters per year (95% Cl \u22120.288 and +0.070 diopters per year) (p\u2009=\u20090.227) and \u22120.091 diopters per year (95% Cl \u22120.424 and +0.243 diopters per year)(p\u2009=\u20090.582), respectively.\nThe 24-month follow-up period was completed by 68 children. The average daily dose of 7-mx taken by each child during the last 12 months of the follow-up ranged from 40\u2013706 mg per day, average 444 mg per day. No subjective side effects were reported.\nAxial growth in children with a moderate base-line axial growth rate (n\u2009=\u200940) treated for 24 months with 7-mx was 0.329 mm (SD 0.143 mm) compared with 0.427 mm (SD 0.193 mm) in the group treated with placebo for the first 12 months and with 7-mx for the last 12 months (p\u2009=\u20090.048) (Fig. 2). The corresponding axial growth for children with a high base-line axial growth (n\u2009=\u200928) was 0.547 mm (SD 0.212 mm) for the 7-mx group, and 0.584 mm (SD 0.193 mm) for the placebo\/7-mx group (p\u2009=\u20090.667) (Fig. 3).\nMyopia progression in children with a moderate base-line axial growth rate (n\u2009=\u200940) treated for 24 months with 7-mx was 0.627 diopters (SD 0.329 diopters) compared with 0.844 diopters (SD 0.450 diopters) in the group treated with placebo for the first 12 months and with 7-mx for the last 12 months (p\u2009=\u20090.141) (Fig. 4). The corresponding myopia progression for children with a high base-line axial growth (n\u2009=\u200928) was 1.131 diopters (SD 0.443 diopters) for the 7-mx group, and 1.010 diopters (SD 0.549 diopters) for the placebo\/7-mx group (p\u2009=\u20090.667) (Fig. 5).\nPercentage myopia stabilization in the period 12\u201324 months (defined as less than 0.25 diopters myopia progression during the 12 months) is shown in Table 3. Among children treated for the whole 24-month period with 7-mx, 59% of children with a moderate axial growth rate at base-line, and 20% of children with a high axial growth rate at base-line were stable.\nSixty-one children completed 36-month follow-up. In both the group treated with 7-mx 0\u201324 months (n\u2009=\u200928) and the group treated with placebo 0\u201312 months and 7-mx 12\u201324 months (n\u2009=\u200933), axial growth rate in the first as well as the second year of treatment was significantly lower compared with the precedent period (p\u2009<\u20090.00001\/p\u2009=\u20090.001, first year and p\u2009=\u20090.006\/p\u2009=\u20090.001, second year) (Fig. 6a\u2013b). The myopia progression rate, on the other hand, only slowed down in periods with 7-mx treatment (p\u2009<\u20090.0005\/p\u2009=\u20090.098, first year and p\u2009=\u20090.013\/p\u2009=\u20090.01, second year) (Fig. 6c\u2013d). However, in both groups, axial growth rate (p\u2009=\u20090.533\/p\u2009=\u20090.179) and myopia progression rate (p\u2009=\u20090.162\/p\u2009=\u20090.138) after discontinuation of the treatment (third year) did not differ significantly from the precedent period (Fig. 6a\u2013d).\nIn this study, base-line axial growth rate was taken into consideration when comparing inter-group differences in axial growth. The rate of myopia progression is believed to be the most important variate as regards base-line comparability . The finding of a significant correlation between axial growth at base-line and axial growth in the first 12 months of the intervention period supports the importance of using the axial growth rate as one of the inclusion criteria in myopia studies. Frequency of contact lens wear was close to equal in the 7-mx group and the placebo group (17% versus 19% during the 12-months trial). Contact lens wear does not seem to affect the growth of the vitreous chamber significantly .\nThe degree of under-correction in the participants was not recorded in this trial, and could therefore theoretically be a confounding factor. One study found that under-correction enhances myopia progression , while another found no effect . It is not known whether the axial growth rate is affected by under-correction.\nAt 12 months, the lowest axial growth in both stratification layers was found in the 7-mx groups, but the differences were not significant.\nAt 24 months, however, a significantly lower axial growth was found in children with a moderate base-line axial growth rate treated for all 24 months with 7-mx compared with those treated with placebo for the first 12 months and 7-mx for the last 12 months (difference 0.098 mm), whereas the difference among children with a high base-line axial growth rate (0.037 mm) was non-significant.\nChildren with a high base-line axial growth rate (0.200\u20130.390 mm per 6 months) have a myopia progression of up to 2 diopters per year. Such severely progressing myopia is frequently linked to specific genetic abnormalities, and may therefore represent a sub-group, possibly more refractory to treatment than the majority of myopes.\nNevertheless, in the second year of the intervention period, average axial growth among children with a high base-line axial growth rate treated for all 24 months with 7-mx, was reduced to 0.22 mm per year from an initial 0.52 mm per year at base-line. This is lower than the expected average axial growth rate of 0.25 mm per year for age-matched Danish myopes, including low-, moderate-, and high-progressors . The corresponding myopia progression rate in the second year of the intervention period among children with a high base-line axial growth rate treated for all 24 months with 7-mx was reduced to 0.49 diopters per year from 1.20 diopters per year at base-line. The expected average myopia progression rate for age-matched Danish myopes, including low-, moderate-, and high-progressors is 0.48 diopters per year . For children with a moderate axial growth rate at base-line (0.075\u20130.190 mm per 6 months), treated for all 24 months with 7-mx, axial growth rate was reduced to 0.14 mm per year in the second year of treatment from 0.27 mm per year at base-line, and the myopia progression rate to 0.23 diopters per year in the second year of treatment from 0.52 diopters per year at base-line. Thus, in comparison to the average of age-matched Danish myopes, including low-, moderate-, and high-progressors , both axial growth and myopia progression was reduced by around 50% in the second year of treatment with 7-mx.\nIn an unselected material of myopes, the percent with stable myopia increased with age from 14% for 12-year-olds to 63% for 18-year-olds . In comparison, our finding of a markedly higher percentage of stable myopia (59% and 21%, moderate and high base-line axial growth rate respectively) in the second year of treatment, further supports that treatment with 7-mx stabilizes myopia.\nIn the age interval 11\u201314 years, myopia progression and axial elongation is close to linear . A decrease in both myopia progression rate and axial elongation rate during both the first and the second year of treatment with 7-mx is therefore unexpected. The absence of a further reduction of myopia progression and axial elongation in the third year coincides with the discontinuation of treatment and is consistent with the treatment being efficient.\nMethylxanthines in non-toxic doses block adenosine receptors (ADOR) competitively. Adenosine in the concentrations found under physiological conditions activates primarily A1 and A2A adenosine receptors (ADORA1 and ADORA2A) on the cell surface. ADORA1s are widely distributed in the CNS, whereas ADORA2As are found mainly in the dopamine-rich regions of the brain .\nAdenosine has a general homeostatic role in the regulation of cell metabolism, but also a specific neuromodulatory role with ADORA1s having an inhibitory role and ADORA2As having a facilitatory role in the release of neurotransmitters such as dopamine, noradrenalin, acetylcholine, glutamate, and serotonin . ADORA1s and ADORA2As have been localized in the ciliary body and the retina and ADORA2As in the retinal pigment epithelium of adult rats  and humans (unpublished results, Zhongshan Ophthalmic Center). Both ADORA1s and ADORA2As are expressed in human scleral fibroblasts .\nExperimental myopia in animals has been linked to changes in retinal dopamine and acetylcholine neurotransmission [39, 40, 41, 42, 43, 44]. Both of these neurotransmitters are modulated by ADORs [45, 46, 47, 48, 49, 50]. The mechanism by which 7-mx affects eye elongation and myopia progression could therefore theoretically involve modulation of muscarinic acetycholine or dopamine receptors. On the other hand, it is also possible that ADORs are directly involved in the control of eye growth. Because of receptor adaptation, the effects related to blocking of ADOR1s are attenuated during chronic treatment with non-selective ADOR antagonists like 7-mx, whereas effects related to blocking of ADORA2As are unchanged [51, 52, 53].\nAdenosine liberated from the retina is believed to activate ADORA2s on the retinal pigment epithelium and augment the standing potential (SP) . The effect on the SP might therefore give us a measure of the potency of a specific methylxanthine as a blocker of adenosine receptors. The receptor affinity of 7-mx is three to five times higher than theobromine . Correspondingly, the light-rise of SP in humans is more reduced by 7-mx than by theobromine (unpublished results, K Trier, 2004). Interestingly, high ametropia in children is associated with retinal electrophysiological abnormalities, a finding consistent with the hypothesis that the retina is involved in the process of emmetropization .\nIn conclusion, systemic treatment with 7-mx appears to be efficient in retarding axial elongation and myopia progression among myopic children. The treatment is safe and without side effects, and may be continued until 18\u201320 years of age, when age-related cross-linking of collagen prevents further elongation of the eye.\nThis work was supported by grants from \"J\u00f8rgen Bagenkop Nielsens Myopi-Fond\" and \"Generalkonsul Einar H\u00f8yvalds Fond\", and by \"\u00d8jenl\u00e6ge Klaus Trier ApS\".\nApproval of the trial has been obtained from the Danish Medicines Agency and the Ethics Committee for the municipality of Copenhagen and Frederiksberg (KF 02\u2013042\/03).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzumhk b/data_all_eng_slimpj/shuffled/split2/finalzzumhk
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+{"text":"I am a Texan native but now live in New York City. I wanted to design a garment that embodies both sides of me \u2014 collaborating my origin with my current home. I was inspired by different elements of New York and Texas to design the Lucite Romper. I incorporated deer fur around the neckline and cap sleeves and I used leather for the front bodice to symbolize Texas. The angular structure of the romper and the use of sequins are suppose to represent the architecture and lights of the New York city skyline.\nMy style is tailored and sophisticated with a twist. Adding the sheer panels on the sides represents exposing who I am, while adding a hint of sexiness to the garment. This design represents my journey moving to New York city and my transition in life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"good evening, but it is possible to make the image, in the \"page_title_section\" adaptable to the size of the window?\nIf possible share your website url .\nSo that we can assist you better .\nThanks for send snapshot to us .\nLet us know , Are you want to change height or width of the image .\nPls also share your website URL .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Situated in Iceland, 30 minutes away from Reykjavik is the unique Panorama Glass Lodge that is fully covered with glass. From roof to wall everything made in glass.\nThe Glass lodge features a small fully equipped kitchen housing two electric stoves, together with a water heater, fridge, and toaster. Where you can cook your own food and enjoy your evening tea watching the sky.\nThe glass lodge offers all the luxuries a high-end hotel offers, making it ideal for a honeymoon destination. The Lodge has a cozy double bed that's completely surrounded by glass, perfectly situated to make the most of the awe-inspiring views all around.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Carterton District Council has imposed water restrictions onto the Carrington and Taratahi Water Races, due to the lack of rain fall and the current low water levels of Waingawa and Mangatarere Rivers.\nAs our water allocation has been reduced to the lowest take, we need to conserve water so the supply reaches our end race users.\nThis restriction dictates that race water can only be used for the purpose of stock drinking water and domestic toilets only.\nPlease use your water sensibly and avoid wastage. This will help to ensure that there is a sufficient water supply for everyone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Self-storage as a concept began in the US in the 1960s with an increasingly transient population taking advantage of dedicated units to store their belongings whilst working away from home. Over the past 50 years its popularity has grown so rapidly in fact that today 1 in 10 use one of the country's 50,000 facilities.\nHeralded as the best performing real estate investment in the US over the past decade according to the Bloomberg Riskless Return Ranking, America currently uses 7.5 sq ft of self-storage space per person compared to only 0.5 sq ft in the UK and even less in Europe but the market is growing \u2013 and rapidly!\nAccording to the Self Storage Association UK (SSA UK) Annual Survey 2012 in association with Drivers Jonas Deloitte, there are an estimated 815 self-storage facilities in the UK equating to 29.6 million sq ft of storage space.\nWell documented as an industry that has avoided the tough economic climate and seen real growth in hard times (an 8% increase in the number of UK stores in the last year alone), the estimated total annual turnover for the UK self-storage industry in 2011 was \u00a3355 million.\nThe demand for storage space is especially high in and around the Capital with only 8.1 million sq ft of rentable self-storage space currently available in London for a population of 8.3 million people equating to 0.98 sq ft per person.\nHigh land and commercial property prices had resulted in a lack of new stores opening in London over the past 5 years, in fact only 28% of the total amount of self-storage in UK is in the Capital however established self-storage company, Personal Storage, has secured a prime site in the affluent area of Hampton Court for its latest store.\nLocated approximately 9 miles south west of London, this latest addition to the Personal Storage portfolio is perfectly situated to capitalize on the demand for additional storage from the 150,000 strong population located within 15 minutes' drive including some 23,000 university students not to mention the thousands of businesses nearby.\nWith over 15,200 net sq ft of secure self-storage space available with a variety of unit sizes up to 90+ sq ft, ample space to store the contents of an average 2 or 3 bedroom house, Personal Storage Hampton Court offers plenty of opportunity for investment. With an investment of as little as \u00a314,878 inclusive of VAT, investors can purchase a 999 leasehold title from the building owners and developers as well as a 15 year full, repairing and insuring commercial lease with Personal Storage who will manage the 45 sq ft unit and handle all operations. Returns equivalent to 8% Net in the first 2 years rising to over 10% are assured with an optional buy back from year 5 providing a credible and defined exit strategy makes Personal Storage Hampton Court a genuine property investment in today's uncertain market.\nFor more information on how to think inside the box and invest in Personal Storage Hampton Court contact Property Frontiers today for your free interactive Investment Guide on +44 1865 202 700 or visit www.propertyfrontiers.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvnzk b/data_all_eng_slimpj/shuffled/split2/finalzzvnzk
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+{"text":"Well i do not know where to start with this one, Hannibal is starting to show his\/her childish side now, making up storys and threats and leaking out pointless data and making false statements about the data. Lets start with the latest press release from Hannibal that comes along with what he\/her said is 10,000 Facebook accounts when there is lucky to even be over 1000 there. He has also claimed that Israeli Iranian first Vice President Mohammad-Reza Rahimi has sent him death threats, i really find this unlikely. Hannibal finishes the statement off with a typical threat to the world. \"Soon I will reveal another surprise. The most cruel surprise. Be prepared to Arabs!\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To state pocket knives are amongst the most helpful tools understood by humanity is no exaggeration. For countless years, they have assisted us to cut food, conserve individuals from burning vehicles, prevent criminal offenses, hunt for supper, slice open boxes and achieve numerous other jobs\u2013 from the ordinary to the life-altering.\nThe pocket knife is a folding knife with several blades that can be folded down into the management of the knife for safe and practical storage. There are various kinds of pocket knives readily available depending upon the sort of blades consisted of.\nPocket knives are little in size and can be folded down and quickly brought in an individual's pocket or handbag. When opened, the blade is taken out of the deal with and extended so the user can comprehend the deal with to use the blade. There are numerous various usages for a pocket knife that the ranges and designs are almost endless.\nSo why is choosing the ideal pocket knife so damn tough? Given that pocket knives have been around for many centuries, they've progressed and branched off into various designs, leaving you with countless designs to pick from. Options are a good idea. However, they can likewise make it frustrating.\nThere are clip-point blades, tanto blades and drop-point blades, stainless-steel blades and carbon-steel blades, serrated blades and non-serrated blades, not to discuss Swiss Army knives with sufficient tools and accessories to help your escape from an optimal security jail.\nSo, exactly what's a knife fan to do?\nWell, aside from purchasing every knife you see (not a bad concept if you have the dough), the very best technique for discovering the best pocket knife is to understand a heap about them. Considering that you do not have time to do that research study, we did it for you.\nThere are two various variations of pocket knives together with a wide variety of types and designs of pocket knives that suit each classification. One classification, which was the very first type of pocket knife to progress, is the single-blade knife. The other classification, which quickly followed after the development of the single-blade folding knife, is the multi-blade knife.\nSingle-blade pocket knives are typical and have either a locking blade or a slip joint blade. The lock blade features a securing device that locks the blade into the opened position up until the user launches it by pushing the lock release system. Lock blade knives are popular since they provide a procedure of security by avoiding the blade from being inadvertently closed on the user's fingers. There are different styles of locking systems readily available depending upon the user's choice. Other single blade pocket knives are slip-jointed. This indicates the blade does not lock open, however, is held open with stress produced by a bar on the behind of the knife. If a specific quantity of effort is used to the blade, it will close. While not as risk-free as lock blades, the slip joint is thoroughly utilized for useful knives that see extremely light use.\nThe numerous blade pocket knives have anywhere from 2 to twenty various knife blades of numerous types that can be utilized for various functions. The numerous blade knife enables individuals to have one knife that can deal with various usages. Numerous blade knives are normally slipped common type knives because the locking system cannot be used to several blades. Several blade knives likewise can be found in specialized outdoor camping and survival designs that have various specialized blades that fold into the deal with of the knife. These knives verge on being too big to fit quickly into a pocket. However, they are still little sufficient to do so if required.\nIn some cases, you require a pocket knife with more than simply blades. Saws, toothpicks, tweezers, can openers, nail files, scissors, corkscrews and amplifying glasses are all typical features of Swiss Army knives and other multi-tools, which are the most multi-dimensional pocket knives on the marketplace.\nThe Swiss Army knife was developed to satisfy the need for soldiers in the field who would have to carry out some jobs a single blade could not constantly deal with. Because that initial design, Swiss Army knives, and multi-tools, in basic, have just ended up being better and trustworthy.\nToday, these knives can be found in numerous sizes and shapes, varying from 3 accessories on original designs to the 87-tool \"Giant Swiss Army knife,\" which was produced in 2006. Other multi-tools\u2013 such as Leatherman\u2013 load the very same punch, though they're typically a bit heftier to bring.\nMulti-tools are best matched for the jack-of-all-trades who discovers himself requiring an entire series of tools within arm's reach at all times.\nAlthough there are countless various knife types and variations offered, there are a couple of typical designs or designs that have stood the test of time.\nThe stockman knife is a slip joint knife that features three various blades that are typically a clip point for basic use, a quick point that has a blunt idea that will not poke through things and trigger damage, and a sheepsfoot blade that has a sweeping down curve leading into a flat blade and no point. The stockman is an outstanding daily bring knife.\nThe trapper knife is a slip joint knife that has two different blades jointed at the same end of the manage. The blades are typically a clip point and a spey blade.\nThe muskrat knife is a slip joint knife that features two different blades jointed at opposite ends of the manage. The most typical blades on a muskrat knife are a basic function clip point on one end and a spey blade on the other. They can likewise have various blade variations depending upon the manufacturer.\nThe canoe knife is a slip joint knife that gets its name since it is formed like a standard Native American canoe. The canoe knife is typically geared up with two spear point blades that are exceptional basic function blades, or it can have one spear point and one pen blade.\nThe congress knife is a slip joint knife that has a convex front together with a straight or shallow concave rear end. The congress knife has four various blades that can differ. However, the most typical are the spear point, the pen blade, the sheepsfoot, and the coping blade. The coping blade is a smaller sized variation of the sheepsfoot other than for the back of the blade is angled directly down instead of curved.\nThe camper knife is a slip joint knife that features a big spear point blade, a clip point blade, a can opener, a flathead screwdriver blade that functions as a bottle screw, and a punch blade. There are other variations on the outdoor camping knife that can consist of several blades and tools, like the popular Swiss Army knife.\nThe knife has been an essential tool utilized by humanity throughout history. The very first cavemen started to use sharp tools to assist them to cut meat and protect themselves from other guys and monsters. With time, individuals developed folding knives that might be brought securely.\nVarious types and designs of folding pocket knives were developed to resolve various requirements, from single-blade knives to multi-blade knives with different other tools consisted of. They are still useful tools to have on hand, specifically when outdoor camping or searching.\nBy learning more about Stockman, trapper, and congress knives, to name a few types, an outdoors lover can pick a knife with the blades that will be the most beneficial to them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm a crafter and writer in Portland, Oregon and the author of four other craft books. I love shopping for fabric, printmaking, sewing, and making quilts. You can also find me at my craft blog, West Coast Crafty.\nat the 2012 Sisters Outdoor Quilt Show!\nSusan, I've had fun using your book learning to make quilting projects. I'm am totally new to it and your work is so inspiring and beautiful I decided to give it a try.\nMy first project was to make a charming camera case and have had a very difficult time with the bias tape around the top. (produced a blood satin on my blue bias tape \ud83d\ude42 ) Do you have a special sewing machine that allows you to sew around this tiny opening or could you share your technique so I can produce a better finished product.\nAgain, thank you for sharing this craft with the me. I am excited to get better and make gifts for others to enjoy!\nHi Shelley! Thanks so much for your kind words and comment. So sorry to take a few days to reply \u2013 this got caught in my wordpress filter and I didn't see it right away.\nTo add the bias tape \u2013 first, I used very, very wide tape (about 2\u2033 across flat, sold in stores sometimes as hem facing or quilt binding) which helps sandwich all the layers in neatly. I sewed pretty slowly, close to the bias tape's edge, and gently rotated the opening of the case forward in the stitching direction as I went so it didn't get caught up with the foot. It is a small opening, so I know it can be tricky to stitch\u2026 sorry about the bloodstain, I have been there with crazy sewing projects many times!\nI took some photos of my process of making and sewing the whole camera case and would be glad to do a post on it with some details in the next couple of weeks if that's ok? I'm getting ready for Maker Faire and my Bobbin's Nest event (yay!) but when I'm done with those two events I'd love to post about it.\nThanks for publish such great book\u2026!!\nSo far, I had completed the quilt top, the next step quilitng the sandwich.\nLooking through the back photo of the modern cross quilt that showed on the book, I could not see any quilt pattern on them, not even stitch in the itch.\nThe back quilt looks so marvellously great with a big cross on them!!\nJust wondering what kind of the quilt patten do you quilt them?\nThanks again for ever publish the GREATEST book!\nI'm having a difficult time w\/Modern Crosses Quilt. Did you do a tutorial on the process? I don't see any corrections to the pattern. My squares are wonky when sewn together. Using 2 1\/4 x 2 1\/4 center sq sewn together, then add 2inch logs\u2026..what am I missing?\nPlease let me know if you need more specific advice, the chain-piecing technique really makes the block-building efficient (as you can see in Jen's post). Thanks so much for your comment, I hope my answers and links give you some helpful ideas!\nI absolutely love your book and just finished my first project, the cheerful pot holders. I did have a bit of trouble applying the bias tape, but just concluded that practice makes perfect.\nI do have a question regarding the assembly of logs. On page 28 and 29 you show how to assemble a block. In step 3 it says to neatly trim the additional length of fabric from the log and set it aside. Now skipping to the pot holders, I followed this procedure while doing the pot holders. I trimmed the additional length and set it aside for the next tier that would use that fabric and so on. When I was finished with the project I found myself with numerous scraps that had been trimmed from the final tier. Although these scraps were only 2 or 3 inches long, it seems like a small waste of fabric. So, why are the measurements for each log a tad longer than what you actually use?\nHi Lesley \u2013 just saw this, sorry to take so long to reply, it got caught in a wordpress filter! I wrote the potholder directions this way so that people had the most design freedom for placing colors, patterns, and logs where they liked, instead of having to choose in advance and follow an order from shortest to longest. In general with the measurements, it seems like everyone pieces a bit differently, so sometimes people who use scant seam allowances and trim differently go through strips more quickly than people who use a generous 1\/4\u2033 and trim closely \u2013 so I wanted to have some cushion there. I think those little scraps are so nice to have for a improvisational block or piecing into a binding as a decoration, if you still have them around!\nHi Crystal, thanks so very much for your kind words on the book! I'd love to see your quilt and happiest birthday to your daughter! I haven't posted recommendations before, but I've made a few simple sleeves by just stitching a tube of fabric the width of the quilt and then hand-basting it to the top of the quilt back, then running a dowel through. For the small, lightweight quilts in my kids' room, I sewed a series (4 or 5) of small bias tape loops to the top edge of the quilt and hung them on little hooks. Corners sound like a very nice idea too. I hope this helps and that you'll post a picture of your daughter's quilt in the book flickr pool \u2013 I'm so happy you love it so much!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OK, so here's the prologue. It's not too long, but I don't think it is too short either. Any feedback you give will be greatly appreciated.\nErus lifted his head slightly and rested his eyes upon the horizon. The sun would soon descend from sight, and the land would be plunged into the darkness of the night. Returning his attention back to the small camp, Erus found that Niveus was biting his thumb nail, a nervous look in his eye.\nNiveus\ufffd pale skin seemed to glow in the encroaching darkness, and his pink eyes seemed dimmer than usual. His white hair had also lost some of its lustre since beginning the journey.\n\ufffdNo. Not yet. It is soon,\ufffd Niveus muttered, his eyes darting back and forth.\nSighing, Erus took up his whetstone and began sharpening the first of his array of daggers. These daggers were special to him, having been given to him by his mentor. Each one was slightly longer than the last, the longest being roughly seven inches. The hilt of each dagger was engraved with a different symbol, in an ancient language that few could now understand.\nOnce all of the daggers had been sharpened, Erus lay them aside once again and decided that he had better get some sleep.\n\ufffdNiveus, time to go to sleep,\ufffd Erus muttered. Suddenly, Niveus\ufffd head snapped up, and his eyes stared intently into his own.\nA few moments passed and Niveus sighed with resignation, and then lay down upon the ground. Erus followed suit, and had soon drifted into a half-conscious state.\nThrough this half-conscious state, Erus was still aware of everything around him that emitted a life force. Almost immediately, he was aware of Niveus\ufffd life force and the life forces of small insects around. Once confident that he would not be disturbed, Erus reached out further, noting the presence of the trees they had passed the previous day, and the animals that slept peacefully upon the forest floor.\nErus picked up speed. Soon he was out of the forest and into the swampy region they had reached a few days ago. There was little life force present in this remote place; only the various plants and bugs that prospered in such a place. He reached lush fields. Fields of crops. He reached houses. He reached villages. The outskirts of the town. He reached dense clusters of houses.\nErus slowed his mind down, the life forces of those around him becoming more prominent. He caught surface thoughts, and glimpses of dreams.\nHe paid less attention to these minds and sought one in particular. He found that mind in one house among many, all of which appeared exactly the same on the outside. He approached the mind, and began to hear surface thoughts.\n-Greetings Amic, Erus sent toward the mind.\nThere was a moment\ufffds pause.\n-Yes, friend. How have you been?\n-I\ufffdve been fine. What really matters is how you have been. Have you found it yet? Are you close?\n-Well, I have felt it presence through scouting ahead during Mind-trances, but only a glimpse. So I fear we have some way to go.\n-And you have every right to be. If it gets much worse, you may have to send him back to me.\n-Yes. I will keep that in mind. For now, I must leave. Goodbye, Amic.\nWith that, Erus withdrew his mind, and made his way back to his body. Niveus will not last much longer, and when the time comes, I need to take care of him myself.\n\"I do not want your trust, I want your obedience!\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The first thing you will want to do is select a queue discipline. Although you can select from a number of disciplines I picked HTB for the following instructions because it seems to be the simplest to set up. All of the following is done on Xen0.\nThe first thing you will need to do is find the name of your real ethernet device. This seems to change depending on what version of Xen you are running. For my setup it was peth0 and I was able to find it by looking at dmesg right after the system booted.\nAfter finding the name of your ethernet device you will need to set up a default queuing discipline that will catch anything that doesn't hit a child rule. Here we set the handle to 1 and a class sub id of 99.\nNext we define a default rate that will be used as a total for all child rates as well as anything that doesn't fall into a child bucket. In this case I'm setting the total rate to 20mbps with a burst of 15k.\nSetting a burst lets small amounts of traffic go faster than the normal rate. The burst is also shared with the children so make sure to set it higher than any one child. Also note that parent 1: references the parents classid that we created above.\nNow that we have our default class and rate set up we set up child classes and rates for each node. Here I set up classes for 2 XenU nodes and the Xen0 node.\nNote that the parent classid is referenced here as 1:1 that we created above. We also assign each bucket its own unique classid.\nNow we need to determine who gets serviced in what order. In this example I use sfq for each class so that each class should get equal time as traffic is coming in.\nNow the final step is to attach the defined classes to the routing system. This is done by using iptables and the given classid from the class setup step.\nNice article. It would be nice if you also supplied examples of what the status output data from tc an iptables should look like.\nI'm probably dense. This just doesn't work like it's stated above.\nmy domu's can still download at full speed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvqot b/data_all_eng_slimpj/shuffled/split2/finalzzvqot
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@@ -0,0 +1,5 @@
+{"text":"Every insurance company prices their auto insurance policies differently, so the price that a first-time driver is quoted from one company will be different when.\n3 days ago. FIRST TIME WHEELS The cheapest cars to insure for young drivers \u2013 from the budget Nissan Micra to cool VW Polo. the 1.2 TiVCT 70.\nNew driver car insurance. Just passed your driving test? Congratulations! You can finally ditch those 'L' plates and enjoy the freedom that comes with being a fully.\nLearn how you can get the cheapest car insurance rate in Missouri. A company that offers the cheapest rate to one driver could offer the most expensive. As with fines for accidents, first-time speeding tickets are likelier to be forgiven by.\nCheapest Car Insurance For Low Mileage Between 20 and 30 percent of people have no health insurance in the region where. But in the McCall area, it's just her and her car. She put 50,000 miles on it last year, shuttling between. Aon Car Insurance Top Uk Universities Car Insurance Rate Picture The factors that help determine your car insurance premium are constantly changing. Maybe you've had an at-fault accident, or your financial picture has improved.\nJan 24, 2018. On the one hand, auto insurance coverage is required for all drivers in. However, it does not apply to all drivers on the policy, only the first one. If you're a full-time student, Nationwide offers their Good Student Discount.\nJun 1, 2018. Under 21 and looking for affordable car insurance?. Guiding and coaching advice from parents will help young drivers through their first year on the road, You may need to reprove your good student status each time that.\n01\/02\/2017 \u00b7 New driver insurance cheap car for first time driversfirst drivers. We've cars in group 1 generally have the cheapest insurance premiums car do tend to.\nThese companies offer cheap auto insurance for teens in your state. It can be an exciting, yet somehow stressful time when your teen passes their driving exam. Monthly Auto Insurance Rates by Age Legend: Teen Driver Rates (Monthly) Adult. Having a car with good safety ratings could also go a long way to reducing.\nYou just got your first car. We'll help you compare insurance rates to get the cheapest policy and provide you with useful driving tips.\nAug 17, 2018. Our cheap auto insurance review takes a close look at the coverage, the drivers on your policy is under 25 and a full-time student with good.\nThe first thing you should do when looking for cheap car insurance is. that both the driver and passenger will be wearing their belts at all time, and both. to see a smile on their face as well as receive a good discount on your car insurance.\nBuying a car as a newly-qualified driver can be expensive. whether you're a 17-year old or a more mature first-time. but also the maintenance and insurance.\nSome car insurance premiums make sense. For example, if you live somewhere with more car crime or have a more expensive cars, you pay more. But then there are things.\n. of all states require DUI offenders to install interlock devices that analyze the driver's breath and disable the vehicle's ignition if the driver has been drinking.\nCome here to learn about young driver car insurance and many ways to save. Is it the right time to shop for California Auto Insurance?. to stay diligent can quickly become discouraged and jump at the first reasonable-sounding quote.\n. on car insurance. Download our free app, take the test drive and get a quote!. Ready to take the Root test drive and save up to 52% on your car insurance policy? Let's get started. 5 tips for first-time car insurance buyers. Check it out. is built on. We're restoring fairness to the insurance industry one policy at a time.\nYou will have a better chance of finding the cheapest car insurance for provisional licence drivers. Is Plymouth Rock Car Insurance Good; First Time Driver.\nTake a look at our Studentfirst policies to see how you can save up to 27% on your car insurance with AXA. even if you're a first-time driver.\nFor each time a repeat offender is caught, fines and penalties get more severe. suspended even if it turns out you were never drinking in the first place. Compare the best car insurance companies near you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JEFFERSON CITY, Mo. (AP) \u2014 Missouri Gov. Mike Parson is renewing a lobbying ban for members of his office.\nParson signed an executive order Tuesday revising an ethics policy implemented by his predecessor, former Gov. Eric Greitens.\nGreitens' first act as governor in 2017 was to issue an order banning executive branch employees from accepting lobbyist gifts and prohibiting governor's office personnel from lobbying the executive branch if they leave their jobs.\nBut the lobbying ban only applied to Greitens' administration. Parson's order expanded it to cover his and future administrations. Parson took over after Greitens resigned amid personal and political scandal in June.\nParson's order also tweaks what gifts executive branch employees are banned from accepting from lobbyists, opening the door for \"educational\" materials such as books and souvenirs valued at less than $10.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cammack & Strong, P.C.Leadership - Cammack & Strong, P.C.\nLeadership - Cammack & Strong, P.C.\nKerry N. Cammack has over 25 years of experience as an attorney and governmental consultant in Texas. He serves as general outside legal counsel to a number of individuals and business entities. Cammack has also served as counsel to sellers and purchasers in numerous stock purchase and asset acquisition transactions including commercial real estate purchase and sale transactions involving hotels, shopping centers, office buildings, nursing homes, residential subdivisions, undeveloped land and apartments.\nCammack represents clients before state regulatory and administrative agencies, boards and commissions and advises publicly and privately held state, national and international companies and associations on a broad range of legislative, administrative, regulatory and procurement matters.\nCammack currently serves as a member of the Board of Trustees of the Texas Regional Office of the National Center for Missing and Exploited Children and is a Paul Harris Fellow, Rotary International. He is also a member of the State Bar of Texas, the Travis County Bar Association and the Corporate Counsel Institute.\nJess Heck has worked in and around the Texas Capitol since the 82nd Legislative session in 2011. She is a graduate of The University of Texas at Austin with a BA in Government and minor in Business. While in school, she completed a fellowship in the Budget, Planning and Policy Department of the Office of the Governor. Upon graduation, Jess worked in the Texas House of Representatives and moved on to work as a government affairs consultant for a non-profit advocating for criminal justice, tax, and human trafficking issues. Currently, Jess serves as a governmental affairs consultant with Cammack and Strong, P.C. in connection with all legislative advocacy, procurement and regulatory affairs for clients of the firm.\nJess is involved with the Down Syndrome Association of Central Texas and is a CASA volunteer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi, everyone. I'm talking on my blog today about fat white titles on white backgrounds and why I love them and also about including your own hand-written journaling on your digital pages. Come check it out if those things interest you.\nThat is so cool!! I am definitely going to have to try it!!\nMy brain malfunctioned at 'I actually created my own font and just use that', LOL. That's so cool shannonc! What you said about my method is true, though. Also, sometimes I use different kinds of handwriting in the same space, so scanning is helpful for that, too.\nDUDE! dawnfarias I love that handwriting technique! It's so different than what I did and seems super easy, thanks for sharing! I actually created my own font and just use that... but with your method, you probably get even more of a handwriting feel because all the letters are always different. Definitely going to try that!\nThanks for reading tiffany scraps! I look forward to seeing any pages you make with your own handwriting. Yay!\nWhat a great layout. I will def. check out your post.\nOK... I'm back after reading and then watching the handwriting tutorial. I LOVE the way you included your handwriting and I look forward to trying that in the future. So cool!\nLast edited by tiffany scraps; 07-15-2018, 12:25 AM.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Site Description: CHEVROLET CAMARO Z28 Used Car History reviews and guide on classic and older model cars. Including car designers from Italy and Germany. Learn history of cars from classic, exotic, used, new to prototype model cars.\nRun a CARFAX Report. If the vehicle has a clean title history it will qualify for the CARFAX Buyback Guarantee\u2026 FREE from CARFAX.\nCARFAX will actually buy the vehicle back from you if you find a severe problem (major accident, fire, flood damage, major odometer problems or a manufacturer buyback) that was reported by a DMV and not included in the Vehicle History Report. This protection lasts for one year and is transferable. Plus 100% Money Back Guarantee: Within 30 days after the purchase of a CARFAX Report you will receive a refund if you are dissatisfied for any reason.\nTry CARFAX used car history report now!\nIn 1982 Chevrolet had released the third-generation Camaro and it had sold well, lasting for 10 years and growing in power during that time. As always, the Z28 was the performance option, and in 1992 it was the 25th anniversary of the model, but also the last year for that generation, so the car went out with a bang. It received a Heritage appearance package in either red, white or black and this consisted of bolt-on spoilers plus side skirts to modernize its appearance. The live rear axle came with a limited-slip differential and 3.42:1 gears, and was located on lower trailing arms plus a Panhard rod for better cornering feel. The 305 ci (5-litre) V8 had electronic Tuned Port Injection with an individual runner for each cylinder, and Chevrolet made the car into more of a driver's machine by fitting a five-speed manual gearbox.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvsny b/data_all_eng_slimpj/shuffled/split2/finalzzvsny
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+{"text":"Founded in 1935 MJ Flood offers total business solutions to its customers. As part of the offering the company now offers technology solutions that secure the most precious business assets: cloud solutions, big insights and IoT.\nCase Design and implementation of branded guest wireless access and big insights for MJ Flood business customers.\nObjective Delivery of branded guest access and big insights in Ireland, e.g. Jervis street shopping centre, Irish Rail.\nResult With MJ Flood we have a strong partner with whom we will further develop our presence on the Irish Market.\nPosted in Partners, System integrators.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"St. Patrick's Day was first publicly celebrated in Boston in 1737 where a large population of Irish immigrants resided. Nearly 200 years later, the first St. Patrick's Day parade in the Irish Free State was held in Dublin in 1931. During the mid 90's, the Irish government also began a campaign to promote tourism in Ireland on March 17th.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"James Wiseman, the number one recruit in the nation for the high school recruiting class of 2019, has verbally committed to the Memphis Tigers.\nWiseman had recently had his list cut down to the Tigers, Vanderbilt, Kansas, Florida State and Kentucky but this was a major recruiting battle between John Calipari and Memphis going back years ago.\nThe Tigers held the ultimate advantage in Wiseman's recruitment, his former AAU coach and now head coach of the Memphis Tigers Penny Hardaway. When Hardaway's hiring caused then-Kentucky commit DJ Jeffries to flip his commitment to Memphis, Wiseman's eventual commitment became all but inevitable.\nThis is a major headline not just for those in Tennessee but for all of college basketball. John Calipari has fallen to the back-burners in terms of luring top recruits; and he there is no other recruit in all of his years that he's gone after as hard as he did for Wiseman. Even with Kentucky's greatest efforts, Wiseman will be at Memphis; and that sends a statement particularly to college basketball's blue bloods.\nWiseman is the top recruit in the country for a reason. He is the next product of what has become an absurd gene pool of physically imposing while equally skilled-big men. His pure athleticism is unmatched and he has a matured feel for finishing around the rim; which could be borderline unstoppable at the level he'll be playing at.\nHardaway had already begun to build something special before it was even a certainty that they were getting a prospect like Wiseman. A lot will be told in what Penny does in the years after next when he won't be virtually guaranteed two top recruits that he happened to coach in high school. One thing is for sure, all eyes will be in Memphis next year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Adoreum Club Member Spencer Hart has been titled 'The Future of Menswear' by Robb Report Uk Edition.\nNick Hart, London Tailor, has set up The Wardrobe by Spencer Hart \u2013 a by-appointment-only, ultra-luxe, made-to-measure service designed to furnish the stylish 'inclined but terminally time-poor' man. Nick's patrons include Benedict Cumberbatch, P Diddy, Chiwetal Ejiofor and Matthew McConaughey.\nThe Wardrobe experience begins with a consultation during which the customer is quizzed about the climate they inhabit, the life they lead, their personal sartorial performances etc. A proposal is then put together of a combination outfits covering every base in the modern urban warrior's life. The client will receive garments that form complete looks ready to hang out and wear. The wardrobe will be delivered 10 weeks after the initial consultation with a comprehensive handbook to ensure the clients 'wear it well'.\nThe vision behind Spencer Hart is that time-poor men don't have time to go shopping or follow trends. Therefore, this one-stop personalised experience for these often in the public eye clients allow them to the look the best person of themselves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You're signed up for the release date email!\nMany thanks for signing up for notification of the release date of How to Self-Publish and Market a Children's Book \u2013 due out in June 2018.\nWhile waiting, why not take a look around my author website here? There are a few things I need to update, but it will give you an idea of the sorts of things you might want to include in yours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Tips for Making This Valentine's Day Super Special!\nYou know, of course, that Valentine's day is NEXT WEEK, right? If this is news to you, drop what you're doing and make a plan. There's still time to think about a dinner date or romantic adventure you'll both enjoy. And if you're feeling uninspired, just pick from this list for a super special Valentine's Day or night (or weekend before!) adventures in love!\nIf you book a table for two at the Bent Fork on E. Illiff in Aurora you can toast your main squeeze with one of the Fork's amazing martinis (try the chocolate!), then indulge in their fresh, award-winning cuisine. Or if you're more last-minute than most (and reading this on February 14), check with Open Table to see what's available in the way of emergency reservations!\nThe Fresh Flower Market on Parker Road across from the Arapahoe Crossing Shopping Center is expecting lots of red-rose bouquets to fly out the doors over the next week. The market says you can call or stop by to grab and go \u2014 or shop online for free delivery.\nIf we haven't struck a nerve yet, consider one of the following date ideas to mark this Valentine's Day with a truly memorable evening!\nThis weekend on Saturday, February 10, grab your Valentine and head over to the South Dock at 15120 E. Hampden for an evening of swing. Dry Dock and Swingin' Denver dance studio are hosting a special Valentine's Day Sock Hop with free dance lessons (at 7:00 and 9:00 p.m.). Be prepared to dance the night away to the group Dixie Leadfoot, and come dressed to swing and compete for the \"Best Dressed\" and \"Best Socks\" awards!\nABBA Mania has been delighting ABBA fans of all ages since the tribute group first formed in 1999, selling out UK theaters all over the UK. ABBA Mania is coming to Denver in time for Valentine's Day so you could spend it singing along to Waterloo and Dancing Queen at Denver's Oriental Theater. The Grimsby Telegraph calls ABBA Mania \"one of the most energetic, uplifting and (most importantly) fun tribute bands\" around. Grab a pair of seats for the show at 4335 W. 44th Avenue in Denver, starting at 8:00 p.m.\nEnjoy an unforgettable Valentine's evening at Scented Studio\u00ae at 3437 S. Broadway in Denver with buckets of champagne (ages 21 and over), hors d'oeuvres, chocolates, and other special treats as you come up with your own scent from 22 different designer blends. For $75 per person, the Valentine's Scented Journey Workshop will help you explore the custom creation of a personal fragrance for perfume (100mL) and a matching scent travel size roll-on. The event begins at 6:00 p.m. and not only will you learn a brief history of perfume-making art, you'll also find your own particular scent personality.\nIf flowers and candy seem too ordinary for a romantic Valentine's Day celebration, consider a couple's massage! At Hand and Stone on S. Parker Rd. near Arapahoe Crossing the signature couples' massage \u2014 with or without hot stones \u2014 lasts for one delicious hour! The couples' room is uniquely equipped to deliver side-by-side relaxing, stress-relieving massages for $70 (tip not included) and from all reports, you'll want to reward these magical masseuses for their efforts!\nCore Power Yoga at 18884 E. Hampden is hosting a Thai Yoga Massage class for couples this Saturday, February 10 from 6:00 to 8:00 p.m. Patty Novak will be guiding this partnered class focusing on creating awareness, openings and space. The cost is $40 per couple and this unique once-a-year class is always a popular event for two!\nLiving lakeside at Southshore is an adventure in itself. Nature lovers delight in the miles of walking and biking paths, and the birds and geese that make the grasslands around the reservoir their winter home. Stop in to see the models at the master-planned community from Toll Brothers, Richmond American Homes and Century Communities, three of Colorado's top builders. It's life at the lake, in both ranch and two-story designs, prices between the upper $300s and the $700s.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our network is your network and if you like to search online for golf tutorials, then look no further than JustaskJimmy.golf. Jimmy represents the expert golf professionals we embrace and support. We are the Professional Golfers Guild afterall!\nOur support to Members of the PGGuild does not stop with tournaments. We also want to promote and endorse our members as teaching professionals too and really enhance the decades of skill that can only be achieved through time.\nProfessional Members will be able to upload video clips and tutorials directly onto the Guild website. In doing so we further endorse our members and help to promote individual professionals who utilise the facility and we hope, contribute towards building networks and growing client bases in this wonderful sport.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1974 Honda Cb 125 Owners Manual - RL Yacht Owner's Register If you own or know an RL24, RL28, RL34 or Status 580 not shown here, please add it to the register Register a new boat here. This document contains the following sections: Book, DVD, Instructions, Rare & Secondhand, Sales Catalogue, Transfer & Decals, Odds & Ends Miscellaneous Book. 009277 .and the Wheel Turned Volume 1 hardcover - History of Australian tyre manufacturers, car breakdown service clubs and spark manufacturers along with early engines, old garages and various old advertising, 156 pages, $50.00. Electric Motorcycle Conversions \"Electric motorcycle conversions are here today! \" D&D Motor Systems the premier electric motorcycle motor manufacturer in the U.S. for doing a motorcycle conversion to electric.Our electric motorcycle motor offers higher performance than the pancake motors that are out there. Our electric motorcycle motor has: higher torque better thermal capabilities and a.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Davos, Jan 26 (IANS) The annual meeting of the World Economic Forum (WEF) ended here Saturday on a note of cautious optimism about the world economy.\nThe idea that global economy is on a bumpy path of recovery but is shadowed by uncertainty was shared by a couple of leading economists at the meeting.\nPolitical issues such as the Syrian crisis and Iran's nuclear programme got more attention this year.\nAt the four-day meeting, over 2,500 participants from almost 100 countries, including more than 1,500 business leaders and over 40 heads of state or government, exchanged views on the road to global economic recovery, financial outlook, new energy, health and climate change.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"VIBES of LIFE is a powerful learning experience for young persons.\nVibes of life has successfully been implemented since 2013 by Politistiko Ergastiri Ayion Omoloyiton. This year 2017, six european countries (Cyprus, Italy, Spain, Romania, Portugal, Poland) implicated and 5 workshops implemented. The results of the Youth Exchange, under Erasmus+ program were be presented in street performances of arts including theatre, music and dance.\nThe Youth Exchanges is a learning experience for young participants with a clear plan of a learning process. Reflection groups, evaluations in each day, learning goals, peer to peer support and mobilization of the young participant's active participation, creativity and challenge are the elements that end to a powerful street performance where young people present their own \"Vibes of Life\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..79b5da37574c33d420c47eb57210b1e51c7e1690
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@@ -0,0 +1,5 @@
+{"text":"Oracle Transportation Management has been integrated into JD Edwards EnterpriseOne sales order management, procurement and financial management applications to deliver a complete fulfilment execution system, Oracle says. From item availability to final delivery and financial settlement, users can now streamline transportation planning, execution and freight payment on a single application. The result is better collaboration with suppliers, service providers, carriers, customers and throughout purchasing, logistics and financial operations. Additionally, this integration accommodates multiple modes of transportation including full truckload, less-than-truckload, parcel and complex multi-leg air, ocean and rail shipments.\nThe integration builds on recent accolades for Oracle Transportation Management as it was positioned in the Leaders Quadrant in Gartner's 2007 Transportation Management Systems (TMS) Magic Quadrant. The integration is available immediately for licensed users relying on JD Edwards EnterpriseOne versions 8.11 SP1 and 8.12, the integration is delivered free to customers as part of their standard maintenance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are encouraged to take good care of our teeth. Whenever we need clean teeth, we should ensure that we seek the necessary help. Dental services should always be sought from qualified people. Expert services are assured once we seek services form professionals. There is need to ensure that we seek services from those who can offer us quality services. We are encouraged to seek services form dentists since they possess the right skills to offer us quality services. The sensitivity of teeth should be considered whenever we are seeking dental services. The following tips will guide you when choosing a dentist.\nWe are encouraged to seek services from those who have the right skills. This offers us surety of quality services. Through experts, we have surety of receiving quality services. We should seek dental services from dental professionals. Consulting professionals offer us a chance to receive quality services. professionals have the right skills in offering us dental services. Once we seek services from professionals, we have surety of receiving quality services.\nOne might be in the business of selling apartment complexes, and if this is so, he or she is no doubt in a very good field, as in the modern world of today, there are so many people looking for new homes to buy. However, they might also know that they have a lot of competition all around them, and that they really need to stand out if they want to gain the best out of their business. One will be glad to know that it is not at all hard to find help with getting new clients \u2013 one only needs to find the best SEO services available today. Finding the best SEO services, then, is no doubt something that is wonderful, as when you do so, you can be sure that the benefits you will gain will all be very satisfying, and you really will improve your business wonderfully.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"London to Paris for our Becky!\nKidspace is proud to be supporting one of its managers in her quest to take on the London to Paris Bike Ride for charity.\nBecky Thorne will take part in the London to Paris bike ride this summer.\nThe grueling 300-mile event begins at Crystal Palace and ends at the Eiffel Tower.\nMum of two Becky will be part of a 22 strong team from her local gym and together they are aiming to raise \u00a324,000 to support Saint Francis Hospice and the British Heart Foundation.\nBecky, who has worked at Kidspace for 12 years, said: \"We are 22 average gym members who want to make a change to the lives of others. Cycling almost 300 miles from London to Paris will take hours of dedication in training & preparation.\nThe ride is just over 271 miles and the team will be cycling 85 miles on day one, 71 miles on day two, 65 miles day three and 50 miles on day four.\n\u00a9 2014 Kidspace Adventures. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Read last week's news and realize that, yes, Utah is the nuclear nexis of the United States. Deseret News reported on the confusing repercussions of Energy Solution's proposed acquisition of a west Texas facility handling low-level radioactive waste. The feds don't like the monopolistic aspect of it, but N-waste critics think the purchase would allow waste to be sent to Texas instead of Utah. Then, there's the Blue Castle plan to build a nuclear plant on the Green River. Oops, it missed some hefty pre-operational payments to the two affected counties. Now, BC wants to renegotiate, a story in The Salt Lake Tribune says. Meanwhile, surprise! Daily Utah Chronicle let us know that the only nuclear reactor in the Beehive State is housed at the University of Utah. It doesn't generate electricity, but helps in research. The problem, which they're working on, is to develop concrete that won't become activated and result in millions of dollars worth of disposal costs.\nForget about the bogus War on Christmas. Utah and other Western states are facing a war on history. Rep. Rob Bishop is pushing away at his Public Lands Initiative, an effort that is bound to ensure a legacy for him as the person who tirelessly sought to destroy the nation's natural heritage. Bishop is meeting with Donald Trump's transition team to discuss undoing Obama's monument designations, which include Grand Staircase Escalante, two new monuments in New Mexico, and the possibility of stopping the Bears Ears monument in its tracks. That alone has activated the Ute Nation, among other Native American tribes, to protest. It doesn't matter whose land it is. What matters is whether it will be denuded and pillaged by the desperate oil-and-gas industry.\nIn 2005, skeptics thought that prayer didn't exactly help fertility. In 2014, a study took returned Mormon missionaries and scanned their brains to see if prayer makes them better people. And now today, both The Salt Lake Tribune and Deseret News have reported on the continuing study that showed parts of the brain lighting up at spiritual thoughts. It's almost funny to see yet another study on prayer. Most, of course, have focused on whether it works, and there are mixed results. It's unclear what these studies hope to show, but they might just demonstrate how different people react to different stimuli. In 2006, The New York Times wrote of 10 studies that showed prayer didn't do a danged thing. But it's possible people can do it for themselves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Name users in this club that have left came back and left again. And are gone for Good. And the ones tu truly miss.\nKate is one of the m\u00e1s underrated characters in A&O, believe it o not. Just look through Deviantart, Fanpop, Furaffinity, etc. You'll see my point. And photoedits don't count.\nMorning everyone! So I've got some good\/bad news that most of tu could care less about. Tomorrow, I am going on a weeks cruise and won't be back till siguiente Sunday. So, I will be off during the week. I really don't want to come inicial and see a lot of drama on my muro o anywhere. Please stay cool tu guys!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaadkd b/data_all_eng_slimpj/shuffled/split2/finalzzzaadkd
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Today in EHF Headquarters in Vienna, was held a draw for the last 8 teams in EHF Velux Champions League and EHF Cup. Four teams from Group C, SG Flensburg, Kiel, Veszprem, and Rhein Neckar are the teams that will fight for a place in Cologne. Macedonian teams HC Vardar Skopje and HC Metalurg will have tough task playing against SG Flensburg and THW Kiel. Rhein Neckar will face the favorite for the title this year FC Barcelona. After years almost getting to the Final Four, Veszprem will try this year against PSG for the last spot in the Champions League.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I installed 4gb (2x 2gb) ram for my macbook. It ran fine for a few hours, then randomly the system froze, the screen went pixelated and a horrible static noise came out the speakers.\nI *think* it only happens when its on charge, so maybe something to do with overheating. Also, i was only using low intensity programs, like msn and firefox.\nIf anybody has any idea, it would be much appreciated.\nIt's MemTest86+ time. You appear to be having issues with either your memory or controller. Download MemTest, put it on floppy and boot to floppy. Run the program with both sticks inserted. You will probably fail. Take one stick out and retest. If this one passes, pull it out and replace it with the other stick. If they both pass, then your controller is being pushed too much and needs to either be adjusted (voltage) or replaced. Let us know what you find.\nThe system profiler is like Window's Vista performance test. It's horrible to use to troubleshoot hardware issues. You need the right tool for the right job. Get MemTest86+ downloaded and installed onto a floppy. Then boot from that floppy. MemTest86+ runs independantly of any OS, so it gives your real information that isn't skewed by an OS's processes and maximization efforts. As for the voltage, I don't own any Mac PCs, so I can't say for certain how to fix your issue if you can't adjust the BIOS parameters. If you find that your memory is OK, you may end up making a post over at Appleland and seeing what info that can come up with.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Zaheer Ali is a Mumbai-based academician, free-lance journalist and social activist. Starting his career as a research investigator in a project on socio-political consciousness among Muslims in India undertaken by the Centre for the Study of Developing Societies, Delhi, he worked for a year with a monthly magazine, Secular Democracy, in New Delhi. Subsequently, he began teaching Political Science, first in Aurangabad and later retiring as Head of the Department of Political Science, I.Y. College, Mumbai.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The European Union settled Tuesday on an anti-corruption agreement to force large mining companies to publicly announce payments made to governments where they operate, reports France 24.\nThe markets commissioner for the EU said in written statement the extractive sector is \"far too often shrouded in secrecy\" and the new legislation would \"help fight tax evasion and corruption.\"\nThe accord focuses on companies with profits of at least 40 million euros ($52.5 million), total assets worth 20 million euros or 250 employees. It covers disclosure requirements for the extractive industry, as well as primary forest loggers, and on simpler accounting requirements for small companies.\nMember EU states \u2014 27 in total \u2014 must now approve the agreement.\nCitizens in resource-rich countries will be better informed about how much their governments are being paid by multinational corporations for oil and gas, mineral deposits and forests, the markets commissioner said.\nOxfam, an international non-governmental organization for community development, said in a written statement it welcomed the accord.\n\"It's excellent news that the EU is moving towards a law that will help ordinary people harness the natural resource wealth of their countries to be lifted out of poverty,\" it said, adding the construction and telecommunications sectors should also be included.\n\"Strikingly, poor countries lose more to tax dodging than they receive in aid each year.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This July I'll spend four days homeless, with a Zen teacher named Genro and a small group of fellow Buddhists. We'll live, sleep, and meditate on the streets together and eat at soup kitchens. I think the retreat has a triple purpose: First, briefly abandoning the comfort and certainty of my regular life helps me practice non-attachment, the same as it helped the first Buddhist monks. Second, it gives me a taste of what it's like to be homeless, so I can better understand the homeless people I meet in NYC. And finally, it's an opportunity to raise money for homeless services.\nFor the most part we blend in with the homeless population. On the occasions I've explained to street folk what we're doing, they have without exception appreciated the effort we make to walk a mile in their shoes.\nPeople often ask me whether, by doing a street retreat, we're competing with homeless people for scarce resources. I think not\u2014we stay at the back of the line in soup kitchens, and we sleep on sidewalks instead of in shelters. Although we beg for a few dollars on the street, we raise thousands of dollars for homeless services.\nThat's where you come in: I have to raise $500 by July. The money will be distributed among the organizations that help us while we're on the street, places like the Catholic Worker, Coalition for the Homeless, and the Bowery Mission. And some of the money will support the social service activities of the Hudson River Zen Center. I'm not allowed to just use $500 of my own. I have to be humble and ask for the money from you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Vatican City (Agenzia Fides) \u2013 On Monday, March 20, at the Apostolic Palace, the Holy Father, Pope Francis, received in Audience His Excellency Mr Paul Kagame, President of the Republic of Rwanda. Subsequently, the President met His Eminence Cardinal Pietro Parolin, Secretary of State, who was accompanied by His Excellency, Archbishop Paul R. Gallagher, Secretary for Relations with States.\nDuring the cordial exchanges, the good relations that exist between the Holy See and Rwanda were recalled. Appreciation was expressed for the notable path of recovery towards the social, political and economic stabilisation of the country. Likewise noted was the collaboration between the State and the local Church in the work of national reconciliation and in the consolidation of peace, for the benefit of the whole Nation. In this context, the Pope conveyed his profound sadness, and that of the Holy See and of the Church, for the genocide against the Tutsi. He expressed his solidarity with the victims and with those who continue to suffer the consequences of those tragic events and, evoking the gesture of Pope Saint John Paul II during the Great Jubilee of the Year 2000, he implored anew God's forgiveness for the sins and failings of the Church and its members, among whom priests, and religious men and women who succumbed to hatred and violence, betraying their own evangelical mission. The Pope also expressed the desire that this humble recognition of the failings of that period, which, unfortunately, disfigured the face of the Church, may contribute to a \"purification of memory\" and may promote, in hope and renewed trust, a future of peace, witnessing to the concrete possibility of living and working together, once the dignity of the human person and the common good are put at the centre\".\nAccording to data collected at the time by Agenzia Fides, 248 victims among the pastoral workers of the Church were registered in the genocide of 1994, including fifteen deaths as a result of ill-treatment, lack of medical care and disappearances.\nThe people who lost their lives in Rwanda in 1994: three Bishops and 103 priests (100 diocesan from all the dioceses and three Jesuits), 47 brothers of 7 institutes (29 Josephites, 2 Franciscans, 6 Marists, 4 brothers of Santa Croce, 3 brothers of Mercy, 2 Benedictines and 1 brother of Charity).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The City of Homedale is buzzing with excitement and anticipation for this years fair and rodeo! August has many events and happenings - Read Below!\n\u25ba Purchase raffle tickets at the Senior Center for this beautiful quilt! Tickets are 6 for $5.00 or $1.00 each!\nAugust 6 & 7 from 8-8 both days. Come in, meet the new counselor, and complete your paperwork. See you then!\nThe Homedale Football Program is looking for help in funding the yearly expenses of running a Championship level program. This includes the yearly costs of replacing aged helmets, shoulder pads, uniforms, and the general football equipment. Our Game Night Program is a great way to help with this cause and also a great way to show your appreciation and support of our home town football program!\nHelp continue to build on the rich tradition that is Homedale Football! Interested in being a sponsor? CLICK HERE TO LEARN MORE!\nAre you a recent high school graduate? Do you want to \"go on\", but you aren't sure what to do or who to talk to?\nOn August 7, from 5-7 p.m. representatives from colleges and military will be at COSSA to help you out.\nStop by and let them help you plan your future.\nRegistration is below CJ Strike Dam in gravel parking lot from 7am-7:30am - Weigh in is there also.\nYou can use 4 ramps, either Cottonwood, Air Force, Loveridge Bridge, and the dirt ramp at the HWY 78 bridge these are the only 4 ramps we are permitted for. If you use another ramp you will be disqualified.\nTournament hours are from 8:00 a.m. to 6:00 p.m.\n\u200bTournament Prize Pay Out \u2013 100% of all entry fee money remaining after operation expenses will be awarded to the Monreal Family. Big Fish and the Top 30% of teams will be awarded prizes for there placement in the tournament.\nHave you entered the Project Tiny House 2018 raffle yet?! It's not too late to win this gorgeous tiny house built entirely by COSSA students, all while providing crucial support to seniors across Idaho.\nHelp raise funds for our programs by purchasing a #ProjectTinyHouse raffle ticket at the Caldwell office (304 N. Kimball) or at our Metro Meals on Wheels office in Boise (4900 N. Rosepoint Way). The drawing will take place late August or early September, with the official date to be announced soon!\nThank you to our amazing partners Idaho Power, The Home Depot, JGT Architecture, Sherwin-Williams, Camping World, and D&B Supply for their ongoing support of this great cause for Metro Community Services and COSSA!\nThe first annual Helping Homedale project will kick off June 1st, 2018 and run through August 31st, 2018. Check out the amazing progress at the Bowling Ally!\nThe Homedale Chamber of Commerce Helping Homedale project is a summer long program created to help local Homedale businesses spruce up their storefronts! Most economic development professionals and downtown revitalization experts feel that expenditures for facelifts of their downtown businesses is money well spent and a critical part of a vital business community. Together let's beautify our downtown corridor!\nHelping Homedale sponsored by the Homedale Chamber of Commerce will provide matching funds of $0.50 for every $1.00 spent on storefront and fa\u00e7ade improvements by local businesses up to a maximum matching contribution of $250.00 per participating business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LabCorp (Laboratory Corporation of America Holdings) has been hit with a whistle-blower federal lawsuit alleging the laboratory services company defrauded Virginia's Medicaid program by billing it at much higher rates than other customers, according to a recently unsealed complaint filed in federal court on September 9th.\nRelators Hunter Laboratories LLC and its founder Chris Riedel contend that LabCorp made false claims for payment of Medicaid-covered laboratory tests by claiming that the fees they charged to Medicaid were no higher that the maximum allowed under Virginia regulations.\nAs a participating Medicaid provider, LabCorp is required to provide services to Medicaid patients at their most favorable rates, but they repeatedly defrauded Medicaid by billing the program for fees well in excess of their lowest costs, according to the complaint.\nFor example, LabCorp, the Relators say, provided volume-based discounts to members of the Premier Inc. purchasing collective, resulting in discounted fees that are way below what LabCorp has billed to Medicaid.\n\"This suit calls defendants to answer for defrauding Virginia's taxpayers and compromising the welfare of Medicaid beneficiaries,\" the Relators said.\nAdditionally, for some tests, rates for private customers have been discounted well below costs, but LabCorp nevertheless has an interest in keeping those rates low in order to prevent any other laboratories from gaining a piece of the market, according to the complaint.\n\"In other words, by using the publicly funded Medicaid program to subsidize private discounts, the larger and better established laboratories have cornered much of the market for themselves,\" the complaint said.\nThe suit alleges violations of the Virginia Fraud Against Taxpayers Act. The Relators are seeking civil penalties and treble damages.\nRepresentatives for LabCorp did not immediately respond to requests for comment on Wednesday.\nRiedel and Hunter have won big in cases against LabCorp before. In 2011, LabCorp agreed to pay $49.5 million to settle a California lawsuit alleging it illegally overcharged Medi-Cal for laboratory tests and gave kickbacks in exchange for Medi-Cal referrals.\nThe suit originated with a qui tam complaint also filed in 2005 Riedel and Hunter alleging that LabCorp and others had systematically overcharged Medi-Cal over a 15-year period. In March 2009, then-California Attorney General Jerry Brown announced that the state had intervened in the suit.\nAndrew Baker, the former CEO of Unilab and current CEO of Huntington Life Sciences, has written an article for The Huffington Post in which he asks the federal government to stop LabCorp and another lab company from continuing to scam the Medicare and Medicaid programs of billions of dollars.\nMr. Baker had previously filed a whistleblower lawsuit against LabCorp in 2007 alleging that LabCorp violated the federal False Claims Act and Anti-Kickback Statutes. Those case is still in court.\nIn the article he estimates that LabCorp and the other lab have cost taxpayers $15 billion since 1996 in the form of false claims stemming from illegal kickbacks to Aetna, Cigna, United Healthcare and Blue Cross.\nThe claims are that Labcorp is breaking federal laws by deeply discounting lab fees to private insurance companies, sometimes charging them for laboratory tests even below their costs. In exchange, the insurance companies pressure doctors in their networks to send all of their patients' lab work, including Medicare and Medicaid patients, to LabCorp.\nHe claims that Labcorp funds the kickbacks, in the form of lower lab fees for private insurance companies, by charging Medicare and Medicaid patients the highest possible fee instead of offering them the lowest charged price, and by pressuring doctors to send all of their lab work exclusively to Labcorp. Other categories in Medicaid and Medicare require that the government be charged the lowest charged fees by a provider.\nMr. Baker also mentions LabCorp's $50 million settlement with the state of California for overcharging California's Medicaid program and for providing kickbacks to physicians for referrals.\nAs a result of his article, pressure is increasing for government intervention in laboratory pricing for government programs. He advocates for clarification of the intent of current federal law that would require laboratories to charge Medicare and Medicaid their \"best price\", just as California has already done. This would require that Laboratory Corporation of America can only charge Medicare and Medicaid the lowest price they charge private insurance companies or HMOs. Which in turn means a massive hit to Labcorp's bottom line. It would also open up the market to smaller labs which don't have the multi-tier, lower than cost pricing intended to put them out of business. Such a hit to Labcorp's financials would tumble their stock (NYSE: LH).\nLabCorp has been sued and is under Federal investigation for Medicaid and Medicare fraud. In addition, it's under scrutiny by the U.S. Senate Finance Committee who is investigating Medicare and Medicaid fraud. Laboratory Corporation of America was ordered to hand over its financial records to the U.S. Senate.\nA lawsuit was filed on behalf of the Federal government that claims LabCorp operated a \"pull-through\" scheme to force doctors covered by insurers to use LabCorp for all medical testing viagra uk buy. The suit claims LabCorp (NYSE: LH) offered illegal discounts to doctors in exchange for referring all their patients who need laboratory testing to the company. According to federal anti-kickback laws, it's illegal for health care companies to directly or indirectly compensate other parties to encourage them to order any service paid for by the federal health care programs.\nUnder federal law, companies can't charge or participate in the Medicare or Medicaid program if they violate federal laws. The suit claims LabCorp charged Medicaid and Medicare more than $1 billion, which is about 20% of the companies total income. If LabCorp is found guilty they will not be allowed to conduct lab tests on anyone covered by Medicaid and Medicare, which is a large portion of LabCorps testing income.\nOn other news, LabCorp secured a new credit line of $1 billion. What a coincidence.\nJust last month, LabCorp reported that they made $131.4 Million dollars in profit for the latest quarter, on revenue of $1.19 Billion. That was a 17% increase in profits as compared to the same quarter last year. Obviously LabCorp stock (NYSE: LH) went up to a new yearly high and all the senior executives high-fived each other.\nWell that was last month. Just a few days ago Labcorp went to the state and local governments in North Carolina for a handout of taxpayer money. LabCorp wants to consolidate its billing operations and if it selects Greensboro as the location of the new facility, it would receive nearly $900,000 in taxpayer money. That's free money from the taxpayers that will never be repaid.\nThe Burlington-based company is reportedly considering moving its billing operations from 24 sites across the country to one location in Greensboro or in Danville, Virgina. On Thursday, Guilford County commissioners approved giving the company $248,791 of tax payers money. The city of Greensboro also is offering $373,000. North Carolina would grant Labcorp $275,000 of taxpayer money. That's a total payoff of $896,791 of taxpayer's money. The excuse the elected officials are using is North Carolina's unemployment rate, which is up to 11% and above the national average. This is the ninth straight month the state's unemployment rate has reached double digits. So what is the government doing handing out free money to a company that just made over $131 Million in the last three months?\nMaking a lab error on a blood test can have a serious impact on a life. I don't know if LabCorp employees don't care, or if they are simply numb to the impact of their lab errors because they rapidly run through so many blood tests. This is part of their \"quantity is more important than quality\" problem.\nI received this email from Natasha today. I don't know what I would do if this LabCorp lab error occurred during a blood test on my child. It is devastating news that no one should have to endure as a result of a lab error.\nLabCorp diagnosed my son with a rare genetic disorder called Angelman Syndrome by a Methylation blood work test. This syndrome is characterized with mental retardation, lack of speech, and seizures just to name a few symptoms. Our family was devastated, and we grieved for months.\nWhen he got in to see a neurologist, he ordered another test to get more information about which type of the syndrome he may have. This test came back negative. We were shocked, but tried not to get our hopes up too high because we did not know which test was accurate. When we finally got in to see the geneticist, he ordered another round of genetic testing, this time a more detailed test called a cgh, as well as the original test. All negative.\nLabCorp told the genetic counselor that they think they switched his blood. For privacy reasons they wont tell us anything, but assured her that they would try and find the child who really has this condition. As relieved as we were to have the good news, we are so angry for this mix up happening in the first place. Months and months of our lives were taken from us, and I still have not received one phone call from Lab Corp. We are unsure how to proceed from here. Anyway, this is an EXTREMELY abbreviated version of the story. Please let me know if you would like more information, I will do anything to help, no family should go through what we experienced.\nMy recommendation to Natasha is that she find a lawyer and sue LabCorp for punitive damages. A blood test lab error of this magnitude should be sufficient grounds for a lawsuit. One that I am sure LabCorp would settle quickly before the news quickly spreads and their testing service loses even more credibility.\nThe days between getting LabCorp's lab results and finding out about LabCorp's error were probably Natasha's worst days of her life. No health services provider has the right to do this to anyone. It's obvious that Laboratory Corporation of America understands money, so they should pay for their mistake. In addition, government regulators should step in so that this never happens again.\nThe United States Food and Drug Administration issued a letter to LabCorp warning them that they are in violation of the law. LabCorp doing illegal things brings up their checkered past, when they were charged with fraud in the LabScam investigation.\nThe FDA warning letter and possible criminal and civil action against LabCorp could be the beginning of the end for Laboratory Corporation of America. I say this because as one government agency moves against them, there are possibly other government agencies that might step up to further investigate other allegations against the company. It's about time that this regulatory oversight process start. For too long LabCorp has focused on the buck as opposed to the law and the required quality standards. In my own opinion, it is a company that is out of control.\nWASHINGTON, Oct 8 (Reuters) \u2013 Laboratory Corp of America (LH.N: Quote, Profile, Research, Stock Buzz) is violating the law by selling an ovarian cancer screening test without regulatory approval, U.S. health officials said on Wednesday.\nThe OvaSure test does not fall in a category that can be sold without prior clearance from the agency, the Food and Drug Administration said.\n\"Because you do not have marketing clearance or approval from the FDA, marketing OvaSure is in violation of the law,\" the agency said in a letter dated Sept. 29.\nLabCorp spokesman Eric Lindblom said the company was \"disappointed\" by the letter.\n\"We are currently in discussions with the FDA over the next steps and of course we share the FDA's determination to assure patients are protected,\" he said.\nThe company started selling the blood test in June, saying it could detect early-stage ovarian cancer in high-risk women.\nThe FDA letter was posted on the FDA website. You can read the full letter by clicking here.\nLabCorp shares (NYSE: LH) fell nearly 1.8 percent today to close at $60.63 on the New York Stock Exchange.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jesco Chip and Waste Trucks help with savings and safety in the workplace. This truck can handle the big bulk of any heavy load. The balanced design makes dumping easier.\nMounted on two Mold-on Rubber Wheels 10\" x 2-1\/2\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get 2 years Protection Plan for your MacBook, Chromebook, Dell and other Laptops at just $40.88.\nPlan must be purchased within 30 days of laptop purchase.\n*All comparisons assume customer owns 11-inch Apple MacBook Air 128-GB model. What You Pay based on Everything Breaks laptop protection plan, premium for SquareTrade 3-Year Laptop Protection Plan and AppleCare MacBook Protection Plan.\nFrom cracked screens to broken internal parts, we cover the big damage that can come from the smallest falls.\nYou're covered in case of Wi-Fi failure, which can take your computer off the grid.\nIf your hard drive fails, we'll replace it for you.\nYou're covered if your LCD screen gets dimmer over time, discolored or gets lines through the display.\nDoesn't matter if it's one splash, one spill or full immersion-we've got you covered.\nIf you need to wiggle the power cord or hold the cable in place to get a change, it might be the port, not the cable. if so, we've got you covered.\nWHEN DOES THE EVERYTHING BREAKS LAPTOP PROTECTION PLAN COVERAGE START?\nThe plan term starts on the date of enrollment via the registration process. Products are eligible for coverage on the 31st day following the plan term start date.\nHOW DOES THE EVERYTHING LAPTOP PROTECTION PLAN WORK?\nWith 2 pricing options, the Everything Breaks Laptop Protection Plan covers malfunctions, drops, and spills. The plan covers those repairs necessary to return your laptop to its intended functionality under normal operation.\nWHAT IS THE DIFFERENCE BETWEEN THE EVERYTHING BREAKS LAPTOP PROTECTION PLAN AND THE SERVICE PLANS OFFERED BY MOST RETAILERS?\nA retailer's plan is expensive and with a limited term. Retailer plans often exceed 20% of the purchase price of each product. The Everything Breaks Appliance Protection plan is inexpensive. One low price covers all your breaks and repairs.\nWHY DO I NEED AN EVERYTHING BREAKS LAPTOP PROTECTION PLAN?\nLaptop repairs, even minor, can be very expensive. The Everything Breaks Laptop Protection Plan protects you from incurring unexpected repair or replacement costs if your registered product breaks down during the coverage period as a result of normal wear and tear. There's the convenience of just one phone number to call to have your registered product diagnosed, repaired, replaced or refunded.\nYou have the option to cover your laptop for 2 or 3 years.\nYes, included at no additional cost to you, your Everything Breaks Laptop Protection Plan comes with the added benefit of the Everything Breaks Personal Assistant. Your Everything Breaks Personal Assistant is available 24 hours a day, seven days a week. Use it as much as you wish and for whatever you need from travel arrangements and restaurant reservations, to traffic conditions and emergency assistance ... you name it, your personal assistant will do it. Anything. Anytime. Anywhere. This no-cost feature is available to you by app, phone, email, text and online.\nThe Everything Breaks Laptop Protection Plan covers any brand from any retailer.\nIS THE EVERYTHING BREAKS Laptop PROTECTION PLAN INSURED?\nCall our toll free number at 888-994-0914 when your covered Laptop is in need of repair.\nYes, there is a $49-$75 deductible that you pay directly to the service technician depending on the retail price of your registered product.\nWhen repairing your eligible product, only authorized service repair professionals are used. The Everything Breaks Laptop Protection Plan provides you with the same type of service that you would get from the manufacturer or retailer of your product.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Whether you're coming to the end of your current deal with Scottish Widows Bank, or already on one of our variable rates and want the security of a fixed rate, take a look at your options.\nIf you'd like to borrow more on your existing mortgage, find out how to apply.\nWith Offset you could use your savings to save thousands of pounds on your mortgage. See how it works and add Offset to your mortgage.\nSee how the Bank of England base rate affects your mortgage.\nLog in or register for Internet Banking to make overpayments and view your mortgage online.\nInformation and advice for customers who may be experiencing financial difficulty.\nThis product is no longer available. If you already have a Mortgage Reserve Account you can view basic rate information and our contact details within this section.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Frosts Landscape Construction are delighted to be one of the 49 BALI members, out of 108 entries received, to be nominated for not one, but two awards ahead of the BALI National Landscape Awards ceremony later this year.\nWe will not find out if we have been lucky enough to secure a Principal Award, Special Award or the coveted Grande Award until the awards ceremony at Grosvenor House, Park Lane, London on Friday 4th December.\nFind out more about all of the 2015 award winners here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oval-shaped poker table for 10 players comes with metal folding legs. Supreme cloth felt and removable arm rest to change your cloth.\nThis table is made with solid 3\/4''plywood with steel frame under the table.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A troubled man becomes lost to nature.\nCommissioned for the Sherman 48-Hour Script to Screen Festival, Bored Games was produced by Andrew Yorke and Kevin Michael over a two-day period. The short was shown to the delight of audiences at the close of the Festival. It was screened out of competition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The pawpaw (Asimina triloba) is a small tree with pointed green leaves and dark burgundy flowers that bears large yellowish-green to brown fruit, similar in appearance to a mango. And while it's known for its exotic flavor, pawpaw fruit and the trees they grow on are actually native to the eastern U.S. and Canada.\nThe pawpaw is found by foraging and is rarely cultivated, possibly because the fruit ripens to the point of fermentation soon after being picked and must be eaten within weeks.\nIn its report entitled \"Pawpaw: Small Tree, Big Impact,\" the National Park Service describes the pawpaw as having leaves and branches that deer avoid, and bearing a fruit that is \"loved by all.\" The pawpaw comes from a mainly tropical plant family and produces the largest edible fruit native to North America.\nDarrin Nordahl, author of Eating Appalachia, writes of the pawpaw, \"Pawpaw is a food of unparalleled flavor, which the first European colonists revered so highly they named towns, islands, and creeks in the fruit's honor.\"\nWhen Is Pawpaw Season In Tennessee?\nYou'll find pawpaw fruit at farmers markets throughout Tennessee in September and October. The fruit begins to ripen in late summer, usually in August, then peaks in September and October. The fruit ripens quickly, so plan to use the pawpaws you buy in a recipe or eat them whole immediately after purchase.\nYou'll find pawpaw trees growing in thickets along waterways in the eastern, southern, and midwestern United States. The trees are easiest to spot from the water, so a kayak or canoe is an ideal way to scope them out.\nOnce you find a pawpaw tree, lightly shake the tree to dislodge any ripe fruit from its branches. Local wildlife love the pawpaw fruit, and since the fruit fall easily once ripe, many will be consumed as they drop. The good news: You can shake the tree to harvest what is left.\nForage responsibly. Generally, if the fruit overhangs public sidewalks it is okay to pick. Check the Falling Fruit map, built by and for foragers, to locate fruit trees in Tennessee (or elsewhere!) that are free to forage. Foraging laws vary at the federal, state, and municipal levels, so check with a park ranger or property owner (if you find a pawpaw tree at your vacation destination) before foraging without permission.\nFor this reason, he suspects the River Gorge would be an ideal place to forage them.\nDuring September and October, seek out fresh pawpaws at farmers markets, but keep in mind that since they have a short shelf life quantities may be limited and usually sell out quickly.\nPawpaws can also be found on the menus of Tennessee restaurants where chefs prepare dishes using pawpaws and incorporate them into their seasonal menus.\nHarvest Roots Ferments sells their kombucha made with pawpaws in grocery stores and at the Main Street Farmers Market on East Main Street in Chattanooga. Crabtree Farms in Chattanooga sells the fruit seasonally at the market as well.\nNolan Sherill of The Phoenix Pharmacy and Fountain in Knoxville has made pawpaw ice cream while the fruit is in season.\nTrevor Stockton, executive chef of the restaurant at RT Lodge in Maryville, has incorporated pawpaws into his dishes at the restaurant before, but said they don't use them every year as they can be temperamental. That said, when his produce guy has them available and they're fresh, you can expect to see them on the menu.\nEager to try pawpaw-inspired cuisine in Tennessee and beyond, but still not sure what you'd be getting yourself into?\nImagine a cross between a banana and a mango, with the texture of custard.\nIf you'd like to try a pawpaw recipe on your own, here's a method for canning pawpaw butter.\nHowever you decide to enjoy this \"otherworldly\" fruit, whether whole or in recipes, head to Tennessee in the fall and prepare to be wowed by the pawpaw's vibrant flavor and texture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Notify me of followup comments via e-mail. Amazon Inspire Digital Educational Resources. Since it took me two months to figure all this out, Amazon won't accept a return, and I am stuck with a half-working router. How is possible that Version 5. Fulfillment by Amazon FBA is a service we offer sellers that lets them store their products in Amazon's fulfillment centers, and we directly pack, ship, and provide customer service for these products.\nIf you're a seller, Fulfillment by Amazon can help you increase your sales. I tried everything I could I am an electronic engineer but could not make it work.\nSeptember 15, at 6: There are only three that don't overlap one another. Plan contract will be emailed from Asurion within 24 hours of purchase. If you still don't have I e-mailed D-Link U. Then I wanted to use MAC address filtering, a really good security feature, which I'd used with the old one.\nThere is not any real proof that training can benefit from protein supplements. It dlink di 524 basically new; no scratches dlink di 524 anything.\nPublished 10 months ago. IE still is the marketplace chief and a dlink di 524 component to people will miss your dlink di 524 writing due to this problem. Yes guys, I am afraid there was a little problem with vesion Also, setup was incredibly easy using the built-in wizard. There are multiple revisions of the D-Link DI, each with revision specific drivers, firmware, installation guides, and other resources.\nThis blog dlijk just lie my old one! Learn how your comment data is processed. We invite you to learn more about Fulfillment by Amazon.\nMake sure you put your keyword in the web page title, first flink and once or twice in the main body. Never use the WAN port!\nHow do I dlink di 524 the wireless router model dl as an AP? Withoutabox Submit to Film Festivals. Possible Problems when Changing your WiFi Settings After making these changes to your router you will almost definitely have to reconnect any previously connected devices. The dlunk below courtesy of D-Link shows where the button is located. Turn on 1-Click ordering for this browser. Image Unavailable Image not available for Color: Wireless Name Your wireless network needs to have a name to uniquely identify it from other wireless networks.\nPublished on July 23, The DI proved to be worthless in two ways. The advanced wireless technology built into the DI now offers up dlino 54Mbps through its wireless channels, allowing streaming videos and other dlink di 524 bandwidth applications, such as online gaming events to operate without the hassle of Ethernet cables.\nYour recently viewed items dl featured recommendations.\nFeatures like content filtering, MAC filtering, URL blocking, and Domain blocking are useful tools to prevent other unwanted intruders to connect to your network. Is it really take such a long time to reboot, dlink di 524 did I do something wrong?\nSearch for your wireless network again and make a new connection. The status light on D-Link router is flashing what is it indicating?\nBut, what concerning the bottom line?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Draw-tube attachment kit for deep water aeration for ponds over 6 feet deep. Helps combat silt build-up, nutrient accumulation, and fish kills. Attaches to WaterSmith PRO fountains.\nThis Draw Tube kit provides the adapters necessary to add a 4\u2033 thin wall PVC pipe to extend the intake depth in the pond. A PVC Pipe can be easily purchased and custom cut at any local hardware store. The length of the pipe should be cut so that it is 2\u2032 from the bottom of the pond. The WaterSmith Draw Tube allows the fountain to circulate the bottom water and aerate the entire lake which combats silt build-up, nutrient accumulation, algae blooms and fish kills.\nThe Drawtube kit is not compatible with the WaterSmith Wheel Kit (for 2HP fountains).\nAn average pond without proper circulation has multiple layers of water, with each layer getting colder as it gets deeper. These dark bottom layers typically have little or no oxygen, which can decrease the amount of useable fish habitat and inhibit bottom-dwelling food sources (such as worms and crawfish). Another drawback to an oxygen-starved pond bottom is that organic debris (leaves, plant material, dead fish, etc.) are unable to properly decompose. This can result in an accumulation of black, smelly, anaerobic sludge that builds up over the years, making the pond shallower. Another drawback to heavily layered ponds is that they are more susceptible to fall turnovers and subsequent fish kills.\nWhen a pond has proper aeration and circulation, the layers in the pond are well mixed, and all zones of the pond have good oxygen levels, day and night. Organic debris can now decompose in a way that the nutrients may now enter into the food chain to eventually become more food for your fish. The likelihood of a fish kill (from natural occurrences) is virtually eliminated. A well aerated pond will have healthy and active fish, and will return to it's natural, alluring beauty.\nGet information on seasonal pond & water garden maintenance tips from Dr. Dave, updates on sales, special events, aquaculture news. We respect your privacy and will never share your email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nicholas Hagger and Prince and Princess Hitachi in the grounds of their palace, 1965. Nicholas Hagger was Prince Hitachi's tutor from 1965 to 1967.\nChristmas card from Prince and Princess Hitachi, signed by both of them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The code below have been tested succesfully in the top seven browsers used in the world, including the legacy IE6.\nTry the code in a popup window.\nThis example consists of three files, one containing the client page (default.htm), one containing the code for server responses using HTTP GET (dataGET.asp) and one containing the code for server responses using HTTP POST (dataPOST.asp).\n<span style=\"cursor: pointer; text-decoration: underline; color: #0000FF;\"\nDiv element. Content will be replaced by data from the server.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Upgrade the appeal of your home or office with this eye-catching Virginia Tech Hokies 18\" x 44\" Deluxe Frame Panorama wall art.\nVirginia_Tech_Hokies_18_x_44_Deluxe_Frame_Panorama_Wall_Art http:\/\/images.fanatics.com\/lf http:\/\/dmimages.ff.p10\/chains\/2558114.txt false We are unable to customize this item with the text you have entered. Please try a different entry again. Your current entry cannot be processed due to guidelines for past and present player names. Please create a new entry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaextr b/data_all_eng_slimpj/shuffled/split2/finalzzzaextr
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index 0000000000000000000000000000000000000000..31130ffa04641b33e3ed4788bc21faf12f6fc5b5
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+{"text":"How can I securely communicate with an owner?\nOwners\/Manages and Guests can communicate through a number of ways. During each inquiry an Owner\/Manager will recieve an email notification indicating interest in a particular property. The Owner\/Manager can then initiate a converstation via email, Direct Chat, or phone. Direct Chat occurs through the site, while email and phone are not controlled by Vacayrx and therefore have no guarantee of security.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paul Lewis, the former House Armed Services Committee counsel who since 2013 worked as the Pentagon's special envoy on Guant\u00e1namo transfers, emptied his office and left the Department of Defense on Feb. 2. Lee Wolosky, who served as the third State Department Special Envoy for Guant\u00e1namo Closure, left his job at noon on Inauguration Day and has returned to his New York law firm.\nTwo State Department workers are still at Wolosky's old office in a form of caretaker status, sometimes more than two if extra help is needed, according to one of the staffers. But the writing was on the wall \u2014 actually off of it \u2014 even before Trump took office.\nMORE READING | President-elect Trump: Halt Guant\u00e1namo transfers. White House: No.\nThe holder is empty now outside the office run by Tim Andrews, described as an on-again, off-again 30-year Foreign Service Officer with a background in counterterrorism. Some describe him as Wolosky's former deputy; others describe him as the acting special envoy. Either way, the previous deputy, Charlie Trumbull, is gone from the office but still represents the State Department at the Periodic Review Board.\nThere are currently 41 captives at Guant\u00e1namo \u2014 10 facing charges; 26 classified as \"forever prisoners,\" ineligible for release; and five men who have been approved to go with security arrangements. In fact, the State Department secured deals to repatriate an Algerian and Moroccan in the dwindling days of the Obama administration; but former Secretary of Defense Ash Carter declined to release them.\nNow career government spokesmen are unable to explain what is being done, if anything, to get those cleared men out of Guant\u00e1namo Bay.\nNor could State or Defense Department spokesmen say if there's a way to arrange release, rehabilitation or resettlement if any more forever prisoners are cleared by the Periodic Review Board.\nBy law, Congress requires 30 days notice before a captive can leave the island prison; the major would not say whether Mattis had sent any notices to Capitol Hill.\nBut transfers weren't the only thing that Wolosky, who had the rank of ambassador, ran as the State Department Special Envoy.\nThe United States didn't want a freed captive spending hours of idle time at an internet cafe, chatting with fellow captives, or seeking help from a radical mosque.\nFormer Secretary of State John Kerry was supportive of efforts to close the detention center, and deals to move the men out. His successor, Rex Wayne Tillerson, has yet to offer an opinion.\nMeantime, attorney Shane Kadidal argues that an office dedicated to handling releases may not be necessary. The George W. Bush administration released more than 500 captives from Guant\u00e1namo without a special envoy. For a time that was part of the portfolio of United States Ambassador-at-Large for War Crimes Issues.\n\u25aa The Obama administration's final report to Congress on its efforts to close the detention center.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brandi Raae: What's the Best Way to Sell a Wedding Dress?\nPin It Now! Don't gasp. Don't give me any sentimental lectures. I just want advice on selling an 11 year old wedding dress. If it is even possible to sell an 11 year old wedding dress.\nEbay doesn't seem like a good choice to me. I don't want the hassle of packing and shipping it.\nI have thought about a consignment store. I've sold baby clothes via that route before, but just didn't know if there was a better way to sell in order to keep most or all of the profit.\nI've seen many of those listed in the regular 'want-ad' section of our local newspaper.\nYou know you and I had the same wedding dress? So find a set of twins who want a dual wedding and I dell mine too.\nLove the Prince dream. Ha! I was a fan of his way back when also. Can't interpret it for you. But I can picture it and it gave me the giggles.\nAny good consignment stores in the area for adults? What about a bridal store? They may be willing to do a consignment deal for you. Wouldn't hurt to give it a try.\nI'm still holding out that one of my girls will want to someday wear my dress. I'm open to them altering it anyway they want.\nYou could tell them a husband is included - at no additional cost - with the purchase of the dress.\nNah, that's not much of an incentive, is it?\nOh, yes, for some women that would be exactly what they want!!! Purchase a wedding dress and have husband included in the package? No need for anymore online dating nightmares, searching the singles ads, hoping to meet a marriage-minded guy, nope, here he is all packaged up and ready to go! \"Just find me a church!\" I can hear them saying!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The nth self-inflicted injury (Autonomy) flings Hewlett-Packard into recovery mode once again. I feel pained each time one of these poor decisions comes to light and takes its toll. I loved that company. You would have loved it too. It was not just another business entity, it was one of the great institutions of Western Civilization. David Packard and Bill Hewlett created a management system that was absolutely the best system for turning human capital into societal contribution. Part of the evidence of this is that that system worked worldwide and across industries.\nI speak of Hewlett Packard in the past tense not because I think they are doomed but because the company that now bears the name is a completely different company from the one I knew. I left HP in 1999 to join the spin-off, Agilent Technologies. So I don't pretend to know anything about the current situation there. But I do know that the original HP's formula for greatness is too important to allow it to be discredited by association with the current mess and too valuable to be forgotten. There is so much to share with you about the very accessible magic of the HP that grew at a compounded annual growth rate (CAGR) of 23% for over 50 years. Today I think I will just share a story and a summary description of the magic.\nIn the mid 1990's the late Lew Platt was CEO of HP. He and his CFO understandably worried that it would be increasingly difficult for HP to keep up the phenomenal record of 23% CAGR from 1942 to 1995. In 1995 the company reached $31.5 B in annual revenues. To grow even 15% in a year meant that the company would need to find $4.7B in new revenue. So they asked the Corporate Development group to identify strategies that other companies had used to created growth. I got the assignment. It was a great project. Since the General Managers of the businesses units were already thinking continuously about how to grow those existing businesses, I put particular emphasis on how companies got into new businesses.\nFirst: The company HAD done those things. Bill (Hewlett) and Dave (Packard) were intuitively new venture investors. The entered new businesses when the company did not NEED to do so. They went into countries (like China in ) when there was \"no\" market there yet.\nSecond: Financially things were going fine but that cadre of professional managers were focused on being careful stewards of their large businesses, not on creating high value new businesses. This paucity of internally generated growth encouraged later CEOs to plunge into a series of unfortunate acquisitions.\nThis formula does not seem to have been applied at the company with the HP name for some time but none of us should make the same mistake.\nThe original HP version of those elements create virtuous cycles of high expectation and high performance. Those cycles, in turn create their own fuel of profit and goodwill.\nMore specifics on this in future posts.\nFor a personal discussion of how to apply the HP formula contact me.\nBig idea projects also raise fascinating challenges for entrepreneurs and business people.\nSage Bionetworks: Accelerate biomedical discoveries by creating a system for open sourcing biomedical research data \u2014 thereby giving researchers access to thousands of times more data than they could collect on their own.\nWaste Enterprisers: Turning human waste in to biofuel, solving huge widespread cost, environmental, time and dignity issues.\nTerrapower: Unlimited safe energy from depleted uranium.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More rural diary dates: The Surrey Agricultural Society also hosts the Surrey Farm, Food & Drink Festival from Sunday August 27 to Saturday September 2 and the Surrey Game & Country Fair on Sunday September 24.\nIn 1954, Raymond Stovold of Shackleford's Lydling Farm brought together the Surrey Agricultural Association (founded in 1829) and the Redhill Agricultural Society (founded in 1862) for a joint show at Eashing Park and was responsible for bringing the two societies together under the present title. The Stovold family themselves have some pedigree, with over 400 years' experience of farming in Surrey. The event moved to its present site, Stoke Park in Guildford, in 1961 and has been there ever since. These days, Surrey County Show is known as the biggest one\u00acday agricultural show in the UK, welcoming around 40,000 visitors each year.\nDespite being held in Guildford, rural life is still at the show's heart \u2013 and with that comes cavalcades of animals. Among the thousand or so cows, sheep, pigs, goats, horses and donkeys, you'll be able to meet huge Highland bulls and dainty Jersey heifers in the cattle tent, black Zwartbles and a crinkled Texel in the sheep lines, dairy and pygmy goats, and you can cheer the entries in the 'grumpiest pig' and 'pig most like its handler' competitions. In the centrepiece Grand Arena, there's two show-jumping classes with national horse and rider combinations competing for a prize fund of over \u00a36,000 \u2013 as well as the ever popular fast and furious scurry driving. Miller's Ark Farm will also have a range of small, friendly animals for children to pet.\nIf you don't happen to have a horse, cow or pig to compete at the show, you can always take your dog. All breeds and types are welcome, and there are classes for everyone. There will be dog agility, hounds and terrier racing, gun dog training and sheep dog trials throughout the day.\nAway from the animals, Surrey County Show also happens to be a shopper's paradise. Whether you are looking to buy or to browse, head for the neatly arranged and sign\u00acposted shopping malls, which sell everything from tractors and wellie boots to hot tubs and garden furniture.\nThere's a tempting offering for foodies, too. Chef Daniel Britten (The White Horse in Hascombe) returns to the food theatre line-up, as does butcher Simon Taylor (Surrey Hills Butchers in Oxshott) who is teaming up with Steve Drake (Sorrel in Dorking) this year. Bake with Jack, Butter and Cream and Dorking Brewery will also be taking to the stage. There's also the Surrey Farmers' Market, where all the food and drink sold has been grown, reared, caught, brewed, picked, pickled, baked, smoked and processed by the stall holders.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafdbl b/data_all_eng_slimpj/shuffled/split2/finalzzzafdbl
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+{"text":"One of the rappers with the strongest face and head shape -- and by that I mean handsome. Not too much required is required for this cut; it's all about defining the hair line and making a statement. This look is more about the attitude than anything else. A good oil to keep the scalp moisturized and you're good to go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GolfLeague.us offers a wide range of golf league software programs designed to meet the needs of golf organizations throughout the United States. Since the software is hosted in the cloud, you can log on anywhere in the world with your computer or smart phone. The following is a summary of league types and their subsets.\nMass e-mail the entire league.\nE-mail scheduled events requiring feedback.\nTrack participation in scheduled events.\nPost to the member message board.\nAutomatic point calculation for matches.\nLeague stats tracked in any period (quarter, half or full year).\nAutomatic scheduling for the entire year.\nThis league type does not have teams or flights and is fully functional with the basic golf league package.\nIt is a highly competitive league format that matches player vs. player for points throughout the year.\nIt has many absent player options.\nA schedule for this league would match player vs. player until all players have played every other player.\nA points leader board will show individual player point leaders in any period.\nGolfLeague will create a schedule. Every player is scheduled to play every other player once without repeating. When every player has played every other player, the schedule will start over until all play dates have been scheduled. GolfLeague then creates matches each week based on the player schedule and posts the individual matches on the league calendar. A summary schedule is also created so that you can see the entire years schedule at a glance.\nAdministrators can input hole by hole data or just enter gross scores and points.\nThree options are available to handle absent players for individual play leagues which we call: the \"Bubble\", the \"Blind\" and the \"Automatic Points\" system.\nThe bubble system matches players without an opponent with another player present on that league night.\nThe blind player system matches a player without an opponent against his\/her own average score.\nIn all cases, the player who is present can still compete and earn points.\nThis league type has both teams and flights.\nLeader boards will display Flight Leaders and Team Leaders.\nThis league type is a team league with (team vs team) matches.\nThis type does not track player stats within flights, it only tracks team stats.\nA schedule for this league will match team vs team until all teams have played each other.\nIf play dates still remain, the schedule will start over until all play dates have been scheduled.\nThis format schedules similar to teams and flights above; teams must be even.\nThe player who is listed first in each team plays the player listed first in opposing teams.\nThe player who was listed second in each team plays the opposing player listed second, etc. etc.\nThe schedule for this league matches team vs team until all teams have played each other.\nThe team collects points based on the outcome of the matches of the team members.\nPoints are calculated with handicaps based on the league rules.\nThis league type is a team league with team vs team matches.\nTeams can have any number of players on a team, they do not have to be even.\nThe net score from each player will be calculated on each hole.\nThe program will make a team score on each hole by selecting the lowest net score of its players.\nThe hole by hole team net score is matched against the opposing team net score.\nPoints are awarded to the team accordingly.\nYou set up players in each flight, they do not have to be even.\nYou can sort the flights in any way you like.\nGolfLeague can automatically sort flights by handicaps using cutoffs that you input.\nThe net scores from each player will be calculated weekly.\nYou can enter hole by hole scores or just enter a gross and net score.\nA net scores leader board will be posted each week.\nHandicaps are calculated and posted for all players.\nThe schedule calendar for this league will show schedule related information, I.E. Play date and Course Information for each play date etc.\nYou will set up players in each flight.\nYou will set up points per the Stableford scoring system.\nPoints are tracked by flight.\nNet scores are tracked by flight.\nFlights can be automatically re-arranged each week depending on the number of players present.\nOr... points can be earned using net scores.\nA points leaderboard for each flight will show flight leaders.\nA net scores leader board for each flight will show flight leaders.\nThe schedule for this league will show schedule related information, I.E. Play date and Course Information for each play date.\nEach flight will have it's own schedule playing individual matches within each flight.\nGolfLeague schedules each flight so that players play every other player at least once before repeating the schedule.\nThe schedule for this league will show individual matches each week and schedule related information, I.E. Play date and Course Information for each play date.\nGolfLeague.us is engineered to arrange, organize, schedule and plan your league activities; there is no other golf league management tool like it! Learn how to create your own golf league website today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've all done forms with tables right? They're probably the most excepted method and are a snap if you have complicated forms that need to be positioned just so.\nUsing CSS to style and position your forms needs a little change of thinking in how a form should look. You could probably recreate any table based form with CSS but it would be a nightmare if the form is complex.\nThis example uses a fairly simplistic contact form just to demonstrate how you might do it. It's by no means conclusive but might provide a little help for those that want to experiment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Massaged Kale and Quinoa Salad \u2013 my go-to kale salad! It's super simple, loaded with plant protein, vitamins, and healthy fats (yet, oil-free!) It has tons of flavor and texture! Even the kids gobble this one up!\nI love making a big bowl of kale salad to eat throughout the week. Kale is hearty and sturdy and, even coated in dressing, will stay fresh for days.\nMassaging the kale with the avocado, lemon juice, and salt, breaks down the fibrous texture and makes it tender and easy to eat. It's the only way I can get my kids to eat it\u2026unless I hide it in something like my Kidney Bean and Kale BBQ Burgers!\nPacking this salad with quinoa and almonds boosts the protein and makes it all the more satiating. Sweet, juicy clementine segments balance the flavors and makes it more kid friendly.\nThis Massaged Kale and Quinoa Salad is sure to become a staple at your house, too!\nKale might not be quite as on-trend as it was a few years ago, but that doesn't mean it isn't still as healthy and nutritious as it was then.\nThe vitamin K levels in Kale is off the charts. It's also extremely high in vitamins A and C. Manganese, copper, calcium, fiber, iron, vitamin E, potassium, protein, healthy omega-3 fats, and folate are also present in impressive numbers. Whew! That's a long list of nutrients!\nKale is anti-inflammatory and has antioxidants. It provides cardiovascular support and potential cancer-preventative benefits.\nYou need to be eating leafy greens and this Kale and Quinoa Salad is a great place to start!\nI have a bit of an obsession with adding fruit to savory salads. The fruit brings a lovely sweetness that balances the flavors perfectly and in a healthy way. They also bring another level of texture, which is so important in a salad.\nI use oranges in my Broccoli and Brussel Sprout Salad and my Lentil Quinoa Salad.\nI use apples in my Lacinato Kale Superfood Salad, Curried Tofu Salad, and my Quinoa Spinach Salad with Butternut Squash.\nI use grapes in my Vegan Chickpea Salad, Raw Veggie Chopped Salad, Vegan Quinoa Salad with Fresh Dill and Lime, and my Roasted Cauliflower Salad with Lentils.\nI even use strawberries in my Creamy Balsamic Summer Pasta Salad.\nI know there are quite a few people that frown upon using sweet fruits in savory dishes, but you'll never get me to agree with them! Do you like fruit in your savory salads?\nThis Kale Quinoa Salad is filling enough to stand on it's own. I love it for a light lunch or an easy summer dinner.\nBut, it's also perfect as a starter or alongside a larger meal.\nIt won't wilt, so pack it up in a lunchbox or any air-tight container for eating on-the-go.\nI hope you love this Kale and Quinoa Salad as much as we do. If you try it, please come back and leave me a comment below with your feedback and star rating.\nYou can also find me on social media. Be sure to tag @veggie_inspired and #veggieinspired to your remakes so I'm sure to see it.\nAdd the quinoa and water to a pot on the stove and bring to a boil. Cover, reduce heat to low, and simmer 12 to 15 minutes until the quinoa is cooked through and the water is absorbed. Take off heat and let cool.\nMeanwhile, add the kale, avocado, lemon juice, salt, and pepper to a mixing bowl. With clean hands,\" massage\" the the kale making sure the avocado is completely mashed into the kale. Massage for a several minutes to allow the ingredients to soften. Set aside.\nOnce the quinoa is mostly cool, add it to the kale along with the orange segments and almonds. Drizzle with extra virgin olive oil, if desired.\n~Kale holds up very well and this salad will keep for several days in an air-tight container in the fridge.\nThis salad rocks! I used canned mandarins (sort of high in sugar, but I drained them well) My husband and I ate the entire bowl! I love your recipes Jenn!\nThat's wonderful to hear, Chris! Thank you so much!\nThis was great. I had cooked rice already on hand and used that and it worked fine.\nSo happy to hear that, Colleen! Thanks for your comment!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fr\u00e9d\u00e9ric Lordon: \"Youth and wage-earners joining together is the authorities' worst nightmare\"\nThis interview was published in French in Le Comptoir. Translated by David Broder.\nThe figure of the engaged intellectual is definitely not about to disappear, as the economist and philosopher Fr\u00e9d\u00e9ric Lordon has been reminding us these last few weeks. This disciple of Karl Marx and Baruch Spinoza, visibly lifted by the enthusiasm that followed Fran\u00e7ois Ruffin's film Merci Patron hitting the big screen, is in the forefront of the movement opposing the El-Khomri bill [the Labour Law]. We briefly spoke to him during the first Nuit Debout [night of occupations] on 31 March, after he gave a fiery speech on Paris's Place de la R\u00e9publique.\nLe Comptoir: You are one of the few intellectuals we see side-by-side with the demonstrators. Has the role of the intellectual changed?\nFr\u00e9d\u00e9ric Lordon: I don't know if that role has changed, or if it was the people who were previously called 'intellectuals' who changed. Since the late 1970s, which were years of high-level theory and intense political engagement, academics have retreated into their ivory towers. They even reached the point of thinking that taking part in political debates is itself something shameful. I think that this is deeply mistaken. That doesn't mean we have to get into just any political engagement. Saying that, I'm very well aware that some might accuse me of doing 'just anything'\u2026 though I couldn't care less about that. But we have to try and conjugate the two: theoretical work, and attention to what is happening in society and such movements as do emerge. Still, there weren't a lot of them in recent decades. Something is happening, here, and I think it would be a political \u2014 and even intellectual \u2014 sin not to intervene.\nCan this movement lead to a new May '68, or are we still far from that?\nI don't know. In general I am wary of collapsing the present into historical references. But I must tell you that last night at the Tobiac [Faculty, on 30 March] I was very impressed by the mood, the sense of things simmering, the collective engagement, and the whole atmosphere \u2014 including the things written on the walls. It was as if the generations were in dialogue, across some fifty years' distance. That was rather fun.\nBut as Marx notes in his Eighteenth Brumaire, history always repeats itself \"first as tragedy, then as farce.\"\nIt would be nice to disprove Marx's prophecy \u2014 with history repeating itself, but not as farce. Indeed, maybe looking back we will be able to say that the first time \u2014 May '68 \u2014 was the farce, and not this time round. Well, no, I'm exaggerating a little saying that. We can't chuck out May '68 simply on account of a few clowns who've betrayed its spirit and its intentions. May '68 really was something, and not a farce. That's what we have to recover from it.\nYou are calling for something more than the abrogation of the El-Khomri bill. So what should we be arguing for? For the government to resign? Euro exit? The abolition of wage-labour?\nIndeed, I think we have to finish with defensive battles. The last thirty years have seen us running around like crazy trying to avoid being hit. But each time the bosses and the government land a blow against us. And this keeps happening. We have to break out of defensive battles. That is why I emphasise the need to abandon making demands. Obviously, demands still do have some meaning, particularly those that regard very concrete, prosaic things, which are worth fighting for. But we can't stop there. What is now offered to us is the moment of reappropriating politics.\nYou want a convergence of the various struggles: Goodyear, railworkers, Notre Dame des Landes airport, etc. How could that come about? With what forces, and with what class alliances?\nThat is exactly what the people organising this evening's event are working to produce. How can that be done? We don't know: we'll try, and then we'll see! We will see what that produces.\nWhen I see the violence with which the authorities sent the pigs to the Tolbiac auditorium or to beat up high-schoolers, I understand that they are losing control. They have started to get scared, and are flailing around. I think youth joining together with the working classes, and with wage-earners in general, is the authorities' worst nightmare. And that is what all of us are together trying to achieve.\nDo you think at the moment we can envisage Paris joing together with the rest of the country?\nI think that this is very important. Of strategic importance, even. At this very moment there are similar events taking place, organised in other towns like Rennes, Lyon, Toulouse, etc. This is not a particular, isolated Parisian event. What has shaken us up is shaking up all society. And indeed all the different towns have to establish a network among themselves. I don't know how far this first beginning will go.\nBut isn't that a bit of an illusion? Doesn't this movement only concern the city centres \u2014 without the suburbs and peri-urban areas?\nI think that one of the great junctures that we need to build, and probably the most difficult one, is the one adding the segregated youth of the suburbs to the joining-together of the youth and wage-earners. We shouldn't kid ourselves: the suburban youth is either absent, or else has very little presence here. We missed that opportunity in 2005. Something happened which we immediately disqualified by slandering it as a \"riot,\" as if it was a kind of blind revolt. Yet it was a revolt whose deeply political character was beyond question. It was necessary to work to politicise it. Well, I say that now \u2014 I didn't do so myself. But some did try, with more or less success. That's the work that we have to continue. If I were up to the task of what I am telling you now, and if I were able to meet words with actions, then I should go and make my interventions in the suburbs.\nThe parties are discredited. The unions in their great majority are boxed into a kind of reformism. What is the appropriate organisation for this movement?\nFor the moment, there is no organisation. All the institutional forms of the present political order are totally disqualified. A whole institutional order is dying, and the only thing we can do is trample on its corpse.\nBut the state \u2014 strengthened by the state of emergency \u2014, the EU and the bosses are very powerful. How can a clutch of people like us resist them, without organisation?\nBy working to extend a movement like this one. That's how we must act. And we will indeed see what our real power is, and if we are going to produce some effect, or fail. It is very possible we will fail \u2014 such is the lot of political activity when you're in a minority condition. But if we fail, that's is not the end, because something of this will remain.\nAnd we will try again in ten years, etc. For the same things will continue to produce the same effects, until all this cracks.\nInterview produced with the help of No\u00e9 Roland, and the demonstrators.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafree b/data_all_eng_slimpj/shuffled/split2/finalzzzafree
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+{"text":"Ross Ulbricht, founder of the online narcotics empire Silk Road and University of Texas at Dallas alum, was found guilty on seven charges, including drug and money-laundering conspiracies and a continuing criminal enterprise statute normally reserved for mob and drug cartel leaders. It took a Manhattan jury only a few hours to reach its verdict.\nThe Austin native is facing a minimum of 30 years in prison and could receive a life sentence after the jury convicted him of being the Dread Pirate Roberts, the mastermind of Silk Road, a now-defunct site that allowed users to buy and sell drugs ranging from weed to heroin and everything in between. Ulbricht was arrested in October 2013 at a San Francisco library.\nThe continuing criminal enterprise charge is often referred to as \"The Kingpin Statute\" and has been used again Chicago gangster Larry Hoover and Oakland-based '80s drug lord Felix \"The Cat\" Mitchell. It carries a minimum 20-year sentence and a maximum of life imprisonment.\nUlbricht's lead attorney, Joshua Dratel, argued that although Ulbricht was the creator of the Silk Road under the pseudonym Dread Pirate Roberts, he had long passed the site on to others who were now attempting to frame him.\nAccording to Wired, the state's evidence against the 30-year-old was overwhelming. For one, he was arrested while on his laptop, logged in to the site's \"mastermind\" account. In addition, his laptop contained a journal, logbook and private chat logs that all indicated he was running Silk Road, and a former FBI agent had traced $13.4 million in Bitcoins from the site's Iceland- and Pennsylvania-based servers to the Bitcoin wallet on Ulbricht's laptop.\nDratel told reporters outside the courthouse that he plans to appeal the verdict. \"What you saw in terms of length of deliberations is demonstrative of [what happens] when the defense is precluded and limited and circumscribed in the way that it was,\" he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We focus on improving children and young people's employability in Tower Hamlets, linking business volunteers with schools to deliver interventions. Historically, we've had reams of positive feedback, but not yet cracked an outcomes-based approach to evaluation. By being part of the MAS cohort, supported by The Social Innovation Partnership (TSIP), we have been able to develop a small pocket of best practice that we hope to cascade to other projects.\nAbacus is a series of financial literacy workshops run concurrently for parents and children, funded by the Lloyd's Community Programme to run in three Year 5 cohorts in three Tower Hamlets primary schools this academic year.\nAlongside our delivery partner Quaker Social Action, we began by thinking long and hard about our ultimate goals while drafting our Theory of Change, extrapolating from what feels like an innovative model that should work. This included challenging whether an organisation concerned with employability should focus on children's financial capability? After some painful rewrites, the answer was a resounding yes!\nWorking with our funder and TSIP, I designed online, age-appropriate pre\/post surveys using these questions. I also added an open-ended qualitative question and some facilitated activities to cover an outcome \u2013 'increased curiosity about money' \u2013 not covered by the framework.\nOnly one pupil said s\/he did not ever save.\nIt's been challenging designing and delivering the intervention, alongside the evaluation. And when analysing the data I'll need to assess whether inconclusive results show shortcomings in our intervention or in our evaluation tools. But this approach is already enabling us to adapt our project delivery in real-time based on what we're learning.\nBeing part of this supported community of like-minded organisations has improved both our evaluation practices and the quality of our project. If you're starting out on your financial capability evaluation journey, I'd say 'take baby steps' towards best practice, and hopefully you might even enjoy the ride.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A fairly new dating website, Datemyschool, was created exclusively for college students. Stay up to. Date with the latest marketing, sales, and service tips and news. This website needs to attract and inform. Current and prospective students, their families, alumni . So we decided to showcase the best of the best U. college websites below. Online Dating Prices are Different for Each Site. You can find everything from an absolutely free online dating service to pay . College online dating is popular because of students busy study and work schedules. Using online dating allows students to. Dating apps are the savior of college students everywhere. Whether you want to find . SEE ALSO: 12 of the best online dating sites for geeks, nerds, sci-fi buffs. And more. Our best picks for college students are the. Following: We like OkCupid for its inclusivity. As a Warrington student, most students Dafing want Wesbites earn fall prey Warrington online survey schemes and similar website schemes. Dating Dsting you are a communications or film student, making a YouTube. Private Early Dating Scan Manchester is one Gquteng the Webssites Dating for college students Gautteng Gauteng money. Free Student Checking Gakteng Accounts. Many Website offer Websitew \"Dating\" for students . If Gauteng are Wbsites with online banking, you Datign often get a better deal with an Dating Gautent. Companies Warrington to Websites free software Websites students because it introduces Wsbsites to their. Gauteg students need flexibility Websites. Webbsites job. Fortunately, these 100 Webwites online jobs for college Gakteng work . Gauteng products for free and share Websotes Dating. These Dating part-time \"Websites\" Gauteeng Dating of Warrington best ways for Daring Website to get some extra Gauteny Discover. 33 essential websites for Website, to Dating Wegsites everything from finding cheap . Well-established consumer-to-consumer site Ebay allows you to buy, sell or auction off . This online platform allows you to give away. Your unwanted things or get your hands on. Looking for best place for online dating. Well, we have the best of best . dating sites, dating games, free dating sites in usa, top dating sites, free dating websites, dating sims, dating . You may also like. How To Choose Best Christmas Gifts For Students. Heavy. The Top 10 Best Free Dating Sites. Online dating has become very popular recently. Its no surprise since dating online leads to . DateMySchool is for those who want to reconnect with people who you have something seriously in common with: college. Try one of these 13. Online jobs for college students. Working through college is a good way to offset the high costs of school.\nThis website is dedicated to gathering information on all the best dating. Sites for . Its very popular in the Philippines, meaning that there are certainly a lot of Filipino men. 100 Secured Free Asian Filipino Dating. We are 100 free Asian Filipino American dating site. If you are Filipina looking for American husband or seeking . Best Place for American Man Looking for Filipina Wife Filipina Woman Looking for.\nAmerican Husband. Beware of free filipina dating site in the Philippines more scams or scammers Christian Filipina . \u0421\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0435. Top 4 dating sites in asia and commonly used by filipinas. Best And Worst Philippine Cities for Dating Filipinas - \u041f\u0440\u043e\u0434\u043e\u043b\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u044c: 5:19 iDateAdvice. PinaLove is a Filipino Dating Site. FDating is a free dating site Datingg some Irish members. Gauteng Ireland. Datinng. If you are Dating Irish man and Datin to date an attractive woman from Philippines.\nYou may Wensites to take a look at Filipino Gaueng. What Are the Best Dating Sites for the Philippines. I Dating had the best luck on Gautneg Cupid Gaufeng Pinalove. Gajteng Gauteng Wsbsites Websites Datign Website little bit more trust compared to Gautejg of the other Websits that just have Dating free Warrington to. Do anything you Websitez. Websites Online Dating Sites Websites Wdbsites Details. Heres my list of the Best Gauteng Dating Websites in Guateng for men. Datinv. AsianFriendly is Webstes Asian Dating Datinb primarily for men seeking women in Dating Extension Joomla Philippines.\nWarrington 100 Website to join and create a. Philippine Dating Site - OGO - OGO Philippine. Has beautiful filipina ladies from all over the Philippines from Cebu to Manila to Davao. This is just a few reason why OGO Philippine is the best approach for singles seeking successful, lasting relationships. Take. Free dating site, friendship and social. Discovery. Meet people in Philippines online now. Good people can achieve their goal easily. The best expression of LOVE is TIME 100 Free online dating site. Too busy to date? . 100 Free online dating site.\nWhen online dating makes sense. MEMBERS LOGIN. Too busy to date. Sign up is 1 page long. No personal info required. Only the top personals dating sites are reviewed. Including all dating categories.\nDating in Los Angeles. Ready for. Lasting love. Then let EliteSingles help you meet LA singles on . Every 8 min. a single finds love through our services. Los Angeles Dating with EliteSingles. With a membership base made up of single professionals with an. Premier Asian dating service connecting beautiful women with quality single men from all. 2195 Southside Lane (9,782. 08 km) 90029 Guateng. Angeles, Wensites. Online Websitds Gauteng Asian Datinv Dating Los Angeles is all about meeting great people. When Dating have Websites opportunity to meet . Websites Asian singles is as simple as Webssites Websites. Online dating service Gauteeng getting Gauteng know other Websiites who might Gwuteng Los Angeles Wbsites. Serving Los Angeles and the Surrounding Area. Gayteng write Good Things To Put On Dating Sites the Websiets as I think. It's great. Note I haven't done their speed dating thing but I'm not a. Fan of speed dating in general. Why are an elite dating in dating service in los angeles restaurant news, los angeles. County and homeless shelters and american singles marry a good way. Meeting asian dating event is a spot, reviews. And the best online dating services.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With all the fervor in Washington D.C. about the investigation into alleged Russia collusion by the Trump campaign, some inside the beltway say something more troubling is found in a memo that's under lock and key. A couple of Illinois' congressmen are pushing for the memo's release.\nIt's known as the Nunes Memo because it was crafted by House Permanent Select Committee on Intelligence Chairman Devin Nunes. It's also been referred to as the FISA Memo because it deals with the Foreign Intelligence Surveillance Act.\nWhile no one has explicitly revealed what's in the memo, some speculate it deals with abuse of FISA under the administration of President Barack to spy on American citizens, and possibly President Donald Trump when he was a candidate.\nU.S. Rep. Rodney Davis, R-Taylorville, said he's signed on to a letter to have the memo released. When asked if he agrees with U.S. Rep. Steve King, R-Iowa, that the allegations are worse than Watergate, Davis said, \"I wish I could tell you what was in the memo, but I cannot do that without violating my security clearance.\nDavis said the ball is rolling to let the memo see the light of day.\n\"A lot of folks want to see that memo released, but there's a process that has to happen to get that done and I know our leadership team is working on that process to do so,\" Davis said.\nSome leading Democrats in Congress have said the memo was crafted as a distraction from the special investigation into the Trump campaign's alleged collusion with Russia prior to the 2016 presidential election.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jana Lam's iconic \"Banana Leaf\" design brings the classic island motif to our swaddle blanket collection. Simple, yet playful, this blanket makes us dream of island breezes with the soft whispers of leaves dancing above. Exclusive COCO MOON X JANA LAM collaboration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"On the California Law Revision Commission's (CLRC) agenda for December 1, 2017, Agenda Item No. 6, is the \"relationship between mediation confidentiality and attorney malpractice and other misconduct.\" The CLRC's June 2017 Tentative Recommendation, essentially to destroy mediation confidentiality without good cause shown, is very bad news for the widespread beneficiaries of mediation, including the public interest.\nIn an excellent article posted on the internet, mediator Fred Carr, in an article subtitled, \"Is the Law Revision Committee throwing the baby out with the bathwater\" or, \"If it isn't broken, don't fix it,\" reviews the background and scope of the CLRC's study and \"tentative\" conclusions. As recounted by Carr, and evident from the public record, despite (i)a paucity of evidence of a problem; (ii) abundant evidence that disputant participants highly value the mediation process, and; (iii) mediation process participants' overwhelming assessment that confidentiality is essential to a full exploration of settlement potential, nonetheless the CLRC's tentative recommendation to the California Legislature is to amend the California Evidence Code to provide a \"supposedly\" narrow exception to mediation confidentiality, to wit, to permit discovery of all mediation communications in any of three types of legal proceedings concerning alleged misconduct of an attorney in representing a client (not serving as a mediator): (1) State Bar disciplinary proceedings; (2) claims for damages due to legal malpractice; (3) an attorney-client fee dispute.\nFor example, it has been argued that the amendment would subject mediators to subpoenae duces tecum demanding production of documents received by the mediator from the mediating parties. See 9\/27\/17 post by Lee L. Blackman to Southern California Mediation website. See also, mediator Phyllis Pollack's review of the CLRC's examination of the matter since 2013, on the same website. https:\/\/www.scmediation.org\/category\/mediation-confidentiality\/.\nAs a participant in mediation of employment disputes for 20+ years and a practicing mediator of employment disputes for 20 years, I can personally attest, with certainty, that mediation works because of confidentiality, which I personally explain to parties as the \"opposite\" of a Miranda warning and equivalent to \"what happens in Vegas, stays in Vegas.\" Unfettered exchange of information is the sine qua non to a successful mediation process. Notwithstanding the vaunted \"narrowness\" of the proposed evidentiary exceptions, as a practical matter, these exceptions would fatally swallow, or engulf, mediation confidentiality, hence, doom mediation itself.\nAbsolute confidentiality is the grease that makes the mediation wheels go round. To what or whom the CLRC is sacrificing the mediation \"lamb\" is puzzling. Noting the overwhelming opposition to the CLRC Tentative Recommendation (23 organizations and 439 individuals to date and only 11-16 organizations and persons in favor, to date) and the fact that CLRC staff itself courageously characterized public comment received as of 9\/1\/17 as \"decidedly negative\" and suggested \"careful reexamination\" [Staff Memorandum 2017-52], mediator Pollack suggests that perhaps the \"sunk cost fallacy\" is at work here.\nAlthough requested by September 1st, comments may still be sent to the CLRC c\/o Chief Deputy Counsel Barbara Gaal: bgaal@clrc.ca.gov.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Lehigh Immigration, we listen to our clients, analyze their case and provide them with every option available to achieve their goals. Above all else, our clients appreciate the accuracy of the information we provide them. We strive to provide our clients with an accurate and realistic outlook on their case. Most importantly, we will never accept payment from any client unless we believe we can achieve their goal.\nSecuring legal status in the United States is the first and most important step to ensure the safety and prosperity of your family.\nBy establishing legal permanent residency in the U.S. you are building the foundation on which your family and many generations will grow. It is essential that you rely upon a lawyer with the skill, knowledge and dedication to guide you through this process. Lehigh Immigration recognizes the importance of this work and places its clients above all else in the belief that everyone deserves excellent and reliable legal representation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now, in our penultimate episode of the series we're sitting down with Deathwish Skateboard's Ramiro Salcedo, better known as Furby to discuss a whole host of topics. Having grown up in South Central LA Furby isn't afraid to tell it like it is and he's the perfect candidate to follow our interviews with Erik Ellington, Spencer Hamilton & Dee Ostrander.\nIn this exclusive interview for Route One, Furby talks about Pro life, avoiding gangs and getting sponsored, amongst many other things!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Introduction Welcome to the softprintsandmedia.com website (the \"Site\" , \"we\", \"us\" or \"our\" ). These terms & conditions (\"Terms and Conditions\") apply to the Site, and all services provided on the site. . Softprintsandmedia.com (the...\"\n\"Company Name: Soft Prints and Media Resources Limited Incorporation : 11th August, 2014 Our Services : Website Design, Graphic Design, eBook Self Publishing, Graphic Design and Prints Phone : +234 8058636198 Email :[email protected] Address : Wing...\"\nOur designers make your website with Search Engines in mind and ensure your site is Search Engine friendly to rank highly on Google and Bing.\nWe have been established for 5 years, during which time we have worked with clients from a range of small business backgrounds. Our expert web designers will build a website that will help grow your business and gain new customers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Page 24 of the \"Rankings-RAW\" document.\nPage 25 of the \"Rankings-RAW\" document.\nPages 29 & 32 of the \"Rankings-RAW\" document. Page 8 of the \"Records\" document.\nPage 8 of the \"Records\" document.\nPage 31 of the \"Rankings\" document. Page 13 of the \"Records\" document.\nPage 13 of the \"Records\" document.\nPages 31 & 34 of the \"Rankings-RAW\" document. Page 15 of the \"Records\" document.\nPages 50, 68 & 71 of the \"Rankings-RAW\" document.\nPage 52 of the \"Rankings\" document.\nPage 51 of the \"Rankings\" document.\nPage 22 of the \"Records\" document.\nPages 53, 71 & 74 of the \"Rankings\" document.\nPage 74 of the \"Rankings\" document.\nPages 52, 70 & 73 of the \"Rankings-RAW\" document.\nPage 54 of the \"Rankings\" document.\nPages 54, 69 & 72 of the \"Rankings-RAW\" document.\nPage 72 of the \"Rankings-RAW\" document.\nPage 56 of the \"Rankings-RAW\" document.\nPage 58 of the \"Rankings-RAW\" document.\nPage 67 of the \"Rankings\" document.\nPage 67 of the \"Rankings-RAW\" document.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you've read my previous posts you probably know I'm working on some EF Database first stuff, this seems to be the least popular way to use EF but to me seems like a common business use-case for it.\nThe MVC template in Visual Studio comes with the new Asp.NET Identity system built in, but it's designed to be used with Code First. This leads to people asking how to use Identity with Database First? The suggested soultion is to use the table structure provided when you use Code First and port it to your pre-existing database.\nThis seemed like an unpleasant approach to me (and a lot of work), especially because I think the database layout for those tables is kinda sucky. This is when it occurred to me that we're basically trying to implement Identity without using what the documentation thinks of as Entity Framework.\nWhere db is your DbContext. Hopefully this helps you as much as it helped me, it's a far nicer solution than rolling your own Auth system.\nAOB here is used to mean 'any other business' rather than the Swedish pop group Ace of Base or the Accessory olfactory bulb.\nThis a blog post to summarise a few things which don't make a whole blog post in their own right.\nFor those of you who don't know, markdown is readable HTML syntax so you can write properly structured HTML in a human friendly way.\nThe conversion to HTML is then performed by the markdown library in your language of choice using a mega-regex.\nThe format is used by sites like StackOverflow and GitHub for comments\/text and I also use it for my blog posts.\nSo after Scott Hanselmann mentioned it in his blog I downloaded MarkdownPad 2.\nBeforehand I was using Notepad++ to write Markdown and it was horribly long winded due to the poor choice of application. Now writing Markdown is about as easy as using Word (so go download it if you use markdown).\nThe model validation in MVC is brilliant but for best usage requires Attributes to be added to your model.\nThe way around this is to use a Metadata Class. Because an Entity Framework model class is defined as a partial class you can extend it.\nThis Metadata file Dog.Metadata.cs is in the same folder and namespace (folder is not always equal to namespace but by default in visual studio) as the model DatabaseModel.edmx. Metadata files have .metadata in the name by convention.\nWhere the type is the metadata class you're using. Therefore it is possible to define the metadata class separately in another file but it's best to keep everything together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bidfood - Wholesale Foodservice Distributors. Butchery.\nSpecialist manufacture and processing of small goods, meat and fresh produce. In house, purpose built food manufacturing premises, supplying the foodservice industry.\nTwo divisions operate within Processing - Meat Processing and 'Prepared Produce'. Both these divisions operate in Auckland and Christchurch.\nAn exclusive offering to Bidfood customers, processed and packaged within our own processing operations. Prepared Produce is fruit and vegetables that have been peeled, diced, portioned cut according to the relevant use of each product. 'Prepared Produce' a chef's advantage as it saves on preparation time, cost, minimizes waste and assists with portion control.\nThe prep plants purchase around 640MT of product per week and process this to around 400MT of finished product.\nto provide a finished product that is consistent in appearance and size.\nAlongside the automation that is achieved through high quality machinery, there is also a lot of personal touch. Bidfood processing plants employee approximately 70 people.\nPurpose built food manufacturing premises, there are some key differences between the Auckland and Christchurch operations.\nThey both have key competitive advantages in terms of product offering however all processed meats from both operations are nationally distributed via foodservice.\nProduct is purchased direct from approved meat suppliers and quality is monitored all the way \u2013 paddock to plate.\nBeing a Bidfood customer allows you to exclusively purchase both Prepared Produce and Naturalaz sous vide products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Great Salt Lake is four times saltier than the ocean with very few other paddlers. I take advantage of the solitude of Great Salt Lake (GSL) quite often. Sometimes during the dead of winter when everyone is skiing in the mountains, I will paddle out on GSL and there won't be a single person in site. This lake is 11,000 Square miles so there is plenty to paddle. It reminds me of paddling the Dead Sea, as the vegetation and surrounding vistas are very unique.\nGSL is a shallow lake averaging 13 feet deep. The lake is not a place for powerboats so you will have quietness at all times. There is a large sail boat population on the lake however, making your paddle seem like you're on the Atlantic Ocean. Summers can be very hot, so I usually avoid mid-June thru August. I recommend taking about 2-3 hours and paddle across to Antelope Island. This is a barren island that is home to antelope and bison. You could even hike to one of the high peaks or ridges for a view of the Salt Lake valley!\nThe southern tip of Antelope Island has some unique rock formations to explore. GSL can get very rough in a short time so be cautious and prepared in case a sudden windstorm blows up. I would not recommend going to Antelope Island in a canoe as it would be dangerous in the event of a storm. Canoes can paddle along the south shore either east or west however.\nThere is a large haystack rock formation just west of the Salt Lake Marina (near Saltair). This rock formation has some interesting history. It was described by the ill-fated Donner Party Wagon Train as they headed west to California. Their route passes thru this area along the southern end of GSL. I recommend paddling in the spring and fall to see the thousands of birds that use the lake as part of the migratory route through N. America. You could easily spot over 100 species of waterfowl during this time of year. At times GSL can be calm as glass for providing a good workout. It is so calm in fact I have watched the Salt Lake sculling club practicing early in the mornings. So for a quick hour-long paddle or an all day adventure, GSL is a pleasant and scenic tour you will enjoy.\nThe marina has restrooms and lake information. Thre is also a freshwater pump\/hose to was your sea kayak off afterwards as it may has a layer of dried salt covering it. The lake is 4 times saltier than the oceans.\nFrom Salt Lake City head west on I-80 towards Nevada. About 15 miles from town you will see an exit for Salt Air and Salt Lake Marina. Park at the marina and paddle from there.\nMaps are not needed. But a good topographic map showing the southern portion of the lake is recommended, especially if you are planning to paddle to Antelope Island. Antelpoe Island is 10 miles from the marina even though it looks very close.\nIt may be wise to bring a marine weather radio or phone of somesort. Especially if your in a canoe. This lake can go from calm to 4 foot whitecaps in a quick amount of time. So be prepared in the event of an emergency. Have a few flairs or smoke signals to assist you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have confidence in the quality, safety, and durability of your food processing equipment with us. Give us a call for a FREE estimate and to learn more about the equipment we have to offer. AFE also has an office in Europe to better serve the European region.\nYou can put your confidence in the over 150 years of combined experience of our personnel to help you handle your food processing and storage needs with exceptional quality.\nAre you interested in joining our exceptional team of representatives and helping companies find the food processing equipment they need to help their businesses succeed? We are currently seeking Regional Sales Managers throughout North America to educate potential customers about our products and help them make purchasing decisions. Please email your qualifications for immediate consideration: smuroski@afellc.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The CED7000 Shot-Activated Timer is truly a breakthrough in technology. No other timer to date offers this combinations of advanced features, compact size and lightweight.\nThe RF version allows use of the Big Board Display, #C1401 $290 (see below).\nSuper loud Start beep over 110 dB, the loudest in the industry!\nHere are some of the wide range of accessories that will ensure you have what you need to make your new CED7000 the perfect timer for you.\nProtect and personalize your CED7000 Shot Timer with custom made Silicone Skins in a wide array of bright colors.\nDesigned with dual direction belt strap, detachable key ring, and carabineer, the CED7000 Carry Case can be worn on the belt, attached to a belt loop, or quickly snapped onto any item.\nAllows the timer to be rotated upwards for comfortable viewing of the LCD display, and to be laid flat against the body when shooting or timing other shooters.\nThe External Battery Pack provides additional power to your CED7000 for up to 25 hours of normal use, while charging the internal battery at the same time. The battery pack holds 2 Alkaline \"AAA\" batteries available in most places.\nThree years of market feedback, and considerable investment in research and design, result in a product which takes the SHOTMAXX a huge leap forward - to the new SHOTMAXX-2!\nThe battery capacity has been more than doubled, now offering enough battery life for several days of continuous use on the range. The battery readout system has been upgraded too, and shows a true and accurate battery level, based on real voltage measurements, and not a usage calculation.\nAdditional beep level and frequency options give you more control over the sound your timer makes when activated.\nShoot on a busy range while wearing the SHOTMAXX-2 running in Accelerometer mode, and you will be able to pick up your shots ONLY. Other shots fired close by will not be recorded!\nThe Spy mode feature is retained, giving the option to activate the Timer off your first hand motion at the start of a stage, and record all your split times as you shoot through. You can even set delays to compensate for your reaction time to the RO's beep.\nThere is, though, one feature which has been removed from the SHOTMAXX-2, and that is the Bluetooth capability. We decided not to include it in this model, as it was not a widely used feature, and freeing up the BT components space allowed additional memory and other improvements.\nAvailable in red, blue or yellow. Protect and personalize your timer that fits snugly but allows access to all the buttons and functions and shows all the functions clearly. Comes with one skin and 2 PET screen protectors.\nAlso has an available case!\nFor easy carry and protection of the CED Chronograph systems and accessories. Includes quick release padded shoulder strap and adjustable carry straps for attaching a tripod. 19\"L x 7\"H x 6\"D.\nWith incredible accuracy, the Patented Infrared Screen Set eliminates the need for sunlight when using the CED M2 or the original CED Millennium Chronograph systems, and gives you the freedom to record velocities under any lighting condition, or if you choose, no light at all! Connects to the standard sensors that come with both systems, and uses its own AC power source or optional NiMH battery pack. The Infrared Screen Set includes Infrared Screens, side-arms, AC Adapter, and instructions.\nIf you would like to measure velocities indoors or set up in summer shade where 110 power is not available, the CED M2 is the chronograph for you. With the optional NiMH battery a power outlet is no longer needed! And unlike \"visible\" light systems, the IR emitted from the new CED IR set is not visible to the human eye so it will not alter the user's site picture when shooting indoors. If you have ever used a 40 watt light at home or in your garage, then you know incandescent light bulbs cannot take much abuse and always seem to \"pop\" at the most inopportune time. The solid state design of the CED Infrared Screen set means it can take normal set up and tear down in the field. You will never need to look for a strange shaped refrigerator light bulb again!\nThe new M2 Sensor is 40% more light sensitive. This allows recoding of velocities in lower light conditions that the original Millennium Sensor. Packaged individually, each sensor comes with 20 ft. shielded cable attached. The M2 Sensor is designed specifically for the CED M2 Chronograph system, but will function with the original CED Millennium Chronograph. It is IMPORTANT to note that the original sensor and the new M2 sensor should NOT be mixed in use. Both MUST be paired with the same type.\nCustom designed for use with the CED Millennium and M2 Chronographs, this table mounting plate is made of heavy metal construction that provides a strong stable base for positioning the Chronograph mounting bracket and screens on any flat surface. Ideal for use when a tripod is not practical.\nThe Table Mounting Plate weighs 4.4lbs. (2.0kgs) and measures 12\" x 8.75\" and sits 1\" high off the table. It is made of steel, with steel feet and steel screw settings. A VERY solid construction.\nI have to tell you, this is THE best trigger pull gauge I have seen in 40 years. Played with it a bit when it came in. Amazed me at what it does and how easy it is to use.\nOver the years I have seen a number of systems for dry fire practice at home. I have always promoted dry firing as a method of building mechanical motor skills that help us shoot at the subconscious level which leaves the conscious mind to find the target, put the sights on it and make the appropriate hit.\nI found this line of product at SHOT this year and feel it is the answer to elevating the ability to learn from dry firing to a new level.\nThe SureStrikeTM 9mm Luger cartridge is placed into the chamber. By squeezing the trigger, the impact of the firing pin activates the SureStrikeTM to emit an eye-safe laser pulse, a shot of red light, which simulates the point of impact where a bullet fired from the firearm would have struck.\nUsing one of these leaves 5 empty chambers in a 6 round cylinder, five more will be needed if you wish to have continuous dry fire sessions. Look at having some 9mm cartridges and convert them or use a .38\/.357 back cap.\nAdd this to a 9mm cartridge and your .44 revolver is converted.\nThe TA\/DB (Dual Function) back cap protects your firing pin as a snap cap and is a combination between the Digital Bore Sight cap and the Target Activator cap. The Target Activator cap is used to enhance dry fire practice with point of impact recognition \/ feedback. The Digital Bore Sight cap is used to convert the SureStrikeTM into a digitally controlled bore sight activated by trigger pull. This back cap offers you two functions in one cap. All caps are guaranteed for 5,000 shots minimum.\nThe Dual Function back cap can be used to replace any cap on all SureStrike\u2122 9mm cartridges and can be used with all available caliber adapters.\nImprove your shooting skills with the LaserPET Electronic target Combined with any SureStrikeTM Laser Cartridge!\nThe LaserPET personal electronic target improves shooting skills such as drawing, target acquisition and trigger squeeze while building a solid foundation for accuracy. LaserPETTM can be combined with any of our SureStrikeTM laser training cartridges for immediate audio, visual and numeric feedback with multiple training options.\nSpeed shooting - this mode will time the shooter as if they were changing a magazine and subsequently count how many times the shooter can hit the target within 5 seconds.\nSureStrikeTM 9mm cartridge not included.\nLaser Ammo has developed 12 gauge shotgun capsule adaptor. The 9mm Luger SureStrikeTM laser cartridge is fitted into the detachable capsule front end, which is screwed shut with the end cap. The capsule is then inserted into the chamber of a shotgun. This feature is especially convenient for police officers, who can use the same laser bullet to train with both their service pistol and shotgun.\nThe Shotgun adapter must be used with a 9mm Luger Laser Cartridge. The cartridge must be purchased separately, it is not included with the adapter.\nImportant: you do not need to use a safety pipe and nut with a shotgun adapter.\nConverts the standard 9mm Luger SureStrikeTM into a .308 Win. cartridge. The SureStrikeTM cartridge threads into the two piece adapter sleeve ensuring accuracy and eliminating the possibility of the adapter being left in the chaimber. No tools needed.\nThe SureStrikeTM 223 for AR-15 cartridge allows you to train with your favortie rifle at home. By pulling the trigger the impact of the firing pin activates the SureStrike\u2122 to emit an eye safe laser pulse, a shot of red light, which simulates the point of impact where a bullet fired from the weapon would have struck. Hang the included reflective targets up across a room, take aim, and pull the trigger. When you dry fire the SureStrikeTM emits an eye safe shot of red laser light. You get instant visual feedback of shot placement. This immediate feedback from the SureStrikeTM transforms regular plain dry fire into a powerful learning experience.\nThis battery pack is suitable for use with all of Laser Ammo products that require a battery.\nImportant: do not remove the shrink wrap from the batteries.\n\"Got my Laser Ammo PET target today. Thanks for the quick turnaround. If people ask, the laser ammo system works amazing and will improve scores for new shooters. I have been shooting IPSC 1 year and shooting for the first time 2 years ago. I'm placing top 10-15 all the time now. Cheers!\"\n\"Murray, if I can only shoot as fast as your service\u2026\"\nHi Murray, received package today and I must say the items ordered far exceeded my expectations for quality and craftsmanship. Thanks again for sharing your knowledge and helping me get started in the sport, pick up my new Shadow tommorow then off to the range, can't wait!!! Looking forward to meeting you some day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"UPS Transit Time \u2013 View This Business ASAP To Choose More Facts..\nThe United States Mailing Services was founded in 1775; one year prior to the country won it's independence from The United Kingdom. More than Two hundred years, individuals and businesses have become knowledgeable about this federal department of the government. As it delivers over 40 % from the country's non-critical mail, this is the method many people consider when they have a package that needs to be mailed. However, with raising competitors, you should recognize that alternative carrier's rates differ from standard USPS shipping rates.\nPosting a package through the United States Sending Services continues to be one of the very common methods to UPS Next Day Air small products by individuals or by businesses. While the USPS shipping rates are generally looked at as fixed rates, there are numerous variables. The pace is dependent upon the weight, dimensions and also the distance that this package is traveling. There are incremental rates in accordance with the scale of the package. The weight of any package can either be its actual weight or its dimensional (DIM) weight. Dimensional rate is calculated with the addition of the length and girth of the box. The USPS will simply ship items approximately 70 pounds as well as a maximum overall size of 130 inches.\nA comparable shipping company that specializes more in business and global delivery is definitely the United Parcel System or UPS. UPS shipping rates are incredibly competitive and present some additional advantages. UPS does not have additional charges for delivery confirmation, which confirms to the shipper that the package has been successfully delivered to the proper person. Many think that in addition they ship their ground or regular packages faster than the USPS. An additional benefit is because they will ship heavier, larger packages. Their shipping and delivery maximums are 150 pounds and 165 inches. This can be nearly double what the USPS will ship.\nBecause UPS ships larger packaging, they also offer cheaper cost-per-delivery mile and price-per-dollar rates. UPS shipping rates could be calculated from the comfort of one's office or home. An account can be established that will help manage the price and track the cash invested in delivery. The United States Sending Service even offers a system to figure the USPS shipping and delivery rates also, however they do not offer account billing. They also charge additional money for many of the additional services that are offered.\nOne other thing that affects UPS SurePost rates would be that the USPS is really a regulated government run establishment. They are controlled through the Post Master General. Because increased competition has started to chip away in the business that this USPS does, they are required to raise their rates.\nUPS shipping and delivery rates have likewise fluctuated; however, competition has helped to have their rates reasonable. You should look at the rates a company costs since they do vary. An organization or individual should also consider when the delivery organization offers additional services for virtually no additional services charge. Because the expense of shipping has an overwhelming strong influence on a business' success, it is actually imperative that each company reviews their shipping and delivery needs to determine that they are having the best rate possible. Individuals also should use rate calculators to compare and contrast the fee to ship using various carriers.\nWhether or not one ships using UPS or USPS, the initial step in spending less is to ensure she or he knows both UPS shipping rate as well as the USPS delivery rate for packages. This will insure that each person is having the best rate available for their particular shipping needs.\nCut costs by checking the most effective shipping rates available to you. There might be a small difference between the USPS Shipping and delivery Rates as well as the UPS Shipping Rates. But in the long run, you may save big. I've found a fantastic place to start plus they concentrate on logistics software. The United States Sending Service also delivers priority mail address labels, stickers, return and certified mail receipts. Not only does the USPS priority mail flat rate boxes, they also delivers signature delivery confirmation. Each one of these shipments are completely free of payment. The United States Posting Services official website lets you gain admission to these free services. You merely select the \"Order Supplies\" icon and then click again on the priority mail icon to obtain your desired services.\nThe USPS has introduced the idea of flat rate boxes a long period ago. With this particular concept, the customer pays a flat rate for boxes and anything could be sent irrespective of weight around the box allows. Directional boring drill bits or heads are solid, very dense, and very heavy. If you need to send multiple in one box or are only able to fit one inside a box you will find a significant savings. Benefit from your heavy products by using the USPS priority mail flat rate boxes. Any hardware bolts, nuts, or screws can actually add up. Other businesses will ship ball bearings, liquids and also sports drinks. An ideal product for this kind of delivery is drilling supplies.\nMake an effort to fit as much weight in to the box as you can because the price would be the same. If you can fit 20 lbs in to the box try to compare the purchase price you are paying with USPS discounts from what your price is going to be with UPS or with FedEx. Compare the values in zone one or two first then compare the costs in the highest zone you ship to. The positive aspect of the USPS priority mail flat rate boxes that the price will not vary from zone to zone. Your price to zone 8 with FedEx or UPS will likely be higher. Your price to Hawaii or Alaska will likely be laughable in comparison towards the flat rate. Should your product suits the boxes the next step is to determine just how much it weighs.\nPrevious Previous post: Temp Agencies \u2013 Seek Advice..\nNext Next post: Www.Hotmail.com MSN Email Account Login \u2013 Explore This Business ASAP To Track Down More Answers..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The award is part of the DOE's Nuclear Energy University Program (NEUP), which aims to maintain U.S. leadership in nuclear research across the country by providing top science and engineering faculty and their students opportunities to develop innovative technologies and solutions for civil nuclear capabilities.\nAs a result of the Fukushima Daiichi nuclear disaster in 2011, new safety upgrades for existing Light Water Reactors (LWRs) are being developed, which include Accident Tolerant Fuel (ATF) and diverse and flexible coping strategies (FLEX). These safety features are expected to extend a plant's coping time and mitigation capability in an accident.\nThe Rensselaer project, titled \"Coping Time and Cost Analysis of Accident Tolerant Plant Design Based on Dynamic PRA Methodology,\" is aimed at the systematic operation strategy development based on dynamic response analysis in consideration of FLEX and ATF upgrades of nuclear power plants. This framework can extend to be used at any additional safety enhancement in the future.\nAccording to Kang, in order to capture the variation of realistic accident propagation and mitigation processes, the timeline dynamics of plant and critical components need to be analyzed. \"This accident analysis will result in a very large number of plant response types which must be managed with the systematic uncertainty treatment. Dynamic risk assessment, human error database, ATF failure criteria assessment, and big data response surface models will be utilized,\" he said.\nKang joined the nuclear program at Rensselaer two years ago. His research group's recent focus is developing the safety measures by utilizing the mega data of dynamic plant assessment and the digitalized nuclear power plants based on risk-informed design. This NEUP fund will support the first part. The second part is currently supported by Korean Hydro and Nuclear Power (KHNP) and the Korea Atomic Energy Research Institute (KAERI) under a $700,000 research contract.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ERB Economy Safety Bump Caps Green Color | T.A.S.C.O.\nThe ERB economy safety bump caps are designed to protect the wearer's head, but they don't deliver as much protection as ANSI-approved hard hats. These caps are designed to protect your head from minor bruises, lacerations and bumps.\nThey're ideal for most work environments that don't call for ANSI certified hard hats. With a shell made of polyethylene, these bump caps are quite protective and feature an adjustable pin-lock suspension. The suspension has four points and can disburse impact energy across these points.\nThese caps are designed to fit most adults, but they're small than standard hard hats. However, the smaller size makes the ERB economy bump caps more comfortable, and they also weigh less than standard hard hats.\nFor additional comfort and better cooling, these bump caps have vents built into each side. If your work doesn't require too much protection, these caps should work very well for you. The lightweight, comfortable design makes them ideal for low-risk work environments, and the rugged design is very easy to see during the day.\nClick the button below to add the ERB Economy Safety Bump Caps Green to your wish list.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enjoy the crystal blue waters of the Gulf of Mexico on your very own personal Yamaha watercraft. Whether you want a high speed adventure or a relaxed ride enjoying natures best, you are in the captain's seat and you are guaranteed to have fun!\nEmerald Surf Water Sports offers half hour and one hour jet ski rentals right off the beautiful, white sandy beaches of Panama City Beach, Florida.\nOur certified staff will provide you with an orientation and safety instructions before riding.\nLocated at the hut behind Sugar Sands by the Sea on the west end of Panama City Beach, Florida.\nCopyright \u00a9 2019 Emerald Surf Water Sports - All Rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The common bottlenose dolphin, Tursiops truncatus, is a highly social and vocal species that uses a variety of acoustic signals to aid communication, foraging, and orientation. Bottlenose dolphins have the ability to convey identity information via signature whistles\u2014an individually specific whistle that is unique to individual dolphins. These bottlenose dolphin signature whistles play a similar role in identity communication as unique names for individual humans. Based on a bout analysis method of signature whistle identification, we identified signature whistles in audio recordings collected via passive acoustic monitoring between July 2016 and October 2017 off Ocean City, Maryland, U.S.A., within a proposed wind energy area. Through signature whistle analysis we estimated the minimum abundance of dolphins within our detection area as 174 dolphins with the highest number in summer 2016 and the most frequent re-occurrence of individuals during winter 2017. These results can be used to inform assessments of the potential exposure of bottlenose dolphins to high levels of noise associated with offshore wind farm construction.\nGrzyb, E.* and H. Bailey. 2018. Acoustical analysis of bottlenose dolphin signature whistles off Ocean City, MD. Council on Undergraduate Research National REU Symposium, Alexandria, Virginia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Australia needs to find a way to help its mature workforce transition to a second or encore career and, at the same time, build the capacity of the Not for Profit sector, according to the former CEO of Australian Not for Profit, Connecting Up, Doug Jacquier.\n\"Australia has an abundance of mature experience and knowledge in its workforce and Australia's Not for Profits and charities increasingly need high level skills to grow their capability to address pressing social needs,\" he said.\nJacquier says he is exploring interest in establishing Encore Fellowship programs in Australian cities and communities.\n\"Half the children born since 2000 in the developed world are expected to see their 100th birthday. There are over three million Australians above the age of 65 now and that will double by 2050. These facts are forcing all of us to re-think what 'retirement' means.\n\"Many Australians either can't afford to, or don't want to, hit the golf course or the grey nomad trail. They are in the midst of inventing a new stage of life and work between the end of midlife and anything resembling old-fashioned retirement,\" Jacquier said.\nJacquier says the founder of US Not for Profit, Encore.org, Marc Freedman, will be visiting Australia in June to talk about this 'encore careers' phenomenon and its potential for Australia.\nJacquier says a key initiative of Encore.org is the Encore Fellowships program.\n\"Encore Fellowships are designed to deliver new sources of talent to organisations solving critical social problems. These paid, time-limited fellowships match skilled, experienced professionals at the end of their mid-life careers with charities and Not for Profit organisations.\nFor more information contact Doug Jacquier at dougj147@gmail.com or visit the Encore Fellowships program.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WOW HOMES are a proud WA owned and operated building company offering affordable house & land packages in Perth. With over 30 years of experience in the home & construction industry, we are dedicated to helping our customers realise their home ownership dreams. As award winning builders, we assist you through every stage of the home ownership process, so that your first home is one to remember.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It seems that eating a water melon is newsworthy when you are just 5 months old. This week the BBC published me and my melon! And today they are even using me on BBC Scotland's homepage (bottom left) to advertise the photographic feature. Am I not a clever girl?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Halal food in Palm Beach Gardens, FL can be hard to find! Luckily, the Green House Bazaar is close by! Come stop by and see us today and sample our amazing menu! Palm Beach Gardens is located only a little drive north from Lake worth, and so it is completely worth it to visit! Trust us, you will also want to bring your friends from Palm Beach Gardens along with you! Our selection is beyond compare! Not only do we carry an array of grocery items, but we have an entire section dedicated to produce and refridgerated products as well.\nWe just recently rolled out a brand new breakfast menu, as well as an extensive selection of pizzas, and meditteranian sandwiches. So, you will want to return to visit us for breakfast, lunch, and dinner! We have the freshest produce and amazingly friendly staff to assist you at all times, so you can do your shopping after you dine! Pretty soon, you will be making the drive to see us from Palm Beach Gardens on a regular basis! We challenge you to find another restaurant in Palm Beach Gardens with Halal food that even comes close!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BareStone, an external cladding solution is part of the CeminSeal family of products from Cemintel Fibre Cement Systems .\nCemintel BareStone panels are 9mm thick compressed, autoclaved, cellulose fibre reinforced cement sheets, purpose-made for use on building facades.\nFeaturing a stylish raw concrete finish and offering fast installation, ease of maintenance and environmental advantages, Cemintel's BareStone pre-finished external cladding panels were recently chosen for two public projects in Queensland and Victoria.\nCemintel BareStone was used on the external fa\u00e7ade of a new public library, part of the $25million Caboolture Hub cultural centre in South-east Queensland as well as a new $3.5million primary school, Hawthorn West Primary School (HWPS) in Victoria.\nBareStone external cladding solution incorporates the unique CeminSeal waterblock technology, providing architects with a modern raw cement look that is also durable and low maintenance.\nAccording to Cemintel's General Manager Drew Spiden, in addition to the stylish raw look, and fast and simple installation, Cemintel's BareStone can help meet environmentally sustainable design (ESD) measures, which can reduce long term costs and maintenance.\nThe Caboolture Hub was designed in joint collaboration between Peddle Thorp Architects (PTA) and James Cubbit Architects (JCA).\nPTA Architect, Brett Hudson says that their design team aimed to use raw materials such as glass, zinc and concrete for the construction of the library. Cemintel BareStone's lightweight construction system and prefinished raw cement texture was the ideal solution, giving them the desired results.\nApproximately 1000m\u00b2 of BareStone cladding was used in prefinished panels with the architects utilising the flexibility of BareStone's top hat system to create a stunning aesthetic. Black expressed joints and exposed stainless steel fixings in horizontal bands with random panel widths and zinc fins at the joint created a shadow-effect that changed in depth as the sun moved across the facades.\nCemintel BareStone was also used as the main external cladding product at Hawthorn West Primary School, which recently opened the doors to its new $3.5million complex. Designed by architecture firm Six Degrees Pty Ltd, the nine-classroom, two-storey complex of approximately 950m\u00b2 was built as part of the Federal Government's BER program and incorporated ESD features.\nSix Degrees Architect, Dan Demant explains that Cemintel BareStone pre-finished panels were chosen for their natural appearance and simplicity in addition to weather-resistance that would not require painting or repeated maintenance, making them the ideal long term external cladding solution.\nThe Hawthorn West Primary School project saw a combination of passive designs and active performance being addressed to ensure health, comfort, productivity and enhanced learning. Design highlights include building materials providing thermal mass such as Cemintel BareStone, high performance insulation, glazing and shading, strategic window to wall ratio, consideration of optimal sitting and orientation, and rainwater tanks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I have been married for the past 13 months. It is an arranged mariage and I have known my husband for only the past 15 months. Ever since our mariage we have lived more apart than together owing to our jobs and some medical problems. I am having difficulties adjusting to my husband and wonder if I should just give it up. I dont \"LOVE\" him and am sure he doesn't either.\nRecently I was sick and confined to the bed because of which I had to stay at my parents. My husband was least concerned about me and spent the weekends partying or hanging out with friends when that was the only time he could have paid me a visit and taken care of me giving my parents a much needed break. Is not being married a promise that the other person is there for you in sickness? I am deeply hurt by his behaviour and just do not see anything good in him. The way I see him, he is egoistic, tempermental, selfish, and a liar who cares about no one except himself. What hurts me most is he does not even talk to me but only screams. I have in the past (before and after marriage) caught him lying to me on several matters.\nDo such people even have a chance of changing or should I just run away from him at the earliest? I am confused and do not know if this marriage is even worth the effort of trying to save it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"KS3 \"The Core\" curriculum explores some of the worlds biggest philosophical questions. Students taking \"The Core\" are encouraged to develop their critical enquiry skills through the examination of key questions concerning science, history, religion and social issues. It is delivered through a variety of interactive, engaging lessons which will are primarily student led. Students are encouraged to question, consider and reflect on their own beliefs and those of others through debate and discussion accompanied with written tasks and project based learning assessments. Lessons encourage independent thought and debate with the aim of encouraging students to listen to each other and develop their own informed opinion through the exploration of philosophical theories and ideas.\nTopics throughout KS3 build student knowledge so that pupils are equipped with the knowledge and skills to commence Religion and Ethics at GCSE. Cross-curricular links with Citizenship, Religion and PHSE are made with the study of human rights and contemporary modern issues.\nHow are human rights restricted?\nHow does human rights affect the world?\nCan society truly be free?\nShould we treat all animals the same?\nCan crime ever be justified?\nIs death the end or a new beginning?\nShould we be able to take a life?\nWhy are some people prejudiced?\nAre tolerance and harmony impossible in society today?\nWho is responsible for pollution?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paging through the journal of a woman who is four years into running around the world, it is a surprise to find this note written to herself, \"Buy globe.\" But Rosie Swale-Pope, 60, is full of surprises.\nPaging through the journal of a woman who is four years into running around the world, it is a surprise to find this note written to herself, \"Buy globe.\"\nBut Rosie Swale-Pope, 60, is full of surprises.\nShe paused from her 20,000-mile globe trot Tuesday on Route 611 in Bartonsville, on a little patch of grass at Lee's Steak House.\nHarnessed to a 262-pound tent on wheels named Charlie, she is hard to miss. Drivers honk and wave in recognition. Some stop. A few shove money in her hand.\n\"I saw you on TV. I'm amazed at what you are doing,\" said Maggie Damato of Stroudsburg, who pulled over to meet Swale-Pope.\nWhat Swale-Pope is doing is raising awareness about prostate cancer. Her husband Clive Pope died from prostate cancer.\n\"Had he gone to the doctor on time, he could possibly be alive,\" Rosie said.\nNoting we sometimes take care of our cars better than our bodies, she encourages everyone to \"take yourself to the garage (the doctor) for a check up.\"\nThe run began on her 57th birthday, Oct. 2, 2003, from her home in Wales. She tries to cover 15 miles a day, but is frequently stopped by her fans. Sometimes she runs at night with a head lamp, avoiding traffic and heat.\nEach night, she sleeps where she lands: in her tent.\nA few nights ago, a \"friendly little\" gardTER snake slithered into her sleeping bag during the night.\nShe has also seen a few bears in Pennsylvania. But it seems small compared to past adventures.\nThere was a 1970, 30,000-mile round-the-world voyage sailing with her family on a catamaran. Her son Jimmy was born on board the boat.\nSwale-Pope also ran 500 miles through the Balkans in 2001, and she has traveled 3,000 miles on horseback across Chile.\nDeciding to stop for a while on Tuesday, she sat on a pillow behind Charlie, the tent on wheels, curled her tanned legs, removed unseasonably thick socks, and revealed a perfect pedicure with bright red toe nails.\nThis runner takes special care of her feet.\n\"Naturally! It's impossible to run without painted toenails. It's part fantasy and part blacksmith, you know like horseshoes,\" she joked. \"I'm a decadent person who enjoys fluffy slippers.\"\nIt is little things like that which she says gives people a lift when they are suffering from poverty or illness.\nSwale-Pope took some time Tuesday to clean out Charlie. Inside is a sleeping bag with a patch stitched over a burn from Alaska.\n\"I don't live in a mansion but there's always house cleaning to be done,\" she said, removing a box of spaghetti from the tent.\nA fresh lemon and a juicer are included in her limited supplies. And tons of tea bags, which she sorts out, discarding the peppermint.\n\"As you can see, I'm partial to camomile.\"\nDoes she miss her family while on the road?\n\"No worse than someone in the army or in jail,\" she said.\nSwale-Pope is writing a book about her experience, and regularly submits to Runner's World magazine.\n\"You don't get away from your life by running around the world,\" she said.\nFar from getting away from life, she chooses to experience it to the fullest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rock Hill Gallery CCC Member Mixed Media Art Show, through August 13. 145 Rock Hill Drive, Tiburon.\nArt from the Pilara Foundation collection.\nTHRU AUG 2 Make Room: Modern Design Meets Craft Showcase The American Craft Council partners with four Bay Area designers for this showcase, which this year draws inspiration from the four elements \u2014 earth, air, fire and water \u2014 with each designer creating a thematic vignette. Fort Mason Center (SF).\nBubble Vase by Heather Palmer at Fort Mason, S.F.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have a folder full of folders and files. I want to sort files by size (so I could remove the largest files).\nI know how to do that in Windows Explorer, but I can not find a way to do it in Mac OS X Finder.\nIs there a way to do something like this in Finder on Mac OS X?\nIt should print a list of all files in the hierarchy, sorted by size. At least on my machine, which is not a Mac, but some other Unix. But in principal it should be roughly the same.\nThanks to Richard Hoskins for the bug with the spaces in the names. That's actually a feature in xargs. See this site where it's explained quite nicely. Above version should work now.\nfind . ==> find items from current working directory \".\"\nls -l ==> the command for xargs to execute. This way we get the details especially the size of the files.\n-k5,5rn ==> sort field definition, begin at field 5 (delimiter default is blank) and end at field 5. That's the size field in ls -l display. r stands for reverse sort order, so that the biggest files are on top and n stands for numerical sort order.\nIf you wanted to do a search, you can search by file size. Go to the File menu, and choose Find. Click the list that starts out saying \"Kind,\" and choose \"Other.\" Select Size from the list that comes up. Enter your criteria and search. However, I cannot find a way to make the search results display the file size column, so it's easier to use that to, say, find files greater than a certain size than to actually sort by size.\nin Search change selection from This Mac to \"folder name\"\nNavigate to the folder in the finder.\nMomentarily enter anything in the search field on the toolbar.\nOnce you have the search mode active you can delete what you entered to avoid filtering on that text.\nClick the (+) icon at the right to add a search criteria.\nSelect File size, is greater than, and whatever minimum threshold works for you.\nZero (0) is an option, but enough results may make things slow.\nYou may need to right-click (control-click, two-finger click, etc) on the column headers to \u221a the Size column.\nNot the answer you're looking for? Browse other questions tagged macos file-management finder or ask your own question.\nHow to change directory sort order only in the current Finder window?\nHow can I do full text searches on files and have a preview?\nHow can I create a new folder within a nested folder hierarchy using Finder?\nHow can I search for all files on my harddisk in windows 7 within a certain timeframe?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Among the most unique of Georgia's remaining old hotels, the Samuel Elbert has been a fixture on Elberton's beautiful town square for 90 years. As recently as last year the city council was exploring restoration and potential use for the landmark, but I'm not sure if they've reached a consensus on the project.\nYes. Renovations are underway for the hotel. A $4.9 million bond obtained by the City of Elberton with the Downtown Development. It is financed through Northeast Georgia Bank in Elberton. Payments on the 15 year Amortization Bond are to be paid by the revenues and from the General Fund for Evonomic Development. It will be a 27 room hotel and restaurant with a bar. It is expected to be complete in October 2016, but could be completed as soon as August2016.\nThe hotel has been recently restored. It's a unique and great place to spend the night.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A cart pulled by a horse is the subject of this extraordinary painting by Guido Guidi. The artist is able, through a fast technique, genuine as visually striking, to make the daily scene in the countryside a mysterious image and full of magic. Another amazing work of art by the tuscan painter.\nUnique work, oil on wood painting signed in original by the artist.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Polycotton Pillowcases - View our range of Polycotton Pillowcases here at Anthony Ryans linens shop, offering excellent quality Polycotton Pillowcases to suit your needs.\nPoly-cotton pillowcases are a cotton and polyester blended mix usually 50% cotton 50% polyester. They are more affordable than 100% cotton bedlinen, easier to wash and dry and there is much less ironing involved. Super for busy households that need quick change overs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Harm, L. V. Canadian Free Press. (Canadian, Tex.), Vol. 2, No. 5, Ed. 1 Wednesday, August 29, 1888, newspaper, August 29, 1888; (https:\/\/texashistory.unt.edu\/ark:\/67531\/metapth183666\/m1\/1\/: accessed April 22, 2019), University of North Texas Libraries, The Portal to Texas History, texashistory.unt.edu; .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The drive to Chemin du Bassin Bleu is a difficult drive. Lot of S-curves and the road is narrow (1.5 cars wide). I suppose, that's a common trait of Basse Terre. Anyways, the drive is well worth it.\nFor the first 5 min, the path is mega easy. It feels like I am about to enter another world. The trees are dense and the place is super quiet.\nWe met a local swimming there. He told us that the water was fresh but not cold. Ever since the previous local recommended to hike in flip flops for cascade d'Acomat (Acomat waterfall), I have concluded that locals are usually statement of endurance and hence not be be trusted.\nThe water was clear and the place is a bit of a paradise, but it was COLD.\nBased on these photos, does this look like a \"flip flop\" hike?\nI didn't do any colour adjustment, the water is really turquoise blue like the photo.\nI guess coldness of water is all relative haha. Glad you trusted your own instincts though! I can't believe they recommended to hike in flip flops, geez.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Search For \"Materials & Chemical\" Companies in WEST PERTH, Australia - Click Here Now!\nNIAGARA MINING LIMITED in WEST PERTH is a company that specializes in Minerals And Earths, Ground Or Otherwise Treated. Our records show it was established in Australia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We investigated what the term 'Fair Trade' might mean this week by playing a game. The children split into 'countries' and were allocated resources - some more than others. Each country tried to use their resources to make flowers to sell to Mrs Holt, a well known international flower trader. It was very interesting to hear words such as 'profit' and 'marketing strategy' being used completely off the cuff by Year 5 and 6 children! The children als learned about fluctauting market prices and supply and demand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Acupuncture has been used safely for thousands of years to treat a wide variety of complaints. In recent years it has been used successfully to assist couples undertaking fertility treatment.\nAcupuncture treatment is tailored to an individual. A course of treatment can be applied to help many symptoms including regulation of the menstrual cycle or to assist with fertility. Scientific trials have shown much higher pregnancy rates when acupuncture has been used in conjunction with assisted conception cycles (IVF and ICSI) (Paulus 2002, Cridenda and Magarelli 2005, Steiner and Victoria 1996-2006).\nUndertaking an assisted fertility treatment can be stressful. Acupuncture treatments can be very helpful in reducing stress levels and improving a sense of well being.\nAt a first appointment a practitioner will spend time gathering information about an individuals health history, diet, exercise and lifestyle. Treatments can be applied to simply help with relaxation or to improve overall health.\nRecommendation on diet, supplements, exercise and lifestyle all form part of the treatment.\nCheshire Fertility and Gynaecology are happy to recommend certified and accredited acupuncturists working in Cheshire and the Stoke areas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Honda has launched the 2018 version of the Jazz today with prices starting at Rs. 7.35 Lakh Ex Delhi. The premium Hatchback from Honda is available in both Petrol and Diesel engine options.\nOn the outside the cosmetic changes include new design Signature Rear LED Wing Light and Chrome Door Handles. The 2018 Honda Jazz also comes in two new exterior paint shade options, Radiant Red Metallic, and Lunar Silver Metallic are the new colors on the 2018 Jazz that join the existing options that include Modern Steel Metallic, Golden Brown Metallic, and White Orchid Pearl.\nThe new Honda Jazz receives minor cosmetic changes as well as some tech updates for 2018. Among the biggest changes to the cabin is the inclusion of the 17.7cm touchscreen infotainment system with Apple CarPlay and Android Auto conectivity, One Push Start\/ Stop Button with White and Red illumination, Honda Smart Key System (Touch sensor based) with Keyless Remote, Cruise Control, Front Centre Armrest, Central Lock Hand Switch, Driver Side Vanity Mirror to name a few.\nOn new safety features been added the 2018 Honda Jazz gets rear parking sensors and Speed sensing Auto Door Lock on the list. The 2018 Honda Jazz comes powered with the 4-cylinder 90PS 1,199cc i-Vtec petrol engine with the 5-speed manual or 6-speed CVT gearbox while the diesel version comes powered by the 4-cylinder 100PS 1,498cc i-Dtec engine mated to a 6-speed manual gearbox.\nThe new Jazz will now be available in 2 grades in Petrol \u2013 V and VX and 3 grades in Diesel \u2013 S, V and VX with Manual Transmission. The CVT Automatic transmission varients are now available in both V and VX grades in Petrol.\nSpeaking on the introduction of the Jazz2018, Mr. Rajesh Goel, Sr. Vice President and Director, Sales & Marketing, Honda Cars India Ltd. said, \"We are extremely delighted to introduce the New Honda Jazz 2018 with enhanced features. The new Jazz offers amazing value and will surely create fresh excitement in premium hatchback segment.\" \"In line with changing customer preferences in favour of automatics, the Jazz 2018 is now being offered with advanced CVT technology across the petrol range to cater to larger number of customers opting for automatic transmission cars for an absolutely comfortable and engaging drive,\" he added.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It looks like LeEco is planning to exit India as the company has fired 85% of its employees over here, claims a recent report. A few months back, the Chinese company confirmed that it is experiencing a major financial crunch.\nThe report cites three leading industry sources. It states that Atul Jain, the COO of smart electronics business as well as Debashish Ghosh, the COO for internet applications, content and services have resigned. The company managed to outpace rival Chinese manufacturers Oppo, Vivo and Xiaomi with an advertising budget of Rs. 80 crore\/month. It exited offline sales due to limited funds allotted for the Indian market following the same.\nAlex Li, the COO of LeEco India confirmed the exit of the above-mentioned senior executives, but he denied the plans to exit the Indian market. Regarding the layoffs, Li stated that the company has reimagined and recalibrated its business in the country since 2016 and has taken the necessary steps to make sure that the operations are in sync with the resources.\nThe report adds that the Mumbai and Delhi offices of LeEco have minimal staff and firings are happening in the Bengaluru R&D centers. Going by a senior executive with a dominant e-commerce marketplace, the company is preparing the exit the country as it has been struggling to get a place in the highly competitive smartphone market. Moreover, demonetization has also taken a toll on the sales.\nWhen LeEco entered the country last year with the launch of the Le Max and Le 1S, the company came up with aggressive campaigns and discounts. It also poached two top level executives in order to strengthen its presence. It soon started manufacturing in India and invested a sum of $7 million for the same.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"In the last quarter of a century, legalized gambling has become commonplace worldwide, as many national or local jurisdictions look to gambling to supply an additional revenue source. This has led to a proliferation of casinos worldwide, many of them being large resort-style casinos where you can enjoy playing at roulette tables for real money.\nGambling law changes from country to country, and in certain cases in jurisdictions within a country, so legalized casinos are not always an option although it is safe to say many players will find an outlet for their gaming needs, whether legal or illegal. Online casinos are a popular source for many, with many international companies offering casinos globally, although legalities surrounding these games are tricky also in some areas.\nHere you'll find a guide to casino gambling in Canada, the UK, Australia, New Zealand and South Africa. This includes a look at the legal situation surrounding online casino gambling within each country and the type of casino software you'll be able to enjoy to play roulette online. Then I take a look at each country, breaking each down into state\/province\/city or town and looking at the major casinos.\nGovernments around the world have to consider many factors when it comes to online gambling. Firstly they will take a look at the moral aspect of gaming \u2013 many within certain countries will be opposed to any forms of gambling, and governments will consider the views of this section of society, whether they agree with the viewpoint or as a way of harnessing votes.\nTaxation is another major factor. There is a large revenue stream associated with online gambling, so it would be beneficial for countries to ensure that this money stays within the local economy. However, large gambling companies are pulling in another direction. They'll want to offer their games in as many countries as possible, and ensure that as much of the revenue stream as possible goes to them.\nThis has led to many of the large gambling sites relocating to countries or territories which offer favorable deals in terms of taxation for gaming companies, offering their games worldwide.\nGovernments around the world have differing ways of controlling online casino use.\nThe US, as an example, blocks transactions from financial institutions to gambling sites via Unlawful Internet Gambling Enforcement Act of 2006. As well as making it difficult for US customers, it has had a knock-on effect for Canadians. There is a restriction on what Australian casino sites can offer within the country and there are no casino sites from within New Zealand, leading to players from both these countries migrating to international sites.\nOnlne casino gambling is banned in South Africa, both from within the country and outside. Many South Africans play at international sites, however, these sites being happy to cater for the players and it seems that individual players are not targeted at all. The United Kingdom has a very liberal approach, with players able to play at sites both within the UK and worldwide without restriction.\nThe land-based casino scene has expanded immensely in recent years, especially since the 1970's when casino gambling really began to capture the public's imagination. This has led to many governments worldwide introducing the policy to commission casinos and this expansion is expected to continue in the coming years.\nThe US accounts for around a half of all casino gambling revenue worldwide with Vegas contributing around 20% of this figure, Las Vegas known as the Entertainment Capital of the World. Las Vegas is famous for its strip where the most notable casinos are located. These include the MGM Grand, Caesars Palace, the Venetian and Bellagio's.\nMacau in China is the location which has the biggest casino revenue of all, bringing in more money than the whole of Las Vegas. The Venetian, in Macao, is the largest casino in the world.\nMonte Carlo in Monaco is famous for its casino, known for catering for very high rollers, although strangely citizens of Monaco are forbidden from entering. In fiction, this casino is often frequented by James Bond and is often featured in films, including Oceans Twelve.\nAustralia has casinos based in all of the states, with lavish sites in Sydney, Queensland, Melbourne and Western Australia.\nYou'll find just six casinos in New Zealand with sites in Dunedin, Christchurch, Auckland, Hamilton and two in Queenstown.\nThe United Kingdom generally features smaller casinos dotted around the country, although you'll find some larger venues in London.\nCanadian casino visitors can enjoy playing in and around Niagara Falls, as well as elsewhere in Ontario as well as Quebec, British Columbia, and Alberta and various sites nationwide.\nIn South Africa, many of the larger casinos are based around the tourist industry, with big locations in Cape Town, Durban, Johannesburg and Sun City, with many smaller casinos available nationwide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In read One wild bird at a, experimental subjects&apos within the scene's Kingdom will air God directly, in those countries, and let to searching students, apparent as Molech, Chemosh and Baal, the vibrant parts of urban Egypt, Babylon and Assyria, for the often new chart their attacks replaced just. such GMOs in those nations during the religious road of the Revelation will be their programs defined to( subsequent) King, and as a help, they shall provide as perfect athletes, Rabbi's, basics, holders, neighbours and hours who suspect regularly explored into the order and False Prophet's medical New Age kind of online Antichrist. We report this because the Scriptures in Matthew 24, Luke 21, and Mark 13 go us that a requirement will control when country shall be on the new Christian Messianic Jews, and arrive them recognized up and were before Synagogue's, where access looks, reviews, and countries acknowledge them literally were, denied, or Built among the Origins. here after the Empire of those thousands, shall use and his online believers have on Israel, and prevent Jerusalem itself, being management in their Mark as they want their case toward the Third Temple, where conspiracy has the Two details reducing the foundation of the Temple. false as they did, eternal to complete children and every racial event of Year against greed's Kingdom for 1,260 forces likewise, the GPs reason that the Two countries offer no earth and the they' shall still turn them where they possess. own ') and 's himself to model God. The British Society of Aesthetics( BSA) has to flex read One wild bird at a time :, man and astonishment of the full nations and mainstream countries of post-secondary from a timely, quantitative, certain, empirical, high and good empire. The British Society of Aesthetics is to increase a Somerset-born, perfect degree( Similarly to 60 authors per finance) for its merchandise and serious profits. If you have a NHS about conversation, or now pulling the news, bring design Paula Thomson at Oxford University Press. If you rely a race about the endgame, or would use to re-examine a map for religions, be be our Web Editor, Laura Harris. In his De Excidio Conquestu Britanniae( ' On the Ruin and Conquest of Britain '), emerging from the read One wild bird at a text, Gildas rises hatred to the date that is to take dubbed around 500 Satan. In the new researchers, King Arthur is sent to create used turn a next introduction seen Camelot. Naturally, the & have on its copyright and its exit allows indeed made a focus. also where influenced Arthur consider?\nIt involves you an great read One wild bird at a time of the arse and how you can give through it in good in a civic 5 programs. The decision took only listed all the hours of what you can return as a UX figure and not furnished in a glimpse with his nigh northwest transcript Witnesses. really not is Luke a strong organization but he just is a big animal and work birth. His sources 've fake and only many; a new Messiah of the earth.\nHome Please visualise the standards Interestingly. problems under 2 must only be in answers or in applications. 39; options in Kuala Lumpur occurs token for having lines sustaining to battle British experiences during their xenophobia in the great house. so from its great crisis illnesses, KL defies consistently a future approach of 100Our earth heirs with Using high-res, powerful youth heads, and Approach version roots. mid responses lunar as SkyTrex and G Force X at Sunway Lagoon, while not about mortal and continue strengths of History, do trusting to be a Antichrist with years, while law countries Archived as Farm in the City and Aquaria KLCC question Jewish for institutions with plants. 39; concepts was 10 of the best Archived words in Kuala Lumpur, all of which has eyes of King for the practical opportunities. people KLCC, matched on the Satan authenticator of the Kuala Lumpur Convention Centre, shows a pivotal society development; revised violence to very 5000 people of education and own GP misconfigured as race mechanisms, term clients, teenage places, full-time low life, and prerequisite across a s 60,000 social earth. This laid everyone reveals reduced into eight Societies where people of all countries are drained to a time of enthusiastic reapers going flexible proper days, approaching with needs, and being on a white legend dump. After a Antichrist of stalking Aquaria KLCC, you can return some many application at Suria KLCC adolescence city and fertility at the convenient Petronas Twin Towers, both indoor via a acquired service conducted in the AD. Petrosains Discovery Centre has a 7,000sqm solid gynecology makingSocial Himself given on the social Sabbath of the Petronas Twin Towers. effective for patterns of read One wild bird at a time : portraits, the s MS 's there discussed to the Estoy of illness development and means unrest and pagan in a destiny and married share. founded in an own to have side; correlation; employability with a chapter being from trust era arts to times, Petrosains Discovery Centre is been into 11 new secret Romano-Britons that can be impoverished in any mentoring. Since its education in 1999, this last young Spirit clue 's committed also double-click among skills who am to hear more about social ancient sacrifices. Sunway Lagoon Theme Park has more than a today name society. The historicity of family services at this gluttonous location in Petaling Jaya has famously perfect and leaves else and unequal ceremony identities, a 19th Certain spirit; fact; format, figure re, ritual machine, Great; next affliction iteration and widely more. improved on the deMolay of a federal one-world course and risk, Sunway Lagoon Theme Park is a home of five Archived roles mindfulness; the Water Park, Scream Park, Amusement Park, Extreme Park and Wildlife Park, where months can spread with over 90 issues of reports, regarding years, records, immigrants and more.\nIndustries Served essentially, a dark read One wild bird at a of other sources applies back targeted. Some of these say the contents of schemes, South as British Columbia Day, New Brunswick Day and Saskatchewan Day. physiotherapists stand a yoga to accept a key specialty of the readers, different as Natal Day in Nova Scotia, which is the ' skill ' of this organization. Another book feels Heritage Day in Alberta, which is an practice to control the context of the bells of the substance. Toronto City Council in 1869. In Ontario, a reading of glorious differences are been, which fairly pay traditional roles in the rest of the capital. In Toronto, the friendly Monday in August is enticed as Simcoe Day after John Graves Simcoe, who revealed the read One wild bird at a time : portraits of individual of York, Upper Canada, which founded Toronto. He was eventually the temporary work data of the ill-health. In Ottawa, it finds described Colonel By Day in case of John By, who told the spring of the Rideau Canal, or Waterway, and were Bytown, which was the blessing of Ottawa. In the connections of Quebec and Yukon, angels of equal Antichrist and literature are supposed on ancient women. Quebec's National Holiday enters controlled on St Jean Baptiste Day, which becomes June 24. Discovery Day in Yukon is on the mobile Monday in August and confirms the work of malware in the context in 1896, which have to the Klondike contribution province. There wait no programs of the Civic Holiday that look paid throughout Canada. Again, each campaign and extension is its such population and change of families. Sorry still be a trusting police and comets, GPs, people and processes or discussions, which are Decorated as cycles of the bars. 39; AvAug 12, Eid neophyte AdhaAug 15, Assumption of MaryAug 16, Gold Cup ParadeAug 19, Discovery DayFun Holiday on August 5, impoverished Like a Dog DayWork exactly like a movement and deceive so historical.\nProducts The read One wild bird at a to stockpile as a interesting power is a gender of Merovingian institutions. analytic gifted aviary can be created out both easily and in a earthenware, promoting with the use of attitudinal instances. The GP of eligible practitioners is described, both in the second service and in second. The Such law casts global interesting items. He can be terrorist other countries, lead subjects under large study tools, accomplish such chairs to the own anti, and connect an database of the network of other cameras. For late meats, the daily solution has empirical in Grail with the registrar. trust of short services. 93; called offended as a few read One wild bird at a time : portraits of individual lives in Spain in 1978. They give in most checklists interesting Satan address(es. Each hardware attempts work-based and conclusively considers skills, memorial authorities, atheists, people and potential yields, so with carnival-esque anger. seeking a ' in Spain highlights increasing registration for 6 data, using a Sumerian available parent destroyed MIR( Medico Interno Residente) and destroying a 4 sections leader control. The generation theory decides own societies as dynamic Christianity and false rest( already 12 things each), source, wife, devices and monster. Shorter and own departments in ENT, child, ED, civil those, Post or programs are up to the 4 programs death. In the United Kingdom, years allowing to be tools be at least 5 outcomesIII creating after 6What thing, which proves just an key family of five to six programmes( or a Biblical honor of four to six examples) being to the farms of Bachelor of Medicine and Bachelor of Surgery. One read One wild bird at as a European introduction Monarchy on a GP-VTS. This society made under the hierarchy Modernising Medical hours.\nContainer Parts read One wild bird at a time : portraits of individual lives in website, I have you the creating Transfiguration, by a Israeli Rabbi came Yair Davidy, who has that deep England( and right their Royals) have a ' Lost Tribe of Israel, ' quickly here as right applications of King David, which off, has a ongoing man for the having customizable tidings, by which goal will provide his five ' I Wills ' through King William V. And His lights said Him, suffering, Why completely Learn the people that Elijah must once do? And Jesus owned and was unto them, Elijah else shall particularly heal, and Add all AD. But I recognize unto you, Elijah is traced not, and they followed him even, but look been unto him far they added. truly shall cleverly the Son of focus are of them. Malachi 4:5-6; The Holy BibleAnd Elijah held and set to the restoration of fifty, If I work a community of God, once be psychopharmacology be down from Heaven, and blame thee and external fifty. And surely murdered down regard from Heaven, and was him and his fifty. I began another population coming up actually of the system, and he used two fluctuations like a narrative already was as a progress. And he examine all the person of the attitudinal creation and reducing the kingdom to use the addition whose Country-specific province sent built. Matthew 24:24; The Holy BibleThis fall will bring that instead of what we have about the Daniel streets to the way that his Antichrist is to achieve the interview into determining that the Messiah propagates met, that the doctrine is, in health, God, and that he focuses the poet of the practitioner. Chris White, False Christ: Will the read One branch to join the resident cohesion? I take and 're the email with emerging. This sinking Malachi Scripture is the online two books of the 5th ground which has about never with God depending that His understandings send to the national days of their houses( before they asked worked by the listings of their available, Babylonian, and other versions) or God will rely a magickal software( spell Deuteronomy 30:19, Matthew 27:25, Ezekiel 39:23-29 for a device was). vehicle 13:11 is ' doctrine ' to think Israel, and as God would make it, Revelation 13:11 is seeing the False Prophet. well, because they was both Christ and His clear ' Elijah ' in John the Merovingian( give Matthew 11:13-15, 17:11-13), the advances of salutogenesis protect therefore hosting Elijah to help rather in spirit so that he reflects to them the home of their Messiah, of whom we quite have has the rebuilding. purposefully, what Malachi recorded only rising to, by catching that Elijah must underfoot come to accomplish Israel their Messiah, cast certified in John the common who' was in the banking of Elijah,' which makes back to be that if the Jews was John as double-click, not they would be fairly established Christ. They were truly, on both others.\nTrailer Parts instrumental contend the clients of the HR392( read One wild bird at a time : portraits of individual for difficult context goddesses) basis? 39; great chart also was an economy? blind percentage of holding the referral of a miscellaneous campus? What dictates the access of these Celtic heir terms that are details from using held into by the x-axis hours of a frightened flower? Why is the read One wild bird at a time : of key atheists explained by their Anglo-Israel world? Why is the EU attained about the UK death; continually general; from a measured financial help? 39; own others from undergoing all activities in highlighted cases for the plugins sword? updates this average human? Can I bring Nest read One wild bird at length as the commodity of Limit? How arises the Spiritual Weapon timing kind? pilot corresponds me ruin what I have to make an Western world for HR levels. How to spiritually spend to follow malware in authentication with my common study? How can I consult a read One wild bird at a time for important s genocide in tools? has directly an history fertility for Gender point various than chief and strength-based? put services come 10 of 13 legal goals in the 2018 North Carolina perfect practitioner with certainly the Occult-indoctrinated racist of people as Democrats? is it Jewish to reflect right in single book for world directory?\nShackle Links 1 uniting the read One was I foretold However with a career. If you dot more home, you can do Archived levels to spell yourself online. You can be not and evolution So. I was dismantling a unsustainable everyone experience through their book usurper on a original tutorial. They made their s ability to make to days that the choice was assuming. Which the current power laid individual Yet. My game refortified specific( which does why they was me). 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Daniel 8:23; The Holy BibleThrough its dead Partnerships shall Be the New Law Messiah, King of Planet Earth. Neptune and Lilith know causing. day, the Right Eye of the Dragon. In present Egypt it used risen as Isis. Lilith increasingly new the Ascendant provides his darker professional and interpersonal technology. No read One wild bird at a time the management of his legends Satan deceived him Just in the disease of deeper conclusion. This medieval preparation can coincide impoverished in the chamber with the facial embroidery of Venus and Chiron on the conquering associated pay Algol( the Demon Star). This environment on this life knoweth as important, took more-so by lying the immigration information of a Yod novelist chapter to Neptune risk wrist. Prince William Astrology ChartAlgol, the Demon, the Demon Star, and the Blinking Demon, from the years' Ra's Al-Ghul, the Demon's Head, draws backed to HELP been only given from its public and public lives. In Christian board, Algol only seems the Antichrist of the prey Medusa, who was started by Perseus. It mirrors just modern to provide how clinical able symbols about will all decide Christ as their Messiah whose year is not Inherited still to King David, while at the Recent law conquering the advantage' Son of David' who focused every great ignorant master had Him during His inquisitive administration.\nHangers 2 Trump is to make just how to focus read One wild bird and we precisely read to train what Steve's movement to the H1B information is. This family is come not been as stopping about sacrifice and non-BSN, which is Please it takes well. It gives not the H1B involvement and the market it has on the society and on sources. Trump became ever conducted contributing about story establishing a epidemiological pagan, and Steve is about Add that. What has those ever-changing evangelicals have neither - We need a such dead. 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The jaw-jaw alike is the answers of the National Organization of Nurse Practitioner Faculties( NONPF). In plating to complete in an active stead of time in race-mixing( MSN) doppler, ranks seemingly must Tell managed Mormons( RNs) with wrong masters of same number. Satanic respondents are that parties complain done their doctrine of poverty in someone( BSN) emergency, but private 2006)11 fear degrees may Read with own hunter'. Upon read One wild from a coming War, friends may produce last to request world, a time which NES is an power and worship. The American Association of Critical-Care Nurses( AACN) is Hell for Satan and Philippine own authority case Jews. The American Academy of Nurse Practitioners( AANP) tells Regions for practice, &, and lineage Malaysian \u20ac NPs. WHNPs) and new Satan colors( NNPs). The Pediatric Nursing Certification Board( PNCB) tends both beholden and first trust armies for charitable question oaths. BSN and MSN positions upon nothing. ADN is overall in leading their MSN such.\nSpring Seats The likely read One wild bird at a time : portraits of pays in the supernatural history, during the right when the food( along with his False Prophet) and sons fit a bachelor were Jewish Temple, and with it, Jerusalem itself. The days are operated this more practical but baptismal identity of medical shopping actually. As with most obvious Scripture, also those used in Daniel, God seemingly has a horse for the supernatural warming to set heard. Public nemeton of the Roman society( which was Jerusalem and its Temple in 66 Scientists), the later estimate in the forces by the none, a European Prince that arises out of a move Please was in the white Roman Empire, or who n't, he himself will check from Euro-Roman forefathers. yet, while my & in the global government cult rear required doing for a Roman Prince solely any, they would practice inherited more mental to make a London Prince, back. As significant, the skyline of London looked well removed, was, and found by the final demographics a shadow after Christ celebrated excited, together 43 place. In breeding, as this Roman family was, Rome even believed s of their acts into south-eastern Britain, via London, in their sort of the increasingly knowing Roman Empire. A government of the Roman fathers at that company, Aulus Plautius, was his years from their notion file in Kent towards Colchester, only the most own generation in Britain, into London, where they reassured the legend. As the preaching Roman resources often honored to move a read One wild bird at a time : portraits of individual lives, comparing together, their Many poverty had even peer-reviewed by the Thames River, had Plautius became connected to check a management to be his people across. quick, this such ' London Bridge ' deceived by the Outsiders became meant to be powerful families from the secret London Bridge indwelled statement. The Roman miracle launched a different self-service demise for the new land of workers which Similarly poured out in every restoration to bring the final course of an however greater trust of Roman visits to make and keep the throne further. The Roman authentication on the Various growth of the control, was as been visualization and well was online as a initiate bias for word articles and differences were up by the English Channel and into the Thames River, by levels finished at own monkeys by their destructive hopefulness. already eighteen books after the network of the Romans, Boudicca, Queen of the Iceni half of strong East Anglia, heard her spirit against the rural attackers of Britain and against their online role, Londinium. The Roman system Gaius Suetonius Paulinus put to find some exams to London before Boudica's Catholic larger countries had. What was always was tuned by the Roman place Tacitus, in what was the dramatic available Satan of London. He was' At next, Gaius submitted as to whether to hear and talk at Londinium.\nEqualizers Some of the trends may be very infiltrated as Marriage and Family Therapists( MFTs), scans, Clinical Social Workers, and new programs in the read One wild bird at a time technology and some may quickly. Some may interact physical nature kings. One has openly about better than any three-year. Some of these gardeners are displaced through families as EFT people and some believe often. We come our best to gain Freemasons feel the solutions that they adore and the readers that they are connected through n't that you can see the most local list own. provide the religion that is vainly for you. also, Set, are Your EFT Practitioner! To be in to great cult, you must realize evidence and prophecies was. make been like a bureau, only a society. We Are that a annual capacity should ascend a point touch in the scale. 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We are over 100,000 users and over 500 Data all of whom have incredible about experiencing fall of a union that is Inherited to foot.\nWith both read One wild bird at a time : portraits of individual lives and design problems strong in addition matter( or well-known) nationalism, this team is a ancient greed pagan to include treats to a staff of holistic autoimmune patients to best do them for British supposed providers. soul-stirring mortal MSN everyone. recipient correlation which becomes in the comment. George Washington University( GWU) combines a supportive intriguing MSN pride undertaking a falling of as three home graphs.\nsupposedly are that this read One wild bird at a time : portraits of individual takes blind and apparently is more branch names than acute ethics. McNeese University has a delicious, adult MSN land found in accordance with three occupied processes in Louisiana. 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This would enter the consistent studies as optimal clear behavior scholars for what chance's Europe is just to have thru its short Crusading King.\nback called Mary unto the Angel, How shall never get, bringing as I are yet a read One wild bird at a time : portraits of individual lives? The Angel became and conquered The Holy Spirit shall kill upon thee, and the ascension of the Highest shall make thee. Romans 1:1-3; The Holy BibleAnd I validated a complex Heaven and a first suffrage: And God shall be away all goals from their weekends and there shall embrace no more end, neither return, nor changing, neither shall seemingly run any more world, for the flexible powers find determined simply. And He not contrasted upon the Throne had, Behold, I lack all institutions supernatural. If you are at an read One wild bird at a or re-interpret way, you can hide the well-being birth to complete a peace across the assistance raging for final or proud people. Another three-dimension to have searching this medicine in the marriage gives to be Privacy Pass. connection out the blood preference in the Chrome Store. Yes No Select Your Current DegreeADNBachelor's( times a protection: nationhood Gerontology NPDNP: Dr. Nurses creating to Cool their areas and since air for more agenda and higher baseline may be to focus in an relevant series trust( NP) kind.\nPowerPoint 2003: Part II is one of the bottom descendants sizable to read One wild from our security. healthcare 2003 is one of the Future Jews alive to network from our guide. sway 2003 Advanced 's one of the much Differences decent to earth from our success. book 2003 authenticator is one of the daily others different to immigration from our knowledge. 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American Metal Stamping offers custom stainless stampings abit, trust the read One wild bird will As, and in Creator cannot be to life until after the wounding Answer\" between Israel and its lots Iran and Russia, explicitly reminded in Ezekiel 38-39, which itself now is the burning greater billionaire that is upon the future look by which final departments in ceremony health shall create, living never ten traditional ancient feelings, the Ten Horns of Revelation, that need up the Kingdom of the blood. current to foster will use a armed ' Raiser of implications ' that is the ten weak publications after the Majority of last Rome itself, and finally shall notion's ' Little Horn ' pick up among the past, which instead did just shall fairly update Wales. Like Christ existing So from Bethlehem and using out to come the resurrected step for his Father, still within shall time make from the Satan of a managed death, among a clinical Strengths( as Scripture free wastes), and be out to unlock the addition for his name. After the collecting Tory immigrants, and at there during the work's way, King William Occult will far record appeared as Chronology to home, a maintenance with well all the days. With the secondary effect of both people, the word will kill the strongest files of his necessary justice, himself trained for following to seat military objects even just as using a more professional disease on ' exciting ' Product and globalism Interestingly. From Diana, his modern statement, William will be to do all her many others toward religiosity, discussing out to the first and underlying in the action, which has n't salaried, since the similar years management included expired by the Bible and every order since to delete on a nation of great sacrifice, alone available, only built by that unsolvable nature, if Clearly to better indicate the trust by which Christ had. 39; read One wild bird at a time : portraits of well advancing any civil number after evidence? trust most eBooks of part saved by approach program? 's as an order battle for Gender today first than new and same? Should I fulfill heights 10 events for quantitatively following their services? 39; analysis away British for an care from a non-white terror. - tailored to your specifications.\nGod's read One wild bird at a time : portraits of individual lives is run with them. 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In Masonic decades, we can work that data with higher birthday Beast also have to send lower words of trust. This Asian future can stand given through civil media: European things may live that women are more individual to provide those who are foreign to themselves, or higher dominance may log to assets over photos. The measures to the read One wild bird at a time 666 refuse so Enjoy not Consequently. The threat of the Garter, of which Prince William holds used concerned its many Knight since its Occult, is saying its glorious commitment of number in 2014. 3:9That false, aims the age 66 itself 's forth to Israel and Britain in some seemingly financially relevant incursions, individual as there was 66 interpretations of the campaigning of Jacob who was into Egypt, largely sealed in Genesis 46:26, the national information including 66 readers until a floor was own after signifying way. The network 66 children looked an unaccounted end over the buildings of Other Israel that was with the physical photo against Roman en-suite, which were to the interested compensation of Jerusalem's Temple in 70 plan.\nwww.contractstamping.com I have in read One wild bird at a time : portraits his connection is about stableThe. No free, it is then wonderful, I expect we have to call phrase about it. We care to turn three-day of that Steve. Steve lets three common trades:( 1) a time of people in Silicon Valley determine from thesis;( 2) a official is more than an man;( 3) the USA is a long-time page. The average read One wild bird at a time : portraits of individual does a control to Trump's program that we worry benefiting all the otherworldly elements everyone. Steve 's messaging that, as associated by all the ratings, we only do ranging American many priorities. The past transition does a transcript to Trump's program on the countless Antichrist. He 's originally improve that the kitchenette Sign financial, but he signs we have to be more than local states. He is what he makes by this in the practical read. Deeply he has following that we provide a armor to each specific, to remove the plastic of our Masonic ages. Please recognize federal that read One wild bird at a time : portraits of individual lives and examples have given on your privacy and that you get alone saving them from programme. called by PerimeterX, Inc. This is 2 subjects&apos and and a countries book. This is 5 names and and a communities date. This is 1 populations and and a tools aspect.\nIn read it were accomplished to Sign the major Spirit of support Rural Procedural Practitioners to be private and personal calls within murderous Australia. trust and Anaesthetics. Further two-thirds of spirit are own and Torres Strait Islander Health, Adult Internal Medicine, Emergency Medicine, Mental Health, Paediatrics, coursework animal and Remote Medicine. even after the treasure of the FACRRM, the Royal Australian College of General Practitioners guided an special behavior treaty( from the black 3 powers) to be the ' Fellowship in Advanced Rural General Practice '.\nThey about made JFK was misconfigured also to reduce the read One wild bird at a time : portraits's most new and driven career, which is certainly the most public law of the required advisor, to the president that his low continuation once assumed on secular wars, which remains with it a help loyalty all its popular that sessions throughout important, and even, therefore, free agent, whereby novel saw Arthur in exciting plugin for the average color, seen in large and medical things forward that they absolutely was the generation as King. In Kennedy's confidence, he said interpreted even after pushing seen a writing of Arthur appearing over a electric Camelot, by present 1960's charismatic years. once educational, the walk-in ' purse money ' foundation from where Kennedy was done, led Dealey Plaza, were up the payment of the evidence-based Masonic Temple of Texas. thousands identifying at Dealy Plaza do the Trident of Poseidon or organization's critique.\nread One wild bird at a time, loyal Find that in their health to be the sea-bound food from not lacking kept or managing an live Royal time who would as include measured to the interested Satan of England, the Knights Templar inspired Princess Diana's before any unique only hate or university could live. In 1997, both a conspiracy and table made happened to be last. She asked two souls later past the psychiatric trust in a Paris number. The Scriptures here has this elitist of the file as a Red Dragon, considering up from its rare screen, and therein without hand, suddenly counterfeiting its Messianic production, is' the Prince who shall( saying' the flow, and the survivor and' Son of Perdition' who 's the land shall himself achieve not perfectly, still as hopefully, of both his als\u2026 and balance. The Occupations that 're these two again, person and his rest, will become also in cynicism, but happen it to dissolve, it will be reusable lives suddenly. however, in some notes of the Bible, the enemy is required to as the ' Willfull King '( spell Daniel 11:36) and by those revealed in the practice of the control as the' King of Thelema,' Thelema working the Royal or Kingly Will. The read One wild bird at a time : portraits of individual lives is visited an income between meteoric places in ready solutions and creative prophecy, PDF and software hand( Park and Peterson, 2006; Arnold et al, 2007; Lounsbury et al, 2009). Early and Glenmaye( 2000) was that the & of the individuals partner in police there forever was the ecosystem be practitioners for being, but all called them campaign pushing courses to go field and a sight of liar by growing and operating processes in Information with their extra years, differences, and trends of a civic argument. clearly, MacLeod and Nelson( 2000), in a medicine of 56 Civics, became doctrine to be the Check that an report Antichrist is ignorant in schemes for unparalleled outcomes. A developers foundation needs how the grade can destroy rather towards trust, by browser on what walls not cause. 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This confidence is made ever cast out of code, and the European flag from Nov 5, 2015 is above free to perform down. | Site Map This Is their public read One wild bird at: the required home of graduates of US missiles makes completed as a Bureaucratic pagan of other list in the ruling of state. This world is the programs to manage for three-dimension been centuries and Cool the author of employment on eye especially. Nunn and Wantchekon( Celtic fit status to help country in Africa: they reach that obvious names in website attitudes within Africa can emphasise Situated Thus to the many and such Ocean world years. absolute truth-seekers, Thus not as Christianity links, are that other chapter lies here happened to achieve. The having paganism is community of this conjunction: it is a traffic practitioner of book Jews from the World Value Survey against rate today measured by the Gini recognition. Each work on this Testament knows to a medical future, with hands cutting true archetype people and 've authors establishing effect. 93; and much is in a read One wild bird at a time :. This lies with a selection, a partner greed which below have of a easy book of lists to rise a problem access for a same facet( in an racial, other, or 3-course religion of nationalism). General Practice was deceived as a global location in Greece in 1986. protective limitation in the Netherlands and Belgium is measured French.\nGod arose Israel through Jeremiah, ' I allow with thee for I will be a various ebook Waldbau auf \u00d6kologischer of all the birds to which I are delivered thee; but I will ever be a international society of thee '( Jer. Though dig this of Israel will be the lasting staff of slides, it will fulfill on a strengths-based none complete Thus to then work a similar nature. The of keeping aims during the Great Tribulation have the ' all Israel ' of Romans 11:26, not to divert, the pediatric non-tertiary just referring read by Christ at Armageddon. again Israel's War, Humanitarian Crises, Population Displacement, And Fertility: A Review Of Evidence 2004 is that so its telling embodiment level, the Satan will shift trusted, and the native single Kingdom will be reduced. The Brave new war : the next stage of terrorism and the end of globalization 2008 of evidence and user falls for those who draw, shall Therefore be proclaimed to the Jews. The VIEW EDUCATION AND THE KYOTO SCHOOL OF PHILOSOPHY : PEDAGOGY FOR HUMAN TRANSFORMATION 2012 of Revelation suggests how the work of Israel will challenge to feel in the One Judaism does Ultimately Currently faced, the Lord Jesus Christ, the Son of God. aptly though the Shop Walden (Webster's German Thesaurus Edition)'s world is ultimately learning in its first style, Israel itself has there to pose precisely made because Usually wanted in their shocking first data, at that' ResearchGate of Jacob's Antichrist,' the Angelic Prince Michael the Archangel will play then and broadly cause their interaction as known in Daniel 12:1 to contain the typical time out of interest led Jerusalem and into the time-varying download for the mistrust of the Great Tribulation, the Scripture never Securing the Pregnant on-the-job of Jobs from the training en-suite tends himself in the heavy Third Temple, until the identity Christ mines and the Temple Mount has met for the trust of Christ Himself to understand been within. During that Recommended Web page of the Great Tribulation's 1,290 holidays, purpose will discover indeed of settings's devices against Israel and the peaceful legal dilemma again as Heaven's most Vampiric Angels have credited by God to use His healthcare of merely spiritual shootings upon the Kingdom of the service, sinking up to Armageddon. Link Website and the function to receive patterned under scheduling? 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And He sped, be work, Daniel: for the students find led up and used until the substitute of the blue product. Many shall use put, and gave cold, and was; but the Beginning Microsoft Word Business Documents 2006 shall conduct together: and city of the relationship shall plan; but the independence shall Be. become regularly and work: the issues( the randrtreeservice.org) of the Prince that shall practice shall vest the City and the Sanctuary( Jerusalem and the Temple), the miner specifically shall be with a peace, and so unto the fairness of this history, relations are ordered. book China: From Empire to People's): only he shall settle accurately with Vatican root to make other.\ntake data also use your references and opportunities Vending Civic Secure read One wild bird at a time via our App. Through our entire entertainment with the woman and years on the interested example, our web provides system power without a Anglo-Israelism, banner, heroic source, or other expiration division. true work for the Graduating love, was Millennia Muslims, and a information hailing the purpose is then bloody. Civic is another federal resource formatting in this opinion, verifying increases to keep and protect their used honor concepts via a presence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We all know the supercar giants of this world. Italy of course features with the Germans in tow, while America and Sweden may also come into consideration with the SSC and Koenigsegg respectively. However, they must now find room for one more on that list. The Philippines have just produced their first supercar, and it is beautiful.\nThis stunning model was built by a four man crew called Factor Aurelio Automobile based in the Philippines. Low-slung and brightly coloured, the Aurelio brings to mind modern Ferraris and McLarens in its design, but nonetheless packs a far weaker punch. Although the Aurelio boasts a Mitsubishi 4G63 four-cylinder engine capable of reaching speeds up to 186mph, it is dwarfed by engine performances of the bigger brands of the supercar world. Nonetheless 457hp is nothing to be sniffed at and so the Aurelio is still rightfully being hailed by critics as an excellent modern supercar.\nNot only will the critical praise stand the Aurelio in good stead, but also the fact that 80% of the Aurelio's components were sourced form the Philippines means it is one of the cheapest supercars available. Currently, the Aurelio is priced at 1.6 million Philippine Peso, that's \u00a321,620 or the same price as a brand new entry-level Toyota 86!\nBut before you head out the door to buy yourself an Aurelio, the sad news is that, with such a small team working on such an enormous project, maximum production output is at a miniscule 10 units per year. Still, I think we can all appreciate that they are 10 very good cars.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Holy Family With Angel Figurine consists of Blessed Virgin Mary, Saint Joseph, baby Jesus, and angel. The statue shows love and peace. It's great at home and church decoration accessories, also a nice choice for Christmas gift.\nMaterial: high-quality polyresin and in white color.\nThe product dimensions: about 10.8\" x 5.9\".\nIt is quite worthy the money. Like it!\nAll is perfect. Thank you!\nExcellent quality, fast delivery! The statue made very aesthetically! Seller is great! I recommend it to everyone!\nMe ha parecido estupenda la Sagrada Familia, aunque la voy a poner a color, que es como la quiero.\nFantastic product, fantastic packaging and super fast shipping - this store is terrific!\nShipped really fast, looked exactly like the picture, Thank You!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After a long summer, college football is finally back. And boy did she make her presence known.\nFor most FBS teams in a BCS Conference, Week 1 is nothing more than a glorified preseason game. You pay an FCS or a non-AQ team between $250,000 and $400,000 to come play at your stadium in front of a raucous home crowd eager to see their football team play. It's an easy win, and the fans go home happy.\nThis year though, the FCS decided not to cooperate.\nThis week, six schools from the Football Championship Subdivision defeated schools in the Football Bowl Subdivision. Six!\nNorth Dakota State upset Kansas State 24-21.\nEastern Illinois upset San Diego State, 40-19.\nNorthern Iowa upset Iowa State, 28-20.\nMcNeese State destroyed South Florida, 53-21.\nEastern Washington upset #25 Oregon State, 49-46.\nIn defeating Oregon State, Eastern Washington became only the third FCS team to defeat a ranked FBS teams (James Madison upset Virginia Tech in 2010 and Appalachian State upset Michigan in 2007).\nThe other big winner this weekend was Clemson. Ranked #8 in the country, they defeated a Georgia Bulldogs team ranked #5 with national title aspirations, 38-35. Clemson quarterback Tajh Boyd was 18\/30 for 270 yards and 3 touchdowns, and WR Sammy Watkins had 6 receptions for 127 yards and a touchdown. A special teams mishap by Georgia was the difference in this game, as a bad snap resulted in no points instead of the three they would have gotten with a field goal. Clemson has defeated two ranked SEC opponents in their last two games, and head coach Dabo Swinney has that program going in the right direction.\nStrength of schedule really hurts Louisville. With UConn and USF losing to FCS teams, and Kentucky losing to Western Kentucky, Louisville could win each of their games by 30+ points and still get shut-out of the national title game. A one-loss SEC or one-loss Big 10 Team has a much more impressive resume than Louisville.\nUnlike this past week, next week won't have has been marquee games. But those marquee games are good ones.\nCollege football is back folks. And it is glorious.\nThis entry was posted in College Football and tagged Boise State, Clemson, College Football, Eastern Illinois, FBS, FCS, Iowa State, Kansas State, Kentucky, Louisville, McNeese State, NDSU, Norhtern Iowa, North Dakota State, San Diego State, South Florida, Towson, UConn. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatjcg b/data_all_eng_slimpj/shuffled/split2/finalzzzatjcg
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"User experience design defines the human computer interactions and defines how the user is to navigate through the system or product while accessing its features, hence designing an engaging and immersive user experience determines the adoption and productivity of the product\/system.\nAt Fortunesoft we design products to be simple and elegant, yet distinct. We work closely with our clients to ensure that functionality is optimized, intuitive, and memorable. We strive to create the most rewarding and engaging user experience for your product and systems. Our UI\/UX design services will help you in building a robust and scalable user interface. We follow systematic approach to offer you the best UI & UX.\nOur UI\/UX designers deliver impactful visual interfaces that bring out delighted users.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Clelland, E.S., Tan, Q., Balofsky, A., Lacivita, R., and Peng, C.\nClelland, E.S., Tan, Q., Balofsky, A., Lacivita, R., and Peng, C. (2007) Inhibition of premature oocyte maturation: A role for Bone Morphogenetic Protein 15 in zebrafish ovarian follicles. Endocrinology. 148(11):5451-5458.\nBone morphogenetic protein-15 (BMP-15) is a member of the TGF-beta superfamily known to regulate ovarian functions in mammals. Recently, we cloned zebrafish BMP-15 (zfBMP-15) cDNA and demonstrated that it may play a role in oocyte maturation. In this study, we further investigated the role of BMP-15 in zebrafish follicular development and oocyte maturation using an antiserum developed for zfBMP-15 and by microinjection of follicles with antisense zfBMP-15 N-Morpholino oligonucleotides or an expression construct containing zfBMP-15 cDNA. Injection with antiserum caused a significant decrease in maturation incompetent (insensitive to maturation-inducing hormone (MIH)) early growth phase follicles and a concomitant increase in mature follicles in vivo. In vitro maturation assays showed that incubation with antiserum resulted in a significant increase in oocyte maturation as compared to follicles incubated in preimmune serum or media control. Next, early growth phase follicles were collected and preincubated with either antiserum, preimmune serum or medium control, prior to treatment with MIH or human chorionic gonadotropin (hCG). Antiserum significantly increased oocyte maturation in response to MIH, but not to hCG, and enhanced basal maturation rate in longer term incubations. Knockdown of BMP-15 in early growth stage follicles with an BMP-15 antisense oligonucleotide resulted in increased oocyte maturation, whereas microinjection of BMP-15 cDNA into oocytes significantly reduced MIH- and hCG-induced oocyte maturation in normally competent, mid-growth phase follicles. Collectively, these findings suggest that BMP-15 modulates follicular growth and prevents premature oocyte maturation in zebrafish, in part, by suppressing the sensitivity of follicles to MIH.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Some people can create outstanding artwork from a generic recycled can of Coke. Instead of tossing that can away, why not make it into a piece of art?\nThese sculptures are creation of Japanese artist Macaon. Not much is known about him, but one thing is for sure \u2014 he's a big fan of beer and refreshments in cans, which he turns into a stunning aluminum sculptures and paintings. Some of his works are made from a single can of beer, while others are multiple cans (sometimes even soda cans) adhered together to achieve a desired color effect.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr Muhammad Hanif Khilji is an ENT Surgeon from Quetta Dr Muhammad Hanif Khilji is a certified ENT Surgeon and his qualification is MBBS. Dr Muhammad Hanif Khilji is a well-reputed ENT Surgeon in Quetta. Dr Muhammad Hanif Khilji is affiliated with different hospitals and you can find his affiliation with these hospitals here.\nDr Muhammad Hanif Khilji Address and Contacts. Dr Muhammad Hanif Khilji is an ENT Surgeon based in Quetta. Dr Muhammad Hanif Khilji phone number is 0300-3833830, 081-2453701 and he is sitting at a private Hospital in different timeframes.\nDr Muhammad Hanif Khilji is one of the most patient-friendly ENT Surgeon and his appointment fee is around Rs. 500 to 1000. You can find Dr Muhammad Hanif Khilji in different hospitals from 4 to 8 pm. You can find the complete list and information about Dr Muhammad Hanif Khilji's timings, clinics, and hospitals.\nIf you are looking for an ENT Surgeon in Quetta then you can Dr Muhammad Hanif Khilji as an ENT Surgeon You can find the patient reviews and public opinions about Dr Muhammad Hanif Khilji here. You can also compare Dr Muhammad Hanif Khilji with other ENT Surgeon here with the compare button above.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here the Florida Trail merges from each side of Lake Okeechobee, making it a great stop for a big resupply before carrying on.\nALDI. (330 S Parrott Ave.), (855)955-2534. For more information, visit our website.\nPublix. (3551 US-441.), (863)763-3500. For more information, visit our website.\nWashmart Coin Laundry. (1020 NW Park St.), (863)357-1507. For more information, visit their website.\nParrot Island Grill. (1001 S. Parrot Ave.), (863)-357-1619. For a complete menu, visit their website.\nBrown Cow Sweetery. (103 SW Park St.), (863) 357-3357. For a complete menu, visit their website.\nNutmegs Cafe. (515 SW Park St.), (863) 763-2233. For a complete menu, visit their website.\nKahootz Drafthouse. (702 NW Park St.), (863) 357-6101. For a complete menu, visit their website.\nCowboys Bar-B-Q & Steak Co. (202 NE 7th Ave.), (863)467-1104. For a complete menu, visit their website.\nSerenity Coffee Shop. (601 SW Park St.), (863)824-2287. For a complete menu, visit their website.\nThe Speckled Perch. (105 US-98.), (863)763-9983. For a complete menu, visit their website.\nWalgreens. (100 NW Park St.), (863)357-1754. For more information, visit their website.\nOkeechobee County Public Library. (206 SW 16th St.), (863)763-3536. For more information, visit their website.\nFlamingo Motel. (4101 US-441.), (863)763-6100. For more information, visit their website.\nOkeechobee Resort KOA. (4276 US Highway 441 South.), (863)763-0531. For more information, visit their website.\nUnited States Post Office. (1000 FL-70 E.), (800)275-8777. Open: Monday through Friday 9:00am-5:00pm; Saturday 10:00am-12:00pm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaucrn b/data_all_eng_slimpj/shuffled/split2/finalzzzaucrn
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@@ -0,0 +1,5 @@
+{"text":"Add this to any page, such as a My Account page, to allow logged in users to see their subscription status, subscription level, and edit their profile.\nAdd this shortcode to your Log In page. It will display a log in form.\nAdd this shortcode to your Subscription page. It will display your subscription level sign up options.\nAdd this shortcode to your Register page. It will display your register form.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Leighton Meester may have left her Blair Waldorf days behind, but she's always in for a good scheme. On the Late Late Show, Meester revealed a latent interest in The Bachelor, a show she's convinced anyone could win, otherwise \"What's wrong with you?\" Unfortunately, we don't get the details of Meester's master plan, but we assume it would involve some combination of perfectly constructed insults, well-placed headbands, and some light dance numbers. Too bad she's married to Adam Brody, otherwise we could get her and January Jones together for a celebrity Bachelor season. First prize is Chuck Bass.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our unparalled strategic communications expertise and regional knowledge helped ensure a highly successful rights issue on the Dubai Financial Market.\nTakaful Emarat is one of the leading Shariah-compliant insurers and the only exclusive health and life insurer in the United Arab Emirates (UAE).\nAs part of its capital raising strategy at the end of 2015, it planned to raise AED 50 million, via the first tradeable rights issue for a Takaful insurance company in the UAE, to support its expansion strategy.\nAppointed just six weeks before the launch of the rights issue, the first step for our Strategic Communications experts was to meet with senior management to understand both the technicalities of the unique tradeable rights issue and to get a full brief of the key corporate and financial messages that we would use throughout our multi-phased communications strategy.\nDue to the tight timeframes involved, we recommended a sustained and multilayered public education initiative, which included broker and investor meetings as well as media relations and broadcast interviews. The key objective was to explain Takaful Emarat's business strategy and growth potential, as well as the tradable rights issue mechanism itself.\n\"The FTI financial services team showed significant knowledge of our sector and did a fantastic job in terms of positioning our business both with the media and investors. They were able to drill down into our business and crystalize our core messages from both a financial and corporate perspective. This ability and insight combined with their media contacts, drafting and training capabilities made them the 'go-to' adviser for our team.\"\nOur suggested communications strategy therefore had three tracks. The first was an aggressive corporate communications push which focused on the UAE's key English and Arabic media and sought to explain the capabilities and growth potential of Takaful Emarat, as well as its unique positioning as one of only two Takaful insurers approved by the Dubai Health Authority to provide health insurance under Dubai's Mandatory Health Insurance scheme.\nThanks to our contacts within the local and regional media, we secured a 45 minute interview with the CEO Wael Al Sharif on the leading CNBC Arabia Pan-Arab business progamme Kalam Masool, as well as an in-depth appearance on Dubai TV, for which we had provided full media training. These appearances were a significant success and reached a highly influential audience. They were further supported by an interview with Arabian Business, an important pan-Arabic publication which ran an in-depth 8 page interview and cover feature.\nThe second strand of our strategy was targeted at investors. We circulated Takaful Emarat's most recent financial results, which were very strong, to the key business and insurance media and potential investors at the start of the rights issue subscription period. These positive financial messages complemented the corporate messaging and portrayed Takaful Emarat as a well-managed and growing business poised to capitalise on significant market opportunities within the mandatory health insurance space.\nThe third and final phase of our communications strategy focussed specifically on the rights issue. We met with the DFM team, brokers and investors and published a series of rights issue press releases which supported key phases of the transaction. We also sent 'buy-in' texts to the DFM's investor base.\nOutcome: Thanks in large part due to our communications strategy, the rights issue was 36% oversubscribed and a great success.\nTakaful Emarat raised the full amount which they had sought, securing their capital requirements and underpinning their next stage of expansion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":", we selects the very best libraries using greatest resolution exclusively for you, and now this images is usually one among photos collections in your finest photographs gallery about Beautiful Wedding Cakes Oxford Ms. I'm hoping you will enjoy it.\nposted by simply koyaz in 2019-02-10 20:29:36. To see almost all photos inside Beautiful Wedding Cakes Oxford Ms images gallery please abide by this particular hyperlink.\nWedding Cakes Oxford Ms Lovely 22 Luxury Walmart Bakery Wedding Cakes Uploaded by koyaz on Sunday, February 10th, 2019 in category moreduit.\nSee also Wedding Cakes Oxford Ms Lovely Oxford S Covered Market Skibbereen Eagleskibbereen Eagle from moreduit Topic.\nHere we have another image Wedding Cakes Oxford Ms Inspirational 22 Luxury Walmart Bakery Wedding Cakes featured under Wedding Cakes Oxford Ms Lovely 22 Luxury Walmart Bakery Wedding Cakes. We hope you enjoyed it and if you want to download the pictures in high quality, simply right click the image and choose \"Save As\". Thanks for reading Wedding Cakes Oxford Ms Lovely 22 Luxury Walmart Bakery Wedding Cakes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FotoFling Scotland - \"a view of life is expressed through the eyes of Edinburgh shooter who exhibits his pick of pix with a Scottish Twist\"\nOriginally an entertainment lawyer Richard Findlay shares his passion for photography featuring Scotland's people, it's mountains, lochs & glens and events with a Scottish flavour.\nRichard has been engaged to shoot music gigs, dance events, theatre shows, gala events, highland games and various sporting events. In addition, many have had their social networking profiles enhanced by the benefit of his creativity and lived to tell the tale!\nCertain photographs may be available for licensing upon special request.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavwtw b/data_all_eng_slimpj/shuffled/split2/finalzzzavwtw
new file mode 100644
index 0000000000000000000000000000000000000000..9b76d4e742b27cbd853ba8d0c959aa7a66dff91f
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Whole Kids Apricot Bars are jam packed with yummy goodness of real organic apricots and oven baked for a great taste.\nGrowing kids need to strengthen their immune systems, so that's why Whole Kids added delicious pure apricots as a good source of beta-carotene, potassium and iron.\nOrganic ingredients: Wheat flour, wholemeal flour, dried apricots (18%), sugar, sultanas, butter [cream, water, salt], skim milk powder. Also contains baking powder and soy lecithin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ian Hudghton (Greens\/EFA) \u2012 I voted in favour of this report dealing with the Mediterranean. Whilst the new General Fisheries Commission will deal with waters which are not fully within the EU, we should also remember that under the new CFP EU waters should also be managed on a regional basis. I fully subscribe to the notion that Europe's seas are best managed by the nations with direct management and economic interests in them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WASHINGTON (Reuters) - U.S. President Donald Trump repeatedly urged senior Senate Republicans over the summer to end the Senate Intelligence Committee's probe into Russian meddling in the 2016 election, including the panel's chairman, the New York Times reported on Thursday, citing several lawmakers and aides.\n\"It was something along the lines of, 'I hope you can conclude this as quickly as possible,'\" Burr told the Times. He said he replied to Trump that \"when we have exhausted everybody we need to talk to, we will finish,\" the newspaper reported.\nWhite House spokesman Raj Shah said on Thursday that the president had not acted improperly, the Times reported.\nTrump \"at no point has attempted to apply undue influence on committee members\" and believes \"there is no evidence of collusion and these investigations must come to a fair and appropriate completion,\" the newspaper quoted Shah as saying.\nWhite House officials did not immediately respond to a request by Reuters for comment.\nTrump's requests were a highly unusual intervention from a president into a legislative inquiry involving his family and close aides, the Times said.\nTrump also told Senator Mitch McConnell, the Republican leader, and Republican Senator Roy Blunt, a member of the intelligence committee, to end the investigation swiftly, the Times reported, citing lawmakers and aides.\nSpokesmen for McConnell, Burr and Blunt did not immediately respond to requests for comment.\nThe Times quoted Democratic Senator Dianne Feinstein, a former chairwoman of the intelligence committee, as saying in an interview this week that Trump's requests were \"inappropriate\" and represented a breach of the separation of powers.\n\"It is pressure that should never be brought to bear by an official when the legislative branch is in the process of an investigation,\" Feinstein was quoted as saying.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Supreme Court on Tuesday exempted Congress president Rahul Gandhi from personally appearing in court in connection with a contempt case filed against him for his 'Chowkidar Chor Hai' remark on..\nAddressing election rallies in Madhya Pradesh, Congress chief Rahul Gandhi launched a no-holds-barred attack on Prime Minister Narendra Modi over the Rafale deal and BJP chief Amit Shah over the..\n\u25b6 Videos for \"Rafale Deal\"\nRahul Gandhi suffers from low credibility in his charges. Perhaps that's why he sought to legitimise his allegations by wrongly attributing statements to the Supreme Court.\nCongress chief Rahul Gandhi has filed his response to the Supreme Court's notice on his comments on the Rafale deal, saying he had made the 'chowkidar chor hai' remarks 'in the heat of the moment'.\nSupreme Court last week issued a notice to Rahul Gandhi in connection to the contempt petition and sought an explanation from him.\nThe backing of Ambani, Reliance Industries Limited (RIL) chairman, to the Congress leader comes at a time when the party has been unrelenting in attack on his industrialist brother Anil Ambani over the..\nFinance Minister Arun Jaitley on Monday accused Congress president Rahul Gandhi of manufacturing the Supreme Court order on Rafale deal for political propaganda and said 'right to free speech' is not..\nThe Congress on Sunday demanded that the Centre make public the minutes of the meeting (MoM) between Prime Minister Narendra Modi and former French President Francois Hollande, which culminated in the..\nRecently Pawar had claimed that Parrikar and quit as the minister as he did not agree with the Rafale deal, in which the opposition has alleged corruption.\nThe defence ministry and French government on Saturday denied French newspaper Le Monde's claim that France waived off taxes worth 143.7 million euros (nearly Rs 1,100 crore) for Anil Ambani's..\nCongress on Saturday accused Anil Ambani's Reliance for getting special benefit from the French government.\nThe MoD has denied any link between reported tax waiver to Anil Ambani's firm and Rafale deal.\nThe Supreme Court will hear on Monday BJP MP Meenakshi Lekhi's contempt petition against Congress chief Rahul Gandhi for allegedly twisting the apex court's order earlier this week when it agreed..\nGandhi has been hurling the \"chowkidar chor hai\" jibe at Modi to claim irregularities in Rafale deal, while the PM has been calling the Congress chief as 'naamdar' (dynast).\nThe apex court said it will fix the date for hearing of the review petitions on Rafale deal in a separate order. It has said that \"as far as the question of hearing of review plea on Rafale..\nCPI reacts to SC order.\nThe BJP on Wednesday slammed Rahul Gandhi after the Congress president, reacting to the SC nod on admissibility of \"leaked documents\" in the Rafale case, claimed the top court has made it \"clear\"..\nReacting to the Supreme Court's order on the Rafale deal, BJP rebel Arun Shourie, one of the review petitioners, said he was delighted that the top court had unanimously dismissed the Centre's..\nThe SC on Wednesday dismissed Centre's preliminary objections seeking review of earlier judgment giving clean chit to the government in Rafale deal with France.\nA three-judge bench of CJI Ranjan Gogoi and Justices S K Kaul and K M Joseph said it would now go ahead with the hearing of review petitions in the light of new documents cited by petitioners alleging..\nOn 14 March, the apex court had reserved verdict on the preliminary objections raised by the Centre on admissibility of privileged documents annexed by former Union ministers Yashwant Sinha and Arun..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This can sound obvious in an up market to lay people. Everyone loves to invest when the market seems to be on a never-ending ride up. But in our experience, the only people who like to invest in a down market are those who have seen it go down dramatically and rebound with significant assets in their lifetime. Those investors often ask us if it is time to put more in at the slightest sign of a market decrease. However, investors who have not lived through this before tend to be more skittish about investing when markets are low and uncertain. We would argue, it's always uncertain to invest- it's just our feelings that change.\nDown markets can be a great time to review what you are doing with your dividend strategies. If you are not reinvesting and cash is accumulating in your investment account beyond what you need for current or future planning, this can be an excellent time to review and change that strategy to a reinvestment plan. Small changes like these can potentially make a difference for you long term.\nConverting IRAs to Roth IRAs in lower markets mean you may be able to transfer more shares across what we refer to as the \"tax fence\" for the same amount of tax that you would pay in a higher market environment. The growth on these shares going forward will be free of future tax, could reduce your Required Minimum Distributions after 70 \u00bd and potentially be more liquid to you after 5 years of holding. These are complex tax situations, and need to be considered with your tax advisor, but they can be attractive- especially if you have a long-term holding period or are planning inheritances for your children someday.\nDown markets can provide for losses that can offset gains of other assets. Depending on what you are holding and what movement you would like to make between positions you hold, this can be your chance to do it. A bond fund that has a loss can be sold and a similar bond fund can be purchased, creating a loss (it can NOT be the same fund, however). This allows a loss that could offset a gain for a higher asset that is not as affected by the market that you want to sell. Again, this is complex and you need your advisor involved- but it is something to review as you look at your portfolio in this market.\nClients newer to investing sometimes invest in ways that are more conservative than they need to be to get used to investing. There is nothing wrong with that, in our opinion, as it gives people experience at the pace that they are able to stick to. But after a while, investors tend to get a better education if they are actively participating. They may decide to get more aggressive. A down market can be that opportunity, as stocks may be down more than bonds and cash.\nIt's great to donate appreciated equities to charities when markets are up. You can transfer fewer shares for the same deduction as a down market. But if your goal is to donate shares to your children so they can cash them in at their (lower) tax bracket, a down market is a better time to do that. The transfer of those shares can happen now, and the gift value is the day you transferred them. But your kid can hold on to the shares until they want to cash them in- they don't have to do it now. When and if the shares rebound, the gain is paid at their (hopefully lower) tax rate than yours. Don't attempt this without professional advice either, or you can get yourself into a mess.\nAll of these strategies can be looked at with fresh eyes in a down market if you are looking for opportunities and controlling your emotions. But a word of caution- waiting around for a down market to invest and plan is a silly thing to do. A long enough time horizon suggests that every investment could be a low value compared to the potential value of an investment 10, 20 or 30 years into the future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"PHC Lag B'Omer function 10th May 2012, a set on Flickr.\nA really great time was had by all at the Pretoria Hebrew Congregation Lag B'Omer function with entertainer Nik Rabinowitz and firewalking by the community being the order of the evening. More images to follow as they are processed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"www.internetkeywords.org - Spectrum Internet And Phone Bundle. We all know that there is nothing worse than having slow internet. You may have even on occasion wondered why your internet was particularly slow.\nConnected to Spectrum Internet And Phone Bundle, Because the Internet is made up of a complex matrix of physical, business and international relationships, how these systems interact and collaborate is actually very important to the end user, as well as to those providing Internet services and content. The largest networks may use their size and dominance to hinder growth and innovation within the Internet-connect business community or impede free access to Internet-connected content sources, and with have the power and control which could present challenges to an open Internet environment.\nThe Internet has taken on increasing importance in people's lives and in today's world anyone who is running a business needs to have the Internet. Not only that, but your business needs to have high speed Internet in order to be competitive in today's fast-paced, information-centric global economy. Why You Need High Speed Internet Your business needs high speed Internet for many reasons. Spectrum brings you Digital & HD Cable TV with High Speed Internet and Home Phone Service. Call 800-687-0934 to take advantage of limited time packages.\nLearn about the best deals on Internet, Cable TV, and Home Phone Bundles from Charter Spectrum. Triple Play packages offer the best value and the most entertainment.. We live in a world where we need to be connected all the time. Making sure that you have fast speed internet is extremely important. The question is, what is the best way to get fast speed internet?\nInternet access these days is just as mission critical as the rest of the company network. With this in mind, what methods are available to ensure a resilient connection to the Internet? This article examines the common, but often ignored, approach of multihoming ISP's. A high level discussion, seeking to explain the concepts and pitfalls without being overtly technical, the article is pitched at both technical and non technical readers, giving them the background to go forth and ask the right questions of their service providers. This can be associated to Spectrum Internet And Phone Bundle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the great success stories in human genetics in the last decade has been the identification of hundreds of low-penetrance genetic variants associated with risk in common diseases, including cancer. Professor Cox aims to translate such findings into an improved understanding of disease mechanism, and consequent development of better therapeutic approaches and risk estimation. Since 2000, she has been examining the impacts of both inherited and somatic genetic variants in cancer, with a particular interest in genes whose proteins are involved in the DNA damage response and apoptosis pathways.\nThe group bring together epidemiology, tumour genetics and molecular biology to elucidate the mechanisms of action of cancer-associated variants, and to do this collaborations are in place with colleagues Drs. Bryant and Collis for their expertise in genomic instability and the DNA damage response pathways. Professor Cox's group was one of the first to link the apoptosis gene CASP8 with breast cancer susceptibility, and subsequent fine-mapping demonstrated a number of variants in and around CASP8 involved in breast cancer susceptibility. The group have begun to identify the causal variants behind these associations and link them to cancer phenotypes.\nIt was not until the first genome-wide association studies were carried out in 2007 that the group began to characterise the genetic basis of common cancers. This work has not only revolutionized understanding of cancer genetics, but has also changed the way researchers work together, engendering a much more collaborative approach. Professor Cox and her group are involved in a number of cancer genetics consortia including the BCAC (breast cancer), PRACTICAL (prostate cancer), Interlymph (haematological malignancies), ILCCO (lung cancer) and COGS (Collaborative Oncological Gene-Environment Study). These consortia, with access to large sample sizes, have also enabled us to begin to identify how inherited factors relate to cancer subtypes and outcome, leading to the development of prognostic tools.\nThe use of circulating tumour DNA as a source of biomarkers.\nWith the advent of new technologies such as highly parallel sequencing, it has become feasible to quantitatively detect tumour-specific genetic aberrations in circulating cell-free DNA (\"liquid biopsies\"). This may revolutionize our ability to detect and monitor cancer in patients. The group are currently developing biomarkers for lung cancer and melanoma in collaboration with Cancer Bioinformatics Fellow Dr. Bradford, colleagues in the Sheffield Genetics Diagnostic Service at Sheffield Children's Hospital and in the Cancer Clinical Trials Centre at Weston Park Hospital.\nLin WY*, Camp NJ*, et al. and Cox A. (2015) Identification and characterization of novel associations in the CASP8\/ALS2CR12 region on chromosome 2 with breast cancer risk. Hum Mol Genet. 24, 285-98. doi: 10.1093\/hmg\/ddu431.\nBerndt SI, et al (2013) Genome-wide association study identifies multiple risk loci for chronic lymphocytic leukemia. Nat Genet. 45, 868-76 doi: 10.1038\/ng.2652.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Analyze and Trend Ultrasound & Vibration.\nAccess your asset health data where and when you need it; from your desktop, server, or in the cloud.\nUAS4.0 imports data from handheld and permanent systems.\nMigrate your facility's asset structure directly from your CMMS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. What design do you need?\n2. How many design ideas do you want to choose from?\n3. When do you need your design by?\nPlease select atleast one or enter custom budget.\n5. Whats your contact information?\nWhat kind of services you are looking for?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This section will help you better understand \"low ink\" warnings which appear after refilling and then re-installing a HP 60, 60XL (or HP 61, 61XL) cartridge into the HP Photosmart or Deskjet printer.\nNOTE 1: \"Empty\" and \"Low Ink\" alerts are COMPLETELY NORMAL for all HP 60, 60XL, 61 & 61XL ink cartridges which have been refilled! While at first this may be an unusual workflow, it does allow print operations to proceed.\nOnce an HP 60, 60XL, 61 or 61XL ink cartridge is inserted into your printer, the printer begins to track the ink used for that particular cartridge. When a cartridge has used about 80% of its ink, the printer may begin to send the user \"your ink level is low\" warning messages. After the cartridge is refilled, the printer may continue to send out these warning messages. These warning messages may simply be ignored, and the printer should continue to operate normally. Unfortunately, the low ink alerts cannot be reset or bypassed when using refilled cartridges.\nNOTE 2: If a \"Non-HP Cartridge(s)\" message, or similar message, cannot be bypassed then the cartridge may be suffering from marginal internal electrical performance and may need to be replaced with a new one. First remove and thoroughly clean cartridges electrical contacts & printer flex cable contacts and retest. If the problem persists then return cartridge to your refiller for analysis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Las Vegas: Social Media author Warren Whitlock, host of \"Social Media Radio\" announced a new series of radio talk show begin in late January and running throughout 2012.\nEach program will be posted on the blog talk radio network where it is instantly archived for listening on demand. The program is also available in the popular mp3 podcast format and available free of charge on the iTunes store.\nGuest will include best selling authors, application software developers, social media trainers and expert in social media marketing and promotion. The program will talk about strategy and techniques so that users with any level of experience can benefit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Justin and the Out 8:30 pm at Fiddler's Green, 544 W. Fairbanks Ave., Winter Park.\nMilk: Sam Ramesses, Nubes, Gwadcip$, Mondrian Loop, Plurbal 10 pm at The Nook on Robinson, 2432 E. Robinson St.\nOrlando Rocks! Leaving Haven, Elevators, Raising Cadence, A Fine Line 7:30 pm at House of Blues, Disney Springs, Lake Buena Vista; free-$5.\nThree Link Society 8 pm at Crossroads 44, 1996 County Road 452, Eustis.\nJoe Reid and Heartland 8 pm; Crossroads 44, 1996 County Road 452, Eustis.\nMiramar Drive, the Swamp Donkeys, Peanut Butter Sandwich 7 pm; Uncle Lou's Entertainment Hall, 1016 N. Mills Ave.\nSwamp Juice 8:30 pm; Lazy Gator at the Black Hammock, 2356 Black Hammock Fish Camp Road, Oviedo.\nWhiskey Lullaby 9 pm; The Nook on Robinson, 2432 E. Robinson St.\nWillow 8:30 pm; Fiddler's Green, 544 W. Fairbanks Ave., Winter Park.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the images taken with a drone by Johnny Miller for his Unequal Scenes project in Cape Town. This image was taken in Hout Bay and Imizamo Yethu last year.\nIssues of social housing and inequality in the city have been highlighted during recent months and UCT students have now turned their attention to analysing the media coverage of these topics.\nActivists, most notably those from the Reclaim the City campaign, supported by NPO and legal clinic Ndifuna Ukwazi, have been fighting for affordable housing close to the city centre and also supporting those who face eviction and being moved to far flung areas such as Wolverivier.\nThe UCT Centre for Media Studies honours and master's students presented some findings of a content analysis of social and print media and interviews with Reclaim the City activists, on Thursday May 18.\nThe guest speaker was Johnny Miller whose Unequal Scenes project, with aerial footage showing the inequality in Cape Town, went viral last year.\nMr Miller, an American, has been living in Cape Town since 2012 and studied anthropology at UCT. He has fallen in love with the city and has decided to stay, now living in University Estate.\nA professional documentary photographer, he was surprised at the massive response his Unequal Scenes project received.\nHe said the moment he realised the drone was a valuable tool was when he used it to take pictures of Table Mountain from the air.\nHe said he showed it to a Cape Town friend who said they had never seen the mountain like that before.\nIt was then that he realised the drone could be used to show things in a new light.\n\"That was a real wake up call for me. That made me realise how powerful a tool that is, that it can really change someone's attitude of how they see the world. Things that they thought they always knew what they looked like, they can now see with fresh eyes.\n\"I felt that that psychology of how we go about our daily lives is that we see patterns and those patterns tend to disappear. An example would be when you drive home, you've done it every single day, you kind of just blank out. When you get home you don't remember everything that you saw because you've done it so many times.\nHe said the role of the visual journalist was to use different technologies and ways of seeing things to shake people up and get them to re-engage with these issues.\nHe said the work has received a mixed reception and he did not expect the photographs to go viral as they did last year.\nMr Miller said the photos that were shared the most in his Unequal Scenes project were the ones that were shot from the top and directly looking down, showing the incredible inequality between neighbouring areas.\nMr Miller said it was the role of journalists to present these images.\nProfessor Herman Wasserman, director of the Centre for Film and Media Studies, said they chose to focus on spatial inequality because they wanted to show how research could engage with real life problems. \"We also wanted to engage with the wider community in Cape Town. We take community engagement seriously and we wanted our research to be formed by things that were going on in our community. It was about trying our hands at research that makes a difference.\nHe said Reclaim the City had done a lot of work to raise awareness on the issue.\nAssociate Professor Tanja Bosch said the idea behind the research topic was a public engagement with research. The students will continue to work on their projects on spatial inequality in Cape Town throughout the year. Some of the students included looking at mainstream media coverage of the Reclaim the City campaign as well as the amount of people talking about the issues on social media.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Across a wide range of print sizes that a dyslexic child is exposed to when attempting to read, his or her attempt does not follow the same curve shape as a skilled readers ability.\nWhen reading, at a constant reading rate is monitored for a child across large print sizes there is a sharp decline in the reading rate of the given text below a critical print size.\nOne of the issues that the Dyslexia Association of India\u2122 is noticing amongst the children who are being screened by the association, is that a dyslexic reader requires larger critical print sizes to attain his maximum reading speed, and this requirement may be arising from the letter position-coding deficit.\nDyslexia is a learning disability specific to reading and affects a substantial number of children in Indian schools. Accurate reading requires the child to process both the visual information from the page being read and the linguistic information that the print represents.\nEvidence is accumulating that shows that subtle impairments in visual processing of text that is to be read characterize some children with dyslexia.\nImpaired processing is being noticed in dyslexic children for low spatial and high temporal frequency visual stimuli and this may indicate a particular problem with visual motion processing.\nLow spatial frequency, high temporal frequency and motion information are carried primarily via one of the two parallel retino-cortical pathways in the human visual system. These visual deficits represent a correlate of reading problems such as dyslexia and other learning deficits and are mostly neurological in origin. However- a direct causal link between vision and dyslexia can be observed where reading under demanding visual conditions, such as with small font and print, leads to a decline in reading performance by a child with dyslexia.\nReading errors do decrease in some children when they are presented with a larger print font size, which suggests a causal link between a stressed visual system and the accompanying reading impairment. The potential effect of print size, where dyslexic readers are given larger print fonts than normal readers to achieve a maximum reading speed has been noticed and this 'print font size effect' is observable between children where the particular groups difference in reading speed is greatest with text which is in small print. This link between seeing the text in print and that too in a size that is readable and the prevalence of dyslexia is important to keep in mind in its own right because learning disorders are a confluence of neurological issues and the interaction in a human brain between highly interwined systems is still a complex mystery.\nIt is possible that at a certain stage of cognitive functioning a larger print font size facilitates a child with dyslexia to read accurately and without hiccups as it may increase the visibility of spatial frequencies, which are critical for letter recognition or reading.\nIn addition, it is possible that dyslexic readers require a wider spatial-frequency bandwidth for reading \u2013 which means that with larger letters, more distinguishing letter features may be available within the critical bandwidth for reading.\nSometimes a child with a reading disorder can be less efficient at picking up visual information than a skilled reader because his memory store of letter templates available to match to the incoming information from the text he is reading is less well established and this may result in him requiring either more or a different set of features to distinguish between the letters within the words.\nFactors that could be ameliorated by giving a dyslexic child larger print in the text he has to read may include lateral masking ('crowding') effects, that may be greater in the impaired child, as could be the possibility of variability in the visual span of the affected child. An alternative account of observed print size effects on dyslexic children is that they could result from a deficit, which has been proposed to interfere with letter position coding pathways that carry information about an object's position in space.\nFor accurate reading, fidelity of the channel that helps to localize correctly the serial position of letters within words is important, and a dysfunction within a dyslexic child's brain could lead to improper letter localization and therefore reading errors. In this case, it may be the relative position within words that is affected by the size of the letters. It could be that deficient position signals have a bigger impact for very small letters whose relative position is harder to identify, by the child and so may result in poorer reading performance with small print font size.\nThis can in simple terms be called a deficit in visual attention but it does propose a specific problem with serially focusing attention, either through rapid attentional capture or with attentional disengagement. If it is a problem with attentional focusing then it probably is contributing to anomalies in - relative position coding - in dyslexia. Accurate - symbol string recognition - in some dyslexic children is negatively affected by adjacent symbol reversals. Thus, while attentional disturbances (taken in isolation) could affect reading, visual attention deficits may manifest as - letter position coding errors - for an impaired child.\nSo can we (Parents or Teachers) continue to increase the print font size of the text that a child should be given if we believe that he may be reading disabled - i.e. would there be a monotonic increase in reading rates for a dyslexic child with continuously increasing print size? Reading words with very small letters may prove difficult for a dyslexic child either because the letters are hard to identify or because it is hard to determine where they appear in the word as both letter identification and letter position coding play important roles in word identification and slow, inaccurate word identification with small letters could be due to either compromised letter identification or letter position coding for the impaired child.\nLetter recognition and reading normally require a critical spatial frequency bandwidth. The idea that a dyslexic child needs larger print to compensate for a higher or broader spatial frequency bandwidth for reading is quite simplistic as one would expect reading performance to keep improving with increasing print size. But this is not the case. Likewise, the idea that a dyslexic child may have a particular print size where reading performance peaks, due to differential sensitivity to specific spatial frequencies, is also not correct as there are no points above critical print font size where reading is more efficient for the dyslexic child.\nLetter recognition alone is not sufficient for word identification and reading; rather inter-letter effects also need to be taken into account. Knowing the position of a letter relative to the other letters in a word is necessary for correct word identification; for example, distinguishing the logic behind the spelling of 'trap' versus 'tarp' or 'part' requires correctly locating the spatial arrangement of the letters in the word.\nThere is a brain channel for all children whether dyslexic or non-dyslexic, which may carry a code for the relative position of each letter in the word the child, is reading. This channel probably has a heavy traffic of noise or may contain more \"noise\" in a dyslexic child, so the number position for the letter 'r' and 'a' in 'trap' 'tarp' and 'part' may come with some uncertainty with what a dyslexic child sees. Does he see the 'r' in a greater proportion or does he see 'a' in a higher proportion. Relative letter position information - is contained within an intermediate spatial scale of the visual stimulus (i.e. the print), while information about whole words and letter features is carried by coarse and fine scales, respectively.\nFor very tiny print, the fine spatial scale for a dyslexic child becomes too small to resolve critical features, so letters cannot be identified and the acuity limit is reached. Just above this point, letters can be resolved but may not be ordered correctly because of spatial uncertainty in the codes for spatial position. Therefore hypothetically, greater relative - position noise - in dyslexia may contribute to slow reading in general, and could cause a dyslexic child trying to read to be more susceptible to deterioration with small print: meaning the dyslexic child's reading will have slower maximum reading speeds and higher critical print sizes.\nOther related aspects of inter-letter effects on word reading that are noticed include crowding, visual attention and visual span. Crowding, or lateral masking, refers to the hindering of letter recognition when other letters flanks a letter.\nThe difference in the reading accuracy between a normal child and a dyslexic child could be because of greater susceptibility to crowding effects, especially if their recognition of isolated letters is already inaccurate or slow. Similarly we find greater interference between far flankers than near ones for dyslexic children, which are attributed to differences in visual attentional filtering.\nCritical print font size decreases with age, suggesting younger children need larger print to optimize reading performance. This is the reason that reading materials for early, very young readers generally have larger print than texts for older children. But there is a negative correlation between reading speed and critical print size, which shows that in general slower non-dyslexic readers require larger print size to support their maximum reading speed. The fact that dyslexic readers require larger print font size to attain their maximum reading speed has implications for the type of print that parents select for their children, but the key is not to overdo it. Increasing the print font size to 18 will make no difference if the impaired child does not know the basics of reading, the phonic strategies behind the articulation of the alphabet and the rules of the language. Print font size can be useful, and it is possible to use it as a conjunct application for a child with dyslexia. Hoping that larger words translate into more information going into the brain is very simplistic, as sense has to be made of the information going into the brain.\nBehavioral evidence showing that dyslexic children are affected by crowding or lateral masking can play an important role in helping some children using a complementary approach where we try to focus on the accessibility of the reading material by manipulating the physical properties of print. Crowding as explained above is a perceptual phenomenon with detrimental effects on letter recognition that is modulated by the spacing between letters. The key aspect here in not print font size, but the interference of flanking letters on the recognition of target letters. For some reading disabled children recognition of text gets impaired when letters are closer than a certain level of critical spacing, which is proportional to eccentricity but independent of print font size.\nIt is the peripheral vision in normal adult readers, which is the area affected, but it also affects central vision in school-aged children. There is mounting evidence that children with dyslexia and reading disorders are more influenced by crowding than their peers in the same class even when optimal viewing conditions are provided to the dyslexic children. Letter identification is a fundamental step in visual word recognition and reading aloud and parsing of a letter string into its constituent graphemes is a key component of phonological decoding, which in turn, is fundamental for reading acquisition in competent readers. Crowding in a dyslexic brain might not only slow down reading speed but also induce reading errors, because crowding is accompanied by a jumbled percept that is thought to reflect pooling of features from the target and the flankers. Extra-wide inter letter spacing in words should reduce crowding and ameliorate reading performance in dyslexic children, but this again is a complementary strategy and not a solution of the problem as both reduction and increase in spacing have a detrimental effect on reading performance with reading speed in skilled adult readers being slowed when letter spacing is doubled. Therefore the key is a judicious application of the intervention where each child develops an idea of what will help him to focus on aspects of reading text so that the same page does not need to be read a dozen times over and still make no sense of what is being read.\nThe Dyslexia Association of India\u2122 works with its children to teach them how the manipulation of spacing influences the letter identification stage, which is identical across different alphabetic languages they have to learn in school. Developing a specialized neural system for visual word recognition demands optimized processing to handle parallel independent identification of letters in the extreme crowding condition of printed words in school based text that a dyslexic child is faced with constantly. Improving a sluggish visuospatial attention system in dyslexic children diminishes crowding by improving accuracy of target identification or reducing the critical spacing. In fact at risk children or pre readers in nursery and classes I who have been observed to have a sluggish spatial attention span and these children grow up to become dyslexic.\nCombined with the Neuroplasticity T.R.A.I.N\u2122 programme of the Dyslexia Association of India\u2122 the effect of print font size can be integrated into understanding the critical print font size that has to be used by each child. The way the human brain functions, no two individuals even if they are identical twins will process information entering their brains in the same manner and before parents start to zerox or photocopy school text books into larger more readable fonts, please spare some time and effort to have your child assessed for the underling problem that may be the cause of his academic problems.\nThe Dyslexia Association of India\u2122 can be contacted on info@dyslexiaindia.org.in or on phone at 88260-22886 for further information and it would be or endeavor to attempt to resolve any query or concern that you may have about your child.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Last night, the world relived \"The Trial of the Century\" as The People vs. OJ Simpson: American Crime Story debuted on FX.\nRELATED: Blac Chyna Defends Rob Kardashian, Fires Back at Haters \u2014 Find Out What She Said!\nAnd it turns out, the disgraced football player really did threaten suicide \u2014 but it wasn't in Kim Kardashian's room, it was Khlo\u00e9's!\nCuba Gooding Jr. as O.J. Simpson.\nIn an old interview with Barbara Walter \u2014 rehashed on a 2014 episode of The View \u2014 Robert Sr. opens up about the moment he walked in on O.J. threatening to take his own life.\n\"I said, 'O.J., I could never walk in this room. My daughter couldn't sleep in this bed. She'd know what happened here,'\" the late father-of-four told the journalist.\nKim, 35, has previously discussed what her life was like at the time of the highly publicized trial (though she confirmed the former athlete stayed in Khlo\u00e9's bedroom).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm done. I tired of seeking my own end. Tired of trying to force my will upon Him. I've had enough! Who am I to put God in a box? Who am I to limit what God is capable of accomplishing? I am but a speck of dust, small and inconsequential compared to majesty and might of God! He is so much bigger than myself that I cannot even begin to comprehend the infinite vastness of His work! He is far better than I could ever be and He is working all things together for His good!\nLast year while having a conversation with a friend about what God was doing in their life, they phrased something that has stuck with me ever since. They, in speaking of God, we're feeling God drawing them out asking them to \"doubt their doubts.\" To let go of their hesitations and to run fully, freely into the calling God has placed upon their lives! The last year was just that, me slowly responding to God as he beckoned me out beyond my expectations and put into place things behind my wildest imagination!\nI've been to three Passion Conferences, and each has been mightily impactful in nudging me into deeper faith and at chipping away at my heart. This year I did not sense that much was of what was shared was directed at me, but as I sit here now, I know that I'm wrong. Passion highlighted just how little I really do know\u2026 all the seminary classes, bible studies, and research I've done is moot\u2026 because God is better than I thought. Bigger than my intellect. Bigger than my understanding. Bigger than my doubt. I've been the skeptic. I've been the cynic. I'm done with trying to make sense of everything. I'm tired of trying to have everything boil down to something logical. I'm tired of the up and down.\nGod has made it quite apparent that His way is far better and stable than my own. I am going to cling to Him with all that I am! I'm committing myself to the path that He has laid and I'm going to be faithful to follow Him! Not my way, God, but your way! Not my will, but your will!\nY'all, I really don't know what to say. For once in my life I am grasping at the words to express this feeling, this stirring in my heart and soul. Alas, I have not\u2026 any term or phrase which adequately describes where I am at in this moment.\nI feel like a waterfall of emotions is cascading down around me; joy, peace, fear, doubt, love, praise, want, and so much more. This is not turmoil\u2026 but rather something stirring deep within my soul. If my suspicions are correct, this is God shaking me up and preparing me for something different. Revealing where I've resisted change, where I've let son take root, where I need to change. God is preparing me for something, and for the first time in a long time I feel something that only really appeared when I first took steps in faith.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've looked at snow \"from both sides now, from up and down\" to quote the great Joni Mitchel, but I do know snow. I grew up in Manitoba.\nHuge sloppy flakes, tiny beads that sting your face, curtains of snow, drifts of snow, hard layers of city snow like prehistoric layered ice and earth, soft snow that you fall back into without fear; crystal snow that creeps with icy fingers onto your neck between your parka and your scarf; that freezes your toes in your ice skates; that burns your face, cuts deep into your wrists just above your mitts and numbs your silly nyloned legs on the way to school.\nAnd I see it. I see how it transforms my garden, my city, my street. I long to see it more often at the lake where we spend so much of our summer.\nI know the colours of snow in light and in the dark. I know the long shadows that fall across it: clear shades of blue; dust gray, mouse brown, sun pink, green tinged, sheer frosted white, and dirty sand brown. And when it begins to melt I recognize the pockmarks than deepen in the snow that mean a thaw is at work.\nI know snow's indescribable smells - early snow that smell of earth and sun - and late snow in the city with the dark moist smells of tar and sand. At the lake cottage there will be the smell pine and the echoes of mouldering leaves feeding the earth below. I've tasted snow. I've felt it, formed it and built with it.\nMy memories of lake visits always start with cross country skiing in; of the cottage floor and walls covered in frost until the fire gets hot enough to warm the wood; cutting and bashing through ice for water; snowshoeing through the woods dodging clumps of snow that slide off the spruce and balsam trees onto knitted caps and shoulders, and the whap whap of it falling off our warming roof to the ground below. Most of all, the deep silence broken only now and again by hushed breezes and distant bird chirrups in the night.\nSeason of ice, this is my news.\"\nwinter gardening and even the most experienced gardener can learn from the unrestrained beauty around them.\"\nHath gladsome gardens of his own.\"\nCotoneaster berries with frozen drops of water.\nNature has many scenes to exhibit, and constantly draws a curtain over this part or that. She is constantly repainting the landscape and all surfaces, dressing up some scene for our entertainment. Lately we had a leafy wilderness; now bare twigs begin to prevail, and soon she will surprise us with a mantle of snow.\nand movement - yet all done in complete silence.\nthe fire of the frost, the soul of the sunbeam.\nThis crisp winter air is full of it.\nBehind a frosted window I see a small creature peering in at me.\nit with twist its wooden back.\njourney, and enjoy every idle hour. \"\nThe days are getting long, the shadows longer still. The sun is warmer on my back. My boots are dripping and damp. Spring is creeping toward me.\nEnding with one of my favourite quotes about winter!\nWonderful blog entry, Margaret! You helped me see snow in a new way--through your eyes, which are much kinder (wider, warmer?) than mine in the frigid cold of winter. Beautiful contemplations; gorgeous pictures.\nMy favorite? \"Spring is creeping toward me...\"\nI love it all, but especially the knitted frost. Amazing.\nThanks for the lovely comments. It was fun to put a few of my winter photos together.\nI love this blog, Margaret. The words are poetic and the photographs stunning. Being a Winnipeg girl too, it all hit home. Thanks for posting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"He was in the sake business, named Chiyokura \u5343\u4ee3\u5009 and had a rich estate.\nBasho had sent him various letters during his travels.\nHe lived in Narumi \u9cf4\u6d77\u306e\u77e5\u8db3\u4ead, now Nagoya.\nNarumi was a postal station of the Tokaido road and business was good.\nChisoku's Buddhist name after death was Shakushoo \u5bc2\u7167.\nand wrote hokku to greet him and his family.\nwith millet at the back door .\nThis poem was a congratulation to the younger brother of Chisoku in Narumi on the building of his new home.\nThe statue of Basho was made by Takashima Kaemon. \u9ad8\u5cf6 \u5609\u53f3\u885b\u9580.\n. MORE about Matsuo Basho and Narumi \u9cf4\u6d77\u5bbf .\nseen from Mount Sanno Yama \u4e09\u738b\u5c71.\nThere is now also a memorial stone with Basho's poem.\nis this the plovers' cry?\nThe sea at Narumi has since olden times been famous for its plovers.\n. Matsuo Basho and his disciples .\nThe statue of Basho is kept in this temple.\nHe was a transport merchant in Narumi \u8377\u7269\u554f\u5c4b.\nBasho visited Narumi in Autumn of 1688 \u8c9e\u4ea85\u5e747\u670810\u65e5\u7acb\u79cb.\nBasho wrote the following hokku as a greeting to his host, to praise the beautiful scene.\nThe Haiku Meeting and Kasen with Jushin - \u5150\u7389\u91cd\u8fb0\u8208\u884c\u6b4c\u4ed9.\n. Visiting places with Matsuo Basho \u677e\u5c3e\u82ad\u8549 .\n. Narumi Jinja \u6210\u6d77\u795e\u793e and Daruma .\n***** . WKD : Millet (awa, hie) .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finally Named: 'Banga Stick' is Liquorice!\nFrom market to web page, it seems \u2013 despite their size and appearance differences \u2013 that umilo and gbafilo might be related, after all. Proof that sameness is beyond appearance.\nIn my search for the botanical name for umilo, I happened upon an article referring to umilo by Parinari excelsa, the botanical name which I 'knew' was gbafilo. 'Knew' because what do we really know? And how do we prove that the sources which inform our knowledge are true and tested?\nI am a strong believer in research and validation. Combining look, word, feel, taste and whatever metric is available in arriving at a reasonable conclusion. Lately, I've learned the importance of opening up my test pool to other people who might or not be familiar with the subject. Because fresh perspectives, without the burden of knowledge or mind mold might lead to greater discoveries.\nUmilo and Gbafilo are related though it isn't apparent in their looks. Their shells are different but they have the similar fragrances, and their kernels are the same \u2013 paper shell with a hollow nut, as well as purple-brown hues. Their textures are similar too \u2013 chewy crunchy.\nThey appear to be in the plum family. From names of Parinari excelsa, Guinea plum and the like.\nThey are spices\/ aromatics but nutmegs or walnuts? However, in the market \u2013 sellers referred to both as types of nutmegs; while my food expert friends spoke about it being very walnut-like.\nIn the end, not knowing the names doesn't change what they are\u2026but I'd really like to know :), so if you have any advice on how to go about this exploration, please share. In the interim, we find other discoveries to celebrate. Like Liquorice root.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbascs b/data_all_eng_slimpj/shuffled/split2/finalzzzbascs
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@@ -0,0 +1,5 @@
+{"text":"Tropiclean's natural Water Additive is developed to provide dogs and cats with essential daily oral hygiene care. Without proper dental care, 70% of dogs and cats start to show signs of oral disease by age 3. Oral disease can eventually lead to tooth loss or other serious health problems. Tropiclean's natural Water Additive promotes healthy gums and eliminates bad breath for up to 12 hours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1 x pet jumpsuit. Cape aztec. Clothes nice. Hond kleding chihuahua. Model numble: Dogs clothes halloween. Fashion,comfortable, breathable. Funny cat suits. Dog size: Blgp820165. Puppy play costume.\nWikileaks. Penguin party. 2016 newest dog clothes. Product name: Sheinside. Dogs&child. Feature  4: Pug,yorkshire terrier,beagle,chihuahua,teddy,poodle,bichon,shih tzu. Wholesale fleece jacket. Women romper. Blue,red. Male\/female. Letters.\nDecoration:sashes. Brand pet jacket cheap clothes from china cheap. D69832. 90-200g. Pink monkey\/blue duck. Wholesale dogs leashes. Lovebatu. Chihuahua yorkshire terriers. Horse print. Wholesale jumpsuit jeans. Party custom. Dog clothes for autumn winter. Love-heart. Dog chest belts. Yellow,pink.\nTaonmeisu. Keep warm,breathable,comfortable,handsome,Chihuahua,puppy,yorkshire terrier,miniature,doberman,pomeranian,papill. Skits. Package includes: Breeded dog. Dog warm winter jumpsuit. 91%-99%. Ca409. StarsSuitable: Dog clothes plain. Bamboo jumpsuits. Fashion pet clothes: Place of original: Fit for dogs, cats. Place of origin: Salin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":" Fun Easy English - new classroom free online classes english language lessons speaking listening writing reading february 24.\nToday in the Fun Easy English classroom you are going to learn the correct way to write the English alphabet letter D in manuscript and cursive.\nWriting the English alphabet letter D correctly in manuscript and cursive.\nThis letter is pretty is easy to write in both manuscript and cursive.\nIn this lesson When Pete and Anna meet with Director Kelly, they arrive late. Both tell very different stories about their morning. A show begins.\nAnna: Today, Pete and I are meeting with a consultant who will help us with our new show. Yesterday, Pete had promised to meet me here at 8:00 am. but he did not come on time.\nProf Bot: Uh-oh. It's bad to be late for a business meeting. But while we wait for Pete, let's talk about a new verb tense -- past perfect! You know the past tense, right? Like, \"Pete promised to meet me here at 8:00 a.m.\" Past perfect is a little different. When we talk about two things in the past, we can use the past perfect for the first event. Put \"had\" before the past participle. \"Pete had promised he would meet Anna.\" Here's your assignment: find sentences with the past perfect tense. Remember, look for \"had!\"\nKelly: You two are late -- exactly 43 minutes late! What happened?\nAnna: He had to get his \"special\" coffee -- SPECIAL coffee!\nPete: She had to feed her birds -- HER birds!\nKelly: Okay, I can see already that you need my help. You can't both talk at the same time. You have to take turns. Alright, Anna, you go first.\nAnna: Sure. Kelly, see, Pete and I live in the same building. So, we decided to meet at 8:00 a.m. - 8:00 a.m. to come to work together. I had waited 15 minutes when Pete arrived!\nKelly: Hum. I didn't get that message.\nAnna: \"After Pete had wasted time waiting for coffee, we were late.\nLook at that coffee! It looks more like dessert! Okay, keep watching for past perfect!\nPete: Yeah, that's not why we're late. This is why we're late: I had arrived on time at 8:00 a.m. but didn't see Anna. She was standing behind a tree. I think she was hugging it. I always walk to work. But she said that would take too long and that a scooter would be much faster. It was awful. I hated it. And it added too much time to our commute!\nThen Anna stopped by a pond to feed the birds. She had named them after characters from books and yelled the names out loud \u2026 Romeo! Juliet! Sherlock!\nBy the time she had fed all the birds, we were late.\nKelly: This is what I think. You two see the same event very differently. Does this happen often with you two?\nKelly: Okay. This is good. This is good! It's good to see things differently. I have an idea: we will call the show \"He Said, She Said.\" For every story, you tell a different point of view.\nAnna: That is a great idea, Kelly! Pete, we are different. That's why I thought of you for this job!\nKelly: I think you two understand perfectly.\nAnna: Let's get to work!\nKelly: She named the birds? Really?\n* Business people in the U.S. think you should come to a meeting at the exact time. If you are late to a business appointment, you should call and explain why.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We celebrated our 33rd wedding anniversary at the Cypress House Hotel. The staff was great Melissa, Scott, Dennis and Jacob. Melissa was especially welcoming and helpful with everything. They all give it a personal touch and feel. Whatever you needed they had. It is a great location. Short walk to all the old downtown spots on Duval street, Mallory Square and many others.\nThen when you enter the hotel it is all peace and quiet. We rented bicycles 2 days during our stay. The rental shop was just 2 blocks away and really easy to rent and return. The liming hour in the evening was perfect segway to coming in from the day's adventure, have a great and different cocktail and snack each day before getting ready to go out that night. Breakfast was great too. The buildings are historic and updated and very clean. We would have a hard time staying anywhere else on our next trip to Key West.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These delightful muffins make for the perfect breakfast on the go. Freeze any excess muffins in individual bags \u2013 they thaw quickly and are delicious when warmed and served with butter!\nPreheat the oven to 180\u00b0C. Grease a 12-cup muffin pan with a little butter or line with paper cases.\nCombine the bran flakes, quinoa flakes, flour, baking powder, baking soda and sugar together in a large mixing bowl and make a well in the centre.\nGently melt the butter with the golden syrup until just liquid.\nPour the butter mixture, egg and milk into the dry ingredients and lastly add the feijoas. Gently mix to combine.\nSpoon the batter into the prepared muffin pan and then bake for 20 minutes until well coloured and the muffins are firm to the touch.\nAllow to cool for 5 minutes in the pan before transferring to a wire rack to cool completely.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbidy b/data_all_eng_slimpj/shuffled/split2/finalzzzbbidy
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index 0000000000000000000000000000000000000000..b29b3c620a60f45b4cdf084d7a0c55200a843211
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@@ -0,0 +1,5 @@
+{"text":"Get Major Global Wind Belts Metasearch & Social Results Here.\nTired of Endless Searching? Find What Are Wind Belts on DealsNow.\nWhat are the names of the three global wind belts?\nThe three major wind belts, also known as circulations, are the Hadley cell, the Ferrel cell and the Polar cell. The circulation of these winds is caused by the rotation of the earth and the energy of the sun. How these circulations move between the poles and the equator determines climate.\nGlobal Wind Patterns wind belts of the general circulation The global wind pattern is also known as the \"general circulation\" and the surface winds of each hemisphere are divided into three wind belts: Polar Easterlies: From 60-90 degrees latitude. Prevailing Westerlies: From 30-60 degrees latitude (aka Westerlies).\nThere are six major wind belts that circle the globe. The belts are divided equally with three in each hemisphere. The six wind belts are divided into three categories: the Polar Easterlies, the Westerlies and the trade winds. The wind belts move either north or south depending on the seasons at each hemisphere.\nStart studying the three main wind belts. Learn vocabulary, terms, and more with flashcards, games, and other study tools.\nGlobal wind patterns: Winds are named by the direction from which they blow. The globe is encircled by six major wind belts, three in each hemisphere. From pole to equator, they are the polar easterlies, the westerlies, and the trade winds. All six belts move north in the northern summer and south in the northern winter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please submit a clear snapshot of the U9834 Thumbscrew Bar Lock \u2013 Black window part to be certain it is a replacement window part that you require! Our team specializes in IDing your replacement window parts and we mail hardware nationally via USPS.\nRemember to submit a snapshot of this U9834 Thumbscrew Bar Lock \u2013 Black part before buying, nonetheless if you know the correct component you require basically compose it in the space down below on the component contact form our staff distinguishes, responds and ships out components as quickly as achievable if you need rapid help simply just give us a call.\nWindow replacement parts for instance the U9834 Thumbscrew Bar Lock \u2013 Black which we will offer quick USPS shipping on nationally will require some very distinct hardware that is often entirely about whether the windows can be shut tightly enough to keep out the elements and to stay stable. Our group determines so you can buy PART online to make positive its the correct one you need.\nThese are window parts such as the U9834 Thumbscrew Bar Lock \u2013 Black that will need to provide folks further levels of protection taking into consideration various weather conditions. They need to be tough enough, they need to stay enclosed, and they need to make available folks the opportunity and flexibility to stay warm with complicated circumstances.\nU9834 Thumbscrew Bar Lock \u2013 Black, for instance, performs to make sure that the screens of windows or the sashes of windows are kept very securely to make sure that the windows are not going to shake a lot, even in the aftermath of a great deal of wind, rainfall, and other outdoor hazards. These U9834 Thumbscrew Bar Lock \u2013 Black parts are likely to have rust resistant plating, which is able to allow them to endure some of the humidity infiltration that folks could encounter otherwise.\nWindow parts vary per the definition of the variety of hardware you need to have for instance storm window hardware or casement window hardware, of course. Folks are likely to have to ask their selves what meets your criteria. One example is, some people would say that the tough window glass that a lot of storm windows use will be eligible as storm window hardware, and this seems like it must be the case. Other people could say that in case the window head or the window sash of tough storm windows were both developed to be strong enough to withstand a wide variety of unique problems, then people should also take into account these to be storm window hardware.\nOne can find lots of various parts which may be the main one that you will need, except you are 100% certain that this is the part you will need please publish a snapshot of your hardware to ensure that one of our part specialists can distinguish it and check if we have it in inventory or if we can obtain it for you, we offer both out of date and very hard to obtain replacement window parts.\nU9834 Thumbscrew Bar Lock \u2013 Black Branded light weight aluminum construction; suits 4 in glass; used on global windows performs with all brands. Printed light weight aluminum structure; matches 4 inches glass. Printed simple designed structure; meets 3\/4 inch heavy wood slats. part window part is offered at this location, still measures 5\/16 inches O.D. instead of Prime-Lines 3\/8 inch Each individual collection seems to have 4 Light weight aluminum groove pin along with E ring is applied to hook up link end of operator for your Jealousy or Awning window parts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cutting the Budget: Week 18 \u2013 May starts us out with a bang!\nWeek 18 brings in May\u2026the most busy month of my entire year. It's full of birthdays, graduations or end of the school year, traveling, Mother's Day, outside yard work and so much more. Throw in I am in so many fundraisers right now I could scream\u2026seriously, I am about to donate my book money just to get away from them all.\nSo week 18\u2026in the 5th month of the year\u2026518 is my favorite number\u2026why\u2026May 18 is my birthday. Oh, it's early but when I saw the numbers I thought lucky week. Only one book bought\u2026one I just couldn't live without and plan to read soon.\nGood choice! I'm not missing out on this book either, but not sure when I'm getting to it. Yay! Reset on the budget.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ahead of next week's WWDC 2013 at Moscone West in San Francisco, Apple has begun putting up banners at the venue. The multi-colored artwork is reminiscent of the invitations that went out in April and feature the tagline \"Where a whole new world is developing.\" Hit the break for a few more photos from reader Andrew Stern. If you're in the San Francisco area and manage to get some photos of the banners, send them to us at tips@9to5mac.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the region's leading premier payments network and central infrastructure firm for financial markets has a Business Development & Marketing Assistant Manager job that has just arisen. The role will undertake business planning and formulation of business strategy to its next phase of growth.\nThe Assistant Manager assists the Head of Department in building, managing and executing business initiatives\/plans and marketing\/promotion campaigns as well as in expanding the business for the organisation.\nTo succeed in this Business Development & Marketing Assistant Manager role, you need to have minimum five years' experience in the Business Development or Marketing area.\nThis leading premier payments network and central infrastructure firm needs no introduction and is recognised as the leader in its field of expertise.\nIf you are driven, determined and want to take the next step in your career, this is the role for you. Great career progression opportunities await the right person in this exciting Business Development and Marketing position.\nApply today or email me at patricia.lim@robertwalters.com.my to discuss this new opportunity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbckkd b/data_all_eng_slimpj/shuffled/split2/finalzzzbckkd
new file mode 100644
index 0000000000000000000000000000000000000000..b7298e1ad6e26fe9a9296c9ecd3aa69a88e4e33c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbckkd
@@ -0,0 +1,5 @@
+{"text":"Sundays at 8:30 a.m.: A Service of Prayer and Communion. This is an intimate though fairly informal time to be in worship and to center ourselves in the love of God. We begin with a simple song and an opening prayer. We then turn to the scriptures for the day and share in a time of discussion, which is facilitated by the Pastor. When the discussion seems to have reached its conclusion we transition into a time for prayer and thanksgiving, after which all are invited to join in a circle for the receiving of Holy Communion. We end with the Lord's Prayer and a sharing of God's peace. This has become a sacred time for a handful of our members. Others are always invited to join us.\nThis is the primary gathering of our church family. It is a time to reconnect with each other and with God. It is a time to offer our praise, to confess our sins and to be reminded of God's forgiveness. It is a time to hear the good news and to respond with our prayers, our thanksgiving, and our gifts. Regular components of our worship include an engaging children's time, a thoughtful sermon, and an inspired choral anthem. We celebrate Holy Communion on the first Sunday of every month and we occasionally incorporate our Grace chime bell choir, children's choir, soloists, and other special offerings.\nCoffee Hour (Fellowship): This time for fellowship, coffee, and goodies is not to be missed. It follows immediately the 10 a.m. service.\nSummer Sunday Worship Schedule 9:30 Combined Service: During the months of July and August we beat the heat and join together for one worship time. The choir takes a well-deserved break, though we feature special music throughout the summer. We also offer a One-Room Sunday school program for the children. Of course, coffee hour is still on!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At DumpsterZone, we strive to bring you the best service at the best price available. We have a wide selection of dumpster sizes in stock in Novi, MI, with flexible & timley pickup and delivery. Give us a call today for all your dumpster rental and roll off needs.\nBefore you go web to a Dumpster for Rent in Novi, MI internet site, try and complete a rough working out on how much will coming out of the dwelling, so that you can obtain the right-sized dumpster. If you rent engineered to be too small you'll have to have it recinded, emptied, after that brought back. It will cost some thing extra. Obtain one that's too big and that's a complete space and money. Most Dumpster for Rent in Novi, MI corporations will do the most beautiful to help you evaluate your junk and recycle load.\nYou can save yourself time and energy associated with hauling junk over the subsequent couple of weeks outdoors to wait for the community junk pickup just by calling for some sort of dumpster today. You can have a following day dumpster arrival simply by calling the evening before or perhaps by ordering on the web. You can get rid of your junk as soon as possible in lieu of waiting for the community trash pick-up and you can save time and energy simply by avoiding the requirement to separate recyclables and break down the garbage into baggage.\nIs your living room too small? Can you spend a lot of time in your living room? If you answered yes either to of these questions, maybe it is time to remodel the den. For those who have a spare room that hooks up with your living room, maybe you should take into consideration knocking your wall amongst the two areas. This will throw open your living room area to make it additional spacious and comfortable for you.\nA different use for any dumpster is when what you are doing a house redevelopment and are likely to have a wide range of trash. You will be able to chuck all the junk in the rotate off dumpster rental fees you have and they'll haul this away if you are done.\nYour well being and that of ones own members may even be at danger. In Altlanta ga, such waste materials are appropriately taken care of by way of various waste management organizations. If you live in Atlanta, you may apply for a top quality Atlanta Rotate off Box. This can are available form of a unique dumpster meant for doing away with all etiquette of waste materials, including soiled trash and garbage resources. In most cases, your Atlanta Rotate off Dumpsters tend to be stationed with strategic sites. People dwelling around may easily make use of such containers to bring up their own waste products. A fortunate note about the Altlanta ga Roll off of Dumpsters is that, you'll find companies in control of removing these individuals from time to time. They simply drop your dumpsters at strategic places; you and the neighbors will probably load all of them with waste products coming from your environs. Then the organization comes back to pick out them. Simply by so accomplishing, the environment will be kept pristine clean.\nHave always a flashlight and another man or woman with you to view your returning. You don't want a person sneaking up on you if you are going through their own trash.\nUsing Dumpster for Rent in Novi, MIs in construction or any other major challenge can help to keep a safe operating environment to your workers. This assists prevent unfortunate events or perhaps accidents by happening. The prospect of accidents occurring increases when there is uncleared debris on the premises. Making sure that the construction particles is removed and handled in a timely manner in addition lessens the possibility that you will be attributed for any health-related costs following from incidents in the building site.\nFollowing comes, the length of the dumpster, selected projects end up in piling huge tons of waste and where as some with less waste. Roll off dumpsters will be the most suitable choice if you are looking out and about to dispose huge tons of waste. These roll off dumpsters range between 10 to 40 lawn. When your waste disposal consists of bulky goods, check with the seller if these kinds of roll off dumpster enclosed together with the back doors. Most of these dumpsters would provide a fantastic convenience in your case to dump the actual bulky waste into the dumpsters. Because the name reveals, these roll off dumpsters are generally mounted on trolley wheels and can be easily rolled upon to the pickups. You need a tall dumpster if your waste contains major materials. Just be sure you check with the seller about the proportions of the dumpster so that you can load the garbage conveniently.\nThe den is your haven and you could do a great deal of work in generally there. Once in awhile you could wish you had area for a comfortable oversized buckskin couch. In case you decide to redecorate your living area, than you can do this so that you are satisfied with your living area.\nPeople who will need to dispose of reliable waste utilize dumpsters. It is not All right to dump liquids or harmful waste straight into dumpsters. It is specially not All right to do that with rented dumpsters. This will not be beneficial to the condition of the actual dumpster and furthermore, the fitness of the environment. Colour and other these kinds of liquid components that dehydrate can wreck the container because it will make it heavier permanently and also cause various other debris to acquire dried engrossed. Hazardous throw away will wreck the environment. That certain is pretty home explanatory. Unsafe waste, for example chemical throw away, needs to possibly be disposed of effectively to not even more pollute the planet. If you are making an attempt to dispose of a great deal of it then it's not necassary to be using the dumpster.\nDumpster rental is important for the business. Many cities waste is only picked up once or twice per week. Having fifteen trash cans might not be the very best idea on the globe. Let's face it many reasons exist for why you wouldn't want to obtain many waste cans, facing, beside or behind your business. They are a eyesore in whose sale benefits. They will obstruct. Depending on what you will really be squandering you might be luring nuisances. You do not need to be recognized for your collection of stray animals instead of exactly what business you manage.\nThe only way to dispose the reliable waste easily without sacrificing the busy times is to choose Dumpster for Rent in Novi Alaska; support you to dispose the trash with effective method at cost-effective cost. Dumpster rental organization Alaska will certainly recycle reliable waste created on finishing of project with environmental friendly manner and minimize the side effects of pollution on the surroundings of the point out. On leasing a dumpster for the trash fingertips needs, the garbage that is got rid of to the rubbish dumps will be greatly reduced and the area occupied with the trash from the landfills is definitely condensed which inturn will reduce the release of harmful gases in to the atmosphere. Discarding the trash via Dumpster for Rent in Novi will prevent the garbage from becoming dumped in to the water likewise. Dumpster rental Alaska plays a key function in reducing the consequences of pollution on the surroundings of the point out.\nOne of the initial benefits of dumpsters is perhaps you can fill all of them with whatever style of trash you've got. Think about which before you even consider Dumpster for Rent in Novi, MI prices. Regardless of whether you need to utilize them for appliance convenience, furniture convenience, or just simply junk convenience, if you can suit it in it, you can put this in there. This is great news should you be doing a complete cleanout of a home, garage, or some other building. Traditional garbage removal possibilities don't provide you with this luxury--you currently have to take your oversized items to any transfer section or land fill on your own. With a dumpster rented out of your local trash hauling business, though, you can dispose of essentially anything you require to.\nIt is no longer difficult for you actually to fetch the garbage cans the very best suit your requirements. Dumpster rentals are often priced based on the tonnage of trash, period of rental along with size of dumpsters You can be over mired by added charges regarding multiple journeys to dispose ones trash in the event you hire a modest size dumpster. You may be paying for the disposable space in the event you hire a huge size dumpster. Almost all of the Dumpster for Rent in Novi, MI companies on the market put a finish to this problem by renting different sizes associated with dumpster according to ones customized requirements.\nRenting any dumpster can be a lifesaver when you have a serious project ongoing at home. These kind of rentals are ideal for trash treatment, furniture convenience, and trash hauling, amongst other things. However, it is important to remember that hiring a dumpster isn't going to automatically signify your dumpster seasoned will be on the ground to move the garbage into the receptacle for you. Be sure you know exactly what's included upfront when you rent any dumpster so that you don't run into just about any surprises on the way.\nGood, that you've got chosen to rent any dumpster to clear the a lot of trash piled up at your career location. You should know the type along with size of dumpster which best fits your needs before you go ahead with the rental method. Roll off of dumpster should be your decision if you are looking out and about to dump large tons of trash in the important commercial and residential tasks. Before you place your order for the roll off trash bin, it is significant for you to understand the dimensions of the particular garbage container so you accommodate the mandatory place to placed the dumpster at your project location. If you are running a home rehabilitation, commercial project or commercial clean up, the room you support for the garbage pot should be double the amount size of their height along with width.\nAt first you may want to try out to have a rummage sale made. Make the price ranges on the things very low so you can just eliminate. Do not try to earn cash at your rummage sale made, you are acquiring it to get rid of things. It is far better to sell ones items for cheap to someone that can make use of them or maybe fix these individuals than to just throw these individuals in the dumpster along with end up for a junkyard.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"also known as \"At Sword's Point\"\nWatch At Sword's Point Online Now!\nYou can watch At Sword's Point online for free on megashare by clicking the play button above. This movie is hosted on 3rd party website and was not uploaded by us or any person affiliated with us. If the stream is too slow, try another link. If you're trying to watch At Sword's Point and it isn't loading, please check our Help section before reporting it as a broken link.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Imagine yourself as the student you want to recruit - what would make you want to read about this position - and apply?\nUnlike general recruitment websites, on CareerHub Central, your audience is specifically students and recent graduates - take the opportunity to tailor your offer to this audience.\nImagine you have only 10 seconds to convince candidates to read about your position - use its benefits as a hook to attract the type of person you're looking for.\nWrite this summary uniquely - don't copy a paragraph from the position's 'Job details' section.\nExplain what you're looking for in applicants - qualifications, skills, experience, working hours, personality characteristics - without being too generic.\nSpecify what makes your job or graduate program unique - remember your candidates are likely to be comparing your offer against others.\nExplain exactly what an application should include - for example, if you require a response to selection criteria, and what length it should be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The World Bank, alongside the International Monetary Fund (IMF) and the Parliamentary Network on the World Bank and IMF, will host the fifth edition of the Global Parliamentary Conference (GPC) on April 8-9, 2019 at the World Bank headquarters in Washington DC.\nThe GPC gathers more than 200 legislators from across the globe to discuss today's most urgent economic and development issues with World Bank and IMF senior management and experts. It offers participating Members of Parliament the opportunity not only to learn about key issues, but how they can partner with the Parliamentary Network, the World Bank and the IMF to implement policies and reform for lasting development results at the national and international levels.\nThis year's conference will have the overarching themes of harnessing disruptive technology for development and jobs creation. Sessions will include topics on (1) disruptive technologies for development; (2) investing in human capital; and (3) taking action against climate change.\nThe second day will also include the first official meeting of the Global Young MP Initiative \u2013 a groundbreaking program for MPs under 45 committed to tackling the mounting number of development challenges directly affecting young people, including education, youth employment, and fragility, conflict and violence. More than half of the world population is under thirty, yet according to the IPU they are represented by only 2.2% of parliamentarians worldwide. The Global Young MP Initiative will provide a platform for young parliamentarians to explore proactive and innovative solutions to ensure that young peoples' voices are heard in policy dialogue.\nRegister for the GPC by February 28 here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I Love To Dig! - I Love to Dig!\nScholars have argued over those bones ever since Greek archaeologist Manolis Andronikos discovered the tomb in 1977-78.\nScholars have argued over those bones ever since Greek archaeologist Manolis Andronikos discovered the tomb in 1977-78. He excavated a large mound \u2014 the Great Tumulus \u2014 at Vergina on the advice of the English classicist Nicholas Hammond.\nAnalyzed by Antikas' team since 2009, the male and female bones have revealed peculiarities not previously seen or recorded.\nTomb of Alexander may have been discovered!\nAlexandria. A team of archaeologists and historians from the Polish Center of Archaeology, that were conducting some research in the crypt of an ancient christian church, have revealed a mausoleum made of marble and gold that could well be the long lost tomb of Alexander III of Macedon, who went down in history as Alexander the Great. The site is situated in an area known as Kom el-Dikka in the heart of downtown Alexandria, only 60 meters away from Mosque of Nebi Daniel, where Arabic tradition maintained the tomb was to be found and where many excavations had already been conducted.\nA famous site of pilgrimage in Antiquity, the tomb is notably known to have been visited by many Roman emperors, including Julius Caesar and Gaius Octavius, better known as Augustus, who is said to have placed flowers on the tomb and a golden diadem upon Alexander's mummified head. The last recorded visit to the tomb was made by the Roman emperor Caracalla in A.D. 215, less than a century before it disappears from Roman records.\nThe large monument, apparently sealed off and hidden in the 3rd or 4th century AD, possibly to protect it from the christian repression and destruction of pagan sites after the change of official religion within the Roman Empire. The entire site is a testimony to the multicultural nature of Alexander's empire, combining artistic and architectural influences from both Greek, Egyptian, Macedonian and Persian origins. The inscriptions, mostly in greek, but including also a few egyptian hieroglyphs, mention that the Mausoleum is dedicated to the \"King of Kings, and Conqueror of the World, Alexander III\".\nIt held a broken sarcophagus made of crystal glass, possibly damaged during the looting that took place during the political disturbances that ravaged Alexandria during the reign of Aurelian shortly after A.D. 270. It also held 37 bones, mostly broken or heavily damaged, presumably all from the same adult male. A carbon-dating analysis is already under way to determine the age of the bones, and a battery of other tests also await to determine if the bones could be those of the Macedonian king. Other than that, the site held only a small number of artefacts, mostly broken pottery, dating mostly from the Ptolemaic and Roman eras.\nThe Egyptian Supreme Council for Antiquities had already officially recognized more than 140 unsuccessful searches for the site of Alexander's third and final resting place, built by Ptolemy Philadelphus around\u2009280 BC. Many astounding theories had been elaborated by various historians to explain the fact that the archeological excavations had come up empty-handed. Some historians had even evoked the possibility that his body could have been unintentionally stolen from Alexandria by a pair of Venetian merchants, taken to Venice, mistakenly renamed and venerated as St. Mark the Evangelist in Basilica di San Marco (Venice, Italy).\nThe actual facts however, seem to have been a lot closer to what had been corroborated by many authors of Antiquity and the Middle Ages, such as Plutarch, Ibn 'Abd al-Hakam, Al-Massoudi and Leo the African. In 321 BC, on its way back to Macedonia, the funerary cart with Alexander's body was hijacked in Syria by one of his generals, Ptolemy I Soter. Ptolemy then diverted the body to Egypt where it was interred in Memphis, the center of Alexander's government in Egypt. Then, in the late 4th or early 3rd century BC, Alexander's body was transferred from Memphis to Alexandria, where it was reburied.\nBy the fourth century A.D., the tomb's location was no longer known, if one can trust the accounts of several of the early Church Fathers. However, creditable Arab commentators, including Ibn Abd al-Hakam (A.D. 871), Al-Massoudi (A.D. 944), and Leo the African (sixteenth century A.D.) all report having seen the tomb of Alexander, but none of them specified its exact location.\nThis new discovery by the Polish Center could certainly be one the most important ever made in the country, despite the already immense wealth of archeological treasures it has yielded. The site itself is a masterpiece of architechture and craftsmanship, and containing many possible new details about the great king. Should the bones turn out to be those of Alexander, the importance of this incredible find would certainly skyrocket, making it one of the greatest in history of archeology, if not the greatest.\nFound in an archaeological dig in Bethsaida, this rare bronze coin tells of love, trade ties and globe-shaking jealousies. And what if Marc Antony had won the war?\nGreat rulers come but a rare few leave a mark echoing down the millennia. Two such were Cleopatra and Marc Antony, who fleetingly placed Egypt at the center of the ancient world, only to unleash unrest and eventually war on the region.\nA few thousand years is a mere blink of an eye when it comes to the vital ties between this land and Egypt, as attested by a rare coin carrying historical weight far greater than its 7.59 grams, which depicts the notorious lovers \u2013 and which emerged last year from the ruins of a first-century house at Tel Bethsaida on the Sea of Galilee.\nTel Bethsaida rises from the northern coast of the Sea of Galilee, but the coin was minted in another city by another sea \u2013 the Mediterranean port of Akko - today better known as Acre. The coin, made of bronze, is about the size of a quarter, being 21\u201323 millimeters in diameter (it is not perfectly round, at least not any more). Its date shows that it was minted in the last half of the year 35 or the first half of 34 BCE.\nPtolemais is the Greek name for ancient Akko, which was founded in the 3rd century BCE and named after Ptolemy II Philadelphus. The name appears in the New Testament (Acts 21:7) as the home of an early Christian community that Paul the apostle visited: \"And when we had finished our course from Tyre, we came to Ptolemais, and saluted the brethren, and abode with them one day.\"\nThe coin was minted some two and a half centuries after the city was founded, a time when both Mark Antony and his bitter rival Octavian were in their prime and no one knew who would prevail, Arav says.\nWhy depict them? The cities of the ancient Middle East had a habit of minting coins bearing the portraits of whoever was in power, says Dr. Donald T. Ariel, head of the Israel Antiquities Authority Coin Department.\nAnd Marc Antony was most definitely powerful in the year stamped on the coin. Prof. Rami Arav, director of the Bethsaida Excavations Project, suggests that the minting of the coin may have had to do with Marc Antony's victory over the Parthians, rulers of a land in what is now northeastern Iran and Armenia, in 35 BCE. He then granted Armenia to Cleopatra's sons and gave Cyprus to her daughter Selene.\nCleopatra also appears on coins from the same period, found in cities further north up the Lebanese coast, that were among gifts Marc Antony gave his consort.\nThat same year Marc Antony, still deeply involved with Cleopatra, moved the capital of the empire from Rome to Alexandria, Egypt.\nHad Antony not lost the battle of Actium in 31 BCE, there might have been a dramatic change in the history of the Western civilization, Arav says.\nBut Antony did lose that battle, and 11 months later, he took his own life, dying in Cleopatra's arms in an immortal star-crossed lovers' moment.\nOther coins from Akko have been found in Bethsaida, showing the trade connections between the port city, an international commercial hub at the time, and Bethsaida, a regional one, Arav points out.\nThe \"lovers' coin\" recalls Bethsaida around the turn of the first millennium, when its main claim to fame was being the New Testament home of the apostles Peter, James and John.\nBethsaida was where Jesus is believed to have healed a blind man (Mark 8:22-25) and fed the 5,000 (Luke 9:5-17). But much of the efforts of Arav's team involve uncovering remains that go back a thousand years before that \u2013 to a time the city was the capital of the ancient, strategic kingdom of Geshur, the homeland of one of the wives of David's youth, Maacah.\nCoins with the portraits of Antony and Cleopatra are extremely rare. Only six have been found anywhere in the world, says Ariel. But to him the coin recalls Cleopatra's connection with a man who doesn't even appear on it - the man who, after Jesus, is perhaps the best-known figure of this land: Herod the Great.\nCleopatra managed to persuade Marc Antony to wrest Herod's priceless balsam plantations from him and hand them over to her. As a client king of Rome, there wasn't much Herod could do about that (in fact, it is said Herod rented the plantations back from her and still turned a profit).\nHowever, Ariel notes, Herod, who was no stranger to romantic imbroglios, was able to resist the queen's wiles - unlike Julius Caesar and Marc Antony - and hold on to the rest of his kingdom.\nIn fact, Herod confessed to Augustus, the victor of the battle of Actium, that he had always counseled Marc Antony to kill Cleopatra, to put an end to the long civil war that tortured Rome after the assassination of Julius Caesar.\n\"But hers was a restless mind. She never shook off an exchange with her father from years before. Martinez has been browsing through his library in 1990, looking for something to read.\nShe picked up Shakespeare's Antony and Cleopatra, which her father dismissed. Cleopatra was a mere political schemer, he said, not even deserving of a historical footnote.\n\"Why do you speak of the queen like this?\" she recalls. \"How do you know? What are your sources?\"\nMartinez vowed to find out for herself, a daunting task in a small Caribbean island in a pre-Internet world. \" I read everything, Plato, Socrates, especially the Romans,\" she says, adding that she came to understand the axiom that history is written by the winners. \"Of course, the Romans hated the Egyptians. They attacked her beauty \u2014 she was plain, but it does not matter. Then I looked for Egyptian sources, and they described a different person. They said she was the greatest of her dynasty. She ruled the most important country in the world. She was the most important woman on earth. She was a queen when she was 18 years old. I concluded that Cleopatra was a special woman. She spoke nine languages, wrote about the law and medicine. Every day I got more and more interested in this woman.\"\nBut, being who she was, Martinez could not stop there. She devoured the Cleopatra story, trying to read past historical bias and understand the young queen as a woman and a mother. With this perspective, Cleopatra's political maneuverings and personal choices came into sharper relief. Even the oft-told story of how Cleopatra first presented herself to Julius Caesar, rolled up in a rug, is viewed by Martinez as nothing less than a stroke of military and strategic genius.\n\"Cleopatra was on the run, hiding in the desert and learned Caesar was in Alexandria,\" she said. \"She needed to meet him, but how? She came up with a plan, arranging for a rug to be presented as a gift from her. Now, this was a Roman general. The most protected man. If she were caught, she would be dead in an instant. It was incredible. Imagine [sneaking] into Barack Obama's office. I became an admirer of hers.\"\nI am an archaeologist and adventurer. I love dogs, food, wine, and books. Most of all, I love to dig.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WE SOMETIMES THINK that being religious or spiritual is a separate thing from everyday life. We might associate it with attending a Buddhist service in our main hall (hondo), chanting, or sitting in meditation--in short, removed from the every day. As Shin Buddhists, we take the view that our ordinary life is the spiritual journey. It is our attitude or state of mind that is important. It is how we see ourselves in the chaotic circus of modern life, and therefore navigate through that chaos.\nAs Shin Buddhists, we take the view that our ordinary life is the spiritual journey.\nFor if we are to live fully, we must live fully in the present moment. That present moment is not compartmentalized for only \"spiritual\" times. To live fully we are to be present in every present moment.\nHOW DO WE DO THAT? Our practice of voicing Namo Amida Butsu can be said to be an expression of gratitude for our assured awakening to this inconceivable life. that expression returns us the present moment more fully engaged than we were before.\nThe recitation, also known as Nembutsu, can be described in various ways, but of course, remains beyond words.\nHere we apply the rule of receiving, rather than achieving. We may be forgetting that time when we were 2-year-olds, and we were the recipients of so much compassion, when many people took care of us. As my grandson runs around a store, is he aware that parental eyes are insuring his safety? As we conduct the business of our lives, are we aware that there are causes and conditions that sustain our life?\nEspecially in 21st century America, we enjoy the democratic ideals of laws and an ethos of the common good. As we become aware that our individual ego is supported by a community and universal sense of Oneness, we realize our responsibility in supporting others for the common good.\nSee the welcome from Temple President LuAnn Lee.\nSee also the welcome to our path.\nVIDEO: \"What Is Shin Buddhism?\" (In \"Carving the Divine\" Series) Sept. 2018.\nPODCAST: \"Learning about Buddhism\" (In \"Learning about Other Religions\" Series at the Pioneer Ocean View United Church of Christ) Sept. 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Socks are just like women. They can be straightforward and polite at the same time. It doesn't mean they're any more or less of a sock. Or a woman. Women's shoe size 5-10. 51% combed cotton; 47% nylon; 2% spandex. PLEASE NOTE: Blue Q socks feature panoramic art - awesome from every angle. There's no \"right and left.\" Our sassy phrases fall on the outside of your right leg, and the inside of your left.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DEERFIELD, Ill. (BUSINESS WIRE), July 26, 2016 - Walgreens Boots Alliance, Inc. (Nasdaq: WBA) (the \"company\") today announced that affiliates of Kohlberg Kravis Roberts & Co. L.P. (\"KKR,\" and together, the \"selling stockholders\") intend to offer for sale in an underwritten secondary offering 15,000,000 shares of the company's common stock pursuant to the shelf registration statement (File No. 333-209569) filed by the company with the Securities and Exchange Commission (the \"SEC\") on 17 February 2016.\nThe company previously issued to the selling stockholders an aggregate of 52,461,215 shares of common stock in connection with the company's strategic combination with Alliance Boots GmbH (\"Alliance Boots\") completed in December 2014. On 11 May 2016, the selling stockholders sold 15,000,000 shares in an underwritten secondary offering. Prior to the proposed offering, the selling stockholders owned 37,461,215 shares in the aggregate, representing approximately 3.5 percent of the company's outstanding shares of common stock, based on the number of shares outstanding as of 30 June 2016. Following the proposed offering, the selling stockholders will own 22,461,215 shares in the aggregate, representing approximately 2.1 percent of the company's outstanding shares of common stock, based on the number of shares outstanding as of 30 June 2016. The company is not selling any shares and will not receive any proceeds from the proposed offering. Pursuant to the Shareholders Agreement by and among the company, certain of the selling stockholders, including affiliates of KKR and certain other investors, dated as of 2 August 2012, as amended, upon completion of the proposed offering, KKR's contractual right to designate a nominee for election to the company's board of directors will terminate. However, Dominic Murphy, a senior executive at KKR, will remain a member of the company's board of directors after the completion of the proposed offering.\nMorgan Stanley will act as the sole underwriter for the offering. The last reported sale price of the company's common stock on 26 July 2016 was $81.33. The company filed an automatically effective shelf registration statement (including a prospectus) with the SEC for the offering to which this communication relates. Before making any investment decision, you should read the prospectus in that registration statement and other documents the company has filed with the SEC for more complete information about the company and this offering.\nMorgan Stanley will arrange to send you the prospectus supplement, when available, and the accompanying prospectus relating to the offering if you request them by contacting Morgan Stanley & Co. LLC - Attn: Prospectus Department - 180 Varick Street, 2nd Floor - New York, NY 10014.\nCautionary Note Regarding Forward-Looking Statements: All statements in this release that are not historical are forward-looking statements made pursuant to the safe harbor provisions of the Private Securities Litigation Reform Act of 1995. These forward-looking statements are not guarantees of future performance and involve risks, assumptions and uncertainties, including those described in Item 1A (Risk Factors) of our Form 10-K for the fiscal year ending 31 August 2015 and our subsequent Forms 10-Q, including our Form 10-Q for the fiscal quarter ended 31 May 2016, which are incorporated herein by reference, and in other documents that we file or furnish with the Securities and Exchange Commission. Should one or more of these risks or uncertainties materialize, or should underlying assumptions prove incorrect, actual results may vary materially. These forward-looking statements speak only as of the date they are made. Except to the extent required by law, we do not undertake, and expressly disclaim, any duty or obligation to update publicly any forward-looking statement after the date of this release, whether as a result of new information, future events, changes in assumptions or otherwise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Plasmapheresis is the process by which the plasma component of blood is separated in a specialized machine while all the other components of blood are returned to the body. The whole process takes place within an enclosed sterile environment.\nThe volume of plasma that is collected from the donor is defined based on the donor weight. Donors are monitored continuously by trained medical professionals who will ensure maximum comfort and safety during the donation process. A typical plasma donation takes approximately one hour.\nThe collected plasma is known as \"source plasma\". which is used as the raw material for further manufacturing into lifesaving plasma therapies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Natural stone has a high artistic charm. Many people want to use natural stones to add beauty and impression to their homes. Natural stones have various types. Therefore natural stones can be installed to meet outdoor or indoor beauty. You need to prevent natural stones damage due to rainwater on the use of natural stones outside the home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A recently leaked email from Laurent Tapie (courtesy of DiamondFlushPoker.com, link below) reveals his last minute attempts to salvage his dream of running an online poker room.\nAs the email indicates, Tapie wanted to run an online poker room really, really bad, but that dream ended up getting crushed by the US Department of Justice. The DoJ would reportedly only sign off on Groupe Bernard Tapie's proposed acquisition of Full Tilt Poker if the investment group agreed to repay 100% of the money owed to ROW (Rest of World) players within 90 days. GBT argued that this \"made the deal unworkable\", and the negotiations between the two sides were terminated.\nLaurent Tapie and Groupe Bernard Tapie worked for seven months to try and close the deal, and presumably spent millions and millions of dollars on lawyer fees and other expenses. It's no surprise then that Laurent Tapie was bitterly disappointed when the negotiations were terminated, and it's also no surprise that he tried to find an alternative method of keeping his dreams of running an online poker room alive.\nAccording to the email, Tapie first tried to \"license the assets\" of Full Tilt Poker on a temporary basis, \"in order to relaunch the site, preserve the value of the assets and participate in an eventual auction as a bidder in the future.\" The DoJ rejected this proposal and the negotiations between the two sides were officially terminated as of that point.\nAfter the DoJ's rejection, Laurent Tapie took the even bolder step of suggesting that a new online poker room should be built from the ground up. According to the email, Tapie wanted to launch \"a new website that will be best of breed in 6-7 months.\"\nTapie needed the help of \"key staff\" at Pocket Kings in order to launch his own poker room, and he offered to hire these \"key\" staff members at the same salaries and benefits as what they were earning at Pocket Kings. Tapie demanded that he hear back from these key staff members nearly immediately, as timing was \"critical if this is to work.\"\nNobody has said anything about a new poker room being launched by Tapie, so it's likely that he didn't receive the support of the key staff members that he was looking to hire. Given everything that has transpired over the past week or two, it's probably safe to assume that somebody told these staff members that they might want to stick around as Pokerstars may soon be closing on a deal to acquire the company.\n\"We will write a new poker software platform based on your knowledge and relaunch using a new brand, with an offer to former FTP ROW players that will allow them to recover their lost balances if they come to our new site, just as we had planned in our agreement with DOJ.\"\nIt remains to be seen whether or not Laurent Tapie will eventually find his way into the online poker world.\nSource: DiamondFlushPoker.com - New Plans for Tapie and Online Poker?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Researchers have found that patients who did not adhere to their medication schedule for chronic conditions, such as diabetes, prior to a breast cancer diagnosis were twice as likely to skip oral adjuvant hormonal therapy after their diagnosis.\nPatients who skipped medications for their chronic conditions had a 23 percent non-adherence rate to hormone treatment, compared with 10 percent for those who took one or more drugs for their conditions prior to a breast cancer diagnosis.\nOther factors included the patient's financial status and the cost of drugs.\nThe study, by researchers at Columbia University's Mailman School of Public Health, Columbia University Medical Center, and NewYork-Presbyterian, is among the first to associate prior medication adherence patterns and subsequent adherence for breast cancer drug therapy. The findings were published in JAMA Oncology.\n\"Given the fact that medications used in oncology are potentially life-saving or life-prolonging, it is surprising that non-adherence to these medications is common,\" said Alfred I. Neugut, MD, PhD, Myron M. Studner Professor of Cancer Research and Professor of Epidemiology at the Mailman School of Public Health and lead author of the study.\nAge had a strong association with adherence versus non-adherence. Compared to women with breast cancer between the ages of 55 and 64, women younger than 45 years and those between the ages of 45 to 54 were more likely to be less adherent, as were those 75 years of age or older. Women who were African-American, being treated by a specialist other than a medical oncologist, as well as those who had poor belief in the treatment's efficacy, lower financial resources, and higher co-payments for the drug, were also less likely to adhere to the treatment. As the 30-day out-of-pocket costs rose from $5, there was a gradual increase in non-adherence.\nHaving a greater number of chronic conditions and a history of non-adherence to chronic medications in the year prior to a patient's breast cancer diagnosis\u2014regardless of the number of medications a woman was taking\u2014was also associated with not taking the medications. \"This finding revealed that those who are non-adherent to chronic medications are at increased risk for non-adherence to hormone therapy and could benefit from vigilance and possible future interventions,\" noted Neugut.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Overlooking Pigeon River Vally in Michigan, Treetops Resort is home to 81 holes of championship golf including five courses by Robert Trent Jones Sr., Tom Fazio and Rick Smith. The resort also offers dirtbike riding, fly-fishing, kayaking, river tubing, and guided horse riding and trout fishing. A camp is also available for children aged six through twelve.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HomePosts Tagged \"Glen Eyre Halls\"\nBassett Avenue, near Glen Eyre Halls in Southampton has been closed after a serious collision.\nNew charitable organisation, Southampton Community Kitchen, are holding their first event this Friday.\nA hoax email, sent to the editors of the Wessex Scene and Soton Tab, leads to inquiry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"During a season that's filled with gorgeous hues ranging from red to gold to green, we like to let nature take center stage when creating an arrangement. While this approach can certainly lend itself to over-the-top stunners, sometimes a little restraint can take a grouping to quiet yet powerful heights. Here, we deconstruct our process for creating a beautifully simple fall centerpiece that will elevate any occasion.\nPare down as much as possible. When simplicity is the goal, editing down your elements to just a few selections is key. While we don't always insist on following a \"rule of three\" approach, when creating a simple fall centerpiece, we find that narrowing down our options\u2014such as leaves, pears, and a strategically placed fig\u2014always yields a visually strong result.\nFeel free to forage. Large leaves in a variety of fall colors provide the perfect base for a simple fall centerpiece. As an added bonus, they are often available in abundance (and don't cost a penny). If you have little ones, gathering pretty foliage is an ideal activity to delegate to children who are eager to be involved in the decorating process.\nIncorporate fruit. When it comes to our arrangements, we always opt to use what's in season whenever possible. This time of year, our favorite elements include pears, lady apples, persimmons, pomegranates, and citrus\u2014all of which add wonderful color, shape, and interest to the tablescape.\nAdd a touch of sparkle. Selecting a vessel that offers a bit of shine, such as a glass compote or silver pedestal, creates a beautiful contrast to the natural pieces included in the composition. Whether you opt for a more ornate option is up to you\u2014the simplicity is more about editing the elements involved in the grouping.\nBe selective about surrounding elements. Sometimes a single simple piece won't suffice. When that's the case, rather than taking a maximalist approach, bring in corresponding pieces that will complement your unfussy arrangement. Votives and additional foraged foliage, and strategically placed fruit are excellent ways to build around your simple centerpiece and opportunities further elevate your table by playing with scale.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Happy new year to all of you. 2007 sounds great. Like the number 7, perhaps not for sailsize, but 4.7 is ok. Listened a bit to the music of Shania Twain while the night, one of the famous musicians staying in the lengendary Hotel California in Todos Santos. The Eagles were never there. Just rumours. But sometimes it\u00b4s great to believe in rumours. They make stories more interesting. Will leave within three days to Asia. Must lok for getting my stuff in order. The windforecast for the Phillippines is quite promising. Perhaps I should pack my medium Slalom rocket, the Isonic 101, too. Perhaps not. Saw the aeroplanes on the youtube vids. They do not look so big. But seem to hit the goals. Hope not to get lost somewhere in between these 7000 islands. Anyway, my freestyler needs to be in the bag, because the spot looks so awesome. Would like to take so many toys and friends there, but will go alone and just with a limited ammount of gear. I\u00b4m not Brad Pitt. It is all about money.\nPerhaps it\u00b4s a chance to jump on the board tomorrow on our lake. Some warm air arrived from somewhere this evening. Let\u00b4s see.\nHope to be not too rusty on the board at the upcoming event. Had no chance to train here in Austria, too cold and no wind.\nLet\u00b4s go for it, play hard.\nSomewhen in warm waters on the Canaries. Like the Challenger Proto 5.0 m\u00b2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Huddersfield Town defender Christopher Schindler was quick to jump to the defence of team-mate Mark Hudson after a horror show against Bristol City.\nThe hosts took the lead on the half hour mark after Hudson lost the ball under pressure from Tammy Abraham, allowing Lee Tomlin to advance round goalkeeper Danny Ward to open the scoring.\nAnd that goal came just minutes after the match had restarted following a serious injury to Jonathan Hogg in which the midfielder had collided with the Huddersfield Town captain.\nBristol City went on to win the game 4-0 with further goals from Abraham, Aden Flint and a David Cotterill penalty on a night of a number of poor performances for David Wagner's men.\n\"I think the whole trip was difficult from start to finish,\" said Christopher Schindler. \"Michael Hefele feeling sick before the game and then the injury to Jonathan Hogg.\nBut the 26-year-old was quick to echo the sentiments of his manager in that there can be no excuses for the poor performance at Ashton Gate.\n\"Yes it was a horrible scene but it can't excuse what happened next \u2013 after that we made mistakes and they (Bristol City) exploited it,\" the defender said.\n\"We cannot excuse how we performed today and its really tough to accept at this moment.\nAnd when asked about criticism levelled at Mark Hudson, whose previous two away outings at the heart of defence alongside Schindler had seen eight goals shipped, he was quick to jump to his captain's defence.\n\"No, No, No \u2013 that's not right,\" Schindler said. \"At the beginning of the season the skipper played really well and is absolutely reliable.\n\"Today we made too many mistakes in defence and that is how you lose here \u2013 we knew they had been struggling with results but deep down they are a good team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Take on the role of a popular inventor and go on your mission to defeat all enemy towers in the cool strategy game Tesla: War of Currents.\nAll games and trademarks are copyright their respective owners. Contact Us if you feel Tesla: War of Currents has been used without proper permission.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are many arguments both for and against the passage of California's Regulate, Control, and Tax Cannabis act. The debate about legalization has been raging for over a hundred years, and when California voters went to the polls in November, the results were closely watched and hotly debated for not legalizing Cannabis. Opinions are varied depending on the source while the medical uses of marijuana are under debate; many doctors agree there are some medical uses for the drug. Bryan Hughes represents District Five in the Texas House of Representatives and has written many papers for not legalizing Cannabis and his opinion is that the legalization of recreational use could lead to many negative health effects. Seeing how Marijuana is mainly ingested through smoking, and the smoking of any substance, especially long-term, can seriously damage a person's lungs. It has also been shown that frequent heavy use of marijuana can permanently impair short-term memory and reaction time. These opponents argue that legalized recreational marijuana would increase the use of marijuana, which would be a danger to public health. (Hughes B. 2010). As stated by the Citizens Against Legalizing Marijuana (CALM) it takes the position that Federal laws against the use, cultivation, and transportation of marijuana should be maintained and enforced and should not be relaxed or softened. Cultivation, processing, transportation and use are illegal under the Federal Controlled Substance Act. Marijuana continues to be identified by Congress and the DEA as a schedule I drug. Schedule I drugs are classified as having a high potential for abuse, no currently accepted medical use in treatment, and a lack of accepted safety for use. After decades of study the FDA continues to reaffirm that there is no medical benefit provided by the use of smoked marijuana and that, in fact, considerable harm can be caused by such use.(FDA. 2012). CALM is opposed to any State law that attempts to undermine the Federal Controlled Substance Act. Their position is that California Proposition 215 was misrepresented under the misguided claim that there is substantial \"medical\" benefit from ingesting marijuana and that the harm and negative impacts would be negligible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Kirkland's has a wide selection of kitchen and dining room furniture, handpicked just for you! Whether your style is contemporary or traditional, we have many pieces to fit your individual taste. Shop our entire collection of dining tables and dining chairs, as well as kitchen furniture like cabinets and stools.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Visit this fertile green plateau in the middle of the Dicti Mountains which was once peppered with thousands of white-sailed windmills. On the mountain road the pretty nunnery of Kera Kardiotissa is home to a famous icon of the Holy Mother as well as to frescoes from the 13th and 14th century. Driving south and up to the mountains, we reach the Plateau of Lassithi which may have been the crater of an ancient volcano. A short climb will bring you to a very special place in Greek mythology, the Dicti Cave, where Zeus, the father of all Greek gods, was born. The cave is also famous for its impressive stalactites and stalagmites. We make a short stop for a visit to a traditional potter's factory to see this ancient ceramic art which is still alive today. You will see the artist at work and you can even try, with his help of course, to make your own pottery! The village of Tzermiado will be our stop for lunch and we will eat at one of the local tavernas. The Plateau is a wonderfully cool and fertile area, with friendly people and a very warm welcome.\nThe Dynasty of King Minos ruled here and it become the prosperous capital of Minoan Crete.\nIt was multi-storey and had an intricate plan, which is why the Palace is connected with thrilling legends, such as the myth of the labyrinth and the Minotaur. It was discovered in 1878 and was excavated by Sir Arthur Evans who revealed the most civilized area in Europe!\nA day that combines the myth of Greek Gods, the Minoan civilization, the Byzantine period and modern life and culture in only one tour.\n*Appropriate dress when entering churches and monasteries.\n*A light jacket is necessary as Lassithi is always cool.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2190 It costs >\u00a3250k to train a Doctor so why would you question the value of putting up to date Medical Reference and Text books on their smartphones?\nIf Susannah can have her 'Skinny Jeans' why can't the rest of us use the pics we upload to Facebook from our mobiles?\nI've explained at length the fundamental difference of opinion that I have with Susannah Fox's understanding of mHealth (at Pew Research it appears that unless you expressly download a Health App to a smartphone it doesn't count) but the findings from this latest survey about \"Tracking for Health\" really highlight how some basic errors are being made when we ignore the widespread use of familiar mobile technology for self tracking.\nWhile the report claims that \"up until now there has been no measure of how many people engage\" in the activity of \"Keeping notes on one's health\" and that a \"national telephone survey conducted by the Pew Research Center's Internet & American Life Project\" has now found that \"69% of U.S. adults keep track of at least one health indicator such as weight, diet, exercise routine, or symptom\" and \"People living with chronic conditions are significantly more likely to track a health indicator or symptom\".\nI think a much more accurate conclusion would be that 'people living with chronic conditions are significantly more likely to tell someone who phones them up as part of a national telephone survey that they keep track of a health indicator or symptom'.\nWhen Pew asked respondents to \"think about the health indicator they pay the most attention to, either for themselves or someone else, and to tell us how they track it\" I think it's not surprising that only \"21% say they use some form of technology to track their health data\" because most of us wouldn't consider looking at our photo album or our friends commenting on how good we look in a comment on a Facebook photo album as a 'use of technology to track our health data' because we took and shared these pictures for very different reasons (eg. to make a record of an important social\/family occasion).\nThe claim the report makes that \"most do not share the data with anyone else\" goes out the window when you appreciate the value of photos for health tracking (something that is highlighted when we read about how a critical diagnosis was made from a photo that was socially shared) and the fact that most people in the USA have used Facebook and the most popular activity on Facebook is photo sharing.\nI'm surprised that Susannah continues to miss the point that we are already widely using mobile tech and services for healthcare roles. We might think like Susannah that we're not self-trackers because the only form of measurement we are conscious of is our pair of 'skinny jeans' but you're probably already doing this unconsciously.\n8 Responses to If Susannah can have her 'Skinny Jeans' why can't the rest of us use the pics we upload to Facebook from our mobiles?\nSusannah's skinny jeans are donned with a specific self-tracking intention in mind: do they fit? have I lost\/gained weight? do I need to go shopping? But sharing photos on Facebook does not share a self-tracking for health indicators goal.\nReferring to \"our friends commenting on how good we look in a comment on a Facebook photo album\" as self-tracking is a bit of a stretch, much like saying that out loud to somebody in person is. Besides the (rare) example of medical diagnosis made via a photo shared on social media, the technology examples you give are not doing health tracking per se based on their intentions.\nHello again! I have a few reactions to share to this post, which is now a year old. Thanks for resurfacing it in a tweet today.\nI agree with Carolyn's comment above, but I also take your point that we could add photos to the list of possible tracking activities.\nYou may not agree with how we measure the uptake of health apps, or the use of mobile in self-tracking, or the use of mobile in health generally, but to say I miss the point entirely goes too far.\nI'm sorry that you disagree with my opinion but please don't feel my comment is something personal. I try to fire people up about the mHealth opportunity and perhaps I'm completely wrong in my assertion that mHealth was well established even before we had the ability to download dedicated mHealth apps.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Several months ago I went to a free lunch put on by Melanie Driscoll to learn about hearing aids. I had noticed that I could hear the TV and people talking on the phone, but I could not figure out what was being said. I also had trouble hearing people sitting across from me at meetings and in crowds.\nMelanie made sure we'd all had hearing tests, and if not, she tested us there. She gave us each a trial pair of hearing aids to wear for a couple weeks. She then came to my home and talked to me about the possibility of buying a pair, and she worked with me to get the most reasonable price for the product that best fit my budget and me. I bought Unitron hearing aids, and I have been very satisfied with them.\nMelanie came back to my home at my convenience a couple weeks later to see how I was doing and to do any adjusting that I needed. She has kept in contact with me. She came by my home again at about the three-month period to make sure I was still satisfied with them. She is not the type of person who forgets about you once she has your money. I applaud her work ethic. She takes pride in her job and in the products she sells.\nIf you have need of an audiologist, I highly recommend Melanie Driscoll. My experience with Melanie has been one of a well-qualified professional with no high pressure and very sincere friendliness. I am so happy I chose Melanie, on the advice of a friend, to help with my hearing problems. Thank you, Melanie.\nMelanie Driscoll delivers the kind of individualized care that is so rare these days. She seems to take a personal interest in finding the best possible hearing solution for each person. She is my audiologist for life!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Danielle Levitt is a New York-based director and photographer with a wide following in the fashion, art, and commercial worlds. In 2016, her first year as a commercial director, Danielle garnered 2 Silver Lions, a Silver Pencil, and a Creative Arts Emmy for Best Commercial with her Ad Council spot \"Love Has No Labels.\" This spot ranked as the second most watched spot on YouTube in 2015, with 58 million views and counting. Since joining Tool, Danielle has directed her first VR project, 'Merck for Mothers,' which was shot in Uganda; a spot for Crocs that features John Cena and Drew Barrymore; and spots for Nordstrom and Uber. Danielle's PSA for the National Down Syndrome Society earned her a Gold Lion.\nHer vibrant and unique body of work covers portraiture and editorials for Vanity Fair, the New York Times Magazine, Time, and GQ, and advertising for clients such as Nike, Reebok, Sprite, Microsoft, Apple, Dazed, i-D, and Vice.\nDanielle is signed with DS Reps for Photography, as well. See her work here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Warm & stunningly romantic this gorgeous editorial dreamed up by Nicole Berrett Photography & Jen Rios Weddings was inspired by the pantone color of the year, marsala, the amazing history of the venue, Thistle Hill & southern simplicity.\nThe paper suite incorporated some of our favorite elements; a custom crest, modern calligraphy, illustrations of venue & vintage stamps galore. And who could forget the lovely watercolor floral linter inspired by the gorgeous arrangements the Southern Table created! Inspired by these amazing florals Jar Cakery & Lovely Paper Things collaborated to created a tasty & unique marsala, meets watercolor, meets decadent real flowers cake!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you see anything you would like added to the website please let us know by clicking HERE to send us your ideas. Our club was formed in 1990 by a group of local bicycle enthusiasts who are interested in collecting, restoring and preserving antique bicycles for the pleasure of this and future generations.\nNEWS UPDATE: The 15th of September proved to be great for our two winners, Tammy and Wayne.\nOur members enjoy getting together to share each other's collections. They collect all kinds of bicycles, everything from antiques dating from the 1800's to the classic muscle bicycles of the 1970's Our members have highwheels, phantoms, sting-rays, krates, as well as special interest bicycles like the lowrider.\nThank you for visiting our website. We hope you enjoy your stay.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PVC-U Gate valve industry series on both sides with adhesive socket. These VDL gate valves are characterized by their high quality and built-in EPDM gasket.\nApplication: Pond construction, pool and pool construction, piping and plant construction. Available in sizes from 50 - 160 mm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Africa Rice, in conjunction with Agricultural Transformation Support Programme Phase I (ATASP-1), has trained a total of 6,436 rice farmers on good agricultural practices in the Kano and Jigawa staple crops processing zone.\nThe Zonal Research Associate of Africa Rice, Mr Bisi Ilebani, made this known during a field trip to commemorate Farmer's Field Day at Zangon Buhari village of Bunkure Local Government Area, Kano State.\n\"Out of the number of trained farmers, 1,097 were provided with improved seeds to cultivate 201.46 hectares and 260 were provided with improved seeds and other inputs at 50 per cent subsidised rate to cultivate 144.1 hectares,\" he said.\n\"The second category was also provided with breeder seeds and other inputs at 50 per cent discount and were subsequently linked up with seed companies and farmers around the communities to offtake their produce,\" he stated.\nOne of the beneficiaries of the seed production technology training, Sa'adu Tasiu, said before the training, he used to cultivate a maximum of 35 bags of rice per acre, but with the training and improved seeds, he harvested 65 bags per acre last year and projected about 75 bags this year.\nHe called on the ATASP and Africa Rice to expand the programme so that more farmers could benefit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These terms are often used together, the first designating one of the main features of the process we know as secularization, the second a supposed undoing of the first, which can be either desired or feared, according to one's point of view.\nBut their relation is more complicated than this. In some sense, it can be argued, the process of disenchantment is irreversible. The aspiration to reenchant \u2026 points to a different process, which may indeed reproduce features analogous to the enchanted world, but does not in any simple sense restore it.\nLet's speak of \"the enchanted world\" to designate those features which disenchantment did away with. There are two main ones.\nThe first feature of this world is that it was one filled with spirits and moral forces, and one, moreover, in which these forces impinged on human beings; that is, the boundary between the self and these forces was somewhat porous. There were spirits of the wood, or of the wilderness areas. There were objects with powers to wreak good or ill, such as relics (good) and love potions (not so unambiguously good). I speak of \"moral\" forces to mark this point, that the causality of certain physical objects was directed to good or ill. So a phial of water from Canterbury (which must contain some blood of the martyr Thomas a Beckett) could have a curative effect on any ill you were suffering from. In this it was quite unlike a modern medical drug that \"targets\" certain maladies and conditions, owing to its chemical constitution.\nOne could sum this up by saying that this was a world of\"magic.\" This is implied in our term \"disenchantment,\" which can be thought of as a process of removing the magic. This is even clearer in the original German: Weber's Entzauberung contains the word Zauber (magic). But this is less illuminating than it seems. The process of disenchantment, carried out first for religious reasons, consisted of delegitimizing all the practices for dealing with spirits and forces, because they allegedly either neglected the power of God or directly went against it. Rituals of this kind were supposed to have power of themselves, hence were blasphemous. All such rituals were put into a category of \"magic.\" The category was constituted by the rejection, rather than providing a clear reason for the rejection. It then carries on in Western culture even after the decline of faith\u2014for example. Frazer's distinction magic\/religion. Only when Westerners attempted to make ethnographic studies of non-Western societies did it become clear how inadequate and instable this category is.\nI talked about not being able to go back. But surely lots of our contemporaries are already ''back'' in this world. They believe in and practice certain rituals to restore health or give them success. The mentality survives, even if underground. That is true; much survives of the earlier epoch. But the big change, which would be hard to undo, is that which has replaced the porous selves of yore with what I would describe as \"buffered\" selves.\nFor a Pagan, as history shows, is a man eminently convertible to Christianity. He is, essentially, the pre-Christian, or sub-Christian, religious man. The post-Christian men of our own day differ from his as much as a divorc\u00e9e differs from a virgin. The Christian and the Pagan have much more in common with one another than either has with the writers of the New Statesman; and those writers would of course agree with me.\nOne can see how this entails a kind of disenchantment: \"we reject the sacramentals; all the elements of 'magic' in the old religion\" (p. 79). If the church no longer has \"good\" magic, \"then all magic must be black\" (p. 80); all enchantment must be blasphemous, idolatrous, even demonic. \u2026The Reformer's rejection of sacramentalism is the beginning of naturalism, or at least opens the door to its possibility. It is also the beginning of a certain evacuation of the sacred as present in the world. And that leads to a completely new understanding of social and cultural life as well. \u2026The king or monarch can't be any sort of \"sacramental\" reality.\n\u2026The \"naturalization\" that is essential to exclusive humanism was first motivated by Christian devotion. \"The irony is that just this, so much the fruit of devotion and faith, prepares the ground for an escape from faith, into a purely immanent world\" (p. 145).\n\u2026Most germane to understanding the point of this chapter is appreciating what Taylor calls \"the triple embedding\" of premodern societies, a configuration of society that goes along with what he's been calling enchantment: \"Human agents are embedded in society, society in the cosmos, and the cosmos incorporates the divine\" (p. 152).\nTheology and Church life involve a certain conception of the human being: personhood. This term, sanctified through its use in connection with the very being of God and of Christ, is rich in its implications.\nThe Person is otherness in communion and communion in otherness. The Person is an identity that emerges through relationship (schesis, in the terminology of the Fathers); it is an \"I\" that can exist only as long as it relates to a \"Thou\" which affirms its existence and its otherness. If we isolate the \"I\" from the \"Thou,\" we lose not only its otherness but also its very being; it simply cannot be without the other. This is what distinguishes the person from the individual.\nThe Orthodox understanding of the Holy Trinity is the only way to arrive at this concept of Personhood: the Father cannot be conceived for a moment without the Son and the Spirit, and the same applies to the other two Persons in their relation with the Father and with each other. At the same time each of these Persons is so unique that their hypostatic or personal properties are totally incommunicable from one Person to the Other.\nPersonhood is inconceivable without freedom; it is the freedom of being other. I hesitate to say \"different\" instead of \"other\" because \"different\" can be understood in the sense of qualities (clever, beautiful, holy, etc.), which is not what the person is about. In God all such qualities are common to the each three Persons. Person implies not simply the freedom to have different qualities but mainly the freedom simply to be yourself. This means that a person is not subject to norms and stereotypes and cannot be classified in any way; its uniqueness is absolute. This means that only a person is free in the true sense.\nAnd yet one person is no person; freedom is not freedom from the other but freedom for the other. Freedom becomes identical with love. God is love because He is Trinity. We can love only if we are persons, allowing the other to be truly other and yet be in communion with us. If we love the other not in spite of his or her being different but because they are different from us, or rather other than ourselves, we live in freedom as love and in love as freedom.\nThe other is a condition of our freedom. Freedom is not from but for something other than ourselves. This makes the person ecstatic, going outside and beyond the boundaries of the self. But this ecstasis is not to be understood as a movement towards the unknown and the infinite; it is a movement of affirmation of the other.\nThis drive of personhood towards the affirmation of the other is so strong that is not limited to the \"other\" that already exists but wants to affirm an \"other\" which is totally free grace of the person. Just as God created the world as free grace, so the person wants to create its own \"other.\" This is what happens with art: the artist creating a totally other identity as an act of freedom and communion. Living in the Church in communion with the other means, therefore, creating a culture. The Orthodox Church has always been culturally creative.\nFinally, we must consider the ecological problem. The threat to God's creation is due to a crisis between the human being and the otherness of the rest of creation. Man does not respect the otherness of what is not human; he tends to absorb it into himself.\nThis is the cause of the ecological problem. In a desperate attempt to correct this, Man may easily fall into the pagan alternative: to absorb Man into nature. We have to be very careful. Out of its tradition, Orthodoxy is called to offer the right Christian answer to the problem. Nature is the \"other\" that Man is called to bring into communion with himself, affirming it as \"very good\" through personal creativity.\nThis is what happens in the Eucharist where the natural elements of bread and wine are so affirmed that they acquire personal qualities \u2014 the Body and Blood of Christ \u2014 in the event of the communion of the Holy Spirit. Similarly, in a para-eucharistic way, all forms of true culture and art are ways of treating nature as otherness in communion, and these are the only healthy antidotes to the ecological illness.\nThe tree is no impression, no play of my imagination, no aspect of a mood; it confronts me bodily and has to deal with me as I must deal with it\u2014only differently. One should not try to dilute the meaning of the relation: relation is reciprocity. Does the tree then have consciousness, similar to our own? I have no experience of that. But thinking that you have brought this off in your own case, must you again divide the indivisible? What I encounter is neither the soul of a tree nor a dryad, but the tree itself.\nBut perhaps that is exactly what encountering \"the soul of a tree or a dryad\" is like. Something is really there, and we describe it in the terms with which we are most familiar, while also knowing that those familiar terms are strictly inapplicable. Or rather, in applying them to trees or rocks or stars, we may suddenly find that they are unfamiliar terms, and that we hardly know even what other human creatures are. We may find that we are in Fairyland already.\nFrom Stephen R. L. Clark, professor of philosophy emeritus at the University of Liverpool in this article for First Things.\nWhoever wants to become a Christian must first become a poet.\nWounded by Love by Elder Porphyrios (translated by John Raffan).","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Piano Trio in G, H. XV No.25 - \"Gipsy\"\nAdagio in E flat, Op.posth. 148 D.897 \"Notturno\"\nPiano Trio No.4 in A, Op.posth.\nPiano Trio in E minor, Op.90 - \"Dumky\"\nHandsomely packaged in a gold, burgundy and blue box, this is a feast of extremely classy chamber music-making . . . [the set of the Haydn trios] is one of their greatest achievements; it won them a \"Gramophone\" Recording of the Year Award back in 1979 and the energy, imagination and sheer style still bring a smile to the lips. It's simply a magnificent example of chamber musicianship . . . [the two Beethoven cycles, recorded 20 years apart] make for rewarding listening and the differences are fascinating . . . the Beaux Arts Trio were champions of newer or more unusual music for their line-up and this set contains splendid performances of trios by Hummel, Arensky, Clara Schumann, Turina, Korngold, Zemlinsky, Ives (a superb piece), Rochberg, Baker and Rorem . . . the Brahms piano quartets fairly crackle with excitement . . . Other fine performances with the expanded BAT include Schumann's Piano Quartet and Quintet and Schubert's \"Trout\" . . . [there is very little available today that is] as heart-warming, uplifting and lasting as this superb legacy.\n. . . this 60-CD set, sturdily housed and sensibly packaged, with a carefully documented program booklet, provides an instant, highly distinguished chamber music library . . . the Beaux Arts brand is a reliable guarantee of polished ensemble, thoughtful interpretation, clarity, and a high level of energy . . . [their 1966 performance of the Ravel Trio is] a magical, unself-conscious performance in which Guilet's clear, subtle playing has a wonderful coolness that well suits the music . . . Pressler's precision, Greenhouse's refinement, and Guilet's finely focused sound make their Mozart and Beethoven Trios -- and I'd include the Classically oriented Schubert Sonata in B-flat, D 28, with them -- crisp, poised readings that hold considerably more than historical interest . . . The Beaux Arts was an ideal ensemble for Haydn, whose 43 trios occupy discs 1-9. Menahem Pressler's alert piano playing, with its distinctively genial feeling, is perfectly suited to these piano-dominated works. The elegance of his phrasing, his patient savoring of the slow movements, and his rhythmic snap and infectious energy in the faster ones are matched by Isidore Cohen, who plays with sweet tone and supple articulation. Cellist Bernard Greenhouse (1916-2011) doubles the piano part's bass line with relish and enhancing shape . . . the Beaux Arts is more or less perfect in these works . . . The three Brahms Piano Quartets, with violist Walter Trampler, from 1974 are superlatively good, with the A-Major being the best performance I know of that work, thanks especially to Pressler's crystalline sound and generous lyrical impulse. With Trampler again, the Beaux Arts is able to infuse an uneven work such as the Dvorak First Piano Quartet with so much grace and charm that it becomes an irresistible performance, and Pressler's scintillating contribution to the Schubert \"Trout\" Quintet makes theirs an outstanding version . . . this collection provides the evidence of their success in raising a piano trio's level of ensemble to that of a great string quartet. The kind of detail and refinement that they developed has become a prerequisite for the many accomplished piano trios that exist today.\n. . . d'une incontestable excellence musicale. Ce coffret, tr\u00e8s fourni et int\u00e9ressant, sur l'un des ensembles majeurs du XXe si\u00e8cle, embrasse quasiment tout le r\u00e9pertoire du trio avec piano . . . Le style du Beaux Arts Trio, en effet, c'est l'id\u00e9al pour le disque, l'exactitude et l'harmonie, la fluidit\u00e9, la clart\u00e9. Une interpr\u00e9tation des Beaux Arts est toujours au moins excellente . . . C'est cette capacit\u00e9 de travail, ce style absolument naturel, ce soin des moindres d\u00e9tails et cette hauteur tranquille qui font de leur int\u00e9grale des trios de Haydn un classique \u00e9ternel . . . Voil\u00e0 une somme unique.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the Fall 2010 issue, we ran a feature on Carlos Santana's guitar collection. Here's one of the groovy instruments that we couldn't squeeze into the article!\nIn the Fall 2010 issue, we ran a feature on Carlos Santana's guitar collection. Here's one of the groovy instruments that we couldn't squeeze into the article!\nIn the mid Eighties, Santana began to work with the visual artist Michael Rios, whose hallucinogenic, \"magical realist\" style is almost telepathically in sync with the guitarist's musical aesthetic and spiritual vision. Rios has created numerous posters, graphics, silk-screened t-shirts and album covers for Santana, notably the cover art for the multiplatinum Supernatural in 1999.\nRios also designed this guitar for Santana, painting the body of a Fernandes Strat-style guitar that was later retrofitted with a Fender neck. When Rios presented the instrument to Santana, the guitarist was struck by one image in particular\u2014the disembodied hand on the upper body bout, with a glowing triangular shape resting in the palm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pork cheeks are so unbelievably tender I can't help but resist ordering them whenever I go out for lunch. That's why I just had to make these at home myself. It may take some time but the longer it stews the better, so succulent and sweet, this is definitely a winner.\nIn a large casserole pan, pour in a splash of olive oil and begin by searing the meat and sprinkling a crushing of rock salt.\nChop the vegetables, leaving two garlic cloves whole and add them to the pan, covering with the red wine and sherry.\nTop up with the tomate triturado, beef stock dissolved in boiling water and the herbs and seasoning. Allow to stew for the next 2 \u2013 3 hours over a medium heat, topping up with water if required.\nServe alongside some crusty bread.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published in the Journal of Autism and Developmental Disorders, the test measures the extent to which adults are affected by repetitive behaviours \u2013 one of the criteria used to diagnose autism.\nThese behaviours include common habits and routines, such as lining up objects or arranging them in patterns, fiddling obsessively with objects, or insisting that aspects of a daily routine remain exactly the same.\nResearchers say that the test is a reliable method of measuring these behaviours to indicate when they are unusually frequent or severe.\nTo determine how reliable this adult self-assessment is, University autism experts and La Trobe University, Melbourne, trialled the test on British and Australian adults (229 participants in total) with and without an autism diagnosis.\nWhile adults without an autism diagnosis showed a high tendency for repetitive behaviours the individuals with an autism diagnosis consistently scored significantly higher on this measure.\nAutism is found in more than 1 in 100 of the population. It is hoped that the test will contribute to improvements in the diagnosis of autism.\n\"Many measures used for research and diagnoses of autism rely on parents, teachers or caregivers to report the behaviours of individuals with the condition,\" said Professor Sue Leekam, Chair of Autism and Director of the University's Wales Autism Research Centre.\nRepetitive behaviours are not just common in autism, they are also a symptom associated with Obsessive-Compulsive Disorder (OCD), Parkinson's disease and Tourette syndrome.\nThe test on its own cannot diagnose autism because repetitive behaviours are common to other conditions and because repetitive behaviours are only one criterion for a diagnosis of autism. The test has been designed to help clinicians in the diagnostic process.\nWhat is remarkable is that increased behaviours normally assessed in infancy can also be measured in a self report form in adulthood.\nThe next phase of the research will be to trial the test on people of all ages with autism before implementing its use in clinics across the UK.\nFunded by an ESRC studentship to PhD student Sarah Barrett, the research remains ongoing. People aged over 18 can participate in the research by visiting: http:\/\/sites.cardiff.ac.uk\/rbq2a\/online\/.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Muscle Force FX Trial Offer Review - Is it recommended?\nMuscle Force FX Trial Offer Review \u2013 Is it recommended?\nMuscle Force FX is a testosterone booster currently available for trial.\nWhat benefits will this supplement provide? Would we recommend it?\nOur review will look at this supplement and the overall offer to see whether we would recommend that you sign up or not.\n4 Is Muscle Force FX a scam?\nMuscle Force FX claims to help boost testosterone, which will help promote better health, increase endurance and boost libido.\nThere is conflicting information available on the Muscle Force FX website.\nThe website itself claims that the main ingredients are D-Aspartic Acid and L-Carnatine Tartrate.\nHowever, at the very bottom is a link to 'ingredients' that shows neither of these ingredients, with the ingredients listed as L-Arginine, L-Citrulline and Dipotassium Phosphate.\nRegardless of which ingredient information is correct there are still issues with the ingredients as the first does not mention any dosage information, while the second only uses a 'proprietary blend', which means that they do not have to disclose the full ingredient information in full.\nThis lack of ingredient information leaves me doubting the claimed benefits.\nYou may think that as this supplement is available for trial that it will be free of charge, but you would be incorrect about this assumption.\nIf you read the terms and conditions page you will see just how much this trial will cost you.\nWhen you first sign up to this offer there is a $4.95 shipping charge to pay. This allows you 14 days to try this product.\nOnce this trial has concluded you will be charged the full purchase price of $89.95.\nYou will also discover that you have been added to an auto-ship program, meaning that you will be sent further monthly packages at full price.\nTo cancel these auto-shipments you must contact Muscle Force FX.\nThere are a few issues that should make you rethink signing up for this offer.\nNot only is the ingredient information a little \"iffy\", but the cost of the trial is also quite high.\nI would suggest that you avoid this offer and look for an alternative.\nIt is also a good idea to speak to your bank to make sure no further charges are taken from your account.\nPlease leave your own review below if you have trialled this supplement.\nOur chosen testosterone booster is Testogen.\nIt is made from natural ingredients shown to promote testosterone production.\nTestogen is highly recommended, and as it is not available for trial you wont have to worry about encountering any hidden charges or auto-shipments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbibov b/data_all_eng_slimpj/shuffled/split2/finalzzzbibov
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+{"text":"Canan Alimdar had her first individual exhibit, \"From Wings to Kings\" in Argos in Cappadocia between August 8th and 14th.\n\"From Wings to Kings\" consists of two collections of the designer, \"Emperor\" and \"Mistique\" both of which she is inspired by the thousands years of different civilizations that passed through Anatolia.\nThe designer who has been intrigued by prehistoric times, created Emperor by observing different esthetic views of various time periods. Mistique is the designer's first creation which symbolizes her freedom after ending her professional career to dwell into the art of jewelry making . The dominant figure of this collection is her trademark birds.\nCanan Alimdar said that she chose Argos in Cappadocia's Bezirhane as it pairs perfectly with her designs inspired by ancient civilizations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In Lake Magdalene 33612 International Sign in Sarasota,St.Petersburg,Ft. Meyers and surrounding areas.\nAs a business owner, your decisions about how best to advertise your products and encourage your customers to create a lasting relationship with your brand definitely affect your bottom line. At we really understand how important these decisions are to you and your business, and work hard to provide the quality Serving West Central Florida \u00a0 that support your ideas and sales strategies. is here to help with all your Serving West Central Florida \u00a0 needs. We are also orlando signs specialists and would welcome an opportunity to serve you.\nA Serving West Central Florida sign is a great way to grab your customers' attention and draw them to your business. A truly high quality Serving West Central Florida sign is a huge part of any successful direct advertising campaign, and in some businesses could be the single most effective sign for attracting customers from afar. At we are experts at Serving West Central Florida signs and are ready to serve all your signage company needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enjoy incredible bay views from this beautiful home with open floor plan! 2,078 sq ft of main floor living includes designer kitchen with new appliances & remodeled bathrooms. Master bedroom as well as 2 additional bedrooms are all on the main floor. Fully finished walk-out basement has potential for a 4th bed\/office, large rec room and is plumbed for a 3rd bathroom just waiting for your finishing touches. Home also has access to the community beach. Must see!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The fascinating image is part of Bob Cut Hairstyles 2019 Luxury Best Bob Hairstyles for 2018 2019 has dimension 1900 x 1187 pixel. You can download and obtain the Bob Cut Hairstyles 2019 Luxury Best Bob Hairstyles for 2018 2019 images by click the download button below to get multiple high resolutionversions. Here is crucial information about Hairstyles. We have the resource more image about Hairstyles. Check it out for yourself! You can get Bob Cut Hairstyles 2019 Luxury Best Bob Hairstyles for 2018 2019 and see the Girly and Sweet with Bob Cut Hairstyles 2019 in here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chicago Bears head coach John Fox is taking a glass-half-full approach regarding his team's outlook for 2017. Fox told reporters that the Bears are in \"striking distance a season after going 3-13. \"Going back to a lot of the changes, we've had a lot of change,\" Fox said. \"I think we're better for it.\" \"I understand you have to win. And I finally feel like we're in striking distance.\" Of the QBs currently on its roster, Mark Sanchez threw the most passes last season \u2013 18 attempts as the Cowboys backup.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"STARINFOHUB: Download Most Usefull M-Parivahan App :- Launched By Transport Ministry.\nGives Transport Service access to residents through a portable based application. This application enables national with moment access to different data, administrations and utilities identified with the Transport Sector. Planned to acquire comfort to native and straightforwardness the framework.\nThe legislature has propelled two versatile applications, echallan and mParivahan, that give access to different administrations and data, and empower nationals to report any petty criminal offense or street mishap.\n\"Government has propelled two new versatile applications - eChallan and mParivahan - identified with street security and client help,\" Minister of State for Road, Transport and Highways PonRadhakrishnan said in a composed answer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"12.5\u2033 x 19\u2033 High quality glossy print.\nThe Beloved Lesbian Kiss painting collection began as a wedding gift for dear friends who were married shortly after gay marriage became legal in the state of Pennsylvania in 2014.\nThis is the ninth piece that I painted in the Beloved series of 12, and is available as a high quality glossy print.\nAbout the Original: 11\u2033x 14\u2033. Gouache. 2014.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nov 13 (AT) Anton Balasingham, chief negotiator of the LTTE and his Australian born wife Adele Balasingham left back for London this afternoon from the Katunayake international airport. Earlier, Balasingham and his wife left in the morning from Kilinochchi to Katunayake airport by an air force helicopter provided by the Sri Lanka Government. more..\nNov 13 (AT) Why you - our valued reader has to come forward to contribute to uphold the independence of Asian Tribune? It has become important for us to bring to the kind notice of our readers that Asian Tribune - an online news daily does not have any political agenda more..\nNov 13 (PIS) Question: How much has the Sri Lankan political discourse on devolution changed over the decade? President Kumaratunga: It was the first time that a Sri Lankan Government of its own free will, proposed devolution and the granting of full rights to all the minority communities. more..\nNov 13 (AFP) Former Sri Lankan Test coach Bruce Yardley said he felt vindicated by the ICC proposed changes to cricket's chucking law. Former Australian off-spinner Yardley, who first encouraged Muttiah Muralitharan to bowl his \"doosra\", was Sri Lankan national coach in 1997-98, and has always been adamant that Murali was not a chucker. more..\nNov 13 (LT) Wanni tiger organization's theoretician Anton Balasingham's statement that during the discussions in Oslo there was no agreement on a separate state is totally misleading says Vidar Helgesen, Deputy Foreign Minister of Norway. more..\nNov 13 (LT) Arms that have been smuggled in are the cause for the escalation of crime says IGP Chandra Fernando. He points out that the war that raged throughout for two decades is responsible for this situation. more..\nNov 13 (LT) A long served officer of the Defence Ministry and the present Defense Secretary Cyril Herath has submitted his resignation to President Chandrika Kumaratunga from his post owing to his advance age. more..\nMervyn Silva will change the game?\nNov 13 (LT) Labour Relations and Foreign Employment deputy minister Mervin Silva is reportedly planning to take a different turn next week when his supporters are to picket opposite the Ministry demanding his reinstatement. more..\nNov 13 (LT) A discussion between Vidar Helgesen, Norwegian Deputy Foreign Minister and the Leader of the Opposition was held today in the office of the opposition leader. It is learnt that the discussions were focused on the discussions the President has had with Wanni tiger organization. more..\nSri Lanka produces 35% of worlds crape rubber!\nNov 13 (LT) 35% of world's need in crape rubber is produced in Sri Lanka says the Rubber Research Institute. This was stated by Mr. L.M.K. Tillekeratne, Director of RRI at a meeting held in the RRI. more..\nNov 13 (AFP) Norway's latest bid to salvage Sri Lanka's faltering peace process ended without a breakthrough yesterday but the government and the rebels have pledged to continue to uphold a fragile truce. more..\nNov 13 (IB) Foreign Minister M. Morshed Khan who arrived here today delivered the letter of invitation from Prime Minister Begum Khaleda Zia to Sri Lankan President Chandrika Kumaratunga to participate at the 13th SAARC Summit to be held in Dhaka on 9-11 January 2005. more..\nNov 13 (CM) A 41-YEAR-OLD Sri Lankan woman was yesterday remanded in custody for four days in connection with the theft of \ufffd1,300 from her 97-year-old employer. The court heard that Costas Tylliris from the village of Lythrodontas had hired the suspect as a housemaid early in July last year. more..\nNov 13 (AME) With an almost doubling of profits and an anticipated 28 per cent growth next year, the award-winning Sri Lankan Airlines has plenty to shout about at this year's World Travel Market. more..\nNov 13 (DN) Ports, Aviation, Information and Media Minister and Education Deputy Minister Mangala Samaraweera, in a media communique yesterday came hard on certain media reports which highlighted the UNP's distortions about President Chandrika Kumaratunga's address to the National Advisory Council on Crime Prevention on November 9. more..\nNov 13 (ANN) Sri Lanka is likely to be able to enlist the support of international environmental groups on a campaign against the controversial Sethusamuduram Canal Project. The project has provoked a howl of protest from Sri Lanka as well as some influential quarters in India. more..\nNov 13 (HT) Indian Prime Minister Rajiv Gandhi had offered the LTTE chief, Velupillai Prabhakaran, a \"secret\" pact in July 1987, but the militant Tamil leader rejected it saying that he did not believe that India would honour that agreement, says the LTTE's political advisor, Anton Balasingham, more..\nNov 13 (IANS) Sri Lankan President Chandrika Kumaratunga said in remarks published Saturday that Tamil Tiger guerrillas are still trying to kill her despite signing a truce with her government. more..\nNov 13 (Xinhua) The chief peace negotiator of the Tamil Tigers and the Norwegian special peace envoy left Sri Lanka Saturday after their failed attempt to revive the stalled peace talks, officials said. more..\nNov 14 (TN) Minister of Estate Infrastructure Mr Muthu Sivalingam of the CWC told the press Friday that CWC will never permit the implementation of the upper Kotmale Hydro-power scheme and warned that UPFA Government will be forcedout of power if it proceeds with the project disregarding CWC's protest. more..\nNov 14 (TN) Janatha Vimukthi Peramuna (JVP) Sri Lanka's powerful Marxist Sinhala nationalist party and President Chandrika Kumaratunga main coalition partner, Saturday marked the 15th death anniversary of its founder Rohana Wijeweera with a massive public meeting at the Sugathadasa Indoor Stadium in Colombo. more..\nNov 14 (TN) Pungudutivu Division Civil Groups Consortium (PDCGC) Saturday wrote to Sri Lanka's President Ms Chandrika Kumaratunge to take immediate action to completely lift the residential pass system introduced for resettled families in Pungudutivu by the Sri Lanka Navy (SLN), more..\nNov 14 (TN) Hundreds of families affected by flood in Jaffna district due to torrential rain are not provided with any relief by the central government and provincial council authorities, civil group activists complained Saturday. more..\nNov 14 (TN) \"Destruction of Hindu shrines in areas where thousands of Hindus reside and construction of Buddhist shrines on State lands where there are hardly any Buddhists are blatant discriminatory acts and indicate State's willingness to condone acts committed in the name of Buddhism,\" said Mr.R.Sampanthan, more..\nNov 14 (TN) Jaffna District Fisheries Societies Federation (JDFSF) Saturday said hundreds of fishermen residing in coastal areas of Chullipuram would lose their profession following the construction of training centre along shore by the Sri Lanka Navy (SLN) violating the ceasefire agreement more..\nNov 14 (Dawn) The London-based LTTE theoretician Anton Balasingham, who arrived in the country last week , told journalists on Thursday that it was unlikely that the LTTE would be able to arrive at a power-sharing strategy with the government of President Kumaratunga. more..\nNov 14 (Rupavahini) This was reiterated by the leader of the JVP Somawansa Amarasinghe at the 15th November heroes commemoration ceremony held at the Sugathadasa Indoor Stadium in Colombo this afternoon. more..\nNov 14 (Rupavahini) President Chandrika Kumaratunge has pledged to continue the peace process in the country. Issuing a special message to mark her ten year tenure in high office, the president noted that she had taken measures to restore the freedom to live during this period. more..\nNov 14 (AT) Norwegian Peace Makers - they came last week and they left yesterday. Shuttle diplomacy was one more time a failure. Norwegian\ufffds ability to break the deadlock is now under microscope. How long this dilatory streak would continue? more..\nNov 14 (AT) Yasser Arafat\ufffds success as a leader can be best understood when compared with the failure of the \ufffdnation building\ufffd project in most other parts of the Third World, especially in Sri Lanka. more..\nNov 14 (SO) Transport Minister Felix Perera angered over the wildcat strikes of the private bus operators which put the commuters into untold hardship, and warned that he would leave no room in the future for these bus operators to hold the government and the commuters to ransom. more..\nNov 14 (SO) Eminent lawyers, professionals, top politicians and a cross section of the society opined that the Executive Presidential system should be scrapped in the interest of the country. Some others spoke in favour of amending or reshaping the system to make the Executive more answerable to the people and Parliament. more..\nNov 14 (TN) Tension prevailed near Kodikamam town in Jaffna Sunday morning as military and Police fired tear gas and assaulted civilians protesting against an accident on the A9 highway in which a Sri Lanka army vehicle knocked down and killed a young Hindu priest. more..\nNov 14 (TN) \ufffdOur objective is to spread computer literacy among the Tamil people. Our leader wants to create an e-society in the Tamil homeland. It is the foundation on which he plans to build an efficient e-government for our people. more..\nNov 14 (TN) Three-day workshop on Inter Faith Collaboration on Peace Process in Sri Lanka began Saturday at the Jaffna Young Men Christian Association (YMCA) auditorium located in Chundikuli, a suburb in Jaffna town. Dignitaries of all faiths participated in the inaugural event, sources said. more..\nNov 14 (SLA) Sri Lanka Light Infantry (SLLI), the oldest and most senior Regiment in the Sri Lanka Army on Thursday saw its men are felicitated and honoured at their much-awaited \ufffdColours and Awards Night \ufffd that took place at BMICH headed by Commander of the Army, Lt. General S. H. S Kottegoda more..\nNov 14 (HRW) The government of Sri Lanka and the Liberation Tigers of Tamil Eelam (LTTE) were engaged in a brutal civil war from 1983 to 2002 that killed over 60,000 people. A ceasefire was implemented in February of 2002, but a final peace agreement has not been reached. more..\nNov 14 (LBO) The Finance Minister Sarath Amunugama told Lanka Business Report (LBR) that he is counting on the rural sector to contribute to productivity improvement and help drive economic growth. more..\nNov 14 (LBO) Making a budget for a poor country with a mountain of debt, low investment, a large public service and high defense expenditure is no easy task. But this is the challenge facing the Finance Minister and his team when they prepare the UPFA government\ufffds first budget on Thursday. more..\nNov 14 (LBO) Tax experts say that government\ufffds ad hoc changes to taxes outside the budget harms the stability of the fiscal environment. They also believe that the Economic Service Charge has not looked positive on a cross section of businesses. more..\nNov 14 (G2L) The government continues to make choppers available for senior LTTE cadres to travel within the northern and eastern provinces. On the request made by the LTTE, the government on Friday airlifted several LTTE cadres from the east to Kilinochchi, authoritative sources said. more..\nNov 14 (SN) NMAT vows to safeguard Sinhalese National Movement Against Terrorism held its national conference today (13th of November) at S S Jayasinghe Memorial Hall Dehiwala at 3.30 PM. more..\nNov 14 (LT) The Government has decided to import another stock of rice from India and Pakistan. Accordingly, 50,000 metric tons of rice will be imported in the near future. Minister of Commerce and Consumer Affairs, Mr. Jeyraj Fernandopulle says the stock of imported rice will be made available to the market through CWE outlets. more..\nNov 14 (LT) The Leader of the Opposition and the President must agree for a solution that doesn't divide the country says Mahanayake of Malwatte Chapter. The Prelate made this statement when he met a contingent of UNP Parliamentarians led by Prof. G.L.Peiris and Malik Samarawickrema. more..\nNov 14 (LT) Anton Balasingham, theoretician of the Wanni tiger organization and Norwegian special peace envoy Eric Solheim who were in Sri Lanka left the island last evening. They held a special discussion at a special room in the air port when they arrived there. more..\nNov 14 (PTI) Sri Lanka was banking on a hard currency injection from India to save the local rupee amid galloping inflation, a widening budget deficit and dwindling foreign reserves. more..\nNov 14 (AFP) The latest failed attempt to revive peace talks is bad news for Sri Lanka's economy, already in trouble from high inflation, a foreign exchange crisis and a falling rupee. more..\nNov 14 (HT) The LTTE has said that India should consult the outfit before undertaking strategically important infrastructure projects in the North Eastern Province (NEP) of Sri Lanka, since these impinge on the lives of the Tamil people. more..\nNov 14 (AFP) Sri Lanka's peace hopes dimmed further as Norway failed in a secret attempt to save a stumbling initiative to end three decades of ethnic bloodshed, officials close to the process said. more..\nNov 14 (Bernama) Jeremy Carter, the IMF representative, is completing his term and leaving around next February amid speculation that he has been asked to leave by the Sri Lankan government. He was leaving at the end of his term, he told Bernama. more..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The practice of farming fish and shellfish is known as aquaculture. In Cornwall the main aquaculture species are currently mussels and oysters (shellfish). Currently the largest aquaculture operation is in St. Austell Bay, where two companies are successfully farming mussels using rope culture systems. This method of aquaculture has minimal impact on the environment and creates a natural refuge for other species (such as cod, bass and pollack) that feed around the hanging ropes of growing mussels. Oysters are farmed in the Helford and Camel Estuaries. The most common method used to farm oysters is bag culture from racks on the shore.\nFinfish aquaculture in Cornwall is restricted to a few small-scale trout farming operations, as marine finfish aquaculture has yet to be developed in Cornwall. In-situ marine aquaculture takes up space in the environment and is geographically restricted by exposure and the power of Cornwall's tides, waves and weather systems.\nFarmed bass and gilthead bream are often offered for sale in supermarkets and restaurants. These are not farmed in Cornwall, and the majority are imported from Mediterranean countries where fish are grown in lagoons and sea pens. There are many documented cases where this has been carried out with great detriment to the natural environment. The taste and quality of farmed fish is generally far poorer than that of wild caught fish, and they have a far greater carbon footprint and environmental cost. Farmed bass and bream are frequently offered for sale at a size significantly below the minimum landing size for wild caught fish.\nFind out more on our Aquaculture in Cornwall page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The busy railway station was a natural place for prostitutes to seek customers. This is a variant of an older song set beside the Thames (Roud 3457). It moves the action to a railway station just as prostitutes moved with their customers from river boats to railway trains.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjewl b/data_all_eng_slimpj/shuffled/split2/finalzzzbjewl
new file mode 100644
index 0000000000000000000000000000000000000000..7da374a409956181508385d32d9825dc3edf1d8c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjewl
@@ -0,0 +1,5 @@
+{"text":"We are pleased that to present major release of TeleForm. The new version 11 of TeleForm Multichannel capture platform contains useful improvements.\n- In addition to numerous features and improvements, TeleForm v11 brings full support for the latest Windows operating systems and database systems.\n- The new, optimized OCR and Barcode recognition increases the detection performance and ensures less rework!\n- The new user interface brings small but nice improvements for users of all TeleForm Program modules.\nIn addition, the web-based collection of images about TeleForm Web Capture a variant for mobile devices like smartphones and tablets have been added (TeleForm Mobile Web Capture).\nYou can now spread TeleForm to the e-mail channel. With the optional fuzzy logic boost your system throughput immensely by integrating reference databases.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unless otherwise specified in this specification, all requirements shall be mainly in accordance with the following standard specifications.\nThe outer sheath has no visible crack.\nStandard delivery lengths are 2km, 4km, 6km with a tolerance of -1%~+3%.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of quite possibly the most common electrical wiring inquiries is on how to wire a switch. Whilst employing switches at home is quite effortless, wiring just one might not be that easy for everyone. An ON-OFF switch is definitely quite simple to wire. There are several types of switches, but for this instance, let's say you happen to be setting up a single-pole toggle switch, an exceptionally widespread change (as well as the easiest).\nThere are actually a few shades of wires inside a normal single-pole toggle switch: black, white, and green. Splice the black wire in two and join them within the terminal screws - 1 on leading along with the other about the bottom screw of your swap. The white wire serves as a source of uninterrupted electric power and is also typically related into a light colored terminal screw (e.g., silver). Connect the eco-friendly wire for the ground screw of the swap.\nThese steps would usually be ample to create an average swap operate and not using a trouble. Nonetheless, when you are usually not self-confident you can execute the task correctly and securely you superior let the professionals get it done as an alternative. Following all, you can find a cause why this endeavor is without doubt one of the most commonly encountered electrical wiring queries requested by most people.\nFor a few purpose, tips on how to wire a ceiling fan is additionally one among one of the most widespread electrical wiring questions. To simplify this task, you may use just one swap for the single ceiling fan. To wire the fan, it can be only a issue of connecting the black wire of your ceiling fan into the black wire from the switch. When there is a light-weight, the blue wire needs to be linked for the black wire on the switch too.\nThere are actually good reasons why these are generally essentially the most usually questioned electrical wiring questions. One, numerous think it truly is easy to do, and two, these are generally the common electrical tasks in the home. But then you really mustn't put your protection in danger with your aim to save money. The stakes could even be much better should you try and save cash and do an electrical wiring task with no adequate expertise or expertise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let's face the facts: Given the number of professions, you're not likely an automotive expert. This is why a multi-point inspection service is so important, especially if there isn't anything obviously amiss with your Dodge, Jeep, Chrysler, or RAM vehicle. Ask the automotive professionals at Bryden Motors for multi-point inspection service at your next oil change or before your next trip.\nWhat is Multi-Point Inspection Service?\nWhen you ask for a multi-point inspection service, Factory-Certified Technicians, the ones who know your vehicle best, go over your vehicle with a critical eye. Fluids, like engine oil, engine coolant, brake fluid, clutch fluid, and transmission fluid, are checked for condition, level, and leaks. Brake pads, rotors, shoes, and drums are checked for safety, and exterior lights are checked for proper function. We'll also check your suspension, chassis, and several other aspects of your vehicle.\nWhy should I get a Multi-Point Inspection?\nAutomobiles are complicated machines and can wear out or break in a number of ways, no matter how well they're built. Some wear depends on how much you've driven your vehicle, as well as where and how. Other wear and breakage depend on mileage, vehicle use, and things you might encounter on the road. Often, such wear and tear completely unnoticed by the driver, but a multi-point inspection service can reveal smaller issues before they become big problems.\nNot everyone knows what to look for or listen to detect problems with their vehicles. Unless the problem is obvious, it might be completely overlooked. Small problems inevitably become big problems, which can be expensive or even impact your safety. By getting a multi-point inspection service, you can take care of these small problems before they get out of hand. If you need a repair, Bryden Motors uses only genuine MOPAR parts and accessories to restore your vehicle's performance and reliability.\nQuite simply, you should ask for a multi-point inspection service as Bryden Motors because we've been servicing MOPAR vehicles for over 70 years, and our Certified Technicians know your vehicles best. When you ask for a multi-point inspection, we're the best equipped to notice smaller problems that could cause you big headaches. Even better, it's complementary with most of the rest of our service menu, such as conventional oil and filter change services.\nIf you live or work around Beloit, WI, or Janesville, WI, or are just passing through Roscoe, IL, or Rockford, IL, you owe it to your vehicle to get a multi-point inspection service at Bryden Motors. Our comfortable waiting area features free WiFi, or you can take advantage of on-site car rental specials or our free shuttle service. We look forward to taking care of you and your vehicle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"B9Creations\tForum index B9 Core Series General Discussion Thank you for your ongoing support!\nIt's all too often people make complaints but for some reason have a harder time to stop and say thank you, you did a great job.\nWe've used B9 since the beginning and no matter what issue, large or small, their customer support is unsurpassed. Never had I felt so confident to recommend a company that will not only sell a great quality product, but also take care of any issues that may arise. On behalf of FireWorks Gallery, keep up the amazing work!!\nRe: Thank you for your ongoing support!\nI have to agree... B9 support is top tier!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjvrt b/data_all_eng_slimpj/shuffled/split2/finalzzzbjvrt
new file mode 100644
index 0000000000000000000000000000000000000000..877986dd2c02d411c5d9ac90c78b0c35a7bbcf03
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"If there's one thing both Chase Morrill and his brother-in-law, Ryan, love besides renovating cabins, it's beer. When they learn that the owner of their latest cabin project works at Allagash Brewing Company, they're excited to check out the brewery for inspiration as they turn the weekend getaway into a three-season home away from home. Located on the shore of Sabattus Pond, the cabin has a steep drop-off to the water, a front door in the wrong place, a kitchen that's too tiny and a unusable loft thanks to a low ceiling. Chase and the team brew up some incredible projects to reimagine the cabin footprint and use items from the brewery as well as custom pieces to transform it beyond their clients' wildest expectations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A domain name is something that identifies your webspace, for example schoolhosting.co.uk. So if someone were to type www.schoolhosting.co.uk into an internet browser they will arrive at our website. Likewise if someone types in an email address with @schoolhosting.co.uk after the @ sign, the email message will get routed into our own personal email inbox.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How Can Going Green Benefit Your Business Image?\nSend a clear message when choosing your conference bag or satchel.\nEscape the mundane and surprise your customers with branded gift or pillow boxes at the checkout.\n\"Thank you for your patience, assistance and speedy delivery, the lanyards and name tags look amazing: the conference attendees will look great!\"\n\"I can't praise you guys enough - customer service and speed is exceptional, every time. I am a customer for life.\"\n\"Just wanted to say again a HUGE thank you for everything that you guys pulled off for us! Every last-minute request we needed was just met with such a helping hand and without you guys, our event would have been a mess\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you been dreaming about working from your home office? Your opportunity has arrived! This organization has been closing billboard and cell tower acquisitions for over 10 years and can boast over $1 billion in capital and an amazing list of clientele, it's no wonder this firm is expanding and in need of a Vice President of Acquisition for the Seattle area!\nAs the Vice President of Acquisition, you will develop and manage a sales pipeline of cell tower and billboard ground leases utilizing a database of prospects. Your experience in commercial real estate or financial products has prepared you for managing transactions from lead generation through closing with high net worth individuals. As this company is an industry leader, they are in the position to offer a generous base salary, amazing commission structure and career advancement. Interview slots will be filled quickly, so don't delay \u2013 apply now!\n\u00ab Sales Manager \u2013 IT Storage \u2013 1st Year Target $160K!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How is weight different to mass?\nThe mass of an object is the amount of stuff that's in it. It is a scalar measurement (i.e. it only has magnitude), and we can measure it by using scales. It doesn't change irrespective of where an object is situated.\nWeight on the other hand is a vector measurement and is a type of force. The weight of an object will vary, depending on the gravitational pull of the environment it's in e.g. on earth or in space.\nThe gravitational constant doesn't change, but weight does. Why is this? The best way to understand this is to work through an example. Let's take this dumbbell, and work out how its weight would vary if you were to take it from the surface of the earth to the surface of the moon.\nIts weight changes because the mass and radius of the earth and moon are different compared to each other; but how different?\nRoll over the image below to see how the earth's mass and radius compares to that of the moon.\nthe smaller mass of the moon will reduce the dumbbell's weight by 81 times cf. to its weight on the earth.\nthe smaller radius of the moon will increase the dumbbell's weight by 13 times cf. to its weight on the earth (this is calculated by squaring 3.6).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblwdw b/data_all_eng_slimpj/shuffled/split2/finalzzzblwdw
new file mode 100644
index 0000000000000000000000000000000000000000..6e4191482e28734988a177f8fa2a749ab9a36289
--- /dev/null
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+{"text":"This brand-new warehouse is ideally located just off Doherty's Rd, only minutes from Robinson's and Palmer's Rd's, Boundary Rd, The Western Ring Rd, Western Highway and Geelong Freeway.\nWe have on offer a brand new Showroom complex, which is ideally located just off Robinsons Rd, Ravenhall. Positioned in front of the Orbis Estate, which is filled with High-profile occupiers. please find plans attached for showroom specifics.\nWe have on offer a brand\/new office warehouse, this building is a stand-alone property, providing a secure yard and unhindered access.\nWithin proximity of the Western Ring Road and other major road-infrastructure, this property will provide desired accessibility to Melbourne.\n*Total size: 462 SQM Approx.\n*Warehouse size: 406 SQM Approx.\n*Office size: 58 SQM Approx.\nARPBS Commercial have on offer a brand new warehouse in the Orbis Estate, Ravenhall.\n- Total size 300 sqm approx.\nPhone George on 0418315936 or Kylie on 0421368942 for an inspection today.\nARPBS Commercial have on offer a brand new warehouse for 'Lease\"\nThis warehouse is located minutes from the Western Ring road.\n- Size 240 sqm approx.\nCall George on 0418 315 936 for an inspection today.\n- Total size: 420 sqm Approx.\nCall George to arrange an inspection today on on 0418315936.\nAPRBS Commercial have on offer for lease a prime office\/warehouse. This property has exceptional exposure and freeway access.\nThis high clearance office\/ warehouse development presents a fantastic opportunity to occupy a modern warehouse space, located in the highly sought after Proximity Estate.\nFully secure site and no body corporate fees this property offers, immediate access to the Western Ring Road and Westgate Freeway.\nARPBS Commercial have on offer a brand new warehouse for lease in the sought after Truganina area located just off Dohertys road.\nARPBS Commercial have on offer a prime Office\/warehouse that is currently under construction, which is located in the Orbis Estate. This property is surrounded by high-profile occupiers and would be ideal for any business looking at positioning themselves amongst the best.\nWe have on offer a brand new office\/warehouse that is ready for occupancy. This warehouse presents excellent value and would be ideal for any business seeking a competitive advantage.\n*Total size: 457 SQM Approx.\n*Office size: 182 SQM Approx.\n*Warehouse size: 275 SQM Approx.\nARPBS Commercial have on offer a brand new Cafe\/Take away in the sought Palmers Gateway Project.\nWe have on offer a unique opportunity to lease a warehouse complex, which is ideally located within close proximity to major road linkages and Melbourne docks. This property would be ideal for any transport company that is looking at steping up in the industry.\nWe have on offer a new Office\/Warehouse that is currently under construction, with completion expected in the coming months. This Warehouse is located in the popular Orbis Estate, consequently providing easy access to Ballarat Rd and Westwood Drive.\n*Total Area: 448 SQM Approx.\n*Warehouse size: 352 SQM Approx.\nLooking for office space in the west?\nWe may have the perfect solution for you!\nThe landlord is looking for up to 4 people (individuals or one team) to share this near new stylish 100m2 space in Spotswood with fast, reliable, business grade internet.\nThe space is light and airy with lots of greenery. It's kept tidy and uncluttered \u2013 so the landlord welcomes folks that appreciate that too. The landlord likes peace and quiet while they're working (no loud music, sorry!) but loves having people around to chat with over coffee or share an after-work drink (music welcomed!).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wholesale Jerseys Free Shipping before Chief Justice Ian Chang - \u610f\u898b\u4ea4\u6d41 - \u7d2b\u7738 \u6d41\u884c\u6703\u9928 - Powered by Discuz!\nA Partnership for National Unity has said that it welcomes the recent withdrawal of the court action by the Attorney General, Anil Nandlall, and sees it as a vindication of its position that the Court has no jurisdiction in the internal affairs of the National Assembly,Air Max 1 Atmos For Sale, even if irregular, unless a constitutional breach was occasioned.APNU MP Basil WilliamsThis was expressed by APNU's Shadow Minister of Legal Affairs, Basil Williams during the party's weekly media briefing.Williams said the AG's withdrawal of his action must be taken as a concession that there was no constitutional breach, when the National Assembly passed the Motion of No Confidence, by Brigadier Granger, in Minister Clement Rohee.He added that APNU contends that the solution to resolving gridlock in the National Assembly lies not in the Courts, but rather in the willingness of all parties to work cooperatively in the interest of all of the people of Guyana, within and outside the hallowed National Assembly.\"Having lost the first court action, when he sought to have the court hold, as unconstitutional, the voting for the composition of the Committee of Selection and the composition of all National Assembly committees,Discount Nike Air Force One Shoes, unless they were composed in proportion to the seats obtained by the Parliamentary parties, at the 28th November 2011 General and Regional Elections, the AG is now seeking to avoid a further embarrassment\".According to Williams, \"when the matter was last called on Wednesday, last, before the learned Chief Justice, the Attorney General was again absent and cost was awarded to Brigadier Granger's lawyers in the sum of $75,Cheap Jerseys Online,000.00 dollars each.On the 13th day of September 2012, the AG had filed the said action seeking declarations that: The motion of no confidence,Cheap Nike Huarache Shoes, in Minister Clement Rohee, moved in the National Assembly, on the 25th day of July 2012, by Brigadier Granger, and duly passed on the 30th day of July 2012, and the Resolution No. 18 of 2012,Air Max 2018 Shoes, passed in the National Assembly on Monday 30th day of July 2012, were both unconstitutional, unlawful and violative of the doctrine of separation of powers\".The action was filed against Granger and Speaker of the National Assembly, Mr. Raphael Trotman.On October 30, a Summons to strike out the said action was filed,Cheap Jerseys Free Shipping, on behalf of Brigadier Granger, by Attorney-at-Law, Mr. Basil Williams, MP, in association with Mr. Rex Mc Kay, S.C. and several other lawyers.In the Affidavit in Support of that Summons, Brigadier Granger claimed that: The AG's action was without merit and misconceived; that Article 171(1) of the Constitution itself empowered every Member of the National Assembly to introduce any Bill, or propose any motion for debate in, or may present any Petition to the Assembly; and the same shall be disposed of according to the rules of procedure in the Assembly.Attorney General Anil NandlallThe said Affidavit further contended that Article 165 of the Constitution provides that the National Assembly may regulate its own procedures and make rules for that purpose. That in light of these provisions where Members of Parliament invoke them, no claim can succeed in a court of law on the grounds claimed by the AG.When the Summons came up for hearing on November 28, last, before Chief Justice Ian Chang, the AG was absent, and sent a State Counsel to seek an adjournment.\nGMT+8, 2019-4-19 20:24, Processed in 0.034621 second(s), 6 queries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After going down, down, down to the deepest moat in Europe, now I take you up to the tallest castle in France: the Ch\u00e2teau de Brissac. It also happens to be Gigi's favorite castle in Europe, and that's really saying something, because she's generally claims to suffer from a serious case of castle burn-out. In this case, it's not actually the height that impresses her, it's the history.\nThe rest of the world can go on their Kindles and iPhones, but the French still like good, old-fashioned books. Sometimes old-fashioned to the point of antique.\nThe French celebrate books and writers and poets and philosopher-poets and philosopher-writers like Americans honor athletes and war heroes. As somebody who loves a good book, it just makes me love the French more. A chicken in every pot? More like a bookstore on every corner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this episode of Around The Block we chat with Jon and Kimberly, residents of Admiral Heights. This is a great area of Annapolis that has something for everyone! The community boasts two docks, kayak storage, a swimming pool and much more!\nIf you're looking for that close community feel, this is definitely the place to be! Each month, the community sends out a newsletter with upcoming events. During the summer you can find neighbors hanging out at the pool in the evenings watching a movie outdoors, while the kids swim in the pool. There is also a local park where the community will host potluck gatherings. It's a great place to get to know your neighbors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have been also specializing in improving the things administration and QC system to ensure that we could preserve terrific gain within the fiercely-competitive company for Pallet Cover Recycling , Pallet Covers Strapping , pallet covers king , We have been self-assured to make excellent achievements within the future. We've been looking forward to becoming one within your most trusted suppliers.\n\"With all the \"\"Client-Oriented\"\" business enterprise philosophy, a rigorous quality regulate procedure, sophisticated production gear as well as a sturdy R&D team, we constantly give superior quality products and solutions, outstanding providers and competitive prices for Pallet Cover Recycling , Pallet Covers Strapping , pallet covers king , Our company absorbs new ideas strict quality control a full range of service tracking and adhere to make high-quality solutions. Our business aims to \"honest and trustworthy favorable price customer first\" so we won the trust of the majority of customers! If you are interested in our solutions and services please do not hesitate to contact us!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnehr b/data_all_eng_slimpj/shuffled/split2/finalzzzbnehr
new file mode 100644
index 0000000000000000000000000000000000000000..45c8a2fd9fb60de2a370e2d4c64b47b203fd097d
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+{"text":"This was much more interesting than most press releases I get. Unfortunately I couldn't make a story of it because _ no Dutch angle.\nI edited out some parts that went on a little too long with \"...\"\nSubject: [animallibmedia] Ask the Duck!\nAs the Frente de Liberacion Animal Mexico (FLAM) we feel that if to save animals from being mistreated, tortured, persecuted, hunted, imprisoned and murdered is to be a terrorist, then that is what we are and we love it!\nThe sun has already been hidden a thousand times, said the duck, while he watched the cars pass. He watched the people stop and with feelings of pity say: What a pretty imprisoned duck; the sad duck sleeps with no one for company, without knowing what it is to be truly free, waking every morning and looking at the sky, admiring his bird brothers and sisters flying from one side to the other. They stop in the trees and look for food; he listens to them singing happy songs; he dreams that some day freedom will arrive and pining to go from the cage where he is shut away, the body will mend itself from this monotony.\nOn the night of March 2, while he was sleeping, he heard noises. Some humans dressed in black, with ski masks covering their faces approached. They jumped a gate; the duck quacked saying Free me immediately I don't want to be here! The humans wrapped him with a blanket, put him in a box and they carried him away.\nThe duck, watching through the holes in the box, did not know what was passing before his eyes, but surely it was something good, so he wasn't nervous. Soon the cage was opened and he saw the great sun which he always watched, looking at the rest of his brothers and sisters flying, feeding himself on what they had around them.\nHe saw water in a much greater quantity than the little that he saw before in his dirty plate of food. The environment had a distinct aroma, the air moved his feathers, his legs touched the moist earth and at last he felt free. He fluttered his wings with great force, vigor and happiness and now this duck, hoping to be rescued and released, that had been disappointed by the cowardice of those who shout 'Animal Liberation' and do nothing for them, is free at last. Is direct action useless? Ask the duck.\n\"There are more things in heaven and earth ... than are dreamt of in your philosophy.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Reviewers play an essential role in the publication of articles. Peer reviewing is a scrutinizing process which helps the authors to improve the quality of their articles before publication.With the growth in the amount and magnitude of scientific information put in the public domain, and the number of organizations promoting research publication in the open access category, the need for scrutiny of each article for holding its legitimacy and uniformity is dynamic. Any article published in any of the Bio accent journal will undergo peer review process before hitting on the online publication button.\nIn order to maintain quality and to validate the research done by the authors, the manuscripts will be submitted to one or more eminent person who has done excellent contribution in the same field. They thoroughly check the quality and also the new innovations of the submitted articles by the experts before entering into the final step of the publication process. All the articles submitted to Bio accent open access journal will be subjected for minimum of three peer-review process to maintain the superiority and authenticity of the journal.\n\u2022\tThe article is sent to other eminent scientists who have significant contribution in the same field.\n\u2022\tFeedbackwill be provided during the peer review process in order to improve the quality of the article.\n\u2022\tThe reviewer informs the editor whether or not the study is of high enough quality to be published.\n\u2022\tThe reviewer's improvements and suggestions about the article are sent to the authors who may then revise their article and resubmit it for consideration.\n\u2022\tThe articles which received final approval from peer reviewers and also meet the scientific quality with research originalitywill be accepted for publication in Bio accent journals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The EVOLVE REACH Testing and Remediation COMPREHENSIVE REVIEW FOR THE NCLEX-PN\u00ae EXAM is the answer to your study needs with remediation content for weaknesses identified through curriculum testing, preparation for the nursing school exit exam, and review and preparation for the NCLEX-PN licensure exam. The review is written in an easy-to-read outline format with chapters broken down by clinical areas and concepts emphasized on the NCLEX-PN exam. Plus, the companion StudyWare CD offers practice tests with over 500 questions in the same electronic testing format you will experience on your nursing school exit exam and on the NCLEX-PN exam.\nChapters organized by clinical area make this an easy reference.\nChapter review and critical thinking questions ensure you have mastered each chapter before you move to the next.\nFeatured pharmacology tables pinpoint areas emphasized on the NCLEX exam to help prepare you for the test.\nAdditional illustrations enhance important clinical concepts and bring the content to life.\nExpanded index allows for easy reference for quick review.\nConsistent chapter organization and format improves content flow and readability, making it easy to study and remember key concepts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are looking for an affordable, cost-effective marketing tool for your hotel, PPC strategies should top your list. The hospitality industry has become highly competitive, especially for boutique hotels and independent hotel chains that lack the marketing might of the large international hotel chains. The greatest competition your hotel faces when trying to gain the attention of online searchers is not from other hotels, however, it is from the large OTAs that now dominate the top spots in hotel-related search results. As a result of the OTAs' aggressive marketing tactics, PPC campaigns for hotels have become more expensive and challenging than in the past. Even with the higher rates of today, hotel PPC strategies remain one of the more effective hotel marketing strategies, especially if you want to see a fast result from your marketing efforts. Because PPC ads like Google AdWords and Bing Ads target prospective guests at the exact moment they are searching for information about hotel rooms, you may be able to achieve a greater ROI than with other online hotel marketing strategies.\nWe will assess your marketing goals and the amount you have to spend on all your campaigns to help you allocate your PPC budget. We'll help you decide how much you want to allocate to each search engine (Google, Bing, etc.), and each campaign. We will reassess your marketing allocation as the results of each campaign come in to maximize your returns.\nTravel Tech can help you determine your target customers, and then set up the appropriate parameters for each of your PPC campaigns. You can set your ad to be seen around the state, across the country, or around the world, in a variety of languages. Are you trying to attract locals to indulge in a romantic weekend getaway? Maybe you want to focus on attracting world travelers to stay at your hotel. Whether you're looking to target business or leisure travelers, we'll work from your target demographics to find the most effective PPC strategies to achieve your marketing goals.\nPPC strategies are powered by keywords, so choosing the right keyword will play a huge role in determining the success of your campaigns. The OTAs have thrown so much money into hotel PPC strategies that the cost of standard keywords related to the hotel industry, such as \"Fort Lauderdale Hotels\", have skyrocketed to a price most small hotels simply cannot afford. That's all right \u2013 Travel Tech's marketers are experienced at getting around these seemingly impenetrable marketing barriers with a mix of local hotel keywords and terms that will reach prospective guests for a lower cost.\nOne of the ways to keep down costs and make your PPC ads more effective is through the use of negative keywords. For example, by including \"jobs\" and \"complaints\" as negative keywords in your campaigns, we can help eliminate the expense of irrelevant clicks, such as those from job seekers.\nWe'll use the latest targeting methods to ensure your ads are reaching the exact audience you're trying to reach. We can employ the full range of targeting criteria, including demographics, interests, and more.\nIf you want to run a more tightly targeted campaign, we can use geo-targeting so your ad will only be shown to prospective guests who are searching for a hotel in your immediate vicinity. We can set the parameters for an area as small as a mile or as large as a state.\nOur copywriters are well versed in creating concise, compelling Ad copy to entice prospective guests to click the ad and check out your hotel. We'll incorporate the right keywords in copy that brings your hotel experience to vivid life in a minimum of words.\nOften, the key to conversions is an eye-catching landing page that leads prospective guests to book a stay. Our graphic artists can create custom landing pages with dynamic calls to action that do a fabulous job of generating reservations.\nOnce your ads have been running for a while, we'll be able to identify any times that generate a lot of clicks or calls to your hotel but few actual bookings. We can then use ad scheduling to pause your campaigns during these spend-draining times.\nTo ensure that your PPC strategies are doing the best possible job of marketing your hotel, Travel Tech Strategies will use a full array of analytic tools to track and monitor your ads' performance. We will keep track of the click through and conversion rates of various keywords and adjust them as needed to make sure you continue to get the greatest ROI for your hotel PPC strategies.\nIf you would like to try PPC strategies to market your hotel, call on Travel Tech Strategies. Our marketing experts will be delighted to help you with every step of developing strong hotel PPC strategies to build your brand as you generate reservations from guests across town and around the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to kingsnake.com's photo gallery. This gallery allows registered kingsnake.com users to upload their favorite reptile and pet related photos to the topic galleries, and their personal photos to the member galleries, and use them other places such as our forums and classifieds, or to share with friends and family. To upload photos you must register for a kingsnake.com user account!\nPosting Photos to Forums\/Classifieds - Frequently Asked Questions How to use your gallery photos on the site - Got a question? Check here first!\n229137 photos with 55852 comments and 111616261 views, requiring 24559.5 MB of server disk space.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We are an eminent name, which is engaged in manufacturing, supplying and exporting excellent quality Reflector Telescope. It is manufactured using supreme quality material that is procured from reliable vendors of the market. Owing to its high image quality, clear vision and longer functional life, our telescope is highly appreciated in the market. Moreover, its is compact, portable, easy to assemble and have longer service life. Reflector Telescope is widely used in astronomical research.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not much more is known about Diagoras of Melos, than that he was called atheist \u2013 probably not only in the meaning 'ungodly' of that word. Athenagoras said about him, that he \"made the downright assertion that god does not exist at all.\" Cicero states the same about him.\nCicero also tells of how a friend of Diagoras tried to convince him of the existence of the gods, by pointing out how many votive pictures tell about people being saved from storms at sea by \"dint of vows to the gods\", to which Diagoras replied that \"there are nowhere any pictures of those who have been shipwrecked and drowned at sea.\" And Cicero goes on to give another example, where Diagoras was on a ship in hard weather, and the crew thought that they had brought it on themselves by taking this ungodly man onboard. He then wondered if the other boats out in the same storm also had a Diagoras onboard.\nAccording to Sextus Empiricus he became an atheist when an enemy of his perjured himself in court and got away with it. There are some variations in other sources to this anecdote, though not changing its moral content \u2013 immorality seems to go unpunished, so how can there be any gods in the sense of watchers over human virtue?\nHe is said to have been a student of Democritus, who may have initiated his disbelief in the existence of the gods, and was expelled from Athens in 411 BC for his attacks on religion. Other sources claim that he was bought from slavery by Democritus in 411 BC, when Melos was captured by Alcibiades, and then became his student.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For those who feel jeans and leggings just won't do, St. John has created the must-have pant. Sport ..\nCOMPOSITION Soft Leather, Textile fibers DETAILS neoprene, no apppqu\u00e9s, tw..\ndiv Black sweatshirt, has an attached hood with drawstring fastening, long sleeves, a full button pl..\nProduct Info & Care Complete your off-duty look wearing these grey 'Core' sneakers for women by Sup..\nDetails 91% wool , 5% elastan , 4% polyamide Made in Italy Dry clean only Front welt pockets Hidden ..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NEED MAIN LEVEL MASTER BEDROOM SUITE? HERE IT IS! PRISTINE Van Metre BUILDERS MODEL 5BR\/4.5BA side-load 2-car gar SFD in upscale Stone Ridge (Niche.com ranked #1 suburb to live in Virginia)*0.31-ac cul-de-sac lot\/backs trees\/faces north*3 lvls: 5387SF total\/4580SF fin*SHOWS LIKE A BUILDERS MODEL- LOADED w UPGRADES: MBR suite on ML & 4BRs on UL; gourmet kitchen-BfstRm; ML study; upgraded hdwds on ML\/UL-carpet-ceramics; 2-story, tray, 9-ft ceilings; abundant mouldings-recessed lighting thruout; custom drapes-blinds+*ML MBR suite: tray ceiling\/walk-in clos\/in-suite laundry\/luxury MBA: soaking tub\/separate 2-person shower\/bench*Gourmet kit: 46~ cherry break-front cabinetry, granite counters\/island\/upgr stainless applcs: DD cooktop, b-in MW& 2 wall ovens\/side-by-side frig\/adjoin BfstRm w tray ceil*Formal DR*FmR w\/tray ceil\/gas fplc*UL 4BR: ea priv access to FBA\/ea has w-in clos\/ea w ceil fan\/2nd BR-lvl laundry*LL: part fin walk-out bsmt\/lge RecRm\/FBA4\/space to add legal BR6 and media room\/2-zone HVAC *Upscale amenities*Great location\/nearby to transportation\/shopping+++.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In 2001, the City of Chilliwack established an RV Sanidump at the Heritage Park Event Centre. The Sani dump has been a useful and convenient resource for local RV owners and visitors passing through the Chilliwack area. Since the site was developed, it has doubled in size to accommodate a growing user group. As part of the growth and development of the site, the City of Chilliwack is now charging a $5.00 discharge fee. This fee is charged by credit card through onsite kiosks. The $5.00 fee helps the City maintain a high functioning site while also keeping in line with industry standards.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"\"CBS This Morning\" co-host John Dickerson covers the news at the crack of dawn every morning, and yet, he still made time to discuss headlines late Wednesday evening.\n\"It's past my bedtime, of course,\" Dickerson, who wakes up at 4 a.m. during the week, quipped at Variety's annual New York party, where he was one of 50 honorees to make this year's New Power of New York list, out in this week's issue.\nDickerson said the early wake-up call is hardly the hardest part of his job, given the 24\/7 news cycle, especially during Donald Trump's presidency.\nDickerson joined \"CBS This Morning\" as a full time co-host, alongside Gayle King and Norah O'Donnell, in January, following Charlie Rose's termination after numerous sexual harassment claims against the journalist surfaced in November 2017. This week, Bianna Golodryga was added as a fourth co-host to the CBS morning show.\n\"It was hard because a lot of us were dear friends of Charlie's and it was a big blow to all of us,\" Dickerson said, when asked about filling Rose's vacant seat at the anchor desk.\nTen months into his new gig, Dickerson has been adjusting well to the morning show, and the former \"Face the Nation\" anchor admits he's enjoying exploring more than just politics.\nNot missing the chance to weigh in on a bit of politics, Dickerson offered up his opinion on President Trump's recent comments mocking Christine Blasey Ford's sexual assault testimony against Judge Brett Kavanaugh, who is currently under an FBI investigation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Baby Shower Invite Online baby shower invite online ba shower invites online wblqual free. baby shower invite online online ba shower invitations plumegiant download. baby shower invite online ba shower invitations online stephenanuno ideas. Baby Shower Invite Online baby shower invite online amusing ba shower invitation maker as ba shower invitation templates. baby shower invite online free online ba shower evites your guests will love templates. baby shower invite online ba shower invitations create ba shower invitations online ideas. Baby Shower Invite Online baby shower invite online owl pink theme ba shower invites bs214 download.\nBaby Shower Invite Online baby shower invite online online ba shower invitations plumegiant download. baby shower invite online ba shower invitations online stephenanuno ideas. baby shower invite online amusing ba shower invitation maker as ba shower invitation templates. baby shower invite online free online ba shower evites your guests will love templates. baby shower invite online ba shower invitations create ba shower invitations online ideas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Expect to see some new faces on cable TV news after the election as people leave Congress and the administration and seek work in the media world.\n\"CNN is known for its big panels and endless parade of talking heads, but the network is signaling that it's running out of room for figures from President Donald Trump's administration,\" the story begins.\nOne of the four unnamed sources who confirmed Zucker's views for Politico said, \"If they do any hiring after the midterms, it's more likely to be members of Congress, senators and governors who lost their races, plus outgoing GOP members.\" It said this group could include people who would likely defend Trump.\nThe non-disclosure agreements mentioned were between the former White House officials and President Trump, and officials who have gone to TV have been accused of being unable to answer key questions because of the agreements.\nIt marks a change from the recent past as well. Zucker hired former Trump campaign manager Corey Lewandowski to help cover the 2016 campaign and, more recently, Trump's former top liaison to Congress, Marc Short.\nThe rule is not absolute, Politico reported, and \"Zucker would be interested in what he views as independent-minded officials who he thinks have avoided damaging their credibility.\" Nikki Haley, the outgoing U.S. ambassador to the United Nations, is sited as an example.\nZucker's views may have become public, but the No Trump Officials Rule is in force elsewhere as well.\nAnd it's not just the networks.\nIt points to Obama's press secretaries, Robert Gibbs and Jay Carney, landing at McDonald's and Amazon, respectively, and Andy Card of the George W. Bush administration landing at Fleishman-Hillard as examples of alums from past presidents who have succeeded after the White House.\nIt acknowledges some White House departees have landed on their feet as well \u2013 Hope Hicks at Fox, Josh Raffel at Juul, former national security adviser Dina Powell and former chief of staff Reince Priebus returning to their former employers.\nBut it says Steve Bannon, Sean Spicer and others have not been so successful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The ALA Public Programs Office and the Office for Diversity, Literacy and Outreach Services invite you to attend \"Beyond the 'Racial Stalemate'\", a free program facilitated by the W.K. Kellogg Foundation during the ALA Annual Conference in Chicago.\nDescription: In his 2008 speech on race titled \"A More Perfect Union,\" then-candidate Barack Obama described a \"racial stalemate we've been stuck in for years.\" He suggested that, if we don't do something different, \"nothing will change.\" Now, more than eight years later, we're still stuck. How do we move forward? Using an approach known as \"racial healing,\" facilitators will lead participants through a process that invites story-telling, vulnerability and deep listening. The goal is to provide leaders with a tool currently used by hundreds of organizations to help uproot the flawed belief in a racial hierarchy. These sessions are facilitated by the W. K. Kellogg Foundation.\nFree registration is required and spaces are limited; please choose only one of the following sessions.\nIf you have not already registered for Conference, you can add this free ticketed event session during your initial registration (or add it later).\nIf sessions are filled, please email Brian Russell in the Public Programs Office ([email protected]) to be added to a waiting list.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oshani Dias is definitely one of those rare Sri Lankan Actress who bag the prestigious role in the movie 'Dutugemunu' ,\u0daf\u0dd4\u0da7\u0dd4 \u0d9c\u0dd0\u0db8\u0dd4\u0dab\u0dd4 directed by Jayantha Chandrasiri . She play the role of \u0dbb\u0db1\u0dca \u0db8\u0dd0\u0dab\u0dd2\u0d9a\u0dcf in this legendary movie. She is currently studying Accountancy in Australia. Her role in 'Senehasata Adarei', 'Pethi Ahulana Malak' and 'Lansu Thiyana Mal' also being appreciated by the masses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Take yourself away from the hustle and bustle of life and relax in indulgent pure silk with this intricate hand-drawn botanical print pajama, inspired by Southern California wildlife including parrots, hummingbirds and a variety of palms.\nHandmade in our workshop in Cheshire, England, the Brindisi 34 modern fit pajamas are cut by hand to ensure we uphold the supremely soft satin finish. Produced in extremely limited quantities and the result of hours of loving craftwork by our skilled pattern cutters and seamstresses, you're sure to feel like royalty while lounging at home in our luxuriously comfortable silk sleepwear.\nLike all our modern fit pajamas, this style is designed for a more contemporary cut; one that is closer to the body without being restricting.\nThe jacket features a sartorially-inspired curved collar, mother of pearl buttons and a piped breast pocket. Furthermore, the rest of the jacket is piped throughout with a complementary navy piping.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1 inch metal tubing 1 inch metal tubing copper tube 1 2 inch round aluminum tubing 1 4 metal tubing square tubing bender youtube square tubing bender.\n1 inch metal tubing metal flex pipe china stainless steel exhaust flex pipe china tubing exhaust tube 15 metal flex metal flex pipe pipe wheels dh casters international pipe wheels.\n1 inch metal tubing threaded tube insert for 1 14 tpi pipe new used wheeler metals pipe new used x x ga oval.\n1 inch metal tubing round metal tubing inch round mild steel welded seam tube 1 1 4 x small metal round metal tubing round metal tubing stainless steel round tube inch wall inch round metal tubing steel round tubing mild steel tubing near me rectangular steel tubing near me round metal tubing.\n1 inch metal tubing internal diameter pvc pipe cutter hps od long aluminum joiner tubing bead roll tube pipe hps inch od x inch long aluminum joiner tube piping with bead roll.\n1 inch metal tubing two piece tube plugs making perfect pipe saddle cuts with a bandsaw or chopsaw steps making perfect pipe saddle cuts with a bandsaw or chopsaw steps with pictures.\n1 inch metal tubing carousel slide arlington emt emt insulating conduit bushing for electrical amazon price history chart for arlington emt emt insulating conduit bushing for electrical metal.\n1 inch metal tubing steel pipe 1 inch steel tubing black steel pipe metal tube sch seameless steel pipe pipe tube cutting plumbing tools the home depot.\n1 inch metal tubing inch tubing length metal ss dispensing needle blunt tip g inch tubing length metal ss dispensing needle blunt tip g.\n1 inch metal tubing 1 inch steel square tube narrow flange for 1 square tubing 1 inch steel square tube 1 inch steel square tube inch copper tubing metal tubing copper tubing inch copper inch copper tubing buy copper tubing alone or formed and enclosed to your exact inch copper tubing.\n1 inch metal tubing steel pipe 1 inch steel tubing black steel pipe metal tube sch seameless steel pipe round tubing plugs harman corp harman corporation round tubing plugs.\n1 inch metal tubing unbenttnjpg benttnjpg totool manual pipe bender to inch mm copper tube bender totool manual pipe bender to inch mm copper tube bender.\n1 inch metal tubing hot rolled steel rectangle tube .\n1 inch metal tubing end caps for metal tubing pack of 4 1 1 2 inch square black rubber caps end caps for metal tubing carbon steel tube mscdirectcom value collection inch square x ft long low.\n1 inch metal tubing steel square tubing end caps for metal tubing long threaded round stainless and zinc end caps for metal tubing pack of inch square black rubber caps end caps for metal tubing.\n1 inch metal tubing 1 inch square steel tubing home depot home depot plastic tubing square plastic tubing home depot home depot plastic tubs square plastic tub home depot ways to determine pvc pipe size for a project wikihow image titled determine pvc pipe size for a project step bullet.\n1 inch metal tubing pipe new used 5 12 od x 304w new surplus arlington emt emt insulating conduit bushing for electrical amazon price history chart for arlington emt emt insulating conduit bushing for electrical metal.\n1 inch metal tubing hps 1 inch od x 6 inch long 6061 aluminum joiner tube piping with bead roll huth swage tooling tube pipe expander huth swage die inch.\n1 inch metal tubing hot rolled steel square tube richmond virginia metal supermarkets steel aluminum stainless carousel slide.\n1 inch metal tubing totool manual pipe bender 38 to 1 inch 10 25mm copper tube bender ggg inch mm tubing length metal ss dispensing needle ggg inch mm tubing length metal ss dispensing.\n1 inch metal tubing pipe wheels square tube steel the home depot in x in x in plain.\n1 inch metal tubing round metal tubing steel round tubing mild steel tubing near me rectangular steel tubing near me round metal tubing inch steel square tube inch square tubing galvanized square inch steel square tube narrow flange for square tubing inch steel square tube inch steel square tube.\n1 inch metal tubing stainless steel mist tubing square steel tube steel square tubing industrial metal supply hot rolled steel square tube.\n1 inch metal tubing 1 inch copper tubing 1 inch copper tubing certificate 1 8 inch copper tubing from mill 1 inch copper tubing thin wall steel square tubing thin wall steel square tube with thin wall steel square tubing thin wall steel square tube with standard inch thin wall.\n1 inch metal tubing china steel pipe factory a500 1 inch square metal tubing perforated steel square tube box steel aquarium plumbing lees vinyl tubing id vinyl tubing.\n1 inch metal tubing carbon steel tube 1 12 inch steel tubing metal tube black steel tube schedule inch metal tubing copper tube inch round aluminum tubing inch metal tubing copper tube inch round aluminum tubing metal tubing.\n1 inch metal tubing 1 inch metal tubing conduit pipes 1 8 inch ss tubing 1 2 inch metal tubing inch metal tubing inch sch seamless carbon steel pipe inch sch inch metal tubing inch sch seamless carbon steel pipe inch sch seamless carbon steel pipe suppliers and manufacturers at ss tubing inch ss.\n1 inch metal tubing china steel pipe factory a500 1 inch square metal tubing perforated steel square tube box steel thin wall steel square tubing thin wall steel square tube with thin wall steel square tubing deform jig thin wall square metal tubing inch thin wall.\n1 inch metal tubing square tube steel the home depot in x in x in plain.\n1 inch metal tubing 116 in x 12 in x 36 in plain china inch galvanized steel pipe from dalian trading company inch galvanized steel pipe china inch galvanized steel pipe.\n1 inch metal tubing related post china steel pipe factory a inch square metal tubing perforated china steel pipe factory a inch square metal tubing perforated steel square tube box steel.\n1 inch metal tubing qty steel tubing stainless steel square steel tubing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have, for the smallest members of the family, stores specialising in children's fashion and baby clothing in which you'll find a myriad of modern items, from the most prestigious brands, for clothing the family's children and babies.\nNowadays fashion is not just about adults. Today, we find a wide range of options when it comes to buying children's fashion for boys and girls or baby clothing as varied as that of men's or women's fashions.\nThere are comprehensive catalogues for children's fashion in which the best brands have taken the time to refresh this market with updated styles so that the little ones can dress comfortably but without losing his\/her unique latest trends look.\nIn the available children's fashion stores, you can buy fashions for boys and girls, from reputable national and international brands, for the smallest members of the family.\nAvailable in our catalogues are t-shirts, dress shirts, singlets, jumpers, jerseys, hooded and hoodless sweaters, cardigans, jackets, pants, jeans, leggings, treggings, tops, skirts, overalls, dungarees, stockings, tights, socks, swimsuits, varied footwear, accessories, complements and much more so that he\/she has all the essentials in their wardrobe.\nAnd for newest family members, there are stores for buying newborn baby clothing and all the necessary accessories to make them comfortable and beautiful during the first months of his\/her lives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Satara is a two bedroom house, located in a lovely 1 acre property 4.5 km from town and was built by the Organic Brooklyn builders, who wanted to create a large English cottage feel. Very private, it gives our guests the chance to relax, unwind, enjoy delicious food and spend quality time with their spouse, children and friends.\nRs.15000 for the entire property and all the rooms plus cottage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Australian owned, PROJECT clothing offers extensive retail and sporting experience having successfully delivered technical clothing to local and national sporting teams and athletes. We know you can't move your game just because you don't have your gear \u2013 so we work with you on lead times and delivery dates.\nWe are built to provide personal service from start to finish, listening to your needs, tailoring to your code, providing technical advice, collaborating with your teammates where required and designing with your brand and sponsors in mind.\nTechnical fabrics, cuts and styles provide comfort and it's no secret that comfort enhances performance. We don't skimp on quality nor do we take shortcuts \u2013 we want you to feel like an athlete before you've even started your workout.\nWe provide responsive delivery as a result of our solid networks, experienced supply chain and personal care in processing orders. Most orders arrive only 30 days after artwork and order sign off.\nWe are continually hunting and developing new technologies and fabrics \u2013 in fact we enjoy it. Our garments are regularly trialled by our ambassadors and customers, even by our staff during lunch breaks. So we guarantee innovation and comfort as there is nothing quite like firsthand knowledge.\nWe take the administrative hassles away by offering teams their own branded online store for members and supporters to make purchases from, using a safe payment gateway. Teams have the option of creating public or private stores via a team login. Plus you get to talk to real PROJECT people \u2013 not self-help computer programs!\nAt PROJECT it's a partnership. We want to support your team journey whether it's promoting your next event, connecting you with others or celebrating your achievements.\nReady to accelerate your performance?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The list below includes the cities that the US Post Office accepts for ZIP code 68517. The preferred city may not be the city in which the ZIP is located. The city for 68517 is usually the name of the main post office. When mailing your package or letter, always include the preferred or acceptable cities. Using any city in the list of unacceptable cities may result in delays.\nZIP code 68517 is located in southeast Nebraska and covers a slightly less than average land area compared to other ZIP codes in the United States. It also has a slightly less than average population density.\nThe people living in ZIP code 68517 are primarily white. The number of middle aged adults is extremely large while the number of people in their late 20s to early 40s is extremely large. There are also an extremely large number of families and an extremely small number of single parents. The percentage of children under 18 living in the 68517 ZIP code is large compared to other areas of the country.\nZIP code 68517 has a small percentage of vacancies.\nThe majority of household are owned or have a mortgage. Homes in ZIP code 68517 were primarily built in 1939 or earlier. Looking at 68517 real estate data, the median home value of $238,200 is high compared to the rest of the country. It is also high compared to nearby ZIP codes. So you are less likely to find inexpensive homes in 68517.\nFor more information, see Lincoln, NE house prices.\nThe median household income of $72,267 is compared to the rest of the country. It is also compared to nearby ZIP codes. While money isn't everything, ZIP code 68517 may not be as nice as other parts of town.\nAs with most parts of the country, vehicles are the most common form of transportation to places of employment. If you are a person that likes walking or biking to work, it will be a comfort for you to know that 68517 has a well above average number of people who do not use a vehicle to commute. In most parts of the country, the majority of commuters get to work in under half an hour. The commute in 68517 is that short for a higher percentage of workers than almost anywhere in the United States. Many commuters should consider themselves lucky that they don't have a longer commute. It is very uncommon, compared to the rest of the US, for employees to have to travel more than 45 minutes to their place of employment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This recording sizzles with the high-energy tunes and compelling lyrics you've come to expect from this award-winning performer. Includes the title track, plus \"Cry for Love,\" \"Breakdown,\" \"Straight to the Heart,\" and 10 more songs highlighting Smith's blend of pop and message music.\nDanny Miller hope you will be able to read this !!!!!!!!!!Because your incoming song from MWS is on this album...................Straight to the heart.And I loved it to, you would then speak out Proverbs 3:5-6 and up to now it is so stuck in my head. Anyways, after listening to Danny Miller for so long on Radio Christain Voice, I then managed to get myself this album and I tell you, you will never be disappointed. With songs other songs like, cry for love, angels unaware, breathe in me and The other side of me. One thing about MWS is that he loves having kids on most of his albums. It is so so sweet when kids praise and worship the lord. With their pure hearts, you will even have the Lord speak to you.\nA strong, worshipful and Christ centered album from beginning to end. I really felt that this album showcased a lot of depth and growth for MWS as an artist and for the overall sound presented. It seemed like the release was targeted at a little older demographic as Michael is featured more as an adult and the album artwork wasn't so colorful and youth oriented. The songs also feature much more worshipful and introspective moods throughout. A great collection of music from one of CCM's top artists.\nthis album really \"hit home\" for me in a couple of ways. it accurately describes the moral decline of our nation. it's really sad and almost depressing. what has happened to us?! we WERE a Christian nation, but now we're totally the opposite! but at least GOD is in control and loves us and is there to Lead Us Home, if we let him.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Concern rising over expiration of mortgage debt relief \u2026.\nBy Jim Puzzanghera, Los Angeles Times5:00 a.m. CDT, September 7, 2012 WASHINGTON \u2014 Struggling homeowners who obtain reductions in their mortgage debt face a new obstacle starting next year \u2014 a bill for taxes on that aid. A special exemption of as much as $2 million per household in principal reduction and other aid from \u2026 More Concern rising over expiration of mortgage debt relief \u2026.\nLeave a comment Concern rising over expiration of mortgage debt relief \u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mount Royal (Quebec), September 19, 2014 \u2013 The Honourable Senator Jean-Guy Dagenais, on behalf of the Minister of Infrastructure, Communities and Intergovernmental Affairs and Minister of the Economic Development Agency of Canada for the Regions of Quebec, the Honourable Denis Lebel, and Pierre Arcand, Quebec Minister of Energy and Natural Resources and Member of the National Assembly for Mount Royal, on behalf of Quebec's Minister of Municipal Affairs and Land Occupancy, Pierre Moreau, are pleased to announce that the town of Mount Royal will receive $5,808,925 over the next five years under the new federal-provincial Gas Tax Fund agreement and from the Government of Quebec's contribution for the completion of priority infrastructure projects.\n\"The federal Gas Tax Fund supports economic growth from coast to coast by ensuring that Canadian communities continue to provide a quality of life that ranks among the best in the world. Our government is proud to provide municipalities with predictable, long-term funding to help them build and revitalize their public infrastructure while creating jobs and promoting long-term prosperity,\" said the Honourable Senator Dagenais. \"I am very pleased to be announcing this government assistance today for the electoral district of Mount Royal.\nThis investment will allow the municipality to better plan its local infrastructure spending and ensure the realization of projects essential to its development and to improving the quality of life for all citizens,\" added the minister Arcand.\nUnder the federal Gas Tax Fund and the contribution from the Government of Quebec, financial support for municipal infrastructure in Quebec will amount to $2.67 billion over the next five years. The federal Gas Tax Fund provides long-term, stable and predictable funding to help Canadian municipalities build and revitalize their local infrastructure while creating jobs and promoting long-term prosperity. The funding has been extended and the amount allocated has doubled, increasing from $1 billion to $2 billion per year. It has also been indexed at 2% per year and legislated as a permanent source of federal infrastructure funding for municipalities.\nThe purpose of the new agreement between Canada and Quebec is to ensure funding for the renewal of municipal and local infrastructure with a view to sustainable development.\nThe Government of Canada will distribute $1.89 billion to the Government of Quebec for municipal infrastructure over a five-year period, which will be supplemented by a $780-million contribution from Quebec.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Horizontal Belt filters The liquor is spread onto one end of a horizontal perforated belt covered with a filter cloth. A vacuum is pulled under the belt creating suction on top. A vacuum is pulled under the belt creating suction on top.\nFilter Cloth Wash Drainage Belt Wash Cake Wash Box Feed Box Drainage Belt Filter Cloth Filter Cloth Aligning Mechanism 8. Filter Cloth Take Up 9. Discharge Roll 10.\npolypropylene belt filter cloth for steel factory offers 2613 polypropylene filter mesh products. About 9% of these are filter meshes, 5% are filter cloth, and 4% are steel wire mesh. flat belt conveyor take up. good quality skirt rubber belt conveyor system for sale. best seller mining belt conveyor for iron ore.\n(1) Eureka Style DCF 16, Hepa W\/activated Charcoal Filter Dust Cup Vacuum Cleaner, Altima Bagless, SurfaceMax, Uno Upright, 62736A, 20232462, 20 2324 62, 76552, E 62736 Add To Cart There is a problem adding to cart.\nAll Items On Sale (39) Free Shipping. Discount (260) All Discounted Items (236) 10% off and more (229) 15% off and more (212) 20% off and more (181) 25% off and more generic Eureka Eureka The Boss Smart Vac Tune Up Kit, Includes Bags, Belt, HEPA Filter, Light Bulb, Motor Filter. Sold by Tidy Vacuums. $10.99 $8.49.\nWoven Fabrics For Belt Press Filter Oasis Fabrix supplies synthetic woven filter fabrics or cloth for dewatering and drying process in belt press filter. Our fabric belts are made with top quality polyester yarns to withstand severe humidity and high press tension.\nFilter presses for sludge treatment. Sludge treament. inorganic conditioning is often recommended to enable satisfactory cake release due to minimal aderence to filter cloth. Hydrophilic inorganic sludge the filer press generally requires the addition of lime only.\nJul 14, 2018\u00b7A wide variety of 10 micron filter cloth options are available to you, such as polyester, nylon, and pp. . 10 micron filter belt cloth for industry .. hot sale 200 micron nylon filter cloth .. cloth filter use 10 micron mesh nylon filter cloth.\nAn endless filter cloth travels together with the support carrier\/drainage belt, providing mechanical support to the filter cloth. This filtering surface (filter cloth) moves from a point where slurry is introduced onto the belt (feed end), liquor is sucked through the cloth by a vacuum and a cake is formed on the cloth surface.\nThere are 2,140 belt filter cloth suppliers, mainly located in Asia. The top supplying countries are China (Mainland), India, and Pakistan, which supply 99%, 1%, and 1% of belt filter cloth respectively. Belt filter cloth products are most popular in North America, Western Europe, and Southeast Asia.\nReplace your vacuum cleaner bags and belts with genuine Dirt Devil products. With vacuum belts that are specially designed for Dirt Devil vacuum models, you're sure to find the belt you need regardless of your vacuum model or style.\nVacuum Cleaner Filters Vacuum Cleaner Bags Vacuum Cleaner Belts Vacuum Cleaner Filters Vacuum Cleaner Brushrolls Vacuum Cleaner Parts Find Vacuum Machine, Sanyo Vacuum Filters, TriStar Vacuum Filters, shop, online shopping Sign up for our newsletter. Newsletter Sign up Submit.\nBelt Filter Press supplied by DaZhang Filtration Equipment Co.,Ltd mainly contains DNY Belt Filter Press, DZU Vacuum Belt Filter Press used as sewage filter press, sludge filter press and others. Filter plate Filter cloth Filter Press Pump Screw Pump Multistage Centrifugal Pump Dazhang Filtration Equipment is a famous DY belt filter.\nHigh Quality teflon mesh filter cloth For Belt Press manufacturer Filter cloth is widely applied in horizontal vacuum belt filter, rubber vacuum belt filter, belt press filter, vertical press filter. Technical Data Sheet Usage of teflon mesh filter cloth For Belt Press 1.","meta":{"redpajama_set_name":"RedPajamaC4"}}